LMFDB - Embedded newform 462.2.a.g.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [462,2,Mod(1,462)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

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")

//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);

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: f = N[0] # Warning: the index may be different

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

Self dual:	yes
Analytic conductor:	$3.68908857338$
Analytic rank:	$0$
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)

$f(q)$

$=$

$q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +2.00000 q^{5} +1.00000 q^{6} -1.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$

\(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +2.00000 q^{5} +1.00000 q^{6} -1.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +2.00000 q^{10} +1.00000 q^{11} +1.00000 q^{12} -2.00000 q^{13} -1.00000 q^{14} +2.00000 q^{15} +1.00000 q^{16} -2.00000 q^{17} +1.00000 q^{18} +2.00000 q^{20} -1.00000 q^{21} +1.00000 q^{22} +1.00000 q^{24} -1.00000 q^{25} -2.00000 q^{26} +1.00000 q^{27} -1.00000 q^{28} -2.00000 q^{29} +2.00000 q^{30} +4.00000 q^{31} +1.00000 q^{32} +1.00000 q^{33} -2.00000 q^{34} -2.00000 q^{35} +1.00000 q^{36} -2.00000 q^{37} -2.00000 q^{39} +2.00000 q^{40} -10.0000 q^{41} -1.00000 q^{42} +4.00000 q^{43} +1.00000 q^{44} +2.00000 q^{45} +4.00000 q^{47} +1.00000 q^{48} +1.00000 q^{49} -1.00000 q^{50} -2.00000 q^{51} -2.00000 q^{52} -2.00000 q^{53} +1.00000 q^{54} +2.00000 q^{55} -1.00000 q^{56} -2.00000 q^{58} -12.0000 q^{59} +2.00000 q^{60} -2.00000 q^{61} +4.00000 q^{62} -1.00000 q^{63} +1.00000 q^{64} -4.00000 q^{65} +1.00000 q^{66} +12.0000 q^{67} -2.00000 q^{68} -2.00000 q^{70} +8.00000 q^{71} +1.00000 q^{72} +6.00000 q^{73} -2.00000 q^{74} -1.00000 q^{75} -1.00000 q^{77} -2.00000 q^{78} -8.00000 q^{79} +2.00000 q^{80} +1.00000 q^{81} -10.0000 q^{82} -8.00000 q^{83} -1.00000 q^{84} -4.00000 q^{85} +4.00000 q^{86} -2.00000 q^{87} +1.00000 q^{88} -14.0000 q^{89} +2.00000 q^{90} +2.00000 q^{91} +4.00000 q^{93} +4.00000 q^{94} +1.00000 q^{96} -14.0000 q^{97} +1.00000 q^{98} +1.00000 q^{99} +O(q^{100})\)

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	$-0.352416\pi$
$5$	2.00000	0.894427	0.447214	+	0.894427i	$0.352416\pi$
$6$	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	$-0.589456\pi$
$13$	−2.00000	−0.554700	−0.277350	+	0.960769i	$0.589456\pi$
$14$	−1.00000	−0.267261
$15$	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
$19$	0	0		−	1.00000i	$-0.5\pi$
$19$	0	0			1.00000i	$0.5\pi$
$20$	2.00000	0.447214
$21$	−1.00000	−0.218218
$22$	1.00000	0.213201
$23$	0	0		−	1.00000i	$-0.5\pi$
$23$	0	0			1.00000i	$0.5\pi$
$24$	1.00000	0.204124
$25$	−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	$-0.559454\pi$
$29$	−2.00000	−0.371391	−0.185695	+	0.982607i	$0.559454\pi$
$30$	2.00000	0.365148
$31$	4.00000	0.718421	0.359211	−	0.933257i	$-0.383046\pi$
$31$	4.00000	0.718421	0.359211	+	0.933257i	$0.383046\pi$
$32$	1.00000	0.176777
$33$	1.00000	0.174078
$34$	−2.00000	−0.342997
$35$	−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	$0.552568\pi$
$38$	0	0
$39$	−2.00000	−0.320256
$40$	2.00000	0.316228
$41$	−10.0000	−1.56174	−0.780869	−	0.624695i	$-0.785223\pi$
$41$	−10.0000	−1.56174	−0.780869	+	0.624695i	$0.785223\pi$
$42$	−1.00000	−0.154303
$43$	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
$46$	0	0
$47$	4.00000	0.583460	0.291730	−	0.956501i	$-0.405769\pi$
$47$	4.00000	0.583460	0.291730	+	0.956501i	$0.405769\pi$
$48$	1.00000	0.144338
$49$	1.00000	0.142857
$50$	−1.00000	−0.141421
$51$	−2.00000	−0.280056
$52$	−2.00000	−0.277350
$53$	−2.00000	−0.274721	−0.137361	−	0.990521i	$-0.543862\pi$
$53$	−2.00000	−0.274721	−0.137361	+	0.990521i	$0.543862\pi$
$54$	1.00000	0.136083
$55$	2.00000	0.269680
$56$	−1.00000	−0.133631
$57$	0	0
$58$	−2.00000	−0.262613
$59$	−12.0000	−1.56227	−0.781133	−	0.624364i	$-0.785358\pi$
$59$	−12.0000	−1.56227	−0.781133	+	0.624364i	$0.785358\pi$
$60$	2.00000	0.258199
$61$	−2.00000	−0.256074	−0.128037	−	0.991769i	$-0.540868\pi$
$61$	−2.00000	−0.256074	−0.128037	+	0.991769i	$0.540868\pi$
$62$	4.00000	0.508001
$63$	−1.00000	−0.125988
$64$	1.00000	0.125000
$65$	−4.00000	−0.496139
$66$	1.00000	0.123091
$67$	12.0000	1.46603	0.733017	−	0.680211i	$-0.238112\pi$
$67$	12.0000	1.46603	0.733017	+	0.680211i	$0.238112\pi$
$68$	−2.00000	−0.242536
$69$	0	0
$70$	−2.00000	−0.239046
$71$	8.00000	0.949425	0.474713	−	0.880141i	$-0.342552\pi$
$71$	8.00000	0.949425	0.474713	+	0.880141i	$0.342552\pi$
$72$	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
$76$	0	0
$77$	−1.00000	−0.113961
$78$	−2.00000	−0.226455
$79$	−8.00000	−0.900070	−0.450035	−	0.893011i	$-0.648589\pi$
$79$	−8.00000	−0.900070	−0.450035	+	0.893011i	$0.648589\pi$
$80$	2.00000	0.223607
$81$	1.00000	0.111111
$82$	−10.0000	−1.10432
$83$	−8.00000	−0.878114	−0.439057	−	0.898459i	$-0.644687\pi$
$83$	−8.00000	−0.878114	−0.439057	+	0.898459i	$0.644687\pi$
$84$	−1.00000	−0.109109
$85$	−4.00000	−0.433861
$86$	4.00000	0.431331
$87$	−2.00000	−0.214423
$88$	1.00000	0.106600
$89$	−14.0000	−1.48400	−0.741999	−	0.670402i	$-0.766122\pi$
$89$	−14.0000	−1.48400	−0.741999	+	0.670402i	$0.766122\pi$
$90$	2.00000	0.210819
$91$	2.00000	0.209657
$92$	0	0
$93$	4.00000	0.414781
$94$	4.00000	0.412568
$95$	0	0
$96$	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	$-0.665755\pi$
$101$	−10.0000	−0.995037	−0.497519	+	0.867453i	$0.665755\pi$
$102$	−2.00000	−0.198030
$103$	4.00000	0.394132	0.197066	−	0.980390i	$-0.436859\pi$
$103$	4.00000	0.394132	0.197066	+	0.980390i	$0.436859\pi$
$104$	−2.00000	−0.196116
$105$	−2.00000	−0.195180
$106$	−2.00000	−0.194257
$107$	−12.0000	−1.16008	−0.580042	−	0.814587i	$-0.696964\pi$
$107$	−12.0000	−1.16008	−0.580042	+	0.814587i	$0.696964\pi$
$108$	1.00000	0.0962250
$109$	14.0000	1.34096	0.670478	−	0.741929i	$-0.266089\pi$
$109$	14.0000	1.34096	0.670478	+	0.741929i	$0.266089\pi$
$110$	2.00000	0.190693
$111$	−2.00000	−0.189832
$112$	−1.00000	−0.0944911
$113$	18.0000	1.69330	0.846649	−	0.532152i	$-0.178617\pi$
$113$	18.0000	1.69330	0.846649	+	0.532152i	$0.178617\pi$
$114$	0	0
$115$	0	0
$116$	−2.00000	−0.185695
$117$	−2.00000	−0.184900
$118$	−12.0000	−1.10469
$119$	2.00000	0.183340
$120$	2.00000	0.182574
$121$	1.00000	0.0909091
$122$	−2.00000	−0.181071
$123$	−10.0000	−0.901670
$124$	4.00000	0.359211
$125$	−12.0000	−1.07331
$126$	−1.00000	−0.0890871
$127$	8.00000	0.709885	0.354943	−	0.934888i	$-0.384500\pi$
$127$	8.00000	0.709885	0.354943	+	0.934888i	$0.384500\pi$
$128$	1.00000	0.0883883
$129$	4.00000	0.352180
$130$	−4.00000	−0.350823
$131$	0	0		−	1.00000i	$-0.5\pi$
$131$	0	0			1.00000i	$0.5\pi$
$132$	1.00000	0.0870388
$133$	0	0
$134$	12.0000	1.03664
$135$	2.00000	0.172133
$136$	−2.00000	−0.171499
$137$	10.0000	0.854358	0.427179	−	0.904167i	$-0.359507\pi$
$137$	10.0000	0.854358	0.427179	+	0.904167i	$0.359507\pi$
$138$	0	0
$139$	0	0		−	1.00000i	$-0.5\pi$
$139$	0	0			1.00000i	$0.5\pi$
$140$	−2.00000	−0.169031
$141$	4.00000	0.336861
$142$	8.00000	0.671345
$143$	−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	$-0.634337\pi$
$149$	−10.0000	−0.819232	−0.409616	+	0.912258i	$0.634337\pi$
$150$	−1.00000	−0.0816497
$151$	8.00000	0.651031	0.325515	−	0.945537i	$-0.394462\pi$
$151$	8.00000	0.651031	0.325515	+	0.945537i	$0.394462\pi$
$152$	0	0
$153$	−2.00000	−0.161690
$154$	−1.00000	−0.0805823
$155$	8.00000	0.642575
$156$	−2.00000	−0.160128
$157$	18.0000	1.43656	0.718278	−	0.695756i	$-0.244931\pi$
$157$	18.0000	1.43656	0.718278	+	0.695756i	$0.244931\pi$
$158$	−8.00000	−0.636446
$159$	−2.00000	−0.158610
$160$	2.00000	0.158114
$161$	0	0
$162$	1.00000	0.0785674
$163$	4.00000	0.313304	0.156652	−	0.987654i	$-0.449930\pi$
$163$	4.00000	0.313304	0.156652	+	0.987654i	$0.449930\pi$
$164$	−10.0000	−0.780869
$165$	2.00000	0.155700
$166$	−8.00000	−0.620920
$167$	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
$170$	−4.00000	−0.306786
$171$	0	0
$172$	4.00000	0.304997
$173$	6.00000	0.456172	0.228086	−	0.973641i	$-0.426753\pi$
$173$	6.00000	0.456172	0.228086	+	0.973641i	$0.426753\pi$
$174$	−2.00000	−0.151620
$175$	1.00000	0.0755929
$176$	1.00000	0.0753778
$177$	−12.0000	−0.901975
$178$	−14.0000	−1.04934
$179$	12.0000	0.896922	0.448461	−	0.893802i	$-0.351972\pi$
$179$	12.0000	0.896922	0.448461	+	0.893802i	$0.351972\pi$
$180$	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	$0.476318\pi$
$182$	2.00000	0.148250
$183$	−2.00000	−0.147844
$184$	0	0
$185$	−4.00000	−0.294086
$186$	4.00000	0.293294
$187$	−2.00000	−0.146254
$188$	4.00000	0.291730
$189$	−1.00000	−0.0727393
$190$	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
$193$	2.00000	0.143963	0.0719816	−	0.997406i	$-0.477068\pi$
$193$	2.00000	0.143963	0.0719816	+	0.997406i	$0.477068\pi$
$194$	−14.0000	−1.00514
$195$	−4.00000	−0.286446
$196$	1.00000	0.0714286
$197$	6.00000	0.427482	0.213741	−	0.976890i	$-0.431435\pi$
$197$	6.00000	0.427482	0.213741	+	0.976890i	$0.431435\pi$
$198$	1.00000	0.0710669
$199$	20.0000	1.41776	0.708881	−	0.705328i	$-0.249200\pi$
$199$	20.0000	1.41776	0.708881	+	0.705328i	$0.249200\pi$
$200$	−1.00000	−0.0707107
$201$	12.0000	0.846415
$202$	−10.0000	−0.703598
$203$	2.00000	0.140372
$204$	−2.00000	−0.140028
$205$	−20.0000	−1.39686
$206$	4.00000	0.278693
$207$	0	0
$208$	−2.00000	−0.138675
$209$	0	0
$210$	−2.00000	−0.138013
$211$	12.0000	0.826114	0.413057	−	0.910705i	$-0.364461\pi$
$211$	12.0000	0.826114	0.413057	+	0.910705i	$0.364461\pi$
$212$	−2.00000	−0.137361
$213$	8.00000	0.548151
$214$	−12.0000	−0.820303
$215$	8.00000	0.545595
$216$	1.00000	0.0680414
$217$	−4.00000	−0.271538
$218$	14.0000	0.948200
$219$	6.00000	0.405442
$220$	2.00000	0.134840
$221$	4.00000	0.269069
$222$	−2.00000	−0.134231
$223$	20.0000	1.33930	0.669650	−	0.742677i	$-0.266444\pi$
$223$	20.0000	1.33930	0.669650	+	0.742677i	$0.266444\pi$
$224$	−1.00000	−0.0668153
$225$	−1.00000	−0.0666667
$226$	18.0000	1.19734
$227$	0	0		−	1.00000i	$-0.5\pi$
$227$	0	0			1.00000i	$0.5\pi$
$228$	0	0
$229$	−14.0000	−0.925146	−0.462573	−	0.886581i	$-0.653074\pi$
$229$	−14.0000	−0.925146	−0.462573	+	0.886581i	$0.653074\pi$
$230$	0	0
$231$	−1.00000	−0.0657952
$232$	−2.00000	−0.131306
$233$	−6.00000	−0.393073	−0.196537	−	0.980497i	$-0.562969\pi$
$233$	−6.00000	−0.393073	−0.196537	+	0.980497i	$0.562969\pi$
$234$	−2.00000	−0.130744
$235$	8.00000	0.521862
$236$	−12.0000	−0.781133
$237$	−8.00000	−0.519656
$238$	2.00000	0.129641
$239$	16.0000	1.03495	0.517477	−	0.855697i	$-0.326871\pi$
$239$	16.0000	1.03495	0.517477	+	0.855697i	$0.326871\pi$
$240$	2.00000	0.129099
$241$	30.0000	1.93247	0.966235	−	0.257663i	$-0.0829523\pi$
$241$	30.0000	1.93247	0.966235	+	0.257663i	$0.0829523\pi$
$242$	1.00000	0.0642824
$243$	1.00000	0.0641500
$244$	−2.00000	−0.128037
$245$	2.00000	0.127775
$246$	−10.0000	−0.637577
$247$	0	0
$248$	4.00000	0.254000
$249$	−8.00000	−0.506979
$250$	−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	$0.459709\pi$
$252$	−1.00000	−0.0629941
$253$	0	0
$254$	8.00000	0.501965
$255$	−4.00000	−0.250490
$256$	1.00000	0.0625000
$257$	−6.00000	−0.374270	−0.187135	−	0.982334i	$-0.559920\pi$
$257$	−6.00000	−0.374270	−0.187135	+	0.982334i	$0.559920\pi$
$258$	4.00000	0.249029
$259$	2.00000	0.124274
$260$	−4.00000	−0.248069
$261$	−2.00000	−0.123797
$262$	0	0
$263$	32.0000	1.97320	0.986602	−	0.163144i	$-0.0521635\pi$
$263$	32.0000	1.97320	0.986602	+	0.163144i	$0.0521635\pi$
$264$	1.00000	0.0615457
$265$	−4.00000	−0.245718
$266$	0	0
$267$	−14.0000	−0.856786
$268$	12.0000	0.733017
$269$	10.0000	0.609711	0.304855	−	0.952399i	$-0.401392\pi$
$269$	10.0000	0.609711	0.304855	+	0.952399i	$0.401392\pi$
$270$	2.00000	0.121716
$271$	−8.00000	−0.485965	−0.242983	−	0.970031i	$-0.578126\pi$
$271$	−8.00000	−0.485965	−0.242983	+	0.970031i	$0.578126\pi$
$272$	−2.00000	−0.121268
$273$	2.00000	0.121046
$274$	10.0000	0.604122
$275$	−1.00000	−0.0603023
$276$	0	0
$277$	−10.0000	−0.600842	−0.300421	−	0.953807i	$-0.597127\pi$
$277$	−10.0000	−0.600842	−0.300421	+	0.953807i	$0.597127\pi$
$278$	0	0
$279$	4.00000	0.239474
$280$	−2.00000	−0.119523
$281$	10.0000	0.596550	0.298275	−	0.954480i	$-0.403589\pi$
$281$	10.0000	0.596550	0.298275	+	0.954480i	$0.403589\pi$
$282$	4.00000	0.238197
$283$	16.0000	0.951101	0.475551	−	0.879688i	$-0.342249\pi$
$283$	16.0000	0.951101	0.475551	+	0.879688i	$0.342249\pi$
$284$	8.00000	0.474713
$285$	0	0
$286$	−2.00000	−0.118262
$287$	10.0000	0.590281
$288$	1.00000	0.0589256
$289$	−13.0000	−0.764706
$290$	−4.00000	−0.234888
$291$	−14.0000	−0.820695
$292$	6.00000	0.351123
$293$	14.0000	0.817889	0.408944	−	0.912559i	$-0.365897\pi$
$293$	14.0000	0.817889	0.408944	+	0.912559i	$0.365897\pi$
$294$	1.00000	0.0583212
$295$	−24.0000	−1.39733
$296$	−2.00000	−0.116248
$297$	1.00000	0.0580259
$298$	−10.0000	−0.579284
$299$	0	0
$300$	−1.00000	−0.0577350
$301$	−4.00000	−0.230556
$302$	8.00000	0.460348
$303$	−10.0000	−0.574485
$304$	0	0
$305$	−4.00000	−0.229039
$306$	−2.00000	−0.114332
$307$	8.00000	0.456584	0.228292	−	0.973593i	$-0.426686\pi$
$307$	8.00000	0.456584	0.228292	+	0.973593i	$0.426686\pi$
$308$	−1.00000	−0.0569803
$309$	4.00000	0.227552
$310$	8.00000	0.454369
$311$	20.0000	1.13410	0.567048	−	0.823685i	$-0.308085\pi$
$311$	20.0000	1.13410	0.567048	+	0.823685i	$0.308085\pi$
$312$	−2.00000	−0.113228
$313$	−30.0000	−1.69570	−0.847850	−	0.530236i	$-0.822103\pi$
$313$	−30.0000	−1.69570	−0.847850	+	0.530236i	$0.822103\pi$
$314$	18.0000	1.01580
$315$	−2.00000	−0.112687
$316$	−8.00000	−0.450035
$317$	22.0000	1.23564	0.617822	−	0.786318i	$-0.288015\pi$
$317$	22.0000	1.23564	0.617822	+	0.786318i	$0.288015\pi$
$318$	−2.00000	−0.112154
$319$	−2.00000	−0.111979
$320$	2.00000	0.111803
$321$	−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
$327$	14.0000	0.774202
$328$	−10.0000	−0.552158
$329$	−4.00000	−0.220527
$330$	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	$0.314761\pi$
$332$	−8.00000	−0.439057
$333$	−2.00000	−0.109599
$334$	0	0
$335$	24.0000	1.31126
$336$	−1.00000	−0.0545545
$337$	−22.0000	−1.19842	−0.599208	−	0.800593i	$-0.704518\pi$
$337$	−22.0000	−1.19842	−0.599208	+	0.800593i	$0.704518\pi$
$338$	−9.00000	−0.489535
$339$	18.0000	0.977626
$340$	−4.00000	−0.216930
$341$	4.00000	0.216612
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	4.00000	0.215666
$345$	0	0
$346$	6.00000	0.322562
$347$	−12.0000	−0.644194	−0.322097	−	0.946707i	$-0.604388\pi$
$347$	−12.0000	−0.644194	−0.322097	+	0.946707i	$0.604388\pi$
$348$	−2.00000	−0.107211
$349$	−26.0000	−1.39175	−0.695874	−	0.718164i	$-0.744983\pi$
$349$	−26.0000	−1.39175	−0.695874	+	0.718164i	$0.744983\pi$
$350$	1.00000	0.0534522
$351$	−2.00000	−0.106752
$352$	1.00000	0.0533002
$353$	−6.00000	−0.319348	−0.159674	−	0.987170i	$-0.551044\pi$
$353$	−6.00000	−0.319348	−0.159674	+	0.987170i	$0.551044\pi$
$354$	−12.0000	−0.637793
$355$	16.0000	0.849192
$356$	−14.0000	−0.741999
$357$	2.00000	0.105851
$358$	12.0000	0.634220
$359$	−32.0000	−1.68890	−0.844448	−	0.535638i	$-0.820071\pi$
$359$	−32.0000	−1.68890	−0.844448	+	0.535638i	$0.820071\pi$
$360$	2.00000	0.105409
$361$	−19.0000	−1.00000
$362$	2.00000	0.105118
$363$	1.00000	0.0524864
$364$	2.00000	0.104828
$365$	12.0000	0.628109
$366$	−2.00000	−0.104542
$367$	−28.0000	−1.46159	−0.730794	−	0.682598i	$-0.760850\pi$
$367$	−28.0000	−1.46159	−0.730794	+	0.682598i	$0.760850\pi$
$368$	0	0
$369$	−10.0000	−0.520579
$370$	−4.00000	−0.207950
$371$	2.00000	0.103835
$372$	4.00000	0.207390
$373$	22.0000	1.13912	0.569558	−	0.821951i	$-0.307114\pi$
$373$	22.0000	1.13912	0.569558	+	0.821951i	$0.307114\pi$
$374$	−2.00000	−0.103418
$375$	−12.0000	−0.619677
$376$	4.00000	0.206284
$377$	4.00000	0.206010
$378$	−1.00000	−0.0514344
$379$	−20.0000	−1.02733	−0.513665	−	0.857991i	$-0.671713\pi$
$379$	−20.0000	−1.02733	−0.513665	+	0.857991i	$0.671713\pi$
$380$	0	0
$381$	8.00000	0.409852
$382$	−8.00000	−0.409316
$383$	12.0000	0.613171	0.306586	−	0.951843i	$-0.400813\pi$
$383$	12.0000	0.613171	0.306586	+	0.951843i	$0.400813\pi$
$384$	1.00000	0.0510310
$385$	−2.00000	−0.101929
$386$	2.00000	0.101797
$387$	4.00000	0.203331
$388$	−14.0000	−0.710742
$389$	22.0000	1.11544	0.557722	−	0.830028i	$-0.311675\pi$
$389$	22.0000	1.11544	0.557722	+	0.830028i	$0.311675\pi$
$390$	−4.00000	−0.202548
$391$	0	0
$392$	1.00000	0.0505076
$393$	0	0
$394$	6.00000	0.302276
$395$	−16.0000	−0.805047
$396$	1.00000	0.0502519
$397$	−30.0000	−1.50566	−0.752828	−	0.658217i	$-0.771311\pi$
$397$	−30.0000	−1.50566	−0.752828	+	0.658217i	$0.771311\pi$
$398$	20.0000	1.00251
$399$	0	0
$400$	−1.00000	−0.0500000
$401$	18.0000	0.898877	0.449439	−	0.893311i	$-0.351624\pi$
$401$	18.0000	0.898877	0.449439	+	0.893311i	$0.351624\pi$
$402$	12.0000	0.598506
$403$	−8.00000	−0.398508
$404$	−10.0000	−0.497519
$405$	2.00000	0.0993808
$406$	2.00000	0.0992583
$407$	−2.00000	−0.0991363
$408$	−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	$0.387496\pi$
$410$	−20.0000	−0.987730
$411$	10.0000	0.493264
$412$	4.00000	0.197066
$413$	12.0000	0.590481
$414$	0	0
$415$	−16.0000	−0.785409
$416$	−2.00000	−0.0980581
$417$	0	0
$418$	0	0
$419$	−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
$421$	6.00000	0.292422	0.146211	−	0.989253i	$-0.453292\pi$
$421$	6.00000	0.292422	0.146211	+	0.989253i	$0.453292\pi$
$422$	12.0000	0.584151
$423$	4.00000	0.194487
$424$	−2.00000	−0.0971286
$425$	2.00000	0.0970143
$426$	8.00000	0.387601
$427$	2.00000	0.0967868
$428$	−12.0000	−0.580042
$429$	−2.00000	−0.0965609
$430$	8.00000	0.385794
$431$	−40.0000	−1.92673	−0.963366	−	0.268190i	$-0.913575\pi$
$431$	−40.0000	−1.92673	−0.963366	+	0.268190i	$0.913575\pi$
$432$	1.00000	0.0481125
$433$	18.0000	0.865025	0.432512	−	0.901628i	$-0.357627\pi$
$433$	18.0000	0.865025	0.432512	+	0.901628i	$0.357627\pi$
$434$	−4.00000	−0.192006
$435$	−4.00000	−0.191785
$436$	14.0000	0.670478
$437$	0	0
$438$	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
$442$	4.00000	0.190261
$443$	−12.0000	−0.570137	−0.285069	−	0.958507i	$-0.592016\pi$
$443$	−12.0000	−0.570137	−0.285069	+	0.958507i	$0.592016\pi$
$444$	−2.00000	−0.0949158
$445$	−28.0000	−1.32733
$446$	20.0000	0.947027
$447$	−10.0000	−0.472984
$448$	−1.00000	−0.0472456
$449$	−30.0000	−1.41579	−0.707894	−	0.706319i	$-0.750354\pi$
$449$	−30.0000	−1.41579	−0.707894	+	0.706319i	$0.750354\pi$
$450$	−1.00000	−0.0471405
$451$	−10.0000	−0.470882
$452$	18.0000	0.846649
$453$	8.00000	0.375873
$454$	0	0
$455$	4.00000	0.187523
$456$	0	0
$457$	−38.0000	−1.77757	−0.888783	−	0.458329i	$-0.848448\pi$
$457$	−38.0000	−1.77757	−0.888783	+	0.458329i	$0.848448\pi$
$458$	−14.0000	−0.654177
$459$	−2.00000	−0.0933520
$460$	0	0
$461$	−18.0000	−0.838344	−0.419172	−	0.907907i	$-0.637680\pi$
$461$	−18.0000	−0.838344	−0.419172	+	0.907907i	$0.637680\pi$
$462$	−1.00000	−0.0465242
$463$	16.0000	0.743583	0.371792	−	0.928316i	$-0.378744\pi$
$463$	16.0000	0.743583	0.371792	+	0.928316i	$0.378744\pi$
$464$	−2.00000	−0.0928477
$465$	8.00000	0.370991
$466$	−6.00000	−0.277945
$467$	4.00000	0.185098	0.0925490	−	0.995708i	$-0.470499\pi$
$467$	4.00000	0.185098	0.0925490	+	0.995708i	$0.470499\pi$
$468$	−2.00000	−0.0924500
$469$	−12.0000	−0.554109
$470$	8.00000	0.369012
$471$	18.0000	0.829396
$472$	−12.0000	−0.552345
$473$	4.00000	0.183920
$474$	−8.00000	−0.367452
$475$	0	0
$476$	2.00000	0.0916698
$477$	−2.00000	−0.0915737
$478$	16.0000	0.731823
$479$	−32.0000	−1.46212	−0.731059	−	0.682315i	$-0.760973\pi$
$479$	−32.0000	−1.46212	−0.731059	+	0.682315i	$0.760973\pi$
$480$	2.00000	0.0912871
$481$	4.00000	0.182384
$482$	30.0000	1.36646
$483$	0	0
$484$	1.00000	0.0454545
$485$	−28.0000	−1.27141
$486$	1.00000	0.0453609
$487$	−32.0000	−1.45006	−0.725029	−	0.688718i	$-0.758174\pi$
$487$	−32.0000	−1.45006	−0.725029	+	0.688718i	$0.758174\pi$
$488$	−2.00000	−0.0905357
$489$	4.00000	0.180886
$490$	2.00000	0.0903508
$491$	−36.0000	−1.62466	−0.812329	−	0.583200i	$-0.801800\pi$
$491$	−36.0000	−1.62466	−0.812329	+	0.583200i	$0.801800\pi$
$492$	−10.0000	−0.450835
$493$	4.00000	0.180151
$494$	0	0
$495$	2.00000	0.0898933
$496$	4.00000	0.179605
$497$	−8.00000	−0.358849
$498$	−8.00000	−0.358489
$499$	28.0000	1.25345	0.626726	−	0.779240i	$-0.284395\pi$
$499$	28.0000	1.25345	0.626726	+	0.779240i	$0.284395\pi$
$500$	−12.0000	−0.536656
$501$	0	0
$502$	4.00000	0.178529
$503$	16.0000	0.713405	0.356702	−	0.934218i	$-0.383901\pi$
$503$	16.0000	0.713405	0.356702	+	0.934218i	$0.383901\pi$
$504$	−1.00000	−0.0445435
$505$	−20.0000	−0.889988
$506$	0	0
$507$	−9.00000	−0.399704
$508$	8.00000	0.354943
$509$	10.0000	0.443242	0.221621	−	0.975133i	$-0.428865\pi$
$509$	10.0000	0.443242	0.221621	+	0.975133i	$0.428865\pi$
$510$	−4.00000	−0.177123
$511$	−6.00000	−0.265424
$512$	1.00000	0.0441942
$513$	0	0
$514$	−6.00000	−0.264649
$515$	8.00000	0.352522
$516$	4.00000	0.176090
$517$	4.00000	0.175920
$518$	2.00000	0.0878750
$519$	6.00000	0.263371
$520$	−4.00000	−0.175412
$521$	−30.0000	−1.31432	−0.657162	−	0.753749i	$-0.728243\pi$
$521$	−30.0000	−1.31432	−0.657162	+	0.753749i	$0.728243\pi$
$522$	−2.00000	−0.0875376
$523$	−16.0000	−0.699631	−0.349816	−	0.936819i	$-0.613756\pi$
$523$	−16.0000	−0.699631	−0.349816	+	0.936819i	$0.613756\pi$
$524$	0	0
$525$	1.00000	0.0436436
$526$	32.0000	1.39527
$527$	−8.00000	−0.348485
$528$	1.00000	0.0435194
$529$	−23.0000	−1.00000
$530$	−4.00000	−0.173749
$531$	−12.0000	−0.520756
$532$	0	0
$533$	20.0000	0.866296
$534$	−14.0000	−0.605839
$535$	−24.0000	−1.03761
$536$	12.0000	0.518321
$537$	12.0000	0.517838
$538$	10.0000	0.431131
$539$	1.00000	0.0430730
$540$	2.00000	0.0860663
$541$	30.0000	1.28980	0.644900	−	0.764267i	$-0.276899\pi$
$541$	30.0000	1.28980	0.644900	+	0.764267i	$0.276899\pi$
$542$	−8.00000	−0.343629
$543$	2.00000	0.0858282
$544$	−2.00000	−0.0857493
$545$	28.0000	1.19939
$546$	2.00000	0.0855921
$547$	−44.0000	−1.88130	−0.940652	−	0.339372i	$-0.889785\pi$
$547$	−44.0000	−1.88130	−0.940652	+	0.339372i	$0.889785\pi$
$548$	10.0000	0.427179
$549$	−2.00000	−0.0853579
$550$	−1.00000	−0.0426401
$551$	0	0
$552$	0	0
$553$	8.00000	0.340195
$554$	−10.0000	−0.424859
$555$	−4.00000	−0.169791
$556$	0	0
$557$	14.0000	0.593199	0.296600	−	0.955002i	$-0.404147\pi$
$557$	14.0000	0.593199	0.296600	+	0.955002i	$0.404147\pi$
$558$	4.00000	0.169334
$559$	−8.00000	−0.338364
$560$	−2.00000	−0.0845154
$561$	−2.00000	−0.0844401
$562$	10.0000	0.421825
$563$	0	0		−	1.00000i	$-0.5\pi$
$563$	0	0			1.00000i	$0.5\pi$
$564$	4.00000	0.168430
$565$	36.0000	1.51453
$566$	16.0000	0.672530
$567$	−1.00000	−0.0419961
$568$	8.00000	0.335673
$569$	−22.0000	−0.922288	−0.461144	−	0.887325i	$-0.652561\pi$
$569$	−22.0000	−0.922288	−0.461144	+	0.887325i	$0.652561\pi$
$570$	0	0
$571$	44.0000	1.84134	0.920671	−	0.390339i	$-0.127642\pi$
$571$	44.0000	1.84134	0.920671	+	0.390339i	$0.127642\pi$
$572$	−2.00000	−0.0836242
$573$	−8.00000	−0.334205
$574$	10.0000	0.417392
$575$	0	0
$576$	1.00000	0.0416667
$577$	10.0000	0.416305	0.208153	−	0.978096i	$-0.433255\pi$
$577$	10.0000	0.416305	0.208153	+	0.978096i	$0.433255\pi$
$578$	−13.0000	−0.540729
$579$	2.00000	0.0831172
$580$	−4.00000	−0.166091
$581$	8.00000	0.331896
$582$	−14.0000	−0.580319
$583$	−2.00000	−0.0828315
$584$	6.00000	0.248282
$585$	−4.00000	−0.165380
$586$	14.0000	0.578335
$587$	−28.0000	−1.15568	−0.577842	−	0.816149i	$-0.696105\pi$
$587$	−28.0000	−1.15568	−0.577842	+	0.816149i	$0.696105\pi$
$588$	1.00000	0.0412393
$589$	0	0
$590$	−24.0000	−0.988064
$591$	6.00000	0.246807
$592$	−2.00000	−0.0821995
$593$	−34.0000	−1.39621	−0.698106	−	0.715994i	$-0.745974\pi$
$593$	−34.0000	−1.39621	−0.698106	+	0.715994i	$0.745974\pi$
$594$	1.00000	0.0410305
$595$	4.00000	0.163984
$596$	−10.0000	−0.409616
$597$	20.0000	0.818546
$598$	0	0
$599$	−24.0000	−0.980613	−0.490307	−	0.871550i	$-0.663115\pi$
$599$	−24.0000	−0.980613	−0.490307	+	0.871550i	$0.663115\pi$
$600$	−1.00000	−0.0408248
$601$	−18.0000	−0.734235	−0.367118	−	0.930175i	$-0.619655\pi$
$601$	−18.0000	−0.734235	−0.367118	+	0.930175i	$0.619655\pi$
$602$	−4.00000	−0.163028
$603$	12.0000	0.488678
$604$	8.00000	0.325515
$605$	2.00000	0.0813116
$606$	−10.0000	−0.406222
$607$	32.0000	1.29884	0.649420	−	0.760430i	$-0.275012\pi$
$607$	32.0000	1.29884	0.649420	+	0.760430i	$0.275012\pi$
$608$	0	0
$609$	2.00000	0.0810441
$610$	−4.00000	−0.161955
$611$	−8.00000	−0.323645
$612$	−2.00000	−0.0808452
$613$	38.0000	1.53481	0.767403	−	0.641165i	$-0.221549\pi$
$613$	38.0000	1.53481	0.767403	+	0.641165i	$0.221549\pi$
$614$	8.00000	0.322854
$615$	−20.0000	−0.806478
$616$	−1.00000	−0.0402911
$617$	26.0000	1.04672	0.523360	−	0.852111i	$-0.324678\pi$
$617$	26.0000	1.04672	0.523360	+	0.852111i	$0.324678\pi$
$618$	4.00000	0.160904
$619$	−20.0000	−0.803868	−0.401934	−	0.915669i	$-0.631662\pi$
$619$	−20.0000	−0.803868	−0.401934	+	0.915669i	$0.631662\pi$
$620$	8.00000	0.321288
$621$	0	0
$622$	20.0000	0.801927
$623$	14.0000	0.560898
$624$	−2.00000	−0.0800641
$625$	−19.0000	−0.760000
$626$	−30.0000	−1.19904
$627$	0	0
$628$	18.0000	0.718278
$629$	4.00000	0.159490
$630$	−2.00000	−0.0796819
$631$	0	0		−	1.00000i	$-0.5\pi$
$631$	0	0			1.00000i	$0.5\pi$
$632$	−8.00000	−0.318223
$633$	12.0000	0.476957
$634$	22.0000	0.873732
$635$	16.0000	0.634941
$636$	−2.00000	−0.0793052
$637$	−2.00000	−0.0792429
$638$	−2.00000	−0.0791808
$639$	8.00000	0.316475
$640$	2.00000	0.0790569
$641$	18.0000	0.710957	0.355479	−	0.934684i	$-0.384318\pi$
$641$	18.0000	0.710957	0.355479	+	0.934684i	$0.384318\pi$
$642$	−12.0000	−0.473602
$643$	44.0000	1.73519	0.867595	−	0.497271i	$-0.165665\pi$
$643$	44.0000	1.73519	0.867595	+	0.497271i	$0.165665\pi$
$644$	0	0
$645$	8.00000	0.315000
$646$	0	0
$647$	−28.0000	−1.10079	−0.550397	−	0.834903i	$-0.685524\pi$
$647$	−28.0000	−1.10079	−0.550397	+	0.834903i	$0.685524\pi$
$648$	1.00000	0.0392837
$649$	−12.0000	−0.471041
$650$	2.00000	0.0784465
$651$	−4.00000	−0.156772
$652$	4.00000	0.156652
$653$	30.0000	1.17399	0.586995	−	0.809590i	$-0.300311\pi$
$653$	30.0000	1.17399	0.586995	+	0.809590i	$0.300311\pi$
$654$	14.0000	0.547443
$655$	0	0
$656$	−10.0000	−0.390434
$657$	6.00000	0.234082
$658$	−4.00000	−0.155936
$659$	−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
$663$	4.00000	0.155347
$664$	−8.00000	−0.310460
$665$	0	0
$666$	−2.00000	−0.0774984
$667$	0	0
$668$	0	0
$669$	20.0000	0.773245
$670$	24.0000	0.927201
$671$	−2.00000	−0.0772091
$672$	−1.00000	−0.0385758
$673$	26.0000	1.00223	0.501113	−	0.865382i	$-0.332924\pi$
$673$	26.0000	1.00223	0.501113	+	0.865382i	$0.332924\pi$
$674$	−22.0000	−0.847408
$675$	−1.00000	−0.0384900
$676$	−9.00000	−0.346154
$677$	14.0000	0.538064	0.269032	−	0.963131i	$-0.413296\pi$
$677$	14.0000	0.538064	0.269032	+	0.963131i	$0.413296\pi$
$678$	18.0000	0.691286
$679$	14.0000	0.537271
$680$	−4.00000	−0.153393
$681$	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	$0.679950\pi$
$684$	0	0
$685$	20.0000	0.764161
$686$	−1.00000	−0.0381802
$687$	−14.0000	−0.534133
$688$	4.00000	0.152499
$689$	4.00000	0.152388
$690$	0	0
$691$	28.0000	1.06517	0.532585	−	0.846376i	$-0.321221\pi$
$691$	28.0000	1.06517	0.532585	+	0.846376i	$0.321221\pi$
$692$	6.00000	0.228086
$693$	−1.00000	−0.0379869
$694$	−12.0000	−0.455514
$695$	0	0
$696$	−2.00000	−0.0758098
$697$	20.0000	0.757554
$698$	−26.0000	−0.984115
$699$	−6.00000	−0.226941
$700$	1.00000	0.0377964
$701$	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
$704$	1.00000	0.0376889
$705$	8.00000	0.301297
$706$	−6.00000	−0.225813
$707$	10.0000	0.376089
$708$	−12.0000	−0.450988
$709$	−2.00000	−0.0751116	−0.0375558	−	0.999295i	$-0.511957\pi$
$709$	−2.00000	−0.0751116	−0.0375558	+	0.999295i	$0.511957\pi$
$710$	16.0000	0.600469
$711$	−8.00000	−0.300023
$712$	−14.0000	−0.524672
$713$	0	0
$714$	2.00000	0.0748481
$715$	−4.00000	−0.149592
$716$	12.0000	0.448461
$717$	16.0000	0.597531
$718$	−32.0000	−1.19423
$719$	28.0000	1.04422	0.522112	−	0.852877i	$-0.325144\pi$
$719$	28.0000	1.04422	0.522112	+	0.852877i	$0.325144\pi$
$720$	2.00000	0.0745356
$721$	−4.00000	−0.148968
$722$	−19.0000	−0.707107
$723$	30.0000	1.11571
$724$	2.00000	0.0743294
$725$	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
$731$	−8.00000	−0.295891
$732$	−2.00000	−0.0739221
$733$	46.0000	1.69905	0.849524	−	0.527549i	$-0.176889\pi$
$733$	46.0000	1.69905	0.849524	+	0.527549i	$0.176889\pi$
$734$	−28.0000	−1.03350
$735$	2.00000	0.0737711
$736$	0	0
$737$	12.0000	0.442026
$738$	−10.0000	−0.368105
$739$	20.0000	0.735712	0.367856	−	0.929883i	$-0.380092\pi$
$739$	20.0000	0.735712	0.367856	+	0.929883i	$0.380092\pi$
$740$	−4.00000	−0.147043
$741$	0	0
$742$	2.00000	0.0734223
$743$	−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
$746$	22.0000	0.805477
$747$	−8.00000	−0.292705
$748$	−2.00000	−0.0731272
$749$	12.0000	0.438470
$750$	−12.0000	−0.438178
$751$	−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
$753$	4.00000	0.145768
$754$	4.00000	0.145671
$755$	16.0000	0.582300
$756$	−1.00000	−0.0363696
$757$	−18.0000	−0.654221	−0.327111	−	0.944986i	$-0.606075\pi$
$757$	−18.0000	−0.654221	−0.327111	+	0.944986i	$0.606075\pi$
$758$	−20.0000	−0.726433
$759$	0	0
$760$	0	0
$761$	−2.00000	−0.0724999	−0.0362500	−	0.999343i	$-0.511541\pi$
$761$	−2.00000	−0.0724999	−0.0362500	+	0.999343i	$0.511541\pi$
$762$	8.00000	0.289809
$763$	−14.0000	−0.506834
$764$	−8.00000	−0.289430
$765$	−4.00000	−0.144620
$766$	12.0000	0.433578
$767$	24.0000	0.866590
$768$	1.00000	0.0360844
$769$	−26.0000	−0.937584	−0.468792	−	0.883309i	$-0.655311\pi$
$769$	−26.0000	−0.937584	−0.468792	+	0.883309i	$0.655311\pi$
$770$	−2.00000	−0.0720750
$771$	−6.00000	−0.216085
$772$	2.00000	0.0719816
$773$	10.0000	0.359675	0.179838	−	0.983696i	$-0.442443\pi$
$773$	10.0000	0.359675	0.179838	+	0.983696i	$0.442443\pi$
$774$	4.00000	0.143777
$775$	−4.00000	−0.143684
$776$	−14.0000	−0.502571
$777$	2.00000	0.0717496
$778$	22.0000	0.788738
$779$	0	0
$780$	−4.00000	−0.143223
$781$	8.00000	0.286263
$782$	0	0
$783$	−2.00000	−0.0714742
$784$	1.00000	0.0357143
$785$	36.0000	1.28490
$786$	0	0
$787$	32.0000	1.14068	0.570338	−	0.821410i	$-0.306812\pi$
$787$	32.0000	1.14068	0.570338	+	0.821410i	$0.306812\pi$
$788$	6.00000	0.213741
$789$	32.0000	1.13923
$790$	−16.0000	−0.569254
$791$	−18.0000	−0.640006
$792$	1.00000	0.0355335
$793$	4.00000	0.142044
$794$	−30.0000	−1.06466
$795$	−4.00000	−0.141865
$796$	20.0000	0.708881
$797$	34.0000	1.20434	0.602171	−	0.798367i	$-0.294303\pi$
$797$	34.0000	1.20434	0.602171	+	0.798367i	$0.294303\pi$
$798$	0	0
$799$	−8.00000	−0.283020
$800$	−1.00000	−0.0353553
$801$	−14.0000	−0.494666
$802$	18.0000	0.635602
$803$	6.00000	0.211735
$804$	12.0000	0.423207
$805$	0	0
$806$	−8.00000	−0.281788
$807$	10.0000	0.352017
$808$	−10.0000	−0.351799
$809$	−14.0000	−0.492214	−0.246107	−	0.969243i	$-0.579151\pi$
$809$	−14.0000	−0.492214	−0.246107	+	0.969243i	$0.579151\pi$
$810$	2.00000	0.0702728
$811$	0	0		−	1.00000i	$-0.5\pi$
$811$	0	0			1.00000i	$0.5\pi$
$812$	2.00000	0.0701862
$813$	−8.00000	−0.280572
$814$	−2.00000	−0.0701000
$815$	8.00000	0.280228
$816$	−2.00000	−0.0700140
$817$	0	0
$818$	14.0000	0.489499
$819$	2.00000	0.0698857
$820$	−20.0000	−0.698430
$821$	22.0000	0.767805	0.383903	−	0.923374i	$-0.374580\pi$
$821$	22.0000	0.767805	0.383903	+	0.923374i	$0.374580\pi$
$822$	10.0000	0.348790
$823$	−48.0000	−1.67317	−0.836587	−	0.547833i	$-0.815453\pi$
$823$	−48.0000	−1.67317	−0.836587	+	0.547833i	$0.815453\pi$
$824$	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	$0.715277\pi$
$828$	0	0
$829$	−6.00000	−0.208389	−0.104194	−	0.994557i	$-0.533226\pi$
$829$	−6.00000	−0.208389	−0.104194	+	0.994557i	$0.533226\pi$
$830$	−16.0000	−0.555368
$831$	−10.0000	−0.346896
$832$	−2.00000	−0.0693375
$833$	−2.00000	−0.0692959
$834$	0	0
$835$	0	0
$836$	0	0
$837$	4.00000	0.138260
$838$	−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	$0.566417\pi$
$840$	−2.00000	−0.0690066
$841$	−25.0000	−0.862069
$842$	6.00000	0.206774
$843$	10.0000	0.344418
$844$	12.0000	0.413057
$845$	−18.0000	−0.619219
$846$	4.00000	0.137523
$847$	−1.00000	−0.0343604
$848$	−2.00000	−0.0686803
$849$	16.0000	0.549119
$850$	2.00000	0.0685994
$851$	0	0
$852$	8.00000	0.274075
$853$	6.00000	0.205436	0.102718	−	0.994711i	$-0.467246\pi$
$853$	6.00000	0.205436	0.102718	+	0.994711i	$0.467246\pi$
$854$	2.00000	0.0684386
$855$	0	0
$856$	−12.0000	−0.410152
$857$	−34.0000	−1.16142	−0.580709	−	0.814111i	$-0.697225\pi$
$857$	−34.0000	−1.16142	−0.580709	+	0.814111i	$0.697225\pi$
$858$	−2.00000	−0.0682789
$859$	4.00000	0.136478	0.0682391	−	0.997669i	$-0.478262\pi$
$859$	4.00000	0.136478	0.0682391	+	0.997669i	$0.478262\pi$
$860$	8.00000	0.272798
$861$	10.0000	0.340799
$862$	−40.0000	−1.36241
$863$	24.0000	0.816970	0.408485	−	0.912765i	$-0.366057\pi$
$863$	24.0000	0.816970	0.408485	+	0.912765i	$0.366057\pi$
$864$	1.00000	0.0340207
$865$	12.0000	0.408012
$866$	18.0000	0.611665
$867$	−13.0000	−0.441503
$868$	−4.00000	−0.135769
$869$	−8.00000	−0.271381
$870$	−4.00000	−0.135613
$871$	−24.0000	−0.813209
$872$	14.0000	0.474100
$873$	−14.0000	−0.473828
$874$	0	0
$875$	12.0000	0.405674
$876$	6.00000	0.202721
$877$	−2.00000	−0.0675352	−0.0337676	−	0.999430i	$-0.510751\pi$
$877$	−2.00000	−0.0675352	−0.0337676	+	0.999430i	$0.510751\pi$
$878$	16.0000	0.539974
$879$	14.0000	0.472208
$880$	2.00000	0.0674200
$881$	−14.0000	−0.471672	−0.235836	−	0.971793i	$-0.575783\pi$
$881$	−14.0000	−0.471672	−0.235836	+	0.971793i	$0.575783\pi$
$882$	1.00000	0.0336718
$883$	−28.0000	−0.942275	−0.471138	−	0.882060i	$-0.656156\pi$
$883$	−28.0000	−0.942275	−0.471138	+	0.882060i	$0.656156\pi$
$884$	4.00000	0.134535
$885$	−24.0000	−0.806751
$886$	−12.0000	−0.403148
$887$	24.0000	0.805841	0.402921	−	0.915235i	$-0.367995\pi$
$887$	24.0000	0.805841	0.402921	+	0.915235i	$0.367995\pi$
$888$	−2.00000	−0.0671156
$889$	−8.00000	−0.268311
$890$	−28.0000	−0.938562
$891$	1.00000	0.0335013
$892$	20.0000	0.669650
$893$	0	0
$894$	−10.0000	−0.334450
$895$	24.0000	0.802232
$896$	−1.00000	−0.0334077
$897$	0	0
$898$	−30.0000	−1.00111
$899$	−8.00000	−0.266815
$900$	−1.00000	−0.0333333
$901$	4.00000	0.133259
$902$	−10.0000	−0.332964
$903$	−4.00000	−0.133112
$904$	18.0000	0.598671
$905$	4.00000	0.132964
$906$	8.00000	0.265782
$907$	−12.0000	−0.398453	−0.199227	−	0.979953i	$-0.563843\pi$
$907$	−12.0000	−0.398453	−0.199227	+	0.979953i	$0.563843\pi$
$908$	0	0
$909$	−10.0000	−0.331679
$910$	4.00000	0.132599
$911$	−40.0000	−1.32526	−0.662630	−	0.748947i	$-0.730560\pi$
$911$	−40.0000	−1.32526	−0.662630	+	0.748947i	$0.730560\pi$
$912$	0	0
$913$	−8.00000	−0.264761
$914$	−38.0000	−1.25693
$915$	−4.00000	−0.132236
$916$	−14.0000	−0.462573
$917$	0	0
$918$	−2.00000	−0.0660098
$919$	−40.0000	−1.31948	−0.659739	−	0.751495i	$-0.729333\pi$
$919$	−40.0000	−1.31948	−0.659739	+	0.751495i	$0.729333\pi$
$920$	0	0
$921$	8.00000	0.263609
$922$	−18.0000	−0.592798
$923$	−16.0000	−0.526646
$924$	−1.00000	−0.0328976
$925$	2.00000	0.0657596
$926$	16.0000	0.525793
$927$	4.00000	0.131377
$928$	−2.00000	−0.0656532
$929$	−54.0000	−1.77168	−0.885841	−	0.463988i	$-0.846418\pi$
$929$	−54.0000	−1.77168	−0.885841	+	0.463988i	$0.846418\pi$
$930$	8.00000	0.262330
$931$	0	0
$932$	−6.00000	−0.196537
$933$	20.0000	0.654771
$934$	4.00000	0.130884
$935$	−4.00000	−0.130814
$936$	−2.00000	−0.0653720
$937$	22.0000	0.718709	0.359354	−	0.933201i	$-0.382997\pi$
$937$	22.0000	0.718709	0.359354	+	0.933201i	$0.382997\pi$
$938$	−12.0000	−0.391814
$939$	−30.0000	−0.979013
$940$	8.00000	0.260931
$941$	6.00000	0.195594	0.0977972	−	0.995206i	$-0.468820\pi$
$941$	6.00000	0.195594	0.0977972	+	0.995206i	$0.468820\pi$
$942$	18.0000	0.586472
$943$	0	0
$944$	−12.0000	−0.390567
$945$	−2.00000	−0.0650600
$946$	4.00000	0.130051
$947$	4.00000	0.129983	0.0649913	−	0.997886i	$-0.479298\pi$
$947$	4.00000	0.129983	0.0649913	+	0.997886i	$0.479298\pi$
$948$	−8.00000	−0.259828
$949$	−12.0000	−0.389536
$950$	0	0
$951$	22.0000	0.713399
$952$	2.00000	0.0648204
$953$	18.0000	0.583077	0.291539	−	0.956559i	$-0.405833\pi$
$953$	18.0000	0.583077	0.291539	+	0.956559i	$0.405833\pi$
$954$	−2.00000	−0.0647524
$955$	−16.0000	−0.517748
$956$	16.0000	0.517477
$957$	−2.00000	−0.0646508
$958$	−32.0000	−1.03387
$959$	−10.0000	−0.322917
$960$	2.00000	0.0645497
$961$	−15.0000	−0.483871
$962$	4.00000	0.128965
$963$	−12.0000	−0.386695
$964$	30.0000	0.966235
$965$	4.00000	0.128765
$966$	0	0
$967$	−40.0000	−1.28631	−0.643157	−	0.765735i	$-0.722376\pi$
$967$	−40.0000	−1.28631	−0.643157	+	0.765735i	$0.722376\pi$
$968$	1.00000	0.0321412
$969$	0	0
$970$	−28.0000	−0.899026
$971$	−52.0000	−1.66876	−0.834380	−	0.551190i	$-0.814174\pi$
$971$	−52.0000	−1.66876	−0.834380	+	0.551190i	$0.814174\pi$
$972$	1.00000	0.0320750
$973$	0	0
$974$	−32.0000	−1.02535
$975$	2.00000	0.0640513
$976$	−2.00000	−0.0640184
$977$	2.00000	0.0639857	0.0319928	−	0.999488i	$-0.489815\pi$
$977$	2.00000	0.0639857	0.0319928	+	0.999488i	$0.489815\pi$
$978$	4.00000	0.127906
$979$	−14.0000	−0.447442
$980$	2.00000	0.0638877
$981$	14.0000	0.446986
$982$	−36.0000	−1.14881
$983$	−4.00000	−0.127580	−0.0637901	−	0.997963i	$-0.520319\pi$
$983$	−4.00000	−0.127580	−0.0637901	+	0.997963i	$0.520319\pi$
$984$	−10.0000	−0.318788
$985$	12.0000	0.382352
$986$	4.00000	0.127386
$987$	−4.00000	−0.127321
$988$	0	0
$989$	0	0
$990$	2.00000	0.0635642
$991$	16.0000	0.508257	0.254128	−	0.967170i	$-0.418211\pi$
$991$	16.0000	0.508257	0.254128	+	0.967170i	$0.418211\pi$
$992$	4.00000	0.127000
$993$	20.0000	0.634681
$994$	−8.00000	−0.253745
$995$	40.0000	1.26809
$996$	−8.00000	−0.253490
$997$	14.0000	0.443384	0.221692	−	0.975117i	$-0.428842\pi$
$997$	14.0000	0.443384	0.221692	+	0.975117i	$0.428842\pi$
$998$	28.0000	0.886325
$999$	−2.00000	−0.0632772

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	462.2.a.g.1.1	✓	1
3.2	odd	2		1386.2.a.a.1.1		1
4.3	odd	2		3696.2.a.m.1.1		1
7.6	odd	2		3234.2.a.p.1.1		1
11.10	odd	2		5082.2.a.n.1.1		1
21.20	even	2		9702.2.a.r.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
462.2.a.g.1.1	✓	1	1.1	even	1	trivial
1386.2.a.a.1.1		1	3.2	odd	2
3234.2.a.p.1.1		1	7.6	odd	2
3696.2.a.m.1.1		1	4.3	odd	2
5082.2.a.n.1.1		1	11.10	odd	2
9702.2.a.r.1.1		1	21.20	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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