LMFDB - Embedded newform 390.2.a.f.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [390,2,Mod(1,390)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(390, 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("390.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 390 = 2 \cdot 3 \cdot 5 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	390.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$3.11416567883$
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	$\chi$	$=$	390.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

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

\(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} -1.00000 q^{6} +1.00000 q^{8} +1.00000 q^{9} +1.00000 q^{10} +4.00000 q^{11} -1.00000 q^{12} +1.00000 q^{13} -1.00000 q^{15} +1.00000 q^{16} -6.00000 q^{17} +1.00000 q^{18} +4.00000 q^{19} +1.00000 q^{20} +4.00000 q^{22} +8.00000 q^{23} -1.00000 q^{24} +1.00000 q^{25} +1.00000 q^{26} -1.00000 q^{27} +6.00000 q^{29} -1.00000 q^{30} -8.00000 q^{31} +1.00000 q^{32} -4.00000 q^{33} -6.00000 q^{34} +1.00000 q^{36} -10.0000 q^{37} +4.00000 q^{38} -1.00000 q^{39} +1.00000 q^{40} -6.00000 q^{41} +4.00000 q^{43} +4.00000 q^{44} +1.00000 q^{45} +8.00000 q^{46} -1.00000 q^{48} -7.00000 q^{49} +1.00000 q^{50} +6.00000 q^{51} +1.00000 q^{52} -10.0000 q^{53} -1.00000 q^{54} +4.00000 q^{55} -4.00000 q^{57} +6.00000 q^{58} +4.00000 q^{59} -1.00000 q^{60} -2.00000 q^{61} -8.00000 q^{62} +1.00000 q^{64} +1.00000 q^{65} -4.00000 q^{66} -12.0000 q^{67} -6.00000 q^{68} -8.00000 q^{69} +16.0000 q^{71} +1.00000 q^{72} +2.00000 q^{73} -10.0000 q^{74} -1.00000 q^{75} +4.00000 q^{76} -1.00000 q^{78} -16.0000 q^{79} +1.00000 q^{80} +1.00000 q^{81} -6.00000 q^{82} -12.0000 q^{83} -6.00000 q^{85} +4.00000 q^{86} -6.00000 q^{87} +4.00000 q^{88} +10.0000 q^{89} +1.00000 q^{90} +8.00000 q^{92} +8.00000 q^{93} +4.00000 q^{95} -1.00000 q^{96} -6.00000 q^{97} -7.00000 q^{98} +4.00000 q^{99} +O(q^{100})\)

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
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	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
$23$	8.00000	1.66812	0.834058	−	0.551677i	$-0.186012\pi$
$23$	8.00000	1.66812	0.834058	+	0.551677i	$0.186012\pi$
$24$	−1.00000	−0.204124
$25$	1.00000	0.200000
$26$	1.00000	0.196116
$27$	−1.00000	−0.192450
$28$	0	0
$29$	6.00000	1.11417	0.557086	−	0.830455i	$-0.311919\pi$
$29$	6.00000	1.11417	0.557086	+	0.830455i	$0.311919\pi$
$30$	−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
$37$	−10.0000	−1.64399	−0.821995	−	0.569495i	$-0.807139\pi$
$37$	−10.0000	−1.64399	−0.821995	+	0.569495i	$0.807139\pi$
$38$	4.00000	0.648886
$39$	−1.00000	−0.160128
$40$	1.00000	0.158114
$41$	−6.00000	−0.937043	−0.468521	−	0.883452i	$-0.655213\pi$
$41$	−6.00000	−0.937043	−0.468521	+	0.883452i	$0.655213\pi$
$42$	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
$46$	8.00000	1.17954
$47$	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
$52$	1.00000	0.138675
$53$	−10.0000	−1.37361	−0.686803	−	0.726844i	$-0.740986\pi$
$53$	−10.0000	−1.37361	−0.686803	+	0.726844i	$0.740986\pi$
$54$	−1.00000	−0.136083
$55$	4.00000	0.539360
$56$	0	0
$57$	−4.00000	−0.529813
$58$	6.00000	0.787839
$59$	4.00000	0.520756	0.260378	−	0.965507i	$-0.416153\pi$
$59$	4.00000	0.520756	0.260378	+	0.965507i	$0.416153\pi$
$60$	−1.00000	−0.129099
$61$	−2.00000	−0.256074	−0.128037	−	0.991769i	$-0.540868\pi$
$61$	−2.00000	−0.256074	−0.128037	+	0.991769i	$0.540868\pi$
$62$	−8.00000	−1.01600
$63$	0	0
$64$	1.00000	0.125000
$65$	1.00000	0.124035
$66$	−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
$69$	−8.00000	−0.963087
$70$	0	0
$71$	16.0000	1.89885	0.949425	−	0.313993i	$-0.101667\pi$
$71$	16.0000	1.89885	0.949425	+	0.313993i	$0.101667\pi$
$72$	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	$0.462659\pi$
$74$	−10.0000	−1.16248
$75$	−1.00000	−0.115470
$76$	4.00000	0.458831
$77$	0	0
$78$	−1.00000	−0.113228
$79$	−16.0000	−1.80014	−0.900070	−	0.435745i	$-0.856485\pi$
$79$	−16.0000	−1.80014	−0.900070	+	0.435745i	$0.856485\pi$
$80$	1.00000	0.111803
$81$	1.00000	0.111111
$82$	−6.00000	−0.662589
$83$	−12.0000	−1.31717	−0.658586	−	0.752506i	$-0.728845\pi$
$83$	−12.0000	−1.31717	−0.658586	+	0.752506i	$0.728845\pi$
$84$	0	0
$85$	−6.00000	−0.650791
$86$	4.00000	0.431331
$87$	−6.00000	−0.643268
$88$	4.00000	0.426401
$89$	10.0000	1.06000	0.529999	−	0.847998i	$-0.322192\pi$
$89$	10.0000	1.06000	0.529999	+	0.847998i	$0.322192\pi$
$90$	1.00000	0.105409
$91$	0	0
$92$	8.00000	0.834058
$93$	8.00000	0.829561
$94$	0	0
$95$	4.00000	0.410391
$96$	−1.00000	−0.102062
$97$	−6.00000	−0.609208	−0.304604	−	0.952479i	$-0.598524\pi$
$97$	−6.00000	−0.609208	−0.304604	+	0.952479i	$0.598524\pi$
$98$	−7.00000	−0.707107
$99$	4.00000	0.402015
$100$	1.00000	0.100000
$101$	−2.00000	−0.199007	−0.0995037	−	0.995037i	$-0.531726\pi$
$101$	−2.00000	−0.199007	−0.0995037	+	0.995037i	$0.531726\pi$
$102$	6.00000	0.594089
$103$	0	0		−	1.00000i	$-0.5\pi$
$103$	0	0			1.00000i	$0.5\pi$
$104$	1.00000	0.0980581
$105$	0	0
$106$	−10.0000	−0.971286
$107$	−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
$111$	10.0000	0.949158
$112$	0	0
$113$	10.0000	0.940721	0.470360	−	0.882474i	$-0.344124\pi$
$113$	10.0000	0.940721	0.470360	+	0.882474i	$0.344124\pi$
$114$	−4.00000	−0.374634
$115$	8.00000	0.746004
$116$	6.00000	0.557086
$117$	1.00000	0.0924500
$118$	4.00000	0.368230
$119$	0	0
$120$	−1.00000	−0.0912871
$121$	5.00000	0.454545
$122$	−2.00000	−0.181071
$123$	6.00000	0.541002
$124$	−8.00000	−0.718421
$125$	1.00000	0.0894427
$126$	0	0
$127$	8.00000	0.709885	0.354943	−	0.934888i	$-0.384500\pi$
$127$	8.00000	0.709885	0.354943	+	0.934888i	$0.384500\pi$
$128$	1.00000	0.0883883
$129$	−4.00000	−0.352180
$130$	1.00000	0.0877058
$131$	12.0000	1.04844	0.524222	−	0.851581i	$-0.324356\pi$
$131$	12.0000	1.04844	0.524222	+	0.851581i	$0.324356\pi$
$132$	−4.00000	−0.348155
$133$	0	0
$134$	−12.0000	−1.03664
$135$	−1.00000	−0.0860663
$136$	−6.00000	−0.514496
$137$	10.0000	0.854358	0.427179	−	0.904167i	$-0.359507\pi$
$137$	10.0000	0.854358	0.427179	+	0.904167i	$0.359507\pi$
$138$	−8.00000	−0.681005
$139$	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
$141$	0	0
$142$	16.0000	1.34269
$143$	4.00000	0.334497
$144$	1.00000	0.0833333
$145$	6.00000	0.498273
$146$	2.00000	0.165521
$147$	7.00000	0.577350
$148$	−10.0000	−0.821995
$149$	6.00000	0.491539	0.245770	−	0.969328i	$-0.420959\pi$
$149$	6.00000	0.491539	0.245770	+	0.969328i	$0.420959\pi$
$150$	−1.00000	−0.0816497
$151$	0	0		−	1.00000i	$-0.5\pi$
$151$	0	0			1.00000i	$0.5\pi$
$152$	4.00000	0.324443
$153$	−6.00000	−0.485071
$154$	0	0
$155$	−8.00000	−0.642575
$156$	−1.00000	−0.0800641
$157$	−2.00000	−0.159617	−0.0798087	−	0.996810i	$-0.525431\pi$
$157$	−2.00000	−0.159617	−0.0798087	+	0.996810i	$0.525431\pi$
$158$	−16.0000	−1.27289
$159$	10.0000	0.793052
$160$	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	$0.449930\pi$
$164$	−6.00000	−0.468521
$165$	−4.00000	−0.311400
$166$	−12.0000	−0.931381
$167$	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
$169$	1.00000	0.0769231
$170$	−6.00000	−0.460179
$171$	4.00000	0.305888
$172$	4.00000	0.304997
$173$	−18.0000	−1.36851	−0.684257	−	0.729241i	$-0.739873\pi$
$173$	−18.0000	−1.36851	−0.684257	+	0.729241i	$0.739873\pi$
$174$	−6.00000	−0.454859
$175$	0	0
$176$	4.00000	0.301511
$177$	−4.00000	−0.300658
$178$	10.0000	0.749532
$179$	12.0000	0.896922	0.448461	−	0.893802i	$-0.351972\pi$
$179$	12.0000	0.896922	0.448461	+	0.893802i	$0.351972\pi$
$180$	1.00000	0.0745356
$181$	6.00000	0.445976	0.222988	−	0.974821i	$-0.428419\pi$
$181$	6.00000	0.445976	0.222988	+	0.974821i	$0.428419\pi$
$182$	0	0
$183$	2.00000	0.147844
$184$	8.00000	0.589768
$185$	−10.0000	−0.735215
$186$	8.00000	0.586588
$187$	−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
$193$	10.0000	0.719816	0.359908	−	0.932988i	$-0.382808\pi$
$193$	10.0000	0.719816	0.359908	+	0.932988i	$0.382808\pi$
$194$	−6.00000	−0.430775
$195$	−1.00000	−0.0716115
$196$	−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
$199$	−8.00000	−0.567105	−0.283552	−	0.958957i	$-0.591513\pi$
$199$	−8.00000	−0.567105	−0.283552	+	0.958957i	$0.591513\pi$
$200$	1.00000	0.0707107
$201$	12.0000	0.846415
$202$	−2.00000	−0.140720
$203$	0	0
$204$	6.00000	0.420084
$205$	−6.00000	−0.419058
$206$	0	0
$207$	8.00000	0.556038
$208$	1.00000	0.0693375
$209$	16.0000	1.10674
$210$	0	0
$211$	−12.0000	−0.826114	−0.413057	−	0.910705i	$-0.635539\pi$
$211$	−12.0000	−0.826114	−0.413057	+	0.910705i	$0.635539\pi$
$212$	−10.0000	−0.686803
$213$	−16.0000	−1.09630
$214$	−12.0000	−0.820303
$215$	4.00000	0.272798
$216$	−1.00000	−0.0680414
$217$	0	0
$218$	−10.0000	−0.677285
$219$	−2.00000	−0.135147
$220$	4.00000	0.269680
$221$	−6.00000	−0.403604
$222$	10.0000	0.671156
$223$	8.00000	0.535720	0.267860	−	0.963458i	$-0.413684\pi$
$223$	8.00000	0.535720	0.267860	+	0.963458i	$0.413684\pi$
$224$	0	0
$225$	1.00000	0.0666667
$226$	10.0000	0.665190
$227$	4.00000	0.265489	0.132745	−	0.991150i	$-0.457621\pi$
$227$	4.00000	0.265489	0.132745	+	0.991150i	$0.457621\pi$
$228$	−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	$0.346926\pi$
$230$	8.00000	0.527504
$231$	0	0
$232$	6.00000	0.393919
$233$	−14.0000	−0.917170	−0.458585	−	0.888650i	$-0.651644\pi$
$233$	−14.0000	−0.917170	−0.458585	+	0.888650i	$0.651644\pi$
$234$	1.00000	0.0653720
$235$	0	0
$236$	4.00000	0.260378
$237$	16.0000	1.03931
$238$	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
$241$	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
$244$	−2.00000	−0.128037
$245$	−7.00000	−0.447214
$246$	6.00000	0.382546
$247$	4.00000	0.254514
$248$	−8.00000	−0.508001
$249$	12.0000	0.760469
$250$	1.00000	0.0632456
$251$	−28.0000	−1.76734	−0.883672	−	0.468106i	$-0.844936\pi$
$251$	−28.0000	−1.76734	−0.883672	+	0.468106i	$0.844936\pi$
$252$	0	0
$253$	32.0000	2.01182
$254$	8.00000	0.501965
$255$	6.00000	0.375735
$256$	1.00000	0.0625000
$257$	−6.00000	−0.374270	−0.187135	−	0.982334i	$-0.559920\pi$
$257$	−6.00000	−0.374270	−0.187135	+	0.982334i	$0.559920\pi$
$258$	−4.00000	−0.249029
$259$	0	0
$260$	1.00000	0.0620174
$261$	6.00000	0.371391
$262$	12.0000	0.741362
$263$	24.0000	1.47990	0.739952	−	0.672660i	$-0.234848\pi$
$263$	24.0000	1.47990	0.739952	+	0.672660i	$0.234848\pi$
$264$	−4.00000	−0.246183
$265$	−10.0000	−0.614295
$266$	0	0
$267$	−10.0000	−0.611990
$268$	−12.0000	−0.733017
$269$	−26.0000	−1.58525	−0.792624	−	0.609711i	$-0.791286\pi$
$269$	−26.0000	−1.58525	−0.792624	+	0.609711i	$0.791286\pi$
$270$	−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
$274$	10.0000	0.604122
$275$	4.00000	0.241209
$276$	−8.00000	−0.481543
$277$	−10.0000	−0.600842	−0.300421	−	0.953807i	$-0.597127\pi$
$277$	−10.0000	−0.600842	−0.300421	+	0.953807i	$0.597127\pi$
$278$	12.0000	0.719712
$279$	−8.00000	−0.478947
$280$	0	0
$281$	10.0000	0.596550	0.298275	−	0.954480i	$-0.403589\pi$
$281$	10.0000	0.596550	0.298275	+	0.954480i	$0.403589\pi$
$282$	0	0
$283$	20.0000	1.18888	0.594438	−	0.804141i	$-0.297374\pi$
$283$	20.0000	1.18888	0.594438	+	0.804141i	$0.297374\pi$
$284$	16.0000	0.949425
$285$	−4.00000	−0.236940
$286$	4.00000	0.236525
$287$	0	0
$288$	1.00000	0.0589256
$289$	19.0000	1.11765
$290$	6.00000	0.352332
$291$	6.00000	0.351726
$292$	2.00000	0.117041
$293$	−26.0000	−1.51894	−0.759468	−	0.650545i	$-0.774541\pi$
$293$	−26.0000	−1.51894	−0.759468	+	0.650545i	$0.774541\pi$
$294$	7.00000	0.408248
$295$	4.00000	0.232889
$296$	−10.0000	−0.581238
$297$	−4.00000	−0.232104
$298$	6.00000	0.347571
$299$	8.00000	0.462652
$300$	−1.00000	−0.0577350
$301$	0	0
$302$	0	0
$303$	2.00000	0.114897
$304$	4.00000	0.229416
$305$	−2.00000	−0.114520
$306$	−6.00000	−0.342997
$307$	−12.0000	−0.684876	−0.342438	−	0.939540i	$-0.611253\pi$
$307$	−12.0000	−0.684876	−0.342438	+	0.939540i	$0.611253\pi$
$308$	0	0
$309$	0	0
$310$	−8.00000	−0.454369
$311$	−24.0000	−1.36092	−0.680458	−	0.732787i	$-0.738219\pi$
$311$	−24.0000	−1.36092	−0.680458	+	0.732787i	$0.738219\pi$
$312$	−1.00000	−0.0566139
$313$	−22.0000	−1.24351	−0.621757	−	0.783210i	$-0.713581\pi$
$313$	−22.0000	−1.24351	−0.621757	+	0.783210i	$0.713581\pi$
$314$	−2.00000	−0.112867
$315$	0	0
$316$	−16.0000	−0.900070
$317$	−2.00000	−0.112331	−0.0561656	−	0.998421i	$-0.517887\pi$
$317$	−2.00000	−0.112331	−0.0561656	+	0.998421i	$0.517887\pi$
$318$	10.0000	0.560772
$319$	24.0000	1.34374
$320$	1.00000	0.0559017
$321$	12.0000	0.669775
$322$	0	0
$323$	−24.0000	−1.33540
$324$	1.00000	0.0555556
$325$	1.00000	0.0554700
$326$	4.00000	0.221540
$327$	10.0000	0.553001
$328$	−6.00000	−0.331295
$329$	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	$0.685239\pi$
$332$	−12.0000	−0.658586
$333$	−10.0000	−0.547997
$334$	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	$0.482652\pi$
$338$	1.00000	0.0543928
$339$	−10.0000	−0.543125
$340$	−6.00000	−0.325396
$341$	−32.0000	−1.73290
$342$	4.00000	0.216295
$343$	0	0
$344$	4.00000	0.215666
$345$	−8.00000	−0.430706
$346$	−18.0000	−0.967686
$347$	20.0000	1.07366	0.536828	−	0.843692i	$-0.319622\pi$
$347$	20.0000	1.07366	0.536828	+	0.843692i	$0.319622\pi$
$348$	−6.00000	−0.321634
$349$	6.00000	0.321173	0.160586	−	0.987022i	$-0.448662\pi$
$349$	6.00000	0.321173	0.160586	+	0.987022i	$0.448662\pi$
$350$	0	0
$351$	−1.00000	−0.0533761
$352$	4.00000	0.213201
$353$	−30.0000	−1.59674	−0.798369	−	0.602168i	$-0.794304\pi$
$353$	−30.0000	−1.59674	−0.798369	+	0.602168i	$0.794304\pi$
$354$	−4.00000	−0.212598
$355$	16.0000	0.849192
$356$	10.0000	0.529999
$357$	0	0
$358$	12.0000	0.634220
$359$	16.0000	0.844448	0.422224	−	0.906492i	$-0.361250\pi$
$359$	16.0000	0.844448	0.422224	+	0.906492i	$0.361250\pi$
$360$	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
$366$	2.00000	0.104542
$367$	−24.0000	−1.25279	−0.626395	−	0.779506i	$-0.715470\pi$
$367$	−24.0000	−1.25279	−0.626395	+	0.779506i	$0.715470\pi$
$368$	8.00000	0.417029
$369$	−6.00000	−0.312348
$370$	−10.0000	−0.519875
$371$	0	0
$372$	8.00000	0.414781
$373$	22.0000	1.13912	0.569558	−	0.821951i	$-0.307114\pi$
$373$	22.0000	1.13912	0.569558	+	0.821951i	$0.307114\pi$
$374$	−24.0000	−1.24101
$375$	−1.00000	−0.0516398
$376$	0	0
$377$	6.00000	0.309016
$378$	0	0
$379$	−4.00000	−0.205466	−0.102733	−	0.994709i	$-0.532759\pi$
$379$	−4.00000	−0.205466	−0.102733	+	0.994709i	$0.532759\pi$
$380$	4.00000	0.205196
$381$	−8.00000	−0.409852
$382$	0	0
$383$	32.0000	1.63512	0.817562	−	0.575841i	$-0.195325\pi$
$383$	32.0000	1.63512	0.817562	+	0.575841i	$0.195325\pi$
$384$	−1.00000	−0.0510310
$385$	0	0
$386$	10.0000	0.508987
$387$	4.00000	0.203331
$388$	−6.00000	−0.304604
$389$	−34.0000	−1.72387	−0.861934	−	0.507020i	$-0.830747\pi$
$389$	−34.0000	−1.72387	−0.861934	+	0.507020i	$0.830747\pi$
$390$	−1.00000	−0.0506370
$391$	−48.0000	−2.42746
$392$	−7.00000	−0.353553
$393$	−12.0000	−0.605320
$394$	6.00000	0.302276
$395$	−16.0000	−0.805047
$396$	4.00000	0.201008
$397$	30.0000	1.50566	0.752828	−	0.658217i	$-0.228689\pi$
$397$	30.0000	1.50566	0.752828	+	0.658217i	$0.228689\pi$
$398$	−8.00000	−0.401004
$399$	0	0
$400$	1.00000	0.0500000
$401$	34.0000	1.69788	0.848939	−	0.528490i	$-0.177242\pi$
$401$	34.0000	1.69788	0.848939	+	0.528490i	$0.177242\pi$
$402$	12.0000	0.598506
$403$	−8.00000	−0.398508
$404$	−2.00000	−0.0995037
$405$	1.00000	0.0496904
$406$	0	0
$407$	−40.0000	−1.98273
$408$	6.00000	0.297044
$409$	−22.0000	−1.08783	−0.543915	−	0.839140i	$-0.683059\pi$
$409$	−22.0000	−1.08783	−0.543915	+	0.839140i	$0.683059\pi$
$410$	−6.00000	−0.296319
$411$	−10.0000	−0.493264
$412$	0	0
$413$	0	0
$414$	8.00000	0.393179
$415$	−12.0000	−0.589057
$416$	1.00000	0.0490290
$417$	−12.0000	−0.587643
$418$	16.0000	0.782586
$419$	−4.00000	−0.195413	−0.0977064	−	0.995215i	$-0.531151\pi$
$419$	−4.00000	−0.195413	−0.0977064	+	0.995215i	$0.531151\pi$
$420$	0	0
$421$	−34.0000	−1.65706	−0.828529	−	0.559946i	$-0.810822\pi$
$421$	−34.0000	−1.65706	−0.828529	+	0.559946i	$0.810822\pi$
$422$	−12.0000	−0.584151
$423$	0	0
$424$	−10.0000	−0.485643
$425$	−6.00000	−0.291043
$426$	−16.0000	−0.775203
$427$	0	0
$428$	−12.0000	−0.580042
$429$	−4.00000	−0.193122
$430$	4.00000	0.192897
$431$	−24.0000	−1.15604	−0.578020	−	0.816023i	$-0.696174\pi$
$431$	−24.0000	−1.15604	−0.578020	+	0.816023i	$0.696174\pi$
$432$	−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
$435$	−6.00000	−0.287678
$436$	−10.0000	−0.478913
$437$	32.0000	1.53077
$438$	−2.00000	−0.0955637
$439$	−40.0000	−1.90910	−0.954548	−	0.298057i	$-0.903661\pi$
$439$	−40.0000	−1.90910	−0.954548	+	0.298057i	$0.903661\pi$
$440$	4.00000	0.190693
$441$	−7.00000	−0.333333
$442$	−6.00000	−0.285391
$443$	36.0000	1.71041	0.855206	−	0.518289i	$-0.173431\pi$
$443$	36.0000	1.71041	0.855206	+	0.518289i	$0.173431\pi$
$444$	10.0000	0.474579
$445$	10.0000	0.474045
$446$	8.00000	0.378811
$447$	−6.00000	−0.283790
$448$	0	0
$449$	18.0000	0.849473	0.424736	−	0.905317i	$-0.360367\pi$
$449$	18.0000	0.849473	0.424736	+	0.905317i	$0.360367\pi$
$450$	1.00000	0.0471405
$451$	−24.0000	−1.13012
$452$	10.0000	0.470360
$453$	0	0
$454$	4.00000	0.187729
$455$	0	0
$456$	−4.00000	−0.187317
$457$	34.0000	1.59045	0.795226	−	0.606313i	$-0.207352\pi$
$457$	34.0000	1.59045	0.795226	+	0.606313i	$0.207352\pi$
$458$	14.0000	0.654177
$459$	6.00000	0.280056
$460$	8.00000	0.373002
$461$	−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
$463$	8.00000	0.371792	0.185896	−	0.982569i	$-0.440481\pi$
$463$	8.00000	0.371792	0.185896	+	0.982569i	$0.440481\pi$
$464$	6.00000	0.278543
$465$	8.00000	0.370991
$466$	−14.0000	−0.648537
$467$	12.0000	0.555294	0.277647	−	0.960683i	$-0.410445\pi$
$467$	12.0000	0.555294	0.277647	+	0.960683i	$0.410445\pi$
$468$	1.00000	0.0462250
$469$	0	0
$470$	0	0
$471$	2.00000	0.0921551
$472$	4.00000	0.184115
$473$	16.0000	0.735681
$474$	16.0000	0.734904
$475$	4.00000	0.183533
$476$	0	0
$477$	−10.0000	−0.457869
$478$	−8.00000	−0.365911
$479$	−8.00000	−0.365529	−0.182765	−	0.983157i	$-0.558505\pi$
$479$	−8.00000	−0.365529	−0.182765	+	0.983157i	$0.558505\pi$
$480$	−1.00000	−0.0456435
$481$	−10.0000	−0.455961
$482$	18.0000	0.819878
$483$	0	0
$484$	5.00000	0.227273
$485$	−6.00000	−0.272446
$486$	−1.00000	−0.0453609
$487$	−16.0000	−0.725029	−0.362515	−	0.931978i	$-0.618082\pi$
$487$	−16.0000	−0.725029	−0.362515	+	0.931978i	$0.618082\pi$
$488$	−2.00000	−0.0905357
$489$	−4.00000	−0.180886
$490$	−7.00000	−0.316228
$491$	4.00000	0.180517	0.0902587	−	0.995918i	$-0.471231\pi$
$491$	4.00000	0.180517	0.0902587	+	0.995918i	$0.471231\pi$
$492$	6.00000	0.270501
$493$	−36.0000	−1.62136
$494$	4.00000	0.179969
$495$	4.00000	0.179787
$496$	−8.00000	−0.359211
$497$	0	0
$498$	12.0000	0.537733
$499$	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
$502$	−28.0000	−1.24970
$503$	8.00000	0.356702	0.178351	−	0.983967i	$-0.442924\pi$
$503$	8.00000	0.356702	0.178351	+	0.983967i	$0.442924\pi$
$504$	0	0
$505$	−2.00000	−0.0889988
$506$	32.0000	1.42257
$507$	−1.00000	−0.0444116
$508$	8.00000	0.354943
$509$	30.0000	1.32973	0.664863	−	0.746965i	$-0.268490\pi$
$509$	30.0000	1.32973	0.664863	+	0.746965i	$0.268490\pi$
$510$	6.00000	0.265684
$511$	0	0
$512$	1.00000	0.0441942
$513$	−4.00000	−0.176604
$514$	−6.00000	−0.264649
$515$	0	0
$516$	−4.00000	−0.176090
$517$	0	0
$518$	0	0
$519$	18.0000	0.790112
$520$	1.00000	0.0438529
$521$	−6.00000	−0.262865	−0.131432	−	0.991325i	$-0.541958\pi$
$521$	−6.00000	−0.262865	−0.131432	+	0.991325i	$0.541958\pi$
$522$	6.00000	0.262613
$523$	36.0000	1.57417	0.787085	−	0.616844i	$-0.211589\pi$
$523$	36.0000	1.57417	0.787085	+	0.616844i	$0.211589\pi$
$524$	12.0000	0.524222
$525$	0	0
$526$	24.0000	1.04645
$527$	48.0000	2.09091
$528$	−4.00000	−0.174078
$529$	41.0000	1.78261
$530$	−10.0000	−0.434372
$531$	4.00000	0.173585
$532$	0	0
$533$	−6.00000	−0.259889
$534$	−10.0000	−0.432742
$535$	−12.0000	−0.518805
$536$	−12.0000	−0.518321
$537$	−12.0000	−0.517838
$538$	−26.0000	−1.12094
$539$	−28.0000	−1.20605
$540$	−1.00000	−0.0430331
$541$	22.0000	0.945854	0.472927	−	0.881102i	$-0.343197\pi$
$541$	22.0000	0.945854	0.472927	+	0.881102i	$0.343197\pi$
$542$	24.0000	1.03089
$543$	−6.00000	−0.257485
$544$	−6.00000	−0.257248
$545$	−10.0000	−0.428353
$546$	0	0
$547$	−20.0000	−0.855138	−0.427569	−	0.903983i	$-0.640630\pi$
$547$	−20.0000	−0.855138	−0.427569	+	0.903983i	$0.640630\pi$
$548$	10.0000	0.427179
$549$	−2.00000	−0.0853579
$550$	4.00000	0.170561
$551$	24.0000	1.02243
$552$	−8.00000	−0.340503
$553$	0	0
$554$	−10.0000	−0.424859
$555$	10.0000	0.424476
$556$	12.0000	0.508913
$557$	−18.0000	−0.762684	−0.381342	−	0.924434i	$-0.624538\pi$
$557$	−18.0000	−0.762684	−0.381342	+	0.924434i	$0.624538\pi$
$558$	−8.00000	−0.338667
$559$	4.00000	0.169182
$560$	0	0
$561$	24.0000	1.01328
$562$	10.0000	0.421825
$563$	12.0000	0.505740	0.252870	−	0.967500i	$-0.418626\pi$
$563$	12.0000	0.505740	0.252870	+	0.967500i	$0.418626\pi$
$564$	0	0
$565$	10.0000	0.420703
$566$	20.0000	0.840663
$567$	0	0
$568$	16.0000	0.671345
$569$	10.0000	0.419222	0.209611	−	0.977785i	$-0.432780\pi$
$569$	10.0000	0.419222	0.209611	+	0.977785i	$0.432780\pi$
$570$	−4.00000	−0.167542
$571$	28.0000	1.17176	0.585882	−	0.810397i	$-0.300748\pi$
$571$	28.0000	1.17176	0.585882	+	0.810397i	$0.300748\pi$
$572$	4.00000	0.167248
$573$	0	0
$574$	0	0
$575$	8.00000	0.333623
$576$	1.00000	0.0416667
$577$	42.0000	1.74848	0.874241	−	0.485491i	$-0.161359\pi$
$577$	42.0000	1.74848	0.874241	+	0.485491i	$0.161359\pi$
$578$	19.0000	0.790296
$579$	−10.0000	−0.415586
$580$	6.00000	0.249136
$581$	0	0
$582$	6.00000	0.248708
$583$	−40.0000	−1.65663
$584$	2.00000	0.0827606
$585$	1.00000	0.0413449
$586$	−26.0000	−1.07405
$587$	12.0000	0.495293	0.247647	−	0.968850i	$-0.420343\pi$
$587$	12.0000	0.495293	0.247647	+	0.968850i	$0.420343\pi$
$588$	7.00000	0.288675
$589$	−32.0000	−1.31854
$590$	4.00000	0.164677
$591$	−6.00000	−0.246807
$592$	−10.0000	−0.410997
$593$	34.0000	1.39621	0.698106	−	0.715994i	$-0.254026\pi$
$593$	34.0000	1.39621	0.698106	+	0.715994i	$0.254026\pi$
$594$	−4.00000	−0.164122
$595$	0	0
$596$	6.00000	0.245770
$597$	8.00000	0.327418
$598$	8.00000	0.327144
$599$	−24.0000	−0.980613	−0.490307	−	0.871550i	$-0.663115\pi$
$599$	−24.0000	−0.980613	−0.490307	+	0.871550i	$0.663115\pi$
$600$	−1.00000	−0.0408248
$601$	−6.00000	−0.244745	−0.122373	−	0.992484i	$-0.539050\pi$
$601$	−6.00000	−0.244745	−0.122373	+	0.992484i	$0.539050\pi$
$602$	0	0
$603$	−12.0000	−0.488678
$604$	0	0
$605$	5.00000	0.203279
$606$	2.00000	0.0812444
$607$	−8.00000	−0.324710	−0.162355	−	0.986732i	$-0.551909\pi$
$607$	−8.00000	−0.324710	−0.162355	+	0.986732i	$0.551909\pi$
$608$	4.00000	0.162221
$609$	0	0
$610$	−2.00000	−0.0809776
$611$	0	0
$612$	−6.00000	−0.242536
$613$	−10.0000	−0.403896	−0.201948	−	0.979396i	$-0.564727\pi$
$613$	−10.0000	−0.403896	−0.201948	+	0.979396i	$0.564727\pi$
$614$	−12.0000	−0.484281
$615$	6.00000	0.241943
$616$	0	0
$617$	42.0000	1.69086	0.845428	−	0.534089i	$-0.179345\pi$
$617$	42.0000	1.69086	0.845428	+	0.534089i	$0.179345\pi$
$618$	0	0
$619$	−4.00000	−0.160774	−0.0803868	−	0.996764i	$-0.525616\pi$
$619$	−4.00000	−0.160774	−0.0803868	+	0.996764i	$0.525616\pi$
$620$	−8.00000	−0.321288
$621$	−8.00000	−0.321029
$622$	−24.0000	−0.962312
$623$	0	0
$624$	−1.00000	−0.0400320
$625$	1.00000	0.0400000
$626$	−22.0000	−0.879297
$627$	−16.0000	−0.638978
$628$	−2.00000	−0.0798087
$629$	60.0000	2.39236
$630$	0	0
$631$	−16.0000	−0.636950	−0.318475	−	0.947931i	$-0.603171\pi$
$631$	−16.0000	−0.636950	−0.318475	+	0.947931i	$0.603171\pi$
$632$	−16.0000	−0.636446
$633$	12.0000	0.476957
$634$	−2.00000	−0.0794301
$635$	8.00000	0.317470
$636$	10.0000	0.396526
$637$	−7.00000	−0.277350
$638$	24.0000	0.950169
$639$	16.0000	0.632950
$640$	1.00000	0.0395285
$641$	−30.0000	−1.18493	−0.592464	−	0.805597i	$-0.701845\pi$
$641$	−30.0000	−1.18493	−0.592464	+	0.805597i	$0.701845\pi$
$642$	12.0000	0.473602
$643$	4.00000	0.157745	0.0788723	−	0.996885i	$-0.474868\pi$
$643$	4.00000	0.157745	0.0788723	+	0.996885i	$0.474868\pi$
$644$	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
$649$	16.0000	0.628055
$650$	1.00000	0.0392232
$651$	0	0
$652$	4.00000	0.156652
$653$	−34.0000	−1.33052	−0.665261	−	0.746611i	$-0.731680\pi$
$653$	−34.0000	−1.33052	−0.665261	+	0.746611i	$0.731680\pi$
$654$	10.0000	0.391031
$655$	12.0000	0.468879
$656$	−6.00000	−0.234261
$657$	2.00000	0.0780274
$658$	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
$661$	14.0000	0.544537	0.272268	−	0.962221i	$-0.412226\pi$
$661$	14.0000	0.544537	0.272268	+	0.962221i	$0.412226\pi$
$662$	−20.0000	−0.777322
$663$	6.00000	0.233021
$664$	−12.0000	−0.465690
$665$	0	0
$666$	−10.0000	−0.387492
$667$	48.0000	1.85857
$668$	8.00000	0.309529
$669$	−8.00000	−0.309298
$670$	−12.0000	−0.463600
$671$	−8.00000	−0.308837
$672$	0	0
$673$	34.0000	1.31060	0.655302	−	0.755367i	$-0.272541\pi$
$673$	34.0000	1.31060	0.655302	+	0.755367i	$0.272541\pi$
$674$	2.00000	0.0770371
$675$	−1.00000	−0.0384900
$676$	1.00000	0.0384615
$677$	22.0000	0.845529	0.422764	−	0.906240i	$-0.361060\pi$
$677$	22.0000	0.845529	0.422764	+	0.906240i	$0.361060\pi$
$678$	−10.0000	−0.384048
$679$	0	0
$680$	−6.00000	−0.230089
$681$	−4.00000	−0.153280
$682$	−32.0000	−1.22534
$683$	28.0000	1.07139	0.535695	−	0.844411i	$-0.320050\pi$
$683$	28.0000	1.07139	0.535695	+	0.844411i	$0.320050\pi$
$684$	4.00000	0.152944
$685$	10.0000	0.382080
$686$	0	0
$687$	−14.0000	−0.534133
$688$	4.00000	0.152499
$689$	−10.0000	−0.380970
$690$	−8.00000	−0.304555
$691$	−12.0000	−0.456502	−0.228251	−	0.973602i	$-0.573301\pi$
$691$	−12.0000	−0.456502	−0.228251	+	0.973602i	$0.573301\pi$
$692$	−18.0000	−0.684257
$693$	0	0
$694$	20.0000	0.759190
$695$	12.0000	0.455186
$696$	−6.00000	−0.227429
$697$	36.0000	1.36360
$698$	6.00000	0.227103
$699$	14.0000	0.529529
$700$	0	0
$701$	22.0000	0.830929	0.415464	−	0.909610i	$-0.363619\pi$
$701$	22.0000	0.830929	0.415464	+	0.909610i	$0.363619\pi$
$702$	−1.00000	−0.0377426
$703$	−40.0000	−1.50863
$704$	4.00000	0.150756
$705$	0	0
$706$	−30.0000	−1.12906
$707$	0	0
$708$	−4.00000	−0.150329
$709$	−50.0000	−1.87779	−0.938895	−	0.344204i	$-0.888149\pi$
$709$	−50.0000	−1.87779	−0.938895	+	0.344204i	$0.888149\pi$
$710$	16.0000	0.600469
$711$	−16.0000	−0.600047
$712$	10.0000	0.374766
$713$	−64.0000	−2.39682
$714$	0	0
$715$	4.00000	0.149592
$716$	12.0000	0.448461
$717$	8.00000	0.298765
$718$	16.0000	0.597115
$719$	0	0		−	1.00000i	$-0.5\pi$
$719$	0	0			1.00000i	$0.5\pi$
$720$	1.00000	0.0372678
$721$	0	0
$722$	−3.00000	−0.111648
$723$	−18.0000	−0.669427
$724$	6.00000	0.222988
$725$	6.00000	0.222834
$726$	−5.00000	−0.185567
$727$	−16.0000	−0.593407	−0.296704	−	0.954970i	$-0.595887\pi$
$727$	−16.0000	−0.593407	−0.296704	+	0.954970i	$0.595887\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	2.00000	0.0740233
$731$	−24.0000	−0.887672
$732$	2.00000	0.0739221
$733$	−50.0000	−1.84679	−0.923396	−	0.383849i	$-0.874598\pi$
$733$	−50.0000	−1.84679	−0.923396	+	0.383849i	$0.874598\pi$
$734$	−24.0000	−0.885856
$735$	7.00000	0.258199
$736$	8.00000	0.294884
$737$	−48.0000	−1.76810
$738$	−6.00000	−0.220863
$739$	20.0000	0.735712	0.367856	−	0.929883i	$-0.380092\pi$
$739$	20.0000	0.735712	0.367856	+	0.929883i	$0.380092\pi$
$740$	−10.0000	−0.367607
$741$	−4.00000	−0.146944
$742$	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
$745$	6.00000	0.219823
$746$	22.0000	0.805477
$747$	−12.0000	−0.439057
$748$	−24.0000	−0.877527
$749$	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	$0.301546\pi$
$752$	0	0
$753$	28.0000	1.02038
$754$	6.00000	0.218507
$755$	0	0
$756$	0	0
$757$	−42.0000	−1.52652	−0.763258	−	0.646094i	$-0.776401\pi$
$757$	−42.0000	−1.52652	−0.763258	+	0.646094i	$0.776401\pi$
$758$	−4.00000	−0.145287
$759$	−32.0000	−1.16153
$760$	4.00000	0.145095
$761$	42.0000	1.52250	0.761249	−	0.648459i	$-0.224586\pi$
$761$	42.0000	1.52250	0.761249	+	0.648459i	$0.224586\pi$
$762$	−8.00000	−0.289809
$763$	0	0
$764$	0	0
$765$	−6.00000	−0.216930
$766$	32.0000	1.15621
$767$	4.00000	0.144432
$768$	−1.00000	−0.0360844
$769$	34.0000	1.22607	0.613036	−	0.790055i	$-0.289948\pi$
$769$	34.0000	1.22607	0.613036	+	0.790055i	$0.289948\pi$
$770$	0	0
$771$	6.00000	0.216085
$772$	10.0000	0.359908
$773$	38.0000	1.36677	0.683383	−	0.730061i	$-0.260508\pi$
$773$	38.0000	1.36677	0.683383	+	0.730061i	$0.260508\pi$
$774$	4.00000	0.143777
$775$	−8.00000	−0.287368
$776$	−6.00000	−0.215387
$777$	0	0
$778$	−34.0000	−1.21896
$779$	−24.0000	−0.859889
$780$	−1.00000	−0.0358057
$781$	64.0000	2.29010
$782$	−48.0000	−1.71648
$783$	−6.00000	−0.214423
$784$	−7.00000	−0.250000
$785$	−2.00000	−0.0713831
$786$	−12.0000	−0.428026
$787$	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
$790$	−16.0000	−0.569254
$791$	0	0
$792$	4.00000	0.142134
$793$	−2.00000	−0.0710221
$794$	30.0000	1.06466
$795$	10.0000	0.354663
$796$	−8.00000	−0.283552
$797$	−2.00000	−0.0708436	−0.0354218	−	0.999372i	$-0.511277\pi$
$797$	−2.00000	−0.0708436	−0.0354218	+	0.999372i	$0.511277\pi$
$798$	0	0
$799$	0	0
$800$	1.00000	0.0353553
$801$	10.0000	0.353333
$802$	34.0000	1.20058
$803$	8.00000	0.282314
$804$	12.0000	0.423207
$805$	0	0
$806$	−8.00000	−0.281788
$807$	26.0000	0.915243
$808$	−2.00000	−0.0703598
$809$	42.0000	1.47664	0.738321	−	0.674450i	$-0.235619\pi$
$809$	42.0000	1.47664	0.738321	+	0.674450i	$0.235619\pi$
$810$	1.00000	0.0351364
$811$	12.0000	0.421377	0.210688	−	0.977553i	$-0.432429\pi$
$811$	12.0000	0.421377	0.210688	+	0.977553i	$0.432429\pi$
$812$	0	0
$813$	−24.0000	−0.841717
$814$	−40.0000	−1.40200
$815$	4.00000	0.140114
$816$	6.00000	0.210042
$817$	16.0000	0.559769
$818$	−22.0000	−0.769212
$819$	0	0
$820$	−6.00000	−0.209529
$821$	6.00000	0.209401	0.104701	−	0.994504i	$-0.466612\pi$
$821$	6.00000	0.209401	0.104701	+	0.994504i	$0.466612\pi$
$822$	−10.0000	−0.348790
$823$	16.0000	0.557725	0.278862	−	0.960331i	$-0.410043\pi$
$823$	16.0000	0.557725	0.278862	+	0.960331i	$0.410043\pi$
$824$	0	0
$825$	−4.00000	−0.139262
$826$	0	0
$827$	−36.0000	−1.25184	−0.625921	−	0.779886i	$-0.715277\pi$
$827$	−36.0000	−1.25184	−0.625921	+	0.779886i	$0.715277\pi$
$828$	8.00000	0.278019
$829$	−2.00000	−0.0694629	−0.0347314	−	0.999397i	$-0.511058\pi$
$829$	−2.00000	−0.0694629	−0.0347314	+	0.999397i	$0.511058\pi$
$830$	−12.0000	−0.416526
$831$	10.0000	0.346896
$832$	1.00000	0.0346688
$833$	42.0000	1.45521
$834$	−12.0000	−0.415526
$835$	8.00000	0.276851
$836$	16.0000	0.553372
$837$	8.00000	0.276520
$838$	−4.00000	−0.138178
$839$	0	0		−	1.00000i	$-0.5\pi$
$839$	0	0			1.00000i	$0.5\pi$
$840$	0	0
$841$	7.00000	0.241379
$842$	−34.0000	−1.17172
$843$	−10.0000	−0.344418
$844$	−12.0000	−0.413057
$845$	1.00000	0.0344010
$846$	0	0
$847$	0	0
$848$	−10.0000	−0.343401
$849$	−20.0000	−0.686398
$850$	−6.00000	−0.205798
$851$	−80.0000	−2.74236
$852$	−16.0000	−0.548151
$853$	−26.0000	−0.890223	−0.445112	−	0.895475i	$-0.646836\pi$
$853$	−26.0000	−0.890223	−0.445112	+	0.895475i	$0.646836\pi$
$854$	0	0
$855$	4.00000	0.136797
$856$	−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	$0.400530\pi$
$858$	−4.00000	−0.136558
$859$	−20.0000	−0.682391	−0.341196	−	0.939992i	$-0.610832\pi$
$859$	−20.0000	−0.682391	−0.341196	+	0.939992i	$0.610832\pi$
$860$	4.00000	0.136399
$861$	0	0
$862$	−24.0000	−0.817443
$863$	0	0		−	1.00000i	$-0.5\pi$
$863$	0	0			1.00000i	$0.5\pi$
$864$	−1.00000	−0.0340207
$865$	−18.0000	−0.612018
$866$	34.0000	1.15537
$867$	−19.0000	−0.645274
$868$	0	0
$869$	−64.0000	−2.17105
$870$	−6.00000	−0.203419
$871$	−12.0000	−0.406604
$872$	−10.0000	−0.338643
$873$	−6.00000	−0.203069
$874$	32.0000	1.08242
$875$	0	0
$876$	−2.00000	−0.0675737
$877$	14.0000	0.472746	0.236373	−	0.971662i	$-0.424041\pi$
$877$	14.0000	0.472746	0.236373	+	0.971662i	$0.424041\pi$
$878$	−40.0000	−1.34993
$879$	26.0000	0.876958
$880$	4.00000	0.134840
$881$	2.00000	0.0673817	0.0336909	−	0.999432i	$-0.489274\pi$
$881$	2.00000	0.0673817	0.0336909	+	0.999432i	$0.489274\pi$
$882$	−7.00000	−0.235702
$883$	−36.0000	−1.21150	−0.605748	−	0.795656i	$-0.707126\pi$
$883$	−36.0000	−1.21150	−0.605748	+	0.795656i	$0.707126\pi$
$884$	−6.00000	−0.201802
$885$	−4.00000	−0.134459
$886$	36.0000	1.20944
$887$	8.00000	0.268614	0.134307	−	0.990940i	$-0.457119\pi$
$887$	8.00000	0.268614	0.134307	+	0.990940i	$0.457119\pi$
$888$	10.0000	0.335578
$889$	0	0
$890$	10.0000	0.335201
$891$	4.00000	0.134005
$892$	8.00000	0.267860
$893$	0	0
$894$	−6.00000	−0.200670
$895$	12.0000	0.401116
$896$	0	0
$897$	−8.00000	−0.267112
$898$	18.0000	0.600668
$899$	−48.0000	−1.60089
$900$	1.00000	0.0333333
$901$	60.0000	1.99889
$902$	−24.0000	−0.799113
$903$	0	0
$904$	10.0000	0.332595
$905$	6.00000	0.199447
$906$	0	0
$907$	−12.0000	−0.398453	−0.199227	−	0.979953i	$-0.563843\pi$
$907$	−12.0000	−0.398453	−0.199227	+	0.979953i	$0.563843\pi$
$908$	4.00000	0.132745
$909$	−2.00000	−0.0663358
$910$	0	0
$911$	0	0		−	1.00000i	$-0.5\pi$
$911$	0	0			1.00000i	$0.5\pi$
$912$	−4.00000	−0.132453
$913$	−48.0000	−1.58857
$914$	34.0000	1.12462
$915$	2.00000	0.0661180
$916$	14.0000	0.462573
$917$	0	0
$918$	6.00000	0.198030
$919$	−56.0000	−1.84727	−0.923635	−	0.383274i	$-0.874797\pi$
$919$	−56.0000	−1.84727	−0.923635	+	0.383274i	$0.874797\pi$
$920$	8.00000	0.263752
$921$	12.0000	0.395413
$922$	−18.0000	−0.592798
$923$	16.0000	0.526646
$924$	0	0
$925$	−10.0000	−0.328798
$926$	8.00000	0.262896
$927$	0	0
$928$	6.00000	0.196960
$929$	−14.0000	−0.459325	−0.229663	−	0.973270i	$-0.573762\pi$
$929$	−14.0000	−0.459325	−0.229663	+	0.973270i	$0.573762\pi$
$930$	8.00000	0.262330
$931$	−28.0000	−0.917663
$932$	−14.0000	−0.458585
$933$	24.0000	0.785725
$934$	12.0000	0.392652
$935$	−24.0000	−0.784884
$936$	1.00000	0.0326860
$937$	−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
$939$	22.0000	0.717943
$940$	0	0
$941$	30.0000	0.977972	0.488986	−	0.872292i	$-0.337367\pi$
$941$	30.0000	0.977972	0.488986	+	0.872292i	$0.337367\pi$
$942$	2.00000	0.0651635
$943$	−48.0000	−1.56310
$944$	4.00000	0.130189
$945$	0	0
$946$	16.0000	0.520205
$947$	52.0000	1.68977	0.844886	−	0.534946i	$-0.179668\pi$
$947$	52.0000	1.68977	0.844886	+	0.534946i	$0.179668\pi$
$948$	16.0000	0.519656
$949$	2.00000	0.0649227
$950$	4.00000	0.129777
$951$	2.00000	0.0648544
$952$	0	0
$953$	34.0000	1.10137	0.550684	−	0.834714i	$-0.314367\pi$
$953$	34.0000	1.10137	0.550684	+	0.834714i	$0.314367\pi$
$954$	−10.0000	−0.323762
$955$	0	0
$956$	−8.00000	−0.258738
$957$	−24.0000	−0.775810
$958$	−8.00000	−0.258468
$959$	0	0
$960$	−1.00000	−0.0322749
$961$	33.0000	1.06452
$962$	−10.0000	−0.322413
$963$	−12.0000	−0.386695
$964$	18.0000	0.579741
$965$	10.0000	0.321911
$966$	0	0
$967$	16.0000	0.514525	0.257263	−	0.966342i	$-0.417179\pi$
$967$	16.0000	0.514525	0.257263	+	0.966342i	$0.417179\pi$
$968$	5.00000	0.160706
$969$	24.0000	0.770991
$970$	−6.00000	−0.192648
$971$	36.0000	1.15529	0.577647	−	0.816286i	$-0.303971\pi$
$971$	36.0000	1.15529	0.577647	+	0.816286i	$0.303971\pi$
$972$	−1.00000	−0.0320750
$973$	0	0
$974$	−16.0000	−0.512673
$975$	−1.00000	−0.0320256
$976$	−2.00000	−0.0640184
$977$	−30.0000	−0.959785	−0.479893	−	0.877327i	$-0.659324\pi$
$977$	−30.0000	−0.959785	−0.479893	+	0.877327i	$0.659324\pi$
$978$	−4.00000	−0.127906
$979$	40.0000	1.27841
$980$	−7.00000	−0.223607
$981$	−10.0000	−0.319275
$982$	4.00000	0.127645
$983$	8.00000	0.255160	0.127580	−	0.991828i	$-0.459279\pi$
$983$	8.00000	0.255160	0.127580	+	0.991828i	$0.459279\pi$
$984$	6.00000	0.191273
$985$	6.00000	0.191176
$986$	−36.0000	−1.14647
$987$	0	0
$988$	4.00000	0.127257
$989$	32.0000	1.01754
$990$	4.00000	0.127128
$991$	−16.0000	−0.508257	−0.254128	−	0.967170i	$-0.581789\pi$
$991$	−16.0000	−0.508257	−0.254128	+	0.967170i	$0.581789\pi$
$992$	−8.00000	−0.254000
$993$	20.0000	0.634681
$994$	0	0
$995$	−8.00000	−0.253617
$996$	12.0000	0.380235
$997$	−58.0000	−1.83688	−0.918439	−	0.395562i	$-0.870550\pi$
$997$	−58.0000	−1.83688	−0.918439	+	0.395562i	$0.870550\pi$
$998$	20.0000	0.633089
$999$	10.0000	0.316386

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	390.2.a.f.1.1	✓	1
3.2	odd	2		1170.2.a.a.1.1		1
4.3	odd	2		3120.2.a.w.1.1		1
5.2	odd	4		1950.2.e.g.1249.2		2
5.3	odd	4		1950.2.e.g.1249.1		2
5.4	even	2		1950.2.a.k.1.1		1
12.11	even	2		9360.2.a.p.1.1		1
13.5	odd	4		5070.2.b.d.1351.1		2
13.8	odd	4		5070.2.b.d.1351.2		2
13.12	even	2		5070.2.a.a.1.1		1
15.2	even	4		5850.2.e.e.5149.1		2
15.8	even	4		5850.2.e.e.5149.2		2
15.14	odd	2		5850.2.a.bo.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
390.2.a.f.1.1	✓	1	1.1	even	1	trivial
1170.2.a.a.1.1		1	3.2	odd	2
1950.2.a.k.1.1		1	5.4	even	2
1950.2.e.g.1249.1		2	5.3	odd	4
1950.2.e.g.1249.2		2	5.2	odd	4
3120.2.a.w.1.1		1	4.3	odd	2
5070.2.a.a.1.1		1	13.12	even	2
5070.2.b.d.1351.1		2	13.5	odd	4
5070.2.b.d.1351.2		2	13.8	odd	4
5850.2.a.bo.1.1		1	15.14	odd	2
5850.2.e.e.5149.1		2	15.2	even	4
5850.2.e.e.5149.2		2	15.8	even	4
9360.2.a.p.1.1		1	12.11	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 390 = 2 \cdot 3 \cdot 5 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	390.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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