LMFDB - Embedded newform 490.2.a.a.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)

$f(q)$

$=$

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

\(q-1.00000 q^{2} -2.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} +2.00000 q^{6} -1.00000 q^{8} +1.00000 q^{9} -1.00000 q^{10} +3.00000 q^{11} -2.00000 q^{12} -1.00000 q^{13} -2.00000 q^{15} +1.00000 q^{16} -6.00000 q^{17} -1.00000 q^{18} -1.00000 q^{19} +1.00000 q^{20} -3.00000 q^{22} +9.00000 q^{23} +2.00000 q^{24} +1.00000 q^{25} +1.00000 q^{26} +4.00000 q^{27} +6.00000 q^{29} +2.00000 q^{30} +8.00000 q^{31} -1.00000 q^{32} -6.00000 q^{33} +6.00000 q^{34} +1.00000 q^{36} -7.00000 q^{37} +1.00000 q^{38} +2.00000 q^{39} -1.00000 q^{40} +3.00000 q^{41} +2.00000 q^{43} +3.00000 q^{44} +1.00000 q^{45} -9.00000 q^{46} +9.00000 q^{47} -2.00000 q^{48} -1.00000 q^{50} +12.0000 q^{51} -1.00000 q^{52} +9.00000 q^{53} -4.00000 q^{54} +3.00000 q^{55} +2.00000 q^{57} -6.00000 q^{58} -2.00000 q^{60} +8.00000 q^{61} -8.00000 q^{62} +1.00000 q^{64} -1.00000 q^{65} +6.00000 q^{66} +8.00000 q^{67} -6.00000 q^{68} -18.0000 q^{69} -1.00000 q^{72} -4.00000 q^{73} +7.00000 q^{74} -2.00000 q^{75} -1.00000 q^{76} -2.00000 q^{78} -10.0000 q^{79} +1.00000 q^{80} -11.0000 q^{81} -3.00000 q^{82} -6.00000 q^{85} -2.00000 q^{86} -12.0000 q^{87} -3.00000 q^{88} +6.00000 q^{89} -1.00000 q^{90} +9.00000 q^{92} -16.0000 q^{93} -9.00000 q^{94} -1.00000 q^{95} +2.00000 q^{96} -10.0000 q^{97} +3.00000 q^{99} +O(q^{100})\)

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
$11$	3.00000	0.904534	0.452267	−	0.891883i	$-0.350615\pi$
$11$	3.00000	0.904534	0.452267	+	0.891883i	$0.350615\pi$
$12$	−2.00000	−0.577350
$13$	−1.00000	−0.277350	−0.138675	−	0.990338i	$-0.544284\pi$
$13$	−1.00000	−0.277350	−0.138675	+	0.990338i	$0.544284\pi$
$14$	0	0
$15$	−2.00000	−0.516398
$16$	1.00000	0.250000
$17$	−6.00000	−1.45521	−0.727607	−	0.685994i	$-0.759367\pi$
$17$	−6.00000	−1.45521	−0.727607	+	0.685994i	$0.759367\pi$
$18$	−1.00000	−0.235702
$19$	−1.00000	−0.229416	−0.114708	−	0.993399i	$-0.536593\pi$
$19$	−1.00000	−0.229416	−0.114708	+	0.993399i	$0.536593\pi$
$20$	1.00000	0.223607
$21$	0	0
$22$	−3.00000	−0.639602
$23$	9.00000	1.87663	0.938315	−	0.345782i	$-0.112386\pi$
$23$	9.00000	1.87663	0.938315	+	0.345782i	$0.112386\pi$
$24$	2.00000	0.408248
$25$	1.00000	0.200000
$26$	1.00000	0.196116
$27$	4.00000	0.769800
$28$	0	0
$29$	6.00000	1.11417	0.557086	−	0.830455i	$-0.311919\pi$
$29$	6.00000	1.11417	0.557086	+	0.830455i	$0.311919\pi$
$30$	2.00000	0.365148
$31$	8.00000	1.43684	0.718421	−	0.695608i	$-0.244865\pi$
$31$	8.00000	1.43684	0.718421	+	0.695608i	$0.244865\pi$
$32$	−1.00000	−0.176777
$33$	−6.00000	−1.04447
$34$	6.00000	1.02899
$35$	0	0
$36$	1.00000	0.166667
$37$	−7.00000	−1.15079	−0.575396	−	0.817875i	$-0.695152\pi$
$37$	−7.00000	−1.15079	−0.575396	+	0.817875i	$0.695152\pi$
$38$	1.00000	0.162221
$39$	2.00000	0.320256
$40$	−1.00000	−0.158114
$41$	3.00000	0.468521	0.234261	−	0.972174i	$-0.424733\pi$
$41$	3.00000	0.468521	0.234261	+	0.972174i	$0.424733\pi$
$42$	0	0
$43$	2.00000	0.304997	0.152499	−	0.988304i	$-0.451268\pi$
$43$	2.00000	0.304997	0.152499	+	0.988304i	$0.451268\pi$
$44$	3.00000	0.452267
$45$	1.00000	0.149071
$46$	−9.00000	−1.32698
$47$	9.00000	1.31278	0.656392	−	0.754420i	$-0.272082\pi$
$47$	9.00000	1.31278	0.656392	+	0.754420i	$0.272082\pi$
$48$	−2.00000	−0.288675
$49$	0	0
$50$	−1.00000	−0.141421
$51$	12.0000	1.68034
$52$	−1.00000	−0.138675
$53$	9.00000	1.23625	0.618123	−	0.786082i	$-0.287894\pi$
$53$	9.00000	1.23625	0.618123	+	0.786082i	$0.287894\pi$
$54$	−4.00000	−0.544331
$55$	3.00000	0.404520
$56$	0	0
$57$	2.00000	0.264906
$58$	−6.00000	−0.787839
$59$	0	0		−	1.00000i	$-0.5\pi$
$59$	0	0			1.00000i	$0.5\pi$
$60$	−2.00000	−0.258199
$61$	8.00000	1.02430	0.512148	−	0.858898i	$-0.328850\pi$
$61$	8.00000	1.02430	0.512148	+	0.858898i	$0.328850\pi$
$62$	−8.00000	−1.01600
$63$	0	0
$64$	1.00000	0.125000
$65$	−1.00000	−0.124035
$66$	6.00000	0.738549
$67$	8.00000	0.977356	0.488678	−	0.872464i	$-0.337479\pi$
$67$	8.00000	0.977356	0.488678	+	0.872464i	$0.337479\pi$
$68$	−6.00000	−0.727607
$69$	−18.0000	−2.16695
$70$	0	0
$71$	0	0		−	1.00000i	$-0.5\pi$
$71$	0	0			1.00000i	$0.5\pi$
$72$	−1.00000	−0.117851
$73$	−4.00000	−0.468165	−0.234082	−	0.972217i	$-0.575209\pi$
$73$	−4.00000	−0.468165	−0.234082	+	0.972217i	$0.575209\pi$
$74$	7.00000	0.813733
$75$	−2.00000	−0.230940
$76$	−1.00000	−0.114708
$77$	0	0
$78$	−2.00000	−0.226455
$79$	−10.0000	−1.12509	−0.562544	−	0.826767i	$-0.690177\pi$
$79$	−10.0000	−1.12509	−0.562544	+	0.826767i	$0.690177\pi$
$80$	1.00000	0.111803
$81$	−11.0000	−1.22222
$82$	−3.00000	−0.331295
$83$	0	0		−	1.00000i	$-0.5\pi$
$83$	0	0			1.00000i	$0.5\pi$
$84$	0	0
$85$	−6.00000	−0.650791
$86$	−2.00000	−0.215666
$87$	−12.0000	−1.28654
$88$	−3.00000	−0.319801
$89$	6.00000	0.635999	0.317999	−	0.948091i	$-0.396989\pi$
$89$	6.00000	0.635999	0.317999	+	0.948091i	$0.396989\pi$
$90$	−1.00000	−0.105409
$91$	0	0
$92$	9.00000	0.938315
$93$	−16.0000	−1.65912
$94$	−9.00000	−0.928279
$95$	−1.00000	−0.102598
$96$	2.00000	0.204124
$97$	−10.0000	−1.01535	−0.507673	−	0.861550i	$-0.669494\pi$
$97$	−10.0000	−1.01535	−0.507673	+	0.861550i	$0.669494\pi$
$98$	0	0
$99$	3.00000	0.301511
$100$	1.00000	0.100000
$101$	12.0000	1.19404	0.597022	−	0.802225i	$-0.296350\pi$
$101$	12.0000	1.19404	0.597022	+	0.802225i	$0.296350\pi$
$102$	−12.0000	−1.18818
$103$	−4.00000	−0.394132	−0.197066	−	0.980390i	$-0.563141\pi$
$103$	−4.00000	−0.394132	−0.197066	+	0.980390i	$0.563141\pi$
$104$	1.00000	0.0980581
$105$	0	0
$106$	−9.00000	−0.874157
$107$	−12.0000	−1.16008	−0.580042	−	0.814587i	$-0.696964\pi$
$107$	−12.0000	−1.16008	−0.580042	+	0.814587i	$0.696964\pi$
$108$	4.00000	0.384900
$109$	−16.0000	−1.53252	−0.766261	−	0.642529i	$-0.777885\pi$
$109$	−16.0000	−1.53252	−0.766261	+	0.642529i	$0.777885\pi$
$110$	−3.00000	−0.286039
$111$	14.0000	1.32882
$112$	0	0
$113$	0	0		−	1.00000i	$-0.5\pi$
$113$	0	0			1.00000i	$0.5\pi$
$114$	−2.00000	−0.187317
$115$	9.00000	0.839254
$116$	6.00000	0.557086
$117$	−1.00000	−0.0924500
$118$	0	0
$119$	0	0
$120$	2.00000	0.182574
$121$	−2.00000	−0.181818
$122$	−8.00000	−0.724286
$123$	−6.00000	−0.541002
$124$	8.00000	0.718421
$125$	1.00000	0.0894427
$126$	0	0
$127$	−1.00000	−0.0887357	−0.0443678	−	0.999015i	$-0.514127\pi$
$127$	−1.00000	−0.0887357	−0.0443678	+	0.999015i	$0.514127\pi$
$128$	−1.00000	−0.0883883
$129$	−4.00000	−0.352180
$130$	1.00000	0.0877058
$131$	3.00000	0.262111	0.131056	−	0.991375i	$-0.458163\pi$
$131$	3.00000	0.262111	0.131056	+	0.991375i	$0.458163\pi$
$132$	−6.00000	−0.522233
$133$	0	0
$134$	−8.00000	−0.691095
$135$	4.00000	0.344265
$136$	6.00000	0.514496
$137$	12.0000	1.02523	0.512615	−	0.858619i	$-0.328677\pi$
$137$	12.0000	1.02523	0.512615	+	0.858619i	$0.328677\pi$
$138$	18.0000	1.53226
$139$	−4.00000	−0.339276	−0.169638	−	0.985506i	$-0.554260\pi$
$139$	−4.00000	−0.339276	−0.169638	+	0.985506i	$0.554260\pi$
$140$	0	0
$141$	−18.0000	−1.51587
$142$	0	0
$143$	−3.00000	−0.250873
$144$	1.00000	0.0833333
$145$	6.00000	0.498273
$146$	4.00000	0.331042
$147$	0	0
$148$	−7.00000	−0.575396
$149$	−6.00000	−0.491539	−0.245770	−	0.969328i	$-0.579041\pi$
$149$	−6.00000	−0.491539	−0.245770	+	0.969328i	$0.579041\pi$
$150$	2.00000	0.163299
$151$	−10.0000	−0.813788	−0.406894	−	0.913475i	$-0.633388\pi$
$151$	−10.0000	−0.813788	−0.406894	+	0.913475i	$0.633388\pi$
$152$	1.00000	0.0811107
$153$	−6.00000	−0.485071
$154$	0	0
$155$	8.00000	0.642575
$156$	2.00000	0.160128
$157$	23.0000	1.83560	0.917800	−	0.397043i	$-0.129964\pi$
$157$	23.0000	1.83560	0.917800	+	0.397043i	$0.129964\pi$
$158$	10.0000	0.795557
$159$	−18.0000	−1.42749
$160$	−1.00000	−0.0790569
$161$	0	0
$162$	11.0000	0.864242
$163$	20.0000	1.56652	0.783260	−	0.621694i	$-0.213555\pi$
$163$	20.0000	1.56652	0.783260	+	0.621694i	$0.213555\pi$
$164$	3.00000	0.234261
$165$	−6.00000	−0.467099
$166$	0	0
$167$	3.00000	0.232147	0.116073	−	0.993241i	$-0.462969\pi$
$167$	3.00000	0.232147	0.116073	+	0.993241i	$0.462969\pi$
$168$	0	0
$169$	−12.0000	−0.923077
$170$	6.00000	0.460179
$171$	−1.00000	−0.0764719
$172$	2.00000	0.152499
$173$	9.00000	0.684257	0.342129	−	0.939653i	$-0.388852\pi$
$173$	9.00000	0.684257	0.342129	+	0.939653i	$0.388852\pi$
$174$	12.0000	0.909718
$175$	0	0
$176$	3.00000	0.226134
$177$	0	0
$178$	−6.00000	−0.449719
$179$	−3.00000	−0.224231	−0.112115	−	0.993695i	$-0.535763\pi$
$179$	−3.00000	−0.224231	−0.112115	+	0.993695i	$0.535763\pi$
$180$	1.00000	0.0745356
$181$	2.00000	0.148659	0.0743294	−	0.997234i	$-0.476318\pi$
$181$	2.00000	0.148659	0.0743294	+	0.997234i	$0.476318\pi$
$182$	0	0
$183$	−16.0000	−1.18275
$184$	−9.00000	−0.663489
$185$	−7.00000	−0.514650
$186$	16.0000	1.17318
$187$	−18.0000	−1.31629
$188$	9.00000	0.656392
$189$	0	0
$190$	1.00000	0.0725476
$191$	12.0000	0.868290	0.434145	−	0.900843i	$-0.357051\pi$
$191$	12.0000	0.868290	0.434145	+	0.900843i	$0.357051\pi$
$192$	−2.00000	−0.144338
$193$	−16.0000	−1.15171	−0.575853	−	0.817554i	$-0.695330\pi$
$193$	−16.0000	−1.15171	−0.575853	+	0.817554i	$0.695330\pi$
$194$	10.0000	0.717958
$195$	2.00000	0.143223
$196$	0	0
$197$	15.0000	1.06871	0.534353	−	0.845262i	$-0.320555\pi$
$197$	15.0000	1.06871	0.534353	+	0.845262i	$0.320555\pi$
$198$	−3.00000	−0.213201
$199$	−16.0000	−1.13421	−0.567105	−	0.823646i	$-0.691937\pi$
$199$	−16.0000	−1.13421	−0.567105	+	0.823646i	$0.691937\pi$
$200$	−1.00000	−0.0707107
$201$	−16.0000	−1.12855
$202$	−12.0000	−0.844317
$203$	0	0
$204$	12.0000	0.840168
$205$	3.00000	0.209529
$206$	4.00000	0.278693
$207$	9.00000	0.625543
$208$	−1.00000	−0.0693375
$209$	−3.00000	−0.207514
$210$	0	0
$211$	23.0000	1.58339	0.791693	−	0.610920i	$-0.209200\pi$
$211$	23.0000	1.58339	0.791693	+	0.610920i	$0.209200\pi$
$212$	9.00000	0.618123
$213$	0	0
$214$	12.0000	0.820303
$215$	2.00000	0.136399
$216$	−4.00000	−0.272166
$217$	0	0
$218$	16.0000	1.08366
$219$	8.00000	0.540590
$220$	3.00000	0.202260
$221$	6.00000	0.403604
$222$	−14.0000	−0.939618
$223$	8.00000	0.535720	0.267860	−	0.963458i	$-0.413684\pi$
$223$	8.00000	0.535720	0.267860	+	0.963458i	$0.413684\pi$
$224$	0	0
$225$	1.00000	0.0666667
$226$	0	0
$227$	−12.0000	−0.796468	−0.398234	−	0.917284i	$-0.630377\pi$
$227$	−12.0000	−0.796468	−0.398234	+	0.917284i	$0.630377\pi$
$228$	2.00000	0.132453
$229$	−4.00000	−0.264327	−0.132164	−	0.991228i	$-0.542192\pi$
$229$	−4.00000	−0.264327	−0.132164	+	0.991228i	$0.542192\pi$
$230$	−9.00000	−0.593442
$231$	0	0
$232$	−6.00000	−0.393919
$233$	−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
$235$	9.00000	0.587095
$236$	0	0
$237$	20.0000	1.29914
$238$	0	0
$239$	−6.00000	−0.388108	−0.194054	−	0.980991i	$-0.562164\pi$
$239$	−6.00000	−0.388108	−0.194054	+	0.980991i	$0.562164\pi$
$240$	−2.00000	−0.129099
$241$	−1.00000	−0.0644157	−0.0322078	−	0.999481i	$-0.510254\pi$
$241$	−1.00000	−0.0644157	−0.0322078	+	0.999481i	$0.510254\pi$
$242$	2.00000	0.128565
$243$	10.0000	0.641500
$244$	8.00000	0.512148
$245$	0	0
$246$	6.00000	0.382546
$247$	1.00000	0.0636285
$248$	−8.00000	−0.508001
$249$	0	0
$250$	−1.00000	−0.0632456
$251$	−15.0000	−0.946792	−0.473396	−	0.880850i	$-0.656972\pi$
$251$	−15.0000	−0.946792	−0.473396	+	0.880850i	$0.656972\pi$
$252$	0	0
$253$	27.0000	1.69748
$254$	1.00000	0.0627456
$255$	12.0000	0.751469
$256$	1.00000	0.0625000
$257$	0	0		−	1.00000i	$-0.5\pi$
$257$	0	0			1.00000i	$0.5\pi$
$258$	4.00000	0.249029
$259$	0	0
$260$	−1.00000	−0.0620174
$261$	6.00000	0.371391
$262$	−3.00000	−0.185341
$263$	0	0		−	1.00000i	$-0.5\pi$
$263$	0	0			1.00000i	$0.5\pi$
$264$	6.00000	0.369274
$265$	9.00000	0.552866
$266$	0	0
$267$	−12.0000	−0.734388
$268$	8.00000	0.488678
$269$	0	0		−	1.00000i	$-0.5\pi$
$269$	0	0			1.00000i	$0.5\pi$
$270$	−4.00000	−0.243432
$271$	−16.0000	−0.971931	−0.485965	−	0.873978i	$-0.661532\pi$
$271$	−16.0000	−0.971931	−0.485965	+	0.873978i	$0.661532\pi$
$272$	−6.00000	−0.363803
$273$	0	0
$274$	−12.0000	−0.724947
$275$	3.00000	0.180907
$276$	−18.0000	−1.08347
$277$	−10.0000	−0.600842	−0.300421	−	0.953807i	$-0.597127\pi$
$277$	−10.0000	−0.600842	−0.300421	+	0.953807i	$0.597127\pi$
$278$	4.00000	0.239904
$279$	8.00000	0.478947
$280$	0	0
$281$	−27.0000	−1.61068	−0.805342	−	0.592810i	$-0.798019\pi$
$281$	−27.0000	−1.61068	−0.805342	+	0.592810i	$0.798019\pi$
$282$	18.0000	1.07188
$283$	14.0000	0.832214	0.416107	−	0.909316i	$-0.363394\pi$
$283$	14.0000	0.832214	0.416107	+	0.909316i	$0.363394\pi$
$284$	0	0
$285$	2.00000	0.118470
$286$	3.00000	0.177394
$287$	0	0
$288$	−1.00000	−0.0589256
$289$	19.0000	1.11765
$290$	−6.00000	−0.352332
$291$	20.0000	1.17242
$292$	−4.00000	−0.234082
$293$	−9.00000	−0.525786	−0.262893	−	0.964825i	$-0.584677\pi$
$293$	−9.00000	−0.525786	−0.262893	+	0.964825i	$0.584677\pi$
$294$	0	0
$295$	0	0
$296$	7.00000	0.406867
$297$	12.0000	0.696311
$298$	6.00000	0.347571
$299$	−9.00000	−0.520483
$300$	−2.00000	−0.115470
$301$	0	0
$302$	10.0000	0.575435
$303$	−24.0000	−1.37876
$304$	−1.00000	−0.0573539
$305$	8.00000	0.458079
$306$	6.00000	0.342997
$307$	14.0000	0.799022	0.399511	−	0.916728i	$-0.369180\pi$
$307$	14.0000	0.799022	0.399511	+	0.916728i	$0.369180\pi$
$308$	0	0
$309$	8.00000	0.455104
$310$	−8.00000	−0.454369
$311$	−24.0000	−1.36092	−0.680458	−	0.732787i	$-0.738219\pi$
$311$	−24.0000	−1.36092	−0.680458	+	0.732787i	$0.738219\pi$
$312$	−2.00000	−0.113228
$313$	−28.0000	−1.58265	−0.791327	−	0.611393i	$-0.790609\pi$
$313$	−28.0000	−1.58265	−0.791327	+	0.611393i	$0.790609\pi$
$314$	−23.0000	−1.29797
$315$	0	0
$316$	−10.0000	−0.562544
$317$	6.00000	0.336994	0.168497	−	0.985702i	$-0.446109\pi$
$317$	6.00000	0.336994	0.168497	+	0.985702i	$0.446109\pi$
$318$	18.0000	1.00939
$319$	18.0000	1.00781
$320$	1.00000	0.0559017
$321$	24.0000	1.33955
$322$	0	0
$323$	6.00000	0.333849
$324$	−11.0000	−0.611111
$325$	−1.00000	−0.0554700
$326$	−20.0000	−1.10770
$327$	32.0000	1.76960
$328$	−3.00000	−0.165647
$329$	0	0
$330$	6.00000	0.330289
$331$	−7.00000	−0.384755	−0.192377	−	0.981321i	$-0.561620\pi$
$331$	−7.00000	−0.384755	−0.192377	+	0.981321i	$0.561620\pi$
$332$	0	0
$333$	−7.00000	−0.383598
$334$	−3.00000	−0.164153
$335$	8.00000	0.437087
$336$	0	0
$337$	−22.0000	−1.19842	−0.599208	−	0.800593i	$-0.704518\pi$
$337$	−22.0000	−1.19842	−0.599208	+	0.800593i	$0.704518\pi$
$338$	12.0000	0.652714
$339$	0	0
$340$	−6.00000	−0.325396
$341$	24.0000	1.29967
$342$	1.00000	0.0540738
$343$	0	0
$344$	−2.00000	−0.107833
$345$	−18.0000	−0.969087
$346$	−9.00000	−0.483843
$347$	0	0		−	1.00000i	$-0.5\pi$
$347$	0	0			1.00000i	$0.5\pi$
$348$	−12.0000	−0.643268
$349$	26.0000	1.39175	0.695874	−	0.718164i	$-0.255017\pi$
$349$	26.0000	1.39175	0.695874	+	0.718164i	$0.255017\pi$
$350$	0	0
$351$	−4.00000	−0.213504
$352$	−3.00000	−0.159901
$353$	12.0000	0.638696	0.319348	−	0.947638i	$-0.396536\pi$
$353$	12.0000	0.638696	0.319348	+	0.947638i	$0.396536\pi$
$354$	0	0
$355$	0	0
$356$	6.00000	0.317999
$357$	0	0
$358$	3.00000	0.158555
$359$	18.0000	0.950004	0.475002	−	0.879985i	$-0.342447\pi$
$359$	18.0000	0.950004	0.475002	+	0.879985i	$0.342447\pi$
$360$	−1.00000	−0.0527046
$361$	−18.0000	−0.947368
$362$	−2.00000	−0.105118
$363$	4.00000	0.209946
$364$	0	0
$365$	−4.00000	−0.209370
$366$	16.0000	0.836333
$367$	−19.0000	−0.991792	−0.495896	−	0.868382i	$-0.665160\pi$
$367$	−19.0000	−0.991792	−0.495896	+	0.868382i	$0.665160\pi$
$368$	9.00000	0.469157
$369$	3.00000	0.156174
$370$	7.00000	0.363913
$371$	0	0
$372$	−16.0000	−0.829561
$373$	2.00000	0.103556	0.0517780	−	0.998659i	$-0.483511\pi$
$373$	2.00000	0.103556	0.0517780	+	0.998659i	$0.483511\pi$
$374$	18.0000	0.930758
$375$	−2.00000	−0.103280
$376$	−9.00000	−0.464140
$377$	−6.00000	−0.309016
$378$	0	0
$379$	23.0000	1.18143	0.590715	−	0.806880i	$-0.298846\pi$
$379$	23.0000	1.18143	0.590715	+	0.806880i	$0.298846\pi$
$380$	−1.00000	−0.0512989
$381$	2.00000	0.102463
$382$	−12.0000	−0.613973
$383$	21.0000	1.07305	0.536525	−	0.843884i	$-0.319737\pi$
$383$	21.0000	1.07305	0.536525	+	0.843884i	$0.319737\pi$
$384$	2.00000	0.102062
$385$	0	0
$386$	16.0000	0.814379
$387$	2.00000	0.101666
$388$	−10.0000	−0.507673
$389$	−12.0000	−0.608424	−0.304212	−	0.952604i	$-0.598393\pi$
$389$	−12.0000	−0.608424	−0.304212	+	0.952604i	$0.598393\pi$
$390$	−2.00000	−0.101274
$391$	−54.0000	−2.73090
$392$	0	0
$393$	−6.00000	−0.302660
$394$	−15.0000	−0.755689
$395$	−10.0000	−0.503155
$396$	3.00000	0.150756
$397$	14.0000	0.702640	0.351320	−	0.936255i	$-0.385733\pi$
$397$	14.0000	0.702640	0.351320	+	0.936255i	$0.385733\pi$
$398$	16.0000	0.802008
$399$	0	0
$400$	1.00000	0.0500000
$401$	−27.0000	−1.34832	−0.674158	−	0.738587i	$-0.735493\pi$
$401$	−27.0000	−1.34832	−0.674158	+	0.738587i	$0.735493\pi$
$402$	16.0000	0.798007
$403$	−8.00000	−0.398508
$404$	12.0000	0.597022
$405$	−11.0000	−0.546594
$406$	0	0
$407$	−21.0000	−1.04093
$408$	−12.0000	−0.594089
$409$	26.0000	1.28562	0.642809	−	0.766027i	$-0.277769\pi$
$409$	26.0000	1.28562	0.642809	+	0.766027i	$0.277769\pi$
$410$	−3.00000	−0.148159
$411$	−24.0000	−1.18383
$412$	−4.00000	−0.197066
$413$	0	0
$414$	−9.00000	−0.442326
$415$	0	0
$416$	1.00000	0.0490290
$417$	8.00000	0.391762
$418$	3.00000	0.146735
$419$	−9.00000	−0.439679	−0.219839	−	0.975536i	$-0.570553\pi$
$419$	−9.00000	−0.439679	−0.219839	+	0.975536i	$0.570553\pi$
$420$	0	0
$421$	2.00000	0.0974740	0.0487370	−	0.998812i	$-0.484480\pi$
$421$	2.00000	0.0974740	0.0487370	+	0.998812i	$0.484480\pi$
$422$	−23.0000	−1.11962
$423$	9.00000	0.437595
$424$	−9.00000	−0.437079
$425$	−6.00000	−0.291043
$426$	0	0
$427$	0	0
$428$	−12.0000	−0.580042
$429$	6.00000	0.289683
$430$	−2.00000	−0.0964486
$431$	12.0000	0.578020	0.289010	−	0.957326i	$-0.406674\pi$
$431$	12.0000	0.578020	0.289010	+	0.957326i	$0.406674\pi$
$432$	4.00000	0.192450
$433$	−40.0000	−1.92228	−0.961139	−	0.276066i	$-0.910969\pi$
$433$	−40.0000	−1.92228	−0.961139	+	0.276066i	$0.910969\pi$
$434$	0	0
$435$	−12.0000	−0.575356
$436$	−16.0000	−0.766261
$437$	−9.00000	−0.430528
$438$	−8.00000	−0.382255
$439$	26.0000	1.24091	0.620456	−	0.784241i	$-0.286947\pi$
$439$	26.0000	1.24091	0.620456	+	0.784241i	$0.286947\pi$
$440$	−3.00000	−0.143019
$441$	0	0
$442$	−6.00000	−0.285391
$443$	12.0000	0.570137	0.285069	−	0.958507i	$-0.407984\pi$
$443$	12.0000	0.570137	0.285069	+	0.958507i	$0.407984\pi$
$444$	14.0000	0.664411
$445$	6.00000	0.284427
$446$	−8.00000	−0.378811
$447$	12.0000	0.567581
$448$	0	0
$449$	21.0000	0.991051	0.495526	−	0.868593i	$-0.334975\pi$
$449$	21.0000	0.991051	0.495526	+	0.868593i	$0.334975\pi$
$450$	−1.00000	−0.0471405
$451$	9.00000	0.423793
$452$	0	0
$453$	20.0000	0.939682
$454$	12.0000	0.563188
$455$	0	0
$456$	−2.00000	−0.0936586
$457$	14.0000	0.654892	0.327446	−	0.944870i	$-0.393812\pi$
$457$	14.0000	0.654892	0.327446	+	0.944870i	$0.393812\pi$
$458$	4.00000	0.186908
$459$	−24.0000	−1.12022
$460$	9.00000	0.419627
$461$	−30.0000	−1.39724	−0.698620	−	0.715493i	$-0.746202\pi$
$461$	−30.0000	−1.39724	−0.698620	+	0.715493i	$0.746202\pi$
$462$	0	0
$463$	−1.00000	−0.0464739	−0.0232370	−	0.999730i	$-0.507397\pi$
$463$	−1.00000	−0.0464739	−0.0232370	+	0.999730i	$0.507397\pi$
$464$	6.00000	0.278543
$465$	−16.0000	−0.741982
$466$	6.00000	0.277945
$467$	−6.00000	−0.277647	−0.138823	−	0.990317i	$-0.544332\pi$
$467$	−6.00000	−0.277647	−0.138823	+	0.990317i	$0.544332\pi$
$468$	−1.00000	−0.0462250
$469$	0	0
$470$	−9.00000	−0.415139
$471$	−46.0000	−2.11957
$472$	0	0
$473$	6.00000	0.275880
$474$	−20.0000	−0.918630
$475$	−1.00000	−0.0458831
$476$	0	0
$477$	9.00000	0.412082
$478$	6.00000	0.274434
$479$	0	0		−	1.00000i	$-0.5\pi$
$479$	0	0			1.00000i	$0.5\pi$
$480$	2.00000	0.0912871
$481$	7.00000	0.319173
$482$	1.00000	0.0455488
$483$	0	0
$484$	−2.00000	−0.0909091
$485$	−10.0000	−0.454077
$486$	−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	$0.618082\pi$
$488$	−8.00000	−0.362143
$489$	−40.0000	−1.80886
$490$	0	0
$491$	36.0000	1.62466	0.812329	−	0.583200i	$-0.198200\pi$
$491$	36.0000	1.62466	0.812329	+	0.583200i	$0.198200\pi$
$492$	−6.00000	−0.270501
$493$	−36.0000	−1.62136
$494$	−1.00000	−0.0449921
$495$	3.00000	0.134840
$496$	8.00000	0.359211
$497$	0	0
$498$	0	0
$499$	−4.00000	−0.179065	−0.0895323	−	0.995984i	$-0.528537\pi$
$499$	−4.00000	−0.179065	−0.0895323	+	0.995984i	$0.528537\pi$
$500$	1.00000	0.0447214
$501$	−6.00000	−0.268060
$502$	15.0000	0.669483
$503$	24.0000	1.07011	0.535054	−	0.844818i	$-0.320291\pi$
$503$	24.0000	1.07011	0.535054	+	0.844818i	$0.320291\pi$
$504$	0	0
$505$	12.0000	0.533993
$506$	−27.0000	−1.20030
$507$	24.0000	1.06588
$508$	−1.00000	−0.0443678
$509$	−42.0000	−1.86162	−0.930809	−	0.365507i	$-0.880896\pi$
$509$	−42.0000	−1.86162	−0.930809	+	0.365507i	$0.880896\pi$
$510$	−12.0000	−0.531369
$511$	0	0
$512$	−1.00000	−0.0441942
$513$	−4.00000	−0.176604
$514$	0	0
$515$	−4.00000	−0.176261
$516$	−4.00000	−0.176090
$517$	27.0000	1.18746
$518$	0	0
$519$	−18.0000	−0.790112
$520$	1.00000	0.0438529
$521$	−15.0000	−0.657162	−0.328581	−	0.944476i	$-0.606570\pi$
$521$	−15.0000	−0.657162	−0.328581	+	0.944476i	$0.606570\pi$
$522$	−6.00000	−0.262613
$523$	−28.0000	−1.22435	−0.612177	−	0.790721i	$-0.709706\pi$
$523$	−28.0000	−1.22435	−0.612177	+	0.790721i	$0.709706\pi$
$524$	3.00000	0.131056
$525$	0	0
$526$	0	0
$527$	−48.0000	−2.09091
$528$	−6.00000	−0.261116
$529$	58.0000	2.52174
$530$	−9.00000	−0.390935
$531$	0	0
$532$	0	0
$533$	−3.00000	−0.129944
$534$	12.0000	0.519291
$535$	−12.0000	−0.518805
$536$	−8.00000	−0.345547
$537$	6.00000	0.258919
$538$	0	0
$539$	0	0
$540$	4.00000	0.172133
$541$	8.00000	0.343947	0.171973	−	0.985102i	$-0.444986\pi$
$541$	8.00000	0.343947	0.171973	+	0.985102i	$0.444986\pi$
$542$	16.0000	0.687259
$543$	−4.00000	−0.171656
$544$	6.00000	0.257248
$545$	−16.0000	−0.685365
$546$	0	0
$547$	8.00000	0.342055	0.171028	−	0.985266i	$-0.445291\pi$
$547$	8.00000	0.342055	0.171028	+	0.985266i	$0.445291\pi$
$548$	12.0000	0.512615
$549$	8.00000	0.341432
$550$	−3.00000	−0.127920
$551$	−6.00000	−0.255609
$552$	18.0000	0.766131
$553$	0	0
$554$	10.0000	0.424859
$555$	14.0000	0.594267
$556$	−4.00000	−0.169638
$557$	−9.00000	−0.381342	−0.190671	−	0.981654i	$-0.561066\pi$
$557$	−9.00000	−0.381342	−0.190671	+	0.981654i	$0.561066\pi$
$558$	−8.00000	−0.338667
$559$	−2.00000	−0.0845910
$560$	0	0
$561$	36.0000	1.51992
$562$	27.0000	1.13893
$563$	42.0000	1.77009	0.885044	−	0.465506i	$-0.154128\pi$
$563$	42.0000	1.77009	0.885044	+	0.465506i	$0.154128\pi$
$564$	−18.0000	−0.757937
$565$	0	0
$566$	−14.0000	−0.588464
$567$	0	0
$568$	0	0
$569$	21.0000	0.880366	0.440183	−	0.897908i	$-0.354914\pi$
$569$	21.0000	0.880366	0.440183	+	0.897908i	$0.354914\pi$
$570$	−2.00000	−0.0837708
$571$	20.0000	0.836974	0.418487	−	0.908223i	$-0.362561\pi$
$571$	20.0000	0.836974	0.418487	+	0.908223i	$0.362561\pi$
$572$	−3.00000	−0.125436
$573$	−24.0000	−1.00261
$574$	0	0
$575$	9.00000	0.375326
$576$	1.00000	0.0416667
$577$	44.0000	1.83174	0.915872	−	0.401470i	$-0.131501\pi$
$577$	44.0000	1.83174	0.915872	+	0.401470i	$0.131501\pi$
$578$	−19.0000	−0.790296
$579$	32.0000	1.32987
$580$	6.00000	0.249136
$581$	0	0
$582$	−20.0000	−0.829027
$583$	27.0000	1.11823
$584$	4.00000	0.165521
$585$	−1.00000	−0.0413449
$586$	9.00000	0.371787
$587$	−24.0000	−0.990586	−0.495293	−	0.868726i	$-0.664939\pi$
$587$	−24.0000	−0.990586	−0.495293	+	0.868726i	$0.664939\pi$
$588$	0	0
$589$	−8.00000	−0.329634
$590$	0	0
$591$	−30.0000	−1.23404
$592$	−7.00000	−0.287698
$593$	24.0000	0.985562	0.492781	−	0.870153i	$-0.335980\pi$
$593$	24.0000	0.985562	0.492781	+	0.870153i	$0.335980\pi$
$594$	−12.0000	−0.492366
$595$	0	0
$596$	−6.00000	−0.245770
$597$	32.0000	1.30967
$598$	9.00000	0.368037
$599$	−42.0000	−1.71607	−0.858037	−	0.513588i	$-0.828316\pi$
$599$	−42.0000	−1.71607	−0.858037	+	0.513588i	$0.828316\pi$
$600$	2.00000	0.0816497
$601$	26.0000	1.06056	0.530281	−	0.847822i	$-0.322086\pi$
$601$	26.0000	1.06056	0.530281	+	0.847822i	$0.322086\pi$
$602$	0	0
$603$	8.00000	0.325785
$604$	−10.0000	−0.406894
$605$	−2.00000	−0.0813116
$606$	24.0000	0.974933
$607$	−1.00000	−0.0405887	−0.0202944	−	0.999794i	$-0.506460\pi$
$607$	−1.00000	−0.0405887	−0.0202944	+	0.999794i	$0.506460\pi$
$608$	1.00000	0.0405554
$609$	0	0
$610$	−8.00000	−0.323911
$611$	−9.00000	−0.364101
$612$	−6.00000	−0.242536
$613$	29.0000	1.17130	0.585649	−	0.810564i	$-0.300840\pi$
$613$	29.0000	1.17130	0.585649	+	0.810564i	$0.300840\pi$
$614$	−14.0000	−0.564994
$615$	−6.00000	−0.241943
$616$	0	0
$617$	−18.0000	−0.724653	−0.362326	−	0.932051i	$-0.618017\pi$
$617$	−18.0000	−0.724653	−0.362326	+	0.932051i	$0.618017\pi$
$618$	−8.00000	−0.321807
$619$	23.0000	0.924448	0.462224	−	0.886763i	$-0.347052\pi$
$619$	23.0000	0.924448	0.462224	+	0.886763i	$0.347052\pi$
$620$	8.00000	0.321288
$621$	36.0000	1.44463
$622$	24.0000	0.962312
$623$	0	0
$624$	2.00000	0.0800641
$625$	1.00000	0.0400000
$626$	28.0000	1.11911
$627$	6.00000	0.239617
$628$	23.0000	0.917800
$629$	42.0000	1.67465
$630$	0	0
$631$	20.0000	0.796187	0.398094	−	0.917345i	$-0.369672\pi$
$631$	20.0000	0.796187	0.398094	+	0.917345i	$0.369672\pi$
$632$	10.0000	0.397779
$633$	−46.0000	−1.82834
$634$	−6.00000	−0.238290
$635$	−1.00000	−0.0396838
$636$	−18.0000	−0.713746
$637$	0	0
$638$	−18.0000	−0.712627
$639$	0	0
$640$	−1.00000	−0.0395285
$641$	27.0000	1.06644	0.533218	−	0.845978i	$-0.320983\pi$
$641$	27.0000	1.06644	0.533218	+	0.845978i	$0.320983\pi$
$642$	−24.0000	−0.947204
$643$	2.00000	0.0788723	0.0394362	−	0.999222i	$-0.487444\pi$
$643$	2.00000	0.0788723	0.0394362	+	0.999222i	$0.487444\pi$
$644$	0	0
$645$	−4.00000	−0.157500
$646$	−6.00000	−0.236067
$647$	33.0000	1.29736	0.648682	−	0.761060i	$-0.275321\pi$
$647$	33.0000	1.29736	0.648682	+	0.761060i	$0.275321\pi$
$648$	11.0000	0.432121
$649$	0	0
$650$	1.00000	0.0392232
$651$	0	0
$652$	20.0000	0.783260
$653$	−9.00000	−0.352197	−0.176099	−	0.984373i	$-0.556348\pi$
$653$	−9.00000	−0.352197	−0.176099	+	0.984373i	$0.556348\pi$
$654$	−32.0000	−1.25130
$655$	3.00000	0.117220
$656$	3.00000	0.117130
$657$	−4.00000	−0.156055
$658$	0	0
$659$	−24.0000	−0.934907	−0.467454	−	0.884018i	$-0.654829\pi$
$659$	−24.0000	−0.934907	−0.467454	+	0.884018i	$0.654829\pi$
$660$	−6.00000	−0.233550
$661$	−28.0000	−1.08907	−0.544537	−	0.838737i	$-0.683295\pi$
$661$	−28.0000	−1.08907	−0.544537	+	0.838737i	$0.683295\pi$
$662$	7.00000	0.272063
$663$	−12.0000	−0.466041
$664$	0	0
$665$	0	0
$666$	7.00000	0.271244
$667$	54.0000	2.09089
$668$	3.00000	0.116073
$669$	−16.0000	−0.618596
$670$	−8.00000	−0.309067
$671$	24.0000	0.926510
$672$	0	0
$673$	−34.0000	−1.31060	−0.655302	−	0.755367i	$-0.727459\pi$
$673$	−34.0000	−1.31060	−0.655302	+	0.755367i	$0.727459\pi$
$674$	22.0000	0.847408
$675$	4.00000	0.153960
$676$	−12.0000	−0.461538
$677$	−9.00000	−0.345898	−0.172949	−	0.984931i	$-0.555330\pi$
$677$	−9.00000	−0.345898	−0.172949	+	0.984931i	$0.555330\pi$
$678$	0	0
$679$	0	0
$680$	6.00000	0.230089
$681$	24.0000	0.919682
$682$	−24.0000	−0.919007
$683$	12.0000	0.459167	0.229584	−	0.973289i	$-0.426264\pi$
$683$	12.0000	0.459167	0.229584	+	0.973289i	$0.426264\pi$
$684$	−1.00000	−0.0382360
$685$	12.0000	0.458496
$686$	0	0
$687$	8.00000	0.305219
$688$	2.00000	0.0762493
$689$	−9.00000	−0.342873
$690$	18.0000	0.685248
$691$	32.0000	1.21734	0.608669	−	0.793424i	$-0.291704\pi$
$691$	32.0000	1.21734	0.608669	+	0.793424i	$0.291704\pi$
$692$	9.00000	0.342129
$693$	0	0
$694$	0	0
$695$	−4.00000	−0.151729
$696$	12.0000	0.454859
$697$	−18.0000	−0.681799
$698$	−26.0000	−0.984115
$699$	12.0000	0.453882
$700$	0	0
$701$	−30.0000	−1.13308	−0.566542	−	0.824033i	$-0.691719\pi$
$701$	−30.0000	−1.13308	−0.566542	+	0.824033i	$0.691719\pi$
$702$	4.00000	0.150970
$703$	7.00000	0.264010
$704$	3.00000	0.113067
$705$	−18.0000	−0.677919
$706$	−12.0000	−0.451626
$707$	0	0
$708$	0	0
$709$	−46.0000	−1.72757	−0.863783	−	0.503864i	$-0.831911\pi$
$709$	−46.0000	−1.72757	−0.863783	+	0.503864i	$0.831911\pi$
$710$	0	0
$711$	−10.0000	−0.375029
$712$	−6.00000	−0.224860
$713$	72.0000	2.69642
$714$	0	0
$715$	−3.00000	−0.112194
$716$	−3.00000	−0.112115
$717$	12.0000	0.448148
$718$	−18.0000	−0.671754
$719$	0	0		−	1.00000i	$-0.5\pi$
$719$	0	0			1.00000i	$0.5\pi$
$720$	1.00000	0.0372678
$721$	0	0
$722$	18.0000	0.669891
$723$	2.00000	0.0743808
$724$	2.00000	0.0743294
$725$	6.00000	0.222834
$726$	−4.00000	−0.148454
$727$	−1.00000	−0.0370879	−0.0185440	−	0.999828i	$-0.505903\pi$
$727$	−1.00000	−0.0370879	−0.0185440	+	0.999828i	$0.505903\pi$
$728$	0	0
$729$	13.0000	0.481481
$730$	4.00000	0.148047
$731$	−12.0000	−0.443836
$732$	−16.0000	−0.591377
$733$	−43.0000	−1.58824	−0.794121	−	0.607760i	$-0.792068\pi$
$733$	−43.0000	−1.58824	−0.794121	+	0.607760i	$0.792068\pi$
$734$	19.0000	0.701303
$735$	0	0
$736$	−9.00000	−0.331744
$737$	24.0000	0.884051
$738$	−3.00000	−0.110432
$739$	35.0000	1.28750	0.643748	−	0.765238i	$-0.277379\pi$
$739$	35.0000	1.28750	0.643748	+	0.765238i	$0.277379\pi$
$740$	−7.00000	−0.257325
$741$	−2.00000	−0.0734718
$742$	0	0
$743$	−45.0000	−1.65089	−0.825445	−	0.564483i	$-0.809076\pi$
$743$	−45.0000	−1.65089	−0.825445	+	0.564483i	$0.809076\pi$
$744$	16.0000	0.586588
$745$	−6.00000	−0.219823
$746$	−2.00000	−0.0732252
$747$	0	0
$748$	−18.0000	−0.658145
$749$	0	0
$750$	2.00000	0.0730297
$751$	−10.0000	−0.364905	−0.182453	−	0.983215i	$-0.558404\pi$
$751$	−10.0000	−0.364905	−0.182453	+	0.983215i	$0.558404\pi$
$752$	9.00000	0.328196
$753$	30.0000	1.09326
$754$	6.00000	0.218507
$755$	−10.0000	−0.363937
$756$	0	0
$757$	38.0000	1.38113	0.690567	−	0.723269i	$-0.257361\pi$
$757$	38.0000	1.38113	0.690567	+	0.723269i	$0.257361\pi$
$758$	−23.0000	−0.835398
$759$	−54.0000	−1.96008
$760$	1.00000	0.0362738
$761$	−27.0000	−0.978749	−0.489375	−	0.872074i	$-0.662775\pi$
$761$	−27.0000	−0.978749	−0.489375	+	0.872074i	$0.662775\pi$
$762$	−2.00000	−0.0724524
$763$	0	0
$764$	12.0000	0.434145
$765$	−6.00000	−0.216930
$766$	−21.0000	−0.758761
$767$	0	0
$768$	−2.00000	−0.0721688
$769$	23.0000	0.829401	0.414701	−	0.909958i	$-0.363886\pi$
$769$	23.0000	0.829401	0.414701	+	0.909958i	$0.363886\pi$
$770$	0	0
$771$	0	0
$772$	−16.0000	−0.575853
$773$	−51.0000	−1.83434	−0.917171	−	0.398493i	$-0.869533\pi$
$773$	−51.0000	−1.83434	−0.917171	+	0.398493i	$0.869533\pi$
$774$	−2.00000	−0.0718885
$775$	8.00000	0.287368
$776$	10.0000	0.358979
$777$	0	0
$778$	12.0000	0.430221
$779$	−3.00000	−0.107486
$780$	2.00000	0.0716115
$781$	0	0
$782$	54.0000	1.93104
$783$	24.0000	0.857690
$784$	0	0
$785$	23.0000	0.820905
$786$	6.00000	0.214013
$787$	−22.0000	−0.784215	−0.392108	−	0.919919i	$-0.628254\pi$
$787$	−22.0000	−0.784215	−0.392108	+	0.919919i	$0.628254\pi$
$788$	15.0000	0.534353
$789$	0	0
$790$	10.0000	0.355784
$791$	0	0
$792$	−3.00000	−0.106600
$793$	−8.00000	−0.284088
$794$	−14.0000	−0.496841
$795$	−18.0000	−0.638394
$796$	−16.0000	−0.567105
$797$	6.00000	0.212531	0.106265	−	0.994338i	$-0.466111\pi$
$797$	6.00000	0.212531	0.106265	+	0.994338i	$0.466111\pi$
$798$	0	0
$799$	−54.0000	−1.91038
$800$	−1.00000	−0.0353553
$801$	6.00000	0.212000
$802$	27.0000	0.953403
$803$	−12.0000	−0.423471
$804$	−16.0000	−0.564276
$805$	0	0
$806$	8.00000	0.281788
$807$	0	0
$808$	−12.0000	−0.422159
$809$	−9.00000	−0.316423	−0.158212	−	0.987405i	$-0.550573\pi$
$809$	−9.00000	−0.316423	−0.158212	+	0.987405i	$0.550573\pi$
$810$	11.0000	0.386501
$811$	−25.0000	−0.877869	−0.438934	−	0.898519i	$-0.644644\pi$
$811$	−25.0000	−0.877869	−0.438934	+	0.898519i	$0.644644\pi$
$812$	0	0
$813$	32.0000	1.12229
$814$	21.0000	0.736050
$815$	20.0000	0.700569
$816$	12.0000	0.420084
$817$	−2.00000	−0.0699711
$818$	−26.0000	−0.909069
$819$	0	0
$820$	3.00000	0.104765
$821$	30.0000	1.04701	0.523504	−	0.852023i	$-0.324625\pi$
$821$	30.0000	1.04701	0.523504	+	0.852023i	$0.324625\pi$
$822$	24.0000	0.837096
$823$	−4.00000	−0.139431	−0.0697156	−	0.997567i	$-0.522209\pi$
$823$	−4.00000	−0.139431	−0.0697156	+	0.997567i	$0.522209\pi$
$824$	4.00000	0.139347
$825$	−6.00000	−0.208893
$826$	0	0
$827$	6.00000	0.208640	0.104320	−	0.994544i	$-0.466733\pi$
$827$	6.00000	0.208640	0.104320	+	0.994544i	$0.466733\pi$
$828$	9.00000	0.312772
$829$	14.0000	0.486240	0.243120	−	0.969996i	$-0.421829\pi$
$829$	14.0000	0.486240	0.243120	+	0.969996i	$0.421829\pi$
$830$	0	0
$831$	20.0000	0.693792
$832$	−1.00000	−0.0346688
$833$	0	0
$834$	−8.00000	−0.277017
$835$	3.00000	0.103819
$836$	−3.00000	−0.103757
$837$	32.0000	1.10608
$838$	9.00000	0.310900
$839$	−30.0000	−1.03572	−0.517858	−	0.855467i	$-0.673270\pi$
$839$	−30.0000	−1.03572	−0.517858	+	0.855467i	$0.673270\pi$
$840$	0	0
$841$	7.00000	0.241379
$842$	−2.00000	−0.0689246
$843$	54.0000	1.85986
$844$	23.0000	0.791693
$845$	−12.0000	−0.412813
$846$	−9.00000	−0.309426
$847$	0	0
$848$	9.00000	0.309061
$849$	−28.0000	−0.960958
$850$	6.00000	0.205798
$851$	−63.0000	−2.15961
$852$	0	0
$853$	−19.0000	−0.650548	−0.325274	−	0.945620i	$-0.605456\pi$
$853$	−19.0000	−0.650548	−0.325274	+	0.945620i	$0.605456\pi$
$854$	0	0
$855$	−1.00000	−0.0341993
$856$	12.0000	0.410152
$857$	−18.0000	−0.614868	−0.307434	−	0.951569i	$-0.599470\pi$
$857$	−18.0000	−0.614868	−0.307434	+	0.951569i	$0.599470\pi$
$858$	−6.00000	−0.204837
$859$	32.0000	1.09183	0.545913	−	0.837842i	$-0.316183\pi$
$859$	32.0000	1.09183	0.545913	+	0.837842i	$0.316183\pi$
$860$	2.00000	0.0681994
$861$	0	0
$862$	−12.0000	−0.408722
$863$	−3.00000	−0.102121	−0.0510606	−	0.998696i	$-0.516260\pi$
$863$	−3.00000	−0.102121	−0.0510606	+	0.998696i	$0.516260\pi$
$864$	−4.00000	−0.136083
$865$	9.00000	0.306009
$866$	40.0000	1.35926
$867$	−38.0000	−1.29055
$868$	0	0
$869$	−30.0000	−1.01768
$870$	12.0000	0.406838
$871$	−8.00000	−0.271070
$872$	16.0000	0.541828
$873$	−10.0000	−0.338449
$874$	9.00000	0.304430
$875$	0	0
$876$	8.00000	0.270295
$877$	−13.0000	−0.438979	−0.219489	−	0.975615i	$-0.570439\pi$
$877$	−13.0000	−0.438979	−0.219489	+	0.975615i	$0.570439\pi$
$878$	−26.0000	−0.877457
$879$	18.0000	0.607125
$880$	3.00000	0.101130
$881$	33.0000	1.11180	0.555899	−	0.831250i	$-0.312374\pi$
$881$	33.0000	1.11180	0.555899	+	0.831250i	$0.312374\pi$
$882$	0	0
$883$	8.00000	0.269221	0.134611	−	0.990899i	$-0.457022\pi$
$883$	8.00000	0.269221	0.134611	+	0.990899i	$0.457022\pi$
$884$	6.00000	0.201802
$885$	0	0
$886$	−12.0000	−0.403148
$887$	−48.0000	−1.61168	−0.805841	−	0.592132i	$-0.798286\pi$
$887$	−48.0000	−1.61168	−0.805841	+	0.592132i	$0.798286\pi$
$888$	−14.0000	−0.469809
$889$	0	0
$890$	−6.00000	−0.201120
$891$	−33.0000	−1.10554
$892$	8.00000	0.267860
$893$	−9.00000	−0.301174
$894$	−12.0000	−0.401340
$895$	−3.00000	−0.100279
$896$	0	0
$897$	18.0000	0.601003
$898$	−21.0000	−0.700779
$899$	48.0000	1.60089
$900$	1.00000	0.0333333
$901$	−54.0000	−1.79900
$902$	−9.00000	−0.299667
$903$	0	0
$904$	0	0
$905$	2.00000	0.0664822
$906$	−20.0000	−0.664455
$907$	−10.0000	−0.332045	−0.166022	−	0.986122i	$-0.553092\pi$
$907$	−10.0000	−0.332045	−0.166022	+	0.986122i	$0.553092\pi$
$908$	−12.0000	−0.398234
$909$	12.0000	0.398015
$910$	0	0
$911$	−30.0000	−0.993944	−0.496972	−	0.867766i	$-0.665555\pi$
$911$	−30.0000	−0.993944	−0.496972	+	0.867766i	$0.665555\pi$
$912$	2.00000	0.0662266
$913$	0	0
$914$	−14.0000	−0.463079
$915$	−16.0000	−0.528944
$916$	−4.00000	−0.132164
$917$	0	0
$918$	24.0000	0.792118
$919$	−22.0000	−0.725713	−0.362857	−	0.931845i	$-0.618198\pi$
$919$	−22.0000	−0.725713	−0.362857	+	0.931845i	$0.618198\pi$
$920$	−9.00000	−0.296721
$921$	−28.0000	−0.922631
$922$	30.0000	0.987997
$923$	0	0
$924$	0	0
$925$	−7.00000	−0.230159
$926$	1.00000	0.0328620
$927$	−4.00000	−0.131377
$928$	−6.00000	−0.196960
$929$	57.0000	1.87011	0.935055	−	0.354504i	$-0.115350\pi$
$929$	57.0000	1.87011	0.935055	+	0.354504i	$0.115350\pi$
$930$	16.0000	0.524661
$931$	0	0
$932$	−6.00000	−0.196537
$933$	48.0000	1.57145
$934$	6.00000	0.196326
$935$	−18.0000	−0.588663
$936$	1.00000	0.0326860
$937$	−10.0000	−0.326686	−0.163343	−	0.986569i	$-0.552228\pi$
$937$	−10.0000	−0.326686	−0.163343	+	0.986569i	$0.552228\pi$
$938$	0	0
$939$	56.0000	1.82749
$940$	9.00000	0.293548
$941$	48.0000	1.56476	0.782378	−	0.622804i	$-0.214007\pi$
$941$	48.0000	1.56476	0.782378	+	0.622804i	$0.214007\pi$
$942$	46.0000	1.49876
$943$	27.0000	0.879241
$944$	0	0
$945$	0	0
$946$	−6.00000	−0.195077
$947$	6.00000	0.194974	0.0974869	−	0.995237i	$-0.468920\pi$
$947$	6.00000	0.194974	0.0974869	+	0.995237i	$0.468920\pi$
$948$	20.0000	0.649570
$949$	4.00000	0.129845
$950$	1.00000	0.0324443
$951$	−12.0000	−0.389127
$952$	0	0
$953$	36.0000	1.16615	0.583077	−	0.812417i	$-0.301849\pi$
$953$	36.0000	1.16615	0.583077	+	0.812417i	$0.301849\pi$
$954$	−9.00000	−0.291386
$955$	12.0000	0.388311
$956$	−6.00000	−0.194054
$957$	−36.0000	−1.16371
$958$	0	0
$959$	0	0
$960$	−2.00000	−0.0645497
$961$	33.0000	1.06452
$962$	−7.00000	−0.225689
$963$	−12.0000	−0.386695
$964$	−1.00000	−0.0322078
$965$	−16.0000	−0.515058
$966$	0	0
$967$	32.0000	1.02905	0.514525	−	0.857475i	$-0.327968\pi$
$967$	32.0000	1.02905	0.514525	+	0.857475i	$0.327968\pi$
$968$	2.00000	0.0642824
$969$	−12.0000	−0.385496
$970$	10.0000	0.321081
$971$	−45.0000	−1.44412	−0.722059	−	0.691831i	$-0.756804\pi$
$971$	−45.0000	−1.44412	−0.722059	+	0.691831i	$0.756804\pi$
$972$	10.0000	0.320750
$973$	0	0
$974$	16.0000	0.512673
$975$	2.00000	0.0640513
$976$	8.00000	0.256074
$977$	−42.0000	−1.34370	−0.671850	−	0.740688i	$-0.734500\pi$
$977$	−42.0000	−1.34370	−0.671850	+	0.740688i	$0.734500\pi$
$978$	40.0000	1.27906
$979$	18.0000	0.575282
$980$	0	0
$981$	−16.0000	−0.510841
$982$	−36.0000	−1.14881
$983$	3.00000	0.0956851	0.0478426	−	0.998855i	$-0.484765\pi$
$983$	3.00000	0.0956851	0.0478426	+	0.998855i	$0.484765\pi$
$984$	6.00000	0.191273
$985$	15.0000	0.477940
$986$	36.0000	1.14647
$987$	0	0
$988$	1.00000	0.0318142
$989$	18.0000	0.572367
$990$	−3.00000	−0.0953463
$991$	44.0000	1.39771	0.698853	−	0.715265i	$-0.253694\pi$
$991$	44.0000	1.39771	0.698853	+	0.715265i	$0.253694\pi$
$992$	−8.00000	−0.254000
$993$	14.0000	0.444277
$994$	0	0
$995$	−16.0000	−0.507234
$996$	0	0
$997$	−22.0000	−0.696747	−0.348373	−	0.937356i	$-0.613266\pi$
$997$	−22.0000	−0.696747	−0.348373	+	0.937356i	$0.613266\pi$
$998$	4.00000	0.126618
$999$	−28.0000	−0.885881

Twists

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

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

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 490 = 2 \cdot 5 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	490.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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