LMFDB - Embedded newform 690.2.a.e.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:	$1$
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)

$f(q)$

$=$

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

\(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} -1.00000 q^{6} -1.00000 q^{8} +1.00000 q^{9} +1.00000 q^{10} -4.00000 q^{11} +1.00000 q^{12} -6.00000 q^{13} -1.00000 q^{15} +1.00000 q^{16} -6.00000 q^{17} -1.00000 q^{18} +4.00000 q^{19} -1.00000 q^{20} +4.00000 q^{22} +1.00000 q^{23} -1.00000 q^{24} +1.00000 q^{25} +6.00000 q^{26} +1.00000 q^{27} -6.00000 q^{29} +1.00000 q^{30} -8.00000 q^{31} -1.00000 q^{32} -4.00000 q^{33} +6.00000 q^{34} +1.00000 q^{36} +6.00000 q^{37} -4.00000 q^{38} -6.00000 q^{39} +1.00000 q^{40} +10.0000 q^{41} +4.00000 q^{43} -4.00000 q^{44} -1.00000 q^{45} -1.00000 q^{46} -8.00000 q^{47} +1.00000 q^{48} -7.00000 q^{49} -1.00000 q^{50} -6.00000 q^{51} -6.00000 q^{52} -14.0000 q^{53} -1.00000 q^{54} +4.00000 q^{55} +4.00000 q^{57} +6.00000 q^{58} -1.00000 q^{60} +10.0000 q^{61} +8.00000 q^{62} +1.00000 q^{64} +6.00000 q^{65} +4.00000 q^{66} +4.00000 q^{67} -6.00000 q^{68} +1.00000 q^{69} +8.00000 q^{71} -1.00000 q^{72} +2.00000 q^{73} -6.00000 q^{74} +1.00000 q^{75} +4.00000 q^{76} +6.00000 q^{78} -12.0000 q^{79} -1.00000 q^{80} +1.00000 q^{81} -10.0000 q^{82} -16.0000 q^{83} +6.00000 q^{85} -4.00000 q^{86} -6.00000 q^{87} +4.00000 q^{88} -2.00000 q^{89} +1.00000 q^{90} +1.00000 q^{92} -8.00000 q^{93} +8.00000 q^{94} -4.00000 q^{95} -1.00000 q^{96} -14.0000 q^{97} +7.00000 q^{98} -4.00000 q^{99} +O(q^{100})\)

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	$-0.706037\pi$
$11$	−4.00000	−1.20605	−0.603023	+	0.797724i	$0.706037\pi$
$12$	1.00000	0.288675
$13$	−6.00000	−1.66410	−0.832050	−	0.554700i	$-0.812833\pi$
$13$	−6.00000	−1.66410	−0.832050	+	0.554700i	$0.812833\pi$
$14$	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
$26$	6.00000	1.17670
$27$	1.00000	0.192450
$28$	0	0
$29$	−6.00000	−1.11417	−0.557086	−	0.830455i	$-0.688081\pi$
$29$	−6.00000	−1.11417	−0.557086	+	0.830455i	$0.688081\pi$
$30$	1.00000	0.182574
$31$	−8.00000	−1.43684	−0.718421	−	0.695608i	$-0.755135\pi$
$31$	−8.00000	−1.43684	−0.718421	+	0.695608i	$0.755135\pi$
$32$	−1.00000	−0.176777
$33$	−4.00000	−0.696311
$34$	6.00000	1.02899
$35$	0	0
$36$	1.00000	0.166667
$37$	6.00000	0.986394	0.493197	−	0.869918i	$-0.335828\pi$
$37$	6.00000	0.986394	0.493197	+	0.869918i	$0.335828\pi$
$38$	−4.00000	−0.648886
$39$	−6.00000	−0.960769
$40$	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	$-0.401344\pi$
$43$	4.00000	0.609994	0.304997	+	0.952353i	$0.401344\pi$
$44$	−4.00000	−0.603023
$45$	−1.00000	−0.149071
$46$	−1.00000	−0.147442
$47$	−8.00000	−1.16692	−0.583460	−	0.812142i	$-0.698301\pi$
$47$	−8.00000	−1.16692	−0.583460	+	0.812142i	$0.698301\pi$
$48$	1.00000	0.144338
$49$	−7.00000	−1.00000
$50$	−1.00000	−0.141421
$51$	−6.00000	−0.840168
$52$	−6.00000	−0.832050
$53$	−14.0000	−1.92305	−0.961524	−	0.274721i	$-0.911414\pi$
$53$	−14.0000	−1.92305	−0.961524	+	0.274721i	$0.911414\pi$
$54$	−1.00000	−0.136083
$55$	4.00000	0.539360
$56$	0	0
$57$	4.00000	0.529813
$58$	6.00000	0.787839
$59$	0	0		−	1.00000i	$-0.5\pi$
$59$	0	0			1.00000i	$0.5\pi$
$60$	−1.00000	−0.129099
$61$	10.0000	1.28037	0.640184	−	0.768221i	$-0.278858\pi$
$61$	10.0000	1.28037	0.640184	+	0.768221i	$0.278858\pi$
$62$	8.00000	1.01600
$63$	0	0
$64$	1.00000	0.125000
$65$	6.00000	0.744208
$66$	4.00000	0.492366
$67$	4.00000	0.488678	0.244339	−	0.969690i	$-0.421429\pi$
$67$	4.00000	0.488678	0.244339	+	0.969690i	$0.421429\pi$
$68$	−6.00000	−0.727607
$69$	1.00000	0.120386
$70$	0	0
$71$	8.00000	0.949425	0.474713	−	0.880141i	$-0.342552\pi$
$71$	8.00000	0.949425	0.474713	+	0.880141i	$0.342552\pi$
$72$	−1.00000	−0.117851
$73$	2.00000	0.234082	0.117041	−	0.993127i	$-0.462659\pi$
$73$	2.00000	0.234082	0.117041	+	0.993127i	$0.462659\pi$
$74$	−6.00000	−0.697486
$75$	1.00000	0.115470
$76$	4.00000	0.458831
$77$	0	0
$78$	6.00000	0.679366
$79$	−12.0000	−1.35011	−0.675053	−	0.737769i	$-0.735879\pi$
$79$	−12.0000	−1.35011	−0.675053	+	0.737769i	$0.735879\pi$
$80$	−1.00000	−0.111803
$81$	1.00000	0.111111
$82$	−10.0000	−1.10432
$83$	−16.0000	−1.75623	−0.878114	−	0.478451i	$-0.841198\pi$
$83$	−16.0000	−1.75623	−0.878114	+	0.478451i	$0.841198\pi$
$84$	0	0
$85$	6.00000	0.650791
$86$	−4.00000	−0.431331
$87$	−6.00000	−0.643268
$88$	4.00000	0.426401
$89$	−2.00000	−0.212000	−0.106000	−	0.994366i	$-0.533804\pi$
$89$	−2.00000	−0.212000	−0.106000	+	0.994366i	$0.533804\pi$
$90$	1.00000	0.105409
$91$	0	0
$92$	1.00000	0.104257
$93$	−8.00000	−0.829561
$94$	8.00000	0.825137
$95$	−4.00000	−0.410391
$96$	−1.00000	−0.102062
$97$	−14.0000	−1.42148	−0.710742	−	0.703452i	$-0.751641\pi$
$97$	−14.0000	−1.42148	−0.710742	+	0.703452i	$0.751641\pi$
$98$	7.00000	0.707107
$99$	−4.00000	−0.402015
$100$	1.00000	0.100000
$101$	18.0000	1.79107	0.895533	−	0.444994i	$-0.146794\pi$
$101$	18.0000	1.79107	0.895533	+	0.444994i	$0.146794\pi$
$102$	6.00000	0.594089
$103$	−16.0000	−1.57653	−0.788263	−	0.615338i	$-0.789020\pi$
$103$	−16.0000	−1.57653	−0.788263	+	0.615338i	$0.789020\pi$
$104$	6.00000	0.588348
$105$	0	0
$106$	14.0000	1.35980
$107$	8.00000	0.773389	0.386695	−	0.922208i	$-0.373617\pi$
$107$	8.00000	0.773389	0.386695	+	0.922208i	$0.373617\pi$
$108$	1.00000	0.0962250
$109$	2.00000	0.191565	0.0957826	−	0.995402i	$-0.469465\pi$
$109$	2.00000	0.191565	0.0957826	+	0.995402i	$0.469465\pi$
$110$	−4.00000	−0.381385
$111$	6.00000	0.569495
$112$	0	0
$113$	−14.0000	−1.31701	−0.658505	−	0.752577i	$-0.728811\pi$
$113$	−14.0000	−1.31701	−0.658505	+	0.752577i	$0.728811\pi$
$114$	−4.00000	−0.374634
$115$	−1.00000	−0.0932505
$116$	−6.00000	−0.557086
$117$	−6.00000	−0.554700
$118$	0	0
$119$	0	0
$120$	1.00000	0.0912871
$121$	5.00000	0.454545
$122$	−10.0000	−0.905357
$123$	10.0000	0.901670
$124$	−8.00000	−0.718421
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	12.0000	1.06483	0.532414	−	0.846484i	$-0.321285\pi$
$127$	12.0000	1.06483	0.532414	+	0.846484i	$0.321285\pi$
$128$	−1.00000	−0.0883883
$129$	4.00000	0.352180
$130$	−6.00000	−0.526235
$131$	8.00000	0.698963	0.349482	−	0.936943i	$-0.386358\pi$
$131$	8.00000	0.698963	0.349482	+	0.936943i	$0.386358\pi$
$132$	−4.00000	−0.348155
$133$	0	0
$134$	−4.00000	−0.345547
$135$	−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
$139$	12.0000	1.01783	0.508913	−	0.860818i	$-0.330047\pi$
$139$	12.0000	1.01783	0.508913	+	0.860818i	$0.330047\pi$
$140$	0	0
$141$	−8.00000	−0.673722
$142$	−8.00000	−0.671345
$143$	24.0000	2.00698
$144$	1.00000	0.0833333
$145$	6.00000	0.498273
$146$	−2.00000	−0.165521
$147$	−7.00000	−0.577350
$148$	6.00000	0.493197
$149$	10.0000	0.819232	0.409616	−	0.912258i	$-0.365663\pi$
$149$	10.0000	0.819232	0.409616	+	0.912258i	$0.365663\pi$
$150$	−1.00000	−0.0816497
$151$	16.0000	1.30206	0.651031	−	0.759051i	$-0.274337\pi$
$151$	16.0000	1.30206	0.651031	+	0.759051i	$0.274337\pi$
$152$	−4.00000	−0.324443
$153$	−6.00000	−0.485071
$154$	0	0
$155$	8.00000	0.642575
$156$	−6.00000	−0.480384
$157$	6.00000	0.478852	0.239426	−	0.970915i	$-0.423041\pi$
$157$	6.00000	0.478852	0.239426	+	0.970915i	$0.423041\pi$
$158$	12.0000	0.954669
$159$	−14.0000	−1.11027
$160$	1.00000	0.0790569
$161$	0	0
$162$	−1.00000	−0.0785674
$163$	4.00000	0.313304	0.156652	−	0.987654i	$-0.449930\pi$
$163$	4.00000	0.313304	0.156652	+	0.987654i	$0.449930\pi$
$164$	10.0000	0.780869
$165$	4.00000	0.311400
$166$	16.0000	1.24184
$167$	−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
$169$	23.0000	1.76923
$170$	−6.00000	−0.460179
$171$	4.00000	0.305888
$172$	4.00000	0.304997
$173$	18.0000	1.36851	0.684257	−	0.729241i	$-0.260127\pi$
$173$	18.0000	1.36851	0.684257	+	0.729241i	$0.260127\pi$
$174$	6.00000	0.454859
$175$	0	0
$176$	−4.00000	−0.301511
$177$	0	0
$178$	2.00000	0.149906
$179$	0	0		−	1.00000i	$-0.5\pi$
$179$	0	0			1.00000i	$0.5\pi$
$180$	−1.00000	−0.0745356
$181$	−6.00000	−0.445976	−0.222988	−	0.974821i	$-0.571581\pi$
$181$	−6.00000	−0.445976	−0.222988	+	0.974821i	$0.571581\pi$
$182$	0	0
$183$	10.0000	0.739221
$184$	−1.00000	−0.0737210
$185$	−6.00000	−0.441129
$186$	8.00000	0.586588
$187$	24.0000	1.75505
$188$	−8.00000	−0.583460
$189$	0	0
$190$	4.00000	0.290191
$191$	−8.00000	−0.578860	−0.289430	−	0.957199i	$-0.593466\pi$
$191$	−8.00000	−0.578860	−0.289430	+	0.957199i	$0.593466\pi$
$192$	1.00000	0.0721688
$193$	10.0000	0.719816	0.359908	−	0.932988i	$-0.382808\pi$
$193$	10.0000	0.719816	0.359908	+	0.932988i	$0.382808\pi$
$194$	14.0000	1.00514
$195$	6.00000	0.429669
$196$	−7.00000	−0.500000
$197$	26.0000	1.85242	0.926212	−	0.377004i	$-0.123046\pi$
$197$	26.0000	1.85242	0.926212	+	0.377004i	$0.123046\pi$
$198$	4.00000	0.284268
$199$	−12.0000	−0.850657	−0.425329	−	0.905039i	$-0.639842\pi$
$199$	−12.0000	−0.850657	−0.425329	+	0.905039i	$0.639842\pi$
$200$	−1.00000	−0.0707107
$201$	4.00000	0.282138
$202$	−18.0000	−1.26648
$203$	0	0
$204$	−6.00000	−0.420084
$205$	−10.0000	−0.698430
$206$	16.0000	1.11477
$207$	1.00000	0.0695048
$208$	−6.00000	−0.416025
$209$	−16.0000	−1.10674
$210$	0	0
$211$	−20.0000	−1.37686	−0.688428	−	0.725304i	$-0.741699\pi$
$211$	−20.0000	−1.37686	−0.688428	+	0.725304i	$0.741699\pi$
$212$	−14.0000	−0.961524
$213$	8.00000	0.548151
$214$	−8.00000	−0.546869
$215$	−4.00000	−0.272798
$216$	−1.00000	−0.0680414
$217$	0	0
$218$	−2.00000	−0.135457
$219$	2.00000	0.135147
$220$	4.00000	0.269680
$221$	36.0000	2.42162
$222$	−6.00000	−0.402694
$223$	−20.0000	−1.33930	−0.669650	−	0.742677i	$-0.733556\pi$
$223$	−20.0000	−1.33930	−0.669650	+	0.742677i	$0.733556\pi$
$224$	0	0
$225$	1.00000	0.0666667
$226$	14.0000	0.931266
$227$	8.00000	0.530979	0.265489	−	0.964114i	$-0.414466\pi$
$227$	8.00000	0.530979	0.265489	+	0.964114i	$0.414466\pi$
$228$	4.00000	0.264906
$229$	−14.0000	−0.925146	−0.462573	−	0.886581i	$-0.653074\pi$
$229$	−14.0000	−0.925146	−0.462573	+	0.886581i	$0.653074\pi$
$230$	1.00000	0.0659380
$231$	0	0
$232$	6.00000	0.393919
$233$	22.0000	1.44127	0.720634	−	0.693316i	$-0.243851\pi$
$233$	22.0000	1.44127	0.720634	+	0.693316i	$0.243851\pi$
$234$	6.00000	0.392232
$235$	8.00000	0.521862
$236$	0	0
$237$	−12.0000	−0.779484
$238$	0	0
$239$	−8.00000	−0.517477	−0.258738	−	0.965947i	$-0.583307\pi$
$239$	−8.00000	−0.517477	−0.258738	+	0.965947i	$0.583307\pi$
$240$	−1.00000	−0.0645497
$241$	2.00000	0.128831	0.0644157	−	0.997923i	$-0.479482\pi$
$241$	2.00000	0.128831	0.0644157	+	0.997923i	$0.479482\pi$
$242$	−5.00000	−0.321412
$243$	1.00000	0.0641500
$244$	10.0000	0.640184
$245$	7.00000	0.447214
$246$	−10.0000	−0.637577
$247$	−24.0000	−1.52708
$248$	8.00000	0.508001
$249$	−16.0000	−1.01396
$250$	1.00000	0.0632456
$251$	−12.0000	−0.757433	−0.378717	−	0.925513i	$-0.623635\pi$
$251$	−12.0000	−0.757433	−0.378717	+	0.925513i	$0.623635\pi$
$252$	0	0
$253$	−4.00000	−0.251478
$254$	−12.0000	−0.752947
$255$	6.00000	0.375735
$256$	1.00000	0.0625000
$257$	6.00000	0.374270	0.187135	−	0.982334i	$-0.440080\pi$
$257$	6.00000	0.374270	0.187135	+	0.982334i	$0.440080\pi$
$258$	−4.00000	−0.249029
$259$	0	0
$260$	6.00000	0.372104
$261$	−6.00000	−0.371391
$262$	−8.00000	−0.494242
$263$	−32.0000	−1.97320	−0.986602	−	0.163144i	$-0.947836\pi$
$263$	−32.0000	−1.97320	−0.986602	+	0.163144i	$0.947836\pi$
$264$	4.00000	0.246183
$265$	14.0000	0.860013
$266$	0	0
$267$	−2.00000	−0.122398
$268$	4.00000	0.244339
$269$	10.0000	0.609711	0.304855	−	0.952399i	$-0.401392\pi$
$269$	10.0000	0.609711	0.304855	+	0.952399i	$0.401392\pi$
$270$	1.00000	0.0608581
$271$	−8.00000	−0.485965	−0.242983	−	0.970031i	$-0.578126\pi$
$271$	−8.00000	−0.485965	−0.242983	+	0.970031i	$0.578126\pi$
$272$	−6.00000	−0.363803
$273$	0	0
$274$	−2.00000	−0.120824
$275$	−4.00000	−0.241209
$276$	1.00000	0.0601929
$277$	−30.0000	−1.80253	−0.901263	−	0.433273i	$-0.857359\pi$
$277$	−30.0000	−1.80253	−0.901263	+	0.433273i	$0.857359\pi$
$278$	−12.0000	−0.719712
$279$	−8.00000	−0.478947
$280$	0	0
$281$	30.0000	1.78965	0.894825	−	0.446417i	$-0.147300\pi$
$281$	30.0000	1.78965	0.894825	+	0.446417i	$0.147300\pi$
$282$	8.00000	0.476393
$283$	4.00000	0.237775	0.118888	−	0.992908i	$-0.462067\pi$
$283$	4.00000	0.237775	0.118888	+	0.992908i	$0.462067\pi$
$284$	8.00000	0.474713
$285$	−4.00000	−0.236940
$286$	−24.0000	−1.41915
$287$	0	0
$288$	−1.00000	−0.0589256
$289$	19.0000	1.11765
$290$	−6.00000	−0.352332
$291$	−14.0000	−0.820695
$292$	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
$295$	0	0
$296$	−6.00000	−0.348743
$297$	−4.00000	−0.232104
$298$	−10.0000	−0.579284
$299$	−6.00000	−0.346989
$300$	1.00000	0.0577350
$301$	0	0
$302$	−16.0000	−0.920697
$303$	18.0000	1.03407
$304$	4.00000	0.229416
$305$	−10.0000	−0.572598
$306$	6.00000	0.342997
$307$	12.0000	0.684876	0.342438	−	0.939540i	$-0.388747\pi$
$307$	12.0000	0.684876	0.342438	+	0.939540i	$0.388747\pi$
$308$	0	0
$309$	−16.0000	−0.910208
$310$	−8.00000	−0.454369
$311$	−8.00000	−0.453638	−0.226819	−	0.973937i	$-0.572833\pi$
$311$	−8.00000	−0.453638	−0.226819	+	0.973937i	$0.572833\pi$
$312$	6.00000	0.339683
$313$	10.0000	0.565233	0.282617	−	0.959233i	$-0.408798\pi$
$313$	10.0000	0.565233	0.282617	+	0.959233i	$0.408798\pi$
$314$	−6.00000	−0.338600
$315$	0	0
$316$	−12.0000	−0.675053
$317$	18.0000	1.01098	0.505490	−	0.862832i	$-0.331312\pi$
$317$	18.0000	1.01098	0.505490	+	0.862832i	$0.331312\pi$
$318$	14.0000	0.785081
$319$	24.0000	1.34374
$320$	−1.00000	−0.0559017
$321$	8.00000	0.446516
$322$	0	0
$323$	−24.0000	−1.33540
$324$	1.00000	0.0555556
$325$	−6.00000	−0.332820
$326$	−4.00000	−0.221540
$327$	2.00000	0.110600
$328$	−10.0000	−0.552158
$329$	0	0
$330$	−4.00000	−0.220193
$331$	−20.0000	−1.09930	−0.549650	−	0.835395i	$-0.685239\pi$
$331$	−20.0000	−1.09930	−0.549650	+	0.835395i	$0.685239\pi$
$332$	−16.0000	−0.878114
$333$	6.00000	0.328798
$334$	16.0000	0.875481
$335$	−4.00000	−0.218543
$336$	0	0
$337$	−14.0000	−0.762629	−0.381314	−	0.924445i	$-0.624528\pi$
$337$	−14.0000	−0.762629	−0.381314	+	0.924445i	$0.624528\pi$
$338$	−23.0000	−1.25104
$339$	−14.0000	−0.760376
$340$	6.00000	0.325396
$341$	32.0000	1.73290
$342$	−4.00000	−0.216295
$343$	0	0
$344$	−4.00000	−0.215666
$345$	−1.00000	−0.0538382
$346$	−18.0000	−0.967686
$347$	−28.0000	−1.50312	−0.751559	−	0.659665i	$-0.770698\pi$
$347$	−28.0000	−1.50312	−0.751559	+	0.659665i	$0.770698\pi$
$348$	−6.00000	−0.321634
$349$	−34.0000	−1.81998	−0.909989	−	0.414632i	$-0.863910\pi$
$349$	−34.0000	−1.81998	−0.909989	+	0.414632i	$0.863910\pi$
$350$	0	0
$351$	−6.00000	−0.320256
$352$	4.00000	0.213201
$353$	6.00000	0.319348	0.159674	−	0.987170i	$-0.448956\pi$
$353$	6.00000	0.319348	0.159674	+	0.987170i	$0.448956\pi$
$354$	0	0
$355$	−8.00000	−0.424596
$356$	−2.00000	−0.106000
$357$	0	0
$358$	0	0
$359$	−8.00000	−0.422224	−0.211112	−	0.977462i	$-0.567708\pi$
$359$	−8.00000	−0.422224	−0.211112	+	0.977462i	$0.567708\pi$
$360$	1.00000	0.0527046
$361$	−3.00000	−0.157895
$362$	6.00000	0.315353
$363$	5.00000	0.262432
$364$	0	0
$365$	−2.00000	−0.104685
$366$	−10.0000	−0.522708
$367$	−32.0000	−1.67039	−0.835193	−	0.549957i	$-0.814644\pi$
$367$	−32.0000	−1.67039	−0.835193	+	0.549957i	$0.814644\pi$
$368$	1.00000	0.0521286
$369$	10.0000	0.520579
$370$	6.00000	0.311925
$371$	0	0
$372$	−8.00000	−0.414781
$373$	−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
$376$	8.00000	0.412568
$377$	36.0000	1.85409
$378$	0	0
$379$	4.00000	0.205466	0.102733	−	0.994709i	$-0.467241\pi$
$379$	4.00000	0.205466	0.102733	+	0.994709i	$0.467241\pi$
$380$	−4.00000	−0.205196
$381$	12.0000	0.614779
$382$	8.00000	0.409316
$383$	16.0000	0.817562	0.408781	−	0.912633i	$-0.365954\pi$
$383$	16.0000	0.817562	0.408781	+	0.912633i	$0.365954\pi$
$384$	−1.00000	−0.0510310
$385$	0	0
$386$	−10.0000	−0.508987
$387$	4.00000	0.203331
$388$	−14.0000	−0.710742
$389$	−6.00000	−0.304212	−0.152106	−	0.988364i	$-0.548606\pi$
$389$	−6.00000	−0.304212	−0.152106	+	0.988364i	$0.548606\pi$
$390$	−6.00000	−0.303822
$391$	−6.00000	−0.303433
$392$	7.00000	0.353553
$393$	8.00000	0.403547
$394$	−26.0000	−1.30986
$395$	12.0000	0.603786
$396$	−4.00000	−0.201008
$397$	2.00000	0.100377	0.0501886	−	0.998740i	$-0.484018\pi$
$397$	2.00000	0.100377	0.0501886	+	0.998740i	$0.484018\pi$
$398$	12.0000	0.601506
$399$	0	0
$400$	1.00000	0.0500000
$401$	6.00000	0.299626	0.149813	−	0.988714i	$-0.452133\pi$
$401$	6.00000	0.299626	0.149813	+	0.988714i	$0.452133\pi$
$402$	−4.00000	−0.199502
$403$	48.0000	2.39105
$404$	18.0000	0.895533
$405$	−1.00000	−0.0496904
$406$	0	0
$407$	−24.0000	−1.18964
$408$	6.00000	0.297044
$409$	10.0000	0.494468	0.247234	−	0.968956i	$-0.420478\pi$
$409$	10.0000	0.494468	0.247234	+	0.968956i	$0.420478\pi$
$410$	10.0000	0.493865
$411$	2.00000	0.0986527
$412$	−16.0000	−0.788263
$413$	0	0
$414$	−1.00000	−0.0491473
$415$	16.0000	0.785409
$416$	6.00000	0.294174
$417$	12.0000	0.587643
$418$	16.0000	0.782586
$419$	−12.0000	−0.586238	−0.293119	−	0.956076i	$-0.594693\pi$
$419$	−12.0000	−0.586238	−0.293119	+	0.956076i	$0.594693\pi$
$420$	0	0
$421$	−38.0000	−1.85201	−0.926003	−	0.377515i	$-0.876779\pi$
$421$	−38.0000	−1.85201	−0.926003	+	0.377515i	$0.876779\pi$
$422$	20.0000	0.973585
$423$	−8.00000	−0.388973
$424$	14.0000	0.679900
$425$	−6.00000	−0.291043
$426$	−8.00000	−0.387601
$427$	0	0
$428$	8.00000	0.386695
$429$	24.0000	1.15873
$430$	4.00000	0.192897
$431$	8.00000	0.385346	0.192673	−	0.981263i	$-0.438284\pi$
$431$	8.00000	0.385346	0.192673	+	0.981263i	$0.438284\pi$
$432$	1.00000	0.0481125
$433$	2.00000	0.0961139	0.0480569	−	0.998845i	$-0.484697\pi$
$433$	2.00000	0.0961139	0.0480569	+	0.998845i	$0.484697\pi$
$434$	0	0
$435$	6.00000	0.287678
$436$	2.00000	0.0957826
$437$	4.00000	0.191346
$438$	−2.00000	−0.0955637
$439$	−24.0000	−1.14546	−0.572729	−	0.819745i	$-0.694115\pi$
$439$	−24.0000	−1.14546	−0.572729	+	0.819745i	$0.694115\pi$
$440$	−4.00000	−0.190693
$441$	−7.00000	−0.333333
$442$	−36.0000	−1.71235
$443$	12.0000	0.570137	0.285069	−	0.958507i	$-0.407984\pi$
$443$	12.0000	0.570137	0.285069	+	0.958507i	$0.407984\pi$
$444$	6.00000	0.284747
$445$	2.00000	0.0948091
$446$	20.0000	0.947027
$447$	10.0000	0.472984
$448$	0	0
$449$	10.0000	0.471929	0.235965	−	0.971762i	$-0.424175\pi$
$449$	10.0000	0.471929	0.235965	+	0.971762i	$0.424175\pi$
$450$	−1.00000	−0.0471405
$451$	−40.0000	−1.88353
$452$	−14.0000	−0.658505
$453$	16.0000	0.751746
$454$	−8.00000	−0.375459
$455$	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
$461$	2.00000	0.0931493	0.0465746	−	0.998915i	$-0.485169\pi$
$461$	2.00000	0.0931493	0.0465746	+	0.998915i	$0.485169\pi$
$462$	0	0
$463$	−20.0000	−0.929479	−0.464739	−	0.885448i	$-0.653852\pi$
$463$	−20.0000	−0.929479	−0.464739	+	0.885448i	$0.653852\pi$
$464$	−6.00000	−0.278543
$465$	8.00000	0.370991
$466$	−22.0000	−1.01913
$467$	8.00000	0.370196	0.185098	−	0.982720i	$-0.440740\pi$
$467$	8.00000	0.370196	0.185098	+	0.982720i	$0.440740\pi$
$468$	−6.00000	−0.277350
$469$	0	0
$470$	−8.00000	−0.369012
$471$	6.00000	0.276465
$472$	0	0
$473$	−16.0000	−0.735681
$474$	12.0000	0.551178
$475$	4.00000	0.183533
$476$	0	0
$477$	−14.0000	−0.641016
$478$	8.00000	0.365911
$479$	24.0000	1.09659	0.548294	−	0.836286i	$-0.315277\pi$
$479$	24.0000	1.09659	0.548294	+	0.836286i	$0.315277\pi$
$480$	1.00000	0.0456435
$481$	−36.0000	−1.64146
$482$	−2.00000	−0.0910975
$483$	0	0
$484$	5.00000	0.227273
$485$	14.0000	0.635707
$486$	−1.00000	−0.0453609
$487$	20.0000	0.906287	0.453143	−	0.891438i	$-0.350303\pi$
$487$	20.0000	0.906287	0.453143	+	0.891438i	$0.350303\pi$
$488$	−10.0000	−0.452679
$489$	4.00000	0.180886
$490$	−7.00000	−0.316228
$491$	0	0		−	1.00000i	$-0.5\pi$
$491$	0	0			1.00000i	$0.5\pi$
$492$	10.0000	0.450835
$493$	36.0000	1.62136
$494$	24.0000	1.07981
$495$	4.00000	0.179787
$496$	−8.00000	−0.359211
$497$	0	0
$498$	16.0000	0.716977
$499$	12.0000	0.537194	0.268597	−	0.963253i	$-0.413440\pi$
$499$	12.0000	0.537194	0.268597	+	0.963253i	$0.413440\pi$
$500$	−1.00000	−0.0447214
$501$	−16.0000	−0.714827
$502$	12.0000	0.535586
$503$	−24.0000	−1.07011	−0.535054	−	0.844818i	$-0.679709\pi$
$503$	−24.0000	−1.07011	−0.535054	+	0.844818i	$0.679709\pi$
$504$	0	0
$505$	−18.0000	−0.800989
$506$	4.00000	0.177822
$507$	23.0000	1.02147
$508$	12.0000	0.532414
$509$	−6.00000	−0.265945	−0.132973	−	0.991120i	$-0.542452\pi$
$509$	−6.00000	−0.265945	−0.132973	+	0.991120i	$0.542452\pi$
$510$	−6.00000	−0.265684
$511$	0	0
$512$	−1.00000	−0.0441942
$513$	4.00000	0.176604
$514$	−6.00000	−0.264649
$515$	16.0000	0.705044
$516$	4.00000	0.176090
$517$	32.0000	1.40736
$518$	0	0
$519$	18.0000	0.790112
$520$	−6.00000	−0.263117
$521$	−42.0000	−1.84005	−0.920027	−	0.391856i	$-0.871833\pi$
$521$	−42.0000	−1.84005	−0.920027	+	0.391856i	$0.871833\pi$
$522$	6.00000	0.262613
$523$	−20.0000	−0.874539	−0.437269	−	0.899331i	$-0.644054\pi$
$523$	−20.0000	−0.874539	−0.437269	+	0.899331i	$0.644054\pi$
$524$	8.00000	0.349482
$525$	0	0
$526$	32.0000	1.39527
$527$	48.0000	2.09091
$528$	−4.00000	−0.174078
$529$	1.00000	0.0434783
$530$	−14.0000	−0.608121
$531$	0	0
$532$	0	0
$533$	−60.0000	−2.59889
$534$	2.00000	0.0865485
$535$	−8.00000	−0.345870
$536$	−4.00000	−0.172774
$537$	0	0
$538$	−10.0000	−0.431131
$539$	28.0000	1.20605
$540$	−1.00000	−0.0430331
$541$	−34.0000	−1.46177	−0.730887	−	0.682498i	$-0.760893\pi$
$541$	−34.0000	−1.46177	−0.730887	+	0.682498i	$0.760893\pi$
$542$	8.00000	0.343629
$543$	−6.00000	−0.257485
$544$	6.00000	0.257248
$545$	−2.00000	−0.0856706
$546$	0	0
$547$	20.0000	0.855138	0.427569	−	0.903983i	$-0.359370\pi$
$547$	20.0000	0.855138	0.427569	+	0.903983i	$0.359370\pi$
$548$	2.00000	0.0854358
$549$	10.0000	0.426790
$550$	4.00000	0.170561
$551$	−24.0000	−1.02243
$552$	−1.00000	−0.0425628
$553$	0	0
$554$	30.0000	1.27458
$555$	−6.00000	−0.254686
$556$	12.0000	0.508913
$557$	18.0000	0.762684	0.381342	−	0.924434i	$-0.375462\pi$
$557$	18.0000	0.762684	0.381342	+	0.924434i	$0.375462\pi$
$558$	8.00000	0.338667
$559$	−24.0000	−1.01509
$560$	0	0
$561$	24.0000	1.01328
$562$	−30.0000	−1.26547
$563$	−8.00000	−0.337160	−0.168580	−	0.985688i	$-0.553918\pi$
$563$	−8.00000	−0.337160	−0.168580	+	0.985688i	$0.553918\pi$
$564$	−8.00000	−0.336861
$565$	14.0000	0.588984
$566$	−4.00000	−0.168133
$567$	0	0
$568$	−8.00000	−0.335673
$569$	−18.0000	−0.754599	−0.377300	−	0.926091i	$-0.623147\pi$
$569$	−18.0000	−0.754599	−0.377300	+	0.926091i	$0.623147\pi$
$570$	4.00000	0.167542
$571$	20.0000	0.836974	0.418487	−	0.908223i	$-0.362561\pi$
$571$	20.0000	0.836974	0.418487	+	0.908223i	$0.362561\pi$
$572$	24.0000	1.00349
$573$	−8.00000	−0.334205
$574$	0	0
$575$	1.00000	0.0417029
$576$	1.00000	0.0416667
$577$	34.0000	1.41544	0.707719	−	0.706494i	$-0.249724\pi$
$577$	34.0000	1.41544	0.707719	+	0.706494i	$0.249724\pi$
$578$	−19.0000	−0.790296
$579$	10.0000	0.415586
$580$	6.00000	0.249136
$581$	0	0
$582$	14.0000	0.580319
$583$	56.0000	2.31928
$584$	−2.00000	−0.0827606
$585$	6.00000	0.248069
$586$	14.0000	0.578335
$587$	−4.00000	−0.165098	−0.0825488	−	0.996587i	$-0.526306\pi$
$587$	−4.00000	−0.165098	−0.0825488	+	0.996587i	$0.526306\pi$
$588$	−7.00000	−0.288675
$589$	−32.0000	−1.31854
$590$	0	0
$591$	26.0000	1.06950
$592$	6.00000	0.246598
$593$	6.00000	0.246390	0.123195	−	0.992382i	$-0.460686\pi$
$593$	6.00000	0.246390	0.123195	+	0.992382i	$0.460686\pi$
$594$	4.00000	0.164122
$595$	0	0
$596$	10.0000	0.409616
$597$	−12.0000	−0.491127
$598$	6.00000	0.245358
$599$	−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
$601$	10.0000	0.407909	0.203954	−	0.978980i	$-0.434621\pi$
$601$	10.0000	0.407909	0.203954	+	0.978980i	$0.434621\pi$
$602$	0	0
$603$	4.00000	0.162893
$604$	16.0000	0.651031
$605$	−5.00000	−0.203279
$606$	−18.0000	−0.731200
$607$	−28.0000	−1.13648	−0.568242	−	0.822861i	$-0.692376\pi$
$607$	−28.0000	−1.13648	−0.568242	+	0.822861i	$0.692376\pi$
$608$	−4.00000	−0.162221
$609$	0	0
$610$	10.0000	0.404888
$611$	48.0000	1.94187
$612$	−6.00000	−0.242536
$613$	14.0000	0.565455	0.282727	−	0.959200i	$-0.408761\pi$
$613$	14.0000	0.565455	0.282727	+	0.959200i	$0.408761\pi$
$614$	−12.0000	−0.484281
$615$	−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	$0.381983\pi$
$618$	16.0000	0.643614
$619$	20.0000	0.803868	0.401934	−	0.915669i	$-0.368338\pi$
$619$	20.0000	0.803868	0.401934	+	0.915669i	$0.368338\pi$
$620$	8.00000	0.321288
$621$	1.00000	0.0401286
$622$	8.00000	0.320771
$623$	0	0
$624$	−6.00000	−0.240192
$625$	1.00000	0.0400000
$626$	−10.0000	−0.399680
$627$	−16.0000	−0.638978
$628$	6.00000	0.239426
$629$	−36.0000	−1.43541
$630$	0	0
$631$	20.0000	0.796187	0.398094	−	0.917345i	$-0.369672\pi$
$631$	20.0000	0.796187	0.398094	+	0.917345i	$0.369672\pi$
$632$	12.0000	0.477334
$633$	−20.0000	−0.794929
$634$	−18.0000	−0.714871
$635$	−12.0000	−0.476205
$636$	−14.0000	−0.555136
$637$	42.0000	1.66410
$638$	−24.0000	−0.950169
$639$	8.00000	0.316475
$640$	1.00000	0.0395285
$641$	−2.00000	−0.0789953	−0.0394976	−	0.999220i	$-0.512576\pi$
$641$	−2.00000	−0.0789953	−0.0394976	+	0.999220i	$0.512576\pi$
$642$	−8.00000	−0.315735
$643$	−44.0000	−1.73519	−0.867595	−	0.497271i	$-0.834335\pi$
$643$	−44.0000	−1.73519	−0.867595	+	0.497271i	$0.834335\pi$
$644$	0	0
$645$	−4.00000	−0.157500
$646$	24.0000	0.944267
$647$	24.0000	0.943537	0.471769	−	0.881722i	$-0.343616\pi$
$647$	24.0000	0.943537	0.471769	+	0.881722i	$0.343616\pi$
$648$	−1.00000	−0.0392837
$649$	0	0
$650$	6.00000	0.235339
$651$	0	0
$652$	4.00000	0.156652
$653$	−14.0000	−0.547862	−0.273931	−	0.961749i	$-0.588324\pi$
$653$	−14.0000	−0.547862	−0.273931	+	0.961749i	$0.588324\pi$
$654$	−2.00000	−0.0782062
$655$	−8.00000	−0.312586
$656$	10.0000	0.390434
$657$	2.00000	0.0780274
$658$	0	0
$659$	−36.0000	−1.40236	−0.701180	−	0.712984i	$-0.747343\pi$
$659$	−36.0000	−1.40236	−0.701180	+	0.712984i	$0.747343\pi$
$660$	4.00000	0.155700
$661$	34.0000	1.32245	0.661223	−	0.750189i	$-0.270038\pi$
$661$	34.0000	1.32245	0.661223	+	0.750189i	$0.270038\pi$
$662$	20.0000	0.777322
$663$	36.0000	1.39812
$664$	16.0000	0.620920
$665$	0	0
$666$	−6.00000	−0.232495
$667$	−6.00000	−0.232321
$668$	−16.0000	−0.619059
$669$	−20.0000	−0.773245
$670$	4.00000	0.154533
$671$	−40.0000	−1.54418
$672$	0	0
$673$	−22.0000	−0.848038	−0.424019	−	0.905653i	$-0.639381\pi$
$673$	−22.0000	−0.848038	−0.424019	+	0.905653i	$0.639381\pi$
$674$	14.0000	0.539260
$675$	1.00000	0.0384900
$676$	23.0000	0.884615
$677$	−6.00000	−0.230599	−0.115299	−	0.993331i	$-0.536783\pi$
$677$	−6.00000	−0.230599	−0.115299	+	0.993331i	$0.536783\pi$
$678$	14.0000	0.537667
$679$	0	0
$680$	−6.00000	−0.230089
$681$	8.00000	0.306561
$682$	−32.0000	−1.22534
$683$	44.0000	1.68361	0.841807	−	0.539779i	$-0.181492\pi$
$683$	44.0000	1.68361	0.841807	+	0.539779i	$0.181492\pi$
$684$	4.00000	0.152944
$685$	−2.00000	−0.0764161
$686$	0	0
$687$	−14.0000	−0.534133
$688$	4.00000	0.152499
$689$	84.0000	3.20015
$690$	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	$0.678779\pi$
$692$	18.0000	0.684257
$693$	0	0
$694$	28.0000	1.06287
$695$	−12.0000	−0.455186
$696$	6.00000	0.227429
$697$	−60.0000	−2.27266
$698$	34.0000	1.28692
$699$	22.0000	0.832116
$700$	0	0
$701$	18.0000	0.679851	0.339925	−	0.940452i	$-0.389598\pi$
$701$	18.0000	0.679851	0.339925	+	0.940452i	$0.389598\pi$
$702$	6.00000	0.226455
$703$	24.0000	0.905177
$704$	−4.00000	−0.150756
$705$	8.00000	0.301297
$706$	−6.00000	−0.225813
$707$	0	0
$708$	0	0
$709$	10.0000	0.375558	0.187779	−	0.982211i	$-0.439871\pi$
$709$	10.0000	0.375558	0.187779	+	0.982211i	$0.439871\pi$
$710$	8.00000	0.300235
$711$	−12.0000	−0.450035
$712$	2.00000	0.0749532
$713$	−8.00000	−0.299602
$714$	0	0
$715$	−24.0000	−0.897549
$716$	0	0
$717$	−8.00000	−0.298765
$718$	8.00000	0.298557
$719$	−40.0000	−1.49175	−0.745874	−	0.666087i	$-0.767968\pi$
$719$	−40.0000	−1.49175	−0.745874	+	0.666087i	$0.767968\pi$
$720$	−1.00000	−0.0372678
$721$	0	0
$722$	3.00000	0.111648
$723$	2.00000	0.0743808
$724$	−6.00000	−0.222988
$725$	−6.00000	−0.222834
$726$	−5.00000	−0.185567
$727$	8.00000	0.296704	0.148352	−	0.988935i	$-0.452603\pi$
$727$	8.00000	0.296704	0.148352	+	0.988935i	$0.452603\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	2.00000	0.0740233
$731$	−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	$0.464656\pi$
$734$	32.0000	1.18114
$735$	7.00000	0.258199
$736$	−1.00000	−0.0368605
$737$	−16.0000	−0.589368
$738$	−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	$0.570839\pi$
$740$	−6.00000	−0.220564
$741$	−24.0000	−0.881662
$742$	0	0
$743$	−24.0000	−0.880475	−0.440237	−	0.897881i	$-0.645106\pi$
$743$	−24.0000	−0.880475	−0.440237	+	0.897881i	$0.645106\pi$
$744$	8.00000	0.293294
$745$	−10.0000	−0.366372
$746$	18.0000	0.659027
$747$	−16.0000	−0.585409
$748$	24.0000	0.877527
$749$	0	0
$750$	1.00000	0.0365148
$751$	−28.0000	−1.02173	−0.510867	−	0.859660i	$-0.670676\pi$
$751$	−28.0000	−1.02173	−0.510867	+	0.859660i	$0.670676\pi$
$752$	−8.00000	−0.291730
$753$	−12.0000	−0.437304
$754$	−36.0000	−1.31104
$755$	−16.0000	−0.582300
$756$	0	0
$757$	54.0000	1.96266	0.981332	−	0.192323i	$-0.0616021\pi$
$757$	54.0000	1.96266	0.981332	+	0.192323i	$0.0616021\pi$
$758$	−4.00000	−0.145287
$759$	−4.00000	−0.145191
$760$	4.00000	0.145095
$761$	10.0000	0.362500	0.181250	−	0.983437i	$-0.441986\pi$
$761$	10.0000	0.362500	0.181250	+	0.983437i	$0.441986\pi$
$762$	−12.0000	−0.434714
$763$	0	0
$764$	−8.00000	−0.289430
$765$	6.00000	0.216930
$766$	−16.0000	−0.578103
$767$	0	0
$768$	1.00000	0.0360844
$769$	−6.00000	−0.216366	−0.108183	−	0.994131i	$-0.534503\pi$
$769$	−6.00000	−0.216366	−0.108183	+	0.994131i	$0.534503\pi$
$770$	0	0
$771$	6.00000	0.216085
$772$	10.0000	0.359908
$773$	2.00000	0.0719350	0.0359675	−	0.999353i	$-0.488549\pi$
$773$	2.00000	0.0719350	0.0359675	+	0.999353i	$0.488549\pi$
$774$	−4.00000	−0.143777
$775$	−8.00000	−0.287368
$776$	14.0000	0.502571
$777$	0	0
$778$	6.00000	0.215110
$779$	40.0000	1.43315
$780$	6.00000	0.214834
$781$	−32.0000	−1.14505
$782$	6.00000	0.214560
$783$	−6.00000	−0.214423
$784$	−7.00000	−0.250000
$785$	−6.00000	−0.214149
$786$	−8.00000	−0.285351
$787$	36.0000	1.28326	0.641631	−	0.767014i	$-0.278258\pi$
$787$	36.0000	1.28326	0.641631	+	0.767014i	$0.278258\pi$
$788$	26.0000	0.926212
$789$	−32.0000	−1.13923
$790$	−12.0000	−0.426941
$791$	0	0
$792$	4.00000	0.142134
$793$	−60.0000	−2.13066
$794$	−2.00000	−0.0709773
$795$	14.0000	0.496529
$796$	−12.0000	−0.425329
$797$	18.0000	0.637593	0.318796	−	0.947823i	$-0.396721\pi$
$797$	18.0000	0.637593	0.318796	+	0.947823i	$0.396721\pi$
$798$	0	0
$799$	48.0000	1.69812
$800$	−1.00000	−0.0353553
$801$	−2.00000	−0.0706665
$802$	−6.00000	−0.211867
$803$	−8.00000	−0.282314
$804$	4.00000	0.141069
$805$	0	0
$806$	−48.0000	−1.69073
$807$	10.0000	0.352017
$808$	−18.0000	−0.633238
$809$	−30.0000	−1.05474	−0.527372	−	0.849635i	$-0.676823\pi$
$809$	−30.0000	−1.05474	−0.527372	+	0.849635i	$0.676823\pi$
$810$	1.00000	0.0351364
$811$	20.0000	0.702295	0.351147	−	0.936320i	$-0.385792\pi$
$811$	20.0000	0.702295	0.351147	+	0.936320i	$0.385792\pi$
$812$	0	0
$813$	−8.00000	−0.280572
$814$	24.0000	0.841200
$815$	−4.00000	−0.140114
$816$	−6.00000	−0.210042
$817$	16.0000	0.559769
$818$	−10.0000	−0.349642
$819$	0	0
$820$	−10.0000	−0.349215
$821$	−46.0000	−1.60541	−0.802706	−	0.596376i	$-0.796607\pi$
$821$	−46.0000	−1.60541	−0.802706	+	0.596376i	$0.796607\pi$
$822$	−2.00000	−0.0697580
$823$	20.0000	0.697156	0.348578	−	0.937280i	$-0.386665\pi$
$823$	20.0000	0.697156	0.348578	+	0.937280i	$0.386665\pi$
$824$	16.0000	0.557386
$825$	−4.00000	−0.139262
$826$	0	0
$827$	−8.00000	−0.278187	−0.139094	−	0.990279i	$-0.544419\pi$
$827$	−8.00000	−0.278187	−0.139094	+	0.990279i	$0.544419\pi$
$828$	1.00000	0.0347524
$829$	−18.0000	−0.625166	−0.312583	−	0.949890i	$-0.601194\pi$
$829$	−18.0000	−0.625166	−0.312583	+	0.949890i	$0.601194\pi$
$830$	−16.0000	−0.555368
$831$	−30.0000	−1.04069
$832$	−6.00000	−0.208013
$833$	42.0000	1.45521
$834$	−12.0000	−0.415526
$835$	16.0000	0.553703
$836$	−16.0000	−0.553372
$837$	−8.00000	−0.276520
$838$	12.0000	0.414533
$839$	−48.0000	−1.65714	−0.828572	−	0.559883i	$-0.810846\pi$
$839$	−48.0000	−1.65714	−0.828572	+	0.559883i	$0.810846\pi$
$840$	0	0
$841$	7.00000	0.241379
$842$	38.0000	1.30957
$843$	30.0000	1.03325
$844$	−20.0000	−0.688428
$845$	−23.0000	−0.791224
$846$	8.00000	0.275046
$847$	0	0
$848$	−14.0000	−0.480762
$849$	4.00000	0.137280
$850$	6.00000	0.205798
$851$	6.00000	0.205677
$852$	8.00000	0.274075
$853$	−22.0000	−0.753266	−0.376633	−	0.926363i	$-0.622918\pi$
$853$	−22.0000	−0.753266	−0.376633	+	0.926363i	$0.622918\pi$
$854$	0	0
$855$	−4.00000	−0.136797
$856$	−8.00000	−0.273434
$857$	−42.0000	−1.43469	−0.717346	−	0.696717i	$-0.754643\pi$
$857$	−42.0000	−1.43469	−0.717346	+	0.696717i	$0.754643\pi$
$858$	−24.0000	−0.819346
$859$	4.00000	0.136478	0.0682391	−	0.997669i	$-0.478262\pi$
$859$	4.00000	0.136478	0.0682391	+	0.997669i	$0.478262\pi$
$860$	−4.00000	−0.136399
$861$	0	0
$862$	−8.00000	−0.272481
$863$	24.0000	0.816970	0.408485	−	0.912765i	$-0.366057\pi$
$863$	24.0000	0.816970	0.408485	+	0.912765i	$0.366057\pi$
$864$	−1.00000	−0.0340207
$865$	−18.0000	−0.612018
$866$	−2.00000	−0.0679628
$867$	19.0000	0.645274
$868$	0	0
$869$	48.0000	1.62829
$870$	−6.00000	−0.203419
$871$	−24.0000	−0.813209
$872$	−2.00000	−0.0677285
$873$	−14.0000	−0.473828
$874$	−4.00000	−0.135302
$875$	0	0
$876$	2.00000	0.0675737
$877$	2.00000	0.0675352	0.0337676	−	0.999430i	$-0.489249\pi$
$877$	2.00000	0.0675352	0.0337676	+	0.999430i	$0.489249\pi$
$878$	24.0000	0.809961
$879$	−14.0000	−0.472208
$880$	4.00000	0.134840
$881$	6.00000	0.202145	0.101073	−	0.994879i	$-0.467773\pi$
$881$	6.00000	0.202145	0.101073	+	0.994879i	$0.467773\pi$
$882$	7.00000	0.235702
$883$	44.0000	1.48072	0.740359	−	0.672212i	$-0.234656\pi$
$883$	44.0000	1.48072	0.740359	+	0.672212i	$0.234656\pi$
$884$	36.0000	1.21081
$885$	0	0
$886$	−12.0000	−0.403148
$887$	8.00000	0.268614	0.134307	−	0.990940i	$-0.457119\pi$
$887$	8.00000	0.268614	0.134307	+	0.990940i	$0.457119\pi$
$888$	−6.00000	−0.201347
$889$	0	0
$890$	−2.00000	−0.0670402
$891$	−4.00000	−0.134005
$892$	−20.0000	−0.669650
$893$	−32.0000	−1.07084
$894$	−10.0000	−0.334450
$895$	0	0
$896$	0	0
$897$	−6.00000	−0.200334
$898$	−10.0000	−0.333704
$899$	48.0000	1.60089
$900$	1.00000	0.0333333
$901$	84.0000	2.79845
$902$	40.0000	1.33185
$903$	0	0
$904$	14.0000	0.465633
$905$	6.00000	0.199447
$906$	−16.0000	−0.531564
$907$	−4.00000	−0.132818	−0.0664089	−	0.997792i	$-0.521154\pi$
$907$	−4.00000	−0.132818	−0.0664089	+	0.997792i	$0.521154\pi$
$908$	8.00000	0.265489
$909$	18.0000	0.597022
$910$	0	0
$911$	8.00000	0.265052	0.132526	−	0.991180i	$-0.457691\pi$
$911$	8.00000	0.265052	0.132526	+	0.991180i	$0.457691\pi$
$912$	4.00000	0.132453
$913$	64.0000	2.11809
$914$	−10.0000	−0.330771
$915$	−10.0000	−0.330590
$916$	−14.0000	−0.462573
$917$	0	0
$918$	6.00000	0.198030
$919$	12.0000	0.395843	0.197922	−	0.980218i	$-0.436581\pi$
$919$	12.0000	0.395843	0.197922	+	0.980218i	$0.436581\pi$
$920$	1.00000	0.0329690
$921$	12.0000	0.395413
$922$	−2.00000	−0.0658665
$923$	−48.0000	−1.57994
$924$	0	0
$925$	6.00000	0.197279
$926$	20.0000	0.657241
$927$	−16.0000	−0.525509
$928$	6.00000	0.196960
$929$	−6.00000	−0.196854	−0.0984268	−	0.995144i	$-0.531381\pi$
$929$	−6.00000	−0.196854	−0.0984268	+	0.995144i	$0.531381\pi$
$930$	−8.00000	−0.262330
$931$	−28.0000	−0.917663
$932$	22.0000	0.720634
$933$	−8.00000	−0.261908
$934$	−8.00000	−0.261768
$935$	−24.0000	−0.784884
$936$	6.00000	0.196116
$937$	10.0000	0.326686	0.163343	−	0.986569i	$-0.447772\pi$
$937$	10.0000	0.326686	0.163343	+	0.986569i	$0.447772\pi$
$938$	0	0
$939$	10.0000	0.326338
$940$	8.00000	0.260931
$941$	−14.0000	−0.456387	−0.228193	−	0.973616i	$-0.573282\pi$
$941$	−14.0000	−0.456387	−0.228193	+	0.973616i	$0.573282\pi$
$942$	−6.00000	−0.195491
$943$	10.0000	0.325645
$944$	0	0
$945$	0	0
$946$	16.0000	0.520205
$947$	−20.0000	−0.649913	−0.324956	−	0.945729i	$-0.605350\pi$
$947$	−20.0000	−0.649913	−0.324956	+	0.945729i	$0.605350\pi$
$948$	−12.0000	−0.389742
$949$	−12.0000	−0.389536
$950$	−4.00000	−0.129777
$951$	18.0000	0.583690
$952$	0	0
$953$	−22.0000	−0.712650	−0.356325	−	0.934362i	$-0.615970\pi$
$953$	−22.0000	−0.712650	−0.356325	+	0.934362i	$0.615970\pi$
$954$	14.0000	0.453267
$955$	8.00000	0.258874
$956$	−8.00000	−0.258738
$957$	24.0000	0.775810
$958$	−24.0000	−0.775405
$959$	0	0
$960$	−1.00000	−0.0322749
$961$	33.0000	1.06452
$962$	36.0000	1.16069
$963$	8.00000	0.257796
$964$	2.00000	0.0644157
$965$	−10.0000	−0.321911
$966$	0	0
$967$	28.0000	0.900419	0.450210	−	0.892923i	$-0.351349\pi$
$967$	28.0000	0.900419	0.450210	+	0.892923i	$0.351349\pi$
$968$	−5.00000	−0.160706
$969$	−24.0000	−0.770991
$970$	−14.0000	−0.449513
$971$	12.0000	0.385098	0.192549	−	0.981287i	$-0.438325\pi$
$971$	12.0000	0.385098	0.192549	+	0.981287i	$0.438325\pi$
$972$	1.00000	0.0320750
$973$	0	0
$974$	−20.0000	−0.640841
$975$	−6.00000	−0.192154
$976$	10.0000	0.320092
$977$	2.00000	0.0639857	0.0319928	−	0.999488i	$-0.489815\pi$
$977$	2.00000	0.0639857	0.0319928	+	0.999488i	$0.489815\pi$
$978$	−4.00000	−0.127906
$979$	8.00000	0.255681
$980$	7.00000	0.223607
$981$	2.00000	0.0638551
$982$	0	0
$983$	−24.0000	−0.765481	−0.382741	−	0.923856i	$-0.625020\pi$
$983$	−24.0000	−0.765481	−0.382741	+	0.923856i	$0.625020\pi$
$984$	−10.0000	−0.318788
$985$	−26.0000	−0.828429
$986$	−36.0000	−1.14647
$987$	0	0
$988$	−24.0000	−0.763542
$989$	4.00000	0.127193
$990$	−4.00000	−0.127128
$991$	0	0		−	1.00000i	$-0.5\pi$
$991$	0	0			1.00000i	$0.5\pi$
$992$	8.00000	0.254000
$993$	−20.0000	−0.634681
$994$	0	0
$995$	12.0000	0.380426
$996$	−16.0000	−0.506979
$997$	−22.0000	−0.696747	−0.348373	−	0.937356i	$-0.613266\pi$
$997$	−22.0000	−0.696747	−0.348373	+	0.937356i	$0.613266\pi$
$998$	−12.0000	−0.379853
$999$	6.00000	0.189832

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
690.2.a.e.1.1	✓	1	1.1	even	1	trivial
2070.2.a.r.1.1		1	3.2	odd	2
3450.2.a.p.1.1		1	5.4	even	2
3450.2.d.a.2899.1		2	5.2	odd	4
3450.2.d.a.2899.2		2	5.3	odd	4
5520.2.a.f.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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