LMFDB - Embedded newform 2352.2.bl.f.31.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2352,2,Mod(31,2352)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2352, base_ring=CyclotomicField(6))

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

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

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

chi := DirichletCharacter("2352.31");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2352 = 2^{4} \cdot 3 \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2352.bl (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:	$18.7808145554$
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:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 336)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			31.1
Root			$0.500000 + 0.866025i$ of defining polynomial
Character	$\chi$	$=$	2352.31
Dual form			2352.2.bl.f.607.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.500000 + 0.866025i) q^{3} +(3.00000 - 1.73205i) q^{5} +(-0.500000 - 0.866025i) q^{9} +O(q^{10})$	$q+(-0.500000 + 0.866025i) q^{3} +(3.00000 - 1.73205i) q^{5} +(-0.500000 - 0.866025i) q^{9} +(3.00000 + 1.73205i) q^{11} -5.19615i q^{13} +3.46410i q^{15} +(6.00000 + 3.46410i) q^{17} +(-3.50000 - 6.06218i) q^{19} +(3.50000 - 6.06218i) q^{25} +1.00000 q^{27} +(-2.50000 + 4.33013i) q^{31} +(-3.00000 + 1.73205i) q^{33} +(-0.500000 - 0.866025i) q^{37} +(4.50000 + 2.59808i) q^{39} -10.3923i q^{41} -1.73205i q^{43} +(-3.00000 - 1.73205i) q^{45} +(3.00000 + 5.19615i) q^{47} +(-6.00000 + 3.46410i) q^{51} +12.0000 q^{55} +7.00000 q^{57} +(-9.00000 - 15.5885i) q^{65} +(-1.50000 - 0.866025i) q^{67} +3.46410i q^{71} +(-7.50000 - 4.33013i) q^{73} +(3.50000 + 6.06218i) q^{75} +(13.5000 - 7.79423i) q^{79} +(-0.500000 + 0.866025i) q^{81} -6.00000 q^{83} +24.0000 q^{85} +(-6.00000 + 3.46410i) q^{89} +(-2.50000 - 4.33013i) q^{93} +(-21.0000 - 12.1244i) q^{95} +6.92820i q^{97} -3.46410i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - q^{3} + 6 q^{5} - q^{9}+O(q^{10}) $ 2 * q - q^3 + 6 * q^5 - q^9	$ 2 q - q^{3} + 6 q^{5} - q^{9} + 6 q^{11} + 12 q^{17} - 7 q^{19} + 7 q^{25} + 2 q^{27} - 5 q^{31} - 6 q^{33} - q^{37} + 9 q^{39} - 6 q^{45} + 6 q^{47} - 12 q^{51} + 24 q^{55} + 14 q^{57} - 18 q^{65} - 3 q^{67} - 15 q^{73} + 7 q^{75} + 27 q^{79} - q^{81} - 12 q^{83} + 48 q^{85} - 12 q^{89} - 5 q^{93} - 42 q^{95}+O(q^{100}) $ 2 * q - q^3 + 6 * q^5 - q^9 + 6 * q^11 + 12 * q^17 - 7 * q^19 + 7 * q^25 + 2 * q^27 - 5 * q^31 - 6 * q^33 - q^37 + 9 * q^39 - 6 * q^45 + 6 * q^47 - 12 * q^51 + 24 * q^55 + 14 * q^57 - 18 * q^65 - 3 * q^67 - 15 * q^73 + 7 * q^75 + 27 * q^79 - q^81 - 12 * q^83 + 48 * q^85 - 12 * q^89 - 5 * q^93 - 42 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2352\mathbb{Z}\right)^\times$.

$n$	$785$	$1471$	$1765$	$2257$
$\chi(n)$	$1$	$-1$	$1$	$e\left(\frac{1}{6}\right)$

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$	−0.500000	+	0.866025i	−0.288675	+	0.500000i
$3$	−0.500000	+	0.866025i	−0.288675	+	0.500000i
$4$		0			0
$5$	3.00000	−	1.73205i	1.34164	−	0.774597i	0.354593	−	0.935021i	$-0.384620\pi$
$5$	3.00000	−	1.73205i	1.34164	−	0.774597i	0.987048	+	0.160424i	$0.0512862\pi$
$6$		0			0
$7$		0			0
$7$		0			0
$8$		0			0
$9$	−0.500000	−	0.866025i	−0.166667	−	0.288675i
$10$		0			0
$11$	3.00000	+	1.73205i	0.904534	+	0.522233i	0.878668	−	0.477432i	$-0.158432\pi$
$11$	3.00000	+	1.73205i	0.904534	+	0.522233i	0.0258656	+	0.999665i	$0.491766\pi$
$12$		0			0
$13$		−	5.19615i		−	1.44115i	−0.693375	−	0.720577i	$-0.743877\pi$
$13$		−	5.19615i		−	1.44115i	0.693375	−	0.720577i	$-0.256123\pi$
$14$		0			0
$15$			3.46410i			0.894427i
$16$		0			0
$17$	6.00000	+	3.46410i	1.45521	+	0.840168i	0.998770	−	0.0495842i	$-0.0157896\pi$
$17$	6.00000	+	3.46410i	1.45521	+	0.840168i	0.456444	+	0.889752i	$0.349123\pi$
$18$		0			0
$19$	−3.50000	−	6.06218i	−0.802955	−	1.39076i	−0.917663	−	0.397360i	$-0.869927\pi$
$19$	−3.50000	−	6.06218i	−0.802955	−	1.39076i	0.114708	−	0.993399i	$-0.463407\pi$
$20$		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
$25$	3.50000	−	6.06218i	0.700000	−	1.21244i
$26$		0			0
$27$	1.00000			0.192450
$28$		0			0
$29$		0			0			−	1.00000i	$-0.5\pi$
$29$		0			0				1.00000i	$0.5\pi$
$30$		0			0
$31$	−2.50000	+	4.33013i	−0.449013	+	0.777714i	−0.998322	−	0.0579057i	$-0.981558\pi$
$31$	−2.50000	+	4.33013i	−0.449013	+	0.777714i	0.549309	+	0.835619i	$0.314891\pi$
$32$		0			0
$33$	−3.00000	+	1.73205i	−0.522233	+	0.301511i
$34$		0			0
$35$		0			0
$36$		0			0
$37$	−0.500000	−	0.866025i	−0.0821995	−	0.142374i	0.821995	−	0.569495i	$-0.192861\pi$
$37$	−0.500000	−	0.866025i	−0.0821995	−	0.142374i	−0.904194	+	0.427121i	$0.859528\pi$
$38$		0			0
$39$	4.50000	+	2.59808i	0.720577	+	0.416025i
$40$		0			0
$41$		−	10.3923i		−	1.62301i	−0.584349	−	0.811503i	$-0.698650\pi$
$41$		−	10.3923i		−	1.62301i	0.584349	−	0.811503i	$-0.301350\pi$
$42$		0			0
$43$		−	1.73205i		−	0.264135i	−0.991241	−	0.132068i	$-0.957838\pi$
$43$		−	1.73205i		−	0.264135i	0.991241	−	0.132068i	$-0.0421616\pi$
$44$		0			0
$45$	−3.00000	−	1.73205i	−0.447214	−	0.258199i
$46$		0			0
$47$	3.00000	+	5.19615i	0.437595	+	0.757937i	0.997503	−	0.0706177i	$-0.0224970\pi$
$47$	3.00000	+	5.19615i	0.437595	+	0.757937i	−0.559908	+	0.828554i	$0.689164\pi$
$48$		0			0
$49$		0			0
$50$		0			0
$51$	−6.00000	+	3.46410i	−0.840168	+	0.485071i
$52$		0			0
$53$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$53$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$54$		0			0
$55$	12.0000			1.61808
$56$		0			0
$57$	7.00000			0.927173
$58$		0			0
$59$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$59$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$60$		0			0
$61$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$61$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−9.00000	−	15.5885i	−1.11631	−	1.93351i
$66$		0			0
$67$	−1.50000	−	0.866025i	−0.183254	−	0.105802i	0.405567	−	0.914066i	$-0.367074\pi$
$67$	−1.50000	−	0.866025i	−0.183254	−	0.105802i	−0.588821	+	0.808264i	$0.700408\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$			3.46410i			0.411113i	0.978645	+	0.205557i	$0.0659005\pi$
$71$			3.46410i			0.411113i	−0.978645	+	0.205557i	$0.934100\pi$
$72$		0			0
$73$	−7.50000	−	4.33013i	−0.877809	−	0.506803i	−0.00787336	−	0.999969i	$-0.502506\pi$
$73$	−7.50000	−	4.33013i	−0.877809	−	0.506803i	−0.869935	+	0.493166i	$0.835840\pi$
$74$		0			0
$75$	3.50000	+	6.06218i	0.404145	+	0.700000i
$76$		0			0
$77$		0			0
$78$		0			0
$79$	13.5000	−	7.79423i	1.51887	−	0.876919i	0.519115	−	0.854704i	$-0.326261\pi$
$79$	13.5000	−	7.79423i	1.51887	−	0.876919i	0.999753	−	0.0222151i	$-0.00707187\pi$
$80$		0			0
$81$	−0.500000	+	0.866025i	−0.0555556	+	0.0962250i
$82$		0			0
$83$	−6.00000			−0.658586			−0.329293	−	0.944228i	$-0.606810\pi$
$83$	−6.00000			−0.658586			−0.329293	+	0.944228i	$0.606810\pi$
$84$		0			0
$85$	24.0000			2.60317
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−6.00000	+	3.46410i	−0.635999	+	0.367194i	−0.783072	−	0.621932i	$-0.786348\pi$
$89$	−6.00000	+	3.46410i	−0.635999	+	0.367194i	0.147073	+	0.989126i	$0.453015\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$	−2.50000	−	4.33013i	−0.259238	−	0.449013i
$94$		0			0
$95$	−21.0000	−	12.1244i	−2.15455	−	1.24393i
$96$		0			0
$97$			6.92820i			0.703452i	0.936103	+	0.351726i	$0.114405\pi$
$97$			6.92820i			0.703452i	−0.936103	+	0.351726i	$0.885595\pi$
$98$		0			0
$99$		−	3.46410i		−	0.348155i
$100$		0			0
$101$	3.00000	+	1.73205i	0.298511	+	0.172345i	0.641774	−	0.766894i	$-0.278199\pi$
$101$	3.00000	+	1.73205i	0.298511	+	0.172345i	−0.343263	+	0.939239i	$0.611532\pi$
$102$		0			0
$103$	−2.50000	−	4.33013i	−0.246332	−	0.426660i	0.716173	−	0.697923i	$-0.245892\pi$
$103$	−2.50000	−	4.33013i	−0.246332	−	0.426660i	−0.962505	+	0.271263i	$0.912559\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	6.00000	−	3.46410i	0.580042	−	0.334887i	−0.181108	−	0.983463i	$-0.557968\pi$
$107$	6.00000	−	3.46410i	0.580042	−	0.334887i	0.761150	+	0.648576i	$0.224635\pi$
$108$		0			0
$109$	−2.50000	+	4.33013i	−0.239457	+	0.414751i	−0.960558	−	0.278078i	$-0.910303\pi$
$109$	−2.50000	+	4.33013i	−0.239457	+	0.414751i	0.721102	+	0.692829i	$0.243636\pi$
$110$		0			0
$111$	1.00000			0.0949158
$112$		0			0
$113$	6.00000			0.564433			0.282216	−	0.959351i	$-0.408930\pi$
$113$	6.00000			0.564433			0.282216	+	0.959351i	$0.408930\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	−4.50000	+	2.59808i	−0.416025	+	0.240192i
$118$		0			0
$119$		0			0
$120$		0			0
$121$	0.500000	+	0.866025i	0.0454545	+	0.0787296i
$122$		0			0
$123$	9.00000	+	5.19615i	0.811503	+	0.468521i
$124$		0			0
$125$		−	6.92820i		−	0.619677i
$126$		0			0
$127$			15.5885i			1.38325i	0.722256	+	0.691626i	$0.243105\pi$
$127$			15.5885i			1.38325i	−0.722256	+	0.691626i	$0.756895\pi$
$128$		0			0
$129$	1.50000	+	0.866025i	0.132068	+	0.0762493i
$130$		0			0
$131$	−9.00000	−	15.5885i	−0.786334	−	1.36197i	−0.928199	−	0.372084i	$-0.878643\pi$
$131$	−9.00000	−	15.5885i	−0.786334	−	1.36197i	0.141865	−	0.989886i	$-0.454690\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$	3.00000	−	1.73205i	0.258199	−	0.149071i
$136$		0			0
$137$	6.00000	−	10.3923i	0.512615	−	0.887875i	−0.487278	−	0.873247i	$-0.662010\pi$
$137$	6.00000	−	10.3923i	0.512615	−	0.887875i	0.999893	−	0.0146279i	$-0.00465636\pi$
$138$		0			0
$139$	5.00000			0.424094			0.212047	−	0.977259i	$-0.431987\pi$
$139$	5.00000			0.424094			0.212047	+	0.977259i	$0.431987\pi$
$140$		0			0
$141$	−6.00000			−0.505291
$142$		0			0
$143$	9.00000	−	15.5885i	0.752618	−	1.30357i
$144$		0			0
$145$		0			0
$146$		0			0
$147$		0			0
$148$		0			0
$149$	6.00000	+	10.3923i	0.491539	+	0.851371i	0.999953	−	0.00974235i	$-0.00310113\pi$
$149$	6.00000	+	10.3923i	0.491539	+	0.851371i	−0.508413	+	0.861113i	$0.669768\pi$
$150$		0			0
$151$	9.00000	+	5.19615i	0.732410	+	0.422857i	0.819303	−	0.573361i	$-0.194361\pi$
$151$	9.00000	+	5.19615i	0.732410	+	0.422857i	−0.0868934	+	0.996218i	$0.527694\pi$
$152$		0			0
$153$		−	6.92820i		−	0.560112i
$154$		0			0
$155$			17.3205i			1.39122i
$156$		0			0
$157$	12.0000	+	6.92820i	0.957704	+	0.552931i	0.895466	−	0.445130i	$-0.146843\pi$
$157$	12.0000	+	6.92820i	0.957704	+	0.552931i	0.0622385	+	0.998061i	$0.480176\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$	3.00000	−	1.73205i	0.234978	−	0.135665i	−0.377888	−	0.925851i	$-0.623350\pi$
$163$	3.00000	−	1.73205i	0.234978	−	0.135665i	0.612866	+	0.790186i	$0.290016\pi$
$164$		0			0
$165$	−6.00000	+	10.3923i	−0.467099	+	0.809040i
$166$		0			0
$167$	6.00000			0.464294			0.232147	−	0.972681i	$-0.425425\pi$
$167$	6.00000			0.464294			0.232147	+	0.972681i	$0.425425\pi$
$168$		0			0
$169$	−14.0000			−1.07692
$170$		0			0
$171$	−3.50000	+	6.06218i	−0.267652	+	0.463586i
$172$		0			0
$173$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$173$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		0			0
$178$		0			0
$179$	15.0000	+	8.66025i	1.12115	+	0.647298i	0.941695	−	0.336468i	$-0.109232\pi$
$179$	15.0000	+	8.66025i	1.12115	+	0.647298i	0.179458	+	0.983766i	$0.442566\pi$
$180$		0			0
$181$		−	15.5885i		−	1.15868i	−0.815086	−	0.579340i	$-0.803310\pi$
$181$		−	15.5885i		−	1.15868i	0.815086	−	0.579340i	$-0.196690\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−3.00000	−	1.73205i	−0.220564	−	0.127343i
$186$		0			0
$187$	12.0000	+	20.7846i	0.877527	+	1.51992i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	3.00000	−	1.73205i	0.217072	−	0.125327i	−0.387522	−	0.921861i	$-0.626669\pi$
$191$	3.00000	−	1.73205i	0.217072	−	0.125327i	0.604594	+	0.796534i	$0.293335\pi$
$192$		0			0
$193$	−6.50000	+	11.2583i	−0.467880	+	0.810392i	−0.999326	−	0.0366998i	$-0.988315\pi$
$193$	−6.50000	+	11.2583i	−0.467880	+	0.810392i	0.531446	+	0.847092i	$0.321649\pi$
$194$		0			0
$195$	18.0000			1.28901
$196$		0			0
$197$	12.0000			0.854965			0.427482	−	0.904024i	$-0.359401\pi$
$197$	12.0000			0.854965			0.427482	+	0.904024i	$0.359401\pi$
$198$		0			0
$199$	−8.00000	+	13.8564i	−0.567105	+	0.982255i	0.429745	+	0.902950i	$0.358603\pi$
$199$	−8.00000	+	13.8564i	−0.567105	+	0.982255i	−0.996850	+	0.0793045i	$0.974730\pi$
$200$		0			0
$201$	1.50000	−	0.866025i	0.105802	−	0.0610847i
$202$		0			0
$203$		0			0
$204$		0			0
$205$	−18.0000	−	31.1769i	−1.25717	−	2.17749i
$206$		0			0
$207$		0			0
$208$		0			0
$209$		−	24.2487i		−	1.67732i
$210$		0			0
$211$		−	17.3205i		−	1.19239i	−0.802839	−	0.596196i	$-0.796678\pi$
$211$		−	17.3205i		−	1.19239i	0.802839	−	0.596196i	$-0.203322\pi$
$212$		0			0
$213$	−3.00000	−	1.73205i	−0.205557	−	0.118678i
$214$		0			0
$215$	−3.00000	−	5.19615i	−0.204598	−	0.354375i
$216$		0			0
$217$		0			0
$218$		0			0
$219$	7.50000	−	4.33013i	0.506803	−	0.292603i
$220$		0			0
$221$	18.0000	−	31.1769i	1.21081	−	2.09719i
$222$		0			0
$223$	−8.00000			−0.535720			−0.267860	−	0.963458i	$-0.586316\pi$
$223$	−8.00000			−0.535720			−0.267860	+	0.963458i	$0.586316\pi$
$224$		0			0
$225$	−7.00000			−0.466667
$226$		0			0
$227$	15.0000	−	25.9808i	0.995585	−	1.72440i	0.416503	−	0.909134i	$-0.363255\pi$
$227$	15.0000	−	25.9808i	0.995585	−	1.72440i	0.579082	−	0.815270i	$-0.303411\pi$
$228$		0			0
$229$	−1.50000	+	0.866025i	−0.0991228	+	0.0572286i	−0.548742	−	0.835992i	$-0.684893\pi$
$229$	−1.50000	+	0.866025i	−0.0991228	+	0.0572286i	0.449619	+	0.893220i	$0.351560\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	9.00000	+	15.5885i	0.589610	+	1.02123i	0.994283	+	0.106773i	$0.0340517\pi$
$233$	9.00000	+	15.5885i	0.589610	+	1.02123i	−0.404674	+	0.914461i	$0.632615\pi$
$234$		0			0
$235$	18.0000	+	10.3923i	1.17419	+	0.677919i
$236$		0			0
$237$			15.5885i			1.01258i
$238$		0			0
$239$			17.3205i			1.12037i	0.828367	+	0.560185i	$0.189270\pi$
$239$			17.3205i			1.12037i	−0.828367	+	0.560185i	$0.810730\pi$
$240$		0			0
$241$	12.0000	+	6.92820i	0.772988	+	0.446285i	0.833939	−	0.551856i	$-0.186080\pi$
$241$	12.0000	+	6.92820i	0.772988	+	0.446285i	−0.0609515	+	0.998141i	$0.519414\pi$
$242$		0			0
$243$	−0.500000	−	0.866025i	−0.0320750	−	0.0555556i
$244$		0			0
$245$		0			0
$246$		0			0
$247$	−31.5000	+	18.1865i	−2.00430	+	1.15718i
$248$		0			0
$249$	3.00000	−	5.19615i	0.190117	−	0.329293i
$250$		0			0
$251$	−24.0000			−1.51487			−0.757433	−	0.652913i	$-0.773547\pi$
$251$	−24.0000			−1.51487			−0.757433	+	0.652913i	$0.773547\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$	−12.0000	+	20.7846i	−0.751469	+	1.30158i
$256$		0			0
$257$	9.00000	−	5.19615i	0.561405	−	0.324127i	−0.192304	−	0.981335i	$-0.561596\pi$
$257$	9.00000	−	5.19615i	0.561405	−	0.324127i	0.753709	+	0.657208i	$0.228263\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−24.0000	−	13.8564i	−1.47990	−	0.854423i	−0.480162	−	0.877180i	$-0.659422\pi$
$263$	−24.0000	−	13.8564i	−1.47990	−	0.854423i	−0.999741	+	0.0227570i	$0.992756\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		−	6.92820i		−	0.423999i
$268$		0			0
$269$	27.0000	+	15.5885i	1.64622	+	0.950445i	0.978556	+	0.205982i	$0.0660387\pi$
$269$	27.0000	+	15.5885i	1.64622	+	0.950445i	0.667663	+	0.744463i	$0.267295\pi$
$270$		0			0
$271$	8.00000	+	13.8564i	0.485965	+	0.841717i	0.999870	−	0.0161307i	$-0.00513477\pi$
$271$	8.00000	+	13.8564i	0.485965	+	0.841717i	−0.513905	+	0.857847i	$0.671801\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	21.0000	−	12.1244i	1.26635	−	0.731126i
$276$		0			0
$277$	8.50000	−	14.7224i	0.510716	−	0.884585i	−0.489207	−	0.872167i	$-0.662714\pi$
$277$	8.50000	−	14.7224i	0.510716	−	0.884585i	0.999923	−	0.0124177i	$-0.00395278\pi$
$278$		0			0
$279$	5.00000			0.299342
$280$		0			0
$281$	12.0000			0.715860			0.357930	−	0.933748i	$-0.383483\pi$
$281$	12.0000			0.715860			0.357930	+	0.933748i	$0.383483\pi$
$282$		0			0
$283$	2.50000	−	4.33013i	0.148610	−	0.257399i	−0.782104	−	0.623148i	$-0.785854\pi$
$283$	2.50000	−	4.33013i	0.148610	−	0.257399i	0.930714	+	0.365748i	$0.119187\pi$
$284$		0			0
$285$	21.0000	−	12.1244i	1.24393	−	0.718185i
$286$		0			0
$287$		0			0
$288$		0			0
$289$	15.5000	+	26.8468i	0.911765	+	1.57922i
$290$		0			0
$291$	−6.00000	−	3.46410i	−0.351726	−	0.203069i
$292$		0			0
$293$			20.7846i			1.21425i	0.794606	+	0.607125i	$0.207677\pi$
$293$			20.7846i			1.21425i	−0.794606	+	0.607125i	$0.792323\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$	3.00000	+	1.73205i	0.174078	+	0.100504i
$298$		0			0
$299$		0			0
$300$		0			0
$301$		0			0
$302$		0			0
$303$	−3.00000	+	1.73205i	−0.172345	+	0.0995037i
$304$		0			0
$305$		0			0
$306$		0			0
$307$	25.0000			1.42683			0.713413	−	0.700744i	$-0.247149\pi$
$307$	25.0000			1.42683			0.713413	+	0.700744i	$0.247149\pi$
$308$		0			0
$309$	5.00000			0.284440
$310$		0			0
$311$	−9.00000	+	15.5885i	−0.510343	+	0.883940i	0.489585	+	0.871956i	$0.337148\pi$
$311$	−9.00000	+	15.5885i	−0.510343	+	0.883940i	−0.999928	+	0.0119847i	$0.996185\pi$
$312$		0			0
$313$	−13.5000	+	7.79423i	−0.763065	+	0.440556i	−0.830395	−	0.557175i	$-0.811885\pi$
$313$	−13.5000	+	7.79423i	−0.763065	+	0.440556i	0.0673300	+	0.997731i	$0.478552\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	6.00000	+	10.3923i	0.336994	+	0.583690i	0.983866	−	0.178908i	$-0.0572566\pi$
$317$	6.00000	+	10.3923i	0.336994	+	0.583690i	−0.646872	+	0.762598i	$0.723923\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$			6.92820i			0.386695i
$322$		0			0
$323$		−	48.4974i		−	2.69847i
$324$		0			0
$325$	−31.5000	−	18.1865i	−1.74731	−	1.00881i
$326$		0			0
$327$	−2.50000	−	4.33013i	−0.138250	−	0.239457i
$328$		0			0
$329$		0			0
$330$		0			0
$331$	−7.50000	+	4.33013i	−0.412237	+	0.238005i	−0.691751	−	0.722137i	$-0.743160\pi$
$331$	−7.50000	+	4.33013i	−0.412237	+	0.238005i	0.279513	+	0.960142i	$0.409827\pi$
$332$		0			0
$333$	−0.500000	+	0.866025i	−0.0273998	+	0.0474579i
$334$		0			0
$335$	−6.00000			−0.327815
$336$		0			0
$337$	−19.0000			−1.03500			−0.517498	−	0.855684i	$-0.673136\pi$
$337$	−19.0000			−1.03500			−0.517498	+	0.855684i	$0.673136\pi$
$338$		0			0
$339$	−3.00000	+	5.19615i	−0.162938	+	0.282216i
$340$		0			0
$341$	−15.0000	+	8.66025i	−0.812296	+	0.468979i
$342$		0			0
$343$		0			0
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−30.0000	−	17.3205i	−1.61048	−	0.929814i	−0.989258	−	0.146183i	$-0.953301\pi$
$347$	−30.0000	−	17.3205i	−1.61048	−	0.929814i	−0.621227	−	0.783631i	$-0.713365\pi$
$348$		0			0
$349$			13.8564i			0.741716i	0.928689	+	0.370858i	$0.120936\pi$
$349$			13.8564i			0.741716i	−0.928689	+	0.370858i	$0.879064\pi$
$350$		0			0
$351$		−	5.19615i		−	0.277350i
$352$		0			0
$353$	−15.0000	−	8.66025i	−0.798369	−	0.460939i	0.0445312	−	0.999008i	$-0.485821\pi$
$353$	−15.0000	−	8.66025i	−0.798369	−	0.460939i	−0.842901	+	0.538069i	$0.819154\pi$
$354$		0			0
$355$	6.00000	+	10.3923i	0.318447	+	0.551566i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−6.00000	+	3.46410i	−0.316668	+	0.182828i	−0.649906	−	0.760014i	$-0.725192\pi$
$359$	−6.00000	+	3.46410i	−0.316668	+	0.182828i	0.333238	+	0.942843i	$0.391859\pi$
$360$		0			0
$361$	−15.0000	+	25.9808i	−0.789474	+	1.36741i
$362$		0			0
$363$	−1.00000			−0.0524864
$364$		0			0
$365$	−30.0000			−1.57027
$366$		0			0
$367$	−18.5000	+	32.0429i	−0.965692	+	1.67263i	−0.257948	+	0.966159i	$0.583046\pi$
$367$	−18.5000	+	32.0429i	−0.965692	+	1.67263i	−0.707744	+	0.706469i	$0.750287\pi$
$368$		0			0
$369$	−9.00000	+	5.19615i	−0.468521	+	0.270501i
$370$		0			0
$371$		0			0
$372$		0			0
$373$	−5.50000	−	9.52628i	−0.284779	−	0.493252i	0.687776	−	0.725923i	$-0.258587\pi$
$373$	−5.50000	−	9.52628i	−0.284779	−	0.493252i	−0.972556	+	0.232671i	$0.925254\pi$
$374$		0			0
$375$	6.00000	+	3.46410i	0.309839	+	0.178885i
$376$		0			0
$377$		0			0
$378$		0			0
$379$			12.1244i			0.622786i	0.950281	+	0.311393i	$0.100796\pi$
$379$			12.1244i			0.622786i	−0.950281	+	0.311393i	$0.899204\pi$
$380$		0			0
$381$	−13.5000	−	7.79423i	−0.691626	−	0.399310i
$382$		0			0
$383$	−18.0000	−	31.1769i	−0.919757	−	1.59307i	−0.799783	−	0.600289i	$-0.795052\pi$
$383$	−18.0000	−	31.1769i	−0.919757	−	1.59307i	−0.119974	−	0.992777i	$-0.538281\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	−1.50000	+	0.866025i	−0.0762493	+	0.0440225i
$388$		0			0
$389$	−15.0000	+	25.9808i	−0.760530	+	1.31728i	0.182047	+	0.983290i	$0.441728\pi$
$389$	−15.0000	+	25.9808i	−0.760530	+	1.31728i	−0.942578	+	0.333987i	$0.891606\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$	18.0000			0.907980
$394$		0			0
$395$	27.0000	−	46.7654i	1.35852	−	2.35302i
$396$		0			0
$397$	−4.50000	+	2.59808i	−0.225849	+	0.130394i	−0.608655	−	0.793435i	$-0.708291\pi$
$397$	−4.50000	+	2.59808i	−0.225849	+	0.130394i	0.382807	+	0.923828i	$0.374957\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−12.0000	−	20.7846i	−0.599251	−	1.03793i	−0.992932	−	0.118686i	$-0.962132\pi$
$401$	−12.0000	−	20.7846i	−0.599251	−	1.03793i	0.393680	−	0.919247i	$-0.371202\pi$
$402$		0			0
$403$	22.5000	+	12.9904i	1.12080	+	0.647097i
$404$		0			0
$405$			3.46410i			0.172133i
$406$		0			0
$407$		−	3.46410i		−	0.171709i
$408$		0			0
$409$	−16.5000	−	9.52628i	−0.815872	−	0.471044i	0.0331186	−	0.999451i	$-0.489456\pi$
$409$	−16.5000	−	9.52628i	−0.815872	−	0.471044i	−0.848991	+	0.528407i	$0.822789\pi$
$410$		0			0
$411$	6.00000	+	10.3923i	0.295958	+	0.512615i
$412$		0			0
$413$		0			0
$414$		0			0
$415$	−18.0000	+	10.3923i	−0.883585	+	0.510138i
$416$		0			0
$417$	−2.50000	+	4.33013i	−0.122426	+	0.212047i
$418$		0			0
$419$	−30.0000			−1.46560			−0.732798	−	0.680446i	$-0.761786\pi$
$419$	−30.0000			−1.46560			−0.732798	+	0.680446i	$0.761786\pi$
$420$		0			0
$421$	−29.0000			−1.41337			−0.706687	−	0.707527i	$-0.749811\pi$
$421$	−29.0000			−1.41337			−0.706687	+	0.707527i	$0.749811\pi$
$422$		0			0
$423$	3.00000	−	5.19615i	0.145865	−	0.252646i
$424$		0			0
$425$	42.0000	−	24.2487i	2.03730	−	1.17624i
$426$		0			0
$427$		0			0
$428$		0			0
$429$	9.00000	+	15.5885i	0.434524	+	0.752618i
$430$		0			0
$431$	−15.0000	−	8.66025i	−0.722525	−	0.417150i	0.0931566	−	0.995651i	$-0.470304\pi$
$431$	−15.0000	−	8.66025i	−0.722525	−	0.417150i	−0.815681	+	0.578502i	$0.803638\pi$
$432$		0			0
$433$			29.4449i			1.41503i	0.706698	+	0.707515i	$0.250184\pi$
$433$			29.4449i			1.41503i	−0.706698	+	0.707515i	$0.749816\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		0			0
$438$		0			0
$439$	−8.00000	−	13.8564i	−0.381819	−	0.661330i	0.609503	−	0.792784i	$-0.291369\pi$
$439$	−8.00000	−	13.8564i	−0.381819	−	0.661330i	−0.991322	+	0.131453i	$0.958036\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−6.00000	+	3.46410i	−0.285069	+	0.164584i	−0.635716	−	0.771923i	$-0.719295\pi$
$443$	−6.00000	+	3.46410i	−0.285069	+	0.164584i	0.350647	+	0.936508i	$0.385962\pi$
$444$		0			0
$445$	−12.0000	+	20.7846i	−0.568855	+	0.985285i
$446$		0			0
$447$	−12.0000			−0.567581
$448$		0			0
$449$	−6.00000			−0.283158			−0.141579	−	0.989927i	$-0.545218\pi$
$449$	−6.00000			−0.283158			−0.141579	+	0.989927i	$0.545218\pi$
$450$		0			0
$451$	18.0000	−	31.1769i	0.847587	−	1.46806i
$452$		0			0
$453$	−9.00000	+	5.19615i	−0.422857	+	0.244137i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−2.50000	−	4.33013i	−0.116945	−	0.202555i	0.801611	−	0.597847i	$-0.203977\pi$
$457$	−2.50000	−	4.33013i	−0.116945	−	0.202555i	−0.918556	+	0.395292i	$0.870643\pi$
$458$		0			0
$459$	6.00000	+	3.46410i	0.280056	+	0.161690i
$460$		0			0
$461$			13.8564i			0.645357i	0.946509	+	0.322679i	$0.104583\pi$
$461$			13.8564i			0.645357i	−0.946509	+	0.322679i	$0.895417\pi$
$462$		0			0
$463$		−	12.1244i		−	0.563467i	−0.959493	−	0.281733i	$-0.909091\pi$
$463$		−	12.1244i		−	0.563467i	0.959493	−	0.281733i	$-0.0909093\pi$
$464$		0			0
$465$	−15.0000	−	8.66025i	−0.695608	−	0.401610i
$466$		0			0
$467$	3.00000	+	5.19615i	0.138823	+	0.240449i	0.927052	−	0.374934i	$-0.122335\pi$
$467$	3.00000	+	5.19615i	0.138823	+	0.240449i	−0.788228	+	0.615383i	$0.789001\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$	−12.0000	+	6.92820i	−0.552931	+	0.319235i
$472$		0			0
$473$	3.00000	−	5.19615i	0.137940	−	0.238919i
$474$		0			0
$475$	−49.0000			−2.24827
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−12.0000	+	20.7846i	−0.548294	+	0.949673i	0.450098	+	0.892979i	$0.351389\pi$
$479$	−12.0000	+	20.7846i	−0.548294	+	0.949673i	−0.998392	+	0.0566937i	$0.981944\pi$
$480$		0			0
$481$	−4.50000	+	2.59808i	−0.205182	+	0.118462i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	12.0000	+	20.7846i	0.544892	+	0.943781i
$486$		0			0
$487$	−10.5000	−	6.06218i	−0.475800	−	0.274703i	0.242864	−	0.970060i	$-0.421913\pi$
$487$	−10.5000	−	6.06218i	−0.475800	−	0.274703i	−0.718665	+	0.695357i	$0.755246\pi$
$488$		0			0
$489$			3.46410i			0.156652i
$490$		0			0
$491$			20.7846i			0.937996i	0.883199	+	0.468998i	$0.155385\pi$
$491$			20.7846i			0.937996i	−0.883199	+	0.468998i	$0.844615\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$	−6.00000	−	10.3923i	−0.269680	−	0.467099i
$496$		0			0
$497$		0			0
$498$		0			0
$499$	10.5000	−	6.06218i	0.470045	−	0.271380i	−0.246214	−	0.969216i	$-0.579187\pi$
$499$	10.5000	−	6.06218i	0.470045	−	0.271380i	0.716258	+	0.697835i	$0.245853\pi$
$500$		0			0
$501$	−3.00000	+	5.19615i	−0.134030	+	0.232147i
$502$		0			0
$503$	−30.0000			−1.33763			−0.668817	−	0.743427i	$-0.733199\pi$
$503$	−30.0000			−1.33763			−0.668817	+	0.743427i	$0.733199\pi$
$504$		0			0
$505$	12.0000			0.533993
$506$		0			0
$507$	7.00000	−	12.1244i	0.310881	−	0.538462i
$508$		0			0
$509$	3.00000	−	1.73205i	0.132973	−	0.0767718i	−0.432038	−	0.901855i	$-0.642205\pi$
$509$	3.00000	−	1.73205i	0.132973	−	0.0767718i	0.565011	+	0.825084i	$0.308872\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$	−3.50000	−	6.06218i	−0.154529	−	0.267652i
$514$		0			0
$515$	−15.0000	−	8.66025i	−0.660979	−	0.381616i
$516$		0			0
$517$			20.7846i			0.914106i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	12.0000	+	6.92820i	0.525730	+	0.303530i	0.739276	−	0.673403i	$-0.235168\pi$
$521$	12.0000	+	6.92820i	0.525730	+	0.303530i	−0.213546	+	0.976933i	$0.568501\pi$
$522$		0			0
$523$	0.500000	+	0.866025i	0.0218635	+	0.0378686i	0.876750	−	0.480946i	$-0.159707\pi$
$523$	0.500000	+	0.866025i	0.0218635	+	0.0378686i	−0.854887	+	0.518815i	$0.826373\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−30.0000	+	17.3205i	−1.30682	+	0.754493i
$528$		0			0
$529$	−11.5000	+	19.9186i	−0.500000	+	0.866025i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−54.0000			−2.33900
$534$		0			0
$535$	12.0000	−	20.7846i	0.518805	−	0.898597i
$536$		0			0
$537$	−15.0000	+	8.66025i	−0.647298	+	0.373718i
$538$		0			0
$539$		0			0
$540$		0			0
$541$	20.5000	+	35.5070i	0.881364	+	1.52657i	0.849825	+	0.527064i	$0.176707\pi$
$541$	20.5000	+	35.5070i	0.881364	+	1.52657i	0.0315385	+	0.999503i	$0.489959\pi$
$542$		0			0
$543$	13.5000	+	7.79423i	0.579340	+	0.334482i
$544$		0			0
$545$			17.3205i			0.741929i
$546$		0			0
$547$		−	10.3923i		−	0.444343i	−0.975008	−	0.222171i	$-0.928686\pi$
$547$		−	10.3923i		−	0.444343i	0.975008	−	0.222171i	$-0.0713145\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		0			0
$552$		0			0
$553$		0			0
$554$		0			0
$555$	3.00000	−	1.73205i	0.127343	−	0.0735215i
$556$		0			0
$557$	−9.00000	+	15.5885i	−0.381342	+	0.660504i	−0.991254	−	0.131965i	$-0.957871\pi$
$557$	−9.00000	+	15.5885i	−0.381342	+	0.660504i	0.609912	+	0.792469i	$0.291205\pi$
$558$		0			0
$559$	−9.00000			−0.380659
$560$		0			0
$561$	−24.0000			−1.01328
$562$		0			0
$563$	−9.00000	+	15.5885i	−0.379305	+	0.656975i	−0.990961	−	0.134148i	$-0.957170\pi$
$563$	−9.00000	+	15.5885i	−0.379305	+	0.656975i	0.611656	+	0.791123i	$0.290503\pi$
$564$		0			0
$565$	18.0000	−	10.3923i	0.757266	−	0.437208i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	3.00000	+	5.19615i	0.125767	+	0.217834i	0.922032	−	0.387113i	$-0.126528\pi$
$569$	3.00000	+	5.19615i	0.125767	+	0.217834i	−0.796266	+	0.604947i	$0.793194\pi$
$570$		0			0
$571$	22.5000	+	12.9904i	0.941596	+	0.543631i	0.890460	−	0.455061i	$-0.150383\pi$
$571$	22.5000	+	12.9904i	0.941596	+	0.543631i	0.0511355	+	0.998692i	$0.483716\pi$
$572$		0			0
$573$			3.46410i			0.144715i
$574$		0			0
$575$		0			0
$576$		0			0
$577$	1.50000	+	0.866025i	0.0624458	+	0.0360531i	0.530898	−	0.847436i	$-0.321855\pi$
$577$	1.50000	+	0.866025i	0.0624458	+	0.0360531i	−0.468452	+	0.883489i	$0.655188\pi$
$578$		0			0
$579$	−6.50000	−	11.2583i	−0.270131	−	0.467880i
$580$		0			0
$581$		0			0
$582$		0			0
$583$		0			0
$584$		0			0
$585$	−9.00000	+	15.5885i	−0.372104	+	0.644503i
$586$		0			0
$587$	12.0000			0.495293			0.247647	−	0.968850i	$-0.420343\pi$
$587$	12.0000			0.495293			0.247647	+	0.968850i	$0.420343\pi$
$588$		0			0
$589$	35.0000			1.44215
$590$		0			0
$591$	−6.00000	+	10.3923i	−0.246807	+	0.427482i
$592$		0			0
$593$	33.0000	−	19.0526i	1.35515	−	0.782395i	0.366182	−	0.930543i	$-0.380665\pi$
$593$	33.0000	−	19.0526i	1.35515	−	0.782395i	0.988965	+	0.148148i	$0.0473313\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	−8.00000	−	13.8564i	−0.327418	−	0.567105i
$598$		0			0
$599$	−6.00000	−	3.46410i	−0.245153	−	0.141539i	0.372390	−	0.928076i	$-0.378539\pi$
$599$	−6.00000	−	3.46410i	−0.245153	−	0.141539i	−0.617543	+	0.786537i	$0.711872\pi$
$600$		0			0
$601$			25.9808i			1.05978i	0.848067	+	0.529889i	$0.177766\pi$
$601$			25.9808i			1.05978i	−0.848067	+	0.529889i	$0.822234\pi$
$602$		0			0
$603$			1.73205i			0.0705346i
$604$		0			0
$605$	3.00000	+	1.73205i	0.121967	+	0.0704179i
$606$		0			0
$607$	−18.5000	−	32.0429i	−0.750892	−	1.30058i	−0.947391	−	0.320079i	$-0.896291\pi$
$607$	−18.5000	−	32.0429i	−0.750892	−	1.30058i	0.196499	−	0.980504i	$-0.437043\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	27.0000	−	15.5885i	1.09230	−	0.630641i
$612$		0			0
$613$	11.0000	−	19.0526i	0.444286	−	0.769526i	−0.553716	−	0.832705i	$-0.686791\pi$
$613$	11.0000	−	19.0526i	0.444286	−	0.769526i	0.998002	+	0.0631797i	$0.0201241\pi$
$614$		0			0
$615$	36.0000			1.45166
$616$		0			0
$617$	−42.0000			−1.69086			−0.845428	−	0.534089i	$-0.820655\pi$
$617$	−42.0000			−1.69086			−0.845428	+	0.534089i	$0.820655\pi$
$618$		0			0
$619$	−17.5000	+	30.3109i	−0.703384	+	1.21830i	0.263887	+	0.964554i	$0.414995\pi$
$619$	−17.5000	+	30.3109i	−0.703384	+	1.21830i	−0.967271	+	0.253744i	$0.918338\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	5.50000	+	9.52628i	0.220000	+	0.381051i
$626$		0			0
$627$	21.0000	+	12.1244i	0.838659	+	0.484200i
$628$		0			0
$629$		−	6.92820i		−	0.276246i
$630$		0			0
$631$		−	3.46410i		−	0.137904i	−0.997620	−	0.0689519i	$-0.978035\pi$
$631$		−	3.46410i		−	0.137904i	0.997620	−	0.0689519i	$-0.0219655\pi$
$632$		0			0
$633$	15.0000	+	8.66025i	0.596196	+	0.344214i
$634$		0			0
$635$	27.0000	+	46.7654i	1.07146	+	1.85583i
$636$		0			0
$637$		0			0
$638$		0			0
$639$	3.00000	−	1.73205i	0.118678	−	0.0685189i
$640$		0			0
$641$	18.0000	−	31.1769i	0.710957	−	1.23141i	−0.253541	−	0.967325i	$-0.581595\pi$
$641$	18.0000	−	31.1769i	0.710957	−	1.23141i	0.964498	−	0.264089i	$-0.0850714\pi$
$642$		0			0
$643$	43.0000			1.69575			0.847877	−	0.530193i	$-0.177880\pi$
$643$	43.0000			1.69575			0.847877	+	0.530193i	$0.177880\pi$
$644$		0			0
$645$	6.00000			0.236250
$646$		0			0
$647$	9.00000	−	15.5885i	0.353827	−	0.612845i	−0.633090	−	0.774078i	$-0.718214\pi$
$647$	9.00000	−	15.5885i	0.353827	−	0.612845i	0.986916	+	0.161233i	$0.0515470\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$		0			0
$652$		0			0
$653$	9.00000	+	15.5885i	0.352197	+	0.610023i	0.986634	−	0.162951i	$-0.0521013\pi$
$653$	9.00000	+	15.5885i	0.352197	+	0.610023i	−0.634437	+	0.772975i	$0.718768\pi$
$654$		0			0
$655$	−54.0000	−	31.1769i	−2.10995	−	1.21818i
$656$		0			0
$657$			8.66025i			0.337869i
$658$		0			0
$659$		−	20.7846i		−	0.809653i	−0.914393	−	0.404827i	$-0.867332\pi$
$659$		−	20.7846i		−	0.809653i	0.914393	−	0.404827i	$-0.132668\pi$
$660$		0			0
$661$	−16.5000	−	9.52628i	−0.641776	−	0.370529i	0.143523	−	0.989647i	$-0.454157\pi$
$661$	−16.5000	−	9.52628i	−0.641776	−	0.370529i	−0.785298	+	0.619118i	$0.787490\pi$
$662$		0			0
$663$	18.0000	+	31.1769i	0.699062	+	1.21081i
$664$		0			0
$665$		0			0
$666$		0			0
$667$		0			0
$668$		0			0
$669$	4.00000	−	6.92820i	0.154649	−	0.267860i
$670$		0			0
$671$		0			0
$672$		0			0
$673$	1.00000			0.0385472			0.0192736	−	0.999814i	$-0.493865\pi$
$673$	1.00000			0.0385472			0.0192736	+	0.999814i	$0.493865\pi$
$674$		0			0
$675$	3.50000	−	6.06218i	0.134715	−	0.233333i
$676$		0			0
$677$	−30.0000	+	17.3205i	−1.15299	+	0.665681i	−0.949615	−	0.313419i	$-0.898526\pi$
$677$	−30.0000	+	17.3205i	−1.15299	+	0.665681i	−0.203379	+	0.979100i	$0.565192\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$	15.0000	+	25.9808i	0.574801	+	0.995585i
$682$		0			0
$683$	36.0000	+	20.7846i	1.37750	+	0.795301i	0.991858	−	0.127347i	$-0.0406461\pi$
$683$	36.0000	+	20.7846i	1.37750	+	0.795301i	0.385643	+	0.922648i	$0.373979\pi$
$684$		0			0
$685$		−	41.5692i		−	1.58828i
$686$		0			0
$687$		−	1.73205i		−	0.0660819i
$688$		0			0
$689$		0			0
$690$		0			0
$691$	6.50000	+	11.2583i	0.247272	+	0.428287i	0.962768	−	0.270330i	$-0.0871327\pi$
$691$	6.50000	+	11.2583i	0.247272	+	0.428287i	−0.715496	+	0.698617i	$0.753799\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	15.0000	−	8.66025i	0.568982	−	0.328502i
$696$		0			0
$697$	36.0000	−	62.3538i	1.36360	−	2.36182i
$698$		0			0
$699$	−18.0000			−0.680823
$700$		0			0
$701$		0			0			−	1.00000i	$-0.5\pi$
$701$		0			0				1.00000i	$0.5\pi$
$702$		0			0
$703$	−3.50000	+	6.06218i	−0.132005	+	0.228639i
$704$		0			0
$705$	−18.0000	+	10.3923i	−0.677919	+	0.391397i
$706$		0			0
$707$		0			0
$708$		0			0
$709$	5.00000	+	8.66025i	0.187779	+	0.325243i	0.944509	−	0.328484i	$-0.106538\pi$
$709$	5.00000	+	8.66025i	0.187779	+	0.325243i	−0.756730	+	0.653727i	$0.773204\pi$
$710$		0			0
$711$	−13.5000	−	7.79423i	−0.506290	−	0.292306i
$712$		0			0
$713$		0			0
$714$		0			0
$715$		−	62.3538i		−	2.33190i
$716$		0			0
$717$	−15.0000	−	8.66025i	−0.560185	−	0.323423i
$718$		0			0
$719$	9.00000	+	15.5885i	0.335643	+	0.581351i	0.983608	−	0.180319i	$-0.0577130\pi$
$719$	9.00000	+	15.5885i	0.335643	+	0.581351i	−0.647965	+	0.761670i	$0.724380\pi$
$720$		0			0
$721$		0			0
$722$		0			0
$723$	−12.0000	+	6.92820i	−0.446285	+	0.257663i
$724$		0			0
$725$		0			0
$726$		0			0
$727$	−1.00000			−0.0370879			−0.0185440	−	0.999828i	$-0.505903\pi$
$727$	−1.00000			−0.0370879			−0.0185440	+	0.999828i	$0.505903\pi$
$728$		0			0
$729$	1.00000			0.0370370
$730$		0			0
$731$	6.00000	−	10.3923i	0.221918	−	0.384373i
$732$		0			0
$733$	31.5000	−	18.1865i	1.16348	−	0.671735i	0.211344	−	0.977412i	$-0.432216\pi$
$733$	31.5000	−	18.1865i	1.16348	−	0.671735i	0.952135	+	0.305677i	$0.0988827\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−3.00000	−	5.19615i	−0.110506	−	0.191403i
$738$		0			0
$739$	−31.5000	−	18.1865i	−1.15875	−	0.669002i	−0.207743	−	0.978183i	$-0.566612\pi$
$739$	−31.5000	−	18.1865i	−1.15875	−	0.669002i	−0.951003	+	0.309181i	$0.899945\pi$
$740$		0			0
$741$		−	36.3731i		−	1.33620i
$742$		0			0
$743$		−	3.46410i		−	0.127086i	−0.997979	−	0.0635428i	$-0.979760\pi$
$743$		−	3.46410i		−	0.127086i	0.997979	−	0.0635428i	$-0.0202399\pi$
$744$		0			0
$745$	36.0000	+	20.7846i	1.31894	+	0.761489i
$746$		0			0
$747$	3.00000	+	5.19615i	0.109764	+	0.190117i
$748$		0			0
$749$		0			0
$750$		0			0
$751$	40.5000	−	23.3827i	1.47787	−	0.853246i	0.478179	−	0.878262i	$-0.341297\pi$
$751$	40.5000	−	23.3827i	1.47787	−	0.853246i	0.999687	+	0.0250161i	$0.00796370\pi$
$752$		0			0
$753$	12.0000	−	20.7846i	0.437304	−	0.757433i
$754$		0			0
$755$	36.0000			1.31017
$756$		0			0
$757$	−22.0000			−0.799604			−0.399802	−	0.916602i	$-0.630921\pi$
$757$	−22.0000			−0.799604			−0.399802	+	0.916602i	$0.630921\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−30.0000	+	17.3205i	−1.08750	+	0.627868i	−0.932910	−	0.360111i	$-0.882739\pi$
$761$	−30.0000	+	17.3205i	−1.08750	+	0.627868i	−0.154590	+	0.987979i	$0.549406\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$	−12.0000	−	20.7846i	−0.433861	−	0.751469i
$766$		0			0
$767$		0			0
$768$		0			0
$769$			29.4449i			1.06181i	0.847432	+	0.530904i	$0.178148\pi$
$769$			29.4449i			1.06181i	−0.847432	+	0.530904i	$0.821852\pi$
$770$		0			0
$771$			10.3923i			0.374270i
$772$		0			0
$773$	27.0000	+	15.5885i	0.971123	+	0.560678i	0.899578	−	0.436760i	$-0.143874\pi$
$773$	27.0000	+	15.5885i	0.971123	+	0.560678i	0.0715442	+	0.997437i	$0.477207\pi$
$774$		0			0
$775$	17.5000	+	30.3109i	0.628619	+	1.08880i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−63.0000	+	36.3731i	−2.25721	+	1.30320i
$780$		0			0
$781$	−6.00000	+	10.3923i	−0.214697	+	0.371866i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	48.0000			1.71319
$786$		0			0
$787$	−4.00000	+	6.92820i	−0.142585	+	0.246964i	−0.928469	−	0.371409i	$-0.878875\pi$
$787$	−4.00000	+	6.92820i	−0.142585	+	0.246964i	0.785885	+	0.618373i	$0.212208\pi$
$788$		0			0
$789$	24.0000	−	13.8564i	0.854423	−	0.493301i
$790$		0			0
$791$		0			0
$792$		0			0
$793$		0			0
$794$		0			0
$795$		0			0
$796$		0			0
$797$		−	41.5692i		−	1.47246i	−0.676733	−	0.736229i	$-0.736605\pi$
$797$		−	41.5692i		−	1.47246i	0.676733	−	0.736229i	$-0.263395\pi$
$798$		0			0
$799$			41.5692i			1.47061i
$800$		0			0
$801$	6.00000	+	3.46410i	0.212000	+	0.122398i
$802$		0			0
$803$	−15.0000	−	25.9808i	−0.529339	−	0.916841i
$804$		0			0
$805$		0			0
$806$		0			0
$807$	−27.0000	+	15.5885i	−0.950445	+	0.548740i
$808$		0			0
$809$	−27.0000	+	46.7654i	−0.949269	+	1.64418i	−0.202301	+	0.979323i	$0.564842\pi$
$809$	−27.0000	+	46.7654i	−0.949269	+	1.64418i	−0.746968	+	0.664860i	$0.768491\pi$
$810$		0			0
$811$	−16.0000			−0.561836			−0.280918	−	0.959732i	$-0.590639\pi$
$811$	−16.0000			−0.561836			−0.280918	+	0.959732i	$0.590639\pi$
$812$		0			0
$813$	−16.0000			−0.561144
$814$		0			0
$815$	6.00000	−	10.3923i	0.210171	−	0.364027i
$816$		0			0
$817$	−10.5000	+	6.06218i	−0.367348	+	0.212089i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	3.00000	+	5.19615i	0.104701	+	0.181347i	0.913616	−	0.406578i	$-0.133278\pi$
$821$	3.00000	+	5.19615i	0.104701	+	0.181347i	−0.808915	+	0.587925i	$0.799945\pi$
$822$		0			0
$823$	15.0000	+	8.66025i	0.522867	+	0.301877i	0.738107	−	0.674684i	$-0.235720\pi$
$823$	15.0000	+	8.66025i	0.522867	+	0.301877i	−0.215240	+	0.976561i	$0.569053\pi$
$824$		0			0
$825$			24.2487i			0.844232i
$826$		0			0
$827$		−	17.3205i		−	0.602293i	−0.953578	−	0.301147i	$-0.902631\pi$
$827$		−	17.3205i		−	0.602293i	0.953578	−	0.301147i	$-0.0973693\pi$
$828$		0			0
$829$	19.5000	+	11.2583i	0.677263	+	0.391018i	0.798823	−	0.601566i	$-0.205456\pi$
$829$	19.5000	+	11.2583i	0.677263	+	0.391018i	−0.121560	+	0.992584i	$0.538790\pi$
$830$		0			0
$831$	8.50000	+	14.7224i	0.294862	+	0.510716i
$832$		0			0
$833$		0			0
$834$		0			0
$835$	18.0000	−	10.3923i	0.622916	−	0.359641i
$836$		0			0
$837$	−2.50000	+	4.33013i	−0.0864126	+	0.149671i
$838$		0			0
$839$	−24.0000			−0.828572			−0.414286	−	0.910147i	$-0.635969\pi$
$839$	−24.0000			−0.828572			−0.414286	+	0.910147i	$0.635969\pi$
$840$		0			0
$841$	−29.0000			−1.00000
$842$		0			0
$843$	−6.00000	+	10.3923i	−0.206651	+	0.357930i
$844$		0			0
$845$	−42.0000	+	24.2487i	−1.44484	+	0.834181i
$846$		0			0
$847$		0			0
$848$		0			0
$849$	2.50000	+	4.33013i	0.0857998	+	0.148610i
$850$		0			0
$851$		0			0
$852$		0			0
$853$		−	1.73205i		−	0.0593043i	−0.999560	−	0.0296521i	$-0.990560\pi$
$853$		−	1.73205i		−	0.0593043i	0.999560	−	0.0296521i	$-0.00943995\pi$
$854$		0			0
$855$			24.2487i			0.829288i
$856$		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
$859$	−20.0000	−	34.6410i	−0.682391	−	1.18194i	−0.974249	−	0.225475i	$-0.927607\pi$
$859$	−20.0000	−	34.6410i	−0.682391	−	1.18194i	0.291858	−	0.956462i	$-0.405727\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	27.0000	−	15.5885i	0.919091	−	0.530637i	0.0357458	−	0.999361i	$-0.488619\pi$
$863$	27.0000	−	15.5885i	0.919091	−	0.530637i	0.883345	+	0.468724i	$0.155286\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	−31.0000			−1.05282
$868$		0			0
$869$	54.0000			1.83182
$870$		0			0
$871$	−4.50000	+	7.79423i	−0.152477	+	0.264097i
$872$		0			0
$873$	6.00000	−	3.46410i	0.203069	−	0.117242i
$874$		0			0
$875$		0			0
$876$		0			0
$877$	11.0000	+	19.0526i	0.371444	+	0.643359i	0.989788	−	0.142548i	$-0.0455296\pi$
$877$	11.0000	+	19.0526i	0.371444	+	0.643359i	−0.618344	+	0.785907i	$0.712196\pi$
$878$		0			0
$879$	−18.0000	−	10.3923i	−0.607125	−	0.350524i
$880$		0			0
$881$			27.7128i			0.933668i	0.884345	+	0.466834i	$0.154606\pi$
$881$			27.7128i			0.933668i	−0.884345	+	0.466834i	$0.845394\pi$
$882$		0			0
$883$			12.1244i			0.408017i	0.978969	+	0.204009i	$0.0653970\pi$
$883$			12.1244i			0.408017i	−0.978969	+	0.204009i	$0.934603\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	15.0000	+	25.9808i	0.503651	+	0.872349i	0.999991	+	0.00422062i	$0.00134347\pi$
$887$	15.0000	+	25.9808i	0.503651	+	0.872349i	−0.496340	+	0.868128i	$0.665323\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$	−3.00000	+	1.73205i	−0.100504	+	0.0580259i
$892$		0			0
$893$	21.0000	−	36.3731i	0.702738	−	1.21718i
$894$		0			0
$895$	60.0000			2.00558
$896$		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
$905$	−27.0000	−	46.7654i	−0.897510	−	1.55453i
$906$		0			0
$907$	13.5000	+	7.79423i	0.448260	+	0.258803i	0.707095	−	0.707118i	$-0.250005\pi$
$907$	13.5000	+	7.79423i	0.448260	+	0.258803i	−0.258835	+	0.965922i	$0.583339\pi$
$908$		0			0
$909$		−	3.46410i		−	0.114897i
$910$		0			0
$911$			20.7846i			0.688625i	0.938855	+	0.344312i	$0.111888\pi$
$911$			20.7846i			0.688625i	−0.938855	+	0.344312i	$0.888112\pi$
$912$		0			0
$913$	−18.0000	−	10.3923i	−0.595713	−	0.343935i
$914$		0			0
$915$		0			0
$916$		0			0
$917$		0			0
$918$		0			0
$919$	−1.50000	+	0.866025i	−0.0494804	+	0.0285675i	−0.524536	−	0.851388i	$-0.675761\pi$
$919$	−1.50000	+	0.866025i	−0.0494804	+	0.0285675i	0.475056	+	0.879956i	$0.342428\pi$
$920$		0			0
$921$	−12.5000	+	21.6506i	−0.411889	+	0.713413i
$922$		0			0
$923$	18.0000			0.592477
$924$		0			0
$925$	−7.00000			−0.230159
$926$		0			0
$927$	−2.50000	+	4.33013i	−0.0821108	+	0.142220i
$928$		0			0
$929$	−3.00000	+	1.73205i	−0.0984268	+	0.0568267i	−0.548405	−	0.836213i	$-0.684765\pi$
$929$	−3.00000	+	1.73205i	−0.0984268	+	0.0568267i	0.449979	+	0.893039i	$0.351432\pi$
$930$		0			0
$931$		0			0
$932$		0			0
$933$	−9.00000	−	15.5885i	−0.294647	−	0.510343i
$934$		0			0
$935$	72.0000	+	41.5692i	2.35465	+	1.35946i
$936$		0			0
$937$		−	29.4449i		−	0.961922i	−0.876742	−	0.480961i	$-0.840288\pi$
$937$		−	29.4449i		−	0.961922i	0.876742	−	0.480961i	$-0.159712\pi$
$938$		0			0
$939$		−	15.5885i		−	0.508710i
$940$		0			0
$941$	−42.0000	−	24.2487i	−1.36916	−	0.790485i	−0.378340	−	0.925667i	$-0.623505\pi$
$941$	−42.0000	−	24.2487i	−1.36916	−	0.790485i	−0.990821	+	0.135181i	$0.956838\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$	15.0000	−	8.66025i	0.487435	−	0.281420i	−0.236075	−	0.971735i	$-0.575861\pi$
$947$	15.0000	−	8.66025i	0.487435	−	0.281420i	0.723510	+	0.690314i	$0.242528\pi$
$948$		0			0
$949$	−22.5000	+	38.9711i	−0.730381	+	1.26506i
$950$		0			0
$951$	−12.0000			−0.389127
$952$		0			0
$953$	36.0000			1.16615			0.583077	−	0.812417i	$-0.301849\pi$
$953$	36.0000			1.16615			0.583077	+	0.812417i	$0.301849\pi$
$954$		0			0
$955$	6.00000	−	10.3923i	0.194155	−	0.336287i
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	3.00000	+	5.19615i	0.0967742	+	0.167618i
$962$		0			0
$963$	−6.00000	−	3.46410i	−0.193347	−	0.111629i
$964$		0			0
$965$			45.0333i			1.44967i
$966$		0			0
$967$			22.5167i			0.724087i	0.932161	+	0.362043i	$0.117921\pi$
$967$			22.5167i			0.724087i	−0.932161	+	0.362043i	$0.882079\pi$
$968$		0			0
$969$	42.0000	+	24.2487i	1.34923	+	0.778981i
$970$		0			0
$971$	−6.00000	−	10.3923i	−0.192549	−	0.333505i	0.753545	−	0.657396i	$-0.228342\pi$
$971$	−6.00000	−	10.3923i	−0.192549	−	0.333505i	−0.946094	+	0.323891i	$0.895009\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$	31.5000	−	18.1865i	1.00881	−	0.582435i
$976$		0			0
$977$	−27.0000	+	46.7654i	−0.863807	+	1.49616i	0.00442082	+	0.999990i	$0.498593\pi$
$977$	−27.0000	+	46.7654i	−0.863807	+	1.49616i	−0.868227	+	0.496167i	$0.834741\pi$
$978$		0			0
$979$	−24.0000			−0.767043
$980$		0			0
$981$	5.00000			0.159638
$982$		0			0
$983$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$983$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$984$		0			0
$985$	36.0000	−	20.7846i	1.14706	−	0.662253i
$986$		0			0
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$	−10.5000	−	6.06218i	−0.333543	−	0.192571i	0.323870	−	0.946102i	$-0.395016\pi$
$991$	−10.5000	−	6.06218i	−0.333543	−	0.192571i	−0.657413	+	0.753530i	$0.728349\pi$
$992$		0			0
$993$		−	8.66025i		−	0.274825i
$994$		0			0
$995$			55.4256i			1.75711i
$996$		0			0
$997$	−37.5000	−	21.6506i	−1.18764	−	0.685682i	−0.229868	−	0.973222i	$-0.573829\pi$
$997$	−37.5000	−	21.6506i	−1.18764	−	0.685682i	−0.957769	+	0.287539i	$0.907163\pi$
$998$		0			0
$999$	−0.500000	−	0.866025i	−0.0158193	−	0.0273998i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2352.2.bl.f.31.1		2
4.3	odd	2		2352.2.bl.l.31.1		2
7.2	even	3		336.2.bl.a.271.1	yes	2
7.3	odd	6		2352.2.b.a.1567.1		2
7.4	even	3		2352.2.b.h.1567.2		2
7.5	odd	6		2352.2.bl.l.607.1		2
7.6	odd	2		336.2.bl.e.31.1	yes	2
21.2	odd	6		1008.2.cs.n.271.1		2
21.11	odd	6		7056.2.b.l.1567.1		2
21.17	even	6		7056.2.b.a.1567.2		2
21.20	even	2		1008.2.cs.m.703.1		2
28.3	even	6		2352.2.b.h.1567.1		2
28.11	odd	6		2352.2.b.a.1567.2		2
28.19	even	6	inner	2352.2.bl.f.607.1		2
28.23	odd	6		336.2.bl.e.271.1	yes	2
28.27	even	2		336.2.bl.a.31.1	✓	2
56.13	odd	2		1344.2.bl.d.703.1		2
56.27	even	2		1344.2.bl.h.703.1		2
56.37	even	6		1344.2.bl.h.1279.1		2
56.51	odd	6		1344.2.bl.d.1279.1		2
84.11	even	6		7056.2.b.a.1567.1		2
84.23	even	6		1008.2.cs.m.271.1		2
84.59	odd	6		7056.2.b.l.1567.2		2
84.83	odd	2		1008.2.cs.n.703.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
336.2.bl.a.31.1	✓	2	28.27	even	2
336.2.bl.a.271.1	yes	2	7.2	even	3
336.2.bl.e.31.1	yes	2	7.6	odd	2
336.2.bl.e.271.1	yes	2	28.23	odd	6
1008.2.cs.m.271.1		2	84.23	even	6
1008.2.cs.m.703.1		2	21.20	even	2
1008.2.cs.n.271.1		2	21.2	odd	6
1008.2.cs.n.703.1		2	84.83	odd	2
1344.2.bl.d.703.1		2	56.13	odd	2
1344.2.bl.d.1279.1		2	56.51	odd	6
1344.2.bl.h.703.1		2	56.27	even	2
1344.2.bl.h.1279.1		2	56.37	even	6
2352.2.b.a.1567.1		2	7.3	odd	6
2352.2.b.a.1567.2		2	28.11	odd	6
2352.2.b.h.1567.1		2	28.3	even	6
2352.2.b.h.1567.2		2	7.4	even	3
2352.2.bl.f.31.1		2	1.1	even	1	trivial
2352.2.bl.f.607.1		2	28.19	even	6	inner
2352.2.bl.l.31.1		2	4.3	odd	2
2352.2.bl.l.607.1		2	7.5	odd	6
7056.2.b.a.1567.1		2	84.11	even	6
7056.2.b.a.1567.2		2	21.17	even	6
7056.2.b.l.1567.1		2	21.11	odd	6
7056.2.b.l.1567.2		2	84.59	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:	\( N \)	\(=\)	\( 2352 = 2^{4} \cdot 3 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2352.bl (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-0.500000 + 0.866025i) q^{3} +(3.00000 - 1.73205i) q^{5} +(-0.500000 - 0.866025i) q^{9} +O(q^{10})\)	\(q+(-0.500000 + 0.866025i) q^{3} +(3.00000 - 1.73205i) q^{5} +(-0.500000 - 0.866025i) q^{9} +(3.00000 + 1.73205i) q^{11} -5.19615i q^{13} +3.46410i q^{15} +(6.00000 + 3.46410i) q^{17} +(-3.50000 - 6.06218i) q^{19} +(3.50000 - 6.06218i) q^{25} +1.00000 q^{27} +(-2.50000 + 4.33013i) q^{31} +(-3.00000 + 1.73205i) q^{33} +(-0.500000 - 0.866025i) q^{37} +(4.50000 + 2.59808i) q^{39} -10.3923i q^{41} -1.73205i q^{43} +(-3.00000 - 1.73205i) q^{45} +(3.00000 + 5.19615i) q^{47} +(-6.00000 + 3.46410i) q^{51} +12.0000 q^{55} +7.00000 q^{57} +(-9.00000 - 15.5885i) q^{65} +(-1.50000 - 0.866025i) q^{67} +3.46410i q^{71} +(-7.50000 - 4.33013i) q^{73} +(3.50000 + 6.06218i) q^{75} +(13.5000 - 7.79423i) q^{79} +(-0.500000 + 0.866025i) q^{81} -6.00000 q^{83} +24.0000 q^{85} +(-6.00000 + 3.46410i) q^{89} +(-2.50000 - 4.33013i) q^{93} +(-21.0000 - 12.1244i) q^{95} +6.92820i q^{97} -3.46410i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - q^{3} + 6 q^{5} - q^{9}+O(q^{10}) \) 2 * q - q^3 + 6 * q^5 - q^9	\( 2 q - q^{3} + 6 q^{5} - q^{9} + 6 q^{11} + 12 q^{17} - 7 q^{19} + 7 q^{25} + 2 q^{27} - 5 q^{31} - 6 q^{33} - q^{37} + 9 q^{39} - 6 q^{45} + 6 q^{47} - 12 q^{51} + 24 q^{55} + 14 q^{57} - 18 q^{65} - 3 q^{67} - 15 q^{73} + 7 q^{75} + 27 q^{79} - q^{81} - 12 q^{83} + 48 q^{85} - 12 q^{89} - 5 q^{93} - 42 q^{95}+O(q^{100}) \) 2 * q - q^3 + 6 * q^5 - q^9 + 6 * q^11 + 12 * q^17 - 7 * q^19 + 7 * q^25 + 2 * q^27 - 5 * q^31 - 6 * q^33 - q^37 + 9 * q^39 - 6 * q^45 + 6 * q^47 - 12 * q^51 + 24 * q^55 + 14 * q^57 - 18 * q^65 - 3 * q^67 - 15 * q^73 + 7 * q^75 + 27 * q^79 - q^81 - 12 * q^83 + 48 * q^85 - 12 * q^89 - 5 * q^93 - 42 * q^95

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