LMFDB - Embedded newform 6048.2.c.e.3025.18

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6048,2,Mod(3025,6048)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6048, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1, 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("6048.3025");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6048 = 2^{5} \cdot 3^{3} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6048.c (of order $2$, degree $1$, not minimal)

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:	no
Analytic conductor:	$48.2935231425$
Analytic rank:	$0$
Dimension:	$20$
Coefficient field:	$\mathbb{Q}[x]/(x^{20} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{20} + x^{18} + 4x^{16} + 8x^{12} + 4x^{10} + 32x^{8} + 256x^{4} + 256x^{2} + 1024 $ x^20 + x^18 + 4x^16 + 8x^12 + 4x^10 + 32x^8 + 256x^4 + 256x^2 + 1024
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{18}\cdot 3^{8} $
Twist minimal:	no (minimal twist has level 1512)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			3025.18
Root			$1.19566 + 0.755240i$ of defining polynomial
Character	$\chi$	$=$	6048.3025
Dual form			6048.2.c.e.3025.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+3.16969i q^{5} +1.00000 q^{7} +O(q^{10})$	$q+3.16969i q^{5} +1.00000 q^{7} -5.44696i q^{11} +3.61205i q^{13} +3.27628 q^{17} -3.20627i q^{19} -0.673274 q^{23} -5.04692 q^{25} -2.85127i q^{29} -3.71845 q^{31} +3.16969i q^{35} -11.9988i q^{37} -7.44602 q^{41} -12.5741i q^{43} +4.06341 q^{47} +1.00000 q^{49} -0.291601i q^{53} +17.2652 q^{55} +0.0209587i q^{59} -5.34034i q^{61} -11.4491 q^{65} +6.20714i q^{67} -15.5050 q^{71} +1.35371 q^{73} -5.44696i q^{77} +11.0846 q^{79} -13.6442i q^{83} +10.3848i q^{85} -5.93965 q^{89} +3.61205i q^{91} +10.1629 q^{95} +9.60073 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q + 20 q^{7}+O(q^{10}) $ 20 * q + 20 * q^7	$ 20 q + 20 q^{7} - 28 q^{25} - 36 q^{31} + 20 q^{49} - 48 q^{55} - 64 q^{79} + 56 q^{97}+O(q^{100}) $ 20 * q + 20 * q^7 - 28 * q^25 - 36 * q^31 + 20 * q^49 - 48 * q^55 - 64 * q^79 + 56 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/6048\mathbb{Z}\right)^\times$.

$n$	$2593$	$3781$	$3809$	$4159$
$\chi(n)$	$1$	$-1$	$1$	$1$

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
$5$	3.16969i	1.41753i	0.705446	+	0.708764i	$0.250747\pi$
$5$	3.16969i	1.41753i	−0.705446	+	0.708764i	$0.749253\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0	0
$9$	0	0
$10$	0	0
$11$	− 5.44696i	− 1.64232i	−0.570699	−	0.821160i	$-0.693327\pi$
$11$	− 5.44696i	− 1.64232i	0.570699	−	0.821160i	$-0.306673\pi$
$12$	0	0
$13$	3.61205i	1.00180i	0.865504	+	0.500902i	$0.166998\pi$
$13$	3.61205i	1.00180i	−0.865504	+	0.500902i	$0.833002\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	3.27628	0.794616	0.397308	−	0.917685i	$-0.369945\pi$
$17$	3.27628	0.794616	0.397308	+	0.917685i	$0.369945\pi$
$18$	0	0
$19$	− 3.20627i	− 0.735569i	−0.929911	−	0.367785i	$-0.880116\pi$
$19$	− 3.20627i	− 0.735569i	0.929911	−	0.367785i	$-0.119884\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−0.673274	−0.140387	−0.0701936	−	0.997533i	$-0.522362\pi$
$23$	−0.673274	−0.140387	−0.0701936	+	0.997533i	$0.522362\pi$
$24$	0	0
$25$	−5.04692	−1.00938
$26$	0	0
$27$	0	0
$28$	0	0
$29$	− 2.85127i	− 0.529468i	−0.964322	−	0.264734i	$-0.914716\pi$
$29$	− 2.85127i	− 0.529468i	0.964322	−	0.264734i	$-0.0852841\pi$
$30$	0	0
$31$	−3.71845	−0.667854	−0.333927	−	0.942599i	$-0.608374\pi$
$31$	−3.71845	−0.667854	−0.333927	+	0.942599i	$0.608374\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	3.16969i	0.535775i
$36$	0	0
$37$	− 11.9988i	− 1.97260i	−0.164971	−	0.986298i	$-0.552753\pi$
$37$	− 11.9988i	− 1.97260i	0.164971	−	0.986298i	$-0.447247\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−7.44602	−1.16287	−0.581436	−	0.813592i	$-0.697509\pi$
$41$	−7.44602	−1.16287	−0.581436	+	0.813592i	$0.697509\pi$
$42$	0	0
$43$	− 12.5741i	− 1.91753i	−0.284195	−	0.958766i	$-0.591726\pi$
$43$	− 12.5741i	− 1.91753i	0.284195	−	0.958766i	$-0.408274\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	4.06341	0.592710	0.296355	−	0.955078i	$-0.404229\pi$
$47$	4.06341	0.592710	0.296355	+	0.955078i	$0.404229\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	0	0
$52$	0	0
$53$	− 0.291601i	− 0.0400545i	−0.999799	−	0.0200272i	$-0.993625\pi$
$53$	− 0.291601i	− 0.0400545i	0.999799	−	0.0200272i	$-0.00637529\pi$
$54$	0	0
$55$	17.2652	2.32803
$56$	0	0
$57$	0	0
$58$	0	0
$59$	0.0209587i	0.00272859i	0.999999	+	0.00136430i	$0.000434269\pi$
$59$	0.0209587i	0.00272859i	−0.999999	+	0.00136430i	$0.999566\pi$
$60$	0	0
$61$	− 5.34034i	− 0.683760i	−0.939744	−	0.341880i	$-0.888936\pi$
$61$	− 5.34034i	− 0.683760i	0.939744	−	0.341880i	$-0.111064\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−11.4491	−1.42008
$66$	0	0
$67$	6.20714i	0.758323i	0.925331	+	0.379161i	$0.123787\pi$
$67$	6.20714i	0.758323i	−0.925331	+	0.379161i	$0.876213\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−15.5050	−1.84010	−0.920050	−	0.391801i	$-0.871852\pi$
$71$	−15.5050	−1.84010	−0.920050	+	0.391801i	$0.871852\pi$
$72$	0	0
$73$	1.35371	0.158440	0.0792198	−	0.996857i	$-0.474757\pi$
$73$	1.35371	0.158440	0.0792198	+	0.996857i	$0.474757\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 5.44696i	− 0.620738i
$78$	0	0
$79$	11.0846	1.24711	0.623555	−	0.781779i	$-0.285688\pi$
$79$	11.0846	1.24711	0.623555	+	0.781779i	$0.285688\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	− 13.6442i	− 1.49765i	−0.662768	−	0.748824i	$-0.730619\pi$
$83$	− 13.6442i	− 1.49765i	0.662768	−	0.748824i	$-0.269381\pi$
$84$	0	0
$85$	10.3848i	1.12639i
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−5.93965	−0.629601	−0.314801	−	0.949158i	$-0.601938\pi$
$89$	−5.93965	−0.629601	−0.314801	+	0.949158i	$0.601938\pi$
$90$	0	0
$91$	3.61205i	0.378646i
$92$	0	0
$93$	0	0
$94$	0	0
$95$	10.1629	1.04269
$96$	0	0
$97$	9.60073	0.974807	0.487403	−	0.873177i	$-0.337944\pi$
$97$	9.60073	0.974807	0.487403	+	0.873177i	$0.337944\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	14.2123i	1.41418i	0.707124	+	0.707090i	$0.249992\pi$
$101$	14.2123i	1.41418i	−0.707124	+	0.707090i	$0.750008\pi$
$102$	0	0
$103$	−1.69321	−0.166837	−0.0834187	−	0.996515i	$-0.526584\pi$
$103$	−1.69321	−0.166837	−0.0834187	+	0.996515i	$0.526584\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 11.6644i	− 1.12764i	−0.825897	−	0.563821i	$-0.809331\pi$
$107$	− 11.6644i	− 1.12764i	0.825897	−	0.563821i	$-0.190669\pi$
$108$	0	0
$109$	− 14.0521i	− 1.34595i	−0.739667	−	0.672973i	$-0.765017\pi$
$109$	− 14.0521i	− 1.34595i	0.739667	−	0.672973i	$-0.234983\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−14.9455	−1.40596	−0.702979	−	0.711211i	$-0.748147\pi$
$113$	−14.9455	−1.40596	−0.702979	+	0.711211i	$0.748147\pi$
$114$	0	0
$115$	− 2.13407i	− 0.199003i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	3.27628	0.300336
$120$	0	0
$121$	−18.6693	−1.69721
$122$	0	0
$123$	0	0
$124$	0	0
$125$	− 0.148729i	− 0.0133028i
$126$	0	0
$127$	−4.07216	−0.361346	−0.180673	−	0.983543i	$-0.557828\pi$
$127$	−4.07216	−0.361346	−0.180673	+	0.983543i	$0.557828\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	17.8643i	1.56081i	0.625275	+	0.780404i	$0.284987\pi$
$131$	17.8643i	1.56081i	−0.625275	+	0.780404i	$0.715013\pi$
$132$	0	0
$133$	− 3.20627i	− 0.278019i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−21.1351	−1.80569	−0.902847	−	0.429963i	$-0.858527\pi$
$137$	−21.1351	−1.80569	−0.902847	+	0.429963i	$0.858527\pi$
$138$	0	0
$139$	− 7.39359i	− 0.627116i	−0.949569	−	0.313558i	$-0.898479\pi$
$139$	− 7.39359i	− 0.627116i	0.949569	−	0.313558i	$-0.101521\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	19.6747	1.64528
$144$	0	0
$145$	9.03764	0.750535
$146$	0	0
$147$	0	0
$148$	0	0
$149$	− 13.0541i	− 1.06944i	−0.845030	−	0.534719i	$-0.820418\pi$
$149$	− 13.0541i	− 1.06944i	0.845030	−	0.534719i	$-0.179582\pi$
$150$	0	0
$151$	17.6477	1.43615	0.718073	−	0.695968i	$-0.245024\pi$
$151$	17.6477	1.43615	0.718073	+	0.695968i	$0.245024\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	− 11.7863i	− 0.946701i
$156$	0	0
$157$	3.00087i	0.239495i	0.992804	+	0.119748i	$0.0382085\pi$
$157$	3.00087i	0.239495i	−0.992804	+	0.119748i	$0.961791\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−0.673274	−0.0530614
$162$	0	0
$163$	− 0.871148i	− 0.0682335i	−0.999418	−	0.0341168i	$-0.989138\pi$
$163$	− 0.871148i	− 0.0682335i	0.999418	−	0.0341168i	$-0.0108618\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−7.49952	−0.580330	−0.290165	−	0.956977i	$-0.593710\pi$
$167$	−7.49952	−0.580330	−0.290165	+	0.956977i	$0.593710\pi$
$168$	0	0
$169$	−0.0469224	−0.00360941
$170$	0	0
$171$	0	0
$172$	0	0
$173$	0.652919i	0.0496405i	0.999692	+	0.0248203i	$0.00790135\pi$
$173$	0.652919i	0.0496405i	−0.999692	+	0.0248203i	$0.992099\pi$
$174$	0	0
$175$	−5.04692	−0.381511
$176$	0	0
$177$	0	0
$178$	0	0
$179$	− 4.43263i	− 0.331310i	−0.986184	−	0.165655i	$-0.947026\pi$
$179$	− 4.43263i	− 0.331310i	0.986184	−	0.165655i	$-0.0529738\pi$
$180$	0	0
$181$	− 21.9779i	− 1.63360i	−0.576920	−	0.816801i	$-0.695746\pi$
$181$	− 21.9779i	− 1.63360i	0.576920	−	0.816801i	$-0.304254\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	38.0326	2.79621
$186$	0	0
$187$	− 17.8458i	− 1.30501i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	6.67327	0.482861	0.241430	−	0.970418i	$-0.422383\pi$
$191$	6.67327	0.482861	0.241430	+	0.970418i	$0.422383\pi$
$192$	0	0
$193$	25.0305	1.80174	0.900868	−	0.434092i	$-0.142931\pi$
$193$	25.0305	1.80174	0.900868	+	0.434092i	$0.142931\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	3.77971i	0.269293i	0.990894	+	0.134646i	$0.0429899\pi$
$197$	3.77971i	0.269293i	−0.990894	+	0.134646i	$0.957010\pi$
$198$	0	0
$199$	7.18059	0.509019	0.254509	−	0.967070i	$-0.418086\pi$
$199$	7.18059	0.509019	0.254509	+	0.967070i	$0.418086\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	− 2.85127i	− 0.200120i
$204$	0	0
$205$	− 23.6016i	− 1.64840i
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−17.4644	−1.20804
$210$	0	0
$211$	5.30442i	0.365171i	0.983190	+	0.182586i	$0.0584467\pi$
$211$	5.30442i	0.365171i	−0.983190	+	0.182586i	$0.941553\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	39.8560	2.71816
$216$	0	0
$217$	−3.71845	−0.252425
$218$	0	0
$219$	0	0
$220$	0	0
$221$	11.8341i	0.796048i
$222$	0	0
$223$	−4.62678	−0.309832	−0.154916	−	0.987928i	$-0.549511\pi$
$223$	−4.62678	−0.309832	−0.154916	+	0.987928i	$0.549511\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	9.92356i	0.658650i	0.944217	+	0.329325i	$0.106821\pi$
$227$	9.92356i	0.658650i	−0.944217	+	0.329325i	$0.893179\pi$
$228$	0	0
$229$	2.32546i	0.153671i	0.997044	+	0.0768355i	$0.0244816\pi$
$229$	2.32546i	0.153671i	−0.997044	+	0.0768355i	$0.975518\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−3.23031	−0.211625	−0.105812	−	0.994386i	$-0.533744\pi$
$233$	−3.23031	−0.211625	−0.105812	+	0.994386i	$0.533744\pi$
$234$	0	0
$235$	12.8797i	0.840182i
$236$	0	0
$237$	0	0
$238$	0	0
$239$	3.48961	0.225724	0.112862	−	0.993611i	$-0.463998\pi$
$239$	3.48961	0.225724	0.112862	+	0.993611i	$0.463998\pi$
$240$	0	0
$241$	9.77778	0.629842	0.314921	−	0.949118i	$-0.398022\pi$
$241$	9.77778	0.629842	0.314921	+	0.949118i	$0.398022\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	3.16969i	0.202504i
$246$	0	0
$247$	11.5812	0.736896
$248$	0	0
$249$	0	0
$250$	0	0
$251$	25.2169i	1.59168i	0.605509	+	0.795838i	$0.292969\pi$
$251$	25.2169i	1.59168i	−0.605509	+	0.795838i	$0.707031\pi$
$252$	0	0
$253$	3.66729i	0.230561i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	6.12920	0.382329	0.191165	−	0.981558i	$-0.438774\pi$
$257$	6.12920	0.382329	0.191165	+	0.981558i	$0.438774\pi$
$258$	0	0
$259$	− 11.9988i	− 0.745572i
$260$	0	0
$261$	0	0
$262$	0	0
$263$	1.27084	0.0783635	0.0391818	−	0.999232i	$-0.487525\pi$
$263$	1.27084	0.0783635	0.0391818	+	0.999232i	$0.487525\pi$
$264$	0	0
$265$	0.924284	0.0567783
$266$	0	0
$267$	0	0
$268$	0	0
$269$	19.8081i	1.20772i	0.797091	+	0.603859i	$0.206371\pi$
$269$	19.8081i	1.20772i	−0.797091	+	0.603859i	$0.793629\pi$
$270$	0	0
$271$	−24.1532	−1.46720	−0.733600	−	0.679581i	$-0.762162\pi$
$271$	−24.1532	−1.46720	−0.733600	+	0.679581i	$0.762162\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	27.4904i	1.65773i
$276$	0	0
$277$	1.31816i	0.0792005i	0.999216	+	0.0396003i	$0.0126084\pi$
$277$	1.31816i	0.0792005i	−0.999216	+	0.0396003i	$0.987392\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	24.7284	1.47517	0.737587	−	0.675252i	$-0.235965\pi$
$281$	24.7284	1.47517	0.737587	+	0.675252i	$0.235965\pi$
$282$	0	0
$283$	− 18.6573i	− 1.10906i	−0.832163	−	0.554532i	$-0.812897\pi$
$283$	− 18.6573i	− 1.10906i	0.832163	−	0.554532i	$-0.187103\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−7.44602	−0.439525
$288$	0	0
$289$	−6.26596	−0.368586
$290$	0	0
$291$	0	0
$292$	0	0
$293$	− 14.4030i	− 0.841431i	−0.907193	−	0.420715i	$-0.861779\pi$
$293$	− 14.4030i	− 0.841431i	0.907193	−	0.420715i	$-0.138221\pi$
$294$	0	0
$295$	−0.0664326	−0.00386786
$296$	0	0
$297$	0	0
$298$	0	0
$299$	− 2.43190i	− 0.140640i
$300$	0	0
$301$	− 12.5741i	− 0.724759i
$302$	0	0
$303$	0	0
$304$	0	0
$305$	16.9272	0.969249
$306$	0	0
$307$	− 27.3734i	− 1.56228i	−0.624353	−	0.781142i	$-0.714637\pi$
$307$	− 27.3734i	− 1.56228i	0.624353	−	0.781142i	$-0.285363\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	27.9767	1.58641	0.793206	−	0.608953i	$-0.208410\pi$
$311$	27.9767	1.58641	0.793206	+	0.608953i	$0.208410\pi$
$312$	0	0
$313$	27.5228	1.55568	0.777841	−	0.628461i	$-0.216315\pi$
$313$	27.5228	1.55568	0.777841	+	0.628461i	$0.216315\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	1.73928i	0.0976879i	0.998806	+	0.0488440i	$0.0155537\pi$
$317$	1.73928i	0.0976879i	−0.998806	+	0.0488440i	$0.984446\pi$
$318$	0	0
$319$	−15.5307	−0.869555
$320$	0	0
$321$	0	0
$322$	0	0
$323$	− 10.5047i	− 0.584495i
$324$	0	0
$325$	− 18.2297i	− 1.01120i
$326$	0	0
$327$	0	0
$328$	0	0
$329$	4.06341	0.224023
$330$	0	0
$331$	− 13.7965i	− 0.758323i	−0.925331	−	0.379161i	$-0.876212\pi$
$331$	− 13.7965i	− 0.758323i	0.925331	−	0.379161i	$-0.123788\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−19.6747	−1.07494
$336$	0	0
$337$	−1.96903	−0.107260	−0.0536300	−	0.998561i	$-0.517079\pi$
$337$	−1.96903	−0.107260	−0.0536300	+	0.998561i	$0.517079\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	20.2542i	1.09683i
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 21.4183i	− 1.14979i	−0.818226	−	0.574897i	$-0.805042\pi$
$347$	− 21.4183i	− 1.14979i	0.818226	−	0.574897i	$-0.194958\pi$
$348$	0	0
$349$	24.2315i	1.29708i	0.761180	+	0.648541i	$0.224620\pi$
$349$	24.2315i	1.29708i	−0.761180	+	0.648541i	$0.775380\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	20.1659	1.07332	0.536662	−	0.843797i	$-0.319685\pi$
$353$	20.1659	1.07332	0.536662	+	0.843797i	$0.319685\pi$
$354$	0	0
$355$	− 49.1459i	− 2.60839i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	19.8643	1.04840	0.524198	−	0.851597i	$-0.324365\pi$
$359$	19.8643	1.04840	0.524198	+	0.851597i	$0.324365\pi$
$360$	0	0
$361$	8.71982	0.458938
$362$	0	0
$363$	0	0
$364$	0	0
$365$	4.29083i	0.224592i
$366$	0	0
$367$	−37.1713	−1.94033	−0.970163	−	0.242454i	$-0.922048\pi$
$367$	−37.1713	−1.94033	−0.970163	+	0.242454i	$0.922048\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	− 0.291601i	− 0.0151392i
$372$	0	0
$373$	23.6106i	1.22251i	0.791433	+	0.611256i	$0.209335\pi$
$373$	23.6106i	1.22251i	−0.791433	+	0.611256i	$0.790665\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	10.2989	0.530422
$378$	0	0
$379$	15.2051i	0.781034i	0.920596	+	0.390517i	$0.127704\pi$
$379$	15.2051i	0.781034i	−0.920596	+	0.390517i	$0.872296\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	1.26331	0.0645522	0.0322761	−	0.999479i	$-0.489724\pi$
$383$	1.26331	0.0645522	0.0322761	+	0.999479i	$0.489724\pi$
$384$	0	0
$385$	17.2652	0.879914
$386$	0	0
$387$	0	0
$388$	0	0
$389$	12.3813i	0.627754i	0.949464	+	0.313877i	$0.101628\pi$
$389$	12.3813i	0.627754i	−0.949464	+	0.313877i	$0.898372\pi$
$390$	0	0
$391$	−2.20584	−0.111554
$392$	0	0
$393$	0	0
$394$	0	0
$395$	35.1346i	1.76781i
$396$	0	0
$397$	19.0272i	0.954948i	0.878646	+	0.477474i	$0.158447\pi$
$397$	19.0272i	0.954948i	−0.878646	+	0.477474i	$0.841553\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	6.06885	0.303064	0.151532	−	0.988452i	$-0.451579\pi$
$401$	6.06885	0.303064	0.151532	+	0.988452i	$0.451579\pi$
$402$	0	0
$403$	− 13.4312i	− 0.669058i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−65.3572	−3.23963
$408$	0	0
$409$	28.9491	1.43144	0.715720	−	0.698387i	$-0.246099\pi$
$409$	28.9491	1.43144	0.715720	+	0.698387i	$0.246099\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	0.0209587i	0.00103131i
$414$	0	0
$415$	43.2480	2.12296
$416$	0	0
$417$	0	0
$418$	0	0
$419$	− 4.30387i	− 0.210258i	−0.994459	−	0.105129i	$-0.966474\pi$
$419$	− 4.30387i	− 0.210258i	0.994459	−	0.105129i	$-0.0335255\pi$
$420$	0	0
$421$	− 18.0243i	− 0.878453i	−0.898376	−	0.439226i	$-0.855253\pi$
$421$	− 18.0243i	− 0.878453i	0.898376	−	0.439226i	$-0.144747\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−16.5351	−0.802073
$426$	0	0
$427$	− 5.34034i	− 0.258437i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	11.0189	0.530760	0.265380	−	0.964144i	$-0.414503\pi$
$431$	11.0189	0.530760	0.265380	+	0.964144i	$0.414503\pi$
$432$	0	0
$433$	11.0881	0.532861	0.266430	−	0.963854i	$-0.414156\pi$
$433$	11.0881	0.532861	0.266430	+	0.963854i	$0.414156\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	2.15870i	0.103265i
$438$	0	0
$439$	32.6928	1.56034	0.780171	−	0.625567i	$-0.215132\pi$
$439$	32.6928	1.56034	0.780171	+	0.625567i	$0.215132\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	− 23.6225i	− 1.12234i	−0.827700	−	0.561171i	$-0.810351\pi$
$443$	− 23.6225i	− 1.12234i	0.827700	−	0.561171i	$-0.189649\pi$
$444$	0	0
$445$	− 18.8268i	− 0.892477i
$446$	0	0
$447$	0	0
$448$	0	0
$449$	8.75599	0.413220	0.206610	−	0.978423i	$-0.433757\pi$
$449$	8.75599	0.413220	0.206610	+	0.978423i	$0.433757\pi$
$450$	0	0
$451$	40.5581i	1.90981i
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−11.4491	−0.536741
$456$	0	0
$457$	15.8326	0.740618	0.370309	−	0.928909i	$-0.379252\pi$
$457$	15.8326	0.740618	0.370309	+	0.928909i	$0.379252\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	− 36.1188i	− 1.68222i	−0.540865	−	0.841109i	$-0.681903\pi$
$461$	− 36.1188i	− 1.68222i	0.540865	−	0.841109i	$-0.318097\pi$
$462$	0	0
$463$	−8.28903	−0.385224	−0.192612	−	0.981275i	$-0.561696\pi$
$463$	−8.28903	−0.385224	−0.192612	+	0.981275i	$0.561696\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 22.7640i	− 1.05339i	−0.850053	−	0.526697i	$-0.823430\pi$
$467$	− 22.7640i	− 1.05339i	0.850053	−	0.526697i	$-0.176570\pi$
$468$	0	0
$469$	6.20714i	0.286619i
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−68.4906	−3.14920
$474$	0	0
$475$	16.1818i	0.742472i
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−38.1931	−1.74509	−0.872543	−	0.488537i	$-0.837531\pi$
$479$	−38.1931	−1.74509	−0.872543	+	0.488537i	$0.837531\pi$
$480$	0	0
$481$	43.3404	1.97615
$482$	0	0
$483$	0	0
$484$	0	0
$485$	30.4313i	1.38182i
$486$	0	0
$487$	−8.84028	−0.400591	−0.200296	−	0.979735i	$-0.564190\pi$
$487$	−8.84028	−0.400591	−0.200296	+	0.979735i	$0.564190\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	− 28.9660i	− 1.30722i	−0.756833	−	0.653608i	$-0.773254\pi$
$491$	− 28.9660i	− 1.30722i	0.756833	−	0.653608i	$-0.226746\pi$
$492$	0	0
$493$	− 9.34157i	− 0.420723i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−15.5050	−0.695492
$498$	0	0
$499$	0.417929i	0.0187091i	0.999956	+	0.00935453i	$0.00297768\pi$
$499$	0.417929i	0.0187091i	−0.999956	+	0.00935453i	$0.997022\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−10.3133	−0.459848	−0.229924	−	0.973209i	$-0.573848\pi$
$503$	−10.3133	−0.459848	−0.229924	+	0.973209i	$0.573848\pi$
$504$	0	0
$505$	−45.0487	−2.00464
$506$	0	0
$507$	0	0
$508$	0	0
$509$	− 22.2340i	− 0.985505i	−0.870169	−	0.492753i	$-0.835991\pi$
$509$	− 22.2340i	− 0.985505i	0.870169	−	0.492753i	$-0.164009\pi$
$510$	0	0
$511$	1.35371	0.0598845
$512$	0	0
$513$	0	0
$514$	0	0
$515$	− 5.36696i	− 0.236497i
$516$	0	0
$517$	− 22.1332i	− 0.973418i
$518$	0	0
$519$	0	0
$520$	0	0
$521$	29.3079	1.28400	0.642001	−	0.766704i	$-0.278104\pi$
$521$	29.3079	1.28400	0.642001	+	0.766704i	$0.278104\pi$
$522$	0	0
$523$	− 1.15302i	− 0.0504181i	−0.999682	−	0.0252090i	$-0.991975\pi$
$523$	− 1.15302i	− 0.0504181i	0.999682	−	0.0252090i	$-0.00802514\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−12.1827	−0.530687
$528$	0	0
$529$	−22.5467	−0.980291
$530$	0	0
$531$	0	0
$532$	0	0
$533$	− 26.8954i	− 1.16497i
$534$	0	0
$535$	36.9726	1.59846
$536$	0	0
$537$	0	0
$538$	0	0
$539$	− 5.44696i	− 0.234617i
$540$	0	0
$541$	− 34.3825i	− 1.47822i	−0.673586	−	0.739109i	$-0.735247\pi$
$541$	− 34.3825i	− 1.47822i	0.673586	−	0.739109i	$-0.264753\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	44.5407	1.90792
$546$	0	0
$547$	17.1388i	0.732802i	0.930457	+	0.366401i	$0.119410\pi$
$547$	17.1388i	0.732802i	−0.930457	+	0.366401i	$0.880590\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−9.14195	−0.389460
$552$	0	0
$553$	11.0846	0.471363
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 29.0976i	− 1.23290i	−0.787393	−	0.616452i	$-0.788570\pi$
$557$	− 29.0976i	− 1.23290i	0.787393	−	0.616452i	$-0.211430\pi$
$558$	0	0
$559$	45.4183	1.92099
$560$	0	0
$561$	0	0
$562$	0	0
$563$	31.6652i	1.33453i	0.744821	+	0.667264i	$0.232535\pi$
$563$	31.6652i	1.33453i	−0.744821	+	0.667264i	$0.767465\pi$
$564$	0	0
$565$	− 47.3727i	− 1.99298i
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−6.16080	−0.258274	−0.129137	−	0.991627i	$-0.541221\pi$
$569$	−6.16080	−0.258274	−0.129137	+	0.991627i	$0.541221\pi$
$570$	0	0
$571$	− 41.0313i	− 1.71711i	−0.512724	−	0.858553i	$-0.671364\pi$
$571$	− 41.0313i	− 1.71711i	0.512724	−	0.858553i	$-0.328636\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	3.39796	0.141705
$576$	0	0
$577$	−25.4290	−1.05862	−0.529311	−	0.848428i	$-0.677550\pi$
$577$	−25.4290	−1.05862	−0.529311	+	0.848428i	$0.677550\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	− 13.6442i	− 0.566058i
$582$	0	0
$583$	−1.58834	−0.0657822
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 3.55639i	− 0.146788i	−0.997303	−	0.0733939i	$-0.976617\pi$
$587$	− 3.55639i	− 0.146788i	0.997303	−	0.0733939i	$-0.0233830\pi$
$588$	0	0
$589$	11.9224i	0.491253i
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−2.57142	−0.105596	−0.0527978	−	0.998605i	$-0.516814\pi$
$593$	−2.57142	−0.105596	−0.0527978	+	0.998605i	$0.516814\pi$
$594$	0	0
$595$	10.3848i	0.425735i
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−21.8400	−0.892357	−0.446178	−	0.894944i	$-0.647215\pi$
$599$	−21.8400	−0.892357	−0.446178	+	0.894944i	$0.647215\pi$
$600$	0	0
$601$	−43.8140	−1.78721	−0.893606	−	0.448853i	$-0.851833\pi$
$601$	−43.8140	−1.78721	−0.893606	+	0.448853i	$0.851833\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	− 59.1760i	− 2.40585i
$606$	0	0
$607$	−17.9012	−0.726588	−0.363294	−	0.931675i	$-0.618348\pi$
$607$	−17.9012	−0.726588	−0.363294	+	0.931675i	$0.618348\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	14.6773i	0.593778i
$612$	0	0
$613$	15.8902i	0.641798i	0.947113	+	0.320899i	$0.103985\pi$
$613$	15.8902i	0.641798i	−0.947113	+	0.320899i	$0.896015\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−20.8402	−0.838994	−0.419497	−	0.907757i	$-0.637794\pi$
$617$	−20.8402	−0.838994	−0.419497	+	0.907757i	$0.637794\pi$
$618$	0	0
$619$	31.4914i	1.26575i	0.774255	+	0.632873i	$0.218125\pi$
$619$	31.4914i	1.26575i	−0.774255	+	0.632873i	$0.781875\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−5.93965	−0.237967
$624$	0	0
$625$	−24.7632	−0.990527
$626$	0	0
$627$	0	0
$628$	0	0
$629$	− 39.3116i	− 1.56746i
$630$	0	0
$631$	10.6786	0.425109	0.212555	−	0.977149i	$-0.431822\pi$
$631$	10.6786	0.425109	0.212555	+	0.977149i	$0.431822\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	− 12.9075i	− 0.512218i
$636$	0	0
$637$	3.61205i	0.143115i
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−24.7669	−0.978232	−0.489116	−	0.872219i	$-0.662681\pi$
$641$	−24.7669	−0.978232	−0.489116	+	0.872219i	$0.662681\pi$
$642$	0	0
$643$	− 11.5834i	− 0.456805i	−0.973567	−	0.228402i	$-0.926650\pi$
$643$	− 11.5834i	− 0.456805i	0.973567	−	0.228402i	$-0.0733502\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−30.1155	−1.18396	−0.591981	−	0.805952i	$-0.701654\pi$
$647$	−30.1155	−1.18396	−0.591981	+	0.805952i	$0.701654\pi$
$648$	0	0
$649$	0.114161	0.00448122
$650$	0	0
$651$	0	0
$652$	0	0
$653$	45.7814i	1.79157i	0.444491	+	0.895783i	$0.353384\pi$
$653$	45.7814i	1.79157i	−0.444491	+	0.895783i	$0.646616\pi$
$654$	0	0
$655$	−56.6242	−2.21249
$656$	0	0
$657$	0	0
$658$	0	0
$659$	34.3446i	1.33788i	0.743318	+	0.668938i	$0.233251\pi$
$659$	34.3446i	1.33788i	−0.743318	+	0.668938i	$0.766749\pi$
$660$	0	0
$661$	26.5119i	1.03120i	0.856831	+	0.515598i	$0.172430\pi$
$661$	26.5119i	1.03120i	−0.856831	+	0.515598i	$0.827570\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	10.1629	0.394100
$666$	0	0
$667$	1.91969i	0.0743305i
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−29.0886	−1.12295
$672$	0	0
$673$	−26.2427	−1.01158	−0.505790	−	0.862657i	$-0.668799\pi$
$673$	−26.2427	−1.01158	−0.505790	+	0.862657i	$0.668799\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	3.23458i	0.124315i	0.998066	+	0.0621576i	$0.0197981\pi$
$677$	3.23458i	0.124315i	−0.998066	+	0.0621576i	$0.980202\pi$
$678$	0	0
$679$	9.60073	0.368442
$680$	0	0
$681$	0	0
$682$	0	0
$683$	0.708911i	0.0271257i	0.999908	+	0.0135629i	$0.00431733\pi$
$683$	0.708911i	0.0271257i	−0.999908	+	0.0135629i	$0.995683\pi$
$684$	0	0
$685$	− 66.9917i	− 2.55962i
$686$	0	0
$687$	0	0
$688$	0	0
$689$	1.05328	0.0401267
$690$	0	0
$691$	39.0071i	1.48390i	0.670454	+	0.741951i	$0.266099\pi$
$691$	39.0071i	1.48390i	−0.670454	+	0.741951i	$0.733901\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	23.4354	0.888955
$696$	0	0
$697$	−24.3953	−0.924037
$698$	0	0
$699$	0	0
$700$	0	0
$701$	16.2448i	0.613559i	0.951781	+	0.306779i	$0.0992514\pi$
$701$	16.2448i	0.613559i	−0.951781	+	0.306779i	$0.900749\pi$
$702$	0	0
$703$	−38.4715	−1.45098
$704$	0	0
$705$	0	0
$706$	0	0
$707$	14.2123i	0.534510i
$708$	0	0
$709$	25.1720i	0.945354i	0.881236	+	0.472677i	$0.156712\pi$
$709$	25.1720i	0.945354i	−0.881236	+	0.472677i	$0.843288\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	2.50354	0.0937581
$714$	0	0
$715$	62.3626i	2.33223i
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−7.92289	−0.295474	−0.147737	−	0.989027i	$-0.547199\pi$
$719$	−7.92289	−0.295474	−0.147737	+	0.989027i	$0.547199\pi$
$720$	0	0
$721$	−1.69321	−0.0630586
$722$	0	0
$723$	0	0
$724$	0	0
$725$	14.3901i	0.534436i
$726$	0	0
$727$	6.10313	0.226353	0.113176	−	0.993575i	$-0.463898\pi$
$727$	6.10313	0.226353	0.113176	+	0.993575i	$0.463898\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	− 41.1963i	− 1.52370i
$732$	0	0
$733$	− 26.6568i	− 0.984591i	−0.870428	−	0.492296i	$-0.836158\pi$
$733$	− 26.6568i	− 0.984591i	0.870428	−	0.492296i	$-0.163842\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	33.8100	1.24541
$738$	0	0
$739$	− 9.27736i	− 0.341273i	−0.985334	−	0.170637i	$-0.945418\pi$
$739$	− 9.27736i	− 0.341273i	0.985334	−	0.170637i	$-0.0545824\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	12.6494	0.464062	0.232031	−	0.972708i	$-0.425463\pi$
$743$	12.6494	0.464062	0.232031	+	0.972708i	$0.425463\pi$
$744$	0	0
$745$	41.3776	1.51596
$746$	0	0
$747$	0	0
$748$	0	0
$749$	− 11.6644i	− 0.426209i
$750$	0	0
$751$	7.10625	0.259311	0.129655	−	0.991559i	$-0.458613\pi$
$751$	7.10625	0.259311	0.129655	+	0.991559i	$0.458613\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	55.9376i	2.03578i
$756$	0	0
$757$	34.6019i	1.25763i	0.777556	+	0.628814i	$0.216459\pi$
$757$	34.6019i	1.25763i	−0.777556	+	0.628814i	$0.783541\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	21.7114	0.787039	0.393520	−	0.919316i	$-0.371257\pi$
$761$	21.7114	0.787039	0.393520	+	0.919316i	$0.371257\pi$
$762$	0	0
$763$	− 14.0521i	− 0.508720i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−0.0757040	−0.00273351
$768$	0	0
$769$	−5.94161	−0.214260	−0.107130	−	0.994245i	$-0.534166\pi$
$769$	−5.94161	−0.214260	−0.107130	+	0.994245i	$0.534166\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	− 1.14102i	− 0.0410398i	−0.999789	−	0.0205199i	$-0.993468\pi$
$773$	− 1.14102i	− 0.0410398i	0.999789	−	0.0205199i	$-0.00653215\pi$
$774$	0	0
$775$	18.7667	0.674121
$776$	0	0
$777$	0	0
$778$	0	0
$779$	23.8740i	0.855374i
$780$	0	0
$781$	84.4548i	3.02203i
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−9.51181	−0.339491
$786$	0	0
$787$	− 42.2224i	− 1.50507i	−0.658555	−	0.752533i	$-0.728832\pi$
$787$	− 42.2224i	− 1.50507i	0.658555	−	0.752533i	$-0.271168\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−14.9455	−0.531402
$792$	0	0
$793$	19.2896	0.684993
$794$	0	0
$795$	0	0
$796$	0	0
$797$	37.0441i	1.31217i	0.754687	+	0.656085i	$0.227789\pi$
$797$	37.0441i	1.31217i	−0.754687	+	0.656085i	$0.772211\pi$
$798$	0	0
$799$	13.3129	0.470976
$800$	0	0
$801$	0	0
$802$	0	0
$803$	− 7.37359i	− 0.260208i
$804$	0	0
$805$	− 2.13407i	− 0.0752160i
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−18.7803	−0.660279	−0.330140	−	0.943932i	$-0.607096\pi$
$809$	−18.7803	−0.660279	−0.330140	+	0.943932i	$0.607096\pi$
$810$	0	0
$811$	− 54.4054i	− 1.91043i	−0.295909	−	0.955216i	$-0.595623\pi$
$811$	− 54.4054i	− 1.91043i	0.295909	−	0.955216i	$-0.404377\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	2.76127	0.0967229
$816$	0	0
$817$	−40.3160	−1.41048
$818$	0	0
$819$	0	0
$820$	0	0
$821$	− 26.7178i	− 0.932458i	−0.884664	−	0.466229i	$-0.845612\pi$
$821$	− 26.7178i	− 0.932458i	0.884664	−	0.466229i	$-0.154388\pi$
$822$	0	0
$823$	15.5033	0.540412	0.270206	−	0.962802i	$-0.412908\pi$
$823$	15.5033	0.540412	0.270206	+	0.962802i	$0.412908\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	16.0053i	0.556559i	0.960500	+	0.278279i	$0.0897641\pi$
$827$	16.0053i	0.556559i	−0.960500	+	0.278279i	$0.910236\pi$
$828$	0	0
$829$	46.1590i	1.60317i	0.597881	+	0.801585i	$0.296009\pi$
$829$	46.1590i	1.60317i	−0.597881	+	0.801585i	$0.703991\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	3.27628	0.113517
$834$	0	0
$835$	− 23.7711i	− 0.822634i
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−4.38306	−0.151320	−0.0756601	−	0.997134i	$-0.524106\pi$
$839$	−4.38306	−0.151320	−0.0756601	+	0.997134i	$0.524106\pi$
$840$	0	0
$841$	20.8703	0.719664
$842$	0	0
$843$	0	0
$844$	0	0
$845$	− 0.148729i	− 0.00511644i
$846$	0	0
$847$	−18.6693	−0.641486
$848$	0	0
$849$	0	0
$850$	0	0
$851$	8.07850i	0.276927i
$852$	0	0
$853$	− 15.5653i	− 0.532946i	−0.963842	−	0.266473i	$-0.914142\pi$
$853$	− 15.5653i	− 0.532946i	0.963842	−	0.266473i	$-0.0858583\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−30.2023	−1.03169	−0.515846	−	0.856681i	$-0.672522\pi$
$857$	−30.2023	−1.03169	−0.515846	+	0.856681i	$0.672522\pi$
$858$	0	0
$859$	− 6.32139i	− 0.215683i	−0.994168	−	0.107841i	$-0.965606\pi$
$859$	− 6.32139i	− 0.215683i	0.994168	−	0.107841i	$-0.0343939\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	35.8902	1.22172	0.610858	−	0.791740i	$-0.290825\pi$
$863$	35.8902	1.22172	0.610858	+	0.791740i	$0.290825\pi$
$864$	0	0
$865$	−2.06955	−0.0703668
$866$	0	0
$867$	0	0
$868$	0	0
$869$	− 60.3771i	− 2.04815i
$870$	0	0
$871$	−22.4205	−0.759690
$872$	0	0
$873$	0	0
$874$	0	0
$875$	− 0.148729i	− 0.00502797i
$876$	0	0
$877$	32.3677i	1.09298i	0.837466	+	0.546490i	$0.184036\pi$
$877$	32.3677i	1.09298i	−0.837466	+	0.546490i	$0.815964\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	33.1052	1.11534	0.557670	−	0.830062i	$-0.311695\pi$
$881$	33.1052	1.11534	0.557670	+	0.830062i	$0.311695\pi$
$882$	0	0
$883$	19.7891i	0.665957i	0.942934	+	0.332979i	$0.108054\pi$
$883$	19.7891i	0.665957i	−0.942934	+	0.332979i	$0.891946\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	0.843244	0.0283133	0.0141567	−	0.999900i	$-0.495494\pi$
$887$	0.843244	0.0283133	0.0141567	+	0.999900i	$0.495494\pi$
$888$	0	0
$889$	−4.07216	−0.136576
$890$	0	0
$891$	0	0
$892$	0	0
$893$	− 13.0284i	− 0.435979i
$894$	0	0
$895$	14.0500	0.469641
$896$	0	0
$897$	0	0
$898$	0	0
$899$	10.6023i	0.353607i
$900$	0	0
$901$	− 0.955367i	− 0.0318279i
$902$	0	0
$903$	0	0
$904$	0	0
$905$	69.6629	2.31567
$906$	0	0
$907$	9.01198i	0.299238i	0.988744	+	0.149619i	$0.0478047\pi$
$907$	9.01198i	0.299238i	−0.988744	+	0.149619i	$0.952195\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	1.53535	0.0508685	0.0254343	−	0.999676i	$-0.491903\pi$
$911$	1.53535	0.0508685	0.0254343	+	0.999676i	$0.491903\pi$
$912$	0	0
$913$	−74.3195	−2.45962
$914$	0	0
$915$	0	0
$916$	0	0
$917$	17.8643i	0.589930i
$918$	0	0
$919$	22.4581	0.740826	0.370413	−	0.928867i	$-0.379216\pi$
$919$	22.4581	0.740826	0.370413	+	0.928867i	$0.379216\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	− 56.0047i	− 1.84342i
$924$	0	0
$925$	60.5572i	1.99111i
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−50.2178	−1.64759	−0.823797	−	0.566885i	$-0.808148\pi$
$929$	−50.2178	−1.64759	−0.823797	+	0.566885i	$0.808148\pi$
$930$	0	0
$931$	− 3.20627i	− 0.105081i
$932$	0	0
$933$	0	0
$934$	0	0
$935$	56.5655	1.84989
$936$	0	0
$937$	11.9114	0.389130	0.194565	−	0.980890i	$-0.437670\pi$
$937$	11.9114	0.389130	0.194565	+	0.980890i	$0.437670\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	59.5916i	1.94263i	0.237795	+	0.971315i	$0.423575\pi$
$941$	59.5916i	1.94263i	−0.237795	+	0.971315i	$0.576425\pi$
$942$	0	0
$943$	5.01321	0.163252
$944$	0	0
$945$	0	0
$946$	0	0
$947$	4.69613i	0.152604i	0.997085	+	0.0763018i	$0.0243113\pi$
$947$	4.69613i	0.152604i	−0.997085	+	0.0763018i	$0.975689\pi$
$948$	0	0
$949$	4.88966i	0.158725i
$950$	0	0
$951$	0	0
$952$	0	0
$953$	38.9727	1.26245	0.631224	−	0.775600i	$-0.282553\pi$
$953$	38.9727	1.26245	0.631224	+	0.775600i	$0.282553\pi$
$954$	0	0
$955$	21.1522i	0.684469i
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−21.1351	−0.682488
$960$	0	0
$961$	−17.1731	−0.553971
$962$	0	0
$963$	0	0
$964$	0	0
$965$	79.3390i	2.55401i
$966$	0	0
$967$	−40.5025	−1.30247	−0.651236	−	0.758875i	$-0.725749\pi$
$967$	−40.5025	−1.30247	−0.651236	+	0.758875i	$0.725749\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	5.76542i	0.185021i	0.995712	+	0.0925105i	$0.0294892\pi$
$971$	5.76542i	0.185021i	−0.995712	+	0.0925105i	$0.970511\pi$
$972$	0	0
$973$	− 7.39359i	− 0.237028i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−10.3362	−0.330684	−0.165342	−	0.986236i	$-0.552873\pi$
$977$	−10.3362	−0.330684	−0.165342	+	0.986236i	$0.552873\pi$
$978$	0	0
$979$	32.3530i	1.03401i
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−22.9663	−0.732512	−0.366256	−	0.930514i	$-0.619361\pi$
$983$	−22.9663	−0.732512	−0.366256	+	0.930514i	$0.619361\pi$
$984$	0	0
$985$	−11.9805	−0.381730
$986$	0	0
$987$	0	0
$988$	0	0
$989$	8.46581i	0.269197i
$990$	0	0
$991$	5.74450	0.182480	0.0912400	−	0.995829i	$-0.470917\pi$
$991$	5.74450	0.182480	0.0912400	+	0.995829i	$0.470917\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	22.7602i	0.721548i
$996$	0	0
$997$	59.7098i	1.89103i	0.325580	+	0.945515i	$0.394441\pi$
$997$	59.7098i	1.89103i	−0.325580	+	0.945515i	$0.605559\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6048.2.c.e.3025.18		20
3.2	odd	2	inner	6048.2.c.e.3025.4		20
4.3	odd	2		1512.2.c.e.757.4	yes	20
8.3	odd	2		1512.2.c.e.757.3	✓	20
8.5	even	2	inner	6048.2.c.e.3025.3		20
12.11	even	2		1512.2.c.e.757.17	yes	20
24.5	odd	2	inner	6048.2.c.e.3025.17		20
24.11	even	2		1512.2.c.e.757.18	yes	20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1512.2.c.e.757.3	✓	20	8.3	odd	2
1512.2.c.e.757.4	yes	20	4.3	odd	2
1512.2.c.e.757.17	yes	20	12.11	even	2
1512.2.c.e.757.18	yes	20	24.11	even	2
6048.2.c.e.3025.3		20	8.5	even	2	inner
6048.2.c.e.3025.4		20	3.2	odd	2	inner
6048.2.c.e.3025.17		20	24.5	odd	2	inner
6048.2.c.e.3025.18		20	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:	\( N \)	\(=\)	\( 6048 = 2^{5} \cdot 3^{3} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6048.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+3.16969i q^{5} +1.00000 q^{7} +O(q^{10})\)	\(q+3.16969i q^{5} +1.00000 q^{7} -5.44696i q^{11} +3.61205i q^{13} +3.27628 q^{17} -3.20627i q^{19} -0.673274 q^{23} -5.04692 q^{25} -2.85127i q^{29} -3.71845 q^{31} +3.16969i q^{35} -11.9988i q^{37} -7.44602 q^{41} -12.5741i q^{43} +4.06341 q^{47} +1.00000 q^{49} -0.291601i q^{53} +17.2652 q^{55} +0.0209587i q^{59} -5.34034i q^{61} -11.4491 q^{65} +6.20714i q^{67} -15.5050 q^{71} +1.35371 q^{73} -5.44696i q^{77} +11.0846 q^{79} -13.6442i q^{83} +10.3848i q^{85} -5.93965 q^{89} +3.61205i q^{91} +10.1629 q^{95} +9.60073 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q + 20 q^{7}+O(q^{10}) \) 20 * q + 20 * q^7	\( 20 q + 20 q^{7} - 28 q^{25} - 36 q^{31} + 20 q^{49} - 48 q^{55} - 64 q^{79} + 56 q^{97}+O(q^{100}) \) 20 * q + 20 * q^7 - 28 * q^25 - 36 * q^31 + 20 * q^49 - 48 * q^55 - 64 * q^79 + 56 * q^97

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