LMFDB - Embedded newform 207.8.a.h.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [207,8,Mod(1,207)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(207, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0]))

N = Newforms(chi, 8, names="a")

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

chi := DirichletCharacter("207.1");

S:= CuspForms(chi, 8);

N := Newforms(S);

Level:	$ N $	$=$	$ 207 = 3^{2} \cdot 23 $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	207.a (trivial)

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:	yes
Analytic conductor:	$64.6637002752$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 4 x^{11} - 1070 x^{10} + 4076 x^{9} + 403334 x^{8} - 1518684 x^{7} - 64710184 x^{6} + \cdots + 90709421512 $ x^12 - 4x^11 - 1070x^10 + 4076x^9 + 403334x^8 - 1518684x^7 - 64710184x^6 + 248484524x^5 + 4244585785x^4 - 14689737568x^3 - 89785869106x^2 + 190549532712*x + 90709421512
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{9}\cdot 3^{7} $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-16.7762$ of defining polynomial
Character	$\chi$	$=$	207.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-15.7762 q^{2} +120.887 q^{4} -308.253 q^{5} +70.1815 q^{7} +112.213 q^{8} +O(q^{10})$	\(q-15.7762 q^{2} +120.887 q^{4} -308.253 q^{5} +70.1815 q^{7} +112.213 q^{8} +4863.05 q^{10} +7820.09 q^{11} -7557.85 q^{13} -1107.19 q^{14} -17243.9 q^{16} +30321.5 q^{17} -965.470 q^{19} -37263.9 q^{20} -123371. q^{22} -12167.0 q^{23} +16895.2 q^{25} +119234. q^{26} +8484.04 q^{28} -119451. q^{29} -20379.0 q^{31} +257678. q^{32} -478357. q^{34} -21633.7 q^{35} +373247. q^{37} +15231.4 q^{38} -34590.2 q^{40} +650588. q^{41} -254154. q^{43} +945349. q^{44} +191949. q^{46} -1.25532e6 q^{47} -818618. q^{49} -266541. q^{50} -913646. q^{52} -1.96789e6 q^{53} -2.41057e6 q^{55} +7875.31 q^{56} +1.88448e6 q^{58} -126336. q^{59} -2.78320e6 q^{61} +321502. q^{62} -1.85796e6 q^{64} +2.32973e6 q^{65} +978803. q^{67} +3.66548e6 q^{68} +341296. q^{70} +4.59470e6 q^{71} +2.87490e6 q^{73} -5.88840e6 q^{74} -116713. q^{76} +548826. q^{77} -3.52715e6 q^{79} +5.31548e6 q^{80} -1.02638e7 q^{82} +3.46038e6 q^{83} -9.34672e6 q^{85} +4.00957e6 q^{86} +877520. q^{88} +8.46884e6 q^{89} -530421. q^{91} -1.47083e6 q^{92} +1.98042e7 q^{94} +297610. q^{95} +6.04661e6 q^{97} +1.29146e7 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q + 16 q^{2} + 640 q^{4} + 500 q^{5} - 228 q^{7} + 3072 q^{8}+O(q^{10}) $ 12 * q + 16 * q^2 + 640 * q^4 + 500 * q^5 - 228 * q^7 + 3072 * q^8	\( 12 q + 16 q^{2} + 640 q^{4} + 500 q^{5} - 228 q^{7} + 3072 q^{8} + 10270 q^{10} - 460 q^{11} - 21060 q^{13} + 4268 q^{14} + 56676 q^{16} + 73124 q^{17} + 8508 q^{19} + 170538 q^{20} + 124754 q^{22} - 146004 q^{23} + 194064 q^{25} + 206080 q^{26} - 390416 q^{28} + 268640 q^{29} - 191880 q^{31} + 1180172 q^{32} - 221436 q^{34} - 487244 q^{35} + 650332 q^{37} + 1432950 q^{38} + 1775722 q^{40} + 980088 q^{41} - 861276 q^{43} + 800666 q^{44} - 194672 q^{46} + 403868 q^{47} + 1699160 q^{49} + 2919092 q^{50} - 2369520 q^{52} - 201948 q^{53} - 1553512 q^{55} - 4848116 q^{56} + 3720672 q^{58} + 1302676 q^{59} + 2141364 q^{61} + 2160944 q^{62} + 9702136 q^{64} + 9099536 q^{65} - 6159260 q^{67} + 18442208 q^{68} - 10891632 q^{70} + 12584184 q^{71} + 7435872 q^{73} + 22491442 q^{74} + 5721386 q^{76} + 16450568 q^{77} + 3658028 q^{79} + 49905778 q^{80} - 5516316 q^{82} + 26137900 q^{83} + 5169556 q^{85} + 30678550 q^{86} + 14753046 q^{88} + 27235908 q^{89} - 7657216 q^{91} - 7786880 q^{92} - 23519352 q^{94} + 63623628 q^{95} + 22454720 q^{97} + 94951532 q^{98}+O(q^{100}) \) 12 * q + 16 * q^2 + 640 * q^4 + 500 * q^5 - 228 * q^7 + 3072 * q^8 + 10270 * q^10 - 460 * q^11 - 21060 * q^13 + 4268 * q^14 + 56676 * q^16 + 73124 * q^17 + 8508 * q^19 + 170538 * q^20 + 124754 * q^22 - 146004 * q^23 + 194064 * q^25 + 206080 * q^26 - 390416 * q^28 + 268640 * q^29 - 191880 * q^31 + 1180172 * q^32 - 221436 * q^34 - 487244 * q^35 + 650332 * q^37 + 1432950 * q^38 + 1775722 * q^40 + 980088 * q^41 - 861276 * q^43 + 800666 * q^44 - 194672 * q^46 + 403868 * q^47 + 1699160 * q^49 + 2919092 * q^50 - 2369520 * q^52 - 201948 * q^53 - 1553512 * q^55 - 4848116 * q^56 + 3720672 * q^58 + 1302676 * q^59 + 2141364 * q^61 + 2160944 * q^62 + 9702136 * q^64 + 9099536 * q^65 - 6159260 * q^67 + 18442208 * q^68 - 10891632 * q^70 + 12584184 * q^71 + 7435872 * q^73 + 22491442 * q^74 + 5721386 * q^76 + 16450568 * q^77 + 3658028 * q^79 + 49905778 * q^80 - 5516316 * q^82 + 26137900 * q^83 + 5169556 * q^85 + 30678550 * q^86 + 14753046 * q^88 + 27235908 * q^89 - 7657216 * q^91 - 7786880 * q^92 - 23519352 * q^94 + 63623628 * q^95 + 22454720 * q^97 + 94951532 * q^98

Display 100 coefficients 10 coefficients

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 $

$2$	−15.7762	−1.39443	−0.697214	−	0.716863i	$-0.745577\pi$
$2$	−15.7762	−1.39443	−0.697214	+	0.716863i	$0.745577\pi$
$3$	0	0
$3$	0	0
$4$	120.887	0.944431
$5$	−308.253	−1.10284	−0.551420	−	0.834227i	$-0.685914\pi$
$5$	−308.253	−1.10284	−0.551420	+	0.834227i	$0.685914\pi$
$6$	0	0
$7$	70.1815	0.0773356	0.0386678	−	0.999252i	$-0.487689\pi$
$7$	70.1815	0.0773356	0.0386678	+	0.999252i	$0.487689\pi$
$8$	112.213	0.0774872
$9$	0	0
$10$	4863.05	1.53783
$11$	7820.09	1.77149	0.885743	−	0.464177i	$-0.153650\pi$
$11$	7820.09	1.77149	0.885743	+	0.464177i	$0.153650\pi$
$12$	0	0
$13$	−7557.85	−0.954105	−0.477053	−	0.878875i	$-0.658295\pi$
$13$	−7557.85	−0.954105	−0.477053	+	0.878875i	$0.658295\pi$
$14$	−1107.19	−0.107839
$15$	0	0
$16$	−17243.9	−1.05248
$17$	30321.5	1.49686	0.748428	−	0.663216i	$-0.230809\pi$
$17$	30321.5	1.49686	0.748428	+	0.663216i	$0.230809\pi$
$18$	0	0
$19$	−965.470	−0.0322925	−0.0161462	−	0.999870i	$-0.505140\pi$
$19$	−965.470	−0.0322925	−0.0161462	+	0.999870i	$0.505140\pi$
$20$	−37263.9	−1.04156
$21$	0	0
$22$	−123371.	−2.47021
$23$	−12167.0	−0.208514
$23$	−12167.0	−0.208514
$24$	0	0
$25$	16895.2	0.216258
$26$	119234.	1.33043
$27$	0	0
$28$	8484.04	0.0730381
$29$	−119451.	−0.909488	−0.454744	−	0.890622i	$-0.650269\pi$
$29$	−119451.	−0.909488	−0.454744	+	0.890622i	$0.650269\pi$
$30$	0	0
$31$	−20379.0	−0.122862	−0.0614308	−	0.998111i	$-0.519566\pi$
$31$	−20379.0	−0.122862	−0.0614308	+	0.998111i	$0.519566\pi$
$32$	257678.	1.39012
$33$	0	0
$34$	−478357.	−2.08726
$35$	−21633.7	−0.0852889
$36$	0	0
$37$	373247.	1.21141	0.605704	−	0.795690i	$-0.292892\pi$
$37$	373247.	1.21141	0.605704	+	0.795690i	$0.292892\pi$
$38$	15231.4	0.0450296
$39$	0	0
$40$	−34590.2	−0.0854561
$41$	650588.	1.47422	0.737110	−	0.675772i	$-0.236190\pi$
$41$	650588.	1.47422	0.737110	+	0.675772i	$0.236190\pi$
$42$	0	0
$43$	−254154.	−0.487480	−0.243740	−	0.969841i	$-0.578374\pi$
$43$	−254154.	−0.487480	−0.243740	+	0.969841i	$0.578374\pi$
$44$	945349.	1.67305
$45$	0	0
$46$	191949.	0.290758
$47$	−1.25532e6	−1.76365	−0.881825	−	0.471576i	$-0.843685\pi$
$47$	−1.25532e6	−1.76365	−0.881825	+	0.471576i	$0.843685\pi$
$48$	0	0
$49$	−818618.	−0.994019
$50$	−266541.	−0.301557
$51$	0	0
$52$	−913646.	−0.901086
$53$	−1.96789e6	−1.81566	−0.907830	−	0.419338i	$-0.862262\pi$
$53$	−1.96789e6	−1.81566	−0.907830	+	0.419338i	$0.862262\pi$
$54$	0	0
$55$	−2.41057e6	−1.95367
$56$	7875.31	0.00599252
$57$	0	0
$58$	1.88448e6	1.26822
$59$	−126336.	−0.0800836	−0.0400418	−	0.999198i	$-0.512749\pi$
$59$	−126336.	−0.0800836	−0.0400418	+	0.999198i	$0.512749\pi$
$60$	0	0
$61$	−2.78320e6	−1.56996	−0.784982	−	0.619518i	$-0.787328\pi$
$61$	−2.78320e6	−1.56996	−0.784982	+	0.619518i	$0.787328\pi$
$62$	321502.	0.171322
$63$	0	0
$64$	−1.85796e6	−0.885945
$65$	2.32973e6	1.05223
$66$	0	0
$67$	978803.	0.397588	0.198794	−	0.980041i	$-0.436298\pi$
$67$	978803.	0.397588	0.198794	+	0.980041i	$0.436298\pi$
$68$	3.66548e6	1.41368
$69$	0	0
$70$	341296.	0.118929
$71$	4.59470e6	1.52354	0.761768	−	0.647850i	$-0.224332\pi$
$71$	4.59470e6	1.52354	0.761768	+	0.647850i	$0.224332\pi$
$72$	0	0
$73$	2.87490e6	0.864953	0.432476	−	0.901645i	$-0.357640\pi$
$73$	2.87490e6	0.864953	0.432476	+	0.901645i	$0.357640\pi$
$74$	−5.88840e6	−1.68922
$75$	0	0
$76$	−116713.	−0.0304980
$77$	548826.	0.136999
$78$	0	0
$79$	−3.52715e6	−0.804876	−0.402438	−	0.915447i	$-0.631837\pi$
$79$	−3.52715e6	−0.804876	−0.402438	+	0.915447i	$0.631837\pi$
$80$	5.31548e6	1.16072
$81$	0	0
$82$	−1.02638e7	−2.05570
$83$	3.46038e6	0.664280	0.332140	−	0.943230i	$-0.392229\pi$
$83$	3.46038e6	0.664280	0.332140	+	0.943230i	$0.392229\pi$
$84$	0	0
$85$	−9.34672e6	−1.65079
$86$	4.00957e6	0.679756
$87$	0	0
$88$	877520.	0.137267
$89$	8.46884e6	1.27338	0.636691	−	0.771119i	$-0.280303\pi$
$89$	8.46884e6	1.27338	0.636691	+	0.771119i	$0.280303\pi$
$90$	0	0
$91$	−530421.	−0.0737863
$92$	−1.47083e6	−0.196927
$93$	0	0
$94$	1.98042e7	2.45928
$95$	297610.	0.0356135
$96$	0	0
$97$	6.04661e6	0.672684	0.336342	−	0.941740i	$-0.390810\pi$
$97$	6.04661e6	0.672684	0.336342	+	0.941740i	$0.390810\pi$
$98$	1.29146e7	1.38609
$99$	0	0
$100$	2.04241e6	0.204241
$101$	−4.95189e6	−0.478241	−0.239120	−	0.970990i	$-0.576859\pi$
$101$	−4.95189e6	−0.478241	−0.239120	+	0.970990i	$0.576859\pi$
$102$	0	0
$103$	2.03183e7	1.83214	0.916069	−	0.401022i	$-0.131345\pi$
$103$	2.03183e7	1.83214	0.916069	+	0.401022i	$0.131345\pi$
$104$	−848092.	−0.0739309
$105$	0	0
$106$	3.10457e7	2.53181
$107$	1.79711e7	1.41818	0.709089	−	0.705119i	$-0.249106\pi$
$107$	1.79711e7	1.41818	0.709089	+	0.705119i	$0.249106\pi$
$108$	0	0
$109$	−8.68325e6	−0.642228	−0.321114	−	0.947041i	$-0.604057\pi$
$109$	−8.68325e6	−0.642228	−0.321114	+	0.947041i	$0.604057\pi$
$110$	3.80295e7	2.72425
$111$	0	0
$112$	−1.21020e6	−0.0813943
$113$	−2.29168e7	−1.49410	−0.747050	−	0.664768i	$-0.768530\pi$
$113$	−2.29168e7	−1.49410	−0.747050	+	0.664768i	$0.768530\pi$
$114$	0	0
$115$	3.75052e6	0.229958
$116$	−1.44401e7	−0.858949
$117$	0	0
$118$	1.99309e6	0.111671
$119$	2.12801e6	0.115760
$120$	0	0
$121$	4.16667e7	2.13816
$122$	4.39082e7	2.18920
$123$	0	0
$124$	−2.46355e6	−0.116034
$125$	1.88743e7	0.864343
$126$	0	0
$127$	−1.64945e6	−0.0714540	−0.0357270	−	0.999362i	$-0.511375\pi$
$127$	−1.64945e6	−0.0714540	−0.0357270	+	0.999362i	$0.511375\pi$
$128$	−3.67133e6	−0.154735
$129$	0	0
$130$	−3.67542e7	−1.46725
$131$	2.72853e7	1.06042	0.530212	−	0.847865i	$-0.322112\pi$
$131$	2.72853e7	1.06042	0.530212	+	0.847865i	$0.322112\pi$
$132$	0	0
$133$	−67758.1	−0.00249736
$134$	−1.54417e7	−0.554408
$135$	0	0
$136$	3.40248e6	0.115987
$137$	−4.33203e6	−0.143936	−0.0719679	−	0.997407i	$-0.522928\pi$
$137$	−4.33203e6	−0.143936	−0.0719679	+	0.997407i	$0.522928\pi$
$138$	0	0
$139$	8.45058e6	0.266891	0.133446	−	0.991056i	$-0.457396\pi$
$139$	8.45058e6	0.266891	0.133446	+	0.991056i	$0.457396\pi$
$140$	−2.61523e6	−0.0805494
$141$	0	0
$142$	−7.24867e7	−2.12446
$143$	−5.91031e7	−1.69018
$144$	0	0
$145$	3.68212e7	1.00302
$146$	−4.53548e7	−1.20611
$147$	0	0
$148$	4.51208e7	1.14409
$149$	−1.83524e7	−0.454507	−0.227253	−	0.973836i	$-0.572975\pi$
$149$	−1.83524e7	−0.454507	−0.227253	+	0.973836i	$0.572975\pi$
$150$	0	0
$151$	1.80735e7	0.427191	0.213596	−	0.976922i	$-0.431482\pi$
$151$	1.80735e7	0.427191	0.213596	+	0.976922i	$0.431482\pi$
$152$	−108339.	−0.00250225
$153$	0	0
$154$	−8.65836e6	−0.191035
$155$	6.28188e6	0.135497
$156$	0	0
$157$	−4.59392e7	−0.947402	−0.473701	−	0.880686i	$-0.657082\pi$
$157$	−4.59392e7	−0.947402	−0.473701	+	0.880686i	$0.657082\pi$
$158$	5.56448e7	1.12234
$159$	0	0
$160$	−7.94303e7	−1.53308
$161$	−853898.	−0.0161256
$162$	0	0
$163$	9.35733e7	1.69237	0.846185	−	0.532889i	$-0.178894\pi$
$163$	9.35733e7	1.69237	0.846185	+	0.532889i	$0.178894\pi$
$164$	7.86477e7	1.39230
$165$	0	0
$166$	−5.45916e7	−0.926290
$167$	3.19994e7	0.531660	0.265830	−	0.964020i	$-0.414354\pi$
$167$	3.19994e7	0.531660	0.265830	+	0.964020i	$0.414354\pi$
$168$	0	0
$169$	−5.62749e6	−0.0896832
$170$	1.47455e8	2.30191
$171$	0	0
$172$	−3.07239e7	−0.460391
$173$	4.39920e7	0.645970	0.322985	−	0.946404i	$-0.395314\pi$
$173$	4.39920e7	0.645970	0.322985	+	0.946404i	$0.395314\pi$
$174$	0	0
$175$	1.18573e6	0.0167245
$176$	−1.34849e8	−1.86445
$177$	0	0
$178$	−1.33606e8	−1.77564
$179$	−1.02267e6	−0.0133275	−0.00666374	−	0.999978i	$-0.502121\pi$
$179$	−1.02267e6	−0.0133275	−0.00666374	+	0.999978i	$0.502121\pi$
$180$	0	0
$181$	−1.06423e8	−1.33401	−0.667005	−	0.745053i	$-0.732424\pi$
$181$	−1.06423e8	−1.33401	−0.667005	+	0.745053i	$0.732424\pi$
$182$	8.36800e6	0.102890
$183$	0	0
$184$	−1.36530e6	−0.0161572
$185$	−1.15055e8	−1.33599
$186$	0	0
$187$	2.37117e8	2.65166
$188$	−1.51752e8	−1.66565
$189$	0	0
$190$	−4.69514e6	−0.0496604
$191$	1.23564e8	1.28314	0.641570	−	0.767065i	$-0.278284\pi$
$191$	1.23564e8	1.28314	0.641570	+	0.767065i	$0.278284\pi$
$192$	0	0
$193$	−4.55571e7	−0.456148	−0.228074	−	0.973644i	$-0.573243\pi$
$193$	−4.55571e7	−0.456148	−0.228074	+	0.973644i	$0.573243\pi$
$194$	−9.53923e7	−0.938010
$195$	0	0
$196$	−9.89603e7	−0.938782
$197$	1.93515e8	1.80337	0.901683	−	0.432398i	$-0.142332\pi$
$197$	1.93515e8	1.80337	0.901683	+	0.432398i	$0.142332\pi$
$198$	0	0
$199$	−1.99087e7	−0.179084	−0.0895421	−	0.995983i	$-0.528540\pi$
$199$	−1.99087e7	−0.179084	−0.0895421	+	0.995983i	$0.528540\pi$
$200$	1.89587e6	0.0167572
$201$	0	0
$202$	7.81219e7	0.666872
$203$	−8.38326e6	−0.0703358
$204$	0	0
$205$	−2.00546e8	−1.62583
$206$	−3.20545e8	−2.55478
$207$	0	0
$208$	1.30326e8	1.00418
$209$	−7.55007e6	−0.0572056
$210$	0	0
$211$	−4.19061e7	−0.307106	−0.153553	−	0.988140i	$-0.549072\pi$
$211$	−4.19061e7	−0.307106	−0.153553	+	0.988140i	$0.549072\pi$
$212$	−2.37892e8	−1.71477
$213$	0	0
$214$	−2.83515e8	−1.97755
$215$	7.83438e7	0.537613
$216$	0	0
$217$	−1.43023e6	−0.00950157
$218$	1.36988e8	0.895541
$219$	0	0
$220$	−2.91407e8	−1.84510
$221$	−2.29165e8	−1.42816
$222$	0	0
$223$	2.83786e8	1.71366	0.856828	−	0.515602i	$-0.172432\pi$
$223$	2.83786e8	1.71366	0.856828	+	0.515602i	$0.172432\pi$
$224$	1.80843e7	0.107506
$225$	0	0
$226$	3.61539e8	2.08341
$227$	−2.41099e8	−1.36806	−0.684028	−	0.729455i	$-0.739774\pi$
$227$	−2.41099e8	−1.36806	−0.684028	+	0.729455i	$0.739774\pi$
$228$	0	0
$229$	1.38130e8	0.760089	0.380045	−	0.924968i	$-0.375909\pi$
$229$	1.38130e8	0.760089	0.380045	+	0.924968i	$0.375909\pi$
$230$	−5.91688e7	−0.320660
$231$	0	0
$232$	−1.34040e7	−0.0704737
$233$	1.24625e8	0.645447	0.322723	−	0.946493i	$-0.395402\pi$
$233$	1.24625e8	0.645447	0.322723	+	0.946493i	$0.395402\pi$
$234$	0	0
$235$	3.86957e8	1.94503
$236$	−1.52723e7	−0.0756334
$237$	0	0
$238$	−3.35718e7	−0.161419
$239$	−3.21702e7	−0.152426	−0.0762132	−	0.997092i	$-0.524283\pi$
$239$	−3.21702e7	−0.152426	−0.0762132	+	0.997092i	$0.524283\pi$
$240$	0	0
$241$	−2.18182e8	−1.00406	−0.502031	−	0.864850i	$-0.667414\pi$
$241$	−2.18182e8	−1.00406	−0.502031	+	0.864850i	$0.667414\pi$
$242$	−6.57340e8	−2.98151
$243$	0	0
$244$	−3.36453e8	−1.48272
$245$	2.52342e8	1.09625
$246$	0	0
$247$	7.29688e6	0.0308104
$248$	−2.28679e6	−0.00952020
$249$	0	0
$250$	−2.97764e8	−1.20526
$251$	−6.18658e7	−0.246941	−0.123470	−	0.992348i	$-0.539402\pi$
$251$	−6.18658e7	−0.246941	−0.123470	+	0.992348i	$0.539402\pi$
$252$	0	0
$253$	−9.51471e7	−0.369380
$254$	2.60220e7	0.0996375
$255$	0	0
$256$	2.95739e8	1.10171
$257$	−3.09024e8	−1.13560	−0.567801	−	0.823166i	$-0.692206\pi$
$257$	−3.09024e8	−1.13560	−0.567801	+	0.823166i	$0.692206\pi$
$258$	0	0
$259$	2.61950e7	0.0936849
$260$	2.81635e8	0.993755
$261$	0	0
$262$	−4.30458e8	−1.47869
$263$	−6.65071e7	−0.225436	−0.112718	−	0.993627i	$-0.535956\pi$
$263$	−6.65071e7	−0.225436	−0.112718	+	0.993627i	$0.535956\pi$
$264$	0	0
$265$	6.06608e8	2.00238
$266$	1.06896e6	0.00348239
$267$	0	0
$268$	1.18325e8	0.375494
$269$	−3.09876e8	−0.970634	−0.485317	−	0.874338i	$-0.661296\pi$
$269$	−3.09876e8	−0.970634	−0.485317	+	0.874338i	$0.661296\pi$
$270$	0	0
$271$	2.92838e8	0.893788	0.446894	−	0.894587i	$-0.352530\pi$
$271$	2.92838e8	0.893788	0.446894	+	0.894587i	$0.352530\pi$
$272$	−5.22860e8	−1.57541
$273$	0	0
$274$	6.83427e7	0.200708
$275$	1.32122e8	0.383098
$276$	0	0
$277$	3.25816e8	0.921072	0.460536	−	0.887641i	$-0.347657\pi$
$277$	3.25816e8	0.921072	0.460536	+	0.887641i	$0.347657\pi$
$278$	−1.33318e8	−0.372161
$279$	0	0
$280$	−2.42759e6	−0.00660879
$281$	5.00645e8	1.34604	0.673020	−	0.739625i	$-0.264997\pi$
$281$	5.00645e8	1.34604	0.673020	+	0.739625i	$0.264997\pi$
$282$	0	0
$283$	−2.81893e7	−0.0739319	−0.0369659	−	0.999317i	$-0.511769\pi$
$283$	−2.81893e7	−0.0739319	−0.0369659	+	0.999317i	$0.511769\pi$
$284$	5.55440e8	1.43887
$285$	0	0
$286$	9.32419e8	2.35684
$287$	4.56592e7	0.114010
$288$	0	0
$289$	5.09057e8	1.24058
$290$	−5.80897e8	−1.39864
$291$	0	0
$292$	3.47538e8	0.816888
$293$	−6.46020e7	−0.150041	−0.0750204	−	0.997182i	$-0.523902\pi$
$293$	−6.46020e7	−0.150041	−0.0750204	+	0.997182i	$0.523902\pi$
$294$	0	0
$295$	3.89434e7	0.0883195
$296$	4.18833e7	0.0938686
$297$	0	0
$298$	2.89530e8	0.633777
$299$	9.19563e7	0.198945
$300$	0	0
$301$	−1.78369e7	−0.0376996
$302$	−2.85130e8	−0.595688
$303$	0	0
$304$	1.66484e7	0.0339872
$305$	8.57931e8	1.73142
$306$	0	0
$307$	−5.03339e8	−0.992833	−0.496416	−	0.868084i	$-0.665351\pi$
$307$	−5.03339e8	−0.992833	−0.496416	+	0.868084i	$0.665351\pi$
$308$	6.63460e7	0.129386
$309$	0	0
$310$	−9.91040e7	−0.188941
$311$	3.23798e8	0.610398	0.305199	−	0.952289i	$-0.401277\pi$
$311$	3.23798e8	0.610398	0.305199	+	0.952289i	$0.401277\pi$
$312$	0	0
$313$	4.43713e8	0.817894	0.408947	−	0.912558i	$-0.365896\pi$
$313$	4.43713e8	0.817894	0.408947	+	0.912558i	$0.365896\pi$
$314$	7.24743e8	1.32108
$315$	0	0
$316$	−4.26387e8	−0.760149
$317$	5.45515e8	0.961831	0.480916	−	0.876767i	$-0.340304\pi$
$317$	5.45515e8	0.961831	0.480916	+	0.876767i	$0.340304\pi$
$318$	0	0
$319$	−9.34119e8	−1.61115
$320$	5.72723e8	0.977057
$321$	0	0
$322$	1.34712e7	0.0224860
$323$	−2.92745e7	−0.0483372
$324$	0	0
$325$	−1.27691e8	−0.206333
$326$	−1.47623e9	−2.35989
$327$	0	0
$328$	7.30047e7	0.114233
$329$	−8.81003e7	−0.136393
$330$	0	0
$331$	2.38474e8	0.361445	0.180723	−	0.983534i	$-0.442156\pi$
$331$	2.38474e8	0.361445	0.180723	+	0.983534i	$0.442156\pi$
$332$	4.18316e8	0.627366
$333$	0	0
$334$	−5.04827e8	−0.741361
$335$	−3.01719e8	−0.438476
$336$	0	0
$337$	−7.15140e8	−1.01786	−0.508928	−	0.860809i	$-0.669958\pi$
$337$	−7.15140e8	−1.01786	−0.508928	+	0.860809i	$0.669958\pi$
$338$	8.87801e7	0.125057
$339$	0	0
$340$	−1.12990e9	−1.55906
$341$	−1.59365e8	−0.217647
$342$	0	0
$343$	−1.15249e8	−0.154209
$344$	−2.85195e7	−0.0377735
$345$	0	0
$346$	−6.94025e8	−0.900759
$347$	7.20495e8	0.925716	0.462858	−	0.886432i	$-0.346824\pi$
$347$	7.20495e8	0.925716	0.462858	+	0.886432i	$0.346824\pi$
$348$	0	0
$349$	−1.32497e9	−1.66846	−0.834232	−	0.551414i	$-0.814088\pi$
$349$	−1.32497e9	−1.66846	−0.834232	+	0.551414i	$0.814088\pi$
$350$	−1.87062e7	−0.0233211
$351$	0	0
$352$	2.01507e9	2.46258
$353$	9.80260e8	1.18612	0.593062	−	0.805157i	$-0.297919\pi$
$353$	9.80260e8	1.18612	0.593062	+	0.805157i	$0.297919\pi$
$354$	0	0
$355$	−1.41633e9	−1.68022
$356$	1.02377e9	1.20262
$357$	0	0
$358$	1.61337e7	0.0185842
$359$	1.74805e8	0.199399	0.0996996	−	0.995018i	$-0.468212\pi$
$359$	1.74805e8	0.199399	0.0996996	+	0.995018i	$0.468212\pi$
$360$	0	0
$361$	−8.92940e8	−0.998957
$362$	1.67894e9	1.86018
$363$	0	0
$364$	−6.41211e7	−0.0696861
$365$	−8.86197e8	−0.953905
$366$	0	0
$367$	4.93002e8	0.520616	0.260308	−	0.965526i	$-0.416176\pi$
$367$	4.93002e8	0.520616	0.260308	+	0.965526i	$0.416176\pi$
$368$	2.09806e8	0.219458
$369$	0	0
$370$	1.81512e9	1.86294
$371$	−1.38109e8	−0.140415
$372$	0	0
$373$	−1.23904e9	−1.23625	−0.618124	−	0.786080i	$-0.712107\pi$
$373$	−1.23904e9	−1.23625	−0.618124	+	0.786080i	$0.712107\pi$
$374$	−3.74080e9	−3.69755
$375$	0	0
$376$	−1.40864e8	−0.136660
$377$	9.02793e8	0.867748
$378$	0	0
$379$	1.12377e9	1.06033	0.530165	−	0.847895i	$-0.322130\pi$
$379$	1.12377e9	1.06033	0.530165	+	0.847895i	$0.322130\pi$
$380$	3.59772e7	0.0336345
$381$	0	0
$382$	−1.94936e9	−1.78925
$383$	−1.35049e9	−1.22827	−0.614136	−	0.789200i	$-0.710495\pi$
$383$	−1.35049e9	−1.22827	−0.614136	+	0.789200i	$0.710495\pi$
$384$	0	0
$385$	−1.69177e8	−0.151088
$386$	7.18716e8	0.636066
$387$	0	0
$388$	7.30958e8	0.635304
$389$	−4.96406e8	−0.427576	−0.213788	−	0.976880i	$-0.568580\pi$
$389$	−4.96406e8	−0.427576	−0.213788	+	0.976880i	$0.568580\pi$
$390$	0	0
$391$	−3.68922e8	−0.312116
$392$	−9.18599e7	−0.0770238
$393$	0	0
$394$	−3.05293e9	−2.51466
$395$	1.08726e9	0.887650
$396$	0	0
$397$	1.07159e9	0.859528	0.429764	−	0.902941i	$-0.358597\pi$
$397$	1.07159e9	0.859528	0.429764	+	0.902941i	$0.358597\pi$
$398$	3.14083e8	0.249720
$399$	0	0
$400$	−2.91338e8	−0.227608
$401$	1.37411e9	1.06418	0.532092	−	0.846686i	$-0.321406\pi$
$401$	1.37411e9	1.06418	0.532092	+	0.846686i	$0.321406\pi$
$402$	0	0
$403$	1.54021e8	0.117223
$404$	−5.98620e8	−0.451665
$405$	0	0
$406$	1.32256e8	0.0980783
$407$	2.91883e9	2.14599
$408$	0	0
$409$	2.71803e8	0.196437	0.0982185	−	0.995165i	$-0.468686\pi$
$409$	2.71803e8	0.196437	0.0982185	+	0.995165i	$0.468686\pi$
$410$	3.16384e9	2.26711
$411$	0	0
$412$	2.45623e9	1.73033
$413$	−8.86642e6	−0.00619331
$414$	0	0
$415$	−1.06668e9	−0.732595
$416$	−1.94749e9	−1.32632
$417$	0	0
$418$	1.19111e8	0.0797692
$419$	1.80796e9	1.20071	0.600356	−	0.799733i	$-0.295025\pi$
$419$	1.80796e9	1.20071	0.600356	+	0.799733i	$0.295025\pi$
$420$	0	0
$421$	1.76990e9	1.15601	0.578005	−	0.816033i	$-0.303831\pi$
$421$	1.76990e9	1.15601	0.578005	+	0.816033i	$0.303831\pi$
$422$	6.61117e8	0.428237
$423$	0	0
$424$	−2.20823e8	−0.140690
$425$	5.12288e8	0.323707
$426$	0	0
$427$	−1.95329e8	−0.121414
$428$	2.17247e9	1.33937
$429$	0	0
$430$	−1.23596e9	−0.749663
$431$	6.80999e8	0.409710	0.204855	−	0.978792i	$-0.434328\pi$
$431$	6.80999e8	0.409710	0.204855	+	0.978792i	$0.434328\pi$
$432$	0	0
$433$	8.27861e8	0.490061	0.245030	−	0.969515i	$-0.421202\pi$
$433$	8.27861e8	0.490061	0.245030	+	0.969515i	$0.421202\pi$
$434$	2.25635e7	0.0132493
$435$	0	0
$436$	−1.04969e9	−0.606540
$437$	1.17469e7	0.00673345
$438$	0	0
$439$	1.07338e8	0.0605518	0.0302759	−	0.999542i	$-0.490361\pi$
$439$	1.07338e8	0.0605518	0.0302759	+	0.999542i	$0.490361\pi$
$440$	−2.70498e8	−0.151384
$441$	0	0
$442$	3.61535e9	1.99146
$443$	5.17218e8	0.282658	0.141329	−	0.989963i	$-0.454863\pi$
$443$	5.17218e8	0.282658	0.141329	+	0.989963i	$0.454863\pi$
$444$	0	0
$445$	−2.61055e9	−1.40434
$446$	−4.47705e9	−2.38957
$447$	0	0
$448$	−1.30395e8	−0.0685151
$449$	7.47831e8	0.389889	0.194945	−	0.980814i	$-0.437547\pi$
$449$	7.47831e8	0.389889	0.194945	+	0.980814i	$0.437547\pi$
$450$	0	0
$451$	5.08766e9	2.61156
$452$	−2.77035e9	−1.41107
$453$	0	0
$454$	3.80361e9	1.90766
$455$	1.63504e8	0.0813746
$456$	0	0
$457$	2.67378e9	1.31045	0.655223	−	0.755436i	$-0.272575\pi$
$457$	2.67378e9	1.31045	0.655223	+	0.755436i	$0.272575\pi$
$458$	−2.17917e9	−1.05989
$459$	0	0
$460$	4.53390e8	0.217180
$461$	9.08392e8	0.431837	0.215919	−	0.976411i	$-0.430725\pi$
$461$	9.08392e8	0.431837	0.215919	+	0.976411i	$0.430725\pi$
$462$	0	0
$463$	−2.46821e9	−1.15571	−0.577856	−	0.816139i	$-0.696110\pi$
$463$	−2.46821e9	−1.15571	−0.577856	+	0.816139i	$0.696110\pi$
$464$	2.05980e9	0.957219
$465$	0	0
$466$	−1.96611e9	−0.900029
$467$	2.53820e9	1.15323	0.576615	−	0.817016i	$-0.304373\pi$
$467$	2.53820e9	1.15323	0.576615	+	0.817016i	$0.304373\pi$
$468$	0	0
$469$	6.86938e7	0.0307477
$470$	−6.10470e9	−2.71220
$471$	0	0
$472$	−1.41765e7	−0.00620545
$473$	−1.98751e9	−0.863564
$474$	0	0
$475$	−1.63118e7	−0.00698351
$476$	2.57249e8	0.109328
$477$	0	0
$478$	5.07521e8	0.212548
$479$	−2.46708e8	−0.102567	−0.0512836	−	0.998684i	$-0.516331\pi$
$479$	−2.46708e8	−0.102567	−0.0512836	+	0.998684i	$0.516331\pi$
$480$	0	0
$481$	−2.82094e9	−1.15581
$482$	3.44208e9	1.40009
$483$	0	0
$484$	5.03697e9	2.01934
$485$	−1.86389e9	−0.741864
$486$	0	0
$487$	5.86826e8	0.230228	0.115114	−	0.993352i	$-0.463277\pi$
$487$	5.86826e8	0.230228	0.115114	+	0.993352i	$0.463277\pi$
$488$	−3.12312e8	−0.121652
$489$	0	0
$490$	−3.98098e9	−1.52864
$491$	−1.60821e9	−0.613138	−0.306569	−	0.951848i	$-0.599181\pi$
$491$	−1.60821e9	−0.613138	−0.306569	+	0.951848i	$0.599181\pi$
$492$	0	0
$493$	−3.62194e9	−1.36137
$494$	−1.15117e8	−0.0429629
$495$	0	0
$496$	3.51412e8	0.129309
$497$	3.22463e8	0.117824
$498$	0	0
$499$	3.01936e9	1.08784	0.543918	−	0.839139i	$-0.316940\pi$
$499$	3.01936e9	1.08784	0.543918	+	0.839139i	$0.316940\pi$
$500$	2.28166e9	0.816312
$501$	0	0
$502$	9.76004e8	0.344341
$503$	−3.83416e8	−0.134333	−0.0671664	−	0.997742i	$-0.521396\pi$
$503$	−3.83416e8	−0.134333	−0.0671664	+	0.997742i	$0.521396\pi$
$504$	0	0
$505$	1.52644e9	0.527423
$506$	1.50106e9	0.515074
$507$	0	0
$508$	−1.99397e8	−0.0674833
$509$	4.76191e9	1.60055	0.800274	−	0.599635i	$-0.204687\pi$
$509$	4.76191e9	1.60055	0.800274	+	0.599635i	$0.204687\pi$
$510$	0	0
$511$	2.01765e8	0.0668916
$512$	−4.19569e9	−1.38152
$513$	0	0
$514$	4.87521e9	1.58352
$515$	−6.26320e9	−2.02056
$516$	0	0
$517$	−9.81673e9	−3.12428
$518$	−4.13257e8	−0.130637
$519$	0	0
$520$	2.61427e8	0.0815341
$521$	−6.27474e8	−0.194385	−0.0971926	−	0.995266i	$-0.530986\pi$
$521$	−6.27474e8	−0.194385	−0.0971926	+	0.995266i	$0.530986\pi$
$522$	0	0
$523$	4.15723e9	1.27071	0.635357	−	0.772218i	$-0.280853\pi$
$523$	4.15723e9	1.27071	0.635357	+	0.772218i	$0.280853\pi$
$524$	3.29845e9	1.00150
$525$	0	0
$526$	1.04923e9	0.314354
$527$	−6.17921e8	−0.183906
$528$	0	0
$529$	1.48036e8	0.0434783
$530$	−9.56994e9	−2.79218
$531$	0	0
$532$	−8.19109e6	−0.00235858
$533$	−4.91704e9	−1.40656
$534$	0	0
$535$	−5.53965e9	−1.56403
$536$	1.09835e8	0.0308080
$537$	0	0
$538$	4.88866e9	1.35348
$539$	−6.40167e9	−1.76089
$540$	0	0
$541$	−7.14802e9	−1.94086	−0.970432	−	0.241375i	$-0.922402\pi$
$541$	−7.14802e9	−1.94086	−0.970432	+	0.241375i	$0.922402\pi$
$542$	−4.61986e9	−1.24632
$543$	0	0
$544$	7.81320e9	2.08081
$545$	2.67664e9	0.708276
$546$	0	0
$547$	1.76073e9	0.459977	0.229988	−	0.973193i	$-0.426131\pi$
$547$	1.76073e9	0.459977	0.229988	+	0.973193i	$0.426131\pi$
$548$	−5.23686e8	−0.135937
$549$	0	0
$550$	−2.08438e9	−0.534203
$551$	1.15327e8	0.0293696
$552$	0	0
$553$	−2.47540e8	−0.0622455
$554$	−5.14013e9	−1.28437
$555$	0	0
$556$	1.02157e9	0.252060
$557$	6.12711e9	1.50232	0.751160	−	0.660120i	$-0.229495\pi$
$557$	6.12711e9	1.50232	0.751160	+	0.660120i	$0.229495\pi$
$558$	0	0
$559$	1.92085e9	0.465107
$560$	3.73048e8	0.0897649
$561$	0	0
$562$	−7.89825e9	−1.87696
$563$	4.46269e9	1.05394	0.526972	−	0.849883i	$-0.323327\pi$
$563$	4.46269e9	1.05394	0.526972	+	0.849883i	$0.323327\pi$
$564$	0	0
$565$	7.06418e9	1.64775
$566$	4.44719e8	0.103093
$567$	0	0
$568$	5.15587e8	0.118055
$569$	−8.22225e8	−0.187110	−0.0935552	−	0.995614i	$-0.529823\pi$
$569$	−8.22225e8	−0.187110	−0.0935552	+	0.995614i	$0.529823\pi$
$570$	0	0
$571$	5.32510e9	1.19702	0.598510	−	0.801116i	$-0.295760\pi$
$571$	5.32510e9	1.19702	0.598510	+	0.801116i	$0.295760\pi$
$572$	−7.14480e9	−1.59626
$573$	0	0
$574$	−7.20327e8	−0.158978
$575$	−2.05564e8	−0.0450930
$576$	0	0
$577$	−2.49509e9	−0.540718	−0.270359	−	0.962760i	$-0.587142\pi$
$577$	−2.49509e9	−0.540718	−0.270359	+	0.962760i	$0.587142\pi$
$578$	−8.03096e9	−1.72990
$579$	0	0
$580$	4.45121e9	0.947284
$581$	2.42855e8	0.0513725
$582$	0	0
$583$	−1.53891e10	−3.21642
$584$	3.22602e8	0.0670227
$585$	0	0
$586$	1.01917e9	0.209221
$587$	6.33563e9	1.29287	0.646437	−	0.762967i	$-0.276258\pi$
$587$	6.33563e9	1.29287	0.646437	+	0.762967i	$0.276258\pi$
$588$	0	0
$589$	1.96753e7	0.00396750
$590$	−6.14377e8	−0.123155
$591$	0	0
$592$	−6.43622e9	−1.27498
$593$	6.13057e9	1.20728	0.603642	−	0.797255i	$-0.293716\pi$
$593$	6.13057e9	1.20728	0.603642	+	0.797255i	$0.293716\pi$
$594$	0	0
$595$	−6.55966e8	−0.127665
$596$	−2.21857e9	−0.429250
$597$	0	0
$598$	−1.45072e9	−0.277414
$599$	8.63125e9	1.64089	0.820446	−	0.571725i	$-0.193725\pi$
$599$	8.63125e9	1.64089	0.820446	+	0.571725i	$0.193725\pi$
$600$	0	0
$601$	−3.85309e9	−0.724016	−0.362008	−	0.932175i	$-0.617909\pi$
$601$	−3.85309e9	−0.724016	−0.362008	+	0.932175i	$0.617909\pi$
$602$	2.81398e8	0.0525693
$603$	0	0
$604$	2.18485e9	0.403453
$605$	−1.28439e10	−2.35805
$606$	0	0
$607$	7.03452e9	1.27666	0.638328	−	0.769765i	$-0.279626\pi$
$607$	7.03452e9	1.27666	0.638328	+	0.769765i	$0.279626\pi$
$608$	−2.48781e8	−0.0448905
$609$	0	0
$610$	−1.35349e10	−2.41434
$611$	9.48753e9	1.68271
$612$	0	0
$613$	3.17986e9	0.557565	0.278783	−	0.960354i	$-0.410069\pi$
$613$	3.17986e9	0.557565	0.278783	+	0.960354i	$0.410069\pi$
$614$	7.94075e9	1.38443
$615$	0	0
$616$	6.15856e7	0.0106157
$617$	−1.83448e9	−0.314423	−0.157211	−	0.987565i	$-0.550250\pi$
$617$	−1.83448e9	−0.314423	−0.157211	+	0.987565i	$0.550250\pi$
$618$	0	0
$619$	−7.16881e9	−1.21487	−0.607435	−	0.794369i	$-0.707802\pi$
$619$	−7.16881e9	−1.21487	−0.607435	+	0.794369i	$0.707802\pi$
$620$	7.59399e8	0.127967
$621$	0	0
$622$	−5.10829e9	−0.851156
$623$	5.94355e8	0.0984777
$624$	0	0
$625$	−7.13800e9	−1.16949
$626$	−7.00008e9	−1.14049
$627$	0	0
$628$	−5.55345e9	−0.894756
$629$	1.13174e10	1.81330
$630$	0	0
$631$	3.01287e9	0.477396	0.238698	−	0.971094i	$-0.423279\pi$
$631$	3.01287e9	0.477396	0.238698	+	0.971094i	$0.423279\pi$
$632$	−3.95793e8	−0.0623675
$633$	0	0
$634$	−8.60612e9	−1.34120
$635$	5.08449e8	0.0788024
$636$	0	0
$637$	6.18699e9	0.948399
$638$	1.47368e10	2.24663
$639$	0	0
$640$	1.13170e9	0.170648
$641$	−5.00983e9	−0.751312	−0.375656	−	0.926759i	$-0.622583\pi$
$641$	−5.00983e9	−0.751312	−0.375656	+	0.926759i	$0.622583\pi$
$642$	0	0
$643$	−7.30760e9	−1.08402	−0.542009	−	0.840373i	$-0.682336\pi$
$643$	−7.30760e9	−1.08402	−0.542009	+	0.840373i	$0.682336\pi$
$644$	−1.03225e8	−0.0152295
$645$	0	0
$646$	4.61840e8	0.0674027
$647$	1.06510e9	0.154606	0.0773032	−	0.997008i	$-0.475369\pi$
$647$	1.06510e9	0.154606	0.0773032	+	0.997008i	$0.475369\pi$
$648$	0	0
$649$	−9.87956e8	−0.141867
$650$	2.01448e9	0.287717
$651$	0	0
$652$	1.13118e10	1.59833
$653$	2.71493e9	0.381560	0.190780	−	0.981633i	$-0.438898\pi$
$653$	2.71493e9	0.381560	0.190780	+	0.981633i	$0.438898\pi$
$654$	0	0
$655$	−8.41080e9	−1.16948
$656$	−1.12186e10	−1.55159
$657$	0	0
$658$	1.38988e9	0.190190
$659$	−1.06610e10	−1.45111	−0.725554	−	0.688166i	$-0.758416\pi$
$659$	−1.06610e10	−1.45111	−0.725554	+	0.688166i	$0.758416\pi$
$660$	0	0
$661$	6.67270e9	0.898663	0.449331	−	0.893365i	$-0.351662\pi$
$661$	6.67270e9	0.898663	0.449331	+	0.893365i	$0.351662\pi$
$662$	−3.76220e9	−0.504010
$663$	0	0
$664$	3.88302e8	0.0514732
$665$	2.08867e7	0.00275419
$666$	0	0
$667$	1.45336e9	0.189641
$668$	3.86831e9	0.502116
$669$	0	0
$670$	4.75997e9	0.611424
$671$	−2.17649e10	−2.78117
$672$	0	0
$673$	1.43749e10	1.81783	0.908914	−	0.416983i	$-0.136913\pi$
$673$	1.43749e10	1.81783	0.908914	+	0.416983i	$0.136913\pi$
$674$	1.12822e10	1.41933
$675$	0	0
$676$	−6.80291e8	−0.0846996
$677$	2.53152e9	0.313561	0.156780	−	0.987634i	$-0.449889\pi$
$677$	2.53152e9	0.313561	0.156780	+	0.987634i	$0.449889\pi$
$678$	0	0
$679$	4.24360e8	0.0520224
$680$	−1.04883e9	−0.127915
$681$	0	0
$682$	2.51417e9	0.303494
$683$	−1.53828e10	−1.84741	−0.923706	−	0.383101i	$-0.874856\pi$
$683$	−1.53828e10	−1.84741	−0.923706	+	0.383101i	$0.874856\pi$
$684$	0	0
$685$	1.33536e9	0.158738
$686$	1.81819e9	0.215033
$687$	0	0
$688$	4.38259e9	0.513064
$689$	1.48730e10	1.73233
$690$	0	0
$691$	2.59688e9	0.299418	0.149709	−	0.988730i	$-0.452166\pi$
$691$	2.59688e9	0.299418	0.149709	+	0.988730i	$0.452166\pi$
$692$	5.31807e9	0.610074
$693$	0	0
$694$	−1.13666e10	−1.29084
$695$	−2.60492e9	−0.294339
$696$	0	0
$697$	1.97268e10	2.20670
$698$	2.09029e10	2.32655
$699$	0	0
$700$	1.43339e8	0.0157951
$701$	6.94971e7	0.00761998	0.00380999	−	0.999993i	$-0.498787\pi$
$701$	6.94971e7	0.00761998	0.00380999	+	0.999993i	$0.498787\pi$
$702$	0	0
$703$	−3.60359e8	−0.0391194
$704$	−1.45294e10	−1.56944
$705$	0	0
$706$	−1.54647e10	−1.65396
$707$	−3.47531e8	−0.0369850
$708$	0	0
$709$	1.16984e10	1.23272	0.616360	−	0.787464i	$-0.288607\pi$
$709$	1.16984e10	1.23272	0.616360	+	0.787464i	$0.288607\pi$
$710$	2.23443e10	2.34294
$711$	0	0
$712$	9.50317e8	0.0986708
$713$	2.47951e8	0.0256184
$714$	0	0
$715$	1.82187e10	1.86400
$716$	−1.23627e8	−0.0125869
$717$	0	0
$718$	−2.75775e9	−0.278048
$719$	5.49649e9	0.551485	0.275743	−	0.961231i	$-0.411076\pi$
$719$	5.49649e9	0.551485	0.275743	+	0.961231i	$0.411076\pi$
$720$	0	0
$721$	1.42597e9	0.141689
$722$	1.40872e10	1.39297
$723$	0	0
$724$	−1.28651e10	−1.25988
$725$	−2.01815e9	−0.196684
$726$	0	0
$727$	1.91280e10	1.84629	0.923143	−	0.384458i	$-0.125612\pi$
$727$	1.91280e10	1.84629	0.923143	+	0.384458i	$0.125612\pi$
$728$	−5.95203e7	−0.00571749
$729$	0	0
$730$	1.39808e10	1.33015
$731$	−7.70633e9	−0.729687
$732$	0	0
$733$	−1.35231e10	−1.26827	−0.634136	−	0.773222i	$-0.718644\pi$
$733$	−1.35231e10	−1.26827	−0.634136	+	0.773222i	$0.718644\pi$
$734$	−7.77767e9	−0.725961
$735$	0	0
$736$	−3.13517e9	−0.289861
$737$	7.65433e9	0.704321
$738$	0	0
$739$	−1.54050e10	−1.40413	−0.702065	−	0.712113i	$-0.747738\pi$
$739$	−1.54050e10	−1.40413	−0.702065	+	0.712113i	$0.747738\pi$
$740$	−1.39086e10	−1.26175
$741$	0	0
$742$	2.17883e9	0.195799
$743$	2.15804e10	1.93019	0.965093	−	0.261907i	$-0.0843512\pi$
$743$	2.15804e10	1.93019	0.965093	+	0.261907i	$0.0843512\pi$
$744$	0	0
$745$	5.65718e9	0.501248
$746$	1.95474e10	1.72386
$747$	0	0
$748$	2.86644e10	2.50431
$749$	1.26124e9	0.109676
$750$	0	0
$751$	−9.56804e9	−0.824296	−0.412148	−	0.911117i	$-0.635221\pi$
$751$	−9.56804e9	−0.824296	−0.412148	+	0.911117i	$0.635221\pi$
$752$	2.16466e10	1.85621
$753$	0	0
$754$	−1.42426e10	−1.21001
$755$	−5.57121e9	−0.471124
$756$	0	0
$757$	3.86591e9	0.323904	0.161952	−	0.986799i	$-0.448221\pi$
$757$	3.86591e9	0.323904	0.161952	+	0.986799i	$0.448221\pi$
$758$	−1.77288e10	−1.47855
$759$	0	0
$760$	3.33958e7	0.00275959
$761$	−9.35754e9	−0.769690	−0.384845	−	0.922981i	$-0.625745\pi$
$761$	−9.35754e9	−0.769690	−0.384845	+	0.922981i	$0.625745\pi$
$762$	0	0
$763$	−6.09403e8	−0.0496671
$764$	1.49373e10	1.21184
$765$	0	0
$766$	2.13055e10	1.71274
$767$	9.54825e8	0.0764082
$768$	0	0
$769$	8.16327e9	0.647324	0.323662	−	0.946173i	$-0.395086\pi$
$769$	8.16327e9	0.647324	0.323662	+	0.946173i	$0.395086\pi$
$770$	2.66897e9	0.210681
$771$	0	0
$772$	−5.50727e9	−0.430800
$773$	−2.22041e10	−1.72904	−0.864521	−	0.502597i	$-0.832378\pi$
$773$	−2.22041e10	−1.72904	−0.864521	+	0.502597i	$0.832378\pi$
$774$	0	0
$775$	−3.44306e8	−0.0265698
$776$	6.78511e8	0.0521244
$777$	0	0
$778$	7.83138e9	0.596224
$779$	−6.28123e8	−0.0476062
$780$	0	0
$781$	3.59310e10	2.69892
$782$	5.82017e9	0.435223
$783$	0	0
$784$	1.41161e10	1.04619
$785$	1.41609e10	1.04483
$786$	0	0
$787$	2.42225e10	1.77136	0.885681	−	0.464294i	$-0.153692\pi$
$787$	2.42225e10	1.77136	0.885681	+	0.464294i	$0.153692\pi$
$788$	2.33935e10	1.70315
$789$	0	0
$790$	−1.71527e10	−1.23776
$791$	−1.60834e9	−0.115547
$792$	0	0
$793$	2.10350e10	1.49791
$794$	−1.69055e10	−1.19855
$795$	0	0
$796$	−2.40671e9	−0.169133
$797$	6.98957e9	0.489042	0.244521	−	0.969644i	$-0.421369\pi$
$797$	6.98957e9	0.489042	0.244521	+	0.969644i	$0.421369\pi$
$798$	0	0
$799$	−3.80633e10	−2.63993
$800$	4.35352e9	0.300626
$801$	0	0
$802$	−2.16782e10	−1.48393
$803$	2.24820e10	1.53225
$804$	0	0
$805$	2.63217e8	0.0177840
$806$	−2.42986e9	−0.163459
$807$	0	0
$808$	−5.55669e8	−0.0370575
$809$	−2.58126e9	−0.171400	−0.0857001	−	0.996321i	$-0.527313\pi$
$809$	−2.58126e9	−0.171400	−0.0857001	+	0.996321i	$0.527313\pi$
$810$	0	0
$811$	−2.00665e10	−1.32099	−0.660495	−	0.750831i	$-0.729653\pi$
$811$	−2.00665e10	−1.32099	−0.660495	+	0.750831i	$0.729653\pi$
$812$	−1.01343e9	−0.0664273
$813$	0	0
$814$	−4.60479e10	−2.99243
$815$	−2.88443e10	−1.86641
$816$	0	0
$817$	2.45378e8	0.0157419
$818$	−4.28801e9	−0.273917
$819$	0	0
$820$	−2.42434e10	−1.53549
$821$	1.53262e10	0.966571	0.483286	−	0.875463i	$-0.339443\pi$
$821$	1.53262e10	0.966571	0.483286	+	0.875463i	$0.339443\pi$
$822$	0	0
$823$	−8.78169e9	−0.549135	−0.274567	−	0.961568i	$-0.588535\pi$
$823$	−8.78169e9	−0.549135	−0.274567	+	0.961568i	$0.588535\pi$
$824$	2.27999e9	0.141967
$825$	0	0
$826$	1.39878e8	0.00863613
$827$	1.94526e10	1.19594	0.597970	−	0.801519i	$-0.295974\pi$
$827$	1.94526e10	1.19594	0.597970	+	0.801519i	$0.295974\pi$
$828$	0	0
$829$	4.62750e9	0.282102	0.141051	−	0.990002i	$-0.454952\pi$
$829$	4.62750e9	0.282102	0.141051	+	0.990002i	$0.454952\pi$
$830$	1.68280e10	1.02155
$831$	0	0
$832$	1.40422e10	0.845285
$833$	−2.48217e10	−1.48790
$834$	0	0
$835$	−9.86391e9	−0.586336
$836$	−9.12706e8	−0.0540268
$837$	0	0
$838$	−2.85226e10	−1.67431
$839$	−1.59754e10	−0.933868	−0.466934	−	0.884292i	$-0.654641\pi$
$839$	−1.59754e10	−0.933868	−0.466934	+	0.884292i	$0.654641\pi$
$840$	0	0
$841$	−2.98131e9	−0.172831
$842$	−2.79222e10	−1.61197
$843$	0	0
$844$	−5.06590e9	−0.290040
$845$	1.73469e9	0.0989063
$846$	0	0
$847$	2.92423e9	0.165356
$848$	3.39340e10	1.91095
$849$	0	0
$850$	−8.08193e9	−0.451387
$851$	−4.54130e9	−0.252596
$852$	0	0
$853$	2.49289e9	0.137525	0.0687624	−	0.997633i	$-0.478095\pi$
$853$	2.49289e9	0.137525	0.0687624	+	0.997633i	$0.478095\pi$
$854$	3.08154e9	0.169303
$855$	0	0
$856$	2.01660e9	0.109891
$857$	2.47858e10	1.34515	0.672573	−	0.740031i	$-0.265189\pi$
$857$	2.47858e10	1.34515	0.672573	+	0.740031i	$0.265189\pi$
$858$	0	0
$859$	5.77120e9	0.310664	0.155332	−	0.987862i	$-0.450355\pi$
$859$	5.77120e9	0.310664	0.155332	+	0.987862i	$0.450355\pi$
$860$	9.47075e9	0.507738
$861$	0	0
$862$	−1.07436e10	−0.571311
$863$	−1.86142e9	−0.0985840	−0.0492920	−	0.998784i	$-0.515696\pi$
$863$	−1.86142e9	−0.0985840	−0.0492920	+	0.998784i	$0.515696\pi$
$864$	0	0
$865$	−1.35607e10	−0.712402
$866$	−1.30605e10	−0.683354
$867$	0	0
$868$	−1.72896e8	−0.00897358
$869$	−2.75826e10	−1.42583
$870$	0	0
$871$	−7.39764e9	−0.379341
$872$	−9.74377e8	−0.0497645
$873$	0	0
$874$	−1.85321e8	−0.00938931
$875$	1.32463e9	0.0668444
$876$	0	0
$877$	2.56518e9	0.128416	0.0642080	−	0.997937i	$-0.479548\pi$
$877$	2.56518e9	0.128416	0.0642080	+	0.997937i	$0.479548\pi$
$878$	−1.69338e9	−0.0844352
$879$	0	0
$880$	4.15675e10	2.05620
$881$	−2.96840e10	−1.46254	−0.731268	−	0.682090i	$-0.761071\pi$
$881$	−2.96840e10	−1.46254	−0.731268	+	0.682090i	$0.761071\pi$
$882$	0	0
$883$	−6.39118e9	−0.312405	−0.156203	−	0.987725i	$-0.549925\pi$
$883$	−6.39118e9	−0.312405	−0.156203	+	0.987725i	$0.549925\pi$
$884$	−2.77032e10	−1.34880
$885$	0	0
$886$	−8.15971e9	−0.394146
$887$	1.95000e10	0.938216	0.469108	−	0.883141i	$-0.344576\pi$
$887$	1.95000e10	0.938216	0.469108	+	0.883141i	$0.344576\pi$
$888$	0	0
$889$	−1.15761e8	−0.00552594
$890$	4.11844e10	1.95825
$891$	0	0
$892$	3.43061e10	1.61843
$893$	1.21198e9	0.0569526
$894$	0	0
$895$	3.15240e8	0.0146981
$896$	−2.57660e8	−0.0119665
$897$	0	0
$898$	−1.17979e10	−0.543672
$899$	2.43429e9	0.111741
$900$	0	0
$901$	−5.96693e10	−2.71778
$902$	−8.02637e10	−3.64163
$903$	0	0
$904$	−2.57157e9	−0.115774
$905$	3.28052e10	1.47120
$906$	0	0
$907$	3.81572e10	1.69805	0.849025	−	0.528352i	$-0.177190\pi$
$907$	3.81572e10	1.69805	0.849025	+	0.528352i	$0.177190\pi$
$908$	−2.91457e10	−1.29204
$909$	0	0
$910$	−2.57947e9	−0.113471
$911$	−1.57478e10	−0.690089	−0.345045	−	0.938586i	$-0.612136\pi$
$911$	−1.57478e10	−0.690089	−0.345045	+	0.938586i	$0.612136\pi$
$912$	0	0
$913$	2.70605e10	1.17676
$914$	−4.21820e10	−1.82732
$915$	0	0
$916$	1.66982e10	0.717852
$917$	1.91493e9	0.0820086
$918$	0	0
$919$	4.62139e8	0.0196412	0.00982060	−	0.999952i	$-0.496874\pi$
$919$	4.62139e8	0.0196412	0.00982060	+	0.999952i	$0.496874\pi$
$920$	4.20859e8	0.0178188
$921$	0	0
$922$	−1.43309e10	−0.602166
$923$	−3.47260e10	−1.45361
$924$	0	0
$925$	6.30607e9	0.261977
$926$	3.89389e10	1.61156
$927$	0	0
$928$	−3.07800e10	−1.26430
$929$	−2.13510e10	−0.873702	−0.436851	−	0.899534i	$-0.643906\pi$
$929$	−2.13510e10	−0.873702	−0.436851	+	0.899534i	$0.643906\pi$
$930$	0	0
$931$	7.90351e8	0.0320993
$932$	1.50656e10	0.609580
$933$	0	0
$934$	−4.00430e10	−1.60810
$935$	−7.30922e10	−2.92436
$936$	0	0
$937$	−8.41470e9	−0.334156	−0.167078	−	0.985944i	$-0.553433\pi$
$937$	−8.41470e9	−0.334156	−0.167078	+	0.985944i	$0.553433\pi$
$938$	−1.08372e9	−0.0428755
$939$	0	0
$940$	4.67782e10	1.83694
$941$	−2.92044e10	−1.14257	−0.571287	−	0.820750i	$-0.693556\pi$
$941$	−2.92044e10	−1.14257	−0.571287	+	0.820750i	$0.693556\pi$
$942$	0	0
$943$	−7.91570e9	−0.307396
$944$	2.17851e9	0.0842865
$945$	0	0
$946$	3.13552e10	1.20418
$947$	−3.62326e10	−1.38635	−0.693177	−	0.720767i	$-0.743790\pi$
$947$	−3.62326e10	−1.38635	−0.693177	+	0.720767i	$0.743790\pi$
$948$	0	0
$949$	−2.17280e10	−0.825256
$950$	2.57337e8	0.00973801
$951$	0	0
$952$	2.38791e8	0.00896993
$953$	−2.82376e10	−1.05682	−0.528412	−	0.848988i	$-0.677212\pi$
$953$	−2.82376e10	−1.05682	−0.528412	+	0.848988i	$0.677212\pi$
$954$	0	0
$955$	−3.80889e10	−1.41510
$956$	−3.88896e9	−0.143956
$957$	0	0
$958$	3.89210e9	0.143023
$959$	−3.04028e8	−0.0111314
$960$	0	0
$961$	−2.70973e10	−0.984905
$962$	4.45036e10	1.61169
$963$	0	0
$964$	−2.63755e10	−0.948266
$965$	1.40431e10	0.503059
$966$	0	0
$967$	3.58758e10	1.27588	0.637938	−	0.770087i	$-0.279787\pi$
$967$	3.58758e10	1.27588	0.637938	+	0.770087i	$0.279787\pi$
$968$	4.67556e9	0.165680
$969$	0	0
$970$	2.94050e10	1.03448
$971$	1.79050e10	0.627634	0.313817	−	0.949483i	$-0.398392\pi$
$971$	1.79050e10	0.627634	0.313817	+	0.949483i	$0.398392\pi$
$972$	0	0
$973$	5.93074e8	0.0206402
$974$	−9.25787e9	−0.321037
$975$	0	0
$976$	4.79931e10	1.65236
$977$	−6.91838e9	−0.237341	−0.118671	−	0.992934i	$-0.537863\pi$
$977$	−6.91838e9	−0.237341	−0.118671	+	0.992934i	$0.537863\pi$
$978$	0	0
$979$	6.62271e10	2.25578
$980$	3.05049e10	1.03533
$981$	0	0
$982$	2.53714e10	0.854978
$983$	5.19842e10	1.74556	0.872779	−	0.488116i	$-0.162316\pi$
$983$	5.19842e10	1.74556	0.872779	+	0.488116i	$0.162316\pi$
$984$	0	0
$985$	−5.96518e10	−1.98883
$986$	5.71403e10	1.89834
$987$	0	0
$988$	8.82099e8	0.0290983
$989$	3.09229e9	0.101647
$990$	0	0
$991$	−3.47566e10	−1.13444	−0.567218	−	0.823568i	$-0.691980\pi$
$991$	−3.47566e10	−1.13444	−0.567218	+	0.823568i	$0.691980\pi$
$992$	−5.25122e9	−0.170793
$993$	0	0
$994$	−5.08722e9	−0.164297
$995$	6.13693e9	0.197501
$996$	0	0
$997$	−1.74989e10	−0.559215	−0.279608	−	0.960114i	$-0.590204\pi$
$997$	−1.74989e10	−0.559215	−0.279608	+	0.960114i	$0.590204\pi$
$998$	−4.76339e10	−1.51691
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	207.8.a.h.1.2	yes	12
3.2	odd	2		207.8.a.g.1.11	✓	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
207.8.a.g.1.11	✓	12	3.2	odd	2
207.8.a.h.1.2	yes	12	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 207 = 3^{2} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	207.a (trivial)

\(f(q)\)	\(=\)	\(q-15.7762 q^{2} +120.887 q^{4} -308.253 q^{5} +70.1815 q^{7} +112.213 q^{8} +O(q^{10})\)	\(q-15.7762 q^{2} +120.887 q^{4} -308.253 q^{5} +70.1815 q^{7} +112.213 q^{8} +4863.05 q^{10} +7820.09 q^{11} -7557.85 q^{13} -1107.19 q^{14} -17243.9 q^{16} +30321.5 q^{17} -965.470 q^{19} -37263.9 q^{20} -123371. q^{22} -12167.0 q^{23} +16895.2 q^{25} +119234. q^{26} +8484.04 q^{28} -119451. q^{29} -20379.0 q^{31} +257678. q^{32} -478357. q^{34} -21633.7 q^{35} +373247. q^{37} +15231.4 q^{38} -34590.2 q^{40} +650588. q^{41} -254154. q^{43} +945349. q^{44} +191949. q^{46} -1.25532e6 q^{47} -818618. q^{49} -266541. q^{50} -913646. q^{52} -1.96789e6 q^{53} -2.41057e6 q^{55} +7875.31 q^{56} +1.88448e6 q^{58} -126336. q^{59} -2.78320e6 q^{61} +321502. q^{62} -1.85796e6 q^{64} +2.32973e6 q^{65} +978803. q^{67} +3.66548e6 q^{68} +341296. q^{70} +4.59470e6 q^{71} +2.87490e6 q^{73} -5.88840e6 q^{74} -116713. q^{76} +548826. q^{77} -3.52715e6 q^{79} +5.31548e6 q^{80} -1.02638e7 q^{82} +3.46038e6 q^{83} -9.34672e6 q^{85} +4.00957e6 q^{86} +877520. q^{88} +8.46884e6 q^{89} -530421. q^{91} -1.47083e6 q^{92} +1.98042e7 q^{94} +297610. q^{95} +6.04661e6 q^{97} +1.29146e7 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 16 q^{2} + 640 q^{4} + 500 q^{5} - 228 q^{7} + 3072 q^{8}+O(q^{10}) \) 12 * q + 16 * q^2 + 640 * q^4 + 500 * q^5 - 228 * q^7 + 3072 * q^8	\( 12 q + 16 q^{2} + 640 q^{4} + 500 q^{5} - 228 q^{7} + 3072 q^{8} + 10270 q^{10} - 460 q^{11} - 21060 q^{13} + 4268 q^{14} + 56676 q^{16} + 73124 q^{17} + 8508 q^{19} + 170538 q^{20} + 124754 q^{22} - 146004 q^{23} + 194064 q^{25} + 206080 q^{26} - 390416 q^{28} + 268640 q^{29} - 191880 q^{31} + 1180172 q^{32} - 221436 q^{34} - 487244 q^{35} + 650332 q^{37} + 1432950 q^{38} + 1775722 q^{40} + 980088 q^{41} - 861276 q^{43} + 800666 q^{44} - 194672 q^{46} + 403868 q^{47} + 1699160 q^{49} + 2919092 q^{50} - 2369520 q^{52} - 201948 q^{53} - 1553512 q^{55} - 4848116 q^{56} + 3720672 q^{58} + 1302676 q^{59} + 2141364 q^{61} + 2160944 q^{62} + 9702136 q^{64} + 9099536 q^{65} - 6159260 q^{67} + 18442208 q^{68} - 10891632 q^{70} + 12584184 q^{71} + 7435872 q^{73} + 22491442 q^{74} + 5721386 q^{76} + 16450568 q^{77} + 3658028 q^{79} + 49905778 q^{80} - 5516316 q^{82} + 26137900 q^{83} + 5169556 q^{85} + 30678550 q^{86} + 14753046 q^{88} + 27235908 q^{89} - 7657216 q^{91} - 7786880 q^{92} - 23519352 q^{94} + 63623628 q^{95} + 22454720 q^{97} + 94951532 q^{98}+O(q^{100}) \) 12 * q + 16 * q^2 + 640 * q^4 + 500 * q^5 - 228 * q^7 + 3072 * q^8 + 10270 * q^10 - 460 * q^11 - 21060 * q^13 + 4268 * q^14 + 56676 * q^16 + 73124 * q^17 + 8508 * q^19 + 170538 * q^20 + 124754 * q^22 - 146004 * q^23 + 194064 * q^25 + 206080 * q^26 - 390416 * q^28 + 268640 * q^29 - 191880 * q^31 + 1180172 * q^32 - 221436 * q^34 - 487244 * q^35 + 650332 * q^37 + 1432950 * q^38 + 1775722 * q^40 + 980088 * q^41 - 861276 * q^43 + 800666 * q^44 - 194672 * q^46 + 403868 * q^47 + 1699160 * q^49 + 2919092 * q^50 - 2369520 * q^52 - 201948 * q^53 - 1553512 * q^55 - 4848116 * q^56 + 3720672 * q^58 + 1302676 * q^59 + 2141364 * q^61 + 2160944 * q^62 + 9702136 * q^64 + 9099536 * q^65 - 6159260 * q^67 + 18442208 * q^68 - 10891632 * q^70 + 12584184 * q^71 + 7435872 * q^73 + 22491442 * q^74 + 5721386 * q^76 + 16450568 * q^77 + 3658028 * q^79 + 49905778 * q^80 - 5516316 * q^82 + 26137900 * q^83 + 5169556 * q^85 + 30678550 * q^86 + 14753046 * q^88 + 27235908 * q^89 - 7657216 * q^91 - 7786880 * q^92 - 23519352 * q^94 + 63623628 * q^95 + 22454720 * q^97 + 94951532 * q^98

Introduction

L-functions

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