LMFDB - Embedded newform 5850.2.e.v.5149.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [5850,2,Mod(5149,5850)] 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("5850.5149"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(5850, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 5850 = 2 \cdot 3^{2} \cdot 5^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5850.e (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [2,0,0,-2,0,0,0,0,0,0,4,0,0,2,0,2,0,0,-12] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(19)] == traces)

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

Self dual:	no
Analytic conductor:	$46.7124851824$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(i)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 1 $ x^2 + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 26)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			5149.1
Root			$1.00000i$ of defining polynomial
Character	$\chi$	$=$	5850.5149
Dual form			5850.2.e.v.5149.2

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{2} -1.00000 q^{4} +1.00000i q^{7} +1.00000i q^{8} +2.00000 q^{11} +1.00000i q^{13} +1.00000 q^{14} +1.00000 q^{16} +3.00000i q^{17} -6.00000 q^{19} -2.00000i q^{22} -4.00000i q^{23} +1.00000 q^{26} -1.00000i q^{28} +2.00000 q^{29} +4.00000 q^{31} -1.00000i q^{32} +3.00000 q^{34} +3.00000i q^{37} +6.00000i q^{38} +5.00000i q^{43} -2.00000 q^{44} -4.00000 q^{46} -13.0000i q^{47} +6.00000 q^{49} -1.00000i q^{52} +12.0000i q^{53} -1.00000 q^{56} -2.00000i q^{58} -10.0000 q^{59} -8.00000 q^{61} -4.00000i q^{62} -1.00000 q^{64} -2.00000i q^{67} -3.00000i q^{68} +5.00000 q^{71} +10.0000i q^{73} +3.00000 q^{74} +6.00000 q^{76} +2.00000i q^{77} +4.00000 q^{79} +5.00000 q^{86} +2.00000i q^{88} +6.00000 q^{89} -1.00000 q^{91} +4.00000i q^{92} -13.0000 q^{94} +14.0000i q^{97} -6.00000i q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{4} + 4 q^{11} + 2 q^{14} + 2 q^{16} - 12 q^{19} + 2 q^{26} + 4 q^{29} + 8 q^{31} + 6 q^{34} - 4 q^{44} - 8 q^{46} + 12 q^{49} - 2 q^{56} - 20 q^{59} - 16 q^{61} - 2 q^{64} + 10 q^{71} + 6 q^{74}+ \cdots - 26 q^{94}+O(q^{100}) $ 2 * q - 2 * q^4 + 4 * q^11 + 2 * q^14 + 2 * q^16 - 12 * q^19 + 2 * q^26 + 4 * q^29 + 8 * q^31 + 6 * q^34 - 4 * q^44 - 8 * q^46 + 12 * q^49 - 2 * q^56 - 20 * q^59 - 16 * q^61 - 2 * q^64 + 10 * q^71 + 6 * q^74 + 12 * q^76 + 8 * q^79 + 10 * q^86 + 12 * q^89 - 2 * q^91 - 26 * q^94

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/5850\mathbb{Z}\right)^\times$.

$n$	$2251$	$3251$	$3277$
$\chi(n)$	$1$	$1$	$-1$

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 $

$2$	− 1.00000i	− 0.707107i
$2$	− 1.00000i	− 0.707107i
$3$	0	0
$3$	0	0
$4$	−1.00000	−0.500000
$5$	0	0
$5$	0	0
$6$	0	0
$7$	1.00000i	0.377964i	0.981981	+	0.188982i	$0.0605189\pi$
$7$	1.00000i	0.377964i	−0.981981	+	0.188982i	$0.939481\pi$
$8$	1.00000i	0.353553i
$9$	0	0
$10$	0	0
$11$	2.00000	0.603023	0.301511	−	0.953463i	$-0.402509\pi$
$11$	2.00000	0.603023	0.301511	+	0.953463i	$0.402509\pi$
$12$	0	0
$13$	1.00000i	0.277350i
$13$	1.00000i	0.277350i
$14$	1.00000	0.267261
$15$	0	0
$16$	1.00000	0.250000
$17$	3.00000i	0.727607i	0.931476	+	0.363803i	$0.118522\pi$
$17$	3.00000i	0.727607i	−0.931476	+	0.363803i	$0.881478\pi$
$18$	0	0
$19$	−6.00000	−1.37649	−0.688247	−	0.725476i	$-0.741620\pi$
$19$	−6.00000	−1.37649	−0.688247	+	0.725476i	$0.741620\pi$
$20$	0	0
$21$	0	0
$22$	− 2.00000i	− 0.426401i
$23$	− 4.00000i	− 0.834058i	−0.908893	−	0.417029i	$-0.863071\pi$
$23$	− 4.00000i	− 0.834058i	0.908893	−	0.417029i	$-0.136929\pi$
$24$	0	0
$25$	0	0
$26$	1.00000	0.196116
$27$	0	0
$28$	− 1.00000i	− 0.188982i
$29$	2.00000	0.371391	0.185695	−	0.982607i	$-0.440546\pi$
$29$	2.00000	0.371391	0.185695	+	0.982607i	$0.440546\pi$
$30$	0	0
$31$	4.00000	0.718421	0.359211	−	0.933257i	$-0.383046\pi$
$31$	4.00000	0.718421	0.359211	+	0.933257i	$0.383046\pi$
$32$	− 1.00000i	− 0.176777i
$33$	0	0
$34$	3.00000	0.514496
$35$	0	0
$36$	0	0
$37$	3.00000i	0.493197i	0.969118	+	0.246598i	$0.0793129\pi$
$37$	3.00000i	0.493197i	−0.969118	+	0.246598i	$0.920687\pi$
$38$	6.00000i	0.973329i
$39$	0	0
$40$	0	0
$41$	0	0		−	1.00000i	$-0.5\pi$
$41$	0	0			1.00000i	$0.5\pi$
$42$	0	0
$43$	5.00000i	0.762493i	0.924473	+	0.381246i	$0.124505\pi$
$43$	5.00000i	0.762493i	−0.924473	+	0.381246i	$0.875495\pi$
$44$	−2.00000	−0.301511
$45$	0	0
$46$	−4.00000	−0.589768
$47$	− 13.0000i	− 1.89624i	−0.317905	−	0.948122i	$-0.602979\pi$
$47$	− 13.0000i	− 1.89624i	0.317905	−	0.948122i	$-0.397021\pi$
$48$	0	0
$49$	6.00000	0.857143
$50$	0	0
$51$	0	0
$52$	− 1.00000i	− 0.138675i
$53$	12.0000i	1.64833i	0.566352	+	0.824163i	$0.308354\pi$
$53$	12.0000i	1.64833i	−0.566352	+	0.824163i	$0.691646\pi$
$54$	0	0
$55$	0	0
$56$	−1.00000	−0.133631
$57$	0	0
$58$	− 2.00000i	− 0.262613i
$59$	−10.0000	−1.30189	−0.650945	−	0.759125i	$-0.725627\pi$
$59$	−10.0000	−1.30189	−0.650945	+	0.759125i	$0.725627\pi$
$60$	0	0
$61$	−8.00000	−1.02430	−0.512148	−	0.858898i	$-0.671150\pi$
$61$	−8.00000	−1.02430	−0.512148	+	0.858898i	$0.671150\pi$
$62$	− 4.00000i	− 0.508001i
$63$	0	0
$64$	−1.00000	−0.125000
$65$	0	0
$66$	0	0
$67$	− 2.00000i	− 0.244339i	−0.992509	−	0.122169i	$-0.961015\pi$
$67$	− 2.00000i	− 0.244339i	0.992509	−	0.122169i	$-0.0389851\pi$
$68$	− 3.00000i	− 0.363803i
$69$	0	0
$70$	0	0
$71$	5.00000	0.593391	0.296695	−	0.954972i	$-0.404115\pi$
$71$	5.00000	0.593391	0.296695	+	0.954972i	$0.404115\pi$
$72$	0	0
$73$	10.0000i	1.17041i	0.810885	+	0.585206i	$0.198986\pi$
$73$	10.0000i	1.17041i	−0.810885	+	0.585206i	$0.801014\pi$
$74$	3.00000	0.348743
$75$	0	0
$76$	6.00000	0.688247
$77$	2.00000i	0.227921i
$78$	0	0
$79$	4.00000	0.450035	0.225018	−	0.974355i	$-0.427756\pi$
$79$	4.00000	0.450035	0.225018	+	0.974355i	$0.427756\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	0	0	1.00000			$0$
$83$	0	0	−1.00000			$\pi$
$84$	0	0
$85$	0	0
$86$	5.00000	0.539164
$87$	0	0
$88$	2.00000i	0.213201i
$89$	6.00000	0.635999	0.317999	−	0.948091i	$-0.396989\pi$
$89$	6.00000	0.635999	0.317999	+	0.948091i	$0.396989\pi$
$90$	0	0
$91$	−1.00000	−0.104828
$92$	4.00000i	0.417029i
$93$	0	0
$94$	−13.0000	−1.34085
$95$	0	0
$96$	0	0
$97$	14.0000i	1.42148i	0.703452	+	0.710742i	$0.251641\pi$
$97$	14.0000i	1.42148i	−0.703452	+	0.710742i	$0.748359\pi$
$98$	− 6.00000i	− 0.606092i
$99$	0	0
$100$	0	0
$101$	−4.00000	−0.398015	−0.199007	−	0.979998i	$-0.563772\pi$
$101$	−4.00000	−0.398015	−0.199007	+	0.979998i	$0.563772\pi$
$102$	0	0
$103$	8.00000i	0.788263i	0.919054	+	0.394132i	$0.128955\pi$
$103$	8.00000i	0.788263i	−0.919054	+	0.394132i	$0.871045\pi$
$104$	−1.00000	−0.0980581
$105$	0	0
$106$	12.0000	1.16554
$107$	4.00000i	0.386695i	0.981130	+	0.193347i	$0.0619344\pi$
$107$	4.00000i	0.386695i	−0.981130	+	0.193347i	$0.938066\pi$
$108$	0	0
$109$	−19.0000	−1.81987	−0.909935	−	0.414751i	$-0.863869\pi$
$109$	−19.0000	−1.81987	−0.909935	+	0.414751i	$0.863869\pi$
$110$	0	0
$111$	0	0
$112$	1.00000i	0.0944911i
$113$	2.00000i	0.188144i	0.995565	+	0.0940721i	$0.0299884\pi$
$113$	2.00000i	0.188144i	−0.995565	+	0.0940721i	$0.970012\pi$
$114$	0	0
$115$	0	0
$116$	−2.00000	−0.185695
$117$	0	0
$118$	10.0000i	0.920575i
$119$	−3.00000	−0.275010
$120$	0	0
$121$	−7.00000	−0.636364
$122$	8.00000i	0.724286i
$123$	0	0
$124$	−4.00000	−0.359211
$125$	0	0
$126$	0	0
$127$	16.0000i	1.41977i	0.704317	+	0.709885i	$0.251253\pi$
$127$	16.0000i	1.41977i	−0.704317	+	0.709885i	$0.748747\pi$
$128$	1.00000i	0.0883883i
$129$	0	0
$130$	0	0
$131$	1.00000	0.0873704	0.0436852	−	0.999045i	$-0.486090\pi$
$131$	1.00000	0.0873704	0.0436852	+	0.999045i	$0.486090\pi$
$132$	0	0
$133$	− 6.00000i	− 0.520266i
$134$	−2.00000	−0.172774
$135$	0	0
$136$	−3.00000	−0.257248
$137$	− 12.0000i	− 1.02523i	−0.858619	−	0.512615i	$-0.828677\pi$
$137$	− 12.0000i	− 1.02523i	0.858619	−	0.512615i	$-0.171323\pi$
$138$	0	0
$139$	−7.00000	−0.593732	−0.296866	−	0.954919i	$-0.595942\pi$
$139$	−7.00000	−0.593732	−0.296866	+	0.954919i	$0.595942\pi$
$140$	0	0
$141$	0	0
$142$	− 5.00000i	− 0.419591i
$143$	2.00000i	0.167248i
$144$	0	0
$145$	0	0
$146$	10.0000	0.827606
$147$	0	0
$148$	− 3.00000i	− 0.246598i
$149$	−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
$151$	−9.00000	−0.732410	−0.366205	−	0.930534i	$-0.619343\pi$
$151$	−9.00000	−0.732410	−0.366205	+	0.930534i	$0.619343\pi$
$152$	− 6.00000i	− 0.486664i
$153$	0	0
$154$	2.00000	0.161165
$155$	0	0
$156$	0	0
$157$	− 10.0000i	− 0.798087i	−0.916932	−	0.399043i	$-0.869342\pi$
$157$	− 10.0000i	− 0.798087i	0.916932	−	0.399043i	$-0.130658\pi$
$158$	− 4.00000i	− 0.318223i
$159$	0	0
$160$	0	0
$161$	4.00000	0.315244
$162$	0	0
$163$	4.00000i	0.313304i	0.987654	+	0.156652i	$0.0500701\pi$
$163$	4.00000i	0.313304i	−0.987654	+	0.156652i	$0.949930\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	0	0	1.00000			$0$
$167$	0	0	−1.00000			$\pi$
$168$	0	0
$169$	−1.00000	−0.0769231
$170$	0	0
$171$	0	0
$172$	− 5.00000i	− 0.381246i
$173$	20.0000i	1.52057i	0.649589	+	0.760286i	$0.274941\pi$
$173$	20.0000i	1.52057i	−0.649589	+	0.760286i	$0.725059\pi$
$174$	0	0
$175$	0	0
$176$	2.00000	0.150756
$177$	0	0
$178$	− 6.00000i	− 0.449719i
$179$	−9.00000	−0.672692	−0.336346	−	0.941739i	$-0.609191\pi$
$179$	−9.00000	−0.672692	−0.336346	+	0.941739i	$0.609191\pi$
$180$	0	0
$181$	0	0		−	1.00000i	$-0.5\pi$
$181$	0	0			1.00000i	$0.5\pi$
$182$	1.00000i	0.0741249i
$183$	0	0
$184$	4.00000	0.294884
$185$	0	0
$186$	0	0
$187$	6.00000i	0.438763i
$188$	13.0000i	0.948122i
$189$	0	0
$190$	0	0
$191$	−10.0000	−0.723575	−0.361787	−	0.932261i	$-0.617833\pi$
$191$	−10.0000	−0.723575	−0.361787	+	0.932261i	$0.617833\pi$
$192$	0	0
$193$	16.0000i	1.15171i	0.817554	+	0.575853i	$0.195330\pi$
$193$	16.0000i	1.15171i	−0.817554	+	0.575853i	$0.804670\pi$
$194$	14.0000	1.00514
$195$	0	0
$196$	−6.00000	−0.428571
$197$	− 9.00000i	− 0.641223i	−0.947211	−	0.320612i	$-0.896112\pi$
$197$	− 9.00000i	− 0.641223i	0.947211	−	0.320612i	$-0.103888\pi$
$198$	0	0
$199$	10.0000	0.708881	0.354441	−	0.935079i	$-0.384671\pi$
$199$	10.0000	0.708881	0.354441	+	0.935079i	$0.384671\pi$
$200$	0	0
$201$	0	0
$202$	4.00000i	0.281439i
$203$	2.00000i	0.140372i
$204$	0	0
$205$	0	0
$206$	8.00000	0.557386
$207$	0	0
$208$	1.00000i	0.0693375i
$209$	−12.0000	−0.830057
$210$	0	0
$211$	23.0000	1.58339	0.791693	−	0.610920i	$-0.209200\pi$
$211$	23.0000	1.58339	0.791693	+	0.610920i	$0.209200\pi$
$212$	− 12.0000i	− 0.824163i
$213$	0	0
$214$	4.00000	0.273434
$215$	0	0
$216$	0	0
$217$	4.00000i	0.271538i
$218$	19.0000i	1.28684i
$219$	0	0
$220$	0	0
$221$	−3.00000	−0.201802
$222$	0	0
$223$	21.0000i	1.40626i	0.711059	+	0.703132i	$0.248216\pi$
$223$	21.0000i	1.40626i	−0.711059	+	0.703132i	$0.751784\pi$
$224$	1.00000	0.0668153
$225$	0	0
$226$	2.00000	0.133038
$227$	24.0000i	1.59294i	0.604681	+	0.796468i	$0.293301\pi$
$227$	24.0000i	1.59294i	−0.604681	+	0.796468i	$0.706699\pi$
$228$	0	0
$229$	15.0000	0.991228	0.495614	−	0.868543i	$-0.334943\pi$
$229$	15.0000	0.991228	0.495614	+	0.868543i	$0.334943\pi$
$230$	0	0
$231$	0	0
$232$	2.00000i	0.131306i
$233$	− 11.0000i	− 0.720634i	−0.932830	−	0.360317i	$-0.882669\pi$
$233$	− 11.0000i	− 0.720634i	0.932830	−	0.360317i	$-0.117331\pi$
$234$	0	0
$235$	0	0
$236$	10.0000	0.650945
$237$	0	0
$238$	3.00000i	0.194461i
$239$	9.00000	0.582162	0.291081	−	0.956698i	$-0.405985\pi$
$239$	9.00000	0.582162	0.291081	+	0.956698i	$0.405985\pi$
$240$	0	0
$241$	18.0000	1.15948	0.579741	−	0.814801i	$-0.303154\pi$
$241$	18.0000	1.15948	0.579741	+	0.814801i	$0.303154\pi$
$242$	7.00000i	0.449977i
$243$	0	0
$244$	8.00000	0.512148
$245$	0	0
$246$	0	0
$247$	− 6.00000i	− 0.381771i
$248$	4.00000i	0.254000i
$249$	0	0
$250$	0	0
$251$	0	0		−	1.00000i	$-0.5\pi$
$251$	0	0			1.00000i	$0.5\pi$
$252$	0	0
$253$	− 8.00000i	− 0.502956i
$254$	16.0000	1.00393
$255$	0	0
$256$	1.00000	0.0625000
$257$	15.0000i	0.935674i	0.883815	+	0.467837i	$0.154967\pi$
$257$	15.0000i	0.935674i	−0.883815	+	0.467837i	$0.845033\pi$
$258$	0	0
$259$	−3.00000	−0.186411
$260$	0	0
$261$	0	0
$262$	− 1.00000i	− 0.0617802i
$263$	12.0000i	0.739952i	0.929041	+	0.369976i	$0.120634\pi$
$263$	12.0000i	0.739952i	−0.929041	+	0.369976i	$0.879366\pi$
$264$	0	0
$265$	0	0
$266$	−6.00000	−0.367884
$267$	0	0
$268$	2.00000i	0.122169i
$269$	−24.0000	−1.46331	−0.731653	−	0.681677i	$-0.761251\pi$
$269$	−24.0000	−1.46331	−0.731653	+	0.681677i	$0.761251\pi$
$270$	0	0
$271$	13.0000	0.789694	0.394847	−	0.918747i	$-0.370798\pi$
$271$	13.0000	0.789694	0.394847	+	0.918747i	$0.370798\pi$
$272$	3.00000i	0.181902i
$273$	0	0
$274$	−12.0000	−0.724947
$275$	0	0
$276$	0	0
$277$	12.0000i	0.721010i	0.932757	+	0.360505i	$0.117396\pi$
$277$	12.0000i	0.721010i	−0.932757	+	0.360505i	$0.882604\pi$
$278$	7.00000i	0.419832i
$279$	0	0
$280$	0	0
$281$	26.0000	1.55103	0.775515	−	0.631329i	$-0.217490\pi$
$281$	26.0000	1.55103	0.775515	+	0.631329i	$0.217490\pi$
$282$	0	0
$283$	− 4.00000i	− 0.237775i	−0.992908	−	0.118888i	$-0.962067\pi$
$283$	− 4.00000i	− 0.237775i	0.992908	−	0.118888i	$-0.0379328\pi$
$284$	−5.00000	−0.296695
$285$	0	0
$286$	2.00000	0.118262
$287$	0	0
$288$	0	0
$289$	8.00000	0.470588
$290$	0	0
$291$	0	0
$292$	− 10.0000i	− 0.585206i
$293$	7.00000i	0.408944i	0.978872	+	0.204472i	$0.0655478\pi$
$293$	7.00000i	0.408944i	−0.978872	+	0.204472i	$0.934452\pi$
$294$	0	0
$295$	0	0
$296$	−3.00000	−0.174371
$297$	0	0
$298$	18.0000i	1.04271i
$299$	4.00000	0.231326
$300$	0	0
$301$	−5.00000	−0.288195
$302$	9.00000i	0.517892i
$303$	0	0
$304$	−6.00000	−0.344124
$305$	0	0
$306$	0	0
$307$	14.0000i	0.799022i	0.916728	+	0.399511i	$0.130820\pi$
$307$	14.0000i	0.799022i	−0.916728	+	0.399511i	$0.869180\pi$
$308$	− 2.00000i	− 0.113961i
$309$	0	0
$310$	0	0
$311$	−18.0000	−1.02069	−0.510343	−	0.859971i	$-0.670482\pi$
$311$	−18.0000	−1.02069	−0.510343	+	0.859971i	$0.670482\pi$
$312$	0	0
$313$	1.00000i	0.0565233i	0.999601	+	0.0282617i	$0.00899717\pi$
$313$	1.00000i	0.0565233i	−0.999601	+	0.0282617i	$0.991003\pi$
$314$	−10.0000	−0.564333
$315$	0	0
$316$	−4.00000	−0.225018
$317$	18.0000i	1.01098i	0.862832	+	0.505490i	$0.168688\pi$
$317$	18.0000i	1.01098i	−0.862832	+	0.505490i	$0.831312\pi$
$318$	0	0
$319$	4.00000	0.223957
$320$	0	0
$321$	0	0
$322$	− 4.00000i	− 0.222911i
$323$	− 18.0000i	− 1.00155i
$324$	0	0
$325$	0	0
$326$	4.00000	0.221540
$327$	0	0
$328$	0	0
$329$	13.0000	0.716713
$330$	0	0
$331$	−4.00000	−0.219860	−0.109930	−	0.993939i	$-0.535063\pi$
$331$	−4.00000	−0.219860	−0.109930	+	0.993939i	$0.535063\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	0	0
$336$	0	0
$337$	23.0000i	1.25289i	0.779466	+	0.626445i	$0.215491\pi$
$337$	23.0000i	1.25289i	−0.779466	+	0.626445i	$0.784509\pi$
$338$	1.00000i	0.0543928i
$339$	0	0
$340$	0	0
$341$	8.00000	0.433224
$342$	0	0
$343$	13.0000i	0.701934i
$344$	−5.00000	−0.269582
$345$	0	0
$346$	20.0000	1.07521
$347$	9.00000i	0.483145i	0.970383	+	0.241573i	$0.0776632\pi$
$347$	9.00000i	0.483145i	−0.970383	+	0.241573i	$0.922337\pi$
$348$	0	0
$349$	−7.00000	−0.374701	−0.187351	−	0.982293i	$-0.559990\pi$
$349$	−7.00000	−0.374701	−0.187351	+	0.982293i	$0.559990\pi$
$350$	0	0
$351$	0	0
$352$	− 2.00000i	− 0.106600i
$353$	4.00000i	0.212899i	0.994318	+	0.106449i	$0.0339482\pi$
$353$	4.00000i	0.212899i	−0.994318	+	0.106449i	$0.966052\pi$
$354$	0	0
$355$	0	0
$356$	−6.00000	−0.317999
$357$	0	0
$358$	9.00000i	0.475665i
$359$	24.0000	1.26667	0.633336	−	0.773877i	$-0.281685\pi$
$359$	24.0000	1.26667	0.633336	+	0.773877i	$0.281685\pi$
$360$	0	0
$361$	17.0000	0.894737
$362$	0	0
$363$	0	0
$364$	1.00000	0.0524142
$365$	0	0
$366$	0	0
$367$	− 10.0000i	− 0.521996i	−0.965339	−	0.260998i	$-0.915948\pi$
$367$	− 10.0000i	− 0.521996i	0.965339	−	0.260998i	$-0.0840516\pi$
$368$	− 4.00000i	− 0.208514i
$369$	0	0
$370$	0	0
$371$	−12.0000	−0.623009
$372$	0	0
$373$	4.00000i	0.207112i	0.994624	+	0.103556i	$0.0330221\pi$
$373$	4.00000i	0.207112i	−0.994624	+	0.103556i	$0.966978\pi$
$374$	6.00000	0.310253
$375$	0	0
$376$	13.0000	0.670424
$377$	2.00000i	0.103005i
$378$	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
$381$	0	0
$382$	10.0000i	0.511645i
$383$	27.0000i	1.37964i	0.723983	+	0.689818i	$0.242309\pi$
$383$	27.0000i	1.37964i	−0.723983	+	0.689818i	$0.757691\pi$
$384$	0	0
$385$	0	0
$386$	16.0000	0.814379
$387$	0	0
$388$	− 14.0000i	− 0.710742i
$389$	−30.0000	−1.52106	−0.760530	−	0.649303i	$-0.775061\pi$
$389$	−30.0000	−1.52106	−0.760530	+	0.649303i	$0.775061\pi$
$390$	0	0
$391$	12.0000	0.606866
$392$	6.00000i	0.303046i
$393$	0	0
$394$	−9.00000	−0.453413
$395$	0	0
$396$	0	0
$397$	− 22.0000i	− 1.10415i	−0.833795	−	0.552074i	$-0.813837\pi$
$397$	− 22.0000i	− 1.10415i	0.833795	−	0.552074i	$-0.186163\pi$
$398$	− 10.0000i	− 0.501255i
$399$	0	0
$400$	0	0
$401$	−24.0000	−1.19850	−0.599251	−	0.800561i	$-0.704535\pi$
$401$	−24.0000	−1.19850	−0.599251	+	0.800561i	$0.704535\pi$
$402$	0	0
$403$	4.00000i	0.199254i
$404$	4.00000	0.199007
$405$	0	0
$406$	2.00000	0.0992583
$407$	6.00000i	0.297409i
$408$	0	0
$409$	−4.00000	−0.197787	−0.0988936	−	0.995098i	$-0.531530\pi$
$409$	−4.00000	−0.197787	−0.0988936	+	0.995098i	$0.531530\pi$
$410$	0	0
$411$	0	0
$412$	− 8.00000i	− 0.394132i
$413$	− 10.0000i	− 0.492068i
$414$	0	0
$415$	0	0
$416$	1.00000	0.0490290
$417$	0	0
$418$	12.0000i	0.586939i
$419$	21.0000	1.02592	0.512959	−	0.858413i	$-0.328549\pi$
$419$	21.0000	1.02592	0.512959	+	0.858413i	$0.328549\pi$
$420$	0	0
$421$	−5.00000	−0.243685	−0.121843	−	0.992549i	$-0.538880\pi$
$421$	−5.00000	−0.243685	−0.121843	+	0.992549i	$0.538880\pi$
$422$	− 23.0000i	− 1.11962i
$423$	0	0
$424$	−12.0000	−0.582772
$425$	0	0
$426$	0	0
$427$	− 8.00000i	− 0.387147i
$428$	− 4.00000i	− 0.193347i
$429$	0	0
$430$	0	0
$431$	−33.0000	−1.58955	−0.794777	−	0.606902i	$-0.792412\pi$
$431$	−33.0000	−1.58955	−0.794777	+	0.606902i	$0.792412\pi$
$432$	0	0
$433$	− 7.00000i	− 0.336399i	−0.985753	−	0.168199i	$-0.946205\pi$
$433$	− 7.00000i	− 0.336399i	0.985753	−	0.168199i	$-0.0537952\pi$
$434$	4.00000	0.192006
$435$	0	0
$436$	19.0000	0.909935
$437$	24.0000i	1.14808i
$438$	0	0
$439$	22.0000	1.05000	0.525001	−	0.851101i	$-0.324065\pi$
$439$	22.0000	1.05000	0.525001	+	0.851101i	$0.324065\pi$
$440$	0	0
$441$	0	0
$442$	3.00000i	0.142695i
$443$	− 39.0000i	− 1.85295i	−0.376361	−	0.926473i	$-0.622825\pi$
$443$	− 39.0000i	− 1.85295i	0.376361	−	0.926473i	$-0.377175\pi$
$444$	0	0
$445$	0	0
$446$	21.0000	0.994379
$447$	0	0
$448$	− 1.00000i	− 0.0472456i
$449$	−26.0000	−1.22702	−0.613508	−	0.789689i	$-0.710242\pi$
$449$	−26.0000	−1.22702	−0.613508	+	0.789689i	$0.710242\pi$
$450$	0	0
$451$	0	0
$452$	− 2.00000i	− 0.0940721i
$453$	0	0
$454$	24.0000	1.12638
$455$	0	0
$456$	0	0
$457$	10.0000i	0.467780i	0.972263	+	0.233890i	$0.0751456\pi$
$457$	10.0000i	0.467780i	−0.972263	+	0.233890i	$0.924854\pi$
$458$	− 15.0000i	− 0.700904i
$459$	0	0
$460$	0	0
$461$	21.0000	0.978068	0.489034	−	0.872265i	$-0.337349\pi$
$461$	21.0000	0.978068	0.489034	+	0.872265i	$0.337349\pi$
$462$	0	0
$463$	− 16.0000i	− 0.743583i	−0.928316	−	0.371792i	$-0.878744\pi$
$463$	− 16.0000i	− 0.743583i	0.928316	−	0.371792i	$-0.121256\pi$
$464$	2.00000	0.0928477
$465$	0	0
$466$	−11.0000	−0.509565
$467$	− 20.0000i	− 0.925490i	−0.886492	−	0.462745i	$-0.846865\pi$
$467$	− 20.0000i	− 0.925490i	0.886492	−	0.462745i	$-0.153135\pi$
$468$	0	0
$469$	2.00000	0.0923514
$470$	0	0
$471$	0	0
$472$	− 10.0000i	− 0.460287i
$473$	10.0000i	0.459800i
$474$	0	0
$475$	0	0
$476$	3.00000	0.137505
$477$	0	0
$478$	− 9.00000i	− 0.411650i
$479$	−3.00000	−0.137073	−0.0685367	−	0.997649i	$-0.521833\pi$
$479$	−3.00000	−0.137073	−0.0685367	+	0.997649i	$0.521833\pi$
$480$	0	0
$481$	−3.00000	−0.136788
$482$	− 18.0000i	− 0.819878i
$483$	0	0
$484$	7.00000	0.318182
$485$	0	0
$486$	0	0
$487$	− 16.0000i	− 0.725029i	−0.931978	−	0.362515i	$-0.881918\pi$
$487$	− 16.0000i	− 0.725029i	0.931978	−	0.362515i	$-0.118082\pi$
$488$	− 8.00000i	− 0.362143i
$489$	0	0
$490$	0	0
$491$	5.00000	0.225647	0.112823	−	0.993615i	$-0.464011\pi$
$491$	5.00000	0.225647	0.112823	+	0.993615i	$0.464011\pi$
$492$	0	0
$493$	6.00000i	0.270226i
$494$	−6.00000	−0.269953
$495$	0	0
$496$	4.00000	0.179605
$497$	5.00000i	0.224281i
$498$	0	0
$499$	32.0000	1.43252	0.716258	−	0.697835i	$-0.245853\pi$
$499$	32.0000	1.43252	0.716258	+	0.697835i	$0.245853\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	− 14.0000i	− 0.624229i	−0.950044	−	0.312115i	$-0.898963\pi$
$503$	− 14.0000i	− 0.624229i	0.950044	−	0.312115i	$-0.101037\pi$
$504$	0	0
$505$	0	0
$506$	−8.00000	−0.355643
$507$	0	0
$508$	− 16.0000i	− 0.709885i
$509$	34.0000	1.50702	0.753512	−	0.657434i	$-0.228358\pi$
$509$	34.0000	1.50702	0.753512	+	0.657434i	$0.228358\pi$
$510$	0	0
$511$	−10.0000	−0.442374
$512$	− 1.00000i	− 0.0441942i
$513$	0	0
$514$	15.0000	0.661622
$515$	0	0
$516$	0	0
$517$	− 26.0000i	− 1.14348i
$518$	3.00000i	0.131812i
$519$	0	0
$520$	0	0
$521$	−39.0000	−1.70862	−0.854311	−	0.519763i	$-0.826020\pi$
$521$	−39.0000	−1.70862	−0.854311	+	0.519763i	$0.826020\pi$
$522$	0	0
$523$	36.0000i	1.57417i	0.616844	+	0.787085i	$0.288411\pi$
$523$	36.0000i	1.57417i	−0.616844	+	0.787085i	$0.711589\pi$
$524$	−1.00000	−0.0436852
$525$	0	0
$526$	12.0000	0.523225
$527$	12.0000i	0.522728i
$528$	0	0
$529$	7.00000	0.304348
$530$	0	0
$531$	0	0
$532$	6.00000i	0.260133i
$533$	0	0
$534$	0	0
$535$	0	0
$536$	2.00000	0.0863868
$537$	0	0
$538$	24.0000i	1.03471i
$539$	12.0000	0.516877
$540$	0	0
$541$	17.0000	0.730887	0.365444	−	0.930834i	$-0.380917\pi$
$541$	17.0000	0.730887	0.365444	+	0.930834i	$0.380917\pi$
$542$	− 13.0000i	− 0.558398i
$543$	0	0
$544$	3.00000	0.128624
$545$	0	0
$546$	0	0
$547$	37.0000i	1.58201i	0.611812	+	0.791003i	$0.290441\pi$
$547$	37.0000i	1.58201i	−0.611812	+	0.791003i	$0.709559\pi$
$548$	12.0000i	0.512615i
$549$	0	0
$550$	0	0
$551$	−12.0000	−0.511217
$552$	0	0
$553$	4.00000i	0.170097i
$554$	12.0000	0.509831
$555$	0	0
$556$	7.00000	0.296866
$557$	− 33.0000i	− 1.39825i	−0.714997	−	0.699127i	$-0.753572\pi$
$557$	− 33.0000i	− 1.39825i	0.714997	−	0.699127i	$-0.246428\pi$
$558$	0	0
$559$	−5.00000	−0.211477
$560$	0	0
$561$	0	0
$562$	− 26.0000i	− 1.09674i
$563$	11.0000i	0.463595i	0.972764	+	0.231797i	$0.0744606\pi$
$563$	11.0000i	0.463595i	−0.972764	+	0.231797i	$0.925539\pi$
$564$	0	0
$565$	0	0
$566$	−4.00000	−0.168133
$567$	0	0
$568$	5.00000i	0.209795i
$569$	31.0000	1.29959	0.649794	−	0.760111i	$-0.274855\pi$
$569$	31.0000	1.29959	0.649794	+	0.760111i	$0.274855\pi$
$570$	0	0
$571$	33.0000	1.38101	0.690504	−	0.723329i	$-0.257389\pi$
$571$	33.0000	1.38101	0.690504	+	0.723329i	$0.257389\pi$
$572$	− 2.00000i	− 0.0836242i
$573$	0	0
$574$	0	0
$575$	0	0
$576$	0	0
$577$	18.0000i	0.749350i	0.927156	+	0.374675i	$0.122246\pi$
$577$	18.0000i	0.749350i	−0.927156	+	0.374675i	$0.877754\pi$
$578$	− 8.00000i	− 0.332756i
$579$	0	0
$580$	0	0
$581$	0	0
$582$	0	0
$583$	24.0000i	0.993978i
$584$	−10.0000	−0.413803
$585$	0	0
$586$	7.00000	0.289167
$587$	28.0000i	1.15568i	0.816149	+	0.577842i	$0.196105\pi$
$587$	28.0000i	1.15568i	−0.816149	+	0.577842i	$0.803895\pi$
$588$	0	0
$589$	−24.0000	−0.988903
$590$	0	0
$591$	0	0
$592$	3.00000i	0.123299i
$593$	− 22.0000i	− 0.903432i	−0.892162	−	0.451716i	$-0.850812\pi$
$593$	− 22.0000i	− 0.903432i	0.892162	−	0.451716i	$-0.149188\pi$
$594$	0	0
$595$	0	0
$596$	18.0000	0.737309
$597$	0	0
$598$	− 4.00000i	− 0.163572i
$599$	−2.00000	−0.0817178	−0.0408589	−	0.999165i	$-0.513009\pi$
$599$	−2.00000	−0.0817178	−0.0408589	+	0.999165i	$0.513009\pi$
$600$	0	0
$601$	−35.0000	−1.42768	−0.713840	−	0.700309i	$-0.753046\pi$
$601$	−35.0000	−1.42768	−0.713840	+	0.700309i	$0.753046\pi$
$602$	5.00000i	0.203785i
$603$	0	0
$604$	9.00000	0.366205
$605$	0	0
$606$	0	0
$607$	6.00000i	0.243532i	0.992559	+	0.121766i	$0.0388558\pi$
$607$	6.00000i	0.243532i	−0.992559	+	0.121766i	$0.961144\pi$
$608$	6.00000i	0.243332i
$609$	0	0
$610$	0	0
$611$	13.0000	0.525924
$612$	0	0
$613$	− 26.0000i	− 1.05013i	−0.851062	−	0.525065i	$-0.824041\pi$
$613$	− 26.0000i	− 1.05013i	0.851062	−	0.525065i	$-0.175959\pi$
$614$	14.0000	0.564994
$615$	0	0
$616$	−2.00000	−0.0805823
$617$	− 16.0000i	− 0.644136i	−0.946717	−	0.322068i	$-0.895622\pi$
$617$	− 16.0000i	− 0.644136i	0.946717	−	0.322068i	$-0.104378\pi$
$618$	0	0
$619$	−4.00000	−0.160774	−0.0803868	−	0.996764i	$-0.525616\pi$
$619$	−4.00000	−0.160774	−0.0803868	+	0.996764i	$0.525616\pi$
$620$	0	0
$621$	0	0
$622$	18.0000i	0.721734i
$623$	6.00000i	0.240385i
$624$	0	0
$625$	0	0
$626$	1.00000	0.0399680
$627$	0	0
$628$	10.0000i	0.399043i
$629$	−9.00000	−0.358854
$630$	0	0
$631$	−5.00000	−0.199047	−0.0995234	−	0.995035i	$-0.531732\pi$
$631$	−5.00000	−0.199047	−0.0995234	+	0.995035i	$0.531732\pi$
$632$	4.00000i	0.159111i
$633$	0	0
$634$	18.0000	0.714871
$635$	0	0
$636$	0	0
$637$	6.00000i	0.237729i
$638$	− 4.00000i	− 0.158362i
$639$	0	0
$640$	0	0
$641$	2.00000	0.0789953	0.0394976	−	0.999220i	$-0.487424\pi$
$641$	2.00000	0.0789953	0.0394976	+	0.999220i	$0.487424\pi$
$642$	0	0
$643$	− 14.0000i	− 0.552106i	−0.961142	−	0.276053i	$-0.910973\pi$
$643$	− 14.0000i	− 0.552106i	0.961142	−	0.276053i	$-0.0890266\pi$
$644$	−4.00000	−0.157622
$645$	0	0
$646$	−18.0000	−0.708201
$647$	38.0000i	1.49393i	0.664861	+	0.746967i	$0.268491\pi$
$647$	38.0000i	1.49393i	−0.664861	+	0.746967i	$0.731509\pi$
$648$	0	0
$649$	−20.0000	−0.785069
$650$	0	0
$651$	0	0
$652$	− 4.00000i	− 0.156652i
$653$	24.0000i	0.939193i	0.882881	+	0.469596i	$0.155601\pi$
$653$	24.0000i	0.939193i	−0.882881	+	0.469596i	$0.844399\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	0	0
$658$	− 13.0000i	− 0.506793i
$659$	−12.0000	−0.467454	−0.233727	−	0.972302i	$-0.575092\pi$
$659$	−12.0000	−0.467454	−0.233727	+	0.972302i	$0.575092\pi$
$660$	0	0
$661$	−10.0000	−0.388955	−0.194477	−	0.980907i	$-0.562301\pi$
$661$	−10.0000	−0.388955	−0.194477	+	0.980907i	$0.562301\pi$
$662$	4.00000i	0.155464i
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	− 8.00000i	− 0.309761i
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−16.0000	−0.617673
$672$	0	0
$673$	− 37.0000i	− 1.42625i	−0.701039	−	0.713123i	$-0.747280\pi$
$673$	− 37.0000i	− 1.42625i	0.701039	−	0.713123i	$-0.252720\pi$
$674$	23.0000	0.885927
$675$	0	0
$676$	1.00000	0.0384615
$677$	36.0000i	1.38359i	0.722093	+	0.691796i	$0.243180\pi$
$677$	36.0000i	1.38359i	−0.722093	+	0.691796i	$0.756820\pi$
$678$	0	0
$679$	−14.0000	−0.537271
$680$	0	0
$681$	0	0
$682$	− 8.00000i	− 0.306336i
$683$	− 44.0000i	− 1.68361i	−0.539779	−	0.841807i	$-0.681492\pi$
$683$	− 44.0000i	− 1.68361i	0.539779	−	0.841807i	$-0.318508\pi$
$684$	0	0
$685$	0	0
$686$	13.0000	0.496342
$687$	0	0
$688$	5.00000i	0.190623i
$689$	−12.0000	−0.457164
$690$	0	0
$691$	−8.00000	−0.304334	−0.152167	−	0.988355i	$-0.548625\pi$
$691$	−8.00000	−0.304334	−0.152167	+	0.988355i	$0.548625\pi$
$692$	− 20.0000i	− 0.760286i
$693$	0	0
$694$	9.00000	0.341635
$695$	0	0
$696$	0	0
$697$	0	0
$698$	7.00000i	0.264954i
$699$	0	0
$700$	0	0
$701$	12.0000	0.453234	0.226617	−	0.973984i	$-0.427233\pi$
$701$	12.0000	0.453234	0.226617	+	0.973984i	$0.427233\pi$
$702$	0	0
$703$	− 18.0000i	− 0.678883i
$704$	−2.00000	−0.0753778
$705$	0	0
$706$	4.00000	0.150542
$707$	− 4.00000i	− 0.150435i
$708$	0	0
$709$	−38.0000	−1.42712	−0.713560	−	0.700594i	$-0.752918\pi$
$709$	−38.0000	−1.42712	−0.713560	+	0.700594i	$0.752918\pi$
$710$	0	0
$711$	0	0
$712$	6.00000i	0.224860i
$713$	− 16.0000i	− 0.599205i
$714$	0	0
$715$	0	0
$716$	9.00000	0.336346
$717$	0	0
$718$	− 24.0000i	− 0.895672i
$719$	−22.0000	−0.820462	−0.410231	−	0.911982i	$-0.634552\pi$
$719$	−22.0000	−0.820462	−0.410231	+	0.911982i	$0.634552\pi$
$720$	0	0
$721$	−8.00000	−0.297936
$722$	− 17.0000i	− 0.632674i
$723$	0	0
$724$	0	0
$725$	0	0
$726$	0	0
$727$	− 14.0000i	− 0.519231i	−0.965712	−	0.259616i	$-0.916404\pi$
$727$	− 14.0000i	− 0.519231i	0.965712	−	0.259616i	$-0.0835959\pi$
$728$	− 1.00000i	− 0.0370625i
$729$	0	0
$730$	0	0
$731$	−15.0000	−0.554795
$732$	0	0
$733$	43.0000i	1.58824i	0.607760	+	0.794121i	$0.292068\pi$
$733$	43.0000i	1.58824i	−0.607760	+	0.794121i	$0.707932\pi$
$734$	−10.0000	−0.369107
$735$	0	0
$736$	−4.00000	−0.147442
$737$	− 4.00000i	− 0.147342i
$738$	0	0
$739$	−12.0000	−0.441427	−0.220714	−	0.975339i	$-0.570839\pi$
$739$	−12.0000	−0.441427	−0.220714	+	0.975339i	$0.570839\pi$
$740$	0	0
$741$	0	0
$742$	12.0000i	0.440534i
$743$	− 47.0000i	− 1.72426i	−0.506685	−	0.862131i	$-0.669129\pi$
$743$	− 47.0000i	− 1.72426i	0.506685	−	0.862131i	$-0.330871\pi$
$744$	0	0
$745$	0	0
$746$	4.00000	0.146450
$747$	0	0
$748$	− 6.00000i	− 0.219382i
$749$	−4.00000	−0.146157
$750$	0	0
$751$	24.0000	0.875772	0.437886	−	0.899030i	$-0.355727\pi$
$751$	24.0000	0.875772	0.437886	+	0.899030i	$0.355727\pi$
$752$	− 13.0000i	− 0.474061i
$753$	0	0
$754$	2.00000	0.0728357
$755$	0	0
$756$	0	0
$757$	− 12.0000i	− 0.436147i	−0.975932	−	0.218074i	$-0.930023\pi$
$757$	− 12.0000i	− 0.436147i	0.975932	−	0.218074i	$-0.0699773\pi$
$758$	16.0000i	0.581146i
$759$	0	0
$760$	0	0
$761$	−6.00000	−0.217500	−0.108750	−	0.994069i	$-0.534685\pi$
$761$	−6.00000	−0.217500	−0.108750	+	0.994069i	$0.534685\pi$
$762$	0	0
$763$	− 19.0000i	− 0.687846i
$764$	10.0000	0.361787
$765$	0	0
$766$	27.0000	0.975550
$767$	− 10.0000i	− 0.361079i
$768$	0	0
$769$	0	0		−	1.00000i	$-0.5\pi$
$769$	0	0			1.00000i	$0.5\pi$
$770$	0	0
$771$	0	0
$772$	− 16.0000i	− 0.575853i
$773$	11.0000i	0.395643i	0.980238	+	0.197821i	$0.0633866\pi$
$773$	11.0000i	0.395643i	−0.980238	+	0.197821i	$0.936613\pi$
$774$	0	0
$775$	0	0
$776$	−14.0000	−0.502571
$777$	0	0
$778$	30.0000i	1.07555i
$779$	0	0
$780$	0	0
$781$	10.0000	0.357828
$782$	− 12.0000i	− 0.429119i
$783$	0	0
$784$	6.00000	0.214286
$785$	0	0
$786$	0	0
$787$	32.0000i	1.14068i	0.821410	+	0.570338i	$0.193188\pi$
$787$	32.0000i	1.14068i	−0.821410	+	0.570338i	$0.806812\pi$
$788$	9.00000i	0.320612i
$789$	0	0
$790$	0	0
$791$	−2.00000	−0.0711118
$792$	0	0
$793$	− 8.00000i	− 0.284088i
$794$	−22.0000	−0.780751
$795$	0	0
$796$	−10.0000	−0.354441
$797$	42.0000i	1.48772i	0.668338	+	0.743858i	$0.267006\pi$
$797$	42.0000i	1.48772i	−0.668338	+	0.743858i	$0.732994\pi$
$798$	0	0
$799$	39.0000	1.37972
$800$	0	0
$801$	0	0
$802$	24.0000i	0.847469i
$803$	20.0000i	0.705785i
$804$	0	0
$805$	0	0
$806$	4.00000	0.140894
$807$	0	0
$808$	− 4.00000i	− 0.140720i
$809$	−9.00000	−0.316423	−0.158212	−	0.987405i	$-0.550573\pi$
$809$	−9.00000	−0.316423	−0.158212	+	0.987405i	$0.550573\pi$
$810$	0	0
$811$	−28.0000	−0.983213	−0.491606	−	0.870817i	$-0.663590\pi$
$811$	−28.0000	−0.983213	−0.491606	+	0.870817i	$0.663590\pi$
$812$	− 2.00000i	− 0.0701862i
$813$	0	0
$814$	6.00000	0.210300
$815$	0	0
$816$	0	0
$817$	− 30.0000i	− 1.04957i
$818$	4.00000i	0.139857i
$819$	0	0
$820$	0	0
$821$	25.0000	0.872506	0.436253	−	0.899824i	$-0.356305\pi$
$821$	25.0000	0.872506	0.436253	+	0.899824i	$0.356305\pi$
$822$	0	0
$823$	− 54.0000i	− 1.88232i	−0.337959	−	0.941161i	$-0.609737\pi$
$823$	− 54.0000i	− 1.88232i	0.337959	−	0.941161i	$-0.390263\pi$
$824$	−8.00000	−0.278693
$825$	0	0
$826$	−10.0000	−0.347945
$827$	− 30.0000i	− 1.04320i	−0.853189	−	0.521601i	$-0.825335\pi$
$827$	− 30.0000i	− 1.04320i	0.853189	−	0.521601i	$-0.174665\pi$
$828$	0	0
$829$	38.0000	1.31979	0.659897	−	0.751356i	$-0.270600\pi$
$829$	38.0000	1.31979	0.659897	+	0.751356i	$0.270600\pi$
$830$	0	0
$831$	0	0
$832$	− 1.00000i	− 0.0346688i
$833$	18.0000i	0.623663i
$834$	0	0
$835$	0	0
$836$	12.0000	0.415029
$837$	0	0
$838$	− 21.0000i	− 0.725433i
$839$	56.0000	1.93333	0.966667	−	0.256036i	$-0.0824164\pi$
$839$	56.0000	1.93333	0.966667	+	0.256036i	$0.0824164\pi$
$840$	0	0
$841$	−25.0000	−0.862069
$842$	5.00000i	0.172311i
$843$	0	0
$844$	−23.0000	−0.791693
$845$	0	0
$846$	0	0
$847$	− 7.00000i	− 0.240523i
$848$	12.0000i	0.412082i
$849$	0	0
$850$	0	0
$851$	12.0000	0.411355
$852$	0	0
$853$	− 49.0000i	− 1.67773i	−0.544341	−	0.838864i	$-0.683220\pi$
$853$	− 49.0000i	− 1.67773i	0.544341	−	0.838864i	$-0.316780\pi$
$854$	−8.00000	−0.273754
$855$	0	0
$856$	−4.00000	−0.136717
$857$	− 46.0000i	− 1.57133i	−0.618652	−	0.785665i	$-0.712321\pi$
$857$	− 46.0000i	− 1.57133i	0.618652	−	0.785665i	$-0.287679\pi$
$858$	0	0
$859$	−20.0000	−0.682391	−0.341196	−	0.939992i	$-0.610832\pi$
$859$	−20.0000	−0.682391	−0.341196	+	0.939992i	$0.610832\pi$
$860$	0	0
$861$	0	0
$862$	33.0000i	1.12398i
$863$	− 11.0000i	− 0.374444i	−0.982318	−	0.187222i	$-0.940052\pi$
$863$	− 11.0000i	− 0.374444i	0.982318	−	0.187222i	$-0.0599484\pi$
$864$	0	0
$865$	0	0
$866$	−7.00000	−0.237870
$867$	0	0
$868$	− 4.00000i	− 0.135769i
$869$	8.00000	0.271381
$870$	0	0
$871$	2.00000	0.0677674
$872$	− 19.0000i	− 0.643421i
$873$	0	0
$874$	24.0000	0.811812
$875$	0	0
$876$	0	0
$877$	− 39.0000i	− 1.31694i	−0.752609	−	0.658468i	$-0.771205\pi$
$877$	− 39.0000i	− 1.31694i	0.752609	−	0.658468i	$-0.228795\pi$
$878$	− 22.0000i	− 0.742464i
$879$	0	0
$880$	0	0
$881$	−21.0000	−0.707508	−0.353754	−	0.935339i	$-0.615095\pi$
$881$	−21.0000	−0.707508	−0.353754	+	0.935339i	$0.615095\pi$
$882$	0	0
$883$	47.0000i	1.58168i	0.612026	+	0.790838i	$0.290355\pi$
$883$	47.0000i	1.58168i	−0.612026	+	0.790838i	$0.709645\pi$
$884$	3.00000	0.100901
$885$	0	0
$886$	−39.0000	−1.31023
$887$	8.00000i	0.268614i	0.990940	+	0.134307i	$0.0428808\pi$
$887$	8.00000i	0.268614i	−0.990940	+	0.134307i	$0.957119\pi$
$888$	0	0
$889$	−16.0000	−0.536623
$890$	0	0
$891$	0	0
$892$	− 21.0000i	− 0.703132i
$893$	78.0000i	2.61017i
$894$	0	0
$895$	0	0
$896$	−1.00000	−0.0334077
$897$	0	0
$898$	26.0000i	0.867631i
$899$	8.00000	0.266815
$900$	0	0
$901$	−36.0000	−1.19933
$902$	0	0
$903$	0	0
$904$	−2.00000	−0.0665190
$905$	0	0
$906$	0	0
$907$	− 9.00000i	− 0.298840i	−0.988774	−	0.149420i	$-0.952259\pi$
$907$	− 9.00000i	− 0.298840i	0.988774	−	0.149420i	$-0.0477407\pi$
$908$	− 24.0000i	− 0.796468i
$909$	0	0
$910$	0	0
$911$	54.0000	1.78910	0.894550	−	0.446968i	$-0.147496\pi$
$911$	54.0000	1.78910	0.894550	+	0.446968i	$0.147496\pi$
$912$	0	0
$913$	0	0
$914$	10.0000	0.330771
$915$	0	0
$916$	−15.0000	−0.495614
$917$	1.00000i	0.0330229i
$918$	0	0
$919$	−24.0000	−0.791687	−0.395843	−	0.918318i	$-0.629548\pi$
$919$	−24.0000	−0.791687	−0.395843	+	0.918318i	$0.629548\pi$
$920$	0	0
$921$	0	0
$922$	− 21.0000i	− 0.691598i
$923$	5.00000i	0.164577i
$924$	0	0
$925$	0	0
$926$	−16.0000	−0.525793
$927$	0	0
$928$	− 2.00000i	− 0.0656532i
$929$	−36.0000	−1.18112	−0.590561	−	0.806993i	$-0.701093\pi$
$929$	−36.0000	−1.18112	−0.590561	+	0.806993i	$0.701093\pi$
$930$	0	0
$931$	−36.0000	−1.17985
$932$	11.0000i	0.360317i
$933$	0	0
$934$	−20.0000	−0.654420
$935$	0	0
$936$	0	0
$937$	− 42.0000i	− 1.37208i	−0.727564	−	0.686040i	$-0.759347\pi$
$937$	− 42.0000i	− 1.37208i	0.727564	−	0.686040i	$-0.240653\pi$
$938$	− 2.00000i	− 0.0653023i
$939$	0	0
$940$	0	0
$941$	−25.0000	−0.814977	−0.407488	−	0.913210i	$-0.633595\pi$
$941$	−25.0000	−0.814977	−0.407488	+	0.913210i	$0.633595\pi$
$942$	0	0
$943$	0	0
$944$	−10.0000	−0.325472
$945$	0	0
$946$	10.0000	0.325128
$947$	18.0000i	0.584921i	0.956278	+	0.292461i	$0.0944741\pi$
$947$	18.0000i	0.584921i	−0.956278	+	0.292461i	$0.905526\pi$
$948$	0	0
$949$	−10.0000	−0.324614
$950$	0	0
$951$	0	0
$952$	− 3.00000i	− 0.0972306i
$953$	23.0000i	0.745043i	0.928024	+	0.372522i	$0.121507\pi$
$953$	23.0000i	0.745043i	−0.928024	+	0.372522i	$0.878493\pi$
$954$	0	0
$955$	0	0
$956$	−9.00000	−0.291081
$957$	0	0
$958$	3.00000i	0.0969256i
$959$	12.0000	0.387500
$960$	0	0
$961$	−15.0000	−0.483871
$962$	3.00000i	0.0967239i
$963$	0	0
$964$	−18.0000	−0.579741
$965$	0	0
$966$	0	0
$967$	23.0000i	0.739630i	0.929105	+	0.369815i	$0.120579\pi$
$967$	23.0000i	0.739630i	−0.929105	+	0.369815i	$0.879421\pi$
$968$	− 7.00000i	− 0.224989i
$969$	0	0
$970$	0	0
$971$	15.0000	0.481373	0.240686	−	0.970603i	$-0.422627\pi$
$971$	15.0000	0.481373	0.240686	+	0.970603i	$0.422627\pi$
$972$	0	0
$973$	− 7.00000i	− 0.224410i
$974$	−16.0000	−0.512673
$975$	0	0
$976$	−8.00000	−0.256074
$977$	30.0000i	0.959785i	0.877327	+	0.479893i	$0.159324\pi$
$977$	30.0000i	0.959785i	−0.877327	+	0.479893i	$0.840676\pi$
$978$	0	0
$979$	12.0000	0.383522
$980$	0	0
$981$	0	0
$982$	− 5.00000i	− 0.159556i
$983$	− 31.0000i	− 0.988746i	−0.869250	−	0.494373i	$-0.835398\pi$
$983$	− 31.0000i	− 0.988746i	0.869250	−	0.494373i	$-0.164602\pi$
$984$	0	0
$985$	0	0
$986$	6.00000	0.191079
$987$	0	0
$988$	6.00000i	0.190885i
$989$	20.0000	0.635963
$990$	0	0
$991$	−30.0000	−0.952981	−0.476491	−	0.879180i	$-0.658091\pi$
$991$	−30.0000	−0.952981	−0.476491	+	0.879180i	$0.658091\pi$
$992$	− 4.00000i	− 0.127000i
$993$	0	0
$994$	5.00000	0.158590
$995$	0	0
$996$	0	0
$997$	− 10.0000i	− 0.316703i	−0.987383	−	0.158352i	$-0.949382\pi$
$997$	− 10.0000i	− 0.316703i	0.987383	−	0.158352i	$-0.0506179\pi$
$998$	− 32.0000i	− 1.01294i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5850.2.e.v.5149.1		2
3.2	odd	2		650.2.b.a.599.2		2
5.2	odd	4		5850.2.a.bn.1.1		1
5.3	odd	4		234.2.a.b.1.1		1
5.4	even	2	inner	5850.2.e.v.5149.2		2
15.2	even	4		650.2.a.g.1.1		1
15.8	even	4		26.2.a.b.1.1	✓	1
15.14	odd	2		650.2.b.a.599.1		2
20.3	even	4		1872.2.a.m.1.1		1
40.3	even	4		7488.2.a.v.1.1		1
40.13	odd	4		7488.2.a.w.1.1		1
45.13	odd	12		2106.2.e.t.703.1		2
45.23	even	12		2106.2.e.h.703.1		2
45.38	even	12		2106.2.e.h.1405.1		2
45.43	odd	12		2106.2.e.t.1405.1		2
60.23	odd	4		208.2.a.d.1.1		1
60.47	odd	4		5200.2.a.c.1.1		1
65.8	even	4		3042.2.b.f.1351.1		2
65.18	even	4		3042.2.b.f.1351.2		2
65.38	odd	4		3042.2.a.l.1.1		1
105.23	even	12		1274.2.f.l.1145.1		2
105.38	odd	12		1274.2.f.a.79.1		2
105.53	even	12		1274.2.f.l.79.1		2
105.68	odd	12		1274.2.f.a.1145.1		2
105.83	odd	4		1274.2.a.o.1.1		1
120.53	even	4		832.2.a.j.1.1		1
120.83	odd	4		832.2.a.a.1.1		1
165.98	odd	4		3146.2.a.a.1.1		1
195.8	odd	4		338.2.b.a.337.2		2
195.23	even	12		338.2.c.g.191.1		2
195.38	even	4		338.2.a.a.1.1		1
195.68	even	12		338.2.c.c.191.1		2
195.77	even	4		8450.2.a.y.1.1		1
195.83	odd	4		338.2.b.a.337.1		2
195.98	odd	12		338.2.e.d.23.2		4
195.113	even	12		338.2.c.c.315.1		2
195.128	odd	12		338.2.e.d.147.1		4
195.158	odd	12		338.2.e.d.147.2		4
195.173	even	12		338.2.c.g.315.1		2
195.188	odd	12		338.2.e.d.23.1		4
240.53	even	4		3328.2.b.g.1665.2		2
240.83	odd	4		3328.2.b.k.1665.2		2
240.173	even	4		3328.2.b.g.1665.1		2
240.203	odd	4		3328.2.b.k.1665.1		2
255.203	even	4		7514.2.a.i.1.1		1
285.113	odd	4		9386.2.a.f.1.1		1
780.83	even	4		2704.2.f.j.337.2		2
780.203	even	4		2704.2.f.j.337.1		2
780.623	odd	4		2704.2.a.n.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
26.2.a.b.1.1	✓	1	15.8	even	4
208.2.a.d.1.1		1	60.23	odd	4
234.2.a.b.1.1		1	5.3	odd	4
338.2.a.a.1.1		1	195.38	even	4
338.2.b.a.337.1		2	195.83	odd	4
338.2.b.a.337.2		2	195.8	odd	4
338.2.c.c.191.1		2	195.68	even	12
338.2.c.c.315.1		2	195.113	even	12
338.2.c.g.191.1		2	195.23	even	12
338.2.c.g.315.1		2	195.173	even	12
338.2.e.d.23.1		4	195.188	odd	12
338.2.e.d.23.2		4	195.98	odd	12
338.2.e.d.147.1		4	195.128	odd	12
338.2.e.d.147.2		4	195.158	odd	12
650.2.a.g.1.1		1	15.2	even	4
650.2.b.a.599.1		2	15.14	odd	2
650.2.b.a.599.2		2	3.2	odd	2
832.2.a.a.1.1		1	120.83	odd	4
832.2.a.j.1.1		1	120.53	even	4
1274.2.a.o.1.1		1	105.83	odd	4
1274.2.f.a.79.1		2	105.38	odd	12
1274.2.f.a.1145.1		2	105.68	odd	12
1274.2.f.l.79.1		2	105.53	even	12
1274.2.f.l.1145.1		2	105.23	even	12
1872.2.a.m.1.1		1	20.3	even	4
2106.2.e.h.703.1		2	45.23	even	12
2106.2.e.h.1405.1		2	45.38	even	12
2106.2.e.t.703.1		2	45.13	odd	12
2106.2.e.t.1405.1		2	45.43	odd	12
2704.2.a.n.1.1		1	780.623	odd	4
2704.2.f.j.337.1		2	780.203	even	4
2704.2.f.j.337.2		2	780.83	even	4
3042.2.a.l.1.1		1	65.38	odd	4
3042.2.b.f.1351.1		2	65.8	even	4
3042.2.b.f.1351.2		2	65.18	even	4
3146.2.a.a.1.1		1	165.98	odd	4
3328.2.b.g.1665.1		2	240.173	even	4
3328.2.b.g.1665.2		2	240.53	even	4
3328.2.b.k.1665.1		2	240.203	odd	4
3328.2.b.k.1665.2		2	240.83	odd	4
5200.2.a.c.1.1		1	60.47	odd	4
5850.2.a.bn.1.1		1	5.2	odd	4
5850.2.e.v.5149.1		2	1.1	even	1	trivial
5850.2.e.v.5149.2		2	5.4	even	2	inner
7488.2.a.v.1.1		1	40.3	even	4
7488.2.a.w.1.1		1	40.13	odd	4
7514.2.a.i.1.1		1	255.203	even	4
8450.2.a.y.1.1		1	195.77	even	4
9386.2.a.f.1.1		1	285.113	odd	4

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:	\( N \)	\(=\)	\( 5850 = 2 \cdot 3^{2} \cdot 5^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5850.e (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.00000i q^{2} -1.00000 q^{4} +1.00000i q^{7} +1.00000i q^{8} +2.00000 q^{11} +1.00000i q^{13} +1.00000 q^{14} +1.00000 q^{16} +3.00000i q^{17} -6.00000 q^{19} -2.00000i q^{22} -4.00000i q^{23} +1.00000 q^{26} -1.00000i q^{28} +2.00000 q^{29} +4.00000 q^{31} -1.00000i q^{32} +3.00000 q^{34} +3.00000i q^{37} +6.00000i q^{38} +5.00000i q^{43} -2.00000 q^{44} -4.00000 q^{46} -13.0000i q^{47} +6.00000 q^{49} -1.00000i q^{52} +12.0000i q^{53} -1.00000 q^{56} -2.00000i q^{58} -10.0000 q^{59} -8.00000 q^{61} -4.00000i q^{62} -1.00000 q^{64} -2.00000i q^{67} -3.00000i q^{68} +5.00000 q^{71} +10.0000i q^{73} +3.00000 q^{74} +6.00000 q^{76} +2.00000i q^{77} +4.00000 q^{79} +5.00000 q^{86} +2.00000i q^{88} +6.00000 q^{89} -1.00000 q^{91} +4.00000i q^{92} -13.0000 q^{94} +14.0000i q^{97} -6.00000i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{4} + 4 q^{11} + 2 q^{14} + 2 q^{16} - 12 q^{19} + 2 q^{26} + 4 q^{29} + 8 q^{31} + 6 q^{34} - 4 q^{44} - 8 q^{46} + 12 q^{49} - 2 q^{56} - 20 q^{59} - 16 q^{61} - 2 q^{64} + 10 q^{71} + 6 q^{74}+ \cdots - 26 q^{94}+O(q^{100}) \) 2 * q - 2 * q^4 + 4 * q^11 + 2 * q^14 + 2 * q^16 - 12 * q^19 + 2 * q^26 + 4 * q^29 + 8 * q^31 + 6 * q^34 - 4 * q^44 - 8 * q^46 + 12 * q^49 - 2 * q^56 - 20 * q^59 - 16 * q^61 - 2 * q^64 + 10 * q^71 + 6 * q^74 + 12 * q^76 + 8 * q^79 + 10 * q^86 + 12 * q^89 - 2 * q^91 - 26 * q^94

Introduction

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