LMFDB - Embedded newform 7440.2.a.m.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [7440,2,Mod(1,7440)] 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("7440.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(7440, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 7440 = 2^{4} \cdot 3 \cdot 5 \cdot 31 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7440.a (trivial)

Newform invariants

comment:select newform

sage:traces = [1,0,-1,0,1,0,2,0,1,0,4,0,-4,0,-1,0,2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(17)] == traces)

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

Self dual:	yes
Analytic conductor:	$59.4086991038$
Analytic rank:	$0$
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	$\chi$	$=$	7440.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$

$=$

$q-1.00000 q^{3} +1.00000 q^{5} +2.00000 q^{7} +1.00000 q^{9} +4.00000 q^{11} -4.00000 q^{13} -1.00000 q^{15} +2.00000 q^{17} +8.00000 q^{19} -2.00000 q^{21} +8.00000 q^{23} +1.00000 q^{25} -1.00000 q^{27} +4.00000 q^{29} +1.00000 q^{31} -4.00000 q^{33} +2.00000 q^{35} -12.0000 q^{37} +4.00000 q^{39} +10.0000 q^{41} -8.00000 q^{43} +1.00000 q^{45} +4.00000 q^{47} -3.00000 q^{49} -2.00000 q^{51} +6.00000 q^{53} +4.00000 q^{55} -8.00000 q^{57} -2.00000 q^{59} +10.0000 q^{61} +2.00000 q^{63} -4.00000 q^{65} +6.00000 q^{67} -8.00000 q^{69} -6.00000 q^{71} -4.00000 q^{73} -1.00000 q^{75} +8.00000 q^{77} +8.00000 q^{79} +1.00000 q^{81} -4.00000 q^{83} +2.00000 q^{85} -4.00000 q^{87} -8.00000 q^{91} -1.00000 q^{93} +8.00000 q^{95} -18.0000 q^{97} +4.00000 q^{99} +O(q^{100})$

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 $

$2$	0	0
$2$	0	0
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	0	0
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	2.00000	0.755929	0.377964	−	0.925820i	$-0.376624\pi$
$7$	2.00000	0.755929	0.377964	+	0.925820i	$0.376624\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	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	$0.687167\pi$
$14$	0	0
$15$	−1.00000	−0.258199
$16$	0	0
$17$	2.00000	0.485071	0.242536	−	0.970143i	$-0.422021\pi$
$17$	2.00000	0.485071	0.242536	+	0.970143i	$0.422021\pi$
$18$	0	0
$19$	8.00000	1.83533	0.917663	−	0.397360i	$-0.130073\pi$
$19$	8.00000	1.83533	0.917663	+	0.397360i	$0.130073\pi$
$20$	0	0
$21$	−2.00000	−0.436436
$22$	0	0
$23$	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
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	4.00000	0.742781	0.371391	−	0.928477i	$-0.378881\pi$
$29$	4.00000	0.742781	0.371391	+	0.928477i	$0.378881\pi$
$30$	0	0
$31$	1.00000	0.179605
$31$	1.00000	0.179605
$32$	0	0
$33$	−4.00000	−0.696311
$34$	0	0
$35$	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	$0.947432\pi$
$38$	0	0
$39$	4.00000	0.640513
$40$	0	0
$41$	10.0000	1.56174	0.780869	−	0.624695i	$-0.214777\pi$
$41$	10.0000	1.56174	0.780869	+	0.624695i	$0.214777\pi$
$42$	0	0
$43$	−8.00000	−1.21999	−0.609994	−	0.792406i	$-0.708828\pi$
$43$	−8.00000	−1.21999	−0.609994	+	0.792406i	$0.708828\pi$
$44$	0	0
$45$	1.00000	0.149071
$46$	0	0
$47$	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
$50$	0	0
$51$	−2.00000	−0.280056
$52$	0	0
$53$	6.00000	0.824163	0.412082	−	0.911147i	$-0.364802\pi$
$53$	6.00000	0.824163	0.412082	+	0.911147i	$0.364802\pi$
$54$	0	0
$55$	4.00000	0.539360
$56$	0	0
$57$	−8.00000	−1.05963
$58$	0	0
$59$	−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	$0.278858\pi$
$62$	0	0
$63$	2.00000	0.251976
$64$	0	0
$65$	−4.00000	−0.496139
$66$	0	0
$67$	6.00000	0.733017	0.366508	−	0.930415i	$-0.380553\pi$
$67$	6.00000	0.733017	0.366508	+	0.930415i	$0.380553\pi$
$68$	0	0
$69$	−8.00000	−0.963087
$70$	0	0
$71$	−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	$0.575209\pi$
$74$	0	0
$75$	−1.00000	−0.115470
$76$	0	0
$77$	8.00000	0.911685
$78$	0	0
$79$	8.00000	0.900070	0.450035	−	0.893011i	$-0.351411\pi$
$79$	8.00000	0.900070	0.450035	+	0.893011i	$0.351411\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−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
$86$	0	0
$87$	−4.00000	−0.428845
$88$	0	0
$89$	0	0		−	1.00000i	$-0.5\pi$
$89$	0	0			1.00000i	$0.5\pi$
$90$	0	0
$91$	−8.00000	−0.838628
$92$	0	0
$93$	−1.00000	−0.103695
$94$	0	0
$95$	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	$0.866875\pi$
$98$	0	0
$99$	4.00000	0.402015
$100$	0	0
$101$	−18.0000	−1.79107	−0.895533	−	0.444994i	$-0.853206\pi$
$101$	−18.0000	−1.79107	−0.895533	+	0.444994i	$0.853206\pi$
$102$	0	0
$103$	14.0000	1.37946	0.689730	−	0.724066i	$-0.257729\pi$
$103$	14.0000	1.37946	0.689730	+	0.724066i	$0.257729\pi$
$104$	0	0
$105$	−2.00000	−0.195180
$106$	0	0
$107$	−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	$0.169184\pi$
$110$	0	0
$111$	12.0000	1.13899
$112$	0	0
$113$	−6.00000	−0.564433	−0.282216	−	0.959351i	$-0.591070\pi$
$113$	−6.00000	−0.564433	−0.282216	+	0.959351i	$0.591070\pi$
$114$	0	0
$115$	8.00000	0.746004
$116$	0	0
$117$	−4.00000	−0.369800
$118$	0	0
$119$	4.00000	0.366679
$120$	0	0
$121$	5.00000	0.454545
$122$	0	0
$123$	−10.0000	−0.901670
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	−4.00000	−0.354943	−0.177471	−	0.984126i	$-0.556792\pi$
$127$	−4.00000	−0.354943	−0.177471	+	0.984126i	$0.556792\pi$
$128$	0	0
$129$	8.00000	0.704361
$130$	0	0
$131$	−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
$134$	0	0
$135$	−1.00000	−0.0860663
$136$	0	0
$137$	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	$0.554260\pi$
$140$	0	0
$141$	−4.00000	−0.336861
$142$	0	0
$143$	−16.0000	−1.33799
$144$	0	0
$145$	4.00000	0.332182
$146$	0	0
$147$	3.00000	0.247436
$148$	0	0
$149$	−18.0000	−1.47462	−0.737309	−	0.675556i	$-0.763904\pi$
$149$	−18.0000	−1.47462	−0.737309	+	0.675556i	$0.763904\pi$
$150$	0	0
$151$	16.0000	1.30206	0.651031	−	0.759051i	$-0.274337\pi$
$151$	16.0000	1.30206	0.651031	+	0.759051i	$0.274337\pi$
$152$	0	0
$153$	2.00000	0.161690
$154$	0	0
$155$	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	$0.841053\pi$
$158$	0	0
$159$	−6.00000	−0.475831
$160$	0	0
$161$	16.0000	1.26098
$162$	0	0
$163$	6.00000	0.469956	0.234978	−	0.972001i	$-0.424498\pi$
$163$	6.00000	0.469956	0.234978	+	0.972001i	$0.424498\pi$
$164$	0	0
$165$	−4.00000	−0.311400
$166$	0	0
$167$	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
$170$	0	0
$171$	8.00000	0.611775
$172$	0	0
$173$	22.0000	1.67263	0.836315	−	0.548250i	$-0.184706\pi$
$173$	22.0000	1.67263	0.836315	+	0.548250i	$0.184706\pi$
$174$	0	0
$175$	2.00000	0.151186
$176$	0	0
$177$	2.00000	0.150329
$178$	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	$0.266738\pi$
$182$	0	0
$183$	−10.0000	−0.739221
$184$	0	0
$185$	−12.0000	−0.882258
$186$	0	0
$187$	8.00000	0.585018
$188$	0	0
$189$	−2.00000	−0.145479
$190$	0	0
$191$	−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	$0.522932\pi$
$194$	0	0
$195$	4.00000	0.286446
$196$	0	0
$197$	10.0000	0.712470	0.356235	−	0.934396i	$-0.384060\pi$
$197$	10.0000	0.712470	0.356235	+	0.934396i	$0.384060\pi$
$198$	0	0
$199$	−16.0000	−1.13421	−0.567105	−	0.823646i	$-0.691937\pi$
$199$	−16.0000	−1.13421	−0.567105	+	0.823646i	$0.691937\pi$
$200$	0	0
$201$	−6.00000	−0.423207
$202$	0	0
$203$	8.00000	0.561490
$204$	0	0
$205$	10.0000	0.698430
$206$	0	0
$207$	8.00000	0.556038
$208$	0	0
$209$	32.0000	2.21349
$210$	0	0
$211$	−24.0000	−1.65223	−0.826114	−	0.563503i	$-0.809453\pi$
$211$	−24.0000	−1.65223	−0.826114	+	0.563503i	$0.809453\pi$
$212$	0	0
$213$	6.00000	0.411113
$214$	0	0
$215$	−8.00000	−0.545595
$216$	0	0
$217$	2.00000	0.135769
$218$	0	0
$219$	4.00000	0.270295
$220$	0	0
$221$	−8.00000	−0.538138
$222$	0	0
$223$	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
$226$	0	0
$227$	−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	$0.240960\pi$
$230$	0	0
$231$	−8.00000	−0.526361
$232$	0	0
$233$	−10.0000	−0.655122	−0.327561	−	0.944830i	$-0.606227\pi$
$233$	−10.0000	−0.655122	−0.327561	+	0.944830i	$0.606227\pi$
$234$	0	0
$235$	4.00000	0.260931
$236$	0	0
$237$	−8.00000	−0.519656
$238$	0	0
$239$	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	$0.395619\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	−3.00000	−0.191663
$246$	0	0
$247$	−32.0000	−2.03611
$248$	0	0
$249$	4.00000	0.253490
$250$	0	0
$251$	−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
$254$	0	0
$255$	−2.00000	−0.125245
$256$	0	0
$257$	−18.0000	−1.12281	−0.561405	−	0.827541i	$-0.689739\pi$
$257$	−18.0000	−1.12281	−0.561405	+	0.827541i	$0.689739\pi$
$258$	0	0
$259$	−24.0000	−1.49129
$260$	0	0
$261$	4.00000	0.247594
$262$	0	0
$263$	0	0		−	1.00000i	$-0.5\pi$
$263$	0	0			1.00000i	$0.5\pi$
$264$	0	0
$265$	6.00000	0.368577
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−12.0000	−0.731653	−0.365826	−	0.930683i	$-0.619214\pi$
$269$	−12.0000	−0.731653	−0.365826	+	0.930683i	$0.619214\pi$
$270$	0	0
$271$	−8.00000	−0.485965	−0.242983	−	0.970031i	$-0.578126\pi$
$271$	−8.00000	−0.485965	−0.242983	+	0.970031i	$0.578126\pi$
$272$	0	0
$273$	8.00000	0.484182
$274$	0	0
$275$	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	$0.422742\pi$
$278$	0	0
$279$	1.00000	0.0598684
$280$	0	0
$281$	6.00000	0.357930	0.178965	−	0.983855i	$-0.442725\pi$
$281$	6.00000	0.357930	0.178965	+	0.983855i	$0.442725\pi$
$282$	0	0
$283$	−2.00000	−0.118888	−0.0594438	−	0.998232i	$-0.518933\pi$
$283$	−2.00000	−0.118888	−0.0594438	+	0.998232i	$0.518933\pi$
$284$	0	0
$285$	−8.00000	−0.473879
$286$	0	0
$287$	20.0000	1.18056
$288$	0	0
$289$	−13.0000	−0.764706
$290$	0	0
$291$	18.0000	1.05518
$292$	0	0
$293$	−6.00000	−0.350524	−0.175262	−	0.984522i	$-0.556077\pi$
$293$	−6.00000	−0.350524	−0.175262	+	0.984522i	$0.556077\pi$
$294$	0	0
$295$	−2.00000	−0.116445
$296$	0	0
$297$	−4.00000	−0.232104
$298$	0	0
$299$	−32.0000	−1.85061
$300$	0	0
$301$	−16.0000	−0.922225
$302$	0	0
$303$	18.0000	1.03407
$304$	0	0
$305$	10.0000	0.572598
$306$	0	0
$307$	2.00000	0.114146	0.0570730	−	0.998370i	$-0.481823\pi$
$307$	2.00000	0.114146	0.0570730	+	0.998370i	$0.481823\pi$
$308$	0	0
$309$	−14.0000	−0.796432
$310$	0	0
$311$	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	$0.308786\pi$
$314$	0	0
$315$	2.00000	0.112687
$316$	0	0
$317$	−22.0000	−1.23564	−0.617822	−	0.786318i	$-0.711985\pi$
$317$	−22.0000	−1.23564	−0.617822	+	0.786318i	$0.711985\pi$
$318$	0	0
$319$	16.0000	0.895828
$320$	0	0
$321$	8.00000	0.446516
$322$	0	0
$323$	16.0000	0.890264
$324$	0	0
$325$	−4.00000	−0.221880
$326$	0	0
$327$	−18.0000	−0.995402
$328$	0	0
$329$	8.00000	0.441054
$330$	0	0
$331$	12.0000	0.659580	0.329790	−	0.944054i	$-0.393022\pi$
$331$	12.0000	0.659580	0.329790	+	0.944054i	$0.393022\pi$
$332$	0	0
$333$	−12.0000	−0.657596
$334$	0	0
$335$	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	$0.836901\pi$
$338$	0	0
$339$	6.00000	0.325875
$340$	0	0
$341$	4.00000	0.216612
$342$	0	0
$343$	−20.0000	−1.07990
$344$	0	0
$345$	−8.00000	−0.430706
$346$	0	0
$347$	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	$0.796728\pi$
$350$	0	0
$351$	4.00000	0.213504
$352$	0	0
$353$	34.0000	1.80964	0.904819	−	0.425797i	$-0.140006\pi$
$353$	34.0000	1.80964	0.904819	+	0.425797i	$0.140006\pi$
$354$	0	0
$355$	−6.00000	−0.318447
$356$	0	0
$357$	−4.00000	−0.211702
$358$	0	0
$359$	−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
$362$	0	0
$363$	−5.00000	−0.262432
$364$	0	0
$365$	−4.00000	−0.209370
$366$	0	0
$367$	8.00000	0.417597	0.208798	−	0.977959i	$-0.433045\pi$
$367$	8.00000	0.417597	0.208798	+	0.977959i	$0.433045\pi$
$368$	0	0
$369$	10.0000	0.520579
$370$	0	0
$371$	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	$0.157390\pi$
$374$	0	0
$375$	−1.00000	−0.0516398
$376$	0	0
$377$	−16.0000	−0.824042
$378$	0	0
$379$	−16.0000	−0.821865	−0.410932	−	0.911666i	$-0.634797\pi$
$379$	−16.0000	−0.821865	−0.410932	+	0.911666i	$0.634797\pi$
$380$	0	0
$381$	4.00000	0.204926
$382$	0	0
$383$	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
$386$	0	0
$387$	−8.00000	−0.406663
$388$	0	0
$389$	−28.0000	−1.41966	−0.709828	−	0.704375i	$-0.751227\pi$
$389$	−28.0000	−1.41966	−0.709828	+	0.704375i	$0.751227\pi$
$390$	0	0
$391$	16.0000	0.809155
$392$	0	0
$393$	10.0000	0.504433
$394$	0	0
$395$	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	$0.825345\pi$
$398$	0	0
$399$	−16.0000	−0.801002
$400$	0	0
$401$	4.00000	0.199750	0.0998752	−	0.995000i	$-0.468156\pi$
$401$	4.00000	0.199750	0.0998752	+	0.995000i	$0.468156\pi$
$402$	0	0
$403$	−4.00000	−0.199254
$404$	0	0
$405$	1.00000	0.0496904
$406$	0	0
$407$	−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	$0.452607\pi$
$410$	0	0
$411$	−6.00000	−0.295958
$412$	0	0
$413$	−4.00000	−0.196827
$414$	0	0
$415$	−4.00000	−0.196352
$416$	0	0
$417$	4.00000	0.195881
$418$	0	0
$419$	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	$0.484480\pi$
$422$	0	0
$423$	4.00000	0.194487
$424$	0	0
$425$	2.00000	0.0970143
$426$	0	0
$427$	20.0000	0.967868
$428$	0	0
$429$	16.0000	0.772487
$430$	0	0
$431$	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	$0.469359\pi$
$434$	0	0
$435$	−4.00000	−0.191785
$436$	0	0
$437$	64.0000	3.06154
$438$	0	0
$439$	40.0000	1.90910	0.954548	−	0.298057i	$-0.0963387\pi$
$439$	40.0000	1.90910	0.954548	+	0.298057i	$0.0963387\pi$
$440$	0	0
$441$	−3.00000	−0.142857
$442$	0	0
$443$	−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
$446$	0	0
$447$	18.0000	0.851371
$448$	0	0
$449$	28.0000	1.32140	0.660701	−	0.750649i	$-0.270259\pi$
$449$	28.0000	1.32140	0.660701	+	0.750649i	$0.270259\pi$
$450$	0	0
$451$	40.0000	1.88353
$452$	0	0
$453$	−16.0000	−0.751746
$454$	0	0
$455$	−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	$0.181376\pi$
$458$	0	0
$459$	−2.00000	−0.0933520
$460$	0	0
$461$	−40.0000	−1.86299	−0.931493	−	0.363760i	$-0.881493\pi$
$461$	−40.0000	−1.86299	−0.931493	+	0.363760i	$0.881493\pi$
$462$	0	0
$463$	36.0000	1.67306	0.836531	−	0.547920i	$-0.184580\pi$
$463$	36.0000	1.67306	0.836531	+	0.547920i	$0.184580\pi$
$464$	0	0
$465$	−1.00000	−0.0463739
$466$	0	0
$467$	0	0		−	1.00000i	$-0.5\pi$
$467$	0	0			1.00000i	$0.5\pi$
$468$	0	0
$469$	12.0000	0.554109
$470$	0	0
$471$	22.0000	1.01371
$472$	0	0
$473$	−32.0000	−1.47136
$474$	0	0
$475$	8.00000	0.367065
$476$	0	0
$477$	6.00000	0.274721
$478$	0	0
$479$	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
$482$	0	0
$483$	−16.0000	−0.728025
$484$	0	0
$485$	−18.0000	−0.817338
$486$	0	0
$487$	12.0000	0.543772	0.271886	−	0.962329i	$-0.412353\pi$
$487$	12.0000	0.543772	0.271886	+	0.962329i	$0.412353\pi$
$488$	0	0
$489$	−6.00000	−0.271329
$490$	0	0
$491$	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
$494$	0	0
$495$	4.00000	0.179787
$496$	0	0
$497$	−12.0000	−0.538274
$498$	0	0
$499$	36.0000	1.61158	0.805791	−	0.592200i	$-0.201741\pi$
$499$	36.0000	1.61158	0.805791	+	0.592200i	$0.201741\pi$
$500$	0	0
$501$	−8.00000	−0.357414
$502$	0	0
$503$	−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
$506$	0	0
$507$	−3.00000	−0.133235
$508$	0	0
$509$	16.0000	0.709188	0.354594	−	0.935020i	$-0.384619\pi$
$509$	16.0000	0.709188	0.354594	+	0.935020i	$0.384619\pi$
$510$	0	0
$511$	−8.00000	−0.353899
$512$	0	0
$513$	−8.00000	−0.353209
$514$	0	0
$515$	14.0000	0.616914
$516$	0	0
$517$	16.0000	0.703679
$518$	0	0
$519$	−22.0000	−0.965693
$520$	0	0
$521$	18.0000	0.788594	0.394297	−	0.918983i	$-0.370988\pi$
$521$	18.0000	0.788594	0.394297	+	0.918983i	$0.370988\pi$
$522$	0	0
$523$	0	0		−	1.00000i	$-0.5\pi$
$523$	0	0			1.00000i	$0.5\pi$
$524$	0	0
$525$	−2.00000	−0.0872872
$526$	0	0
$527$	2.00000	0.0871214
$528$	0	0
$529$	41.0000	1.78261
$530$	0	0
$531$	−2.00000	−0.0867926
$532$	0	0
$533$	−40.0000	−1.73259
$534$	0	0
$535$	−8.00000	−0.345870
$536$	0	0
$537$	−20.0000	−0.863064
$538$	0	0
$539$	−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	$0.486311\pi$
$542$	0	0
$543$	−18.0000	−0.772454
$544$	0	0
$545$	18.0000	0.771035
$546$	0	0
$547$	−2.00000	−0.0855138	−0.0427569	−	0.999086i	$-0.513614\pi$
$547$	−2.00000	−0.0855138	−0.0427569	+	0.999086i	$0.513614\pi$
$548$	0	0
$549$	10.0000	0.426790
$550$	0	0
$551$	32.0000	1.36325
$552$	0	0
$553$	16.0000	0.680389
$554$	0	0
$555$	12.0000	0.509372
$556$	0	0
$557$	6.00000	0.254228	0.127114	−	0.991888i	$-0.459429\pi$
$557$	6.00000	0.254228	0.127114	+	0.991888i	$0.459429\pi$
$558$	0	0
$559$	32.0000	1.35346
$560$	0	0
$561$	−8.00000	−0.337760
$562$	0	0
$563$	0	0		−	1.00000i	$-0.5\pi$
$563$	0	0			1.00000i	$0.5\pi$
$564$	0	0
$565$	−6.00000	−0.252422
$566$	0	0
$567$	2.00000	0.0839921
$568$	0	0
$569$	−28.0000	−1.17382	−0.586911	−	0.809652i	$-0.699656\pi$
$569$	−28.0000	−1.17382	−0.586911	+	0.809652i	$0.699656\pi$
$570$	0	0
$571$	12.0000	0.502184	0.251092	−	0.967963i	$-0.419210\pi$
$571$	12.0000	0.502184	0.251092	+	0.967963i	$0.419210\pi$
$572$	0	0
$573$	18.0000	0.751961
$574$	0	0
$575$	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	$0.539858\pi$
$578$	0	0
$579$	2.00000	0.0831172
$580$	0	0
$581$	−8.00000	−0.331896
$582$	0	0
$583$	24.0000	0.993978
$584$	0	0
$585$	−4.00000	−0.165380
$586$	0	0
$587$	−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
$590$	0	0
$591$	−10.0000	−0.411345
$592$	0	0
$593$	−30.0000	−1.23195	−0.615976	−	0.787765i	$-0.711238\pi$
$593$	−30.0000	−1.23195	−0.615976	+	0.787765i	$0.711238\pi$
$594$	0	0
$595$	4.00000	0.163984
$596$	0	0
$597$	16.0000	0.654836
$598$	0	0
$599$	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	$0.677914\pi$
$602$	0	0
$603$	6.00000	0.244339
$604$	0	0
$605$	5.00000	0.203279
$606$	0	0
$607$	14.0000	0.568242	0.284121	−	0.958788i	$-0.408298\pi$
$607$	14.0000	0.568242	0.284121	+	0.958788i	$0.408298\pi$
$608$	0	0
$609$	−8.00000	−0.324176
$610$	0	0
$611$	−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	$0.448348\pi$
$614$	0	0
$615$	−10.0000	−0.403239
$616$	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
$619$	20.0000	0.803868	0.401934	−	0.915669i	$-0.368338\pi$
$619$	20.0000	0.803868	0.401934	+	0.915669i	$0.368338\pi$
$620$	0	0
$621$	−8.00000	−0.321029
$622$	0	0
$623$	0	0
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	−32.0000	−1.27796
$628$	0	0
$629$	−24.0000	−0.956943
$630$	0	0
$631$	−8.00000	−0.318475	−0.159237	−	0.987240i	$-0.550904\pi$
$631$	−8.00000	−0.318475	−0.159237	+	0.987240i	$0.550904\pi$
$632$	0	0
$633$	24.0000	0.953914
$634$	0	0
$635$	−4.00000	−0.158735
$636$	0	0
$637$	12.0000	0.475457
$638$	0	0
$639$	−6.00000	−0.237356
$640$	0	0
$641$	−4.00000	−0.157991	−0.0789953	−	0.996875i	$-0.525171\pi$
$641$	−4.00000	−0.157991	−0.0789953	+	0.996875i	$0.525171\pi$
$642$	0	0
$643$	8.00000	0.315489	0.157745	−	0.987480i	$-0.449578\pi$
$643$	8.00000	0.315489	0.157745	+	0.987480i	$0.449578\pi$
$644$	0	0
$645$	8.00000	0.315000
$646$	0	0
$647$	−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
$650$	0	0
$651$	−2.00000	−0.0783862
$652$	0	0
$653$	−6.00000	−0.234798	−0.117399	−	0.993085i	$-0.537456\pi$
$653$	−6.00000	−0.234798	−0.117399	+	0.993085i	$0.537456\pi$
$654$	0	0
$655$	−10.0000	−0.390732
$656$	0	0
$657$	−4.00000	−0.156055
$658$	0	0
$659$	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	$0.147463\pi$
$662$	0	0
$663$	8.00000	0.310694
$664$	0	0
$665$	16.0000	0.620453
$666$	0	0
$667$	32.0000	1.23904
$668$	0	0
$669$	−8.00000	−0.309298
$670$	0	0
$671$	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	$0.450724\pi$
$674$	0	0
$675$	−1.00000	−0.0384900
$676$	0	0
$677$	6.00000	0.230599	0.115299	−	0.993331i	$-0.463217\pi$
$677$	6.00000	0.230599	0.115299	+	0.993331i	$0.463217\pi$
$678$	0	0
$679$	−36.0000	−1.38155
$680$	0	0
$681$	4.00000	0.153280
$682$	0	0
$683$	−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
$686$	0	0
$687$	−22.0000	−0.839352
$688$	0	0
$689$	−24.0000	−0.914327
$690$	0	0
$691$	−28.0000	−1.06517	−0.532585	−	0.846376i	$-0.678779\pi$
$691$	−28.0000	−1.06517	−0.532585	+	0.846376i	$0.678779\pi$
$692$	0	0
$693$	8.00000	0.303895
$694$	0	0
$695$	−4.00000	−0.151729
$696$	0	0
$697$	20.0000	0.757554
$698$	0	0
$699$	10.0000	0.378235
$700$	0	0
$701$	2.00000	0.0755390	0.0377695	−	0.999286i	$-0.487975\pi$
$701$	2.00000	0.0755390	0.0377695	+	0.999286i	$0.487975\pi$
$702$	0	0
$703$	−96.0000	−3.62071
$704$	0	0
$705$	−4.00000	−0.150649
$706$	0	0
$707$	−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	$0.789231\pi$
$710$	0	0
$711$	8.00000	0.300023
$712$	0	0
$713$	8.00000	0.299602
$714$	0	0
$715$	−16.0000	−0.598366
$716$	0	0
$717$	−4.00000	−0.149383
$718$	0	0
$719$	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
$722$	0	0
$723$	−10.0000	−0.371904
$724$	0	0
$725$	4.00000	0.148556
$726$	0	0
$727$	2.00000	0.0741759	0.0370879	−	0.999312i	$-0.488192\pi$
$727$	2.00000	0.0741759	0.0370879	+	0.999312i	$0.488192\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−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	$0.464656\pi$
$734$	0	0
$735$	3.00000	0.110657
$736$	0	0
$737$	24.0000	0.884051
$738$	0	0
$739$	−20.0000	−0.735712	−0.367856	−	0.929883i	$-0.619908\pi$
$739$	−20.0000	−0.735712	−0.367856	+	0.929883i	$0.619908\pi$
$740$	0	0
$741$	32.0000	1.17555
$742$	0	0
$743$	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
$746$	0	0
$747$	−4.00000	−0.146352
$748$	0	0
$749$	−16.0000	−0.584627
$750$	0	0
$751$	−16.0000	−0.583848	−0.291924	−	0.956441i	$-0.594295\pi$
$751$	−16.0000	−0.583848	−0.291924	+	0.956441i	$0.594295\pi$
$752$	0	0
$753$	20.0000	0.728841
$754$	0	0
$755$	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	$0.272997\pi$
$758$	0	0
$759$	−32.0000	−1.16153
$760$	0	0
$761$	−20.0000	−0.724999	−0.362500	−	0.931984i	$-0.618077\pi$
$761$	−20.0000	−0.724999	−0.362500	+	0.931984i	$0.618077\pi$
$762$	0	0
$763$	36.0000	1.30329
$764$	0	0
$765$	2.00000	0.0723102
$766$	0	0
$767$	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	$0.710052\pi$
$770$	0	0
$771$	18.0000	0.648254
$772$	0	0
$773$	14.0000	0.503545	0.251773	−	0.967786i	$-0.418987\pi$
$773$	14.0000	0.503545	0.251773	+	0.967786i	$0.418987\pi$
$774$	0	0
$775$	1.00000	0.0359211
$776$	0	0
$777$	24.0000	0.860995
$778$	0	0
$779$	80.0000	2.86630
$780$	0	0
$781$	−24.0000	−0.858788
$782$	0	0
$783$	−4.00000	−0.142948
$784$	0	0
$785$	−22.0000	−0.785214
$786$	0	0
$787$	−28.0000	−0.998092	−0.499046	−	0.866575i	$-0.666316\pi$
$787$	−28.0000	−0.998092	−0.499046	+	0.866575i	$0.666316\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−12.0000	−0.426671
$792$	0	0
$793$	−40.0000	−1.42044
$794$	0	0
$795$	−6.00000	−0.212798
$796$	0	0
$797$	−30.0000	−1.06265	−0.531327	−	0.847167i	$-0.678307\pi$
$797$	−30.0000	−1.06265	−0.531327	+	0.847167i	$0.678307\pi$
$798$	0	0
$799$	8.00000	0.283020
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−16.0000	−0.564628
$804$	0	0
$805$	16.0000	0.563926
$806$	0	0
$807$	12.0000	0.422420
$808$	0	0
$809$	28.0000	0.984428	0.492214	−	0.870474i	$-0.336188\pi$
$809$	28.0000	0.984428	0.492214	+	0.870474i	$0.336188\pi$
$810$	0	0
$811$	16.0000	0.561836	0.280918	−	0.959732i	$-0.409361\pi$
$811$	16.0000	0.561836	0.280918	+	0.959732i	$0.409361\pi$
$812$	0	0
$813$	8.00000	0.280572
$814$	0	0
$815$	6.00000	0.210171
$816$	0	0
$817$	−64.0000	−2.23908
$818$	0	0
$819$	−8.00000	−0.279543
$820$	0	0
$821$	−4.00000	−0.139601	−0.0698005	−	0.997561i	$-0.522236\pi$
$821$	−4.00000	−0.139601	−0.0698005	+	0.997561i	$0.522236\pi$
$822$	0	0
$823$	−28.0000	−0.976019	−0.488009	−	0.872838i	$-0.662277\pi$
$823$	−28.0000	−0.976019	−0.488009	+	0.872838i	$0.662277\pi$
$824$	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	$0.715277\pi$
$828$	0	0
$829$	−46.0000	−1.59765	−0.798823	−	0.601566i	$-0.794544\pi$
$829$	−46.0000	−1.59765	−0.798823	+	0.601566i	$0.794544\pi$
$830$	0	0
$831$	−8.00000	−0.277517
$832$	0	0
$833$	−6.00000	−0.207888
$834$	0	0
$835$	8.00000	0.276851
$836$	0	0
$837$	−1.00000	−0.0345651
$838$	0	0
$839$	−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
$842$	0	0
$843$	−6.00000	−0.206651
$844$	0	0
$845$	3.00000	0.103203
$846$	0	0
$847$	10.0000	0.343604
$848$	0	0
$849$	2.00000	0.0686398
$850$	0	0
$851$	−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	$0.422959\pi$
$854$	0	0
$855$	8.00000	0.273594
$856$	0	0
$857$	10.0000	0.341593	0.170797	−	0.985306i	$-0.445366\pi$
$857$	10.0000	0.341593	0.170797	+	0.985306i	$0.445366\pi$
$858$	0	0
$859$	20.0000	0.682391	0.341196	−	0.939992i	$-0.389168\pi$
$859$	20.0000	0.682391	0.341196	+	0.939992i	$0.389168\pi$
$860$	0	0
$861$	−20.0000	−0.681598
$862$	0	0
$863$	−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
$866$	0	0
$867$	13.0000	0.441503
$868$	0	0
$869$	32.0000	1.08553
$870$	0	0
$871$	−24.0000	−0.813209
$872$	0	0
$873$	−18.0000	−0.609208
$874$	0	0
$875$	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	$0.510751\pi$
$878$	0	0
$879$	6.00000	0.202375
$880$	0	0
$881$	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	$-0.609252\pi$
$883$	−20.0000	−0.673054	−0.336527	+	0.941674i	$0.609252\pi$
$884$	0	0
$885$	2.00000	0.0672293
$886$	0	0
$887$	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
$891$	4.00000	0.134005
$892$	0	0
$893$	32.0000	1.07084
$894$	0	0
$895$	20.0000	0.668526
$896$	0	0
$897$	32.0000	1.06845
$898$	0	0
$899$	4.00000	0.133407
$900$	0	0
$901$	12.0000	0.399778
$902$	0	0
$903$	16.0000	0.532447
$904$	0	0
$905$	18.0000	0.598340
$906$	0	0
$907$	10.0000	0.332045	0.166022	−	0.986122i	$-0.446908\pi$
$907$	10.0000	0.332045	0.166022	+	0.986122i	$0.446908\pi$
$908$	0	0
$909$	−18.0000	−0.597022
$910$	0	0
$911$	−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
$914$	0	0
$915$	−10.0000	−0.330590
$916$	0	0
$917$	−20.0000	−0.660458
$918$	0	0
$919$	−4.00000	−0.131948	−0.0659739	−	0.997821i	$-0.521015\pi$
$919$	−4.00000	−0.131948	−0.0659739	+	0.997821i	$0.521015\pi$
$920$	0	0
$921$	−2.00000	−0.0659022
$922$	0	0
$923$	24.0000	0.789970
$924$	0	0
$925$	−12.0000	−0.394558
$926$	0	0
$927$	14.0000	0.459820
$928$	0	0
$929$	32.0000	1.04989	0.524943	−	0.851137i	$-0.324087\pi$
$929$	32.0000	1.04989	0.524943	+	0.851137i	$0.324087\pi$
$930$	0	0
$931$	−24.0000	−0.786568
$932$	0	0
$933$	−30.0000	−0.982156
$934$	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	$0.510401\pi$
$938$	0	0
$939$	−20.0000	−0.652675
$940$	0	0
$941$	12.0000	0.391189	0.195594	−	0.980685i	$-0.437336\pi$
$941$	12.0000	0.391189	0.195594	+	0.980685i	$0.437336\pi$
$942$	0	0
$943$	80.0000	2.60516
$944$	0	0
$945$	−2.00000	−0.0650600
$946$	0	0
$947$	−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
$950$	0	0
$951$	22.0000	0.713399
$952$	0	0
$953$	22.0000	0.712650	0.356325	−	0.934362i	$-0.384030\pi$
$953$	22.0000	0.712650	0.356325	+	0.934362i	$0.384030\pi$
$954$	0	0
$955$	−18.0000	−0.582466
$956$	0	0
$957$	−16.0000	−0.517207
$958$	0	0
$959$	12.0000	0.387500
$960$	0	0
$961$	1.00000	0.0322581
$962$	0	0
$963$	−8.00000	−0.257796
$964$	0	0
$965$	−2.00000	−0.0643823
$966$	0	0
$967$	0	0		−	1.00000i	$-0.5\pi$
$967$	0	0			1.00000i	$0.5\pi$
$968$	0	0
$969$	−16.0000	−0.513994
$970$	0	0
$971$	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
$974$	0	0
$975$	4.00000	0.128103
$976$	0	0
$977$	−46.0000	−1.47167	−0.735835	−	0.677161i	$-0.763210\pi$
$977$	−46.0000	−1.47167	−0.735835	+	0.677161i	$0.763210\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	18.0000	0.574696
$982$	0	0
$983$	0	0		−	1.00000i	$-0.5\pi$
$983$	0	0			1.00000i	$0.5\pi$
$984$	0	0
$985$	10.0000	0.318626
$986$	0	0
$987$	−8.00000	−0.254643
$988$	0	0
$989$	−64.0000	−2.03508
$990$	0	0
$991$	56.0000	1.77890	0.889449	−	0.457034i	$-0.151088\pi$
$991$	56.0000	1.77890	0.889449	+	0.457034i	$0.151088\pi$
$992$	0	0
$993$	−12.0000	−0.380808
$994$	0	0
$995$	−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	$0.407995\pi$
$998$	0	0
$999$	12.0000	0.379663

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7440.2.a.m.1.1		1
4.3	odd	2		930.2.a.h.1.1	✓	1
12.11	even	2		2790.2.a.p.1.1		1
20.3	even	4		4650.2.d.b.3349.2		2
20.7	even	4		4650.2.d.b.3349.1		2
20.19	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	4.3	odd	2
2790.2.a.p.1.1		1	12.11	even	2
4650.2.a.bg.1.1		1	20.19	odd	2
4650.2.d.b.3349.1		2	20.7	even	4
4650.2.d.b.3349.2		2	20.3	even	4
7440.2.a.m.1.1		1	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 7440 = 2^{4} \cdot 3 \cdot 5 \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7440.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

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