LMFDB - Embedded newform 1080.4.f.a.649.9

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1080,4,Mod(649,1080)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1080, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0, 1]))

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("1080.649");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1080 = 2^{3} \cdot 3^{3} \cdot 5 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1080.f (of order $2$, degree $1$, 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:	$63.7220628062$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 216 x^{14} + 19078 x^{12} - 888840 x^{10} + 23532927 x^{8} - 353885448 x^{6} + \cdots + 15738957025 $ x^16 - 216x^14 + 19078x^12 - 888840x^10 + 23532927x^8 - 353885448x^6 + 2835058534x^4 - 10430598936*x^2 + 15738957025
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{24}\cdot 3^{8} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			649.9
Root			$-5.58918 - 0.500000i$ of defining polynomial
Character	$\chi$	$=$	1080.649
Dual form			1080.4.f.a.649.10

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.319421 - 11.1758i) q^{5} -17.5714i q^{7} +O(q^{10})$	$q+(0.319421 - 11.1758i) q^{5} -17.5714i q^{7} -59.7658 q^{11} -35.3743i q^{13} +57.2613i q^{17} -51.0310 q^{19} +27.5818i q^{23} +(-124.796 - 7.13956i) q^{25} -76.2910 q^{29} -142.972 q^{31} +(-196.374 - 5.61266i) q^{35} +73.3502i q^{37} +464.564 q^{41} +212.944i q^{43} -166.720i q^{47} +34.2475 q^{49} +744.723i q^{53} +(-19.0904 + 667.929i) q^{55} -18.9197 q^{59} +83.2183 q^{61} +(-395.335 - 11.2993i) q^{65} -644.765i q^{67} +1030.82 q^{71} -1084.29i q^{73} +1050.17i q^{77} +512.603 q^{79} +69.4147i q^{83} +(639.939 + 18.2905i) q^{85} -864.817 q^{89} -621.574 q^{91} +(-16.3004 + 570.311i) q^{95} +686.142i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q+O(q^{10}) $ 16 * q	$ 16 q + 100 q^{19} - 186 q^{25} - 288 q^{31} - 2124 q^{49} + 1222 q^{55} + 188 q^{61} + 4964 q^{79} + 636 q^{85} - 4144 q^{91}+O(q^{100}) $ 16 * q + 100 * q^19 - 186 * q^25 - 288 * q^31 - 2124 * q^49 + 1222 * q^55 + 188 * q^61 + 4964 * q^79 + 636 * q^85 - 4144 * q^91

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1080\mathbb{Z}\right)^\times$.

$n$	$217$	$271$	$541$	$1001$
$\chi(n)$	$-1$	$1$	$1$	$1$

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
$3$		0			0
$3$		0			0
$4$		0			0
$5$	0.319421	−	11.1758i	0.0285699	−	0.999592i
$5$	0.319421	−	11.1758i	0.0285699	−	0.999592i
$6$		0			0
$7$		−	17.5714i		−	0.948764i	−0.880319	−	0.474382i	$-0.842672\pi$
$7$		−	17.5714i		−	0.948764i	0.880319	−	0.474382i	$-0.157328\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−59.7658			−1.63819			−0.819093	−	0.573660i	$-0.805523\pi$
$11$	−59.7658			−1.63819			−0.819093	+	0.573660i	$0.805523\pi$
$12$		0			0
$13$		−	35.3743i		−	0.754697i	−0.926071	−	0.377348i	$-0.876836\pi$
$13$		−	35.3743i		−	0.754697i	0.926071	−	0.377348i	$-0.123164\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$			57.2613i			0.816935i	0.912773	+	0.408468i	$0.133937\pi$
$17$			57.2613i			0.816935i	−0.912773	+	0.408468i	$0.866063\pi$
$18$		0			0
$19$	−51.0310			−0.616174			−0.308087	−	0.951358i	$-0.599689\pi$
$19$	−51.0310			−0.616174			−0.308087	+	0.951358i	$0.599689\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$			27.5818i			0.250053i	0.992153	+	0.125026i	$0.0399015\pi$
$23$			27.5818i			0.250053i	−0.992153	+	0.125026i	$0.960098\pi$
$24$		0			0
$25$	−124.796	−	7.13956i	−0.998368	−	0.0571165i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−76.2910			−0.488513			−0.244257	−	0.969711i	$-0.578544\pi$
$29$	−76.2910			−0.488513			−0.244257	+	0.969711i	$0.578544\pi$
$30$		0			0
$31$	−142.972			−0.828338			−0.414169	−	0.910200i	$-0.635928\pi$
$31$	−142.972			−0.828338			−0.414169	+	0.910200i	$0.635928\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−196.374	−	5.61266i	−0.948377	−	0.0271061i
$36$		0			0
$37$			73.3502i			0.325911i	0.986633	+	0.162955i	$0.0521026\pi$
$37$			73.3502i			0.325911i	−0.986633	+	0.162955i	$0.947897\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	464.564			1.76958			0.884788	−	0.465993i	$-0.154303\pi$
$41$	464.564			1.76958			0.884788	+	0.465993i	$0.154303\pi$
$42$		0			0
$43$			212.944i			0.755203i	0.925968	+	0.377601i	$0.123251\pi$
$43$			212.944i			0.755203i	−0.925968	+	0.377601i	$0.876749\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		−	166.720i		−	0.517418i	−0.965955	−	0.258709i	$-0.916703\pi$
$47$		−	166.720i		−	0.517418i	0.965955	−	0.258709i	$-0.0832970\pi$
$48$		0			0
$49$	34.2475			0.0998470
$50$		0			0
$51$		0			0
$52$		0			0
$53$			744.723i			1.93010i	0.262057	+	0.965052i	$0.415599\pi$
$53$			744.723i			1.93010i	−0.262057	+	0.965052i	$0.584401\pi$
$54$		0			0
$55$	−19.0904	+	667.929i	−0.0468028	+	1.63752i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−18.9197			−0.0417481			−0.0208741	−	0.999782i	$-0.506645\pi$
$59$	−18.9197			−0.0417481			−0.0208741	+	0.999782i	$0.506645\pi$
$60$		0			0
$61$	83.2183			0.174672			0.0873362	−	0.996179i	$-0.472165\pi$
$61$	83.2183			0.174672			0.0873362	+	0.996179i	$0.472165\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−395.335	−	11.2993i	−0.754389	−	0.0215616i
$66$		0			0
$67$		−	644.765i		−	1.17568i	−0.808977	−	0.587840i	$-0.799978\pi$
$67$		−	644.765i		−	1.17568i	0.808977	−	0.587840i	$-0.200022\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	1030.82			1.72304			0.861522	−	0.507719i	$-0.169511\pi$
$71$	1030.82			1.72304			0.861522	+	0.507719i	$0.169511\pi$
$72$		0			0
$73$		−	1084.29i		−	1.73845i	−0.494416	−	0.869225i	$-0.664618\pi$
$73$		−	1084.29i		−	1.73845i	0.494416	−	0.869225i	$-0.335382\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$			1050.17i			1.55425i
$78$		0			0
$79$	512.603			0.730030			0.365015	−	0.931002i	$-0.381064\pi$
$79$	512.603			0.730030			0.365015	+	0.931002i	$0.381064\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$			69.4147i			0.0917983i	0.998946	+	0.0458991i	$0.0146153\pi$
$83$			69.4147i			0.0917983i	−0.998946	+	0.0458991i	$0.985385\pi$
$84$		0			0
$85$	639.939	+	18.2905i	0.816602	+	0.0233397i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−864.817			−1.03000			−0.515002	−	0.857189i	$-0.672209\pi$
$89$	−864.817			−1.03000			−0.515002	+	0.857189i	$0.672209\pi$
$90$		0			0
$91$	−621.574			−0.716029
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−16.3004	+	570.311i	−0.0176040	+	0.615923i
$96$		0			0
$97$			686.142i			0.718218i	0.933296	+	0.359109i	$0.116919\pi$
$97$			686.142i			0.718218i	−0.933296	+	0.359109i	$0.883081\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	745.975			0.734924			0.367462	−	0.930039i	$-0.380227\pi$
$101$	745.975			0.734924			0.367462	+	0.930039i	$0.380227\pi$
$102$		0			0
$103$		−	774.575i		−	0.740982i	−0.928836	−	0.370491i	$-0.879189\pi$
$103$		−	774.575i		−	0.740982i	0.928836	−	0.370491i	$-0.120811\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$			1380.95i			1.24768i	0.781552	+	0.623840i	$0.214428\pi$
$107$			1380.95i			1.24768i	−0.781552	+	0.623840i	$0.785572\pi$
$108$		0			0
$109$	323.696			0.284444			0.142222	−	0.989835i	$-0.454575\pi$
$109$	323.696			0.284444			0.142222	+	0.989835i	$0.454575\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$			711.632i			0.592431i	0.955121	+	0.296215i	$0.0957246\pi$
$113$			711.632i			0.592431i	−0.955121	+	0.296215i	$0.904275\pi$
$114$		0			0
$115$	308.248	+	8.81021i	0.249950	+	0.00714397i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	1006.16			0.775079
$120$		0			0
$121$	2240.95			1.68366
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−119.653	+	1392.41i	−0.0856164	+	0.996328i
$126$		0			0
$127$			468.609i			0.327420i	0.986508	+	0.163710i	$0.0523462\pi$
$127$			468.609i			0.327420i	−0.986508	+	0.163710i	$0.947654\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−2579.11			−1.72014			−0.860069	−	0.510177i	$-0.829580\pi$
$131$	−2579.11			−1.72014			−0.860069	+	0.510177i	$0.829580\pi$
$132$		0			0
$133$			896.684i			0.584604i
$134$		0			0
$135$		0			0
$136$		0			0
$137$			112.007i			0.0698494i	0.999390	+	0.0349247i	$0.0111191\pi$
$137$			112.007i			0.0698494i	−0.999390	+	0.0349247i	$0.988881\pi$
$138$		0			0
$139$	−1455.71			−0.888283			−0.444142	−	0.895957i	$-0.646491\pi$
$139$	−1455.71			−0.888283			−0.444142	+	0.895957i	$0.646491\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$			2114.17i			1.23633i
$144$		0			0
$145$	−24.3690	+	852.611i	−0.0139568	+	0.488314i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−717.240			−0.394353			−0.197177	−	0.980368i	$-0.563177\pi$
$149$	−717.240			−0.394353			−0.197177	+	0.980368i	$0.563177\pi$
$150$		0			0
$151$	−1685.27			−0.908245			−0.454122	−	0.890939i	$-0.650047\pi$
$151$	−1685.27			−0.908245			−0.454122	+	0.890939i	$0.650047\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−45.6682	+	1597.82i	−0.0236655	+	0.828000i
$156$		0			0
$157$			2832.78i			1.44000i	0.693972	+	0.720002i	$0.255859\pi$
$157$			2832.78i			1.44000i	−0.693972	+	0.720002i	$0.744141\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	484.650			0.237241
$162$		0			0
$163$		−	1872.09i		−	0.899591i	−0.893132	−	0.449796i	$-0.851497\pi$
$163$		−	1872.09i		−	0.899591i	0.893132	−	0.449796i	$-0.148503\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$			2616.97i			1.21262i	0.795230	+	0.606308i	$0.207350\pi$
$167$			2616.97i			1.21262i	−0.795230	+	0.606308i	$0.792650\pi$
$168$		0			0
$169$	945.661			0.430433
$170$		0			0
$171$		0			0
$172$		0			0
$173$			1476.86i			0.649040i	0.945879	+	0.324520i	$0.105203\pi$
$173$			1476.86i			0.649040i	−0.945879	+	0.324520i	$0.894797\pi$
$174$		0			0
$175$	−125.452	+	2192.83i	−0.0541900	+	0.947215i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−109.556			−0.0457464			−0.0228732	−	0.999738i	$-0.507281\pi$
$179$	−109.556			−0.0457464			−0.0228732	+	0.999738i	$0.507281\pi$
$180$		0			0
$181$	−2782.02			−1.14246			−0.571231	−	0.820789i	$-0.693534\pi$
$181$	−2782.02			−1.14246			−0.571231	+	0.820789i	$0.693534\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	819.745	+	23.4296i	0.325778	+	0.00931123i
$186$		0			0
$187$		−	3422.26i		−	1.33829i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−2454.38			−0.929806			−0.464903	−	0.885362i	$-0.653911\pi$
$191$	−2454.38			−0.929806			−0.464903	+	0.885362i	$0.653911\pi$
$192$		0			0
$193$		−	3119.11i		−	1.16331i	−0.813436	−	0.581655i	$-0.802405\pi$
$193$		−	3119.11i		−	1.16331i	0.813436	−	0.581655i	$-0.197595\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$			3096.46i			1.11987i	0.828538	+	0.559933i	$0.189173\pi$
$197$			3096.46i			1.11987i	−0.828538	+	0.559933i	$0.810827\pi$
$198$		0			0
$199$	−4167.11			−1.48442			−0.742208	−	0.670169i	$-0.766222\pi$
$199$	−4167.11			−1.48442			−0.742208	+	0.670169i	$0.766222\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$			1340.54i			0.463484i
$204$		0			0
$205$	148.391	−	5191.86i	0.0505566	−	1.76885i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	3049.91			1.00941
$210$		0			0
$211$	−2610.21			−0.851631			−0.425815	−	0.904810i	$-0.640013\pi$
$211$	−2610.21			−0.851631			−0.425815	+	0.904810i	$0.640013\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	2379.82	+	68.0189i	0.754895	+	0.0215761i
$216$		0			0
$217$			2512.21i			0.785897i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	2025.58			0.616538
$222$		0			0
$223$			3000.95i			0.901159i	0.892736	+	0.450579i	$0.148783\pi$
$223$			3000.95i			0.901159i	−0.892736	+	0.450579i	$0.851217\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$			3593.44i			1.05068i	0.850891	+	0.525342i	$0.176062\pi$
$227$			3593.44i			1.05068i	−0.850891	+	0.525342i	$0.823938\pi$
$228$		0			0
$229$	6569.02			1.89560			0.947801	−	0.318862i	$-0.103301\pi$
$229$	6569.02			1.89560			0.947801	+	0.318862i	$0.103301\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		−	2158.00i		−	0.606761i	−0.952869	−	0.303381i	$-0.901885\pi$
$233$		−	2158.00i		−	0.606761i	0.952869	−	0.303381i	$-0.0981153\pi$
$234$		0			0
$235$	−1863.23	−	53.2539i	−0.517207	−	0.0147826i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	4228.53			1.14444			0.572219	−	0.820101i	$-0.306083\pi$
$239$	4228.53			1.14444			0.572219	+	0.820101i	$0.306083\pi$
$240$		0			0
$241$	847.470			0.226516			0.113258	−	0.993566i	$-0.463871\pi$
$241$	847.470			0.226516			0.113258	+	0.993566i	$0.463871\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	10.9394	−	382.743i	0.00285262	−	0.0998062i
$246$		0			0
$247$			1805.18i			0.465025i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−1478.79			−0.371873			−0.185937	−	0.982562i	$-0.559532\pi$
$251$	−1478.79			−0.371873			−0.185937	+	0.982562i	$0.559532\pi$
$252$		0			0
$253$		−	1648.45i		−	0.409633i
$254$		0			0
$255$		0			0
$256$		0			0
$257$		−	771.166i		−	0.187175i	−0.995611	−	0.0935876i	$-0.970166\pi$
$257$		−	771.166i		−	0.187175i	0.995611	−	0.0935876i	$-0.0298335\pi$
$258$		0			0
$259$	1288.86			0.309212
$260$		0			0
$261$		0			0
$262$		0			0
$263$			7372.46i			1.72854i	0.503031	+	0.864268i	$0.332218\pi$
$263$			7372.46i			1.72854i	−0.503031	+	0.864268i	$0.667782\pi$
$264$		0			0
$265$	8322.86	+	237.880i	1.92932	+	0.0551429i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−4228.47			−0.958417			−0.479209	−	0.877701i	$-0.659076\pi$
$269$	−4228.47			−0.958417			−0.479209	+	0.877701i	$0.659076\pi$
$270$		0			0
$271$	−7450.12			−1.66997			−0.834986	−	0.550271i	$-0.814524\pi$
$271$	−7450.12			−1.66997			−0.834986	+	0.550271i	$0.814524\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	7458.52	+	426.701i	1.63551	+	0.0935674i
$276$		0			0
$277$		−	2672.71i		−	0.579738i	−0.957066	−	0.289869i	$-0.906388\pi$
$277$		−	2672.71i		−	0.579738i	0.957066	−	0.289869i	$-0.0936117\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−8260.59			−1.75369			−0.876843	−	0.480778i	$-0.840354\pi$
$281$	−8260.59			−1.75369			−0.876843	+	0.480778i	$0.840354\pi$
$282$		0			0
$283$			7322.40i			1.53806i	0.639211	+	0.769031i	$0.279261\pi$
$283$			7322.40i			1.53806i	−0.639211	+	0.769031i	$0.720739\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		−	8163.01i		−	1.67891i
$288$		0			0
$289$	1634.15			0.332617
$290$		0			0
$291$		0			0
$292$		0			0
$293$			3552.78i			0.708382i	0.935173	+	0.354191i	$0.115244\pi$
$293$			3552.78i			0.708382i	−0.935173	+	0.354191i	$0.884756\pi$
$294$		0			0
$295$	−6.04336	+	211.443i	−0.00119274	+	0.0417311i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	975.687			0.188714
$300$		0			0
$301$	3741.72			0.716509
$302$		0			0
$303$		0			0
$304$		0			0
$305$	26.5817	−	930.029i	0.00499037	−	0.174601i
$306$		0			0
$307$			1703.91i			0.316766i	0.987378	+	0.158383i	$0.0506281\pi$
$307$			1703.91i			0.316766i	−0.987378	+	0.158383i	$0.949372\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−7353.82			−1.34083			−0.670413	−	0.741988i	$-0.733883\pi$
$311$	−7353.82			−1.34083			−0.670413	+	0.741988i	$0.733883\pi$
$312$		0			0
$313$			8250.26i			1.48988i	0.667132	+	0.744940i	$0.267522\pi$
$313$			8250.26i			1.48988i	−0.667132	+	0.744940i	$0.732478\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$			2708.83i			0.479946i	0.970780	+	0.239973i	$0.0771387\pi$
$317$			2708.83i			0.479946i	−0.970780	+	0.239973i	$0.922861\pi$
$318$		0			0
$319$	4559.59			0.800276
$320$		0			0
$321$		0			0
$322$		0			0
$323$		−	2922.10i		−	0.503375i
$324$		0			0
$325$	−252.557	+	4414.57i	−0.0431056	+	0.753465i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−2929.50			−0.490907
$330$		0			0
$331$	−1188.08			−0.197289			−0.0986446	−	0.995123i	$-0.531451\pi$
$331$	−1188.08			−0.197289			−0.0986446	+	0.995123i	$0.531451\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−7205.75	−	205.952i	−1.17520	−	0.0335891i
$336$		0			0
$337$		−	10808.0i		−	1.74703i	−0.486797	−	0.873515i	$-0.661835\pi$
$337$		−	10808.0i		−	1.74703i	0.486797	−	0.873515i	$-0.338165\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	8544.81			1.35697
$342$		0			0
$343$		−	6628.75i		−	1.04350i
$344$		0			0
$345$		0			0
$346$		0			0
$347$		−	6496.98i		−	1.00512i	−0.864543	−	0.502559i	$-0.832392\pi$
$347$		−	6496.98i		−	1.00512i	0.864543	−	0.502559i	$-0.167608\pi$
$348$		0			0
$349$	5386.62			0.826187			0.413094	−	0.910689i	$-0.364448\pi$
$349$	5386.62			0.826187			0.413094	+	0.910689i	$0.364448\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$		−	4940.57i		−	0.744929i	−0.928046	−	0.372465i	$-0.878513\pi$
$353$		−	4940.57i		−	0.744929i	0.928046	−	0.372465i	$-0.121487\pi$
$354$		0			0
$355$	329.267	−	11520.2i	0.0492272	−	1.72234i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−9085.56			−1.33570			−0.667851	−	0.744295i	$-0.732786\pi$
$359$	−9085.56			−1.33570			−0.667851	+	0.744295i	$0.732786\pi$
$360$		0			0
$361$	−4254.84			−0.620329
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−12117.8	−	346.346i	−1.73774	−	0.0496673i
$366$		0			0
$367$			391.218i			0.0556441i	0.999613	+	0.0278221i	$0.00885718\pi$
$367$			391.218i			0.0556441i	−0.999613	+	0.0278221i	$0.991143\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	13085.8			1.83121
$372$		0			0
$373$			2453.60i			0.340596i	0.985393	+	0.170298i	$0.0544731\pi$
$373$			2453.60i			0.340596i	−0.985393	+	0.170298i	$0.945527\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$			2698.74i			0.368679i
$378$		0			0
$379$	−13981.3			−1.89491			−0.947453	−	0.319896i	$-0.896352\pi$
$379$	−13981.3			−1.89491			−0.947453	+	0.319896i	$0.896352\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		−	13334.3i		−	1.77898i	−0.456953	−	0.889491i	$-0.651059\pi$
$383$		−	13334.3i		−	1.77898i	0.456953	−	0.889491i	$-0.348941\pi$
$384$		0			0
$385$	11736.4	+	335.445i	1.55362	+	0.0444048i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	10992.2			1.43272			0.716360	−	0.697731i	$-0.245807\pi$
$389$	10992.2			1.43272			0.716360	+	0.697731i	$0.245807\pi$
$390$		0			0
$391$	−1579.37			−0.204277
$392$		0			0
$393$		0			0
$394$		0			0
$395$	163.736	−	5728.74i	0.0208569	−	0.729732i
$396$		0			0
$397$		−	9393.78i		−	1.18756i	−0.804628	−	0.593779i	$-0.797636\pi$
$397$		−	9393.78i		−	1.18756i	0.804628	−	0.593779i	$-0.202364\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	13195.9			1.64332			0.821659	−	0.569979i	$-0.193049\pi$
$401$	13195.9			1.64332			0.821659	+	0.569979i	$0.193049\pi$
$402$		0			0
$403$			5057.52i			0.625144i
$404$		0			0
$405$		0			0
$406$		0			0
$407$		−	4383.83i		−	0.533903i
$408$		0			0
$409$	12366.3			1.49505			0.747526	−	0.664233i	$-0.231242\pi$
$409$	12366.3			1.49505			0.747526	+	0.664233i	$0.231242\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$			332.445i			0.0396091i
$414$		0			0
$415$	775.764	+	22.1725i	0.0917608	+	0.00262267i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−8531.20			−0.994693			−0.497347	−	0.867552i	$-0.665692\pi$
$419$	−8531.20			−0.994693			−0.497347	+	0.867552i	$0.665692\pi$
$420$		0			0
$421$	−5545.75			−0.642003			−0.321001	−	0.947079i	$-0.604019\pi$
$421$	−5545.75			−0.642003			−0.321001	+	0.947079i	$0.604019\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	408.820	−	7145.97i	0.0466604	−	0.815602i
$426$		0			0
$427$		−	1462.26i		−	0.165723i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	3182.36			0.355659			0.177829	−	0.984061i	$-0.443092\pi$
$431$	3182.36			0.355659			0.177829	+	0.984061i	$0.443092\pi$
$432$		0			0
$433$			10826.4i			1.20158i	0.799406	+	0.600791i	$0.205148\pi$
$433$			10826.4i			1.20158i	−0.799406	+	0.600791i	$0.794852\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		−	1407.53i		−	0.154076i
$438$		0			0
$439$	−10130.0			−1.10132			−0.550659	−	0.834730i	$-0.685624\pi$
$439$	−10130.0			−1.10132			−0.550659	+	0.834730i	$0.685624\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$		−	6249.98i		−	0.670305i	−0.942164	−	0.335153i	$-0.891212\pi$
$443$		−	6249.98i		−	0.670305i	0.942164	−	0.335153i	$-0.108788\pi$
$444$		0			0
$445$	−276.241	+	9665.00i	−0.0294271	+	1.02958i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	5941.11			0.624451			0.312225	−	0.950008i	$-0.398926\pi$
$449$	5941.11			0.624451			0.312225	+	0.950008i	$0.398926\pi$
$450$		0			0
$451$	−27765.0			−2.89890
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−198.544	+	6946.57i	−0.0204569	+	0.715737i
$456$		0			0
$457$			4865.48i			0.498026i	0.968500	+	0.249013i	$0.0801061\pi$
$457$			4865.48i			0.498026i	−0.968500	+	0.249013i	$0.919894\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−4460.28			−0.450621			−0.225310	−	0.974287i	$-0.572340\pi$
$461$	−4460.28			−0.450621			−0.225310	+	0.974287i	$0.572340\pi$
$462$		0			0
$463$		−	10024.2i		−	1.00619i	−0.864231	−	0.503095i	$-0.832195\pi$
$463$		−	10024.2i		−	1.00619i	0.864231	−	0.503095i	$-0.167805\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		−	6812.49i		−	0.675041i	−0.941318	−	0.337521i	$-0.890412\pi$
$467$		−	6812.49i		−	0.675041i	0.941318	−	0.337521i	$-0.109588\pi$
$468$		0			0
$469$	−11329.4			−1.11544
$470$		0			0
$471$		0			0
$472$		0			0
$473$		−	12726.8i		−	1.23716i
$474$		0			0
$475$	6368.46	+	364.339i	0.615169	+	0.0351937i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	5647.51			0.538709			0.269354	−	0.963041i	$-0.413190\pi$
$479$	5647.51			0.538709			0.269354	+	0.963041i	$0.413190\pi$
$480$		0			0
$481$	2594.71			0.245964
$482$		0			0
$483$		0			0
$484$		0			0
$485$	7668.17	+	219.168i	0.717925	+	0.0205194i
$486$		0			0
$487$			11282.6i			1.04982i	0.851157	+	0.524911i	$0.175901\pi$
$487$			11282.6i			1.04982i	−0.851157	+	0.524911i	$0.824099\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	5131.48			0.471650			0.235825	−	0.971795i	$-0.424221\pi$
$491$	5131.48			0.471650			0.235825	+	0.971795i	$0.424221\pi$
$492$		0			0
$493$		−	4368.52i		−	0.399084i
$494$		0			0
$495$		0			0
$496$		0			0
$497$		−	18113.0i		−	1.63476i
$498$		0			0
$499$	18920.1			1.69735			0.848677	−	0.528911i	$-0.177399\pi$
$499$	18920.1			1.69735			0.848677	+	0.528911i	$0.177399\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		−	8669.16i		−	0.768467i	−0.923236	−	0.384233i	$-0.874466\pi$
$503$		−	8669.16i		−	0.768467i	0.923236	−	0.384233i	$-0.125534\pi$
$504$		0			0
$505$	238.280	−	8336.85i	0.0209967	−	0.734624i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−11310.8			−0.984953			−0.492476	−	0.870326i	$-0.663908\pi$
$509$	−11310.8			−0.984953			−0.492476	+	0.870326i	$0.663908\pi$
$510$		0			0
$511$	−19052.5			−1.64938
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−8656.48	−	247.416i	−0.740680	−	0.0211698i
$516$		0			0
$517$			9964.16i			0.847627i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−13621.9			−1.14546			−0.572732	−	0.819743i	$-0.694116\pi$
$521$	−13621.9			−1.14546			−0.572732	+	0.819743i	$0.694116\pi$
$522$		0			0
$523$			17394.3i			1.45430i	0.686479	+	0.727149i	$0.259155\pi$
$523$			17394.3i			1.45430i	−0.686479	+	0.727149i	$0.740845\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		−	8186.74i		−	0.676698i
$528$		0			0
$529$	11406.2			0.937474
$530$		0			0
$531$		0			0
$532$		0			0
$533$		−	16433.6i		−	1.33549i
$534$		0			0
$535$	15433.2	+	441.106i	1.24717	+	0.0356461i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−2046.83			−0.163568
$540$		0			0
$541$	−20329.2			−1.61556			−0.807781	−	0.589483i	$-0.799331\pi$
$541$	−20329.2			−1.61556			−0.807781	+	0.589483i	$0.799331\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	103.395	−	3617.55i	0.00812655	−	0.284328i
$546$		0			0
$547$			8478.87i			0.662761i	0.943497	+	0.331381i	$0.107514\pi$
$547$			8478.87i			0.662761i	−0.943497	+	0.331381i	$0.892486\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	3893.21			0.301009
$552$		0			0
$553$		−	9007.14i		−	0.692626i
$554$		0			0
$555$		0			0
$556$		0			0
$557$		−	92.4691i		−	0.00703419i	−0.999994	−	0.00351709i	$-0.998880\pi$
$557$		−	92.4691i		−	0.00703419i	0.999994	−	0.00351709i	$-0.00111953\pi$
$558$		0			0
$559$	7532.76			0.569949
$560$		0			0
$561$		0			0
$562$		0			0
$563$			1527.66i			0.114357i	0.998364	+	0.0571787i	$0.0182105\pi$
$563$			1527.66i			0.114357i	−0.998364	+	0.0571787i	$0.981790\pi$
$564$		0			0
$565$	7953.04	+	227.310i	0.592189	+	0.0169257i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−14372.3			−1.05891			−0.529453	−	0.848339i	$-0.677603\pi$
$569$	−14372.3			−1.05891			−0.529453	+	0.848339i	$0.677603\pi$
$570$		0			0
$571$	2454.57			0.179896			0.0899481	−	0.995946i	$-0.471330\pi$
$571$	2454.57			0.179896			0.0899481	+	0.995946i	$0.471330\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	196.922	−	3442.10i	0.0142821	−	0.249644i
$576$		0			0
$577$		−	15197.5i		−	1.09650i	−0.836314	−	0.548251i	$-0.815294\pi$
$577$		−	15197.5i		−	1.09650i	0.836314	−	0.548251i	$-0.184706\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	1219.71			0.0870949
$582$		0			0
$583$		−	44508.9i		−	3.16187i
$584$		0			0
$585$		0			0
$586$		0			0
$587$			623.123i			0.0438144i	0.999760	+	0.0219072i	$0.00697384\pi$
$587$			623.123i			0.0438144i	−0.999760	+	0.0219072i	$0.993026\pi$
$588$		0			0
$589$	7295.99			0.510401
$590$		0			0
$591$		0			0
$592$		0			0
$593$			13435.7i			0.930417i	0.885201	+	0.465208i	$0.154021\pi$
$593$			13435.7i			0.930417i	−0.885201	+	0.465208i	$0.845979\pi$
$594$		0			0
$595$	321.388	−	11244.6i	0.0221439	−	0.774762i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	19129.4			1.30485			0.652426	−	0.757853i	$-0.273751\pi$
$599$	19129.4			1.30485			0.652426	+	0.757853i	$0.273751\pi$
$600$		0			0
$601$	12887.8			0.874715			0.437357	−	0.899288i	$-0.355914\pi$
$601$	12887.8			0.874715			0.437357	+	0.899288i	$0.355914\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	715.806	−	25044.3i	0.0481019	−	1.68297i
$606$		0			0
$607$		−	3815.46i		−	0.255131i	−0.991830	−	0.127566i	$-0.959284\pi$
$607$		−	3815.46i		−	0.255131i	0.991830	−	0.127566i	$-0.0407164\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−5897.61			−0.390494
$612$		0			0
$613$			2074.38i			0.136678i	0.997662	+	0.0683389i	$0.0217699\pi$
$613$			2074.38i			0.136678i	−0.997662	+	0.0683389i	$0.978230\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$			17539.8i			1.14445i	0.820097	+	0.572224i	$0.193919\pi$
$617$			17539.8i			1.14445i	−0.820097	+	0.572224i	$0.806081\pi$
$618$		0			0
$619$	13847.5			0.899155			0.449577	−	0.893241i	$-0.351575\pi$
$619$	13847.5			0.899155			0.449577	+	0.893241i	$0.351575\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$			15196.0i			0.977231i
$624$		0			0
$625$	15523.1	+	1781.98i	0.993475	+	0.114046i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−4200.12			−0.266248
$630$		0			0
$631$	−17212.4			−1.08592			−0.542958	−	0.839760i	$-0.682696\pi$
$631$	−17212.4			−1.08592			−0.542958	+	0.839760i	$0.682696\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	5237.07	+	149.684i	0.327287	+	0.00935436i
$636$		0			0
$637$		−	1211.48i		−	0.0753542i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−11443.5			−0.705134			−0.352567	−	0.935787i	$-0.614691\pi$
$641$	−11443.5			−0.705134			−0.352567	+	0.935787i	$0.614691\pi$
$642$		0			0
$643$			24782.2i			1.51993i	0.649964	+	0.759965i	$0.274784\pi$
$643$			24782.2i			1.51993i	−0.649964	+	0.759965i	$0.725216\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		−	11330.4i		−	0.688478i	−0.938882	−	0.344239i	$-0.888137\pi$
$647$		−	11330.4i		−	0.688478i	0.938882	−	0.344239i	$-0.111863\pi$
$648$		0			0
$649$	1130.75			0.0683912
$650$		0			0
$651$		0			0
$652$		0			0
$653$		−	16116.3i		−	0.965818i	−0.875671	−	0.482909i	$-0.839580\pi$
$653$		−	16116.3i		−	0.965818i	0.875671	−	0.482909i	$-0.160420\pi$
$654$		0			0
$655$	−823.823	+	28823.6i	−0.0491442	+	1.71944i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−15790.1			−0.933374			−0.466687	−	0.884423i	$-0.654553\pi$
$659$	−15790.1			−0.933374			−0.466687	+	0.884423i	$0.654553\pi$
$660$		0			0
$661$	−28056.4			−1.65093			−0.825466	−	0.564452i	$-0.809088\pi$
$661$	−28056.4			−1.65093			−0.825466	+	0.564452i	$0.809088\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	10021.1	+	286.420i	0.584365	+	0.0167021i
$666$		0			0
$667$		−	2104.24i		−	0.122154i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−4973.61			−0.286146
$672$		0			0
$673$		−	19234.1i		−	1.10166i	−0.834617	−	0.550831i	$-0.814311\pi$
$673$		−	19234.1i		−	1.10166i	0.834617	−	0.550831i	$-0.185689\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$		−	23286.9i		−	1.32199i	−0.750390	−	0.660995i	$-0.770135\pi$
$677$		−	23286.9i		−	1.32199i	0.750390	−	0.660995i	$-0.229865\pi$
$678$		0			0
$679$	12056.4			0.681420
$680$		0			0
$681$		0			0
$682$		0			0
$683$			1061.98i			0.0594957i	0.999557	+	0.0297478i	$0.00947043\pi$
$683$			1061.98i			0.0594957i	−0.999557	+	0.0297478i	$0.990530\pi$
$684$		0			0
$685$	1251.76	+	35.7773i	0.0698209	+	0.00199559i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	26344.0			1.45664
$690$		0			0
$691$	1387.05			0.0763614			0.0381807	−	0.999271i	$-0.487844\pi$
$691$	1387.05			0.0763614			0.0381807	+	0.999271i	$0.487844\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−464.983	+	16268.6i	−0.0253782	+	0.887921i
$696$		0			0
$697$			26601.5i			1.44563i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−16082.2			−0.866499			−0.433250	−	0.901274i	$-0.642633\pi$
$701$	−16082.2			−0.866499			−0.433250	+	0.901274i	$0.642633\pi$
$702$		0			0
$703$		−	3743.13i		−	0.200818i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		−	13107.8i		−	0.697269i
$708$		0			0
$709$	5094.37			0.269849			0.134925	−	0.990856i	$-0.456921\pi$
$709$	5094.37			0.269849			0.134925	+	0.990856i	$0.456921\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$		−	3943.42i		−	0.207128i
$714$		0			0
$715$	23627.5	+	675.311i	1.23583	+	0.0353219i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	15625.6			0.810482			0.405241	−	0.914210i	$-0.367188\pi$
$719$	15625.6			0.810482			0.405241	+	0.914210i	$0.367188\pi$
$720$		0			0
$721$	−13610.3			−0.703017
$722$		0			0
$723$		0			0
$724$		0			0
$725$	9520.81	+	544.684i	0.487716	+	0.0279021i
$726$		0			0
$727$			16821.1i			0.858129i	0.903274	+	0.429065i	$0.141157\pi$
$727$			16821.1i			0.858129i	−0.903274	+	0.429065i	$0.858843\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−12193.5			−0.616952
$732$		0			0
$733$		−	17831.0i		−	0.898504i	−0.893405	−	0.449252i	$-0.851691\pi$
$733$		−	17831.0i		−	0.898504i	0.893405	−	0.449252i	$-0.148309\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$			38534.9i			1.92598i
$738$		0			0
$739$	−5976.93			−0.297517			−0.148758	−	0.988874i	$-0.547528\pi$
$739$	−5976.93			−0.297517			−0.148758	+	0.988874i	$0.547528\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$			9474.06i			0.467792i	0.972262	+	0.233896i	$0.0751476\pi$
$743$			9474.06i			0.467792i	−0.972262	+	0.233896i	$0.924852\pi$
$744$		0			0
$745$	−229.102	+	8015.72i	−0.0112666	+	0.394192i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	24265.2			1.18375
$750$		0			0
$751$	11803.3			0.573514			0.286757	−	0.958003i	$-0.407423\pi$
$751$	11803.3			0.573514			0.286757	+	0.958003i	$0.407423\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−538.309	+	18834.2i	−0.0259485	+	0.907874i
$756$		0			0
$757$			21612.0i			1.03765i	0.854880	+	0.518826i	$0.173631\pi$
$757$			21612.0i			1.03765i	−0.854880	+	0.518826i	$0.826369\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−14784.0			−0.704229			−0.352114	−	0.935957i	$-0.614537\pi$
$761$	−14784.0			−0.704229			−0.352114	+	0.935957i	$0.614537\pi$
$762$		0			0
$763$		−	5687.78i		−	0.269871i
$764$		0			0
$765$		0			0
$766$		0			0
$767$			669.272i			0.0315072i
$768$		0			0
$769$	6574.29			0.308290			0.154145	−	0.988048i	$-0.450738\pi$
$769$	6574.29			0.308290			0.154145	+	0.988048i	$0.450738\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$			7714.13i			0.358936i	0.983764	+	0.179468i	$0.0574377\pi$
$773$			7714.13i			0.358936i	−0.983764	+	0.179468i	$0.942562\pi$
$774$		0			0
$775$	17842.3	+	1020.75i	0.826986	+	0.0473117i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−23707.1			−1.09037
$780$		0			0
$781$	−61607.9			−2.82267
$782$		0			0
$783$		0			0
$784$		0			0
$785$	31658.5	+	904.850i	1.43942	+	0.0411408i
$786$		0			0
$787$		−	10561.6i		−	0.478376i	−0.970973	−	0.239188i	$-0.923119\pi$
$787$		−	10561.6i		−	0.478376i	0.970973	−	0.239188i	$-0.0768812\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	12504.3			0.562077
$792$		0			0
$793$		−	2943.79i		−	0.131825i
$794$		0			0
$795$		0			0
$796$		0			0
$797$			26719.8i			1.18753i	0.804637	+	0.593767i	$0.202360\pi$
$797$			26719.8i			1.18753i	−0.804637	+	0.593767i	$0.797640\pi$
$798$		0			0
$799$	9546.61			0.422697
$800$		0			0
$801$		0			0
$802$		0			0
$803$			64803.6i			2.84791i
$804$		0			0
$805$	154.807	−	5416.34i	0.00677794	−	0.237144i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	9918.80			0.431058			0.215529	−	0.976497i	$-0.430852\pi$
$809$	9918.80			0.431058			0.215529	+	0.976497i	$0.430852\pi$
$810$		0			0
$811$	−29650.5			−1.28381			−0.641905	−	0.766784i	$-0.721856\pi$
$811$	−29650.5			−1.28381			−0.641905	+	0.766784i	$0.721856\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−20922.1	−	597.985i	−0.899224	−	0.0257012i
$816$		0			0
$817$		−	10866.8i		−	0.465337i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	9238.87			0.392739			0.196370	−	0.980530i	$-0.437085\pi$
$821$	9238.87			0.392739			0.196370	+	0.980530i	$0.437085\pi$
$822$		0			0
$823$			16961.4i			0.718391i	0.933262	+	0.359196i	$0.116949\pi$
$823$			16961.4i			0.718391i	−0.933262	+	0.359196i	$0.883051\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		−	38281.8i		−	1.60966i	−0.593506	−	0.804830i	$-0.702257\pi$
$827$		−	38281.8i		−	1.60966i	0.593506	−	0.804830i	$-0.297743\pi$
$828$		0			0
$829$	−34050.0			−1.42654			−0.713271	−	0.700888i	$-0.752787\pi$
$829$	−34050.0			−1.42654			−0.713271	+	0.700888i	$0.752787\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$			1961.06i			0.0815685i
$834$		0			0
$835$	29246.6	+	835.915i	1.21212	+	0.0346443i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	19298.5			0.794108			0.397054	−	0.917795i	$-0.370033\pi$
$839$	19298.5			0.794108			0.397054	+	0.917795i	$0.370033\pi$
$840$		0			0
$841$	−18568.7			−0.761355
$842$		0			0
$843$		0			0
$844$		0			0
$845$	302.064	−	10568.5i	0.0122974	−	0.430257i
$846$		0			0
$847$		−	39376.5i		−	1.59739i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−2023.13			−0.0814948
$852$		0			0
$853$			19978.0i			0.801916i	0.916096	+	0.400958i	$0.131323\pi$
$853$			19978.0i			0.801916i	−0.916096	+	0.400958i	$0.868677\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$			35885.5i			1.43037i	0.698936	+	0.715184i	$0.253657\pi$
$857$			35885.5i			1.43037i	−0.698936	+	0.715184i	$0.746343\pi$
$858$		0			0
$859$	−9370.87			−0.372212			−0.186106	−	0.982530i	$-0.559587\pi$
$859$	−9370.87			−0.372212			−0.186106	+	0.982530i	$0.559587\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		−	29899.3i		−	1.17936i	−0.807638	−	0.589679i	$-0.799254\pi$
$863$		−	29899.3i		−	1.17936i	0.807638	−	0.589679i	$-0.200746\pi$
$864$		0			0
$865$	16505.1	+	471.741i	0.648775	+	0.0185430i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−30636.1			−1.19593
$870$		0			0
$871$	−22808.1			−0.887283
$872$		0			0
$873$		0			0
$874$		0			0
$875$	24466.5	+	2102.46i	0.945280	+	0.0812297i
$876$		0			0
$877$		−	40439.0i		−	1.55705i	−0.627616	−	0.778523i	$-0.715969\pi$
$877$		−	40439.0i		−	1.55705i	0.627616	−	0.778523i	$-0.284031\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−14337.8			−0.548302			−0.274151	−	0.961687i	$-0.588397\pi$
$881$	−14337.8			−0.548302			−0.274151	+	0.961687i	$0.588397\pi$
$882$		0			0
$883$		−	12611.1i		−	0.480632i	−0.970695	−	0.240316i	$-0.922749\pi$
$883$		−	12611.1i		−	0.480632i	0.970695	−	0.240316i	$-0.0772511\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$			32322.1i			1.22353i	0.791040	+	0.611765i	$0.209540\pi$
$887$			32322.1i			1.22353i	−0.791040	+	0.611765i	$0.790460\pi$
$888$		0			0
$889$	8234.10			0.310645
$890$		0			0
$891$		0			0
$892$		0			0
$893$			8507.90i			0.318820i
$894$		0			0
$895$	−34.9945	+	1224.37i	−0.00130697	+	0.0457277i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	10907.5			0.404654
$900$		0			0
$901$	−42643.8			−1.57677
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−888.635	+	31091.2i	−0.0326400	+	1.14200i
$906$		0			0
$907$		−	35377.9i		−	1.29515i	−0.762000	−	0.647576i	$-0.775783\pi$
$907$		−	35377.9i		−	1.29515i	0.762000	−	0.647576i	$-0.224217\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	10102.4			0.367408			0.183704	−	0.982982i	$-0.441191\pi$
$911$	10102.4			0.367408			0.183704	+	0.982982i	$0.441191\pi$
$912$		0			0
$913$		−	4148.63i		−	0.150383i
$914$		0			0
$915$		0			0
$916$		0			0
$917$			45318.5i			1.63201i
$918$		0			0
$919$	44307.4			1.59039			0.795194	−	0.606356i	$-0.207369\pi$
$919$	44307.4			1.59039			0.795194	+	0.606356i	$0.207369\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		−	36464.6i		−	1.30038i
$924$		0			0
$925$	523.688	−	9153.80i	0.0186149	−	0.325379i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−51178.0			−1.80742			−0.903711	−	0.428144i	$-0.859168\pi$
$929$	−51178.0			−1.80742			−0.903711	+	0.428144i	$0.859168\pi$
$930$		0			0
$931$	−1747.68			−0.0615232
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−38246.5	−	1093.14i	−1.33775	−	0.0382349i
$936$		0			0
$937$			47970.2i			1.67248i	0.548361	+	0.836242i	$0.315252\pi$
$937$			47970.2i			1.67248i	−0.548361	+	0.836242i	$0.684748\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	9535.97			0.330355			0.165177	−	0.986264i	$-0.447180\pi$
$941$	9535.97			0.330355			0.165177	+	0.986264i	$0.447180\pi$
$942$		0			0
$943$			12813.5i			0.442487i
$944$		0			0
$945$		0			0
$946$		0			0
$947$			30046.3i			1.03102i	0.856885	+	0.515508i	$0.172397\pi$
$947$			30046.3i			1.03102i	−0.856885	+	0.515508i	$0.827603\pi$
$948$		0			0
$949$	−38356.1			−1.31200
$950$		0			0
$951$		0			0
$952$		0			0
$953$			35291.1i			1.19957i	0.800160	+	0.599786i	$0.204748\pi$
$953$			35291.1i			1.19957i	−0.800160	+	0.599786i	$0.795252\pi$
$954$		0			0
$955$	−783.981	+	27429.6i	−0.0265644	+	0.929426i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	1968.11			0.0662706
$960$		0			0
$961$	−9350.09			−0.313856
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−34858.5	−	996.311i	−1.16283	−	0.0332356i
$966$		0			0
$967$		−	8977.12i		−	0.298537i	−0.988797	−	0.149268i	$-0.952308\pi$
$967$		−	8977.12i		−	0.298537i	0.988797	−	0.149268i	$-0.0476918\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	35752.7			1.18163			0.590813	−	0.806809i	$-0.298807\pi$
$971$	35752.7			1.18163			0.590813	+	0.806809i	$0.298807\pi$
$972$		0			0
$973$			25578.7i			0.842771i
$974$		0			0
$975$		0			0
$976$		0			0
$977$			36796.6i			1.20494i	0.798141	+	0.602470i	$0.205817\pi$
$977$			36796.6i			1.20494i	−0.798141	+	0.602470i	$0.794183\pi$
$978$		0			0
$979$	51686.4			1.68734
$980$		0			0
$981$		0			0
$982$		0			0
$983$			53869.0i			1.74787i	0.486043	+	0.873935i	$0.338440\pi$
$983$			53869.0i			1.74787i	−0.486043	+	0.873935i	$0.661560\pi$
$984$		0			0
$985$	34605.3	+	989.074i	1.11941	+	0.0319944i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−5873.40			−0.188840
$990$		0			0
$991$	45440.7			1.45658			0.728291	−	0.685268i	$-0.240315\pi$
$991$	45440.7			1.45658			0.728291	+	0.685268i	$0.240315\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−1331.06	+	46570.7i	−0.0424096	+	1.48381i
$996$		0			0
$997$			39968.7i			1.26963i	0.772664	+	0.634815i	$0.218924\pi$
$997$			39968.7i			1.26963i	−0.772664	+	0.634815i	$0.781076\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1080.4.f.a.649.9	yes	16
3.2	odd	2	inner	1080.4.f.a.649.8	yes	16
5.4	even	2	inner	1080.4.f.a.649.10	yes	16
15.14	odd	2	inner	1080.4.f.a.649.7	✓	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1080.4.f.a.649.7	✓	16	15.14	odd	2	inner
1080.4.f.a.649.8	yes	16	3.2	odd	2	inner
1080.4.f.a.649.9	yes	16	1.1	even	1	trivial
1080.4.f.a.649.10	yes	16	5.4	even	2	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:	\( N \)	\(=\)	\( 1080 = 2^{3} \cdot 3^{3} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1080.f (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(0.319421 - 11.1758i) q^{5} -17.5714i q^{7} +O(q^{10})\)	\(q+(0.319421 - 11.1758i) q^{5} -17.5714i q^{7} -59.7658 q^{11} -35.3743i q^{13} +57.2613i q^{17} -51.0310 q^{19} +27.5818i q^{23} +(-124.796 - 7.13956i) q^{25} -76.2910 q^{29} -142.972 q^{31} +(-196.374 - 5.61266i) q^{35} +73.3502i q^{37} +464.564 q^{41} +212.944i q^{43} -166.720i q^{47} +34.2475 q^{49} +744.723i q^{53} +(-19.0904 + 667.929i) q^{55} -18.9197 q^{59} +83.2183 q^{61} +(-395.335 - 11.2993i) q^{65} -644.765i q^{67} +1030.82 q^{71} -1084.29i q^{73} +1050.17i q^{77} +512.603 q^{79} +69.4147i q^{83} +(639.939 + 18.2905i) q^{85} -864.817 q^{89} -621.574 q^{91} +(-16.3004 + 570.311i) q^{95} +686.142i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q+O(q^{10}) \) 16 * q	\( 16 q + 100 q^{19} - 186 q^{25} - 288 q^{31} - 2124 q^{49} + 1222 q^{55} + 188 q^{61} + 4964 q^{79} + 636 q^{85} - 4144 q^{91}+O(q^{100}) \) 16 * q + 100 * q^19 - 186 * q^25 - 288 * q^31 - 2124 * q^49 + 1222 * q^55 + 188 * q^61 + 4964 * q^79 + 636 * q^85 - 4144 * q^91

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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