LMFDB - Embedded newform 2160.4.a.ba.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2160,4,Mod(1,2160)]

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

lf = mfeigenbasis(mf)

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")

//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);

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: f = N[0] # Warning: the index may be different

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

Self dual:	yes
Analytic conductor:	$127.444125612$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{69}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 17 $ x^2 - x - 17
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2\cdot 3 $
Twist minimal:	no (minimal twist has level 540)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$4.65331$ 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)

$f(q)$	$=$	$q+5.00000 q^{5} -29.9199 q^{7} +O(q^{10})$	$q+5.00000 q^{5} -29.9199 q^{7} +15.9199 q^{11} +41.9199 q^{13} +36.9199 q^{17} +10.9199 q^{19} -114.679 q^{23} +25.0000 q^{25} -145.599 q^{29} +46.4391 q^{31} -149.599 q^{35} +74.0000 q^{37} -335.279 q^{41} -84.4006 q^{43} +255.199 q^{47} +552.199 q^{49} +399.638 q^{53} +79.5994 q^{55} +2.56090 q^{59} +568.519 q^{61} +209.599 q^{65} -769.519 q^{67} +441.837 q^{71} +966.317 q^{73} -476.321 q^{77} -503.240 q^{79} -1270.76 q^{83} +184.599 q^{85} +0.240385 q^{89} -1254.24 q^{91} +54.5994 q^{95} -6.71795 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 10 q^{5} - 10 q^{7}+O(q^{10}) $ 2 * q + 10 * q^5 - 10 * q^7	$ 2 q + 10 q^{5} - 10 q^{7} - 18 q^{11} + 34 q^{13} + 24 q^{17} - 28 q^{19} - 30 q^{23} + 50 q^{25} - 42 q^{29} - 256 q^{31} - 50 q^{35} + 148 q^{37} - 222 q^{41} - 418 q^{43} + 12 q^{47} + 606 q^{49} - 48 q^{53} - 90 q^{55} + 354 q^{59} + 838 q^{61} + 170 q^{65} - 1240 q^{67} - 462 q^{71} + 886 q^{73} - 1152 q^{77} - 1156 q^{79} - 1146 q^{83} + 120 q^{85} + 150 q^{89} - 1412 q^{91} - 140 q^{95} + 784 q^{97}+O(q^{100}) $ 2 * q + 10 * q^5 - 10 * q^7 - 18 * q^11 + 34 * q^13 + 24 * q^17 - 28 * q^19 - 30 * q^23 + 50 * q^25 - 42 * q^29 - 256 * q^31 - 50 * q^35 + 148 * q^37 - 222 * q^41 - 418 * q^43 + 12 * q^47 + 606 * q^49 - 48 * q^53 - 90 * q^55 + 354 * q^59 + 838 * q^61 + 170 * q^65 - 1240 * q^67 - 462 * q^71 + 886 * q^73 - 1152 * q^77 - 1156 * q^79 - 1146 * q^83 + 120 * q^85 + 150 * q^89 - 1412 * q^91 - 140 * q^95 + 784 * 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
$7$	−29.9199	−1.61552	−0.807761	−	0.589511i	$-0.799321\pi$
$7$	−29.9199	−1.61552	−0.807761	+	0.589511i	$0.799321\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	15.9199	0.436366	0.218183	−	0.975908i	$-0.429987\pi$
$11$	15.9199	0.436366	0.218183	+	0.975908i	$0.429987\pi$
$12$	0	0
$13$	41.9199	0.894345	0.447172	−	0.894448i	$-0.352431\pi$
$13$	41.9199	0.894345	0.447172	+	0.894448i	$0.352431\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	36.9199	0.526728	0.263364	−	0.964696i	$-0.415168\pi$
$17$	36.9199	0.526728	0.263364	+	0.964696i	$0.415168\pi$
$18$	0	0
$19$	10.9199	0.131852	0.0659261	−	0.997825i	$-0.479000\pi$
$19$	10.9199	0.131852	0.0659261	+	0.997825i	$0.479000\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−114.679	−1.03967	−0.519833	−	0.854268i	$-0.674006\pi$
$23$	−114.679	−1.03967	−0.519833	+	0.854268i	$0.674006\pi$
$24$	0	0
$25$	25.0000	0.200000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−145.599	−0.932315	−0.466157	−	0.884702i	$-0.654362\pi$
$29$	−145.599	−0.932315	−0.466157	+	0.884702i	$0.654362\pi$
$30$	0	0
$31$	46.4391	0.269055	0.134528	−	0.990910i	$-0.457048\pi$
$31$	46.4391	0.269055	0.134528	+	0.990910i	$0.457048\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−149.599	−0.722483
$36$	0	0
$37$	74.0000	0.328798	0.164399	−	0.986394i	$-0.447432\pi$
$37$	74.0000	0.328798	0.164399	+	0.986394i	$0.447432\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−335.279	−1.27712	−0.638558	−	0.769574i	$-0.720469\pi$
$41$	−335.279	−1.27712	−0.638558	+	0.769574i	$0.720469\pi$
$42$	0	0
$43$	−84.4006	−0.299325	−0.149663	−	0.988737i	$-0.547819\pi$
$43$	−84.4006	−0.299325	−0.149663	+	0.988737i	$0.547819\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	255.199	0.792012	0.396006	−	0.918248i	$-0.370396\pi$
$47$	255.199	0.792012	0.396006	+	0.918248i	$0.370396\pi$
$48$	0	0
$49$	552.199	1.60991
$50$	0	0
$51$	0	0
$52$	0	0
$53$	399.638	1.03574	0.517872	−	0.855458i	$-0.326724\pi$
$53$	399.638	1.03574	0.517872	+	0.855458i	$0.326724\pi$
$54$	0	0
$55$	79.5994	0.195149
$56$	0	0
$57$	0	0
$58$	0	0
$59$	2.56090	0.00565086	0.00282543	−	0.999996i	$-0.499101\pi$
$59$	2.56090	0.00565086	0.00282543	+	0.999996i	$0.499101\pi$
$60$	0	0
$61$	568.519	1.19330	0.596651	−	0.802501i	$-0.296498\pi$
$61$	568.519	1.19330	0.596651	+	0.802501i	$0.296498\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	209.599	0.399963
$66$	0	0
$67$	−769.519	−1.40316	−0.701580	−	0.712591i	$-0.747522\pi$
$67$	−769.519	−1.40316	−0.701580	+	0.712591i	$0.747522\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	441.837	0.738540	0.369270	−	0.929322i	$-0.379608\pi$
$71$	441.837	0.738540	0.369270	+	0.929322i	$0.379608\pi$
$72$	0	0
$73$	966.317	1.54930	0.774650	−	0.632390i	$-0.217926\pi$
$73$	966.317	1.54930	0.774650	+	0.632390i	$0.217926\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−476.321	−0.704958
$78$	0	0
$79$	−503.240	−0.716696	−0.358348	−	0.933588i	$-0.616660\pi$
$79$	−503.240	−0.716696	−0.358348	+	0.933588i	$0.616660\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−1270.76	−1.68053	−0.840263	−	0.542179i	$-0.817599\pi$
$83$	−1270.76	−1.68053	−0.840263	+	0.542179i	$0.817599\pi$
$84$	0	0
$85$	184.599	0.235560
$86$	0	0
$87$	0	0
$88$	0	0
$89$	0.240385	0.000286301 0	0.000143150	−	1.00000i	$-0.499954\pi$
$89$	0.240385	0.000286301 0	0.000143150		1.00000i	$0.499954\pi$
$90$	0	0
$91$	−1254.24	−1.44483
$92$	0	0
$93$	0	0
$94$	0	0
$95$	54.5994	0.0589661
$96$	0	0
$97$	−6.71795	−0.00703200	−0.00351600	−	0.999994i	$-0.501119\pi$
$97$	−6.71795	−0.00703200	−0.00351600	+	0.999994i	$0.501119\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	1195.36	1.17765	0.588825	−	0.808261i	$-0.299591\pi$
$101$	1195.36	1.17765	0.588825	+	0.808261i	$0.299591\pi$
$102$	0	0
$103$	456.154	0.436371	0.218185	−	0.975907i	$-0.429986\pi$
$103$	456.154	0.436371	0.218185	+	0.975907i	$0.429986\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	1516.47	1.37012	0.685061	−	0.728485i	$-0.259775\pi$
$107$	1516.47	1.37012	0.685061	+	0.728485i	$0.259775\pi$
$108$	0	0
$109$	−1485.55	−1.30541	−0.652706	−	0.757611i	$-0.726366\pi$
$109$	−1485.55	−1.30541	−0.652706	+	0.757611i	$0.726366\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−1762.96	−1.46765	−0.733827	−	0.679337i	$-0.762268\pi$
$113$	−1762.96	−1.46765	−0.733827	+	0.679337i	$0.762268\pi$
$114$	0	0
$115$	−573.397	−0.464953
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−1104.64	−0.850941
$120$	0	0
$121$	−1077.56	−0.809585
$122$	0	0
$123$	0	0
$124$	0	0
$125$	125.000	0.0894427
$126$	0	0
$127$	−698.481	−0.488033	−0.244016	−	0.969771i	$-0.578465\pi$
$127$	−698.481	−0.488033	−0.244016	+	0.969771i	$0.578465\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−1180.87	−0.787582	−0.393791	−	0.919200i	$-0.628837\pi$
$131$	−1180.87	−0.787582	−0.393791	+	0.919200i	$0.628837\pi$
$132$	0	0
$133$	−326.721	−0.213010
$134$	0	0
$135$	0	0
$136$	0	0
$137$	1965.95	1.22600	0.613002	−	0.790081i	$-0.289962\pi$
$137$	1965.95	1.22600	0.613002	+	0.790081i	$0.289962\pi$
$138$	0	0
$139$	−3132.31	−1.91136	−0.955679	−	0.294409i	$-0.904877\pi$
$139$	−3132.31	−1.91136	−0.955679	+	0.294409i	$0.904877\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	667.359	0.390261
$144$	0	0
$145$	−727.997	−0.416944
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−3452.87	−1.89846	−0.949230	−	0.314583i	$-0.898135\pi$
$149$	−3452.87	−1.89846	−0.949230	+	0.314583i	$0.898135\pi$
$150$	0	0
$151$	−644.000	−0.347073	−0.173536	−	0.984827i	$-0.555519\pi$
$151$	−644.000	−0.347073	−0.173536	+	0.984827i	$0.555519\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	232.196	0.120325
$156$	0	0
$157$	−1523.12	−0.774255	−0.387128	−	0.922026i	$-0.626533\pi$
$157$	−1523.12	−0.774255	−0.387128	+	0.922026i	$0.626533\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	3431.20	1.67960
$162$	0	0
$163$	−3805.67	−1.82873	−0.914364	−	0.404892i	$-0.867309\pi$
$163$	−3805.67	−1.82873	−0.914364	+	0.404892i	$0.867309\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	3102.28	1.43750	0.718748	−	0.695271i	$-0.244716\pi$
$167$	3102.28	1.43750	0.718748	+	0.695271i	$0.244716\pi$
$168$	0	0
$169$	−439.724	−0.200148
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−3228.90	−1.41901	−0.709505	−	0.704700i	$-0.751081\pi$
$173$	−3228.90	−1.41901	−0.709505	+	0.704700i	$0.751081\pi$
$174$	0	0
$175$	−747.997	−0.323104
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−313.276	−0.130812	−0.0654059	−	0.997859i	$-0.520834\pi$
$179$	−313.276	−0.130812	−0.0654059	+	0.997859i	$0.520834\pi$
$180$	0	0
$181$	−542.526	−0.222793	−0.111397	−	0.993776i	$-0.535532\pi$
$181$	−542.526	−0.222793	−0.111397	+	0.993776i	$0.535532\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	370.000	0.147043
$186$	0	0
$187$	587.760	0.229846
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−4569.68	−1.73115	−0.865577	−	0.500776i	$-0.833048\pi$
$191$	−4569.68	−1.73115	−0.865577	+	0.500776i	$0.833048\pi$
$192$	0	0
$193$	−4732.95	−1.76521	−0.882604	−	0.470116i	$-0.844212\pi$
$193$	−4732.95	−1.76521	−0.882604	+	0.470116i	$0.844212\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	3335.06	1.20616	0.603079	−	0.797681i	$-0.293940\pi$
$197$	3335.06	1.20616	0.603079	+	0.797681i	$0.293940\pi$
$198$	0	0
$199$	4861.26	1.73169	0.865844	−	0.500315i	$-0.166782\pi$
$199$	4861.26	1.73169	0.865844	+	0.500315i	$0.166782\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	4356.31	1.50617
$204$	0	0
$205$	−1676.39	−0.571144
$206$	0	0
$207$	0	0
$208$	0	0
$209$	173.843	0.0575357
$210$	0	0
$211$	2065.87	0.674030	0.337015	−	0.941499i	$-0.390583\pi$
$211$	2065.87	0.674030	0.337015	+	0.941499i	$0.390583\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−422.003	−0.133862
$216$	0	0
$217$	−1389.45	−0.434664
$218$	0	0
$219$	0	0
$220$	0	0
$221$	1547.68	0.471077
$222$	0	0
$223$	4062.95	1.22007	0.610034	−	0.792375i	$-0.291156\pi$
$223$	4062.95	1.22007	0.610034	+	0.792375i	$0.291156\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	2693.87	0.787657	0.393828	−	0.919184i	$-0.371150\pi$
$227$	2693.87	0.787657	0.393828	+	0.919184i	$0.371150\pi$
$228$	0	0
$229$	−2048.04	−0.590998	−0.295499	−	0.955343i	$-0.595486\pi$
$229$	−2048.04	−0.590998	−0.295499	+	0.955343i	$0.595486\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	799.923	0.224913	0.112456	−	0.993657i	$-0.464128\pi$
$233$	799.923	0.224913	0.112456	+	0.993657i	$0.464128\pi$
$234$	0	0
$235$	1275.99	0.354198
$236$	0	0
$237$	0	0
$238$	0	0
$239$	1073.85	0.290635	0.145317	−	0.989385i	$-0.453580\pi$
$239$	1073.85	0.290635	0.145317	+	0.989385i	$0.453580\pi$
$240$	0	0
$241$	−4787.39	−1.27960	−0.639799	−	0.768543i	$-0.720982\pi$
$241$	−4787.39	−1.27960	−0.639799	+	0.768543i	$0.720982\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	2760.99	0.719973
$246$	0	0
$247$	457.760	0.117921
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−5498.07	−1.38261	−0.691305	−	0.722563i	$-0.742964\pi$
$251$	−5498.07	−1.38261	−0.691305	+	0.722563i	$0.742964\pi$
$252$	0	0
$253$	−1825.68	−0.453675
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−6901.23	−1.67505	−0.837524	−	0.546401i	$-0.815997\pi$
$257$	−6901.23	−1.67505	−0.837524	+	0.546401i	$0.815997\pi$
$258$	0	0
$259$	−2214.07	−0.531180
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−6464.30	−1.51561	−0.757806	−	0.652480i	$-0.773729\pi$
$263$	−6464.30	−1.51561	−0.757806	+	0.652480i	$0.773729\pi$
$264$	0	0
$265$	1998.19	0.463199
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−5349.66	−1.21254	−0.606272	−	0.795257i	$-0.707336\pi$
$269$	−5349.66	−1.21254	−0.606272	+	0.795257i	$0.707336\pi$
$270$	0	0
$271$	2189.80	0.490852	0.245426	−	0.969415i	$-0.421072\pi$
$271$	2189.80	0.490852	0.245426	+	0.969415i	$0.421072\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	397.997	0.0872731
$276$	0	0
$277$	1303.91	0.282831	0.141415	−	0.989950i	$-0.454835\pi$
$277$	1303.91	0.282831	0.141415	+	0.989950i	$0.454835\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	5405.45	1.14755	0.573776	−	0.819013i	$-0.305478\pi$
$281$	5405.45	1.14755	0.573776	+	0.819013i	$0.305478\pi$
$282$	0	0
$283$	−1287.60	−0.270459	−0.135229	−	0.990814i	$-0.543177\pi$
$283$	−1287.60	−0.270459	−0.135229	+	0.990814i	$0.543177\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	10031.5	2.06321
$288$	0	0
$289$	−3549.92	−0.722557
$290$	0	0
$291$	0	0
$292$	0	0
$293$	1557.45	0.310537	0.155268	−	0.987872i	$-0.450376\pi$
$293$	1557.45	0.310537	0.155268	+	0.987872i	$0.450376\pi$
$294$	0	0
$295$	12.8045	0.00252714
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−4807.35	−0.929820
$300$	0	0
$301$	2525.26	0.483566
$302$	0	0
$303$	0	0
$304$	0	0
$305$	2842.60	0.533661
$306$	0	0
$307$	2343.27	0.435627	0.217813	−	0.975990i	$-0.430108\pi$
$307$	2343.27	0.435627	0.217813	+	0.975990i	$0.430108\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	6107.64	1.11361	0.556804	−	0.830644i	$-0.312027\pi$
$311$	6107.64	1.11361	0.556804	+	0.830644i	$0.312027\pi$
$312$	0	0
$313$	−2081.78	−0.375939	−0.187970	−	0.982175i	$-0.560191\pi$
$313$	−2081.78	−0.375939	−0.187970	+	0.982175i	$0.560191\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−3611.73	−0.639921	−0.319960	−	0.947431i	$-0.603670\pi$
$317$	−3611.73	−0.639921	−0.319960	+	0.947431i	$0.603670\pi$
$318$	0	0
$319$	−2317.92	−0.406830
$320$	0	0
$321$	0	0
$322$	0	0
$323$	403.160	0.0694503
$324$	0	0
$325$	1048.00	0.178869
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−7635.51	−1.27951
$330$	0	0
$331$	−6373.75	−1.05841	−0.529204	−	0.848495i	$-0.677509\pi$
$331$	−6373.75	−1.05841	−0.529204	+	0.848495i	$0.677509\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−3847.60	−0.627512
$336$	0	0
$337$	−9547.17	−1.54323	−0.771613	−	0.636092i	$-0.780550\pi$
$337$	−9547.17	−1.54323	−0.771613	+	0.636092i	$0.780550\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	739.305	0.117406
$342$	0	0
$343$	−6259.20	−0.985321
$344$	0	0
$345$	0	0
$346$	0	0
$347$	9275.04	1.43490	0.717450	−	0.696610i	$-0.245309\pi$
$347$	9275.04	1.43490	0.717450	+	0.696610i	$0.245309\pi$
$348$	0	0
$349$	−4459.46	−0.683981	−0.341991	−	0.939703i	$-0.611101\pi$
$349$	−4459.46	−0.683981	−0.341991	+	0.939703i	$0.611101\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	9666.44	1.45749	0.728743	−	0.684787i	$-0.240105\pi$
$353$	9666.44	1.45749	0.728743	+	0.684787i	$0.240105\pi$
$354$	0	0
$355$	2209.18	0.330285
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−3789.49	−0.557108	−0.278554	−	0.960421i	$-0.589855\pi$
$359$	−3789.49	−0.557108	−0.278554	+	0.960421i	$0.589855\pi$
$360$	0	0
$361$	−6739.76	−0.982615
$362$	0	0
$363$	0	0
$364$	0	0
$365$	4831.59	0.692868
$366$	0	0
$367$	−3816.82	−0.542878	−0.271439	−	0.962456i	$-0.587500\pi$
$367$	−3816.82	−0.542878	−0.271439	+	0.962456i	$0.587500\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−11957.1	−1.67327
$372$	0	0
$373$	2082.72	0.289114	0.144557	−	0.989497i	$-0.453824\pi$
$373$	2082.72	0.289114	0.144557	+	0.989497i	$0.453824\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−6103.51	−0.833811
$378$	0	0
$379$	−8074.43	−1.09434	−0.547171	−	0.837021i	$-0.684295\pi$
$379$	−8074.43	−1.09434	−0.547171	+	0.837021i	$0.684295\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−1372.74	−0.183143	−0.0915713	−	0.995799i	$-0.529189\pi$
$383$	−1372.74	−0.183143	−0.0915713	+	0.995799i	$0.529189\pi$
$384$	0	0
$385$	−2381.60	−0.315267
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−10377.0	−1.35253	−0.676267	−	0.736656i	$-0.736404\pi$
$389$	−10377.0	−1.35253	−0.676267	+	0.736656i	$0.736404\pi$
$390$	0	0
$391$	−4233.95	−0.547622
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−2516.20	−0.320516
$396$	0	0
$397$	−1717.90	−0.217176	−0.108588	−	0.994087i	$-0.534633\pi$
$397$	−1717.90	−0.217176	−0.108588	+	0.994087i	$0.534633\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	5159.77	0.642560	0.321280	−	0.946984i	$-0.395887\pi$
$401$	5159.77	0.642560	0.321280	+	0.946984i	$0.395887\pi$
$402$	0	0
$403$	1946.72	0.240628
$404$	0	0
$405$	0	0
$406$	0	0
$407$	1178.07	0.143476
$408$	0	0
$409$	−7603.29	−0.919215	−0.459607	−	0.888122i	$-0.652010\pi$
$409$	−7603.29	−0.919215	−0.459607	+	0.888122i	$0.652010\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−76.6218	−0.00912908
$414$	0	0
$415$	−6353.78	−0.751554
$416$	0	0
$417$	0	0
$418$	0	0
$419$	7162.45	0.835103	0.417552	−	0.908653i	$-0.362888\pi$
$419$	7162.45	0.835103	0.417552	+	0.908653i	$0.362888\pi$
$420$	0	0
$421$	2587.19	0.299506	0.149753	−	0.988723i	$-0.452152\pi$
$421$	2587.19	0.299506	0.149753	+	0.988723i	$0.452152\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	922.997	0.105346
$426$	0	0
$427$	−17010.0	−1.92780
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−11053.4	−1.23532	−0.617661	−	0.786444i	$-0.711920\pi$
$431$	−11053.4	−1.23532	−0.617661	+	0.786444i	$0.711920\pi$
$432$	0	0
$433$	−2510.68	−0.278650	−0.139325	−	0.990247i	$-0.544493\pi$
$433$	−2510.68	−0.278650	−0.139325	+	0.990247i	$0.544493\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−1252.29	−0.137082
$438$	0	0
$439$	−6839.49	−0.743579	−0.371789	−	0.928317i	$-0.621256\pi$
$439$	−6839.49	−0.743579	−0.371789	+	0.928317i	$0.621256\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	12307.0	1.31991	0.659956	−	0.751304i	$-0.270575\pi$
$443$	12307.0	1.31991	0.659956	+	0.751304i	$0.270575\pi$
$444$	0	0
$445$	1.20193	0.000128038 0
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−1073.61	−0.112844	−0.0564219	−	0.998407i	$-0.517969\pi$
$449$	−1073.61	−0.112844	−0.0564219	+	0.998407i	$0.517969\pi$
$450$	0	0
$451$	−5337.60	−0.557290
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−6271.19	−0.646149
$456$	0	0
$457$	7320.29	0.749297	0.374648	−	0.927167i	$-0.377763\pi$
$457$	7320.29	0.749297	0.374648	+	0.927167i	$0.377763\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	173.058	0.0174840	0.00874198	−	0.999962i	$-0.497217\pi$
$461$	173.058	0.0174840	0.00874198	+	0.999962i	$0.497217\pi$
$462$	0	0
$463$	−15598.6	−1.56572	−0.782860	−	0.622198i	$-0.786240\pi$
$463$	−15598.6	−1.56572	−0.782860	+	0.622198i	$0.786240\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−1561.75	−0.154752	−0.0773760	−	0.997002i	$-0.524654\pi$
$467$	−1561.75	−0.154752	−0.0773760	+	0.997002i	$0.524654\pi$
$468$	0	0
$469$	23023.9	2.26684
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−1343.65	−0.130615
$474$	0	0
$475$	272.997	0.0263704
$476$	0	0
$477$	0	0
$478$	0	0
$479$	4459.10	0.425347	0.212674	−	0.977123i	$-0.431783\pi$
$479$	4459.10	0.425347	0.212674	+	0.977123i	$0.431783\pi$
$480$	0	0
$481$	3102.07	0.294059
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−33.5897	−0.00314481
$486$	0	0
$487$	−9005.17	−0.837912	−0.418956	−	0.908007i	$-0.637604\pi$
$487$	−9005.17	−0.837912	−0.418956	+	0.908007i	$0.637604\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−20596.2	−1.89306	−0.946530	−	0.322615i	$-0.895438\pi$
$491$	−20596.2	−1.89306	−0.946530	+	0.322615i	$0.895438\pi$
$492$	0	0
$493$	−5375.51	−0.491077
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−13219.7	−1.19313
$498$	0	0
$499$	8146.55	0.730841	0.365420	−	0.930843i	$-0.380925\pi$
$499$	8146.55	0.730841	0.365420	+	0.930843i	$0.380925\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	9277.97	0.822434	0.411217	−	0.911537i	$-0.365104\pi$
$503$	9277.97	0.822434	0.411217	+	0.911537i	$0.365104\pi$
$504$	0	0
$505$	5976.79	0.526661
$506$	0	0
$507$	0	0
$508$	0	0
$509$	9999.77	0.870790	0.435395	−	0.900240i	$-0.356609\pi$
$509$	9999.77	0.870790	0.435395	+	0.900240i	$0.356609\pi$
$510$	0	0
$511$	−28912.1	−2.50293
$512$	0	0
$513$	0	0
$514$	0	0
$515$	2280.77	0.195151
$516$	0	0
$517$	4062.73	0.345607
$518$	0	0
$519$	0	0
$520$	0	0
$521$	17101.2	1.43804	0.719018	−	0.694992i	$-0.244592\pi$
$521$	17101.2	1.43804	0.719018	+	0.694992i	$0.244592\pi$
$522$	0	0
$523$	−766.452	−0.0640815	−0.0320407	−	0.999487i	$-0.510201\pi$
$523$	−766.452	−0.0640815	−0.0320407	+	0.999487i	$0.510201\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	1714.53	0.141719
$528$	0	0
$529$	984.385	0.0809061
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−14054.8	−1.14218
$534$	0	0
$535$	7582.37	0.612737
$536$	0	0
$537$	0	0
$538$	0	0
$539$	8790.93	0.702509
$540$	0	0
$541$	−19609.4	−1.55837	−0.779183	−	0.626797i	$-0.784365\pi$
$541$	−19609.4	−1.55837	−0.779183	+	0.626797i	$0.784365\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−7427.76	−0.583798
$546$	0	0
$547$	8819.86	0.689415	0.344707	−	0.938710i	$-0.387978\pi$
$547$	8819.86	0.689415	0.344707	+	0.938710i	$0.387978\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−1589.93	−0.122928
$552$	0	0
$553$	15056.9	1.15784
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−3398.63	−0.258536	−0.129268	−	0.991610i	$-0.541263\pi$
$557$	−3398.63	−0.258536	−0.129268	+	0.991610i	$0.541263\pi$
$558$	0	0
$559$	−3538.06	−0.267700
$560$	0	0
$561$	0	0
$562$	0	0
$563$	17081.5	1.27869	0.639343	−	0.768921i	$-0.279206\pi$
$563$	17081.5	1.27869	0.639343	+	0.768921i	$0.279206\pi$
$564$	0	0
$565$	−8814.78	−0.656355
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−9073.79	−0.668529	−0.334264	−	0.942479i	$-0.608488\pi$
$569$	−9073.79	−0.668529	−0.334264	+	0.942479i	$0.608488\pi$
$570$	0	0
$571$	24887.1	1.82398	0.911992	−	0.410209i	$-0.134544\pi$
$571$	24887.1	1.82398	0.911992	+	0.410209i	$0.134544\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−2866.99	−0.207933
$576$	0	0
$577$	−15279.3	−1.10240	−0.551199	−	0.834374i	$-0.685830\pi$
$577$	−15279.3	−1.10240	−0.551199	+	0.834374i	$0.685830\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	38020.9	2.71492
$582$	0	0
$583$	6362.18	0.451963
$584$	0	0
$585$	0	0
$586$	0	0
$587$	369.039	0.0259486	0.0129743	−	0.999916i	$-0.495870\pi$
$587$	369.039	0.0259486	0.0129743	+	0.999916i	$0.495870\pi$
$588$	0	0
$589$	507.109	0.0354755
$590$	0	0
$591$	0	0
$592$	0	0
$593$	7306.60	0.505980	0.252990	−	0.967469i	$-0.418586\pi$
$593$	7306.60	0.505980	0.252990	+	0.967469i	$0.418586\pi$
$594$	0	0
$595$	−5523.19	−0.380552
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−3554.93	−0.242489	−0.121244	−	0.992623i	$-0.538688\pi$
$599$	−3554.93	−0.242489	−0.121244	+	0.992623i	$0.538688\pi$
$600$	0	0
$601$	7111.24	0.482651	0.241326	−	0.970444i	$-0.422418\pi$
$601$	7111.24	0.482651	0.241326	+	0.970444i	$0.422418\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−5387.79	−0.362057
$606$	0	0
$607$	250.971	0.0167819	0.00839094	−	0.999965i	$-0.497329\pi$
$607$	250.971	0.0167819	0.00839094	+	0.999965i	$0.497329\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	10697.9	0.708332
$612$	0	0
$613$	8375.38	0.551841	0.275921	−	0.961180i	$-0.411017\pi$
$613$	8375.38	0.551841	0.275921	+	0.961180i	$0.411017\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−7740.30	−0.505045	−0.252522	−	0.967591i	$-0.581260\pi$
$617$	−7740.30	−0.505045	−0.252522	+	0.967591i	$0.581260\pi$
$618$	0	0
$619$	11300.3	0.733759	0.366880	−	0.930268i	$-0.380426\pi$
$619$	11300.3	0.733759	0.366880	+	0.930268i	$0.380426\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−7.19230	−0.000462525 0
$624$	0	0
$625$	625.000	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	2732.07	0.173187
$630$	0	0
$631$	13549.6	0.854835	0.427417	−	0.904054i	$-0.359423\pi$
$631$	13549.6	0.854835	0.427417	+	0.904054i	$0.359423\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−3492.40	−0.218255
$636$	0	0
$637$	23148.1	1.43981
$638$	0	0
$639$	0	0
$640$	0	0
$641$	4063.62	0.250395	0.125197	−	0.992132i	$-0.460044\pi$
$641$	4063.62	0.250395	0.125197	+	0.992132i	$0.460044\pi$
$642$	0	0
$643$	−3591.18	−0.220253	−0.110126	−	0.993918i	$-0.535126\pi$
$643$	−3591.18	−0.220253	−0.110126	+	0.993918i	$0.535126\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	11040.5	0.670860	0.335430	−	0.942065i	$-0.391118\pi$
$647$	11040.5	0.670860	0.335430	+	0.942065i	$0.391118\pi$
$648$	0	0
$649$	40.7692	0.00246584
$650$	0	0
$651$	0	0
$652$	0	0
$653$	18857.7	1.13011	0.565054	−	0.825054i	$-0.308856\pi$
$653$	18857.7	1.13011	0.565054	+	0.825054i	$0.308856\pi$
$654$	0	0
$655$	−5904.36	−0.352217
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−24191.7	−1.43001	−0.715005	−	0.699119i	$-0.753576\pi$
$659$	−24191.7	−1.43001	−0.715005	+	0.699119i	$0.753576\pi$
$660$	0	0
$661$	32666.8	1.92223	0.961113	−	0.276155i	$-0.0890603\pi$
$661$	32666.8	1.92223	0.961113	+	0.276155i	$0.0890603\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−1633.61	−0.0952609
$666$	0	0
$667$	16697.3	0.969296
$668$	0	0
$669$	0	0
$670$	0	0
$671$	9050.75	0.520716
$672$	0	0
$673$	−27740.8	−1.58890	−0.794450	−	0.607330i	$-0.792241\pi$
$673$	−27740.8	−1.58890	−0.794450	+	0.607330i	$0.792241\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−18946.0	−1.07556	−0.537779	−	0.843086i	$-0.680736\pi$
$677$	−18946.0	−1.07556	−0.537779	+	0.843086i	$0.680736\pi$
$678$	0	0
$679$	201.000	0.0113603
$680$	0	0
$681$	0	0
$682$	0	0
$683$	19889.1	1.11425	0.557125	−	0.830428i	$-0.311904\pi$
$683$	19889.1	1.11425	0.557125	+	0.830428i	$0.311904\pi$
$684$	0	0
$685$	9829.76	0.548286
$686$	0	0
$687$	0	0
$688$	0	0
$689$	16752.8	0.926313
$690$	0	0
$691$	−4930.30	−0.271429	−0.135714	−	0.990748i	$-0.543333\pi$
$691$	−4930.30	−0.271429	−0.135714	+	0.990748i	$0.543333\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−15661.5	−0.854786
$696$	0	0
$697$	−12378.5	−0.672693
$698$	0	0
$699$	0	0
$700$	0	0
$701$	8173.40	0.440378	0.220189	−	0.975457i	$-0.429333\pi$
$701$	8173.40	0.440378	0.220189	+	0.975457i	$0.429333\pi$
$702$	0	0
$703$	808.070	0.0433527
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−35765.0	−1.90252
$708$	0	0
$709$	7698.47	0.407789	0.203894	−	0.978993i	$-0.434640\pi$
$709$	7698.47	0.407789	0.203894	+	0.978993i	$0.434640\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−5325.61	−0.279728
$714$	0	0
$715$	3336.79	0.174530
$716$	0	0
$717$	0	0
$718$	0	0
$719$	30407.2	1.57719	0.788593	−	0.614915i	$-0.210810\pi$
$719$	30407.2	1.57719	0.788593	+	0.614915i	$0.210810\pi$
$720$	0	0
$721$	−13648.1	−0.704966
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−3639.98	−0.186463
$726$	0	0
$727$	15263.1	0.778650	0.389325	−	0.921101i	$-0.372708\pi$
$727$	15263.1	0.778650	0.389325	+	0.921101i	$0.372708\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−3116.06	−0.157663
$732$	0	0
$733$	−7064.23	−0.355966	−0.177983	−	0.984034i	$-0.556957\pi$
$733$	−7064.23	−0.355966	−0.177983	+	0.984034i	$0.556957\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−12250.6	−0.612291
$738$	0	0
$739$	20382.0	1.01457	0.507284	−	0.861779i	$-0.330650\pi$
$739$	20382.0	1.01457	0.507284	+	0.861779i	$0.330650\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−30935.0	−1.52745	−0.763724	−	0.645543i	$-0.776631\pi$
$743$	−30935.0	−1.52745	−0.763724	+	0.645543i	$0.776631\pi$
$744$	0	0
$745$	−17264.4	−0.849017
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−45372.7	−2.21346
$750$	0	0
$751$	29084.9	1.41321	0.706606	−	0.707607i	$-0.250225\pi$
$751$	29084.9	1.41321	0.706606	+	0.707607i	$0.250225\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−3220.00	−0.155216
$756$	0	0
$757$	5538.75	0.265930	0.132965	−	0.991121i	$-0.457550\pi$
$757$	5538.75	0.265930	0.132965	+	0.991121i	$0.457550\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−25304.2	−1.20535	−0.602677	−	0.797985i	$-0.705899\pi$
$761$	−25304.2	−1.20535	−0.602677	+	0.797985i	$0.705899\pi$
$762$	0	0
$763$	44447.5	2.10892
$764$	0	0
$765$	0	0
$766$	0	0
$767$	107.353	0.00505381
$768$	0	0
$769$	−35063.9	−1.64426	−0.822129	−	0.569301i	$-0.807214\pi$
$769$	−35063.9	−1.64426	−0.822129	+	0.569301i	$0.807214\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	32958.1	1.53353	0.766766	−	0.641927i	$-0.221865\pi$
$773$	32958.1	1.53353	0.766766	+	0.641927i	$0.221865\pi$
$774$	0	0
$775$	1160.98	0.0538110
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−3661.20	−0.168390
$780$	0	0
$781$	7033.98	0.322274
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−7615.59	−0.346258
$786$	0	0
$787$	2230.57	0.101031	0.0505153	−	0.998723i	$-0.483914\pi$
$787$	2230.57	0.101031	0.0505153	+	0.998723i	$0.483914\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	52747.4	2.37103
$792$	0	0
$793$	23832.3	1.06722
$794$	0	0
$795$	0	0
$796$	0	0
$797$	21953.8	0.975714	0.487857	−	0.872924i	$-0.337779\pi$
$797$	21953.8	0.975714	0.487857	+	0.872924i	$0.337779\pi$
$798$	0	0
$799$	9421.90	0.417175
$800$	0	0
$801$	0	0
$802$	0	0
$803$	15383.6	0.676061
$804$	0	0
$805$	17156.0	0.751141
$806$	0	0
$807$	0	0
$808$	0	0
$809$	27001.6	1.17345	0.586727	−	0.809785i	$-0.300416\pi$
$809$	27001.6	1.17345	0.586727	+	0.809785i	$0.300416\pi$
$810$	0	0
$811$	37870.0	1.63970	0.819850	−	0.572579i	$-0.194057\pi$
$811$	37870.0	1.63970	0.819850	+	0.572579i	$0.194057\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−19028.3	−0.817832
$816$	0	0
$817$	−921.644	−0.0394666
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−37717.5	−1.60335	−0.801674	−	0.597761i	$-0.796057\pi$
$821$	−37717.5	−1.60335	−0.801674	+	0.597761i	$0.796057\pi$
$822$	0	0
$823$	17358.8	0.735222	0.367611	−	0.929980i	$-0.380176\pi$
$823$	17358.8	0.735222	0.367611	+	0.929980i	$0.380176\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−4418.15	−0.185773	−0.0928864	−	0.995677i	$-0.529609\pi$
$827$	−4418.15	−0.185773	−0.0928864	+	0.995677i	$0.529609\pi$
$828$	0	0
$829$	−19161.9	−0.802797	−0.401399	−	0.915903i	$-0.631476\pi$
$829$	−19161.9	−0.802797	−0.401399	+	0.915903i	$0.631476\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	20387.1	0.847985
$834$	0	0
$835$	15511.4	0.642868
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−5693.37	−0.234275	−0.117138	−	0.993116i	$-0.537372\pi$
$839$	−5693.37	−0.234275	−0.117138	+	0.993116i	$0.537372\pi$
$840$	0	0
$841$	−3189.83	−0.130790
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−2198.62	−0.0895087
$846$	0	0
$847$	32240.4	1.30790
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−8486.28	−0.341840
$852$	0	0
$853$	−1897.67	−0.0761724	−0.0380862	−	0.999274i	$-0.512126\pi$
$853$	−1897.67	−0.0761724	−0.0380862	+	0.999274i	$0.512126\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−22301.5	−0.888919	−0.444459	−	0.895799i	$-0.646604\pi$
$857$	−22301.5	−0.888919	−0.444459	+	0.895799i	$0.646604\pi$
$858$	0	0
$859$	−29752.8	−1.18178	−0.590892	−	0.806750i	$-0.701224\pi$
$859$	−29752.8	−1.18178	−0.590892	+	0.806750i	$0.701224\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−18461.6	−0.728203	−0.364102	−	0.931359i	$-0.618624\pi$
$863$	−18461.6	−0.728203	−0.364102	+	0.931359i	$0.618624\pi$
$864$	0	0
$865$	−16144.5	−0.634601
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−8011.52	−0.312741
$870$	0	0
$871$	−32258.1	−1.25491
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−3739.98	−0.144497
$876$	0	0
$877$	−27223.6	−1.04820	−0.524102	−	0.851655i	$-0.675599\pi$
$877$	−27223.6	−1.04820	−0.524102	+	0.851655i	$0.675599\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	24637.3	0.942169	0.471085	−	0.882088i	$-0.343863\pi$
$881$	24637.3	0.942169	0.471085	+	0.882088i	$0.343863\pi$
$882$	0	0
$883$	−37662.2	−1.43537	−0.717686	−	0.696367i	$-0.754799\pi$
$883$	−37662.2	−1.43537	−0.717686	+	0.696367i	$0.754799\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	2381.72	0.0901584	0.0450792	−	0.998983i	$-0.485646\pi$
$887$	2381.72	0.0901584	0.0450792	+	0.998983i	$0.485646\pi$
$888$	0	0
$889$	20898.5	0.788427
$890$	0	0
$891$	0	0
$892$	0	0
$893$	2786.74	0.104428
$894$	0	0
$895$	−1566.38	−0.0585008
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−6761.50	−0.250844
$900$	0	0
$901$	14754.6	0.545556
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−2712.63	−0.0996363
$906$	0	0
$907$	−34865.9	−1.27641	−0.638204	−	0.769867i	$-0.720323\pi$
$907$	−34865.9	−1.27641	−0.638204	+	0.769867i	$0.720323\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	53141.1	1.93265	0.966325	−	0.257326i	$-0.0828415\pi$
$911$	53141.1	1.93265	0.966325	+	0.257326i	$0.0828415\pi$
$912$	0	0
$913$	−20230.3	−0.733324
$914$	0	0
$915$	0	0
$916$	0	0
$917$	35331.5	1.27236
$918$	0	0
$919$	9861.81	0.353984	0.176992	−	0.984212i	$-0.443363\pi$
$919$	9861.81	0.353984	0.176992	+	0.984212i	$0.443363\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	18521.7	0.660509
$924$	0	0
$925$	1850.00	0.0657596
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−20683.2	−0.730455	−0.365228	−	0.930918i	$-0.619009\pi$
$929$	−20683.2	−0.730455	−0.365228	+	0.930918i	$0.619009\pi$
$930$	0	0
$931$	6029.94	0.212270
$932$	0	0
$933$	0	0
$934$	0	0
$935$	2938.80	0.102790
$936$	0	0
$937$	6795.89	0.236939	0.118470	−	0.992958i	$-0.462201\pi$
$937$	6795.89	0.236939	0.118470	+	0.992958i	$0.462201\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−6643.17	−0.230139	−0.115070	−	0.993357i	$-0.536709\pi$
$941$	−6643.17	−0.230139	−0.115070	+	0.993357i	$0.536709\pi$
$942$	0	0
$943$	38449.6	1.32777
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−26434.2	−0.907070	−0.453535	−	0.891239i	$-0.649837\pi$
$947$	−26434.2	−0.907070	−0.453535	+	0.891239i	$0.649837\pi$
$948$	0	0
$949$	40507.9	1.38561
$950$	0	0
$951$	0	0
$952$	0	0
$953$	8451.56	0.287275	0.143637	−	0.989630i	$-0.454120\pi$
$953$	8451.56	0.287275	0.143637	+	0.989630i	$0.454120\pi$
$954$	0	0
$955$	−22848.4	−0.774196
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−58821.0	−1.98064
$960$	0	0
$961$	−27634.4	−0.927609
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−23664.8	−0.789425
$966$	0	0
$967$	15324.8	0.509632	0.254816	−	0.966990i	$-0.417985\pi$
$967$	15324.8	0.509632	0.254816	+	0.966990i	$0.417985\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	1686.74	0.0557468	0.0278734	−	0.999611i	$-0.491126\pi$
$971$	1686.74	0.0557468	0.0278734	+	0.999611i	$0.491126\pi$
$972$	0	0
$973$	93718.2	3.08784
$974$	0	0
$975$	0	0
$976$	0	0
$977$	9701.06	0.317671	0.158835	−	0.987305i	$-0.449226\pi$
$977$	9701.06	0.317671	0.158835	+	0.987305i	$0.449226\pi$
$978$	0	0
$979$	3.82690	0.000124932 0
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−36456.3	−1.18288	−0.591442	−	0.806348i	$-0.701441\pi$
$983$	−36456.3	−1.18288	−0.591442	+	0.806348i	$0.701441\pi$
$984$	0	0
$985$	16675.3	0.539411
$986$	0	0
$987$	0	0
$988$	0	0
$989$	9679.02	0.311198
$990$	0	0
$991$	13556.5	0.434547	0.217274	−	0.976111i	$-0.430284\pi$
$991$	13556.5	0.434547	0.217274	+	0.976111i	$0.430284\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	24306.3	0.774434
$996$	0	0
$997$	61853.3	1.96481	0.982404	−	0.186768i	$-0.0598013\pi$
$997$	61853.3	1.96481	0.982404	+	0.186768i	$0.0598013\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2160.4.a.ba.1.1		2
3.2	odd	2		2160.4.a.v.1.1		2
4.3	odd	2		540.4.a.i.1.2	yes	2
12.11	even	2		540.4.a.f.1.2	✓	2
36.7	odd	6		1620.4.i.n.1081.1		4
36.11	even	6		1620.4.i.q.1081.1		4
36.23	even	6		1620.4.i.q.541.1		4
36.31	odd	6		1620.4.i.n.541.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
540.4.a.f.1.2	✓	2	12.11	even	2
540.4.a.i.1.2	yes	2	4.3	odd	2
1620.4.i.n.541.1		4	36.31	odd	6
1620.4.i.n.1081.1		4	36.7	odd	6
1620.4.i.q.541.1		4	36.23	even	6
1620.4.i.q.1081.1		4	36.11	even	6
2160.4.a.v.1.1		2	3.2	odd	2
2160.4.a.ba.1.1		2	1.1	even	1	trivial

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

\(f(q)\)	\(=\)	\(q+5.00000 q^{5} -29.9199 q^{7} +O(q^{10})\)	\(q+5.00000 q^{5} -29.9199 q^{7} +15.9199 q^{11} +41.9199 q^{13} +36.9199 q^{17} +10.9199 q^{19} -114.679 q^{23} +25.0000 q^{25} -145.599 q^{29} +46.4391 q^{31} -149.599 q^{35} +74.0000 q^{37} -335.279 q^{41} -84.4006 q^{43} +255.199 q^{47} +552.199 q^{49} +399.638 q^{53} +79.5994 q^{55} +2.56090 q^{59} +568.519 q^{61} +209.599 q^{65} -769.519 q^{67} +441.837 q^{71} +966.317 q^{73} -476.321 q^{77} -503.240 q^{79} -1270.76 q^{83} +184.599 q^{85} +0.240385 q^{89} -1254.24 q^{91} +54.5994 q^{95} -6.71795 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 10 q^{5} - 10 q^{7}+O(q^{10}) \) 2 * q + 10 * q^5 - 10 * q^7	\( 2 q + 10 q^{5} - 10 q^{7} - 18 q^{11} + 34 q^{13} + 24 q^{17} - 28 q^{19} - 30 q^{23} + 50 q^{25} - 42 q^{29} - 256 q^{31} - 50 q^{35} + 148 q^{37} - 222 q^{41} - 418 q^{43} + 12 q^{47} + 606 q^{49} - 48 q^{53} - 90 q^{55} + 354 q^{59} + 838 q^{61} + 170 q^{65} - 1240 q^{67} - 462 q^{71} + 886 q^{73} - 1152 q^{77} - 1156 q^{79} - 1146 q^{83} + 120 q^{85} + 150 q^{89} - 1412 q^{91} - 140 q^{95} + 784 q^{97}+O(q^{100}) \) 2 * q + 10 * q^5 - 10 * q^7 - 18 * q^11 + 34 * q^13 + 24 * q^17 - 28 * q^19 - 30 * q^23 + 50 * q^25 - 42 * q^29 - 256 * q^31 - 50 * q^35 + 148 * q^37 - 222 * q^41 - 418 * q^43 + 12 * q^47 + 606 * q^49 - 48 * q^53 - 90 * q^55 + 354 * q^59 + 838 * q^61 + 170 * q^65 - 1240 * q^67 - 462 * q^71 + 886 * q^73 - 1152 * q^77 - 1156 * q^79 - 1146 * q^83 + 120 * q^85 + 150 * q^89 - 1412 * q^91 - 140 * q^95 + 784 * q^97

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