LMFDB - Embedded newform 3264.2.c.i.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 102)
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.i.577.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{3} -2.00000i q^{5} +2.00000i q^{7} -1.00000 q^{9} +6.00000 q^{13} -2.00000 q^{15} +(-1.00000 + 4.00000i) q^{17} +2.00000 q^{21} -6.00000i q^{23} +1.00000 q^{25} +1.00000i q^{27} +6.00000i q^{29} +10.0000i q^{31} +4.00000 q^{35} +2.00000i q^{37} -6.00000i q^{39} +4.00000 q^{43} +2.00000i q^{45} -8.00000 q^{47} +3.00000 q^{49} +(4.00000 + 1.00000i) q^{51} +6.00000 q^{53} -10.0000i q^{61} -2.00000i q^{63} -12.0000i q^{65} +8.00000 q^{67} -6.00000 q^{69} +10.0000i q^{71} +16.0000i q^{73} -1.00000i q^{75} +6.00000i q^{79} +1.00000 q^{81} -16.0000 q^{83} +(8.00000 + 2.00000i) q^{85} +6.00000 q^{87} +10.0000 q^{89} +12.0000i q^{91} +10.0000 q^{93} -12.0000i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{9} + 12 q^{13} - 4 q^{15} - 2 q^{17} + 4 q^{21} + 2 q^{25} + 8 q^{35} + 8 q^{43} - 16 q^{47} + 6 q^{49} + 8 q^{51} + 12 q^{53} + 16 q^{67} - 12 q^{69} + 2 q^{81} - 32 q^{83} + 16 q^{85} + 12 q^{87}+ \cdots + 20 q^{93}+O(q^{100}) $ 2 * q - 2 * q^9 + 12 * q^13 - 4 * q^15 - 2 * q^17 + 4 * q^21 + 2 * q^25 + 8 * q^35 + 8 * q^43 - 16 * q^47 + 6 * q^49 + 8 * q^51 + 12 * q^53 + 16 * q^67 - 12 * q^69 + 2 * q^81 - 32 * q^83 + 16 * q^85 + 12 * q^87 + 20 * q^89 + 20 * 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
$5$		−	2.00000i		−	0.894427i	−0.894427	−	0.447214i	$-0.852416\pi$
$5$		−	2.00000i		−	0.894427i	0.894427	−	0.447214i	$-0.147584\pi$
$6$		0			0
$7$			2.00000i			0.755929i	0.925820	+	0.377964i	$0.123376\pi$
$7$			2.00000i			0.755929i	−0.925820	+	0.377964i	$0.876624\pi$
$8$		0			0
$9$	−1.00000			−0.333333
$10$		0			0
$11$		0			0		1.00000			$0$
$11$		0			0		−1.00000			$\pi$
$12$		0			0
$13$	6.00000			1.66410			0.832050	−	0.554700i	$-0.187167\pi$
$13$	6.00000			1.66410			0.832050	+	0.554700i	$0.187167\pi$
$14$		0			0
$15$	−2.00000			−0.516398
$16$		0			0
$17$	−1.00000	+	4.00000i	−0.242536	+	0.970143i
$17$	−1.00000	+	4.00000i	−0.242536	+	0.970143i
$18$		0			0
$19$		0			0			−	1.00000i	$-0.5\pi$
$19$		0			0				1.00000i	$0.5\pi$
$20$		0			0
$21$	2.00000			0.436436
$22$		0			0
$23$		−	6.00000i		−	1.25109i	−0.780189	−	0.625543i	$-0.784877\pi$
$23$		−	6.00000i		−	1.25109i	0.780189	−	0.625543i	$-0.215123\pi$
$24$		0			0
$25$	1.00000			0.200000
$26$		0			0
$27$			1.00000i			0.192450i
$28$		0			0
$29$			6.00000i			1.11417i	0.830455	+	0.557086i	$0.188081\pi$
$29$			6.00000i			1.11417i	−0.830455	+	0.557086i	$0.811919\pi$
$30$		0			0
$31$			10.0000i			1.79605i	0.439941	+	0.898027i	$0.354999\pi$
$31$			10.0000i			1.79605i	−0.439941	+	0.898027i	$0.645001\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	4.00000			0.676123
$36$		0			0
$37$			2.00000i			0.328798i	0.986394	+	0.164399i	$0.0525685\pi$
$37$			2.00000i			0.328798i	−0.986394	+	0.164399i	$0.947432\pi$
$38$		0			0
$39$		−	6.00000i		−	0.960769i
$40$		0			0
$41$		0			0		1.00000			$0$
$41$		0			0		−1.00000			$\pi$
$42$		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
$45$			2.00000i			0.298142i
$46$		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
$49$	3.00000			0.428571
$50$		0			0
$51$	4.00000	+	1.00000i	0.560112	+	0.140028i
$52$		0			0
$53$	6.00000			0.824163			0.412082	−	0.911147i	$-0.364802\pi$
$53$	6.00000			0.824163			0.412082	+	0.911147i	$0.364802\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$		0			0
$58$		0			0
$59$		0			0			−	1.00000i	$-0.5\pi$
$59$		0			0				1.00000i	$0.5\pi$
$60$		0			0
$61$		−	10.0000i		−	1.28037i	−0.768221	−	0.640184i	$-0.778858\pi$
$61$		−	10.0000i		−	1.28037i	0.768221	−	0.640184i	$-0.221142\pi$
$62$		0			0
$63$		−	2.00000i		−	0.251976i
$64$		0			0
$65$		−	12.0000i		−	1.48842i
$66$		0			0
$67$	8.00000			0.977356			0.488678	−	0.872464i	$-0.337479\pi$
$67$	8.00000			0.977356			0.488678	+	0.872464i	$0.337479\pi$
$68$		0			0
$69$	−6.00000			−0.722315
$70$		0			0
$71$			10.0000i			1.18678i	0.804914	+	0.593391i	$0.202211\pi$
$71$			10.0000i			1.18678i	−0.804914	+	0.593391i	$0.797789\pi$
$72$		0			0
$73$			16.0000i			1.87266i	0.351123	+	0.936329i	$0.385800\pi$
$73$			16.0000i			1.87266i	−0.351123	+	0.936329i	$0.614200\pi$
$74$		0			0
$75$		−	1.00000i		−	0.115470i
$76$		0			0
$77$		0			0
$78$		0			0
$79$			6.00000i			0.675053i	0.941316	+	0.337526i	$0.109590\pi$
$79$			6.00000i			0.675053i	−0.941316	+	0.337526i	$0.890410\pi$
$80$		0			0
$81$	1.00000			0.111111
$82$		0			0
$83$	−16.0000			−1.75623			−0.878114	−	0.478451i	$-0.841198\pi$
$83$	−16.0000			−1.75623			−0.878114	+	0.478451i	$0.841198\pi$
$84$		0			0
$85$	8.00000	+	2.00000i	0.867722	+	0.216930i
$86$		0			0
$87$	6.00000			0.643268
$88$		0			0
$89$	10.0000			1.06000			0.529999	−	0.847998i	$-0.322192\pi$
$89$	10.0000			1.06000			0.529999	+	0.847998i	$0.322192\pi$
$90$		0			0
$91$			12.0000i			1.25794i
$92$		0			0
$93$	10.0000			1.03695
$94$		0			0
$95$		0			0
$96$		0			0
$97$		−	12.0000i		−	1.21842i	−0.793011	−	0.609208i	$-0.791488\pi$
$97$		−	12.0000i		−	1.21842i	0.793011	−	0.609208i	$-0.208512\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	18.0000			1.79107			0.895533	−	0.444994i	$-0.146794\pi$
$101$	18.0000			1.79107			0.895533	+	0.444994i	$0.146794\pi$
$102$		0			0
$103$	16.0000			1.57653			0.788263	−	0.615338i	$-0.210980\pi$
$103$	16.0000			1.57653			0.788263	+	0.615338i	$0.210980\pi$
$104$		0			0
$105$		−	4.00000i		−	0.390360i
$106$		0			0
$107$			8.00000i			0.773389i	0.922208	+	0.386695i	$0.126383\pi$
$107$			8.00000i			0.773389i	−0.922208	+	0.386695i	$0.873617\pi$
$108$		0			0
$109$		−	14.0000i		−	1.34096i	−0.741929	−	0.670478i	$-0.766089\pi$
$109$		−	14.0000i		−	1.34096i	0.741929	−	0.670478i	$-0.233911\pi$
$110$		0			0
$111$	2.00000			0.189832
$112$		0			0
$113$		−	4.00000i		−	0.376288i	−0.982141	−	0.188144i	$-0.939753\pi$
$113$		−	4.00000i		−	0.376288i	0.982141	−	0.188144i	$-0.0602472\pi$
$114$		0			0
$115$	−12.0000			−1.11901
$116$		0			0
$117$	−6.00000			−0.554700
$118$		0			0
$119$	−8.00000	−	2.00000i	−0.733359	−	0.183340i
$120$		0			0
$121$	11.0000			1.00000
$122$		0			0
$123$		0			0
$124$		0			0
$125$		−	12.0000i		−	1.07331i
$126$		0			0
$127$	−8.00000			−0.709885			−0.354943	−	0.934888i	$-0.615500\pi$
$127$	−8.00000			−0.709885			−0.354943	+	0.934888i	$0.615500\pi$
$128$		0			0
$129$		−	4.00000i		−	0.352180i
$130$		0			0
$131$		0			0		1.00000			$0$
$131$		0			0		−1.00000			$\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$	2.00000			0.172133
$136$		0			0
$137$	−2.00000			−0.170872			−0.0854358	−	0.996344i	$-0.527228\pi$
$137$	−2.00000			−0.170872			−0.0854358	+	0.996344i	$0.527228\pi$
$138$		0			0
$139$		−	16.0000i		−	1.35710i	−0.734553	−	0.678551i	$-0.762608\pi$
$139$		−	16.0000i		−	1.35710i	0.734553	−	0.678551i	$-0.237392\pi$
$140$		0			0
$141$			8.00000i			0.673722i
$142$		0			0
$143$		0			0
$144$		0			0
$145$	12.0000			0.996546
$146$		0			0
$147$		−	3.00000i		−	0.247436i
$148$		0			0
$149$	−10.0000			−0.819232			−0.409616	−	0.912258i	$-0.634337\pi$
$149$	−10.0000			−0.819232			−0.409616	+	0.912258i	$0.634337\pi$
$150$		0			0
$151$	8.00000			0.651031			0.325515	−	0.945537i	$-0.394462\pi$
$151$	8.00000			0.651031			0.325515	+	0.945537i	$0.394462\pi$
$152$		0			0
$153$	1.00000	−	4.00000i	0.0808452	−	0.323381i
$154$		0			0
$155$	20.0000			1.60644
$156$		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
$161$	12.0000			0.945732
$162$		0			0
$163$		−	4.00000i		−	0.313304i	−0.987654	−	0.156652i	$-0.949930\pi$
$163$		−	4.00000i		−	0.313304i	0.987654	−	0.156652i	$-0.0500701\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		−	18.0000i		−	1.39288i	−0.717614	−	0.696441i	$-0.754766\pi$
$167$		−	18.0000i		−	1.39288i	0.717614	−	0.696441i	$-0.245234\pi$
$168$		0			0
$169$	23.0000			1.76923
$170$		0			0
$171$		0			0
$172$		0			0
$173$			14.0000i			1.06440i	0.846619	+	0.532200i	$0.178635\pi$
$173$			14.0000i			1.06440i	−0.846619	+	0.532200i	$0.821365\pi$
$174$		0			0
$175$			2.00000i			0.151186i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−20.0000			−1.49487			−0.747435	−	0.664335i	$-0.768715\pi$
$179$	−20.0000			−1.49487			−0.747435	+	0.664335i	$0.768715\pi$
$180$		0			0
$181$		−	10.0000i		−	0.743294i	−0.928374	−	0.371647i	$-0.878793\pi$
$181$		−	10.0000i		−	0.743294i	0.928374	−	0.371647i	$-0.121207\pi$
$182$		0			0
$183$	−10.0000			−0.739221
$184$		0			0
$185$	4.00000			0.294086
$186$		0			0
$187$		0			0
$188$		0			0
$189$	−2.00000			−0.145479
$190$		0			0
$191$	8.00000			0.578860			0.289430	−	0.957199i	$-0.406534\pi$
$191$	8.00000			0.578860			0.289430	+	0.957199i	$0.406534\pi$
$192$		0			0
$193$		−	4.00000i		−	0.287926i	−0.989583	−	0.143963i	$-0.954015\pi$
$193$		−	4.00000i		−	0.287926i	0.989583	−	0.143963i	$-0.0459847\pi$
$194$		0			0
$195$	−12.0000			−0.859338
$196$		0			0
$197$		−	18.0000i		−	1.28245i	−0.767354	−	0.641223i	$-0.778427\pi$
$197$		−	18.0000i		−	1.28245i	0.767354	−	0.641223i	$-0.221573\pi$
$198$		0			0
$199$			6.00000i			0.425329i	0.977125	+	0.212664i	$0.0682141\pi$
$199$			6.00000i			0.425329i	−0.977125	+	0.212664i	$0.931786\pi$
$200$		0			0
$201$		−	8.00000i		−	0.564276i
$202$		0			0
$203$	−12.0000			−0.842235
$204$		0			0
$205$		0			0
$206$		0			0
$207$			6.00000i			0.417029i
$208$		0			0
$209$		0			0
$210$		0			0
$211$		0			0		1.00000			$0$
$211$		0			0		−1.00000			$\pi$
$212$		0			0
$213$	10.0000			0.685189
$214$		0			0
$215$		−	8.00000i		−	0.545595i
$216$		0			0
$217$	−20.0000			−1.35769
$218$		0			0
$219$	16.0000			1.08118
$220$		0			0
$221$	−6.00000	+	24.0000i	−0.403604	+	1.61441i
$222$		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
$225$	−1.00000			−0.0666667
$226$		0			0
$227$		−	12.0000i		−	0.796468i	−0.917284	−	0.398234i	$-0.869623\pi$
$227$		−	12.0000i		−	0.796468i	0.917284	−	0.398234i	$-0.130377\pi$
$228$		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
$231$		0			0
$232$		0			0
$233$		−	4.00000i		−	0.262049i	−0.991379	−	0.131024i	$-0.958173\pi$
$233$		−	4.00000i		−	0.262049i	0.991379	−	0.131024i	$-0.0418266\pi$
$234$		0			0
$235$			16.0000i			1.04372i
$236$		0			0
$237$	6.00000			0.389742
$238$		0			0
$239$		0			0			−	1.00000i	$-0.5\pi$
$239$		0			0				1.00000i	$0.5\pi$
$240$		0			0
$241$			20.0000i			1.28831i	0.764894	+	0.644157i	$0.222792\pi$
$241$			20.0000i			1.28831i	−0.764894	+	0.644157i	$0.777208\pi$
$242$		0			0
$243$		−	1.00000i		−	0.0641500i
$244$		0			0
$245$		−	6.00000i		−	0.383326i
$246$		0			0
$247$		0			0
$248$		0			0
$249$			16.0000i			1.01396i
$250$		0			0
$251$	12.0000			0.757433			0.378717	−	0.925513i	$-0.376365\pi$
$251$	12.0000			0.757433			0.378717	+	0.925513i	$0.376365\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$	2.00000	−	8.00000i	0.125245	−	0.500979i
$256$		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
$259$	−4.00000			−0.248548
$260$		0			0
$261$		−	6.00000i		−	0.371391i
$262$		0			0
$263$	16.0000			0.986602			0.493301	−	0.869859i	$-0.335790\pi$
$263$	16.0000			0.986602			0.493301	+	0.869859i	$0.335790\pi$
$264$		0			0
$265$		−	12.0000i		−	0.737154i
$266$		0			0
$267$		−	10.0000i		−	0.611990i
$268$		0			0
$269$			6.00000i			0.365826i	0.983129	+	0.182913i	$0.0585527\pi$
$269$			6.00000i			0.365826i	−0.983129	+	0.182913i	$0.941447\pi$
$270$		0			0
$271$	−32.0000			−1.94386			−0.971931	−	0.235267i	$-0.924404\pi$
$271$	−32.0000			−1.94386			−0.971931	+	0.235267i	$0.924404\pi$
$272$		0			0
$273$	12.0000			0.726273
$274$		0			0
$275$		0			0
$276$		0			0
$277$			22.0000i			1.32185i	0.750451	+	0.660926i	$0.229836\pi$
$277$			22.0000i			1.32185i	−0.750451	+	0.660926i	$0.770164\pi$
$278$		0			0
$279$		−	10.0000i		−	0.598684i
$280$		0			0
$281$	22.0000			1.31241			0.656205	−	0.754583i	$-0.272161\pi$
$281$	22.0000			1.31241			0.656205	+	0.754583i	$0.272161\pi$
$282$		0			0
$283$			16.0000i			0.951101i	0.879688	+	0.475551i	$0.157751\pi$
$283$			16.0000i			0.951101i	−0.879688	+	0.475551i	$0.842249\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	−15.0000	−	8.00000i	−0.882353	−	0.470588i
$290$		0			0
$291$	−12.0000			−0.703452
$292$		0			0
$293$	6.00000			0.350524			0.175262	−	0.984522i	$-0.443923\pi$
$293$	6.00000			0.350524			0.175262	+	0.984522i	$0.443923\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$		−	36.0000i		−	2.08193i
$300$		0			0
$301$			8.00000i			0.461112i
$302$		0			0
$303$		−	18.0000i		−	1.03407i
$304$		0			0
$305$	−20.0000			−1.14520
$306$		0			0
$307$	−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
$309$		−	16.0000i		−	0.910208i
$310$		0			0
$311$		−	10.0000i		−	0.567048i	−0.958965	−	0.283524i	$-0.908496\pi$
$311$		−	10.0000i		−	0.567048i	0.958965	−	0.283524i	$-0.0915036\pi$
$312$		0			0
$313$		−	4.00000i		−	0.226093i	−0.993590	−	0.113047i	$-0.963939\pi$
$313$		−	4.00000i		−	0.226093i	0.993590	−	0.113047i	$-0.0360610\pi$
$314$		0			0
$315$	−4.00000			−0.225374
$316$		0			0
$317$		−	18.0000i		−	1.01098i	−0.862832	−	0.505490i	$-0.831312\pi$
$317$		−	18.0000i		−	1.01098i	0.862832	−	0.505490i	$-0.168688\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$	8.00000			0.446516
$322$		0			0
$323$		0			0
$324$		0			0
$325$	6.00000			0.332820
$326$		0			0
$327$	−14.0000			−0.774202
$328$		0			0
$329$		−	16.0000i		−	0.882109i
$330$		0			0
$331$	12.0000			0.659580			0.329790	−	0.944054i	$-0.393022\pi$
$331$	12.0000			0.659580			0.329790	+	0.944054i	$0.393022\pi$
$332$		0			0
$333$		−	2.00000i		−	0.109599i
$334$		0			0
$335$		−	16.0000i		−	0.874173i
$336$		0			0
$337$			8.00000i			0.435788i	0.975972	+	0.217894i	$0.0699187\pi$
$337$			8.00000i			0.435788i	−0.975972	+	0.217894i	$0.930081\pi$
$338$		0			0
$339$	−4.00000			−0.217250
$340$		0			0
$341$		0			0
$342$		0			0
$343$			20.0000i			1.07990i
$344$		0			0
$345$			12.0000i			0.646058i
$346$		0			0
$347$		−	12.0000i		−	0.644194i	−0.946707	−	0.322097i	$-0.895612\pi$
$347$		−	12.0000i		−	0.644194i	0.946707	−	0.322097i	$-0.104388\pi$
$348$		0			0
$349$	10.0000			0.535288			0.267644	−	0.963518i	$-0.413755\pi$
$349$	10.0000			0.535288			0.267644	+	0.963518i	$0.413755\pi$
$350$		0			0
$351$			6.00000i			0.320256i
$352$		0			0
$353$	14.0000			0.745145			0.372572	−	0.928003i	$-0.378476\pi$
$353$	14.0000			0.745145			0.372572	+	0.928003i	$0.378476\pi$
$354$		0			0
$355$	20.0000			1.06149
$356$		0			0
$357$	−2.00000	+	8.00000i	−0.105851	+	0.423405i
$358$		0			0
$359$		0			0			−	1.00000i	$-0.5\pi$
$359$		0			0				1.00000i	$0.5\pi$
$360$		0			0
$361$	−19.0000			−1.00000
$362$		0			0
$363$		−	11.0000i		−	0.577350i
$364$		0			0
$365$	32.0000			1.67496
$366$		0			0
$367$			22.0000i			1.14839i	0.818718	+	0.574195i	$0.194685\pi$
$367$			22.0000i			1.14839i	−0.818718	+	0.574195i	$0.805315\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$			12.0000i			0.623009i
$372$		0			0
$373$	6.00000			0.310668			0.155334	−	0.987862i	$-0.450355\pi$
$373$	6.00000			0.310668			0.155334	+	0.987862i	$0.450355\pi$
$374$		0			0
$375$	−12.0000			−0.619677
$376$		0			0
$377$			36.0000i			1.85409i
$378$		0			0
$379$			4.00000i			0.205466i	0.994709	+	0.102733i	$0.0327588\pi$
$379$			4.00000i			0.205466i	−0.994709	+	0.102733i	$0.967241\pi$
$380$		0			0
$381$			8.00000i			0.409852i
$382$		0			0
$383$	−24.0000			−1.22634			−0.613171	−	0.789950i	$-0.710106\pi$
$383$	−24.0000			−1.22634			−0.613171	+	0.789950i	$0.710106\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	−4.00000			−0.203331
$388$		0			0
$389$	10.0000			0.507020			0.253510	−	0.967333i	$-0.418415\pi$
$389$	10.0000			0.507020			0.253510	+	0.967333i	$0.418415\pi$
$390$		0			0
$391$	24.0000	+	6.00000i	1.21373	+	0.303433i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	12.0000			0.603786
$396$		0			0
$397$			22.0000i			1.10415i	0.833795	+	0.552074i	$0.186163\pi$
$397$			22.0000i			1.10415i	−0.833795	+	0.552074i	$0.813837\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$		0			0		1.00000			$0$
$401$		0			0		−1.00000			$\pi$
$402$		0			0
$403$			60.0000i			2.98881i
$404$		0			0
$405$		−	2.00000i		−	0.0993808i
$406$		0			0
$407$		0			0
$408$		0			0
$409$	−10.0000			−0.494468			−0.247234	−	0.968956i	$-0.579522\pi$
$409$	−10.0000			−0.494468			−0.247234	+	0.968956i	$0.579522\pi$
$410$		0			0
$411$			2.00000i			0.0986527i
$412$		0			0
$413$		0			0
$414$		0			0
$415$			32.0000i			1.57082i
$416$		0			0
$417$	−16.0000			−0.783523
$418$		0			0
$419$			4.00000i			0.195413i	0.995215	+	0.0977064i	$0.0311506\pi$
$419$			4.00000i			0.195413i	−0.995215	+	0.0977064i	$0.968849\pi$
$420$		0			0
$421$	−2.00000			−0.0974740			−0.0487370	−	0.998812i	$-0.515520\pi$
$421$	−2.00000			−0.0974740			−0.0487370	+	0.998812i	$0.515520\pi$
$422$		0			0
$423$	8.00000			0.388973
$424$		0			0
$425$	−1.00000	+	4.00000i	−0.0485071	+	0.194029i
$426$		0			0
$427$	20.0000			0.967868
$428$		0			0
$429$		0			0
$430$		0			0
$431$			10.0000i			0.481683i	0.970564	+	0.240842i	$0.0774234\pi$
$431$			10.0000i			0.481683i	−0.970564	+	0.240842i	$0.922577\pi$
$432$		0			0
$433$	34.0000			1.63394			0.816968	−	0.576683i	$-0.195653\pi$
$433$	34.0000			1.63394			0.816968	+	0.576683i	$0.195653\pi$
$434$		0			0
$435$		−	12.0000i		−	0.575356i
$436$		0			0
$437$		0			0
$438$		0			0
$439$			6.00000i			0.286364i	0.989696	+	0.143182i	$0.0457335\pi$
$439$			6.00000i			0.286364i	−0.989696	+	0.143182i	$0.954267\pi$
$440$		0			0
$441$	−3.00000			−0.142857
$442$		0			0
$443$	4.00000			0.190046			0.0950229	−	0.995475i	$-0.469708\pi$
$443$	4.00000			0.190046			0.0950229	+	0.995475i	$0.469708\pi$
$444$		0			0
$445$		−	20.0000i		−	0.948091i
$446$		0			0
$447$			10.0000i			0.472984i
$448$		0			0
$449$			24.0000i			1.13263i	0.824189	+	0.566315i	$0.191631\pi$
$449$			24.0000i			1.13263i	−0.824189	+	0.566315i	$0.808369\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$		−	8.00000i		−	0.375873i
$454$		0			0
$455$	24.0000			1.12514
$456$		0			0
$457$	−22.0000			−1.02912			−0.514558	−	0.857455i	$-0.672044\pi$
$457$	−22.0000			−1.02912			−0.514558	+	0.857455i	$0.672044\pi$
$458$		0			0
$459$	−4.00000	−	1.00000i	−0.186704	−	0.0466760i
$460$		0			0
$461$	−2.00000			−0.0931493			−0.0465746	−	0.998915i	$-0.514831\pi$
$461$	−2.00000			−0.0931493			−0.0465746	+	0.998915i	$0.514831\pi$
$462$		0			0
$463$	−24.0000			−1.11537			−0.557687	−	0.830051i	$-0.688311\pi$
$463$	−24.0000			−1.11537			−0.557687	+	0.830051i	$0.688311\pi$
$464$		0			0
$465$		−	20.0000i		−	0.927478i
$466$		0			0
$467$	28.0000			1.29569			0.647843	−	0.761774i	$-0.275671\pi$
$467$	28.0000			1.29569			0.647843	+	0.761774i	$0.275671\pi$
$468$		0			0
$469$			16.0000i			0.738811i
$470$		0			0
$471$		−	2.00000i		−	0.0921551i
$472$		0			0
$473$		0			0
$474$		0			0
$475$		0			0
$476$		0			0
$477$	−6.00000			−0.274721
$478$		0			0
$479$		−	14.0000i		−	0.639676i	−0.947472	−	0.319838i	$-0.896371\pi$
$479$		−	14.0000i		−	0.639676i	0.947472	−	0.319838i	$-0.103629\pi$
$480$		0			0
$481$			12.0000i			0.547153i
$482$		0			0
$483$		−	12.0000i		−	0.546019i
$484$		0			0
$485$	−24.0000			−1.08978
$486$		0			0
$487$		−	18.0000i		−	0.815658i	−0.913058	−	0.407829i	$-0.866286\pi$
$487$		−	18.0000i		−	0.815658i	0.913058	−	0.407829i	$-0.133714\pi$
$488$		0			0
$489$	−4.00000			−0.180886
$490$		0			0
$491$	12.0000			0.541552			0.270776	−	0.962642i	$-0.412720\pi$
$491$	12.0000			0.541552			0.270776	+	0.962642i	$0.412720\pi$
$492$		0			0
$493$	−24.0000	−	6.00000i	−1.08091	−	0.270226i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−20.0000			−0.897123
$498$		0			0
$499$			24.0000i			1.07439i	0.843459	+	0.537194i	$0.180516\pi$
$499$			24.0000i			1.07439i	−0.843459	+	0.537194i	$0.819484\pi$
$500$		0			0
$501$	−18.0000			−0.804181
$502$		0			0
$503$			34.0000i			1.51599i	0.652263	+	0.757993i	$0.273820\pi$
$503$			34.0000i			1.51599i	−0.652263	+	0.757993i	$0.726180\pi$
$504$		0			0
$505$		−	36.0000i		−	1.60198i
$506$		0			0
$507$		−	23.0000i		−	1.02147i
$508$		0			0
$509$	−10.0000			−0.443242			−0.221621	−	0.975133i	$-0.571135\pi$
$509$	−10.0000			−0.443242			−0.221621	+	0.975133i	$0.571135\pi$
$510$		0			0
$511$	−32.0000			−1.41560
$512$		0			0
$513$		0			0
$514$		0			0
$515$		−	32.0000i		−	1.41009i
$516$		0			0
$517$		0			0
$518$		0			0
$519$	14.0000			0.614532
$520$		0			0
$521$		0			0		1.00000			$0$
$521$		0			0		−1.00000			$\pi$
$522$		0			0
$523$	−16.0000			−0.699631			−0.349816	−	0.936819i	$-0.613756\pi$
$523$	−16.0000			−0.699631			−0.349816	+	0.936819i	$0.613756\pi$
$524$		0			0
$525$	2.00000			0.0872872
$526$		0			0
$527$	−40.0000	−	10.0000i	−1.74243	−	0.435607i
$528$		0			0
$529$	−13.0000			−0.565217
$530$		0			0
$531$		0			0
$532$		0			0
$533$		0			0
$534$		0			0
$535$	16.0000			0.691740
$536$		0			0
$537$			20.0000i			0.863064i
$538$		0			0
$539$		0			0
$540$		0			0
$541$			10.0000i			0.429934i	0.976621	+	0.214967i	$0.0689643\pi$
$541$			10.0000i			0.429934i	−0.976621	+	0.214967i	$0.931036\pi$
$542$		0			0
$543$	−10.0000			−0.429141
$544$		0			0
$545$	−28.0000			−1.19939
$546$		0			0
$547$			8.00000i			0.342055i	0.985266	+	0.171028i	$0.0547087\pi$
$547$			8.00000i			0.342055i	−0.985266	+	0.171028i	$0.945291\pi$
$548$		0			0
$549$			10.0000i			0.426790i
$550$		0			0
$551$		0			0
$552$		0			0
$553$	−12.0000			−0.510292
$554$		0			0
$555$		−	4.00000i		−	0.169791i
$556$		0			0
$557$	−38.0000			−1.61011			−0.805056	−	0.593199i	$-0.797865\pi$
$557$	−38.0000			−1.61011			−0.805056	+	0.593199i	$0.797865\pi$
$558$		0			0
$559$	24.0000			1.01509
$560$		0			0
$561$		0			0
$562$		0			0
$563$	24.0000			1.01148			0.505740	−	0.862686i	$-0.331220\pi$
$563$	24.0000			1.01148			0.505740	+	0.862686i	$0.331220\pi$
$564$		0			0
$565$	−8.00000			−0.336563
$566$		0			0
$567$			2.00000i			0.0839921i
$568$		0			0
$569$	−30.0000			−1.25767			−0.628833	−	0.777541i	$-0.716467\pi$
$569$	−30.0000			−1.25767			−0.628833	+	0.777541i	$0.716467\pi$
$570$		0			0
$571$			20.0000i			0.836974i	0.908223	+	0.418487i	$0.137439\pi$
$571$			20.0000i			0.836974i	−0.908223	+	0.418487i	$0.862561\pi$
$572$		0			0
$573$		−	8.00000i		−	0.334205i
$574$		0			0
$575$		−	6.00000i		−	0.250217i
$576$		0			0
$577$	−2.00000			−0.0832611			−0.0416305	−	0.999133i	$-0.513255\pi$
$577$	−2.00000			−0.0832611			−0.0416305	+	0.999133i	$0.513255\pi$
$578$		0			0
$579$	−4.00000			−0.166234
$580$		0			0
$581$		−	32.0000i		−	1.32758i
$582$		0			0
$583$		0			0
$584$		0			0
$585$			12.0000i			0.496139i
$586$		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
$589$		0			0
$590$		0			0
$591$	−18.0000			−0.740421
$592$		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
$595$	−4.00000	+	16.0000i	−0.163984	+	0.655936i
$596$		0			0
$597$	6.00000			0.245564
$598$		0			0
$599$	40.0000			1.63436			0.817178	−	0.576386i	$-0.195537\pi$
$599$	40.0000			1.63436			0.817178	+	0.576386i	$0.195537\pi$
$600$		0			0
$601$		−	40.0000i		−	1.63163i	−0.578310	−	0.815817i	$-0.696288\pi$
$601$		−	40.0000i		−	1.63163i	0.578310	−	0.815817i	$-0.303712\pi$
$602$		0			0
$603$	−8.00000			−0.325785
$604$		0			0
$605$		−	22.0000i		−	0.894427i
$606$		0			0
$607$		−	38.0000i		−	1.54237i	−0.636610	−	0.771186i	$-0.719664\pi$
$607$		−	38.0000i		−	1.54237i	0.636610	−	0.771186i	$-0.280336\pi$
$608$		0			0
$609$			12.0000i			0.486265i
$610$		0			0
$611$	−48.0000			−1.94187
$612$		0			0
$613$	−14.0000			−0.565455			−0.282727	−	0.959200i	$-0.591239\pi$
$613$	−14.0000			−0.565455			−0.282727	+	0.959200i	$0.591239\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		−	12.0000i		−	0.483102i	−0.970388	−	0.241551i	$-0.922344\pi$
$617$		−	12.0000i		−	0.483102i	0.970388	−	0.241551i	$-0.0776561\pi$
$618$		0			0
$619$			4.00000i			0.160774i	0.996764	+	0.0803868i	$0.0256155\pi$
$619$			4.00000i			0.160774i	−0.996764	+	0.0803868i	$0.974384\pi$
$620$		0			0
$621$	6.00000			0.240772
$622$		0			0
$623$			20.0000i			0.801283i
$624$		0			0
$625$	−19.0000			−0.760000
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−8.00000	−	2.00000i	−0.318981	−	0.0797452i
$630$		0			0
$631$	8.00000			0.318475			0.159237	−	0.987240i	$-0.449096\pi$
$631$	8.00000			0.318475			0.159237	+	0.987240i	$0.449096\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$			16.0000i			0.634941i
$636$		0			0
$637$	18.0000			0.713186
$638$		0			0
$639$		−	10.0000i		−	0.395594i
$640$		0			0
$641$		−	40.0000i		−	1.57991i	−0.613168	−	0.789953i	$-0.710105\pi$
$641$		−	40.0000i		−	1.57991i	0.613168	−	0.789953i	$-0.289895\pi$
$642$		0			0
$643$			36.0000i			1.41970i	0.704352	+	0.709851i	$0.251238\pi$
$643$			36.0000i			1.41970i	−0.704352	+	0.709851i	$0.748762\pi$
$644$		0			0
$645$	−8.00000			−0.315000
$646$		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
$649$		0			0
$650$		0			0
$651$			20.0000i			0.783862i
$652$		0			0
$653$			34.0000i			1.33052i	0.746611	+	0.665261i	$0.231680\pi$
$653$			34.0000i			1.33052i	−0.746611	+	0.665261i	$0.768320\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$		−	16.0000i		−	0.624219i
$658$		0			0
$659$	−40.0000			−1.55818			−0.779089	−	0.626913i	$-0.784318\pi$
$659$	−40.0000			−1.55818			−0.779089	+	0.626913i	$0.784318\pi$
$660$		0			0
$661$	−22.0000			−0.855701			−0.427850	−	0.903850i	$-0.640729\pi$
$661$	−22.0000			−0.855701			−0.427850	+	0.903850i	$0.640729\pi$
$662$		0			0
$663$	24.0000	+	6.00000i	0.932083	+	0.233021i
$664$		0			0
$665$		0			0
$666$		0			0
$667$	36.0000			1.39393
$668$		0			0
$669$		−	16.0000i		−	0.618596i
$670$		0			0
$671$		0			0
$672$		0			0
$673$			36.0000i			1.38770i	0.720121	+	0.693849i	$0.244086\pi$
$673$			36.0000i			1.38770i	−0.720121	+	0.693849i	$0.755914\pi$
$674$		0			0
$675$			1.00000i			0.0384900i
$676$		0			0
$677$			42.0000i			1.61419i	0.590421	+	0.807096i	$0.298962\pi$
$677$			42.0000i			1.61419i	−0.590421	+	0.807096i	$0.701038\pi$
$678$		0			0
$679$	24.0000			0.921035
$680$		0			0
$681$	−12.0000			−0.459841
$682$		0			0
$683$		−	44.0000i		−	1.68361i	−0.539779	−	0.841807i	$-0.681492\pi$
$683$		−	44.0000i		−	1.68361i	0.539779	−	0.841807i	$-0.318508\pi$
$684$		0			0
$685$			4.00000i			0.152832i
$686$		0			0
$687$			10.0000i			0.381524i
$688$		0			0
$689$	36.0000			1.37149
$690$		0			0
$691$		−	40.0000i		−	1.52167i	−0.648944	−	0.760836i	$-0.724789\pi$
$691$		−	40.0000i		−	1.52167i	0.648944	−	0.760836i	$-0.275211\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−32.0000			−1.21383
$696$		0			0
$697$		0			0
$698$		0			0
$699$	−4.00000			−0.151294
$700$		0			0
$701$	−22.0000			−0.830929			−0.415464	−	0.909610i	$-0.636381\pi$
$701$	−22.0000			−0.830929			−0.415464	+	0.909610i	$0.636381\pi$
$702$		0			0
$703$		0			0
$704$		0			0
$705$	16.0000			0.602595
$706$		0			0
$707$			36.0000i			1.35392i
$708$		0			0
$709$			6.00000i			0.225335i	0.993633	+	0.112667i	$0.0359394\pi$
$709$			6.00000i			0.225335i	−0.993633	+	0.112667i	$0.964061\pi$
$710$		0			0
$711$		−	6.00000i		−	0.225018i
$712$		0			0
$713$	60.0000			2.24702
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$			26.0000i			0.969636i	0.874615	+	0.484818i	$0.161114\pi$
$719$			26.0000i			0.969636i	−0.874615	+	0.484818i	$0.838886\pi$
$720$		0			0
$721$			32.0000i			1.19174i
$722$		0			0
$723$	20.0000			0.743808
$724$		0			0
$725$			6.00000i			0.222834i
$726$		0			0
$727$	−48.0000			−1.78022			−0.890111	−	0.455744i	$-0.849373\pi$
$727$	−48.0000			−1.78022			−0.890111	+	0.455744i	$0.849373\pi$
$728$		0			0
$729$	−1.00000			−0.0370370
$730$		0			0
$731$	−4.00000	+	16.0000i	−0.147945	+	0.591781i
$732$		0			0
$733$	−14.0000			−0.517102			−0.258551	−	0.965998i	$-0.583245\pi$
$733$	−14.0000			−0.517102			−0.258551	+	0.965998i	$0.583245\pi$
$734$		0			0
$735$	−6.00000			−0.221313
$736$		0			0
$737$		0			0
$738$		0			0
$739$	−40.0000			−1.47142			−0.735712	−	0.677295i	$-0.763152\pi$
$739$	−40.0000			−1.47142			−0.735712	+	0.677295i	$0.763152\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		−	26.0000i		−	0.953847i	−0.878945	−	0.476924i	$-0.841752\pi$
$743$		−	26.0000i		−	0.953847i	0.878945	−	0.476924i	$-0.158248\pi$
$744$		0			0
$745$			20.0000i			0.732743i
$746$		0			0
$747$	16.0000			0.585409
$748$		0			0
$749$	−16.0000			−0.584627
$750$		0			0
$751$		−	10.0000i		−	0.364905i	−0.983215	−	0.182453i	$-0.941596\pi$
$751$		−	10.0000i		−	0.364905i	0.983215	−	0.182453i	$-0.0584036\pi$
$752$		0			0
$753$		−	12.0000i		−	0.437304i
$754$		0			0
$755$		−	16.0000i		−	0.582300i
$756$		0			0
$757$	22.0000			0.799604			0.399802	−	0.916602i	$-0.369079\pi$
$757$	22.0000			0.799604			0.399802	+	0.916602i	$0.369079\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−18.0000			−0.652499			−0.326250	−	0.945284i	$-0.605785\pi$
$761$	−18.0000			−0.652499			−0.326250	+	0.945284i	$0.605785\pi$
$762$		0			0
$763$	28.0000			1.01367
$764$		0			0
$765$	−8.00000	−	2.00000i	−0.289241	−	0.0723102i
$766$		0			0
$767$		0			0
$768$		0			0
$769$	30.0000			1.08183			0.540914	−	0.841078i	$-0.318079\pi$
$769$	30.0000			1.08183			0.540914	+	0.841078i	$0.318079\pi$
$770$		0			0
$771$			2.00000i			0.0720282i
$772$		0			0
$773$	−14.0000			−0.503545			−0.251773	−	0.967786i	$-0.581013\pi$
$773$	−14.0000			−0.503545			−0.251773	+	0.967786i	$0.581013\pi$
$774$		0			0
$775$			10.0000i			0.359211i
$776$		0			0
$777$			4.00000i			0.143499i
$778$		0			0
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$	−6.00000			−0.214423
$784$		0			0
$785$		−	4.00000i		−	0.142766i
$786$		0			0
$787$		−	32.0000i		−	1.14068i	−0.821410	−	0.570338i	$-0.806812\pi$
$787$		−	32.0000i		−	1.14068i	0.821410	−	0.570338i	$-0.193188\pi$
$788$		0			0
$789$		−	16.0000i		−	0.569615i
$790$		0			0
$791$	8.00000			0.284447
$792$		0			0
$793$		−	60.0000i		−	2.13066i
$794$		0			0
$795$	−12.0000			−0.425596
$796$		0			0
$797$	−18.0000			−0.637593			−0.318796	−	0.947823i	$-0.603279\pi$
$797$	−18.0000			−0.637593			−0.318796	+	0.947823i	$0.603279\pi$
$798$		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
$805$		−	24.0000i		−	0.845889i
$806$		0			0
$807$	6.00000			0.211210
$808$		0			0
$809$			24.0000i			0.843795i	0.906644	+	0.421898i	$0.138636\pi$
$809$			24.0000i			0.843795i	−0.906644	+	0.421898i	$0.861364\pi$
$810$		0			0
$811$		−	20.0000i		−	0.702295i	−0.936320	−	0.351147i	$-0.885792\pi$
$811$		−	20.0000i		−	0.702295i	0.936320	−	0.351147i	$-0.114208\pi$
$812$		0			0
$813$			32.0000i			1.12229i
$814$		0			0
$815$	−8.00000			−0.280228
$816$		0			0
$817$		0			0
$818$		0			0
$819$		−	12.0000i		−	0.419314i
$820$		0			0
$821$			10.0000i			0.349002i	0.984657	+	0.174501i	$0.0558313\pi$
$821$			10.0000i			0.349002i	−0.984657	+	0.174501i	$0.944169\pi$
$822$		0			0
$823$			14.0000i			0.488009i	0.969774	+	0.244005i	$0.0784612\pi$
$823$			14.0000i			0.488009i	−0.969774	+	0.244005i	$0.921539\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		−	32.0000i		−	1.11275i	−0.830932	−	0.556375i	$-0.812192\pi$
$827$		−	32.0000i		−	1.11275i	0.830932	−	0.556375i	$-0.187808\pi$
$828$		0			0
$829$	−30.0000			−1.04194			−0.520972	−	0.853574i	$-0.674430\pi$
$829$	−30.0000			−1.04194			−0.520972	+	0.853574i	$0.674430\pi$
$830$		0			0
$831$	22.0000			0.763172
$832$		0			0
$833$	−3.00000	+	12.0000i	−0.103944	+	0.415775i
$834$		0			0
$835$	−36.0000			−1.24583
$836$		0			0
$837$	−10.0000			−0.345651
$838$		0			0
$839$		−	34.0000i		−	1.17381i	−0.809656	−	0.586905i	$-0.800346\pi$
$839$		−	34.0000i		−	1.17381i	0.809656	−	0.586905i	$-0.199654\pi$
$840$		0			0
$841$	−7.00000			−0.241379
$842$		0			0
$843$		−	22.0000i		−	0.757720i
$844$		0			0
$845$		−	46.0000i		−	1.58245i
$846$		0			0
$847$			22.0000i			0.755929i
$848$		0			0
$849$	16.0000			0.549119
$850$		0			0
$851$	12.0000			0.411355
$852$		0			0
$853$			14.0000i			0.479351i	0.970853	+	0.239675i	$0.0770410\pi$
$853$			14.0000i			0.479351i	−0.970853	+	0.239675i	$0.922959\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$			48.0000i			1.63965i	0.572615	+	0.819824i	$0.305929\pi$
$857$			48.0000i			1.63965i	−0.572615	+	0.819824i	$0.694071\pi$
$858$		0			0
$859$	20.0000			0.682391			0.341196	−	0.939992i	$-0.389168\pi$
$859$	20.0000			0.682391			0.341196	+	0.939992i	$0.389168\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	16.0000			0.544646			0.272323	−	0.962206i	$-0.412208\pi$
$863$	16.0000			0.544646			0.272323	+	0.962206i	$0.412208\pi$
$864$		0			0
$865$	28.0000			0.952029
$866$		0			0
$867$	−8.00000	+	15.0000i	−0.271694	+	0.509427i
$868$		0			0
$869$		0			0
$870$		0			0
$871$	48.0000			1.62642
$872$		0			0
$873$			12.0000i			0.406138i
$874$		0			0
$875$	24.0000			0.811348
$876$		0			0
$877$		−	38.0000i		−	1.28317i	−0.767052	−	0.641584i	$-0.778277\pi$
$877$		−	38.0000i		−	1.28317i	0.767052	−	0.641584i	$-0.221723\pi$
$878$		0			0
$879$		−	6.00000i		−	0.202375i
$880$		0			0
$881$			20.0000i			0.673817i	0.941537	+	0.336909i	$0.109381\pi$
$881$			20.0000i			0.673817i	−0.941537	+	0.336909i	$0.890619\pi$
$882$		0			0
$883$	−16.0000			−0.538443			−0.269221	−	0.963078i	$-0.586766\pi$
$883$	−16.0000			−0.538443			−0.269221	+	0.963078i	$0.586766\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$			42.0000i			1.41022i	0.709097	+	0.705111i	$0.249103\pi$
$887$			42.0000i			1.41022i	−0.709097	+	0.705111i	$0.750897\pi$
$888$		0			0
$889$		−	16.0000i		−	0.536623i
$890$		0			0
$891$		0			0
$892$		0			0
$893$		0			0
$894$		0			0
$895$			40.0000i			1.33705i
$896$		0			0
$897$	−36.0000			−1.20201
$898$		0			0
$899$	−60.0000			−2.00111
$900$		0			0
$901$	−6.00000	+	24.0000i	−0.199889	+	0.799556i
$902$		0			0
$903$	8.00000			0.266223
$904$		0			0
$905$	−20.0000			−0.664822
$906$		0			0
$907$			8.00000i			0.265636i	0.991140	+	0.132818i	$0.0424025\pi$
$907$			8.00000i			0.265636i	−0.991140	+	0.132818i	$0.957597\pi$
$908$		0			0
$909$	−18.0000			−0.597022
$910$		0			0
$911$		−	50.0000i		−	1.65657i	−0.560304	−	0.828287i	$-0.689316\pi$
$911$		−	50.0000i		−	1.65657i	0.560304	−	0.828287i	$-0.310684\pi$
$912$		0			0
$913$		0			0
$914$		0			0
$915$			20.0000i			0.661180i
$916$		0			0
$917$		0			0
$918$		0			0
$919$		0			0			−	1.00000i	$-0.5\pi$
$919$		0			0				1.00000i	$0.5\pi$
$920$		0			0
$921$			12.0000i			0.395413i
$922$		0			0
$923$			60.0000i			1.97492i
$924$		0			0
$925$			2.00000i			0.0657596i
$926$		0			0
$927$	−16.0000			−0.525509
$928$		0			0
$929$		−	16.0000i		−	0.524943i	−0.964940	−	0.262471i	$-0.915462\pi$
$929$		−	16.0000i		−	0.524943i	0.964940	−	0.262471i	$-0.0845376\pi$
$930$		0			0
$931$		0			0
$932$		0			0
$933$	−10.0000			−0.327385
$934$		0			0
$935$		0			0
$936$		0			0
$937$	−2.00000			−0.0653372			−0.0326686	−	0.999466i	$-0.510401\pi$
$937$	−2.00000			−0.0653372			−0.0326686	+	0.999466i	$0.510401\pi$
$938$		0			0
$939$	−4.00000			−0.130535
$940$		0			0
$941$			30.0000i			0.977972i	0.872292	+	0.488986i	$0.162633\pi$
$941$			30.0000i			0.977972i	−0.872292	+	0.488986i	$0.837367\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$			4.00000i			0.130120i
$946$		0			0
$947$			8.00000i			0.259965i	0.991516	+	0.129983i	$0.0414921\pi$
$947$			8.00000i			0.259965i	−0.991516	+	0.129983i	$0.958508\pi$
$948$		0			0
$949$			96.0000i			3.11629i
$950$		0			0
$951$	−18.0000			−0.583690
$952$		0			0
$953$	−6.00000			−0.194359			−0.0971795	−	0.995267i	$-0.530982\pi$
$953$	−6.00000			−0.194359			−0.0971795	+	0.995267i	$0.530982\pi$
$954$		0			0
$955$		−	16.0000i		−	0.517748i
$956$		0			0
$957$		0			0
$958$		0			0
$959$		−	4.00000i		−	0.129167i
$960$		0			0
$961$	−69.0000			−2.22581
$962$		0			0
$963$		−	8.00000i		−	0.257796i
$964$		0			0
$965$	−8.00000			−0.257529
$966$		0			0
$967$	−8.00000			−0.257263			−0.128631	−	0.991692i	$-0.541058\pi$
$967$	−8.00000			−0.257263			−0.128631	+	0.991692i	$0.541058\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	32.0000			1.02693			0.513464	−	0.858111i	$-0.328362\pi$
$971$	32.0000			1.02693			0.513464	+	0.858111i	$0.328362\pi$
$972$		0			0
$973$	32.0000			1.02587
$974$		0			0
$975$		−	6.00000i		−	0.192154i
$976$		0			0
$977$	−22.0000			−0.703842			−0.351921	−	0.936030i	$-0.614471\pi$
$977$	−22.0000			−0.703842			−0.351921	+	0.936030i	$0.614471\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$			14.0000i			0.446986i
$982$		0			0
$983$		−	46.0000i		−	1.46717i	−0.679597	−	0.733586i	$-0.737845\pi$
$983$		−	46.0000i		−	1.46717i	0.679597	−	0.733586i	$-0.262155\pi$
$984$		0			0
$985$	−36.0000			−1.14706
$986$		0			0
$987$	−16.0000			−0.509286
$988$		0			0
$989$		−	24.0000i		−	0.763156i
$990$		0			0
$991$		−	10.0000i		−	0.317660i	−0.987306	−	0.158830i	$-0.949228\pi$
$991$		−	10.0000i		−	0.317660i	0.987306	−	0.158830i	$-0.0507723\pi$
$992$		0			0
$993$		−	12.0000i		−	0.380808i
$994$		0			0
$995$	12.0000			0.380426
$996$		0			0
$997$		−	18.0000i		−	0.570066i	−0.958518	−	0.285033i	$-0.907995\pi$
$997$		−	18.0000i		−	0.570066i	0.958518	−	0.285033i	$-0.0920045\pi$
$998$		0			0
$999$	−2.00000			−0.0632772

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3264.2.c.i.577.1		2
4.3	odd	2		3264.2.c.j.577.2		2
8.3	odd	2		102.2.b.a.67.1	✓	2
8.5	even	2		816.2.c.a.577.2		2
17.16	even	2	inner	3264.2.c.i.577.2		2
24.5	odd	2		2448.2.c.a.577.1		2
24.11	even	2		306.2.b.a.271.1		2
40.3	even	4		2550.2.f.j.1699.1		2
40.19	odd	2		2550.2.c.f.1801.2		2
40.27	even	4		2550.2.f.e.1699.2		2
68.67	odd	2		3264.2.c.j.577.1		2
136.19	odd	8		1734.2.f.h.829.1		4
136.43	odd	8		1734.2.f.h.1483.2		4
136.59	odd	8		1734.2.f.h.1483.1		4
136.67	odd	2		102.2.b.a.67.2	yes	2
136.83	odd	8		1734.2.f.h.829.2		4
136.101	even	2		816.2.c.a.577.1		2
136.115	odd	4		1734.2.a.e.1.1		1
136.123	odd	4		1734.2.a.d.1.1		1
408.101	odd	2		2448.2.c.a.577.2		2
408.203	even	2		306.2.b.a.271.2		2
408.251	even	4		5202.2.a.n.1.1		1
408.395	even	4		5202.2.a.h.1.1		1
680.67	even	4		2550.2.f.j.1699.2		2
680.203	even	4		2550.2.f.e.1699.1		2
680.339	odd	2		2550.2.c.f.1801.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
102.2.b.a.67.1	✓	2	8.3	odd	2
102.2.b.a.67.2	yes	2	136.67	odd	2
306.2.b.a.271.1		2	24.11	even	2
306.2.b.a.271.2		2	408.203	even	2
816.2.c.a.577.1		2	136.101	even	2
816.2.c.a.577.2		2	8.5	even	2
1734.2.a.d.1.1		1	136.123	odd	4
1734.2.a.e.1.1		1	136.115	odd	4
1734.2.f.h.829.1		4	136.19	odd	8
1734.2.f.h.829.2		4	136.83	odd	8
1734.2.f.h.1483.1		4	136.59	odd	8
1734.2.f.h.1483.2		4	136.43	odd	8
2448.2.c.a.577.1		2	24.5	odd	2
2448.2.c.a.577.2		2	408.101	odd	2
2550.2.c.f.1801.1		2	680.339	odd	2
2550.2.c.f.1801.2		2	40.19	odd	2
2550.2.f.e.1699.1		2	680.203	even	4
2550.2.f.e.1699.2		2	40.27	even	4
2550.2.f.j.1699.1		2	40.3	even	4
2550.2.f.j.1699.2		2	680.67	even	4
3264.2.c.i.577.1		2	1.1	even	1	trivial
3264.2.c.i.577.2		2	17.16	even	2	inner
3264.2.c.j.577.1		2	68.67	odd	2
3264.2.c.j.577.2		2	4.3	odd	2
5202.2.a.h.1.1		1	408.395	even	4
5202.2.a.n.1.1		1	408.251	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 $\F_{q}$

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

\(f(q)\)	\(=\)	\(q-1.00000i q^{3} -2.00000i q^{5} +2.00000i q^{7} -1.00000 q^{9} +6.00000 q^{13} -2.00000 q^{15} +(-1.00000 + 4.00000i) q^{17} +2.00000 q^{21} -6.00000i q^{23} +1.00000 q^{25} +1.00000i q^{27} +6.00000i q^{29} +10.0000i q^{31} +4.00000 q^{35} +2.00000i q^{37} -6.00000i q^{39} +4.00000 q^{43} +2.00000i q^{45} -8.00000 q^{47} +3.00000 q^{49} +(4.00000 + 1.00000i) q^{51} +6.00000 q^{53} -10.0000i q^{61} -2.00000i q^{63} -12.0000i q^{65} +8.00000 q^{67} -6.00000 q^{69} +10.0000i q^{71} +16.0000i q^{73} -1.00000i q^{75} +6.00000i q^{79} +1.00000 q^{81} -16.0000 q^{83} +(8.00000 + 2.00000i) q^{85} +6.00000 q^{87} +10.0000 q^{89} +12.0000i q^{91} +10.0000 q^{93} -12.0000i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{9} + 12 q^{13} - 4 q^{15} - 2 q^{17} + 4 q^{21} + 2 q^{25} + 8 q^{35} + 8 q^{43} - 16 q^{47} + 6 q^{49} + 8 q^{51} + 12 q^{53} + 16 q^{67} - 12 q^{69} + 2 q^{81} - 32 q^{83} + 16 q^{85} + 12 q^{87}+ \cdots + 20 q^{93}+O(q^{100}) \) 2 * q - 2 * q^9 + 12 * q^13 - 4 * q^15 - 2 * q^17 + 4 * q^21 + 2 * q^25 + 8 * q^35 + 8 * q^43 - 16 * q^47 + 6 * q^49 + 8 * q^51 + 12 * q^53 + 16 * q^67 - 12 * q^69 + 2 * q^81 - 32 * q^83 + 16 * q^85 + 12 * q^87 + 20 * q^89 + 20 * q^93

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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