LMFDB - Embedded newform 54.10.c.a.19.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [54,10,Mod(19,54)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(54, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([4]))

N = Newforms(chi, 10, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("54.19");

S:= CuspForms(chi, 10);

N := Newforms(S);

Level:	$ N $	$=$	$ 54 = 2 \cdot 3^{3} $
Weight:	$ k $	$=$	$ 10 $
Character orbit:	$[\chi]$	$=$	54.c (of order $3$, degree $2$, not minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$27.8119351528$
Analytic rank:	$0$
Dimension:	$8$
Relative dimension:	$4$ over $\Q(\zeta_{3})$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 656x^{6} - 4002x^{5} + 415806x^{4} - 1312656x^{3} + 13535681x^{2} + 29074530x + 211120900 $ x^8 + 656x^6 - 4002x^5 + 415806x^4 - 1312656x^3 + 13535681x^2 + 29074530x + 211120900
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{6}\cdot 3^{18} $
Twist minimal:	no (minimal twist has level 18)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			19.1
Root			$11.6557 + 20.1883i$ of defining polynomial
Character	$\chi$	$=$	54.19
Dual form			54.10.c.a.37.1

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-8.00000 + 13.8564i) q^{2} +(-128.000 - 221.703i) q^{4} +(-897.758 - 1554.96i) q^{5} +(5590.83 - 9683.61i) q^{7} +4096.00 q^{8} +O(q^{10})$	\(q+(-8.00000 + 13.8564i) q^{2} +(-128.000 - 221.703i) q^{4} +(-897.758 - 1554.96i) q^{5} +(5590.83 - 9683.61i) q^{7} +4096.00 q^{8} +28728.3 q^{10} +(-9213.10 + 15957.6i) q^{11} +(-10588.5 - 18339.7i) q^{13} +(89453.3 + 154938. i) q^{14} +(-32768.0 + 56755.8i) q^{16} -660849. q^{17} +522089. q^{19} +(-229826. + 398070. i) q^{20} +(-147410. - 255321. i) q^{22} +(524295. + 908105. i) q^{23} +(-635377. + 1.10051e6i) q^{25} +338831. q^{26} -2.86251e6 q^{28} +(1.30249e6 - 2.25598e6i) q^{29} +(26962.3 + 46700.1i) q^{31} +(-524288. - 908093. i) q^{32} +(5.28679e6 - 9.15699e6i) q^{34} -2.00769e7 q^{35} -1.93749e7 q^{37} +(-4.17672e6 + 7.23428e6i) q^{38} +(-3.67722e6 - 6.36913e6i) q^{40} +(-648885. - 1.12390e6i) q^{41} +(7.35308e6 - 1.27359e7i) q^{43} +4.71711e6 q^{44} -1.67774e7 q^{46} +(-1.53291e7 + 2.65507e7i) q^{47} +(-4.23380e7 - 7.33316e7i) q^{49} +(-1.01660e7 - 1.76081e7i) q^{50} +(-2.71064e6 + 4.69497e6i) q^{52} -5.86557e7 q^{53} +3.30845e7 q^{55} +(2.29001e7 - 3.96641e7i) q^{56} +(2.08398e7 + 3.60957e7i) q^{58} +(1.14839e7 + 1.98906e7i) q^{59} +(-9.03978e7 + 1.56574e8i) q^{61} -862794. q^{62} +1.67772e7 q^{64} +(-1.90117e7 + 3.29293e7i) q^{65} +(1.18973e8 + 2.06067e8i) q^{67} +(8.45886e7 + 1.46512e8i) q^{68} +(1.60615e8 - 2.78193e8i) q^{70} -4.73186e7 q^{71} -7.05287e7 q^{73} +(1.55000e8 - 2.68467e8i) q^{74} +(-6.68275e7 - 1.15749e8i) q^{76} +(1.03018e8 + 1.78432e8i) q^{77} +(1.28832e8 - 2.23143e8i) q^{79} +1.17671e8 q^{80} +2.07643e7 q^{82} +(1.46745e8 - 2.54170e8i) q^{83} +(5.93282e8 + 1.02760e9i) q^{85} +(1.17649e8 + 2.03774e8i) q^{86} +(-3.77368e7 + 6.53621e7i) q^{88} -1.41522e8 q^{89} -2.36793e8 q^{91} +(1.34219e8 - 2.32475e8i) q^{92} +(-2.45265e8 - 4.24812e8i) q^{94} +(-4.68710e8 - 8.11830e8i) q^{95} +(-1.64320e8 + 2.84610e8i) q^{97} +1.35482e9 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 64 q^{2} - 1024 q^{4} - 171 q^{5} + 7135 q^{7} + 32768 q^{8}+O(q^{10}) $ 8 * q - 64 * q^2 - 1024 * q^4 - 171 * q^5 + 7135 * q^7 + 32768 * q^8	\( 8 q - 64 q^{2} - 1024 q^{4} - 171 q^{5} + 7135 q^{7} + 32768 q^{8} + 5472 q^{10} + 26130 q^{11} - 4163 q^{13} + 114160 q^{14} - 262144 q^{16} - 1098510 q^{17} - 436382 q^{19} - 43776 q^{20} + 418080 q^{22} + 289383 q^{23} + 2947937 q^{25} + 133216 q^{26} - 3653120 q^{28} + 601707 q^{29} + 5671315 q^{31} - 4194304 q^{32} + 8788080 q^{34} - 29787930 q^{35} - 38737148 q^{37} + 3491056 q^{38} - 700416 q^{40} + 18418410 q^{41} + 35096140 q^{43} - 13378560 q^{44} - 9260256 q^{46} + 79830825 q^{47} - 40540299 q^{49} + 47166992 q^{50} - 1065728 q^{52} - 330697236 q^{53} + 56528442 q^{55} + 29224960 q^{56} + 9627312 q^{58} + 90704166 q^{59} - 122811677 q^{61} - 181482080 q^{62} + 134217728 q^{64} + 116600103 q^{65} + 221601736 q^{67} + 140609280 q^{68} + 238303440 q^{70} - 276408240 q^{71} - 988917014 q^{73} + 309897184 q^{74} + 55856896 q^{76} + 548139525 q^{77} + 592840885 q^{79} + 22413312 q^{80} - 589389120 q^{82} + 478410747 q^{83} + 1468792818 q^{85} + 561538240 q^{86} + 107028480 q^{88} - 875519952 q^{89} - 3695513062 q^{91} + 74082048 q^{92} + 1277293200 q^{94} - 813906756 q^{95} + 2679512242 q^{97} + 1297289568 q^{98}+O(q^{100}) \) 8 * q - 64 * q^2 - 1024 * q^4 - 171 * q^5 + 7135 * q^7 + 32768 * q^8 + 5472 * q^10 + 26130 * q^11 - 4163 * q^13 + 114160 * q^14 - 262144 * q^16 - 1098510 * q^17 - 436382 * q^19 - 43776 * q^20 + 418080 * q^22 + 289383 * q^23 + 2947937 * q^25 + 133216 * q^26 - 3653120 * q^28 + 601707 * q^29 + 5671315 * q^31 - 4194304 * q^32 + 8788080 * q^34 - 29787930 * q^35 - 38737148 * q^37 + 3491056 * q^38 - 700416 * q^40 + 18418410 * q^41 + 35096140 * q^43 - 13378560 * q^44 - 9260256 * q^46 + 79830825 * q^47 - 40540299 * q^49 + 47166992 * q^50 - 1065728 * q^52 - 330697236 * q^53 + 56528442 * q^55 + 29224960 * q^56 + 9627312 * q^58 + 90704166 * q^59 - 122811677 * q^61 - 181482080 * q^62 + 134217728 * q^64 + 116600103 * q^65 + 221601736 * q^67 + 140609280 * q^68 + 238303440 * q^70 - 276408240 * q^71 - 988917014 * q^73 + 309897184 * q^74 + 55856896 * q^76 + 548139525 * q^77 + 592840885 * q^79 + 22413312 * q^80 - 589389120 * q^82 + 478410747 * q^83 + 1468792818 * q^85 + 561538240 * q^86 + 107028480 * q^88 - 875519952 * q^89 - 3695513062 * q^91 + 74082048 * q^92 + 1277293200 * q^94 - 813906756 * q^95 + 2679512242 * q^97 + 1297289568 * q^98

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/54\mathbb{Z}\right)^\times$.

$n$	$29$
$\chi(n)$	$e\left(\frac{2}{3}\right)$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$	−8.00000	+	13.8564i	−0.353553	+	0.612372i
$2$	−8.00000	+	13.8564i	−0.353553	+	0.612372i
$3$		0			0
$3$		0			0
$4$	−128.000	−	221.703i	−0.250000	−	0.433013i
$5$	−897.758	−	1554.96i	−0.642383	−	1.11264i	−0.984899	−	0.173129i	$-0.944612\pi$
$5$	−897.758	−	1554.96i	−0.642383	−	1.11264i	0.342516	−	0.939512i	$-0.388721\pi$
$6$		0			0
$7$	5590.83	−	9683.61i	0.880107	−	1.52439i	0.0288856	−	0.999583i	$-0.490804\pi$
$7$	5590.83	−	9683.61i	0.880107	−	1.52439i	0.851221	−	0.524807i	$-0.175863\pi$
$8$	4096.00			0.353553
$9$		0			0
$10$	28728.3			0.908467
$11$	−9213.10	+	15957.6i	−0.189731	+	0.328624i	−0.945161	−	0.326606i	$-0.894095\pi$
$11$	−9213.10	+	15957.6i	−0.189731	+	0.328624i	0.755429	+	0.655230i	$0.227428\pi$
$12$		0			0
$13$	−10588.5	−	18339.7i	−0.102822	−	0.178094i	0.810024	−	0.586397i	$-0.199454\pi$
$13$	−10588.5	−	18339.7i	−0.102822	−	0.178094i	−0.912846	+	0.408303i	$0.866121\pi$
$14$	89453.3	+	154938.i	0.622330	+	1.07791i
$15$		0			0
$16$	−32768.0	+	56755.8i	−0.125000	+	0.216506i
$17$	−660849.			−1.91903			−0.959515	−	0.281656i	$-0.909116\pi$
$17$	−660849.			−1.91903			−0.959515	+	0.281656i	$0.909116\pi$
$18$		0			0
$19$	522089.			0.919081			0.459540	−	0.888157i	$-0.348014\pi$
$19$	522089.			0.919081			0.459540	+	0.888157i	$0.348014\pi$
$20$	−229826.	+	398070.i	−0.321192	+	0.556320i
$21$		0			0
$22$	−147410.	−	255321.i	−0.134160	−	0.232372i
$23$	524295.	+	908105.i	0.390661	+	0.676645i	0.992537	−	0.121945i	$-0.0389131\pi$
$23$	524295.	+	908105.i	0.390661	+	0.676645i	−0.601876	+	0.798590i	$0.705580\pi$
$24$		0			0
$25$	−635377.	+	1.10051e6i	−0.325313	+	0.563459i
$26$	338831.			0.145413
$27$		0			0
$28$	−2.86251e6			−0.880107
$29$	1.30249e6	−	2.25598e6i	0.341966	−	0.592303i	−0.642831	−	0.766008i	$-0.722240\pi$
$29$	1.30249e6	−	2.25598e6i	0.341966	−	0.592303i	0.984798	+	0.173705i	$0.0555738\pi$
$30$		0			0
$31$	26962.3	+	46700.1i	0.00524360	+	0.00908218i	0.868635	−	0.495452i	$-0.164998\pi$
$31$	26962.3	+	46700.1i	0.00524360	+	0.00908218i	−0.863392	+	0.504534i	$0.831664\pi$
$32$	−524288.	−	908093.i	−0.0883883	−	0.153093i
$33$		0			0
$34$	5.28679e6	−	9.15699e6i	0.678480	−	1.17516i
$35$	−2.00769e7			−2.26146
$36$		0			0
$37$	−1.93749e7			−1.69955			−0.849773	−	0.527150i	$-0.823261\pi$
$37$	−1.93749e7			−1.69955			−0.849773	+	0.527150i	$0.823261\pi$
$38$	−4.17672e6	+	7.23428e6i	−0.324944	+	0.562820i
$39$		0			0
$40$	−3.67722e6	−	6.36913e6i	−0.227117	−	0.393378i
$41$	−648885.	−	1.12390e6i	−0.0358625	−	0.0621156i	0.847537	−	0.530736i	$-0.178084\pi$
$41$	−648885.	−	1.12390e6i	−0.0358625	−	0.0621156i	−0.883400	+	0.468621i	$0.844751\pi$
$42$		0			0
$43$	7.35308e6	−	1.27359e7i	0.327990	−	0.568096i	−0.654123	−	0.756388i	$-0.726962\pi$
$43$	7.35308e6	−	1.27359e7i	0.327990	−	0.568096i	0.982113	+	0.188293i	$0.0602953\pi$
$44$	4.71711e6			0.189731
$45$		0			0
$46$	−1.67774e7			−0.552478
$47$	−1.53291e7	+	2.65507e7i	−0.458222	+	0.793663i	−0.998867	−	0.0475872i	$-0.984847\pi$
$47$	−1.53291e7	+	2.65507e7i	−0.458222	+	0.793663i	0.540645	+	0.841251i	$0.318180\pi$
$48$		0			0
$49$	−4.23380e7	−	7.33316e7i	−1.04918	−	1.81723i
$50$	−1.01660e7	−	1.76081e7i	−0.230031	−	0.398425i
$51$		0			0
$52$	−2.71064e6	+	4.69497e6i	−0.0514112	+	0.0890468i
$53$	−5.86557e7			−1.02110			−0.510551	−	0.859848i	$-0.670558\pi$
$53$	−5.86557e7			−1.02110			−0.510551	+	0.859848i	$0.670558\pi$
$54$		0			0
$55$	3.30845e7			0.487521
$56$	2.29001e7	−	3.96641e7i	0.311165	−	0.538953i
$57$		0			0
$58$	2.08398e7	+	3.60957e7i	0.241807	+	0.418821i
$59$	1.14839e7	+	1.98906e7i	0.123383	+	0.213705i	0.921100	−	0.389327i	$-0.127292\pi$
$59$	1.14839e7	+	1.98906e7i	0.123383	+	0.213705i	−0.797717	+	0.603032i	$0.793959\pi$
$60$		0			0
$61$	−9.03978e7	+	1.56574e8i	−0.835937	+	1.44789i	0.0573274	+	0.998355i	$0.481742\pi$
$61$	−9.03978e7	+	1.56574e8i	−0.835937	+	1.44789i	−0.893265	+	0.449531i	$0.851591\pi$
$62$	−862794.			−0.00741557
$63$		0			0
$64$	1.67772e7			0.125000
$65$	−1.90117e7	+	3.29293e7i	−0.132103	+	0.228809i
$66$		0			0
$67$	1.18973e8	+	2.06067e8i	0.721294	+	1.24932i	0.960481	+	0.278344i	$0.0897855\pi$
$67$	1.18973e8	+	2.06067e8i	0.721294	+	1.24932i	−0.239188	+	0.970973i	$0.576881\pi$
$68$	8.45886e7	+	1.46512e8i	0.479758	+	0.830965i
$69$		0			0
$70$	1.60615e8	−	2.78193e8i	0.799548	−	1.38486i
$71$	−4.73186e7			−0.220988			−0.110494	−	0.993877i	$-0.535243\pi$
$71$	−4.73186e7			−0.220988			−0.110494	+	0.993877i	$0.535243\pi$
$72$		0			0
$73$	−7.05287e7			−0.290679			−0.145339	−	0.989382i	$-0.546427\pi$
$73$	−7.05287e7			−0.290679			−0.145339	+	0.989382i	$0.546427\pi$
$74$	1.55000e8	−	2.68467e8i	0.600880	−	1.04075i
$75$		0			0
$76$	−6.68275e7	−	1.15749e8i	−0.229770	−	0.397974i
$77$	1.03018e8	+	1.78432e8i	0.333967	+	0.578448i
$78$		0			0
$79$	1.28832e8	−	2.23143e8i	0.372136	−	0.644558i	−0.617758	−	0.786368i	$-0.711959\pi$
$79$	1.28832e8	−	2.23143e8i	0.372136	−	0.644558i	0.989894	+	0.141810i	$0.0452922\pi$
$80$	1.17671e8			0.321192
$81$		0			0
$82$	2.07643e7			0.0507172
$83$	1.46745e8	−	2.54170e8i	0.339401	−	0.587860i	−0.644919	−	0.764251i	$-0.723109\pi$
$83$	1.46745e8	−	2.54170e8i	0.339401	−	0.587860i	0.984320	+	0.176391i	$0.0564423\pi$
$84$		0			0
$85$	5.93282e8	+	1.02760e9i	1.23275	+	2.13519i
$86$	1.17649e8	+	2.03774e8i	0.231924	+	0.401704i
$87$		0			0
$88$	−3.77368e7	+	6.53621e7i	−0.0670801	+	0.116186i
$89$	−1.41522e8			−0.239093			−0.119547	−	0.992829i	$-0.538144\pi$
$89$	−1.41522e8			−0.239093			−0.119547	+	0.992829i	$0.538144\pi$
$90$		0			0
$91$	−2.36793e8			−0.361979
$92$	1.34219e8	−	2.32475e8i	0.195331	−	0.338322i
$93$		0			0
$94$	−2.45265e8	−	4.24812e8i	−0.324012	−	0.561205i
$95$	−4.68710e8	−	8.11830e8i	−0.590402	−	1.02261i
$96$		0			0
$97$	−1.64320e8	+	2.84610e8i	−0.188459	+	0.326420i	−0.944737	−	0.327830i	$-0.893683\pi$
$97$	−1.64320e8	+	2.84610e8i	−0.188459	+	0.326420i	0.756278	+	0.654251i	$0.227016\pi$
$98$	1.35482e9			1.48376
$99$		0			0
$100$	3.25313e8			0.325313
$101$	2.47471e8	−	4.28632e8i	0.236635	−	0.409863i	−0.723112	−	0.690731i	$-0.757289\pi$
$101$	2.47471e8	−	4.28632e8i	0.236635	−	0.409863i	0.959746	+	0.280868i	$0.0906223\pi$
$102$		0			0
$103$	1.99528e8	+	3.45593e8i	0.174677	+	0.302550i	0.940050	−	0.341038i	$-0.110778\pi$
$103$	1.99528e8	+	3.45593e8i	0.174677	+	0.302550i	−0.765372	+	0.643588i	$0.777445\pi$
$104$	−4.33703e7	−	7.51196e7i	−0.0363532	−	0.0629656i
$105$		0			0
$106$	4.69246e8	−	8.12757e8i	0.361014	−	0.625294i
$107$	1.26980e8			0.0936502			0.0468251	−	0.998903i	$-0.485090\pi$
$107$	1.26980e8			0.0936502			0.0468251	+	0.998903i	$0.485090\pi$
$108$		0			0
$109$	−2.21053e9			−1.49995			−0.749974	−	0.661467i	$-0.769934\pi$
$109$	−2.21053e9			−1.49995			−0.749974	+	0.661467i	$0.769934\pi$
$110$	−2.64676e8	+	4.58433e8i	−0.172365	+	0.298544i
$111$		0			0
$112$	3.66401e8	+	6.34625e8i	0.220027	+	0.381097i
$113$	3.33955e8	+	5.78427e8i	0.192679	+	0.333730i	0.946137	−	0.323766i	$-0.104949\pi$
$113$	3.33955e8	+	5.78427e8i	0.192679	+	0.333730i	−0.753458	+	0.657496i	$0.771616\pi$
$114$		0			0
$115$	9.41380e8	−	1.63052e9i	0.501908	−	0.869331i
$116$	−6.66875e8			−0.341966
$117$		0			0
$118$	−3.67484e8			−0.174489
$119$	−3.69470e9	+	6.39940e9i	−1.68895	+	2.92535i
$120$		0			0
$121$	1.00921e9	+	1.74801e9i	0.428004	+	0.741325i
$122$	−1.44637e9	−	2.50518e9i	−0.591097	−	1.02381i
$123$		0			0
$124$	6.90235e6	−	1.19552e7i	0.00262180	−	0.00454109i
$125$	−1.22521e9			−0.448864
$126$		0			0
$127$	−2.12304e9			−0.724171			−0.362085	−	0.932145i	$-0.617935\pi$
$127$	−2.12304e9			−0.724171			−0.362085	+	0.932145i	$0.617935\pi$
$128$	−1.34218e8	+	2.32472e8i	−0.0441942	+	0.0765466i
$129$		0			0
$130$	−3.04188e8	−	5.26869e8i	−0.0934108	−	0.161792i
$131$	−2.48713e7	−	4.30783e7i	−0.00737865	−	0.0127802i	0.862313	−	0.506376i	$-0.169015\pi$
$131$	−2.48713e7	−	4.30783e7i	−0.00737865	−	0.0127802i	−0.869691	+	0.493596i	$0.835682\pi$
$132$		0			0
$133$	2.91892e9	−	5.05571e9i	0.808889	−	1.40104i
$134$	−3.80714e9			−1.02006
$135$		0			0
$136$	−2.70684e9			−0.678480
$137$	3.38969e9	−	5.87112e9i	0.822087	−	1.42390i	−0.0820385	−	0.996629i	$-0.526143\pi$
$137$	3.38969e9	−	5.87112e9i	0.822087	−	1.42390i	0.904125	−	0.427267i	$-0.140524\pi$
$138$		0			0
$139$	−2.37656e9	−	4.11632e9i	−0.539985	−	0.935282i	−0.998904	−	0.0468036i	$-0.985097\pi$
$139$	−2.37656e9	−	4.11632e9i	−0.539985	−	0.935282i	0.458919	−	0.888478i	$-0.348237\pi$
$140$	2.56984e9	+	4.45109e9i	0.565366	+	0.979243i
$141$		0			0
$142$	3.78549e8	−	6.55666e8i	0.0781312	−	0.135327i
$143$	3.90210e8			0.0780344
$144$		0			0
$145$	−4.67728e9			−0.878694
$146$	5.64230e8	−	9.77275e8i	0.102770	−	0.178004i
$147$		0			0
$148$	2.47999e9	+	4.29547e9i	0.424886	+	0.735925i
$149$	−3.60147e9	−	6.23793e9i	−0.598606	−	1.03682i	−0.993027	−	0.117887i	$-0.962388\pi$
$149$	−3.60147e9	−	6.23793e9i	−0.598606	−	1.03682i	0.394421	−	0.918930i	$-0.370945\pi$
$150$		0			0
$151$	−2.48682e9	+	4.30729e9i	−0.389267	+	0.674230i	−0.992351	−	0.123448i	$-0.960605\pi$
$151$	−2.48682e9	+	4.30729e9i	−0.389267	+	0.674230i	0.603084	+	0.797678i	$0.293938\pi$
$152$	2.13848e9			0.324944
$153$		0			0
$154$	−3.29657e9			−0.472301
$155$	4.84112e7	−	8.38507e7i	0.00673680	−	0.0116685i
$156$		0			0
$157$	3.46761e8	+	6.00607e8i	0.0455493	+	0.0788936i	0.887901	−	0.460034i	$-0.152163\pi$
$157$	3.46761e8	+	6.00607e8i	0.0455493	+	0.0788936i	−0.842352	+	0.538928i	$0.818830\pi$
$158$	2.06131e9	+	3.57029e9i	0.263140	+	0.455771i
$159$		0			0
$160$	−9.41368e8	+	1.63050e9i	−0.113558	+	0.196689i
$161$	1.17250e10			1.37529
$162$		0			0
$163$	7.96784e9			0.884090			0.442045	−	0.896993i	$-0.354253\pi$
$163$	7.96784e9			0.884090			0.442045	+	0.896993i	$0.354253\pi$
$164$	−1.66114e8	+	2.87719e8i	−0.0179312	+	0.0310578i
$165$		0			0
$166$	2.34793e9	+	4.06673e9i	0.239993	+	0.415680i
$167$	−3.06914e9	−	5.31590e9i	−0.305346	−	0.528875i	0.671992	−	0.740558i	$-0.265439\pi$
$167$	−3.06914e9	−	5.31590e9i	−0.305346	−	0.528875i	−0.977338	+	0.211683i	$0.932106\pi$
$168$		0			0
$169$	5.07802e9	−	8.79539e9i	0.478855	−	0.829401i
$170$	−1.89850e10			−1.74338
$171$		0			0
$172$	−3.76477e9			−0.327990
$173$	−1.87019e9	+	3.23926e9i	−0.158737	+	0.274940i	−0.934413	−	0.356190i	$-0.884075\pi$
$173$	−1.87019e9	+	3.23926e9i	−0.158737	+	0.274940i	0.775677	+	0.631131i	$0.217409\pi$
$174$		0			0
$175$	7.10457e9	+	1.23055e10i	0.572620	+	0.991808i
$176$	−6.03790e8	−	1.04579e9i	−0.0474328	−	0.0821560i
$177$		0			0
$178$	1.13217e9	−	1.96098e9i	0.0845323	−	0.146414i
$179$	2.42118e10			1.76274			0.881369	−	0.472429i	$-0.156623\pi$
$179$	2.42118e10			1.76274			0.881369	+	0.472429i	$0.156623\pi$
$180$		0			0
$181$	1.15322e10			0.798657			0.399328	−	0.916808i	$-0.369243\pi$
$181$	1.15322e10			0.798657			0.399328	+	0.916808i	$0.369243\pi$
$182$	1.89435e9	−	3.28110e9i	0.127979	−	0.221666i
$183$		0			0
$184$	2.14751e9	+	3.71960e9i	0.138120	+	0.239230i
$185$	1.73940e10	+	3.01273e10i	1.09176	+	1.89098i
$186$		0			0
$187$	6.08846e9	−	1.05455e10i	0.364100	−	0.630639i
$188$	7.84849e9			0.458222
$189$		0			0
$190$	1.49987e10			0.834955
$191$	9.08575e9	−	1.57370e10i	0.493981	−	0.855601i	−0.505995	−	0.862537i	$-0.668874\pi$
$191$	9.08575e9	−	1.57370e10i	0.493981	−	0.855601i	0.999976	+	0.00693591i	$0.00220779\pi$
$192$		0			0
$193$	9.56069e9	+	1.65596e10i	0.496000	+	0.859097i	0.999989	−	0.00461309i	$-0.00146840\pi$
$193$	9.56069e9	+	1.65596e10i	0.496000	+	0.859097i	−0.503990	+	0.863710i	$0.668135\pi$
$194$	−2.62911e9	−	4.55376e9i	−0.133261	−	0.230814i
$195$		0			0
$196$	−1.08385e10	+	1.87729e10i	−0.524588	+	0.908613i
$197$	−1.18643e10			−0.561233			−0.280617	−	0.959820i	$-0.590539\pi$
$197$	−1.18643e10			−0.561233			−0.280617	+	0.959820i	$0.590539\pi$
$198$		0			0
$199$	−2.22347e10			−1.00506			−0.502530	−	0.864560i	$-0.667598\pi$
$199$	−2.22347e10			−1.00506			−0.502530	+	0.864560i	$0.667598\pi$
$200$	−2.60250e9	+	4.50767e9i	−0.115016	+	0.199213i
$201$		0			0
$202$	3.95954e9	+	6.85812e9i	0.167326	+	0.289817i
$203$	−1.45640e10	−	2.52256e10i	−0.601934	−	1.04258i
$204$		0			0
$205$	−1.16508e9	+	2.01798e9i	−0.0460749	+	0.0798041i
$206$	−6.38490e9			−0.247031
$207$		0			0
$208$	1.38785e9			0.0514112
$209$	−4.81006e9	+	8.33127e9i	−0.174378	+	0.302032i
$210$		0			0
$211$	−2.80343e10	−	4.85568e10i	−0.973686	−	1.68647i	−0.684206	−	0.729289i	$-0.739851\pi$
$211$	−2.80343e10	−	4.85568e10i	−0.973686	−	1.68647i	−0.289479	−	0.957184i	$-0.593482\pi$
$212$	7.50793e9	+	1.30041e10i	0.255275	+	0.442150i
$213$		0			0
$214$	−1.01584e9	+	1.75949e9i	−0.0331103	+	0.0573488i
$215$	−2.64051e10			−0.842782
$216$		0			0
$217$	6.02967e8			0.0184597
$218$	1.76842e10	−	3.06299e10i	0.530312	−	0.918527i
$219$		0			0
$220$	−4.23482e9	−	7.33492e9i	−0.121880	−	0.211103i
$221$	6.99737e9	+	1.21198e10i	0.197319	+	0.341767i
$222$		0			0
$223$	−3.21706e10	+	5.57212e10i	−0.871139	+	1.50886i	−0.0103203	+	0.999947i	$0.503285\pi$
$223$	−3.21706e10	+	5.57212e10i	−0.871139	+	1.50886i	−0.860819	+	0.508911i	$0.830048\pi$
$224$	−1.17248e10			−0.311165
$225$		0			0
$226$	−1.06866e10			−0.272490
$227$	6.18839e9	−	1.07186e10i	0.154690	−	0.267930i	−0.778256	−	0.627947i	$-0.783896\pi$
$227$	6.18839e9	−	1.07186e10i	0.154690	−	0.267930i	0.932946	+	0.360016i	$0.117229\pi$
$228$		0			0
$229$	−2.12375e10	−	3.67844e10i	−0.510321	−	0.883901i	−0.999928	−	0.0119584i	$-0.996193\pi$
$229$	−2.12375e10	−	3.67844e10i	−0.510321	−	0.883901i	0.489608	−	0.871943i	$-0.337140\pi$
$230$	1.50621e10	+	2.60883e10i	0.354903	+	0.614710i
$231$		0			0
$232$	5.33500e9	−	9.24049e9i	0.120903	−	0.209411i
$233$	−4.56572e9			−0.101486			−0.0507431	−	0.998712i	$-0.516159\pi$
$233$	−4.56572e9			−0.101486			−0.0507431	+	0.998712i	$0.516159\pi$
$234$		0			0
$235$	5.50472e10			1.17742
$236$	2.93987e9	−	5.09200e9i	0.0616913	−	0.106852i
$237$		0			0
$238$	−5.91151e10	−	1.02390e11i	−1.19427	−	2.06854i
$239$	−3.22318e10	−	5.58270e10i	−0.638989	−	1.10676i	−0.985655	−	0.168773i	$-0.946020\pi$
$239$	−3.22318e10	−	5.58270e10i	−0.638989	−	1.10676i	0.346666	−	0.937989i	$-0.387314\pi$
$240$		0			0
$241$	3.08142e10	−	5.33717e10i	0.588401	−	1.01914i	−0.406041	−	0.913855i	$-0.633091\pi$
$241$	3.08142e10	−	5.33717e10i	0.588401	−	1.01914i	0.994442	−	0.105286i	$-0.0335757\pi$
$242$	−3.22948e10			−0.605289
$243$		0			0
$244$	4.62837e10			0.835937
$245$	−7.60186e10	+	1.31668e11i	−1.34795	+	2.33471i
$246$		0			0
$247$	−5.52812e9	−	9.57499e9i	−0.0945021	−	0.163682i
$248$	1.10438e8	+	1.91283e8i	0.00185389	+	0.00321103i
$249$		0			0
$250$	9.80167e9	−	1.69770e10i	0.158697	−	0.274872i
$251$	−1.50155e10			−0.238785			−0.119393	−	0.992847i	$-0.538095\pi$
$251$	−1.50155e10			−0.238785			−0.119393	+	0.992847i	$0.538095\pi$
$252$		0			0
$253$	−1.93215e10			−0.296482
$254$	1.69843e10	−	2.94177e10i	0.256033	−	0.443462i
$255$		0			0
$256$	−2.14748e9	−	3.71955e9i	−0.0312500	−	0.0541266i
$257$	−4.31146e10	−	7.46766e10i	−0.616489	−	1.06779i	−0.990121	−	0.140213i	$-0.955221\pi$
$257$	−4.31146e10	−	7.46766e10i	−0.616489	−	1.06779i	0.373633	−	0.927577i	$-0.378112\pi$
$258$		0			0
$259$	−1.08322e11	+	1.87619e11i	−1.49578	+	2.59077i
$260$	9.73401e9			0.132103
$261$		0			0
$262$	7.95880e8			0.0104350
$263$	−5.72954e10	+	9.92386e10i	−0.738446	+	1.27903i	0.214748	+	0.976669i	$0.431107\pi$
$263$	−5.72954e10	+	9.92386e10i	−0.738446	+	1.27903i	−0.953195	+	0.302357i	$0.902226\pi$
$264$		0			0
$265$	5.26586e10	+	9.12074e10i	0.655938	+	1.13612i
$266$	4.67026e10	+	8.08914e10i	0.571971	+	0.990683i
$267$		0			0
$268$	3.04571e10	−	5.27533e10i	0.360647	−	0.624659i
$269$	1.28269e11			1.49361			0.746806	−	0.665042i	$-0.231586\pi$
$269$	1.28269e11			1.49361			0.746806	+	0.665042i	$0.231586\pi$
$270$		0			0
$271$	−7.44512e10			−0.838514			−0.419257	−	0.907868i	$-0.637709\pi$
$271$	−7.44512e10			−0.838514			−0.419257	+	0.907868i	$0.637709\pi$
$272$	2.16547e10	−	3.75070e10i	0.239879	−	0.415482i
$273$		0			0
$274$	5.42351e10	+	9.39379e10i	0.581303	+	1.00685i
$275$	−1.17076e10	−	2.02781e10i	−0.123444	−	0.213811i
$276$		0			0
$277$	1.83059e10	−	3.17067e10i	0.186823	−	0.323588i	−0.757366	−	0.652991i	$-0.773514\pi$
$277$	1.83059e10	−	3.17067e10i	0.186823	−	0.323588i	0.944189	+	0.329403i	$0.106847\pi$
$278$	7.60499e10			0.763654
$279$		0			0
$280$	−8.22348e10			−0.799548
$281$	3.75847e10	−	6.50987e10i	0.359611	−	0.622865i	−0.628285	−	0.777984i	$-0.716243\pi$
$281$	3.75847e10	−	6.50987e10i	0.359611	−	0.622865i	0.987896	+	0.155119i	$0.0495760\pi$
$282$		0			0
$283$	−1.69742e10	−	2.94001e10i	−0.157308	−	0.272465i	0.776589	−	0.630007i	$-0.216948\pi$
$283$	−1.69742e10	−	2.94001e10i	−0.157308	−	0.272465i	−0.933897	+	0.357542i	$0.883615\pi$
$284$	6.05678e9	+	1.04907e10i	0.0552471	+	0.0956908i
$285$		0			0
$286$	−3.12168e9	+	5.40691e9i	−0.0275893	+	0.0477861i
$287$	−1.45112e10			−0.126251
$288$		0			0
$289$	3.18133e11			2.68268
$290$	3.74183e10	−	6.48103e10i	0.310665	−	0.538088i
$291$		0			0
$292$	9.02768e9	+	1.56364e10i	0.0726697	+	0.125868i
$293$	−4.22072e10	−	7.31050e10i	−0.334566	−	0.579486i	0.648835	−	0.760929i	$-0.275257\pi$
$293$	−4.22072e10	−	7.31050e10i	−0.334566	−	0.579486i	−0.983401	+	0.181443i	$0.941923\pi$
$294$		0			0
$295$	2.06195e10	−	3.57140e10i	0.158518	−	0.274561i
$296$	−7.93598e10			−0.600880
$297$		0			0
$298$	1.15247e11			0.846557
$299$	1.11029e10	−	1.92309e10i	0.0803374	−	0.139148i
$300$		0			0
$301$	−8.22196e10	−	1.42409e11i	−0.577333	−	0.999970i
$302$	−3.97891e10	−	6.89167e10i	−0.275253	−	0.476753i
$303$		0			0
$304$	−1.71078e10	+	2.96316e10i	−0.114885	+	0.198987i
$305$	3.24622e11			2.14797
$306$		0			0
$307$	−2.60554e10			−0.167408			−0.0837039	−	0.996491i	$-0.526675\pi$
$307$	−2.60554e10			−0.167408			−0.0837039	+	0.996491i	$0.526675\pi$
$308$	2.63726e10	−	4.56786e10i	0.166984	−	0.289224i
$309$		0			0
$310$	7.74580e8	+	1.34161e9i	0.00476364	+	0.00825086i
$311$	7.50063e10	+	1.29915e11i	0.454648	+	0.787474i	0.998668	−	0.0515984i	$-0.0164316\pi$
$311$	7.50063e10	+	1.29915e11i	0.454648	+	0.787474i	−0.544019	+	0.839073i	$0.683098\pi$
$312$		0			0
$313$	−8.67447e10	+	1.50246e11i	−0.510850	+	0.884818i	0.489071	+	0.872244i	$0.337336\pi$
$313$	−8.67447e10	+	1.50246e11i	−0.510850	+	0.884818i	−0.999921	+	0.0125741i	$0.995997\pi$
$314$	−1.10963e10			−0.0644164
$315$		0			0
$316$	−6.59619e10			−0.372136
$317$	−9.58489e9	+	1.66015e10i	−0.0533115	+	0.0923382i	−0.891450	−	0.453120i	$-0.850311\pi$
$317$	−9.58489e9	+	1.66015e10i	−0.0533115	+	0.0923382i	0.838138	+	0.545458i	$0.183644\pi$
$318$		0			0
$319$	2.39999e10	+	4.15691e10i	0.129763	+	0.224757i
$320$	−1.50619e10	−	2.60879e10i	−0.0802979	−	0.139080i
$321$		0			0
$322$	−9.37998e10	+	1.62466e11i	−0.486240	+	0.842192i
$323$	−3.45022e11			−1.76374
$324$		0			0
$325$	2.69106e10			0.133798
$326$	−6.37428e10	+	1.10406e11i	−0.312573	+	0.541393i
$327$		0			0
$328$	−2.65783e9	−	4.60350e9i	−0.0126793	−	0.0219612i
$329$	1.71405e11	+	2.96882e11i	0.806568	+	1.39702i
$330$		0			0
$331$	1.01835e11	−	1.76383e11i	0.466305	−	0.807663i	−0.532955	−	0.846144i	$-0.678918\pi$
$331$	1.01835e11	−	1.76383e11i	0.466305	−	0.807663i	0.999259	+	0.0384806i	$0.0122518\pi$
$332$	−7.51336e10			−0.339401
$333$		0			0
$334$	9.82125e10			0.431825
$335$	2.13618e11	−	3.69997e11i	0.926694	−	1.60508i
$336$		0			0
$337$	9.54368e10	+	1.65301e11i	0.403071	+	0.698139i	0.994095	−	0.108515i	$-0.0346095\pi$
$337$	9.54368e10	+	1.65301e11i	0.403071	+	0.698139i	−0.591024	+	0.806654i	$0.701276\pi$
$338$	8.12483e10	+	1.40726e11i	0.338602	+	0.586475i
$339$		0			0
$340$	1.51880e11	−	2.63064e11i	0.616377	−	1.06760i
$341$	−9.93625e8			−0.00397949
$342$		0			0
$343$	−4.95599e11			−1.93333
$344$	3.01182e10	−	5.21663e10i	0.115962	−	0.200852i
$345$		0			0
$346$	−2.99230e10	−	5.18282e10i	−0.112244	−	0.194412i
$347$	−2.51960e11	−	4.36407e11i	−0.932928	−	1.61588i	−0.778287	−	0.627908i	$-0.783911\pi$
$347$	−2.51960e11	−	4.36407e11i	−0.932928	−	1.61588i	−0.154641	−	0.987971i	$-0.549422\pi$
$348$		0			0
$349$	1.33958e11	−	2.32022e11i	0.483341	−	0.837171i	−0.516476	−	0.856302i	$-0.672756\pi$
$349$	1.33958e11	−	2.32022e11i	0.483341	−	0.837171i	0.999817	+	0.0191305i	$0.00608980\pi$
$350$	−2.27346e11			−0.809808
$351$		0			0
$352$	1.93213e10			0.0670801
$353$	−2.71507e10	+	4.70264e10i	−0.0930668	+	0.161196i	−0.908800	−	0.417232i	$-0.863000\pi$
$353$	−2.71507e10	+	4.70264e10i	−0.0930668	+	0.161196i	0.815733	+	0.578428i	$0.196334\pi$
$354$		0			0
$355$	4.24807e10	+	7.35787e10i	0.141959	+	0.245881i
$356$	1.81148e10	+	3.13757e10i	0.0597733	+	0.103530i
$357$		0			0
$358$	−1.93694e11	+	3.35488e11i	−0.623222	+	1.07945i
$359$	2.39369e11			0.760577			0.380289	−	0.924868i	$-0.375825\pi$
$359$	2.39369e11			0.760577			0.380289	+	0.924868i	$0.375825\pi$
$360$		0			0
$361$	−5.01103e10			−0.155290
$362$	−9.22579e10	+	1.59795e11i	−0.282368	+	0.489075i
$363$		0			0
$364$	3.03095e10	+	5.24976e10i	0.0904947	+	0.156741i
$365$	6.33178e10	+	1.09670e11i	0.186727	+	0.323421i
$366$		0			0
$367$	2.12436e11	−	3.67950e11i	0.611267	−	1.05875i	−0.379760	−	0.925085i	$-0.623993\pi$
$367$	2.12436e11	−	3.67950e11i	0.611267	−	1.05875i	0.991027	−	0.133661i	$-0.0426733\pi$
$368$	−6.87203e10			−0.195331
$369$		0			0
$370$	−5.56608e11			−1.54398
$371$	−3.27934e11	+	5.67999e11i	−0.898678	+	1.55656i
$372$		0			0
$373$	1.10701e11	+	1.91740e11i	0.296117	+	0.512889i	0.975244	−	0.221131i	$-0.0709749\pi$
$373$	1.10701e11	+	1.91740e11i	0.296117	+	0.512889i	−0.679127	+	0.734021i	$0.737642\pi$
$374$	9.74154e10	+	1.68728e11i	0.257457	+	0.445929i
$375$		0			0
$376$	−6.27879e10	+	1.08752e11i	−0.162006	+	0.280602i
$377$	−5.51654e10			−0.140647
$378$		0			0
$379$	−7.92760e10			−0.197363			−0.0986814	−	0.995119i	$-0.531462\pi$
$379$	−7.92760e10			−0.197363			−0.0986814	+	0.995119i	$0.531462\pi$
$380$	−1.19990e11	+	2.07828e11i	−0.295201	+	0.511303i
$381$		0			0
$382$	1.45372e11	+	2.51792e11i	0.349298	+	0.605001i
$383$	−1.74198e11	−	3.01720e11i	−0.413666	−	0.716490i	0.581622	−	0.813460i	$-0.302418\pi$
$383$	−1.74198e11	−	3.01720e11i	−0.413666	−	0.716490i	−0.995287	+	0.0969692i	$0.969085\pi$
$384$		0			0
$385$	1.84970e11	−	3.20378e11i	0.429070	−	0.743171i
$386$	−3.05942e11			−0.701449
$387$		0			0
$388$	8.41316e10			0.188459
$389$	−4.65041e10	+	8.05475e10i	−0.102972	+	0.178352i	−0.912908	−	0.408166i	$-0.866168\pi$
$389$	−4.65041e10	+	8.05475e10i	−0.102972	+	0.178352i	0.809936	+	0.586518i	$0.199502\pi$
$390$		0			0
$391$	−3.46479e11	−	6.00120e11i	−0.749691	−	1.29850i
$392$	−1.73417e11	−	3.00366e11i	−0.370940	−	0.642487i
$393$		0			0
$394$	9.49142e10	−	1.64396e11i	0.198426	−	0.343684i
$395$	−4.62640e11			−0.956216
$396$		0			0
$397$	−2.16735e11			−0.437896			−0.218948	−	0.975737i	$-0.570262\pi$
$397$	−2.16735e11			−0.437896			−0.218948	+	0.975737i	$0.570262\pi$
$398$	1.77878e11	−	3.08093e11i	0.355343	−	0.615472i
$399$		0			0
$400$	−4.16401e10	−	7.21227e10i	−0.0813283	−	0.140865i
$401$	2.98152e11	+	5.16415e11i	0.575822	+	0.997353i	0.995952	+	0.0898892i	$0.0286513\pi$
$401$	2.98152e11	+	5.16415e11i	0.575822	+	0.997353i	−0.420130	+	0.907464i	$0.638015\pi$
$402$		0			0
$403$	5.70978e8	−	9.88963e8i	0.00107832	−	0.00186770i
$404$	−1.26705e11			−0.236635
$405$		0			0
$406$	4.66048e11			0.851263
$407$	1.78503e11	−	3.09177e11i	0.322457	−	0.558511i
$408$		0			0
$409$	−2.45553e11	−	4.25310e11i	−0.433901	−	0.751538i	0.563304	−	0.826249i	$-0.309530\pi$
$409$	−2.45553e11	−	4.25310e11i	−0.433901	−	0.751538i	−0.997205	+	0.0747111i	$0.976197\pi$
$410$	−1.86413e10	−	3.22877e10i	−0.0325799	−	0.0564300i
$411$		0			0
$412$	5.10792e10	−	8.84717e10i	0.0873386	−	0.151275i
$413$	2.56817e11			0.434360
$414$		0			0
$415$	−5.26967e11			−0.872102
$416$	−1.11028e10	+	1.92306e10i	−0.0181766	+	0.0314828i
$417$		0			0
$418$	−7.69610e10	−	1.33300e11i	−0.123304	−	0.213569i
$419$	−1.47003e11	−	2.54617e11i	−0.233004	−	0.403575i	0.725687	−	0.688025i	$-0.241522\pi$
$419$	−1.47003e11	−	2.54617e11i	−0.233004	−	0.403575i	−0.958691	+	0.284450i	$0.908189\pi$
$420$		0			0
$421$	4.84670e10	−	8.39473e10i	0.0751929	−	0.130238i	−0.825977	−	0.563703i	$-0.809376\pi$
$421$	4.84670e10	−	8.39473e10i	0.0751929	−	0.130238i	0.901170	+	0.433466i	$0.142709\pi$
$422$	8.97098e11			1.37700
$423$		0			0
$424$	−2.40254e11			−0.361014
$425$	4.19888e11	−	7.27268e11i	0.624286	−	1.08129i
$426$		0			0
$427$	1.01080e12	+	1.75075e12i	1.47143	+	2.54859i
$428$	−1.62534e10	−	2.81518e10i	−0.0234125	−	0.0405517i
$429$		0			0
$430$	2.11241e11	−	3.65880e11i	0.297968	−	0.516097i
$431$	1.11086e12			1.55064			0.775318	−	0.631571i	$-0.217590\pi$
$431$	1.11086e12			1.55064			0.775318	+	0.631571i	$0.217590\pi$
$432$		0			0
$433$	2.51266e11			0.343509			0.171755	−	0.985140i	$-0.445056\pi$
$433$	2.51266e11			0.343509			0.171755	+	0.985140i	$0.445056\pi$
$434$	−4.82374e9	+	8.35495e9i	−0.00652649	+	0.0113042i
$435$		0			0
$436$	2.82947e11	+	4.90079e11i	0.374987	+	0.649497i
$437$	2.73729e11	+	4.74112e11i	0.359049	+	0.621891i
$438$		0			0
$439$	−5.52531e11	+	9.57012e11i	−0.710013	+	1.22978i	0.254838	+	0.966984i	$0.417978\pi$
$439$	−5.52531e11	+	9.57012e11i	−0.710013	+	1.22978i	−0.964852	+	0.262795i	$0.915356\pi$
$440$	1.35514e11			0.172365
$441$		0			0
$442$	−2.23916e11			−0.279052
$443$	7.66592e9	−	1.32778e10i	0.00945687	−	0.0163798i	−0.861258	−	0.508168i	$-0.830323\pi$
$443$	7.66592e9	−	1.32778e10i	0.00945687	−	0.0163798i	0.870715	+	0.491788i	$0.163656\pi$
$444$		0			0
$445$	1.27052e11	+	2.20061e11i	0.153590	+	0.266025i
$446$	−5.14730e11	−	8.91539e11i	−0.615989	−	1.06692i
$447$		0			0
$448$	9.37986e10	−	1.62464e11i	0.110013	−	0.190549i
$449$	−1.10134e12			−1.27883			−0.639415	−	0.768862i	$-0.720823\pi$
$449$	−1.10134e12			−1.27883			−0.639415	+	0.768862i	$0.720823\pi$
$450$		0			0
$451$	2.39129e10			0.0272169
$452$	8.54925e10	−	1.48077e11i	0.0963396	−	0.166865i
$453$		0			0
$454$	9.90143e10	+	1.71498e11i	0.109382	+	0.189455i
$455$	2.12583e11	+	3.68205e11i	0.232529	+	0.402752i
$456$		0			0
$457$	1.74031e11	−	3.01431e11i	0.186640	−	0.323270i	−0.757488	−	0.652849i	$-0.773574\pi$
$457$	1.74031e11	−	3.01431e11i	0.186640	−	0.323270i	0.944128	+	0.329579i	$0.106907\pi$
$458$	6.79599e11			0.721702
$459$		0			0
$460$	−4.81986e11			−0.501908
$461$	8.85382e10	−	1.53353e11i	0.0913013	−	0.158138i	−0.816758	−	0.576981i	$-0.804231\pi$
$461$	8.85382e10	−	1.53353e11i	0.0913013	−	0.158138i	0.908059	+	0.418842i	$0.137564\pi$
$462$		0			0
$463$	−2.14420e10	−	3.71386e10i	−0.0216846	−	0.0375588i	0.854979	−	0.518662i	$-0.173570\pi$
$463$	−2.14420e10	−	3.71386e10i	−0.0216846	−	0.0375588i	−0.876664	+	0.481103i	$0.840236\pi$
$464$	8.53600e10	+	1.47848e11i	0.0854916	+	0.148076i
$465$		0			0
$466$	3.65257e10	−	6.32644e10i	0.0358808	−	0.0621474i
$467$	−1.23771e12			−1.20419			−0.602093	−	0.798426i	$-0.705666\pi$
$467$	−1.23771e12			−1.20419			−0.602093	+	0.798426i	$0.705666\pi$
$468$		0			0
$469$	2.66064e12			2.53926
$470$	−4.40378e11	+	7.62757e11i	−0.416280	+	0.721017i
$471$		0			0
$472$	4.70379e10	+	8.14720e10i	0.0436223	+	0.0755561i
$473$	1.35489e11	+	2.34674e11i	0.124460	+	0.215571i
$474$		0			0
$475$	−3.31724e11	+	5.74562e11i	−0.298989	+	0.517864i
$476$	1.89168e12			1.68895
$477$		0			0
$478$	1.03142e12			0.903667
$479$	−4.46116e11	+	7.72695e11i	−0.387202	+	0.670654i	−0.992072	−	0.125670i	$-0.959892\pi$
$479$	−4.46116e11	+	7.72695e11i	−0.387202	+	0.670654i	0.604870	+	0.796324i	$0.293225\pi$
$480$		0			0
$481$	2.05151e11	+	3.55332e11i	0.174751	+	0.302678i
$482$	4.93026e11	+	8.53947e11i	0.416062	+	0.720641i
$483$		0			0
$484$	2.58358e11	−	4.47489e11i	0.214002	−	0.370663i
$485$	5.90077e11			0.484251
$486$		0			0
$487$	−4.38243e11			−0.353049			−0.176524	−	0.984296i	$-0.556485\pi$
$487$	−4.38243e11			−0.353049			−0.176524	+	0.984296i	$0.556485\pi$
$488$	−3.70270e11	+	6.41326e11i	−0.295549	+	0.511905i
$489$		0			0
$490$	−1.21630e12	−	2.10669e12i	−0.953142	−	1.65089i
$491$	−6.10106e11	−	1.05673e12i	−0.473738	−	0.820539i	0.525810	−	0.850602i	$-0.323762\pi$
$491$	−6.10106e11	−	1.05673e12i	−0.473738	−	0.820539i	−0.999548	+	0.0300635i	$0.990429\pi$
$492$		0			0
$493$	−8.60749e11	+	1.49086e12i	−0.656244	+	1.13665i
$494$	1.76900e11			0.133646
$495$		0			0
$496$	−3.53400e9			−0.00262180
$497$	−2.64551e11	+	4.58215e11i	−0.194493	+	0.336872i
$498$		0			0
$499$	−1.02247e12	−	1.77097e12i	−0.738242	−	1.27867i	−0.953286	−	0.302068i	$-0.902323\pi$
$499$	−1.02247e12	−	1.77097e12i	−0.738242	−	1.27867i	0.215045	−	0.976604i	$-0.431010\pi$
$500$	1.56827e11	+	2.71632e11i	0.112216	+	0.194364i
$501$		0			0
$502$	1.20124e11	−	2.08060e11i	0.0844233	−	0.146225i
$503$	−2.19466e11			−0.152866			−0.0764329	−	0.997075i	$-0.524353\pi$
$503$	−2.19466e11			−0.152866			−0.0764329	+	0.997075i	$0.524353\pi$
$504$		0			0
$505$	−8.88676e11			−0.608040
$506$	1.54572e11	−	2.67727e11i	0.104822	−	0.181558i
$507$		0			0
$508$	2.71749e11	+	4.70683e11i	0.181043	+	0.313575i
$509$	−3.39312e11	−	5.87706e11i	−0.224063	−	0.388088i	0.731975	−	0.681331i	$-0.238599\pi$
$509$	−3.39312e11	−	5.87706e11i	−0.224063	−	0.388088i	−0.956038	+	0.293243i	$0.905265\pi$
$510$		0			0
$511$	−3.94314e11	+	6.82973e11i	−0.255828	+	0.443108i
$512$	6.87195e10			0.0441942
$513$		0			0
$514$	1.37967e12			0.871847
$515$	3.58256e11	−	6.20517e11i	0.224420	−	0.388706i
$516$		0			0
$517$	−2.82457e11	−	4.89229e11i	−0.173878	−	0.301165i
$518$	−1.73315e12	−	3.00191e12i	−1.05768	−	1.83195i
$519$		0			0
$520$	−7.78721e10	+	1.34878e11i	−0.0467054	+	0.0808961i
$521$	−3.00159e12			−1.78477			−0.892383	−	0.451279i	$-0.850968\pi$
$521$	−3.00159e12			−1.78477			−0.892383	+	0.451279i	$0.850968\pi$
$522$		0			0
$523$	−1.25398e12			−0.732881			−0.366440	−	0.930442i	$-0.619424\pi$
$523$	−1.25398e12			−0.732881			−0.366440	+	0.930442i	$0.619424\pi$
$524$	−6.36704e9	+	1.10280e10i	−0.00368932	+	0.00639010i
$525$		0			0
$526$	−9.16727e11	−	1.58782e12i	−0.522160	−	0.904408i
$527$	−1.78180e10	−	3.08617e10i	−0.0100626	−	0.0174290i
$528$		0			0
$529$	3.50807e11	−	6.07615e11i	0.194768	−	0.337348i
$530$	−1.68508e12			−0.927637
$531$		0			0
$532$	−1.49448e12			−0.808889
$533$	−1.37414e10	+	2.38008e10i	−0.00737493	+	0.0127737i
$534$		0			0
$535$	−1.13997e11	−	1.97449e11i	−0.0601593	−	0.104199i
$536$	4.87314e11	+	8.44052e11i	0.255016	+	0.441700i
$537$		0			0
$538$	−1.02615e12	+	1.77735e12i	−0.528071	+	0.914647i
$539$	1.56026e12			0.796245
$540$		0			0
$541$	1.80539e11			0.0906113			0.0453056	−	0.998973i	$-0.485574\pi$
$541$	1.80539e11			0.0906113			0.0453056	+	0.998973i	$0.485574\pi$
$542$	5.95610e11	−	1.03163e12i	0.296459	−	0.513483i
$543$		0			0
$544$	3.46475e11	+	6.00112e11i	0.169620	+	0.293790i
$545$	1.98452e12	+	3.43729e12i	0.963542	+	1.66890i
$546$		0			0
$547$	7.51495e11	−	1.30163e12i	0.358908	−	0.621647i	−0.628871	−	0.777510i	$-0.716483\pi$
$547$	7.51495e11	−	1.30163e12i	0.358908	−	0.621647i	0.987779	+	0.155863i	$0.0498159\pi$
$548$	−1.73552e12			−0.822087
$549$		0			0
$550$	3.74643e11			0.174576
$551$	6.80016e11	−	1.17782e12i	0.314295	−	0.544374i
$552$		0			0
$553$	−1.44056e12	−	2.49511e12i	−0.655039	−	1.13456i
$554$	2.92894e11	+	5.07307e11i	0.132104	+	0.228811i
$555$		0			0
$556$	−6.08399e11	+	1.05378e12i	−0.269993	+	0.467641i
$557$	−1.28134e12			−0.564047			−0.282023	−	0.959408i	$-0.591006\pi$
$557$	−1.28134e12			−0.564047			−0.282023	+	0.959408i	$0.591006\pi$
$558$		0			0
$559$	−3.11431e11			−0.134899
$560$	6.57879e11	−	1.13948e12i	0.282683	−	0.489621i
$561$		0			0
$562$	6.01356e11	+	1.04158e12i	0.254283	+	0.440432i
$563$	1.18810e12	+	2.05784e12i	0.498384	+	0.863227i	0.999998	−	0.00186484i	$-0.000593596\pi$
$563$	1.18810e12	+	2.05784e12i	0.498384	+	0.863227i	−0.501614	+	0.865091i	$0.667260\pi$
$564$		0			0
$565$	5.99622e11	−	1.03858e12i	0.247548	−	0.428766i
$566$	5.43174e11			0.222467
$567$		0			0
$568$	−1.93817e11			−0.0781312
$569$	1.47643e11	−	2.55724e11i	0.0590481	−	0.102274i	−0.834990	−	0.550265i	$-0.814527\pi$
$569$	1.47643e11	−	2.55724e11i	0.0590481	−	0.102274i	0.894038	+	0.447990i	$0.147860\pi$
$570$		0			0
$571$	−3.77029e11	−	6.53034e11i	−0.148427	−	0.257083i	0.782219	−	0.623003i	$-0.214088\pi$
$571$	−3.77029e11	−	6.53034e11i	−0.148427	−	0.257083i	−0.930646	+	0.365920i	$0.880754\pi$
$572$	−4.99469e10	−	8.65105e10i	−0.0195086	−	0.0337899i
$573$		0			0
$574$	1.16090e11	−	2.01073e11i	0.0446365	−	0.0773127i
$575$	−1.33250e12			−0.508349
$576$		0			0
$577$	−2.25656e12			−0.847532			−0.423766	−	0.905772i	$-0.639292\pi$
$577$	−2.25656e12			−0.847532			−0.423766	+	0.905772i	$0.639292\pi$
$578$	−2.54507e12	+	4.40818e12i	−0.948470	+	1.64280i
$579$		0			0
$580$	5.98692e11	+	1.03697e12i	0.219674	+	0.380486i
$581$	−1.64086e12	−	2.84205e12i	−0.597418	−	1.03476i
$582$		0			0
$583$	5.40401e11	−	9.36001e11i	0.193735	−	0.335558i
$584$	−2.88886e11			−0.102770
$585$		0			0
$586$	1.35063e12			0.473148
$587$	5.51711e11	−	9.55592e11i	0.191796	−	0.332201i	−0.754049	−	0.656818i	$-0.771902\pi$
$587$	5.51711e11	−	9.55592e11i	0.191796	−	0.332201i	0.945846	+	0.324617i	$0.105235\pi$
$588$		0			0
$589$	1.40767e10	+	2.43816e10i	0.00481929	+	0.00834725i
$590$	3.29911e11	+	5.71423e11i	0.112089	+	0.194144i
$591$		0			0
$592$	6.34878e11	−	1.09964e12i	0.212443	−	0.367962i
$593$	−8.96636e11			−0.297763			−0.148881	−	0.988855i	$-0.547567\pi$
$593$	−8.96636e11			−0.297763			−0.148881	+	0.988855i	$0.547567\pi$
$594$		0			0
$595$	1.32678e13			4.33982
$596$	−9.21976e11	+	1.59691e12i	−0.299303	+	0.518408i
$597$		0			0
$598$	1.77647e11	+	3.07694e11i	0.0568071	+	0.0983928i
$599$	1.76747e12	+	3.06135e12i	0.560959	+	0.971610i	0.997413	+	0.0718830i	$0.0229008\pi$
$599$	1.76747e12	+	3.06135e12i	0.560959	+	0.971610i	−0.436454	+	0.899727i	$0.643766\pi$
$600$		0			0
$601$	−2.23158e12	+	3.86522e12i	−0.697715	+	1.20848i	0.271541	+	0.962427i	$0.412467\pi$
$601$	−2.23158e12	+	3.86522e12i	−0.697715	+	1.20848i	−0.969257	+	0.246052i	$0.920867\pi$
$602$	2.63103e12			0.816472
$603$		0			0
$604$	1.27325e12			0.389267
$605$	1.81206e12	−	3.13857e12i	0.549886	−	0.952430i
$606$		0			0
$607$	9.84361e11	+	1.70496e12i	0.294310	+	0.509760i	0.974824	−	0.222975i	$-0.0715768\pi$
$607$	9.84361e11	+	1.70496e12i	0.294310	+	0.509760i	−0.680514	+	0.732735i	$0.738243\pi$
$608$	−2.73725e11	−	4.74106e11i	−0.0812360	−	0.140705i
$609$		0			0
$610$	−2.59697e12	+	4.49809e12i	−0.759422	+	1.31536i
$611$	6.49245e11			0.188462
$612$		0			0
$613$	−5.68828e12			−1.62708			−0.813540	−	0.581509i	$-0.802463\pi$
$613$	−5.68828e12			−1.62708			−0.813540	+	0.581509i	$0.802463\pi$
$614$	2.08444e11	−	3.61035e11i	0.0591876	−	0.102516i
$615$		0			0
$616$	4.21961e11	+	7.30858e11i	0.118075	+	0.204512i
$617$	−1.36075e12	−	2.35689e12i	−0.378003	−	0.654720i	0.612769	−	0.790262i	$-0.290056\pi$
$617$	−1.36075e12	−	2.35689e12i	−0.378003	−	0.654720i	−0.990772	+	0.135542i	$0.956722\pi$
$618$		0			0
$619$	4.27121e11	−	7.39795e11i	0.116935	−	0.202537i	−0.801617	−	0.597838i	$-0.796027\pi$
$619$	4.27121e11	−	7.39795e11i	0.116935	−	0.202537i	0.918551	+	0.395301i	$0.129360\pi$
$620$	−2.47866e10			−0.00673680
$621$		0			0
$622$	−2.40020e12			−0.642970
$623$	−7.91223e11	+	1.37044e12i	−0.210428	+	0.364471i
$624$		0			0
$625$	2.34091e12	+	4.05458e12i	0.613656	+	1.06288i
$626$	−1.38791e12	−	2.40394e12i	−0.361225	−	0.625661i
$627$		0			0
$628$	8.87707e10	−	1.53755e11i	0.0227746	−	0.0394468i
$629$	1.28039e13			3.26148
$630$		0			0
$631$	1.82693e12			0.458764			0.229382	−	0.973336i	$-0.426330\pi$
$631$	1.82693e12			0.458764			0.229382	+	0.973336i	$0.426330\pi$
$632$	5.27695e11	−	9.13995e11i	0.131570	−	0.227886i
$633$		0			0
$634$	−1.53358e11	−	2.65624e11i	−0.0376969	−	0.0652929i
$635$	1.90597e12	+	3.30124e12i	0.465195	+	0.805742i
$636$		0			0
$637$	−8.96589e11	+	1.55294e12i	−0.215758	+	0.373703i
$638$	−7.67998e11			−0.183513
$639$		0			0
$640$	4.81980e11			0.113558
$641$	3.56045e12	−	6.16687e12i	0.832997	−	1.44279i	−0.0626545	−	0.998035i	$-0.519957\pi$
$641$	3.56045e12	−	6.16687e12i	0.832997	−	1.44279i	0.895651	−	0.444757i	$-0.146710\pi$
$642$		0			0
$643$	−3.01320e12	−	5.21902e12i	−0.695151	−	1.20404i	−0.970130	−	0.242586i	$-0.922004\pi$
$643$	−3.01320e12	−	5.21902e12i	−0.695151	−	1.20404i	0.274979	−	0.961450i	$-0.411329\pi$
$644$	−1.50080e12	−	2.59946e12i	−0.343823	−	0.595520i
$645$		0			0
$646$	2.76018e12	−	4.78077e12i	0.623578	−	1.08007i
$647$	5.11089e12			1.14664			0.573320	−	0.819332i	$-0.305655\pi$
$647$	5.11089e12			1.14664			0.573320	+	0.819332i	$0.305655\pi$
$648$		0			0
$649$	−4.23208e11			−0.0936381
$650$	−2.15285e11	+	3.72885e11i	−0.0473047	+	0.0819341i
$651$		0			0
$652$	−1.01988e12	−	1.76649e12i	−0.221023	−	0.382822i
$653$	−3.26957e12	−	5.66306e12i	−0.703689	−	1.21883i	−0.967163	−	0.254159i	$-0.918201\pi$
$653$	−3.26957e12	−	5.66306e12i	−0.703689	−	1.21883i	0.263473	−	0.964667i	$-0.415132\pi$
$654$		0			0
$655$	−4.46567e10	+	7.73478e10i	−0.00947984	+	0.0164196i
$656$	8.50506e10			0.0179312
$657$		0			0
$658$	−5.48495e12			−1.14066
$659$	−9.73090e11	+	1.68544e12i	−0.200987	+	0.348120i	−0.948847	−	0.315737i	$-0.897748\pi$
$659$	−9.73090e11	+	1.68544e12i	−0.200987	+	0.348120i	0.747860	+	0.663857i	$0.231082\pi$
$660$		0			0
$661$	8.94843e11	+	1.54991e12i	0.182323	+	0.315792i	0.942671	−	0.333723i	$-0.108305\pi$
$661$	8.94843e11	+	1.54991e12i	0.182323	+	0.315792i	−0.760348	+	0.649515i	$0.774972\pi$
$662$	1.62935e12	+	2.82212e12i	0.329727	+	0.571104i
$663$		0			0
$664$	6.01069e11	−	1.04108e12i	0.119996	−	0.207840i
$665$	−1.04819e13			−2.07847
$666$		0			0
$667$	2.73155e12			0.534372
$668$	−7.85700e11	+	1.36087e12i	−0.152673	+	0.264438i
$669$		0			0
$670$	3.41789e12	+	5.91996e12i	0.655272	+	1.13496i
$671$	−1.66569e12	−	2.88506e12i	−0.317207	−	0.549418i
$672$		0			0
$673$	2.62493e12	−	4.54651e12i	0.493230	−	0.854300i	−0.506740	−	0.862099i	$-0.669149\pi$
$673$	2.62493e12	−	4.54651e12i	0.493230	−	0.854300i	0.999970	+	0.00779968i	$0.00248274\pi$
$674$	−3.05398e12			−0.570028
$675$		0			0
$676$	−2.59995e12			−0.478855
$677$	−5.01631e12	+	8.68851e12i	−0.917774	+	1.58963i	−0.114984	+	0.993367i	$0.536682\pi$
$677$	−5.01631e12	+	8.68851e12i	−0.917774	+	1.58963i	−0.802789	+	0.596263i	$0.796652\pi$
$678$		0			0
$679$	1.83737e12	+	3.18241e12i	0.331728	+	0.574569i
$680$	2.43008e12	+	4.20903e12i	0.435844	+	0.754904i
$681$		0			0
$682$	7.94900e9	−	1.37681e10i	0.00140696	−	0.00243693i
$683$	2.26372e12			0.398043			0.199021	−	0.979995i	$-0.436224\pi$
$683$	2.26372e12			0.398043			0.199021	+	0.979995i	$0.436224\pi$
$684$		0			0
$685$	−1.21725e13			−2.11238
$686$	3.96479e12	−	6.86722e12i	0.683537	−	1.18392i
$687$		0			0
$688$	4.81891e11	+	8.34660e11i	0.0819976	+	0.142024i
$689$	6.21073e11	+	1.07573e12i	0.104992	+	0.181852i
$690$		0			0
$691$	1.70919e12	−	2.96041e12i	0.285193	−	0.493970i	−0.687463	−	0.726220i	$-0.741275\pi$
$691$	1.70919e12	−	2.96041e12i	0.285193	−	0.493970i	0.972656	+	0.232250i	$0.0746088\pi$
$692$	9.57536e11			0.158737
$693$		0			0
$694$	8.06271e12			1.31936
$695$	−4.26715e12	+	7.39092e12i	−0.693755	+	1.20162i
$696$		0			0
$697$	4.28815e11	+	7.42729e11i	0.0688212	+	0.119202i
$698$	2.14333e12	+	3.71235e12i	0.341774	+	0.591969i
$699$		0			0
$700$	1.81877e12	−	3.15020e12i	0.286310	−	0.495904i
$701$	−6.70506e12			−1.04875			−0.524374	−	0.851488i	$-0.675701\pi$
$701$	−6.70506e12			−1.04875			−0.524374	+	0.851488i	$0.675701\pi$
$702$		0			0
$703$	−1.01155e13			−1.56202
$704$	−1.54570e11	+	2.67723e11i	−0.0237164	+	0.0410780i
$705$		0			0
$706$	−4.34411e11	−	7.52422e11i	−0.0658082	−	0.113983i
$707$	−2.76714e12	−	4.79282e12i	−0.416527	−	0.721447i
$708$		0			0
$709$	2.50944e12	−	4.34647e12i	0.372965	−	0.645995i	−0.617055	−	0.786920i	$-0.711674\pi$
$709$	2.50944e12	−	4.34647e12i	0.372965	−	0.645995i	0.990020	+	0.140925i	$0.0450077\pi$
$710$	−1.35938e12			−0.200761
$711$		0			0
$712$	−5.79672e11			−0.0845323
$713$	−2.82724e10	+	4.89692e10i	−0.00409694	+	0.00709611i
$714$		0			0
$715$	−3.50314e11	−	6.06762e11i	−0.0501280	−	0.0868243i
$716$	−3.09910e12	−	5.36781e12i	−0.440684	−	0.763288i
$717$		0			0
$718$	−1.91495e12	+	3.31680e12i	−0.268905	+	0.465756i
$719$	9.03791e11			0.126121			0.0630606	−	0.998010i	$-0.479914\pi$
$719$	9.03791e11			0.126121			0.0630606	+	0.998010i	$0.479914\pi$
$720$		0			0
$721$	4.46211e12			0.614939
$722$	4.00882e11	−	6.94348e11i	0.0549034	−	0.0950955i
$723$		0			0
$724$	−1.47613e12	−	2.55673e12i	−0.199664	−	0.345829i
$725$	1.65514e12	+	2.86679e12i	0.222492	+	0.385368i
$726$		0			0
$727$	7.29367e12	−	1.26330e13i	0.968369	−	1.67727i	0.268094	−	0.963393i	$-0.413606\pi$
$727$	7.29367e12	−	1.26330e13i	0.968369	−	1.67727i	0.700276	−	0.713872i	$-0.253060\pi$
$728$	−9.69905e11			−0.127979
$729$		0			0
$730$	−2.02617e12			−0.264072
$731$	−4.85927e12	+	8.41650e12i	−0.629423	+	1.09019i
$732$		0			0
$733$	1.46865e12	+	2.54378e12i	0.187911	+	0.325471i	0.944553	−	0.328358i	$-0.106495\pi$
$733$	1.46865e12	+	2.54378e12i	0.187911	+	0.325471i	−0.756643	+	0.653828i	$0.773162\pi$
$734$	3.39898e12	+	5.88720e12i	0.432231	+	0.748646i
$735$		0			0
$736$	5.49763e11	−	9.52217e11i	0.0690598	−	0.119615i
$737$	−4.38444e12			−0.547407
$738$		0			0
$739$	−1.26838e13			−1.56441			−0.782206	−	0.623020i	$-0.785905\pi$
$739$	−1.26838e13			−1.56441			−0.782206	+	0.623020i	$0.785905\pi$
$740$	4.45287e12	−	7.71259e12i	0.545880	−	0.945492i
$741$		0			0
$742$	−5.24695e12	−	9.08798e12i	−0.635461	−	1.10065i
$743$	5.84236e11	+	1.01193e12i	0.0703296	+	0.121814i	0.899046	−	0.437855i	$-0.144262\pi$
$743$	5.84236e11	+	1.01193e12i	0.0703296	+	0.121814i	−0.828716	+	0.559669i	$0.810928\pi$
$744$		0			0
$745$	−6.46649e12	+	1.12003e13i	−0.769070	+	1.33207i
$746$	−3.54244e12			−0.418772
$747$		0			0
$748$	−3.11729e12			−0.364100
$749$	7.09924e11	−	1.22963e12i	0.0824221	−	0.142759i
$750$		0			0
$751$	5.94877e12	+	1.03036e13i	0.682413	+	1.18197i	0.974242	+	0.225504i	$0.0724029\pi$
$751$	5.94877e12	+	1.03036e13i	0.682413	+	1.18197i	−0.291829	+	0.956471i	$0.594264\pi$
$752$	−1.00461e12	−	1.74003e12i	−0.114555	−	0.198416i
$753$		0			0
$754$	4.41323e11	−	7.64395e11i	0.0497263	−	0.0861284i
$755$	8.93024e12			1.00023
$756$		0			0
$757$	5.40853e12			0.598615			0.299308	−	0.954157i	$-0.403244\pi$
$757$	5.40853e12			0.598615			0.299308	+	0.954157i	$0.403244\pi$
$758$	6.34208e11	−	1.09848e12i	0.0697783	−	0.120859i
$759$		0			0
$760$	−1.91984e12	−	3.32525e12i	−0.208739	−	0.361546i
$761$	−1.70400e12	−	2.95141e12i	−0.184178	−	0.319006i	0.759121	−	0.650949i	$-0.225629\pi$
$761$	−1.70400e12	−	2.95141e12i	−0.184178	−	0.319006i	−0.943299	+	0.331944i	$0.892296\pi$
$762$		0			0
$763$	−1.23587e13	+	2.14059e13i	−1.32011	+	2.28651i
$764$	−4.65190e12			−0.493981
$765$		0			0
$766$	5.57435e12			0.585012
$767$	2.43193e11	−	4.21222e11i	0.0253730	−	0.0439473i
$768$		0			0
$769$	1.25469e12	+	2.17319e12i	0.129381	+	0.224094i	0.923437	−	0.383751i	$-0.125368\pi$
$769$	1.25469e12	+	2.17319e12i	0.129381	+	0.224094i	−0.794056	+	0.607844i	$0.792034\pi$
$770$	2.95952e12	+	5.12604e12i	0.303398	+	0.525501i
$771$		0			0
$772$	2.44754e12	−	4.23926e12i	0.248000	−	0.429548i
$773$	−1.35883e13			−1.36885			−0.684427	−	0.729082i	$-0.739948\pi$
$773$	−1.35883e13			−1.36885			−0.684427	+	0.729082i	$0.739948\pi$
$774$		0			0
$775$	−6.85249e10			−0.00682324
$776$	−6.73053e11	+	1.16576e12i	−0.0666303	+	0.115407i
$777$		0			0
$778$	−7.44066e11	−	1.28876e12i	−0.0728120	−	0.126114i
$779$	−3.38776e11	−	5.86777e11i	−0.0329605	−	0.0570893i
$780$		0			0
$781$	4.35951e11	−	7.55089e11i	0.0419284	−	0.0726221i
$782$	1.10873e13			1.06022
$783$		0			0
$784$	5.54933e12			0.524588
$785$	6.22614e11	−	1.07840e12i	0.0585202	−	0.101360i
$786$		0			0
$787$	6.70949e10	+	1.16212e11i	0.00623452	+	0.0107985i	0.869126	−	0.494591i	$-0.164682\pi$
$787$	6.70949e10	+	1.16212e11i	0.00623452	+	0.0107985i	−0.862891	+	0.505390i	$0.831349\pi$
$788$	1.51863e12	+	2.63034e12i	0.140308	+	0.243021i
$789$		0			0
$790$	3.70112e12	−	6.41052e12i	0.338073	−	0.585560i
$791$	7.46835e12			0.678313
$792$		0			0
$793$	3.82869e12			0.343812
$794$	1.73388e12	−	3.00316e12i	0.154820	−	0.268155i
$795$		0			0
$796$	2.84604e12	+	4.92949e12i	0.251265	+	0.435204i
$797$	−7.58270e12	−	1.31336e13i	−0.665674	−	1.15298i	−0.979102	−	0.203369i	$-0.934811\pi$
$797$	−7.58270e12	−	1.31336e13i	−0.665674	−	1.15298i	0.313428	−	0.949612i	$-0.398522\pi$
$798$		0			0
$799$	1.01302e13	−	1.75460e13i	0.879342	−	1.52306i
$800$	1.33248e12			0.115016
$801$		0			0
$802$	−9.54087e12			−0.814336
$803$	6.49788e11	−	1.12547e12i	0.0551508	−	0.0955240i
$804$		0			0
$805$	−1.05262e13	−	1.82319e13i	−0.883466	−	1.53021i
$806$	9.13565e9	+	1.58234e10i	0.000762486	+	0.00132066i
$807$		0			0
$808$	1.01364e12	−	1.75568e12i	0.0836629	−	0.144908i
$809$	4.97475e12			0.408322			0.204161	−	0.978937i	$-0.434553\pi$
$809$	4.97475e12			0.408322			0.204161	+	0.978937i	$0.434553\pi$
$810$		0			0
$811$	1.46659e13			1.19046			0.595230	−	0.803555i	$-0.297061\pi$
$811$	1.46659e13			1.19046			0.595230	+	0.803555i	$0.297061\pi$
$812$	−3.72839e12	+	6.45775e12i	−0.300967	+	0.521290i
$813$		0			0
$814$	2.85605e12	+	4.94683e12i	0.228011	+	0.394927i
$815$	−7.15320e12	−	1.23897e13i	−0.567925	−	0.983675i
$816$		0			0
$817$	3.83896e12	−	6.64928e12i	0.301450	−	0.522126i
$818$	7.85770e12			0.613628
$819$		0			0
$820$	5.96522e11			0.0460749
$821$	1.22211e13	−	2.11675e13i	0.938784	−	1.62602i	0.171040	−	0.985264i	$-0.445287\pi$
$821$	1.22211e13	−	2.11675e13i	0.938784	−	1.62602i	0.767744	−	0.640757i	$-0.221379\pi$
$822$		0			0
$823$	8.82302e12	+	1.52819e13i	0.670375	+	1.16112i	0.977798	+	0.209551i	$0.0672003\pi$
$823$	8.82302e12	+	1.52819e13i	0.670375	+	1.16112i	−0.307422	+	0.951573i	$0.599466\pi$
$824$	8.17267e11	+	1.41555e12i	0.0617577	+	0.106968i
$825$		0			0
$826$	−2.05454e12	+	3.55857e12i	−0.153569	+	0.265990i
$827$	1.53876e13			1.14392			0.571959	−	0.820282i	$-0.306184\pi$
$827$	1.53876e13			1.14392			0.571959	+	0.820282i	$0.306184\pi$
$828$		0			0
$829$	−1.22874e13			−0.903579			−0.451789	−	0.892125i	$-0.649214\pi$
$829$	−1.22874e13			−0.903579			−0.451789	+	0.892125i	$0.649214\pi$
$830$	4.21574e12	−	7.30188e12i	0.308335	−	0.534051i
$831$		0			0
$832$	−1.77645e11	−	3.07690e11i	−0.0128528	−	0.0222617i
$833$	2.79790e13	+	4.84611e13i	2.01340	+	3.48731i
$834$		0			0
$835$	−5.51069e12	+	9.54479e12i	−0.392299	+	0.679481i
$836$	2.46275e12			0.174378
$837$		0			0
$838$	4.70410e12			0.329518
$839$	−2.51427e12	+	4.35484e12i	−0.175179	+	0.303419i	−0.940223	−	0.340559i	$-0.889384\pi$
$839$	−2.51427e12	+	4.35484e12i	−0.175179	+	0.303419i	0.765044	+	0.643978i	$0.222717\pi$
$840$		0			0
$841$	3.86061e12	+	6.68678e12i	0.266118	+	0.460930i
$842$	7.75472e11	+	1.34316e12i	0.0531694	+	0.0920921i
$843$		0			0
$844$	−7.17678e12	+	1.24306e13i	−0.486843	+	0.843236i
$845$	−1.82353e13			−1.23043
$846$		0			0
$847$	2.25693e13			1.50676
$848$	1.92203e12	−	3.32905e12i	0.127638	−	0.221075i
$849$		0			0
$850$	6.71821e12	+	1.16363e13i	0.441437	+	0.764591i
$851$	−1.01582e13	−	1.75945e13i	−0.663946	−	1.14999i
$852$		0			0
$853$	−5.96108e12	+	1.03249e13i	−0.385526	+	0.667751i	−0.991842	−	0.127473i	$-0.959313\pi$
$853$	−5.96108e12	+	1.03249e13i	−0.385526	+	0.667751i	0.606316	+	0.795224i	$0.292647\pi$
$854$	−3.23456e13			−2.08091
$855$		0			0
$856$	5.20110e11			0.0331103
$857$	2.85974e12	−	4.95321e12i	0.181097	−	0.313670i	−0.761157	−	0.648568i	$-0.775368\pi$
$857$	2.85974e12	−	4.95321e12i	0.181097	−	0.313670i	0.942255	+	0.334898i	$0.108702\pi$
$858$		0			0
$859$	2.91056e12	+	5.04124e12i	0.182393	+	0.315914i	0.942695	−	0.333656i	$-0.108282\pi$
$859$	2.91056e12	+	5.04124e12i	0.182393	+	0.315914i	−0.760302	+	0.649570i	$0.774949\pi$
$860$	3.37986e12	+	5.85408e12i	0.210696	+	0.364935i
$861$		0			0
$862$	−8.88685e12	+	1.53925e13i	−0.548233	+	0.949567i
$863$	1.65530e13			1.01585			0.507924	−	0.861402i	$-0.330413\pi$
$863$	1.65530e13			1.01585			0.507924	+	0.861402i	$0.330413\pi$
$864$		0			0
$865$	6.71591e12			0.407880
$866$	−2.01013e12	+	3.48164e12i	−0.121449	+	0.210356i
$867$		0			0
$868$	−7.71798e10	−	1.33679e11i	−0.00461493	−	0.00799329i
$869$	2.37388e12	+	4.11168e12i	0.141211	+	0.244585i
$870$		0			0
$871$	2.51948e12	−	4.36387e12i	0.148330	−	0.256915i
$872$	−9.05431e12			−0.530312
$873$		0			0
$874$	−8.75932e12			−0.507772
$875$	−6.84994e12	+	1.18644e13i	−0.395048	+	0.684244i
$876$		0			0
$877$	−1.53953e13	−	2.66654e13i	−0.878799	−	1.52213i	−0.852659	−	0.522467i	$-0.825012\pi$
$877$	−1.53953e13	−	2.66654e13i	−0.878799	−	1.52213i	−0.0261400	−	0.999658i	$-0.508322\pi$
$878$	−8.84050e12	−	1.53122e13i	−0.502055	−	0.869585i
$879$		0			0
$880$	−1.08411e12	+	1.87774e12i	−0.0609401	+	0.105551i
$881$	2.43879e13			1.36390			0.681949	−	0.731399i	$-0.261132\pi$
$881$	2.43879e13			1.36390			0.681949	+	0.731399i	$0.261132\pi$
$882$		0			0
$883$	−5.39949e12			−0.298902			−0.149451	−	0.988769i	$-0.547751\pi$
$883$	−5.39949e12			−0.298902			−0.149451	+	0.988769i	$0.547751\pi$
$884$	1.79133e12	−	3.10267e12i	0.0986596	−	0.170883i
$885$		0			0
$886$	1.22655e11	+	2.12444e11i	0.00668702	+	0.0115823i
$887$	1.32250e12	+	2.29063e12i	0.0717361	+	0.124251i	0.899662	−	0.436587i	$-0.143813\pi$
$887$	1.32250e12	+	2.29063e12i	0.0717361	+	0.124251i	−0.827926	+	0.560837i	$0.810479\pi$
$888$		0			0
$889$	−1.18696e13	+	2.05587e13i	−0.637348	+	1.10392i
$890$	−4.06567e12			−0.217209
$891$		0			0
$892$	1.64714e13			0.871139
$893$	−8.00315e12	+	1.38619e13i	−0.421143	+	0.729441i
$894$		0			0
$895$	−2.17363e13	−	3.76484e13i	−1.13235	−	1.96129i
$896$	1.50078e12	+	2.59942e12i	0.0777912	+	0.134738i
$897$		0			0
$898$	8.81072e12	−	1.52606e13i	0.452135	−	0.783120i
$899$	1.40472e11			0.00717253
$900$		0			0
$901$	3.87625e13			1.95952
$902$	−1.91304e11	+	3.31347e11i	−0.00962263	+	0.0166669i
$903$		0			0
$904$	1.36788e12	+	2.36924e12i	0.0681224	+	0.117991i
$905$	−1.03532e13	−	1.79322e13i	−0.513044	−	0.888618i
$906$		0			0
$907$	−2.85590e12	+	4.94657e12i	−0.140123	+	0.242701i	−0.927543	−	0.373717i	$-0.878083\pi$
$907$	−2.85590e12	+	4.94657e12i	−0.140123	+	0.242701i	0.787420	+	0.616417i	$0.211417\pi$
$908$	−3.16846e12			−0.154690
$909$		0			0
$910$	−6.80266e12			−0.328846
$911$	7.23101e12	−	1.25245e13i	0.347829	−	0.602458i	−0.638034	−	0.770008i	$-0.720252\pi$
$911$	7.23101e12	−	1.25245e13i	0.347829	−	0.602458i	0.985864	+	0.167550i	$0.0535855\pi$
$912$		0			0
$913$	2.70396e12	+	4.68339e12i	0.128790	+	0.223071i
$914$	2.78450e12	+	4.82290e12i	0.131974	+	0.228586i
$915$		0			0
$916$	−5.43679e12	+	9.41680e12i	−0.255160	+	0.441951i
$917$	−5.56204e11			−0.0259760
$918$		0			0
$919$	−3.39599e13			−1.57053			−0.785266	−	0.619158i	$-0.787474\pi$
$919$	−3.39599e13			−1.57053			−0.785266	+	0.619158i	$0.787474\pi$
$920$	3.85589e12	−	6.67860e12i	0.177451	−	0.307355i
$921$		0			0
$922$	1.41661e12	+	2.45364e12i	0.0645597	+	0.111821i
$923$	5.01031e11	+	8.67811e11i	0.0227226	+	0.0393566i
$924$		0			0
$925$	1.23104e13	−	2.13222e13i	0.552884	−	0.957623i
$926$	6.86144e11			0.0306666
$927$		0			0
$928$	−2.73152e12			−0.120903
$929$	−1.14626e13	+	1.98538e13i	−0.504908	+	0.874526i	0.495076	+	0.868849i	$0.335140\pi$
$929$	−1.14626e13	+	1.98538e13i	−0.504908	+	0.874526i	−0.999984	+	0.00567608i	$0.998193\pi$
$930$		0			0
$931$	−2.21042e13	−	3.82857e13i	−0.964278	−	1.67018i
$932$	5.84412e11	+	1.01223e12i	0.0253716	+	0.0439448i
$933$		0			0
$934$	9.90169e12	−	1.71502e13i	0.425744	−	0.737410i
$935$	−2.18639e13			−0.935567
$936$		0			0
$937$	2.69013e13			1.14010			0.570052	−	0.821608i	$-0.306923\pi$
$937$	2.69013e13			1.14010			0.570052	+	0.821608i	$0.306923\pi$
$938$	−2.12851e13	+	3.68668e13i	−0.897765	+	1.55497i
$939$		0			0
$940$	−7.04605e12	−	1.22041e13i	−0.294354	−	0.509836i
$941$	8.91220e12	+	1.54364e13i	0.370537	+	0.641789i	0.989648	−	0.143514i	$-0.0458403\pi$
$941$	8.91220e12	+	1.54364e13i	0.370537	+	0.641789i	−0.619111	+	0.785303i	$0.712507\pi$
$942$		0			0
$943$	6.80413e11	−	1.17851e12i	0.0280201	−	0.0485323i
$944$	−1.50521e12			−0.0616913
$945$		0			0
$946$	−4.33565e12			−0.176013
$947$	−8.37750e12	+	1.45103e13i	−0.338485	+	0.586274i	−0.984148	−	0.177349i	$-0.943248\pi$
$947$	−8.37750e12	+	1.45103e13i	−0.338485	+	0.586274i	0.645663	+	0.763623i	$0.276581\pi$
$948$		0			0
$949$	7.46790e11	+	1.29348e12i	0.0298883	+	0.0517680i
$950$	−5.30758e12	−	9.19300e12i	−0.211417	−	0.366185i
$951$		0			0
$952$	−1.51335e13	+	2.62119e13i	−0.597135	+	1.03427i
$953$	−1.77293e13			−0.696263			−0.348131	−	0.937446i	$-0.613184\pi$
$953$	−1.77293e13			−0.696263			−0.348131	+	0.937446i	$0.613184\pi$
$954$		0			0
$955$	−3.26272e13			−1.26930
$956$	−8.25133e12	+	1.42917e13i	−0.319495	+	0.553381i
$957$		0			0
$958$	−7.13785e12	−	1.23631e13i	−0.273793	−	0.474224i
$959$	−3.79024e13	−	6.56489e13i	−1.44705	−	2.50636i
$960$		0			0
$961$	1.32184e13	−	2.28949e13i	0.499945	−	0.865930i
$962$	−6.56482e12			−0.247136
$963$		0			0
$964$	−1.57768e13			−0.588401
$965$	1.71664e13	−	2.97330e13i	0.637244	−	1.10374i
$966$		0			0
$967$	1.92667e13	+	3.33709e13i	0.708578	+	1.22729i	0.965385	+	0.260831i	$0.0839964\pi$
$967$	1.92667e13	+	3.33709e13i	0.708578	+	1.22729i	−0.256806	+	0.966463i	$0.582670\pi$
$968$	4.13373e12	+	7.15983e12i	0.151322	+	0.262098i
$969$		0			0
$970$	−4.72062e12	+	8.17635e12i	−0.171209	+	0.296542i
$971$	1.91351e13			0.690786			0.345393	−	0.938458i	$-0.387746\pi$
$971$	1.91351e13			0.690786			0.345393	+	0.938458i	$0.387746\pi$
$972$		0			0
$973$	−5.31478e13			−1.90098
$974$	3.50595e12	−	6.07248e12i	0.124822	−	0.216197i
$975$		0			0
$976$	−5.92431e12	−	1.02612e13i	−0.208984	−	0.361972i
$977$	−5.65903e12	−	9.80173e12i	−0.198708	−	0.344173i	0.749401	−	0.662116i	$-0.230341\pi$
$977$	−5.65903e12	−	9.80173e12i	−0.198708	−	0.344173i	−0.948110	+	0.317943i	$0.897008\pi$
$978$		0			0
$979$	1.30385e12	−	2.25834e12i	0.0453634	−	0.0785718i
$980$	3.89215e13			1.34795
$981$		0			0
$982$	1.95234e13			0.669967
$983$	−1.36426e13	+	2.36297e13i	−0.466023	+	0.807175i	−0.999247	−	0.0387990i	$-0.987647\pi$
$983$	−1.36426e13	+	2.36297e13i	−0.466023	+	0.807175i	0.533224	+	0.845974i	$0.320980\pi$
$984$		0			0
$985$	1.06513e13	+	1.84485e13i	0.360527	+	0.624451i
$986$	−1.37720e13	−	2.38538e13i	−0.464034	−	0.803731i
$987$		0			0
$988$	−1.41520e12	+	2.45120e12i	−0.0472510	+	0.0818412i
$989$	1.54207e13			0.512532
$990$		0			0
$991$	2.96159e13			0.975423			0.487711	−	0.873005i	$-0.337832\pi$
$991$	2.96159e13			0.975423			0.487711	+	0.873005i	$0.337832\pi$
$992$	2.82720e10	−	4.89686e10i	0.000926946	−	0.00160552i
$993$		0			0
$994$	−4.23281e12	−	7.33144e12i	−0.137528	−	0.238205i
$995$	1.99614e13	+	3.45741e13i	0.645634	+	1.11827i
$996$		0			0
$997$	−1.12148e13	+	1.94247e13i	−0.359471	+	0.622623i	−0.987873	−	0.155267i	$-0.950376\pi$
$997$	−1.12148e13	+	1.94247e13i	−0.359471	+	0.622623i	0.628401	+	0.777889i	$0.283710\pi$
$998$	3.27191e13			1.04403
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	54.10.c.a.19.1		8
3.2	odd	2		18.10.c.a.7.1	✓	8
9.2	odd	6		162.10.a.e.1.1		4
9.4	even	3	inner	54.10.c.a.37.1		8
9.5	odd	6		18.10.c.a.13.1	yes	8
9.7	even	3		162.10.a.h.1.4		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
18.10.c.a.7.1	✓	8	3.2	odd	2
18.10.c.a.13.1	yes	8	9.5	odd	6
54.10.c.a.19.1		8	1.1	even	1	trivial
54.10.c.a.37.1		8	9.4	even	3	inner
162.10.a.e.1.1		4	9.2	odd	6
162.10.a.h.1.4		4	9.7	even	3

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 54 = 2 \cdot 3^{3} \)
Weight:	\( k \)	\(=\)	\( 10 \)
Character orbit:	\([\chi]\)	\(=\)	54.c (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-8.00000 + 13.8564i) q^{2} +(-128.000 - 221.703i) q^{4} +(-897.758 - 1554.96i) q^{5} +(5590.83 - 9683.61i) q^{7} +4096.00 q^{8} +O(q^{10})\)	\(q+(-8.00000 + 13.8564i) q^{2} +(-128.000 - 221.703i) q^{4} +(-897.758 - 1554.96i) q^{5} +(5590.83 - 9683.61i) q^{7} +4096.00 q^{8} +28728.3 q^{10} +(-9213.10 + 15957.6i) q^{11} +(-10588.5 - 18339.7i) q^{13} +(89453.3 + 154938. i) q^{14} +(-32768.0 + 56755.8i) q^{16} -660849. q^{17} +522089. q^{19} +(-229826. + 398070. i) q^{20} +(-147410. - 255321. i) q^{22} +(524295. + 908105. i) q^{23} +(-635377. + 1.10051e6i) q^{25} +338831. q^{26} -2.86251e6 q^{28} +(1.30249e6 - 2.25598e6i) q^{29} +(26962.3 + 46700.1i) q^{31} +(-524288. - 908093. i) q^{32} +(5.28679e6 - 9.15699e6i) q^{34} -2.00769e7 q^{35} -1.93749e7 q^{37} +(-4.17672e6 + 7.23428e6i) q^{38} +(-3.67722e6 - 6.36913e6i) q^{40} +(-648885. - 1.12390e6i) q^{41} +(7.35308e6 - 1.27359e7i) q^{43} +4.71711e6 q^{44} -1.67774e7 q^{46} +(-1.53291e7 + 2.65507e7i) q^{47} +(-4.23380e7 - 7.33316e7i) q^{49} +(-1.01660e7 - 1.76081e7i) q^{50} +(-2.71064e6 + 4.69497e6i) q^{52} -5.86557e7 q^{53} +3.30845e7 q^{55} +(2.29001e7 - 3.96641e7i) q^{56} +(2.08398e7 + 3.60957e7i) q^{58} +(1.14839e7 + 1.98906e7i) q^{59} +(-9.03978e7 + 1.56574e8i) q^{61} -862794. q^{62} +1.67772e7 q^{64} +(-1.90117e7 + 3.29293e7i) q^{65} +(1.18973e8 + 2.06067e8i) q^{67} +(8.45886e7 + 1.46512e8i) q^{68} +(1.60615e8 - 2.78193e8i) q^{70} -4.73186e7 q^{71} -7.05287e7 q^{73} +(1.55000e8 - 2.68467e8i) q^{74} +(-6.68275e7 - 1.15749e8i) q^{76} +(1.03018e8 + 1.78432e8i) q^{77} +(1.28832e8 - 2.23143e8i) q^{79} +1.17671e8 q^{80} +2.07643e7 q^{82} +(1.46745e8 - 2.54170e8i) q^{83} +(5.93282e8 + 1.02760e9i) q^{85} +(1.17649e8 + 2.03774e8i) q^{86} +(-3.77368e7 + 6.53621e7i) q^{88} -1.41522e8 q^{89} -2.36793e8 q^{91} +(1.34219e8 - 2.32475e8i) q^{92} +(-2.45265e8 - 4.24812e8i) q^{94} +(-4.68710e8 - 8.11830e8i) q^{95} +(-1.64320e8 + 2.84610e8i) q^{97} +1.35482e9 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 64 q^{2} - 1024 q^{4} - 171 q^{5} + 7135 q^{7} + 32768 q^{8}+O(q^{10}) \) 8 * q - 64 * q^2 - 1024 * q^4 - 171 * q^5 + 7135 * q^7 + 32768 * q^8	\( 8 q - 64 q^{2} - 1024 q^{4} - 171 q^{5} + 7135 q^{7} + 32768 q^{8} + 5472 q^{10} + 26130 q^{11} - 4163 q^{13} + 114160 q^{14} - 262144 q^{16} - 1098510 q^{17} - 436382 q^{19} - 43776 q^{20} + 418080 q^{22} + 289383 q^{23} + 2947937 q^{25} + 133216 q^{26} - 3653120 q^{28} + 601707 q^{29} + 5671315 q^{31} - 4194304 q^{32} + 8788080 q^{34} - 29787930 q^{35} - 38737148 q^{37} + 3491056 q^{38} - 700416 q^{40} + 18418410 q^{41} + 35096140 q^{43} - 13378560 q^{44} - 9260256 q^{46} + 79830825 q^{47} - 40540299 q^{49} + 47166992 q^{50} - 1065728 q^{52} - 330697236 q^{53} + 56528442 q^{55} + 29224960 q^{56} + 9627312 q^{58} + 90704166 q^{59} - 122811677 q^{61} - 181482080 q^{62} + 134217728 q^{64} + 116600103 q^{65} + 221601736 q^{67} + 140609280 q^{68} + 238303440 q^{70} - 276408240 q^{71} - 988917014 q^{73} + 309897184 q^{74} + 55856896 q^{76} + 548139525 q^{77} + 592840885 q^{79} + 22413312 q^{80} - 589389120 q^{82} + 478410747 q^{83} + 1468792818 q^{85} + 561538240 q^{86} + 107028480 q^{88} - 875519952 q^{89} - 3695513062 q^{91} + 74082048 q^{92} + 1277293200 q^{94} - 813906756 q^{95} + 2679512242 q^{97} + 1297289568 q^{98}+O(q^{100}) \) 8 * q - 64 * q^2 - 1024 * q^4 - 171 * q^5 + 7135 * q^7 + 32768 * q^8 + 5472 * q^10 + 26130 * q^11 - 4163 * q^13 + 114160 * q^14 - 262144 * q^16 - 1098510 * q^17 - 436382 * q^19 - 43776 * q^20 + 418080 * q^22 + 289383 * q^23 + 2947937 * q^25 + 133216 * q^26 - 3653120 * q^28 + 601707 * q^29 + 5671315 * q^31 - 4194304 * q^32 + 8788080 * q^34 - 29787930 * q^35 - 38737148 * q^37 + 3491056 * q^38 - 700416 * q^40 + 18418410 * q^41 + 35096140 * q^43 - 13378560 * q^44 - 9260256 * q^46 + 79830825 * q^47 - 40540299 * q^49 + 47166992 * q^50 - 1065728 * q^52 - 330697236 * q^53 + 56528442 * q^55 + 29224960 * q^56 + 9627312 * q^58 + 90704166 * q^59 - 122811677 * q^61 - 181482080 * q^62 + 134217728 * q^64 + 116600103 * q^65 + 221601736 * q^67 + 140609280 * q^68 + 238303440 * q^70 - 276408240 * q^71 - 988917014 * q^73 + 309897184 * q^74 + 55856896 * q^76 + 548139525 * q^77 + 592840885 * q^79 + 22413312 * q^80 - 589389120 * q^82 + 478410747 * q^83 + 1468792818 * q^85 + 561538240 * q^86 + 107028480 * q^88 - 875519952 * q^89 - 3695513062 * q^91 + 74082048 * q^92 + 1277293200 * q^94 - 813906756 * q^95 + 2679512242 * q^97 + 1297289568 * q^98

Introduction

L-functions

Modular forms

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