LMFDB - Embedded newform 3264.2.c.e.577.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3264,2,Mod(577,3264)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3264, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0, 1]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("3264.577");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3264 = 2^{6} \cdot 3 \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3264.c (of order $2$, degree $1$, not minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	no
Analytic conductor:	$26.0631712197$
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:	$ 1 $
Twist minimal:	no (minimal twist has level 51)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			577.1
Root			$1.00000i$ of defining polynomial
Character	$\chi$	$=$	3264.577
Dual form			3264.2.c.e.577.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{3} -4.00000i q^{7} -1.00000 q^{9} -4.00000i q^{11} -2.00000 q^{13} +(1.00000 + 4.00000i) q^{17} +4.00000 q^{19} -4.00000 q^{21} -4.00000i q^{23} +5.00000 q^{25} +1.00000i q^{27} -4.00000i q^{31} -4.00000 q^{33} -8.00000i q^{37} +2.00000i q^{39} -8.00000i q^{41} +4.00000 q^{43} -8.00000 q^{47} -9.00000 q^{49} +(4.00000 - 1.00000i) q^{51} -6.00000 q^{53} -4.00000i q^{57} +12.0000 q^{59} +8.00000i q^{61} +4.00000i q^{63} -12.0000 q^{67} -4.00000 q^{69} +12.0000i q^{71} -5.00000i q^{75} -16.0000 q^{77} -4.00000i q^{79} +1.00000 q^{81} -12.0000 q^{83} -10.0000 q^{89} +8.00000i q^{91} -4.00000 q^{93} +16.0000i q^{97} +4.00000i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{9} - 4 q^{13} + 2 q^{17} + 8 q^{19} - 8 q^{21} + 10 q^{25} - 8 q^{33} + 8 q^{43} - 16 q^{47} - 18 q^{49} + 8 q^{51} - 12 q^{53} + 24 q^{59} - 24 q^{67} - 8 q^{69} - 32 q^{77} + 2 q^{81} - 24 q^{83}+ \cdots - 8 q^{93}+O(q^{100}) $ 2 * q - 2 * q^9 - 4 * q^13 + 2 * q^17 + 8 * q^19 - 8 * q^21 + 10 * q^25 - 8 * q^33 + 8 * q^43 - 16 * q^47 - 18 * q^49 + 8 * q^51 - 12 * q^53 + 24 * q^59 - 24 * q^67 - 8 * q^69 - 32 * q^77 + 2 * q^81 - 24 * q^83 - 20 * q^89 - 8 * q^93

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3264\mathbb{Z}\right)^\times$.

$n$	$511$	$2177$	$2245$	$2689$
$\chi(n)$	$1$	$1$	$1$	$-1$

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 $

$2$		0			0
$2$		0			0
$3$		−	1.00000i		−	0.577350i
$3$		−	1.00000i		−	0.577350i
$4$		0			0
$5$		0			0		1.00000			$0$
$5$		0			0		−1.00000			$\pi$
$6$		0			0
$7$		−	4.00000i		−	1.51186i	−0.654654	−	0.755929i	$-0.727186\pi$
$7$		−	4.00000i		−	1.51186i	0.654654	−	0.755929i	$-0.272814\pi$
$8$		0			0
$9$	−1.00000			−0.333333
$10$		0			0
$11$		−	4.00000i		−	1.20605i	−0.797724	−	0.603023i	$-0.793963\pi$
$11$		−	4.00000i		−	1.20605i	0.797724	−	0.603023i	$-0.206037\pi$
$12$		0			0
$13$	−2.00000			−0.554700			−0.277350	−	0.960769i	$-0.589456\pi$
$13$	−2.00000			−0.554700			−0.277350	+	0.960769i	$0.589456\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	1.00000	+	4.00000i	0.242536	+	0.970143i
$17$	1.00000	+	4.00000i	0.242536	+	0.970143i
$18$		0			0
$19$	4.00000			0.917663			0.458831	−	0.888523i	$-0.348268\pi$
$19$	4.00000			0.917663			0.458831	+	0.888523i	$0.348268\pi$
$20$		0			0
$21$	−4.00000			−0.872872
$22$		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
$25$	5.00000			1.00000
$26$		0			0
$27$			1.00000i			0.192450i
$28$		0			0
$29$		0			0		1.00000			$0$
$29$		0			0		−1.00000			$\pi$
$30$		0			0
$31$		−	4.00000i		−	0.718421i	−0.933257	−	0.359211i	$-0.883046\pi$
$31$		−	4.00000i		−	0.718421i	0.933257	−	0.359211i	$-0.116954\pi$
$32$		0			0
$33$	−4.00000			−0.696311
$34$		0			0
$35$		0			0
$36$		0			0
$37$		−	8.00000i		−	1.31519i	−0.753371	−	0.657596i	$-0.771573\pi$
$37$		−	8.00000i		−	1.31519i	0.753371	−	0.657596i	$-0.228427\pi$
$38$		0			0
$39$			2.00000i			0.320256i
$40$		0			0
$41$		−	8.00000i		−	1.24939i	−0.780869	−	0.624695i	$-0.785223\pi$
$41$		−	8.00000i		−	1.24939i	0.780869	−	0.624695i	$-0.214777\pi$
$42$		0			0
$43$	4.00000			0.609994			0.304997	−	0.952353i	$-0.401344\pi$
$43$	4.00000			0.609994			0.304997	+	0.952353i	$0.401344\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−8.00000			−1.16692			−0.583460	−	0.812142i	$-0.698301\pi$
$47$	−8.00000			−1.16692			−0.583460	+	0.812142i	$0.698301\pi$
$48$		0			0
$49$	−9.00000			−1.28571
$50$		0			0
$51$	4.00000	−	1.00000i	0.560112	−	0.140028i
$52$		0			0
$53$	−6.00000			−0.824163			−0.412082	−	0.911147i	$-0.635198\pi$
$53$	−6.00000			−0.824163			−0.412082	+	0.911147i	$0.635198\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$		−	4.00000i		−	0.529813i
$58$		0			0
$59$	12.0000			1.56227			0.781133	−	0.624364i	$-0.214642\pi$
$59$	12.0000			1.56227			0.781133	+	0.624364i	$0.214642\pi$
$60$		0			0
$61$			8.00000i			1.02430i	0.858898	+	0.512148i	$0.171150\pi$
$61$			8.00000i			1.02430i	−0.858898	+	0.512148i	$0.828850\pi$
$62$		0			0
$63$			4.00000i			0.503953i
$64$		0			0
$65$		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	$0.761888\pi$
$68$		0			0
$69$	−4.00000			−0.481543
$70$		0			0
$71$			12.0000i			1.42414i	0.702109	+	0.712069i	$0.252242\pi$
$71$			12.0000i			1.42414i	−0.702109	+	0.712069i	$0.747758\pi$
$72$		0			0
$73$		0			0		1.00000			$0$
$73$		0			0		−1.00000			$\pi$
$74$		0			0
$75$		−	5.00000i		−	0.577350i
$76$		0			0
$77$	−16.0000			−1.82337
$78$		0			0
$79$		−	4.00000i		−	0.450035i	−0.974355	−	0.225018i	$-0.927756\pi$
$79$		−	4.00000i		−	0.450035i	0.974355	−	0.225018i	$-0.0722440\pi$
$80$		0			0
$81$	1.00000			0.111111
$82$		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
$86$		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
$91$			8.00000i			0.838628i
$92$		0			0
$93$	−4.00000			−0.414781
$94$		0			0
$95$		0			0
$96$		0			0
$97$			16.0000i			1.62455i	0.583272	+	0.812277i	$0.301772\pi$
$97$			16.0000i			1.62455i	−0.583272	+	0.812277i	$0.698228\pi$
$98$		0			0
$99$			4.00000i			0.402015i
$100$		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
$103$		0			0			−	1.00000i	$-0.5\pi$
$103$		0			0				1.00000i	$0.5\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$			12.0000i			1.16008i	0.814587	+	0.580042i	$0.196964\pi$
$107$			12.0000i			1.16008i	−0.814587	+	0.580042i	$0.803036\pi$
$108$		0			0
$109$		−	8.00000i		−	0.766261i	−0.923694	−	0.383131i	$-0.874846\pi$
$109$		−	8.00000i		−	0.766261i	0.923694	−	0.383131i	$-0.125154\pi$
$110$		0			0
$111$	−8.00000			−0.759326
$112$		0			0
$113$		−	8.00000i		−	0.752577i	−0.926503	−	0.376288i	$-0.877200\pi$
$113$		−	8.00000i		−	0.752577i	0.926503	−	0.376288i	$-0.122800\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	2.00000			0.184900
$118$		0			0
$119$	16.0000	−	4.00000i	1.46672	−	0.366679i
$120$		0			0
$121$	−5.00000			−0.454545
$122$		0			0
$123$	−8.00000			−0.721336
$124$		0			0
$125$		0			0
$126$		0			0
$127$	8.00000			0.709885			0.354943	−	0.934888i	$-0.384500\pi$
$127$	8.00000			0.709885			0.354943	+	0.934888i	$0.384500\pi$
$128$		0			0
$129$		−	4.00000i		−	0.352180i
$130$		0			0
$131$		−	4.00000i		−	0.349482i	−0.984614	−	0.174741i	$-0.944091\pi$
$131$		−	4.00000i		−	0.349482i	0.984614	−	0.174741i	$-0.0559088\pi$
$132$		0			0
$133$		−	16.0000i		−	1.38738i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−10.0000			−0.854358			−0.427179	−	0.904167i	$-0.640493\pi$
$137$	−10.0000			−0.854358			−0.427179	+	0.904167i	$0.640493\pi$
$138$		0			0
$139$		−	4.00000i		−	0.339276i	−0.985506	−	0.169638i	$-0.945740\pi$
$139$		−	4.00000i		−	0.339276i	0.985506	−	0.169638i	$-0.0542598\pi$
$140$		0			0
$141$			8.00000i			0.673722i
$142$		0			0
$143$			8.00000i			0.668994i
$144$		0			0
$145$		0			0
$146$		0			0
$147$			9.00000i			0.742307i
$148$		0			0
$149$	−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
$151$		0			0			−	1.00000i	$-0.5\pi$
$151$		0			0				1.00000i	$0.5\pi$
$152$		0			0
$153$	−1.00000	−	4.00000i	−0.0808452	−	0.323381i
$154$		0			0
$155$		0			0
$156$		0			0
$157$	2.00000			0.159617			0.0798087	−	0.996810i	$-0.474569\pi$
$157$	2.00000			0.159617			0.0798087	+	0.996810i	$0.474569\pi$
$158$		0			0
$159$			6.00000i			0.475831i
$160$		0			0
$161$	−16.0000			−1.26098
$162$		0			0
$163$		−	20.0000i		−	1.56652i	−0.621694	−	0.783260i	$-0.713555\pi$
$163$		−	20.0000i		−	1.56652i	0.621694	−	0.783260i	$-0.286445\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$			12.0000i			0.928588i	0.885681	+	0.464294i	$0.153692\pi$
$167$			12.0000i			0.928588i	−0.885681	+	0.464294i	$0.846308\pi$
$168$		0			0
$169$	−9.00000			−0.692308
$170$		0			0
$171$	−4.00000			−0.305888
$172$		0			0
$173$			16.0000i			1.21646i	0.793762	+	0.608229i	$0.208120\pi$
$173$			16.0000i			1.21646i	−0.793762	+	0.608229i	$0.791880\pi$
$174$		0			0
$175$		−	20.0000i		−	1.51186i
$176$		0			0
$177$		−	12.0000i		−	0.901975i
$178$		0			0
$179$	−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
$181$			8.00000i			0.594635i	0.954779	+	0.297318i	$0.0960920\pi$
$181$			8.00000i			0.594635i	−0.954779	+	0.297318i	$0.903908\pi$
$182$		0			0
$183$	8.00000			0.591377
$184$		0			0
$185$		0			0
$186$		0			0
$187$	16.0000	−	4.00000i	1.17004	−	0.292509i
$188$		0			0
$189$	4.00000			0.290957
$190$		0			0
$191$	−8.00000			−0.578860			−0.289430	−	0.957199i	$-0.593466\pi$
$191$	−8.00000			−0.578860			−0.289430	+	0.957199i	$0.593466\pi$
$192$		0			0
$193$		0			0		1.00000			$0$
$193$		0			0		−1.00000			$\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		−	16.0000i		−	1.13995i	−0.821661	−	0.569976i	$-0.806952\pi$
$197$		−	16.0000i		−	1.13995i	0.821661	−	0.569976i	$-0.193048\pi$
$198$		0			0
$199$		−	20.0000i		−	1.41776i	−0.705328	−	0.708881i	$-0.749200\pi$
$199$		−	20.0000i		−	1.41776i	0.705328	−	0.708881i	$-0.250800\pi$
$200$		0			0
$201$			12.0000i			0.846415i
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$			4.00000i			0.278019i
$208$		0			0
$209$		−	16.0000i		−	1.10674i
$210$		0			0
$211$		−	4.00000i		−	0.275371i	−0.990476	−	0.137686i	$-0.956034\pi$
$211$		−	4.00000i		−	0.275371i	0.990476	−	0.137686i	$-0.0439664\pi$
$212$		0			0
$213$	12.0000			0.822226
$214$		0			0
$215$		0			0
$216$		0			0
$217$	−16.0000			−1.08615
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−2.00000	−	8.00000i	−0.134535	−	0.538138i
$222$		0			0
$223$	16.0000			1.07144			0.535720	−	0.844396i	$-0.320040\pi$
$223$	16.0000			1.07144			0.535720	+	0.844396i	$0.320040\pi$
$224$		0			0
$225$	−5.00000			−0.333333
$226$		0			0
$227$			12.0000i			0.796468i	0.917284	+	0.398234i	$0.130377\pi$
$227$			12.0000i			0.796468i	−0.917284	+	0.398234i	$0.869623\pi$
$228$		0			0
$229$	−10.0000			−0.660819			−0.330409	−	0.943838i	$-0.607187\pi$
$229$	−10.0000			−0.660819			−0.330409	+	0.943838i	$0.607187\pi$
$230$		0			0
$231$			16.0000i			1.05272i
$232$		0			0
$233$			8.00000i			0.524097i	0.965055	+	0.262049i	$0.0843981\pi$
$233$			8.00000i			0.524097i	−0.965055	+	0.262049i	$0.915602\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	−4.00000			−0.259828
$238$		0			0
$239$	−8.00000			−0.517477			−0.258738	−	0.965947i	$-0.583307\pi$
$239$	−8.00000			−0.517477			−0.258738	+	0.965947i	$0.583307\pi$
$240$		0			0
$241$		−	16.0000i		−	1.03065i	−0.856995	−	0.515325i	$-0.827671\pi$
$241$		−	16.0000i		−	1.03065i	0.856995	−	0.515325i	$-0.172329\pi$
$242$		0			0
$243$		−	1.00000i		−	0.0641500i
$244$		0			0
$245$		0			0
$246$		0			0
$247$	−8.00000			−0.509028
$248$		0			0
$249$			12.0000i			0.760469i
$250$		0			0
$251$	−12.0000			−0.757433			−0.378717	−	0.925513i	$-0.623635\pi$
$251$	−12.0000			−0.757433			−0.378717	+	0.925513i	$0.623635\pi$
$252$		0			0
$253$	−16.0000			−1.00591
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−2.00000			−0.124757			−0.0623783	−	0.998053i	$-0.519869\pi$
$257$	−2.00000			−0.124757			−0.0623783	+	0.998053i	$0.519869\pi$
$258$		0			0
$259$	−32.0000			−1.98838
$260$		0			0
$261$		0			0
$262$		0			0
$263$	24.0000			1.47990			0.739952	−	0.672660i	$-0.234848\pi$
$263$	24.0000			1.47990			0.739952	+	0.672660i	$0.234848\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$			10.0000i			0.611990i
$268$		0			0
$269$		0			0		1.00000			$0$
$269$		0			0		−1.00000			$\pi$
$270$		0			0
$271$		0			0			−	1.00000i	$-0.5\pi$
$271$		0			0				1.00000i	$0.5\pi$
$272$		0			0
$273$	8.00000			0.484182
$274$		0			0
$275$		−	20.0000i		−	1.20605i
$276$		0			0
$277$			8.00000i			0.480673i	0.970690	+	0.240337i	$0.0772579\pi$
$277$			8.00000i			0.480673i	−0.970690	+	0.240337i	$0.922742\pi$
$278$		0			0
$279$			4.00000i			0.239474i
$280$		0			0
$281$	10.0000			0.596550			0.298275	−	0.954480i	$-0.403589\pi$
$281$	10.0000			0.596550			0.298275	+	0.954480i	$0.403589\pi$
$282$		0			0
$283$		−	4.00000i		−	0.237775i	−0.992908	−	0.118888i	$-0.962067\pi$
$283$		−	4.00000i		−	0.237775i	0.992908	−	0.118888i	$-0.0379328\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−32.0000			−1.88890
$288$		0			0
$289$	−15.0000	+	8.00000i	−0.882353	+	0.470588i
$290$		0			0
$291$	16.0000			0.937937
$292$		0			0
$293$	10.0000			0.584206			0.292103	−	0.956387i	$-0.405645\pi$
$293$	10.0000			0.584206			0.292103	+	0.956387i	$0.405645\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$	4.00000			0.232104
$298$		0			0
$299$			8.00000i			0.462652i
$300$		0			0
$301$		−	16.0000i		−	0.922225i
$302$		0			0
$303$		−	6.00000i		−	0.344691i
$304$		0			0
$305$		0			0
$306$		0			0
$307$	12.0000			0.684876			0.342438	−	0.939540i	$-0.388747\pi$
$307$	12.0000			0.684876			0.342438	+	0.939540i	$0.388747\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$			12.0000i			0.680458i	0.940343	+	0.340229i	$0.110505\pi$
$311$			12.0000i			0.680458i	−0.940343	+	0.340229i	$0.889495\pi$
$312$		0			0
$313$		−	16.0000i		−	0.904373i	−0.891923	−	0.452187i	$-0.850644\pi$
$313$		−	16.0000i		−	0.904373i	0.891923	−	0.452187i	$-0.149356\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$			32.0000i			1.79730i	0.438667	+	0.898650i	$0.355451\pi$
$317$			32.0000i			1.79730i	−0.438667	+	0.898650i	$0.644549\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$	12.0000			0.669775
$322$		0			0
$323$	4.00000	+	16.0000i	0.222566	+	0.890264i
$324$		0			0
$325$	−10.0000			−0.554700
$326$		0			0
$327$	−8.00000			−0.442401
$328$		0			0
$329$			32.0000i			1.76422i
$330$		0			0
$331$	20.0000			1.09930			0.549650	−	0.835395i	$-0.314761\pi$
$331$	20.0000			1.09930			0.549650	+	0.835395i	$0.314761\pi$
$332$		0			0
$333$			8.00000i			0.438397i
$334$		0			0
$335$		0			0
$336$		0			0
$337$		−	16.0000i		−	0.871576i	−0.900049	−	0.435788i	$-0.856470\pi$
$337$		−	16.0000i		−	0.871576i	0.900049	−	0.435788i	$-0.143530\pi$
$338$		0			0
$339$	−8.00000			−0.434500
$340$		0			0
$341$	−16.0000			−0.866449
$342$		0			0
$343$			8.00000i			0.431959i
$344$		0			0
$345$		0			0
$346$		0			0
$347$		−	4.00000i		−	0.214731i	−0.994220	−	0.107366i	$-0.965758\pi$
$347$		−	4.00000i		−	0.214731i	0.994220	−	0.107366i	$-0.0342415\pi$
$348$		0			0
$349$	2.00000			0.107058			0.0535288	−	0.998566i	$-0.482953\pi$
$349$	2.00000			0.107058			0.0535288	+	0.998566i	$0.482953\pi$
$350$		0			0
$351$		−	2.00000i		−	0.106752i
$352$		0			0
$353$	18.0000			0.958043			0.479022	−	0.877803i	$-0.340992\pi$
$353$	18.0000			0.958043			0.479022	+	0.877803i	$0.340992\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$	−4.00000	−	16.0000i	−0.211702	−	0.846810i
$358$		0			0
$359$	32.0000			1.68890			0.844448	−	0.535638i	$-0.179929\pi$
$359$	32.0000			1.68890			0.844448	+	0.535638i	$0.179929\pi$
$360$		0			0
$361$	−3.00000			−0.157895
$362$		0			0
$363$			5.00000i			0.262432i
$364$		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	$0.898600\pi$
$368$		0			0
$369$			8.00000i			0.416463i
$370$		0			0
$371$			24.0000i			1.24602i
$372$		0			0
$373$	−26.0000			−1.34623			−0.673114	−	0.739538i	$-0.735044\pi$
$373$	−26.0000			−1.34623			−0.673114	+	0.739538i	$0.735044\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$		−	20.0000i		−	1.02733i	−0.857991	−	0.513665i	$-0.828287\pi$
$379$		−	20.0000i		−	1.02733i	0.857991	−	0.513665i	$-0.171713\pi$
$380$		0			0
$381$		−	8.00000i		−	0.409852i
$382$		0			0
$383$	−16.0000			−0.817562			−0.408781	−	0.912633i	$-0.634046\pi$
$383$	−16.0000			−0.817562			−0.408781	+	0.912633i	$0.634046\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	−4.00000			−0.203331
$388$		0			0
$389$	6.00000			0.304212			0.152106	−	0.988364i	$-0.451394\pi$
$389$	6.00000			0.304212			0.152106	+	0.988364i	$0.451394\pi$
$390$		0			0
$391$	16.0000	−	4.00000i	0.809155	−	0.202289i
$392$		0			0
$393$	−4.00000			−0.201773
$394$		0			0
$395$		0			0
$396$		0			0
$397$			8.00000i			0.401508i	0.979642	+	0.200754i	$0.0643393\pi$
$397$			8.00000i			0.401508i	−0.979642	+	0.200754i	$0.935661\pi$
$398$		0			0
$399$	−16.0000			−0.801002
$400$		0			0
$401$		−	24.0000i		−	1.19850i	−0.800561	−	0.599251i	$-0.795465\pi$
$401$		−	24.0000i		−	1.19850i	0.800561	−	0.599251i	$-0.204535\pi$
$402$		0			0
$403$			8.00000i			0.398508i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−32.0000			−1.58618
$408$		0			0
$409$	10.0000			0.494468			0.247234	−	0.968956i	$-0.420478\pi$
$409$	10.0000			0.494468			0.247234	+	0.968956i	$0.420478\pi$
$410$		0			0
$411$			10.0000i			0.493264i
$412$		0			0
$413$		−	48.0000i		−	2.36193i
$414$		0			0
$415$		0			0
$416$		0			0
$417$	−4.00000			−0.195881
$418$		0			0
$419$		−	4.00000i		−	0.195413i	−0.995215	−	0.0977064i	$-0.968849\pi$
$419$		−	4.00000i		−	0.195413i	0.995215	−	0.0977064i	$-0.0311506\pi$
$420$		0			0
$421$	22.0000			1.07221			0.536107	−	0.844150i	$-0.319894\pi$
$421$	22.0000			1.07221			0.536107	+	0.844150i	$0.319894\pi$
$422$		0			0
$423$	8.00000			0.388973
$424$		0			0
$425$	5.00000	+	20.0000i	0.242536	+	0.970143i
$426$		0			0
$427$	32.0000			1.54859
$428$		0			0
$429$	8.00000			0.386244
$430$		0			0
$431$		−	20.0000i		−	0.963366i	−0.876346	−	0.481683i	$-0.840026\pi$
$431$		−	20.0000i		−	0.963366i	0.876346	−	0.481683i	$-0.159974\pi$
$432$		0			0
$433$	−2.00000			−0.0961139			−0.0480569	−	0.998845i	$-0.515303\pi$
$433$	−2.00000			−0.0961139			−0.0480569	+	0.998845i	$0.515303\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		−	16.0000i		−	0.765384i
$438$		0			0
$439$		−	36.0000i		−	1.71819i	−0.511819	−	0.859093i	$-0.671028\pi$
$439$		−	36.0000i		−	1.71819i	0.511819	−	0.859093i	$-0.328972\pi$
$440$		0			0
$441$	9.00000			0.428571
$442$		0			0
$443$	20.0000			0.950229			0.475114	−	0.879924i	$-0.342407\pi$
$443$	20.0000			0.950229			0.475114	+	0.879924i	$0.342407\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$			6.00000i			0.283790i
$448$		0			0
$449$			8.00000i			0.377543i	0.982021	+	0.188772i	$0.0604506\pi$
$449$			8.00000i			0.377543i	−0.982021	+	0.188772i	$0.939549\pi$
$450$		0			0
$451$	−32.0000			−1.50682
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	38.0000			1.77757			0.888783	−	0.458329i	$-0.151552\pi$
$457$	38.0000			1.77757			0.888783	+	0.458329i	$0.151552\pi$
$458$		0			0
$459$	−4.00000	+	1.00000i	−0.186704	+	0.0466760i
$460$		0			0
$461$	34.0000			1.58354			0.791769	−	0.610821i	$-0.209160\pi$
$461$	34.0000			1.58354			0.791769	+	0.610821i	$0.209160\pi$
$462$		0			0
$463$	−40.0000			−1.85896			−0.929479	−	0.368875i	$-0.879743\pi$
$463$	−40.0000			−1.85896			−0.929479	+	0.368875i	$0.879743\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−4.00000			−0.185098			−0.0925490	−	0.995708i	$-0.529501\pi$
$467$	−4.00000			−0.185098			−0.0925490	+	0.995708i	$0.529501\pi$
$468$		0			0
$469$			48.0000i			2.21643i
$470$		0			0
$471$		−	2.00000i		−	0.0921551i
$472$		0			0
$473$		−	16.0000i		−	0.735681i
$474$		0			0
$475$	20.0000			0.917663
$476$		0			0
$477$	6.00000			0.274721
$478$		0			0
$479$			12.0000i			0.548294i	0.961688	+	0.274147i	$0.0883955\pi$
$479$			12.0000i			0.548294i	−0.961688	+	0.274147i	$0.911605\pi$
$480$		0			0
$481$			16.0000i			0.729537i
$482$		0			0
$483$			16.0000i			0.728025i
$484$		0			0
$485$		0			0
$486$		0			0
$487$		−	4.00000i		−	0.181257i	−0.995885	−	0.0906287i	$-0.971112\pi$
$487$		−	4.00000i		−	0.181257i	0.995885	−	0.0906287i	$-0.0288876\pi$
$488$		0			0
$489$	−20.0000			−0.904431
$490$		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
$493$		0			0
$494$		0			0
$495$		0			0
$496$		0			0
$497$	48.0000			2.15309
$498$		0			0
$499$		−	36.0000i		−	1.61158i	−0.592200	−	0.805791i	$-0.701741\pi$
$499$		−	36.0000i		−	1.61158i	0.592200	−	0.805791i	$-0.298259\pi$
$500$		0			0
$501$	12.0000			0.536120
$502$		0			0
$503$			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
$507$			9.00000i			0.399704i
$508$		0			0
$509$	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
$512$		0			0
$513$			4.00000i			0.176604i
$514$		0			0
$515$		0			0
$516$		0			0
$517$			32.0000i			1.40736i
$518$		0			0
$519$	16.0000			0.702322
$520$		0			0
$521$			24.0000i			1.05146i	0.850652	+	0.525730i	$0.176208\pi$
$521$			24.0000i			1.05146i	−0.850652	+	0.525730i	$0.823792\pi$
$522$		0			0
$523$	12.0000			0.524723			0.262362	−	0.964970i	$-0.415499\pi$
$523$	12.0000			0.524723			0.262362	+	0.964970i	$0.415499\pi$
$524$		0			0
$525$	−20.0000			−0.872872
$526$		0			0
$527$	16.0000	−	4.00000i	0.696971	−	0.174243i
$528$		0			0
$529$	7.00000			0.304348
$530$		0			0
$531$	−12.0000			−0.520756
$532$		0			0
$533$			16.0000i			0.693037i
$534$		0			0
$535$		0			0
$536$		0			0
$537$			4.00000i			0.172613i
$538$		0			0
$539$			36.0000i			1.55063i
$540$		0			0
$541$		−	40.0000i		−	1.71973i	−0.510518	−	0.859867i	$-0.670546\pi$
$541$		−	40.0000i		−	1.71973i	0.510518	−	0.859867i	$-0.329454\pi$
$542$		0			0
$543$	8.00000			0.343313
$544$		0			0
$545$		0			0
$546$		0			0
$547$		−	36.0000i		−	1.53925i	−0.638497	−	0.769624i	$-0.720443\pi$
$547$		−	36.0000i		−	1.53925i	0.638497	−	0.769624i	$-0.279557\pi$
$548$		0			0
$549$		−	8.00000i		−	0.341432i
$550$		0			0
$551$		0			0
$552$		0			0
$553$	−16.0000			−0.680389
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−18.0000			−0.762684			−0.381342	−	0.924434i	$-0.624538\pi$
$557$	−18.0000			−0.762684			−0.381342	+	0.924434i	$0.624538\pi$
$558$		0			0
$559$	−8.00000			−0.338364
$560$		0			0
$561$	−4.00000	−	16.0000i	−0.168880	−	0.675521i
$562$		0			0
$563$	−44.0000			−1.85438			−0.927189	−	0.374593i	$-0.877783\pi$
$563$	−44.0000			−1.85438			−0.927189	+	0.374593i	$0.877783\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$		−	4.00000i		−	0.167984i
$568$		0			0
$569$	26.0000			1.08998			0.544988	−	0.838444i	$-0.316534\pi$
$569$	26.0000			1.08998			0.544988	+	0.838444i	$0.316534\pi$
$570$		0			0
$571$		−	20.0000i		−	0.836974i	−0.908223	−	0.418487i	$-0.862561\pi$
$571$		−	20.0000i		−	0.836974i	0.908223	−	0.418487i	$-0.137439\pi$
$572$		0			0
$573$			8.00000i			0.334205i
$574$		0			0
$575$		−	20.0000i		−	0.834058i
$576$		0			0
$577$	30.0000			1.24892			0.624458	−	0.781058i	$-0.285320\pi$
$577$	30.0000			1.24892			0.624458	+	0.781058i	$0.285320\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$			48.0000i			1.99138i
$582$		0			0
$583$			24.0000i			0.993978i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−12.0000			−0.495293			−0.247647	−	0.968850i	$-0.579657\pi$
$587$	−12.0000			−0.495293			−0.247647	+	0.968850i	$0.579657\pi$
$588$		0			0
$589$		−	16.0000i		−	0.659269i
$590$		0			0
$591$	−16.0000			−0.658152
$592$		0			0
$593$	−46.0000			−1.88899			−0.944497	−	0.328521i	$-0.893450\pi$
$593$	−46.0000			−1.88899			−0.944497	+	0.328521i	$0.893450\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	−20.0000			−0.818546
$598$		0			0
$599$	−8.00000			−0.326871			−0.163436	−	0.986554i	$-0.552258\pi$
$599$	−8.00000			−0.326871			−0.163436	+	0.986554i	$0.552258\pi$
$600$		0			0
$601$			16.0000i			0.652654i	0.945257	+	0.326327i	$0.105811\pi$
$601$			16.0000i			0.652654i	−0.945257	+	0.326327i	$0.894189\pi$
$602$		0			0
$603$	12.0000			0.488678
$604$		0			0
$605$		0			0
$606$		0			0
$607$		−	4.00000i		−	0.162355i	−0.996700	−	0.0811775i	$-0.974132\pi$
$607$		−	4.00000i		−	0.162355i	0.996700	−	0.0811775i	$-0.0258681\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	16.0000			0.647291
$612$		0			0
$613$	−6.00000			−0.242338			−0.121169	−	0.992632i	$-0.538664\pi$
$613$	−6.00000			−0.242338			−0.121169	+	0.992632i	$0.538664\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		−	24.0000i		−	0.966204i	−0.875564	−	0.483102i	$-0.839510\pi$
$617$		−	24.0000i		−	0.966204i	0.875564	−	0.483102i	$-0.160490\pi$
$618$		0			0
$619$		−	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
$621$	4.00000			0.160514
$622$		0			0
$623$			40.0000i			1.60257i
$624$		0			0
$625$	25.0000			1.00000
$626$		0			0
$627$	−16.0000			−0.638978
$628$		0			0
$629$	32.0000	−	8.00000i	1.27592	−	0.318981i
$630$		0			0
$631$	−8.00000			−0.318475			−0.159237	−	0.987240i	$-0.550904\pi$
$631$	−8.00000			−0.318475			−0.159237	+	0.987240i	$0.550904\pi$
$632$		0			0
$633$	−4.00000			−0.158986
$634$		0			0
$635$		0			0
$636$		0			0
$637$	18.0000			0.713186
$638$		0			0
$639$		−	12.0000i		−	0.474713i
$640$		0			0
$641$			8.00000i			0.315981i	0.987441	+	0.157991i	$0.0505015\pi$
$641$			8.00000i			0.315981i	−0.987441	+	0.157991i	$0.949498\pi$
$642$		0			0
$643$		−	4.00000i		−	0.157745i	−0.996885	−	0.0788723i	$-0.974868\pi$
$643$		−	4.00000i		−	0.157745i	0.996885	−	0.0788723i	$-0.0251319\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	32.0000			1.25805			0.629025	−	0.777385i	$-0.283454\pi$
$647$	32.0000			1.25805			0.629025	+	0.777385i	$0.283454\pi$
$648$		0			0
$649$		−	48.0000i		−	1.88416i
$650$		0			0
$651$			16.0000i			0.627089i
$652$		0			0
$653$			32.0000i			1.25226i	0.779720	+	0.626128i	$0.215361\pi$
$653$			32.0000i			1.25226i	−0.779720	+	0.626128i	$0.784639\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$		0			0
$658$		0			0
$659$	4.00000			0.155818			0.0779089	−	0.996960i	$-0.475176\pi$
$659$	4.00000			0.155818			0.0779089	+	0.996960i	$0.475176\pi$
$660$		0			0
$661$	42.0000			1.63361			0.816805	−	0.576913i	$-0.195743\pi$
$661$	42.0000			1.63361			0.816805	+	0.576913i	$0.195743\pi$
$662$		0			0
$663$	−8.00000	+	2.00000i	−0.310694	+	0.0776736i
$664$		0			0
$665$		0			0
$666$		0			0
$667$		0			0
$668$		0			0
$669$		−	16.0000i		−	0.618596i
$670$		0			0
$671$	32.0000			1.23535
$672$		0			0
$673$			32.0000i			1.23351i	0.787155	+	0.616755i	$0.211553\pi$
$673$			32.0000i			1.23351i	−0.787155	+	0.616755i	$0.788447\pi$
$674$		0			0
$675$			5.00000i			0.192450i
$676$		0			0
$677$		−	48.0000i		−	1.84479i	−0.386248	−	0.922395i	$-0.626229\pi$
$677$		−	48.0000i		−	1.84479i	0.386248	−	0.922395i	$-0.373771\pi$
$678$		0			0
$679$	64.0000			2.45609
$680$		0			0
$681$	12.0000			0.459841
$682$		0			0
$683$			12.0000i			0.459167i	0.973289	+	0.229584i	$0.0737364\pi$
$683$			12.0000i			0.459167i	−0.973289	+	0.229584i	$0.926264\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$			10.0000i			0.381524i
$688$		0			0
$689$	12.0000			0.457164
$690$		0			0
$691$		−	20.0000i		−	0.760836i	−0.924815	−	0.380418i	$-0.875780\pi$
$691$		−	20.0000i		−	0.760836i	0.924815	−	0.380418i	$-0.124220\pi$
$692$		0			0
$693$	16.0000			0.607790
$694$		0			0
$695$		0			0
$696$		0			0
$697$	32.0000	−	8.00000i	1.21209	−	0.303022i
$698$		0			0
$699$	8.00000			0.302588
$700$		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
$703$		−	32.0000i		−	1.20690i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		−	24.0000i		−	0.902613i
$708$		0			0
$709$			40.0000i			1.50223i	0.660171	+	0.751116i	$0.270484\pi$
$709$			40.0000i			1.50223i	−0.660171	+	0.751116i	$0.729516\pi$
$710$		0			0
$711$			4.00000i			0.150012i
$712$		0			0
$713$	−16.0000			−0.599205
$714$		0			0
$715$		0			0
$716$		0			0
$717$			8.00000i			0.298765i
$718$		0			0
$719$		−	36.0000i		−	1.34257i	−0.741198	−	0.671287i	$-0.765742\pi$
$719$		−	36.0000i		−	1.34257i	0.741198	−	0.671287i	$-0.234258\pi$
$720$		0			0
$721$		0			0
$722$		0			0
$723$	−16.0000			−0.595046
$724$		0			0
$725$		0			0
$726$		0			0
$727$	8.00000			0.296704			0.148352	−	0.988935i	$-0.452603\pi$
$727$	8.00000			0.296704			0.148352	+	0.988935i	$0.452603\pi$
$728$		0			0
$729$	−1.00000			−0.0370370
$730$		0			0
$731$	4.00000	+	16.0000i	0.147945	+	0.591781i
$732$		0			0
$733$	34.0000			1.25582			0.627909	−	0.778287i	$-0.283911\pi$
$733$	34.0000			1.25582			0.627909	+	0.778287i	$0.283911\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$			48.0000i			1.76810i
$738$		0			0
$739$	4.00000			0.147142			0.0735712	−	0.997290i	$-0.476560\pi$
$739$	4.00000			0.147142			0.0735712	+	0.997290i	$0.476560\pi$
$740$		0			0
$741$			8.00000i			0.293887i
$742$		0			0
$743$		−	36.0000i		−	1.32071i	−0.750953	−	0.660356i	$-0.770405\pi$
$743$		−	36.0000i		−	1.32071i	0.750953	−	0.660356i	$-0.229595\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$	12.0000			0.439057
$748$		0			0
$749$	48.0000			1.75388
$750$		0			0
$751$		−	20.0000i		−	0.729810i	−0.931045	−	0.364905i	$-0.881101\pi$
$751$		−	20.0000i		−	0.729810i	0.931045	−	0.364905i	$-0.118899\pi$
$752$		0			0
$753$			12.0000i			0.437304i
$754$		0			0
$755$		0			0
$756$		0			0
$757$	38.0000			1.38113			0.690567	−	0.723269i	$-0.257361\pi$
$757$	38.0000			1.38113			0.690567	+	0.723269i	$0.257361\pi$
$758$		0			0
$759$			16.0000i			0.580763i
$760$		0			0
$761$	6.00000			0.217500			0.108750	−	0.994069i	$-0.465315\pi$
$761$	6.00000			0.217500			0.108750	+	0.994069i	$0.465315\pi$
$762$		0			0
$763$	−32.0000			−1.15848
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−24.0000			−0.866590
$768$		0			0
$769$	−34.0000			−1.22607			−0.613036	−	0.790055i	$-0.710052\pi$
$769$	−34.0000			−1.22607			−0.613036	+	0.790055i	$0.710052\pi$
$770$		0			0
$771$			2.00000i			0.0720282i
$772$		0			0
$773$	6.00000			0.215805			0.107903	−	0.994161i	$-0.465587\pi$
$773$	6.00000			0.215805			0.107903	+	0.994161i	$0.465587\pi$
$774$		0			0
$775$		−	20.0000i		−	0.718421i
$776$		0			0
$777$			32.0000i			1.14799i
$778$		0			0
$779$		−	32.0000i		−	1.14652i
$780$		0			0
$781$	48.0000			1.71758
$782$		0			0
$783$		0			0
$784$		0			0
$785$		0			0
$786$		0			0
$787$			12.0000i			0.427754i	0.976861	+	0.213877i	$0.0686091\pi$
$787$			12.0000i			0.427754i	−0.976861	+	0.213877i	$0.931391\pi$
$788$		0			0
$789$		−	24.0000i		−	0.854423i
$790$		0			0
$791$	−32.0000			−1.13779
$792$		0			0
$793$		−	16.0000i		−	0.568177i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	18.0000			0.637593			0.318796	−	0.947823i	$-0.396721\pi$
$797$	18.0000			0.637593			0.318796	+	0.947823i	$0.396721\pi$
$798$		0			0
$799$	−8.00000	−	32.0000i	−0.283020	−	1.13208i
$800$		0			0
$801$	10.0000			0.353333
$802$		0			0
$803$		0			0
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$		−	40.0000i		−	1.40633i	−0.711029	−	0.703163i	$-0.751771\pi$
$809$		−	40.0000i		−	1.40633i	0.711029	−	0.703163i	$-0.248229\pi$
$810$		0			0
$811$		−	4.00000i		−	0.140459i	−0.997531	−	0.0702295i	$-0.977627\pi$
$811$		−	4.00000i		−	0.140459i	0.997531	−	0.0702295i	$-0.0223732\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		0			0
$816$		0			0
$817$	16.0000			0.559769
$818$		0			0
$819$		−	8.00000i		−	0.279543i
$820$		0			0
$821$		−	32.0000i		−	1.11681i	−0.829569	−	0.558404i	$-0.811414\pi$
$821$		−	32.0000i		−	1.11681i	0.829569	−	0.558404i	$-0.188586\pi$
$822$		0			0
$823$			44.0000i			1.53374i	0.641800	+	0.766872i	$0.278188\pi$
$823$			44.0000i			1.53374i	−0.641800	+	0.766872i	$0.721812\pi$
$824$		0			0
$825$	−20.0000			−0.696311
$826$		0			0
$827$			12.0000i			0.417281i	0.977992	+	0.208640i	$0.0669038\pi$
$827$			12.0000i			0.417281i	−0.977992	+	0.208640i	$0.933096\pi$
$828$		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
$831$	8.00000			0.277517
$832$		0			0
$833$	−9.00000	−	36.0000i	−0.311832	−	1.24733i
$834$		0			0
$835$		0			0
$836$		0			0
$837$	4.00000			0.138260
$838$		0			0
$839$			44.0000i			1.51905i	0.650479	+	0.759524i	$0.274568\pi$
$839$			44.0000i			1.51905i	−0.650479	+	0.759524i	$0.725432\pi$
$840$		0			0
$841$	29.0000			1.00000
$842$		0			0
$843$		−	10.0000i		−	0.344418i
$844$		0			0
$845$		0			0
$846$		0			0
$847$			20.0000i			0.687208i
$848$		0			0
$849$	−4.00000			−0.137280
$850$		0			0
$851$	−32.0000			−1.09695
$852$		0			0
$853$			24.0000i			0.821744i	0.911693	+	0.410872i	$0.134776\pi$
$853$			24.0000i			0.821744i	−0.911693	+	0.410872i	$0.865224\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$			8.00000i			0.273275i	0.990621	+	0.136637i	$0.0436295\pi$
$857$			8.00000i			0.273275i	−0.990621	+	0.136637i	$0.956370\pi$
$858$		0			0
$859$	4.00000			0.136478			0.0682391	−	0.997669i	$-0.478262\pi$
$859$	4.00000			0.136478			0.0682391	+	0.997669i	$0.478262\pi$
$860$		0			0
$861$			32.0000i			1.09056i
$862$		0			0
$863$		0			0			−	1.00000i	$-0.5\pi$
$863$		0			0				1.00000i	$0.5\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	8.00000	+	15.0000i	0.271694	+	0.509427i
$868$		0			0
$869$	−16.0000			−0.542763
$870$		0			0
$871$	24.0000			0.813209
$872$		0			0
$873$		−	16.0000i		−	0.541518i
$874$		0			0
$875$		0			0
$876$		0			0
$877$			40.0000i			1.35070i	0.737496	+	0.675352i	$0.236008\pi$
$877$			40.0000i			1.35070i	−0.737496	+	0.675352i	$0.763992\pi$
$878$		0			0
$879$		−	10.0000i		−	0.337292i
$880$		0			0
$881$			24.0000i			0.808581i	0.914631	+	0.404290i	$0.132481\pi$
$881$			24.0000i			0.808581i	−0.914631	+	0.404290i	$0.867519\pi$
$882$		0			0
$883$	20.0000			0.673054			0.336527	−	0.941674i	$-0.390748\pi$
$883$	20.0000			0.673054			0.336527	+	0.941674i	$0.390748\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		−	4.00000i		−	0.134307i	−0.997743	−	0.0671534i	$-0.978608\pi$
$887$		−	4.00000i		−	0.134307i	0.997743	−	0.0671534i	$-0.0213917\pi$
$888$		0			0
$889$		−	32.0000i		−	1.07325i
$890$		0			0
$891$		−	4.00000i		−	0.134005i
$892$		0			0
$893$	−32.0000			−1.07084
$894$		0			0
$895$		0			0
$896$		0			0
$897$	8.00000			0.267112
$898$		0			0
$899$		0			0
$900$		0			0
$901$	−6.00000	−	24.0000i	−0.199889	−	0.799556i
$902$		0			0
$903$	−16.0000			−0.532447
$904$		0			0
$905$		0			0
$906$		0			0
$907$			28.0000i			0.929725i	0.885383	+	0.464862i	$0.153896\pi$
$907$			28.0000i			0.929725i	−0.885383	+	0.464862i	$0.846104\pi$
$908$		0			0
$909$	−6.00000			−0.199007
$910$		0			0
$911$			28.0000i			0.927681i	0.885919	+	0.463841i	$0.153529\pi$
$911$			28.0000i			0.927681i	−0.885919	+	0.463841i	$0.846471\pi$
$912$		0			0
$913$			48.0000i			1.58857i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−16.0000			−0.528367
$918$		0			0
$919$	24.0000			0.791687			0.395843	−	0.918318i	$-0.370452\pi$
$919$	24.0000			0.791687			0.395843	+	0.918318i	$0.370452\pi$
$920$		0			0
$921$		−	12.0000i		−	0.395413i
$922$		0			0
$923$		−	24.0000i		−	0.789970i
$924$		0			0
$925$		−	40.0000i		−	1.31519i
$926$		0			0
$927$		0			0
$928$		0			0
$929$			24.0000i			0.787414i	0.919236	+	0.393707i	$0.128808\pi$
$929$			24.0000i			0.787414i	−0.919236	+	0.393707i	$0.871192\pi$
$930$		0			0
$931$	−36.0000			−1.17985
$932$		0			0
$933$	12.0000			0.392862
$934$		0			0
$935$		0			0
$936$		0			0
$937$	26.0000			0.849383			0.424691	−	0.905338i	$-0.360383\pi$
$937$	26.0000			0.849383			0.424691	+	0.905338i	$0.360383\pi$
$938$		0			0
$939$	−16.0000			−0.522140
$940$		0			0
$941$			48.0000i			1.56476i	0.622804	+	0.782378i	$0.285993\pi$
$941$			48.0000i			1.56476i	−0.622804	+	0.782378i	$0.714007\pi$
$942$		0			0
$943$	−32.0000			−1.04206
$944$		0			0
$945$		0			0
$946$		0			0
$947$			12.0000i			0.389948i	0.980808	+	0.194974i	$0.0624622\pi$
$947$			12.0000i			0.389948i	−0.980808	+	0.194974i	$0.937538\pi$
$948$		0			0
$949$		0			0
$950$		0			0
$951$	32.0000			1.03767
$952$		0			0
$953$	−42.0000			−1.36051			−0.680257	−	0.732974i	$-0.738132\pi$
$953$	−42.0000			−1.36051			−0.680257	+	0.732974i	$0.738132\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$			40.0000i			1.29167i
$960$		0			0
$961$	15.0000			0.483871
$962$		0			0
$963$		−	12.0000i		−	0.386695i
$964$		0			0
$965$		0			0
$966$		0			0
$967$	24.0000			0.771788			0.385894	−	0.922543i	$-0.373893\pi$
$967$	24.0000			0.771788			0.385894	+	0.922543i	$0.373893\pi$
$968$		0			0
$969$	16.0000	−	4.00000i	0.513994	−	0.128499i
$970$		0			0
$971$	−36.0000			−1.15529			−0.577647	−	0.816286i	$-0.696029\pi$
$971$	−36.0000			−1.15529			−0.577647	+	0.816286i	$0.696029\pi$
$972$		0			0
$973$	−16.0000			−0.512936
$974$		0			0
$975$			10.0000i			0.320256i
$976$		0			0
$977$	−46.0000			−1.47167			−0.735835	−	0.677161i	$-0.763210\pi$
$977$	−46.0000			−1.47167			−0.735835	+	0.677161i	$0.763210\pi$
$978$		0			0
$979$			40.0000i			1.27841i
$980$		0			0
$981$			8.00000i			0.255420i
$982$		0			0
$983$		−	20.0000i		−	0.637901i	−0.947771	−	0.318950i	$-0.896670\pi$
$983$		−	20.0000i		−	0.637901i	0.947771	−	0.318950i	$-0.103330\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$	32.0000			1.01857
$988$		0			0
$989$		−	16.0000i		−	0.508770i
$990$		0			0
$991$			12.0000i			0.381193i	0.981669	+	0.190596i	$0.0610421\pi$
$991$			12.0000i			0.381193i	−0.981669	+	0.190596i	$0.938958\pi$
$992$		0			0
$993$		−	20.0000i		−	0.634681i
$994$		0			0
$995$		0			0
$996$		0			0
$997$		−	8.00000i		−	0.253363i	−0.991943	−	0.126681i	$-0.959567\pi$
$997$		−	8.00000i		−	0.253363i	0.991943	−	0.126681i	$-0.0404325\pi$
$998$		0			0
$999$	8.00000			0.253109

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3264.2.c.e.577.1		2
4.3	odd	2		3264.2.c.d.577.2		2
8.3	odd	2		816.2.c.c.577.1		2
8.5	even	2		51.2.d.b.16.2	yes	2
17.16	even	2	inner	3264.2.c.e.577.2		2
24.5	odd	2		153.2.d.a.118.1		2
24.11	even	2		2448.2.c.j.577.2		2
40.13	odd	4		1275.2.d.b.424.1		2
40.29	even	2		1275.2.g.a.526.1		2
40.37	odd	4		1275.2.d.d.424.2		2
68.67	odd	2		3264.2.c.d.577.1		2
136.5	odd	16		867.2.h.d.757.1		8
136.13	even	4		867.2.a.a.1.1		1
136.21	even	4		867.2.a.b.1.1		1
136.29	odd	16		867.2.h.d.757.2		8
136.37	odd	16		867.2.h.d.688.1		8
136.45	odd	16		867.2.h.d.712.1		8
136.53	even	8		867.2.e.d.829.2		4
136.61	odd	16		867.2.h.d.733.1		8
136.67	odd	2		816.2.c.c.577.2		2
136.77	even	8		867.2.e.d.616.1		4
136.93	even	8		867.2.e.d.616.2		4
136.101	even	2		51.2.d.b.16.1	✓	2
136.109	odd	16		867.2.h.d.733.2		8
136.117	even	8		867.2.e.d.829.1		4
136.125	odd	16		867.2.h.d.712.2		8
136.133	odd	16		867.2.h.d.688.2		8
408.101	odd	2		153.2.d.a.118.2		2
408.149	odd	4		2601.2.a.i.1.1		1
408.203	even	2		2448.2.c.j.577.1		2
408.293	odd	4		2601.2.a.j.1.1		1
680.237	odd	4		1275.2.d.b.424.2		2
680.373	odd	4		1275.2.d.d.424.1		2
680.509	even	2		1275.2.g.a.526.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
51.2.d.b.16.1	✓	2	136.101	even	2
51.2.d.b.16.2	yes	2	8.5	even	2
153.2.d.a.118.1		2	24.5	odd	2
153.2.d.a.118.2		2	408.101	odd	2
816.2.c.c.577.1		2	8.3	odd	2
816.2.c.c.577.2		2	136.67	odd	2
867.2.a.a.1.1		1	136.13	even	4
867.2.a.b.1.1		1	136.21	even	4
867.2.e.d.616.1		4	136.77	even	8
867.2.e.d.616.2		4	136.93	even	8
867.2.e.d.829.1		4	136.117	even	8
867.2.e.d.829.2		4	136.53	even	8
867.2.h.d.688.1		8	136.37	odd	16
867.2.h.d.688.2		8	136.133	odd	16
867.2.h.d.712.1		8	136.45	odd	16
867.2.h.d.712.2		8	136.125	odd	16
867.2.h.d.733.1		8	136.61	odd	16
867.2.h.d.733.2		8	136.109	odd	16
867.2.h.d.757.1		8	136.5	odd	16
867.2.h.d.757.2		8	136.29	odd	16
1275.2.d.b.424.1		2	40.13	odd	4
1275.2.d.b.424.2		2	680.237	odd	4
1275.2.d.d.424.1		2	680.373	odd	4
1275.2.d.d.424.2		2	40.37	odd	4
1275.2.g.a.526.1		2	40.29	even	2
1275.2.g.a.526.2		2	680.509	even	2
2448.2.c.j.577.1		2	408.203	even	2
2448.2.c.j.577.2		2	24.11	even	2
2601.2.a.i.1.1		1	408.149	odd	4
2601.2.a.j.1.1		1	408.293	odd	4
3264.2.c.d.577.1		2	68.67	odd	2
3264.2.c.d.577.2		2	4.3	odd	2
3264.2.c.e.577.1		2	1.1	even	1	trivial
3264.2.c.e.577.2		2	17.16	even	2	inner

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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 3264 = 2^{6} \cdot 3 \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3264.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.00000i q^{3} -4.00000i q^{7} -1.00000 q^{9} -4.00000i q^{11} -2.00000 q^{13} +(1.00000 + 4.00000i) q^{17} +4.00000 q^{19} -4.00000 q^{21} -4.00000i q^{23} +5.00000 q^{25} +1.00000i q^{27} -4.00000i q^{31} -4.00000 q^{33} -8.00000i q^{37} +2.00000i q^{39} -8.00000i q^{41} +4.00000 q^{43} -8.00000 q^{47} -9.00000 q^{49} +(4.00000 - 1.00000i) q^{51} -6.00000 q^{53} -4.00000i q^{57} +12.0000 q^{59} +8.00000i q^{61} +4.00000i q^{63} -12.0000 q^{67} -4.00000 q^{69} +12.0000i q^{71} -5.00000i q^{75} -16.0000 q^{77} -4.00000i q^{79} +1.00000 q^{81} -12.0000 q^{83} -10.0000 q^{89} +8.00000i q^{91} -4.00000 q^{93} +16.0000i q^{97} +4.00000i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{9} - 4 q^{13} + 2 q^{17} + 8 q^{19} - 8 q^{21} + 10 q^{25} - 8 q^{33} + 8 q^{43} - 16 q^{47} - 18 q^{49} + 8 q^{51} - 12 q^{53} + 24 q^{59} - 24 q^{67} - 8 q^{69} - 32 q^{77} + 2 q^{81} - 24 q^{83}+ \cdots - 8 q^{93}+O(q^{100}) \) 2 * q - 2 * q^9 - 4 * q^13 + 2 * q^17 + 8 * q^19 - 8 * q^21 + 10 * q^25 - 8 * q^33 + 8 * q^43 - 16 * q^47 - 18 * q^49 + 8 * q^51 - 12 * q^53 + 24 * q^59 - 24 * q^67 - 8 * q^69 - 32 * q^77 + 2 * q^81 - 24 * q^83 - 20 * q^89 - 8 * q^93

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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