LMFDB - Embedded newform 6370.2.a.l.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6370,2,Mod(1,6370)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6370, 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("6370.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6370 = 2 \cdot 5 \cdot 7^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6370.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:	$50.8647060876$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 130)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	6370.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

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

\(q+1.00000 q^{2} -2.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} -2.00000 q^{6} +1.00000 q^{8} +1.00000 q^{9} +1.00000 q^{10} -2.00000 q^{11} -2.00000 q^{12} +1.00000 q^{13} -2.00000 q^{15} +1.00000 q^{16} -2.00000 q^{17} +1.00000 q^{18} -6.00000 q^{19} +1.00000 q^{20} -2.00000 q^{22} +6.00000 q^{23} -2.00000 q^{24} +1.00000 q^{25} +1.00000 q^{26} +4.00000 q^{27} +2.00000 q^{29} -2.00000 q^{30} +6.00000 q^{31} +1.00000 q^{32} +4.00000 q^{33} -2.00000 q^{34} +1.00000 q^{36} -2.00000 q^{37} -6.00000 q^{38} -2.00000 q^{39} +1.00000 q^{40} -10.0000 q^{41} -10.0000 q^{43} -2.00000 q^{44} +1.00000 q^{45} +6.00000 q^{46} +12.0000 q^{47} -2.00000 q^{48} +1.00000 q^{50} +4.00000 q^{51} +1.00000 q^{52} +2.00000 q^{53} +4.00000 q^{54} -2.00000 q^{55} +12.0000 q^{57} +2.00000 q^{58} -10.0000 q^{59} -2.00000 q^{60} -2.00000 q^{61} +6.00000 q^{62} +1.00000 q^{64} +1.00000 q^{65} +4.00000 q^{66} -12.0000 q^{67} -2.00000 q^{68} -12.0000 q^{69} +10.0000 q^{71} +1.00000 q^{72} -10.0000 q^{73} -2.00000 q^{74} -2.00000 q^{75} -6.00000 q^{76} -2.00000 q^{78} -4.00000 q^{79} +1.00000 q^{80} -11.0000 q^{81} -10.0000 q^{82} -2.00000 q^{85} -10.0000 q^{86} -4.00000 q^{87} -2.00000 q^{88} +14.0000 q^{89} +1.00000 q^{90} +6.00000 q^{92} -12.0000 q^{93} +12.0000 q^{94} -6.00000 q^{95} -2.00000 q^{96} -14.0000 q^{97} -2.00000 q^{99} +O(q^{100})\)

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
$3$	−2.00000	−1.15470	−0.577350	−	0.816497i	$-0.695913\pi$
$3$	−2.00000	−1.15470	−0.577350	+	0.816497i	$0.695913\pi$
$4$	1.00000	0.500000
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	−2.00000	−0.816497
$7$	0	0
$7$	0	0
$8$	1.00000	0.353553
$9$	1.00000	0.333333
$10$	1.00000	0.316228
$11$	−2.00000	−0.603023	−0.301511	−	0.953463i	$-0.597491\pi$
$11$	−2.00000	−0.603023	−0.301511	+	0.953463i	$0.597491\pi$
$12$	−2.00000	−0.577350
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	0	0
$15$	−2.00000	−0.516398
$16$	1.00000	0.250000
$17$	−2.00000	−0.485071	−0.242536	−	0.970143i	$-0.577979\pi$
$17$	−2.00000	−0.485071	−0.242536	+	0.970143i	$0.577979\pi$
$18$	1.00000	0.235702
$19$	−6.00000	−1.37649	−0.688247	−	0.725476i	$-0.741620\pi$
$19$	−6.00000	−1.37649	−0.688247	+	0.725476i	$0.741620\pi$
$20$	1.00000	0.223607
$21$	0	0
$22$	−2.00000	−0.426401
$23$	6.00000	1.25109	0.625543	−	0.780189i	$-0.284877\pi$
$23$	6.00000	1.25109	0.625543	+	0.780189i	$0.284877\pi$
$24$	−2.00000	−0.408248
$25$	1.00000	0.200000
$26$	1.00000	0.196116
$27$	4.00000	0.769800
$28$	0	0
$29$	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
$31$	6.00000	1.07763	0.538816	−	0.842424i	$-0.318872\pi$
$31$	6.00000	1.07763	0.538816	+	0.842424i	$0.318872\pi$
$32$	1.00000	0.176777
$33$	4.00000	0.696311
$34$	−2.00000	−0.342997
$35$	0	0
$36$	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	$0.552568\pi$
$38$	−6.00000	−0.973329
$39$	−2.00000	−0.320256
$40$	1.00000	0.158114
$41$	−10.0000	−1.56174	−0.780869	−	0.624695i	$-0.785223\pi$
$41$	−10.0000	−1.56174	−0.780869	+	0.624695i	$0.785223\pi$
$42$	0	0
$43$	−10.0000	−1.52499	−0.762493	−	0.646997i	$-0.776025\pi$
$43$	−10.0000	−1.52499	−0.762493	+	0.646997i	$0.776025\pi$
$44$	−2.00000	−0.301511
$45$	1.00000	0.149071
$46$	6.00000	0.884652
$47$	12.0000	1.75038	0.875190	−	0.483779i	$-0.160736\pi$
$47$	12.0000	1.75038	0.875190	+	0.483779i	$0.160736\pi$
$48$	−2.00000	−0.288675
$49$	0	0
$50$	1.00000	0.141421
$51$	4.00000	0.560112
$52$	1.00000	0.138675
$53$	2.00000	0.274721	0.137361	−	0.990521i	$-0.456138\pi$
$53$	2.00000	0.274721	0.137361	+	0.990521i	$0.456138\pi$
$54$	4.00000	0.544331
$55$	−2.00000	−0.269680
$56$	0	0
$57$	12.0000	1.58944
$58$	2.00000	0.262613
$59$	−10.0000	−1.30189	−0.650945	−	0.759125i	$-0.725627\pi$
$59$	−10.0000	−1.30189	−0.650945	+	0.759125i	$0.725627\pi$
$60$	−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	$0.540868\pi$
$62$	6.00000	0.762001
$63$	0	0
$64$	1.00000	0.125000
$65$	1.00000	0.124035
$66$	4.00000	0.492366
$67$	−12.0000	−1.46603	−0.733017	−	0.680211i	$-0.761888\pi$
$67$	−12.0000	−1.46603	−0.733017	+	0.680211i	$0.761888\pi$
$68$	−2.00000	−0.242536
$69$	−12.0000	−1.44463
$70$	0	0
$71$	10.0000	1.18678	0.593391	−	0.804914i	$-0.297789\pi$
$71$	10.0000	1.18678	0.593391	+	0.804914i	$0.297789\pi$
$72$	1.00000	0.117851
$73$	−10.0000	−1.17041	−0.585206	−	0.810885i	$-0.698986\pi$
$73$	−10.0000	−1.17041	−0.585206	+	0.810885i	$0.698986\pi$
$74$	−2.00000	−0.232495
$75$	−2.00000	−0.230940
$76$	−6.00000	−0.688247
$77$	0	0
$78$	−2.00000	−0.226455
$79$	−4.00000	−0.450035	−0.225018	−	0.974355i	$-0.572244\pi$
$79$	−4.00000	−0.450035	−0.225018	+	0.974355i	$0.572244\pi$
$80$	1.00000	0.111803
$81$	−11.0000	−1.22222
$82$	−10.0000	−1.10432
$83$	0	0		−	1.00000i	$-0.5\pi$
$83$	0	0			1.00000i	$0.5\pi$
$84$	0	0
$85$	−2.00000	−0.216930
$86$	−10.0000	−1.07833
$87$	−4.00000	−0.428845
$88$	−2.00000	−0.213201
$89$	14.0000	1.48400	0.741999	−	0.670402i	$-0.233878\pi$
$89$	14.0000	1.48400	0.741999	+	0.670402i	$0.233878\pi$
$90$	1.00000	0.105409
$91$	0	0
$92$	6.00000	0.625543
$93$	−12.0000	−1.24434
$94$	12.0000	1.23771
$95$	−6.00000	−0.615587
$96$	−2.00000	−0.204124
$97$	−14.0000	−1.42148	−0.710742	−	0.703452i	$-0.751641\pi$
$97$	−14.0000	−1.42148	−0.710742	+	0.703452i	$0.751641\pi$
$98$	0	0
$99$	−2.00000	−0.201008
$100$	1.00000	0.100000
$101$	−14.0000	−1.39305	−0.696526	−	0.717532i	$-0.745272\pi$
$101$	−14.0000	−1.39305	−0.696526	+	0.717532i	$0.745272\pi$
$102$	4.00000	0.396059
$103$	18.0000	1.77359	0.886796	−	0.462160i	$-0.152926\pi$
$103$	18.0000	1.77359	0.886796	+	0.462160i	$0.152926\pi$
$104$	1.00000	0.0980581
$105$	0	0
$106$	2.00000	0.194257
$107$	6.00000	0.580042	0.290021	−	0.957020i	$-0.406338\pi$
$107$	6.00000	0.580042	0.290021	+	0.957020i	$0.406338\pi$
$108$	4.00000	0.384900
$109$	−6.00000	−0.574696	−0.287348	−	0.957826i	$-0.592774\pi$
$109$	−6.00000	−0.574696	−0.287348	+	0.957826i	$0.592774\pi$
$110$	−2.00000	−0.190693
$111$	4.00000	0.379663
$112$	0	0
$113$	2.00000	0.188144	0.0940721	−	0.995565i	$-0.470012\pi$
$113$	2.00000	0.188144	0.0940721	+	0.995565i	$0.470012\pi$
$114$	12.0000	1.12390
$115$	6.00000	0.559503
$116$	2.00000	0.185695
$117$	1.00000	0.0924500
$118$	−10.0000	−0.920575
$119$	0	0
$120$	−2.00000	−0.182574
$121$	−7.00000	−0.636364
$122$	−2.00000	−0.181071
$123$	20.0000	1.80334
$124$	6.00000	0.538816
$125$	1.00000	0.0894427
$126$	0	0
$127$	−14.0000	−1.24230	−0.621150	−	0.783692i	$-0.713334\pi$
$127$	−14.0000	−1.24230	−0.621150	+	0.783692i	$0.713334\pi$
$128$	1.00000	0.0883883
$129$	20.0000	1.76090
$130$	1.00000	0.0877058
$131$	−4.00000	−0.349482	−0.174741	−	0.984614i	$-0.555909\pi$
$131$	−4.00000	−0.349482	−0.174741	+	0.984614i	$0.555909\pi$
$132$	4.00000	0.348155
$133$	0	0
$134$	−12.0000	−1.03664
$135$	4.00000	0.344265
$136$	−2.00000	−0.171499
$137$	−18.0000	−1.53784	−0.768922	−	0.639343i	$-0.779207\pi$
$137$	−18.0000	−1.53784	−0.768922	+	0.639343i	$0.779207\pi$
$138$	−12.0000	−1.02151
$139$	8.00000	0.678551	0.339276	−	0.940687i	$-0.389818\pi$
$139$	8.00000	0.678551	0.339276	+	0.940687i	$0.389818\pi$
$140$	0	0
$141$	−24.0000	−2.02116
$142$	10.0000	0.839181
$143$	−2.00000	−0.167248
$144$	1.00000	0.0833333
$145$	2.00000	0.166091
$146$	−10.0000	−0.827606
$147$	0	0
$148$	−2.00000	−0.164399
$149$	2.00000	0.163846	0.0819232	−	0.996639i	$-0.473894\pi$
$149$	2.00000	0.163846	0.0819232	+	0.996639i	$0.473894\pi$
$150$	−2.00000	−0.163299
$151$	6.00000	0.488273	0.244137	−	0.969741i	$-0.421495\pi$
$151$	6.00000	0.488273	0.244137	+	0.969741i	$0.421495\pi$
$152$	−6.00000	−0.486664
$153$	−2.00000	−0.161690
$154$	0	0
$155$	6.00000	0.481932
$156$	−2.00000	−0.160128
$157$	−10.0000	−0.798087	−0.399043	−	0.916932i	$-0.630658\pi$
$157$	−10.0000	−0.798087	−0.399043	+	0.916932i	$0.630658\pi$
$158$	−4.00000	−0.318223
$159$	−4.00000	−0.317221
$160$	1.00000	0.0790569
$161$	0	0
$162$	−11.0000	−0.864242
$163$	−4.00000	−0.313304	−0.156652	−	0.987654i	$-0.550070\pi$
$163$	−4.00000	−0.313304	−0.156652	+	0.987654i	$0.550070\pi$
$164$	−10.0000	−0.780869
$165$	4.00000	0.311400
$166$	0	0
$167$	−20.0000	−1.54765	−0.773823	−	0.633402i	$-0.781658\pi$
$167$	−20.0000	−1.54765	−0.773823	+	0.633402i	$0.781658\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	−2.00000	−0.153393
$171$	−6.00000	−0.458831
$172$	−10.0000	−0.762493
$173$	−10.0000	−0.760286	−0.380143	−	0.924928i	$-0.624125\pi$
$173$	−10.0000	−0.760286	−0.380143	+	0.924928i	$0.624125\pi$
$174$	−4.00000	−0.303239
$175$	0	0
$176$	−2.00000	−0.150756
$177$	20.0000	1.50329
$178$	14.0000	1.04934
$179$	−4.00000	−0.298974	−0.149487	−	0.988764i	$-0.547762\pi$
$179$	−4.00000	−0.298974	−0.149487	+	0.988764i	$0.547762\pi$
$180$	1.00000	0.0745356
$181$	−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
$183$	4.00000	0.295689
$184$	6.00000	0.442326
$185$	−2.00000	−0.147043
$186$	−12.0000	−0.879883
$187$	4.00000	0.292509
$188$	12.0000	0.875190
$189$	0	0
$190$	−6.00000	−0.435286
$191$	0	0		−	1.00000i	$-0.5\pi$
$191$	0	0			1.00000i	$0.5\pi$
$192$	−2.00000	−0.144338
$193$	14.0000	1.00774	0.503871	−	0.863779i	$-0.331909\pi$
$193$	14.0000	1.00774	0.503871	+	0.863779i	$0.331909\pi$
$194$	−14.0000	−1.00514
$195$	−2.00000	−0.143223
$196$	0	0
$197$	−6.00000	−0.427482	−0.213741	−	0.976890i	$-0.568565\pi$
$197$	−6.00000	−0.427482	−0.213741	+	0.976890i	$0.568565\pi$
$198$	−2.00000	−0.142134
$199$	0	0		−	1.00000i	$-0.5\pi$
$199$	0	0			1.00000i	$0.5\pi$
$200$	1.00000	0.0707107
$201$	24.0000	1.69283
$202$	−14.0000	−0.985037
$203$	0	0
$204$	4.00000	0.280056
$205$	−10.0000	−0.698430
$206$	18.0000	1.25412
$207$	6.00000	0.417029
$208$	1.00000	0.0693375
$209$	12.0000	0.830057
$210$	0	0
$211$	28.0000	1.92760	0.963800	−	0.266627i	$-0.0859092\pi$
$211$	28.0000	1.92760	0.963800	+	0.266627i	$0.0859092\pi$
$212$	2.00000	0.137361
$213$	−20.0000	−1.37038
$214$	6.00000	0.410152
$215$	−10.0000	−0.681994
$216$	4.00000	0.272166
$217$	0	0
$218$	−6.00000	−0.406371
$219$	20.0000	1.35147
$220$	−2.00000	−0.134840
$221$	−2.00000	−0.134535
$222$	4.00000	0.268462
$223$	−4.00000	−0.267860	−0.133930	−	0.990991i	$-0.542760\pi$
$223$	−4.00000	−0.267860	−0.133930	+	0.990991i	$0.542760\pi$
$224$	0	0
$225$	1.00000	0.0666667
$226$	2.00000	0.133038
$227$	4.00000	0.265489	0.132745	−	0.991150i	$-0.457621\pi$
$227$	4.00000	0.265489	0.132745	+	0.991150i	$0.457621\pi$
$228$	12.0000	0.794719
$229$	−10.0000	−0.660819	−0.330409	−	0.943838i	$-0.607187\pi$
$229$	−10.0000	−0.660819	−0.330409	+	0.943838i	$0.607187\pi$
$230$	6.00000	0.395628
$231$	0	0
$232$	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	$0.562969\pi$
$234$	1.00000	0.0653720
$235$	12.0000	0.782794
$236$	−10.0000	−0.650945
$237$	8.00000	0.519656
$238$	0	0
$239$	−26.0000	−1.68180	−0.840900	−	0.541190i	$-0.817974\pi$
$239$	−26.0000	−1.68180	−0.840900	+	0.541190i	$0.817974\pi$
$240$	−2.00000	−0.129099
$241$	22.0000	1.41714	0.708572	−	0.705638i	$-0.249340\pi$
$241$	22.0000	1.41714	0.708572	+	0.705638i	$0.249340\pi$
$242$	−7.00000	−0.449977
$243$	10.0000	0.641500
$244$	−2.00000	−0.128037
$245$	0	0
$246$	20.0000	1.27515
$247$	−6.00000	−0.381771
$248$	6.00000	0.381000
$249$	0	0
$250$	1.00000	0.0632456
$251$	0	0		−	1.00000i	$-0.5\pi$
$251$	0	0			1.00000i	$0.5\pi$
$252$	0	0
$253$	−12.0000	−0.754434
$254$	−14.0000	−0.878438
$255$	4.00000	0.250490
$256$	1.00000	0.0625000
$257$	30.0000	1.87135	0.935674	−	0.352865i	$-0.114792\pi$
$257$	30.0000	1.87135	0.935674	+	0.352865i	$0.114792\pi$
$258$	20.0000	1.24515
$259$	0	0
$260$	1.00000	0.0620174
$261$	2.00000	0.123797
$262$	−4.00000	−0.247121
$263$	2.00000	0.123325	0.0616626	−	0.998097i	$-0.480360\pi$
$263$	2.00000	0.123325	0.0616626	+	0.998097i	$0.480360\pi$
$264$	4.00000	0.246183
$265$	2.00000	0.122859
$266$	0	0
$267$	−28.0000	−1.71357
$268$	−12.0000	−0.733017
$269$	−6.00000	−0.365826	−0.182913	−	0.983129i	$-0.558553\pi$
$269$	−6.00000	−0.365826	−0.182913	+	0.983129i	$0.558553\pi$
$270$	4.00000	0.243432
$271$	2.00000	0.121491	0.0607457	−	0.998153i	$-0.480652\pi$
$271$	2.00000	0.121491	0.0607457	+	0.998153i	$0.480652\pi$
$272$	−2.00000	−0.121268
$273$	0	0
$274$	−18.0000	−1.08742
$275$	−2.00000	−0.120605
$276$	−12.0000	−0.722315
$277$	2.00000	0.120168	0.0600842	−	0.998193i	$-0.480863\pi$
$277$	2.00000	0.120168	0.0600842	+	0.998193i	$0.480863\pi$
$278$	8.00000	0.479808
$279$	6.00000	0.359211
$280$	0	0
$281$	−6.00000	−0.357930	−0.178965	−	0.983855i	$-0.557275\pi$
$281$	−6.00000	−0.357930	−0.178965	+	0.983855i	$0.557275\pi$
$282$	−24.0000	−1.42918
$283$	−14.0000	−0.832214	−0.416107	−	0.909316i	$-0.636606\pi$
$283$	−14.0000	−0.832214	−0.416107	+	0.909316i	$0.636606\pi$
$284$	10.0000	0.593391
$285$	12.0000	0.710819
$286$	−2.00000	−0.118262
$287$	0	0
$288$	1.00000	0.0589256
$289$	−13.0000	−0.764706
$290$	2.00000	0.117444
$291$	28.0000	1.64139
$292$	−10.0000	−0.585206
$293$	−22.0000	−1.28525	−0.642627	−	0.766179i	$-0.722155\pi$
$293$	−22.0000	−1.28525	−0.642627	+	0.766179i	$0.722155\pi$
$294$	0	0
$295$	−10.0000	−0.582223
$296$	−2.00000	−0.116248
$297$	−8.00000	−0.464207
$298$	2.00000	0.115857
$299$	6.00000	0.346989
$300$	−2.00000	−0.115470
$301$	0	0
$302$	6.00000	0.345261
$303$	28.0000	1.60856
$304$	−6.00000	−0.344124
$305$	−2.00000	−0.114520
$306$	−2.00000	−0.114332
$307$	−24.0000	−1.36975	−0.684876	−	0.728659i	$-0.740144\pi$
$307$	−24.0000	−1.36975	−0.684876	+	0.728659i	$0.740144\pi$
$308$	0	0
$309$	−36.0000	−2.04797
$310$	6.00000	0.340777
$311$	12.0000	0.680458	0.340229	−	0.940343i	$-0.389495\pi$
$311$	12.0000	0.680458	0.340229	+	0.940343i	$0.389495\pi$
$312$	−2.00000	−0.113228
$313$	6.00000	0.339140	0.169570	−	0.985518i	$-0.445762\pi$
$313$	6.00000	0.339140	0.169570	+	0.985518i	$0.445762\pi$
$314$	−10.0000	−0.564333
$315$	0	0
$316$	−4.00000	−0.225018
$317$	−18.0000	−1.01098	−0.505490	−	0.862832i	$-0.668688\pi$
$317$	−18.0000	−1.01098	−0.505490	+	0.862832i	$0.668688\pi$
$318$	−4.00000	−0.224309
$319$	−4.00000	−0.223957
$320$	1.00000	0.0559017
$321$	−12.0000	−0.669775
$322$	0	0
$323$	12.0000	0.667698
$324$	−11.0000	−0.611111
$325$	1.00000	0.0554700
$326$	−4.00000	−0.221540
$327$	12.0000	0.663602
$328$	−10.0000	−0.552158
$329$	0	0
$330$	4.00000	0.220193
$331$	−14.0000	−0.769510	−0.384755	−	0.923019i	$-0.625714\pi$
$331$	−14.0000	−0.769510	−0.384755	+	0.923019i	$0.625714\pi$
$332$	0	0
$333$	−2.00000	−0.109599
$334$	−20.0000	−1.09435
$335$	−12.0000	−0.655630
$336$	0	0
$337$	−22.0000	−1.19842	−0.599208	−	0.800593i	$-0.704518\pi$
$337$	−22.0000	−1.19842	−0.599208	+	0.800593i	$0.704518\pi$
$338$	1.00000	0.0543928
$339$	−4.00000	−0.217250
$340$	−2.00000	−0.108465
$341$	−12.0000	−0.649836
$342$	−6.00000	−0.324443
$343$	0	0
$344$	−10.0000	−0.539164
$345$	−12.0000	−0.646058
$346$	−10.0000	−0.537603
$347$	6.00000	0.322097	0.161048	−	0.986947i	$-0.448512\pi$
$347$	6.00000	0.322097	0.161048	+	0.986947i	$0.448512\pi$
$348$	−4.00000	−0.214423
$349$	−2.00000	−0.107058	−0.0535288	−	0.998566i	$-0.517047\pi$
$349$	−2.00000	−0.107058	−0.0535288	+	0.998566i	$0.517047\pi$
$350$	0	0
$351$	4.00000	0.213504
$352$	−2.00000	−0.106600
$353$	−34.0000	−1.80964	−0.904819	−	0.425797i	$-0.859994\pi$
$353$	−34.0000	−1.80964	−0.904819	+	0.425797i	$0.859994\pi$
$354$	20.0000	1.06299
$355$	10.0000	0.530745
$356$	14.0000	0.741999
$357$	0	0
$358$	−4.00000	−0.211407
$359$	−6.00000	−0.316668	−0.158334	−	0.987386i	$-0.550612\pi$
$359$	−6.00000	−0.316668	−0.158334	+	0.987386i	$0.550612\pi$
$360$	1.00000	0.0527046
$361$	17.0000	0.894737
$362$	−10.0000	−0.525588
$363$	14.0000	0.734809
$364$	0	0
$365$	−10.0000	−0.523424
$366$	4.00000	0.209083
$367$	−30.0000	−1.56599	−0.782994	−	0.622030i	$-0.786308\pi$
$367$	−30.0000	−1.56599	−0.782994	+	0.622030i	$0.786308\pi$
$368$	6.00000	0.312772
$369$	−10.0000	−0.520579
$370$	−2.00000	−0.103975
$371$	0	0
$372$	−12.0000	−0.622171
$373$	−14.0000	−0.724893	−0.362446	−	0.932005i	$-0.618058\pi$
$373$	−14.0000	−0.724893	−0.362446	+	0.932005i	$0.618058\pi$
$374$	4.00000	0.206835
$375$	−2.00000	−0.103280
$376$	12.0000	0.618853
$377$	2.00000	0.103005
$378$	0	0
$379$	6.00000	0.308199	0.154100	−	0.988055i	$-0.450752\pi$
$379$	6.00000	0.308199	0.154100	+	0.988055i	$0.450752\pi$
$380$	−6.00000	−0.307794
$381$	28.0000	1.43448
$382$	0	0
$383$	−12.0000	−0.613171	−0.306586	−	0.951843i	$-0.599187\pi$
$383$	−12.0000	−0.613171	−0.306586	+	0.951843i	$0.599187\pi$
$384$	−2.00000	−0.102062
$385$	0	0
$386$	14.0000	0.712581
$387$	−10.0000	−0.508329
$388$	−14.0000	−0.710742
$389$	−10.0000	−0.507020	−0.253510	−	0.967333i	$-0.581585\pi$
$389$	−10.0000	−0.507020	−0.253510	+	0.967333i	$0.581585\pi$
$390$	−2.00000	−0.101274
$391$	−12.0000	−0.606866
$392$	0	0
$393$	8.00000	0.403547
$394$	−6.00000	−0.302276
$395$	−4.00000	−0.201262
$396$	−2.00000	−0.100504
$397$	−38.0000	−1.90717	−0.953583	−	0.301131i	$-0.902636\pi$
$397$	−38.0000	−1.90717	−0.953583	+	0.301131i	$0.902636\pi$
$398$	0	0
$399$	0	0
$400$	1.00000	0.0500000
$401$	−6.00000	−0.299626	−0.149813	−	0.988714i	$-0.547867\pi$
$401$	−6.00000	−0.299626	−0.149813	+	0.988714i	$0.547867\pi$
$402$	24.0000	1.19701
$403$	6.00000	0.298881
$404$	−14.0000	−0.696526
$405$	−11.0000	−0.546594
$406$	0	0
$407$	4.00000	0.198273
$408$	4.00000	0.198030
$409$	−34.0000	−1.68119	−0.840596	−	0.541663i	$-0.817795\pi$
$409$	−34.0000	−1.68119	−0.840596	+	0.541663i	$0.817795\pi$
$410$	−10.0000	−0.493865
$411$	36.0000	1.77575
$412$	18.0000	0.886796
$413$	0	0
$414$	6.00000	0.294884
$415$	0	0
$416$	1.00000	0.0490290
$417$	−16.0000	−0.783523
$418$	12.0000	0.586939
$419$	−16.0000	−0.781651	−0.390826	−	0.920465i	$-0.627810\pi$
$419$	−16.0000	−0.781651	−0.390826	+	0.920465i	$0.627810\pi$
$420$	0	0
$421$	10.0000	0.487370	0.243685	−	0.969854i	$-0.421644\pi$
$421$	10.0000	0.487370	0.243685	+	0.969854i	$0.421644\pi$
$422$	28.0000	1.36302
$423$	12.0000	0.583460
$424$	2.00000	0.0971286
$425$	−2.00000	−0.0970143
$426$	−20.0000	−0.969003
$427$	0	0
$428$	6.00000	0.290021
$429$	4.00000	0.193122
$430$	−10.0000	−0.482243
$431$	18.0000	0.867029	0.433515	−	0.901146i	$-0.357273\pi$
$431$	18.0000	0.867029	0.433515	+	0.901146i	$0.357273\pi$
$432$	4.00000	0.192450
$433$	38.0000	1.82616	0.913082	−	0.407777i	$-0.133696\pi$
$433$	38.0000	1.82616	0.913082	+	0.407777i	$0.133696\pi$
$434$	0	0
$435$	−4.00000	−0.191785
$436$	−6.00000	−0.287348
$437$	−36.0000	−1.72211
$438$	20.0000	0.955637
$439$	32.0000	1.52728	0.763638	−	0.645644i	$-0.223411\pi$
$439$	32.0000	1.52728	0.763638	+	0.645644i	$0.223411\pi$
$440$	−2.00000	−0.0953463
$441$	0	0
$442$	−2.00000	−0.0951303
$443$	−14.0000	−0.665160	−0.332580	−	0.943075i	$-0.607919\pi$
$443$	−14.0000	−0.665160	−0.332580	+	0.943075i	$0.607919\pi$
$444$	4.00000	0.189832
$445$	14.0000	0.663664
$446$	−4.00000	−0.189405
$447$	−4.00000	−0.189194
$448$	0	0
$449$	−6.00000	−0.283158	−0.141579	−	0.989927i	$-0.545218\pi$
$449$	−6.00000	−0.283158	−0.141579	+	0.989927i	$0.545218\pi$
$450$	1.00000	0.0471405
$451$	20.0000	0.941763
$452$	2.00000	0.0940721
$453$	−12.0000	−0.563809
$454$	4.00000	0.187729
$455$	0	0
$456$	12.0000	0.561951
$457$	−10.0000	−0.467780	−0.233890	−	0.972263i	$-0.575146\pi$
$457$	−10.0000	−0.467780	−0.233890	+	0.972263i	$0.575146\pi$
$458$	−10.0000	−0.467269
$459$	−8.00000	−0.373408
$460$	6.00000	0.279751
$461$	6.00000	0.279448	0.139724	−	0.990190i	$-0.455378\pi$
$461$	6.00000	0.279448	0.139724	+	0.990190i	$0.455378\pi$
$462$	0	0
$463$	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
$465$	−12.0000	−0.556487
$466$	−6.00000	−0.277945
$467$	−10.0000	−0.462745	−0.231372	−	0.972865i	$-0.574322\pi$
$467$	−10.0000	−0.462745	−0.231372	+	0.972865i	$0.574322\pi$
$468$	1.00000	0.0462250
$469$	0	0
$470$	12.0000	0.553519
$471$	20.0000	0.921551
$472$	−10.0000	−0.460287
$473$	20.0000	0.919601
$474$	8.00000	0.367452
$475$	−6.00000	−0.275299
$476$	0	0
$477$	2.00000	0.0915737
$478$	−26.0000	−1.18921
$479$	−2.00000	−0.0913823	−0.0456912	−	0.998956i	$-0.514549\pi$
$479$	−2.00000	−0.0913823	−0.0456912	+	0.998956i	$0.514549\pi$
$480$	−2.00000	−0.0912871
$481$	−2.00000	−0.0911922
$482$	22.0000	1.00207
$483$	0	0
$484$	−7.00000	−0.318182
$485$	−14.0000	−0.635707
$486$	10.0000	0.453609
$487$	−16.0000	−0.725029	−0.362515	−	0.931978i	$-0.618082\pi$
$487$	−16.0000	−0.725029	−0.362515	+	0.931978i	$0.618082\pi$
$488$	−2.00000	−0.0905357
$489$	8.00000	0.361773
$490$	0	0
$491$	0	0		−	1.00000i	$-0.5\pi$
$491$	0	0			1.00000i	$0.5\pi$
$492$	20.0000	0.901670
$493$	−4.00000	−0.180151
$494$	−6.00000	−0.269953
$495$	−2.00000	−0.0898933
$496$	6.00000	0.269408
$497$	0	0
$498$	0	0
$499$	38.0000	1.70111	0.850557	−	0.525883i	$-0.176265\pi$
$499$	38.0000	1.70111	0.850557	+	0.525883i	$0.176265\pi$
$500$	1.00000	0.0447214
$501$	40.0000	1.78707
$502$	0	0
$503$	14.0000	0.624229	0.312115	−	0.950044i	$-0.398963\pi$
$503$	14.0000	0.624229	0.312115	+	0.950044i	$0.398963\pi$
$504$	0	0
$505$	−14.0000	−0.622992
$506$	−12.0000	−0.533465
$507$	−2.00000	−0.0888231
$508$	−14.0000	−0.621150
$509$	6.00000	0.265945	0.132973	−	0.991120i	$-0.457548\pi$
$509$	6.00000	0.265945	0.132973	+	0.991120i	$0.457548\pi$
$510$	4.00000	0.177123
$511$	0	0
$512$	1.00000	0.0441942
$513$	−24.0000	−1.05963
$514$	30.0000	1.32324
$515$	18.0000	0.793175
$516$	20.0000	0.880451
$517$	−24.0000	−1.05552
$518$	0	0
$519$	20.0000	0.877903
$520$	1.00000	0.0438529
$521$	26.0000	1.13908	0.569540	−	0.821963i	$-0.307121\pi$
$521$	26.0000	1.13908	0.569540	+	0.821963i	$0.307121\pi$
$522$	2.00000	0.0875376
$523$	6.00000	0.262362	0.131181	−	0.991358i	$-0.458123\pi$
$523$	6.00000	0.262362	0.131181	+	0.991358i	$0.458123\pi$
$524$	−4.00000	−0.174741
$525$	0	0
$526$	2.00000	0.0872041
$527$	−12.0000	−0.522728
$528$	4.00000	0.174078
$529$	13.0000	0.565217
$530$	2.00000	0.0868744
$531$	−10.0000	−0.433963
$532$	0	0
$533$	−10.0000	−0.433148
$534$	−28.0000	−1.21168
$535$	6.00000	0.259403
$536$	−12.0000	−0.518321
$537$	8.00000	0.345225
$538$	−6.00000	−0.258678
$539$	0	0
$540$	4.00000	0.172133
$541$	−38.0000	−1.63375	−0.816874	−	0.576816i	$-0.804295\pi$
$541$	−38.0000	−1.63375	−0.816874	+	0.576816i	$0.804295\pi$
$542$	2.00000	0.0859074
$543$	20.0000	0.858282
$544$	−2.00000	−0.0857493
$545$	−6.00000	−0.257012
$546$	0	0
$547$	22.0000	0.940652	0.470326	−	0.882493i	$-0.344136\pi$
$547$	22.0000	0.940652	0.470326	+	0.882493i	$0.344136\pi$
$548$	−18.0000	−0.768922
$549$	−2.00000	−0.0853579
$550$	−2.00000	−0.0852803
$551$	−12.0000	−0.511217
$552$	−12.0000	−0.510754
$553$	0	0
$554$	2.00000	0.0849719
$555$	4.00000	0.169791
$556$	8.00000	0.339276
$557$	38.0000	1.61011	0.805056	−	0.593199i	$-0.202135\pi$
$557$	38.0000	1.61011	0.805056	+	0.593199i	$0.202135\pi$
$558$	6.00000	0.254000
$559$	−10.0000	−0.422955
$560$	0	0
$561$	−8.00000	−0.337760
$562$	−6.00000	−0.253095
$563$	−6.00000	−0.252870	−0.126435	−	0.991975i	$-0.540353\pi$
$563$	−6.00000	−0.252870	−0.126435	+	0.991975i	$0.540353\pi$
$564$	−24.0000	−1.01058
$565$	2.00000	0.0841406
$566$	−14.0000	−0.588464
$567$	0	0
$568$	10.0000	0.419591
$569$	6.00000	0.251533	0.125767	−	0.992060i	$-0.459861\pi$
$569$	6.00000	0.251533	0.125767	+	0.992060i	$0.459861\pi$
$570$	12.0000	0.502625
$571$	−32.0000	−1.33916	−0.669579	−	0.742741i	$-0.733526\pi$
$571$	−32.0000	−1.33916	−0.669579	+	0.742741i	$0.733526\pi$
$572$	−2.00000	−0.0836242
$573$	0	0
$574$	0	0
$575$	6.00000	0.250217
$576$	1.00000	0.0416667
$577$	−18.0000	−0.749350	−0.374675	−	0.927156i	$-0.622246\pi$
$577$	−18.0000	−0.749350	−0.374675	+	0.927156i	$0.622246\pi$
$578$	−13.0000	−0.540729
$579$	−28.0000	−1.16364
$580$	2.00000	0.0830455
$581$	0	0
$582$	28.0000	1.16064
$583$	−4.00000	−0.165663
$584$	−10.0000	−0.413803
$585$	1.00000	0.0413449
$586$	−22.0000	−0.908812
$587$	−12.0000	−0.495293	−0.247647	−	0.968850i	$-0.579657\pi$
$587$	−12.0000	−0.495293	−0.247647	+	0.968850i	$0.579657\pi$
$588$	0	0
$589$	−36.0000	−1.48335
$590$	−10.0000	−0.411693
$591$	12.0000	0.493614
$592$	−2.00000	−0.0821995
$593$	2.00000	0.0821302	0.0410651	−	0.999156i	$-0.486925\pi$
$593$	2.00000	0.0821302	0.0410651	+	0.999156i	$0.486925\pi$
$594$	−8.00000	−0.328244
$595$	0	0
$596$	2.00000	0.0819232
$597$	0	0
$598$	6.00000	0.245358
$599$	−12.0000	−0.490307	−0.245153	−	0.969484i	$-0.578838\pi$
$599$	−12.0000	−0.490307	−0.245153	+	0.969484i	$0.578838\pi$
$600$	−2.00000	−0.0816497
$601$	−10.0000	−0.407909	−0.203954	−	0.978980i	$-0.565379\pi$
$601$	−10.0000	−0.407909	−0.203954	+	0.978980i	$0.565379\pi$
$602$	0	0
$603$	−12.0000	−0.488678
$604$	6.00000	0.244137
$605$	−7.00000	−0.284590
$606$	28.0000	1.13742
$607$	34.0000	1.38002	0.690009	−	0.723801i	$-0.257607\pi$
$607$	34.0000	1.38002	0.690009	+	0.723801i	$0.257607\pi$
$608$	−6.00000	−0.243332
$609$	0	0
$610$	−2.00000	−0.0809776
$611$	12.0000	0.485468
$612$	−2.00000	−0.0808452
$613$	26.0000	1.05013	0.525065	−	0.851062i	$-0.324041\pi$
$613$	26.0000	1.05013	0.525065	+	0.851062i	$0.324041\pi$
$614$	−24.0000	−0.968561
$615$	20.0000	0.806478
$616$	0	0
$617$	6.00000	0.241551	0.120775	−	0.992680i	$-0.461462\pi$
$617$	6.00000	0.241551	0.120775	+	0.992680i	$0.461462\pi$
$618$	−36.0000	−1.44813
$619$	46.0000	1.84890	0.924448	−	0.381308i	$-0.124526\pi$
$619$	46.0000	1.84890	0.924448	+	0.381308i	$0.124526\pi$
$620$	6.00000	0.240966
$621$	24.0000	0.963087
$622$	12.0000	0.481156
$623$	0	0
$624$	−2.00000	−0.0800641
$625$	1.00000	0.0400000
$626$	6.00000	0.239808
$627$	−24.0000	−0.958468
$628$	−10.0000	−0.399043
$629$	4.00000	0.159490
$630$	0	0
$631$	30.0000	1.19428	0.597141	−	0.802137i	$-0.296303\pi$
$631$	30.0000	1.19428	0.597141	+	0.802137i	$0.296303\pi$
$632$	−4.00000	−0.159111
$633$	−56.0000	−2.22580
$634$	−18.0000	−0.714871
$635$	−14.0000	−0.555573
$636$	−4.00000	−0.158610
$637$	0	0
$638$	−4.00000	−0.158362
$639$	10.0000	0.395594
$640$	1.00000	0.0395285
$641$	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
$643$	16.0000	0.630978	0.315489	−	0.948929i	$-0.397831\pi$
$643$	16.0000	0.630978	0.315489	+	0.948929i	$0.397831\pi$
$644$	0	0
$645$	20.0000	0.787499
$646$	12.0000	0.472134
$647$	−42.0000	−1.65119	−0.825595	−	0.564263i	$-0.809160\pi$
$647$	−42.0000	−1.65119	−0.825595	+	0.564263i	$0.809160\pi$
$648$	−11.0000	−0.432121
$649$	20.0000	0.785069
$650$	1.00000	0.0392232
$651$	0	0
$652$	−4.00000	−0.156652
$653$	−6.00000	−0.234798	−0.117399	−	0.993085i	$-0.537456\pi$
$653$	−6.00000	−0.234798	−0.117399	+	0.993085i	$0.537456\pi$
$654$	12.0000	0.469237
$655$	−4.00000	−0.156293
$656$	−10.0000	−0.390434
$657$	−10.0000	−0.390137
$658$	0	0
$659$	8.00000	0.311636	0.155818	−	0.987786i	$-0.450199\pi$
$659$	8.00000	0.311636	0.155818	+	0.987786i	$0.450199\pi$
$660$	4.00000	0.155700
$661$	30.0000	1.16686	0.583432	−	0.812162i	$-0.301709\pi$
$661$	30.0000	1.16686	0.583432	+	0.812162i	$0.301709\pi$
$662$	−14.0000	−0.544125
$663$	4.00000	0.155347
$664$	0	0
$665$	0	0
$666$	−2.00000	−0.0774984
$667$	12.0000	0.464642
$668$	−20.0000	−0.773823
$669$	8.00000	0.309298
$670$	−12.0000	−0.463600
$671$	4.00000	0.154418
$672$	0	0
$673$	2.00000	0.0770943	0.0385472	−	0.999257i	$-0.487727\pi$
$673$	2.00000	0.0770943	0.0385472	+	0.999257i	$0.487727\pi$
$674$	−22.0000	−0.847408
$675$	4.00000	0.153960
$676$	1.00000	0.0384615
$677$	6.00000	0.230599	0.115299	−	0.993331i	$-0.463217\pi$
$677$	6.00000	0.230599	0.115299	+	0.993331i	$0.463217\pi$
$678$	−4.00000	−0.153619
$679$	0	0
$680$	−2.00000	−0.0766965
$681$	−8.00000	−0.306561
$682$	−12.0000	−0.459504
$683$	−44.0000	−1.68361	−0.841807	−	0.539779i	$-0.818508\pi$
$683$	−44.0000	−1.68361	−0.841807	+	0.539779i	$0.818508\pi$
$684$	−6.00000	−0.229416
$685$	−18.0000	−0.687745
$686$	0	0
$687$	20.0000	0.763048
$688$	−10.0000	−0.381246
$689$	2.00000	0.0761939
$690$	−12.0000	−0.456832
$691$	38.0000	1.44559	0.722794	−	0.691063i	$-0.242858\pi$
$691$	38.0000	1.44559	0.722794	+	0.691063i	$0.242858\pi$
$692$	−10.0000	−0.380143
$693$	0	0
$694$	6.00000	0.227757
$695$	8.00000	0.303457
$696$	−4.00000	−0.151620
$697$	20.0000	0.757554
$698$	−2.00000	−0.0757011
$699$	12.0000	0.453882
$700$	0	0
$701$	38.0000	1.43524	0.717620	−	0.696435i	$-0.245231\pi$
$701$	38.0000	1.43524	0.717620	+	0.696435i	$0.245231\pi$
$702$	4.00000	0.150970
$703$	12.0000	0.452589
$704$	−2.00000	−0.0753778
$705$	−24.0000	−0.903892
$706$	−34.0000	−1.27961
$707$	0	0
$708$	20.0000	0.751646
$709$	−22.0000	−0.826227	−0.413114	−	0.910679i	$-0.635559\pi$
$709$	−22.0000	−0.826227	−0.413114	+	0.910679i	$0.635559\pi$
$710$	10.0000	0.375293
$711$	−4.00000	−0.150012
$712$	14.0000	0.524672
$713$	36.0000	1.34821
$714$	0	0
$715$	−2.00000	−0.0747958
$716$	−4.00000	−0.149487
$717$	52.0000	1.94198
$718$	−6.00000	−0.223918
$719$	−48.0000	−1.79010	−0.895049	−	0.445968i	$-0.852860\pi$
$719$	−48.0000	−1.79010	−0.895049	+	0.445968i	$0.852860\pi$
$720$	1.00000	0.0372678
$721$	0	0
$722$	17.0000	0.632674
$723$	−44.0000	−1.63638
$724$	−10.0000	−0.371647
$725$	2.00000	0.0742781
$726$	14.0000	0.519589
$727$	14.0000	0.519231	0.259616	−	0.965712i	$-0.416404\pi$
$727$	14.0000	0.519231	0.259616	+	0.965712i	$0.416404\pi$
$728$	0	0
$729$	13.0000	0.481481
$730$	−10.0000	−0.370117
$731$	20.0000	0.739727
$732$	4.00000	0.147844
$733$	−2.00000	−0.0738717	−0.0369358	−	0.999318i	$-0.511760\pi$
$733$	−2.00000	−0.0738717	−0.0369358	+	0.999318i	$0.511760\pi$
$734$	−30.0000	−1.10732
$735$	0	0
$736$	6.00000	0.221163
$737$	24.0000	0.884051
$738$	−10.0000	−0.368105
$739$	42.0000	1.54499	0.772497	−	0.635018i	$-0.219007\pi$
$739$	42.0000	1.54499	0.772497	+	0.635018i	$0.219007\pi$
$740$	−2.00000	−0.0735215
$741$	12.0000	0.440831
$742$	0	0
$743$	−12.0000	−0.440237	−0.220119	−	0.975473i	$-0.570644\pi$
$743$	−12.0000	−0.440237	−0.220119	+	0.975473i	$0.570644\pi$
$744$	−12.0000	−0.439941
$745$	2.00000	0.0732743
$746$	−14.0000	−0.512576
$747$	0	0
$748$	4.00000	0.146254
$749$	0	0
$750$	−2.00000	−0.0730297
$751$	44.0000	1.60558	0.802791	−	0.596260i	$-0.203347\pi$
$751$	44.0000	1.60558	0.802791	+	0.596260i	$0.203347\pi$
$752$	12.0000	0.437595
$753$	0	0
$754$	2.00000	0.0728357
$755$	6.00000	0.218362
$756$	0	0
$757$	18.0000	0.654221	0.327111	−	0.944986i	$-0.393925\pi$
$757$	18.0000	0.654221	0.327111	+	0.944986i	$0.393925\pi$
$758$	6.00000	0.217930
$759$	24.0000	0.871145
$760$	−6.00000	−0.217643
$761$	14.0000	0.507500	0.253750	−	0.967270i	$-0.418336\pi$
$761$	14.0000	0.507500	0.253750	+	0.967270i	$0.418336\pi$
$762$	28.0000	1.01433
$763$	0	0
$764$	0	0
$765$	−2.00000	−0.0723102
$766$	−12.0000	−0.433578
$767$	−10.0000	−0.361079
$768$	−2.00000	−0.0721688
$769$	−10.0000	−0.360609	−0.180305	−	0.983611i	$-0.557708\pi$
$769$	−10.0000	−0.360609	−0.180305	+	0.983611i	$0.557708\pi$
$770$	0	0
$771$	−60.0000	−2.16085
$772$	14.0000	0.503871
$773$	34.0000	1.22290	0.611448	−	0.791285i	$-0.290588\pi$
$773$	34.0000	1.22290	0.611448	+	0.791285i	$0.290588\pi$
$774$	−10.0000	−0.359443
$775$	6.00000	0.215526
$776$	−14.0000	−0.502571
$777$	0	0
$778$	−10.0000	−0.358517
$779$	60.0000	2.14972
$780$	−2.00000	−0.0716115
$781$	−20.0000	−0.715656
$782$	−12.0000	−0.429119
$783$	8.00000	0.285897
$784$	0	0
$785$	−10.0000	−0.356915
$786$	8.00000	0.285351
$787$	8.00000	0.285169	0.142585	−	0.989783i	$-0.454459\pi$
$787$	8.00000	0.285169	0.142585	+	0.989783i	$0.454459\pi$
$788$	−6.00000	−0.213741
$789$	−4.00000	−0.142404
$790$	−4.00000	−0.142314
$791$	0	0
$792$	−2.00000	−0.0710669
$793$	−2.00000	−0.0710221
$794$	−38.0000	−1.34857
$795$	−4.00000	−0.141865
$796$	0	0
$797$	22.0000	0.779280	0.389640	−	0.920967i	$-0.372599\pi$
$797$	22.0000	0.779280	0.389640	+	0.920967i	$0.372599\pi$
$798$	0	0
$799$	−24.0000	−0.849059
$800$	1.00000	0.0353553
$801$	14.0000	0.494666
$802$	−6.00000	−0.211867
$803$	20.0000	0.705785
$804$	24.0000	0.846415
$805$	0	0
$806$	6.00000	0.211341
$807$	12.0000	0.422420
$808$	−14.0000	−0.492518
$809$	−34.0000	−1.19538	−0.597688	−	0.801729i	$-0.703914\pi$
$809$	−34.0000	−1.19538	−0.597688	+	0.801729i	$0.703914\pi$
$810$	−11.0000	−0.386501
$811$	38.0000	1.33436	0.667180	−	0.744896i	$-0.267501\pi$
$811$	38.0000	1.33436	0.667180	+	0.744896i	$0.267501\pi$
$812$	0	0
$813$	−4.00000	−0.140286
$814$	4.00000	0.140200
$815$	−4.00000	−0.140114
$816$	4.00000	0.140028
$817$	60.0000	2.09913
$818$	−34.0000	−1.18878
$819$	0	0
$820$	−10.0000	−0.349215
$821$	50.0000	1.74501	0.872506	−	0.488603i	$-0.162493\pi$
$821$	50.0000	1.74501	0.872506	+	0.488603i	$0.162493\pi$
$822$	36.0000	1.25564
$823$	14.0000	0.488009	0.244005	−	0.969774i	$-0.421539\pi$
$823$	14.0000	0.488009	0.244005	+	0.969774i	$0.421539\pi$
$824$	18.0000	0.627060
$825$	4.00000	0.139262
$826$	0	0
$827$	0	0		−	1.00000i	$-0.5\pi$
$827$	0	0			1.00000i	$0.5\pi$
$828$	6.00000	0.208514
$829$	−2.00000	−0.0694629	−0.0347314	−	0.999397i	$-0.511058\pi$
$829$	−2.00000	−0.0694629	−0.0347314	+	0.999397i	$0.511058\pi$
$830$	0	0
$831$	−4.00000	−0.138758
$832$	1.00000	0.0346688
$833$	0	0
$834$	−16.0000	−0.554035
$835$	−20.0000	−0.692129
$836$	12.0000	0.415029
$837$	24.0000	0.829561
$838$	−16.0000	−0.552711
$839$	−26.0000	−0.897620	−0.448810	−	0.893627i	$-0.648152\pi$
$839$	−26.0000	−0.897620	−0.448810	+	0.893627i	$0.648152\pi$
$840$	0	0
$841$	−25.0000	−0.862069
$842$	10.0000	0.344623
$843$	12.0000	0.413302
$844$	28.0000	0.963800
$845$	1.00000	0.0344010
$846$	12.0000	0.412568
$847$	0	0
$848$	2.00000	0.0686803
$849$	28.0000	0.960958
$850$	−2.00000	−0.0685994
$851$	−12.0000	−0.411355
$852$	−20.0000	−0.685189
$853$	26.0000	0.890223	0.445112	−	0.895475i	$-0.353164\pi$
$853$	26.0000	0.890223	0.445112	+	0.895475i	$0.353164\pi$
$854$	0	0
$855$	−6.00000	−0.205196
$856$	6.00000	0.205076
$857$	14.0000	0.478231	0.239115	−	0.970991i	$-0.423143\pi$
$857$	14.0000	0.478231	0.239115	+	0.970991i	$0.423143\pi$
$858$	4.00000	0.136558
$859$	0	0		−	1.00000i	$-0.5\pi$
$859$	0	0			1.00000i	$0.5\pi$
$860$	−10.0000	−0.340997
$861$	0	0
$862$	18.0000	0.613082
$863$	−36.0000	−1.22545	−0.612727	−	0.790295i	$-0.709928\pi$
$863$	−36.0000	−1.22545	−0.612727	+	0.790295i	$0.709928\pi$
$864$	4.00000	0.136083
$865$	−10.0000	−0.340010
$866$	38.0000	1.29129
$867$	26.0000	0.883006
$868$	0	0
$869$	8.00000	0.271381
$870$	−4.00000	−0.135613
$871$	−12.0000	−0.406604
$872$	−6.00000	−0.203186
$873$	−14.0000	−0.473828
$874$	−36.0000	−1.21772
$875$	0	0
$876$	20.0000	0.675737
$877$	−14.0000	−0.472746	−0.236373	−	0.971662i	$-0.575959\pi$
$877$	−14.0000	−0.472746	−0.236373	+	0.971662i	$0.575959\pi$
$878$	32.0000	1.07995
$879$	44.0000	1.48408
$880$	−2.00000	−0.0674200
$881$	−6.00000	−0.202145	−0.101073	−	0.994879i	$-0.532227\pi$
$881$	−6.00000	−0.202145	−0.101073	+	0.994879i	$0.532227\pi$
$882$	0	0
$883$	38.0000	1.27880	0.639401	−	0.768874i	$-0.279182\pi$
$883$	38.0000	1.27880	0.639401	+	0.768874i	$0.279182\pi$
$884$	−2.00000	−0.0672673
$885$	20.0000	0.672293
$886$	−14.0000	−0.470339
$887$	−22.0000	−0.738688	−0.369344	−	0.929293i	$-0.620418\pi$
$887$	−22.0000	−0.738688	−0.369344	+	0.929293i	$0.620418\pi$
$888$	4.00000	0.134231
$889$	0	0
$890$	14.0000	0.469281
$891$	22.0000	0.737028
$892$	−4.00000	−0.133930
$893$	−72.0000	−2.40939
$894$	−4.00000	−0.133780
$895$	−4.00000	−0.133705
$896$	0	0
$897$	−12.0000	−0.400668
$898$	−6.00000	−0.200223
$899$	12.0000	0.400222
$900$	1.00000	0.0333333
$901$	−4.00000	−0.133259
$902$	20.0000	0.665927
$903$	0	0
$904$	2.00000	0.0665190
$905$	−10.0000	−0.332411
$906$	−12.0000	−0.398673
$907$	−14.0000	−0.464862	−0.232431	−	0.972613i	$-0.574668\pi$
$907$	−14.0000	−0.464862	−0.232431	+	0.972613i	$0.574668\pi$
$908$	4.00000	0.132745
$909$	−14.0000	−0.464351
$910$	0	0
$911$	16.0000	0.530104	0.265052	−	0.964234i	$-0.414611\pi$
$911$	16.0000	0.530104	0.265052	+	0.964234i	$0.414611\pi$
$912$	12.0000	0.397360
$913$	0	0
$914$	−10.0000	−0.330771
$915$	4.00000	0.132236
$916$	−10.0000	−0.330409
$917$	0	0
$918$	−8.00000	−0.264039
$919$	4.00000	0.131948	0.0659739	−	0.997821i	$-0.478985\pi$
$919$	4.00000	0.131948	0.0659739	+	0.997821i	$0.478985\pi$
$920$	6.00000	0.197814
$921$	48.0000	1.58165
$922$	6.00000	0.197599
$923$	10.0000	0.329154
$924$	0	0
$925$	−2.00000	−0.0657596
$926$	16.0000	0.525793
$927$	18.0000	0.591198
$928$	2.00000	0.0656532
$929$	6.00000	0.196854	0.0984268	−	0.995144i	$-0.468619\pi$
$929$	6.00000	0.196854	0.0984268	+	0.995144i	$0.468619\pi$
$930$	−12.0000	−0.393496
$931$	0	0
$932$	−6.00000	−0.196537
$933$	−24.0000	−0.785725
$934$	−10.0000	−0.327210
$935$	4.00000	0.130814
$936$	1.00000	0.0326860
$937$	−18.0000	−0.588034	−0.294017	−	0.955800i	$-0.594992\pi$
$937$	−18.0000	−0.588034	−0.294017	+	0.955800i	$0.594992\pi$
$938$	0	0
$939$	−12.0000	−0.391605
$940$	12.0000	0.391397
$941$	−50.0000	−1.62995	−0.814977	−	0.579494i	$-0.803250\pi$
$941$	−50.0000	−1.62995	−0.814977	+	0.579494i	$0.803250\pi$
$942$	20.0000	0.651635
$943$	−60.0000	−1.95387
$944$	−10.0000	−0.325472
$945$	0	0
$946$	20.0000	0.650256
$947$	−8.00000	−0.259965	−0.129983	−	0.991516i	$-0.541492\pi$
$947$	−8.00000	−0.259965	−0.129983	+	0.991516i	$0.541492\pi$
$948$	8.00000	0.259828
$949$	−10.0000	−0.324614
$950$	−6.00000	−0.194666
$951$	36.0000	1.16738
$952$	0	0
$953$	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
$955$	0	0
$956$	−26.0000	−0.840900
$957$	8.00000	0.258603
$958$	−2.00000	−0.0646171
$959$	0	0
$960$	−2.00000	−0.0645497
$961$	5.00000	0.161290
$962$	−2.00000	−0.0644826
$963$	6.00000	0.193347
$964$	22.0000	0.708572
$965$	14.0000	0.450676
$966$	0	0
$967$	−32.0000	−1.02905	−0.514525	−	0.857475i	$-0.672032\pi$
$967$	−32.0000	−1.02905	−0.514525	+	0.857475i	$0.672032\pi$
$968$	−7.00000	−0.224989
$969$	−24.0000	−0.770991
$970$	−14.0000	−0.449513
$971$	−20.0000	−0.641831	−0.320915	−	0.947108i	$-0.603990\pi$
$971$	−20.0000	−0.641831	−0.320915	+	0.947108i	$0.603990\pi$
$972$	10.0000	0.320750
$973$	0	0
$974$	−16.0000	−0.512673
$975$	−2.00000	−0.0640513
$976$	−2.00000	−0.0640184
$977$	−30.0000	−0.959785	−0.479893	−	0.877327i	$-0.659324\pi$
$977$	−30.0000	−0.959785	−0.479893	+	0.877327i	$0.659324\pi$
$978$	8.00000	0.255812
$979$	−28.0000	−0.894884
$980$	0	0
$981$	−6.00000	−0.191565
$982$	0	0
$983$	16.0000	0.510321	0.255160	−	0.966899i	$-0.417872\pi$
$983$	16.0000	0.510321	0.255160	+	0.966899i	$0.417872\pi$
$984$	20.0000	0.637577
$985$	−6.00000	−0.191176
$986$	−4.00000	−0.127386
$987$	0	0
$988$	−6.00000	−0.190885
$989$	−60.0000	−1.90789
$990$	−2.00000	−0.0635642
$991$	0	0		−	1.00000i	$-0.5\pi$
$991$	0	0			1.00000i	$0.5\pi$
$992$	6.00000	0.190500
$993$	28.0000	0.888553
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−10.0000	−0.316703	−0.158352	−	0.987383i	$-0.550618\pi$
$997$	−10.0000	−0.316703	−0.158352	+	0.987383i	$0.550618\pi$
$998$	38.0000	1.20287
$999$	−8.00000	−0.253109

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6370.2.a.l.1.1		1
7.6	odd	2		130.2.a.c.1.1	✓	1
21.20	even	2		1170.2.a.d.1.1		1
28.27	even	2		1040.2.a.b.1.1		1
35.13	even	4		650.2.b.g.599.1		2
35.27	even	4		650.2.b.g.599.2		2
35.34	odd	2		650.2.a.c.1.1		1
56.13	odd	2		4160.2.a.c.1.1		1
56.27	even	2		4160.2.a.t.1.1		1
84.83	odd	2		9360.2.a.by.1.1		1
91.6	even	12		1690.2.l.a.361.1		4
91.20	even	12		1690.2.l.a.361.2		4
91.34	even	4		1690.2.d.e.1351.2		2
91.41	even	12		1690.2.l.a.1161.2		4
91.48	odd	6		1690.2.e.a.991.1		2
91.55	odd	6		1690.2.e.a.191.1		2
91.62	odd	6		1690.2.e.g.191.1		2
91.69	odd	6		1690.2.e.g.991.1		2
91.76	even	12		1690.2.l.a.1161.1		4
91.83	even	4		1690.2.d.e.1351.1		2
91.90	odd	2		1690.2.a.e.1.1		1
105.62	odd	4		5850.2.e.u.5149.1		2
105.83	odd	4		5850.2.e.u.5149.2		2
105.104	even	2		5850.2.a.cb.1.1		1
140.139	even	2		5200.2.a.bd.1.1		1
455.454	odd	2		8450.2.a.n.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
130.2.a.c.1.1	✓	1	7.6	odd	2
650.2.a.c.1.1		1	35.34	odd	2
650.2.b.g.599.1		2	35.13	even	4
650.2.b.g.599.2		2	35.27	even	4
1040.2.a.b.1.1		1	28.27	even	2
1170.2.a.d.1.1		1	21.20	even	2
1690.2.a.e.1.1		1	91.90	odd	2
1690.2.d.e.1351.1		2	91.83	even	4
1690.2.d.e.1351.2		2	91.34	even	4
1690.2.e.a.191.1		2	91.55	odd	6
1690.2.e.a.991.1		2	91.48	odd	6
1690.2.e.g.191.1		2	91.62	odd	6
1690.2.e.g.991.1		2	91.69	odd	6
1690.2.l.a.361.1		4	91.6	even	12
1690.2.l.a.361.2		4	91.20	even	12
1690.2.l.a.1161.1		4	91.76	even	12
1690.2.l.a.1161.2		4	91.41	even	12
4160.2.a.c.1.1		1	56.13	odd	2
4160.2.a.t.1.1		1	56.27	even	2
5200.2.a.bd.1.1		1	140.139	even	2
5850.2.a.cb.1.1		1	105.104	even	2
5850.2.e.u.5149.1		2	105.62	odd	4
5850.2.e.u.5149.2		2	105.83	odd	4
6370.2.a.l.1.1		1	1.1	even	1	trivial
8450.2.a.n.1.1		1	455.454	odd	2
9360.2.a.by.1.1		1	84.83	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 6370 = 2 \cdot 5 \cdot 7^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6370.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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