LMFDB - Embedded newform 2160.4.a.bf.1.3

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:	3.3.47977.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 60x - 44 $ x^3 - x^2 - 60*x - 44
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 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			1.3
Root			$-0.749725$ 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)

$f(q)$	$=$	$q-5.00000 q^{5} +24.3916 q^{7} +26.8932 q^{11} -68.2815 q^{13} -61.8899 q^{17} +75.1747 q^{19} +43.3916 q^{23} +25.0000 q^{25} +174.951 q^{29} -222.666 q^{31} -121.958 q^{35} +67.1813 q^{37} -22.2070 q^{41} +84.7442 q^{43} +585.650 q^{47} +251.948 q^{49} +38.3625 q^{53} -134.466 q^{55} -92.0000 q^{59} -226.300 q^{61} +341.407 q^{65} -858.035 q^{67} -116.912 q^{71} +911.769 q^{73} +655.967 q^{77} +285.129 q^{79} +999.099 q^{83} +309.450 q^{85} -374.848 q^{89} -1665.49 q^{91} -375.873 q^{95} -227.413 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 15 q^{5} - 9 q^{7} + 18 q^{11} - 21 q^{13} - 84 q^{17} - 21 q^{19} + 48 q^{23} + 75 q^{25} + 36 q^{29} - 324 q^{31} + 45 q^{35} + 33 q^{37} - 114 q^{41} - 282 q^{43} + 282 q^{47} + 228 q^{49} - 222 q^{53}+ \cdots + 1191 q^{97}+O(q^{100}) $ 3 * q - 15 * q^5 - 9 * q^7 + 18 * q^11 - 21 * q^13 - 84 * q^17 - 21 * q^19 + 48 * q^23 + 75 * q^25 + 36 * q^29 - 324 * q^31 + 45 * q^35 + 33 * q^37 - 114 * q^41 - 282 * q^43 + 282 * q^47 + 228 * q^49 - 222 * q^53 - 90 * q^55 - 276 * q^59 + 303 * q^61 + 105 * q^65 - 1035 * q^67 + 510 * q^71 + 447 * q^73 + 1578 * q^77 - 777 * q^79 + 78 * q^83 + 420 * q^85 + 324 * q^89 - 1995 * q^91 + 105 * q^95 + 1191 * 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
$7$	24.3916	1.31702	0.658510	−	0.752572i	$-0.271187\pi$
$7$	24.3916	1.31702	0.658510	+	0.752572i	$0.271187\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	26.8932	0.737146	0.368573	−	0.929599i	$-0.379846\pi$
$11$	26.8932	0.737146	0.368573	+	0.929599i	$0.379846\pi$
$12$	0	0
$13$	−68.2815	−1.45676	−0.728380	−	0.685174i	$-0.759726\pi$
$13$	−68.2815	−1.45676	−0.728380	+	0.685174i	$0.759726\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−61.8899	−0.882971	−0.441485	−	0.897268i	$-0.645548\pi$
$17$	−61.8899	−0.882971	−0.441485	+	0.897268i	$0.645548\pi$
$18$	0	0
$19$	75.1747	0.907697	0.453849	−	0.891079i	$-0.350051\pi$
$19$	75.1747	0.907697	0.453849	+	0.891079i	$0.350051\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	43.3916	0.393381	0.196691	−	0.980466i	$-0.436981\pi$
$23$	43.3916	0.393381	0.196691	+	0.980466i	$0.436981\pi$
$24$	0	0
$25$	25.0000	0.200000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	174.951	1.12026	0.560131	−	0.828404i	$-0.310751\pi$
$29$	174.951	1.12026	0.560131	+	0.828404i	$0.310751\pi$
$30$	0	0
$31$	−222.666	−1.29007	−0.645033	−	0.764154i	$-0.723157\pi$
$31$	−222.666	−1.29007	−0.645033	+	0.764154i	$0.723157\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−121.958	−0.588989
$36$	0	0
$37$	67.1813	0.298501	0.149250	−	0.988799i	$-0.452314\pi$
$37$	67.1813	0.298501	0.149250	+	0.988799i	$0.452314\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−22.2070	−0.0845890	−0.0422945	−	0.999105i	$-0.513467\pi$
$41$	−22.2070	−0.0845890	−0.0422945	+	0.999105i	$0.513467\pi$
$42$	0	0
$43$	84.7442	0.300543	0.150272	−	0.988645i	$-0.451985\pi$
$43$	84.7442	0.300543	0.150272	+	0.988645i	$0.451985\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	585.650	1.81757	0.908785	−	0.417264i	$-0.137011\pi$
$47$	585.650	1.81757	0.908785	+	0.417264i	$0.137011\pi$
$48$	0	0
$49$	251.948	0.734542
$50$	0	0
$51$	0	0
$52$	0	0
$53$	38.3625	0.0994245	0.0497122	−	0.998764i	$-0.484170\pi$
$53$	38.3625	0.0994245	0.0497122	+	0.998764i	$0.484170\pi$
$54$	0	0
$55$	−134.466	−0.329662
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−92.0000	−0.203006	−0.101503	−	0.994835i	$-0.532365\pi$
$59$	−92.0000	−0.203006	−0.101503	+	0.994835i	$0.532365\pi$
$60$	0	0
$61$	−226.300	−0.474997	−0.237498	−	0.971388i	$-0.576327\pi$
$61$	−226.300	−0.474997	−0.237498	+	0.971388i	$0.576327\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	341.407	0.651482
$66$	0	0
$67$	−858.035	−1.56456	−0.782281	−	0.622925i	$-0.785944\pi$
$67$	−858.035	−1.56456	−0.782281	+	0.622925i	$0.785944\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−116.912	−0.195422	−0.0977108	−	0.995215i	$-0.531152\pi$
$71$	−116.912	−0.195422	−0.0977108	+	0.995215i	$0.531152\pi$
$72$	0	0
$73$	911.769	1.46184	0.730921	−	0.682462i	$-0.239091\pi$
$73$	911.769	1.46184	0.730921	+	0.682462i	$0.239091\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	655.967	0.970836
$78$	0	0
$79$	285.129	0.406070	0.203035	−	0.979171i	$-0.434919\pi$
$79$	285.129	0.406070	0.203035	+	0.979171i	$0.434919\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	999.099	1.32127	0.660635	−	0.750707i	$-0.270287\pi$
$83$	999.099	1.32127	0.660635	+	0.750707i	$0.270287\pi$
$84$	0	0
$85$	309.450	0.394877
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−374.848	−0.446447	−0.223223	−	0.974767i	$-0.571658\pi$
$89$	−374.848	−0.446447	−0.223223	+	0.974767i	$0.571658\pi$
$90$	0	0
$91$	−1665.49	−1.91858
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−375.873	−0.405935
$96$	0	0
$97$	−227.413	−0.238044	−0.119022	−	0.992892i	$-0.537976\pi$
$97$	−227.413	−0.238044	−0.119022	+	0.992892i	$0.537976\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	1857.75	1.83023	0.915115	−	0.403193i	$-0.132100\pi$
$101$	1857.75	1.83023	0.915115	+	0.403193i	$0.132100\pi$
$102$	0	0
$103$	−172.787	−0.165294	−0.0826468	−	0.996579i	$-0.526337\pi$
$103$	−172.787	−0.165294	−0.0826468	+	0.996579i	$0.526337\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	1806.79	1.63242	0.816209	−	0.577756i	$-0.196072\pi$
$107$	1806.79	1.63242	0.816209	+	0.577756i	$0.196072\pi$
$108$	0	0
$109$	287.644	0.252764	0.126382	−	0.991982i	$-0.459663\pi$
$109$	287.644	0.252764	0.126382	+	0.991982i	$0.459663\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−671.875	−0.559333	−0.279667	−	0.960097i	$-0.590224\pi$
$113$	−671.875	−0.559333	−0.279667	+	0.960097i	$0.590224\pi$
$114$	0	0
$115$	−216.958	−0.175925
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−1509.59	−1.16289
$120$	0	0
$121$	−607.756	−0.456616
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−125.000	−0.0894427
$126$	0	0
$127$	−1818.41	−1.27053	−0.635265	−	0.772294i	$-0.719109\pi$
$127$	−1818.41	−1.27053	−0.635265	+	0.772294i	$0.719109\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−1768.99	−1.17983	−0.589915	−	0.807465i	$-0.700839\pi$
$131$	−1768.99	−1.17983	−0.589915	+	0.807465i	$0.700839\pi$
$132$	0	0
$133$	1833.63	1.19546
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−1842.85	−1.14924	−0.574618	−	0.818422i	$-0.694849\pi$
$137$	−1842.85	−1.14924	−0.574618	+	0.818422i	$0.694849\pi$
$138$	0	0
$139$	−1371.14	−0.836678	−0.418339	−	0.908291i	$-0.637388\pi$
$139$	−1371.14	−0.836678	−0.418339	+	0.908291i	$0.637388\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−1836.31	−1.07384
$144$	0	0
$145$	−874.756	−0.500997
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−2139.65	−1.17642	−0.588212	−	0.808707i	$-0.700168\pi$
$149$	−2139.65	−1.17642	−0.588212	+	0.808707i	$0.700168\pi$
$150$	0	0
$151$	1199.81	0.646616	0.323308	−	0.946294i	$-0.395205\pi$
$151$	1199.81	0.646616	0.323308	+	0.946294i	$0.395205\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	1113.33	0.576935
$156$	0	0
$157$	1847.13	0.938960	0.469480	−	0.882943i	$-0.344441\pi$
$157$	1847.13	0.938960	0.469480	+	0.882943i	$0.344441\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	1058.39	0.518091
$162$	0	0
$163$	3203.12	1.53919	0.769595	−	0.638532i	$-0.220458\pi$
$163$	3203.12	1.53919	0.769595	+	0.638532i	$0.220458\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−590.884	−0.273796	−0.136898	−	0.990585i	$-0.543713\pi$
$167$	−590.884	−0.273796	−0.136898	+	0.990585i	$0.543713\pi$
$168$	0	0
$169$	2465.36	1.12215
$170$	0	0
$171$	0	0
$172$	0	0
$173$	3110.68	1.36705	0.683527	−	0.729925i	$-0.260445\pi$
$173$	3110.68	1.36705	0.683527	+	0.729925i	$0.260445\pi$
$174$	0	0
$175$	609.789	0.263404
$176$	0	0
$177$	0	0
$178$	0	0
$179$	2448.94	1.02258	0.511291	−	0.859408i	$-0.329168\pi$
$179$	2448.94	1.02258	0.511291	+	0.859408i	$0.329168\pi$
$180$	0	0
$181$	2416.13	0.992206	0.496103	−	0.868264i	$-0.334764\pi$
$181$	2416.13	0.992206	0.496103	+	0.868264i	$0.334764\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−335.906	−0.133494
$186$	0	0
$187$	−1664.42	−0.650879
$188$	0	0
$189$	0	0
$190$	0	0
$191$	1652.25	0.625929	0.312964	−	0.949765i	$-0.398678\pi$
$191$	1652.25	0.625929	0.312964	+	0.949765i	$0.398678\pi$
$192$	0	0
$193$	4736.65	1.76659	0.883295	−	0.468818i	$-0.155320\pi$
$193$	4736.65	1.76659	0.883295	+	0.468818i	$0.155320\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−2106.79	−0.761944	−0.380972	−	0.924587i	$-0.624411\pi$
$197$	−2106.79	−0.761944	−0.380972	+	0.924587i	$0.624411\pi$
$198$	0	0
$199$	−891.149	−0.317447	−0.158723	−	0.987323i	$-0.550738\pi$
$199$	−891.149	−0.317447	−0.158723	+	0.987323i	$0.550738\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	4267.33	1.47541
$204$	0	0
$205$	111.035	0.0378294
$206$	0	0
$207$	0	0
$208$	0	0
$209$	2021.69	0.669105
$210$	0	0
$211$	−3607.11	−1.17689	−0.588444	−	0.808538i	$-0.700259\pi$
$211$	−3607.11	−1.17689	−0.588444	+	0.808538i	$0.700259\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−423.721	−0.134407
$216$	0	0
$217$	−5431.18	−1.69904
$218$	0	0
$219$	0	0
$220$	0	0
$221$	4225.93	1.28628
$222$	0	0
$223$	−2507.12	−0.752865	−0.376432	−	0.926444i	$-0.622849\pi$
$223$	−2507.12	−0.752865	−0.376432	+	0.926444i	$0.622849\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	3970.82	1.16102	0.580512	−	0.814252i	$-0.302853\pi$
$227$	3970.82	1.16102	0.580512	+	0.814252i	$0.302853\pi$
$228$	0	0
$229$	5494.88	1.58564	0.792820	−	0.609455i	$-0.208612\pi$
$229$	5494.88	1.58564	0.792820	+	0.609455i	$0.208612\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	6895.27	1.93873	0.969365	−	0.245625i	$-0.0789931\pi$
$233$	6895.27	1.93873	0.969365	+	0.245625i	$0.0789931\pi$
$234$	0	0
$235$	−2928.25	−0.812842
$236$	0	0
$237$	0	0
$238$	0	0
$239$	4308.62	1.16612	0.583058	−	0.812431i	$-0.301856\pi$
$239$	4308.62	1.16612	0.583058	+	0.812431i	$0.301856\pi$
$240$	0	0
$241$	3448.92	0.921843	0.460922	−	0.887441i	$-0.347519\pi$
$241$	3448.92	0.921843	0.460922	+	0.887441i	$0.347519\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−1259.74	−0.328497
$246$	0	0
$247$	−5133.04	−1.32230
$248$	0	0
$249$	0	0
$250$	0	0
$251$	6184.80	1.55530	0.777652	−	0.628695i	$-0.216411\pi$
$251$	6184.80	1.55530	0.777652	+	0.628695i	$0.216411\pi$
$252$	0	0
$253$	1166.94	0.289979
$254$	0	0
$255$	0	0
$256$	0	0
$257$	4616.98	1.12062	0.560310	−	0.828283i	$-0.310682\pi$
$257$	4616.98	1.12062	0.560310	+	0.828283i	$0.310682\pi$
$258$	0	0
$259$	1638.66	0.393132
$260$	0	0
$261$	0	0
$262$	0	0
$263$	728.557	0.170816	0.0854082	−	0.996346i	$-0.472781\pi$
$263$	728.557	0.170816	0.0854082	+	0.996346i	$0.472781\pi$
$264$	0	0
$265$	−191.813	−0.0444640
$266$	0	0
$267$	0	0
$268$	0	0
$269$	6718.17	1.52273	0.761364	−	0.648325i	$-0.224530\pi$
$269$	6718.17	1.52273	0.761364	+	0.648325i	$0.224530\pi$
$270$	0	0
$271$	2435.89	0.546015	0.273007	−	0.962012i	$-0.411982\pi$
$271$	2435.89	0.546015	0.273007	+	0.962012i	$0.411982\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	672.330	0.147429
$276$	0	0
$277$	−4808.18	−1.04295	−0.521473	−	0.853268i	$-0.674617\pi$
$277$	−4808.18	−1.04295	−0.521473	+	0.853268i	$0.674617\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−3535.67	−0.750606	−0.375303	−	0.926902i	$-0.622461\pi$
$281$	−3535.67	−0.750606	−0.375303	+	0.926902i	$0.622461\pi$
$282$	0	0
$283$	−3070.64	−0.644984	−0.322492	−	0.946572i	$-0.604521\pi$
$283$	−3070.64	−0.644984	−0.322492	+	0.946572i	$0.604521\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−541.663	−0.111405
$288$	0	0
$289$	−1082.64	−0.220362
$290$	0	0
$291$	0	0
$292$	0	0
$293$	1245.18	0.248273	0.124136	−	0.992265i	$-0.460384\pi$
$293$	1245.18	0.248273	0.124136	+	0.992265i	$0.460384\pi$
$294$	0	0
$295$	460.000	0.0907872
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−2962.84	−0.573061
$300$	0	0
$301$	2067.04	0.395822
$302$	0	0
$303$	0	0
$304$	0	0
$305$	1131.50	0.212425
$306$	0	0
$307$	4269.79	0.793778	0.396889	−	0.917867i	$-0.370090\pi$
$307$	4269.79	0.793778	0.396889	+	0.917867i	$0.370090\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	2584.39	0.471213	0.235607	−	0.971849i	$-0.424292\pi$
$311$	2584.39	0.471213	0.235607	+	0.971849i	$0.424292\pi$
$312$	0	0
$313$	−8899.44	−1.60711	−0.803556	−	0.595229i	$-0.797061\pi$
$313$	−8899.44	−1.60711	−0.803556	+	0.595229i	$0.797061\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	6168.58	1.09294	0.546470	−	0.837479i	$-0.315971\pi$
$317$	6168.58	1.09294	0.546470	+	0.837479i	$0.315971\pi$
$318$	0	0
$319$	4705.00	0.825797
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−4652.55	−0.801470
$324$	0	0
$325$	−1707.04	−0.291352
$326$	0	0
$327$	0	0
$328$	0	0
$329$	14284.9	2.39378
$330$	0	0
$331$	9996.33	1.65996	0.829982	−	0.557791i	$-0.188351\pi$
$331$	9996.33	1.65996	0.829982	+	0.557791i	$0.188351\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	4290.18	0.699694
$336$	0	0
$337$	−2497.40	−0.403685	−0.201843	−	0.979418i	$-0.564693\pi$
$337$	−2497.40	−0.403685	−0.201843	+	0.979418i	$0.564693\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−5988.21	−0.950967
$342$	0	0
$343$	−2220.90	−0.349614
$344$	0	0
$345$	0	0
$346$	0	0
$347$	9001.22	1.39254	0.696269	−	0.717780i	$-0.254842\pi$
$347$	9001.22	1.39254	0.696269	+	0.717780i	$0.254842\pi$
$348$	0	0
$349$	−10707.4	−1.64227	−0.821135	−	0.570733i	$-0.806659\pi$
$349$	−10707.4	−1.64227	−0.821135	+	0.570733i	$0.806659\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−2843.35	−0.428714	−0.214357	−	0.976755i	$-0.568766\pi$
$353$	−2843.35	−0.428714	−0.214357	+	0.976755i	$0.568766\pi$
$354$	0	0
$355$	584.561	0.0873952
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−9328.89	−1.37148	−0.685738	−	0.727849i	$-0.740520\pi$
$359$	−9328.89	−1.37148	−0.685738	+	0.727849i	$0.740520\pi$
$360$	0	0
$361$	−1207.77	−0.176086
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−4558.85	−0.653756
$366$	0	0
$367$	−46.7367	−0.00664751	−0.00332376	−	0.999994i	$-0.501058\pi$
$367$	−46.7367	−0.00664751	−0.00332376	+	0.999994i	$0.501058\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	935.721	0.130944
$372$	0	0
$373$	−3791.98	−0.526384	−0.263192	−	0.964743i	$-0.584775\pi$
$373$	−3791.98	−0.526384	−0.263192	+	0.964743i	$0.584775\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−11945.9	−1.63195
$378$	0	0
$379$	−1881.74	−0.255035	−0.127518	−	0.991836i	$-0.540701\pi$
$379$	−1881.74	−0.255035	−0.127518	+	0.991836i	$0.540701\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	12011.9	1.60256	0.801280	−	0.598289i	$-0.204153\pi$
$383$	12011.9	1.60256	0.801280	+	0.598289i	$0.204153\pi$
$384$	0	0
$385$	−3279.83	−0.434171
$386$	0	0
$387$	0	0
$388$	0	0
$389$	7758.91	1.01129	0.505645	−	0.862741i	$-0.331254\pi$
$389$	7758.91	1.01129	0.505645	+	0.862741i	$0.331254\pi$
$390$	0	0
$391$	−2685.50	−0.347344
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−1425.65	−0.181600
$396$	0	0
$397$	14841.9	1.87630	0.938152	−	0.346223i	$-0.112536\pi$
$397$	14841.9	1.87630	0.938152	+	0.346223i	$0.112536\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−435.446	−0.0542273	−0.0271136	−	0.999632i	$-0.508632\pi$
$401$	−435.446	−0.0542273	−0.0271136	+	0.999632i	$0.508632\pi$
$402$	0	0
$403$	15204.0	1.87932
$404$	0	0
$405$	0	0
$406$	0	0
$407$	1806.72	0.220039
$408$	0	0
$409$	10550.6	1.27554	0.637770	−	0.770227i	$-0.279857\pi$
$409$	10550.6	1.27554	0.637770	+	0.770227i	$0.279857\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−2244.02	−0.267364
$414$	0	0
$415$	−4995.50	−0.590890
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−4161.11	−0.485164	−0.242582	−	0.970131i	$-0.577994\pi$
$419$	−4161.11	−0.485164	−0.242582	+	0.970131i	$0.577994\pi$
$420$	0	0
$421$	154.745	0.0179140	0.00895701	−	0.999960i	$-0.497149\pi$
$421$	154.745	0.0179140	0.00895701	+	0.999960i	$0.497149\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−1547.25	−0.176594
$426$	0	0
$427$	−5519.82	−0.625580
$428$	0	0
$429$	0	0
$430$	0	0
$431$	4393.72	0.491040	0.245520	−	0.969392i	$-0.421041\pi$
$431$	4393.72	0.491040	0.245520	+	0.969392i	$0.421041\pi$
$432$	0	0
$433$	4437.08	0.492454	0.246227	−	0.969212i	$-0.420809\pi$
$433$	4437.08	0.492454	0.246227	+	0.969212i	$0.420809\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	3261.95	0.357071
$438$	0	0
$439$	7046.68	0.766104	0.383052	−	0.923727i	$-0.374873\pi$
$439$	7046.68	0.766104	0.383052	+	0.923727i	$0.374873\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−7335.09	−0.786683	−0.393341	−	0.919392i	$-0.628681\pi$
$443$	−7335.09	−0.786683	−0.393341	+	0.919392i	$0.628681\pi$
$444$	0	0
$445$	1874.24	0.199657
$446$	0	0
$447$	0	0
$448$	0	0
$449$	731.914	0.0769290	0.0384645	−	0.999260i	$-0.487753\pi$
$449$	731.914	0.0769290	0.0384645	+	0.999260i	$0.487753\pi$
$450$	0	0
$451$	−597.217	−0.0623545
$452$	0	0
$453$	0	0
$454$	0	0
$455$	8327.45	0.858016
$456$	0	0
$457$	−12611.7	−1.29092	−0.645462	−	0.763792i	$-0.723335\pi$
$457$	−12611.7	−1.29092	−0.645462	+	0.763792i	$0.723335\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	12971.9	1.31055	0.655275	−	0.755391i	$-0.272553\pi$
$461$	12971.9	1.31055	0.655275	+	0.755391i	$0.272553\pi$
$462$	0	0
$463$	−9899.48	−0.993667	−0.496833	−	0.867846i	$-0.665504\pi$
$463$	−9899.48	−0.993667	−0.496833	+	0.867846i	$0.665504\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	2180.60	0.216073	0.108037	−	0.994147i	$-0.465544\pi$
$467$	2180.60	0.216073	0.108037	+	0.994147i	$0.465544\pi$
$468$	0	0
$469$	−20928.8	−2.06056
$470$	0	0
$471$	0	0
$472$	0	0
$473$	2279.04	0.221544
$474$	0	0
$475$	1879.37	0.181539
$476$	0	0
$477$	0	0
$478$	0	0
$479$	4132.54	0.394197	0.197099	−	0.980384i	$-0.436848\pi$
$479$	4132.54	0.394197	0.197099	+	0.980384i	$0.436848\pi$
$480$	0	0
$481$	−4587.23	−0.434844
$482$	0	0
$483$	0	0
$484$	0	0
$485$	1137.07	0.106457
$486$	0	0
$487$	19277.8	1.79376	0.896880	−	0.442275i	$-0.145828\pi$
$487$	19277.8	1.79376	0.896880	+	0.442275i	$0.145828\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−1315.88	−0.120946	−0.0604732	−	0.998170i	$-0.519261\pi$
$491$	−1315.88	−0.120946	−0.0604732	+	0.998170i	$0.519261\pi$
$492$	0	0
$493$	−10827.7	−0.989159
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−2851.67	−0.257374
$498$	0	0
$499$	−10477.9	−0.939988	−0.469994	−	0.882670i	$-0.655744\pi$
$499$	−10477.9	−0.939988	−0.469994	+	0.882670i	$0.655744\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−9312.29	−0.825476	−0.412738	−	0.910850i	$-0.635428\pi$
$503$	−9312.29	−0.825476	−0.412738	+	0.910850i	$0.635428\pi$
$504$	0	0
$505$	−9288.76	−0.818504
$506$	0	0
$507$	0	0
$508$	0	0
$509$	21595.3	1.88054	0.940270	−	0.340429i	$-0.110572\pi$
$509$	21595.3	1.88054	0.940270	+	0.340429i	$0.110572\pi$
$510$	0	0
$511$	22239.5	1.92528
$512$	0	0
$513$	0	0
$514$	0	0
$515$	863.936	0.0739215
$516$	0	0
$517$	15750.0	1.33981
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−9386.88	−0.789341	−0.394670	−	0.918823i	$-0.629141\pi$
$521$	−9386.88	−0.789341	−0.394670	+	0.918823i	$0.629141\pi$
$522$	0	0
$523$	19068.9	1.59431	0.797156	−	0.603773i	$-0.206337\pi$
$523$	19068.9	1.59431	0.797156	+	0.603773i	$0.206337\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	13780.8	1.13909
$528$	0	0
$529$	−10284.2	−0.845251
$530$	0	0
$531$	0	0
$532$	0	0
$533$	1516.33	0.123226
$534$	0	0
$535$	−9033.94	−0.730040
$536$	0	0
$537$	0	0
$538$	0	0
$539$	6775.69	0.541465
$540$	0	0
$541$	−22209.4	−1.76499	−0.882493	−	0.470325i	$-0.844137\pi$
$541$	−22209.4	−1.76499	−0.882493	+	0.470325i	$0.844137\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−1438.22	−0.113040
$546$	0	0
$547$	2233.14	0.174556	0.0872782	−	0.996184i	$-0.472183\pi$
$547$	2233.14	0.174556	0.0872782	+	0.996184i	$0.472183\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	13151.9	1.01686
$552$	0	0
$553$	6954.74	0.534802
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−19727.7	−1.50070	−0.750351	−	0.661039i	$-0.770116\pi$
$557$	−19727.7	−1.50070	−0.750351	+	0.661039i	$0.770116\pi$
$558$	0	0
$559$	−5786.46	−0.437819
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−12025.7	−0.900216	−0.450108	−	0.892974i	$-0.648615\pi$
$563$	−12025.7	−0.900216	−0.450108	+	0.892974i	$0.648615\pi$
$564$	0	0
$565$	3359.37	0.250142
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−19708.8	−1.45209	−0.726044	−	0.687648i	$-0.758643\pi$
$569$	−19708.8	−1.45209	−0.726044	+	0.687648i	$0.758643\pi$
$570$	0	0
$571$	−3965.65	−0.290643	−0.145322	−	0.989384i	$-0.546422\pi$
$571$	−3965.65	−0.290643	−0.145322	+	0.989384i	$0.546422\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	1084.79	0.0786762
$576$	0	0
$577$	4134.92	0.298334	0.149167	−	0.988812i	$-0.452341\pi$
$577$	4134.92	0.298334	0.149167	+	0.988812i	$0.452341\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	24369.6	1.74014
$582$	0	0
$583$	1031.69	0.0732904
$584$	0	0
$585$	0	0
$586$	0	0
$587$	10949.0	0.769873	0.384937	−	0.922943i	$-0.374223\pi$
$587$	10949.0	0.769873	0.384937	+	0.922943i	$0.374223\pi$
$588$	0	0
$589$	−16738.9	−1.17099
$590$	0	0
$591$	0	0
$592$	0	0
$593$	17834.7	1.23505	0.617523	−	0.786553i	$-0.288136\pi$
$593$	17834.7	1.23505	0.617523	+	0.786553i	$0.288136\pi$
$594$	0	0
$595$	7547.95	0.520060
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−6052.79	−0.412872	−0.206436	−	0.978460i	$-0.566186\pi$
$599$	−6052.79	−0.412872	−0.206436	+	0.978460i	$0.566186\pi$
$600$	0	0
$601$	5233.02	0.355174	0.177587	−	0.984105i	$-0.443171\pi$
$601$	5233.02	0.355174	0.177587	+	0.984105i	$0.443171\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	3038.78	0.204205
$606$	0	0
$607$	24209.4	1.61883	0.809413	−	0.587240i	$-0.199785\pi$
$607$	24209.4	1.61883	0.809413	+	0.587240i	$0.199785\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−39989.0	−2.64776
$612$	0	0
$613$	7255.45	0.478050	0.239025	−	0.971013i	$-0.423172\pi$
$613$	7255.45	0.478050	0.239025	+	0.971013i	$0.423172\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−29694.7	−1.93754	−0.968772	−	0.247951i	$-0.920243\pi$
$617$	−29694.7	−1.93754	−0.968772	+	0.247951i	$0.920243\pi$
$618$	0	0
$619$	−10775.2	−0.699665	−0.349833	−	0.936812i	$-0.613762\pi$
$619$	−10775.2	−0.699665	−0.349833	+	0.936812i	$0.613762\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−9143.11	−0.587979
$624$	0	0
$625$	625.000	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−4157.84	−0.263568
$630$	0	0
$631$	3661.18	0.230981	0.115491	−	0.993309i	$-0.463156\pi$
$631$	3661.18	0.230981	0.115491	+	0.993309i	$0.463156\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	9092.03	0.568199
$636$	0	0
$637$	−17203.4	−1.07005
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−5627.43	−0.346755	−0.173378	−	0.984855i	$-0.555468\pi$
$641$	−5627.43	−0.346755	−0.173378	+	0.984855i	$0.555468\pi$
$642$	0	0
$643$	10732.5	0.658243	0.329122	−	0.944288i	$-0.393247\pi$
$643$	10732.5	0.658243	0.329122	+	0.944288i	$0.393247\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	28753.1	1.74714	0.873571	−	0.486697i	$-0.161798\pi$
$647$	28753.1	1.74714	0.873571	+	0.486697i	$0.161798\pi$
$648$	0	0
$649$	−2474.17	−0.149645
$650$	0	0
$651$	0	0
$652$	0	0
$653$	27398.6	1.64195	0.820974	−	0.570966i	$-0.193431\pi$
$653$	27398.6	1.64195	0.820974	+	0.570966i	$0.193431\pi$
$654$	0	0
$655$	8844.97	0.527636
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−26446.7	−1.56331	−0.781653	−	0.623714i	$-0.785623\pi$
$659$	−26446.7	−1.56331	−0.781653	+	0.623714i	$0.785623\pi$
$660$	0	0
$661$	25643.7	1.50897	0.754483	−	0.656320i	$-0.227888\pi$
$661$	25643.7	1.50897	0.754483	+	0.656320i	$0.227888\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−9168.13	−0.534624
$666$	0	0
$667$	7591.40	0.440690
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−6085.94	−0.350142
$672$	0	0
$673$	10796.0	0.618357	0.309179	−	0.951004i	$-0.399946\pi$
$673$	10796.0	0.618357	0.309179	+	0.951004i	$0.399946\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	13183.2	0.748406	0.374203	−	0.927347i	$-0.377916\pi$
$677$	13183.2	0.748406	0.374203	+	0.927347i	$0.377916\pi$
$678$	0	0
$679$	−5546.96	−0.313509
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−1349.06	−0.0755786	−0.0377893	−	0.999286i	$-0.512032\pi$
$683$	−1349.06	−0.0755786	−0.0377893	+	0.999286i	$0.512032\pi$
$684$	0	0
$685$	9214.26	0.513954
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−2619.45	−0.144838
$690$	0	0
$691$	−25133.8	−1.38370	−0.691849	−	0.722042i	$-0.743204\pi$
$691$	−25133.8	−1.38370	−0.691849	+	0.722042i	$0.743204\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	6855.68	0.374174
$696$	0	0
$697$	1374.39	0.0746896
$698$	0	0
$699$	0	0
$700$	0	0
$701$	14808.9	0.797895	0.398948	−	0.916974i	$-0.369375\pi$
$701$	14808.9	0.797895	0.398948	+	0.916974i	$0.369375\pi$
$702$	0	0
$703$	5050.33	0.270948
$704$	0	0
$705$	0	0
$706$	0	0
$707$	45313.5	2.41045
$708$	0	0
$709$	−13857.2	−0.734019	−0.367009	−	0.930217i	$-0.619618\pi$
$709$	−13857.2	−0.734019	−0.367009	+	0.930217i	$0.619618\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−9661.84	−0.507488
$714$	0	0
$715$	9181.54	0.480238
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−21853.5	−1.13352	−0.566759	−	0.823884i	$-0.691803\pi$
$719$	−21853.5	−1.13352	−0.566759	+	0.823884i	$0.691803\pi$
$720$	0	0
$721$	−4214.55	−0.217695
$722$	0	0
$723$	0	0
$724$	0	0
$725$	4373.78	0.224053
$726$	0	0
$727$	1442.23	0.0735757	0.0367878	−	0.999323i	$-0.488287\pi$
$727$	1442.23	0.0735757	0.0367878	+	0.999323i	$0.488287\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−5244.81	−0.265371
$732$	0	0
$733$	2606.50	0.131342	0.0656708	−	0.997841i	$-0.479081\pi$
$733$	2606.50	0.131342	0.0656708	+	0.997841i	$0.479081\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−23075.3	−1.15331
$738$	0	0
$739$	−9072.37	−0.451600	−0.225800	−	0.974174i	$-0.572500\pi$
$739$	−9072.37	−0.451600	−0.225800	+	0.974174i	$0.572500\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−10093.2	−0.498364	−0.249182	−	0.968457i	$-0.580162\pi$
$743$	−10093.2	−0.498364	−0.249182	+	0.968457i	$0.580162\pi$
$744$	0	0
$745$	10698.3	0.526113
$746$	0	0
$747$	0	0
$748$	0	0
$749$	44070.4	2.14993
$750$	0	0
$751$	−22033.4	−1.07059	−0.535294	−	0.844666i	$-0.679799\pi$
$751$	−22033.4	−1.07059	−0.535294	+	0.844666i	$0.679799\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−5999.04	−0.289175
$756$	0	0
$757$	8362.75	0.401518	0.200759	−	0.979641i	$-0.435659\pi$
$757$	8362.75	0.401518	0.200759	+	0.979641i	$0.435659\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	11405.1	0.543280	0.271640	−	0.962399i	$-0.412434\pi$
$761$	11405.1	0.543280	0.271640	+	0.962399i	$0.412434\pi$
$762$	0	0
$763$	7016.09	0.332896
$764$	0	0
$765$	0	0
$766$	0	0
$767$	6281.89	0.295731
$768$	0	0
$769$	1447.34	0.0678704	0.0339352	−	0.999424i	$-0.489196\pi$
$769$	1447.34	0.0678704	0.0339352	+	0.999424i	$0.489196\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−25835.9	−1.20214	−0.601070	−	0.799196i	$-0.705259\pi$
$773$	−25835.9	−1.20214	−0.601070	+	0.799196i	$0.705259\pi$
$774$	0	0
$775$	−5566.66	−0.258013
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−1669.40	−0.0767812
$780$	0	0
$781$	−3144.14	−0.144054
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−9235.63	−0.419916
$786$	0	0
$787$	−15859.9	−0.718355	−0.359178	−	0.933269i	$-0.616943\pi$
$787$	−15859.9	−0.718355	−0.359178	+	0.933269i	$0.616943\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−16388.1	−0.736653
$792$	0	0
$793$	15452.1	0.691956
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−32805.2	−1.45799	−0.728997	−	0.684517i	$-0.760013\pi$
$797$	−32805.2	−1.45799	−0.728997	+	0.684517i	$0.760013\pi$
$798$	0	0
$799$	−36245.8	−1.60486
$800$	0	0
$801$	0	0
$802$	0	0
$803$	24520.4	1.07759
$804$	0	0
$805$	−5291.94	−0.231697
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−17068.4	−0.741772	−0.370886	−	0.928678i	$-0.620946\pi$
$809$	−17068.4	−0.741772	−0.370886	+	0.928678i	$0.620946\pi$
$810$	0	0
$811$	−12079.5	−0.523018	−0.261509	−	0.965201i	$-0.584220\pi$
$811$	−12079.5	−0.523018	−0.261509	+	0.965201i	$0.584220\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−16015.6	−0.688347
$816$	0	0
$817$	6370.61	0.272802
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−34364.3	−1.46081	−0.730403	−	0.683016i	$-0.760668\pi$
$821$	−34364.3	−1.46081	−0.730403	+	0.683016i	$0.760668\pi$
$822$	0	0
$823$	−12983.0	−0.549891	−0.274946	−	0.961460i	$-0.588660\pi$
$823$	−12983.0	−0.549891	−0.274946	+	0.961460i	$0.588660\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−25503.7	−1.07237	−0.536186	−	0.844100i	$-0.680135\pi$
$827$	−25503.7	−1.07237	−0.536186	+	0.844100i	$0.680135\pi$
$828$	0	0
$829$	26016.9	1.08999	0.544997	−	0.838438i	$-0.316531\pi$
$829$	26016.9	1.08999	0.544997	+	0.838438i	$0.316531\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−15593.0	−0.648579
$834$	0	0
$835$	2954.42	0.122445
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−11090.3	−0.456353	−0.228177	−	0.973620i	$-0.573276\pi$
$839$	−11090.3	−0.456353	−0.228177	+	0.973620i	$0.573276\pi$
$840$	0	0
$841$	6218.91	0.254988
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−12326.8	−0.501839
$846$	0	0
$847$	−14824.1	−0.601372
$848$	0	0
$849$	0	0
$850$	0	0
$851$	2915.10	0.117425
$852$	0	0
$853$	41006.8	1.64601	0.823004	−	0.568036i	$-0.192296\pi$
$853$	41006.8	1.64601	0.823004	+	0.568036i	$0.192296\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−38890.1	−1.55013	−0.775065	−	0.631881i	$-0.782283\pi$
$857$	−38890.1	−1.55013	−0.775065	+	0.631881i	$0.782283\pi$
$858$	0	0
$859$	−43062.9	−1.71046	−0.855232	−	0.518246i	$-0.826585\pi$
$859$	−43062.9	−1.71046	−0.855232	+	0.518246i	$0.826585\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−2294.97	−0.0905232	−0.0452616	−	0.998975i	$-0.514412\pi$
$863$	−2294.97	−0.0905232	−0.0452616	+	0.998975i	$0.514412\pi$
$864$	0	0
$865$	−15553.4	−0.611365
$866$	0	0
$867$	0	0
$868$	0	0
$869$	7668.03	0.299333
$870$	0	0
$871$	58587.9	2.27919
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−3048.94	−0.117798
$876$	0	0
$877$	33278.9	1.28136	0.640678	−	0.767810i	$-0.278653\pi$
$877$	33278.9	1.28136	0.640678	+	0.767810i	$0.278653\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	9821.12	0.375575	0.187788	−	0.982210i	$-0.439868\pi$
$881$	9821.12	0.375575	0.187788	+	0.982210i	$0.439868\pi$
$882$	0	0
$883$	13745.6	0.523869	0.261934	−	0.965086i	$-0.415640\pi$
$883$	13745.6	0.523869	0.261934	+	0.965086i	$0.415640\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−24565.8	−0.929922	−0.464961	−	0.885331i	$-0.653932\pi$
$887$	−24565.8	−0.929922	−0.464961	+	0.885331i	$0.653932\pi$
$888$	0	0
$889$	−44353.7	−1.67331
$890$	0	0
$891$	0	0
$892$	0	0
$893$	44026.0	1.64980
$894$	0	0
$895$	−12244.7	−0.457312
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−38955.7	−1.44521
$900$	0	0
$901$	−2374.25	−0.0877889
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−12080.6	−0.443728
$906$	0	0
$907$	−20218.5	−0.740181	−0.370091	−	0.928996i	$-0.620673\pi$
$907$	−20218.5	−0.740181	−0.370091	+	0.928996i	$0.620673\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−4552.96	−0.165583	−0.0827915	−	0.996567i	$-0.526384\pi$
$911$	−4552.96	−0.165583	−0.0827915	+	0.996567i	$0.526384\pi$
$912$	0	0
$913$	26869.0	0.973969
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−43148.5	−1.55386
$918$	0	0
$919$	−19814.0	−0.711211	−0.355605	−	0.934636i	$-0.615725\pi$
$919$	−19814.0	−0.711211	−0.355605	+	0.934636i	$0.615725\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	7982.94	0.284682
$924$	0	0
$925$	1679.53	0.0597002
$926$	0	0
$927$	0	0
$928$	0	0
$929$	40202.9	1.41982	0.709911	−	0.704291i	$-0.248735\pi$
$929$	40202.9	1.41982	0.709911	+	0.704291i	$0.248735\pi$
$930$	0	0
$931$	18940.1	0.666742
$932$	0	0
$933$	0	0
$934$	0	0
$935$	8322.09	0.291082
$936$	0	0
$937$	24587.9	0.857260	0.428630	−	0.903480i	$-0.358996\pi$
$937$	24587.9	0.857260	0.428630	+	0.903480i	$0.358996\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−33462.1	−1.15923	−0.579614	−	0.814891i	$-0.696797\pi$
$941$	−33462.1	−1.15923	−0.579614	+	0.814891i	$0.696797\pi$
$942$	0	0
$943$	−963.596	−0.0332757
$944$	0	0
$945$	0	0
$946$	0	0
$947$	8385.17	0.287731	0.143866	−	0.989597i	$-0.454047\pi$
$947$	8385.17	0.287731	0.143866	+	0.989597i	$0.454047\pi$
$948$	0	0
$949$	−62256.9	−2.12955
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−55532.0	−1.88758	−0.943788	−	0.330552i	$-0.892765\pi$
$953$	−55532.0	−1.88758	−0.943788	+	0.330552i	$0.892765\pi$
$954$	0	0
$955$	−8261.23	−0.279924
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−44950.0	−1.51357
$960$	0	0
$961$	19789.3	0.664272
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−23683.3	−0.790043
$966$	0	0
$967$	44080.8	1.46592	0.732960	−	0.680272i	$-0.238138\pi$
$967$	44080.8	1.46592	0.732960	+	0.680272i	$0.238138\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−29721.4	−0.982291	−0.491146	−	0.871077i	$-0.663422\pi$
$971$	−29721.4	−0.982291	−0.491146	+	0.871077i	$0.663422\pi$
$972$	0	0
$973$	−33444.2	−1.10192
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−462.881	−0.0151575	−0.00757876	−	0.999971i	$-0.502412\pi$
$977$	−462.881	−0.0151575	−0.00757876	+	0.999971i	$0.502412\pi$
$978$	0	0
$979$	−10080.9	−0.329096
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−27389.7	−0.888703	−0.444351	−	0.895853i	$-0.646566\pi$
$983$	−27389.7	−0.888703	−0.444351	+	0.895853i	$0.646566\pi$
$984$	0	0
$985$	10534.0	0.340752
$986$	0	0
$987$	0	0
$988$	0	0
$989$	3677.18	0.118228
$990$	0	0
$991$	−32596.6	−1.04487	−0.522435	−	0.852679i	$-0.674976\pi$
$991$	−32596.6	−1.04487	−0.522435	+	0.852679i	$0.674976\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	4455.75	0.141966
$996$	0	0
$997$	−26630.6	−0.845936	−0.422968	−	0.906145i	$-0.639012\pi$
$997$	−26630.6	−0.845936	−0.422968	+	0.906145i	$0.639012\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2160.4.a.bf.1.3		3
3.2	odd	2		2160.4.a.bn.1.3		3
4.3	odd	2		1080.4.a.h.1.1	✓	3
12.11	even	2		1080.4.a.n.1.1	yes	3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1080.4.a.h.1.1	✓	3	4.3	odd	2
1080.4.a.n.1.1	yes	3	12.11	even	2
2160.4.a.bf.1.3		3	1.1	even	1	trivial
2160.4.a.bn.1.3		3	3.2	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

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

\(f(q)\)	\(=\)	\(q-5.00000 q^{5} +24.3916 q^{7} +26.8932 q^{11} -68.2815 q^{13} -61.8899 q^{17} +75.1747 q^{19} +43.3916 q^{23} +25.0000 q^{25} +174.951 q^{29} -222.666 q^{31} -121.958 q^{35} +67.1813 q^{37} -22.2070 q^{41} +84.7442 q^{43} +585.650 q^{47} +251.948 q^{49} +38.3625 q^{53} -134.466 q^{55} -92.0000 q^{59} -226.300 q^{61} +341.407 q^{65} -858.035 q^{67} -116.912 q^{71} +911.769 q^{73} +655.967 q^{77} +285.129 q^{79} +999.099 q^{83} +309.450 q^{85} -374.848 q^{89} -1665.49 q^{91} -375.873 q^{95} -227.413 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 15 q^{5} - 9 q^{7} + 18 q^{11} - 21 q^{13} - 84 q^{17} - 21 q^{19} + 48 q^{23} + 75 q^{25} + 36 q^{29} - 324 q^{31} + 45 q^{35} + 33 q^{37} - 114 q^{41} - 282 q^{43} + 282 q^{47} + 228 q^{49} - 222 q^{53}+ \cdots + 1191 q^{97}+O(q^{100}) \) 3 * q - 15 * q^5 - 9 * q^7 + 18 * q^11 - 21 * q^13 - 84 * q^17 - 21 * q^19 + 48 * q^23 + 75 * q^25 + 36 * q^29 - 324 * q^31 + 45 * q^35 + 33 * q^37 - 114 * q^41 - 282 * q^43 + 282 * q^47 + 228 * q^49 - 222 * q^53 - 90 * q^55 - 276 * q^59 + 303 * q^61 + 105 * q^65 - 1035 * q^67 + 510 * q^71 + 447 * q^73 + 1578 * q^77 - 777 * q^79 + 78 * q^83 + 420 * q^85 + 324 * q^89 - 1995 * q^91 + 105 * q^95 + 1191 * q^97

Introduction

L-functions

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