LMFDB - Embedded newform 3234.2.a.t.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:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 66)
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)

$f(q)$

$=$

$q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -2.00000 q^{5} +1.00000 q^{6} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$

\(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -2.00000 q^{5} +1.00000 q^{6} +1.00000 q^{8} +1.00000 q^{9} -2.00000 q^{10} -1.00000 q^{11} +1.00000 q^{12} +6.00000 q^{13} -2.00000 q^{15} +1.00000 q^{16} -2.00000 q^{17} +1.00000 q^{18} -4.00000 q^{19} -2.00000 q^{20} -1.00000 q^{22} +4.00000 q^{23} +1.00000 q^{24} -1.00000 q^{25} +6.00000 q^{26} +1.00000 q^{27} +6.00000 q^{29} -2.00000 q^{30} +1.00000 q^{32} -1.00000 q^{33} -2.00000 q^{34} +1.00000 q^{36} +6.00000 q^{37} -4.00000 q^{38} +6.00000 q^{39} -2.00000 q^{40} +6.00000 q^{41} +4.00000 q^{43} -1.00000 q^{44} -2.00000 q^{45} +4.00000 q^{46} +12.0000 q^{47} +1.00000 q^{48} -1.00000 q^{50} -2.00000 q^{51} +6.00000 q^{52} +2.00000 q^{53} +1.00000 q^{54} +2.00000 q^{55} -4.00000 q^{57} +6.00000 q^{58} -12.0000 q^{59} -2.00000 q^{60} +14.0000 q^{61} +1.00000 q^{64} -12.0000 q^{65} -1.00000 q^{66} +4.00000 q^{67} -2.00000 q^{68} +4.00000 q^{69} -12.0000 q^{71} +1.00000 q^{72} +6.00000 q^{73} +6.00000 q^{74} -1.00000 q^{75} -4.00000 q^{76} +6.00000 q^{78} -4.00000 q^{79} -2.00000 q^{80} +1.00000 q^{81} +6.00000 q^{82} -4.00000 q^{83} +4.00000 q^{85} +4.00000 q^{86} +6.00000 q^{87} -1.00000 q^{88} -10.0000 q^{89} -2.00000 q^{90} +4.00000 q^{92} +12.0000 q^{94} +8.00000 q^{95} +1.00000 q^{96} +14.0000 q^{97} -1.00000 q^{99} +O(q^{100})\)

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	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	$-0.647584\pi$
$5$	−2.00000	−0.894427	−0.447214	+	0.894427i	$0.647584\pi$
$6$	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
$13$	6.00000	1.66410	0.832050	−	0.554700i	$-0.187167\pi$
$13$	6.00000	1.66410	0.832050	+	0.554700i	$0.187167\pi$
$14$	0	0
$15$	−2.00000	−0.516398
$16$	1.00000	0.250000
$17$	−2.00000	−0.485071	−0.242536	−	0.970143i	$-0.577979\pi$
$17$	−2.00000	−0.485071	−0.242536	+	0.970143i	$0.577979\pi$
$18$	1.00000	0.235702
$19$	−4.00000	−0.917663	−0.458831	−	0.888523i	$-0.651732\pi$
$19$	−4.00000	−0.917663	−0.458831	+	0.888523i	$0.651732\pi$
$20$	−2.00000	−0.447214
$21$	0	0
$22$	−1.00000	−0.213201
$23$	4.00000	0.834058	0.417029	−	0.908893i	$-0.363071\pi$
$23$	4.00000	0.834058	0.417029	+	0.908893i	$0.363071\pi$
$24$	1.00000	0.204124
$25$	−1.00000	−0.200000
$26$	6.00000	1.17670
$27$	1.00000	0.192450
$28$	0	0
$29$	6.00000	1.11417	0.557086	−	0.830455i	$-0.311919\pi$
$29$	6.00000	1.11417	0.557086	+	0.830455i	$0.311919\pi$
$30$	−2.00000	−0.365148
$31$	0	0		−	1.00000i	$-0.5\pi$
$31$	0	0			1.00000i	$0.5\pi$
$32$	1.00000	0.176777
$33$	−1.00000	−0.174078
$34$	−2.00000	−0.342997
$35$	0	0
$36$	1.00000	0.166667
$37$	6.00000	0.986394	0.493197	−	0.869918i	$-0.335828\pi$
$37$	6.00000	0.986394	0.493197	+	0.869918i	$0.335828\pi$
$38$	−4.00000	−0.648886
$39$	6.00000	0.960769
$40$	−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
$43$	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
$45$	−2.00000	−0.298142
$46$	4.00000	0.589768
$47$	12.0000	1.75038	0.875190	−	0.483779i	$-0.160736\pi$
$47$	12.0000	1.75038	0.875190	+	0.483779i	$0.160736\pi$
$48$	1.00000	0.144338
$49$	0	0
$50$	−1.00000	−0.141421
$51$	−2.00000	−0.280056
$52$	6.00000	0.832050
$53$	2.00000	0.274721	0.137361	−	0.990521i	$-0.456138\pi$
$53$	2.00000	0.274721	0.137361	+	0.990521i	$0.456138\pi$
$54$	1.00000	0.136083
$55$	2.00000	0.269680
$56$	0	0
$57$	−4.00000	−0.529813
$58$	6.00000	0.787839
$59$	−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	$0.146275\pi$
$62$	0	0
$63$	0	0
$64$	1.00000	0.125000
$65$	−12.0000	−1.48842
$66$	−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	$0.421429\pi$
$68$	−2.00000	−0.242536
$69$	4.00000	0.481543
$70$	0	0
$71$	−12.0000	−1.42414	−0.712069	−	0.702109i	$-0.752242\pi$
$71$	−12.0000	−1.42414	−0.712069	+	0.702109i	$0.752242\pi$
$72$	1.00000	0.117851
$73$	6.00000	0.702247	0.351123	−	0.936329i	$-0.385800\pi$
$73$	6.00000	0.702247	0.351123	+	0.936329i	$0.385800\pi$
$74$	6.00000	0.697486
$75$	−1.00000	−0.115470
$76$	−4.00000	−0.458831
$77$	0	0
$78$	6.00000	0.679366
$79$	−4.00000	−0.450035	−0.225018	−	0.974355i	$-0.572244\pi$
$79$	−4.00000	−0.450035	−0.225018	+	0.974355i	$0.572244\pi$
$80$	−2.00000	−0.223607
$81$	1.00000	0.111111
$82$	6.00000	0.662589
$83$	−4.00000	−0.439057	−0.219529	−	0.975606i	$-0.570452\pi$
$83$	−4.00000	−0.439057	−0.219529	+	0.975606i	$0.570452\pi$
$84$	0	0
$85$	4.00000	0.433861
$86$	4.00000	0.431331
$87$	6.00000	0.643268
$88$	−1.00000	−0.106600
$89$	−10.0000	−1.06000	−0.529999	−	0.847998i	$-0.677808\pi$
$89$	−10.0000	−1.06000	−0.529999	+	0.847998i	$0.677808\pi$
$90$	−2.00000	−0.210819
$91$	0	0
$92$	4.00000	0.417029
$93$	0	0
$94$	12.0000	1.23771
$95$	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
$101$	−14.0000	−1.39305	−0.696526	−	0.717532i	$-0.745272\pi$
$101$	−14.0000	−1.39305	−0.696526	+	0.717532i	$0.745272\pi$
$102$	−2.00000	−0.198030
$103$	0	0		−	1.00000i	$-0.5\pi$
$103$	0	0			1.00000i	$0.5\pi$
$104$	6.00000	0.588348
$105$	0	0
$106$	2.00000	0.194257
$107$	4.00000	0.386695	0.193347	−	0.981130i	$-0.438066\pi$
$107$	4.00000	0.386695	0.193347	+	0.981130i	$0.438066\pi$
$108$	1.00000	0.0962250
$109$	−6.00000	−0.574696	−0.287348	−	0.957826i	$-0.592774\pi$
$109$	−6.00000	−0.574696	−0.287348	+	0.957826i	$0.592774\pi$
$110$	2.00000	0.190693
$111$	6.00000	0.569495
$112$	0	0
$113$	2.00000	0.188144	0.0940721	−	0.995565i	$-0.470012\pi$
$113$	2.00000	0.188144	0.0940721	+	0.995565i	$0.470012\pi$
$114$	−4.00000	−0.374634
$115$	−8.00000	−0.746004
$116$	6.00000	0.557086
$117$	6.00000	0.554700
$118$	−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
$124$	0	0
$125$	12.0000	1.07331
$126$	0	0
$127$	12.0000	1.06483	0.532414	−	0.846484i	$-0.321285\pi$
$127$	12.0000	1.06483	0.532414	+	0.846484i	$0.321285\pi$
$128$	1.00000	0.0883883
$129$	4.00000	0.352180
$130$	−12.0000	−1.05247
$131$	−4.00000	−0.349482	−0.174741	−	0.984614i	$-0.555909\pi$
$131$	−4.00000	−0.349482	−0.174741	+	0.984614i	$0.555909\pi$
$132$	−1.00000	−0.0870388
$133$	0	0
$134$	4.00000	0.345547
$135$	−2.00000	−0.172133
$136$	−2.00000	−0.171499
$137$	2.00000	0.170872	0.0854358	−	0.996344i	$-0.472772\pi$
$137$	2.00000	0.170872	0.0854358	+	0.996344i	$0.472772\pi$
$138$	4.00000	0.340503
$139$	4.00000	0.339276	0.169638	−	0.985506i	$-0.445740\pi$
$139$	4.00000	0.339276	0.169638	+	0.985506i	$0.445740\pi$
$140$	0	0
$141$	12.0000	1.01058
$142$	−12.0000	−1.00702
$143$	−6.00000	−0.501745
$144$	1.00000	0.0833333
$145$	−12.0000	−0.996546
$146$	6.00000	0.496564
$147$	0	0
$148$	6.00000	0.493197
$149$	−10.0000	−0.819232	−0.409616	−	0.912258i	$-0.634337\pi$
$149$	−10.0000	−0.819232	−0.409616	+	0.912258i	$0.634337\pi$
$150$	−1.00000	−0.0816497
$151$	4.00000	0.325515	0.162758	−	0.986666i	$-0.447961\pi$
$151$	4.00000	0.325515	0.162758	+	0.986666i	$0.447961\pi$
$152$	−4.00000	−0.324443
$153$	−2.00000	−0.161690
$154$	0	0
$155$	0	0
$156$	6.00000	0.480384
$157$	10.0000	0.798087	0.399043	−	0.916932i	$-0.369342\pi$
$157$	10.0000	0.798087	0.399043	+	0.916932i	$0.369342\pi$
$158$	−4.00000	−0.318223
$159$	2.00000	0.158610
$160$	−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
$166$	−4.00000	−0.310460
$167$	0	0		−	1.00000i	$-0.5\pi$
$167$	0	0			1.00000i	$0.5\pi$
$168$	0	0
$169$	23.0000	1.76923
$170$	4.00000	0.306786
$171$	−4.00000	−0.305888
$172$	4.00000	0.304997
$173$	10.0000	0.760286	0.380143	−	0.924928i	$-0.375875\pi$
$173$	10.0000	0.760286	0.380143	+	0.924928i	$0.375875\pi$
$174$	6.00000	0.454859
$175$	0	0
$176$	−1.00000	−0.0753778
$177$	−12.0000	−0.901975
$178$	−10.0000	−0.749532
$179$	20.0000	1.49487	0.747435	−	0.664335i	$-0.231285\pi$
$179$	20.0000	1.49487	0.747435	+	0.664335i	$0.231285\pi$
$180$	−2.00000	−0.149071
$181$	2.00000	0.148659	0.0743294	−	0.997234i	$-0.476318\pi$
$181$	2.00000	0.148659	0.0743294	+	0.997234i	$0.476318\pi$
$182$	0	0
$183$	14.0000	1.03491
$184$	4.00000	0.294884
$185$	−12.0000	−0.882258
$186$	0	0
$187$	2.00000	0.146254
$188$	12.0000	0.875190
$189$	0	0
$190$	8.00000	0.580381
$191$	−12.0000	−0.868290	−0.434145	−	0.900843i	$-0.642949\pi$
$191$	−12.0000	−0.868290	−0.434145	+	0.900843i	$0.642949\pi$
$192$	1.00000	0.0721688
$193$	10.0000	0.719816	0.359908	−	0.932988i	$-0.382808\pi$
$193$	10.0000	0.719816	0.359908	+	0.932988i	$0.382808\pi$
$194$	14.0000	1.00514
$195$	−12.0000	−0.859338
$196$	0	0
$197$	−2.00000	−0.142494	−0.0712470	−	0.997459i	$-0.522698\pi$
$197$	−2.00000	−0.142494	−0.0712470	+	0.997459i	$0.522698\pi$
$198$	−1.00000	−0.0710669
$199$	16.0000	1.13421	0.567105	−	0.823646i	$-0.308063\pi$
$199$	16.0000	1.13421	0.567105	+	0.823646i	$0.308063\pi$
$200$	−1.00000	−0.0707107
$201$	4.00000	0.282138
$202$	−14.0000	−0.985037
$203$	0	0
$204$	−2.00000	−0.140028
$205$	−12.0000	−0.838116
$206$	0	0
$207$	4.00000	0.278019
$208$	6.00000	0.416025
$209$	4.00000	0.276686
$210$	0	0
$211$	−4.00000	−0.275371	−0.137686	−	0.990476i	$-0.543966\pi$
$211$	−4.00000	−0.275371	−0.137686	+	0.990476i	$0.543966\pi$
$212$	2.00000	0.137361
$213$	−12.0000	−0.822226
$214$	4.00000	0.273434
$215$	−8.00000	−0.545595
$216$	1.00000	0.0680414
$217$	0	0
$218$	−6.00000	−0.406371
$219$	6.00000	0.405442
$220$	2.00000	0.134840
$221$	−12.0000	−0.807207
$222$	6.00000	0.402694
$223$	0	0		−	1.00000i	$-0.5\pi$
$223$	0	0			1.00000i	$0.5\pi$
$224$	0	0
$225$	−1.00000	−0.0666667
$226$	2.00000	0.133038
$227$	−12.0000	−0.796468	−0.398234	−	0.917284i	$-0.630377\pi$
$227$	−12.0000	−0.796468	−0.398234	+	0.917284i	$0.630377\pi$
$228$	−4.00000	−0.264906
$229$	−14.0000	−0.925146	−0.462573	−	0.886581i	$-0.653074\pi$
$229$	−14.0000	−0.925146	−0.462573	+	0.886581i	$0.653074\pi$
$230$	−8.00000	−0.527504
$231$	0	0
$232$	6.00000	0.393919
$233$	10.0000	0.655122	0.327561	−	0.944830i	$-0.393773\pi$
$233$	10.0000	0.655122	0.327561	+	0.944830i	$0.393773\pi$
$234$	6.00000	0.392232
$235$	−24.0000	−1.56559
$236$	−12.0000	−0.781133
$237$	−4.00000	−0.259828
$238$	0	0
$239$	8.00000	0.517477	0.258738	−	0.965947i	$-0.416693\pi$
$239$	8.00000	0.517477	0.258738	+	0.965947i	$0.416693\pi$
$240$	−2.00000	−0.129099
$241$	−10.0000	−0.644157	−0.322078	−	0.946713i	$-0.604381\pi$
$241$	−10.0000	−0.644157	−0.322078	+	0.946713i	$0.604381\pi$
$242$	1.00000	0.0642824
$243$	1.00000	0.0641500
$244$	14.0000	0.896258
$245$	0	0
$246$	6.00000	0.382546
$247$	−24.0000	−1.52708
$248$	0	0
$249$	−4.00000	−0.253490
$250$	12.0000	0.758947
$251$	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
$253$	−4.00000	−0.251478
$254$	12.0000	0.752947
$255$	4.00000	0.250490
$256$	1.00000	0.0625000
$257$	−2.00000	−0.124757	−0.0623783	−	0.998053i	$-0.519869\pi$
$257$	−2.00000	−0.124757	−0.0623783	+	0.998053i	$0.519869\pi$
$258$	4.00000	0.249029
$259$	0	0
$260$	−12.0000	−0.744208
$261$	6.00000	0.371391
$262$	−4.00000	−0.247121
$263$	−24.0000	−1.47990	−0.739952	−	0.672660i	$-0.765152\pi$
$263$	−24.0000	−1.47990	−0.739952	+	0.672660i	$0.765152\pi$
$264$	−1.00000	−0.0615457
$265$	−4.00000	−0.245718
$266$	0	0
$267$	−10.0000	−0.611990
$268$	4.00000	0.244339
$269$	−26.0000	−1.58525	−0.792624	−	0.609711i	$-0.791286\pi$
$269$	−26.0000	−1.58525	−0.792624	+	0.609711i	$0.791286\pi$
$270$	−2.00000	−0.121716
$271$	−20.0000	−1.21491	−0.607457	−	0.794353i	$-0.707810\pi$
$271$	−20.0000	−1.21491	−0.607457	+	0.794353i	$0.707810\pi$
$272$	−2.00000	−0.121268
$273$	0	0
$274$	2.00000	0.120824
$275$	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	$0.214662\pi$
$278$	4.00000	0.239904
$279$	0	0
$280$	0	0
$281$	−22.0000	−1.31241	−0.656205	−	0.754583i	$-0.727839\pi$
$281$	−22.0000	−1.31241	−0.656205	+	0.754583i	$0.727839\pi$
$282$	12.0000	0.714590
$283$	−4.00000	−0.237775	−0.118888	−	0.992908i	$-0.537933\pi$
$283$	−4.00000	−0.237775	−0.118888	+	0.992908i	$0.537933\pi$
$284$	−12.0000	−0.712069
$285$	8.00000	0.473879
$286$	−6.00000	−0.354787
$287$	0	0
$288$	1.00000	0.0589256
$289$	−13.0000	−0.764706
$290$	−12.0000	−0.704664
$291$	14.0000	0.820695
$292$	6.00000	0.351123
$293$	−22.0000	−1.28525	−0.642627	−	0.766179i	$-0.722155\pi$
$293$	−22.0000	−1.28525	−0.642627	+	0.766179i	$0.722155\pi$
$294$	0	0
$295$	24.0000	1.39733
$296$	6.00000	0.348743
$297$	−1.00000	−0.0580259
$298$	−10.0000	−0.579284
$299$	24.0000	1.38796
$300$	−1.00000	−0.0577350
$301$	0	0
$302$	4.00000	0.230174
$303$	−14.0000	−0.804279
$304$	−4.00000	−0.229416
$305$	−28.0000	−1.60328
$306$	−2.00000	−0.114332
$307$	−4.00000	−0.228292	−0.114146	−	0.993464i	$-0.536413\pi$
$307$	−4.00000	−0.228292	−0.114146	+	0.993464i	$0.536413\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−4.00000	−0.226819	−0.113410	−	0.993548i	$-0.536177\pi$
$311$	−4.00000	−0.226819	−0.113410	+	0.993548i	$0.536177\pi$
$312$	6.00000	0.339683
$313$	−26.0000	−1.46961	−0.734803	−	0.678280i	$-0.762726\pi$
$313$	−26.0000	−1.46961	−0.734803	+	0.678280i	$0.762726\pi$
$314$	10.0000	0.564333
$315$	0	0
$316$	−4.00000	−0.225018
$317$	18.0000	1.01098	0.505490	−	0.862832i	$-0.331312\pi$
$317$	18.0000	1.01098	0.505490	+	0.862832i	$0.331312\pi$
$318$	2.00000	0.112154
$319$	−6.00000	−0.335936
$320$	−2.00000	−0.111803
$321$	4.00000	0.223258
$322$	0	0
$323$	8.00000	0.445132
$324$	1.00000	0.0555556
$325$	−6.00000	−0.332820
$326$	−20.0000	−1.10770
$327$	−6.00000	−0.331801
$328$	6.00000	0.331295
$329$	0	0
$330$	2.00000	0.110096
$331$	20.0000	1.09930	0.549650	−	0.835395i	$-0.314761\pi$
$331$	20.0000	1.09930	0.549650	+	0.835395i	$0.314761\pi$
$332$	−4.00000	−0.219529
$333$	6.00000	0.328798
$334$	0	0
$335$	−8.00000	−0.437087
$336$	0	0
$337$	18.0000	0.980522	0.490261	−	0.871576i	$-0.336901\pi$
$337$	18.0000	0.980522	0.490261	+	0.871576i	$0.336901\pi$
$338$	23.0000	1.25104
$339$	2.00000	0.108625
$340$	4.00000	0.216930
$341$	0	0
$342$	−4.00000	−0.216295
$343$	0	0
$344$	4.00000	0.215666
$345$	−8.00000	−0.430706
$346$	10.0000	0.537603
$347$	−4.00000	−0.214731	−0.107366	−	0.994220i	$-0.534242\pi$
$347$	−4.00000	−0.214731	−0.107366	+	0.994220i	$0.534242\pi$
$348$	6.00000	0.321634
$349$	6.00000	0.321173	0.160586	−	0.987022i	$-0.448662\pi$
$349$	6.00000	0.321173	0.160586	+	0.987022i	$0.448662\pi$
$350$	0	0
$351$	6.00000	0.320256
$352$	−1.00000	−0.0533002
$353$	−18.0000	−0.958043	−0.479022	−	0.877803i	$-0.659008\pi$
$353$	−18.0000	−0.958043	−0.479022	+	0.877803i	$0.659008\pi$
$354$	−12.0000	−0.637793
$355$	24.0000	1.27379
$356$	−10.0000	−0.529999
$357$	0	0
$358$	20.0000	1.05703
$359$	0	0		−	1.00000i	$-0.5\pi$
$359$	0	0			1.00000i	$0.5\pi$
$360$	−2.00000	−0.105409
$361$	−3.00000	−0.157895
$362$	2.00000	0.105118
$363$	1.00000	0.0524864
$364$	0	0
$365$	−12.0000	−0.628109
$366$	14.0000	0.731792
$367$	16.0000	0.835193	0.417597	−	0.908633i	$-0.362873\pi$
$367$	16.0000	0.835193	0.417597	+	0.908633i	$0.362873\pi$
$368$	4.00000	0.208514
$369$	6.00000	0.312348
$370$	−12.0000	−0.623850
$371$	0	0
$372$	0	0
$373$	−14.0000	−0.724893	−0.362446	−	0.932005i	$-0.618058\pi$
$373$	−14.0000	−0.724893	−0.362446	+	0.932005i	$0.618058\pi$
$374$	2.00000	0.103418
$375$	12.0000	0.619677
$376$	12.0000	0.618853
$377$	36.0000	1.85409
$378$	0	0
$379$	−28.0000	−1.43826	−0.719132	−	0.694874i	$-0.755460\pi$
$379$	−28.0000	−1.43826	−0.719132	+	0.694874i	$0.755460\pi$
$380$	8.00000	0.410391
$381$	12.0000	0.614779
$382$	−12.0000	−0.613973
$383$	4.00000	0.204390	0.102195	−	0.994764i	$-0.467413\pi$
$383$	4.00000	0.204390	0.102195	+	0.994764i	$0.467413\pi$
$384$	1.00000	0.0510310
$385$	0	0
$386$	10.0000	0.508987
$387$	4.00000	0.203331
$388$	14.0000	0.710742
$389$	−30.0000	−1.52106	−0.760530	−	0.649303i	$-0.775061\pi$
$389$	−30.0000	−1.52106	−0.760530	+	0.649303i	$0.775061\pi$
$390$	−12.0000	−0.607644
$391$	−8.00000	−0.404577
$392$	0	0
$393$	−4.00000	−0.201773
$394$	−2.00000	−0.100759
$395$	8.00000	0.402524
$396$	−1.00000	−0.0502519
$397$	−22.0000	−1.10415	−0.552074	−	0.833795i	$-0.686163\pi$
$397$	−22.0000	−1.10415	−0.552074	+	0.833795i	$0.686163\pi$
$398$	16.0000	0.802008
$399$	0	0
$400$	−1.00000	−0.0500000
$401$	−38.0000	−1.89763	−0.948815	−	0.315833i	$-0.897716\pi$
$401$	−38.0000	−1.89763	−0.948815	+	0.315833i	$0.897716\pi$
$402$	4.00000	0.199502
$403$	0	0
$404$	−14.0000	−0.696526
$405$	−2.00000	−0.0993808
$406$	0	0
$407$	−6.00000	−0.297409
$408$	−2.00000	−0.0990148
$409$	14.0000	0.692255	0.346128	−	0.938187i	$-0.387496\pi$
$409$	14.0000	0.692255	0.346128	+	0.938187i	$0.387496\pi$
$410$	−12.0000	−0.592638
$411$	2.00000	0.0986527
$412$	0	0
$413$	0	0
$414$	4.00000	0.196589
$415$	8.00000	0.392705
$416$	6.00000	0.294174
$417$	4.00000	0.195881
$418$	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
$421$	−10.0000	−0.487370	−0.243685	−	0.969854i	$-0.578356\pi$
$421$	−10.0000	−0.487370	−0.243685	+	0.969854i	$0.578356\pi$
$422$	−4.00000	−0.194717
$423$	12.0000	0.583460
$424$	2.00000	0.0971286
$425$	2.00000	0.0970143
$426$	−12.0000	−0.581402
$427$	0	0
$428$	4.00000	0.193347
$429$	−6.00000	−0.289683
$430$	−8.00000	−0.385794
$431$	0	0		−	1.00000i	$-0.5\pi$
$431$	0	0			1.00000i	$0.5\pi$
$432$	1.00000	0.0481125
$433$	−2.00000	−0.0961139	−0.0480569	−	0.998845i	$-0.515303\pi$
$433$	−2.00000	−0.0961139	−0.0480569	+	0.998845i	$0.515303\pi$
$434$	0	0
$435$	−12.0000	−0.575356
$436$	−6.00000	−0.287348
$437$	−16.0000	−0.765384
$438$	6.00000	0.286691
$439$	−4.00000	−0.190910	−0.0954548	−	0.995434i	$-0.530431\pi$
$439$	−4.00000	−0.190910	−0.0954548	+	0.995434i	$0.530431\pi$
$440$	2.00000	0.0953463
$441$	0	0
$442$	−12.0000	−0.570782
$443$	28.0000	1.33032	0.665160	−	0.746701i	$-0.268363\pi$
$443$	28.0000	1.33032	0.665160	+	0.746701i	$0.268363\pi$
$444$	6.00000	0.284747
$445$	20.0000	0.948091
$446$	0	0
$447$	−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
$452$	2.00000	0.0940721
$453$	4.00000	0.187936
$454$	−12.0000	−0.563188
$455$	0	0
$456$	−4.00000	−0.187317
$457$	34.0000	1.59045	0.795226	−	0.606313i	$-0.207352\pi$
$457$	34.0000	1.59045	0.795226	+	0.606313i	$0.207352\pi$
$458$	−14.0000	−0.654177
$459$	−2.00000	−0.0933520
$460$	−8.00000	−0.373002
$461$	−14.0000	−0.652045	−0.326023	−	0.945362i	$-0.605709\pi$
$461$	−14.0000	−0.652045	−0.326023	+	0.945362i	$0.605709\pi$
$462$	0	0
$463$	−24.0000	−1.11537	−0.557687	−	0.830051i	$-0.688311\pi$
$463$	−24.0000	−1.11537	−0.557687	+	0.830051i	$0.688311\pi$
$464$	6.00000	0.278543
$465$	0	0
$466$	10.0000	0.463241
$467$	−12.0000	−0.555294	−0.277647	−	0.960683i	$-0.589555\pi$
$467$	−12.0000	−0.555294	−0.277647	+	0.960683i	$0.589555\pi$
$468$	6.00000	0.277350
$469$	0	0
$470$	−24.0000	−1.10704
$471$	10.0000	0.460776
$472$	−12.0000	−0.552345
$473$	−4.00000	−0.183920
$474$	−4.00000	−0.183726
$475$	4.00000	0.183533
$476$	0	0
$477$	2.00000	0.0915737
$478$	8.00000	0.365911
$479$	0	0		−	1.00000i	$-0.5\pi$
$479$	0	0			1.00000i	$0.5\pi$
$480$	−2.00000	−0.0912871
$481$	36.0000	1.64146
$482$	−10.0000	−0.455488
$483$	0	0
$484$	1.00000	0.0454545
$485$	−28.0000	−1.27141
$486$	1.00000	0.0453609
$487$	−16.0000	−0.725029	−0.362515	−	0.931978i	$-0.618082\pi$
$487$	−16.0000	−0.725029	−0.362515	+	0.931978i	$0.618082\pi$
$488$	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
$494$	−24.0000	−1.07981
$495$	2.00000	0.0898933
$496$	0	0
$497$	0	0
$498$	−4.00000	−0.179244
$499$	−4.00000	−0.179065	−0.0895323	−	0.995984i	$-0.528537\pi$
$499$	−4.00000	−0.179065	−0.0895323	+	0.995984i	$0.528537\pi$
$500$	12.0000	0.536656
$501$	0	0
$502$	4.00000	0.178529
$503$	−32.0000	−1.42681	−0.713405	−	0.700752i	$-0.752848\pi$
$503$	−32.0000	−1.42681	−0.713405	+	0.700752i	$0.752848\pi$
$504$	0	0
$505$	28.0000	1.24598
$506$	−4.00000	−0.177822
$507$	23.0000	1.02147
$508$	12.0000	0.532414
$509$	22.0000	0.975133	0.487566	−	0.873086i	$-0.337885\pi$
$509$	22.0000	0.975133	0.487566	+	0.873086i	$0.337885\pi$
$510$	4.00000	0.177123
$511$	0	0
$512$	1.00000	0.0441942
$513$	−4.00000	−0.176604
$514$	−2.00000	−0.0882162
$515$	0	0
$516$	4.00000	0.176090
$517$	−12.0000	−0.527759
$518$	0	0
$519$	10.0000	0.438951
$520$	−12.0000	−0.526235
$521$	−18.0000	−0.788594	−0.394297	−	0.918983i	$-0.629012\pi$
$521$	−18.0000	−0.788594	−0.394297	+	0.918983i	$0.629012\pi$
$522$	6.00000	0.262613
$523$	−20.0000	−0.874539	−0.437269	−	0.899331i	$-0.644054\pi$
$523$	−20.0000	−0.874539	−0.437269	+	0.899331i	$0.644054\pi$
$524$	−4.00000	−0.174741
$525$	0	0
$526$	−24.0000	−1.04645
$527$	0	0
$528$	−1.00000	−0.0435194
$529$	−7.00000	−0.304348
$530$	−4.00000	−0.173749
$531$	−12.0000	−0.520756
$532$	0	0
$533$	36.0000	1.55933
$534$	−10.0000	−0.432742
$535$	−8.00000	−0.345870
$536$	4.00000	0.172774
$537$	20.0000	0.863064
$538$	−26.0000	−1.12094
$539$	0	0
$540$	−2.00000	−0.0860663
$541$	−38.0000	−1.63375	−0.816874	−	0.576816i	$-0.804295\pi$
$541$	−38.0000	−1.63375	−0.816874	+	0.576816i	$0.804295\pi$
$542$	−20.0000	−0.859074
$543$	2.00000	0.0858282
$544$	−2.00000	−0.0857493
$545$	12.0000	0.514024
$546$	0	0
$547$	−28.0000	−1.19719	−0.598597	−	0.801050i	$-0.704275\pi$
$547$	−28.0000	−1.19719	−0.598597	+	0.801050i	$0.704275\pi$
$548$	2.00000	0.0854358
$549$	14.0000	0.597505
$550$	1.00000	0.0426401
$551$	−24.0000	−1.02243
$552$	4.00000	0.170251
$553$	0	0
$554$	26.0000	1.10463
$555$	−12.0000	−0.509372
$556$	4.00000	0.169638
$557$	30.0000	1.27114	0.635570	−	0.772043i	$-0.280765\pi$
$557$	30.0000	1.27114	0.635570	+	0.772043i	$0.280765\pi$
$558$	0	0
$559$	24.0000	1.01509
$560$	0	0
$561$	2.00000	0.0844401
$562$	−22.0000	−0.928014
$563$	20.0000	0.842900	0.421450	−	0.906852i	$-0.361521\pi$
$563$	20.0000	0.842900	0.421450	+	0.906852i	$0.361521\pi$
$564$	12.0000	0.505291
$565$	−4.00000	−0.168281
$566$	−4.00000	−0.168133
$567$	0	0
$568$	−12.0000	−0.503509
$569$	10.0000	0.419222	0.209611	−	0.977785i	$-0.432780\pi$
$569$	10.0000	0.419222	0.209611	+	0.977785i	$0.432780\pi$
$570$	8.00000	0.335083
$571$	−20.0000	−0.836974	−0.418487	−	0.908223i	$-0.637439\pi$
$571$	−20.0000	−0.836974	−0.418487	+	0.908223i	$0.637439\pi$
$572$	−6.00000	−0.250873
$573$	−12.0000	−0.501307
$574$	0	0
$575$	−4.00000	−0.166812
$576$	1.00000	0.0416667
$577$	46.0000	1.91501	0.957503	−	0.288425i	$-0.0931316\pi$
$577$	46.0000	1.91501	0.957503	+	0.288425i	$0.0931316\pi$
$578$	−13.0000	−0.540729
$579$	10.0000	0.415586
$580$	−12.0000	−0.498273
$581$	0	0
$582$	14.0000	0.580319
$583$	−2.00000	−0.0828315
$584$	6.00000	0.248282
$585$	−12.0000	−0.496139
$586$	−22.0000	−0.908812
$587$	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
$590$	24.0000	0.988064
$591$	−2.00000	−0.0822690
$592$	6.00000	0.246598
$593$	30.0000	1.23195	0.615976	−	0.787765i	$-0.288762\pi$
$593$	30.0000	1.23195	0.615976	+	0.787765i	$0.288762\pi$
$594$	−1.00000	−0.0410305
$595$	0	0
$596$	−10.0000	−0.409616
$597$	16.0000	0.654836
$598$	24.0000	0.981433
$599$	36.0000	1.47092	0.735460	−	0.677568i	$-0.236966\pi$
$599$	36.0000	1.47092	0.735460	+	0.677568i	$0.236966\pi$
$600$	−1.00000	−0.0408248
$601$	−10.0000	−0.407909	−0.203954	−	0.978980i	$-0.565379\pi$
$601$	−10.0000	−0.407909	−0.203954	+	0.978980i	$0.565379\pi$
$602$	0	0
$603$	4.00000	0.162893
$604$	4.00000	0.162758
$605$	−2.00000	−0.0813116
$606$	−14.0000	−0.568711
$607$	−28.0000	−1.13648	−0.568242	−	0.822861i	$-0.692376\pi$
$607$	−28.0000	−1.13648	−0.568242	+	0.822861i	$0.692376\pi$
$608$	−4.00000	−0.162221
$609$	0	0
$610$	−28.0000	−1.13369
$611$	72.0000	2.91281
$612$	−2.00000	−0.0808452
$613$	−6.00000	−0.242338	−0.121169	−	0.992632i	$-0.538664\pi$
$613$	−6.00000	−0.242338	−0.121169	+	0.992632i	$0.538664\pi$
$614$	−4.00000	−0.161427
$615$	−12.0000	−0.483887
$616$	0	0
$617$	−22.0000	−0.885687	−0.442843	−	0.896599i	$-0.646030\pi$
$617$	−22.0000	−0.885687	−0.442843	+	0.896599i	$0.646030\pi$
$618$	0	0
$619$	4.00000	0.160774	0.0803868	−	0.996764i	$-0.474384\pi$
$619$	4.00000	0.160774	0.0803868	+	0.996764i	$0.474384\pi$
$620$	0	0
$621$	4.00000	0.160514
$622$	−4.00000	−0.160385
$623$	0	0
$624$	6.00000	0.240192
$625$	−19.0000	−0.760000
$626$	−26.0000	−1.03917
$627$	4.00000	0.159745
$628$	10.0000	0.399043
$629$	−12.0000	−0.478471
$630$	0	0
$631$	−8.00000	−0.318475	−0.159237	−	0.987240i	$-0.550904\pi$
$631$	−8.00000	−0.318475	−0.159237	+	0.987240i	$0.550904\pi$
$632$	−4.00000	−0.159111
$633$	−4.00000	−0.158986
$634$	18.0000	0.714871
$635$	−24.0000	−0.952411
$636$	2.00000	0.0793052
$637$	0	0
$638$	−6.00000	−0.237542
$639$	−12.0000	−0.474713
$640$	−2.00000	−0.0790569
$641$	42.0000	1.65890	0.829450	−	0.558581i	$-0.188654\pi$
$641$	42.0000	1.65890	0.829450	+	0.558581i	$0.188654\pi$
$642$	4.00000	0.157867
$643$	28.0000	1.10421	0.552106	−	0.833774i	$-0.313824\pi$
$643$	28.0000	1.10421	0.552106	+	0.833774i	$0.313824\pi$
$644$	0	0
$645$	−8.00000	−0.315000
$646$	8.00000	0.314756
$647$	28.0000	1.10079	0.550397	−	0.834903i	$-0.314476\pi$
$647$	28.0000	1.10079	0.550397	+	0.834903i	$0.314476\pi$
$648$	1.00000	0.0392837
$649$	12.0000	0.471041
$650$	−6.00000	−0.235339
$651$	0	0
$652$	−20.0000	−0.783260
$653$	18.0000	0.704394	0.352197	−	0.935926i	$-0.385435\pi$
$653$	18.0000	0.704394	0.352197	+	0.935926i	$0.385435\pi$
$654$	−6.00000	−0.234619
$655$	8.00000	0.312586
$656$	6.00000	0.234261
$657$	6.00000	0.234082
$658$	0	0
$659$	−36.0000	−1.40236	−0.701180	−	0.712984i	$-0.747343\pi$
$659$	−36.0000	−1.40236	−0.701180	+	0.712984i	$0.747343\pi$
$660$	2.00000	0.0778499
$661$	18.0000	0.700119	0.350059	−	0.936727i	$-0.386161\pi$
$661$	18.0000	0.700119	0.350059	+	0.936727i	$0.386161\pi$
$662$	20.0000	0.777322
$663$	−12.0000	−0.466041
$664$	−4.00000	−0.155230
$665$	0	0
$666$	6.00000	0.232495
$667$	24.0000	0.929284
$668$	0	0
$669$	0	0
$670$	−8.00000	−0.309067
$671$	−14.0000	−0.540464
$672$	0	0
$673$	26.0000	1.00223	0.501113	−	0.865382i	$-0.332924\pi$
$673$	26.0000	1.00223	0.501113	+	0.865382i	$0.332924\pi$
$674$	18.0000	0.693334
$675$	−1.00000	−0.0384900
$676$	23.0000	0.884615
$677$	−46.0000	−1.76792	−0.883962	−	0.467559i	$-0.845134\pi$
$677$	−46.0000	−1.76792	−0.883962	+	0.467559i	$0.845134\pi$
$678$	2.00000	0.0768095
$679$	0	0
$680$	4.00000	0.153393
$681$	−12.0000	−0.459841
$682$	0	0
$683$	12.0000	0.459167	0.229584	−	0.973289i	$-0.426264\pi$
$683$	12.0000	0.459167	0.229584	+	0.973289i	$0.426264\pi$
$684$	−4.00000	−0.152944
$685$	−4.00000	−0.152832
$686$	0	0
$687$	−14.0000	−0.534133
$688$	4.00000	0.152499
$689$	12.0000	0.457164
$690$	−8.00000	−0.304555
$691$	12.0000	0.456502	0.228251	−	0.973602i	$-0.426699\pi$
$691$	12.0000	0.456502	0.228251	+	0.973602i	$0.426699\pi$
$692$	10.0000	0.380143
$693$	0	0
$694$	−4.00000	−0.151838
$695$	−8.00000	−0.303457
$696$	6.00000	0.227429
$697$	−12.0000	−0.454532
$698$	6.00000	0.227103
$699$	10.0000	0.378235
$700$	0	0
$701$	30.0000	1.13308	0.566542	−	0.824033i	$-0.308281\pi$
$701$	30.0000	1.13308	0.566542	+	0.824033i	$0.308281\pi$
$702$	6.00000	0.226455
$703$	−24.0000	−0.905177
$704$	−1.00000	−0.0376889
$705$	−24.0000	−0.903892
$706$	−18.0000	−0.677439
$707$	0	0
$708$	−12.0000	−0.450988
$709$	−18.0000	−0.676004	−0.338002	−	0.941145i	$-0.609751\pi$
$709$	−18.0000	−0.676004	−0.338002	+	0.941145i	$0.609751\pi$
$710$	24.0000	0.900704
$711$	−4.00000	−0.150012
$712$	−10.0000	−0.374766
$713$	0	0
$714$	0	0
$715$	12.0000	0.448775
$716$	20.0000	0.747435
$717$	8.00000	0.298765
$718$	0	0
$719$	−12.0000	−0.447524	−0.223762	−	0.974644i	$-0.571834\pi$
$719$	−12.0000	−0.447524	−0.223762	+	0.974644i	$0.571834\pi$
$720$	−2.00000	−0.0745356
$721$	0	0
$722$	−3.00000	−0.111648
$723$	−10.0000	−0.371904
$724$	2.00000	0.0743294
$725$	−6.00000	−0.222834
$726$	1.00000	0.0371135
$727$	0	0		−	1.00000i	$-0.5\pi$
$727$	0	0			1.00000i	$0.5\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	−12.0000	−0.444140
$731$	−8.00000	−0.295891
$732$	14.0000	0.517455
$733$	22.0000	0.812589	0.406294	−	0.913742i	$-0.366821\pi$
$733$	22.0000	0.812589	0.406294	+	0.913742i	$0.366821\pi$
$734$	16.0000	0.590571
$735$	0	0
$736$	4.00000	0.147442
$737$	−4.00000	−0.147342
$738$	6.00000	0.220863
$739$	44.0000	1.61857	0.809283	−	0.587419i	$-0.199856\pi$
$739$	44.0000	1.61857	0.809283	+	0.587419i	$0.199856\pi$
$740$	−12.0000	−0.441129
$741$	−24.0000	−0.881662
$742$	0	0
$743$	−32.0000	−1.17397	−0.586983	−	0.809599i	$-0.699684\pi$
$743$	−32.0000	−1.17397	−0.586983	+	0.809599i	$0.699684\pi$
$744$	0	0
$745$	20.0000	0.732743
$746$	−14.0000	−0.512576
$747$	−4.00000	−0.146352
$748$	2.00000	0.0731272
$749$	0	0
$750$	12.0000	0.438178
$751$	32.0000	1.16770	0.583848	−	0.811863i	$-0.301546\pi$
$751$	32.0000	1.16770	0.583848	+	0.811863i	$0.301546\pi$
$752$	12.0000	0.437595
$753$	4.00000	0.145768
$754$	36.0000	1.31104
$755$	−8.00000	−0.291150
$756$	0	0
$757$	38.0000	1.38113	0.690567	−	0.723269i	$-0.257361\pi$
$757$	38.0000	1.38113	0.690567	+	0.723269i	$0.257361\pi$
$758$	−28.0000	−1.01701
$759$	−4.00000	−0.145191
$760$	8.00000	0.290191
$761$	−42.0000	−1.52250	−0.761249	−	0.648459i	$-0.775414\pi$
$761$	−42.0000	−1.52250	−0.761249	+	0.648459i	$0.775414\pi$
$762$	12.0000	0.434714
$763$	0	0
$764$	−12.0000	−0.434145
$765$	4.00000	0.144620
$766$	4.00000	0.144526
$767$	−72.0000	−2.59977
$768$	1.00000	0.0360844
$769$	−10.0000	−0.360609	−0.180305	−	0.983611i	$-0.557708\pi$
$769$	−10.0000	−0.360609	−0.180305	+	0.983611i	$0.557708\pi$
$770$	0	0
$771$	−2.00000	−0.0720282
$772$	10.0000	0.359908
$773$	−18.0000	−0.647415	−0.323708	−	0.946157i	$-0.604929\pi$
$773$	−18.0000	−0.647415	−0.323708	+	0.946157i	$0.604929\pi$
$774$	4.00000	0.143777
$775$	0	0
$776$	14.0000	0.502571
$777$	0	0
$778$	−30.0000	−1.07555
$779$	−24.0000	−0.859889
$780$	−12.0000	−0.429669
$781$	12.0000	0.429394
$782$	−8.00000	−0.286079
$783$	6.00000	0.214423
$784$	0	0
$785$	−20.0000	−0.713831
$786$	−4.00000	−0.142675
$787$	20.0000	0.712923	0.356462	−	0.934310i	$-0.383983\pi$
$787$	20.0000	0.712923	0.356462	+	0.934310i	$0.383983\pi$
$788$	−2.00000	−0.0712470
$789$	−24.0000	−0.854423
$790$	8.00000	0.284627
$791$	0	0
$792$	−1.00000	−0.0355335
$793$	84.0000	2.98293
$794$	−22.0000	−0.780751
$795$	−4.00000	−0.141865
$796$	16.0000	0.567105
$797$	54.0000	1.91278	0.956389	−	0.292096i	$-0.0943526\pi$
$797$	54.0000	1.91278	0.956389	+	0.292096i	$0.0943526\pi$
$798$	0	0
$799$	−24.0000	−0.849059
$800$	−1.00000	−0.0353553
$801$	−10.0000	−0.353333
$802$	−38.0000	−1.34183
$803$	−6.00000	−0.211735
$804$	4.00000	0.141069
$805$	0	0
$806$	0	0
$807$	−26.0000	−0.915243
$808$	−14.0000	−0.492518
$809$	10.0000	0.351581	0.175791	−	0.984428i	$-0.443752\pi$
$809$	10.0000	0.351581	0.175791	+	0.984428i	$0.443752\pi$
$810$	−2.00000	−0.0702728
$811$	28.0000	0.983213	0.491606	−	0.870817i	$-0.336410\pi$
$811$	28.0000	0.983213	0.491606	+	0.870817i	$0.336410\pi$
$812$	0	0
$813$	−20.0000	−0.701431
$814$	−6.00000	−0.210300
$815$	40.0000	1.40114
$816$	−2.00000	−0.0700140
$817$	−16.0000	−0.559769
$818$	14.0000	0.489499
$819$	0	0
$820$	−12.0000	−0.419058
$821$	−2.00000	−0.0698005	−0.0349002	−	0.999391i	$-0.511111\pi$
$821$	−2.00000	−0.0698005	−0.0349002	+	0.999391i	$0.511111\pi$
$822$	2.00000	0.0697580
$823$	−40.0000	−1.39431	−0.697156	−	0.716919i	$-0.745552\pi$
$823$	−40.0000	−1.39431	−0.697156	+	0.716919i	$0.745552\pi$
$824$	0	0
$825$	1.00000	0.0348155
$826$	0	0
$827$	−52.0000	−1.80822	−0.904109	−	0.427303i	$-0.859464\pi$
$827$	−52.0000	−1.80822	−0.904109	+	0.427303i	$0.859464\pi$
$828$	4.00000	0.139010
$829$	−46.0000	−1.59765	−0.798823	−	0.601566i	$-0.794544\pi$
$829$	−46.0000	−1.59765	−0.798823	+	0.601566i	$0.794544\pi$
$830$	8.00000	0.277684
$831$	26.0000	0.901930
$832$	6.00000	0.208013
$833$	0	0
$834$	4.00000	0.138509
$835$	0	0
$836$	4.00000	0.138343
$837$	0	0
$838$	12.0000	0.414533
$839$	−12.0000	−0.414286	−0.207143	−	0.978311i	$-0.566417\pi$
$839$	−12.0000	−0.414286	−0.207143	+	0.978311i	$0.566417\pi$
$840$	0	0
$841$	7.00000	0.241379
$842$	−10.0000	−0.344623
$843$	−22.0000	−0.757720
$844$	−4.00000	−0.137686
$845$	−46.0000	−1.58245
$846$	12.0000	0.412568
$847$	0	0
$848$	2.00000	0.0686803
$849$	−4.00000	−0.137280
$850$	2.00000	0.0685994
$851$	24.0000	0.822709
$852$	−12.0000	−0.411113
$853$	6.00000	0.205436	0.102718	−	0.994711i	$-0.467246\pi$
$853$	6.00000	0.205436	0.102718	+	0.994711i	$0.467246\pi$
$854$	0	0
$855$	8.00000	0.273594
$856$	4.00000	0.136717
$857$	−26.0000	−0.888143	−0.444072	−	0.895991i	$-0.646466\pi$
$857$	−26.0000	−0.888143	−0.444072	+	0.895991i	$0.646466\pi$
$858$	−6.00000	−0.204837
$859$	−4.00000	−0.136478	−0.0682391	−	0.997669i	$-0.521738\pi$
$859$	−4.00000	−0.136478	−0.0682391	+	0.997669i	$0.521738\pi$
$860$	−8.00000	−0.272798
$861$	0	0
$862$	0	0
$863$	20.0000	0.680808	0.340404	−	0.940279i	$-0.389436\pi$
$863$	20.0000	0.680808	0.340404	+	0.940279i	$0.389436\pi$
$864$	1.00000	0.0340207
$865$	−20.0000	−0.680020
$866$	−2.00000	−0.0679628
$867$	−13.0000	−0.441503
$868$	0	0
$869$	4.00000	0.135691
$870$	−12.0000	−0.406838
$871$	24.0000	0.813209
$872$	−6.00000	−0.203186
$873$	14.0000	0.473828
$874$	−16.0000	−0.541208
$875$	0	0
$876$	6.00000	0.202721
$877$	42.0000	1.41824	0.709120	−	0.705088i	$-0.249093\pi$
$877$	42.0000	1.41824	0.709120	+	0.705088i	$0.249093\pi$
$878$	−4.00000	−0.134993
$879$	−22.0000	−0.742042
$880$	2.00000	0.0674200
$881$	14.0000	0.471672	0.235836	−	0.971793i	$-0.424217\pi$
$881$	14.0000	0.471672	0.235836	+	0.971793i	$0.424217\pi$
$882$	0	0
$883$	4.00000	0.134611	0.0673054	−	0.997732i	$-0.478560\pi$
$883$	4.00000	0.134611	0.0673054	+	0.997732i	$0.478560\pi$
$884$	−12.0000	−0.403604
$885$	24.0000	0.806751
$886$	28.0000	0.940678
$887$	0	0		−	1.00000i	$-0.5\pi$
$887$	0	0			1.00000i	$0.5\pi$
$888$	6.00000	0.201347
$889$	0	0
$890$	20.0000	0.670402
$891$	−1.00000	−0.0335013
$892$	0	0
$893$	−48.0000	−1.60626
$894$	−10.0000	−0.334450
$895$	−40.0000	−1.33705
$896$	0	0
$897$	24.0000	0.801337
$898$	−22.0000	−0.734150
$899$	0	0
$900$	−1.00000	−0.0333333
$901$	−4.00000	−0.133259
$902$	−6.00000	−0.199778
$903$	0	0
$904$	2.00000	0.0665190
$905$	−4.00000	−0.132964
$906$	4.00000	0.132891
$907$	28.0000	0.929725	0.464862	−	0.885383i	$-0.346104\pi$
$907$	28.0000	0.929725	0.464862	+	0.885383i	$0.346104\pi$
$908$	−12.0000	−0.398234
$909$	−14.0000	−0.464351
$910$	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
$913$	4.00000	0.132381
$914$	34.0000	1.12462
$915$	−28.0000	−0.925651
$916$	−14.0000	−0.462573
$917$	0	0
$918$	−2.00000	−0.0660098
$919$	−4.00000	−0.131948	−0.0659739	−	0.997821i	$-0.521015\pi$
$919$	−4.00000	−0.131948	−0.0659739	+	0.997821i	$0.521015\pi$
$920$	−8.00000	−0.263752
$921$	−4.00000	−0.131804
$922$	−14.0000	−0.461065
$923$	−72.0000	−2.36991
$924$	0	0
$925$	−6.00000	−0.197279
$926$	−24.0000	−0.788689
$927$	0	0
$928$	6.00000	0.196960
$929$	−42.0000	−1.37798	−0.688988	−	0.724773i	$-0.741945\pi$
$929$	−42.0000	−1.37798	−0.688988	+	0.724773i	$0.741945\pi$
$930$	0	0
$931$	0	0
$932$	10.0000	0.327561
$933$	−4.00000	−0.130954
$934$	−12.0000	−0.392652
$935$	−4.00000	−0.130814
$936$	6.00000	0.196116
$937$	38.0000	1.24141	0.620703	−	0.784046i	$-0.286847\pi$
$937$	38.0000	1.24141	0.620703	+	0.784046i	$0.286847\pi$
$938$	0	0
$939$	−26.0000	−0.848478
$940$	−24.0000	−0.782794
$941$	42.0000	1.36916	0.684580	−	0.728937i	$-0.259985\pi$
$941$	42.0000	1.36916	0.684580	+	0.728937i	$0.259985\pi$
$942$	10.0000	0.325818
$943$	24.0000	0.781548
$944$	−12.0000	−0.390567
$945$	0	0
$946$	−4.00000	−0.130051
$947$	12.0000	0.389948	0.194974	−	0.980808i	$-0.437538\pi$
$947$	12.0000	0.389948	0.194974	+	0.980808i	$0.437538\pi$
$948$	−4.00000	−0.129914
$949$	36.0000	1.16861
$950$	4.00000	0.129777
$951$	18.0000	0.583690
$952$	0	0
$953$	−6.00000	−0.194359	−0.0971795	−	0.995267i	$-0.530982\pi$
$953$	−6.00000	−0.194359	−0.0971795	+	0.995267i	$0.530982\pi$
$954$	2.00000	0.0647524
$955$	24.0000	0.776622
$956$	8.00000	0.258738
$957$	−6.00000	−0.193952
$958$	0	0
$959$	0	0
$960$	−2.00000	−0.0645497
$961$	−31.0000	−1.00000
$962$	36.0000	1.16069
$963$	4.00000	0.128898
$964$	−10.0000	−0.322078
$965$	−20.0000	−0.643823
$966$	0	0
$967$	−44.0000	−1.41494	−0.707472	−	0.706741i	$-0.750165\pi$
$967$	−44.0000	−1.41494	−0.707472	+	0.706741i	$0.750165\pi$
$968$	1.00000	0.0321412
$969$	8.00000	0.256997
$970$	−28.0000	−0.899026
$971$	−12.0000	−0.385098	−0.192549	−	0.981287i	$-0.561675\pi$
$971$	−12.0000	−0.385098	−0.192549	+	0.981287i	$0.561675\pi$
$972$	1.00000	0.0320750
$973$	0	0
$974$	−16.0000	−0.512673
$975$	−6.00000	−0.192154
$976$	14.0000	0.448129
$977$	26.0000	0.831814	0.415907	−	0.909407i	$-0.363464\pi$
$977$	26.0000	0.831814	0.415907	+	0.909407i	$0.363464\pi$
$978$	−20.0000	−0.639529
$979$	10.0000	0.319601
$980$	0	0
$981$	−6.00000	−0.191565
$982$	−28.0000	−0.893516
$983$	−36.0000	−1.14822	−0.574111	−	0.818778i	$-0.694652\pi$
$983$	−36.0000	−1.14822	−0.574111	+	0.818778i	$0.694652\pi$
$984$	6.00000	0.191273
$985$	4.00000	0.127451
$986$	−12.0000	−0.382158
$987$	0	0
$988$	−24.0000	−0.763542
$989$	16.0000	0.508770
$990$	2.00000	0.0635642
$991$	32.0000	1.01651	0.508257	−	0.861206i	$-0.330290\pi$
$991$	32.0000	1.01651	0.508257	+	0.861206i	$0.330290\pi$
$992$	0	0
$993$	20.0000	0.634681
$994$	0	0
$995$	−32.0000	−1.01447
$996$	−4.00000	−0.126745
$997$	14.0000	0.443384	0.221692	−	0.975117i	$-0.428842\pi$
$997$	14.0000	0.443384	0.221692	+	0.975117i	$0.428842\pi$
$998$	−4.00000	−0.126618
$999$	6.00000	0.189832

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3234.2.a.t.1.1		1
3.2	odd	2		9702.2.a.x.1.1		1
7.6	odd	2		66.2.a.b.1.1	✓	1
21.20	even	2		198.2.a.a.1.1		1
28.27	even	2		528.2.a.j.1.1		1
35.13	even	4		1650.2.c.e.199.1		2
35.27	even	4		1650.2.c.e.199.2		2
35.34	odd	2		1650.2.a.k.1.1		1
56.13	odd	2		2112.2.a.r.1.1		1
56.27	even	2		2112.2.a.e.1.1		1
63.13	odd	6		1782.2.e.e.1189.1		2
63.20	even	6		1782.2.e.v.595.1		2
63.34	odd	6		1782.2.e.e.595.1		2
63.41	even	6		1782.2.e.v.1189.1		2
77.6	even	10		726.2.e.o.487.1		4
77.13	even	10		726.2.e.o.565.1		4
77.20	odd	10		726.2.e.g.565.1		4
77.27	odd	10		726.2.e.g.487.1		4
77.41	even	10		726.2.e.o.493.1		4
77.48	odd	10		726.2.e.g.511.1		4
77.62	even	10		726.2.e.o.511.1		4
77.69	odd	10		726.2.e.g.493.1		4
77.76	even	2		726.2.a.c.1.1		1
84.83	odd	2		1584.2.a.f.1.1		1
105.62	odd	4		4950.2.c.p.199.1		2
105.83	odd	4		4950.2.c.p.199.2		2
105.104	even	2		4950.2.a.bu.1.1		1
168.83	odd	2		6336.2.a.cj.1.1		1
168.125	even	2		6336.2.a.bw.1.1		1
231.230	odd	2		2178.2.a.g.1.1		1
308.307	odd	2		5808.2.a.bc.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
66.2.a.b.1.1	✓	1	7.6	odd	2
198.2.a.a.1.1		1	21.20	even	2
528.2.a.j.1.1		1	28.27	even	2
726.2.a.c.1.1		1	77.76	even	2
726.2.e.g.487.1		4	77.27	odd	10
726.2.e.g.493.1		4	77.69	odd	10
726.2.e.g.511.1		4	77.48	odd	10
726.2.e.g.565.1		4	77.20	odd	10
726.2.e.o.487.1		4	77.6	even	10
726.2.e.o.493.1		4	77.41	even	10
726.2.e.o.511.1		4	77.62	even	10
726.2.e.o.565.1		4	77.13	even	10
1584.2.a.f.1.1		1	84.83	odd	2
1650.2.a.k.1.1		1	35.34	odd	2
1650.2.c.e.199.1		2	35.13	even	4
1650.2.c.e.199.2		2	35.27	even	4
1782.2.e.e.595.1		2	63.34	odd	6
1782.2.e.e.1189.1		2	63.13	odd	6
1782.2.e.v.595.1		2	63.20	even	6
1782.2.e.v.1189.1		2	63.41	even	6
2112.2.a.e.1.1		1	56.27	even	2
2112.2.a.r.1.1		1	56.13	odd	2
2178.2.a.g.1.1		1	231.230	odd	2
3234.2.a.t.1.1		1	1.1	even	1	trivial
4950.2.a.bu.1.1		1	105.104	even	2
4950.2.c.p.199.1		2	105.62	odd	4
4950.2.c.p.199.2		2	105.83	odd	4
5808.2.a.bc.1.1		1	308.307	odd	2
6336.2.a.bw.1.1		1	168.125	even	2
6336.2.a.cj.1.1		1	168.83	odd	2
9702.2.a.x.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 $\F_{q}$

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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