LMFDB - Embedded newform 3234.2.a.e.1.1

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:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
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)

$f(q)$

$=$

$q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{6} -1.00000 q^{8} +1.00000 q^{9} +1.00000 q^{11} -1.00000 q^{12} -4.00000 q^{13} +1.00000 q^{16} +4.00000 q^{17} -1.00000 q^{18} -1.00000 q^{22} +1.00000 q^{24} -5.00000 q^{25} +4.00000 q^{26} -1.00000 q^{27} +2.00000 q^{29} -4.00000 q^{31} -1.00000 q^{32} -1.00000 q^{33} -4.00000 q^{34} +1.00000 q^{36} +10.0000 q^{37} +4.00000 q^{39} -12.0000 q^{41} +4.00000 q^{43} +1.00000 q^{44} +4.00000 q^{47} -1.00000 q^{48} +5.00000 q^{50} -4.00000 q^{51} -4.00000 q^{52} -10.0000 q^{53} +1.00000 q^{54} -2.00000 q^{58} -4.00000 q^{59} +4.00000 q^{61} +4.00000 q^{62} +1.00000 q^{64} +1.00000 q^{66} +4.00000 q^{67} +4.00000 q^{68} -8.00000 q^{71} -1.00000 q^{72} +4.00000 q^{73} -10.0000 q^{74} +5.00000 q^{75} -4.00000 q^{78} +8.00000 q^{79} +1.00000 q^{81} +12.0000 q^{82} -16.0000 q^{83} -4.00000 q^{86} -2.00000 q^{87} -1.00000 q^{88} +8.00000 q^{89} +4.00000 q^{93} -4.00000 q^{94} +1.00000 q^{96} -8.00000 q^{97} +1.00000 q^{99} +O(q^{100})$

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
$13$	−4.00000	−1.10940	−0.554700	−	0.832050i	$-0.687167\pi$
$13$	−4.00000	−1.10940	−0.554700	+	0.832050i	$0.687167\pi$
$14$	0	0
$15$	0	0
$16$	1.00000	0.250000
$17$	4.00000	0.970143	0.485071	−	0.874475i	$-0.338794\pi$
$17$	4.00000	0.970143	0.485071	+	0.874475i	$0.338794\pi$
$18$	−1.00000	−0.235702
$19$	0	0		−	1.00000i	$-0.5\pi$
$19$	0	0			1.00000i	$0.5\pi$
$20$	0	0
$21$	0	0
$22$	−1.00000	−0.213201
$23$	0	0		−	1.00000i	$-0.5\pi$
$23$	0	0			1.00000i	$0.5\pi$
$24$	1.00000	0.204124
$25$	−5.00000	−1.00000
$26$	4.00000	0.784465
$27$	−1.00000	−0.192450
$28$	0	0
$29$	2.00000	0.371391	0.185695	−	0.982607i	$-0.440546\pi$
$29$	2.00000	0.371391	0.185695	+	0.982607i	$0.440546\pi$
$30$	0	0
$31$	−4.00000	−0.718421	−0.359211	−	0.933257i	$-0.616954\pi$
$31$	−4.00000	−0.718421	−0.359211	+	0.933257i	$0.616954\pi$
$32$	−1.00000	−0.176777
$33$	−1.00000	−0.174078
$34$	−4.00000	−0.685994
$35$	0	0
$36$	1.00000	0.166667
$37$	10.0000	1.64399	0.821995	−	0.569495i	$-0.192861\pi$
$37$	10.0000	1.64399	0.821995	+	0.569495i	$0.192861\pi$
$38$	0	0
$39$	4.00000	0.640513
$40$	0	0
$41$	−12.0000	−1.87409	−0.937043	−	0.349215i	$-0.886448\pi$
$41$	−12.0000	−1.87409	−0.937043	+	0.349215i	$0.886448\pi$
$42$	0	0
$43$	4.00000	0.609994	0.304997	−	0.952353i	$-0.401344\pi$
$43$	4.00000	0.609994	0.304997	+	0.952353i	$0.401344\pi$
$44$	1.00000	0.150756
$45$	0	0
$46$	0	0
$47$	4.00000	0.583460	0.291730	−	0.956501i	$-0.405769\pi$
$47$	4.00000	0.583460	0.291730	+	0.956501i	$0.405769\pi$
$48$	−1.00000	−0.144338
$49$	0	0
$50$	5.00000	0.707107
$51$	−4.00000	−0.560112
$52$	−4.00000	−0.554700
$53$	−10.0000	−1.37361	−0.686803	−	0.726844i	$-0.740986\pi$
$53$	−10.0000	−1.37361	−0.686803	+	0.726844i	$0.740986\pi$
$54$	1.00000	0.136083
$55$	0	0
$56$	0	0
$57$	0	0
$58$	−2.00000	−0.262613
$59$	−4.00000	−0.520756	−0.260378	−	0.965507i	$-0.583847\pi$
$59$	−4.00000	−0.520756	−0.260378	+	0.965507i	$0.583847\pi$
$60$	0	0
$61$	4.00000	0.512148	0.256074	−	0.966657i	$-0.417571\pi$
$61$	4.00000	0.512148	0.256074	+	0.966657i	$0.417571\pi$
$62$	4.00000	0.508001
$63$	0	0
$64$	1.00000	0.125000
$65$	0	0
$66$	1.00000	0.123091
$67$	4.00000	0.488678	0.244339	−	0.969690i	$-0.421429\pi$
$67$	4.00000	0.488678	0.244339	+	0.969690i	$0.421429\pi$
$68$	4.00000	0.485071
$69$	0	0
$70$	0	0
$71$	−8.00000	−0.949425	−0.474713	−	0.880141i	$-0.657448\pi$
$71$	−8.00000	−0.949425	−0.474713	+	0.880141i	$0.657448\pi$
$72$	−1.00000	−0.117851
$73$	4.00000	0.468165	0.234082	−	0.972217i	$-0.424791\pi$
$73$	4.00000	0.468165	0.234082	+	0.972217i	$0.424791\pi$
$74$	−10.0000	−1.16248
$75$	5.00000	0.577350
$76$	0	0
$77$	0	0
$78$	−4.00000	−0.452911
$79$	8.00000	0.900070	0.450035	−	0.893011i	$-0.351411\pi$
$79$	8.00000	0.900070	0.450035	+	0.893011i	$0.351411\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	12.0000	1.32518
$83$	−16.0000	−1.75623	−0.878114	−	0.478451i	$-0.841198\pi$
$83$	−16.0000	−1.75623	−0.878114	+	0.478451i	$0.841198\pi$
$84$	0	0
$85$	0	0
$86$	−4.00000	−0.431331
$87$	−2.00000	−0.214423
$88$	−1.00000	−0.106600
$89$	8.00000	0.847998	0.423999	−	0.905663i	$-0.360626\pi$
$89$	8.00000	0.847998	0.423999	+	0.905663i	$0.360626\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	4.00000	0.414781
$94$	−4.00000	−0.412568
$95$	0	0
$96$	1.00000	0.102062
$97$	−8.00000	−0.812277	−0.406138	−	0.913812i	$-0.633125\pi$
$97$	−8.00000	−0.812277	−0.406138	+	0.913812i	$0.633125\pi$
$98$	0	0
$99$	1.00000	0.100504
$100$	−5.00000	−0.500000
$101$	−12.0000	−1.19404	−0.597022	−	0.802225i	$-0.703650\pi$
$101$	−12.0000	−1.19404	−0.597022	+	0.802225i	$0.703650\pi$
$102$	4.00000	0.396059
$103$	4.00000	0.394132	0.197066	−	0.980390i	$-0.436859\pi$
$103$	4.00000	0.394132	0.197066	+	0.980390i	$0.436859\pi$
$104$	4.00000	0.392232
$105$	0	0
$106$	10.0000	0.971286
$107$	−12.0000	−1.16008	−0.580042	−	0.814587i	$-0.696964\pi$
$107$	−12.0000	−1.16008	−0.580042	+	0.814587i	$0.696964\pi$
$108$	−1.00000	−0.0962250
$109$	18.0000	1.72409	0.862044	−	0.506834i	$-0.169184\pi$
$109$	18.0000	1.72409	0.862044	+	0.506834i	$0.169184\pi$
$110$	0	0
$111$	−10.0000	−0.949158
$112$	0	0
$113$	2.00000	0.188144	0.0940721	−	0.995565i	$-0.470012\pi$
$113$	2.00000	0.188144	0.0940721	+	0.995565i	$0.470012\pi$
$114$	0	0
$115$	0	0
$116$	2.00000	0.185695
$117$	−4.00000	−0.369800
$118$	4.00000	0.368230
$119$	0	0
$120$	0	0
$121$	1.00000	0.0909091
$122$	−4.00000	−0.362143
$123$	12.0000	1.08200
$124$	−4.00000	−0.359211
$125$	0	0
$126$	0	0
$127$	0	0		−	1.00000i	$-0.5\pi$
$127$	0	0			1.00000i	$0.5\pi$
$128$	−1.00000	−0.0883883
$129$	−4.00000	−0.352180
$130$	0	0
$131$	−8.00000	−0.698963	−0.349482	−	0.936943i	$-0.613642\pi$
$131$	−8.00000	−0.698963	−0.349482	+	0.936943i	$0.613642\pi$
$132$	−1.00000	−0.0870388
$133$	0	0
$134$	−4.00000	−0.345547
$135$	0	0
$136$	−4.00000	−0.342997
$137$	6.00000	0.512615	0.256307	−	0.966595i	$-0.417494\pi$
$137$	6.00000	0.512615	0.256307	+	0.966595i	$0.417494\pi$
$138$	0	0
$139$	−16.0000	−1.35710	−0.678551	−	0.734553i	$-0.737392\pi$
$139$	−16.0000	−1.35710	−0.678551	+	0.734553i	$0.737392\pi$
$140$	0	0
$141$	−4.00000	−0.336861
$142$	8.00000	0.671345
$143$	−4.00000	−0.334497
$144$	1.00000	0.0833333
$145$	0	0
$146$	−4.00000	−0.331042
$147$	0	0
$148$	10.0000	0.821995
$149$	−10.0000	−0.819232	−0.409616	−	0.912258i	$-0.634337\pi$
$149$	−10.0000	−0.819232	−0.409616	+	0.912258i	$0.634337\pi$
$150$	−5.00000	−0.408248
$151$	−8.00000	−0.651031	−0.325515	−	0.945537i	$-0.605538\pi$
$151$	−8.00000	−0.651031	−0.325515	+	0.945537i	$0.605538\pi$
$152$	0	0
$153$	4.00000	0.323381
$154$	0	0
$155$	0	0
$156$	4.00000	0.320256
$157$	8.00000	0.638470	0.319235	−	0.947676i	$-0.396574\pi$
$157$	8.00000	0.638470	0.319235	+	0.947676i	$0.396574\pi$
$158$	−8.00000	−0.636446
$159$	10.0000	0.793052
$160$	0	0
$161$	0	0
$162$	−1.00000	−0.0785674
$163$	12.0000	0.939913	0.469956	−	0.882690i	$-0.344270\pi$
$163$	12.0000	0.939913	0.469956	+	0.882690i	$0.344270\pi$
$164$	−12.0000	−0.937043
$165$	0	0
$166$	16.0000	1.24184
$167$	0	0		−	1.00000i	$-0.5\pi$
$167$	0	0			1.00000i	$0.5\pi$
$168$	0	0
$169$	3.00000	0.230769
$170$	0	0
$171$	0	0
$172$	4.00000	0.304997
$173$	4.00000	0.304114	0.152057	−	0.988372i	$-0.451410\pi$
$173$	4.00000	0.304114	0.152057	+	0.988372i	$0.451410\pi$
$174$	2.00000	0.151620
$175$	0	0
$176$	1.00000	0.0753778
$177$	4.00000	0.300658
$178$	−8.00000	−0.599625
$179$	−4.00000	−0.298974	−0.149487	−	0.988764i	$-0.547762\pi$
$179$	−4.00000	−0.298974	−0.149487	+	0.988764i	$0.547762\pi$
$180$	0	0
$181$	0	0		−	1.00000i	$-0.5\pi$
$181$	0	0			1.00000i	$0.5\pi$
$182$	0	0
$183$	−4.00000	−0.295689
$184$	0	0
$185$	0	0
$186$	−4.00000	−0.293294
$187$	4.00000	0.292509
$188$	4.00000	0.291730
$189$	0	0
$190$	0	0
$191$	−16.0000	−1.15772	−0.578860	−	0.815427i	$-0.696502\pi$
$191$	−16.0000	−1.15772	−0.578860	+	0.815427i	$0.696502\pi$
$192$	−1.00000	−0.0721688
$193$	−14.0000	−1.00774	−0.503871	−	0.863779i	$-0.668091\pi$
$193$	−14.0000	−1.00774	−0.503871	+	0.863779i	$0.668091\pi$
$194$	8.00000	0.574367
$195$	0	0
$196$	0	0
$197$	6.00000	0.427482	0.213741	−	0.976890i	$-0.431435\pi$
$197$	6.00000	0.427482	0.213741	+	0.976890i	$0.431435\pi$
$198$	−1.00000	−0.0710669
$199$	−12.0000	−0.850657	−0.425329	−	0.905039i	$-0.639842\pi$
$199$	−12.0000	−0.850657	−0.425329	+	0.905039i	$0.639842\pi$
$200$	5.00000	0.353553
$201$	−4.00000	−0.282138
$202$	12.0000	0.844317
$203$	0	0
$204$	−4.00000	−0.280056
$205$	0	0
$206$	−4.00000	−0.278693
$207$	0	0
$208$	−4.00000	−0.277350
$209$	0	0
$210$	0	0
$211$	−12.0000	−0.826114	−0.413057	−	0.910705i	$-0.635539\pi$
$211$	−12.0000	−0.826114	−0.413057	+	0.910705i	$0.635539\pi$
$212$	−10.0000	−0.686803
$213$	8.00000	0.548151
$214$	12.0000	0.820303
$215$	0	0
$216$	1.00000	0.0680414
$217$	0	0
$218$	−18.0000	−1.21911
$219$	−4.00000	−0.270295
$220$	0	0
$221$	−16.0000	−1.07628
$222$	10.0000	0.671156
$223$	−12.0000	−0.803579	−0.401790	−	0.915732i	$-0.631612\pi$
$223$	−12.0000	−0.803579	−0.401790	+	0.915732i	$0.631612\pi$
$224$	0	0
$225$	−5.00000	−0.333333
$226$	−2.00000	−0.133038
$227$	−8.00000	−0.530979	−0.265489	−	0.964114i	$-0.585534\pi$
$227$	−8.00000	−0.530979	−0.265489	+	0.964114i	$0.585534\pi$
$228$	0	0
$229$	−16.0000	−1.05731	−0.528655	−	0.848837i	$-0.677303\pi$
$229$	−16.0000	−1.05731	−0.528655	+	0.848837i	$0.677303\pi$
$230$	0	0
$231$	0	0
$232$	−2.00000	−0.131306
$233$	6.00000	0.393073	0.196537	−	0.980497i	$-0.437031\pi$
$233$	6.00000	0.393073	0.196537	+	0.980497i	$0.437031\pi$
$234$	4.00000	0.261488
$235$	0	0
$236$	−4.00000	−0.260378
$237$	−8.00000	−0.519656
$238$	0	0
$239$	8.00000	0.517477	0.258738	−	0.965947i	$-0.416693\pi$
$239$	8.00000	0.517477	0.258738	+	0.965947i	$0.416693\pi$
$240$	0	0
$241$	−20.0000	−1.28831	−0.644157	−	0.764894i	$-0.722792\pi$
$241$	−20.0000	−1.28831	−0.644157	+	0.764894i	$0.722792\pi$
$242$	−1.00000	−0.0642824
$243$	−1.00000	−0.0641500
$244$	4.00000	0.256074
$245$	0	0
$246$	−12.0000	−0.765092
$247$	0	0
$248$	4.00000	0.254000
$249$	16.0000	1.01396
$250$	0	0
$251$	4.00000	0.252478	0.126239	−	0.992000i	$-0.459709\pi$
$251$	4.00000	0.252478	0.126239	+	0.992000i	$0.459709\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	0	0
$256$	1.00000	0.0625000
$257$	−24.0000	−1.49708	−0.748539	−	0.663090i	$-0.769245\pi$
$257$	−24.0000	−1.49708	−0.748539	+	0.663090i	$0.769245\pi$
$258$	4.00000	0.249029
$259$	0	0
$260$	0	0
$261$	2.00000	0.123797
$262$	8.00000	0.494242
$263$	−8.00000	−0.493301	−0.246651	−	0.969104i	$-0.579330\pi$
$263$	−8.00000	−0.493301	−0.246651	+	0.969104i	$0.579330\pi$
$264$	1.00000	0.0615457
$265$	0	0
$266$	0	0
$267$	−8.00000	−0.489592
$268$	4.00000	0.244339
$269$	−8.00000	−0.487769	−0.243884	−	0.969804i	$-0.578422\pi$
$269$	−8.00000	−0.487769	−0.243884	+	0.969804i	$0.578422\pi$
$270$	0	0
$271$	−24.0000	−1.45790	−0.728948	−	0.684569i	$-0.759990\pi$
$271$	−24.0000	−1.45790	−0.728948	+	0.684569i	$0.759990\pi$
$272$	4.00000	0.242536
$273$	0	0
$274$	−6.00000	−0.362473
$275$	−5.00000	−0.301511
$276$	0	0
$277$	−10.0000	−0.600842	−0.300421	−	0.953807i	$-0.597127\pi$
$277$	−10.0000	−0.600842	−0.300421	+	0.953807i	$0.597127\pi$
$278$	16.0000	0.959616
$279$	−4.00000	−0.239474
$280$	0	0
$281$	−10.0000	−0.596550	−0.298275	−	0.954480i	$-0.596411\pi$
$281$	−10.0000	−0.596550	−0.298275	+	0.954480i	$0.596411\pi$
$282$	4.00000	0.238197
$283$	−24.0000	−1.42665	−0.713326	−	0.700832i	$-0.752812\pi$
$283$	−24.0000	−1.42665	−0.713326	+	0.700832i	$0.752812\pi$
$284$	−8.00000	−0.474713
$285$	0	0
$286$	4.00000	0.236525
$287$	0	0
$288$	−1.00000	−0.0589256
$289$	−1.00000	−0.0588235
$290$	0	0
$291$	8.00000	0.468968
$292$	4.00000	0.234082
$293$	−4.00000	−0.233682	−0.116841	−	0.993151i	$-0.537277\pi$
$293$	−4.00000	−0.233682	−0.116841	+	0.993151i	$0.537277\pi$
$294$	0	0
$295$	0	0
$296$	−10.0000	−0.581238
$297$	−1.00000	−0.0580259
$298$	10.0000	0.579284
$299$	0	0
$300$	5.00000	0.288675
$301$	0	0
$302$	8.00000	0.460348
$303$	12.0000	0.689382
$304$	0	0
$305$	0	0
$306$	−4.00000	−0.228665
$307$	−8.00000	−0.456584	−0.228292	−	0.973593i	$-0.573314\pi$
$307$	−8.00000	−0.456584	−0.228292	+	0.973593i	$0.573314\pi$
$308$	0	0
$309$	−4.00000	−0.227552
$310$	0	0
$311$	4.00000	0.226819	0.113410	−	0.993548i	$-0.463823\pi$
$311$	4.00000	0.226819	0.113410	+	0.993548i	$0.463823\pi$
$312$	−4.00000	−0.226455
$313$	32.0000	1.80875	0.904373	−	0.426742i	$-0.140339\pi$
$313$	32.0000	1.80875	0.904373	+	0.426742i	$0.140339\pi$
$314$	−8.00000	−0.451466
$315$	0	0
$316$	8.00000	0.450035
$317$	−18.0000	−1.01098	−0.505490	−	0.862832i	$-0.668688\pi$
$317$	−18.0000	−1.01098	−0.505490	+	0.862832i	$0.668688\pi$
$318$	−10.0000	−0.560772
$319$	2.00000	0.111979
$320$	0	0
$321$	12.0000	0.669775
$322$	0	0
$323$	0	0
$324$	1.00000	0.0555556
$325$	20.0000	1.10940
$326$	−12.0000	−0.664619
$327$	−18.0000	−0.995402
$328$	12.0000	0.662589
$329$	0	0
$330$	0	0
$331$	20.0000	1.09930	0.549650	−	0.835395i	$-0.314761\pi$
$331$	20.0000	1.09930	0.549650	+	0.835395i	$0.314761\pi$
$332$	−16.0000	−0.878114
$333$	10.0000	0.547997
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−2.00000	−0.108947	−0.0544735	−	0.998515i	$-0.517348\pi$
$337$	−2.00000	−0.108947	−0.0544735	+	0.998515i	$0.517348\pi$
$338$	−3.00000	−0.163178
$339$	−2.00000	−0.108625
$340$	0	0
$341$	−4.00000	−0.216612
$342$	0	0
$343$	0	0
$344$	−4.00000	−0.215666
$345$	0	0
$346$	−4.00000	−0.215041
$347$	12.0000	0.644194	0.322097	−	0.946707i	$-0.395612\pi$
$347$	12.0000	0.644194	0.322097	+	0.946707i	$0.395612\pi$
$348$	−2.00000	−0.107211
$349$	−28.0000	−1.49881	−0.749403	−	0.662114i	$-0.769659\pi$
$349$	−28.0000	−1.49881	−0.749403	+	0.662114i	$0.769659\pi$
$350$	0	0
$351$	4.00000	0.213504
$352$	−1.00000	−0.0533002
$353$	24.0000	1.27739	0.638696	−	0.769460i	$-0.279474\pi$
$353$	24.0000	1.27739	0.638696	+	0.769460i	$0.279474\pi$
$354$	−4.00000	−0.212598
$355$	0	0
$356$	8.00000	0.423999
$357$	0	0
$358$	4.00000	0.211407
$359$	−32.0000	−1.68890	−0.844448	−	0.535638i	$-0.820071\pi$
$359$	−32.0000	−1.68890	−0.844448	+	0.535638i	$0.820071\pi$
$360$	0	0
$361$	−19.0000	−1.00000
$362$	0	0
$363$	−1.00000	−0.0524864
$364$	0	0
$365$	0	0
$366$	4.00000	0.209083
$367$	−12.0000	−0.626395	−0.313197	−	0.949688i	$-0.601400\pi$
$367$	−12.0000	−0.626395	−0.313197	+	0.949688i	$0.601400\pi$
$368$	0	0
$369$	−12.0000	−0.624695
$370$	0	0
$371$	0	0
$372$	4.00000	0.207390
$373$	22.0000	1.13912	0.569558	−	0.821951i	$-0.307114\pi$
$373$	22.0000	1.13912	0.569558	+	0.821951i	$0.307114\pi$
$374$	−4.00000	−0.206835
$375$	0	0
$376$	−4.00000	−0.206284
$377$	−8.00000	−0.412021
$378$	0	0
$379$	4.00000	0.205466	0.102733	−	0.994709i	$-0.467241\pi$
$379$	4.00000	0.205466	0.102733	+	0.994709i	$0.467241\pi$
$380$	0	0
$381$	0	0
$382$	16.0000	0.818631
$383$	−4.00000	−0.204390	−0.102195	−	0.994764i	$-0.532587\pi$
$383$	−4.00000	−0.204390	−0.102195	+	0.994764i	$0.532587\pi$
$384$	1.00000	0.0510310
$385$	0	0
$386$	14.0000	0.712581
$387$	4.00000	0.203331
$388$	−8.00000	−0.406138
$389$	10.0000	0.507020	0.253510	−	0.967333i	$-0.418415\pi$
$389$	10.0000	0.507020	0.253510	+	0.967333i	$0.418415\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	8.00000	0.403547
$394$	−6.00000	−0.302276
$395$	0	0
$396$	1.00000	0.0502519
$397$	−8.00000	−0.401508	−0.200754	−	0.979642i	$-0.564339\pi$
$397$	−8.00000	−0.401508	−0.200754	+	0.979642i	$0.564339\pi$
$398$	12.0000	0.601506
$399$	0	0
$400$	−5.00000	−0.250000
$401$	−34.0000	−1.69788	−0.848939	−	0.528490i	$-0.822758\pi$
$401$	−34.0000	−1.69788	−0.848939	+	0.528490i	$0.822758\pi$
$402$	4.00000	0.199502
$403$	16.0000	0.797017
$404$	−12.0000	−0.597022
$405$	0	0
$406$	0	0
$407$	10.0000	0.495682
$408$	4.00000	0.198030
$409$	20.0000	0.988936	0.494468	−	0.869196i	$-0.335363\pi$
$409$	20.0000	0.988936	0.494468	+	0.869196i	$0.335363\pi$
$410$	0	0
$411$	−6.00000	−0.295958
$412$	4.00000	0.197066
$413$	0	0
$414$	0	0
$415$	0	0
$416$	4.00000	0.196116
$417$	16.0000	0.783523
$418$	0	0
$419$	−36.0000	−1.75872	−0.879358	−	0.476162i	$-0.842028\pi$
$419$	−36.0000	−1.75872	−0.879358	+	0.476162i	$0.842028\pi$
$420$	0	0
$421$	22.0000	1.07221	0.536107	−	0.844150i	$-0.319894\pi$
$421$	22.0000	1.07221	0.536107	+	0.844150i	$0.319894\pi$
$422$	12.0000	0.584151
$423$	4.00000	0.194487
$424$	10.0000	0.485643
$425$	−20.0000	−0.970143
$426$	−8.00000	−0.387601
$427$	0	0
$428$	−12.0000	−0.580042
$429$	4.00000	0.193122
$430$	0	0
$431$	−40.0000	−1.92673	−0.963366	−	0.268190i	$-0.913575\pi$
$431$	−40.0000	−1.92673	−0.963366	+	0.268190i	$0.913575\pi$
$432$	−1.00000	−0.0481125
$433$	40.0000	1.92228	0.961139	−	0.276066i	$-0.0890309\pi$
$433$	40.0000	1.92228	0.961139	+	0.276066i	$0.0890309\pi$
$434$	0	0
$435$	0	0
$436$	18.0000	0.862044
$437$	0	0
$438$	4.00000	0.191127
$439$	−40.0000	−1.90910	−0.954548	−	0.298057i	$-0.903661\pi$
$439$	−40.0000	−1.90910	−0.954548	+	0.298057i	$0.903661\pi$
$440$	0	0
$441$	0	0
$442$	16.0000	0.761042
$443$	4.00000	0.190046	0.0950229	−	0.995475i	$-0.469708\pi$
$443$	4.00000	0.190046	0.0950229	+	0.995475i	$0.469708\pi$
$444$	−10.0000	−0.474579
$445$	0	0
$446$	12.0000	0.568216
$447$	10.0000	0.472984
$448$	0	0
$449$	18.0000	0.849473	0.424736	−	0.905317i	$-0.360367\pi$
$449$	18.0000	0.849473	0.424736	+	0.905317i	$0.360367\pi$
$450$	5.00000	0.235702
$451$	−12.0000	−0.565058
$452$	2.00000	0.0940721
$453$	8.00000	0.375873
$454$	8.00000	0.375459
$455$	0	0
$456$	0	0
$457$	26.0000	1.21623	0.608114	−	0.793849i	$-0.291926\pi$
$457$	26.0000	1.21623	0.608114	+	0.793849i	$0.291926\pi$
$458$	16.0000	0.747631
$459$	−4.00000	−0.186704
$460$	0	0
$461$	28.0000	1.30409	0.652045	−	0.758180i	$-0.273911\pi$
$461$	28.0000	1.30409	0.652045	+	0.758180i	$0.273911\pi$
$462$	0	0
$463$	−40.0000	−1.85896	−0.929479	−	0.368875i	$-0.879743\pi$
$463$	−40.0000	−1.85896	−0.929479	+	0.368875i	$0.879743\pi$
$464$	2.00000	0.0928477
$465$	0	0
$466$	−6.00000	−0.277945
$467$	36.0000	1.66588	0.832941	−	0.553362i	$-0.186655\pi$
$467$	36.0000	1.66588	0.832941	+	0.553362i	$0.186655\pi$
$468$	−4.00000	−0.184900
$469$	0	0
$470$	0	0
$471$	−8.00000	−0.368621
$472$	4.00000	0.184115
$473$	4.00000	0.183920
$474$	8.00000	0.367452
$475$	0	0
$476$	0	0
$477$	−10.0000	−0.457869
$478$	−8.00000	−0.365911
$479$	24.0000	1.09659	0.548294	−	0.836286i	$-0.315277\pi$
$479$	24.0000	1.09659	0.548294	+	0.836286i	$0.315277\pi$
$480$	0	0
$481$	−40.0000	−1.82384
$482$	20.0000	0.910975
$483$	0	0
$484$	1.00000	0.0454545
$485$	0	0
$486$	1.00000	0.0453609
$487$	−8.00000	−0.362515	−0.181257	−	0.983436i	$-0.558017\pi$
$487$	−8.00000	−0.362515	−0.181257	+	0.983436i	$0.558017\pi$
$488$	−4.00000	−0.181071
$489$	−12.0000	−0.542659
$490$	0	0
$491$	20.0000	0.902587	0.451294	−	0.892375i	$-0.350963\pi$
$491$	20.0000	0.902587	0.451294	+	0.892375i	$0.350963\pi$
$492$	12.0000	0.541002
$493$	8.00000	0.360302
$494$	0	0
$495$	0	0
$496$	−4.00000	−0.179605
$497$	0	0
$498$	−16.0000	−0.716977
$499$	20.0000	0.895323	0.447661	−	0.894203i	$-0.352257\pi$
$499$	20.0000	0.895323	0.447661	+	0.894203i	$0.352257\pi$
$500$	0	0
$501$	0	0
$502$	−4.00000	−0.178529
$503$	−24.0000	−1.07011	−0.535054	−	0.844818i	$-0.679709\pi$
$503$	−24.0000	−1.07011	−0.535054	+	0.844818i	$0.679709\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	−3.00000	−0.133235
$508$	0	0
$509$	−24.0000	−1.06378	−0.531891	−	0.846813i	$-0.678518\pi$
$509$	−24.0000	−1.06378	−0.531891	+	0.846813i	$0.678518\pi$
$510$	0	0
$511$	0	0
$512$	−1.00000	−0.0441942
$513$	0	0
$514$	24.0000	1.05859
$515$	0	0
$516$	−4.00000	−0.176090
$517$	4.00000	0.175920
$518$	0	0
$519$	−4.00000	−0.175581
$520$	0	0
$521$	0	0		−	1.00000i	$-0.5\pi$
$521$	0	0			1.00000i	$0.5\pi$
$522$	−2.00000	−0.0875376
$523$	8.00000	0.349816	0.174908	−	0.984585i	$-0.444037\pi$
$523$	8.00000	0.349816	0.174908	+	0.984585i	$0.444037\pi$
$524$	−8.00000	−0.349482
$525$	0	0
$526$	8.00000	0.348817
$527$	−16.0000	−0.696971
$528$	−1.00000	−0.0435194
$529$	−23.0000	−1.00000
$530$	0	0
$531$	−4.00000	−0.173585
$532$	0	0
$533$	48.0000	2.07911
$534$	8.00000	0.346194
$535$	0	0
$536$	−4.00000	−0.172774
$537$	4.00000	0.172613
$538$	8.00000	0.344904
$539$	0	0
$540$	0	0
$541$	−34.0000	−1.46177	−0.730887	−	0.682498i	$-0.760893\pi$
$541$	−34.0000	−1.46177	−0.730887	+	0.682498i	$0.760893\pi$
$542$	24.0000	1.03089
$543$	0	0
$544$	−4.00000	−0.171499
$545$	0	0
$546$	0	0
$547$	−36.0000	−1.53925	−0.769624	−	0.638497i	$-0.779557\pi$
$547$	−36.0000	−1.53925	−0.769624	+	0.638497i	$0.779557\pi$
$548$	6.00000	0.256307
$549$	4.00000	0.170716
$550$	5.00000	0.213201
$551$	0	0
$552$	0	0
$553$	0	0
$554$	10.0000	0.424859
$555$	0	0
$556$	−16.0000	−0.678551
$557$	14.0000	0.593199	0.296600	−	0.955002i	$-0.404147\pi$
$557$	14.0000	0.593199	0.296600	+	0.955002i	$0.404147\pi$
$558$	4.00000	0.169334
$559$	−16.0000	−0.676728
$560$	0	0
$561$	−4.00000	−0.168880
$562$	10.0000	0.421825
$563$	0	0		−	1.00000i	$-0.5\pi$
$563$	0	0			1.00000i	$0.5\pi$
$564$	−4.00000	−0.168430
$565$	0	0
$566$	24.0000	1.00880
$567$	0	0
$568$	8.00000	0.335673
$569$	6.00000	0.251533	0.125767	−	0.992060i	$-0.459861\pi$
$569$	6.00000	0.251533	0.125767	+	0.992060i	$0.459861\pi$
$570$	0	0
$571$	−12.0000	−0.502184	−0.251092	−	0.967963i	$-0.580790\pi$
$571$	−12.0000	−0.502184	−0.251092	+	0.967963i	$0.580790\pi$
$572$	−4.00000	−0.167248
$573$	16.0000	0.668410
$574$	0	0
$575$	0	0
$576$	1.00000	0.0416667
$577$	40.0000	1.66522	0.832611	−	0.553858i	$-0.186845\pi$
$577$	40.0000	1.66522	0.832611	+	0.553858i	$0.186845\pi$
$578$	1.00000	0.0415945
$579$	14.0000	0.581820
$580$	0	0
$581$	0	0
$582$	−8.00000	−0.331611
$583$	−10.0000	−0.414158
$584$	−4.00000	−0.165521
$585$	0	0
$586$	4.00000	0.165238
$587$	36.0000	1.48588	0.742940	−	0.669359i	$-0.233431\pi$
$587$	36.0000	1.48588	0.742940	+	0.669359i	$0.233431\pi$
$588$	0	0
$589$	0	0
$590$	0	0
$591$	−6.00000	−0.246807
$592$	10.0000	0.410997
$593$	−36.0000	−1.47834	−0.739171	−	0.673517i	$-0.764783\pi$
$593$	−36.0000	−1.47834	−0.739171	+	0.673517i	$0.764783\pi$
$594$	1.00000	0.0410305
$595$	0	0
$596$	−10.0000	−0.409616
$597$	12.0000	0.491127
$598$	0	0
$599$	−24.0000	−0.980613	−0.490307	−	0.871550i	$-0.663115\pi$
$599$	−24.0000	−0.980613	−0.490307	+	0.871550i	$0.663115\pi$
$600$	−5.00000	−0.204124
$601$	28.0000	1.14214	0.571072	−	0.820900i	$-0.306528\pi$
$601$	28.0000	1.14214	0.571072	+	0.820900i	$0.306528\pi$
$602$	0	0
$603$	4.00000	0.162893
$604$	−8.00000	−0.325515
$605$	0	0
$606$	−12.0000	−0.487467
$607$	16.0000	0.649420	0.324710	−	0.945814i	$-0.394733\pi$
$607$	16.0000	0.649420	0.324710	+	0.945814i	$0.394733\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−16.0000	−0.647291
$612$	4.00000	0.161690
$613$	26.0000	1.05013	0.525065	−	0.851062i	$-0.324041\pi$
$613$	26.0000	1.05013	0.525065	+	0.851062i	$0.324041\pi$
$614$	8.00000	0.322854
$615$	0	0
$616$	0	0
$617$	22.0000	0.885687	0.442843	−	0.896599i	$-0.353970\pi$
$617$	22.0000	0.885687	0.442843	+	0.896599i	$0.353970\pi$
$618$	4.00000	0.160904
$619$	36.0000	1.44696	0.723481	−	0.690344i	$-0.242541\pi$
$619$	36.0000	1.44696	0.723481	+	0.690344i	$0.242541\pi$
$620$	0	0
$621$	0	0
$622$	−4.00000	−0.160385
$623$	0	0
$624$	4.00000	0.160128
$625$	25.0000	1.00000
$626$	−32.0000	−1.27898
$627$	0	0
$628$	8.00000	0.319235
$629$	40.0000	1.59490
$630$	0	0
$631$	−16.0000	−0.636950	−0.318475	−	0.947931i	$-0.603171\pi$
$631$	−16.0000	−0.636950	−0.318475	+	0.947931i	$0.603171\pi$
$632$	−8.00000	−0.318223
$633$	12.0000	0.476957
$634$	18.0000	0.714871
$635$	0	0
$636$	10.0000	0.396526
$637$	0	0
$638$	−2.00000	−0.0791808
$639$	−8.00000	−0.316475
$640$	0	0
$641$	−2.00000	−0.0789953	−0.0394976	−	0.999220i	$-0.512576\pi$
$641$	−2.00000	−0.0789953	−0.0394976	+	0.999220i	$0.512576\pi$
$642$	−12.0000	−0.473602
$643$	20.0000	0.788723	0.394362	−	0.918955i	$-0.370966\pi$
$643$	20.0000	0.788723	0.394362	+	0.918955i	$0.370966\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−20.0000	−0.786281	−0.393141	−	0.919478i	$-0.628611\pi$
$647$	−20.0000	−0.786281	−0.393141	+	0.919478i	$0.628611\pi$
$648$	−1.00000	−0.0392837
$649$	−4.00000	−0.157014
$650$	−20.0000	−0.784465
$651$	0	0
$652$	12.0000	0.469956
$653$	−30.0000	−1.17399	−0.586995	−	0.809590i	$-0.699689\pi$
$653$	−30.0000	−1.17399	−0.586995	+	0.809590i	$0.699689\pi$
$654$	18.0000	0.703856
$655$	0	0
$656$	−12.0000	−0.468521
$657$	4.00000	0.156055
$658$	0	0
$659$	4.00000	0.155818	0.0779089	−	0.996960i	$-0.475176\pi$
$659$	4.00000	0.155818	0.0779089	+	0.996960i	$0.475176\pi$
$660$	0	0
$661$	40.0000	1.55582	0.777910	−	0.628376i	$-0.216280\pi$
$661$	40.0000	1.55582	0.777910	+	0.628376i	$0.216280\pi$
$662$	−20.0000	−0.777322
$663$	16.0000	0.621389
$664$	16.0000	0.620920
$665$	0	0
$666$	−10.0000	−0.387492
$667$	0	0
$668$	0	0
$669$	12.0000	0.463947
$670$	0	0
$671$	4.00000	0.154418
$672$	0	0
$673$	−2.00000	−0.0770943	−0.0385472	−	0.999257i	$-0.512273\pi$
$673$	−2.00000	−0.0770943	−0.0385472	+	0.999257i	$0.512273\pi$
$674$	2.00000	0.0770371
$675$	5.00000	0.192450
$676$	3.00000	0.115385
$677$	4.00000	0.153732	0.0768662	−	0.997041i	$-0.475509\pi$
$677$	4.00000	0.153732	0.0768662	+	0.997041i	$0.475509\pi$
$678$	2.00000	0.0768095
$679$	0	0
$680$	0	0
$681$	8.00000	0.306561
$682$	4.00000	0.153168
$683$	4.00000	0.153056	0.0765279	−	0.997067i	$-0.475617\pi$
$683$	4.00000	0.153056	0.0765279	+	0.997067i	$0.475617\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	16.0000	0.610438
$688$	4.00000	0.152499
$689$	40.0000	1.52388
$690$	0	0
$691$	28.0000	1.06517	0.532585	−	0.846376i	$-0.321221\pi$
$691$	28.0000	1.06517	0.532585	+	0.846376i	$0.321221\pi$
$692$	4.00000	0.152057
$693$	0	0
$694$	−12.0000	−0.455514
$695$	0	0
$696$	2.00000	0.0758098
$697$	−48.0000	−1.81813
$698$	28.0000	1.05982
$699$	−6.00000	−0.226941
$700$	0	0
$701$	2.00000	0.0755390	0.0377695	−	0.999286i	$-0.487975\pi$
$701$	2.00000	0.0755390	0.0377695	+	0.999286i	$0.487975\pi$
$702$	−4.00000	−0.150970
$703$	0	0
$704$	1.00000	0.0376889
$705$	0	0
$706$	−24.0000	−0.903252
$707$	0	0
$708$	4.00000	0.150329
$709$	26.0000	0.976450	0.488225	−	0.872718i	$-0.337644\pi$
$709$	26.0000	0.976450	0.488225	+	0.872718i	$0.337644\pi$
$710$	0	0
$711$	8.00000	0.300023
$712$	−8.00000	−0.299813
$713$	0	0
$714$	0	0
$715$	0	0
$716$	−4.00000	−0.149487
$717$	−8.00000	−0.298765
$718$	32.0000	1.19423
$719$	−36.0000	−1.34257	−0.671287	−	0.741198i	$-0.734258\pi$
$719$	−36.0000	−1.34257	−0.671287	+	0.741198i	$0.734258\pi$
$720$	0	0
$721$	0	0
$722$	19.0000	0.707107
$723$	20.0000	0.743808
$724$	0	0
$725$	−10.0000	−0.371391
$726$	1.00000	0.0371135
$727$	12.0000	0.445055	0.222528	−	0.974926i	$-0.428569\pi$
$727$	12.0000	0.445055	0.222528	+	0.974926i	$0.428569\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	16.0000	0.591781
$732$	−4.00000	−0.147844
$733$	4.00000	0.147743	0.0738717	−	0.997268i	$-0.476464\pi$
$733$	4.00000	0.147743	0.0738717	+	0.997268i	$0.476464\pi$
$734$	12.0000	0.442928
$735$	0	0
$736$	0	0
$737$	4.00000	0.147342
$738$	12.0000	0.441726
$739$	28.0000	1.03000	0.514998	−	0.857191i	$-0.327793\pi$
$739$	28.0000	1.03000	0.514998	+	0.857191i	$0.327793\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	32.0000	1.17397	0.586983	−	0.809599i	$-0.300316\pi$
$743$	32.0000	1.17397	0.586983	+	0.809599i	$0.300316\pi$
$744$	−4.00000	−0.146647
$745$	0	0
$746$	−22.0000	−0.805477
$747$	−16.0000	−0.585409
$748$	4.00000	0.146254
$749$	0	0
$750$	0	0
$751$	32.0000	1.16770	0.583848	−	0.811863i	$-0.301546\pi$
$751$	32.0000	1.16770	0.583848	+	0.811863i	$0.301546\pi$
$752$	4.00000	0.145865
$753$	−4.00000	−0.145768
$754$	8.00000	0.291343
$755$	0	0
$756$	0	0
$757$	−22.0000	−0.799604	−0.399802	−	0.916602i	$-0.630921\pi$
$757$	−22.0000	−0.799604	−0.399802	+	0.916602i	$0.630921\pi$
$758$	−4.00000	−0.145287
$759$	0	0
$760$	0	0
$761$	44.0000	1.59500	0.797499	−	0.603320i	$-0.206156\pi$
$761$	44.0000	1.59500	0.797499	+	0.603320i	$0.206156\pi$
$762$	0	0
$763$	0	0
$764$	−16.0000	−0.578860
$765$	0	0
$766$	4.00000	0.144526
$767$	16.0000	0.577727
$768$	−1.00000	−0.0360844
$769$	20.0000	0.721218	0.360609	−	0.932717i	$-0.382569\pi$
$769$	20.0000	0.721218	0.360609	+	0.932717i	$0.382569\pi$
$770$	0	0
$771$	24.0000	0.864339
$772$	−14.0000	−0.503871
$773$	40.0000	1.43870	0.719350	−	0.694648i	$-0.244440\pi$
$773$	40.0000	1.43870	0.719350	+	0.694648i	$0.244440\pi$
$774$	−4.00000	−0.143777
$775$	20.0000	0.718421
$776$	8.00000	0.287183
$777$	0	0
$778$	−10.0000	−0.358517
$779$	0	0
$780$	0	0
$781$	−8.00000	−0.286263
$782$	0	0
$783$	−2.00000	−0.0714742
$784$	0	0
$785$	0	0
$786$	−8.00000	−0.285351
$787$	32.0000	1.14068	0.570338	−	0.821410i	$-0.306812\pi$
$787$	32.0000	1.14068	0.570338	+	0.821410i	$0.306812\pi$
$788$	6.00000	0.213741
$789$	8.00000	0.284808
$790$	0	0
$791$	0	0
$792$	−1.00000	−0.0355335
$793$	−16.0000	−0.568177
$794$	8.00000	0.283909
$795$	0	0
$796$	−12.0000	−0.425329
$797$	16.0000	0.566749	0.283375	−	0.959009i	$-0.408546\pi$
$797$	16.0000	0.566749	0.283375	+	0.959009i	$0.408546\pi$
$798$	0	0
$799$	16.0000	0.566039
$800$	5.00000	0.176777
$801$	8.00000	0.282666
$802$	34.0000	1.20058
$803$	4.00000	0.141157
$804$	−4.00000	−0.141069
$805$	0	0
$806$	−16.0000	−0.563576
$807$	8.00000	0.281613
$808$	12.0000	0.422159
$809$	10.0000	0.351581	0.175791	−	0.984428i	$-0.443752\pi$
$809$	10.0000	0.351581	0.175791	+	0.984428i	$0.443752\pi$
$810$	0	0
$811$	−8.00000	−0.280918	−0.140459	−	0.990086i	$-0.544858\pi$
$811$	−8.00000	−0.280918	−0.140459	+	0.990086i	$0.544858\pi$
$812$	0	0
$813$	24.0000	0.841717
$814$	−10.0000	−0.350500
$815$	0	0
$816$	−4.00000	−0.140028
$817$	0	0
$818$	−20.0000	−0.699284
$819$	0	0
$820$	0	0
$821$	54.0000	1.88461	0.942306	−	0.334751i	$-0.108652\pi$
$821$	54.0000	1.88461	0.942306	+	0.334751i	$0.108652\pi$
$822$	6.00000	0.209274
$823$	0	0		−	1.00000i	$-0.5\pi$
$823$	0	0			1.00000i	$0.5\pi$
$824$	−4.00000	−0.139347
$825$	5.00000	0.174078
$826$	0	0
$827$	−28.0000	−0.973655	−0.486828	−	0.873498i	$-0.661846\pi$
$827$	−28.0000	−0.973655	−0.486828	+	0.873498i	$0.661846\pi$
$828$	0	0
$829$	−24.0000	−0.833554	−0.416777	−	0.909009i	$-0.636840\pi$
$829$	−24.0000	−0.833554	−0.416777	+	0.909009i	$0.636840\pi$
$830$	0	0
$831$	10.0000	0.346896
$832$	−4.00000	−0.138675
$833$	0	0
$834$	−16.0000	−0.554035
$835$	0	0
$836$	0	0
$837$	4.00000	0.138260
$838$	36.0000	1.24360
$839$	−36.0000	−1.24286	−0.621429	−	0.783470i	$-0.713448\pi$
$839$	−36.0000	−1.24286	−0.621429	+	0.783470i	$0.713448\pi$
$840$	0	0
$841$	−25.0000	−0.862069
$842$	−22.0000	−0.758170
$843$	10.0000	0.344418
$844$	−12.0000	−0.413057
$845$	0	0
$846$	−4.00000	−0.137523
$847$	0	0
$848$	−10.0000	−0.343401
$849$	24.0000	0.823678
$850$	20.0000	0.685994
$851$	0	0
$852$	8.00000	0.274075
$853$	−4.00000	−0.136957	−0.0684787	−	0.997653i	$-0.521815\pi$
$853$	−4.00000	−0.136957	−0.0684787	+	0.997653i	$0.521815\pi$
$854$	0	0
$855$	0	0
$856$	12.0000	0.410152
$857$	−4.00000	−0.136637	−0.0683187	−	0.997664i	$-0.521763\pi$
$857$	−4.00000	−0.136637	−0.0683187	+	0.997664i	$0.521763\pi$
$858$	−4.00000	−0.136558
$859$	52.0000	1.77422	0.887109	−	0.461561i	$-0.152710\pi$
$859$	52.0000	1.77422	0.887109	+	0.461561i	$0.152710\pi$
$860$	0	0
$861$	0	0
$862$	40.0000	1.36241
$863$	−32.0000	−1.08929	−0.544646	−	0.838666i	$-0.683336\pi$
$863$	−32.0000	−1.08929	−0.544646	+	0.838666i	$0.683336\pi$
$864$	1.00000	0.0340207
$865$	0	0
$866$	−40.0000	−1.35926
$867$	1.00000	0.0339618
$868$	0	0
$869$	8.00000	0.271381
$870$	0	0
$871$	−16.0000	−0.542139
$872$	−18.0000	−0.609557
$873$	−8.00000	−0.270759
$874$	0	0
$875$	0	0
$876$	−4.00000	−0.135147
$877$	−46.0000	−1.55331	−0.776655	−	0.629926i	$-0.783085\pi$
$877$	−46.0000	−1.55331	−0.776655	+	0.629926i	$0.783085\pi$
$878$	40.0000	1.34993
$879$	4.00000	0.134917
$880$	0	0
$881$	16.0000	0.539054	0.269527	−	0.962993i	$-0.413133\pi$
$881$	16.0000	0.539054	0.269527	+	0.962993i	$0.413133\pi$
$882$	0	0
$883$	4.00000	0.134611	0.0673054	−	0.997732i	$-0.478560\pi$
$883$	4.00000	0.134611	0.0673054	+	0.997732i	$0.478560\pi$
$884$	−16.0000	−0.538138
$885$	0	0
$886$	−4.00000	−0.134383
$887$	−8.00000	−0.268614	−0.134307	−	0.990940i	$-0.542881\pi$
$887$	−8.00000	−0.268614	−0.134307	+	0.990940i	$0.542881\pi$
$888$	10.0000	0.335578
$889$	0	0
$890$	0	0
$891$	1.00000	0.0335013
$892$	−12.0000	−0.401790
$893$	0	0
$894$	−10.0000	−0.334450
$895$	0	0
$896$	0	0
$897$	0	0
$898$	−18.0000	−0.600668
$899$	−8.00000	−0.266815
$900$	−5.00000	−0.166667
$901$	−40.0000	−1.33259
$902$	12.0000	0.399556
$903$	0	0
$904$	−2.00000	−0.0665190
$905$	0	0
$906$	−8.00000	−0.265782
$907$	12.0000	0.398453	0.199227	−	0.979953i	$-0.436157\pi$
$907$	12.0000	0.398453	0.199227	+	0.979953i	$0.436157\pi$
$908$	−8.00000	−0.265489
$909$	−12.0000	−0.398015
$910$	0	0
$911$	−40.0000	−1.32526	−0.662630	−	0.748947i	$-0.730560\pi$
$911$	−40.0000	−1.32526	−0.662630	+	0.748947i	$0.730560\pi$
$912$	0	0
$913$	−16.0000	−0.529523
$914$	−26.0000	−0.860004
$915$	0	0
$916$	−16.0000	−0.528655
$917$	0	0
$918$	4.00000	0.132020
$919$	−16.0000	−0.527791	−0.263896	−	0.964551i	$-0.585007\pi$
$919$	−16.0000	−0.527791	−0.263896	+	0.964551i	$0.585007\pi$
$920$	0	0
$921$	8.00000	0.263609
$922$	−28.0000	−0.922131
$923$	32.0000	1.05329
$924$	0	0
$925$	−50.0000	−1.64399
$926$	40.0000	1.31448
$927$	4.00000	0.131377
$928$	−2.00000	−0.0656532
$929$	48.0000	1.57483	0.787414	−	0.616424i	$-0.211419\pi$
$929$	48.0000	1.57483	0.787414	+	0.616424i	$0.211419\pi$
$930$	0	0
$931$	0	0
$932$	6.00000	0.196537
$933$	−4.00000	−0.130954
$934$	−36.0000	−1.17796
$935$	0	0
$936$	4.00000	0.130744
$937$	36.0000	1.17607	0.588034	−	0.808836i	$-0.299902\pi$
$937$	36.0000	1.17607	0.588034	+	0.808836i	$0.299902\pi$
$938$	0	0
$939$	−32.0000	−1.04428
$940$	0	0
$941$	28.0000	0.912774	0.456387	−	0.889781i	$-0.349143\pi$
$941$	28.0000	0.912774	0.456387	+	0.889781i	$0.349143\pi$
$942$	8.00000	0.260654
$943$	0	0
$944$	−4.00000	−0.130189
$945$	0	0
$946$	−4.00000	−0.130051
$947$	−12.0000	−0.389948	−0.194974	−	0.980808i	$-0.562462\pi$
$947$	−12.0000	−0.389948	−0.194974	+	0.980808i	$0.562462\pi$
$948$	−8.00000	−0.259828
$949$	−16.0000	−0.519382
$950$	0	0
$951$	18.0000	0.583690
$952$	0	0
$953$	−54.0000	−1.74923	−0.874616	−	0.484817i	$-0.838886\pi$
$953$	−54.0000	−1.74923	−0.874616	+	0.484817i	$0.838886\pi$
$954$	10.0000	0.323762
$955$	0	0
$956$	8.00000	0.258738
$957$	−2.00000	−0.0646508
$958$	−24.0000	−0.775405
$959$	0	0
$960$	0	0
$961$	−15.0000	−0.483871
$962$	40.0000	1.28965
$963$	−12.0000	−0.386695
$964$	−20.0000	−0.644157
$965$	0	0
$966$	0	0
$967$	−8.00000	−0.257263	−0.128631	−	0.991692i	$-0.541058\pi$
$967$	−8.00000	−0.257263	−0.128631	+	0.991692i	$0.541058\pi$
$968$	−1.00000	−0.0321412
$969$	0	0
$970$	0	0
$971$	20.0000	0.641831	0.320915	−	0.947108i	$-0.396010\pi$
$971$	20.0000	0.641831	0.320915	+	0.947108i	$0.396010\pi$
$972$	−1.00000	−0.0320750
$973$	0	0
$974$	8.00000	0.256337
$975$	−20.0000	−0.640513
$976$	4.00000	0.128037
$977$	−50.0000	−1.59964	−0.799821	−	0.600239i	$-0.795072\pi$
$977$	−50.0000	−1.59964	−0.799821	+	0.600239i	$0.795072\pi$
$978$	12.0000	0.383718
$979$	8.00000	0.255681
$980$	0	0
$981$	18.0000	0.574696
$982$	−20.0000	−0.638226
$983$	−60.0000	−1.91370	−0.956851	−	0.290578i	$-0.906153\pi$
$983$	−60.0000	−1.91370	−0.956851	+	0.290578i	$0.906153\pi$
$984$	−12.0000	−0.382546
$985$	0	0
$986$	−8.00000	−0.254772
$987$	0	0
$988$	0	0
$989$	0	0
$990$	0	0
$991$	56.0000	1.77890	0.889449	−	0.457034i	$-0.151088\pi$
$991$	56.0000	1.77890	0.889449	+	0.457034i	$0.151088\pi$
$992$	4.00000	0.127000
$993$	−20.0000	−0.634681
$994$	0	0
$995$	0	0
$996$	16.0000	0.506979
$997$	28.0000	0.886769	0.443384	−	0.896332i	$-0.353778\pi$
$997$	28.0000	0.886769	0.443384	+	0.896332i	$0.353778\pi$
$998$	−20.0000	−0.633089
$999$	−10.0000	−0.316386

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3234.2.a.e.1.1	✓	1
3.2	odd	2		9702.2.a.bk.1.1		1
7.6	odd	2		3234.2.a.m.1.1	yes	1
21.20	even	2		9702.2.a.bo.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3234.2.a.e.1.1	✓	1	1.1	even	1	trivial
3234.2.a.m.1.1	yes	1	7.6	odd	2
9702.2.a.bk.1.1		1	3.2	odd	2
9702.2.a.bo.1.1		1	21.20	even	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)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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