LMFDB - Embedded newform 3234.2.a.z.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3234,2,Mod(1,3234)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3234, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("3234.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3234 = 2 \cdot 3 \cdot 7^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3234.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:	$25.8236200137$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{8})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 2 $ x^2 - 2
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-1.41421$ of defining polynomial
Character	$\chi$	$=$	3234.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{6} -1.00000 q^{8} +1.00000 q^{9} +1.00000 q^{11} +1.00000 q^{12} +1.41421 q^{13} +1.00000 q^{16} -4.00000 q^{17} -1.00000 q^{18} -2.58579 q^{19} -1.00000 q^{22} -3.65685 q^{23} -1.00000 q^{24} -5.00000 q^{25} -1.41421 q^{26} +1.00000 q^{27} +5.65685 q^{29} -11.0711 q^{31} -1.00000 q^{32} +1.00000 q^{33} +4.00000 q^{34} +1.00000 q^{36} -7.65685 q^{37} +2.58579 q^{38} +1.41421 q^{39} +1.65685 q^{41} -10.0000 q^{43} +1.00000 q^{44} +3.65685 q^{46} -1.41421 q^{47} +1.00000 q^{48} +5.00000 q^{50} -4.00000 q^{51} +1.41421 q^{52} +7.65685 q^{53} -1.00000 q^{54} -2.58579 q^{57} -5.65685 q^{58} -1.17157 q^{59} -1.41421 q^{61} +11.0711 q^{62} +1.00000 q^{64} -1.00000 q^{66} +11.3137 q^{67} -4.00000 q^{68} -3.65685 q^{69} +2.34315 q^{71} -1.00000 q^{72} -4.00000 q^{73} +7.65685 q^{74} -5.00000 q^{75} -2.58579 q^{76} -1.41421 q^{78} -9.65685 q^{79} +1.00000 q^{81} -1.65685 q^{82} +0.928932 q^{83} +10.0000 q^{86} +5.65685 q^{87} -1.00000 q^{88} -5.41421 q^{89} -3.65685 q^{92} -11.0711 q^{93} +1.41421 q^{94} -1.00000 q^{96} +12.7279 q^{97} +1.00000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{2} + 2 q^{3} + 2 q^{4} - 2 q^{6} - 2 q^{8} + 2 q^{9} + 2 q^{11} + 2 q^{12} + 2 q^{16} - 8 q^{17} - 2 q^{18} - 8 q^{19} - 2 q^{22} + 4 q^{23} - 2 q^{24} - 10 q^{25} + 2 q^{27} - 8 q^{31} - 2 q^{32}+ \cdots + 2 q^{99}+O(q^{100}) $ 2 * q - 2 * q^2 + 2 * q^3 + 2 * q^4 - 2 * q^6 - 2 * q^8 + 2 * q^9 + 2 * q^11 + 2 * q^12 + 2 * q^16 - 8 * q^17 - 2 * q^18 - 8 * q^19 - 2 * q^22 + 4 * q^23 - 2 * q^24 - 10 * q^25 + 2 * q^27 - 8 * q^31 - 2 * q^32 + 2 * q^33 + 8 * q^34 + 2 * q^36 - 4 * q^37 + 8 * q^38 - 8 * q^41 - 20 * q^43 + 2 * q^44 - 4 * q^46 + 2 * q^48 + 10 * q^50 - 8 * q^51 + 4 * q^53 - 2 * q^54 - 8 * q^57 - 8 * q^59 + 8 * q^62 + 2 * q^64 - 2 * q^66 - 8 * q^68 + 4 * q^69 + 16 * q^71 - 2 * q^72 - 8 * q^73 + 4 * q^74 - 10 * q^75 - 8 * q^76 - 8 * q^79 + 2 * q^81 + 8 * q^82 + 16 * q^83 + 20 * q^86 - 2 * q^88 - 8 * q^89 + 4 * q^92 - 8 * q^93 - 2 * q^96 + 2 * q^99

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 $

$2$	−1.00000	−0.707107
$2$	−1.00000	−0.707107
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	1.00000	0.500000
$5$	0	0		−	1.00000i	$-0.5\pi$
$5$	0	0			1.00000i	$0.5\pi$
$6$	−1.00000	−0.408248
$7$	0	0
$7$	0	0
$8$	−1.00000	−0.353553
$9$	1.00000	0.333333
$10$	0	0
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	1.00000	0.288675
$13$	1.41421	0.392232	0.196116	−	0.980581i	$-0.437167\pi$
$13$	1.41421	0.392232	0.196116	+	0.980581i	$0.437167\pi$
$14$	0	0
$15$	0	0
$16$	1.00000	0.250000
$17$	−4.00000	−0.970143	−0.485071	−	0.874475i	$-0.661206\pi$
$17$	−4.00000	−0.970143	−0.485071	+	0.874475i	$0.661206\pi$
$18$	−1.00000	−0.235702
$19$	−2.58579	−0.593220	−0.296610	−	0.954999i	$-0.595856\pi$
$19$	−2.58579	−0.593220	−0.296610	+	0.954999i	$0.595856\pi$
$20$	0	0
$21$	0	0
$22$	−1.00000	−0.213201
$23$	−3.65685	−0.762507	−0.381253	−	0.924471i	$-0.624507\pi$
$23$	−3.65685	−0.762507	−0.381253	+	0.924471i	$0.624507\pi$
$24$	−1.00000	−0.204124
$25$	−5.00000	−1.00000
$26$	−1.41421	−0.277350
$27$	1.00000	0.192450
$28$	0	0
$29$	5.65685	1.05045	0.525226	−	0.850963i	$-0.323981\pi$
$29$	5.65685	1.05045	0.525226	+	0.850963i	$0.323981\pi$
$30$	0	0
$31$	−11.0711	−1.98842	−0.994211	−	0.107443i	$-0.965734\pi$
$31$	−11.0711	−1.98842	−0.994211	+	0.107443i	$0.965734\pi$
$32$	−1.00000	−0.176777
$33$	1.00000	0.174078
$34$	4.00000	0.685994
$35$	0	0
$36$	1.00000	0.166667
$37$	−7.65685	−1.25878	−0.629390	−	0.777090i	$-0.716695\pi$
$37$	−7.65685	−1.25878	−0.629390	+	0.777090i	$0.716695\pi$
$38$	2.58579	0.419470
$39$	1.41421	0.226455
$40$	0	0
$41$	1.65685	0.258757	0.129379	−	0.991595i	$-0.458702\pi$
$41$	1.65685	0.258757	0.129379	+	0.991595i	$0.458702\pi$
$42$	0	0
$43$	−10.0000	−1.52499	−0.762493	−	0.646997i	$-0.776025\pi$
$43$	−10.0000	−1.52499	−0.762493	+	0.646997i	$0.776025\pi$
$44$	1.00000	0.150756
$45$	0	0
$46$	3.65685	0.539174
$47$	−1.41421	−0.206284	−0.103142	−	0.994667i	$-0.532890\pi$
$47$	−1.41421	−0.206284	−0.103142	+	0.994667i	$0.532890\pi$
$48$	1.00000	0.144338
$49$	0	0
$50$	5.00000	0.707107
$51$	−4.00000	−0.560112
$52$	1.41421	0.196116
$53$	7.65685	1.05175	0.525875	−	0.850562i	$-0.323738\pi$
$53$	7.65685	1.05175	0.525875	+	0.850562i	$0.323738\pi$
$54$	−1.00000	−0.136083
$55$	0	0
$56$	0	0
$57$	−2.58579	−0.342496
$58$	−5.65685	−0.742781
$59$	−1.17157	−0.152526	−0.0762629	−	0.997088i	$-0.524299\pi$
$59$	−1.17157	−0.152526	−0.0762629	+	0.997088i	$0.524299\pi$
$60$	0	0
$61$	−1.41421	−0.181071	−0.0905357	−	0.995893i	$-0.528858\pi$
$61$	−1.41421	−0.181071	−0.0905357	+	0.995893i	$0.528858\pi$
$62$	11.0711	1.40603
$63$	0	0
$64$	1.00000	0.125000
$65$	0	0
$66$	−1.00000	−0.123091
$67$	11.3137	1.38219	0.691095	−	0.722764i	$-0.257129\pi$
$67$	11.3137	1.38219	0.691095	+	0.722764i	$0.257129\pi$
$68$	−4.00000	−0.485071
$69$	−3.65685	−0.440234
$70$	0	0
$71$	2.34315	0.278080	0.139040	−	0.990287i	$-0.455598\pi$
$71$	2.34315	0.278080	0.139040	+	0.990287i	$0.455598\pi$
$72$	−1.00000	−0.117851
$73$	−4.00000	−0.468165	−0.234082	−	0.972217i	$-0.575209\pi$
$73$	−4.00000	−0.468165	−0.234082	+	0.972217i	$0.575209\pi$
$74$	7.65685	0.890091
$75$	−5.00000	−0.577350
$76$	−2.58579	−0.296610
$77$	0	0
$78$	−1.41421	−0.160128
$79$	−9.65685	−1.08648	−0.543240	−	0.839577i	$-0.682803\pi$
$79$	−9.65685	−1.08648	−0.543240	+	0.839577i	$0.682803\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	−1.65685	−0.182969
$83$	0.928932	0.101964	0.0509818	−	0.998700i	$-0.483765\pi$
$83$	0.928932	0.101964	0.0509818	+	0.998700i	$0.483765\pi$
$84$	0	0
$85$	0	0
$86$	10.0000	1.07833
$87$	5.65685	0.606478
$88$	−1.00000	−0.106600
$89$	−5.41421	−0.573905	−0.286953	−	0.957945i	$-0.592642\pi$
$89$	−5.41421	−0.573905	−0.286953	+	0.957945i	$0.592642\pi$
$90$	0	0
$91$	0	0
$92$	−3.65685	−0.381253
$93$	−11.0711	−1.14802
$94$	1.41421	0.145865
$95$	0	0
$96$	−1.00000	−0.102062
$97$	12.7279	1.29232	0.646162	−	0.763200i	$-0.276373\pi$
$97$	12.7279	1.29232	0.646162	+	0.763200i	$0.276373\pi$
$98$	0	0
$99$	1.00000	0.100504
$100$	−5.00000	−0.500000
$101$	−8.24264	−0.820173	−0.410087	−	0.912047i	$-0.634502\pi$
$101$	−8.24264	−0.820173	−0.410087	+	0.912047i	$0.634502\pi$
$102$	4.00000	0.396059
$103$	5.89949	0.581295	0.290647	−	0.956830i	$-0.406129\pi$
$103$	5.89949	0.581295	0.290647	+	0.956830i	$0.406129\pi$
$104$	−1.41421	−0.138675
$105$	0	0
$106$	−7.65685	−0.743699
$107$	−5.31371	−0.513696	−0.256848	−	0.966452i	$-0.582684\pi$
$107$	−5.31371	−0.513696	−0.256848	+	0.966452i	$0.582684\pi$
$108$	1.00000	0.0962250
$109$	−10.0000	−0.957826	−0.478913	−	0.877862i	$-0.658969\pi$
$109$	−10.0000	−0.957826	−0.478913	+	0.877862i	$0.658969\pi$
$110$	0	0
$111$	−7.65685	−0.726756
$112$	0	0
$113$	5.65685	0.532152	0.266076	−	0.963952i	$-0.414273\pi$
$113$	5.65685	0.532152	0.266076	+	0.963952i	$0.414273\pi$
$114$	2.58579	0.242181
$115$	0	0
$116$	5.65685	0.525226
$117$	1.41421	0.130744
$118$	1.17157	0.107852
$119$	0	0
$120$	0	0
$121$	1.00000	0.0909091
$122$	1.41421	0.128037
$123$	1.65685	0.149394
$124$	−11.0711	−0.994211
$125$	0	0
$126$	0	0
$127$	7.31371	0.648987	0.324493	−	0.945888i	$-0.394806\pi$
$127$	7.31371	0.648987	0.324493	+	0.945888i	$0.394806\pi$
$128$	−1.00000	−0.0883883
$129$	−10.0000	−0.880451
$130$	0	0
$131$	−12.2426	−1.06964	−0.534822	−	0.844965i	$-0.679621\pi$
$131$	−12.2426	−1.06964	−0.534822	+	0.844965i	$0.679621\pi$
$132$	1.00000	0.0870388
$133$	0	0
$134$	−11.3137	−0.977356
$135$	0	0
$136$	4.00000	0.342997
$137$	13.3137	1.13747	0.568733	−	0.822522i	$-0.307434\pi$
$137$	13.3137	1.13747	0.568733	+	0.822522i	$0.307434\pi$
$138$	3.65685	0.311292
$139$	−21.8995	−1.85749	−0.928745	−	0.370718i	$-0.879112\pi$
$139$	−21.8995	−1.85749	−0.928745	+	0.370718i	$0.879112\pi$
$140$	0	0
$141$	−1.41421	−0.119098
$142$	−2.34315	−0.196632
$143$	1.41421	0.118262
$144$	1.00000	0.0833333
$145$	0	0
$146$	4.00000	0.331042
$147$	0	0
$148$	−7.65685	−0.629390
$149$	−20.9706	−1.71798	−0.858988	−	0.511996i	$-0.828906\pi$
$149$	−20.9706	−1.71798	−0.858988	+	0.511996i	$0.828906\pi$
$150$	5.00000	0.408248
$151$	9.65685	0.785864	0.392932	−	0.919568i	$-0.371461\pi$
$151$	9.65685	0.785864	0.392932	+	0.919568i	$0.371461\pi$
$152$	2.58579	0.209735
$153$	−4.00000	−0.323381
$154$	0	0
$155$	0	0
$156$	1.41421	0.113228
$157$	−2.82843	−0.225733	−0.112867	−	0.993610i	$-0.536003\pi$
$157$	−2.82843	−0.225733	−0.112867	+	0.993610i	$0.536003\pi$
$158$	9.65685	0.768258
$159$	7.65685	0.607228
$160$	0	0
$161$	0	0
$162$	−1.00000	−0.0785674
$163$	−5.65685	−0.443079	−0.221540	−	0.975151i	$-0.571108\pi$
$163$	−5.65685	−0.443079	−0.221540	+	0.975151i	$0.571108\pi$
$164$	1.65685	0.129379
$165$	0	0
$166$	−0.928932	−0.0720991
$167$	−5.17157	−0.400188	−0.200094	−	0.979777i	$-0.564125\pi$
$167$	−5.17157	−0.400188	−0.200094	+	0.979777i	$0.564125\pi$
$168$	0	0
$169$	−11.0000	−0.846154
$170$	0	0
$171$	−2.58579	−0.197740
$172$	−10.0000	−0.762493
$173$	−19.0711	−1.44995	−0.724973	−	0.688777i	$-0.758148\pi$
$173$	−19.0711	−1.44995	−0.724973	+	0.688777i	$0.758148\pi$
$174$	−5.65685	−0.428845
$175$	0	0
$176$	1.00000	0.0753778
$177$	−1.17157	−0.0880608
$178$	5.41421	0.405812
$179$	13.6569	1.02076	0.510381	−	0.859949i	$-0.329505\pi$
$179$	13.6569	1.02076	0.510381	+	0.859949i	$0.329505\pi$
$180$	0	0
$181$	10.3431	0.768800	0.384400	−	0.923167i	$-0.374408\pi$
$181$	10.3431	0.768800	0.384400	+	0.923167i	$0.374408\pi$
$182$	0	0
$183$	−1.41421	−0.104542
$184$	3.65685	0.269587
$185$	0	0
$186$	11.0711	0.811770
$187$	−4.00000	−0.292509
$188$	−1.41421	−0.103142
$189$	0	0
$190$	0	0
$191$	22.9706	1.66209	0.831046	−	0.556204i	$-0.187743\pi$
$191$	22.9706	1.66209	0.831046	+	0.556204i	$0.187743\pi$
$192$	1.00000	0.0721688
$193$	−3.65685	−0.263226	−0.131613	−	0.991301i	$-0.542016\pi$
$193$	−3.65685	−0.263226	−0.131613	+	0.991301i	$0.542016\pi$
$194$	−12.7279	−0.913812
$195$	0	0
$196$	0	0
$197$	−1.31371	−0.0935979	−0.0467989	−	0.998904i	$-0.514902\pi$
$197$	−1.31371	−0.0935979	−0.0467989	+	0.998904i	$0.514902\pi$
$198$	−1.00000	−0.0710669
$199$	−13.4142	−0.950908	−0.475454	−	0.879740i	$-0.657716\pi$
$199$	−13.4142	−0.950908	−0.475454	+	0.879740i	$0.657716\pi$
$200$	5.00000	0.353553
$201$	11.3137	0.798007
$202$	8.24264	0.579950
$203$	0	0
$204$	−4.00000	−0.280056
$205$	0	0
$206$	−5.89949	−0.411037
$207$	−3.65685	−0.254169
$208$	1.41421	0.0980581
$209$	−2.58579	−0.178863
$210$	0	0
$211$	−5.31371	−0.365811	−0.182905	−	0.983131i	$-0.558550\pi$
$211$	−5.31371	−0.365811	−0.182905	+	0.983131i	$0.558550\pi$
$212$	7.65685	0.525875
$213$	2.34315	0.160550
$214$	5.31371	0.363238
$215$	0	0
$216$	−1.00000	−0.0680414
$217$	0	0
$218$	10.0000	0.677285
$219$	−4.00000	−0.270295
$220$	0	0
$221$	−5.65685	−0.380521
$222$	7.65685	0.513894
$223$	11.5563	0.773870	0.386935	−	0.922107i	$-0.373534\pi$
$223$	11.5563	0.773870	0.386935	+	0.922107i	$0.373534\pi$
$224$	0	0
$225$	−5.00000	−0.333333
$226$	−5.65685	−0.376288
$227$	7.55635	0.501533	0.250766	−	0.968048i	$-0.419317\pi$
$227$	7.55635	0.501533	0.250766	+	0.968048i	$0.419317\pi$
$228$	−2.58579	−0.171248
$229$	5.65685	0.373815	0.186908	−	0.982377i	$-0.440153\pi$
$229$	5.65685	0.373815	0.186908	+	0.982377i	$0.440153\pi$
$230$	0	0
$231$	0	0
$232$	−5.65685	−0.371391
$233$	23.6569	1.54981	0.774906	−	0.632076i	$-0.217797\pi$
$233$	23.6569	1.54981	0.774906	+	0.632076i	$0.217797\pi$
$234$	−1.41421	−0.0924500
$235$	0	0
$236$	−1.17157	−0.0762629
$237$	−9.65685	−0.627280
$238$	0	0
$239$	25.6569	1.65960	0.829802	−	0.558058i	$-0.188453\pi$
$239$	25.6569	1.65960	0.829802	+	0.558058i	$0.188453\pi$
$240$	0	0
$241$	−15.3137	−0.986443	−0.493221	−	0.869904i	$-0.664181\pi$
$241$	−15.3137	−0.986443	−0.493221	+	0.869904i	$0.664181\pi$
$242$	−1.00000	−0.0642824
$243$	1.00000	0.0641500
$244$	−1.41421	−0.0905357
$245$	0	0
$246$	−1.65685	−0.105637
$247$	−3.65685	−0.232680
$248$	11.0711	0.703014
$249$	0.928932	0.0588687
$250$	0	0
$251$	−9.17157	−0.578905	−0.289452	−	0.957192i	$-0.593473\pi$
$251$	−9.17157	−0.578905	−0.289452	+	0.957192i	$0.593473\pi$
$252$	0	0
$253$	−3.65685	−0.229904
$254$	−7.31371	−0.458903
$255$	0	0
$256$	1.00000	0.0625000
$257$	−8.72792	−0.544433	−0.272216	−	0.962236i	$-0.587757\pi$
$257$	−8.72792	−0.544433	−0.272216	+	0.962236i	$0.587757\pi$
$258$	10.0000	0.622573
$259$	0	0
$260$	0	0
$261$	5.65685	0.350150
$262$	12.2426	0.756353
$263$	2.34315	0.144485	0.0722423	−	0.997387i	$-0.476985\pi$
$263$	2.34315	0.144485	0.0722423	+	0.997387i	$0.476985\pi$
$264$	−1.00000	−0.0615457
$265$	0	0
$266$	0	0
$267$	−5.41421	−0.331344
$268$	11.3137	0.691095
$269$	22.6274	1.37962	0.689809	−	0.723991i	$-0.257694\pi$
$269$	22.6274	1.37962	0.689809	+	0.723991i	$0.257694\pi$
$270$	0	0
$271$	−26.8284	−1.62971	−0.814855	−	0.579664i	$-0.803184\pi$
$271$	−26.8284	−1.62971	−0.814855	+	0.579664i	$0.803184\pi$
$272$	−4.00000	−0.242536
$273$	0	0
$274$	−13.3137	−0.804311
$275$	−5.00000	−0.301511
$276$	−3.65685	−0.220117
$277$	4.00000	0.240337	0.120168	−	0.992754i	$-0.461657\pi$
$277$	4.00000	0.240337	0.120168	+	0.992754i	$0.461657\pi$
$278$	21.8995	1.31344
$279$	−11.0711	−0.662807
$280$	0	0
$281$	14.9706	0.893069	0.446534	−	0.894766i	$-0.352658\pi$
$281$	14.9706	0.893069	0.446534	+	0.894766i	$0.352658\pi$
$282$	1.41421	0.0842152
$283$	−24.2426	−1.44108	−0.720538	−	0.693416i	$-0.756105\pi$
$283$	−24.2426	−1.44108	−0.720538	+	0.693416i	$0.756105\pi$
$284$	2.34315	0.139040
$285$	0	0
$286$	−1.41421	−0.0836242
$287$	0	0
$288$	−1.00000	−0.0589256
$289$	−1.00000	−0.0588235
$290$	0	0
$291$	12.7279	0.746124
$292$	−4.00000	−0.234082
$293$	−11.0711	−0.646779	−0.323389	−	0.946266i	$-0.604822\pi$
$293$	−11.0711	−0.646779	−0.323389	+	0.946266i	$0.604822\pi$
$294$	0	0
$295$	0	0
$296$	7.65685	0.445046
$297$	1.00000	0.0580259
$298$	20.9706	1.21479
$299$	−5.17157	−0.299080
$300$	−5.00000	−0.288675
$301$	0	0
$302$	−9.65685	−0.555690
$303$	−8.24264	−0.473527
$304$	−2.58579	−0.148305
$305$	0	0
$306$	4.00000	0.228665
$307$	−24.7279	−1.41130	−0.705649	−	0.708562i	$-0.749344\pi$
$307$	−24.7279	−1.41130	−0.705649	+	0.708562i	$0.749344\pi$
$308$	0	0
$309$	5.89949	0.335611
$310$	0	0
$311$	−11.7574	−0.666699	−0.333349	−	0.942803i	$-0.608179\pi$
$311$	−11.7574	−0.666699	−0.333349	+	0.942803i	$0.608179\pi$
$312$	−1.41421	−0.0800641
$313$	3.75736	0.212379	0.106189	−	0.994346i	$-0.466135\pi$
$313$	3.75736	0.212379	0.106189	+	0.994346i	$0.466135\pi$
$314$	2.82843	0.159617
$315$	0	0
$316$	−9.65685	−0.543240
$317$	17.3137	0.972435	0.486217	−	0.873838i	$-0.338376\pi$
$317$	17.3137	0.972435	0.486217	+	0.873838i	$0.338376\pi$
$318$	−7.65685	−0.429375
$319$	5.65685	0.316723
$320$	0	0
$321$	−5.31371	−0.296582
$322$	0	0
$323$	10.3431	0.575508
$324$	1.00000	0.0555556
$325$	−7.07107	−0.392232
$326$	5.65685	0.313304
$327$	−10.0000	−0.553001
$328$	−1.65685	−0.0914845
$329$	0	0
$330$	0	0
$331$	−4.97056	−0.273207	−0.136603	−	0.990626i	$-0.543619\pi$
$331$	−4.97056	−0.273207	−0.136603	+	0.990626i	$0.543619\pi$
$332$	0.928932	0.0509818
$333$	−7.65685	−0.419593
$334$	5.17157	0.282976
$335$	0	0
$336$	0	0
$337$	5.31371	0.289456	0.144728	−	0.989471i	$-0.453769\pi$
$337$	5.31371	0.289456	0.144728	+	0.989471i	$0.453769\pi$
$338$	11.0000	0.598321
$339$	5.65685	0.307238
$340$	0	0
$341$	−11.0711	−0.599532
$342$	2.58579	0.139823
$343$	0	0
$344$	10.0000	0.539164
$345$	0	0
$346$	19.0711	1.02527
$347$	−2.00000	−0.107366	−0.0536828	−	0.998558i	$-0.517096\pi$
$347$	−2.00000	−0.107366	−0.0536828	+	0.998558i	$0.517096\pi$
$348$	5.65685	0.303239
$349$	−25.4142	−1.36039	−0.680196	−	0.733030i	$-0.738105\pi$
$349$	−25.4142	−1.36039	−0.680196	+	0.733030i	$0.738105\pi$
$350$	0	0
$351$	1.41421	0.0754851
$352$	−1.00000	−0.0533002
$353$	−5.89949	−0.313998	−0.156999	−	0.987599i	$-0.550182\pi$
$353$	−5.89949	−0.313998	−0.156999	+	0.987599i	$0.550182\pi$
$354$	1.17157	0.0622684
$355$	0	0
$356$	−5.41421	−0.286953
$357$	0	0
$358$	−13.6569	−0.721787
$359$	−4.00000	−0.211112	−0.105556	−	0.994413i	$-0.533662\pi$
$359$	−4.00000	−0.211112	−0.105556	+	0.994413i	$0.533662\pi$
$360$	0	0
$361$	−12.3137	−0.648090
$362$	−10.3431	−0.543624
$363$	1.00000	0.0524864
$364$	0	0
$365$	0	0
$366$	1.41421	0.0739221
$367$	−22.8701	−1.19381	−0.596904	−	0.802313i	$-0.703603\pi$
$367$	−22.8701	−1.19381	−0.596904	+	0.802313i	$0.703603\pi$
$368$	−3.65685	−0.190627
$369$	1.65685	0.0862524
$370$	0	0
$371$	0	0
$372$	−11.0711	−0.574008
$373$	8.62742	0.446711	0.223355	−	0.974737i	$-0.428299\pi$
$373$	8.62742	0.446711	0.223355	+	0.974737i	$0.428299\pi$
$374$	4.00000	0.206835
$375$	0	0
$376$	1.41421	0.0729325
$377$	8.00000	0.412021
$378$	0	0
$379$	−6.34315	−0.325826	−0.162913	−	0.986640i	$-0.552089\pi$
$379$	−6.34315	−0.325826	−0.162913	+	0.986640i	$0.552089\pi$
$380$	0	0
$381$	7.31371	0.374693
$382$	−22.9706	−1.17528
$383$	−23.5563	−1.20367	−0.601837	−	0.798619i	$-0.705564\pi$
$383$	−23.5563	−1.20367	−0.601837	+	0.798619i	$0.705564\pi$
$384$	−1.00000	−0.0510310
$385$	0	0
$386$	3.65685	0.186129
$387$	−10.0000	−0.508329
$388$	12.7279	0.646162
$389$	−32.6274	−1.65428	−0.827138	−	0.561999i	$-0.810032\pi$
$389$	−32.6274	−1.65428	−0.827138	+	0.561999i	$0.810032\pi$
$390$	0	0
$391$	14.6274	0.739740
$392$	0	0
$393$	−12.2426	−0.617560
$394$	1.31371	0.0661837
$395$	0	0
$396$	1.00000	0.0502519
$397$	27.7990	1.39519	0.697596	−	0.716492i	$-0.254253\pi$
$397$	27.7990	1.39519	0.697596	+	0.716492i	$0.254253\pi$
$398$	13.4142	0.672394
$399$	0	0
$400$	−5.00000	−0.250000
$401$	−6.00000	−0.299626	−0.149813	−	0.988714i	$-0.547867\pi$
$401$	−6.00000	−0.299626	−0.149813	+	0.988714i	$0.547867\pi$
$402$	−11.3137	−0.564276
$403$	−15.6569	−0.779923
$404$	−8.24264	−0.410087
$405$	0	0
$406$	0	0
$407$	−7.65685	−0.379536
$408$	4.00000	0.198030
$409$	−4.48528	−0.221783	−0.110891	−	0.993833i	$-0.535371\pi$
$409$	−4.48528	−0.221783	−0.110891	+	0.993833i	$0.535371\pi$
$410$	0	0
$411$	13.3137	0.656717
$412$	5.89949	0.290647
$413$	0	0
$414$	3.65685	0.179725
$415$	0	0
$416$	−1.41421	−0.0693375
$417$	−21.8995	−1.07242
$418$	2.58579	0.126475
$419$	40.2843	1.96802	0.984008	−	0.178126i	$-0.0570034\pi$
$419$	40.2843	1.96802	0.984008	+	0.178126i	$0.0570034\pi$
$420$	0	0
$421$	18.9706	0.924569	0.462284	−	0.886732i	$-0.347030\pi$
$421$	18.9706	0.924569	0.462284	+	0.886732i	$0.347030\pi$
$422$	5.31371	0.258667
$423$	−1.41421	−0.0687614
$424$	−7.65685	−0.371850
$425$	20.0000	0.970143
$426$	−2.34315	−0.113526
$427$	0	0
$428$	−5.31371	−0.256848
$429$	1.41421	0.0682789
$430$	0	0
$431$	16.0000	0.770693	0.385346	−	0.922772i	$-0.374082\pi$
$431$	16.0000	0.770693	0.385346	+	0.922772i	$0.374082\pi$
$432$	1.00000	0.0481125
$433$	5.21320	0.250531	0.125265	−	0.992123i	$-0.460022\pi$
$433$	5.21320	0.250531	0.125265	+	0.992123i	$0.460022\pi$
$434$	0	0
$435$	0	0
$436$	−10.0000	−0.478913
$437$	9.45584	0.452334
$438$	4.00000	0.191127
$439$	3.79899	0.181316	0.0906579	−	0.995882i	$-0.471103\pi$
$439$	3.79899	0.181316	0.0906579	+	0.995882i	$0.471103\pi$
$440$	0	0
$441$	0	0
$442$	5.65685	0.269069
$443$	−28.2843	−1.34383	−0.671913	−	0.740630i	$-0.734527\pi$
$443$	−28.2843	−1.34383	−0.671913	+	0.740630i	$0.734527\pi$
$444$	−7.65685	−0.363378
$445$	0	0
$446$	−11.5563	−0.547209
$447$	−20.9706	−0.991874
$448$	0	0
$449$	21.6569	1.02205	0.511025	−	0.859566i	$-0.329266\pi$
$449$	21.6569	1.02205	0.511025	+	0.859566i	$0.329266\pi$
$450$	5.00000	0.235702
$451$	1.65685	0.0780182
$452$	5.65685	0.266076
$453$	9.65685	0.453719
$454$	−7.55635	−0.354637
$455$	0	0
$456$	2.58579	0.121091
$457$	8.34315	0.390276	0.195138	−	0.980776i	$-0.437485\pi$
$457$	8.34315	0.390276	0.195138	+	0.980776i	$0.437485\pi$
$458$	−5.65685	−0.264327
$459$	−4.00000	−0.186704
$460$	0	0
$461$	−18.1005	−0.843025	−0.421512	−	0.906823i	$-0.638501\pi$
$461$	−18.1005	−0.843025	−0.421512	+	0.906823i	$0.638501\pi$
$462$	0	0
$463$	−4.68629	−0.217790	−0.108895	−	0.994053i	$-0.534731\pi$
$463$	−4.68629	−0.217790	−0.108895	+	0.994053i	$0.534731\pi$
$464$	5.65685	0.262613
$465$	0	0
$466$	−23.6569	−1.09588
$467$	−29.9411	−1.38551	−0.692755	−	0.721173i	$-0.743603\pi$
$467$	−29.9411	−1.38551	−0.692755	+	0.721173i	$0.743603\pi$
$468$	1.41421	0.0653720
$469$	0	0
$470$	0	0
$471$	−2.82843	−0.130327
$472$	1.17157	0.0539260
$473$	−10.0000	−0.459800
$474$	9.65685	0.443554
$475$	12.9289	0.593220
$476$	0	0
$477$	7.65685	0.350583
$478$	−25.6569	−1.17352
$479$	0.970563	0.0443461	0.0221731	−	0.999754i	$-0.492942\pi$
$479$	0.970563	0.0443461	0.0221731	+	0.999754i	$0.492942\pi$
$480$	0	0
$481$	−10.8284	−0.493734
$482$	15.3137	0.697520
$483$	0	0
$484$	1.00000	0.0454545
$485$	0	0
$486$	−1.00000	−0.0453609
$487$	−11.6569	−0.528222	−0.264111	−	0.964492i	$-0.585079\pi$
$487$	−11.6569	−0.528222	−0.264111	+	0.964492i	$0.585079\pi$
$488$	1.41421	0.0640184
$489$	−5.65685	−0.255812
$490$	0	0
$491$	20.0000	0.902587	0.451294	−	0.892375i	$-0.350963\pi$
$491$	20.0000	0.902587	0.451294	+	0.892375i	$0.350963\pi$
$492$	1.65685	0.0746968
$493$	−22.6274	−1.01909
$494$	3.65685	0.164530
$495$	0	0
$496$	−11.0711	−0.497106
$497$	0	0
$498$	−0.928932	−0.0416264
$499$	34.6274	1.55014	0.775068	−	0.631878i	$-0.217716\pi$
$499$	34.6274	1.55014	0.775068	+	0.631878i	$0.217716\pi$
$500$	0	0
$501$	−5.17157	−0.231049
$502$	9.17157	0.409347
$503$	−16.4853	−0.735042	−0.367521	−	0.930015i	$-0.619793\pi$
$503$	−16.4853	−0.735042	−0.367521	+	0.930015i	$0.619793\pi$
$504$	0	0
$505$	0	0
$506$	3.65685	0.162567
$507$	−11.0000	−0.488527
$508$	7.31371	0.324493
$509$	−25.9411	−1.14982	−0.574910	−	0.818217i	$-0.694963\pi$
$509$	−25.9411	−1.14982	−0.574910	+	0.818217i	$0.694963\pi$
$510$	0	0
$511$	0	0
$512$	−1.00000	−0.0441942
$513$	−2.58579	−0.114165
$514$	8.72792	0.384972
$515$	0	0
$516$	−10.0000	−0.440225
$517$	−1.41421	−0.0621970
$518$	0	0
$519$	−19.0711	−0.837127
$520$	0	0
$521$	32.7279	1.43384	0.716918	−	0.697157i	$-0.245552\pi$
$521$	32.7279	1.43384	0.716918	+	0.697157i	$0.245552\pi$
$522$	−5.65685	−0.247594
$523$	40.2426	1.75969	0.879844	−	0.475263i	$-0.157647\pi$
$523$	40.2426	1.75969	0.879844	+	0.475263i	$0.157647\pi$
$524$	−12.2426	−0.534822
$525$	0	0
$526$	−2.34315	−0.102166
$527$	44.2843	1.92905
$528$	1.00000	0.0435194
$529$	−9.62742	−0.418583
$530$	0	0
$531$	−1.17157	−0.0508419
$532$	0	0
$533$	2.34315	0.101493
$534$	5.41421	0.234296
$535$	0	0
$536$	−11.3137	−0.488678
$537$	13.6569	0.589337
$538$	−22.6274	−0.975537
$539$	0	0
$540$	0	0
$541$	36.6274	1.57474	0.787368	−	0.616483i	$-0.211443\pi$
$541$	36.6274	1.57474	0.787368	+	0.616483i	$0.211443\pi$
$542$	26.8284	1.15238
$543$	10.3431	0.443867
$544$	4.00000	0.171499
$545$	0	0
$546$	0	0
$547$	9.65685	0.412897	0.206449	−	0.978457i	$-0.433809\pi$
$547$	9.65685	0.412897	0.206449	+	0.978457i	$0.433809\pi$
$548$	13.3137	0.568733
$549$	−1.41421	−0.0603572
$550$	5.00000	0.213201
$551$	−14.6274	−0.623149
$552$	3.65685	0.155646
$553$	0	0
$554$	−4.00000	−0.169944
$555$	0	0
$556$	−21.8995	−0.928745
$557$	−35.9411	−1.52287	−0.761437	−	0.648239i	$-0.775506\pi$
$557$	−35.9411	−1.52287	−0.761437	+	0.648239i	$0.775506\pi$
$558$	11.0711	0.468676
$559$	−14.1421	−0.598149
$560$	0	0
$561$	−4.00000	−0.168880
$562$	−14.9706	−0.631495
$563$	25.4142	1.07108	0.535541	−	0.844509i	$-0.320108\pi$
$563$	25.4142	1.07108	0.535541	+	0.844509i	$0.320108\pi$
$564$	−1.41421	−0.0595491
$565$	0	0
$566$	24.2426	1.01899
$567$	0	0
$568$	−2.34315	−0.0983162
$569$	−29.3137	−1.22889	−0.614447	−	0.788958i	$-0.710621\pi$
$569$	−29.3137	−1.22889	−0.614447	+	0.788958i	$0.710621\pi$
$570$	0	0
$571$	−16.2843	−0.681476	−0.340738	−	0.940158i	$-0.610677\pi$
$571$	−16.2843	−0.681476	−0.340738	+	0.940158i	$0.610677\pi$
$572$	1.41421	0.0591312
$573$	22.9706	0.959609
$574$	0	0
$575$	18.2843	0.762507
$576$	1.00000	0.0416667
$577$	36.2426	1.50880	0.754400	−	0.656414i	$-0.227928\pi$
$577$	36.2426	1.50880	0.754400	+	0.656414i	$0.227928\pi$
$578$	1.00000	0.0415945
$579$	−3.65685	−0.151974
$580$	0	0
$581$	0	0
$582$	−12.7279	−0.527589
$583$	7.65685	0.317115
$584$	4.00000	0.165521
$585$	0	0
$586$	11.0711	0.457342
$587$	−20.4853	−0.845518	−0.422759	−	0.906242i	$-0.638938\pi$
$587$	−20.4853	−0.845518	−0.422759	+	0.906242i	$0.638938\pi$
$588$	0	0
$589$	28.6274	1.17957
$590$	0	0
$591$	−1.31371	−0.0540387
$592$	−7.65685	−0.314695
$593$	0.686292	0.0281826	0.0140913	−	0.999901i	$-0.495514\pi$
$593$	0.686292	0.0281826	0.0140913	+	0.999901i	$0.495514\pi$
$594$	−1.00000	−0.0410305
$595$	0	0
$596$	−20.9706	−0.858988
$597$	−13.4142	−0.549007
$598$	5.17157	0.211481
$599$	0.970563	0.0396561	0.0198281	−	0.999803i	$-0.493688\pi$
$599$	0.970563	0.0396561	0.0198281	+	0.999803i	$0.493688\pi$
$600$	5.00000	0.204124
$601$	37.4558	1.52786	0.763928	−	0.645302i	$-0.223268\pi$
$601$	37.4558	1.52786	0.763928	+	0.645302i	$0.223268\pi$
$602$	0	0
$603$	11.3137	0.460730
$604$	9.65685	0.392932
$605$	0	0
$606$	8.24264	0.334834
$607$	19.3137	0.783919	0.391960	−	0.919982i	$-0.371797\pi$
$607$	19.3137	0.783919	0.391960	+	0.919982i	$0.371797\pi$
$608$	2.58579	0.104867
$609$	0	0
$610$	0	0
$611$	−2.00000	−0.0809113
$612$	−4.00000	−0.161690
$613$	12.0000	0.484675	0.242338	−	0.970192i	$-0.422086\pi$
$613$	12.0000	0.484675	0.242338	+	0.970192i	$0.422086\pi$
$614$	24.7279	0.997938
$615$	0	0
$616$	0	0
$617$	25.6569	1.03291	0.516453	−	0.856316i	$-0.327252\pi$
$617$	25.6569	1.03291	0.516453	+	0.856316i	$0.327252\pi$
$618$	−5.89949	−0.237312
$619$	−15.3137	−0.615510	−0.307755	−	0.951466i	$-0.599578\pi$
$619$	−15.3137	−0.615510	−0.307755	+	0.951466i	$0.599578\pi$
$620$	0	0
$621$	−3.65685	−0.146745
$622$	11.7574	0.471427
$623$	0	0
$624$	1.41421	0.0566139
$625$	25.0000	1.00000
$626$	−3.75736	−0.150174
$627$	−2.58579	−0.103266
$628$	−2.82843	−0.112867
$629$	30.6274	1.22120
$630$	0	0
$631$	−41.5980	−1.65599	−0.827995	−	0.560736i	$-0.810518\pi$
$631$	−41.5980	−1.65599	−0.827995	+	0.560736i	$0.810518\pi$
$632$	9.65685	0.384129
$633$	−5.31371	−0.211201
$634$	−17.3137	−0.687615
$635$	0	0
$636$	7.65685	0.303614
$637$	0	0
$638$	−5.65685	−0.223957
$639$	2.34315	0.0926934
$640$	0	0
$641$	−48.2843	−1.90711	−0.953557	−	0.301213i	$-0.902609\pi$
$641$	−48.2843	−1.90711	−0.953557	+	0.301213i	$0.902609\pi$
$642$	5.31371	0.209715
$643$	41.1716	1.62365	0.811824	−	0.583902i	$-0.198475\pi$
$643$	41.1716	1.62365	0.811824	+	0.583902i	$0.198475\pi$
$644$	0	0
$645$	0	0
$646$	−10.3431	−0.406946
$647$	1.89949	0.0746769	0.0373384	−	0.999303i	$-0.488112\pi$
$647$	1.89949	0.0746769	0.0373384	+	0.999303i	$0.488112\pi$
$648$	−1.00000	−0.0392837
$649$	−1.17157	−0.0459883
$650$	7.07107	0.277350
$651$	0	0
$652$	−5.65685	−0.221540
$653$	12.6274	0.494149	0.247075	−	0.968996i	$-0.420531\pi$
$653$	12.6274	0.494149	0.247075	+	0.968996i	$0.420531\pi$
$654$	10.0000	0.391031
$655$	0	0
$656$	1.65685	0.0646893
$657$	−4.00000	−0.156055
$658$	0	0
$659$	−2.68629	−0.104643	−0.0523215	−	0.998630i	$-0.516662\pi$
$659$	−2.68629	−0.104643	−0.0523215	+	0.998630i	$0.516662\pi$
$660$	0	0
$661$	−8.97056	−0.348914	−0.174457	−	0.984665i	$-0.555817\pi$
$661$	−8.97056	−0.348914	−0.174457	+	0.984665i	$0.555817\pi$
$662$	4.97056	0.193186
$663$	−5.65685	−0.219694
$664$	−0.928932	−0.0360496
$665$	0	0
$666$	7.65685	0.296697
$667$	−20.6863	−0.800976
$668$	−5.17157	−0.200094
$669$	11.5563	0.446794
$670$	0	0
$671$	−1.41421	−0.0545951
$672$	0	0
$673$	8.34315	0.321605	0.160802	−	0.986987i	$-0.448592\pi$
$673$	8.34315	0.321605	0.160802	+	0.986987i	$0.448592\pi$
$674$	−5.31371	−0.204676
$675$	−5.00000	−0.192450
$676$	−11.0000	−0.423077
$677$	21.4142	0.823015	0.411508	−	0.911406i	$-0.365002\pi$
$677$	21.4142	0.823015	0.411508	+	0.911406i	$0.365002\pi$
$678$	−5.65685	−0.217250
$679$	0	0
$680$	0	0
$681$	7.55635	0.289560
$682$	11.0711	0.423933
$683$	8.28427	0.316989	0.158494	−	0.987360i	$-0.449336\pi$
$683$	8.28427	0.316989	0.158494	+	0.987360i	$0.449336\pi$
$684$	−2.58579	−0.0988700
$685$	0	0
$686$	0	0
$687$	5.65685	0.215822
$688$	−10.0000	−0.381246
$689$	10.8284	0.412530
$690$	0	0
$691$	12.4853	0.474962	0.237481	−	0.971392i	$-0.423678\pi$
$691$	12.4853	0.474962	0.237481	+	0.971392i	$0.423678\pi$
$692$	−19.0711	−0.724973
$693$	0	0
$694$	2.00000	0.0759190
$695$	0	0
$696$	−5.65685	−0.214423
$697$	−6.62742	−0.251031
$698$	25.4142	0.961942
$699$	23.6569	0.894784
$700$	0	0
$701$	−18.6863	−0.705771	−0.352886	−	0.935666i	$-0.614800\pi$
$701$	−18.6863	−0.705771	−0.352886	+	0.935666i	$0.614800\pi$
$702$	−1.41421	−0.0533761
$703$	19.7990	0.746733
$704$	1.00000	0.0376889
$705$	0	0
$706$	5.89949	0.222030
$707$	0	0
$708$	−1.17157	−0.0440304
$709$	−44.6274	−1.67602	−0.838009	−	0.545657i	$-0.816280\pi$
$709$	−44.6274	−1.67602	−0.838009	+	0.545657i	$0.816280\pi$
$710$	0	0
$711$	−9.65685	−0.362160
$712$	5.41421	0.202906
$713$	40.4853	1.51619
$714$	0	0
$715$	0	0
$716$	13.6569	0.510381
$717$	25.6569	0.958173
$718$	4.00000	0.149279
$719$	−17.4142	−0.649441	−0.324720	−	0.945810i	$-0.605270\pi$
$719$	−17.4142	−0.649441	−0.324720	+	0.945810i	$0.605270\pi$
$720$	0	0
$721$	0	0
$722$	12.3137	0.458269
$723$	−15.3137	−0.569523
$724$	10.3431	0.384400
$725$	−28.2843	−1.05045
$726$	−1.00000	−0.0371135
$727$	−31.3553	−1.16291	−0.581453	−	0.813580i	$-0.697515\pi$
$727$	−31.3553	−1.16291	−0.581453	+	0.813580i	$0.697515\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	40.0000	1.47945
$732$	−1.41421	−0.0522708
$733$	44.2426	1.63414	0.817070	−	0.576539i	$-0.195597\pi$
$733$	44.2426	1.63414	0.817070	+	0.576539i	$0.195597\pi$
$734$	22.8701	0.844149
$735$	0	0
$736$	3.65685	0.134793
$737$	11.3137	0.416746
$738$	−1.65685	−0.0609896
$739$	17.6569	0.649518	0.324759	−	0.945797i	$-0.394717\pi$
$739$	17.6569	0.649518	0.324759	+	0.945797i	$0.394717\pi$
$740$	0	0
$741$	−3.65685	−0.134338
$742$	0	0
$743$	28.9706	1.06283	0.531413	−	0.847113i	$-0.321661\pi$
$743$	28.9706	1.06283	0.531413	+	0.847113i	$0.321661\pi$
$744$	11.0711	0.405885
$745$	0	0
$746$	−8.62742	−0.315872
$747$	0.928932	0.0339879
$748$	−4.00000	−0.146254
$749$	0	0
$750$	0	0
$751$	−6.97056	−0.254359	−0.127180	−	0.991880i	$-0.540592\pi$
$751$	−6.97056	−0.254359	−0.127180	+	0.991880i	$0.540592\pi$
$752$	−1.41421	−0.0515711
$753$	−9.17157	−0.334231
$754$	−8.00000	−0.291343
$755$	0	0
$756$	0	0
$757$	−29.3137	−1.06542	−0.532712	−	0.846296i	$-0.678827\pi$
$757$	−29.3137	−1.06542	−0.532712	+	0.846296i	$0.678827\pi$
$758$	6.34315	0.230393
$759$	−3.65685	−0.132735
$760$	0	0
$761$	42.1421	1.52765	0.763826	−	0.645423i	$-0.223319\pi$
$761$	42.1421	1.52765	0.763826	+	0.645423i	$0.223319\pi$
$762$	−7.31371	−0.264948
$763$	0	0
$764$	22.9706	0.831046
$765$	0	0
$766$	23.5563	0.851125
$767$	−1.65685	−0.0598255
$768$	1.00000	0.0360844
$769$	−0.201010	−0.00724861	−0.00362431	−	0.999993i	$-0.501154\pi$
$769$	−0.201010	−0.00724861	−0.00362431	+	0.999993i	$0.501154\pi$
$770$	0	0
$771$	−8.72792	−0.314328
$772$	−3.65685	−0.131613
$773$	−14.1421	−0.508657	−0.254329	−	0.967118i	$-0.581854\pi$
$773$	−14.1421	−0.508657	−0.254329	+	0.967118i	$0.581854\pi$
$774$	10.0000	0.359443
$775$	55.3553	1.98842
$776$	−12.7279	−0.456906
$777$	0	0
$778$	32.6274	1.16975
$779$	−4.28427	−0.153500
$780$	0	0
$781$	2.34315	0.0838443
$782$	−14.6274	−0.523075
$783$	5.65685	0.202159
$784$	0	0
$785$	0	0
$786$	12.2426	0.436681
$787$	1.61522	0.0575765	0.0287883	−	0.999586i	$-0.490835\pi$
$787$	1.61522	0.0575765	0.0287883	+	0.999586i	$0.490835\pi$
$788$	−1.31371	−0.0467989
$789$	2.34315	0.0834182
$790$	0	0
$791$	0	0
$792$	−1.00000	−0.0355335
$793$	−2.00000	−0.0710221
$794$	−27.7990	−0.986549
$795$	0	0
$796$	−13.4142	−0.475454
$797$	45.1716	1.60006	0.800030	−	0.599961i	$-0.204817\pi$
$797$	45.1716	1.60006	0.800030	+	0.599961i	$0.204817\pi$
$798$	0	0
$799$	5.65685	0.200125
$800$	5.00000	0.176777
$801$	−5.41421	−0.191302
$802$	6.00000	0.211867
$803$	−4.00000	−0.141157
$804$	11.3137	0.399004
$805$	0	0
$806$	15.6569	0.551489
$807$	22.6274	0.796523
$808$	8.24264	0.289975
$809$	−14.9706	−0.526337	−0.263168	−	0.964750i	$-0.584768\pi$
$809$	−14.9706	−0.526337	−0.263168	+	0.964750i	$0.584768\pi$
$810$	0	0
$811$	−54.8701	−1.92675	−0.963374	−	0.268161i	$-0.913584\pi$
$811$	−54.8701	−1.92675	−0.963374	+	0.268161i	$0.913584\pi$
$812$	0	0
$813$	−26.8284	−0.940914
$814$	7.65685	0.268373
$815$	0	0
$816$	−4.00000	−0.140028
$817$	25.8579	0.904652
$818$	4.48528	0.156824
$819$	0	0
$820$	0	0
$821$	36.9706	1.29028	0.645141	−	0.764064i	$-0.276799\pi$
$821$	36.9706	1.29028	0.645141	+	0.764064i	$0.276799\pi$
$822$	−13.3137	−0.464369
$823$	25.5980	0.892289	0.446145	−	0.894961i	$-0.352797\pi$
$823$	25.5980	0.892289	0.446145	+	0.894961i	$0.352797\pi$
$824$	−5.89949	−0.205519
$825$	−5.00000	−0.174078
$826$	0	0
$827$	28.0000	0.973655	0.486828	−	0.873498i	$-0.338154\pi$
$827$	28.0000	0.973655	0.486828	+	0.873498i	$0.338154\pi$
$828$	−3.65685	−0.127084
$829$	3.31371	0.115090	0.0575449	−	0.998343i	$-0.481673\pi$
$829$	3.31371	0.115090	0.0575449	+	0.998343i	$0.481673\pi$
$830$	0	0
$831$	4.00000	0.138758
$832$	1.41421	0.0490290
$833$	0	0
$834$	21.8995	0.758317
$835$	0	0
$836$	−2.58579	−0.0894313
$837$	−11.0711	−0.382672
$838$	−40.2843	−1.39160
$839$	12.7279	0.439417	0.219708	−	0.975566i	$-0.429489\pi$
$839$	12.7279	0.439417	0.219708	+	0.975566i	$0.429489\pi$
$840$	0	0
$841$	3.00000	0.103448
$842$	−18.9706	−0.653769
$843$	14.9706	0.515614
$844$	−5.31371	−0.182905
$845$	0	0
$846$	1.41421	0.0486217
$847$	0	0
$848$	7.65685	0.262937
$849$	−24.2426	−0.832005
$850$	−20.0000	−0.685994
$851$	28.0000	0.959828
$852$	2.34315	0.0802749
$853$	27.2721	0.933778	0.466889	−	0.884316i	$-0.345375\pi$
$853$	27.2721	0.933778	0.466889	+	0.884316i	$0.345375\pi$
$854$	0	0
$855$	0	0
$856$	5.31371	0.181619
$857$	9.17157	0.313295	0.156647	−	0.987655i	$-0.449931\pi$
$857$	9.17157	0.313295	0.156647	+	0.987655i	$0.449931\pi$
$858$	−1.41421	−0.0482805
$859$	38.4264	1.31109	0.655546	−	0.755155i	$-0.272439\pi$
$859$	38.4264	1.31109	0.655546	+	0.755155i	$0.272439\pi$
$860$	0	0
$861$	0	0
$862$	−16.0000	−0.544962
$863$	−29.5980	−1.00753	−0.503763	−	0.863842i	$-0.668052\pi$
$863$	−29.5980	−1.00753	−0.503763	+	0.863842i	$0.668052\pi$
$864$	−1.00000	−0.0340207
$865$	0	0
$866$	−5.21320	−0.177152
$867$	−1.00000	−0.0339618
$868$	0	0
$869$	−9.65685	−0.327586
$870$	0	0
$871$	16.0000	0.542139
$872$	10.0000	0.338643
$873$	12.7279	0.430775
$874$	−9.45584	−0.319849
$875$	0	0
$876$	−4.00000	−0.135147
$877$	30.6863	1.03620	0.518101	−	0.855319i	$-0.326639\pi$
$877$	30.6863	1.03620	0.518101	+	0.855319i	$0.326639\pi$
$878$	−3.79899	−0.128210
$879$	−11.0711	−0.373418
$880$	0	0
$881$	2.10051	0.0707678	0.0353839	−	0.999374i	$-0.488735\pi$
$881$	2.10051	0.0707678	0.0353839	+	0.999374i	$0.488735\pi$
$882$	0	0
$883$	18.6274	0.626862	0.313431	−	0.949611i	$-0.398521\pi$
$883$	18.6274	0.626862	0.313431	+	0.949611i	$0.398521\pi$
$884$	−5.65685	−0.190261
$885$	0	0
$886$	28.2843	0.950229
$887$	−52.2843	−1.75553	−0.877767	−	0.479088i	$-0.840968\pi$
$887$	−52.2843	−1.75553	−0.877767	+	0.479088i	$0.840968\pi$
$888$	7.65685	0.256947
$889$	0	0
$890$	0	0
$891$	1.00000	0.0335013
$892$	11.5563	0.386935
$893$	3.65685	0.122372
$894$	20.9706	0.701361
$895$	0	0
$896$	0	0
$897$	−5.17157	−0.172674
$898$	−21.6569	−0.722699
$899$	−62.6274	−2.08874
$900$	−5.00000	−0.166667
$901$	−30.6274	−1.02035
$902$	−1.65685	−0.0551672
$903$	0	0
$904$	−5.65685	−0.188144
$905$	0	0
$906$	−9.65685	−0.320827
$907$	33.9411	1.12700	0.563498	−	0.826117i	$-0.309455\pi$
$907$	33.9411	1.12700	0.563498	+	0.826117i	$0.309455\pi$
$908$	7.55635	0.250766
$909$	−8.24264	−0.273391
$910$	0	0
$911$	−37.5980	−1.24568	−0.622838	−	0.782351i	$-0.714021\pi$
$911$	−37.5980	−1.24568	−0.622838	+	0.782351i	$0.714021\pi$
$912$	−2.58579	−0.0856239
$913$	0.928932	0.0307432
$914$	−8.34315	−0.275967
$915$	0	0
$916$	5.65685	0.186908
$917$	0	0
$918$	4.00000	0.132020
$919$	−30.6274	−1.01031	−0.505153	−	0.863030i	$-0.668564\pi$
$919$	−30.6274	−1.01031	−0.505153	+	0.863030i	$0.668564\pi$
$920$	0	0
$921$	−24.7279	−0.814813
$922$	18.1005	0.596108
$923$	3.31371	0.109072
$924$	0	0
$925$	38.2843	1.25878
$926$	4.68629	0.154001
$927$	5.89949	0.193765
$928$	−5.65685	−0.185695
$929$	20.9289	0.686656	0.343328	−	0.939216i	$-0.388446\pi$
$929$	20.9289	0.686656	0.343328	+	0.939216i	$0.388446\pi$
$930$	0	0
$931$	0	0
$932$	23.6569	0.774906
$933$	−11.7574	−0.384919
$934$	29.9411	0.979704
$935$	0	0
$936$	−1.41421	−0.0462250
$937$	5.37258	0.175515	0.0877573	−	0.996142i	$-0.472030\pi$
$937$	5.37258	0.175515	0.0877573	+	0.996142i	$0.472030\pi$
$938$	0	0
$939$	3.75736	0.122617
$940$	0	0
$941$	57.6985	1.88092	0.940458	−	0.339909i	$-0.110396\pi$
$941$	57.6985	1.88092	0.940458	+	0.339909i	$0.110396\pi$
$942$	2.82843	0.0921551
$943$	−6.05887	−0.197304
$944$	−1.17157	−0.0381314
$945$	0	0
$946$	10.0000	0.325128
$947$	−47.3137	−1.53749	−0.768744	−	0.639556i	$-0.779118\pi$
$947$	−47.3137	−1.53749	−0.768744	+	0.639556i	$0.779118\pi$
$948$	−9.65685	−0.313640
$949$	−5.65685	−0.183629
$950$	−12.9289	−0.419470
$951$	17.3137	0.561435
$952$	0	0
$953$	−26.0000	−0.842223	−0.421111	−	0.907009i	$-0.638360\pi$
$953$	−26.0000	−0.842223	−0.421111	+	0.907009i	$0.638360\pi$
$954$	−7.65685	−0.247900
$955$	0	0
$956$	25.6569	0.829802
$957$	5.65685	0.182860
$958$	−0.970563	−0.0313575
$959$	0	0
$960$	0	0
$961$	91.5685	2.95382
$962$	10.8284	0.349123
$963$	−5.31371	−0.171232
$964$	−15.3137	−0.493221
$965$	0	0
$966$	0	0
$967$	−12.2843	−0.395036	−0.197518	−	0.980299i	$-0.563288\pi$
$967$	−12.2843	−0.395036	−0.197518	+	0.980299i	$0.563288\pi$
$968$	−1.00000	−0.0321412
$969$	10.3431	0.332270
$970$	0	0
$971$	54.9117	1.76220	0.881100	−	0.472930i	$-0.156804\pi$
$971$	54.9117	1.76220	0.881100	+	0.472930i	$0.156804\pi$
$972$	1.00000	0.0320750
$973$	0	0
$974$	11.6569	0.373510
$975$	−7.07107	−0.226455
$976$	−1.41421	−0.0452679
$977$	37.6569	1.20475	0.602375	−	0.798213i	$-0.294221\pi$
$977$	37.6569	1.20475	0.602375	+	0.798213i	$0.294221\pi$
$978$	5.65685	0.180886
$979$	−5.41421	−0.173039
$980$	0	0
$981$	−10.0000	−0.319275
$982$	−20.0000	−0.638226
$983$	41.0122	1.30809	0.654043	−	0.756457i	$-0.273072\pi$
$983$	41.0122	1.30809	0.654043	+	0.756457i	$0.273072\pi$
$984$	−1.65685	−0.0528186
$985$	0	0
$986$	22.6274	0.720604
$987$	0	0
$988$	−3.65685	−0.116340
$989$	36.5685	1.16281
$990$	0	0
$991$	20.6863	0.657122	0.328561	−	0.944483i	$-0.393436\pi$
$991$	20.6863	0.657122	0.328561	+	0.944483i	$0.393436\pi$
$992$	11.0711	0.351507
$993$	−4.97056	−0.157736
$994$	0	0
$995$	0	0
$996$	0.928932	0.0294343
$997$	9.89949	0.313520	0.156760	−	0.987637i	$-0.449895\pi$
$997$	9.89949	0.313520	0.156760	+	0.987637i	$0.449895\pi$
$998$	−34.6274	−1.09611
$999$	−7.65685	−0.242252

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3234.2.a.z.1.2	yes	2
3.2	odd	2		9702.2.a.dl.1.2		2
7.6	odd	2		3234.2.a.y.1.1	✓	2
21.20	even	2		9702.2.a.de.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3234.2.a.y.1.1	✓	2	7.6	odd	2
3234.2.a.z.1.2	yes	2	1.1	even	1	trivial
9702.2.a.de.1.1		2	21.20	even	2
9702.2.a.dl.1.2		2	3.2	odd	2

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:	\( N \)	\(=\)	\( 3234 = 2 \cdot 3 \cdot 7^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3234.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{6} -1.00000 q^{8} +1.00000 q^{9} +1.00000 q^{11} +1.00000 q^{12} +1.41421 q^{13} +1.00000 q^{16} -4.00000 q^{17} -1.00000 q^{18} -2.58579 q^{19} -1.00000 q^{22} -3.65685 q^{23} -1.00000 q^{24} -5.00000 q^{25} -1.41421 q^{26} +1.00000 q^{27} +5.65685 q^{29} -11.0711 q^{31} -1.00000 q^{32} +1.00000 q^{33} +4.00000 q^{34} +1.00000 q^{36} -7.65685 q^{37} +2.58579 q^{38} +1.41421 q^{39} +1.65685 q^{41} -10.0000 q^{43} +1.00000 q^{44} +3.65685 q^{46} -1.41421 q^{47} +1.00000 q^{48} +5.00000 q^{50} -4.00000 q^{51} +1.41421 q^{52} +7.65685 q^{53} -1.00000 q^{54} -2.58579 q^{57} -5.65685 q^{58} -1.17157 q^{59} -1.41421 q^{61} +11.0711 q^{62} +1.00000 q^{64} -1.00000 q^{66} +11.3137 q^{67} -4.00000 q^{68} -3.65685 q^{69} +2.34315 q^{71} -1.00000 q^{72} -4.00000 q^{73} +7.65685 q^{74} -5.00000 q^{75} -2.58579 q^{76} -1.41421 q^{78} -9.65685 q^{79} +1.00000 q^{81} -1.65685 q^{82} +0.928932 q^{83} +10.0000 q^{86} +5.65685 q^{87} -1.00000 q^{88} -5.41421 q^{89} -3.65685 q^{92} -11.0711 q^{93} +1.41421 q^{94} -1.00000 q^{96} +12.7279 q^{97} +1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{2} + 2 q^{3} + 2 q^{4} - 2 q^{6} - 2 q^{8} + 2 q^{9} + 2 q^{11} + 2 q^{12} + 2 q^{16} - 8 q^{17} - 2 q^{18} - 8 q^{19} - 2 q^{22} + 4 q^{23} - 2 q^{24} - 10 q^{25} + 2 q^{27} - 8 q^{31} - 2 q^{32}+ \cdots + 2 q^{99}+O(q^{100}) \) 2 * q - 2 * q^2 + 2 * q^3 + 2 * q^4 - 2 * q^6 - 2 * q^8 + 2 * q^9 + 2 * q^11 + 2 * q^12 + 2 * q^16 - 8 * q^17 - 2 * q^18 - 8 * q^19 - 2 * q^22 + 4 * q^23 - 2 * q^24 - 10 * q^25 + 2 * q^27 - 8 * q^31 - 2 * q^32 + 2 * q^33 + 8 * q^34 + 2 * q^36 - 4 * q^37 + 8 * q^38 - 8 * q^41 - 20 * q^43 + 2 * q^44 - 4 * q^46 + 2 * q^48 + 10 * q^50 - 8 * q^51 + 4 * q^53 - 2 * q^54 - 8 * q^57 - 8 * q^59 + 8 * q^62 + 2 * q^64 - 2 * q^66 - 8 * q^68 + 4 * q^69 + 16 * q^71 - 2 * q^72 - 8 * q^73 + 4 * q^74 - 10 * q^75 - 8 * q^76 - 8 * q^79 + 2 * q^81 + 8 * q^82 + 16 * q^83 + 20 * q^86 - 2 * q^88 - 8 * q^89 + 4 * q^92 - 8 * q^93 - 2 * q^96 + 2 * q^99

Introduction

L-functions

Modular forms

Varieties

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Groups

Database

Properties

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