LMFDB - Embedded newform 690.2.a.j.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)

$f(q)$

$=$

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

\(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} +1.00000 q^{6} +1.00000 q^{8} +1.00000 q^{9} +1.00000 q^{10} -2.00000 q^{11} +1.00000 q^{12} +4.00000 q^{13} +1.00000 q^{15} +1.00000 q^{16} +6.00000 q^{17} +1.00000 q^{18} -8.00000 q^{19} +1.00000 q^{20} -2.00000 q^{22} -1.00000 q^{23} +1.00000 q^{24} +1.00000 q^{25} +4.00000 q^{26} +1.00000 q^{27} +4.00000 q^{29} +1.00000 q^{30} +1.00000 q^{32} -2.00000 q^{33} +6.00000 q^{34} +1.00000 q^{36} -2.00000 q^{37} -8.00000 q^{38} +4.00000 q^{39} +1.00000 q^{40} -2.00000 q^{41} +2.00000 q^{43} -2.00000 q^{44} +1.00000 q^{45} -1.00000 q^{46} -12.0000 q^{47} +1.00000 q^{48} -7.00000 q^{49} +1.00000 q^{50} +6.00000 q^{51} +4.00000 q^{52} -6.00000 q^{53} +1.00000 q^{54} -2.00000 q^{55} -8.00000 q^{57} +4.00000 q^{58} +8.00000 q^{59} +1.00000 q^{60} +2.00000 q^{61} +1.00000 q^{64} +4.00000 q^{65} -2.00000 q^{66} -6.00000 q^{67} +6.00000 q^{68} -1.00000 q^{69} +10.0000 q^{71} +1.00000 q^{72} -2.00000 q^{73} -2.00000 q^{74} +1.00000 q^{75} -8.00000 q^{76} +4.00000 q^{78} -8.00000 q^{79} +1.00000 q^{80} +1.00000 q^{81} -2.00000 q^{82} +8.00000 q^{83} +6.00000 q^{85} +2.00000 q^{86} +4.00000 q^{87} -2.00000 q^{88} -12.0000 q^{89} +1.00000 q^{90} -1.00000 q^{92} -12.0000 q^{94} -8.00000 q^{95} +1.00000 q^{96} -16.0000 q^{97} -7.00000 q^{98} -2.00000 q^{99} +O(q^{100})\)

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	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
$11$	−2.00000	−0.603023	−0.301511	−	0.953463i	$-0.597491\pi$
$11$	−2.00000	−0.603023	−0.301511	+	0.953463i	$0.597491\pi$
$12$	1.00000	0.288675
$13$	4.00000	1.10940	0.554700	−	0.832050i	$-0.312833\pi$
$13$	4.00000	1.10940	0.554700	+	0.832050i	$0.312833\pi$
$14$	0	0
$15$	1.00000	0.258199
$16$	1.00000	0.250000
$17$	6.00000	1.45521	0.727607	−	0.685994i	$-0.240633\pi$
$17$	6.00000	1.45521	0.727607	+	0.685994i	$0.240633\pi$
$18$	1.00000	0.235702
$19$	−8.00000	−1.83533	−0.917663	−	0.397360i	$-0.869927\pi$
$19$	−8.00000	−1.83533	−0.917663	+	0.397360i	$0.869927\pi$
$20$	1.00000	0.223607
$21$	0	0
$22$	−2.00000	−0.426401
$23$	−1.00000	−0.208514
$23$	−1.00000	−0.208514
$24$	1.00000	0.204124
$25$	1.00000	0.200000
$26$	4.00000	0.784465
$27$	1.00000	0.192450
$28$	0	0
$29$	4.00000	0.742781	0.371391	−	0.928477i	$-0.378881\pi$
$29$	4.00000	0.742781	0.371391	+	0.928477i	$0.378881\pi$
$30$	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
$33$	−2.00000	−0.348155
$34$	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	$0.552568\pi$
$38$	−8.00000	−1.29777
$39$	4.00000	0.640513
$40$	1.00000	0.158114
$41$	−2.00000	−0.312348	−0.156174	−	0.987730i	$-0.549916\pi$
$41$	−2.00000	−0.312348	−0.156174	+	0.987730i	$0.549916\pi$
$42$	0	0
$43$	2.00000	0.304997	0.152499	−	0.988304i	$-0.451268\pi$
$43$	2.00000	0.304997	0.152499	+	0.988304i	$0.451268\pi$
$44$	−2.00000	−0.301511
$45$	1.00000	0.149071
$46$	−1.00000	−0.147442
$47$	−12.0000	−1.75038	−0.875190	−	0.483779i	$-0.839264\pi$
$47$	−12.0000	−1.75038	−0.875190	+	0.483779i	$0.839264\pi$
$48$	1.00000	0.144338
$49$	−7.00000	−1.00000
$50$	1.00000	0.141421
$51$	6.00000	0.840168
$52$	4.00000	0.554700
$53$	−6.00000	−0.824163	−0.412082	−	0.911147i	$-0.635198\pi$
$53$	−6.00000	−0.824163	−0.412082	+	0.911147i	$0.635198\pi$
$54$	1.00000	0.136083
$55$	−2.00000	−0.269680
$56$	0	0
$57$	−8.00000	−1.05963
$58$	4.00000	0.525226
$59$	8.00000	1.04151	0.520756	−	0.853706i	$-0.325650\pi$
$59$	8.00000	1.04151	0.520756	+	0.853706i	$0.325650\pi$
$60$	1.00000	0.129099
$61$	2.00000	0.256074	0.128037	−	0.991769i	$-0.459132\pi$
$61$	2.00000	0.256074	0.128037	+	0.991769i	$0.459132\pi$
$62$	0	0
$63$	0	0
$64$	1.00000	0.125000
$65$	4.00000	0.496139
$66$	−2.00000	−0.246183
$67$	−6.00000	−0.733017	−0.366508	−	0.930415i	$-0.619447\pi$
$67$	−6.00000	−0.733017	−0.366508	+	0.930415i	$0.619447\pi$
$68$	6.00000	0.727607
$69$	−1.00000	−0.120386
$70$	0	0
$71$	10.0000	1.18678	0.593391	−	0.804914i	$-0.297789\pi$
$71$	10.0000	1.18678	0.593391	+	0.804914i	$0.297789\pi$
$72$	1.00000	0.117851
$73$	−2.00000	−0.234082	−0.117041	−	0.993127i	$-0.537341\pi$
$73$	−2.00000	−0.234082	−0.117041	+	0.993127i	$0.537341\pi$
$74$	−2.00000	−0.232495
$75$	1.00000	0.115470
$76$	−8.00000	−0.917663
$77$	0	0
$78$	4.00000	0.452911
$79$	−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
$82$	−2.00000	−0.220863
$83$	8.00000	0.878114	0.439057	−	0.898459i	$-0.355313\pi$
$83$	8.00000	0.878114	0.439057	+	0.898459i	$0.355313\pi$
$84$	0	0
$85$	6.00000	0.650791
$86$	2.00000	0.215666
$87$	4.00000	0.428845
$88$	−2.00000	−0.213201
$89$	−12.0000	−1.27200	−0.635999	−	0.771690i	$-0.719412\pi$
$89$	−12.0000	−1.27200	−0.635999	+	0.771690i	$0.719412\pi$
$90$	1.00000	0.105409
$91$	0	0
$92$	−1.00000	−0.104257
$93$	0	0
$94$	−12.0000	−1.23771
$95$	−8.00000	−0.820783
$96$	1.00000	0.102062
$97$	−16.0000	−1.62455	−0.812277	−	0.583272i	$-0.801772\pi$
$97$	−16.0000	−1.62455	−0.812277	+	0.583272i	$0.801772\pi$
$98$	−7.00000	−0.707107
$99$	−2.00000	−0.201008
$100$	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	$0.296350\pi$
$102$	6.00000	0.594089
$103$	0	0		−	1.00000i	$-0.5\pi$
$103$	0	0			1.00000i	$0.5\pi$
$104$	4.00000	0.392232
$105$	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	$0.658969\pi$
$110$	−2.00000	−0.190693
$111$	−2.00000	−0.189832
$112$	0	0
$113$	18.0000	1.69330	0.846649	−	0.532152i	$-0.178617\pi$
$113$	18.0000	1.69330	0.846649	+	0.532152i	$0.178617\pi$
$114$	−8.00000	−0.749269
$115$	−1.00000	−0.0932505
$116$	4.00000	0.371391
$117$	4.00000	0.369800
$118$	8.00000	0.736460
$119$	0	0
$120$	1.00000	0.0912871
$121$	−7.00000	−0.636364
$122$	2.00000	0.181071
$123$	−2.00000	−0.180334
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	2.00000	0.177471	0.0887357	−	0.996055i	$-0.471717\pi$
$127$	2.00000	0.177471	0.0887357	+	0.996055i	$0.471717\pi$
$128$	1.00000	0.0883883
$129$	2.00000	0.176090
$130$	4.00000	0.350823
$131$	−4.00000	−0.349482	−0.174741	−	0.984614i	$-0.555909\pi$
$131$	−4.00000	−0.349482	−0.174741	+	0.984614i	$0.555909\pi$
$132$	−2.00000	−0.174078
$133$	0	0
$134$	−6.00000	−0.518321
$135$	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
$139$	4.00000	0.339276	0.169638	−	0.985506i	$-0.445740\pi$
$139$	4.00000	0.339276	0.169638	+	0.985506i	$0.445740\pi$
$140$	0	0
$141$	−12.0000	−1.01058
$142$	10.0000	0.839181
$143$	−8.00000	−0.668994
$144$	1.00000	0.0833333
$145$	4.00000	0.332182
$146$	−2.00000	−0.165521
$147$	−7.00000	−0.577350
$148$	−2.00000	−0.164399
$149$	14.0000	1.14692	0.573462	−	0.819232i	$-0.305600\pi$
$149$	14.0000	1.14692	0.573462	+	0.819232i	$0.305600\pi$
$150$	1.00000	0.0816497
$151$	−4.00000	−0.325515	−0.162758	−	0.986666i	$-0.552039\pi$
$151$	−4.00000	−0.325515	−0.162758	+	0.986666i	$0.552039\pi$
$152$	−8.00000	−0.648886
$153$	6.00000	0.485071
$154$	0	0
$155$	0	0
$156$	4.00000	0.320256
$157$	−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
$163$	−16.0000	−1.25322	−0.626608	−	0.779334i	$-0.715557\pi$
$163$	−16.0000	−1.25322	−0.626608	+	0.779334i	$0.715557\pi$
$164$	−2.00000	−0.156174
$165$	−2.00000	−0.155700
$166$	8.00000	0.620920
$167$	−16.0000	−1.23812	−0.619059	−	0.785345i	$-0.712486\pi$
$167$	−16.0000	−1.23812	−0.619059	+	0.785345i	$0.712486\pi$
$168$	0	0
$169$	3.00000	0.230769
$170$	6.00000	0.460179
$171$	−8.00000	−0.611775
$172$	2.00000	0.152499
$173$	−10.0000	−0.760286	−0.380143	−	0.924928i	$-0.624125\pi$
$173$	−10.0000	−0.760286	−0.380143	+	0.924928i	$0.624125\pi$
$174$	4.00000	0.303239
$175$	0	0
$176$	−2.00000	−0.150756
$177$	8.00000	0.601317
$178$	−12.0000	−0.899438
$179$	16.0000	1.19590	0.597948	−	0.801535i	$-0.295983\pi$
$179$	16.0000	1.19590	0.597948	+	0.801535i	$0.295983\pi$
$180$	1.00000	0.0745356
$181$	10.0000	0.743294	0.371647	−	0.928374i	$-0.378793\pi$
$181$	10.0000	0.743294	0.371647	+	0.928374i	$0.378793\pi$
$182$	0	0
$183$	2.00000	0.147844
$184$	−1.00000	−0.0737210
$185$	−2.00000	−0.147043
$186$	0	0
$187$	−12.0000	−0.877527
$188$	−12.0000	−0.875190
$189$	0	0
$190$	−8.00000	−0.580381
$191$	24.0000	1.73658	0.868290	−	0.496058i	$-0.165220\pi$
$191$	24.0000	1.73658	0.868290	+	0.496058i	$0.165220\pi$
$192$	1.00000	0.0721688
$193$	26.0000	1.87152	0.935760	−	0.352636i	$-0.114715\pi$
$193$	26.0000	1.87152	0.935760	+	0.352636i	$0.114715\pi$
$194$	−16.0000	−1.14873
$195$	4.00000	0.286446
$196$	−7.00000	−0.500000
$197$	−18.0000	−1.28245	−0.641223	−	0.767354i	$-0.721573\pi$
$197$	−18.0000	−1.28245	−0.641223	+	0.767354i	$0.721573\pi$
$198$	−2.00000	−0.142134
$199$	−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
$201$	−6.00000	−0.423207
$202$	12.0000	0.844317
$203$	0	0
$204$	6.00000	0.420084
$205$	−2.00000	−0.139686
$206$	0	0
$207$	−1.00000	−0.0695048
$208$	4.00000	0.277350
$209$	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
$213$	10.0000	0.685189
$214$	−12.0000	−0.820303
$215$	2.00000	0.136399
$216$	1.00000	0.0680414
$217$	0	0
$218$	−10.0000	−0.677285
$219$	−2.00000	−0.135147
$220$	−2.00000	−0.134840
$221$	24.0000	1.61441
$222$	−2.00000	−0.134231
$223$	−2.00000	−0.133930	−0.0669650	−	0.997755i	$-0.521332\pi$
$223$	−2.00000	−0.133930	−0.0669650	+	0.997755i	$0.521332\pi$
$224$	0	0
$225$	1.00000	0.0666667
$226$	18.0000	1.19734
$227$	28.0000	1.85843	0.929213	−	0.369546i	$-0.120487\pi$
$227$	28.0000	1.85843	0.929213	+	0.369546i	$0.120487\pi$
$228$	−8.00000	−0.529813
$229$	−10.0000	−0.660819	−0.330409	−	0.943838i	$-0.607187\pi$
$229$	−10.0000	−0.660819	−0.330409	+	0.943838i	$0.607187\pi$
$230$	−1.00000	−0.0659380
$231$	0	0
$232$	4.00000	0.262613
$233$	6.00000	0.393073	0.196537	−	0.980497i	$-0.437031\pi$
$233$	6.00000	0.393073	0.196537	+	0.980497i	$0.437031\pi$
$234$	4.00000	0.261488
$235$	−12.0000	−0.782794
$236$	8.00000	0.520756
$237$	−8.00000	−0.519656
$238$	0	0
$239$	−14.0000	−0.905585	−0.452792	−	0.891616i	$-0.649572\pi$
$239$	−14.0000	−0.905585	−0.452792	+	0.891616i	$0.649572\pi$
$240$	1.00000	0.0645497
$241$	−6.00000	−0.386494	−0.193247	−	0.981150i	$-0.561902\pi$
$241$	−6.00000	−0.386494	−0.193247	+	0.981150i	$0.561902\pi$
$242$	−7.00000	−0.449977
$243$	1.00000	0.0641500
$244$	2.00000	0.128037
$245$	−7.00000	−0.447214
$246$	−2.00000	−0.127515
$247$	−32.0000	−2.03611
$248$	0	0
$249$	8.00000	0.506979
$250$	1.00000	0.0632456
$251$	2.00000	0.126239	0.0631194	−	0.998006i	$-0.479895\pi$
$251$	2.00000	0.126239	0.0631194	+	0.998006i	$0.479895\pi$
$252$	0	0
$253$	2.00000	0.125739
$254$	2.00000	0.125491
$255$	6.00000	0.375735
$256$	1.00000	0.0625000
$257$	22.0000	1.37232	0.686161	−	0.727450i	$-0.259294\pi$
$257$	22.0000	1.37232	0.686161	+	0.727450i	$0.259294\pi$
$258$	2.00000	0.124515
$259$	0	0
$260$	4.00000	0.248069
$261$	4.00000	0.247594
$262$	−4.00000	−0.247121
$263$	0	0		−	1.00000i	$-0.5\pi$
$263$	0	0			1.00000i	$0.5\pi$
$264$	−2.00000	−0.123091
$265$	−6.00000	−0.368577
$266$	0	0
$267$	−12.0000	−0.734388
$268$	−6.00000	−0.366508
$269$	8.00000	0.487769	0.243884	−	0.969804i	$-0.421578\pi$
$269$	8.00000	0.487769	0.243884	+	0.969804i	$0.421578\pi$
$270$	1.00000	0.0608581
$271$	24.0000	1.45790	0.728948	−	0.684569i	$-0.240010\pi$
$271$	24.0000	1.45790	0.728948	+	0.684569i	$0.240010\pi$
$272$	6.00000	0.363803
$273$	0	0
$274$	2.00000	0.120824
$275$	−2.00000	−0.120605
$276$	−1.00000	−0.0601929
$277$	−28.0000	−1.68236	−0.841178	−	0.540758i	$-0.818138\pi$
$277$	−28.0000	−1.68236	−0.841178	+	0.540758i	$0.818138\pi$
$278$	4.00000	0.239904
$279$	0	0
$280$	0	0
$281$	12.0000	0.715860	0.357930	−	0.933748i	$-0.383483\pi$
$281$	12.0000	0.715860	0.357930	+	0.933748i	$0.383483\pi$
$282$	−12.0000	−0.714590
$283$	14.0000	0.832214	0.416107	−	0.909316i	$-0.363394\pi$
$283$	14.0000	0.832214	0.416107	+	0.909316i	$0.363394\pi$
$284$	10.0000	0.593391
$285$	−8.00000	−0.473879
$286$	−8.00000	−0.473050
$287$	0	0
$288$	1.00000	0.0589256
$289$	19.0000	1.11765
$290$	4.00000	0.234888
$291$	−16.0000	−0.937937
$292$	−2.00000	−0.117041
$293$	−14.0000	−0.817889	−0.408944	−	0.912559i	$-0.634103\pi$
$293$	−14.0000	−0.817889	−0.408944	+	0.912559i	$0.634103\pi$
$294$	−7.00000	−0.408248
$295$	8.00000	0.465778
$296$	−2.00000	−0.116248
$297$	−2.00000	−0.116052
$298$	14.0000	0.810998
$299$	−4.00000	−0.231326
$300$	1.00000	0.0577350
$301$	0	0
$302$	−4.00000	−0.230174
$303$	12.0000	0.689382
$304$	−8.00000	−0.458831
$305$	2.00000	0.114520
$306$	6.00000	0.342997
$307$	28.0000	1.59804	0.799022	−	0.601302i	$-0.205351\pi$
$307$	28.0000	1.59804	0.799022	+	0.601302i	$0.205351\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	26.0000	1.47432	0.737162	−	0.675716i	$-0.236165\pi$
$311$	26.0000	1.47432	0.737162	+	0.675716i	$0.236165\pi$
$312$	4.00000	0.226455
$313$	28.0000	1.58265	0.791327	−	0.611393i	$-0.209391\pi$
$313$	28.0000	1.58265	0.791327	+	0.611393i	$0.209391\pi$
$314$	−10.0000	−0.564333
$315$	0	0
$316$	−8.00000	−0.450035
$317$	18.0000	1.01098	0.505490	−	0.862832i	$-0.331312\pi$
$317$	18.0000	1.01098	0.505490	+	0.862832i	$0.331312\pi$
$318$	−6.00000	−0.336463
$319$	−8.00000	−0.447914
$320$	1.00000	0.0559017
$321$	−12.0000	−0.669775
$322$	0	0
$323$	−48.0000	−2.67079
$324$	1.00000	0.0555556
$325$	4.00000	0.221880
$326$	−16.0000	−0.886158
$327$	−10.0000	−0.553001
$328$	−2.00000	−0.110432
$329$	0	0
$330$	−2.00000	−0.110096
$331$	28.0000	1.53902	0.769510	−	0.638635i	$-0.220501\pi$
$331$	28.0000	1.53902	0.769510	+	0.638635i	$0.220501\pi$
$332$	8.00000	0.439057
$333$	−2.00000	−0.109599
$334$	−16.0000	−0.875481
$335$	−6.00000	−0.327815
$336$	0	0
$337$	−28.0000	−1.52526	−0.762629	−	0.646837i	$-0.776092\pi$
$337$	−28.0000	−1.52526	−0.762629	+	0.646837i	$0.776092\pi$
$338$	3.00000	0.163178
$339$	18.0000	0.977626
$340$	6.00000	0.325396
$341$	0	0
$342$	−8.00000	−0.432590
$343$	0	0
$344$	2.00000	0.107833
$345$	−1.00000	−0.0538382
$346$	−10.0000	−0.537603
$347$	−12.0000	−0.644194	−0.322097	−	0.946707i	$-0.604388\pi$
$347$	−12.0000	−0.644194	−0.322097	+	0.946707i	$0.604388\pi$
$348$	4.00000	0.214423
$349$	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
$352$	−2.00000	−0.106600
$353$	10.0000	0.532246	0.266123	−	0.963939i	$-0.414257\pi$
$353$	10.0000	0.532246	0.266123	+	0.963939i	$0.414257\pi$
$354$	8.00000	0.425195
$355$	10.0000	0.530745
$356$	−12.0000	−0.635999
$357$	0	0
$358$	16.0000	0.845626
$359$	−16.0000	−0.844448	−0.422224	−	0.906492i	$-0.638750\pi$
$359$	−16.0000	−0.844448	−0.422224	+	0.906492i	$0.638750\pi$
$360$	1.00000	0.0527046
$361$	45.0000	2.36842
$362$	10.0000	0.525588
$363$	−7.00000	−0.367405
$364$	0	0
$365$	−2.00000	−0.104685
$366$	2.00000	0.104542
$367$	16.0000	0.835193	0.417597	−	0.908633i	$-0.362873\pi$
$367$	16.0000	0.835193	0.417597	+	0.908633i	$0.362873\pi$
$368$	−1.00000	−0.0521286
$369$	−2.00000	−0.104116
$370$	−2.00000	−0.103975
$371$	0	0
$372$	0	0
$373$	6.00000	0.310668	0.155334	−	0.987862i	$-0.450355\pi$
$373$	6.00000	0.310668	0.155334	+	0.987862i	$0.450355\pi$
$374$	−12.0000	−0.620505
$375$	1.00000	0.0516398
$376$	−12.0000	−0.618853
$377$	16.0000	0.824042
$378$	0	0
$379$	12.0000	0.616399	0.308199	−	0.951322i	$-0.400274\pi$
$379$	12.0000	0.616399	0.308199	+	0.951322i	$0.400274\pi$
$380$	−8.00000	−0.410391
$381$	2.00000	0.102463
$382$	24.0000	1.22795
$383$	−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
$386$	26.0000	1.32337
$387$	2.00000	0.101666
$388$	−16.0000	−0.812277
$389$	26.0000	1.31825	0.659126	−	0.752032i	$-0.270926\pi$
$389$	26.0000	1.31825	0.659126	+	0.752032i	$0.270926\pi$
$390$	4.00000	0.202548
$391$	−6.00000	−0.303433
$392$	−7.00000	−0.353553
$393$	−4.00000	−0.201773
$394$	−18.0000	−0.906827
$395$	−8.00000	−0.402524
$396$	−2.00000	−0.100504
$397$	28.0000	1.40528	0.702640	−	0.711546i	$-0.252005\pi$
$397$	28.0000	1.40528	0.702640	+	0.711546i	$0.252005\pi$
$398$	−16.0000	−0.802008
$399$	0	0
$400$	1.00000	0.0500000
$401$	−8.00000	−0.399501	−0.199750	−	0.979847i	$-0.564013\pi$
$401$	−8.00000	−0.399501	−0.199750	+	0.979847i	$0.564013\pi$
$402$	−6.00000	−0.299253
$403$	0	0
$404$	12.0000	0.597022
$405$	1.00000	0.0496904
$406$	0	0
$407$	4.00000	0.198273
$408$	6.00000	0.297044
$409$	−10.0000	−0.494468	−0.247234	−	0.968956i	$-0.579522\pi$
$409$	−10.0000	−0.494468	−0.247234	+	0.968956i	$0.579522\pi$
$410$	−2.00000	−0.0987730
$411$	2.00000	0.0986527
$412$	0	0
$413$	0	0
$414$	−1.00000	−0.0491473
$415$	8.00000	0.392705
$416$	4.00000	0.196116
$417$	4.00000	0.195881
$418$	16.0000	0.782586
$419$	30.0000	1.46560	0.732798	−	0.680446i	$-0.238214\pi$
$419$	30.0000	1.46560	0.732798	+	0.680446i	$0.238214\pi$
$420$	0	0
$421$	−14.0000	−0.682318	−0.341159	−	0.940006i	$-0.610819\pi$
$421$	−14.0000	−0.682318	−0.341159	+	0.940006i	$0.610819\pi$
$422$	4.00000	0.194717
$423$	−12.0000	−0.583460
$424$	−6.00000	−0.291386
$425$	6.00000	0.291043
$426$	10.0000	0.484502
$427$	0	0
$428$	−12.0000	−0.580042
$429$	−8.00000	−0.386244
$430$	2.00000	0.0964486
$431$	−8.00000	−0.385346	−0.192673	−	0.981263i	$-0.561716\pi$
$431$	−8.00000	−0.385346	−0.192673	+	0.981263i	$0.561716\pi$
$432$	1.00000	0.0481125
$433$	16.0000	0.768911	0.384455	−	0.923144i	$-0.374389\pi$
$433$	16.0000	0.768911	0.384455	+	0.923144i	$0.374389\pi$
$434$	0	0
$435$	4.00000	0.191785
$436$	−10.0000	−0.478913
$437$	8.00000	0.382692
$438$	−2.00000	−0.0955637
$439$	28.0000	1.33637	0.668184	−	0.743996i	$-0.267072\pi$
$439$	28.0000	1.33637	0.668184	+	0.743996i	$0.267072\pi$
$440$	−2.00000	−0.0953463
$441$	−7.00000	−0.333333
$442$	24.0000	1.14156
$443$	−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
$445$	−12.0000	−0.568855
$446$	−2.00000	−0.0947027
$447$	14.0000	0.662177
$448$	0	0
$449$	30.0000	1.41579	0.707894	−	0.706319i	$-0.249646\pi$
$449$	30.0000	1.41579	0.707894	+	0.706319i	$0.249646\pi$
$450$	1.00000	0.0471405
$451$	4.00000	0.188353
$452$	18.0000	0.846649
$453$	−4.00000	−0.187936
$454$	28.0000	1.31411
$455$	0	0
$456$	−8.00000	−0.374634
$457$	−8.00000	−0.374224	−0.187112	−	0.982339i	$-0.559913\pi$
$457$	−8.00000	−0.374224	−0.187112	+	0.982339i	$0.559913\pi$
$458$	−10.0000	−0.467269
$459$	6.00000	0.280056
$460$	−1.00000	−0.0466252
$461$	−12.0000	−0.558896	−0.279448	−	0.960161i	$-0.590151\pi$
$461$	−12.0000	−0.558896	−0.279448	+	0.960161i	$0.590151\pi$
$462$	0	0
$463$	−26.0000	−1.20832	−0.604161	−	0.796862i	$-0.706492\pi$
$463$	−26.0000	−1.20832	−0.604161	+	0.796862i	$0.706492\pi$
$464$	4.00000	0.185695
$465$	0	0
$466$	6.00000	0.277945
$467$	−24.0000	−1.11059	−0.555294	−	0.831654i	$-0.687394\pi$
$467$	−24.0000	−1.11059	−0.555294	+	0.831654i	$0.687394\pi$
$468$	4.00000	0.184900
$469$	0	0
$470$	−12.0000	−0.553519
$471$	−10.0000	−0.460776
$472$	8.00000	0.368230
$473$	−4.00000	−0.183920
$474$	−8.00000	−0.367452
$475$	−8.00000	−0.367065
$476$	0	0
$477$	−6.00000	−0.274721
$478$	−14.0000	−0.640345
$479$	−16.0000	−0.731059	−0.365529	−	0.930800i	$-0.619112\pi$
$479$	−16.0000	−0.731059	−0.365529	+	0.930800i	$0.619112\pi$
$480$	1.00000	0.0456435
$481$	−8.00000	−0.364769
$482$	−6.00000	−0.273293
$483$	0	0
$484$	−7.00000	−0.318182
$485$	−16.0000	−0.726523
$486$	1.00000	0.0453609
$487$	−26.0000	−1.17817	−0.589086	−	0.808070i	$-0.700512\pi$
$487$	−26.0000	−1.17817	−0.589086	+	0.808070i	$0.700512\pi$
$488$	2.00000	0.0905357
$489$	−16.0000	−0.723545
$490$	−7.00000	−0.316228
$491$	20.0000	0.902587	0.451294	−	0.892375i	$-0.350963\pi$
$491$	20.0000	0.902587	0.451294	+	0.892375i	$0.350963\pi$
$492$	−2.00000	−0.0901670
$493$	24.0000	1.08091
$494$	−32.0000	−1.43975
$495$	−2.00000	−0.0898933
$496$	0	0
$497$	0	0
$498$	8.00000	0.358489
$499$	−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
$501$	−16.0000	−0.714827
$502$	2.00000	0.0892644
$503$	−16.0000	−0.713405	−0.356702	−	0.934218i	$-0.616099\pi$
$503$	−16.0000	−0.713405	−0.356702	+	0.934218i	$0.616099\pi$
$504$	0	0
$505$	12.0000	0.533993
$506$	2.00000	0.0889108
$507$	3.00000	0.133235
$508$	2.00000	0.0887357
$509$	20.0000	0.886484	0.443242	−	0.896402i	$-0.353828\pi$
$509$	20.0000	0.886484	0.443242	+	0.896402i	$0.353828\pi$
$510$	6.00000	0.265684
$511$	0	0
$512$	1.00000	0.0441942
$513$	−8.00000	−0.353209
$514$	22.0000	0.970378
$515$	0	0
$516$	2.00000	0.0880451
$517$	24.0000	1.05552
$518$	0	0
$519$	−10.0000	−0.438951
$520$	4.00000	0.175412
$521$	−20.0000	−0.876216	−0.438108	−	0.898922i	$-0.644351\pi$
$521$	−20.0000	−0.876216	−0.438108	+	0.898922i	$0.644351\pi$
$522$	4.00000	0.175075
$523$	30.0000	1.31181	0.655904	−	0.754844i	$-0.272288\pi$
$523$	30.0000	1.31181	0.655904	+	0.754844i	$0.272288\pi$
$524$	−4.00000	−0.174741
$525$	0	0
$526$	0	0
$527$	0	0
$528$	−2.00000	−0.0870388
$529$	1.00000	0.0434783
$530$	−6.00000	−0.260623
$531$	8.00000	0.347170
$532$	0	0
$533$	−8.00000	−0.346518
$534$	−12.0000	−0.519291
$535$	−12.0000	−0.518805
$536$	−6.00000	−0.259161
$537$	16.0000	0.690451
$538$	8.00000	0.344904
$539$	14.0000	0.603023
$540$	1.00000	0.0430331
$541$	−22.0000	−0.945854	−0.472927	−	0.881102i	$-0.656803\pi$
$541$	−22.0000	−0.945854	−0.472927	+	0.881102i	$0.656803\pi$
$542$	24.0000	1.03089
$543$	10.0000	0.429141
$544$	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
$549$	2.00000	0.0853579
$550$	−2.00000	−0.0852803
$551$	−32.0000	−1.36325
$552$	−1.00000	−0.0425628
$553$	0	0
$554$	−28.0000	−1.18961
$555$	−2.00000	−0.0848953
$556$	4.00000	0.169638
$557$	38.0000	1.61011	0.805056	−	0.593199i	$-0.202135\pi$
$557$	38.0000	1.61011	0.805056	+	0.593199i	$0.202135\pi$
$558$	0	0
$559$	8.00000	0.338364
$560$	0	0
$561$	−12.0000	−0.506640
$562$	12.0000	0.506189
$563$	−24.0000	−1.01148	−0.505740	−	0.862686i	$-0.668780\pi$
$563$	−24.0000	−1.01148	−0.505740	+	0.862686i	$0.668780\pi$
$564$	−12.0000	−0.505291
$565$	18.0000	0.757266
$566$	14.0000	0.588464
$567$	0	0
$568$	10.0000	0.419591
$569$	−24.0000	−1.00613	−0.503066	−	0.864248i	$-0.667795\pi$
$569$	−24.0000	−1.00613	−0.503066	+	0.864248i	$0.667795\pi$
$570$	−8.00000	−0.335083
$571$	20.0000	0.836974	0.418487	−	0.908223i	$-0.362561\pi$
$571$	20.0000	0.836974	0.418487	+	0.908223i	$0.362561\pi$
$572$	−8.00000	−0.334497
$573$	24.0000	1.00261
$574$	0	0
$575$	−1.00000	−0.0417029
$576$	1.00000	0.0416667
$577$	38.0000	1.58196	0.790980	−	0.611842i	$-0.209571\pi$
$577$	38.0000	1.58196	0.790980	+	0.611842i	$0.209571\pi$
$578$	19.0000	0.790296
$579$	26.0000	1.08052
$580$	4.00000	0.166091
$581$	0	0
$582$	−16.0000	−0.663221
$583$	12.0000	0.496989
$584$	−2.00000	−0.0827606
$585$	4.00000	0.165380
$586$	−14.0000	−0.578335
$587$	4.00000	0.165098	0.0825488	−	0.996587i	$-0.473694\pi$
$587$	4.00000	0.165098	0.0825488	+	0.996587i	$0.473694\pi$
$588$	−7.00000	−0.288675
$589$	0	0
$590$	8.00000	0.329355
$591$	−18.0000	−0.740421
$592$	−2.00000	−0.0821995
$593$	6.00000	0.246390	0.123195	−	0.992382i	$-0.460686\pi$
$593$	6.00000	0.246390	0.123195	+	0.992382i	$0.460686\pi$
$594$	−2.00000	−0.0820610
$595$	0	0
$596$	14.0000	0.573462
$597$	−16.0000	−0.654836
$598$	−4.00000	−0.163572
$599$	−34.0000	−1.38920	−0.694601	−	0.719395i	$-0.744419\pi$
$599$	−34.0000	−1.38920	−0.694601	+	0.719395i	$0.744419\pi$
$600$	1.00000	0.0408248
$601$	−26.0000	−1.06056	−0.530281	−	0.847822i	$-0.677914\pi$
$601$	−26.0000	−1.06056	−0.530281	+	0.847822i	$0.677914\pi$
$602$	0	0
$603$	−6.00000	−0.244339
$604$	−4.00000	−0.162758
$605$	−7.00000	−0.284590
$606$	12.0000	0.487467
$607$	−10.0000	−0.405887	−0.202944	−	0.979190i	$-0.565051\pi$
$607$	−10.0000	−0.405887	−0.202944	+	0.979190i	$0.565051\pi$
$608$	−8.00000	−0.324443
$609$	0	0
$610$	2.00000	0.0809776
$611$	−48.0000	−1.94187
$612$	6.00000	0.242536
$613$	−30.0000	−1.21169	−0.605844	−	0.795583i	$-0.707165\pi$
$613$	−30.0000	−1.21169	−0.605844	+	0.795583i	$0.707165\pi$
$614$	28.0000	1.12999
$615$	−2.00000	−0.0806478
$616$	0	0
$617$	−18.0000	−0.724653	−0.362326	−	0.932051i	$-0.618017\pi$
$617$	−18.0000	−0.724653	−0.362326	+	0.932051i	$0.618017\pi$
$618$	0	0
$619$	32.0000	1.28619	0.643094	−	0.765787i	$-0.277650\pi$
$619$	32.0000	1.28619	0.643094	+	0.765787i	$0.277650\pi$
$620$	0	0
$621$	−1.00000	−0.0401286
$622$	26.0000	1.04251
$623$	0	0
$624$	4.00000	0.160128
$625$	1.00000	0.0400000
$626$	28.0000	1.11911
$627$	16.0000	0.638978
$628$	−10.0000	−0.399043
$629$	−12.0000	−0.478471
$630$	0	0
$631$	−32.0000	−1.27390	−0.636950	−	0.770905i	$-0.719804\pi$
$631$	−32.0000	−1.27390	−0.636950	+	0.770905i	$0.719804\pi$
$632$	−8.00000	−0.318223
$633$	4.00000	0.158986
$634$	18.0000	0.714871
$635$	2.00000	0.0793676
$636$	−6.00000	−0.237915
$637$	−28.0000	−1.10940
$638$	−8.00000	−0.316723
$639$	10.0000	0.395594
$640$	1.00000	0.0395285
$641$	−4.00000	−0.157991	−0.0789953	−	0.996875i	$-0.525171\pi$
$641$	−4.00000	−0.157991	−0.0789953	+	0.996875i	$0.525171\pi$
$642$	−12.0000	−0.473602
$643$	−10.0000	−0.394362	−0.197181	−	0.980367i	$-0.563179\pi$
$643$	−10.0000	−0.394362	−0.197181	+	0.980367i	$0.563179\pi$
$644$	0	0
$645$	2.00000	0.0787499
$646$	−48.0000	−1.88853
$647$	−28.0000	−1.10079	−0.550397	−	0.834903i	$-0.685524\pi$
$647$	−28.0000	−1.10079	−0.550397	+	0.834903i	$0.685524\pi$
$648$	1.00000	0.0392837
$649$	−16.0000	−0.628055
$650$	4.00000	0.156893
$651$	0	0
$652$	−16.0000	−0.626608
$653$	2.00000	0.0782660	0.0391330	−	0.999234i	$-0.487540\pi$
$653$	2.00000	0.0782660	0.0391330	+	0.999234i	$0.487540\pi$
$654$	−10.0000	−0.391031
$655$	−4.00000	−0.156293
$656$	−2.00000	−0.0780869
$657$	−2.00000	−0.0780274
$658$	0	0
$659$	−26.0000	−1.01282	−0.506408	−	0.862294i	$-0.669027\pi$
$659$	−26.0000	−1.01282	−0.506408	+	0.862294i	$0.669027\pi$
$660$	−2.00000	−0.0778499
$661$	−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
$663$	24.0000	0.932083
$664$	8.00000	0.310460
$665$	0	0
$666$	−2.00000	−0.0774984
$667$	−4.00000	−0.154881
$668$	−16.0000	−0.619059
$669$	−2.00000	−0.0773245
$670$	−6.00000	−0.231800
$671$	−4.00000	−0.154418
$672$	0	0
$673$	−10.0000	−0.385472	−0.192736	−	0.981251i	$-0.561736\pi$
$673$	−10.0000	−0.385472	−0.192736	+	0.981251i	$0.561736\pi$
$674$	−28.0000	−1.07852
$675$	1.00000	0.0384900
$676$	3.00000	0.115385
$677$	−10.0000	−0.384331	−0.192166	−	0.981363i	$-0.561551\pi$
$677$	−10.0000	−0.384331	−0.192166	+	0.981363i	$0.561551\pi$
$678$	18.0000	0.691286
$679$	0	0
$680$	6.00000	0.230089
$681$	28.0000	1.07296
$682$	0	0
$683$	−12.0000	−0.459167	−0.229584	−	0.973289i	$-0.573736\pi$
$683$	−12.0000	−0.459167	−0.229584	+	0.973289i	$0.573736\pi$
$684$	−8.00000	−0.305888
$685$	2.00000	0.0764161
$686$	0	0
$687$	−10.0000	−0.381524
$688$	2.00000	0.0762493
$689$	−24.0000	−0.914327
$690$	−1.00000	−0.0380693
$691$	20.0000	0.760836	0.380418	−	0.924815i	$-0.375780\pi$
$691$	20.0000	0.760836	0.380418	+	0.924815i	$0.375780\pi$
$692$	−10.0000	−0.380143
$693$	0	0
$694$	−12.0000	−0.455514
$695$	4.00000	0.151729
$696$	4.00000	0.151620
$697$	−12.0000	−0.454532
$698$	26.0000	0.984115
$699$	6.00000	0.226941
$700$	0	0
$701$	−14.0000	−0.528773	−0.264386	−	0.964417i	$-0.585169\pi$
$701$	−14.0000	−0.528773	−0.264386	+	0.964417i	$0.585169\pi$
$702$	4.00000	0.150970
$703$	16.0000	0.603451
$704$	−2.00000	−0.0753778
$705$	−12.0000	−0.451946
$706$	10.0000	0.376355
$707$	0	0
$708$	8.00000	0.300658
$709$	38.0000	1.42712	0.713560	−	0.700594i	$-0.247082\pi$
$709$	38.0000	1.42712	0.713560	+	0.700594i	$0.247082\pi$
$710$	10.0000	0.375293
$711$	−8.00000	−0.300023
$712$	−12.0000	−0.449719
$713$	0	0
$714$	0	0
$715$	−8.00000	−0.299183
$716$	16.0000	0.597948
$717$	−14.0000	−0.522840
$718$	−16.0000	−0.597115
$719$	−2.00000	−0.0745874	−0.0372937	−	0.999304i	$-0.511874\pi$
$719$	−2.00000	−0.0745874	−0.0372937	+	0.999304i	$0.511874\pi$
$720$	1.00000	0.0372678
$721$	0	0
$722$	45.0000	1.67473
$723$	−6.00000	−0.223142
$724$	10.0000	0.371647
$725$	4.00000	0.148556
$726$	−7.00000	−0.259794
$727$	28.0000	1.03846	0.519231	−	0.854634i	$-0.326218\pi$
$727$	28.0000	1.03846	0.519231	+	0.854634i	$0.326218\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	−2.00000	−0.0740233
$731$	12.0000	0.443836
$732$	2.00000	0.0739221
$733$	−42.0000	−1.55131	−0.775653	−	0.631160i	$-0.782579\pi$
$733$	−42.0000	−1.55131	−0.775653	+	0.631160i	$0.782579\pi$
$734$	16.0000	0.590571
$735$	−7.00000	−0.258199
$736$	−1.00000	−0.0368605
$737$	12.0000	0.442026
$738$	−2.00000	−0.0736210
$739$	−36.0000	−1.32428	−0.662141	−	0.749380i	$-0.730352\pi$
$739$	−36.0000	−1.32428	−0.662141	+	0.749380i	$0.730352\pi$
$740$	−2.00000	−0.0735215
$741$	−32.0000	−1.17555
$742$	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
$745$	14.0000	0.512920
$746$	6.00000	0.219676
$747$	8.00000	0.292705
$748$	−12.0000	−0.438763
$749$	0	0
$750$	1.00000	0.0365148
$751$	32.0000	1.16770	0.583848	−	0.811863i	$-0.301546\pi$
$751$	32.0000	1.16770	0.583848	+	0.811863i	$0.301546\pi$
$752$	−12.0000	−0.437595
$753$	2.00000	0.0728841
$754$	16.0000	0.582686
$755$	−4.00000	−0.145575
$756$	0	0
$757$	−6.00000	−0.218074	−0.109037	−	0.994038i	$-0.534777\pi$
$757$	−6.00000	−0.218074	−0.109037	+	0.994038i	$0.534777\pi$
$758$	12.0000	0.435860
$759$	2.00000	0.0725954
$760$	−8.00000	−0.290191
$761$	−22.0000	−0.797499	−0.398750	−	0.917060i	$-0.630556\pi$
$761$	−22.0000	−0.797499	−0.398750	+	0.917060i	$0.630556\pi$
$762$	2.00000	0.0724524
$763$	0	0
$764$	24.0000	0.868290
$765$	6.00000	0.216930
$766$	−32.0000	−1.15621
$767$	32.0000	1.15545
$768$	1.00000	0.0360844
$769$	−2.00000	−0.0721218	−0.0360609	−	0.999350i	$-0.511481\pi$
$769$	−2.00000	−0.0721218	−0.0360609	+	0.999350i	$0.511481\pi$
$770$	0	0
$771$	22.0000	0.792311
$772$	26.0000	0.935760
$773$	−6.00000	−0.215805	−0.107903	−	0.994161i	$-0.534413\pi$
$773$	−6.00000	−0.215805	−0.107903	+	0.994161i	$0.534413\pi$
$774$	2.00000	0.0718885
$775$	0	0
$776$	−16.0000	−0.574367
$777$	0	0
$778$	26.0000	0.932145
$779$	16.0000	0.573259
$780$	4.00000	0.143223
$781$	−20.0000	−0.715656
$782$	−6.00000	−0.214560
$783$	4.00000	0.142948
$784$	−7.00000	−0.250000
$785$	−10.0000	−0.356915
$786$	−4.00000	−0.142675
$787$	22.0000	0.784215	0.392108	−	0.919919i	$-0.371746\pi$
$787$	22.0000	0.784215	0.392108	+	0.919919i	$0.371746\pi$
$788$	−18.0000	−0.641223
$789$	0	0
$790$	−8.00000	−0.284627
$791$	0	0
$792$	−2.00000	−0.0710669
$793$	8.00000	0.284088
$794$	28.0000	0.993683
$795$	−6.00000	−0.212798
$796$	−16.0000	−0.567105
$797$	14.0000	0.495905	0.247953	−	0.968772i	$-0.420242\pi$
$797$	14.0000	0.495905	0.247953	+	0.968772i	$0.420242\pi$
$798$	0	0
$799$	−72.0000	−2.54718
$800$	1.00000	0.0353553
$801$	−12.0000	−0.423999
$802$	−8.00000	−0.282490
$803$	4.00000	0.141157
$804$	−6.00000	−0.211604
$805$	0	0
$806$	0	0
$807$	8.00000	0.281613
$808$	12.0000	0.422159
$809$	−46.0000	−1.61727	−0.808637	−	0.588308i	$-0.799794\pi$
$809$	−46.0000	−1.61727	−0.808637	+	0.588308i	$0.799794\pi$
$810$	1.00000	0.0351364
$811$	4.00000	0.140459	0.0702295	−	0.997531i	$-0.477627\pi$
$811$	4.00000	0.140459	0.0702295	+	0.997531i	$0.477627\pi$
$812$	0	0
$813$	24.0000	0.841717
$814$	4.00000	0.140200
$815$	−16.0000	−0.560456
$816$	6.00000	0.210042
$817$	−16.0000	−0.559769
$818$	−10.0000	−0.349642
$819$	0	0
$820$	−2.00000	−0.0698430
$821$	0	0		−	1.00000i	$-0.5\pi$
$821$	0	0			1.00000i	$0.5\pi$
$822$	2.00000	0.0697580
$823$	−22.0000	−0.766872	−0.383436	−	0.923567i	$-0.625259\pi$
$823$	−22.0000	−0.766872	−0.383436	+	0.923567i	$0.625259\pi$
$824$	0	0
$825$	−2.00000	−0.0696311
$826$	0	0
$827$	12.0000	0.417281	0.208640	−	0.977992i	$-0.433096\pi$
$827$	12.0000	0.417281	0.208640	+	0.977992i	$0.433096\pi$
$828$	−1.00000	−0.0347524
$829$	−38.0000	−1.31979	−0.659897	−	0.751356i	$-0.729400\pi$
$829$	−38.0000	−1.31979	−0.659897	+	0.751356i	$0.729400\pi$
$830$	8.00000	0.277684
$831$	−28.0000	−0.971309
$832$	4.00000	0.138675
$833$	−42.0000	−1.45521
$834$	4.00000	0.138509
$835$	−16.0000	−0.553703
$836$	16.0000	0.553372
$837$	0	0
$838$	30.0000	1.03633
$839$	−12.0000	−0.414286	−0.207143	−	0.978311i	$-0.566417\pi$
$839$	−12.0000	−0.414286	−0.207143	+	0.978311i	$0.566417\pi$
$840$	0	0
$841$	−13.0000	−0.448276
$842$	−14.0000	−0.482472
$843$	12.0000	0.413302
$844$	4.00000	0.137686
$845$	3.00000	0.103203
$846$	−12.0000	−0.412568
$847$	0	0
$848$	−6.00000	−0.206041
$849$	14.0000	0.480479
$850$	6.00000	0.205798
$851$	2.00000	0.0685591
$852$	10.0000	0.342594
$853$	36.0000	1.23262	0.616308	−	0.787505i	$-0.288628\pi$
$853$	36.0000	1.23262	0.616308	+	0.787505i	$0.288628\pi$
$854$	0	0
$855$	−8.00000	−0.273594
$856$	−12.0000	−0.410152
$857$	18.0000	0.614868	0.307434	−	0.951569i	$-0.400530\pi$
$857$	18.0000	0.614868	0.307434	+	0.951569i	$0.400530\pi$
$858$	−8.00000	−0.273115
$859$	−28.0000	−0.955348	−0.477674	−	0.878537i	$-0.658520\pi$
$859$	−28.0000	−0.955348	−0.477674	+	0.878537i	$0.658520\pi$
$860$	2.00000	0.0681994
$861$	0	0
$862$	−8.00000	−0.272481
$863$	−20.0000	−0.680808	−0.340404	−	0.940279i	$-0.610564\pi$
$863$	−20.0000	−0.680808	−0.340404	+	0.940279i	$0.610564\pi$
$864$	1.00000	0.0340207
$865$	−10.0000	−0.340010
$866$	16.0000	0.543702
$867$	19.0000	0.645274
$868$	0	0
$869$	16.0000	0.542763
$870$	4.00000	0.135613
$871$	−24.0000	−0.813209
$872$	−10.0000	−0.338643
$873$	−16.0000	−0.541518
$874$	8.00000	0.270604
$875$	0	0
$876$	−2.00000	−0.0675737
$877$	−20.0000	−0.675352	−0.337676	−	0.941262i	$-0.609641\pi$
$877$	−20.0000	−0.675352	−0.337676	+	0.941262i	$0.609641\pi$
$878$	28.0000	0.944954
$879$	−14.0000	−0.472208
$880$	−2.00000	−0.0674200
$881$	4.00000	0.134763	0.0673817	−	0.997727i	$-0.478535\pi$
$881$	4.00000	0.134763	0.0673817	+	0.997727i	$0.478535\pi$
$882$	−7.00000	−0.235702
$883$	32.0000	1.07689	0.538443	−	0.842662i	$-0.319013\pi$
$883$	32.0000	1.07689	0.538443	+	0.842662i	$0.319013\pi$
$884$	24.0000	0.807207
$885$	8.00000	0.268917
$886$	−4.00000	−0.134383
$887$	36.0000	1.20876	0.604381	−	0.796696i	$-0.293421\pi$
$887$	36.0000	1.20876	0.604381	+	0.796696i	$0.293421\pi$
$888$	−2.00000	−0.0671156
$889$	0	0
$890$	−12.0000	−0.402241
$891$	−2.00000	−0.0670025
$892$	−2.00000	−0.0669650
$893$	96.0000	3.21252
$894$	14.0000	0.468230
$895$	16.0000	0.534821
$896$	0	0
$897$	−4.00000	−0.133556
$898$	30.0000	1.00111
$899$	0	0
$900$	1.00000	0.0333333
$901$	−36.0000	−1.19933
$902$	4.00000	0.133185
$903$	0	0
$904$	18.0000	0.598671
$905$	10.0000	0.332411
$906$	−4.00000	−0.132891
$907$	−22.0000	−0.730498	−0.365249	−	0.930910i	$-0.619016\pi$
$907$	−22.0000	−0.730498	−0.365249	+	0.930910i	$0.619016\pi$
$908$	28.0000	0.929213
$909$	12.0000	0.398015
$910$	0	0
$911$	−4.00000	−0.132526	−0.0662630	−	0.997802i	$-0.521108\pi$
$911$	−4.00000	−0.132526	−0.0662630	+	0.997802i	$0.521108\pi$
$912$	−8.00000	−0.264906
$913$	−16.0000	−0.529523
$914$	−8.00000	−0.264616
$915$	2.00000	0.0661180
$916$	−10.0000	−0.330409
$917$	0	0
$918$	6.00000	0.198030
$919$	40.0000	1.31948	0.659739	−	0.751495i	$-0.270667\pi$
$919$	40.0000	1.31948	0.659739	+	0.751495i	$0.270667\pi$
$920$	−1.00000	−0.0329690
$921$	28.0000	0.922631
$922$	−12.0000	−0.395199
$923$	40.0000	1.31662
$924$	0	0
$925$	−2.00000	−0.0657596
$926$	−26.0000	−0.854413
$927$	0	0
$928$	4.00000	0.131306
$929$	30.0000	0.984268	0.492134	−	0.870519i	$-0.336217\pi$
$929$	30.0000	0.984268	0.492134	+	0.870519i	$0.336217\pi$
$930$	0	0
$931$	56.0000	1.83533
$932$	6.00000	0.196537
$933$	26.0000	0.851202
$934$	−24.0000	−0.785304
$935$	−12.0000	−0.392442
$936$	4.00000	0.130744
$937$	32.0000	1.04539	0.522697	−	0.852518i	$-0.324926\pi$
$937$	32.0000	1.04539	0.522697	+	0.852518i	$0.324926\pi$
$938$	0	0
$939$	28.0000	0.913745
$940$	−12.0000	−0.391397
$941$	38.0000	1.23876	0.619382	−	0.785090i	$-0.287383\pi$
$941$	38.0000	1.23876	0.619382	+	0.785090i	$0.287383\pi$
$942$	−10.0000	−0.325818
$943$	2.00000	0.0651290
$944$	8.00000	0.260378
$945$	0	0
$946$	−4.00000	−0.130051
$947$	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
$949$	−8.00000	−0.259691
$950$	−8.00000	−0.259554
$951$	18.0000	0.583690
$952$	0	0
$953$	26.0000	0.842223	0.421111	−	0.907009i	$-0.361640\pi$
$953$	26.0000	0.842223	0.421111	+	0.907009i	$0.361640\pi$
$954$	−6.00000	−0.194257
$955$	24.0000	0.776622
$956$	−14.0000	−0.452792
$957$	−8.00000	−0.258603
$958$	−16.0000	−0.516937
$959$	0	0
$960$	1.00000	0.0322749
$961$	−31.0000	−1.00000
$962$	−8.00000	−0.257930
$963$	−12.0000	−0.386695
$964$	−6.00000	−0.193247
$965$	26.0000	0.836970
$966$	0	0
$967$	−42.0000	−1.35063	−0.675314	−	0.737530i	$-0.735992\pi$
$967$	−42.0000	−1.35063	−0.675314	+	0.737530i	$0.735992\pi$
$968$	−7.00000	−0.224989
$969$	−48.0000	−1.54198
$970$	−16.0000	−0.513729
$971$	−14.0000	−0.449281	−0.224641	−	0.974442i	$-0.572121\pi$
$971$	−14.0000	−0.449281	−0.224641	+	0.974442i	$0.572121\pi$
$972$	1.00000	0.0320750
$973$	0	0
$974$	−26.0000	−0.833094
$975$	4.00000	0.128103
$976$	2.00000	0.0640184
$977$	−42.0000	−1.34370	−0.671850	−	0.740688i	$-0.734500\pi$
$977$	−42.0000	−1.34370	−0.671850	+	0.740688i	$0.734500\pi$
$978$	−16.0000	−0.511624
$979$	24.0000	0.767043
$980$	−7.00000	−0.223607
$981$	−10.0000	−0.319275
$982$	20.0000	0.638226
$983$	48.0000	1.53096	0.765481	−	0.643458i	$-0.222501\pi$
$983$	48.0000	1.53096	0.765481	+	0.643458i	$0.222501\pi$
$984$	−2.00000	−0.0637577
$985$	−18.0000	−0.573528
$986$	24.0000	0.764316
$987$	0	0
$988$	−32.0000	−1.01806
$989$	−2.00000	−0.0635963
$990$	−2.00000	−0.0635642
$991$	12.0000	0.381193	0.190596	−	0.981669i	$-0.438958\pi$
$991$	12.0000	0.381193	0.190596	+	0.981669i	$0.438958\pi$
$992$	0	0
$993$	28.0000	0.888553
$994$	0	0
$995$	−16.0000	−0.507234
$996$	8.00000	0.253490
$997$	−4.00000	−0.126681	−0.0633406	−	0.997992i	$-0.520175\pi$
$997$	−4.00000	−0.126681	−0.0633406	+	0.997992i	$0.520175\pi$
$998$	−4.00000	−0.126618
$999$	−2.00000	−0.0632772

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
690.2.a.j.1.1	✓	1	1.1	even	1	trivial
2070.2.a.c.1.1		1	3.2	odd	2
3450.2.a.b.1.1		1	5.4	even	2
3450.2.d.m.2899.1		2	5.3	odd	4
3450.2.d.m.2899.2		2	5.2	odd	4
5520.2.a.l.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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