LMFDB - Embedded newform 1296.3.q.b.593.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1296,3,Mod(593,1296)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1296, base_ring=CyclotomicField(6))

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

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("1296.593");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 1296 = 2^{4} \cdot 3^{4} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1296.q (of order $6$, degree $2$, 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:	$35.3134422611$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{6})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x + 1 $ x^2 - x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{25}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 12)
Sato-Tate group:	$\mathrm{U}(1)[D_{6}]$

Embedding invariants

Embedding label			593.1
Root			$0.500000 + 0.866025i$ of defining polynomial
Character	$\chi$	$=$	1296.593
Dual form			1296.3.q.b.1025.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.00000 - 1.73205i) q^{7} +O(q^{10})$	$q+(1.00000 - 1.73205i) q^{7} +(11.0000 + 19.0526i) q^{13} -26.0000 q^{19} +(-12.5000 + 21.6506i) q^{25} +(-23.0000 - 39.8372i) q^{31} +26.0000 q^{37} +(-11.0000 + 19.0526i) q^{43} +(22.5000 + 38.9711i) q^{49} +(-37.0000 + 64.0859i) q^{61} +(61.0000 + 105.655i) q^{67} -46.0000 q^{73} +(-71.0000 + 122.976i) q^{79} +44.0000 q^{91} +(-1.00000 + 1.73205i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{7}+O(q^{10}) $ 2 * q + 2 * q^7	$ 2 q + 2 q^{7} + 22 q^{13} - 52 q^{19} - 25 q^{25} - 46 q^{31} + 52 q^{37} - 22 q^{43} + 45 q^{49} - 74 q^{61} + 122 q^{67} - 92 q^{73} - 142 q^{79} + 88 q^{91} - 2 q^{97}+O(q^{100}) $ 2 * q + 2 * q^7 + 22 * q^13 - 52 * q^19 - 25 * q^25 - 46 * q^31 + 52 * q^37 - 22 * q^43 + 45 * q^49 - 74 * q^61 + 122 * q^67 - 92 * q^73 - 142 * q^79 + 88 * q^91 - 2 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1296\mathbb{Z}\right)^\times$.

$n$	$325$	$1135$	$1217$
$\chi(n)$	$1$	$1$	$e\left(\frac{1}{6}\right)$

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
$3$		0			0
$3$		0			0
$4$		0			0
$5$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$5$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$6$		0			0
$7$	1.00000	−	1.73205i	0.142857	−	0.247436i	−0.785714	−	0.618590i	$-0.787704\pi$
$7$	1.00000	−	1.73205i	0.142857	−	0.247436i	0.928571	+	0.371154i	$0.121038\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$11$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$12$		0			0
$13$	11.0000	+	19.0526i	0.846154	+	1.46558i	0.884615	+	0.466321i	$0.154421\pi$
$13$	11.0000	+	19.0526i	0.846154	+	1.46558i	−0.0384615	+	0.999260i	$0.512246\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$		0			0		1.00000			$0$
$17$		0			0		−1.00000			$\pi$
$18$		0			0
$19$	−26.0000			−1.36842			−0.684211	−	0.729285i	$-0.739853\pi$
$19$	−26.0000			−1.36842			−0.684211	+	0.729285i	$0.739853\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$23$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$24$		0			0
$25$	−12.5000	+	21.6506i	−0.500000	+	0.866025i
$26$		0			0
$27$		0			0
$28$		0			0
$29$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$29$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$30$		0			0
$31$	−23.0000	−	39.8372i	−0.741935	−	1.28507i	−0.951613	−	0.307299i	$-0.900575\pi$
$31$	−23.0000	−	39.8372i	−0.741935	−	1.28507i	0.209677	−	0.977771i	$-0.432759\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$		0			0
$37$	26.0000			0.702703			0.351351	−	0.936244i	$-0.385722\pi$
$37$	26.0000			0.702703			0.351351	+	0.936244i	$0.385722\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$41$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$42$		0			0
$43$	−11.0000	+	19.0526i	−0.255814	+	0.443083i	−0.965116	−	0.261822i	$-0.915677\pi$
$43$	−11.0000	+	19.0526i	−0.255814	+	0.443083i	0.709302	+	0.704904i	$0.249010\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$47$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$48$		0			0
$49$	22.5000	+	38.9711i	0.459184	+	0.795329i
$50$		0			0
$51$		0			0
$52$		0			0
$53$		0			0		1.00000			$0$
$53$		0			0		−1.00000			$\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$		0			0
$58$		0			0
$59$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$59$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$60$		0			0
$61$	−37.0000	+	64.0859i	−0.606557	+	1.05059i	0.385246	+	0.922814i	$0.374117\pi$
$61$	−37.0000	+	64.0859i	−0.606557	+	1.05059i	−0.991803	+	0.127774i	$0.959217\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$		0			0
$66$		0			0
$67$	61.0000	+	105.655i	0.910448	+	1.57694i	0.813433	+	0.581659i	$0.197596\pi$
$67$	61.0000	+	105.655i	0.910448	+	1.57694i	0.0970149	+	0.995283i	$0.469071\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		0			0		1.00000			$0$
$71$		0			0		−1.00000			$\pi$
$72$		0			0
$73$	−46.0000			−0.630137			−0.315068	−	0.949069i	$-0.602027\pi$
$73$	−46.0000			−0.630137			−0.315068	+	0.949069i	$0.602027\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$		0			0
$78$		0			0
$79$	−71.0000	+	122.976i	−0.898734	+	1.55665i	−0.0696203	+	0.997574i	$0.522179\pi$
$79$	−71.0000	+	122.976i	−0.898734	+	1.55665i	−0.829114	+	0.559080i	$0.811155\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$83$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$		0			0		1.00000			$0$
$89$		0			0		−1.00000			$\pi$
$90$		0			0
$91$	44.0000			0.483516
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$	−1.00000	+	1.73205i	−0.0103093	+	0.0178562i	−0.871134	−	0.491045i	$-0.836615\pi$
$97$	−1.00000	+	1.73205i	−0.0103093	+	0.0178562i	0.860825	+	0.508902i	$0.169948\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$101$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$102$		0			0
$103$	97.0000	+	168.009i	0.941748	+	1.63115i	0.762136	+	0.647417i	$0.224151\pi$
$103$	97.0000	+	168.009i	0.941748	+	1.63115i	0.179612	+	0.983738i	$0.442516\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		0			0		1.00000			$0$
$107$		0			0		−1.00000			$\pi$
$108$		0			0
$109$	−214.000			−1.96330			−0.981651	−	0.190684i	$-0.938929\pi$
$109$	−214.000			−1.96330			−0.981651	+	0.190684i	$0.938929\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$113$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−60.5000	−	104.789i	−0.500000	−	0.866025i
$122$		0			0
$123$		0			0
$124$		0			0
$125$		0			0
$126$		0			0
$127$	−146.000			−1.14961			−0.574803	−	0.818292i	$-0.694921\pi$
$127$	−146.000			−1.14961			−0.574803	+	0.818292i	$0.694921\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$131$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$132$		0			0
$133$	−26.0000	+	45.0333i	−0.195489	+	0.338596i
$134$		0			0
$135$		0			0
$136$		0			0
$137$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$137$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$138$		0			0
$139$	−11.0000	−	19.0526i	−0.0791367	−	0.137069i	0.823741	−	0.566966i	$-0.191883\pi$
$139$	−11.0000	−	19.0526i	−0.0791367	−	0.137069i	−0.902878	+	0.429898i	$0.858550\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$		0			0
$145$		0			0
$146$		0			0
$147$		0			0
$148$		0			0
$149$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$149$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$150$		0			0
$151$	−143.000	+	247.683i	−0.947020	+	1.64029i	−0.195364	+	0.980731i	$0.562589\pi$
$151$	−143.000	+	247.683i	−0.947020	+	1.64029i	−0.751656	+	0.659556i	$0.770744\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$		0			0
$157$	59.0000	+	102.191i	0.375796	+	0.650898i	0.990446	−	0.137902i	$-0.0440359\pi$
$157$	59.0000	+	102.191i	0.375796	+	0.650898i	−0.614650	+	0.788800i	$0.710703\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$	262.000			1.60736			0.803681	−	0.595060i	$-0.202872\pi$
$163$	262.000			1.60736			0.803681	+	0.595060i	$0.202872\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$167$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$168$		0			0
$169$	−157.500	+	272.798i	−0.931953	+	1.61419i
$170$		0			0
$171$		0			0
$172$		0			0
$173$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$173$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$174$		0			0
$175$	25.0000	+	43.3013i	0.142857	+	0.247436i
$176$		0			0
$177$		0			0
$178$		0			0
$179$		0			0		1.00000			$0$
$179$		0			0		−1.00000			$\pi$
$180$		0			0
$181$	314.000			1.73481			0.867403	−	0.497606i	$-0.165787\pi$
$181$	314.000			1.73481			0.867403	+	0.497606i	$0.165787\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$		0			0
$186$		0			0
$187$		0			0
$188$		0			0
$189$		0			0
$190$		0			0
$191$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$191$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$192$		0			0
$193$	191.000	+	330.822i	0.989637	+	1.71410i	0.619171	+	0.785256i	$0.287469\pi$
$193$	191.000	+	330.822i	0.989637	+	1.71410i	0.370466	+	0.928846i	$0.379198\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		0			0		1.00000			$0$
$197$		0			0		−1.00000			$\pi$
$198$		0			0
$199$	−386.000			−1.93970			−0.969849	−	0.243706i	$-0.921637\pi$
$199$	−386.000			−1.93970			−0.969849	+	0.243706i	$0.921637\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$		0			0
$209$		0			0
$210$		0			0
$211$	−83.0000	−	143.760i	−0.393365	−	0.681328i	0.599526	−	0.800355i	$-0.295356\pi$
$211$	−83.0000	−	143.760i	−0.393365	−	0.681328i	−0.992891	+	0.119027i	$0.962022\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$	−92.0000			−0.423963
$218$		0			0
$219$		0			0
$220$		0			0
$221$		0			0
$222$		0			0
$223$	169.000	−	292.717i	0.757848	−	1.31263i	−0.186099	−	0.982531i	$-0.559584\pi$
$223$	169.000	−	292.717i	0.757848	−	1.31263i	0.943946	−	0.330099i	$-0.107082\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$227$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$228$		0			0
$229$	−13.0000	−	22.5167i	−0.0567686	−	0.0983260i	0.836245	−	0.548357i	$-0.184746\pi$
$229$	−13.0000	−	22.5167i	−0.0567686	−	0.0983260i	−0.893013	+	0.450031i	$0.851413\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		0			0		1.00000			$0$
$233$		0			0		−1.00000			$\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$239$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$240$		0			0
$241$	143.000	−	247.683i	0.593361	−	1.02773i	−0.400415	−	0.916334i	$-0.631134\pi$
$241$	143.000	−	247.683i	0.593361	−	1.02773i	0.993776	−	0.111397i	$-0.0355327\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$		0			0
$246$		0			0
$247$	−286.000	−	495.367i	−1.15789	−	2.00553i
$248$		0			0
$249$		0			0
$250$		0			0
$251$		0			0		1.00000			$0$
$251$		0			0		−1.00000			$\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$		0			0
$256$		0			0
$257$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$257$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$258$		0			0
$259$	26.0000	−	45.0333i	0.100386	−	0.173874i
$260$		0			0
$261$		0			0
$262$		0			0
$263$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$263$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$		0			0
$269$		0			0		1.00000			$0$
$269$		0			0		−1.00000			$\pi$
$270$		0			0
$271$	−242.000			−0.892989			−0.446494	−	0.894786i	$-0.647328\pi$
$271$	−242.000			−0.892989			−0.446494	+	0.894786i	$0.647328\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$		0			0
$276$		0			0
$277$	−61.0000	+	105.655i	−0.220217	+	0.381426i	−0.954874	−	0.297012i	$-0.904010\pi$
$277$	−61.0000	+	105.655i	−0.220217	+	0.381426i	0.734657	+	0.678439i	$0.237343\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$281$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$282$		0			0
$283$	229.000	+	396.640i	0.809187	+	1.40155i	0.913428	+	0.407001i	$0.133426\pi$
$283$	229.000	+	396.640i	0.809187	+	1.40155i	−0.104240	+	0.994552i	$0.533241\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	289.000			1.00000
$290$		0			0
$291$		0			0
$292$		0			0
$293$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$293$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$		0			0
$300$		0			0
$301$	22.0000	+	38.1051i	0.0730897	+	0.126595i
$302$		0			0
$303$		0			0
$304$		0			0
$305$		0			0
$306$		0			0
$307$	358.000			1.16612			0.583062	−	0.812428i	$-0.301855\pi$
$307$	358.000			1.16612			0.583062	+	0.812428i	$0.301855\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$311$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$312$		0			0
$313$	71.0000	−	122.976i	0.226837	−	0.392893i	−0.730032	−	0.683413i	$-0.760495\pi$
$313$	71.0000	−	122.976i	0.226837	−	0.392893i	0.956869	+	0.290520i	$0.0938282\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$317$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$		0			0
$322$		0			0
$323$		0			0
$324$		0			0
$325$	−550.000			−1.69231
$326$		0			0
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$	181.000	−	313.501i	0.546828	−	0.947134i	−0.451662	−	0.892189i	$-0.649169\pi$
$331$	181.000	−	313.501i	0.546828	−	0.947134i	0.998489	−	0.0549442i	$-0.0174981\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$		0			0
$336$		0			0
$337$	−241.000	−	417.424i	−0.715134	−	1.23865i	−0.962908	−	0.269830i	$-0.913033\pi$
$337$	−241.000	−	417.424i	−0.715134	−	1.23865i	0.247774	−	0.968818i	$-0.420301\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$	188.000			0.548105
$344$		0			0
$345$		0			0
$346$		0			0
$347$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$347$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$348$		0			0
$349$	251.000	−	434.745i	0.719198	−	1.24569i	−0.242120	−	0.970246i	$-0.577843\pi$
$349$	251.000	−	434.745i	0.719198	−	1.24569i	0.961318	−	0.275441i	$-0.0888238\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$353$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$		0			0		1.00000			$0$
$359$		0			0		−1.00000			$\pi$
$360$		0			0
$361$	315.000			0.872576
$362$		0			0
$363$		0			0
$364$		0			0
$365$		0			0
$366$		0			0
$367$	−359.000	+	621.806i	−0.978202	+	1.69429i	−0.309264	+	0.950976i	$0.600083\pi$
$367$	−359.000	+	621.806i	−0.978202	+	1.69429i	−0.668937	+	0.743319i	$0.733251\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$	−349.000	−	604.486i	−0.935657	−	1.62061i	−0.773458	−	0.633847i	$-0.781475\pi$
$373$	−349.000	−	604.486i	−0.935657	−	1.62061i	−0.162198	−	0.986758i	$-0.551858\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$	694.000			1.83113			0.915567	−	0.402165i	$-0.131742\pi$
$379$	694.000			1.83113			0.915567	+	0.402165i	$0.131742\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$383$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$		0			0
$389$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$389$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$		0			0
$394$		0			0
$395$		0			0
$396$		0			0
$397$	362.000			0.911839			0.455919	−	0.890021i	$-0.349311\pi$
$397$	362.000			0.911839			0.455919	+	0.890021i	$0.349311\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$401$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$402$		0			0
$403$	506.000	−	876.418i	1.25558	−	2.17473i
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$	−313.000	−	542.132i	−0.765281	−	1.32551i	−0.940098	−	0.340905i	$-0.889267\pi$
$409$	−313.000	−	542.132i	−0.765281	−	1.32551i	0.174817	−	0.984601i	$-0.444067\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$		0			0
$418$		0			0
$419$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$419$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$420$		0			0
$421$	179.000	−	310.037i	0.425178	−	0.736430i	−0.571259	−	0.820770i	$-0.693545\pi$
$421$	179.000	−	310.037i	0.425178	−	0.736430i	0.996437	+	0.0843398i	$0.0268781\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		0			0
$426$		0			0
$427$	74.0000	+	128.172i	0.173302	+	0.300168i
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0		1.00000			$0$
$431$		0			0		−1.00000			$\pi$
$432$		0			0
$433$	−862.000			−1.99076			−0.995381	−	0.0960028i	$-0.969394\pi$
$433$	−862.000			−1.99076			−0.995381	+	0.0960028i	$0.969394\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		0			0
$438$		0			0
$439$	−47.0000	+	81.4064i	−0.107062	+	0.185436i	−0.914579	−	0.404408i	$-0.867478\pi$
$439$	−47.0000	+	81.4064i	−0.107062	+	0.185436i	0.807517	+	0.589844i	$0.200811\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$443$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$		0			0
$449$		0			0		1.00000			$0$
$449$		0			0		−1.00000			$\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	407.000	−	704.945i	0.890591	−	1.54255i	0.0514223	−	0.998677i	$-0.483625\pi$
$457$	407.000	−	704.945i	0.890591	−	1.54255i	0.839168	−	0.543872i	$-0.183042\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$461$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$462$		0			0
$463$	−263.000	−	455.529i	−0.568035	−	0.983865i	−0.996760	−	0.0804300i	$-0.974371\pi$
$463$	−263.000	−	455.529i	−0.568035	−	0.983865i	0.428726	−	0.903435i	$-0.358963\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		0			0		1.00000			$0$
$467$		0			0		−1.00000			$\pi$
$468$		0			0
$469$	244.000			0.520256
$470$		0			0
$471$		0			0
$472$		0			0
$473$		0			0
$474$		0			0
$475$	325.000	−	562.917i	0.684211	−	1.18509i
$476$		0			0
$477$		0			0
$478$		0			0
$479$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$479$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$480$		0			0
$481$	286.000	+	495.367i	0.594595	+	1.02987i
$482$		0			0
$483$		0			0
$484$		0			0
$485$		0			0
$486$		0			0
$487$	−962.000			−1.97536			−0.987680	−	0.156489i	$-0.949982\pi$
$487$	−962.000			−1.97536			−0.987680	+	0.156489i	$0.949982\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$491$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$	13.0000	+	22.5167i	0.0260521	+	0.0451236i	0.878758	−	0.477269i	$-0.158373\pi$
$499$	13.0000	+	22.5167i	0.0260521	+	0.0451236i	−0.852705	+	0.522392i	$0.825040\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		0			0		1.00000			$0$
$503$		0			0		−1.00000			$\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$		0			0
$508$		0			0
$509$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$509$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$510$		0			0
$511$	−46.0000	+	79.6743i	−0.0900196	+	0.155918i
$512$		0			0
$513$		0			0
$514$		0			0
$515$		0			0
$516$		0			0
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$		0			0		1.00000			$0$
$521$		0			0		−1.00000			$\pi$
$522$		0			0
$523$	982.000			1.87763			0.938815	−	0.344423i	$-0.111925\pi$
$523$	982.000			1.87763			0.938815	+	0.344423i	$0.111925\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		0			0
$528$		0			0
$529$	−264.500	+	458.127i	−0.500000	+	0.866025i
$530$		0			0
$531$		0			0
$532$		0			0
$533$		0			0
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$	1034.00			1.91128			0.955638	−	0.294545i	$-0.0951680\pi$
$541$	1034.00			1.91128			0.955638	+	0.294545i	$0.0951680\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		0			0
$546$		0			0
$547$	253.000	−	438.209i	0.462523	−	0.801113i	−0.536563	−	0.843860i	$-0.680278\pi$
$547$	253.000	−	438.209i	0.462523	−	0.801113i	0.999086	+	0.0427471i	$0.0136110\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		0			0
$552$		0			0
$553$	142.000	+	245.951i	0.256781	+	0.444758i
$554$		0			0
$555$		0			0
$556$		0			0
$557$		0			0		1.00000			$0$
$557$		0			0		−1.00000			$\pi$
$558$		0			0
$559$	−484.000			−0.865832
$560$		0			0
$561$		0			0
$562$		0			0
$563$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$563$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$		0			0
$568$		0			0
$569$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$569$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$570$		0			0
$571$	−443.000	−	767.299i	−0.775832	−	1.34378i	−0.934326	−	0.356420i	$-0.883997\pi$
$571$	−443.000	−	767.299i	−0.775832	−	1.34378i	0.158494	−	0.987360i	$-0.449336\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$		0			0
$577$	962.000			1.66724			0.833622	−	0.552335i	$-0.186263\pi$
$577$	962.000			1.66724			0.833622	+	0.552335i	$0.186263\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$		0			0
$584$		0			0
$585$		0			0
$586$		0			0
$587$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$587$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$588$		0			0
$589$	598.000	+	1035.77i	1.01528	+	1.75852i
$590$		0			0
$591$		0			0
$592$		0			0
$593$		0			0		1.00000			$0$
$593$		0			0		−1.00000			$\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$599$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$600$		0			0
$601$	263.000	−	455.529i	0.437604	−	0.757952i	−0.559900	−	0.828560i	$-0.689161\pi$
$601$	263.000	−	455.529i	0.437604	−	0.757952i	0.997504	+	0.0706077i	$0.0224939\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$		0			0
$606$		0			0
$607$	−407.000	−	704.945i	−0.670511	−	1.16136i	−0.977759	−	0.209729i	$-0.932742\pi$
$607$	−407.000	−	704.945i	−0.670511	−	1.16136i	0.307249	−	0.951629i	$-0.400592\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$	−1126.00			−1.83687			−0.918434	−	0.395574i	$-0.870546\pi$
$613$	−1126.00			−1.83687			−0.918434	+	0.395574i	$0.870546\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$617$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$618$		0			0
$619$	−107.000	+	185.329i	−0.172859	+	0.299401i	−0.939418	−	0.342773i	$-0.888634\pi$
$619$	−107.000	+	185.329i	−0.172859	+	0.299401i	0.766559	+	0.642174i	$0.221967\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	−312.500	−	541.266i	−0.500000	−	0.866025i
$626$		0			0
$627$		0			0
$628$		0			0
$629$		0			0
$630$		0			0
$631$	−674.000			−1.06815			−0.534073	−	0.845438i	$-0.679339\pi$
$631$	−674.000			−1.06815			−0.534073	+	0.845438i	$0.679339\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$		0			0
$636$		0			0
$637$	−495.000	+	857.365i	−0.777080	+	1.34594i
$638$		0			0
$639$		0			0
$640$		0			0
$641$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$641$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$642$		0			0
$643$	157.000	+	271.932i	0.244168	+	0.422911i	0.961897	−	0.273411i	$-0.0881518\pi$
$643$	157.000	+	271.932i	0.244168	+	0.422911i	−0.717729	+	0.696322i	$0.754819\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0		1.00000			$0$
$647$		0			0		−1.00000			$\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$		0			0
$652$		0			0
$653$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$653$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$		0			0
$658$		0			0
$659$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$659$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$660$		0			0
$661$	−61.0000	−	105.655i	−0.0922844	−	0.159841i	0.816188	−	0.577787i	$-0.196084\pi$
$661$	−61.0000	−	105.655i	−0.0922844	−	0.159841i	−0.908472	+	0.417946i	$0.862750\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$		0			0
$668$		0			0
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$	−577.000	+	999.393i	−0.857355	+	1.48498i	0.0170877	+	0.999854i	$0.494561\pi$
$673$	−577.000	+	999.393i	−0.857355	+	1.48498i	−0.874443	+	0.485129i	$0.838773\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$677$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$678$		0			0
$679$	2.00000	+	3.46410i	0.00294551	+	0.00510177i
$680$		0			0
$681$		0			0
$682$		0			0
$683$		0			0		1.00000			$0$
$683$		0			0		−1.00000			$\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$		0			0
$688$		0			0
$689$		0			0
$690$		0			0
$691$	−659.000	+	1141.42i	−0.953690	+	1.65184i	−0.216353	+	0.976315i	$0.569416\pi$
$691$	−659.000	+	1141.42i	−0.953690	+	1.65184i	−0.737337	+	0.675525i	$0.763917\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$		0			0
$701$		0			0		1.00000			$0$
$701$		0			0		−1.00000			$\pi$
$702$		0			0
$703$	−676.000			−0.961593
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	467.000	−	808.868i	0.658674	−	1.14086i	−0.322285	−	0.946643i	$-0.604451\pi$
$709$	467.000	−	808.868i	0.658674	−	1.14086i	0.980959	−	0.194214i	$-0.0622158\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$		0			0		1.00000			$0$
$719$		0			0		−1.00000			$\pi$
$720$		0			0
$721$	388.000			0.538141
$722$		0			0
$723$		0			0
$724$		0			0
$725$		0			0
$726$		0			0
$727$	241.000	−	417.424i	0.331499	−	0.574174i	−0.651307	−	0.758815i	$-0.725779\pi$
$727$	241.000	−	417.424i	0.331499	−	0.574174i	0.982806	+	0.184641i	$0.0591122\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$		0			0
$732$		0			0
$733$	−517.000	−	895.470i	−0.705321	−	1.22165i	−0.966576	−	0.256381i	$-0.917470\pi$
$733$	−517.000	−	895.470i	−0.705321	−	1.22165i	0.261255	−	0.965270i	$-0.415864\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$		0			0
$739$	1222.00			1.65359			0.826793	−	0.562506i	$-0.190163\pi$
$739$	1222.00			1.65359			0.826793	+	0.562506i	$0.190163\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$743$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	601.000	+	1040.96i	0.800266	+	1.38610i	0.919441	+	0.393229i	$0.128642\pi$
$751$	601.000	+	1040.96i	0.800266	+	1.38610i	−0.119174	+	0.992873i	$0.538025\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$	−838.000			−1.10700			−0.553501	−	0.832849i	$-0.686708\pi$
$757$	−838.000			−1.10700			−0.553501	+	0.832849i	$0.686708\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$761$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$762$		0			0
$763$	−214.000	+	370.659i	−0.280472	+	0.485791i
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$		0			0
$769$	767.000	+	1328.48i	0.997399	+	1.72755i	0.561118	+	0.827736i	$0.310371\pi$
$769$	767.000	+	1328.48i	0.997399	+	1.72755i	0.436281	+	0.899811i	$0.356295\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$		0			0		1.00000			$0$
$773$		0			0		−1.00000			$\pi$
$774$		0			0
$775$	1150.00			1.48387
$776$		0			0
$777$		0			0
$778$		0			0
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$		0			0
$785$		0			0
$786$		0			0
$787$	781.000	+	1352.73i	0.992376	+	1.71885i	0.602922	+	0.797800i	$0.294003\pi$
$787$	781.000	+	1352.73i	0.992376	+	1.71885i	0.389454	+	0.921046i	$0.372664\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	−1628.00			−2.05296
$794$		0			0
$795$		0			0
$796$		0			0
$797$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$797$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$		0			0
$802$		0			0
$803$		0			0
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$		0			0		1.00000			$0$
$809$		0			0		−1.00000			$\pi$
$810$		0			0
$811$	−1514.00			−1.86683			−0.933416	−	0.358797i	$-0.883187\pi$
$811$	−1514.00			−1.86683			−0.933416	+	0.358797i	$0.883187\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		0			0
$816$		0			0
$817$	286.000	−	495.367i	0.350061	−	0.606324i
$818$		0			0
$819$		0			0
$820$		0			0
$821$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$821$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$822$		0			0
$823$	529.000	+	916.255i	0.642770	+	1.11331i	0.984812	+	0.173626i	$0.0555484\pi$
$823$	529.000	+	916.255i	0.642770	+	1.11331i	−0.342041	+	0.939685i	$0.611118\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		0			0		1.00000			$0$
$827$		0			0		−1.00000			$\pi$
$828$		0			0
$829$	458.000			0.552473			0.276236	−	0.961090i	$-0.410913\pi$
$829$	458.000			0.552473			0.276236	+	0.961090i	$0.410913\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		0			0
$834$		0			0
$835$		0			0
$836$		0			0
$837$		0			0
$838$		0			0
$839$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$839$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$840$		0			0
$841$	−420.500	−	728.327i	−0.500000	−	0.866025i
$842$		0			0
$843$		0			0
$844$		0			0
$845$		0			0
$846$		0			0
$847$	−242.000			−0.285714
$848$		0			0
$849$		0			0
$850$		0			0
$851$		0			0
$852$		0			0
$853$	−829.000	+	1435.87i	−0.971864	+	1.68332i	−0.281946	+	0.959430i	$0.590980\pi$
$853$	−829.000	+	1435.87i	−0.971864	+	1.68332i	−0.689918	+	0.723888i	$0.742353\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$857$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$858$		0			0
$859$	709.000	+	1228.02i	0.825378	+	1.42960i	0.901630	+	0.432509i	$0.142371\pi$
$859$	709.000	+	1228.02i	0.825378	+	1.42960i	−0.0762515	+	0.997089i	$0.524295\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		0			0		1.00000			$0$
$863$		0			0		−1.00000			$\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$		0			0
$868$		0			0
$869$		0			0
$870$		0			0
$871$	−1342.00	+	2324.41i	−1.54076	+	2.66867i
$872$		0			0
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$	299.000	+	517.883i	0.340935	+	0.590517i	0.984607	−	0.174785i	$-0.0559231\pi$
$877$	299.000	+	517.883i	0.340935	+	0.590517i	−0.643672	+	0.765302i	$0.722590\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		0			0		1.00000			$0$
$881$		0			0		−1.00000			$\pi$
$882$		0			0
$883$	1702.00			1.92752			0.963760	−	0.266771i	$-0.0859568\pi$
$883$	1702.00			1.92752			0.963760	+	0.266771i	$0.0859568\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$887$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$888$		0			0
$889$	−146.000	+	252.879i	−0.164229	+	0.284454i
$890$		0			0
$891$		0			0
$892$		0			0
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$		0			0
$899$		0			0
$900$		0			0
$901$		0			0
$902$		0			0
$903$		0			0
$904$		0			0
$905$		0			0
$906$		0			0
$907$	−107.000	+	185.329i	−0.117971	+	0.204332i	−0.918964	−	0.394342i	$-0.870972\pi$
$907$	−107.000	+	185.329i	−0.117971	+	0.204332i	0.800992	+	0.598675i	$0.204306\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$911$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$912$		0			0
$913$		0			0
$914$		0			0
$915$		0			0
$916$		0			0
$917$		0			0
$918$		0			0
$919$	−866.000			−0.942329			−0.471164	−	0.882045i	$-0.656166\pi$
$919$	−866.000			−0.942329			−0.471164	+	0.882045i	$0.656166\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		0			0
$924$		0			0
$925$	−325.000	+	562.917i	−0.351351	+	0.608558i
$926$		0			0
$927$		0			0
$928$		0			0
$929$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$929$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$930$		0			0
$931$	−585.000	−	1013.25i	−0.628357	−	1.08835i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$	−1198.00			−1.27855			−0.639274	−	0.768979i	$-0.720765\pi$
$937$	−1198.00			−1.27855			−0.639274	+	0.768979i	$0.720765\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$941$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$947$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$948$		0			0
$949$	−506.000	−	876.418i	−0.533193	−	0.923517i
$950$		0			0
$951$		0			0
$952$		0			0
$953$		0			0		1.00000			$0$
$953$		0			0		−1.00000			$\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	−577.500	+	1000.26i	−0.600937	+	1.04085i
$962$		0			0
$963$		0			0
$964$		0			0
$965$		0			0
$966$		0			0
$967$	−767.000	−	1328.48i	−0.793175	−	1.37382i	−0.923992	−	0.382412i	$-0.875093\pi$
$967$	−767.000	−	1328.48i	−0.793175	−	1.37382i	0.130817	−	0.991407i	$-0.458240\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		0			0		1.00000			$0$
$971$		0			0		−1.00000			$\pi$
$972$		0			0
$973$	−44.0000			−0.0452210
$974$		0			0
$975$		0			0
$976$		0			0
$977$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$977$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$		0			0
$982$		0			0
$983$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$983$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$	46.0000			0.0464178			0.0232089	−	0.999731i	$-0.492612\pi$
$991$	46.0000			0.0464178			0.0232089	+	0.999731i	$0.492612\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		0			0
$996$		0			0
$997$	947.000	−	1640.25i	0.949850	−	1.64519i	0.204112	−	0.978947i	$-0.434569\pi$
$997$	947.000	−	1640.25i	0.949850	−	1.64519i	0.745737	−	0.666240i	$-0.232097\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1296.3.q.b.593.1		2
3.2	odd	2	CM	1296.3.q.b.593.1		2
4.3	odd	2		324.3.g.b.269.1		2
9.2	odd	6		48.3.e.a.17.1		1
9.4	even	3	inner	1296.3.q.b.1025.1		2
9.5	odd	6	inner	1296.3.q.b.1025.1		2
9.7	even	3		48.3.e.a.17.1		1
12.11	even	2		324.3.g.b.269.1		2
36.7	odd	6		12.3.c.a.5.1	✓	1
36.11	even	6		12.3.c.a.5.1	✓	1
36.23	even	6		324.3.g.b.53.1		2
36.31	odd	6		324.3.g.b.53.1		2
45.2	even	12		1200.3.c.c.449.1		2
45.7	odd	12		1200.3.c.c.449.1		2
45.29	odd	6		1200.3.l.b.401.1		1
45.34	even	6		1200.3.l.b.401.1		1
45.38	even	12		1200.3.c.c.449.2		2
45.43	odd	12		1200.3.c.c.449.2		2
72.11	even	6		192.3.e.b.65.1		1
72.29	odd	6		192.3.e.a.65.1		1
72.43	odd	6		192.3.e.b.65.1		1
72.61	even	6		192.3.e.a.65.1		1
144.11	even	12		768.3.h.a.641.2		2
144.29	odd	12		768.3.h.b.641.2		2
144.43	odd	12		768.3.h.a.641.2		2
144.61	even	12		768.3.h.b.641.2		2
144.83	even	12		768.3.h.a.641.1		2
144.101	odd	12		768.3.h.b.641.1		2
144.115	odd	12		768.3.h.a.641.1		2
144.133	even	12		768.3.h.b.641.1		2
180.7	even	12		300.3.b.a.149.2		2
180.43	even	12		300.3.b.a.149.1		2
180.47	odd	12		300.3.b.a.149.2		2
180.79	odd	6		300.3.g.b.101.1		1
180.83	odd	12		300.3.b.a.149.1		2
180.119	even	6		300.3.g.b.101.1		1
252.11	even	6		588.3.p.c.569.1		2
252.47	odd	6		588.3.p.b.557.1		2
252.79	odd	6		588.3.p.c.557.1		2
252.83	odd	6		588.3.c.c.197.1		1
252.115	even	6		588.3.p.b.569.1		2
252.151	odd	6		588.3.p.c.569.1		2
252.187	even	6		588.3.p.b.557.1		2
252.191	even	6		588.3.p.c.557.1		2
252.223	even	6		588.3.c.c.197.1		1
252.227	odd	6		588.3.p.b.569.1		2
396.43	even	6		1452.3.e.b.485.1		1
396.263	odd	6		1452.3.e.b.485.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
12.3.c.a.5.1	✓	1	36.7	odd	6
12.3.c.a.5.1	✓	1	36.11	even	6
48.3.e.a.17.1		1	9.2	odd	6
48.3.e.a.17.1		1	9.7	even	3
192.3.e.a.65.1		1	72.29	odd	6
192.3.e.a.65.1		1	72.61	even	6
192.3.e.b.65.1		1	72.11	even	6
192.3.e.b.65.1		1	72.43	odd	6
300.3.b.a.149.1		2	180.43	even	12
300.3.b.a.149.1		2	180.83	odd	12
300.3.b.a.149.2		2	180.7	even	12
300.3.b.a.149.2		2	180.47	odd	12
300.3.g.b.101.1		1	180.79	odd	6
300.3.g.b.101.1		1	180.119	even	6
324.3.g.b.53.1		2	36.23	even	6
324.3.g.b.53.1		2	36.31	odd	6
324.3.g.b.269.1		2	4.3	odd	2
324.3.g.b.269.1		2	12.11	even	2
588.3.c.c.197.1		1	252.83	odd	6
588.3.c.c.197.1		1	252.223	even	6
588.3.p.b.557.1		2	252.47	odd	6
588.3.p.b.557.1		2	252.187	even	6
588.3.p.b.569.1		2	252.115	even	6
588.3.p.b.569.1		2	252.227	odd	6
588.3.p.c.557.1		2	252.79	odd	6
588.3.p.c.557.1		2	252.191	even	6
588.3.p.c.569.1		2	252.11	even	6
588.3.p.c.569.1		2	252.151	odd	6
768.3.h.a.641.1		2	144.83	even	12
768.3.h.a.641.1		2	144.115	odd	12
768.3.h.a.641.2		2	144.11	even	12
768.3.h.a.641.2		2	144.43	odd	12
768.3.h.b.641.1		2	144.101	odd	12
768.3.h.b.641.1		2	144.133	even	12
768.3.h.b.641.2		2	144.29	odd	12
768.3.h.b.641.2		2	144.61	even	12
1200.3.c.c.449.1		2	45.2	even	12
1200.3.c.c.449.1		2	45.7	odd	12
1200.3.c.c.449.2		2	45.38	even	12
1200.3.c.c.449.2		2	45.43	odd	12
1200.3.l.b.401.1		1	45.29	odd	6
1200.3.l.b.401.1		1	45.34	even	6
1296.3.q.b.593.1		2	1.1	even	1	trivial
1296.3.q.b.593.1		2	3.2	odd	2	CM
1296.3.q.b.1025.1		2	9.4	even	3	inner
1296.3.q.b.1025.1		2	9.5	odd	6	inner
1452.3.e.b.485.1		1	396.43	even	6
1452.3.e.b.485.1		1	396.263	odd	6

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:	\( N \)	\(=\)	\( 1296 = 2^{4} \cdot 3^{4} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1296.q (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(1.00000 - 1.73205i) q^{7} +O(q^{10})\)	\(q+(1.00000 - 1.73205i) q^{7} +(11.0000 + 19.0526i) q^{13} -26.0000 q^{19} +(-12.5000 + 21.6506i) q^{25} +(-23.0000 - 39.8372i) q^{31} +26.0000 q^{37} +(-11.0000 + 19.0526i) q^{43} +(22.5000 + 38.9711i) q^{49} +(-37.0000 + 64.0859i) q^{61} +(61.0000 + 105.655i) q^{67} -46.0000 q^{73} +(-71.0000 + 122.976i) q^{79} +44.0000 q^{91} +(-1.00000 + 1.73205i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{7}+O(q^{10}) \) 2 * q + 2 * q^7	\( 2 q + 2 q^{7} + 22 q^{13} - 52 q^{19} - 25 q^{25} - 46 q^{31} + 52 q^{37} - 22 q^{43} + 45 q^{49} - 74 q^{61} + 122 q^{67} - 92 q^{73} - 142 q^{79} + 88 q^{91} - 2 q^{97}+O(q^{100}) \) 2 * q + 2 * q^7 + 22 * q^13 - 52 * q^19 - 25 * q^25 - 46 * q^31 + 52 * q^37 - 22 * q^43 + 45 * q^49 - 74 * q^61 + 122 * q^67 - 92 * q^73 - 142 * q^79 + 88 * q^91 - 2 * q^97

Introduction

L-functions

Modular forms

Varieties

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Representations

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Database

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