LMFDB - Embedded newform 650.2.b.e.599.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [650,2,Mod(599,650)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(650, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 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("650.599"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 650 = 2 \cdot 5^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	650.b (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [2,0,0,-2,0,0,0,0,6,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

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

Self dual:	no
Analytic conductor:	$5.19027613138$
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 130)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			599.1
Root			$-1.00000i$ of defining polynomial
Character	$\chi$	$=$	650.599
Dual form			650.2.b.e.599.2

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{2} -1.00000 q^{4} +1.00000i q^{8} +3.00000 q^{9} +1.00000i q^{13} +1.00000 q^{16} -2.00000i q^{17} -3.00000i q^{18} +8.00000 q^{19} -4.00000i q^{23} +1.00000 q^{26} +2.00000 q^{29} -4.00000 q^{31} -1.00000i q^{32} -2.00000 q^{34} -3.00000 q^{36} -6.00000i q^{37} -8.00000i q^{38} +10.0000 q^{41} -4.00000 q^{46} -8.00000i q^{47} +7.00000 q^{49} -1.00000i q^{52} +6.00000i q^{53} -2.00000i q^{58} -8.00000 q^{59} -2.00000 q^{61} +4.00000i q^{62} -1.00000 q^{64} -4.00000i q^{67} +2.00000i q^{68} -12.0000 q^{71} +3.00000i q^{72} +10.0000i q^{73} -6.00000 q^{74} -8.00000 q^{76} +8.00000 q^{79} +9.00000 q^{81} -10.0000i q^{82} +12.0000i q^{83} -10.0000 q^{89} +4.00000i q^{92} -8.00000 q^{94} +14.0000i q^{97} -7.00000i q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{4} + 6 q^{9} + 2 q^{16} + 16 q^{19} + 2 q^{26} + 4 q^{29} - 8 q^{31} - 4 q^{34} - 6 q^{36} + 20 q^{41} - 8 q^{46} + 14 q^{49} - 16 q^{59} - 4 q^{61} - 2 q^{64} - 24 q^{71} - 12 q^{74} - 16 q^{76}+ \cdots - 16 q^{94}+O(q^{100}) $ 2 * q - 2 * q^4 + 6 * q^9 + 2 * q^16 + 16 * q^19 + 2 * q^26 + 4 * q^29 - 8 * q^31 - 4 * q^34 - 6 * q^36 + 20 * q^41 - 8 * q^46 + 14 * q^49 - 16 * q^59 - 4 * q^61 - 2 * q^64 - 24 * q^71 - 12 * q^74 - 16 * q^76 + 16 * q^79 + 18 * q^81 - 20 * q^89 - 16 * q^94

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/650\mathbb{Z}\right)^\times$.

$n$	$27$	$301$
$\chi(n)$	$-1$	$1$

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	1.00000			$0$
$3$	0	0	−1.00000			$\pi$
$4$	−1.00000	−0.500000
$5$	0	0
$5$	0	0
$6$	0	0
$7$	0	0	1.00000			$0$
$7$	0	0	−1.00000			$\pi$
$8$	1.00000i	0.353553i
$9$	3.00000	1.00000
$10$	0	0
$11$	0	0		−	1.00000i	$-0.5\pi$
$11$	0	0			1.00000i	$0.5\pi$
$12$	0	0
$13$	1.00000i	0.277350i
$13$	1.00000i	0.277350i
$14$	0	0
$15$	0	0
$16$	1.00000	0.250000
$17$	− 2.00000i	− 0.485071i	−0.970143	−	0.242536i	$-0.922021\pi$
$17$	− 2.00000i	− 0.485071i	0.970143	−	0.242536i	$-0.0779791\pi$
$18$	− 3.00000i	− 0.707107i
$19$	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
$21$	0	0
$22$	0	0
$23$	− 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
$28$	0	0
$29$	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	$-0.616954\pi$
$31$	−4.00000	−0.718421	−0.359211	+	0.933257i	$0.616954\pi$
$32$	− 1.00000i	− 0.176777i
$33$	0	0
$34$	−2.00000	−0.342997
$35$	0	0
$36$	−3.00000	−0.500000
$37$	− 6.00000i	− 0.986394i	−0.869918	−	0.493197i	$-0.835828\pi$
$37$	− 6.00000i	− 0.986394i	0.869918	−	0.493197i	$-0.164172\pi$
$38$	− 8.00000i	− 1.29777i
$39$	0	0
$40$	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
$43$	0	0	1.00000			$0$
$43$	0	0	−1.00000			$\pi$
$44$	0	0
$45$	0	0
$46$	−4.00000	−0.589768
$47$	− 8.00000i	− 1.16692i	−0.812142	−	0.583460i	$-0.801699\pi$
$47$	− 8.00000i	− 1.16692i	0.812142	−	0.583460i	$-0.198301\pi$
$48$	0	0
$49$	7.00000	1.00000
$50$	0	0
$51$	0	0
$52$	− 1.00000i	− 0.138675i
$53$	6.00000i	0.824163i	0.911147	+	0.412082i	$0.135198\pi$
$53$	6.00000i	0.824163i	−0.911147	+	0.412082i	$0.864802\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	0	0
$58$	− 2.00000i	− 0.262613i
$59$	−8.00000	−1.04151	−0.520756	−	0.853706i	$-0.674350\pi$
$59$	−8.00000	−1.04151	−0.520756	+	0.853706i	$0.674350\pi$
$60$	0	0
$61$	−2.00000	−0.256074	−0.128037	−	0.991769i	$-0.540868\pi$
$61$	−2.00000	−0.256074	−0.128037	+	0.991769i	$0.540868\pi$
$62$	4.00000i	0.508001i
$63$	0	0
$64$	−1.00000	−0.125000
$65$	0	0
$66$	0	0
$67$	− 4.00000i	− 0.488678i	−0.969690	−	0.244339i	$-0.921429\pi$
$67$	− 4.00000i	− 0.488678i	0.969690	−	0.244339i	$-0.0785709\pi$
$68$	2.00000i	0.242536i
$69$	0	0
$70$	0	0
$71$	−12.0000	−1.42414	−0.712069	−	0.702109i	$-0.752242\pi$
$71$	−12.0000	−1.42414	−0.712069	+	0.702109i	$0.752242\pi$
$72$	3.00000i	0.353553i
$73$	10.0000i	1.17041i	0.810885	+	0.585206i	$0.198986\pi$
$73$	10.0000i	1.17041i	−0.810885	+	0.585206i	$0.801014\pi$
$74$	−6.00000	−0.697486
$75$	0	0
$76$	−8.00000	−0.917663
$77$	0	0
$78$	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
$81$	9.00000	1.00000
$82$	− 10.0000i	− 1.10432i
$83$	12.0000i	1.31717i	0.752506	+	0.658586i	$0.228845\pi$
$83$	12.0000i	1.31717i	−0.752506	+	0.658586i	$0.771155\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−10.0000	−1.06000	−0.529999	−	0.847998i	$-0.677808\pi$
$89$	−10.0000	−1.06000	−0.529999	+	0.847998i	$0.677808\pi$
$90$	0	0
$91$	0	0
$92$	4.00000i	0.417029i
$93$	0	0
$94$	−8.00000	−0.825137
$95$	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	$0.748359\pi$
$98$	− 7.00000i	− 0.707107i
$99$	0	0
$100$	0	0
$101$	6.00000	0.597022	0.298511	−	0.954406i	$-0.403510\pi$
$101$	6.00000	0.597022	0.298511	+	0.954406i	$0.403510\pi$
$102$	0	0
$103$	− 4.00000i	− 0.394132i	−0.980390	−	0.197066i	$-0.936859\pi$
$103$	− 4.00000i	− 0.394132i	0.980390	−	0.197066i	$-0.0631413\pi$
$104$	−1.00000	−0.0980581
$105$	0	0
$106$	6.00000	0.582772
$107$	0	0	1.00000			$0$
$107$	0	0	−1.00000			$\pi$
$108$	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
$112$	0	0
$113$	− 14.0000i	− 1.31701i	−0.752577	−	0.658505i	$-0.771189\pi$
$113$	− 14.0000i	− 1.31701i	0.752577	−	0.658505i	$-0.228811\pi$
$114$	0	0
$115$	0	0
$116$	−2.00000	−0.185695
$117$	3.00000i	0.277350i
$118$	8.00000i	0.736460i
$119$	0	0
$120$	0	0
$121$	−11.0000	−1.00000
$122$	2.00000i	0.181071i
$123$	0	0
$124$	4.00000	0.359211
$125$	0	0
$126$	0	0
$127$	20.0000i	1.77471i	0.461084	+	0.887357i	$0.347461\pi$
$127$	20.0000i	1.77471i	−0.461084	+	0.887357i	$0.652539\pi$
$128$	1.00000i	0.0883883i
$129$	0	0
$130$	0	0
$131$	4.00000	0.349482	0.174741	−	0.984614i	$-0.444091\pi$
$131$	4.00000	0.349482	0.174741	+	0.984614i	$0.444091\pi$
$132$	0	0
$133$	0	0
$134$	−4.00000	−0.345547
$135$	0	0
$136$	2.00000	0.171499
$137$	6.00000i	0.512615i	0.966595	+	0.256307i	$0.0825059\pi$
$137$	6.00000i	0.512615i	−0.966595	+	0.256307i	$0.917494\pi$
$138$	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
$141$	0	0
$142$	12.0000i	1.00702i
$143$	0	0
$144$	3.00000	0.250000
$145$	0	0
$146$	10.0000	0.827606
$147$	0	0
$148$	6.00000i	0.493197i
$149$	−6.00000	−0.491539	−0.245770	−	0.969328i	$-0.579041\pi$
$149$	−6.00000	−0.491539	−0.245770	+	0.969328i	$0.579041\pi$
$150$	0	0
$151$	−20.0000	−1.62758	−0.813788	−	0.581161i	$-0.802599\pi$
$151$	−20.0000	−1.62758	−0.813788	+	0.581161i	$0.802599\pi$
$152$	8.00000i	0.648886i
$153$	− 6.00000i	− 0.485071i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	18.0000i	1.43656i	0.695756	+	0.718278i	$0.255069\pi$
$157$	18.0000i	1.43656i	−0.695756	+	0.718278i	$0.744931\pi$
$158$	− 8.00000i	− 0.636446i
$159$	0	0
$160$	0	0
$161$	0	0
$162$	− 9.00000i	− 0.707107i
$163$	20.0000i	1.56652i	0.621694	+	0.783260i	$0.286445\pi$
$163$	20.0000i	1.56652i	−0.621694	+	0.783260i	$0.713555\pi$
$164$	−10.0000	−0.780869
$165$	0	0
$166$	12.0000	0.931381
$167$	− 16.0000i	− 1.23812i	−0.785345	−	0.619059i	$-0.787514\pi$
$167$	− 16.0000i	− 1.23812i	0.785345	−	0.619059i	$-0.212486\pi$
$168$	0	0
$169$	−1.00000	−0.0769231
$170$	0	0
$171$	24.0000	1.83533
$172$	0	0
$173$	− 18.0000i	− 1.36851i	−0.729241	−	0.684257i	$-0.760127\pi$
$173$	− 18.0000i	− 1.36851i	0.729241	−	0.684257i	$-0.239873\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	0	0
$178$	10.0000i	0.749532i
$179$	−4.00000	−0.298974	−0.149487	−	0.988764i	$-0.547762\pi$
$179$	−4.00000	−0.298974	−0.149487	+	0.988764i	$0.547762\pi$
$180$	0	0
$181$	22.0000	1.63525	0.817624	−	0.575753i	$-0.195291\pi$
$181$	22.0000	1.63525	0.817624	+	0.575753i	$0.195291\pi$
$182$	0	0
$183$	0	0
$184$	4.00000	0.294884
$185$	0	0
$186$	0	0
$187$	0	0
$188$	8.00000i	0.583460i
$189$	0	0
$190$	0	0
$191$	0	0		−	1.00000i	$-0.5\pi$
$191$	0	0			1.00000i	$0.5\pi$
$192$	0	0
$193$	18.0000i	1.29567i	0.761781	+	0.647834i	$0.224325\pi$
$193$	18.0000i	1.29567i	−0.761781	+	0.647834i	$0.775675\pi$
$194$	14.0000	1.00514
$195$	0	0
$196$	−7.00000	−0.500000
$197$	26.0000i	1.85242i	0.377004	+	0.926212i	$0.376954\pi$
$197$	26.0000i	1.85242i	−0.377004	+	0.926212i	$0.623046\pi$
$198$	0	0
$199$	−24.0000	−1.70131	−0.850657	−	0.525720i	$-0.823796\pi$
$199$	−24.0000	−1.70131	−0.850657	+	0.525720i	$0.823796\pi$
$200$	0	0
$201$	0	0
$202$	− 6.00000i	− 0.422159i
$203$	0	0
$204$	0	0
$205$	0	0
$206$	−4.00000	−0.278693
$207$	− 12.0000i	− 0.834058i
$208$	1.00000i	0.0693375i
$209$	0	0
$210$	0	0
$211$	4.00000	0.275371	0.137686	−	0.990476i	$-0.456034\pi$
$211$	4.00000	0.275371	0.137686	+	0.990476i	$0.456034\pi$
$212$	− 6.00000i	− 0.412082i
$213$	0	0
$214$	0	0
$215$	0	0
$216$	0	0
$217$	0	0
$218$	− 2.00000i	− 0.135457i
$219$	0	0
$220$	0	0
$221$	2.00000	0.134535
$222$	0	0
$223$	− 8.00000i	− 0.535720i	−0.963458	−	0.267860i	$-0.913684\pi$
$223$	− 8.00000i	− 0.535720i	0.963458	−	0.267860i	$-0.0863164\pi$
$224$	0	0
$225$	0	0
$226$	−14.0000	−0.931266
$227$	− 20.0000i	− 1.32745i	−0.747978	−	0.663723i	$-0.768975\pi$
$227$	− 20.0000i	− 1.32745i	0.747978	−	0.663723i	$-0.231025\pi$
$228$	0	0
$229$	10.0000	0.660819	0.330409	−	0.943838i	$-0.392813\pi$
$229$	10.0000	0.660819	0.330409	+	0.943838i	$0.392813\pi$
$230$	0	0
$231$	0	0
$232$	2.00000i	0.131306i
$233$	− 6.00000i	− 0.393073i	−0.980497	−	0.196537i	$-0.937031\pi$
$233$	− 6.00000i	− 0.393073i	0.980497	−	0.196537i	$-0.0629694\pi$
$234$	3.00000	0.196116
$235$	0	0
$236$	8.00000	0.520756
$237$	0	0
$238$	0	0
$239$	−20.0000	−1.29369	−0.646846	−	0.762620i	$-0.723912\pi$
$239$	−20.0000	−1.29369	−0.646846	+	0.762620i	$0.723912\pi$
$240$	0	0
$241$	2.00000	0.128831	0.0644157	−	0.997923i	$-0.479482\pi$
$241$	2.00000	0.128831	0.0644157	+	0.997923i	$0.479482\pi$
$242$	11.0000i	0.707107i
$243$	0	0
$244$	2.00000	0.128037
$245$	0	0
$246$	0	0
$247$	8.00000i	0.509028i
$248$	− 4.00000i	− 0.254000i
$249$	0	0
$250$	0	0
$251$	−28.0000	−1.76734	−0.883672	−	0.468106i	$-0.844936\pi$
$251$	−28.0000	−1.76734	−0.883672	+	0.468106i	$0.844936\pi$
$252$	0	0
$253$	0	0
$254$	20.0000	1.25491
$255$	0	0
$256$	1.00000	0.0625000
$257$	− 2.00000i	− 0.124757i	−0.998053	−	0.0623783i	$-0.980131\pi$
$257$	− 2.00000i	− 0.124757i	0.998053	−	0.0623783i	$-0.0198685\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	6.00000	0.371391
$262$	− 4.00000i	− 0.247121i
$263$	− 28.0000i	− 1.72655i	−0.504730	−	0.863277i	$-0.668408\pi$
$263$	− 28.0000i	− 1.72655i	0.504730	−	0.863277i	$-0.331592\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	0	0
$268$	4.00000i	0.244339i
$269$	18.0000	1.09748	0.548740	−	0.835993i	$-0.315108\pi$
$269$	18.0000	1.09748	0.548740	+	0.835993i	$0.315108\pi$
$270$	0	0
$271$	−28.0000	−1.70088	−0.850439	−	0.526073i	$-0.823664\pi$
$271$	−28.0000	−1.70088	−0.850439	+	0.526073i	$0.823664\pi$
$272$	− 2.00000i	− 0.121268i
$273$	0	0
$274$	6.00000	0.362473
$275$	0	0
$276$	0	0
$277$	− 22.0000i	− 1.32185i	−0.750451	−	0.660926i	$-0.770164\pi$
$277$	− 22.0000i	− 1.32185i	0.750451	−	0.660926i	$-0.229836\pi$
$278$	4.00000i	0.239904i
$279$	−12.0000	−0.718421
$280$	0	0
$281$	−22.0000	−1.31241	−0.656205	−	0.754583i	$-0.727839\pi$
$281$	−22.0000	−1.31241	−0.656205	+	0.754583i	$0.727839\pi$
$282$	0	0
$283$	16.0000i	0.951101i	0.879688	+	0.475551i	$0.157751\pi$
$283$	16.0000i	0.951101i	−0.879688	+	0.475551i	$0.842249\pi$
$284$	12.0000	0.712069
$285$	0	0
$286$	0	0
$287$	0	0
$288$	− 3.00000i	− 0.176777i
$289$	13.0000	0.764706
$290$	0	0
$291$	0	0
$292$	− 10.0000i	− 0.585206i
$293$	6.00000i	0.350524i	0.984522	+	0.175262i	$0.0560772\pi$
$293$	6.00000i	0.350524i	−0.984522	+	0.175262i	$0.943923\pi$
$294$	0	0
$295$	0	0
$296$	6.00000	0.348743
$297$	0	0
$298$	6.00000i	0.347571i
$299$	4.00000	0.231326
$300$	0	0
$301$	0	0
$302$	20.0000i	1.15087i
$303$	0	0
$304$	8.00000	0.458831
$305$	0	0
$306$	−6.00000	−0.342997
$307$	− 28.0000i	− 1.59804i	−0.601302	−	0.799022i	$-0.705351\pi$
$307$	− 28.0000i	− 1.59804i	0.601302	−	0.799022i	$-0.294649\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	0	0		−	1.00000i	$-0.5\pi$
$311$	0	0			1.00000i	$0.5\pi$
$312$	0	0
$313$	− 6.00000i	− 0.339140i	−0.985518	−	0.169570i	$-0.945762\pi$
$313$	− 6.00000i	− 0.339140i	0.985518	−	0.169570i	$-0.0542379\pi$
$314$	18.0000	1.01580
$315$	0	0
$316$	−8.00000	−0.450035
$317$	2.00000i	0.112331i	0.998421	+	0.0561656i	$0.0178875\pi$
$317$	2.00000i	0.112331i	−0.998421	+	0.0561656i	$0.982113\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	0	0
$322$	0	0
$323$	− 16.0000i	− 0.890264i
$324$	−9.00000	−0.500000
$325$	0	0
$326$	20.0000	1.10770
$327$	0	0
$328$	10.0000i	0.552158i
$329$	0	0
$330$	0	0
$331$	−8.00000	−0.439720	−0.219860	−	0.975531i	$-0.570560\pi$
$331$	−8.00000	−0.439720	−0.219860	+	0.975531i	$0.570560\pi$
$332$	− 12.0000i	− 0.658586i
$333$	− 18.0000i	− 0.986394i
$334$	−16.0000	−0.875481
$335$	0	0
$336$	0	0
$337$	− 18.0000i	− 0.980522i	−0.871576	−	0.490261i	$-0.836901\pi$
$337$	− 18.0000i	− 0.980522i	0.871576	−	0.490261i	$-0.163099\pi$
$338$	1.00000i	0.0543928i
$339$	0	0
$340$	0	0
$341$	0	0
$342$	− 24.0000i	− 1.29777i
$343$	0	0
$344$	0	0
$345$	0	0
$346$	−18.0000	−0.967686
$347$	− 32.0000i	− 1.71785i	−0.512101	−	0.858925i	$-0.671133\pi$
$347$	− 32.0000i	− 1.71785i	0.512101	−	0.858925i	$-0.328867\pi$
$348$	0	0
$349$	−14.0000	−0.749403	−0.374701	−	0.927146i	$-0.622255\pi$
$349$	−14.0000	−0.749403	−0.374701	+	0.927146i	$0.622255\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	− 30.0000i	− 1.59674i	−0.602168	−	0.798369i	$-0.705696\pi$
$353$	− 30.0000i	− 1.59674i	0.602168	−	0.798369i	$-0.294304\pi$
$354$	0	0
$355$	0	0
$356$	10.0000	0.529999
$357$	0	0
$358$	4.00000i	0.211407i
$359$	−4.00000	−0.211112	−0.105556	−	0.994413i	$-0.533662\pi$
$359$	−4.00000	−0.211112	−0.105556	+	0.994413i	$0.533662\pi$
$360$	0	0
$361$	45.0000	2.36842
$362$	− 22.0000i	− 1.15629i
$363$	0	0
$364$	0	0
$365$	0	0
$366$	0	0
$367$	12.0000i	0.626395i	0.949688	+	0.313197i	$0.101400\pi$
$367$	12.0000i	0.626395i	−0.949688	+	0.313197i	$0.898600\pi$
$368$	− 4.00000i	− 0.208514i
$369$	30.0000	1.56174
$370$	0	0
$371$	0	0
$372$	0	0
$373$	6.00000i	0.310668i	0.987862	+	0.155334i	$0.0496454\pi$
$373$	6.00000i	0.310668i	−0.987862	+	0.155334i	$0.950355\pi$
$374$	0	0
$375$	0	0
$376$	8.00000	0.412568
$377$	2.00000i	0.103005i
$378$	0	0
$379$	0	0		−	1.00000i	$-0.5\pi$
$379$	0	0			1.00000i	$0.5\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	8.00000i	0.408781i	0.978889	+	0.204390i	$0.0655212\pi$
$383$	8.00000i	0.408781i	−0.978889	+	0.204390i	$0.934479\pi$
$384$	0	0
$385$	0	0
$386$	18.0000	0.916176
$387$	0	0
$388$	− 14.0000i	− 0.710742i
$389$	−6.00000	−0.304212	−0.152106	−	0.988364i	$-0.548606\pi$
$389$	−6.00000	−0.304212	−0.152106	+	0.988364i	$0.548606\pi$
$390$	0	0
$391$	−8.00000	−0.404577
$392$	7.00000i	0.353553i
$393$	0	0
$394$	26.0000	1.30986
$395$	0	0
$396$	0	0
$397$	18.0000i	0.903394i	0.892171	+	0.451697i	$0.149181\pi$
$397$	18.0000i	0.903394i	−0.892171	+	0.451697i	$0.850819\pi$
$398$	24.0000i	1.20301i
$399$	0	0
$400$	0	0
$401$	−30.0000	−1.49813	−0.749064	−	0.662497i	$-0.769497\pi$
$401$	−30.0000	−1.49813	−0.749064	+	0.662497i	$0.769497\pi$
$402$	0	0
$403$	− 4.00000i	− 0.199254i
$404$	−6.00000	−0.298511
$405$	0	0
$406$	0	0
$407$	0	0
$408$	0	0
$409$	−26.0000	−1.28562	−0.642809	−	0.766027i	$-0.722231\pi$
$409$	−26.0000	−1.28562	−0.642809	+	0.766027i	$0.722231\pi$
$410$	0	0
$411$	0	0
$412$	4.00000i	0.197066i
$413$	0	0
$414$	−12.0000	−0.589768
$415$	0	0
$416$	1.00000	0.0490290
$417$	0	0
$418$	0	0
$419$	36.0000	1.75872	0.879358	−	0.476162i	$-0.157972\pi$
$419$	36.0000	1.75872	0.879358	+	0.476162i	$0.157972\pi$
$420$	0	0
$421$	−26.0000	−1.26716	−0.633581	−	0.773676i	$-0.718416\pi$
$421$	−26.0000	−1.26716	−0.633581	+	0.773676i	$0.718416\pi$
$422$	− 4.00000i	− 0.194717i
$423$	− 24.0000i	− 1.16692i
$424$	−6.00000	−0.291386
$425$	0	0
$426$	0	0
$427$	0	0
$428$	0	0
$429$	0	0
$430$	0	0
$431$	12.0000	0.578020	0.289010	−	0.957326i	$-0.406674\pi$
$431$	12.0000	0.578020	0.289010	+	0.957326i	$0.406674\pi$
$432$	0	0
$433$	34.0000i	1.63394i	0.576683	+	0.816968i	$0.304347\pi$
$433$	34.0000i	1.63394i	−0.576683	+	0.816968i	$0.695653\pi$
$434$	0	0
$435$	0	0
$436$	−2.00000	−0.0957826
$437$	− 32.0000i	− 1.53077i
$438$	0	0
$439$	24.0000	1.14546	0.572729	−	0.819745i	$-0.305885\pi$
$439$	24.0000	1.14546	0.572729	+	0.819745i	$0.305885\pi$
$440$	0	0
$441$	21.0000	1.00000
$442$	− 2.00000i	− 0.0951303i
$443$	24.0000i	1.14027i	0.821549	+	0.570137i	$0.193110\pi$
$443$	24.0000i	1.14027i	−0.821549	+	0.570137i	$0.806890\pi$
$444$	0	0
$445$	0	0
$446$	−8.00000	−0.378811
$447$	0	0
$448$	0	0
$449$	−18.0000	−0.849473	−0.424736	−	0.905317i	$-0.639633\pi$
$449$	−18.0000	−0.849473	−0.424736	+	0.905317i	$0.639633\pi$
$450$	0	0
$451$	0	0
$452$	14.0000i	0.658505i
$453$	0	0
$454$	−20.0000	−0.938647
$455$	0	0
$456$	0	0
$457$	38.0000i	1.77757i	0.458329	+	0.888783i	$0.348448\pi$
$457$	38.0000i	1.77757i	−0.458329	+	0.888783i	$0.651552\pi$
$458$	− 10.0000i	− 0.467269i
$459$	0	0
$460$	0	0
$461$	14.0000	0.652045	0.326023	−	0.945362i	$-0.394291\pi$
$461$	14.0000	0.652045	0.326023	+	0.945362i	$0.394291\pi$
$462$	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
$466$	−6.00000	−0.277945
$467$	16.0000i	0.740392i	0.928954	+	0.370196i	$0.120709\pi$
$467$	16.0000i	0.740392i	−0.928954	+	0.370196i	$0.879291\pi$
$468$	− 3.00000i	− 0.138675i
$469$	0	0
$470$	0	0
$471$	0	0
$472$	− 8.00000i	− 0.368230i
$473$	0	0
$474$	0	0
$475$	0	0
$476$	0	0
$477$	18.0000i	0.824163i
$478$	20.0000i	0.914779i
$479$	4.00000	0.182765	0.0913823	−	0.995816i	$-0.470871\pi$
$479$	4.00000	0.182765	0.0913823	+	0.995816i	$0.470871\pi$
$480$	0	0
$481$	6.00000	0.273576
$482$	− 2.00000i	− 0.0910975i
$483$	0	0
$484$	11.0000	0.500000
$485$	0	0
$486$	0	0
$487$	8.00000i	0.362515i	0.983436	+	0.181257i	$0.0580167\pi$
$487$	8.00000i	0.362515i	−0.983436	+	0.181257i	$0.941983\pi$
$488$	− 2.00000i	− 0.0905357i
$489$	0	0
$490$	0	0
$491$	20.0000	0.902587	0.451294	−	0.892375i	$-0.350963\pi$
$491$	20.0000	0.902587	0.451294	+	0.892375i	$0.350963\pi$
$492$	0	0
$493$	− 4.00000i	− 0.180151i
$494$	8.00000	0.359937
$495$	0	0
$496$	−4.00000	−0.179605
$497$	0	0
$498$	0	0
$499$	−8.00000	−0.358129	−0.179065	−	0.983837i	$-0.557307\pi$
$499$	−8.00000	−0.358129	−0.179065	+	0.983837i	$0.557307\pi$
$500$	0	0
$501$	0	0
$502$	28.0000i	1.24970i
$503$	20.0000i	0.891756i	0.895094	+	0.445878i	$0.147108\pi$
$503$	20.0000i	0.891756i	−0.895094	+	0.445878i	$0.852892\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	0	0
$508$	− 20.0000i	− 0.887357i
$509$	18.0000	0.797836	0.398918	−	0.916987i	$-0.369386\pi$
$509$	18.0000	0.797836	0.398918	+	0.916987i	$0.369386\pi$
$510$	0	0
$511$	0	0
$512$	− 1.00000i	− 0.0441942i
$513$	0	0
$514$	−2.00000	−0.0882162
$515$	0	0
$516$	0	0
$517$	0	0
$518$	0	0
$519$	0	0
$520$	0	0
$521$	26.0000	1.13908	0.569540	−	0.821963i	$-0.307121\pi$
$521$	26.0000	1.13908	0.569540	+	0.821963i	$0.307121\pi$
$522$	− 6.00000i	− 0.262613i
$523$	− 8.00000i	− 0.349816i	−0.984585	−	0.174908i	$-0.944037\pi$
$523$	− 8.00000i	− 0.349816i	0.984585	−	0.174908i	$-0.0559627\pi$
$524$	−4.00000	−0.174741
$525$	0	0
$526$	−28.0000	−1.22086
$527$	8.00000i	0.348485i
$528$	0	0
$529$	7.00000	0.304348
$530$	0	0
$531$	−24.0000	−1.04151
$532$	0	0
$533$	10.0000i	0.433148i
$534$	0	0
$535$	0	0
$536$	4.00000	0.172774
$537$	0	0
$538$	− 18.0000i	− 0.776035i
$539$	0	0
$540$	0	0
$541$	−2.00000	−0.0859867	−0.0429934	−	0.999075i	$-0.513689\pi$
$541$	−2.00000	−0.0859867	−0.0429934	+	0.999075i	$0.513689\pi$
$542$	28.0000i	1.20270i
$543$	0	0
$544$	−2.00000	−0.0857493
$545$	0	0
$546$	0	0
$547$	40.0000i	1.71028i	0.518400	+	0.855138i	$0.326528\pi$
$547$	40.0000i	1.71028i	−0.518400	+	0.855138i	$0.673472\pi$
$548$	− 6.00000i	− 0.256307i
$549$	−6.00000	−0.256074
$550$	0	0
$551$	16.0000	0.681623
$552$	0	0
$553$	0	0
$554$	−22.0000	−0.934690
$555$	0	0
$556$	4.00000	0.169638
$557$	− 14.0000i	− 0.593199i	−0.955002	−	0.296600i	$-0.904147\pi$
$557$	− 14.0000i	− 0.593199i	0.955002	−	0.296600i	$-0.0958526\pi$
$558$	12.0000i	0.508001i
$559$	0	0
$560$	0	0
$561$	0	0
$562$	22.0000i	0.928014i
$563$	24.0000i	1.01148i	0.862686	+	0.505740i	$0.168780\pi$
$563$	24.0000i	1.01148i	−0.862686	+	0.505740i	$0.831220\pi$
$564$	0	0
$565$	0	0
$566$	16.0000	0.672530
$567$	0	0
$568$	− 12.0000i	− 0.503509i
$569$	−10.0000	−0.419222	−0.209611	−	0.977785i	$-0.567220\pi$
$569$	−10.0000	−0.419222	−0.209611	+	0.977785i	$0.567220\pi$
$570$	0	0
$571$	−28.0000	−1.17176	−0.585882	−	0.810397i	$-0.699252\pi$
$571$	−28.0000	−1.17176	−0.585882	+	0.810397i	$0.699252\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	0	0
$576$	−3.00000	−0.125000
$577$	− 2.00000i	− 0.0832611i	−0.999133	−	0.0416305i	$-0.986745\pi$
$577$	− 2.00000i	− 0.0832611i	0.999133	−	0.0416305i	$-0.0132552\pi$
$578$	− 13.0000i	− 0.540729i
$579$	0	0
$580$	0	0
$581$	0	0
$582$	0	0
$583$	0	0
$584$	−10.0000	−0.413803
$585$	0	0
$586$	6.00000	0.247858
$587$	20.0000i	0.825488i	0.910847	+	0.412744i	$0.135430\pi$
$587$	20.0000i	0.825488i	−0.910847	+	0.412744i	$0.864570\pi$
$588$	0	0
$589$	−32.0000	−1.31854
$590$	0	0
$591$	0	0
$592$	− 6.00000i	− 0.246598i
$593$	2.00000i	0.0821302i	0.999156	+	0.0410651i	$0.0130751\pi$
$593$	2.00000i	0.0821302i	−0.999156	+	0.0410651i	$0.986925\pi$
$594$	0	0
$595$	0	0
$596$	6.00000	0.245770
$597$	0	0
$598$	− 4.00000i	− 0.163572i
$599$	−32.0000	−1.30748	−0.653742	−	0.756717i	$-0.726802\pi$
$599$	−32.0000	−1.30748	−0.653742	+	0.756717i	$0.726802\pi$
$600$	0	0
$601$	26.0000	1.06056	0.530281	−	0.847822i	$-0.322086\pi$
$601$	26.0000	1.06056	0.530281	+	0.847822i	$0.322086\pi$
$602$	0	0
$603$	− 12.0000i	− 0.488678i
$604$	20.0000	0.813788
$605$	0	0
$606$	0	0
$607$	− 36.0000i	− 1.46119i	−0.682808	−	0.730597i	$-0.739242\pi$
$607$	− 36.0000i	− 1.46119i	0.682808	−	0.730597i	$-0.260758\pi$
$608$	− 8.00000i	− 0.324443i
$609$	0	0
$610$	0	0
$611$	8.00000	0.323645
$612$	6.00000i	0.242536i
$613$	38.0000i	1.53481i	0.641165	+	0.767403i	$0.278451\pi$
$613$	38.0000i	1.53481i	−0.641165	+	0.767403i	$0.721549\pi$
$614$	−28.0000	−1.12999
$615$	0	0
$616$	0	0
$617$	− 26.0000i	− 1.04672i	−0.852111	−	0.523360i	$-0.824678\pi$
$617$	− 26.0000i	− 1.04672i	0.852111	−	0.523360i	$-0.175322\pi$
$618$	0	0
$619$	24.0000	0.964641	0.482321	−	0.875995i	$-0.339794\pi$
$619$	24.0000	0.964641	0.482321	+	0.875995i	$0.339794\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	0	0
$624$	0	0
$625$	0	0
$626$	−6.00000	−0.239808
$627$	0	0
$628$	− 18.0000i	− 0.718278i
$629$	−12.0000	−0.478471
$630$	0	0
$631$	12.0000	0.477712	0.238856	−	0.971055i	$-0.423228\pi$
$631$	12.0000	0.477712	0.238856	+	0.971055i	$0.423228\pi$
$632$	8.00000i	0.318223i
$633$	0	0
$634$	2.00000	0.0794301
$635$	0	0
$636$	0	0
$637$	7.00000i	0.277350i
$638$	0	0
$639$	−36.0000	−1.42414
$640$	0	0
$641$	−30.0000	−1.18493	−0.592464	−	0.805597i	$-0.701845\pi$
$641$	−30.0000	−1.18493	−0.592464	+	0.805597i	$0.701845\pi$
$642$	0	0
$643$	28.0000i	1.10421i	0.833774	+	0.552106i	$0.186176\pi$
$643$	28.0000i	1.10421i	−0.833774	+	0.552106i	$0.813824\pi$
$644$	0	0
$645$	0	0
$646$	−16.0000	−0.629512
$647$	28.0000i	1.10079i	0.834903	+	0.550397i	$0.185524\pi$
$647$	28.0000i	1.10079i	−0.834903	+	0.550397i	$0.814476\pi$
$648$	9.00000i	0.353553i
$649$	0	0
$650$	0	0
$651$	0	0
$652$	− 20.0000i	− 0.783260i
$653$	46.0000i	1.80012i	0.435767	+	0.900060i	$0.356477\pi$
$653$	46.0000i	1.80012i	−0.435767	+	0.900060i	$0.643523\pi$
$654$	0	0
$655$	0	0
$656$	10.0000	0.390434
$657$	30.0000i	1.17041i
$658$	0	0
$659$	36.0000	1.40236	0.701180	−	0.712984i	$-0.252657\pi$
$659$	36.0000	1.40236	0.701180	+	0.712984i	$0.252657\pi$
$660$	0	0
$661$	22.0000	0.855701	0.427850	−	0.903850i	$-0.359271\pi$
$661$	22.0000	0.855701	0.427850	+	0.903850i	$0.359271\pi$
$662$	8.00000i	0.310929i
$663$	0	0
$664$	−12.0000	−0.465690
$665$	0	0
$666$	−18.0000	−0.697486
$667$	− 8.00000i	− 0.309761i
$668$	16.0000i	0.619059i
$669$	0	0
$670$	0	0
$671$	0	0
$672$	0	0
$673$	− 14.0000i	− 0.539660i	−0.962908	−	0.269830i	$-0.913032\pi$
$673$	− 14.0000i	− 0.539660i	0.962908	−	0.269830i	$-0.0869676\pi$
$674$	−18.0000	−0.693334
$675$	0	0
$676$	1.00000	0.0384615
$677$	− 22.0000i	− 0.845529i	−0.906240	−	0.422764i	$-0.861060\pi$
$677$	− 22.0000i	− 0.845529i	0.906240	−	0.422764i	$-0.138940\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	0	0
$682$	0	0
$683$	28.0000i	1.07139i	0.844411	+	0.535695i	$0.179950\pi$
$683$	28.0000i	1.07139i	−0.844411	+	0.535695i	$0.820050\pi$
$684$	−24.0000	−0.917663
$685$	0	0
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−6.00000	−0.228582
$690$	0	0
$691$	32.0000	1.21734	0.608669	−	0.793424i	$-0.291704\pi$
$691$	32.0000	1.21734	0.608669	+	0.793424i	$0.291704\pi$
$692$	18.0000i	0.684257i
$693$	0	0
$694$	−32.0000	−1.21470
$695$	0	0
$696$	0	0
$697$	− 20.0000i	− 0.757554i
$698$	14.0000i	0.529908i
$699$	0	0
$700$	0	0
$701$	−34.0000	−1.28416	−0.642081	−	0.766637i	$-0.721929\pi$
$701$	−34.0000	−1.28416	−0.642081	+	0.766637i	$0.721929\pi$
$702$	0	0
$703$	− 48.0000i	− 1.81035i
$704$	0	0
$705$	0	0
$706$	−30.0000	−1.12906
$707$	0	0
$708$	0	0
$709$	10.0000	0.375558	0.187779	−	0.982211i	$-0.439871\pi$
$709$	10.0000	0.375558	0.187779	+	0.982211i	$0.439871\pi$
$710$	0	0
$711$	24.0000	0.900070
$712$	− 10.0000i	− 0.374766i
$713$	16.0000i	0.599205i
$714$	0	0
$715$	0	0
$716$	4.00000	0.149487
$717$	0	0
$718$	4.00000i	0.149279i
$719$	−16.0000	−0.596699	−0.298350	−	0.954457i	$-0.596436\pi$
$719$	−16.0000	−0.596699	−0.298350	+	0.954457i	$0.596436\pi$
$720$	0	0
$721$	0	0
$722$	− 45.0000i	− 1.67473i
$723$	0	0
$724$	−22.0000	−0.817624
$725$	0	0
$726$	0	0
$727$	28.0000i	1.03846i	0.854634	+	0.519231i	$0.173782\pi$
$727$	28.0000i	1.03846i	−0.854634	+	0.519231i	$0.826218\pi$
$728$	0	0
$729$	27.0000	1.00000
$730$	0	0
$731$	0	0
$732$	0	0
$733$	− 34.0000i	− 1.25582i	−0.778287	−	0.627909i	$-0.783911\pi$
$733$	− 34.0000i	− 1.25582i	0.778287	−	0.627909i	$-0.216089\pi$
$734$	12.0000	0.442928
$735$	0	0
$736$	−4.00000	−0.147442
$737$	0	0
$738$	− 30.0000i	− 1.10432i
$739$	0	0		−	1.00000i	$-0.5\pi$
$739$	0	0			1.00000i	$0.5\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	16.0000i	0.586983i	0.955962	+	0.293492i	$0.0948173\pi$
$743$	16.0000i	0.586983i	−0.955962	+	0.293492i	$0.905183\pi$
$744$	0	0
$745$	0	0
$746$	6.00000	0.219676
$747$	36.0000i	1.31717i
$748$	0	0
$749$	0	0
$750$	0	0
$751$	−40.0000	−1.45962	−0.729810	−	0.683650i	$-0.760392\pi$
$751$	−40.0000	−1.45962	−0.729810	+	0.683650i	$0.760392\pi$
$752$	− 8.00000i	− 0.291730i
$753$	0	0
$754$	2.00000	0.0728357
$755$	0	0
$756$	0	0
$757$	− 22.0000i	− 0.799604i	−0.916602	−	0.399802i	$-0.869079\pi$
$757$	− 22.0000i	− 0.799604i	0.916602	−	0.399802i	$-0.130921\pi$
$758$	0	0
$759$	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
$763$	0	0
$764$	0	0
$765$	0	0
$766$	8.00000	0.289052
$767$	− 8.00000i	− 0.288863i
$768$	0	0
$769$	14.0000	0.504853	0.252426	−	0.967616i	$-0.418771\pi$
$769$	14.0000	0.504853	0.252426	+	0.967616i	$0.418771\pi$
$770$	0	0
$771$	0	0
$772$	− 18.0000i	− 0.647834i
$773$	− 26.0000i	− 0.935155i	−0.883952	−	0.467578i	$-0.845127\pi$
$773$	− 26.0000i	− 0.935155i	0.883952	−	0.467578i	$-0.154873\pi$
$774$	0	0
$775$	0	0
$776$	−14.0000	−0.502571
$777$	0	0
$778$	6.00000i	0.215110i
$779$	80.0000	2.86630
$780$	0	0
$781$	0	0
$782$	8.00000i	0.286079i
$783$	0	0
$784$	7.00000	0.250000
$785$	0	0
$786$	0	0
$787$	4.00000i	0.142585i	0.997455	+	0.0712923i	$0.0227123\pi$
$787$	4.00000i	0.142585i	−0.997455	+	0.0712923i	$0.977288\pi$
$788$	− 26.0000i	− 0.926212i
$789$	0	0
$790$	0	0
$791$	0	0
$792$	0	0
$793$	− 2.00000i	− 0.0710221i
$794$	18.0000	0.638796
$795$	0	0
$796$	24.0000	0.850657
$797$	34.0000i	1.20434i	0.798367	+	0.602171i	$0.205697\pi$
$797$	34.0000i	1.20434i	−0.798367	+	0.602171i	$0.794303\pi$
$798$	0	0
$799$	−16.0000	−0.566039
$800$	0	0
$801$	−30.0000	−1.06000
$802$	30.0000i	1.05934i
$803$	0	0
$804$	0	0
$805$	0	0
$806$	−4.00000	−0.140894
$807$	0	0
$808$	6.00000i	0.211079i
$809$	−26.0000	−0.914111	−0.457056	−	0.889438i	$-0.651096\pi$
$809$	−26.0000	−0.914111	−0.457056	+	0.889438i	$0.651096\pi$
$810$	0	0
$811$	24.0000	0.842754	0.421377	−	0.906886i	$-0.361547\pi$
$811$	24.0000	0.842754	0.421377	+	0.906886i	$0.361547\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	0	0
$816$	0	0
$817$	0	0
$818$	26.0000i	0.909069i
$819$	0	0
$820$	0	0
$821$	−42.0000	−1.46581	−0.732905	−	0.680331i	$-0.761836\pi$
$821$	−42.0000	−1.46581	−0.732905	+	0.680331i	$0.761836\pi$
$822$	0	0
$823$	− 4.00000i	− 0.139431i	−0.997567	−	0.0697156i	$-0.977791\pi$
$823$	− 4.00000i	− 0.139431i	0.997567	−	0.0697156i	$-0.0222092\pi$
$824$	4.00000	0.139347
$825$	0	0
$826$	0	0
$827$	− 36.0000i	− 1.25184i	−0.779886	−	0.625921i	$-0.784723\pi$
$827$	− 36.0000i	− 1.25184i	0.779886	−	0.625921i	$-0.215277\pi$
$828$	12.0000i	0.417029i
$829$	34.0000	1.18087	0.590434	−	0.807086i	$-0.298956\pi$
$829$	34.0000	1.18087	0.590434	+	0.807086i	$0.298956\pi$
$830$	0	0
$831$	0	0
$832$	− 1.00000i	− 0.0346688i
$833$	− 14.0000i	− 0.485071i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	0	0
$838$	− 36.0000i	− 1.24360i
$839$	−4.00000	−0.138095	−0.0690477	−	0.997613i	$-0.521996\pi$
$839$	−4.00000	−0.138095	−0.0690477	+	0.997613i	$0.521996\pi$
$840$	0	0
$841$	−25.0000	−0.862069
$842$	26.0000i	0.896019i
$843$	0	0
$844$	−4.00000	−0.137686
$845$	0	0
$846$	−24.0000	−0.825137
$847$	0	0
$848$	6.00000i	0.206041i
$849$	0	0
$850$	0	0
$851$	−24.0000	−0.822709
$852$	0	0
$853$	22.0000i	0.753266i	0.926363	+	0.376633i	$0.122918\pi$
$853$	22.0000i	0.753266i	−0.926363	+	0.376633i	$0.877082\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	6.00000i	0.204956i	0.994735	+	0.102478i	$0.0326771\pi$
$857$	6.00000i	0.204956i	−0.994735	+	0.102478i	$0.967323\pi$
$858$	0	0
$859$	28.0000	0.955348	0.477674	−	0.878537i	$-0.341480\pi$
$859$	28.0000	0.955348	0.477674	+	0.878537i	$0.341480\pi$
$860$	0	0
$861$	0	0
$862$	− 12.0000i	− 0.408722i
$863$	40.0000i	1.36162i	0.732462	+	0.680808i	$0.238371\pi$
$863$	40.0000i	1.36162i	−0.732462	+	0.680808i	$0.761629\pi$
$864$	0	0
$865$	0	0
$866$	34.0000	1.15537
$867$	0	0
$868$	0	0
$869$	0	0
$870$	0	0
$871$	4.00000	0.135535
$872$	2.00000i	0.0677285i
$873$	42.0000i	1.42148i
$874$	−32.0000	−1.08242
$875$	0	0
$876$	0	0
$877$	− 14.0000i	− 0.472746i	−0.971662	−	0.236373i	$-0.924041\pi$
$877$	− 14.0000i	− 0.472746i	0.971662	−	0.236373i	$-0.0759588\pi$
$878$	− 24.0000i	− 0.809961i
$879$	0	0
$880$	0	0
$881$	2.00000	0.0673817	0.0336909	−	0.999432i	$-0.489274\pi$
$881$	2.00000	0.0673817	0.0336909	+	0.999432i	$0.489274\pi$
$882$	− 21.0000i	− 0.707107i
$883$	− 8.00000i	− 0.269221i	−0.990899	−	0.134611i	$-0.957022\pi$
$883$	− 8.00000i	− 0.269221i	0.990899	−	0.134611i	$-0.0429784\pi$
$884$	−2.00000	−0.0672673
$885$	0	0
$886$	24.0000	0.806296
$887$	− 44.0000i	− 1.47738i	−0.674048	−	0.738688i	$-0.735446\pi$
$887$	− 44.0000i	− 1.47738i	0.674048	−	0.738688i	$-0.264554\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	0	0
$892$	8.00000i	0.267860i
$893$	− 64.0000i	− 2.14168i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	0	0
$898$	18.0000i	0.600668i
$899$	−8.00000	−0.266815
$900$	0	0
$901$	12.0000	0.399778
$902$	0	0
$903$	0	0
$904$	14.0000	0.465633
$905$	0	0
$906$	0	0
$907$	− 8.00000i	− 0.265636i	−0.991140	−	0.132818i	$-0.957597\pi$
$907$	− 8.00000i	− 0.265636i	0.991140	−	0.132818i	$-0.0424025\pi$
$908$	20.0000i	0.663723i
$909$	18.0000	0.597022
$910$	0	0
$911$	48.0000	1.59031	0.795155	−	0.606406i	$-0.207389\pi$
$911$	48.0000	1.59031	0.795155	+	0.606406i	$0.207389\pi$
$912$	0	0
$913$	0	0
$914$	38.0000	1.25693
$915$	0	0
$916$	−10.0000	−0.330409
$917$	0	0
$918$	0	0
$919$	−32.0000	−1.05558	−0.527791	−	0.849374i	$-0.676980\pi$
$919$	−32.0000	−1.05558	−0.527791	+	0.849374i	$0.676980\pi$
$920$	0	0
$921$	0	0
$922$	− 14.0000i	− 0.461065i
$923$	− 12.0000i	− 0.394985i
$924$	0	0
$925$	0	0
$926$	−16.0000	−0.525793
$927$	− 12.0000i	− 0.394132i
$928$	− 2.00000i	− 0.0656532i
$929$	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
$931$	56.0000	1.83533
$932$	6.00000i	0.196537i
$933$	0	0
$934$	16.0000	0.523536
$935$	0	0
$936$	−3.00000	−0.0980581
$937$	− 26.0000i	− 0.849383i	−0.905338	−	0.424691i	$-0.860383\pi$
$937$	− 26.0000i	− 0.849383i	0.905338	−	0.424691i	$-0.139617\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	30.0000	0.977972	0.488986	−	0.872292i	$-0.337367\pi$
$941$	30.0000	0.977972	0.488986	+	0.872292i	$0.337367\pi$
$942$	0	0
$943$	− 40.0000i	− 1.30258i
$944$	−8.00000	−0.260378
$945$	0	0
$946$	0	0
$947$	4.00000i	0.129983i	0.997886	+	0.0649913i	$0.0207020\pi$
$947$	4.00000i	0.129983i	−0.997886	+	0.0649913i	$0.979298\pi$
$948$	0	0
$949$	−10.0000	−0.324614
$950$	0	0
$951$	0	0
$952$	0	0
$953$	26.0000i	0.842223i	0.907009	+	0.421111i	$0.138360\pi$
$953$	26.0000i	0.842223i	−0.907009	+	0.421111i	$0.861640\pi$
$954$	18.0000	0.582772
$955$	0	0
$956$	20.0000	0.646846
$957$	0	0
$958$	− 4.00000i	− 0.129234i
$959$	0	0
$960$	0	0
$961$	−15.0000	−0.483871
$962$	− 6.00000i	− 0.193448i
$963$	0	0
$964$	−2.00000	−0.0644157
$965$	0	0
$966$	0	0
$967$	− 24.0000i	− 0.771788i	−0.922543	−	0.385894i	$-0.873893\pi$
$967$	− 24.0000i	− 0.771788i	0.922543	−	0.385894i	$-0.126107\pi$
$968$	− 11.0000i	− 0.353553i
$969$	0	0
$970$	0	0
$971$	−4.00000	−0.128366	−0.0641831	−	0.997938i	$-0.520444\pi$
$971$	−4.00000	−0.128366	−0.0641831	+	0.997938i	$0.520444\pi$
$972$	0	0
$973$	0	0
$974$	8.00000	0.256337
$975$	0	0
$976$	−2.00000	−0.0640184
$977$	− 18.0000i	− 0.575871i	−0.957650	−	0.287936i	$-0.907031\pi$
$977$	− 18.0000i	− 0.575871i	0.957650	−	0.287936i	$-0.0929689\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	6.00000	0.191565
$982$	− 20.0000i	− 0.638226i
$983$	− 24.0000i	− 0.765481i	−0.923856	−	0.382741i	$-0.874980\pi$
$983$	− 24.0000i	− 0.765481i	0.923856	−	0.382741i	$-0.125020\pi$
$984$	0	0
$985$	0	0
$986$	−4.00000	−0.127386
$987$	0	0
$988$	− 8.00000i	− 0.254514i
$989$	0	0
$990$	0	0
$991$	32.0000	1.01651	0.508257	−	0.861206i	$-0.330290\pi$
$991$	32.0000	1.01651	0.508257	+	0.861206i	$0.330290\pi$
$992$	4.00000i	0.127000i
$993$	0	0
$994$	0	0
$995$	0	0
$996$	0	0
$997$	− 22.0000i	− 0.696747i	−0.937356	−	0.348373i	$-0.886734\pi$
$997$	− 22.0000i	− 0.696747i	0.937356	−	0.348373i	$-0.113266\pi$
$998$	8.00000i	0.253236i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	650.2.b.e.599.1		2
3.2	odd	2		5850.2.e.q.5149.2		2
5.2	odd	4		130.2.a.b.1.1	✓	1
5.3	odd	4		650.2.a.d.1.1		1
5.4	even	2	inner	650.2.b.e.599.2		2
15.2	even	4		1170.2.a.b.1.1		1
15.8	even	4		5850.2.a.bq.1.1		1
15.14	odd	2		5850.2.e.q.5149.1		2
20.3	even	4		5200.2.a.r.1.1		1
20.7	even	4		1040.2.a.e.1.1		1
35.27	even	4		6370.2.a.r.1.1		1
40.27	even	4		4160.2.a.h.1.1		1
40.37	odd	4		4160.2.a.i.1.1		1
60.47	odd	4		9360.2.a.l.1.1		1
65.2	even	12		1690.2.l.f.1161.2		4
65.7	even	12		1690.2.l.f.361.2		4
65.12	odd	4		1690.2.a.b.1.1		1
65.17	odd	12		1690.2.e.i.991.1		2
65.22	odd	12		1690.2.e.c.991.1		2
65.32	even	12		1690.2.l.f.361.1		4
65.37	even	12		1690.2.l.f.1161.1		4
65.38	odd	4		8450.2.a.r.1.1		1
65.42	odd	12		1690.2.e.c.191.1		2
65.47	even	4		1690.2.d.d.1351.2		2
65.57	even	4		1690.2.d.d.1351.1		2
65.62	odd	12		1690.2.e.i.191.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
130.2.a.b.1.1	✓	1	5.2	odd	4
650.2.a.d.1.1		1	5.3	odd	4
650.2.b.e.599.1		2	1.1	even	1	trivial
650.2.b.e.599.2		2	5.4	even	2	inner
1040.2.a.e.1.1		1	20.7	even	4
1170.2.a.b.1.1		1	15.2	even	4
1690.2.a.b.1.1		1	65.12	odd	4
1690.2.d.d.1351.1		2	65.57	even	4
1690.2.d.d.1351.2		2	65.47	even	4
1690.2.e.c.191.1		2	65.42	odd	12
1690.2.e.c.991.1		2	65.22	odd	12
1690.2.e.i.191.1		2	65.62	odd	12
1690.2.e.i.991.1		2	65.17	odd	12
1690.2.l.f.361.1		4	65.32	even	12
1690.2.l.f.361.2		4	65.7	even	12
1690.2.l.f.1161.1		4	65.37	even	12
1690.2.l.f.1161.2		4	65.2	even	12
4160.2.a.h.1.1		1	40.27	even	4
4160.2.a.i.1.1		1	40.37	odd	4
5200.2.a.r.1.1		1	20.3	even	4
5850.2.a.bq.1.1		1	15.8	even	4
5850.2.e.q.5149.1		2	15.14	odd	2
5850.2.e.q.5149.2		2	3.2	odd	2
6370.2.a.r.1.1		1	35.27	even	4
8450.2.a.r.1.1		1	65.38	odd	4
9360.2.a.l.1.1		1	60.47	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:	\( N \)	\(=\)	\( 650 = 2 \cdot 5^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	650.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.00000i q^{2} -1.00000 q^{4} +1.00000i q^{8} +3.00000 q^{9} +1.00000i q^{13} +1.00000 q^{16} -2.00000i q^{17} -3.00000i q^{18} +8.00000 q^{19} -4.00000i q^{23} +1.00000 q^{26} +2.00000 q^{29} -4.00000 q^{31} -1.00000i q^{32} -2.00000 q^{34} -3.00000 q^{36} -6.00000i q^{37} -8.00000i q^{38} +10.0000 q^{41} -4.00000 q^{46} -8.00000i q^{47} +7.00000 q^{49} -1.00000i q^{52} +6.00000i q^{53} -2.00000i q^{58} -8.00000 q^{59} -2.00000 q^{61} +4.00000i q^{62} -1.00000 q^{64} -4.00000i q^{67} +2.00000i q^{68} -12.0000 q^{71} +3.00000i q^{72} +10.0000i q^{73} -6.00000 q^{74} -8.00000 q^{76} +8.00000 q^{79} +9.00000 q^{81} -10.0000i q^{82} +12.0000i q^{83} -10.0000 q^{89} +4.00000i q^{92} -8.00000 q^{94} +14.0000i q^{97} -7.00000i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{4} + 6 q^{9} + 2 q^{16} + 16 q^{19} + 2 q^{26} + 4 q^{29} - 8 q^{31} - 4 q^{34} - 6 q^{36} + 20 q^{41} - 8 q^{46} + 14 q^{49} - 16 q^{59} - 4 q^{61} - 2 q^{64} - 24 q^{71} - 12 q^{74} - 16 q^{76}+ \cdots - 16 q^{94}+O(q^{100}) \) 2 * q - 2 * q^4 + 6 * q^9 + 2 * q^16 + 16 * q^19 + 2 * q^26 + 4 * q^29 - 8 * q^31 - 4 * q^34 - 6 * q^36 + 20 * q^41 - 8 * q^46 + 14 * q^49 - 16 * q^59 - 4 * q^61 - 2 * q^64 - 24 * q^71 - 12 * q^74 - 16 * q^76 + 16 * q^79 + 18 * q^81 - 20 * q^89 - 16 * q^94

Introduction

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