LMFDB - Embedded newform 252.9.bg.a.29.15

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [252,9,Mod(29,252)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(252, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 1, 0]))

N = Newforms(chi, 9, names="a")

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

chi := DirichletCharacter("252.29");

S:= CuspForms(chi, 9);

N := Newforms(S);

Level:	$ N $	$=$	$ 252 = 2^{2} \cdot 3^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 9 $
Character orbit:	$[\chi]$	$=$	252.bg (of order $6$, degree $2$, 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:	$102.659409735$
Analytic rank:	$0$
Dimension:	$96$
Relative dimension:	$48$ over $\Q(\zeta_{6})$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			29.15
Character	$\chi$	$=$	252.29
Dual form			252.9.bg.a.113.15

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-45.2321 - 67.1942i) q^{3} +(-4.45748 + 2.57353i) q^{5} +(-453.746 + 785.912i) q^{7} +(-2469.12 + 6078.67i) q^{9} +O(q^{10})$	\(q+(-45.2321 - 67.1942i) q^{3} +(-4.45748 + 2.57353i) q^{5} +(-453.746 + 785.912i) q^{7} +(-2469.12 + 6078.67i) q^{9} +(23525.6 + 13582.5i) q^{11} +(3068.09 + 5314.08i) q^{13} +(374.547 + 183.111i) q^{15} +33588.4i q^{17} +39146.7 q^{19} +(73332.6 - 5059.31i) q^{21} +(218910. - 126388. i) q^{23} +(-195299. + 338268. i) q^{25} +(520134. - 109040. i) q^{27} +(-1.19671e6 - 690919. i) q^{29} +(-84814.9 - 146904. i) q^{31} +(-151446. - 2.19514e6i) q^{33} -4670.92i q^{35} +921691. q^{37} +(218299. - 446525. i) q^{39} +(-2.33616e6 + 1.34878e6i) q^{41} +(-2.04077e6 + 3.53472e6i) q^{43} +(-4637.58 - 33449.9i) q^{45} +(-5.46498e6 - 3.15521e6i) q^{47} +(-411772. - 713209. i) q^{49} +(2.25695e6 - 1.51928e6i) q^{51} -781046. i q^{53} -139820. q^{55} +(-1.77069e6 - 2.63043e6i) q^{57} +(1.44811e7 - 8.36069e6i) q^{59} +(-2.45661e6 + 4.25498e6i) q^{61} +(-3.65694e6 - 4.69868e6i) q^{63} +(-27351.9 - 15791.6i) q^{65} +(-4.23026e6 - 7.32703e6i) q^{67} +(-1.83943e7 - 8.99270e6i) q^{69} -1.88049e7i q^{71} -6.69574e6 q^{73} +(3.15635e7 - 2.17760e6i) q^{75} +(-2.13493e7 + 1.23260e7i) q^{77} +(-9.43518e6 + 1.63422e7i) q^{79} +(-3.08536e7 - 3.00179e7i) q^{81} +(5.58539e6 + 3.22473e6i) q^{83} +(-86440.8 - 149720. i) q^{85} +(7.70381e6 + 1.11663e8i) q^{87} +9.36716e7i q^{89} -5.56853e6 q^{91} +(-6.03472e6 + 1.23438e7i) q^{93} +(-174496. + 100745. i) q^{95} +(5.67963e7 - 9.83741e7i) q^{97} +(-1.40651e8 + 1.09467e8i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 96 q - 42 q^{3} - 882 q^{5} - 14642 q^{9}+O(q^{10}) $ 96 * q - 42 * q^3 - 882 * q^5 - 14642 * q^9	$ 96 q - 42 q^{3} - 882 q^{5} - 14642 q^{9} - 6102 q^{11} - 63218 q^{15} - 354144 q^{19} + 81634 q^{21} - 689760 q^{23} + 4088394 q^{25} - 2939076 q^{27} - 1902474 q^{29} + 613830 q^{31} - 3732526 q^{33} + 4437300 q^{37} - 2690876 q^{39} + 8275176 q^{41} - 2941680 q^{43} + 7299362 q^{45} - 7663950 q^{47} - 39530064 q^{49} - 23625052 q^{51} + 8608908 q^{55} + 28697652 q^{57} + 38291778 q^{59} + 7577556 q^{63} + 42391494 q^{65} + 47903562 q^{67} - 52586968 q^{69} - 32396448 q^{73} + 245976220 q^{75} + 11461314 q^{79} - 16224230 q^{81} - 104964174 q^{83} + 108387294 q^{85} - 213493700 q^{87} - 12590844 q^{91} - 88124258 q^{93} + 293841792 q^{95} + 9277590 q^{97} - 77959808 q^{99}+O(q^{100}) $ 96 * q - 42 * q^3 - 882 * q^5 - 14642 * q^9 - 6102 * q^11 - 63218 * q^15 - 354144 * q^19 + 81634 * q^21 - 689760 * q^23 + 4088394 * q^25 - 2939076 * q^27 - 1902474 * q^29 + 613830 * q^31 - 3732526 * q^33 + 4437300 * q^37 - 2690876 * q^39 + 8275176 * q^41 - 2941680 * q^43 + 7299362 * q^45 - 7663950 * q^47 - 39530064 * q^49 - 23625052 * q^51 + 8608908 * q^55 + 28697652 * q^57 + 38291778 * q^59 + 7577556 * q^63 + 42391494 * q^65 + 47903562 * q^67 - 52586968 * q^69 - 32396448 * q^73 + 245976220 * q^75 + 11461314 * q^79 - 16224230 * q^81 - 104964174 * q^83 + 108387294 * q^85 - 213493700 * q^87 - 12590844 * q^91 - 88124258 * q^93 + 293841792 * q^95 + 9277590 * q^97 - 77959808 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/252\mathbb{Z}\right)^\times$.

$n$	$29$	$73$	$127$
$\chi(n)$	$e\left(\frac{1}{6}\right)$	$1$	$1$

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 $

$2$		0			0
$2$		0			0
$3$	−45.2321	−	67.1942i	−0.558421	−	0.829558i
$3$	−45.2321	−	67.1942i	−0.558421	−	0.829558i
$4$		0			0
$5$	−4.45748	+	2.57353i	−0.00713197	+	0.00411765i	−0.503562	−	0.863959i	$-0.667977\pi$
$5$	−4.45748	+	2.57353i	−0.00713197	+	0.00411765i	0.496430	+	0.868077i	$0.334644\pi$
$6$		0			0
$7$	−453.746	+	785.912i	−0.188982	+	0.327327i
$7$	−453.746	+	785.912i	−0.188982	+	0.327327i
$8$		0			0
$9$	−2469.12	+	6078.67i	−0.376332	+	0.926485i
$10$		0			0
$11$	23525.6	+	13582.5i	1.60683	+	0.927702i	0.990074	+	0.140545i	$0.0448855\pi$
$11$	23525.6	+	13582.5i	1.60683	+	0.927702i	0.616753	+	0.787157i	$0.288448\pi$
$12$		0			0
$13$	3068.09	+	5314.08i	0.107422	+	0.186061i	0.914725	−	0.404076i	$-0.132407\pi$
$13$	3068.09	+	5314.08i	0.107422	+	0.186061i	−0.807303	+	0.590137i	$0.799074\pi$
$14$		0			0
$15$	374.547	+	183.111i	0.00739847	+	0.00361700i
$16$		0			0
$17$			33588.4i			0.402156i	0.979575	+	0.201078i	$0.0644444\pi$
$17$			33588.4i			0.402156i	−0.979575	+	0.201078i	$0.935556\pi$
$18$		0			0
$19$	39146.7			0.300387			0.150193	−	0.988657i	$-0.452010\pi$
$19$	39146.7			0.300387			0.150193	+	0.988657i	$0.452010\pi$
$20$		0			0
$21$	73332.6	−	5059.31i	0.377068	−	0.0260144i
$22$		0			0
$23$	218910.	−	126388.i	0.782266	−	0.451641i	−0.0549670	−	0.998488i	$-0.517505\pi$
$23$	218910.	−	126388.i	0.782266	−	0.451641i	0.837233	+	0.546847i	$0.184172\pi$
$24$		0			0
$25$	−195299.	+	338268.i	−0.499966	+	0.865967i
$26$		0			0
$27$	520134.	−	109040.i	0.978724	−	0.205179i
$28$		0			0
$29$	−1.19671e6	−	690919.i	−1.69198	−	0.976867i	−0.952916	−	0.303234i	$-0.901934\pi$
$29$	−1.19671e6	−	690919.i	−1.69198	−	0.976867i	−0.739067	−	0.673632i	$-0.764733\pi$
$30$		0			0
$31$	−84814.9	−	146904.i	−0.0918386	−	0.159069i	0.816446	−	0.577421i	$-0.195941\pi$
$31$	−84814.9	−	146904.i	−0.0918386	−	0.159069i	−0.908285	+	0.418352i	$0.862608\pi$
$32$		0			0
$33$	−151446.	−	2.19514e6i	−0.127703	−	1.85100i
$34$		0			0
$35$		−	4670.92i		−	0.00311265i
$36$		0			0
$37$	921691.			0.491789			0.245894	−	0.969297i	$-0.420918\pi$
$37$	921691.			0.491789			0.245894	+	0.969297i	$0.420918\pi$
$38$		0			0
$39$	218299.	−	446525.i	0.0943614	−	0.193013i
$40$		0			0
$41$	−2.33616e6	+	1.34878e6i	−0.826738	+	0.477317i	−0.852734	−	0.522345i	$-0.825057\pi$
$41$	−2.33616e6	+	1.34878e6i	−0.826738	+	0.477317i	0.0259965	+	0.999662i	$0.491724\pi$
$42$		0			0
$43$	−2.04077e6	+	3.53472e6i	−0.596926	+	1.03391i	0.396346	+	0.918101i	$0.370278\pi$
$43$	−2.04077e6	+	3.53472e6i	−0.596926	+	1.03391i	−0.993272	+	0.115805i	$0.963055\pi$
$44$		0			0
$45$	−4637.58	−	33449.9i	−0.00113094	−	0.00815727i
$46$		0			0
$47$	−5.46498e6	−	3.15521e6i	−1.11995	−	0.646601i	−0.178559	−	0.983929i	$-0.557143\pi$
$47$	−5.46498e6	−	3.15521e6i	−1.11995	−	0.646601i	−0.941387	+	0.337328i	$0.890477\pi$
$48$		0			0
$49$	−411772.	−	713209.i	−0.0714286	−	0.123718i
$50$		0			0
$51$	2.25695e6	−	1.51928e6i	0.333611	−	0.224572i
$52$		0			0
$53$		−	781046.i		−	0.0989858i	−0.998774	−	0.0494929i	$-0.984239\pi$
$53$		−	781046.i		−	0.0989858i	0.998774	−	0.0494929i	$-0.0157605\pi$
$54$		0			0
$55$	−139820.			−0.0152798
$56$		0			0
$57$	−1.77069e6	−	2.63043e6i	−0.167742	−	0.249188i
$58$		0			0
$59$	1.44811e7	−	8.36069e6i	1.19507	−	0.689976i	0.235620	−	0.971845i	$-0.424288\pi$
$59$	1.44811e7	−	8.36069e6i	1.19507	−	0.689976i	0.959453	+	0.281869i	$0.0909545\pi$
$60$		0			0
$61$	−2.45661e6	+	4.25498e6i	−0.177426	+	0.307311i	−0.940998	−	0.338412i	$-0.890110\pi$
$61$	−2.45661e6	+	4.25498e6i	−0.177426	+	0.307311i	0.763572	+	0.645722i	$0.223444\pi$
$62$		0			0
$63$	−3.65694e6	−	4.69868e6i	−0.232143	−	0.298273i
$64$		0			0
$65$	−27351.9	−	15791.6i	−0.00153227	−	0.000884654i
$66$		0			0
$67$	−4.23026e6	−	7.32703e6i	−0.209927	−	0.363604i	0.741764	−	0.670661i	$-0.233989\pi$
$67$	−4.23026e6	−	7.32703e6i	−0.209927	−	0.363604i	−0.951691	+	0.307057i	$0.900656\pi$
$68$		0			0
$69$	−1.83943e7	−	8.99270e6i	−0.811496	−	0.396729i
$70$		0			0
$71$		−	1.88049e7i		−	0.740009i	−0.929030	−	0.370005i	$-0.879356\pi$
$71$		−	1.88049e7i		−	0.740009i	0.929030	−	0.370005i	$-0.120644\pi$
$72$		0			0
$73$	−6.69574e6			−0.235780			−0.117890	−	0.993027i	$-0.537613\pi$
$73$	−6.69574e6			−0.235780			−0.117890	+	0.993027i	$0.537613\pi$
$74$		0			0
$75$	3.15635e7	−	2.17760e6i	0.997561	−	0.0688230i
$76$		0			0
$77$	−2.13493e7	+	1.23260e7i	−0.607324	+	0.350638i
$78$		0			0
$79$	−9.43518e6	+	1.63422e7i	−0.242238	+	0.419568i	−0.961351	−	0.275324i	$-0.911215\pi$
$79$	−9.43518e6	+	1.63422e7i	−0.242238	+	0.419568i	0.719114	+	0.694892i	$0.244548\pi$
$80$		0			0
$81$	−3.08536e7	−	3.00179e7i	−0.716748	−	0.697332i
$82$		0			0
$83$	5.58539e6	+	3.22473e6i	0.117690	+	0.0679486i	0.557690	−	0.830050i	$-0.311688\pi$
$83$	5.58539e6	+	3.22473e6i	0.117690	+	0.0679486i	−0.439999	+	0.897998i	$0.645021\pi$
$84$		0			0
$85$	−86440.8	−	149720.i	−0.00165593	−	0.00286816i
$86$		0			0
$87$	7.70381e6	+	1.11663e8i	0.134471	+	1.94910i
$88$		0			0
$89$			9.36716e7i			1.49296i	0.665408	+	0.746480i	$0.268258\pi$
$89$			9.36716e7i			1.49296i	−0.665408	+	0.746480i	$0.731742\pi$
$90$		0			0
$91$	−5.56853e6			−0.0812036
$92$		0			0
$93$	−6.03472e6	+	1.23438e7i	−0.0806724	+	0.165013i
$94$		0			0
$95$	−174496.	+	100745.i	−0.00214235	+	0.00123689i
$96$		0			0
$97$	5.67963e7	−	9.83741e7i	0.641554	−	1.11120i	−0.343532	−	0.939141i	$-0.611623\pi$
$97$	5.67963e7	−	9.83741e7i	0.641554	−	1.11120i	0.985086	−	0.172063i	$-0.0550434\pi$
$98$		0			0
$99$	−1.40651e8	+	1.09467e8i	−1.46420	+	1.13958i
$100$		0			0
$101$	1.39642e8	+	8.06226e7i	1.34194	+	0.774767i	0.987091	−	0.160158i	$-0.0512003\pi$
$101$	1.39642e8	+	8.06226e7i	1.34194	+	0.774767i	0.354845	+	0.934925i	$0.384534\pi$
$102$		0			0
$103$	2.46124e7	+	4.26300e7i	0.218678	+	0.378762i	0.954404	−	0.298518i	$-0.0964922\pi$
$103$	2.46124e7	+	4.26300e7i	0.218678	+	0.378762i	−0.735726	+	0.677279i	$0.763159\pi$
$104$		0			0
$105$	−313858.	+	211275.i	−0.00258212	+	0.00173817i
$106$		0			0
$107$			7.46133e7i			0.569221i	0.958643	+	0.284610i	$0.0918643\pi$
$107$			7.46133e7i			0.569221i	−0.958643	+	0.284610i	$0.908136\pi$
$108$		0			0
$109$	−1.25910e8			−0.891975			−0.445987	−	0.895039i	$-0.647147\pi$
$109$	−1.25910e8			−0.891975			−0.445987	+	0.895039i	$0.647147\pi$
$110$		0			0
$111$	−4.16900e7	−	6.19323e7i	−0.274625	−	0.407967i
$112$		0			0
$113$	3.84744e7	−	2.22132e7i	0.235971	−	0.136238i	−0.377353	−	0.926070i	$-0.623166\pi$
$113$	3.84744e7	−	2.22132e7i	0.235971	−	0.136238i	0.613323	+	0.789832i	$0.289832\pi$
$114$		0			0
$115$	−650525.	+	1.12674e6i	−0.00371940	+	0.00644219i
$116$		0			0
$117$	−3.98780e7	+	5.52879e6i	−0.212809	+	0.0295044i
$118$		0			0
$119$	−2.63975e7	−	1.52406e7i	−0.131636	−	0.0760003i
$120$		0			0
$121$	2.61788e8	+	4.53431e8i	1.22126	+	2.11529i
$122$		0			0
$123$	1.96300e8	+	9.59683e7i	0.857630	+	0.419283i
$124$		0			0
$125$		−	4.02100e6i		−	0.0164700i
$126$		0			0
$127$	−4.62012e8			−1.77598			−0.887991	−	0.459861i	$-0.847899\pi$
$127$	−4.62012e8			−1.77598			−0.887991	+	0.459861i	$0.847899\pi$
$128$		0			0
$129$	3.29821e8	−	2.27548e7i	1.19102	−	0.0821701i
$130$		0			0
$131$	−3.53576e8	+	2.04137e8i	−1.20060	+	0.693166i	−0.960688	−	0.277629i	$-0.910451\pi$
$131$	−3.53576e8	+	2.04137e8i	−1.20060	+	0.693166i	−0.239910	+	0.970795i	$0.577118\pi$
$132$		0			0
$133$	−1.77627e7	+	3.07658e7i	−0.0567677	+	0.0983246i
$134$		0			0
$135$	−2.03787e6	+	1.82463e6i	−0.00613538	+	0.00549337i
$136$		0			0
$137$	−3.64576e8	−	2.10488e8i	−1.03492	−	0.597509i	−0.116527	−	0.993187i	$-0.537176\pi$
$137$	−3.64576e8	−	2.10488e8i	−1.03492	−	0.597509i	−0.918389	+	0.395678i	$0.870510\pi$
$138$		0			0
$139$	1.86788e8	+	3.23526e8i	0.500368	+	0.866663i	1.00000		0.000425358i	$0.000135396\pi$
$139$	1.86788e8	+	3.23526e8i	0.500368	+	0.866663i	−0.499632	+	0.866238i	$0.666531\pi$
$140$		0			0
$141$	3.51808e7	+	5.09931e8i	0.0890081	+	1.29013i
$142$		0			0
$143$			1.66689e8i			0.398623i
$144$		0			0
$145$	7.11240e6			0.0160896
$146$		0			0
$147$	−2.92982e7	+	5.99286e7i	−0.0627440	+	0.128341i
$148$		0			0
$149$	−4.56874e8	+	2.63776e8i	−0.926939	+	0.535169i	−0.885842	−	0.463987i	$-0.846419\pi$
$149$	−4.56874e8	+	2.63776e8i	−0.926939	+	0.535169i	−0.0410970	+	0.999155i	$0.513085\pi$
$150$		0			0
$151$	−2.75935e8	+	4.77933e8i	−0.530760	+	0.919304i	0.468596	+	0.883413i	$0.344760\pi$
$151$	−2.75935e8	+	4.77933e8i	−0.530760	+	0.919304i	−0.999356	+	0.0358908i	$0.988573\pi$
$152$		0			0
$153$	−2.04173e8	−	8.29338e7i	−0.372591	−	0.151344i
$154$		0			0
$155$	756122.	+	436547.i	0.00130998	+	0.000756318i
$156$		0			0
$157$	−1.55377e8	−	2.69122e8i	−0.255735	−	0.442945i	0.709360	−	0.704846i	$-0.248984\pi$
$157$	−1.55377e8	−	2.69122e8i	−0.255735	−	0.442945i	−0.965095	+	0.261901i	$0.915651\pi$
$158$		0			0
$159$	−5.24817e7	+	3.53283e7i	−0.0821145	+	0.0552757i
$160$		0			0
$161$			2.29392e8i			0.341409i
$162$		0			0
$163$	−1.26460e8			−0.179144			−0.0895722	−	0.995980i	$-0.528550\pi$
$163$	−1.26460e8			−0.179144			−0.0895722	+	0.995980i	$0.528550\pi$
$164$		0			0
$165$	6.32434e6	+	9.39507e6i	0.00853256	+	0.0126755i
$166$		0			0
$167$	−2.19953e8	+	1.26990e8i	−0.282791	+	0.163269i	−0.634686	−	0.772770i	$-0.718871\pi$
$167$	−2.19953e8	+	1.26990e8i	−0.282791	+	0.163269i	0.351895	+	0.936039i	$0.385537\pi$
$168$		0			0
$169$	3.89039e8	−	6.73835e8i	0.476921	−	0.826051i
$170$		0			0
$171$	−9.66577e7	+	2.37960e8i	−0.113045	+	0.278304i
$172$		0			0
$173$	1.68238e8	+	9.71323e7i	0.187819	+	0.108437i	0.590961	−	0.806700i	$-0.298749\pi$
$173$	1.68238e8	+	9.71323e7i	0.187819	+	0.108437i	−0.403142	+	0.915137i	$0.632082\pi$
$174$		0			0
$175$	−1.77233e8	−	3.06976e8i	−0.188969	−	0.327305i
$176$		0			0
$177$	−1.21680e9	−	5.94877e8i	−1.23973	−	0.606085i
$178$		0			0
$179$			2.28518e8i			0.222592i	0.993787	+	0.111296i	$0.0355001\pi$
$179$			2.28518e8i			0.222592i	−0.993787	+	0.111296i	$0.964500\pi$
$180$		0			0
$181$	4.75740e8			0.443257			0.221628	−	0.975131i	$-0.428863\pi$
$181$	4.75740e8			0.443257			0.221628	+	0.975131i	$0.428863\pi$
$182$		0			0
$183$	3.97027e8	−	2.73914e7i	0.354010	−	0.0244237i
$184$		0			0
$185$	−4.10842e6	+	2.37200e6i	−0.00350742	+	0.00202501i
$186$		0			0
$187$	−4.56214e8	+	7.90187e8i	−0.373081	+	0.646194i
$188$		0			0
$189$	−1.50313e8	+	4.58256e8i	−0.117801	+	0.359138i
$190$		0			0
$191$	−9.50924e8	−	5.49016e8i	−0.714517	−	0.412527i	0.0982143	−	0.995165i	$-0.468687\pi$
$191$	−9.50924e8	−	5.49016e8i	−0.714517	−	0.412527i	−0.812731	+	0.582639i	$0.802020\pi$
$192$		0			0
$193$	−2.81262e8	−	4.87159e8i	−0.202713	−	0.351109i	0.746689	−	0.665174i	$-0.231642\pi$
$193$	−2.81262e8	−	4.87159e8i	−0.202713	−	0.351109i	−0.949402	+	0.314065i	$0.898309\pi$
$194$		0			0
$195$	176078.	+	2.55218e6i	0.000121777	+	0.00176511i
$196$		0			0
$197$			2.06311e9i			1.36980i	0.728637	+	0.684901i	$0.240154\pi$
$197$			2.06311e9i			1.36980i	−0.728637	+	0.684901i	$0.759846\pi$
$198$		0			0
$199$	−2.85343e9			−1.81951			−0.909756	−	0.415143i	$-0.863732\pi$
$199$	−2.85343e9			−1.81951			−0.909756	+	0.415143i	$0.863732\pi$
$200$		0			0
$201$	−3.00990e8	+	6.15666e8i	−0.184403	+	0.377191i
$202$		0			0
$203$	1.08600e9	−	6.27004e8i	0.639509	−	0.369221i
$204$		0			0
$205$	6.94227e6	−	1.20244e7i	0.00393085	−	0.00680843i
$206$		0			0
$207$	2.27755e8	+	1.64275e9i	0.124047	+	0.894724i
$208$		0			0
$209$	9.20947e8	+	5.31709e8i	0.482669	+	0.278669i
$210$		0			0
$211$	1.27894e9	+	2.21519e9i	0.645239	+	1.11759i	0.984246	+	0.176803i	$0.0565755\pi$
$211$	1.27894e9	+	2.21519e9i	0.645239	+	1.11759i	−0.339008	+	0.940784i	$0.610091\pi$
$212$		0			0
$213$	−1.26358e9	+	8.50584e8i	−0.613880	+	0.413237i
$214$		0			0
$215$		−	2.10079e7i		−	0.00983172i
$216$		0			0
$217$	1.53938e8			0.0694234
$218$		0			0
$219$	3.02862e8	+	4.49915e8i	0.131664	+	0.195593i
$220$		0			0
$221$	−1.78492e8	+	1.03052e8i	−0.0748254	+	0.0432005i
$222$		0			0
$223$	−5.46286e8	+	9.46196e8i	−0.220903	+	0.382615i	−0.955082	−	0.296341i	$-0.904234\pi$
$223$	−5.46286e8	+	9.46196e8i	−0.220903	+	0.382615i	0.734180	+	0.678955i	$0.237567\pi$
$224$		0			0
$225$	−1.57400e9	−	2.02238e9i	−0.614151	−	0.789102i
$226$		0			0
$227$	1.07010e9	+	6.17822e8i	0.403014	+	0.232680i	0.687784	−	0.725916i	$-0.258584\pi$
$227$	1.07010e9	+	6.17822e8i	0.403014	+	0.232680i	−0.284769	+	0.958596i	$0.591917\pi$
$228$		0			0
$229$	−2.20846e9	−	3.82516e9i	−0.803059	−	1.39094i	−0.917594	−	0.397520i	$-0.869871\pi$
$229$	−2.20846e9	−	3.82516e9i	−0.803059	−	1.39094i	0.114535	−	0.993419i	$-0.463462\pi$
$230$		0			0
$231$	1.79391e9	+	8.77016e8i	0.630017	+	0.308006i
$232$		0			0
$233$			2.60832e9i			0.884986i	0.896772	+	0.442493i	$0.145906\pi$
$233$			2.60832e9i			0.884986i	−0.896772	+	0.442493i	$0.854094\pi$
$234$		0			0
$235$	3.24801e7			0.0106499
$236$		0			0
$237$	1.52487e9	−	1.05203e8i	0.483327	−	0.0333453i
$238$		0			0
$239$	−1.53484e9	+	8.86139e8i	−0.470404	+	0.271588i	−0.716409	−	0.697681i	$-0.754215\pi$
$239$	−1.53484e9	+	8.86139e8i	−0.470404	+	0.271588i	0.246005	+	0.969269i	$0.420882\pi$
$240$		0			0
$241$	7.31649e8	−	1.26725e9i	0.216888	−	0.375660i	−0.736967	−	0.675928i	$-0.763743\pi$
$241$	7.31649e8	−	1.26725e9i	0.216888	−	0.375660i	0.953855	+	0.300268i	$0.0970762\pi$
$242$		0			0
$243$	−6.21452e8	+	3.43096e9i	−0.178231	+	0.983989i
$244$		0			0
$245$	3.67093e6	+	2.11941e6i	0.00101885	+	0.000588235i
$246$		0			0
$247$	1.20105e8	+	2.08029e8i	0.0322682	+	0.0558902i
$248$		0			0
$249$	−3.59560e7	−	5.21167e8i	−0.00935349	−	0.135575i
$250$		0			0
$251$			4.17990e9i			1.05310i	0.850143	+	0.526552i	$0.176516\pi$
$251$			4.17990e9i			1.05310i	−0.850143	+	0.526552i	$0.823484\pi$
$252$		0			0
$253$	6.86664e9			1.67595
$254$		0			0
$255$	−6.15041e6	+	1.25805e7i	−0.00145460	+	0.00297534i
$256$		0			0
$257$	4.02433e9	−	2.32345e9i	0.922489	−	0.532599i	0.0380604	−	0.999275i	$-0.487882\pi$
$257$	4.02433e9	−	2.32345e9i	0.922489	−	0.532599i	0.884428	+	0.466676i	$0.154549\pi$
$258$		0			0
$259$	−4.18214e8	+	7.24368e8i	−0.0929393	+	0.160976i
$260$		0			0
$261$	7.15468e9	−	5.56842e9i	1.54180	−	1.19997i
$262$		0			0
$263$	−7.84081e8	−	4.52689e8i	−0.163884	−	0.0946187i	0.415815	−	0.909449i	$-0.363497\pi$
$263$	−7.84081e8	−	4.52689e8i	−0.163884	−	0.0946187i	−0.579699	+	0.814831i	$0.696830\pi$
$264$		0			0
$265$	2.01004e6	+	3.48150e6i	0.000407589	+	0.000705964i
$266$		0			0
$267$	6.29419e9	−	4.23696e9i	1.23850	−	0.833700i
$268$		0			0
$269$		−	5.08763e9i		−	0.971642i	−0.874058	−	0.485821i	$-0.838521\pi$
$269$		−	5.08763e9i		−	0.971642i	0.874058	−	0.485821i	$-0.161479\pi$
$270$		0			0
$271$	−6.98111e9			−1.29434			−0.647168	−	0.762347i	$-0.724047\pi$
$271$	−6.98111e9			−1.29434			−0.647168	+	0.762347i	$0.724047\pi$
$272$		0			0
$273$	2.51876e8	+	3.74173e8i	0.0453458	+	0.0673631i
$274$		0			0
$275$	−9.18905e9	+	5.30530e9i	−1.60672	+	0.927639i
$276$		0			0
$277$	−4.52036e9	+	7.82949e9i	−0.767810	+	1.32989i	0.170939	+	0.985282i	$0.445320\pi$
$277$	−4.52036e9	+	7.82949e9i	−0.767810	+	1.32989i	−0.938748	+	0.344604i	$0.888013\pi$
$278$		0			0
$279$	1.10240e9	−	1.52839e8i	0.181937	−	0.0252242i
$280$		0			0
$281$	−3.06336e9	−	1.76863e9i	−0.491330	−	0.283669i	0.233796	−	0.972286i	$-0.424885\pi$
$281$	−3.06336e9	−	1.76863e9i	−0.491330	−	0.283669i	−0.725126	+	0.688616i	$0.758218\pi$
$282$		0			0
$283$	1.13059e9	+	1.95824e9i	0.176262	+	0.305295i	0.940597	−	0.339524i	$-0.110266\pi$
$283$	1.13059e9	+	1.95824e9i	0.176262	+	0.305295i	−0.764335	+	0.644819i	$0.776933\pi$
$284$		0			0
$285$	1.46623e7	+	7.16818e6i	0.00222240	+	0.00108650i
$286$		0			0
$287$		−	2.44802e9i		−	0.360818i
$288$		0			0
$289$	5.84757e9			0.838271
$290$		0			0
$291$	−9.17919e9	+	6.33284e8i	−1.28007	+	0.0883134i
$292$		0			0
$293$	−4.14727e9	+	2.39443e9i	−0.562719	+	0.324886i	−0.754236	−	0.656603i	$-0.771993\pi$
$293$	−4.14727e9	+	2.39443e9i	−0.562719	+	0.324886i	0.191517	+	0.981489i	$0.438659\pi$
$294$		0			0
$295$	−4.30329e7	+	7.45352e7i	−0.00568215	+	0.00984178i
$296$		0			0
$297$	1.37175e10	+	4.49948e9i	1.76299	+	0.578278i
$298$		0			0
$299$	1.34327e9	+	7.75537e8i	0.168065	+	0.0970326i
$300$		0			0
$301$	−1.85198e9	−	3.20773e9i	−0.225617	−	0.390780i
$302$		0			0
$303$	−8.98949e8	−	1.30299e10i	−0.106651	−	1.54586i
$304$		0			0
$305$		−	2.52886e7i		−	0.00292231i
$306$		0			0
$307$	7.91791e9			0.891368			0.445684	−	0.895190i	$-0.352961\pi$
$307$	7.91791e9			0.891368			0.445684	+	0.895190i	$0.352961\pi$
$308$		0			0
$309$	1.75121e9	−	3.58205e9i	0.192090	−	0.392915i
$310$		0			0
$311$	8.01447e9	−	4.62715e9i	0.856708	−	0.494621i	−0.00620019	−	0.999981i	$-0.501974\pi$
$311$	8.01447e9	−	4.62715e9i	0.856708	−	0.494621i	0.862909	+	0.505360i	$0.168640\pi$
$312$		0			0
$313$	4.66235e9	−	8.07542e9i	0.485766	−	0.841372i	−0.514100	−	0.857730i	$-0.671874\pi$
$313$	4.66235e9	−	8.07542e9i	0.485766	−	0.841372i	0.999866	+	0.0163586i	$0.00520735\pi$
$314$		0			0
$315$	2.83929e7	+	1.15330e7i	0.00288382	+	0.00117139i
$316$		0			0
$317$	−6.30815e9	−	3.64201e9i	−0.624691	−	0.360665i	0.154002	−	0.988070i	$-0.450784\pi$
$317$	−6.30815e9	−	3.64201e9i	−0.624691	−	0.360665i	−0.778693	+	0.627405i	$0.784117\pi$
$318$		0			0
$319$	−1.87688e10	−	3.25085e10i	−1.81248	−	3.13931i
$320$		0			0
$321$	5.01358e9	−	3.37491e9i	0.472202	−	0.317865i
$322$		0			0
$323$			1.31488e9i			0.120802i
$324$		0			0
$325$	−2.39678e9			−0.214830
$326$		0			0
$327$	5.69515e9	+	8.46039e9i	0.498097	+	0.739945i
$328$		0			0
$329$	4.95943e9	−	2.86333e9i	0.423300	−	0.244392i
$330$		0			0
$331$	9.44071e9	−	1.63518e10i	0.786489	−	1.36224i	−0.141617	−	0.989922i	$-0.545230\pi$
$331$	9.44071e9	−	1.63518e10i	0.786489	−	1.36224i	0.928106	−	0.372317i	$-0.121437\pi$
$332$		0			0
$333$	−2.27576e9	+	5.60265e9i	−0.185076	+	0.455635i
$334$		0			0
$335$	3.77126e7	+	2.17734e7i	0.00299439	+	0.00172881i
$336$		0			0
$337$	−1.11108e10	−	1.92444e10i	−0.861438	−	1.49205i	−0.870541	−	0.492096i	$-0.836231\pi$
$337$	−1.11108e10	−	1.92444e10i	−0.861438	−	1.49205i	0.00910282	−	0.999959i	$-0.497102\pi$
$338$		0			0
$339$	−3.23288e9	−	1.58051e9i	−0.244788	−	0.119673i
$340$		0			0
$341$		−	4.60799e9i		−	0.340795i
$342$		0			0
$343$	7.47359e8			0.0539949
$344$		0			0
$345$	1.05135e8	−	7.25341e6i	0.00742115	−	0.000511995i
$346$		0			0
$347$	1.26598e9	−	7.30914e8i	0.0873190	−	0.0504137i	−0.455705	−	0.890131i	$-0.650613\pi$
$347$	1.26598e9	−	7.30914e8i	0.0873190	−	0.0504137i	0.543024	+	0.839717i	$0.317279\pi$
$348$		0			0
$349$	4.72183e9	−	8.17845e9i	0.318280	−	0.551276i	−0.661850	−	0.749637i	$-0.730228\pi$
$349$	4.72183e9	−	8.17845e9i	0.318280	−	0.551276i	0.980129	+	0.198360i	$0.0635616\pi$
$350$		0			0
$351$	2.17527e9	+	2.42949e9i	0.143313	+	0.160061i
$352$		0			0
$353$	1.85659e10	+	1.07190e10i	1.19569	+	0.690330i	0.959591	−	0.281400i	$-0.0907987\pi$
$353$	1.85659e10	+	1.07190e10i	1.19569	+	0.690330i	0.236096	+	0.971730i	$0.424132\pi$
$354$		0			0
$355$	4.83949e7	+	8.38224e7i	0.00304710	+	0.00527773i
$356$		0			0
$357$	1.69934e8	+	2.46313e9i	0.0104618	+	0.151640i
$358$		0			0
$359$			2.71745e10i			1.63600i	0.575218	+	0.818000i	$0.304917\pi$
$359$			2.71745e10i			1.63600i	−0.575218	+	0.818000i	$0.695083\pi$
$360$		0			0
$361$	−1.54511e10			−0.909768
$362$		0			0
$363$	1.86267e10	−	3.81003e10i	1.07278	−	2.19433i
$364$		0			0
$365$	2.98461e7	−	1.72317e7i	0.00168158	−	0.000970859i
$366$		0			0
$367$	1.21367e10	−	2.10213e10i	0.669013	−	1.15877i	−0.309167	−	0.951008i	$-0.600050\pi$
$367$	1.21367e10	−	2.10213e10i	0.669013	−	1.15877i	0.978180	−	0.207757i	$-0.0666164\pi$
$368$		0			0
$369$	−2.43055e9	−	1.75311e10i	−0.131099	−	0.945590i
$370$		0			0
$371$	6.13833e8	+	3.54397e8i	0.0324007	+	0.0187066i
$372$		0			0
$373$	1.23498e10	+	2.13905e10i	0.638007	+	1.10506i	0.985870	+	0.167515i	$0.0535743\pi$
$373$	1.23498e10	+	2.13905e10i	0.638007	+	1.10506i	−0.347862	+	0.937546i	$0.613092\pi$
$374$		0			0
$375$	−2.70188e8	+	1.81878e8i	−0.0136628	+	0.00919721i
$376$		0			0
$377$		−	8.47920e9i		−	0.419749i
$378$		0			0
$379$	−1.21625e10			−0.589475			−0.294738	−	0.955578i	$-0.595232\pi$
$379$	−1.21625e10			−0.589475			−0.294738	+	0.955578i	$0.595232\pi$
$380$		0			0
$381$	2.08978e10	+	3.10445e10i	0.991745	+	1.47328i
$382$		0			0
$383$	−1.18286e10	+	6.82922e9i	−0.549714	+	0.317378i	−0.749007	−	0.662562i	$-0.769469\pi$
$383$	−1.18286e10	+	6.82922e9i	−0.549714	+	0.317378i	0.199293	+	0.979940i	$0.436136\pi$
$384$		0			0
$385$	6.34427e7	−	1.09886e8i	0.00288761	−	0.00500149i
$386$		0			0
$387$	−1.64475e10	−	2.11328e10i	−0.733255	−	0.942135i
$388$		0			0
$389$	−1.12109e10	−	6.47262e9i	−0.489601	−	0.282671i	0.234808	−	0.972042i	$-0.424554\pi$
$389$	−1.12109e10	−	6.47262e9i	−0.489601	−	0.282671i	−0.724409	+	0.689371i	$0.757887\pi$
$390$		0			0
$391$	4.24517e9	+	7.35284e9i	0.181630	+	0.314592i
$392$		0			0
$393$	2.97098e10	+	1.45247e10i	1.24546	+	0.608888i
$394$		0			0
$395$		−	9.71268e7i		−	0.00398980i
$396$		0			0
$397$	−4.16720e10			−1.67758			−0.838789	−	0.544457i	$-0.816736\pi$
$397$	−4.16720e10			−1.67758			−0.838789	+	0.544457i	$0.816736\pi$
$398$		0			0
$399$	2.87073e9	−	1.98055e8i	0.113266	−	0.00781438i
$400$		0			0
$401$	−2.10791e10	+	1.21700e10i	−0.815221	+	0.470668i	−0.848766	−	0.528769i	$-0.822654\pi$
$401$	−2.10791e10	+	1.21700e10i	−0.815221	+	0.470668i	0.0335447	+	0.999437i	$0.489320\pi$
$402$		0			0
$403$	5.20439e8	−	9.01426e8i	0.0197310	−	0.0341751i
$404$		0			0
$405$	2.14781e8	+	5.44014e7i	0.00798319	+	0.00202204i
$406$		0			0
$407$	2.16833e10	+	1.25189e10i	0.790219	+	0.456233i
$408$		0			0
$409$	−1.12821e10	−	1.95411e10i	−0.403177	−	0.698324i	0.590930	−	0.806723i	$-0.298761\pi$
$409$	−1.12821e10	−	1.95411e10i	−0.403177	−	0.698324i	−0.994107	+	0.108399i	$0.965428\pi$
$410$		0			0
$411$	2.34696e9	+	3.40182e10i	0.0822504	+	1.19219i
$412$		0			0
$413$			1.51745e10i			0.521573i
$414$		0			0
$415$	−3.31957e7			−0.00111915
$416$		0			0
$417$	1.32903e10	−	2.71848e10i	0.439531	−	0.899047i
$418$		0			0
$419$	−2.68720e10	+	1.55146e10i	−0.871855	+	0.503366i	−0.867964	−	0.496626i	$-0.834572\pi$
$419$	−2.68720e10	+	1.55146e10i	−0.871855	+	0.503366i	−0.00389105	+	0.999992i	$0.501239\pi$
$420$		0			0
$421$	2.84521e9	−	4.92805e9i	0.0905704	−	0.156873i	−0.817181	−	0.576381i	$-0.804464\pi$
$421$	2.84521e9	−	4.92805e9i	0.0905704	−	0.156873i	0.907751	+	0.419509i	$0.137798\pi$
$422$		0			0
$423$	3.26731e10	−	2.54292e10i	1.02054	−	0.794276i
$424$		0			0
$425$	−1.13619e10	−	6.55980e9i	−0.348253	−	0.201064i
$426$		0			0
$427$	−2.22936e9	−	3.86136e9i	−0.0670607	−	0.116153i
$428$		0			0
$429$	1.12005e10	−	7.53969e9i	0.330681	−	0.222600i
$430$		0			0
$431$			2.10204e9i			0.0609160i	0.999536	+	0.0304580i	$0.00969658\pi$
$431$			2.10204e9i			0.0609160i	−0.999536	+	0.0304580i	$0.990303\pi$
$432$		0			0
$433$	3.98416e10			1.13341			0.566703	−	0.823922i	$-0.308219\pi$
$433$	3.98416e10			1.13341			0.566703	+	0.823922i	$0.308219\pi$
$434$		0			0
$435$	−3.21709e8	−	4.77912e8i	−0.00898475	−	0.0133472i
$436$		0			0
$437$	8.56960e9	−	4.94766e9i	0.234982	−	0.135667i
$438$		0			0
$439$	7.52796e9	−	1.30388e10i	0.202684	−	0.351059i	−0.746708	−	0.665151i	$-0.768367\pi$
$439$	7.52796e9	−	1.30388e10i	0.202684	−	0.351059i	0.949392	+	0.314093i	$0.101700\pi$
$440$		0			0
$441$	5.35207e9	−	7.42025e8i	0.141504	−	0.0196184i
$442$		0			0
$443$	−3.26685e10	−	1.88611e10i	−0.848230	−	0.489726i	0.0118231	−	0.999930i	$-0.496237\pi$
$443$	−3.26685e10	−	1.88611e10i	−0.848230	−	0.489726i	−0.860053	+	0.510204i	$0.829570\pi$
$444$		0			0
$445$	−2.41067e8	−	4.17540e8i	−0.00614748	−	0.0106477i
$446$		0			0
$447$	3.83896e10	+	1.87681e10i	0.961575	+	0.470100i
$448$		0			0
$449$			3.43010e10i			0.843959i	0.906605	+	0.421979i	$0.138665\pi$
$449$			3.43010e10i			0.843959i	−0.906605	+	0.421979i	$0.861335\pi$
$450$		0			0
$451$	−7.32794e10			−1.77123
$452$		0			0
$453$	4.45954e10	−	3.07670e9i	1.05900	−	0.0730620i
$454$		0			0
$455$	2.48216e7	−	1.43308e7i	0.000579142	−	0.000334368i
$456$		0			0
$457$	−1.67695e10	+	2.90457e10i	−0.384464	+	0.665912i	−0.991695	−	0.128614i	$-0.958947\pi$
$457$	−1.67695e10	+	2.90457e10i	−0.384464	+	0.665912i	0.607230	+	0.794526i	$0.292281\pi$
$458$		0			0
$459$	3.66250e9	+	1.74705e10i	0.0825139	+	0.393600i
$460$		0			0
$461$	7.49918e10	+	4.32965e10i	1.66039	+	0.958627i	0.972528	+	0.232785i	$0.0747838\pi$
$461$	7.49918e10	+	4.32965e10i	1.66039	+	0.958627i	0.687862	+	0.725842i	$0.258550\pi$
$462$		0			0
$463$	2.71493e10	+	4.70239e10i	0.590792	+	1.02328i	0.994126	+	0.108228i	$0.0345178\pi$
$463$	2.71493e10	+	4.70239e10i	0.590792	+	1.02328i	−0.403335	+	0.915053i	$0.632149\pi$
$464$		0			0
$465$	−4.86754e6	−	7.05529e7i	−0.000104111	−	0.00150905i
$466$		0			0
$467$			7.03645e10i			1.47940i	0.672936	+	0.739701i	$0.265033\pi$
$467$			7.03645e10i			1.47940i	−0.672936	+	0.739701i	$0.734967\pi$
$468$		0			0
$469$	7.67786e9			0.158690
$470$		0			0
$471$	−1.10554e10	+	2.26134e10i	−0.224641	+	0.459497i
$472$		0			0
$473$	−9.60205e10	+	5.54375e10i	−1.91831	+	1.10754i
$474$		0			0
$475$	−7.64532e9	+	1.32421e10i	−0.150183	+	0.260125i
$476$		0			0
$477$	4.74772e9	+	1.92849e9i	0.0917089	+	0.0372516i
$478$		0			0
$479$	−3.74181e10	−	2.16033e10i	−0.710787	−	0.410373i	0.100565	−	0.994930i	$-0.467935\pi$
$479$	−3.74181e10	−	2.16033e10i	−0.710787	−	0.410373i	−0.811352	+	0.584557i	$0.801268\pi$
$480$		0			0
$481$	2.82783e9	+	4.89794e9i	0.0528290	+	0.0915026i
$482$		0			0
$483$	1.54138e10	−	1.03759e10i	0.283218	−	0.190650i
$484$		0			0
$485$			5.84668e8i			0.0105668i
$486$		0			0
$487$	−4.35690e10			−0.774572			−0.387286	−	0.921960i	$-0.626587\pi$
$487$	−4.35690e10			−0.774572			−0.387286	+	0.921960i	$0.626587\pi$
$488$		0			0
$489$	5.72006e9	+	8.49739e9i	0.100038	+	0.148611i
$490$		0			0
$491$	−1.27466e10	+	7.35924e9i	−0.219315	+	0.126621i	−0.605633	−	0.795744i	$-0.707080\pi$
$491$	−1.27466e10	+	7.35924e9i	−0.219315	+	0.126621i	0.386318	+	0.922366i	$0.373747\pi$
$492$		0			0
$493$	2.32069e10	−	4.01955e10i	0.392852	−	0.680440i
$494$		0			0
$495$	3.45231e8	−	8.49917e8i	0.00575028	−	0.0141565i
$496$		0			0
$497$	1.47790e10	+	8.53264e9i	0.242225	+	0.139849i
$498$		0			0
$499$	−6.01261e10	−	1.04141e11i	−0.969752	−	1.67966i	−0.696265	−	0.717785i	$-0.745156\pi$
$499$	−6.01261e10	−	1.04141e11i	−0.969752	−	1.67966i	−0.273487	−	0.961876i	$-0.588177\pi$
$500$		0			0
$501$	1.84820e10	+	9.03556e9i	0.293357	+	0.143418i
$502$		0			0
$503$			3.05585e10i			0.477375i	0.971096	+	0.238687i	$0.0767171\pi$
$503$			3.05585e10i			0.477375i	−0.971096	+	0.238687i	$0.923283\pi$
$504$		0			0
$505$	−8.29938e8			−0.0127609
$506$		0			0
$507$	−6.28749e10	+	4.33782e9i	−0.951580	+	0.0656508i
$508$		0			0
$509$	−1.43145e10	+	8.26450e9i	−0.213258	+	0.123125i	−0.602825	−	0.797874i	$-0.705958\pi$
$509$	−1.43145e10	+	8.26450e9i	−0.213258	+	0.123125i	0.389566	+	0.920998i	$0.372625\pi$
$510$		0			0
$511$	3.03817e9	−	5.26226e9i	0.0445582	−	0.0771771i
$512$		0			0
$513$	2.03615e10	−	4.26857e9i	0.293996	−	0.0616330i
$514$		0			0
$515$	−2.19419e8	−	1.26682e8i	−0.00311921	−	0.00180088i
$516$		0			0
$517$	−8.57111e10	−	1.48456e11i	−1.19971	−	2.07795i
$518$		0			0
$519$	−1.08303e9	−	1.56981e10i	−0.0149270	−	0.216361i
$520$		0			0
$521$			1.32694e11i			1.80094i	0.434919	+	0.900470i	$0.356777\pi$
$521$			1.32694e11i			1.80094i	−0.434919	+	0.900470i	$0.643223\pi$
$522$		0			0
$523$	−9.57373e10			−1.27960			−0.639800	−	0.768541i	$-0.720983\pi$
$523$	−9.57373e10			−1.27960			−0.639800	+	0.768541i	$0.720983\pi$
$524$		0			0
$525$	−1.26104e10	+	2.57942e10i	−0.165994	+	0.339535i
$526$		0			0
$527$	4.93426e9	−	2.84880e9i	0.0639705	−	0.0369334i
$528$		0			0
$529$	−7.20777e9	+	1.24842e10i	−0.0920404	+	0.159419i
$530$		0			0
$531$	1.50662e10	+	1.08669e11i	0.189507	+	1.36688i
$532$		0			0
$533$	−1.43351e10	−	8.27638e9i	−0.177620	−	0.102549i
$534$		0			0
$535$	−1.92019e8	−	3.32587e8i	−0.00234385	−	0.00405967i
$536$		0			0
$537$	1.53551e10	−	1.03364e10i	0.184653	−	0.124300i
$538$		0			0
$539$		−	2.23715e10i		−	0.265058i
$540$		0			0
$541$	−9.46951e10			−1.10545			−0.552724	−	0.833364i	$-0.686412\pi$
$541$	−9.46951e10			−1.10545			−0.552724	+	0.833364i	$0.686412\pi$
$542$		0			0
$543$	−2.15187e10	−	3.19670e10i	−0.247524	−	0.367707i
$544$		0			0
$545$	5.61239e8	−	3.24032e8i	0.00636154	−	0.00367284i
$546$		0			0
$547$	3.83925e10	−	6.64978e10i	0.428842	−	0.742776i	−0.567928	−	0.823078i	$-0.692255\pi$
$547$	3.83925e10	−	6.64978e10i	0.428842	−	0.742776i	0.996771	+	0.0803015i	$0.0255883\pi$
$548$		0			0
$549$	−1.97989e10	−	2.54390e10i	−0.217948	−	0.280033i
$550$		0			0
$551$	−4.68471e10	−	2.70472e10i	−0.508249	−	0.293438i
$552$		0			0
$553$	−8.56236e9	−	1.48304e10i	−0.0915573	−	0.158582i
$554$		0			0
$555$	3.45217e8	+	1.68772e8i	0.00363848	+	0.00177880i
$556$		0			0
$557$			5.69962e10i			0.592141i	0.955166	+	0.296070i	$0.0956763\pi$
$557$			5.69962e10i			0.592141i	−0.955166	+	0.296070i	$0.904324\pi$
$558$		0			0
$559$	−2.50450e10			−0.256492
$560$		0			0
$561$	7.37315e10	−	5.08683e9i	0.744392	−	0.0513566i
$562$		0			0
$563$	−1.28349e11	+	7.41021e10i	−1.27749	+	0.737560i	−0.976386	−	0.216032i	$-0.930689\pi$
$563$	−1.28349e11	+	7.41021e10i	−1.27749	+	0.737560i	−0.301104	+	0.953591i	$0.597355\pi$
$564$		0			0
$565$	−1.14333e8	+	1.98030e8i	−0.00112196	+	0.00194329i
$566$		0			0
$567$	3.75911e10	−	1.06277e10i	0.363708	−	0.102827i
$568$		0			0
$569$	9.32686e10	+	5.38486e10i	0.889788	+	0.513719i	0.873873	−	0.486154i	$-0.161601\pi$
$569$	9.32686e10	+	5.38486e10i	0.889788	+	0.513719i	0.0159146	+	0.999873i	$0.494934\pi$
$570$		0			0
$571$	−4.25071e10	−	7.36245e10i	−0.399869	−	0.692593i	0.593841	−	0.804583i	$-0.297611\pi$
$571$	−4.25071e10	−	7.36245e10i	−0.399869	−	0.692593i	−0.993709	+	0.111990i	$0.964278\pi$
$572$		0			0
$573$	6.12158e9	+	8.87297e10i	0.0567865	+	0.823096i
$574$		0			0
$575$			9.87337e10i			0.903221i
$576$		0			0
$577$	2.06637e11			1.86425			0.932125	−	0.362138i	$-0.117953\pi$
$577$	2.06637e11			1.86425			0.932125	+	0.362138i	$0.117953\pi$
$578$		0			0
$579$	−2.00122e10	+	4.09344e10i	−0.178066	+	0.364228i
$580$		0			0
$581$	−5.06870e9	+	2.92642e9i	−0.0444828	+	0.0256822i
$582$		0			0
$583$	1.06085e10	−	1.83745e10i	0.0918294	−	0.159053i
$584$		0			0
$585$	1.63527e8	−	1.27272e8i	0.00139626	−	0.00108670i
$586$		0			0
$587$	−6.54041e10	−	3.77611e10i	−0.550874	−	0.318047i	0.198600	−	0.980081i	$-0.436360\pi$
$587$	−6.54041e10	−	3.77611e10i	−0.550874	−	0.318047i	−0.749474	+	0.662033i	$0.769694\pi$
$588$		0			0
$589$	−3.32022e9	−	5.75079e9i	−0.0275871	−	0.0477822i
$590$		0			0
$591$	1.38629e11	−	9.33188e10i	1.13633	−	0.764925i
$592$		0			0
$593$			1.62123e11i			1.31107i	0.755166	+	0.655534i	$0.227556\pi$
$593$			1.62123e11i			1.31107i	−0.755166	+	0.655534i	$0.772444\pi$
$594$		0			0
$595$	1.56889e8			0.00125177
$596$		0			0
$597$	1.29067e11	+	1.91734e11i	1.01605	+	1.50939i
$598$		0			0
$599$	−3.18123e10	+	1.83668e10i	−0.247109	+	0.142668i	−0.618440	−	0.785832i	$-0.712235\pi$
$599$	−3.18123e10	+	1.83668e10i	−0.247109	+	0.142668i	0.371331	+	0.928501i	$0.378902\pi$
$600$		0			0
$601$	−9.35809e10	+	1.62087e11i	−0.717281	+	1.24237i	0.244792	+	0.969576i	$0.421280\pi$
$601$	−9.35809e10	+	1.62087e11i	−0.717281	+	1.24237i	−0.962073	+	0.272792i	$0.912053\pi$
$602$		0			0
$603$	5.49836e10	−	7.62306e9i	0.415876	−	0.0576581i
$604$		0			0
$605$	−2.33383e9	−	1.34744e9i	−0.0174200	−	0.0100574i
$606$		0			0
$607$	6.69644e10	+	1.15986e11i	0.493275	+	0.854378i	0.999970	−	0.00774754i	$-0.00246614\pi$
$607$	6.69644e10	+	1.15986e11i	0.493275	+	0.854378i	−0.506695	+	0.862126i	$0.669133\pi$
$608$		0			0
$609$	−9.12532e10	−	4.46124e10i	−0.663405	−	0.324329i
$610$		0			0
$611$		−	3.87218e10i		−	0.277837i
$612$		0			0
$613$	6.60603e10			0.467842			0.233921	−	0.972256i	$-0.424844\pi$
$613$	6.60603e10			0.467842			0.233921	+	0.972256i	$0.424844\pi$
$614$		0			0
$615$	−1.12198e9	+	7.74070e7i	−0.00784305	+	0.000541103i
$616$		0			0
$617$	−8.14672e10	+	4.70351e10i	−0.562137	+	0.324550i	−0.754003	−	0.656871i	$-0.771879\pi$
$617$	−8.14672e10	+	4.70351e10i	−0.562137	+	0.324550i	0.191866	+	0.981421i	$0.438546\pi$
$618$		0			0
$619$	−1.09510e11	+	1.89677e11i	−0.745918	+	1.29197i	0.203847	+	0.979003i	$0.434656\pi$
$619$	−1.09510e11	+	1.89677e11i	−0.745918	+	1.29197i	−0.949765	+	0.312965i	$0.898678\pi$
$620$		0			0
$621$	1.00081e11	−	8.96087e10i	0.672955	−	0.602537i
$622$		0			0
$623$	−7.36176e10	−	4.25032e10i	−0.488686	−	0.282143i
$624$		0			0
$625$	−7.62784e10	−	1.32118e11i	−0.499898	−	0.865849i
$626$		0			0
$627$	−5.92860e9	−	8.59326e10i	−0.0383603	−	0.556017i
$628$		0			0
$629$			3.09582e10i			0.197776i
$630$		0			0
$631$	−2.59542e11			−1.63715			−0.818577	−	0.574397i	$-0.805237\pi$
$631$	−2.59542e11			−1.63715			−0.818577	+	0.574397i	$0.805237\pi$
$632$		0			0
$633$	9.09987e10	−	1.86135e11i	0.566788	−	1.15935i
$634$		0			0
$635$	2.05941e9	−	1.18900e9i	0.0126663	−	0.00731286i
$636$		0			0
$637$	2.52670e9	−	4.37638e9i	0.0153460	−	0.0265801i
$638$		0			0
$639$	1.14309e11	+	4.64314e10i	0.685607	+	0.278489i
$640$		0			0
$641$	−1.46968e11	−	8.48520e10i	−0.870544	−	0.502609i	−0.00301493	−	0.999995i	$-0.500960\pi$
$641$	−1.46968e11	−	8.48520e10i	−0.870544	−	0.502609i	−0.867529	+	0.497387i	$0.834293\pi$
$642$		0			0
$643$	−4.14650e10	−	7.18195e10i	−0.242571	−	0.420144i	0.718875	−	0.695139i	$-0.244657\pi$
$643$	−4.14650e10	−	7.18195e10i	−0.242571	−	0.420144i	−0.961446	+	0.274995i	$0.911324\pi$
$644$		0			0
$645$	−1.41161e9	+	9.50232e8i	−0.00815598	+	0.00549024i
$646$		0			0
$647$		−	1.90109e10i		−	0.108489i	−0.998528	−	0.0542445i	$-0.982725\pi$
$647$		−	1.90109e10i		−	0.108489i	0.998528	−	0.0542445i	$-0.0172750\pi$
$648$		0			0
$649$	4.54236e11			2.56037
$650$		0			0
$651$	−6.96292e9	−	1.03437e10i	−0.0387675	−	0.0575908i
$652$		0			0
$653$	2.17474e11	−	1.25559e11i	1.19607	−	0.690550i	0.236391	−	0.971658i	$-0.424035\pi$
$653$	2.17474e11	−	1.25559e11i	1.19607	−	0.690550i	0.959676	+	0.281108i	$0.0907020\pi$
$654$		0			0
$655$	1.05071e9	−	1.81988e9i	0.00570843	−	0.00988728i
$656$		0			0
$657$	1.65326e10	−	4.07012e10i	0.0887316	−	0.218447i
$658$		0			0
$659$	4.15120e10	+	2.39670e10i	0.220106	+	0.127078i	0.605999	−	0.795465i	$-0.292773\pi$
$659$	4.15120e10	+	2.39670e10i	0.220106	+	0.127078i	−0.385893	+	0.922543i	$0.626107\pi$
$660$		0			0
$661$	−5.45597e10	−	9.45001e10i	−0.285802	−	0.495024i	0.687001	−	0.726656i	$-0.258927\pi$
$661$	−5.45597e10	−	9.45001e10i	−0.285802	−	0.495024i	−0.972803	+	0.231632i	$0.925593\pi$
$662$		0			0
$663$	1.49981e10	+	7.33234e9i	0.0776213	+	0.0379480i
$664$		0			0
$665$		−	1.82851e8i		−	0.000934998i
$666$		0			0
$667$	−3.49295e11			−1.76477
$668$		0			0
$669$	8.82885e10	−	6.09114e9i	0.440758	−	0.0304084i
$670$		0			0
$671$	−1.15586e11	+	6.67338e10i	−0.570186	+	0.329197i
$672$		0			0
$673$	6.25543e10	−	1.08347e11i	0.304928	−	0.528150i	−0.672318	−	0.740263i	$-0.734701\pi$
$673$	6.25543e10	−	1.08347e11i	0.304928	−	0.528150i	0.977245	+	0.212113i	$0.0680344\pi$
$674$		0			0
$675$	−6.46969e10	+	1.97240e11i	−0.311651	+	0.950125i
$676$		0			0
$677$	2.60813e11	+	1.50580e11i	1.24158	+	0.716826i	0.969416	−	0.245425i	$-0.0789275\pi$
$677$	2.60813e11	+	1.50580e11i	1.24158	+	0.716826i	0.272164	+	0.962251i	$0.412261\pi$
$678$		0			0
$679$	5.15423e10	+	8.92738e10i	0.242485	+	0.419996i
$680$		0			0
$681$	−6.88877e9	−	9.98498e10i	−0.0320297	−	0.464257i
$682$		0			0
$683$			8.43479e10i			0.387607i	0.981040	+	0.193803i	$0.0620824\pi$
$683$			8.43479e10i			0.387607i	−0.981040	+	0.193803i	$0.937918\pi$
$684$		0			0
$685$	2.16679e9			0.00984133
$686$		0			0
$687$	−1.57136e11	+	3.21416e11i	−0.705419	+	1.44291i
$688$		0			0
$689$	4.15054e9	−	2.39632e9i	0.0184174	−	0.0106333i
$690$		0			0
$691$	2.68065e10	−	4.64302e10i	0.117578	−	0.203652i	−0.801229	−	0.598358i	$-0.795820\pi$
$691$	2.68065e10	−	4.64302e10i	0.117578	−	0.203652i	0.918807	+	0.394706i	$0.129154\pi$
$692$		0			0
$693$	−2.22118e10	−	1.60209e11i	−0.0963056	−	0.694633i
$694$		0			0
$695$	−1.66521e9	−	9.61409e8i	−0.00713723	−	0.00412068i
$696$		0			0
$697$	−4.53036e10	−	7.84681e10i	−0.191956	−	0.332477i
$698$		0			0
$699$	1.75264e11	−	1.17980e11i	0.734147	−	0.494195i
$700$		0			0
$701$		−	1.07710e11i		−	0.446050i	−0.974813	−	0.223025i	$-0.928407\pi$
$701$		−	1.07710e11i		−	0.446050i	0.974813	−	0.223025i	$-0.0715931\pi$
$702$		0			0
$703$	3.60811e10			0.147727
$704$		0			0
$705$	−1.46914e9	−	2.18247e9i	−0.00594712	−	0.00883470i
$706$		0			0
$707$	−1.26725e11	+	7.31644e10i	−0.507204	+	0.292835i
$708$		0			0
$709$	1.76936e11	−	3.06462e11i	0.700214	−	1.21281i	−0.268177	−	0.963370i	$-0.586421\pi$
$709$	1.76936e11	−	3.06462e11i	0.700214	−	1.21281i	0.968391	−	0.249437i	$-0.0802455\pi$
$710$		0			0
$711$	−7.60423e10	−	9.77041e10i	−0.297561	−	0.382327i
$712$		0			0
$713$	−3.71336e10	−	2.14391e10i	−0.143684	−	0.0829562i
$714$		0			0
$715$	−4.28979e8	−	7.43013e8i	−0.00164139	−	0.00284297i
$716$		0			0
$717$	1.28967e11	+	6.30503e10i	0.487981	+	0.238567i
$718$		0			0
$719$			2.15378e11i			0.805907i	0.915220	+	0.402953i	$0.132016\pi$
$719$			2.15378e11i			0.805907i	−0.915220	+	0.402953i	$0.867984\pi$
$720$		0			0
$721$	−4.46712e10			−0.165305
$722$		0			0
$723$	−1.18246e11	+	8.15795e9i	−0.432746	+	0.0298557i
$724$		0			0
$725$	4.67432e11	−	2.69872e11i	1.69187	−	0.976800i
$726$		0			0
$727$	4.81617e10	−	8.34185e10i	0.172411	−	0.298624i	−0.766851	−	0.641825i	$-0.778178\pi$
$727$	4.81617e10	−	8.34185e10i	0.172411	−	0.298624i	0.939262	+	0.343200i	$0.111511\pi$
$728$		0			0
$729$	2.58650e11	−	1.13431e11i	0.915803	−	0.401627i
$730$		0			0
$731$	−1.18726e11	−	6.85463e10i	−0.415791	−	0.240057i
$732$		0			0
$733$	1.71492e10	+	2.97033e10i	0.0594058	+	0.102894i	0.894199	−	0.447670i	$-0.147746\pi$
$733$	1.71492e10	+	2.97033e10i	0.0594058	+	0.102894i	−0.834793	+	0.550564i	$0.814413\pi$
$734$		0			0
$735$	−2.36316e7	−	3.42530e8i	−8.09738e−5	−	0.00117368i
$736$		0			0
$737$		−	2.29830e11i		−	0.778998i
$738$		0			0
$739$	3.06331e11			1.02710			0.513551	−	0.858059i	$-0.328330\pi$
$739$	3.06331e11			1.02710			0.513551	+	0.858059i	$0.328330\pi$
$740$		0			0
$741$	8.54570e9	−	1.74800e10i	0.0283449	−	0.0579786i
$742$		0			0
$743$	3.93004e11	−	2.26901e11i	1.28956	−	0.744527i	0.310984	−	0.950415i	$-0.399342\pi$
$743$	3.93004e11	−	2.26901e11i	1.28956	−	0.744527i	0.978576	+	0.205888i	$0.0660082\pi$
$744$		0			0
$745$	1.35767e9	−	2.35156e9i	0.00440727	−	0.00763362i
$746$		0			0
$747$	−3.33930e10	+	2.59895e10i	−0.107244	+	0.0834671i
$748$		0			0
$749$	−5.86394e10	−	3.38555e10i	−0.186321	−	0.107573i
$750$		0			0
$751$	1.89842e10	+	3.28815e10i	0.0596804	+	0.103370i	0.894322	−	0.447424i	$-0.147659\pi$
$751$	1.89842e10	+	3.28815e10i	0.0596804	+	0.103370i	−0.834642	+	0.550794i	$0.814325\pi$
$752$		0			0
$753$	2.80865e11	−	1.89066e11i	0.873611	−	0.588075i
$754$		0			0
$755$		−	2.84050e9i		−	0.00874193i
$756$		0			0
$757$	−3.34296e11			−1.01800			−0.509000	−	0.860767i	$-0.669985\pi$
$757$	−3.34296e11			−1.01800			−0.509000	+	0.860767i	$0.669985\pi$
$758$		0			0
$759$	−3.10592e11	−	4.61398e11i	−0.935888	−	1.39030i
$760$		0			0
$761$	−1.62741e11	+	9.39583e10i	−0.485241	+	0.280154i	−0.722598	−	0.691269i	$-0.757052\pi$
$761$	−1.62741e11	+	9.39583e10i	−0.485241	+	0.280154i	0.237357	+	0.971422i	$0.423719\pi$
$762$		0			0
$763$	5.71310e10	−	9.89538e10i	0.168567	−	0.291967i
$764$		0			0
$765$	1.12353e9	−	1.55769e8i	0.00328049	−	0.000454816i
$766$		0			0
$767$	8.88587e10	+	5.13026e10i	0.256755	+	0.148238i
$768$		0			0
$769$	−6.67525e10	−	1.15619e11i	−0.190881	−	0.330615i	0.754662	−	0.656114i	$-0.227801\pi$
$769$	−6.67525e10	−	1.15619e11i	−0.190881	−	0.330615i	−0.945542	+	0.325499i	$0.894468\pi$
$770$		0			0
$771$	−3.38151e11	−	1.65317e11i	−0.956959	−	0.467843i
$772$		0			0
$773$		−	3.27997e11i		−	0.918654i	−0.888267	−	0.459327i	$-0.848091\pi$
$773$		−	3.27997e11i		−	0.918654i	0.888267	−	0.459327i	$-0.151909\pi$
$774$		0			0
$775$	6.62571e10			0.183665
$776$		0			0
$777$	6.75900e10	−	4.66312e9i	0.185438	−	0.0127936i
$778$		0			0
$779$	−9.14531e10	+	5.28004e10i	−0.248341	+	0.143380i
$780$		0			0
$781$	2.55417e11	−	4.42395e11i	0.686508	−	1.18907i
$782$		0			0
$783$	−6.97787e11	−	2.28881e11i	−1.85642	−	0.608924i
$784$		0			0
$785$	1.38519e9	+	7.99737e8i	0.00364778	+	0.00210605i
$786$		0			0
$787$	−2.97548e11	−	5.15367e11i	−0.775635	−	1.34344i	−0.934437	−	0.356129i	$-0.884096\pi$
$787$	−2.97548e11	−	5.15367e11i	−0.775635	−	1.34344i	0.158802	−	0.987310i	$-0.449237\pi$
$788$		0			0
$789$	5.04752e9	+	7.31617e10i	0.0130248	+	0.188789i
$790$		0			0
$791$			4.03166e10i			0.102986i
$792$		0			0
$793$	−3.01484e10			−0.0762380
$794$		0			0
$795$	1.43018e8	−	2.92539e8i	0.000358032	−	0.000732343i
$796$		0			0
$797$	5.98955e11	−	3.45807e11i	1.48444	−	0.857039i	0.484592	−	0.874740i	$-0.338968\pi$
$797$	5.98955e11	−	3.45807e11i	1.48444	−	0.857039i	0.999843	+	0.0177015i	$0.00563486\pi$
$798$		0			0
$799$	1.05978e11	−	1.83560e11i	0.260034	−	0.450392i
$800$		0			0
$801$	−5.69399e11	−	2.31286e11i	−1.38320	−	0.561849i
$802$		0			0
$803$	−1.57521e11	−	9.09447e10i	−0.378858	−	0.218734i
$804$		0			0
$805$	−5.90347e8	−	1.02251e9i	−0.00140580	−	0.00243492i
$806$		0			0
$807$	−3.41859e11	+	2.30124e11i	−0.806033	+	0.542585i
$808$		0			0
$809$		−	2.02609e11i		−	0.473005i	−0.971631	−	0.236502i	$-0.923999\pi$
$809$		−	2.02609e11i		−	0.473005i	0.971631	−	0.236502i	$-0.0760011\pi$
$810$		0			0
$811$	2.65632e11			0.614041			0.307020	−	0.951703i	$-0.400668\pi$
$811$	2.65632e11			0.614041			0.307020	+	0.951703i	$0.400668\pi$
$812$		0			0
$813$	3.15770e11	+	4.69090e11i	0.722784	+	1.07373i
$814$		0			0
$815$	5.63694e8	−	3.25449e8i	0.00127765	−	0.000737653i
$816$		0			0
$817$	−7.98894e10	+	1.38372e11i	−0.179308	+	0.310571i
$818$		0			0
$819$	1.37494e10	−	3.38493e10i	0.0305595	−	0.0752339i
$820$		0			0
$821$	−2.71451e11	−	1.56722e11i	−0.597473	−	0.344951i	0.170574	−	0.985345i	$-0.445438\pi$
$821$	−2.71451e11	−	1.56722e11i	−0.597473	−	0.344951i	−0.768047	+	0.640394i	$0.778771\pi$
$822$		0			0
$823$	−1.59021e10	−	2.75433e10i	−0.0346621	−	0.0600366i	0.848174	−	0.529718i	$-0.177702\pi$
$823$	−1.59021e10	−	2.75433e10i	−0.0346621	−	0.0600366i	−0.882836	+	0.469681i	$0.844369\pi$
$824$		0			0
$825$	7.72125e11	+	3.77481e11i	1.66676	+	0.814853i
$826$		0			0
$827$			2.68494e11i			0.574001i	0.957930	+	0.287000i	$0.0926581\pi$
$827$			2.68494e11i			0.574001i	−0.957930	+	0.287000i	$0.907342\pi$
$828$		0			0
$829$	−1.62759e11			−0.344608			−0.172304	−	0.985044i	$-0.555121\pi$
$829$	−1.62759e11			−0.344608			−0.172304	+	0.985044i	$0.555121\pi$
$830$		0			0
$831$	7.30561e11	−	5.04024e10i	1.53198	−	0.105693i
$832$		0			0
$833$	2.39556e10	−	1.38308e10i	0.0497539	−	0.0287254i
$834$		0			0
$835$	6.53626e8	−	1.13211e9i	0.00134457	−	0.00232886i
$836$		0			0
$837$	−6.01336e10	−	6.71614e10i	−0.122522	−	0.136841i
$838$		0			0
$839$	−2.43872e11	−	1.40800e11i	−0.492169	−	0.284154i	0.233305	−	0.972404i	$-0.425046\pi$
$839$	−2.43872e11	−	1.40800e11i	−0.492169	−	0.284154i	−0.725474	+	0.688250i	$0.758379\pi$
$840$		0			0
$841$	7.04615e11	+	1.22043e12i	1.40854	+	2.43966i
$842$		0			0
$843$	1.97204e10	+	2.85839e11i	0.0390486	+	0.565993i
$844$		0			0
$845$			4.00481e9i			0.00785517i
$846$		0			0
$847$	−4.75142e11			−0.923187
$848$		0			0
$849$	8.04432e10	−	1.64544e11i	0.154831	−	0.316703i
$850$		0			0
$851$	2.01767e11	−	1.16490e11i	0.384709	−	0.222112i
$852$		0			0
$853$	−2.08935e11	+	3.61887e11i	−0.394653	+	0.683560i	−0.993057	−	0.117635i	$-0.962469\pi$
$853$	−2.08935e11	+	3.61887e11i	−0.394653	+	0.683560i	0.598404	+	0.801195i	$0.295802\pi$
$854$		0			0
$855$	−1.81546e8	−	1.30945e9i	−0.000339720	−	0.00245033i
$856$		0			0
$857$	−7.79919e11	−	4.50287e11i	−1.44586	−	0.834768i	−0.447629	−	0.894219i	$-0.647732\pi$
$857$	−7.79919e11	−	4.50287e11i	−1.44586	−	0.834768i	−0.998231	+	0.0594511i	$0.981065\pi$
$858$		0			0
$859$	5.18005e11	+	8.97210e11i	0.951395	+	1.64786i	0.742410	+	0.669946i	$0.233683\pi$
$859$	5.18005e11	+	8.97210e11i	0.951395	+	1.64786i	0.208985	+	0.977919i	$0.432984\pi$
$860$		0			0
$861$	−1.64493e11	+	1.10729e11i	−0.299319	+	0.201488i
$862$		0			0
$863$		−	5.32215e11i		−	0.959497i	−0.877406	−	0.479749i	$-0.840728\pi$
$863$		−	5.32215e11i		−	0.959497i	0.877406	−	0.479749i	$-0.159272\pi$
$864$		0			0
$865$	−9.99891e8			−0.00178603
$866$		0			0
$867$	−2.64498e11	−	3.92923e11i	−0.468108	−	0.695394i
$868$		0			0
$869$	−4.43936e11	+	2.56306e11i	−0.778468	+	0.449449i
$870$		0			0
$871$	2.59576e10	−	4.49599e10i	0.0451016	−	0.0781183i
$872$		0			0
$873$	4.57747e11	+	5.88143e11i	0.788076	+	1.01257i
$874$		0			0
$875$	3.16015e9	+	1.82452e9i	0.00539108	+	0.00311254i
$876$		0			0
$877$	4.02785e11	+	6.97643e11i	0.680886	+	1.17933i	0.974711	+	0.223470i	$0.0717385\pi$
$877$	4.02785e11	+	6.97643e11i	0.680886	+	1.17933i	−0.293825	+	0.955859i	$0.594928\pi$
$878$		0			0
$879$	3.48481e11	+	1.70367e11i	0.583746	+	0.285385i
$880$		0			0
$881$			3.97305e11i			0.659509i	0.944067	+	0.329754i	$0.106966\pi$
$881$			3.97305e11i			0.659509i	−0.944067	+	0.329754i	$0.893034\pi$
$882$		0			0
$883$	3.00206e11			0.493829			0.246914	−	0.969037i	$-0.420583\pi$
$883$	3.00206e11			0.493829			0.246914	+	0.969037i	$0.420583\pi$
$884$		0			0
$885$	6.95480e9	−	4.79821e8i	0.0113374	−	0.000782179i
$886$		0			0
$887$	6.71810e11	−	3.87869e11i	1.08530	−	0.626601i	0.152982	−	0.988229i	$-0.451112\pi$
$887$	6.71810e11	−	3.87869e11i	1.08530	−	0.626601i	0.932323	+	0.361628i	$0.117779\pi$
$888$		0			0
$889$	2.09636e11	−	3.63101e11i	0.335629	−	0.581326i
$890$		0			0
$891$	−3.18132e11	−	1.12526e12i	−0.504773	−	1.78542i
$892$		0			0
$893$	−2.13936e11	−	1.23516e11i	−0.336417	−	0.194230i
$894$		0			0
$895$	−5.88099e8	−	1.01862e9i	−0.000916554	−	0.00158752i
$896$		0			0
$897$	−8.64731e9	−	1.25339e11i	−0.0133571	−	0.193605i
$898$		0			0
$899$			2.34401e11i			0.358856i
$900$		0			0
$901$	2.62341e10			0.0398077
$902$		0			0
$903$	−1.31772e11	+	2.69535e11i	−0.198185	+	0.405382i
$904$		0			0
$905$	−2.12060e9	+	1.22433e9i	−0.00316129	+	0.00182517i
$906$		0			0
$907$	−4.44026e11	+	7.69075e11i	−0.656113	+	1.13642i	0.325500	+	0.945542i	$0.394467\pi$
$907$	−4.44026e11	+	7.69075e11i	−0.656113	+	1.13642i	−0.981613	+	0.190880i	$0.938866\pi$
$908$		0			0
$909$	−8.34871e11	+	6.49773e11i	−1.22282	+	0.951714i
$910$		0			0
$911$	1.65853e11	+	9.57554e10i	0.240797	+	0.139024i	0.615543	−	0.788103i	$-0.288937\pi$
$911$	1.65853e11	+	9.57554e10i	0.240797	+	0.139024i	−0.374746	+	0.927127i	$0.622270\pi$
$912$		0			0
$913$	8.75996e10	+	1.51727e11i	0.126072	+	0.218363i
$914$		0			0
$915$	−1.69925e9	+	1.14386e9i	−0.00242423	+	0.00163188i
$916$		0			0
$917$		−	3.70506e11i		−	0.523984i
$918$		0			0
$919$	5.22923e11			0.733121			0.366560	−	0.930394i	$-0.380535\pi$
$919$	5.22923e11			0.733121			0.366560	+	0.930394i	$0.380535\pi$
$920$		0			0
$921$	−3.58143e11	−	5.32037e11i	−0.497758	−	0.739441i
$922$		0			0
$923$	9.99307e10	−	5.76950e10i	0.137687	−	0.0794935i
$924$		0			0
$925$	−1.80006e11	+	3.11779e11i	−0.245878	+	0.425873i
$926$		0			0
$927$	−3.19904e11	+	4.43523e10i	−0.433213	+	0.0600617i
$928$		0			0
$929$	1.26571e12	+	7.30757e11i	1.69930	+	0.981093i	0.946421	+	0.322937i	$0.104670\pi$
$929$	1.26571e12	+	7.30757e11i	1.69930	+	0.981093i	0.752882	+	0.658156i	$0.228663\pi$
$930$		0			0
$931$	−1.61195e10	−	2.79198e10i	−0.0214562	−	0.0371632i
$932$		0			0
$933$	−6.73429e11	−	3.29230e11i	−0.888720	−	0.434483i
$934$		0			0
$935$		−	4.69632e9i		−	0.00614485i
$936$		0			0
$937$	9.95202e11			1.29108			0.645539	−	0.763727i	$-0.276633\pi$
$937$	9.95202e11			1.29108			0.645539	+	0.763727i	$0.276633\pi$
$938$		0			0
$939$	−7.53509e11	+	5.19856e10i	−0.969228	+	0.0668683i
$940$		0			0
$941$	1.23211e11	−	7.11358e10i	0.157141	−	0.0907256i	−0.419367	−	0.907817i	$-0.637748\pi$
$941$	1.23211e11	−	7.11358e10i	0.157141	−	0.0907256i	0.576509	+	0.817091i	$0.304415\pi$
$942$		0			0
$943$	−3.40940e11	+	5.90525e11i	−0.431152	+	0.746778i
$944$		0			0
$945$	−5.09319e8	−	2.42950e9i	−0.000638650	−	0.00304642i
$946$		0			0
$947$	−5.83528e11	−	3.36900e11i	−0.725541	−	0.418891i	0.0912479	−	0.995828i	$-0.470914\pi$
$947$	−5.83528e11	−	3.36900e11i	−0.725541	−	0.418891i	−0.816789	+	0.576937i	$0.804248\pi$
$948$		0			0
$949$	−2.05431e10	−	3.55817e10i	−0.0253280	−	0.0438694i
$950$		0			0
$951$	4.06088e10	+	5.88607e11i	0.0496475	+	0.719620i
$952$		0			0
$953$			3.41253e11i			0.413719i	0.978371	+	0.206859i	$0.0663243\pi$
$953$			3.41253e11i			0.413719i	−0.978371	+	0.206859i	$0.933676\pi$
$954$		0			0
$955$	5.65164e9			0.00679455
$956$		0			0
$957$	−1.33543e12	+	2.73158e12i	−1.59211	+	3.25662i
$958$		0			0
$959$	3.30850e11	−	1.91016e11i	0.391162	−	0.225837i
$960$		0			0
$961$	4.12058e11	−	7.13706e11i	0.483131	−	0.836808i
$962$		0			0
$963$	−4.53549e11	−	1.84229e11i	−0.527374	−	0.214216i
$964$		0			0
$965$	2.50744e9	+	1.44767e9i	0.00289148	+	0.00166940i
$966$		0			0
$967$	−2.91086e11	−	5.04176e11i	−0.332902	−	0.576602i	0.650178	−	0.759782i	$-0.274694\pi$
$967$	−2.91086e11	−	5.04176e11i	−0.332902	−	0.576602i	−0.983079	+	0.183180i	$0.941361\pi$
$968$		0			0
$969$	8.83520e10	−	5.94746e10i	0.100212	−	0.0674584i
$970$		0			0
$971$		−	3.11410e11i		−	0.350313i	−0.984541	−	0.175156i	$-0.943957\pi$
$971$		−	3.11410e11i		−	0.350313i	0.984541	−	0.175156i	$-0.0560431\pi$
$972$		0			0
$973$	−3.39018e11			−0.378243
$974$		0			0
$975$	1.08411e11	+	1.61050e11i	0.119966	+	0.178214i
$976$		0			0
$977$	4.02728e11	−	2.32515e11i	0.442012	−	0.255196i	−0.262439	−	0.964949i	$-0.584527\pi$
$977$	4.02728e11	−	2.32515e11i	0.442012	−	0.255196i	0.704451	+	0.709753i	$0.251193\pi$
$978$		0			0
$979$	−1.27229e12	+	2.20368e12i	−1.38502	+	2.39893i
$980$		0			0
$981$	3.10885e11	−	7.65362e11i	0.335679	−	0.826401i
$982$		0			0
$983$	1.48449e12	+	8.57074e11i	1.58988	+	0.917918i	0.993325	+	0.115347i	$0.0367979\pi$
$983$	1.48449e12	+	8.57074e11i	1.58988	+	0.917918i	0.596556	+	0.802572i	$0.296535\pi$
$984$		0			0
$985$	−5.30947e9	−	9.19628e9i	−0.00564036	−	0.00976938i
$986$		0			0
$987$	−4.16724e11	−	2.03730e11i	−0.439117	−	0.214678i
$988$		0			0
$989$			1.03171e12i			1.07839i
$990$		0			0
$991$	−7.50469e9			−0.00778105			−0.00389053	−	0.999992i	$-0.501238\pi$
$991$	−7.50469e9			−0.00778105			−0.00389053	+	0.999992i	$0.501238\pi$
$992$		0			0
$993$	−1.52577e12	+	1.05265e11i	−1.56925	+	0.108264i
$994$		0			0
$995$	1.27191e10	−	7.34339e9i	0.0129767	−	0.00749211i
$996$		0			0
$997$	−2.61604e11	+	4.53112e11i	−0.264767	+	0.458590i	−0.967503	−	0.252861i	$-0.918628\pi$
$997$	−2.61604e11	+	4.53112e11i	−0.264767	+	0.458590i	0.702735	+	0.711451i	$0.251962\pi$
$998$		0			0
$999$	4.79403e11	−	1.00502e11i	0.481326	−	0.100905i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	252.9.bg.a.29.15	✓	96
9.5	odd	6	inner	252.9.bg.a.113.15	yes	96

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
252.9.bg.a.29.15	✓	96	1.1	even	1	trivial
252.9.bg.a.113.15	yes	96	9.5	odd	6	inner

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:	\( N \)	\(=\)	\( 252 = 2^{2} \cdot 3^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 9 \)
Character orbit:	\([\chi]\)	\(=\)	252.bg (of order \(6\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(-45.2321 - 67.1942i) q^{3} +(-4.45748 + 2.57353i) q^{5} +(-453.746 + 785.912i) q^{7} +(-2469.12 + 6078.67i) q^{9} +O(q^{10})\)	\(q+(-45.2321 - 67.1942i) q^{3} +(-4.45748 + 2.57353i) q^{5} +(-453.746 + 785.912i) q^{7} +(-2469.12 + 6078.67i) q^{9} +(23525.6 + 13582.5i) q^{11} +(3068.09 + 5314.08i) q^{13} +(374.547 + 183.111i) q^{15} +33588.4i q^{17} +39146.7 q^{19} +(73332.6 - 5059.31i) q^{21} +(218910. - 126388. i) q^{23} +(-195299. + 338268. i) q^{25} +(520134. - 109040. i) q^{27} +(-1.19671e6 - 690919. i) q^{29} +(-84814.9 - 146904. i) q^{31} +(-151446. - 2.19514e6i) q^{33} -4670.92i q^{35} +921691. q^{37} +(218299. - 446525. i) q^{39} +(-2.33616e6 + 1.34878e6i) q^{41} +(-2.04077e6 + 3.53472e6i) q^{43} +(-4637.58 - 33449.9i) q^{45} +(-5.46498e6 - 3.15521e6i) q^{47} +(-411772. - 713209. i) q^{49} +(2.25695e6 - 1.51928e6i) q^{51} -781046. i q^{53} -139820. q^{55} +(-1.77069e6 - 2.63043e6i) q^{57} +(1.44811e7 - 8.36069e6i) q^{59} +(-2.45661e6 + 4.25498e6i) q^{61} +(-3.65694e6 - 4.69868e6i) q^{63} +(-27351.9 - 15791.6i) q^{65} +(-4.23026e6 - 7.32703e6i) q^{67} +(-1.83943e7 - 8.99270e6i) q^{69} -1.88049e7i q^{71} -6.69574e6 q^{73} +(3.15635e7 - 2.17760e6i) q^{75} +(-2.13493e7 + 1.23260e7i) q^{77} +(-9.43518e6 + 1.63422e7i) q^{79} +(-3.08536e7 - 3.00179e7i) q^{81} +(5.58539e6 + 3.22473e6i) q^{83} +(-86440.8 - 149720. i) q^{85} +(7.70381e6 + 1.11663e8i) q^{87} +9.36716e7i q^{89} -5.56853e6 q^{91} +(-6.03472e6 + 1.23438e7i) q^{93} +(-174496. + 100745. i) q^{95} +(5.67963e7 - 9.83741e7i) q^{97} +(-1.40651e8 + 1.09467e8i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 96 q - 42 q^{3} - 882 q^{5} - 14642 q^{9}+O(q^{10}) \) 96 * q - 42 * q^3 - 882 * q^5 - 14642 * q^9	\( 96 q - 42 q^{3} - 882 q^{5} - 14642 q^{9} - 6102 q^{11} - 63218 q^{15} - 354144 q^{19} + 81634 q^{21} - 689760 q^{23} + 4088394 q^{25} - 2939076 q^{27} - 1902474 q^{29} + 613830 q^{31} - 3732526 q^{33} + 4437300 q^{37} - 2690876 q^{39} + 8275176 q^{41} - 2941680 q^{43} + 7299362 q^{45} - 7663950 q^{47} - 39530064 q^{49} - 23625052 q^{51} + 8608908 q^{55} + 28697652 q^{57} + 38291778 q^{59} + 7577556 q^{63} + 42391494 q^{65} + 47903562 q^{67} - 52586968 q^{69} - 32396448 q^{73} + 245976220 q^{75} + 11461314 q^{79} - 16224230 q^{81} - 104964174 q^{83} + 108387294 q^{85} - 213493700 q^{87} - 12590844 q^{91} - 88124258 q^{93} + 293841792 q^{95} + 9277590 q^{97} - 77959808 q^{99}+O(q^{100}) \) 96 * q - 42 * q^3 - 882 * q^5 - 14642 * q^9 - 6102 * q^11 - 63218 * q^15 - 354144 * q^19 + 81634 * q^21 - 689760 * q^23 + 4088394 * q^25 - 2939076 * q^27 - 1902474 * q^29 + 613830 * q^31 - 3732526 * q^33 + 4437300 * q^37 - 2690876 * q^39 + 8275176 * q^41 - 2941680 * q^43 + 7299362 * q^45 - 7663950 * q^47 - 39530064 * q^49 - 23625052 * q^51 + 8608908 * q^55 + 28697652 * q^57 + 38291778 * q^59 + 7577556 * q^63 + 42391494 * q^65 + 47903562 * q^67 - 52586968 * q^69 - 32396448 * q^73 + 245976220 * q^75 + 11461314 * q^79 - 16224230 * q^81 - 104964174 * q^83 + 108387294 * q^85 - 213493700 * q^87 - 12590844 * q^91 - 88124258 * q^93 + 293841792 * q^95 + 9277590 * q^97 - 77959808 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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