LMFDB - Embedded newform 2160.4.a.bp.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:	$0$
Dimension:	$3$
Coefficient field:	3.3.1765.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 11x + 16 $ x^3 - x^2 - 11*x + 16
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{2}\cdot 3^{2} $
Twist minimal:	no (minimal twist has level 1080)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-3.48079$ 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)

$f(q)$	$=$	$q+5.00000 q^{5} -24.7900 q^{7} -8.16892 q^{11} -46.7694 q^{13} -60.3173 q^{17} +111.802 q^{19} +36.9268 q^{23} +25.0000 q^{25} -33.2627 q^{29} +124.274 q^{31} -123.950 q^{35} -438.610 q^{37} -508.071 q^{41} +48.5247 q^{43} -248.030 q^{47} +271.543 q^{49} +320.404 q^{53} -40.8446 q^{55} -652.268 q^{59} +693.019 q^{61} -233.847 q^{65} +12.0308 q^{67} +1168.30 q^{71} -122.195 q^{73} +202.507 q^{77} +441.669 q^{79} +428.581 q^{83} -301.586 q^{85} +1549.24 q^{89} +1159.41 q^{91} +559.008 q^{95} -500.136 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 15 q^{5} + 27 q^{11} - 3 q^{13} - 15 q^{17} + 78 q^{19} + 105 q^{23} + 75 q^{25} - 117 q^{29} + 207 q^{31} - 120 q^{37} - 300 q^{41} + 483 q^{43} + 303 q^{47} - 15 q^{49} + 492 q^{53} + 135 q^{55}+ \cdots - 1392 q^{97}+O(q^{100}) $ 3 * q + 15 * q^5 + 27 * q^11 - 3 * q^13 - 15 * q^17 + 78 * q^19 + 105 * q^23 + 75 * q^25 - 117 * q^29 + 207 * q^31 - 120 * q^37 - 300 * q^41 + 483 * q^43 + 303 * q^47 - 15 * q^49 + 492 * q^53 + 135 * q^55 + 240 * q^59 - 444 * q^61 - 15 * q^65 + 522 * q^67 - 168 * q^71 - 876 * q^73 + 150 * q^77 + 2103 * q^79 - 42 * q^83 - 75 * q^85 + 2268 * q^89 + 1596 * q^91 + 390 * q^95 - 1392 * 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
$7$	−24.7900	−1.33853	−0.669267	−	0.743022i	$-0.733392\pi$
$7$	−24.7900	−1.33853	−0.669267	+	0.743022i	$0.733392\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−8.16892	−0.223911	−0.111956	−	0.993713i	$-0.535711\pi$
$11$	−8.16892	−0.223911	−0.111956	+	0.993713i	$0.535711\pi$
$12$	0	0
$13$	−46.7694	−0.997808	−0.498904	−	0.866657i	$-0.666264\pi$
$13$	−46.7694	−0.997808	−0.498904	+	0.866657i	$0.666264\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−60.3173	−0.860535	−0.430267	−	0.902701i	$-0.641581\pi$
$17$	−60.3173	−0.860535	−0.430267	+	0.902701i	$0.641581\pi$
$18$	0	0
$19$	111.802	1.34995	0.674975	−	0.737841i	$-0.264155\pi$
$19$	111.802	1.34995	0.674975	+	0.737841i	$0.264155\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	36.9268	0.334772	0.167386	−	0.985891i	$-0.446467\pi$
$23$	36.9268	0.334772	0.167386	+	0.985891i	$0.446467\pi$
$24$	0	0
$25$	25.0000	0.200000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−33.2627	−0.212991	−0.106495	−	0.994313i	$-0.533963\pi$
$29$	−33.2627	−0.212991	−0.106495	+	0.994313i	$0.533963\pi$
$30$	0	0
$31$	124.274	0.720010	0.360005	−	0.932950i	$-0.382775\pi$
$31$	124.274	0.720010	0.360005	+	0.932950i	$0.382775\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−123.950	−0.598610
$36$	0	0
$37$	−438.610	−1.94884	−0.974419	−	0.224741i	$-0.927846\pi$
$37$	−438.610	−1.94884	−0.974419	+	0.224741i	$0.927846\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−508.071	−1.93530	−0.967651	−	0.252291i	$-0.918816\pi$
$41$	−508.071	−1.93530	−0.967651	+	0.252291i	$0.918816\pi$
$42$	0	0
$43$	48.5247	0.172092	0.0860459	−	0.996291i	$-0.472577\pi$
$43$	48.5247	0.172092	0.0860459	+	0.996291i	$0.472577\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−248.030	−0.769763	−0.384881	−	0.922966i	$-0.625758\pi$
$47$	−248.030	−0.769763	−0.384881	+	0.922966i	$0.625758\pi$
$48$	0	0
$49$	271.543	0.791672
$50$	0	0
$51$	0	0
$52$	0	0
$53$	320.404	0.830394	0.415197	−	0.909732i	$-0.363713\pi$
$53$	320.404	0.830394	0.415197	+	0.909732i	$0.363713\pi$
$54$	0	0
$55$	−40.8446	−0.100136
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−652.268	−1.43929	−0.719645	−	0.694343i	$-0.755695\pi$
$59$	−652.268	−1.43929	−0.719645	+	0.694343i	$0.755695\pi$
$60$	0	0
$61$	693.019	1.45462	0.727311	−	0.686308i	$-0.240770\pi$
$61$	693.019	1.45462	0.727311	+	0.686308i	$0.240770\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−233.847	−0.446233
$66$	0	0
$67$	12.0308	0.0219373	0.0109686	−	0.999940i	$-0.496509\pi$
$67$	12.0308	0.0219373	0.0109686	+	0.999940i	$0.496509\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	1168.30	1.95284	0.976421	−	0.215877i	$-0.0692609\pi$
$71$	1168.30	1.95284	0.976421	+	0.215877i	$0.0692609\pi$
$72$	0	0
$73$	−122.195	−0.195915	−0.0979575	−	0.995191i	$-0.531231\pi$
$73$	−122.195	−0.195915	−0.0979575	+	0.995191i	$0.531231\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	202.507	0.299712
$78$	0	0
$79$	441.669	0.629008	0.314504	−	0.949256i	$-0.398162\pi$
$79$	441.669	0.629008	0.314504	+	0.949256i	$0.398162\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	428.581	0.566782	0.283391	−	0.959004i	$-0.408541\pi$
$83$	428.581	0.566782	0.283391	+	0.959004i	$0.408541\pi$
$84$	0	0
$85$	−301.586	−0.384843
$86$	0	0
$87$	0	0
$88$	0	0
$89$	1549.24	1.84515	0.922577	−	0.385813i	$-0.126079\pi$
$89$	1549.24	1.84515	0.922577	+	0.385813i	$0.126079\pi$
$90$	0	0
$91$	1159.41	1.33560
$92$	0	0
$93$	0	0
$94$	0	0
$95$	559.008	0.603716
$96$	0	0
$97$	−500.136	−0.523516	−0.261758	−	0.965134i	$-0.584302\pi$
$97$	−500.136	−0.523516	−0.261758	+	0.965134i	$0.584302\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	535.936	0.527997	0.263998	−	0.964523i	$-0.414959\pi$
$101$	535.936	0.527997	0.263998	+	0.964523i	$0.414959\pi$
$102$	0	0
$103$	134.905	0.129055	0.0645273	−	0.997916i	$-0.479446\pi$
$103$	134.905	0.129055	0.0645273	+	0.997916i	$0.479446\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−808.631	−0.730592	−0.365296	−	0.930891i	$-0.619032\pi$
$107$	−808.631	−0.730592	−0.365296	+	0.930891i	$0.619032\pi$
$108$	0	0
$109$	−1073.96	−0.943730	−0.471865	−	0.881671i	$-0.656419\pi$
$109$	−1073.96	−0.943730	−0.471865	+	0.881671i	$0.656419\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−523.376	−0.435709	−0.217854	−	0.975981i	$-0.569906\pi$
$113$	−523.376	−0.435709	−0.217854	+	0.975981i	$0.569906\pi$
$114$	0	0
$115$	184.634	0.149715
$116$	0	0
$117$	0	0
$118$	0	0
$119$	1495.26	1.15185
$120$	0	0
$121$	−1264.27	−0.949864
$122$	0	0
$123$	0	0
$124$	0	0
$125$	125.000	0.0894427
$126$	0	0
$127$	114.730	0.0801622	0.0400811	−	0.999196i	$-0.487238\pi$
$127$	114.730	0.0801622	0.0400811	+	0.999196i	$0.487238\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−499.073	−0.332857	−0.166428	−	0.986054i	$-0.553223\pi$
$131$	−499.073	−0.332857	−0.166428	+	0.986054i	$0.553223\pi$
$132$	0	0
$133$	−2771.56	−1.80695
$134$	0	0
$135$	0	0
$136$	0	0
$137$	1887.38	1.17701	0.588504	−	0.808495i	$-0.299717\pi$
$137$	1887.38	1.17701	0.588504	+	0.808495i	$0.299717\pi$
$138$	0	0
$139$	1784.90	1.08916	0.544580	−	0.838709i	$-0.316689\pi$
$139$	1784.90	1.08916	0.544580	+	0.838709i	$0.316689\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	382.056	0.223420
$144$	0	0
$145$	−166.313	−0.0952522
$146$	0	0
$147$	0	0
$148$	0	0
$149$	996.321	0.547797	0.273899	−	0.961759i	$-0.411687\pi$
$149$	996.321	0.547797	0.273899	+	0.961759i	$0.411687\pi$
$150$	0	0
$151$	3487.14	1.87933	0.939666	−	0.342093i	$-0.111136\pi$
$151$	3487.14	1.87933	0.939666	+	0.342093i	$0.111136\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	621.371	0.321998
$156$	0	0
$157$	−1785.99	−0.907880	−0.453940	−	0.891032i	$-0.649982\pi$
$157$	−1785.99	−0.907880	−0.453940	+	0.891032i	$0.649982\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−915.415	−0.448104
$162$	0	0
$163$	396.237	0.190403	0.0952014	−	0.995458i	$-0.469650\pi$
$163$	396.237	0.190403	0.0952014	+	0.995458i	$0.469650\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	1749.48	0.810654	0.405327	−	0.914172i	$-0.367158\pi$
$167$	1749.48	0.810654	0.405327	+	0.914172i	$0.367158\pi$
$168$	0	0
$169$	−9.61953	−0.00437849
$170$	0	0
$171$	0	0
$172$	0	0
$173$	2998.95	1.31796	0.658978	−	0.752162i	$-0.270989\pi$
$173$	2998.95	1.31796	0.658978	+	0.752162i	$0.270989\pi$
$174$	0	0
$175$	−619.750	−0.267707
$176$	0	0
$177$	0	0
$178$	0	0
$179$	682.749	0.285090	0.142545	−	0.989788i	$-0.454471\pi$
$179$	682.749	0.285090	0.142545	+	0.989788i	$0.454471\pi$
$180$	0	0
$181$	59.3943	0.0243909	0.0121954	−	0.999926i	$-0.496118\pi$
$181$	59.3943	0.0243909	0.0121954	+	0.999926i	$0.496118\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−2193.05	−0.871546
$186$	0	0
$187$	492.727	0.192683
$188$	0	0
$189$	0	0
$190$	0	0
$191$	3618.13	1.37067	0.685336	−	0.728227i	$-0.259655\pi$
$191$	3618.13	1.37067	0.685336	+	0.728227i	$0.259655\pi$
$192$	0	0
$193$	1536.37	0.573007	0.286504	−	0.958079i	$-0.407507\pi$
$193$	1536.37	0.573007	0.286504	+	0.958079i	$0.407507\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−136.471	−0.0493561	−0.0246780	−	0.999695i	$-0.507856\pi$
$197$	−136.471	−0.0493561	−0.0246780	+	0.999695i	$0.507856\pi$
$198$	0	0
$199$	123.255	0.0439061	0.0219531	−	0.999759i	$-0.493012\pi$
$199$	123.255	0.0439061	0.0219531	+	0.999759i	$0.493012\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	824.581	0.285095
$204$	0	0
$205$	−2540.36	−0.865494
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−913.298	−0.302269
$210$	0	0
$211$	−4128.55	−1.34702	−0.673509	−	0.739179i	$-0.735214\pi$
$211$	−4128.55	−1.34702	−0.673509	+	0.739179i	$0.735214\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	242.624	0.0769618
$216$	0	0
$217$	−3080.76	−0.963758
$218$	0	0
$219$	0	0
$220$	0	0
$221$	2821.01	0.858649
$222$	0	0
$223$	3094.00	0.929102	0.464551	−	0.885546i	$-0.346216\pi$
$223$	3094.00	0.929102	0.464551	+	0.885546i	$0.346216\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	1399.26	0.409128	0.204564	−	0.978853i	$-0.434422\pi$
$227$	1399.26	0.409128	0.204564	+	0.978853i	$0.434422\pi$
$228$	0	0
$229$	4787.68	1.38157	0.690784	−	0.723061i	$-0.257265\pi$
$229$	4787.68	1.38157	0.690784	+	0.723061i	$0.257265\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	5117.74	1.43894	0.719472	−	0.694521i	$-0.244384\pi$
$233$	5117.74	1.43894	0.719472	+	0.694521i	$0.244384\pi$
$234$	0	0
$235$	−1240.15	−0.344248
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−2813.92	−0.761580	−0.380790	−	0.924662i	$-0.624348\pi$
$239$	−2813.92	−0.761580	−0.380790	+	0.924662i	$0.624348\pi$
$240$	0	0
$241$	−3543.20	−0.947045	−0.473522	−	0.880782i	$-0.657018\pi$
$241$	−3543.20	−0.947045	−0.473522	+	0.880782i	$0.657018\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	1357.72	0.354046
$246$	0	0
$247$	−5228.90	−1.34699
$248$	0	0
$249$	0	0
$250$	0	0
$251$	3446.10	0.866597	0.433299	−	0.901250i	$-0.357350\pi$
$251$	3446.10	0.866597	0.433299	+	0.901250i	$0.357350\pi$
$252$	0	0
$253$	−301.652	−0.0749593
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−1354.83	−0.328841	−0.164421	−	0.986390i	$-0.552575\pi$
$257$	−1354.83	−0.328841	−0.164421	+	0.986390i	$0.552575\pi$
$258$	0	0
$259$	10873.1	2.60858
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−344.218	−0.0807050	−0.0403525	−	0.999186i	$-0.512848\pi$
$263$	−344.218	−0.0807050	−0.0403525	+	0.999186i	$0.512848\pi$
$264$	0	0
$265$	1602.02	0.371363
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−203.341	−0.0460889	−0.0230445	−	0.999734i	$-0.507336\pi$
$269$	−203.341	−0.0460889	−0.0230445	+	0.999734i	$0.507336\pi$
$270$	0	0
$271$	1586.34	0.355584	0.177792	−	0.984068i	$-0.443105\pi$
$271$	1586.34	0.355584	0.177792	+	0.984068i	$0.443105\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−204.223	−0.0447822
$276$	0	0
$277$	−4330.18	−0.939260	−0.469630	−	0.882863i	$-0.655613\pi$
$277$	−4330.18	−0.939260	−0.469630	+	0.882863i	$0.655613\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	6569.00	1.39457	0.697284	−	0.716795i	$-0.254391\pi$
$281$	6569.00	1.39457	0.697284	+	0.716795i	$0.254391\pi$
$282$	0	0
$283$	997.534	0.209531	0.104765	−	0.994497i	$-0.466591\pi$
$283$	997.534	0.209531	0.104765	+	0.994497i	$0.466591\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	12595.1	2.59047
$288$	0	0
$289$	−1274.82	−0.259480
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−182.221	−0.0363326	−0.0181663	−	0.999835i	$-0.505783\pi$
$293$	−182.221	−0.0363326	−0.0181663	+	0.999835i	$0.505783\pi$
$294$	0	0
$295$	−3261.34	−0.643670
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−1727.05	−0.334039
$300$	0	0
$301$	−1202.93	−0.230351
$302$	0	0
$303$	0	0
$304$	0	0
$305$	3465.09	0.650527
$306$	0	0
$307$	−1671.12	−0.310670	−0.155335	−	0.987862i	$-0.549646\pi$
$307$	−1671.12	−0.310670	−0.155335	+	0.987862i	$0.549646\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−9271.81	−1.69053	−0.845267	−	0.534345i	$-0.820558\pi$
$311$	−9271.81	−1.69053	−0.845267	+	0.534345i	$0.820558\pi$
$312$	0	0
$313$	−572.410	−0.103369	−0.0516845	−	0.998663i	$-0.516459\pi$
$313$	−572.410	−0.103369	−0.0516845	+	0.998663i	$0.516459\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−6048.37	−1.07164	−0.535820	−	0.844332i	$-0.679998\pi$
$317$	−6048.37	−1.07164	−0.535820	+	0.844332i	$0.679998\pi$
$318$	0	0
$319$	271.720	0.0476909
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−6743.57	−1.16168
$324$	0	0
$325$	−1169.24	−0.199562
$326$	0	0
$327$	0	0
$328$	0	0
$329$	6148.65	1.03035
$330$	0	0
$331$	−8537.53	−1.41772	−0.708860	−	0.705350i	$-0.750790\pi$
$331$	−8537.53	−1.41772	−0.708860	+	0.705350i	$0.750790\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	60.1541	0.00981065
$336$	0	0
$337$	3638.91	0.588202	0.294101	−	0.955774i	$-0.404980\pi$
$337$	3638.91	0.588202	0.294101	+	0.955774i	$0.404980\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−1015.19	−0.161218
$342$	0	0
$343$	1771.41	0.278854
$344$	0	0
$345$	0	0
$346$	0	0
$347$	7965.23	1.23226	0.616132	−	0.787643i	$-0.288699\pi$
$347$	7965.23	1.23226	0.616132	+	0.787643i	$0.288699\pi$
$348$	0	0
$349$	7699.35	1.18091	0.590454	−	0.807072i	$-0.298949\pi$
$349$	7699.35	1.18091	0.590454	+	0.807072i	$0.298949\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−11674.7	−1.76029	−0.880144	−	0.474706i	$-0.842554\pi$
$353$	−11674.7	−1.76029	−0.880144	+	0.474706i	$0.842554\pi$
$354$	0	0
$355$	5841.50	0.873337
$356$	0	0
$357$	0	0
$358$	0	0
$359$	2547.12	0.374463	0.187231	−	0.982316i	$-0.440049\pi$
$359$	2547.12	0.374463	0.187231	+	0.982316i	$0.440049\pi$
$360$	0	0
$361$	5640.59	0.822363
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−610.973	−0.0876158
$366$	0	0
$367$	11020.0	1.56742	0.783708	−	0.621130i	$-0.213326\pi$
$367$	11020.0	1.56742	0.783708	+	0.621130i	$0.213326\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−7942.81	−1.11151
$372$	0	0
$373$	8430.05	1.17022	0.585109	−	0.810955i	$-0.301052\pi$
$373$	8430.05	1.17022	0.585109	+	0.810955i	$0.301052\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	1555.68	0.212524
$378$	0	0
$379$	−1675.32	−0.227059	−0.113529	−	0.993535i	$-0.536216\pi$
$379$	−1675.32	−0.227059	−0.113529	+	0.993535i	$0.536216\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	12124.4	1.61757	0.808783	−	0.588108i	$-0.200127\pi$
$383$	12124.4	1.61757	0.808783	+	0.588108i	$0.200127\pi$
$384$	0	0
$385$	1012.54	0.134035
$386$	0	0
$387$	0	0
$388$	0	0
$389$	15159.1	1.97583	0.987916	−	0.154992i	$-0.0495351\pi$
$389$	15159.1	1.97583	0.987916	+	0.154992i	$0.0495351\pi$
$390$	0	0
$391$	−2227.32	−0.288083
$392$	0	0
$393$	0	0
$394$	0	0
$395$	2208.34	0.281301
$396$	0	0
$397$	−3884.64	−0.491095	−0.245547	−	0.969385i	$-0.578968\pi$
$397$	−3884.64	−0.491095	−0.245547	+	0.969385i	$0.578968\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	7056.99	0.878825	0.439413	−	0.898285i	$-0.355187\pi$
$401$	7056.99	0.878825	0.439413	+	0.898285i	$0.355187\pi$
$402$	0	0
$403$	−5812.24	−0.718432
$404$	0	0
$405$	0	0
$406$	0	0
$407$	3582.97	0.436366
$408$	0	0
$409$	7034.26	0.850420	0.425210	−	0.905095i	$-0.360200\pi$
$409$	7034.26	0.850420	0.425210	+	0.905095i	$0.360200\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	16169.7	1.92654
$414$	0	0
$415$	2142.91	0.253473
$416$	0	0
$417$	0	0
$418$	0	0
$419$	6153.56	0.717473	0.358737	−	0.933439i	$-0.383208\pi$
$419$	6153.56	0.717473	0.358737	+	0.933439i	$0.383208\pi$
$420$	0	0
$421$	−13024.0	−1.50772	−0.753862	−	0.657033i	$-0.771811\pi$
$421$	−13024.0	−1.50772	−0.753862	+	0.657033i	$0.771811\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−1507.93	−0.172107
$426$	0	0
$427$	−17179.9	−1.94706
$428$	0	0
$429$	0	0
$430$	0	0
$431$	11614.6	1.29805	0.649023	−	0.760769i	$-0.275178\pi$
$431$	11614.6	1.29805	0.649023	+	0.760769i	$0.275178\pi$
$432$	0	0
$433$	1245.75	0.138260	0.0691301	−	0.997608i	$-0.477978\pi$
$433$	1245.75	0.138260	0.0691301	+	0.997608i	$0.477978\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	4128.47	0.451926
$438$	0	0
$439$	−7984.30	−0.868041	−0.434020	−	0.900903i	$-0.642905\pi$
$439$	−7984.30	−0.868041	−0.434020	+	0.900903i	$0.642905\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	13840.8	1.48442	0.742210	−	0.670167i	$-0.233778\pi$
$443$	13840.8	1.48442	0.742210	+	0.670167i	$0.233778\pi$
$444$	0	0
$445$	7746.18	0.825178
$446$	0	0
$447$	0	0
$448$	0	0
$449$	7756.38	0.815247	0.407624	−	0.913150i	$-0.366358\pi$
$449$	7756.38	0.815247	0.407624	+	0.913150i	$0.366358\pi$
$450$	0	0
$451$	4150.39	0.433336
$452$	0	0
$453$	0	0
$454$	0	0
$455$	5797.07	0.597298
$456$	0	0
$457$	16107.6	1.64875	0.824377	−	0.566041i	$-0.191526\pi$
$457$	16107.6	1.64875	0.824377	+	0.566041i	$0.191526\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	6272.71	0.633730	0.316865	−	0.948471i	$-0.397370\pi$
$461$	6272.71	0.633730	0.316865	+	0.948471i	$0.397370\pi$
$462$	0	0
$463$	−18657.9	−1.87280	−0.936399	−	0.350937i	$-0.885863\pi$
$463$	−18657.9	−1.87280	−0.936399	+	0.350937i	$0.885863\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	16289.5	1.61411	0.807054	−	0.590478i	$-0.201061\pi$
$467$	16289.5	1.61411	0.807054	+	0.590478i	$0.201061\pi$
$468$	0	0
$469$	−298.244	−0.0293638
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−396.394	−0.0385333
$474$	0	0
$475$	2795.04	0.269990
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−11328.9	−1.08065	−0.540324	−	0.841457i	$-0.681698\pi$
$479$	−11328.9	−1.08065	−0.540324	+	0.841457i	$0.681698\pi$
$480$	0	0
$481$	20513.5	1.94457
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−2500.68	−0.234124
$486$	0	0
$487$	−14122.4	−1.31406	−0.657029	−	0.753866i	$-0.728187\pi$
$487$	−14122.4	−1.31406	−0.657029	+	0.753866i	$0.728187\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	5220.81	0.479861	0.239931	−	0.970790i	$-0.422875\pi$
$491$	5220.81	0.479861	0.239931	+	0.970790i	$0.422875\pi$
$492$	0	0
$493$	2006.31	0.183286
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−28962.1	−2.61394
$498$	0	0
$499$	−17934.3	−1.60892	−0.804459	−	0.594008i	$-0.797545\pi$
$499$	−17934.3	−1.60892	−0.804459	+	0.594008i	$0.797545\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−7305.37	−0.647576	−0.323788	−	0.946130i	$-0.604956\pi$
$503$	−7305.37	−0.647576	−0.323788	+	0.946130i	$0.604956\pi$
$504$	0	0
$505$	2679.68	0.236127
$506$	0	0
$507$	0	0
$508$	0	0
$509$	6083.17	0.529728	0.264864	−	0.964286i	$-0.414673\pi$
$509$	6083.17	0.529728	0.264864	+	0.964286i	$0.414673\pi$
$510$	0	0
$511$	3029.20	0.262239
$512$	0	0
$513$	0	0
$514$	0	0
$515$	674.527	0.0577150
$516$	0	0
$517$	2026.13	0.172358
$518$	0	0
$519$	0	0
$520$	0	0
$521$	10005.0	0.841316	0.420658	−	0.907219i	$-0.361799\pi$
$521$	10005.0	0.841316	0.420658	+	0.907219i	$0.361799\pi$
$522$	0	0
$523$	12931.9	1.08121	0.540607	−	0.841276i	$-0.318195\pi$
$523$	12931.9	1.08121	0.540607	+	0.841276i	$0.318195\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−7495.89	−0.619594
$528$	0	0
$529$	−10803.4	−0.887927
$530$	0	0
$531$	0	0
$532$	0	0
$533$	23762.2	1.93106
$534$	0	0
$535$	−4043.15	−0.326730
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−2218.22	−0.177264
$540$	0	0
$541$	13088.9	1.04017	0.520087	−	0.854113i	$-0.325900\pi$
$541$	13088.9	1.04017	0.520087	+	0.854113i	$0.325900\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−5369.79	−0.422049
$546$	0	0
$547$	4522.72	0.353524	0.176762	−	0.984254i	$-0.443438\pi$
$547$	4522.72	0.353524	0.176762	+	0.984254i	$0.443438\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−3718.82	−0.287526
$552$	0	0
$553$	−10949.0	−0.841948
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−5477.36	−0.416667	−0.208333	−	0.978058i	$-0.566804\pi$
$557$	−5477.36	−0.416667	−0.208333	+	0.978058i	$0.566804\pi$
$558$	0	0
$559$	−2269.47	−0.171715
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−17369.7	−1.30026	−0.650129	−	0.759824i	$-0.725285\pi$
$563$	−17369.7	−1.30026	−0.650129	+	0.759824i	$0.725285\pi$
$564$	0	0
$565$	−2616.88	−0.194855
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−7997.66	−0.589243	−0.294622	−	0.955614i	$-0.595194\pi$
$569$	−7997.66	−0.589243	−0.294622	+	0.955614i	$0.595194\pi$
$570$	0	0
$571$	−14623.1	−1.07173	−0.535864	−	0.844305i	$-0.680014\pi$
$571$	−14623.1	−1.07173	−0.535864	+	0.844305i	$0.680014\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	923.170	0.0669545
$576$	0	0
$577$	−3818.29	−0.275490	−0.137745	−	0.990468i	$-0.543985\pi$
$577$	−3818.29	−0.275490	−0.137745	+	0.990468i	$0.543985\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−10624.5	−0.758657
$582$	0	0
$583$	−2617.35	−0.185934
$584$	0	0
$585$	0	0
$586$	0	0
$587$	19568.8	1.37597	0.687983	−	0.725726i	$-0.258496\pi$
$587$	19568.8	1.37597	0.687983	+	0.725726i	$0.258496\pi$
$588$	0	0
$589$	13894.1	0.971977
$590$	0	0
$591$	0	0
$592$	0	0
$593$	5170.99	0.358090	0.179045	−	0.983841i	$-0.442699\pi$
$593$	5170.99	0.358090	0.179045	+	0.983841i	$0.442699\pi$
$594$	0	0
$595$	7476.32	0.515125
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−10252.3	−0.699327	−0.349663	−	0.936875i	$-0.613704\pi$
$599$	−10252.3	−0.699327	−0.349663	+	0.936875i	$0.613704\pi$
$600$	0	0
$601$	−21251.0	−1.44234	−0.721168	−	0.692760i	$-0.756395\pi$
$601$	−21251.0	−1.44234	−0.721168	+	0.692760i	$0.756395\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−6321.34	−0.424792
$606$	0	0
$607$	18653.8	1.24734	0.623669	−	0.781689i	$-0.285641\pi$
$607$	18653.8	1.24734	0.623669	+	0.781689i	$0.285641\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	11600.2	0.768076
$612$	0	0
$613$	6156.56	0.405646	0.202823	−	0.979215i	$-0.434988\pi$
$613$	6156.56	0.405646	0.202823	+	0.979215i	$0.434988\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−16106.4	−1.05092	−0.525461	−	0.850818i	$-0.676107\pi$
$617$	−16106.4	−1.05092	−0.525461	+	0.850818i	$0.676107\pi$
$618$	0	0
$619$	27462.5	1.78322	0.891608	−	0.452807i	$-0.149577\pi$
$619$	27462.5	1.78322	0.891608	+	0.452807i	$0.149577\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−38405.6	−2.46980
$624$	0	0
$625$	625.000	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	26455.7	1.67704
$630$	0	0
$631$	9282.80	0.585646	0.292823	−	0.956167i	$-0.405405\pi$
$631$	9282.80	0.585646	0.292823	+	0.956167i	$0.405405\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	573.648	0.0358496
$636$	0	0
$637$	−12699.9	−0.789937
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−22076.4	−1.36032	−0.680160	−	0.733064i	$-0.738090\pi$
$641$	−22076.4	−1.36032	−0.680160	+	0.733064i	$0.738090\pi$
$642$	0	0
$643$	−25416.1	−1.55881	−0.779403	−	0.626523i	$-0.784478\pi$
$643$	−25416.1	−1.55881	−0.779403	+	0.626523i	$0.784478\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−15054.7	−0.914777	−0.457389	−	0.889267i	$-0.651215\pi$
$647$	−15054.7	−0.914777	−0.457389	+	0.889267i	$0.651215\pi$
$648$	0	0
$649$	5328.32	0.322273
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−14629.6	−0.876726	−0.438363	−	0.898798i	$-0.644442\pi$
$653$	−14629.6	−0.876726	−0.438363	+	0.898798i	$0.644442\pi$
$654$	0	0
$655$	−2495.37	−0.148858
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−5302.96	−0.313466	−0.156733	−	0.987641i	$-0.550096\pi$
$659$	−5302.96	−0.313466	−0.156733	+	0.987641i	$0.550096\pi$
$660$	0	0
$661$	7326.18	0.431098	0.215549	−	0.976493i	$-0.430846\pi$
$661$	7326.18	0.431098	0.215549	+	0.976493i	$0.430846\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−13857.8	−0.808094
$666$	0	0
$667$	−1228.28	−0.0713034
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−5661.22	−0.325706
$672$	0	0
$673$	−27965.1	−1.60174	−0.800872	−	0.598836i	$-0.795630\pi$
$673$	−27965.1	−1.60174	−0.800872	+	0.598836i	$0.795630\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	17423.1	0.989103	0.494551	−	0.869148i	$-0.335332\pi$
$677$	17423.1	0.989103	0.494551	+	0.869148i	$0.335332\pi$
$678$	0	0
$679$	12398.4	0.700744
$680$	0	0
$681$	0	0
$682$	0	0
$683$	18318.0	1.02624	0.513118	−	0.858318i	$-0.328490\pi$
$683$	18318.0	1.02624	0.513118	+	0.858318i	$0.328490\pi$
$684$	0	0
$685$	9436.91	0.526374
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−14985.1	−0.828574
$690$	0	0
$691$	−27603.8	−1.51968	−0.759838	−	0.650112i	$-0.774722\pi$
$691$	−27603.8	−1.51968	−0.759838	+	0.650112i	$0.774722\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	8924.50	0.487087
$696$	0	0
$697$	30645.5	1.66540
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−8448.22	−0.455185	−0.227593	−	0.973756i	$-0.573085\pi$
$701$	−8448.22	−0.455185	−0.227593	+	0.973756i	$0.573085\pi$
$702$	0	0
$703$	−49037.2	−2.63083
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−13285.9	−0.706741
$708$	0	0
$709$	14440.0	0.764889	0.382444	−	0.923979i	$-0.375082\pi$
$709$	14440.0	0.764889	0.382444	+	0.923979i	$0.375082\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	4589.05	0.241040
$714$	0	0
$715$	1910.28	0.0999166
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−24213.5	−1.25593	−0.627964	−	0.778242i	$-0.716112\pi$
$719$	−24213.5	−1.25593	−0.627964	+	0.778242i	$0.716112\pi$
$720$	0	0
$721$	−3344.31	−0.172744
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−831.567	−0.0425981
$726$	0	0
$727$	10688.4	0.545271	0.272636	−	0.962117i	$-0.412105\pi$
$727$	10688.4	0.545271	0.272636	+	0.962117i	$0.412105\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−2926.88	−0.148091
$732$	0	0
$733$	−18715.7	−0.943085	−0.471542	−	0.881843i	$-0.656303\pi$
$733$	−18715.7	−0.943085	−0.471542	+	0.881843i	$0.656303\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−98.2788	−0.00491200
$738$	0	0
$739$	−13155.2	−0.654832	−0.327416	−	0.944880i	$-0.606178\pi$
$739$	−13155.2	−0.654832	−0.327416	+	0.944880i	$0.606178\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−20680.3	−1.02111	−0.510557	−	0.859844i	$-0.670561\pi$
$743$	−20680.3	−1.02111	−0.510557	+	0.859844i	$0.670561\pi$
$744$	0	0
$745$	4981.60	0.244982
$746$	0	0
$747$	0	0
$748$	0	0
$749$	20045.9	0.977921
$750$	0	0
$751$	76.2198	0.00370346	0.00185173	−	0.999998i	$-0.499411\pi$
$751$	76.2198	0.00370346	0.00185173	+	0.999998i	$0.499411\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	17435.7	0.840463
$756$	0	0
$757$	15781.3	0.757701	0.378851	−	0.925458i	$-0.376319\pi$
$757$	15781.3	0.757701	0.378851	+	0.925458i	$0.376319\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−32298.0	−1.53851	−0.769253	−	0.638944i	$-0.779371\pi$
$761$	−32298.0	−1.53851	−0.769253	+	0.638944i	$0.779371\pi$
$762$	0	0
$763$	26623.4	1.26321
$764$	0	0
$765$	0	0
$766$	0	0
$767$	30506.2	1.43613
$768$	0	0
$769$	−6530.51	−0.306237	−0.153119	−	0.988208i	$-0.548932\pi$
$769$	−6530.51	−0.306237	−0.153119	+	0.988208i	$0.548932\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	8952.60	0.416562	0.208281	−	0.978069i	$-0.433213\pi$
$773$	8952.60	0.416562	0.208281	+	0.978069i	$0.433213\pi$
$774$	0	0
$775$	3106.86	0.144002
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−56803.2	−2.61256
$780$	0	0
$781$	−9543.75	−0.437263
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−8929.93	−0.406016
$786$	0	0
$787$	15275.5	0.691886	0.345943	−	0.938256i	$-0.387559\pi$
$787$	15275.5	0.691886	0.345943	+	0.938256i	$0.387559\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	12974.5	0.583211
$792$	0	0
$793$	−32412.1	−1.45143
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−42475.4	−1.88777	−0.943887	−	0.330267i	$-0.892861\pi$
$797$	−42475.4	−1.88777	−0.943887	+	0.330267i	$0.892861\pi$
$798$	0	0
$799$	14960.5	0.662408
$800$	0	0
$801$	0	0
$802$	0	0
$803$	998.197	0.0438675
$804$	0	0
$805$	−4577.07	−0.200398
$806$	0	0
$807$	0	0
$808$	0	0
$809$	39893.7	1.73373	0.866864	−	0.498544i	$-0.166132\pi$
$809$	39893.7	1.73373	0.866864	+	0.498544i	$0.166132\pi$
$810$	0	0
$811$	8645.69	0.374342	0.187171	−	0.982327i	$-0.440068\pi$
$811$	8645.69	0.374342	0.187171	+	0.982327i	$0.440068\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	1981.18	0.0851508
$816$	0	0
$817$	5425.14	0.232315
$818$	0	0
$819$	0	0
$820$	0	0
$821$	42123.4	1.79064	0.895320	−	0.445424i	$-0.146947\pi$
$821$	42123.4	1.79064	0.895320	+	0.445424i	$0.146947\pi$
$822$	0	0
$823$	−17950.5	−0.760287	−0.380143	−	0.924928i	$-0.624125\pi$
$823$	−17950.5	−0.760287	−0.380143	+	0.924928i	$0.624125\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	19430.2	0.816994	0.408497	−	0.912760i	$-0.366053\pi$
$827$	19430.2	0.816994	0.408497	+	0.912760i	$0.366053\pi$
$828$	0	0
$829$	−23552.0	−0.986724	−0.493362	−	0.869824i	$-0.664232\pi$
$829$	−23552.0	−0.986724	−0.493362	+	0.869824i	$0.664232\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−16378.8	−0.681261
$834$	0	0
$835$	8747.42	0.362535
$836$	0	0
$837$	0	0
$838$	0	0
$839$	15526.2	0.638883	0.319442	−	0.947606i	$-0.396505\pi$
$839$	15526.2	0.638883	0.319442	+	0.947606i	$0.396505\pi$
$840$	0	0
$841$	−23282.6	−0.954635
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−48.0977	−0.00195812
$846$	0	0
$847$	31341.2	1.27142
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−16196.4	−0.652417
$852$	0	0
$853$	−6582.11	−0.264205	−0.132103	−	0.991236i	$-0.542173\pi$
$853$	−6582.11	−0.264205	−0.132103	+	0.991236i	$0.542173\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−41261.8	−1.64466	−0.822332	−	0.569009i	$-0.807327\pi$
$857$	−41261.8	−1.64466	−0.822332	+	0.569009i	$0.807327\pi$
$858$	0	0
$859$	2460.09	0.0977149	0.0488575	−	0.998806i	$-0.484442\pi$
$859$	2460.09	0.0977149	0.0488575	+	0.998806i	$0.484442\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	38478.8	1.51777	0.758884	−	0.651226i	$-0.225745\pi$
$863$	38478.8	1.51777	0.758884	+	0.651226i	$0.225745\pi$
$864$	0	0
$865$	14994.8	0.589408
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−3607.95	−0.140842
$870$	0	0
$871$	−562.675	−0.0218892
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−3098.75	−0.119722
$876$	0	0
$877$	28139.7	1.08348	0.541739	−	0.840547i	$-0.317766\pi$
$877$	28139.7	1.08348	0.541739	+	0.840547i	$0.317766\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−14885.8	−0.569257	−0.284629	−	0.958638i	$-0.591870\pi$
$881$	−14885.8	−0.569257	−0.284629	+	0.958638i	$0.591870\pi$
$882$	0	0
$883$	22144.8	0.843979	0.421989	−	0.906601i	$-0.361332\pi$
$883$	22144.8	0.843979	0.421989	+	0.906601i	$0.361332\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	24626.1	0.932202	0.466101	−	0.884732i	$-0.345658\pi$
$887$	24626.1	0.932202	0.466101	+	0.884732i	$0.345658\pi$
$888$	0	0
$889$	−2844.14	−0.107300
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−27730.1	−1.03914
$894$	0	0
$895$	3413.75	0.127496
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−4133.69	−0.153355
$900$	0	0
$901$	−19325.9	−0.714583
$902$	0	0
$903$	0	0
$904$	0	0
$905$	296.972	0.0109079
$906$	0	0
$907$	18461.8	0.675871	0.337935	−	0.941169i	$-0.390271\pi$
$907$	18461.8	0.675871	0.337935	+	0.941169i	$0.390271\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−44797.5	−1.62921	−0.814603	−	0.580018i	$-0.803045\pi$
$911$	−44797.5	−1.62921	−0.814603	+	0.580018i	$0.803045\pi$
$912$	0	0
$913$	−3501.05	−0.126909
$914$	0	0
$915$	0	0
$916$	0	0
$917$	12372.0	0.445540
$918$	0	0
$919$	46730.6	1.67737	0.838684	−	0.544618i	$-0.183325\pi$
$919$	46730.6	1.67737	0.838684	+	0.544618i	$0.183325\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−54640.7	−1.94856
$924$	0	0
$925$	−10965.2	−0.389767
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−8092.99	−0.285815	−0.142908	−	0.989736i	$-0.545645\pi$
$929$	−8092.99	−0.285815	−0.142908	+	0.989736i	$0.545645\pi$
$930$	0	0
$931$	30359.0	1.06872
$932$	0	0
$933$	0	0
$934$	0	0
$935$	2463.64	0.0861706
$936$	0	0
$937$	−40337.2	−1.40636	−0.703180	−	0.711012i	$-0.748237\pi$
$937$	−40337.2	−1.40636	−0.703180	+	0.711012i	$0.748237\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	17800.7	0.616668	0.308334	−	0.951278i	$-0.400229\pi$
$941$	17800.7	0.616668	0.308334	+	0.951278i	$0.400229\pi$
$942$	0	0
$943$	−18761.4	−0.647886
$944$	0	0
$945$	0	0
$946$	0	0
$947$	3077.80	0.105612	0.0528062	−	0.998605i	$-0.483183\pi$
$947$	3077.80	0.105612	0.0528062	+	0.998605i	$0.483183\pi$
$948$	0	0
$949$	5714.97	0.195486
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−15297.9	−0.519986	−0.259993	−	0.965611i	$-0.583720\pi$
$953$	−15297.9	−0.519986	−0.259993	+	0.965611i	$0.583720\pi$
$954$	0	0
$955$	18090.6	0.612983
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−46788.2	−1.57546
$960$	0	0
$961$	−14346.9	−0.481585
$962$	0	0
$963$	0	0
$964$	0	0
$965$	7681.85	0.256257
$966$	0	0
$967$	22338.3	0.742865	0.371433	−	0.928460i	$-0.378867\pi$
$967$	22338.3	0.742865	0.371433	+	0.928460i	$0.378867\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−14686.2	−0.485378	−0.242689	−	0.970104i	$-0.578030\pi$
$971$	−14686.2	−0.485378	−0.242689	+	0.970104i	$0.578030\pi$
$972$	0	0
$973$	−44247.6	−1.45788
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−31945.7	−1.04609	−0.523047	−	0.852304i	$-0.675205\pi$
$977$	−31945.7	−1.04609	−0.523047	+	0.852304i	$0.675205\pi$
$978$	0	0
$979$	−12655.6	−0.413151
$980$	0	0
$981$	0	0
$982$	0	0
$983$	18534.7	0.601389	0.300694	−	0.953721i	$-0.402782\pi$
$983$	18534.7	0.601389	0.300694	+	0.953721i	$0.402782\pi$
$984$	0	0
$985$	−682.354	−0.0220727
$986$	0	0
$987$	0	0
$988$	0	0
$989$	1791.86	0.0576116
$990$	0	0
$991$	31869.1	1.02155	0.510775	−	0.859714i	$-0.329359\pi$
$991$	31869.1	1.02155	0.510775	+	0.859714i	$0.329359\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	616.275	0.0196354
$996$	0	0
$997$	7832.72	0.248811	0.124406	−	0.992231i	$-0.460298\pi$
$997$	7832.72	0.248811	0.124406	+	0.992231i	$0.460298\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2160.4.a.bp.1.1		3
3.2	odd	2		2160.4.a.bh.1.1		3
4.3	odd	2		1080.4.a.l.1.3	yes	3
12.11	even	2		1080.4.a.f.1.3	✓	3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1080.4.a.f.1.3	✓	3	12.11	even	2
1080.4.a.l.1.3	yes	3	4.3	odd	2
2160.4.a.bh.1.1		3	3.2	odd	2
2160.4.a.bp.1.1		3	1.1	even	1	trivial

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

\(f(q)\)	\(=\)	\(q+5.00000 q^{5} -24.7900 q^{7} -8.16892 q^{11} -46.7694 q^{13} -60.3173 q^{17} +111.802 q^{19} +36.9268 q^{23} +25.0000 q^{25} -33.2627 q^{29} +124.274 q^{31} -123.950 q^{35} -438.610 q^{37} -508.071 q^{41} +48.5247 q^{43} -248.030 q^{47} +271.543 q^{49} +320.404 q^{53} -40.8446 q^{55} -652.268 q^{59} +693.019 q^{61} -233.847 q^{65} +12.0308 q^{67} +1168.30 q^{71} -122.195 q^{73} +202.507 q^{77} +441.669 q^{79} +428.581 q^{83} -301.586 q^{85} +1549.24 q^{89} +1159.41 q^{91} +559.008 q^{95} -500.136 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 15 q^{5} + 27 q^{11} - 3 q^{13} - 15 q^{17} + 78 q^{19} + 105 q^{23} + 75 q^{25} - 117 q^{29} + 207 q^{31} - 120 q^{37} - 300 q^{41} + 483 q^{43} + 303 q^{47} - 15 q^{49} + 492 q^{53} + 135 q^{55}+ \cdots - 1392 q^{97}+O(q^{100}) \) 3 * q + 15 * q^5 + 27 * q^11 - 3 * q^13 - 15 * q^17 + 78 * q^19 + 105 * q^23 + 75 * q^25 - 117 * q^29 + 207 * q^31 - 120 * q^37 - 300 * q^41 + 483 * q^43 + 303 * q^47 - 15 * q^49 + 492 * q^53 + 135 * q^55 + 240 * q^59 - 444 * q^61 - 15 * q^65 + 522 * q^67 - 168 * q^71 - 876 * q^73 + 150 * q^77 + 2103 * q^79 - 42 * q^83 - 75 * q^85 + 2268 * q^89 + 1596 * q^91 + 390 * q^95 - 1392 * q^97

Introduction

L-functions

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