LMFDB - Embedded newform 650.2.c.b.649.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [650,2,Mod(649,650)] 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("650.649"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(650, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 1])) N = Newforms(chi, 2, names="a")

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

Newform invariants

comment:select newform

sage:traces = [2,-2,0,2,0,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == 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:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 130)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			649.1
Root			$-1.00000i$ of defining polynomial
Character	$\chi$	$=$	650.649
Dual form			650.2.c.b.649.2

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} -2.00000i q^{3} +1.00000 q^{4} +2.00000i q^{6} -1.00000 q^{8} -1.00000 q^{9} -2.00000i q^{12} +(-3.00000 - 2.00000i) q^{13} +1.00000 q^{16} -6.00000i q^{17} +1.00000 q^{18} +2.00000i q^{24} +(3.00000 + 2.00000i) q^{26} -4.00000i q^{27} -6.00000 q^{29} -6.00000i q^{31} -1.00000 q^{32} +6.00000i q^{34} -1.00000 q^{36} +6.00000 q^{37} +(-4.00000 + 6.00000i) q^{39} +10.0000i q^{43} -12.0000 q^{47} -2.00000i q^{48} -7.00000 q^{49} -12.0000 q^{51} +(-3.00000 - 2.00000i) q^{52} +4.00000i q^{54} +6.00000 q^{58} -12.0000i q^{59} +10.0000 q^{61} +6.00000i q^{62} +1.00000 q^{64} -12.0000 q^{67} -6.00000i q^{68} +6.00000i q^{71} +1.00000 q^{72} +6.00000 q^{73} -6.00000 q^{74} +(4.00000 - 6.00000i) q^{78} +8.00000 q^{79} -11.0000 q^{81} -12.0000 q^{83} -10.0000i q^{86} +12.0000i q^{87} -12.0000i q^{89} -12.0000 q^{93} +12.0000 q^{94} +2.00000i q^{96} +18.0000 q^{97} +7.00000 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{2} + 2 q^{4} - 2 q^{8} - 2 q^{9} - 6 q^{13} + 2 q^{16} + 2 q^{18} + 6 q^{26} - 12 q^{29} - 2 q^{32} - 2 q^{36} + 12 q^{37} - 8 q^{39} - 24 q^{47} - 14 q^{49} - 24 q^{51} - 6 q^{52} + 12 q^{58}+ \cdots + 14 q^{98}+O(q^{100}) $ 2 * q - 2 * q^2 + 2 * q^4 - 2 * q^8 - 2 * q^9 - 6 * q^13 + 2 * q^16 + 2 * q^18 + 6 * q^26 - 12 * q^29 - 2 * q^32 - 2 * q^36 + 12 * q^37 - 8 * q^39 - 24 * q^47 - 14 * q^49 - 24 * q^51 - 6 * q^52 + 12 * q^58 + 20 * q^61 + 2 * q^64 - 24 * q^67 + 2 * q^72 + 12 * q^73 - 12 * q^74 + 8 * q^78 + 16 * q^79 - 22 * q^81 - 24 * q^83 - 24 * q^93 + 24 * q^94 + 36 * q^97 + 14 * q^98

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 $

$2$	−1.00000			−0.707107
$2$	−1.00000			−0.707107
$3$		−	2.00000i		−	1.15470i	−0.816497	−	0.577350i	$-0.804087\pi$
$3$		−	2.00000i		−	1.15470i	0.816497	−	0.577350i	$-0.195913\pi$
$4$	1.00000			0.500000
$5$		0			0
$5$		0			0
$6$			2.00000i			0.816497i
$7$		0			0			−	1.00000i	$-0.5\pi$
$7$		0			0				1.00000i	$0.5\pi$
$8$	−1.00000			−0.353553
$9$	−1.00000			−0.333333
$10$		0			0
$11$		0			0		1.00000			$0$
$11$		0			0		−1.00000			$\pi$
$12$		−	2.00000i		−	0.577350i
$13$	−3.00000	−	2.00000i	−0.832050	−	0.554700i
$13$	−3.00000	−	2.00000i	−0.832050	−	0.554700i
$14$		0			0
$15$		0			0
$16$	1.00000			0.250000
$17$		−	6.00000i		−	1.45521i	−0.685994	−	0.727607i	$-0.740633\pi$
$17$		−	6.00000i		−	1.45521i	0.685994	−	0.727607i	$-0.259367\pi$
$18$	1.00000			0.235702
$19$		0			0		1.00000			$0$
$19$		0			0		−1.00000			$\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$		0			0		1.00000			$0$
$23$		0			0		−1.00000			$\pi$
$24$			2.00000i			0.408248i
$25$		0			0
$26$	3.00000	+	2.00000i	0.588348	+	0.392232i
$27$		−	4.00000i		−	0.769800i
$28$		0			0
$29$	−6.00000			−1.11417			−0.557086	−	0.830455i	$-0.688081\pi$
$29$	−6.00000			−1.11417			−0.557086	+	0.830455i	$0.688081\pi$
$30$		0			0
$31$		−	6.00000i		−	1.07763i	−0.842424	−	0.538816i	$-0.818872\pi$
$31$		−	6.00000i		−	1.07763i	0.842424	−	0.538816i	$-0.181128\pi$
$32$	−1.00000			−0.176777
$33$		0			0
$34$			6.00000i			1.02899i
$35$		0			0
$36$	−1.00000			−0.166667
$37$	6.00000			0.986394			0.493197	−	0.869918i	$-0.335828\pi$
$37$	6.00000			0.986394			0.493197	+	0.869918i	$0.335828\pi$
$38$		0			0
$39$	−4.00000	+	6.00000i	−0.640513	+	0.960769i
$40$		0			0
$41$		0			0		1.00000			$0$
$41$		0			0		−1.00000			$\pi$
$42$		0			0
$43$			10.0000i			1.52499i	0.646997	+	0.762493i	$0.276025\pi$
$43$			10.0000i			1.52499i	−0.646997	+	0.762493i	$0.723975\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−12.0000			−1.75038			−0.875190	−	0.483779i	$-0.839264\pi$
$47$	−12.0000			−1.75038			−0.875190	+	0.483779i	$0.839264\pi$
$48$		−	2.00000i		−	0.288675i
$49$	−7.00000			−1.00000
$50$		0			0
$51$	−12.0000			−1.68034
$52$	−3.00000	−	2.00000i	−0.416025	−	0.277350i
$53$		0			0		1.00000			$0$
$53$		0			0		−1.00000			$\pi$
$54$			4.00000i			0.544331i
$55$		0			0
$56$		0			0
$57$		0			0
$58$	6.00000			0.787839
$59$		−	12.0000i		−	1.56227i	−0.624364	−	0.781133i	$-0.714642\pi$
$59$		−	12.0000i		−	1.56227i	0.624364	−	0.781133i	$-0.285358\pi$
$60$		0			0
$61$	10.0000			1.28037			0.640184	−	0.768221i	$-0.278858\pi$
$61$	10.0000			1.28037			0.640184	+	0.768221i	$0.278858\pi$
$62$			6.00000i			0.762001i
$63$		0			0
$64$	1.00000			0.125000
$65$		0			0
$66$		0			0
$67$	−12.0000			−1.46603			−0.733017	−	0.680211i	$-0.761888\pi$
$67$	−12.0000			−1.46603			−0.733017	+	0.680211i	$0.761888\pi$
$68$		−	6.00000i		−	0.727607i
$69$		0			0
$70$		0			0
$71$			6.00000i			0.712069i	0.934473	+	0.356034i	$0.115871\pi$
$71$			6.00000i			0.712069i	−0.934473	+	0.356034i	$0.884129\pi$
$72$	1.00000			0.117851
$73$	6.00000			0.702247			0.351123	−	0.936329i	$-0.385800\pi$
$73$	6.00000			0.702247			0.351123	+	0.936329i	$0.385800\pi$
$74$	−6.00000			−0.697486
$75$		0			0
$76$		0			0
$77$		0			0
$78$	4.00000	−	6.00000i	0.452911	−	0.679366i
$79$	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
$81$	−11.0000			−1.22222
$82$		0			0
$83$	−12.0000			−1.31717			−0.658586	−	0.752506i	$-0.728845\pi$
$83$	−12.0000			−1.31717			−0.658586	+	0.752506i	$0.728845\pi$
$84$		0			0
$85$		0			0
$86$		−	10.0000i		−	1.07833i
$87$			12.0000i			1.28654i
$88$		0			0
$89$		−	12.0000i		−	1.27200i	−0.771690	−	0.635999i	$-0.780588\pi$
$89$		−	12.0000i		−	1.27200i	0.771690	−	0.635999i	$-0.219412\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$	−12.0000			−1.24434
$94$	12.0000			1.23771
$95$		0			0
$96$			2.00000i			0.204124i
$97$	18.0000			1.82762			0.913812	−	0.406138i	$-0.133125\pi$
$97$	18.0000			1.82762			0.913812	+	0.406138i	$0.133125\pi$
$98$	7.00000			0.707107
$99$		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	$0.403510\pi$
$102$	12.0000			1.18818
$103$			4.00000i			0.394132i	0.980390	+	0.197066i	$0.0631413\pi$
$103$			4.00000i			0.394132i	−0.980390	+	0.197066i	$0.936859\pi$
$104$	3.00000	+	2.00000i	0.294174	+	0.196116i
$105$		0			0
$106$		0			0
$107$		−	6.00000i		−	0.580042i	−0.957020	−	0.290021i	$-0.906338\pi$
$107$		−	6.00000i		−	0.580042i	0.957020	−	0.290021i	$-0.0936623\pi$
$108$		−	4.00000i		−	0.384900i
$109$		−	6.00000i		−	0.574696i	−0.957826	−	0.287348i	$-0.907226\pi$
$109$		−	6.00000i		−	0.574696i	0.957826	−	0.287348i	$-0.0927736\pi$
$110$		0			0
$111$		−	12.0000i		−	1.13899i
$112$		0			0
$113$		−	6.00000i		−	0.564433i	−0.959351	−	0.282216i	$-0.908930\pi$
$113$		−	6.00000i		−	0.564433i	0.959351	−	0.282216i	$-0.0910696\pi$
$114$		0			0
$115$		0			0
$116$	−6.00000			−0.557086
$117$	3.00000	+	2.00000i	0.277350	+	0.184900i
$118$			12.0000i			1.10469i
$119$		0			0
$120$		0			0
$121$	11.0000			1.00000
$122$	−10.0000			−0.905357
$123$		0			0
$124$		−	6.00000i		−	0.538816i
$125$		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	$0.748747\pi$
$128$	−1.00000			−0.0883883
$129$	20.0000			1.76090
$130$		0			0
$131$		0			0			−	1.00000i	$-0.5\pi$
$131$		0			0				1.00000i	$0.5\pi$
$132$		0			0
$133$		0			0
$134$	12.0000			1.03664
$135$		0			0
$136$			6.00000i			0.514496i
$137$	18.0000			1.53784			0.768922	−	0.639343i	$-0.220793\pi$
$137$	18.0000			1.53784			0.768922	+	0.639343i	$0.220793\pi$
$138$		0			0
$139$	4.00000			0.339276			0.169638	−	0.985506i	$-0.445740\pi$
$139$	4.00000			0.339276			0.169638	+	0.985506i	$0.445740\pi$
$140$		0			0
$141$			24.0000i			2.02116i
$142$		−	6.00000i		−	0.503509i
$143$		0			0
$144$	−1.00000			−0.0833333
$145$		0			0
$146$	−6.00000			−0.496564
$147$			14.0000i			1.15470i
$148$	6.00000			0.493197
$149$			6.00000i			0.491539i	0.969328	+	0.245770i	$0.0790407\pi$
$149$			6.00000i			0.491539i	−0.969328	+	0.245770i	$0.920959\pi$
$150$		0			0
$151$		−	18.0000i		−	1.46482i	−0.680864	−	0.732410i	$-0.738396\pi$
$151$		−	18.0000i		−	1.46482i	0.680864	−	0.732410i	$-0.261604\pi$
$152$		0			0
$153$			6.00000i			0.485071i
$154$		0			0
$155$		0			0
$156$	−4.00000	+	6.00000i	−0.320256	+	0.480384i
$157$		−	4.00000i		−	0.319235i	−0.987179	−	0.159617i	$-0.948974\pi$
$157$		−	4.00000i		−	0.319235i	0.987179	−	0.159617i	$-0.0510260\pi$
$158$	−8.00000			−0.636446
$159$		0			0
$160$		0			0
$161$		0			0
$162$	11.0000			0.864242
$163$	12.0000			0.939913			0.469956	−	0.882690i	$-0.344270\pi$
$163$	12.0000			0.939913			0.469956	+	0.882690i	$0.344270\pi$
$164$		0			0
$165$		0			0
$166$	12.0000			0.931381
$167$	12.0000			0.928588			0.464294	−	0.885681i	$-0.346308\pi$
$167$	12.0000			0.928588			0.464294	+	0.885681i	$0.346308\pi$
$168$		0			0
$169$	5.00000	+	12.0000i	0.384615	+	0.923077i
$170$		0			0
$171$		0			0
$172$			10.0000i			0.762493i
$173$			24.0000i			1.82469i	0.409426	+	0.912343i	$0.365729\pi$
$173$			24.0000i			1.82469i	−0.409426	+	0.912343i	$0.634271\pi$
$174$		−	12.0000i		−	0.909718i
$175$		0			0
$176$		0			0
$177$	−24.0000			−1.80395
$178$			12.0000i			0.899438i
$179$		0			0			−	1.00000i	$-0.5\pi$
$179$		0			0				1.00000i	$0.5\pi$
$180$		0			0
$181$	2.00000			0.148659			0.0743294	−	0.997234i	$-0.476318\pi$
$181$	2.00000			0.148659			0.0743294	+	0.997234i	$0.476318\pi$
$182$		0			0
$183$		−	20.0000i		−	1.47844i
$184$		0			0
$185$		0			0
$186$	12.0000			0.879883
$187$		0			0
$188$	−12.0000			−0.875190
$189$		0			0
$190$		0			0
$191$		0			0			−	1.00000i	$-0.5\pi$
$191$		0			0				1.00000i	$0.5\pi$
$192$		−	2.00000i		−	0.144338i
$193$	18.0000			1.29567			0.647834	−	0.761781i	$-0.275675\pi$
$193$	18.0000			1.29567			0.647834	+	0.761781i	$0.275675\pi$
$194$	−18.0000			−1.29232
$195$		0			0
$196$	−7.00000			−0.500000
$197$	6.00000			0.427482			0.213741	−	0.976890i	$-0.431435\pi$
$197$	6.00000			0.427482			0.213741	+	0.976890i	$0.431435\pi$
$198$		0			0
$199$	−16.0000			−1.13421			−0.567105	−	0.823646i	$-0.691937\pi$
$199$	−16.0000			−1.13421			−0.567105	+	0.823646i	$0.691937\pi$
$200$		0			0
$201$			24.0000i			1.69283i
$202$	−6.00000			−0.422159
$203$		0			0
$204$	−12.0000			−0.840168
$205$		0			0
$206$		−	4.00000i		−	0.278693i
$207$		0			0
$208$	−3.00000	−	2.00000i	−0.208013	−	0.138675i
$209$		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	$0.456034\pi$
$212$		0			0
$213$	12.0000			0.822226
$214$			6.00000i			0.410152i
$215$		0			0
$216$			4.00000i			0.272166i
$217$		0			0
$218$			6.00000i			0.406371i
$219$		−	12.0000i		−	0.810885i
$220$		0			0
$221$	−12.0000	+	18.0000i	−0.807207	+	1.21081i
$222$			12.0000i			0.805387i
$223$	12.0000			0.803579			0.401790	−	0.915732i	$-0.368388\pi$
$223$	12.0000			0.803579			0.401790	+	0.915732i	$0.368388\pi$
$224$		0			0
$225$		0			0
$226$			6.00000i			0.399114i
$227$	12.0000			0.796468			0.398234	−	0.917284i	$-0.369623\pi$
$227$	12.0000			0.796468			0.398234	+	0.917284i	$0.369623\pi$
$228$		0			0
$229$		−	6.00000i		−	0.396491i	−0.980152	−	0.198246i	$-0.936476\pi$
$229$		−	6.00000i		−	0.396491i	0.980152	−	0.198246i	$-0.0635244\pi$
$230$		0			0
$231$		0			0
$232$	6.00000			0.393919
$233$		−	18.0000i		−	1.17922i	−0.807688	−	0.589610i	$-0.799282\pi$
$233$		−	18.0000i		−	1.17922i	0.807688	−	0.589610i	$-0.200718\pi$
$234$	−3.00000	−	2.00000i	−0.196116	−	0.130744i
$235$		0			0
$236$		−	12.0000i		−	0.781133i
$237$		−	16.0000i		−	1.03931i
$238$		0			0
$239$			18.0000i			1.16432i	0.813073	+	0.582162i	$0.197793\pi$
$239$			18.0000i			1.16432i	−0.813073	+	0.582162i	$0.802207\pi$
$240$		0			0
$241$		−	24.0000i		−	1.54598i	−0.634421	−	0.772988i	$-0.718761\pi$
$241$		−	24.0000i		−	1.54598i	0.634421	−	0.772988i	$-0.281239\pi$
$242$	−11.0000			−0.707107
$243$			10.0000i			0.641500i
$244$	10.0000			0.640184
$245$		0			0
$246$		0			0
$247$		0			0
$248$			6.00000i			0.381000i
$249$			24.0000i			1.52094i
$250$		0			0
$251$	−24.0000			−1.51487			−0.757433	−	0.652913i	$-0.773547\pi$
$251$	−24.0000			−1.51487			−0.757433	+	0.652913i	$0.773547\pi$
$252$		0			0
$253$		0			0
$254$		−	16.0000i		−	1.00393i
$255$		0			0
$256$	1.00000			0.0625000
$257$		−	30.0000i		−	1.87135i	−0.352865	−	0.935674i	$-0.614792\pi$
$257$		−	30.0000i		−	1.87135i	0.352865	−	0.935674i	$-0.385208\pi$
$258$	−20.0000			−1.24515
$259$		0			0
$260$		0			0
$261$	6.00000			0.371391
$262$		0			0
$263$		−	24.0000i		−	1.47990i	−0.672660	−	0.739952i	$-0.734848\pi$
$263$		−	24.0000i		−	1.47990i	0.672660	−	0.739952i	$-0.265152\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$	−24.0000			−1.46878
$268$	−12.0000			−0.733017
$269$	30.0000			1.82913			0.914566	−	0.404436i	$-0.132532\pi$
$269$	30.0000			1.82913			0.914566	+	0.404436i	$0.132532\pi$
$270$		0			0
$271$			6.00000i			0.364474i	0.983255	+	0.182237i	$0.0583338\pi$
$271$			6.00000i			0.364474i	−0.983255	+	0.182237i	$0.941666\pi$
$272$		−	6.00000i		−	0.363803i
$273$		0			0
$274$	−18.0000			−1.08742
$275$		0			0
$276$		0			0
$277$			8.00000i			0.480673i	0.970690	+	0.240337i	$0.0772579\pi$
$277$			8.00000i			0.480673i	−0.970690	+	0.240337i	$0.922742\pi$
$278$	−4.00000			−0.239904
$279$			6.00000i			0.359211i
$280$		0			0
$281$		−	12.0000i		−	0.715860i	−0.933748	−	0.357930i	$-0.883483\pi$
$281$		−	12.0000i		−	0.715860i	0.933748	−	0.357930i	$-0.116517\pi$
$282$		−	24.0000i		−	1.42918i
$283$			14.0000i			0.832214i	0.909316	+	0.416107i	$0.136606\pi$
$283$			14.0000i			0.832214i	−0.909316	+	0.416107i	$0.863394\pi$
$284$			6.00000i			0.356034i
$285$		0			0
$286$		0			0
$287$		0			0
$288$	1.00000			0.0589256
$289$	−19.0000			−1.11765
$290$		0			0
$291$		−	36.0000i		−	2.11036i
$292$	6.00000			0.351123
$293$	−18.0000			−1.05157			−0.525786	−	0.850617i	$-0.676229\pi$
$293$	−18.0000			−1.05157			−0.525786	+	0.850617i	$0.676229\pi$
$294$		−	14.0000i		−	0.816497i
$295$		0			0
$296$	−6.00000			−0.348743
$297$		0			0
$298$		−	6.00000i		−	0.347571i
$299$		0			0
$300$		0			0
$301$		0			0
$302$			18.0000i			1.03578i
$303$		−	12.0000i		−	0.689382i
$304$		0			0
$305$		0			0
$306$		−	6.00000i		−	0.342997i
$307$	−12.0000			−0.684876			−0.342438	−	0.939540i	$-0.611253\pi$
$307$	−12.0000			−0.684876			−0.342438	+	0.939540i	$0.611253\pi$
$308$		0			0
$309$	8.00000			0.455104
$310$		0			0
$311$	24.0000			1.36092			0.680458	−	0.732787i	$-0.261781\pi$
$311$	24.0000			1.36092			0.680458	+	0.732787i	$0.261781\pi$
$312$	4.00000	−	6.00000i	0.226455	−	0.339683i
$313$		−	10.0000i		−	0.565233i	−0.959233	−	0.282617i	$-0.908798\pi$
$313$		−	10.0000i		−	0.565233i	0.959233	−	0.282617i	$-0.0912024\pi$
$314$			4.00000i			0.225733i
$315$		0			0
$316$	8.00000			0.450035
$317$	−18.0000			−1.01098			−0.505490	−	0.862832i	$-0.668688\pi$
$317$	−18.0000			−1.01098			−0.505490	+	0.862832i	$0.668688\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$	−12.0000			−0.669775
$322$		0			0
$323$		0			0
$324$	−11.0000			−0.611111
$325$		0			0
$326$	−12.0000			−0.664619
$327$	−12.0000			−0.663602
$328$		0			0
$329$		0			0
$330$		0			0
$331$			12.0000i			0.659580i	0.944054	+	0.329790i	$0.106978\pi$
$331$			12.0000i			0.659580i	−0.944054	+	0.329790i	$0.893022\pi$
$332$	−12.0000			−0.658586
$333$	−6.00000			−0.328798
$334$	−12.0000			−0.656611
$335$		0			0
$336$		0			0
$337$			22.0000i			1.19842i	0.800593	+	0.599208i	$0.204518\pi$
$337$			22.0000i			1.19842i	−0.800593	+	0.599208i	$0.795482\pi$
$338$	−5.00000	−	12.0000i	−0.271964	−	0.652714i
$339$	−12.0000			−0.651751
$340$		0			0
$341$		0			0
$342$		0			0
$343$		0			0
$344$		−	10.0000i		−	0.539164i
$345$		0			0
$346$		−	24.0000i		−	1.29025i
$347$		−	6.00000i		−	0.322097i	−0.986947	−	0.161048i	$-0.948512\pi$
$347$		−	6.00000i		−	0.322097i	0.986947	−	0.161048i	$-0.0514875\pi$
$348$			12.0000i			0.643268i
$349$			30.0000i			1.60586i	0.596071	+	0.802932i	$0.296728\pi$
$349$			30.0000i			1.60586i	−0.596071	+	0.802932i	$0.703272\pi$
$350$		0			0
$351$	−8.00000	+	12.0000i	−0.427008	+	0.640513i
$352$		0			0
$353$	6.00000			0.319348			0.159674	−	0.987170i	$-0.448956\pi$
$353$	6.00000			0.319348			0.159674	+	0.987170i	$0.448956\pi$
$354$	24.0000			1.27559
$355$		0			0
$356$		−	12.0000i		−	0.635999i
$357$		0			0
$358$		0			0
$359$		−	6.00000i		−	0.316668i	−0.987386	−	0.158334i	$-0.949388\pi$
$359$		−	6.00000i		−	0.316668i	0.987386	−	0.158334i	$-0.0506123\pi$
$360$		0			0
$361$	19.0000			1.00000
$362$	−2.00000			−0.105118
$363$		−	22.0000i		−	1.15470i
$364$		0			0
$365$		0			0
$366$			20.0000i			1.04542i
$367$		−	28.0000i		−	1.46159i	−0.682598	−	0.730794i	$-0.739150\pi$
$367$		−	28.0000i		−	1.46159i	0.682598	−	0.730794i	$-0.260850\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$	−12.0000			−0.622171
$373$		−	4.00000i		−	0.207112i	−0.994624	−	0.103556i	$-0.966978\pi$
$373$		−	4.00000i		−	0.207112i	0.994624	−	0.103556i	$-0.0330221\pi$
$374$		0			0
$375$		0			0
$376$	12.0000			0.618853
$377$	18.0000	+	12.0000i	0.927047	+	0.618031i
$378$		0			0
$379$		−	24.0000i		−	1.23280i	−0.787434	−	0.616399i	$-0.788591\pi$
$379$		−	24.0000i		−	1.23280i	0.787434	−	0.616399i	$-0.211409\pi$
$380$		0			0
$381$	32.0000			1.63941
$382$		0			0
$383$		0			0			−	1.00000i	$-0.5\pi$
$383$		0			0				1.00000i	$0.5\pi$
$384$			2.00000i			0.102062i
$385$		0			0
$386$	−18.0000			−0.916176
$387$		−	10.0000i		−	0.508329i
$388$	18.0000			0.913812
$389$	−18.0000			−0.912636			−0.456318	−	0.889817i	$-0.650832\pi$
$389$	−18.0000			−0.912636			−0.456318	+	0.889817i	$0.650832\pi$
$390$		0			0
$391$		0			0
$392$	7.00000			0.353553
$393$		0			0
$394$	−6.00000			−0.302276
$395$		0			0
$396$		0			0
$397$	−30.0000			−1.50566			−0.752828	−	0.658217i	$-0.771311\pi$
$397$	−30.0000			−1.50566			−0.752828	+	0.658217i	$0.771311\pi$
$398$	16.0000			0.802008
$399$		0			0
$400$		0			0
$401$			12.0000i			0.599251i	0.954057	+	0.299626i	$0.0968618\pi$
$401$			12.0000i			0.599251i	−0.954057	+	0.299626i	$0.903138\pi$
$402$		−	24.0000i		−	1.19701i
$403$	−12.0000	+	18.0000i	−0.597763	+	0.896644i
$404$	6.00000			0.298511
$405$		0			0
$406$		0			0
$407$		0			0
$408$	12.0000			0.594089
$409$			12.0000i			0.593362i	0.954977	+	0.296681i	$0.0958798\pi$
$409$			12.0000i			0.593362i	−0.954977	+	0.296681i	$0.904120\pi$
$410$		0			0
$411$		−	36.0000i		−	1.77575i
$412$			4.00000i			0.197066i
$413$		0			0
$414$		0			0
$415$		0			0
$416$	3.00000	+	2.00000i	0.147087	+	0.0980581i
$417$		−	8.00000i		−	0.391762i
$418$		0			0
$419$	−12.0000			−0.586238			−0.293119	−	0.956076i	$-0.594693\pi$
$419$	−12.0000			−0.586238			−0.293119	+	0.956076i	$0.594693\pi$
$420$		0			0
$421$			30.0000i			1.46211i	0.682318	+	0.731055i	$0.260972\pi$
$421$			30.0000i			1.46211i	−0.682318	+	0.731055i	$0.739028\pi$
$422$	−4.00000			−0.194717
$423$	12.0000			0.583460
$424$		0			0
$425$		0			0
$426$	−12.0000			−0.581402
$427$		0			0
$428$		−	6.00000i		−	0.290021i
$429$		0			0
$430$		0			0
$431$			30.0000i			1.44505i	0.691345	+	0.722525i	$0.257018\pi$
$431$			30.0000i			1.44505i	−0.691345	+	0.722525i	$0.742982\pi$
$432$		−	4.00000i		−	0.192450i
$433$			2.00000i			0.0961139i	0.998845	+	0.0480569i	$0.0153029\pi$
$433$			2.00000i			0.0961139i	−0.998845	+	0.0480569i	$0.984697\pi$
$434$		0			0
$435$		0			0
$436$		−	6.00000i		−	0.287348i
$437$		0			0
$438$			12.0000i			0.573382i
$439$	−8.00000			−0.381819			−0.190910	−	0.981608i	$-0.561144\pi$
$439$	−8.00000			−0.381819			−0.190910	+	0.981608i	$0.561144\pi$
$440$		0			0
$441$	7.00000			0.333333
$442$	12.0000	−	18.0000i	0.570782	−	0.856173i
$443$		−	6.00000i		−	0.285069i	−0.989790	−	0.142534i	$-0.954475\pi$
$443$		−	6.00000i		−	0.285069i	0.989790	−	0.142534i	$-0.0455251\pi$
$444$		−	12.0000i		−	0.569495i
$445$		0			0
$446$	−12.0000			−0.568216
$447$	12.0000			0.567581
$448$		0			0
$449$			12.0000i			0.566315i	0.959073	+	0.283158i	$0.0913819\pi$
$449$			12.0000i			0.566315i	−0.959073	+	0.283158i	$0.908618\pi$
$450$		0			0
$451$		0			0
$452$		−	6.00000i		−	0.282216i
$453$	−36.0000			−1.69143
$454$	−12.0000			−0.563188
$455$		0			0
$456$		0			0
$457$	18.0000			0.842004			0.421002	−	0.907060i	$-0.361678\pi$
$457$	18.0000			0.842004			0.421002	+	0.907060i	$0.361678\pi$
$458$			6.00000i			0.280362i
$459$	−24.0000			−1.12022
$460$		0			0
$461$			30.0000i			1.39724i	0.715493	+	0.698620i	$0.246202\pi$
$461$			30.0000i			1.39724i	−0.715493	+	0.698620i	$0.753798\pi$
$462$		0			0
$463$	12.0000			0.557687			0.278844	−	0.960337i	$-0.410049\pi$
$463$	12.0000			0.557687			0.278844	+	0.960337i	$0.410049\pi$
$464$	−6.00000			−0.278543
$465$		0			0
$466$			18.0000i			0.833834i
$467$			18.0000i			0.832941i	0.909149	+	0.416470i	$0.136733\pi$
$467$			18.0000i			0.832941i	−0.909149	+	0.416470i	$0.863267\pi$
$468$	3.00000	+	2.00000i	0.138675	+	0.0924500i
$469$		0			0
$470$		0			0
$471$	−8.00000			−0.368621
$472$			12.0000i			0.552345i
$473$		0			0
$474$			16.0000i			0.734904i
$475$		0			0
$476$		0			0
$477$		0			0
$478$		−	18.0000i		−	0.823301i
$479$			42.0000i			1.91903i	0.281659	+	0.959514i	$0.409115\pi$
$479$			42.0000i			1.91903i	−0.281659	+	0.959514i	$0.590885\pi$
$480$		0			0
$481$	−18.0000	−	12.0000i	−0.820729	−	0.547153i
$482$			24.0000i			1.09317i
$483$		0			0
$484$	11.0000			0.500000
$485$		0			0
$486$		−	10.0000i		−	0.453609i
$487$	24.0000			1.08754			0.543772	−	0.839233i	$-0.316996\pi$
$487$	24.0000			1.08754			0.543772	+	0.839233i	$0.316996\pi$
$488$	−10.0000			−0.452679
$489$		−	24.0000i		−	1.08532i
$490$		0			0
$491$	−12.0000			−0.541552			−0.270776	−	0.962642i	$-0.587280\pi$
$491$	−12.0000			−0.541552			−0.270776	+	0.962642i	$0.587280\pi$
$492$		0			0
$493$			36.0000i			1.62136i
$494$		0			0
$495$		0			0
$496$		−	6.00000i		−	0.269408i
$497$		0			0
$498$		−	24.0000i		−	1.07547i
$499$		−	12.0000i		−	0.537194i	−0.963253	−	0.268597i	$-0.913440\pi$
$499$		−	12.0000i		−	0.537194i	0.963253	−	0.268597i	$-0.0865599\pi$
$500$		0			0
$501$		−	24.0000i		−	1.07224i
$502$	24.0000			1.07117
$503$			12.0000i			0.535054i	0.963550	+	0.267527i	$0.0862064\pi$
$503$			12.0000i			0.535054i	−0.963550	+	0.267527i	$0.913794\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	24.0000	−	10.0000i	1.06588	−	0.444116i
$508$			16.0000i			0.709885i
$509$			6.00000i			0.265945i	0.991120	+	0.132973i	$0.0424523\pi$
$509$			6.00000i			0.265945i	−0.991120	+	0.132973i	$0.957548\pi$
$510$		0			0
$511$		0			0
$512$	−1.00000			−0.0441942
$513$		0			0
$514$			30.0000i			1.32324i
$515$		0			0
$516$	20.0000			0.880451
$517$		0			0
$518$		0			0
$519$	48.0000			2.10697
$520$		0			0
$521$	6.00000			0.262865			0.131432	−	0.991325i	$-0.458042\pi$
$521$	6.00000			0.262865			0.131432	+	0.991325i	$0.458042\pi$
$522$	−6.00000			−0.262613
$523$			34.0000i			1.48672i	0.668894	+	0.743358i	$0.266768\pi$
$523$			34.0000i			1.48672i	−0.668894	+	0.743358i	$0.733232\pi$
$524$		0			0
$525$		0			0
$526$			24.0000i			1.04645i
$527$	−36.0000			−1.56818
$528$		0			0
$529$	23.0000			1.00000
$530$		0			0
$531$			12.0000i			0.520756i
$532$		0			0
$533$		0			0
$534$	24.0000			1.03858
$535$		0			0
$536$	12.0000			0.518321
$537$		0			0
$538$	−30.0000			−1.29339
$539$		0			0
$540$		0			0
$541$		−	18.0000i		−	0.773880i	−0.922105	−	0.386940i	$-0.873532\pi$
$541$		−	18.0000i		−	0.773880i	0.922105	−	0.386940i	$-0.126468\pi$
$542$		−	6.00000i		−	0.257722i
$543$		−	4.00000i		−	0.171656i
$544$			6.00000i			0.257248i
$545$		0			0
$546$		0			0
$547$		−	10.0000i		−	0.427569i	−0.976881	−	0.213785i	$-0.931421\pi$
$547$		−	10.0000i		−	0.427569i	0.976881	−	0.213785i	$-0.0685791\pi$
$548$	18.0000			0.768922
$549$	−10.0000			−0.426790
$550$		0			0
$551$		0			0
$552$		0			0
$553$		0			0
$554$		−	8.00000i		−	0.339887i
$555$		0			0
$556$	4.00000			0.169638
$557$	18.0000			0.762684			0.381342	−	0.924434i	$-0.375462\pi$
$557$	18.0000			0.762684			0.381342	+	0.924434i	$0.375462\pi$
$558$		−	6.00000i		−	0.254000i
$559$	20.0000	−	30.0000i	0.845910	−	1.26886i
$560$		0			0
$561$		0			0
$562$			12.0000i			0.506189i
$563$		−	6.00000i		−	0.252870i	−0.991975	−	0.126435i	$-0.959647\pi$
$563$		−	6.00000i		−	0.252870i	0.991975	−	0.126435i	$-0.0403535\pi$
$564$			24.0000i			1.01058i
$565$		0			0
$566$		−	14.0000i		−	0.588464i
$567$		0			0
$568$		−	6.00000i		−	0.251754i
$569$	6.00000			0.251533			0.125767	−	0.992060i	$-0.459861\pi$
$569$	6.00000			0.251533			0.125767	+	0.992060i	$0.459861\pi$
$570$		0			0
$571$	−32.0000			−1.33916			−0.669579	−	0.742741i	$-0.733526\pi$
$571$	−32.0000			−1.33916			−0.669579	+	0.742741i	$0.733526\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$	−1.00000			−0.0416667
$577$	−18.0000			−0.749350			−0.374675	−	0.927156i	$-0.622246\pi$
$577$	−18.0000			−0.749350			−0.374675	+	0.927156i	$0.622246\pi$
$578$	19.0000			0.790296
$579$		−	36.0000i		−	1.49611i
$580$		0			0
$581$		0			0
$582$			36.0000i			1.49225i
$583$		0			0
$584$	−6.00000			−0.248282
$585$		0			0
$586$	18.0000			0.743573
$587$	36.0000			1.48588			0.742940	−	0.669359i	$-0.233431\pi$
$587$	36.0000			1.48588			0.742940	+	0.669359i	$0.233431\pi$
$588$			14.0000i			0.577350i
$589$		0			0
$590$		0			0
$591$		−	12.0000i		−	0.493614i
$592$	6.00000			0.246598
$593$	18.0000			0.739171			0.369586	−	0.929197i	$-0.379500\pi$
$593$	18.0000			0.739171			0.369586	+	0.929197i	$0.379500\pi$
$594$		0			0
$595$		0			0
$596$			6.00000i			0.245770i
$597$			32.0000i			1.30967i
$598$		0			0
$599$		0			0			−	1.00000i	$-0.5\pi$
$599$		0			0				1.00000i	$0.5\pi$
$600$		0			0
$601$	−26.0000			−1.06056			−0.530281	−	0.847822i	$-0.677914\pi$
$601$	−26.0000			−1.06056			−0.530281	+	0.847822i	$0.677914\pi$
$602$		0			0
$603$	12.0000			0.488678
$604$		−	18.0000i		−	0.732410i
$605$		0			0
$606$			12.0000i			0.487467i
$607$			32.0000i			1.29884i	0.760430	+	0.649420i	$0.224988\pi$
$607$			32.0000i			1.29884i	−0.760430	+	0.649420i	$0.775012\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	36.0000	+	24.0000i	1.45640	+	0.970936i
$612$			6.00000i			0.242536i
$613$	6.00000			0.242338			0.121169	−	0.992632i	$-0.461336\pi$
$613$	6.00000			0.242338			0.121169	+	0.992632i	$0.461336\pi$
$614$	12.0000			0.484281
$615$		0			0
$616$		0			0
$617$	−6.00000			−0.241551			−0.120775	−	0.992680i	$-0.538538\pi$
$617$	−6.00000			−0.241551			−0.120775	+	0.992680i	$0.538538\pi$
$618$	−8.00000			−0.321807
$619$		−	36.0000i		−	1.44696i	−0.690344	−	0.723481i	$-0.742541\pi$
$619$		−	36.0000i		−	1.44696i	0.690344	−	0.723481i	$-0.257459\pi$
$620$		0			0
$621$		0			0
$622$	−24.0000			−0.962312
$623$		0			0
$624$	−4.00000	+	6.00000i	−0.160128	+	0.240192i
$625$		0			0
$626$			10.0000i			0.399680i
$627$		0			0
$628$		−	4.00000i		−	0.159617i
$629$		−	36.0000i		−	1.43541i
$630$		0			0
$631$			6.00000i			0.238856i	0.992843	+	0.119428i	$0.0381061\pi$
$631$			6.00000i			0.238856i	−0.992843	+	0.119428i	$0.961894\pi$
$632$	−8.00000			−0.318223
$633$		−	8.00000i		−	0.317971i
$634$	18.0000			0.714871
$635$		0			0
$636$		0			0
$637$	21.0000	+	14.0000i	0.832050	+	0.554700i
$638$		0			0
$639$		−	6.00000i		−	0.237356i
$640$		0			0
$641$	6.00000			0.236986			0.118493	−	0.992955i	$-0.462194\pi$
$641$	6.00000			0.236986			0.118493	+	0.992955i	$0.462194\pi$
$642$	12.0000			0.473602
$643$	−36.0000			−1.41970			−0.709851	−	0.704352i	$-0.751238\pi$
$643$	−36.0000			−1.41970			−0.709851	+	0.704352i	$0.751238\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$			12.0000i			0.471769i	0.971781	+	0.235884i	$0.0757987\pi$
$647$			12.0000i			0.471769i	−0.971781	+	0.235884i	$0.924201\pi$
$648$	11.0000			0.432121
$649$		0			0
$650$		0			0
$651$		0			0
$652$	12.0000			0.469956
$653$		−	36.0000i		−	1.40879i	−0.709809	−	0.704394i	$-0.751219\pi$
$653$		−	36.0000i		−	1.40879i	0.709809	−	0.704394i	$-0.248781\pi$
$654$	12.0000			0.469237
$655$		0			0
$656$		0			0
$657$	−6.00000			−0.234082
$658$		0			0
$659$	12.0000			0.467454			0.233727	−	0.972302i	$-0.424908\pi$
$659$	12.0000			0.467454			0.233727	+	0.972302i	$0.424908\pi$
$660$		0			0
$661$		−	42.0000i		−	1.63361i	−0.576913	−	0.816805i	$-0.695743\pi$
$661$		−	42.0000i		−	1.63361i	0.576913	−	0.816805i	$-0.304257\pi$
$662$		−	12.0000i		−	0.466393i
$663$	36.0000	+	24.0000i	1.39812	+	0.932083i
$664$	12.0000			0.465690
$665$		0			0
$666$	6.00000			0.232495
$667$		0			0
$668$	12.0000			0.464294
$669$		−	24.0000i		−	0.927894i
$670$		0			0
$671$		0			0
$672$		0			0
$673$		−	26.0000i		−	1.00223i	−0.865382	−	0.501113i	$-0.832924\pi$
$673$		−	26.0000i		−	1.00223i	0.865382	−	0.501113i	$-0.167076\pi$
$674$		−	22.0000i		−	0.847408i
$675$		0			0
$676$	5.00000	+	12.0000i	0.192308	+	0.461538i
$677$			12.0000i			0.461197i	0.973049	+	0.230599i	$0.0740685\pi$
$677$			12.0000i			0.461197i	−0.973049	+	0.230599i	$0.925932\pi$
$678$	12.0000			0.460857
$679$		0			0
$680$		0			0
$681$		−	24.0000i		−	0.919682i
$682$		0			0
$683$	−12.0000			−0.459167			−0.229584	−	0.973289i	$-0.573736\pi$
$683$	−12.0000			−0.459167			−0.229584	+	0.973289i	$0.573736\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$	−12.0000			−0.457829
$688$			10.0000i			0.381246i
$689$		0			0
$690$		0			0
$691$		−	36.0000i		−	1.36950i	−0.728776	−	0.684752i	$-0.759910\pi$
$691$		−	36.0000i		−	1.36950i	0.728776	−	0.684752i	$-0.240090\pi$
$692$			24.0000i			0.912343i
$693$		0			0
$694$			6.00000i			0.227757i
$695$		0			0
$696$		−	12.0000i		−	0.454859i
$697$		0			0
$698$		−	30.0000i		−	1.13552i
$699$	−36.0000			−1.36165
$700$		0			0
$701$	−18.0000			−0.679851			−0.339925	−	0.940452i	$-0.610402\pi$
$701$	−18.0000			−0.679851			−0.339925	+	0.940452i	$0.610402\pi$
$702$	8.00000	−	12.0000i	0.301941	−	0.452911i
$703$		0			0
$704$		0			0
$705$		0			0
$706$	−6.00000			−0.225813
$707$		0			0
$708$	−24.0000			−0.901975
$709$			18.0000i			0.676004i	0.941145	+	0.338002i	$0.109751\pi$
$709$			18.0000i			0.676004i	−0.941145	+	0.338002i	$0.890249\pi$
$710$		0			0
$711$	−8.00000			−0.300023
$712$			12.0000i			0.449719i
$713$		0			0
$714$		0			0
$715$		0			0
$716$		0			0
$717$	36.0000			1.34444
$718$			6.00000i			0.223918i
$719$	48.0000			1.79010			0.895049	−	0.445968i	$-0.147140\pi$
$719$	48.0000			1.79010			0.895049	+	0.445968i	$0.147140\pi$
$720$		0			0
$721$		0			0
$722$	−19.0000			−0.707107
$723$	−48.0000			−1.78514
$724$	2.00000			0.0743294
$725$		0			0
$726$			22.0000i			0.816497i
$727$		−	8.00000i		−	0.296704i	−0.988935	−	0.148352i	$-0.952603\pi$
$727$		−	8.00000i		−	0.296704i	0.988935	−	0.148352i	$-0.0473968\pi$
$728$		0			0
$729$	−13.0000			−0.481481
$730$		0			0
$731$	60.0000			2.21918
$732$		−	20.0000i		−	0.739221i
$733$	18.0000			0.664845			0.332423	−	0.943131i	$-0.392134\pi$
$733$	18.0000			0.664845			0.332423	+	0.943131i	$0.392134\pi$
$734$			28.0000i			1.03350i
$735$		0			0
$736$		0			0
$737$		0			0
$738$		0			0
$739$		0			0		1.00000			$0$
$739$		0			0		−1.00000			$\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	24.0000			0.880475			0.440237	−	0.897881i	$-0.354894\pi$
$743$	24.0000			0.880475			0.440237	+	0.897881i	$0.354894\pi$
$744$	12.0000			0.439941
$745$		0			0
$746$			4.00000i			0.146450i
$747$	12.0000			0.439057
$748$		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	$0.760392\pi$
$752$	−12.0000			−0.437595
$753$			48.0000i			1.74922i
$754$	−18.0000	−	12.0000i	−0.655521	−	0.437014i
$755$		0			0
$756$		0			0
$757$			16.0000i			0.581530i	0.956795	+	0.290765i	$0.0939098\pi$
$757$			16.0000i			0.581530i	−0.956795	+	0.290765i	$0.906090\pi$
$758$			24.0000i			0.871719i
$759$		0			0
$760$		0			0
$761$		−	12.0000i		−	0.435000i	−0.976060	−	0.217500i	$-0.930210\pi$
$761$		−	12.0000i		−	0.435000i	0.976060	−	0.217500i	$-0.0697902\pi$
$762$	−32.0000			−1.15924
$763$		0			0
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−24.0000	+	36.0000i	−0.866590	+	1.29988i
$768$		−	2.00000i		−	0.0721688i
$769$		−	48.0000i		−	1.73092i	−0.500974	−	0.865462i	$-0.667025\pi$
$769$		−	48.0000i		−	1.73092i	0.500974	−	0.865462i	$-0.332975\pi$
$770$		0			0
$771$	−60.0000			−2.16085
$772$	18.0000			0.647834
$773$	−6.00000			−0.215805			−0.107903	−	0.994161i	$-0.534413\pi$
$773$	−6.00000			−0.215805			−0.107903	+	0.994161i	$0.534413\pi$
$774$			10.0000i			0.359443i
$775$		0			0
$776$	−18.0000			−0.646162
$777$		0			0
$778$	18.0000			0.645331
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$			24.0000i			0.857690i
$784$	−7.00000			−0.250000
$785$		0			0
$786$		0			0
$787$	−36.0000			−1.28326			−0.641631	−	0.767014i	$-0.721742\pi$
$787$	−36.0000			−1.28326			−0.641631	+	0.767014i	$0.721742\pi$
$788$	6.00000			0.213741
$789$	−48.0000			−1.70885
$790$		0			0
$791$		0			0
$792$		0			0
$793$	−30.0000	−	20.0000i	−1.06533	−	0.710221i
$794$	30.0000			1.06466
$795$		0			0
$796$	−16.0000			−0.567105
$797$		−	48.0000i		−	1.70025i	−0.526583	−	0.850124i	$-0.676527\pi$
$797$		−	48.0000i		−	1.70025i	0.526583	−	0.850124i	$-0.323473\pi$
$798$		0			0
$799$			72.0000i			2.54718i
$800$		0			0
$801$			12.0000i			0.423999i
$802$		−	12.0000i		−	0.423735i
$803$		0			0
$804$			24.0000i			0.846415i
$805$		0			0
$806$	12.0000	−	18.0000i	0.422682	−	0.634023i
$807$		−	60.0000i		−	2.11210i
$808$	−6.00000			−0.211079
$809$	−42.0000			−1.47664			−0.738321	−	0.674450i	$-0.764381\pi$
$809$	−42.0000			−1.47664			−0.738321	+	0.674450i	$0.764381\pi$
$810$		0			0
$811$			24.0000i			0.842754i	0.906886	+	0.421377i	$0.138453\pi$
$811$			24.0000i			0.842754i	−0.906886	+	0.421377i	$0.861547\pi$
$812$		0			0
$813$	12.0000			0.420858
$814$		0			0
$815$		0			0
$816$	−12.0000			−0.420084
$817$		0			0
$818$		−	12.0000i		−	0.419570i
$819$		0			0
$820$		0			0
$821$		−	18.0000i		−	0.628204i	−0.949389	−	0.314102i	$-0.898297\pi$
$821$		−	18.0000i		−	0.628204i	0.949389	−	0.314102i	$-0.101703\pi$
$822$			36.0000i			1.25564i
$823$		−	32.0000i		−	1.11545i	−0.830026	−	0.557725i	$-0.811674\pi$
$823$		−	32.0000i		−	1.11545i	0.830026	−	0.557725i	$-0.188326\pi$
$824$		−	4.00000i		−	0.139347i
$825$		0			0
$826$		0			0
$827$	36.0000			1.25184			0.625921	−	0.779886i	$-0.284723\pi$
$827$	36.0000			1.25184			0.625921	+	0.779886i	$0.284723\pi$
$828$		0			0
$829$	2.00000			0.0694629			0.0347314	−	0.999397i	$-0.488942\pi$
$829$	2.00000			0.0694629			0.0347314	+	0.999397i	$0.488942\pi$
$830$		0			0
$831$	16.0000			0.555034
$832$	−3.00000	−	2.00000i	−0.104006	−	0.0693375i
$833$			42.0000i			1.45521i
$834$			8.00000i			0.277017i
$835$		0			0
$836$		0			0
$837$	−24.0000			−0.829561
$838$	12.0000			0.414533
$839$			30.0000i			1.03572i	0.855467	+	0.517858i	$0.173270\pi$
$839$			30.0000i			1.03572i	−0.855467	+	0.517858i	$0.826730\pi$
$840$		0			0
$841$	7.00000			0.241379
$842$		−	30.0000i		−	1.03387i
$843$	−24.0000			−0.826604
$844$	4.00000			0.137686
$845$		0			0
$846$	−12.0000			−0.412568
$847$		0			0
$848$		0			0
$849$	28.0000			0.960958
$850$		0			0
$851$		0			0
$852$	12.0000			0.411113
$853$	−6.00000			−0.205436			−0.102718	−	0.994711i	$-0.532754\pi$
$853$	−6.00000			−0.205436			−0.102718	+	0.994711i	$0.532754\pi$
$854$		0			0
$855$		0			0
$856$			6.00000i			0.205076i
$857$			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
$859$	−40.0000			−1.36478			−0.682391	−	0.730987i	$-0.739060\pi$
$859$	−40.0000			−1.36478			−0.682391	+	0.730987i	$0.739060\pi$
$860$		0			0
$861$		0			0
$862$		−	30.0000i		−	1.02180i
$863$	−12.0000			−0.408485			−0.204242	−	0.978920i	$-0.565473\pi$
$863$	−12.0000			−0.408485			−0.204242	+	0.978920i	$0.565473\pi$
$864$			4.00000i			0.136083i
$865$		0			0
$866$		−	2.00000i		−	0.0679628i
$867$			38.0000i			1.29055i
$868$		0			0
$869$		0			0
$870$		0			0
$871$	36.0000	+	24.0000i	1.21981	+	0.813209i
$872$			6.00000i			0.203186i
$873$	−18.0000			−0.609208
$874$		0			0
$875$		0			0
$876$		−	12.0000i		−	0.405442i
$877$	−18.0000			−0.607817			−0.303908	−	0.952701i	$-0.598292\pi$
$877$	−18.0000			−0.607817			−0.303908	+	0.952701i	$0.598292\pi$
$878$	8.00000			0.269987
$879$			36.0000i			1.21425i
$880$		0			0
$881$	−30.0000			−1.01073			−0.505363	−	0.862907i	$-0.668641\pi$
$881$	−30.0000			−1.01073			−0.505363	+	0.862907i	$0.668641\pi$
$882$	−7.00000			−0.235702
$883$			2.00000i			0.0673054i	0.999434	+	0.0336527i	$0.0107140\pi$
$883$			2.00000i			0.0673054i	−0.999434	+	0.0336527i	$0.989286\pi$
$884$	−12.0000	+	18.0000i	−0.403604	+	0.605406i
$885$		0			0
$886$			6.00000i			0.201574i
$887$		−	48.0000i		−	1.61168i	−0.592132	−	0.805841i	$-0.701714\pi$
$887$		−	48.0000i		−	1.61168i	0.592132	−	0.805841i	$-0.298286\pi$
$888$			12.0000i			0.402694i
$889$		0			0
$890$		0			0
$891$		0			0
$892$	12.0000			0.401790
$893$		0			0
$894$	−12.0000			−0.401340
$895$		0			0
$896$		0			0
$897$		0			0
$898$		−	12.0000i		−	0.400445i
$899$			36.0000i			1.20067i
$900$		0			0
$901$		0			0
$902$		0			0
$903$		0			0
$904$			6.00000i			0.199557i
$905$		0			0
$906$	36.0000			1.19602
$907$			10.0000i			0.332045i	0.986122	+	0.166022i	$0.0530924\pi$
$907$			10.0000i			0.332045i	−0.986122	+	0.166022i	$0.946908\pi$
$908$	12.0000			0.398234
$909$	−6.00000			−0.199007
$910$		0			0
$911$	24.0000			0.795155			0.397578	−	0.917568i	$-0.369851\pi$
$911$	24.0000			0.795155			0.397578	+	0.917568i	$0.369851\pi$
$912$		0			0
$913$		0			0
$914$	−18.0000			−0.595387
$915$		0			0
$916$		−	6.00000i		−	0.198246i
$917$		0			0
$918$	24.0000			0.792118
$919$	−16.0000			−0.527791			−0.263896	−	0.964551i	$-0.585007\pi$
$919$	−16.0000			−0.527791			−0.263896	+	0.964551i	$0.585007\pi$
$920$		0			0
$921$			24.0000i			0.790827i
$922$		−	30.0000i		−	0.987997i
$923$	12.0000	−	18.0000i	0.394985	−	0.592477i
$924$		0			0
$925$		0			0
$926$	−12.0000			−0.394344
$927$		−	4.00000i		−	0.131377i
$928$	6.00000			0.196960
$929$		−	36.0000i		−	1.18112i	−0.806993	−	0.590561i	$-0.798907\pi$
$929$		−	36.0000i		−	1.18112i	0.806993	−	0.590561i	$-0.201093\pi$
$930$		0			0
$931$		0			0
$932$		−	18.0000i		−	0.589610i
$933$		−	48.0000i		−	1.57145i
$934$		−	18.0000i		−	0.588978i
$935$		0			0
$936$	−3.00000	−	2.00000i	−0.0980581	−	0.0653720i
$937$		−	38.0000i		−	1.24141i	−0.784046	−	0.620703i	$-0.786847\pi$
$937$		−	38.0000i		−	1.24141i	0.784046	−	0.620703i	$-0.213153\pi$
$938$		0			0
$939$	−20.0000			−0.652675
$940$		0			0
$941$			18.0000i			0.586783i	0.955992	+	0.293392i	$0.0947840\pi$
$941$			18.0000i			0.586783i	−0.955992	+	0.293392i	$0.905216\pi$
$942$	8.00000			0.260654
$943$		0			0
$944$		−	12.0000i		−	0.390567i
$945$		0			0
$946$		0			0
$947$	−12.0000			−0.389948			−0.194974	−	0.980808i	$-0.562462\pi$
$947$	−12.0000			−0.389948			−0.194974	+	0.980808i	$0.562462\pi$
$948$		−	16.0000i		−	0.519656i
$949$	−18.0000	−	12.0000i	−0.584305	−	0.389536i
$950$		0			0
$951$			36.0000i			1.16738i
$952$		0			0
$953$			18.0000i			0.583077i	0.956559	+	0.291539i	$0.0941672\pi$
$953$			18.0000i			0.583077i	−0.956559	+	0.291539i	$0.905833\pi$
$954$		0			0
$955$		0			0
$956$			18.0000i			0.582162i
$957$		0			0
$958$		−	42.0000i		−	1.35696i
$959$		0			0
$960$		0			0
$961$	−5.00000			−0.161290
$962$	18.0000	+	12.0000i	0.580343	+	0.386896i
$963$			6.00000i			0.193347i
$964$		−	24.0000i		−	0.772988i
$965$		0			0
$966$		0			0
$967$	−12.0000			−0.385894			−0.192947	−	0.981209i	$-0.561805\pi$
$967$	−12.0000			−0.385894			−0.192947	+	0.981209i	$0.561805\pi$
$968$	−11.0000			−0.353553
$969$		0			0
$970$		0			0
$971$	36.0000			1.15529			0.577647	−	0.816286i	$-0.303971\pi$
$971$	36.0000			1.15529			0.577647	+	0.816286i	$0.303971\pi$
$972$			10.0000i			0.320750i
$973$		0			0
$974$	−24.0000			−0.769010
$975$		0			0
$976$	10.0000			0.320092
$977$	−42.0000			−1.34370			−0.671850	−	0.740688i	$-0.734500\pi$
$977$	−42.0000			−1.34370			−0.671850	+	0.740688i	$0.734500\pi$
$978$			24.0000i			0.767435i
$979$		0			0
$980$		0			0
$981$			6.00000i			0.191565i
$982$	12.0000			0.382935
$983$	36.0000			1.14822			0.574111	−	0.818778i	$-0.305348\pi$
$983$	36.0000			1.14822			0.574111	+	0.818778i	$0.305348\pi$
$984$		0			0
$985$		0			0
$986$		−	36.0000i		−	1.14647i
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$	56.0000			1.77890			0.889449	−	0.457034i	$-0.151088\pi$
$991$	56.0000			1.77890			0.889449	+	0.457034i	$0.151088\pi$
$992$			6.00000i			0.190500i
$993$	24.0000			0.761617
$994$		0			0
$995$		0			0
$996$			24.0000i			0.760469i
$997$		−	28.0000i		−	0.886769i	−0.896332	−	0.443384i	$-0.853778\pi$
$997$		−	28.0000i		−	0.886769i	0.896332	−	0.443384i	$-0.146222\pi$
$998$			12.0000i			0.379853i
$999$		−	24.0000i		−	0.759326i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	650.2.c.b.649.1		2
5.2	odd	4		650.2.d.a.51.1		2
5.3	odd	4		130.2.d.a.51.2	yes	2
5.4	even	2		650.2.c.c.649.2		2
13.12	even	2		650.2.c.c.649.1		2
15.8	even	4		1170.2.b.a.181.1		2
20.3	even	4		1040.2.k.a.961.2		2
65.3	odd	12		1690.2.l.b.1161.1		4
65.8	even	4		1690.2.a.d.1.1		1
65.12	odd	4		650.2.d.a.51.2		2
65.18	even	4		1690.2.a.i.1.1		1
65.23	odd	12		1690.2.l.b.1161.2		4
65.28	even	12		1690.2.e.b.191.1		2
65.33	even	12		1690.2.e.f.991.1		2
65.38	odd	4		130.2.d.a.51.1	✓	2
65.43	odd	12		1690.2.l.b.361.1		4
65.47	even	4		8450.2.a.o.1.1		1
65.48	odd	12		1690.2.l.b.361.2		4
65.57	even	4		8450.2.a.b.1.1		1
65.58	even	12		1690.2.e.b.991.1		2
65.63	even	12		1690.2.e.f.191.1		2
65.64	even	2	inner	650.2.c.b.649.2		2
195.38	even	4		1170.2.b.a.181.2		2
260.103	even	4		1040.2.k.a.961.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
130.2.d.a.51.1	✓	2	65.38	odd	4
130.2.d.a.51.2	yes	2	5.3	odd	4
650.2.c.b.649.1		2	1.1	even	1	trivial
650.2.c.b.649.2		2	65.64	even	2	inner
650.2.c.c.649.1		2	13.12	even	2
650.2.c.c.649.2		2	5.4	even	2
650.2.d.a.51.1		2	5.2	odd	4
650.2.d.a.51.2		2	65.12	odd	4
1040.2.k.a.961.1		2	260.103	even	4
1040.2.k.a.961.2		2	20.3	even	4
1170.2.b.a.181.1		2	15.8	even	4
1170.2.b.a.181.2		2	195.38	even	4
1690.2.a.d.1.1		1	65.8	even	4
1690.2.a.i.1.1		1	65.18	even	4
1690.2.e.b.191.1		2	65.28	even	12
1690.2.e.b.991.1		2	65.58	even	12
1690.2.e.f.191.1		2	65.63	even	12
1690.2.e.f.991.1		2	65.33	even	12
1690.2.l.b.361.1		4	65.43	odd	12
1690.2.l.b.361.2		4	65.48	odd	12
1690.2.l.b.1161.1		4	65.3	odd	12
1690.2.l.b.1161.2		4	65.23	odd	12
8450.2.a.b.1.1		1	65.57	even	4
8450.2.a.o.1.1		1	65.47	even	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.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} -2.00000i q^{3} +1.00000 q^{4} +2.00000i q^{6} -1.00000 q^{8} -1.00000 q^{9} -2.00000i q^{12} +(-3.00000 - 2.00000i) q^{13} +1.00000 q^{16} -6.00000i q^{17} +1.00000 q^{18} +2.00000i q^{24} +(3.00000 + 2.00000i) q^{26} -4.00000i q^{27} -6.00000 q^{29} -6.00000i q^{31} -1.00000 q^{32} +6.00000i q^{34} -1.00000 q^{36} +6.00000 q^{37} +(-4.00000 + 6.00000i) q^{39} +10.0000i q^{43} -12.0000 q^{47} -2.00000i q^{48} -7.00000 q^{49} -12.0000 q^{51} +(-3.00000 - 2.00000i) q^{52} +4.00000i q^{54} +6.00000 q^{58} -12.0000i q^{59} +10.0000 q^{61} +6.00000i q^{62} +1.00000 q^{64} -12.0000 q^{67} -6.00000i q^{68} +6.00000i q^{71} +1.00000 q^{72} +6.00000 q^{73} -6.00000 q^{74} +(4.00000 - 6.00000i) q^{78} +8.00000 q^{79} -11.0000 q^{81} -12.0000 q^{83} -10.0000i q^{86} +12.0000i q^{87} -12.0000i q^{89} -12.0000 q^{93} +12.0000 q^{94} +2.00000i q^{96} +18.0000 q^{97} +7.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{2} + 2 q^{4} - 2 q^{8} - 2 q^{9} - 6 q^{13} + 2 q^{16} + 2 q^{18} + 6 q^{26} - 12 q^{29} - 2 q^{32} - 2 q^{36} + 12 q^{37} - 8 q^{39} - 24 q^{47} - 14 q^{49} - 24 q^{51} - 6 q^{52} + 12 q^{58}+ \cdots + 14 q^{98}+O(q^{100}) \) 2 * q - 2 * q^2 + 2 * q^4 - 2 * q^8 - 2 * q^9 - 6 * q^13 + 2 * q^16 + 2 * q^18 + 6 * q^26 - 12 * q^29 - 2 * q^32 - 2 * q^36 + 12 * q^37 - 8 * q^39 - 24 * q^47 - 14 * q^49 - 24 * q^51 - 6 * q^52 + 12 * q^58 + 20 * q^61 + 2 * q^64 - 24 * q^67 + 2 * q^72 + 12 * q^73 - 12 * q^74 + 8 * q^78 + 16 * q^79 - 22 * q^81 - 24 * q^83 - 24 * q^93 + 24 * q^94 + 36 * q^97 + 14 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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