LMFDB - Embedded newform 3234.2.a.n.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [3234,2,Mod(1,3234)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://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);

sage: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")

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:traces = [1,-1,1,1,2,-1,0,-1,1,-2,1,1,-2,0,2,1,6] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(17)] == traces)

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

Self dual:	yes
Analytic conductor:	$25.8236200137$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 462)
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)

$f(q)$

$=$

\(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +2.00000 q^{5} -1.00000 q^{6} -1.00000 q^{8} +1.00000 q^{9} -2.00000 q^{10} +1.00000 q^{11} +1.00000 q^{12} -2.00000 q^{13} +2.00000 q^{15} +1.00000 q^{16} +6.00000 q^{17} -1.00000 q^{18} +4.00000 q^{19} +2.00000 q^{20} -1.00000 q^{22} -4.00000 q^{23} -1.00000 q^{24} -1.00000 q^{25} +2.00000 q^{26} +1.00000 q^{27} +2.00000 q^{29} -2.00000 q^{30} +4.00000 q^{31} -1.00000 q^{32} +1.00000 q^{33} -6.00000 q^{34} +1.00000 q^{36} -2.00000 q^{37} -4.00000 q^{38} -2.00000 q^{39} -2.00000 q^{40} +6.00000 q^{41} +1.00000 q^{44} +2.00000 q^{45} +4.00000 q^{46} +8.00000 q^{47} +1.00000 q^{48} +1.00000 q^{50} +6.00000 q^{51} -2.00000 q^{52} -14.0000 q^{53} -1.00000 q^{54} +2.00000 q^{55} +4.00000 q^{57} -2.00000 q^{58} -12.0000 q^{59} +2.00000 q^{60} +14.0000 q^{61} -4.00000 q^{62} +1.00000 q^{64} -4.00000 q^{65} -1.00000 q^{66} +4.00000 q^{67} +6.00000 q^{68} -4.00000 q^{69} +12.0000 q^{71} -1.00000 q^{72} -6.00000 q^{73} +2.00000 q^{74} -1.00000 q^{75} +4.00000 q^{76} +2.00000 q^{78} +2.00000 q^{80} +1.00000 q^{81} -6.00000 q^{82} +12.0000 q^{85} +2.00000 q^{87} -1.00000 q^{88} +6.00000 q^{89} -2.00000 q^{90} -4.00000 q^{92} +4.00000 q^{93} -8.00000 q^{94} +8.00000 q^{95} -1.00000 q^{96} +14.0000 q^{97} +1.00000 q^{99} +O(q^{100})\)

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
$5$	2.00000	0.894427	0.447214	−	0.894427i	$-0.352416\pi$
$5$	2.00000	0.894427	0.447214	+	0.894427i	$0.352416\pi$
$6$	−1.00000	−0.408248
$7$	0	0
$7$	0	0
$8$	−1.00000	−0.353553
$9$	1.00000	0.333333
$10$	−2.00000	−0.632456
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	1.00000	0.288675
$13$	−2.00000	−0.554700	−0.277350	−	0.960769i	$-0.589456\pi$
$13$	−2.00000	−0.554700	−0.277350	+	0.960769i	$0.589456\pi$
$14$	0	0
$15$	2.00000	0.516398
$16$	1.00000	0.250000
$17$	6.00000	1.45521	0.727607	−	0.685994i	$-0.240633\pi$
$17$	6.00000	1.45521	0.727607	+	0.685994i	$0.240633\pi$
$18$	−1.00000	−0.235702
$19$	4.00000	0.917663	0.458831	−	0.888523i	$-0.348268\pi$
$19$	4.00000	0.917663	0.458831	+	0.888523i	$0.348268\pi$
$20$	2.00000	0.447214
$21$	0	0
$22$	−1.00000	−0.213201
$23$	−4.00000	−0.834058	−0.417029	−	0.908893i	$-0.636929\pi$
$23$	−4.00000	−0.834058	−0.417029	+	0.908893i	$0.636929\pi$
$24$	−1.00000	−0.204124
$25$	−1.00000	−0.200000
$26$	2.00000	0.392232
$27$	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	$0.440546\pi$
$30$	−2.00000	−0.365148
$31$	4.00000	0.718421	0.359211	−	0.933257i	$-0.383046\pi$
$31$	4.00000	0.718421	0.359211	+	0.933257i	$0.383046\pi$
$32$	−1.00000	−0.176777
$33$	1.00000	0.174078
$34$	−6.00000	−1.02899
$35$	0	0
$36$	1.00000	0.166667
$37$	−2.00000	−0.328798	−0.164399	−	0.986394i	$-0.552568\pi$
$37$	−2.00000	−0.328798	−0.164399	+	0.986394i	$0.552568\pi$
$38$	−4.00000	−0.648886
$39$	−2.00000	−0.320256
$40$	−2.00000	−0.316228
$41$	6.00000	0.937043	0.468521	−	0.883452i	$-0.344787\pi$
$41$	6.00000	0.937043	0.468521	+	0.883452i	$0.344787\pi$
$42$	0	0
$43$	0	0		−	1.00000i	$-0.5\pi$
$43$	0	0			1.00000i	$0.5\pi$
$44$	1.00000	0.150756
$45$	2.00000	0.298142
$46$	4.00000	0.589768
$47$	8.00000	1.16692	0.583460	−	0.812142i	$-0.301699\pi$
$47$	8.00000	1.16692	0.583460	+	0.812142i	$0.301699\pi$
$48$	1.00000	0.144338
$49$	0	0
$50$	1.00000	0.141421
$51$	6.00000	0.840168
$52$	−2.00000	−0.277350
$53$	−14.0000	−1.92305	−0.961524	−	0.274721i	$-0.911414\pi$
$53$	−14.0000	−1.92305	−0.961524	+	0.274721i	$0.911414\pi$
$54$	−1.00000	−0.136083
$55$	2.00000	0.269680
$56$	0	0
$57$	4.00000	0.529813
$58$	−2.00000	−0.262613
$59$	−12.0000	−1.56227	−0.781133	−	0.624364i	$-0.785358\pi$
$59$	−12.0000	−1.56227	−0.781133	+	0.624364i	$0.785358\pi$
$60$	2.00000	0.258199
$61$	14.0000	1.79252	0.896258	−	0.443533i	$-0.146275\pi$
$61$	14.0000	1.79252	0.896258	+	0.443533i	$0.146275\pi$
$62$	−4.00000	−0.508001
$63$	0	0
$64$	1.00000	0.125000
$65$	−4.00000	−0.496139
$66$	−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	$0.421429\pi$
$68$	6.00000	0.727607
$69$	−4.00000	−0.481543
$70$	0	0
$71$	12.0000	1.42414	0.712069	−	0.702109i	$-0.247758\pi$
$71$	12.0000	1.42414	0.712069	+	0.702109i	$0.247758\pi$
$72$	−1.00000	−0.117851
$73$	−6.00000	−0.702247	−0.351123	−	0.936329i	$-0.614200\pi$
$73$	−6.00000	−0.702247	−0.351123	+	0.936329i	$0.614200\pi$
$74$	2.00000	0.232495
$75$	−1.00000	−0.115470
$76$	4.00000	0.458831
$77$	0	0
$78$	2.00000	0.226455
$79$	0	0		−	1.00000i	$-0.5\pi$
$79$	0	0			1.00000i	$0.5\pi$
$80$	2.00000	0.223607
$81$	1.00000	0.111111
$82$	−6.00000	−0.662589
$83$	0	0		−	1.00000i	$-0.5\pi$
$83$	0	0			1.00000i	$0.5\pi$
$84$	0	0
$85$	12.0000	1.30158
$86$	0	0
$87$	2.00000	0.214423
$88$	−1.00000	−0.106600
$89$	6.00000	0.635999	0.317999	−	0.948091i	$-0.396989\pi$
$89$	6.00000	0.635999	0.317999	+	0.948091i	$0.396989\pi$
$90$	−2.00000	−0.210819
$91$	0	0
$92$	−4.00000	−0.417029
$93$	4.00000	0.414781
$94$	−8.00000	−0.825137
$95$	8.00000	0.820783
$96$	−1.00000	−0.102062
$97$	14.0000	1.42148	0.710742	−	0.703452i	$-0.248359\pi$
$97$	14.0000	1.42148	0.710742	+	0.703452i	$0.248359\pi$
$98$	0	0
$99$	1.00000	0.100504
$100$	−1.00000	−0.100000
$101$	−10.0000	−0.995037	−0.497519	−	0.867453i	$-0.665755\pi$
$101$	−10.0000	−0.995037	−0.497519	+	0.867453i	$0.665755\pi$
$102$	−6.00000	−0.594089
$103$	4.00000	0.394132	0.197066	−	0.980390i	$-0.436859\pi$
$103$	4.00000	0.394132	0.197066	+	0.980390i	$0.436859\pi$
$104$	2.00000	0.196116
$105$	0	0
$106$	14.0000	1.35980
$107$	12.0000	1.16008	0.580042	−	0.814587i	$-0.303036\pi$
$107$	12.0000	1.16008	0.580042	+	0.814587i	$0.303036\pi$
$108$	1.00000	0.0962250
$109$	10.0000	0.957826	0.478913	−	0.877862i	$-0.341031\pi$
$109$	10.0000	0.957826	0.478913	+	0.877862i	$0.341031\pi$
$110$	−2.00000	−0.190693
$111$	−2.00000	−0.189832
$112$	0	0
$113$	−6.00000	−0.564433	−0.282216	−	0.959351i	$-0.591070\pi$
$113$	−6.00000	−0.564433	−0.282216	+	0.959351i	$0.591070\pi$
$114$	−4.00000	−0.374634
$115$	−8.00000	−0.746004
$116$	2.00000	0.185695
$117$	−2.00000	−0.184900
$118$	12.0000	1.10469
$119$	0	0
$120$	−2.00000	−0.182574
$121$	1.00000	0.0909091
$122$	−14.0000	−1.26750
$123$	6.00000	0.541002
$124$	4.00000	0.359211
$125$	−12.0000	−1.07331
$126$	0	0
$127$	16.0000	1.41977	0.709885	−	0.704317i	$-0.248747\pi$
$127$	16.0000	1.41977	0.709885	+	0.704317i	$0.248747\pi$
$128$	−1.00000	−0.0883883
$129$	0	0
$130$	4.00000	0.350823
$131$	0	0		−	1.00000i	$-0.5\pi$
$131$	0	0			1.00000i	$0.5\pi$
$132$	1.00000	0.0870388
$133$	0	0
$134$	−4.00000	−0.345547
$135$	2.00000	0.172133
$136$	−6.00000	−0.514496
$137$	−22.0000	−1.87959	−0.939793	−	0.341743i	$-0.888983\pi$
$137$	−22.0000	−1.87959	−0.939793	+	0.341743i	$0.888983\pi$
$138$	4.00000	0.340503
$139$	−12.0000	−1.01783	−0.508913	−	0.860818i	$-0.669953\pi$
$139$	−12.0000	−1.01783	−0.508913	+	0.860818i	$0.669953\pi$
$140$	0	0
$141$	8.00000	0.673722
$142$	−12.0000	−1.00702
$143$	−2.00000	−0.167248
$144$	1.00000	0.0833333
$145$	4.00000	0.332182
$146$	6.00000	0.496564
$147$	0	0
$148$	−2.00000	−0.164399
$149$	10.0000	0.819232	0.409616	−	0.912258i	$-0.365663\pi$
$149$	10.0000	0.819232	0.409616	+	0.912258i	$0.365663\pi$
$150$	1.00000	0.0816497
$151$	0	0		−	1.00000i	$-0.5\pi$
$151$	0	0			1.00000i	$0.5\pi$
$152$	−4.00000	−0.324443
$153$	6.00000	0.485071
$154$	0	0
$155$	8.00000	0.642575
$156$	−2.00000	−0.160128
$157$	−2.00000	−0.159617	−0.0798087	−	0.996810i	$-0.525431\pi$
$157$	−2.00000	−0.159617	−0.0798087	+	0.996810i	$0.525431\pi$
$158$	0	0
$159$	−14.0000	−1.11027
$160$	−2.00000	−0.158114
$161$	0	0
$162$	−1.00000	−0.0785674
$163$	−20.0000	−1.56652	−0.783260	−	0.621694i	$-0.786445\pi$
$163$	−20.0000	−1.56652	−0.783260	+	0.621694i	$0.786445\pi$
$164$	6.00000	0.468521
$165$	2.00000	0.155700
$166$	0	0
$167$	0	0		−	1.00000i	$-0.5\pi$
$167$	0	0			1.00000i	$0.5\pi$
$168$	0	0
$169$	−9.00000	−0.692308
$170$	−12.0000	−0.920358
$171$	4.00000	0.305888
$172$	0	0
$173$	−10.0000	−0.760286	−0.380143	−	0.924928i	$-0.624125\pi$
$173$	−10.0000	−0.760286	−0.380143	+	0.924928i	$0.624125\pi$
$174$	−2.00000	−0.151620
$175$	0	0
$176$	1.00000	0.0753778
$177$	−12.0000	−0.901975
$178$	−6.00000	−0.449719
$179$	−4.00000	−0.298974	−0.149487	−	0.988764i	$-0.547762\pi$
$179$	−4.00000	−0.298974	−0.149487	+	0.988764i	$0.547762\pi$
$180$	2.00000	0.149071
$181$	14.0000	1.04061	0.520306	−	0.853980i	$-0.325818\pi$
$181$	14.0000	1.04061	0.520306	+	0.853980i	$0.325818\pi$
$182$	0	0
$183$	14.0000	1.03491
$184$	4.00000	0.294884
$185$	−4.00000	−0.294086
$186$	−4.00000	−0.293294
$187$	6.00000	0.438763
$188$	8.00000	0.583460
$189$	0	0
$190$	−8.00000	−0.580381
$191$	4.00000	0.289430	0.144715	−	0.989473i	$-0.453773\pi$
$191$	4.00000	0.289430	0.144715	+	0.989473i	$0.453773\pi$
$192$	1.00000	0.0721688
$193$	−6.00000	−0.431889	−0.215945	−	0.976406i	$-0.569283\pi$
$193$	−6.00000	−0.431889	−0.215945	+	0.976406i	$0.569283\pi$
$194$	−14.0000	−1.00514
$195$	−4.00000	−0.286446
$196$	0	0
$197$	−6.00000	−0.427482	−0.213741	−	0.976890i	$-0.568565\pi$
$197$	−6.00000	−0.427482	−0.213741	+	0.976890i	$0.568565\pi$
$198$	−1.00000	−0.0710669
$199$	12.0000	0.850657	0.425329	−	0.905039i	$-0.360158\pi$
$199$	12.0000	0.850657	0.425329	+	0.905039i	$0.360158\pi$
$200$	1.00000	0.0707107
$201$	4.00000	0.282138
$202$	10.0000	0.703598
$203$	0	0
$204$	6.00000	0.420084
$205$	12.0000	0.838116
$206$	−4.00000	−0.278693
$207$	−4.00000	−0.278019
$208$	−2.00000	−0.138675
$209$	4.00000	0.276686
$210$	0	0
$211$	0	0		−	1.00000i	$-0.5\pi$
$211$	0	0			1.00000i	$0.5\pi$
$212$	−14.0000	−0.961524
$213$	12.0000	0.822226
$214$	−12.0000	−0.820303
$215$	0	0
$216$	−1.00000	−0.0680414
$217$	0	0
$218$	−10.0000	−0.677285
$219$	−6.00000	−0.405442
$220$	2.00000	0.134840
$221$	−12.0000	−0.807207
$222$	2.00000	0.134231
$223$	12.0000	0.803579	0.401790	−	0.915732i	$-0.368388\pi$
$223$	12.0000	0.803579	0.401790	+	0.915732i	$0.368388\pi$
$224$	0	0
$225$	−1.00000	−0.0666667
$226$	6.00000	0.399114
$227$	0	0		−	1.00000i	$-0.5\pi$
$227$	0	0			1.00000i	$0.5\pi$
$228$	4.00000	0.264906
$229$	14.0000	0.925146	0.462573	−	0.886581i	$-0.346926\pi$
$229$	14.0000	0.925146	0.462573	+	0.886581i	$0.346926\pi$
$230$	8.00000	0.527504
$231$	0	0
$232$	−2.00000	−0.131306
$233$	−2.00000	−0.131024	−0.0655122	−	0.997852i	$-0.520868\pi$
$233$	−2.00000	−0.131024	−0.0655122	+	0.997852i	$0.520868\pi$
$234$	2.00000	0.130744
$235$	16.0000	1.04372
$236$	−12.0000	−0.781133
$237$	0	0
$238$	0	0
$239$	24.0000	1.55243	0.776215	−	0.630468i	$-0.217137\pi$
$239$	24.0000	1.55243	0.776215	+	0.630468i	$0.217137\pi$
$240$	2.00000	0.129099
$241$	26.0000	1.67481	0.837404	−	0.546585i	$-0.184072\pi$
$241$	26.0000	1.67481	0.837404	+	0.546585i	$0.184072\pi$
$242$	−1.00000	−0.0642824
$243$	1.00000	0.0641500
$244$	14.0000	0.896258
$245$	0	0
$246$	−6.00000	−0.382546
$247$	−8.00000	−0.509028
$248$	−4.00000	−0.254000
$249$	0	0
$250$	12.0000	0.758947
$251$	−20.0000	−1.26239	−0.631194	−	0.775625i	$-0.717435\pi$
$251$	−20.0000	−1.26239	−0.631194	+	0.775625i	$0.717435\pi$
$252$	0	0
$253$	−4.00000	−0.251478
$254$	−16.0000	−1.00393
$255$	12.0000	0.751469
$256$	1.00000	0.0625000
$257$	−18.0000	−1.12281	−0.561405	−	0.827541i	$-0.689739\pi$
$257$	−18.0000	−1.12281	−0.561405	+	0.827541i	$0.689739\pi$
$258$	0	0
$259$	0	0
$260$	−4.00000	−0.248069
$261$	2.00000	0.123797
$262$	0	0
$263$	0	0		−	1.00000i	$-0.5\pi$
$263$	0	0			1.00000i	$0.5\pi$
$264$	−1.00000	−0.0615457
$265$	−28.0000	−1.72003
$266$	0	0
$267$	6.00000	0.367194
$268$	4.00000	0.244339
$269$	−30.0000	−1.82913	−0.914566	−	0.404436i	$-0.867468\pi$
$269$	−30.0000	−1.82913	−0.914566	+	0.404436i	$0.867468\pi$
$270$	−2.00000	−0.121716
$271$	0	0		−	1.00000i	$-0.5\pi$
$271$	0	0			1.00000i	$0.5\pi$
$272$	6.00000	0.363803
$273$	0	0
$274$	22.0000	1.32907
$275$	−1.00000	−0.0603023
$276$	−4.00000	−0.240772
$277$	26.0000	1.56219	0.781094	−	0.624413i	$-0.214662\pi$
$277$	26.0000	1.56219	0.781094	+	0.624413i	$0.214662\pi$
$278$	12.0000	0.719712
$279$	4.00000	0.239474
$280$	0	0
$281$	−18.0000	−1.07379	−0.536895	−	0.843649i	$-0.680403\pi$
$281$	−18.0000	−1.07379	−0.536895	+	0.843649i	$0.680403\pi$
$282$	−8.00000	−0.476393
$283$	4.00000	0.237775	0.118888	−	0.992908i	$-0.462067\pi$
$283$	4.00000	0.237775	0.118888	+	0.992908i	$0.462067\pi$
$284$	12.0000	0.712069
$285$	8.00000	0.473879
$286$	2.00000	0.118262
$287$	0	0
$288$	−1.00000	−0.0589256
$289$	19.0000	1.11765
$290$	−4.00000	−0.234888
$291$	14.0000	0.820695
$292$	−6.00000	−0.351123
$293$	14.0000	0.817889	0.408944	−	0.912559i	$-0.365897\pi$
$293$	14.0000	0.817889	0.408944	+	0.912559i	$0.365897\pi$
$294$	0	0
$295$	−24.0000	−1.39733
$296$	2.00000	0.116248
$297$	1.00000	0.0580259
$298$	−10.0000	−0.579284
$299$	8.00000	0.462652
$300$	−1.00000	−0.0577350
$301$	0	0
$302$	0	0
$303$	−10.0000	−0.574485
$304$	4.00000	0.229416
$305$	28.0000	1.60328
$306$	−6.00000	−0.342997
$307$	12.0000	0.684876	0.342438	−	0.939540i	$-0.388747\pi$
$307$	12.0000	0.684876	0.342438	+	0.939540i	$0.388747\pi$
$308$	0	0
$309$	4.00000	0.227552
$310$	−8.00000	−0.454369
$311$	−8.00000	−0.453638	−0.226819	−	0.973937i	$-0.572833\pi$
$311$	−8.00000	−0.453638	−0.226819	+	0.973937i	$0.572833\pi$
$312$	2.00000	0.113228
$313$	14.0000	0.791327	0.395663	−	0.918396i	$-0.370515\pi$
$313$	14.0000	0.791327	0.395663	+	0.918396i	$0.370515\pi$
$314$	2.00000	0.112867
$315$	0	0
$316$	0	0
$317$	10.0000	0.561656	0.280828	−	0.959758i	$-0.409391\pi$
$317$	10.0000	0.561656	0.280828	+	0.959758i	$0.409391\pi$
$318$	14.0000	0.785081
$319$	2.00000	0.111979
$320$	2.00000	0.111803
$321$	12.0000	0.669775
$322$	0	0
$323$	24.0000	1.33540
$324$	1.00000	0.0555556
$325$	2.00000	0.110940
$326$	20.0000	1.10770
$327$	10.0000	0.553001
$328$	−6.00000	−0.331295
$329$	0	0
$330$	−2.00000	−0.110096
$331$	−20.0000	−1.09930	−0.549650	−	0.835395i	$-0.685239\pi$
$331$	−20.0000	−1.09930	−0.549650	+	0.835395i	$0.685239\pi$
$332$	0	0
$333$	−2.00000	−0.109599
$334$	0	0
$335$	8.00000	0.437087
$336$	0	0
$337$	−14.0000	−0.762629	−0.381314	−	0.924445i	$-0.624528\pi$
$337$	−14.0000	−0.762629	−0.381314	+	0.924445i	$0.624528\pi$
$338$	9.00000	0.489535
$339$	−6.00000	−0.325875
$340$	12.0000	0.650791
$341$	4.00000	0.216612
$342$	−4.00000	−0.216295
$343$	0	0
$344$	0	0
$345$	−8.00000	−0.430706
$346$	10.0000	0.537603
$347$	−28.0000	−1.50312	−0.751559	−	0.659665i	$-0.770698\pi$
$347$	−28.0000	−1.50312	−0.751559	+	0.659665i	$0.770698\pi$
$348$	2.00000	0.107211
$349$	−34.0000	−1.81998	−0.909989	−	0.414632i	$-0.863910\pi$
$349$	−34.0000	−1.81998	−0.909989	+	0.414632i	$0.863910\pi$
$350$	0	0
$351$	−2.00000	−0.106752
$352$	−1.00000	−0.0533002
$353$	14.0000	0.745145	0.372572	−	0.928003i	$-0.378476\pi$
$353$	14.0000	0.745145	0.372572	+	0.928003i	$0.378476\pi$
$354$	12.0000	0.637793
$355$	24.0000	1.27379
$356$	6.00000	0.317999
$357$	0	0
$358$	4.00000	0.211407
$359$	32.0000	1.68890	0.844448	−	0.535638i	$-0.179929\pi$
$359$	32.0000	1.68890	0.844448	+	0.535638i	$0.179929\pi$
$360$	−2.00000	−0.105409
$361$	−3.00000	−0.157895
$362$	−14.0000	−0.735824
$363$	1.00000	0.0524864
$364$	0	0
$365$	−12.0000	−0.628109
$366$	−14.0000	−0.731792
$367$	28.0000	1.46159	0.730794	−	0.682598i	$-0.239150\pi$
$367$	28.0000	1.46159	0.730794	+	0.682598i	$0.239150\pi$
$368$	−4.00000	−0.208514
$369$	6.00000	0.312348
$370$	4.00000	0.207950
$371$	0	0
$372$	4.00000	0.207390
$373$	26.0000	1.34623	0.673114	−	0.739538i	$-0.264956\pi$
$373$	26.0000	1.34623	0.673114	+	0.739538i	$0.264956\pi$
$374$	−6.00000	−0.310253
$375$	−12.0000	−0.619677
$376$	−8.00000	−0.412568
$377$	−4.00000	−0.206010
$378$	0	0
$379$	36.0000	1.84920	0.924598	−	0.380945i	$-0.124401\pi$
$379$	36.0000	1.84920	0.924598	+	0.380945i	$0.124401\pi$
$380$	8.00000	0.410391
$381$	16.0000	0.819705
$382$	−4.00000	−0.204658
$383$	−16.0000	−0.817562	−0.408781	−	0.912633i	$-0.634046\pi$
$383$	−16.0000	−0.817562	−0.408781	+	0.912633i	$0.634046\pi$
$384$	−1.00000	−0.0510310
$385$	0	0
$386$	6.00000	0.305392
$387$	0	0
$388$	14.0000	0.710742
$389$	18.0000	0.912636	0.456318	−	0.889817i	$-0.349168\pi$
$389$	18.0000	0.912636	0.456318	+	0.889817i	$0.349168\pi$
$390$	4.00000	0.202548
$391$	−24.0000	−1.21373
$392$	0	0
$393$	0	0
$394$	6.00000	0.302276
$395$	0	0
$396$	1.00000	0.0502519
$397$	−18.0000	−0.903394	−0.451697	−	0.892171i	$-0.649181\pi$
$397$	−18.0000	−0.903394	−0.451697	+	0.892171i	$0.649181\pi$
$398$	−12.0000	−0.601506
$399$	0	0
$400$	−1.00000	−0.0500000
$401$	−14.0000	−0.699127	−0.349563	−	0.936913i	$-0.613670\pi$
$401$	−14.0000	−0.699127	−0.349563	+	0.936913i	$0.613670\pi$
$402$	−4.00000	−0.199502
$403$	−8.00000	−0.398508
$404$	−10.0000	−0.497519
$405$	2.00000	0.0993808
$406$	0	0
$407$	−2.00000	−0.0991363
$408$	−6.00000	−0.297044
$409$	−38.0000	−1.87898	−0.939490	−	0.342578i	$-0.888700\pi$
$409$	−38.0000	−1.87898	−0.939490	+	0.342578i	$0.888700\pi$
$410$	−12.0000	−0.592638
$411$	−22.0000	−1.08518
$412$	4.00000	0.197066
$413$	0	0
$414$	4.00000	0.196589
$415$	0	0
$416$	2.00000	0.0980581
$417$	−12.0000	−0.587643
$418$	−4.00000	−0.195646
$419$	12.0000	0.586238	0.293119	−	0.956076i	$-0.405307\pi$
$419$	12.0000	0.586238	0.293119	+	0.956076i	$0.405307\pi$
$420$	0	0
$421$	−26.0000	−1.26716	−0.633581	−	0.773676i	$-0.718416\pi$
$421$	−26.0000	−1.26716	−0.633581	+	0.773676i	$0.718416\pi$
$422$	0	0
$423$	8.00000	0.388973
$424$	14.0000	0.679900
$425$	−6.00000	−0.291043
$426$	−12.0000	−0.581402
$427$	0	0
$428$	12.0000	0.580042
$429$	−2.00000	−0.0965609
$430$	0	0
$431$	−8.00000	−0.385346	−0.192673	−	0.981263i	$-0.561716\pi$
$431$	−8.00000	−0.385346	−0.192673	+	0.981263i	$0.561716\pi$
$432$	1.00000	0.0481125
$433$	−34.0000	−1.63394	−0.816968	−	0.576683i	$-0.804347\pi$
$433$	−34.0000	−1.63394	−0.816968	+	0.576683i	$0.804347\pi$
$434$	0	0
$435$	4.00000	0.191785
$436$	10.0000	0.478913
$437$	−16.0000	−0.765384
$438$	6.00000	0.286691
$439$	−16.0000	−0.763638	−0.381819	−	0.924237i	$-0.624702\pi$
$439$	−16.0000	−0.763638	−0.381819	+	0.924237i	$0.624702\pi$
$440$	−2.00000	−0.0953463
$441$	0	0
$442$	12.0000	0.570782
$443$	12.0000	0.570137	0.285069	−	0.958507i	$-0.407984\pi$
$443$	12.0000	0.570137	0.285069	+	0.958507i	$0.407984\pi$
$444$	−2.00000	−0.0949158
$445$	12.0000	0.568855
$446$	−12.0000	−0.568216
$447$	10.0000	0.472984
$448$	0	0
$449$	−22.0000	−1.03824	−0.519122	−	0.854700i	$-0.673741\pi$
$449$	−22.0000	−1.03824	−0.519122	+	0.854700i	$0.673741\pi$
$450$	1.00000	0.0471405
$451$	6.00000	0.282529
$452$	−6.00000	−0.282216
$453$	0	0
$454$	0	0
$455$	0	0
$456$	−4.00000	−0.187317
$457$	26.0000	1.21623	0.608114	−	0.793849i	$-0.291926\pi$
$457$	26.0000	1.21623	0.608114	+	0.793849i	$0.291926\pi$
$458$	−14.0000	−0.654177
$459$	6.00000	0.280056
$460$	−8.00000	−0.373002
$461$	30.0000	1.39724	0.698620	−	0.715493i	$-0.253798\pi$
$461$	30.0000	1.39724	0.698620	+	0.715493i	$0.253798\pi$
$462$	0	0
$463$	40.0000	1.85896	0.929479	−	0.368875i	$-0.120257\pi$
$463$	40.0000	1.85896	0.929479	+	0.368875i	$0.120257\pi$
$464$	2.00000	0.0928477
$465$	8.00000	0.370991
$466$	2.00000	0.0926482
$467$	−20.0000	−0.925490	−0.462745	−	0.886492i	$-0.653135\pi$
$467$	−20.0000	−0.925490	−0.462745	+	0.886492i	$0.653135\pi$
$468$	−2.00000	−0.0924500
$469$	0	0
$470$	−16.0000	−0.738025
$471$	−2.00000	−0.0921551
$472$	12.0000	0.552345
$473$	0	0
$474$	0	0
$475$	−4.00000	−0.183533
$476$	0	0
$477$	−14.0000	−0.641016
$478$	−24.0000	−1.09773
$479$	24.0000	1.09659	0.548294	−	0.836286i	$-0.315277\pi$
$479$	24.0000	1.09659	0.548294	+	0.836286i	$0.315277\pi$
$480$	−2.00000	−0.0912871
$481$	4.00000	0.182384
$482$	−26.0000	−1.18427
$483$	0	0
$484$	1.00000	0.0454545
$485$	28.0000	1.27141
$486$	−1.00000	−0.0453609
$487$	0	0		−	1.00000i	$-0.5\pi$
$487$	0	0			1.00000i	$0.5\pi$
$488$	−14.0000	−0.633750
$489$	−20.0000	−0.904431
$490$	0	0
$491$	−28.0000	−1.26362	−0.631811	−	0.775122i	$-0.717688\pi$
$491$	−28.0000	−1.26362	−0.631811	+	0.775122i	$0.717688\pi$
$492$	6.00000	0.270501
$493$	12.0000	0.540453
$494$	8.00000	0.359937
$495$	2.00000	0.0898933
$496$	4.00000	0.179605
$497$	0	0
$498$	0	0
$499$	12.0000	0.537194	0.268597	−	0.963253i	$-0.413440\pi$
$499$	12.0000	0.537194	0.268597	+	0.963253i	$0.413440\pi$
$500$	−12.0000	−0.536656
$501$	0	0
$502$	20.0000	0.892644
$503$	16.0000	0.713405	0.356702	−	0.934218i	$-0.383901\pi$
$503$	16.0000	0.713405	0.356702	+	0.934218i	$0.383901\pi$
$504$	0	0
$505$	−20.0000	−0.889988
$506$	4.00000	0.177822
$507$	−9.00000	−0.399704
$508$	16.0000	0.709885
$509$	18.0000	0.797836	0.398918	−	0.916987i	$-0.369386\pi$
$509$	18.0000	0.797836	0.398918	+	0.916987i	$0.369386\pi$
$510$	−12.0000	−0.531369
$511$	0	0
$512$	−1.00000	−0.0441942
$513$	4.00000	0.176604
$514$	18.0000	0.793946
$515$	8.00000	0.352522
$516$	0	0
$517$	8.00000	0.351840
$518$	0	0
$519$	−10.0000	−0.438951
$520$	4.00000	0.175412
$521$	6.00000	0.262865	0.131432	−	0.991325i	$-0.458042\pi$
$521$	6.00000	0.262865	0.131432	+	0.991325i	$0.458042\pi$
$522$	−2.00000	−0.0875376
$523$	36.0000	1.57417	0.787085	−	0.616844i	$-0.211589\pi$
$523$	36.0000	1.57417	0.787085	+	0.616844i	$0.211589\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	24.0000	1.04546
$528$	1.00000	0.0435194
$529$	−7.00000	−0.304348
$530$	28.0000	1.21624
$531$	−12.0000	−0.520756
$532$	0	0
$533$	−12.0000	−0.519778
$534$	−6.00000	−0.259645
$535$	24.0000	1.03761
$536$	−4.00000	−0.172774
$537$	−4.00000	−0.172613
$538$	30.0000	1.29339
$539$	0	0
$540$	2.00000	0.0860663
$541$	10.0000	0.429934	0.214967	−	0.976621i	$-0.431036\pi$
$541$	10.0000	0.429934	0.214967	+	0.976621i	$0.431036\pi$
$542$	0	0
$543$	14.0000	0.600798
$544$	−6.00000	−0.257248
$545$	20.0000	0.856706
$546$	0	0
$547$	−24.0000	−1.02617	−0.513083	−	0.858339i	$-0.671497\pi$
$547$	−24.0000	−1.02617	−0.513083	+	0.858339i	$0.671497\pi$
$548$	−22.0000	−0.939793
$549$	14.0000	0.597505
$550$	1.00000	0.0426401
$551$	8.00000	0.340811
$552$	4.00000	0.170251
$553$	0	0
$554$	−26.0000	−1.10463
$555$	−4.00000	−0.169791
$556$	−12.0000	−0.508913
$557$	−46.0000	−1.94908	−0.974541	−	0.224208i	$-0.928020\pi$
$557$	−46.0000	−1.94908	−0.974541	+	0.224208i	$0.928020\pi$
$558$	−4.00000	−0.169334
$559$	0	0
$560$	0	0
$561$	6.00000	0.253320
$562$	18.0000	0.759284
$563$	−24.0000	−1.01148	−0.505740	−	0.862686i	$-0.668780\pi$
$563$	−24.0000	−1.01148	−0.505740	+	0.862686i	$0.668780\pi$
$564$	8.00000	0.336861
$565$	−12.0000	−0.504844
$566$	−4.00000	−0.168133
$567$	0	0
$568$	−12.0000	−0.503509
$569$	22.0000	0.922288	0.461144	−	0.887325i	$-0.347439\pi$
$569$	22.0000	0.922288	0.461144	+	0.887325i	$0.347439\pi$
$570$	−8.00000	−0.335083
$571$	32.0000	1.33916	0.669579	−	0.742741i	$-0.266474\pi$
$571$	32.0000	1.33916	0.669579	+	0.742741i	$0.266474\pi$
$572$	−2.00000	−0.0836242
$573$	4.00000	0.167102
$574$	0	0
$575$	4.00000	0.166812
$576$	1.00000	0.0416667
$577$	6.00000	0.249783	0.124892	−	0.992170i	$-0.460142\pi$
$577$	6.00000	0.249783	0.124892	+	0.992170i	$0.460142\pi$
$578$	−19.0000	−0.790296
$579$	−6.00000	−0.249351
$580$	4.00000	0.166091
$581$	0	0
$582$	−14.0000	−0.580319
$583$	−14.0000	−0.579821
$584$	6.00000	0.248282
$585$	−4.00000	−0.165380
$586$	−14.0000	−0.578335
$587$	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
$589$	16.0000	0.659269
$590$	24.0000	0.988064
$591$	−6.00000	−0.246807
$592$	−2.00000	−0.0821995
$593$	−18.0000	−0.739171	−0.369586	−	0.929197i	$-0.620500\pi$
$593$	−18.0000	−0.739171	−0.369586	+	0.929197i	$0.620500\pi$
$594$	−1.00000	−0.0410305
$595$	0	0
$596$	10.0000	0.409616
$597$	12.0000	0.491127
$598$	−8.00000	−0.327144
$599$	−44.0000	−1.79779	−0.898896	−	0.438163i	$-0.855629\pi$
$599$	−44.0000	−1.79779	−0.898896	+	0.438163i	$0.855629\pi$
$600$	1.00000	0.0408248
$601$	26.0000	1.06056	0.530281	−	0.847822i	$-0.322086\pi$
$601$	26.0000	1.06056	0.530281	+	0.847822i	$0.322086\pi$
$602$	0	0
$603$	4.00000	0.162893
$604$	0	0
$605$	2.00000	0.0813116
$606$	10.0000	0.406222
$607$	−8.00000	−0.324710	−0.162355	−	0.986732i	$-0.551909\pi$
$607$	−8.00000	−0.324710	−0.162355	+	0.986732i	$0.551909\pi$
$608$	−4.00000	−0.162221
$609$	0	0
$610$	−28.0000	−1.13369
$611$	−16.0000	−0.647291
$612$	6.00000	0.242536
$613$	−46.0000	−1.85792	−0.928961	−	0.370177i	$-0.879297\pi$
$613$	−46.0000	−1.85792	−0.928961	+	0.370177i	$0.879297\pi$
$614$	−12.0000	−0.484281
$615$	12.0000	0.483887
$616$	0	0
$617$	−30.0000	−1.20775	−0.603877	−	0.797077i	$-0.706378\pi$
$617$	−30.0000	−1.20775	−0.603877	+	0.797077i	$0.706378\pi$
$618$	−4.00000	−0.160904
$619$	−4.00000	−0.160774	−0.0803868	−	0.996764i	$-0.525616\pi$
$619$	−4.00000	−0.160774	−0.0803868	+	0.996764i	$0.525616\pi$
$620$	8.00000	0.321288
$621$	−4.00000	−0.160514
$622$	8.00000	0.320771
$623$	0	0
$624$	−2.00000	−0.0800641
$625$	−19.0000	−0.760000
$626$	−14.0000	−0.559553
$627$	4.00000	0.159745
$628$	−2.00000	−0.0798087
$629$	−12.0000	−0.478471
$630$	0	0
$631$	−40.0000	−1.59237	−0.796187	−	0.605050i	$-0.793153\pi$
$631$	−40.0000	−1.59237	−0.796187	+	0.605050i	$0.793153\pi$
$632$	0	0
$633$	0	0
$634$	−10.0000	−0.397151
$635$	32.0000	1.26988
$636$	−14.0000	−0.555136
$637$	0	0
$638$	−2.00000	−0.0791808
$639$	12.0000	0.474713
$640$	−2.00000	−0.0790569
$641$	10.0000	0.394976	0.197488	−	0.980305i	$-0.436722\pi$
$641$	10.0000	0.394976	0.197488	+	0.980305i	$0.436722\pi$
$642$	−12.0000	−0.473602
$643$	−36.0000	−1.41970	−0.709851	−	0.704352i	$-0.751238\pi$
$643$	−36.0000	−1.41970	−0.709851	+	0.704352i	$0.751238\pi$
$644$	0	0
$645$	0	0
$646$	−24.0000	−0.944267
$647$	−40.0000	−1.57256	−0.786281	−	0.617869i	$-0.787996\pi$
$647$	−40.0000	−1.57256	−0.786281	+	0.617869i	$0.787996\pi$
$648$	−1.00000	−0.0392837
$649$	−12.0000	−0.471041
$650$	−2.00000	−0.0784465
$651$	0	0
$652$	−20.0000	−0.783260
$653$	−6.00000	−0.234798	−0.117399	−	0.993085i	$-0.537456\pi$
$653$	−6.00000	−0.234798	−0.117399	+	0.993085i	$0.537456\pi$
$654$	−10.0000	−0.391031
$655$	0	0
$656$	6.00000	0.234261
$657$	−6.00000	−0.234082
$658$	0	0
$659$	−20.0000	−0.779089	−0.389545	−	0.921008i	$-0.627368\pi$
$659$	−20.0000	−0.779089	−0.389545	+	0.921008i	$0.627368\pi$
$660$	2.00000	0.0778499
$661$	−18.0000	−0.700119	−0.350059	−	0.936727i	$-0.613839\pi$
$661$	−18.0000	−0.700119	−0.350059	+	0.936727i	$0.613839\pi$
$662$	20.0000	0.777322
$663$	−12.0000	−0.466041
$664$	0	0
$665$	0	0
$666$	2.00000	0.0774984
$667$	−8.00000	−0.309761
$668$	0	0
$669$	12.0000	0.463947
$670$	−8.00000	−0.309067
$671$	14.0000	0.540464
$672$	0	0
$673$	−14.0000	−0.539660	−0.269830	−	0.962908i	$-0.586968\pi$
$673$	−14.0000	−0.539660	−0.269830	+	0.962908i	$0.586968\pi$
$674$	14.0000	0.539260
$675$	−1.00000	−0.0384900
$676$	−9.00000	−0.346154
$677$	−18.0000	−0.691796	−0.345898	−	0.938272i	$-0.612426\pi$
$677$	−18.0000	−0.691796	−0.345898	+	0.938272i	$0.612426\pi$
$678$	6.00000	0.230429
$679$	0	0
$680$	−12.0000	−0.460179
$681$	0	0
$682$	−4.00000	−0.153168
$683$	−28.0000	−1.07139	−0.535695	−	0.844411i	$-0.679950\pi$
$683$	−28.0000	−1.07139	−0.535695	+	0.844411i	$0.679950\pi$
$684$	4.00000	0.152944
$685$	−44.0000	−1.68115
$686$	0	0
$687$	14.0000	0.534133
$688$	0	0
$689$	28.0000	1.06672
$690$	8.00000	0.304555
$691$	44.0000	1.67384	0.836919	−	0.547326i	$-0.184354\pi$
$691$	44.0000	1.67384	0.836919	+	0.547326i	$0.184354\pi$
$692$	−10.0000	−0.380143
$693$	0	0
$694$	28.0000	1.06287
$695$	−24.0000	−0.910372
$696$	−2.00000	−0.0758098
$697$	36.0000	1.36360
$698$	34.0000	1.28692
$699$	−2.00000	−0.0756469
$700$	0	0
$701$	2.00000	0.0755390	0.0377695	−	0.999286i	$-0.487975\pi$
$701$	2.00000	0.0755390	0.0377695	+	0.999286i	$0.487975\pi$
$702$	2.00000	0.0754851
$703$	−8.00000	−0.301726
$704$	1.00000	0.0376889
$705$	16.0000	0.602595
$706$	−14.0000	−0.526897
$707$	0	0
$708$	−12.0000	−0.450988
$709$	−10.0000	−0.375558	−0.187779	−	0.982211i	$-0.560129\pi$
$709$	−10.0000	−0.375558	−0.187779	+	0.982211i	$0.560129\pi$
$710$	−24.0000	−0.900704
$711$	0	0
$712$	−6.00000	−0.224860
$713$	−16.0000	−0.599205
$714$	0	0
$715$	−4.00000	−0.149592
$716$	−4.00000	−0.149487
$717$	24.0000	0.896296
$718$	−32.0000	−1.19423
$719$	−24.0000	−0.895049	−0.447524	−	0.894272i	$-0.647694\pi$
$719$	−24.0000	−0.895049	−0.447524	+	0.894272i	$0.647694\pi$
$720$	2.00000	0.0745356
$721$	0	0
$722$	3.00000	0.111648
$723$	26.0000	0.966950
$724$	14.0000	0.520306
$725$	−2.00000	−0.0742781
$726$	−1.00000	−0.0371135
$727$	−28.0000	−1.03846	−0.519231	−	0.854634i	$-0.673782\pi$
$727$	−28.0000	−1.03846	−0.519231	+	0.854634i	$0.673782\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	12.0000	0.444140
$731$	0	0
$732$	14.0000	0.517455
$733$	−34.0000	−1.25582	−0.627909	−	0.778287i	$-0.716089\pi$
$733$	−34.0000	−1.25582	−0.627909	+	0.778287i	$0.716089\pi$
$734$	−28.0000	−1.03350
$735$	0	0
$736$	4.00000	0.147442
$737$	4.00000	0.147342
$738$	−6.00000	−0.220863
$739$	−48.0000	−1.76571	−0.882854	−	0.469647i	$-0.844381\pi$
$739$	−48.0000	−1.76571	−0.882854	+	0.469647i	$0.844381\pi$
$740$	−4.00000	−0.147043
$741$	−8.00000	−0.293887
$742$	0	0
$743$	−40.0000	−1.46746	−0.733729	−	0.679442i	$-0.762222\pi$
$743$	−40.0000	−1.46746	−0.733729	+	0.679442i	$0.762222\pi$
$744$	−4.00000	−0.146647
$745$	20.0000	0.732743
$746$	−26.0000	−0.951928
$747$	0	0
$748$	6.00000	0.219382
$749$	0	0
$750$	12.0000	0.438178
$751$	16.0000	0.583848	0.291924	−	0.956441i	$-0.405705\pi$
$751$	16.0000	0.583848	0.291924	+	0.956441i	$0.405705\pi$
$752$	8.00000	0.291730
$753$	−20.0000	−0.728841
$754$	4.00000	0.145671
$755$	0	0
$756$	0	0
$757$	−2.00000	−0.0726912	−0.0363456	−	0.999339i	$-0.511572\pi$
$757$	−2.00000	−0.0726912	−0.0363456	+	0.999339i	$0.511572\pi$
$758$	−36.0000	−1.30758
$759$	−4.00000	−0.145191
$760$	−8.00000	−0.290191
$761$	−26.0000	−0.942499	−0.471250	−	0.882000i	$-0.656197\pi$
$761$	−26.0000	−0.942499	−0.471250	+	0.882000i	$0.656197\pi$
$762$	−16.0000	−0.579619
$763$	0	0
$764$	4.00000	0.144715
$765$	12.0000	0.433861
$766$	16.0000	0.578103
$767$	24.0000	0.866590
$768$	1.00000	0.0360844
$769$	−38.0000	−1.37032	−0.685158	−	0.728395i	$-0.740267\pi$
$769$	−38.0000	−1.37032	−0.685158	+	0.728395i	$0.740267\pi$
$770$	0	0
$771$	−18.0000	−0.648254
$772$	−6.00000	−0.215945
$773$	−14.0000	−0.503545	−0.251773	−	0.967786i	$-0.581013\pi$
$773$	−14.0000	−0.503545	−0.251773	+	0.967786i	$0.581013\pi$
$774$	0	0
$775$	−4.00000	−0.143684
$776$	−14.0000	−0.502571
$777$	0	0
$778$	−18.0000	−0.645331
$779$	24.0000	0.859889
$780$	−4.00000	−0.143223
$781$	12.0000	0.429394
$782$	24.0000	0.858238
$783$	2.00000	0.0714742
$784$	0	0
$785$	−4.00000	−0.142766
$786$	0	0
$787$	−12.0000	−0.427754	−0.213877	−	0.976861i	$-0.568609\pi$
$787$	−12.0000	−0.427754	−0.213877	+	0.976861i	$0.568609\pi$
$788$	−6.00000	−0.213741
$789$	0	0
$790$	0	0
$791$	0	0
$792$	−1.00000	−0.0355335
$793$	−28.0000	−0.994309
$794$	18.0000	0.638796
$795$	−28.0000	−0.993058
$796$	12.0000	0.425329
$797$	18.0000	0.637593	0.318796	−	0.947823i	$-0.396721\pi$
$797$	18.0000	0.637593	0.318796	+	0.947823i	$0.396721\pi$
$798$	0	0
$799$	48.0000	1.69812
$800$	1.00000	0.0353553
$801$	6.00000	0.212000
$802$	14.0000	0.494357
$803$	−6.00000	−0.211735
$804$	4.00000	0.141069
$805$	0	0
$806$	8.00000	0.281788
$807$	−30.0000	−1.05605
$808$	10.0000	0.351799
$809$	−50.0000	−1.75791	−0.878953	−	0.476908i	$-0.841757\pi$
$809$	−50.0000	−1.75791	−0.878953	+	0.476908i	$0.841757\pi$
$810$	−2.00000	−0.0702728
$811$	44.0000	1.54505	0.772524	−	0.634985i	$-0.218994\pi$
$811$	44.0000	1.54505	0.772524	+	0.634985i	$0.218994\pi$
$812$	0	0
$813$	0	0
$814$	2.00000	0.0701000
$815$	−40.0000	−1.40114
$816$	6.00000	0.210042
$817$	0	0
$818$	38.0000	1.32864
$819$	0	0
$820$	12.0000	0.419058
$821$	−22.0000	−0.767805	−0.383903	−	0.923374i	$-0.625420\pi$
$821$	−22.0000	−0.767805	−0.383903	+	0.923374i	$0.625420\pi$
$822$	22.0000	0.767338
$823$	32.0000	1.11545	0.557725	−	0.830026i	$-0.311674\pi$
$823$	32.0000	1.11545	0.557725	+	0.830026i	$0.311674\pi$
$824$	−4.00000	−0.139347
$825$	−1.00000	−0.0348155
$826$	0	0
$827$	−36.0000	−1.25184	−0.625921	−	0.779886i	$-0.715277\pi$
$827$	−36.0000	−1.25184	−0.625921	+	0.779886i	$0.715277\pi$
$828$	−4.00000	−0.139010
$829$	14.0000	0.486240	0.243120	−	0.969996i	$-0.421829\pi$
$829$	14.0000	0.486240	0.243120	+	0.969996i	$0.421829\pi$
$830$	0	0
$831$	26.0000	0.901930
$832$	−2.00000	−0.0693375
$833$	0	0
$834$	12.0000	0.415526
$835$	0	0
$836$	4.00000	0.138343
$837$	4.00000	0.138260
$838$	−12.0000	−0.414533
$839$	16.0000	0.552381	0.276191	−	0.961103i	$-0.410928\pi$
$839$	16.0000	0.552381	0.276191	+	0.961103i	$0.410928\pi$
$840$	0	0
$841$	−25.0000	−0.862069
$842$	26.0000	0.896019
$843$	−18.0000	−0.619953
$844$	0	0
$845$	−18.0000	−0.619219
$846$	−8.00000	−0.275046
$847$	0	0
$848$	−14.0000	−0.480762
$849$	4.00000	0.137280
$850$	6.00000	0.205798
$851$	8.00000	0.274236
$852$	12.0000	0.411113
$853$	−42.0000	−1.43805	−0.719026	−	0.694983i	$-0.755412\pi$
$853$	−42.0000	−1.43805	−0.719026	+	0.694983i	$0.755412\pi$
$854$	0	0
$855$	8.00000	0.273594
$856$	−12.0000	−0.410152
$857$	22.0000	0.751506	0.375753	−	0.926720i	$-0.377384\pi$
$857$	22.0000	0.751506	0.375753	+	0.926720i	$0.377384\pi$
$858$	2.00000	0.0682789
$859$	−20.0000	−0.682391	−0.341196	−	0.939992i	$-0.610832\pi$
$859$	−20.0000	−0.682391	−0.341196	+	0.939992i	$0.610832\pi$
$860$	0	0
$861$	0	0
$862$	8.00000	0.272481
$863$	−36.0000	−1.22545	−0.612727	−	0.790295i	$-0.709928\pi$
$863$	−36.0000	−1.22545	−0.612727	+	0.790295i	$0.709928\pi$
$864$	−1.00000	−0.0340207
$865$	−20.0000	−0.680020
$866$	34.0000	1.15537
$867$	19.0000	0.645274
$868$	0	0
$869$	0	0
$870$	−4.00000	−0.135613
$871$	−8.00000	−0.271070
$872$	−10.0000	−0.338643
$873$	14.0000	0.473828
$874$	16.0000	0.541208
$875$	0	0
$876$	−6.00000	−0.202721
$877$	−14.0000	−0.472746	−0.236373	−	0.971662i	$-0.575959\pi$
$877$	−14.0000	−0.472746	−0.236373	+	0.971662i	$0.575959\pi$
$878$	16.0000	0.539974
$879$	14.0000	0.472208
$880$	2.00000	0.0674200
$881$	−34.0000	−1.14549	−0.572745	−	0.819734i	$-0.694121\pi$
$881$	−34.0000	−1.14549	−0.572745	+	0.819734i	$0.694121\pi$
$882$	0	0
$883$	28.0000	0.942275	0.471138	−	0.882060i	$-0.343844\pi$
$883$	28.0000	0.942275	0.471138	+	0.882060i	$0.343844\pi$
$884$	−12.0000	−0.403604
$885$	−24.0000	−0.806751
$886$	−12.0000	−0.403148
$887$	−16.0000	−0.537227	−0.268614	−	0.963248i	$-0.586566\pi$
$887$	−16.0000	−0.537227	−0.268614	+	0.963248i	$0.586566\pi$
$888$	2.00000	0.0671156
$889$	0	0
$890$	−12.0000	−0.402241
$891$	1.00000	0.0335013
$892$	12.0000	0.401790
$893$	32.0000	1.07084
$894$	−10.0000	−0.334450
$895$	−8.00000	−0.267411
$896$	0	0
$897$	8.00000	0.267112
$898$	22.0000	0.734150
$899$	8.00000	0.266815
$900$	−1.00000	−0.0333333
$901$	−84.0000	−2.79845
$902$	−6.00000	−0.199778
$903$	0	0
$904$	6.00000	0.199557
$905$	28.0000	0.930751
$906$	0	0
$907$	−44.0000	−1.46100	−0.730498	−	0.682915i	$-0.760712\pi$
$907$	−44.0000	−1.46100	−0.730498	+	0.682915i	$0.760712\pi$
$908$	0	0
$909$	−10.0000	−0.331679
$910$	0	0
$911$	12.0000	0.397578	0.198789	−	0.980042i	$-0.436299\pi$
$911$	12.0000	0.397578	0.198789	+	0.980042i	$0.436299\pi$
$912$	4.00000	0.132453
$913$	0	0
$914$	−26.0000	−0.860004
$915$	28.0000	0.925651
$916$	14.0000	0.462573
$917$	0	0
$918$	−6.00000	−0.198030
$919$	−16.0000	−0.527791	−0.263896	−	0.964551i	$-0.585007\pi$
$919$	−16.0000	−0.527791	−0.263896	+	0.964551i	$0.585007\pi$
$920$	8.00000	0.263752
$921$	12.0000	0.395413
$922$	−30.0000	−0.987997
$923$	−24.0000	−0.789970
$924$	0	0
$925$	2.00000	0.0657596
$926$	−40.0000	−1.31448
$927$	4.00000	0.131377
$928$	−2.00000	−0.0656532
$929$	30.0000	0.984268	0.492134	−	0.870519i	$-0.336217\pi$
$929$	30.0000	0.984268	0.492134	+	0.870519i	$0.336217\pi$
$930$	−8.00000	−0.262330
$931$	0	0
$932$	−2.00000	−0.0655122
$933$	−8.00000	−0.261908
$934$	20.0000	0.654420
$935$	12.0000	0.392442
$936$	2.00000	0.0653720
$937$	−14.0000	−0.457360	−0.228680	−	0.973502i	$-0.573441\pi$
$937$	−14.0000	−0.457360	−0.228680	+	0.973502i	$0.573441\pi$
$938$	0	0
$939$	14.0000	0.456873
$940$	16.0000	0.521862
$941$	38.0000	1.23876	0.619382	−	0.785090i	$-0.287383\pi$
$941$	38.0000	1.23876	0.619382	+	0.785090i	$0.287383\pi$
$942$	2.00000	0.0651635
$943$	−24.0000	−0.781548
$944$	−12.0000	−0.390567
$945$	0	0
$946$	0	0
$947$	44.0000	1.42981	0.714904	−	0.699223i	$-0.246470\pi$
$947$	44.0000	1.42981	0.714904	+	0.699223i	$0.246470\pi$
$948$	0	0
$949$	12.0000	0.389536
$950$	4.00000	0.129777
$951$	10.0000	0.324272
$952$	0	0
$953$	−2.00000	−0.0647864	−0.0323932	−	0.999475i	$-0.510313\pi$
$953$	−2.00000	−0.0647864	−0.0323932	+	0.999475i	$0.510313\pi$
$954$	14.0000	0.453267
$955$	8.00000	0.258874
$956$	24.0000	0.776215
$957$	2.00000	0.0646508
$958$	−24.0000	−0.775405
$959$	0	0
$960$	2.00000	0.0645497
$961$	−15.0000	−0.483871
$962$	−4.00000	−0.128965
$963$	12.0000	0.386695
$964$	26.0000	0.837404
$965$	−12.0000	−0.386294
$966$	0	0
$967$	0	0		−	1.00000i	$-0.5\pi$
$967$	0	0			1.00000i	$0.5\pi$
$968$	−1.00000	−0.0321412
$969$	24.0000	0.770991
$970$	−28.0000	−0.899026
$971$	12.0000	0.385098	0.192549	−	0.981287i	$-0.438325\pi$
$971$	12.0000	0.385098	0.192549	+	0.981287i	$0.438325\pi$
$972$	1.00000	0.0320750
$973$	0	0
$974$	0	0
$975$	2.00000	0.0640513
$976$	14.0000	0.448129
$977$	18.0000	0.575871	0.287936	−	0.957650i	$-0.407031\pi$
$977$	18.0000	0.575871	0.287936	+	0.957650i	$0.407031\pi$
$978$	20.0000	0.639529
$979$	6.00000	0.191761
$980$	0	0
$981$	10.0000	0.319275
$982$	28.0000	0.893516
$983$	32.0000	1.02064	0.510321	−	0.859984i	$-0.329527\pi$
$983$	32.0000	1.02064	0.510321	+	0.859984i	$0.329527\pi$
$984$	−6.00000	−0.191273
$985$	−12.0000	−0.382352
$986$	−12.0000	−0.382158
$987$	0	0
$988$	−8.00000	−0.254514
$989$	0	0
$990$	−2.00000	−0.0635642
$991$	8.00000	0.254128	0.127064	−	0.991894i	$-0.459445\pi$
$991$	8.00000	0.254128	0.127064	+	0.991894i	$0.459445\pi$
$992$	−4.00000	−0.127000
$993$	−20.0000	−0.634681
$994$	0	0
$995$	24.0000	0.760851
$996$	0	0
$997$	−26.0000	−0.823428	−0.411714	−	0.911313i	$-0.635070\pi$
$997$	−26.0000	−0.823428	−0.411714	+	0.911313i	$0.635070\pi$
$998$	−12.0000	−0.379853
$999$	−2.00000	−0.0632772

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3234.2.a.n.1.1		1
3.2	odd	2		9702.2.a.bf.1.1		1
7.6	odd	2		462.2.a.a.1.1	✓	1
21.20	even	2		1386.2.a.k.1.1		1
28.27	even	2		3696.2.a.s.1.1		1
77.76	even	2		5082.2.a.q.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
462.2.a.a.1.1	✓	1	7.6	odd	2
1386.2.a.k.1.1		1	21.20	even	2
3234.2.a.n.1.1		1	1.1	even	1	trivial
3696.2.a.s.1.1		1	28.27	even	2
5082.2.a.q.1.1		1	77.76	even	2
9702.2.a.bf.1.1		1	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 $\F_{q}$
Belyi maps

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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