LMFDB - Embedded newform 980.2.i.b.361.1

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			361.1
Root			$0.500000 + 0.866025i$ of defining polynomial
Character	$\chi$	$=$	980.361
Dual form			980.2.i.b.961.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+(-1.50000 + 2.59808i) q^{3} +(0.500000 + 0.866025i) q^{5} +(-3.00000 - 5.19615i) q^{9} +(2.50000 - 4.33013i) q^{11} -3.00000 q^{13} -3.00000 q^{15} +(0.500000 - 0.866025i) q^{17} +(-3.00000 - 5.19615i) q^{19} +(-3.00000 - 5.19615i) q^{23} +(-0.500000 + 0.866025i) q^{25} +9.00000 q^{27} -9.00000 q^{29} +(2.00000 - 3.46410i) q^{31} +(7.50000 + 12.9904i) q^{33} +(-1.00000 - 1.73205i) q^{37} +(4.50000 - 7.79423i) q^{39} -4.00000 q^{41} +10.0000 q^{43} +(3.00000 - 5.19615i) q^{45} +(0.500000 + 0.866025i) q^{47} +(1.50000 + 2.59808i) q^{51} +(-2.00000 + 3.46410i) q^{53} +5.00000 q^{55} +18.0000 q^{57} +(4.00000 - 6.92820i) q^{59} +(4.00000 + 6.92820i) q^{61} +(-1.50000 - 2.59808i) q^{65} +(-6.00000 + 10.3923i) q^{67} +18.0000 q^{69} +8.00000 q^{71} +(-1.00000 + 1.73205i) q^{73} +(-1.50000 - 2.59808i) q^{75} +(-6.50000 - 11.2583i) q^{79} +(-4.50000 + 7.79423i) q^{81} -4.00000 q^{83} +1.00000 q^{85} +(13.5000 - 23.3827i) q^{87} +(-2.00000 - 3.46410i) q^{89} +(6.00000 + 10.3923i) q^{93} +(3.00000 - 5.19615i) q^{95} -13.0000 q^{97} -30.0000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 3 q^{3} + q^{5} - 6 q^{9} + 5 q^{11} - 6 q^{13} - 6 q^{15} + q^{17} - 6 q^{19} - 6 q^{23} - q^{25} + 18 q^{27} - 18 q^{29} + 4 q^{31} + 15 q^{33} - 2 q^{37} + 9 q^{39} - 8 q^{41} + 20 q^{43} + 6 q^{45}+ \cdots - 60 q^{99}+O(q^{100}) $ 2 * q - 3 * q^3 + q^5 - 6 * q^9 + 5 * q^11 - 6 * q^13 - 6 * q^15 + q^17 - 6 * q^19 - 6 * q^23 - q^25 + 18 * q^27 - 18 * q^29 + 4 * q^31 + 15 * q^33 - 2 * q^37 + 9 * q^39 - 8 * q^41 + 20 * q^43 + 6 * q^45 + q^47 + 3 * q^51 - 4 * q^53 + 10 * q^55 + 36 * q^57 + 8 * q^59 + 8 * q^61 - 3 * q^65 - 12 * q^67 + 36 * q^69 + 16 * q^71 - 2 * q^73 - 3 * q^75 - 13 * q^79 - 9 * q^81 - 8 * q^83 + 2 * q^85 + 27 * q^87 - 4 * q^89 + 12 * q^93 + 6 * q^95 - 26 * q^97 - 60 * q^99

Character values

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

$n$	$101$	$197$	$491$
$\chi(n)$	$e\left(\frac{2}{3}\right)$	$1$	$1$

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			1.00000i	$0.5\pi$
$3$	−1.50000	+	2.59808i	−0.866025	+	1.50000i	−0.866025	+	0.500000i	$0.833333\pi$
$4$		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
$11$	2.50000	−	4.33013i	0.753778	−	1.30558i	−0.192201	−	0.981356i	$-0.561563\pi$
$11$	2.50000	−	4.33013i	0.753778	−	1.30558i	0.945979	−	0.324227i	$-0.105104\pi$
$12$		0			0
$13$	−3.00000			−0.832050			−0.416025	−	0.909353i	$-0.636577\pi$
$13$	−3.00000			−0.832050			−0.416025	+	0.909353i	$0.636577\pi$
$14$		0			0
$15$	−3.00000			−0.774597
$16$		0			0
$17$	0.500000	−	0.866025i	0.121268	−	0.210042i	−0.799000	−	0.601331i	$-0.794637\pi$
$17$	0.500000	−	0.866025i	0.121268	−	0.210042i	0.920268	+	0.391289i	$0.127971\pi$
$18$		0			0
$19$	−3.00000	−	5.19615i	−0.688247	−	1.19208i	−0.972404	−	0.233301i	$-0.925047\pi$
$19$	−3.00000	−	5.19615i	−0.688247	−	1.19208i	0.284157	−	0.958778i	$-0.408286\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−3.00000	−	5.19615i	−0.625543	−	1.08347i	−0.988436	−	0.151642i	$-0.951544\pi$
$23$	−3.00000	−	5.19615i	−0.625543	−	1.08347i	0.362892	−	0.931831i	$-0.381789\pi$
$24$		0			0
$25$	−0.500000	+	0.866025i	−0.100000	+	0.173205i
$26$		0			0
$27$	9.00000			1.73205
$28$		0			0
$29$	−9.00000			−1.67126			−0.835629	−	0.549294i	$-0.814897\pi$
$29$	−9.00000			−1.67126			−0.835629	+	0.549294i	$0.814897\pi$
$30$		0			0
$31$	2.00000	−	3.46410i	0.359211	−	0.622171i	−0.628619	−	0.777714i	$-0.716379\pi$
$31$	2.00000	−	3.46410i	0.359211	−	0.622171i	0.987829	+	0.155543i	$0.0497126\pi$
$32$		0			0
$33$	7.50000	+	12.9904i	1.30558	+	2.26134i
$34$		0			0
$35$		0			0
$36$		0			0
$37$	−1.00000	−	1.73205i	−0.164399	−	0.284747i	0.772043	−	0.635571i	$-0.219235\pi$
$37$	−1.00000	−	1.73205i	−0.164399	−	0.284747i	−0.936442	+	0.350823i	$0.885902\pi$
$38$		0			0
$39$	4.50000	−	7.79423i	0.720577	−	1.24808i
$40$		0			0
$41$	−4.00000			−0.624695			−0.312348	−	0.949968i	$-0.601115\pi$
$41$	−4.00000			−0.624695			−0.312348	+	0.949968i	$0.601115\pi$
$42$		0			0
$43$	10.0000			1.52499			0.762493	−	0.646997i	$-0.223975\pi$
$43$	10.0000			1.52499			0.762493	+	0.646997i	$0.223975\pi$
$44$		0			0
$45$	3.00000	−	5.19615i	0.447214	−	0.774597i
$46$		0			0
$47$	0.500000	+	0.866025i	0.0729325	+	0.126323i	0.900185	−	0.435507i	$-0.143431\pi$
$47$	0.500000	+	0.866025i	0.0729325	+	0.126323i	−0.827253	+	0.561830i	$0.810098\pi$
$48$		0			0
$49$		0			0
$50$		0			0
$51$	1.50000	+	2.59808i	0.210042	+	0.363803i
$52$		0			0
$53$	−2.00000	+	3.46410i	−0.274721	+	0.475831i	−0.970065	−	0.242846i	$-0.921919\pi$
$53$	−2.00000	+	3.46410i	−0.274721	+	0.475831i	0.695344	+	0.718677i	$0.255252\pi$
$54$		0			0
$55$	5.00000			0.674200
$56$		0			0
$57$	18.0000			2.38416
$58$		0			0
$59$	4.00000	−	6.92820i	0.520756	−	0.901975i	−0.478953	−	0.877841i	$-0.658984\pi$
$59$	4.00000	−	6.92820i	0.520756	−	0.901975i	0.999709	−	0.0241347i	$-0.00768307\pi$
$60$		0			0
$61$	4.00000	+	6.92820i	0.512148	+	0.887066i	0.999901	+	0.0140840i	$0.00448323\pi$
$61$	4.00000	+	6.92820i	0.512148	+	0.887066i	−0.487753	+	0.872982i	$0.662183\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−1.50000	−	2.59808i	−0.186052	−	0.322252i
$66$		0			0
$67$	−6.00000	+	10.3923i	−0.733017	+	1.26962i	0.222571	+	0.974916i	$0.428555\pi$
$67$	−6.00000	+	10.3923i	−0.733017	+	1.26962i	−0.955588	+	0.294706i	$0.904778\pi$
$68$		0			0
$69$	18.0000			2.16695
$70$		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
$73$	−1.00000	+	1.73205i	−0.117041	+	0.202721i	−0.918594	−	0.395203i	$-0.870674\pi$
$73$	−1.00000	+	1.73205i	−0.117041	+	0.202721i	0.801553	+	0.597924i	$0.204008\pi$
$74$		0			0
$75$	−1.50000	−	2.59808i	−0.173205	−	0.300000i
$76$		0			0
$77$		0			0
$78$		0			0
$79$	−6.50000	−	11.2583i	−0.731307	−	1.26666i	−0.956325	−	0.292306i	$-0.905577\pi$
$79$	−6.50000	−	11.2583i	−0.731307	−	1.26666i	0.225018	−	0.974355i	$-0.427756\pi$
$80$		0			0
$81$	−4.50000	+	7.79423i	−0.500000	+	0.866025i
$82$		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
$85$	1.00000			0.108465
$86$		0			0
$87$	13.5000	−	23.3827i	1.44735	−	2.50689i
$88$		0			0
$89$	−2.00000	−	3.46410i	−0.212000	−	0.367194i	0.740341	−	0.672232i	$-0.234664\pi$
$89$	−2.00000	−	3.46410i	−0.212000	−	0.367194i	−0.952340	+	0.305038i	$0.901331\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$	6.00000	+	10.3923i	0.622171	+	1.07763i
$94$		0			0
$95$	3.00000	−	5.19615i	0.307794	−	0.533114i
$96$		0			0
$97$	−13.0000			−1.31995			−0.659975	−	0.751288i	$-0.729433\pi$
$97$	−13.0000			−1.31995			−0.659975	+	0.751288i	$0.729433\pi$
$98$		0			0
$99$	−30.0000			−3.01511
$100$		0			0
$101$	−3.00000	+	5.19615i	−0.298511	+	0.517036i	−0.975796	−	0.218685i	$-0.929823\pi$
$101$	−3.00000	+	5.19615i	−0.298511	+	0.517036i	0.677284	+	0.735721i	$0.263157\pi$
$102$		0			0
$103$	−9.50000	−	16.4545i	−0.936063	−	1.62131i	−0.772728	−	0.634738i	$-0.781108\pi$
$103$	−9.50000	−	16.4545i	−0.936063	−	1.62131i	−0.163335	−	0.986571i	$-0.552225\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	3.00000	+	5.19615i	0.290021	+	0.502331i	0.973814	−	0.227345i	$-0.0730044\pi$
$107$	3.00000	+	5.19615i	0.290021	+	0.502331i	−0.683793	+	0.729676i	$0.739671\pi$
$108$		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
$111$	6.00000			0.569495
$112$		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
$115$	3.00000	−	5.19615i	0.279751	−	0.484544i
$116$		0			0
$117$	9.00000	+	15.5885i	0.832050	+	1.44115i
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−7.00000	−	12.1244i	−0.636364	−	1.10221i
$122$		0			0
$123$	6.00000	−	10.3923i	0.541002	−	0.937043i
$124$		0			0
$125$	−1.00000			−0.0894427
$126$		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
$129$	−15.0000	+	25.9808i	−1.32068	+	2.28748i
$130$		0			0
$131$	5.00000	+	8.66025i	0.436852	+	0.756650i	0.997445	−	0.0714417i	$-0.0227600\pi$
$131$	5.00000	+	8.66025i	0.436852	+	0.756650i	−0.560593	+	0.828092i	$0.689427\pi$
$132$		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	−0.487278	−	0.873247i	$-0.662010\pi$
$137$	6.00000	−	10.3923i	0.512615	−	0.887875i	0.999893	−	0.0146279i	$-0.00465636\pi$
$138$		0			0
$139$	−14.0000			−1.18746			−0.593732	−	0.804663i	$-0.702346\pi$
$139$	−14.0000			−1.18746			−0.593732	+	0.804663i	$0.702346\pi$
$140$		0			0
$141$	−3.00000			−0.252646
$142$		0			0
$143$	−7.50000	+	12.9904i	−0.627182	+	1.08631i
$144$		0			0
$145$	−4.50000	−	7.79423i	−0.373705	−	0.647275i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	3.00000	+	5.19615i	0.245770	+	0.425685i	0.962348	−	0.271821i	$-0.0876260\pi$
$149$	3.00000	+	5.19615i	0.245770	+	0.425685i	−0.716578	+	0.697507i	$0.754293\pi$
$150$		0			0
$151$	−2.50000	+	4.33013i	−0.203447	+	0.352381i	−0.949637	−	0.313353i	$-0.898548\pi$
$151$	−2.50000	+	4.33013i	−0.203447	+	0.352381i	0.746190	+	0.665733i	$0.231881\pi$
$152$		0			0
$153$	−6.00000			−0.485071
$154$		0			0
$155$	4.00000			0.321288
$156$		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
$159$	−6.00000	−	10.3923i	−0.475831	−	0.824163i
$160$		0			0
$161$		0			0
$162$		0			0
$163$	5.00000	+	8.66025i	0.391630	+	0.678323i	0.992665	−	0.120900i	$-0.0385779\pi$
$163$	5.00000	+	8.66025i	0.391630	+	0.678323i	−0.601035	+	0.799223i	$0.705245\pi$
$164$		0			0
$165$	−7.50000	+	12.9904i	−0.583874	+	1.01130i
$166$		0			0
$167$	3.00000			0.232147			0.116073	−	0.993241i	$-0.462969\pi$
$167$	3.00000			0.232147			0.116073	+	0.993241i	$0.462969\pi$
$168$		0			0
$169$	−4.00000			−0.307692
$170$		0			0
$171$	−18.0000	+	31.1769i	−1.37649	+	2.38416i
$172$		0			0
$173$	0.500000	+	0.866025i	0.0380143	+	0.0658427i	0.884407	−	0.466717i	$-0.154563\pi$
$173$	0.500000	+	0.866025i	0.0380143	+	0.0658427i	−0.846392	+	0.532560i	$0.821230\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$	12.0000	+	20.7846i	0.901975	+	1.56227i
$178$		0			0
$179$	6.00000	−	10.3923i	0.448461	−	0.776757i	−0.549825	−	0.835280i	$-0.685306\pi$
$179$	6.00000	−	10.3923i	0.448461	−	0.776757i	0.998286	+	0.0585225i	$0.0186389\pi$
$180$		0			0
$181$	−20.0000			−1.48659			−0.743294	−	0.668965i	$-0.766738\pi$
$181$	−20.0000			−1.48659			−0.743294	+	0.668965i	$0.766738\pi$
$182$		0			0
$183$	−24.0000			−1.77413
$184$		0			0
$185$	1.00000	−	1.73205i	0.0735215	−	0.127343i
$186$		0			0
$187$	−2.50000	−	4.33013i	−0.182818	−	0.316650i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−1.50000	−	2.59808i	−0.108536	−	0.187990i	0.806641	−	0.591041i	$-0.201283\pi$
$191$	−1.50000	−	2.59808i	−0.108536	−	0.187990i	−0.915177	+	0.403051i	$0.867950\pi$
$192$		0			0
$193$	2.00000	−	3.46410i	0.143963	−	0.249351i	−0.785022	−	0.619467i	$-0.787349\pi$
$193$	2.00000	−	3.46410i	0.143963	−	0.249351i	0.928986	+	0.370116i	$0.120682\pi$
$194$		0			0
$195$	9.00000			0.644503
$196$		0			0
$197$	8.00000			0.569976			0.284988	−	0.958531i	$-0.408010\pi$
$197$	8.00000			0.569976			0.284988	+	0.958531i	$0.408010\pi$
$198$		0			0
$199$	−4.00000	+	6.92820i	−0.283552	+	0.491127i	−0.972257	−	0.233915i	$-0.924846\pi$
$199$	−4.00000	+	6.92820i	−0.283552	+	0.491127i	0.688705	+	0.725042i	$0.258180\pi$
$200$		0			0
$201$	−18.0000	−	31.1769i	−1.26962	−	2.19905i
$202$		0			0
$203$		0			0
$204$		0			0
$205$	−2.00000	−	3.46410i	−0.139686	−	0.241943i
$206$		0			0
$207$	−18.0000	+	31.1769i	−1.25109	+	2.16695i
$208$		0			0
$209$	−30.0000			−2.07514
$210$		0			0
$211$	−11.0000			−0.757271			−0.378636	−	0.925546i	$-0.623607\pi$
$211$	−11.0000			−0.757271			−0.378636	+	0.925546i	$0.623607\pi$
$212$		0			0
$213$	−12.0000	+	20.7846i	−0.822226	+	1.42414i
$214$		0			0
$215$	5.00000	+	8.66025i	0.340997	+	0.590624i
$216$		0			0
$217$		0			0
$218$		0			0
$219$	−3.00000	−	5.19615i	−0.202721	−	0.351123i
$220$		0			0
$221$	−1.50000	+	2.59808i	−0.100901	+	0.174766i
$222$		0			0
$223$	−5.00000			−0.334825			−0.167412	−	0.985887i	$-0.553541\pi$
$223$	−5.00000			−0.334825			−0.167412	+	0.985887i	$0.553541\pi$
$224$		0			0
$225$	6.00000			0.400000
$226$		0			0
$227$	0.500000	−	0.866025i	0.0331862	−	0.0574801i	−0.848955	−	0.528465i	$-0.822768\pi$
$227$	0.500000	−	0.866025i	0.0331862	−	0.0574801i	0.882141	+	0.470985i	$0.156101\pi$
$228$		0			0
$229$	−2.00000	−	3.46410i	−0.132164	−	0.228914i	0.792347	−	0.610071i	$-0.208859\pi$
$229$	−2.00000	−	3.46410i	−0.132164	−	0.228914i	−0.924510	+	0.381157i	$0.875526\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−12.0000	−	20.7846i	−0.786146	−	1.36165i	−0.928312	−	0.371802i	$-0.878740\pi$
$233$	−12.0000	−	20.7846i	−0.786146	−	1.36165i	0.142166	−	0.989843i	$-0.454593\pi$
$234$		0			0
$235$	−0.500000	+	0.866025i	−0.0326164	+	0.0564933i
$236$		0			0
$237$	39.0000			2.53332
$238$		0			0
$239$	1.00000			0.0646846			0.0323423	−	0.999477i	$-0.489703\pi$
$239$	1.00000			0.0646846			0.0323423	+	0.999477i	$0.489703\pi$
$240$		0			0
$241$	13.0000	−	22.5167i	0.837404	−	1.45043i	−0.0546547	−	0.998505i	$-0.517406\pi$
$241$	13.0000	−	22.5167i	0.837404	−	1.45043i	0.892058	−	0.451920i	$-0.149261\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$		0			0
$246$		0			0
$247$	9.00000	+	15.5885i	0.572656	+	0.991870i
$248$		0			0
$249$	6.00000	−	10.3923i	0.380235	−	0.658586i
$250$		0			0
$251$	30.0000			1.89358			0.946792	−	0.321847i	$-0.104304\pi$
$251$	30.0000			1.89358			0.946792	+	0.321847i	$0.104304\pi$
$252$		0			0
$253$	−30.0000			−1.88608
$254$		0			0
$255$	−1.50000	+	2.59808i	−0.0939336	+	0.162698i
$256$		0			0
$257$	−7.00000	−	12.1244i	−0.436648	−	0.756297i	0.560781	−	0.827964i	$-0.310501\pi$
$257$	−7.00000	−	12.1244i	−0.436648	−	0.756297i	−0.997429	+	0.0716680i	$0.977168\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$	27.0000	+	46.7654i	1.67126	+	2.89470i
$262$		0			0
$263$	−1.00000	+	1.73205i	−0.0616626	+	0.106803i	−0.895209	−	0.445647i	$-0.852974\pi$
$263$	−1.00000	+	1.73205i	−0.0616626	+	0.106803i	0.833546	+	0.552450i	$0.186307\pi$
$264$		0			0
$265$	−4.00000			−0.245718
$266$		0			0
$267$	12.0000			0.734388
$268$		0			0
$269$	−9.00000	+	15.5885i	−0.548740	+	0.950445i	0.449622	+	0.893219i	$0.351559\pi$
$269$	−9.00000	+	15.5885i	−0.548740	+	0.950445i	−0.998361	+	0.0572259i	$0.981774\pi$
$270$		0			0
$271$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$271$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	2.50000	+	4.33013i	0.150756	+	0.261116i
$276$		0			0
$277$	−3.00000	+	5.19615i	−0.180253	+	0.312207i	−0.941966	−	0.335707i	$-0.891025\pi$
$277$	−3.00000	+	5.19615i	−0.180253	+	0.312207i	0.761714	+	0.647913i	$0.224358\pi$
$278$		0			0
$279$	−24.0000			−1.43684
$280$		0			0
$281$	11.0000			0.656205			0.328102	−	0.944642i	$-0.393591\pi$
$281$	11.0000			0.656205			0.328102	+	0.944642i	$0.393591\pi$
$282$		0			0
$283$	15.5000	−	26.8468i	0.921379	−	1.59588i	0.124096	−	0.992270i	$-0.460397\pi$
$283$	15.5000	−	26.8468i	0.921379	−	1.59588i	0.797283	−	0.603606i	$-0.206270\pi$
$284$		0			0
$285$	9.00000	+	15.5885i	0.533114	+	0.923381i
$286$		0			0
$287$		0			0
$288$		0			0
$289$	8.00000	+	13.8564i	0.470588	+	0.815083i
$290$		0			0
$291$	19.5000	−	33.7750i	1.14311	−	1.97993i
$292$		0			0
$293$	−5.00000			−0.292103			−0.146052	−	0.989277i	$-0.546657\pi$
$293$	−5.00000			−0.292103			−0.146052	+	0.989277i	$0.546657\pi$
$294$		0			0
$295$	8.00000			0.465778
$296$		0			0
$297$	22.5000	−	38.9711i	1.30558	−	2.26134i
$298$		0			0
$299$	9.00000	+	15.5885i	0.520483	+	0.901504i
$300$		0			0
$301$		0			0
$302$		0			0
$303$	−9.00000	−	15.5885i	−0.517036	−	0.895533i
$304$		0			0
$305$	−4.00000	+	6.92820i	−0.229039	+	0.396708i
$306$		0			0
$307$	−23.0000			−1.31268			−0.656340	−	0.754466i	$-0.727896\pi$
$307$	−23.0000			−1.31268			−0.656340	+	0.754466i	$0.727896\pi$
$308$		0			0
$309$	57.0000			3.24262
$310$		0			0
$311$	9.00000	−	15.5885i	0.510343	−	0.883940i	−0.489585	−	0.871956i	$-0.662852\pi$
$311$	9.00000	−	15.5885i	0.510343	−	0.883940i	0.999928	−	0.0119847i	$-0.00381495\pi$
$312$		0			0
$313$	3.50000	+	6.06218i	0.197832	+	0.342655i	0.947825	−	0.318791i	$-0.103277\pi$
$313$	3.50000	+	6.06218i	0.197832	+	0.342655i	−0.749993	+	0.661445i	$0.769943\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	13.0000	+	22.5167i	0.730153	+	1.26466i	0.956818	+	0.290689i	$0.0938844\pi$
$317$	13.0000	+	22.5167i	0.730153	+	1.26466i	−0.226665	+	0.973973i	$0.572782\pi$
$318$		0			0
$319$	−22.5000	+	38.9711i	−1.25976	+	2.18197i
$320$		0			0
$321$	−18.0000			−1.00466
$322$		0			0
$323$	−6.00000			−0.333849
$324$		0			0
$325$	1.50000	−	2.59808i	0.0832050	−	0.144115i
$326$		0			0
$327$	4.50000	+	7.79423i	0.248851	+	0.431022i
$328$		0			0
$329$		0			0
$330$		0			0
$331$	−14.0000	−	24.2487i	−0.769510	−	1.33283i	−0.937829	−	0.347097i	$-0.887167\pi$
$331$	−14.0000	−	24.2487i	−0.769510	−	1.33283i	0.168320	−	0.985732i	$-0.446166\pi$
$332$		0			0
$333$	−6.00000	+	10.3923i	−0.328798	+	0.569495i
$334$		0			0
$335$	−12.0000			−0.655630
$336$		0			0
$337$	22.0000			1.19842			0.599208	−	0.800593i	$-0.295482\pi$
$337$	22.0000			1.19842			0.599208	+	0.800593i	$0.295482\pi$
$338$		0			0
$339$	−21.0000	+	36.3731i	−1.14056	+	1.97551i
$340$		0			0
$341$	−10.0000	−	17.3205i	−0.541530	−	0.937958i
$342$		0			0
$343$		0			0
$344$		0			0
$345$	9.00000	+	15.5885i	0.484544	+	0.839254i
$346$		0			0
$347$	−5.00000	+	8.66025i	−0.268414	+	0.464907i	−0.968452	−	0.249198i	$-0.919833\pi$
$347$	−5.00000	+	8.66025i	−0.268414	+	0.464907i	0.700038	+	0.714105i	$0.253166\pi$
$348$		0			0
$349$	10.0000			0.535288			0.267644	−	0.963518i	$-0.413755\pi$
$349$	10.0000			0.535288			0.267644	+	0.963518i	$0.413755\pi$
$350$		0			0
$351$	−27.0000			−1.44115
$352$		0			0
$353$	−1.50000	+	2.59808i	−0.0798369	+	0.138282i	−0.903179	−	0.429263i	$-0.858773\pi$
$353$	−1.50000	+	2.59808i	−0.0798369	+	0.138282i	0.823343	+	0.567545i	$0.192107\pi$
$354$		0			0
$355$	4.00000	+	6.92820i	0.212298	+	0.367711i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	8.00000	+	13.8564i	0.422224	+	0.731313i	0.996157	−	0.0875892i	$-0.0279163\pi$
$359$	8.00000	+	13.8564i	0.422224	+	0.731313i	−0.573933	+	0.818902i	$0.694583\pi$
$360$		0			0
$361$	−8.50000	+	14.7224i	−0.447368	+	0.774865i
$362$		0			0
$363$	42.0000			2.20443
$364$		0			0
$365$	−2.00000			−0.104685
$366$		0			0
$367$	8.50000	−	14.7224i	0.443696	−	0.768505i	−0.554264	−	0.832341i	$-0.687000\pi$
$367$	8.50000	−	14.7224i	0.443696	−	0.768505i	0.997960	+	0.0638362i	$0.0203335\pi$
$368$		0			0
$369$	12.0000	+	20.7846i	0.624695	+	1.08200i
$370$		0			0
$371$		0			0
$372$		0			0
$373$	2.00000	+	3.46410i	0.103556	+	0.179364i	0.913147	−	0.407630i	$-0.133645\pi$
$373$	2.00000	+	3.46410i	0.103556	+	0.179364i	−0.809591	+	0.586994i	$0.800311\pi$
$374$		0			0
$375$	1.50000	−	2.59808i	0.0774597	−	0.134164i
$376$		0			0
$377$	27.0000			1.39057
$378$		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
$381$	12.0000	−	20.7846i	0.614779	−	1.06483i
$382$		0			0
$383$	−18.0000	−	31.1769i	−0.919757	−	1.59307i	−0.799783	−	0.600289i	$-0.795052\pi$
$383$	−18.0000	−	31.1769i	−0.919757	−	1.59307i	−0.119974	−	0.992777i	$-0.538281\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	−30.0000	−	51.9615i	−1.52499	−	2.64135i
$388$		0			0
$389$	15.5000	−	26.8468i	0.785881	−	1.36119i	−0.142590	−	0.989782i	$-0.545543\pi$
$389$	15.5000	−	26.8468i	0.785881	−	1.36119i	0.928471	−	0.371404i	$-0.121124\pi$
$390$		0			0
$391$	−6.00000			−0.303433
$392$		0			0
$393$	−30.0000			−1.51330
$394$		0			0
$395$	6.50000	−	11.2583i	0.327050	−	0.566468i
$396$		0			0
$397$	−3.50000	−	6.06218i	−0.175660	−	0.304252i	0.764730	−	0.644351i	$-0.222873\pi$
$397$	−3.50000	−	6.06218i	−0.175660	−	0.304252i	−0.940389	+	0.340099i	$0.889539\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−12.5000	−	21.6506i	−0.624220	−	1.08118i	−0.988691	−	0.149966i	$-0.952083\pi$
$401$	−12.5000	−	21.6506i	−0.624220	−	1.08118i	0.364471	−	0.931215i	$-0.381250\pi$
$402$		0			0
$403$	−6.00000	+	10.3923i	−0.298881	+	0.517678i
$404$		0			0
$405$	−9.00000			−0.447214
$406$		0			0
$407$	−10.0000			−0.495682
$408$		0			0
$409$	−3.00000	+	5.19615i	−0.148340	+	0.256933i	−0.930614	−	0.366002i	$-0.880726\pi$
$409$	−3.00000	+	5.19615i	−0.148340	+	0.256933i	0.782274	+	0.622935i	$0.214060\pi$
$410$		0			0
$411$	18.0000	+	31.1769i	0.887875	+	1.53784i
$412$		0			0
$413$		0			0
$414$		0			0
$415$	−2.00000	−	3.46410i	−0.0981761	−	0.170046i
$416$		0			0
$417$	21.0000	−	36.3731i	1.02837	−	1.78120i
$418$		0			0
$419$	36.0000			1.75872			0.879358	−	0.476162i	$-0.157972\pi$
$419$	36.0000			1.75872			0.879358	+	0.476162i	$0.157972\pi$
$420$		0			0
$421$	−19.0000			−0.926003			−0.463002	−	0.886357i	$-0.653228\pi$
$421$	−19.0000			−0.926003			−0.463002	+	0.886357i	$0.653228\pi$
$422$		0			0
$423$	3.00000	−	5.19615i	0.145865	−	0.252646i
$424$		0			0
$425$	0.500000	+	0.866025i	0.0242536	+	0.0420084i
$426$		0			0
$427$		0			0
$428$		0			0
$429$	−22.5000	−	38.9711i	−1.08631	−	1.88154i
$430$		0			0
$431$	4.50000	−	7.79423i	0.216757	−	0.375435i	−0.737057	−	0.675830i	$-0.763785\pi$
$431$	4.50000	−	7.79423i	0.216757	−	0.375435i	0.953815	+	0.300395i	$0.0971186\pi$
$432$		0			0
$433$	2.00000			0.0961139			0.0480569	−	0.998845i	$-0.484697\pi$
$433$	2.00000			0.0961139			0.0480569	+	0.998845i	$0.484697\pi$
$434$		0			0
$435$	27.0000			1.29455
$436$		0			0
$437$	−18.0000	+	31.1769i	−0.861057	+	1.49139i
$438$		0			0
$439$	1.00000	+	1.73205i	0.0477274	+	0.0826663i	0.888902	−	0.458097i	$-0.151469\pi$
$439$	1.00000	+	1.73205i	0.0477274	+	0.0826663i	−0.841175	+	0.540763i	$0.818135\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	7.00000	+	12.1244i	0.332580	+	0.576046i	0.983017	−	0.183515i	$-0.0587475\pi$
$443$	7.00000	+	12.1244i	0.332580	+	0.576046i	−0.650437	+	0.759560i	$0.725414\pi$
$444$		0			0
$445$	2.00000	−	3.46410i	0.0948091	−	0.164214i
$446$		0			0
$447$	−18.0000			−0.851371
$448$		0			0
$449$	−9.00000			−0.424736			−0.212368	−	0.977190i	$-0.568118\pi$
$449$	−9.00000			−0.424736			−0.212368	+	0.977190i	$0.568118\pi$
$450$		0			0
$451$	−10.0000	+	17.3205i	−0.470882	+	0.815591i
$452$		0			0
$453$	−7.50000	−	12.9904i	−0.352381	−	0.610341i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−8.00000	−	13.8564i	−0.374224	−	0.648175i	0.615986	−	0.787757i	$-0.288758\pi$
$457$	−8.00000	−	13.8564i	−0.374224	−	0.648175i	−0.990211	+	0.139581i	$0.955424\pi$
$458$		0			0
$459$	4.50000	−	7.79423i	0.210042	−	0.363803i
$460$		0			0
$461$	40.0000			1.86299			0.931493	−	0.363760i	$-0.118507\pi$
$461$	40.0000			1.86299			0.931493	+	0.363760i	$0.118507\pi$
$462$		0			0
$463$	−16.0000			−0.743583			−0.371792	−	0.928316i	$-0.621256\pi$
$463$	−16.0000			−0.743583			−0.371792	+	0.928316i	$0.621256\pi$
$464$		0			0
$465$	−6.00000	+	10.3923i	−0.278243	+	0.481932i
$466$		0			0
$467$	−2.50000	−	4.33013i	−0.115686	−	0.200374i	0.802368	−	0.596830i	$-0.203573\pi$
$467$	−2.50000	−	4.33013i	−0.115686	−	0.200374i	−0.918054	+	0.396456i	$0.870240\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$	3.00000	+	5.19615i	0.138233	+	0.239426i
$472$		0			0
$473$	25.0000	−	43.3013i	1.14950	−	1.99099i
$474$		0			0
$475$	6.00000			0.275299
$476$		0			0
$477$	24.0000			1.09888
$478$		0			0
$479$	−3.00000	+	5.19615i	−0.137073	+	0.237418i	−0.926388	−	0.376571i	$-0.877103\pi$
$479$	−3.00000	+	5.19615i	−0.137073	+	0.237418i	0.789314	+	0.613990i	$0.210436\pi$
$480$		0			0
$481$	3.00000	+	5.19615i	0.136788	+	0.236924i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−6.50000	−	11.2583i	−0.295150	−	0.511214i
$486$		0			0
$487$	−13.0000	+	22.5167i	−0.589086	+	1.02033i	0.405266	+	0.914199i	$0.367179\pi$
$487$	−13.0000	+	22.5167i	−0.589086	+	1.02033i	−0.994352	+	0.106129i	$0.966154\pi$
$488$		0			0
$489$	−30.0000			−1.35665
$490$		0			0
$491$	−15.0000			−0.676941			−0.338470	−	0.940977i	$-0.609909\pi$
$491$	−15.0000			−0.676941			−0.338470	+	0.940977i	$0.609909\pi$
$492$		0			0
$493$	−4.50000	+	7.79423i	−0.202670	+	0.351034i
$494$		0			0
$495$	−15.0000	−	25.9808i	−0.674200	−	1.16775i
$496$		0			0
$497$		0			0
$498$		0			0
$499$	4.50000	+	7.79423i	0.201448	+	0.348918i	0.948995	−	0.315291i	$-0.102102\pi$
$499$	4.50000	+	7.79423i	0.201448	+	0.348918i	−0.747547	+	0.664208i	$0.768769\pi$
$500$		0			0
$501$	−4.50000	+	7.79423i	−0.201045	+	0.348220i
$502$		0			0
$503$	−3.00000			−0.133763			−0.0668817	−	0.997761i	$-0.521305\pi$
$503$	−3.00000			−0.133763			−0.0668817	+	0.997761i	$0.521305\pi$
$504$		0			0
$505$	−6.00000			−0.266996
$506$		0			0
$507$	6.00000	−	10.3923i	0.266469	−	0.461538i
$508$		0			0
$509$	7.00000	+	12.1244i	0.310270	+	0.537403i	0.978421	−	0.206623i	$-0.0662474\pi$
$509$	7.00000	+	12.1244i	0.310270	+	0.537403i	−0.668151	+	0.744026i	$0.732914\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$	−27.0000	−	46.7654i	−1.19208	−	2.06474i
$514$		0			0
$515$	9.50000	−	16.4545i	0.418620	−	0.725071i
$516$		0			0
$517$	5.00000			0.219900
$518$		0			0
$519$	−3.00000			−0.131685
$520$		0			0
$521$	−7.00000	+	12.1244i	−0.306676	+	0.531178i	−0.977633	−	0.210318i	$-0.932550\pi$
$521$	−7.00000	+	12.1244i	−0.306676	+	0.531178i	0.670957	+	0.741496i	$0.265883\pi$
$522$		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
$527$	−2.00000	−	3.46410i	−0.0871214	−	0.150899i
$528$		0			0
$529$	−6.50000	+	11.2583i	−0.282609	+	0.489493i
$530$		0			0
$531$	−48.0000			−2.08302
$532$		0			0
$533$	12.0000			0.519778
$534$		0			0
$535$	−3.00000	+	5.19615i	−0.129701	+	0.224649i
$536$		0			0
$537$	18.0000	+	31.1769i	0.776757	+	1.34538i
$538$		0			0
$539$		0			0
$540$		0			0
$541$	−19.5000	−	33.7750i	−0.838370	−	1.45210i	−0.891256	−	0.453500i	$-0.850175\pi$
$541$	−19.5000	−	33.7750i	−0.838370	−	1.45210i	0.0528859	−	0.998601i	$-0.483158\pi$
$542$		0			0
$543$	30.0000	−	51.9615i	1.28742	−	2.22988i
$544$		0			0
$545$	3.00000			0.128506
$546$		0			0
$547$	−24.0000			−1.02617			−0.513083	−	0.858339i	$-0.671497\pi$
$547$	−24.0000			−1.02617			−0.513083	+	0.858339i	$0.671497\pi$
$548$		0			0
$549$	24.0000	−	41.5692i	1.02430	−	1.77413i
$550$		0			0
$551$	27.0000	+	46.7654i	1.15024	+	1.99227i
$552$		0			0
$553$		0			0
$554$		0			0
$555$	3.00000	+	5.19615i	0.127343	+	0.220564i
$556$		0			0
$557$	−20.0000	+	34.6410i	−0.847427	+	1.46779i	0.0360693	+	0.999349i	$0.488516\pi$
$557$	−20.0000	+	34.6410i	−0.847427	+	1.46779i	−0.883497	+	0.468438i	$0.844817\pi$
$558$		0			0
$559$	−30.0000			−1.26886
$560$		0			0
$561$	15.0000			0.633300
$562$		0			0
$563$	14.0000	−	24.2487i	0.590030	−	1.02196i	−0.404198	−	0.914671i	$-0.632449\pi$
$563$	14.0000	−	24.2487i	0.590030	−	1.02196i	0.994228	−	0.107290i	$-0.0342173\pi$
$564$		0			0
$565$	7.00000	+	12.1244i	0.294492	+	0.510075i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	15.0000	+	25.9808i	0.628833	+	1.08917i	0.987786	+	0.155815i	$0.0498003\pi$
$569$	15.0000	+	25.9808i	0.628833	+	1.08917i	−0.358954	+	0.933355i	$0.616866\pi$
$570$		0			0
$571$	6.00000	−	10.3923i	0.251092	−	0.434904i	−0.712735	−	0.701434i	$-0.752544\pi$
$571$	6.00000	−	10.3923i	0.251092	−	0.434904i	0.963827	+	0.266529i	$0.0858769\pi$
$572$		0			0
$573$	9.00000			0.375980
$574$		0			0
$575$	6.00000			0.250217
$576$		0			0
$577$	−6.50000	+	11.2583i	−0.270599	+	0.468690i	−0.969015	−	0.247001i	$-0.920555\pi$
$577$	−6.50000	+	11.2583i	−0.270599	+	0.468690i	0.698417	+	0.715691i	$0.253888\pi$
$578$		0			0
$579$	6.00000	+	10.3923i	0.249351	+	0.431889i
$580$		0			0
$581$		0			0
$582$		0			0
$583$	10.0000	+	17.3205i	0.414158	+	0.717342i
$584$		0			0
$585$	−9.00000	+	15.5885i	−0.372104	+	0.644503i
$586$		0			0
$587$	−16.0000			−0.660391			−0.330195	−	0.943913i	$-0.607115\pi$
$587$	−16.0000			−0.660391			−0.330195	+	0.943913i	$0.607115\pi$
$588$		0			0
$589$	−24.0000			−0.988903
$590$		0			0
$591$	−12.0000	+	20.7846i	−0.493614	+	0.854965i
$592$		0			0
$593$	13.5000	+	23.3827i	0.554379	+	0.960212i	0.997952	+	0.0639736i	$0.0203773\pi$
$593$	13.5000	+	23.3827i	0.554379	+	0.960212i	−0.443573	+	0.896238i	$0.646289\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	−12.0000	−	20.7846i	−0.491127	−	0.850657i
$598$		0			0
$599$	−7.50000	+	12.9904i	−0.306442	+	0.530773i	−0.977581	−	0.210558i	$-0.932472\pi$
$599$	−7.50000	+	12.9904i	−0.306442	+	0.530773i	0.671140	+	0.741331i	$0.265805\pi$
$600$		0			0
$601$	30.0000			1.22373			0.611863	−	0.790964i	$-0.290420\pi$
$601$	30.0000			1.22373			0.611863	+	0.790964i	$0.290420\pi$
$602$		0			0
$603$	72.0000			2.93207
$604$		0			0
$605$	7.00000	−	12.1244i	0.284590	−	0.492925i
$606$		0			0
$607$	−6.50000	−	11.2583i	−0.263827	−	0.456962i	0.703429	−	0.710766i	$-0.251651\pi$
$607$	−6.50000	−	11.2583i	−0.263827	−	0.456962i	−0.967256	+	0.253804i	$0.918318\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−1.50000	−	2.59808i	−0.0606835	−	0.105107i
$612$		0			0
$613$	−21.0000	+	36.3731i	−0.848182	+	1.46909i	0.0346469	+	0.999400i	$0.488969\pi$
$613$	−21.0000	+	36.3731i	−0.848182	+	1.46909i	−0.882829	+	0.469695i	$0.844364\pi$
$614$		0			0
$615$	12.0000			0.483887
$616$		0			0
$617$	−46.0000			−1.85189			−0.925945	−	0.377658i	$-0.876729\pi$
$617$	−46.0000			−1.85189			−0.925945	+	0.377658i	$0.876729\pi$
$618$		0			0
$619$	5.00000	−	8.66025i	0.200967	−	0.348085i	−0.747873	−	0.663842i	$-0.768925\pi$
$619$	5.00000	−	8.66025i	0.200967	−	0.348085i	0.948840	+	0.315757i	$0.102258\pi$
$620$		0			0
$621$	−27.0000	−	46.7654i	−1.08347	−	1.87663i
$622$		0			0
$623$		0			0
$624$		0			0
$625$	−0.500000	−	0.866025i	−0.0200000	−	0.0346410i
$626$		0			0
$627$	45.0000	−	77.9423i	1.79713	−	3.11272i
$628$		0			0
$629$	−2.00000			−0.0797452
$630$		0			0
$631$	47.0000			1.87104			0.935520	−	0.353273i	$-0.114931\pi$
$631$	47.0000			1.87104			0.935520	+	0.353273i	$0.114931\pi$
$632$		0			0
$633$	16.5000	−	28.5788i	0.655816	−	1.13591i
$634$		0			0
$635$	−4.00000	−	6.92820i	−0.158735	−	0.274937i
$636$		0			0
$637$		0			0
$638$		0			0
$639$	−24.0000	−	41.5692i	−0.949425	−	1.64445i
$640$		0			0
$641$	−1.00000	+	1.73205i	−0.0394976	+	0.0684119i	−0.885098	−	0.465404i	$-0.845909\pi$
$641$	−1.00000	+	1.73205i	−0.0394976	+	0.0684119i	0.845601	+	0.533816i	$0.179242\pi$
$642$		0			0
$643$	19.0000			0.749287			0.374643	−	0.927169i	$-0.377765\pi$
$643$	19.0000			0.749287			0.374643	+	0.927169i	$0.377765\pi$
$644$		0			0
$645$	−30.0000			−1.18125
$646$		0			0
$647$	4.00000	−	6.92820i	0.157256	−	0.272376i	−0.776622	−	0.629967i	$-0.783068\pi$
$647$	4.00000	−	6.92820i	0.157256	−	0.272376i	0.933878	+	0.357591i	$0.116402\pi$
$648$		0			0
$649$	−20.0000	−	34.6410i	−0.785069	−	1.35978i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−1.00000	−	1.73205i	−0.0391330	−	0.0677804i	0.845796	−	0.533507i	$-0.179126\pi$
$653$	−1.00000	−	1.73205i	−0.0391330	−	0.0677804i	−0.884929	+	0.465727i	$0.845793\pi$
$654$		0			0
$655$	−5.00000	+	8.66025i	−0.195366	+	0.338384i
$656$		0			0
$657$	12.0000			0.468165
$658$		0			0
$659$	31.0000			1.20759			0.603794	−	0.797140i	$-0.293655\pi$
$659$	31.0000			1.20759			0.603794	+	0.797140i	$0.293655\pi$
$660$		0			0
$661$	16.0000	−	27.7128i	0.622328	−	1.07790i	−0.366723	−	0.930330i	$-0.619520\pi$
$661$	16.0000	−	27.7128i	0.622328	−	1.07790i	0.989051	−	0.147573i	$-0.0471463\pi$
$662$		0			0
$663$	−4.50000	−	7.79423i	−0.174766	−	0.302703i
$664$		0			0
$665$		0			0
$666$		0			0
$667$	27.0000	+	46.7654i	1.04544	+	1.81076i
$668$		0			0
$669$	7.50000	−	12.9904i	0.289967	−	0.502237i
$670$		0			0
$671$	40.0000			1.54418
$672$		0			0
$673$	4.00000			0.154189			0.0770943	−	0.997024i	$-0.475436\pi$
$673$	4.00000			0.154189			0.0770943	+	0.997024i	$0.475436\pi$
$674$		0			0
$675$	−4.50000	+	7.79423i	−0.173205	+	0.300000i
$676$		0			0
$677$	13.5000	+	23.3827i	0.518847	+	0.898670i	0.999760	+	0.0219013i	$0.00697196\pi$
$677$	13.5000	+	23.3827i	0.518847	+	0.898670i	−0.480913	+	0.876768i	$0.659695\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$	1.50000	+	2.59808i	0.0574801	+	0.0995585i
$682$		0			0
$683$	−4.00000	+	6.92820i	−0.153056	+	0.265100i	−0.932349	−	0.361559i	$-0.882245\pi$
$683$	−4.00000	+	6.92820i	−0.153056	+	0.265100i	0.779294	+	0.626659i	$0.215578\pi$
$684$		0			0
$685$	12.0000			0.458496
$686$		0			0
$687$	12.0000			0.457829
$688$		0			0
$689$	6.00000	−	10.3923i	0.228582	−	0.395915i
$690$		0			0
$691$	−14.0000	−	24.2487i	−0.532585	−	0.922464i	−0.999276	−	0.0380440i	$-0.987887\pi$
$691$	−14.0000	−	24.2487i	−0.532585	−	0.922464i	0.466691	−	0.884420i	$-0.345446\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−7.00000	−	12.1244i	−0.265525	−	0.459903i
$696$		0			0
$697$	−2.00000	+	3.46410i	−0.0757554	+	0.131212i
$698$		0			0
$699$	72.0000			2.72329
$700$		0			0
$701$	−5.00000			−0.188847			−0.0944237	−	0.995532i	$-0.530101\pi$
$701$	−5.00000			−0.188847			−0.0944237	+	0.995532i	$0.530101\pi$
$702$		0			0
$703$	−6.00000	+	10.3923i	−0.226294	+	0.391953i
$704$		0			0
$705$	−1.50000	−	2.59808i	−0.0564933	−	0.0978492i
$706$		0			0
$707$		0			0
$708$		0			0
$709$	−15.5000	−	26.8468i	−0.582115	−	1.00825i	−0.995228	−	0.0975728i	$-0.968892\pi$
$709$	−15.5000	−	26.8468i	−0.582115	−	1.00825i	0.413114	−	0.910679i	$-0.364441\pi$
$710$		0			0
$711$	−39.0000	+	67.5500i	−1.46261	+	2.53332i
$712$		0			0
$713$	−24.0000			−0.898807
$714$		0			0
$715$	−15.0000			−0.560968
$716$		0			0
$717$	−1.50000	+	2.59808i	−0.0560185	+	0.0970269i
$718$		0			0
$719$	−3.00000	−	5.19615i	−0.111881	−	0.193784i	0.804648	−	0.593753i	$-0.202354\pi$
$719$	−3.00000	−	5.19615i	−0.111881	−	0.193784i	−0.916529	+	0.399969i	$0.869021\pi$
$720$		0			0
$721$		0			0
$722$		0			0
$723$	39.0000	+	67.5500i	1.45043	+	2.51221i
$724$		0			0
$725$	4.50000	−	7.79423i	0.167126	−	0.289470i
$726$		0			0
$727$	24.0000			0.890111			0.445055	−	0.895503i	$-0.353184\pi$
$727$	24.0000			0.890111			0.445055	+	0.895503i	$0.353184\pi$
$728$		0			0
$729$	−27.0000			−1.00000
$730$		0			0
$731$	5.00000	−	8.66025i	0.184932	−	0.320311i
$732$		0			0
$733$	23.5000	+	40.7032i	0.867992	+	1.50341i	0.864045	+	0.503415i	$0.167923\pi$
$733$	23.5000	+	40.7032i	0.867992	+	1.50341i	0.00394730	+	0.999992i	$0.498744\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	30.0000	+	51.9615i	1.10506	+	1.91403i
$738$		0			0
$739$	2.50000	−	4.33013i	0.0919640	−	0.159286i	−0.816373	−	0.577524i	$-0.804019\pi$
$739$	2.50000	−	4.33013i	0.0919640	−	0.159286i	0.908337	+	0.418238i	$0.137352\pi$
$740$		0			0
$741$	−54.0000			−1.98374
$742$		0			0
$743$	36.0000			1.32071			0.660356	−	0.750953i	$-0.270405\pi$
$743$	36.0000			1.32071			0.660356	+	0.750953i	$0.270405\pi$
$744$		0			0
$745$	−3.00000	+	5.19615i	−0.109911	+	0.190372i
$746$		0			0
$747$	12.0000	+	20.7846i	0.439057	+	0.760469i
$748$		0			0
$749$		0			0
$750$		0			0
$751$	9.50000	+	16.4545i	0.346660	+	0.600433i	0.985654	−	0.168779i	$-0.0539825\pi$
$751$	9.50000	+	16.4545i	0.346660	+	0.600433i	−0.638994	+	0.769212i	$0.720649\pi$
$752$		0			0
$753$	−45.0000	+	77.9423i	−1.63989	+	2.84037i
$754$		0			0
$755$	−5.00000			−0.181969
$756$		0			0
$757$	−32.0000			−1.16306			−0.581530	−	0.813525i	$-0.697546\pi$
$757$	−32.0000			−1.16306			−0.581530	+	0.813525i	$0.697546\pi$
$758$		0			0
$759$	45.0000	−	77.9423i	1.63340	−	2.82913i
$760$		0			0
$761$	9.00000	+	15.5885i	0.326250	+	0.565081i	0.981764	−	0.190101i	$-0.0608816\pi$
$761$	9.00000	+	15.5885i	0.326250	+	0.565081i	−0.655515	+	0.755182i	$0.727548\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$	−3.00000	−	5.19615i	−0.108465	−	0.187867i
$766$		0			0
$767$	−12.0000	+	20.7846i	−0.433295	+	0.750489i
$768$		0			0
$769$	22.0000			0.793340			0.396670	−	0.917961i	$-0.370166\pi$
$769$	22.0000			0.793340			0.396670	+	0.917961i	$0.370166\pi$
$770$		0			0
$771$	42.0000			1.51259
$772$		0			0
$773$	−22.5000	+	38.9711i	−0.809269	+	1.40169i	0.104102	+	0.994567i	$0.466803\pi$
$773$	−22.5000	+	38.9711i	−0.809269	+	1.40169i	−0.913371	+	0.407128i	$0.866530\pi$
$774$		0			0
$775$	2.00000	+	3.46410i	0.0718421	+	0.124434i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	12.0000	+	20.7846i	0.429945	+	0.744686i
$780$		0			0
$781$	20.0000	−	34.6410i	0.715656	−	1.23955i
$782$		0			0
$783$	−81.0000			−2.89470
$784$		0			0
$785$	2.00000			0.0713831
$786$		0			0
$787$	−15.5000	+	26.8468i	−0.552515	+	0.956985i	0.445577	+	0.895244i	$0.352999\pi$
$787$	−15.5000	+	26.8468i	−0.552515	+	0.956985i	−0.998092	+	0.0617409i	$0.980335\pi$
$788$		0			0
$789$	−3.00000	−	5.19615i	−0.106803	−	0.184988i
$790$		0			0
$791$		0			0
$792$		0			0
$793$	−12.0000	−	20.7846i	−0.426132	−	0.738083i

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)

\(f(q)\)	\(=\)	\(q+(-1.50000 + 2.59808i) q^{3} +(0.500000 + 0.866025i) q^{5} +(-3.00000 - 5.19615i) q^{9} +(2.50000 - 4.33013i) q^{11} -3.00000 q^{13} -3.00000 q^{15} +(0.500000 - 0.866025i) q^{17} +(-3.00000 - 5.19615i) q^{19} +(-3.00000 - 5.19615i) q^{23} +(-0.500000 + 0.866025i) q^{25} +9.00000 q^{27} -9.00000 q^{29} +(2.00000 - 3.46410i) q^{31} +(7.50000 + 12.9904i) q^{33} +(-1.00000 - 1.73205i) q^{37} +(4.50000 - 7.79423i) q^{39} -4.00000 q^{41} +10.0000 q^{43} +(3.00000 - 5.19615i) q^{45} +(0.500000 + 0.866025i) q^{47} +(1.50000 + 2.59808i) q^{51} +(-2.00000 + 3.46410i) q^{53} +5.00000 q^{55} +18.0000 q^{57} +(4.00000 - 6.92820i) q^{59} +(4.00000 + 6.92820i) q^{61} +(-1.50000 - 2.59808i) q^{65} +(-6.00000 + 10.3923i) q^{67} +18.0000 q^{69} +8.00000 q^{71} +(-1.00000 + 1.73205i) q^{73} +(-1.50000 - 2.59808i) q^{75} +(-6.50000 - 11.2583i) q^{79} +(-4.50000 + 7.79423i) q^{81} -4.00000 q^{83} +1.00000 q^{85} +(13.5000 - 23.3827i) q^{87} +(-2.00000 - 3.46410i) q^{89} +(6.00000 + 10.3923i) q^{93} +(3.00000 - 5.19615i) q^{95} -13.0000 q^{97} -30.0000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 3 q^{3} + q^{5} - 6 q^{9} + 5 q^{11} - 6 q^{13} - 6 q^{15} + q^{17} - 6 q^{19} - 6 q^{23} - q^{25} + 18 q^{27} - 18 q^{29} + 4 q^{31} + 15 q^{33} - 2 q^{37} + 9 q^{39} - 8 q^{41} + 20 q^{43} + 6 q^{45}+ \cdots - 60 q^{99}+O(q^{100}) \) 2 * q - 3 * q^3 + q^5 - 6 * q^9 + 5 * q^11 - 6 * q^13 - 6 * q^15 + q^17 - 6 * q^19 - 6 * q^23 - q^25 + 18 * q^27 - 18 * q^29 + 4 * q^31 + 15 * q^33 - 2 * q^37 + 9 * q^39 - 8 * q^41 + 20 * q^43 + 6 * q^45 + q^47 + 3 * q^51 - 4 * q^53 + 10 * q^55 + 36 * q^57 + 8 * q^59 + 8 * q^61 - 3 * q^65 - 12 * q^67 + 36 * q^69 + 16 * q^71 - 2 * q^73 - 3 * q^75 - 13 * q^79 - 9 * q^81 - 8 * q^83 + 2 * q^85 + 27 * q^87 - 4 * q^89 + 12 * q^93 + 6 * q^95 - 26 * q^97 - 60 * q^99

Introduction

L-functions

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Database

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