LMFDB - Embedded newform 8024.2.a.bb.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8024,2,Mod(1,8024)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8024, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("8024.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8024 = 2^{3} \cdot 17 \cdot 59 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8024.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:	$64.0719625819$
Analytic rank:	$0$
Dimension:	$32$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	8024.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.16628 q^{3} -1.06664 q^{5} -0.836052 q^{7} +7.02534 q^{9} +O(q^{10})$	$q-3.16628 q^{3} -1.06664 q^{5} -0.836052 q^{7} +7.02534 q^{9} -3.32563 q^{11} -1.60499 q^{13} +3.37729 q^{15} +1.00000 q^{17} -6.50007 q^{19} +2.64717 q^{21} -1.48772 q^{23} -3.86227 q^{25} -12.7454 q^{27} +7.50404 q^{29} -4.45792 q^{31} +10.5299 q^{33} +0.891769 q^{35} -3.56122 q^{37} +5.08184 q^{39} +4.62536 q^{41} -4.37602 q^{43} -7.49353 q^{45} +2.49637 q^{47} -6.30102 q^{49} -3.16628 q^{51} -12.6743 q^{53} +3.54726 q^{55} +20.5810 q^{57} +1.00000 q^{59} +1.68445 q^{61} -5.87355 q^{63} +1.71195 q^{65} -3.14546 q^{67} +4.71053 q^{69} +1.93648 q^{71} +1.89004 q^{73} +12.2290 q^{75} +2.78040 q^{77} +0.220848 q^{79} +19.2794 q^{81} -5.92117 q^{83} -1.06664 q^{85} -23.7599 q^{87} +2.30235 q^{89} +1.34185 q^{91} +14.1150 q^{93} +6.93326 q^{95} -0.988209 q^{97} -23.3637 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 32 q + 8 q^{5} - 3 q^{7} + 40 q^{9}+O(q^{10}) $ 32 * q + 8 * q^5 - 3 * q^7 + 40 * q^9	$ 32 q + 8 q^{5} - 3 q^{7} + 40 q^{9} + 3 q^{11} + 13 q^{13} + 4 q^{15} + 32 q^{17} + 14 q^{19} - 7 q^{21} + 7 q^{23} + 38 q^{25} + 9 q^{27} + 17 q^{29} + 15 q^{31} + 18 q^{33} + 6 q^{35} + 21 q^{37} + 16 q^{39} + 49 q^{41} - 7 q^{43} + 14 q^{45} - 25 q^{47} + 37 q^{49} + 12 q^{53} + 15 q^{55} + 45 q^{57} + 32 q^{59} + 5 q^{61} - 12 q^{63} + 39 q^{65} + 12 q^{69} - 13 q^{71} + 70 q^{73} - 47 q^{75} - 10 q^{77} - q^{79} + 84 q^{81} - 17 q^{83} + 8 q^{85} + 20 q^{87} + 42 q^{89} + 36 q^{91} + 2 q^{93} - q^{95} + 58 q^{97} + 7 q^{99}+O(q^{100}) $ 32 * q + 8 * q^5 - 3 * q^7 + 40 * q^9 + 3 * q^11 + 13 * q^13 + 4 * q^15 + 32 * q^17 + 14 * q^19 - 7 * q^21 + 7 * q^23 + 38 * q^25 + 9 * q^27 + 17 * q^29 + 15 * q^31 + 18 * q^33 + 6 * q^35 + 21 * q^37 + 16 * q^39 + 49 * q^41 - 7 * q^43 + 14 * q^45 - 25 * q^47 + 37 * q^49 + 12 * q^53 + 15 * q^55 + 45 * q^57 + 32 * q^59 + 5 * q^61 - 12 * q^63 + 39 * q^65 + 12 * q^69 - 13 * q^71 + 70 * q^73 - 47 * q^75 - 10 * q^77 - q^79 + 84 * q^81 - 17 * q^83 + 8 * q^85 + 20 * q^87 + 42 * q^89 + 36 * q^91 + 2 * q^93 - q^95 + 58 * q^97 + 7 * q^99

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
$3$	−3.16628	−1.82805	−0.914027	−	0.405654i	$-0.867044\pi$
$3$	−3.16628	−1.82805	−0.914027	+	0.405654i	$0.867044\pi$
$4$	0	0
$5$	−1.06664	−0.477018	−0.238509	−	0.971140i	$-0.576659\pi$
$5$	−1.06664	−0.477018	−0.238509	+	0.971140i	$0.576659\pi$
$6$	0	0
$7$	−0.836052	−0.315998	−0.157999	−	0.987439i	$-0.550504\pi$
$7$	−0.836052	−0.315998	−0.157999	+	0.987439i	$0.550504\pi$
$8$	0	0
$9$	7.02534	2.34178
$10$	0	0
$11$	−3.32563	−1.00272	−0.501358	−	0.865240i	$-0.667166\pi$
$11$	−3.32563	−1.00272	−0.501358	+	0.865240i	$0.667166\pi$
$12$	0	0
$13$	−1.60499	−0.445143	−0.222572	−	0.974916i	$-0.571445\pi$
$13$	−1.60499	−0.445143	−0.222572	+	0.974916i	$0.571445\pi$
$14$	0	0
$15$	3.37729	0.872014
$16$	0	0
$17$	1.00000	0.242536
$17$	1.00000	0.242536
$18$	0	0
$19$	−6.50007	−1.49122	−0.745609	−	0.666384i	$-0.767841\pi$
$19$	−6.50007	−1.49122	−0.745609	+	0.666384i	$0.767841\pi$
$20$	0	0
$21$	2.64717	0.577661
$22$	0	0
$23$	−1.48772	−0.310210	−0.155105	−	0.987898i	$-0.549572\pi$
$23$	−1.48772	−0.310210	−0.155105	+	0.987898i	$0.549572\pi$
$24$	0	0
$25$	−3.86227	−0.772454
$26$	0	0
$27$	−12.7454	−2.45284
$28$	0	0
$29$	7.50404	1.39347	0.696733	−	0.717331i	$-0.254636\pi$
$29$	7.50404	1.39347	0.696733	+	0.717331i	$0.254636\pi$
$30$	0	0
$31$	−4.45792	−0.800665	−0.400333	−	0.916370i	$-0.631105\pi$
$31$	−4.45792	−0.800665	−0.400333	+	0.916370i	$0.631105\pi$
$32$	0	0
$33$	10.5299	1.83302
$34$	0	0
$35$	0.891769	0.150737
$36$	0	0
$37$	−3.56122	−0.585462	−0.292731	−	0.956195i	$-0.594564\pi$
$37$	−3.56122	−0.585462	−0.292731	+	0.956195i	$0.594564\pi$
$38$	0	0
$39$	5.08184	0.813746
$40$	0	0
$41$	4.62536	0.722360	0.361180	−	0.932496i	$-0.382374\pi$
$41$	4.62536	0.722360	0.361180	+	0.932496i	$0.382374\pi$
$42$	0	0
$43$	−4.37602	−0.667337	−0.333668	−	0.942691i	$-0.608287\pi$
$43$	−4.37602	−0.667337	−0.333668	+	0.942691i	$0.608287\pi$
$44$	0	0
$45$	−7.49353	−1.11707
$46$	0	0
$47$	2.49637	0.364134	0.182067	−	0.983286i	$-0.441721\pi$
$47$	2.49637	0.364134	0.182067	+	0.983286i	$0.441721\pi$
$48$	0	0
$49$	−6.30102	−0.900145
$50$	0	0
$51$	−3.16628	−0.443368
$52$	0	0
$53$	−12.6743	−1.74095	−0.870473	−	0.492216i	$-0.836187\pi$
$53$	−12.6743	−1.74095	−0.870473	+	0.492216i	$0.836187\pi$
$54$	0	0
$55$	3.54726	0.478313
$56$	0	0
$57$	20.5810	2.72603
$58$	0	0
$59$	1.00000	0.130189
$59$	1.00000	0.130189
$60$	0	0
$61$	1.68445	0.215671	0.107836	−	0.994169i	$-0.465608\pi$
$61$	1.68445	0.215671	0.107836	+	0.994169i	$0.465608\pi$
$62$	0	0
$63$	−5.87355	−0.739997
$64$	0	0
$65$	1.71195	0.212341
$66$	0	0
$67$	−3.14546	−0.384279	−0.192139	−	0.981368i	$-0.561543\pi$
$67$	−3.14546	−0.384279	−0.192139	+	0.981368i	$0.561543\pi$
$68$	0	0
$69$	4.71053	0.567081
$70$	0	0
$71$	1.93648	0.229818	0.114909	−	0.993376i	$-0.463342\pi$
$71$	1.93648	0.229818	0.114909	+	0.993376i	$0.463342\pi$
$72$	0	0
$73$	1.89004	0.221213	0.110606	−	0.993864i	$-0.464721\pi$
$73$	1.89004	0.221213	0.110606	+	0.993864i	$0.464721\pi$
$74$	0	0
$75$	12.2290	1.41209
$76$	0	0
$77$	2.78040	0.316856
$78$	0	0
$79$	0.220848	0.0248474	0.0124237	−	0.999923i	$-0.496045\pi$
$79$	0.220848	0.0248474	0.0124237	+	0.999923i	$0.496045\pi$
$80$	0	0
$81$	19.2794	2.14215
$82$	0	0
$83$	−5.92117	−0.649933	−0.324966	−	0.945726i	$-0.605353\pi$
$83$	−5.92117	−0.649933	−0.324966	+	0.945726i	$0.605353\pi$
$84$	0	0
$85$	−1.06664	−0.115694
$86$	0	0
$87$	−23.7599	−2.54733
$88$	0	0
$89$	2.30235	0.244049	0.122024	−	0.992527i	$-0.461061\pi$
$89$	2.30235	0.244049	0.122024	+	0.992527i	$0.461061\pi$
$90$	0	0
$91$	1.34185	0.140664
$92$	0	0
$93$	14.1150	1.46366
$94$	0	0
$95$	6.93326	0.711337
$96$	0	0
$97$	−0.988209	−0.100337	−0.0501687	−	0.998741i	$-0.515976\pi$
$97$	−0.988209	−0.100337	−0.0501687	+	0.998741i	$0.515976\pi$
$98$	0	0
$99$	−23.3637	−2.34814
$100$	0	0
$101$	−10.1936	−1.01430	−0.507152	−	0.861857i	$-0.669302\pi$
$101$	−10.1936	−1.01430	−0.507152	+	0.861857i	$0.669302\pi$
$102$	0	0
$103$	−10.3507	−1.01989	−0.509943	−	0.860208i	$-0.670333\pi$
$103$	−10.3507	−1.01989	−0.509943	+	0.860208i	$0.670333\pi$
$104$	0	0
$105$	−2.82359	−0.275554
$106$	0	0
$107$	−16.3291	−1.57859	−0.789297	−	0.614012i	$-0.789555\pi$
$107$	−16.3291	−1.57859	−0.789297	+	0.614012i	$0.789555\pi$
$108$	0	0
$109$	−11.1335	−1.06639	−0.533197	−	0.845991i	$-0.679010\pi$
$109$	−11.1335	−1.06639	−0.533197	+	0.845991i	$0.679010\pi$
$110$	0	0
$111$	11.2758	1.07025
$112$	0	0
$113$	−5.01329	−0.471610	−0.235805	−	0.971800i	$-0.575773\pi$
$113$	−5.01329	−0.471610	−0.235805	+	0.971800i	$0.575773\pi$
$114$	0	0
$115$	1.58686	0.147976
$116$	0	0
$117$	−11.2756	−1.04243
$118$	0	0
$119$	−0.836052	−0.0766407
$120$	0	0
$121$	0.0598298	0.00543907
$122$	0	0
$123$	−14.6452	−1.32051
$124$	0	0
$125$	9.45289	0.845492
$126$	0	0
$127$	−7.62690	−0.676778	−0.338389	−	0.941006i	$-0.609882\pi$
$127$	−7.62690	−0.676778	−0.338389	+	0.941006i	$0.609882\pi$
$128$	0	0
$129$	13.8557	1.21993
$130$	0	0
$131$	3.14134	0.274460	0.137230	−	0.990539i	$-0.456180\pi$
$131$	3.14134	0.274460	0.137230	+	0.990539i	$0.456180\pi$
$132$	0	0
$133$	5.43439	0.471222
$134$	0	0
$135$	13.5947	1.17005
$136$	0	0
$137$	−19.1031	−1.63209	−0.816043	−	0.577991i	$-0.803837\pi$
$137$	−19.1031	−1.63209	−0.816043	+	0.577991i	$0.803837\pi$
$138$	0	0
$139$	−14.2746	−1.21076	−0.605378	−	0.795938i	$-0.706978\pi$
$139$	−14.2746	−1.21076	−0.605378	+	0.795938i	$0.706978\pi$
$140$	0	0
$141$	−7.90422	−0.665656
$142$	0	0
$143$	5.33760	0.446352
$144$	0	0
$145$	−8.00414	−0.664707
$146$	0	0
$147$	19.9508	1.64551
$148$	0	0
$149$	6.83597	0.560025	0.280012	−	0.959996i	$-0.409661\pi$
$149$	6.83597	0.560025	0.280012	+	0.959996i	$0.409661\pi$
$150$	0	0
$151$	1.46566	0.119273	0.0596367	−	0.998220i	$-0.481006\pi$
$151$	1.46566	0.119273	0.0596367	+	0.998220i	$0.481006\pi$
$152$	0	0
$153$	7.02534	0.567965
$154$	0	0
$155$	4.75501	0.381931
$156$	0	0
$157$	3.99113	0.318527	0.159263	−	0.987236i	$-0.449088\pi$
$157$	3.99113	0.318527	0.159263	+	0.987236i	$0.449088\pi$
$158$	0	0
$159$	40.1303	3.18254
$160$	0	0
$161$	1.24381	0.0980257
$162$	0	0
$163$	6.37124	0.499034	0.249517	−	0.968370i	$-0.419728\pi$
$163$	6.37124	0.499034	0.249517	+	0.968370i	$0.419728\pi$
$164$	0	0
$165$	−11.2316	−0.874382
$166$	0	0
$167$	12.3668	0.956969	0.478485	−	0.878096i	$-0.341186\pi$
$167$	12.3668	0.956969	0.478485	+	0.878096i	$0.341186\pi$
$168$	0	0
$169$	−10.4240	−0.801847
$170$	0	0
$171$	−45.6652	−3.49210
$172$	0	0
$173$	−3.92953	−0.298756	−0.149378	−	0.988780i	$-0.547727\pi$
$173$	−3.92953	−0.298756	−0.149378	+	0.988780i	$0.547727\pi$
$174$	0	0
$175$	3.22906	0.244094
$176$	0	0
$177$	−3.16628	−0.237992
$178$	0	0
$179$	4.74032	0.354308	0.177154	−	0.984183i	$-0.443311\pi$
$179$	4.74032	0.354308	0.177154	+	0.984183i	$0.443311\pi$
$180$	0	0
$181$	−16.0430	−1.19246	−0.596232	−	0.802812i	$-0.703336\pi$
$181$	−16.0430	−1.19246	−0.596232	+	0.802812i	$0.703336\pi$
$182$	0	0
$183$	−5.33343	−0.394259
$184$	0	0
$185$	3.79856	0.279275
$186$	0	0
$187$	−3.32563	−0.243194
$188$	0	0
$189$	10.6558	0.775093
$190$	0	0
$191$	−4.77799	−0.345723	−0.172861	−	0.984946i	$-0.555301\pi$
$191$	−4.77799	−0.345723	−0.172861	+	0.984946i	$0.555301\pi$
$192$	0	0
$193$	6.11849	0.440418	0.220209	−	0.975453i	$-0.429326\pi$
$193$	6.11849	0.440418	0.220209	+	0.975453i	$0.429326\pi$
$194$	0	0
$195$	−5.42051	−0.388171
$196$	0	0
$197$	2.10799	0.150188	0.0750939	−	0.997176i	$-0.476074\pi$
$197$	2.10799	0.150188	0.0750939	+	0.997176i	$0.476074\pi$
$198$	0	0
$199$	0.391057	0.0277213	0.0138606	−	0.999904i	$-0.495588\pi$
$199$	0.391057	0.0277213	0.0138606	+	0.999904i	$0.495588\pi$
$200$	0	0
$201$	9.95940	0.702482
$202$	0	0
$203$	−6.27376	−0.440332
$204$	0	0
$205$	−4.93361	−0.344578
$206$	0	0
$207$	−10.4517	−0.726444
$208$	0	0
$209$	21.6168	1.49527
$210$	0	0
$211$	−21.9045	−1.50797	−0.753985	−	0.656891i	$-0.771871\pi$
$211$	−21.9045	−1.50797	−0.753985	+	0.656891i	$0.771871\pi$
$212$	0	0
$213$	−6.13145	−0.420120
$214$	0	0
$215$	4.66765	0.318331
$216$	0	0
$217$	3.72705	0.253008
$218$	0	0
$219$	−5.98441	−0.404389
$220$	0	0
$221$	−1.60499	−0.107963
$222$	0	0
$223$	7.56821	0.506805	0.253402	−	0.967361i	$-0.418450\pi$
$223$	7.56821	0.506805	0.253402	+	0.967361i	$0.418450\pi$
$224$	0	0
$225$	−27.1338	−1.80892
$226$	0	0
$227$	−3.46432	−0.229935	−0.114968	−	0.993369i	$-0.536676\pi$
$227$	−3.46432	−0.229935	−0.114968	+	0.993369i	$0.536676\pi$
$228$	0	0
$229$	−9.08344	−0.600250	−0.300125	−	0.953900i	$-0.597028\pi$
$229$	−9.08344	−0.600250	−0.300125	+	0.953900i	$0.597028\pi$
$230$	0	0
$231$	−8.80353	−0.579230
$232$	0	0
$233$	−9.95001	−0.651847	−0.325924	−	0.945396i	$-0.605675\pi$
$233$	−9.95001	−0.651847	−0.325924	+	0.945396i	$0.605675\pi$
$234$	0	0
$235$	−2.66274	−0.173698
$236$	0	0
$237$	−0.699268	−0.0454224
$238$	0	0
$239$	−20.0639	−1.29783	−0.648914	−	0.760862i	$-0.724776\pi$
$239$	−20.0639	−1.29783	−0.648914	+	0.760862i	$0.724776\pi$
$240$	0	0
$241$	−0.199983	−0.0128820	−0.00644101	−	0.999979i	$-0.502050\pi$
$241$	−0.199983	−0.0128820	−0.00644101	+	0.999979i	$0.502050\pi$
$242$	0	0
$243$	−22.8078	−1.46312
$244$	0	0
$245$	6.72094	0.429385
$246$	0	0
$247$	10.4325	0.663806
$248$	0	0
$249$	18.7481	1.18811
$250$	0	0
$251$	−5.01806	−0.316737	−0.158368	−	0.987380i	$-0.550623\pi$
$251$	−5.01806	−0.316737	−0.158368	+	0.987380i	$0.550623\pi$
$252$	0	0
$253$	4.94760	0.311053
$254$	0	0
$255$	3.37729	0.211494
$256$	0	0
$257$	−9.47294	−0.590906	−0.295453	−	0.955357i	$-0.595471\pi$
$257$	−9.47294	−0.590906	−0.295453	+	0.955357i	$0.595471\pi$
$258$	0	0
$259$	2.97737	0.185005
$260$	0	0
$261$	52.7184	3.26319
$262$	0	0
$263$	−17.2416	−1.06316	−0.531580	−	0.847008i	$-0.678402\pi$
$263$	−17.2416	−1.06316	−0.531580	+	0.847008i	$0.678402\pi$
$264$	0	0
$265$	13.5189	0.830462
$266$	0	0
$267$	−7.28990	−0.446134
$268$	0	0
$269$	12.1225	0.739122	0.369561	−	0.929206i	$-0.379508\pi$
$269$	12.1225	0.739122	0.369561	+	0.929206i	$0.379508\pi$
$270$	0	0
$271$	28.0124	1.70163	0.850817	−	0.525462i	$-0.176107\pi$
$271$	28.0124	1.70163	0.850817	+	0.525462i	$0.176107\pi$
$272$	0	0
$273$	−4.24868	−0.257142
$274$	0	0
$275$	12.8445	0.774552
$276$	0	0
$277$	18.5584	1.11506	0.557532	−	0.830155i	$-0.311748\pi$
$277$	18.5584	1.11506	0.557532	+	0.830155i	$0.311748\pi$
$278$	0	0
$279$	−31.3184	−1.87498
$280$	0	0
$281$	18.3575	1.09511	0.547557	−	0.836768i	$-0.315558\pi$
$281$	18.3575	1.09511	0.547557	+	0.836768i	$0.315558\pi$
$282$	0	0
$283$	4.85773	0.288762	0.144381	−	0.989522i	$-0.453881\pi$
$283$	4.85773	0.288762	0.144381	+	0.989522i	$0.453881\pi$
$284$	0	0
$285$	−21.9526	−1.30036
$286$	0	0
$287$	−3.86704	−0.228264
$288$	0	0
$289$	1.00000	0.0588235
$290$	0	0
$291$	3.12895	0.183422
$292$	0	0
$293$	−2.53080	−0.147851	−0.0739255	−	0.997264i	$-0.523553\pi$
$293$	−2.53080	−0.147851	−0.0739255	+	0.997264i	$0.523553\pi$
$294$	0	0
$295$	−1.06664	−0.0621024
$296$	0	0
$297$	42.3863	2.45951
$298$	0	0
$299$	2.38776	0.138088
$300$	0	0
$301$	3.65858	0.210877
$302$	0	0
$303$	32.2759	1.85420
$304$	0	0
$305$	−1.79670	−0.102879
$306$	0	0
$307$	−23.3203	−1.33096	−0.665478	−	0.746417i	$-0.731772\pi$
$307$	−23.3203	−1.33096	−0.665478	+	0.746417i	$0.731772\pi$
$308$	0	0
$309$	32.7733	1.86441
$310$	0	0
$311$	23.4312	1.32866	0.664330	−	0.747439i	$-0.268717\pi$
$311$	23.4312	1.32866	0.664330	+	0.747439i	$0.268717\pi$
$312$	0	0
$313$	14.0433	0.793771	0.396886	−	0.917868i	$-0.370091\pi$
$313$	14.0433	0.793771	0.396886	+	0.917868i	$0.370091\pi$
$314$	0	0
$315$	6.26498	0.352992
$316$	0	0
$317$	−13.8937	−0.780348	−0.390174	−	0.920741i	$-0.627585\pi$
$317$	−13.8937	−0.780348	−0.390174	+	0.920741i	$0.627585\pi$
$318$	0	0
$319$	−24.9557	−1.39725
$320$	0	0
$321$	51.7025	2.88575
$322$	0	0
$323$	−6.50007	−0.361674
$324$	0	0
$325$	6.19890	0.343853
$326$	0	0
$327$	35.2517	1.94942
$328$	0	0
$329$	−2.08710	−0.115065
$330$	0	0
$331$	−6.10774	−0.335712	−0.167856	−	0.985812i	$-0.553684\pi$
$331$	−6.10774	−0.335712	−0.167856	+	0.985812i	$0.553684\pi$
$332$	0	0
$333$	−25.0188	−1.37102
$334$	0	0
$335$	3.35508	0.183308
$336$	0	0
$337$	−19.5600	−1.06550	−0.532750	−	0.846272i	$-0.678842\pi$
$337$	−19.5600	−1.06550	−0.532750	+	0.846272i	$0.678842\pi$
$338$	0	0
$339$	15.8735	0.862129
$340$	0	0
$341$	14.8254	0.802840
$342$	0	0
$343$	11.1203	0.600442
$344$	0	0
$345$	−5.02445	−0.270507
$346$	0	0
$347$	5.47604	0.293969	0.146985	−	0.989139i	$-0.453043\pi$
$347$	5.47604	0.293969	0.146985	+	0.989139i	$0.453043\pi$
$348$	0	0
$349$	7.30751	0.391162	0.195581	−	0.980688i	$-0.437341\pi$
$349$	7.30751	0.391162	0.195581	+	0.980688i	$0.437341\pi$
$350$	0	0
$351$	20.4561	1.09187
$352$	0	0
$353$	−4.75986	−0.253342	−0.126671	−	0.991945i	$-0.540429\pi$
$353$	−4.75986	−0.253342	−0.126671	+	0.991945i	$0.540429\pi$
$354$	0	0
$355$	−2.06554	−0.109627
$356$	0	0
$357$	2.64717	0.140103
$358$	0	0
$359$	3.60563	0.190298	0.0951490	−	0.995463i	$-0.469667\pi$
$359$	3.60563	0.190298	0.0951490	+	0.995463i	$0.469667\pi$
$360$	0	0
$361$	23.2509	1.22373
$362$	0	0
$363$	−0.189438	−0.00994291
$364$	0	0
$365$	−2.01600	−0.105522
$366$	0	0
$367$	−7.97047	−0.416055	−0.208028	−	0.978123i	$-0.566704\pi$
$367$	−7.97047	−0.416055	−0.208028	+	0.978123i	$0.566704\pi$
$368$	0	0
$369$	32.4947	1.69161
$370$	0	0
$371$	10.5964	0.550135
$372$	0	0
$373$	27.7469	1.43668	0.718340	−	0.695692i	$-0.244902\pi$
$373$	27.7469	1.43668	0.718340	+	0.695692i	$0.244902\pi$
$374$	0	0
$375$	−29.9305	−1.54560
$376$	0	0
$377$	−12.0439	−0.620292
$378$	0	0
$379$	33.6008	1.72596	0.862978	−	0.505241i	$-0.168596\pi$
$379$	33.6008	1.72596	0.862978	+	0.505241i	$0.168596\pi$
$380$	0	0
$381$	24.1489	1.23719
$382$	0	0
$383$	−14.3797	−0.734768	−0.367384	−	0.930069i	$-0.619746\pi$
$383$	−14.3797	−0.734768	−0.367384	+	0.930069i	$0.619746\pi$
$384$	0	0
$385$	−2.96570	−0.151146
$386$	0	0
$387$	−30.7430	−1.56276
$388$	0	0
$389$	−28.7909	−1.45976	−0.729878	−	0.683578i	$-0.760423\pi$
$389$	−28.7909	−1.45976	−0.729878	+	0.683578i	$0.760423\pi$
$390$	0	0
$391$	−1.48772	−0.0752370
$392$	0	0
$393$	−9.94635	−0.501727
$394$	0	0
$395$	−0.235567	−0.0118526
$396$	0	0
$397$	21.5308	1.08060	0.540301	−	0.841472i	$-0.318311\pi$
$397$	21.5308	1.08060	0.540301	+	0.841472i	$0.318311\pi$
$398$	0	0
$399$	−17.2068	−0.861418
$400$	0	0
$401$	1.74784	0.0872828	0.0436414	−	0.999047i	$-0.486104\pi$
$401$	1.74784	0.0872828	0.0436414	+	0.999047i	$0.486104\pi$
$402$	0	0
$403$	7.15490	0.356411
$404$	0	0
$405$	−20.5642	−1.02184
$406$	0	0
$407$	11.8433	0.587052
$408$	0	0
$409$	34.2359	1.69286	0.846429	−	0.532502i	$-0.178748\pi$
$409$	34.2359	1.69286	0.846429	+	0.532502i	$0.178748\pi$
$410$	0	0
$411$	60.4857	2.98354
$412$	0	0
$413$	−0.836052	−0.0411394
$414$	0	0
$415$	6.31578	0.310029
$416$	0	0
$417$	45.1974	2.21333
$418$	0	0
$419$	14.4827	0.707524	0.353762	−	0.935335i	$-0.384902\pi$
$419$	14.4827	0.707524	0.353762	+	0.935335i	$0.384902\pi$
$420$	0	0
$421$	1.88624	0.0919299	0.0459650	−	0.998943i	$-0.485364\pi$
$421$	1.88624	0.0919299	0.0459650	+	0.998943i	$0.485364\pi$
$422$	0	0
$423$	17.5379	0.852720
$424$	0	0
$425$	−3.86227	−0.187348
$426$	0	0
$427$	−1.40828	−0.0681516
$428$	0	0
$429$	−16.9003	−0.815956
$430$	0	0
$431$	−23.5121	−1.13254	−0.566269	−	0.824221i	$-0.691614\pi$
$431$	−23.5121	−1.13254	−0.566269	+	0.824221i	$0.691614\pi$
$432$	0	0
$433$	35.6606	1.71374	0.856870	−	0.515533i	$-0.172406\pi$
$433$	35.6606	1.71374	0.856870	+	0.515533i	$0.172406\pi$
$434$	0	0
$435$	25.3433	1.21512
$436$	0	0
$437$	9.67025	0.462591
$438$	0	0
$439$	16.3621	0.780920	0.390460	−	0.920620i	$-0.372316\pi$
$439$	16.3621	0.780920	0.390460	+	0.920620i	$0.372316\pi$
$440$	0	0
$441$	−44.2668	−2.10794
$442$	0	0
$443$	14.6448	0.695794	0.347897	−	0.937533i	$-0.386896\pi$
$443$	14.6448	0.695794	0.347897	+	0.937533i	$0.386896\pi$
$444$	0	0
$445$	−2.45579	−0.116416
$446$	0	0
$447$	−21.6446	−1.02375
$448$	0	0
$449$	−4.17268	−0.196921	−0.0984604	−	0.995141i	$-0.531392\pi$
$449$	−4.17268	−0.196921	−0.0984604	+	0.995141i	$0.531392\pi$
$450$	0	0
$451$	−15.3822	−0.724322
$452$	0	0
$453$	−4.64068	−0.218038
$454$	0	0
$455$	−1.43128	−0.0670994
$456$	0	0
$457$	27.9156	1.30584	0.652919	−	0.757427i	$-0.273544\pi$
$457$	27.9156	1.30584	0.652919	+	0.757427i	$0.273544\pi$
$458$	0	0
$459$	−12.7454	−0.594902
$460$	0	0
$461$	8.29293	0.386240	0.193120	−	0.981175i	$-0.438139\pi$
$461$	8.29293	0.386240	0.193120	+	0.981175i	$0.438139\pi$
$462$	0	0
$463$	23.2037	1.07837	0.539183	−	0.842188i	$-0.318733\pi$
$463$	23.2037	1.07837	0.539183	+	0.842188i	$0.318733\pi$
$464$	0	0
$465$	−15.0557	−0.698191
$466$	0	0
$467$	−41.4026	−1.91588	−0.957942	−	0.286963i	$-0.907354\pi$
$467$	−41.4026	−1.91588	−0.957942	+	0.286963i	$0.907354\pi$
$468$	0	0
$469$	2.62977	0.121431
$470$	0	0
$471$	−12.6370	−0.582283
$472$	0	0
$473$	14.5530	0.669149
$474$	0	0
$475$	25.1050	1.15190
$476$	0	0
$477$	−89.0411	−4.07691
$478$	0	0
$479$	14.5816	0.666251	0.333125	−	0.942883i	$-0.391897\pi$
$479$	14.5816	0.666251	0.333125	+	0.942883i	$0.391897\pi$
$480$	0	0
$481$	5.71572	0.260614
$482$	0	0
$483$	−3.93824	−0.179196
$484$	0	0
$485$	1.05407	0.0478627
$486$	0	0
$487$	−5.91329	−0.267957	−0.133978	−	0.990984i	$-0.542775\pi$
$487$	−5.91329	−0.267957	−0.133978	+	0.990984i	$0.542775\pi$
$488$	0	0
$489$	−20.1731	−0.912261
$490$	0	0
$491$	−21.4335	−0.967279	−0.483640	−	0.875267i	$-0.660685\pi$
$491$	−21.4335	−0.967279	−0.483640	+	0.875267i	$0.660685\pi$
$492$	0	0
$493$	7.50404	0.337965
$494$	0	0
$495$	24.9207	1.12010
$496$	0	0
$497$	−1.61900	−0.0726221
$498$	0	0
$499$	28.6354	1.28190	0.640949	−	0.767583i	$-0.278541\pi$
$499$	28.6354	1.28190	0.640949	+	0.767583i	$0.278541\pi$
$500$	0	0
$501$	−39.1567	−1.74939
$502$	0	0
$503$	9.36809	0.417703	0.208851	−	0.977947i	$-0.433028\pi$
$503$	9.36809	0.417703	0.208851	+	0.977947i	$0.433028\pi$
$504$	0	0
$505$	10.8730	0.483840
$506$	0	0
$507$	33.0054	1.46582
$508$	0	0
$509$	10.7481	0.476400	0.238200	−	0.971216i	$-0.423443\pi$
$509$	10.7481	0.476400	0.238200	+	0.971216i	$0.423443\pi$
$510$	0	0
$511$	−1.58017	−0.0699027
$512$	0	0
$513$	82.8457	3.65773
$514$	0	0
$515$	11.0405	0.486503
$516$	0	0
$517$	−8.30202	−0.365123
$518$	0	0
$519$	12.4420	0.546142
$520$	0	0
$521$	−2.12230	−0.0929796	−0.0464898	−	0.998919i	$-0.514803\pi$
$521$	−2.12230	−0.0929796	−0.0464898	+	0.998919i	$0.514803\pi$
$522$	0	0
$523$	28.3451	1.23944	0.619722	−	0.784821i	$-0.287245\pi$
$523$	28.3451	1.23944	0.619722	+	0.784821i	$0.287245\pi$
$524$	0	0
$525$	−10.2241	−0.446217
$526$	0	0
$527$	−4.45792	−0.194190
$528$	0	0
$529$	−20.7867	−0.903770
$530$	0	0
$531$	7.02534	0.304874
$532$	0	0
$533$	−7.42364	−0.321554
$534$	0	0
$535$	17.4173	0.753017
$536$	0	0
$537$	−15.0092	−0.647695
$538$	0	0
$539$	20.9549	0.902590
$540$	0	0
$541$	−32.1215	−1.38101	−0.690505	−	0.723327i	$-0.742612\pi$
$541$	−32.1215	−1.38101	−0.690505	+	0.723327i	$0.742612\pi$
$542$	0	0
$543$	50.7965	2.17989
$544$	0	0
$545$	11.8755	0.508689
$546$	0	0
$547$	−7.50387	−0.320842	−0.160421	−	0.987049i	$-0.551285\pi$
$547$	−7.50387	−0.320842	−0.160421	+	0.987049i	$0.551285\pi$
$548$	0	0
$549$	11.8338	0.505054
$550$	0	0
$551$	−48.7768	−2.07796
$552$	0	0
$553$	−0.184641	−0.00785172
$554$	0	0
$555$	−12.0273	−0.510530
$556$	0	0
$557$	14.9053	0.631558	0.315779	−	0.948833i	$-0.397734\pi$
$557$	14.9053	0.631558	0.315779	+	0.948833i	$0.397734\pi$
$558$	0	0
$559$	7.02346	0.297061
$560$	0	0
$561$	10.5299	0.444572
$562$	0	0
$563$	−6.92273	−0.291758	−0.145879	−	0.989302i	$-0.546601\pi$
$563$	−6.92273	−0.291758	−0.145879	+	0.989302i	$0.546601\pi$
$564$	0	0
$565$	5.34739	0.224966
$566$	0	0
$567$	−16.1185	−0.676915
$568$	0	0
$569$	36.1854	1.51697	0.758486	−	0.651690i	$-0.225940\pi$
$569$	36.1854	1.51697	0.758486	+	0.651690i	$0.225940\pi$
$570$	0	0
$571$	0.752604	0.0314955	0.0157478	−	0.999876i	$-0.494987\pi$
$571$	0.752604	0.0314955	0.0157478	+	0.999876i	$0.494987\pi$
$572$	0	0
$573$	15.1284	0.632000
$574$	0	0
$575$	5.74596	0.239623
$576$	0	0
$577$	−13.3793	−0.556986	−0.278493	−	0.960438i	$-0.589835\pi$
$577$	−13.3793	−0.556986	−0.278493	+	0.960438i	$0.589835\pi$
$578$	0	0
$579$	−19.3729	−0.805108
$580$	0	0
$581$	4.95040	0.205377
$582$	0	0
$583$	42.1500	1.74567
$584$	0	0
$585$	12.0270	0.497256
$586$	0	0
$587$	−6.40584	−0.264397	−0.132199	−	0.991223i	$-0.542204\pi$
$587$	−6.40584	−0.264397	−0.132199	+	0.991223i	$0.542204\pi$
$588$	0	0
$589$	28.9768	1.19397
$590$	0	0
$591$	−6.67448	−0.274551
$592$	0	0
$593$	−40.1557	−1.64900	−0.824498	−	0.565865i	$-0.808542\pi$
$593$	−40.1557	−1.64900	−0.824498	+	0.565865i	$0.808542\pi$
$594$	0	0
$595$	0.891769	0.0365590
$596$	0	0
$597$	−1.23820	−0.0506760
$598$	0	0
$599$	−11.2081	−0.457951	−0.228975	−	0.973432i	$-0.573538\pi$
$599$	−11.2081	−0.457951	−0.228975	+	0.973432i	$0.573538\pi$
$600$	0	0
$601$	15.3077	0.624414	0.312207	−	0.950014i	$-0.398932\pi$
$601$	15.3077	0.624414	0.312207	+	0.950014i	$0.398932\pi$
$602$	0	0
$603$	−22.0979	−0.899896
$604$	0	0
$605$	−0.0638171	−0.00259453
$606$	0	0
$607$	27.3494	1.11008	0.555039	−	0.831824i	$-0.312703\pi$
$607$	27.3494	1.11008	0.555039	+	0.831824i	$0.312703\pi$
$608$	0	0
$609$	19.8645	0.804950
$610$	0	0
$611$	−4.00665	−0.162092
$612$	0	0
$613$	−39.8293	−1.60869	−0.804345	−	0.594162i	$-0.797484\pi$
$613$	−39.8293	−1.60869	−0.804345	+	0.594162i	$0.797484\pi$
$614$	0	0
$615$	15.6212	0.629908
$616$	0	0
$617$	14.1539	0.569814	0.284907	−	0.958555i	$-0.408037\pi$
$617$	14.1539	0.569814	0.284907	+	0.958555i	$0.408037\pi$
$618$	0	0
$619$	30.6550	1.23213	0.616064	−	0.787696i	$-0.288726\pi$
$619$	30.6550	1.23213	0.616064	+	0.787696i	$0.288726\pi$
$620$	0	0
$621$	18.9615	0.760897
$622$	0	0
$623$	−1.92489	−0.0771189
$624$	0	0
$625$	9.22849	0.369140
$626$	0	0
$627$	−68.4450	−2.73343
$628$	0	0
$629$	−3.56122	−0.141995
$630$	0	0
$631$	−24.3186	−0.968107	−0.484054	−	0.875038i	$-0.660836\pi$
$631$	−24.3186	−0.968107	−0.484054	+	0.875038i	$0.660836\pi$
$632$	0	0
$633$	69.3559	2.75665
$634$	0	0
$635$	8.13519	0.322835
$636$	0	0
$637$	10.1131	0.400694
$638$	0	0
$639$	13.6044	0.538184
$640$	0	0
$641$	15.7055	0.620331	0.310166	−	0.950683i	$-0.399616\pi$
$641$	15.7055	0.620331	0.310166	+	0.950683i	$0.399616\pi$
$642$	0	0
$643$	2.85618	0.112637	0.0563184	−	0.998413i	$-0.482064\pi$
$643$	2.85618	0.112637	0.0563184	+	0.998413i	$0.482064\pi$
$644$	0	0
$645$	−14.7791	−0.581927
$646$	0	0
$647$	−7.03922	−0.276740	−0.138370	−	0.990381i	$-0.544186\pi$
$647$	−7.03922	−0.276740	−0.138370	+	0.990381i	$0.544186\pi$
$648$	0	0
$649$	−3.32563	−0.130542
$650$	0	0
$651$	−11.8009	−0.462513
$652$	0	0
$653$	−1.51121	−0.0591381	−0.0295691	−	0.999563i	$-0.509413\pi$
$653$	−1.51121	−0.0591381	−0.0295691	+	0.999563i	$0.509413\pi$
$654$	0	0
$655$	−3.35069	−0.130922
$656$	0	0
$657$	13.2782	0.518031
$658$	0	0
$659$	23.7571	0.925444	0.462722	−	0.886503i	$-0.346873\pi$
$659$	23.7571	0.925444	0.462722	+	0.886503i	$0.346873\pi$
$660$	0	0
$661$	−24.8204	−0.965400	−0.482700	−	0.875786i	$-0.660344\pi$
$661$	−24.8204	−0.965400	−0.482700	+	0.875786i	$0.660344\pi$
$662$	0	0
$663$	5.08184	0.197362
$664$	0	0
$665$	−5.79656	−0.224781
$666$	0	0
$667$	−11.1639	−0.432267
$668$	0	0
$669$	−23.9631	−0.926466
$670$	0	0
$671$	−5.60185	−0.216257
$672$	0	0
$673$	19.4374	0.749255	0.374628	−	0.927175i	$-0.377771\pi$
$673$	19.4374	0.749255	0.374628	+	0.927175i	$0.377771\pi$
$674$	0	0
$675$	49.2260	1.89471
$676$	0	0
$677$	−22.0352	−0.846883	−0.423441	−	0.905923i	$-0.639178\pi$
$677$	−22.0352	−0.846883	−0.423441	+	0.905923i	$0.639178\pi$
$678$	0	0
$679$	0.826193	0.0317064
$680$	0	0
$681$	10.9690	0.420334
$682$	0	0
$683$	−12.7978	−0.489693	−0.244846	−	0.969562i	$-0.578738\pi$
$683$	−12.7978	−0.489693	−0.244846	+	0.969562i	$0.578738\pi$
$684$	0	0
$685$	20.3762	0.778534
$686$	0	0
$687$	28.7607	1.09729
$688$	0	0
$689$	20.3421	0.774971
$690$	0	0
$691$	23.2609	0.884887	0.442444	−	0.896796i	$-0.354112\pi$
$691$	23.2609	0.884887	0.442444	+	0.896796i	$0.354112\pi$
$692$	0	0
$693$	19.5333	0.742007
$694$	0	0
$695$	15.2259	0.577552
$696$	0	0
$697$	4.62536	0.175198
$698$	0	0
$699$	31.5045	1.19161
$700$	0	0
$701$	34.8105	1.31477	0.657387	−	0.753553i	$-0.271662\pi$
$701$	34.8105	1.31477	0.657387	+	0.753553i	$0.271662\pi$
$702$	0	0
$703$	23.1482	0.873051
$704$	0	0
$705$	8.43099	0.317529
$706$	0	0
$707$	8.52239	0.320518
$708$	0	0
$709$	0.555903	0.0208774	0.0104387	−	0.999946i	$-0.496677\pi$
$709$	0.555903	0.0208774	0.0104387	+	0.999946i	$0.496677\pi$
$710$	0	0
$711$	1.55154	0.0581871
$712$	0	0
$713$	6.63211	0.248375
$714$	0	0
$715$	−5.69331	−0.212918
$716$	0	0
$717$	63.5280	2.37250
$718$	0	0
$719$	47.8661	1.78511	0.892553	−	0.450943i	$-0.148912\pi$
$719$	47.8661	1.78511	0.892553	+	0.450943i	$0.148912\pi$
$720$	0	0
$721$	8.65373	0.322282
$722$	0	0
$723$	0.633201	0.0235490
$724$	0	0
$725$	−28.9826	−1.07639
$726$	0	0
$727$	7.57824	0.281062	0.140531	−	0.990076i	$-0.455119\pi$
$727$	7.57824	0.281062	0.140531	+	0.990076i	$0.455119\pi$
$728$	0	0
$729$	14.3779	0.532514
$730$	0	0
$731$	−4.37602	−0.161853
$732$	0	0
$733$	24.0102	0.886835	0.443418	−	0.896315i	$-0.353766\pi$
$733$	24.0102	0.886835	0.443418	+	0.896315i	$0.353766\pi$
$734$	0	0
$735$	−21.2804	−0.784939
$736$	0	0
$737$	10.4606	0.385322
$738$	0	0
$739$	−38.5158	−1.41683	−0.708414	−	0.705797i	$-0.750589\pi$
$739$	−38.5158	−1.41683	−0.708414	+	0.705797i	$0.750589\pi$
$740$	0	0
$741$	−33.0323	−1.21347
$742$	0	0
$743$	6.19895	0.227417	0.113709	−	0.993514i	$-0.463727\pi$
$743$	6.19895	0.227417	0.113709	+	0.993514i	$0.463727\pi$
$744$	0	0
$745$	−7.29155	−0.267142
$746$	0	0
$747$	−41.5982	−1.52200
$748$	0	0
$749$	13.6520	0.498832
$750$	0	0
$751$	−11.1844	−0.408124	−0.204062	−	0.978958i	$-0.565414\pi$
$751$	−11.1844	−0.408124	−0.204062	+	0.978958i	$0.565414\pi$
$752$	0	0
$753$	15.8886	0.579012
$754$	0	0
$755$	−1.56333	−0.0568955
$756$	0	0
$757$	−25.8192	−0.938414	−0.469207	−	0.883088i	$-0.655460\pi$
$757$	−25.8192	−0.938414	−0.469207	+	0.883088i	$0.655460\pi$
$758$	0	0
$759$	−15.6655	−0.568621
$760$	0	0
$761$	−30.8344	−1.11774	−0.558872	−	0.829254i	$-0.688766\pi$
$761$	−30.8344	−1.11774	−0.558872	+	0.829254i	$0.688766\pi$
$762$	0	0
$763$	9.30816	0.336978
$764$	0	0
$765$	−7.49353	−0.270929
$766$	0	0
$767$	−1.60499	−0.0579527
$768$	0	0
$769$	−11.5358	−0.415993	−0.207996	−	0.978130i	$-0.566694\pi$
$769$	−11.5358	−0.415993	−0.207996	+	0.978130i	$0.566694\pi$
$770$	0	0
$771$	29.9940	1.08021
$772$	0	0
$773$	33.1702	1.19305	0.596525	−	0.802595i	$-0.296548\pi$
$773$	33.1702	1.19305	0.596525	+	0.802595i	$0.296548\pi$
$774$	0	0
$775$	17.2177	0.618477
$776$	0	0
$777$	−9.42718	−0.338198
$778$	0	0
$779$	−30.0652	−1.07720
$780$	0	0
$781$	−6.44003	−0.230442
$782$	0	0
$783$	−95.6416	−3.41795
$784$	0	0
$785$	−4.25711	−0.151943
$786$	0	0
$787$	−23.7659	−0.847163	−0.423581	−	0.905858i	$-0.639227\pi$
$787$	−23.7659	−0.847163	−0.423581	+	0.905858i	$0.639227\pi$
$788$	0	0
$789$	54.5917	1.94351
$790$	0	0
$791$	4.19137	0.149028
$792$	0	0
$793$	−2.70351	−0.0960046
$794$	0	0
$795$	−42.8048	−1.51813
$796$	0	0
$797$	−45.2035	−1.60119	−0.800595	−	0.599205i	$-0.795483\pi$
$797$	−45.2035	−1.60119	−0.800595	+	0.599205i	$0.795483\pi$
$798$	0	0
$799$	2.49637	0.0883154
$800$	0	0
$801$	16.1748	0.571509
$802$	0	0
$803$	−6.28559	−0.221813
$804$	0	0
$805$	−1.32670	−0.0467600
$806$	0	0
$807$	−38.3833	−1.35115
$808$	0	0
$809$	−21.0785	−0.741082	−0.370541	−	0.928816i	$-0.620828\pi$
$809$	−21.0785	−0.741082	−0.370541	+	0.928816i	$0.620828\pi$
$810$	0	0
$811$	36.0091	1.26445	0.632225	−	0.774785i	$-0.282142\pi$
$811$	36.0091	1.26445	0.632225	+	0.774785i	$0.282142\pi$
$812$	0	0
$813$	−88.6953	−3.11068
$814$	0	0
$815$	−6.79585	−0.238048
$816$	0	0
$817$	28.4444	0.995145
$818$	0	0
$819$	9.42696	0.329405
$820$	0	0
$821$	−24.4075	−0.851829	−0.425915	−	0.904763i	$-0.640048\pi$
$821$	−24.4075	−0.851829	−0.425915	+	0.904763i	$0.640048\pi$
$822$	0	0
$823$	−26.7916	−0.933896	−0.466948	−	0.884285i	$-0.654646\pi$
$823$	−26.7916	−0.933896	−0.466948	+	0.884285i	$0.654646\pi$
$824$	0	0
$825$	−40.6693	−1.41592
$826$	0	0
$827$	−21.2990	−0.740640	−0.370320	−	0.928904i	$-0.620752\pi$
$827$	−21.2990	−0.740640	−0.370320	+	0.928904i	$0.620752\pi$
$828$	0	0
$829$	32.3120	1.12224	0.561120	−	0.827734i	$-0.310370\pi$
$829$	32.3120	1.12224	0.561120	+	0.827734i	$0.310370\pi$
$830$	0	0
$831$	−58.7610	−2.03840
$832$	0	0
$833$	−6.30102	−0.218317
$834$	0	0
$835$	−13.1909	−0.456491
$836$	0	0
$837$	56.8177	1.96391
$838$	0	0
$839$	−9.79718	−0.338236	−0.169118	−	0.985596i	$-0.554092\pi$
$839$	−9.79718	−0.338236	−0.169118	+	0.985596i	$0.554092\pi$
$840$	0	0
$841$	27.3106	0.941745
$842$	0	0
$843$	−58.1249	−2.00193
$844$	0	0
$845$	11.1187	0.382495
$846$	0	0
$847$	−0.0500208	−0.00171873
$848$	0	0
$849$	−15.3809	−0.527872
$850$	0	0
$851$	5.29809	0.181616
$852$	0	0
$853$	53.3904	1.82805	0.914025	−	0.405657i	$-0.132957\pi$
$853$	53.3904	1.82805	0.914025	+	0.405657i	$0.132957\pi$
$854$	0	0
$855$	48.7085	1.66579
$856$	0	0
$857$	−4.57106	−0.156144	−0.0780721	−	0.996948i	$-0.524876\pi$
$857$	−4.57106	−0.156144	−0.0780721	+	0.996948i	$0.524876\pi$
$858$	0	0
$859$	21.5415	0.734988	0.367494	−	0.930026i	$-0.380216\pi$
$859$	21.5415	0.734988	0.367494	+	0.930026i	$0.380216\pi$
$860$	0	0
$861$	12.2441	0.417279
$862$	0	0
$863$	−20.0761	−0.683397	−0.341699	−	0.939810i	$-0.611002\pi$
$863$	−20.0761	−0.683397	−0.341699	+	0.939810i	$0.611002\pi$
$864$	0	0
$865$	4.19140	0.142512
$866$	0	0
$867$	−3.16628	−0.107533
$868$	0	0
$869$	−0.734461	−0.0249149
$870$	0	0
$871$	5.04842	0.171059
$872$	0	0
$873$	−6.94250	−0.234968
$874$	0	0
$875$	−7.90310	−0.267174
$876$	0	0
$877$	26.3028	0.888184	0.444092	−	0.895981i	$-0.353526\pi$
$877$	26.3028	0.888184	0.444092	+	0.895981i	$0.353526\pi$
$878$	0	0
$879$	8.01323	0.270280
$880$	0	0
$881$	21.0844	0.710350	0.355175	−	0.934800i	$-0.384421\pi$
$881$	21.0844	0.710350	0.355175	+	0.934800i	$0.384421\pi$
$882$	0	0
$883$	−12.7277	−0.428322	−0.214161	−	0.976798i	$-0.568702\pi$
$883$	−12.7277	−0.428322	−0.214161	+	0.976798i	$0.568702\pi$
$884$	0	0
$885$	3.37729	0.113527
$886$	0	0
$887$	−28.1459	−0.945048	−0.472524	−	0.881318i	$-0.656657\pi$
$887$	−28.1459	−0.945048	−0.472524	+	0.881318i	$0.656657\pi$
$888$	0	0
$889$	6.37649	0.213860
$890$	0	0
$891$	−64.1160	−2.14797
$892$	0	0
$893$	−16.2266	−0.543003
$894$	0	0
$895$	−5.05624	−0.169011
$896$	0	0
$897$	−7.56033	−0.252432
$898$	0	0
$899$	−33.4524	−1.11570
$900$	0	0
$901$	−12.6743	−0.422241
$902$	0	0
$903$	−11.5841	−0.385494
$904$	0	0
$905$	17.1121	0.568826
$906$	0	0
$907$	−44.6222	−1.48166	−0.740828	−	0.671694i	$-0.765567\pi$
$907$	−44.6222	−1.48166	−0.740828	+	0.671694i	$0.765567\pi$
$908$	0	0
$909$	−71.6136	−2.37527
$910$	0	0
$911$	13.4366	0.445175	0.222588	−	0.974913i	$-0.428550\pi$
$911$	13.4366	0.445175	0.222588	+	0.974913i	$0.428550\pi$
$912$	0	0
$913$	19.6916	0.651698
$914$	0	0
$915$	5.68887	0.188068
$916$	0	0
$917$	−2.62632	−0.0867287
$918$	0	0
$919$	37.5456	1.23852	0.619258	−	0.785188i	$-0.287433\pi$
$919$	37.5456	1.23852	0.619258	+	0.785188i	$0.287433\pi$
$920$	0	0
$921$	73.8385	2.43306
$922$	0	0
$923$	−3.10803	−0.102302
$924$	0	0
$925$	13.7544	0.452242
$926$	0	0
$927$	−72.7172	−2.38835
$928$	0	0
$929$	−46.1061	−1.51269	−0.756347	−	0.654171i	$-0.773018\pi$
$929$	−46.1061	−1.51269	−0.756347	+	0.654171i	$0.773018\pi$
$930$	0	0
$931$	40.9570	1.34231
$932$	0	0
$933$	−74.1897	−2.42886
$934$	0	0
$935$	3.54726	0.116008
$936$	0	0
$937$	−35.8085	−1.16981	−0.584906	−	0.811101i	$-0.698869\pi$
$937$	−35.8085	−1.16981	−0.584906	+	0.811101i	$0.698869\pi$
$938$	0	0
$939$	−44.4649	−1.45106
$940$	0	0
$941$	31.0081	1.01084	0.505418	−	0.862875i	$-0.331338\pi$
$941$	31.0081	1.01084	0.505418	+	0.862875i	$0.331338\pi$
$942$	0	0
$943$	−6.88122	−0.224083
$944$	0	0
$945$	−11.3659	−0.369733
$946$	0	0
$947$	−2.00454	−0.0651389	−0.0325695	−	0.999469i	$-0.510369\pi$
$947$	−2.00454	−0.0651389	−0.0325695	+	0.999469i	$0.510369\pi$
$948$	0	0
$949$	−3.03349	−0.0984714
$950$	0	0
$951$	43.9914	1.42652
$952$	0	0
$953$	−0.522488	−0.0169251	−0.00846253	−	0.999964i	$-0.502694\pi$
$953$	−0.522488	−0.0169251	−0.00846253	+	0.999964i	$0.502694\pi$
$954$	0	0
$955$	5.09641	0.164916
$956$	0	0
$957$	79.0167	2.55425
$958$	0	0
$959$	15.9712	0.515736
$960$	0	0
$961$	−11.1270	−0.358935
$962$	0	0
$963$	−114.717	−3.69672
$964$	0	0
$965$	−6.52625	−0.210087
$966$	0	0
$967$	−11.5992	−0.373004	−0.186502	−	0.982455i	$-0.559715\pi$
$967$	−11.5992	−0.373004	−0.186502	+	0.982455i	$0.559715\pi$
$968$	0	0
$969$	20.5810	0.661158
$970$	0	0
$971$	31.2396	1.00253	0.501263	−	0.865295i	$-0.332869\pi$
$971$	31.2396	1.00253	0.501263	+	0.865295i	$0.332869\pi$
$972$	0	0
$973$	11.9343	0.382596
$974$	0	0
$975$	−19.6274	−0.628581
$976$	0	0
$977$	42.4397	1.35777	0.678883	−	0.734247i	$-0.262464\pi$
$977$	42.4397	1.35777	0.678883	+	0.734247i	$0.262464\pi$
$978$	0	0
$979$	−7.65678	−0.244712
$980$	0	0
$981$	−78.2164	−2.49726
$982$	0	0
$983$	−43.9347	−1.40130	−0.700650	−	0.713505i	$-0.747107\pi$
$983$	−43.9347	−1.40130	−0.700650	+	0.713505i	$0.747107\pi$
$984$	0	0
$985$	−2.24847	−0.0716423
$986$	0	0
$987$	6.60834	0.210346
$988$	0	0
$989$	6.51027	0.207015
$990$	0	0
$991$	−14.4331	−0.458483	−0.229242	−	0.973370i	$-0.573625\pi$
$991$	−14.4331	−0.458483	−0.229242	+	0.973370i	$0.573625\pi$
$992$	0	0
$993$	19.3388	0.613699
$994$	0	0
$995$	−0.417118	−0.0132235
$996$	0	0
$997$	−40.9395	−1.29657	−0.648284	−	0.761399i	$-0.724513\pi$
$997$	−40.9395	−1.29657	−0.648284	+	0.761399i	$0.724513\pi$
$998$	0	0
$999$	45.3890	1.43605

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8024.2.a.bb.1.1	✓	32

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8024.2.a.bb.1.1	✓	32	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:	\( N \)	\(=\)	\( 8024 = 2^{3} \cdot 17 \cdot 59 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8024.a (trivial)

\(f(q)\)	\(=\)	\(q-3.16628 q^{3} -1.06664 q^{5} -0.836052 q^{7} +7.02534 q^{9} +O(q^{10})\)	\(q-3.16628 q^{3} -1.06664 q^{5} -0.836052 q^{7} +7.02534 q^{9} -3.32563 q^{11} -1.60499 q^{13} +3.37729 q^{15} +1.00000 q^{17} -6.50007 q^{19} +2.64717 q^{21} -1.48772 q^{23} -3.86227 q^{25} -12.7454 q^{27} +7.50404 q^{29} -4.45792 q^{31} +10.5299 q^{33} +0.891769 q^{35} -3.56122 q^{37} +5.08184 q^{39} +4.62536 q^{41} -4.37602 q^{43} -7.49353 q^{45} +2.49637 q^{47} -6.30102 q^{49} -3.16628 q^{51} -12.6743 q^{53} +3.54726 q^{55} +20.5810 q^{57} +1.00000 q^{59} +1.68445 q^{61} -5.87355 q^{63} +1.71195 q^{65} -3.14546 q^{67} +4.71053 q^{69} +1.93648 q^{71} +1.89004 q^{73} +12.2290 q^{75} +2.78040 q^{77} +0.220848 q^{79} +19.2794 q^{81} -5.92117 q^{83} -1.06664 q^{85} -23.7599 q^{87} +2.30235 q^{89} +1.34185 q^{91} +14.1150 q^{93} +6.93326 q^{95} -0.988209 q^{97} -23.3637 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q + 8 q^{5} - 3 q^{7} + 40 q^{9}+O(q^{10}) \) 32 * q + 8 * q^5 - 3 * q^7 + 40 * q^9	\( 32 q + 8 q^{5} - 3 q^{7} + 40 q^{9} + 3 q^{11} + 13 q^{13} + 4 q^{15} + 32 q^{17} + 14 q^{19} - 7 q^{21} + 7 q^{23} + 38 q^{25} + 9 q^{27} + 17 q^{29} + 15 q^{31} + 18 q^{33} + 6 q^{35} + 21 q^{37} + 16 q^{39} + 49 q^{41} - 7 q^{43} + 14 q^{45} - 25 q^{47} + 37 q^{49} + 12 q^{53} + 15 q^{55} + 45 q^{57} + 32 q^{59} + 5 q^{61} - 12 q^{63} + 39 q^{65} + 12 q^{69} - 13 q^{71} + 70 q^{73} - 47 q^{75} - 10 q^{77} - q^{79} + 84 q^{81} - 17 q^{83} + 8 q^{85} + 20 q^{87} + 42 q^{89} + 36 q^{91} + 2 q^{93} - q^{95} + 58 q^{97} + 7 q^{99}+O(q^{100}) \) 32 * q + 8 * q^5 - 3 * q^7 + 40 * q^9 + 3 * q^11 + 13 * q^13 + 4 * q^15 + 32 * q^17 + 14 * q^19 - 7 * q^21 + 7 * q^23 + 38 * q^25 + 9 * q^27 + 17 * q^29 + 15 * q^31 + 18 * q^33 + 6 * q^35 + 21 * q^37 + 16 * q^39 + 49 * q^41 - 7 * q^43 + 14 * q^45 - 25 * q^47 + 37 * q^49 + 12 * q^53 + 15 * q^55 + 45 * q^57 + 32 * q^59 + 5 * q^61 - 12 * q^63 + 39 * q^65 + 12 * q^69 - 13 * q^71 + 70 * q^73 - 47 * q^75 - 10 * q^77 - q^79 + 84 * q^81 - 17 * q^83 + 8 * q^85 + 20 * q^87 + 42 * q^89 + 36 * q^91 + 2 * q^93 - q^95 + 58 * q^97 + 7 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more