LMFDB - Embedded newform 6422.2.a.c.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6422,2,Mod(1,6422)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6422, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("6422.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6422 = 2 \cdot 13^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6422.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:	$51.2799281781$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 494)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	6422.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +3.00000 q^{5} -1.00000 q^{6} -1.00000 q^{7} -1.00000 q^{8} -2.00000 q^{9} +O(q^{10})$

\(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +3.00000 q^{5} -1.00000 q^{6} -1.00000 q^{7} -1.00000 q^{8} -2.00000 q^{9} -3.00000 q^{10} -2.00000 q^{11} +1.00000 q^{12} +1.00000 q^{14} +3.00000 q^{15} +1.00000 q^{16} +1.00000 q^{17} +2.00000 q^{18} +1.00000 q^{19} +3.00000 q^{20} -1.00000 q^{21} +2.00000 q^{22} -4.00000 q^{23} -1.00000 q^{24} +4.00000 q^{25} -5.00000 q^{27} -1.00000 q^{28} +2.00000 q^{29} -3.00000 q^{30} +4.00000 q^{31} -1.00000 q^{32} -2.00000 q^{33} -1.00000 q^{34} -3.00000 q^{35} -2.00000 q^{36} -3.00000 q^{37} -1.00000 q^{38} -3.00000 q^{40} -10.0000 q^{41} +1.00000 q^{42} -11.0000 q^{43} -2.00000 q^{44} -6.00000 q^{45} +4.00000 q^{46} +3.00000 q^{47} +1.00000 q^{48} -6.00000 q^{49} -4.00000 q^{50} +1.00000 q^{51} +5.00000 q^{54} -6.00000 q^{55} +1.00000 q^{56} +1.00000 q^{57} -2.00000 q^{58} +6.00000 q^{59} +3.00000 q^{60} +6.00000 q^{61} -4.00000 q^{62} +2.00000 q^{63} +1.00000 q^{64} +2.00000 q^{66} +2.00000 q^{67} +1.00000 q^{68} -4.00000 q^{69} +3.00000 q^{70} +3.00000 q^{71} +2.00000 q^{72} +4.00000 q^{73} +3.00000 q^{74} +4.00000 q^{75} +1.00000 q^{76} +2.00000 q^{77} -6.00000 q^{79} +3.00000 q^{80} +1.00000 q^{81} +10.0000 q^{82} +14.0000 q^{83} -1.00000 q^{84} +3.00000 q^{85} +11.0000 q^{86} +2.00000 q^{87} +2.00000 q^{88} -8.00000 q^{89} +6.00000 q^{90} -4.00000 q^{92} +4.00000 q^{93} -3.00000 q^{94} +3.00000 q^{95} -1.00000 q^{96} -16.0000 q^{97} +6.00000 q^{98} +4.00000 q^{99} +O(q^{100})\)

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	0.288675	−	0.957427i	$-0.406785\pi$
$3$	1.00000	0.577350	0.288675	+	0.957427i	$0.406785\pi$
$4$	1.00000	0.500000
$5$	3.00000	1.34164	0.670820	−	0.741620i	$-0.265942\pi$
$5$	3.00000	1.34164	0.670820	+	0.741620i	$0.265942\pi$
$6$	−1.00000	−0.408248
$7$	−1.00000	−0.377964	−0.188982	−	0.981981i	$-0.560519\pi$
$7$	−1.00000	−0.377964	−0.188982	+	0.981981i	$0.560519\pi$
$8$	−1.00000	−0.353553
$9$	−2.00000	−0.666667
$10$	−3.00000	−0.948683
$11$	−2.00000	−0.603023	−0.301511	−	0.953463i	$-0.597491\pi$
$11$	−2.00000	−0.603023	−0.301511	+	0.953463i	$0.597491\pi$
$12$	1.00000	0.288675
$13$	0	0
$13$	0	0
$14$	1.00000	0.267261
$15$	3.00000	0.774597
$16$	1.00000	0.250000
$17$	1.00000	0.242536	0.121268	−	0.992620i	$-0.461304\pi$
$17$	1.00000	0.242536	0.121268	+	0.992620i	$0.461304\pi$
$18$	2.00000	0.471405
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	3.00000	0.670820
$21$	−1.00000	−0.218218
$22$	2.00000	0.426401
$23$	−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
$25$	4.00000	0.800000
$26$	0	0
$27$	−5.00000	−0.962250
$28$	−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	$0.440546\pi$
$30$	−3.00000	−0.547723
$31$	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
$33$	−2.00000	−0.348155
$34$	−1.00000	−0.171499
$35$	−3.00000	−0.507093
$36$	−2.00000	−0.333333
$37$	−3.00000	−0.493197	−0.246598	−	0.969118i	$-0.579313\pi$
$37$	−3.00000	−0.493197	−0.246598	+	0.969118i	$0.579313\pi$
$38$	−1.00000	−0.162221
$39$	0	0
$40$	−3.00000	−0.474342
$41$	−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
$43$	−11.0000	−1.67748	−0.838742	−	0.544529i	$-0.816708\pi$
$43$	−11.0000	−1.67748	−0.838742	+	0.544529i	$0.816708\pi$
$44$	−2.00000	−0.301511
$45$	−6.00000	−0.894427
$46$	4.00000	0.589768
$47$	3.00000	0.437595	0.218797	−	0.975770i	$-0.429787\pi$
$47$	3.00000	0.437595	0.218797	+	0.975770i	$0.429787\pi$
$48$	1.00000	0.144338
$49$	−6.00000	−0.857143
$50$	−4.00000	−0.565685
$51$	1.00000	0.140028
$52$	0	0
$53$	0	0		−	1.00000i	$-0.5\pi$
$53$	0	0			1.00000i	$0.5\pi$
$54$	5.00000	0.680414
$55$	−6.00000	−0.809040
$56$	1.00000	0.133631
$57$	1.00000	0.132453
$58$	−2.00000	−0.262613
$59$	6.00000	0.781133	0.390567	−	0.920575i	$-0.372279\pi$
$59$	6.00000	0.781133	0.390567	+	0.920575i	$0.372279\pi$
$60$	3.00000	0.387298
$61$	6.00000	0.768221	0.384111	−	0.923287i	$-0.374508\pi$
$61$	6.00000	0.768221	0.384111	+	0.923287i	$0.374508\pi$
$62$	−4.00000	−0.508001
$63$	2.00000	0.251976
$64$	1.00000	0.125000
$65$	0	0
$66$	2.00000	0.246183
$67$	2.00000	0.244339	0.122169	−	0.992509i	$-0.461015\pi$
$67$	2.00000	0.244339	0.122169	+	0.992509i	$0.461015\pi$
$68$	1.00000	0.121268
$69$	−4.00000	−0.481543
$70$	3.00000	0.358569
$71$	3.00000	0.356034	0.178017	−	0.984027i	$-0.443032\pi$
$71$	3.00000	0.356034	0.178017	+	0.984027i	$0.443032\pi$
$72$	2.00000	0.235702
$73$	4.00000	0.468165	0.234082	−	0.972217i	$-0.424791\pi$
$73$	4.00000	0.468165	0.234082	+	0.972217i	$0.424791\pi$
$74$	3.00000	0.348743
$75$	4.00000	0.461880
$76$	1.00000	0.114708
$77$	2.00000	0.227921
$78$	0	0
$79$	−6.00000	−0.675053	−0.337526	−	0.941316i	$-0.609590\pi$
$79$	−6.00000	−0.675053	−0.337526	+	0.941316i	$0.609590\pi$
$80$	3.00000	0.335410
$81$	1.00000	0.111111
$82$	10.0000	1.10432
$83$	14.0000	1.53670	0.768350	−	0.640030i	$-0.221078\pi$
$83$	14.0000	1.53670	0.768350	+	0.640030i	$0.221078\pi$
$84$	−1.00000	−0.109109
$85$	3.00000	0.325396
$86$	11.0000	1.18616
$87$	2.00000	0.214423
$88$	2.00000	0.213201
$89$	−8.00000	−0.847998	−0.423999	−	0.905663i	$-0.639374\pi$
$89$	−8.00000	−0.847998	−0.423999	+	0.905663i	$0.639374\pi$
$90$	6.00000	0.632456
$91$	0	0
$92$	−4.00000	−0.417029
$93$	4.00000	0.414781
$94$	−3.00000	−0.309426
$95$	3.00000	0.307794
$96$	−1.00000	−0.102062
$97$	−16.0000	−1.62455	−0.812277	−	0.583272i	$-0.801772\pi$
$97$	−16.0000	−1.62455	−0.812277	+	0.583272i	$0.801772\pi$
$98$	6.00000	0.606092
$99$	4.00000	0.402015
$100$	4.00000	0.400000
$101$	−6.00000	−0.597022	−0.298511	−	0.954406i	$-0.596490\pi$
$101$	−6.00000	−0.597022	−0.298511	+	0.954406i	$0.596490\pi$
$102$	−1.00000	−0.0990148
$103$	−8.00000	−0.788263	−0.394132	−	0.919054i	$-0.628955\pi$
$103$	−8.00000	−0.788263	−0.394132	+	0.919054i	$0.628955\pi$
$104$	0	0
$105$	−3.00000	−0.292770
$106$	0	0
$107$	−12.0000	−1.16008	−0.580042	−	0.814587i	$-0.696964\pi$
$107$	−12.0000	−1.16008	−0.580042	+	0.814587i	$0.696964\pi$
$108$	−5.00000	−0.481125
$109$	9.00000	0.862044	0.431022	−	0.902342i	$-0.358153\pi$
$109$	9.00000	0.862044	0.431022	+	0.902342i	$0.358153\pi$
$110$	6.00000	0.572078
$111$	−3.00000	−0.284747
$112$	−1.00000	−0.0944911
$113$	−18.0000	−1.69330	−0.846649	−	0.532152i	$-0.821383\pi$
$113$	−18.0000	−1.69330	−0.846649	+	0.532152i	$0.821383\pi$
$114$	−1.00000	−0.0936586
$115$	−12.0000	−1.11901
$116$	2.00000	0.185695
$117$	0	0
$118$	−6.00000	−0.552345
$119$	−1.00000	−0.0916698
$120$	−3.00000	−0.273861
$121$	−7.00000	−0.636364
$122$	−6.00000	−0.543214
$123$	−10.0000	−0.901670
$124$	4.00000	0.359211
$125$	−3.00000	−0.268328
$126$	−2.00000	−0.178174
$127$	0	0		−	1.00000i	$-0.5\pi$
$127$	0	0			1.00000i	$0.5\pi$
$128$	−1.00000	−0.0883883
$129$	−11.0000	−0.968496
$130$	0	0
$131$	−15.0000	−1.31056	−0.655278	−	0.755388i	$-0.727449\pi$
$131$	−15.0000	−1.31056	−0.655278	+	0.755388i	$0.727449\pi$
$132$	−2.00000	−0.174078
$133$	−1.00000	−0.0867110
$134$	−2.00000	−0.172774
$135$	−15.0000	−1.29099
$136$	−1.00000	−0.0857493
$137$	−2.00000	−0.170872	−0.0854358	−	0.996344i	$-0.527228\pi$
$137$	−2.00000	−0.170872	−0.0854358	+	0.996344i	$0.527228\pi$
$138$	4.00000	0.340503
$139$	1.00000	0.0848189	0.0424094	−	0.999100i	$-0.486497\pi$
$139$	1.00000	0.0848189	0.0424094	+	0.999100i	$0.486497\pi$
$140$	−3.00000	−0.253546
$141$	3.00000	0.252646
$142$	−3.00000	−0.251754
$143$	0	0
$144$	−2.00000	−0.166667
$145$	6.00000	0.498273
$146$	−4.00000	−0.331042
$147$	−6.00000	−0.494872
$148$	−3.00000	−0.246598
$149$	−6.00000	−0.491539	−0.245770	−	0.969328i	$-0.579041\pi$
$149$	−6.00000	−0.491539	−0.245770	+	0.969328i	$0.579041\pi$
$150$	−4.00000	−0.326599
$151$	−1.00000	−0.0813788	−0.0406894	−	0.999172i	$-0.512955\pi$
$151$	−1.00000	−0.0813788	−0.0406894	+	0.999172i	$0.512955\pi$
$152$	−1.00000	−0.0811107
$153$	−2.00000	−0.161690
$154$	−2.00000	−0.161165
$155$	12.0000	0.963863
$156$	0	0
$157$	14.0000	1.11732	0.558661	−	0.829396i	$-0.311315\pi$
$157$	14.0000	1.11732	0.558661	+	0.829396i	$0.311315\pi$
$158$	6.00000	0.477334
$159$	0	0
$160$	−3.00000	−0.237171
$161$	4.00000	0.315244
$162$	−1.00000	−0.0785674
$163$	−12.0000	−0.939913	−0.469956	−	0.882690i	$-0.655730\pi$
$163$	−12.0000	−0.939913	−0.469956	+	0.882690i	$0.655730\pi$
$164$	−10.0000	−0.780869
$165$	−6.00000	−0.467099
$166$	−14.0000	−1.08661
$167$	24.0000	1.85718	0.928588	−	0.371113i	$-0.121024\pi$
$167$	24.0000	1.85718	0.928588	+	0.371113i	$0.121024\pi$
$168$	1.00000	0.0771517
$169$	0	0
$170$	−3.00000	−0.230089
$171$	−2.00000	−0.152944
$172$	−11.0000	−0.838742
$173$	−16.0000	−1.21646	−0.608229	−	0.793762i	$-0.708120\pi$
$173$	−16.0000	−1.21646	−0.608229	+	0.793762i	$0.708120\pi$
$174$	−2.00000	−0.151620
$175$	−4.00000	−0.302372
$176$	−2.00000	−0.150756
$177$	6.00000	0.450988
$178$	8.00000	0.599625
$179$	19.0000	1.42013	0.710063	−	0.704138i	$-0.248666\pi$
$179$	19.0000	1.42013	0.710063	+	0.704138i	$0.248666\pi$
$180$	−6.00000	−0.447214
$181$	−10.0000	−0.743294	−0.371647	−	0.928374i	$-0.621207\pi$
$181$	−10.0000	−0.743294	−0.371647	+	0.928374i	$0.621207\pi$
$182$	0	0
$183$	6.00000	0.443533
$184$	4.00000	0.294884
$185$	−9.00000	−0.661693
$186$	−4.00000	−0.293294
$187$	−2.00000	−0.146254
$188$	3.00000	0.218797
$189$	5.00000	0.363696
$190$	−3.00000	−0.217643
$191$	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
$193$	−2.00000	−0.143963	−0.0719816	−	0.997406i	$-0.522932\pi$
$193$	−2.00000	−0.143963	−0.0719816	+	0.997406i	$0.522932\pi$
$194$	16.0000	1.14873
$195$	0	0
$196$	−6.00000	−0.428571
$197$	−15.0000	−1.06871	−0.534353	−	0.845262i	$-0.679445\pi$
$197$	−15.0000	−1.06871	−0.534353	+	0.845262i	$0.679445\pi$
$198$	−4.00000	−0.284268
$199$	−16.0000	−1.13421	−0.567105	−	0.823646i	$-0.691937\pi$
$199$	−16.0000	−1.13421	−0.567105	+	0.823646i	$0.691937\pi$
$200$	−4.00000	−0.282843
$201$	2.00000	0.141069
$202$	6.00000	0.422159
$203$	−2.00000	−0.140372
$204$	1.00000	0.0700140
$205$	−30.0000	−2.09529
$206$	8.00000	0.557386
$207$	8.00000	0.556038
$208$	0	0
$209$	−2.00000	−0.138343
$210$	3.00000	0.207020
$211$	−1.00000	−0.0688428	−0.0344214	−	0.999407i	$-0.510959\pi$
$211$	−1.00000	−0.0688428	−0.0344214	+	0.999407i	$0.510959\pi$
$212$	0	0
$213$	3.00000	0.205557
$214$	12.0000	0.820303
$215$	−33.0000	−2.25058
$216$	5.00000	0.340207
$217$	−4.00000	−0.271538
$218$	−9.00000	−0.609557
$219$	4.00000	0.270295
$220$	−6.00000	−0.404520
$221$	0	0
$222$	3.00000	0.201347
$223$	23.0000	1.54019	0.770097	−	0.637927i	$-0.220208\pi$
$223$	23.0000	1.54019	0.770097	+	0.637927i	$0.220208\pi$
$224$	1.00000	0.0668153
$225$	−8.00000	−0.533333
$226$	18.0000	1.19734
$227$	10.0000	0.663723	0.331862	−	0.943328i	$-0.392323\pi$
$227$	10.0000	0.663723	0.331862	+	0.943328i	$0.392323\pi$
$228$	1.00000	0.0662266
$229$	−7.00000	−0.462573	−0.231287	−	0.972886i	$-0.574293\pi$
$229$	−7.00000	−0.462573	−0.231287	+	0.972886i	$0.574293\pi$
$230$	12.0000	0.791257
$231$	2.00000	0.131590
$232$	−2.00000	−0.131306
$233$	−19.0000	−1.24473	−0.622366	−	0.782727i	$-0.713828\pi$
$233$	−19.0000	−1.24473	−0.622366	+	0.782727i	$0.713828\pi$
$234$	0	0
$235$	9.00000	0.587095
$236$	6.00000	0.390567
$237$	−6.00000	−0.389742
$238$	1.00000	0.0648204
$239$	−5.00000	−0.323423	−0.161712	−	0.986838i	$-0.551701\pi$
$239$	−5.00000	−0.323423	−0.161712	+	0.986838i	$0.551701\pi$
$240$	3.00000	0.193649
$241$	−10.0000	−0.644157	−0.322078	−	0.946713i	$-0.604381\pi$
$241$	−10.0000	−0.644157	−0.322078	+	0.946713i	$0.604381\pi$
$242$	7.00000	0.449977
$243$	16.0000	1.02640
$244$	6.00000	0.384111
$245$	−18.0000	−1.14998
$246$	10.0000	0.637577
$247$	0	0
$248$	−4.00000	−0.254000
$249$	14.0000	0.887214
$250$	3.00000	0.189737
$251$	0	0		−	1.00000i	$-0.5\pi$
$251$	0	0			1.00000i	$0.5\pi$
$252$	2.00000	0.125988
$253$	8.00000	0.502956
$254$	0	0
$255$	3.00000	0.187867
$256$	1.00000	0.0625000
$257$	−21.0000	−1.30994	−0.654972	−	0.755653i	$-0.727320\pi$
$257$	−21.0000	−1.30994	−0.654972	+	0.755653i	$0.727320\pi$
$258$	11.0000	0.684830
$259$	3.00000	0.186411
$260$	0	0
$261$	−4.00000	−0.247594
$262$	15.0000	0.926703
$263$	−6.00000	−0.369976	−0.184988	−	0.982741i	$-0.559225\pi$
$263$	−6.00000	−0.369976	−0.184988	+	0.982741i	$0.559225\pi$
$264$	2.00000	0.123091
$265$	0	0
$266$	1.00000	0.0613139
$267$	−8.00000	−0.489592
$268$	2.00000	0.122169
$269$	30.0000	1.82913	0.914566	−	0.404436i	$-0.132532\pi$
$269$	30.0000	1.82913	0.914566	+	0.404436i	$0.132532\pi$
$270$	15.0000	0.912871
$271$	7.00000	0.425220	0.212610	−	0.977137i	$-0.431804\pi$
$271$	7.00000	0.425220	0.212610	+	0.977137i	$0.431804\pi$
$272$	1.00000	0.0606339
$273$	0	0
$274$	2.00000	0.120824
$275$	−8.00000	−0.482418
$276$	−4.00000	−0.240772
$277$	−6.00000	−0.360505	−0.180253	−	0.983620i	$-0.557691\pi$
$277$	−6.00000	−0.360505	−0.180253	+	0.983620i	$0.557691\pi$
$278$	−1.00000	−0.0599760
$279$	−8.00000	−0.478947
$280$	3.00000	0.179284
$281$	10.0000	0.596550	0.298275	−	0.954480i	$-0.403589\pi$
$281$	10.0000	0.596550	0.298275	+	0.954480i	$0.403589\pi$
$282$	−3.00000	−0.178647
$283$	−32.0000	−1.90220	−0.951101	−	0.308879i	$-0.900046\pi$
$283$	−32.0000	−1.90220	−0.951101	+	0.308879i	$0.900046\pi$
$284$	3.00000	0.178017
$285$	3.00000	0.177705
$286$	0	0
$287$	10.0000	0.590281
$288$	2.00000	0.117851
$289$	−16.0000	−0.941176
$290$	−6.00000	−0.352332
$291$	−16.0000	−0.937937
$292$	4.00000	0.234082
$293$	13.0000	0.759468	0.379734	−	0.925096i	$-0.376015\pi$
$293$	13.0000	0.759468	0.379734	+	0.925096i	$0.376015\pi$
$294$	6.00000	0.349927
$295$	18.0000	1.04800
$296$	3.00000	0.174371
$297$	10.0000	0.580259
$298$	6.00000	0.347571
$299$	0	0
$300$	4.00000	0.230940
$301$	11.0000	0.634029
$302$	1.00000	0.0575435
$303$	−6.00000	−0.344691
$304$	1.00000	0.0573539
$305$	18.0000	1.03068
$306$	2.00000	0.114332
$307$	−16.0000	−0.913168	−0.456584	−	0.889680i	$-0.650927\pi$
$307$	−16.0000	−0.913168	−0.456584	+	0.889680i	$0.650927\pi$
$308$	2.00000	0.113961
$309$	−8.00000	−0.455104
$310$	−12.0000	−0.681554
$311$	34.0000	1.92796	0.963982	−	0.265969i	$-0.0856919\pi$
$311$	34.0000	1.92796	0.963982	+	0.265969i	$0.0856919\pi$
$312$	0	0
$313$	23.0000	1.30004	0.650018	−	0.759918i	$-0.274761\pi$
$313$	23.0000	1.30004	0.650018	+	0.759918i	$0.274761\pi$
$314$	−14.0000	−0.790066
$315$	6.00000	0.338062
$316$	−6.00000	−0.337526
$317$	−18.0000	−1.01098	−0.505490	−	0.862832i	$-0.668688\pi$
$317$	−18.0000	−1.01098	−0.505490	+	0.862832i	$0.668688\pi$
$318$	0	0
$319$	−4.00000	−0.223957
$320$	3.00000	0.167705
$321$	−12.0000	−0.669775
$322$	−4.00000	−0.222911
$323$	1.00000	0.0556415
$324$	1.00000	0.0555556
$325$	0	0
$326$	12.0000	0.664619
$327$	9.00000	0.497701
$328$	10.0000	0.552158
$329$	−3.00000	−0.165395
$330$	6.00000	0.330289
$331$	6.00000	0.329790	0.164895	−	0.986311i	$-0.447272\pi$
$331$	6.00000	0.329790	0.164895	+	0.986311i	$0.447272\pi$
$332$	14.0000	0.768350
$333$	6.00000	0.328798
$334$	−24.0000	−1.31322
$335$	6.00000	0.327815
$336$	−1.00000	−0.0545545
$337$	5.00000	0.272367	0.136184	−	0.990684i	$-0.456516\pi$
$337$	5.00000	0.272367	0.136184	+	0.990684i	$0.456516\pi$
$338$	0	0
$339$	−18.0000	−0.977626
$340$	3.00000	0.162698
$341$	−8.00000	−0.433224
$342$	2.00000	0.108148
$343$	13.0000	0.701934
$344$	11.0000	0.593080
$345$	−12.0000	−0.646058
$346$	16.0000	0.860165
$347$	−3.00000	−0.161048	−0.0805242	−	0.996753i	$-0.525659\pi$
$347$	−3.00000	−0.161048	−0.0805242	+	0.996753i	$0.525659\pi$
$348$	2.00000	0.107211
$349$	35.0000	1.87351	0.936754	−	0.349990i	$-0.113815\pi$
$349$	35.0000	1.87351	0.936754	+	0.349990i	$0.113815\pi$
$350$	4.00000	0.213809
$351$	0	0
$352$	2.00000	0.106600
$353$	−14.0000	−0.745145	−0.372572	−	0.928003i	$-0.621524\pi$
$353$	−14.0000	−0.745145	−0.372572	+	0.928003i	$0.621524\pi$
$354$	−6.00000	−0.318896
$355$	9.00000	0.477670
$356$	−8.00000	−0.423999
$357$	−1.00000	−0.0529256
$358$	−19.0000	−1.00418
$359$	−32.0000	−1.68890	−0.844448	−	0.535638i	$-0.820071\pi$
$359$	−32.0000	−1.68890	−0.844448	+	0.535638i	$0.820071\pi$
$360$	6.00000	0.316228
$361$	1.00000	0.0526316
$362$	10.0000	0.525588
$363$	−7.00000	−0.367405
$364$	0	0
$365$	12.0000	0.628109
$366$	−6.00000	−0.313625
$367$	−18.0000	−0.939592	−0.469796	−	0.882775i	$-0.655673\pi$
$367$	−18.0000	−0.939592	−0.469796	+	0.882775i	$0.655673\pi$
$368$	−4.00000	−0.208514
$369$	20.0000	1.04116
$370$	9.00000	0.467888
$371$	0	0
$372$	4.00000	0.207390
$373$	−10.0000	−0.517780	−0.258890	−	0.965907i	$-0.583357\pi$
$373$	−10.0000	−0.517780	−0.258890	+	0.965907i	$0.583357\pi$
$374$	2.00000	0.103418
$375$	−3.00000	−0.154919
$376$	−3.00000	−0.154713
$377$	0	0
$378$	−5.00000	−0.257172
$379$	8.00000	0.410932	0.205466	−	0.978664i	$-0.434129\pi$
$379$	8.00000	0.410932	0.205466	+	0.978664i	$0.434129\pi$
$380$	3.00000	0.153897
$381$	0	0
$382$	−4.00000	−0.204658
$383$	7.00000	0.357683	0.178842	−	0.983878i	$-0.442765\pi$
$383$	7.00000	0.357683	0.178842	+	0.983878i	$0.442765\pi$
$384$	−1.00000	−0.0510310
$385$	6.00000	0.305788
$386$	2.00000	0.101797
$387$	22.0000	1.11832
$388$	−16.0000	−0.812277
$389$	36.0000	1.82527	0.912636	−	0.408773i	$-0.134043\pi$
$389$	36.0000	1.82527	0.912636	+	0.408773i	$0.134043\pi$
$390$	0	0
$391$	−4.00000	−0.202289
$392$	6.00000	0.303046
$393$	−15.0000	−0.756650
$394$	15.0000	0.755689
$395$	−18.0000	−0.905678
$396$	4.00000	0.201008
$397$	10.0000	0.501886	0.250943	−	0.968002i	$-0.419259\pi$
$397$	10.0000	0.501886	0.250943	+	0.968002i	$0.419259\pi$
$398$	16.0000	0.802008
$399$	−1.00000	−0.0500626
$400$	4.00000	0.200000
$401$	12.0000	0.599251	0.299626	−	0.954057i	$-0.403138\pi$
$401$	12.0000	0.599251	0.299626	+	0.954057i	$0.403138\pi$
$402$	−2.00000	−0.0997509
$403$	0	0
$404$	−6.00000	−0.298511
$405$	3.00000	0.149071
$406$	2.00000	0.0992583
$407$	6.00000	0.297409
$408$	−1.00000	−0.0495074
$409$	−38.0000	−1.87898	−0.939490	−	0.342578i	$-0.888700\pi$
$409$	−38.0000	−1.87898	−0.939490	+	0.342578i	$0.888700\pi$
$410$	30.0000	1.48159
$411$	−2.00000	−0.0986527
$412$	−8.00000	−0.394132
$413$	−6.00000	−0.295241
$414$	−8.00000	−0.393179
$415$	42.0000	2.06170
$416$	0	0
$417$	1.00000	0.0489702
$418$	2.00000	0.0978232
$419$	−5.00000	−0.244266	−0.122133	−	0.992514i	$-0.538973\pi$
$419$	−5.00000	−0.244266	−0.122133	+	0.992514i	$0.538973\pi$
$420$	−3.00000	−0.146385
$421$	5.00000	0.243685	0.121843	−	0.992549i	$-0.461120\pi$
$421$	5.00000	0.243685	0.121843	+	0.992549i	$0.461120\pi$
$422$	1.00000	0.0486792
$423$	−6.00000	−0.291730
$424$	0	0
$425$	4.00000	0.194029
$426$	−3.00000	−0.145350
$427$	−6.00000	−0.290360
$428$	−12.0000	−0.580042
$429$	0	0
$430$	33.0000	1.59140
$431$	−11.0000	−0.529851	−0.264926	−	0.964269i	$-0.585347\pi$
$431$	−11.0000	−0.529851	−0.264926	+	0.964269i	$0.585347\pi$
$432$	−5.00000	−0.240563
$433$	37.0000	1.77811	0.889053	−	0.457804i	$-0.151364\pi$
$433$	37.0000	1.77811	0.889053	+	0.457804i	$0.151364\pi$
$434$	4.00000	0.192006
$435$	6.00000	0.287678
$436$	9.00000	0.431022
$437$	−4.00000	−0.191346
$438$	−4.00000	−0.191127
$439$	−10.0000	−0.477274	−0.238637	−	0.971109i	$-0.576701\pi$
$439$	−10.0000	−0.477274	−0.238637	+	0.971109i	$0.576701\pi$
$440$	6.00000	0.286039
$441$	12.0000	0.571429
$442$	0	0
$443$	3.00000	0.142534	0.0712672	−	0.997457i	$-0.477296\pi$
$443$	3.00000	0.142534	0.0712672	+	0.997457i	$0.477296\pi$
$444$	−3.00000	−0.142374
$445$	−24.0000	−1.13771
$446$	−23.0000	−1.08908
$447$	−6.00000	−0.283790
$448$	−1.00000	−0.0472456
$449$	−8.00000	−0.377543	−0.188772	−	0.982021i	$-0.560451\pi$
$449$	−8.00000	−0.377543	−0.188772	+	0.982021i	$0.560451\pi$
$450$	8.00000	0.377124
$451$	20.0000	0.941763
$452$	−18.0000	−0.846649
$453$	−1.00000	−0.0469841
$454$	−10.0000	−0.469323
$455$	0	0
$456$	−1.00000	−0.0468293
$457$	26.0000	1.21623	0.608114	−	0.793849i	$-0.291926\pi$
$457$	26.0000	1.21623	0.608114	+	0.793849i	$0.291926\pi$
$458$	7.00000	0.327089
$459$	−5.00000	−0.233380
$460$	−12.0000	−0.559503
$461$	−13.0000	−0.605470	−0.302735	−	0.953075i	$-0.597900\pi$
$461$	−13.0000	−0.605470	−0.302735	+	0.953075i	$0.597900\pi$
$462$	−2.00000	−0.0930484
$463$	4.00000	0.185896	0.0929479	−	0.995671i	$-0.470371\pi$
$463$	4.00000	0.185896	0.0929479	+	0.995671i	$0.470371\pi$
$464$	2.00000	0.0928477
$465$	12.0000	0.556487
$466$	19.0000	0.880158
$467$	−20.0000	−0.925490	−0.462745	−	0.886492i	$-0.653135\pi$
$467$	−20.0000	−0.925490	−0.462745	+	0.886492i	$0.653135\pi$
$468$	0	0
$469$	−2.00000	−0.0923514
$470$	−9.00000	−0.415139
$471$	14.0000	0.645086
$472$	−6.00000	−0.276172
$473$	22.0000	1.01156
$474$	6.00000	0.275589
$475$	4.00000	0.183533
$476$	−1.00000	−0.0458349
$477$	0	0
$478$	5.00000	0.228695
$479$	15.0000	0.685367	0.342684	−	0.939451i	$-0.388664\pi$
$479$	15.0000	0.685367	0.342684	+	0.939451i	$0.388664\pi$
$480$	−3.00000	−0.136931
$481$	0	0
$482$	10.0000	0.455488
$483$	4.00000	0.182006
$484$	−7.00000	−0.318182
$485$	−48.0000	−2.17957
$486$	−16.0000	−0.725775
$487$	−8.00000	−0.362515	−0.181257	−	0.983436i	$-0.558017\pi$
$487$	−8.00000	−0.362515	−0.181257	+	0.983436i	$0.558017\pi$
$488$	−6.00000	−0.271607
$489$	−12.0000	−0.542659
$490$	18.0000	0.813157
$491$	29.0000	1.30875	0.654376	−	0.756169i	$-0.272931\pi$
$491$	29.0000	1.30875	0.654376	+	0.756169i	$0.272931\pi$
$492$	−10.0000	−0.450835
$493$	2.00000	0.0900755
$494$	0	0
$495$	12.0000	0.539360
$496$	4.00000	0.179605
$497$	−3.00000	−0.134568
$498$	−14.0000	−0.627355
$499$	10.0000	0.447661	0.223831	−	0.974628i	$-0.428144\pi$
$499$	10.0000	0.447661	0.223831	+	0.974628i	$0.428144\pi$
$500$	−3.00000	−0.134164
$501$	24.0000	1.07224
$502$	0	0
$503$	44.0000	1.96186	0.980932	−	0.194354i	$-0.0622609\pi$
$503$	44.0000	1.96186	0.980932	+	0.194354i	$0.0622609\pi$
$504$	−2.00000	−0.0890871
$505$	−18.0000	−0.800989
$506$	−8.00000	−0.355643
$507$	0	0
$508$	0	0
$509$	−38.0000	−1.68432	−0.842160	−	0.539227i	$-0.818716\pi$
$509$	−38.0000	−1.68432	−0.842160	+	0.539227i	$0.818716\pi$
$510$	−3.00000	−0.132842
$511$	−4.00000	−0.176950
$512$	−1.00000	−0.0441942
$513$	−5.00000	−0.220755
$514$	21.0000	0.926270
$515$	−24.0000	−1.05757
$516$	−11.0000	−0.484248
$517$	−6.00000	−0.263880
$518$	−3.00000	−0.131812
$519$	−16.0000	−0.702322
$520$	0	0
$521$	−31.0000	−1.35813	−0.679067	−	0.734076i	$-0.737616\pi$
$521$	−31.0000	−1.35813	−0.679067	+	0.734076i	$0.737616\pi$
$522$	4.00000	0.175075
$523$	24.0000	1.04945	0.524723	−	0.851273i	$-0.324169\pi$
$523$	24.0000	1.04945	0.524723	+	0.851273i	$0.324169\pi$
$524$	−15.0000	−0.655278
$525$	−4.00000	−0.174574
$526$	6.00000	0.261612
$527$	4.00000	0.174243
$528$	−2.00000	−0.0870388
$529$	−7.00000	−0.304348
$530$	0	0
$531$	−12.0000	−0.520756
$532$	−1.00000	−0.0433555
$533$	0	0
$534$	8.00000	0.346194
$535$	−36.0000	−1.55642
$536$	−2.00000	−0.0863868
$537$	19.0000	0.819911
$538$	−30.0000	−1.29339
$539$	12.0000	0.516877
$540$	−15.0000	−0.645497
$541$	−3.00000	−0.128980	−0.0644900	−	0.997918i	$-0.520542\pi$
$541$	−3.00000	−0.128980	−0.0644900	+	0.997918i	$0.520542\pi$
$542$	−7.00000	−0.300676
$543$	−10.0000	−0.429141
$544$	−1.00000	−0.0428746
$545$	27.0000	1.15655
$546$	0	0
$547$	−27.0000	−1.15444	−0.577218	−	0.816590i	$-0.695862\pi$
$547$	−27.0000	−1.15444	−0.577218	+	0.816590i	$0.695862\pi$
$548$	−2.00000	−0.0854358
$549$	−12.0000	−0.512148
$550$	8.00000	0.341121
$551$	2.00000	0.0852029
$552$	4.00000	0.170251
$553$	6.00000	0.255146
$554$	6.00000	0.254916
$555$	−9.00000	−0.382029
$556$	1.00000	0.0424094
$557$	−3.00000	−0.127114	−0.0635570	−	0.997978i	$-0.520244\pi$
$557$	−3.00000	−0.127114	−0.0635570	+	0.997978i	$0.520244\pi$
$558$	8.00000	0.338667
$559$	0	0
$560$	−3.00000	−0.126773
$561$	−2.00000	−0.0844401
$562$	−10.0000	−0.421825
$563$	−25.0000	−1.05362	−0.526812	−	0.849982i	$-0.676613\pi$
$563$	−25.0000	−1.05362	−0.526812	+	0.849982i	$0.676613\pi$
$564$	3.00000	0.126323
$565$	−54.0000	−2.27180
$566$	32.0000	1.34506
$567$	−1.00000	−0.0419961
$568$	−3.00000	−0.125877
$569$	−11.0000	−0.461144	−0.230572	−	0.973055i	$-0.574060\pi$
$569$	−11.0000	−0.461144	−0.230572	+	0.973055i	$0.574060\pi$
$570$	−3.00000	−0.125656
$571$	27.0000	1.12991	0.564957	−	0.825120i	$-0.308893\pi$
$571$	27.0000	1.12991	0.564957	+	0.825120i	$0.308893\pi$
$572$	0	0
$573$	4.00000	0.167102
$574$	−10.0000	−0.417392
$575$	−16.0000	−0.667246
$576$	−2.00000	−0.0833333
$577$	22.0000	0.915872	0.457936	−	0.888985i	$-0.348589\pi$
$577$	22.0000	0.915872	0.457936	+	0.888985i	$0.348589\pi$
$578$	16.0000	0.665512
$579$	−2.00000	−0.0831172
$580$	6.00000	0.249136
$581$	−14.0000	−0.580818
$582$	16.0000	0.663221
$583$	0	0
$584$	−4.00000	−0.165521
$585$	0	0
$586$	−13.0000	−0.537025
$587$	−38.0000	−1.56843	−0.784214	−	0.620491i	$-0.786934\pi$
$587$	−38.0000	−1.56843	−0.784214	+	0.620491i	$0.786934\pi$
$588$	−6.00000	−0.247436
$589$	4.00000	0.164817
$590$	−18.0000	−0.741048
$591$	−15.0000	−0.617018
$592$	−3.00000	−0.123299
$593$	−46.0000	−1.88899	−0.944497	−	0.328521i	$-0.893450\pi$
$593$	−46.0000	−1.88899	−0.944497	+	0.328521i	$0.893450\pi$
$594$	−10.0000	−0.410305
$595$	−3.00000	−0.122988
$596$	−6.00000	−0.245770
$597$	−16.0000	−0.654836
$598$	0	0
$599$	−12.0000	−0.490307	−0.245153	−	0.969484i	$-0.578838\pi$
$599$	−12.0000	−0.490307	−0.245153	+	0.969484i	$0.578838\pi$
$600$	−4.00000	−0.163299
$601$	−29.0000	−1.18293	−0.591467	−	0.806329i	$-0.701451\pi$
$601$	−29.0000	−1.18293	−0.591467	+	0.806329i	$0.701451\pi$
$602$	−11.0000	−0.448327
$603$	−4.00000	−0.162893
$604$	−1.00000	−0.0406894
$605$	−21.0000	−0.853771
$606$	6.00000	0.243733
$607$	28.0000	1.13648	0.568242	−	0.822861i	$-0.307624\pi$
$607$	28.0000	1.13648	0.568242	+	0.822861i	$0.307624\pi$
$608$	−1.00000	−0.0405554
$609$	−2.00000	−0.0810441
$610$	−18.0000	−0.728799
$611$	0	0
$612$	−2.00000	−0.0808452
$613$	−22.0000	−0.888572	−0.444286	−	0.895885i	$-0.646543\pi$
$613$	−22.0000	−0.888572	−0.444286	+	0.895885i	$0.646543\pi$
$614$	16.0000	0.645707
$615$	−30.0000	−1.20972
$616$	−2.00000	−0.0805823
$617$	18.0000	0.724653	0.362326	−	0.932051i	$-0.381983\pi$
$617$	18.0000	0.724653	0.362326	+	0.932051i	$0.381983\pi$
$618$	8.00000	0.321807
$619$	0	0		−	1.00000i	$-0.5\pi$
$619$	0	0			1.00000i	$0.5\pi$
$620$	12.0000	0.481932
$621$	20.0000	0.802572
$622$	−34.0000	−1.36328
$623$	8.00000	0.320513
$624$	0	0
$625$	−29.0000	−1.16000
$626$	−23.0000	−0.919265
$627$	−2.00000	−0.0798723
$628$	14.0000	0.558661
$629$	−3.00000	−0.119618
$630$	−6.00000	−0.239046
$631$	−3.00000	−0.119428	−0.0597141	−	0.998216i	$-0.519019\pi$
$631$	−3.00000	−0.119428	−0.0597141	+	0.998216i	$0.519019\pi$
$632$	6.00000	0.238667
$633$	−1.00000	−0.0397464
$634$	18.0000	0.714871
$635$	0	0
$636$	0	0
$637$	0	0
$638$	4.00000	0.158362
$639$	−6.00000	−0.237356
$640$	−3.00000	−0.118585
$641$	30.0000	1.18493	0.592464	−	0.805597i	$-0.298155\pi$
$641$	30.0000	1.18493	0.592464	+	0.805597i	$0.298155\pi$
$642$	12.0000	0.473602
$643$	22.0000	0.867595	0.433798	−	0.901010i	$-0.357173\pi$
$643$	22.0000	0.867595	0.433798	+	0.901010i	$0.357173\pi$
$644$	4.00000	0.157622
$645$	−33.0000	−1.29937
$646$	−1.00000	−0.0393445
$647$	38.0000	1.49393	0.746967	−	0.664861i	$-0.231509\pi$
$647$	38.0000	1.49393	0.746967	+	0.664861i	$0.231509\pi$
$648$	−1.00000	−0.0392837
$649$	−12.0000	−0.471041
$650$	0	0
$651$	−4.00000	−0.156772
$652$	−12.0000	−0.469956
$653$	8.00000	0.313064	0.156532	−	0.987673i	$-0.449969\pi$
$653$	8.00000	0.313064	0.156532	+	0.987673i	$0.449969\pi$
$654$	−9.00000	−0.351928
$655$	−45.0000	−1.75830
$656$	−10.0000	−0.390434
$657$	−8.00000	−0.312110
$658$	3.00000	0.116952
$659$	4.00000	0.155818	0.0779089	−	0.996960i	$-0.475176\pi$
$659$	4.00000	0.155818	0.0779089	+	0.996960i	$0.475176\pi$
$660$	−6.00000	−0.233550
$661$	−14.0000	−0.544537	−0.272268	−	0.962221i	$-0.587774\pi$
$661$	−14.0000	−0.544537	−0.272268	+	0.962221i	$0.587774\pi$
$662$	−6.00000	−0.233197
$663$	0	0
$664$	−14.0000	−0.543305
$665$	−3.00000	−0.116335
$666$	−6.00000	−0.232495
$667$	−8.00000	−0.309761
$668$	24.0000	0.928588
$669$	23.0000	0.889231
$670$	−6.00000	−0.231800
$671$	−12.0000	−0.463255
$672$	1.00000	0.0385758
$673$	−29.0000	−1.11787	−0.558934	−	0.829212i	$-0.688789\pi$
$673$	−29.0000	−1.11787	−0.558934	+	0.829212i	$0.688789\pi$
$674$	−5.00000	−0.192593
$675$	−20.0000	−0.769800
$676$	0	0
$677$	42.0000	1.61419	0.807096	−	0.590421i	$-0.201038\pi$
$677$	42.0000	1.61419	0.807096	+	0.590421i	$0.201038\pi$
$678$	18.0000	0.691286
$679$	16.0000	0.614024
$680$	−3.00000	−0.115045
$681$	10.0000	0.383201
$682$	8.00000	0.306336
$683$	44.0000	1.68361	0.841807	−	0.539779i	$-0.181492\pi$
$683$	44.0000	1.68361	0.841807	+	0.539779i	$0.181492\pi$
$684$	−2.00000	−0.0764719
$685$	−6.00000	−0.229248
$686$	−13.0000	−0.496342
$687$	−7.00000	−0.267067
$688$	−11.0000	−0.419371
$689$	0	0
$690$	12.0000	0.456832
$691$	12.0000	0.456502	0.228251	−	0.973602i	$-0.426699\pi$
$691$	12.0000	0.456502	0.228251	+	0.973602i	$0.426699\pi$
$692$	−16.0000	−0.608229
$693$	−4.00000	−0.151947
$694$	3.00000	0.113878
$695$	3.00000	0.113796
$696$	−2.00000	−0.0758098
$697$	−10.0000	−0.378777
$698$	−35.0000	−1.32477
$699$	−19.0000	−0.718646
$700$	−4.00000	−0.151186
$701$	−8.00000	−0.302156	−0.151078	−	0.988522i	$-0.548274\pi$
$701$	−8.00000	−0.302156	−0.151078	+	0.988522i	$0.548274\pi$
$702$	0	0
$703$	−3.00000	−0.113147
$704$	−2.00000	−0.0753778
$705$	9.00000	0.338960
$706$	14.0000	0.526897
$707$	6.00000	0.225653
$708$	6.00000	0.225494
$709$	−26.0000	−0.976450	−0.488225	−	0.872718i	$-0.662356\pi$
$709$	−26.0000	−0.976450	−0.488225	+	0.872718i	$0.662356\pi$
$710$	−9.00000	−0.337764
$711$	12.0000	0.450035
$712$	8.00000	0.299813
$713$	−16.0000	−0.599205
$714$	1.00000	0.0374241
$715$	0	0
$716$	19.0000	0.710063
$717$	−5.00000	−0.186728
$718$	32.0000	1.19423
$719$	36.0000	1.34257	0.671287	−	0.741198i	$-0.265742\pi$
$719$	36.0000	1.34257	0.671287	+	0.741198i	$0.265742\pi$
$720$	−6.00000	−0.223607
$721$	8.00000	0.297936
$722$	−1.00000	−0.0372161
$723$	−10.0000	−0.371904
$724$	−10.0000	−0.371647
$725$	8.00000	0.297113
$726$	7.00000	0.259794
$727$	−26.0000	−0.964287	−0.482143	−	0.876092i	$-0.660142\pi$
$727$	−26.0000	−0.964287	−0.482143	+	0.876092i	$0.660142\pi$
$728$	0	0
$729$	13.0000	0.481481
$730$	−12.0000	−0.444140
$731$	−11.0000	−0.406850
$732$	6.00000	0.221766
$733$	33.0000	1.21888	0.609441	−	0.792831i	$-0.291394\pi$
$733$	33.0000	1.21888	0.609441	+	0.792831i	$0.291394\pi$
$734$	18.0000	0.664392
$735$	−18.0000	−0.663940
$736$	4.00000	0.147442
$737$	−4.00000	−0.147342
$738$	−20.0000	−0.736210
$739$	40.0000	1.47142	0.735712	−	0.677295i	$-0.236848\pi$
$739$	40.0000	1.47142	0.735712	+	0.677295i	$0.236848\pi$
$740$	−9.00000	−0.330847
$741$	0	0
$742$	0	0
$743$	1.00000	0.0366864	0.0183432	−	0.999832i	$-0.494161\pi$
$743$	1.00000	0.0366864	0.0183432	+	0.999832i	$0.494161\pi$
$744$	−4.00000	−0.146647
$745$	−18.0000	−0.659469
$746$	10.0000	0.366126
$747$	−28.0000	−1.02447
$748$	−2.00000	−0.0731272
$749$	12.0000	0.438470
$750$	3.00000	0.109545
$751$	−24.0000	−0.875772	−0.437886	−	0.899030i	$-0.644273\pi$
$751$	−24.0000	−0.875772	−0.437886	+	0.899030i	$0.644273\pi$
$752$	3.00000	0.109399
$753$	0	0
$754$	0	0
$755$	−3.00000	−0.109181
$756$	5.00000	0.181848
$757$	0	0		−	1.00000i	$-0.5\pi$
$757$	0	0			1.00000i	$0.5\pi$
$758$	−8.00000	−0.290573
$759$	8.00000	0.290382
$760$	−3.00000	−0.108821
$761$	−6.00000	−0.217500	−0.108750	−	0.994069i	$-0.534685\pi$
$761$	−6.00000	−0.217500	−0.108750	+	0.994069i	$0.534685\pi$
$762$	0	0
$763$	−9.00000	−0.325822
$764$	4.00000	0.144715
$765$	−6.00000	−0.216930
$766$	−7.00000	−0.252920
$767$	0	0
$768$	1.00000	0.0360844
$769$	−30.0000	−1.08183	−0.540914	−	0.841078i	$-0.681921\pi$
$769$	−30.0000	−1.08183	−0.540914	+	0.841078i	$0.681921\pi$
$770$	−6.00000	−0.216225
$771$	−21.0000	−0.756297
$772$	−2.00000	−0.0719816
$773$	−19.0000	−0.683383	−0.341691	−	0.939812i	$-0.611000\pi$
$773$	−19.0000	−0.683383	−0.341691	+	0.939812i	$0.611000\pi$
$774$	−22.0000	−0.790774
$775$	16.0000	0.574737
$776$	16.0000	0.574367
$777$	3.00000	0.107624
$778$	−36.0000	−1.29066
$779$	−10.0000	−0.358287
$780$	0	0
$781$	−6.00000	−0.214697
$782$	4.00000	0.143040
$783$	−10.0000	−0.357371
$784$	−6.00000	−0.214286
$785$	42.0000	1.49904
$786$	15.0000	0.535032
$787$	−14.0000	−0.499046	−0.249523	−	0.968369i	$-0.580274\pi$
$787$	−14.0000	−0.499046	−0.249523	+	0.968369i	$0.580274\pi$
$788$	−15.0000	−0.534353
$789$	−6.00000	−0.213606
$790$	18.0000	0.640411
$791$	18.0000	0.640006
$792$	−4.00000	−0.142134
$793$	0	0
$794$	−10.0000	−0.354887
$795$	0	0
$796$	−16.0000	−0.567105
$797$	30.0000	1.06265	0.531327	−	0.847167i	$-0.321693\pi$
$797$	30.0000	1.06265	0.531327	+	0.847167i	$0.321693\pi$
$798$	1.00000	0.0353996
$799$	3.00000	0.106132
$800$	−4.00000	−0.141421
$801$	16.0000	0.565332
$802$	−12.0000	−0.423735
$803$	−8.00000	−0.282314
$804$	2.00000	0.0705346
$805$	12.0000	0.422944
$806$	0	0
$807$	30.0000	1.05605
$808$	6.00000	0.211079
$809$	39.0000	1.37117	0.685583	−	0.727994i	$-0.259547\pi$
$809$	39.0000	1.37117	0.685583	+	0.727994i	$0.259547\pi$
$810$	−3.00000	−0.105409
$811$	−54.0000	−1.89620	−0.948098	−	0.317978i	$-0.896996\pi$
$811$	−54.0000	−1.89620	−0.948098	+	0.317978i	$0.896996\pi$
$812$	−2.00000	−0.0701862
$813$	7.00000	0.245501
$814$	−6.00000	−0.210300
$815$	−36.0000	−1.26102
$816$	1.00000	0.0350070
$817$	−11.0000	−0.384841
$818$	38.0000	1.32864
$819$	0	0
$820$	−30.0000	−1.04765
$821$	39.0000	1.36111	0.680555	−	0.732697i	$-0.261739\pi$
$821$	39.0000	1.36111	0.680555	+	0.732697i	$0.261739\pi$
$822$	2.00000	0.0697580
$823$	20.0000	0.697156	0.348578	−	0.937280i	$-0.386665\pi$
$823$	20.0000	0.697156	0.348578	+	0.937280i	$0.386665\pi$
$824$	8.00000	0.278693
$825$	−8.00000	−0.278524
$826$	6.00000	0.208767
$827$	−18.0000	−0.625921	−0.312961	−	0.949766i	$-0.601321\pi$
$827$	−18.0000	−0.625921	−0.312961	+	0.949766i	$0.601321\pi$
$828$	8.00000	0.278019
$829$	38.0000	1.31979	0.659897	−	0.751356i	$-0.270600\pi$
$829$	38.0000	1.31979	0.659897	+	0.751356i	$0.270600\pi$
$830$	−42.0000	−1.45784
$831$	−6.00000	−0.208138
$832$	0	0
$833$	−6.00000	−0.207888
$834$	−1.00000	−0.0346272
$835$	72.0000	2.49166
$836$	−2.00000	−0.0691714
$837$	−20.0000	−0.691301
$838$	5.00000	0.172722
$839$	−24.0000	−0.828572	−0.414286	−	0.910147i	$-0.635969\pi$
$839$	−24.0000	−0.828572	−0.414286	+	0.910147i	$0.635969\pi$
$840$	3.00000	0.103510
$841$	−25.0000	−0.862069
$842$	−5.00000	−0.172311
$843$	10.0000	0.344418
$844$	−1.00000	−0.0344214
$845$	0	0
$846$	6.00000	0.206284
$847$	7.00000	0.240523
$848$	0	0
$849$	−32.0000	−1.09824
$850$	−4.00000	−0.137199
$851$	12.0000	0.411355
$852$	3.00000	0.102778
$853$	5.00000	0.171197	0.0855984	−	0.996330i	$-0.472720\pi$
$853$	5.00000	0.171197	0.0855984	+	0.996330i	$0.472720\pi$
$854$	6.00000	0.205316
$855$	−6.00000	−0.205196
$856$	12.0000	0.410152
$857$	6.00000	0.204956	0.102478	−	0.994735i	$-0.467323\pi$
$857$	6.00000	0.204956	0.102478	+	0.994735i	$0.467323\pi$
$858$	0	0
$859$	24.0000	0.818869	0.409435	−	0.912339i	$-0.365726\pi$
$859$	24.0000	0.818869	0.409435	+	0.912339i	$0.365726\pi$
$860$	−33.0000	−1.12529
$861$	10.0000	0.340799
$862$	11.0000	0.374661
$863$	33.0000	1.12333	0.561667	−	0.827364i	$-0.310160\pi$
$863$	33.0000	1.12333	0.561667	+	0.827364i	$0.310160\pi$
$864$	5.00000	0.170103
$865$	−48.0000	−1.63205
$866$	−37.0000	−1.25731
$867$	−16.0000	−0.543388
$868$	−4.00000	−0.135769
$869$	12.0000	0.407072
$870$	−6.00000	−0.203419
$871$	0	0
$872$	−9.00000	−0.304778
$873$	32.0000	1.08304
$874$	4.00000	0.135302
$875$	3.00000	0.101419
$876$	4.00000	0.135147
$877$	−45.0000	−1.51954	−0.759771	−	0.650191i	$-0.774689\pi$
$877$	−45.0000	−1.51954	−0.759771	+	0.650191i	$0.774689\pi$
$878$	10.0000	0.337484
$879$	13.0000	0.438479
$880$	−6.00000	−0.202260
$881$	21.0000	0.707508	0.353754	−	0.935339i	$-0.384905\pi$
$881$	21.0000	0.707508	0.353754	+	0.935339i	$0.384905\pi$
$882$	−12.0000	−0.404061
$883$	27.0000	0.908622	0.454311	−	0.890843i	$-0.349885\pi$
$883$	27.0000	0.908622	0.454311	+	0.890843i	$0.349885\pi$
$884$	0	0
$885$	18.0000	0.605063
$886$	−3.00000	−0.100787
$887$	44.0000	1.47738	0.738688	−	0.674048i	$-0.235446\pi$
$887$	44.0000	1.47738	0.738688	+	0.674048i	$0.235446\pi$
$888$	3.00000	0.100673
$889$	0	0
$890$	24.0000	0.804482
$891$	−2.00000	−0.0670025
$892$	23.0000	0.770097
$893$	3.00000	0.100391
$894$	6.00000	0.200670
$895$	57.0000	1.90530
$896$	1.00000	0.0334077
$897$	0	0
$898$	8.00000	0.266963
$899$	8.00000	0.266815
$900$	−8.00000	−0.266667
$901$	0	0
$902$	−20.0000	−0.665927
$903$	11.0000	0.366057
$904$	18.0000	0.598671
$905$	−30.0000	−0.997234
$906$	1.00000	0.0332228
$907$	−53.0000	−1.75984	−0.879918	−	0.475125i	$-0.842403\pi$
$907$	−53.0000	−1.75984	−0.879918	+	0.475125i	$0.842403\pi$
$908$	10.0000	0.331862
$909$	12.0000	0.398015
$910$	0	0
$911$	−44.0000	−1.45779	−0.728893	−	0.684628i	$-0.759965\pi$
$911$	−44.0000	−1.45779	−0.728893	+	0.684628i	$0.759965\pi$
$912$	1.00000	0.0331133
$913$	−28.0000	−0.926665
$914$	−26.0000	−0.860004
$915$	18.0000	0.595062
$916$	−7.00000	−0.231287
$917$	15.0000	0.495344
$918$	5.00000	0.165025
$919$	−4.00000	−0.131948	−0.0659739	−	0.997821i	$-0.521015\pi$
$919$	−4.00000	−0.131948	−0.0659739	+	0.997821i	$0.521015\pi$
$920$	12.0000	0.395628
$921$	−16.0000	−0.527218
$922$	13.0000	0.428132
$923$	0	0
$924$	2.00000	0.0657952
$925$	−12.0000	−0.394558
$926$	−4.00000	−0.131448
$927$	16.0000	0.525509
$928$	−2.00000	−0.0656532
$929$	−12.0000	−0.393707	−0.196854	−	0.980433i	$-0.563072\pi$
$929$	−12.0000	−0.393707	−0.196854	+	0.980433i	$0.563072\pi$
$930$	−12.0000	−0.393496
$931$	−6.00000	−0.196642
$932$	−19.0000	−0.622366
$933$	34.0000	1.11311
$934$	20.0000	0.654420
$935$	−6.00000	−0.196221
$936$	0	0
$937$	−54.0000	−1.76410	−0.882052	−	0.471153i	$-0.843838\pi$
$937$	−54.0000	−1.76410	−0.882052	+	0.471153i	$0.843838\pi$
$938$	2.00000	0.0653023
$939$	23.0000	0.750577
$940$	9.00000	0.293548
$941$	−25.0000	−0.814977	−0.407488	−	0.913210i	$-0.633595\pi$
$941$	−25.0000	−0.814977	−0.407488	+	0.913210i	$0.633595\pi$
$942$	−14.0000	−0.456145
$943$	40.0000	1.30258
$944$	6.00000	0.195283
$945$	15.0000	0.487950
$946$	−22.0000	−0.715282
$947$	56.0000	1.81976	0.909878	−	0.414876i	$-0.136175\pi$
$947$	56.0000	1.81976	0.909878	+	0.414876i	$0.136175\pi$
$948$	−6.00000	−0.194871
$949$	0	0
$950$	−4.00000	−0.129777
$951$	−18.0000	−0.583690
$952$	1.00000	0.0324102
$953$	25.0000	0.809829	0.404915	−	0.914354i	$-0.367301\pi$
$953$	25.0000	0.809829	0.404915	+	0.914354i	$0.367301\pi$
$954$	0	0
$955$	12.0000	0.388311
$956$	−5.00000	−0.161712
$957$	−4.00000	−0.129302
$958$	−15.0000	−0.484628
$959$	2.00000	0.0645834
$960$	3.00000	0.0968246
$961$	−15.0000	−0.483871
$962$	0	0
$963$	24.0000	0.773389
$964$	−10.0000	−0.322078
$965$	−6.00000	−0.193147
$966$	−4.00000	−0.128698
$967$	−11.0000	−0.353736	−0.176868	−	0.984235i	$-0.556597\pi$
$967$	−11.0000	−0.353736	−0.176868	+	0.984235i	$0.556597\pi$
$968$	7.00000	0.224989
$969$	1.00000	0.0321246
$970$	48.0000	1.54119
$971$	−3.00000	−0.0962746	−0.0481373	−	0.998841i	$-0.515328\pi$
$971$	−3.00000	−0.0962746	−0.0481373	+	0.998841i	$0.515328\pi$
$972$	16.0000	0.513200
$973$	−1.00000	−0.0320585
$974$	8.00000	0.256337
$975$	0	0
$976$	6.00000	0.192055
$977$	−48.0000	−1.53566	−0.767828	−	0.640656i	$-0.778662\pi$
$977$	−48.0000	−1.53566	−0.767828	+	0.640656i	$0.778662\pi$
$978$	12.0000	0.383718
$979$	16.0000	0.511362
$980$	−18.0000	−0.574989
$981$	−18.0000	−0.574696
$982$	−29.0000	−0.925427
$983$	−39.0000	−1.24391	−0.621953	−	0.783054i	$-0.713661\pi$
$983$	−39.0000	−1.24391	−0.621953	+	0.783054i	$0.713661\pi$
$984$	10.0000	0.318788
$985$	−45.0000	−1.43382
$986$	−2.00000	−0.0636930
$987$	−3.00000	−0.0954911
$988$	0	0
$989$	44.0000	1.39912
$990$	−12.0000	−0.381385
$991$	6.00000	0.190596	0.0952981	−	0.995449i	$-0.469620\pi$
$991$	6.00000	0.190596	0.0952981	+	0.995449i	$0.469620\pi$
$992$	−4.00000	−0.127000
$993$	6.00000	0.190404
$994$	3.00000	0.0951542
$995$	−48.0000	−1.52170
$996$	14.0000	0.443607
$997$	16.0000	0.506725	0.253363	−	0.967371i	$-0.418463\pi$
$997$	16.0000	0.506725	0.253363	+	0.967371i	$0.418463\pi$
$998$	−10.0000	−0.316544
$999$	15.0000	0.474579

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6422.2.a.c.1.1		1
13.5	odd	4		494.2.d.a.77.2	yes	2
13.8	odd	4		494.2.d.a.77.1	✓	2
13.12	even	2		6422.2.a.g.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
494.2.d.a.77.1	✓	2	13.8	odd	4
494.2.d.a.77.2	yes	2	13.5	odd	4
6422.2.a.c.1.1		1	1.1	even	1	trivial
6422.2.a.g.1.1		1	13.12	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:	\( N \)	\(=\)	\( 6422 = 2 \cdot 13^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6422.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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