LMFDB - Embedded newform 6048.2.h.i.2591.20

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6048,2,Mod(2591,6048)]

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([1, 0, 1, 0]))

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

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

chi := DirichletCharacter("6048.2591");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6048 = 2^{5} \cdot 3^{3} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6048.h (of order $2$, degree $1$, 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:	$24$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			2591.20
Character	$\chi$	$=$	6048.2591
Dual form			6048.2.h.i.2591.19

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.64388i q^{5} -1.00000i q^{7} +O(q^{10})$	$q+2.64388i q^{5} -1.00000i q^{7} -4.12083 q^{11} -4.86886 q^{13} -2.03749i q^{17} +4.16827i q^{19} -9.48754 q^{23} -1.99011 q^{25} +1.54751i q^{29} +0.248395i q^{31} +2.64388 q^{35} +7.88909 q^{37} +2.00821i q^{41} -0.547808i q^{43} -2.59422 q^{47} -1.00000 q^{49} -11.5632i q^{53} -10.8950i q^{55} +12.4330 q^{59} +14.2962 q^{61} -12.8727i q^{65} +6.52832i q^{67} -5.95273 q^{71} +11.7079 q^{73} +4.12083i q^{77} -7.47832i q^{79} -3.94011 q^{83} +5.38690 q^{85} -12.9230i q^{89} +4.86886i q^{91} -11.0204 q^{95} -5.64630 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q+O(q^{10}) $ 24 * q	$ 24 q - 8 q^{13} - 24 q^{25} + 48 q^{37} - 24 q^{49} - 48 q^{61} - 32 q^{73} - 80 q^{85} + 64 q^{97}+O(q^{100}) $ 24 * q - 8 * q^13 - 24 * q^25 + 48 * q^37 - 24 * q^49 - 48 * q^61 - 32 * q^73 - 80 * q^85 + 64 * 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
$5$	2.64388i	1.18238i	0.806532	+	0.591190i	$0.201342\pi$
$5$	2.64388i	1.18238i	−0.806532	+	0.591190i	$0.798658\pi$
$6$	0	0
$7$	− 1.00000i	− 0.377964i
$7$	− 1.00000i	− 0.377964i
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−4.12083	−1.24248	−0.621238	−	0.783622i	$-0.713370\pi$
$11$	−4.12083	−1.24248	−0.621238	+	0.783622i	$0.713370\pi$
$12$	0	0
$13$	−4.86886	−1.35038	−0.675190	−	0.737644i	$-0.735938\pi$
$13$	−4.86886	−1.35038	−0.675190	+	0.737644i	$0.735938\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 2.03749i	− 0.494165i	−0.968994	−	0.247083i	$-0.920528\pi$
$17$	− 2.03749i	− 0.494165i	0.968994	−	0.247083i	$-0.0794719\pi$
$18$	0	0
$19$	4.16827i	0.956268i	0.878287	+	0.478134i	$0.158687\pi$
$19$	4.16827i	0.956268i	−0.878287	+	0.478134i	$0.841313\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−9.48754	−1.97829	−0.989145	−	0.146945i	$-0.953056\pi$
$23$	−9.48754	−1.97829	−0.989145	+	0.146945i	$0.953056\pi$
$24$	0	0
$25$	−1.99011	−0.398022
$26$	0	0
$27$	0	0
$28$	0	0
$29$	1.54751i	0.287365i	0.989624	+	0.143682i	$0.0458943\pi$
$29$	1.54751i	0.287365i	−0.989624	+	0.143682i	$0.954106\pi$
$30$	0	0
$31$	0.248395i	0.0446131i	0.999751	+	0.0223066i	$0.00710099\pi$
$31$	0.248395i	0.0446131i	−0.999751	+	0.0223066i	$0.992899\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	2.64388	0.446898
$36$	0	0
$37$	7.88909	1.29696	0.648479	−	0.761232i	$-0.275405\pi$
$37$	7.88909	1.29696	0.648479	+	0.761232i	$0.275405\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	2.00821i	0.313629i	0.987628	+	0.156815i	$0.0501225\pi$
$41$	2.00821i	0.313629i	−0.987628	+	0.156815i	$0.949877\pi$
$42$	0	0
$43$	− 0.547808i	− 0.0835399i	−0.999127	−	0.0417700i	$-0.986700\pi$
$43$	− 0.547808i	− 0.0835399i	0.999127	−	0.0417700i	$-0.0132997\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−2.59422	−0.378406	−0.189203	−	0.981938i	$-0.560590\pi$
$47$	−2.59422	−0.378406	−0.189203	+	0.981938i	$0.560590\pi$
$48$	0	0
$49$	−1.00000	−0.142857
$50$	0	0
$51$	0	0
$52$	0	0
$53$	− 11.5632i	− 1.58832i	−0.607707	−	0.794161i	$-0.707910\pi$
$53$	− 11.5632i	− 1.58832i	0.607707	−	0.794161i	$-0.292090\pi$
$54$	0	0
$55$	− 10.8950i	− 1.46908i
$56$	0	0
$57$	0	0
$58$	0	0
$59$	12.4330	1.61864	0.809322	−	0.587365i	$-0.199835\pi$
$59$	12.4330	1.61864	0.809322	+	0.587365i	$0.199835\pi$
$60$	0	0
$61$	14.2962	1.83044	0.915218	−	0.402959i	$-0.132018\pi$
$61$	14.2962	1.83044	0.915218	+	0.402959i	$0.132018\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	− 12.8727i	− 1.59666i
$66$	0	0
$67$	6.52832i	0.797561i	0.917046	+	0.398781i	$0.130567\pi$
$67$	6.52832i	0.797561i	−0.917046	+	0.398781i	$0.869433\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−5.95273	−0.706459	−0.353229	−	0.935537i	$-0.614916\pi$
$71$	−5.95273	−0.706459	−0.353229	+	0.935537i	$0.614916\pi$
$72$	0	0
$73$	11.7079	1.37031	0.685156	−	0.728396i	$-0.259734\pi$
$73$	11.7079	1.37031	0.685156	+	0.728396i	$0.259734\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	4.12083i	0.469612i
$78$	0	0
$79$	− 7.47832i	− 0.841376i	−0.907205	−	0.420688i	$-0.861789\pi$
$79$	− 7.47832i	− 0.841376i	0.907205	−	0.420688i	$-0.138211\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−3.94011	−0.432483	−0.216241	−	0.976340i	$-0.569380\pi$
$83$	−3.94011	−0.432483	−0.216241	+	0.976340i	$0.569380\pi$
$84$	0	0
$85$	5.38690	0.584291
$86$	0	0
$87$	0	0
$88$	0	0
$89$	− 12.9230i	− 1.36984i	−0.728619	−	0.684920i	$-0.759837\pi$
$89$	− 12.9230i	− 1.36984i	0.728619	−	0.684920i	$-0.240163\pi$
$90$	0	0
$91$	4.86886i	0.510395i
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−11.0204	−1.13067
$96$	0	0
$97$	−5.64630	−0.573295	−0.286648	−	0.958036i	$-0.592541\pi$
$97$	−5.64630	−0.573295	−0.286648	+	0.958036i	$0.592541\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	− 4.71066i	− 0.468729i	−0.972149	−	0.234364i	$-0.924699\pi$
$101$	− 4.71066i	− 0.468729i	0.972149	−	0.234364i	$-0.0753008\pi$
$102$	0	0
$103$	− 19.6158i	− 1.93280i	−0.257038	−	0.966401i	$-0.582747\pi$
$103$	− 19.6158i	− 1.93280i	0.257038	−	0.966401i	$-0.417253\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	4.10360	0.396710	0.198355	−	0.980130i	$-0.436440\pi$
$107$	4.10360	0.396710	0.198355	+	0.980130i	$0.436440\pi$
$108$	0	0
$109$	8.48792	0.812995	0.406497	−	0.913652i	$-0.366750\pi$
$109$	8.48792	0.812995	0.406497	+	0.913652i	$0.366750\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	5.60719i	0.527480i	0.964594	+	0.263740i	$0.0849561\pi$
$113$	5.60719i	0.527480i	−0.964594	+	0.263740i	$0.915044\pi$
$114$	0	0
$115$	− 25.0839i	− 2.33909i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−2.03749	−0.186777
$120$	0	0
$121$	5.98124	0.543749
$122$	0	0
$123$	0	0
$124$	0	0
$125$	7.95779i	0.711766i
$126$	0	0
$127$	16.0322i	1.42263i	0.702875	+	0.711313i	$0.251899\pi$
$127$	16.0322i	1.42263i	−0.702875	+	0.711313i	$0.748101\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−2.05016	−0.179123	−0.0895615	−	0.995981i	$-0.528547\pi$
$131$	−2.05016	−0.179123	−0.0895615	+	0.995981i	$0.528547\pi$
$132$	0	0
$133$	4.16827	0.361435
$134$	0	0
$135$	0	0
$136$	0	0
$137$	10.4115i	0.889517i	0.895650	+	0.444759i	$0.146711\pi$
$137$	10.4115i	0.889517i	−0.895650	+	0.444759i	$0.853289\pi$
$138$	0	0
$139$	− 13.0177i	− 1.10415i	−0.833795	−	0.552074i	$-0.813837\pi$
$139$	− 13.0177i	− 1.10415i	0.833795	−	0.552074i	$-0.186163\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	20.0637	1.67781
$144$	0	0
$145$	−4.09142	−0.339774
$146$	0	0
$147$	0	0
$148$	0	0
$149$	− 22.5842i	− 1.85017i	−0.379758	−	0.925086i	$-0.623993\pi$
$149$	− 22.5842i	− 1.85017i	0.379758	−	0.925086i	$-0.376007\pi$
$150$	0	0
$151$	− 23.7533i	− 1.93302i	−0.256638	−	0.966508i	$-0.582615\pi$
$151$	− 23.7533i	− 1.93302i	0.256638	−	0.966508i	$-0.417385\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−0.656728	−0.0527497
$156$	0	0
$157$	1.78100	0.142139	0.0710696	−	0.997471i	$-0.477359\pi$
$157$	1.78100	0.142139	0.0710696	+	0.997471i	$0.477359\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	9.48754i	0.747723i
$162$	0	0
$163$	8.78773i	0.688308i	0.938913	+	0.344154i	$0.111834\pi$
$163$	8.78773i	0.688308i	−0.938913	+	0.344154i	$0.888166\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−3.35719	−0.259787	−0.129893	−	0.991528i	$-0.541464\pi$
$167$	−3.35719	−0.259787	−0.129893	+	0.991528i	$0.541464\pi$
$168$	0	0
$169$	10.7058	0.823524
$170$	0	0
$171$	0	0
$172$	0	0
$173$	16.7408i	1.27278i	0.771369	+	0.636388i	$0.219572\pi$
$173$	16.7408i	1.27278i	−0.771369	+	0.636388i	$0.780428\pi$
$174$	0	0
$175$	1.99011i	0.150438i
$176$	0	0
$177$	0	0
$178$	0	0
$179$	13.1125	0.980074	0.490037	−	0.871702i	$-0.336983\pi$
$179$	13.1125	0.980074	0.490037	+	0.871702i	$0.336983\pi$
$180$	0	0
$181$	−10.1190	−0.752136	−0.376068	−	0.926592i	$-0.622724\pi$
$181$	−10.1190	−0.752136	−0.376068	+	0.926592i	$0.622724\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	20.8578i	1.53350i
$186$	0	0
$187$	8.39617i	0.613989i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	7.11278	0.514663	0.257331	−	0.966323i	$-0.417157\pi$
$191$	7.11278	0.514663	0.257331	+	0.966323i	$0.417157\pi$
$192$	0	0
$193$	9.21790	0.663519	0.331759	−	0.943364i	$-0.392358\pi$
$193$	9.21790	0.663519	0.331759	+	0.943364i	$0.392358\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	8.43379i	0.600883i	0.953800	+	0.300441i	$0.0971340\pi$
$197$	8.43379i	0.600883i	−0.953800	+	0.300441i	$0.902866\pi$
$198$	0	0
$199$	− 0.742006i	− 0.0525994i	−0.999654	−	0.0262997i	$-0.991628\pi$
$199$	− 0.742006i	− 0.0525994i	0.999654	−	0.0262997i	$-0.00837242\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	1.54751	0.108614
$204$	0	0
$205$	−5.30946	−0.370829
$206$	0	0
$207$	0	0
$208$	0	0
$209$	− 17.1767i	− 1.18814i
$210$	0	0
$211$	− 7.66439i	− 0.527639i	−0.964572	−	0.263819i	$-0.915018\pi$
$211$	− 7.66439i	− 0.527639i	0.964572	−	0.263819i	$-0.0849823\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	1.44834	0.0987759
$216$	0	0
$217$	0.248395	0.0168622
$218$	0	0
$219$	0	0
$220$	0	0
$221$	9.92028i	0.667310i
$222$	0	0
$223$	− 16.3982i	− 1.09810i	−0.835788	−	0.549052i	$-0.814989\pi$
$223$	− 16.3982i	− 1.09810i	0.835788	−	0.549052i	$-0.185011\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	14.6099	0.969692	0.484846	−	0.874599i	$-0.338876\pi$
$227$	14.6099	0.969692	0.484846	+	0.874599i	$0.338876\pi$
$228$	0	0
$229$	−20.1027	−1.32842	−0.664212	−	0.747544i	$-0.731233\pi$
$229$	−20.1027	−1.32842	−0.664212	+	0.747544i	$0.731233\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	− 4.91623i	− 0.322073i	−0.986948	−	0.161036i	$-0.948516\pi$
$233$	− 4.91623i	− 0.322073i	0.986948	−	0.161036i	$-0.0514837\pi$
$234$	0	0
$235$	− 6.85881i	− 0.447419i
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−21.3618	−1.38178	−0.690890	−	0.722960i	$-0.742781\pi$
$239$	−21.3618	−1.38178	−0.690890	+	0.722960i	$0.742781\pi$
$240$	0	0
$241$	5.87880	0.378687	0.189344	−	0.981911i	$-0.439364\pi$
$241$	5.87880	0.378687	0.189344	+	0.981911i	$0.439364\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	− 2.64388i	− 0.168911i
$246$	0	0
$247$	− 20.2947i	− 1.29132i
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−11.0745	−0.699016	−0.349508	−	0.936933i	$-0.613651\pi$
$251$	−11.0745	−0.699016	−0.349508	+	0.936933i	$0.613651\pi$
$252$	0	0
$253$	39.0965	2.45798
$254$	0	0
$255$	0	0
$256$	0	0
$257$	11.7076i	0.730299i	0.930949	+	0.365150i	$0.118982\pi$
$257$	11.7076i	0.730299i	−0.930949	+	0.365150i	$0.881018\pi$
$258$	0	0
$259$	− 7.88909i	− 0.490204i
$260$	0	0
$261$	0	0
$262$	0	0
$263$	11.5702	0.713451	0.356726	−	0.934209i	$-0.383893\pi$
$263$	11.5702	0.713451	0.356726	+	0.934209i	$0.383893\pi$
$264$	0	0
$265$	30.5716	1.87800
$266$	0	0
$267$	0	0
$268$	0	0
$269$	− 18.6358i	− 1.13625i	−0.822944	−	0.568123i	$-0.807670\pi$
$269$	− 18.6358i	− 1.13625i	0.822944	−	0.568123i	$-0.192330\pi$
$270$	0	0
$271$	− 11.2036i	− 0.680571i	−0.940322	−	0.340286i	$-0.889476\pi$
$271$	− 11.2036i	− 0.680571i	0.940322	−	0.340286i	$-0.110524\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	8.20091	0.494533
$276$	0	0
$277$	−29.2418	−1.75697	−0.878485	−	0.477769i	$-0.841446\pi$
$277$	−29.2418	−1.75697	−0.878485	+	0.477769i	$0.841446\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	− 11.6928i	− 0.697534i	−0.937209	−	0.348767i	$-0.886600\pi$
$281$	− 11.6928i	− 0.697534i	0.937209	−	0.348767i	$-0.113400\pi$
$282$	0	0
$283$	18.3784i	1.09248i	0.837628	+	0.546242i	$0.183942\pi$
$283$	18.3784i	1.09248i	−0.837628	+	0.546242i	$0.816058\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	2.00821	0.118541
$288$	0	0
$289$	12.8486	0.755801
$290$	0	0
$291$	0	0
$292$	0	0
$293$	22.2299i	1.29868i	0.760497	+	0.649341i	$0.224955\pi$
$293$	22.2299i	1.29868i	−0.760497	+	0.649341i	$0.775045\pi$
$294$	0	0
$295$	32.8715i	1.91385i
$296$	0	0
$297$	0	0
$298$	0	0
$299$	46.1935	2.67144
$300$	0	0
$301$	−0.547808	−0.0315751
$302$	0	0
$303$	0	0
$304$	0	0
$305$	37.7974i	2.16427i
$306$	0	0
$307$	− 17.2640i	− 0.985306i	−0.870226	−	0.492653i	$-0.836027\pi$
$307$	− 17.2640i	− 0.985306i	0.870226	−	0.492653i	$-0.163973\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−3.80882	−0.215978	−0.107989	−	0.994152i	$-0.534441\pi$
$311$	−3.80882	−0.215978	−0.107989	+	0.994152i	$0.534441\pi$
$312$	0	0
$313$	25.6882	1.45198	0.725991	−	0.687705i	$-0.241382\pi$
$313$	25.6882	1.45198	0.725991	+	0.687705i	$0.241382\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	6.32114i	0.355030i	0.984118	+	0.177515i	$0.0568059\pi$
$317$	6.32114i	0.355030i	−0.984118	+	0.177515i	$0.943194\pi$
$318$	0	0
$319$	− 6.37701i	− 0.357044i
$320$	0	0
$321$	0	0
$322$	0	0
$323$	8.49284	0.472554
$324$	0	0
$325$	9.68958	0.537481
$326$	0	0
$327$	0	0
$328$	0	0
$329$	2.59422i	0.143024i
$330$	0	0
$331$	3.46340i	0.190366i	0.995460	+	0.0951829i	$0.0303436\pi$
$331$	3.46340i	0.190366i	−0.995460	+	0.0951829i	$0.969656\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−17.2601	−0.943020
$336$	0	0
$337$	−5.11720	−0.278752	−0.139376	−	0.990240i	$-0.544510\pi$
$337$	−5.11720	−0.278752	−0.139376	+	0.990240i	$0.544510\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	− 1.02359i	− 0.0554308i
$342$	0	0
$343$	1.00000i	0.0539949i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−26.8628	−1.44207	−0.721034	−	0.692900i	$-0.756333\pi$
$347$	−26.8628	−1.44207	−0.721034	+	0.692900i	$0.756333\pi$
$348$	0	0
$349$	−19.9241	−1.06651	−0.533256	−	0.845954i	$-0.679032\pi$
$349$	−19.9241	−1.06651	−0.533256	+	0.845954i	$0.679032\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	− 28.8604i	− 1.53608i	−0.640401	−	0.768041i	$-0.721232\pi$
$353$	− 28.8604i	− 1.53608i	0.640401	−	0.768041i	$-0.278768\pi$
$354$	0	0
$355$	− 15.7383i	− 0.835302i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−26.1252	−1.37883	−0.689417	−	0.724365i	$-0.742133\pi$
$359$	−26.1252	−1.37883	−0.689417	+	0.724365i	$0.742133\pi$
$360$	0	0
$361$	1.62549	0.0855522
$362$	0	0
$363$	0	0
$364$	0	0
$365$	30.9544i	1.62023i
$366$	0	0
$367$	17.4489i	0.910827i	0.890280	+	0.455414i	$0.150509\pi$
$367$	17.4489i	0.910827i	−0.890280	+	0.455414i	$0.849491\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−11.5632	−0.600330
$372$	0	0
$373$	21.0226	1.08851	0.544254	−	0.838920i	$-0.316813\pi$
$373$	21.0226	1.08851	0.544254	+	0.838920i	$0.316813\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 7.53459i	− 0.388051i
$378$	0	0
$379$	− 28.6392i	− 1.47110i	−0.677471	−	0.735549i	$-0.736924\pi$
$379$	− 28.6392i	− 1.47110i	0.677471	−	0.735549i	$-0.263076\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−8.16718	−0.417323	−0.208662	−	0.977988i	$-0.566911\pi$
$383$	−8.16718	−0.417323	−0.208662	+	0.977988i	$0.566911\pi$
$384$	0	0
$385$	−10.8950	−0.555260
$386$	0	0
$387$	0	0
$388$	0	0
$389$	26.2176i	1.32929i	0.747161	+	0.664643i	$0.231416\pi$
$389$	26.2176i	1.32929i	−0.747161	+	0.664643i	$0.768584\pi$
$390$	0	0
$391$	19.3308i	0.977601i
$392$	0	0
$393$	0	0
$394$	0	0
$395$	19.7718	0.994827
$396$	0	0
$397$	16.3245	0.819304	0.409652	−	0.912242i	$-0.365650\pi$
$397$	16.3245	0.819304	0.409652	+	0.912242i	$0.365650\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	20.4029i	1.01887i	0.860509	+	0.509435i	$0.170146\pi$
$401$	20.4029i	1.01887i	−0.860509	+	0.509435i	$0.829854\pi$
$402$	0	0
$403$	− 1.20940i	− 0.0602446i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−32.5096	−1.61144
$408$	0	0
$409$	24.0523	1.18931	0.594654	−	0.803982i	$-0.297289\pi$
$409$	24.0523	1.18931	0.594654	+	0.803982i	$0.297289\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	− 12.4330i	− 0.611790i
$414$	0	0
$415$	− 10.4172i	− 0.511359i
$416$	0	0
$417$	0	0
$418$	0	0
$419$	7.79478	0.380800	0.190400	−	0.981707i	$-0.439022\pi$
$419$	7.79478	0.380800	0.190400	+	0.981707i	$0.439022\pi$
$420$	0	0
$421$	30.7294	1.49766	0.748830	−	0.662762i	$-0.230616\pi$
$421$	30.7294	1.49766	0.748830	+	0.662762i	$0.230616\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	4.05484i	0.196689i
$426$	0	0
$427$	− 14.2962i	− 0.691840i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−16.3139	−0.785814	−0.392907	−	0.919578i	$-0.628531\pi$
$431$	−16.3139	−0.785814	−0.392907	+	0.919578i	$0.628531\pi$
$432$	0	0
$433$	−6.02836	−0.289705	−0.144852	−	0.989453i	$-0.546271\pi$
$433$	−6.02836	−0.289705	−0.144852	+	0.989453i	$0.546271\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 39.5467i	− 1.89177i
$438$	0	0
$439$	− 25.2529i	− 1.20526i	−0.798022	−	0.602629i	$-0.794120\pi$
$439$	− 25.2529i	− 1.20526i	0.798022	−	0.602629i	$-0.205880\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	26.6419	1.26580	0.632898	−	0.774235i	$-0.281865\pi$
$443$	26.6419	1.26580	0.632898	+	0.774235i	$0.281865\pi$
$444$	0	0
$445$	34.1670	1.61967
$446$	0	0
$447$	0	0
$448$	0	0
$449$	7.22348i	0.340897i	0.985367	+	0.170449i	$0.0545217\pi$
$449$	7.22348i	0.340897i	−0.985367	+	0.170449i	$0.945478\pi$
$450$	0	0
$451$	− 8.27548i	− 0.389677i
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−12.8727	−0.603481
$456$	0	0
$457$	−39.2788	−1.83738	−0.918692	−	0.394974i	$-0.870754\pi$
$457$	−39.2788	−1.83738	−0.918692	+	0.394974i	$0.870754\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	− 18.0976i	− 0.842892i	−0.906854	−	0.421446i	$-0.861523\pi$
$461$	− 18.0976i	− 0.842892i	0.906854	−	0.421446i	$-0.138477\pi$
$462$	0	0
$463$	− 28.4913i	− 1.32410i	−0.749458	−	0.662052i	$-0.769686\pi$
$463$	− 28.4913i	− 1.32410i	0.749458	−	0.662052i	$-0.230314\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	31.9125	1.47673	0.738366	−	0.674400i	$-0.235598\pi$
$467$	31.9125	1.47673	0.738366	+	0.674400i	$0.235598\pi$
$468$	0	0
$469$	6.52832	0.301450
$470$	0	0
$471$	0	0
$472$	0	0
$473$	2.25742i	0.103796i
$474$	0	0
$475$	− 8.29533i	− 0.380616i
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−3.73828	−0.170806	−0.0854031	−	0.996346i	$-0.527218\pi$
$479$	−3.73828	−0.170806	−0.0854031	+	0.996346i	$0.527218\pi$
$480$	0	0
$481$	−38.4109	−1.75139
$482$	0	0
$483$	0	0
$484$	0	0
$485$	− 14.9282i	− 0.677853i
$486$	0	0
$487$	3.85953i	0.174892i	0.996169	+	0.0874461i	$0.0278706\pi$
$487$	3.85953i	0.174892i	−0.996169	+	0.0874461i	$0.972129\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	0.428753	0.0193494	0.00967468	−	0.999953i	$-0.496920\pi$
$491$	0.428753	0.0193494	0.00967468	+	0.999953i	$0.496920\pi$
$492$	0	0
$493$	3.15303	0.142006
$494$	0	0
$495$	0	0
$496$	0	0
$497$	5.95273i	0.267016i
$498$	0	0
$499$	− 39.9296i	− 1.78749i	−0.448571	−	0.893747i	$-0.648067\pi$
$499$	− 39.9296i	− 1.78749i	0.448571	−	0.893747i	$-0.351933\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	2.19899	0.0980479	0.0490239	−	0.998798i	$-0.484389\pi$
$503$	2.19899	0.0980479	0.0490239	+	0.998798i	$0.484389\pi$
$504$	0	0
$505$	12.4544	0.554215
$506$	0	0
$507$	0	0
$508$	0	0
$509$	− 16.5978i	− 0.735684i	−0.929888	−	0.367842i	$-0.880097\pi$
$509$	− 16.5978i	− 0.735684i	0.929888	−	0.367842i	$-0.119903\pi$
$510$	0	0
$511$	− 11.7079i	− 0.517929i
$512$	0	0
$513$	0	0
$514$	0	0
$515$	51.8619	2.28531
$516$	0	0
$517$	10.6903	0.470161
$518$	0	0
$519$	0	0
$520$	0	0
$521$	− 41.1482i	− 1.80274i	−0.433053	−	0.901368i	$-0.642564\pi$
$521$	− 41.1482i	− 1.80274i	0.433053	−	0.901368i	$-0.357436\pi$
$522$	0	0
$523$	− 14.5857i	− 0.637790i	−0.947790	−	0.318895i	$-0.896688\pi$
$523$	− 14.5857i	− 0.637790i	0.947790	−	0.318895i	$-0.103312\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	0.506104	0.0220462
$528$	0	0
$529$	67.0135	2.91363
$530$	0	0
$531$	0	0
$532$	0	0
$533$	− 9.77768i	− 0.423518i
$534$	0	0
$535$	10.8494i	0.469062i
$536$	0	0
$537$	0	0
$538$	0	0
$539$	4.12083	0.177497
$540$	0	0
$541$	27.2868	1.17315	0.586575	−	0.809895i	$-0.300476\pi$
$541$	27.2868	1.17315	0.586575	+	0.809895i	$0.300476\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	22.4410i	0.961269i
$546$	0	0
$547$	31.2511i	1.33620i	0.744071	+	0.668100i	$0.232892\pi$
$547$	31.2511i	1.33620i	−0.744071	+	0.668100i	$0.767108\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−6.45043	−0.274797
$552$	0	0
$553$	−7.47832	−0.318010
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 32.1278i	− 1.36130i	−0.732610	−	0.680648i	$-0.761698\pi$
$557$	− 32.1278i	− 1.36130i	0.732610	−	0.680648i	$-0.238302\pi$
$558$	0	0
$559$	2.66720i	0.112811i
$560$	0	0
$561$	0	0
$562$	0	0
$563$	4.62528	0.194933	0.0974663	−	0.995239i	$-0.468926\pi$
$563$	4.62528	0.194933	0.0974663	+	0.995239i	$0.468926\pi$
$564$	0	0
$565$	−14.8248	−0.623682
$566$	0	0
$567$	0	0
$568$	0	0
$569$	− 26.5383i	− 1.11254i	−0.831000	−	0.556272i	$-0.812231\pi$
$569$	− 26.5383i	− 1.11254i	0.831000	−	0.556272i	$-0.187769\pi$
$570$	0	0
$571$	30.6374i	1.28213i	0.767485	+	0.641067i	$0.221508\pi$
$571$	30.6374i	1.28213i	−0.767485	+	0.641067i	$0.778492\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	18.8813	0.787403
$576$	0	0
$577$	−39.0588	−1.62604	−0.813019	−	0.582238i	$-0.802177\pi$
$577$	−39.0588	−1.62604	−0.813019	+	0.582238i	$0.802177\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	3.94011i	0.163463i
$582$	0	0
$583$	47.6498i	1.97345i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−7.80587	−0.322183	−0.161091	−	0.986940i	$-0.551501\pi$
$587$	−7.80587	−0.322183	−0.161091	+	0.986940i	$0.551501\pi$
$588$	0	0
$589$	−1.03538	−0.0426621
$590$	0	0
$591$	0	0
$592$	0	0
$593$	− 41.6924i	− 1.71210i	−0.516892	−	0.856051i	$-0.672911\pi$
$593$	− 41.6924i	− 1.71210i	0.516892	−	0.856051i	$-0.327089\pi$
$594$	0	0
$595$	− 5.38690i	− 0.220841i
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−8.38483	−0.342595	−0.171297	−	0.985219i	$-0.554796\pi$
$599$	−8.38483	−0.342595	−0.171297	+	0.985219i	$0.554796\pi$
$600$	0	0
$601$	20.3188	0.828820	0.414410	−	0.910090i	$-0.363988\pi$
$601$	20.3188	0.828820	0.414410	+	0.910090i	$0.363988\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	15.8137i	0.642918i
$606$	0	0
$607$	27.8127i	1.12888i	0.825473	+	0.564442i	$0.190909\pi$
$607$	27.8127i	1.12888i	−0.825473	+	0.564442i	$0.809091\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	12.6309	0.510991
$612$	0	0
$613$	−3.65547	−0.147643	−0.0738214	−	0.997271i	$-0.523519\pi$
$613$	−3.65547	−0.147643	−0.0738214	+	0.997271i	$0.523519\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 21.3637i	− 0.860070i	−0.902812	−	0.430035i	$-0.858501\pi$
$617$	− 21.3637i	− 0.860070i	0.902812	−	0.430035i	$-0.141499\pi$
$618$	0	0
$619$	− 1.48661i	− 0.0597520i	−0.999554	−	0.0298760i	$-0.990489\pi$
$619$	− 1.48661i	− 0.0597520i	0.999554	−	0.0298760i	$-0.00951124\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−12.9230	−0.517750
$624$	0	0
$625$	−30.9900	−1.23960
$626$	0	0
$627$	0	0
$628$	0	0
$629$	− 16.0740i	− 0.640912i
$630$	0	0
$631$	10.2236i	0.406995i	0.979075	+	0.203497i	$0.0652308\pi$
$631$	10.2236i	0.406995i	−0.979075	+	0.203497i	$0.934769\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−42.3872	−1.68208
$636$	0	0
$637$	4.86886	0.192911
$638$	0	0
$639$	0	0
$640$	0	0
$641$	7.36460i	0.290884i	0.989367	+	0.145442i	$0.0464605\pi$
$641$	7.36460i	0.290884i	−0.989367	+	0.145442i	$0.953540\pi$
$642$	0	0
$643$	− 20.0823i	− 0.791967i	−0.918258	−	0.395983i	$-0.870404\pi$
$643$	− 20.0823i	− 0.791967i	0.918258	−	0.395983i	$-0.129596\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−21.1276	−0.830610	−0.415305	−	0.909682i	$-0.636325\pi$
$647$	−21.1276	−0.830610	−0.415305	+	0.909682i	$0.636325\pi$
$648$	0	0
$649$	−51.2345	−2.01113
$650$	0	0
$651$	0	0
$652$	0	0
$653$	48.3880i	1.89357i	0.321867	+	0.946785i	$0.395690\pi$
$653$	48.3880i	1.89357i	−0.321867	+	0.946785i	$0.604310\pi$
$654$	0	0
$655$	− 5.42037i	− 0.211791i
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−16.8453	−0.656198	−0.328099	−	0.944643i	$-0.606408\pi$
$659$	−16.8453	−0.656198	−0.328099	+	0.944643i	$0.606408\pi$
$660$	0	0
$661$	−25.7271	−1.00067	−0.500334	−	0.865832i	$-0.666790\pi$
$661$	−25.7271	−1.00067	−0.500334	+	0.865832i	$0.666790\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	11.0204i	0.427354i
$666$	0	0
$667$	− 14.6820i	− 0.568490i
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−58.9120	−2.27427
$672$	0	0
$673$	−0.545897	−0.0210428	−0.0105214	−	0.999945i	$-0.503349\pi$
$673$	−0.545897	−0.0210428	−0.0105214	+	0.999945i	$0.503349\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 40.6066i	− 1.56064i	−0.625382	−	0.780318i	$-0.715057\pi$
$677$	− 40.6066i	− 1.56064i	0.625382	−	0.780318i	$-0.284943\pi$
$678$	0	0
$679$	5.64630i	0.216685i
$680$	0	0
$681$	0	0
$682$	0	0
$683$	3.25283	0.124466	0.0622330	−	0.998062i	$-0.480178\pi$
$683$	3.25283	0.124466	0.0622330	+	0.998062i	$0.480178\pi$
$684$	0	0
$685$	−27.5269	−1.05175
$686$	0	0
$687$	0	0
$688$	0	0
$689$	56.2994i	2.14484i
$690$	0	0
$691$	− 34.9406i	− 1.32920i	−0.747198	−	0.664602i	$-0.768601\pi$
$691$	− 34.9406i	− 1.32920i	0.747198	−	0.664602i	$-0.231399\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	34.4173	1.30552
$696$	0	0
$697$	4.09171	0.154985
$698$	0	0
$699$	0	0
$700$	0	0
$701$	7.09751i	0.268069i	0.990977	+	0.134035i	$0.0427934\pi$
$701$	7.09751i	0.268069i	−0.990977	+	0.134035i	$0.957207\pi$
$702$	0	0
$703$	32.8839i	1.24024i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−4.71066	−0.177163
$708$	0	0
$709$	−43.0565	−1.61702	−0.808510	−	0.588482i	$-0.799726\pi$
$709$	−43.0565	−1.61702	−0.808510	+	0.588482i	$0.799726\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	− 2.35666i	− 0.0882577i
$714$	0	0
$715$	53.0462i	1.98381i
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−10.9861	−0.409714	−0.204857	−	0.978792i	$-0.565673\pi$
$719$	−10.9861	−0.409714	−0.204857	+	0.978792i	$0.565673\pi$
$720$	0	0
$721$	−19.6158	−0.730531
$722$	0	0
$723$	0	0
$724$	0	0
$725$	− 3.07971i	− 0.114377i
$726$	0	0
$727$	10.6945i	0.396637i	0.980138	+	0.198319i	$0.0635481\pi$
$727$	10.6945i	0.396637i	−0.980138	+	0.198319i	$0.936452\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−1.11616	−0.0412825
$732$	0	0
$733$	16.3535	0.604029	0.302015	−	0.953303i	$-0.402341\pi$
$733$	16.3535	0.604029	0.302015	+	0.953303i	$0.402341\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 26.9021i	− 0.990951i
$738$	0	0
$739$	33.0891i	1.21720i	0.793476	+	0.608601i	$0.208269\pi$
$739$	33.0891i	1.21720i	−0.793476	+	0.608601i	$0.791731\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−28.9244	−1.06113	−0.530566	−	0.847644i	$-0.678021\pi$
$743$	−28.9244	−1.06113	−0.530566	+	0.847644i	$0.678021\pi$
$744$	0	0
$745$	59.7100	2.18761
$746$	0	0
$747$	0	0
$748$	0	0
$749$	− 4.10360i	− 0.149942i
$750$	0	0
$751$	− 18.3341i	− 0.669020i	−0.942392	−	0.334510i	$-0.891429\pi$
$751$	− 18.3341i	− 0.669020i	0.942392	−	0.334510i	$-0.108571\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	62.8009	2.28556
$756$	0	0
$757$	14.8508	0.539761	0.269880	−	0.962894i	$-0.413016\pi$
$757$	14.8508	0.539761	0.269880	+	0.962894i	$0.413016\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	7.52709i	0.272857i	0.990650	+	0.136428i	$0.0435624\pi$
$761$	7.52709i	0.272857i	−0.990650	+	0.136428i	$0.956438\pi$
$762$	0	0
$763$	− 8.48792i	− 0.307283i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−60.5348	−2.18578
$768$	0	0
$769$	49.6002	1.78863	0.894314	−	0.447440i	$-0.147664\pi$
$769$	49.6002	1.78863	0.894314	+	0.447440i	$0.147664\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	− 45.5457i	− 1.63817i	−0.573676	−	0.819083i	$-0.694483\pi$
$773$	− 45.5457i	− 1.63817i	0.573676	−	0.819083i	$-0.305517\pi$
$774$	0	0
$775$	− 0.494334i	− 0.0177570i
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−8.37076	−0.299914
$780$	0	0
$781$	24.5302	0.877758
$782$	0	0
$783$	0	0
$784$	0	0
$785$	4.70875i	0.168063i
$786$	0	0
$787$	24.5359i	0.874610i	0.899313	+	0.437305i	$0.144067\pi$
$787$	24.5359i	0.874610i	−0.899313	+	0.437305i	$0.855933\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	5.60719	0.199369
$792$	0	0
$793$	−69.6060	−2.47178
$794$	0	0
$795$	0	0
$796$	0	0
$797$	25.2974i	0.896080i	0.894013	+	0.448040i	$0.147878\pi$
$797$	25.2974i	0.896080i	−0.894013	+	0.448040i	$0.852122\pi$
$798$	0	0
$799$	5.28571i	0.186995i
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−48.2465	−1.70258
$804$	0	0
$805$	−25.0839	−0.884093
$806$	0	0
$807$	0	0
$808$	0	0
$809$	− 23.3260i	− 0.820099i	−0.912063	−	0.410050i	$-0.865511\pi$
$809$	− 23.3260i	− 0.820099i	0.912063	−	0.410050i	$-0.134489\pi$
$810$	0	0
$811$	− 36.8400i	− 1.29363i	−0.762648	−	0.646814i	$-0.776101\pi$
$811$	− 36.8400i	− 1.29363i	0.762648	−	0.646814i	$-0.223899\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−23.2337	−0.813841
$816$	0	0
$817$	2.28341	0.0798865
$818$	0	0
$819$	0	0
$820$	0	0
$821$	54.7705i	1.91150i	0.294175	+	0.955752i	$0.404955\pi$
$821$	54.7705i	1.91150i	−0.294175	+	0.955752i	$0.595045\pi$
$822$	0	0
$823$	43.9701i	1.53270i	0.642423	+	0.766350i	$0.277929\pi$
$823$	43.9701i	1.53270i	−0.642423	+	0.766350i	$0.722071\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	24.0167	0.835142	0.417571	−	0.908644i	$-0.362882\pi$
$827$	24.0167	0.835142	0.417571	+	0.908644i	$0.362882\pi$
$828$	0	0
$829$	6.23391	0.216513	0.108256	−	0.994123i	$-0.465473\pi$
$829$	6.23391	0.216513	0.108256	+	0.994123i	$0.465473\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	2.03749i	0.0705950i
$834$	0	0
$835$	− 8.87601i	− 0.307167i
$836$	0	0
$837$	0	0
$838$	0	0
$839$	13.7221	0.473738	0.236869	−	0.971542i	$-0.423879\pi$
$839$	13.7221	0.473738	0.236869	+	0.971542i	$0.423879\pi$
$840$	0	0
$841$	26.6052	0.917422
$842$	0	0
$843$	0	0
$844$	0	0
$845$	28.3049i	0.973718i
$846$	0	0
$847$	− 5.98124i	− 0.205518i
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−74.8481	−2.56576
$852$	0	0
$853$	−36.8815	−1.26280	−0.631400	−	0.775457i	$-0.717519\pi$
$853$	−36.8815	−1.26280	−0.631400	+	0.775457i	$0.717519\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	6.35611i	0.217120i	0.994090	+	0.108560i	$0.0346240\pi$
$857$	6.35611i	0.217120i	−0.994090	+	0.108560i	$0.965376\pi$
$858$	0	0
$859$	− 25.3449i	− 0.864755i	−0.901693	−	0.432378i	$-0.857675\pi$
$859$	− 25.3449i	− 0.864755i	0.901693	−	0.432378i	$-0.142325\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	17.9486	0.610976	0.305488	−	0.952196i	$-0.401180\pi$
$863$	17.9486	0.610976	0.305488	+	0.952196i	$0.401180\pi$
$864$	0	0
$865$	−44.2606	−1.50491
$866$	0	0
$867$	0	0
$868$	0	0
$869$	30.8169i	1.04539i
$870$	0	0
$871$	− 31.7855i	− 1.07701i
$872$	0	0
$873$	0	0
$874$	0	0
$875$	7.95779	0.269022
$876$	0	0
$877$	−24.0001	−0.810427	−0.405213	−	0.914222i	$-0.632803\pi$
$877$	−24.0001	−0.810427	−0.405213	+	0.914222i	$0.632803\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	37.1921i	1.25303i	0.779408	+	0.626516i	$0.215520\pi$
$881$	37.1921i	1.25303i	−0.779408	+	0.626516i	$0.784480\pi$
$882$	0	0
$883$	48.2212i	1.62277i	0.584511	+	0.811386i	$0.301286\pi$
$883$	48.2212i	1.62277i	−0.584511	+	0.811386i	$0.698714\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	0.900487	0.0302354	0.0151177	−	0.999886i	$-0.495188\pi$
$887$	0.900487	0.0302354	0.0151177	+	0.999886i	$0.495188\pi$
$888$	0	0
$889$	16.0322	0.537702
$890$	0	0
$891$	0	0
$892$	0	0
$893$	− 10.8134i	− 0.361857i
$894$	0	0
$895$	34.6679i	1.15882i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−0.384393	−0.0128202
$900$	0	0
$901$	−23.5599	−0.784894
$902$	0	0
$903$	0	0
$904$	0	0
$905$	− 26.7533i	− 0.889311i
$906$	0	0
$907$	7.11861i	0.236370i	0.992992	+	0.118185i	$0.0377075\pi$
$907$	7.11861i	0.236370i	−0.992992	+	0.118185i	$0.962292\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	53.7843	1.78195	0.890977	−	0.454049i	$-0.150021\pi$
$911$	53.7843	1.78195	0.890977	+	0.454049i	$0.150021\pi$
$912$	0	0
$913$	16.2365	0.537350
$914$	0	0
$915$	0	0
$916$	0	0
$917$	2.05016i	0.0677021i
$918$	0	0
$919$	− 59.4104i	− 1.95977i	−0.199563	−	0.979885i	$-0.563952\pi$
$919$	− 59.4104i	− 1.95977i	0.199563	−	0.979885i	$-0.436048\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	28.9830	0.953987
$924$	0	0
$925$	−15.7002	−0.516218
$926$	0	0
$927$	0	0
$928$	0	0
$929$	32.3649i	1.06186i	0.847416	+	0.530930i	$0.178157\pi$
$929$	32.3649i	1.06186i	−0.847416	+	0.530930i	$0.821843\pi$
$930$	0	0
$931$	− 4.16827i	− 0.136610i
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−22.1985	−0.725968
$936$	0	0
$937$	−14.8314	−0.484521	−0.242260	−	0.970211i	$-0.577889\pi$
$937$	−14.8314	−0.484521	−0.242260	+	0.970211i	$0.577889\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	22.1324i	0.721494i	0.932664	+	0.360747i	$0.117478\pi$
$941$	22.1324i	0.721494i	−0.932664	+	0.360747i	$0.882522\pi$
$942$	0	0
$943$	− 19.0530i	− 0.620450i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	9.12874	0.296644	0.148322	−	0.988939i	$-0.452613\pi$
$947$	9.12874	0.296644	0.148322	+	0.988939i	$0.452613\pi$
$948$	0	0
$949$	−57.0044	−1.85044
$950$	0	0
$951$	0	0
$952$	0	0
$953$	− 4.40579i	− 0.142718i	−0.997451	−	0.0713588i	$-0.977266\pi$
$953$	− 4.40579i	− 0.142718i	0.997451	−	0.0713588i	$-0.0227335\pi$
$954$	0	0
$955$	18.8054i	0.608527i
$956$	0	0
$957$	0	0
$958$	0	0
$959$	10.4115	0.336206
$960$	0	0
$961$	30.9383	0.998010
$962$	0	0
$963$	0	0
$964$	0	0
$965$	24.3710i	0.784531i
$966$	0	0
$967$	− 19.7941i	− 0.636536i	−0.948001	−	0.318268i	$-0.896899\pi$
$967$	− 19.7941i	− 0.636536i	0.948001	−	0.318268i	$-0.103101\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	4.61141	0.147987	0.0739936	−	0.997259i	$-0.476426\pi$
$971$	4.61141	0.147987	0.0739936	+	0.997259i	$0.476426\pi$
$972$	0	0
$973$	−13.0177	−0.417328
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 15.0782i	− 0.482395i	−0.970476	−	0.241197i	$-0.922460\pi$
$977$	− 15.0782i	− 0.482395i	0.970476	−	0.241197i	$-0.0775401\pi$
$978$	0	0
$979$	53.2536i	1.70199i
$980$	0	0
$981$	0	0
$982$	0	0
$983$	35.7616	1.14062	0.570308	−	0.821431i	$-0.306824\pi$
$983$	35.7616	1.14062	0.570308	+	0.821431i	$0.306824\pi$
$984$	0	0
$985$	−22.2980	−0.710472
$986$	0	0
$987$	0	0
$988$	0	0
$989$	5.19735i	0.165266i
$990$	0	0
$991$	− 51.8448i	− 1.64690i	−0.567387	−	0.823451i	$-0.692046\pi$
$991$	− 51.8448i	− 1.64690i	0.567387	−	0.823451i	$-0.307954\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	1.96178	0.0621925
$996$	0	0
$997$	42.7850	1.35501	0.677507	−	0.735516i	$-0.263060\pi$
$997$	42.7850	1.35501	0.677507	+	0.735516i	$0.263060\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6048.2.h.i.2591.20	yes	24
3.2	odd	2	inner	6048.2.h.i.2591.5	✓	24
4.3	odd	2	inner	6048.2.h.i.2591.6	yes	24
12.11	even	2	inner	6048.2.h.i.2591.19	yes	24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6048.2.h.i.2591.5	✓	24	3.2	odd	2	inner
6048.2.h.i.2591.6	yes	24	4.3	odd	2	inner
6048.2.h.i.2591.19	yes	24	12.11	even	2	inner
6048.2.h.i.2591.20	yes	24	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.h (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+2.64388i q^{5} -1.00000i q^{7} +O(q^{10})\)	\(q+2.64388i q^{5} -1.00000i q^{7} -4.12083 q^{11} -4.86886 q^{13} -2.03749i q^{17} +4.16827i q^{19} -9.48754 q^{23} -1.99011 q^{25} +1.54751i q^{29} +0.248395i q^{31} +2.64388 q^{35} +7.88909 q^{37} +2.00821i q^{41} -0.547808i q^{43} -2.59422 q^{47} -1.00000 q^{49} -11.5632i q^{53} -10.8950i q^{55} +12.4330 q^{59} +14.2962 q^{61} -12.8727i q^{65} +6.52832i q^{67} -5.95273 q^{71} +11.7079 q^{73} +4.12083i q^{77} -7.47832i q^{79} -3.94011 q^{83} +5.38690 q^{85} -12.9230i q^{89} +4.86886i q^{91} -11.0204 q^{95} -5.64630 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q+O(q^{10}) \) 24 * q	\( 24 q - 8 q^{13} - 24 q^{25} + 48 q^{37} - 24 q^{49} - 48 q^{61} - 32 q^{73} - 80 q^{85} + 64 q^{97}+O(q^{100}) \) 24 * q - 8 * q^13 - 24 * q^25 + 48 * q^37 - 24 * q^49 - 48 * q^61 - 32 * q^73 - 80 * q^85 + 64 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more