LMFDB - Embedded newform 490.2.a.k.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [490,2,Mod(1,490)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(490, 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("490.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 490 = 2 \cdot 5 \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	490.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:	$3.91266969904$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 70)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	490.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q+1.00000 q^{2} +3.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} +3.00000 q^{6} +1.00000 q^{8} +6.00000 q^{9} +O(q^{10})$

\(q+1.00000 q^{2} +3.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} +3.00000 q^{6} +1.00000 q^{8} +6.00000 q^{9} -1.00000 q^{10} -2.00000 q^{11} +3.00000 q^{12} -3.00000 q^{15} +1.00000 q^{16} -4.00000 q^{17} +6.00000 q^{18} -6.00000 q^{19} -1.00000 q^{20} -2.00000 q^{22} +3.00000 q^{23} +3.00000 q^{24} +1.00000 q^{25} +9.00000 q^{27} +9.00000 q^{29} -3.00000 q^{30} -4.00000 q^{31} +1.00000 q^{32} -6.00000 q^{33} -4.00000 q^{34} +6.00000 q^{36} -4.00000 q^{37} -6.00000 q^{38} -1.00000 q^{40} -7.00000 q^{41} -5.00000 q^{43} -2.00000 q^{44} -6.00000 q^{45} +3.00000 q^{46} +8.00000 q^{47} +3.00000 q^{48} +1.00000 q^{50} -12.0000 q^{51} -2.00000 q^{53} +9.00000 q^{54} +2.00000 q^{55} -18.0000 q^{57} +9.00000 q^{58} +10.0000 q^{59} -3.00000 q^{60} +1.00000 q^{61} -4.00000 q^{62} +1.00000 q^{64} -6.00000 q^{66} -9.00000 q^{67} -4.00000 q^{68} +9.00000 q^{69} +2.00000 q^{71} +6.00000 q^{72} -4.00000 q^{73} -4.00000 q^{74} +3.00000 q^{75} -6.00000 q^{76} +10.0000 q^{79} -1.00000 q^{80} +9.00000 q^{81} -7.00000 q^{82} -7.00000 q^{83} +4.00000 q^{85} -5.00000 q^{86} +27.0000 q^{87} -2.00000 q^{88} +1.00000 q^{89} -6.00000 q^{90} +3.00000 q^{92} -12.0000 q^{93} +8.00000 q^{94} +6.00000 q^{95} +3.00000 q^{96} +14.0000 q^{97} -12.0000 q^{99} +O(q^{100})\)

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
$3$	3.00000	1.73205	0.866025	−	0.500000i	$-0.166667\pi$
$3$	3.00000	1.73205	0.866025	+	0.500000i	$0.166667\pi$
$4$	1.00000	0.500000
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	3.00000	1.22474
$7$	0	0
$7$	0	0
$8$	1.00000	0.353553
$9$	6.00000	2.00000
$10$	−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	$0.597491\pi$
$12$	3.00000	0.866025
$13$	0	0		−	1.00000i	$-0.5\pi$
$13$	0	0			1.00000i	$0.5\pi$
$14$	0	0
$15$	−3.00000	−0.774597
$16$	1.00000	0.250000
$17$	−4.00000	−0.970143	−0.485071	−	0.874475i	$-0.661206\pi$
$17$	−4.00000	−0.970143	−0.485071	+	0.874475i	$0.661206\pi$
$18$	6.00000	1.41421
$19$	−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
$23$	3.00000	0.625543	0.312772	−	0.949828i	$-0.398743\pi$
$23$	3.00000	0.625543	0.312772	+	0.949828i	$0.398743\pi$
$24$	3.00000	0.612372
$25$	1.00000	0.200000
$26$	0	0
$27$	9.00000	1.73205
$28$	0	0
$29$	9.00000	1.67126	0.835629	−	0.549294i	$-0.185103\pi$
$29$	9.00000	1.67126	0.835629	+	0.549294i	$0.185103\pi$
$30$	−3.00000	−0.547723
$31$	−4.00000	−0.718421	−0.359211	−	0.933257i	$-0.616954\pi$
$31$	−4.00000	−0.718421	−0.359211	+	0.933257i	$0.616954\pi$
$32$	1.00000	0.176777
$33$	−6.00000	−1.04447
$34$	−4.00000	−0.685994
$35$	0	0
$36$	6.00000	1.00000
$37$	−4.00000	−0.657596	−0.328798	−	0.944400i	$-0.606644\pi$
$37$	−4.00000	−0.657596	−0.328798	+	0.944400i	$0.606644\pi$
$38$	−6.00000	−0.973329
$39$	0	0
$40$	−1.00000	−0.158114
$41$	−7.00000	−1.09322	−0.546608	−	0.837389i	$-0.684081\pi$
$41$	−7.00000	−1.09322	−0.546608	+	0.837389i	$0.684081\pi$
$42$	0	0
$43$	−5.00000	−0.762493	−0.381246	−	0.924473i	$-0.624505\pi$
$43$	−5.00000	−0.762493	−0.381246	+	0.924473i	$0.624505\pi$
$44$	−2.00000	−0.301511
$45$	−6.00000	−0.894427
$46$	3.00000	0.442326
$47$	8.00000	1.16692	0.583460	−	0.812142i	$-0.301699\pi$
$47$	8.00000	1.16692	0.583460	+	0.812142i	$0.301699\pi$
$48$	3.00000	0.433013
$49$	0	0
$50$	1.00000	0.141421
$51$	−12.0000	−1.68034
$52$	0	0
$53$	−2.00000	−0.274721	−0.137361	−	0.990521i	$-0.543862\pi$
$53$	−2.00000	−0.274721	−0.137361	+	0.990521i	$0.543862\pi$
$54$	9.00000	1.22474
$55$	2.00000	0.269680
$56$	0	0
$57$	−18.0000	−2.38416
$58$	9.00000	1.18176
$59$	10.0000	1.30189	0.650945	−	0.759125i	$-0.274373\pi$
$59$	10.0000	1.30189	0.650945	+	0.759125i	$0.274373\pi$
$60$	−3.00000	−0.387298
$61$	1.00000	0.128037	0.0640184	−	0.997949i	$-0.479608\pi$
$61$	1.00000	0.128037	0.0640184	+	0.997949i	$0.479608\pi$
$62$	−4.00000	−0.508001
$63$	0	0
$64$	1.00000	0.125000
$65$	0	0
$66$	−6.00000	−0.738549
$67$	−9.00000	−1.09952	−0.549762	−	0.835321i	$-0.685282\pi$
$67$	−9.00000	−1.09952	−0.549762	+	0.835321i	$0.685282\pi$
$68$	−4.00000	−0.485071
$69$	9.00000	1.08347
$70$	0	0
$71$	2.00000	0.237356	0.118678	−	0.992933i	$-0.462134\pi$
$71$	2.00000	0.237356	0.118678	+	0.992933i	$0.462134\pi$
$72$	6.00000	0.707107
$73$	−4.00000	−0.468165	−0.234082	−	0.972217i	$-0.575209\pi$
$73$	−4.00000	−0.468165	−0.234082	+	0.972217i	$0.575209\pi$
$74$	−4.00000	−0.464991
$75$	3.00000	0.346410
$76$	−6.00000	−0.688247
$77$	0	0
$78$	0	0
$79$	10.0000	1.12509	0.562544	−	0.826767i	$-0.309823\pi$
$79$	10.0000	1.12509	0.562544	+	0.826767i	$0.309823\pi$
$80$	−1.00000	−0.111803
$81$	9.00000	1.00000
$82$	−7.00000	−0.773021
$83$	−7.00000	−0.768350	−0.384175	−	0.923260i	$-0.625514\pi$
$83$	−7.00000	−0.768350	−0.384175	+	0.923260i	$0.625514\pi$
$84$	0	0
$85$	4.00000	0.433861
$86$	−5.00000	−0.539164
$87$	27.0000	2.89470
$88$	−2.00000	−0.213201
$89$	1.00000	0.106000	0.0529999	−	0.998595i	$-0.483122\pi$
$89$	1.00000	0.106000	0.0529999	+	0.998595i	$0.483122\pi$
$90$	−6.00000	−0.632456
$91$	0	0
$92$	3.00000	0.312772
$93$	−12.0000	−1.24434
$94$	8.00000	0.825137
$95$	6.00000	0.615587
$96$	3.00000	0.306186
$97$	14.0000	1.42148	0.710742	−	0.703452i	$-0.248359\pi$
$97$	14.0000	1.42148	0.710742	+	0.703452i	$0.248359\pi$
$98$	0	0
$99$	−12.0000	−1.20605
$100$	1.00000	0.100000
$101$	3.00000	0.298511	0.149256	−	0.988799i	$-0.452312\pi$
$101$	3.00000	0.298511	0.149256	+	0.988799i	$0.452312\pi$
$102$	−12.0000	−1.18818
$103$	1.00000	0.0985329	0.0492665	−	0.998786i	$-0.484312\pi$
$103$	1.00000	0.0985329	0.0492665	+	0.998786i	$0.484312\pi$
$104$	0	0
$105$	0	0
$106$	−2.00000	−0.194257
$107$	3.00000	0.290021	0.145010	−	0.989430i	$-0.453678\pi$
$107$	3.00000	0.290021	0.145010	+	0.989430i	$0.453678\pi$
$108$	9.00000	0.866025
$109$	−9.00000	−0.862044	−0.431022	−	0.902342i	$-0.641847\pi$
$109$	−9.00000	−0.862044	−0.431022	+	0.902342i	$0.641847\pi$
$110$	2.00000	0.190693
$111$	−12.0000	−1.13899
$112$	0	0
$113$	2.00000	0.188144	0.0940721	−	0.995565i	$-0.470012\pi$
$113$	2.00000	0.188144	0.0940721	+	0.995565i	$0.470012\pi$
$114$	−18.0000	−1.68585
$115$	−3.00000	−0.279751
$116$	9.00000	0.835629
$117$	0	0
$118$	10.0000	0.920575
$119$	0	0
$120$	−3.00000	−0.273861
$121$	−7.00000	−0.636364
$122$	1.00000	0.0905357
$123$	−21.0000	−1.89351
$124$	−4.00000	−0.359211
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	16.0000	1.41977	0.709885	−	0.704317i	$-0.248747\pi$
$127$	16.0000	1.41977	0.709885	+	0.704317i	$0.248747\pi$
$128$	1.00000	0.0883883
$129$	−15.0000	−1.32068
$130$	0	0
$131$	8.00000	0.698963	0.349482	−	0.936943i	$-0.386358\pi$
$131$	8.00000	0.698963	0.349482	+	0.936943i	$0.386358\pi$
$132$	−6.00000	−0.522233
$133$	0	0
$134$	−9.00000	−0.777482
$135$	−9.00000	−0.774597
$136$	−4.00000	−0.342997
$137$	12.0000	1.02523	0.512615	−	0.858619i	$-0.328677\pi$
$137$	12.0000	1.02523	0.512615	+	0.858619i	$0.328677\pi$
$138$	9.00000	0.766131
$139$	−14.0000	−1.18746	−0.593732	−	0.804663i	$-0.702346\pi$
$139$	−14.0000	−1.18746	−0.593732	+	0.804663i	$0.702346\pi$
$140$	0	0
$141$	24.0000	2.02116
$142$	2.00000	0.167836
$143$	0	0
$144$	6.00000	0.500000
$145$	−9.00000	−0.747409
$146$	−4.00000	−0.331042
$147$	0	0
$148$	−4.00000	−0.328798
$149$	3.00000	0.245770	0.122885	−	0.992421i	$-0.460785\pi$
$149$	3.00000	0.245770	0.122885	+	0.992421i	$0.460785\pi$
$150$	3.00000	0.244949
$151$	−16.0000	−1.30206	−0.651031	−	0.759051i	$-0.725663\pi$
$151$	−16.0000	−1.30206	−0.651031	+	0.759051i	$0.725663\pi$
$152$	−6.00000	−0.486664
$153$	−24.0000	−1.94029
$154$	0	0
$155$	4.00000	0.321288
$156$	0	0
$157$	10.0000	0.798087	0.399043	−	0.916932i	$-0.369342\pi$
$157$	10.0000	0.798087	0.399043	+	0.916932i	$0.369342\pi$
$158$	10.0000	0.795557
$159$	−6.00000	−0.475831
$160$	−1.00000	−0.0790569
$161$	0	0
$162$	9.00000	0.707107
$163$	−4.00000	−0.313304	−0.156652	−	0.987654i	$-0.550070\pi$
$163$	−4.00000	−0.313304	−0.156652	+	0.987654i	$0.550070\pi$
$164$	−7.00000	−0.546608
$165$	6.00000	0.467099
$166$	−7.00000	−0.543305
$167$	−21.0000	−1.62503	−0.812514	−	0.582941i	$-0.801902\pi$
$167$	−21.0000	−1.62503	−0.812514	+	0.582941i	$0.801902\pi$
$168$	0	0
$169$	−13.0000	−1.00000
$170$	4.00000	0.306786
$171$	−36.0000	−2.75299
$172$	−5.00000	−0.381246
$173$	8.00000	0.608229	0.304114	−	0.952636i	$-0.401639\pi$
$173$	8.00000	0.608229	0.304114	+	0.952636i	$0.401639\pi$
$174$	27.0000	2.04686
$175$	0	0
$176$	−2.00000	−0.150756
$177$	30.0000	2.25494
$178$	1.00000	0.0749532
$179$	12.0000	0.896922	0.448461	−	0.893802i	$-0.351972\pi$
$179$	12.0000	0.896922	0.448461	+	0.893802i	$0.351972\pi$
$180$	−6.00000	−0.447214
$181$	7.00000	0.520306	0.260153	−	0.965567i	$-0.416227\pi$
$181$	7.00000	0.520306	0.260153	+	0.965567i	$0.416227\pi$
$182$	0	0
$183$	3.00000	0.221766
$184$	3.00000	0.221163
$185$	4.00000	0.294086
$186$	−12.0000	−0.879883
$187$	8.00000	0.585018
$188$	8.00000	0.583460
$189$	0	0
$190$	6.00000	0.435286
$191$	−18.0000	−1.30243	−0.651217	−	0.758891i	$-0.725741\pi$
$191$	−18.0000	−1.30243	−0.651217	+	0.758891i	$0.725741\pi$
$192$	3.00000	0.216506
$193$	26.0000	1.87152	0.935760	−	0.352636i	$-0.114715\pi$
$193$	26.0000	1.87152	0.935760	+	0.352636i	$0.114715\pi$
$194$	14.0000	1.00514
$195$	0	0
$196$	0	0
$197$	2.00000	0.142494	0.0712470	−	0.997459i	$-0.477302\pi$
$197$	2.00000	0.142494	0.0712470	+	0.997459i	$0.477302\pi$
$198$	−12.0000	−0.852803
$199$	−4.00000	−0.283552	−0.141776	−	0.989899i	$-0.545281\pi$
$199$	−4.00000	−0.283552	−0.141776	+	0.989899i	$0.545281\pi$
$200$	1.00000	0.0707107
$201$	−27.0000	−1.90443
$202$	3.00000	0.211079
$203$	0	0
$204$	−12.0000	−0.840168
$205$	7.00000	0.488901
$206$	1.00000	0.0696733
$207$	18.0000	1.25109
$208$	0	0
$209$	12.0000	0.830057
$210$	0	0
$211$	−26.0000	−1.78991	−0.894957	−	0.446153i	$-0.852794\pi$
$211$	−26.0000	−1.78991	−0.894957	+	0.446153i	$0.852794\pi$
$212$	−2.00000	−0.137361
$213$	6.00000	0.411113
$214$	3.00000	0.205076
$215$	5.00000	0.340997
$216$	9.00000	0.612372
$217$	0	0
$218$	−9.00000	−0.609557
$219$	−12.0000	−0.810885
$220$	2.00000	0.134840
$221$	0	0
$222$	−12.0000	−0.805387
$223$	28.0000	1.87502	0.937509	−	0.347960i	$-0.113126\pi$
$223$	28.0000	1.87502	0.937509	+	0.347960i	$0.113126\pi$
$224$	0	0
$225$	6.00000	0.400000
$226$	2.00000	0.133038
$227$	−4.00000	−0.265489	−0.132745	−	0.991150i	$-0.542379\pi$
$227$	−4.00000	−0.265489	−0.132745	+	0.991150i	$0.542379\pi$
$228$	−18.0000	−1.19208
$229$	22.0000	1.45380	0.726900	−	0.686743i	$-0.240960\pi$
$229$	22.0000	1.45380	0.726900	+	0.686743i	$0.240960\pi$
$230$	−3.00000	−0.197814
$231$	0	0
$232$	9.00000	0.590879
$233$	24.0000	1.57229	0.786146	−	0.618041i	$-0.212073\pi$
$233$	24.0000	1.57229	0.786146	+	0.618041i	$0.212073\pi$
$234$	0	0
$235$	−8.00000	−0.521862
$236$	10.0000	0.650945
$237$	30.0000	1.94871
$238$	0	0
$239$	16.0000	1.03495	0.517477	−	0.855697i	$-0.326871\pi$
$239$	16.0000	1.03495	0.517477	+	0.855697i	$0.326871\pi$
$240$	−3.00000	−0.193649
$241$	10.0000	0.644157	0.322078	−	0.946713i	$-0.395619\pi$
$241$	10.0000	0.644157	0.322078	+	0.946713i	$0.395619\pi$
$242$	−7.00000	−0.449977
$243$	0	0
$244$	1.00000	0.0640184
$245$	0	0
$246$	−21.0000	−1.33891
$247$	0	0
$248$	−4.00000	−0.254000
$249$	−21.0000	−1.33082
$250$	−1.00000	−0.0632456
$251$	0	0		−	1.00000i	$-0.5\pi$
$251$	0	0			1.00000i	$0.5\pi$
$252$	0	0
$253$	−6.00000	−0.377217
$254$	16.0000	1.00393
$255$	12.0000	0.751469
$256$	1.00000	0.0625000
$257$	8.00000	0.499026	0.249513	−	0.968371i	$-0.419729\pi$
$257$	8.00000	0.499026	0.249513	+	0.968371i	$0.419729\pi$
$258$	−15.0000	−0.933859
$259$	0	0
$260$	0	0
$261$	54.0000	3.34252
$262$	8.00000	0.494242
$263$	5.00000	0.308313	0.154157	−	0.988046i	$-0.450734\pi$
$263$	5.00000	0.308313	0.154157	+	0.988046i	$0.450734\pi$
$264$	−6.00000	−0.369274
$265$	2.00000	0.122859
$266$	0	0
$267$	3.00000	0.183597
$268$	−9.00000	−0.549762
$269$	3.00000	0.182913	0.0914566	−	0.995809i	$-0.470848\pi$
$269$	3.00000	0.182913	0.0914566	+	0.995809i	$0.470848\pi$
$270$	−9.00000	−0.547723
$271$	−6.00000	−0.364474	−0.182237	−	0.983255i	$-0.558334\pi$
$271$	−6.00000	−0.364474	−0.182237	+	0.983255i	$0.558334\pi$
$272$	−4.00000	−0.242536
$273$	0	0
$274$	12.0000	0.724947
$275$	−2.00000	−0.120605
$276$	9.00000	0.541736
$277$	12.0000	0.721010	0.360505	−	0.932757i	$-0.382604\pi$
$277$	12.0000	0.721010	0.360505	+	0.932757i	$0.382604\pi$
$278$	−14.0000	−0.839664
$279$	−24.0000	−1.43684
$280$	0	0
$281$	2.00000	0.119310	0.0596550	−	0.998219i	$-0.481000\pi$
$281$	2.00000	0.119310	0.0596550	+	0.998219i	$0.481000\pi$
$282$	24.0000	1.42918
$283$	−4.00000	−0.237775	−0.118888	−	0.992908i	$-0.537933\pi$
$283$	−4.00000	−0.237775	−0.118888	+	0.992908i	$0.537933\pi$
$284$	2.00000	0.118678
$285$	18.0000	1.06623
$286$	0	0
$287$	0	0
$288$	6.00000	0.353553
$289$	−1.00000	−0.0588235
$290$	−9.00000	−0.528498
$291$	42.0000	2.46208
$292$	−4.00000	−0.234082
$293$	28.0000	1.63578	0.817889	−	0.575376i	$-0.195144\pi$
$293$	28.0000	1.63578	0.817889	+	0.575376i	$0.195144\pi$
$294$	0	0
$295$	−10.0000	−0.582223
$296$	−4.00000	−0.232495
$297$	−18.0000	−1.04447
$298$	3.00000	0.173785
$299$	0	0
$300$	3.00000	0.173205
$301$	0	0
$302$	−16.0000	−0.920697
$303$	9.00000	0.517036
$304$	−6.00000	−0.344124
$305$	−1.00000	−0.0572598
$306$	−24.0000	−1.37199
$307$	7.00000	0.399511	0.199756	−	0.979846i	$-0.435985\pi$
$307$	7.00000	0.399511	0.199756	+	0.979846i	$0.435985\pi$
$308$	0	0
$309$	3.00000	0.170664
$310$	4.00000	0.227185
$311$	−18.0000	−1.02069	−0.510343	−	0.859971i	$-0.670482\pi$
$311$	−18.0000	−1.02069	−0.510343	+	0.859971i	$0.670482\pi$
$312$	0	0
$313$	8.00000	0.452187	0.226093	−	0.974106i	$-0.427405\pi$
$313$	8.00000	0.452187	0.226093	+	0.974106i	$0.427405\pi$
$314$	10.0000	0.564333
$315$	0	0
$316$	10.0000	0.562544
$317$	−32.0000	−1.79730	−0.898650	−	0.438667i	$-0.855451\pi$
$317$	−32.0000	−1.79730	−0.898650	+	0.438667i	$0.855451\pi$
$318$	−6.00000	−0.336463
$319$	−18.0000	−1.00781
$320$	−1.00000	−0.0559017
$321$	9.00000	0.502331
$322$	0	0
$323$	24.0000	1.33540
$324$	9.00000	0.500000
$325$	0	0
$326$	−4.00000	−0.221540
$327$	−27.0000	−1.49310
$328$	−7.00000	−0.386510
$329$	0	0
$330$	6.00000	0.330289
$331$	−32.0000	−1.75888	−0.879440	−	0.476011i	$-0.842082\pi$
$331$	−32.0000	−1.75888	−0.879440	+	0.476011i	$0.842082\pi$
$332$	−7.00000	−0.384175
$333$	−24.0000	−1.31519
$334$	−21.0000	−1.14907
$335$	9.00000	0.491723
$336$	0	0
$337$	−26.0000	−1.41631	−0.708155	−	0.706057i	$-0.750472\pi$
$337$	−26.0000	−1.41631	−0.708155	+	0.706057i	$0.750472\pi$
$338$	−13.0000	−0.707107
$339$	6.00000	0.325875
$340$	4.00000	0.216930
$341$	8.00000	0.433224
$342$	−36.0000	−1.94666
$343$	0	0
$344$	−5.00000	−0.269582
$345$	−9.00000	−0.484544
$346$	8.00000	0.430083
$347$	19.0000	1.01997	0.509987	−	0.860182i	$-0.329650\pi$
$347$	19.0000	1.01997	0.509987	+	0.860182i	$0.329650\pi$
$348$	27.0000	1.44735
$349$	−35.0000	−1.87351	−0.936754	−	0.349990i	$-0.886185\pi$
$349$	−35.0000	−1.87351	−0.936754	+	0.349990i	$0.886185\pi$
$350$	0	0
$351$	0	0
$352$	−2.00000	−0.106600
$353$	−18.0000	−0.958043	−0.479022	−	0.877803i	$-0.659008\pi$
$353$	−18.0000	−0.958043	−0.479022	+	0.877803i	$0.659008\pi$
$354$	30.0000	1.59448
$355$	−2.00000	−0.106149
$356$	1.00000	0.0529999
$357$	0	0
$358$	12.0000	0.634220
$359$	−4.00000	−0.211112	−0.105556	−	0.994413i	$-0.533662\pi$
$359$	−4.00000	−0.211112	−0.105556	+	0.994413i	$0.533662\pi$
$360$	−6.00000	−0.316228
$361$	17.0000	0.894737
$362$	7.00000	0.367912
$363$	−21.0000	−1.10221
$364$	0	0
$365$	4.00000	0.209370
$366$	3.00000	0.156813
$367$	−11.0000	−0.574195	−0.287098	−	0.957901i	$-0.592690\pi$
$367$	−11.0000	−0.574195	−0.287098	+	0.957901i	$0.592690\pi$
$368$	3.00000	0.156386
$369$	−42.0000	−2.18643
$370$	4.00000	0.207950
$371$	0	0
$372$	−12.0000	−0.622171
$373$	−4.00000	−0.207112	−0.103556	−	0.994624i	$-0.533022\pi$
$373$	−4.00000	−0.207112	−0.103556	+	0.994624i	$0.533022\pi$
$374$	8.00000	0.413670
$375$	−3.00000	−0.154919
$376$	8.00000	0.412568
$377$	0	0
$378$	0	0
$379$	30.0000	1.54100	0.770498	−	0.637442i	$-0.220007\pi$
$379$	30.0000	1.54100	0.770498	+	0.637442i	$0.220007\pi$
$380$	6.00000	0.307794
$381$	48.0000	2.45911
$382$	−18.0000	−0.920960
$383$	15.0000	0.766464	0.383232	−	0.923652i	$-0.374811\pi$
$383$	15.0000	0.766464	0.383232	+	0.923652i	$0.374811\pi$
$384$	3.00000	0.153093
$385$	0	0
$386$	26.0000	1.32337
$387$	−30.0000	−1.52499
$388$	14.0000	0.710742
$389$	26.0000	1.31825	0.659126	−	0.752032i	$-0.270926\pi$
$389$	26.0000	1.31825	0.659126	+	0.752032i	$0.270926\pi$
$390$	0	0
$391$	−12.0000	−0.606866
$392$	0	0
$393$	24.0000	1.21064
$394$	2.00000	0.100759
$395$	−10.0000	−0.503155
$396$	−12.0000	−0.603023
$397$	22.0000	1.10415	0.552074	−	0.833795i	$-0.313837\pi$
$397$	22.0000	1.10415	0.552074	+	0.833795i	$0.313837\pi$
$398$	−4.00000	−0.200502
$399$	0	0
$400$	1.00000	0.0500000
$401$	31.0000	1.54807	0.774033	−	0.633145i	$-0.218236\pi$
$401$	31.0000	1.54807	0.774033	+	0.633145i	$0.218236\pi$
$402$	−27.0000	−1.34664
$403$	0	0
$404$	3.00000	0.149256
$405$	−9.00000	−0.447214
$406$	0	0
$407$	8.00000	0.396545
$408$	−12.0000	−0.594089
$409$	3.00000	0.148340	0.0741702	−	0.997246i	$-0.476369\pi$
$409$	3.00000	0.148340	0.0741702	+	0.997246i	$0.476369\pi$
$410$	7.00000	0.345705
$411$	36.0000	1.77575
$412$	1.00000	0.0492665
$413$	0	0
$414$	18.0000	0.884652
$415$	7.00000	0.343616
$416$	0	0
$417$	−42.0000	−2.05675
$418$	12.0000	0.586939
$419$	0	0		−	1.00000i	$-0.5\pi$
$419$	0	0			1.00000i	$0.5\pi$
$420$	0	0
$421$	−19.0000	−0.926003	−0.463002	−	0.886357i	$-0.653228\pi$
$421$	−19.0000	−0.926003	−0.463002	+	0.886357i	$0.653228\pi$
$422$	−26.0000	−1.26566
$423$	48.0000	2.33384
$424$	−2.00000	−0.0971286
$425$	−4.00000	−0.194029
$426$	6.00000	0.290701
$427$	0	0
$428$	3.00000	0.145010
$429$	0	0
$430$	5.00000	0.241121
$431$	−30.0000	−1.44505	−0.722525	−	0.691345i	$-0.757018\pi$
$431$	−30.0000	−1.44505	−0.722525	+	0.691345i	$0.757018\pi$
$432$	9.00000	0.433013
$433$	14.0000	0.672797	0.336399	−	0.941720i	$-0.390791\pi$
$433$	14.0000	0.672797	0.336399	+	0.941720i	$0.390791\pi$
$434$	0	0
$435$	−27.0000	−1.29455
$436$	−9.00000	−0.431022
$437$	−18.0000	−0.861057
$438$	−12.0000	−0.573382
$439$	−20.0000	−0.954548	−0.477274	−	0.878755i	$-0.658375\pi$
$439$	−20.0000	−0.954548	−0.477274	+	0.878755i	$0.658375\pi$
$440$	2.00000	0.0953463
$441$	0	0
$442$	0	0
$443$	31.0000	1.47285	0.736427	−	0.676517i	$-0.236511\pi$
$443$	31.0000	1.47285	0.736427	+	0.676517i	$0.236511\pi$
$444$	−12.0000	−0.569495
$445$	−1.00000	−0.0474045
$446$	28.0000	1.32584
$447$	9.00000	0.425685
$448$	0	0
$449$	−33.0000	−1.55737	−0.778683	−	0.627417i	$-0.784112\pi$
$449$	−33.0000	−1.55737	−0.778683	+	0.627417i	$0.784112\pi$
$450$	6.00000	0.282843
$451$	14.0000	0.659234
$452$	2.00000	0.0940721
$453$	−48.0000	−2.25524
$454$	−4.00000	−0.187729
$455$	0	0
$456$	−18.0000	−0.842927
$457$	−32.0000	−1.49690	−0.748448	−	0.663193i	$-0.769201\pi$
$457$	−32.0000	−1.49690	−0.748448	+	0.663193i	$0.769201\pi$
$458$	22.0000	1.02799
$459$	−36.0000	−1.68034
$460$	−3.00000	−0.139876
$461$	−14.0000	−0.652045	−0.326023	−	0.945362i	$-0.605709\pi$
$461$	−14.0000	−0.652045	−0.326023	+	0.945362i	$0.605709\pi$
$462$	0	0
$463$	−19.0000	−0.883005	−0.441502	−	0.897260i	$-0.645554\pi$
$463$	−19.0000	−0.883005	−0.441502	+	0.897260i	$0.645554\pi$
$464$	9.00000	0.417815
$465$	12.0000	0.556487
$466$	24.0000	1.11178
$467$	−13.0000	−0.601568	−0.300784	−	0.953692i	$-0.597248\pi$
$467$	−13.0000	−0.601568	−0.300784	+	0.953692i	$0.597248\pi$
$468$	0	0
$469$	0	0
$470$	−8.00000	−0.369012
$471$	30.0000	1.38233
$472$	10.0000	0.460287
$473$	10.0000	0.459800
$474$	30.0000	1.37795
$475$	−6.00000	−0.275299
$476$	0	0
$477$	−12.0000	−0.549442
$478$	16.0000	0.731823
$479$	24.0000	1.09659	0.548294	−	0.836286i	$-0.315277\pi$
$479$	24.0000	1.09659	0.548294	+	0.836286i	$0.315277\pi$
$480$	−3.00000	−0.136931
$481$	0	0
$482$	10.0000	0.455488
$483$	0	0
$484$	−7.00000	−0.318182
$485$	−14.0000	−0.635707
$486$	0	0
$487$	−16.0000	−0.725029	−0.362515	−	0.931978i	$-0.618082\pi$
$487$	−16.0000	−0.725029	−0.362515	+	0.931978i	$0.618082\pi$
$488$	1.00000	0.0452679
$489$	−12.0000	−0.542659
$490$	0	0
$491$	−12.0000	−0.541552	−0.270776	−	0.962642i	$-0.587280\pi$
$491$	−12.0000	−0.541552	−0.270776	+	0.962642i	$0.587280\pi$
$492$	−21.0000	−0.946753
$493$	−36.0000	−1.62136
$494$	0	0
$495$	12.0000	0.539360
$496$	−4.00000	−0.179605
$497$	0	0
$498$	−21.0000	−0.941033
$499$	−18.0000	−0.805791	−0.402895	−	0.915246i	$-0.631996\pi$
$499$	−18.0000	−0.805791	−0.402895	+	0.915246i	$0.631996\pi$
$500$	−1.00000	−0.0447214
$501$	−63.0000	−2.81463
$502$	0	0
$503$	−21.0000	−0.936344	−0.468172	−	0.883637i	$-0.655087\pi$
$503$	−21.0000	−0.936344	−0.468172	+	0.883637i	$0.655087\pi$
$504$	0	0
$505$	−3.00000	−0.133498
$506$	−6.00000	−0.266733
$507$	−39.0000	−1.73205
$508$	16.0000	0.709885
$509$	1.00000	0.0443242	0.0221621	−	0.999754i	$-0.492945\pi$
$509$	1.00000	0.0443242	0.0221621	+	0.999754i	$0.492945\pi$
$510$	12.0000	0.531369
$511$	0	0
$512$	1.00000	0.0441942
$513$	−54.0000	−2.38416
$514$	8.00000	0.352865
$515$	−1.00000	−0.0440653
$516$	−15.0000	−0.660338
$517$	−16.0000	−0.703679
$518$	0	0
$519$	24.0000	1.05348
$520$	0	0
$521$	38.0000	1.66481	0.832405	−	0.554168i	$-0.186963\pi$
$521$	38.0000	1.66481	0.832405	+	0.554168i	$0.186963\pi$
$522$	54.0000	2.36352
$523$	−20.0000	−0.874539	−0.437269	−	0.899331i	$-0.644054\pi$
$523$	−20.0000	−0.874539	−0.437269	+	0.899331i	$0.644054\pi$
$524$	8.00000	0.349482
$525$	0	0
$526$	5.00000	0.218010
$527$	16.0000	0.696971
$528$	−6.00000	−0.261116
$529$	−14.0000	−0.608696
$530$	2.00000	0.0868744
$531$	60.0000	2.60378
$532$	0	0
$533$	0	0
$534$	3.00000	0.129823
$535$	−3.00000	−0.129701
$536$	−9.00000	−0.388741
$537$	36.0000	1.55351
$538$	3.00000	0.129339
$539$	0	0
$540$	−9.00000	−0.387298
$541$	3.00000	0.128980	0.0644900	−	0.997918i	$-0.479458\pi$
$541$	3.00000	0.128980	0.0644900	+	0.997918i	$0.479458\pi$
$542$	−6.00000	−0.257722
$543$	21.0000	0.901196
$544$	−4.00000	−0.171499
$545$	9.00000	0.385518
$546$	0	0
$547$	−33.0000	−1.41098	−0.705489	−	0.708721i	$-0.749273\pi$
$547$	−33.0000	−1.41098	−0.705489	+	0.708721i	$0.749273\pi$
$548$	12.0000	0.512615
$549$	6.00000	0.256074
$550$	−2.00000	−0.0852803
$551$	−54.0000	−2.30048
$552$	9.00000	0.383065
$553$	0	0
$554$	12.0000	0.509831
$555$	12.0000	0.509372
$556$	−14.0000	−0.593732
$557$	−2.00000	−0.0847427	−0.0423714	−	0.999102i	$-0.513491\pi$
$557$	−2.00000	−0.0847427	−0.0423714	+	0.999102i	$0.513491\pi$
$558$	−24.0000	−1.01600
$559$	0	0
$560$	0	0
$561$	24.0000	1.01328
$562$	2.00000	0.0843649
$563$	17.0000	0.716465	0.358232	−	0.933632i	$-0.383380\pi$
$563$	17.0000	0.716465	0.358232	+	0.933632i	$0.383380\pi$
$564$	24.0000	1.01058
$565$	−2.00000	−0.0841406
$566$	−4.00000	−0.168133
$567$	0	0
$568$	2.00000	0.0839181
$569$	−18.0000	−0.754599	−0.377300	−	0.926091i	$-0.623147\pi$
$569$	−18.0000	−0.754599	−0.377300	+	0.926091i	$0.623147\pi$
$570$	18.0000	0.753937
$571$	−30.0000	−1.25546	−0.627730	−	0.778431i	$-0.716016\pi$
$571$	−30.0000	−1.25546	−0.627730	+	0.778431i	$0.716016\pi$
$572$	0	0
$573$	−54.0000	−2.25588
$574$	0	0
$575$	3.00000	0.125109
$576$	6.00000	0.250000
$577$	10.0000	0.416305	0.208153	−	0.978096i	$-0.433255\pi$
$577$	10.0000	0.416305	0.208153	+	0.978096i	$0.433255\pi$
$578$	−1.00000	−0.0415945
$579$	78.0000	3.24157
$580$	−9.00000	−0.373705
$581$	0	0
$582$	42.0000	1.74096
$583$	4.00000	0.165663
$584$	−4.00000	−0.165521
$585$	0	0
$586$	28.0000	1.15667
$587$	−28.0000	−1.15568	−0.577842	−	0.816149i	$-0.696105\pi$
$587$	−28.0000	−1.15568	−0.577842	+	0.816149i	$0.696105\pi$
$588$	0	0
$589$	24.0000	0.988903
$590$	−10.0000	−0.411693
$591$	6.00000	0.246807
$592$	−4.00000	−0.164399
$593$	−6.00000	−0.246390	−0.123195	−	0.992382i	$-0.539314\pi$
$593$	−6.00000	−0.246390	−0.123195	+	0.992382i	$0.539314\pi$
$594$	−18.0000	−0.738549
$595$	0	0
$596$	3.00000	0.122885
$597$	−12.0000	−0.491127
$598$	0	0
$599$	12.0000	0.490307	0.245153	−	0.969484i	$-0.421162\pi$
$599$	12.0000	0.490307	0.245153	+	0.969484i	$0.421162\pi$
$600$	3.00000	0.122474
$601$	42.0000	1.71322	0.856608	−	0.515968i	$-0.172568\pi$
$601$	42.0000	1.71322	0.856608	+	0.515968i	$0.172568\pi$
$602$	0	0
$603$	−54.0000	−2.19905
$604$	−16.0000	−0.651031
$605$	7.00000	0.284590
$606$	9.00000	0.365600
$607$	1.00000	0.0405887	0.0202944	−	0.999794i	$-0.493540\pi$
$607$	1.00000	0.0405887	0.0202944	+	0.999794i	$0.493540\pi$
$608$	−6.00000	−0.243332
$609$	0	0
$610$	−1.00000	−0.0404888
$611$	0	0
$612$	−24.0000	−0.970143
$613$	12.0000	0.484675	0.242338	−	0.970192i	$-0.422086\pi$
$613$	12.0000	0.484675	0.242338	+	0.970192i	$0.422086\pi$
$614$	7.00000	0.282497
$615$	21.0000	0.846802
$616$	0	0
$617$	44.0000	1.77137	0.885687	−	0.464283i	$-0.153688\pi$
$617$	44.0000	1.77137	0.885687	+	0.464283i	$0.153688\pi$
$618$	3.00000	0.120678
$619$	−46.0000	−1.84890	−0.924448	−	0.381308i	$-0.875474\pi$
$619$	−46.0000	−1.84890	−0.924448	+	0.381308i	$0.875474\pi$
$620$	4.00000	0.160644
$621$	27.0000	1.08347
$622$	−18.0000	−0.721734
$623$	0	0
$624$	0	0
$625$	1.00000	0.0400000
$626$	8.00000	0.319744
$627$	36.0000	1.43770
$628$	10.0000	0.399043
$629$	16.0000	0.637962
$630$	0	0
$631$	2.00000	0.0796187	0.0398094	−	0.999207i	$-0.487325\pi$
$631$	2.00000	0.0796187	0.0398094	+	0.999207i	$0.487325\pi$
$632$	10.0000	0.397779
$633$	−78.0000	−3.10022
$634$	−32.0000	−1.27088
$635$	−16.0000	−0.634941
$636$	−6.00000	−0.237915
$637$	0	0
$638$	−18.0000	−0.712627
$639$	12.0000	0.474713
$640$	−1.00000	−0.0395285
$641$	5.00000	0.197488	0.0987441	−	0.995113i	$-0.468517\pi$
$641$	5.00000	0.197488	0.0987441	+	0.995113i	$0.468517\pi$
$642$	9.00000	0.355202
$643$	28.0000	1.10421	0.552106	−	0.833774i	$-0.313824\pi$
$643$	28.0000	1.10421	0.552106	+	0.833774i	$0.313824\pi$
$644$	0	0
$645$	15.0000	0.590624
$646$	24.0000	0.944267
$647$	−11.0000	−0.432455	−0.216227	−	0.976343i	$-0.569375\pi$
$647$	−11.0000	−0.432455	−0.216227	+	0.976343i	$0.569375\pi$
$648$	9.00000	0.353553
$649$	−20.0000	−0.785069
$650$	0	0
$651$	0	0
$652$	−4.00000	−0.156652
$653$	−4.00000	−0.156532	−0.0782660	−	0.996933i	$-0.524938\pi$
$653$	−4.00000	−0.156532	−0.0782660	+	0.996933i	$0.524938\pi$
$654$	−27.0000	−1.05578
$655$	−8.00000	−0.312586
$656$	−7.00000	−0.273304
$657$	−24.0000	−0.936329
$658$	0	0
$659$	−26.0000	−1.01282	−0.506408	−	0.862294i	$-0.669027\pi$
$659$	−26.0000	−1.01282	−0.506408	+	0.862294i	$0.669027\pi$
$660$	6.00000	0.233550
$661$	−11.0000	−0.427850	−0.213925	−	0.976850i	$-0.568625\pi$
$661$	−11.0000	−0.427850	−0.213925	+	0.976850i	$0.568625\pi$
$662$	−32.0000	−1.24372
$663$	0	0
$664$	−7.00000	−0.271653
$665$	0	0
$666$	−24.0000	−0.929981
$667$	27.0000	1.04544
$668$	−21.0000	−0.812514
$669$	84.0000	3.24763
$670$	9.00000	0.347700
$671$	−2.00000	−0.0772091
$672$	0	0
$673$	16.0000	0.616755	0.308377	−	0.951264i	$-0.400214\pi$
$673$	16.0000	0.616755	0.308377	+	0.951264i	$0.400214\pi$
$674$	−26.0000	−1.00148
$675$	9.00000	0.346410
$676$	−13.0000	−0.500000
$677$	−48.0000	−1.84479	−0.922395	−	0.386248i	$-0.873771\pi$
$677$	−48.0000	−1.84479	−0.922395	+	0.386248i	$0.873771\pi$
$678$	6.00000	0.230429
$679$	0	0
$680$	4.00000	0.153393
$681$	−12.0000	−0.459841
$682$	8.00000	0.306336
$683$	−37.0000	−1.41577	−0.707883	−	0.706330i	$-0.750350\pi$
$683$	−37.0000	−1.41577	−0.707883	+	0.706330i	$0.750350\pi$
$684$	−36.0000	−1.37649
$685$	−12.0000	−0.458496
$686$	0	0
$687$	66.0000	2.51806
$688$	−5.00000	−0.190623
$689$	0	0
$690$	−9.00000	−0.342624
$691$	22.0000	0.836919	0.418460	−	0.908235i	$-0.362570\pi$
$691$	22.0000	0.836919	0.418460	+	0.908235i	$0.362570\pi$
$692$	8.00000	0.304114
$693$	0	0
$694$	19.0000	0.721230
$695$	14.0000	0.531050
$696$	27.0000	1.02343
$697$	28.0000	1.06058
$698$	−35.0000	−1.32477
$699$	72.0000	2.72329
$700$	0	0
$701$	−47.0000	−1.77517	−0.887583	−	0.460648i	$-0.847617\pi$
$701$	−47.0000	−1.77517	−0.887583	+	0.460648i	$0.847617\pi$
$702$	0	0
$703$	24.0000	0.905177
$704$	−2.00000	−0.0753778
$705$	−24.0000	−0.903892
$706$	−18.0000	−0.677439
$707$	0	0
$708$	30.0000	1.12747
$709$	−11.0000	−0.413114	−0.206557	−	0.978435i	$-0.566226\pi$
$709$	−11.0000	−0.413114	−0.206557	+	0.978435i	$0.566226\pi$
$710$	−2.00000	−0.0750587
$711$	60.0000	2.25018
$712$	1.00000	0.0374766
$713$	−12.0000	−0.449404
$714$	0	0
$715$	0	0
$716$	12.0000	0.448461
$717$	48.0000	1.79259
$718$	−4.00000	−0.149279
$719$	−6.00000	−0.223762	−0.111881	−	0.993722i	$-0.535688\pi$
$719$	−6.00000	−0.223762	−0.111881	+	0.993722i	$0.535688\pi$
$720$	−6.00000	−0.223607
$721$	0	0
$722$	17.0000	0.632674
$723$	30.0000	1.11571
$724$	7.00000	0.260153
$725$	9.00000	0.334252
$726$	−21.0000	−0.779383
$727$	−21.0000	−0.778847	−0.389423	−	0.921059i	$-0.627326\pi$
$727$	−21.0000	−0.778847	−0.389423	+	0.921059i	$0.627326\pi$
$728$	0	0
$729$	−27.0000	−1.00000
$730$	4.00000	0.148047
$731$	20.0000	0.739727
$732$	3.00000	0.110883
$733$	22.0000	0.812589	0.406294	−	0.913742i	$-0.366821\pi$
$733$	22.0000	0.812589	0.406294	+	0.913742i	$0.366821\pi$
$734$	−11.0000	−0.406017
$735$	0	0
$736$	3.00000	0.110581
$737$	18.0000	0.663039
$738$	−42.0000	−1.54604
$739$	−2.00000	−0.0735712	−0.0367856	−	0.999323i	$-0.511712\pi$
$739$	−2.00000	−0.0735712	−0.0367856	+	0.999323i	$0.511712\pi$
$740$	4.00000	0.147043
$741$	0	0
$742$	0	0
$743$	9.00000	0.330178	0.165089	−	0.986279i	$-0.447209\pi$
$743$	9.00000	0.330178	0.165089	+	0.986279i	$0.447209\pi$
$744$	−12.0000	−0.439941
$745$	−3.00000	−0.109911
$746$	−4.00000	−0.146450
$747$	−42.0000	−1.53670
$748$	8.00000	0.292509
$749$	0	0
$750$	−3.00000	−0.109545
$751$	−4.00000	−0.145962	−0.0729810	−	0.997333i	$-0.523251\pi$
$751$	−4.00000	−0.145962	−0.0729810	+	0.997333i	$0.523251\pi$
$752$	8.00000	0.291730
$753$	0	0
$754$	0	0
$755$	16.0000	0.582300
$756$	0	0
$757$	16.0000	0.581530	0.290765	−	0.956795i	$-0.406090\pi$
$757$	16.0000	0.581530	0.290765	+	0.956795i	$0.406090\pi$
$758$	30.0000	1.08965
$759$	−18.0000	−0.653359
$760$	6.00000	0.217643
$761$	−6.00000	−0.217500	−0.108750	−	0.994069i	$-0.534685\pi$
$761$	−6.00000	−0.217500	−0.108750	+	0.994069i	$0.534685\pi$
$762$	48.0000	1.73886
$763$	0	0
$764$	−18.0000	−0.651217
$765$	24.0000	0.867722
$766$	15.0000	0.541972
$767$	0	0
$768$	3.00000	0.108253
$769$	−14.0000	−0.504853	−0.252426	−	0.967616i	$-0.581229\pi$
$769$	−14.0000	−0.504853	−0.252426	+	0.967616i	$0.581229\pi$
$770$	0	0
$771$	24.0000	0.864339
$772$	26.0000	0.935760
$773$	24.0000	0.863220	0.431610	−	0.902060i	$-0.357946\pi$
$773$	24.0000	0.863220	0.431610	+	0.902060i	$0.357946\pi$
$774$	−30.0000	−1.07833
$775$	−4.00000	−0.143684
$776$	14.0000	0.502571
$777$	0	0
$778$	26.0000	0.932145
$779$	42.0000	1.50481
$780$	0	0
$781$	−4.00000	−0.143131
$782$	−12.0000	−0.429119
$783$	81.0000	2.89470
$784$	0	0
$785$	−10.0000	−0.356915
$786$	24.0000	0.856052
$787$	31.0000	1.10503	0.552515	−	0.833503i	$-0.313668\pi$
$787$	31.0000	1.10503	0.552515	+	0.833503i	$0.313668\pi$
$788$	2.00000	0.0712470
$789$	15.0000	0.534014
$790$	−10.0000	−0.355784
$791$	0	0
$792$	−12.0000	−0.426401
$793$	0	0
$794$	22.0000	0.780751
$795$	6.00000	0.212798
$796$	−4.00000	−0.141776
$797$	0	0		−	1.00000i	$-0.5\pi$
$797$	0	0			1.00000i	$0.5\pi$
$798$	0	0
$799$	−32.0000	−1.13208
$800$	1.00000	0.0353553
$801$	6.00000	0.212000
$802$	31.0000	1.09465
$803$	8.00000	0.282314
$804$	−27.0000	−0.952217
$805$	0	0
$806$	0	0
$807$	9.00000	0.316815
$808$	3.00000	0.105540
$809$	−51.0000	−1.79306	−0.896532	−	0.442978i	$-0.853922\pi$
$809$	−51.0000	−1.79306	−0.896532	+	0.442978i	$0.853922\pi$
$810$	−9.00000	−0.316228
$811$	−28.0000	−0.983213	−0.491606	−	0.870817i	$-0.663590\pi$
$811$	−28.0000	−0.983213	−0.491606	+	0.870817i	$0.663590\pi$
$812$	0	0
$813$	−18.0000	−0.631288
$814$	8.00000	0.280400
$815$	4.00000	0.140114
$816$	−12.0000	−0.420084
$817$	30.0000	1.04957
$818$	3.00000	0.104893
$819$	0	0
$820$	7.00000	0.244451
$821$	−18.0000	−0.628204	−0.314102	−	0.949389i	$-0.601703\pi$
$821$	−18.0000	−0.628204	−0.314102	+	0.949389i	$0.601703\pi$
$822$	36.0000	1.25564
$823$	19.0000	0.662298	0.331149	−	0.943578i	$-0.392564\pi$
$823$	19.0000	0.662298	0.331149	+	0.943578i	$0.392564\pi$
$824$	1.00000	0.0348367
$825$	−6.00000	−0.208893
$826$	0	0
$827$	−19.0000	−0.660695	−0.330347	−	0.943859i	$-0.607166\pi$
$827$	−19.0000	−0.660695	−0.330347	+	0.943859i	$0.607166\pi$
$828$	18.0000	0.625543
$829$	−46.0000	−1.59765	−0.798823	−	0.601566i	$-0.794544\pi$
$829$	−46.0000	−1.59765	−0.798823	+	0.601566i	$0.794544\pi$
$830$	7.00000	0.242974
$831$	36.0000	1.24883
$832$	0	0
$833$	0	0
$834$	−42.0000	−1.45434
$835$	21.0000	0.726735
$836$	12.0000	0.415029
$837$	−36.0000	−1.24434
$838$	0	0
$839$	14.0000	0.483334	0.241667	−	0.970359i	$-0.422306\pi$
$839$	14.0000	0.483334	0.241667	+	0.970359i	$0.422306\pi$
$840$	0	0
$841$	52.0000	1.79310
$842$	−19.0000	−0.654783
$843$	6.00000	0.206651
$844$	−26.0000	−0.894957
$845$	13.0000	0.447214
$846$	48.0000	1.65027
$847$	0	0
$848$	−2.00000	−0.0686803
$849$	−12.0000	−0.411839
$850$	−4.00000	−0.137199
$851$	−12.0000	−0.411355
$852$	6.00000	0.205557
$853$	14.0000	0.479351	0.239675	−	0.970853i	$-0.422959\pi$
$853$	14.0000	0.479351	0.239675	+	0.970853i	$0.422959\pi$
$854$	0	0
$855$	36.0000	1.23117
$856$	3.00000	0.102538
$857$	−18.0000	−0.614868	−0.307434	−	0.951569i	$-0.599470\pi$
$857$	−18.0000	−0.614868	−0.307434	+	0.951569i	$0.599470\pi$
$858$	0	0
$859$	−48.0000	−1.63774	−0.818869	−	0.573980i	$-0.805399\pi$
$859$	−48.0000	−1.63774	−0.818869	+	0.573980i	$0.805399\pi$
$860$	5.00000	0.170499
$861$	0	0
$862$	−30.0000	−1.02180
$863$	−11.0000	−0.374444	−0.187222	−	0.982318i	$-0.559948\pi$
$863$	−11.0000	−0.374444	−0.187222	+	0.982318i	$0.559948\pi$
$864$	9.00000	0.306186
$865$	−8.00000	−0.272008
$866$	14.0000	0.475739
$867$	−3.00000	−0.101885
$868$	0	0
$869$	−20.0000	−0.678454
$870$	−27.0000	−0.915386
$871$	0	0
$872$	−9.00000	−0.304778
$873$	84.0000	2.84297
$874$	−18.0000	−0.608859
$875$	0	0
$876$	−12.0000	−0.405442
$877$	38.0000	1.28317	0.641584	−	0.767052i	$-0.278277\pi$
$877$	38.0000	1.28317	0.641584	+	0.767052i	$0.278277\pi$
$878$	−20.0000	−0.674967
$879$	84.0000	2.83325
$880$	2.00000	0.0674200
$881$	−7.00000	−0.235836	−0.117918	−	0.993023i	$-0.537622\pi$
$881$	−7.00000	−0.235836	−0.117918	+	0.993023i	$0.537622\pi$
$882$	0	0
$883$	−12.0000	−0.403832	−0.201916	−	0.979403i	$-0.564717\pi$
$883$	−12.0000	−0.403832	−0.201916	+	0.979403i	$0.564717\pi$
$884$	0	0
$885$	−30.0000	−1.00844
$886$	31.0000	1.04147
$887$	29.0000	0.973725	0.486862	−	0.873479i	$-0.338141\pi$
$887$	29.0000	0.973725	0.486862	+	0.873479i	$0.338141\pi$
$888$	−12.0000	−0.402694
$889$	0	0
$890$	−1.00000	−0.0335201
$891$	−18.0000	−0.603023
$892$	28.0000	0.937509
$893$	−48.0000	−1.60626
$894$	9.00000	0.301005
$895$	−12.0000	−0.401116
$896$	0	0
$897$	0	0
$898$	−33.0000	−1.10122
$899$	−36.0000	−1.20067
$900$	6.00000	0.200000
$901$	8.00000	0.266519
$902$	14.0000	0.466149
$903$	0	0
$904$	2.00000	0.0665190
$905$	−7.00000	−0.232688
$906$	−48.0000	−1.59469
$907$	5.00000	0.166022	0.0830111	−	0.996549i	$-0.473546\pi$
$907$	5.00000	0.166022	0.0830111	+	0.996549i	$0.473546\pi$
$908$	−4.00000	−0.132745
$909$	18.0000	0.597022
$910$	0	0
$911$	30.0000	0.993944	0.496972	−	0.867766i	$-0.334445\pi$
$911$	30.0000	0.993944	0.496972	+	0.867766i	$0.334445\pi$
$912$	−18.0000	−0.596040
$913$	14.0000	0.463332
$914$	−32.0000	−1.05847
$915$	−3.00000	−0.0991769
$916$	22.0000	0.726900
$917$	0	0
$918$	−36.0000	−1.18818
$919$	38.0000	1.25350	0.626752	−	0.779219i	$-0.284384\pi$
$919$	38.0000	1.25350	0.626752	+	0.779219i	$0.284384\pi$
$920$	−3.00000	−0.0989071
$921$	21.0000	0.691974
$922$	−14.0000	−0.461065
$923$	0	0
$924$	0	0
$925$	−4.00000	−0.131519
$926$	−19.0000	−0.624379
$927$	6.00000	0.197066
$928$	9.00000	0.295439
$929$	43.0000	1.41078	0.705392	−	0.708817i	$-0.250771\pi$
$929$	43.0000	1.41078	0.705392	+	0.708817i	$0.250771\pi$
$930$	12.0000	0.393496
$931$	0	0
$932$	24.0000	0.786146
$933$	−54.0000	−1.76788
$934$	−13.0000	−0.425373
$935$	−8.00000	−0.261628
$936$	0	0
$937$	−28.0000	−0.914720	−0.457360	−	0.889282i	$-0.651205\pi$
$937$	−28.0000	−0.914720	−0.457360	+	0.889282i	$0.651205\pi$
$938$	0	0
$939$	24.0000	0.783210
$940$	−8.00000	−0.260931
$941$	−46.0000	−1.49956	−0.749779	−	0.661689i	$-0.769840\pi$
$941$	−46.0000	−1.49956	−0.749779	+	0.661689i	$0.769840\pi$
$942$	30.0000	0.977453
$943$	−21.0000	−0.683854
$944$	10.0000	0.325472
$945$	0	0
$946$	10.0000	0.325128
$947$	−25.0000	−0.812391	−0.406195	−	0.913786i	$-0.633145\pi$
$947$	−25.0000	−0.812391	−0.406195	+	0.913786i	$0.633145\pi$
$948$	30.0000	0.974355
$949$	0	0
$950$	−6.00000	−0.194666
$951$	−96.0000	−3.11301
$952$	0	0
$953$	−12.0000	−0.388718	−0.194359	−	0.980930i	$-0.562263\pi$
$953$	−12.0000	−0.388718	−0.194359	+	0.980930i	$0.562263\pi$
$954$	−12.0000	−0.388514
$955$	18.0000	0.582466
$956$	16.0000	0.517477
$957$	−54.0000	−1.74557
$958$	24.0000	0.775405
$959$	0	0
$960$	−3.00000	−0.0968246
$961$	−15.0000	−0.483871
$962$	0	0
$963$	18.0000	0.580042
$964$	10.0000	0.322078
$965$	−26.0000	−0.836970
$966$	0	0
$967$	37.0000	1.18984	0.594920	−	0.803785i	$-0.297184\pi$
$967$	37.0000	1.18984	0.594920	+	0.803785i	$0.297184\pi$
$968$	−7.00000	−0.224989
$969$	72.0000	2.31297
$970$	−14.0000	−0.449513
$971$	−48.0000	−1.54039	−0.770197	−	0.637806i	$-0.779842\pi$
$971$	−48.0000	−1.54039	−0.770197	+	0.637806i	$0.779842\pi$
$972$	0	0
$973$	0	0
$974$	−16.0000	−0.512673
$975$	0	0
$976$	1.00000	0.0320092
$977$	−30.0000	−0.959785	−0.479893	−	0.877327i	$-0.659324\pi$
$977$	−30.0000	−0.959785	−0.479893	+	0.877327i	$0.659324\pi$
$978$	−12.0000	−0.383718
$979$	−2.00000	−0.0639203
$980$	0	0
$981$	−54.0000	−1.72409
$982$	−12.0000	−0.382935
$983$	17.0000	0.542216	0.271108	−	0.962549i	$-0.412610\pi$
$983$	17.0000	0.542216	0.271108	+	0.962549i	$0.412610\pi$
$984$	−21.0000	−0.669456
$985$	−2.00000	−0.0637253
$986$	−36.0000	−1.14647
$987$	0	0
$988$	0	0
$989$	−15.0000	−0.476972
$990$	12.0000	0.381385
$991$	40.0000	1.27064	0.635321	−	0.772248i	$-0.280868\pi$
$991$	40.0000	1.27064	0.635321	+	0.772248i	$0.280868\pi$
$992$	−4.00000	−0.127000
$993$	−96.0000	−3.04647
$994$	0	0
$995$	4.00000	0.126809
$996$	−21.0000	−0.665410
$997$	−46.0000	−1.45683	−0.728417	−	0.685134i	$-0.759744\pi$
$997$	−46.0000	−1.45683	−0.728417	+	0.685134i	$0.759744\pi$
$998$	−18.0000	−0.569780
$999$	−36.0000	−1.13899

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	490.2.a.k.1.1		1
3.2	odd	2		4410.2.a.r.1.1		1
4.3	odd	2		3920.2.a.b.1.1		1
5.2	odd	4		2450.2.c.s.99.2		2
5.3	odd	4		2450.2.c.s.99.1		2
5.4	even	2		2450.2.a.b.1.1		1
7.2	even	3		70.2.e.a.11.1	✓	2
7.3	odd	6		490.2.e.f.471.1		2
7.4	even	3		70.2.e.a.51.1	yes	2
7.5	odd	6		490.2.e.f.361.1		2
7.6	odd	2		490.2.a.e.1.1		1
21.2	odd	6		630.2.k.f.361.1		2
21.11	odd	6		630.2.k.f.541.1		2
21.20	even	2		4410.2.a.h.1.1		1
28.11	odd	6		560.2.q.i.401.1		2
28.23	odd	6		560.2.q.i.81.1		2
28.27	even	2		3920.2.a.bk.1.1		1
35.2	odd	12		350.2.j.f.249.2		4
35.4	even	6		350.2.e.l.51.1		2
35.9	even	6		350.2.e.l.151.1		2
35.13	even	4		2450.2.c.a.99.1		2
35.18	odd	12		350.2.j.f.149.2		4
35.23	odd	12		350.2.j.f.249.1		4
35.27	even	4		2450.2.c.a.99.2		2
35.32	odd	12		350.2.j.f.149.1		4
35.34	odd	2		2450.2.a.q.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
70.2.e.a.11.1	✓	2	7.2	even	3
70.2.e.a.51.1	yes	2	7.4	even	3
350.2.e.l.51.1		2	35.4	even	6
350.2.e.l.151.1		2	35.9	even	6
350.2.j.f.149.1		4	35.32	odd	12
350.2.j.f.149.2		4	35.18	odd	12
350.2.j.f.249.1		4	35.23	odd	12
350.2.j.f.249.2		4	35.2	odd	12
490.2.a.e.1.1		1	7.6	odd	2
490.2.a.k.1.1		1	1.1	even	1	trivial
490.2.e.f.361.1		2	7.5	odd	6
490.2.e.f.471.1		2	7.3	odd	6
560.2.q.i.81.1		2	28.23	odd	6
560.2.q.i.401.1		2	28.11	odd	6
630.2.k.f.361.1		2	21.2	odd	6
630.2.k.f.541.1		2	21.11	odd	6
2450.2.a.b.1.1		1	5.4	even	2
2450.2.a.q.1.1		1	35.34	odd	2
2450.2.c.a.99.1		2	35.13	even	4
2450.2.c.a.99.2		2	35.27	even	4
2450.2.c.s.99.1		2	5.3	odd	4
2450.2.c.s.99.2		2	5.2	odd	4
3920.2.a.b.1.1		1	4.3	odd	2
3920.2.a.bk.1.1		1	28.27	even	2
4410.2.a.h.1.1		1	21.20	even	2
4410.2.a.r.1.1		1	3.2	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:	\( N \)	\(=\)	\( 490 = 2 \cdot 5 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	490.a (trivial)

Introduction

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Modular forms

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Fields

Representations

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