LMFDB - Embedded newform 324.5.g.a.269.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [324,5,Mod(53,324)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(324, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 5]))

N = Newforms(chi, 5, names="a")

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

chi := DirichletCharacter("324.53");

S:= CuspForms(chi, 5);

N := Newforms(S);

Level:	$ N $	$=$	$ 324 = 2^{2} \cdot 3^{4} $
Weight:	$ k $	$=$	$ 5 $
Character orbit:	$[\chi]$	$=$	324.g (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:	$33.4918680392$
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			269.1
Root			$0.500000 + 0.866025i$ of defining polynomial
Character	$\chi$	$=$	324.269
Dual form			324.5.g.a.53.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-11.5000 + 19.9186i) q^{7} +O(q^{10})$	$q+(-11.5000 + 19.9186i) q^{7} +(-95.5000 - 165.411i) q^{13} +647.000 q^{19} +(-312.500 + 541.266i) q^{25} +(-97.0000 - 168.009i) q^{31} +2591.00 q^{37} +(1607.00 - 2783.41i) q^{43} +(936.000 + 1621.20i) q^{49} +(2616.50 - 4531.91i) q^{61} +(4404.50 + 7628.82i) q^{67} +9791.00 q^{73} +(6180.50 - 10704.9i) q^{79} +4393.00 q^{91} +(-4871.50 + 8437.69i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 23 q^{7}+O(q^{10}) $ 2 * q - 23 * q^7	$ 2 q - 23 q^{7} - 191 q^{13} + 1294 q^{19} - 625 q^{25} - 194 q^{31} + 5182 q^{37} + 3214 q^{43} + 1872 q^{49} + 5233 q^{61} + 8809 q^{67} + 19582 q^{73} + 12361 q^{79} + 8786 q^{91} - 9743 q^{97}+O(q^{100}) $ 2 * q - 23 * q^7 - 191 * q^13 + 1294 * q^19 - 625 * q^25 - 194 * q^31 + 5182 * q^37 + 3214 * q^43 + 1872 * q^49 + 5233 * q^61 + 8809 * q^67 + 19582 * q^73 + 12361 * q^79 + 8786 * q^91 - 9743 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/324\mathbb{Z}\right)^\times$.

$n$	$163$	$245$
$\chi(n)$	$1$	$e\left(\frac{1}{6}\right)$

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
$7$	−11.5000	+	19.9186i	−0.234694	+	0.406502i	−0.959184	−	0.282784i	$-0.908742\pi$
$7$	−11.5000	+	19.9186i	−0.234694	+	0.406502i	0.724490	+	0.689286i	$0.242075\pi$
$8$		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
$13$	−95.5000	−	165.411i	−0.565089	−	0.978762i	−0.997041	−	0.0768662i	$-0.975509\pi$
$13$	−95.5000	−	165.411i	−0.565089	−	0.978762i	0.431953	−	0.901896i	$-0.357825\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$		0			0		1.00000			$0$
$17$		0			0		−1.00000			$\pi$
$18$		0			0
$19$	647.000			1.79224			0.896122	−	0.443808i	$-0.146373\pi$
$19$	647.000			1.79224			0.896122	+	0.443808i	$0.146373\pi$
$20$		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
$25$	−312.500	+	541.266i	−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
$31$	−97.0000	−	168.009i	−0.100937	−	0.174827i	0.811134	−	0.584860i	$-0.198851\pi$
$31$	−97.0000	−	168.009i	−0.100937	−	0.174827i	−0.912071	+	0.410033i	$0.865517\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$		0			0
$37$	2591.00			1.89262			0.946311	−	0.323257i	$-0.104778\pi$
$37$	2591.00			1.89262			0.946311	+	0.323257i	$0.104778\pi$
$38$		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
$43$	1607.00	−	2783.41i	0.869118	−	1.50536i	0.00621958	−	0.999981i	$-0.498020\pi$
$43$	1607.00	−	2783.41i	0.869118	−	1.50536i	0.862899	−	0.505377i	$-0.168646\pi$
$44$		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
$49$	936.000	+	1621.20i	0.389838	+	0.675218i
$50$		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
$61$	2616.50	−	4531.91i	0.703171	−	1.21793i	−0.264176	−	0.964474i	$-0.585100\pi$
$61$	2616.50	−	4531.91i	0.703171	−	1.21793i	0.967347	−	0.253454i	$-0.0815666\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$		0			0
$66$		0			0
$67$	4404.50	+	7628.82i	0.981176	+	1.69945i	0.657830	+	0.753166i	$0.271474\pi$
$67$	4404.50	+	7628.82i	0.981176	+	1.69945i	0.323346	+	0.946281i	$0.395192\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		0			0		1.00000			$0$
$71$		0			0		−1.00000			$\pi$
$72$		0			0
$73$	9791.00			1.83731			0.918653	−	0.395066i	$-0.129278\pi$
$73$	9791.00			1.83731			0.918653	+	0.395066i	$0.129278\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$		0			0
$78$		0			0
$79$	6180.50	−	10704.9i	0.990306	−	1.71526i	0.374860	−	0.927082i	$-0.377691\pi$
$79$	6180.50	−	10704.9i	0.990306	−	1.71526i	0.615446	−	0.788179i	$-0.288976\pi$
$80$		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
$91$	4393.00			0.530491
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$	−4871.50	+	8437.69i	−0.517749	+	0.896768i	0.482038	+	0.876150i	$0.339897\pi$
$97$	−4871.50	+	8437.69i	−0.517749	+	0.896768i	−0.999787	+	0.0206175i	$0.993437\pi$
$98$		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
$103$	−1715.50	−	2971.33i	−0.161702	−	0.280077i	0.773777	−	0.633458i	$-0.218365\pi$
$103$	−1715.50	−	2971.33i	−0.161702	−	0.280077i	−0.935479	+	0.353381i	$0.885032\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		0			0		1.00000			$0$
$107$		0			0		−1.00000			$\pi$
$108$		0			0
$109$	22034.0			1.85456			0.927279	−	0.374371i	$-0.122141\pi$
$109$	22034.0			1.85456			0.927279	+	0.374371i	$0.122141\pi$
$110$		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
$121$	−7320.50	−	12679.5i	−0.500000	−	0.866025i
$122$		0			0
$123$		0			0
$124$		0			0
$125$		0			0
$126$		0			0
$127$	−10942.0			−0.678405			−0.339203	−	0.940713i	$-0.610157\pi$
$127$	−10942.0			−0.678405			−0.339203	+	0.940713i	$0.610157\pi$
$128$		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
$133$	−7440.50	+	12887.3i	−0.420629	+	0.728550i
$134$		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
$139$	−6899.50	−	11950.3i	−0.357098	−	0.618513i	0.630376	−	0.776290i	$-0.282901\pi$
$139$	−6899.50	−	11950.3i	−0.357098	−	0.618513i	−0.987475	+	0.157777i	$0.949567\pi$
$140$		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
$151$	−2963.50	+	5132.93i	−0.129972	+	0.225119i	−0.923666	−	0.383199i	$-0.874822\pi$
$151$	−2963.50	+	5132.93i	−0.129972	+	0.225119i	0.793693	+	0.608318i	$0.208156\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$		0			0
$157$	17687.0	+	30634.8i	0.717554	+	1.24284i	0.961966	+	0.273169i	$0.0880719\pi$
$157$	17687.0	+	30634.8i	0.717554	+	1.24284i	−0.244412	+	0.969672i	$0.578595\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$	−51769.0			−1.94847			−0.974237	−	0.225527i	$-0.927590\pi$
$163$	−51769.0			−1.94847			−0.974237	+	0.225527i	$0.927590\pi$
$164$		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
$169$	−3960.00	+	6858.92i	−0.138651	+	0.240150i
$170$		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
$175$	−7187.50	−	12449.1i	−0.234694	−	0.406502i
$176$		0			0
$177$		0			0
$178$		0			0
$179$		0			0		1.00000			$0$
$179$		0			0		−1.00000			$\pi$
$180$		0			0
$181$	−65521.0			−1.99997			−0.999985	−	0.00552484i	$-0.998241\pi$
$181$	−65521.0			−1.99997			−0.999985	+	0.00552484i	$0.998241\pi$
$182$		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
$193$	8688.50	+	15048.9i	0.233255	+	0.404009i	0.958764	−	0.284203i	$-0.0917291\pi$
$193$	8688.50	+	15048.9i	0.233255	+	0.404009i	−0.725509	+	0.688212i	$0.758396\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		0			0		1.00000			$0$
$197$		0			0		−1.00000			$\pi$
$198$		0			0
$199$	−67321.0			−1.69998			−0.849991	−	0.526797i	$-0.823393\pi$
$199$	−67321.0			−1.69998			−0.849991	+	0.526797i	$0.823393\pi$
$200$		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
$211$	−43259.5	−	74927.7i	−0.971665	−	1.68297i	−0.690528	−	0.723306i	$-0.742622\pi$
$211$	−43259.5	−	74927.7i	−0.971665	−	1.68297i	−0.281137	−	0.959668i	$-0.590712\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$	4462.00			0.0947567
$218$		0			0
$219$		0			0
$220$		0			0
$221$		0			0
$222$		0			0
$223$	−7393.00	+	12805.1i	−0.148666	+	0.257497i	−0.930735	−	0.365695i	$-0.880831\pi$
$223$	−7393.00	+	12805.1i	−0.148666	+	0.257497i	0.782069	+	0.623192i	$0.214165\pi$
$224$		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
$229$	52103.0	+	90245.0i	0.993555	+	1.72089i	0.594945	+	0.803767i	$0.297174\pi$
$229$	52103.0	+	90245.0i	0.993555	+	1.72089i	0.398610	+	0.917121i	$0.369493\pi$
$230$		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
$241$	−56639.5	+	98102.5i	−0.975181	+	1.68906i	−0.295845	+	0.955236i	$0.595601\pi$
$241$	−56639.5	+	98102.5i	−0.975181	+	1.68906i	−0.679336	+	0.733828i	$0.737732\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$		0			0
$246$		0			0
$247$	−61788.5	−	107021.i	−1.01278	−	1.75418i
$248$		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
$259$	−29796.5	+	51609.1i	−0.444187	+	0.769354i
$260$		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
$271$	145799.			1.98525			0.992627	−	0.121211i	$-0.0386778\pi$
$271$	145799.			1.98525			0.992627	+	0.121211i	$0.0386778\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$		0			0
$276$		0			0
$277$	69287.0	−	120009.i	0.903009	−	1.56406i	0.0794419	−	0.996839i	$-0.474686\pi$
$277$	69287.0	−	120009.i	0.903009	−	1.56406i	0.823567	−	0.567218i	$-0.191980\pi$
$278$		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
$283$	−24793.0	−	42942.7i	−0.309568	−	0.536188i	0.668700	−	0.743532i	$-0.266851\pi$
$283$	−24793.0	−	42942.7i	−0.309568	−	0.536188i	−0.978268	+	0.207345i	$0.933518\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	83521.0			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
$301$	36961.0	+	64018.3i	0.407954	+	0.706596i
$302$		0			0
$303$		0			0
$304$		0			0
$305$		0			0
$306$		0			0
$307$	−60334.0			−0.640155			−0.320078	−	0.947391i	$-0.603709\pi$
$307$	−60334.0			−0.640155			−0.320078	+	0.947391i	$0.603709\pi$
$308$		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
$313$	−6455.50	+	11181.3i	−0.0658933	+	0.114131i	−0.897090	−	0.441848i	$-0.854323\pi$
$313$	−6455.50	+	11181.3i	−0.0658933	+	0.114131i	0.831197	+	0.555979i	$0.187656\pi$
$314$		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
$325$	119375.			1.13018
$326$		0			0
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$	64860.5	−	112342.i	0.592004	−	1.02538i	−0.401959	−	0.915658i	$-0.631670\pi$
$331$	64860.5	−	112342.i	0.592004	−	1.02538i	0.993962	−	0.109722i	$-0.0349962\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$		0			0
$336$		0			0
$337$	99624.5	+	172555.i	0.877216	+	1.51938i	0.854384	+	0.519643i	$0.173935\pi$
$337$	99624.5	+	172555.i	0.877216	+	1.51938i	0.0228319	+	0.999739i	$0.492732\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$	−98279.0			−0.835358
$344$		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
$349$	107520.	−	186231.i	0.882755	−	1.52898i	0.0344907	−	0.999405i	$-0.489019\pi$
$349$	107520.	−	186231.i	0.882755	−	1.52898i	0.848265	−	0.529572i	$-0.177648\pi$
$350$		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
$361$	288288.			2.21214
$362$		0			0
$363$		0			0
$364$		0			0
$365$		0			0
$366$		0			0
$367$	108924.	−	188663.i	0.808711	−	1.40073i	−0.105046	−	0.994467i	$-0.533499\pi$
$367$	108924.	−	188663.i	0.808711	−	1.40073i	0.913757	−	0.406261i	$-0.133168\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$	131808.	+	228299.i	0.947383	+	1.64092i	0.750907	+	0.660407i	$0.229616\pi$
$373$	131808.	+	228299.i	0.947383	+	1.64092i	0.196476	+	0.980509i	$0.437050\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$	86039.0			0.598986			0.299493	−	0.954098i	$-0.403182\pi$
$379$	86039.0			0.598986			0.299493	+	0.954098i	$0.403182\pi$
$380$		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
$397$	−184174.			−1.16855			−0.584275	−	0.811556i	$-0.698621\pi$
$397$	−184174.			−1.16855			−0.584275	+	0.811556i	$0.698621\pi$
$398$		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
$403$	−18527.0	+	32089.7i	−0.114076	+	0.197586i
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$	−128400.	−	222394.i	−0.767568	−	1.32947i	−0.938878	−	0.344249i	$-0.888133\pi$
$409$	−128400.	−	222394.i	−0.767568	−	1.32947i	0.171311	−	0.985217i	$-0.445200\pi$
$410$		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
$421$	−174720.	+	302623.i	−0.985774	+	1.70741i	−0.347327	+	0.937744i	$0.612910\pi$
$421$	−174720.	+	302623.i	−0.985774	+	1.70741i	−0.638447	+	0.769666i	$0.720423\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		0			0
$426$		0			0
$427$	60179.5	+	104234.i	0.330060	+	0.571681i
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0		1.00000			$0$
$431$		0			0		−1.00000			$\pi$
$432$		0			0
$433$	368066.			1.96313			0.981567	−	0.191119i	$-0.0612116\pi$
$433$	368066.			1.96313			0.981567	+	0.191119i	$0.0612116\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		0			0
$438$		0			0
$439$	188303.	−	326150.i	0.977076	−	1.69234i	0.304168	−	0.952619i	$-0.401622\pi$
$439$	188303.	−	326150.i	0.977076	−	1.69234i	0.672908	−	0.739726i	$-0.265045\pi$
$440$		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
$457$	−122449.	+	212088.i	−0.586304	+	1.01551i	0.408408	+	0.912800i	$0.366084\pi$
$457$	−122449.	+	212088.i	−0.586304	+	1.01551i	−0.994711	+	0.102709i	$0.967249\pi$
$458$		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
$463$	−211596.	−	366494.i	−0.987062	−	1.70964i	−0.632389	−	0.774651i	$-0.717925\pi$
$463$	−211596.	−	366494.i	−0.987062	−	1.70964i	−0.354673	−	0.934990i	$-0.615408\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		0			0		1.00000			$0$
$467$		0			0		−1.00000			$\pi$
$468$		0			0
$469$	−202607.			−0.921104
$470$		0			0
$471$		0			0
$472$		0			0
$473$		0			0
$474$		0			0
$475$	−202188.	+	350199.i	−0.896122	+	1.55213i
$476$		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
$481$	−247440.	−	428580.i	−1.06950	−	1.85243i
$482$		0			0
$483$		0			0
$484$		0			0
$485$		0			0
$486$		0			0
$487$	−98569.0			−0.415607			−0.207803	−	0.978171i	$-0.566631\pi$
$487$	−98569.0			−0.415607			−0.207803	+	0.978171i	$0.566631\pi$
$488$		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
$499$	248663.	+	430697.i	0.998643	+	1.72970i	0.544430	+	0.838807i	$0.316746\pi$
$499$	248663.	+	430697.i	0.998643	+	1.72970i	0.454213	+	0.890893i	$0.349920\pi$
$500$		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
$511$	−112596.	+	195023.i	−0.431204	+	0.746868i
$512$		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
$523$	−515017.			−1.88286			−0.941430	−	0.337208i	$-0.890518\pi$
$523$	−515017.			−1.88286			−0.941430	+	0.337208i	$0.890518\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		0			0
$528$		0			0
$529$	−139920.	+	242349.i	−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
$541$	43487.0			0.148582			0.0742908	−	0.997237i	$-0.476331\pi$
$541$	43487.0			0.148582			0.0742908	+	0.997237i	$0.476331\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		0			0
$546$		0			0
$547$	126924.	−	219840.i	0.424200	−	0.734736i	−0.572145	−	0.820152i	$-0.693889\pi$
$547$	126924.	−	219840.i	0.424200	−	0.734736i	0.996345	+	0.0854161i	$0.0272220\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		0			0
$552$		0			0
$553$	142152.	+	246214.i	0.464838	+	0.805122i
$554$		0			0
$555$		0			0
$556$		0			0
$557$		0			0		1.00000			$0$
$557$		0			0		−1.00000			$\pi$
$558$		0			0
$559$	−613874.			−1.96452
$560$		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
$571$	−243203.	−	421241.i	−0.745929	−	1.29199i	−0.949759	−	0.312981i	$-0.898672\pi$
$571$	−243203.	−	421241.i	−0.745929	−	1.29199i	0.203830	−	0.979006i	$-0.434661\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$		0			0
$577$	−660817.			−1.98486			−0.992429	−	0.122817i	$-0.960807\pi$
$577$	−660817.			−1.98486			−0.992429	+	0.122817i	$0.960807\pi$
$578$		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
$589$	−62759.0	−	108702.i	−0.180903	−	0.313333i
$590$		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
$601$	222863.	−	386010.i	0.617005	−	1.06868i	−0.373024	−	0.927822i	$-0.621679\pi$
$601$	222863.	−	386010.i	0.617005	−	1.06868i	0.990029	−	0.140863i	$-0.0449877\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$		0			0
$606$		0			0
$607$	298884.	+	517683.i	0.811196	+	1.40503i	0.912027	+	0.410130i	$0.134517\pi$
$607$	298884.	+	517683.i	0.811196	+	1.40503i	−0.100831	+	0.994904i	$0.532150\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$	−731089.			−1.94558			−0.972790	−	0.231687i	$-0.925576\pi$
$613$	−731089.			−1.94558			−0.972790	+	0.231687i	$0.925576\pi$
$614$		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
$619$	−293124.	+	507705.i	−0.765014	+	1.32504i	0.175225	+	0.984528i	$0.443935\pi$
$619$	−293124.	+	507705.i	−0.765014	+	1.32504i	−0.940239	+	0.340515i	$0.889399\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	−195312.	−	338291.i	−0.500000	−	0.866025i
$626$		0			0
$627$		0			0
$628$		0			0
$629$		0			0
$630$		0			0
$631$	793799.			1.99366			0.996832	−	0.0795399i	$-0.0253451\pi$
$631$	793799.			1.99366			0.996832	+	0.0795399i	$0.0253451\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$		0			0
$636$		0			0
$637$	178776.	−	309649.i	0.440586	−	0.763117i
$638$		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
$643$	364151.	+	630728.i	0.880764	+	1.52553i	0.850493	+	0.525986i	$0.176304\pi$
$643$	364151.	+	630728.i	0.880764	+	1.52553i	0.0302710	+	0.999542i	$0.490363\pi$
$644$		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
$661$	−284280.	−	492387.i	−0.650643	−	1.12695i	−0.982967	−	0.183781i	$-0.941166\pi$
$661$	−284280.	−	492387.i	−0.650643	−	1.12695i	0.332324	−	0.943165i	$-0.392167\pi$
$662$		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
$673$	−239736.	+	415234.i	−0.529300	+	0.916775i	0.470116	+	0.882605i	$0.344212\pi$
$673$	−239736.	+	415234.i	−0.529300	+	0.916775i	−0.999416	+	0.0341703i	$0.989121\pi$
$674$		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
$679$	−112044.	−	194067.i	−0.243025	−	0.420932i
$680$		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
$691$	−391081.	+	677372.i	−0.819050	+	1.41864i	0.0873323	+	0.996179i	$0.472166\pi$
$691$	−391081.	+	677372.i	−0.819050	+	1.41864i	−0.906383	+	0.422458i	$0.861168\pi$
$692$		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
$703$	1.67638e6			3.39204
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	398256.	−	689800.i	0.792265	−	1.37224i	−0.132297	−	0.991210i	$-0.542235\pi$
$709$	398256.	−	689800.i	0.792265	−	1.37224i	0.924562	−	0.381033i	$-0.124432\pi$
$710$		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
$721$	78913.0			0.151802
$722$		0			0
$723$		0			0
$724$		0			0
$725$		0			0
$726$		0			0
$727$	412367.	−	714241.i	0.780216	−	1.35137i	−0.151599	−	0.988442i	$-0.548442\pi$
$727$	412367.	−	714241.i	0.780216	−	1.35137i	0.931815	−	0.362932i	$-0.118224\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$		0			0
$732$		0			0
$733$	2711.00	+	4695.59i	0.00504570	+	0.00873941i	0.868537	−	0.495624i	$-0.165061\pi$
$733$	2711.00	+	4695.59i	0.00504570	+	0.00873941i	−0.863492	+	0.504363i	$0.831727\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$		0			0
$739$	401042.			0.734346			0.367173	−	0.930153i	$-0.380326\pi$
$739$	401042.			0.734346			0.367173	+	0.930153i	$0.380326\pi$
$740$		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
$751$	−389580.	−	674771.i	−0.690743	−	1.19640i	−0.971595	−	0.236650i	$-0.923950\pi$
$751$	−389580.	−	674771.i	−0.690743	−	1.19640i	0.280852	−	0.959751i	$-0.409383\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$	1.13702e6			1.98416			0.992082	−	0.125593i	$-0.0400834\pi$
$757$	1.13702e6			1.98416			0.992082	+	0.125593i	$0.0400834\pi$
$758$		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
$763$	−253391.	+	438886.i	−0.435253	+	0.753881i
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$		0			0
$769$	366240.	+	634347.i	0.619318	+	1.07269i	0.989610	+	0.143775i	$0.0459241\pi$
$769$	366240.	+	634347.i	0.619318	+	1.07269i	−0.370292	+	0.928915i	$0.620743\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$		0			0		1.00000			$0$
$773$		0			0		−1.00000			$\pi$
$774$		0			0
$775$	121250.			0.201873
$776$		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
$787$	169068.	+	292835.i	0.272969	+	0.472796i	0.969621	−	0.244613i	$-0.0786610\pi$
$787$	169068.	+	292835.i	0.272969	+	0.472796i	−0.696652	+	0.717409i	$0.745328\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	−999503.			−1.58942
$794$		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
$811$	976754.			1.48506			0.742529	−	0.669814i	$-0.233626\pi$
$811$	976754.			1.48506			0.742529	+	0.669814i	$0.233626\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		0			0
$816$		0			0
$817$	1.03973e6	−	1.80086e6i	1.55767	−	2.69797i
$818$		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
$823$	518844.	+	898665.i	0.766015	+	1.32678i	0.939708	+	0.341978i	$0.111097\pi$
$823$	518844.	+	898665.i	0.766015	+	1.32678i	−0.173693	+	0.984800i	$0.555570\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		0			0		1.00000			$0$
$827$		0			0		−1.00000			$\pi$
$828$		0			0
$829$	−49681.0			−0.0722905			−0.0361453	−	0.999347i	$-0.511508\pi$
$829$	−49681.0			−0.0722905			−0.0361453	+	0.999347i	$0.511508\pi$
$830$		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
$841$	−353640.	−	612523.i	−0.500000	−	0.866025i
$842$		0			0
$843$		0			0
$844$		0			0
$845$		0			0
$846$		0			0
$847$	336743.			0.469388
$848$		0			0
$849$		0			0
$850$		0			0
$851$		0			0
$852$		0			0
$853$	34944.5	−	60525.6i	0.0480265	−	0.0831843i	−0.841013	−	0.541015i	$-0.818040\pi$
$853$	34944.5	−	60525.6i	0.0480265	−	0.0831843i	0.889039	+	0.457831i	$0.151373\pi$
$854$		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
$859$	729300.	+	1.26319e6i	0.988371	+	1.71191i	0.625873	+	0.779925i	$0.284743\pi$
$859$	729300.	+	1.26319e6i	0.988371	+	1.71191i	0.362499	+	0.931984i	$0.381924\pi$
$860$		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
$871$	841260.	−	1.45710e6i	1.10890	−	1.92068i
$872$		0			0
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$	131808.	+	228299.i	0.171374	+	0.296828i	0.938900	−	0.344189i	$-0.111846\pi$
$877$	131808.	+	228299.i	0.171374	+	0.296828i	−0.767527	+	0.641017i	$0.778513\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		0			0		1.00000			$0$
$881$		0			0		−1.00000			$\pi$
$882$		0			0
$883$	25703.0			0.0329657			0.0164829	−	0.999864i	$-0.494753\pi$
$883$	25703.0			0.0329657			0.0164829	+	0.999864i	$0.494753\pi$
$884$		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
$889$	125833.	−	217949.i	0.159218	−	0.275773i
$890$		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
$907$	−566796.	+	981719.i	−0.688988	+	1.19336i	0.283177	+	0.959068i	$0.408612\pi$
$907$	−566796.	+	981719.i	−0.688988	+	1.19336i	−0.972166	+	0.234295i	$0.924722\pi$
$908$		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
$919$	−939166.			−1.11202			−0.556008	−	0.831177i	$-0.687668\pi$
$919$	−939166.			−1.11202			−0.556008	+	0.831177i	$0.687668\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		0			0
$924$		0			0
$925$	−809688.	+	1.40242e6i	−0.946311	+	1.63906i
$926$		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
$931$	605592.	+	1.04892e6i	0.698684	+	1.21016i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$	−1.33474e6			−1.52026			−0.760128	−	0.649774i	$-0.774864\pi$
$937$	−1.33474e6			−1.52026			−0.760128	+	0.649774i	$0.774864\pi$
$938$		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
$949$	−935040.	−	1.61954e6i	−1.03824	−	1.79829i
$950$		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
$961$	442942.	−	767199.i	0.479624	−	0.830733i
$962$		0			0
$963$		0			0
$964$		0			0
$965$		0			0
$966$		0			0
$967$	−661596.	−	1.14592e6i	−0.707521	−	1.22546i	−0.965774	−	0.259386i	$-0.916480\pi$
$967$	−661596.	−	1.14592e6i	−0.707521	−	1.22546i	0.258252	−	0.966077i	$-0.416853\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		0			0		1.00000			$0$
$971$		0			0		−1.00000			$\pi$
$972$		0			0
$973$	317377.			0.335235
$974$		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
$991$	902087.			0.918546			0.459273	−	0.888295i	$-0.348110\pi$
$991$	902087.			0.918546			0.459273	+	0.888295i	$0.348110\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		0			0
$996$		0			0
$997$	−799609.	+	1.38496e6i	−0.804428	+	1.39331i	0.112248	+	0.993680i	$0.464195\pi$
$997$	−799609.	+	1.38496e6i	−0.804428	+	1.39331i	−0.916676	+	0.399631i	$0.869138\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	324.5.g.a.269.1		2
3.2	odd	2	CM	324.5.g.a.269.1		2
9.2	odd	6		108.5.c.a.53.1	✓	1
9.4	even	3	inner	324.5.g.a.53.1		2
9.5	odd	6	inner	324.5.g.a.53.1		2
9.7	even	3		108.5.c.a.53.1	✓	1
36.7	odd	6		432.5.e.b.161.1		1
36.11	even	6		432.5.e.b.161.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
108.5.c.a.53.1	✓	1	9.2	odd	6
108.5.c.a.53.1	✓	1	9.7	even	3
324.5.g.a.53.1		2	9.4	even	3	inner
324.5.g.a.53.1		2	9.5	odd	6	inner
324.5.g.a.269.1		2	1.1	even	1	trivial
324.5.g.a.269.1		2	3.2	odd	2	CM
432.5.e.b.161.1		1	36.7	odd	6
432.5.e.b.161.1		1	36.11	even	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:	\( N \)	\(=\)	\( 324 = 2^{2} \cdot 3^{4} \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	324.g (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-11.5000 + 19.9186i) q^{7} +O(q^{10})\)	\(q+(-11.5000 + 19.9186i) q^{7} +(-95.5000 - 165.411i) q^{13} +647.000 q^{19} +(-312.500 + 541.266i) q^{25} +(-97.0000 - 168.009i) q^{31} +2591.00 q^{37} +(1607.00 - 2783.41i) q^{43} +(936.000 + 1621.20i) q^{49} +(2616.50 - 4531.91i) q^{61} +(4404.50 + 7628.82i) q^{67} +9791.00 q^{73} +(6180.50 - 10704.9i) q^{79} +4393.00 q^{91} +(-4871.50 + 8437.69i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 23 q^{7}+O(q^{10}) \) 2 * q - 23 * q^7	\( 2 q - 23 q^{7} - 191 q^{13} + 1294 q^{19} - 625 q^{25} - 194 q^{31} + 5182 q^{37} + 3214 q^{43} + 1872 q^{49} + 5233 q^{61} + 8809 q^{67} + 19582 q^{73} + 12361 q^{79} + 8786 q^{91} - 9743 q^{97}+O(q^{100}) \) 2 * q - 23 * q^7 - 191 * q^13 + 1294 * q^19 - 625 * q^25 - 194 * q^31 + 5182 * q^37 + 3214 * q^43 + 1872 * q^49 + 5233 * q^61 + 8809 * q^67 + 19582 * q^73 + 12361 * q^79 + 8786 * q^91 - 9743 * q^97

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