LMFDB - Embedded newform 690.2.a.k.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [690,2,Mod(1,690)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(690, 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("690.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 690 = 2 \cdot 3 \cdot 5 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	690.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:	$5.50967773947$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	690.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

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

\(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} +1.00000 q^{6} +1.00000 q^{8} +1.00000 q^{9} +1.00000 q^{10} +4.00000 q^{11} +1.00000 q^{12} -2.00000 q^{13} +1.00000 q^{15} +1.00000 q^{16} -6.00000 q^{17} +1.00000 q^{18} +4.00000 q^{19} +1.00000 q^{20} +4.00000 q^{22} -1.00000 q^{23} +1.00000 q^{24} +1.00000 q^{25} -2.00000 q^{26} +1.00000 q^{27} -2.00000 q^{29} +1.00000 q^{30} +1.00000 q^{32} +4.00000 q^{33} -6.00000 q^{34} +1.00000 q^{36} -2.00000 q^{37} +4.00000 q^{38} -2.00000 q^{39} +1.00000 q^{40} +10.0000 q^{41} -4.00000 q^{43} +4.00000 q^{44} +1.00000 q^{45} -1.00000 q^{46} +1.00000 q^{48} -7.00000 q^{49} +1.00000 q^{50} -6.00000 q^{51} -2.00000 q^{52} +6.00000 q^{53} +1.00000 q^{54} +4.00000 q^{55} +4.00000 q^{57} -2.00000 q^{58} -4.00000 q^{59} +1.00000 q^{60} -10.0000 q^{61} +1.00000 q^{64} -2.00000 q^{65} +4.00000 q^{66} -12.0000 q^{67} -6.00000 q^{68} -1.00000 q^{69} -8.00000 q^{71} +1.00000 q^{72} +10.0000 q^{73} -2.00000 q^{74} +1.00000 q^{75} +4.00000 q^{76} -2.00000 q^{78} -8.00000 q^{79} +1.00000 q^{80} +1.00000 q^{81} +10.0000 q^{82} -4.00000 q^{83} -6.00000 q^{85} -4.00000 q^{86} -2.00000 q^{87} +4.00000 q^{88} +18.0000 q^{89} +1.00000 q^{90} -1.00000 q^{92} +4.00000 q^{95} +1.00000 q^{96} +2.00000 q^{97} -7.00000 q^{98} +4.00000 q^{99} +O(q^{100})\)

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	1.00000	0.500000
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	1.00000	0.408248
$7$	0	0		−	1.00000i	$-0.5\pi$
$7$	0	0			1.00000i	$0.5\pi$
$8$	1.00000	0.353553
$9$	1.00000	0.333333
$10$	1.00000	0.316228
$11$	4.00000	1.20605	0.603023	−	0.797724i	$-0.293963\pi$
$11$	4.00000	1.20605	0.603023	+	0.797724i	$0.293963\pi$
$12$	1.00000	0.288675
$13$	−2.00000	−0.554700	−0.277350	−	0.960769i	$-0.589456\pi$
$13$	−2.00000	−0.554700	−0.277350	+	0.960769i	$0.589456\pi$
$14$	0	0
$15$	1.00000	0.258199
$16$	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
$19$	4.00000	0.917663	0.458831	−	0.888523i	$-0.348268\pi$
$19$	4.00000	0.917663	0.458831	+	0.888523i	$0.348268\pi$
$20$	1.00000	0.223607
$21$	0	0
$22$	4.00000	0.852803
$23$	−1.00000	−0.208514
$23$	−1.00000	−0.208514
$24$	1.00000	0.204124
$25$	1.00000	0.200000
$26$	−2.00000	−0.392232
$27$	1.00000	0.192450
$28$	0	0
$29$	−2.00000	−0.371391	−0.185695	−	0.982607i	$-0.559454\pi$
$29$	−2.00000	−0.371391	−0.185695	+	0.982607i	$0.559454\pi$
$30$	1.00000	0.182574
$31$	0	0		−	1.00000i	$-0.5\pi$
$31$	0	0			1.00000i	$0.5\pi$
$32$	1.00000	0.176777
$33$	4.00000	0.696311
$34$	−6.00000	−1.02899
$35$	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	$0.552568\pi$
$38$	4.00000	0.648886
$39$	−2.00000	−0.320256
$40$	1.00000	0.158114
$41$	10.0000	1.56174	0.780869	−	0.624695i	$-0.214777\pi$
$41$	10.0000	1.56174	0.780869	+	0.624695i	$0.214777\pi$
$42$	0	0
$43$	−4.00000	−0.609994	−0.304997	−	0.952353i	$-0.598656\pi$
$43$	−4.00000	−0.609994	−0.304997	+	0.952353i	$0.598656\pi$
$44$	4.00000	0.603023
$45$	1.00000	0.149071
$46$	−1.00000	−0.147442
$47$	0	0		−	1.00000i	$-0.5\pi$
$47$	0	0			1.00000i	$0.5\pi$
$48$	1.00000	0.144338
$49$	−7.00000	−1.00000
$50$	1.00000	0.141421
$51$	−6.00000	−0.840168
$52$	−2.00000	−0.277350
$53$	6.00000	0.824163	0.412082	−	0.911147i	$-0.364802\pi$
$53$	6.00000	0.824163	0.412082	+	0.911147i	$0.364802\pi$
$54$	1.00000	0.136083
$55$	4.00000	0.539360
$56$	0	0
$57$	4.00000	0.529813
$58$	−2.00000	−0.262613
$59$	−4.00000	−0.520756	−0.260378	−	0.965507i	$-0.583847\pi$
$59$	−4.00000	−0.520756	−0.260378	+	0.965507i	$0.583847\pi$
$60$	1.00000	0.129099
$61$	−10.0000	−1.28037	−0.640184	−	0.768221i	$-0.721142\pi$
$61$	−10.0000	−1.28037	−0.640184	+	0.768221i	$0.721142\pi$
$62$	0	0
$63$	0	0
$64$	1.00000	0.125000
$65$	−2.00000	−0.248069
$66$	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	$0.761888\pi$
$68$	−6.00000	−0.727607
$69$	−1.00000	−0.120386
$70$	0	0
$71$	−8.00000	−0.949425	−0.474713	−	0.880141i	$-0.657448\pi$
$71$	−8.00000	−0.949425	−0.474713	+	0.880141i	$0.657448\pi$
$72$	1.00000	0.117851
$73$	10.0000	1.17041	0.585206	−	0.810885i	$-0.301014\pi$
$73$	10.0000	1.17041	0.585206	+	0.810885i	$0.301014\pi$
$74$	−2.00000	−0.232495
$75$	1.00000	0.115470
$76$	4.00000	0.458831
$77$	0	0
$78$	−2.00000	−0.226455
$79$	−8.00000	−0.900070	−0.450035	−	0.893011i	$-0.648589\pi$
$79$	−8.00000	−0.900070	−0.450035	+	0.893011i	$0.648589\pi$
$80$	1.00000	0.111803
$81$	1.00000	0.111111
$82$	10.0000	1.10432
$83$	−4.00000	−0.439057	−0.219529	−	0.975606i	$-0.570452\pi$
$83$	−4.00000	−0.439057	−0.219529	+	0.975606i	$0.570452\pi$
$84$	0	0
$85$	−6.00000	−0.650791
$86$	−4.00000	−0.431331
$87$	−2.00000	−0.214423
$88$	4.00000	0.426401
$89$	18.0000	1.90800	0.953998	−	0.299813i	$-0.0969242\pi$
$89$	18.0000	1.90800	0.953998	+	0.299813i	$0.0969242\pi$
$90$	1.00000	0.105409
$91$	0	0
$92$	−1.00000	−0.104257
$93$	0	0
$94$	0	0
$95$	4.00000	0.410391
$96$	1.00000	0.102062
$97$	2.00000	0.203069	0.101535	−	0.994832i	$-0.467625\pi$
$97$	2.00000	0.203069	0.101535	+	0.994832i	$0.467625\pi$
$98$	−7.00000	−0.707107
$99$	4.00000	0.402015
$100$	1.00000	0.100000
$101$	6.00000	0.597022	0.298511	−	0.954406i	$-0.403510\pi$
$101$	6.00000	0.597022	0.298511	+	0.954406i	$0.403510\pi$
$102$	−6.00000	−0.594089
$103$	0	0		−	1.00000i	$-0.5\pi$
$103$	0	0			1.00000i	$0.5\pi$
$104$	−2.00000	−0.196116
$105$	0	0
$106$	6.00000	0.582772
$107$	−12.0000	−1.16008	−0.580042	−	0.814587i	$-0.696964\pi$
$107$	−12.0000	−1.16008	−0.580042	+	0.814587i	$0.696964\pi$
$108$	1.00000	0.0962250
$109$	−10.0000	−0.957826	−0.478913	−	0.877862i	$-0.658969\pi$
$109$	−10.0000	−0.957826	−0.478913	+	0.877862i	$0.658969\pi$
$110$	4.00000	0.381385
$111$	−2.00000	−0.189832
$112$	0	0
$113$	−6.00000	−0.564433	−0.282216	−	0.959351i	$-0.591070\pi$
$113$	−6.00000	−0.564433	−0.282216	+	0.959351i	$0.591070\pi$
$114$	4.00000	0.374634
$115$	−1.00000	−0.0932505
$116$	−2.00000	−0.185695
$117$	−2.00000	−0.184900
$118$	−4.00000	−0.368230
$119$	0	0
$120$	1.00000	0.0912871
$121$	5.00000	0.454545
$122$	−10.0000	−0.905357
$123$	10.0000	0.901670
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	−16.0000	−1.41977	−0.709885	−	0.704317i	$-0.751253\pi$
$127$	−16.0000	−1.41977	−0.709885	+	0.704317i	$0.751253\pi$
$128$	1.00000	0.0883883
$129$	−4.00000	−0.352180
$130$	−2.00000	−0.175412
$131$	20.0000	1.74741	0.873704	−	0.486458i	$-0.161711\pi$
$131$	20.0000	1.74741	0.873704	+	0.486458i	$0.161711\pi$
$132$	4.00000	0.348155
$133$	0	0
$134$	−12.0000	−1.03664
$135$	1.00000	0.0860663
$136$	−6.00000	−0.514496
$137$	2.00000	0.170872	0.0854358	−	0.996344i	$-0.472772\pi$
$137$	2.00000	0.170872	0.0854358	+	0.996344i	$0.472772\pi$
$138$	−1.00000	−0.0851257
$139$	−20.0000	−1.69638	−0.848189	−	0.529694i	$-0.822307\pi$
$139$	−20.0000	−1.69638	−0.848189	+	0.529694i	$0.822307\pi$
$140$	0	0
$141$	0	0
$142$	−8.00000	−0.671345
$143$	−8.00000	−0.668994
$144$	1.00000	0.0833333
$145$	−2.00000	−0.166091
$146$	10.0000	0.827606
$147$	−7.00000	−0.577350
$148$	−2.00000	−0.164399
$149$	−10.0000	−0.819232	−0.409616	−	0.912258i	$-0.634337\pi$
$149$	−10.0000	−0.819232	−0.409616	+	0.912258i	$0.634337\pi$
$150$	1.00000	0.0816497
$151$	8.00000	0.651031	0.325515	−	0.945537i	$-0.394462\pi$
$151$	8.00000	0.651031	0.325515	+	0.945537i	$0.394462\pi$
$152$	4.00000	0.324443
$153$	−6.00000	−0.485071
$154$	0	0
$155$	0	0
$156$	−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	$0.630658\pi$
$158$	−8.00000	−0.636446
$159$	6.00000	0.475831
$160$	1.00000	0.0790569
$161$	0	0
$162$	1.00000	0.0785674
$163$	20.0000	1.56652	0.783260	−	0.621694i	$-0.213555\pi$
$163$	20.0000	1.56652	0.783260	+	0.621694i	$0.213555\pi$
$164$	10.0000	0.780869
$165$	4.00000	0.311400
$166$	−4.00000	−0.310460
$167$	8.00000	0.619059	0.309529	−	0.950890i	$-0.399829\pi$
$167$	8.00000	0.619059	0.309529	+	0.950890i	$0.399829\pi$
$168$	0	0
$169$	−9.00000	−0.692308
$170$	−6.00000	−0.460179
$171$	4.00000	0.305888
$172$	−4.00000	−0.304997
$173$	14.0000	1.06440	0.532200	−	0.846619i	$-0.321365\pi$
$173$	14.0000	1.06440	0.532200	+	0.846619i	$0.321365\pi$
$174$	−2.00000	−0.151620
$175$	0	0
$176$	4.00000	0.301511
$177$	−4.00000	−0.300658
$178$	18.0000	1.34916
$179$	4.00000	0.298974	0.149487	−	0.988764i	$-0.452238\pi$
$179$	4.00000	0.298974	0.149487	+	0.988764i	$0.452238\pi$
$180$	1.00000	0.0745356
$181$	−2.00000	−0.148659	−0.0743294	−	0.997234i	$-0.523682\pi$
$181$	−2.00000	−0.148659	−0.0743294	+	0.997234i	$0.523682\pi$
$182$	0	0
$183$	−10.0000	−0.739221
$184$	−1.00000	−0.0737210
$185$	−2.00000	−0.147043
$186$	0	0
$187$	−24.0000	−1.75505
$188$	0	0
$189$	0	0
$190$	4.00000	0.290191
$191$	0	0		−	1.00000i	$-0.5\pi$
$191$	0	0			1.00000i	$0.5\pi$
$192$	1.00000	0.0721688
$193$	2.00000	0.143963	0.0719816	−	0.997406i	$-0.477068\pi$
$193$	2.00000	0.143963	0.0719816	+	0.997406i	$0.477068\pi$
$194$	2.00000	0.143592
$195$	−2.00000	−0.143223
$196$	−7.00000	−0.500000
$197$	6.00000	0.427482	0.213741	−	0.976890i	$-0.431435\pi$
$197$	6.00000	0.427482	0.213741	+	0.976890i	$0.431435\pi$
$198$	4.00000	0.284268
$199$	−16.0000	−1.13421	−0.567105	−	0.823646i	$-0.691937\pi$
$199$	−16.0000	−1.13421	−0.567105	+	0.823646i	$0.691937\pi$
$200$	1.00000	0.0707107
$201$	−12.0000	−0.846415
$202$	6.00000	0.422159
$203$	0	0
$204$	−6.00000	−0.420084
$205$	10.0000	0.698430
$206$	0	0
$207$	−1.00000	−0.0695048
$208$	−2.00000	−0.138675
$209$	16.0000	1.10674
$210$	0	0
$211$	4.00000	0.275371	0.137686	−	0.990476i	$-0.456034\pi$
$211$	4.00000	0.275371	0.137686	+	0.990476i	$0.456034\pi$
$212$	6.00000	0.412082
$213$	−8.00000	−0.548151
$214$	−12.0000	−0.820303
$215$	−4.00000	−0.272798
$216$	1.00000	0.0680414
$217$	0	0
$218$	−10.0000	−0.677285
$219$	10.0000	0.675737
$220$	4.00000	0.269680
$221$	12.0000	0.807207
$222$	−2.00000	−0.134231
$223$	16.0000	1.07144	0.535720	−	0.844396i	$-0.320040\pi$
$223$	16.0000	1.07144	0.535720	+	0.844396i	$0.320040\pi$
$224$	0	0
$225$	1.00000	0.0666667
$226$	−6.00000	−0.399114
$227$	−20.0000	−1.32745	−0.663723	−	0.747978i	$-0.731025\pi$
$227$	−20.0000	−1.32745	−0.663723	+	0.747978i	$0.731025\pi$
$228$	4.00000	0.264906
$229$	14.0000	0.925146	0.462573	−	0.886581i	$-0.346926\pi$
$229$	14.0000	0.925146	0.462573	+	0.886581i	$0.346926\pi$
$230$	−1.00000	−0.0659380
$231$	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	$0.562969\pi$
$234$	−2.00000	−0.130744
$235$	0	0
$236$	−4.00000	−0.260378
$237$	−8.00000	−0.519656
$238$	0	0
$239$	16.0000	1.03495	0.517477	−	0.855697i	$-0.326871\pi$
$239$	16.0000	1.03495	0.517477	+	0.855697i	$0.326871\pi$
$240$	1.00000	0.0645497
$241$	18.0000	1.15948	0.579741	−	0.814801i	$-0.303154\pi$
$241$	18.0000	1.15948	0.579741	+	0.814801i	$0.303154\pi$
$242$	5.00000	0.321412
$243$	1.00000	0.0641500
$244$	−10.0000	−0.640184
$245$	−7.00000	−0.447214
$246$	10.0000	0.637577
$247$	−8.00000	−0.509028
$248$	0	0
$249$	−4.00000	−0.253490
$250$	1.00000	0.0632456
$251$	20.0000	1.26239	0.631194	−	0.775625i	$-0.282565\pi$
$251$	20.0000	1.26239	0.631194	+	0.775625i	$0.282565\pi$
$252$	0	0
$253$	−4.00000	−0.251478
$254$	−16.0000	−1.00393
$255$	−6.00000	−0.375735
$256$	1.00000	0.0625000
$257$	−14.0000	−0.873296	−0.436648	−	0.899632i	$-0.643834\pi$
$257$	−14.0000	−0.873296	−0.436648	+	0.899632i	$0.643834\pi$
$258$	−4.00000	−0.249029
$259$	0	0
$260$	−2.00000	−0.124035
$261$	−2.00000	−0.123797
$262$	20.0000	1.23560
$263$	−24.0000	−1.47990	−0.739952	−	0.672660i	$-0.765152\pi$
$263$	−24.0000	−1.47990	−0.739952	+	0.672660i	$0.765152\pi$
$264$	4.00000	0.246183
$265$	6.00000	0.368577
$266$	0	0
$267$	18.0000	1.10158
$268$	−12.0000	−0.733017
$269$	14.0000	0.853595	0.426798	−	0.904347i	$-0.359642\pi$
$269$	14.0000	0.853595	0.426798	+	0.904347i	$0.359642\pi$
$270$	1.00000	0.0608581
$271$	0	0		−	1.00000i	$-0.5\pi$
$271$	0	0			1.00000i	$0.5\pi$
$272$	−6.00000	−0.363803
$273$	0	0
$274$	2.00000	0.120824
$275$	4.00000	0.241209
$276$	−1.00000	−0.0601929
$277$	−10.0000	−0.600842	−0.300421	−	0.953807i	$-0.597127\pi$
$277$	−10.0000	−0.600842	−0.300421	+	0.953807i	$0.597127\pi$
$278$	−20.0000	−1.19952
$279$	0	0
$280$	0	0
$281$	18.0000	1.07379	0.536895	−	0.843649i	$-0.319597\pi$
$281$	18.0000	1.07379	0.536895	+	0.843649i	$0.319597\pi$
$282$	0	0
$283$	−4.00000	−0.237775	−0.118888	−	0.992908i	$-0.537933\pi$
$283$	−4.00000	−0.237775	−0.118888	+	0.992908i	$0.537933\pi$
$284$	−8.00000	−0.474713
$285$	4.00000	0.236940
$286$	−8.00000	−0.473050
$287$	0	0
$288$	1.00000	0.0589256
$289$	19.0000	1.11765
$290$	−2.00000	−0.117444
$291$	2.00000	0.117242
$292$	10.0000	0.585206
$293$	22.0000	1.28525	0.642627	−	0.766179i	$-0.277845\pi$
$293$	22.0000	1.28525	0.642627	+	0.766179i	$0.277845\pi$
$294$	−7.00000	−0.408248
$295$	−4.00000	−0.232889
$296$	−2.00000	−0.116248
$297$	4.00000	0.232104
$298$	−10.0000	−0.579284
$299$	2.00000	0.115663
$300$	1.00000	0.0577350
$301$	0	0
$302$	8.00000	0.460348
$303$	6.00000	0.344691
$304$	4.00000	0.229416
$305$	−10.0000	−0.572598
$306$	−6.00000	−0.342997
$307$	4.00000	0.228292	0.114146	−	0.993464i	$-0.463587\pi$
$307$	4.00000	0.228292	0.114146	+	0.993464i	$0.463587\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	8.00000	0.453638	0.226819	−	0.973937i	$-0.427167\pi$
$311$	8.00000	0.453638	0.226819	+	0.973937i	$0.427167\pi$
$312$	−2.00000	−0.113228
$313$	10.0000	0.565233	0.282617	−	0.959233i	$-0.408798\pi$
$313$	10.0000	0.565233	0.282617	+	0.959233i	$0.408798\pi$
$314$	−10.0000	−0.564333
$315$	0	0
$316$	−8.00000	−0.450035
$317$	30.0000	1.68497	0.842484	−	0.538721i	$-0.181092\pi$
$317$	30.0000	1.68497	0.842484	+	0.538721i	$0.181092\pi$
$318$	6.00000	0.336463
$319$	−8.00000	−0.447914
$320$	1.00000	0.0559017
$321$	−12.0000	−0.669775
$322$	0	0
$323$	−24.0000	−1.33540
$324$	1.00000	0.0555556
$325$	−2.00000	−0.110940
$326$	20.0000	1.10770
$327$	−10.0000	−0.553001
$328$	10.0000	0.552158
$329$	0	0
$330$	4.00000	0.220193
$331$	28.0000	1.53902	0.769510	−	0.638635i	$-0.220501\pi$
$331$	28.0000	1.53902	0.769510	+	0.638635i	$0.220501\pi$
$332$	−4.00000	−0.219529
$333$	−2.00000	−0.109599
$334$	8.00000	0.437741
$335$	−12.0000	−0.655630
$336$	0	0
$337$	2.00000	0.108947	0.0544735	−	0.998515i	$-0.482652\pi$
$337$	2.00000	0.108947	0.0544735	+	0.998515i	$0.482652\pi$
$338$	−9.00000	−0.489535
$339$	−6.00000	−0.325875
$340$	−6.00000	−0.325396
$341$	0	0
$342$	4.00000	0.216295
$343$	0	0
$344$	−4.00000	−0.215666
$345$	−1.00000	−0.0538382
$346$	14.0000	0.752645
$347$	12.0000	0.644194	0.322097	−	0.946707i	$-0.395612\pi$
$347$	12.0000	0.644194	0.322097	+	0.946707i	$0.395612\pi$
$348$	−2.00000	−0.107211
$349$	14.0000	0.749403	0.374701	−	0.927146i	$-0.377745\pi$
$349$	14.0000	0.749403	0.374701	+	0.927146i	$0.377745\pi$
$350$	0	0
$351$	−2.00000	−0.106752
$352$	4.00000	0.213201
$353$	34.0000	1.80964	0.904819	−	0.425797i	$-0.140006\pi$
$353$	34.0000	1.80964	0.904819	+	0.425797i	$0.140006\pi$
$354$	−4.00000	−0.212598
$355$	−8.00000	−0.424596
$356$	18.0000	0.953998
$357$	0	0
$358$	4.00000	0.211407
$359$	8.00000	0.422224	0.211112	−	0.977462i	$-0.432292\pi$
$359$	8.00000	0.422224	0.211112	+	0.977462i	$0.432292\pi$
$360$	1.00000	0.0527046
$361$	−3.00000	−0.157895
$362$	−2.00000	−0.105118
$363$	5.00000	0.262432
$364$	0	0
$365$	10.0000	0.523424
$366$	−10.0000	−0.522708
$367$	−8.00000	−0.417597	−0.208798	−	0.977959i	$-0.566955\pi$
$367$	−8.00000	−0.417597	−0.208798	+	0.977959i	$0.566955\pi$
$368$	−1.00000	−0.0521286
$369$	10.0000	0.520579
$370$	−2.00000	−0.103975
$371$	0	0
$372$	0	0
$373$	−18.0000	−0.932005	−0.466002	−	0.884783i	$-0.654306\pi$
$373$	−18.0000	−0.932005	−0.466002	+	0.884783i	$0.654306\pi$
$374$	−24.0000	−1.24101
$375$	1.00000	0.0516398
$376$	0	0
$377$	4.00000	0.206010
$378$	0	0
$379$	12.0000	0.616399	0.308199	−	0.951322i	$-0.400274\pi$
$379$	12.0000	0.616399	0.308199	+	0.951322i	$0.400274\pi$
$380$	4.00000	0.205196
$381$	−16.0000	−0.819705
$382$	0	0
$383$	−32.0000	−1.63512	−0.817562	−	0.575841i	$-0.804675\pi$
$383$	−32.0000	−1.63512	−0.817562	+	0.575841i	$0.804675\pi$
$384$	1.00000	0.0510310
$385$	0	0
$386$	2.00000	0.101797
$387$	−4.00000	−0.203331
$388$	2.00000	0.101535
$389$	38.0000	1.92668	0.963338	−	0.268290i	$-0.0864585\pi$
$389$	38.0000	1.92668	0.963338	+	0.268290i	$0.0864585\pi$
$390$	−2.00000	−0.101274
$391$	6.00000	0.303433
$392$	−7.00000	−0.353553
$393$	20.0000	1.00887
$394$	6.00000	0.302276
$395$	−8.00000	−0.402524
$396$	4.00000	0.201008
$397$	−2.00000	−0.100377	−0.0501886	−	0.998740i	$-0.515982\pi$
$397$	−2.00000	−0.100377	−0.0501886	+	0.998740i	$0.515982\pi$
$398$	−16.0000	−0.802008
$399$	0	0
$400$	1.00000	0.0500000
$401$	10.0000	0.499376	0.249688	−	0.968326i	$-0.419672\pi$
$401$	10.0000	0.499376	0.249688	+	0.968326i	$0.419672\pi$
$402$	−12.0000	−0.598506
$403$	0	0
$404$	6.00000	0.298511
$405$	1.00000	0.0496904
$406$	0	0
$407$	−8.00000	−0.396545
$408$	−6.00000	−0.297044
$409$	26.0000	1.28562	0.642809	−	0.766027i	$-0.277769\pi$
$409$	26.0000	1.28562	0.642809	+	0.766027i	$0.277769\pi$
$410$	10.0000	0.493865
$411$	2.00000	0.0986527
$412$	0	0
$413$	0	0
$414$	−1.00000	−0.0491473
$415$	−4.00000	−0.196352
$416$	−2.00000	−0.0980581
$417$	−20.0000	−0.979404
$418$	16.0000	0.782586
$419$	12.0000	0.586238	0.293119	−	0.956076i	$-0.405307\pi$
$419$	12.0000	0.586238	0.293119	+	0.956076i	$0.405307\pi$
$420$	0	0
$421$	−2.00000	−0.0974740	−0.0487370	−	0.998812i	$-0.515520\pi$
$421$	−2.00000	−0.0974740	−0.0487370	+	0.998812i	$0.515520\pi$
$422$	4.00000	0.194717
$423$	0	0
$424$	6.00000	0.291386
$425$	−6.00000	−0.291043
$426$	−8.00000	−0.387601
$427$	0	0
$428$	−12.0000	−0.580042
$429$	−8.00000	−0.386244
$430$	−4.00000	−0.192897
$431$	−32.0000	−1.54139	−0.770693	−	0.637207i	$-0.780090\pi$
$431$	−32.0000	−1.54139	−0.770693	+	0.637207i	$0.780090\pi$
$432$	1.00000	0.0481125
$433$	34.0000	1.63394	0.816968	−	0.576683i	$-0.195653\pi$
$433$	34.0000	1.63394	0.816968	+	0.576683i	$0.195653\pi$
$434$	0	0
$435$	−2.00000	−0.0958927
$436$	−10.0000	−0.478913
$437$	−4.00000	−0.191346
$438$	10.0000	0.477818
$439$	−8.00000	−0.381819	−0.190910	−	0.981608i	$-0.561144\pi$
$439$	−8.00000	−0.381819	−0.190910	+	0.981608i	$0.561144\pi$
$440$	4.00000	0.190693
$441$	−7.00000	−0.333333
$442$	12.0000	0.570782
$443$	−4.00000	−0.190046	−0.0950229	−	0.995475i	$-0.530292\pi$
$443$	−4.00000	−0.190046	−0.0950229	+	0.995475i	$0.530292\pi$
$444$	−2.00000	−0.0949158
$445$	18.0000	0.853282
$446$	16.0000	0.757622
$447$	−10.0000	−0.472984
$448$	0	0
$449$	−30.0000	−1.41579	−0.707894	−	0.706319i	$-0.750354\pi$
$449$	−30.0000	−1.41579	−0.707894	+	0.706319i	$0.750354\pi$
$450$	1.00000	0.0471405
$451$	40.0000	1.88353
$452$	−6.00000	−0.282216
$453$	8.00000	0.375873
$454$	−20.0000	−0.938647
$455$	0	0
$456$	4.00000	0.187317
$457$	10.0000	0.467780	0.233890	−	0.972263i	$-0.424854\pi$
$457$	10.0000	0.467780	0.233890	+	0.972263i	$0.424854\pi$
$458$	14.0000	0.654177
$459$	−6.00000	−0.280056
$460$	−1.00000	−0.0466252
$461$	−18.0000	−0.838344	−0.419172	−	0.907907i	$-0.637680\pi$
$461$	−18.0000	−0.838344	−0.419172	+	0.907907i	$0.637680\pi$
$462$	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
$465$	0	0
$466$	−6.00000	−0.277945
$467$	12.0000	0.555294	0.277647	−	0.960683i	$-0.410445\pi$
$467$	12.0000	0.555294	0.277647	+	0.960683i	$0.410445\pi$
$468$	−2.00000	−0.0924500
$469$	0	0
$470$	0	0
$471$	−10.0000	−0.460776
$472$	−4.00000	−0.184115
$473$	−16.0000	−0.735681
$474$	−8.00000	−0.367452
$475$	4.00000	0.183533
$476$	0	0
$477$	6.00000	0.274721
$478$	16.0000	0.731823
$479$	32.0000	1.46212	0.731059	−	0.682315i	$-0.239027\pi$
$479$	32.0000	1.46212	0.731059	+	0.682315i	$0.239027\pi$
$480$	1.00000	0.0456435
$481$	4.00000	0.182384
$482$	18.0000	0.819878
$483$	0	0
$484$	5.00000	0.227273
$485$	2.00000	0.0908153
$486$	1.00000	0.0453609
$487$	40.0000	1.81257	0.906287	−	0.422664i	$-0.138905\pi$
$487$	40.0000	1.81257	0.906287	+	0.422664i	$0.138905\pi$
$488$	−10.0000	−0.452679
$489$	20.0000	0.904431
$490$	−7.00000	−0.316228
$491$	−4.00000	−0.180517	−0.0902587	−	0.995918i	$-0.528769\pi$
$491$	−4.00000	−0.180517	−0.0902587	+	0.995918i	$0.528769\pi$
$492$	10.0000	0.450835
$493$	12.0000	0.540453
$494$	−8.00000	−0.359937
$495$	4.00000	0.179787
$496$	0	0
$497$	0	0
$498$	−4.00000	−0.179244
$499$	20.0000	0.895323	0.447661	−	0.894203i	$-0.352257\pi$
$499$	20.0000	0.895323	0.447661	+	0.894203i	$0.352257\pi$
$500$	1.00000	0.0447214
$501$	8.00000	0.357414
$502$	20.0000	0.892644
$503$	−40.0000	−1.78351	−0.891756	−	0.452517i	$-0.850526\pi$
$503$	−40.0000	−1.78351	−0.891756	+	0.452517i	$0.850526\pi$
$504$	0	0
$505$	6.00000	0.266996
$506$	−4.00000	−0.177822
$507$	−9.00000	−0.399704
$508$	−16.0000	−0.709885
$509$	−34.0000	−1.50702	−0.753512	−	0.657434i	$-0.771642\pi$
$509$	−34.0000	−1.50702	−0.753512	+	0.657434i	$0.771642\pi$
$510$	−6.00000	−0.265684
$511$	0	0
$512$	1.00000	0.0441942
$513$	4.00000	0.176604
$514$	−14.0000	−0.617514
$515$	0	0
$516$	−4.00000	−0.176090
$517$	0	0
$518$	0	0
$519$	14.0000	0.614532
$520$	−2.00000	−0.0877058
$521$	−14.0000	−0.613351	−0.306676	−	0.951814i	$-0.599217\pi$
$521$	−14.0000	−0.613351	−0.306676	+	0.951814i	$0.599217\pi$
$522$	−2.00000	−0.0875376
$523$	−36.0000	−1.57417	−0.787085	−	0.616844i	$-0.788411\pi$
$523$	−36.0000	−1.57417	−0.787085	+	0.616844i	$0.788411\pi$
$524$	20.0000	0.873704
$525$	0	0
$526$	−24.0000	−1.04645
$527$	0	0
$528$	4.00000	0.174078
$529$	1.00000	0.0434783
$530$	6.00000	0.260623
$531$	−4.00000	−0.173585
$532$	0	0
$533$	−20.0000	−0.866296
$534$	18.0000	0.778936
$535$	−12.0000	−0.518805
$536$	−12.0000	−0.518321
$537$	4.00000	0.172613
$538$	14.0000	0.603583
$539$	−28.0000	−1.20605
$540$	1.00000	0.0430331
$541$	14.0000	0.601907	0.300954	−	0.953639i	$-0.402695\pi$
$541$	14.0000	0.601907	0.300954	+	0.953639i	$0.402695\pi$
$542$	0	0
$543$	−2.00000	−0.0858282
$544$	−6.00000	−0.257248
$545$	−10.0000	−0.428353
$546$	0	0
$547$	4.00000	0.171028	0.0855138	−	0.996337i	$-0.472747\pi$
$547$	4.00000	0.171028	0.0855138	+	0.996337i	$0.472747\pi$
$548$	2.00000	0.0854358
$549$	−10.0000	−0.426790
$550$	4.00000	0.170561
$551$	−8.00000	−0.340811
$552$	−1.00000	−0.0425628
$553$	0	0
$554$	−10.0000	−0.424859
$555$	−2.00000	−0.0848953
$556$	−20.0000	−0.848189
$557$	14.0000	0.593199	0.296600	−	0.955002i	$-0.404147\pi$
$557$	14.0000	0.593199	0.296600	+	0.955002i	$0.404147\pi$
$558$	0	0
$559$	8.00000	0.338364
$560$	0	0
$561$	−24.0000	−1.01328
$562$	18.0000	0.759284
$563$	−36.0000	−1.51722	−0.758610	−	0.651546i	$-0.774121\pi$
$563$	−36.0000	−1.51722	−0.758610	+	0.651546i	$0.774121\pi$
$564$	0	0
$565$	−6.00000	−0.252422
$566$	−4.00000	−0.168133
$567$	0	0
$568$	−8.00000	−0.335673
$569$	18.0000	0.754599	0.377300	−	0.926091i	$-0.376853\pi$
$569$	18.0000	0.754599	0.377300	+	0.926091i	$0.376853\pi$
$570$	4.00000	0.167542
$571$	−4.00000	−0.167395	−0.0836974	−	0.996491i	$-0.526673\pi$
$571$	−4.00000	−0.167395	−0.0836974	+	0.996491i	$0.526673\pi$
$572$	−8.00000	−0.334497
$573$	0	0
$574$	0	0
$575$	−1.00000	−0.0417029
$576$	1.00000	0.0416667
$577$	2.00000	0.0832611	0.0416305	−	0.999133i	$-0.486745\pi$
$577$	2.00000	0.0832611	0.0416305	+	0.999133i	$0.486745\pi$
$578$	19.0000	0.790296
$579$	2.00000	0.0831172
$580$	−2.00000	−0.0830455
$581$	0	0
$582$	2.00000	0.0829027
$583$	24.0000	0.993978
$584$	10.0000	0.413803
$585$	−2.00000	−0.0826898
$586$	22.0000	0.908812
$587$	28.0000	1.15568	0.577842	−	0.816149i	$-0.303895\pi$
$587$	28.0000	1.15568	0.577842	+	0.816149i	$0.303895\pi$
$588$	−7.00000	−0.288675
$589$	0	0
$590$	−4.00000	−0.164677
$591$	6.00000	0.246807
$592$	−2.00000	−0.0821995
$593$	−30.0000	−1.23195	−0.615976	−	0.787765i	$-0.711238\pi$
$593$	−30.0000	−1.23195	−0.615976	+	0.787765i	$0.711238\pi$
$594$	4.00000	0.164122
$595$	0	0
$596$	−10.0000	−0.409616
$597$	−16.0000	−0.654836
$598$	2.00000	0.0817861
$599$	−40.0000	−1.63436	−0.817178	−	0.576386i	$-0.804463\pi$
$599$	−40.0000	−1.63436	−0.817178	+	0.576386i	$0.804463\pi$
$600$	1.00000	0.0408248
$601$	−38.0000	−1.55005	−0.775026	−	0.631929i	$-0.782263\pi$
$601$	−38.0000	−1.55005	−0.775026	+	0.631929i	$0.782263\pi$
$602$	0	0
$603$	−12.0000	−0.488678
$604$	8.00000	0.325515
$605$	5.00000	0.203279
$606$	6.00000	0.243733
$607$	−16.0000	−0.649420	−0.324710	−	0.945814i	$-0.605267\pi$
$607$	−16.0000	−0.649420	−0.324710	+	0.945814i	$0.605267\pi$
$608$	4.00000	0.162221
$609$	0	0
$610$	−10.0000	−0.404888
$611$	0	0
$612$	−6.00000	−0.242536
$613$	−18.0000	−0.727013	−0.363507	−	0.931592i	$-0.618421\pi$
$613$	−18.0000	−0.727013	−0.363507	+	0.931592i	$0.618421\pi$
$614$	4.00000	0.161427
$615$	10.0000	0.403239
$616$	0	0
$617$	18.0000	0.724653	0.362326	−	0.932051i	$-0.381983\pi$
$617$	18.0000	0.724653	0.362326	+	0.932051i	$0.381983\pi$
$618$	0	0
$619$	−4.00000	−0.160774	−0.0803868	−	0.996764i	$-0.525616\pi$
$619$	−4.00000	−0.160774	−0.0803868	+	0.996764i	$0.525616\pi$
$620$	0	0
$621$	−1.00000	−0.0401286
$622$	8.00000	0.320771
$623$	0	0
$624$	−2.00000	−0.0800641
$625$	1.00000	0.0400000
$626$	10.0000	0.399680
$627$	16.0000	0.638978
$628$	−10.0000	−0.399043
$629$	12.0000	0.478471
$630$	0	0
$631$	16.0000	0.636950	0.318475	−	0.947931i	$-0.396829\pi$
$631$	16.0000	0.636950	0.318475	+	0.947931i	$0.396829\pi$
$632$	−8.00000	−0.318223
$633$	4.00000	0.158986
$634$	30.0000	1.19145
$635$	−16.0000	−0.634941
$636$	6.00000	0.237915
$637$	14.0000	0.554700
$638$	−8.00000	−0.316723
$639$	−8.00000	−0.316475
$640$	1.00000	0.0395285
$641$	26.0000	1.02694	0.513469	−	0.858108i	$-0.328360\pi$
$641$	26.0000	1.02694	0.513469	+	0.858108i	$0.328360\pi$
$642$	−12.0000	−0.473602
$643$	20.0000	0.788723	0.394362	−	0.918955i	$-0.370966\pi$
$643$	20.0000	0.788723	0.394362	+	0.918955i	$0.370966\pi$
$644$	0	0
$645$	−4.00000	−0.157500
$646$	−24.0000	−0.944267
$647$	8.00000	0.314512	0.157256	−	0.987558i	$-0.449735\pi$
$647$	8.00000	0.314512	0.157256	+	0.987558i	$0.449735\pi$
$648$	1.00000	0.0392837
$649$	−16.0000	−0.628055
$650$	−2.00000	−0.0784465
$651$	0	0
$652$	20.0000	0.783260
$653$	14.0000	0.547862	0.273931	−	0.961749i	$-0.411676\pi$
$653$	14.0000	0.547862	0.273931	+	0.961749i	$0.411676\pi$
$654$	−10.0000	−0.391031
$655$	20.0000	0.781465
$656$	10.0000	0.390434
$657$	10.0000	0.390137
$658$	0	0
$659$	−20.0000	−0.779089	−0.389545	−	0.921008i	$-0.627368\pi$
$659$	−20.0000	−0.779089	−0.389545	+	0.921008i	$0.627368\pi$
$660$	4.00000	0.155700
$661$	−50.0000	−1.94477	−0.972387	−	0.233373i	$-0.925024\pi$
$661$	−50.0000	−1.94477	−0.972387	+	0.233373i	$0.925024\pi$
$662$	28.0000	1.08825
$663$	12.0000	0.466041
$664$	−4.00000	−0.155230
$665$	0	0
$666$	−2.00000	−0.0774984
$667$	2.00000	0.0774403
$668$	8.00000	0.309529
$669$	16.0000	0.618596
$670$	−12.0000	−0.463600
$671$	−40.0000	−1.54418
$672$	0	0
$673$	2.00000	0.0770943	0.0385472	−	0.999257i	$-0.487727\pi$
$673$	2.00000	0.0770943	0.0385472	+	0.999257i	$0.487727\pi$
$674$	2.00000	0.0770371
$675$	1.00000	0.0384900
$676$	−9.00000	−0.346154
$677$	38.0000	1.46046	0.730229	−	0.683202i	$-0.239413\pi$
$677$	38.0000	1.46046	0.730229	+	0.683202i	$0.239413\pi$
$678$	−6.00000	−0.230429
$679$	0	0
$680$	−6.00000	−0.230089
$681$	−20.0000	−0.766402
$682$	0	0
$683$	−36.0000	−1.37750	−0.688751	−	0.724998i	$-0.741841\pi$
$683$	−36.0000	−1.37750	−0.688751	+	0.724998i	$0.741841\pi$
$684$	4.00000	0.152944
$685$	2.00000	0.0764161
$686$	0	0
$687$	14.0000	0.534133
$688$	−4.00000	−0.152499
$689$	−12.0000	−0.457164
$690$	−1.00000	−0.0380693
$691$	−28.0000	−1.06517	−0.532585	−	0.846376i	$-0.678779\pi$
$691$	−28.0000	−1.06517	−0.532585	+	0.846376i	$0.678779\pi$
$692$	14.0000	0.532200
$693$	0	0
$694$	12.0000	0.455514
$695$	−20.0000	−0.758643
$696$	−2.00000	−0.0758098
$697$	−60.0000	−2.27266
$698$	14.0000	0.529908
$699$	−6.00000	−0.226941
$700$	0	0
$701$	−50.0000	−1.88847	−0.944237	−	0.329267i	$-0.893198\pi$
$701$	−50.0000	−1.88847	−0.944237	+	0.329267i	$0.893198\pi$
$702$	−2.00000	−0.0754851
$703$	−8.00000	−0.301726
$704$	4.00000	0.150756
$705$	0	0
$706$	34.0000	1.27961
$707$	0	0
$708$	−4.00000	−0.150329
$709$	−34.0000	−1.27690	−0.638448	−	0.769665i	$-0.720423\pi$
$709$	−34.0000	−1.27690	−0.638448	+	0.769665i	$0.720423\pi$
$710$	−8.00000	−0.300235
$711$	−8.00000	−0.300023
$712$	18.0000	0.674579
$713$	0	0
$714$	0	0
$715$	−8.00000	−0.299183
$716$	4.00000	0.149487
$717$	16.0000	0.597531
$718$	8.00000	0.298557
$719$	−32.0000	−1.19340	−0.596699	−	0.802465i	$-0.703521\pi$
$719$	−32.0000	−1.19340	−0.596699	+	0.802465i	$0.703521\pi$
$720$	1.00000	0.0372678
$721$	0	0
$722$	−3.00000	−0.111648
$723$	18.0000	0.669427
$724$	−2.00000	−0.0743294
$725$	−2.00000	−0.0742781
$726$	5.00000	0.185567
$727$	16.0000	0.593407	0.296704	−	0.954970i	$-0.404113\pi$
$727$	16.0000	0.593407	0.296704	+	0.954970i	$0.404113\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	10.0000	0.370117
$731$	24.0000	0.887672
$732$	−10.0000	−0.369611
$733$	6.00000	0.221615	0.110808	−	0.993842i	$-0.464656\pi$
$733$	6.00000	0.221615	0.110808	+	0.993842i	$0.464656\pi$
$734$	−8.00000	−0.295285
$735$	−7.00000	−0.258199
$736$	−1.00000	−0.0368605
$737$	−48.0000	−1.76810
$738$	10.0000	0.368105
$739$	−12.0000	−0.441427	−0.220714	−	0.975339i	$-0.570839\pi$
$739$	−12.0000	−0.441427	−0.220714	+	0.975339i	$0.570839\pi$
$740$	−2.00000	−0.0735215
$741$	−8.00000	−0.293887
$742$	0	0
$743$	−24.0000	−0.880475	−0.440237	−	0.897881i	$-0.645106\pi$
$743$	−24.0000	−0.880475	−0.440237	+	0.897881i	$0.645106\pi$
$744$	0	0
$745$	−10.0000	−0.366372
$746$	−18.0000	−0.659027
$747$	−4.00000	−0.146352
$748$	−24.0000	−0.877527
$749$	0	0
$750$	1.00000	0.0365148
$751$	−40.0000	−1.45962	−0.729810	−	0.683650i	$-0.760392\pi$
$751$	−40.0000	−1.45962	−0.729810	+	0.683650i	$0.760392\pi$
$752$	0	0
$753$	20.0000	0.728841
$754$	4.00000	0.145671
$755$	8.00000	0.291150
$756$	0	0
$757$	−18.0000	−0.654221	−0.327111	−	0.944986i	$-0.606075\pi$
$757$	−18.0000	−0.654221	−0.327111	+	0.944986i	$0.606075\pi$
$758$	12.0000	0.435860
$759$	−4.00000	−0.145191
$760$	4.00000	0.145095
$761$	−22.0000	−0.797499	−0.398750	−	0.917060i	$-0.630556\pi$
$761$	−22.0000	−0.797499	−0.398750	+	0.917060i	$0.630556\pi$
$762$	−16.0000	−0.579619
$763$	0	0
$764$	0	0
$765$	−6.00000	−0.216930
$766$	−32.0000	−1.15621
$767$	8.00000	0.288863
$768$	1.00000	0.0360844
$769$	−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
$771$	−14.0000	−0.504198
$772$	2.00000	0.0719816
$773$	54.0000	1.94225	0.971123	−	0.238581i	$-0.0766824\pi$
$773$	54.0000	1.94225	0.971123	+	0.238581i	$0.0766824\pi$
$774$	−4.00000	−0.143777
$775$	0	0
$776$	2.00000	0.0717958
$777$	0	0
$778$	38.0000	1.36237
$779$	40.0000	1.43315
$780$	−2.00000	−0.0716115
$781$	−32.0000	−1.14505
$782$	6.00000	0.214560
$783$	−2.00000	−0.0714742
$784$	−7.00000	−0.250000
$785$	−10.0000	−0.356915
$786$	20.0000	0.713376
$787$	4.00000	0.142585	0.0712923	−	0.997455i	$-0.477288\pi$
$787$	4.00000	0.142585	0.0712923	+	0.997455i	$0.477288\pi$
$788$	6.00000	0.213741
$789$	−24.0000	−0.854423
$790$	−8.00000	−0.284627
$791$	0	0
$792$	4.00000	0.142134
$793$	20.0000	0.710221
$794$	−2.00000	−0.0709773
$795$	6.00000	0.212798
$796$	−16.0000	−0.567105
$797$	−34.0000	−1.20434	−0.602171	−	0.798367i	$-0.705697\pi$
$797$	−34.0000	−1.20434	−0.602171	+	0.798367i	$0.705697\pi$
$798$	0	0
$799$	0	0
$800$	1.00000	0.0353553
$801$	18.0000	0.635999
$802$	10.0000	0.353112
$803$	40.0000	1.41157
$804$	−12.0000	−0.423207
$805$	0	0
$806$	0	0
$807$	14.0000	0.492823
$808$	6.00000	0.211079
$809$	26.0000	0.914111	0.457056	−	0.889438i	$-0.348904\pi$
$809$	26.0000	0.914111	0.457056	+	0.889438i	$0.348904\pi$
$810$	1.00000	0.0351364
$811$	28.0000	0.983213	0.491606	−	0.870817i	$-0.336410\pi$
$811$	28.0000	0.983213	0.491606	+	0.870817i	$0.336410\pi$
$812$	0	0
$813$	0	0
$814$	−8.00000	−0.280400
$815$	20.0000	0.700569
$816$	−6.00000	−0.210042
$817$	−16.0000	−0.559769
$818$	26.0000	0.909069
$819$	0	0
$820$	10.0000	0.349215
$821$	−42.0000	−1.46581	−0.732905	−	0.680331i	$-0.761836\pi$
$821$	−42.0000	−1.46581	−0.732905	+	0.680331i	$0.761836\pi$
$822$	2.00000	0.0697580
$823$	−40.0000	−1.39431	−0.697156	−	0.716919i	$-0.745552\pi$
$823$	−40.0000	−1.39431	−0.697156	+	0.716919i	$0.745552\pi$
$824$	0	0
$825$	4.00000	0.139262
$826$	0	0
$827$	−12.0000	−0.417281	−0.208640	−	0.977992i	$-0.566904\pi$
$827$	−12.0000	−0.417281	−0.208640	+	0.977992i	$0.566904\pi$
$828$	−1.00000	−0.0347524
$829$	46.0000	1.59765	0.798823	−	0.601566i	$-0.205456\pi$
$829$	46.0000	1.59765	0.798823	+	0.601566i	$0.205456\pi$
$830$	−4.00000	−0.138842
$831$	−10.0000	−0.346896
$832$	−2.00000	−0.0693375
$833$	42.0000	1.45521
$834$	−20.0000	−0.692543
$835$	8.00000	0.276851
$836$	16.0000	0.553372
$837$	0	0
$838$	12.0000	0.414533
$839$	24.0000	0.828572	0.414286	−	0.910147i	$-0.364031\pi$
$839$	24.0000	0.828572	0.414286	+	0.910147i	$0.364031\pi$
$840$	0	0
$841$	−25.0000	−0.862069
$842$	−2.00000	−0.0689246
$843$	18.0000	0.619953
$844$	4.00000	0.137686
$845$	−9.00000	−0.309609
$846$	0	0
$847$	0	0
$848$	6.00000	0.206041
$849$	−4.00000	−0.137280
$850$	−6.00000	−0.205798
$851$	2.00000	0.0685591
$852$	−8.00000	−0.274075
$853$	6.00000	0.205436	0.102718	−	0.994711i	$-0.467246\pi$
$853$	6.00000	0.205436	0.102718	+	0.994711i	$0.467246\pi$
$854$	0	0
$855$	4.00000	0.136797
$856$	−12.0000	−0.410152
$857$	42.0000	1.43469	0.717346	−	0.696717i	$-0.245357\pi$
$857$	42.0000	1.43469	0.717346	+	0.696717i	$0.245357\pi$
$858$	−8.00000	−0.273115
$859$	44.0000	1.50126	0.750630	−	0.660722i	$-0.229750\pi$
$859$	44.0000	1.50126	0.750630	+	0.660722i	$0.229750\pi$
$860$	−4.00000	−0.136399
$861$	0	0
$862$	−32.0000	−1.08992
$863$	−32.0000	−1.08929	−0.544646	−	0.838666i	$-0.683336\pi$
$863$	−32.0000	−1.08929	−0.544646	+	0.838666i	$0.683336\pi$
$864$	1.00000	0.0340207
$865$	14.0000	0.476014
$866$	34.0000	1.15537
$867$	19.0000	0.645274
$868$	0	0
$869$	−32.0000	−1.08553
$870$	−2.00000	−0.0678064
$871$	24.0000	0.813209
$872$	−10.0000	−0.338643
$873$	2.00000	0.0676897
$874$	−4.00000	−0.135302
$875$	0	0
$876$	10.0000	0.337869
$877$	−50.0000	−1.68838	−0.844190	−	0.536044i	$-0.819918\pi$
$877$	−50.0000	−1.68838	−0.844190	+	0.536044i	$0.819918\pi$
$878$	−8.00000	−0.269987
$879$	22.0000	0.742042
$880$	4.00000	0.134840
$881$	10.0000	0.336909	0.168454	−	0.985709i	$-0.446122\pi$
$881$	10.0000	0.336909	0.168454	+	0.985709i	$0.446122\pi$
$882$	−7.00000	−0.235702
$883$	20.0000	0.673054	0.336527	−	0.941674i	$-0.390748\pi$
$883$	20.0000	0.673054	0.336527	+	0.941674i	$0.390748\pi$
$884$	12.0000	0.403604
$885$	−4.00000	−0.134459
$886$	−4.00000	−0.134383
$887$	−24.0000	−0.805841	−0.402921	−	0.915235i	$-0.632005\pi$
$887$	−24.0000	−0.805841	−0.402921	+	0.915235i	$0.632005\pi$
$888$	−2.00000	−0.0671156
$889$	0	0
$890$	18.0000	0.603361
$891$	4.00000	0.134005
$892$	16.0000	0.535720
$893$	0	0
$894$	−10.0000	−0.334450
$895$	4.00000	0.133705
$896$	0	0
$897$	2.00000	0.0667781
$898$	−30.0000	−1.00111
$899$	0	0
$900$	1.00000	0.0333333
$901$	−36.0000	−1.19933
$902$	40.0000	1.33185
$903$	0	0
$904$	−6.00000	−0.199557
$905$	−2.00000	−0.0664822
$906$	8.00000	0.265782
$907$	44.0000	1.46100	0.730498	−	0.682915i	$-0.239288\pi$
$907$	44.0000	1.46100	0.730498	+	0.682915i	$0.239288\pi$
$908$	−20.0000	−0.663723
$909$	6.00000	0.199007
$910$	0	0
$911$	32.0000	1.06021	0.530104	−	0.847933i	$-0.322153\pi$
$911$	32.0000	1.06021	0.530104	+	0.847933i	$0.322153\pi$
$912$	4.00000	0.132453
$913$	−16.0000	−0.529523
$914$	10.0000	0.330771
$915$	−10.0000	−0.330590
$916$	14.0000	0.462573
$917$	0	0
$918$	−6.00000	−0.198030
$919$	−32.0000	−1.05558	−0.527791	−	0.849374i	$-0.676980\pi$
$919$	−32.0000	−1.05558	−0.527791	+	0.849374i	$0.676980\pi$
$920$	−1.00000	−0.0329690
$921$	4.00000	0.131804
$922$	−18.0000	−0.592798
$923$	16.0000	0.526646
$924$	0	0
$925$	−2.00000	−0.0657596
$926$	16.0000	0.525793
$927$	0	0
$928$	−2.00000	−0.0656532
$929$	−30.0000	−0.984268	−0.492134	−	0.870519i	$-0.663783\pi$
$929$	−30.0000	−0.984268	−0.492134	+	0.870519i	$0.663783\pi$
$930$	0	0
$931$	−28.0000	−0.917663
$932$	−6.00000	−0.196537
$933$	8.00000	0.261908
$934$	12.0000	0.392652
$935$	−24.0000	−0.784884
$936$	−2.00000	−0.0653720
$937$	−22.0000	−0.718709	−0.359354	−	0.933201i	$-0.617003\pi$
$937$	−22.0000	−0.718709	−0.359354	+	0.933201i	$0.617003\pi$
$938$	0	0
$939$	10.0000	0.326338
$940$	0	0
$941$	14.0000	0.456387	0.228193	−	0.973616i	$-0.426718\pi$
$941$	14.0000	0.456387	0.228193	+	0.973616i	$0.426718\pi$
$942$	−10.0000	−0.325818
$943$	−10.0000	−0.325645
$944$	−4.00000	−0.130189
$945$	0	0
$946$	−16.0000	−0.520205
$947$	4.00000	0.129983	0.0649913	−	0.997886i	$-0.479298\pi$
$947$	4.00000	0.129983	0.0649913	+	0.997886i	$0.479298\pi$
$948$	−8.00000	−0.259828
$949$	−20.0000	−0.649227
$950$	4.00000	0.129777
$951$	30.0000	0.972817
$952$	0	0
$953$	2.00000	0.0647864	0.0323932	−	0.999475i	$-0.489687\pi$
$953$	2.00000	0.0647864	0.0323932	+	0.999475i	$0.489687\pi$
$954$	6.00000	0.194257
$955$	0	0
$956$	16.0000	0.517477
$957$	−8.00000	−0.258603
$958$	32.0000	1.03387
$959$	0	0
$960$	1.00000	0.0322749
$961$	−31.0000	−1.00000
$962$	4.00000	0.128965
$963$	−12.0000	−0.386695
$964$	18.0000	0.579741
$965$	2.00000	0.0643823
$966$	0	0
$967$	−24.0000	−0.771788	−0.385894	−	0.922543i	$-0.626107\pi$
$967$	−24.0000	−0.771788	−0.385894	+	0.922543i	$0.626107\pi$
$968$	5.00000	0.160706
$969$	−24.0000	−0.770991
$970$	2.00000	0.0642161
$971$	52.0000	1.66876	0.834380	−	0.551190i	$-0.185826\pi$
$971$	52.0000	1.66876	0.834380	+	0.551190i	$0.185826\pi$
$972$	1.00000	0.0320750
$973$	0	0
$974$	40.0000	1.28168
$975$	−2.00000	−0.0640513
$976$	−10.0000	−0.320092
$977$	42.0000	1.34370	0.671850	−	0.740688i	$-0.265500\pi$
$977$	42.0000	1.34370	0.671850	+	0.740688i	$0.265500\pi$
$978$	20.0000	0.639529
$979$	72.0000	2.30113
$980$	−7.00000	−0.223607
$981$	−10.0000	−0.319275
$982$	−4.00000	−0.127645
$983$	24.0000	0.765481	0.382741	−	0.923856i	$-0.374980\pi$
$983$	24.0000	0.765481	0.382741	+	0.923856i	$0.374980\pi$
$984$	10.0000	0.318788
$985$	6.00000	0.191176
$986$	12.0000	0.382158
$987$	0	0
$988$	−8.00000	−0.254514
$989$	4.00000	0.127193
$990$	4.00000	0.127128
$991$	0	0		−	1.00000i	$-0.5\pi$
$991$	0	0			1.00000i	$0.5\pi$
$992$	0	0
$993$	28.0000	0.888553
$994$	0	0
$995$	−16.0000	−0.507234
$996$	−4.00000	−0.126745
$997$	38.0000	1.20347	0.601736	−	0.798695i	$-0.294476\pi$
$997$	38.0000	1.20347	0.601736	+	0.798695i	$0.294476\pi$
$998$	20.0000	0.633089
$999$	−2.00000	−0.0632772

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	690.2.a.k.1.1	✓	1
3.2	odd	2		2070.2.a.b.1.1		1
4.3	odd	2		5520.2.a.i.1.1		1
5.2	odd	4		3450.2.d.t.2899.2		2
5.3	odd	4		3450.2.d.t.2899.1		2
5.4	even	2		3450.2.a.d.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
690.2.a.k.1.1	✓	1	1.1	even	1	trivial
2070.2.a.b.1.1		1	3.2	odd	2
3450.2.a.d.1.1		1	5.4	even	2
3450.2.d.t.2899.1		2	5.3	odd	4
3450.2.d.t.2899.2		2	5.2	odd	4
5520.2.a.i.1.1		1	4.3	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:	\( N \)	\(=\)	\( 690 = 2 \cdot 3 \cdot 5 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	690.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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