LMFDB - Embedded newform 8450.2.a.q.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [8450,2,Mod(1,8450)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(8450, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("8450.1"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 8450 = 2 \cdot 5^{2} \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8450.a (trivial)

Newform invariants

comment:select newform

sage:traces = [1,1,0,1,0,0,-3,1,-3,0,3,0,0,-3,0,1,4] 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:	$67.4735897080$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 130)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	8450.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$

$=$

$q+1.00000 q^{2} +1.00000 q^{4} -3.00000 q^{7} +1.00000 q^{8} -3.00000 q^{9} +3.00000 q^{11} -3.00000 q^{14} +1.00000 q^{16} +4.00000 q^{17} -3.00000 q^{18} -7.00000 q^{19} +3.00000 q^{22} +4.00000 q^{23} -3.00000 q^{28} -8.00000 q^{29} +10.0000 q^{31} +1.00000 q^{32} +4.00000 q^{34} -3.00000 q^{36} +3.00000 q^{37} -7.00000 q^{38} +2.00000 q^{41} -6.00000 q^{43} +3.00000 q^{44} +4.00000 q^{46} -1.00000 q^{47} +2.00000 q^{49} +9.00000 q^{53} -3.00000 q^{56} -8.00000 q^{58} +4.00000 q^{59} -14.0000 q^{61} +10.0000 q^{62} +9.00000 q^{63} +1.00000 q^{64} +4.00000 q^{67} +4.00000 q^{68} -6.00000 q^{71} -3.00000 q^{72} +4.00000 q^{73} +3.00000 q^{74} -7.00000 q^{76} -9.00000 q^{77} +10.0000 q^{79} +9.00000 q^{81} +2.00000 q^{82} -6.00000 q^{86} +3.00000 q^{88} -1.00000 q^{89} +4.00000 q^{92} -1.00000 q^{94} +16.0000 q^{97} +2.00000 q^{98} -9.00000 q^{99} +O(q^{100})$

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 $

$2$	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
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−3.00000	−1.13389	−0.566947	−	0.823754i	$-0.691875\pi$
$7$	−3.00000	−1.13389	−0.566947	+	0.823754i	$0.691875\pi$
$8$	1.00000	0.353553
$9$	−3.00000	−1.00000
$10$	0	0
$11$	3.00000	0.904534	0.452267	−	0.891883i	$-0.350615\pi$
$11$	3.00000	0.904534	0.452267	+	0.891883i	$0.350615\pi$
$12$	0	0
$13$	0	0
$13$	0	0
$14$	−3.00000	−0.801784
$15$	0	0
$16$	1.00000	0.250000
$17$	4.00000	0.970143	0.485071	−	0.874475i	$-0.338794\pi$
$17$	4.00000	0.970143	0.485071	+	0.874475i	$0.338794\pi$
$18$	−3.00000	−0.707107
$19$	−7.00000	−1.60591	−0.802955	−	0.596040i	$-0.796740\pi$
$19$	−7.00000	−1.60591	−0.802955	+	0.596040i	$0.796740\pi$
$20$	0	0
$21$	0	0
$22$	3.00000	0.639602
$23$	4.00000	0.834058	0.417029	−	0.908893i	$-0.363071\pi$
$23$	4.00000	0.834058	0.417029	+	0.908893i	$0.363071\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	0	0
$28$	−3.00000	−0.566947
$29$	−8.00000	−1.48556	−0.742781	−	0.669534i	$-0.766494\pi$
$29$	−8.00000	−1.48556	−0.742781	+	0.669534i	$0.766494\pi$
$30$	0	0
$31$	10.0000	1.79605	0.898027	−	0.439941i	$-0.145001\pi$
$31$	10.0000	1.79605	0.898027	+	0.439941i	$0.145001\pi$
$32$	1.00000	0.176777
$33$	0	0
$34$	4.00000	0.685994
$35$	0	0
$36$	−3.00000	−0.500000
$37$	3.00000	0.493197	0.246598	−	0.969118i	$-0.420687\pi$
$37$	3.00000	0.493197	0.246598	+	0.969118i	$0.420687\pi$
$38$	−7.00000	−1.13555
$39$	0	0
$40$	0	0
$41$	2.00000	0.312348	0.156174	−	0.987730i	$-0.450084\pi$
$41$	2.00000	0.312348	0.156174	+	0.987730i	$0.450084\pi$
$42$	0	0
$43$	−6.00000	−0.914991	−0.457496	−	0.889212i	$-0.651253\pi$
$43$	−6.00000	−0.914991	−0.457496	+	0.889212i	$0.651253\pi$
$44$	3.00000	0.452267
$45$	0	0
$46$	4.00000	0.589768
$47$	−1.00000	−0.145865	−0.0729325	−	0.997337i	$-0.523236\pi$
$47$	−1.00000	−0.145865	−0.0729325	+	0.997337i	$0.523236\pi$
$48$	0	0
$49$	2.00000	0.285714
$50$	0	0
$51$	0	0
$52$	0	0
$53$	9.00000	1.23625	0.618123	−	0.786082i	$-0.287894\pi$
$53$	9.00000	1.23625	0.618123	+	0.786082i	$0.287894\pi$
$54$	0	0
$55$	0	0
$56$	−3.00000	−0.400892
$57$	0	0
$58$	−8.00000	−1.05045
$59$	4.00000	0.520756	0.260378	−	0.965507i	$-0.416153\pi$
$59$	4.00000	0.520756	0.260378	+	0.965507i	$0.416153\pi$
$60$	0	0
$61$	−14.0000	−1.79252	−0.896258	−	0.443533i	$-0.853725\pi$
$61$	−14.0000	−1.79252	−0.896258	+	0.443533i	$0.853725\pi$
$62$	10.0000	1.27000
$63$	9.00000	1.13389
$64$	1.00000	0.125000
$65$	0	0
$66$	0	0
$67$	4.00000	0.488678	0.244339	−	0.969690i	$-0.421429\pi$
$67$	4.00000	0.488678	0.244339	+	0.969690i	$0.421429\pi$
$68$	4.00000	0.485071
$69$	0	0
$70$	0	0
$71$	−6.00000	−0.712069	−0.356034	−	0.934473i	$-0.615871\pi$
$71$	−6.00000	−0.712069	−0.356034	+	0.934473i	$0.615871\pi$
$72$	−3.00000	−0.353553
$73$	4.00000	0.468165	0.234082	−	0.972217i	$-0.424791\pi$
$73$	4.00000	0.468165	0.234082	+	0.972217i	$0.424791\pi$
$74$	3.00000	0.348743
$75$	0	0
$76$	−7.00000	−0.802955
$77$	−9.00000	−1.02565
$78$	0	0
$79$	10.0000	1.12509	0.562544	−	0.826767i	$-0.309823\pi$
$79$	10.0000	1.12509	0.562544	+	0.826767i	$0.309823\pi$
$80$	0	0
$81$	9.00000	1.00000
$82$	2.00000	0.220863
$83$	0	0		−	1.00000i	$-0.5\pi$
$83$	0	0			1.00000i	$0.5\pi$
$84$	0	0
$85$	0	0
$86$	−6.00000	−0.646997
$87$	0	0
$88$	3.00000	0.319801
$89$	−1.00000	−0.106000	−0.0529999	−	0.998595i	$-0.516878\pi$
$89$	−1.00000	−0.106000	−0.0529999	+	0.998595i	$0.516878\pi$
$90$	0	0
$91$	0	0
$92$	4.00000	0.417029
$93$	0	0
$94$	−1.00000	−0.103142
$95$	0	0
$96$	0	0
$97$	16.0000	1.62455	0.812277	−	0.583272i	$-0.198228\pi$
$97$	16.0000	1.62455	0.812277	+	0.583272i	$0.198228\pi$
$98$	2.00000	0.202031
$99$	−9.00000	−0.904534
$100$	0	0
$101$	0	0		−	1.00000i	$-0.5\pi$
$101$	0	0			1.00000i	$0.5\pi$
$102$	0	0
$103$	1.00000	0.0985329	0.0492665	−	0.998786i	$-0.484312\pi$
$103$	1.00000	0.0985329	0.0492665	+	0.998786i	$0.484312\pi$
$104$	0	0
$105$	0	0
$106$	9.00000	0.874157
$107$	18.0000	1.74013	0.870063	−	0.492941i	$-0.164078\pi$
$107$	18.0000	1.74013	0.870063	+	0.492941i	$0.164078\pi$
$108$	0	0
$109$	2.00000	0.191565	0.0957826	−	0.995402i	$-0.469465\pi$
$109$	2.00000	0.191565	0.0957826	+	0.995402i	$0.469465\pi$
$110$	0	0
$111$	0	0
$112$	−3.00000	−0.283473
$113$	8.00000	0.752577	0.376288	−	0.926503i	$-0.377200\pi$
$113$	8.00000	0.752577	0.376288	+	0.926503i	$0.377200\pi$
$114$	0	0
$115$	0	0
$116$	−8.00000	−0.742781
$117$	0	0
$118$	4.00000	0.368230
$119$	−12.0000	−1.10004
$120$	0	0
$121$	−2.00000	−0.181818
$122$	−14.0000	−1.26750
$123$	0	0
$124$	10.0000	0.898027
$125$	0	0
$126$	9.00000	0.801784
$127$	−13.0000	−1.15356	−0.576782	−	0.816898i	$-0.695692\pi$
$127$	−13.0000	−1.15356	−0.576782	+	0.816898i	$0.695692\pi$
$128$	1.00000	0.0883883
$129$	0	0
$130$	0	0
$131$	13.0000	1.13582	0.567908	−	0.823092i	$-0.307753\pi$
$131$	13.0000	1.13582	0.567908	+	0.823092i	$0.307753\pi$
$132$	0	0
$133$	21.0000	1.82093
$134$	4.00000	0.345547
$135$	0	0
$136$	4.00000	0.342997
$137$	12.0000	1.02523	0.512615	−	0.858619i	$-0.328677\pi$
$137$	12.0000	1.02523	0.512615	+	0.858619i	$0.328677\pi$
$138$	0	0
$139$	−5.00000	−0.424094	−0.212047	−	0.977259i	$-0.568013\pi$
$139$	−5.00000	−0.424094	−0.212047	+	0.977259i	$0.568013\pi$
$140$	0	0
$141$	0	0
$142$	−6.00000	−0.503509
$143$	0	0
$144$	−3.00000	−0.250000
$145$	0	0
$146$	4.00000	0.331042
$147$	0	0
$148$	3.00000	0.246598
$149$	6.00000	0.491539	0.245770	−	0.969328i	$-0.420959\pi$
$149$	6.00000	0.491539	0.245770	+	0.969328i	$0.420959\pi$
$150$	0	0
$151$	2.00000	0.162758	0.0813788	−	0.996683i	$-0.474068\pi$
$151$	2.00000	0.162758	0.0813788	+	0.996683i	$0.474068\pi$
$152$	−7.00000	−0.567775
$153$	−12.0000	−0.970143
$154$	−9.00000	−0.725241
$155$	0	0
$156$	0	0
$157$	3.00000	0.239426	0.119713	−	0.992809i	$-0.461803\pi$
$157$	3.00000	0.239426	0.119713	+	0.992809i	$0.461803\pi$
$158$	10.0000	0.795557
$159$	0	0
$160$	0	0
$161$	−12.0000	−0.945732
$162$	9.00000	0.707107
$163$	8.00000	0.626608	0.313304	−	0.949653i	$-0.398564\pi$
$163$	8.00000	0.626608	0.313304	+	0.949653i	$0.398564\pi$
$164$	2.00000	0.156174
$165$	0	0
$166$	0	0
$167$	19.0000	1.47026	0.735132	−	0.677924i	$-0.237120\pi$
$167$	19.0000	1.47026	0.735132	+	0.677924i	$0.237120\pi$
$168$	0	0
$169$	0	0
$170$	0	0
$171$	21.0000	1.60591
$172$	−6.00000	−0.457496
$173$	21.0000	1.59660	0.798300	−	0.602260i	$-0.205733\pi$
$173$	21.0000	1.59660	0.798300	+	0.602260i	$0.205733\pi$
$174$	0	0
$175$	0	0
$176$	3.00000	0.226134
$177$	0	0
$178$	−1.00000	−0.0749532
$179$	4.00000	0.298974	0.149487	−	0.988764i	$-0.452238\pi$
$179$	4.00000	0.298974	0.149487	+	0.988764i	$0.452238\pi$
$180$	0	0
$181$	−14.0000	−1.04061	−0.520306	−	0.853980i	$-0.674182\pi$
$181$	−14.0000	−1.04061	−0.520306	+	0.853980i	$0.674182\pi$
$182$	0	0
$183$	0	0
$184$	4.00000	0.294884
$185$	0	0
$186$	0	0
$187$	12.0000	0.877527
$188$	−1.00000	−0.0729325
$189$	0	0
$190$	0	0
$191$	−6.00000	−0.434145	−0.217072	−	0.976156i	$-0.569651\pi$
$191$	−6.00000	−0.434145	−0.217072	+	0.976156i	$0.569651\pi$
$192$	0	0
$193$	−24.0000	−1.72756	−0.863779	−	0.503871i	$-0.831909\pi$
$193$	−24.0000	−1.72756	−0.863779	+	0.503871i	$0.831909\pi$
$194$	16.0000	1.14873
$195$	0	0
$196$	2.00000	0.142857
$197$	13.0000	0.926212	0.463106	−	0.886303i	$-0.346735\pi$
$197$	13.0000	0.926212	0.463106	+	0.886303i	$0.346735\pi$
$198$	−9.00000	−0.639602
$199$	−18.0000	−1.27599	−0.637993	−	0.770042i	$-0.720235\pi$
$199$	−18.0000	−1.27599	−0.637993	+	0.770042i	$0.720235\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	24.0000	1.68447
$204$	0	0
$205$	0	0
$206$	1.00000	0.0696733
$207$	−12.0000	−0.834058
$208$	0	0
$209$	−21.0000	−1.45260
$210$	0	0
$211$	13.0000	0.894957	0.447478	−	0.894295i	$-0.352322\pi$
$211$	13.0000	0.894957	0.447478	+	0.894295i	$0.352322\pi$
$212$	9.00000	0.618123
$213$	0	0
$214$	18.0000	1.23045
$215$	0	0
$216$	0	0
$217$	−30.0000	−2.03653
$218$	2.00000	0.135457
$219$	0	0
$220$	0	0
$221$	0	0
$222$	0	0
$223$	19.0000	1.27233	0.636167	−	0.771551i	$-0.280519\pi$
$223$	19.0000	1.27233	0.636167	+	0.771551i	$0.280519\pi$
$224$	−3.00000	−0.200446
$225$	0	0
$226$	8.00000	0.532152
$227$	−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
$231$	0	0
$232$	−8.00000	−0.525226
$233$	18.0000	1.17922	0.589610	−	0.807688i	$-0.299282\pi$
$233$	18.0000	1.17922	0.589610	+	0.807688i	$0.299282\pi$
$234$	0	0
$235$	0	0
$236$	4.00000	0.260378
$237$	0	0
$238$	−12.0000	−0.777844
$239$	−2.00000	−0.129369	−0.0646846	−	0.997906i	$-0.520604\pi$
$239$	−2.00000	−0.129369	−0.0646846	+	0.997906i	$0.520604\pi$
$240$	0	0
$241$	25.0000	1.61039	0.805196	−	0.593009i	$-0.202060\pi$
$241$	25.0000	1.61039	0.805196	+	0.593009i	$0.202060\pi$
$242$	−2.00000	−0.128565
$243$	0	0
$244$	−14.0000	−0.896258
$245$	0	0
$246$	0	0
$247$	0	0
$248$	10.0000	0.635001
$249$	0	0
$250$	0	0
$251$	23.0000	1.45175	0.725874	−	0.687828i	$-0.241436\pi$
$251$	23.0000	1.45175	0.725874	+	0.687828i	$0.241436\pi$
$252$	9.00000	0.566947
$253$	12.0000	0.754434
$254$	−13.0000	−0.815693
$255$	0	0
$256$	1.00000	0.0625000
$257$	−8.00000	−0.499026	−0.249513	−	0.968371i	$-0.580271\pi$
$257$	−8.00000	−0.499026	−0.249513	+	0.968371i	$0.580271\pi$
$258$	0	0
$259$	−9.00000	−0.559233
$260$	0	0
$261$	24.0000	1.48556
$262$	13.0000	0.803143
$263$	−11.0000	−0.678289	−0.339145	−	0.940734i	$-0.610138\pi$
$263$	−11.0000	−0.678289	−0.339145	+	0.940734i	$0.610138\pi$
$264$	0	0
$265$	0	0
$266$	21.0000	1.28759
$267$	0	0
$268$	4.00000	0.244339
$269$	24.0000	1.46331	0.731653	−	0.681677i	$-0.238749\pi$
$269$	24.0000	1.46331	0.731653	+	0.681677i	$0.238749\pi$
$270$	0	0
$271$	−8.00000	−0.485965	−0.242983	−	0.970031i	$-0.578126\pi$
$271$	−8.00000	−0.485965	−0.242983	+	0.970031i	$0.578126\pi$
$272$	4.00000	0.242536
$273$	0	0
$274$	12.0000	0.724947
$275$	0	0
$276$	0	0
$277$	−19.0000	−1.14160	−0.570800	−	0.821089i	$-0.693367\pi$
$277$	−19.0000	−1.14160	−0.570800	+	0.821089i	$0.693367\pi$
$278$	−5.00000	−0.299880
$279$	−30.0000	−1.79605
$280$	0	0
$281$	−2.00000	−0.119310	−0.0596550	−	0.998219i	$-0.519000\pi$
$281$	−2.00000	−0.119310	−0.0596550	+	0.998219i	$0.519000\pi$
$282$	0	0
$283$	14.0000	0.832214	0.416107	−	0.909316i	$-0.363394\pi$
$283$	14.0000	0.832214	0.416107	+	0.909316i	$0.363394\pi$
$284$	−6.00000	−0.356034
$285$	0	0
$286$	0	0
$287$	−6.00000	−0.354169
$288$	−3.00000	−0.176777
$289$	−1.00000	−0.0588235
$290$	0	0
$291$	0	0
$292$	4.00000	0.234082
$293$	9.00000	0.525786	0.262893	−	0.964825i	$-0.415323\pi$
$293$	9.00000	0.525786	0.262893	+	0.964825i	$0.415323\pi$
$294$	0	0
$295$	0	0
$296$	3.00000	0.174371
$297$	0	0
$298$	6.00000	0.347571
$299$	0	0
$300$	0	0
$301$	18.0000	1.03750
$302$	2.00000	0.115087
$303$	0	0
$304$	−7.00000	−0.401478
$305$	0	0
$306$	−12.0000	−0.685994
$307$	−14.0000	−0.799022	−0.399511	−	0.916728i	$-0.630820\pi$
$307$	−14.0000	−0.799022	−0.399511	+	0.916728i	$0.630820\pi$
$308$	−9.00000	−0.512823
$309$	0	0
$310$	0	0
$311$	−24.0000	−1.36092	−0.680458	−	0.732787i	$-0.738219\pi$
$311$	−24.0000	−1.36092	−0.680458	+	0.732787i	$0.738219\pi$
$312$	0	0
$313$	−6.00000	−0.339140	−0.169570	−	0.985518i	$-0.554238\pi$
$313$	−6.00000	−0.339140	−0.169570	+	0.985518i	$0.554238\pi$
$314$	3.00000	0.169300
$315$	0	0
$316$	10.0000	0.562544
$317$	31.0000	1.74113	0.870567	−	0.492050i	$-0.163752\pi$
$317$	31.0000	1.74113	0.870567	+	0.492050i	$0.163752\pi$
$318$	0	0
$319$	−24.0000	−1.34374
$320$	0	0
$321$	0	0
$322$	−12.0000	−0.668734
$323$	−28.0000	−1.55796
$324$	9.00000	0.500000
$325$	0	0
$326$	8.00000	0.443079
$327$	0	0
$328$	2.00000	0.110432
$329$	3.00000	0.165395
$330$	0	0
$331$	20.0000	1.09930	0.549650	−	0.835395i	$-0.314761\pi$
$331$	20.0000	1.09930	0.549650	+	0.835395i	$0.314761\pi$
$332$	0	0
$333$	−9.00000	−0.493197
$334$	19.0000	1.03963
$335$	0	0
$336$	0	0
$337$	−30.0000	−1.63420	−0.817102	−	0.576493i	$-0.804421\pi$
$337$	−30.0000	−1.63420	−0.817102	+	0.576493i	$0.804421\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	30.0000	1.62459
$342$	21.0000	1.13555
$343$	15.0000	0.809924
$344$	−6.00000	−0.323498
$345$	0	0
$346$	21.0000	1.12897
$347$	16.0000	0.858925	0.429463	−	0.903085i	$-0.358703\pi$
$347$	16.0000	0.858925	0.429463	+	0.903085i	$0.358703\pi$
$348$	0	0
$349$	16.0000	0.856460	0.428230	−	0.903670i	$-0.359137\pi$
$349$	16.0000	0.856460	0.428230	+	0.903670i	$0.359137\pi$
$350$	0	0
$351$	0	0
$352$	3.00000	0.159901
$353$	0	0		−	1.00000i	$-0.5\pi$
$353$	0	0			1.00000i	$0.5\pi$
$354$	0	0
$355$	0	0
$356$	−1.00000	−0.0529999
$357$	0	0
$358$	4.00000	0.211407
$359$	−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
$361$	30.0000	1.57895
$362$	−14.0000	−0.735824
$363$	0	0
$364$	0	0
$365$	0	0
$366$	0	0
$367$	0	0		−	1.00000i	$-0.5\pi$
$367$	0	0			1.00000i	$0.5\pi$
$368$	4.00000	0.208514
$369$	−6.00000	−0.312348
$370$	0	0
$371$	−27.0000	−1.40177
$372$	0	0
$373$	6.00000	0.310668	0.155334	−	0.987862i	$-0.450355\pi$
$373$	6.00000	0.310668	0.155334	+	0.987862i	$0.450355\pi$
$374$	12.0000	0.620505
$375$	0	0
$376$	−1.00000	−0.0515711
$377$	0	0
$378$	0	0
$379$	−27.0000	−1.38690	−0.693448	−	0.720506i	$-0.743909\pi$
$379$	−27.0000	−1.38690	−0.693448	+	0.720506i	$0.743909\pi$
$380$	0	0
$381$	0	0
$382$	−6.00000	−0.306987
$383$	−28.0000	−1.43073	−0.715367	−	0.698749i	$-0.753740\pi$
$383$	−28.0000	−1.43073	−0.715367	+	0.698749i	$0.753740\pi$
$384$	0	0
$385$	0	0
$386$	−24.0000	−1.22157
$387$	18.0000	0.914991
$388$	16.0000	0.812277
$389$	12.0000	0.608424	0.304212	−	0.952604i	$-0.401607\pi$
$389$	12.0000	0.608424	0.304212	+	0.952604i	$0.401607\pi$
$390$	0	0
$391$	16.0000	0.809155
$392$	2.00000	0.101015
$393$	0	0
$394$	13.0000	0.654931
$395$	0	0
$396$	−9.00000	−0.452267
$397$	−21.0000	−1.05396	−0.526980	−	0.849878i	$-0.676676\pi$
$397$	−21.0000	−1.05396	−0.526980	+	0.849878i	$0.676676\pi$
$398$	−18.0000	−0.902258
$399$	0	0
$400$	0	0
$401$	39.0000	1.94757	0.973784	−	0.227477i	$-0.0730475\pi$
$401$	39.0000	1.94757	0.973784	+	0.227477i	$0.0730475\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	0	0
$406$	24.0000	1.19110
$407$	9.00000	0.446113
$408$	0	0
$409$	−17.0000	−0.840596	−0.420298	−	0.907386i	$-0.638074\pi$
$409$	−17.0000	−0.840596	−0.420298	+	0.907386i	$0.638074\pi$
$410$	0	0
$411$	0	0
$412$	1.00000	0.0492665
$413$	−12.0000	−0.590481
$414$	−12.0000	−0.589768
$415$	0	0
$416$	0	0
$417$	0	0
$418$	−21.0000	−1.02714
$419$	12.0000	0.586238	0.293119	−	0.956076i	$-0.405307\pi$
$419$	12.0000	0.586238	0.293119	+	0.956076i	$0.405307\pi$
$420$	0	0
$421$	−16.0000	−0.779792	−0.389896	−	0.920859i	$-0.627489\pi$
$421$	−16.0000	−0.779792	−0.389896	+	0.920859i	$0.627489\pi$
$422$	13.0000	0.632830
$423$	3.00000	0.145865
$424$	9.00000	0.437079
$425$	0	0
$426$	0	0
$427$	42.0000	2.03252
$428$	18.0000	0.870063
$429$	0	0
$430$	0	0
$431$	24.0000	1.15604	0.578020	−	0.816023i	$-0.303826\pi$
$431$	24.0000	1.15604	0.578020	+	0.816023i	$0.303826\pi$
$432$	0	0
$433$	−16.0000	−0.768911	−0.384455	−	0.923144i	$-0.625611\pi$
$433$	−16.0000	−0.768911	−0.384455	+	0.923144i	$0.625611\pi$
$434$	−30.0000	−1.44005
$435$	0	0
$436$	2.00000	0.0957826
$437$	−28.0000	−1.33942
$438$	0	0
$439$	−6.00000	−0.286364	−0.143182	−	0.989696i	$-0.545733\pi$
$439$	−6.00000	−0.286364	−0.143182	+	0.989696i	$0.545733\pi$
$440$	0	0
$441$	−6.00000	−0.285714
$442$	0	0
$443$	18.0000	0.855206	0.427603	−	0.903967i	$-0.359358\pi$
$443$	18.0000	0.855206	0.427603	+	0.903967i	$0.359358\pi$
$444$	0	0
$445$	0	0
$446$	19.0000	0.899676
$447$	0	0
$448$	−3.00000	−0.141737
$449$	15.0000	0.707894	0.353947	−	0.935266i	$-0.384839\pi$
$449$	15.0000	0.707894	0.353947	+	0.935266i	$0.384839\pi$
$450$	0	0
$451$	6.00000	0.282529
$452$	8.00000	0.376288
$453$	0	0
$454$	−4.00000	−0.187729
$455$	0	0
$456$	0	0
$457$	10.0000	0.467780	0.233890	−	0.972263i	$-0.424854\pi$
$457$	10.0000	0.467780	0.233890	+	0.972263i	$0.424854\pi$
$458$	22.0000	1.02799
$459$	0	0
$460$	0	0
$461$	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
$463$	−40.0000	−1.85896	−0.929479	−	0.368875i	$-0.879743\pi$
$463$	−40.0000	−1.85896	−0.929479	+	0.368875i	$0.879743\pi$
$464$	−8.00000	−0.371391
$465$	0	0
$466$	18.0000	0.833834
$467$	−20.0000	−0.925490	−0.462745	−	0.886492i	$-0.653135\pi$
$467$	−20.0000	−0.925490	−0.462745	+	0.886492i	$0.653135\pi$
$468$	0	0
$469$	−12.0000	−0.554109
$470$	0	0
$471$	0	0
$472$	4.00000	0.184115
$473$	−18.0000	−0.827641
$474$	0	0
$475$	0	0
$476$	−12.0000	−0.550019
$477$	−27.0000	−1.23625
$478$	−2.00000	−0.0914779
$479$	28.0000	1.27935	0.639676	−	0.768644i	$-0.279068\pi$
$479$	28.0000	1.27935	0.639676	+	0.768644i	$0.279068\pi$
$480$	0	0
$481$	0	0
$482$	25.0000	1.13872
$483$	0	0
$484$	−2.00000	−0.0909091
$485$	0	0
$486$	0	0
$487$	19.0000	0.860972	0.430486	−	0.902597i	$-0.358342\pi$
$487$	19.0000	0.860972	0.430486	+	0.902597i	$0.358342\pi$
$488$	−14.0000	−0.633750
$489$	0	0
$490$	0	0
$491$	−43.0000	−1.94056	−0.970281	−	0.241979i	$-0.922203\pi$
$491$	−43.0000	−1.94056	−0.970281	+	0.241979i	$0.922203\pi$
$492$	0	0
$493$	−32.0000	−1.44121
$494$	0	0
$495$	0	0
$496$	10.0000	0.449013
$497$	18.0000	0.807410
$498$	0	0
$499$	4.00000	0.179065	0.0895323	−	0.995984i	$-0.471463\pi$
$499$	4.00000	0.179065	0.0895323	+	0.995984i	$0.471463\pi$
$500$	0	0
$501$	0	0
$502$	23.0000	1.02654
$503$	−23.0000	−1.02552	−0.512760	−	0.858532i	$-0.671377\pi$
$503$	−23.0000	−1.02552	−0.512760	+	0.858532i	$0.671377\pi$
$504$	9.00000	0.400892
$505$	0	0
$506$	12.0000	0.533465
$507$	0	0
$508$	−13.0000	−0.576782
$509$	6.00000	0.265945	0.132973	−	0.991120i	$-0.457548\pi$
$509$	6.00000	0.265945	0.132973	+	0.991120i	$0.457548\pi$
$510$	0	0
$511$	−12.0000	−0.530849
$512$	1.00000	0.0441942
$513$	0	0
$514$	−8.00000	−0.352865
$515$	0	0
$516$	0	0
$517$	−3.00000	−0.131940
$518$	−9.00000	−0.395437
$519$	0	0
$520$	0	0
$521$	−31.0000	−1.35813	−0.679067	−	0.734076i	$-0.737616\pi$
$521$	−31.0000	−1.35813	−0.679067	+	0.734076i	$0.737616\pi$
$522$	24.0000	1.05045
$523$	−22.0000	−0.961993	−0.480996	−	0.876723i	$-0.659725\pi$
$523$	−22.0000	−0.961993	−0.480996	+	0.876723i	$0.659725\pi$
$524$	13.0000	0.567908
$525$	0	0
$526$	−11.0000	−0.479623
$527$	40.0000	1.74243
$528$	0	0
$529$	−7.00000	−0.304348
$530$	0	0
$531$	−12.0000	−0.520756
$532$	21.0000	0.910465
$533$	0	0
$534$	0	0
$535$	0	0
$536$	4.00000	0.172774
$537$	0	0
$538$	24.0000	1.03471
$539$	6.00000	0.258438
$540$	0	0
$541$	−10.0000	−0.429934	−0.214967	−	0.976621i	$-0.568964\pi$
$541$	−10.0000	−0.429934	−0.214967	+	0.976621i	$0.568964\pi$
$542$	−8.00000	−0.343629
$543$	0	0
$544$	4.00000	0.171499
$545$	0	0
$546$	0	0
$547$	−2.00000	−0.0855138	−0.0427569	−	0.999086i	$-0.513614\pi$
$547$	−2.00000	−0.0855138	−0.0427569	+	0.999086i	$0.513614\pi$
$548$	12.0000	0.512615
$549$	42.0000	1.79252
$550$	0	0
$551$	56.0000	2.38568
$552$	0	0
$553$	−30.0000	−1.27573
$554$	−19.0000	−0.807233
$555$	0	0
$556$	−5.00000	−0.212047
$557$	−31.0000	−1.31351	−0.656756	−	0.754103i	$-0.728072\pi$
$557$	−31.0000	−1.31351	−0.656756	+	0.754103i	$0.728072\pi$
$558$	−30.0000	−1.27000
$559$	0	0
$560$	0	0
$561$	0	0
$562$	−2.00000	−0.0843649
$563$	−24.0000	−1.01148	−0.505740	−	0.862686i	$-0.668780\pi$
$563$	−24.0000	−1.01148	−0.505740	+	0.862686i	$0.668780\pi$
$564$	0	0
$565$	0	0
$566$	14.0000	0.588464
$567$	−27.0000	−1.13389
$568$	−6.00000	−0.251754
$569$	−11.0000	−0.461144	−0.230572	−	0.973055i	$-0.574060\pi$
$569$	−11.0000	−0.461144	−0.230572	+	0.973055i	$0.574060\pi$
$570$	0	0
$571$	−37.0000	−1.54840	−0.774201	−	0.632940i	$-0.781848\pi$
$571$	−37.0000	−1.54840	−0.774201	+	0.632940i	$0.781848\pi$
$572$	0	0
$573$	0	0
$574$	−6.00000	−0.250435
$575$	0	0
$576$	−3.00000	−0.125000
$577$	−10.0000	−0.416305	−0.208153	−	0.978096i	$-0.566745\pi$
$577$	−10.0000	−0.416305	−0.208153	+	0.978096i	$0.566745\pi$
$578$	−1.00000	−0.0415945
$579$	0	0
$580$	0	0
$581$	0	0
$582$	0	0
$583$	27.0000	1.11823
$584$	4.00000	0.165521
$585$	0	0
$586$	9.00000	0.371787
$587$	−2.00000	−0.0825488	−0.0412744	−	0.999148i	$-0.513142\pi$
$587$	−2.00000	−0.0825488	−0.0412744	+	0.999148i	$0.513142\pi$
$588$	0	0
$589$	−70.0000	−2.88430
$590$	0	0
$591$	0	0
$592$	3.00000	0.123299
$593$	20.0000	0.821302	0.410651	−	0.911793i	$-0.365302\pi$
$593$	20.0000	0.821302	0.410651	+	0.911793i	$0.365302\pi$
$594$	0	0
$595$	0	0
$596$	6.00000	0.245770
$597$	0	0
$598$	0	0
$599$	26.0000	1.06233	0.531166	−	0.847268i	$-0.321754\pi$
$599$	26.0000	1.06233	0.531166	+	0.847268i	$0.321754\pi$
$600$	0	0
$601$	5.00000	0.203954	0.101977	−	0.994787i	$-0.467483\pi$
$601$	5.00000	0.203954	0.101977	+	0.994787i	$0.467483\pi$
$602$	18.0000	0.733625
$603$	−12.0000	−0.488678
$604$	2.00000	0.0813788
$605$	0	0
$606$	0	0
$607$	−21.0000	−0.852364	−0.426182	−	0.904638i	$-0.640142\pi$
$607$	−21.0000	−0.852364	−0.426182	+	0.904638i	$0.640142\pi$
$608$	−7.00000	−0.283887
$609$	0	0
$610$	0	0
$611$	0	0
$612$	−12.0000	−0.485071
$613$	11.0000	0.444286	0.222143	−	0.975014i	$-0.428695\pi$
$613$	11.0000	0.444286	0.222143	+	0.975014i	$0.428695\pi$
$614$	−14.0000	−0.564994
$615$	0	0
$616$	−9.00000	−0.362620
$617$	−22.0000	−0.885687	−0.442843	−	0.896599i	$-0.646030\pi$
$617$	−22.0000	−0.885687	−0.442843	+	0.896599i	$0.646030\pi$
$618$	0	0
$619$	3.00000	0.120580	0.0602901	−	0.998181i	$-0.480797\pi$
$619$	3.00000	0.120580	0.0602901	+	0.998181i	$0.480797\pi$
$620$	0	0
$621$	0	0
$622$	−24.0000	−0.962312
$623$	3.00000	0.120192
$624$	0	0
$625$	0	0
$626$	−6.00000	−0.239808
$627$	0	0
$628$	3.00000	0.119713
$629$	12.0000	0.478471
$630$	0	0
$631$	36.0000	1.43314	0.716569	−	0.697517i	$-0.245712\pi$
$631$	36.0000	1.43314	0.716569	+	0.697517i	$0.245712\pi$
$632$	10.0000	0.397779
$633$	0	0
$634$	31.0000	1.23117
$635$	0	0
$636$	0	0
$637$	0	0
$638$	−24.0000	−0.950169
$639$	18.0000	0.712069
$640$	0	0
$641$	−9.00000	−0.355479	−0.177739	−	0.984078i	$-0.556878\pi$
$641$	−9.00000	−0.355479	−0.177739	+	0.984078i	$0.556878\pi$
$642$	0	0
$643$	16.0000	0.630978	0.315489	−	0.948929i	$-0.397831\pi$
$643$	16.0000	0.630978	0.315489	+	0.948929i	$0.397831\pi$
$644$	−12.0000	−0.472866
$645$	0	0
$646$	−28.0000	−1.10165
$647$	7.00000	0.275198	0.137599	−	0.990488i	$-0.456061\pi$
$647$	7.00000	0.275198	0.137599	+	0.990488i	$0.456061\pi$
$648$	9.00000	0.353553
$649$	12.0000	0.471041
$650$	0	0
$651$	0	0
$652$	8.00000	0.313304
$653$	11.0000	0.430463	0.215232	−	0.976563i	$-0.430949\pi$
$653$	11.0000	0.430463	0.215232	+	0.976563i	$0.430949\pi$
$654$	0	0
$655$	0	0
$656$	2.00000	0.0780869
$657$	−12.0000	−0.468165
$658$	3.00000	0.116952
$659$	−36.0000	−1.40236	−0.701180	−	0.712984i	$-0.747343\pi$
$659$	−36.0000	−1.40236	−0.701180	+	0.712984i	$0.747343\pi$
$660$	0	0
$661$	14.0000	0.544537	0.272268	−	0.962221i	$-0.412226\pi$
$661$	14.0000	0.544537	0.272268	+	0.962221i	$0.412226\pi$
$662$	20.0000	0.777322
$663$	0	0
$664$	0	0
$665$	0	0
$666$	−9.00000	−0.348743
$667$	−32.0000	−1.23904
$668$	19.0000	0.735132
$669$	0	0
$670$	0	0
$671$	−42.0000	−1.62139
$672$	0	0
$673$	8.00000	0.308377	0.154189	−	0.988041i	$-0.450724\pi$
$673$	8.00000	0.308377	0.154189	+	0.988041i	$0.450724\pi$
$674$	−30.0000	−1.15556
$675$	0	0
$676$	0	0
$677$	50.0000	1.92166	0.960828	−	0.277145i	$-0.0893883\pi$
$677$	50.0000	1.92166	0.960828	+	0.277145i	$0.0893883\pi$
$678$	0	0
$679$	−48.0000	−1.84207
$680$	0	0
$681$	0	0
$682$	30.0000	1.14876
$683$	−2.00000	−0.0765279	−0.0382639	−	0.999268i	$-0.512183\pi$
$683$	−2.00000	−0.0765279	−0.0382639	+	0.999268i	$0.512183\pi$
$684$	21.0000	0.802955
$685$	0	0
$686$	15.0000	0.572703
$687$	0	0
$688$	−6.00000	−0.228748
$689$	0	0
$690$	0	0
$691$	−29.0000	−1.10321	−0.551606	−	0.834105i	$-0.685985\pi$
$691$	−29.0000	−1.10321	−0.551606	+	0.834105i	$0.685985\pi$
$692$	21.0000	0.798300
$693$	27.0000	1.02565
$694$	16.0000	0.607352
$695$	0	0
$696$	0	0
$697$	8.00000	0.303022
$698$	16.0000	0.605609
$699$	0	0
$700$	0	0
$701$	−40.0000	−1.51078	−0.755390	−	0.655276i	$-0.772552\pi$
$701$	−40.0000	−1.51078	−0.755390	+	0.655276i	$0.772552\pi$
$702$	0	0
$703$	−21.0000	−0.792030
$704$	3.00000	0.113067
$705$	0	0
$706$	0	0
$707$	0	0
$708$	0	0
$709$	−8.00000	−0.300446	−0.150223	−	0.988652i	$-0.547999\pi$
$709$	−8.00000	−0.300446	−0.150223	+	0.988652i	$0.547999\pi$
$710$	0	0
$711$	−30.0000	−1.12509
$712$	−1.00000	−0.0374766
$713$	40.0000	1.49801
$714$	0	0
$715$	0	0
$716$	4.00000	0.149487
$717$	0	0
$718$	−10.0000	−0.373197
$719$	28.0000	1.04422	0.522112	−	0.852877i	$-0.325144\pi$
$719$	28.0000	1.04422	0.522112	+	0.852877i	$0.325144\pi$
$720$	0	0
$721$	−3.00000	−0.111726
$722$	30.0000	1.11648
$723$	0	0
$724$	−14.0000	−0.520306
$725$	0	0
$726$	0	0
$727$	−35.0000	−1.29808	−0.649039	−	0.760755i	$-0.724829\pi$
$727$	−35.0000	−1.29808	−0.649039	+	0.760755i	$0.724829\pi$
$728$	0	0
$729$	−27.0000	−1.00000
$730$	0	0
$731$	−24.0000	−0.887672
$732$	0	0
$733$	−1.00000	−0.0369358	−0.0184679	−	0.999829i	$-0.505879\pi$
$733$	−1.00000	−0.0369358	−0.0184679	+	0.999829i	$0.505879\pi$
$734$	0	0
$735$	0	0
$736$	4.00000	0.147442
$737$	12.0000	0.442026
$738$	−6.00000	−0.220863
$739$	−39.0000	−1.43464	−0.717319	−	0.696745i	$-0.754631\pi$
$739$	−39.0000	−1.43464	−0.717319	+	0.696745i	$0.754631\pi$
$740$	0	0
$741$	0	0
$742$	−27.0000	−0.991201
$743$	−8.00000	−0.293492	−0.146746	−	0.989174i	$-0.546880\pi$
$743$	−8.00000	−0.293492	−0.146746	+	0.989174i	$0.546880\pi$
$744$	0	0
$745$	0	0
$746$	6.00000	0.219676
$747$	0	0
$748$	12.0000	0.438763
$749$	−54.0000	−1.97312
$750$	0	0
$751$	−4.00000	−0.145962	−0.0729810	−	0.997333i	$-0.523251\pi$
$751$	−4.00000	−0.145962	−0.0729810	+	0.997333i	$0.523251\pi$
$752$	−1.00000	−0.0364662
$753$	0	0
$754$	0	0
$755$	0	0
$756$	0	0
$757$	29.0000	1.05402	0.527011	−	0.849858i	$-0.323312\pi$
$757$	29.0000	1.05402	0.527011	+	0.849858i	$0.323312\pi$
$758$	−27.0000	−0.980684
$759$	0	0
$760$	0	0
$761$	−21.0000	−0.761249	−0.380625	−	0.924730i	$-0.624291\pi$
$761$	−21.0000	−0.761249	−0.380625	+	0.924730i	$0.624291\pi$
$762$	0	0
$763$	−6.00000	−0.217215
$764$	−6.00000	−0.217072
$765$	0	0
$766$	−28.0000	−1.01168
$767$	0	0
$768$	0	0
$769$	2.00000	0.0721218	0.0360609	−	0.999350i	$-0.488519\pi$
$769$	2.00000	0.0721218	0.0360609	+	0.999350i	$0.488519\pi$
$770$	0	0
$771$	0	0
$772$	−24.0000	−0.863779
$773$	31.0000	1.11499	0.557496	−	0.830179i	$-0.311762\pi$
$773$	31.0000	1.11499	0.557496	+	0.830179i	$0.311762\pi$
$774$	18.0000	0.646997
$775$	0	0
$776$	16.0000	0.574367
$777$	0	0
$778$	12.0000	0.430221
$779$	−14.0000	−0.501602
$780$	0	0
$781$	−18.0000	−0.644091
$782$	16.0000	0.572159
$783$	0	0
$784$	2.00000	0.0714286
$785$	0	0
$786$	0	0
$787$	−4.00000	−0.142585	−0.0712923	−	0.997455i	$-0.522712\pi$
$787$	−4.00000	−0.142585	−0.0712923	+	0.997455i	$0.522712\pi$
$788$	13.0000	0.463106
$789$	0	0
$790$	0	0
$791$	−24.0000	−0.853342
$792$	−9.00000	−0.319801
$793$	0	0
$794$	−21.0000	−0.745262
$795$	0	0
$796$	−18.0000	−0.637993
$797$	−2.00000	−0.0708436	−0.0354218	−	0.999372i	$-0.511277\pi$
$797$	−2.00000	−0.0708436	−0.0354218	+	0.999372i	$0.511277\pi$
$798$	0	0
$799$	−4.00000	−0.141510
$800$	0	0
$801$	3.00000	0.106000
$802$	39.0000	1.37714
$803$	12.0000	0.423471
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−10.0000	−0.351581	−0.175791	−	0.984428i	$-0.556248\pi$
$809$	−10.0000	−0.351581	−0.175791	+	0.984428i	$0.556248\pi$
$810$	0	0
$811$	−21.0000	−0.737410	−0.368705	−	0.929547i	$-0.620199\pi$
$811$	−21.0000	−0.737410	−0.368705	+	0.929547i	$0.620199\pi$
$812$	24.0000	0.842235
$813$	0	0
$814$	9.00000	0.315450
$815$	0	0
$816$	0	0
$817$	42.0000	1.46939
$818$	−17.0000	−0.594391
$819$	0	0
$820$	0	0
$821$	18.0000	0.628204	0.314102	−	0.949389i	$-0.398297\pi$
$821$	18.0000	0.628204	0.314102	+	0.949389i	$0.398297\pi$
$822$	0	0
$823$	37.0000	1.28974	0.644869	−	0.764293i	$-0.276912\pi$
$823$	37.0000	1.28974	0.644869	+	0.764293i	$0.276912\pi$
$824$	1.00000	0.0348367
$825$	0	0
$826$	−12.0000	−0.417533
$827$	30.0000	1.04320	0.521601	−	0.853189i	$-0.325335\pi$
$827$	30.0000	1.04320	0.521601	+	0.853189i	$0.325335\pi$
$828$	−12.0000	−0.417029
$829$	−28.0000	−0.972480	−0.486240	−	0.873825i	$-0.661632\pi$
$829$	−28.0000	−0.972480	−0.486240	+	0.873825i	$0.661632\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	8.00000	0.277184
$834$	0	0
$835$	0	0
$836$	−21.0000	−0.726300
$837$	0	0
$838$	12.0000	0.414533
$839$	26.0000	0.897620	0.448810	−	0.893627i	$-0.351848\pi$
$839$	26.0000	0.897620	0.448810	+	0.893627i	$0.351848\pi$
$840$	0	0
$841$	35.0000	1.20690
$842$	−16.0000	−0.551396
$843$	0	0
$844$	13.0000	0.447478
$845$	0	0
$846$	3.00000	0.103142
$847$	6.00000	0.206162
$848$	9.00000	0.309061
$849$	0	0
$850$	0	0
$851$	12.0000	0.411355
$852$	0	0
$853$	−14.0000	−0.479351	−0.239675	−	0.970853i	$-0.577041\pi$
$853$	−14.0000	−0.479351	−0.239675	+	0.970853i	$0.577041\pi$
$854$	42.0000	1.43721
$855$	0	0
$856$	18.0000	0.615227
$857$	−18.0000	−0.614868	−0.307434	−	0.951569i	$-0.599470\pi$
$857$	−18.0000	−0.614868	−0.307434	+	0.951569i	$0.599470\pi$
$858$	0	0
$859$	−1.00000	−0.0341196	−0.0170598	−	0.999854i	$-0.505431\pi$
$859$	−1.00000	−0.0341196	−0.0170598	+	0.999854i	$0.505431\pi$
$860$	0	0
$861$	0	0
$862$	24.0000	0.817443
$863$	−32.0000	−1.08929	−0.544646	−	0.838666i	$-0.683336\pi$
$863$	−32.0000	−1.08929	−0.544646	+	0.838666i	$0.683336\pi$
$864$	0	0
$865$	0	0
$866$	−16.0000	−0.543702
$867$	0	0
$868$	−30.0000	−1.01827
$869$	30.0000	1.01768
$870$	0	0
$871$	0	0
$872$	2.00000	0.0677285
$873$	−48.0000	−1.62455
$874$	−28.0000	−0.947114
$875$	0	0
$876$	0	0
$877$	−34.0000	−1.14810	−0.574049	−	0.818821i	$-0.694628\pi$
$877$	−34.0000	−1.14810	−0.574049	+	0.818821i	$0.694628\pi$
$878$	−6.00000	−0.202490
$879$	0	0
$880$	0	0
$881$	5.00000	0.168454	0.0842271	−	0.996447i	$-0.473158\pi$
$881$	5.00000	0.168454	0.0842271	+	0.996447i	$0.473158\pi$
$882$	−6.00000	−0.202031
$883$	26.0000	0.874970	0.437485	−	0.899226i	$-0.355869\pi$
$883$	26.0000	0.874970	0.437485	+	0.899226i	$0.355869\pi$
$884$	0	0
$885$	0	0
$886$	18.0000	0.604722
$887$	13.0000	0.436497	0.218249	−	0.975893i	$-0.429966\pi$
$887$	13.0000	0.436497	0.218249	+	0.975893i	$0.429966\pi$
$888$	0	0
$889$	39.0000	1.30802
$890$	0	0
$891$	27.0000	0.904534
$892$	19.0000	0.636167
$893$	7.00000	0.234246
$894$	0	0
$895$	0	0
$896$	−3.00000	−0.100223
$897$	0	0
$898$	15.0000	0.500556
$899$	−80.0000	−2.66815
$900$	0	0
$901$	36.0000	1.19933
$902$	6.00000	0.199778
$903$	0	0
$904$	8.00000	0.266076
$905$	0	0
$906$	0	0
$907$	10.0000	0.332045	0.166022	−	0.986122i	$-0.446908\pi$
$907$	10.0000	0.332045	0.166022	+	0.986122i	$0.446908\pi$
$908$	−4.00000	−0.132745
$909$	0	0
$910$	0	0
$911$	−12.0000	−0.397578	−0.198789	−	0.980042i	$-0.563701\pi$
$911$	−12.0000	−0.397578	−0.198789	+	0.980042i	$0.563701\pi$
$912$	0	0
$913$	0	0
$914$	10.0000	0.330771
$915$	0	0
$916$	22.0000	0.726900
$917$	−39.0000	−1.28789
$918$	0	0
$919$	50.0000	1.64935	0.824674	−	0.565608i	$-0.191359\pi$
$919$	50.0000	1.64935	0.824674	+	0.565608i	$0.191359\pi$
$920$	0	0
$921$	0	0
$922$	40.0000	1.31733
$923$	0	0
$924$	0	0
$925$	0	0
$926$	−40.0000	−1.31448
$927$	−3.00000	−0.0985329
$928$	−8.00000	−0.262613
$929$	14.0000	0.459325	0.229663	−	0.973270i	$-0.426238\pi$
$929$	14.0000	0.459325	0.229663	+	0.973270i	$0.426238\pi$
$930$	0	0
$931$	−14.0000	−0.458831
$932$	18.0000	0.589610
$933$	0	0
$934$	−20.0000	−0.654420
$935$	0	0
$936$	0	0
$937$	−14.0000	−0.457360	−0.228680	−	0.973502i	$-0.573441\pi$
$937$	−14.0000	−0.457360	−0.228680	+	0.973502i	$0.573441\pi$
$938$	−12.0000	−0.391814
$939$	0	0
$940$	0	0
$941$	24.0000	0.782378	0.391189	−	0.920310i	$-0.372064\pi$
$941$	24.0000	0.782378	0.391189	+	0.920310i	$0.372064\pi$
$942$	0	0
$943$	8.00000	0.260516
$944$	4.00000	0.130189
$945$	0	0
$946$	−18.0000	−0.585230
$947$	38.0000	1.23483	0.617417	−	0.786636i	$-0.288179\pi$
$947$	38.0000	1.23483	0.617417	+	0.786636i	$0.288179\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	0	0
$952$	−12.0000	−0.388922
$953$	22.0000	0.712650	0.356325	−	0.934362i	$-0.384030\pi$
$953$	22.0000	0.712650	0.356325	+	0.934362i	$0.384030\pi$
$954$	−27.0000	−0.874157
$955$	0	0
$956$	−2.00000	−0.0646846
$957$	0	0
$958$	28.0000	0.904639
$959$	−36.0000	−1.16250
$960$	0	0
$961$	69.0000	2.22581
$962$	0	0
$963$	−54.0000	−1.74013
$964$	25.0000	0.805196
$965$	0	0
$966$	0	0
$967$	−33.0000	−1.06121	−0.530604	−	0.847620i	$-0.678035\pi$
$967$	−33.0000	−1.06121	−0.530604	+	0.847620i	$0.678035\pi$
$968$	−2.00000	−0.0642824
$969$	0	0
$970$	0	0
$971$	−49.0000	−1.57248	−0.786242	−	0.617918i	$-0.787976\pi$
$971$	−49.0000	−1.57248	−0.786242	+	0.617918i	$0.787976\pi$
$972$	0	0
$973$	15.0000	0.480878
$974$	19.0000	0.608799
$975$	0	0
$976$	−14.0000	−0.448129
$977$	−24.0000	−0.767828	−0.383914	−	0.923369i	$-0.625424\pi$
$977$	−24.0000	−0.767828	−0.383914	+	0.923369i	$0.625424\pi$
$978$	0	0
$979$	−3.00000	−0.0958804
$980$	0	0
$981$	−6.00000	−0.191565
$982$	−43.0000	−1.37219
$983$	−3.00000	−0.0956851	−0.0478426	−	0.998855i	$-0.515235\pi$
$983$	−3.00000	−0.0956851	−0.0478426	+	0.998855i	$0.515235\pi$
$984$	0	0
$985$	0	0
$986$	−32.0000	−1.01909
$987$	0	0
$988$	0	0
$989$	−24.0000	−0.763156
$990$	0	0
$991$	−10.0000	−0.317660	−0.158830	−	0.987306i	$-0.550772\pi$
$991$	−10.0000	−0.317660	−0.158830	+	0.987306i	$0.550772\pi$
$992$	10.0000	0.317500
$993$	0	0
$994$	18.0000	0.570925
$995$	0	0
$996$	0	0
$997$	11.0000	0.348373	0.174187	−	0.984713i	$-0.444270\pi$
$997$	11.0000	0.348373	0.174187	+	0.984713i	$0.444270\pi$
$998$	4.00000	0.126618
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8450.2.a.q.1.1		1
5.4	even	2		1690.2.a.c.1.1		1
13.4	even	6		650.2.e.b.601.1		2
13.10	even	6		650.2.e.b.451.1		2
13.12	even	2		8450.2.a.g.1.1		1
65.4	even	6		130.2.e.a.81.1	yes	2
65.9	even	6		1690.2.e.h.991.1		2
65.17	odd	12		650.2.o.d.549.1		4
65.19	odd	12		1690.2.l.e.361.2		4
65.23	odd	12		650.2.o.d.399.1		4
65.24	odd	12		1690.2.l.e.1161.2		4
65.29	even	6		1690.2.e.h.191.1		2
65.34	odd	4		1690.2.d.c.1351.1		2
65.43	odd	12		650.2.o.d.549.2		4
65.44	odd	4		1690.2.d.c.1351.2		2
65.49	even	6		130.2.e.a.61.1	✓	2
65.54	odd	12		1690.2.l.e.1161.1		4
65.59	odd	12		1690.2.l.e.361.1		4
65.62	odd	12		650.2.o.d.399.2		4
65.64	even	2		1690.2.a.h.1.1		1
195.134	odd	6		1170.2.i.i.991.1		2
195.179	odd	6		1170.2.i.i.451.1		2
260.179	odd	6		1040.2.q.h.321.1		2
260.199	odd	6		1040.2.q.h.81.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
130.2.e.a.61.1	✓	2	65.49	even	6
130.2.e.a.81.1	yes	2	65.4	even	6
650.2.e.b.451.1		2	13.10	even	6
650.2.e.b.601.1		2	13.4	even	6
650.2.o.d.399.1		4	65.23	odd	12
650.2.o.d.399.2		4	65.62	odd	12
650.2.o.d.549.1		4	65.17	odd	12
650.2.o.d.549.2		4	65.43	odd	12
1040.2.q.h.81.1		2	260.199	odd	6
1040.2.q.h.321.1		2	260.179	odd	6
1170.2.i.i.451.1		2	195.179	odd	6
1170.2.i.i.991.1		2	195.134	odd	6
1690.2.a.c.1.1		1	5.4	even	2
1690.2.a.h.1.1		1	65.64	even	2
1690.2.d.c.1351.1		2	65.34	odd	4
1690.2.d.c.1351.2		2	65.44	odd	4
1690.2.e.h.191.1		2	65.29	even	6
1690.2.e.h.991.1		2	65.9	even	6
1690.2.l.e.361.1		4	65.59	odd	12
1690.2.l.e.361.2		4	65.19	odd	12
1690.2.l.e.1161.1		4	65.54	odd	12
1690.2.l.e.1161.2		4	65.24	odd	12
8450.2.a.g.1.1		1	13.12	even	2
8450.2.a.q.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:	\( N \)	\(=\)	\( 8450 = 2 \cdot 5^{2} \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8450.a (trivial)

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