LMFDB - Embedded newform 756.2.t.c.593.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [756,2,Mod(269,756)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(756, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 3, 1]))

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("756.269");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 756 = 2^{2} \cdot 3^{3} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	756.t (of order $6$, degree $2$, 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:	$6.03669039281$
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:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{6}]$

Embedding invariants

Embedding label			593.1
Root			$0.500000 - 0.866025i$ of defining polynomial
Character	$\chi$	$=$	756.593
Dual form			756.2.t.c.269.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(2.00000 + 1.73205i) q^{7} +O(q^{10})$	$q+(2.00000 + 1.73205i) q^{7} +1.73205i q^{13} +(3.00000 + 1.73205i) q^{19} +(2.50000 + 4.33013i) q^{25} +(-1.50000 + 0.866025i) q^{31} +(-0.500000 + 0.866025i) q^{37} +13.0000 q^{43} +(1.00000 + 6.92820i) q^{49} +(13.5000 + 7.79423i) q^{61} +(-5.50000 - 9.52628i) q^{67} +(-12.0000 + 6.92820i) q^{73} +(6.50000 - 11.2583i) q^{79} +(-3.00000 + 3.46410i) q^{91} +5.19615i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 4 q^{7}+O(q^{10}) $ 2 * q + 4 * q^7	$ 2 q + 4 q^{7} + 6 q^{19} + 5 q^{25} - 3 q^{31} - q^{37} + 26 q^{43} + 2 q^{49} + 27 q^{61} - 11 q^{67} - 24 q^{73} + 13 q^{79} - 6 q^{91}+O(q^{100}) $ 2 * q + 4 * q^7 + 6 * q^19 + 5 * q^25 - 3 * q^31 - q^37 + 26 * q^43 + 2 * q^49 + 27 * q^61 - 11 * q^67 - 24 * q^73 + 13 * q^79 - 6 * q^91

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/756\mathbb{Z}\right)^\times$.

$n$	$29$	$325$	$379$
$\chi(n)$	$-1$	$e\left(\frac{5}{6}\right)$	$1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$		0			0
$2$		0			0
$3$		0			0
$3$		0			0
$4$		0			0
$5$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$5$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$6$		0			0
$7$	2.00000	+	1.73205i	0.755929	+	0.654654i
$7$	2.00000	+	1.73205i	0.755929	+	0.654654i
$8$		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
$13$			1.73205i			0.480384i	0.970725	+	0.240192i	$0.0772105\pi$
$13$			1.73205i			0.480384i	−0.970725	+	0.240192i	$0.922790\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$17$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$18$		0			0
$19$	3.00000	+	1.73205i	0.688247	+	0.397360i	0.802955	−	0.596040i	$-0.203260\pi$
$19$	3.00000	+	1.73205i	0.688247	+	0.397360i	−0.114708	+	0.993399i	$0.536593\pi$
$20$		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
$25$	2.50000	+	4.33013i	0.500000	+	0.866025i
$26$		0			0
$27$		0			0
$28$		0			0
$29$		0			0		1.00000			$0$
$29$		0			0		−1.00000			$\pi$
$30$		0			0
$31$	−1.50000	+	0.866025i	−0.269408	+	0.155543i	−0.628619	−	0.777714i	$-0.716379\pi$
$31$	−1.50000	+	0.866025i	−0.269408	+	0.155543i	0.359211	+	0.933257i	$0.383046\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$		0			0
$37$	−0.500000	+	0.866025i	−0.0821995	+	0.142374i	−0.904194	−	0.427121i	$-0.859528\pi$
$37$	−0.500000	+	0.866025i	−0.0821995	+	0.142374i	0.821995	+	0.569495i	$0.192861\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		0			0			−	1.00000i	$-0.5\pi$
$41$		0			0				1.00000i	$0.5\pi$
$42$		0			0
$43$	13.0000			1.98248			0.991241	−	0.132068i	$-0.0421616\pi$
$43$	13.0000			1.98248			0.991241	+	0.132068i	$0.0421616\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$47$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$48$		0			0
$49$	1.00000	+	6.92820i	0.142857	+	0.989743i
$50$		0			0
$51$		0			0
$52$		0			0
$53$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$53$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$		0			0
$58$		0			0
$59$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$59$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$60$		0			0
$61$	13.5000	+	7.79423i	1.72850	+	0.997949i	0.896258	+	0.443533i	$0.146275\pi$
$61$	13.5000	+	7.79423i	1.72850	+	0.997949i	0.832240	+	0.554416i	$0.187058\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$		0			0
$66$		0			0
$67$	−5.50000	−	9.52628i	−0.671932	−	1.16382i	−0.977356	−	0.211604i	$-0.932131\pi$
$67$	−5.50000	−	9.52628i	−0.671932	−	1.16382i	0.305424	−	0.952217i	$-0.401202\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		0			0		1.00000			$0$
$71$		0			0		−1.00000			$\pi$
$72$		0			0
$73$	−12.0000	+	6.92820i	−1.40449	+	0.810885i	−0.994850	−	0.101361i	$-0.967680\pi$
$73$	−12.0000	+	6.92820i	−1.40449	+	0.810885i	−0.409644	+	0.912245i	$0.634347\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$		0			0
$78$		0			0
$79$	6.50000	−	11.2583i	0.731307	−	1.26666i	−0.225018	−	0.974355i	$-0.572244\pi$
$79$	6.50000	−	11.2583i	0.731307	−	1.26666i	0.956325	−	0.292306i	$-0.0944227\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$		0			0			−	1.00000i	$-0.5\pi$
$83$		0			0				1.00000i	$0.5\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$89$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$90$		0			0
$91$	−3.00000	+	3.46410i	−0.314485	+	0.363137i
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$			5.19615i			0.527589i	0.964579	+	0.263795i	$0.0849741\pi$
$97$			5.19615i			0.527589i	−0.964579	+	0.263795i	$0.915026\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$101$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$102$		0			0
$103$	−16.5000	−	9.52628i	−1.62579	−	0.938652i	−0.985329	−	0.170664i	$-0.945409\pi$
$103$	−16.5000	−	9.52628i	−1.62579	−	0.938652i	−0.640464	−	0.767988i	$-0.721258\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$107$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$108$		0			0
$109$	−9.50000	−	16.4545i	−0.909935	−	1.57605i	−0.814152	−	0.580651i	$-0.802798\pi$
$109$	−9.50000	−	16.4545i	−0.909935	−	1.57605i	−0.0957826	−	0.995402i	$-0.530535\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$		0			0		1.00000			$0$
$113$		0			0		−1.00000			$\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−5.50000	+	9.52628i	−0.500000	+	0.866025i
$122$		0			0
$123$		0			0
$124$		0			0
$125$		0			0
$126$		0			0
$127$	−1.00000			−0.0887357			−0.0443678	−	0.999015i	$-0.514127\pi$
$127$	−1.00000			−0.0887357			−0.0443678	+	0.999015i	$0.514127\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$131$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$132$		0			0
$133$	3.00000	+	8.66025i	0.260133	+	0.750939i
$134$		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
$139$		−	22.5167i		−	1.90984i	−0.296866	−	0.954919i	$-0.595942\pi$
$139$		−	22.5167i		−	1.90984i	0.296866	−	0.954919i	$-0.404058\pi$
$140$		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
$151$	−9.50000	−	16.4545i	−0.773099	−	1.33905i	−0.935857	−	0.352381i	$-0.885372\pi$
$151$	−9.50000	−	16.4545i	−0.773099	−	1.33905i	0.162758	−	0.986666i	$-0.447961\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$		0			0
$157$	−18.0000	+	10.3923i	−1.43656	+	0.829396i	−0.997609	−	0.0691164i	$-0.977982\pi$
$157$	−18.0000	+	10.3923i	−1.43656	+	0.829396i	−0.438948	+	0.898513i	$0.644649\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$	12.5000	−	21.6506i	0.979076	−	1.69581i	0.313304	−	0.949653i	$-0.398564\pi$
$163$	12.5000	−	21.6506i	0.979076	−	1.69581i	0.665771	−	0.746156i	$-0.268103\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		0			0			−	1.00000i	$-0.5\pi$
$167$		0			0				1.00000i	$0.5\pi$
$168$		0			0
$169$	10.0000			0.769231
$170$		0			0
$171$		0			0
$172$		0			0
$173$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$173$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$174$		0			0
$175$	−2.50000	+	12.9904i	−0.188982	+	0.981981i
$176$		0			0
$177$		0			0
$178$		0			0
$179$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$179$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$180$		0			0
$181$		−	6.92820i		−	0.514969i	−0.966282	−	0.257485i	$-0.917106\pi$
$181$		−	6.92820i		−	0.514969i	0.966282	−	0.257485i	$-0.0828937\pi$
$182$		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
$193$	−12.5000	−	21.6506i	−0.899770	−	1.55845i	−0.827788	−	0.561041i	$-0.810401\pi$
$193$	−12.5000	−	21.6506i	−0.899770	−	1.55845i	−0.0719816	−	0.997406i	$-0.522932\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		0			0		1.00000			$0$
$197$		0			0		−1.00000			$\pi$
$198$		0			0
$199$	−22.5000	+	12.9904i	−1.59498	+	0.920864i	−0.602549	+	0.798082i	$0.705848\pi$
$199$	−22.5000	+	12.9904i	−1.59498	+	0.920864i	−0.992434	+	0.122782i	$0.960818\pi$
$200$		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
$211$	13.0000			0.894957			0.447478	−	0.894295i	$-0.352322\pi$
$211$	13.0000			0.894957			0.447478	+	0.894295i	$0.352322\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$	−4.50000	−	0.866025i	−0.305480	−	0.0587896i
$218$		0			0
$219$		0			0
$220$		0			0
$221$		0			0
$222$		0			0
$223$		−	10.3923i		−	0.695920i	−0.937509	−	0.347960i	$-0.886874\pi$
$223$		−	10.3923i		−	0.695920i	0.937509	−	0.347960i	$-0.113126\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$227$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$228$		0			0
$229$	−7.50000	−	4.33013i	−0.495614	−	0.286143i	0.231287	−	0.972886i	$-0.425707\pi$
$229$	−7.50000	−	4.33013i	−0.495614	−	0.286143i	−0.726900	+	0.686743i	$0.759040\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$233$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$		0			0		1.00000			$0$
$239$		0			0		−1.00000			$\pi$
$240$		0			0
$241$	−22.5000	+	12.9904i	−1.44935	+	0.836784i	−0.998443	−	0.0557856i	$-0.982234\pi$
$241$	−22.5000	+	12.9904i	−1.44935	+	0.836784i	−0.450910	+	0.892570i	$0.648900\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$		0			0
$246$		0			0
$247$	−3.00000	+	5.19615i	−0.190885	+	0.330623i
$248$		0			0
$249$		0			0
$250$		0			0
$251$		0			0			−	1.00000i	$-0.5\pi$
$251$		0			0				1.00000i	$0.5\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$		0			0
$256$		0			0
$257$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$257$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$258$		0			0
$259$	−2.50000	+	0.866025i	−0.155342	+	0.0538122i
$260$		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		0.866025	−	0.500000i	$-0.166667\pi$
$269$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$270$		0			0
$271$	13.5000	+	7.79423i	0.820067	+	0.473466i	0.850439	−	0.526073i	$-0.176336\pi$
$271$	13.5000	+	7.79423i	0.820067	+	0.473466i	−0.0303728	+	0.999539i	$0.509669\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$		0			0
$276$		0			0
$277$	−2.50000	−	4.33013i	−0.150210	−	0.260172i	0.781094	−	0.624413i	$-0.214662\pi$
$277$	−2.50000	−	4.33013i	−0.150210	−	0.260172i	−0.931305	+	0.364241i	$0.881328\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$		0			0		1.00000			$0$
$281$		0			0		−1.00000			$\pi$
$282$		0			0
$283$	19.5000	−	11.2583i	1.15915	−	0.669238i	0.208053	−	0.978117i	$-0.433287\pi$
$283$	19.5000	−	11.2583i	1.15915	−	0.669238i	0.951101	+	0.308879i	$0.0999539\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	8.50000	−	14.7224i	0.500000	−	0.866025i
$290$		0			0
$291$		0			0
$292$		0			0
$293$		0			0			−	1.00000i	$-0.5\pi$
$293$		0			0				1.00000i	$0.5\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$		0			0
$300$		0			0
$301$	26.0000	+	22.5167i	1.49862	+	1.29784i
$302$		0			0
$303$		0			0
$304$		0			0
$305$		0			0
$306$		0			0
$307$			29.4449i			1.68051i	0.542194	+	0.840254i	$0.317594\pi$
$307$			29.4449i			1.68051i	−0.542194	+	0.840254i	$0.682406\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$311$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$312$		0			0
$313$	24.0000	+	13.8564i	1.35656	+	0.783210i	0.989158	−	0.146852i	$-0.0469141\pi$
$313$	24.0000	+	13.8564i	1.35656	+	0.783210i	0.367402	+	0.930062i	$0.380247\pi$
$314$		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
$325$	−7.50000	+	4.33013i	−0.416025	+	0.240192i
$326$		0			0
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$	16.0000	−	27.7128i	0.879440	−	1.52323i	0.0274825	−	0.999622i	$-0.491251\pi$
$331$	16.0000	−	27.7128i	0.879440	−	1.52323i	0.851957	−	0.523612i	$-0.175416\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$		0			0
$336$		0			0
$337$	34.0000			1.85210			0.926049	−	0.377403i	$-0.123183\pi$
$337$	34.0000			1.85210			0.926049	+	0.377403i	$0.123183\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$	−10.0000	+	15.5885i	−0.539949	+	0.841698i
$344$		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
$349$			5.19615i			0.278144i	0.990282	+	0.139072i	$0.0444119\pi$
$349$			5.19615i			0.278144i	−0.990282	+	0.139072i	$0.955588\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$353$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$359$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$360$		0			0
$361$	−3.50000	−	6.06218i	−0.184211	−	0.319062i
$362$		0			0
$363$		0			0
$364$		0			0
$365$		0			0
$366$		0			0
$367$	−33.0000	+	19.0526i	−1.72259	+	0.994535i	−0.809093	+	0.587680i	$0.800041\pi$
$367$	−33.0000	+	19.0526i	−1.72259	+	0.994535i	−0.913493	+	0.406855i	$0.866625\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$	−19.0000	+	32.9090i	−0.983783	+	1.70396i	−0.336557	+	0.941663i	$0.609263\pi$
$373$	−19.0000	+	32.9090i	−0.983783	+	1.70396i	−0.647225	+	0.762299i	$0.724071\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$	37.0000			1.90056			0.950281	−	0.311393i	$-0.100796\pi$
$379$	37.0000			1.90056			0.950281	+	0.311393i	$0.100796\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$383$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$384$		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
$397$	34.5000	+	19.9186i	1.73151	+	0.999685i	0.878300	+	0.478110i	$0.158678\pi$
$397$	34.5000	+	19.9186i	1.73151	+	0.999685i	0.853206	+	0.521575i	$0.174655\pi$
$398$		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
$403$	−1.50000	−	2.59808i	−0.0747203	−	0.129419i
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$	34.5000	−	19.9186i	1.70592	−	0.984911i	0.766426	−	0.642333i	$-0.222033\pi$
$409$	34.5000	−	19.9186i	1.70592	−	0.984911i	0.939490	−	0.342578i	$-0.111300\pi$
$410$		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			−	1.00000i	$-0.5\pi$
$419$		0			0				1.00000i	$0.5\pi$
$420$		0			0
$421$	−22.0000			−1.07221			−0.536107	−	0.844150i	$-0.680106\pi$
$421$	−22.0000			−1.07221			−0.536107	+	0.844150i	$0.680106\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		0			0
$426$		0			0
$427$	13.5000	+	38.9711i	0.653311	+	1.88595i
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$431$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$432$		0			0
$433$		−	19.0526i		−	0.915608i	−0.889053	−	0.457804i	$-0.848636\pi$
$433$		−	19.0526i		−	0.915608i	0.889053	−	0.457804i	$-0.151364\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		0			0
$438$		0			0
$439$	−27.0000	−	15.5885i	−1.28864	−	0.743996i	−0.310228	−	0.950662i	$-0.600405\pi$
$439$	−27.0000	−	15.5885i	−1.28864	−	0.743996i	−0.978412	+	0.206666i	$0.933739\pi$
$440$		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
$457$	−15.5000	+	26.8468i	−0.725059	+	1.25584i	0.233890	+	0.972263i	$0.424854\pi$
$457$	−15.5000	+	26.8468i	−0.725059	+	1.25584i	−0.958950	+	0.283577i	$0.908479\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		0			0			−	1.00000i	$-0.5\pi$
$461$		0			0				1.00000i	$0.5\pi$
$462$		0			0
$463$	20.0000			0.929479			0.464739	−	0.885448i	$-0.346148\pi$
$463$	20.0000			0.929479			0.464739	+	0.885448i	$0.346148\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$467$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$468$		0			0
$469$	5.50000	−	28.5788i	0.253966	−	1.31965i
$470$		0			0
$471$		0			0
$472$		0			0
$473$		0			0
$474$		0			0
$475$			17.3205i			0.794719i
$476$		0			0
$477$		0			0
$478$		0			0
$479$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$479$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$480$		0			0
$481$	−1.50000	−	0.866025i	−0.0683941	−	0.0394874i
$482$		0			0
$483$		0			0
$484$		0			0
$485$		0			0
$486$		0			0
$487$	22.0000	+	38.1051i	0.996915	+	1.72671i	0.566429	+	0.824110i	$0.308325\pi$
$487$	22.0000	+	38.1051i	0.996915	+	1.72671i	0.430486	+	0.902597i	$0.358342\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$		0			0		1.00000			$0$
$491$		0			0		−1.00000			$\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$	5.50000	−	9.52628i	0.246214	−	0.426455i	−0.716258	−	0.697835i	$-0.754147\pi$
$499$	5.50000	−	9.52628i	0.246214	−	0.426455i	0.962472	+	0.271380i	$0.0874801\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		0			0			−	1.00000i	$-0.5\pi$
$503$		0			0				1.00000i	$0.5\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$		0			0
$508$		0			0
$509$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$509$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$510$		0			0
$511$	−36.0000	−	6.92820i	−1.59255	−	0.306486i
$512$		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		0.866025	−	0.500000i	$-0.166667\pi$
$521$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$522$		0			0
$523$	25.5000	+	14.7224i	1.11504	+	0.643767i	0.940129	−	0.340818i	$-0.110704\pi$
$523$	25.5000	+	14.7224i	1.11504	+	0.643767i	0.174908	+	0.984585i	$0.444037\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		0			0
$528$		0			0
$529$	−11.5000	−	19.9186i	−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
$541$	23.0000	−	39.8372i	0.988847	−	1.71273i	0.365444	−	0.930834i	$-0.380917\pi$
$541$	23.0000	−	39.8372i	0.988847	−	1.71273i	0.623404	−	0.781900i	$-0.285749\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		0			0
$546$		0			0
$547$	−1.00000			−0.0427569			−0.0213785	−	0.999771i	$-0.506805\pi$
$547$	−1.00000			−0.0427569			−0.0213785	+	0.999771i	$0.506805\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		0			0
$552$		0			0
$553$	32.5000	−	11.2583i	1.38204	−	0.478753i
$554$		0			0
$555$		0			0
$556$		0			0
$557$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$557$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$558$		0			0
$559$			22.5167i			0.952353i
$560$		0			0
$561$		0			0
$562$		0			0
$563$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$563$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$564$		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
$571$	8.00000	+	13.8564i	0.334790	+	0.579873i	0.983444	−	0.181210i	$-0.0580014\pi$
$571$	8.00000	+	13.8564i	0.334790	+	0.579873i	−0.648655	+	0.761083i	$0.724668\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$		0			0
$577$	−28.5000	+	16.4545i	−1.18647	+	0.685009i	−0.957503	−	0.288425i	$-0.906868\pi$
$577$	−28.5000	+	16.4545i	−1.18647	+	0.685009i	−0.228968	+	0.973434i	$0.573535\pi$
$578$		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			−	1.00000i	$-0.5\pi$
$587$		0			0				1.00000i	$0.5\pi$
$588$		0			0
$589$	−6.00000			−0.247226
$590$		0			0
$591$		0			0
$592$		0			0
$593$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$593$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$594$		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
$601$		−	43.3013i		−	1.76630i	−0.469095	−	0.883148i	$-0.655420\pi$
$601$		−	43.3013i		−	1.76630i	0.469095	−	0.883148i	$-0.344580\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$		0			0
$606$		0			0
$607$	−39.0000	−	22.5167i	−1.58296	−	0.913923i	−0.994424	−	0.105453i	$-0.966371\pi$
$607$	−39.0000	−	22.5167i	−1.58296	−	0.913923i	−0.588537	−	0.808470i	$-0.700296\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$	−23.5000	−	40.7032i	−0.949156	−	1.64399i	−0.747208	−	0.664590i	$-0.768606\pi$
$613$	−23.5000	−	40.7032i	−0.949156	−	1.64399i	−0.201948	−	0.979396i	$-0.564727\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		0			0		1.00000			$0$
$617$		0			0		−1.00000			$\pi$
$618$		0			0
$619$	−7.50000	+	4.33013i	−0.301450	+	0.174042i	−0.643094	−	0.765787i	$-0.722350\pi$
$619$	−7.50000	+	4.33013i	−0.301450	+	0.174042i	0.341644	+	0.939829i	$0.389016\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	−12.5000	+	21.6506i	−0.500000	+	0.866025i
$626$		0			0
$627$		0			0
$628$		0			0
$629$		0			0
$630$		0			0
$631$	−43.0000			−1.71180			−0.855901	−	0.517139i	$-0.826997\pi$
$631$	−43.0000			−1.71180			−0.855901	+	0.517139i	$0.826997\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$		0			0
$636$		0			0
$637$	−12.0000	+	1.73205i	−0.475457	+	0.0686264i
$638$		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
$643$		−	19.0526i		−	0.751360i	−0.926750	−	0.375680i	$-0.877409\pi$
$643$		−	19.0526i		−	0.751360i	0.926750	−	0.375680i	$-0.122591\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$647$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$648$		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		1.00000			$0$
$659$		0			0		−1.00000			$\pi$
$660$		0			0
$661$	30.0000	−	17.3205i	1.16686	−	0.673690i	0.213925	−	0.976850i	$-0.431375\pi$
$661$	30.0000	−	17.3205i	1.16686	−	0.673690i	0.952940	+	0.303160i	$0.0980418\pi$
$662$		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
$673$	−50.0000			−1.92736			−0.963679	−	0.267063i	$-0.913947\pi$
$673$	−50.0000			−1.92736			−0.963679	+	0.267063i	$0.913947\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$677$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$678$		0			0
$679$	−9.00000	+	10.3923i	−0.345388	+	0.398820i
$680$		0			0
$681$		0			0
$682$		0			0
$683$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$683$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$		0			0
$688$		0			0
$689$		0			0
$690$		0			0
$691$	−28.5000	−	16.4545i	−1.08419	−	0.625958i	−0.152167	−	0.988355i	$-0.548625\pi$
$691$	−28.5000	−	16.4545i	−1.08419	−	0.625958i	−0.932024	+	0.362397i	$0.881959\pi$
$692$		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
$703$	−3.00000	+	1.73205i	−0.113147	+	0.0653255i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	26.5000	−	45.8993i	0.995228	−	1.72379i	0.413114	−	0.910679i	$-0.364441\pi$
$709$	26.5000	−	45.8993i	0.995228	−	1.72379i	0.582115	−	0.813107i	$-0.302225\pi$
$710$		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		−0.866025	−	0.500000i	$-0.833333\pi$
$719$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$720$		0			0
$721$	−16.5000	−	47.6314i	−0.614492	−	1.77389i
$722$		0			0
$723$		0			0
$724$		0			0
$725$		0			0
$726$		0			0
$727$			53.6936i			1.99138i	0.0927199	+	0.995692i	$0.470444\pi$
$727$			53.6936i			1.99138i	−0.0927199	+	0.995692i	$0.529556\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$		0			0
$732$		0			0
$733$	−28.5000	−	16.4545i	−1.05267	−	0.607760i	−0.129275	−	0.991609i	$-0.541265\pi$
$733$	−28.5000	−	16.4545i	−1.05267	−	0.607760i	−0.923396	+	0.383849i	$0.874598\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$		0			0
$739$	18.5000	+	32.0429i	0.680534	+	1.17872i	0.974818	+	0.223001i	$0.0715853\pi$
$739$	18.5000	+	32.0429i	0.680534	+	1.17872i	−0.294285	+	0.955718i	$0.595081\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		0			0		1.00000			$0$
$743$		0			0		−1.00000			$\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	−26.0000	+	45.0333i	−0.948753	+	1.64329i	−0.200698	+	0.979653i	$0.564321\pi$
$751$	−26.0000	+	45.0333i	−0.948753	+	1.64329i	−0.748056	+	0.663636i	$0.769012\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$	−29.0000			−1.05402			−0.527011	−	0.849858i	$-0.676688\pi$
$757$	−29.0000			−1.05402			−0.527011	+	0.849858i	$0.676688\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$761$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$762$		0			0
$763$	9.50000	−	49.3634i	0.343923	−	1.78708i
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$		0			0
$769$		−	55.4256i		−	1.99870i	−0.0360609	−	0.999350i	$-0.511481\pi$
$769$		−	55.4256i		−	1.99870i	0.0360609	−	0.999350i	$-0.488519\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$773$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$774$		0			0
$775$	−7.50000	−	4.33013i	−0.269408	−	0.155543i
$776$		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
$787$	40.5000	−	23.3827i	1.44367	−	0.833503i	0.445577	−	0.895244i	$-0.352999\pi$
$787$	40.5000	−	23.3827i	1.44367	−	0.833503i	0.998092	+	0.0617409i	$0.0196653\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	−13.5000	+	23.3827i	−0.479399	+	0.830344i
$794$		0			0
$795$		0			0
$796$		0			0
$797$		0			0			−	1.00000i	$-0.5\pi$
$797$		0			0				1.00000i	$0.5\pi$
$798$		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		−0.500000	−	0.866025i	$-0.666667\pi$
$809$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$810$		0			0
$811$		−	10.3923i		−	0.364923i	−0.983213	−	0.182462i	$-0.941593\pi$
$811$		−	10.3923i		−	0.364923i	0.983213	−	0.182462i	$-0.0584065\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		0			0
$816$		0			0
$817$	39.0000	+	22.5167i	1.36444	+	0.787758i
$818$		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
$823$	−2.50000	−	4.33013i	−0.0871445	−	0.150939i	0.819159	−	0.573567i	$-0.194441\pi$
$823$	−2.50000	−	4.33013i	−0.0871445	−	0.150939i	−0.906303	+	0.422628i	$0.861108\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		0			0		1.00000			$0$
$827$		0			0		−1.00000			$\pi$
$828$		0			0
$829$	30.0000	−	17.3205i	1.04194	−	0.601566i	0.121560	−	0.992584i	$-0.461210\pi$
$829$	30.0000	−	17.3205i	1.04194	−	0.601566i	0.920383	+	0.391018i	$0.127877\pi$
$830$		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			−	1.00000i	$-0.5\pi$
$839$		0			0				1.00000i	$0.5\pi$
$840$		0			0
$841$	29.0000			1.00000
$842$		0			0
$843$		0			0
$844$		0			0
$845$		0			0
$846$		0			0
$847$	−27.5000	+	9.52628i	−0.944911	+	0.327327i
$848$		0			0
$849$		0			0
$850$		0			0
$851$		0			0
$852$		0			0
$853$		−	6.92820i		−	0.237217i	−0.992941	−	0.118609i	$-0.962157\pi$
$853$		−	6.92820i		−	0.237217i	0.992941	−	0.118609i	$-0.0378434\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$857$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$858$		0			0
$859$	−49.5000	−	28.5788i	−1.68892	−	0.975097i	−0.955348	−	0.295484i	$-0.904519\pi$
$859$	−49.5000	−	28.5788i	−1.68892	−	0.975097i	−0.733571	−	0.679613i	$-0.762148\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$863$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$		0			0
$868$		0			0
$869$		0			0
$870$		0			0
$871$	16.5000	−	9.52628i	0.559081	−	0.322786i
$872$		0			0
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$	−29.5000	+	51.0955i	−0.996144	+	1.72537i	−0.422095	+	0.906552i	$0.638705\pi$
$877$	−29.5000	+	51.0955i	−0.996144	+	1.72537i	−0.574049	+	0.818821i	$0.694628\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		0			0			−	1.00000i	$-0.5\pi$
$881$		0			0				1.00000i	$0.5\pi$
$882$		0			0
$883$	−8.00000			−0.269221			−0.134611	−	0.990899i	$-0.542978\pi$
$883$	−8.00000			−0.269221			−0.134611	+	0.990899i	$0.542978\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$887$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$888$		0			0
$889$	−2.00000	−	1.73205i	−0.0670778	−	0.0580911i
$890$		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
$907$	29.5000	+	51.0955i	0.979531	+	1.69660i	0.664089	+	0.747653i	$0.268820\pi$
$907$	29.5000	+	51.0955i	0.979531	+	1.69660i	0.315442	+	0.948945i	$0.397847\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		0			0		1.00000			$0$
$911$		0			0		−1.00000			$\pi$
$912$		0			0
$913$		0			0
$914$		0			0
$915$		0			0
$916$		0			0
$917$		0			0
$918$		0			0
$919$	26.5000	−	45.8993i	0.874154	−	1.51408i	0.0164935	−	0.999864i	$-0.494750\pi$
$919$	26.5000	−	45.8993i	0.874154	−	1.51408i	0.857661	−	0.514216i	$-0.171917\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		0			0
$924$		0			0
$925$	−5.00000			−0.164399
$926$		0			0
$927$		0			0
$928$		0			0
$929$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$929$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$930$		0			0
$931$	−9.00000	+	22.5167i	−0.294963	+	0.737954i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$			50.2295i			1.64093i	0.571700	+	0.820463i	$0.306284\pi$
$937$			50.2295i			1.64093i	−0.571700	+	0.820463i	$0.693716\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$941$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$942$		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
$949$	−12.0000	−	20.7846i	−0.389536	−	0.674697i
$950$		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
$961$	−14.0000	+	24.2487i	−0.451613	+	0.782216i
$962$		0			0
$963$		0			0
$964$		0			0
$965$		0			0
$966$		0			0
$967$	−61.0000			−1.96163			−0.980814	−	0.194946i	$-0.937547\pi$
$967$	−61.0000			−1.96163			−0.980814	+	0.194946i	$0.937547\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$971$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$972$		0			0
$973$	39.0000	−	45.0333i	1.25028	−	1.44370i
$974$		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		0.866025	−	0.500000i	$-0.166667\pi$
$983$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$	8.50000	+	14.7224i	0.270011	+	0.467673i	0.968864	−	0.247592i	$-0.0796392\pi$
$991$	8.50000	+	14.7224i	0.270011	+	0.467673i	−0.698853	+	0.715265i	$0.746306\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		0			0
$996$		0			0
$997$	19.5000	−	11.2583i	0.617571	−	0.356555i	−0.158352	−	0.987383i	$-0.550618\pi$
$997$	19.5000	−	11.2583i	0.617571	−	0.356555i	0.775923	+	0.630828i	$0.217285\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	756.2.t.c.593.1	yes	2
3.2	odd	2	CM	756.2.t.c.593.1	yes	2
7.2	even	3		5292.2.f.b.2645.2		2
7.3	odd	6	inner	756.2.t.c.269.1	✓	2
7.5	odd	6		5292.2.f.b.2645.1		2
9.2	odd	6		2268.2.w.d.1349.1		2
9.4	even	3		2268.2.bm.b.593.1		2
9.5	odd	6		2268.2.bm.b.593.1		2
9.7	even	3		2268.2.w.d.1349.1		2
21.2	odd	6		5292.2.f.b.2645.2		2
21.5	even	6		5292.2.f.b.2645.1		2
21.17	even	6	inner	756.2.t.c.269.1	✓	2
63.31	odd	6		2268.2.w.d.269.1		2
63.38	even	6		2268.2.bm.b.1025.1		2
63.52	odd	6		2268.2.bm.b.1025.1		2
63.59	even	6		2268.2.w.d.269.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
756.2.t.c.269.1	✓	2	7.3	odd	6	inner
756.2.t.c.269.1	✓	2	21.17	even	6	inner
756.2.t.c.593.1	yes	2	1.1	even	1	trivial
756.2.t.c.593.1	yes	2	3.2	odd	2	CM
2268.2.w.d.269.1		2	63.31	odd	6
2268.2.w.d.269.1		2	63.59	even	6
2268.2.w.d.1349.1		2	9.2	odd	6
2268.2.w.d.1349.1		2	9.7	even	3
2268.2.bm.b.593.1		2	9.4	even	3
2268.2.bm.b.593.1		2	9.5	odd	6
2268.2.bm.b.1025.1		2	63.38	even	6
2268.2.bm.b.1025.1		2	63.52	odd	6
5292.2.f.b.2645.1		2	7.5	odd	6
5292.2.f.b.2645.1		2	21.5	even	6
5292.2.f.b.2645.2		2	7.2	even	3
5292.2.f.b.2645.2		2	21.2	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:	\( N \)	\(=\)	\( 756 = 2^{2} \cdot 3^{3} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	756.t (of order \(6\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(2.00000 + 1.73205i) q^{7} +O(q^{10})\)	\(q+(2.00000 + 1.73205i) q^{7} +1.73205i q^{13} +(3.00000 + 1.73205i) q^{19} +(2.50000 + 4.33013i) q^{25} +(-1.50000 + 0.866025i) q^{31} +(-0.500000 + 0.866025i) q^{37} +13.0000 q^{43} +(1.00000 + 6.92820i) q^{49} +(13.5000 + 7.79423i) q^{61} +(-5.50000 - 9.52628i) q^{67} +(-12.0000 + 6.92820i) q^{73} +(6.50000 - 11.2583i) q^{79} +(-3.00000 + 3.46410i) q^{91} +5.19615i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 4 q^{7}+O(q^{10}) \) 2 * q + 4 * q^7	\( 2 q + 4 q^{7} + 6 q^{19} + 5 q^{25} - 3 q^{31} - q^{37} + 26 q^{43} + 2 q^{49} + 27 q^{61} - 11 q^{67} - 24 q^{73} + 13 q^{79} - 6 q^{91}+O(q^{100}) \) 2 * q + 4 * q^7 + 6 * q^19 + 5 * q^25 - 3 * q^31 - q^37 + 26 * q^43 + 2 * q^49 + 27 * q^61 - 11 * q^67 - 24 * q^73 + 13 * q^79 - 6 * q^91

Introduction

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