LMFDB - Embedded newform 980.2.i.a.961.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [980,2,Mod(361,980)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(980, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([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("980.361");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 980 = 2^{2} \cdot 5 \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	980.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:	$7.82533939809$
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:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 140)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			961.1
Root			$0.500000 - 0.866025i$ of defining polynomial
Character	$\chi$	$=$	980.961
Dual form			980.2.i.a.361.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.50000 - 2.59808i) q^{3} +(-0.500000 + 0.866025i) q^{5} +(-3.00000 + 5.19615i) q^{9} +O(q^{10})$	$q+(-1.50000 - 2.59808i) q^{3} +(-0.500000 + 0.866025i) q^{5} +(-3.00000 + 5.19615i) q^{9} +(1.00000 + 1.73205i) q^{11} +6.00000 q^{13} +3.00000 q^{15} +(1.00000 + 1.73205i) q^{17} +(4.50000 - 7.79423i) q^{23} +(-0.500000 - 0.866025i) q^{25} +9.00000 q^{27} +3.00000 q^{29} +(1.00000 + 1.73205i) q^{31} +(3.00000 - 5.19615i) q^{33} +(-4.00000 + 6.92820i) q^{37} +(-9.00000 - 15.5885i) q^{39} -5.00000 q^{41} +1.00000 q^{43} +(-3.00000 - 5.19615i) q^{45} +(4.00000 - 6.92820i) q^{47} +(3.00000 - 5.19615i) q^{51} +(-2.00000 - 3.46410i) q^{53} -2.00000 q^{55} +(-4.00000 - 6.92820i) q^{59} +(3.50000 - 6.06218i) q^{61} +(-3.00000 + 5.19615i) q^{65} +(1.50000 + 2.59808i) q^{67} -27.0000 q^{69} +8.00000 q^{71} +(7.00000 + 12.1244i) q^{73} +(-1.50000 + 2.59808i) q^{75} +(-2.00000 + 3.46410i) q^{79} +(-4.50000 - 7.79423i) q^{81} +1.00000 q^{83} -2.00000 q^{85} +(-4.50000 - 7.79423i) q^{87} +(6.50000 - 11.2583i) q^{89} +(3.00000 - 5.19615i) q^{93} +10.0000 q^{97} -12.0000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 3 q^{3} - q^{5} - 6 q^{9}+O(q^{10}) $ 2 * q - 3 * q^3 - q^5 - 6 * q^9	$ 2 q - 3 q^{3} - q^{5} - 6 q^{9} + 2 q^{11} + 12 q^{13} + 6 q^{15} + 2 q^{17} + 9 q^{23} - q^{25} + 18 q^{27} + 6 q^{29} + 2 q^{31} + 6 q^{33} - 8 q^{37} - 18 q^{39} - 10 q^{41} + 2 q^{43} - 6 q^{45} + 8 q^{47} + 6 q^{51} - 4 q^{53} - 4 q^{55} - 8 q^{59} + 7 q^{61} - 6 q^{65} + 3 q^{67} - 54 q^{69} + 16 q^{71} + 14 q^{73} - 3 q^{75} - 4 q^{79} - 9 q^{81} + 2 q^{83} - 4 q^{85} - 9 q^{87} + 13 q^{89} + 6 q^{93} + 20 q^{97} - 24 q^{99}+O(q^{100}) $ 2 * q - 3 * q^3 - q^5 - 6 * q^9 + 2 * q^11 + 12 * q^13 + 6 * q^15 + 2 * q^17 + 9 * q^23 - q^25 + 18 * q^27 + 6 * q^29 + 2 * q^31 + 6 * q^33 - 8 * q^37 - 18 * q^39 - 10 * q^41 + 2 * q^43 - 6 * q^45 + 8 * q^47 + 6 * q^51 - 4 * q^53 - 4 * q^55 - 8 * q^59 + 7 * q^61 - 6 * q^65 + 3 * q^67 - 54 * q^69 + 16 * q^71 + 14 * q^73 - 3 * q^75 - 4 * q^79 - 9 * q^81 + 2 * q^83 - 4 * q^85 - 9 * q^87 + 13 * q^89 + 6 * q^93 + 20 * q^97 - 24 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/980\mathbb{Z}\right)^\times$.

$n$	$101$	$197$	$491$
$\chi(n)$	$e\left(\frac{1}{3}\right)$	$1$	$1$

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
$3$	−1.50000	−	2.59808i	−0.866025	−	1.50000i	−0.866025	−	0.500000i	$-0.833333\pi$
$3$	−1.50000	−	2.59808i	−0.866025	−	1.50000i		−	1.00000i	$-0.5\pi$
$4$		0			0
$5$	−0.500000	+	0.866025i	−0.223607	+	0.387298i
$5$	−0.500000	+	0.866025i	−0.223607	+	0.387298i
$6$		0			0
$7$		0			0
$7$		0			0
$8$		0			0
$9$	−3.00000	+	5.19615i	−1.00000	+	1.73205i
$10$		0			0
$11$	1.00000	+	1.73205i	0.301511	+	0.522233i	0.976478	−	0.215615i	$-0.0691756\pi$
$11$	1.00000	+	1.73205i	0.301511	+	0.522233i	−0.674967	+	0.737848i	$0.735842\pi$
$12$		0			0
$13$	6.00000			1.66410			0.832050	−	0.554700i	$-0.187167\pi$
$13$	6.00000			1.66410			0.832050	+	0.554700i	$0.187167\pi$
$14$		0			0
$15$	3.00000			0.774597
$16$		0			0
$17$	1.00000	+	1.73205i	0.242536	+	0.420084i	0.961436	−	0.275029i	$-0.0886875\pi$
$17$	1.00000	+	1.73205i	0.242536	+	0.420084i	−0.718900	+	0.695113i	$0.755354\pi$
$18$		0			0
$19$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$19$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	4.50000	−	7.79423i	0.938315	−	1.62521i	0.169701	−	0.985496i	$-0.445720\pi$
$23$	4.50000	−	7.79423i	0.938315	−	1.62521i	0.768613	−	0.639713i	$-0.220947\pi$
$24$		0			0
$25$	−0.500000	−	0.866025i	−0.100000	−	0.173205i
$26$		0			0
$27$	9.00000			1.73205
$28$		0			0
$29$	3.00000			0.557086			0.278543	−	0.960424i	$-0.410149\pi$
$29$	3.00000			0.557086			0.278543	+	0.960424i	$0.410149\pi$
$30$		0			0
$31$	1.00000	+	1.73205i	0.179605	+	0.311086i	0.941745	−	0.336327i	$-0.109185\pi$
$31$	1.00000	+	1.73205i	0.179605	+	0.311086i	−0.762140	+	0.647412i	$0.775851\pi$
$32$		0			0
$33$	3.00000	−	5.19615i	0.522233	−	0.904534i
$34$		0			0
$35$		0			0
$36$		0			0
$37$	−4.00000	+	6.92820i	−0.657596	+	1.13899i	0.323640	+	0.946180i	$0.395093\pi$
$37$	−4.00000	+	6.92820i	−0.657596	+	1.13899i	−0.981236	+	0.192809i	$0.938240\pi$
$38$		0			0
$39$	−9.00000	−	15.5885i	−1.44115	−	2.49615i
$40$		0			0
$41$	−5.00000			−0.780869			−0.390434	−	0.920631i	$-0.627675\pi$
$41$	−5.00000			−0.780869			−0.390434	+	0.920631i	$0.627675\pi$
$42$		0			0
$43$	1.00000			0.152499			0.0762493	−	0.997089i	$-0.475706\pi$
$43$	1.00000			0.152499			0.0762493	+	0.997089i	$0.475706\pi$
$44$		0			0
$45$	−3.00000	−	5.19615i	−0.447214	−	0.774597i
$46$		0			0
$47$	4.00000	−	6.92820i	0.583460	−	1.01058i	−0.411606	−	0.911362i	$-0.635032\pi$
$47$	4.00000	−	6.92820i	0.583460	−	1.01058i	0.995066	−	0.0992202i	$-0.0316348\pi$
$48$		0			0
$49$		0			0
$50$		0			0
$51$	3.00000	−	5.19615i	0.420084	−	0.727607i
$52$		0			0
$53$	−2.00000	−	3.46410i	−0.274721	−	0.475831i	0.695344	−	0.718677i	$-0.255252\pi$
$53$	−2.00000	−	3.46410i	−0.274721	−	0.475831i	−0.970065	+	0.242846i	$0.921919\pi$
$54$		0			0
$55$	−2.00000			−0.269680
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−4.00000	−	6.92820i	−0.520756	−	0.901975i	−0.999709	−	0.0241347i	$-0.992317\pi$
$59$	−4.00000	−	6.92820i	−0.520756	−	0.901975i	0.478953	−	0.877841i	$-0.341016\pi$
$60$		0			0
$61$	3.50000	−	6.06218i	0.448129	−	0.776182i	−0.550135	−	0.835076i	$-0.685424\pi$
$61$	3.50000	−	6.06218i	0.448129	−	0.776182i	0.998264	+	0.0588933i	$0.0187572\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−3.00000	+	5.19615i	−0.372104	+	0.644503i
$66$		0			0
$67$	1.50000	+	2.59808i	0.183254	+	0.317406i	0.942987	−	0.332830i	$-0.108004\pi$
$67$	1.50000	+	2.59808i	0.183254	+	0.317406i	−0.759733	+	0.650236i	$0.774670\pi$
$68$		0			0
$69$	−27.0000			−3.25042
$70$		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
$73$	7.00000	+	12.1244i	0.819288	+	1.41905i	0.906208	+	0.422833i	$0.138964\pi$
$73$	7.00000	+	12.1244i	0.819288	+	1.41905i	−0.0869195	+	0.996215i	$0.527702\pi$
$74$		0			0
$75$	−1.50000	+	2.59808i	−0.173205	+	0.300000i
$76$		0			0
$77$		0			0
$78$		0			0
$79$	−2.00000	+	3.46410i	−0.225018	+	0.389742i	−0.956325	−	0.292306i	$-0.905577\pi$
$79$	−2.00000	+	3.46410i	−0.225018	+	0.389742i	0.731307	+	0.682048i	$0.238911\pi$
$80$		0			0
$81$	−4.50000	−	7.79423i	−0.500000	−	0.866025i
$82$		0			0
$83$	1.00000			0.109764			0.0548821	−	0.998493i	$-0.482522\pi$
$83$	1.00000			0.109764			0.0548821	+	0.998493i	$0.482522\pi$
$84$		0			0
$85$	−2.00000			−0.216930
$86$		0			0
$87$	−4.50000	−	7.79423i	−0.482451	−	0.835629i
$88$		0			0
$89$	6.50000	−	11.2583i	0.688999	−	1.19338i	−0.283164	−	0.959072i	$-0.591384\pi$
$89$	6.50000	−	11.2583i	0.688999	−	1.19338i	0.972162	−	0.234309i	$-0.0752827\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$	3.00000	−	5.19615i	0.311086	−	0.538816i
$94$		0			0
$95$		0			0
$96$		0			0
$97$	10.0000			1.01535			0.507673	−	0.861550i	$-0.330506\pi$
$97$	10.0000			1.01535			0.507673	+	0.861550i	$0.330506\pi$
$98$		0			0
$99$	−12.0000			−1.20605
$100$		0			0
$101$	−1.50000	−	2.59808i	−0.149256	−	0.258518i	0.781697	−	0.623658i	$-0.214354\pi$
$101$	−1.50000	−	2.59808i	−0.149256	−	0.258518i	−0.930953	+	0.365140i	$0.881021\pi$
$102$		0			0
$103$	6.50000	−	11.2583i	0.640464	−	1.10932i	−0.344865	−	0.938652i	$-0.612075\pi$
$103$	6.50000	−	11.2583i	0.640464	−	1.10932i	0.985329	−	0.170664i	$-0.0545913\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	7.50000	−	12.9904i	0.725052	−	1.25583i	−0.233900	−	0.972261i	$-0.575149\pi$
$107$	7.50000	−	12.9904i	0.725052	−	1.25583i	0.958952	−	0.283567i	$-0.0915178\pi$
$108$		0			0
$109$	−4.50000	−	7.79423i	−0.431022	−	0.746552i	0.565940	−	0.824447i	$-0.308513\pi$
$109$	−4.50000	−	7.79423i	−0.431022	−	0.746552i	−0.996962	+	0.0778949i	$0.975180\pi$
$110$		0			0
$111$	24.0000			2.27798
$112$		0			0
$113$	−4.00000			−0.376288			−0.188144	−	0.982141i	$-0.560247\pi$
$113$	−4.00000			−0.376288			−0.188144	+	0.982141i	$0.560247\pi$
$114$		0			0
$115$	4.50000	+	7.79423i	0.419627	+	0.726816i
$116$		0			0
$117$	−18.0000	+	31.1769i	−1.66410	+	2.88231i
$118$		0			0
$119$		0			0
$120$		0			0
$121$	3.50000	−	6.06218i	0.318182	−	0.551107i
$122$		0			0
$123$	7.50000	+	12.9904i	0.676252	+	1.17130i
$124$		0			0
$125$	1.00000			0.0894427
$126$		0			0
$127$	16.0000			1.41977			0.709885	−	0.704317i	$-0.248747\pi$
$127$	16.0000			1.41977			0.709885	+	0.704317i	$0.248747\pi$
$128$		0			0
$129$	−1.50000	−	2.59808i	−0.132068	−	0.228748i
$130$		0			0
$131$	−2.00000	+	3.46410i	−0.174741	+	0.302660i	−0.940072	−	0.340977i	$-0.889242\pi$
$131$	−2.00000	+	3.46410i	−0.174741	+	0.302660i	0.765331	+	0.643637i	$0.222575\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$	−4.50000	+	7.79423i	−0.387298	+	0.670820i
$136$		0			0
$137$	6.00000	+	10.3923i	0.512615	+	0.887875i	0.999893	+	0.0146279i	$0.00465636\pi$
$137$	6.00000	+	10.3923i	0.512615	+	0.887875i	−0.487278	+	0.873247i	$0.662010\pi$
$138$		0			0
$139$	−10.0000			−0.848189			−0.424094	−	0.905618i	$-0.639408\pi$
$139$	−10.0000			−0.848189			−0.424094	+	0.905618i	$0.639408\pi$
$140$		0			0
$141$	−24.0000			−2.02116
$142$		0			0
$143$	6.00000	+	10.3923i	0.501745	+	0.869048i
$144$		0			0
$145$	−1.50000	+	2.59808i	−0.124568	+	0.215758i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−4.50000	+	7.79423i	−0.368654	+	0.638528i	−0.989355	−	0.145519i	$-0.953515\pi$
$149$	−4.50000	+	7.79423i	−0.368654	+	0.638528i	0.620701	+	0.784047i	$0.286848\pi$
$150$		0			0
$151$	5.00000	+	8.66025i	0.406894	+	0.704761i	0.994540	−	0.104357i	$-0.0332784\pi$
$151$	5.00000	+	8.66025i	0.406894	+	0.704761i	−0.587646	+	0.809118i	$0.699945\pi$
$152$		0			0
$153$	−12.0000			−0.970143
$154$		0			0
$155$	−2.00000			−0.160644
$156$		0			0
$157$	−1.00000	−	1.73205i	−0.0798087	−	0.138233i	0.823359	−	0.567521i	$-0.192098\pi$
$157$	−1.00000	−	1.73205i	−0.0798087	−	0.138233i	−0.903167	+	0.429289i	$0.858764\pi$
$158$		0			0
$159$	−6.00000	+	10.3923i	−0.475831	+	0.824163i
$160$		0			0
$161$		0			0
$162$		0			0
$163$	−4.00000	+	6.92820i	−0.313304	+	0.542659i	−0.979076	−	0.203497i	$-0.934769\pi$
$163$	−4.00000	+	6.92820i	−0.313304	+	0.542659i	0.665771	+	0.746156i	$0.268103\pi$
$164$		0			0
$165$	3.00000	+	5.19615i	0.233550	+	0.404520i
$166$		0			0
$167$	9.00000			0.696441			0.348220	−	0.937413i	$-0.386786\pi$
$167$	9.00000			0.696441			0.348220	+	0.937413i	$0.386786\pi$
$168$		0			0
$169$	23.0000			1.76923
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−8.00000	+	13.8564i	−0.608229	+	1.05348i	0.383304	+	0.923622i	$0.374786\pi$
$173$	−8.00000	+	13.8564i	−0.608229	+	1.05348i	−0.991532	+	0.129861i	$0.958547\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$	−12.0000	+	20.7846i	−0.901975	+	1.56227i
$178$		0			0
$179$	3.00000	+	5.19615i	0.224231	+	0.388379i	0.956088	−	0.293079i	$-0.0946798\pi$
$179$	3.00000	+	5.19615i	0.224231	+	0.388379i	−0.731858	+	0.681457i	$0.761346\pi$
$180$		0			0
$181$	−1.00000			−0.0743294			−0.0371647	−	0.999309i	$-0.511833\pi$
$181$	−1.00000			−0.0743294			−0.0371647	+	0.999309i	$0.511833\pi$
$182$		0			0
$183$	−21.0000			−1.55236
$184$		0			0
$185$	−4.00000	−	6.92820i	−0.294086	−	0.509372i
$186$		0			0
$187$	−2.00000	+	3.46410i	−0.146254	+	0.253320i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−3.00000	+	5.19615i	−0.217072	+	0.375980i	−0.953912	−	0.300088i	$-0.902984\pi$
$191$	−3.00000	+	5.19615i	−0.217072	+	0.375980i	0.736839	+	0.676068i	$0.236317\pi$
$192$		0			0
$193$	−1.00000	−	1.73205i	−0.0719816	−	0.124676i	0.827788	−	0.561041i	$-0.189599\pi$
$193$	−1.00000	−	1.73205i	−0.0719816	−	0.124676i	−0.899770	+	0.436365i	$0.856266\pi$
$194$		0			0
$195$	18.0000			1.28901
$196$		0			0
$197$	14.0000			0.997459			0.498729	−	0.866758i	$-0.333800\pi$
$197$	14.0000			0.997459			0.498729	+	0.866758i	$0.333800\pi$
$198$		0			0
$199$	10.0000	+	17.3205i	0.708881	+	1.22782i	0.965272	+	0.261245i	$0.0841331\pi$
$199$	10.0000	+	17.3205i	0.708881	+	1.22782i	−0.256391	+	0.966573i	$0.582534\pi$
$200$		0			0
$201$	4.50000	−	7.79423i	0.317406	−	0.549762i
$202$		0			0
$203$		0			0
$204$		0			0
$205$	2.50000	−	4.33013i	0.174608	−	0.302429i
$206$		0			0
$207$	27.0000	+	46.7654i	1.87663	+	3.25042i
$208$		0			0
$209$		0			0
$210$		0			0
$211$	4.00000			0.275371			0.137686	−	0.990476i	$-0.456034\pi$
$211$	4.00000			0.275371			0.137686	+	0.990476i	$0.456034\pi$
$212$		0			0
$213$	−12.0000	−	20.7846i	−0.822226	−	1.42414i
$214$		0			0
$215$	−0.500000	+	0.866025i	−0.0340997	+	0.0590624i
$216$		0			0
$217$		0			0
$218$		0			0
$219$	21.0000	−	36.3731i	1.41905	−	2.45786i
$220$		0			0
$221$	6.00000	+	10.3923i	0.403604	+	0.699062i
$222$		0			0
$223$	−16.0000			−1.07144			−0.535720	−	0.844396i	$-0.679960\pi$
$223$	−16.0000			−1.07144			−0.535720	+	0.844396i	$0.679960\pi$
$224$		0			0
$225$	6.00000			0.400000
$226$		0			0
$227$	−2.00000	−	3.46410i	−0.132745	−	0.229920i	0.791989	−	0.610535i	$-0.209046\pi$
$227$	−2.00000	−	3.46410i	−0.132745	−	0.229920i	−0.924734	+	0.380615i	$0.875712\pi$
$228$		0			0
$229$	−7.00000	+	12.1244i	−0.462573	+	0.801200i	−0.999088	−	0.0426906i	$-0.986407\pi$
$229$	−7.00000	+	12.1244i	−0.462573	+	0.801200i	0.536515	+	0.843891i	$0.319740\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	9.00000	−	15.5885i	0.589610	−	1.02123i	−0.404674	−	0.914461i	$-0.632615\pi$
$233$	9.00000	−	15.5885i	0.589610	−	1.02123i	0.994283	−	0.106773i	$-0.0340517\pi$
$234$		0			0
$235$	4.00000	+	6.92820i	0.260931	+	0.451946i
$236$		0			0
$237$	12.0000			0.779484
$238$		0			0
$239$	−26.0000			−1.68180			−0.840900	−	0.541190i	$-0.817974\pi$
$239$	−26.0000			−1.68180			−0.840900	+	0.541190i	$0.817974\pi$
$240$		0			0
$241$	5.00000	+	8.66025i	0.322078	+	0.557856i	0.980917	−	0.194429i	$-0.0622852\pi$
$241$	5.00000	+	8.66025i	0.322078	+	0.557856i	−0.658838	+	0.752285i	$0.728952\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$		0			0
$246$		0			0
$247$		0			0
$248$		0			0
$249$	−1.50000	−	2.59808i	−0.0950586	−	0.164646i
$250$		0			0
$251$	−30.0000			−1.89358			−0.946792	−	0.321847i	$-0.895696\pi$
$251$	−30.0000			−1.89358			−0.946792	+	0.321847i	$0.895696\pi$
$252$		0			0
$253$	18.0000			1.13165
$254$		0			0
$255$	3.00000	+	5.19615i	0.187867	+	0.325396i
$256$		0			0
$257$	4.00000	−	6.92820i	0.249513	−	0.432169i	−0.713878	−	0.700270i	$-0.753063\pi$
$257$	4.00000	−	6.92820i	0.249513	−	0.432169i	0.963391	+	0.268101i	$0.0863961\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$	−9.00000	+	15.5885i	−0.557086	+	0.964901i
$262$		0			0
$263$	−8.50000	−	14.7224i	−0.524132	−	0.907824i	−0.999605	−	0.0280936i	$-0.991056\pi$
$263$	−8.50000	−	14.7224i	−0.524132	−	0.907824i	0.475473	−	0.879730i	$-0.342277\pi$
$264$		0			0
$265$	4.00000			0.245718
$266$		0			0
$267$	−39.0000			−2.38676
$268$		0			0
$269$	4.50000	+	7.79423i	0.274370	+	0.475223i	0.969976	−	0.243201i	$-0.0781974\pi$
$269$	4.50000	+	7.79423i	0.274370	+	0.475223i	−0.695606	+	0.718423i	$0.744864\pi$
$270$		0			0
$271$	−12.0000	+	20.7846i	−0.728948	+	1.26258i	0.228380	+	0.973572i	$0.426657\pi$
$271$	−12.0000	+	20.7846i	−0.728948	+	1.26258i	−0.957328	+	0.289003i	$0.906676\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	1.00000	−	1.73205i	0.0603023	−	0.104447i
$276$		0			0
$277$	−9.00000	−	15.5885i	−0.540758	−	0.936620i	−0.998861	−	0.0477206i	$-0.984804\pi$
$277$	−9.00000	−	15.5885i	−0.540758	−	0.936620i	0.458103	−	0.888899i	$-0.348529\pi$
$278$		0			0
$279$	−12.0000			−0.718421
$280$		0			0
$281$	−22.0000			−1.31241			−0.656205	−	0.754583i	$-0.727839\pi$
$281$	−22.0000			−1.31241			−0.656205	+	0.754583i	$0.727839\pi$
$282$		0			0
$283$	−2.00000	−	3.46410i	−0.118888	−	0.205919i	0.800439	−	0.599414i	$-0.204600\pi$
$283$	−2.00000	−	3.46410i	−0.118888	−	0.205919i	−0.919327	+	0.393494i	$0.871266\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	6.50000	−	11.2583i	0.382353	−	0.662255i
$290$		0			0
$291$	−15.0000	−	25.9808i	−0.879316	−	1.52302i
$292$		0			0
$293$	−4.00000			−0.233682			−0.116841	−	0.993151i	$-0.537277\pi$
$293$	−4.00000			−0.233682			−0.116841	+	0.993151i	$0.537277\pi$
$294$		0			0
$295$	8.00000			0.465778
$296$		0			0
$297$	9.00000	+	15.5885i	0.522233	+	0.904534i
$298$		0			0
$299$	27.0000	−	46.7654i	1.56145	−	2.70451i
$300$		0			0
$301$		0			0
$302$		0			0
$303$	−4.50000	+	7.79423i	−0.258518	+	0.447767i
$304$		0			0
$305$	3.50000	+	6.06218i	0.200409	+	0.347119i
$306$		0			0
$307$	−1.00000			−0.0570730			−0.0285365	−	0.999593i	$-0.509085\pi$
$307$	−1.00000			−0.0570730			−0.0285365	+	0.999593i	$0.509085\pi$
$308$		0			0
$309$	−39.0000			−2.21863
$310$		0			0
$311$	12.0000	+	20.7846i	0.680458	+	1.17859i	0.974841	+	0.222900i	$0.0715523\pi$
$311$	12.0000	+	20.7846i	0.680458	+	1.17859i	−0.294384	+	0.955687i	$0.595114\pi$
$312$		0			0
$313$	−2.00000	+	3.46410i	−0.113047	+	0.195803i	−0.916997	−	0.398894i	$-0.869394\pi$
$313$	−2.00000	+	3.46410i	−0.113047	+	0.195803i	0.803951	+	0.594696i	$0.202728\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−5.00000	+	8.66025i	−0.280828	+	0.486408i	−0.971589	−	0.236675i	$-0.923942\pi$
$317$	−5.00000	+	8.66025i	−0.280828	+	0.486408i	0.690761	+	0.723083i	$0.257276\pi$
$318$		0			0
$319$	3.00000	+	5.19615i	0.167968	+	0.290929i
$320$		0			0
$321$	−45.0000			−2.51166
$322$		0			0
$323$		0			0
$324$		0			0
$325$	−3.00000	−	5.19615i	−0.166410	−	0.288231i
$326$		0			0
$327$	−13.5000	+	23.3827i	−0.746552	+	1.29307i
$328$		0			0
$329$		0			0
$330$		0			0
$331$	−5.00000	+	8.66025i	−0.274825	+	0.476011i	−0.970091	−	0.242742i	$-0.921953\pi$
$331$	−5.00000	+	8.66025i	−0.274825	+	0.476011i	0.695266	+	0.718752i	$0.255287\pi$
$332$		0			0
$333$	−24.0000	−	41.5692i	−1.31519	−	2.27798i
$334$		0			0
$335$	−3.00000			−0.163908
$336$		0			0
$337$	28.0000			1.52526			0.762629	−	0.646837i	$-0.223908\pi$
$337$	28.0000			1.52526			0.762629	+	0.646837i	$0.223908\pi$
$338$		0			0
$339$	6.00000	+	10.3923i	0.325875	+	0.564433i
$340$		0			0
$341$	−2.00000	+	3.46410i	−0.108306	+	0.187592i
$342$		0			0
$343$		0			0
$344$		0			0
$345$	13.5000	−	23.3827i	0.726816	−	1.25888i
$346$		0			0
$347$	−12.5000	−	21.6506i	−0.671035	−	1.16227i	−0.977611	−	0.210421i	$-0.932517\pi$
$347$	−12.5000	−	21.6506i	−0.671035	−	1.16227i	0.306576	−	0.951846i	$-0.400817\pi$
$348$		0			0
$349$	17.0000			0.909989			0.454995	−	0.890494i	$-0.349641\pi$
$349$	17.0000			0.909989			0.454995	+	0.890494i	$0.349641\pi$
$350$		0			0
$351$	54.0000			2.88231
$352$		0			0
$353$	−18.0000	−	31.1769i	−0.958043	−	1.65938i	−0.727245	−	0.686378i	$-0.759200\pi$
$353$	−18.0000	−	31.1769i	−0.958043	−	1.65938i	−0.230799	−	0.973002i	$-0.574134\pi$
$354$		0			0
$355$	−4.00000	+	6.92820i	−0.212298	+	0.367711i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	5.00000	−	8.66025i	0.263890	−	0.457071i	−0.703382	−	0.710812i	$-0.748328\pi$
$359$	5.00000	−	8.66025i	0.263890	−	0.457071i	0.967272	+	0.253741i	$0.0816611\pi$
$360$		0			0
$361$	9.50000	+	16.4545i	0.500000	+	0.866025i
$362$		0			0
$363$	−21.0000			−1.10221
$364$		0			0
$365$	−14.0000			−0.732793
$366$		0			0
$367$	0.500000	+	0.866025i	0.0260998	+	0.0452062i	0.878780	−	0.477227i	$-0.158358\pi$
$367$	0.500000	+	0.866025i	0.0260998	+	0.0452062i	−0.852680	+	0.522433i	$0.825025\pi$
$368$		0			0
$369$	15.0000	−	25.9808i	0.780869	−	1.35250i
$370$		0			0
$371$		0			0
$372$		0			0
$373$	−16.0000	+	27.7128i	−0.828449	+	1.43492i	0.0708063	+	0.997490i	$0.477443\pi$
$373$	−16.0000	+	27.7128i	−0.828449	+	1.43492i	−0.899255	+	0.437425i	$0.855891\pi$
$374$		0			0
$375$	−1.50000	−	2.59808i	−0.0774597	−	0.134164i
$376$		0			0
$377$	18.0000			0.927047
$378$		0			0
$379$		0			0			−	1.00000i	$-0.5\pi$
$379$		0			0				1.00000i	$0.5\pi$
$380$		0			0
$381$	−24.0000	−	41.5692i	−1.22956	−	2.12966i
$382$		0			0
$383$	−4.50000	+	7.79423i	−0.229939	+	0.398266i	−0.957790	−	0.287469i	$-0.907186\pi$
$383$	−4.50000	+	7.79423i	−0.229939	+	0.398266i	0.727851	+	0.685736i	$0.240519\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	−3.00000	+	5.19615i	−0.152499	+	0.264135i
$388$		0			0
$389$	5.00000	+	8.66025i	0.253510	+	0.439092i	0.964490	−	0.264120i	$-0.0850816\pi$
$389$	5.00000	+	8.66025i	0.253510	+	0.439092i	−0.710980	+	0.703213i	$0.751748\pi$
$390$		0			0
$391$	18.0000			0.910299
$392$		0			0
$393$	12.0000			0.605320
$394$		0			0
$395$	−2.00000	−	3.46410i	−0.100631	−	0.174298i
$396$		0			0
$397$	5.00000	−	8.66025i	0.250943	−	0.434646i	−0.712843	−	0.701324i	$-0.752593\pi$
$397$	5.00000	−	8.66025i	0.250943	−	0.434646i	0.963786	+	0.266678i	$0.0859261\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	14.5000	−	25.1147i	0.724095	−	1.25417i	−0.235250	−	0.971935i	$-0.575591\pi$
$401$	14.5000	−	25.1147i	0.724095	−	1.25417i	0.959345	−	0.282235i	$-0.0910758\pi$
$402$		0			0
$403$	6.00000	+	10.3923i	0.298881	+	0.517678i
$404$		0			0
$405$	9.00000			0.447214
$406$		0			0
$407$	−16.0000			−0.793091
$408$		0			0
$409$	1.50000	+	2.59808i	0.0741702	+	0.128467i	0.900725	−	0.434389i	$-0.143036\pi$
$409$	1.50000	+	2.59808i	0.0741702	+	0.128467i	−0.826555	+	0.562856i	$0.809703\pi$
$410$		0			0
$411$	18.0000	−	31.1769i	0.887875	−	1.53784i
$412$		0			0
$413$		0			0
$414$		0			0
$415$	−0.500000	+	0.866025i	−0.0245440	+	0.0425115i
$416$		0			0
$417$	15.0000	+	25.9808i	0.734553	+	1.27228i
$418$		0			0
$419$	24.0000			1.17248			0.586238	−	0.810139i	$-0.300608\pi$
$419$	24.0000			1.17248			0.586238	+	0.810139i	$0.300608\pi$
$420$		0			0
$421$	−1.00000			−0.0487370			−0.0243685	−	0.999703i	$-0.507758\pi$
$421$	−1.00000			−0.0487370			−0.0243685	+	0.999703i	$0.507758\pi$
$422$		0			0
$423$	24.0000	+	41.5692i	1.16692	+	2.02116i
$424$		0			0
$425$	1.00000	−	1.73205i	0.0485071	−	0.0840168i
$426$		0			0
$427$		0			0
$428$		0			0
$429$	18.0000	−	31.1769i	0.869048	−	1.50524i
$430$		0			0
$431$	−3.00000	−	5.19615i	−0.144505	−	0.250290i	0.784683	−	0.619897i	$-0.212826\pi$
$431$	−3.00000	−	5.19615i	−0.144505	−	0.250290i	−0.929188	+	0.369607i	$0.879492\pi$
$432$		0			0
$433$	4.00000			0.192228			0.0961139	−	0.995370i	$-0.469359\pi$
$433$	4.00000			0.192228			0.0961139	+	0.995370i	$0.469359\pi$
$434$		0			0
$435$	9.00000			0.431517
$436$		0			0
$437$		0			0
$438$		0			0
$439$	2.00000	−	3.46410i	0.0954548	−	0.165333i	−0.814344	−	0.580383i	$-0.802903\pi$
$439$	2.00000	−	3.46410i	0.0954548	−	0.165333i	0.909798	+	0.415051i	$0.136236\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−18.5000	+	32.0429i	−0.878962	+	1.52241i	−0.0264796	+	0.999649i	$0.508430\pi$
$443$	−18.5000	+	32.0429i	−0.878962	+	1.52241i	−0.852482	+	0.522757i	$0.824904\pi$
$444$		0			0
$445$	6.50000	+	11.2583i	0.308130	+	0.533696i
$446$		0			0
$447$	27.0000			1.27706
$448$		0			0
$449$	15.0000			0.707894			0.353947	−	0.935266i	$-0.384839\pi$
$449$	15.0000			0.707894			0.353947	+	0.935266i	$0.384839\pi$
$450$		0			0
$451$	−5.00000	−	8.66025i	−0.235441	−	0.407795i
$452$		0			0
$453$	15.0000	−	25.9808i	0.704761	−	1.22068i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−14.0000	+	24.2487i	−0.654892	+	1.13431i	0.327028	+	0.945015i	$0.393953\pi$
$457$	−14.0000	+	24.2487i	−0.654892	+	1.13431i	−0.981921	+	0.189292i	$0.939381\pi$
$458$		0			0
$459$	9.00000	+	15.5885i	0.420084	+	0.727607i
$460$		0			0
$461$	2.00000			0.0931493			0.0465746	−	0.998915i	$-0.485169\pi$
$461$	2.00000			0.0931493			0.0465746	+	0.998915i	$0.485169\pi$
$462$		0			0
$463$	17.0000			0.790057			0.395029	−	0.918669i	$-0.370735\pi$
$463$	17.0000			0.790057			0.395029	+	0.918669i	$0.370735\pi$
$464$		0			0
$465$	3.00000	+	5.19615i	0.139122	+	0.240966i
$466$		0			0
$467$	2.50000	−	4.33013i	0.115686	−	0.200374i	−0.802368	−	0.596830i	$-0.796427\pi$
$467$	2.50000	−	4.33013i	0.115686	−	0.200374i	0.918054	+	0.396456i	$0.129760\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$	−3.00000	+	5.19615i	−0.138233	+	0.239426i
$472$		0			0
$473$	1.00000	+	1.73205i	0.0459800	+	0.0796398i
$474$		0			0
$475$		0			0
$476$		0			0
$477$	24.0000			1.09888
$478$		0			0
$479$	−15.0000	−	25.9808i	−0.685367	−	1.18709i	−0.973321	−	0.229447i	$-0.926308\pi$
$479$	−15.0000	−	25.9808i	−0.685367	−	1.18709i	0.287954	−	0.957644i	$-0.407025\pi$
$480$		0			0
$481$	−24.0000	+	41.5692i	−1.09431	+	1.89539i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−5.00000	+	8.66025i	−0.227038	+	0.393242i
$486$		0			0
$487$	−16.0000	−	27.7128i	−0.725029	−	1.25579i	−0.958962	−	0.283535i	$-0.908493\pi$
$487$	−16.0000	−	27.7128i	−0.725029	−	1.25579i	0.233933	−	0.972253i	$-0.424840\pi$
$488$		0			0
$489$	24.0000			1.08532
$490$		0			0
$491$	6.00000			0.270776			0.135388	−	0.990793i	$-0.456772\pi$
$491$	6.00000			0.270776			0.135388	+	0.990793i	$0.456772\pi$
$492$		0			0
$493$	3.00000	+	5.19615i	0.135113	+	0.234023i
$494$		0			0
$495$	6.00000	−	10.3923i	0.269680	−	0.467099i
$496$		0			0
$497$		0			0
$498$		0			0
$499$	3.00000	−	5.19615i	0.134298	−	0.232612i	−0.791031	−	0.611776i	$-0.790455\pi$
$499$	3.00000	−	5.19615i	0.134298	−	0.232612i	0.925329	+	0.379165i	$0.123789\pi$
$500$		0			0
$501$	−13.5000	−	23.3827i	−0.603136	−	1.04466i
$502$		0			0
$503$	−27.0000			−1.20387			−0.601935	−	0.798545i	$-0.705603\pi$
$503$	−27.0000			−1.20387			−0.601935	+	0.798545i	$0.705603\pi$
$504$		0			0
$505$	3.00000			0.133498
$506$		0			0
$507$	−34.5000	−	59.7558i	−1.53220	−	2.65385i
$508$		0			0
$509$	−20.5000	+	35.5070i	−0.908647	+	1.57382i	−0.0927004	+	0.995694i	$0.529550\pi$
$509$	−20.5000	+	35.5070i	−0.908647	+	1.57382i	−0.815946	+	0.578128i	$0.803783\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$		0			0
$514$		0			0
$515$	6.50000	+	11.2583i	0.286424	+	0.496101i
$516$		0			0
$517$	16.0000			0.703679
$518$		0			0
$519$	48.0000			2.10697
$520$		0			0
$521$	7.00000	+	12.1244i	0.306676	+	0.531178i	0.977633	−	0.210318i	$-0.0674500\pi$
$521$	7.00000	+	12.1244i	0.306676	+	0.531178i	−0.670957	+	0.741496i	$0.734117\pi$
$522$		0			0
$523$	2.00000	−	3.46410i	0.0874539	−	0.151475i	−0.818980	−	0.573822i	$-0.805460\pi$
$523$	2.00000	−	3.46410i	0.0874539	−	0.151475i	0.906434	+	0.422347i	$0.138794\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−2.00000	+	3.46410i	−0.0871214	+	0.150899i
$528$		0			0
$529$	−29.0000	−	50.2295i	−1.26087	−	2.18389i
$530$		0			0
$531$	48.0000			2.08302
$532$		0			0
$533$	−30.0000			−1.29944
$534$		0			0
$535$	7.50000	+	12.9904i	0.324253	+	0.561623i
$536$		0			0
$537$	9.00000	−	15.5885i	0.388379	−	0.672692i
$538$		0			0
$539$		0			0
$540$		0			0
$541$	−16.5000	+	28.5788i	−0.709390	+	1.22870i	0.255693	+	0.966758i	$0.417696\pi$
$541$	−16.5000	+	28.5788i	−0.709390	+	1.22870i	−0.965084	+	0.261942i	$0.915637\pi$
$542$		0			0
$543$	1.50000	+	2.59808i	0.0643712	+	0.111494i
$544$		0			0
$545$	9.00000			0.385518
$546$		0			0
$547$	−15.0000			−0.641354			−0.320677	−	0.947189i	$-0.603910\pi$
$547$	−15.0000			−0.641354			−0.320677	+	0.947189i	$0.603910\pi$
$548$		0			0
$549$	21.0000	+	36.3731i	0.896258	+	1.55236i
$550$		0			0
$551$		0			0
$552$		0			0
$553$		0			0
$554$		0			0
$555$	−12.0000	+	20.7846i	−0.509372	+	0.882258i
$556$		0			0
$557$	1.00000	+	1.73205i	0.0423714	+	0.0733893i	0.886433	−	0.462856i	$-0.153175\pi$
$557$	1.00000	+	1.73205i	0.0423714	+	0.0733893i	−0.844062	+	0.536246i	$0.819842\pi$
$558$		0			0
$559$	6.00000			0.253773
$560$		0			0
$561$	12.0000			0.506640
$562$		0			0
$563$	−6.50000	−	11.2583i	−0.273942	−	0.474482i	0.695925	−	0.718114i	$-0.254994\pi$
$563$	−6.50000	−	11.2583i	−0.273942	−	0.474482i	−0.969868	+	0.243632i	$0.921661\pi$
$564$		0			0
$565$	2.00000	−	3.46410i	0.0841406	−	0.145736i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−15.0000	+	25.9808i	−0.628833	+	1.08917i	0.358954	+	0.933355i	$0.383134\pi$
$569$	−15.0000	+	25.9808i	−0.628833	+	1.08917i	−0.987786	+	0.155815i	$0.950200\pi$
$570$		0			0
$571$	−12.0000	−	20.7846i	−0.502184	−	0.869809i	−0.999997	−	0.00252413i	$-0.999197\pi$
$571$	−12.0000	−	20.7846i	−0.502184	−	0.869809i	0.497812	−	0.867285i	$-0.334137\pi$
$572$		0			0
$573$	18.0000			0.751961
$574$		0			0
$575$	−9.00000			−0.375326
$576$		0			0
$577$	11.0000	+	19.0526i	0.457936	+	0.793168i	0.998852	−	0.0479084i	$-0.0152556\pi$
$577$	11.0000	+	19.0526i	0.457936	+	0.793168i	−0.540916	+	0.841077i	$0.681922\pi$
$578$		0			0
$579$	−3.00000	+	5.19615i	−0.124676	+	0.215945i
$580$		0			0
$581$		0			0
$582$		0			0
$583$	4.00000	−	6.92820i	0.165663	−	0.286937i
$584$		0			0
$585$	−18.0000	−	31.1769i	−0.744208	−	1.28901i
$586$		0			0
$587$	−20.0000			−0.825488			−0.412744	−	0.910847i	$-0.635430\pi$
$587$	−20.0000			−0.825488			−0.412744	+	0.910847i	$0.635430\pi$
$588$		0			0
$589$		0			0
$590$		0			0
$591$	−21.0000	−	36.3731i	−0.863825	−	1.49619i
$592$		0			0
$593$	−18.0000	+	31.1769i	−0.739171	+	1.28028i	0.213697	+	0.976900i	$0.431449\pi$
$593$	−18.0000	+	31.1769i	−0.739171	+	1.28028i	−0.952869	+	0.303383i	$0.901884\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	30.0000	−	51.9615i	1.22782	−	2.12664i
$598$		0			0
$599$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$599$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$600$		0			0
$601$	6.00000			0.244745			0.122373	−	0.992484i	$-0.460950\pi$
$601$	6.00000			0.244745			0.122373	+	0.992484i	$0.460950\pi$
$602$		0			0
$603$	−18.0000			−0.733017
$604$		0			0
$605$	3.50000	+	6.06218i	0.142295	+	0.246463i
$606$		0			0
$607$	0.500000	−	0.866025i	0.0202944	−	0.0351509i	−0.855700	−	0.517472i	$-0.826873\pi$
$607$	0.500000	−	0.866025i	0.0202944	−	0.0351509i	0.875994	+	0.482322i	$0.160206\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	24.0000	−	41.5692i	0.970936	−	1.68171i
$612$		0			0
$613$	3.00000	+	5.19615i	0.121169	+	0.209871i	0.920229	−	0.391381i	$-0.128002\pi$
$613$	3.00000	+	5.19615i	0.121169	+	0.209871i	−0.799060	+	0.601251i	$0.794669\pi$
$614$		0			0
$615$	−15.0000			−0.604858
$616$		0			0
$617$	20.0000			0.805170			0.402585	−	0.915383i	$-0.368112\pi$
$617$	20.0000			0.805170			0.402585	+	0.915383i	$0.368112\pi$
$618$		0			0
$619$	−5.00000	−	8.66025i	−0.200967	−	0.348085i	0.747873	−	0.663842i	$-0.231075\pi$
$619$	−5.00000	−	8.66025i	−0.200967	−	0.348085i	−0.948840	+	0.315757i	$0.897742\pi$
$620$		0			0
$621$	40.5000	−	70.1481i	1.62521	−	2.81494i
$622$		0			0
$623$		0			0
$624$		0			0
$625$	−0.500000	+	0.866025i	−0.0200000	+	0.0346410i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−16.0000			−0.637962
$630$		0			0
$631$	−34.0000			−1.35352			−0.676759	−	0.736204i	$-0.736616\pi$
$631$	−34.0000			−1.35352			−0.676759	+	0.736204i	$0.736616\pi$
$632$		0			0
$633$	−6.00000	−	10.3923i	−0.238479	−	0.413057i
$634$		0			0
$635$	−8.00000	+	13.8564i	−0.317470	+	0.549875i
$636$		0			0
$637$		0			0
$638$		0			0
$639$	−24.0000	+	41.5692i	−0.949425	+	1.64445i
$640$		0			0
$641$	15.5000	+	26.8468i	0.612213	+	1.06038i	0.990867	+	0.134846i	$0.0430539\pi$
$641$	15.5000	+	26.8468i	0.612213	+	1.06038i	−0.378653	+	0.925539i	$0.623613\pi$
$642$		0			0
$643$	20.0000			0.788723			0.394362	−	0.918955i	$-0.370966\pi$
$643$	20.0000			0.788723			0.394362	+	0.918955i	$0.370966\pi$
$644$		0			0
$645$	3.00000			0.118125
$646$		0			0
$647$	0.500000	+	0.866025i	0.0196570	+	0.0340470i	0.875687	−	0.482880i	$-0.160409\pi$
$647$	0.500000	+	0.866025i	0.0196570	+	0.0340470i	−0.856030	+	0.516927i	$0.827076\pi$
$648$		0			0
$649$	8.00000	−	13.8564i	0.314027	−	0.543912i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	17.0000	−	29.4449i	0.665261	−	1.15227i	−0.313953	−	0.949439i	$-0.601653\pi$
$653$	17.0000	−	29.4449i	0.665261	−	1.15227i	0.979214	−	0.202828i	$-0.0650132\pi$
$654$		0			0
$655$	−2.00000	−	3.46410i	−0.0781465	−	0.135354i
$656$		0			0
$657$	−84.0000			−3.27715
$658$		0			0
$659$	4.00000			0.155818			0.0779089	−	0.996960i	$-0.475176\pi$
$659$	4.00000			0.155818			0.0779089	+	0.996960i	$0.475176\pi$
$660$		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
$663$	18.0000	−	31.1769i	0.699062	−	1.21081i
$664$		0			0
$665$		0			0
$666$		0			0
$667$	13.5000	−	23.3827i	0.522722	−	0.905381i
$668$		0			0
$669$	24.0000	+	41.5692i	0.927894	+	1.60716i
$670$		0			0
$671$	14.0000			0.540464
$672$		0			0
$673$	16.0000			0.616755			0.308377	−	0.951264i	$-0.400214\pi$
$673$	16.0000			0.616755			0.308377	+	0.951264i	$0.400214\pi$
$674$		0			0
$675$	−4.50000	−	7.79423i	−0.173205	−	0.300000i
$676$		0			0
$677$	3.00000	−	5.19615i	0.115299	−	0.199704i	−0.802600	−	0.596518i	$-0.796551\pi$
$677$	3.00000	−	5.19615i	0.115299	−	0.199704i	0.917899	+	0.396813i	$0.129884\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$	−6.00000	+	10.3923i	−0.229920	+	0.398234i
$682$		0			0
$683$	−14.5000	−	25.1147i	−0.554827	−	0.960989i	−0.997917	−	0.0645115i	$-0.979451\pi$
$683$	−14.5000	−	25.1147i	−0.554827	−	0.960989i	0.443090	−	0.896477i	$-0.353882\pi$
$684$		0			0
$685$	−12.0000			−0.458496
$686$		0			0
$687$	42.0000			1.60240
$688$		0			0
$689$	−12.0000	−	20.7846i	−0.457164	−	0.791831i
$690$		0			0
$691$	−13.0000	+	22.5167i	−0.494543	+	0.856574i	−0.999980	−	0.00628943i	$-0.997998\pi$
$691$	−13.0000	+	22.5167i	−0.494543	+	0.856574i	0.505437	+	0.862864i	$0.331331\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	5.00000	−	8.66025i	0.189661	−	0.328502i
$696$		0			0
$697$	−5.00000	−	8.66025i	−0.189389	−	0.328031i
$698$		0			0
$699$	−54.0000			−2.04247
$700$		0			0
$701$	−29.0000			−1.09531			−0.547657	−	0.836703i	$-0.684480\pi$
$701$	−29.0000			−1.09531			−0.547657	+	0.836703i	$0.684480\pi$
$702$		0			0
$703$		0			0
$704$		0			0
$705$	12.0000	−	20.7846i	0.451946	−	0.782794i
$706$		0			0
$707$		0			0
$708$		0			0
$709$	−15.5000	+	26.8468i	−0.582115	+	1.00825i	0.413114	+	0.910679i	$0.364441\pi$
$709$	−15.5000	+	26.8468i	−0.582115	+	1.00825i	−0.995228	+	0.0975728i	$0.968892\pi$
$710$		0			0
$711$	−12.0000	−	20.7846i	−0.450035	−	0.779484i
$712$		0			0
$713$	18.0000			0.674105
$714$		0			0
$715$	−12.0000			−0.448775
$716$		0			0
$717$	39.0000	+	67.5500i	1.45648	+	2.52270i
$718$		0			0
$719$	−3.00000	+	5.19615i	−0.111881	+	0.193784i	−0.916529	−	0.399969i	$-0.869021\pi$
$719$	−3.00000	+	5.19615i	−0.111881	+	0.193784i	0.804648	+	0.593753i	$0.202354\pi$
$720$		0			0
$721$		0			0
$722$		0			0
$723$	15.0000	−	25.9808i	0.557856	−	0.966235i
$724$		0			0
$725$	−1.50000	−	2.59808i	−0.0557086	−	0.0964901i
$726$		0			0
$727$	−3.00000			−0.111264			−0.0556319	−	0.998451i	$-0.517717\pi$
$727$	−3.00000			−0.111264			−0.0556319	+	0.998451i	$0.517717\pi$
$728$		0			0
$729$	−27.0000			−1.00000
$730$		0			0
$731$	1.00000	+	1.73205i	0.0369863	+	0.0640622i
$732$		0			0
$733$	17.0000	−	29.4449i	0.627909	−	1.08757i	−0.360061	−	0.932929i	$-0.617244\pi$
$733$	17.0000	−	29.4449i	0.627909	−	1.08757i	0.987971	−	0.154642i	$-0.0494225\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−3.00000	+	5.19615i	−0.110506	+	0.191403i
$738$		0			0
$739$	−5.00000	−	8.66025i	−0.183928	−	0.318573i	0.759287	−	0.650756i	$-0.225548\pi$
$739$	−5.00000	−	8.66025i	−0.183928	−	0.318573i	−0.943215	+	0.332184i	$0.892215\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−3.00000			−0.110059			−0.0550297	−	0.998485i	$-0.517525\pi$
$743$	−3.00000			−0.110059			−0.0550297	+	0.998485i	$0.517525\pi$
$744$		0			0
$745$	−4.50000	−	7.79423i	−0.164867	−	0.285558i
$746$		0			0
$747$	−3.00000	+	5.19615i	−0.109764	+	0.190117i
$748$		0			0
$749$		0			0
$750$		0			0
$751$	−10.0000	+	17.3205i	−0.364905	+	0.632034i	−0.988761	−	0.149505i	$-0.952232\pi$
$751$	−10.0000	+	17.3205i	−0.364905	+	0.632034i	0.623856	+	0.781540i	$0.285565\pi$
$752$		0			0
$753$	45.0000	+	77.9423i	1.63989	+	2.84037i
$754$		0			0
$755$	−10.0000			−0.363937
$756$		0			0
$757$	−38.0000			−1.38113			−0.690567	−	0.723269i	$-0.742639\pi$
$757$	−38.0000			−1.38113			−0.690567	+	0.723269i	$0.742639\pi$
$758$		0			0
$759$	−27.0000	−	46.7654i	−0.980038	−	1.69748i
$760$		0			0
$761$	15.0000	−	25.9808i	0.543750	−	0.941802i	−0.454935	−	0.890525i	$-0.650337\pi$
$761$	15.0000	−	25.9808i	0.543750	−	0.941802i	0.998684	−	0.0512772i	$-0.0163292\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$	6.00000	−	10.3923i	0.216930	−	0.375735i
$766$		0			0
$767$	−24.0000	−	41.5692i	−0.866590	−	1.50098i
$768$		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
$771$	−24.0000			−0.864339
$772$		0			0
$773$	−9.00000	−	15.5885i	−0.323708	−	0.560678i	0.657542	−	0.753418i	$-0.271596\pi$
$773$	−9.00000	−	15.5885i	−0.323708	−	0.560678i	−0.981250	+	0.192740i	$0.938263\pi$
$774$		0			0
$775$	1.00000	−	1.73205i	0.0359211	−	0.0622171i
$776$		0			0
$777$		0			0
$778$		0			0
$779$		0			0
$780$		0			0
$781$	8.00000	+	13.8564i	0.286263	+	0.495821i
$782$		0			0
$783$	27.0000			0.964901
$784$		0			0
$785$	2.00000			0.0713831
$786$		0			0
$787$	−5.50000	−	9.52628i	−0.196054	−	0.339575i	0.751192	−	0.660084i	$-0.229479\pi$
$787$	−5.50000	−	9.52628i	−0.196054	−	0.339575i	−0.947245	+	0.320509i	$0.896146\pi$
$788$		0			0
$789$	−25.5000	+	44.1673i	−0.907824	+	1.57240i
$790$		0			0
$791$		0			0
$792$		0			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:	\( N \)	\(=\)	\( 980 = 2^{2} \cdot 5 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	980.i (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-1.50000 - 2.59808i) q^{3} +(-0.500000 + 0.866025i) q^{5} +(-3.00000 + 5.19615i) q^{9} +O(q^{10})\)	\(q+(-1.50000 - 2.59808i) q^{3} +(-0.500000 + 0.866025i) q^{5} +(-3.00000 + 5.19615i) q^{9} +(1.00000 + 1.73205i) q^{11} +6.00000 q^{13} +3.00000 q^{15} +(1.00000 + 1.73205i) q^{17} +(4.50000 - 7.79423i) q^{23} +(-0.500000 - 0.866025i) q^{25} +9.00000 q^{27} +3.00000 q^{29} +(1.00000 + 1.73205i) q^{31} +(3.00000 - 5.19615i) q^{33} +(-4.00000 + 6.92820i) q^{37} +(-9.00000 - 15.5885i) q^{39} -5.00000 q^{41} +1.00000 q^{43} +(-3.00000 - 5.19615i) q^{45} +(4.00000 - 6.92820i) q^{47} +(3.00000 - 5.19615i) q^{51} +(-2.00000 - 3.46410i) q^{53} -2.00000 q^{55} +(-4.00000 - 6.92820i) q^{59} +(3.50000 - 6.06218i) q^{61} +(-3.00000 + 5.19615i) q^{65} +(1.50000 + 2.59808i) q^{67} -27.0000 q^{69} +8.00000 q^{71} +(7.00000 + 12.1244i) q^{73} +(-1.50000 + 2.59808i) q^{75} +(-2.00000 + 3.46410i) q^{79} +(-4.50000 - 7.79423i) q^{81} +1.00000 q^{83} -2.00000 q^{85} +(-4.50000 - 7.79423i) q^{87} +(6.50000 - 11.2583i) q^{89} +(3.00000 - 5.19615i) q^{93} +10.0000 q^{97} -12.0000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 3 q^{3} - q^{5} - 6 q^{9}+O(q^{10}) \) 2 * q - 3 * q^3 - q^5 - 6 * q^9	\( 2 q - 3 q^{3} - q^{5} - 6 q^{9} + 2 q^{11} + 12 q^{13} + 6 q^{15} + 2 q^{17} + 9 q^{23} - q^{25} + 18 q^{27} + 6 q^{29} + 2 q^{31} + 6 q^{33} - 8 q^{37} - 18 q^{39} - 10 q^{41} + 2 q^{43} - 6 q^{45} + 8 q^{47} + 6 q^{51} - 4 q^{53} - 4 q^{55} - 8 q^{59} + 7 q^{61} - 6 q^{65} + 3 q^{67} - 54 q^{69} + 16 q^{71} + 14 q^{73} - 3 q^{75} - 4 q^{79} - 9 q^{81} + 2 q^{83} - 4 q^{85} - 9 q^{87} + 13 q^{89} + 6 q^{93} + 20 q^{97} - 24 q^{99}+O(q^{100}) \) 2 * q - 3 * q^3 - q^5 - 6 * q^9 + 2 * q^11 + 12 * q^13 + 6 * q^15 + 2 * q^17 + 9 * q^23 - q^25 + 18 * q^27 + 6 * q^29 + 2 * q^31 + 6 * q^33 - 8 * q^37 - 18 * q^39 - 10 * q^41 + 2 * q^43 - 6 * q^45 + 8 * q^47 + 6 * q^51 - 4 * q^53 - 4 * q^55 - 8 * q^59 + 7 * q^61 - 6 * q^65 + 3 * q^67 - 54 * q^69 + 16 * q^71 + 14 * q^73 - 3 * q^75 - 4 * q^79 - 9 * q^81 + 2 * q^83 - 4 * q^85 - 9 * q^87 + 13 * q^89 + 6 * q^93 + 20 * q^97 - 24 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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