LMFDB - Embedded newform 2790.2.a.p.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2790,2,Mod(1,2790)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2790.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2790, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 2790 = 2 \cdot 3^{2} \cdot 5 \cdot 31 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2790.a (trivial)

Newform invariants

comment:select newform

sage:traces = [1,1,0,1,-1,0,-2,1,0,-1,4,0,-4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)

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

Self dual:	yes
Analytic conductor:	$22.2782621639$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 930)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	2790.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$

$=$

\(q+1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} -2.00000 q^{7} +1.00000 q^{8} -1.00000 q^{10} +4.00000 q^{11} -4.00000 q^{13} -2.00000 q^{14} +1.00000 q^{16} -2.00000 q^{17} -8.00000 q^{19} -1.00000 q^{20} +4.00000 q^{22} +8.00000 q^{23} +1.00000 q^{25} -4.00000 q^{26} -2.00000 q^{28} -4.00000 q^{29} -1.00000 q^{31} +1.00000 q^{32} -2.00000 q^{34} +2.00000 q^{35} -12.0000 q^{37} -8.00000 q^{38} -1.00000 q^{40} -10.0000 q^{41} +8.00000 q^{43} +4.00000 q^{44} +8.00000 q^{46} +4.00000 q^{47} -3.00000 q^{49} +1.00000 q^{50} -4.00000 q^{52} -6.00000 q^{53} -4.00000 q^{55} -2.00000 q^{56} -4.00000 q^{58} -2.00000 q^{59} +10.0000 q^{61} -1.00000 q^{62} +1.00000 q^{64} +4.00000 q^{65} -6.00000 q^{67} -2.00000 q^{68} +2.00000 q^{70} -6.00000 q^{71} -4.00000 q^{73} -12.0000 q^{74} -8.00000 q^{76} -8.00000 q^{77} -8.00000 q^{79} -1.00000 q^{80} -10.0000 q^{82} -4.00000 q^{83} +2.00000 q^{85} +8.00000 q^{86} +4.00000 q^{88} +8.00000 q^{91} +8.00000 q^{92} +4.00000 q^{94} +8.00000 q^{95} -18.0000 q^{97} -3.00000 q^{98} +O(q^{100})\)

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
$3$	0	0
$4$	1.00000	0.500000
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	−2.00000	−0.755929	−0.377964	−	0.925820i	$-0.623376\pi$
$7$	−2.00000	−0.755929	−0.377964	+	0.925820i	$0.623376\pi$
$8$	1.00000	0.353553
$9$	0	0
$10$	−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	$0.293963\pi$
$12$	0	0
$13$	−4.00000	−1.10940	−0.554700	−	0.832050i	$-0.687167\pi$
$13$	−4.00000	−1.10940	−0.554700	+	0.832050i	$0.687167\pi$
$14$	−2.00000	−0.534522
$15$	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	$0.577979\pi$
$18$	0	0
$19$	−8.00000	−1.83533	−0.917663	−	0.397360i	$-0.869927\pi$
$19$	−8.00000	−1.83533	−0.917663	+	0.397360i	$0.869927\pi$
$20$	−1.00000	−0.223607
$21$	0	0
$22$	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	$0.186012\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	−4.00000	−0.784465
$27$	0	0
$28$	−2.00000	−0.377964
$29$	−4.00000	−0.742781	−0.371391	−	0.928477i	$-0.621119\pi$
$29$	−4.00000	−0.742781	−0.371391	+	0.928477i	$0.621119\pi$
$30$	0	0
$31$	−1.00000	−0.179605
$31$	−1.00000	−0.179605
$32$	1.00000	0.176777
$33$	0	0
$34$	−2.00000	−0.342997
$35$	2.00000	0.338062
$36$	0	0
$37$	−12.0000	−1.97279	−0.986394	−	0.164399i	$-0.947432\pi$
$37$	−12.0000	−1.97279	−0.986394	+	0.164399i	$0.947432\pi$
$38$	−8.00000	−1.29777
$39$	0	0
$40$	−1.00000	−0.158114
$41$	−10.0000	−1.56174	−0.780869	−	0.624695i	$-0.785223\pi$
$41$	−10.0000	−1.56174	−0.780869	+	0.624695i	$0.785223\pi$
$42$	0	0
$43$	8.00000	1.21999	0.609994	−	0.792406i	$-0.291172\pi$
$43$	8.00000	1.21999	0.609994	+	0.792406i	$0.291172\pi$
$44$	4.00000	0.603023
$45$	0	0
$46$	8.00000	1.17954
$47$	4.00000	0.583460	0.291730	−	0.956501i	$-0.405769\pi$
$47$	4.00000	0.583460	0.291730	+	0.956501i	$0.405769\pi$
$48$	0	0
$49$	−3.00000	−0.428571
$50$	1.00000	0.141421
$51$	0	0
$52$	−4.00000	−0.554700
$53$	−6.00000	−0.824163	−0.412082	−	0.911147i	$-0.635198\pi$
$53$	−6.00000	−0.824163	−0.412082	+	0.911147i	$0.635198\pi$
$54$	0	0
$55$	−4.00000	−0.539360
$56$	−2.00000	−0.267261
$57$	0	0
$58$	−4.00000	−0.525226
$59$	−2.00000	−0.260378	−0.130189	−	0.991489i	$-0.541558\pi$
$59$	−2.00000	−0.260378	−0.130189	+	0.991489i	$0.541558\pi$
$60$	0	0
$61$	10.0000	1.28037	0.640184	−	0.768221i	$-0.278858\pi$
$61$	10.0000	1.28037	0.640184	+	0.768221i	$0.278858\pi$
$62$	−1.00000	−0.127000
$63$	0	0
$64$	1.00000	0.125000
$65$	4.00000	0.496139
$66$	0	0
$67$	−6.00000	−0.733017	−0.366508	−	0.930415i	$-0.619447\pi$
$67$	−6.00000	−0.733017	−0.366508	+	0.930415i	$0.619447\pi$
$68$	−2.00000	−0.242536
$69$	0	0
$70$	2.00000	0.239046
$71$	−6.00000	−0.712069	−0.356034	−	0.934473i	$-0.615871\pi$
$71$	−6.00000	−0.712069	−0.356034	+	0.934473i	$0.615871\pi$
$72$	0	0
$73$	−4.00000	−0.468165	−0.234082	−	0.972217i	$-0.575209\pi$
$73$	−4.00000	−0.468165	−0.234082	+	0.972217i	$0.575209\pi$
$74$	−12.0000	−1.39497
$75$	0	0
$76$	−8.00000	−0.917663
$77$	−8.00000	−0.911685
$78$	0	0
$79$	−8.00000	−0.900070	−0.450035	−	0.893011i	$-0.648589\pi$
$79$	−8.00000	−0.900070	−0.450035	+	0.893011i	$0.648589\pi$
$80$	−1.00000	−0.111803
$81$	0	0
$82$	−10.0000	−1.10432
$83$	−4.00000	−0.439057	−0.219529	−	0.975606i	$-0.570452\pi$
$83$	−4.00000	−0.439057	−0.219529	+	0.975606i	$0.570452\pi$
$84$	0	0
$85$	2.00000	0.216930
$86$	8.00000	0.862662
$87$	0	0
$88$	4.00000	0.426401
$89$	0	0		−	1.00000i	$-0.5\pi$
$89$	0	0			1.00000i	$0.5\pi$
$90$	0	0
$91$	8.00000	0.838628
$92$	8.00000	0.834058
$93$	0	0
$94$	4.00000	0.412568
$95$	8.00000	0.820783
$96$	0	0
$97$	−18.0000	−1.82762	−0.913812	−	0.406138i	$-0.866875\pi$
$97$	−18.0000	−1.82762	−0.913812	+	0.406138i	$0.866875\pi$
$98$	−3.00000	−0.303046
$99$	0	0
$100$	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	$0.146794\pi$
$102$	0	0
$103$	−14.0000	−1.37946	−0.689730	−	0.724066i	$-0.742271\pi$
$103$	−14.0000	−1.37946	−0.689730	+	0.724066i	$0.742271\pi$
$104$	−4.00000	−0.392232
$105$	0	0
$106$	−6.00000	−0.582772
$107$	−8.00000	−0.773389	−0.386695	−	0.922208i	$-0.626383\pi$
$107$	−8.00000	−0.773389	−0.386695	+	0.922208i	$0.626383\pi$
$108$	0	0
$109$	18.0000	1.72409	0.862044	−	0.506834i	$-0.169184\pi$
$109$	18.0000	1.72409	0.862044	+	0.506834i	$0.169184\pi$
$110$	−4.00000	−0.381385
$111$	0	0
$112$	−2.00000	−0.188982
$113$	6.00000	0.564433	0.282216	−	0.959351i	$-0.408930\pi$
$113$	6.00000	0.564433	0.282216	+	0.959351i	$0.408930\pi$
$114$	0	0
$115$	−8.00000	−0.746004
$116$	−4.00000	−0.371391
$117$	0	0
$118$	−2.00000	−0.184115
$119$	4.00000	0.366679
$120$	0	0
$121$	5.00000	0.454545
$122$	10.0000	0.905357
$123$	0	0
$124$	−1.00000	−0.0898027
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	4.00000	0.354943	0.177471	−	0.984126i	$-0.443208\pi$
$127$	4.00000	0.354943	0.177471	+	0.984126i	$0.443208\pi$
$128$	1.00000	0.0883883
$129$	0	0
$130$	4.00000	0.350823
$131$	−10.0000	−0.873704	−0.436852	−	0.899533i	$-0.643907\pi$
$131$	−10.0000	−0.873704	−0.436852	+	0.899533i	$0.643907\pi$
$132$	0	0
$133$	16.0000	1.38738
$134$	−6.00000	−0.518321
$135$	0	0
$136$	−2.00000	−0.171499
$137$	−6.00000	−0.512615	−0.256307	−	0.966595i	$-0.582506\pi$
$137$	−6.00000	−0.512615	−0.256307	+	0.966595i	$0.582506\pi$
$138$	0	0
$139$	4.00000	0.339276	0.169638	−	0.985506i	$-0.445740\pi$
$139$	4.00000	0.339276	0.169638	+	0.985506i	$0.445740\pi$
$140$	2.00000	0.169031
$141$	0	0
$142$	−6.00000	−0.503509
$143$	−16.0000	−1.33799
$144$	0	0
$145$	4.00000	0.332182
$146$	−4.00000	−0.331042
$147$	0	0
$148$	−12.0000	−0.986394
$149$	18.0000	1.47462	0.737309	−	0.675556i	$-0.236096\pi$
$149$	18.0000	1.47462	0.737309	+	0.675556i	$0.236096\pi$
$150$	0	0
$151$	−16.0000	−1.30206	−0.651031	−	0.759051i	$-0.725663\pi$
$151$	−16.0000	−1.30206	−0.651031	+	0.759051i	$0.725663\pi$
$152$	−8.00000	−0.648886
$153$	0	0
$154$	−8.00000	−0.644658
$155$	1.00000	0.0803219
$156$	0	0
$157$	−22.0000	−1.75579	−0.877896	−	0.478852i	$-0.841053\pi$
$157$	−22.0000	−1.75579	−0.877896	+	0.478852i	$0.841053\pi$
$158$	−8.00000	−0.636446
$159$	0	0
$160$	−1.00000	−0.0790569
$161$	−16.0000	−1.26098
$162$	0	0
$163$	−6.00000	−0.469956	−0.234978	−	0.972001i	$-0.575502\pi$
$163$	−6.00000	−0.469956	−0.234978	+	0.972001i	$0.575502\pi$
$164$	−10.0000	−0.780869
$165$	0	0
$166$	−4.00000	−0.310460
$167$	8.00000	0.619059	0.309529	−	0.950890i	$-0.399829\pi$
$167$	8.00000	0.619059	0.309529	+	0.950890i	$0.399829\pi$
$168$	0	0
$169$	3.00000	0.230769
$170$	2.00000	0.153393
$171$	0	0
$172$	8.00000	0.609994
$173$	−22.0000	−1.67263	−0.836315	−	0.548250i	$-0.815294\pi$
$173$	−22.0000	−1.67263	−0.836315	+	0.548250i	$0.815294\pi$
$174$	0	0
$175$	−2.00000	−0.151186
$176$	4.00000	0.301511
$177$	0	0
$178$	0	0
$179$	20.0000	1.49487	0.747435	−	0.664335i	$-0.231285\pi$
$179$	20.0000	1.49487	0.747435	+	0.664335i	$0.231285\pi$
$180$	0	0
$181$	18.0000	1.33793	0.668965	−	0.743294i	$-0.266738\pi$
$181$	18.0000	1.33793	0.668965	+	0.743294i	$0.266738\pi$
$182$	8.00000	0.592999
$183$	0	0
$184$	8.00000	0.589768
$185$	12.0000	0.882258
$186$	0	0
$187$	−8.00000	−0.585018
$188$	4.00000	0.291730
$189$	0	0
$190$	8.00000	0.580381
$191$	−18.0000	−1.30243	−0.651217	−	0.758891i	$-0.725741\pi$
$191$	−18.0000	−1.30243	−0.651217	+	0.758891i	$0.725741\pi$
$192$	0	0
$193$	−2.00000	−0.143963	−0.0719816	−	0.997406i	$-0.522932\pi$
$193$	−2.00000	−0.143963	−0.0719816	+	0.997406i	$0.522932\pi$
$194$	−18.0000	−1.29232
$195$	0	0
$196$	−3.00000	−0.214286
$197$	−10.0000	−0.712470	−0.356235	−	0.934396i	$-0.615940\pi$
$197$	−10.0000	−0.712470	−0.356235	+	0.934396i	$0.615940\pi$
$198$	0	0
$199$	16.0000	1.13421	0.567105	−	0.823646i	$-0.308063\pi$
$199$	16.0000	1.13421	0.567105	+	0.823646i	$0.308063\pi$
$200$	1.00000	0.0707107
$201$	0	0
$202$	18.0000	1.26648
$203$	8.00000	0.561490
$204$	0	0
$205$	10.0000	0.698430
$206$	−14.0000	−0.975426
$207$	0	0
$208$	−4.00000	−0.277350
$209$	−32.0000	−2.21349
$210$	0	0
$211$	24.0000	1.65223	0.826114	−	0.563503i	$-0.190547\pi$
$211$	24.0000	1.65223	0.826114	+	0.563503i	$0.190547\pi$
$212$	−6.00000	−0.412082
$213$	0	0
$214$	−8.00000	−0.546869
$215$	−8.00000	−0.545595
$216$	0	0
$217$	2.00000	0.135769
$218$	18.0000	1.21911
$219$	0	0
$220$	−4.00000	−0.269680
$221$	8.00000	0.538138
$222$	0	0
$223$	−8.00000	−0.535720	−0.267860	−	0.963458i	$-0.586316\pi$
$223$	−8.00000	−0.535720	−0.267860	+	0.963458i	$0.586316\pi$
$224$	−2.00000	−0.133631
$225$	0	0
$226$	6.00000	0.399114
$227$	−4.00000	−0.265489	−0.132745	−	0.991150i	$-0.542379\pi$
$227$	−4.00000	−0.265489	−0.132745	+	0.991150i	$0.542379\pi$
$228$	0	0
$229$	22.0000	1.45380	0.726900	−	0.686743i	$-0.240960\pi$
$229$	22.0000	1.45380	0.726900	+	0.686743i	$0.240960\pi$
$230$	−8.00000	−0.527504
$231$	0	0
$232$	−4.00000	−0.262613
$233$	10.0000	0.655122	0.327561	−	0.944830i	$-0.393773\pi$
$233$	10.0000	0.655122	0.327561	+	0.944830i	$0.393773\pi$
$234$	0	0
$235$	−4.00000	−0.260931
$236$	−2.00000	−0.130189
$237$	0	0
$238$	4.00000	0.259281
$239$	4.00000	0.258738	0.129369	−	0.991596i	$-0.458705\pi$
$239$	4.00000	0.258738	0.129369	+	0.991596i	$0.458705\pi$
$240$	0	0
$241$	10.0000	0.644157	0.322078	−	0.946713i	$-0.395619\pi$
$241$	10.0000	0.644157	0.322078	+	0.946713i	$0.395619\pi$
$242$	5.00000	0.321412
$243$	0	0
$244$	10.0000	0.640184
$245$	3.00000	0.191663
$246$	0	0
$247$	32.0000	2.03611
$248$	−1.00000	−0.0635001
$249$	0	0
$250$	−1.00000	−0.0632456
$251$	−20.0000	−1.26239	−0.631194	−	0.775625i	$-0.717435\pi$
$251$	−20.0000	−1.26239	−0.631194	+	0.775625i	$0.717435\pi$
$252$	0	0
$253$	32.0000	2.01182
$254$	4.00000	0.250982
$255$	0	0
$256$	1.00000	0.0625000
$257$	18.0000	1.12281	0.561405	−	0.827541i	$-0.310261\pi$
$257$	18.0000	1.12281	0.561405	+	0.827541i	$0.310261\pi$
$258$	0	0
$259$	24.0000	1.49129
$260$	4.00000	0.248069
$261$	0	0
$262$	−10.0000	−0.617802
$263$	0	0		−	1.00000i	$-0.5\pi$
$263$	0	0			1.00000i	$0.5\pi$
$264$	0	0
$265$	6.00000	0.368577
$266$	16.0000	0.981023
$267$	0	0
$268$	−6.00000	−0.366508
$269$	12.0000	0.731653	0.365826	−	0.930683i	$-0.380786\pi$
$269$	12.0000	0.731653	0.365826	+	0.930683i	$0.380786\pi$
$270$	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
$275$	4.00000	0.241209
$276$	0	0
$277$	8.00000	0.480673	0.240337	−	0.970690i	$-0.422742\pi$
$277$	8.00000	0.480673	0.240337	+	0.970690i	$0.422742\pi$
$278$	4.00000	0.239904
$279$	0	0
$280$	2.00000	0.119523
$281$	−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
$283$	2.00000	0.118888	0.0594438	−	0.998232i	$-0.481067\pi$
$283$	2.00000	0.118888	0.0594438	+	0.998232i	$0.481067\pi$
$284$	−6.00000	−0.356034
$285$	0	0
$286$	−16.0000	−0.946100
$287$	20.0000	1.18056
$288$	0	0
$289$	−13.0000	−0.764706
$290$	4.00000	0.234888
$291$	0	0
$292$	−4.00000	−0.234082
$293$	6.00000	0.350524	0.175262	−	0.984522i	$-0.443923\pi$
$293$	6.00000	0.350524	0.175262	+	0.984522i	$0.443923\pi$
$294$	0	0
$295$	2.00000	0.116445
$296$	−12.0000	−0.697486
$297$	0	0
$298$	18.0000	1.04271
$299$	−32.0000	−1.85061
$300$	0	0
$301$	−16.0000	−0.922225
$302$	−16.0000	−0.920697
$303$	0	0
$304$	−8.00000	−0.458831
$305$	−10.0000	−0.572598
$306$	0	0
$307$	−2.00000	−0.114146	−0.0570730	−	0.998370i	$-0.518177\pi$
$307$	−2.00000	−0.114146	−0.0570730	+	0.998370i	$0.518177\pi$
$308$	−8.00000	−0.455842
$309$	0	0
$310$	1.00000	0.0567962
$311$	30.0000	1.70114	0.850572	−	0.525859i	$-0.176256\pi$
$311$	30.0000	1.70114	0.850572	+	0.525859i	$0.176256\pi$
$312$	0	0
$313$	20.0000	1.13047	0.565233	−	0.824931i	$-0.308786\pi$
$313$	20.0000	1.13047	0.565233	+	0.824931i	$0.308786\pi$
$314$	−22.0000	−1.24153
$315$	0	0
$316$	−8.00000	−0.450035
$317$	22.0000	1.23564	0.617822	−	0.786318i	$-0.288015\pi$
$317$	22.0000	1.23564	0.617822	+	0.786318i	$0.288015\pi$
$318$	0	0
$319$	−16.0000	−0.895828
$320$	−1.00000	−0.0559017
$321$	0	0
$322$	−16.0000	−0.891645
$323$	16.0000	0.890264
$324$	0	0
$325$	−4.00000	−0.221880
$326$	−6.00000	−0.332309
$327$	0	0
$328$	−10.0000	−0.552158
$329$	−8.00000	−0.441054
$330$	0	0
$331$	−12.0000	−0.659580	−0.329790	−	0.944054i	$-0.606978\pi$
$331$	−12.0000	−0.659580	−0.329790	+	0.944054i	$0.606978\pi$
$332$	−4.00000	−0.219529
$333$	0	0
$334$	8.00000	0.437741
$335$	6.00000	0.327815
$336$	0	0
$337$	−32.0000	−1.74315	−0.871576	−	0.490261i	$-0.836901\pi$
$337$	−32.0000	−1.74315	−0.871576	+	0.490261i	$0.836901\pi$
$338$	3.00000	0.163178
$339$	0	0
$340$	2.00000	0.108465
$341$	−4.00000	−0.216612
$342$	0	0
$343$	20.0000	1.07990
$344$	8.00000	0.431331
$345$	0	0
$346$	−22.0000	−1.18273
$347$	4.00000	0.214731	0.107366	−	0.994220i	$-0.465758\pi$
$347$	4.00000	0.214731	0.107366	+	0.994220i	$0.465758\pi$
$348$	0	0
$349$	−30.0000	−1.60586	−0.802932	−	0.596071i	$-0.796728\pi$
$349$	−30.0000	−1.60586	−0.802932	+	0.596071i	$0.796728\pi$
$350$	−2.00000	−0.106904
$351$	0	0
$352$	4.00000	0.213201
$353$	−34.0000	−1.80964	−0.904819	−	0.425797i	$-0.859994\pi$
$353$	−34.0000	−1.80964	−0.904819	+	0.425797i	$0.859994\pi$
$354$	0	0
$355$	6.00000	0.318447
$356$	0	0
$357$	0	0
$358$	20.0000	1.05703
$359$	−10.0000	−0.527780	−0.263890	−	0.964553i	$-0.585006\pi$
$359$	−10.0000	−0.527780	−0.263890	+	0.964553i	$0.585006\pi$
$360$	0	0
$361$	45.0000	2.36842
$362$	18.0000	0.946059
$363$	0	0
$364$	8.00000	0.419314
$365$	4.00000	0.209370
$366$	0	0
$367$	−8.00000	−0.417597	−0.208798	−	0.977959i	$-0.566955\pi$
$367$	−8.00000	−0.417597	−0.208798	+	0.977959i	$0.566955\pi$
$368$	8.00000	0.417029
$369$	0	0
$370$	12.0000	0.623850
$371$	12.0000	0.623009
$372$	0	0
$373$	34.0000	1.76045	0.880227	−	0.474554i	$-0.157390\pi$
$373$	34.0000	1.76045	0.880227	+	0.474554i	$0.157390\pi$
$374$	−8.00000	−0.413670
$375$	0	0
$376$	4.00000	0.206284
$377$	16.0000	0.824042
$378$	0	0
$379$	16.0000	0.821865	0.410932	−	0.911666i	$-0.365203\pi$
$379$	16.0000	0.821865	0.410932	+	0.911666i	$0.365203\pi$
$380$	8.00000	0.410391
$381$	0	0
$382$	−18.0000	−0.920960
$383$	16.0000	0.817562	0.408781	−	0.912633i	$-0.365954\pi$
$383$	16.0000	0.817562	0.408781	+	0.912633i	$0.365954\pi$
$384$	0	0
$385$	8.00000	0.407718
$386$	−2.00000	−0.101797
$387$	0	0
$388$	−18.0000	−0.913812
$389$	28.0000	1.41966	0.709828	−	0.704375i	$-0.248773\pi$
$389$	28.0000	1.41966	0.709828	+	0.704375i	$0.248773\pi$
$390$	0	0
$391$	−16.0000	−0.809155
$392$	−3.00000	−0.151523
$393$	0	0
$394$	−10.0000	−0.503793
$395$	8.00000	0.402524
$396$	0	0
$397$	−34.0000	−1.70641	−0.853206	−	0.521575i	$-0.825345\pi$
$397$	−34.0000	−1.70641	−0.853206	+	0.521575i	$0.825345\pi$
$398$	16.0000	0.802008
$399$	0	0
$400$	1.00000	0.0500000
$401$	−4.00000	−0.199750	−0.0998752	−	0.995000i	$-0.531844\pi$
$401$	−4.00000	−0.199750	−0.0998752	+	0.995000i	$0.531844\pi$
$402$	0	0
$403$	4.00000	0.199254
$404$	18.0000	0.895533
$405$	0	0
$406$	8.00000	0.397033
$407$	−48.0000	−2.37927
$408$	0	0
$409$	6.00000	0.296681	0.148340	−	0.988936i	$-0.452607\pi$
$409$	6.00000	0.296681	0.148340	+	0.988936i	$0.452607\pi$
$410$	10.0000	0.493865
$411$	0	0
$412$	−14.0000	−0.689730
$413$	4.00000	0.196827
$414$	0	0
$415$	4.00000	0.196352
$416$	−4.00000	−0.196116
$417$	0	0
$418$	−32.0000	−1.56517
$419$	10.0000	0.488532	0.244266	−	0.969708i	$-0.421453\pi$
$419$	10.0000	0.488532	0.244266	+	0.969708i	$0.421453\pi$
$420$	0	0
$421$	2.00000	0.0974740	0.0487370	−	0.998812i	$-0.484480\pi$
$421$	2.00000	0.0974740	0.0487370	+	0.998812i	$0.484480\pi$
$422$	24.0000	1.16830
$423$	0	0
$424$	−6.00000	−0.291386
$425$	−2.00000	−0.0970143
$426$	0	0
$427$	−20.0000	−0.967868
$428$	−8.00000	−0.386695
$429$	0	0
$430$	−8.00000	−0.385794
$431$	30.0000	1.44505	0.722525	−	0.691345i	$-0.242982\pi$
$431$	30.0000	1.44505	0.722525	+	0.691345i	$0.242982\pi$
$432$	0	0
$433$	4.00000	0.192228	0.0961139	−	0.995370i	$-0.469359\pi$
$433$	4.00000	0.192228	0.0961139	+	0.995370i	$0.469359\pi$
$434$	2.00000	0.0960031
$435$	0	0
$436$	18.0000	0.862044
$437$	−64.0000	−3.06154
$438$	0	0
$439$	−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
$441$	0	0
$442$	8.00000	0.380521
$443$	−4.00000	−0.190046	−0.0950229	−	0.995475i	$-0.530292\pi$
$443$	−4.00000	−0.190046	−0.0950229	+	0.995475i	$0.530292\pi$
$444$	0	0
$445$	0	0
$446$	−8.00000	−0.378811
$447$	0	0
$448$	−2.00000	−0.0944911
$449$	−28.0000	−1.32140	−0.660701	−	0.750649i	$-0.729741\pi$
$449$	−28.0000	−1.32140	−0.660701	+	0.750649i	$0.729741\pi$
$450$	0	0
$451$	−40.0000	−1.88353
$452$	6.00000	0.282216
$453$	0	0
$454$	−4.00000	−0.187729
$455$	−8.00000	−0.375046
$456$	0	0
$457$	36.0000	1.68401	0.842004	−	0.539471i	$-0.181376\pi$
$457$	36.0000	1.68401	0.842004	+	0.539471i	$0.181376\pi$
$458$	22.0000	1.02799
$459$	0	0
$460$	−8.00000	−0.373002
$461$	40.0000	1.86299	0.931493	−	0.363760i	$-0.118507\pi$
$461$	40.0000	1.86299	0.931493	+	0.363760i	$0.118507\pi$
$462$	0	0
$463$	−36.0000	−1.67306	−0.836531	−	0.547920i	$-0.815420\pi$
$463$	−36.0000	−1.67306	−0.836531	+	0.547920i	$0.815420\pi$
$464$	−4.00000	−0.185695
$465$	0	0
$466$	10.0000	0.463241
$467$	0	0		−	1.00000i	$-0.5\pi$
$467$	0	0			1.00000i	$0.5\pi$
$468$	0	0
$469$	12.0000	0.554109
$470$	−4.00000	−0.184506
$471$	0	0
$472$	−2.00000	−0.0920575
$473$	32.0000	1.47136
$474$	0	0
$475$	−8.00000	−0.367065
$476$	4.00000	0.183340
$477$	0	0
$478$	4.00000	0.182956
$479$	10.0000	0.456912	0.228456	−	0.973554i	$-0.426632\pi$
$479$	10.0000	0.456912	0.228456	+	0.973554i	$0.426632\pi$
$480$	0	0
$481$	48.0000	2.18861
$482$	10.0000	0.455488
$483$	0	0
$484$	5.00000	0.227273
$485$	18.0000	0.817338
$486$	0	0
$487$	−12.0000	−0.543772	−0.271886	−	0.962329i	$-0.587647\pi$
$487$	−12.0000	−0.543772	−0.271886	+	0.962329i	$0.587647\pi$
$488$	10.0000	0.452679
$489$	0	0
$490$	3.00000	0.135526
$491$	8.00000	0.361035	0.180517	−	0.983572i	$-0.442223\pi$
$491$	8.00000	0.361035	0.180517	+	0.983572i	$0.442223\pi$
$492$	0	0
$493$	8.00000	0.360302
$494$	32.0000	1.43975
$495$	0	0
$496$	−1.00000	−0.0449013
$497$	12.0000	0.538274
$498$	0	0
$499$	−36.0000	−1.61158	−0.805791	−	0.592200i	$-0.798259\pi$
$499$	−36.0000	−1.61158	−0.805791	+	0.592200i	$0.798259\pi$
$500$	−1.00000	−0.0447214
$501$	0	0
$502$	−20.0000	−0.892644
$503$	−12.0000	−0.535054	−0.267527	−	0.963550i	$-0.586206\pi$
$503$	−12.0000	−0.535054	−0.267527	+	0.963550i	$0.586206\pi$
$504$	0	0
$505$	−18.0000	−0.800989
$506$	32.0000	1.42257
$507$	0	0
$508$	4.00000	0.177471
$509$	−16.0000	−0.709188	−0.354594	−	0.935020i	$-0.615381\pi$
$509$	−16.0000	−0.709188	−0.354594	+	0.935020i	$0.615381\pi$
$510$	0	0
$511$	8.00000	0.353899
$512$	1.00000	0.0441942
$513$	0	0
$514$	18.0000	0.793946
$515$	14.0000	0.616914
$516$	0	0
$517$	16.0000	0.703679
$518$	24.0000	1.05450
$519$	0	0
$520$	4.00000	0.175412
$521$	−18.0000	−0.788594	−0.394297	−	0.918983i	$-0.629012\pi$
$521$	−18.0000	−0.788594	−0.394297	+	0.918983i	$0.629012\pi$
$522$	0	0
$523$	0	0		−	1.00000i	$-0.5\pi$
$523$	0	0			1.00000i	$0.5\pi$
$524$	−10.0000	−0.436852
$525$	0	0
$526$	0	0
$527$	2.00000	0.0871214
$528$	0	0
$529$	41.0000	1.78261
$530$	6.00000	0.260623
$531$	0	0
$532$	16.0000	0.693688
$533$	40.0000	1.73259
$534$	0	0
$535$	8.00000	0.345870
$536$	−6.00000	−0.259161
$537$	0	0
$538$	12.0000	0.517357
$539$	−12.0000	−0.516877
$540$	0	0
$541$	2.00000	0.0859867	0.0429934	−	0.999075i	$-0.486311\pi$
$541$	2.00000	0.0859867	0.0429934	+	0.999075i	$0.486311\pi$
$542$	8.00000	0.343629
$543$	0	0
$544$	−2.00000	−0.0857493
$545$	−18.0000	−0.771035
$546$	0	0
$547$	2.00000	0.0855138	0.0427569	−	0.999086i	$-0.486386\pi$
$547$	2.00000	0.0855138	0.0427569	+	0.999086i	$0.486386\pi$
$548$	−6.00000	−0.256307
$549$	0	0
$550$	4.00000	0.170561
$551$	32.0000	1.36325
$552$	0	0
$553$	16.0000	0.680389
$554$	8.00000	0.339887
$555$	0	0
$556$	4.00000	0.169638
$557$	−6.00000	−0.254228	−0.127114	−	0.991888i	$-0.540571\pi$
$557$	−6.00000	−0.254228	−0.127114	+	0.991888i	$0.540571\pi$
$558$	0	0
$559$	−32.0000	−1.35346
$560$	2.00000	0.0845154
$561$	0	0
$562$	−6.00000	−0.253095
$563$	0	0		−	1.00000i	$-0.5\pi$
$563$	0	0			1.00000i	$0.5\pi$
$564$	0	0
$565$	−6.00000	−0.252422
$566$	2.00000	0.0840663
$567$	0	0
$568$	−6.00000	−0.251754
$569$	28.0000	1.17382	0.586911	−	0.809652i	$-0.300344\pi$
$569$	28.0000	1.17382	0.586911	+	0.809652i	$0.300344\pi$
$570$	0	0
$571$	−12.0000	−0.502184	−0.251092	−	0.967963i	$-0.580790\pi$
$571$	−12.0000	−0.502184	−0.251092	+	0.967963i	$0.580790\pi$
$572$	−16.0000	−0.668994
$573$	0	0
$574$	20.0000	0.834784
$575$	8.00000	0.333623
$576$	0	0
$577$	−6.00000	−0.249783	−0.124892	−	0.992170i	$-0.539858\pi$
$577$	−6.00000	−0.249783	−0.124892	+	0.992170i	$0.539858\pi$
$578$	−13.0000	−0.540729
$579$	0	0
$580$	4.00000	0.166091
$581$	8.00000	0.331896
$582$	0	0
$583$	−24.0000	−0.993978
$584$	−4.00000	−0.165521
$585$	0	0
$586$	6.00000	0.247858
$587$	−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
$589$	8.00000	0.329634
$590$	2.00000	0.0823387
$591$	0	0
$592$	−12.0000	−0.493197
$593$	30.0000	1.23195	0.615976	−	0.787765i	$-0.288762\pi$
$593$	30.0000	1.23195	0.615976	+	0.787765i	$0.288762\pi$
$594$	0	0
$595$	−4.00000	−0.163984
$596$	18.0000	0.737309
$597$	0	0
$598$	−32.0000	−1.30858
$599$	30.0000	1.22577	0.612883	−	0.790173i	$-0.290010\pi$
$599$	30.0000	1.22577	0.612883	+	0.790173i	$0.290010\pi$
$600$	0	0
$601$	−26.0000	−1.06056	−0.530281	−	0.847822i	$-0.677914\pi$
$601$	−26.0000	−1.06056	−0.530281	+	0.847822i	$0.677914\pi$
$602$	−16.0000	−0.652111
$603$	0	0
$604$	−16.0000	−0.651031
$605$	−5.00000	−0.203279
$606$	0	0
$607$	−14.0000	−0.568242	−0.284121	−	0.958788i	$-0.591702\pi$
$607$	−14.0000	−0.568242	−0.284121	+	0.958788i	$0.591702\pi$
$608$	−8.00000	−0.324443
$609$	0	0
$610$	−10.0000	−0.404888
$611$	−16.0000	−0.647291
$612$	0	0
$613$	8.00000	0.323117	0.161558	−	0.986863i	$-0.448348\pi$
$613$	8.00000	0.323117	0.161558	+	0.986863i	$0.448348\pi$
$614$	−2.00000	−0.0807134
$615$	0	0
$616$	−8.00000	−0.322329
$617$	−42.0000	−1.69086	−0.845428	−	0.534089i	$-0.820655\pi$
$617$	−42.0000	−1.69086	−0.845428	+	0.534089i	$0.820655\pi$
$618$	0	0
$619$	−20.0000	−0.803868	−0.401934	−	0.915669i	$-0.631662\pi$
$619$	−20.0000	−0.803868	−0.401934	+	0.915669i	$0.631662\pi$
$620$	1.00000	0.0401610
$621$	0	0
$622$	30.0000	1.20289
$623$	0	0
$624$	0	0
$625$	1.00000	0.0400000
$626$	20.0000	0.799361
$627$	0	0
$628$	−22.0000	−0.877896
$629$	24.0000	0.956943
$630$	0	0
$631$	8.00000	0.318475	0.159237	−	0.987240i	$-0.449096\pi$
$631$	8.00000	0.318475	0.159237	+	0.987240i	$0.449096\pi$
$632$	−8.00000	−0.318223
$633$	0	0
$634$	22.0000	0.873732
$635$	−4.00000	−0.158735
$636$	0	0
$637$	12.0000	0.475457
$638$	−16.0000	−0.633446
$639$	0	0
$640$	−1.00000	−0.0395285
$641$	4.00000	0.157991	0.0789953	−	0.996875i	$-0.474829\pi$
$641$	4.00000	0.157991	0.0789953	+	0.996875i	$0.474829\pi$
$642$	0	0
$643$	−8.00000	−0.315489	−0.157745	−	0.987480i	$-0.550422\pi$
$643$	−8.00000	−0.315489	−0.157745	+	0.987480i	$0.550422\pi$
$644$	−16.0000	−0.630488
$645$	0	0
$646$	16.0000	0.629512
$647$	−8.00000	−0.314512	−0.157256	−	0.987558i	$-0.550265\pi$
$647$	−8.00000	−0.314512	−0.157256	+	0.987558i	$0.550265\pi$
$648$	0	0
$649$	−8.00000	−0.314027
$650$	−4.00000	−0.156893
$651$	0	0
$652$	−6.00000	−0.234978
$653$	6.00000	0.234798	0.117399	−	0.993085i	$-0.462544\pi$
$653$	6.00000	0.234798	0.117399	+	0.993085i	$0.462544\pi$
$654$	0	0
$655$	10.0000	0.390732
$656$	−10.0000	−0.390434
$657$	0	0
$658$	−8.00000	−0.311872
$659$	2.00000	0.0779089	0.0389545	−	0.999241i	$-0.487597\pi$
$659$	2.00000	0.0779089	0.0389545	+	0.999241i	$0.487597\pi$
$660$	0	0
$661$	46.0000	1.78919	0.894596	−	0.446875i	$-0.147463\pi$
$661$	46.0000	1.78919	0.894596	+	0.446875i	$0.147463\pi$
$662$	−12.0000	−0.466393
$663$	0	0
$664$	−4.00000	−0.155230
$665$	−16.0000	−0.620453
$666$	0	0
$667$	−32.0000	−1.23904
$668$	8.00000	0.309529
$669$	0	0
$670$	6.00000	0.231800
$671$	40.0000	1.54418
$672$	0	0
$673$	8.00000	0.308377	0.154189	−	0.988041i	$-0.450724\pi$
$673$	8.00000	0.308377	0.154189	+	0.988041i	$0.450724\pi$
$674$	−32.0000	−1.23259
$675$	0	0
$676$	3.00000	0.115385
$677$	−6.00000	−0.230599	−0.115299	−	0.993331i	$-0.536783\pi$
$677$	−6.00000	−0.230599	−0.115299	+	0.993331i	$0.536783\pi$
$678$	0	0
$679$	36.0000	1.38155
$680$	2.00000	0.0766965
$681$	0	0
$682$	−4.00000	−0.153168
$683$	−4.00000	−0.153056	−0.0765279	−	0.997067i	$-0.524383\pi$
$683$	−4.00000	−0.153056	−0.0765279	+	0.997067i	$0.524383\pi$
$684$	0	0
$685$	6.00000	0.229248
$686$	20.0000	0.763604
$687$	0	0
$688$	8.00000	0.304997
$689$	24.0000	0.914327
$690$	0	0
$691$	28.0000	1.06517	0.532585	−	0.846376i	$-0.321221\pi$
$691$	28.0000	1.06517	0.532585	+	0.846376i	$0.321221\pi$
$692$	−22.0000	−0.836315
$693$	0	0
$694$	4.00000	0.151838
$695$	−4.00000	−0.151729
$696$	0	0
$697$	20.0000	0.757554
$698$	−30.0000	−1.13552
$699$	0	0
$700$	−2.00000	−0.0755929
$701$	−2.00000	−0.0755390	−0.0377695	−	0.999286i	$-0.512025\pi$
$701$	−2.00000	−0.0755390	−0.0377695	+	0.999286i	$0.512025\pi$
$702$	0	0
$703$	96.0000	3.62071
$704$	4.00000	0.150756
$705$	0	0
$706$	−34.0000	−1.27961
$707$	−36.0000	−1.35392
$708$	0	0
$709$	−42.0000	−1.57734	−0.788672	−	0.614815i	$-0.789231\pi$
$709$	−42.0000	−1.57734	−0.788672	+	0.614815i	$0.789231\pi$
$710$	6.00000	0.225176
$711$	0	0
$712$	0	0
$713$	−8.00000	−0.299602
$714$	0	0
$715$	16.0000	0.598366
$716$	20.0000	0.747435
$717$	0	0
$718$	−10.0000	−0.373197
$719$	24.0000	0.895049	0.447524	−	0.894272i	$-0.352306\pi$
$719$	24.0000	0.895049	0.447524	+	0.894272i	$0.352306\pi$
$720$	0	0
$721$	28.0000	1.04277
$722$	45.0000	1.67473
$723$	0	0
$724$	18.0000	0.668965
$725$	−4.00000	−0.148556
$726$	0	0
$727$	−2.00000	−0.0741759	−0.0370879	−	0.999312i	$-0.511808\pi$
$727$	−2.00000	−0.0741759	−0.0370879	+	0.999312i	$0.511808\pi$
$728$	8.00000	0.296500
$729$	0	0
$730$	4.00000	0.148047
$731$	−16.0000	−0.591781
$732$	0	0
$733$	6.00000	0.221615	0.110808	−	0.993842i	$-0.464656\pi$
$733$	6.00000	0.221615	0.110808	+	0.993842i	$0.464656\pi$
$734$	−8.00000	−0.295285
$735$	0	0
$736$	8.00000	0.294884
$737$	−24.0000	−0.884051
$738$	0	0
$739$	20.0000	0.735712	0.367856	−	0.929883i	$-0.380092\pi$
$739$	20.0000	0.735712	0.367856	+	0.929883i	$0.380092\pi$
$740$	12.0000	0.441129
$741$	0	0
$742$	12.0000	0.440534
$743$	16.0000	0.586983	0.293492	−	0.955962i	$-0.405183\pi$
$743$	16.0000	0.586983	0.293492	+	0.955962i	$0.405183\pi$
$744$	0	0
$745$	−18.0000	−0.659469
$746$	34.0000	1.24483
$747$	0	0
$748$	−8.00000	−0.292509
$749$	16.0000	0.584627
$750$	0	0
$751$	16.0000	0.583848	0.291924	−	0.956441i	$-0.405705\pi$
$751$	16.0000	0.583848	0.291924	+	0.956441i	$0.405705\pi$
$752$	4.00000	0.145865
$753$	0	0
$754$	16.0000	0.582686
$755$	16.0000	0.582300
$756$	0	0
$757$	36.0000	1.30844	0.654221	−	0.756303i	$-0.272997\pi$
$757$	36.0000	1.30844	0.654221	+	0.756303i	$0.272997\pi$
$758$	16.0000	0.581146
$759$	0	0
$760$	8.00000	0.290191
$761$	20.0000	0.724999	0.362500	−	0.931984i	$-0.381923\pi$
$761$	20.0000	0.724999	0.362500	+	0.931984i	$0.381923\pi$
$762$	0	0
$763$	−36.0000	−1.30329
$764$	−18.0000	−0.651217
$765$	0	0
$766$	16.0000	0.578103
$767$	8.00000	0.288863
$768$	0	0
$769$	−34.0000	−1.22607	−0.613036	−	0.790055i	$-0.710052\pi$
$769$	−34.0000	−1.22607	−0.613036	+	0.790055i	$0.710052\pi$
$770$	8.00000	0.288300
$771$	0	0
$772$	−2.00000	−0.0719816
$773$	−14.0000	−0.503545	−0.251773	−	0.967786i	$-0.581013\pi$
$773$	−14.0000	−0.503545	−0.251773	+	0.967786i	$0.581013\pi$
$774$	0	0
$775$	−1.00000	−0.0359211
$776$	−18.0000	−0.646162
$777$	0	0
$778$	28.0000	1.00385
$779$	80.0000	2.86630
$780$	0	0
$781$	−24.0000	−0.858788
$782$	−16.0000	−0.572159
$783$	0	0
$784$	−3.00000	−0.107143
$785$	22.0000	0.785214
$786$	0	0
$787$	28.0000	0.998092	0.499046	−	0.866575i	$-0.333684\pi$
$787$	28.0000	0.998092	0.499046	+	0.866575i	$0.333684\pi$
$788$	−10.0000	−0.356235
$789$	0	0
$790$	8.00000	0.284627
$791$	−12.0000	−0.426671
$792$	0	0
$793$	−40.0000	−1.42044
$794$	−34.0000	−1.20661
$795$	0	0
$796$	16.0000	0.567105
$797$	30.0000	1.06265	0.531327	−	0.847167i	$-0.321693\pi$
$797$	30.0000	1.06265	0.531327	+	0.847167i	$0.321693\pi$
$798$	0	0
$799$	−8.00000	−0.283020
$800$	1.00000	0.0353553
$801$	0	0
$802$	−4.00000	−0.141245
$803$	−16.0000	−0.564628
$804$	0	0
$805$	16.0000	0.563926
$806$	4.00000	0.140894
$807$	0	0
$808$	18.0000	0.633238
$809$	−28.0000	−0.984428	−0.492214	−	0.870474i	$-0.663812\pi$
$809$	−28.0000	−0.984428	−0.492214	+	0.870474i	$0.663812\pi$
$810$	0	0
$811$	−16.0000	−0.561836	−0.280918	−	0.959732i	$-0.590639\pi$
$811$	−16.0000	−0.561836	−0.280918	+	0.959732i	$0.590639\pi$
$812$	8.00000	0.280745
$813$	0	0
$814$	−48.0000	−1.68240
$815$	6.00000	0.210171
$816$	0	0
$817$	−64.0000	−2.23908
$818$	6.00000	0.209785
$819$	0	0
$820$	10.0000	0.349215
$821$	4.00000	0.139601	0.0698005	−	0.997561i	$-0.477764\pi$
$821$	4.00000	0.139601	0.0698005	+	0.997561i	$0.477764\pi$
$822$	0	0
$823$	28.0000	0.976019	0.488009	−	0.872838i	$-0.337723\pi$
$823$	28.0000	0.976019	0.488009	+	0.872838i	$0.337723\pi$
$824$	−14.0000	−0.487713
$825$	0	0
$826$	4.00000	0.139178
$827$	−36.0000	−1.25184	−0.625921	−	0.779886i	$-0.715277\pi$
$827$	−36.0000	−1.25184	−0.625921	+	0.779886i	$0.715277\pi$
$828$	0	0
$829$	−46.0000	−1.59765	−0.798823	−	0.601566i	$-0.794544\pi$
$829$	−46.0000	−1.59765	−0.798823	+	0.601566i	$0.794544\pi$
$830$	4.00000	0.138842
$831$	0	0
$832$	−4.00000	−0.138675
$833$	6.00000	0.207888
$834$	0	0
$835$	−8.00000	−0.276851
$836$	−32.0000	−1.10674
$837$	0	0
$838$	10.0000	0.345444
$839$	−42.0000	−1.45000	−0.725001	−	0.688748i	$-0.758161\pi$
$839$	−42.0000	−1.45000	−0.725001	+	0.688748i	$0.758161\pi$
$840$	0	0
$841$	−13.0000	−0.448276
$842$	2.00000	0.0689246
$843$	0	0
$844$	24.0000	0.826114
$845$	−3.00000	−0.103203
$846$	0	0
$847$	−10.0000	−0.343604
$848$	−6.00000	−0.206041
$849$	0	0
$850$	−2.00000	−0.0685994
$851$	−96.0000	−3.29084
$852$	0	0
$853$	14.0000	0.479351	0.239675	−	0.970853i	$-0.422959\pi$
$853$	14.0000	0.479351	0.239675	+	0.970853i	$0.422959\pi$
$854$	−20.0000	−0.684386
$855$	0	0
$856$	−8.00000	−0.273434
$857$	−10.0000	−0.341593	−0.170797	−	0.985306i	$-0.554634\pi$
$857$	−10.0000	−0.341593	−0.170797	+	0.985306i	$0.554634\pi$
$858$	0	0
$859$	−20.0000	−0.682391	−0.341196	−	0.939992i	$-0.610832\pi$
$859$	−20.0000	−0.682391	−0.341196	+	0.939992i	$0.610832\pi$
$860$	−8.00000	−0.272798
$861$	0	0
$862$	30.0000	1.02180
$863$	−16.0000	−0.544646	−0.272323	−	0.962206i	$-0.587792\pi$
$863$	−16.0000	−0.544646	−0.272323	+	0.962206i	$0.587792\pi$
$864$	0	0
$865$	22.0000	0.748022
$866$	4.00000	0.135926
$867$	0	0
$868$	2.00000	0.0678844
$869$	−32.0000	−1.08553
$870$	0	0
$871$	24.0000	0.813209
$872$	18.0000	0.609557
$873$	0	0
$874$	−64.0000	−2.16483
$875$	2.00000	0.0676123
$876$	0	0
$877$	−2.00000	−0.0675352	−0.0337676	−	0.999430i	$-0.510751\pi$
$877$	−2.00000	−0.0675352	−0.0337676	+	0.999430i	$0.510751\pi$
$878$	−40.0000	−1.34993
$879$	0	0
$880$	−4.00000	−0.134840
$881$	0	0		−	1.00000i	$-0.5\pi$
$881$	0	0			1.00000i	$0.5\pi$
$882$	0	0
$883$	20.0000	0.673054	0.336527	−	0.941674i	$-0.390748\pi$
$883$	20.0000	0.673054	0.336527	+	0.941674i	$0.390748\pi$
$884$	8.00000	0.269069
$885$	0	0
$886$	−4.00000	−0.134383
$887$	56.0000	1.88030	0.940148	−	0.340766i	$-0.110687\pi$
$887$	56.0000	1.88030	0.940148	+	0.340766i	$0.110687\pi$
$888$	0	0
$889$	−8.00000	−0.268311
$890$	0	0
$891$	0	0
$892$	−8.00000	−0.267860
$893$	−32.0000	−1.07084
$894$	0	0
$895$	−20.0000	−0.668526
$896$	−2.00000	−0.0668153
$897$	0	0
$898$	−28.0000	−0.934372
$899$	4.00000	0.133407
$900$	0	0
$901$	12.0000	0.399778
$902$	−40.0000	−1.33185
$903$	0	0
$904$	6.00000	0.199557
$905$	−18.0000	−0.598340
$906$	0	0
$907$	−10.0000	−0.332045	−0.166022	−	0.986122i	$-0.553092\pi$
$907$	−10.0000	−0.332045	−0.166022	+	0.986122i	$0.553092\pi$
$908$	−4.00000	−0.132745
$909$	0	0
$910$	−8.00000	−0.265197
$911$	−24.0000	−0.795155	−0.397578	−	0.917568i	$-0.630149\pi$
$911$	−24.0000	−0.795155	−0.397578	+	0.917568i	$0.630149\pi$
$912$	0	0
$913$	−16.0000	−0.529523
$914$	36.0000	1.19077
$915$	0	0
$916$	22.0000	0.726900
$917$	20.0000	0.660458
$918$	0	0
$919$	4.00000	0.131948	0.0659739	−	0.997821i	$-0.478985\pi$
$919$	4.00000	0.131948	0.0659739	+	0.997821i	$0.478985\pi$
$920$	−8.00000	−0.263752
$921$	0	0
$922$	40.0000	1.31733
$923$	24.0000	0.789970
$924$	0	0
$925$	−12.0000	−0.394558
$926$	−36.0000	−1.18303
$927$	0	0
$928$	−4.00000	−0.131306
$929$	−32.0000	−1.04989	−0.524943	−	0.851137i	$-0.675913\pi$
$929$	−32.0000	−1.04989	−0.524943	+	0.851137i	$0.675913\pi$
$930$	0	0
$931$	24.0000	0.786568
$932$	10.0000	0.327561
$933$	0	0
$934$	0	0
$935$	8.00000	0.261628
$936$	0	0
$937$	−2.00000	−0.0653372	−0.0326686	−	0.999466i	$-0.510401\pi$
$937$	−2.00000	−0.0653372	−0.0326686	+	0.999466i	$0.510401\pi$
$938$	12.0000	0.391814
$939$	0	0
$940$	−4.00000	−0.130466
$941$	−12.0000	−0.391189	−0.195594	−	0.980685i	$-0.562664\pi$
$941$	−12.0000	−0.391189	−0.195594	+	0.980685i	$0.562664\pi$
$942$	0	0
$943$	−80.0000	−2.60516
$944$	−2.00000	−0.0650945
$945$	0	0
$946$	32.0000	1.04041
$947$	−28.0000	−0.909878	−0.454939	−	0.890523i	$-0.650339\pi$
$947$	−28.0000	−0.909878	−0.454939	+	0.890523i	$0.650339\pi$
$948$	0	0
$949$	16.0000	0.519382
$950$	−8.00000	−0.259554
$951$	0	0
$952$	4.00000	0.129641
$953$	−22.0000	−0.712650	−0.356325	−	0.934362i	$-0.615970\pi$
$953$	−22.0000	−0.712650	−0.356325	+	0.934362i	$0.615970\pi$
$954$	0	0
$955$	18.0000	0.582466
$956$	4.00000	0.129369
$957$	0	0
$958$	10.0000	0.323085
$959$	12.0000	0.387500
$960$	0	0
$961$	1.00000	0.0322581
$962$	48.0000	1.54758
$963$	0	0
$964$	10.0000	0.322078
$965$	2.00000	0.0643823
$966$	0	0
$967$	0	0		−	1.00000i	$-0.5\pi$
$967$	0	0			1.00000i	$0.5\pi$
$968$	5.00000	0.160706
$969$	0	0
$970$	18.0000	0.577945
$971$	2.00000	0.0641831	0.0320915	−	0.999485i	$-0.489783\pi$
$971$	2.00000	0.0641831	0.0320915	+	0.999485i	$0.489783\pi$
$972$	0	0
$973$	−8.00000	−0.256468
$974$	−12.0000	−0.384505
$975$	0	0
$976$	10.0000	0.320092
$977$	46.0000	1.47167	0.735835	−	0.677161i	$-0.236790\pi$
$977$	46.0000	1.47167	0.735835	+	0.677161i	$0.236790\pi$
$978$	0	0
$979$	0	0
$980$	3.00000	0.0958315
$981$	0	0
$982$	8.00000	0.255290
$983$	0	0		−	1.00000i	$-0.5\pi$
$983$	0	0			1.00000i	$0.5\pi$
$984$	0	0
$985$	10.0000	0.318626
$986$	8.00000	0.254772
$987$	0	0
$988$	32.0000	1.01806
$989$	64.0000	2.03508
$990$	0	0
$991$	−56.0000	−1.77890	−0.889449	−	0.457034i	$-0.848912\pi$
$991$	−56.0000	−1.77890	−0.889449	+	0.457034i	$0.848912\pi$
$992$	−1.00000	−0.0317500
$993$	0	0
$994$	12.0000	0.380617
$995$	−16.0000	−0.507234
$996$	0	0
$997$	18.0000	0.570066	0.285033	−	0.958518i	$-0.407995\pi$
$997$	18.0000	0.570066	0.285033	+	0.958518i	$0.407995\pi$
$998$	−36.0000	−1.13956
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2790.2.a.p.1.1		1
3.2	odd	2		930.2.a.h.1.1	✓	1
12.11	even	2		7440.2.a.m.1.1		1
15.2	even	4		4650.2.d.b.3349.1		2
15.8	even	4		4650.2.d.b.3349.2		2
15.14	odd	2		4650.2.a.bg.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
930.2.a.h.1.1	✓	1	3.2	odd	2
2790.2.a.p.1.1		1	1.1	even	1	trivial
4650.2.a.bg.1.1		1	15.14	odd	2
4650.2.d.b.3349.1		2	15.2	even	4
4650.2.d.b.3349.2		2	15.8	even	4
7440.2.a.m.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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 2790 = 2 \cdot 3^{2} \cdot 5 \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2790.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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