LMFDB - Embedded newform 462.2.a.a.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [462,2,Mod(1,462)] 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("462.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(462, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 462 = 2 \cdot 3 \cdot 7 \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	462.a (trivial)

Newform invariants

comment:select newform

sage:traces = [1,-1,-1,1,-2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)

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

Self dual:	yes
Analytic conductor:	$3.68908857338$
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	$\chi$	$=$	462.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$

$=$

\(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -2.00000 q^{5} +1.00000 q^{6} +1.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +2.00000 q^{10} +1.00000 q^{11} -1.00000 q^{12} +2.00000 q^{13} -1.00000 q^{14} +2.00000 q^{15} +1.00000 q^{16} -6.00000 q^{17} -1.00000 q^{18} -4.00000 q^{19} -2.00000 q^{20} -1.00000 q^{21} -1.00000 q^{22} -4.00000 q^{23} +1.00000 q^{24} -1.00000 q^{25} -2.00000 q^{26} -1.00000 q^{27} +1.00000 q^{28} +2.00000 q^{29} -2.00000 q^{30} -4.00000 q^{31} -1.00000 q^{32} -1.00000 q^{33} +6.00000 q^{34} -2.00000 q^{35} +1.00000 q^{36} -2.00000 q^{37} +4.00000 q^{38} -2.00000 q^{39} +2.00000 q^{40} -6.00000 q^{41} +1.00000 q^{42} +1.00000 q^{44} -2.00000 q^{45} +4.00000 q^{46} -8.00000 q^{47} -1.00000 q^{48} +1.00000 q^{49} +1.00000 q^{50} +6.00000 q^{51} +2.00000 q^{52} -14.0000 q^{53} +1.00000 q^{54} -2.00000 q^{55} -1.00000 q^{56} +4.00000 q^{57} -2.00000 q^{58} +12.0000 q^{59} +2.00000 q^{60} -14.0000 q^{61} +4.00000 q^{62} +1.00000 q^{63} +1.00000 q^{64} -4.00000 q^{65} +1.00000 q^{66} +4.00000 q^{67} -6.00000 q^{68} +4.00000 q^{69} +2.00000 q^{70} +12.0000 q^{71} -1.00000 q^{72} +6.00000 q^{73} +2.00000 q^{74} +1.00000 q^{75} -4.00000 q^{76} +1.00000 q^{77} +2.00000 q^{78} -2.00000 q^{80} +1.00000 q^{81} +6.00000 q^{82} -1.00000 q^{84} +12.0000 q^{85} -2.00000 q^{87} -1.00000 q^{88} -6.00000 q^{89} +2.00000 q^{90} +2.00000 q^{91} -4.00000 q^{92} +4.00000 q^{93} +8.00000 q^{94} +8.00000 q^{95} +1.00000 q^{96} -14.0000 q^{97} -1.00000 q^{98} +1.00000 q^{99} +O(q^{100})\)

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	$-0.647584\pi$
$5$	−2.00000	−0.894427	−0.447214	+	0.894427i	$0.647584\pi$
$6$	1.00000	0.408248
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	−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	$-0.410544\pi$
$13$	2.00000	0.554700	0.277350	+	0.960769i	$0.410544\pi$
$14$	−1.00000	−0.267261
$15$	2.00000	0.516398
$16$	1.00000	0.250000
$17$	−6.00000	−1.45521	−0.727607	−	0.685994i	$-0.759367\pi$
$17$	−6.00000	−1.45521	−0.727607	+	0.685994i	$0.759367\pi$
$18$	−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
$21$	−1.00000	−0.218218
$22$	−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
$28$	1.00000	0.188982
$29$	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	$-0.616954\pi$
$31$	−4.00000	−0.718421	−0.359211	+	0.933257i	$0.616954\pi$
$32$	−1.00000	−0.176777
$33$	−1.00000	−0.174078
$34$	6.00000	1.02899
$35$	−2.00000	−0.338062
$36$	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	$-0.655213\pi$
$41$	−6.00000	−0.937043	−0.468521	+	0.883452i	$0.655213\pi$
$42$	1.00000	0.154303
$43$	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	$-0.698301\pi$
$47$	−8.00000	−1.16692	−0.583460	+	0.812142i	$0.698301\pi$
$48$	−1.00000	−0.144338
$49$	1.00000	0.142857
$50$	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
$56$	−1.00000	−0.133631
$57$	4.00000	0.529813
$58$	−2.00000	−0.262613
$59$	12.0000	1.56227	0.781133	−	0.624364i	$-0.214642\pi$
$59$	12.0000	1.56227	0.781133	+	0.624364i	$0.214642\pi$
$60$	2.00000	0.258199
$61$	−14.0000	−1.79252	−0.896258	−	0.443533i	$-0.853725\pi$
$61$	−14.0000	−1.79252	−0.896258	+	0.443533i	$0.853725\pi$
$62$	4.00000	0.508001
$63$	1.00000	0.125988
$64$	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
$70$	2.00000	0.239046
$71$	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	$-0.385800\pi$
$73$	6.00000	0.702247	0.351123	+	0.936329i	$0.385800\pi$
$74$	2.00000	0.232495
$75$	1.00000	0.115470
$76$	−4.00000	−0.458831
$77$	1.00000	0.113961
$78$	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	$0.5\pi$
$84$	−1.00000	−0.109109
$85$	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	$-0.603011\pi$
$89$	−6.00000	−0.635999	−0.317999	+	0.948091i	$0.603011\pi$
$90$	2.00000	0.210819
$91$	2.00000	0.209657
$92$	−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	$-0.751641\pi$
$97$	−14.0000	−1.42148	−0.710742	+	0.703452i	$0.751641\pi$
$98$	−1.00000	−0.101015
$99$	1.00000	0.100504
$100$	−1.00000	−0.100000
$101$	10.0000	0.995037	0.497519	−	0.867453i	$-0.334245\pi$
$101$	10.0000	0.995037	0.497519	+	0.867453i	$0.334245\pi$
$102$	−6.00000	−0.594089
$103$	−4.00000	−0.394132	−0.197066	−	0.980390i	$-0.563141\pi$
$103$	−4.00000	−0.394132	−0.197066	+	0.980390i	$0.563141\pi$
$104$	−2.00000	−0.196116
$105$	2.00000	0.195180
$106$	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
$112$	1.00000	0.0944911
$113$	−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
$119$	−6.00000	−0.550019
$120$	−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
$126$	−1.00000	−0.0890871
$127$	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
$133$	−4.00000	−0.346844
$134$	−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	$-0.330047\pi$
$139$	12.0000	1.01783	0.508913	+	0.860818i	$0.330047\pi$
$140$	−2.00000	−0.169031
$141$	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
$147$	−1.00000	−0.0824786
$148$	−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
$154$	−1.00000	−0.0805823
$155$	8.00000	0.642575
$156$	−2.00000	−0.160128
$157$	2.00000	0.159617	0.0798087	−	0.996810i	$-0.474569\pi$
$157$	2.00000	0.159617	0.0798087	+	0.996810i	$0.474569\pi$
$158$	0	0
$159$	14.0000	1.11027
$160$	2.00000	0.158114
$161$	−4.00000	−0.315244
$162$	−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	$0.5\pi$
$168$	1.00000	0.0771517
$169$	−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	$-0.375875\pi$
$173$	10.0000	0.760286	0.380143	+	0.924928i	$0.375875\pi$
$174$	2.00000	0.151620
$175$	−1.00000	−0.0755929
$176$	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	$-0.674182\pi$
$181$	−14.0000	−1.04061	−0.520306	+	0.853980i	$0.674182\pi$
$182$	−2.00000	−0.148250
$183$	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
$189$	−1.00000	−0.0727393
$190$	−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
$196$	1.00000	0.0714286
$197$	−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	$-0.639842\pi$
$199$	−12.0000	−0.850657	−0.425329	+	0.905039i	$0.639842\pi$
$200$	1.00000	0.0707107
$201$	−4.00000	−0.282138
$202$	−10.0000	−0.703598
$203$	2.00000	0.140372
$204$	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
$210$	−2.00000	−0.138013
$211$	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
$217$	−4.00000	−0.271538
$218$	−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	$-0.631612\pi$
$223$	−12.0000	−0.803579	−0.401790	+	0.915732i	$0.631612\pi$
$224$	−1.00000	−0.0668153
$225$	−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	$-0.653074\pi$
$229$	−14.0000	−0.925146	−0.462573	+	0.886581i	$0.653074\pi$
$230$	−8.00000	−0.527504
$231$	−1.00000	−0.0657952
$232$	−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
$238$	6.00000	0.388922
$239$	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	$-0.815928\pi$
$241$	−26.0000	−1.67481	−0.837404	+	0.546585i	$0.815928\pi$
$242$	−1.00000	−0.0642824
$243$	−1.00000	−0.0641500
$244$	−14.0000	−0.896258
$245$	−2.00000	−0.127775
$246$	−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	$-0.282565\pi$
$251$	20.0000	1.26239	0.631194	+	0.775625i	$0.282565\pi$
$252$	1.00000	0.0629941
$253$	−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	$-0.310261\pi$
$257$	18.0000	1.12281	0.561405	+	0.827541i	$0.310261\pi$
$258$	0	0
$259$	−2.00000	−0.124274
$260$	−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
$266$	4.00000	0.245256
$267$	6.00000	0.367194
$268$	4.00000	0.244339
$269$	30.0000	1.82913	0.914566	−	0.404436i	$-0.132532\pi$
$269$	30.0000	1.82913	0.914566	+	0.404436i	$0.132532\pi$
$270$	−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
$273$	−2.00000	−0.121046
$274$	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
$280$	2.00000	0.119523
$281$	−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	$-0.537933\pi$
$283$	−4.00000	−0.237775	−0.118888	+	0.992908i	$0.537933\pi$
$284$	12.0000	0.712069
$285$	−8.00000	−0.473879
$286$	−2.00000	−0.118262
$287$	−6.00000	−0.354169
$288$	−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	$-0.634103\pi$
$293$	−14.0000	−0.817889	−0.408944	+	0.912559i	$0.634103\pi$
$294$	1.00000	0.0583212
$295$	−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	$-0.611253\pi$
$307$	−12.0000	−0.684876	−0.342438	+	0.939540i	$0.611253\pi$
$308$	1.00000	0.0569803
$309$	4.00000	0.227552
$310$	−8.00000	−0.454369
$311$	8.00000	0.453638	0.226819	−	0.973937i	$-0.427167\pi$
$311$	8.00000	0.453638	0.226819	+	0.973937i	$0.427167\pi$
$312$	2.00000	0.113228
$313$	−14.0000	−0.791327	−0.395663	−	0.918396i	$-0.629485\pi$
$313$	−14.0000	−0.791327	−0.395663	+	0.918396i	$0.629485\pi$
$314$	−2.00000	−0.112867
$315$	−2.00000	−0.112687
$316$	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
$322$	4.00000	0.222911
$323$	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
$329$	−8.00000	−0.441054
$330$	−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
$336$	−1.00000	−0.0545545
$337$	−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
$343$	1.00000	0.0539949
$344$	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	$-0.136090\pi$
$349$	34.0000	1.81998	0.909989	+	0.414632i	$0.136090\pi$
$350$	1.00000	0.0534522
$351$	−2.00000	−0.106752
$352$	−1.00000	−0.0533002
$353$	−14.0000	−0.745145	−0.372572	−	0.928003i	$-0.621524\pi$
$353$	−14.0000	−0.745145	−0.372572	+	0.928003i	$0.621524\pi$
$354$	12.0000	0.637793
$355$	−24.0000	−1.27379
$356$	−6.00000	−0.317999
$357$	6.00000	0.317554
$358$	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
$364$	2.00000	0.104828
$365$	−12.0000	−0.628109
$366$	−14.0000	−0.731792
$367$	−28.0000	−1.46159	−0.730794	−	0.682598i	$-0.760850\pi$
$367$	−28.0000	−1.46159	−0.730794	+	0.682598i	$0.760850\pi$
$368$	−4.00000	−0.208514
$369$	−6.00000	−0.312348
$370$	−4.00000	−0.207950
$371$	−14.0000	−0.726844
$372$	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
$378$	1.00000	0.0514344
$379$	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	$-0.365954\pi$
$383$	16.0000	0.817562	0.408781	+	0.912633i	$0.365954\pi$
$384$	1.00000	0.0510310
$385$	−2.00000	−0.101929
$386$	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
$392$	−1.00000	−0.0505076
$393$	0	0
$394$	6.00000	0.302276
$395$	0	0
$396$	1.00000	0.0502519
$397$	18.0000	0.903394	0.451697	−	0.892171i	$-0.350819\pi$
$397$	18.0000	0.903394	0.451697	+	0.892171i	$0.350819\pi$
$398$	12.0000	0.601506
$399$	4.00000	0.200250
$400$	−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
$406$	−2.00000	−0.0992583
$407$	−2.00000	−0.0991363
$408$	−6.00000	−0.297044
$409$	38.0000	1.87898	0.939490	−	0.342578i	$-0.111300\pi$
$409$	38.0000	1.87898	0.939490	+	0.342578i	$0.111300\pi$
$410$	−12.0000	−0.592638
$411$	22.0000	1.08518
$412$	−4.00000	−0.197066
$413$	12.0000	0.590481
$414$	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	$-0.594693\pi$
$419$	−12.0000	−0.586238	−0.293119	+	0.956076i	$0.594693\pi$
$420$	2.00000	0.0975900
$421$	−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
$427$	−14.0000	−0.677507
$428$	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	$-0.195653\pi$
$433$	34.0000	1.63394	0.816968	+	0.576683i	$0.195653\pi$
$434$	4.00000	0.192006
$435$	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	$-0.375298\pi$
$439$	16.0000	0.763638	0.381819	+	0.924237i	$0.375298\pi$
$440$	2.00000	0.0953463
$441$	1.00000	0.0476190
$442$	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
$448$	1.00000	0.0472456
$449$	−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
$455$	−4.00000	−0.187523
$456$	−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	$-0.746202\pi$
$461$	−30.0000	−1.39724	−0.698620	+	0.715493i	$0.746202\pi$
$462$	1.00000	0.0465242
$463$	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	$-0.346865\pi$
$467$	20.0000	0.925490	0.462745	+	0.886492i	$0.346865\pi$
$468$	2.00000	0.0924500
$469$	4.00000	0.184703
$470$	−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
$476$	−6.00000	−0.275010
$477$	−14.0000	−0.641016
$478$	−24.0000	−1.09773
$479$	−24.0000	−1.09659	−0.548294	−	0.836286i	$-0.684723\pi$
$479$	−24.0000	−1.09659	−0.548294	+	0.836286i	$0.684723\pi$
$480$	−2.00000	−0.0912871
$481$	−4.00000	−0.182384
$482$	26.0000	1.18427
$483$	4.00000	0.182006
$484$	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
$490$	2.00000	0.0903508
$491$	−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
$497$	12.0000	0.538274
$498$	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	$-0.616099\pi$
$503$	−16.0000	−0.713405	−0.356702	+	0.934218i	$0.616099\pi$
$504$	−1.00000	−0.0445435
$505$	−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	$-0.630614\pi$
$509$	−18.0000	−0.797836	−0.398918	+	0.916987i	$0.630614\pi$
$510$	12.0000	0.531369
$511$	6.00000	0.265424
$512$	−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
$518$	2.00000	0.0878750
$519$	−10.0000	−0.438951
$520$	4.00000	0.175412
$521$	−6.00000	−0.262865	−0.131432	−	0.991325i	$-0.541958\pi$
$521$	−6.00000	−0.262865	−0.131432	+	0.991325i	$0.541958\pi$
$522$	−2.00000	−0.0875376
$523$	−36.0000	−1.57417	−0.787085	−	0.616844i	$-0.788411\pi$
$523$	−36.0000	−1.57417	−0.787085	+	0.616844i	$0.788411\pi$
$524$	0	0
$525$	1.00000	0.0436436
$526$	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
$532$	−4.00000	−0.173422
$533$	−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
$539$	1.00000	0.0430730
$540$	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
$546$	2.00000	0.0855921
$547$	−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
$560$	−2.00000	−0.0845154
$561$	6.00000	0.253320
$562$	18.0000	0.759284
$563$	24.0000	1.01148	0.505740	−	0.862686i	$-0.331220\pi$
$563$	24.0000	1.01148	0.505740	+	0.862686i	$0.331220\pi$
$564$	8.00000	0.336861
$565$	12.0000	0.504844
$566$	4.00000	0.168133
$567$	1.00000	0.0419961
$568$	−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
$574$	6.00000	0.250435
$575$	4.00000	0.166812
$576$	1.00000	0.0416667
$577$	−6.00000	−0.249783	−0.124892	−	0.992170i	$-0.539858\pi$
$577$	−6.00000	−0.249783	−0.124892	+	0.992170i	$0.539858\pi$
$578$	−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	$-0.766569\pi$
$587$	−36.0000	−1.48588	−0.742940	+	0.669359i	$0.766569\pi$
$588$	−1.00000	−0.0412393
$589$	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	$-0.379500\pi$
$593$	18.0000	0.739171	0.369586	+	0.929197i	$0.379500\pi$
$594$	1.00000	0.0410305
$595$	12.0000	0.491952
$596$	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	$-0.677914\pi$
$601$	−26.0000	−1.06056	−0.530281	+	0.847822i	$0.677914\pi$
$602$	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	$-0.448091\pi$
$607$	8.00000	0.324710	0.162355	+	0.986732i	$0.448091\pi$
$608$	4.00000	0.162221
$609$	−2.00000	−0.0810441
$610$	−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
$616$	−1.00000	−0.0402911
$617$	−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	$-0.474384\pi$
$619$	4.00000	0.160774	0.0803868	+	0.996764i	$0.474384\pi$
$620$	8.00000	0.321288
$621$	4.00000	0.160514
$622$	−8.00000	−0.320771
$623$	−6.00000	−0.240385
$624$	−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
$630$	2.00000	0.0796819
$631$	−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
$637$	2.00000	0.0792429
$638$	−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	$-0.248762\pi$
$643$	36.0000	1.41970	0.709851	+	0.704352i	$0.248762\pi$
$644$	−4.00000	−0.157622
$645$	0	0
$646$	−24.0000	−0.944267
$647$	40.0000	1.57256	0.786281	−	0.617869i	$-0.212004\pi$
$647$	40.0000	1.57256	0.786281	+	0.617869i	$0.212004\pi$
$648$	−1.00000	−0.0392837
$649$	12.0000	0.471041
$650$	2.00000	0.0784465
$651$	4.00000	0.156772
$652$	−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
$658$	8.00000	0.311872
$659$	−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	$-0.386161\pi$
$661$	18.0000	0.700119	0.350059	+	0.936727i	$0.386161\pi$
$662$	20.0000	0.777322
$663$	12.0000	0.466041
$664$	0	0
$665$	8.00000	0.310227
$666$	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
$672$	1.00000	0.0385758
$673$	−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	$-0.387574\pi$
$677$	18.0000	0.691796	0.345898	+	0.938272i	$0.387574\pi$
$678$	−6.00000	−0.230429
$679$	−14.0000	−0.537271
$680$	−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
$686$	−1.00000	−0.0381802
$687$	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	$-0.815646\pi$
$691$	−44.0000	−1.67384	−0.836919	+	0.547326i	$0.815646\pi$
$692$	10.0000	0.380143
$693$	1.00000	0.0379869
$694$	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
$700$	−1.00000	−0.0377964
$701$	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
$707$	10.0000	0.376089
$708$	−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
$714$	−6.00000	−0.224544
$715$	−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	$-0.352306\pi$
$719$	24.0000	0.895049	0.447524	+	0.894272i	$0.352306\pi$
$720$	−2.00000	−0.0745356
$721$	−4.00000	−0.148968
$722$	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	$-0.326218\pi$
$727$	28.0000	1.03846	0.519231	+	0.854634i	$0.326218\pi$
$728$	−2.00000	−0.0741249
$729$	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	$-0.283911\pi$
$733$	34.0000	1.25582	0.627909	+	0.778287i	$0.283911\pi$
$734$	28.0000	1.03350
$735$	2.00000	0.0737711
$736$	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
$742$	14.0000	0.513956
$743$	−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
$749$	12.0000	0.438470
$750$	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
$756$	−1.00000	−0.0363696
$757$	−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	$-0.343803\pi$
$761$	26.0000	0.942499	0.471250	+	0.882000i	$0.343803\pi$
$762$	16.0000	0.579619
$763$	10.0000	0.362024
$764$	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	$-0.259733\pi$
$769$	38.0000	1.37032	0.685158	+	0.728395i	$0.259733\pi$
$770$	2.00000	0.0720750
$771$	−18.0000	−0.648254
$772$	−6.00000	−0.215945
$773$	14.0000	0.503545	0.251773	−	0.967786i	$-0.418987\pi$
$773$	14.0000	0.503545	0.251773	+	0.967786i	$0.418987\pi$
$774$	0	0
$775$	4.00000	0.143684
$776$	14.0000	0.502571
$777$	2.00000	0.0717496
$778$	−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
$784$	1.00000	0.0357143
$785$	−4.00000	−0.142766
$786$	0	0
$787$	12.0000	0.427754	0.213877	−	0.976861i	$-0.431391\pi$
$787$	12.0000	0.427754	0.213877	+	0.976861i	$0.431391\pi$
$788$	−6.00000	−0.213741
$789$	0	0
$790$	0	0
$791$	−6.00000	−0.213335
$792$	−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	$-0.603279\pi$
$797$	−18.0000	−0.637593	−0.318796	+	0.947823i	$0.603279\pi$
$798$	−4.00000	−0.141598
$799$	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
$805$	8.00000	0.281963
$806$	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	$-0.781006\pi$
$811$	−44.0000	−1.54505	−0.772524	+	0.634985i	$0.781006\pi$
$812$	2.00000	0.0701862
$813$	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
$819$	2.00000	0.0698857
$820$	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
$826$	−12.0000	−0.417533
$827$	−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	$-0.578171\pi$
$829$	−14.0000	−0.486240	−0.243120	+	0.969996i	$0.578171\pi$
$830$	0	0
$831$	−26.0000	−0.901930
$832$	2.00000	0.0693375
$833$	−6.00000	−0.207888
$834$	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	$-0.589072\pi$
$839$	−16.0000	−0.552381	−0.276191	+	0.961103i	$0.589072\pi$
$840$	−2.00000	−0.0690066
$841$	−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
$847$	1.00000	0.0343604
$848$	−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	$-0.244588\pi$
$853$	42.0000	1.43805	0.719026	+	0.694983i	$0.244588\pi$
$854$	14.0000	0.479070
$855$	8.00000	0.273594
$856$	−12.0000	−0.410152
$857$	−22.0000	−0.751506	−0.375753	−	0.926720i	$-0.622616\pi$
$857$	−22.0000	−0.751506	−0.375753	+	0.926720i	$0.622616\pi$
$858$	2.00000	0.0682789
$859$	20.0000	0.682391	0.341196	−	0.939992i	$-0.389168\pi$
$859$	20.0000	0.682391	0.341196	+	0.939992i	$0.389168\pi$
$860$	0	0
$861$	6.00000	0.204479
$862$	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
$868$	−4.00000	−0.135769
$869$	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
$875$	12.0000	0.405674
$876$	−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	$-0.305879\pi$
$881$	34.0000	1.14549	0.572745	+	0.819734i	$0.305879\pi$
$882$	−1.00000	−0.0336718
$883$	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	$-0.413434\pi$
$887$	16.0000	0.537227	0.268614	+	0.963248i	$0.413434\pi$
$888$	−2.00000	−0.0671156
$889$	16.0000	0.536623
$890$	−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
$896$	−1.00000	−0.0334077
$897$	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
$910$	4.00000	0.132599
$911$	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
$924$	−1.00000	−0.0328976
$925$	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	$-0.663783\pi$
$929$	−30.0000	−0.984268	−0.492134	+	0.870519i	$0.663783\pi$
$930$	8.00000	0.262330
$931$	−4.00000	−0.131095
$932$	−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	$-0.426559\pi$
$937$	14.0000	0.457360	0.228680	+	0.973502i	$0.426559\pi$
$938$	−4.00000	−0.130605
$939$	14.0000	0.456873
$940$	16.0000	0.521862
$941$	−38.0000	−1.23876	−0.619382	−	0.785090i	$-0.712617\pi$
$941$	−38.0000	−1.23876	−0.619382	+	0.785090i	$0.712617\pi$
$942$	2.00000	0.0651635
$943$	24.0000	0.781548
$944$	12.0000	0.390567
$945$	2.00000	0.0650600
$946$	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
$952$	6.00000	0.194461
$953$	−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
$959$	−22.0000	−0.710417
$960$	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
$966$	−4.00000	−0.128698
$967$	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	$-0.561675\pi$
$971$	−12.0000	−0.385098	−0.192549	+	0.981287i	$0.561675\pi$
$972$	−1.00000	−0.0320750
$973$	12.0000	0.384702
$974$	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
$980$	−2.00000	−0.0638877
$981$	10.0000	0.319275
$982$	28.0000	0.893516
$983$	−32.0000	−1.02064	−0.510321	−	0.859984i	$-0.670473\pi$
$983$	−32.0000	−1.02064	−0.510321	+	0.859984i	$0.670473\pi$
$984$	−6.00000	−0.191273
$985$	12.0000	0.382352
$986$	12.0000	0.382158
$987$	8.00000	0.254643
$988$	−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
$994$	−12.0000	−0.380617
$995$	24.0000	0.760851
$996$	0	0
$997$	26.0000	0.823428	0.411714	−	0.911313i	$-0.364930\pi$
$997$	26.0000	0.823428	0.411714	+	0.911313i	$0.364930\pi$
$998$	−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	462.2.a.a.1.1	✓	1
3.2	odd	2		1386.2.a.k.1.1		1
4.3	odd	2		3696.2.a.s.1.1		1
7.6	odd	2		3234.2.a.n.1.1		1
11.10	odd	2		5082.2.a.q.1.1		1
21.20	even	2		9702.2.a.bf.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
462.2.a.a.1.1	✓	1	1.1	even	1	trivial
1386.2.a.k.1.1		1	3.2	odd	2
3234.2.a.n.1.1		1	7.6	odd	2
3696.2.a.s.1.1		1	4.3	odd	2
5082.2.a.q.1.1		1	11.10	odd	2
9702.2.a.bf.1.1		1	21.20	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 462 = 2 \cdot 3 \cdot 7 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	462.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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