LMFDB - Embedded newform 46.2.a.a.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [46,2,Mod(1,46)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(46, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("46.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 46 = 2 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	46.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:	$0.367311849298$
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	$\chi$	$=$	46.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q-1.00000 q^{2} +1.00000 q^{4} +4.00000 q^{5} -4.00000 q^{7} -1.00000 q^{8} -3.00000 q^{9} +O(q^{10})$

\(q-1.00000 q^{2} +1.00000 q^{4} +4.00000 q^{5} -4.00000 q^{7} -1.00000 q^{8} -3.00000 q^{9} -4.00000 q^{10} +2.00000 q^{11} -2.00000 q^{13} +4.00000 q^{14} +1.00000 q^{16} -2.00000 q^{17} +3.00000 q^{18} -2.00000 q^{19} +4.00000 q^{20} -2.00000 q^{22} +1.00000 q^{23} +11.0000 q^{25} +2.00000 q^{26} -4.00000 q^{28} +2.00000 q^{29} -1.00000 q^{32} +2.00000 q^{34} -16.0000 q^{35} -3.00000 q^{36} -4.00000 q^{37} +2.00000 q^{38} -4.00000 q^{40} +6.00000 q^{41} +10.0000 q^{43} +2.00000 q^{44} -12.0000 q^{45} -1.00000 q^{46} +9.00000 q^{49} -11.0000 q^{50} -2.00000 q^{52} -4.00000 q^{53} +8.00000 q^{55} +4.00000 q^{56} -2.00000 q^{58} +12.0000 q^{59} -8.00000 q^{61} +12.0000 q^{63} +1.00000 q^{64} -8.00000 q^{65} -10.0000 q^{67} -2.00000 q^{68} +16.0000 q^{70} +3.00000 q^{72} +6.00000 q^{73} +4.00000 q^{74} -2.00000 q^{76} -8.00000 q^{77} -12.0000 q^{79} +4.00000 q^{80} +9.00000 q^{81} -6.00000 q^{82} +14.0000 q^{83} -8.00000 q^{85} -10.0000 q^{86} -2.00000 q^{88} -6.00000 q^{89} +12.0000 q^{90} +8.00000 q^{91} +1.00000 q^{92} -8.00000 q^{95} +6.00000 q^{97} -9.00000 q^{98} -6.00000 q^{99} +O(q^{100})\)

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
$3$	0	0		−	1.00000i	$-0.5\pi$
$3$	0	0			1.00000i	$0.5\pi$
$4$	1.00000	0.500000
$5$	4.00000	1.78885	0.894427	−	0.447214i	$-0.147584\pi$
$5$	4.00000	1.78885	0.894427	+	0.447214i	$0.147584\pi$
$6$	0	0
$7$	−4.00000	−1.51186	−0.755929	−	0.654654i	$-0.772814\pi$
$7$	−4.00000	−1.51186	−0.755929	+	0.654654i	$0.772814\pi$
$8$	−1.00000	−0.353553
$9$	−3.00000	−1.00000
$10$	−4.00000	−1.26491
$11$	2.00000	0.603023	0.301511	−	0.953463i	$-0.402509\pi$
$11$	2.00000	0.603023	0.301511	+	0.953463i	$0.402509\pi$
$12$	0	0
$13$	−2.00000	−0.554700	−0.277350	−	0.960769i	$-0.589456\pi$
$13$	−2.00000	−0.554700	−0.277350	+	0.960769i	$0.589456\pi$
$14$	4.00000	1.06904
$15$	0	0
$16$	1.00000	0.250000
$17$	−2.00000	−0.485071	−0.242536	−	0.970143i	$-0.577979\pi$
$17$	−2.00000	−0.485071	−0.242536	+	0.970143i	$0.577979\pi$
$18$	3.00000	0.707107
$19$	−2.00000	−0.458831	−0.229416	−	0.973329i	$-0.573682\pi$
$19$	−2.00000	−0.458831	−0.229416	+	0.973329i	$0.573682\pi$
$20$	4.00000	0.894427
$21$	0	0
$22$	−2.00000	−0.426401
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	0	0
$25$	11.0000	2.20000
$26$	2.00000	0.392232
$27$	0	0
$28$	−4.00000	−0.755929
$29$	2.00000	0.371391	0.185695	−	0.982607i	$-0.440546\pi$
$29$	2.00000	0.371391	0.185695	+	0.982607i	$0.440546\pi$
$30$	0	0
$31$	0	0		−	1.00000i	$-0.5\pi$
$31$	0	0			1.00000i	$0.5\pi$
$32$	−1.00000	−0.176777
$33$	0	0
$34$	2.00000	0.342997
$35$	−16.0000	−2.70449
$36$	−3.00000	−0.500000
$37$	−4.00000	−0.657596	−0.328798	−	0.944400i	$-0.606644\pi$
$37$	−4.00000	−0.657596	−0.328798	+	0.944400i	$0.606644\pi$
$38$	2.00000	0.324443
$39$	0	0
$40$	−4.00000	−0.632456
$41$	6.00000	0.937043	0.468521	−	0.883452i	$-0.344787\pi$
$41$	6.00000	0.937043	0.468521	+	0.883452i	$0.344787\pi$
$42$	0	0
$43$	10.0000	1.52499	0.762493	−	0.646997i	$-0.223975\pi$
$43$	10.0000	1.52499	0.762493	+	0.646997i	$0.223975\pi$
$44$	2.00000	0.301511
$45$	−12.0000	−1.78885
$46$	−1.00000	−0.147442
$47$	0	0		−	1.00000i	$-0.5\pi$
$47$	0	0			1.00000i	$0.5\pi$
$48$	0	0
$49$	9.00000	1.28571
$50$	−11.0000	−1.55563
$51$	0	0
$52$	−2.00000	−0.277350
$53$	−4.00000	−0.549442	−0.274721	−	0.961524i	$-0.588586\pi$
$53$	−4.00000	−0.549442	−0.274721	+	0.961524i	$0.588586\pi$
$54$	0	0
$55$	8.00000	1.07872
$56$	4.00000	0.534522
$57$	0	0
$58$	−2.00000	−0.262613
$59$	12.0000	1.56227	0.781133	−	0.624364i	$-0.214642\pi$
$59$	12.0000	1.56227	0.781133	+	0.624364i	$0.214642\pi$
$60$	0	0
$61$	−8.00000	−1.02430	−0.512148	−	0.858898i	$-0.671150\pi$
$61$	−8.00000	−1.02430	−0.512148	+	0.858898i	$0.671150\pi$
$62$	0	0
$63$	12.0000	1.51186
$64$	1.00000	0.125000
$65$	−8.00000	−0.992278
$66$	0	0
$67$	−10.0000	−1.22169	−0.610847	−	0.791748i	$-0.709171\pi$
$67$	−10.0000	−1.22169	−0.610847	+	0.791748i	$0.709171\pi$
$68$	−2.00000	−0.242536
$69$	0	0
$70$	16.0000	1.91237
$71$	0	0		−	1.00000i	$-0.5\pi$
$71$	0	0			1.00000i	$0.5\pi$
$72$	3.00000	0.353553
$73$	6.00000	0.702247	0.351123	−	0.936329i	$-0.385800\pi$
$73$	6.00000	0.702247	0.351123	+	0.936329i	$0.385800\pi$
$74$	4.00000	0.464991
$75$	0	0
$76$	−2.00000	−0.229416
$77$	−8.00000	−0.911685
$78$	0	0
$79$	−12.0000	−1.35011	−0.675053	−	0.737769i	$-0.735879\pi$
$79$	−12.0000	−1.35011	−0.675053	+	0.737769i	$0.735879\pi$
$80$	4.00000	0.447214
$81$	9.00000	1.00000
$82$	−6.00000	−0.662589
$83$	14.0000	1.53670	0.768350	−	0.640030i	$-0.221078\pi$
$83$	14.0000	1.53670	0.768350	+	0.640030i	$0.221078\pi$
$84$	0	0
$85$	−8.00000	−0.867722
$86$	−10.0000	−1.07833
$87$	0	0
$88$	−2.00000	−0.213201
$89$	−6.00000	−0.635999	−0.317999	−	0.948091i	$-0.603011\pi$
$89$	−6.00000	−0.635999	−0.317999	+	0.948091i	$0.603011\pi$
$90$	12.0000	1.26491
$91$	8.00000	0.838628
$92$	1.00000	0.104257
$93$	0	0
$94$	0	0
$95$	−8.00000	−0.820783
$96$	0	0
$97$	6.00000	0.609208	0.304604	−	0.952479i	$-0.401476\pi$
$97$	6.00000	0.609208	0.304604	+	0.952479i	$0.401476\pi$
$98$	−9.00000	−0.909137
$99$	−6.00000	−0.603023
$100$	11.0000	1.10000
$101$	−10.0000	−0.995037	−0.497519	−	0.867453i	$-0.665755\pi$
$101$	−10.0000	−0.995037	−0.497519	+	0.867453i	$0.665755\pi$
$102$	0	0
$103$	−8.00000	−0.788263	−0.394132	−	0.919054i	$-0.628955\pi$
$103$	−8.00000	−0.788263	−0.394132	+	0.919054i	$0.628955\pi$
$104$	2.00000	0.196116
$105$	0	0
$106$	4.00000	0.388514
$107$	−10.0000	−0.966736	−0.483368	−	0.875417i	$-0.660587\pi$
$107$	−10.0000	−0.966736	−0.483368	+	0.875417i	$0.660587\pi$
$108$	0	0
$109$	0	0		−	1.00000i	$-0.5\pi$
$109$	0	0			1.00000i	$0.5\pi$
$110$	−8.00000	−0.762770
$111$	0	0
$112$	−4.00000	−0.377964
$113$	−14.0000	−1.31701	−0.658505	−	0.752577i	$-0.728811\pi$
$113$	−14.0000	−1.31701	−0.658505	+	0.752577i	$0.728811\pi$
$114$	0	0
$115$	4.00000	0.373002
$116$	2.00000	0.185695
$117$	6.00000	0.554700
$118$	−12.0000	−1.10469
$119$	8.00000	0.733359
$120$	0	0
$121$	−7.00000	−0.636364
$122$	8.00000	0.724286
$123$	0	0
$124$	0	0
$125$	24.0000	2.14663
$126$	−12.0000	−1.06904
$127$	16.0000	1.41977	0.709885	−	0.704317i	$-0.248747\pi$
$127$	16.0000	1.41977	0.709885	+	0.704317i	$0.248747\pi$
$128$	−1.00000	−0.0883883
$129$	0	0
$130$	8.00000	0.701646
$131$	12.0000	1.04844	0.524222	−	0.851581i	$-0.324356\pi$
$131$	12.0000	1.04844	0.524222	+	0.851581i	$0.324356\pi$
$132$	0	0
$133$	8.00000	0.693688
$134$	10.0000	0.863868
$135$	0	0
$136$	2.00000	0.171499
$137$	6.00000	0.512615	0.256307	−	0.966595i	$-0.417494\pi$
$137$	6.00000	0.512615	0.256307	+	0.966595i	$0.417494\pi$
$138$	0	0
$139$	−4.00000	−0.339276	−0.169638	−	0.985506i	$-0.554260\pi$
$139$	−4.00000	−0.339276	−0.169638	+	0.985506i	$0.554260\pi$
$140$	−16.0000	−1.35225
$141$	0	0
$142$	0	0
$143$	−4.00000	−0.334497
$144$	−3.00000	−0.250000
$145$	8.00000	0.664364
$146$	−6.00000	−0.496564
$147$	0	0
$148$	−4.00000	−0.328798
$149$	−4.00000	−0.327693	−0.163846	−	0.986486i	$-0.552390\pi$
$149$	−4.00000	−0.327693	−0.163846	+	0.986486i	$0.552390\pi$
$150$	0	0
$151$	−8.00000	−0.651031	−0.325515	−	0.945537i	$-0.605538\pi$
$151$	−8.00000	−0.651031	−0.325515	+	0.945537i	$0.605538\pi$
$152$	2.00000	0.162221
$153$	6.00000	0.485071
$154$	8.00000	0.644658
$155$	0	0
$156$	0	0
$157$	12.0000	0.957704	0.478852	−	0.877896i	$-0.341053\pi$
$157$	12.0000	0.957704	0.478852	+	0.877896i	$0.341053\pi$
$158$	12.0000	0.954669
$159$	0	0
$160$	−4.00000	−0.316228
$161$	−4.00000	−0.315244
$162$	−9.00000	−0.707107
$163$	−8.00000	−0.626608	−0.313304	−	0.949653i	$-0.601436\pi$
$163$	−8.00000	−0.626608	−0.313304	+	0.949653i	$0.601436\pi$
$164$	6.00000	0.468521
$165$	0	0
$166$	−14.0000	−1.08661
$167$	16.0000	1.23812	0.619059	−	0.785345i	$-0.287514\pi$
$167$	16.0000	1.23812	0.619059	+	0.785345i	$0.287514\pi$
$168$	0	0
$169$	−9.00000	−0.692308
$170$	8.00000	0.613572
$171$	6.00000	0.458831
$172$	10.0000	0.762493
$173$	−6.00000	−0.456172	−0.228086	−	0.973641i	$-0.573247\pi$
$173$	−6.00000	−0.456172	−0.228086	+	0.973641i	$0.573247\pi$
$174$	0	0
$175$	−44.0000	−3.32609
$176$	2.00000	0.150756
$177$	0	0
$178$	6.00000	0.449719
$179$	−16.0000	−1.19590	−0.597948	−	0.801535i	$-0.704017\pi$
$179$	−16.0000	−1.19590	−0.597948	+	0.801535i	$0.704017\pi$
$180$	−12.0000	−0.894427
$181$	−24.0000	−1.78391	−0.891953	−	0.452128i	$-0.850665\pi$
$181$	−24.0000	−1.78391	−0.891953	+	0.452128i	$0.850665\pi$
$182$	−8.00000	−0.592999
$183$	0	0
$184$	−1.00000	−0.0737210
$185$	−16.0000	−1.17634
$186$	0	0
$187$	−4.00000	−0.292509
$188$	0	0
$189$	0	0
$190$	8.00000	0.580381
$191$	20.0000	1.44715	0.723575	−	0.690246i	$-0.242498\pi$
$191$	20.0000	1.44715	0.723575	+	0.690246i	$0.242498\pi$
$192$	0	0
$193$	26.0000	1.87152	0.935760	−	0.352636i	$-0.114715\pi$
$193$	26.0000	1.87152	0.935760	+	0.352636i	$0.114715\pi$
$194$	−6.00000	−0.430775
$195$	0	0
$196$	9.00000	0.642857
$197$	18.0000	1.28245	0.641223	−	0.767354i	$-0.278427\pi$
$197$	18.0000	1.28245	0.641223	+	0.767354i	$0.278427\pi$
$198$	6.00000	0.426401
$199$	−4.00000	−0.283552	−0.141776	−	0.989899i	$-0.545281\pi$
$199$	−4.00000	−0.283552	−0.141776	+	0.989899i	$0.545281\pi$
$200$	−11.0000	−0.777817
$201$	0	0
$202$	10.0000	0.703598
$203$	−8.00000	−0.561490
$204$	0	0
$205$	24.0000	1.67623
$206$	8.00000	0.557386
$207$	−3.00000	−0.208514
$208$	−2.00000	−0.138675
$209$	−4.00000	−0.276686
$210$	0	0
$211$	−20.0000	−1.37686	−0.688428	−	0.725304i	$-0.741699\pi$
$211$	−20.0000	−1.37686	−0.688428	+	0.725304i	$0.741699\pi$
$212$	−4.00000	−0.274721
$213$	0	0
$214$	10.0000	0.683586
$215$	40.0000	2.72798
$216$	0	0
$217$	0	0
$218$	0	0
$219$	0	0
$220$	8.00000	0.539360
$221$	4.00000	0.269069
$222$	0	0
$223$	−16.0000	−1.07144	−0.535720	−	0.844396i	$-0.679960\pi$
$223$	−16.0000	−1.07144	−0.535720	+	0.844396i	$0.679960\pi$
$224$	4.00000	0.267261
$225$	−33.0000	−2.20000
$226$	14.0000	0.931266
$227$	−6.00000	−0.398234	−0.199117	−	0.979976i	$-0.563807\pi$
$227$	−6.00000	−0.398234	−0.199117	+	0.979976i	$0.563807\pi$
$228$	0	0
$229$	8.00000	0.528655	0.264327	−	0.964433i	$-0.414850\pi$
$229$	8.00000	0.528655	0.264327	+	0.964433i	$0.414850\pi$
$230$	−4.00000	−0.263752
$231$	0	0
$232$	−2.00000	−0.131306
$233$	−6.00000	−0.393073	−0.196537	−	0.980497i	$-0.562969\pi$
$233$	−6.00000	−0.393073	−0.196537	+	0.980497i	$0.562969\pi$
$234$	−6.00000	−0.392232
$235$	0	0
$236$	12.0000	0.781133
$237$	0	0
$238$	−8.00000	−0.518563
$239$	−24.0000	−1.55243	−0.776215	−	0.630468i	$-0.782863\pi$
$239$	−24.0000	−1.55243	−0.776215	+	0.630468i	$0.782863\pi$
$240$	0	0
$241$	22.0000	1.41714	0.708572	−	0.705638i	$-0.249340\pi$
$241$	22.0000	1.41714	0.708572	+	0.705638i	$0.249340\pi$
$242$	7.00000	0.449977
$243$	0	0
$244$	−8.00000	−0.512148
$245$	36.0000	2.29996
$246$	0	0
$247$	4.00000	0.254514
$248$	0	0
$249$	0	0
$250$	−24.0000	−1.51789
$251$	14.0000	0.883672	0.441836	−	0.897096i	$-0.354327\pi$
$251$	14.0000	0.883672	0.441836	+	0.897096i	$0.354327\pi$
$252$	12.0000	0.755929
$253$	2.00000	0.125739
$254$	−16.0000	−1.00393
$255$	0	0
$256$	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	$0.259294\pi$
$258$	0	0
$259$	16.0000	0.994192
$260$	−8.00000	−0.496139
$261$	−6.00000	−0.371391
$262$	−12.0000	−0.741362
$263$	−16.0000	−0.986602	−0.493301	−	0.869859i	$-0.664210\pi$
$263$	−16.0000	−0.986602	−0.493301	+	0.869859i	$0.664210\pi$
$264$	0	0
$265$	−16.0000	−0.982872
$266$	−8.00000	−0.490511
$267$	0	0
$268$	−10.0000	−0.610847
$269$	−14.0000	−0.853595	−0.426798	−	0.904347i	$-0.640358\pi$
$269$	−14.0000	−0.853595	−0.426798	+	0.904347i	$0.640358\pi$
$270$	0	0
$271$	8.00000	0.485965	0.242983	−	0.970031i	$-0.421874\pi$
$271$	8.00000	0.485965	0.242983	+	0.970031i	$0.421874\pi$
$272$	−2.00000	−0.121268
$273$	0	0
$274$	−6.00000	−0.362473
$275$	22.0000	1.32665
$276$	0	0
$277$	−14.0000	−0.841178	−0.420589	−	0.907251i	$-0.638177\pi$
$277$	−14.0000	−0.841178	−0.420589	+	0.907251i	$0.638177\pi$
$278$	4.00000	0.239904
$279$	0	0
$280$	16.0000	0.956183
$281$	−6.00000	−0.357930	−0.178965	−	0.983855i	$-0.557275\pi$
$281$	−6.00000	−0.357930	−0.178965	+	0.983855i	$0.557275\pi$
$282$	0	0
$283$	−14.0000	−0.832214	−0.416107	−	0.909316i	$-0.636606\pi$
$283$	−14.0000	−0.832214	−0.416107	+	0.909316i	$0.636606\pi$
$284$	0	0
$285$	0	0
$286$	4.00000	0.236525
$287$	−24.0000	−1.41668
$288$	3.00000	0.176777
$289$	−13.0000	−0.764706
$290$	−8.00000	−0.469776
$291$	0	0
$292$	6.00000	0.351123
$293$	4.00000	0.233682	0.116841	−	0.993151i	$-0.462723\pi$
$293$	4.00000	0.233682	0.116841	+	0.993151i	$0.462723\pi$
$294$	0	0
$295$	48.0000	2.79467
$296$	4.00000	0.232495
$297$	0	0
$298$	4.00000	0.231714
$299$	−2.00000	−0.115663
$300$	0	0
$301$	−40.0000	−2.30556
$302$	8.00000	0.460348
$303$	0	0
$304$	−2.00000	−0.114708
$305$	−32.0000	−1.83231
$306$	−6.00000	−0.342997
$307$	−16.0000	−0.913168	−0.456584	−	0.889680i	$-0.650927\pi$
$307$	−16.0000	−0.913168	−0.456584	+	0.889680i	$0.650927\pi$
$308$	−8.00000	−0.455842
$309$	0	0
$310$	0	0
$311$	32.0000	1.81455	0.907277	−	0.420534i	$-0.138157\pi$
$311$	32.0000	1.81455	0.907277	+	0.420534i	$0.138157\pi$
$312$	0	0
$313$	22.0000	1.24351	0.621757	−	0.783210i	$-0.286419\pi$
$313$	22.0000	1.24351	0.621757	+	0.783210i	$0.286419\pi$
$314$	−12.0000	−0.677199
$315$	48.0000	2.70449
$316$	−12.0000	−0.675053
$317$	2.00000	0.112331	0.0561656	−	0.998421i	$-0.482113\pi$
$317$	2.00000	0.112331	0.0561656	+	0.998421i	$0.482113\pi$
$318$	0	0
$319$	4.00000	0.223957
$320$	4.00000	0.223607
$321$	0	0
$322$	4.00000	0.222911
$323$	4.00000	0.222566
$324$	9.00000	0.500000
$325$	−22.0000	−1.22034
$326$	8.00000	0.443079
$327$	0	0
$328$	−6.00000	−0.331295
$329$	0	0
$330$	0	0
$331$	−4.00000	−0.219860	−0.109930	−	0.993939i	$-0.535063\pi$
$331$	−4.00000	−0.219860	−0.109930	+	0.993939i	$0.535063\pi$
$332$	14.0000	0.768350
$333$	12.0000	0.657596
$334$	−16.0000	−0.875481
$335$	−40.0000	−2.18543
$336$	0	0
$337$	10.0000	0.544735	0.272367	−	0.962193i	$-0.412193\pi$
$337$	10.0000	0.544735	0.272367	+	0.962193i	$0.412193\pi$
$338$	9.00000	0.489535
$339$	0	0
$340$	−8.00000	−0.433861
$341$	0	0
$342$	−6.00000	−0.324443
$343$	−8.00000	−0.431959
$344$	−10.0000	−0.539164
$345$	0	0
$346$	6.00000	0.322562
$347$	8.00000	0.429463	0.214731	−	0.976673i	$-0.431112\pi$
$347$	8.00000	0.429463	0.214731	+	0.976673i	$0.431112\pi$
$348$	0	0
$349$	26.0000	1.39175	0.695874	−	0.718164i	$-0.255017\pi$
$349$	26.0000	1.39175	0.695874	+	0.718164i	$0.255017\pi$
$350$	44.0000	2.35190
$351$	0	0
$352$	−2.00000	−0.106600
$353$	2.00000	0.106449	0.0532246	−	0.998583i	$-0.483050\pi$
$353$	2.00000	0.106449	0.0532246	+	0.998583i	$0.483050\pi$
$354$	0	0
$355$	0	0
$356$	−6.00000	−0.317999
$357$	0	0
$358$	16.0000	0.845626
$359$	−12.0000	−0.633336	−0.316668	−	0.948536i	$-0.602564\pi$
$359$	−12.0000	−0.633336	−0.316668	+	0.948536i	$0.602564\pi$
$360$	12.0000	0.632456
$361$	−15.0000	−0.789474
$362$	24.0000	1.26141
$363$	0	0
$364$	8.00000	0.419314
$365$	24.0000	1.25622
$366$	0	0
$367$	12.0000	0.626395	0.313197	−	0.949688i	$-0.398600\pi$
$367$	12.0000	0.626395	0.313197	+	0.949688i	$0.398600\pi$
$368$	1.00000	0.0521286
$369$	−18.0000	−0.937043
$370$	16.0000	0.831800
$371$	16.0000	0.830679
$372$	0	0
$373$	−4.00000	−0.207112	−0.103556	−	0.994624i	$-0.533022\pi$
$373$	−4.00000	−0.207112	−0.103556	+	0.994624i	$0.533022\pi$
$374$	4.00000	0.206835
$375$	0	0
$376$	0	0
$377$	−4.00000	−0.206010
$378$	0	0
$379$	22.0000	1.13006	0.565032	−	0.825069i	$-0.308864\pi$
$379$	22.0000	1.13006	0.565032	+	0.825069i	$0.308864\pi$
$380$	−8.00000	−0.410391
$381$	0	0
$382$	−20.0000	−1.02329
$383$	−24.0000	−1.22634	−0.613171	−	0.789950i	$-0.710106\pi$
$383$	−24.0000	−1.22634	−0.613171	+	0.789950i	$0.710106\pi$
$384$	0	0
$385$	−32.0000	−1.63087
$386$	−26.0000	−1.32337
$387$	−30.0000	−1.52499
$388$	6.00000	0.304604
$389$	0	0		−	1.00000i	$-0.5\pi$
$389$	0	0			1.00000i	$0.5\pi$
$390$	0	0
$391$	−2.00000	−0.101144
$392$	−9.00000	−0.454569
$393$	0	0
$394$	−18.0000	−0.906827
$395$	−48.0000	−2.41514
$396$	−6.00000	−0.301511
$397$	14.0000	0.702640	0.351320	−	0.936255i	$-0.385733\pi$
$397$	14.0000	0.702640	0.351320	+	0.936255i	$0.385733\pi$
$398$	4.00000	0.200502
$399$	0	0
$400$	11.0000	0.550000
$401$	−18.0000	−0.898877	−0.449439	−	0.893311i	$-0.648376\pi$
$401$	−18.0000	−0.898877	−0.449439	+	0.893311i	$0.648376\pi$
$402$	0	0
$403$	0	0
$404$	−10.0000	−0.497519
$405$	36.0000	1.78885
$406$	8.00000	0.397033
$407$	−8.00000	−0.396545
$408$	0	0
$409$	34.0000	1.68119	0.840596	−	0.541663i	$-0.182205\pi$
$409$	34.0000	1.68119	0.840596	+	0.541663i	$0.182205\pi$
$410$	−24.0000	−1.18528
$411$	0	0
$412$	−8.00000	−0.394132
$413$	−48.0000	−2.36193
$414$	3.00000	0.147442
$415$	56.0000	2.74893
$416$	2.00000	0.0980581
$417$	0	0
$418$	4.00000	0.195646
$419$	−18.0000	−0.879358	−0.439679	−	0.898155i	$-0.644908\pi$
$419$	−18.0000	−0.879358	−0.439679	+	0.898155i	$0.644908\pi$
$420$	0	0
$421$	28.0000	1.36464	0.682318	−	0.731055i	$-0.260972\pi$
$421$	28.0000	1.36464	0.682318	+	0.731055i	$0.260972\pi$
$422$	20.0000	0.973585
$423$	0	0
$424$	4.00000	0.194257
$425$	−22.0000	−1.06716
$426$	0	0
$427$	32.0000	1.54859
$428$	−10.0000	−0.483368
$429$	0	0
$430$	−40.0000	−1.92897
$431$	−12.0000	−0.578020	−0.289010	−	0.957326i	$-0.593326\pi$
$431$	−12.0000	−0.578020	−0.289010	+	0.957326i	$0.593326\pi$
$432$	0	0
$433$	−26.0000	−1.24948	−0.624740	−	0.780833i	$-0.714795\pi$
$433$	−26.0000	−1.24948	−0.624740	+	0.780833i	$0.714795\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−2.00000	−0.0956730
$438$	0	0
$439$	8.00000	0.381819	0.190910	−	0.981608i	$-0.438856\pi$
$439$	8.00000	0.381819	0.190910	+	0.981608i	$0.438856\pi$
$440$	−8.00000	−0.381385
$441$	−27.0000	−1.28571
$442$	−4.00000	−0.190261
$443$	8.00000	0.380091	0.190046	−	0.981775i	$-0.439136\pi$
$443$	8.00000	0.380091	0.190046	+	0.981775i	$0.439136\pi$
$444$	0	0
$445$	−24.0000	−1.13771
$446$	16.0000	0.757622
$447$	0	0
$448$	−4.00000	−0.188982
$449$	14.0000	0.660701	0.330350	−	0.943858i	$-0.392833\pi$
$449$	14.0000	0.660701	0.330350	+	0.943858i	$0.392833\pi$
$450$	33.0000	1.55563
$451$	12.0000	0.565058
$452$	−14.0000	−0.658505
$453$	0	0
$454$	6.00000	0.281594
$455$	32.0000	1.50018
$456$	0	0
$457$	−10.0000	−0.467780	−0.233890	−	0.972263i	$-0.575146\pi$
$457$	−10.0000	−0.467780	−0.233890	+	0.972263i	$0.575146\pi$
$458$	−8.00000	−0.373815
$459$	0	0
$460$	4.00000	0.186501
$461$	18.0000	0.838344	0.419172	−	0.907907i	$-0.362320\pi$
$461$	18.0000	0.838344	0.419172	+	0.907907i	$0.362320\pi$
$462$	0	0
$463$	40.0000	1.85896	0.929479	−	0.368875i	$-0.120257\pi$
$463$	40.0000	1.85896	0.929479	+	0.368875i	$0.120257\pi$
$464$	2.00000	0.0928477
$465$	0	0
$466$	6.00000	0.277945
$467$	−2.00000	−0.0925490	−0.0462745	−	0.998929i	$-0.514735\pi$
$467$	−2.00000	−0.0925490	−0.0462745	+	0.998929i	$0.514735\pi$
$468$	6.00000	0.277350
$469$	40.0000	1.84703
$470$	0	0
$471$	0	0
$472$	−12.0000	−0.552345
$473$	20.0000	0.919601
$474$	0	0
$475$	−22.0000	−1.00943
$476$	8.00000	0.366679
$477$	12.0000	0.549442
$478$	24.0000	1.09773
$479$	−8.00000	−0.365529	−0.182765	−	0.983157i	$-0.558505\pi$
$479$	−8.00000	−0.365529	−0.182765	+	0.983157i	$0.558505\pi$
$480$	0	0
$481$	8.00000	0.364769
$482$	−22.0000	−1.00207
$483$	0	0
$484$	−7.00000	−0.318182
$485$	24.0000	1.08978
$486$	0	0
$487$	−8.00000	−0.362515	−0.181257	−	0.983436i	$-0.558017\pi$
$487$	−8.00000	−0.362515	−0.181257	+	0.983436i	$0.558017\pi$
$488$	8.00000	0.362143
$489$	0	0
$490$	−36.0000	−1.62631
$491$	24.0000	1.08310	0.541552	−	0.840667i	$-0.317837\pi$
$491$	24.0000	1.08310	0.541552	+	0.840667i	$0.317837\pi$
$492$	0	0
$493$	−4.00000	−0.180151
$494$	−4.00000	−0.179969
$495$	−24.0000	−1.07872
$496$	0	0
$497$	0	0
$498$	0	0
$499$	16.0000	0.716258	0.358129	−	0.933672i	$-0.383415\pi$
$499$	16.0000	0.716258	0.358129	+	0.933672i	$0.383415\pi$
$500$	24.0000	1.07331
$501$	0	0
$502$	−14.0000	−0.624851
$503$	12.0000	0.535054	0.267527	−	0.963550i	$-0.413794\pi$
$503$	12.0000	0.535054	0.267527	+	0.963550i	$0.413794\pi$
$504$	−12.0000	−0.534522
$505$	−40.0000	−1.77998
$506$	−2.00000	−0.0889108
$507$	0	0
$508$	16.0000	0.709885
$509$	18.0000	0.797836	0.398918	−	0.916987i	$-0.369386\pi$
$509$	18.0000	0.797836	0.398918	+	0.916987i	$0.369386\pi$
$510$	0	0
$511$	−24.0000	−1.06170
$512$	−1.00000	−0.0441942
$513$	0	0
$514$	−22.0000	−0.970378
$515$	−32.0000	−1.41009
$516$	0	0
$517$	0	0
$518$	−16.0000	−0.703000
$519$	0	0
$520$	8.00000	0.350823
$521$	38.0000	1.66481	0.832405	−	0.554168i	$-0.186963\pi$
$521$	38.0000	1.66481	0.832405	+	0.554168i	$0.186963\pi$
$522$	6.00000	0.262613
$523$	−34.0000	−1.48672	−0.743358	−	0.668894i	$-0.766768\pi$
$523$	−34.0000	−1.48672	−0.743358	+	0.668894i	$0.766768\pi$
$524$	12.0000	0.524222
$525$	0	0
$526$	16.0000	0.697633
$527$	0	0
$528$	0	0
$529$	1.00000	0.0434783
$530$	16.0000	0.694996
$531$	−36.0000	−1.56227
$532$	8.00000	0.346844
$533$	−12.0000	−0.519778
$534$	0	0
$535$	−40.0000	−1.72935
$536$	10.0000	0.431934
$537$	0	0
$538$	14.0000	0.603583
$539$	18.0000	0.775315
$540$	0	0
$541$	−6.00000	−0.257960	−0.128980	−	0.991647i	$-0.541170\pi$
$541$	−6.00000	−0.257960	−0.128980	+	0.991647i	$0.541170\pi$
$542$	−8.00000	−0.343629
$543$	0	0
$544$	2.00000	0.0857493
$545$	0	0
$546$	0	0
$547$	−36.0000	−1.53925	−0.769624	−	0.638497i	$-0.779557\pi$
$547$	−36.0000	−1.53925	−0.769624	+	0.638497i	$0.779557\pi$
$548$	6.00000	0.256307
$549$	24.0000	1.02430
$550$	−22.0000	−0.938083
$551$	−4.00000	−0.170406
$552$	0	0
$553$	48.0000	2.04117
$554$	14.0000	0.594803
$555$	0	0
$556$	−4.00000	−0.169638
$557$	16.0000	0.677942	0.338971	−	0.940797i	$-0.389921\pi$
$557$	16.0000	0.677942	0.338971	+	0.940797i	$0.389921\pi$
$558$	0	0
$559$	−20.0000	−0.845910
$560$	−16.0000	−0.676123
$561$	0	0
$562$	6.00000	0.253095
$563$	−14.0000	−0.590030	−0.295015	−	0.955493i	$-0.595325\pi$
$563$	−14.0000	−0.590030	−0.295015	+	0.955493i	$0.595325\pi$
$564$	0	0
$565$	−56.0000	−2.35594
$566$	14.0000	0.588464
$567$	−36.0000	−1.51186
$568$	0	0
$569$	14.0000	0.586911	0.293455	−	0.955973i	$-0.405195\pi$
$569$	14.0000	0.586911	0.293455	+	0.955973i	$0.405195\pi$
$570$	0	0
$571$	14.0000	0.585882	0.292941	−	0.956131i	$-0.405366\pi$
$571$	14.0000	0.585882	0.292941	+	0.956131i	$0.405366\pi$
$572$	−4.00000	−0.167248
$573$	0	0
$574$	24.0000	1.00174
$575$	11.0000	0.458732
$576$	−3.00000	−0.125000
$577$	30.0000	1.24892	0.624458	−	0.781058i	$-0.285320\pi$
$577$	30.0000	1.24892	0.624458	+	0.781058i	$0.285320\pi$
$578$	13.0000	0.540729
$579$	0	0
$580$	8.00000	0.332182
$581$	−56.0000	−2.32327
$582$	0	0
$583$	−8.00000	−0.331326
$584$	−6.00000	−0.248282
$585$	24.0000	0.992278
$586$	−4.00000	−0.165238
$587$	−28.0000	−1.15568	−0.577842	−	0.816149i	$-0.696105\pi$
$587$	−28.0000	−1.15568	−0.577842	+	0.816149i	$0.696105\pi$
$588$	0	0
$589$	0	0
$590$	−48.0000	−1.97613
$591$	0	0
$592$	−4.00000	−0.164399
$593$	6.00000	0.246390	0.123195	−	0.992382i	$-0.460686\pi$
$593$	6.00000	0.246390	0.123195	+	0.992382i	$0.460686\pi$
$594$	0	0
$595$	32.0000	1.31187
$596$	−4.00000	−0.163846
$597$	0	0
$598$	2.00000	0.0817861
$599$	16.0000	0.653742	0.326871	−	0.945069i	$-0.394006\pi$
$599$	16.0000	0.653742	0.326871	+	0.945069i	$0.394006\pi$
$600$	0	0
$601$	−26.0000	−1.06056	−0.530281	−	0.847822i	$-0.677914\pi$
$601$	−26.0000	−1.06056	−0.530281	+	0.847822i	$0.677914\pi$
$602$	40.0000	1.63028
$603$	30.0000	1.22169
$604$	−8.00000	−0.325515
$605$	−28.0000	−1.13836
$606$	0	0
$607$	32.0000	1.29884	0.649420	−	0.760430i	$-0.275012\pi$
$607$	32.0000	1.29884	0.649420	+	0.760430i	$0.275012\pi$
$608$	2.00000	0.0811107
$609$	0	0
$610$	32.0000	1.29564
$611$	0	0
$612$	6.00000	0.242536
$613$	−16.0000	−0.646234	−0.323117	−	0.946359i	$-0.604731\pi$
$613$	−16.0000	−0.646234	−0.323117	+	0.946359i	$0.604731\pi$
$614$	16.0000	0.645707
$615$	0	0
$616$	8.00000	0.322329
$617$	18.0000	0.724653	0.362326	−	0.932051i	$-0.381983\pi$
$617$	18.0000	0.724653	0.362326	+	0.932051i	$0.381983\pi$
$618$	0	0
$619$	−14.0000	−0.562708	−0.281354	−	0.959604i	$-0.590783\pi$
$619$	−14.0000	−0.562708	−0.281354	+	0.959604i	$0.590783\pi$
$620$	0	0
$621$	0	0
$622$	−32.0000	−1.28308
$623$	24.0000	0.961540
$624$	0	0
$625$	41.0000	1.64000
$626$	−22.0000	−0.879297
$627$	0	0
$628$	12.0000	0.478852
$629$	8.00000	0.318981
$630$	−48.0000	−1.91237
$631$	−40.0000	−1.59237	−0.796187	−	0.605050i	$-0.793153\pi$
$631$	−40.0000	−1.59237	−0.796187	+	0.605050i	$0.793153\pi$
$632$	12.0000	0.477334
$633$	0	0
$634$	−2.00000	−0.0794301
$635$	64.0000	2.53976
$636$	0	0
$637$	−18.0000	−0.713186
$638$	−4.00000	−0.158362
$639$	0	0
$640$	−4.00000	−0.158114
$641$	14.0000	0.552967	0.276483	−	0.961019i	$-0.410831\pi$
$641$	14.0000	0.552967	0.276483	+	0.961019i	$0.410831\pi$
$642$	0	0
$643$	10.0000	0.394362	0.197181	−	0.980367i	$-0.436821\pi$
$643$	10.0000	0.394362	0.197181	+	0.980367i	$0.436821\pi$
$644$	−4.00000	−0.157622
$645$	0	0
$646$	−4.00000	−0.157378
$647$	−40.0000	−1.57256	−0.786281	−	0.617869i	$-0.787996\pi$
$647$	−40.0000	−1.57256	−0.786281	+	0.617869i	$0.787996\pi$
$648$	−9.00000	−0.353553
$649$	24.0000	0.942082
$650$	22.0000	0.862911
$651$	0	0
$652$	−8.00000	−0.313304
$653$	−42.0000	−1.64359	−0.821794	−	0.569785i	$-0.807026\pi$
$653$	−42.0000	−1.64359	−0.821794	+	0.569785i	$0.807026\pi$
$654$	0	0
$655$	48.0000	1.87552
$656$	6.00000	0.234261
$657$	−18.0000	−0.702247
$658$	0	0
$659$	30.0000	1.16863	0.584317	−	0.811525i	$-0.301362\pi$
$659$	30.0000	1.16863	0.584317	+	0.811525i	$0.301362\pi$
$660$	0	0
$661$	−24.0000	−0.933492	−0.466746	−	0.884391i	$-0.654574\pi$
$661$	−24.0000	−0.933492	−0.466746	+	0.884391i	$0.654574\pi$
$662$	4.00000	0.155464
$663$	0	0
$664$	−14.0000	−0.543305
$665$	32.0000	1.24091
$666$	−12.0000	−0.464991
$667$	2.00000	0.0774403
$668$	16.0000	0.619059
$669$	0	0
$670$	40.0000	1.54533
$671$	−16.0000	−0.617673
$672$	0	0
$673$	−2.00000	−0.0770943	−0.0385472	−	0.999257i	$-0.512273\pi$
$673$	−2.00000	−0.0770943	−0.0385472	+	0.999257i	$0.512273\pi$
$674$	−10.0000	−0.385186
$675$	0	0
$676$	−9.00000	−0.346154
$677$	−12.0000	−0.461197	−0.230599	−	0.973049i	$-0.574068\pi$
$677$	−12.0000	−0.461197	−0.230599	+	0.973049i	$0.574068\pi$
$678$	0	0
$679$	−24.0000	−0.921035
$680$	8.00000	0.306786
$681$	0	0
$682$	0	0
$683$	12.0000	0.459167	0.229584	−	0.973289i	$-0.426264\pi$
$683$	12.0000	0.459167	0.229584	+	0.973289i	$0.426264\pi$
$684$	6.00000	0.229416
$685$	24.0000	0.916993
$686$	8.00000	0.305441
$687$	0	0
$688$	10.0000	0.381246
$689$	8.00000	0.304776
$690$	0	0
$691$	−4.00000	−0.152167	−0.0760836	−	0.997101i	$-0.524242\pi$
$691$	−4.00000	−0.152167	−0.0760836	+	0.997101i	$0.524242\pi$
$692$	−6.00000	−0.228086
$693$	24.0000	0.911685
$694$	−8.00000	−0.303676
$695$	−16.0000	−0.606915
$696$	0	0
$697$	−12.0000	−0.454532
$698$	−26.0000	−0.984115
$699$	0	0
$700$	−44.0000	−1.66304
$701$	0	0		−	1.00000i	$-0.5\pi$
$701$	0	0			1.00000i	$0.5\pi$
$702$	0	0
$703$	8.00000	0.301726
$704$	2.00000	0.0753778
$705$	0	0
$706$	−2.00000	−0.0752710
$707$	40.0000	1.50435
$708$	0	0
$709$	−28.0000	−1.05156	−0.525781	−	0.850620i	$-0.676227\pi$
$709$	−28.0000	−1.05156	−0.525781	+	0.850620i	$0.676227\pi$
$710$	0	0
$711$	36.0000	1.35011
$712$	6.00000	0.224860
$713$	0	0
$714$	0	0
$715$	−16.0000	−0.598366
$716$	−16.0000	−0.597948
$717$	0	0
$718$	12.0000	0.447836
$719$	48.0000	1.79010	0.895049	−	0.445968i	$-0.147140\pi$
$719$	48.0000	1.79010	0.895049	+	0.445968i	$0.147140\pi$
$720$	−12.0000	−0.447214
$721$	32.0000	1.19174
$722$	15.0000	0.558242
$723$	0	0
$724$	−24.0000	−0.891953
$725$	22.0000	0.817059
$726$	0	0
$727$	−36.0000	−1.33517	−0.667583	−	0.744535i	$-0.732671\pi$
$727$	−36.0000	−1.33517	−0.667583	+	0.744535i	$0.732671\pi$
$728$	−8.00000	−0.296500
$729$	−27.0000	−1.00000
$730$	−24.0000	−0.888280
$731$	−20.0000	−0.739727
$732$	0	0
$733$	−44.0000	−1.62518	−0.812589	−	0.582838i	$-0.801942\pi$
$733$	−44.0000	−1.62518	−0.812589	+	0.582838i	$0.801942\pi$
$734$	−12.0000	−0.442928
$735$	0	0
$736$	−1.00000	−0.0368605
$737$	−20.0000	−0.736709
$738$	18.0000	0.662589
$739$	−32.0000	−1.17714	−0.588570	−	0.808447i	$-0.700309\pi$
$739$	−32.0000	−1.17714	−0.588570	+	0.808447i	$0.700309\pi$
$740$	−16.0000	−0.588172
$741$	0	0
$742$	−16.0000	−0.587378
$743$	−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
$745$	−16.0000	−0.586195
$746$	4.00000	0.146450
$747$	−42.0000	−1.53670
$748$	−4.00000	−0.146254
$749$	40.0000	1.46157
$750$	0	0
$751$	48.0000	1.75154	0.875772	−	0.482724i	$-0.160353\pi$
$751$	48.0000	1.75154	0.875772	+	0.482724i	$0.160353\pi$
$752$	0	0
$753$	0	0
$754$	4.00000	0.145671
$755$	−32.0000	−1.16460
$756$	0	0
$757$	−48.0000	−1.74459	−0.872295	−	0.488980i	$-0.837369\pi$
$757$	−48.0000	−1.74459	−0.872295	+	0.488980i	$0.837369\pi$
$758$	−22.0000	−0.799076
$759$	0	0
$760$	8.00000	0.290191
$761$	−30.0000	−1.08750	−0.543750	−	0.839248i	$-0.682996\pi$
$761$	−30.0000	−1.08750	−0.543750	+	0.839248i	$0.682996\pi$
$762$	0	0
$763$	0	0
$764$	20.0000	0.723575
$765$	24.0000	0.867722
$766$	24.0000	0.867155
$767$	−24.0000	−0.866590
$768$	0	0
$769$	22.0000	0.793340	0.396670	−	0.917961i	$-0.370166\pi$
$769$	22.0000	0.793340	0.396670	+	0.917961i	$0.370166\pi$
$770$	32.0000	1.15320
$771$	0	0
$772$	26.0000	0.935760
$773$	−40.0000	−1.43870	−0.719350	−	0.694648i	$-0.755560\pi$
$773$	−40.0000	−1.43870	−0.719350	+	0.694648i	$0.755560\pi$
$774$	30.0000	1.07833
$775$	0	0
$776$	−6.00000	−0.215387
$777$	0	0
$778$	0	0
$779$	−12.0000	−0.429945
$780$	0	0
$781$	0	0
$782$	2.00000	0.0715199
$783$	0	0
$784$	9.00000	0.321429
$785$	48.0000	1.71319
$786$	0	0
$787$	18.0000	0.641631	0.320815	−	0.947142i	$-0.396043\pi$
$787$	18.0000	0.641631	0.320815	+	0.947142i	$0.396043\pi$
$788$	18.0000	0.641223
$789$	0	0
$790$	48.0000	1.70776
$791$	56.0000	1.99113
$792$	6.00000	0.213201
$793$	16.0000	0.568177
$794$	−14.0000	−0.496841
$795$	0	0
$796$	−4.00000	−0.141776
$797$	8.00000	0.283375	0.141687	−	0.989911i	$-0.454747\pi$
$797$	8.00000	0.283375	0.141687	+	0.989911i	$0.454747\pi$
$798$	0	0
$799$	0	0
$800$	−11.0000	−0.388909
$801$	18.0000	0.635999
$802$	18.0000	0.635602
$803$	12.0000	0.423471
$804$	0	0
$805$	−16.0000	−0.563926
$806$	0	0
$807$	0	0
$808$	10.0000	0.351799
$809$	−30.0000	−1.05474	−0.527372	−	0.849635i	$-0.676823\pi$
$809$	−30.0000	−1.05474	−0.527372	+	0.849635i	$0.676823\pi$
$810$	−36.0000	−1.26491
$811$	20.0000	0.702295	0.351147	−	0.936320i	$-0.385792\pi$
$811$	20.0000	0.702295	0.351147	+	0.936320i	$0.385792\pi$
$812$	−8.00000	−0.280745
$813$	0	0
$814$	8.00000	0.280400
$815$	−32.0000	−1.12091
$816$	0	0
$817$	−20.0000	−0.699711
$818$	−34.0000	−1.18878
$819$	−24.0000	−0.838628
$820$	24.0000	0.838116
$821$	30.0000	1.04701	0.523504	−	0.852023i	$-0.324625\pi$
$821$	30.0000	1.04701	0.523504	+	0.852023i	$0.324625\pi$
$822$	0	0
$823$	24.0000	0.836587	0.418294	−	0.908312i	$-0.362628\pi$
$823$	24.0000	0.836587	0.418294	+	0.908312i	$0.362628\pi$
$824$	8.00000	0.278693
$825$	0	0
$826$	48.0000	1.67013
$827$	6.00000	0.208640	0.104320	−	0.994544i	$-0.466733\pi$
$827$	6.00000	0.208640	0.104320	+	0.994544i	$0.466733\pi$
$828$	−3.00000	−0.104257
$829$	−30.0000	−1.04194	−0.520972	−	0.853574i	$-0.674430\pi$
$829$	−30.0000	−1.04194	−0.520972	+	0.853574i	$0.674430\pi$
$830$	−56.0000	−1.94379
$831$	0	0
$832$	−2.00000	−0.0693375
$833$	−18.0000	−0.623663
$834$	0	0
$835$	64.0000	2.21481
$836$	−4.00000	−0.138343
$837$	0	0
$838$	18.0000	0.621800
$839$	24.0000	0.828572	0.414286	−	0.910147i	$-0.364031\pi$
$839$	24.0000	0.828572	0.414286	+	0.910147i	$0.364031\pi$
$840$	0	0
$841$	−25.0000	−0.862069
$842$	−28.0000	−0.964944
$843$	0	0
$844$	−20.0000	−0.688428
$845$	−36.0000	−1.23844
$846$	0	0
$847$	28.0000	0.962091
$848$	−4.00000	−0.137361
$849$	0	0
$850$	22.0000	0.754594
$851$	−4.00000	−0.137118
$852$	0	0
$853$	46.0000	1.57501	0.787505	−	0.616308i	$-0.211372\pi$
$853$	46.0000	1.57501	0.787505	+	0.616308i	$0.211372\pi$
$854$	−32.0000	−1.09502
$855$	24.0000	0.820783
$856$	10.0000	0.341793
$857$	42.0000	1.43469	0.717346	−	0.696717i	$-0.245357\pi$
$857$	42.0000	1.43469	0.717346	+	0.696717i	$0.245357\pi$
$858$	0	0
$859$	40.0000	1.36478	0.682391	−	0.730987i	$-0.260940\pi$
$859$	40.0000	1.36478	0.682391	+	0.730987i	$0.260940\pi$
$860$	40.0000	1.36399
$861$	0	0
$862$	12.0000	0.408722
$863$	−8.00000	−0.272323	−0.136162	−	0.990687i	$-0.543477\pi$
$863$	−8.00000	−0.272323	−0.136162	+	0.990687i	$0.543477\pi$
$864$	0	0
$865$	−24.0000	−0.816024
$866$	26.0000	0.883516
$867$	0	0
$868$	0	0
$869$	−24.0000	−0.814144
$870$	0	0
$871$	20.0000	0.677674
$872$	0	0
$873$	−18.0000	−0.609208
$874$	2.00000	0.0676510
$875$	−96.0000	−3.24539
$876$	0	0
$877$	18.0000	0.607817	0.303908	−	0.952701i	$-0.401708\pi$
$877$	18.0000	0.607817	0.303908	+	0.952701i	$0.401708\pi$
$878$	−8.00000	−0.269987
$879$	0	0
$880$	8.00000	0.269680
$881$	18.0000	0.606435	0.303218	−	0.952921i	$-0.401939\pi$
$881$	18.0000	0.606435	0.303218	+	0.952921i	$0.401939\pi$
$882$	27.0000	0.909137
$883$	−36.0000	−1.21150	−0.605748	−	0.795656i	$-0.707126\pi$
$883$	−36.0000	−1.21150	−0.605748	+	0.795656i	$0.707126\pi$
$884$	4.00000	0.134535
$885$	0	0
$886$	−8.00000	−0.268765
$887$	24.0000	0.805841	0.402921	−	0.915235i	$-0.367995\pi$
$887$	24.0000	0.805841	0.402921	+	0.915235i	$0.367995\pi$
$888$	0	0
$889$	−64.0000	−2.14649
$890$	24.0000	0.804482
$891$	18.0000	0.603023
$892$	−16.0000	−0.535720
$893$	0	0
$894$	0	0
$895$	−64.0000	−2.13928
$896$	4.00000	0.133631
$897$	0	0
$898$	−14.0000	−0.467186
$899$	0	0
$900$	−33.0000	−1.10000
$901$	8.00000	0.266519
$902$	−12.0000	−0.399556
$903$	0	0
$904$	14.0000	0.465633
$905$	−96.0000	−3.19115
$906$	0	0
$907$	10.0000	0.332045	0.166022	−	0.986122i	$-0.446908\pi$
$907$	10.0000	0.332045	0.166022	+	0.986122i	$0.446908\pi$
$908$	−6.00000	−0.199117
$909$	30.0000	0.995037
$910$	−32.0000	−1.06079
$911$	20.0000	0.662630	0.331315	−	0.943520i	$-0.392508\pi$
$911$	20.0000	0.662630	0.331315	+	0.943520i	$0.392508\pi$
$912$	0	0
$913$	28.0000	0.926665
$914$	10.0000	0.330771
$915$	0	0
$916$	8.00000	0.264327
$917$	−48.0000	−1.58510
$918$	0	0
$919$	−4.00000	−0.131948	−0.0659739	−	0.997821i	$-0.521015\pi$
$919$	−4.00000	−0.131948	−0.0659739	+	0.997821i	$0.521015\pi$
$920$	−4.00000	−0.131876
$921$	0	0
$922$	−18.0000	−0.592798
$923$	0	0
$924$	0	0
$925$	−44.0000	−1.44671
$926$	−40.0000	−1.31448
$927$	24.0000	0.788263
$928$	−2.00000	−0.0656532
$929$	−18.0000	−0.590561	−0.295280	−	0.955411i	$-0.595413\pi$
$929$	−18.0000	−0.590561	−0.295280	+	0.955411i	$0.595413\pi$
$930$	0	0
$931$	−18.0000	−0.589926
$932$	−6.00000	−0.196537
$933$	0	0
$934$	2.00000	0.0654420
$935$	−16.0000	−0.523256
$936$	−6.00000	−0.196116
$937$	58.0000	1.89478	0.947389	−	0.320085i	$-0.103712\pi$
$937$	58.0000	1.89478	0.947389	+	0.320085i	$0.103712\pi$
$938$	−40.0000	−1.30605
$939$	0	0
$940$	0	0
$941$	36.0000	1.17357	0.586783	−	0.809744i	$-0.300394\pi$
$941$	36.0000	1.17357	0.586783	+	0.809744i	$0.300394\pi$
$942$	0	0
$943$	6.00000	0.195387
$944$	12.0000	0.390567
$945$	0	0
$946$	−20.0000	−0.650256
$947$	48.0000	1.55979	0.779895	−	0.625910i	$-0.215272\pi$
$947$	48.0000	1.55979	0.779895	+	0.625910i	$0.215272\pi$
$948$	0	0
$949$	−12.0000	−0.389536
$950$	22.0000	0.713774
$951$	0	0
$952$	−8.00000	−0.259281
$953$	46.0000	1.49009	0.745043	−	0.667016i	$-0.232429\pi$
$953$	46.0000	1.49009	0.745043	+	0.667016i	$0.232429\pi$
$954$	−12.0000	−0.388514
$955$	80.0000	2.58874
$956$	−24.0000	−0.776215
$957$	0	0
$958$	8.00000	0.258468
$959$	−24.0000	−0.775000
$960$	0	0
$961$	−31.0000	−1.00000
$962$	−8.00000	−0.257930
$963$	30.0000	0.966736
$964$	22.0000	0.708572
$965$	104.000	3.34788
$966$	0	0
$967$	−16.0000	−0.514525	−0.257263	−	0.966342i	$-0.582821\pi$
$967$	−16.0000	−0.514525	−0.257263	+	0.966342i	$0.582821\pi$
$968$	7.00000	0.224989
$969$	0	0
$970$	−24.0000	−0.770594
$971$	−38.0000	−1.21948	−0.609739	−	0.792602i	$-0.708726\pi$
$971$	−38.0000	−1.21948	−0.609739	+	0.792602i	$0.708726\pi$
$972$	0	0
$973$	16.0000	0.512936
$974$	8.00000	0.256337
$975$	0	0
$976$	−8.00000	−0.256074
$977$	6.00000	0.191957	0.0959785	−	0.995383i	$-0.469402\pi$
$977$	6.00000	0.191957	0.0959785	+	0.995383i	$0.469402\pi$
$978$	0	0
$979$	−12.0000	−0.383522
$980$	36.0000	1.14998
$981$	0	0
$982$	−24.0000	−0.765871
$983$	−36.0000	−1.14822	−0.574111	−	0.818778i	$-0.694652\pi$
$983$	−36.0000	−1.14822	−0.574111	+	0.818778i	$0.694652\pi$
$984$	0	0
$985$	72.0000	2.29411
$986$	4.00000	0.127386
$987$	0	0
$988$	4.00000	0.127257
$989$	10.0000	0.317982
$990$	24.0000	0.762770
$991$	−56.0000	−1.77890	−0.889449	−	0.457034i	$-0.848912\pi$
$991$	−56.0000	−1.77890	−0.889449	+	0.457034i	$0.848912\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−16.0000	−0.507234
$996$	0	0
$997$	10.0000	0.316703	0.158352	−	0.987383i	$-0.449382\pi$
$997$	10.0000	0.316703	0.158352	+	0.987383i	$0.449382\pi$
$998$	−16.0000	−0.506471
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	46.2.a.a.1.1	✓	1
3.2	odd	2		414.2.a.b.1.1		1
4.3	odd	2		368.2.a.d.1.1		1
5.2	odd	4		1150.2.b.d.599.1		2
5.3	odd	4		1150.2.b.d.599.2		2
5.4	even	2		1150.2.a.h.1.1		1
7.6	odd	2		2254.2.a.c.1.1		1
8.3	odd	2		1472.2.a.g.1.1		1
8.5	even	2		1472.2.a.f.1.1		1
11.10	odd	2		5566.2.a.h.1.1		1
12.11	even	2		3312.2.a.b.1.1		1
13.12	even	2		7774.2.a.d.1.1		1
20.19	odd	2		9200.2.a.p.1.1		1
23.22	odd	2		1058.2.a.c.1.1		1
69.68	even	2		9522.2.a.p.1.1		1
92.91	even	2		8464.2.a.g.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
46.2.a.a.1.1	✓	1	1.1	even	1	trivial
368.2.a.d.1.1		1	4.3	odd	2
414.2.a.b.1.1		1	3.2	odd	2
1058.2.a.c.1.1		1	23.22	odd	2
1150.2.a.h.1.1		1	5.4	even	2
1150.2.b.d.599.1		2	5.2	odd	4
1150.2.b.d.599.2		2	5.3	odd	4
1472.2.a.f.1.1		1	8.5	even	2
1472.2.a.g.1.1		1	8.3	odd	2
2254.2.a.c.1.1		1	7.6	odd	2
3312.2.a.b.1.1		1	12.11	even	2
5566.2.a.h.1.1		1	11.10	odd	2
7774.2.a.d.1.1		1	13.12	even	2
8464.2.a.g.1.1		1	92.91	even	2
9200.2.a.p.1.1		1	20.19	odd	2
9522.2.a.p.1.1		1	69.68	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:	\( N \)	\(=\)	\( 46 = 2 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	46.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more