LMFDB - Embedded newform 2160.4.a.bc.1.2

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:	$2$
Coefficient field:	$\Q(\sqrt{21}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 5 $ x^2 - x - 5
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			1.2
Root			$2.79129$ 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)

$f(q)$	$=$	$q+5.00000 q^{5} +14.7477 q^{7} +O(q^{10})$	$q+5.00000 q^{5} +14.7477 q^{7} +7.25227 q^{11} +61.7386 q^{13} -108.234 q^{17} +56.2523 q^{19} -46.9818 q^{23} +25.0000 q^{25} +214.225 q^{29} +261.711 q^{31} +73.7386 q^{35} -286.000 q^{37} -255.702 q^{41} +361.234 q^{43} -5.53182 q^{47} -125.505 q^{49} +595.693 q^{53} +36.2614 q^{55} +315.234 q^{59} +276.405 q^{61} +308.693 q^{65} -117.405 q^{67} +192.784 q^{71} -756.189 q^{73} +106.955 q^{77} +1150.70 q^{79} +141.936 q^{83} -541.170 q^{85} -719.107 q^{89} +910.505 q^{91} +281.261 q^{95} -1043.82 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 10 q^{5} + 2 q^{7}+O(q^{10}) $ 2 * q + 10 * q^5 + 2 * q^7	$ 2 q + 10 q^{5} + 2 q^{7} + 42 q^{11} - 14 q^{13} - 24 q^{17} + 140 q^{19} + 126 q^{23} + 50 q^{25} + 126 q^{29} + 56 q^{31} + 10 q^{35} - 572 q^{37} + 66 q^{41} + 530 q^{43} - 396 q^{47} - 306 q^{49} + 504 q^{53} + 210 q^{55} + 438 q^{59} - 602 q^{61} - 70 q^{65} + 920 q^{67} + 798 q^{71} - 770 q^{73} - 336 q^{77} + 1724 q^{79} - 486 q^{83} - 120 q^{85} + 294 q^{89} + 1876 q^{91} + 700 q^{95} + 112 q^{97}+O(q^{100}) $ 2 * q + 10 * q^5 + 2 * q^7 + 42 * q^11 - 14 * q^13 - 24 * q^17 + 140 * q^19 + 126 * q^23 + 50 * q^25 + 126 * q^29 + 56 * q^31 + 10 * q^35 - 572 * q^37 + 66 * q^41 + 530 * q^43 - 396 * q^47 - 306 * q^49 + 504 * q^53 + 210 * q^55 + 438 * q^59 - 602 * q^61 - 70 * q^65 + 920 * q^67 + 798 * q^71 - 770 * q^73 - 336 * q^77 + 1724 * q^79 - 486 * q^83 - 120 * q^85 + 294 * q^89 + 1876 * q^91 + 700 * q^95 + 112 * 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
$7$	14.7477	0.796302	0.398151	−	0.917320i	$-0.369652\pi$
$7$	14.7477	0.796302	0.398151	+	0.917320i	$0.369652\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	7.25227	0.198786	0.0993928	−	0.995048i	$-0.468310\pi$
$11$	7.25227	0.198786	0.0993928	+	0.995048i	$0.468310\pi$
$12$	0	0
$13$	61.7386	1.31717	0.658585	−	0.752506i	$-0.271155\pi$
$13$	61.7386	1.31717	0.658585	+	0.752506i	$0.271155\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−108.234	−1.54415	−0.772077	−	0.635529i	$-0.780782\pi$
$17$	−108.234	−1.54415	−0.772077	+	0.635529i	$0.780782\pi$
$18$	0	0
$19$	56.2523	0.679219	0.339609	−	0.940567i	$-0.389705\pi$
$19$	56.2523	0.679219	0.339609	+	0.940567i	$0.389705\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−46.9818	−0.425930	−0.212965	−	0.977060i	$-0.568312\pi$
$23$	−46.9818	−0.425930	−0.212965	+	0.977060i	$0.568312\pi$
$24$	0	0
$25$	25.0000	0.200000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	214.225	1.37174	0.685872	−	0.727722i	$-0.259421\pi$
$29$	214.225	1.37174	0.685872	+	0.727722i	$0.259421\pi$
$30$	0	0
$31$	261.711	1.51628	0.758141	−	0.652091i	$-0.226108\pi$
$31$	261.711	1.51628	0.758141	+	0.652091i	$0.226108\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	73.7386	0.356117
$36$	0	0
$37$	−286.000	−1.27076	−0.635380	−	0.772200i	$-0.719156\pi$
$37$	−286.000	−1.27076	−0.635380	+	0.772200i	$0.719156\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−255.702	−0.974000	−0.487000	−	0.873402i	$-0.661909\pi$
$41$	−255.702	−0.974000	−0.487000	+	0.873402i	$0.661909\pi$
$42$	0	0
$43$	361.234	1.28111	0.640554	−	0.767913i	$-0.278705\pi$
$43$	361.234	1.28111	0.640554	+	0.767913i	$0.278705\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−5.53182	−0.0171681	−0.00858403	−	0.999963i	$-0.502732\pi$
$47$	−5.53182	−0.0171681	−0.00858403	+	0.999963i	$0.502732\pi$
$48$	0	0
$49$	−125.505	−0.365902
$50$	0	0
$51$	0	0
$52$	0	0
$53$	595.693	1.54386	0.771932	−	0.635706i	$-0.219291\pi$
$53$	595.693	1.54386	0.771932	+	0.635706i	$0.219291\pi$
$54$	0	0
$55$	36.2614	0.0888997
$56$	0	0
$57$	0	0
$58$	0	0
$59$	315.234	0.695593	0.347796	−	0.937570i	$-0.386930\pi$
$59$	315.234	0.695593	0.347796	+	0.937570i	$0.386930\pi$
$60$	0	0
$61$	276.405	0.580164	0.290082	−	0.957002i	$-0.406317\pi$
$61$	276.405	0.580164	0.290082	+	0.957002i	$0.406317\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	308.693	0.589057
$66$	0	0
$67$	−117.405	−0.214078	−0.107039	−	0.994255i	$-0.534137\pi$
$67$	−117.405	−0.214078	−0.107039	+	0.994255i	$0.534137\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	192.784	0.322243	0.161122	−	0.986935i	$-0.448489\pi$
$71$	192.784	0.322243	0.161122	+	0.986935i	$0.448489\pi$
$72$	0	0
$73$	−756.189	−1.21240	−0.606200	−	0.795312i	$-0.707307\pi$
$73$	−756.189	−1.21240	−0.606200	+	0.795312i	$0.707307\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	106.955	0.158294
$78$	0	0
$79$	1150.70	1.63879	0.819393	−	0.573232i	$-0.194311\pi$
$79$	1150.70	1.63879	0.819393	+	0.573232i	$0.194311\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	141.936	0.187705	0.0938526	−	0.995586i	$-0.470082\pi$
$83$	141.936	0.187705	0.0938526	+	0.995586i	$0.470082\pi$
$84$	0	0
$85$	−541.170	−0.690567
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−719.107	−0.856463	−0.428231	−	0.903669i	$-0.640863\pi$
$89$	−719.107	−0.856463	−0.428231	+	0.903669i	$0.640863\pi$
$90$	0	0
$91$	910.505	1.04887
$92$	0	0
$93$	0	0
$94$	0	0
$95$	281.261	0.303756
$96$	0	0
$97$	−1043.82	−1.09262	−0.546308	−	0.837585i	$-0.683967\pi$
$97$	−1043.82	−1.09262	−0.546308	+	0.837585i	$0.683967\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−685.873	−0.675712	−0.337856	−	0.941198i	$-0.609702\pi$
$101$	−685.873	−0.675712	−0.337856	+	0.941198i	$0.609702\pi$
$102$	0	0
$103$	370.055	0.354005	0.177003	−	0.984210i	$-0.443360\pi$
$103$	370.055	0.354005	0.177003	+	0.984210i	$0.443360\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−1354.85	−1.22409	−0.612046	−	0.790822i	$-0.709653\pi$
$107$	−1354.85	−1.22409	−0.612046	+	0.790822i	$0.709653\pi$
$108$	0	0
$109$	882.873	0.775815	0.387908	−	0.921698i	$-0.373198\pi$
$109$	882.873	0.775815	0.387908	+	0.921698i	$0.373198\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−986.559	−0.821307	−0.410653	−	0.911792i	$-0.634699\pi$
$113$	−986.559	−0.821307	−0.410653	+	0.911792i	$0.634699\pi$
$114$	0	0
$115$	−234.909	−0.190482
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−1596.21	−1.22961
$120$	0	0
$121$	−1278.40	−0.960484
$122$	0	0
$123$	0	0
$124$	0	0
$125$	125.000	0.0894427
$126$	0	0
$127$	809.405	0.565536	0.282768	−	0.959188i	$-0.408747\pi$
$127$	809.405	0.565536	0.282768	+	0.959188i	$0.408747\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	1288.13	0.859116	0.429558	−	0.903039i	$-0.358669\pi$
$131$	1288.13	0.859116	0.429558	+	0.903039i	$0.358669\pi$
$132$	0	0
$133$	829.593	0.540864
$134$	0	0
$135$	0	0
$136$	0	0
$137$	2300.16	1.43442	0.717212	−	0.696855i	$-0.245418\pi$
$137$	2300.16	1.43442	0.717212	+	0.696855i	$0.245418\pi$
$138$	0	0
$139$	1359.89	0.829816	0.414908	−	0.909863i	$-0.363814\pi$
$139$	1359.89	0.829816	0.414908	+	0.909863i	$0.363814\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	447.745	0.261835
$144$	0	0
$145$	1071.12	0.613463
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−233.270	−0.128257	−0.0641284	−	0.997942i	$-0.520427\pi$
$149$	−233.270	−0.128257	−0.0641284	+	0.997942i	$0.520427\pi$
$150$	0	0
$151$	76.0000	0.0409589	0.0204794	−	0.999790i	$-0.493481\pi$
$151$	76.0000	0.0409589	0.0204794	+	0.999790i	$0.493481\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	1308.56	0.678102
$156$	0	0
$157$	1348.60	0.685543	0.342771	−	0.939419i	$-0.388634\pi$
$157$	1348.60	0.685543	0.342771	+	0.939419i	$0.388634\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−692.875	−0.339169
$162$	0	0
$163$	−216.127	−0.103855	−0.0519276	−	0.998651i	$-0.516537\pi$
$163$	−216.127	−0.103855	−0.0519276	+	0.998651i	$0.516537\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	2601.55	1.20548	0.602738	−	0.797940i	$-0.294077\pi$
$167$	2601.55	1.20548	0.602738	+	0.797940i	$0.294077\pi$
$168$	0	0
$169$	1614.66	0.734938
$170$	0	0
$171$	0	0
$172$	0	0
$173$	2563.64	1.12665	0.563323	−	0.826237i	$-0.309523\pi$
$173$	2563.64	1.12665	0.563323	+	0.826237i	$0.309523\pi$
$174$	0	0
$175$	368.693	0.159260
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−3791.98	−1.58338	−0.791692	−	0.610921i	$-0.790799\pi$
$179$	−3791.98	−1.58338	−0.791692	+	0.610921i	$0.790799\pi$
$180$	0	0
$181$	1245.18	0.511346	0.255673	−	0.966763i	$-0.417703\pi$
$181$	1245.18	0.511346	0.255673	+	0.966763i	$0.417703\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−1430.00	−0.568301
$186$	0	0
$187$	−784.943	−0.306956
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−3466.57	−1.31326	−0.656629	−	0.754213i	$-0.728018\pi$
$191$	−3466.57	−1.31326	−0.656629	+	0.754213i	$0.728018\pi$
$192$	0	0
$193$	3572.51	1.33241	0.666205	−	0.745769i	$-0.267918\pi$
$193$	3572.51	1.33241	0.666205	+	0.745769i	$0.267918\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	761.048	0.275241	0.137620	−	0.990485i	$-0.456055\pi$
$197$	761.048	0.275241	0.137620	+	0.990485i	$0.456055\pi$
$198$	0	0
$199$	−764.195	−0.272223	−0.136112	−	0.990694i	$-0.543461\pi$
$199$	−764.195	−0.272223	−0.136112	+	0.990694i	$0.543461\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	3159.33	1.09232
$204$	0	0
$205$	−1278.51	−0.435586
$206$	0	0
$207$	0	0
$208$	0	0
$209$	407.957	0.135019
$210$	0	0
$211$	−4213.38	−1.37470	−0.687349	−	0.726327i	$-0.741226\pi$
$211$	−4213.38	−1.37470	−0.687349	+	0.726327i	$0.741226\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	1806.17	0.572929
$216$	0	0
$217$	3859.65	1.20742
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−6682.22	−2.03391
$222$	0	0
$223$	3407.55	1.02326	0.511628	−	0.859207i	$-0.329043\pi$
$223$	3407.55	1.02326	0.511628	+	0.859207i	$0.329043\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−5638.40	−1.64861	−0.824304	−	0.566147i	$-0.808433\pi$
$227$	−5638.40	−1.64861	−0.824304	+	0.566147i	$0.808433\pi$
$228$	0	0
$229$	−73.0863	−0.0210903	−0.0105452	−	0.999944i	$-0.503357\pi$
$229$	−73.0863	−0.0210903	−0.0105452	+	0.999944i	$0.503357\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−1865.78	−0.524598	−0.262299	−	0.964987i	$-0.584481\pi$
$233$	−1865.78	−0.524598	−0.262299	+	0.964987i	$0.584481\pi$
$234$	0	0
$235$	−27.6591	−0.00767779
$236$	0	0
$237$	0	0
$238$	0	0
$239$	3147.60	0.851890	0.425945	−	0.904749i	$-0.359942\pi$
$239$	3147.60	0.851890	0.425945	+	0.904749i	$0.359942\pi$
$240$	0	0
$241$	5098.59	1.36278	0.681388	−	0.731923i	$-0.261377\pi$
$241$	5098.59	1.36278	0.681388	+	0.731923i	$0.261377\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−627.523	−0.163637
$246$	0	0
$247$	3472.94	0.894647
$248$	0	0
$249$	0	0
$250$	0	0
$251$	6566.90	1.65139	0.825695	−	0.564117i	$-0.190783\pi$
$251$	6566.90	1.65139	0.825695	+	0.564117i	$0.190783\pi$
$252$	0	0
$253$	−340.725	−0.0846688
$254$	0	0
$255$	0	0
$256$	0	0
$257$	6118.62	1.48509	0.742547	−	0.669794i	$-0.233618\pi$
$257$	6118.62	1.48509	0.742547	+	0.669794i	$0.233618\pi$
$258$	0	0
$259$	−4217.85	−1.01191
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−7268.01	−1.70405	−0.852025	−	0.523502i	$-0.824625\pi$
$263$	−7268.01	−1.70405	−0.852025	+	0.523502i	$0.824625\pi$
$264$	0	0
$265$	2978.47	0.690437
$266$	0	0
$267$	0	0
$268$	0	0
$269$	4292.72	0.972981	0.486491	−	0.873686i	$-0.338277\pi$
$269$	4292.72	0.972981	0.486491	+	0.873686i	$0.338277\pi$
$270$	0	0
$271$	3847.73	0.862483	0.431242	−	0.902237i	$-0.358076\pi$
$271$	3847.73	0.862483	0.431242	+	0.902237i	$0.358076\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	181.307	0.0397571
$276$	0	0
$277$	7770.26	1.68545	0.842725	−	0.538345i	$-0.180950\pi$
$277$	7770.26	1.68545	0.842725	+	0.538345i	$0.180950\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−6325.03	−1.34278	−0.671388	−	0.741106i	$-0.734301\pi$
$281$	−6325.03	−1.34278	−0.671388	+	0.741106i	$0.734301\pi$
$282$	0	0
$283$	−2662.96	−0.559352	−0.279676	−	0.960094i	$-0.590227\pi$
$283$	−2662.96	−0.559352	−0.279676	+	0.960094i	$0.590227\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−3771.03	−0.775598
$288$	0	0
$289$	6801.62	1.38441
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−3510.32	−0.699915	−0.349958	−	0.936766i	$-0.613804\pi$
$293$	−3510.32	−0.699915	−0.349958	+	0.936766i	$0.613804\pi$
$294$	0	0
$295$	1576.17	0.311079
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−2900.59	−0.561022
$300$	0	0
$301$	5327.38	1.02015
$302$	0	0
$303$	0	0
$304$	0	0
$305$	1382.02	0.259457
$306$	0	0
$307$	3451.19	0.641596	0.320798	−	0.947148i	$-0.396049\pi$
$307$	3451.19	0.641596	0.320798	+	0.947148i	$0.396049\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	9390.04	1.71209	0.856046	−	0.516900i	$-0.172914\pi$
$311$	9390.04	1.71209	0.856046	+	0.516900i	$0.172914\pi$
$312$	0	0
$313$	7116.66	1.28517	0.642584	−	0.766216i	$-0.277863\pi$
$313$	7116.66	1.28517	0.642584	+	0.766216i	$0.277863\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	6219.53	1.10197	0.550984	−	0.834516i	$-0.314253\pi$
$317$	6219.53	1.10197	0.550984	+	0.834516i	$0.314253\pi$
$318$	0	0
$319$	1553.62	0.272683
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−6088.41	−1.04882
$324$	0	0
$325$	1543.47	0.263434
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−81.5818	−0.0136710
$330$	0	0
$331$	8443.07	1.40203	0.701017	−	0.713145i	$-0.252730\pi$
$331$	8443.07	1.40203	0.701017	+	0.713145i	$0.252730\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−587.023	−0.0957387
$336$	0	0
$337$	1647.83	0.266359	0.133180	−	0.991092i	$-0.457481\pi$
$337$	1647.83	0.266359	0.133180	+	0.991092i	$0.457481\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	1898.00	0.301415
$342$	0	0
$343$	−6909.38	−1.08767
$344$	0	0
$345$	0	0
$346$	0	0
$347$	2462.48	0.380960	0.190480	−	0.981691i	$-0.438996\pi$
$347$	2462.48	0.380960	0.190480	+	0.981691i	$0.438996\pi$
$348$	0	0
$349$	−8126.99	−1.24650	−0.623249	−	0.782023i	$-0.714188\pi$
$349$	−8126.99	−1.24650	−0.623249	+	0.782023i	$0.714188\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−5376.64	−0.810679	−0.405340	−	0.914166i	$-0.632847\pi$
$353$	−5376.64	−0.810679	−0.405340	+	0.914166i	$0.632847\pi$
$354$	0	0
$355$	963.920	0.144112
$356$	0	0
$357$	0	0
$358$	0	0
$359$	4860.14	0.714508	0.357254	−	0.934007i	$-0.383713\pi$
$359$	4860.14	0.714508	0.357254	+	0.934007i	$0.383713\pi$
$360$	0	0
$361$	−3694.68	−0.538662
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−3780.94	−0.542202
$366$	0	0
$367$	964.293	0.137154	0.0685772	−	0.997646i	$-0.478154\pi$
$367$	964.293	0.137154	0.0685772	+	0.997646i	$0.478154\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	8785.12	1.22938
$372$	0	0
$373$	−6114.13	−0.848734	−0.424367	−	0.905490i	$-0.639503\pi$
$373$	−6114.13	−0.848734	−0.424367	+	0.905490i	$0.639503\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	13226.0	1.80682
$378$	0	0
$379$	12063.6	1.63501	0.817504	−	0.575924i	$-0.195357\pi$
$379$	12063.6	1.63501	0.817504	+	0.575924i	$0.195357\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−9275.13	−1.23743	−0.618717	−	0.785614i	$-0.712347\pi$
$383$	−9275.13	−1.23743	−0.618717	+	0.785614i	$0.712347\pi$
$384$	0	0
$385$	534.773	0.0707910
$386$	0	0
$387$	0	0
$388$	0	0
$389$	11309.4	1.47406	0.737030	−	0.675860i	$-0.236228\pi$
$389$	11309.4	1.47406	0.737030	+	0.675860i	$0.236228\pi$
$390$	0	0
$391$	5085.03	0.657701
$392$	0	0
$393$	0	0
$394$	0	0
$395$	5753.51	0.732888
$396$	0	0
$397$	−6235.73	−0.788319	−0.394159	−	0.919042i	$-0.628964\pi$
$397$	−6235.73	−0.788319	−0.394159	+	0.919042i	$0.628964\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−13997.3	−1.74313	−0.871564	−	0.490282i	$-0.836894\pi$
$401$	−13997.3	−1.74313	−0.871564	+	0.490282i	$0.836894\pi$
$402$	0	0
$403$	16157.7	1.99720
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−2074.15	−0.252609
$408$	0	0
$409$	6772.52	0.818776	0.409388	−	0.912360i	$-0.365742\pi$
$409$	6772.52	0.818776	0.409388	+	0.912360i	$0.365742\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	4648.99	0.553902
$414$	0	0
$415$	709.682	0.0839444
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−4636.81	−0.540628	−0.270314	−	0.962772i	$-0.587127\pi$
$419$	−4636.81	−0.540628	−0.270314	+	0.962772i	$0.587127\pi$
$420$	0	0
$421$	9177.47	1.06243	0.531215	−	0.847237i	$-0.321736\pi$
$421$	9177.47	1.06243	0.531215	+	0.847237i	$0.321736\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−2705.85	−0.308831
$426$	0	0
$427$	4076.34	0.461986
$428$	0	0
$429$	0	0
$430$	0	0
$431$	16939.0	1.89310	0.946549	−	0.322561i	$-0.104544\pi$
$431$	16939.0	1.89310	0.946549	+	0.322561i	$0.104544\pi$
$432$	0	0
$433$	7368.61	0.817812	0.408906	−	0.912576i	$-0.365910\pi$
$433$	7368.61	0.817812	0.408906	+	0.912576i	$0.365910\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−2642.83	−0.289300
$438$	0	0
$439$	−7899.77	−0.858850	−0.429425	−	0.903102i	$-0.641284\pi$
$439$	−7899.77	−0.858850	−0.429425	+	0.903102i	$0.641284\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−3730.12	−0.400053	−0.200027	−	0.979790i	$-0.564103\pi$
$443$	−3730.12	−0.400053	−0.200027	+	0.979790i	$0.564103\pi$
$444$	0	0
$445$	−3595.53	−0.383022
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−742.811	−0.0780745	−0.0390372	−	0.999238i	$-0.512429\pi$
$449$	−742.811	−0.0780745	−0.0390372	+	0.999238i	$0.512429\pi$
$450$	0	0
$451$	−1854.42	−0.193617
$452$	0	0
$453$	0	0
$454$	0	0
$455$	4552.52	0.469067
$456$	0	0
$457$	−3281.02	−0.335842	−0.167921	−	0.985800i	$-0.553705\pi$
$457$	−3281.02	−0.335842	−0.167921	+	0.985800i	$0.553705\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−2708.55	−0.273644	−0.136822	−	0.990596i	$-0.543689\pi$
$461$	−2708.55	−0.273644	−0.136822	+	0.990596i	$0.543689\pi$
$462$	0	0
$463$	12296.2	1.23424	0.617120	−	0.786869i	$-0.288299\pi$
$463$	12296.2	1.23424	0.617120	+	0.786869i	$0.288299\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−347.168	−0.0344005	−0.0172002	−	0.999852i	$-0.505475\pi$
$467$	−347.168	−0.0344005	−0.0172002	+	0.999852i	$0.505475\pi$
$468$	0	0
$469$	−1731.45	−0.170471
$470$	0	0
$471$	0	0
$472$	0	0
$473$	2619.77	0.254666
$474$	0	0
$475$	1406.31	0.135844
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−11968.0	−1.14162	−0.570808	−	0.821084i	$-0.693370\pi$
$479$	−11968.0	−1.14162	−0.570808	+	0.821084i	$0.693370\pi$
$480$	0	0
$481$	−17657.2	−1.67381
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−5219.09	−0.488632
$486$	0	0
$487$	827.189	0.0769682	0.0384841	−	0.999259i	$-0.487747\pi$
$487$	827.189	0.0769682	0.0384841	+	0.999259i	$0.487747\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−10268.9	−0.943848	−0.471924	−	0.881639i	$-0.656440\pi$
$491$	−10268.9	−0.943848	−0.471924	+	0.881639i	$0.656440\pi$
$492$	0	0
$493$	−23186.4	−2.11818
$494$	0	0
$495$	0	0
$496$	0	0
$497$	2843.13	0.256603
$498$	0	0
$499$	8313.30	0.745800	0.372900	−	0.927871i	$-0.378363\pi$
$499$	8313.30	0.745800	0.372900	+	0.927871i	$0.378363\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	7575.17	0.671492	0.335746	−	0.941953i	$-0.391012\pi$
$503$	7575.17	0.671492	0.335746	+	0.941953i	$0.391012\pi$
$504$	0	0
$505$	−3429.36	−0.302187
$506$	0	0
$507$	0	0
$508$	0	0
$509$	10638.4	0.926401	0.463200	−	0.886254i	$-0.346701\pi$
$509$	10638.4	0.926401	0.463200	+	0.886254i	$0.346701\pi$
$510$	0	0
$511$	−11152.1	−0.965437
$512$	0	0
$513$	0	0
$514$	0	0
$515$	1850.27	0.158316
$516$	0	0
$517$	−40.1183	−0.00341277
$518$	0	0
$519$	0	0
$520$	0	0
$521$	289.152	0.0243148	0.0121574	−	0.999926i	$-0.496130\pi$
$521$	289.152	0.0243148	0.0121574	+	0.999926i	$0.496130\pi$
$522$	0	0
$523$	−5708.40	−0.477268	−0.238634	−	0.971110i	$-0.576700\pi$
$523$	−5708.40	−0.477268	−0.238634	+	0.971110i	$0.576700\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−28326.1	−2.34137
$528$	0	0
$529$	−9959.71	−0.818584
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−15786.7	−1.28292
$534$	0	0
$535$	−6774.23	−0.547431
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−910.193	−0.0727362
$540$	0	0
$541$	−7904.89	−0.628203	−0.314101	−	0.949389i	$-0.601703\pi$
$541$	−7904.89	−0.628203	−0.314101	+	0.949389i	$0.601703\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	4414.36	0.346955
$546$	0	0
$547$	4563.11	0.356681	0.178340	−	0.983969i	$-0.442927\pi$
$547$	4563.11	0.356681	0.178340	+	0.983969i	$0.442927\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	12050.6	0.931714
$552$	0	0
$553$	16970.2	1.30497
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−1729.19	−0.131540	−0.0657702	−	0.997835i	$-0.520950\pi$
$557$	−1729.19	−0.131540	−0.0657702	+	0.997835i	$0.520950\pi$
$558$	0	0
$559$	22302.1	1.68744
$560$	0	0
$561$	0	0
$562$	0	0
$563$	15656.0	1.17198	0.585988	−	0.810320i	$-0.300707\pi$
$563$	15656.0	1.17198	0.585988	+	0.810320i	$0.300707\pi$
$564$	0	0
$565$	−4932.80	−0.367300
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−15868.5	−1.16914	−0.584570	−	0.811343i	$-0.698737\pi$
$569$	−15868.5	−1.16914	−0.584570	+	0.811343i	$0.698737\pi$
$570$	0	0
$571$	12989.1	0.951976	0.475988	−	0.879452i	$-0.342091\pi$
$571$	12989.1	0.951976	0.475988	+	0.879452i	$0.342091\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−1174.55	−0.0851860
$576$	0	0
$577$	−6404.62	−0.462093	−0.231046	−	0.972943i	$-0.574215\pi$
$577$	−6404.62	−0.462093	−0.231046	+	0.972943i	$0.574215\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	2093.24	0.149470
$582$	0	0
$583$	4320.13	0.306898
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−21081.5	−1.48233	−0.741165	−	0.671323i	$-0.765726\pi$
$587$	−21081.5	−1.48233	−0.741165	+	0.671323i	$0.765726\pi$
$588$	0	0
$589$	14721.9	1.02989
$590$	0	0
$591$	0	0
$592$	0	0
$593$	1379.98	0.0955629	0.0477814	−	0.998858i	$-0.484785\pi$
$593$	1379.98	0.0955629	0.0477814	+	0.998858i	$0.484785\pi$
$594$	0	0
$595$	−7981.03	−0.549900
$596$	0	0
$597$	0	0
$598$	0	0
$599$	12952.1	0.883484	0.441742	−	0.897142i	$-0.354361\pi$
$599$	12952.1	0.883484	0.441742	+	0.897142i	$0.354361\pi$
$600$	0	0
$601$	23140.5	1.57059	0.785293	−	0.619124i	$-0.212512\pi$
$601$	23140.5	1.57059	0.785293	+	0.619124i	$0.212512\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−6392.02	−0.429542
$606$	0	0
$607$	1479.65	0.0989411	0.0494705	−	0.998776i	$-0.484247\pi$
$607$	1479.65	0.0989411	0.0494705	+	0.998776i	$0.484247\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−341.527	−0.0226133
$612$	0	0
$613$	20327.3	1.33934	0.669668	−	0.742661i	$-0.266436\pi$
$613$	20327.3	1.33934	0.669668	+	0.742661i	$0.266436\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	14362.8	0.937158	0.468579	−	0.883422i	$-0.344766\pi$
$617$	14362.8	0.937158	0.468579	+	0.883422i	$0.344766\pi$
$618$	0	0
$619$	−3115.20	−0.202278	−0.101139	−	0.994872i	$-0.532249\pi$
$619$	−3115.20	−0.202278	−0.101139	+	0.994872i	$0.532249\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−10605.2	−0.682003
$624$	0	0
$625$	625.000	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	30954.9	1.96225
$630$	0	0
$631$	23024.2	1.45258	0.726292	−	0.687386i	$-0.241242\pi$
$631$	23024.2	1.45258	0.726292	+	0.687386i	$0.241242\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	4047.02	0.252915
$636$	0	0
$637$	−7748.48	−0.481956
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−27234.2	−1.67814	−0.839068	−	0.544027i	$-0.816899\pi$
$641$	−27234.2	−1.67814	−0.839068	+	0.544027i	$0.816899\pi$
$642$	0	0
$643$	−2245.50	−0.137720	−0.0688600	−	0.997626i	$-0.521936\pi$
$643$	−2245.50	−0.137720	−0.0688600	+	0.997626i	$0.521936\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−7245.60	−0.440269	−0.220135	−	0.975470i	$-0.570650\pi$
$647$	−7245.60	−0.440269	−0.220135	+	0.975470i	$0.570650\pi$
$648$	0	0
$649$	2286.16	0.138274
$650$	0	0
$651$	0	0
$652$	0	0
$653$	12061.0	0.722795	0.361398	−	0.932412i	$-0.382300\pi$
$653$	12061.0	0.722795	0.361398	+	0.932412i	$0.382300\pi$
$654$	0	0
$655$	6440.64	0.384208
$656$	0	0
$657$	0	0
$658$	0	0
$659$	31632.7	1.86986	0.934929	−	0.354836i	$-0.115463\pi$
$659$	31632.7	1.86986	0.934929	+	0.354836i	$0.115463\pi$
$660$	0	0
$661$	−9312.26	−0.547965	−0.273983	−	0.961735i	$-0.588341\pi$
$661$	−9312.26	−0.547965	−0.273983	+	0.961735i	$0.588341\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	4147.97	0.241882
$666$	0	0
$667$	−10064.7	−0.584267
$668$	0	0
$669$	0	0
$670$	0	0
$671$	2004.56	0.115328
$672$	0	0
$673$	−3761.02	−0.215418	−0.107709	−	0.994182i	$-0.534352\pi$
$673$	−3761.02	−0.215418	−0.107709	+	0.994182i	$0.534352\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−29897.7	−1.69728	−0.848642	−	0.528968i	$-0.822579\pi$
$677$	−29897.7	−1.69728	−0.848642	+	0.528968i	$0.822579\pi$
$678$	0	0
$679$	−15393.9	−0.870052
$680$	0	0
$681$	0	0
$682$	0	0
$683$	7931.11	0.444327	0.222164	−	0.975009i	$-0.428688\pi$
$683$	7931.11	0.444327	0.222164	+	0.975009i	$0.428688\pi$
$684$	0	0
$685$	11500.8	0.641494
$686$	0	0
$687$	0	0
$688$	0	0
$689$	36777.3	2.03353
$690$	0	0
$691$	−19436.9	−1.07007	−0.535033	−	0.844831i	$-0.679701\pi$
$691$	−19436.9	−1.07007	−0.535033	+	0.844831i	$0.679701\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	6799.45	0.371105
$696$	0	0
$697$	27675.7	1.50401
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−26442.4	−1.42470	−0.712351	−	0.701823i	$-0.752370\pi$
$701$	−26442.4	−1.42470	−0.712351	+	0.701823i	$0.752370\pi$
$702$	0	0
$703$	−16088.2	−0.863124
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−10115.1	−0.538071
$708$	0	0
$709$	22397.5	1.18640	0.593199	−	0.805056i	$-0.297865\pi$
$709$	22397.5	1.18640	0.593199	+	0.805056i	$0.297865\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−12295.7	−0.645830
$714$	0	0
$715$	2238.73	0.117096
$716$	0	0
$717$	0	0
$718$	0	0
$719$	29828.0	1.54715	0.773573	−	0.633707i	$-0.218467\pi$
$719$	29828.0	1.54715	0.773573	+	0.633707i	$0.218467\pi$
$720$	0	0
$721$	5457.46	0.281895
$722$	0	0
$723$	0	0
$724$	0	0
$725$	5355.62	0.274349
$726$	0	0
$727$	−37300.9	−1.90291	−0.951454	−	0.307791i	$-0.900410\pi$
$727$	−37300.9	−1.90291	−0.951454	+	0.307791i	$0.900410\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−39097.8	−1.97823
$732$	0	0
$733$	−15835.9	−0.797969	−0.398985	−	0.916958i	$-0.630637\pi$
$733$	−15835.9	−0.797969	−0.398985	+	0.916958i	$0.630637\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−851.450	−0.0425557
$738$	0	0
$739$	−26946.7	−1.34134	−0.670669	−	0.741756i	$-0.733993\pi$
$739$	−26946.7	−1.34134	−0.670669	+	0.741756i	$0.733993\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−37380.0	−1.84568	−0.922839	−	0.385187i	$-0.874137\pi$
$743$	−37380.0	−1.84568	−0.922839	+	0.385187i	$0.874137\pi$
$744$	0	0
$745$	−1166.35	−0.0573582
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−19980.9	−0.974747
$750$	0	0
$751$	−39652.4	−1.92668	−0.963339	−	0.268285i	$-0.913543\pi$
$751$	−39652.4	−1.92668	−0.963339	+	0.268285i	$0.913543\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	380.000	0.0183174
$756$	0	0
$757$	−27952.6	−1.34208	−0.671041	−	0.741420i	$-0.734153\pi$
$757$	−27952.6	−1.34208	−0.671041	+	0.741420i	$0.734153\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	25619.6	1.22038	0.610190	−	0.792255i	$-0.291093\pi$
$761$	25619.6	1.22038	0.610190	+	0.792255i	$0.291093\pi$
$762$	0	0
$763$	13020.4	0.617784
$764$	0	0
$765$	0	0
$766$	0	0
$767$	19462.1	0.916214
$768$	0	0
$769$	−16666.0	−0.781521	−0.390761	−	0.920492i	$-0.627788\pi$
$769$	−16666.0	−0.781521	−0.390761	+	0.920492i	$0.627788\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−4266.62	−0.198525	−0.0992623	−	0.995061i	$-0.531648\pi$
$773$	−4266.62	−0.198525	−0.0992623	+	0.995061i	$0.531648\pi$
$774$	0	0
$775$	6542.78	0.303256
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−14383.8	−0.661559
$780$	0	0
$781$	1398.12	0.0640573
$782$	0	0
$783$	0	0
$784$	0	0
$785$	6743.01	0.306584
$786$	0	0
$787$	12095.0	0.547829	0.273914	−	0.961754i	$-0.411682\pi$
$787$	12095.0	0.547829	0.273914	+	0.961754i	$0.411682\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−14549.5	−0.654009
$792$	0	0
$793$	17064.8	0.764174
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−15082.0	−0.670303	−0.335152	−	0.942164i	$-0.608788\pi$
$797$	−15082.0	−0.670303	−0.335152	+	0.942164i	$0.608788\pi$
$798$	0	0
$799$	598.732	0.0265101
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−5484.09	−0.241008
$804$	0	0
$805$	−3464.38	−0.151681
$806$	0	0
$807$	0	0
$808$	0	0
$809$	29078.6	1.26372	0.631859	−	0.775083i	$-0.282292\pi$
$809$	29078.6	1.26372	0.631859	+	0.775083i	$0.282292\pi$
$810$	0	0
$811$	−10900.8	−0.471985	−0.235992	−	0.971755i	$-0.575834\pi$
$811$	−10900.8	−0.471985	−0.235992	+	0.971755i	$0.575834\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−1080.64	−0.0464455
$816$	0	0
$817$	20320.2	0.870153
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−24577.1	−1.04476	−0.522378	−	0.852714i	$-0.674955\pi$
$821$	−24577.1	−1.04476	−0.522378	+	0.852714i	$0.674955\pi$
$822$	0	0
$823$	−36000.8	−1.52480	−0.762399	−	0.647107i	$-0.775979\pi$
$823$	−36000.8	−1.52480	−0.762399	+	0.647107i	$0.775979\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	31061.1	1.30605	0.653024	−	0.757337i	$-0.273500\pi$
$827$	31061.1	1.30605	0.653024	+	0.757337i	$0.273500\pi$
$828$	0	0
$829$	−30816.6	−1.29108	−0.645540	−	0.763727i	$-0.723368\pi$
$829$	−30816.6	−1.29108	−0.645540	+	0.763727i	$0.723368\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	13583.9	0.565010
$834$	0	0
$835$	13007.8	0.539105
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−27778.0	−1.14303	−0.571515	−	0.820592i	$-0.693644\pi$
$839$	−27778.0	−1.14303	−0.571515	+	0.820592i	$0.693644\pi$
$840$	0	0
$841$	21503.3	0.881682
$842$	0	0
$843$	0	0
$844$	0	0
$845$	8073.30	0.328674
$846$	0	0
$847$	−18853.6	−0.764836
$848$	0	0
$849$	0	0
$850$	0	0
$851$	13436.8	0.541254
$852$	0	0
$853$	−46700.6	−1.87456	−0.937279	−	0.348580i	$-0.886664\pi$
$853$	−46700.6	−1.87456	−0.937279	+	0.348580i	$0.886664\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−25745.8	−1.02621	−0.513103	−	0.858327i	$-0.671504\pi$
$857$	−25745.8	−1.02621	−0.513103	+	0.858327i	$0.671504\pi$
$858$	0	0
$859$	−9974.80	−0.396200	−0.198100	−	0.980182i	$-0.563477\pi$
$859$	−9974.80	−0.396200	−0.198100	+	0.980182i	$0.563477\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−9723.99	−0.383555	−0.191778	−	0.981438i	$-0.561425\pi$
$863$	−9723.99	−0.383555	−0.191778	+	0.981438i	$0.561425\pi$
$864$	0	0
$865$	12818.2	0.503852
$866$	0	0
$867$	0	0
$868$	0	0
$869$	8345.21	0.325767
$870$	0	0
$871$	−7248.40	−0.281978
$872$	0	0
$873$	0	0
$874$	0	0
$875$	1843.47	0.0712235
$876$	0	0
$877$	−13654.9	−0.525763	−0.262882	−	0.964828i	$-0.584673\pi$
$877$	−13654.9	−0.525763	−0.262882	+	0.964828i	$0.584673\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−9893.62	−0.378348	−0.189174	−	0.981944i	$-0.560581\pi$
$881$	−9893.62	−0.378348	−0.189174	+	0.981944i	$0.560581\pi$
$882$	0	0
$883$	−41598.9	−1.58541	−0.792703	−	0.609607i	$-0.791327\pi$
$883$	−41598.9	−1.58541	−0.792703	+	0.609607i	$0.791327\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	16369.6	0.619660	0.309830	−	0.950792i	$-0.399728\pi$
$887$	16369.6	0.619660	0.309830	+	0.950792i	$0.399728\pi$
$888$	0	0
$889$	11936.9	0.450337
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−311.177	−0.0116609
$894$	0	0
$895$	−18959.9	−0.708111
$896$	0	0
$897$	0	0
$898$	0	0
$899$	56065.1	2.07995
$900$	0	0
$901$	−64474.3	−2.38396
$902$	0	0
$903$	0	0
$904$	0	0
$905$	6225.91	0.228681
$906$	0	0
$907$	−39211.0	−1.43548	−0.717740	−	0.696312i	$-0.754823\pi$
$907$	−39211.0	−1.43548	−0.717740	+	0.696312i	$0.754823\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	10858.8	0.394917	0.197459	−	0.980311i	$-0.436731\pi$
$911$	10858.8	0.394917	0.197459	+	0.980311i	$0.436731\pi$
$912$	0	0
$913$	1029.36	0.0373131
$914$	0	0
$915$	0	0
$916$	0	0
$917$	18996.9	0.684116
$918$	0	0
$919$	−50169.6	−1.80081	−0.900403	−	0.435057i	$-0.856728\pi$
$919$	−50169.6	−1.80081	−0.900403	+	0.435057i	$0.856728\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	11902.2	0.424449
$924$	0	0
$925$	−7150.00	−0.254152
$926$	0	0
$927$	0	0
$928$	0	0
$929$	12666.4	0.447330	0.223665	−	0.974666i	$-0.428198\pi$
$929$	12666.4	0.447330	0.223665	+	0.974666i	$0.428198\pi$
$930$	0	0
$931$	−7059.92	−0.248528
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−3924.72	−0.137275
$936$	0	0
$937$	6348.26	0.221333	0.110666	−	0.993858i	$-0.464702\pi$
$937$	6348.26	0.221333	0.110666	+	0.993858i	$0.464702\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	39141.4	1.35598	0.677988	−	0.735073i	$-0.262852\pi$
$941$	39141.4	1.35598	0.677988	+	0.735073i	$0.262852\pi$
$942$	0	0
$943$	12013.4	0.414855
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−26580.2	−0.912082	−0.456041	−	0.889959i	$-0.650733\pi$
$947$	−26580.2	−0.912082	−0.456041	+	0.889959i	$0.650733\pi$
$948$	0	0
$949$	−46686.1	−1.59694
$950$	0	0
$951$	0	0
$952$	0	0
$953$	406.982	0.0138336	0.00691680	−	0.999976i	$-0.497798\pi$
$953$	406.982	0.0138336	0.00691680	+	0.999976i	$0.497798\pi$
$954$	0	0
$955$	−17332.9	−0.587307
$956$	0	0
$957$	0	0
$958$	0	0
$959$	33922.2	1.14224
$960$	0	0
$961$	38701.8	1.29911
$962$	0	0
$963$	0	0
$964$	0	0
$965$	17862.6	0.595872
$966$	0	0
$967$	58873.6	1.95786	0.978928	−	0.204204i	$-0.0654607\pi$
$967$	58873.6	1.95786	0.978928	+	0.204204i	$0.0654607\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−52987.1	−1.75122	−0.875611	−	0.483016i	$-0.839541\pi$
$971$	−52987.1	−1.75122	−0.875611	+	0.483016i	$0.839541\pi$
$972$	0	0
$973$	20055.3	0.660785
$974$	0	0
$975$	0	0
$976$	0	0
$977$	6382.49	0.209001	0.104500	−	0.994525i	$-0.466676\pi$
$977$	6382.49	0.209001	0.104500	+	0.994525i	$0.466676\pi$
$978$	0	0
$979$	−5215.16	−0.170253
$980$	0	0
$981$	0	0
$982$	0	0
$983$	25662.0	0.832646	0.416323	−	0.909217i	$-0.363319\pi$
$983$	25662.0	0.832646	0.416323	+	0.909217i	$0.363319\pi$
$984$	0	0
$985$	3805.24	0.123091
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−16971.4	−0.545662
$990$	0	0
$991$	49058.5	1.57255	0.786275	−	0.617877i	$-0.212007\pi$
$991$	49058.5	1.57255	0.786275	+	0.617877i	$0.212007\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−3820.98	−0.121742
$996$	0	0
$997$	−4731.80	−0.150309	−0.0751543	−	0.997172i	$-0.523945\pi$
$997$	−4731.80	−0.150309	−0.0751543	+	0.997172i	$0.523945\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2160.4.a.bc.1.2		2
3.2	odd	2		2160.4.a.x.1.2		2
4.3	odd	2		540.4.a.h.1.1	yes	2
12.11	even	2		540.4.a.e.1.1	✓	2
36.7	odd	6		1620.4.i.o.1081.2		4
36.11	even	6		1620.4.i.r.1081.2		4
36.23	even	6		1620.4.i.r.541.2		4
36.31	odd	6		1620.4.i.o.541.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
540.4.a.e.1.1	✓	2	12.11	even	2
540.4.a.h.1.1	yes	2	4.3	odd	2
1620.4.i.o.541.2		4	36.31	odd	6
1620.4.i.o.1081.2		4	36.7	odd	6
1620.4.i.r.541.2		4	36.23	even	6
1620.4.i.r.1081.2		4	36.11	even	6
2160.4.a.x.1.2		2	3.2	odd	2
2160.4.a.bc.1.2		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)

\(f(q)\)	\(=\)	\(q+5.00000 q^{5} +14.7477 q^{7} +O(q^{10})\)	\(q+5.00000 q^{5} +14.7477 q^{7} +7.25227 q^{11} +61.7386 q^{13} -108.234 q^{17} +56.2523 q^{19} -46.9818 q^{23} +25.0000 q^{25} +214.225 q^{29} +261.711 q^{31} +73.7386 q^{35} -286.000 q^{37} -255.702 q^{41} +361.234 q^{43} -5.53182 q^{47} -125.505 q^{49} +595.693 q^{53} +36.2614 q^{55} +315.234 q^{59} +276.405 q^{61} +308.693 q^{65} -117.405 q^{67} +192.784 q^{71} -756.189 q^{73} +106.955 q^{77} +1150.70 q^{79} +141.936 q^{83} -541.170 q^{85} -719.107 q^{89} +910.505 q^{91} +281.261 q^{95} -1043.82 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 10 q^{5} + 2 q^{7}+O(q^{10}) \) 2 * q + 10 * q^5 + 2 * q^7	\( 2 q + 10 q^{5} + 2 q^{7} + 42 q^{11} - 14 q^{13} - 24 q^{17} + 140 q^{19} + 126 q^{23} + 50 q^{25} + 126 q^{29} + 56 q^{31} + 10 q^{35} - 572 q^{37} + 66 q^{41} + 530 q^{43} - 396 q^{47} - 306 q^{49} + 504 q^{53} + 210 q^{55} + 438 q^{59} - 602 q^{61} - 70 q^{65} + 920 q^{67} + 798 q^{71} - 770 q^{73} - 336 q^{77} + 1724 q^{79} - 486 q^{83} - 120 q^{85} + 294 q^{89} + 1876 q^{91} + 700 q^{95} + 112 q^{97}+O(q^{100}) \) 2 * q + 10 * q^5 + 2 * q^7 + 42 * q^11 - 14 * q^13 - 24 * q^17 + 140 * q^19 + 126 * q^23 + 50 * q^25 + 126 * q^29 + 56 * q^31 + 10 * q^35 - 572 * q^37 + 66 * q^41 + 530 * q^43 - 396 * q^47 - 306 * q^49 + 504 * q^53 + 210 * q^55 + 438 * q^59 - 602 * q^61 - 70 * q^65 + 920 * q^67 + 798 * q^71 - 770 * q^73 - 336 * q^77 + 1724 * q^79 - 486 * q^83 - 120 * q^85 + 294 * q^89 + 1876 * q^91 + 700 * q^95 + 112 * q^97

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