LMFDB - Embedded newform 930.2.a.l.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [930,2,Mod(1,930)] 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("930.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(930, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 930 = 2 \cdot 3 \cdot 5 \cdot 31 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	930.a (trivial)

Newform invariants

comment:select newform

sage:traces = [1,1,-1,1,-1,-1,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

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

Self dual:	yes
Analytic conductor:	$7.42608738798$
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	$\chi$	$=$	930.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$

$=$

\(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} -1.00000 q^{6} +1.00000 q^{8} +1.00000 q^{9} -1.00000 q^{10} -6.00000 q^{11} -1.00000 q^{12} -2.00000 q^{13} +1.00000 q^{15} +1.00000 q^{16} -4.00000 q^{17} +1.00000 q^{18} -1.00000 q^{20} -6.00000 q^{22} +2.00000 q^{23} -1.00000 q^{24} +1.00000 q^{25} -2.00000 q^{26} -1.00000 q^{27} -8.00000 q^{29} +1.00000 q^{30} +1.00000 q^{31} +1.00000 q^{32} +6.00000 q^{33} -4.00000 q^{34} +1.00000 q^{36} -6.00000 q^{37} +2.00000 q^{39} -1.00000 q^{40} -2.00000 q^{41} +4.00000 q^{43} -6.00000 q^{44} -1.00000 q^{45} +2.00000 q^{46} +4.00000 q^{47} -1.00000 q^{48} -7.00000 q^{49} +1.00000 q^{50} +4.00000 q^{51} -2.00000 q^{52} -6.00000 q^{53} -1.00000 q^{54} +6.00000 q^{55} -8.00000 q^{58} +1.00000 q^{60} +4.00000 q^{61} +1.00000 q^{62} +1.00000 q^{64} +2.00000 q^{65} +6.00000 q^{66} -4.00000 q^{67} -4.00000 q^{68} -2.00000 q^{69} -8.00000 q^{71} +1.00000 q^{72} -4.00000 q^{73} -6.00000 q^{74} -1.00000 q^{75} +2.00000 q^{78} -4.00000 q^{79} -1.00000 q^{80} +1.00000 q^{81} -2.00000 q^{82} +4.00000 q^{85} +4.00000 q^{86} +8.00000 q^{87} -6.00000 q^{88} -2.00000 q^{89} -1.00000 q^{90} +2.00000 q^{92} -1.00000 q^{93} +4.00000 q^{94} -1.00000 q^{96} +14.0000 q^{97} -7.00000 q^{98} -6.00000 q^{99} +O(q^{100})\)

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
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	−1.00000	−0.408248
$7$	0	0		−	1.00000i	$-0.5\pi$
$7$	0	0			1.00000i	$0.5\pi$
$8$	1.00000	0.353553
$9$	1.00000	0.333333
$10$	−1.00000	−0.316228
$11$	−6.00000	−1.80907	−0.904534	−	0.426401i	$-0.859781\pi$
$11$	−6.00000	−1.80907	−0.904534	+	0.426401i	$0.859781\pi$
$12$	−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
$15$	1.00000	0.258199
$16$	1.00000	0.250000
$17$	−4.00000	−0.970143	−0.485071	−	0.874475i	$-0.661206\pi$
$17$	−4.00000	−0.970143	−0.485071	+	0.874475i	$0.661206\pi$
$18$	1.00000	0.235702
$19$	0	0		−	1.00000i	$-0.5\pi$
$19$	0	0			1.00000i	$0.5\pi$
$20$	−1.00000	−0.223607
$21$	0	0
$22$	−6.00000	−1.27920
$23$	2.00000	0.417029	0.208514	−	0.978019i	$-0.433137\pi$
$23$	2.00000	0.417029	0.208514	+	0.978019i	$0.433137\pi$
$24$	−1.00000	−0.204124
$25$	1.00000	0.200000
$26$	−2.00000	−0.392232
$27$	−1.00000	−0.192450
$28$	0	0
$29$	−8.00000	−1.48556	−0.742781	−	0.669534i	$-0.766494\pi$
$29$	−8.00000	−1.48556	−0.742781	+	0.669534i	$0.766494\pi$
$30$	1.00000	0.182574
$31$	1.00000	0.179605
$31$	1.00000	0.179605
$32$	1.00000	0.176777
$33$	6.00000	1.04447
$34$	−4.00000	−0.685994
$35$	0	0
$36$	1.00000	0.166667
$37$	−6.00000	−0.986394	−0.493197	−	0.869918i	$-0.664172\pi$
$37$	−6.00000	−0.986394	−0.493197	+	0.869918i	$0.664172\pi$
$38$	0	0
$39$	2.00000	0.320256
$40$	−1.00000	−0.158114
$41$	−2.00000	−0.312348	−0.156174	−	0.987730i	$-0.549916\pi$
$41$	−2.00000	−0.312348	−0.156174	+	0.987730i	$0.549916\pi$
$42$	0	0
$43$	4.00000	0.609994	0.304997	−	0.952353i	$-0.401344\pi$
$43$	4.00000	0.609994	0.304997	+	0.952353i	$0.401344\pi$
$44$	−6.00000	−0.904534
$45$	−1.00000	−0.149071
$46$	2.00000	0.294884
$47$	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
$49$	−7.00000	−1.00000
$50$	1.00000	0.141421
$51$	4.00000	0.560112
$52$	−2.00000	−0.277350
$53$	−6.00000	−0.824163	−0.412082	−	0.911147i	$-0.635198\pi$
$53$	−6.00000	−0.824163	−0.412082	+	0.911147i	$0.635198\pi$
$54$	−1.00000	−0.136083
$55$	6.00000	0.809040
$56$	0	0
$57$	0	0
$58$	−8.00000	−1.05045
$59$	0	0		−	1.00000i	$-0.5\pi$
$59$	0	0			1.00000i	$0.5\pi$
$60$	1.00000	0.129099
$61$	4.00000	0.512148	0.256074	−	0.966657i	$-0.417571\pi$
$61$	4.00000	0.512148	0.256074	+	0.966657i	$0.417571\pi$
$62$	1.00000	0.127000
$63$	0	0
$64$	1.00000	0.125000
$65$	2.00000	0.248069
$66$	6.00000	0.738549
$67$	−4.00000	−0.488678	−0.244339	−	0.969690i	$-0.578571\pi$
$67$	−4.00000	−0.488678	−0.244339	+	0.969690i	$0.578571\pi$
$68$	−4.00000	−0.485071
$69$	−2.00000	−0.240772
$70$	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	$-0.575209\pi$
$73$	−4.00000	−0.468165	−0.234082	+	0.972217i	$0.575209\pi$
$74$	−6.00000	−0.697486
$75$	−1.00000	−0.115470
$76$	0	0
$77$	0	0
$78$	2.00000	0.226455
$79$	−4.00000	−0.450035	−0.225018	−	0.974355i	$-0.572244\pi$
$79$	−4.00000	−0.450035	−0.225018	+	0.974355i	$0.572244\pi$
$80$	−1.00000	−0.111803
$81$	1.00000	0.111111
$82$	−2.00000	−0.220863
$83$	0	0		−	1.00000i	$-0.5\pi$
$83$	0	0			1.00000i	$0.5\pi$
$84$	0	0
$85$	4.00000	0.433861
$86$	4.00000	0.431331
$87$	8.00000	0.857690
$88$	−6.00000	−0.639602
$89$	−2.00000	−0.212000	−0.106000	−	0.994366i	$-0.533804\pi$
$89$	−2.00000	−0.212000	−0.106000	+	0.994366i	$0.533804\pi$
$90$	−1.00000	−0.105409
$91$	0	0
$92$	2.00000	0.208514
$93$	−1.00000	−0.103695
$94$	4.00000	0.412568
$95$	0	0
$96$	−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	$0.248359\pi$
$98$	−7.00000	−0.707107
$99$	−6.00000	−0.603023
$100$	1.00000	0.100000
$101$	2.00000	0.199007	0.0995037	−	0.995037i	$-0.468274\pi$
$101$	2.00000	0.199007	0.0995037	+	0.995037i	$0.468274\pi$
$102$	4.00000	0.396059
$103$	16.0000	1.57653	0.788263	−	0.615338i	$-0.210980\pi$
$103$	16.0000	1.57653	0.788263	+	0.615338i	$0.210980\pi$
$104$	−2.00000	−0.196116
$105$	0	0
$106$	−6.00000	−0.582772
$107$	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	$-0.658969\pi$
$109$	−10.0000	−0.957826	−0.478913	+	0.877862i	$0.658969\pi$
$110$	6.00000	0.572078
$111$	6.00000	0.569495
$112$	0	0
$113$	6.00000	0.564433	0.282216	−	0.959351i	$-0.408930\pi$
$113$	6.00000	0.564433	0.282216	+	0.959351i	$0.408930\pi$
$114$	0	0
$115$	−2.00000	−0.186501
$116$	−8.00000	−0.742781
$117$	−2.00000	−0.184900
$118$	0	0
$119$	0	0
$120$	1.00000	0.0912871
$121$	25.0000	2.27273
$122$	4.00000	0.362143
$123$	2.00000	0.180334
$124$	1.00000	0.0898027
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	14.0000	1.24230	0.621150	−	0.783692i	$-0.286666\pi$
$127$	14.0000	1.24230	0.621150	+	0.783692i	$0.286666\pi$
$128$	1.00000	0.0883883
$129$	−4.00000	−0.352180
$130$	2.00000	0.175412
$131$	0	0		−	1.00000i	$-0.5\pi$
$131$	0	0			1.00000i	$0.5\pi$
$132$	6.00000	0.522233
$133$	0	0
$134$	−4.00000	−0.345547
$135$	1.00000	0.0860663
$136$	−4.00000	−0.342997
$137$	8.00000	0.683486	0.341743	−	0.939793i	$-0.388983\pi$
$137$	8.00000	0.683486	0.341743	+	0.939793i	$0.388983\pi$
$138$	−2.00000	−0.170251
$139$	−2.00000	−0.169638	−0.0848189	−	0.996396i	$-0.527031\pi$
$139$	−2.00000	−0.169638	−0.0848189	+	0.996396i	$0.527031\pi$
$140$	0	0
$141$	−4.00000	−0.336861
$142$	−8.00000	−0.671345
$143$	12.0000	1.00349
$144$	1.00000	0.0833333
$145$	8.00000	0.664364
$146$	−4.00000	−0.331042
$147$	7.00000	0.577350
$148$	−6.00000	−0.493197
$149$	−14.0000	−1.14692	−0.573462	−	0.819232i	$-0.694400\pi$
$149$	−14.0000	−1.14692	−0.573462	+	0.819232i	$0.694400\pi$
$150$	−1.00000	−0.0816497
$151$	12.0000	0.976546	0.488273	−	0.872691i	$-0.337627\pi$
$151$	12.0000	0.976546	0.488273	+	0.872691i	$0.337627\pi$
$152$	0	0
$153$	−4.00000	−0.323381
$154$	0	0
$155$	−1.00000	−0.0803219
$156$	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	$0.525431\pi$
$158$	−4.00000	−0.318223
$159$	6.00000	0.475831
$160$	−1.00000	−0.0790569
$161$	0	0
$162$	1.00000	0.0785674
$163$	−4.00000	−0.313304	−0.156652	−	0.987654i	$-0.550070\pi$
$163$	−4.00000	−0.313304	−0.156652	+	0.987654i	$0.550070\pi$
$164$	−2.00000	−0.156174
$165$	−6.00000	−0.467099
$166$	0	0
$167$	−10.0000	−0.773823	−0.386912	−	0.922117i	$-0.626458\pi$
$167$	−10.0000	−0.773823	−0.386912	+	0.922117i	$0.626458\pi$
$168$	0	0
$169$	−9.00000	−0.692308
$170$	4.00000	0.306786
$171$	0	0
$172$	4.00000	0.304997
$173$	14.0000	1.06440	0.532200	−	0.846619i	$-0.321365\pi$
$173$	14.0000	1.06440	0.532200	+	0.846619i	$0.321365\pi$
$174$	8.00000	0.606478
$175$	0	0
$176$	−6.00000	−0.452267
$177$	0	0
$178$	−2.00000	−0.149906
$179$	−2.00000	−0.149487	−0.0747435	−	0.997203i	$-0.523814\pi$
$179$	−2.00000	−0.149487	−0.0747435	+	0.997203i	$0.523814\pi$
$180$	−1.00000	−0.0745356
$181$	8.00000	0.594635	0.297318	−	0.954779i	$-0.403908\pi$
$181$	8.00000	0.594635	0.297318	+	0.954779i	$0.403908\pi$
$182$	0	0
$183$	−4.00000	−0.295689
$184$	2.00000	0.147442
$185$	6.00000	0.441129
$186$	−1.00000	−0.0733236
$187$	24.0000	1.75505
$188$	4.00000	0.291730
$189$	0	0
$190$	0	0
$191$	−8.00000	−0.578860	−0.289430	−	0.957199i	$-0.593466\pi$
$191$	−8.00000	−0.578860	−0.289430	+	0.957199i	$0.593466\pi$
$192$	−1.00000	−0.0721688
$193$	−18.0000	−1.29567	−0.647834	−	0.761781i	$-0.724325\pi$
$193$	−18.0000	−1.29567	−0.647834	+	0.761781i	$0.724325\pi$
$194$	14.0000	1.00514
$195$	−2.00000	−0.143223
$196$	−7.00000	−0.500000
$197$	−18.0000	−1.28245	−0.641223	−	0.767354i	$-0.721573\pi$
$197$	−18.0000	−1.28245	−0.641223	+	0.767354i	$0.721573\pi$
$198$	−6.00000	−0.426401
$199$	−20.0000	−1.41776	−0.708881	−	0.705328i	$-0.750800\pi$
$199$	−20.0000	−1.41776	−0.708881	+	0.705328i	$0.750800\pi$
$200$	1.00000	0.0707107
$201$	4.00000	0.282138
$202$	2.00000	0.140720
$203$	0	0
$204$	4.00000	0.280056
$205$	2.00000	0.139686
$206$	16.0000	1.11477
$207$	2.00000	0.139010
$208$	−2.00000	−0.138675
$209$	0	0
$210$	0	0
$211$	−4.00000	−0.275371	−0.137686	−	0.990476i	$-0.543966\pi$
$211$	−4.00000	−0.275371	−0.137686	+	0.990476i	$0.543966\pi$
$212$	−6.00000	−0.412082
$213$	8.00000	0.548151
$214$	12.0000	0.820303
$215$	−4.00000	−0.272798
$216$	−1.00000	−0.0680414
$217$	0	0
$218$	−10.0000	−0.677285
$219$	4.00000	0.270295
$220$	6.00000	0.404520
$221$	8.00000	0.538138
$222$	6.00000	0.402694
$223$	18.0000	1.20537	0.602685	−	0.797980i	$-0.294098\pi$
$223$	18.0000	1.20537	0.602685	+	0.797980i	$0.294098\pi$
$224$	0	0
$225$	1.00000	0.0666667
$226$	6.00000	0.399114
$227$	12.0000	0.796468	0.398234	−	0.917284i	$-0.369623\pi$
$227$	12.0000	0.796468	0.398234	+	0.917284i	$0.369623\pi$
$228$	0	0
$229$	16.0000	1.05731	0.528655	−	0.848837i	$-0.322697\pi$
$229$	16.0000	1.05731	0.528655	+	0.848837i	$0.322697\pi$
$230$	−2.00000	−0.131876
$231$	0	0
$232$	−8.00000	−0.525226
$233$	6.00000	0.393073	0.196537	−	0.980497i	$-0.437031\pi$
$233$	6.00000	0.393073	0.196537	+	0.980497i	$0.437031\pi$
$234$	−2.00000	−0.130744
$235$	−4.00000	−0.260931
$236$	0	0
$237$	4.00000	0.259828
$238$	0	0
$239$	−24.0000	−1.55243	−0.776215	−	0.630468i	$-0.782863\pi$
$239$	−24.0000	−1.55243	−0.776215	+	0.630468i	$0.782863\pi$
$240$	1.00000	0.0645497
$241$	18.0000	1.15948	0.579741	−	0.814801i	$-0.303154\pi$
$241$	18.0000	1.15948	0.579741	+	0.814801i	$0.303154\pi$
$242$	25.0000	1.60706
$243$	−1.00000	−0.0641500
$244$	4.00000	0.256074
$245$	7.00000	0.447214
$246$	2.00000	0.127515
$247$	0	0
$248$	1.00000	0.0635001
$249$	0	0
$250$	−1.00000	−0.0632456
$251$	18.0000	1.13615	0.568075	−	0.822977i	$-0.307688\pi$
$251$	18.0000	1.13615	0.568075	+	0.822977i	$0.307688\pi$
$252$	0	0
$253$	−12.0000	−0.754434
$254$	14.0000	0.878438
$255$	−4.00000	−0.250490
$256$	1.00000	0.0625000
$257$	−26.0000	−1.62184	−0.810918	−	0.585160i	$-0.801032\pi$
$257$	−26.0000	−1.62184	−0.810918	+	0.585160i	$0.801032\pi$
$258$	−4.00000	−0.249029
$259$	0	0
$260$	2.00000	0.124035
$261$	−8.00000	−0.495188
$262$	0	0
$263$	18.0000	1.10993	0.554964	−	0.831875i	$-0.312732\pi$
$263$	18.0000	1.10993	0.554964	+	0.831875i	$0.312732\pi$
$264$	6.00000	0.369274
$265$	6.00000	0.368577
$266$	0	0
$267$	2.00000	0.122398
$268$	−4.00000	−0.244339
$269$	−16.0000	−0.975537	−0.487769	−	0.872973i	$-0.662189\pi$
$269$	−16.0000	−0.975537	−0.487769	+	0.872973i	$0.662189\pi$
$270$	1.00000	0.0608581
$271$	16.0000	0.971931	0.485965	−	0.873978i	$-0.338468\pi$
$271$	16.0000	0.971931	0.485965	+	0.873978i	$0.338468\pi$
$272$	−4.00000	−0.242536
$273$	0	0
$274$	8.00000	0.483298
$275$	−6.00000	−0.361814
$276$	−2.00000	−0.120386
$277$	18.0000	1.08152	0.540758	−	0.841178i	$-0.318138\pi$
$277$	18.0000	1.08152	0.540758	+	0.841178i	$0.318138\pi$
$278$	−2.00000	−0.119952
$279$	1.00000	0.0598684
$280$	0	0
$281$	−26.0000	−1.55103	−0.775515	−	0.631329i	$-0.782510\pi$
$281$	−26.0000	−1.55103	−0.775515	+	0.631329i	$0.782510\pi$
$282$	−4.00000	−0.238197
$283$	−12.0000	−0.713326	−0.356663	−	0.934233i	$-0.616086\pi$
$283$	−12.0000	−0.713326	−0.356663	+	0.934233i	$0.616086\pi$
$284$	−8.00000	−0.474713
$285$	0	0
$286$	12.0000	0.709575
$287$	0	0
$288$	1.00000	0.0589256
$289$	−1.00000	−0.0588235
$290$	8.00000	0.469776
$291$	−14.0000	−0.820695
$292$	−4.00000	−0.234082
$293$	−30.0000	−1.75262	−0.876309	−	0.481749i	$-0.840002\pi$
$293$	−30.0000	−1.75262	−0.876309	+	0.481749i	$0.840002\pi$
$294$	7.00000	0.408248
$295$	0	0
$296$	−6.00000	−0.348743
$297$	6.00000	0.348155
$298$	−14.0000	−0.810998
$299$	−4.00000	−0.231326
$300$	−1.00000	−0.0577350
$301$	0	0
$302$	12.0000	0.690522
$303$	−2.00000	−0.114897
$304$	0	0
$305$	−4.00000	−0.229039
$306$	−4.00000	−0.228665
$307$	4.00000	0.228292	0.114146	−	0.993464i	$-0.463587\pi$
$307$	4.00000	0.228292	0.114146	+	0.993464i	$0.463587\pi$
$308$	0	0
$309$	−16.0000	−0.910208
$310$	−1.00000	−0.0567962
$311$	−24.0000	−1.36092	−0.680458	−	0.732787i	$-0.738219\pi$
$311$	−24.0000	−1.36092	−0.680458	+	0.732787i	$0.738219\pi$
$312$	2.00000	0.113228
$313$	−12.0000	−0.678280	−0.339140	−	0.940736i	$-0.610136\pi$
$313$	−12.0000	−0.678280	−0.339140	+	0.940736i	$0.610136\pi$
$314$	−2.00000	−0.112867
$315$	0	0
$316$	−4.00000	−0.225018
$317$	−26.0000	−1.46031	−0.730153	−	0.683284i	$-0.760551\pi$
$317$	−26.0000	−1.46031	−0.730153	+	0.683284i	$0.760551\pi$
$318$	6.00000	0.336463
$319$	48.0000	2.68748
$320$	−1.00000	−0.0559017
$321$	−12.0000	−0.669775
$322$	0	0
$323$	0	0
$324$	1.00000	0.0555556
$325$	−2.00000	−0.110940
$326$	−4.00000	−0.221540
$327$	10.0000	0.553001
$328$	−2.00000	−0.110432
$329$	0	0
$330$	−6.00000	−0.330289
$331$	−18.0000	−0.989369	−0.494685	−	0.869072i	$-0.664716\pi$
$331$	−18.0000	−0.989369	−0.494685	+	0.869072i	$0.664716\pi$
$332$	0	0
$333$	−6.00000	−0.328798
$334$	−10.0000	−0.547176
$335$	4.00000	0.218543
$336$	0	0
$337$	−8.00000	−0.435788	−0.217894	−	0.975972i	$-0.569919\pi$
$337$	−8.00000	−0.435788	−0.217894	+	0.975972i	$0.569919\pi$
$338$	−9.00000	−0.489535
$339$	−6.00000	−0.325875
$340$	4.00000	0.216930
$341$	−6.00000	−0.324918
$342$	0	0
$343$	0	0
$344$	4.00000	0.215666
$345$	2.00000	0.107676
$346$	14.0000	0.752645
$347$	16.0000	0.858925	0.429463	−	0.903085i	$-0.358703\pi$
$347$	16.0000	0.858925	0.429463	+	0.903085i	$0.358703\pi$
$348$	8.00000	0.428845
$349$	−30.0000	−1.60586	−0.802932	−	0.596071i	$-0.796728\pi$
$349$	−30.0000	−1.60586	−0.802932	+	0.596071i	$0.796728\pi$
$350$	0	0
$351$	2.00000	0.106752
$352$	−6.00000	−0.319801
$353$	0	0		−	1.00000i	$-0.5\pi$
$353$	0	0			1.00000i	$0.5\pi$
$354$	0	0
$355$	8.00000	0.424596
$356$	−2.00000	−0.106000
$357$	0	0
$358$	−2.00000	−0.105703
$359$	24.0000	1.26667	0.633336	−	0.773877i	$-0.281685\pi$
$359$	24.0000	1.26667	0.633336	+	0.773877i	$0.281685\pi$
$360$	−1.00000	−0.0527046
$361$	−19.0000	−1.00000
$362$	8.00000	0.420471
$363$	−25.0000	−1.31216
$364$	0	0
$365$	4.00000	0.209370
$366$	−4.00000	−0.209083
$367$	−10.0000	−0.521996	−0.260998	−	0.965339i	$-0.584052\pi$
$367$	−10.0000	−0.521996	−0.260998	+	0.965339i	$0.584052\pi$
$368$	2.00000	0.104257
$369$	−2.00000	−0.104116
$370$	6.00000	0.311925
$371$	0	0
$372$	−1.00000	−0.0518476
$373$	18.0000	0.932005	0.466002	−	0.884783i	$-0.345694\pi$
$373$	18.0000	0.932005	0.466002	+	0.884783i	$0.345694\pi$
$374$	24.0000	1.24101
$375$	1.00000	0.0516398
$376$	4.00000	0.206284
$377$	16.0000	0.824042
$378$	0	0
$379$	−24.0000	−1.23280	−0.616399	−	0.787434i	$-0.711409\pi$
$379$	−24.0000	−1.23280	−0.616399	+	0.787434i	$0.711409\pi$
$380$	0	0
$381$	−14.0000	−0.717242
$382$	−8.00000	−0.409316
$383$	6.00000	0.306586	0.153293	−	0.988181i	$-0.451012\pi$
$383$	6.00000	0.306586	0.153293	+	0.988181i	$0.451012\pi$
$384$	−1.00000	−0.0510310
$385$	0	0
$386$	−18.0000	−0.916176
$387$	4.00000	0.203331
$388$	14.0000	0.710742
$389$	−20.0000	−1.01404	−0.507020	−	0.861934i	$-0.669253\pi$
$389$	−20.0000	−1.01404	−0.507020	+	0.861934i	$0.669253\pi$
$390$	−2.00000	−0.101274
$391$	−8.00000	−0.404577
$392$	−7.00000	−0.353553
$393$	0	0
$394$	−18.0000	−0.906827
$395$	4.00000	0.201262
$396$	−6.00000	−0.301511
$397$	38.0000	1.90717	0.953583	−	0.301131i	$-0.0973643\pi$
$397$	38.0000	1.90717	0.953583	+	0.301131i	$0.0973643\pi$
$398$	−20.0000	−1.00251
$399$	0	0
$400$	1.00000	0.0500000
$401$	−26.0000	−1.29838	−0.649189	−	0.760627i	$-0.724892\pi$
$401$	−26.0000	−1.29838	−0.649189	+	0.760627i	$0.724892\pi$
$402$	4.00000	0.199502
$403$	−2.00000	−0.0996271
$404$	2.00000	0.0995037
$405$	−1.00000	−0.0496904
$406$	0	0
$407$	36.0000	1.78445
$408$	4.00000	0.198030
$409$	14.0000	0.692255	0.346128	−	0.938187i	$-0.387496\pi$
$409$	14.0000	0.692255	0.346128	+	0.938187i	$0.387496\pi$
$410$	2.00000	0.0987730
$411$	−8.00000	−0.394611
$412$	16.0000	0.788263
$413$	0	0
$414$	2.00000	0.0982946
$415$	0	0
$416$	−2.00000	−0.0980581
$417$	2.00000	0.0979404
$418$	0	0
$419$	0	0		−	1.00000i	$-0.5\pi$
$419$	0	0			1.00000i	$0.5\pi$
$420$	0	0
$421$	26.0000	1.26716	0.633581	−	0.773676i	$-0.281584\pi$
$421$	26.0000	1.26716	0.633581	+	0.773676i	$0.281584\pi$
$422$	−4.00000	−0.194717
$423$	4.00000	0.194487
$424$	−6.00000	−0.291386
$425$	−4.00000	−0.194029
$426$	8.00000	0.387601
$427$	0	0
$428$	12.0000	0.580042
$429$	−12.0000	−0.579365
$430$	−4.00000	−0.192897
$431$	40.0000	1.92673	0.963366	−	0.268190i	$-0.0864254\pi$
$431$	40.0000	1.92673	0.963366	+	0.268190i	$0.0864254\pi$
$432$	−1.00000	−0.0481125
$433$	−28.0000	−1.34559	−0.672797	−	0.739827i	$-0.734907\pi$
$433$	−28.0000	−1.34559	−0.672797	+	0.739827i	$0.734907\pi$
$434$	0	0
$435$	−8.00000	−0.383571
$436$	−10.0000	−0.478913
$437$	0	0
$438$	4.00000	0.191127
$439$	0	0		−	1.00000i	$-0.5\pi$
$439$	0	0			1.00000i	$0.5\pi$
$440$	6.00000	0.286039
$441$	−7.00000	−0.333333
$442$	8.00000	0.380521
$443$	12.0000	0.570137	0.285069	−	0.958507i	$-0.407984\pi$
$443$	12.0000	0.570137	0.285069	+	0.958507i	$0.407984\pi$
$444$	6.00000	0.284747
$445$	2.00000	0.0948091
$446$	18.0000	0.852325
$447$	14.0000	0.662177
$448$	0	0
$449$	14.0000	0.660701	0.330350	−	0.943858i	$-0.392833\pi$
$449$	14.0000	0.660701	0.330350	+	0.943858i	$0.392833\pi$
$450$	1.00000	0.0471405
$451$	12.0000	0.565058
$452$	6.00000	0.282216
$453$	−12.0000	−0.563809
$454$	12.0000	0.563188
$455$	0	0
$456$	0	0
$457$	−8.00000	−0.374224	−0.187112	−	0.982339i	$-0.559913\pi$
$457$	−8.00000	−0.374224	−0.187112	+	0.982339i	$0.559913\pi$
$458$	16.0000	0.747631
$459$	4.00000	0.186704
$460$	−2.00000	−0.0932505
$461$	0	0		−	1.00000i	$-0.5\pi$
$461$	0	0			1.00000i	$0.5\pi$
$462$	0	0
$463$	−14.0000	−0.650635	−0.325318	−	0.945605i	$-0.605471\pi$
$463$	−14.0000	−0.650635	−0.325318	+	0.945605i	$0.605471\pi$
$464$	−8.00000	−0.371391
$465$	1.00000	0.0463739
$466$	6.00000	0.277945
$467$	−36.0000	−1.66588	−0.832941	−	0.553362i	$-0.813345\pi$
$467$	−36.0000	−1.66588	−0.832941	+	0.553362i	$0.813345\pi$
$468$	−2.00000	−0.0924500
$469$	0	0
$470$	−4.00000	−0.184506
$471$	2.00000	0.0921551
$472$	0	0
$473$	−24.0000	−1.10352
$474$	4.00000	0.183726
$475$	0	0
$476$	0	0
$477$	−6.00000	−0.274721
$478$	−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	$0.315277\pi$
$480$	1.00000	0.0456435
$481$	12.0000	0.547153
$482$	18.0000	0.819878
$483$	0	0
$484$	25.0000	1.13636
$485$	−14.0000	−0.635707
$486$	−1.00000	−0.0453609
$487$	−34.0000	−1.54069	−0.770344	−	0.637629i	$-0.779915\pi$
$487$	−34.0000	−1.54069	−0.770344	+	0.637629i	$0.779915\pi$
$488$	4.00000	0.181071
$489$	4.00000	0.180886
$490$	7.00000	0.316228
$491$	−34.0000	−1.53440	−0.767199	−	0.641409i	$-0.778350\pi$
$491$	−34.0000	−1.53440	−0.767199	+	0.641409i	$0.778350\pi$
$492$	2.00000	0.0901670
$493$	32.0000	1.44121
$494$	0	0
$495$	6.00000	0.269680
$496$	1.00000	0.0449013
$497$	0	0
$498$	0	0
$499$	−38.0000	−1.70111	−0.850557	−	0.525883i	$-0.823735\pi$
$499$	−38.0000	−1.70111	−0.850557	+	0.525883i	$0.823735\pi$
$500$	−1.00000	−0.0447214
$501$	10.0000	0.446767
$502$	18.0000	0.803379
$503$	−16.0000	−0.713405	−0.356702	−	0.934218i	$-0.616099\pi$
$503$	−16.0000	−0.713405	−0.356702	+	0.934218i	$0.616099\pi$
$504$	0	0
$505$	−2.00000	−0.0889988
$506$	−12.0000	−0.533465
$507$	9.00000	0.399704
$508$	14.0000	0.621150
$509$	−36.0000	−1.59567	−0.797836	−	0.602875i	$-0.794022\pi$
$509$	−36.0000	−1.59567	−0.797836	+	0.602875i	$0.794022\pi$
$510$	−4.00000	−0.177123
$511$	0	0
$512$	1.00000	0.0441942
$513$	0	0
$514$	−26.0000	−1.14681
$515$	−16.0000	−0.705044
$516$	−4.00000	−0.176090
$517$	−24.0000	−1.05552
$518$	0	0
$519$	−14.0000	−0.614532
$520$	2.00000	0.0877058
$521$	6.00000	0.262865	0.131432	−	0.991325i	$-0.458042\pi$
$521$	6.00000	0.262865	0.131432	+	0.991325i	$0.458042\pi$
$522$	−8.00000	−0.350150
$523$	20.0000	0.874539	0.437269	−	0.899331i	$-0.355946\pi$
$523$	20.0000	0.874539	0.437269	+	0.899331i	$0.355946\pi$
$524$	0	0
$525$	0	0
$526$	18.0000	0.784837
$527$	−4.00000	−0.174243
$528$	6.00000	0.261116
$529$	−19.0000	−0.826087
$530$	6.00000	0.260623
$531$	0	0
$532$	0	0
$533$	4.00000	0.173259
$534$	2.00000	0.0865485
$535$	−12.0000	−0.518805
$536$	−4.00000	−0.172774
$537$	2.00000	0.0863064
$538$	−16.0000	−0.689809
$539$	42.0000	1.80907
$540$	1.00000	0.0430331
$541$	−2.00000	−0.0859867	−0.0429934	−	0.999075i	$-0.513689\pi$
$541$	−2.00000	−0.0859867	−0.0429934	+	0.999075i	$0.513689\pi$
$542$	16.0000	0.687259
$543$	−8.00000	−0.343313
$544$	−4.00000	−0.171499
$545$	10.0000	0.428353
$546$	0	0
$547$	−44.0000	−1.88130	−0.940652	−	0.339372i	$-0.889785\pi$
$547$	−44.0000	−1.88130	−0.940652	+	0.339372i	$0.889785\pi$
$548$	8.00000	0.341743
$549$	4.00000	0.170716
$550$	−6.00000	−0.255841
$551$	0	0
$552$	−2.00000	−0.0851257
$553$	0	0
$554$	18.0000	0.764747
$555$	−6.00000	−0.254686
$556$	−2.00000	−0.0848189
$557$	38.0000	1.61011	0.805056	−	0.593199i	$-0.202135\pi$
$557$	38.0000	1.61011	0.805056	+	0.593199i	$0.202135\pi$
$558$	1.00000	0.0423334
$559$	−8.00000	−0.338364
$560$	0	0
$561$	−24.0000	−1.01328
$562$	−26.0000	−1.09674
$563$	44.0000	1.85438	0.927189	−	0.374593i	$-0.122217\pi$
$563$	44.0000	1.85438	0.927189	+	0.374593i	$0.122217\pi$
$564$	−4.00000	−0.168430
$565$	−6.00000	−0.252422
$566$	−12.0000	−0.504398
$567$	0	0
$568$	−8.00000	−0.335673
$569$	−2.00000	−0.0838444	−0.0419222	−	0.999121i	$-0.513348\pi$
$569$	−2.00000	−0.0838444	−0.0419222	+	0.999121i	$0.513348\pi$
$570$	0	0
$571$	−38.0000	−1.59025	−0.795125	−	0.606445i	$-0.792595\pi$
$571$	−38.0000	−1.59025	−0.795125	+	0.606445i	$0.792595\pi$
$572$	12.0000	0.501745
$573$	8.00000	0.334205
$574$	0	0
$575$	2.00000	0.0834058
$576$	1.00000	0.0416667
$577$	−38.0000	−1.58196	−0.790980	−	0.611842i	$-0.790429\pi$
$577$	−38.0000	−1.58196	−0.790980	+	0.611842i	$0.790429\pi$
$578$	−1.00000	−0.0415945
$579$	18.0000	0.748054
$580$	8.00000	0.332182
$581$	0	0
$582$	−14.0000	−0.580319
$583$	36.0000	1.49097
$584$	−4.00000	−0.165521
$585$	2.00000	0.0826898
$586$	−30.0000	−1.23929
$587$	32.0000	1.32078	0.660391	−	0.750922i	$-0.270391\pi$
$587$	32.0000	1.32078	0.660391	+	0.750922i	$0.270391\pi$
$588$	7.00000	0.288675
$589$	0	0
$590$	0	0
$591$	18.0000	0.740421
$592$	−6.00000	−0.246598
$593$	34.0000	1.39621	0.698106	−	0.715994i	$-0.254026\pi$
$593$	34.0000	1.39621	0.698106	+	0.715994i	$0.254026\pi$
$594$	6.00000	0.246183
$595$	0	0
$596$	−14.0000	−0.573462
$597$	20.0000	0.818546
$598$	−4.00000	−0.163572
$599$	48.0000	1.96123	0.980613	−	0.195952i	$-0.0627798\pi$
$599$	48.0000	1.96123	0.980613	+	0.195952i	$0.0627798\pi$
$600$	−1.00000	−0.0408248
$601$	30.0000	1.22373	0.611863	−	0.790964i	$-0.290420\pi$
$601$	30.0000	1.22373	0.611863	+	0.790964i	$0.290420\pi$
$602$	0	0
$603$	−4.00000	−0.162893
$604$	12.0000	0.488273
$605$	−25.0000	−1.01639
$606$	−2.00000	−0.0812444
$607$	−28.0000	−1.13648	−0.568242	−	0.822861i	$-0.692376\pi$
$607$	−28.0000	−1.13648	−0.568242	+	0.822861i	$0.692376\pi$
$608$	0	0
$609$	0	0
$610$	−4.00000	−0.161955
$611$	−8.00000	−0.323645
$612$	−4.00000	−0.161690
$613$	−22.0000	−0.888572	−0.444286	−	0.895885i	$-0.646543\pi$
$613$	−22.0000	−0.888572	−0.444286	+	0.895885i	$0.646543\pi$
$614$	4.00000	0.161427
$615$	−2.00000	−0.0806478
$616$	0	0
$617$	−6.00000	−0.241551	−0.120775	−	0.992680i	$-0.538538\pi$
$617$	−6.00000	−0.241551	−0.120775	+	0.992680i	$0.538538\pi$
$618$	−16.0000	−0.643614
$619$	−26.0000	−1.04503	−0.522514	−	0.852631i	$-0.675006\pi$
$619$	−26.0000	−1.04503	−0.522514	+	0.852631i	$0.675006\pi$
$620$	−1.00000	−0.0401610
$621$	−2.00000	−0.0802572
$622$	−24.0000	−0.962312
$623$	0	0
$624$	2.00000	0.0800641
$625$	1.00000	0.0400000
$626$	−12.0000	−0.479616
$627$	0	0
$628$	−2.00000	−0.0798087
$629$	24.0000	0.956943
$630$	0	0
$631$	8.00000	0.318475	0.159237	−	0.987240i	$-0.449096\pi$
$631$	8.00000	0.318475	0.159237	+	0.987240i	$0.449096\pi$
$632$	−4.00000	−0.159111
$633$	4.00000	0.158986
$634$	−26.0000	−1.03259
$635$	−14.0000	−0.555573
$636$	6.00000	0.237915
$637$	14.0000	0.554700
$638$	48.0000	1.90034
$639$	−8.00000	−0.316475
$640$	−1.00000	−0.0395285
$641$	−22.0000	−0.868948	−0.434474	−	0.900684i	$-0.643066\pi$
$641$	−22.0000	−0.868948	−0.434474	+	0.900684i	$0.643066\pi$
$642$	−12.0000	−0.473602
$643$	−44.0000	−1.73519	−0.867595	−	0.497271i	$-0.834335\pi$
$643$	−44.0000	−1.73519	−0.867595	+	0.497271i	$0.834335\pi$
$644$	0	0
$645$	4.00000	0.157500
$646$	0	0
$647$	−18.0000	−0.707653	−0.353827	−	0.935311i	$-0.615120\pi$
$647$	−18.0000	−0.707653	−0.353827	+	0.935311i	$0.615120\pi$
$648$	1.00000	0.0392837
$649$	0	0
$650$	−2.00000	−0.0784465
$651$	0	0
$652$	−4.00000	−0.156652
$653$	34.0000	1.33052	0.665261	−	0.746611i	$-0.268320\pi$
$653$	34.0000	1.33052	0.665261	+	0.746611i	$0.268320\pi$
$654$	10.0000	0.391031
$655$	0	0
$656$	−2.00000	−0.0780869
$657$	−4.00000	−0.156055
$658$	0	0
$659$	−16.0000	−0.623272	−0.311636	−	0.950202i	$-0.600877\pi$
$659$	−16.0000	−0.623272	−0.311636	+	0.950202i	$0.600877\pi$
$660$	−6.00000	−0.233550
$661$	−26.0000	−1.01128	−0.505641	−	0.862744i	$-0.668744\pi$
$661$	−26.0000	−1.01128	−0.505641	+	0.862744i	$0.668744\pi$
$662$	−18.0000	−0.699590
$663$	−8.00000	−0.310694
$664$	0	0
$665$	0	0
$666$	−6.00000	−0.232495
$667$	−16.0000	−0.619522
$668$	−10.0000	−0.386912
$669$	−18.0000	−0.695920
$670$	4.00000	0.154533
$671$	−24.0000	−0.926510
$672$	0	0
$673$	32.0000	1.23351	0.616755	−	0.787155i	$-0.288447\pi$
$673$	32.0000	1.23351	0.616755	+	0.787155i	$0.288447\pi$
$674$	−8.00000	−0.308148
$675$	−1.00000	−0.0384900
$676$	−9.00000	−0.346154
$677$	6.00000	0.230599	0.115299	−	0.993331i	$-0.463217\pi$
$677$	6.00000	0.230599	0.115299	+	0.993331i	$0.463217\pi$
$678$	−6.00000	−0.230429
$679$	0	0
$680$	4.00000	0.153393
$681$	−12.0000	−0.459841
$682$	−6.00000	−0.229752
$683$	−4.00000	−0.153056	−0.0765279	−	0.997067i	$-0.524383\pi$
$683$	−4.00000	−0.153056	−0.0765279	+	0.997067i	$0.524383\pi$
$684$	0	0
$685$	−8.00000	−0.305664
$686$	0	0
$687$	−16.0000	−0.610438
$688$	4.00000	0.152499
$689$	12.0000	0.457164
$690$	2.00000	0.0761387
$691$	44.0000	1.67384	0.836919	−	0.547326i	$-0.184354\pi$
$691$	44.0000	1.67384	0.836919	+	0.547326i	$0.184354\pi$
$692$	14.0000	0.532200
$693$	0	0
$694$	16.0000	0.607352
$695$	2.00000	0.0758643
$696$	8.00000	0.303239
$697$	8.00000	0.303022
$698$	−30.0000	−1.13552
$699$	−6.00000	−0.226941
$700$	0	0
$701$	30.0000	1.13308	0.566542	−	0.824033i	$-0.308281\pi$
$701$	30.0000	1.13308	0.566542	+	0.824033i	$0.308281\pi$
$702$	2.00000	0.0754851
$703$	0	0
$704$	−6.00000	−0.226134
$705$	4.00000	0.150649
$706$	0	0
$707$	0	0
$708$	0	0
$709$	−4.00000	−0.150223	−0.0751116	−	0.997175i	$-0.523931\pi$
$709$	−4.00000	−0.150223	−0.0751116	+	0.997175i	$0.523931\pi$
$710$	8.00000	0.300235
$711$	−4.00000	−0.150012
$712$	−2.00000	−0.0749532
$713$	2.00000	0.0749006
$714$	0	0
$715$	−12.0000	−0.448775
$716$	−2.00000	−0.0747435
$717$	24.0000	0.896296
$718$	24.0000	0.895672
$719$	20.0000	0.745874	0.372937	−	0.927857i	$-0.378351\pi$
$719$	20.0000	0.745874	0.372937	+	0.927857i	$0.378351\pi$
$720$	−1.00000	−0.0372678
$721$	0	0
$722$	−19.0000	−0.707107
$723$	−18.0000	−0.669427
$724$	8.00000	0.297318
$725$	−8.00000	−0.297113
$726$	−25.0000	−0.927837
$727$	−12.0000	−0.445055	−0.222528	−	0.974926i	$-0.571431\pi$
$727$	−12.0000	−0.445055	−0.222528	+	0.974926i	$0.571431\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	4.00000	0.148047
$731$	−16.0000	−0.591781
$732$	−4.00000	−0.147844
$733$	6.00000	0.221615	0.110808	−	0.993842i	$-0.464656\pi$
$733$	6.00000	0.221615	0.110808	+	0.993842i	$0.464656\pi$
$734$	−10.0000	−0.369107
$735$	−7.00000	−0.258199
$736$	2.00000	0.0737210
$737$	24.0000	0.884051
$738$	−2.00000	−0.0736210
$739$	−2.00000	−0.0735712	−0.0367856	−	0.999323i	$-0.511712\pi$
$739$	−2.00000	−0.0735712	−0.0367856	+	0.999323i	$0.511712\pi$
$740$	6.00000	0.220564
$741$	0	0
$742$	0	0
$743$	−22.0000	−0.807102	−0.403551	−	0.914957i	$-0.632224\pi$
$743$	−22.0000	−0.807102	−0.403551	+	0.914957i	$0.632224\pi$
$744$	−1.00000	−0.0366618
$745$	14.0000	0.512920
$746$	18.0000	0.659027
$747$	0	0
$748$	24.0000	0.877527
$749$	0	0
$750$	1.00000	0.0365148
$751$	−8.00000	−0.291924	−0.145962	−	0.989290i	$-0.546628\pi$
$751$	−8.00000	−0.291924	−0.145962	+	0.989290i	$0.546628\pi$
$752$	4.00000	0.145865
$753$	−18.0000	−0.655956
$754$	16.0000	0.582686
$755$	−12.0000	−0.436725
$756$	0	0
$757$	2.00000	0.0726912	0.0363456	−	0.999339i	$-0.488428\pi$
$757$	2.00000	0.0726912	0.0363456	+	0.999339i	$0.488428\pi$
$758$	−24.0000	−0.871719
$759$	12.0000	0.435572
$760$	0	0
$761$	−18.0000	−0.652499	−0.326250	−	0.945284i	$-0.605785\pi$
$761$	−18.0000	−0.652499	−0.326250	+	0.945284i	$0.605785\pi$
$762$	−14.0000	−0.507166
$763$	0	0
$764$	−8.00000	−0.289430
$765$	4.00000	0.144620
$766$	6.00000	0.216789
$767$	0	0
$768$	−1.00000	−0.0360844
$769$	30.0000	1.08183	0.540914	−	0.841078i	$-0.318079\pi$
$769$	30.0000	1.08183	0.540914	+	0.841078i	$0.318079\pi$
$770$	0	0
$771$	26.0000	0.936367
$772$	−18.0000	−0.647834
$773$	26.0000	0.935155	0.467578	−	0.883952i	$-0.345127\pi$
$773$	26.0000	0.935155	0.467578	+	0.883952i	$0.345127\pi$
$774$	4.00000	0.143777
$775$	1.00000	0.0359211
$776$	14.0000	0.502571
$777$	0	0
$778$	−20.0000	−0.717035
$779$	0	0
$780$	−2.00000	−0.0716115
$781$	48.0000	1.71758
$782$	−8.00000	−0.286079
$783$	8.00000	0.285897
$784$	−7.00000	−0.250000
$785$	2.00000	0.0713831
$786$	0	0
$787$	−8.00000	−0.285169	−0.142585	−	0.989783i	$-0.545541\pi$
$787$	−8.00000	−0.285169	−0.142585	+	0.989783i	$0.545541\pi$
$788$	−18.0000	−0.641223
$789$	−18.0000	−0.640817
$790$	4.00000	0.142314
$791$	0	0
$792$	−6.00000	−0.213201
$793$	−8.00000	−0.284088
$794$	38.0000	1.34857
$795$	−6.00000	−0.212798
$796$	−20.0000	−0.708881
$797$	2.00000	0.0708436	0.0354218	−	0.999372i	$-0.488723\pi$
$797$	2.00000	0.0708436	0.0354218	+	0.999372i	$0.488723\pi$
$798$	0	0
$799$	−16.0000	−0.566039
$800$	1.00000	0.0353553
$801$	−2.00000	−0.0706665
$802$	−26.0000	−0.918092
$803$	24.0000	0.846942
$804$	4.00000	0.141069
$805$	0	0
$806$	−2.00000	−0.0704470
$807$	16.0000	0.563227
$808$	2.00000	0.0703598
$809$	−46.0000	−1.61727	−0.808637	−	0.588308i	$-0.799794\pi$
$809$	−46.0000	−1.61727	−0.808637	+	0.588308i	$0.799794\pi$
$810$	−1.00000	−0.0351364
$811$	24.0000	0.842754	0.421377	−	0.906886i	$-0.361547\pi$
$811$	24.0000	0.842754	0.421377	+	0.906886i	$0.361547\pi$
$812$	0	0
$813$	−16.0000	−0.561144
$814$	36.0000	1.26180
$815$	4.00000	0.140114
$816$	4.00000	0.140028
$817$	0	0
$818$	14.0000	0.489499
$819$	0	0
$820$	2.00000	0.0698430
$821$	16.0000	0.558404	0.279202	−	0.960232i	$-0.409930\pi$
$821$	16.0000	0.558404	0.279202	+	0.960232i	$0.409930\pi$
$822$	−8.00000	−0.279032
$823$	6.00000	0.209147	0.104573	−	0.994517i	$-0.466652\pi$
$823$	6.00000	0.209147	0.104573	+	0.994517i	$0.466652\pi$
$824$	16.0000	0.557386
$825$	6.00000	0.208893
$826$	0	0
$827$	32.0000	1.11275	0.556375	−	0.830932i	$-0.312192\pi$
$827$	32.0000	1.11275	0.556375	+	0.830932i	$0.312192\pi$
$828$	2.00000	0.0695048
$829$	−32.0000	−1.11141	−0.555703	−	0.831381i	$-0.687551\pi$
$829$	−32.0000	−1.11141	−0.555703	+	0.831381i	$0.687551\pi$
$830$	0	0
$831$	−18.0000	−0.624413
$832$	−2.00000	−0.0693375
$833$	28.0000	0.970143
$834$	2.00000	0.0692543
$835$	10.0000	0.346064
$836$	0	0
$837$	−1.00000	−0.0345651
$838$	0	0
$839$	0	0		−	1.00000i	$-0.5\pi$
$839$	0	0			1.00000i	$0.5\pi$
$840$	0	0
$841$	35.0000	1.20690
$842$	26.0000	0.896019
$843$	26.0000	0.895488
$844$	−4.00000	−0.137686
$845$	9.00000	0.309609
$846$	4.00000	0.137523
$847$	0	0
$848$	−6.00000	−0.206041
$849$	12.0000	0.411839
$850$	−4.00000	−0.137199
$851$	−12.0000	−0.411355
$852$	8.00000	0.274075
$853$	10.0000	0.342393	0.171197	−	0.985237i	$-0.445237\pi$
$853$	10.0000	0.342393	0.171197	+	0.985237i	$0.445237\pi$
$854$	0	0
$855$	0	0
$856$	12.0000	0.410152
$857$	−18.0000	−0.614868	−0.307434	−	0.951569i	$-0.599470\pi$
$857$	−18.0000	−0.614868	−0.307434	+	0.951569i	$0.599470\pi$
$858$	−12.0000	−0.409673
$859$	−22.0000	−0.750630	−0.375315	−	0.926897i	$-0.622466\pi$
$859$	−22.0000	−0.750630	−0.375315	+	0.926897i	$0.622466\pi$
$860$	−4.00000	−0.136399
$861$	0	0
$862$	40.0000	1.36241
$863$	22.0000	0.748889	0.374444	−	0.927249i	$-0.377833\pi$
$863$	22.0000	0.748889	0.374444	+	0.927249i	$0.377833\pi$
$864$	−1.00000	−0.0340207
$865$	−14.0000	−0.476014
$866$	−28.0000	−0.951479
$867$	1.00000	0.0339618
$868$	0	0
$869$	24.0000	0.814144
$870$	−8.00000	−0.271225
$871$	8.00000	0.271070
$872$	−10.0000	−0.338643
$873$	14.0000	0.473828
$874$	0	0
$875$	0	0
$876$	4.00000	0.135147
$877$	−50.0000	−1.68838	−0.844190	−	0.536044i	$-0.819918\pi$
$877$	−50.0000	−1.68838	−0.844190	+	0.536044i	$0.819918\pi$
$878$	0	0
$879$	30.0000	1.01187
$880$	6.00000	0.202260
$881$	38.0000	1.28025	0.640126	−	0.768270i	$-0.278882\pi$
$881$	38.0000	1.28025	0.640126	+	0.768270i	$0.278882\pi$
$882$	−7.00000	−0.235702
$883$	20.0000	0.673054	0.336527	−	0.941674i	$-0.390748\pi$
$883$	20.0000	0.673054	0.336527	+	0.941674i	$0.390748\pi$
$884$	8.00000	0.269069
$885$	0	0
$886$	12.0000	0.403148
$887$	−4.00000	−0.134307	−0.0671534	−	0.997743i	$-0.521392\pi$
$887$	−4.00000	−0.134307	−0.0671534	+	0.997743i	$0.521392\pi$
$888$	6.00000	0.201347
$889$	0	0
$890$	2.00000	0.0670402
$891$	−6.00000	−0.201008
$892$	18.0000	0.602685
$893$	0	0
$894$	14.0000	0.468230
$895$	2.00000	0.0668526
$896$	0	0
$897$	4.00000	0.133556
$898$	14.0000	0.467186
$899$	−8.00000	−0.266815
$900$	1.00000	0.0333333
$901$	24.0000	0.799556
$902$	12.0000	0.399556
$903$	0	0
$904$	6.00000	0.199557
$905$	−8.00000	−0.265929
$906$	−12.0000	−0.398673
$907$	44.0000	1.46100	0.730498	−	0.682915i	$-0.239288\pi$
$907$	44.0000	1.46100	0.730498	+	0.682915i	$0.239288\pi$
$908$	12.0000	0.398234
$909$	2.00000	0.0663358
$910$	0	0
$911$	−16.0000	−0.530104	−0.265052	−	0.964234i	$-0.585389\pi$
$911$	−16.0000	−0.530104	−0.265052	+	0.964234i	$0.585389\pi$
$912$	0	0
$913$	0	0
$914$	−8.00000	−0.264616
$915$	4.00000	0.132236
$916$	16.0000	0.528655
$917$	0	0
$918$	4.00000	0.132020
$919$	16.0000	0.527791	0.263896	−	0.964551i	$-0.414993\pi$
$919$	16.0000	0.527791	0.263896	+	0.964551i	$0.414993\pi$
$920$	−2.00000	−0.0659380
$921$	−4.00000	−0.131804
$922$	0	0
$923$	16.0000	0.526646
$924$	0	0
$925$	−6.00000	−0.197279
$926$	−14.0000	−0.460069
$927$	16.0000	0.525509
$928$	−8.00000	−0.262613
$929$	46.0000	1.50921	0.754606	−	0.656179i	$-0.227828\pi$
$929$	46.0000	1.50921	0.754606	+	0.656179i	$0.227828\pi$
$930$	1.00000	0.0327913
$931$	0	0
$932$	6.00000	0.196537
$933$	24.0000	0.785725
$934$	−36.0000	−1.17796
$935$	−24.0000	−0.784884
$936$	−2.00000	−0.0653720
$937$	14.0000	0.457360	0.228680	−	0.973502i	$-0.426559\pi$
$937$	14.0000	0.457360	0.228680	+	0.973502i	$0.426559\pi$
$938$	0	0
$939$	12.0000	0.391605
$940$	−4.00000	−0.130466
$941$	52.0000	1.69515	0.847576	−	0.530674i	$-0.178061\pi$
$941$	52.0000	1.69515	0.847576	+	0.530674i	$0.178061\pi$
$942$	2.00000	0.0651635
$943$	−4.00000	−0.130258
$944$	0	0
$945$	0	0
$946$	−24.0000	−0.780307
$947$	−12.0000	−0.389948	−0.194974	−	0.980808i	$-0.562462\pi$
$947$	−12.0000	−0.389948	−0.194974	+	0.980808i	$0.562462\pi$
$948$	4.00000	0.129914
$949$	8.00000	0.259691
$950$	0	0
$951$	26.0000	0.843108
$952$	0	0
$953$	8.00000	0.259145	0.129573	−	0.991570i	$-0.458639\pi$
$953$	8.00000	0.259145	0.129573	+	0.991570i	$0.458639\pi$
$954$	−6.00000	−0.194257
$955$	8.00000	0.258874
$956$	−24.0000	−0.776215
$957$	−48.0000	−1.55162
$958$	24.0000	0.775405
$959$	0	0
$960$	1.00000	0.0322749
$961$	1.00000	0.0322581
$962$	12.0000	0.386896
$963$	12.0000	0.386695
$964$	18.0000	0.579741
$965$	18.0000	0.579441
$966$	0	0
$967$	6.00000	0.192947	0.0964735	−	0.995336i	$-0.469244\pi$
$967$	6.00000	0.192947	0.0964735	+	0.995336i	$0.469244\pi$
$968$	25.0000	0.803530
$969$	0	0
$970$	−14.0000	−0.449513
$971$	36.0000	1.15529	0.577647	−	0.816286i	$-0.303971\pi$
$971$	36.0000	1.15529	0.577647	+	0.816286i	$0.303971\pi$
$972$	−1.00000	−0.0320750
$973$	0	0
$974$	−34.0000	−1.08943
$975$	2.00000	0.0640513
$976$	4.00000	0.128037
$977$	26.0000	0.831814	0.415907	−	0.909407i	$-0.363464\pi$
$977$	26.0000	0.831814	0.415907	+	0.909407i	$0.363464\pi$
$978$	4.00000	0.127906
$979$	12.0000	0.383522
$980$	7.00000	0.223607
$981$	−10.0000	−0.319275
$982$	−34.0000	−1.08498
$983$	10.0000	0.318950	0.159475	−	0.987202i	$-0.449020\pi$
$983$	10.0000	0.318950	0.159475	+	0.987202i	$0.449020\pi$
$984$	2.00000	0.0637577
$985$	18.0000	0.573528
$986$	32.0000	1.01909
$987$	0	0
$988$	0	0
$989$	8.00000	0.254385
$990$	6.00000	0.190693
$991$	20.0000	0.635321	0.317660	−	0.948205i	$-0.397103\pi$
$991$	20.0000	0.635321	0.317660	+	0.948205i	$0.397103\pi$
$992$	1.00000	0.0317500
$993$	18.0000	0.571213
$994$	0	0
$995$	20.0000	0.634043
$996$	0	0
$997$	18.0000	0.570066	0.285033	−	0.958518i	$-0.407995\pi$
$997$	18.0000	0.570066	0.285033	+	0.958518i	$0.407995\pi$
$998$	−38.0000	−1.20287
$999$	6.00000	0.189832

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	930.2.a.l.1.1	✓	1
3.2	odd	2		2790.2.a.j.1.1		1
4.3	odd	2		7440.2.a.r.1.1		1
5.2	odd	4		4650.2.d.a.3349.2		2
5.3	odd	4		4650.2.d.a.3349.1		2
5.4	even	2		4650.2.a.r.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
930.2.a.l.1.1	✓	1	1.1	even	1	trivial
2790.2.a.j.1.1		1	3.2	odd	2
4650.2.a.r.1.1		1	5.4	even	2
4650.2.d.a.3349.1		2	5.3	odd	4
4650.2.d.a.3349.2		2	5.2	odd	4
7440.2.a.r.1.1		1	4.3	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:	\( N \)	\(=\)	\( 930 = 2 \cdot 3 \cdot 5 \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	930.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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