LMFDB - Embedded newform 1296.3.q.c.1025.1

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:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 108)
Sato-Tate group:	$\mathrm{U}(1)[D_{6}]$

Embedding invariants

Embedding label			1025.1
Root			$0.500000 - 0.866025i$ of defining polynomial
Character	$\chi$	$=$	1296.1025
Dual form			1296.3.q.c.593.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(5.50000 + 9.52628i) q^{7} +O(q^{10})$	$q+(5.50000 + 9.52628i) q^{7} +(-11.5000 + 19.9186i) q^{13} +37.0000 q^{19} +(-12.5000 - 21.6506i) q^{25} +(-23.0000 + 39.8372i) q^{31} -73.0000 q^{37} +(-11.0000 - 19.0526i) q^{43} +(-36.0000 + 62.3538i) q^{49} +(-23.5000 - 40.7032i) q^{61} +(-6.50000 + 11.2583i) q^{67} +143.000 q^{73} +(5.50000 + 9.52628i) q^{79} -253.000 q^{91} +(84.5000 + 146.358i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 11 q^{7}+O(q^{10}) $ 2 * q + 11 * q^7	$ 2 q + 11 q^{7} - 23 q^{13} + 74 q^{19} - 25 q^{25} - 46 q^{31} - 146 q^{37} - 22 q^{43} - 72 q^{49} - 47 q^{61} - 13 q^{67} + 286 q^{73} + 11 q^{79} - 506 q^{91} + 169 q^{97}+O(q^{100}) $ 2 * q + 11 * q^7 - 23 * q^13 + 74 * q^19 - 25 * q^25 - 46 * q^31 - 146 * q^37 - 22 * q^43 - 72 * q^49 - 47 * q^61 - 13 * q^67 + 286 * q^73 + 11 * q^79 - 506 * q^91 + 169 * q^97

Display 100 coefficients 10 coefficients

Character values

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

$n$	$325$	$1135$	$1217$
$\chi(n)$	$1$	$1$	$e\left(\frac{5}{6}\right)$

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	$-0.333333\pi$
$5$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$6$		0			0
$7$	5.50000	+	9.52628i	0.785714	+	1.36090i	0.928571	+	0.371154i	$0.121038\pi$
$7$	5.50000	+	9.52628i	0.785714	+	1.36090i	−0.142857	+	0.989743i	$0.545629\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$11$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$12$		0			0
$13$	−11.5000	+	19.9186i	−0.884615	+	1.53220i	−0.0384615	+	0.999260i	$0.512246\pi$
$13$	−11.5000	+	19.9186i	−0.884615	+	1.53220i	−0.846154	+	0.532939i	$0.821088\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$		0			0		1.00000			$0$
$17$		0			0		−1.00000			$\pi$
$18$		0			0
$19$	37.0000			1.94737			0.973684	−	0.227901i	$-0.0731864\pi$
$19$	37.0000			1.94737			0.973684	+	0.227901i	$0.0731864\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$23$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$24$		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	$-0.666667\pi$
$29$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$30$		0			0
$31$	−23.0000	+	39.8372i	−0.741935	+	1.28507i	0.209677	+	0.977771i	$0.432759\pi$
$31$	−23.0000	+	39.8372i	−0.741935	+	1.28507i	−0.951613	+	0.307299i	$0.900575\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$		0			0
$37$	−73.0000			−1.97297			−0.986486	−	0.163843i	$-0.947611\pi$
$37$	−73.0000			−1.97297			−0.986486	+	0.163843i	$0.947611\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$41$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$42$		0			0
$43$	−11.0000	−	19.0526i	−0.255814	−	0.443083i	0.709302	−	0.704904i	$-0.249010\pi$
$43$	−11.0000	−	19.0526i	−0.255814	−	0.443083i	−0.965116	+	0.261822i	$0.915677\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$47$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$48$		0			0
$49$	−36.0000	+	62.3538i	−0.734694	+	1.27253i
$50$		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	$-0.333333\pi$
$59$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$60$		0			0
$61$	−23.5000	−	40.7032i	−0.385246	−	0.667265i	0.606557	−	0.795040i	$-0.292550\pi$
$61$	−23.5000	−	40.7032i	−0.385246	−	0.667265i	−0.991803	+	0.127774i	$0.959217\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$		0			0
$66$		0			0
$67$	−6.50000	+	11.2583i	−0.0970149	+	0.168035i	−0.910448	−	0.413624i	$-0.864263\pi$
$67$	−6.50000	+	11.2583i	−0.0970149	+	0.168035i	0.813433	+	0.581659i	$0.197596\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		0			0		1.00000			$0$
$71$		0			0		−1.00000			$\pi$
$72$		0			0
$73$	143.000			1.95890			0.979452	−	0.201677i	$-0.0646392\pi$
$73$	143.000			1.95890			0.979452	+	0.201677i	$0.0646392\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$		0			0
$78$		0			0
$79$	5.50000	+	9.52628i	0.0696203	+	0.120586i	0.898734	−	0.438494i	$-0.144488\pi$
$79$	5.50000	+	9.52628i	0.0696203	+	0.120586i	−0.829114	+	0.559080i	$0.811155\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$83$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$84$		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
$91$	−253.000			−2.78022
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$	84.5000	+	146.358i	0.871134	+	1.50885i	0.860825	+	0.508902i	$0.169948\pi$
$97$	84.5000	+	146.358i	0.871134	+	1.50885i	0.0103093	+	0.999947i	$0.496718\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$101$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$102$		0			0
$103$	−78.5000	+	135.966i	−0.762136	+	1.32006i	0.179612	+	0.983738i	$0.442516\pi$
$103$	−78.5000	+	135.966i	−0.762136	+	1.32006i	−0.941748	+	0.336321i	$0.890817\pi$
$104$		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	$-0.333333\pi$
$113$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$114$		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	$-0.333333\pi$
$131$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$132$		0			0
$133$	203.500	+	352.472i	1.53008	+	2.65017i
$134$		0			0
$135$		0			0
$136$		0			0
$137$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$137$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$138$		0			0
$139$	−114.500	+	198.320i	−0.823741	+	1.42676i	0.0791367	+	0.996864i	$0.474784\pi$
$139$	−114.500	+	198.320i	−0.823741	+	1.42676i	−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	$-0.333333\pi$
$149$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$150$		0			0
$151$	113.500	+	196.588i	0.751656	+	1.30191i	0.947020	+	0.321175i	$0.104078\pi$
$151$	113.500	+	196.588i	0.751656	+	1.30191i	−0.195364	+	0.980731i	$0.562589\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$		0			0
$157$	59.0000	−	102.191i	0.375796	−	0.650898i	−0.614650	−	0.788800i	$-0.710703\pi$
$157$	59.0000	−	102.191i	0.375796	−	0.650898i	0.990446	+	0.137902i	$0.0440359\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$	37.0000			0.226994			0.113497	−	0.993538i	$-0.463795\pi$
$163$	37.0000			0.226994			0.113497	+	0.993538i	$0.463795\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$167$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$168$		0			0
$169$	−180.000	−	311.769i	−1.06509	−	1.84479i
$170$		0			0
$171$		0			0
$172$		0			0
$173$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$173$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$174$		0			0
$175$	137.500	−	238.157i	0.785714	−	1.36090i
$176$		0			0
$177$		0			0
$178$		0			0
$179$		0			0		1.00000			$0$
$179$		0			0		−1.00000			$\pi$
$180$		0			0
$181$	−1.00000			−0.00552486			−0.00276243	−	0.999996i	$-0.500879\pi$
$181$	−1.00000			−0.00552486			−0.00276243	+	0.999996i	$0.500879\pi$
$182$		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	$-0.666667\pi$
$191$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$192$		0			0
$193$	−119.500	+	206.980i	−0.619171	+	1.07244i	0.370466	+	0.928846i	$0.379198\pi$
$193$	−119.500	+	206.980i	−0.619171	+	1.07244i	−0.989637	+	0.143590i	$0.954135\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		0			0		1.00000			$0$
$197$		0			0		−1.00000			$\pi$
$198$		0			0
$199$	109.000			0.547739			0.273869	−	0.961767i	$-0.411696\pi$
$199$	109.000			0.547739			0.273869	+	0.961767i	$0.411696\pi$
$200$		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
$211$	209.500	−	362.865i	0.992891	−	1.71974i	0.393365	−	0.919382i	$-0.371311\pi$
$211$	209.500	−	362.865i	0.992891	−	1.71974i	0.599526	−	0.800355i	$-0.295356\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$	−506.000			−2.33180
$218$		0			0
$219$		0			0
$220$		0			0
$221$		0			0
$222$		0			0
$223$	169.000	+	292.717i	0.757848	+	1.31263i	0.943946	+	0.330099i	$0.107082\pi$
$223$	169.000	+	292.717i	0.757848	+	1.31263i	−0.186099	+	0.982531i	$0.559584\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$227$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$228$		0			0
$229$	−13.0000	+	22.5167i	−0.0567686	+	0.0983260i	−0.893013	−	0.450031i	$-0.851413\pi$
$229$	−13.0000	+	22.5167i	−0.0567686	+	0.0983260i	0.836245	+	0.548357i	$0.184746\pi$
$230$		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	$-0.333333\pi$
$239$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$240$		0			0
$241$	−239.500	−	414.826i	−0.993776	−	1.72127i	−0.593361	−	0.804936i	$-0.702199\pi$
$241$	−239.500	−	414.826i	−0.993776	−	1.72127i	−0.400415	−	0.916334i	$-0.631134\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$		0			0
$246$		0			0
$247$	−425.500	+	736.988i	−1.72267	+	2.98376i
$248$		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	$-0.333333\pi$
$257$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$258$		0			0
$259$	−401.500	−	695.418i	−1.55019	−	2.68501i
$260$		0			0
$261$		0			0
$262$		0			0
$263$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$263$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$264$		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
$271$	541.000			1.99631			0.998155	−	0.0607176i	$-0.0193389\pi$
$271$	541.000			1.99631			0.998155	+	0.0607176i	$0.0193389\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$		0			0
$276$		0			0
$277$	−61.0000	−	105.655i	−0.220217	−	0.381426i	0.734657	−	0.678439i	$-0.237343\pi$
$277$	−61.0000	−	105.655i	−0.220217	−	0.381426i	−0.954874	+	0.297012i	$0.904010\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$281$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$282$		0			0
$283$	229.000	−	396.640i	0.809187	−	1.40155i	−0.104240	−	0.994552i	$-0.533241\pi$
$283$	229.000	−	396.640i	0.809187	−	1.40155i	0.913428	−	0.407001i	$-0.133426\pi$
$284$		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	$-0.333333\pi$
$293$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$		0			0
$300$		0			0
$301$	121.000	−	209.578i	0.401993	−	0.696273i
$302$		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	$-0.333333\pi$
$311$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$312$		0			0
$313$	228.500	+	395.774i	0.730032	+	1.26445i	0.956869	+	0.290520i	$0.0938282\pi$
$313$	228.500	+	395.774i	0.730032	+	1.26445i	−0.226837	+	0.973933i	$0.572838\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$317$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$		0			0
$322$		0			0
$323$		0			0
$324$		0			0
$325$	575.000			1.76923
$326$		0			0
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$	149.500	+	258.942i	0.451662	+	0.782301i	0.998489	−	0.0549442i	$-0.0174981\pi$
$331$	149.500	+	258.942i	0.451662	+	0.782301i	−0.546828	+	0.837245i	$0.684165\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$		0			0
$336$		0			0
$337$	−83.5000	+	144.626i	−0.247774	+	0.429158i	−0.962908	−	0.269830i	$-0.913033\pi$
$337$	−83.5000	+	144.626i	−0.247774	+	0.429158i	0.715134	+	0.698988i	$0.246366\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$	−253.000			−0.737609
$344$		0			0
$345$		0			0
$346$		0			0
$347$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$347$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$348$		0			0
$349$	84.5000	+	146.358i	0.242120	+	0.419365i	0.961318	−	0.275441i	$-0.0888238\pi$
$349$	84.5000	+	146.358i	0.242120	+	0.419365i	−0.719198	+	0.694805i	$0.755490\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$353$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$354$		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
$361$	1008.00			2.79224
$362$		0			0
$363$		0			0
$364$		0			0
$365$		0			0
$366$		0			0
$367$	113.500	+	196.588i	0.309264	+	0.535661i	0.978202	−	0.207657i	$-0.0665839\pi$
$367$	113.500	+	196.588i	0.309264	+	0.535661i	−0.668937	+	0.743319i	$0.733251\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$	60.5000	−	104.789i	0.162198	−	0.280936i	−0.773458	−	0.633847i	$-0.781475\pi$
$373$	60.5000	−	104.789i	0.162198	−	0.280936i	0.935657	+	0.352911i	$0.114808\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$	−611.000			−1.61214			−0.806069	−	0.591822i	$-0.798409\pi$
$379$	−611.000			−1.61214			−0.806069	+	0.591822i	$0.798409\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$383$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$		0			0
$389$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$389$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$390$		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	$-0.333333\pi$
$401$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$402$		0			0
$403$	−529.000	−	916.255i	−1.31266	−	2.27359i
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$	384.500	−	665.974i	0.940098	−	1.62830i	0.174817	−	0.984601i	$-0.444067\pi$
$409$	384.500	−	665.974i	0.940098	−	1.62830i	0.765281	−	0.643696i	$-0.222600\pi$
$410$		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	$-0.333333\pi$
$419$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$420$		0			0
$421$	−419.500	−	726.595i	−0.996437	−	1.72588i	−0.571259	−	0.820770i	$-0.693545\pi$
$421$	−419.500	−	726.595i	−0.996437	−	1.72588i	−0.425178	−	0.905110i	$-0.639789\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		0			0
$426$		0			0
$427$	258.500	−	447.735i	0.605386	−	1.04856i
$428$		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	0.807517	−	0.589844i	$-0.200811\pi$
$439$	−47.0000	−	81.4064i	−0.107062	−	0.185436i	−0.914579	+	0.404408i	$0.867478\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$443$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$444$		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	0.839168	+	0.543872i	$0.183042\pi$
$457$	407.000	+	704.945i	0.890591	+	1.54255i	0.0514223	+	0.998677i	$0.483625\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$461$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$462$		0			0
$463$	461.500	−	799.341i	0.996760	−	1.72644i	0.428726	−	0.903435i	$-0.358963\pi$
$463$	461.500	−	799.341i	0.996760	−	1.72644i	0.568035	−	0.823005i	$-0.307704\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		0			0		1.00000			$0$
$467$		0			0		−1.00000			$\pi$
$468$		0			0
$469$	−143.000			−0.304904
$470$		0			0
$471$		0			0
$472$		0			0
$473$		0			0
$474$		0			0
$475$	−462.500	−	801.073i	−0.973684	−	1.68647i
$476$		0			0
$477$		0			0
$478$		0			0
$479$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$479$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$480$		0			0
$481$	839.500	−	1454.06i	1.74532	−	3.02299i
$482$		0			0
$483$		0			0
$484$		0			0
$485$		0			0
$486$		0			0
$487$	613.000			1.25873			0.629363	−	0.777111i	$-0.283316\pi$
$487$	613.000			1.25873			0.629363	+	0.777111i	$0.283316\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$491$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$492$		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	−0.852705	−	0.522392i	$-0.825040\pi$
$499$	13.0000	−	22.5167i	0.0260521	−	0.0451236i	0.878758	+	0.477269i	$0.158373\pi$
$500$		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	$-0.333333\pi$
$509$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$510$		0			0
$511$	786.500	+	1362.26i	1.53914	+	2.66587i
$512$		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
$523$	−179.000			−0.342256			−0.171128	−	0.985249i	$-0.554741\pi$
$523$	−179.000			−0.342256			−0.171128	+	0.985249i	$0.554741\pi$
$524$		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
$541$	−793.000			−1.46580			−0.732902	−	0.680334i	$-0.761835\pi$
$541$	−793.000			−1.46580			−0.732902	+	0.680334i	$0.761835\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		0			0
$546$		0			0
$547$	293.500	+	508.357i	0.536563	+	0.929355i	0.999086	+	0.0427471i	$0.0136110\pi$
$547$	293.500	+	508.357i	0.536563	+	0.929355i	−0.462523	+	0.886607i	$0.653056\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		0			0
$552$		0			0
$553$	−60.5000	+	104.789i	−0.109403	+	0.189492i
$554$		0			0
$555$		0			0
$556$		0			0
$557$		0			0		1.00000			$0$
$557$		0			0		−1.00000			$\pi$
$558$		0			0
$559$	506.000			0.905188
$560$		0			0
$561$		0			0
$562$		0			0
$563$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$563$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$		0			0
$568$		0			0
$569$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$569$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$570$		0			0
$571$	533.500	−	924.049i	0.934326	−	1.61830i	0.158494	−	0.987360i	$-0.449336\pi$
$571$	533.500	−	924.049i	0.934326	−	1.61830i	0.775832	−	0.630940i	$-0.217330\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$		0			0
$577$	71.0000			0.123050			0.0615251	−	0.998106i	$-0.480404\pi$
$577$	71.0000			0.123050			0.0615251	+	0.998106i	$0.480404\pi$
$578$		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	$-0.666667\pi$
$587$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$588$		0			0
$589$	−851.000	+	1473.98i	−1.44482	+	2.50250i
$590$		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	$-0.333333\pi$
$599$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$600$		0			0
$601$	263.000	+	455.529i	0.437604	+	0.757952i	0.997504	−	0.0706077i	$-0.0224939\pi$
$601$	263.000	+	455.529i	0.437604	+	0.757952i	−0.559900	+	0.828560i	$0.689161\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$		0			0
$606$		0			0
$607$	−186.500	+	323.027i	−0.307249	+	0.532170i	−0.977759	−	0.209729i	$-0.932742\pi$
$607$	−186.500	+	323.027i	−0.307249	+	0.532170i	0.670511	+	0.741900i	$0.266075\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$	143.000			0.233279			0.116639	−	0.993174i	$-0.462788\pi$
$613$	143.000			0.233279			0.116639	+	0.993174i	$0.462788\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$617$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$618$		0			0
$619$	581.500	+	1007.19i	0.939418	+	1.62712i	0.766559	+	0.642174i	$0.221967\pi$
$619$	581.500	+	1007.19i	0.939418	+	1.62712i	0.172859	+	0.984947i	$0.444699\pi$
$620$		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
$631$	1261.00			1.99842			0.999208	−	0.0398015i	$-0.0126726\pi$
$631$	1261.00			1.99842			0.999208	+	0.0398015i	$0.0126726\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$		0			0
$636$		0			0
$637$	−828.000	−	1434.14i	−1.29984	−	2.25139i
$638$		0			0
$639$		0			0
$640$		0			0
$641$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$641$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$642$		0			0
$643$	157.000	−	271.932i	0.244168	−	0.422911i	−0.717729	−	0.696322i	$-0.754819\pi$
$643$	157.000	−	271.932i	0.244168	−	0.422911i	0.961897	+	0.273411i	$0.0881518\pi$
$644$		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	$-0.333333\pi$
$653$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$		0			0
$658$		0			0
$659$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$659$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$660$		0			0
$661$	600.500	−	1040.10i	0.908472	−	1.57352i	0.0922844	−	0.995733i	$-0.470583\pi$
$661$	600.500	−	1040.10i	0.908472	−	1.57352i	0.816188	−	0.577787i	$-0.196084\pi$
$662$		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
$673$	588.500	+	1019.31i	0.874443	+	1.51458i	0.857355	+	0.514725i	$0.172106\pi$
$673$	588.500	+	1019.31i	0.874443	+	1.51458i	0.0170877	+	0.999854i	$0.494561\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$677$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$678$		0			0
$679$	−929.500	+	1609.94i	−1.36892	+	2.37105i
$680$		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	−0.737337	−	0.675525i	$-0.763917\pi$
$691$	−659.000	−	1141.42i	−0.953690	−	1.65184i	−0.216353	−	0.976315i	$-0.569416\pi$
$692$		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
$703$	−2701.00			−3.84211
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	228.500	+	395.774i	0.322285	+	0.558214i	0.980959	−	0.194214i	$-0.0622158\pi$
$709$	228.500	+	395.774i	0.322285	+	0.558214i	−0.658674	+	0.752428i	$0.728882\pi$
$710$		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
$721$	−1727.00			−2.39528
$722$		0			0
$723$		0			0
$724$		0			0
$725$		0			0
$726$		0			0
$727$	241.000	+	417.424i	0.331499	+	0.574174i	0.982806	−	0.184641i	$-0.0591122\pi$
$727$	241.000	+	417.424i	0.331499	+	0.574174i	−0.651307	+	0.758815i	$0.725779\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$		0			0
$732$		0			0
$733$	−517.000	+	895.470i	−0.705321	+	1.22165i	0.261255	+	0.965270i	$0.415864\pi$
$733$	−517.000	+	895.470i	−0.705321	+	1.22165i	−0.966576	+	0.256381i	$0.917470\pi$
$734$		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	$-0.333333\pi$
$743$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	−690.500	+	1195.98i	−0.919441	+	1.59252i	−0.119174	+	0.992873i	$0.538025\pi$
$751$	−690.500	+	1195.98i	−0.919441	+	1.59252i	−0.800266	+	0.599645i	$0.795309\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$	1511.00			1.99604			0.998018	−	0.0629213i	$-0.0200417\pi$
$757$	1511.00			1.99604			0.998018	+	0.0629213i	$0.0200417\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$761$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$762$		0			0
$763$	−1177.00	−	2038.62i	−1.54260	−	2.67185i
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$		0			0
$769$	−335.500	+	581.103i	−0.436281	+	0.755661i	−0.997399	−	0.0720749i	$-0.977038\pi$
$769$	−335.500	+	581.103i	−0.436281	+	0.755661i	0.561118	+	0.827736i	$0.310371\pi$
$770$		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
$787$	−474.500	+	821.858i	−0.602922	+	1.04429i	0.389454	+	0.921046i	$0.372664\pi$
$787$	−474.500	+	821.858i	−0.602922	+	1.04429i	−0.992376	+	0.123246i	$0.960669\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	1081.00			1.36318
$794$		0			0
$795$		0			0
$796$		0			0
$797$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$797$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$798$		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
$817$	−407.000	−	704.945i	−0.498164	−	0.862845i
$818$		0			0
$819$		0			0
$820$		0			0
$821$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$821$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$822$		0			0
$823$	281.500	−	487.572i	0.342041	−	0.592433i	−0.642770	−	0.766059i	$-0.722215\pi$
$823$	281.500	−	487.572i	0.342041	−	0.592433i	0.984812	+	0.173626i	$0.0555484\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		0			0		1.00000			$0$
$827$		0			0		−1.00000			$\pi$
$828$		0			0
$829$	1151.00			1.38842			0.694210	−	0.719773i	$-0.255754\pi$
$829$	1151.00			1.38842			0.694210	+	0.719773i	$0.255754\pi$
$830$		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	$-0.666667\pi$
$839$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$840$		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
$847$	−1331.00			−1.57143
$848$		0			0
$849$		0			0
$850$		0			0
$851$		0			0
$852$		0			0
$853$	588.500	+	1019.31i	0.689918	+	1.19497i	0.971864	+	0.235543i	$0.0756867\pi$
$853$	588.500	+	1019.31i	0.689918	+	1.19497i	−0.281946	+	0.959430i	$0.590980\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$857$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$858$		0			0
$859$	65.5000	−	113.449i	0.0762515	−	0.132071i	−0.825378	−	0.564580i	$-0.809038\pi$
$859$	65.5000	−	113.449i	0.0762515	−	0.132071i	0.901630	+	0.432509i	$0.142371\pi$
$860$		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
$871$	−149.500	−	258.942i	−0.171642	−	0.297292i
$872$		0			0
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$	564.500	−	977.743i	0.643672	−	1.11487i	−0.340935	−	0.940087i	$-0.610744\pi$
$877$	564.500	−	977.743i	0.643672	−	1.11487i	0.984607	−	0.174785i	$-0.0559231\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		0			0		1.00000			$0$
$881$		0			0		−1.00000			$\pi$
$882$		0			0
$883$	−1259.00			−1.42582			−0.712911	−	0.701255i	$-0.752623\pi$
$883$	−1259.00			−1.42582			−0.712911	+	0.701255i	$0.752623\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$887$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$888$		0			0
$889$	−803.000	−	1390.84i	−0.903262	−	1.56450i
$890$		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
$907$	833.500	+	1443.66i	0.918964	+	1.59169i	0.800992	+	0.598675i	$0.204306\pi$
$907$	833.500	+	1443.66i	0.918964	+	1.59169i	0.117971	+	0.993017i	$0.462361\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$911$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$912$		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
$925$	912.500	+	1580.50i	0.986486	+	1.70864i
$926$		0			0
$927$		0			0
$928$		0			0
$929$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$929$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$930$		0			0
$931$	−1332.00	+	2307.09i	−1.43072	+	2.47808i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$	−649.000			−0.692636			−0.346318	−	0.938117i	$-0.612568\pi$
$937$	−649.000			−0.692636			−0.346318	+	0.938117i	$0.612568\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$941$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$947$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$948$		0			0
$949$	−1644.50	+	2848.36i	−1.73288	+	3.00143i
$950$		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
$967$	893.500	−	1547.59i	0.923992	−	1.60040i	0.130817	−	0.991407i	$-0.458240\pi$
$967$	893.500	−	1547.59i	0.923992	−	1.60040i	0.793175	−	0.608994i	$-0.208427\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		0			0		1.00000			$0$
$971$		0			0		−1.00000			$\pi$
$972$		0			0
$973$	−2519.00			−2.58890
$974$		0			0
$975$		0			0
$976$		0			0
$977$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$977$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$		0			0
$982$		0			0
$983$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$983$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$	1693.00			1.70838			0.854188	−	0.519965i	$-0.174055\pi$
$991$	1693.00			1.70838			0.854188	+	0.519965i	$0.174055\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		0			0
$996$		0			0
$997$	947.000	+	1640.25i	0.949850	+	1.64519i	0.745737	+	0.666240i	$0.232097\pi$
$997$	947.000	+	1640.25i	0.949850	+	1.64519i	0.204112	+	0.978947i	$0.434569\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1296.3.q.c.1025.1		2
3.2	odd	2	CM	1296.3.q.c.1025.1		2
4.3	odd	2		324.3.g.a.53.1		2
9.2	odd	6	inner	1296.3.q.c.593.1		2
9.4	even	3		432.3.e.a.161.1		1
9.5	odd	6		432.3.e.a.161.1		1
9.7	even	3	inner	1296.3.q.c.593.1		2
12.11	even	2		324.3.g.a.53.1		2
36.7	odd	6		324.3.g.a.269.1		2
36.11	even	6		324.3.g.a.269.1		2
36.23	even	6		108.3.c.a.53.1	✓	1
36.31	odd	6		108.3.c.a.53.1	✓	1
72.5	odd	6		1728.3.e.b.1025.1		1
72.13	even	6		1728.3.e.b.1025.1		1
72.59	even	6		1728.3.e.c.1025.1		1
72.67	odd	6		1728.3.e.c.1025.1		1
180.23	odd	12		2700.3.b.d.1349.1		2
180.59	even	6		2700.3.g.b.701.1		1
180.67	even	12		2700.3.b.d.1349.2		2
180.103	even	12		2700.3.b.d.1349.1		2
180.139	odd	6		2700.3.g.b.701.1		1
180.167	odd	12		2700.3.b.d.1349.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
108.3.c.a.53.1	✓	1	36.23	even	6
108.3.c.a.53.1	✓	1	36.31	odd	6
324.3.g.a.53.1		2	4.3	odd	2
324.3.g.a.53.1		2	12.11	even	2
324.3.g.a.269.1		2	36.7	odd	6
324.3.g.a.269.1		2	36.11	even	6
432.3.e.a.161.1		1	9.4	even	3
432.3.e.a.161.1		1	9.5	odd	6
1296.3.q.c.593.1		2	9.2	odd	6	inner
1296.3.q.c.593.1		2	9.7	even	3	inner
1296.3.q.c.1025.1		2	1.1	even	1	trivial
1296.3.q.c.1025.1		2	3.2	odd	2	CM
1728.3.e.b.1025.1		1	72.5	odd	6
1728.3.e.b.1025.1		1	72.13	even	6
1728.3.e.c.1025.1		1	72.59	even	6
1728.3.e.c.1025.1		1	72.67	odd	6
2700.3.b.d.1349.1		2	180.23	odd	12
2700.3.b.d.1349.1		2	180.103	even	12
2700.3.b.d.1349.2		2	180.67	even	12
2700.3.b.d.1349.2		2	180.167	odd	12
2700.3.g.b.701.1		1	180.59	even	6
2700.3.g.b.701.1		1	180.139	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)

\(f(q)\)	\(=\)	\(q+(5.50000 + 9.52628i) q^{7} +O(q^{10})\)	\(q+(5.50000 + 9.52628i) q^{7} +(-11.5000 + 19.9186i) q^{13} +37.0000 q^{19} +(-12.5000 - 21.6506i) q^{25} +(-23.0000 + 39.8372i) q^{31} -73.0000 q^{37} +(-11.0000 - 19.0526i) q^{43} +(-36.0000 + 62.3538i) q^{49} +(-23.5000 - 40.7032i) q^{61} +(-6.50000 + 11.2583i) q^{67} +143.000 q^{73} +(5.50000 + 9.52628i) q^{79} -253.000 q^{91} +(84.5000 + 146.358i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 11 q^{7}+O(q^{10}) \) 2 * q + 11 * q^7	\( 2 q + 11 q^{7} - 23 q^{13} + 74 q^{19} - 25 q^{25} - 46 q^{31} - 146 q^{37} - 22 q^{43} - 72 q^{49} - 47 q^{61} - 13 q^{67} + 286 q^{73} + 11 q^{79} - 506 q^{91} + 169 q^{97}+O(q^{100}) \) 2 * q + 11 * q^7 - 23 * q^13 + 74 * q^19 - 25 * q^25 - 46 * q^31 - 146 * q^37 - 22 * q^43 - 72 * q^49 - 47 * q^61 - 13 * q^67 + 286 * q^73 + 11 * q^79 - 506 * q^91 + 169 * q^97

Introduction

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