LMFDB - Embedded newform 324.5.g.b.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:	$\Z[a_1, \ldots, a_{25}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 12)
Sato-Tate group:	$\mathrm{U}(1)[D_{6}]$

Embedding invariants

Embedding label			269.1
Root			$0.500000 + 0.866025i$ of defining polynomial
Character	$\chi$	$=$	324.269
Dual form			324.5.g.b.53.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(47.0000 - 81.4064i) q^{7} +O(q^{10})$	$q+(47.0000 - 81.4064i) q^{7} +(-73.0000 - 126.440i) q^{13} -46.0000 q^{19} +(-312.500 + 541.266i) q^{25} +(-97.0000 - 168.009i) q^{31} -2062.00 q^{37} +(1607.00 - 2783.41i) q^{43} +(-3217.50 - 5572.87i) q^{49} +(983.000 - 1702.61i) q^{61} +(-2953.00 - 5114.75i) q^{67} -8542.00 q^{73} +(-3841.00 + 6652.81i) q^{79} -13724.0 q^{91} +(9407.00 - 16293.4i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 94 q^{7}+O(q^{10}) $ 2 * q + 94 * q^7	$ 2 q + 94 q^{7} - 146 q^{13} - 92 q^{19} - 625 q^{25} - 194 q^{31} - 4124 q^{37} + 3214 q^{43} - 6435 q^{49} + 1966 q^{61} - 5906 q^{67} - 17084 q^{73} - 7682 q^{79} - 27448 q^{91} + 18814 q^{97}+O(q^{100}) $ 2 * q + 94 * q^7 - 146 * q^13 - 92 * q^19 - 625 * q^25 - 194 * q^31 - 4124 * q^37 + 3214 * q^43 - 6435 * q^49 + 1966 * q^61 - 5906 * q^67 - 17084 * q^73 - 7682 * q^79 - 27448 * q^91 + 18814 * 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
$7$	47.0000	−	81.4064i	0.959184	−	1.66135i	0.234694	−	0.972069i	$-0.424591\pi$
$7$	47.0000	−	81.4064i	0.959184	−	1.66135i	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
$13$	−73.0000	−	126.440i	−0.431953	−	0.748164i	0.565089	−	0.825030i	$-0.308842\pi$
$13$	−73.0000	−	126.440i	−0.431953	−	0.748164i	−0.997041	+	0.0768662i	$0.975509\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$		0			0		1.00000			$0$
$17$		0			0		−1.00000			$\pi$
$18$		0			0
$19$	−46.0000			−0.127424			−0.0637119	−	0.997968i	$-0.520294\pi$
$19$	−46.0000			−0.127424			−0.0637119	+	0.997968i	$0.520294\pi$
$20$		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
$37$	−2062.00			−1.50621			−0.753104	−	0.657901i	$-0.771445\pi$
$37$	−2062.00			−1.50621			−0.753104	+	0.657901i	$0.771445\pi$
$38$		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
$49$	−3217.50	−	5572.87i	−1.34007	−	2.32106i
$50$		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
$61$	983.000	−	1702.61i	0.264176	−	0.457567i	−0.703171	−	0.711021i	$-0.748233\pi$
$61$	983.000	−	1702.61i	0.264176	−	0.457567i	0.967347	+	0.253454i	$0.0815666\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$		0			0
$66$		0			0
$67$	−2953.00	−	5114.75i	−0.657830	−	1.13940i	−0.981176	−	0.193115i	$-0.938141\pi$
$67$	−2953.00	−	5114.75i	−0.657830	−	1.13940i	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
$73$	−8542.00			−1.60293			−0.801464	−	0.598043i	$-0.795945\pi$
$73$	−8542.00			−1.60293			−0.801464	+	0.598043i	$0.795945\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$		0			0
$78$		0			0
$79$	−3841.00	+	6652.81i	−0.615446	+	1.06598i	0.374860	+	0.927082i	$0.377691\pi$
$79$	−3841.00	+	6652.81i	−0.615446	+	1.06598i	−0.990306	+	0.138903i	$0.955642\pi$
$80$		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
$91$	−13724.0			−1.65729
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$	9407.00	−	16293.4i	0.999787	−	1.73168i	0.482038	−	0.876150i	$-0.339897\pi$
$97$	9407.00	−	16293.4i	0.999787	−	1.73168i	0.517749	−	0.855533i	$-0.326770\pi$
$98$		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
$103$	−8209.00	−	14218.4i	−0.773777	−	1.34022i	−0.935479	−	0.353381i	$-0.885032\pi$
$103$	−8209.00	−	14218.4i	−0.773777	−	1.34022i	0.161702	−	0.986840i	$-0.448302\pi$
$104$		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
$133$	−2162.00	+	3744.69i	−0.122223	+	0.211696i
$134$		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
$139$	19079.0	+	33045.8i	0.987475	+	1.71036i	0.630376	+	0.776290i	$0.282901\pi$
$139$	19079.0	+	33045.8i	0.987475	+	1.71036i	0.357098	+	0.934067i	$0.383766\pi$
$140$		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
$151$	−18097.0	+	31344.9i	−0.793693	+	1.37472i	0.129972	+	0.991518i	$0.458511\pi$
$151$	−18097.0	+	31344.9i	−0.793693	+	1.37472i	−0.923666	+	0.383199i	$0.874822\pi$
$152$		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
$163$	15506.0			0.583612			0.291806	−	0.956477i	$-0.405744\pi$
$163$	15506.0			0.583612			0.291806	+	0.956477i	$0.405744\pi$
$164$		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
$169$	3622.50	−	6274.35i	0.126834	−	0.219683i
$170$		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
$175$	29375.0	+	50879.0i	0.959184	+	1.66135i
$176$		0			0
$177$		0			0
$178$		0			0
$179$		0			0		1.00000			$0$
$179$		0			0		−1.00000			$\pi$
$180$		0			0
$181$	33074.0			1.00955			0.504777	−	0.863250i	$-0.331575\pi$
$181$	33074.0			1.00955			0.504777	+	0.863250i	$0.331575\pi$
$182$		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
$193$	−35713.0	−	61856.7i	−0.958764	−	1.66063i	−0.725509	−	0.688212i	$-0.758396\pi$
$193$	−35713.0	−	61856.7i	−0.958764	−	1.66063i	−0.233255	−	0.972416i	$-0.574938\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		0			0		1.00000			$0$
$197$		0			0		−1.00000			$\pi$
$198$		0			0
$199$	69794.0			1.76243			0.881215	−	0.472715i	$-0.156726\pi$
$199$	69794.0			1.76243			0.881215	+	0.472715i	$0.156726\pi$
$200$		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
$211$	30743.0	+	53248.4i	0.690528	+	1.19603i	0.971665	+	0.236362i	$0.0759551\pi$
$211$	30743.0	+	53248.4i	0.690528	+	1.19603i	−0.281137	+	0.959668i	$0.590712\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$	−18236.0			−0.387267
$218$		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
$241$	17183.0	−	29761.8i	0.295845	−	0.512419i	−0.679336	−	0.733828i	$-0.737732\pi$
$241$	17183.0	−	29761.8i	0.295845	−	0.512419i	0.975181	+	0.221408i	$0.0710653\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$		0			0
$246$		0			0
$247$	3358.00	+	5816.23i	0.0550411	+	0.0953339i
$248$		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
$259$	−96914.0	+	167860.i	−1.44473	+	2.50235i
$260$		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
$271$	−88318.0			−1.20257			−0.601285	−	0.799034i	$-0.705345\pi$
$271$	−88318.0			−1.20257			−0.601285	+	0.799034i	$0.705345\pi$
$272$		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
$301$	−151058.	−	261640.i	−1.66729	−	2.88783i
$302$		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
$313$	87887.0	−	152225.i	0.897090	−	1.55381i	0.0658933	−	0.997827i	$-0.479010\pi$
$313$	87887.0	−	152225.i	0.897090	−	1.55381i	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
$325$	91250.0			0.863905
$326$		0			0
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$	44039.0	−	76277.8i	0.401959	−	0.696213i	−0.592004	−	0.805935i	$-0.701663\pi$
$331$	44039.0	−	76277.8i	0.401959	−	0.696213i	0.993962	+	0.109722i	$0.0349962\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$		0			0
$336$		0			0
$337$	−2593.00	−	4491.21i	−0.0228319	−	0.0395461i	0.854384	−	0.519643i	$-0.173935\pi$
$337$	−2593.00	−	4491.21i	−0.0228319	−	0.0395461i	−0.877216	+	0.480097i	$0.840602\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$	−379196.			−3.22311
$344$		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
$349$	−4201.00	+	7276.35i	−0.0344907	+	0.0597396i	−0.882755	−	0.469833i	$-0.844314\pi$
$349$	−4201.00	+	7276.35i	−0.0344907	+	0.0597396i	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
$361$	−128205.			−0.983763
$362$		0			0
$363$		0			0
$364$		0			0
$365$		0			0
$366$		0			0
$367$	−123073.	+	213169.i	−0.913757	+	1.58267i	−0.105046	+	0.994467i	$0.533499\pi$
$367$	−123073.	+	213169.i	−0.913757	+	1.58267i	−0.808711	+	0.588206i	$0.799834\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$	−104473.	−	180953.i	−0.750907	−	1.30061i	−0.947383	−	0.320101i	$-0.896283\pi$
$373$	−104473.	−	180953.i	−0.750907	−	1.30061i	0.196476	−	0.980509i	$-0.437050\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$	194354.			1.35305			0.676527	−	0.736418i	$-0.263484\pi$
$379$	194354.			1.35305			0.676527	+	0.736418i	$0.263484\pi$
$380$		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
$403$	−14162.0	+	24529.3i	−0.0871996	+	0.151034i
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$	−28657.0	−	49635.4i	−0.171311	−	0.296719i	0.767568	−	0.640968i	$-0.221467\pi$
$409$	−28657.0	−	49635.4i	−0.171311	−	0.296719i	−0.938878	+	0.344249i	$0.888133\pi$
$410$		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
$421$	113159.	−	195997.i	0.638447	−	1.10582i	−0.347327	−	0.937744i	$-0.612910\pi$
$421$	113159.	−	195997.i	0.638447	−	1.10582i	0.985774	−	0.168079i	$-0.0537562\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		0			0
$426$		0			0
$427$	−92402.0	−	160045.i	−0.506787	−	0.877781i
$428$		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
$463$	76031.0	+	131690.i	0.354673	+	0.614312i	0.987062	−	0.160339i	$-0.0512587\pi$
$463$	76031.0	+	131690.i	0.354673	+	0.614312i	−0.632389	+	0.774651i	$0.717925\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		0			0		1.00000			$0$
$467$		0			0		−1.00000			$\pi$
$468$		0			0
$469$	−555164.			−2.52392
$470$		0			0
$471$		0			0
$472$		0			0
$473$		0			0
$474$		0			0
$475$	14375.0	−	24898.2i	0.0637119	−	0.110352i
$476$		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
$481$	150526.	+	260719.i	0.650611	+	1.12689i
$482$		0			0
$483$		0			0
$484$		0			0
$485$		0			0
$486$		0			0
$487$	451106.			1.90204			0.951022	−	0.309122i	$-0.100035\pi$
$487$	451106.			1.90204			0.951022	+	0.309122i	$0.100035\pi$
$488$		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
$511$	−401474.	+	695373.i	−1.53750	+	2.66303i
$512$		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
$523$	417266.			1.52549			0.762745	−	0.646699i	$-0.223851\pi$
$523$	417266.			1.52549			0.762745	+	0.646699i	$0.223851\pi$
$524$		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
$541$	483794.			1.65297			0.826487	−	0.562956i	$-0.190336\pi$
$541$	483794.			1.65297			0.826487	+	0.562956i	$0.190336\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		0			0
$546$		0			0
$547$	171191.	−	296512.i	0.572145	−	0.990985i	−0.424200	−	0.905568i	$-0.639445\pi$
$547$	171191.	−	296512.i	0.572145	−	0.990985i	0.996345	−	0.0854161i	$-0.0272220\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		0			0
$552$		0			0
$553$	361054.	+	625364.i	1.18065	+	2.04495i
$554$		0			0
$555$		0			0
$556$		0			0
$557$		0			0		1.00000			$0$
$557$		0			0		−1.00000			$\pi$
$558$		0			0
$559$	−469244.			−1.50167
$560$		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
$571$	−66457.0	−	115107.i	−0.203830	−	0.353044i	0.745929	−	0.666025i	$-0.232006\pi$
$571$	−66457.0	−	115107.i	−0.203830	−	0.353044i	−0.949759	+	0.312981i	$0.898672\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$		0			0
$577$	259586.			0.779704			0.389852	−	0.920878i	$-0.372526\pi$
$577$	259586.			0.779704			0.389852	+	0.920878i	$0.372526\pi$
$578$		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
$589$	4462.00	+	7728.41i	0.0128617	+	0.0222771i
$590$		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
$607$	37151.0	+	64347.4i	0.100831	+	0.174644i	0.912027	−	0.410130i	$-0.134517\pi$
$607$	37151.0	+	64347.4i	0.100831	+	0.174644i	−0.811196	+	0.584774i	$0.801183\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$	516338.			1.37408			0.687042	−	0.726618i	$-0.258909\pi$
$613$	516338.			1.37408			0.687042	+	0.726618i	$0.258909\pi$
$614$		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
$619$	360263.	−	623994.i	0.940239	−	1.62854i	0.175225	−	0.984528i	$-0.443935\pi$
$619$	360263.	−	623994.i	0.940239	−	1.62854i	0.765014	−	0.644014i	$-0.222732\pi$
$620$		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
$631$	−342046.			−0.859065			−0.429532	−	0.903052i	$-0.641322\pi$
$631$	−342046.			−0.859065			−0.429532	+	0.903052i	$0.641322\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$		0			0
$636$		0			0
$637$	−469755.	+	813640.i	−1.15769	+	2.00518i
$638$		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
$661$	429479.	+	743879.i	0.982967	+	1.70255i	0.650643	+	0.759384i	$0.274500\pi$
$661$	429479.	+	743879.i	0.982967	+	1.70255i	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
$673$	−212929.	+	368804.i	−0.470116	+	0.814264i	−0.999416	−	0.0341703i	$-0.989121\pi$
$673$	−212929.	+	368804.i	−0.470116	+	0.814264i	0.529300	+	0.848434i	$0.322454\pi$
$674$		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
$679$	−884258.	−	1.53158e6i	−1.91796	−	3.32200i
$680$		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
$703$	94852.0			0.191927
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	66503.0	−	115187.i	0.132297	−	0.229144i	−0.792265	−	0.610177i	$-0.791098\pi$
$709$	66503.0	−	115187.i	0.132297	−	0.229144i	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
$721$	−1.54329e6			−2.96878
$722$		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
$751$	−158401.	−	274359.i	−0.280852	−	0.486451i	0.690743	−	0.723101i	$-0.257284\pi$
$751$	−158401.	−	274359.i	−0.280852	−	0.486451i	−0.971595	+	0.236650i	$0.923950\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$	−443854.			−0.774548			−0.387274	−	0.921965i	$-0.626583\pi$
$757$	−443854.			−0.774548			−0.387274	+	0.921965i	$0.626583\pi$
$758$		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
$763$	1.03560e6	−	1.79371e6i	1.77886	−	3.08108i
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$		0			0
$769$	−585217.	−	1.01363e6i	−0.989610	−	1.71406i	−0.619318	−	0.785140i	$-0.712591\pi$
$769$	−585217.	−	1.01363e6i	−0.989610	−	1.71406i	−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
$787$	−600553.	−	1.04019e6i	−0.969621	−	1.67943i	−0.696652	−	0.717409i	$-0.745328\pi$
$787$	−600553.	−	1.04019e6i	−0.969621	−	1.67943i	−0.272969	−	0.962023i	$-0.588006\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	−287036.			−0.456447
$794$		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
$817$	−73922.0	+	128037.i	−0.110746	+	0.191818i
$818$		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
$823$	117647.	+	203771.i	0.173693	+	0.300844i	0.939708	−	0.341978i	$-0.111097\pi$
$823$	117647.	+	203771.i	0.173693	+	0.300844i	−0.766015	+	0.642822i	$0.777763\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		0			0		1.00000			$0$
$827$		0			0		−1.00000			$\pi$
$828$		0			0
$829$	−1.16472e6			−1.69477			−0.847387	−	0.530976i	$-0.821825\pi$
$829$	−1.16472e6			−1.69477			−0.847387	+	0.530976i	$0.821825\pi$
$830$		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
$847$	−1.37625e6			−1.91837
$848$		0			0
$849$		0			0
$850$		0			0
$851$		0			0
$852$		0			0
$853$	−646873.	+	1.12042e6i	−0.889039	+	1.53986i	−0.0480265	+	0.998846i	$0.515293\pi$
$853$	−646873.	+	1.12042e6i	−0.889039	+	1.53986i	−0.841013	+	0.541015i	$0.818040\pi$
$854$		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
$859$	−267481.	−	463291.i	−0.362499	−	0.627866i	0.625873	−	0.779925i	$-0.284743\pi$
$859$	−267481.	−	463291.i	−0.362499	−	0.627866i	−0.988371	+	0.152059i	$0.951410\pi$
$860$		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
$871$	−431138.	+	746753.i	−0.568303	+	0.984330i
$872$		0			0
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$	590327.	+	1.02248e6i	0.767527	+	1.32940i	0.938900	+	0.344189i	$0.111846\pi$
$877$	590327.	+	1.02248e6i	0.767527	+	1.32940i	−0.171374	+	0.985206i	$0.554821\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		0			0		1.00000			$0$
$881$		0			0		−1.00000			$\pi$
$882$		0			0
$883$	1.33743e6			1.71533			0.857666	−	0.514207i	$-0.171914\pi$
$883$	1.33743e6			1.71533			0.857666	+	0.514207i	$0.171914\pi$
$884$		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
$889$	−514274.	+	890749.i	−0.650715	+	1.12707i
$890$		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
$907$	799751.	−	1.38521e6i	0.972166	−	1.68384i	0.283177	−	0.959068i	$-0.408612\pi$
$907$	799751.	−	1.38521e6i	0.972166	−	1.68384i	0.688988	−	0.724773i	$-0.258055\pi$
$908$		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
$925$	644375.	−	1.11609e6i	0.753104	−	1.30442i
$926$		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
$931$	148005.	+	256352.i	0.170756	+	0.295759i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$	−320734.			−0.365314			−0.182657	−	0.983177i	$-0.558470\pi$
$937$	−320734.			−0.365314			−0.182657	+	0.983177i	$0.558470\pi$
$938$		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
$949$	623566.	+	1.08005e6i	0.692389	+	1.19925i
$950$		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
$967$	−241489.	−	418271.i	−0.258252	−	0.447306i	0.707521	−	0.706692i	$-0.249813\pi$
$967$	−241489.	−	418271.i	−0.258252	−	0.447306i	−0.965774	+	0.259386i	$0.916480\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		0			0		1.00000			$0$
$971$		0			0		−1.00000			$\pi$
$972$		0			0
$973$	3.58685e6			3.78868
$974$		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
$991$	−1.96205e6			−1.99785			−0.998923	−	0.0464053i	$-0.985223\pi$
$991$	−1.96205e6			−1.99785			−0.998923	+	0.0464053i	$0.985223\pi$
$992$		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.b.269.1		2
3.2	odd	2	CM	324.5.g.b.269.1		2
9.2	odd	6		12.5.c.a.5.1	✓	1
9.4	even	3	inner	324.5.g.b.53.1		2
9.5	odd	6	inner	324.5.g.b.53.1		2
9.7	even	3		12.5.c.a.5.1	✓	1
36.7	odd	6		48.5.e.a.17.1		1
36.11	even	6		48.5.e.a.17.1		1
45.2	even	12		300.5.b.a.149.1		2
45.7	odd	12		300.5.b.a.149.1		2
45.29	odd	6		300.5.g.b.101.1		1
45.34	even	6		300.5.g.b.101.1		1
45.38	even	12		300.5.b.a.149.2		2
45.43	odd	12		300.5.b.a.149.2		2
63.20	even	6		588.5.c.a.197.1		1
63.34	odd	6		588.5.c.a.197.1		1
72.11	even	6		192.5.e.b.65.1		1
72.29	odd	6		192.5.e.a.65.1		1
72.43	odd	6		192.5.e.b.65.1		1
72.61	even	6		192.5.e.a.65.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
12.5.c.a.5.1	✓	1	9.2	odd	6
12.5.c.a.5.1	✓	1	9.7	even	3
48.5.e.a.17.1		1	36.7	odd	6
48.5.e.a.17.1		1	36.11	even	6
192.5.e.a.65.1		1	72.29	odd	6
192.5.e.a.65.1		1	72.61	even	6
192.5.e.b.65.1		1	72.11	even	6
192.5.e.b.65.1		1	72.43	odd	6
300.5.b.a.149.1		2	45.2	even	12
300.5.b.a.149.1		2	45.7	odd	12
300.5.b.a.149.2		2	45.38	even	12
300.5.b.a.149.2		2	45.43	odd	12
300.5.g.b.101.1		1	45.29	odd	6
300.5.g.b.101.1		1	45.34	even	6
324.5.g.b.53.1		2	9.4	even	3	inner
324.5.g.b.53.1		2	9.5	odd	6	inner
324.5.g.b.269.1		2	1.1	even	1	trivial
324.5.g.b.269.1		2	3.2	odd	2	CM
588.5.c.a.197.1		1	63.20	even	6
588.5.c.a.197.1		1	63.34	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:	\( N \)	\(=\)	\( 324 = 2^{2} \cdot 3^{4} \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	324.g (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(47.0000 - 81.4064i) q^{7} +O(q^{10})\)	\(q+(47.0000 - 81.4064i) q^{7} +(-73.0000 - 126.440i) q^{13} -46.0000 q^{19} +(-312.500 + 541.266i) q^{25} +(-97.0000 - 168.009i) q^{31} -2062.00 q^{37} +(1607.00 - 2783.41i) q^{43} +(-3217.50 - 5572.87i) q^{49} +(983.000 - 1702.61i) q^{61} +(-2953.00 - 5114.75i) q^{67} -8542.00 q^{73} +(-3841.00 + 6652.81i) q^{79} -13724.0 q^{91} +(9407.00 - 16293.4i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 94 q^{7}+O(q^{10}) \) 2 * q + 94 * q^7	\( 2 q + 94 q^{7} - 146 q^{13} - 92 q^{19} - 625 q^{25} - 194 q^{31} - 4124 q^{37} + 3214 q^{43} - 6435 q^{49} + 1966 q^{61} - 5906 q^{67} - 17084 q^{73} - 7682 q^{79} - 27448 q^{91} + 18814 q^{97}+O(q^{100}) \) 2 * q + 94 * q^7 - 146 * q^13 - 92 * q^19 - 625 * q^25 - 194 * q^31 - 4124 * q^37 + 3214 * q^43 - 6435 * q^49 + 1966 * q^61 - 5906 * q^67 - 17084 * q^73 - 7682 * q^79 - 27448 * q^91 + 18814 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more