LMFDB - Embedded newform 1620.4.a.k.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1620,4,Mod(1,1620)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1620, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("1620.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1620 = 2^{2} \cdot 3^{4} \cdot 5 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1620.a (trivial)

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:	yes
Analytic conductor:	$95.5830942093$
Analytic rank:	$0$
Dimension:	$7$
Coefficient field:	$\mathbb{Q}[x]/(x^{7} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{7} - 2x^{6} - 167x^{5} + 940x^{4} + 1153x^{3} - 13196x^{2} + 16629x + 714 $ x^7 - 2x^6 - 167x^5 + 940x^4 + 1153x^3 - 13196x^2 + 16629x + 714
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{5}\cdot 3^{7} $
Twist minimal:	no (minimal twist has level 180)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$-13.9701$ of defining polynomial
Character	$\chi$	$=$	1620.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-5.00000 q^{5} +8.37861 q^{7} +O(q^{10})$	$q-5.00000 q^{5} +8.37861 q^{7} -9.42075 q^{11} +43.8332 q^{13} +73.3552 q^{17} +54.5132 q^{19} -181.040 q^{23} +25.0000 q^{25} +147.099 q^{29} +83.2355 q^{31} -41.8930 q^{35} -219.405 q^{37} -220.325 q^{41} +223.206 q^{43} -254.584 q^{47} -272.799 q^{49} +394.172 q^{53} +47.1038 q^{55} -230.588 q^{59} +899.238 q^{61} -219.166 q^{65} -643.620 q^{67} -132.733 q^{71} +543.219 q^{73} -78.9328 q^{77} -390.403 q^{79} +434.327 q^{83} -366.776 q^{85} +218.569 q^{89} +367.262 q^{91} -272.566 q^{95} +1777.14 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 7 q - 35 q^{5} + 8 q^{7}+O(q^{10}) $ 7 * q - 35 * q^5 + 8 * q^7	$ 7 q - 35 q^{5} + 8 q^{7} - 27 q^{11} + 32 q^{13} - 123 q^{17} - 67 q^{19} - 42 q^{23} + 175 q^{25} - 324 q^{29} + 98 q^{31} - 40 q^{35} + 356 q^{37} - 339 q^{41} + 119 q^{43} + 96 q^{47} + 813 q^{49} - 858 q^{53} + 135 q^{55} - 549 q^{59} + 260 q^{61} - 160 q^{65} + 881 q^{67} - 342 q^{71} + 737 q^{73} + 456 q^{77} + 1886 q^{79} + 132 q^{83} + 615 q^{85} - 396 q^{89} + 1264 q^{91} + 335 q^{95} + 1991 q^{97}+O(q^{100}) $ 7 * q - 35 * q^5 + 8 * q^7 - 27 * q^11 + 32 * q^13 - 123 * q^17 - 67 * q^19 - 42 * q^23 + 175 * q^25 - 324 * q^29 + 98 * q^31 - 40 * q^35 + 356 * q^37 - 339 * q^41 + 119 * q^43 + 96 * q^47 + 813 * q^49 - 858 * q^53 + 135 * q^55 - 549 * q^59 + 260 * q^61 - 160 * q^65 + 881 * q^67 - 342 * q^71 + 737 * q^73 + 456 * q^77 + 1886 * q^79 + 132 * q^83 + 615 * q^85 - 396 * q^89 + 1264 * q^91 + 335 * q^95 + 1991 * q^97

Display 100 coefficients 10 coefficients

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
$7$	8.37861	0.452402	0.226201	−	0.974081i	$-0.427369\pi$
$7$	8.37861	0.452402	0.226201	+	0.974081i	$0.427369\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−9.42075	−0.258224	−0.129112	−	0.991630i	$-0.541213\pi$
$11$	−9.42075	−0.258224	−0.129112	+	0.991630i	$0.541213\pi$
$12$	0	0
$13$	43.8332	0.935166	0.467583	−	0.883949i	$-0.345125\pi$
$13$	43.8332	0.935166	0.467583	+	0.883949i	$0.345125\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	73.3552	1.04654	0.523272	−	0.852166i	$-0.324711\pi$
$17$	73.3552	1.04654	0.523272	+	0.852166i	$0.324711\pi$
$18$	0	0
$19$	54.5132	0.658220	0.329110	−	0.944292i	$-0.393251\pi$
$19$	54.5132	0.658220	0.329110	+	0.944292i	$0.393251\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−181.040	−1.64128	−0.820640	−	0.571446i	$-0.806383\pi$
$23$	−181.040	−1.64128	−0.820640	+	0.571446i	$0.806383\pi$
$24$	0	0
$25$	25.0000	0.200000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	147.099	0.941915	0.470957	−	0.882156i	$-0.343909\pi$
$29$	147.099	0.941915	0.470957	+	0.882156i	$0.343909\pi$
$30$	0	0
$31$	83.2355	0.482243	0.241122	−	0.970495i	$-0.422485\pi$
$31$	83.2355	0.482243	0.241122	+	0.970495i	$0.422485\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−41.8930	−0.202320
$36$	0	0
$37$	−219.405	−0.974863	−0.487431	−	0.873161i	$-0.662066\pi$
$37$	−219.405	−0.974863	−0.487431	+	0.873161i	$0.662066\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−220.325	−0.839242	−0.419621	−	0.907699i	$-0.637837\pi$
$41$	−220.325	−0.839242	−0.419621	+	0.907699i	$0.637837\pi$
$42$	0	0
$43$	223.206	0.791594	0.395797	−	0.918338i	$-0.370468\pi$
$43$	223.206	0.791594	0.395797	+	0.918338i	$0.370468\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−254.584	−0.790105	−0.395052	−	0.918659i	$-0.629274\pi$
$47$	−254.584	−0.790105	−0.395052	+	0.918659i	$0.629274\pi$
$48$	0	0
$49$	−272.799	−0.795332
$50$	0	0
$51$	0	0
$52$	0	0
$53$	394.172	1.02158	0.510790	−	0.859706i	$-0.329353\pi$
$53$	394.172	1.02158	0.510790	+	0.859706i	$0.329353\pi$
$54$	0	0
$55$	47.1038	0.115481
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−230.588	−0.508814	−0.254407	−	0.967097i	$-0.581880\pi$
$59$	−230.588	−0.508814	−0.254407	+	0.967097i	$0.581880\pi$
$60$	0	0
$61$	899.238	1.88747	0.943735	−	0.330702i	$-0.107286\pi$
$61$	899.238	1.88747	0.943735	+	0.330702i	$0.107286\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−219.166	−0.418219
$66$	0	0
$67$	−643.620	−1.17359	−0.586797	−	0.809734i	$-0.699611\pi$
$67$	−643.620	−1.17359	−0.586797	+	0.809734i	$0.699611\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−132.733	−0.221866	−0.110933	−	0.993828i	$-0.535384\pi$
$71$	−132.733	−0.221866	−0.110933	+	0.993828i	$0.535384\pi$
$72$	0	0
$73$	543.219	0.870944	0.435472	−	0.900202i	$-0.356581\pi$
$73$	543.219	0.870944	0.435472	+	0.900202i	$0.356581\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−78.9328	−0.116821
$78$	0	0
$79$	−390.403	−0.555997	−0.277999	−	0.960581i	$-0.589671\pi$
$79$	−390.403	−0.555997	−0.277999	+	0.960581i	$0.589671\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	434.327	0.574381	0.287190	−	0.957873i	$-0.407279\pi$
$83$	434.327	0.574381	0.287190	+	0.957873i	$0.407279\pi$
$84$	0	0
$85$	−366.776	−0.468028
$86$	0	0
$87$	0	0
$88$	0	0
$89$	218.569	0.260317	0.130159	−	0.991493i	$-0.458451\pi$
$89$	218.569	0.260317	0.130159	+	0.991493i	$0.458451\pi$
$90$	0	0
$91$	367.262	0.423071
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−272.566	−0.294365
$96$	0	0
$97$	1777.14	1.86022	0.930108	−	0.367286i	$-0.119713\pi$
$97$	1777.14	1.86022	0.930108	+	0.367286i	$0.119713\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	328.805	0.323933	0.161967	−	0.986796i	$-0.448216\pi$
$101$	328.805	0.323933	0.161967	+	0.986796i	$0.448216\pi$
$102$	0	0
$103$	287.133	0.274680	0.137340	−	0.990524i	$-0.456145\pi$
$103$	287.133	0.274680	0.137340	+	0.990524i	$0.456145\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	1751.54	1.58250	0.791251	−	0.611492i	$-0.209430\pi$
$107$	1751.54	1.58250	0.791251	+	0.611492i	$0.209430\pi$
$108$	0	0
$109$	−131.439	−0.115501	−0.0577504	−	0.998331i	$-0.518393\pi$
$109$	−131.439	−0.115501	−0.0577504	+	0.998331i	$0.518393\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−178.824	−0.148870	−0.0744351	−	0.997226i	$-0.523715\pi$
$113$	−178.824	−0.148870	−0.0744351	+	0.997226i	$0.523715\pi$
$114$	0	0
$115$	905.199	0.734002
$116$	0	0
$117$	0	0
$118$	0	0
$119$	614.614	0.473459
$120$	0	0
$121$	−1242.25	−0.933320
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−125.000	−0.0894427
$126$	0	0
$127$	795.324	0.555698	0.277849	−	0.960625i	$-0.410379\pi$
$127$	795.324	0.555698	0.277849	+	0.960625i	$0.410379\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−451.066	−0.300838	−0.150419	−	0.988622i	$-0.548062\pi$
$131$	−451.066	−0.300838	−0.150419	+	0.988622i	$0.548062\pi$
$132$	0	0
$133$	456.744	0.297780
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−1543.04	−0.962267	−0.481133	−	0.876647i	$-0.659775\pi$
$137$	−1543.04	−0.962267	−0.481133	+	0.876647i	$0.659775\pi$
$138$	0	0
$139$	2556.83	1.56020	0.780098	−	0.625657i	$-0.215169\pi$
$139$	2556.83	1.56020	0.780098	+	0.625657i	$0.215169\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−412.942	−0.241482
$144$	0	0
$145$	−735.493	−0.421237
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−2755.80	−1.51519	−0.757596	−	0.652724i	$-0.773626\pi$
$149$	−2755.80	−1.51519	−0.757596	+	0.652724i	$0.773626\pi$
$150$	0	0
$151$	−313.671	−0.169047	−0.0845237	−	0.996421i	$-0.526937\pi$
$151$	−313.671	−0.169047	−0.0845237	+	0.996421i	$0.526937\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−416.178	−0.215666
$156$	0	0
$157$	3116.54	1.58425	0.792123	−	0.610361i	$-0.208976\pi$
$157$	3116.54	1.58425	0.792123	+	0.610361i	$0.208976\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−1516.86	−0.742518
$162$	0	0
$163$	891.274	0.428282	0.214141	−	0.976803i	$-0.431305\pi$
$163$	891.274	0.428282	0.214141	+	0.976803i	$0.431305\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	2738.33	1.26885	0.634427	−	0.772983i	$-0.281236\pi$
$167$	2738.33	1.26885	0.634427	+	0.772983i	$0.281236\pi$
$168$	0	0
$169$	−275.647	−0.125465
$170$	0	0
$171$	0	0
$172$	0	0
$173$	1538.68	0.676204	0.338102	−	0.941109i	$-0.390215\pi$
$173$	1538.68	0.676204	0.338102	+	0.941109i	$0.390215\pi$
$174$	0	0
$175$	209.465	0.0904805
$176$	0	0
$177$	0	0
$178$	0	0
$179$	107.794	0.0450108	0.0225054	−	0.999747i	$-0.492836\pi$
$179$	107.794	0.0450108	0.0225054	+	0.999747i	$0.492836\pi$
$180$	0	0
$181$	−483.341	−0.198489	−0.0992443	−	0.995063i	$-0.531643\pi$
$181$	−483.341	−0.198489	−0.0992443	+	0.995063i	$0.531643\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	1097.02	0.435972
$186$	0	0
$187$	−691.061	−0.270243
$188$	0	0
$189$	0	0
$190$	0	0
$191$	3289.22	1.24607	0.623036	−	0.782193i	$-0.285899\pi$
$191$	3289.22	1.24607	0.623036	+	0.782193i	$0.285899\pi$
$192$	0	0
$193$	2731.08	1.01859	0.509293	−	0.860593i	$-0.329907\pi$
$193$	2731.08	1.01859	0.509293	+	0.860593i	$0.329907\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	2666.54	0.964381	0.482191	−	0.876066i	$-0.339841\pi$
$197$	2666.54	0.964381	0.482191	+	0.876066i	$0.339841\pi$
$198$	0	0
$199$	5421.50	1.93126	0.965628	−	0.259926i	$-0.0836983\pi$
$199$	5421.50	1.93126	0.965628	+	0.259926i	$0.0836983\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	1232.48	0.426124
$204$	0	0
$205$	1101.62	0.375321
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−513.555	−0.169968
$210$	0	0
$211$	323.759	0.105633	0.0528164	−	0.998604i	$-0.483180\pi$
$211$	323.759	0.105633	0.0528164	+	0.998604i	$0.483180\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−1116.03	−0.354012
$216$	0	0
$217$	697.398	0.218168
$218$	0	0
$219$	0	0
$220$	0	0
$221$	3215.39	0.978692
$222$	0	0
$223$	−1569.34	−0.471258	−0.235629	−	0.971843i	$-0.575715\pi$
$223$	−1569.34	−0.471258	−0.235629	+	0.971843i	$0.575715\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−4071.24	−1.19039	−0.595194	−	0.803582i	$-0.702925\pi$
$227$	−4071.24	−1.19039	−0.595194	+	0.803582i	$0.702925\pi$
$228$	0	0
$229$	5147.56	1.48542	0.742708	−	0.669616i	$-0.233541\pi$
$229$	5147.56	1.48542	0.742708	+	0.669616i	$0.233541\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−2640.93	−0.742546	−0.371273	−	0.928524i	$-0.621079\pi$
$233$	−2640.93	−0.742546	−0.371273	+	0.928524i	$0.621079\pi$
$234$	0	0
$235$	1272.92	0.353346
$236$	0	0
$237$	0	0
$238$	0	0
$239$	5000.70	1.35342	0.676712	−	0.736248i	$-0.263404\pi$
$239$	5000.70	1.35342	0.676712	+	0.736248i	$0.263404\pi$
$240$	0	0
$241$	6677.46	1.78479	0.892393	−	0.451260i	$-0.149025\pi$
$241$	6677.46	1.78479	0.892393	+	0.451260i	$0.149025\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	1363.99	0.355683
$246$	0	0
$247$	2389.49	0.615545
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−4227.22	−1.06303	−0.531513	−	0.847050i	$-0.678376\pi$
$251$	−4227.22	−1.06303	−0.531513	+	0.847050i	$0.678376\pi$
$252$	0	0
$253$	1705.53	0.423818
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−2193.36	−0.532366	−0.266183	−	0.963923i	$-0.585763\pi$
$257$	−2193.36	−0.532366	−0.266183	+	0.963923i	$0.585763\pi$
$258$	0	0
$259$	−1838.31	−0.441030
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−8109.22	−1.90128	−0.950639	−	0.310300i	$-0.899571\pi$
$263$	−8109.22	−1.90128	−0.950639	+	0.310300i	$0.899571\pi$
$264$	0	0
$265$	−1970.86	−0.456864
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−5721.90	−1.29692	−0.648458	−	0.761251i	$-0.724586\pi$
$269$	−5721.90	−1.29692	−0.648458	+	0.761251i	$0.724586\pi$
$270$	0	0
$271$	4717.66	1.05748	0.528741	−	0.848783i	$-0.322664\pi$
$271$	4717.66	1.05748	0.528741	+	0.848783i	$0.322664\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−235.519	−0.0516448
$276$	0	0
$277$	−5861.75	−1.27147	−0.635737	−	0.771906i	$-0.719304\pi$
$277$	−5861.75	−1.27147	−0.635737	+	0.771906i	$0.719304\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	6936.41	1.47257	0.736284	−	0.676673i	$-0.236579\pi$
$281$	6936.41	1.47257	0.736284	+	0.676673i	$0.236579\pi$
$282$	0	0
$283$	2684.67	0.563912	0.281956	−	0.959427i	$-0.409017\pi$
$283$	2684.67	0.563912	0.281956	+	0.959427i	$0.409017\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−1846.01	−0.379675
$288$	0	0
$289$	467.978	0.0952531
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−1453.38	−0.289787	−0.144893	−	0.989447i	$-0.546284\pi$
$293$	−1453.38	−0.289787	−0.144893	+	0.989447i	$0.546284\pi$
$294$	0	0
$295$	1152.94	0.227548
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−7935.56	−1.53487
$300$	0	0
$301$	1870.15	0.358119
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−4496.19	−0.844102
$306$	0	0
$307$	4622.83	0.859410	0.429705	−	0.902969i	$-0.358618\pi$
$307$	4622.83	0.859410	0.429705	+	0.902969i	$0.358618\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	9055.50	1.65109	0.825547	−	0.564333i	$-0.190866\pi$
$311$	9055.50	1.65109	0.825547	+	0.564333i	$0.190866\pi$
$312$	0	0
$313$	553.967	0.100039	0.0500193	−	0.998748i	$-0.484072\pi$
$313$	553.967	0.100039	0.0500193	+	0.998748i	$0.484072\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−445.230	−0.0788853	−0.0394426	−	0.999222i	$-0.512558\pi$
$317$	−445.230	−0.0788853	−0.0394426	+	0.999222i	$0.512558\pi$
$318$	0	0
$319$	−1385.78	−0.243225
$320$	0	0
$321$	0	0
$322$	0	0
$323$	3998.82	0.688856
$324$	0	0
$325$	1095.83	0.187033
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−2133.06	−0.357445
$330$	0	0
$331$	8684.44	1.44211	0.721057	−	0.692876i	$-0.243657\pi$
$331$	8684.44	1.44211	0.721057	+	0.692876i	$0.243657\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	3218.10	0.524847
$336$	0	0
$337$	10871.6	1.75732	0.878659	−	0.477450i	$-0.158439\pi$
$337$	10871.6	1.75732	0.878659	+	0.477450i	$0.158439\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−784.141	−0.124527
$342$	0	0
$343$	−5159.54	−0.812212
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−2177.07	−0.336805	−0.168403	−	0.985718i	$-0.553861\pi$
$347$	−2177.07	−0.336805	−0.168403	+	0.985718i	$0.553861\pi$
$348$	0	0
$349$	4339.24	0.665542	0.332771	−	0.943008i	$-0.392016\pi$
$349$	4339.24	0.665542	0.332771	+	0.943008i	$0.392016\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	7251.13	1.09331	0.546655	−	0.837358i	$-0.315901\pi$
$353$	7251.13	1.09331	0.546655	+	0.837358i	$0.315901\pi$
$354$	0	0
$355$	663.663	0.0992214
$356$	0	0
$357$	0	0
$358$	0	0
$359$	12945.8	1.90322	0.951608	−	0.307313i	$-0.0994300\pi$
$359$	12945.8	1.90322	0.951608	+	0.307313i	$0.0994300\pi$
$360$	0	0
$361$	−3887.31	−0.566746
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−2716.09	−0.389498
$366$	0	0
$367$	−13353.5	−1.89932	−0.949658	−	0.313288i	$-0.898570\pi$
$367$	−13353.5	−1.89932	−0.949658	+	0.313288i	$0.898570\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	3302.61	0.462165
$372$	0	0
$373$	−1236.93	−0.171705	−0.0858523	−	0.996308i	$-0.527361\pi$
$373$	−1236.93	−0.171705	−0.0858523	+	0.996308i	$0.527361\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	6447.81	0.880847
$378$	0	0
$379$	−7278.48	−0.986466	−0.493233	−	0.869897i	$-0.664185\pi$
$379$	−7278.48	−0.986466	−0.493233	+	0.869897i	$0.664185\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−5009.86	−0.668387	−0.334193	−	0.942505i	$-0.608464\pi$
$383$	−5009.86	−0.668387	−0.334193	+	0.942505i	$0.608464\pi$
$384$	0	0
$385$	394.664	0.0522440
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−1607.90	−0.209573	−0.104787	−	0.994495i	$-0.533416\pi$
$389$	−1607.90	−0.209573	−0.104787	+	0.994495i	$0.533416\pi$
$390$	0	0
$391$	−13280.2	−1.71767
$392$	0	0
$393$	0	0
$394$	0	0
$395$	1952.02	0.248649
$396$	0	0
$397$	10.6225	0.00134289	0.000671444	−	1.00000i	$-0.499786\pi$
$397$	10.6225	0.00134289	0.000671444		1.00000i	$0.499786\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−9803.91	−1.22091	−0.610454	−	0.792052i	$-0.709013\pi$
$401$	−9803.91	−1.22091	−0.610454	+	0.792052i	$0.709013\pi$
$402$	0	0
$403$	3648.48	0.450977
$404$	0	0
$405$	0	0
$406$	0	0
$407$	2066.96	0.251733
$408$	0	0
$409$	1277.19	0.154408	0.0772041	−	0.997015i	$-0.475401\pi$
$409$	1277.19	0.154408	0.0772041	+	0.997015i	$0.475401\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−1932.01	−0.230189
$414$	0	0
$415$	−2171.64	−0.256871
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−742.758	−0.0866017	−0.0433008	−	0.999062i	$-0.513787\pi$
$419$	−742.758	−0.0866017	−0.0433008	+	0.999062i	$0.513787\pi$
$420$	0	0
$421$	14299.4	1.65537	0.827684	−	0.561194i	$-0.189658\pi$
$421$	14299.4	1.65537	0.827684	+	0.561194i	$0.189658\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	1833.88	0.209309
$426$	0	0
$427$	7534.37	0.853896
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−13923.5	−1.55608	−0.778040	−	0.628215i	$-0.783786\pi$
$431$	−13923.5	−1.55608	−0.778040	+	0.628215i	$0.783786\pi$
$432$	0	0
$433$	17480.4	1.94008	0.970038	−	0.242953i	$-0.0781160\pi$
$433$	17480.4	1.94008	0.970038	+	0.242953i	$0.0781160\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−9869.06	−1.08032
$438$	0	0
$439$	−5618.39	−0.610823	−0.305411	−	0.952220i	$-0.598794\pi$
$439$	−5618.39	−0.610823	−0.305411	+	0.952220i	$0.598794\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−17134.4	−1.83766	−0.918828	−	0.394659i	$-0.870863\pi$
$443$	−17134.4	−1.83766	−0.918828	+	0.394659i	$0.870863\pi$
$444$	0	0
$445$	−1092.84	−0.116417
$446$	0	0
$447$	0	0
$448$	0	0
$449$	17900.8	1.88150	0.940748	−	0.339107i	$-0.110125\pi$
$449$	17900.8	1.88150	0.940748	+	0.339107i	$0.110125\pi$
$450$	0	0
$451$	2075.62	0.216712
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−1836.31	−0.189203
$456$	0	0
$457$	5763.63	0.589959	0.294979	−	0.955504i	$-0.404687\pi$
$457$	5763.63	0.589959	0.294979	+	0.955504i	$0.404687\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	15004.7	1.51592	0.757958	−	0.652304i	$-0.226197\pi$
$461$	15004.7	1.51592	0.757958	+	0.652304i	$0.226197\pi$
$462$	0	0
$463$	−5405.41	−0.542572	−0.271286	−	0.962499i	$-0.587449\pi$
$463$	−5405.41	−0.542572	−0.271286	+	0.962499i	$0.587449\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	15900.9	1.57560	0.787802	−	0.615928i	$-0.211219\pi$
$467$	15900.9	1.57560	0.787802	+	0.615928i	$0.211219\pi$
$468$	0	0
$469$	−5392.64	−0.530936
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−2102.77	−0.204409
$474$	0	0
$475$	1362.83	0.131644
$476$	0	0
$477$	0	0
$478$	0	0
$479$	4329.76	0.413010	0.206505	−	0.978446i	$-0.433791\pi$
$479$	4329.76	0.413010	0.206505	+	0.978446i	$0.433791\pi$
$480$	0	0
$481$	−9617.22	−0.911658
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−8885.69	−0.831914
$486$	0	0
$487$	7831.85	0.728737	0.364369	−	0.931255i	$-0.381285\pi$
$487$	7831.85	0.728737	0.364369	+	0.931255i	$0.381285\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	7443.34	0.684141	0.342070	−	0.939674i	$-0.388872\pi$
$491$	7443.34	0.684141	0.342070	+	0.939674i	$0.388872\pi$
$492$	0	0
$493$	10790.4	0.985755
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−1112.12	−0.100373
$498$	0	0
$499$	−11603.8	−1.04100	−0.520500	−	0.853862i	$-0.674254\pi$
$499$	−11603.8	−1.04100	−0.520500	+	0.853862i	$0.674254\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−9141.05	−0.810297	−0.405148	−	0.914251i	$-0.632780\pi$
$503$	−9141.05	−0.810297	−0.405148	+	0.914251i	$0.632780\pi$
$504$	0	0
$505$	−1644.02	−0.144867
$506$	0	0
$507$	0	0
$508$	0	0
$509$	10028.2	0.873267	0.436633	−	0.899640i	$-0.356171\pi$
$509$	10028.2	0.873267	0.436633	+	0.899640i	$0.356171\pi$
$510$	0	0
$511$	4551.41	0.394017
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−1435.66	−0.122841
$516$	0	0
$517$	2398.37	0.204024
$518$	0	0
$519$	0	0
$520$	0	0
$521$	11375.9	0.956597	0.478299	−	0.878197i	$-0.341254\pi$
$521$	11375.9	0.956597	0.478299	+	0.878197i	$0.341254\pi$
$522$	0	0
$523$	−15476.8	−1.29398	−0.646990	−	0.762498i	$-0.723973\pi$
$523$	−15476.8	−1.29398	−0.646990	+	0.762498i	$0.723973\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	6105.75	0.504689
$528$	0	0
$529$	20608.4	1.69380
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−9657.55	−0.784831
$534$	0	0
$535$	−8757.69	−0.707716
$536$	0	0
$537$	0	0
$538$	0	0
$539$	2569.97	0.205374
$540$	0	0
$541$	−11290.1	−0.897228	−0.448614	−	0.893726i	$-0.648082\pi$
$541$	−11290.1	−0.897228	−0.448614	+	0.893726i	$0.648082\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	657.195	0.0516535
$546$	0	0
$547$	−13942.1	−1.08980	−0.544901	−	0.838500i	$-0.683433\pi$
$547$	−13942.1	−1.08980	−0.544901	+	0.838500i	$0.683433\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	8018.82	0.619987
$552$	0	0
$553$	−3271.03	−0.251534
$554$	0	0
$555$	0	0
$556$	0	0
$557$	7078.85	0.538492	0.269246	−	0.963071i	$-0.413225\pi$
$557$	7078.85	0.538492	0.269246	+	0.963071i	$0.413225\pi$
$558$	0	0
$559$	9783.83	0.740272
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−14086.7	−1.05450	−0.527250	−	0.849710i	$-0.676777\pi$
$563$	−14086.7	−1.05450	−0.527250	+	0.849710i	$0.676777\pi$
$564$	0	0
$565$	894.119	0.0665768
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−11769.6	−0.867148	−0.433574	−	0.901118i	$-0.642748\pi$
$569$	−11769.6	−0.867148	−0.433574	+	0.901118i	$0.642748\pi$
$570$	0	0
$571$	−20985.7	−1.53804	−0.769022	−	0.639222i	$-0.779257\pi$
$571$	−20985.7	−1.53804	−0.769022	+	0.639222i	$0.779257\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−4526.00	−0.328256
$576$	0	0
$577$	2441.88	0.176182	0.0880908	−	0.996112i	$-0.471923\pi$
$577$	2441.88	0.176182	0.0880908	+	0.996112i	$0.471923\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	3639.06	0.259851
$582$	0	0
$583$	−3713.40	−0.263796
$584$	0	0
$585$	0	0
$586$	0	0
$587$	6591.80	0.463497	0.231748	−	0.972776i	$-0.425555\pi$
$587$	6591.80	0.463497	0.231748	+	0.972776i	$0.425555\pi$
$588$	0	0
$589$	4537.43	0.317422
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−2453.72	−0.169919	−0.0849597	−	0.996384i	$-0.527076\pi$
$593$	−2453.72	−0.169919	−0.0849597	+	0.996384i	$0.527076\pi$
$594$	0	0
$595$	−3073.07	−0.211737
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−12699.8	−0.866280	−0.433140	−	0.901327i	$-0.642594\pi$
$599$	−12699.8	−0.866280	−0.433140	+	0.901327i	$0.642594\pi$
$600$	0	0
$601$	−7275.44	−0.493796	−0.246898	−	0.969041i	$-0.579411\pi$
$601$	−7275.44	−0.493796	−0.246898	+	0.969041i	$0.579411\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	6211.25	0.417394
$606$	0	0
$607$	−9593.69	−0.641509	−0.320754	−	0.947162i	$-0.603936\pi$
$607$	−9593.69	−0.641509	−0.320754	+	0.947162i	$0.603936\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−11159.3	−0.738879
$612$	0	0
$613$	−15008.6	−0.988895	−0.494448	−	0.869207i	$-0.664630\pi$
$613$	−15008.6	−0.988895	−0.494448	+	0.869207i	$0.664630\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	11089.0	0.723546	0.361773	−	0.932266i	$-0.382172\pi$
$617$	11089.0	0.723546	0.361773	+	0.932266i	$0.382172\pi$
$618$	0	0
$619$	−28740.2	−1.86618	−0.933090	−	0.359644i	$-0.882898\pi$
$619$	−28740.2	−1.86618	−0.933090	+	0.359644i	$0.882898\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	1831.30	0.117768
$624$	0	0
$625$	625.000	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−16094.5	−1.02024
$630$	0	0
$631$	−6353.32	−0.400827	−0.200414	−	0.979711i	$-0.564229\pi$
$631$	−6353.32	−0.400827	−0.200414	+	0.979711i	$0.564229\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−3976.62	−0.248516
$636$	0	0
$637$	−11957.7	−0.743767
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−6335.16	−0.390365	−0.195182	−	0.980767i	$-0.562530\pi$
$641$	−6335.16	−0.390365	−0.195182	+	0.980767i	$0.562530\pi$
$642$	0	0
$643$	7179.89	0.440354	0.220177	−	0.975460i	$-0.429337\pi$
$643$	7179.89	0.440354	0.220177	+	0.975460i	$0.429337\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	14209.0	0.863393	0.431696	−	0.902019i	$-0.357915\pi$
$647$	14209.0	0.863393	0.431696	+	0.902019i	$0.357915\pi$
$648$	0	0
$649$	2172.31	0.131388
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−14994.6	−0.898596	−0.449298	−	0.893382i	$-0.648326\pi$
$653$	−14994.6	−0.898596	−0.449298	+	0.893382i	$0.648326\pi$
$654$	0	0
$655$	2255.33	0.134539
$656$	0	0
$657$	0	0
$658$	0	0
$659$	10305.9	0.609200	0.304600	−	0.952480i	$-0.401477\pi$
$659$	10305.9	0.609200	0.304600	+	0.952480i	$0.401477\pi$
$660$	0	0
$661$	6434.54	0.378630	0.189315	−	0.981916i	$-0.439373\pi$
$661$	6434.54	0.378630	0.189315	+	0.981916i	$0.439373\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−2283.72	−0.133171
$666$	0	0
$667$	−26630.7	−1.54595
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−8471.50	−0.487390
$672$	0	0
$673$	−33034.8	−1.89212	−0.946061	−	0.323988i	$-0.894976\pi$
$673$	−33034.8	−1.89212	−0.946061	+	0.323988i	$0.894976\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−7217.06	−0.409711	−0.204855	−	0.978792i	$-0.565672\pi$
$677$	−7217.06	−0.409711	−0.204855	+	0.978792i	$0.565672\pi$
$678$	0	0
$679$	14889.9	0.841566
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−5868.66	−0.328782	−0.164391	−	0.986395i	$-0.552566\pi$
$683$	−5868.66	−0.328782	−0.164391	+	0.986395i	$0.552566\pi$
$684$	0	0
$685$	7715.18	0.430339
$686$	0	0
$687$	0	0
$688$	0	0
$689$	17277.8	0.955346
$690$	0	0
$691$	−13651.0	−0.751530	−0.375765	−	0.926715i	$-0.622620\pi$
$691$	−13651.0	−0.751530	−0.375765	+	0.926715i	$0.622620\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−12784.1	−0.697741
$696$	0	0
$697$	−16162.0	−0.878303
$698$	0	0
$699$	0	0
$700$	0	0
$701$	9374.18	0.505076	0.252538	−	0.967587i	$-0.418735\pi$
$701$	9374.18	0.505076	0.252538	+	0.967587i	$0.418735\pi$
$702$	0	0
$703$	−11960.5	−0.641674
$704$	0	0
$705$	0	0
$706$	0	0
$707$	2754.92	0.146548
$708$	0	0
$709$	−23608.1	−1.25052	−0.625261	−	0.780415i	$-0.715008\pi$
$709$	−23608.1	−1.25052	−0.625261	+	0.780415i	$0.715008\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−15068.9	−0.791496
$714$	0	0
$715$	2064.71	0.107994
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−6884.80	−0.357107	−0.178553	−	0.983930i	$-0.557142\pi$
$719$	−6884.80	−0.357107	−0.178553	+	0.983930i	$0.557142\pi$
$720$	0	0
$721$	2405.77	0.124266
$722$	0	0
$723$	0	0
$724$	0	0
$725$	3677.47	0.188383
$726$	0	0
$727$	−1960.86	−0.100033	−0.0500166	−	0.998748i	$-0.515927\pi$
$727$	−1960.86	−0.100033	−0.0500166	+	0.998748i	$0.515927\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	16373.3	0.828438
$732$	0	0
$733$	−4372.69	−0.220340	−0.110170	−	0.993913i	$-0.535139\pi$
$733$	−4372.69	−0.220340	−0.110170	+	0.993913i	$0.535139\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	6063.39	0.303050
$738$	0	0
$739$	−3139.32	−0.156268	−0.0781339	−	0.996943i	$-0.524896\pi$
$739$	−3139.32	−0.156268	−0.0781339	+	0.996943i	$0.524896\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	5681.61	0.280536	0.140268	−	0.990114i	$-0.455204\pi$
$743$	5681.61	0.280536	0.140268	+	0.990114i	$0.455204\pi$
$744$	0	0
$745$	13779.0	0.677615
$746$	0	0
$747$	0	0
$748$	0	0
$749$	14675.5	0.715927
$750$	0	0
$751$	19371.1	0.941227	0.470613	−	0.882340i	$-0.344033\pi$
$751$	19371.1	0.941227	0.470613	+	0.882340i	$0.344033\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	1568.35	0.0756003
$756$	0	0
$757$	−21528.2	−1.03363	−0.516815	−	0.856097i	$-0.672882\pi$
$757$	−21528.2	−1.03363	−0.516815	+	0.856097i	$0.672882\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	31554.2	1.50307	0.751537	−	0.659691i	$-0.229313\pi$
$761$	31554.2	1.50307	0.751537	+	0.659691i	$0.229313\pi$
$762$	0	0
$763$	−1101.28	−0.0522528
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−10107.4	−0.475825
$768$	0	0
$769$	8841.52	0.414608	0.207304	−	0.978277i	$-0.433531\pi$
$769$	8841.52	0.414608	0.207304	+	0.978277i	$0.433531\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−36891.5	−1.71655	−0.858276	−	0.513189i	$-0.828464\pi$
$773$	−36891.5	−1.71655	−0.858276	+	0.513189i	$0.828464\pi$
$774$	0	0
$775$	2080.89	0.0964487
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−12010.6	−0.552406
$780$	0	0
$781$	1250.44	0.0572911
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−15582.7	−0.708496
$786$	0	0
$787$	7476.33	0.338631	0.169315	−	0.985562i	$-0.445844\pi$
$787$	7476.33	0.338631	0.169315	+	0.985562i	$0.445844\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−1498.30	−0.0673492
$792$	0	0
$793$	39416.5	1.76510
$794$	0	0
$795$	0	0
$796$	0	0
$797$	25834.3	1.14818	0.574089	−	0.818793i	$-0.305356\pi$
$797$	25834.3	1.14818	0.574089	+	0.818793i	$0.305356\pi$
$798$	0	0
$799$	−18675.1	−0.826879
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−5117.53	−0.224899
$804$	0	0
$805$	7584.31	0.332064
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−4632.87	−0.201339	−0.100669	−	0.994920i	$-0.532098\pi$
$809$	−4632.87	−0.201339	−0.100669	+	0.994920i	$0.532098\pi$
$810$	0	0
$811$	−11882.9	−0.514507	−0.257253	−	0.966344i	$-0.582818\pi$
$811$	−11882.9	−0.514507	−0.257253	+	0.966344i	$0.582818\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−4456.37	−0.191533
$816$	0	0
$817$	12167.7	0.521043
$818$	0	0
$819$	0	0
$820$	0	0
$821$	27585.4	1.17264	0.586320	−	0.810080i	$-0.300576\pi$
$821$	27585.4	1.17264	0.586320	+	0.810080i	$0.300576\pi$
$822$	0	0
$823$	−4125.81	−0.174747	−0.0873735	−	0.996176i	$-0.527847\pi$
$823$	−4125.81	−0.174747	−0.0873735	+	0.996176i	$0.527847\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−39352.4	−1.65468	−0.827338	−	0.561704i	$-0.810146\pi$
$827$	−39352.4	−1.65468	−0.827338	+	0.561704i	$0.810146\pi$
$828$	0	0
$829$	−12875.6	−0.539431	−0.269716	−	0.962940i	$-0.586930\pi$
$829$	−12875.6	−0.539431	−0.269716	+	0.962940i	$0.586930\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−20011.2	−0.832350
$834$	0	0
$835$	−13691.7	−0.567448
$836$	0	0
$837$	0	0
$838$	0	0
$839$	25008.9	1.02908	0.514542	−	0.857465i	$-0.327962\pi$
$839$	25008.9	1.02908	0.514542	+	0.857465i	$0.327962\pi$
$840$	0	0
$841$	−2750.99	−0.112796
$842$	0	0
$843$	0	0
$844$	0	0
$845$	1378.23	0.0561097
$846$	0	0
$847$	−10408.3	−0.422236
$848$	0	0
$849$	0	0
$850$	0	0
$851$	39721.0	1.60002
$852$	0	0
$853$	−47646.2	−1.91252	−0.956258	−	0.292524i	$-0.905505\pi$
$853$	−47646.2	−1.91252	−0.956258	+	0.292524i	$0.905505\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−32401.5	−1.29150	−0.645749	−	0.763549i	$-0.723455\pi$
$857$	−32401.5	−1.29150	−0.645749	+	0.763549i	$0.723455\pi$
$858$	0	0
$859$	−46288.3	−1.83858	−0.919288	−	0.393585i	$-0.871235\pi$
$859$	−46288.3	−1.83858	−0.919288	+	0.393585i	$0.871235\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−23841.4	−0.940405	−0.470203	−	0.882558i	$-0.655819\pi$
$863$	−23841.4	−0.940405	−0.470203	+	0.882558i	$0.655819\pi$
$864$	0	0
$865$	−7693.38	−0.302408
$866$	0	0
$867$	0	0
$868$	0	0
$869$	3677.89	0.143572
$870$	0	0
$871$	−28212.0	−1.09750
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−1047.33	−0.0404641
$876$	0	0
$877$	−10683.3	−0.411345	−0.205672	−	0.978621i	$-0.565938\pi$
$877$	−10683.3	−0.411345	−0.205672	+	0.978621i	$0.565938\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	36663.4	1.40207	0.701033	−	0.713128i	$-0.252722\pi$
$881$	36663.4	1.40207	0.701033	+	0.713128i	$0.252722\pi$
$882$	0	0
$883$	−18727.5	−0.713737	−0.356869	−	0.934155i	$-0.616156\pi$
$883$	−18727.5	−0.713737	−0.356869	+	0.934155i	$0.616156\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	38451.9	1.45557	0.727784	−	0.685806i	$-0.240550\pi$
$887$	38451.9	1.45557	0.727784	+	0.685806i	$0.240550\pi$
$888$	0	0
$889$	6663.71	0.251399
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−13878.2	−0.520063
$894$	0	0
$895$	−538.972	−0.0201294
$896$	0	0
$897$	0	0
$898$	0	0
$899$	12243.8	0.454232
$900$	0	0
$901$	28914.6	1.06913
$902$	0	0
$903$	0	0
$904$	0	0
$905$	2416.70	0.0887668
$906$	0	0
$907$	29143.3	1.06691	0.533455	−	0.845829i	$-0.320893\pi$
$907$	29143.3	1.06691	0.533455	+	0.845829i	$0.320893\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−12449.7	−0.452774	−0.226387	−	0.974037i	$-0.572691\pi$
$911$	−12449.7	−0.452774	−0.226387	+	0.974037i	$0.572691\pi$
$912$	0	0
$913$	−4091.69	−0.148319
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−3779.31	−0.136100
$918$	0	0
$919$	−13465.5	−0.483338	−0.241669	−	0.970359i	$-0.577695\pi$
$919$	−13465.5	−0.483338	−0.241669	+	0.970359i	$0.577695\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−5818.10	−0.207481
$924$	0	0
$925$	−5485.12	−0.194973
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−8808.96	−0.311101	−0.155550	−	0.987828i	$-0.549715\pi$
$929$	−8808.96	−0.311101	−0.155550	+	0.987828i	$0.549715\pi$
$930$	0	0
$931$	−14871.1	−0.523504
$932$	0	0
$933$	0	0
$934$	0	0
$935$	3455.30	0.120856
$936$	0	0
$937$	46042.9	1.60529	0.802644	−	0.596459i	$-0.203426\pi$
$937$	46042.9	1.60529	0.802644	+	0.596459i	$0.203426\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	26177.4	0.906862	0.453431	−	0.891291i	$-0.350200\pi$
$941$	26177.4	0.906862	0.453431	+	0.891291i	$0.350200\pi$
$942$	0	0
$943$	39887.5	1.37743
$944$	0	0
$945$	0	0
$946$	0	0
$947$	26655.3	0.914659	0.457329	−	0.889297i	$-0.348806\pi$
$947$	26655.3	0.914659	0.457329	+	0.889297i	$0.348806\pi$
$948$	0	0
$949$	23811.0	0.814477
$950$	0	0
$951$	0	0
$952$	0	0
$953$	6645.34	0.225880	0.112940	−	0.993602i	$-0.463973\pi$
$953$	6645.34	0.225880	0.112940	+	0.993602i	$0.463973\pi$
$954$	0	0
$955$	−16446.1	−0.557260
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−12928.5	−0.435332
$960$	0	0
$961$	−22862.8	−0.767441
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−13655.4	−0.455526
$966$	0	0
$967$	−42169.6	−1.40236	−0.701180	−	0.712984i	$-0.747343\pi$
$967$	−42169.6	−1.40236	−0.701180	+	0.712984i	$0.747343\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	49330.3	1.63037	0.815183	−	0.579203i	$-0.196636\pi$
$971$	49330.3	1.63037	0.815183	+	0.579203i	$0.196636\pi$
$972$	0	0
$973$	21422.7	0.705837
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−45153.6	−1.47860	−0.739300	−	0.673376i	$-0.764843\pi$
$977$	−45153.6	−1.47860	−0.739300	+	0.673376i	$0.764843\pi$
$978$	0	0
$979$	−2059.08	−0.0672201
$980$	0	0
$981$	0	0
$982$	0	0
$983$	15898.3	0.515848	0.257924	−	0.966165i	$-0.416962\pi$
$983$	15898.3	0.515848	0.257924	+	0.966165i	$0.416962\pi$
$984$	0	0
$985$	−13332.7	−0.431284
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−40409.1	−1.29923
$990$	0	0
$991$	−38449.5	−1.23248	−0.616241	−	0.787558i	$-0.711345\pi$
$991$	−38449.5	−1.23248	−0.616241	+	0.787558i	$0.711345\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−27107.5	−0.863684
$996$	0	0
$997$	15177.2	0.482113	0.241057	−	0.970511i	$-0.422506\pi$
$997$	15177.2	0.482113	0.241057	+	0.970511i	$0.422506\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1620.4.a.k.1.4		7
3.2	odd	2		1620.4.a.l.1.4		7
9.2	odd	6		180.4.i.c.121.3	yes	14
9.4	even	3		540.4.i.c.181.4		14
9.5	odd	6		180.4.i.c.61.3	✓	14
9.7	even	3		540.4.i.c.361.4		14

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
180.4.i.c.61.3	✓	14	9.5	odd	6
180.4.i.c.121.3	yes	14	9.2	odd	6
540.4.i.c.181.4		14	9.4	even	3
540.4.i.c.361.4		14	9.7	even	3
1620.4.a.k.1.4		7	1.1	even	1	trivial
1620.4.a.l.1.4		7	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 1620 = 2^{2} \cdot 3^{4} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1620.a (trivial)

\(f(q)\)	\(=\)	\(q-5.00000 q^{5} +8.37861 q^{7} +O(q^{10})\)	\(q-5.00000 q^{5} +8.37861 q^{7} -9.42075 q^{11} +43.8332 q^{13} +73.3552 q^{17} +54.5132 q^{19} -181.040 q^{23} +25.0000 q^{25} +147.099 q^{29} +83.2355 q^{31} -41.8930 q^{35} -219.405 q^{37} -220.325 q^{41} +223.206 q^{43} -254.584 q^{47} -272.799 q^{49} +394.172 q^{53} +47.1038 q^{55} -230.588 q^{59} +899.238 q^{61} -219.166 q^{65} -643.620 q^{67} -132.733 q^{71} +543.219 q^{73} -78.9328 q^{77} -390.403 q^{79} +434.327 q^{83} -366.776 q^{85} +218.569 q^{89} +367.262 q^{91} -272.566 q^{95} +1777.14 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q - 35 q^{5} + 8 q^{7}+O(q^{10}) \) 7 * q - 35 * q^5 + 8 * q^7	\( 7 q - 35 q^{5} + 8 q^{7} - 27 q^{11} + 32 q^{13} - 123 q^{17} - 67 q^{19} - 42 q^{23} + 175 q^{25} - 324 q^{29} + 98 q^{31} - 40 q^{35} + 356 q^{37} - 339 q^{41} + 119 q^{43} + 96 q^{47} + 813 q^{49} - 858 q^{53} + 135 q^{55} - 549 q^{59} + 260 q^{61} - 160 q^{65} + 881 q^{67} - 342 q^{71} + 737 q^{73} + 456 q^{77} + 1886 q^{79} + 132 q^{83} + 615 q^{85} - 396 q^{89} + 1264 q^{91} + 335 q^{95} + 1991 q^{97}+O(q^{100}) \) 7 * q - 35 * q^5 + 8 * q^7 - 27 * q^11 + 32 * q^13 - 123 * q^17 - 67 * q^19 - 42 * q^23 + 175 * q^25 - 324 * q^29 + 98 * q^31 - 40 * q^35 + 356 * q^37 - 339 * q^41 + 119 * q^43 + 96 * q^47 + 813 * q^49 - 858 * q^53 + 135 * q^55 - 549 * q^59 + 260 * q^61 - 160 * q^65 + 881 * q^67 - 342 * q^71 + 737 * q^73 + 456 * q^77 + 1886 * q^79 + 132 * q^83 + 615 * q^85 - 396 * q^89 + 1264 * q^91 + 335 * q^95 + 1991 * q^97

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