LMFDB - Embedded newform 1176.2.q.i.361.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1176,2,Mod(361,1176)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1176, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 0, 0, 4]))

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

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

chi := DirichletCharacter("1176.361");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1176 = 2^{3} \cdot 3 \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1176.q (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:	$9.39040727770$
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 24)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			361.1
Root			$0.500000 + 0.866025i$ of defining polynomial
Character	$\chi$	$=$	1176.361
Dual form			1176.2.q.i.961.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+(0.500000 - 0.866025i) q^{3} +(1.00000 + 1.73205i) q^{5} +(-0.500000 - 0.866025i) q^{9} +(-2.00000 + 3.46410i) q^{11} -2.00000 q^{13} +2.00000 q^{15} +(-1.00000 + 1.73205i) q^{17} +(2.00000 + 3.46410i) q^{19} +(4.00000 + 6.92820i) q^{23} +(0.500000 - 0.866025i) q^{25} -1.00000 q^{27} +6.00000 q^{29} +(-4.00000 + 6.92820i) q^{31} +(2.00000 + 3.46410i) q^{33} +(-3.00000 - 5.19615i) q^{37} +(-1.00000 + 1.73205i) q^{39} -6.00000 q^{41} +4.00000 q^{43} +(1.00000 - 1.73205i) q^{45} +(1.00000 + 1.73205i) q^{51} +(1.00000 - 1.73205i) q^{53} -8.00000 q^{55} +4.00000 q^{57} +(-2.00000 + 3.46410i) q^{59} +(1.00000 + 1.73205i) q^{61} +(-2.00000 - 3.46410i) q^{65} +(2.00000 - 3.46410i) q^{67} +8.00000 q^{69} +8.00000 q^{71} +(-5.00000 + 8.66025i) q^{73} +(-0.500000 - 0.866025i) q^{75} +(4.00000 + 6.92820i) q^{79} +(-0.500000 + 0.866025i) q^{81} -4.00000 q^{83} -4.00000 q^{85} +(3.00000 - 5.19615i) q^{87} +(3.00000 + 5.19615i) q^{89} +(4.00000 + 6.92820i) q^{93} +(-4.00000 + 6.92820i) q^{95} +2.00000 q^{97} +4.00000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + q^{3} + 2 q^{5} - q^{9} - 4 q^{11} - 4 q^{13} + 4 q^{15} - 2 q^{17} + 4 q^{19} + 8 q^{23} + q^{25} - 2 q^{27} + 12 q^{29} - 8 q^{31} + 4 q^{33} - 6 q^{37} - 2 q^{39} - 12 q^{41} + 8 q^{43} + 2 q^{45}+ \cdots + 8 q^{99}+O(q^{100}) $ 2 * q + q^3 + 2 * q^5 - q^9 - 4 * q^11 - 4 * q^13 + 4 * q^15 - 2 * q^17 + 4 * q^19 + 8 * q^23 + q^25 - 2 * q^27 + 12 * q^29 - 8 * q^31 + 4 * q^33 - 6 * q^37 - 2 * q^39 - 12 * q^41 + 8 * q^43 + 2 * q^45 + 2 * q^51 + 2 * q^53 - 16 * q^55 + 8 * q^57 - 4 * q^59 + 2 * q^61 - 4 * q^65 + 4 * q^67 + 16 * q^69 + 16 * q^71 - 10 * q^73 - q^75 + 8 * q^79 - q^81 - 8 * q^83 - 8 * q^85 + 6 * q^87 + 6 * q^89 + 8 * q^93 - 8 * q^95 + 4 * q^97 + 8 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1176\mathbb{Z}\right)^\times$.

$n$	$295$	$589$	$785$	$1081$
$\chi(n)$	$1$	$1$	$1$	$e\left(\frac{2}{3}\right)$

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
$5$	1.00000	+	1.73205i	0.447214	+	0.774597i	0.998203	−	0.0599153i	$-0.0190830\pi$
$5$	1.00000	+	1.73205i	0.447214	+	0.774597i	−0.550990	+	0.834512i	$0.685750\pi$
$6$		0			0
$7$		0			0
$7$		0			0
$8$		0			0
$9$	−0.500000	−	0.866025i	−0.166667	−	0.288675i
$10$		0			0
$11$	−2.00000	+	3.46410i	−0.603023	+	1.04447i	0.389338	+	0.921095i	$0.372704\pi$
$11$	−2.00000	+	3.46410i	−0.603023	+	1.04447i	−0.992361	+	0.123371i	$0.960630\pi$
$12$		0			0
$13$	−2.00000			−0.554700			−0.277350	−	0.960769i	$-0.589456\pi$
$13$	−2.00000			−0.554700			−0.277350	+	0.960769i	$0.589456\pi$
$14$		0			0
$15$	2.00000			0.516398
$16$		0			0
$17$	−1.00000	+	1.73205i	−0.242536	+	0.420084i	−0.961436	−	0.275029i	$-0.911312\pi$
$17$	−1.00000	+	1.73205i	−0.242536	+	0.420084i	0.718900	+	0.695113i	$0.244646\pi$
$18$		0			0
$19$	2.00000	+	3.46410i	0.458831	+	0.794719i	0.998899	−	0.0469020i	$-0.0149348\pi$
$19$	2.00000	+	3.46410i	0.458831	+	0.794719i	−0.540068	+	0.841621i	$0.681602\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	4.00000	+	6.92820i	0.834058	+	1.44463i	0.894795	+	0.446476i	$0.147321\pi$
$23$	4.00000	+	6.92820i	0.834058	+	1.44463i	−0.0607377	+	0.998154i	$0.519345\pi$
$24$		0			0
$25$	0.500000	−	0.866025i	0.100000	−	0.173205i
$26$		0			0
$27$	−1.00000			−0.192450
$28$		0			0
$29$	6.00000			1.11417			0.557086	−	0.830455i	$-0.311919\pi$
$29$	6.00000			1.11417			0.557086	+	0.830455i	$0.311919\pi$
$30$		0			0
$31$	−4.00000	+	6.92820i	−0.718421	+	1.24434i	0.243204	+	0.969975i	$0.421802\pi$
$31$	−4.00000	+	6.92820i	−0.718421	+	1.24434i	−0.961625	+	0.274367i	$0.911532\pi$
$32$		0			0
$33$	2.00000	+	3.46410i	0.348155	+	0.603023i
$34$		0			0
$35$		0			0
$36$		0			0
$37$	−3.00000	−	5.19615i	−0.493197	−	0.854242i	0.506772	−	0.862080i	$-0.330838\pi$
$37$	−3.00000	−	5.19615i	−0.493197	−	0.854242i	−0.999969	+	0.00783774i	$0.997505\pi$
$38$		0			0
$39$	−1.00000	+	1.73205i	−0.160128	+	0.277350i
$40$		0			0
$41$	−6.00000			−0.937043			−0.468521	−	0.883452i	$-0.655213\pi$
$41$	−6.00000			−0.937043			−0.468521	+	0.883452i	$0.655213\pi$
$42$		0			0
$43$	4.00000			0.609994			0.304997	−	0.952353i	$-0.401344\pi$
$43$	4.00000			0.609994			0.304997	+	0.952353i	$0.401344\pi$
$44$		0			0
$45$	1.00000	−	1.73205i	0.149071	−	0.258199i
$46$		0			0
$47$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$47$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$48$		0			0
$49$		0			0
$50$		0			0
$51$	1.00000	+	1.73205i	0.140028	+	0.242536i
$52$		0			0
$53$	1.00000	−	1.73205i	0.137361	−	0.237915i	−0.789136	−	0.614218i	$-0.789471\pi$
$53$	1.00000	−	1.73205i	0.137361	−	0.237915i	0.926497	+	0.376303i	$0.122805\pi$
$54$		0			0
$55$	−8.00000			−1.07872
$56$		0			0
$57$	4.00000			0.529813
$58$		0			0
$59$	−2.00000	+	3.46410i	−0.260378	+	0.450988i	−0.966342	−	0.257260i	$-0.917180\pi$
$59$	−2.00000	+	3.46410i	−0.260378	+	0.450988i	0.705965	+	0.708247i	$0.250514\pi$
$60$		0			0
$61$	1.00000	+	1.73205i	0.128037	+	0.221766i	0.922916	−	0.385002i	$-0.125799\pi$
$61$	1.00000	+	1.73205i	0.128037	+	0.221766i	−0.794879	+	0.606768i	$0.792466\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−2.00000	−	3.46410i	−0.248069	−	0.429669i
$66$		0			0
$67$	2.00000	−	3.46410i	0.244339	−	0.423207i	−0.717607	−	0.696449i	$-0.754762\pi$
$67$	2.00000	−	3.46410i	0.244339	−	0.423207i	0.961946	+	0.273241i	$0.0880957\pi$
$68$		0			0
$69$	8.00000			0.963087
$70$		0			0
$71$	8.00000			0.949425			0.474713	−	0.880141i	$-0.342552\pi$
$71$	8.00000			0.949425			0.474713	+	0.880141i	$0.342552\pi$
$72$		0			0
$73$	−5.00000	+	8.66025i	−0.585206	+	1.01361i	0.409644	+	0.912245i	$0.365653\pi$
$73$	−5.00000	+	8.66025i	−0.585206	+	1.01361i	−0.994850	+	0.101361i	$0.967680\pi$
$74$		0			0
$75$	−0.500000	−	0.866025i	−0.0577350	−	0.100000i
$76$		0			0
$77$		0			0
$78$		0			0
$79$	4.00000	+	6.92820i	0.450035	+	0.779484i	0.998388	−	0.0567635i	$-0.0180781\pi$
$79$	4.00000	+	6.92820i	0.450035	+	0.779484i	−0.548352	+	0.836247i	$0.684745\pi$
$80$		0			0
$81$	−0.500000	+	0.866025i	−0.0555556	+	0.0962250i
$82$		0			0
$83$	−4.00000			−0.439057			−0.219529	−	0.975606i	$-0.570452\pi$
$83$	−4.00000			−0.439057			−0.219529	+	0.975606i	$0.570452\pi$
$84$		0			0
$85$	−4.00000			−0.433861
$86$		0			0
$87$	3.00000	−	5.19615i	0.321634	−	0.557086i
$88$		0			0
$89$	3.00000	+	5.19615i	0.317999	+	0.550791i	0.980071	−	0.198650i	$-0.0636557\pi$
$89$	3.00000	+	5.19615i	0.317999	+	0.550791i	−0.662071	+	0.749441i	$0.730322\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$	4.00000	+	6.92820i	0.414781	+	0.718421i
$94$		0			0
$95$	−4.00000	+	6.92820i	−0.410391	+	0.710819i
$96$		0			0
$97$	2.00000			0.203069			0.101535	−	0.994832i	$-0.467625\pi$
$97$	2.00000			0.203069			0.101535	+	0.994832i	$0.467625\pi$
$98$		0			0
$99$	4.00000			0.402015
$100$		0			0
$101$	9.00000	−	15.5885i	0.895533	−	1.55111i	0.0623905	−	0.998052i	$-0.480128\pi$
$101$	9.00000	−	15.5885i	0.895533	−	1.55111i	0.833143	−	0.553058i	$-0.186539\pi$
$102$		0			0
$103$	−8.00000	−	13.8564i	−0.788263	−	1.36531i	−0.927030	−	0.374987i	$-0.877647\pi$
$103$	−8.00000	−	13.8564i	−0.788263	−	1.36531i	0.138767	−	0.990325i	$-0.455686\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	6.00000	+	10.3923i	0.580042	+	1.00466i	0.995474	+	0.0950377i	$0.0302972\pi$
$107$	6.00000	+	10.3923i	0.580042	+	1.00466i	−0.415432	+	0.909624i	$0.636370\pi$
$108$		0			0
$109$	1.00000	−	1.73205i	0.0957826	−	0.165900i	−0.814152	−	0.580651i	$-0.802798\pi$
$109$	1.00000	−	1.73205i	0.0957826	−	0.165900i	0.909935	+	0.414751i	$0.136131\pi$
$110$		0			0
$111$	−6.00000			−0.569495
$112$		0			0
$113$	18.0000			1.69330			0.846649	−	0.532152i	$-0.178617\pi$
$113$	18.0000			1.69330			0.846649	+	0.532152i	$0.178617\pi$
$114$		0			0
$115$	−8.00000	+	13.8564i	−0.746004	+	1.29212i
$116$		0			0
$117$	1.00000	+	1.73205i	0.0924500	+	0.160128i
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−2.50000	−	4.33013i	−0.227273	−	0.393648i
$122$		0			0
$123$	−3.00000	+	5.19615i	−0.270501	+	0.468521i
$124$		0			0
$125$	12.0000			1.07331
$126$		0			0
$127$	−8.00000			−0.709885			−0.354943	−	0.934888i	$-0.615500\pi$
$127$	−8.00000			−0.709885			−0.354943	+	0.934888i	$0.615500\pi$
$128$		0			0
$129$	2.00000	−	3.46410i	0.176090	−	0.304997i
$130$		0			0
$131$	2.00000	+	3.46410i	0.174741	+	0.302660i	0.940072	−	0.340977i	$-0.110758\pi$
$131$	2.00000	+	3.46410i	0.174741	+	0.302660i	−0.765331	+	0.643637i	$0.777425\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$	−1.00000	−	1.73205i	−0.0860663	−	0.149071i
$136$		0			0
$137$	3.00000	−	5.19615i	0.256307	−	0.443937i	−0.708942	−	0.705266i	$-0.750827\pi$
$137$	3.00000	−	5.19615i	0.256307	−	0.443937i	0.965250	+	0.261329i	$0.0841608\pi$
$138$		0			0
$139$	−12.0000			−1.01783			−0.508913	−	0.860818i	$-0.669953\pi$
$139$	−12.0000			−1.01783			−0.508913	+	0.860818i	$0.669953\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	4.00000	−	6.92820i	0.334497	−	0.579365i
$144$		0			0
$145$	6.00000	+	10.3923i	0.498273	+	0.863034i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−7.00000	−	12.1244i	−0.573462	−	0.993266i	−0.996207	−	0.0870170i	$-0.972267\pi$
$149$	−7.00000	−	12.1244i	−0.573462	−	0.993266i	0.422744	−	0.906249i	$-0.361067\pi$
$150$		0			0
$151$	8.00000	−	13.8564i	0.651031	−	1.12762i	−0.331842	−	0.943335i	$-0.607670\pi$
$151$	8.00000	−	13.8564i	0.651031	−	1.12762i	0.982873	−	0.184284i	$-0.0589965\pi$
$152$		0			0
$153$	2.00000			0.161690
$154$		0			0
$155$	−16.0000			−1.28515
$156$		0			0
$157$	1.00000	−	1.73205i	0.0798087	−	0.138233i	−0.823359	−	0.567521i	$-0.807902\pi$
$157$	1.00000	−	1.73205i	0.0798087	−	0.138233i	0.903167	+	0.429289i	$0.141236\pi$
$158$		0			0
$159$	−1.00000	−	1.73205i	−0.0793052	−	0.137361i
$160$		0			0
$161$		0			0
$162$		0			0
$163$	−6.00000	−	10.3923i	−0.469956	−	0.813988i	0.529454	−	0.848339i	$-0.322397\pi$
$163$	−6.00000	−	10.3923i	−0.469956	−	0.813988i	−0.999410	+	0.0343508i	$0.989064\pi$
$164$		0			0
$165$	−4.00000	+	6.92820i	−0.311400	+	0.539360i
$166$		0			0
$167$	24.0000			1.85718			0.928588	−	0.371113i	$-0.121024\pi$
$167$	24.0000			1.85718			0.928588	+	0.371113i	$0.121024\pi$
$168$		0			0
$169$	−9.00000			−0.692308
$170$		0			0
$171$	2.00000	−	3.46410i	0.152944	−	0.264906i
$172$		0			0
$173$	−3.00000	−	5.19615i	−0.228086	−	0.395056i	0.729155	−	0.684349i	$-0.239913\pi$
$173$	−3.00000	−	5.19615i	−0.228086	−	0.395056i	−0.957241	+	0.289292i	$0.906580\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$	2.00000	+	3.46410i	0.150329	+	0.260378i
$178$		0			0
$179$	−6.00000	+	10.3923i	−0.448461	+	0.776757i	−0.998286	−	0.0585225i	$-0.981361\pi$
$179$	−6.00000	+	10.3923i	−0.448461	+	0.776757i	0.549825	+	0.835280i	$0.314694\pi$
$180$		0			0
$181$	6.00000			0.445976			0.222988	−	0.974821i	$-0.428419\pi$
$181$	6.00000			0.445976			0.222988	+	0.974821i	$0.428419\pi$
$182$		0			0
$183$	2.00000			0.147844
$184$		0			0
$185$	6.00000	−	10.3923i	0.441129	−	0.764057i
$186$		0			0
$187$	−4.00000	−	6.92820i	−0.292509	−	0.506640i
$188$		0			0
$189$		0			0
$190$		0			0
$191$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$191$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$192$		0			0
$193$	−1.00000	+	1.73205i	−0.0719816	+	0.124676i	−0.899770	−	0.436365i	$-0.856266\pi$
$193$	−1.00000	+	1.73205i	−0.0719816	+	0.124676i	0.827788	+	0.561041i	$0.189599\pi$
$194$		0			0
$195$	−4.00000			−0.286446
$196$		0			0
$197$	−18.0000			−1.28245			−0.641223	−	0.767354i	$-0.721573\pi$
$197$	−18.0000			−1.28245			−0.641223	+	0.767354i	$0.721573\pi$
$198$		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
$201$	−2.00000	−	3.46410i	−0.141069	−	0.244339i
$202$		0			0
$203$		0			0
$204$		0			0
$205$	−6.00000	−	10.3923i	−0.419058	−	0.725830i
$206$		0			0
$207$	4.00000	−	6.92820i	0.278019	−	0.481543i
$208$		0			0
$209$	−16.0000			−1.10674
$210$		0			0
$211$	−20.0000			−1.37686			−0.688428	−	0.725304i	$-0.741699\pi$
$211$	−20.0000			−1.37686			−0.688428	+	0.725304i	$0.741699\pi$
$212$		0			0
$213$	4.00000	−	6.92820i	0.274075	−	0.474713i
$214$		0			0
$215$	4.00000	+	6.92820i	0.272798	+	0.472500i
$216$		0			0
$217$		0			0
$218$		0			0
$219$	5.00000	+	8.66025i	0.337869	+	0.585206i
$220$		0			0
$221$	2.00000	−	3.46410i	0.134535	−	0.233021i
$222$		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
$225$	−1.00000			−0.0666667
$226$		0			0
$227$	−6.00000	+	10.3923i	−0.398234	+	0.689761i	−0.993508	−	0.113761i	$-0.963710\pi$
$227$	−6.00000	+	10.3923i	−0.398234	+	0.689761i	0.595274	+	0.803523i	$0.297043\pi$
$228$		0			0
$229$	−11.0000	−	19.0526i	−0.726900	−	1.25903i	−0.958187	−	0.286143i	$-0.907627\pi$
$229$	−11.0000	−	19.0526i	−0.726900	−	1.25903i	0.231287	−	0.972886i	$-0.425707\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−5.00000	−	8.66025i	−0.327561	−	0.567352i	0.654466	−	0.756091i	$-0.272893\pi$
$233$	−5.00000	−	8.66025i	−0.327561	−	0.567352i	−0.982027	+	0.188739i	$0.939560\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	8.00000			0.519656
$238$		0			0
$239$	−16.0000			−1.03495			−0.517477	−	0.855697i	$-0.673129\pi$
$239$	−16.0000			−1.03495			−0.517477	+	0.855697i	$0.673129\pi$
$240$		0			0
$241$	−9.00000	+	15.5885i	−0.579741	+	1.00414i	0.415768	+	0.909471i	$0.363513\pi$
$241$	−9.00000	+	15.5885i	−0.579741	+	1.00414i	−0.995509	+	0.0946700i	$0.969820\pi$
$242$		0			0
$243$	0.500000	+	0.866025i	0.0320750	+	0.0555556i
$244$		0			0
$245$		0			0
$246$		0			0
$247$	−4.00000	−	6.92820i	−0.254514	−	0.440831i
$248$		0			0
$249$	−2.00000	+	3.46410i	−0.126745	+	0.219529i
$250$		0			0
$251$	20.0000			1.26239			0.631194	−	0.775625i	$-0.282565\pi$
$251$	20.0000			1.26239			0.631194	+	0.775625i	$0.282565\pi$
$252$		0			0
$253$	−32.0000			−2.01182
$254$		0			0
$255$	−2.00000	+	3.46410i	−0.125245	+	0.216930i
$256$		0			0
$257$	−1.00000	−	1.73205i	−0.0623783	−	0.108042i	0.833150	−	0.553047i	$-0.186535\pi$
$257$	−1.00000	−	1.73205i	−0.0623783	−	0.108042i	−0.895528	+	0.445005i	$0.853202\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$	−3.00000	−	5.19615i	−0.185695	−	0.321634i
$262$		0			0
$263$	4.00000	−	6.92820i	0.246651	−	0.427211i	−0.715944	−	0.698158i	$-0.754003\pi$
$263$	4.00000	−	6.92820i	0.246651	−	0.427211i	0.962594	+	0.270947i	$0.0873367\pi$
$264$		0			0
$265$	4.00000			0.245718
$266$		0			0
$267$	6.00000			0.367194
$268$		0			0
$269$	5.00000	−	8.66025i	0.304855	−	0.528025i	−0.672374	−	0.740212i	$-0.734725\pi$
$269$	5.00000	−	8.66025i	0.304855	−	0.528025i	0.977229	+	0.212187i	$0.0680585\pi$
$270$		0			0
$271$	−4.00000	−	6.92820i	−0.242983	−	0.420858i	0.718580	−	0.695444i	$-0.244792\pi$
$271$	−4.00000	−	6.92820i	−0.242983	−	0.420858i	−0.961563	+	0.274586i	$0.911459\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	2.00000	+	3.46410i	0.120605	+	0.208893i
$276$		0			0
$277$	13.0000	−	22.5167i	0.781094	−	1.35290i	−0.150210	−	0.988654i	$-0.547995\pi$
$277$	13.0000	−	22.5167i	0.781094	−	1.35290i	0.931305	−	0.364241i	$-0.118672\pi$
$278$		0			0
$279$	8.00000			0.478947
$280$		0			0
$281$	26.0000			1.55103			0.775515	−	0.631329i	$-0.217490\pi$
$281$	26.0000			1.55103			0.775515	+	0.631329i	$0.217490\pi$
$282$		0			0
$283$	14.0000	−	24.2487i	0.832214	−	1.44144i	−0.0640654	−	0.997946i	$-0.520407\pi$
$283$	14.0000	−	24.2487i	0.832214	−	1.44144i	0.896279	−	0.443491i	$-0.146260\pi$
$284$		0			0
$285$	4.00000	+	6.92820i	0.236940	+	0.410391i
$286$		0			0
$287$		0			0
$288$		0			0
$289$	6.50000	+	11.2583i	0.382353	+	0.662255i
$290$		0			0
$291$	1.00000	−	1.73205i	0.0586210	−	0.101535i
$292$		0			0
$293$	−18.0000			−1.05157			−0.525786	−	0.850617i	$-0.676229\pi$
$293$	−18.0000			−1.05157			−0.525786	+	0.850617i	$0.676229\pi$
$294$		0			0
$295$	−8.00000			−0.465778
$296$		0			0
$297$	2.00000	−	3.46410i	0.116052	−	0.201008i
$298$		0			0
$299$	−8.00000	−	13.8564i	−0.462652	−	0.801337i
$300$		0			0
$301$		0			0
$302$		0			0
$303$	−9.00000	−	15.5885i	−0.517036	−	0.895533i
$304$		0			0
$305$	−2.00000	+	3.46410i	−0.114520	+	0.198354i
$306$		0			0
$307$	12.0000			0.684876			0.342438	−	0.939540i	$-0.388747\pi$
$307$	12.0000			0.684876			0.342438	+	0.939540i	$0.388747\pi$
$308$		0			0
$309$	−16.0000			−0.910208
$310$		0			0
$311$	12.0000	−	20.7846i	0.680458	−	1.17859i	−0.294384	−	0.955687i	$-0.595114\pi$
$311$	12.0000	−	20.7846i	0.680458	−	1.17859i	0.974841	−	0.222900i	$-0.0715523\pi$
$312$		0			0
$313$	3.00000	+	5.19615i	0.169570	+	0.293704i	0.938269	−	0.345907i	$-0.112429\pi$
$313$	3.00000	+	5.19615i	0.169570	+	0.293704i	−0.768699	+	0.639611i	$0.779095\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−3.00000	−	5.19615i	−0.168497	−	0.291845i	0.769395	−	0.638774i	$-0.220558\pi$
$317$	−3.00000	−	5.19615i	−0.168497	−	0.291845i	−0.937892	+	0.346929i	$0.887225\pi$
$318$		0			0
$319$	−12.0000	+	20.7846i	−0.671871	+	1.16371i
$320$		0			0
$321$	12.0000			0.669775
$322$		0			0
$323$	−8.00000			−0.445132
$324$		0			0
$325$	−1.00000	+	1.73205i	−0.0554700	+	0.0960769i
$326$		0			0
$327$	−1.00000	−	1.73205i	−0.0553001	−	0.0957826i
$328$		0			0
$329$		0			0
$330$		0			0
$331$	−10.0000	−	17.3205i	−0.549650	−	0.952021i	−0.998298	−	0.0583130i	$-0.981428\pi$
$331$	−10.0000	−	17.3205i	−0.549650	−	0.952021i	0.448649	−	0.893708i	$-0.351905\pi$
$332$		0			0
$333$	−3.00000	+	5.19615i	−0.164399	+	0.284747i
$334$		0			0
$335$	8.00000			0.437087
$336$		0			0
$337$	18.0000			0.980522			0.490261	−	0.871576i	$-0.336901\pi$
$337$	18.0000			0.980522			0.490261	+	0.871576i	$0.336901\pi$
$338$		0			0
$339$	9.00000	−	15.5885i	0.488813	−	0.846649i
$340$		0			0
$341$	−16.0000	−	27.7128i	−0.866449	−	1.50073i
$342$		0			0
$343$		0			0
$344$		0			0
$345$	8.00000	+	13.8564i	0.430706	+	0.746004i
$346$		0			0
$347$	6.00000	−	10.3923i	0.322097	−	0.557888i	−0.658824	−	0.752297i	$-0.728946\pi$
$347$	6.00000	−	10.3923i	0.322097	−	0.557888i	0.980921	+	0.194409i	$0.0622790\pi$
$348$		0			0
$349$	30.0000			1.60586			0.802932	−	0.596071i	$-0.203272\pi$
$349$	30.0000			1.60586			0.802932	+	0.596071i	$0.203272\pi$
$350$		0			0
$351$	2.00000			0.106752
$352$		0			0
$353$	−1.00000	+	1.73205i	−0.0532246	+	0.0921878i	−0.891410	−	0.453197i	$-0.850283\pi$
$353$	−1.00000	+	1.73205i	−0.0532246	+	0.0921878i	0.838186	+	0.545385i	$0.183617\pi$
$354$		0			0
$355$	8.00000	+	13.8564i	0.424596	+	0.735422i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	12.0000	+	20.7846i	0.633336	+	1.09697i	0.986865	+	0.161546i	$0.0516481\pi$
$359$	12.0000	+	20.7846i	0.633336	+	1.09697i	−0.353529	+	0.935423i	$0.615019\pi$
$360$		0			0
$361$	1.50000	−	2.59808i	0.0789474	−	0.136741i
$362$		0			0
$363$	−5.00000			−0.262432
$364$		0			0
$365$	−20.0000			−1.04685
$366$		0			0
$367$	4.00000	−	6.92820i	0.208798	−	0.361649i	−0.742538	−	0.669804i	$-0.766378\pi$
$367$	4.00000	−	6.92820i	0.208798	−	0.361649i	0.951336	+	0.308155i	$0.0997115\pi$
$368$		0			0
$369$	3.00000	+	5.19615i	0.156174	+	0.270501i
$370$		0			0
$371$		0			0
$372$		0			0
$373$	5.00000	+	8.66025i	0.258890	+	0.448411i	0.965945	−	0.258748i	$-0.0833099\pi$
$373$	5.00000	+	8.66025i	0.258890	+	0.448411i	−0.707055	+	0.707159i	$0.749977\pi$
$374$		0			0
$375$	6.00000	−	10.3923i	0.309839	−	0.536656i
$376$		0			0
$377$	−12.0000			−0.618031
$378$		0			0
$379$	20.0000			1.02733			0.513665	−	0.857991i	$-0.328287\pi$
$379$	20.0000			1.02733			0.513665	+	0.857991i	$0.328287\pi$
$380$		0			0
$381$	−4.00000	+	6.92820i	−0.204926	+	0.354943i
$382$		0			0
$383$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$383$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	−2.00000	−	3.46410i	−0.101666	−	0.176090i
$388$		0			0
$389$	1.00000	−	1.73205i	0.0507020	−	0.0878185i	−0.839561	−	0.543266i	$-0.817187\pi$
$389$	1.00000	−	1.73205i	0.0507020	−	0.0878185i	0.890263	+	0.455448i	$0.150521\pi$
$390$		0			0
$391$	−16.0000			−0.809155
$392$		0			0
$393$	4.00000			0.201773
$394$		0			0
$395$	−8.00000	+	13.8564i	−0.402524	+	0.697191i
$396$		0			0
$397$	−7.00000	−	12.1244i	−0.351320	−	0.608504i	0.635161	−	0.772380i	$-0.280934\pi$
$397$	−7.00000	−	12.1244i	−0.351320	−	0.608504i	−0.986481	+	0.163876i	$0.947600\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	15.0000	+	25.9808i	0.749064	+	1.29742i	0.948272	+	0.317460i	$0.102830\pi$
$401$	15.0000	+	25.9808i	0.749064	+	1.29742i	−0.199207	+	0.979957i	$0.563837\pi$
$402$		0			0
$403$	8.00000	−	13.8564i	0.398508	−	0.690237i
$404$		0			0
$405$	−2.00000			−0.0993808
$406$		0			0
$407$	24.0000			1.18964
$408$		0			0
$409$	3.00000	−	5.19615i	0.148340	−	0.256933i	−0.782274	−	0.622935i	$-0.785940\pi$
$409$	3.00000	−	5.19615i	0.148340	−	0.256933i	0.930614	+	0.366002i	$0.119274\pi$
$410$		0			0
$411$	−3.00000	−	5.19615i	−0.147979	−	0.256307i
$412$		0			0
$413$		0			0
$414$		0			0
$415$	−4.00000	−	6.92820i	−0.196352	−	0.340092i
$416$		0			0
$417$	−6.00000	+	10.3923i	−0.293821	+	0.508913i
$418$		0			0
$419$	12.0000			0.586238			0.293119	−	0.956076i	$-0.405307\pi$
$419$	12.0000			0.586238			0.293119	+	0.956076i	$0.405307\pi$
$420$		0			0
$421$	−10.0000			−0.487370			−0.243685	−	0.969854i	$-0.578356\pi$
$421$	−10.0000			−0.487370			−0.243685	+	0.969854i	$0.578356\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	1.00000	+	1.73205i	0.0485071	+	0.0840168i
$426$		0			0
$427$		0			0
$428$		0			0
$429$	−4.00000	−	6.92820i	−0.193122	−	0.334497i
$430$		0			0
$431$	−16.0000	+	27.7128i	−0.770693	+	1.33488i	0.166491	+	0.986043i	$0.446756\pi$
$431$	−16.0000	+	27.7128i	−0.770693	+	1.33488i	−0.937184	+	0.348836i	$0.886577\pi$
$432$		0			0
$433$	−14.0000			−0.672797			−0.336399	−	0.941720i	$-0.609209\pi$
$433$	−14.0000			−0.672797			−0.336399	+	0.941720i	$0.609209\pi$
$434$		0			0
$435$	12.0000			0.575356
$436$		0			0
$437$	−16.0000	+	27.7128i	−0.765384	+	1.32568i
$438$		0			0
$439$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$439$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−10.0000	−	17.3205i	−0.475114	−	0.822922i	0.524479	−	0.851423i	$-0.324260\pi$
$443$	−10.0000	−	17.3205i	−0.475114	−	0.822922i	−0.999594	+	0.0285009i	$0.990927\pi$
$444$		0			0
$445$	−6.00000	+	10.3923i	−0.284427	+	0.492642i
$446$		0			0
$447$	−14.0000			−0.662177
$448$		0			0
$449$	−14.0000			−0.660701			−0.330350	−	0.943858i	$-0.607167\pi$
$449$	−14.0000			−0.660701			−0.330350	+	0.943858i	$0.607167\pi$
$450$		0			0
$451$	12.0000	−	20.7846i	0.565058	−	0.978709i
$452$		0			0
$453$	−8.00000	−	13.8564i	−0.375873	−	0.651031i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	11.0000	+	19.0526i	0.514558	+	0.891241i	0.999857	+	0.0168929i	$0.00537742\pi$
$457$	11.0000	+	19.0526i	0.514558	+	0.891241i	−0.485299	+	0.874348i	$0.661289\pi$
$458$		0			0
$459$	1.00000	−	1.73205i	0.0466760	−	0.0808452i
$460$		0			0
$461$	−26.0000			−1.21094			−0.605470	−	0.795868i	$-0.707015\pi$
$461$	−26.0000			−1.21094			−0.605470	+	0.795868i	$0.707015\pi$
$462$		0			0
$463$	8.00000			0.371792			0.185896	−	0.982569i	$-0.440481\pi$
$463$	8.00000			0.371792			0.185896	+	0.982569i	$0.440481\pi$
$464$		0			0
$465$	−8.00000	+	13.8564i	−0.370991	+	0.642575i
$466$		0			0
$467$	18.0000	+	31.1769i	0.832941	+	1.44270i	0.895696	+	0.444667i	$0.146678\pi$
$467$	18.0000	+	31.1769i	0.832941	+	1.44270i	−0.0627555	+	0.998029i	$0.519989\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$	−1.00000	−	1.73205i	−0.0460776	−	0.0798087i
$472$		0			0
$473$	−8.00000	+	13.8564i	−0.367840	+	0.637118i
$474$		0			0
$475$	4.00000			0.183533
$476$		0			0
$477$	−2.00000			−0.0915737
$478$		0			0
$479$	8.00000	−	13.8564i	0.365529	−	0.633115i	−0.623332	−	0.781958i	$-0.714221\pi$
$479$	8.00000	−	13.8564i	0.365529	−	0.633115i	0.988861	+	0.148842i	$0.0475547\pi$
$480$		0			0
$481$	6.00000	+	10.3923i	0.273576	+	0.473848i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	2.00000	+	3.46410i	0.0908153	+	0.157297i
$486$		0			0
$487$	16.0000	−	27.7128i	0.725029	−	1.25579i	−0.233933	−	0.972253i	$-0.575160\pi$
$487$	16.0000	−	27.7128i	0.725029	−	1.25579i	0.958962	−	0.283535i	$-0.0915071\pi$
$488$		0			0
$489$	−12.0000			−0.542659
$490$		0			0
$491$	−12.0000			−0.541552			−0.270776	−	0.962642i	$-0.587280\pi$
$491$	−12.0000			−0.541552			−0.270776	+	0.962642i	$0.587280\pi$
$492$		0			0
$493$	−6.00000	+	10.3923i	−0.270226	+	0.468046i
$494$		0			0
$495$	4.00000	+	6.92820i	0.179787	+	0.311400i
$496$		0			0
$497$		0			0
$498$		0			0
$499$	−6.00000	−	10.3923i	−0.268597	−	0.465223i	0.699903	−	0.714238i	$-0.253227\pi$
$499$	−6.00000	−	10.3923i	−0.268597	−	0.465223i	−0.968500	+	0.249015i	$0.919893\pi$
$500$		0			0
$501$	12.0000	−	20.7846i	0.536120	−	0.928588i
$502$		0			0
$503$	24.0000			1.07011			0.535054	−	0.844818i	$-0.320291\pi$
$503$	24.0000			1.07011			0.535054	+	0.844818i	$0.320291\pi$
$504$		0			0
$505$	36.0000			1.60198
$506$		0			0
$507$	−4.50000	+	7.79423i	−0.199852	+	0.346154i
$508$		0			0
$509$	−3.00000	−	5.19615i	−0.132973	−	0.230315i	0.791849	−	0.610718i	$-0.209119\pi$
$509$	−3.00000	−	5.19615i	−0.132973	−	0.230315i	−0.924821	+	0.380402i	$0.875786\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$	−2.00000	−	3.46410i	−0.0883022	−	0.152944i
$514$		0			0
$515$	16.0000	−	27.7128i	0.705044	−	1.22117i
$516$		0			0
$517$		0			0
$518$		0			0
$519$	−6.00000			−0.263371
$520$		0			0
$521$	−13.0000	+	22.5167i	−0.569540	+	0.986473i	0.427071	+	0.904218i	$0.359545\pi$
$521$	−13.0000	+	22.5167i	−0.569540	+	0.986473i	−0.996611	+	0.0822547i	$0.973788\pi$
$522$		0			0
$523$	−2.00000	−	3.46410i	−0.0874539	−	0.151475i	0.818980	−	0.573822i	$-0.194540\pi$
$523$	−2.00000	−	3.46410i	−0.0874539	−	0.151475i	−0.906434	+	0.422347i	$0.861206\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−8.00000	−	13.8564i	−0.348485	−	0.603595i
$528$		0			0
$529$	−20.5000	+	35.5070i	−0.891304	+	1.54378i
$530$		0			0
$531$	4.00000			0.173585
$532$		0			0
$533$	12.0000			0.519778
$534$		0			0
$535$	−12.0000	+	20.7846i	−0.518805	+	0.898597i
$536$		0			0
$537$	6.00000	+	10.3923i	0.258919	+	0.448461i
$538$		0			0
$539$		0			0
$540$		0			0
$541$	9.00000	+	15.5885i	0.386940	+	0.670200i	0.992036	−	0.125952i	$-0.0401986\pi$
$541$	9.00000	+	15.5885i	0.386940	+	0.670200i	−0.605096	+	0.796152i	$0.706865\pi$
$542$		0			0
$543$	3.00000	−	5.19615i	0.128742	−	0.222988i
$544$		0			0
$545$	4.00000			0.171341
$546$		0			0
$547$	44.0000			1.88130			0.940652	−	0.339372i	$-0.110215\pi$
$547$	44.0000			1.88130			0.940652	+	0.339372i	$0.110215\pi$
$548$		0			0
$549$	1.00000	−	1.73205i	0.0426790	−	0.0739221i
$550$		0			0
$551$	12.0000	+	20.7846i	0.511217	+	0.885454i
$552$		0			0
$553$		0			0
$554$		0			0
$555$	−6.00000	−	10.3923i	−0.254686	−	0.441129i
$556$		0			0
$557$	13.0000	−	22.5167i	0.550828	−	0.954062i	−0.447387	−	0.894340i	$-0.647645\pi$
$557$	13.0000	−	22.5167i	0.550828	−	0.954062i	0.998215	−	0.0597213i	$-0.0190212\pi$
$558$		0			0
$559$	−8.00000			−0.338364
$560$		0			0
$561$	−8.00000			−0.337760
$562$		0			0
$563$	−14.0000	+	24.2487i	−0.590030	+	1.02196i	0.404198	+	0.914671i	$0.367551\pi$
$563$	−14.0000	+	24.2487i	−0.590030	+	1.02196i	−0.994228	+	0.107290i	$0.965783\pi$
$564$		0			0
$565$	18.0000	+	31.1769i	0.757266	+	1.31162i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−5.00000	−	8.66025i	−0.209611	−	0.363057i	0.741981	−	0.670421i	$-0.233886\pi$
$569$	−5.00000	−	8.66025i	−0.209611	−	0.363057i	−0.951592	+	0.307364i	$0.900553\pi$
$570$		0			0
$571$	−18.0000	+	31.1769i	−0.753277	+	1.30471i	0.192950	+	0.981209i	$0.438194\pi$
$571$	−18.0000	+	31.1769i	−0.753277	+	1.30471i	−0.946227	+	0.323505i	$0.895139\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	8.00000			0.333623
$576$		0			0
$577$	−1.00000	+	1.73205i	−0.0416305	+	0.0721062i	−0.886090	−	0.463513i	$-0.846589\pi$
$577$	−1.00000	+	1.73205i	−0.0416305	+	0.0721062i	0.844459	+	0.535620i	$0.179922\pi$
$578$		0			0
$579$	1.00000	+	1.73205i	0.0415586	+	0.0719816i
$580$		0			0
$581$		0			0
$582$		0			0
$583$	4.00000	+	6.92820i	0.165663	+	0.286937i
$584$		0			0
$585$	−2.00000	+	3.46410i	−0.0826898	+	0.143223i
$586$		0			0
$587$	−44.0000			−1.81607			−0.908037	−	0.418890i	$-0.862419\pi$
$587$	−44.0000			−1.81607			−0.908037	+	0.418890i	$0.862419\pi$
$588$		0			0
$589$	−32.0000			−1.31854
$590$		0			0
$591$	−9.00000	+	15.5885i	−0.370211	+	0.641223i
$592$		0			0
$593$	7.00000	+	12.1244i	0.287456	+	0.497888i	0.973202	−	0.229953i	$-0.0738573\pi$
$593$	7.00000	+	12.1244i	0.287456	+	0.497888i	−0.685746	+	0.727841i	$0.740524\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	8.00000	+	13.8564i	0.327418	+	0.567105i
$598$		0			0
$599$	−12.0000	+	20.7846i	−0.490307	+	0.849236i	−0.999938	−	0.0111569i	$-0.996449\pi$
$599$	−12.0000	+	20.7846i	−0.490307	+	0.849236i	0.509631	+	0.860393i	$0.329782\pi$
$600$		0			0
$601$	−38.0000			−1.55005			−0.775026	−	0.631929i	$-0.782263\pi$
$601$	−38.0000			−1.55005			−0.775026	+	0.631929i	$0.782263\pi$
$602$		0			0
$603$	−4.00000			−0.162893
$604$		0			0
$605$	5.00000	−	8.66025i	0.203279	−	0.352089i
$606$		0			0
$607$	20.0000	+	34.6410i	0.811775	+	1.40604i	0.911621	+	0.411033i	$0.134832\pi$
$607$	20.0000	+	34.6410i	0.811775	+	1.40604i	−0.0998457	+	0.995003i	$0.531835\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$	−19.0000	+	32.9090i	−0.767403	+	1.32918i	0.171564	+	0.985173i	$0.445118\pi$
$613$	−19.0000	+	32.9090i	−0.767403	+	1.32918i	−0.938967	+	0.344008i	$0.888215\pi$
$614$		0			0
$615$	−12.0000			−0.483887
$616$		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
$619$	22.0000	−	38.1051i	0.884255	−	1.53157i	0.0376891	−	0.999290i	$-0.488000\pi$
$619$	22.0000	−	38.1051i	0.884255	−	1.53157i	0.846566	−	0.532284i	$-0.178666\pi$
$620$		0			0
$621$	−4.00000	−	6.92820i	−0.160514	−	0.278019i
$622$		0			0
$623$		0			0
$624$		0			0
$625$	9.50000	+	16.4545i	0.380000	+	0.658179i
$626$		0			0
$627$	−8.00000	+	13.8564i	−0.319489	+	0.553372i
$628$		0			0
$629$	12.0000			0.478471
$630$		0			0
$631$	16.0000			0.636950			0.318475	−	0.947931i	$-0.396829\pi$
$631$	16.0000			0.636950			0.318475	+	0.947931i	$0.396829\pi$
$632$		0			0
$633$	−10.0000	+	17.3205i	−0.397464	+	0.688428i
$634$		0			0
$635$	−8.00000	−	13.8564i	−0.317470	−	0.549875i
$636$		0			0
$637$		0			0
$638$		0			0
$639$	−4.00000	−	6.92820i	−0.158238	−	0.274075i
$640$		0			0
$641$	7.00000	−	12.1244i	0.276483	−	0.478883i	−0.694025	−	0.719951i	$-0.744164\pi$
$641$	7.00000	−	12.1244i	0.276483	−	0.478883i	0.970508	+	0.241068i	$0.0774976\pi$
$642$		0			0
$643$	12.0000			0.473234			0.236617	−	0.971603i	$-0.423961\pi$
$643$	12.0000			0.473234			0.236617	+	0.971603i	$0.423961\pi$
$644$		0			0
$645$	8.00000			0.315000
$646$		0			0
$647$	−4.00000	+	6.92820i	−0.157256	+	0.272376i	−0.933878	−	0.357591i	$-0.883598\pi$
$647$	−4.00000	+	6.92820i	−0.157256	+	0.272376i	0.776622	+	0.629967i	$0.216932\pi$
$648$		0			0
$649$	−8.00000	−	13.8564i	−0.314027	−	0.543912i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−3.00000	−	5.19615i	−0.117399	−	0.203341i	0.801337	−	0.598213i	$-0.204122\pi$
$653$	−3.00000	−	5.19615i	−0.117399	−	0.203341i	−0.918736	+	0.394872i	$0.870789\pi$
$654$		0			0
$655$	−4.00000	+	6.92820i	−0.156293	+	0.270707i
$656$		0			0
$657$	10.0000			0.390137
$658$		0			0
$659$	12.0000			0.467454			0.233727	−	0.972302i	$-0.424908\pi$
$659$	12.0000			0.467454			0.233727	+	0.972302i	$0.424908\pi$
$660$		0			0
$661$	5.00000	−	8.66025i	0.194477	−	0.336845i	−0.752252	−	0.658876i	$-0.771032\pi$
$661$	5.00000	−	8.66025i	0.194477	−	0.336845i	0.946729	+	0.322031i	$0.104366\pi$
$662$		0			0
$663$	−2.00000	−	3.46410i	−0.0776736	−	0.134535i
$664$		0			0
$665$		0			0
$666$		0			0
$667$	24.0000	+	41.5692i	0.929284	+	1.60957i
$668$		0			0
$669$	−4.00000	+	6.92820i	−0.154649	+	0.267860i
$670$		0			0
$671$	−8.00000			−0.308837
$672$		0			0
$673$	34.0000			1.31060			0.655302	−	0.755367i	$-0.272541\pi$
$673$	34.0000			1.31060			0.655302	+	0.755367i	$0.272541\pi$
$674$		0			0
$675$	−0.500000	+	0.866025i	−0.0192450	+	0.0333333i
$676$		0			0
$677$	1.00000	+	1.73205i	0.0384331	+	0.0665681i	0.884602	−	0.466347i	$-0.154430\pi$
$677$	1.00000	+	1.73205i	0.0384331	+	0.0665681i	−0.846169	+	0.532915i	$0.821097\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$	6.00000	+	10.3923i	0.229920	+	0.398234i
$682$		0			0
$683$	−2.00000	+	3.46410i	−0.0765279	+	0.132550i	−0.901750	−	0.432259i	$-0.857717\pi$
$683$	−2.00000	+	3.46410i	−0.0765279	+	0.132550i	0.825222	+	0.564809i	$0.191050\pi$
$684$		0			0
$685$	12.0000			0.458496
$686$		0			0
$687$	−22.0000			−0.839352
$688$		0			0
$689$	−2.00000	+	3.46410i	−0.0761939	+	0.131972i
$690$		0			0
$691$	2.00000	+	3.46410i	0.0760836	+	0.131781i	0.901557	−	0.432660i	$-0.142425\pi$
$691$	2.00000	+	3.46410i	0.0760836	+	0.131781i	−0.825473	+	0.564441i	$0.809092\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−12.0000	−	20.7846i	−0.455186	−	0.788405i
$696$		0			0
$697$	6.00000	−	10.3923i	0.227266	−	0.393637i
$698$		0			0
$699$	−10.0000			−0.378235
$700$		0			0
$701$	6.00000			0.226617			0.113308	−	0.993560i	$-0.463855\pi$
$701$	6.00000			0.226617			0.113308	+	0.993560i	$0.463855\pi$
$702$		0			0
$703$	12.0000	−	20.7846i	0.452589	−	0.783906i
$704$		0			0
$705$		0			0
$706$		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
$711$	4.00000	−	6.92820i	0.150012	−	0.259828i
$712$		0			0
$713$	−64.0000			−2.39682
$714$		0			0
$715$	16.0000			0.598366
$716$		0			0
$717$	−8.00000	+	13.8564i	−0.298765	+	0.517477i
$718$		0			0
$719$	16.0000	+	27.7128i	0.596699	+	1.03351i	0.993305	+	0.115524i	$0.0368548\pi$
$719$	16.0000	+	27.7128i	0.596699	+	1.03351i	−0.396605	+	0.917989i	$0.629812\pi$
$720$		0			0
$721$		0			0
$722$		0			0
$723$	9.00000	+	15.5885i	0.334714	+	0.579741i
$724$		0			0
$725$	3.00000	−	5.19615i	0.111417	−	0.192980i
$726$		0			0
$727$	48.0000			1.78022			0.890111	−	0.455744i	$-0.150627\pi$
$727$	48.0000			1.78022			0.890111	+	0.455744i	$0.150627\pi$
$728$		0			0
$729$	1.00000			0.0370370
$730$		0			0
$731$	−4.00000	+	6.92820i	−0.147945	+	0.256249i
$732$		0			0
$733$	−7.00000	−	12.1244i	−0.258551	−	0.447823i	0.707303	−	0.706910i	$-0.249912\pi$
$733$	−7.00000	−	12.1244i	−0.258551	−	0.447823i	−0.965854	+	0.259087i	$0.916578\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	8.00000	+	13.8564i	0.294684	+	0.510407i
$738$		0			0
$739$	2.00000	−	3.46410i	0.0735712	−	0.127429i	−0.826893	−	0.562360i	$-0.809894\pi$
$739$	2.00000	−	3.46410i	0.0735712	−	0.127429i	0.900464	+	0.434930i	$0.143227\pi$
$740$		0			0
$741$	−8.00000			−0.293887
$742$		0			0
$743$	−8.00000			−0.293492			−0.146746	−	0.989174i	$-0.546880\pi$
$743$	−8.00000			−0.293492			−0.146746	+	0.989174i	$0.546880\pi$
$744$		0			0
$745$	14.0000	−	24.2487i	0.512920	−	0.888404i
$746$		0			0
$747$	2.00000	+	3.46410i	0.0731762	+	0.126745i
$748$		0			0
$749$		0			0
$750$		0			0
$751$	−12.0000	−	20.7846i	−0.437886	−	0.758441i	0.559640	−	0.828736i	$-0.310939\pi$
$751$	−12.0000	−	20.7846i	−0.437886	−	0.758441i	−0.997526	+	0.0702946i	$0.977606\pi$
$752$		0			0
$753$	10.0000	−	17.3205i	0.364420	−	0.631194i
$754$		0			0
$755$	32.0000			1.16460
$756$		0			0
$757$	38.0000			1.38113			0.690567	−	0.723269i	$-0.257361\pi$
$757$	38.0000			1.38113			0.690567	+	0.723269i	$0.257361\pi$
$758$		0			0
$759$	−16.0000	+	27.7128i	−0.580763	+	1.00591i
$760$		0			0
$761$	11.0000	+	19.0526i	0.398750	+	0.690655i	0.993572	−	0.113203i	$-0.0361109\pi$
$761$	11.0000	+	19.0526i	0.398750	+	0.690655i	−0.594822	+	0.803857i	$0.702778\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$	2.00000	+	3.46410i	0.0723102	+	0.125245i
$766$		0			0
$767$	4.00000	−	6.92820i	0.144432	−	0.250163i
$768$		0			0
$769$	2.00000			0.0721218			0.0360609	−	0.999350i	$-0.488519\pi$
$769$	2.00000			0.0721218			0.0360609	+	0.999350i	$0.488519\pi$
$770$		0			0
$771$	−2.00000			−0.0720282
$772$		0			0
$773$	9.00000	−	15.5885i	0.323708	−	0.560678i	−0.657542	−	0.753418i	$-0.728404\pi$
$773$	9.00000	−	15.5885i	0.323708	−	0.560678i	0.981250	+	0.192740i	$0.0617373\pi$
$774$		0			0
$775$	4.00000	+	6.92820i	0.143684	+	0.248868i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−12.0000	−	20.7846i	−0.429945	−	0.744686i
$780$		0			0
$781$	−16.0000	+	27.7128i	−0.572525	+	0.991642i
$782$		0			0
$783$	−6.00000			−0.214423
$784$		0			0
$785$	4.00000			0.142766
$786$		0			0
$787$	−14.0000	+	24.2487i	−0.499046	+	0.864373i	−0.999999	−	0.00110111i	$-0.999650\pi$
$787$	−14.0000	+	24.2487i	−0.499046	+	0.864373i	0.500953	+	0.865474i	$0.332983\pi$
$788$		0			0
$789$	−4.00000	−	6.92820i	−0.142404	−	0.246651i
$790$		0			0
$791$		0			0
$792$		0			0
$793$	−2.00000	−	3.46410i	−0.0710221	−	0.123014i
$794$		0			0
$795$	2.00000	−	3.46410i	0.0709327	−	0.122859i

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:	\( N \)	\(=\)	\( 1176 = 2^{3} \cdot 3 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1176.q (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(0.500000 - 0.866025i) q^{3} +(1.00000 + 1.73205i) q^{5} +(-0.500000 - 0.866025i) q^{9} +(-2.00000 + 3.46410i) q^{11} -2.00000 q^{13} +2.00000 q^{15} +(-1.00000 + 1.73205i) q^{17} +(2.00000 + 3.46410i) q^{19} +(4.00000 + 6.92820i) q^{23} +(0.500000 - 0.866025i) q^{25} -1.00000 q^{27} +6.00000 q^{29} +(-4.00000 + 6.92820i) q^{31} +(2.00000 + 3.46410i) q^{33} +(-3.00000 - 5.19615i) q^{37} +(-1.00000 + 1.73205i) q^{39} -6.00000 q^{41} +4.00000 q^{43} +(1.00000 - 1.73205i) q^{45} +(1.00000 + 1.73205i) q^{51} +(1.00000 - 1.73205i) q^{53} -8.00000 q^{55} +4.00000 q^{57} +(-2.00000 + 3.46410i) q^{59} +(1.00000 + 1.73205i) q^{61} +(-2.00000 - 3.46410i) q^{65} +(2.00000 - 3.46410i) q^{67} +8.00000 q^{69} +8.00000 q^{71} +(-5.00000 + 8.66025i) q^{73} +(-0.500000 - 0.866025i) q^{75} +(4.00000 + 6.92820i) q^{79} +(-0.500000 + 0.866025i) q^{81} -4.00000 q^{83} -4.00000 q^{85} +(3.00000 - 5.19615i) q^{87} +(3.00000 + 5.19615i) q^{89} +(4.00000 + 6.92820i) q^{93} +(-4.00000 + 6.92820i) q^{95} +2.00000 q^{97} +4.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + q^{3} + 2 q^{5} - q^{9} - 4 q^{11} - 4 q^{13} + 4 q^{15} - 2 q^{17} + 4 q^{19} + 8 q^{23} + q^{25} - 2 q^{27} + 12 q^{29} - 8 q^{31} + 4 q^{33} - 6 q^{37} - 2 q^{39} - 12 q^{41} + 8 q^{43} + 2 q^{45}+ \cdots + 8 q^{99}+O(q^{100}) \) 2 * q + q^3 + 2 * q^5 - q^9 - 4 * q^11 - 4 * q^13 + 4 * q^15 - 2 * q^17 + 4 * q^19 + 8 * q^23 + q^25 - 2 * q^27 + 12 * q^29 - 8 * q^31 + 4 * q^33 - 6 * q^37 - 2 * q^39 - 12 * q^41 + 8 * q^43 + 2 * q^45 + 2 * q^51 + 2 * q^53 - 16 * q^55 + 8 * q^57 - 4 * q^59 + 2 * q^61 - 4 * q^65 + 4 * q^67 + 16 * q^69 + 16 * q^71 - 10 * q^73 - q^75 + 8 * q^79 - q^81 - 8 * q^83 - 8 * q^85 + 6 * q^87 + 6 * q^89 + 8 * q^93 - 8 * q^95 + 4 * q^97 + 8 * q^99

Introduction

L-functions

Modular forms

Varieties

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Database

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