LMFDB - Embedded newform 8028.2.a.i.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8028,2,Mod(1,8028)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8028, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("8028.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8028 = 2^{2} \cdot 3^{2} \cdot 223 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8028.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:	$64.1039027427$
Analytic rank:	$0$
Dimension:	$5$
Coefficient field:	5.5.1710888.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - x^{4} - 10x^{3} + 3x^{2} + 12x + 4 $ x^5 - x^4 - 10x^3 + 3x^2 + 12*x + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 2676)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-2.64396$ of defining polynomial
Character	$\chi$	$=$	8028.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.756442 q^{5} +1.69379 q^{7} +O(q^{10})$	$q+0.756442 q^{5} +1.69379 q^{7} +2.88419 q^{17} -2.99051 q^{19} +1.38758 q^{23} -4.42780 q^{25} -4.15684 q^{29} +3.13107 q^{31} +1.28125 q^{35} +4.52814 q^{37} +2.30621 q^{41} -4.79978 q^{43} -1.41566 q^{47} -4.13107 q^{49} +8.51006 q^{53} +15.0629 q^{59} +6.78182 q^{61} +12.8410 q^{67} +5.41322 q^{71} +9.35967 q^{73} +6.04022 q^{79} -13.8509 q^{83} +2.18172 q^{85} +16.4756 q^{89} -2.26215 q^{95} -18.7317 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q + 6 q^{5} + 3 q^{7}+O(q^{10}) $ 5 * q + 6 * q^5 + 3 * q^7	$ 5 q + 6 q^{5} + 3 q^{7} + 13 q^{17} - q^{19} - 4 q^{23} + 5 q^{25} - 11 q^{29} - 3 q^{31} + 8 q^{35} + 7 q^{37} + 17 q^{41} + 15 q^{43} + 25 q^{47} - 2 q^{49} + 3 q^{53} + 14 q^{59} - 18 q^{61} + 24 q^{67} + 14 q^{71} - 23 q^{73} + 14 q^{79} + 15 q^{83} + 6 q^{85} + 25 q^{89} + 26 q^{95} - 12 q^{97}+O(q^{100}) $ 5 * q + 6 * q^5 + 3 * q^7 + 13 * q^17 - q^19 - 4 * q^23 + 5 * q^25 - 11 * q^29 - 3 * q^31 + 8 * q^35 + 7 * q^37 + 17 * q^41 + 15 * q^43 + 25 * q^47 - 2 * q^49 + 3 * q^53 + 14 * q^59 - 18 * q^61 + 24 * q^67 + 14 * q^71 - 23 * q^73 + 14 * q^79 + 15 * q^83 + 6 * q^85 + 25 * q^89 + 26 * q^95 - 12 * 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
$5$	0.756442	0.338291	0.169146	−	0.985591i	$-0.445899\pi$
$5$	0.756442	0.338291	0.169146	+	0.985591i	$0.445899\pi$
$6$	0	0
$7$	1.69379	0.640193	0.320096	−	0.947385i	$-0.396285\pi$
$7$	1.69379	0.640193	0.320096	+	0.947385i	$0.396285\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	0	0		−	1.00000i	$-0.5\pi$
$11$	0	0			1.00000i	$0.5\pi$
$12$	0	0
$13$	0	0		−	1.00000i	$-0.5\pi$
$13$	0	0			1.00000i	$0.5\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	2.88419	0.699518	0.349759	−	0.936840i	$-0.386264\pi$
$17$	2.88419	0.699518	0.349759	+	0.936840i	$0.386264\pi$
$18$	0	0
$19$	−2.99051	−0.686071	−0.343035	−	0.939323i	$-0.611455\pi$
$19$	−2.99051	−0.686071	−0.343035	+	0.939323i	$0.611455\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	1.38758	0.289331	0.144665	−	0.989481i	$-0.453789\pi$
$23$	1.38758	0.289331	0.144665	+	0.989481i	$0.453789\pi$
$24$	0	0
$25$	−4.42780	−0.885559
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−4.15684	−0.771906	−0.385953	−	0.922518i	$-0.626127\pi$
$29$	−4.15684	−0.771906	−0.385953	+	0.922518i	$0.626127\pi$
$30$	0	0
$31$	3.13107	0.562358	0.281179	−	0.959655i	$-0.409275\pi$
$31$	3.13107	0.562358	0.281179	+	0.959655i	$0.409275\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	1.28125	0.216571
$36$	0	0
$37$	4.52814	0.744422	0.372211	−	0.928148i	$-0.378600\pi$
$37$	4.52814	0.744422	0.372211	+	0.928148i	$0.378600\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	2.30621	0.360169	0.180085	−	0.983651i	$-0.442363\pi$
$41$	2.30621	0.360169	0.180085	+	0.983651i	$0.442363\pi$
$42$	0	0
$43$	−4.79978	−0.731959	−0.365980	−	0.930623i	$-0.619266\pi$
$43$	−4.79978	−0.731959	−0.365980	+	0.930623i	$0.619266\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−1.41566	−0.206495	−0.103248	−	0.994656i	$-0.532923\pi$
$47$	−1.41566	−0.206495	−0.103248	+	0.994656i	$0.532923\pi$
$48$	0	0
$49$	−4.13107	−0.590153
$50$	0	0
$51$	0	0
$52$	0	0
$53$	8.51006	1.16895	0.584473	−	0.811413i	$-0.301301\pi$
$53$	8.51006	1.16895	0.584473	+	0.811413i	$0.301301\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	0	0
$58$	0	0
$59$	15.0629	1.96103	0.980514	−	0.196448i	$-0.0629405\pi$
$59$	15.0629	1.96103	0.980514	+	0.196448i	$0.0629405\pi$
$60$	0	0
$61$	6.78182	0.868323	0.434162	−	0.900835i	$-0.357045\pi$
$61$	6.78182	0.868323	0.434162	+	0.900835i	$0.357045\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	0	0
$66$	0	0
$67$	12.8410	1.56878	0.784390	−	0.620268i	$-0.212976\pi$
$67$	12.8410	1.56878	0.784390	+	0.620268i	$0.212976\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	5.41322	0.642431	0.321215	−	0.947006i	$-0.395909\pi$
$71$	5.41322	0.642431	0.321215	+	0.947006i	$0.395909\pi$
$72$	0	0
$73$	9.35967	1.09547	0.547733	−	0.836653i	$-0.315491\pi$
$73$	9.35967	1.09547	0.547733	+	0.836653i	$0.315491\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	0	0
$78$	0	0
$79$	6.04022	0.679577	0.339789	−	0.940502i	$-0.389644\pi$
$79$	6.04022	0.679577	0.339789	+	0.940502i	$0.389644\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−13.8509	−1.52034	−0.760168	−	0.649726i	$-0.774884\pi$
$83$	−13.8509	−1.52034	−0.760168	+	0.649726i	$0.774884\pi$
$84$	0	0
$85$	2.18172	0.236641
$86$	0	0
$87$	0	0
$88$	0	0
$89$	16.4756	1.74641	0.873206	−	0.487352i	$-0.162037\pi$
$89$	16.4756	1.74641	0.873206	+	0.487352i	$0.162037\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−2.26215	−0.232092
$96$	0	0
$97$	−18.7317	−1.90191	−0.950956	−	0.309328i	$-0.899896\pi$
$97$	−18.7317	−1.90191	−0.950956	+	0.309328i	$0.899896\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−6.75674	−0.672321	−0.336160	−	0.941805i	$-0.609128\pi$
$101$	−6.75674	−0.672321	−0.336160	+	0.941805i	$0.609128\pi$
$102$	0	0
$103$	−14.2096	−1.40012	−0.700058	−	0.714086i	$-0.746842\pi$
$103$	−14.2096	−1.40012	−0.700058	+	0.714086i	$0.746842\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	16.6822	1.61273	0.806363	−	0.591422i	$-0.201433\pi$
$107$	16.6822	1.61273	0.806363	+	0.591422i	$0.201433\pi$
$108$	0	0
$109$	−4.90013	−0.469347	−0.234673	−	0.972074i	$-0.575402\pi$
$109$	−4.90013	−0.469347	−0.234673	+	0.972074i	$0.575402\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	8.84893	0.832437	0.416219	−	0.909265i	$-0.363355\pi$
$113$	8.84893	0.832437	0.416219	+	0.909265i	$0.363355\pi$
$114$	0	0
$115$	1.04962	0.0978779
$116$	0	0
$117$	0	0
$118$	0	0
$119$	4.88521	0.447826
$120$	0	0
$121$	−11.0000	−1.00000
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−7.13158	−0.637868
$126$	0	0
$127$	−5.51019	−0.488950	−0.244475	−	0.969656i	$-0.578616\pi$
$127$	−5.51019	−0.488950	−0.244475	+	0.969656i	$0.578616\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−1.74481	−0.152445	−0.0762223	−	0.997091i	$-0.524286\pi$
$131$	−1.74481	−0.152445	−0.0762223	+	0.997091i	$0.524286\pi$
$132$	0	0
$133$	−5.06530	−0.439217
$134$	0	0
$135$	0	0
$136$	0	0
$137$	21.2136	1.81240	0.906201	−	0.422846i	$-0.138969\pi$
$137$	21.2136	1.81240	0.906201	+	0.422846i	$0.138969\pi$
$138$	0	0
$139$	−3.97491	−0.337148	−0.168574	−	0.985689i	$-0.553916\pi$
$139$	−3.97491	−0.337148	−0.168574	+	0.985689i	$0.553916\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	0	0
$144$	0	0
$145$	−3.14441	−0.261129
$146$	0	0
$147$	0	0
$148$	0	0
$149$	9.56238	0.783381	0.391690	−	0.920097i	$-0.371890\pi$
$149$	9.56238	0.783381	0.391690	+	0.920097i	$0.371890\pi$
$150$	0	0
$151$	15.9541	1.29833	0.649164	−	0.760648i	$-0.275119\pi$
$151$	15.9541	1.29833	0.649164	+	0.760648i	$0.275119\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	2.36848	0.190241
$156$	0	0
$157$	−2.49404	−0.199046	−0.0995229	−	0.995035i	$-0.531732\pi$
$157$	−2.49404	−0.199046	−0.0995229	+	0.995035i	$0.531732\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	2.35027	0.185227
$162$	0	0
$163$	−0.240909	−0.0188694	−0.00943472	−	0.999955i	$-0.503003\pi$
$163$	−0.240909	−0.0188694	−0.00943472	+	0.999955i	$0.503003\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−1.60023	−0.123830	−0.0619149	−	0.998081i	$-0.519721\pi$
$167$	−1.60023	−0.123830	−0.0619149	+	0.998081i	$0.519721\pi$
$168$	0	0
$169$	−13.0000	−1.00000
$170$	0	0
$171$	0	0
$172$	0	0
$173$	15.2000	1.15564	0.577819	−	0.816165i	$-0.303904\pi$
$173$	15.2000	1.15564	0.577819	+	0.816165i	$0.303904\pi$
$174$	0	0
$175$	−7.49976	−0.566928
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−0.625029	−0.0467169	−0.0233584	−	0.999727i	$-0.507436\pi$
$179$	−0.625029	−0.0467169	−0.0233584	+	0.999727i	$0.507436\pi$
$180$	0	0
$181$	−8.22961	−0.611702	−0.305851	−	0.952079i	$-0.598941\pi$
$181$	−8.22961	−0.611702	−0.305851	+	0.952079i	$0.598941\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	3.42528	0.251831
$186$	0	0
$187$	0	0
$188$	0	0
$189$	0	0
$190$	0	0
$191$	26.2256	1.89761	0.948807	−	0.315855i	$-0.102291\pi$
$191$	26.2256	1.89761	0.948807	+	0.315855i	$0.102291\pi$
$192$	0	0
$193$	25.5201	1.83698	0.918490	−	0.395445i	$-0.129409\pi$
$193$	25.5201	1.83698	0.918490	+	0.395445i	$0.129409\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−7.57515	−0.539707	−0.269854	−	0.962901i	$-0.586975\pi$
$197$	−7.57515	−0.539707	−0.269854	+	0.962901i	$0.586975\pi$
$198$	0	0
$199$	14.3757	1.01907	0.509535	−	0.860450i	$-0.329818\pi$
$199$	14.3757	1.01907	0.509535	+	0.860450i	$0.329818\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−7.04082	−0.494169
$204$	0	0
$205$	1.74451	0.121842
$206$	0	0
$207$	0	0
$208$	0	0
$209$	0	0
$210$	0	0
$211$	10.1159	0.696410	0.348205	−	0.937418i	$-0.386791\pi$
$211$	10.1159	0.696410	0.348205	+	0.937418i	$0.386791\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−3.63075	−0.247615
$216$	0	0
$217$	5.30338	0.360017
$218$	0	0
$219$	0	0
$220$	0	0
$221$	0	0
$222$	0	0
$223$	1.00000	0.0669650
$223$	1.00000	0.0669650
$224$	0	0
$225$	0	0
$226$	0	0
$227$	22.3506	1.48346	0.741731	−	0.670698i	$-0.234005\pi$
$227$	22.3506	1.48346	0.741731	+	0.670698i	$0.234005\pi$
$228$	0	0
$229$	−29.7019	−1.96275	−0.981377	−	0.192092i	$-0.938473\pi$
$229$	−29.7019	−1.96275	−0.981377	+	0.192092i	$0.938473\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−12.5000	−0.818904	−0.409452	−	0.912332i	$-0.634280\pi$
$233$	−12.5000	−0.818904	−0.409452	+	0.912332i	$0.634280\pi$
$234$	0	0
$235$	−1.07086	−0.0698554
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−24.5580	−1.58852	−0.794261	−	0.607576i	$-0.792142\pi$
$239$	−24.5580	−1.58852	−0.794261	+	0.607576i	$0.792142\pi$
$240$	0	0
$241$	−16.9703	−1.09315	−0.546576	−	0.837410i	$-0.684069\pi$
$241$	−16.9703	−1.09315	−0.546576	+	0.837410i	$0.684069\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−3.12492	−0.199644
$246$	0	0
$247$	0	0
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−9.84468	−0.621391	−0.310695	−	0.950510i	$-0.600562\pi$
$251$	−9.84468	−0.621391	−0.310695	+	0.950510i	$0.600562\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	0	0
$256$	0	0
$257$	9.41084	0.587032	0.293516	−	0.955954i	$-0.405175\pi$
$257$	9.41084	0.587032	0.293516	+	0.955954i	$0.405175\pi$
$258$	0	0
$259$	7.66972	0.476574
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−13.2433	−0.816617	−0.408308	−	0.912844i	$-0.633881\pi$
$263$	−13.2433	−0.816617	−0.408308	+	0.912844i	$0.633881\pi$
$264$	0	0
$265$	6.43736	0.395444
$266$	0	0
$267$	0	0
$268$	0	0
$269$	20.4820	1.24881	0.624404	−	0.781101i	$-0.285342\pi$
$269$	20.4820	1.24881	0.624404	+	0.781101i	$0.285342\pi$
$270$	0	0
$271$	−24.3747	−1.48066	−0.740329	−	0.672244i	$-0.765330\pi$
$271$	−24.3747	−1.48066	−0.740329	+	0.672244i	$0.765330\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	0	0
$276$	0	0
$277$	14.4439	0.867848	0.433924	−	0.900950i	$-0.357129\pi$
$277$	14.4439	0.867848	0.433924	+	0.900950i	$0.357129\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	5.18791	0.309485	0.154742	−	0.987955i	$-0.450545\pi$
$281$	5.18791	0.309485	0.154742	+	0.987955i	$0.450545\pi$
$282$	0	0
$283$	16.5494	0.983759	0.491879	−	0.870663i	$-0.336310\pi$
$283$	16.5494	0.983759	0.491879	+	0.870663i	$0.336310\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	3.90624	0.230578
$288$	0	0
$289$	−8.68148	−0.510675
$290$	0	0
$291$	0	0
$292$	0	0
$293$	18.5136	1.08158	0.540788	−	0.841159i	$-0.318126\pi$
$293$	18.5136	1.08158	0.540788	+	0.841159i	$0.318126\pi$
$294$	0	0
$295$	11.3942	0.663398
$296$	0	0
$297$	0	0
$298$	0	0
$299$	0	0
$300$	0	0
$301$	−8.12982	−0.468595
$302$	0	0
$303$	0	0
$304$	0	0
$305$	5.13005	0.293746
$306$	0	0
$307$	15.7089	0.896555	0.448278	−	0.893894i	$-0.352038\pi$
$307$	15.7089	0.896555	0.448278	+	0.893894i	$0.352038\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	28.1190	1.59448	0.797240	−	0.603662i	$-0.206292\pi$
$311$	28.1190	1.59448	0.797240	+	0.603662i	$0.206292\pi$
$312$	0	0
$313$	−7.13209	−0.403130	−0.201565	−	0.979475i	$-0.564603\pi$
$313$	−7.13209	−0.403130	−0.201565	+	0.979475i	$0.564603\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−26.5708	−1.49236	−0.746182	−	0.665742i	$-0.768115\pi$
$317$	−26.5708	−1.49236	−0.746182	+	0.665742i	$0.768115\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−8.62519	−0.479918
$324$	0	0
$325$	0	0
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−2.39783	−0.132197
$330$	0	0
$331$	32.0434	1.76127	0.880633	−	0.473799i	$-0.157118\pi$
$331$	32.0434	1.76127	0.880633	+	0.473799i	$0.157118\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	9.71348	0.530704
$336$	0	0
$337$	8.56225	0.466415	0.233208	−	0.972427i	$-0.425078\pi$
$337$	8.56225	0.466415	0.233208	+	0.972427i	$0.425078\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	0	0
$342$	0	0
$343$	−18.8537	−1.01800
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−5.05401	−0.271313	−0.135657	−	0.990756i	$-0.543314\pi$
$347$	−5.05401	−0.271313	−0.135657	+	0.990756i	$0.543314\pi$
$348$	0	0
$349$	21.8580	1.17003	0.585016	−	0.811022i	$-0.301088\pi$
$349$	21.8580	1.17003	0.585016	+	0.811022i	$0.301088\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−18.8941	−1.00563	−0.502814	−	0.864394i	$-0.667702\pi$
$353$	−18.8941	−1.00563	−0.502814	+	0.864394i	$0.667702\pi$
$354$	0	0
$355$	4.09478	0.217329
$356$	0	0
$357$	0	0
$358$	0	0
$359$	13.3523	0.704709	0.352354	−	0.935867i	$-0.385381\pi$
$359$	13.3523	0.704709	0.352354	+	0.935867i	$0.385381\pi$
$360$	0	0
$361$	−10.0568	−0.529307
$362$	0	0
$363$	0	0
$364$	0	0
$365$	7.08004	0.370586
$366$	0	0
$367$	−3.36509	−0.175656	−0.0878280	−	0.996136i	$-0.527993\pi$
$367$	−3.36509	−0.175656	−0.0878280	+	0.996136i	$0.527993\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	14.4143	0.748351
$372$	0	0
$373$	0.337957	0.0174987	0.00874937	−	0.999962i	$-0.497215\pi$
$373$	0.337957	0.0174987	0.00874937	+	0.999962i	$0.497215\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	0	0
$378$	0	0
$379$	30.3378	1.55835	0.779174	−	0.626808i	$-0.215639\pi$
$379$	30.3378	1.55835	0.779174	+	0.626808i	$0.215639\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−17.3938	−0.888783	−0.444391	−	0.895833i	$-0.646580\pi$
$383$	−17.3938	−0.888783	−0.444391	+	0.895833i	$0.646580\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	0	0
$388$	0	0
$389$	9.90644	0.502277	0.251138	−	0.967951i	$-0.419195\pi$
$389$	9.90644	0.502277	0.251138	+	0.967951i	$0.419195\pi$
$390$	0	0
$391$	4.00204	0.202392
$392$	0	0
$393$	0	0
$394$	0	0
$395$	4.56907	0.229895
$396$	0	0
$397$	21.8381	1.09602	0.548012	−	0.836471i	$-0.315385\pi$
$397$	21.8381	1.09602	0.548012	+	0.836471i	$0.315385\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	3.75875	0.187703	0.0938516	−	0.995586i	$-0.470082\pi$
$401$	3.75875	0.187703	0.0938516	+	0.995586i	$0.470082\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	0	0
$406$	0	0
$407$	0	0
$408$	0	0
$409$	27.5012	1.35984	0.679922	−	0.733284i	$-0.262013\pi$
$409$	27.5012	1.35984	0.679922	+	0.733284i	$0.262013\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	25.5135	1.25544
$414$	0	0
$415$	−10.4774	−0.514316
$416$	0	0
$417$	0	0
$418$	0	0
$419$	16.1455	0.788762	0.394381	−	0.918947i	$-0.370959\pi$
$419$	16.1455	0.788762	0.394381	+	0.918947i	$0.370959\pi$
$420$	0	0
$421$	−20.4828	−0.998272	−0.499136	−	0.866524i	$-0.666349\pi$
$421$	−20.4828	−0.998272	−0.499136	+	0.866524i	$0.666349\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−12.7706	−0.619464
$426$	0	0
$427$	11.4870	0.555894
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−27.2241	−1.31134	−0.655669	−	0.755049i	$-0.727613\pi$
$431$	−27.2241	−1.31134	−0.655669	+	0.755049i	$0.727613\pi$
$432$	0	0
$433$	−32.8234	−1.57739	−0.788697	−	0.614782i	$-0.789244\pi$
$433$	−32.8234	−1.57739	−0.788697	+	0.614782i	$0.789244\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−4.14958	−0.198501
$438$	0	0
$439$	32.7529	1.56321	0.781605	−	0.623774i	$-0.214401\pi$
$439$	32.7529	1.56321	0.781605	+	0.623774i	$0.214401\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−34.1133	−1.62077	−0.810386	−	0.585897i	$-0.800742\pi$
$443$	−34.1133	−1.62077	−0.810386	+	0.585897i	$0.800742\pi$
$444$	0	0
$445$	12.4628	0.590795
$446$	0	0
$447$	0	0
$448$	0	0
$449$	29.5703	1.39551	0.697755	−	0.716337i	$-0.254183\pi$
$449$	29.5703	1.39551	0.697755	+	0.716337i	$0.254183\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	0	0
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−24.4091	−1.14181	−0.570904	−	0.821017i	$-0.693407\pi$
$457$	−24.4091	−1.14181	−0.570904	+	0.821017i	$0.693407\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−24.1938	−1.12682	−0.563408	−	0.826179i	$-0.690510\pi$
$461$	−24.1938	−1.12682	−0.563408	+	0.826179i	$0.690510\pi$
$462$	0	0
$463$	−26.3343	−1.22386	−0.611929	−	0.790913i	$-0.709606\pi$
$463$	−26.3343	−1.22386	−0.611929	+	0.790913i	$0.709606\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−6.90671	−0.319604	−0.159802	−	0.987149i	$-0.551086\pi$
$467$	−6.90671	−0.319604	−0.159802	+	0.987149i	$0.551086\pi$
$468$	0	0
$469$	21.7500	1.00432
$470$	0	0
$471$	0	0
$472$	0	0
$473$	0	0
$474$	0	0
$475$	13.2414	0.607556
$476$	0	0
$477$	0	0
$478$	0	0
$479$	10.8423	0.495396	0.247698	−	0.968837i	$-0.420326\pi$
$479$	10.8423	0.495396	0.247698	+	0.968837i	$0.420326\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−14.1694	−0.643399
$486$	0	0
$487$	−16.0155	−0.725732	−0.362866	−	0.931841i	$-0.618202\pi$
$487$	−16.0155	−0.725732	−0.362866	+	0.931841i	$0.618202\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−15.3873	−0.694420	−0.347210	−	0.937787i	$-0.612871\pi$
$491$	−15.3873	−0.694420	−0.347210	+	0.937787i	$0.612871\pi$
$492$	0	0
$493$	−11.9891	−0.539962
$494$	0	0
$495$	0	0
$496$	0	0
$497$	9.16886	0.411279
$498$	0	0
$499$	6.80025	0.304421	0.152210	−	0.988348i	$-0.451361\pi$
$499$	6.80025	0.304421	0.152210	+	0.988348i	$0.451361\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−22.5805	−1.00681	−0.503406	−	0.864050i	$-0.667920\pi$
$503$	−22.5805	−1.00681	−0.503406	+	0.864050i	$0.667920\pi$
$504$	0	0
$505$	−5.11108	−0.227440
$506$	0	0
$507$	0	0
$508$	0	0
$509$	4.80417	0.212941	0.106471	−	0.994316i	$-0.466045\pi$
$509$	4.80417	0.212941	0.106471	+	0.994316i	$0.466045\pi$
$510$	0	0
$511$	15.8533	0.701309
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−10.7487	−0.473646
$516$	0	0
$517$	0	0
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−8.84175	−0.387364	−0.193682	−	0.981064i	$-0.562043\pi$
$521$	−8.84175	−0.387364	−0.193682	+	0.981064i	$0.562043\pi$
$522$	0	0
$523$	19.6206	0.857948	0.428974	−	0.903317i	$-0.358875\pi$
$523$	19.6206	0.857948	0.428974	+	0.903317i	$0.358875\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	9.03060	0.393379
$528$	0	0
$529$	−21.0746	−0.916288
$530$	0	0
$531$	0	0
$532$	0	0
$533$	0	0
$534$	0	0
$535$	12.6191	0.545570
$536$	0	0
$537$	0	0
$538$	0	0
$539$	0	0
$540$	0	0
$541$	37.3128	1.60420	0.802101	−	0.597189i	$-0.203716\pi$
$541$	37.3128	1.60420	0.802101	+	0.597189i	$0.203716\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−3.70666	−0.158776
$546$	0	0
$547$	0.0435096	0.00186034	0.000930169	−	1.00000i	$-0.499704\pi$
$547$	0.0435096	0.00186034	0.000930169		1.00000i	$0.499704\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	12.4311	0.529582
$552$	0	0
$553$	10.2309	0.435060
$554$	0	0
$555$	0	0
$556$	0	0
$557$	4.73951	0.200819	0.100410	−	0.994946i	$-0.467985\pi$
$557$	4.73951	0.200819	0.100410	+	0.994946i	$0.467985\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	0	0
$562$	0	0
$563$	45.5766	1.92082	0.960411	−	0.278586i	$-0.0898657\pi$
$563$	45.5766	1.92082	0.960411	+	0.278586i	$0.0898657\pi$
$564$	0	0
$565$	6.69370	0.281606
$566$	0	0
$567$	0	0
$568$	0	0
$569$	46.3886	1.94471	0.972355	−	0.233509i	$-0.0750207\pi$
$569$	46.3886	1.94471	0.972355	+	0.233509i	$0.0750207\pi$
$570$	0	0
$571$	−1.62284	−0.0679136	−0.0339568	−	0.999423i	$-0.510811\pi$
$571$	−1.62284	−0.0679136	−0.0339568	+	0.999423i	$0.510811\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−6.14392	−0.256219
$576$	0	0
$577$	−28.2129	−1.17452	−0.587259	−	0.809399i	$-0.699793\pi$
$577$	−28.2129	−1.17452	−0.587259	+	0.809399i	$0.699793\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−23.4606	−0.973308
$582$	0	0
$583$	0	0
$584$	0	0
$585$	0	0
$586$	0	0
$587$	1.15637	0.0477284	0.0238642	−	0.999715i	$-0.492403\pi$
$587$	1.15637	0.0477284	0.0238642	+	0.999715i	$0.492403\pi$
$588$	0	0
$589$	−9.36352	−0.385817
$590$	0	0
$591$	0	0
$592$	0	0
$593$	4.10717	0.168661	0.0843307	−	0.996438i	$-0.473125\pi$
$593$	4.10717	0.168661	0.0843307	+	0.996438i	$0.473125\pi$
$594$	0	0
$595$	3.69537	0.151496
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−10.9063	−0.445620	−0.222810	−	0.974862i	$-0.571523\pi$
$599$	−10.9063	−0.445620	−0.222810	+	0.974862i	$0.571523\pi$
$600$	0	0
$601$	24.0588	0.981378	0.490689	−	0.871335i	$-0.336745\pi$
$601$	24.0588	0.981378	0.490689	+	0.871335i	$0.336745\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−8.32086	−0.338291
$606$	0	0
$607$	22.5031	0.913371	0.456685	−	0.889628i	$-0.349036\pi$
$607$	22.5031	0.913371	0.456685	+	0.889628i	$0.349036\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	0	0
$612$	0	0
$613$	−33.7009	−1.36117	−0.680583	−	0.732671i	$-0.738274\pi$
$613$	−33.7009	−1.36117	−0.680583	+	0.732671i	$0.738274\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−37.2572	−1.49992	−0.749959	−	0.661485i	$-0.769927\pi$
$617$	−37.2572	−1.49992	−0.749959	+	0.661485i	$0.769927\pi$
$618$	0	0
$619$	2.73495	0.109927	0.0549634	−	0.998488i	$-0.482496\pi$
$619$	2.73495	0.109927	0.0549634	+	0.998488i	$0.482496\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	27.9062	1.11804
$624$	0	0
$625$	16.7444	0.669774
$626$	0	0
$627$	0	0
$628$	0	0
$629$	13.0600	0.520736
$630$	0	0
$631$	40.6413	1.61790	0.808952	−	0.587875i	$-0.200035\pi$
$631$	40.6413	1.61790	0.808952	+	0.587875i	$0.200035\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−4.16814	−0.165407
$636$	0	0
$637$	0	0
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−42.1187	−1.66359	−0.831795	−	0.555083i	$-0.812686\pi$
$641$	−42.1187	−1.66359	−0.831795	+	0.555083i	$0.812686\pi$
$642$	0	0
$643$	20.3366	0.801998	0.400999	−	0.916078i	$-0.368663\pi$
$643$	20.3366	0.801998	0.400999	+	0.916078i	$0.368663\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	26.8198	1.05440	0.527198	−	0.849743i	$-0.323243\pi$
$647$	26.8198	1.05440	0.527198	+	0.849743i	$0.323243\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−18.5516	−0.725978	−0.362989	−	0.931793i	$-0.618244\pi$
$653$	−18.5516	−0.725978	−0.362989	+	0.931793i	$0.618244\pi$
$654$	0	0
$655$	−1.31985	−0.0515707
$656$	0	0
$657$	0	0
$658$	0	0
$659$	10.4183	0.405838	0.202919	−	0.979196i	$-0.434957\pi$
$659$	10.4183	0.405838	0.202919	+	0.979196i	$0.434957\pi$
$660$	0	0
$661$	26.9805	1.04942	0.524709	−	0.851281i	$-0.324174\pi$
$661$	26.9805	1.04942	0.524709	+	0.851281i	$0.324174\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−3.83160	−0.148583
$666$	0	0
$667$	−5.76795	−0.223336
$668$	0	0
$669$	0	0
$670$	0	0
$671$	0	0
$672$	0	0
$673$	35.3466	1.36251	0.681255	−	0.732046i	$-0.261434\pi$
$673$	35.3466	1.36251	0.681255	+	0.732046i	$0.261434\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	47.0609	1.80870	0.904349	−	0.426793i	$-0.140357\pi$
$677$	47.0609	1.80870	0.904349	+	0.426793i	$0.140357\pi$
$678$	0	0
$679$	−31.7275	−1.21759
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−5.93575	−0.227125	−0.113563	−	0.993531i	$-0.536226\pi$
$683$	−5.93575	−0.227125	−0.113563	+	0.993531i	$0.536226\pi$
$684$	0	0
$685$	16.0469	0.613120
$686$	0	0
$687$	0	0
$688$	0	0
$689$	0	0
$690$	0	0
$691$	0.526655	0.0200349	0.0100174	−	0.999950i	$-0.496811\pi$
$691$	0.526655	0.0200349	0.0100174	+	0.999950i	$0.496811\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−3.00679	−0.114054
$696$	0	0
$697$	6.65154	0.251945
$698$	0	0
$699$	0	0
$700$	0	0
$701$	28.7782	1.08694	0.543469	−	0.839429i	$-0.317111\pi$
$701$	28.7782	1.08694	0.543469	+	0.839429i	$0.317111\pi$
$702$	0	0
$703$	−13.5415	−0.510726
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−11.4445	−0.430415
$708$	0	0
$709$	13.2796	0.498727	0.249364	−	0.968410i	$-0.419779\pi$
$709$	13.2796	0.498727	0.249364	+	0.968410i	$0.419779\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	4.34462	0.162707
$714$	0	0
$715$	0	0
$716$	0	0
$717$	0	0
$718$	0	0
$719$	21.1082	0.787204	0.393602	−	0.919281i	$-0.371229\pi$
$719$	21.1082	0.787204	0.393602	+	0.919281i	$0.371229\pi$
$720$	0	0
$721$	−24.0681	−0.896343
$722$	0	0
$723$	0	0
$724$	0	0
$725$	18.4056	0.683568
$726$	0	0
$727$	25.5596	0.947952	0.473976	−	0.880538i	$-0.342818\pi$
$727$	25.5596	0.947952	0.473976	+	0.880538i	$0.342818\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−13.8435	−0.512019
$732$	0	0
$733$	20.3180	0.750461	0.375231	−	0.926932i	$-0.377564\pi$
$733$	20.3180	0.750461	0.375231	+	0.926932i	$0.377564\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	0	0
$738$	0	0
$739$	−21.1052	−0.776367	−0.388184	−	0.921582i	$-0.626897\pi$
$739$	−21.1052	−0.776367	−0.388184	+	0.921582i	$0.626897\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−27.7888	−1.01947	−0.509735	−	0.860331i	$-0.670257\pi$
$743$	−27.7888	−1.01947	−0.509735	+	0.860331i	$0.670257\pi$
$744$	0	0
$745$	7.23338	0.265011
$746$	0	0
$747$	0	0
$748$	0	0
$749$	28.2561	1.03245
$750$	0	0
$751$	16.2418	0.592673	0.296337	−	0.955084i	$-0.404235\pi$
$751$	16.2418	0.592673	0.296337	+	0.955084i	$0.404235\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	12.0684	0.439213
$756$	0	0
$757$	46.6880	1.69691	0.848453	−	0.529271i	$-0.177534\pi$
$757$	46.6880	1.69691	0.848453	+	0.529271i	$0.177534\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	4.91022	0.177995	0.0889977	−	0.996032i	$-0.471634\pi$
$761$	4.91022	0.177995	0.0889977	+	0.996032i	$0.471634\pi$
$762$	0	0
$763$	−8.29979	−0.300472
$764$	0	0
$765$	0	0
$766$	0	0
$767$	0	0
$768$	0	0
$769$	−21.8683	−0.788590	−0.394295	−	0.918984i	$-0.629011\pi$
$769$	−21.8683	−0.788590	−0.394295	+	0.918984i	$0.629011\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−9.09414	−0.327094	−0.163547	−	0.986536i	$-0.552293\pi$
$773$	−9.09414	−0.327094	−0.163547	+	0.986536i	$0.552293\pi$
$774$	0	0
$775$	−13.8638	−0.498001
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−6.89675	−0.247102
$780$	0	0
$781$	0	0
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−1.88659	−0.0673354
$786$	0	0
$787$	39.3484	1.40262	0.701309	−	0.712857i	$-0.252599\pi$
$787$	39.3484	1.40262	0.701309	+	0.712857i	$0.252599\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	14.9882	0.532920
$792$	0	0
$793$	0	0
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−39.7961	−1.40965	−0.704825	−	0.709381i	$-0.748974\pi$
$797$	−39.7961	−1.40965	−0.704825	+	0.709381i	$0.748974\pi$
$798$	0	0
$799$	−4.08302	−0.144447
$800$	0	0
$801$	0	0
$802$	0	0
$803$	0	0
$804$	0	0
$805$	1.77784	0.0626607
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−28.2815	−0.994323	−0.497162	−	0.867658i	$-0.665624\pi$
$809$	−28.2815	−0.994323	−0.497162	+	0.867658i	$0.665624\pi$
$810$	0	0
$811$	−15.3148	−0.537776	−0.268888	−	0.963171i	$-0.586656\pi$
$811$	−15.3148	−0.537776	−0.268888	+	0.963171i	$0.586656\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−0.182234	−0.00638336
$816$	0	0
$817$	14.3538	0.502176
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−41.1333	−1.43556	−0.717782	−	0.696268i	$-0.754842\pi$
$821$	−41.1333	−1.43556	−0.717782	+	0.696268i	$0.754842\pi$
$822$	0	0
$823$	22.2665	0.776160	0.388080	−	0.921626i	$-0.373138\pi$
$823$	22.2665	0.776160	0.388080	+	0.921626i	$0.373138\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−19.6749	−0.684165	−0.342083	−	0.939670i	$-0.611132\pi$
$827$	−19.6749	−0.684165	−0.342083	+	0.939670i	$0.611132\pi$
$828$	0	0
$829$	−40.5318	−1.40773	−0.703864	−	0.710335i	$-0.748544\pi$
$829$	−40.5318	−1.40773	−0.703864	+	0.710335i	$0.748544\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−11.9148	−0.412823
$834$	0	0
$835$	−1.21048	−0.0418905
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−8.53950	−0.294816	−0.147408	−	0.989076i	$-0.547093\pi$
$839$	−8.53950	−0.294816	−0.147408	+	0.989076i	$0.547093\pi$
$840$	0	0
$841$	−11.7207	−0.404161
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−9.83374	−0.338291
$846$	0	0
$847$	−18.6317	−0.640193
$848$	0	0
$849$	0	0
$850$	0	0
$851$	6.28316	0.215384
$852$	0	0
$853$	−52.7762	−1.80702	−0.903511	−	0.428565i	$-0.859019\pi$
$853$	−52.7762	−1.80702	−0.903511	+	0.428565i	$0.859019\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−5.63334	−0.192431	−0.0962157	−	0.995361i	$-0.530674\pi$
$857$	−5.63334	−0.192431	−0.0962157	+	0.995361i	$0.530674\pi$
$858$	0	0
$859$	9.82714	0.335298	0.167649	−	0.985847i	$-0.446383\pi$
$859$	9.82714	0.335298	0.167649	+	0.985847i	$0.446383\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−5.94580	−0.202397	−0.101199	−	0.994866i	$-0.532268\pi$
$863$	−5.94580	−0.202397	−0.101199	+	0.994866i	$0.532268\pi$
$864$	0	0
$865$	11.4980	0.390942
$866$	0	0
$867$	0	0
$868$	0	0
$869$	0	0
$870$	0	0
$871$	0	0
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−12.0794	−0.408358
$876$	0	0
$877$	−14.1515	−0.477863	−0.238932	−	0.971036i	$-0.576797\pi$
$877$	−14.1515	−0.477863	−0.238932	+	0.971036i	$0.576797\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	11.4681	0.386370	0.193185	−	0.981162i	$-0.438118\pi$
$881$	11.4681	0.386370	0.193185	+	0.981162i	$0.438118\pi$
$882$	0	0
$883$	6.60450	0.222259	0.111130	−	0.993806i	$-0.464553\pi$
$883$	6.60450	0.222259	0.111130	+	0.993806i	$0.464553\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	34.9830	1.17461	0.587307	−	0.809364i	$-0.300188\pi$
$887$	34.9830	1.17461	0.587307	+	0.809364i	$0.300188\pi$
$888$	0	0
$889$	−9.33310	−0.313022
$890$	0	0
$891$	0	0
$892$	0	0
$893$	4.23354	0.141670
$894$	0	0
$895$	−0.472798	−0.0158039
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−13.0154	−0.434087
$900$	0	0
$901$	24.5446	0.817699
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−6.22522	−0.206933
$906$	0	0
$907$	−57.5902	−1.91225	−0.956126	−	0.292956i	$-0.905361\pi$
$907$	−57.5902	−1.91225	−0.956126	+	0.292956i	$0.905361\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	34.8815	1.15568	0.577838	−	0.816151i	$-0.303896\pi$
$911$	34.8815	1.15568	0.577838	+	0.816151i	$0.303896\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−2.95534	−0.0975939
$918$	0	0
$919$	−56.8477	−1.87523	−0.937617	−	0.347670i	$-0.886973\pi$
$919$	−56.8477	−1.87523	−0.937617	+	0.347670i	$0.886973\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	0	0
$924$	0	0
$925$	−20.0497	−0.659230
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−10.5818	−0.347177	−0.173589	−	0.984818i	$-0.555536\pi$
$929$	−10.5818	−0.347177	−0.173589	+	0.984818i	$0.555536\pi$
$930$	0	0
$931$	12.3540	0.404887
$932$	0	0
$933$	0	0
$934$	0	0
$935$	0	0
$936$	0	0
$937$	4.77720	0.156064	0.0780322	−	0.996951i	$-0.475136\pi$
$937$	4.77720	0.156064	0.0780322	+	0.996951i	$0.475136\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	5.85246	0.190785	0.0953924	−	0.995440i	$-0.469589\pi$
$941$	5.85246	0.190785	0.0953924	+	0.995440i	$0.469589\pi$
$942$	0	0
$943$	3.20005	0.104208
$944$	0	0
$945$	0	0
$946$	0	0
$947$	38.8455	1.26231	0.631154	−	0.775658i	$-0.282582\pi$
$947$	38.8455	1.26231	0.631154	+	0.775658i	$0.282582\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−12.6809	−0.410774	−0.205387	−	0.978681i	$-0.565845\pi$
$953$	−12.6809	−0.410774	−0.205387	+	0.978681i	$0.565845\pi$
$954$	0	0
$955$	19.8381	0.641946
$956$	0	0
$957$	0	0
$958$	0	0
$959$	35.9314	1.16029
$960$	0	0
$961$	−21.1964	−0.683754
$962$	0	0
$963$	0	0
$964$	0	0
$965$	19.3045	0.621434
$966$	0	0
$967$	40.5498	1.30399	0.651997	−	0.758221i	$-0.273931\pi$
$967$	40.5498	1.30399	0.651997	+	0.758221i	$0.273931\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	34.2891	1.10039	0.550194	−	0.835037i	$-0.314554\pi$
$971$	34.2891	1.10039	0.550194	+	0.835037i	$0.314554\pi$
$972$	0	0
$973$	−6.73267	−0.215840
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−58.3363	−1.86634	−0.933171	−	0.359433i	$-0.882970\pi$
$977$	−58.3363	−1.86634	−0.933171	+	0.359433i	$0.882970\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	0	0
$982$	0	0
$983$	14.5422	0.463823	0.231912	−	0.972737i	$-0.425502\pi$
$983$	14.5422	0.463823	0.231912	+	0.972737i	$0.425502\pi$
$984$	0	0
$985$	−5.73016	−0.182578
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−6.66008	−0.211778
$990$	0	0
$991$	2.40591	0.0764264	0.0382132	−	0.999270i	$-0.487833\pi$
$991$	2.40591	0.0764264	0.0382132	+	0.999270i	$0.487833\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	10.8744	0.344742
$996$	0	0
$997$	−3.72115	−0.117850	−0.0589250	−	0.998262i	$-0.518767\pi$
$997$	−3.72115	−0.117850	−0.0589250	+	0.998262i	$0.518767\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8028.2.a.i.1.3		5
3.2	odd	2		2676.2.a.c.1.3	✓	5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2676.2.a.c.1.3	✓	5	3.2	odd	2
8028.2.a.i.1.3		5	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:	\( N \)	\(=\)	\( 8028 = 2^{2} \cdot 3^{2} \cdot 223 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8028.a (trivial)

\(f(q)\)	\(=\)	\(q+0.756442 q^{5} +1.69379 q^{7} +O(q^{10})\)	\(q+0.756442 q^{5} +1.69379 q^{7} +2.88419 q^{17} -2.99051 q^{19} +1.38758 q^{23} -4.42780 q^{25} -4.15684 q^{29} +3.13107 q^{31} +1.28125 q^{35} +4.52814 q^{37} +2.30621 q^{41} -4.79978 q^{43} -1.41566 q^{47} -4.13107 q^{49} +8.51006 q^{53} +15.0629 q^{59} +6.78182 q^{61} +12.8410 q^{67} +5.41322 q^{71} +9.35967 q^{73} +6.04022 q^{79} -13.8509 q^{83} +2.18172 q^{85} +16.4756 q^{89} -2.26215 q^{95} -18.7317 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q + 6 q^{5} + 3 q^{7}+O(q^{10}) \) 5 * q + 6 * q^5 + 3 * q^7	\( 5 q + 6 q^{5} + 3 q^{7} + 13 q^{17} - q^{19} - 4 q^{23} + 5 q^{25} - 11 q^{29} - 3 q^{31} + 8 q^{35} + 7 q^{37} + 17 q^{41} + 15 q^{43} + 25 q^{47} - 2 q^{49} + 3 q^{53} + 14 q^{59} - 18 q^{61} + 24 q^{67} + 14 q^{71} - 23 q^{73} + 14 q^{79} + 15 q^{83} + 6 q^{85} + 25 q^{89} + 26 q^{95} - 12 q^{97}+O(q^{100}) \) 5 * q + 6 * q^5 + 3 * q^7 + 13 * q^17 - q^19 - 4 * q^23 + 5 * q^25 - 11 * q^29 - 3 * q^31 + 8 * q^35 + 7 * q^37 + 17 * q^41 + 15 * q^43 + 25 * q^47 - 2 * q^49 + 3 * q^53 + 14 * q^59 - 18 * q^61 + 24 * q^67 + 14 * q^71 - 23 * q^73 + 14 * q^79 + 15 * q^83 + 6 * q^85 + 25 * q^89 + 26 * q^95 - 12 * q^97

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