LMFDB - Embedded newform 1216.2.i.c.961.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1216,2,Mod(577,1216)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1216, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 0, 2]))

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

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

chi := DirichletCharacter("1216.577");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1216 = 2^{6} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1216.i (of order $3$, 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:	$9.70980888579$
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 76)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			961.1
Root			$0.500000 + 0.866025i$ of defining polynomial
Character	$\chi$	$=$	1216.961
Dual form			1216.2.i.c.577.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.500000 + 0.866025i) q^{3} +(-0.500000 + 0.866025i) q^{5} +(1.00000 + 1.73205i) q^{9} +O(q^{10})$	\(q+(-0.500000 + 0.866025i) q^{3} +(-0.500000 + 0.866025i) q^{5} +(1.00000 + 1.73205i) q^{9} +4.00000 q^{11} +(-0.500000 - 0.866025i) q^{13} +(-0.500000 - 0.866025i) q^{15} +(-1.50000 + 2.59808i) q^{17} +(4.00000 - 1.73205i) q^{19} +(-2.50000 - 4.33013i) q^{23} +(2.00000 + 3.46410i) q^{25} -5.00000 q^{27} +(3.50000 + 6.06218i) q^{29} +4.00000 q^{31} +(-2.00000 + 3.46410i) q^{33} -10.0000 q^{37} +1.00000 q^{39} +(2.50000 - 4.33013i) q^{41} +(-2.50000 + 4.33013i) q^{43} -2.00000 q^{45} +(3.50000 + 6.06218i) q^{47} -7.00000 q^{49} +(-1.50000 - 2.59808i) q^{51} +(5.50000 + 9.52628i) q^{53} +(-2.00000 + 3.46410i) q^{55} +(-0.500000 + 4.33013i) q^{57} +(1.50000 - 2.59808i) q^{59} +(5.50000 + 9.52628i) q^{61} +1.00000 q^{65} +(-1.50000 - 2.59808i) q^{67} +5.00000 q^{69} +(-5.50000 + 9.52628i) q^{71} +(-7.50000 + 12.9904i) q^{73} -4.00000 q^{75} +(6.50000 - 11.2583i) q^{79} +(-0.500000 + 0.866025i) q^{81} +(-1.50000 - 2.59808i) q^{85} -7.00000 q^{87} +(-1.50000 - 2.59808i) q^{89} +(-2.00000 + 3.46410i) q^{93} +(-0.500000 + 4.33013i) q^{95} +(2.50000 - 4.33013i) q^{97} +(4.00000 + 6.92820i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - q^{3} - q^{5} + 2 q^{9}+O(q^{10}) $ 2 * q - q^3 - q^5 + 2 * q^9	$ 2 q - q^{3} - q^{5} + 2 q^{9} + 8 q^{11} - q^{13} - q^{15} - 3 q^{17} + 8 q^{19} - 5 q^{23} + 4 q^{25} - 10 q^{27} + 7 q^{29} + 8 q^{31} - 4 q^{33} - 20 q^{37} + 2 q^{39} + 5 q^{41} - 5 q^{43} - 4 q^{45} + 7 q^{47} - 14 q^{49} - 3 q^{51} + 11 q^{53} - 4 q^{55} - q^{57} + 3 q^{59} + 11 q^{61} + 2 q^{65} - 3 q^{67} + 10 q^{69} - 11 q^{71} - 15 q^{73} - 8 q^{75} + 13 q^{79} - q^{81} - 3 q^{85} - 14 q^{87} - 3 q^{89} - 4 q^{93} - q^{95} + 5 q^{97} + 8 q^{99}+O(q^{100}) $ 2 * q - q^3 - q^5 + 2 * q^9 + 8 * q^11 - q^13 - q^15 - 3 * q^17 + 8 * q^19 - 5 * q^23 + 4 * q^25 - 10 * q^27 + 7 * q^29 + 8 * q^31 - 4 * q^33 - 20 * q^37 + 2 * q^39 + 5 * q^41 - 5 * q^43 - 4 * q^45 + 7 * q^47 - 14 * q^49 - 3 * q^51 + 11 * q^53 - 4 * q^55 - q^57 + 3 * q^59 + 11 * q^61 + 2 * q^65 - 3 * q^67 + 10 * q^69 - 11 * q^71 - 15 * q^73 - 8 * q^75 + 13 * q^79 - q^81 - 3 * q^85 - 14 * q^87 - 3 * q^89 - 4 * q^93 - q^95 + 5 * q^97 + 8 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1216\mathbb{Z}\right)^\times$.

$n$	$191$	$705$	$837$
$\chi(n)$	$1$	$e\left(\frac{2}{3}\right)$	$1$

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$	−0.500000	+	0.866025i	−0.288675	+	0.500000i	−0.973494	−	0.228714i	$-0.926548\pi$
$3$	−0.500000	+	0.866025i	−0.288675	+	0.500000i	0.684819	+	0.728714i	$0.259881\pi$
$4$		0			0
$5$	−0.500000	+	0.866025i	−0.223607	+	0.387298i	−0.955901	−	0.293691i	$-0.905116\pi$
$5$	−0.500000	+	0.866025i	−0.223607	+	0.387298i	0.732294	+	0.680989i	$0.238450\pi$
$6$		0			0
$7$		0			0			−	1.00000i	$-0.5\pi$
$7$		0			0				1.00000i	$0.5\pi$
$8$		0			0
$9$	1.00000	+	1.73205i	0.333333	+	0.577350i
$10$		0			0
$11$	4.00000			1.20605			0.603023	−	0.797724i	$-0.293963\pi$
$11$	4.00000			1.20605			0.603023	+	0.797724i	$0.293963\pi$
$12$		0			0
$13$	−0.500000	−	0.866025i	−0.138675	−	0.240192i	0.788320	−	0.615265i	$-0.210951\pi$
$13$	−0.500000	−	0.866025i	−0.138675	−	0.240192i	−0.926995	+	0.375073i	$0.877618\pi$
$14$		0			0
$15$	−0.500000	−	0.866025i	−0.129099	−	0.223607i
$16$		0			0
$17$	−1.50000	+	2.59808i	−0.363803	+	0.630126i	−0.988583	−	0.150675i	$-0.951855\pi$
$17$	−1.50000	+	2.59808i	−0.363803	+	0.630126i	0.624780	+	0.780801i	$0.285189\pi$
$18$		0			0
$19$	4.00000	−	1.73205i	0.917663	−	0.397360i
$19$	4.00000	−	1.73205i	0.917663	−	0.397360i
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−2.50000	−	4.33013i	−0.521286	−	0.902894i	−0.999694	−	0.0247559i	$-0.992119\pi$
$23$	−2.50000	−	4.33013i	−0.521286	−	0.902894i	0.478407	−	0.878138i	$-0.341214\pi$
$24$		0			0
$25$	2.00000	+	3.46410i	0.400000	+	0.692820i
$26$		0			0
$27$	−5.00000			−0.962250
$28$		0			0
$29$	3.50000	+	6.06218i	0.649934	+	1.12572i	0.983138	+	0.182864i	$0.0585367\pi$
$29$	3.50000	+	6.06218i	0.649934	+	1.12572i	−0.333205	+	0.942855i	$0.608130\pi$
$30$		0			0
$31$	4.00000			0.718421			0.359211	−	0.933257i	$-0.383046\pi$
$31$	4.00000			0.718421			0.359211	+	0.933257i	$0.383046\pi$
$32$		0			0
$33$	−2.00000	+	3.46410i	−0.348155	+	0.603023i
$34$		0			0
$35$		0			0
$36$		0			0
$37$	−10.0000			−1.64399			−0.821995	−	0.569495i	$-0.807139\pi$
$37$	−10.0000			−1.64399			−0.821995	+	0.569495i	$0.807139\pi$
$38$		0			0
$39$	1.00000			0.160128
$40$		0			0
$41$	2.50000	−	4.33013i	0.390434	−	0.676252i	−0.602072	−	0.798441i	$-0.705658\pi$
$41$	2.50000	−	4.33013i	0.390434	−	0.676252i	0.992507	+	0.122189i	$0.0389915\pi$
$42$		0			0
$43$	−2.50000	+	4.33013i	−0.381246	+	0.660338i	−0.991241	−	0.132068i	$-0.957838\pi$
$43$	−2.50000	+	4.33013i	−0.381246	+	0.660338i	0.609994	+	0.792406i	$0.291172\pi$
$44$		0			0
$45$	−2.00000			−0.298142
$46$		0			0
$47$	3.50000	+	6.06218i	0.510527	+	0.884260i	0.999926	+	0.0121990i	$0.00388317\pi$
$47$	3.50000	+	6.06218i	0.510527	+	0.884260i	−0.489398	+	0.872060i	$0.662783\pi$
$48$		0			0
$49$	−7.00000			−1.00000
$50$		0			0
$51$	−1.50000	−	2.59808i	−0.210042	−	0.363803i
$52$		0			0
$53$	5.50000	+	9.52628i	0.755483	+	1.30854i	0.945134	+	0.326683i	$0.105931\pi$
$53$	5.50000	+	9.52628i	0.755483	+	1.30854i	−0.189651	+	0.981852i	$0.560736\pi$
$54$		0			0
$55$	−2.00000	+	3.46410i	−0.269680	+	0.467099i
$56$		0			0
$57$	−0.500000	+	4.33013i	−0.0662266	+	0.573539i
$58$		0			0
$59$	1.50000	−	2.59808i	0.195283	−	0.338241i	−0.751710	−	0.659494i	$-0.770771\pi$
$59$	1.50000	−	2.59808i	0.195283	−	0.338241i	0.946993	+	0.321253i	$0.104104\pi$
$60$		0			0
$61$	5.50000	+	9.52628i	0.704203	+	1.21972i	0.966978	+	0.254858i	$0.0820288\pi$
$61$	5.50000	+	9.52628i	0.704203	+	1.21972i	−0.262776	+	0.964857i	$0.584638\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	1.00000			0.124035
$66$		0			0
$67$	−1.50000	−	2.59808i	−0.183254	−	0.317406i	0.759733	−	0.650236i	$-0.225330\pi$
$67$	−1.50000	−	2.59808i	−0.183254	−	0.317406i	−0.942987	+	0.332830i	$0.891996\pi$
$68$		0			0
$69$	5.00000			0.601929
$70$		0			0
$71$	−5.50000	+	9.52628i	−0.652730	+	1.13056i	0.329728	+	0.944076i	$0.393043\pi$
$71$	−5.50000	+	9.52628i	−0.652730	+	1.13056i	−0.982458	+	0.186485i	$0.940290\pi$
$72$		0			0
$73$	−7.50000	+	12.9904i	−0.877809	+	1.52041i	−0.0240681	+	0.999710i	$0.507662\pi$
$73$	−7.50000	+	12.9904i	−0.877809	+	1.52041i	−0.853740	+	0.520699i	$0.825671\pi$
$74$		0			0
$75$	−4.00000			−0.461880
$76$		0			0
$77$		0			0
$78$		0			0
$79$	6.50000	−	11.2583i	0.731307	−	1.26666i	−0.225018	−	0.974355i	$-0.572244\pi$
$79$	6.50000	−	11.2583i	0.731307	−	1.26666i	0.956325	−	0.292306i	$-0.0944227\pi$
$80$		0			0
$81$	−0.500000	+	0.866025i	−0.0555556	+	0.0962250i
$82$		0			0
$83$		0			0			−	1.00000i	$-0.5\pi$
$83$		0			0				1.00000i	$0.5\pi$
$84$		0			0
$85$	−1.50000	−	2.59808i	−0.162698	−	0.281801i
$86$		0			0
$87$	−7.00000			−0.750479
$88$		0			0
$89$	−1.50000	−	2.59808i	−0.159000	−	0.275396i	0.775509	−	0.631337i	$-0.217494\pi$
$89$	−1.50000	−	2.59808i	−0.159000	−	0.275396i	−0.934508	+	0.355942i	$0.884160\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$	−2.00000	+	3.46410i	−0.207390	+	0.359211i
$94$		0			0
$95$	−0.500000	+	4.33013i	−0.0512989	+	0.444262i
$96$		0			0
$97$	2.50000	−	4.33013i	0.253837	−	0.439658i	−0.710742	−	0.703452i	$-0.751641\pi$
$97$	2.50000	−	4.33013i	0.253837	−	0.439658i	0.964579	+	0.263795i	$0.0849741\pi$
$98$		0			0
$99$	4.00000	+	6.92820i	0.402015	+	0.696311i
$100$		0			0
$101$	−0.500000	−	0.866025i	−0.0497519	−	0.0861727i	0.840077	−	0.542467i	$-0.182510\pi$
$101$	−0.500000	−	0.866025i	−0.0497519	−	0.0861727i	−0.889829	+	0.456294i	$0.849176\pi$
$102$		0			0
$103$		0			0			−	1.00000i	$-0.5\pi$
$103$		0			0				1.00000i	$0.5\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−20.0000			−1.93347			−0.966736	−	0.255774i	$-0.917670\pi$
$107$	−20.0000			−1.93347			−0.966736	+	0.255774i	$0.917670\pi$
$108$		0			0
$109$	1.50000	−	2.59808i	0.143674	−	0.248851i	−0.785203	−	0.619238i	$-0.787442\pi$
$109$	1.50000	−	2.59808i	0.143674	−	0.248851i	0.928877	+	0.370387i	$0.120775\pi$
$110$		0			0
$111$	5.00000	−	8.66025i	0.474579	−	0.821995i
$112$		0			0
$113$	14.0000			1.31701			0.658505	−	0.752577i	$-0.271189\pi$
$113$	14.0000			1.31701			0.658505	+	0.752577i	$0.271189\pi$
$114$		0			0
$115$	5.00000			0.466252
$116$		0			0
$117$	1.00000	−	1.73205i	0.0924500	−	0.160128i
$118$		0			0
$119$		0			0
$120$		0			0
$121$	5.00000			0.454545
$122$		0			0
$123$	2.50000	+	4.33013i	0.225417	+	0.390434i
$124$		0			0
$125$	−9.00000			−0.804984
$126$		0			0
$127$	1.50000	+	2.59808i	0.133103	+	0.230542i	0.924871	−	0.380280i	$-0.124172\pi$
$127$	1.50000	+	2.59808i	0.133103	+	0.230542i	−0.791768	+	0.610822i	$0.790839\pi$
$128$		0			0
$129$	−2.50000	−	4.33013i	−0.220113	−	0.381246i
$130$		0			0
$131$	7.50000	−	12.9904i	0.655278	−	1.13497i	−0.326546	−	0.945181i	$-0.605885\pi$
$131$	7.50000	−	12.9904i	0.655278	−	1.13497i	0.981824	−	0.189794i	$-0.0607819\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$	2.50000	−	4.33013i	0.215166	−	0.372678i
$136$		0			0
$137$	2.50000	+	4.33013i	0.213589	+	0.369948i	0.952835	−	0.303488i	$-0.0981512\pi$
$137$	2.50000	+	4.33013i	0.213589	+	0.369948i	−0.739246	+	0.673436i	$0.764818\pi$
$138$		0			0
$139$	4.50000	+	7.79423i	0.381685	+	0.661098i	0.991303	−	0.131597i	$-0.0420106\pi$
$139$	4.50000	+	7.79423i	0.381685	+	0.661098i	−0.609618	+	0.792695i	$0.708677\pi$
$140$		0			0
$141$	−7.00000			−0.589506
$142$		0			0
$143$	−2.00000	−	3.46410i	−0.167248	−	0.289683i
$144$		0			0
$145$	−7.00000			−0.581318
$146$		0			0
$147$	3.50000	−	6.06218i	0.288675	−	0.500000i
$148$		0			0
$149$	1.50000	−	2.59808i	0.122885	−	0.212843i	−0.798019	−	0.602632i	$-0.794119\pi$
$149$	1.50000	−	2.59808i	0.122885	−	0.212843i	0.920904	+	0.389789i	$0.127452\pi$
$150$		0			0
$151$	−16.0000			−1.30206			−0.651031	−	0.759051i	$-0.725663\pi$
$151$	−16.0000			−1.30206			−0.651031	+	0.759051i	$0.725663\pi$
$152$		0			0
$153$	−6.00000			−0.485071
$154$		0			0
$155$	−2.00000	+	3.46410i	−0.160644	+	0.278243i
$156$		0			0
$157$	3.50000	−	6.06218i	0.279330	−	0.483814i	−0.691888	−	0.722005i	$-0.743221\pi$
$157$	3.50000	−	6.06218i	0.279330	−	0.483814i	0.971219	+	0.238190i	$0.0765542\pi$
$158$		0			0
$159$	−11.0000			−0.872357
$160$		0			0
$161$		0			0
$162$		0			0
$163$	4.00000			0.313304			0.156652	−	0.987654i	$-0.449930\pi$
$163$	4.00000			0.313304			0.156652	+	0.987654i	$0.449930\pi$
$164$		0			0
$165$	−2.00000	−	3.46410i	−0.155700	−	0.269680i
$166$		0			0
$167$	7.50000	+	12.9904i	0.580367	+	1.00523i	0.995436	+	0.0954356i	$0.0304244\pi$
$167$	7.50000	+	12.9904i	0.580367	+	1.00523i	−0.415068	+	0.909790i	$0.636242\pi$
$168$		0			0
$169$	6.00000	−	10.3923i	0.461538	−	0.799408i
$170$		0			0
$171$	7.00000	+	5.19615i	0.535303	+	0.397360i
$172$		0			0
$173$	7.50000	−	12.9904i	0.570214	−	0.987640i	−0.426329	−	0.904568i	$-0.640193\pi$
$173$	7.50000	−	12.9904i	0.570214	−	0.987640i	0.996544	−	0.0830722i	$-0.0264732\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$	1.50000	+	2.59808i	0.112747	+	0.195283i
$178$		0			0
$179$	12.0000			0.896922			0.448461	−	0.893802i	$-0.351972\pi$
$179$	12.0000			0.896922			0.448461	+	0.893802i	$0.351972\pi$
$180$		0			0
$181$	−2.50000	−	4.33013i	−0.185824	−	0.321856i	0.758030	−	0.652219i	$-0.226162\pi$
$181$	−2.50000	−	4.33013i	−0.185824	−	0.321856i	−0.943854	+	0.330364i	$0.892829\pi$
$182$		0			0
$183$	−11.0000			−0.813143
$184$		0			0
$185$	5.00000	−	8.66025i	0.367607	−	0.636715i
$186$		0			0
$187$	−6.00000	+	10.3923i	−0.438763	+	0.759961i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	16.0000			1.15772			0.578860	−	0.815427i	$-0.303498\pi$
$191$	16.0000			1.15772			0.578860	+	0.815427i	$0.303498\pi$
$192$		0			0
$193$	−7.50000	+	12.9904i	−0.539862	+	0.935068i	0.459049	+	0.888411i	$0.348190\pi$
$193$	−7.50000	+	12.9904i	−0.539862	+	0.935068i	−0.998911	+	0.0466572i	$0.985143\pi$
$194$		0			0
$195$	−0.500000	+	0.866025i	−0.0358057	+	0.0620174i
$196$		0			0
$197$	−2.00000			−0.142494			−0.0712470	−	0.997459i	$-0.522698\pi$
$197$	−2.00000			−0.142494			−0.0712470	+	0.997459i	$0.522698\pi$
$198$		0			0
$199$	3.50000	+	6.06218i	0.248108	+	0.429736i	0.963001	−	0.269498i	$-0.0868577\pi$
$199$	3.50000	+	6.06218i	0.248108	+	0.429736i	−0.714893	+	0.699234i	$0.753524\pi$
$200$		0			0
$201$	3.00000			0.211604
$202$		0			0
$203$		0			0
$204$		0			0
$205$	2.50000	+	4.33013i	0.174608	+	0.302429i
$206$		0			0
$207$	5.00000	−	8.66025i	0.347524	−	0.601929i
$208$		0			0
$209$	16.0000	−	6.92820i	1.10674	−	0.479234i
$210$		0			0
$211$	−4.50000	+	7.79423i	−0.309793	+	0.536577i	−0.978317	−	0.207114i	$-0.933593\pi$
$211$	−4.50000	+	7.79423i	−0.309793	+	0.536577i	0.668524	+	0.743690i	$0.266926\pi$
$212$		0			0
$213$	−5.50000	−	9.52628i	−0.376854	−	0.652730i
$214$		0			0
$215$	−2.50000	−	4.33013i	−0.170499	−	0.295312i
$216$		0			0
$217$		0			0
$218$		0			0
$219$	−7.50000	−	12.9904i	−0.506803	−	0.877809i
$220$		0			0
$221$	3.00000			0.201802
$222$		0			0
$223$	12.5000	−	21.6506i	0.837062	−	1.44983i	−0.0552786	−	0.998471i	$-0.517605\pi$
$223$	12.5000	−	21.6506i	0.837062	−	1.44983i	0.892341	−	0.451363i	$-0.149062\pi$
$224$		0			0
$225$	−4.00000	+	6.92820i	−0.266667	+	0.461880i
$226$		0			0
$227$	20.0000			1.32745			0.663723	−	0.747978i	$-0.268975\pi$
$227$	20.0000			1.32745			0.663723	+	0.747978i	$0.268975\pi$
$228$		0			0
$229$	−2.00000			−0.132164			−0.0660819	−	0.997814i	$-0.521050\pi$
$229$	−2.00000			−0.132164			−0.0660819	+	0.997814i	$0.521050\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	10.5000	−	18.1865i	0.687878	−	1.19144i	−0.284645	−	0.958633i	$-0.591876\pi$
$233$	10.5000	−	18.1865i	0.687878	−	1.19144i	0.972523	−	0.232806i	$-0.0747909\pi$
$234$		0			0
$235$	−7.00000			−0.456630
$236$		0			0
$237$	6.50000	+	11.2583i	0.422220	+	0.731307i
$238$		0			0
$239$	12.0000			0.776215			0.388108	−	0.921614i	$-0.373129\pi$
$239$	12.0000			0.776215			0.388108	+	0.921614i	$0.373129\pi$
$240$		0			0
$241$	−9.50000	−	16.4545i	−0.611949	−	1.05993i	−0.990912	−	0.134515i	$-0.957053\pi$
$241$	−9.50000	−	16.4545i	−0.611949	−	1.05993i	0.378963	−	0.925412i	$-0.376281\pi$
$242$		0			0
$243$	−8.00000	−	13.8564i	−0.513200	−	0.888889i
$244$		0			0
$245$	3.50000	−	6.06218i	0.223607	−	0.387298i
$246$		0			0
$247$	−3.50000	−	2.59808i	−0.222700	−	0.165312i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−15.5000	−	26.8468i	−0.978351	−	1.69455i	−0.668400	−	0.743802i	$-0.733021\pi$
$251$	−15.5000	−	26.8468i	−0.978351	−	1.69455i	−0.309951	−	0.950753i	$-0.600313\pi$
$252$		0			0
$253$	−10.0000	−	17.3205i	−0.628695	−	1.08893i
$254$		0			0
$255$	3.00000			0.187867
$256$		0			0
$257$	−11.5000	−	19.9186i	−0.717350	−	1.24249i	−0.962046	−	0.272887i	$-0.912021\pi$
$257$	−11.5000	−	19.9186i	−0.717350	−	1.24249i	0.244696	−	0.969600i	$-0.421312\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$	−7.00000	+	12.1244i	−0.433289	+	0.750479i
$262$		0			0
$263$	4.50000	−	7.79423i	0.277482	−	0.480613i	−0.693276	−	0.720672i	$-0.743833\pi$
$263$	4.50000	−	7.79423i	0.277482	−	0.480613i	0.970758	+	0.240059i	$0.0771668\pi$
$264$		0			0
$265$	−11.0000			−0.675725
$266$		0			0
$267$	3.00000			0.183597
$268$		0			0
$269$	13.5000	−	23.3827i	0.823110	−	1.42567i	−0.0802460	−	0.996775i	$-0.525571\pi$
$269$	13.5000	−	23.3827i	0.823110	−	1.42567i	0.903356	−	0.428892i	$-0.141096\pi$
$270$		0			0
$271$	−15.5000	+	26.8468i	−0.941558	+	1.63083i	−0.179057	+	0.983839i	$0.557305\pi$
$271$	−15.5000	+	26.8468i	−0.941558	+	1.63083i	−0.762501	+	0.646988i	$0.776029\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	8.00000	+	13.8564i	0.482418	+	0.835573i
$276$		0			0
$277$	−10.0000			−0.600842			−0.300421	−	0.953807i	$-0.597127\pi$
$277$	−10.0000			−0.600842			−0.300421	+	0.953807i	$0.597127\pi$
$278$		0			0
$279$	4.00000	+	6.92820i	0.239474	+	0.414781i
$280$		0			0
$281$	−3.50000	−	6.06218i	−0.208792	−	0.361639i	0.742542	−	0.669800i	$-0.233620\pi$
$281$	−3.50000	−	6.06218i	−0.208792	−	0.361639i	−0.951334	+	0.308160i	$0.900287\pi$
$282$		0			0
$283$	−4.50000	+	7.79423i	−0.267497	+	0.463319i	−0.968215	−	0.250120i	$-0.919530\pi$
$283$	−4.50000	+	7.79423i	−0.267497	+	0.463319i	0.700718	+	0.713439i	$0.252863\pi$
$284$		0			0
$285$	−3.50000	−	2.59808i	−0.207322	−	0.153897i
$286$		0			0
$287$		0			0
$288$		0			0
$289$	4.00000	+	6.92820i	0.235294	+	0.407541i
$290$		0			0
$291$	2.50000	+	4.33013i	0.146553	+	0.253837i
$292$		0			0
$293$	30.0000			1.75262			0.876309	−	0.481749i	$-0.159998\pi$
$293$	30.0000			1.75262			0.876309	+	0.481749i	$0.159998\pi$
$294$		0			0
$295$	1.50000	+	2.59808i	0.0873334	+	0.151266i
$296$		0			0
$297$	−20.0000			−1.16052
$298$		0			0
$299$	−2.50000	+	4.33013i	−0.144579	+	0.250418i
$300$		0			0
$301$		0			0
$302$		0			0
$303$	1.00000			0.0574485
$304$		0			0
$305$	−11.0000			−0.629858
$306$		0			0
$307$	13.5000	−	23.3827i	0.770486	−	1.33452i	−0.166811	−	0.985989i	$-0.553347\pi$
$307$	13.5000	−	23.3827i	0.770486	−	1.33452i	0.937297	−	0.348532i	$-0.113320\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−20.0000			−1.13410			−0.567048	−	0.823685i	$-0.691915\pi$
$311$	−20.0000			−1.13410			−0.567048	+	0.823685i	$0.691915\pi$
$312$		0			0
$313$	−5.50000	−	9.52628i	−0.310878	−	0.538457i	0.667674	−	0.744453i	$-0.267290\pi$
$313$	−5.50000	−	9.52628i	−0.310878	−	0.538457i	−0.978553	+	0.205996i	$0.933957\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	7.50000	+	12.9904i	0.421242	+	0.729612i	0.996061	−	0.0886679i	$-0.0282610\pi$
$317$	7.50000	+	12.9904i	0.421242	+	0.729612i	−0.574819	+	0.818280i	$0.694928\pi$
$318$		0			0
$319$	14.0000	+	24.2487i	0.783850	+	1.35767i
$320$		0			0
$321$	10.0000	−	17.3205i	0.558146	−	0.966736i
$322$		0			0
$323$	−1.50000	+	12.9904i	−0.0834622	+	0.722804i
$324$		0			0
$325$	2.00000	−	3.46410i	0.110940	−	0.192154i
$326$		0			0
$327$	1.50000	+	2.59808i	0.0829502	+	0.143674i
$328$		0			0
$329$		0			0
$330$		0			0
$331$	−20.0000			−1.09930			−0.549650	−	0.835395i	$-0.685239\pi$
$331$	−20.0000			−1.09930			−0.549650	+	0.835395i	$0.685239\pi$
$332$		0			0
$333$	−10.0000	−	17.3205i	−0.547997	−	0.949158i
$334$		0			0
$335$	3.00000			0.163908
$336$		0			0
$337$	2.50000	−	4.33013i	0.136184	−	0.235877i	−0.789865	−	0.613280i	$-0.789850\pi$
$337$	2.50000	−	4.33013i	0.136184	−	0.235877i	0.926049	+	0.377403i	$0.123183\pi$
$338$		0			0
$339$	−7.00000	+	12.1244i	−0.380188	+	0.658505i
$340$		0			0
$341$	16.0000			0.866449
$342$		0			0
$343$		0			0
$344$		0			0
$345$	−2.50000	+	4.33013i	−0.134595	+	0.233126i
$346$		0			0
$347$	−2.50000	+	4.33013i	−0.134207	+	0.232453i	−0.925294	−	0.379250i	$-0.876182\pi$
$347$	−2.50000	+	4.33013i	−0.134207	+	0.232453i	0.791087	+	0.611703i	$0.209515\pi$
$348$		0			0
$349$	−10.0000			−0.535288			−0.267644	−	0.963518i	$-0.586245\pi$
$349$	−10.0000			−0.535288			−0.267644	+	0.963518i	$0.586245\pi$
$350$		0			0
$351$	2.50000	+	4.33013i	0.133440	+	0.231125i
$352$		0			0
$353$	30.0000			1.59674			0.798369	−	0.602168i	$-0.205696\pi$
$353$	30.0000			1.59674			0.798369	+	0.602168i	$0.205696\pi$
$354$		0			0
$355$	−5.50000	−	9.52628i	−0.291910	−	0.505602i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−7.50000	+	12.9904i	−0.395835	+	0.685606i	−0.993207	−	0.116358i	$-0.962878\pi$
$359$	−7.50000	+	12.9904i	−0.395835	+	0.685606i	0.597372	+	0.801964i	$0.296211\pi$
$360$		0			0
$361$	13.0000	−	13.8564i	0.684211	−	0.729285i
$362$		0			0
$363$	−2.50000	+	4.33013i	−0.131216	+	0.227273i
$364$		0			0
$365$	−7.50000	−	12.9904i	−0.392568	−	0.679948i
$366$		0			0
$367$	−12.5000	−	21.6506i	−0.652495	−	1.13015i	−0.982516	−	0.186180i	$-0.940389\pi$
$367$	−12.5000	−	21.6506i	−0.652495	−	1.13015i	0.330021	−	0.943974i	$-0.392944\pi$
$368$		0			0
$369$	10.0000			0.520579
$370$		0			0
$371$		0			0
$372$		0			0
$373$	10.0000			0.517780			0.258890	−	0.965907i	$-0.416643\pi$
$373$	10.0000			0.517780			0.258890	+	0.965907i	$0.416643\pi$
$374$		0			0
$375$	4.50000	−	7.79423i	0.232379	−	0.402492i
$376$		0			0
$377$	3.50000	−	6.06218i	0.180259	−	0.312218i
$378$		0			0
$379$	12.0000			0.616399			0.308199	−	0.951322i	$-0.400274\pi$
$379$	12.0000			0.616399			0.308199	+	0.951322i	$0.400274\pi$
$380$		0			0
$381$	−3.00000			−0.153695
$382$		0			0
$383$	14.5000	−	25.1147i	0.740915	−	1.28330i	−0.211164	−	0.977451i	$-0.567725\pi$
$383$	14.5000	−	25.1147i	0.740915	−	1.28330i	0.952079	−	0.305852i	$-0.0989414\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	−10.0000			−0.508329
$388$		0			0
$389$	1.50000	+	2.59808i	0.0760530	+	0.131728i	0.901544	−	0.432688i	$-0.142435\pi$
$389$	1.50000	+	2.59808i	0.0760530	+	0.131728i	−0.825491	+	0.564416i	$0.809102\pi$
$390$		0			0
$391$	15.0000			0.758583
$392$		0			0
$393$	7.50000	+	12.9904i	0.378325	+	0.655278i
$394$		0			0
$395$	6.50000	+	11.2583i	0.327050	+	0.566468i
$396$		0			0
$397$	−12.5000	+	21.6506i	−0.627357	+	1.08661i	0.360723	+	0.932673i	$0.382530\pi$
$397$	−12.5000	+	21.6506i	−0.627357	+	1.08661i	−0.988080	+	0.153941i	$0.950803\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−9.50000	+	16.4545i	−0.474407	+	0.821698i	−0.999571	−	0.0293039i	$-0.990671\pi$
$401$	−9.50000	+	16.4545i	−0.474407	+	0.821698i	0.525163	+	0.851002i	$0.324004\pi$
$402$		0			0
$403$	−2.00000	−	3.46410i	−0.0996271	−	0.172559i
$404$		0			0
$405$	−0.500000	−	0.866025i	−0.0248452	−	0.0430331i
$406$		0			0
$407$	−40.0000			−1.98273
$408$		0			0
$409$	8.50000	+	14.7224i	0.420298	+	0.727977i	0.995968	−	0.0897044i	$-0.0285922\pi$
$409$	8.50000	+	14.7224i	0.420298	+	0.727977i	−0.575670	+	0.817682i	$0.695259\pi$
$410$		0			0
$411$	−5.00000			−0.246632
$412$		0			0
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$	−9.00000			−0.440732
$418$		0			0
$419$	−12.0000			−0.586238			−0.293119	−	0.956076i	$-0.594693\pi$
$419$	−12.0000			−0.586238			−0.293119	+	0.956076i	$0.594693\pi$
$420$		0			0
$421$	−0.500000	+	0.866025i	−0.0243685	+	0.0422075i	−0.877952	−	0.478748i	$-0.841091\pi$
$421$	−0.500000	+	0.866025i	−0.0243685	+	0.0422075i	0.853584	+	0.520955i	$0.174424\pi$
$422$		0			0
$423$	−7.00000	+	12.1244i	−0.340352	+	0.589506i
$424$		0			0
$425$	−12.0000			−0.582086
$426$		0			0
$427$		0			0
$428$		0			0
$429$	4.00000			0.193122
$430$		0			0
$431$	−10.5000	−	18.1865i	−0.505767	−	0.876014i	−0.999978	−	0.00667224i	$-0.997876\pi$
$431$	−10.5000	−	18.1865i	−0.505767	−	0.876014i	0.494211	−	0.869342i	$-0.335457\pi$
$432$		0			0
$433$	12.5000	+	21.6506i	0.600712	+	1.04046i	0.992713	+	0.120499i	$0.0384494\pi$
$433$	12.5000	+	21.6506i	0.600712	+	1.04046i	−0.392002	+	0.919964i	$0.628217\pi$
$434$		0			0
$435$	3.50000	−	6.06218i	0.167812	−	0.290659i
$436$		0			0
$437$	−17.5000	−	12.9904i	−0.837139	−	0.621414i
$438$		0			0
$439$	6.50000	−	11.2583i	0.310228	−	0.537331i	−0.668184	−	0.743996i	$-0.732928\pi$
$439$	6.50000	−	11.2583i	0.310228	−	0.537331i	0.978412	+	0.206666i	$0.0662612\pi$
$440$		0			0
$441$	−7.00000	−	12.1244i	−0.333333	−	0.577350i
$442$		0			0
$443$	12.5000	+	21.6506i	0.593893	+	1.02865i	0.993702	+	0.112054i	$0.0357431\pi$
$443$	12.5000	+	21.6506i	0.593893	+	1.02865i	−0.399809	+	0.916598i	$0.630924\pi$
$444$		0			0
$445$	3.00000			0.142214
$446$		0			0
$447$	1.50000	+	2.59808i	0.0709476	+	0.122885i
$448$		0			0
$449$	22.0000			1.03824			0.519122	−	0.854700i	$-0.326259\pi$
$449$	22.0000			1.03824			0.519122	+	0.854700i	$0.326259\pi$
$450$		0			0
$451$	10.0000	−	17.3205i	0.470882	−	0.815591i
$452$		0			0
$453$	8.00000	−	13.8564i	0.375873	−	0.651031i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−22.0000			−1.02912			−0.514558	−	0.857455i	$-0.672044\pi$
$457$	−22.0000			−1.02912			−0.514558	+	0.857455i	$0.672044\pi$
$458$		0			0
$459$	7.50000	−	12.9904i	0.350070	−	0.606339i
$460$		0			0
$461$	5.50000	−	9.52628i	0.256161	−	0.443683i	−0.709050	−	0.705159i	$-0.750876\pi$
$461$	5.50000	−	9.52628i	0.256161	−	0.443683i	0.965210	+	0.261476i	$0.0842091\pi$
$462$		0			0
$463$	−20.0000			−0.929479			−0.464739	−	0.885448i	$-0.653852\pi$
$463$	−20.0000			−0.929479			−0.464739	+	0.885448i	$0.653852\pi$
$464$		0			0
$465$	−2.00000	−	3.46410i	−0.0927478	−	0.160644i
$466$		0			0
$467$	20.0000			0.925490			0.462745	−	0.886492i	$-0.346865\pi$
$467$	20.0000			0.925490			0.462745	+	0.886492i	$0.346865\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$	3.50000	+	6.06218i	0.161271	+	0.279330i
$472$		0			0
$473$	−10.0000	+	17.3205i	−0.459800	+	0.796398i
$474$		0			0
$475$	14.0000	+	10.3923i	0.642364	+	0.476832i
$476$		0			0
$477$	−11.0000	+	19.0526i	−0.503655	+	0.872357i
$478$		0			0
$479$	11.5000	+	19.9186i	0.525448	+	0.910103i	0.999561	+	0.0296389i	$0.00943575\pi$
$479$	11.5000	+	19.9186i	0.525448	+	0.910103i	−0.474112	+	0.880464i	$0.657231\pi$
$480$		0			0
$481$	5.00000	+	8.66025i	0.227980	+	0.394874i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	2.50000	+	4.33013i	0.113519	+	0.196621i
$486$		0			0
$487$	−8.00000			−0.362515			−0.181257	−	0.983436i	$-0.558017\pi$
$487$	−8.00000			−0.362515			−0.181257	+	0.983436i	$0.558017\pi$
$488$		0			0
$489$	−2.00000	+	3.46410i	−0.0904431	+	0.156652i
$490$		0			0
$491$	−0.500000	+	0.866025i	−0.0225647	+	0.0390832i	−0.877087	−	0.480331i	$-0.840517\pi$
$491$	−0.500000	+	0.866025i	−0.0225647	+	0.0390832i	0.854523	+	0.519414i	$0.173850\pi$
$492$		0			0
$493$	−21.0000			−0.945792
$494$		0			0
$495$	−8.00000			−0.359573
$496$		0			0
$497$		0			0
$498$		0			0
$499$	−2.50000	+	4.33013i	−0.111915	+	0.193843i	−0.916542	−	0.399937i	$-0.869032\pi$
$499$	−2.50000	+	4.33013i	−0.111915	+	0.193843i	0.804627	+	0.593780i	$0.202365\pi$
$500$		0			0
$501$	−15.0000			−0.670151
$502$		0			0
$503$	−10.5000	−	18.1865i	−0.468172	−	0.810897i	0.531167	−	0.847267i	$-0.321754\pi$
$503$	−10.5000	−	18.1865i	−0.468172	−	0.810897i	−0.999338	+	0.0363700i	$0.988421\pi$
$504$		0			0
$505$	1.00000			0.0444994
$506$		0			0
$507$	6.00000	+	10.3923i	0.266469	+	0.461538i
$508$		0			0
$509$	7.50000	+	12.9904i	0.332432	+	0.575789i	0.982988	−	0.183669i	$-0.0587976\pi$
$509$	7.50000	+	12.9904i	0.332432	+	0.575789i	−0.650556	+	0.759458i	$0.725464\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$	−20.0000	+	8.66025i	−0.883022	+	0.382360i
$514$		0			0
$515$		0			0
$516$		0			0
$517$	14.0000	+	24.2487i	0.615719	+	1.06646i
$518$		0			0
$519$	7.50000	+	12.9904i	0.329213	+	0.570214i
$520$		0			0
$521$	26.0000			1.13908			0.569540	−	0.821963i	$-0.307121\pi$
$521$	26.0000			1.13908			0.569540	+	0.821963i	$0.307121\pi$
$522$		0			0
$523$	−9.50000	−	16.4545i	−0.415406	−	0.719504i	0.580065	−	0.814570i	$-0.303027\pi$
$523$	−9.50000	−	16.4545i	−0.415406	−	0.719504i	−0.995471	+	0.0950659i	$0.969694\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−6.00000	+	10.3923i	−0.261364	+	0.452696i
$528$		0			0
$529$	−1.00000	+	1.73205i	−0.0434783	+	0.0753066i
$530$		0			0
$531$	6.00000			0.260378
$532$		0			0
$533$	−5.00000			−0.216574
$534$		0			0
$535$	10.0000	−	17.3205i	0.432338	−	0.748831i
$536$		0			0
$537$	−6.00000	+	10.3923i	−0.258919	+	0.448461i
$538$		0			0
$539$	−28.0000			−1.20605
$540$		0			0
$541$	−0.500000	−	0.866025i	−0.0214967	−	0.0372333i	0.855077	−	0.518501i	$-0.173510\pi$
$541$	−0.500000	−	0.866025i	−0.0214967	−	0.0372333i	−0.876574	+	0.481268i	$0.840176\pi$
$542$		0			0
$543$	5.00000			0.214571
$544$		0			0
$545$	1.50000	+	2.59808i	0.0642529	+	0.111289i
$546$		0			0
$547$	12.5000	+	21.6506i	0.534461	+	0.925714i	0.999189	+	0.0402607i	$0.0128188\pi$
$547$	12.5000	+	21.6506i	0.534461	+	0.925714i	−0.464728	+	0.885454i	$0.653848\pi$
$548$		0			0
$549$	−11.0000	+	19.0526i	−0.469469	+	0.813143i
$550$		0			0
$551$	24.5000	+	18.1865i	1.04374	+	0.774772i
$552$		0			0
$553$		0			0
$554$		0			0
$555$	5.00000	+	8.66025i	0.212238	+	0.367607i
$556$		0			0
$557$	−2.50000	−	4.33013i	−0.105928	−	0.183473i	0.808189	−	0.588924i	$-0.200448\pi$
$557$	−2.50000	−	4.33013i	−0.105928	−	0.183473i	−0.914117	+	0.405450i	$0.867115\pi$
$558$		0			0
$559$	5.00000			0.211477
$560$		0			0
$561$	−6.00000	−	10.3923i	−0.253320	−	0.438763i
$562$		0			0
$563$	−36.0000			−1.51722			−0.758610	−	0.651546i	$-0.774121\pi$
$563$	−36.0000			−1.51722			−0.758610	+	0.651546i	$0.774121\pi$
$564$		0			0
$565$	−7.00000	+	12.1244i	−0.294492	+	0.510075i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	22.0000			0.922288			0.461144	−	0.887325i	$-0.347439\pi$
$569$	22.0000			0.922288			0.461144	+	0.887325i	$0.347439\pi$
$570$		0			0
$571$	4.00000			0.167395			0.0836974	−	0.996491i	$-0.473327\pi$
$571$	4.00000			0.167395			0.0836974	+	0.996491i	$0.473327\pi$
$572$		0			0
$573$	−8.00000	+	13.8564i	−0.334205	+	0.578860i
$574$		0			0
$575$	10.0000	−	17.3205i	0.417029	−	0.722315i
$576$		0			0
$577$	−10.0000			−0.416305			−0.208153	−	0.978096i	$-0.566745\pi$
$577$	−10.0000			−0.416305			−0.208153	+	0.978096i	$0.566745\pi$
$578$		0			0
$579$	−7.50000	−	12.9904i	−0.311689	−	0.539862i
$580$		0			0
$581$		0			0
$582$		0			0
$583$	22.0000	+	38.1051i	0.911147	+	1.57815i
$584$		0			0
$585$	1.00000	+	1.73205i	0.0413449	+	0.0716115i
$586$		0			0
$587$	7.50000	−	12.9904i	0.309558	−	0.536170i	−0.668708	−	0.743525i	$-0.733152\pi$
$587$	7.50000	−	12.9904i	0.309558	−	0.536170i	0.978266	+	0.207355i	$0.0664855\pi$
$588$		0			0
$589$	16.0000	−	6.92820i	0.659269	−	0.285472i
$590$		0			0
$591$	1.00000	−	1.73205i	0.0411345	−	0.0712470i
$592$		0			0
$593$	2.50000	+	4.33013i	0.102663	+	0.177817i	0.912781	−	0.408450i	$-0.133930\pi$
$593$	2.50000	+	4.33013i	0.102663	+	0.177817i	−0.810118	+	0.586267i	$0.800597\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	−7.00000			−0.286491
$598$		0			0
$599$	−22.5000	−	38.9711i	−0.919325	−	1.59232i	−0.800443	−	0.599409i	$-0.795402\pi$
$599$	−22.5000	−	38.9711i	−0.919325	−	1.59232i	−0.118882	−	0.992908i	$-0.537931\pi$
$600$		0			0
$601$	−26.0000			−1.06056			−0.530281	−	0.847822i	$-0.677914\pi$
$601$	−26.0000			−1.06056			−0.530281	+	0.847822i	$0.677914\pi$
$602$		0			0
$603$	3.00000	−	5.19615i	0.122169	−	0.211604i
$604$		0			0
$605$	−2.50000	+	4.33013i	−0.101639	+	0.176045i
$606$		0			0
$607$	32.0000			1.29884			0.649420	−	0.760430i	$-0.275012\pi$
$607$	32.0000			1.29884			0.649420	+	0.760430i	$0.275012\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	3.50000	−	6.06218i	0.141595	−	0.245249i
$612$		0			0
$613$	−14.5000	+	25.1147i	−0.585649	+	1.01437i	0.409145	+	0.912470i	$0.365827\pi$
$613$	−14.5000	+	25.1147i	−0.585649	+	1.01437i	−0.994794	+	0.101905i	$0.967506\pi$
$614$		0			0
$615$	−5.00000			−0.201619
$616$		0			0
$617$	22.5000	+	38.9711i	0.905816	+	1.56892i	0.819818	+	0.572624i	$0.194074\pi$
$617$	22.5000	+	38.9711i	0.905816	+	1.56892i	0.0859976	+	0.996295i	$0.472592\pi$
$618$		0			0
$619$	20.0000			0.803868			0.401934	−	0.915669i	$-0.368338\pi$
$619$	20.0000			0.803868			0.401934	+	0.915669i	$0.368338\pi$
$620$		0			0
$621$	12.5000	+	21.6506i	0.501608	+	0.868810i
$622$		0			0
$623$		0			0
$624$		0			0
$625$	−5.50000	+	9.52628i	−0.220000	+	0.381051i
$626$		0			0
$627$	−2.00000	+	17.3205i	−0.0798723	+	0.691714i
$628$		0			0
$629$	15.0000	−	25.9808i	0.598089	−	1.03592i
$630$		0			0
$631$	−20.5000	−	35.5070i	−0.816092	−	1.41351i	−0.908541	−	0.417796i	$-0.862803\pi$
$631$	−20.5000	−	35.5070i	−0.816092	−	1.41351i	0.0924489	−	0.995717i	$-0.470531\pi$
$632$		0			0
$633$	−4.50000	−	7.79423i	−0.178859	−	0.309793i
$634$		0			0
$635$	−3.00000			−0.119051
$636$		0			0
$637$	3.50000	+	6.06218i	0.138675	+	0.240192i
$638$		0			0
$639$	−22.0000			−0.870307
$640$		0			0
$641$	−19.5000	+	33.7750i	−0.770204	+	1.33403i	0.167247	+	0.985915i	$0.446512\pi$
$641$	−19.5000	+	33.7750i	−0.770204	+	1.33403i	−0.937451	+	0.348117i	$0.886821\pi$
$642$		0			0
$643$	9.50000	−	16.4545i	0.374643	−	0.648901i	−0.615630	−	0.788035i	$-0.711098\pi$
$643$	9.50000	−	16.4545i	0.374643	−	0.648901i	0.990274	+	0.139134i	$0.0444318\pi$
$644$		0			0
$645$	5.00000			0.196875
$646$		0			0
$647$	32.0000			1.25805			0.629025	−	0.777385i	$-0.283454\pi$
$647$	32.0000			1.25805			0.629025	+	0.777385i	$0.283454\pi$
$648$		0			0
$649$	6.00000	−	10.3923i	0.235521	−	0.407934i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	30.0000			1.17399			0.586995	−	0.809590i	$-0.300311\pi$
$653$	30.0000			1.17399			0.586995	+	0.809590i	$0.300311\pi$
$654$		0			0
$655$	7.50000	+	12.9904i	0.293049	+	0.507576i
$656$		0			0
$657$	−30.0000			−1.17041
$658$		0			0
$659$	2.50000	+	4.33013i	0.0973862	+	0.168678i	0.910602	−	0.413284i	$-0.135618\pi$
$659$	2.50000	+	4.33013i	0.0973862	+	0.168678i	−0.813216	+	0.581962i	$0.802285\pi$
$660$		0			0
$661$	15.5000	+	26.8468i	0.602880	+	1.04422i	0.992383	+	0.123194i	$0.0393136\pi$
$661$	15.5000	+	26.8468i	0.602880	+	1.04422i	−0.389503	+	0.921025i	$0.627353\pi$
$662$		0			0
$663$	−1.50000	+	2.59808i	−0.0582552	+	0.100901i
$664$		0			0
$665$		0			0
$666$		0			0
$667$	17.5000	−	30.3109i	0.677603	−	1.17364i
$668$		0			0
$669$	12.5000	+	21.6506i	0.483278	+	0.837062i
$670$		0			0
$671$	22.0000	+	38.1051i	0.849301	+	1.47103i
$672$		0			0
$673$	−10.0000			−0.385472			−0.192736	−	0.981251i	$-0.561736\pi$
$673$	−10.0000			−0.385472			−0.192736	+	0.981251i	$0.561736\pi$
$674$		0			0
$675$	−10.0000	−	17.3205i	−0.384900	−	0.666667i
$676$		0			0
$677$	−10.0000			−0.384331			−0.192166	−	0.981363i	$-0.561551\pi$
$677$	−10.0000			−0.384331			−0.192166	+	0.981363i	$0.561551\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$	−10.0000	+	17.3205i	−0.383201	+	0.663723i
$682$		0			0
$683$	−16.0000			−0.612223			−0.306111	−	0.951996i	$-0.599028\pi$
$683$	−16.0000			−0.612223			−0.306111	+	0.951996i	$0.599028\pi$
$684$		0			0
$685$	−5.00000			−0.191040
$686$		0			0
$687$	1.00000	−	1.73205i	0.0381524	−	0.0660819i
$688$		0			0
$689$	5.50000	−	9.52628i	0.209533	−	0.362922i
$690$		0			0
$691$	−36.0000			−1.36950			−0.684752	−	0.728776i	$-0.740090\pi$
$691$	−36.0000			−1.36950			−0.684752	+	0.728776i	$0.740090\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−9.00000			−0.341389
$696$		0			0
$697$	7.50000	+	12.9904i	0.284083	+	0.492046i
$698$		0			0
$699$	10.5000	+	18.1865i	0.397146	+	0.687878i
$700$		0			0
$701$	17.5000	−	30.3109i	0.660966	−	1.14483i	−0.319396	−	0.947621i	$-0.603480\pi$
$701$	17.5000	−	30.3109i	0.660966	−	1.14483i	0.980362	−	0.197205i	$-0.0631865\pi$
$702$		0			0
$703$	−40.0000	+	17.3205i	−1.50863	+	0.653255i
$704$		0			0
$705$	3.50000	−	6.06218i	0.131818	−	0.228315i
$706$		0			0
$707$		0			0
$708$		0			0
$709$	1.50000	+	2.59808i	0.0563337	+	0.0975728i	0.892817	−	0.450420i	$-0.148726\pi$
$709$	1.50000	+	2.59808i	0.0563337	+	0.0975728i	−0.836483	+	0.547992i	$0.815392\pi$
$710$		0			0
$711$	26.0000			0.975076
$712$		0			0
$713$	−10.0000	−	17.3205i	−0.374503	−	0.648658i
$714$		0			0
$715$	4.00000			0.149592
$716$		0			0
$717$	−6.00000	+	10.3923i	−0.224074	+	0.388108i
$718$		0			0
$719$	−3.50000	+	6.06218i	−0.130528	+	0.226081i	−0.923880	−	0.382682i	$-0.875001\pi$
$719$	−3.50000	+	6.06218i	−0.130528	+	0.226081i	0.793352	+	0.608763i	$0.208334\pi$
$720$		0			0
$721$		0			0
$722$		0			0
$723$	19.0000			0.706618
$724$		0			0
$725$	−14.0000	+	24.2487i	−0.519947	+	0.900575i
$726$		0			0
$727$	−3.50000	+	6.06218i	−0.129808	+	0.224834i	−0.923602	−	0.383353i	$-0.874769\pi$
$727$	−3.50000	+	6.06218i	−0.129808	+	0.224834i	0.793794	+	0.608186i	$0.208103\pi$
$728$		0			0
$729$	13.0000			0.481481
$730$		0			0
$731$	−7.50000	−	12.9904i	−0.277398	−	0.480467i
$732$		0			0
$733$	30.0000			1.10808			0.554038	−	0.832492i	$-0.313086\pi$
$733$	30.0000			1.10808			0.554038	+	0.832492i	$0.313086\pi$
$734$		0			0
$735$	3.50000	+	6.06218i	0.129099	+	0.223607i
$736$		0			0
$737$	−6.00000	−	10.3923i	−0.221013	−	0.382805i
$738$		0			0
$739$	−6.50000	+	11.2583i	−0.239106	+	0.414144i	−0.960458	−	0.278425i	$-0.910188\pi$
$739$	−6.50000	+	11.2583i	−0.239106	+	0.414144i	0.721352	+	0.692569i	$0.243521\pi$
$740$		0			0
$741$	4.00000	−	1.73205i	0.146944	−	0.0636285i
$742$		0			0
$743$	12.5000	−	21.6506i	0.458581	−	0.794285i	−0.540306	−	0.841469i	$-0.681691\pi$
$743$	12.5000	−	21.6506i	0.458581	−	0.794285i	0.998886	+	0.0471840i	$0.0150247\pi$
$744$		0			0
$745$	1.50000	+	2.59808i	0.0549557	+	0.0951861i
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	−14.5000	−	25.1147i	−0.529113	−	0.916450i	−0.999424	−	0.0339490i	$-0.989192\pi$
$751$	−14.5000	−	25.1147i	−0.529113	−	0.916450i	0.470311	−	0.882501i	$-0.344142\pi$
$752$		0			0
$753$	31.0000			1.12970
$754$		0			0
$755$	8.00000	−	13.8564i	0.291150	−	0.504286i
$756$		0			0
$757$	−8.50000	+	14.7224i	−0.308938	+	0.535096i	−0.978130	−	0.207993i	$-0.933307\pi$
$757$	−8.50000	+	14.7224i	−0.308938	+	0.535096i	0.669193	+	0.743089i	$0.266640\pi$
$758$		0			0
$759$	20.0000			0.725954
$760$		0			0
$761$	30.0000			1.08750			0.543750	−	0.839248i	$-0.317004\pi$
$761$	30.0000			1.08750			0.543750	+	0.839248i	$0.317004\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$	3.00000	−	5.19615i	0.108465	−	0.187867i
$766$		0			0
$767$	−3.00000			−0.108324
$768$		0			0
$769$	−3.50000	−	6.06218i	−0.126213	−	0.218608i	0.795993	−	0.605305i	$-0.206949\pi$
$769$	−3.50000	−	6.06218i	−0.126213	−	0.218608i	−0.922207	+	0.386698i	$0.873616\pi$
$770$		0			0
$771$	23.0000			0.828325
$772$		0			0
$773$	−10.5000	−	18.1865i	−0.377659	−	0.654124i	0.613062	−	0.790034i	$-0.289937\pi$
$773$	−10.5000	−	18.1865i	−0.377659	−	0.654124i	−0.990721	+	0.135910i	$0.956604\pi$
$774$		0			0
$775$	8.00000	+	13.8564i	0.287368	+	0.497737i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	2.50000	−	21.6506i	0.0895718	−	0.775715i
$780$		0			0
$781$	−22.0000	+	38.1051i	−0.787222	+	1.36351i
$782$		0			0
$783$	−17.5000	−	30.3109i	−0.625399	−	1.08322i
$784$		0			0
$785$	3.50000	+	6.06218i	0.124920	+	0.216368i
$786$		0			0
$787$	−52.0000			−1.85360			−0.926800	−	0.375555i	$-0.877452\pi$
$787$	−52.0000			−1.85360			−0.926800	+	0.375555i	$0.877452\pi$
$788$		0			0
$789$	4.50000	+	7.79423i	0.160204	+	0.277482i
$790$		0			0
$791$		0			0
$792$		0

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:	\( N \)	\(=\)	\( 1216 = 2^{6} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1216.i (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-0.500000 + 0.866025i) q^{3} +(-0.500000 + 0.866025i) q^{5} +(1.00000 + 1.73205i) q^{9} +O(q^{10})\)	\(q+(-0.500000 + 0.866025i) q^{3} +(-0.500000 + 0.866025i) q^{5} +(1.00000 + 1.73205i) q^{9} +4.00000 q^{11} +(-0.500000 - 0.866025i) q^{13} +(-0.500000 - 0.866025i) q^{15} +(-1.50000 + 2.59808i) q^{17} +(4.00000 - 1.73205i) q^{19} +(-2.50000 - 4.33013i) q^{23} +(2.00000 + 3.46410i) q^{25} -5.00000 q^{27} +(3.50000 + 6.06218i) q^{29} +4.00000 q^{31} +(-2.00000 + 3.46410i) q^{33} -10.0000 q^{37} +1.00000 q^{39} +(2.50000 - 4.33013i) q^{41} +(-2.50000 + 4.33013i) q^{43} -2.00000 q^{45} +(3.50000 + 6.06218i) q^{47} -7.00000 q^{49} +(-1.50000 - 2.59808i) q^{51} +(5.50000 + 9.52628i) q^{53} +(-2.00000 + 3.46410i) q^{55} +(-0.500000 + 4.33013i) q^{57} +(1.50000 - 2.59808i) q^{59} +(5.50000 + 9.52628i) q^{61} +1.00000 q^{65} +(-1.50000 - 2.59808i) q^{67} +5.00000 q^{69} +(-5.50000 + 9.52628i) q^{71} +(-7.50000 + 12.9904i) q^{73} -4.00000 q^{75} +(6.50000 - 11.2583i) q^{79} +(-0.500000 + 0.866025i) q^{81} +(-1.50000 - 2.59808i) q^{85} -7.00000 q^{87} +(-1.50000 - 2.59808i) q^{89} +(-2.00000 + 3.46410i) q^{93} +(-0.500000 + 4.33013i) q^{95} +(2.50000 - 4.33013i) q^{97} +(4.00000 + 6.92820i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - q^{3} - q^{5} + 2 q^{9}+O(q^{10}) \) 2 * q - q^3 - q^5 + 2 * q^9	\( 2 q - q^{3} - q^{5} + 2 q^{9} + 8 q^{11} - q^{13} - q^{15} - 3 q^{17} + 8 q^{19} - 5 q^{23} + 4 q^{25} - 10 q^{27} + 7 q^{29} + 8 q^{31} - 4 q^{33} - 20 q^{37} + 2 q^{39} + 5 q^{41} - 5 q^{43} - 4 q^{45} + 7 q^{47} - 14 q^{49} - 3 q^{51} + 11 q^{53} - 4 q^{55} - q^{57} + 3 q^{59} + 11 q^{61} + 2 q^{65} - 3 q^{67} + 10 q^{69} - 11 q^{71} - 15 q^{73} - 8 q^{75} + 13 q^{79} - q^{81} - 3 q^{85} - 14 q^{87} - 3 q^{89} - 4 q^{93} - q^{95} + 5 q^{97} + 8 q^{99}+O(q^{100}) \) 2 * q - q^3 - q^5 + 2 * q^9 + 8 * q^11 - q^13 - q^15 - 3 * q^17 + 8 * q^19 - 5 * q^23 + 4 * q^25 - 10 * q^27 + 7 * q^29 + 8 * q^31 - 4 * q^33 - 20 * q^37 + 2 * q^39 + 5 * q^41 - 5 * q^43 - 4 * q^45 + 7 * q^47 - 14 * q^49 - 3 * q^51 + 11 * q^53 - 4 * q^55 - q^57 + 3 * q^59 + 11 * q^61 + 2 * q^65 - 3 * q^67 + 10 * q^69 - 11 * q^71 - 15 * q^73 - 8 * q^75 + 13 * q^79 - q^81 - 3 * q^85 - 14 * q^87 - 3 * q^89 - 4 * q^93 - q^95 + 5 * q^97 + 8 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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