LMFDB - Embedded newform 1008.2.cs.n.271.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1008,2,Mod(271,1008)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1008, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([3, 0, 0, 5]))

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

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

chi := DirichletCharacter("1008.271");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1008 = 2^{4} \cdot 3^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1008.cs (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:	$8.04892052375$
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:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 336)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			271.1
Root			$0.500000 - 0.866025i$ of defining polynomial
Character	$\chi$	$=$	1008.271
Dual form			1008.2.cs.n.703.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(3.00000 + 1.73205i) q^{5} +(0.500000 + 2.59808i) q^{7} +O(q^{10})$	$q+(3.00000 + 1.73205i) q^{5} +(0.500000 + 2.59808i) q^{7} +(3.00000 - 1.73205i) q^{11} -5.19615i q^{13} +(6.00000 - 3.46410i) q^{17} +(-3.50000 + 6.06218i) q^{19} +(3.50000 + 6.06218i) q^{25} +(-2.50000 - 4.33013i) q^{31} +(-3.00000 + 8.66025i) q^{35} +(-0.500000 + 0.866025i) q^{37} +10.3923i q^{41} -1.73205i q^{43} +(-3.00000 + 5.19615i) q^{47} +(-6.50000 + 2.59808i) q^{49} +12.0000 q^{55} +(9.00000 - 15.5885i) q^{65} +(1.50000 - 0.866025i) q^{67} -3.46410i q^{71} +(7.50000 - 4.33013i) q^{73} +(6.00000 + 6.92820i) q^{77} +(-13.5000 - 7.79423i) q^{79} +6.00000 q^{83} +24.0000 q^{85} +(-6.00000 - 3.46410i) q^{89} +(13.5000 - 2.59808i) q^{91} +(-21.0000 + 12.1244i) q^{95} +6.92820i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 6 q^{5} + q^{7}+O(q^{10}) $ 2 * q + 6 * q^5 + q^7	$ 2 q + 6 q^{5} + q^{7} + 6 q^{11} + 12 q^{17} - 7 q^{19} + 7 q^{25} - 5 q^{31} - 6 q^{35} - q^{37} - 6 q^{47} - 13 q^{49} + 24 q^{55} + 18 q^{65} + 3 q^{67} + 15 q^{73} + 12 q^{77} - 27 q^{79} + 12 q^{83} + 48 q^{85} - 12 q^{89} + 27 q^{91} - 42 q^{95}+O(q^{100}) $ 2 * q + 6 * q^5 + q^7 + 6 * q^11 + 12 * q^17 - 7 * q^19 + 7 * q^25 - 5 * q^31 - 6 * q^35 - q^37 - 6 * q^47 - 13 * q^49 + 24 * q^55 + 18 * q^65 + 3 * q^67 + 15 * q^73 + 12 * q^77 - 27 * q^79 + 12 * q^83 + 48 * q^85 - 12 * q^89 + 27 * q^91 - 42 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1008\mathbb{Z}\right)^\times$.

$n$	$127$	$577$	$757$	$785$
$\chi(n)$	$-1$	$e\left(\frac{5}{6}\right)$	$1$	$1$

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
$5$	3.00000	+	1.73205i	1.34164	+	0.774597i	0.987048	−	0.160424i	$-0.0512862\pi$
$5$	3.00000	+	1.73205i	1.34164	+	0.774597i	0.354593	+	0.935021i	$0.384620\pi$
$6$		0			0
$7$	0.500000	+	2.59808i	0.188982	+	0.981981i
$7$	0.500000	+	2.59808i	0.188982	+	0.981981i
$8$		0			0
$9$		0			0
$10$		0			0
$11$	3.00000	−	1.73205i	0.904534	−	0.522233i	0.0258656	−	0.999665i	$-0.491766\pi$
$11$	3.00000	−	1.73205i	0.904534	−	0.522233i	0.878668	+	0.477432i	$0.158432\pi$
$12$		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
$15$		0			0
$16$		0			0
$17$	6.00000	−	3.46410i	1.45521	−	0.840168i	0.456444	−	0.889752i	$-0.349123\pi$
$17$	6.00000	−	3.46410i	1.45521	−	0.840168i	0.998770	+	0.0495842i	$0.0157896\pi$
$18$		0			0
$19$	−3.50000	+	6.06218i	−0.802955	+	1.39076i	0.114708	+	0.993399i	$0.463407\pi$
$19$	−3.50000	+	6.06218i	−0.802955	+	1.39076i	−0.917663	+	0.397360i	$0.869927\pi$
$20$		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
$25$	3.50000	+	6.06218i	0.700000	+	1.21244i
$26$		0			0
$27$		0			0
$28$		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	0.549309	−	0.835619i	$-0.314891\pi$
$31$	−2.50000	−	4.33013i	−0.449013	−	0.777714i	−0.998322	+	0.0579057i	$0.981558\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−3.00000	+	8.66025i	−0.507093	+	1.46385i
$36$		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
$41$			10.3923i			1.62301i	0.584349	+	0.811503i	$0.301350\pi$
$41$			10.3923i			1.62301i	−0.584349	+	0.811503i	$0.698650\pi$
$42$		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
$45$		0			0
$46$		0			0
$47$	−3.00000	+	5.19615i	−0.437595	+	0.757937i	−0.997503	−	0.0706177i	$-0.977503\pi$
$47$	−3.00000	+	5.19615i	−0.437595	+	0.757937i	0.559908	+	0.828554i	$0.310836\pi$
$48$		0			0
$49$	−6.50000	+	2.59808i	−0.928571	+	0.371154i
$50$		0			0
$51$		0			0
$52$		0			0
$53$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$53$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$54$		0			0
$55$	12.0000			1.61808
$56$		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
$61$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$61$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$62$		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	$-0.632926\pi$
$67$	1.50000	−	0.866025i	0.183254	−	0.105802i	0.588821	+	0.808264i	$0.299592\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		−	3.46410i		−	0.411113i	−0.978645	−	0.205557i	$-0.934100\pi$
$71$		−	3.46410i		−	0.411113i	0.978645	−	0.205557i	$-0.0659005\pi$
$72$		0			0
$73$	7.50000	−	4.33013i	0.877809	−	0.506803i	0.00787336	−	0.999969i	$-0.497494\pi$
$73$	7.50000	−	4.33013i	0.877809	−	0.506803i	0.869935	+	0.493166i	$0.164160\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	6.00000	+	6.92820i	0.683763	+	0.789542i
$78$		0			0
$79$	−13.5000	−	7.79423i	−1.51887	−	0.876919i	−0.999753	−	0.0222151i	$-0.992928\pi$
$79$	−13.5000	−	7.79423i	−1.51887	−	0.876919i	−0.519115	−	0.854704i	$-0.673739\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	6.00000			0.658586			0.329293	−	0.944228i	$-0.393190\pi$
$83$	6.00000			0.658586			0.329293	+	0.944228i	$0.393190\pi$
$84$		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	0.147073	−	0.989126i	$-0.453015\pi$
$89$	−6.00000	−	3.46410i	−0.635999	−	0.367194i	−0.783072	+	0.621932i	$0.786348\pi$
$90$		0			0
$91$	13.5000	−	2.59808i	1.41518	−	0.272352i
$92$		0			0
$93$		0			0
$94$		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
$99$		0			0
$100$		0			0
$101$	3.00000	−	1.73205i	0.298511	−	0.172345i	−0.343263	−	0.939239i	$-0.611532\pi$
$101$	3.00000	−	1.73205i	0.298511	−	0.172345i	0.641774	+	0.766894i	$0.278199\pi$
$102$		0			0
$103$	−2.50000	+	4.33013i	−0.246332	+	0.426660i	−0.962505	−	0.271263i	$-0.912559\pi$
$103$	−2.50000	+	4.33013i	−0.246332	+	0.426660i	0.716173	+	0.697923i	$0.245892\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	6.00000	+	3.46410i	0.580042	+	0.334887i	0.761150	−	0.648576i	$-0.224635\pi$
$107$	6.00000	+	3.46410i	0.580042	+	0.334887i	−0.181108	+	0.983463i	$0.557968\pi$
$108$		0			0
$109$	−2.50000	−	4.33013i	−0.239457	−	0.414751i	0.721102	−	0.692829i	$-0.243636\pi$
$109$	−2.50000	−	4.33013i	−0.239457	−	0.414751i	−0.960558	+	0.278078i	$0.910303\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−6.00000			−0.564433			−0.282216	−	0.959351i	$-0.591070\pi$
$113$	−6.00000			−0.564433			−0.282216	+	0.959351i	$0.591070\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$		0			0
$118$		0			0
$119$	12.0000	+	13.8564i	1.10004	+	1.27021i
$120$		0			0
$121$	0.500000	−	0.866025i	0.0454545	−	0.0787296i
$122$		0			0
$123$		0			0
$124$		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
$129$		0			0
$130$		0			0
$131$	9.00000	−	15.5885i	0.786334	−	1.36197i	−0.141865	−	0.989886i	$-0.545310\pi$
$131$	9.00000	−	15.5885i	0.786334	−	1.36197i	0.928199	−	0.372084i	$-0.121357\pi$
$132$		0			0
$133$	−17.5000	−	6.06218i	−1.51744	−	0.525657i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−6.00000	−	10.3923i	−0.512615	−	0.887875i	−0.999893	−	0.0146279i	$-0.995344\pi$
$137$	−6.00000	−	10.3923i	−0.512615	−	0.887875i	0.487278	−	0.873247i	$-0.337990\pi$
$138$		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
$141$		0			0
$142$		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	$-0.996899\pi$
$149$	−6.00000	+	10.3923i	−0.491539	+	0.851371i	0.508413	+	0.861113i	$0.330232\pi$
$150$		0			0
$151$	−9.00000	+	5.19615i	−0.732410	+	0.422857i	−0.819303	−	0.573361i	$-0.805639\pi$
$151$	−9.00000	+	5.19615i	−0.732410	+	0.422857i	0.0868934	+	0.996218i	$0.472306\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		−	17.3205i		−	1.39122i
$156$		0			0
$157$	−12.0000	+	6.92820i	−0.957704	+	0.552931i	−0.895466	−	0.445130i	$-0.853157\pi$
$157$	−12.0000	+	6.92820i	−0.957704	+	0.552931i	−0.0622385	+	0.998061i	$0.519824\pi$
$158$		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	$-0.376650\pi$
$163$	−3.00000	−	1.73205i	−0.234978	−	0.135665i	−0.612866	+	0.790186i	$0.709984\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−6.00000			−0.464294			−0.232147	−	0.972681i	$-0.574575\pi$
$167$	−6.00000			−0.464294			−0.232147	+	0.972681i	$0.574575\pi$
$168$		0			0
$169$	−14.0000			−1.07692
$170$		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
$175$	−14.0000	+	12.1244i	−1.05830	+	0.916515i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	15.0000	−	8.66025i	1.12115	−	0.647298i	0.179458	−	0.983766i	$-0.442566\pi$
$179$	15.0000	−	8.66025i	1.12115	−	0.647298i	0.941695	+	0.336468i	$0.109232\pi$
$180$		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	0.604594	−	0.796534i	$-0.293335\pi$
$191$	3.00000	+	1.73205i	0.217072	+	0.125327i	−0.387522	+	0.921861i	$0.626669\pi$
$192$		0			0
$193$	−6.50000	−	11.2583i	−0.467880	−	0.810392i	0.531446	−	0.847092i	$-0.321649\pi$
$193$	−6.50000	−	11.2583i	−0.467880	−	0.810392i	−0.999326	+	0.0366998i	$0.988315\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−12.0000			−0.854965			−0.427482	−	0.904024i	$-0.640599\pi$
$197$	−12.0000			−0.854965			−0.427482	+	0.904024i	$0.640599\pi$
$198$		0			0
$199$	−8.00000	−	13.8564i	−0.567105	−	0.982255i	−0.996850	−	0.0793045i	$-0.974730\pi$
$199$	−8.00000	−	13.8564i	−0.567105	−	0.982255i	0.429745	−	0.902950i	$-0.358603\pi$
$200$		0			0
$201$		0			0
$202$		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
$213$		0			0
$214$		0			0
$215$	3.00000	−	5.19615i	0.204598	−	0.354375i
$216$		0			0
$217$	10.0000	−	8.66025i	0.678844	−	0.587896i
$218$		0			0
$219$		0			0
$220$		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
$225$		0			0
$226$		0			0
$227$	−15.0000	−	25.9808i	−0.995585	−	1.72440i	−0.579082	−	0.815270i	$-0.696589\pi$
$227$	−15.0000	−	25.9808i	−0.995585	−	1.72440i	−0.416503	−	0.909134i	$-0.636745\pi$
$228$		0			0
$229$	1.50000	+	0.866025i	0.0991228	+	0.0572286i	0.548742	−	0.835992i	$-0.315107\pi$
$229$	1.50000	+	0.866025i	0.0991228	+	0.0572286i	−0.449619	+	0.893220i	$0.648440\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−9.00000	+	15.5885i	−0.589610	+	1.02123i	0.404674	+	0.914461i	$0.367385\pi$
$233$	−9.00000	+	15.5885i	−0.589610	+	1.02123i	−0.994283	+	0.106773i	$0.965948\pi$
$234$		0			0
$235$	−18.0000	+	10.3923i	−1.17419	+	0.677919i
$236$		0			0
$237$		0			0
$238$		0			0
$239$		−	17.3205i		−	1.12037i	−0.828367	−	0.560185i	$-0.810730\pi$
$239$		−	17.3205i		−	1.12037i	0.828367	−	0.560185i	$-0.189270\pi$
$240$		0			0
$241$	−12.0000	+	6.92820i	−0.772988	+	0.446285i	−0.833939	−	0.551856i	$-0.813920\pi$
$241$	−12.0000	+	6.92820i	−0.772988	+	0.446285i	0.0609515	+	0.998141i	$0.480586\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−24.0000	−	3.46410i	−1.53330	−	0.221313i
$246$		0			0
$247$	31.5000	+	18.1865i	2.00430	+	1.15718i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	24.0000			1.51487			0.757433	−	0.652913i	$-0.226453\pi$
$251$	24.0000			1.51487			0.757433	+	0.652913i	$0.226453\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$		0			0
$256$		0			0
$257$	9.00000	+	5.19615i	0.561405	+	0.324127i	0.753709	−	0.657208i	$-0.228263\pi$
$257$	9.00000	+	5.19615i	0.561405	+	0.324127i	−0.192304	+	0.981335i	$0.561596\pi$
$258$		0			0
$259$	−2.50000	−	0.866025i	−0.155342	−	0.0538122i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−24.0000	+	13.8564i	−1.47990	+	0.854423i	−0.999741	−	0.0227570i	$-0.992756\pi$
$263$	−24.0000	+	13.8564i	−1.47990	+	0.854423i	−0.480162	+	0.877180i	$0.659422\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$		0			0
$269$	27.0000	−	15.5885i	1.64622	−	0.950445i	0.667663	−	0.744463i	$-0.267295\pi$
$269$	27.0000	−	15.5885i	1.64622	−	0.950445i	0.978556	−	0.205982i	$-0.0660387\pi$
$270$		0			0
$271$	8.00000	−	13.8564i	0.485965	−	0.841717i	−0.513905	−	0.857847i	$-0.671801\pi$
$271$	8.00000	−	13.8564i	0.485965	−	0.841717i	0.999870	+	0.0161307i	$0.00513477\pi$
$272$		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	0.999923	+	0.0124177i	$0.00395278\pi$
$277$	8.50000	+	14.7224i	0.510716	+	0.884585i	−0.489207	+	0.872167i	$0.662714\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−12.0000			−0.715860			−0.357930	−	0.933748i	$-0.616517\pi$
$281$	−12.0000			−0.715860			−0.357930	+	0.933748i	$0.616517\pi$
$282$		0			0
$283$	2.50000	+	4.33013i	0.148610	+	0.257399i	0.930714	−	0.365748i	$-0.119187\pi$
$283$	2.50000	+	4.33013i	0.148610	+	0.257399i	−0.782104	+	0.623148i	$0.785854\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−27.0000	+	5.19615i	−1.59376	+	0.306719i
$288$		0			0
$289$	15.5000	−	26.8468i	0.911765	−	1.57922i
$290$		0			0
$291$		0			0
$292$		0			0
$293$		−	20.7846i		−	1.21425i	−0.794606	−	0.607125i	$-0.792323\pi$
$293$		−	20.7846i		−	1.21425i	0.794606	−	0.607125i	$-0.207677\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$		0			0
$300$		0			0
$301$	4.50000	−	0.866025i	0.259376	−	0.0499169i
$302$		0			0
$303$		0			0
$304$		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
$309$		0			0
$310$		0			0
$311$	9.00000	+	15.5885i	0.510343	+	0.883940i	0.999928	+	0.0119847i	$0.00381495\pi$
$311$	9.00000	+	15.5885i	0.510343	+	0.883940i	−0.489585	+	0.871956i	$0.662852\pi$
$312$		0			0
$313$	13.5000	+	7.79423i	0.763065	+	0.440556i	0.830395	−	0.557175i	$-0.188115\pi$
$313$	13.5000	+	7.79423i	0.763065	+	0.440556i	−0.0673300	+	0.997731i	$0.521448\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−6.00000	+	10.3923i	−0.336994	+	0.583690i	−0.983866	−	0.178908i	$-0.942743\pi$
$317$	−6.00000	+	10.3923i	−0.336994	+	0.583690i	0.646872	+	0.762598i	$0.276077\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$		0			0
$322$		0			0
$323$			48.4974i			2.69847i
$324$		0			0
$325$	31.5000	−	18.1865i	1.74731	−	1.00881i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−15.0000	−	5.19615i	−0.826977	−	0.286473i
$330$		0			0
$331$	7.50000	+	4.33013i	0.412237	+	0.238005i	0.691751	−	0.722137i	$-0.256840\pi$
$331$	7.50000	+	4.33013i	0.412237	+	0.238005i	−0.279513	+	0.960142i	$0.590173\pi$
$332$		0			0
$333$		0			0
$334$		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
$339$		0			0
$340$		0			0
$341$	−15.0000	−	8.66025i	−0.812296	−	0.468979i
$342$		0			0
$343$	−10.0000	−	15.5885i	−0.539949	−	0.841698i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−30.0000	+	17.3205i	−1.61048	+	0.929814i	−0.621227	+	0.783631i	$0.713365\pi$
$347$	−30.0000	+	17.3205i	−1.61048	+	0.929814i	−0.989258	+	0.146183i	$0.953301\pi$
$348$		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
$351$		0			0
$352$		0			0
$353$	−15.0000	+	8.66025i	−0.798369	+	0.460939i	−0.842901	−	0.538069i	$-0.819154\pi$
$353$	−15.0000	+	8.66025i	−0.798369	+	0.460939i	0.0445312	+	0.999008i	$0.485821\pi$
$354$		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	0.333238	−	0.942843i	$-0.391859\pi$
$359$	−6.00000	−	3.46410i	−0.316668	−	0.182828i	−0.649906	+	0.760014i	$0.725192\pi$
$360$		0			0
$361$	−15.0000	−	25.9808i	−0.789474	−	1.36741i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	30.0000			1.57027
$366$		0			0
$367$	−18.5000	−	32.0429i	−0.965692	−	1.67263i	−0.707744	−	0.706469i	$-0.750287\pi$
$367$	−18.5000	−	32.0429i	−0.965692	−	1.67263i	−0.257948	−	0.966159i	$-0.583046\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$	−5.50000	+	9.52628i	−0.284779	+	0.493252i	−0.972556	−	0.232671i	$-0.925254\pi$
$373$	−5.50000	+	9.52628i	−0.284779	+	0.493252i	0.687776	+	0.725923i	$0.258587\pi$
$374$		0			0
$375$		0			0
$376$		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
$381$		0			0
$382$		0			0
$383$	18.0000	−	31.1769i	0.919757	−	1.59307i	0.119974	−	0.992777i	$-0.461719\pi$
$383$	18.0000	−	31.1769i	0.919757	−	1.59307i	0.799783	−	0.600289i	$-0.204948\pi$
$384$		0			0
$385$	6.00000	+	31.1769i	0.305788	+	1.58892i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	15.0000	+	25.9808i	0.760530	+	1.31728i	0.942578	+	0.333987i	$0.108394\pi$
$389$	15.0000	+	25.9808i	0.760530	+	1.31728i	−0.182047	+	0.983290i	$0.558272\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$		0			0
$394$		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	$-0.291709\pi$
$397$	4.50000	+	2.59808i	0.225849	+	0.130394i	−0.382807	+	0.923828i	$0.625043\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	12.0000	−	20.7846i	0.599251	−	1.03793i	−0.393680	−	0.919247i	$-0.628798\pi$
$401$	12.0000	−	20.7846i	0.599251	−	1.03793i	0.992932	−	0.118686i	$-0.0378683\pi$
$402$		0			0
$403$	−22.5000	+	12.9904i	−1.12080	+	0.647097i
$404$		0			0
$405$		0			0
$406$		0			0
$407$			3.46410i			0.171709i
$408$		0			0
$409$	16.5000	−	9.52628i	0.815872	−	0.471044i	−0.0331186	−	0.999451i	$-0.510544\pi$
$409$	16.5000	−	9.52628i	0.815872	−	0.471044i	0.848991	+	0.528407i	$0.177211\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		0			0
$414$		0			0
$415$	18.0000	+	10.3923i	0.883585	+	0.510138i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	30.0000			1.46560			0.732798	−	0.680446i	$-0.238214\pi$
$419$	30.0000			1.46560			0.732798	+	0.680446i	$0.238214\pi$
$420$		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
$423$		0			0
$424$		0			0
$425$	42.0000	+	24.2487i	2.03730	+	1.17624i
$426$		0			0
$427$		0			0
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−15.0000	+	8.66025i	−0.722525	+	0.417150i	−0.815681	−	0.578502i	$-0.803638\pi$
$431$	−15.0000	+	8.66025i	−0.722525	+	0.417150i	0.0931566	+	0.995651i	$0.470304\pi$
$432$		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	−0.991322	−	0.131453i	$-0.958036\pi$
$439$	−8.00000	+	13.8564i	−0.381819	+	0.661330i	0.609503	+	0.792784i	$0.291369\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−6.00000	−	3.46410i	−0.285069	−	0.164584i	0.350647	−	0.936508i	$-0.385962\pi$
$443$	−6.00000	−	3.46410i	−0.285069	−	0.164584i	−0.635716	+	0.771923i	$0.719295\pi$
$444$		0			0
$445$	−12.0000	−	20.7846i	−0.568855	−	0.985285i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	6.00000			0.283158			0.141579	−	0.989927i	$-0.454782\pi$
$449$	6.00000			0.283158			0.141579	+	0.989927i	$0.454782\pi$
$450$		0			0
$451$	18.0000	+	31.1769i	0.847587	+	1.46806i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	45.0000	+	15.5885i	2.10963	+	0.730798i
$456$		0			0
$457$	−2.50000	+	4.33013i	−0.116945	+	0.202555i	−0.918556	−	0.395292i	$-0.870643\pi$
$457$	−2.50000	+	4.33013i	−0.116945	+	0.202555i	0.801611	+	0.597847i	$0.203977\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		−	13.8564i		−	0.645357i	−0.946509	−	0.322679i	$-0.895417\pi$
$461$		−	13.8564i		−	0.645357i	0.946509	−	0.322679i	$-0.104583\pi$
$462$		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
$465$		0			0
$466$		0			0
$467$	−3.00000	+	5.19615i	−0.138823	+	0.240449i	−0.927052	−	0.374934i	$-0.877665\pi$
$467$	−3.00000	+	5.19615i	−0.138823	+	0.240449i	0.788228	+	0.615383i	$0.210999\pi$
$468$		0			0
$469$	3.00000	+	3.46410i	0.138527	+	0.159957i
$470$		0			0
$471$		0			0
$472$		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	0.998392	+	0.0566937i	$0.0180558\pi$
$479$	12.0000	+	20.7846i	0.548294	+	0.949673i	−0.450098	+	0.892979i	$0.648611\pi$
$480$		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	$-0.578087\pi$
$487$	10.5000	−	6.06218i	0.475800	−	0.274703i	0.718665	+	0.695357i	$0.244754\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$		−	20.7846i		−	0.937996i	−0.883199	−	0.468998i	$-0.844615\pi$
$491$		−	20.7846i		−	0.937996i	0.883199	−	0.468998i	$-0.155385\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$		0			0
$497$	9.00000	−	1.73205i	0.403705	−	0.0776931i
$498$		0			0
$499$	−10.5000	−	6.06218i	−0.470045	−	0.271380i	0.246214	−	0.969216i	$-0.420813\pi$
$499$	−10.5000	−	6.06218i	−0.470045	−	0.271380i	−0.716258	+	0.697835i	$0.754147\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	30.0000			1.33763			0.668817	−	0.743427i	$-0.266801\pi$
$503$	30.0000			1.33763			0.668817	+	0.743427i	$0.266801\pi$
$504$		0			0
$505$	12.0000			0.533993
$506$		0			0
$507$		0			0
$508$		0			0
$509$	3.00000	+	1.73205i	0.132973	+	0.0767718i	0.565011	−	0.825084i	$-0.308872\pi$
$509$	3.00000	+	1.73205i	0.132973	+	0.0767718i	−0.432038	+	0.901855i	$0.642205\pi$
$510$		0			0
$511$	15.0000	+	17.3205i	0.663561	+	0.766214i
$512$		0			0
$513$		0			0
$514$		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	−0.213546	−	0.976933i	$-0.568501\pi$
$521$	12.0000	−	6.92820i	0.525730	−	0.303530i	0.739276	+	0.673403i	$0.235168\pi$
$522$		0			0
$523$	0.500000	−	0.866025i	0.0218635	−	0.0378686i	−0.854887	−	0.518815i	$-0.826373\pi$
$523$	0.500000	−	0.866025i	0.0218635	−	0.0378686i	0.876750	+	0.480946i	$0.159707\pi$
$524$		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
$537$		0			0
$538$		0			0
$539$	−15.0000	+	19.0526i	−0.646096	+	0.820652i
$540$		0			0
$541$	20.5000	−	35.5070i	0.881364	−	1.52657i	0.0315385	−	0.999503i	$-0.489959\pi$
$541$	20.5000	−	35.5070i	0.881364	−	1.52657i	0.849825	−	0.527064i	$-0.176707\pi$
$542$		0			0
$543$		0			0
$544$		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
$553$	13.5000	−	38.9711i	0.574078	−	1.65722i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	9.00000	+	15.5885i	0.381342	+	0.660504i	0.991254	−	0.131965i	$-0.0421286\pi$
$557$	9.00000	+	15.5885i	0.381342	+	0.660504i	−0.609912	+	0.792469i	$0.708795\pi$
$558$		0			0
$559$	−9.00000			−0.380659
$560$		0			0
$561$		0			0
$562$		0			0
$563$	9.00000	+	15.5885i	0.379305	+	0.656975i	0.990961	−	0.134148i	$-0.0428299\pi$
$563$	9.00000	+	15.5885i	0.379305	+	0.656975i	−0.611656	+	0.791123i	$0.709497\pi$
$564$		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	$-0.873472\pi$
$569$	−3.00000	+	5.19615i	−0.125767	+	0.217834i	0.796266	+	0.604947i	$0.206806\pi$
$570$		0			0
$571$	−22.5000	+	12.9904i	−0.941596	+	0.543631i	−0.890460	−	0.455061i	$-0.849617\pi$
$571$	−22.5000	+	12.9904i	−0.941596	+	0.543631i	−0.0511355	+	0.998692i	$0.516284\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$		0			0
$577$	−1.50000	+	0.866025i	−0.0624458	+	0.0360531i	−0.530898	−	0.847436i	$-0.678145\pi$
$577$	−1.50000	+	0.866025i	−0.0624458	+	0.0360531i	0.468452	+	0.883489i	$0.344812\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	3.00000	+	15.5885i	0.124461	+	0.646718i
$582$		0			0
$583$		0			0
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−12.0000			−0.495293			−0.247647	−	0.968850i	$-0.579657\pi$
$587$	−12.0000			−0.495293			−0.247647	+	0.968850i	$0.579657\pi$
$588$		0			0
$589$	35.0000			1.44215
$590$		0			0
$591$		0			0
$592$		0			0
$593$	33.0000	+	19.0526i	1.35515	+	0.782395i	0.988965	−	0.148148i	$-0.0473313\pi$
$593$	33.0000	+	19.0526i	1.35515	+	0.782395i	0.366182	+	0.930543i	$0.380665\pi$
$594$		0			0
$595$	12.0000	+	62.3538i	0.491952	+	2.55626i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−6.00000	+	3.46410i	−0.245153	+	0.141539i	−0.617543	−	0.786537i	$-0.711872\pi$
$599$	−6.00000	+	3.46410i	−0.245153	+	0.141539i	0.372390	+	0.928076i	$0.378539\pi$
$600$		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
$603$		0			0
$604$		0			0
$605$	3.00000	−	1.73205i	0.121967	−	0.0704179i
$606$		0			0
$607$	−18.5000	+	32.0429i	−0.750892	+	1.30058i	0.196499	+	0.980504i	$0.437043\pi$
$607$	−18.5000	+	32.0429i	−0.750892	+	1.30058i	−0.947391	+	0.320079i	$0.896291\pi$
$608$		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	0.998002	−	0.0631797i	$-0.0201241\pi$
$613$	11.0000	+	19.0526i	0.444286	+	0.769526i	−0.553716	+	0.832705i	$0.686791\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	42.0000			1.69086			0.845428	−	0.534089i	$-0.179345\pi$
$617$	42.0000			1.69086			0.845428	+	0.534089i	$0.179345\pi$
$618$		0			0
$619$	−17.5000	−	30.3109i	−0.703384	−	1.21830i	−0.967271	−	0.253744i	$-0.918338\pi$
$619$	−17.5000	−	30.3109i	−0.703384	−	1.21830i	0.263887	−	0.964554i	$-0.414995\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	6.00000	−	17.3205i	0.240385	−	0.693932i
$624$		0			0
$625$	5.50000	−	9.52628i	0.220000	−	0.381051i
$626$		0			0
$627$		0			0
$628$		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
$633$		0			0
$634$		0			0
$635$	−27.0000	+	46.7654i	−1.07146	+	1.85583i
$636$		0			0
$637$	13.5000	+	33.7750i	0.534889	+	1.33821i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−18.0000	−	31.1769i	−0.710957	−	1.23141i	−0.964498	−	0.264089i	$-0.914929\pi$
$641$	−18.0000	−	31.1769i	−0.710957	−	1.23141i	0.253541	−	0.967325i	$-0.418405\pi$
$642$		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
$645$		0			0
$646$		0			0
$647$	−9.00000	−	15.5885i	−0.353827	−	0.612845i	0.633090	−	0.774078i	$-0.281786\pi$
$647$	−9.00000	−	15.5885i	−0.353827	−	0.612845i	−0.986916	+	0.161233i	$0.948453\pi$
$648$		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	$-0.947899\pi$
$653$	−9.00000	+	15.5885i	−0.352197	+	0.610023i	0.634437	+	0.772975i	$0.281232\pi$
$654$		0			0
$655$	54.0000	−	31.1769i	2.10995	−	1.21818i
$656$		0			0
$657$		0			0
$658$		0			0
$659$			20.7846i			0.809653i	0.914393	+	0.404827i	$0.132668\pi$
$659$			20.7846i			0.809653i	−0.914393	+	0.404827i	$0.867332\pi$
$660$		0			0
$661$	16.5000	−	9.52628i	0.641776	−	0.370529i	−0.143523	−	0.989647i	$-0.545843\pi$
$661$	16.5000	−	9.52628i	0.641776	−	0.370529i	0.785298	+	0.619118i	$0.212510\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−42.0000	−	48.4974i	−1.62869	−	1.88065i
$666$		0			0
$667$		0			0
$668$		0			0
$669$		0			0
$670$		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
$675$		0			0
$676$		0			0
$677$	−30.0000	−	17.3205i	−1.15299	−	0.665681i	−0.203379	−	0.979100i	$-0.565192\pi$
$677$	−30.0000	−	17.3205i	−1.15299	−	0.665681i	−0.949615	+	0.313419i	$0.898526\pi$
$678$		0			0
$679$	−18.0000	+	3.46410i	−0.690777	+	0.132940i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	36.0000	−	20.7846i	1.37750	−	0.795301i	0.385643	−	0.922648i	$-0.373979\pi$
$683$	36.0000	−	20.7846i	1.37750	−	0.795301i	0.991858	+	0.127347i	$0.0406461\pi$
$684$		0			0
$685$		−	41.5692i		−	1.58828i
$686$		0			0
$687$		0			0
$688$		0			0
$689$		0			0
$690$		0			0
$691$	6.50000	−	11.2583i	0.247272	−	0.428287i	−0.715496	−	0.698617i	$-0.753799\pi$
$691$	6.50000	−	11.2583i	0.247272	−	0.428287i	0.962768	+	0.270330i	$0.0871327\pi$
$692$		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
$699$		0			0
$700$		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
$705$		0			0
$706$		0			0
$707$	6.00000	+	6.92820i	0.225653	+	0.260562i
$708$		0			0
$709$	5.00000	−	8.66025i	0.187779	−	0.325243i	−0.756730	−	0.653727i	$-0.773204\pi$
$709$	5.00000	−	8.66025i	0.187779	−	0.325243i	0.944509	+	0.328484i	$0.106538\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$		0			0
$714$		0			0
$715$		−	62.3538i		−	2.33190i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−9.00000	+	15.5885i	−0.335643	+	0.581351i	−0.983608	−	0.180319i	$-0.942287\pi$
$719$	−9.00000	+	15.5885i	−0.335643	+	0.581351i	0.647965	+	0.761670i	$0.275620\pi$
$720$		0			0
$721$	−12.5000	−	4.33013i	−0.465524	−	0.161262i
$722$		0			0
$723$		0			0
$724$		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
$729$		0			0
$730$		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	$-0.567784\pi$
$733$	−31.5000	−	18.1865i	−1.16348	−	0.671735i	−0.952135	+	0.305677i	$0.901117\pi$
$734$		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	$-0.433388\pi$
$739$	31.5000	−	18.1865i	1.15875	−	0.669002i	0.951003	+	0.309181i	$0.100055\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$			3.46410i			0.127086i	0.997979	+	0.0635428i	$0.0202399\pi$
$743$			3.46410i			0.127086i	−0.997979	+	0.0635428i	$0.979760\pi$
$744$		0			0
$745$	−36.0000	+	20.7846i	−1.31894	+	0.761489i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−6.00000	+	17.3205i	−0.219235	+	0.632878i
$750$		0			0
$751$	−40.5000	−	23.3827i	−1.47787	−	0.853246i	−0.478179	−	0.878262i	$-0.658703\pi$
$751$	−40.5000	−	23.3827i	−1.47787	−	0.853246i	−0.999687	+	0.0250161i	$0.992036\pi$
$752$		0			0
$753$		0			0
$754$		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	−0.154590	−	0.987979i	$-0.549406\pi$
$761$	−30.0000	−	17.3205i	−1.08750	−	0.627868i	−0.932910	+	0.360111i	$0.882739\pi$
$762$		0			0
$763$	10.0000	−	8.66025i	0.362024	−	0.313522i
$764$		0			0
$765$		0			0
$766$		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
$771$		0			0
$772$		0			0
$773$	27.0000	−	15.5885i	0.971123	−	0.560678i	0.0715442	−	0.997437i	$-0.477207\pi$
$773$	27.0000	−	15.5885i	0.971123	−	0.560678i	0.899578	+	0.436760i	$0.143874\pi$
$774$		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	0.785885	−	0.618373i	$-0.212208\pi$
$787$	−4.00000	−	6.92820i	−0.142585	−	0.246964i	−0.928469	+	0.371409i	$0.878875\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−3.00000	−	15.5885i	−0.106668	−	0.554262i
$792$		0			0
$793$		0			0
$794$		0			0
$795$		0			0
$796$		0			0
$797$			41.5692i			1.47246i	0.676733	+	0.736229i	$0.263395\pi$
$797$			41.5692i			1.47246i	−0.676733	+	0.736229i	$0.736605\pi$
$798$		0			0
$799$			41.5692i			1.47061i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	15.0000	−	25.9808i	0.529339	−	0.916841i
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$	27.0000	+	46.7654i	0.949269	+	1.64418i	0.746968	+	0.664860i	$0.231509\pi$
$809$	27.0000	+	46.7654i	0.949269	+	1.64418i	0.202301	+	0.979323i	$0.435158\pi$
$810$		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
$813$		0			0
$814$		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	$-0.866722\pi$
$821$	−3.00000	+	5.19615i	−0.104701	+	0.181347i	0.808915	+	0.587925i	$0.200055\pi$
$822$		0			0
$823$	−15.0000	+	8.66025i	−0.522867	+	0.301877i	−0.738107	−	0.674684i	$-0.764280\pi$
$823$	−15.0000	+	8.66025i	−0.522867	+	0.301877i	0.215240	+	0.976561i	$0.430947\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$			17.3205i			0.602293i	0.953578	+	0.301147i	$0.0973693\pi$
$827$			17.3205i			0.602293i	−0.953578	+	0.301147i	$0.902631\pi$
$828$		0			0
$829$	−19.5000	+	11.2583i	−0.677263	+	0.391018i	−0.798823	−	0.601566i	$-0.794544\pi$
$829$	−19.5000	+	11.2583i	−0.677263	+	0.391018i	0.121560	+	0.992584i	$0.461210\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−30.0000	+	38.1051i	−1.03944	+	1.32026i
$834$		0			0
$835$	−18.0000	−	10.3923i	−0.622916	−	0.359641i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	24.0000			0.828572			0.414286	−	0.910147i	$-0.364031\pi$
$839$	24.0000			0.828572			0.414286	+	0.910147i	$0.364031\pi$
$840$		0			0
$841$	−29.0000			−1.00000
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−42.0000	−	24.2487i	−1.44484	−	0.834181i
$846$		0			0
$847$	2.50000	+	0.866025i	0.0859010	+	0.0297570i
$848$		0			0
$849$		0			0
$850$		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
$855$		0			0
$856$		0			0
$857$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$857$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$858$		0			0
$859$	−20.0000	+	34.6410i	−0.682391	+	1.18194i	0.291858	+	0.956462i	$0.405727\pi$
$859$	−20.0000	+	34.6410i	−0.682391	+	1.18194i	−0.974249	+	0.225475i	$0.927607\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	27.0000	+	15.5885i	0.919091	+	0.530637i	0.883345	−	0.468724i	$-0.155286\pi$
$863$	27.0000	+	15.5885i	0.919091	+	0.530637i	0.0357458	+	0.999361i	$0.488619\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−54.0000			−1.83182
$870$		0			0
$871$	−4.50000	−	7.79423i	−0.152477	−	0.264097i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−18.0000	+	3.46410i	−0.608511	+	0.117108i
$876$		0			0
$877$	11.0000	−	19.0526i	0.371444	−	0.643359i	−0.618344	−	0.785907i	$-0.712196\pi$
$877$	11.0000	−	19.0526i	0.371444	−	0.643359i	0.989788	+	0.142548i	$0.0455296\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		−	27.7128i		−	0.933668i	−0.884345	−	0.466834i	$-0.845394\pi$
$881$		−	27.7128i		−	0.933668i	0.884345	−	0.466834i	$-0.154606\pi$
$882$		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	0.496340	+	0.868128i	$0.334677\pi$
$887$	−15.0000	+	25.9808i	−0.503651	+	0.872349i	−0.999991	+	0.00422062i	$0.998657\pi$
$888$		0			0
$889$	−40.5000	+	7.79423i	−1.35833	+	0.261410i
$890$		0			0
$891$		0			0
$892$		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	$-0.749995\pi$
$907$	−13.5000	+	7.79423i	−0.448260	+	0.258803i	0.258835	+	0.965922i	$0.416661\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		−	20.7846i		−	0.688625i	−0.938855	−	0.344312i	$-0.888112\pi$
$911$		−	20.7846i		−	0.688625i	0.938855	−	0.344312i	$-0.111888\pi$
$912$		0			0
$913$	18.0000	−	10.3923i	0.595713	−	0.343935i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	45.0000	+	15.5885i	1.48603	+	0.514776i
$918$		0			0
$919$	1.50000	+	0.866025i	0.0494804	+	0.0285675i	0.524536	−	0.851388i	$-0.324239\pi$
$919$	1.50000	+	0.866025i	0.0494804	+	0.0285675i	−0.475056	+	0.879956i	$0.657572\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−18.0000			−0.592477
$924$		0			0
$925$	−7.00000			−0.230159
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−3.00000	−	1.73205i	−0.0984268	−	0.0568267i	0.449979	−	0.893039i	$-0.351432\pi$
$929$	−3.00000	−	1.73205i	−0.0984268	−	0.0568267i	−0.548405	+	0.836213i	$0.684765\pi$
$930$		0			0
$931$	7.00000	−	48.4974i	0.229416	−	1.58944i
$932$		0			0
$933$		0			0
$934$		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
$939$		0			0
$940$		0			0
$941$	−42.0000	+	24.2487i	−1.36916	+	0.790485i	−0.990821	−	0.135181i	$-0.956838\pi$
$941$	−42.0000	+	24.2487i	−1.36916	+	0.790485i	−0.378340	+	0.925667i	$0.623505\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$	15.0000	+	8.66025i	0.487435	+	0.281420i	0.723510	−	0.690314i	$-0.242528\pi$
$947$	15.0000	+	8.66025i	0.487435	+	0.281420i	−0.236075	+	0.971735i	$0.575861\pi$
$948$		0			0
$949$	−22.5000	−	38.9711i	−0.730381	−	1.26506i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−36.0000			−1.16615			−0.583077	−	0.812417i	$-0.698151\pi$
$953$	−36.0000			−1.16615			−0.583077	+	0.812417i	$0.698151\pi$
$954$		0			0
$955$	6.00000	+	10.3923i	0.194155	+	0.336287i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	24.0000	−	20.7846i	0.775000	−	0.671170i
$960$		0			0
$961$	3.00000	−	5.19615i	0.0967742	−	0.167618i
$962$		0			0
$963$		0			0
$964$		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
$969$		0			0
$970$		0			0
$971$	6.00000	−	10.3923i	0.192549	−	0.333505i	−0.753545	−	0.657396i	$-0.771658\pi$
$971$	6.00000	−	10.3923i	0.192549	−	0.333505i	0.946094	+	0.323891i	$0.104991\pi$
$972$		0			0
$973$	2.50000	+	12.9904i	0.0801463	+	0.416452i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	27.0000	+	46.7654i	0.863807	+	1.49616i	0.868227	+	0.496167i	$0.165259\pi$
$977$	27.0000	+	46.7654i	0.863807	+	1.49616i	−0.00442082	+	0.999990i	$0.501407\pi$
$978$		0			0
$979$	−24.0000			−0.767043
$980$		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
$985$	−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	$-0.604984\pi$
$991$	10.5000	−	6.06218i	0.333543	−	0.192571i	0.657413	+	0.753530i	$0.271651\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		−	55.4256i		−	1.75711i
$996$		0			0
$997$	37.5000	−	21.6506i	1.18764	−	0.685682i	0.229868	−	0.973222i	$-0.426171\pi$
$997$	37.5000	−	21.6506i	1.18764	−	0.685682i	0.957769	+	0.287539i	$0.0928372\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1008.2.cs.n.271.1		2
3.2	odd	2		336.2.bl.a.271.1	yes	2
4.3	odd	2		1008.2.cs.m.271.1		2
7.2	even	3		7056.2.b.l.1567.1		2
7.3	odd	6		1008.2.cs.m.703.1		2
7.5	odd	6		7056.2.b.a.1567.2		2
12.11	even	2		336.2.bl.e.271.1	yes	2
21.2	odd	6		2352.2.b.h.1567.2		2
21.5	even	6		2352.2.b.a.1567.1		2
21.11	odd	6		2352.2.bl.f.31.1		2
21.17	even	6		336.2.bl.e.31.1	yes	2
21.20	even	2		2352.2.bl.l.607.1		2
24.5	odd	2		1344.2.bl.h.1279.1		2
24.11	even	2		1344.2.bl.d.1279.1		2
28.3	even	6	inner	1008.2.cs.n.703.1		2
28.19	even	6		7056.2.b.l.1567.2		2
28.23	odd	6		7056.2.b.a.1567.1		2
84.11	even	6		2352.2.bl.l.31.1		2
84.23	even	6		2352.2.b.a.1567.2		2
84.47	odd	6		2352.2.b.h.1567.1		2
84.59	odd	6		336.2.bl.a.31.1	✓	2
84.83	odd	2		2352.2.bl.f.607.1		2
168.59	odd	6		1344.2.bl.h.703.1		2
168.101	even	6		1344.2.bl.d.703.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
336.2.bl.a.31.1	✓	2	84.59	odd	6
336.2.bl.a.271.1	yes	2	3.2	odd	2
336.2.bl.e.31.1	yes	2	21.17	even	6
336.2.bl.e.271.1	yes	2	12.11	even	2
1008.2.cs.m.271.1		2	4.3	odd	2
1008.2.cs.m.703.1		2	7.3	odd	6
1008.2.cs.n.271.1		2	1.1	even	1	trivial
1008.2.cs.n.703.1		2	28.3	even	6	inner
1344.2.bl.d.703.1		2	168.101	even	6
1344.2.bl.d.1279.1		2	24.11	even	2
1344.2.bl.h.703.1		2	168.59	odd	6
1344.2.bl.h.1279.1		2	24.5	odd	2
2352.2.b.a.1567.1		2	21.5	even	6
2352.2.b.a.1567.2		2	84.23	even	6
2352.2.b.h.1567.1		2	84.47	odd	6
2352.2.b.h.1567.2		2	21.2	odd	6
2352.2.bl.f.31.1		2	21.11	odd	6
2352.2.bl.f.607.1		2	84.83	odd	2
2352.2.bl.l.31.1		2	84.11	even	6
2352.2.bl.l.607.1		2	21.20	even	2
7056.2.b.a.1567.1		2	28.23	odd	6
7056.2.b.a.1567.2		2	7.5	odd	6
7056.2.b.l.1567.1		2	7.2	even	3
7056.2.b.l.1567.2		2	28.19	even	6

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 1008 = 2^{4} \cdot 3^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1008.cs (of order \(6\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(3.00000 + 1.73205i) q^{5} +(0.500000 + 2.59808i) q^{7} +O(q^{10})\)	\(q+(3.00000 + 1.73205i) q^{5} +(0.500000 + 2.59808i) q^{7} +(3.00000 - 1.73205i) q^{11} -5.19615i q^{13} +(6.00000 - 3.46410i) q^{17} +(-3.50000 + 6.06218i) q^{19} +(3.50000 + 6.06218i) q^{25} +(-2.50000 - 4.33013i) q^{31} +(-3.00000 + 8.66025i) q^{35} +(-0.500000 + 0.866025i) q^{37} +10.3923i q^{41} -1.73205i q^{43} +(-3.00000 + 5.19615i) q^{47} +(-6.50000 + 2.59808i) q^{49} +12.0000 q^{55} +(9.00000 - 15.5885i) q^{65} +(1.50000 - 0.866025i) q^{67} -3.46410i q^{71} +(7.50000 - 4.33013i) q^{73} +(6.00000 + 6.92820i) q^{77} +(-13.5000 - 7.79423i) q^{79} +6.00000 q^{83} +24.0000 q^{85} +(-6.00000 - 3.46410i) q^{89} +(13.5000 - 2.59808i) q^{91} +(-21.0000 + 12.1244i) q^{95} +6.92820i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 6 q^{5} + q^{7}+O(q^{10}) \) 2 * q + 6 * q^5 + q^7	\( 2 q + 6 q^{5} + q^{7} + 6 q^{11} + 12 q^{17} - 7 q^{19} + 7 q^{25} - 5 q^{31} - 6 q^{35} - q^{37} - 6 q^{47} - 13 q^{49} + 24 q^{55} + 18 q^{65} + 3 q^{67} + 15 q^{73} + 12 q^{77} - 27 q^{79} + 12 q^{83} + 48 q^{85} - 12 q^{89} + 27 q^{91} - 42 q^{95}+O(q^{100}) \) 2 * q + 6 * q^5 + q^7 + 6 * q^11 + 12 * q^17 - 7 * q^19 + 7 * q^25 - 5 * q^31 - 6 * q^35 - q^37 - 6 * q^47 - 13 * q^49 + 24 * q^55 + 18 * q^65 + 3 * q^67 + 15 * q^73 + 12 * q^77 - 27 * q^79 + 12 * q^83 + 48 * q^85 - 12 * q^89 + 27 * q^91 - 42 * q^95

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