LMFDB - Embedded newform 2160.4.a.bk.1.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2160,4,Mod(1,2160)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2160.1"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2160, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 2160 = 2^{4} \cdot 3^{3} \cdot 5 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	2160.a (trivial)

Newform invariants

comment:select newform

sage:traces = [3,0,0,0,-15,0,10,0,0,0,28] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$127.444125612$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	3.3.1257.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 8x + 9 $ x^3 - x^2 - 8*x + 9
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{3}\cdot 3 $
Twist minimal:	no (minimal twist has level 1080)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-2.87370$ of defining polynomial
Character	$\chi$	$=$	2160.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-5.00000 q^{5} +30.2207 q^{7} +67.3185 q^{11} -60.7911 q^{13} -61.1347 q^{17} +27.8771 q^{19} -40.4655 q^{23} +25.0000 q^{25} -212.108 q^{29} -167.318 q^{31} -151.103 q^{35} -366.539 q^{37} +363.385 q^{41} -153.839 q^{43} -434.212 q^{47} +570.291 q^{49} -79.6550 q^{53} -336.593 q^{55} +339.414 q^{59} -525.856 q^{61} +303.956 q^{65} -131.916 q^{67} -296.263 q^{71} -1230.34 q^{73} +2034.41 q^{77} +621.772 q^{79} -76.3372 q^{83} +305.673 q^{85} +192.406 q^{89} -1837.15 q^{91} -139.386 q^{95} -874.771 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 15 q^{5} + 10 q^{7} + 28 q^{11} - 78 q^{13} - 11 q^{17} + 71 q^{19} - 25 q^{23} + 75 q^{25} + 118 q^{29} + 107 q^{31} - 50 q^{35} - 410 q^{37} + 592 q^{41} - 52 q^{43} - 580 q^{47} + 479 q^{49} + 169 q^{53}+ \cdots - 840 q^{97}+O(q^{100}) $ 3 * q - 15 * q^5 + 10 * q^7 + 28 * q^11 - 78 * q^13 - 11 * q^17 + 71 * q^19 - 25 * q^23 + 75 * q^25 + 118 * q^29 + 107 * q^31 - 50 * q^35 - 410 * q^37 + 592 * q^41 - 52 * q^43 - 580 * q^47 + 479 * q^49 + 169 * q^53 - 140 * q^55 + 234 * q^59 - 673 * q^61 + 390 * q^65 - 386 * q^67 + 16 * q^71 - 892 * q^73 + 1800 * q^77 - 1263 * q^79 - 1815 * q^83 + 55 * q^85 + 1800 * q^89 - 1284 * q^91 - 355 * q^95 - 840 * q^97

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
$5$	−5.00000	−0.447214
$5$	−5.00000	−0.447214
$6$	0	0
$7$	30.2207	1.63176	0.815882	−	0.578218i	$-0.196252\pi$
$7$	30.2207	1.63176	0.815882	+	0.578218i	$0.196252\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	67.3185	1.84521	0.922604	−	0.385748i	$-0.126056\pi$
$11$	67.3185	1.84521	0.922604	+	0.385748i	$0.126056\pi$
$12$	0	0
$13$	−60.7911	−1.29696	−0.648478	−	0.761234i	$-0.724594\pi$
$13$	−60.7911	−1.29696	−0.648478	+	0.761234i	$0.724594\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−61.1347	−0.872196	−0.436098	−	0.899899i	$-0.643640\pi$
$17$	−61.1347	−0.872196	−0.436098	+	0.899899i	$0.643640\pi$
$18$	0	0
$19$	27.8771	0.336603	0.168301	−	0.985736i	$-0.446172\pi$
$19$	27.8771	0.336603	0.168301	+	0.985736i	$0.446172\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−40.4655	−0.366854	−0.183427	−	0.983033i	$-0.558719\pi$
$23$	−40.4655	−0.366854	−0.183427	+	0.983033i	$0.558719\pi$
$24$	0	0
$25$	25.0000	0.200000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−212.108	−1.35819	−0.679095	−	0.734051i	$-0.737627\pi$
$29$	−212.108	−1.35819	−0.679095	+	0.734051i	$0.737627\pi$
$30$	0	0
$31$	−167.318	−0.969394	−0.484697	−	0.874682i	$-0.661070\pi$
$31$	−167.318	−0.969394	−0.484697	+	0.874682i	$0.661070\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−151.103	−0.729747
$36$	0	0
$37$	−366.539	−1.62861	−0.814305	−	0.580437i	$-0.802882\pi$
$37$	−366.539	−1.62861	−0.814305	+	0.580437i	$0.802882\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	363.385	1.38418	0.692088	−	0.721813i	$-0.256691\pi$
$41$	363.385	1.38418	0.692088	+	0.721813i	$0.256691\pi$
$42$	0	0
$43$	−153.839	−0.545586	−0.272793	−	0.962073i	$-0.587947\pi$
$43$	−153.839	−0.545586	−0.272793	+	0.962073i	$0.587947\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−434.212	−1.34758	−0.673791	−	0.738922i	$-0.735335\pi$
$47$	−434.212	−1.34758	−0.673791	+	0.738922i	$0.735335\pi$
$48$	0	0
$49$	570.291	1.66265
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−79.6550	−0.206442	−0.103221	−	0.994658i	$-0.532915\pi$
$53$	−79.6550	−0.206442	−0.103221	+	0.994658i	$0.532915\pi$
$54$	0	0
$55$	−336.593	−0.825202
$56$	0	0
$57$	0	0
$58$	0	0
$59$	339.414	0.748948	0.374474	−	0.927237i	$-0.377823\pi$
$59$	339.414	0.748948	0.374474	+	0.927237i	$0.377823\pi$
$60$	0	0
$61$	−525.856	−1.10375	−0.551877	−	0.833926i	$-0.686088\pi$
$61$	−525.856	−1.10375	−0.551877	+	0.833926i	$0.686088\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	303.956	0.580016
$66$	0	0
$67$	−131.916	−0.240539	−0.120269	−	0.992741i	$-0.538376\pi$
$67$	−131.916	−0.240539	−0.120269	+	0.992741i	$0.538376\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−296.263	−0.495210	−0.247605	−	0.968861i	$-0.579644\pi$
$71$	−296.263	−0.495210	−0.247605	+	0.968861i	$0.579644\pi$
$72$	0	0
$73$	−1230.34	−1.97261	−0.986305	−	0.164932i	$-0.947259\pi$
$73$	−1230.34	−1.97261	−0.986305	+	0.164932i	$0.947259\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	2034.41	3.01095
$78$	0	0
$79$	621.772	0.885504	0.442752	−	0.896644i	$-0.354002\pi$
$79$	621.772	0.885504	0.442752	+	0.896644i	$0.354002\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−76.3372	−0.100953	−0.0504765	−	0.998725i	$-0.516074\pi$
$83$	−76.3372	−0.100953	−0.0504765	+	0.998725i	$0.516074\pi$
$84$	0	0
$85$	305.673	0.390058
$86$	0	0
$87$	0	0
$88$	0	0
$89$	192.406	0.229157	0.114579	−	0.993414i	$-0.463448\pi$
$89$	192.406	0.229157	0.114579	+	0.993414i	$0.463448\pi$
$90$	0	0
$91$	−1837.15	−2.11633
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−139.386	−0.150533
$96$	0	0
$97$	−874.771	−0.915666	−0.457833	−	0.889038i	$-0.651374\pi$
$97$	−874.771	−0.915666	−0.457833	+	0.889038i	$0.651374\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	1093.02	1.07682	0.538412	−	0.842682i	$-0.319025\pi$
$101$	1093.02	1.07682	0.538412	+	0.842682i	$0.319025\pi$
$102$	0	0
$103$	−228.932	−0.219003	−0.109502	−	0.993987i	$-0.534925\pi$
$103$	−228.932	−0.219003	−0.109502	+	0.993987i	$0.534925\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	479.745	0.433446	0.216723	−	0.976233i	$-0.430463\pi$
$107$	479.745	0.433446	0.216723	+	0.976233i	$0.430463\pi$
$108$	0	0
$109$	−116.246	−0.102150	−0.0510751	−	0.998695i	$-0.516265\pi$
$109$	−116.246	−0.102150	−0.0510751	+	0.998695i	$0.516265\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−741.222	−0.617065	−0.308532	−	0.951214i	$-0.599838\pi$
$113$	−741.222	−0.617065	−0.308532	+	0.951214i	$0.599838\pi$
$114$	0	0
$115$	202.327	0.164062
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−1847.53	−1.42322
$120$	0	0
$121$	3200.78	2.40479
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−125.000	−0.0894427
$126$	0	0
$127$	2595.38	1.81340	0.906702	−	0.421771i	$-0.138591\pi$
$127$	2595.38	1.81340	0.906702	+	0.421771i	$0.138591\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	1405.62	0.937476	0.468738	−	0.883337i	$-0.344709\pi$
$131$	1405.62	0.937476	0.468738	+	0.883337i	$0.344709\pi$
$132$	0	0
$133$	842.466	0.549256
$134$	0	0
$135$	0	0
$136$	0	0
$137$	2457.39	1.53248	0.766239	−	0.642556i	$-0.222126\pi$
$137$	2457.39	1.53248	0.766239	+	0.642556i	$0.222126\pi$
$138$	0	0
$139$	585.391	0.357210	0.178605	−	0.983921i	$-0.442842\pi$
$139$	585.391	0.357210	0.178605	+	0.983921i	$0.442842\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−4092.37	−2.39315
$144$	0	0
$145$	1060.54	0.607401
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−926.505	−0.509411	−0.254705	−	0.967019i	$-0.581979\pi$
$149$	−926.505	−0.509411	−0.254705	+	0.967019i	$0.581979\pi$
$150$	0	0
$151$	−411.900	−0.221987	−0.110993	−	0.993821i	$-0.535403\pi$
$151$	−411.900	−0.221987	−0.110993	+	0.993821i	$0.535403\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	836.590	0.433526
$156$	0	0
$157$	−2959.97	−1.50466	−0.752328	−	0.658789i	$-0.771069\pi$
$157$	−2959.97	−1.50466	−0.752328	+	0.658789i	$0.771069\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−1222.89	−0.598619
$162$	0	0
$163$	−2504.50	−1.20348	−0.601741	−	0.798691i	$-0.705526\pi$
$163$	−2504.50	−1.20348	−0.601741	+	0.798691i	$0.705526\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	2055.42	0.952413	0.476207	−	0.879333i	$-0.342011\pi$
$167$	2055.42	0.952413	0.476207	+	0.879333i	$0.342011\pi$
$168$	0	0
$169$	1498.56	0.682093
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−1724.17	−0.757722	−0.378861	−	0.925454i	$-0.623684\pi$
$173$	−1724.17	−0.757722	−0.378861	+	0.925454i	$0.623684\pi$
$174$	0	0
$175$	755.517	0.326353
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−3601.35	−1.50379	−0.751893	−	0.659285i	$-0.770859\pi$
$179$	−3601.35	−1.50379	−0.751893	+	0.659285i	$0.770859\pi$
$180$	0	0
$181$	−1257.17	−0.516267	−0.258134	−	0.966109i	$-0.583107\pi$
$181$	−1257.17	−0.516267	−0.258134	+	0.966109i	$0.583107\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	1832.69	0.728337
$186$	0	0
$187$	−4115.50	−1.60938
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−4677.47	−1.77199	−0.885994	−	0.463697i	$-0.846523\pi$
$191$	−4677.47	−1.77199	−0.885994	+	0.463697i	$0.846523\pi$
$192$	0	0
$193$	−2647.88	−0.987556	−0.493778	−	0.869588i	$-0.664384\pi$
$193$	−2647.88	−0.987556	−0.493778	+	0.869588i	$0.664384\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−1368.91	−0.495082	−0.247541	−	0.968877i	$-0.579622\pi$
$197$	−1368.91	−0.495082	−0.247541	+	0.968877i	$0.579622\pi$
$198$	0	0
$199$	5411.43	1.92767	0.963835	−	0.266501i	$-0.0858675\pi$
$199$	5411.43	1.92767	0.963835	+	0.266501i	$0.0858675\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−6410.06	−2.21625
$204$	0	0
$205$	−1816.93	−0.619022
$206$	0	0
$207$	0	0
$208$	0	0
$209$	1876.65	0.621102
$210$	0	0
$211$	−1987.12	−0.648337	−0.324168	−	0.945999i	$-0.605084\pi$
$211$	−1987.12	−0.648337	−0.324168	+	0.945999i	$0.605084\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	769.194	0.243993
$216$	0	0
$217$	−5056.47	−1.58182
$218$	0	0
$219$	0	0
$220$	0	0
$221$	3716.45	1.13120
$222$	0	0
$223$	−3687.95	−1.10746	−0.553730	−	0.832697i	$-0.686796\pi$
$223$	−3687.95	−1.10746	−0.553730	+	0.832697i	$0.686796\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−859.424	−0.251286	−0.125643	−	0.992076i	$-0.540099\pi$
$227$	−859.424	−0.251286	−0.125643	+	0.992076i	$0.540099\pi$
$228$	0	0
$229$	−1862.67	−0.537504	−0.268752	−	0.963209i	$-0.586611\pi$
$229$	−1862.67	−0.537504	−0.268752	+	0.963209i	$0.586611\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	5011.22	1.40900	0.704498	−	0.709706i	$-0.251172\pi$
$233$	5011.22	1.40900	0.704498	+	0.709706i	$0.251172\pi$
$234$	0	0
$235$	2171.06	0.602657
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−5057.92	−1.36891	−0.684455	−	0.729055i	$-0.739960\pi$
$239$	−5057.92	−1.36891	−0.684455	+	0.729055i	$0.739960\pi$
$240$	0	0
$241$	−7276.37	−1.94486	−0.972431	−	0.233189i	$-0.925084\pi$
$241$	−7276.37	−1.94486	−0.972431	+	0.233189i	$0.925084\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−2851.45	−0.743562
$246$	0	0
$247$	−1694.68	−0.436559
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−204.208	−0.0513525	−0.0256762	−	0.999670i	$-0.508174\pi$
$251$	−204.208	−0.0513525	−0.0256762	+	0.999670i	$0.508174\pi$
$252$	0	0
$253$	−2724.07	−0.676921
$254$	0	0
$255$	0	0
$256$	0	0
$257$	4591.80	1.11451	0.557254	−	0.830342i	$-0.311855\pi$
$257$	4591.80	1.11451	0.557254	+	0.830342i	$0.311855\pi$
$258$	0	0
$259$	−11077.1	−2.65751
$260$	0	0
$261$	0	0
$262$	0	0
$263$	7689.01	1.80276	0.901378	−	0.433033i	$-0.142557\pi$
$263$	7689.01	1.80276	0.901378	+	0.433033i	$0.142557\pi$
$264$	0	0
$265$	398.275	0.0923239
$266$	0	0
$267$	0	0
$268$	0	0
$269$	5790.73	1.31252	0.656258	−	0.754536i	$-0.272138\pi$
$269$	5790.73	1.31252	0.656258	+	0.754536i	$0.272138\pi$
$270$	0	0
$271$	−6146.43	−1.37775	−0.688873	−	0.724882i	$-0.741894\pi$
$271$	−6146.43	−1.37775	−0.688873	+	0.724882i	$0.741894\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	1682.96	0.369042
$276$	0	0
$277$	4811.02	1.04356	0.521781	−	0.853080i	$-0.325268\pi$
$277$	4811.02	1.04356	0.521781	+	0.853080i	$0.325268\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	2333.19	0.495325	0.247663	−	0.968846i	$-0.420338\pi$
$281$	2333.19	0.495325	0.247663	+	0.968846i	$0.420338\pi$
$282$	0	0
$283$	−8697.84	−1.82697	−0.913486	−	0.406870i	$-0.866620\pi$
$283$	−8697.84	−1.82697	−0.913486	+	0.406870i	$0.866620\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	10981.8	2.25865
$288$	0	0
$289$	−1175.55	−0.239273
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−1838.13	−0.366501	−0.183251	−	0.983066i	$-0.558662\pi$
$293$	−1838.13	−0.366501	−0.183251	+	0.983066i	$0.558662\pi$
$294$	0	0
$295$	−1697.07	−0.334940
$296$	0	0
$297$	0	0
$298$	0	0
$299$	2459.94	0.475793
$300$	0	0
$301$	−4649.12	−0.890268
$302$	0	0
$303$	0	0
$304$	0	0
$305$	2629.28	0.493614
$306$	0	0
$307$	4152.70	0.772010	0.386005	−	0.922497i	$-0.373855\pi$
$307$	4152.70	0.772010	0.386005	+	0.922497i	$0.373855\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−2861.14	−0.521674	−0.260837	−	0.965383i	$-0.583999\pi$
$311$	−2861.14	−0.521674	−0.260837	+	0.965383i	$0.583999\pi$
$312$	0	0
$313$	−438.494	−0.0791858	−0.0395929	−	0.999216i	$-0.512606\pi$
$313$	−438.494	−0.0791858	−0.0395929	+	0.999216i	$0.512606\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−6279.80	−1.11265	−0.556323	−	0.830966i	$-0.687788\pi$
$317$	−6279.80	−1.11265	−0.556323	+	0.830966i	$0.687788\pi$
$318$	0	0
$319$	−14278.8	−2.50614
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−1704.26	−0.293584
$324$	0	0
$325$	−1519.78	−0.259391
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−13122.2	−2.19894
$330$	0	0
$331$	7564.20	1.25609	0.628046	−	0.778176i	$-0.283855\pi$
$331$	7564.20	1.25609	0.628046	+	0.778176i	$0.283855\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	659.580	0.107572
$336$	0	0
$337$	3468.51	0.560658	0.280329	−	0.959904i	$-0.409556\pi$
$337$	3468.51	0.560658	0.280329	+	0.959904i	$0.409556\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−11263.6	−1.78873
$342$	0	0
$343$	6868.88	1.08130
$344$	0	0
$345$	0	0
$346$	0	0
$347$	6415.10	0.992452	0.496226	−	0.868193i	$-0.334719\pi$
$347$	6415.10	0.992452	0.496226	+	0.868193i	$0.334719\pi$
$348$	0	0
$349$	−2248.97	−0.344941	−0.172470	−	0.985015i	$-0.555175\pi$
$349$	−2248.97	−0.344941	−0.172470	+	0.985015i	$0.555175\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−7197.94	−1.08529	−0.542645	−	0.839962i	$-0.682577\pi$
$353$	−7197.94	−1.08529	−0.542645	+	0.839962i	$0.682577\pi$
$354$	0	0
$355$	1481.31	0.221465
$356$	0	0
$357$	0	0
$358$	0	0
$359$	8618.50	1.26704	0.633520	−	0.773726i	$-0.281609\pi$
$359$	8618.50	1.26704	0.633520	+	0.773726i	$0.281609\pi$
$360$	0	0
$361$	−6081.87	−0.886699
$362$	0	0
$363$	0	0
$364$	0	0
$365$	6151.70	0.882178
$366$	0	0
$367$	9025.95	1.28379	0.641895	−	0.766793i	$-0.278149\pi$
$367$	9025.95	1.28379	0.641895	+	0.766793i	$0.278149\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−2407.23	−0.336865
$372$	0	0
$373$	7549.69	1.04801	0.524005	−	0.851715i	$-0.324437\pi$
$373$	7549.69	1.04801	0.524005	+	0.851715i	$0.324437\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	12894.3	1.76151
$378$	0	0
$379$	−11849.7	−1.60601	−0.803007	−	0.595970i	$-0.796768\pi$
$379$	−11849.7	−1.60601	−0.803007	+	0.595970i	$0.796768\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	8613.06	1.14910	0.574552	−	0.818468i	$-0.305176\pi$
$383$	8613.06	1.14910	0.574552	+	0.818468i	$0.305176\pi$
$384$	0	0
$385$	−10172.1	−1.34654
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−2529.23	−0.329658	−0.164829	−	0.986322i	$-0.552707\pi$
$389$	−2529.23	−0.329658	−0.164829	+	0.986322i	$0.552707\pi$
$390$	0	0
$391$	2473.84	0.319968
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−3108.86	−0.396009
$396$	0	0
$397$	−14195.7	−1.79461	−0.897306	−	0.441410i	$-0.854479\pi$
$397$	−14195.7	−1.79461	−0.897306	+	0.441410i	$0.854479\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−11005.5	−1.37054	−0.685270	−	0.728289i	$-0.740316\pi$
$401$	−11005.5	−1.37054	−0.685270	+	0.728289i	$0.740316\pi$
$402$	0	0
$403$	10171.4	1.25726
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−24674.8	−3.00513
$408$	0	0
$409$	14339.1	1.73356	0.866779	−	0.498692i	$-0.166186\pi$
$409$	14339.1	1.73356	0.866779	+	0.498692i	$0.166186\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	10257.3	1.22211
$414$	0	0
$415$	381.686	0.0451475
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−5352.51	−0.624075	−0.312037	−	0.950070i	$-0.601011\pi$
$419$	−5352.51	−0.624075	−0.312037	+	0.950070i	$0.601011\pi$
$420$	0	0
$421$	−2683.89	−0.310701	−0.155350	−	0.987859i	$-0.549651\pi$
$421$	−2683.89	−0.310701	−0.155350	+	0.987859i	$0.549651\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−1528.37	−0.174439
$426$	0	0
$427$	−15891.7	−1.80107
$428$	0	0
$429$	0	0
$430$	0	0
$431$	10019.0	1.11972	0.559861	−	0.828587i	$-0.310855\pi$
$431$	10019.0	1.11972	0.559861	+	0.828587i	$0.310855\pi$
$432$	0	0
$433$	−5292.20	−0.587361	−0.293680	−	0.955904i	$-0.594880\pi$
$433$	−5292.20	−0.587361	−0.293680	+	0.955904i	$0.594880\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−1128.06	−0.123484
$438$	0	0
$439$	−9453.33	−1.02775	−0.513875	−	0.857865i	$-0.671791\pi$
$439$	−9453.33	−1.02775	−0.513875	+	0.857865i	$0.671791\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−2858.23	−0.306543	−0.153272	−	0.988184i	$-0.548981\pi$
$443$	−2858.23	−0.306543	−0.153272	+	0.988184i	$0.548981\pi$
$444$	0	0
$445$	−962.030	−0.102482
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−9322.81	−0.979890	−0.489945	−	0.871753i	$-0.662983\pi$
$449$	−9322.81	−0.979890	−0.489945	+	0.871753i	$0.662983\pi$
$450$	0	0
$451$	24462.5	2.55409
$452$	0	0
$453$	0	0
$454$	0	0
$455$	9185.75	0.946449
$456$	0	0
$457$	4960.54	0.507755	0.253878	−	0.967236i	$-0.418294\pi$
$457$	4960.54	0.507755	0.253878	+	0.967236i	$0.418294\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−5235.83	−0.528974	−0.264487	−	0.964389i	$-0.585203\pi$
$461$	−5235.83	−0.528974	−0.264487	+	0.964389i	$0.585203\pi$
$462$	0	0
$463$	−7121.96	−0.714871	−0.357436	−	0.933938i	$-0.616349\pi$
$463$	−7121.96	−0.714871	−0.357436	+	0.933938i	$0.616349\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−1991.23	−0.197308	−0.0986541	−	0.995122i	$-0.531454\pi$
$467$	−1991.23	−0.197308	−0.0986541	+	0.995122i	$0.531454\pi$
$468$	0	0
$469$	−3986.59	−0.392503
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−10356.2	−1.00672
$474$	0	0
$475$	696.928	0.0673205
$476$	0	0
$477$	0	0
$478$	0	0
$479$	9377.15	0.894474	0.447237	−	0.894415i	$-0.352408\pi$
$479$	9377.15	0.894474	0.447237	+	0.894415i	$0.352408\pi$
$480$	0	0
$481$	22282.3	2.11224
$482$	0	0
$483$	0	0
$484$	0	0
$485$	4373.86	0.409498
$486$	0	0
$487$	4855.98	0.451838	0.225919	−	0.974146i	$-0.427461\pi$
$487$	4855.98	0.451838	0.225919	+	0.974146i	$0.427461\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	4048.75	0.372133	0.186067	−	0.982537i	$-0.440426\pi$
$491$	4048.75	0.372133	0.186067	+	0.982537i	$0.440426\pi$
$492$	0	0
$493$	12967.2	1.18461
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−8953.27	−0.808066
$498$	0	0
$499$	−6259.64	−0.561563	−0.280781	−	0.959772i	$-0.590594\pi$
$499$	−6259.64	−0.561563	−0.280781	+	0.959772i	$0.590594\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	9127.86	0.809128	0.404564	−	0.914510i	$-0.367423\pi$
$503$	9127.86	0.809128	0.404564	+	0.914510i	$0.367423\pi$
$504$	0	0
$505$	−5465.08	−0.481570
$506$	0	0
$507$	0	0
$508$	0	0
$509$	9495.89	0.826911	0.413456	−	0.910524i	$-0.364322\pi$
$509$	9495.89	0.826911	0.413456	+	0.910524i	$0.364322\pi$
$510$	0	0
$511$	−37181.8	−3.21883
$512$	0	0
$513$	0	0
$514$	0	0
$515$	1144.66	0.0979412
$516$	0	0
$517$	−29230.5	−2.48657
$518$	0	0
$519$	0	0
$520$	0	0
$521$	12282.6	1.03284	0.516421	−	0.856335i	$-0.327264\pi$
$521$	12282.6	1.03284	0.516421	+	0.856335i	$0.327264\pi$
$522$	0	0
$523$	−8010.57	−0.669747	−0.334873	−	0.942263i	$-0.608694\pi$
$523$	−8010.57	−0.669747	−0.334873	+	0.942263i	$0.608694\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	10228.9	0.845502
$528$	0	0
$529$	−10529.5	−0.865418
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−22090.6	−1.79521
$534$	0	0
$535$	−2398.73	−0.193843
$536$	0	0
$537$	0	0
$538$	0	0
$539$	38391.1	3.06794
$540$	0	0
$541$	−15476.8	−1.22995	−0.614974	−	0.788548i	$-0.710833\pi$
$541$	−15476.8	−1.22995	−0.614974	+	0.788548i	$0.710833\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	581.231	0.0456829
$546$	0	0
$547$	22837.0	1.78508	0.892540	−	0.450968i	$-0.148921\pi$
$547$	22837.0	1.78508	0.892540	+	0.450968i	$0.148921\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−5912.96	−0.457170
$552$	0	0
$553$	18790.4	1.44493
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−15800.3	−1.20194	−0.600971	−	0.799271i	$-0.705219\pi$
$557$	−15800.3	−1.20194	−0.600971	+	0.799271i	$0.705219\pi$
$558$	0	0
$559$	9352.03	0.707601
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−13515.7	−1.01176	−0.505879	−	0.862605i	$-0.668832\pi$
$563$	−13515.7	−1.01176	−0.505879	+	0.862605i	$0.668832\pi$
$564$	0	0
$565$	3706.11	0.275960
$566$	0	0
$567$	0	0
$568$	0	0
$569$	19632.0	1.44642	0.723211	−	0.690627i	$-0.242665\pi$
$569$	19632.0	1.44642	0.723211	+	0.690627i	$0.242665\pi$
$570$	0	0
$571$	12800.7	0.938166	0.469083	−	0.883154i	$-0.344584\pi$
$571$	12800.7	0.938166	0.469083	+	0.883154i	$0.344584\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−1011.64	−0.0733707
$576$	0	0
$577$	−9214.08	−0.664796	−0.332398	−	0.943139i	$-0.607858\pi$
$577$	−9214.08	−0.664796	−0.332398	+	0.943139i	$0.607858\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−2306.96	−0.164731
$582$	0	0
$583$	−5362.25	−0.380929
$584$	0	0
$585$	0	0
$586$	0	0
$587$	275.136	0.0193460	0.00967298	−	0.999953i	$-0.496921\pi$
$587$	275.136	0.0193460	0.00967298	+	0.999953i	$0.496921\pi$
$588$	0	0
$589$	−4664.34	−0.326300
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−15753.5	−1.09093	−0.545464	−	0.838135i	$-0.683646\pi$
$593$	−15753.5	−1.09093	−0.545464	+	0.838135i	$0.683646\pi$
$594$	0	0
$595$	9237.66	0.636483
$596$	0	0
$597$	0	0
$598$	0	0
$599$	13557.3	0.924770	0.462385	−	0.886679i	$-0.346994\pi$
$599$	13557.3	0.924770	0.462385	+	0.886679i	$0.346994\pi$
$600$	0	0
$601$	23976.7	1.62734	0.813668	−	0.581330i	$-0.197467\pi$
$601$	23976.7	1.62734	0.813668	+	0.581330i	$0.197467\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−16003.9	−1.07546
$606$	0	0
$607$	12766.6	0.853677	0.426839	−	0.904328i	$-0.359627\pi$
$607$	12766.6	0.853677	0.426839	+	0.904328i	$0.359627\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	26396.2	1.74775
$612$	0	0
$613$	−16776.3	−1.10537	−0.552684	−	0.833391i	$-0.686396\pi$
$613$	−16776.3	−1.10537	−0.552684	+	0.833391i	$0.686396\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−17941.4	−1.17065	−0.585327	−	0.810798i	$-0.699034\pi$
$617$	−17941.4	−1.17065	−0.585327	+	0.810798i	$0.699034\pi$
$618$	0	0
$619$	987.096	0.0640949	0.0320475	−	0.999486i	$-0.489797\pi$
$619$	987.096	0.0640949	0.0320475	+	0.999486i	$0.489797\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	5814.64	0.373930
$624$	0	0
$625$	625.000	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	22408.2	1.42047
$630$	0	0
$631$	−18504.9	−1.16746	−0.583730	−	0.811948i	$-0.698407\pi$
$631$	−18504.9	−1.16746	−0.583730	+	0.811948i	$0.698407\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−12976.9	−0.810979
$636$	0	0
$637$	−34668.6	−2.15639
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−10260.2	−0.632219	−0.316110	−	0.948723i	$-0.602377\pi$
$641$	−10260.2	−0.632219	−0.316110	+	0.948723i	$0.602377\pi$
$642$	0	0
$643$	−6221.29	−0.381561	−0.190781	−	0.981633i	$-0.561102\pi$
$643$	−6221.29	−0.381561	−0.190781	+	0.981633i	$0.561102\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−21446.7	−1.30318	−0.651590	−	0.758572i	$-0.725898\pi$
$647$	−21446.7	−1.30318	−0.651590	+	0.758572i	$0.725898\pi$
$648$	0	0
$649$	22848.8	1.38196
$650$	0	0
$651$	0	0
$652$	0	0
$653$	25051.0	1.50126	0.750628	−	0.660725i	$-0.229751\pi$
$653$	25051.0	1.50126	0.750628	+	0.660725i	$0.229751\pi$
$654$	0	0
$655$	−7028.09	−0.419252
$656$	0	0
$657$	0	0
$658$	0	0
$659$	20.7125	0.00122435	0.000612174	−	1.00000i	$-0.499805\pi$
$659$	20.7125	0.00122435	0.000612174		1.00000i	$0.499805\pi$
$660$	0	0
$661$	23294.7	1.37074	0.685371	−	0.728194i	$-0.259640\pi$
$661$	23294.7	1.37074	0.685371	+	0.728194i	$0.259640\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−4212.33	−0.245635
$666$	0	0
$667$	8583.05	0.498257
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−35399.9	−2.03666
$672$	0	0
$673$	25764.3	1.47569	0.737845	−	0.674970i	$-0.235843\pi$
$673$	25764.3	1.47569	0.737845	+	0.674970i	$0.235843\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−26667.3	−1.51390	−0.756948	−	0.653475i	$-0.773311\pi$
$677$	−26667.3	−1.51390	−0.756948	+	0.653475i	$0.773311\pi$
$678$	0	0
$679$	−26436.2	−1.49415
$680$	0	0
$681$	0	0
$682$	0	0
$683$	14.8616	0.000832599 0	0.000416299	−	1.00000i	$-0.499867\pi$
$683$	14.8616	0.000832599 0	0.000416299		1.00000i	$0.499867\pi$
$684$	0	0
$685$	−12287.0	−0.685345
$686$	0	0
$687$	0	0
$688$	0	0
$689$	4842.31	0.267747
$690$	0	0
$691$	−26645.5	−1.46692	−0.733460	−	0.679733i	$-0.762096\pi$
$691$	−26645.5	−1.46692	−0.733460	+	0.679733i	$0.762096\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−2926.96	−0.159749
$696$	0	0
$697$	−22215.4	−1.20727
$698$	0	0
$699$	0	0
$700$	0	0
$701$	431.952	0.0232733	0.0116367	−	0.999932i	$-0.496296\pi$
$701$	431.952	0.0232733	0.0116367	+	0.999932i	$0.496296\pi$
$702$	0	0
$703$	−10218.0	−0.548195
$704$	0	0
$705$	0	0
$706$	0	0
$707$	33031.7	1.75712
$708$	0	0
$709$	35463.0	1.87848	0.939240	−	0.343261i	$-0.111532\pi$
$709$	35463.0	1.87848	0.939240	+	0.343261i	$0.111532\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	6770.60	0.355626
$714$	0	0
$715$	20461.8	1.07025
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−26340.5	−1.36625	−0.683125	−	0.730302i	$-0.739379\pi$
$719$	−26340.5	−1.36625	−0.683125	+	0.730302i	$0.739379\pi$
$720$	0	0
$721$	−6918.48	−0.357362
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−5302.70	−0.271638
$726$	0	0
$727$	38279.1	1.95281	0.976404	−	0.215952i	$-0.0692853\pi$
$727$	38279.1	1.95281	0.976404	+	0.215952i	$0.0692853\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	9404.89	0.475858
$732$	0	0
$733$	−16445.7	−0.828700	−0.414350	−	0.910118i	$-0.635991\pi$
$733$	−16445.7	−0.828700	−0.414350	+	0.910118i	$0.635991\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−8880.39	−0.443844
$738$	0	0
$739$	5150.65	0.256387	0.128193	−	0.991749i	$-0.459082\pi$
$739$	5150.65	0.256387	0.128193	+	0.991749i	$0.459082\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−8844.91	−0.436727	−0.218364	−	0.975868i	$-0.570072\pi$
$743$	−8844.91	−0.436727	−0.218364	+	0.975868i	$0.570072\pi$
$744$	0	0
$745$	4632.52	0.227816
$746$	0	0
$747$	0	0
$748$	0	0
$749$	14498.2	0.707282
$750$	0	0
$751$	−5774.81	−0.280594	−0.140297	−	0.990109i	$-0.544806\pi$
$751$	−5774.81	−0.280594	−0.140297	+	0.990109i	$0.544806\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	2059.50	0.0992754
$756$	0	0
$757$	14267.1	0.685003	0.342501	−	0.939517i	$-0.388726\pi$
$757$	14267.1	0.685003	0.342501	+	0.939517i	$0.388726\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	21823.0	1.03953	0.519765	−	0.854309i	$-0.326020\pi$
$761$	21823.0	1.03953	0.519765	+	0.854309i	$0.326020\pi$
$762$	0	0
$763$	−3513.04	−0.166685
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−20633.3	−0.971352
$768$	0	0
$769$	−3080.07	−0.144435	−0.0722173	−	0.997389i	$-0.523008\pi$
$769$	−3080.07	−0.144435	−0.0722173	+	0.997389i	$0.523008\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	8834.10	0.411049	0.205524	−	0.978652i	$-0.434110\pi$
$773$	8834.10	0.411049	0.205524	+	0.978652i	$0.434110\pi$
$774$	0	0
$775$	−4182.95	−0.193879
$776$	0	0
$777$	0	0
$778$	0	0
$779$	10130.1	0.465917
$780$	0	0
$781$	−19944.0	−0.913766
$782$	0	0
$783$	0	0
$784$	0	0
$785$	14799.8	0.672903
$786$	0	0
$787$	−11060.1	−0.500952	−0.250476	−	0.968123i	$-0.580587\pi$
$787$	−11060.1	−0.500952	−0.250476	+	0.968123i	$0.580587\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−22400.3	−1.00690
$792$	0	0
$793$	31967.4	1.43152
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−3379.95	−0.150218	−0.0751091	−	0.997175i	$-0.523930\pi$
$797$	−3379.95	−0.150218	−0.0751091	+	0.997175i	$0.523930\pi$
$798$	0	0
$799$	26545.4	1.17536
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−82824.7	−3.63988
$804$	0	0
$805$	6114.47	0.267710
$806$	0	0
$807$	0	0
$808$	0	0
$809$	38358.8	1.66703	0.833513	−	0.552499i	$-0.186326\pi$
$809$	38358.8	1.66703	0.833513	+	0.552499i	$0.186326\pi$
$810$	0	0
$811$	9110.42	0.394464	0.197232	−	0.980357i	$-0.436805\pi$
$811$	9110.42	0.394464	0.197232	+	0.980357i	$0.436805\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	12522.5	0.538214
$816$	0	0
$817$	−4288.58	−0.183646
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−8177.12	−0.347605	−0.173802	−	0.984781i	$-0.555605\pi$
$821$	−8177.12	−0.347605	−0.173802	+	0.984781i	$0.555605\pi$
$822$	0	0
$823$	4238.40	0.179515	0.0897577	−	0.995964i	$-0.471391\pi$
$823$	4238.40	0.179515	0.0897577	+	0.995964i	$0.471391\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−12021.9	−0.505492	−0.252746	−	0.967533i	$-0.581334\pi$
$827$	−12021.9	−0.505492	−0.252746	+	0.967533i	$0.581334\pi$
$828$	0	0
$829$	6564.14	0.275009	0.137504	−	0.990501i	$-0.456092\pi$
$829$	6564.14	0.275009	0.137504	+	0.990501i	$0.456092\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−34864.5	−1.45016
$834$	0	0
$835$	−10277.1	−0.425932
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−3645.41	−0.150004	−0.0750021	−	0.997183i	$-0.523896\pi$
$839$	−3645.41	−0.150004	−0.0750021	+	0.997183i	$0.523896\pi$
$840$	0	0
$841$	20600.9	0.844679
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−7492.79	−0.305041
$846$	0	0
$847$	96729.9	3.92406
$848$	0	0
$849$	0	0
$850$	0	0
$851$	14832.2	0.597462
$852$	0	0
$853$	−45697.7	−1.83430	−0.917151	−	0.398540i	$-0.869517\pi$
$853$	−45697.7	−1.83430	−0.917151	+	0.398540i	$0.869517\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	13766.3	0.548716	0.274358	−	0.961628i	$-0.411535\pi$
$857$	13766.3	0.548716	0.274358	+	0.961628i	$0.411535\pi$
$858$	0	0
$859$	−15422.0	−0.612563	−0.306282	−	0.951941i	$-0.599085\pi$
$859$	−15422.0	−0.612563	−0.306282	+	0.951941i	$0.599085\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	7382.54	0.291199	0.145599	−	0.989344i	$-0.453489\pi$
$863$	7382.54	0.291199	0.145599	+	0.989344i	$0.453489\pi$
$864$	0	0
$865$	8620.83	0.338863
$866$	0	0
$867$	0	0
$868$	0	0
$869$	41856.8	1.63394
$870$	0	0
$871$	8019.32	0.311968
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−3777.59	−0.145949
$876$	0	0
$877$	−12079.7	−0.465112	−0.232556	−	0.972583i	$-0.574709\pi$
$877$	−12079.7	−0.465112	−0.232556	+	0.972583i	$0.574709\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	25501.7	0.975227	0.487613	−	0.873060i	$-0.337868\pi$
$881$	25501.7	0.975227	0.487613	+	0.873060i	$0.337868\pi$
$882$	0	0
$883$	29353.2	1.11870	0.559351	−	0.828931i	$-0.311050\pi$
$883$	29353.2	1.11870	0.559351	+	0.828931i	$0.311050\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−17185.8	−0.650556	−0.325278	−	0.945618i	$-0.605458\pi$
$887$	−17185.8	−0.650556	−0.325278	+	0.945618i	$0.605458\pi$
$888$	0	0
$889$	78434.1	2.95905
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−12104.6	−0.453600
$894$	0	0
$895$	18006.8	0.672513
$896$	0	0
$897$	0	0
$898$	0	0
$899$	35489.5	1.31662
$900$	0	0
$901$	4869.68	0.180058
$902$	0	0
$903$	0	0
$904$	0	0
$905$	6285.83	0.230882
$906$	0	0
$907$	38511.6	1.40988	0.704938	−	0.709269i	$-0.250975\pi$
$907$	38511.6	1.40988	0.704938	+	0.709269i	$0.250975\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	11451.4	0.416467	0.208233	−	0.978079i	$-0.433229\pi$
$911$	11451.4	0.416467	0.208233	+	0.978079i	$0.433229\pi$
$912$	0	0
$913$	−5138.90	−0.186279
$914$	0	0
$915$	0	0
$916$	0	0
$917$	42478.8	1.52974
$918$	0	0
$919$	−14096.0	−0.505968	−0.252984	−	0.967470i	$-0.581412\pi$
$919$	−14096.0	−0.505968	−0.252984	+	0.967470i	$0.581412\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	18010.1	0.642265
$924$	0	0
$925$	−9163.47	−0.325722
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−15020.3	−0.530462	−0.265231	−	0.964185i	$-0.585448\pi$
$929$	−15020.3	−0.530462	−0.265231	+	0.964185i	$0.585448\pi$
$930$	0	0
$931$	15898.1	0.559654
$932$	0	0
$933$	0	0
$934$	0	0
$935$	20577.5	0.719739
$936$	0	0
$937$	18430.5	0.642582	0.321291	−	0.946981i	$-0.395883\pi$
$937$	18430.5	0.642582	0.321291	+	0.946981i	$0.395883\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	55141.3	1.91026	0.955131	−	0.296184i	$-0.0957141\pi$
$941$	55141.3	1.91026	0.955131	+	0.296184i	$0.0957141\pi$
$942$	0	0
$943$	−14704.5	−0.507790
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−24231.3	−0.831480	−0.415740	−	0.909484i	$-0.636477\pi$
$947$	−24231.3	−0.831480	−0.415740	+	0.909484i	$0.636477\pi$
$948$	0	0
$949$	74793.8	2.55839
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−19107.5	−0.649479	−0.324739	−	0.945804i	$-0.605277\pi$
$953$	−19107.5	−0.649479	−0.324739	+	0.945804i	$0.605277\pi$
$954$	0	0
$955$	23387.3	0.792457
$956$	0	0
$957$	0	0
$958$	0	0
$959$	74264.2	2.50064
$960$	0	0
$961$	−1795.67	−0.0602756
$962$	0	0
$963$	0	0
$964$	0	0
$965$	13239.4	0.441649
$966$	0	0
$967$	−21993.6	−0.731401	−0.365701	−	0.930732i	$-0.619171\pi$
$967$	−21993.6	−0.731401	−0.365701	+	0.930732i	$0.619171\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−19409.3	−0.641477	−0.320738	−	0.947168i	$-0.603931\pi$
$971$	−19409.3	−0.641477	−0.320738	+	0.947168i	$0.603931\pi$
$972$	0	0
$973$	17690.9	0.582883
$974$	0	0
$975$	0	0
$976$	0	0
$977$	36384.1	1.19143	0.595717	−	0.803194i	$-0.296868\pi$
$977$	36384.1	1.19143	0.595717	+	0.803194i	$0.296868\pi$
$978$	0	0
$979$	12952.5	0.422843
$980$	0	0
$981$	0	0
$982$	0	0
$983$	4202.43	0.136355	0.0681774	−	0.997673i	$-0.478282\pi$
$983$	4202.43	0.136355	0.0681774	+	0.997673i	$0.478282\pi$
$984$	0	0
$985$	6844.57	0.221407
$986$	0	0
$987$	0	0
$988$	0	0
$989$	6225.16	0.200150
$990$	0	0
$991$	17753.0	0.569064	0.284532	−	0.958666i	$-0.408162\pi$
$991$	17753.0	0.569064	0.284532	+	0.958666i	$0.408162\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−27057.2	−0.862080
$996$	0	0
$997$	12519.0	0.397673	0.198837	−	0.980033i	$-0.436284\pi$
$997$	12519.0	0.397673	0.198837	+	0.980033i	$0.436284\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2160.4.a.bk.1.3		3
3.2	odd	2		2160.4.a.bs.1.3		3
4.3	odd	2		1080.4.a.d.1.1	✓	3
12.11	even	2		1080.4.a.j.1.1	yes	3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1080.4.a.d.1.1	✓	3	4.3	odd	2
1080.4.a.j.1.1	yes	3	12.11	even	2
2160.4.a.bk.1.3		3	1.1	even	1	trivial
2160.4.a.bs.1.3		3	3.2	odd	2

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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 2160 = 2^{4} \cdot 3^{3} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2160.a (trivial)

\(f(q)\)	\(=\)	\(q-5.00000 q^{5} +30.2207 q^{7} +67.3185 q^{11} -60.7911 q^{13} -61.1347 q^{17} +27.8771 q^{19} -40.4655 q^{23} +25.0000 q^{25} -212.108 q^{29} -167.318 q^{31} -151.103 q^{35} -366.539 q^{37} +363.385 q^{41} -153.839 q^{43} -434.212 q^{47} +570.291 q^{49} -79.6550 q^{53} -336.593 q^{55} +339.414 q^{59} -525.856 q^{61} +303.956 q^{65} -131.916 q^{67} -296.263 q^{71} -1230.34 q^{73} +2034.41 q^{77} +621.772 q^{79} -76.3372 q^{83} +305.673 q^{85} +192.406 q^{89} -1837.15 q^{91} -139.386 q^{95} -874.771 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 15 q^{5} + 10 q^{7} + 28 q^{11} - 78 q^{13} - 11 q^{17} + 71 q^{19} - 25 q^{23} + 75 q^{25} + 118 q^{29} + 107 q^{31} - 50 q^{35} - 410 q^{37} + 592 q^{41} - 52 q^{43} - 580 q^{47} + 479 q^{49} + 169 q^{53}+ \cdots - 840 q^{97}+O(q^{100}) \) 3 * q - 15 * q^5 + 10 * q^7 + 28 * q^11 - 78 * q^13 - 11 * q^17 + 71 * q^19 - 25 * q^23 + 75 * q^25 + 118 * q^29 + 107 * q^31 - 50 * q^35 - 410 * q^37 + 592 * q^41 - 52 * q^43 - 580 * q^47 + 479 * q^49 + 169 * q^53 - 140 * q^55 + 234 * q^59 - 673 * q^61 + 390 * q^65 - 386 * q^67 + 16 * q^71 - 892 * q^73 + 1800 * q^77 - 1263 * q^79 - 1815 * q^83 + 55 * q^85 + 1800 * q^89 - 1284 * q^91 - 355 * q^95 - 840 * q^97

Introduction

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