LMFDB - Embedded newform 507.2.x.a.11.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [507,2,Mod(2,507)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(507, base_ring=CyclotomicField(156))

chi = DirichletCharacter(H, H._module([78, 1]))

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

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

chi := DirichletCharacter("507.2");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 507 = 3 \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	507.x (of order $156$, degree $48$, 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:	$4.04841538248$
Analytic rank:	$0$
Dimension:	$48$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{156}]$

Embedding invariants

Embedding label			11.1
Character	$\chi$	$=$	507.11
Dual form			507.2.x.a.461.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.59345 - 0.678906i) q^{3} +(-1.06893 - 1.69038i) q^{4} +(1.78351 - 3.57116i) q^{7} +(2.07817 - 2.16361i) q^{9} +O(q^{10})$	$q+(1.59345 - 0.678906i) q^{3} +(-1.06893 - 1.69038i) q^{4} +(1.78351 - 3.57116i) q^{7} +(2.07817 - 2.16361i) q^{9} +(-2.85090 - 1.96783i) q^{12} +(-2.59808 + 2.50000i) q^{13} +(-1.71477 + 3.61380i) q^{16} +(-6.86118 + 1.83845i) q^{19} +(0.417452 - 6.90130i) q^{21} +(4.67508 - 1.77302i) q^{25} +(1.84258 - 4.85849i) q^{27} +(-7.94306 + 0.802517i) q^{28} +(9.43875 - 4.24804i) q^{31} +(-5.87874 - 1.20015i) q^{36} +(5.59004 - 7.75956i) q^{37} +(-2.44264 + 5.74748i) q^{39} +(-1.17487 + 7.22929i) q^{43} +(-0.278971 + 6.92258i) q^{48} +(-5.36707 - 7.14227i) q^{49} +(7.00312 + 1.71941i) q^{52} +(-9.68482 + 7.58757i) q^{57} +(2.74229 + 13.4326i) q^{61} +(-4.02014 - 11.2803i) q^{63} +(7.94167 - 0.964293i) q^{64} +(3.42113 + 15.1903i) q^{67} +(8.48917 - 14.0428i) q^{73} +(6.24580 - 5.99917i) q^{75} +(10.4418 + 9.63283i) q^{76} +(-10.1330 + 5.31819i) q^{79} +(-0.362393 - 8.99270i) q^{81} +(-12.1120 + 6.67136i) q^{84} +(4.29420 + 13.7369i) q^{91} +(12.1562 - 13.1771i) q^{93} +(-12.9550 - 11.0196i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 48 q + 10 q^{7} + 6 q^{9}+O(q^{10}) $ 48 * q + 10 * q^7 + 6 * q^9	$ 48 q + 10 q^{7} + 6 q^{9} - 8 q^{16} - 14 q^{19} - 18 q^{21} + 20 q^{28} + 14 q^{31} + 2 q^{37} + 24 q^{39} + 6 q^{43} - 18 q^{49} - 28 q^{52} - 12 q^{57} - 24 q^{63} - 32 q^{67} + 34 q^{73} + 30 q^{75} + 28 q^{76} + 18 q^{81} + 12 q^{84} - 2 q^{91} - 6 q^{93} + 38 q^{97}+O(q^{100}) $ 48 * q + 10 * q^7 + 6 * q^9 - 8 * q^16 - 14 * q^19 - 18 * q^21 + 20 * q^28 + 14 * q^31 + 2 * q^37 + 24 * q^39 + 6 * q^43 - 18 * q^49 - 28 * q^52 - 12 * q^57 - 24 * q^63 - 32 * q^67 + 34 * q^73 + 30 * q^75 + 28 * q^76 + 18 * q^81 + 12 * q^84 - 2 * q^91 - 6 * q^93 + 38 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/507\mathbb{Z}\right)^\times$.

$n$	$170$	$340$
$\chi(n)$	$-1$	$e\left(\frac{103}{156}\right)$

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		0.482459	−	0.875918i	$-0.339744\pi$
$2$		0			0		−0.482459	+	0.875918i	$0.660256\pi$
$3$	1.59345	−	0.678906i	0.919979	−	0.391967i
$3$	1.59345	−	0.678906i	0.919979	−	0.391967i
$4$	−1.06893	−	1.69038i	−0.534466	−	0.845190i
$5$		0			0		0.983620	−	0.180255i	$-0.0576923\pi$
$5$		0			0		−0.983620	+	0.180255i	$0.942308\pi$
$6$		0			0
$7$	1.78351	−	3.57116i	0.674102	−	1.34977i	−0.250232	−	0.968186i	$-0.580507\pi$
$7$	1.78351	−	3.57116i	0.674102	−	1.34977i	0.924334	−	0.381584i	$-0.124621\pi$
$8$		0			0
$9$	2.07817	−	2.16361i	0.692724	−	0.721202i
$10$		0			0
$11$		0			0		−0.999797	−	0.0201371i	$-0.993590\pi$
$11$		0			0		0.999797	+	0.0201371i	$0.00641026\pi$
$12$	−2.85090	−	1.96783i	−0.822984	−	0.568065i
$13$	−2.59808	+	2.50000i	−0.720577	+	0.693375i
$13$	−2.59808	+	2.50000i	−0.720577	+	0.693375i
$14$		0			0
$15$		0			0
$16$	−1.71477	+	3.61380i	−0.428693	+	0.903450i
$17$		0			0		−0.316668	−	0.948536i	$-0.602564\pi$
$17$		0			0		0.316668	+	0.948536i	$0.397436\pi$
$18$		0			0
$19$	−6.86118	+	1.83845i	−1.57406	+	0.421769i	−0.937082	−	0.349111i	$-0.886484\pi$
$19$	−6.86118	+	1.83845i	−1.57406	+	0.421769i	−0.636981	+	0.770879i	$0.719817\pi$
$20$		0			0
$21$	0.417452	−	6.90130i	0.0910954	−	1.50599i
$22$		0			0
$23$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$23$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$24$		0			0
$25$	4.67508	−	1.77302i	0.935016	−	0.354605i
$26$		0			0
$27$	1.84258	−	4.85849i	0.354605	−	0.935016i
$28$	−7.94306	+	0.802517i	−1.50110	+	0.151661i
$29$		0			0		−0.278217	−	0.960518i	$-0.589744\pi$
$29$		0			0		0.278217	+	0.960518i	$0.410256\pi$
$30$		0			0
$31$	9.43875	−	4.24804i	1.69525	−	0.762970i	0.696272	−	0.717778i	$-0.254841\pi$
$31$	9.43875	−	4.24804i	1.69525	−	0.762970i	0.998978	−	0.0451919i	$-0.0143899\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$	−5.87874	−	1.20015i	−0.979791	−	0.200026i
$37$	5.59004	−	7.75956i	0.918997	−	1.27566i	−0.0420032	−	0.999117i	$-0.513374\pi$
$37$	5.59004	−	7.75956i	0.918997	−	1.27566i	0.961000	−	0.276547i	$-0.0891901\pi$
$38$		0			0
$39$	−2.44264	+	5.74748i	−0.391136	+	0.920333i
$40$		0			0
$41$		0			0		−0.927686	−	0.373361i	$-0.878205\pi$
$41$		0			0		0.927686	+	0.373361i	$0.121795\pi$
$42$		0			0
$43$	−1.17487	+	7.22929i	−0.179167	+	1.10246i	0.729206	+	0.684294i	$0.239890\pi$
$43$	−1.17487	+	7.22929i	−0.179167	+	1.10246i	−0.908372	+	0.418162i	$0.862674\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		0			0		0.297503	−	0.954721i	$-0.403846\pi$
$47$		0			0		−0.297503	+	0.954721i	$0.596154\pi$
$48$	−0.278971	+	6.92258i	−0.0402659	+	0.999189i
$49$	−5.36707	−	7.14227i	−0.766724	−	1.02032i
$50$		0			0
$51$		0			0
$52$	7.00312	+	1.71941i	0.971157	+	0.238439i
$53$		0			0		−0.748511	−	0.663123i	$-0.769231\pi$
$53$		0			0		0.748511	+	0.663123i	$0.230769\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	−9.68482	+	7.58757i	−1.28279	+	1.00500i
$58$		0			0
$59$		0			0		−0.941967	−	0.335705i	$-0.891026\pi$
$59$		0			0		0.941967	+	0.335705i	$0.108974\pi$
$60$		0			0
$61$	2.74229	+	13.4326i	0.351114	+	1.71987i	0.645133	+	0.764070i	$0.276802\pi$
$61$	2.74229	+	13.4326i	0.351114	+	1.71987i	−0.294019	+	0.955800i	$0.594993\pi$
$62$		0			0
$63$	−4.02014	−	11.2803i	−0.506491	−	1.42118i
$64$	7.94167	−	0.964293i	0.992709	−	0.120537i
$65$		0			0
$66$		0			0
$67$	3.42113	+	15.1903i	0.417957	+	1.85578i	0.510211	+	0.860049i	$0.329567\pi$
$67$	3.42113	+	15.1903i	0.417957	+	1.85578i	−0.0922537	+	0.995736i	$0.529407\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		0			0		−0.140502	−	0.990080i	$-0.544872\pi$
$71$		0			0		0.140502	+	0.990080i	$0.455128\pi$
$72$		0			0
$73$	8.48917	−	14.0428i	0.993582	−	1.64358i	0.258005	−	0.966144i	$-0.416935\pi$
$73$	8.48917	−	14.0428i	0.993582	−	1.64358i	0.735577	−	0.677441i	$-0.236911\pi$
$74$		0			0
$75$	6.24580	−	5.99917i	0.721202	−	0.692724i
$76$	10.4418	+	9.63283i	1.19776	+	1.10496i
$77$		0			0
$78$		0			0
$79$	−10.1330	+	5.31819i	−1.14005	+	0.598343i	−0.925642	−	0.378401i	$-0.876474\pi$
$79$	−10.1330	+	5.31819i	−1.14005	+	0.598343i	−0.214406	+	0.976745i	$0.568782\pi$
$80$		0			0
$81$	−0.362393	−	8.99270i	−0.0402659	−	0.999189i
$82$		0			0
$83$		0			0		0.616719	−	0.787183i	$-0.288462\pi$
$83$		0			0		−0.616719	+	0.787183i	$0.711538\pi$
$84$	−12.1120	+	6.67136i	−1.32153	+	0.727905i
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$		0			0		−0.965926	−	0.258819i	$-0.916667\pi$
$89$		0			0		0.965926	+	0.258819i	$0.0833333\pi$
$90$		0			0
$91$	4.29420	+	13.7369i	0.450155	+	1.44002i
$92$		0			0
$93$	12.1562	−	13.1771i	1.26054	−	1.36640i
$94$		0			0
$95$		0			0
$96$		0			0
$97$	−12.9550	−	11.0196i	−1.31538	−	1.11887i	−0.984536	−	0.175182i	$-0.943948\pi$
$97$	−12.9550	−	11.0196i	−1.31538	−	1.11887i	−0.330844	−	0.943685i	$-0.607334\pi$
$98$		0			0
$99$		0			0
$100$	−7.99443	−	6.00742i	−0.799443	−	0.600742i
$101$		0			0		−0.774605	−	0.632445i	$-0.782051\pi$
$101$		0			0		0.774605	+	0.632445i	$0.217949\pi$
$102$		0			0
$103$	19.4115	+	2.35698i	1.91267	+	0.232240i	0.988665	−	0.150141i	$-0.0479728\pi$
$103$	19.4115	+	2.35698i	1.91267	+	0.232240i	0.924005	+	0.382381i	$0.124896\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		0			0		−0.996757	−	0.0804666i	$-0.974359\pi$
$107$		0			0		0.996757	+	0.0804666i	$0.0256410\pi$
$108$	−10.1823	+	2.07873i	−0.979791	+	0.200026i
$109$	−4.90166	+	10.8910i	−0.469494	+	1.04317i	0.513256	+	0.858235i	$0.328439\pi$
$109$	−4.90166	+	10.8910i	−0.469494	+	1.04317i	−0.982750	+	0.184937i	$0.940792\pi$
$110$		0			0
$111$	3.63944	−	16.1596i	0.345441	−	1.53380i
$112$	9.84715	+	12.5690i	0.930468	+	1.18765i
$113$		0			0		−0.919979	−	0.391967i	$-0.871795\pi$
$113$		0			0		0.919979	+	0.391967i	$0.128205\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	0.00976636	+	10.8166i	0.000902901	+	1.00000i
$118$		0			0
$119$		0			0
$120$		0			0
$121$	10.9911	+	0.442925i	0.999189	+	0.0402659i
$122$		0			0
$123$		0			0
$124$	−17.2702	−	11.4142i	−1.55091	−	1.02503i
$125$		0			0
$126$		0			0
$127$	−10.2077	+	16.1422i	−0.905788	+	1.43239i	−0.00410555	+	0.999992i	$0.501307\pi$
$127$	−10.2077	+	16.1422i	−0.905788	+	1.43239i	−0.901683	+	0.432398i	$0.857668\pi$
$128$		0			0
$129$	3.03590	+	12.3171i	0.267297	+	1.08446i
$130$		0			0
$131$		0			0		−0.970942	−	0.239316i	$-0.923077\pi$
$131$		0			0		0.970942	+	0.239316i	$0.0769231\pi$
$132$		0			0
$133$	−5.67158	+	27.7812i	−0.491788	+	2.40894i
$134$		0			0
$135$		0			0
$136$		0			0
$137$		0			0		0.811378	−	0.584522i	$-0.198718\pi$
$137$		0			0		−0.811378	+	0.584522i	$0.801282\pi$
$138$		0			0
$139$	18.5110	−	1.49436i	1.57008	−	0.126750i	0.735458	−	0.677571i	$-0.236967\pi$
$139$	18.5110	−	1.49436i	1.57008	−	0.126750i	0.834624	+	0.550820i	$0.185685\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$	4.25526	+	11.2202i	0.354605	+	0.935016i
$145$		0			0
$146$		0			0
$147$	−13.4011	−	7.73712i	−1.10530	−	0.638147i
$148$	−19.0920	−	1.15485i	−1.56935	−	0.0949283i
$149$		0			0		0.735006	−	0.678061i	$-0.237179\pi$
$149$		0			0		−0.735006	+	0.678061i	$0.762821\pi$
$150$		0			0
$151$	−6.01893	+	1.87557i	−0.489813	+	0.152632i	−0.532455	−	0.846458i	$-0.678730\pi$
$151$	−6.01893	+	1.87557i	−0.489813	+	0.152632i	0.0426416	+	0.999090i	$0.486423\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$	12.3264	−	2.01467i	0.986905	−	0.161302i
$157$	0.411278	−	0.595838i	0.0328235	−	0.0475531i	−0.806229	−	0.591604i	$-0.798495\pi$
$157$	0.411278	−	0.595838i	0.0328235	−	0.0475531i	0.839052	+	0.544051i	$0.183110\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$	−20.5603	−	2.07728i	−1.61040	−	0.162705i	−0.746156	−	0.665771i	$-0.768103\pi$
$163$	−20.5603	−	2.07728i	−1.61040	−	0.162705i	−0.864248	+	0.503066i	$0.832205\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		0			0		−0.875918	−	0.482459i	$-0.839744\pi$
$167$		0			0		0.875918	+	0.482459i	$0.160256\pi$
$168$		0			0
$169$	0.500000	−	12.9904i	0.0384615	−	0.999260i
$170$		0			0
$171$	−10.2810	+	18.6655i	−0.786211	+	1.42739i
$172$	13.4761	−	5.74163i	1.02754	−	0.437795i
$173$		0			0		−0.534466	−	0.845190i	$-0.679487\pi$
$173$		0			0		0.534466	+	0.845190i	$0.320513\pi$
$174$		0			0
$175$	2.00629	−	19.8576i	0.151661	−	1.50110i
$176$		0			0
$177$		0			0
$178$		0			0
$179$		0			0		0.316668	−	0.948536i	$-0.397436\pi$
$179$		0			0		−0.316668	+	0.948536i	$0.602564\pi$
$180$		0			0
$181$	22.0290	+	15.2055i	1.63740	+	1.13022i	0.876584	+	0.481249i	$0.159817\pi$
$181$	22.0290	+	15.2055i	1.63740	+	1.13022i	0.760817	+	0.648967i	$0.224799\pi$
$182$		0			0
$183$	13.4892	+	19.5425i	0.997149	+	1.44462i
$184$		0			0
$185$		0			0
$186$		0			0
$187$		0			0
$188$		0			0
$189$	−14.0642	−	15.2453i	−1.02302	−	1.10893i
$190$		0			0
$191$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$191$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$192$	12.0000	−	6.92820i	0.866025	−	0.500000i
$193$	15.1629	−	7.57266i	1.09145	−	0.545092i	0.191681	−	0.981457i	$-0.438606\pi$
$193$	15.1629	−	7.57266i	1.09145	−	0.545092i	0.899770	+	0.436365i	$0.143734\pi$
$194$		0			0
$195$		0			0
$196$	−6.33612	+	16.7070i	−0.452580	+	1.19336i
$197$		0			0		0.994935	−	0.100522i	$-0.0320513\pi$
$197$		0			0		−0.994935	+	0.100522i	$0.967949\pi$
$198$		0			0
$199$	1.76178	+	21.8236i	0.124890	+	1.54703i	0.691091	+	0.722768i	$0.257130\pi$
$199$	1.76178	+	21.8236i	0.124890	+	1.54703i	−0.566202	+	0.824267i	$0.691588\pi$
$200$		0			0
$201$	15.7642	+	21.8823i	1.11192	+	1.54346i
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$	−4.57940	−	13.6759i	−0.317524	−	0.948250i
$209$		0			0
$210$		0			0
$211$	−18.0601	−	11.4205i	−1.24331	−	0.786222i	−0.259553	−	0.965729i	$-0.583575\pi$
$211$	−18.0601	−	11.4205i	−1.24331	−	0.786222i	−0.983757	+	0.179507i	$0.942550\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$	1.66368	−	41.2837i	0.112938	−	2.80252i
$218$		0			0
$219$	3.99333	−	28.1399i	0.269844	−	1.90152i
$220$		0			0
$221$		0			0
$222$		0			0
$223$	−8.06688	−	9.48371i	−0.540199	−	0.635077i	0.422270	−	0.906470i	$-0.361233\pi$
$223$	−8.06688	−	9.48371i	−0.540199	−	0.635077i	−0.962468	+	0.271394i	$0.912515\pi$
$224$		0			0
$225$	5.87950	−	13.7997i	0.391967	−	0.919979i
$226$		0			0
$227$		0			0		−0.975564	−	0.219715i	$-0.929487\pi$
$227$		0			0		0.975564	+	0.219715i	$0.0705128\pi$
$228$	23.1783	+	8.26043i	1.53502	+	0.547061i
$229$	−15.0268	−	6.76301i	−0.992998	−	0.446912i	−0.152252	−	0.988342i	$-0.548652\pi$
$229$	−15.0268	−	6.76301i	−0.992998	−	0.446912i	−0.840746	+	0.541430i	$0.817883\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		0			0		0.992709	−	0.120537i	$-0.0384615\pi$
$233$		0			0		−0.992709	+	0.120537i	$0.961538\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	−12.5358	+	15.3536i	−0.814290	+	0.997324i
$238$		0			0
$239$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$239$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$240$		0			0
$241$	−3.31826	+	3.90107i	−0.213748	+	0.251290i	−0.858152	−	0.513395i	$-0.828388\pi$
$241$	−3.31826	+	3.90107i	−0.213748	+	0.251290i	0.644404	+	0.764685i	$0.277105\pi$
$242$		0			0
$243$	−6.68266	−	14.0834i	−0.428693	−	0.903450i
$244$	19.7749	−	18.9941i	1.26596	−	1.21597i
$245$		0			0
$246$		0			0
$247$	13.2297	−	21.9294i	0.841789	−	1.39533i
$248$		0			0
$249$		0			0
$250$		0			0
$251$		0			0		0.774605	−	0.632445i	$-0.217949\pi$
$251$		0			0		−0.774605	+	0.632445i	$0.782051\pi$
$252$	−14.7707	+	18.8534i	−0.930468	+	1.18765i
$253$		0			0
$254$		0			0
$255$		0			0
$256$	−10.1191	−	12.3937i	−0.632445	−	0.774605i
$257$		0			0		0.999189	−	0.0402659i	$-0.0128205\pi$
$257$		0			0		−0.999189	+	0.0402659i	$0.987179\pi$
$258$		0			0
$259$	−17.7407	−	33.8022i	−1.10236	−	2.10036i
$260$		0			0
$261$		0			0
$262$		0			0
$263$		0			0		−0.692724	−	0.721202i	$-0.743590\pi$
$263$		0			0		0.692724	+	0.721202i	$0.256410\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$	22.0204	−	22.0204i	1.34511	−	1.34511i
$269$		0			0		−0.799443	−	0.600742i	$-0.794872\pi$
$269$		0			0		0.799443	+	0.600742i	$0.205128\pi$
$270$		0			0
$271$	−22.4861	+	5.06428i	−1.36593	+	0.307633i	−0.839858	−	0.542806i	$-0.817362\pi$
$271$	−22.4861	+	5.06428i	−1.36593	+	0.307633i	−0.526073	+	0.850439i	$0.676336\pi$
$272$		0			0
$273$	16.1687	+	18.9737i	0.978573	+	1.14834i
$274$		0			0
$275$		0			0
$276$		0			0
$277$	−27.7435	+	5.66388i	−1.66695	+	0.340309i	−0.938628	−	0.344931i	$-0.887902\pi$
$277$	−27.7435	+	5.66388i	−1.66695	+	0.340309i	−0.728317	+	0.685240i	$0.759697\pi$
$278$		0			0
$279$	10.4243	−	29.2499i	0.624085	−	1.75115i
$280$		0			0
$281$		0			0		−0.616719	−	0.787183i	$-0.711538\pi$
$281$		0			0		0.616719	+	0.787183i	$0.288462\pi$
$282$		0			0
$283$	−0.694457	−	4.27317i	−0.0412812	−	0.254013i	0.958204	−	0.286085i	$-0.0923538\pi$
$283$	−0.694457	−	4.27317i	−0.0412812	−	0.254013i	−0.999486	+	0.0320711i	$0.989790\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	−13.5905	+	10.2126i	−0.799443	+	0.600742i
$290$		0			0
$291$	−28.1244	−	8.76392i	−1.64868	−	0.513750i
$292$	−32.8120	+	0.660871i	−1.92018	+	0.0386746i
$293$		0			0		−0.834256	−	0.551377i	$-0.814103\pi$
$293$		0			0		0.834256	+	0.551377i	$0.185897\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$		0			0
$300$	−16.8172	−	4.14507i	−0.970942	−	0.239316i
$301$	23.7215	+	17.0892i	1.36729	+	0.985002i
$302$		0			0
$303$		0			0
$304$	5.12156	−	27.9475i	0.293742	−	1.60290i
$305$		0			0
$306$		0			0
$307$	−0.157745	−	0.350495i	−0.00900298	−	0.0200038i	0.907344	−	0.420390i	$-0.138107\pi$
$307$	−0.157745	−	0.350495i	−0.00900298	−	0.0200038i	−0.916346	+	0.400386i	$0.868876\pi$
$308$		0			0
$309$	32.5314	−	9.42284i	1.85065	−	0.536046i
$310$		0			0
$311$		0			0		−0.935016	−	0.354605i	$-0.884615\pi$
$311$		0			0		0.935016	+	0.354605i	$0.115385\pi$
$312$		0			0
$313$	5.98288	+	15.7755i	0.338172	+	0.891687i	0.990961	+	0.134147i	$0.0428295\pi$
$313$	5.98288	+	15.7755i	0.338172	+	0.891687i	−0.652789	+	0.757540i	$0.726401\pi$
$314$		0			0
$315$		0			0
$316$	19.8212	+	11.4438i	1.11503	+	0.643763i
$317$		0			0		−0.998176	−	0.0603785i	$-0.980769\pi$
$317$		0			0		0.998176	+	0.0603785i	$0.0192308\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$		0			0
$322$		0			0
$323$		0			0
$324$	−14.8137	+	10.2252i	−0.822984	+	0.568065i
$325$	−7.71366	+	16.2942i	−0.427877	+	0.903837i
$326$		0			0
$327$	−0.416561	+	20.6821i	−0.0230359	+	1.14372i
$328$		0			0
$329$		0			0
$330$		0			0
$331$	−4.16115	−	2.07816i	−0.228717	−	0.114226i	0.328774	−	0.944408i	$-0.393364\pi$
$331$	−4.16115	−	2.07816i	−0.228717	−	0.114226i	−0.557492	+	0.830182i	$0.688236\pi$
$332$		0			0
$333$	−5.17158	−	28.2204i	−0.283401	−	1.54647i
$334$		0			0
$335$		0			0
$336$	24.2241	+	13.3427i	1.32153	+	0.727905i
$337$			6.77550i			0.369085i	0.982825	+	0.184543i	$0.0590804\pi$
$337$			6.77550i			0.369085i	−0.982825	+	0.184543i	$0.940920\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$	−7.59381	+	1.39162i	−0.410027	+	0.0751403i
$344$		0			0
$345$		0			0
$346$		0			0
$347$		0			0		0.692724	−	0.721202i	$-0.256410\pi$
$347$		0			0		−0.692724	+	0.721202i	$0.743590\pi$
$348$		0			0
$349$	4.55628	+	0.0917687i	0.243892	+	0.00491227i	0.141933	−	0.989876i	$-0.454668\pi$
$349$	4.55628	+	0.0917687i	0.243892	+	0.00491227i	0.101959	+	0.994789i	$0.467489\pi$
$350$		0			0
$351$	7.35905	+	17.2292i	0.392797	+	0.919625i
$352$		0			0
$353$		0			0		0.834256	−	0.551377i	$-0.185897\pi$
$353$		0			0		−0.834256	+	0.551377i	$0.814103\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$		0			0		0.0603785	−	0.998176i	$-0.480769\pi$
$359$		0			0		−0.0603785	+	0.998176i	$0.519231\pi$
$360$		0			0
$361$	27.2414	−	15.7278i	1.43376	−	0.827781i
$362$		0			0
$363$	17.8145	−	6.75613i	0.935016	−	0.354605i
$364$	18.6304	−	21.9427i	0.976497	−	1.15011i
$365$		0			0
$366$		0			0
$367$	7.25845	+	25.0591i	0.378888	+	1.30807i	0.892886	+	0.450282i	$0.148677\pi$
$367$	7.25845	+	25.0591i	0.378888	+	1.30807i	−0.513998	+	0.857791i	$0.671836\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$	−35.2684	−	6.46317i	−1.82858	−	0.335100i
$373$	7.27966	−	25.1323i	0.376926	−	1.30130i	−0.518112	−	0.855313i	$-0.673365\pi$
$373$	7.27966	−	25.1323i	0.376926	−	1.30130i	0.895039	−	0.445988i	$-0.147148\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$	−23.5554	−	9.48021i	−1.20996	−	0.486966i	−0.321933	−	0.946762i	$-0.604333\pi$
$379$	−23.5554	−	9.48021i	−1.20996	−	0.486966i	−0.888024	+	0.459797i	$0.847922\pi$
$380$		0			0
$381$	−5.30645	+	32.6519i	−0.271858	+	1.67281i
$382$		0			0
$383$		0			0		0.551377	−	0.834256i	$-0.314103\pi$
$383$		0			0		−0.551377	+	0.834256i	$0.685897\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	13.1998	+	17.5657i	0.670981	+	0.892914i
$388$	−4.77926	+	33.6780i	−0.242630	+	1.70974i
$389$		0			0		−0.663123	−	0.748511i	$-0.730769\pi$
$389$		0			0		0.663123	+	0.748511i	$0.269231\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$		0			0
$394$		0			0
$395$		0			0
$396$		0			0
$397$	12.0632	+	4.29917i	0.605435	+	0.215769i	0.620615	−	0.784116i	$-0.286883\pi$
$397$	12.0632	+	4.29917i	0.605435	+	0.215769i	−0.0151793	+	0.999885i	$0.504832\pi$
$398$		0			0
$399$	9.82346	+	48.1185i	0.491788	+	2.40894i
$400$	−1.60933	+	19.9351i	−0.0804666	+	0.996757i
$401$		0			0		−0.335705	−	0.941967i	$-0.608974\pi$
$401$		0			0		0.335705	+	0.941967i	$0.391026\pi$
$402$		0			0
$403$	−13.9025	+	34.6336i	−0.692533	+	1.72522i
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$	−1.83895	−	12.9586i	−0.0909303	−	0.640759i	−0.982277	−	0.187437i	$-0.939982\pi$
$409$	−1.83895	−	12.9586i	−0.0909303	−	0.640759i	0.891346	−	0.453323i	$-0.149762\pi$
$410$		0			0
$411$		0			0
$412$	−16.7653	−	35.3322i	−0.825969	−	1.74069i
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$	28.4818	−	14.9484i	1.39476	−	0.732027i
$418$		0			0
$419$		0			0		−0.0402659	−	0.999189i	$-0.512821\pi$
$419$		0			0		0.0402659	+	0.999189i	$0.487179\pi$
$420$		0			0
$421$	−7.48225	+	9.55039i	−0.364663	+	0.465457i	−0.934325	−	0.356421i	$-0.883997\pi$
$421$	−7.48225	+	9.55039i	−0.364663	+	0.465457i	0.569663	+	0.821878i	$0.307074\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		0			0
$426$		0			0
$427$	52.8609	+	14.1640i	2.55812	+	0.685445i
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0		0.678061	−	0.735006i	$-0.262821\pi$
$431$		0			0		−0.678061	+	0.735006i	$0.737179\pi$
$432$	14.3980	+	14.9899i	0.692724	+	0.721202i
$433$	−25.8336	+	12.2582i	−1.24148	+	0.589091i	−0.932222	−	0.361886i	$-0.882133\pi$
$433$	−25.8336	+	12.2582i	−1.24148	+	0.589091i	−0.309261	+	0.950977i	$0.600082\pi$
$434$		0			0
$435$		0			0
$436$	23.6495	−	3.35611i	1.13261	−	0.160729i
$437$		0			0
$438$		0			0
$439$	−32.3713	−	26.4303i	−1.54500	−	1.26145i	−0.819113	−	0.573633i	$-0.805534\pi$
$439$	−32.3713	−	26.4303i	−1.54500	−	1.26145i	−0.725884	−	0.687817i	$-0.758569\pi$
$440$		0			0
$441$	−26.6068	−	3.23065i	−1.26699	−	0.153840i
$442$		0			0
$443$		0			0		−0.120537	−	0.992709i	$-0.538462\pi$
$443$		0			0		0.120537	+	0.992709i	$0.461538\pi$
$444$	−31.2062	+	11.1215i	−1.48098	+	0.527801i
$445$		0			0
$446$		0			0
$447$		0			0
$448$	10.7204	−	30.0808i	0.506491	−	1.42118i
$449$		0			0		0.219715	−	0.975564i	$-0.429487\pi$
$449$		0			0		−0.219715	+	0.975564i	$0.570513\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$	−8.31753	+	7.07492i	−0.390791	+	0.332409i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	4.49031	+	0.637221i	0.210048	+	0.0298079i	0.244734	−	0.969590i	$-0.421299\pi$
$457$	4.49031	+	0.637221i	0.210048	+	0.0298079i	−0.0346858	+	0.999398i	$0.511043\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		0			0		0.999797	−	0.0201371i	$-0.00641026\pi$
$461$		0			0		−0.999797	+	0.0201371i	$0.993590\pi$
$462$		0			0
$463$	−32.2584	+	19.5009i	−1.49917	+	0.906282i	−0.499984	+	0.866035i	$0.666661\pi$
$463$	−32.2584	+	19.5009i	−1.49917	+	0.906282i	−0.999190	+	0.0402476i	$0.987185\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		0			0		−0.239316	−	0.970942i	$-0.576923\pi$
$467$		0			0		0.239316	+	0.970942i	$0.423077\pi$
$468$	18.2738	−	11.5788i	0.844707	−	0.535229i
$469$	60.3484	+	14.8745i	2.78663	+	0.686842i
$470$		0			0
$471$	0.250832	−	1.22866i	0.0115577	−	0.0566136i
$472$		0			0
$473$		0			0
$474$		0			0
$475$	−28.8170	+	20.7599i	−1.32221	+	0.952531i
$476$		0			0
$477$		0			0
$478$		0			0
$479$		0			0		−0.100522	−	0.994935i	$-0.532051\pi$
$479$		0			0		0.100522	+	0.994935i	$0.467949\pi$
$480$		0			0
$481$	4.87556	+	34.1350i	0.222306	+	1.55642i
$482$		0			0
$483$		0			0
$484$	−11.0000	−	19.0526i	−0.500000	−	0.866025i
$485$		0			0
$486$		0			0
$487$	−30.1184	+	27.7850i	−1.36480	+	1.25906i	−0.430486	+	0.902597i	$0.641658\pi$
$487$	−30.1184	+	27.7850i	−1.36480	+	1.25906i	−0.934310	+	0.356461i	$0.883983\pi$
$488$		0			0
$489$	−34.1720	+	10.6484i	−1.54531	+	0.481539i
$490$		0			0
$491$		0			0		−0.903450	−	0.428693i	$-0.858974\pi$
$491$		0			0		0.903450	+	0.428693i	$0.141026\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$	−0.833726	+	41.3942i	−0.0374354	+	1.85865i
$497$		0			0
$498$		0			0
$499$	−2.23978	−	37.0280i	−0.100266	−	1.65760i	−0.606412	−	0.795151i	$-0.707392\pi$
$499$	−2.23978	−	37.0280i	−0.100266	−	1.65760i	0.506146	−	0.862448i	$-0.331070\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		0			0		0.845190	−	0.534466i	$-0.179487\pi$
$503$		0			0		−0.845190	+	0.534466i	$0.820513\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	−8.02252	−	21.0390i	−0.356293	−	0.934374i
$508$	38.1978			1.69475
$509$		0			0		0.482459	−	0.875918i	$-0.339744\pi$
$509$		0			0		−0.482459	+	0.875918i	$0.660256\pi$
$510$		0			0
$511$	−35.0085	−	55.3616i	−1.54869	−	2.44905i
$512$		0			0
$513$	−3.71021	+	36.7224i	−0.163810	+	1.62134i
$514$		0			0
$515$		0			0
$516$	17.5755	−	18.2980i	0.773718	−	0.805526i
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$		0			0		−0.568065	−	0.822984i	$-0.692308\pi$
$521$		0			0		0.568065	+	0.822984i	$0.307692\pi$
$522$		0			0
$523$	7.94998	−	16.7542i	0.347628	−	0.732611i	−0.652045	−	0.758181i	$-0.726089\pi$
$523$	7.94998	−	16.7542i	0.347628	−	0.732611i	0.999673	+	0.0255692i	$0.00813983\pi$
$524$		0			0
$525$	−10.2845	−	33.0043i	−0.448855	−	1.44042i
$526$		0			0
$527$		0			0
$528$		0			0
$529$	11.5000	−	19.9186i	0.500000	−	0.866025i
$530$		0			0
$531$		0			0
$532$	53.0234	−	20.1091i	2.29885	−	0.871841i
$533$		0			0
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$	32.7287	+	5.99775i	1.40712	+	0.257863i	0.829528	−	0.558465i	$-0.188610\pi$
$541$	32.7287	+	5.99775i	1.40712	+	0.257863i	0.577587	+	0.816329i	$0.303994\pi$
$542$		0			0
$543$	45.4252	+	9.27362i	1.94938	+	0.397969i
$544$		0			0
$545$		0			0
$546$		0			0
$547$	−45.4087	+	11.1922i	−1.94154	+	0.478546i	−0.966246	+	0.257619i	$0.917062\pi$
$547$	−45.4087	+	11.1922i	−1.94154	+	0.478546i	−0.975290	+	0.220926i	$0.929092\pi$
$548$		0			0
$549$	34.7618	+	21.9821i	1.48360	+	0.938171i
$550$		0			0
$551$		0			0
$552$		0			0
$553$	0.919875	+	45.6715i	0.0391171	+	1.94215i
$554$		0			0
$555$		0			0
$556$	−22.3130	−	29.6932i	−0.946283	−	1.25927i
$557$		0			0		0.140502	−	0.990080i	$-0.455128\pi$
$557$		0			0		−0.140502	+	0.990080i	$0.544872\pi$
$558$		0			0
$559$	−15.0208	−	21.7194i	−0.635313	−	0.918634i
$560$		0			0
$561$		0			0
$562$		0			0
$563$		0			0		0.391967	−	0.919979i	$-0.371795\pi$
$563$		0			0		−0.391967	+	0.919979i	$0.628205\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	−32.7607	−	14.7444i	−1.37582	−	0.619206i
$568$		0			0
$569$		0			0		0.0804666	−	0.996757i	$-0.474359\pi$
$569$		0			0		−0.0804666	+	0.996757i	$0.525641\pi$
$570$		0			0
$571$	−42.4645	+	5.15612i	−1.77708	+	0.215777i	−0.942293	−	0.334790i	$-0.891335\pi$
$571$	−42.4645	+	5.15612i	−1.77708	+	0.215777i	−0.834791	+	0.550567i	$0.814412\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$	14.4178	−	19.1866i	0.600742	−	0.799443i
$577$	33.6825	+	33.6825i	1.40222	+	1.40222i	0.793004	+	0.609217i	$0.208516\pi$
$577$	33.6825	+	33.6825i	1.40222	+	1.40222i	0.609217	+	0.793004i	$0.291484\pi$
$578$		0			0
$579$	19.0202	−	22.3609i	0.790454	−	0.929286i
$580$		0			0
$581$		0			0
$582$		0			0
$583$		0			0
$584$		0			0
$585$		0			0
$586$		0			0
$587$		0			0		0.258819	−	0.965926i	$-0.416667\pi$
$587$		0			0		−0.258819	+	0.965926i	$0.583333\pi$
$588$	1.24617	+	30.9234i	0.0513912	+	1.27526i
$589$	−56.9512	+	46.4992i	−2.34663	+	1.91597i
$590$		0			0
$591$		0			0
$592$	18.4559	+	33.5072i	0.758532	+	1.37714i
$593$		0			0		−0.787183	−	0.616719i	$-0.788462\pi$
$593$		0			0		0.787183	+	0.616719i	$0.211538\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	17.6235	+	33.5788i	0.721282	+	1.37429i
$598$		0			0
$599$		0			0		−0.885456	−	0.464723i	$-0.846154\pi$
$599$		0			0		0.885456	+	0.464723i	$0.153846\pi$
$600$		0			0
$601$	−33.4975	−	34.8746i	−1.36639	−	1.42256i	−0.802403	−	0.596783i	$-0.796445\pi$
$601$	−33.4975	−	34.8746i	−1.36639	−	1.42256i	−0.563989	−	0.825782i	$-0.690734\pi$
$602$		0			0
$603$	39.9754	+	24.1660i	1.62793	+	0.984115i
$604$	9.60425	+	8.16941i	0.390791	+	0.332409i
$605$		0			0
$606$		0			0
$607$	35.8413	+	26.9330i	1.45475	+	1.09318i	0.976641	+	0.214877i	$0.0689350\pi$
$607$	35.8413	+	26.9330i	1.45475	+	1.09318i	0.478112	+	0.878299i	$0.341321\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$	36.6609	−	13.0654i	1.48072	−	0.527708i	0.532806	−	0.846237i	$-0.321137\pi$
$613$	36.6609	−	13.0654i	1.48072	−	0.527708i	0.947913	+	0.318529i	$0.103189\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		0			0		0.335705	−	0.941967i	$-0.391026\pi$
$617$		0			0		−0.335705	+	0.941967i	$0.608974\pi$
$618$		0			0
$619$	−12.2065	−	15.5804i	−0.490620	−	0.626230i	0.476921	−	0.878946i	$-0.341753\pi$
$619$	−12.2065	−	15.5804i	−0.490620	−	0.626230i	−0.967540	+	0.252717i	$0.918676\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$	−16.5817	−	18.6828i	−0.663798	−	0.747912i
$625$	18.7128	−	16.5781i	0.748511	−	0.663123i
$626$		0			0
$627$		0			0
$628$	−1.44682	−	0.0583049i	−0.0577344	−	0.00232662i
$629$		0			0
$630$		0			0
$631$	−41.7774	−	27.6116i	−1.66313	−	1.09920i	−0.875806	−	0.482663i	$-0.839670\pi$
$631$	−41.7774	−	27.6116i	−1.66313	−	1.09920i	−0.787327	−	0.616536i	$-0.788536\pi$
$632$		0			0
$633$	−36.5314	−	5.93693i	−1.45199	−	0.235972i
$634$		0			0
$635$		0			0
$636$		0			0
$637$	31.7997	+	5.13849i	1.25995	+	0.203594i
$638$		0			0
$639$		0			0
$640$		0			0
$641$		0			0		−0.960518	−	0.278217i	$-0.910256\pi$
$641$		0			0		0.960518	+	0.278217i	$0.0897436\pi$
$642$		0			0
$643$	6.04099	+	15.0100i	0.238233	+	0.591935i	0.998323	−	0.0578826i	$-0.0184349\pi$
$643$	6.04099	+	15.0100i	0.238233	+	0.591935i	−0.760090	+	0.649818i	$0.774845\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0		0.960518	−	0.278217i	$-0.0897436\pi$
$647$		0			0		−0.960518	+	0.278217i	$0.910256\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$	−25.3768	−	66.9130i	−0.994593	−	2.62253i
$652$	18.4661	+	36.9751i	0.723189	+	1.44806i
$653$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$653$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	−12.7411	−	47.5506i	−0.497079	−	1.85512i
$658$		0			0
$659$		0			0		0.948536	−	0.316668i	$-0.102564\pi$
$659$		0			0		−0.948536	+	0.316668i	$0.897436\pi$
$660$		0			0
$661$	27.5676	+	41.7110i	1.07226	+	1.62237i	0.730971	+	0.682408i	$0.239067\pi$
$661$	27.5676	+	41.7110i	1.07226	+	1.62237i	0.341286	+	0.939960i	$0.389138\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$		0			0
$668$		0			0
$669$	−19.2927	−	9.63518i	−0.745900	−	0.372518i
$670$		0			0
$671$		0			0
$672$		0			0
$673$	17.8967	+	42.0052i	0.689868	+	1.61918i	0.781885	+	0.623422i	$0.214258\pi$
$673$	17.8967	+	42.0052i	0.689868	+	1.61918i	−0.0920175	+	0.995757i	$0.529332\pi$
$674$		0			0
$675$		−	25.9808i		−	1.00000i
$676$	−22.4931	+	13.0406i	−0.865121	+	0.501563i
$677$		0			0		1.00000			$0$
$677$		0			0		−1.00000			$\pi$
$678$		0			0
$679$	−62.4579	+	26.6108i	−2.39692	+	1.02123i
$680$		0			0
$681$		0			0
$682$		0			0
$683$		0			0		0.446798	−	0.894635i	$-0.352564\pi$
$683$		0			0		−0.446798	+	0.894635i	$0.647436\pi$
$684$	42.5415	−	2.57329i	1.62662	−	0.0983922i
$685$		0			0
$686$		0			0
$687$	−28.5359	−	0.574745i	−1.08871	−	0.0219279i
$688$	−24.1106	−	16.6423i	−0.919207	−	0.634483i
$689$		0			0
$690$		0			0
$691$	16.0354	−	10.5982i	0.610017	−	0.403173i	−0.208197	−	0.978087i	$-0.566759\pi$
$691$	16.0354	−	10.5982i	0.610017	−	0.403173i	0.818214	+	0.574914i	$0.194965\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$	−35.7116	+	17.8351i	−1.34977	+	0.674102i
$701$		0			0		0.935016	−	0.354605i	$-0.115385\pi$
$701$		0			0		−0.935016	+	0.354605i	$0.884615\pi$
$702$		0			0
$703$	−24.0887	+	63.5168i	−0.908524	+	2.39558i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	−24.5746	+	9.89044i	−0.922920	+	0.371443i	−0.785352	−	0.619049i	$-0.787518\pi$
$709$	−24.5746	+	9.89044i	−0.922920	+	0.371443i	−0.137567	+	0.990492i	$0.543928\pi$
$710$		0			0
$711$	−9.55158	+	32.9759i	−0.358212	+	1.23669i
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$		0			0		0.160411	−	0.987050i	$-0.448718\pi$
$719$		0			0		−0.160411	+	0.987050i	$0.551282\pi$
$720$		0			0
$721$	43.0376	−	65.1177i	1.60281	−	2.42511i
$722$		0			0
$723$	−2.63903	+	8.46895i	−0.0981466	+	0.314964i
$724$	2.15561	−	53.4910i	0.0801127	−	1.98798i
$725$		0			0
$726$		0			0
$727$	34.2227	+	38.6294i	1.26925	+	1.43268i	0.851700	+	0.524030i	$0.175572\pi$
$727$	34.2227	+	38.6294i	1.26925	+	1.43268i	0.417548	+	0.908655i	$0.362889\pi$
$728$		0			0
$729$	−20.2098	−	17.9043i	−0.748511	−	0.663123i
$730$		0			0
$731$		0			0
$732$	18.6152	−	43.6914i	0.688036	−	1.61488i
$733$	8.05223	−	6.30852i	0.297416	−	0.233010i	−0.456906	−	0.889515i	$-0.651043\pi$
$733$	8.05223	−	6.30852i	0.297416	−	0.233010i	0.754322	+	0.656504i	$0.227966\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$		0			0
$739$	−10.3176	−	28.9507i	−0.379540	−	1.06497i	−0.966792	−	0.255563i	$-0.917739\pi$
$739$	−10.3176	−	28.9507i	−0.379540	−	1.06497i	0.587252	−	0.809404i	$-0.300210\pi$
$740$		0			0
$741$	6.19297	−	43.9251i	0.227504	−	1.61363i
$742$		0			0
$743$		0			0		−0.219715	−	0.975564i	$-0.570513\pi$
$743$		0			0		0.219715	+	0.975564i	$0.429487\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	37.9600	−	36.4611i	1.38518	−	1.33048i	0.499406	−	0.866368i	$-0.333552\pi$
$751$	37.9600	−	36.4611i	1.38518	−	1.33048i	0.885775	−	0.464116i	$-0.153628\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$	−10.7367	+	40.0700i	−0.390490	+	1.45733i
$757$	−0.767455	−	19.0442i	−0.0278936	−	0.692173i	−0.951285	−	0.308313i	$-0.900236\pi$
$757$	−0.767455	−	19.0442i	−0.0278936	−	0.692173i	0.923391	−	0.383860i	$-0.125405\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		0			0		−0.482459	−	0.875918i	$-0.660256\pi$
$761$		0			0		0.482459	+	0.875918i	$0.339744\pi$
$762$		0			0
$763$	30.1515	+	36.9288i	1.09156	+	1.33691i
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$	−24.5385	−	12.8788i	−0.885456	−	0.464723i
$769$	34.8275	−	37.7524i	1.25591	−	1.36139i	0.350667	−	0.936500i	$-0.385955\pi$
$769$	34.8275	−	37.7524i	1.25591	−	1.36139i	0.905246	−	0.424887i	$-0.139686\pi$
$770$		0			0
$771$		0			0
$772$	−29.0088	−	17.5364i	−1.04405	−	0.631150i
$773$		0			0		−0.761712	−	0.647915i	$-0.775641\pi$
$773$		0			0		0.761712	+	0.647915i	$0.224359\pi$
$774$		0			0
$775$	36.5951	−	36.5951i	1.31453	−	1.31453i
$776$		0			0
$777$	−51.2175	−	41.8178i	−1.83742	−	1.50020i
$778$		0			0
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$	35.0140	−	7.14816i	1.25050	−	0.255292i
$785$		0			0
$786$		0			0
$787$	9.86529	−	43.8032i	0.351660	−	1.56141i	−0.409168	−	0.912459i	$-0.634181\pi$
$787$	9.86529	−	43.8032i	0.351660	−	1.56141i	0.760827	−	0.648955i	$-0.224793\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	−40.7062	−	28.0432i	−1.44552	−	0.995844i
$794$		0			0
$795$		0			0
$796$	35.0070	−	26.3060i	1.24079	−	0.932392i
$797$		0			0		−0.999189	−	0.0402659i	$-0.987179\pi$
$797$		0			0		0.999189	+	0.0402659i	$0.0128205\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$		0			0
$802$		0			0
$803$		0			0
$804$	20.1386	−	50.0381i	0.710234	−	1.76471i
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$		0			0		0.200026	−	0.979791i	$-0.435897\pi$
$809$		0			0		−0.200026	+	0.979791i	$0.564103\pi$
$810$		0			0
$811$	−2.07615	+	11.3292i	−0.0729036	+	0.397822i	0.926846	+	0.375442i	$0.122509\pi$
$811$	−2.07615	+	11.3292i	−0.0729036	+	0.397822i	−0.999750	+	0.0223803i	$0.992876\pi$
$812$		0			0
$813$	−32.3923	+	23.3356i	−1.13605	+	0.818415i
$814$		0			0
$815$		0			0
$816$		0			0
$817$	−5.22965	−	51.7614i	−0.182962	−	1.81090i
$818$		0			0
$819$	38.6454	+	19.2567i	1.35038	+	0.672883i
$820$		0			0
$821$		0			0		−0.446798	−	0.894635i	$-0.647436\pi$
$821$		0			0		0.446798	+	0.894635i	$0.352564\pi$
$822$		0			0
$823$	31.1670	+	17.9943i	1.08641	+	0.627240i	0.932619	−	0.360863i	$-0.117518\pi$
$823$	31.1670	+	17.9943i	1.08641	+	0.627240i	0.153793	+	0.988103i	$0.450851\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		0			0		0.954721	−	0.297503i	$-0.0961538\pi$
$827$		0			0		−0.954721	+	0.297503i	$0.903846\pi$
$828$		0			0
$829$	−43.9423	−	20.8509i	−1.52618	−	0.724181i	−0.533594	−	0.845741i	$-0.679159\pi$
$829$	−43.9423	−	20.8509i	−1.52618	−	0.724181i	−0.992584	+	0.121560i	$0.961210\pi$
$830$		0			0
$831$	−40.3627	+	27.8603i	−1.40017	+	0.966464i
$832$	−18.2223	+	22.3595i	−0.631746	+	0.775176i
$833$		0			0
$834$		0			0
$835$		0			0
$836$		0			0
$837$	−3.24737	−	53.6854i	−0.112245	−	1.85564i
$838$		0			0
$839$		0			0		−0.994935	−	0.100522i	$-0.967949\pi$
$839$		0			0		0.994935	+	0.100522i	$0.0320513\pi$
$840$		0			0
$841$	−24.5105	+	15.4995i	−0.845190	+	0.534466i
$842$		0			0
$843$		0			0
$844$			42.7362i			1.47104i
$845$		0			0
$846$		0			0
$847$	21.1844	−	38.4609i	0.727905	−	1.32153i
$848$		0			0
$849$	−4.00766	−	6.33761i	−0.137543	−	0.217506i
$850$		0			0
$851$		0			0
$852$		0			0
$853$	52.2128	−	3.15830i	1.78773	−	0.108138i	0.866995	−	0.498317i	$-0.166048\pi$
$853$	52.2128	−	3.15830i	1.78773	−	0.108138i	0.920739	+	0.390179i	$0.127587\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		0			0		−0.822984	−	0.568065i	$-0.807692\pi$
$857$		0			0		0.822984	+	0.568065i	$0.192308\pi$
$858$		0			0
$859$	−3.41726	−	4.95076i	−0.116595	−	0.168918i	0.760341	−	0.649524i	$-0.225032\pi$
$859$	−3.41726	−	4.95076i	−0.116595	−	0.168918i	−0.876936	+	0.480607i	$0.840417\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		0			0		−0.297503	−	0.954721i	$-0.596154\pi$
$863$		0			0		0.297503	+	0.954721i	$0.403846\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	−14.7224	+	25.5000i	−0.500000	+	0.866025i
$868$	−71.5635	+	41.3172i	−2.42902	+	1.40240i
$869$		0			0
$870$		0			0
$871$	−46.8640	−	30.9126i	−1.58793	−	1.04743i
$872$		0			0
$873$	−50.7647	+	5.12895i	−1.71813	+	0.173589i
$874$		0			0
$875$		0			0
$876$	−51.8357	+	23.3293i	−1.75136	+	0.788225i
$877$	29.4960	+	40.9436i	0.996010	+	1.38257i	0.922812	+	0.385250i	$0.125885\pi$
$877$	29.4960	+	40.9436i	0.996010	+	1.38257i	0.0731981	+	0.997317i	$0.476679\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		0			0		−0.979791	−	0.200026i	$-0.935897\pi$
$881$		0			0		0.979791	+	0.200026i	$0.0641026\pi$
$882$		0			0
$883$	6.27887	−	25.4744i	0.211301	−	0.857281i	−0.765553	−	0.643373i	$-0.777534\pi$
$883$	6.27887	−	25.4744i	0.211301	−	0.857281i	0.976854	−	0.213908i	$-0.0686194\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		0			0		−0.845190	−	0.534466i	$-0.820513\pi$
$887$		0			0		0.845190	+	0.534466i	$0.179487\pi$
$888$		0			0
$889$	39.4408	+	65.2431i	1.32280	+	2.18818i
$890$		0			0
$891$		0			0
$892$	−7.40814	+	23.7735i	−0.248043	+	0.795997i
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$		0			0
$899$		0			0
$900$	−29.6115	+	4.81234i	−0.987050	+	0.160411i
$901$		0			0
$902$		0			0
$903$	49.4010	+	11.1260i	1.64396	+	0.370251i
$904$		0			0
$905$		0			0
$906$		0			0
$907$	−2.24684	+	27.8321i	−0.0746052	+	0.924151i	0.845880	+	0.533373i	$0.179076\pi$
$907$	−2.24684	+	27.8321i	−0.0746052	+	0.924151i	−0.920485	+	0.390778i	$0.872206\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		0			0		0.120537	−	0.992709i	$-0.461538\pi$
$911$		0			0		−0.120537	+	0.992709i	$0.538462\pi$
$912$	−10.8127	−	48.0100i	−0.358046	−	1.58977i
$913$		0			0
$914$		0			0
$915$		0			0
$916$	4.63055	+	32.6302i	0.152998	+	1.07813i
$917$		0			0
$918$		0			0
$919$	−16.1040	−	33.9384i	−0.531222	−	1.11953i	−0.974757	−	0.223269i	$-0.928327\pi$
$919$	−16.1040	−	33.9384i	−0.531222	−	1.11953i	0.443535	−	0.896257i	$-0.353724\pi$
$920$		0			0
$921$	−0.489312	−	0.451402i	−0.0161234	−	0.0148742i
$922$		0			0
$923$		0			0
$924$		0			0
$925$	12.3760	−	46.1879i	0.406920	−	1.51865i
$926$		0			0
$927$	45.4400	−	37.1006i	1.49244	−	1.21854i
$928$		0			0
$929$		0			0		0.875918	−	0.482459i	$-0.160256\pi$
$929$		0			0		−0.875918	+	0.482459i	$0.839744\pi$
$930$		0			0
$931$	49.9551	+	39.1373i	1.63721	+	1.28267i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$	42.1414	+	22.1175i	1.37670	+	0.722547i	0.980666	−	0.195689i	$-0.0626944\pi$
$937$	42.1414	+	22.1175i	1.37670	+	0.722547i	0.396033	+	0.918236i	$0.370387\pi$
$938$		0			0
$939$	20.2435	+	21.0758i	0.660623	+	0.687781i
$940$		0			0
$941$		0			0		−0.855781	−	0.517338i	$-0.826923\pi$
$941$		0			0		0.855781	+	0.517338i	$0.173077\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$		0			0		0.975564	−	0.219715i	$-0.0705128\pi$
$947$		0			0		−0.975564	+	0.219715i	$0.929487\pi$
$948$	39.3534	+	4.77837i	1.27814	+	0.155194i
$949$	13.0515	+	57.7072i	0.423669	+	1.87325i
$950$		0			0
$951$		0			0
$952$		0			0
$953$		0			0		0.979791	−	0.200026i	$-0.0641026\pi$
$953$		0			0		−0.979791	+	0.200026i	$0.935897\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	50.4874	−	56.9885i	1.62863	−	1.83834i
$962$		0			0
$963$		0			0
$964$	10.1413	+	1.43915i	0.326629	+	0.0463519i
$965$		0			0
$966$		0			0
$967$	47.9969	+	14.9564i	1.54348	+	0.480966i	0.946883	−	0.321578i	$-0.104213\pi$
$967$	47.9969	+	14.9564i	1.54348	+	0.480966i	0.596592	+	0.802545i	$0.296521\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		0			0		−0.987050	−	0.160411i	$-0.948718\pi$
$971$		0			0		0.987050	+	0.160411i	$0.0512821\pi$
$972$	−16.6630	+	26.3504i	−0.534466	+	0.845190i
$973$	27.6779	−	68.7708i	0.887312	−	2.20469i
$974$		0			0
$975$	−1.22913	+	31.2008i	−0.0393638	+	0.999225i
$976$	−53.2452	−	13.1238i	−1.70434	−	0.420081i
$977$		0			0		−0.811378	−	0.584522i	$-0.801282\pi$
$977$		0			0		0.811378	+	0.584522i	$0.198718\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$	13.3774	+	33.2387i	0.427109	+	1.06123i
$982$		0			0
$983$		0			0		−0.410413	−	0.911900i	$-0.634615\pi$
$983$		0			0		0.410413	+	0.911900i	$0.365385\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$	−51.2107	+	1.07770i	−1.62923	+	0.0342862i
$989$		0			0
$990$		0			0
$991$	15.9721	+	27.6645i	0.507371	+	0.878793i	0.999964	+	0.00853283i	$0.00271612\pi$
$991$	15.9721	+	27.6645i	0.507371	+	0.878793i	−0.492592	+	0.870260i	$0.663951\pi$
$992$		0			0
$993$	−8.04147	−	0.486419i	−0.255188	−	0.0154360i
$994$		0			0
$995$		0			0
$996$		0			0
$997$	−0.124974	+	0.0417226i	−0.00395798	+	0.00132137i	−0.318646	−	0.947874i	$-0.603228\pi$
$997$	−0.124974	+	0.0417226i	−0.00395798	+	0.00132137i	0.314688	+	0.949195i	$0.398100\pi$
$998$		0			0
$999$	−27.3996	−	41.4568i	−0.866886	−	1.31163i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	507.2.x.a.11.1	✓	48
3.2	odd	2	CM	507.2.x.a.11.1	✓	48
169.123	odd	156	inner	507.2.x.a.461.1	yes	48
507.461	even	156	inner	507.2.x.a.461.1	yes	48

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
507.2.x.a.11.1	✓	48	1.1	even	1	trivial
507.2.x.a.11.1	✓	48	3.2	odd	2	CM
507.2.x.a.461.1	yes	48	169.123	odd	156	inner
507.2.x.a.461.1	yes	48	507.461	even	156	inner

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:	\( N \)	\(=\)	\( 507 = 3 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	507.x (of order \(156\), degree \(48\), minimal)

\(f(q)\)	\(=\)	\(q+(1.59345 - 0.678906i) q^{3} +(-1.06893 - 1.69038i) q^{4} +(1.78351 - 3.57116i) q^{7} +(2.07817 - 2.16361i) q^{9} +O(q^{10})\)	\(q+(1.59345 - 0.678906i) q^{3} +(-1.06893 - 1.69038i) q^{4} +(1.78351 - 3.57116i) q^{7} +(2.07817 - 2.16361i) q^{9} +(-2.85090 - 1.96783i) q^{12} +(-2.59808 + 2.50000i) q^{13} +(-1.71477 + 3.61380i) q^{16} +(-6.86118 + 1.83845i) q^{19} +(0.417452 - 6.90130i) q^{21} +(4.67508 - 1.77302i) q^{25} +(1.84258 - 4.85849i) q^{27} +(-7.94306 + 0.802517i) q^{28} +(9.43875 - 4.24804i) q^{31} +(-5.87874 - 1.20015i) q^{36} +(5.59004 - 7.75956i) q^{37} +(-2.44264 + 5.74748i) q^{39} +(-1.17487 + 7.22929i) q^{43} +(-0.278971 + 6.92258i) q^{48} +(-5.36707 - 7.14227i) q^{49} +(7.00312 + 1.71941i) q^{52} +(-9.68482 + 7.58757i) q^{57} +(2.74229 + 13.4326i) q^{61} +(-4.02014 - 11.2803i) q^{63} +(7.94167 - 0.964293i) q^{64} +(3.42113 + 15.1903i) q^{67} +(8.48917 - 14.0428i) q^{73} +(6.24580 - 5.99917i) q^{75} +(10.4418 + 9.63283i) q^{76} +(-10.1330 + 5.31819i) q^{79} +(-0.362393 - 8.99270i) q^{81} +(-12.1120 + 6.67136i) q^{84} +(4.29420 + 13.7369i) q^{91} +(12.1562 - 13.1771i) q^{93} +(-12.9550 - 11.0196i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q + 10 q^{7} + 6 q^{9}+O(q^{10}) \) 48 * q + 10 * q^7 + 6 * q^9	\( 48 q + 10 q^{7} + 6 q^{9} - 8 q^{16} - 14 q^{19} - 18 q^{21} + 20 q^{28} + 14 q^{31} + 2 q^{37} + 24 q^{39} + 6 q^{43} - 18 q^{49} - 28 q^{52} - 12 q^{57} - 24 q^{63} - 32 q^{67} + 34 q^{73} + 30 q^{75} + 28 q^{76} + 18 q^{81} + 12 q^{84} - 2 q^{91} - 6 q^{93} + 38 q^{97}+O(q^{100}) \) 48 * q + 10 * q^7 + 6 * q^9 - 8 * q^16 - 14 * q^19 - 18 * q^21 + 20 * q^28 + 14 * q^31 + 2 * q^37 + 24 * q^39 + 6 * q^43 - 18 * q^49 - 28 * q^52 - 12 * q^57 - 24 * q^63 - 32 * q^67 + 34 * q^73 + 30 * q^75 + 28 * q^76 + 18 * q^81 + 12 * q^84 - 2 * q^91 - 6 * q^93 + 38 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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