LMFDB - Embedded newform 6048.2.h.i.2591.2

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			2591.2
Character	$\chi$	$=$	6048.2591
Dual form			6048.2.h.i.2591.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+4.29730i q^{5} -1.00000i q^{7} +O(q^{10})$	$q+4.29730i q^{5} -1.00000i q^{7} +4.77104 q^{11} -0.0836472 q^{13} +2.60889i q^{17} -4.85636i q^{19} -4.78044 q^{23} -13.4668 q^{25} -0.263176i q^{29} -0.304131i q^{31} +4.29730 q^{35} +9.56782 q^{37} +10.5898i q^{41} +4.24414i q^{43} +9.85829 q^{47} -1.00000 q^{49} +11.9022i q^{53} +20.5026i q^{55} -5.50796 q^{59} -8.62260 q^{61} -0.359457i q^{65} -1.80727i q^{67} +10.8966 q^{71} -4.88339 q^{73} -4.77104i q^{77} +14.3421i q^{79} -10.4067 q^{83} -11.2112 q^{85} +7.85368i q^{89} +0.0836472i q^{91} +20.8693 q^{95} -1.29824 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$	4.29730i	1.92181i	0.276875	+	0.960906i	$0.410701\pi$
$5$	4.29730i	1.92181i	−0.276875	+	0.960906i	$0.589299\pi$
$6$	0	0
$7$	− 1.00000i	− 0.377964i
$7$	− 1.00000i	− 0.377964i
$8$	0	0
$9$	0	0
$10$	0	0
$11$	4.77104	1.43852	0.719262	−	0.694739i	$-0.244480\pi$
$11$	4.77104	1.43852	0.719262	+	0.694739i	$0.244480\pi$
$12$	0	0
$13$	−0.0836472	−0.0231996	−0.0115998	−	0.999933i	$-0.503692\pi$
$13$	−0.0836472	−0.0231996	−0.0115998	+	0.999933i	$0.503692\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	2.60889i	0.632748i	0.948635	+	0.316374i	$0.102465\pi$
$17$	2.60889i	0.632748i	−0.948635	+	0.316374i	$0.897535\pi$
$18$	0	0
$19$	− 4.85636i	− 1.11413i	−0.830470	−	0.557063i	$-0.811928\pi$
$19$	− 4.85636i	− 1.11413i	0.830470	−	0.557063i	$-0.188072\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−4.78044	−0.996790	−0.498395	−	0.866950i	$-0.666077\pi$
$23$	−4.78044	−0.996790	−0.498395	+	0.866950i	$0.666077\pi$
$24$	0	0
$25$	−13.4668	−2.69336
$26$	0	0
$27$	0	0
$28$	0	0
$29$	− 0.263176i	− 0.0488706i	−0.999701	−	0.0244353i	$-0.992221\pi$
$29$	− 0.263176i	− 0.0488706i	0.999701	−	0.0244353i	$-0.00777877\pi$
$30$	0	0
$31$	− 0.304131i	− 0.0546236i	−0.999627	−	0.0273118i	$-0.991305\pi$
$31$	− 0.304131i	− 0.0546236i	0.999627	−	0.0273118i	$-0.00869469\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	4.29730	0.726377
$36$	0	0
$37$	9.56782	1.57294	0.786470	−	0.617628i	$-0.211906\pi$
$37$	9.56782	1.57294	0.786470	+	0.617628i	$0.211906\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	10.5898i	1.65385i	0.562310	+	0.826927i	$0.309913\pi$
$41$	10.5898i	1.65385i	−0.562310	+	0.826927i	$0.690087\pi$
$42$	0	0
$43$	4.24414i	0.647225i	0.946190	+	0.323613i	$0.104897\pi$
$43$	4.24414i	0.647225i	−0.946190	+	0.323613i	$0.895103\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	9.85829	1.43798	0.718990	−	0.695021i	$-0.244605\pi$
$47$	9.85829	1.43798	0.718990	+	0.695021i	$0.244605\pi$
$48$	0	0
$49$	−1.00000	−0.142857
$50$	0	0
$51$	0	0
$52$	0	0
$53$	11.9022i	1.63489i	0.576009	+	0.817443i	$0.304609\pi$
$53$	11.9022i	1.63489i	−0.576009	+	0.817443i	$0.695391\pi$
$54$	0	0
$55$	20.5026i	2.76457i
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−5.50796	−0.717076	−0.358538	−	0.933515i	$-0.616725\pi$
$59$	−5.50796	−0.717076	−0.358538	+	0.933515i	$0.616725\pi$
$60$	0	0
$61$	−8.62260	−1.10401	−0.552005	−	0.833841i	$-0.686137\pi$
$61$	−8.62260	−1.10401	−0.552005	+	0.833841i	$0.686137\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	− 0.359457i	− 0.0445852i
$66$	0	0
$67$	− 1.80727i	− 0.220793i	−0.993888	−	0.110396i	$-0.964788\pi$
$67$	− 1.80727i	− 0.220793i	0.993888	−	0.110396i	$-0.0352120\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	10.8966	1.29319	0.646597	−	0.762832i	$-0.276192\pi$
$71$	10.8966	1.29319	0.646597	+	0.762832i	$0.276192\pi$
$72$	0	0
$73$	−4.88339	−0.571557	−0.285779	−	0.958296i	$-0.592252\pi$
$73$	−4.88339	−0.571557	−0.285779	+	0.958296i	$0.592252\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 4.77104i	− 0.543711i
$78$	0	0
$79$	14.3421i	1.61362i	0.590814	+	0.806808i	$0.298807\pi$
$79$	14.3421i	1.61362i	−0.590814	+	0.806808i	$0.701193\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−10.4067	−1.14228	−0.571142	−	0.820852i	$-0.693499\pi$
$83$	−10.4067	−1.14228	−0.571142	+	0.820852i	$0.693499\pi$
$84$	0	0
$85$	−11.2112	−1.21602
$86$	0	0
$87$	0	0
$88$	0	0
$89$	7.85368i	0.832488i	0.909253	+	0.416244i	$0.136654\pi$
$89$	7.85368i	0.832488i	−0.909253	+	0.416244i	$0.863346\pi$
$90$	0	0
$91$	0.0836472i	0.00876861i
$92$	0	0
$93$	0	0
$94$	0	0
$95$	20.8693	2.14114
$96$	0	0
$97$	−1.29824	−0.131816	−0.0659082	−	0.997826i	$-0.520994\pi$
$97$	−1.29824	−0.131816	−0.0659082	+	0.997826i	$0.520994\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	11.5103i	1.14532i	0.819794	+	0.572659i	$0.194088\pi$
$101$	11.5103i	1.14532i	−0.819794	+	0.572659i	$0.805912\pi$
$102$	0	0
$103$	1.07843i	0.106261i	0.998588	+	0.0531306i	$0.0169200\pi$
$103$	1.07843i	0.106261i	−0.998588	+	0.0531306i	$0.983080\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	3.95223	0.382077	0.191038	−	0.981583i	$-0.438814\pi$
$107$	3.95223	0.382077	0.191038	+	0.981583i	$0.438814\pi$
$108$	0	0
$109$	1.68780	0.161662	0.0808311	−	0.996728i	$-0.474243\pi$
$109$	1.68780	0.161662	0.0808311	+	0.996728i	$0.474243\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	− 8.49874i	− 0.799494i	−0.916625	−	0.399747i	$-0.869098\pi$
$113$	− 8.49874i	− 0.799494i	0.916625	−	0.399747i	$-0.130902\pi$
$114$	0	0
$115$	− 20.5430i	− 1.91564i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	2.60889	0.239156
$120$	0	0
$121$	11.7629	1.06935
$122$	0	0
$123$	0	0
$124$	0	0
$125$	− 36.3844i	− 3.25432i
$126$	0	0
$127$	12.2578i	1.08770i	0.839181	+	0.543852i	$0.183035\pi$
$127$	12.2578i	1.08770i	−0.839181	+	0.543852i	$0.816965\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−10.2250	−0.893362	−0.446681	−	0.894693i	$-0.647394\pi$
$131$	−10.2250	−0.893362	−0.446681	+	0.894693i	$0.647394\pi$
$132$	0	0
$133$	−4.85636	−0.421100
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 22.2500i	− 1.90095i	−0.310806	−	0.950473i	$-0.600599\pi$
$137$	− 22.2500i	− 1.90095i	0.310806	−	0.950473i	$-0.399401\pi$
$138$	0	0
$139$	− 10.1276i	− 0.859010i	−0.903064	−	0.429505i	$-0.858688\pi$
$139$	− 10.1276i	− 0.859010i	0.903064	−	0.429505i	$-0.141312\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−0.399084	−0.0333731
$144$	0	0
$145$	1.13095	0.0939201
$146$	0	0
$147$	0	0
$148$	0	0
$149$	− 8.83161i	− 0.723514i	−0.932272	−	0.361757i	$-0.882177\pi$
$149$	− 8.83161i	− 0.723514i	0.932272	−	0.361757i	$-0.117823\pi$
$150$	0	0
$151$	− 4.18526i	− 0.340592i	−0.985393	−	0.170296i	$-0.945528\pi$
$151$	− 4.18526i	− 0.340592i	0.985393	−	0.170296i	$-0.0544723\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	1.30694	0.104976
$156$	0	0
$157$	−12.0045	−0.958062	−0.479031	−	0.877798i	$-0.659012\pi$
$157$	−12.0045	−0.958062	−0.479031	+	0.877798i	$0.659012\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	4.78044i	0.376751i
$162$	0	0
$163$	12.7022i	0.994909i	0.867490	+	0.497455i	$0.165732\pi$
$163$	12.7022i	0.994909i	−0.867490	+	0.497455i	$0.834268\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	20.2726	1.56874	0.784372	−	0.620290i	$-0.212985\pi$
$167$	20.2726	1.56874	0.784372	+	0.620290i	$0.212985\pi$
$168$	0	0
$169$	−12.9930	−0.999462
$170$	0	0
$171$	0	0
$172$	0	0
$173$	20.6754i	1.57192i	0.618278	+	0.785959i	$0.287830\pi$
$173$	20.6754i	1.57192i	−0.618278	+	0.785959i	$0.712170\pi$
$174$	0	0
$175$	13.4668i	1.01799i
$176$	0	0
$177$	0	0
$178$	0	0
$179$	4.67796	0.349647	0.174824	−	0.984600i	$-0.444064\pi$
$179$	4.67796	0.349647	0.174824	+	0.984600i	$0.444064\pi$
$180$	0	0
$181$	23.9336	1.77897	0.889485	−	0.456965i	$-0.151063\pi$
$181$	23.9336	1.77897	0.889485	+	0.456965i	$0.151063\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	41.1158i	3.02290i
$186$	0	0
$187$	12.4471i	0.910223i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−24.0841	−1.74267	−0.871334	−	0.490691i	$-0.836744\pi$
$191$	−24.0841	−1.74267	−0.871334	+	0.490691i	$0.836744\pi$
$192$	0	0
$193$	−7.68081	−0.552876	−0.276438	−	0.961032i	$-0.589154\pi$
$193$	−7.68081	−0.552876	−0.276438	+	0.961032i	$0.589154\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 22.5136i	− 1.60403i	−0.597305	−	0.802014i	$-0.703762\pi$
$197$	− 22.5136i	− 1.60403i	0.597305	−	0.802014i	$-0.296238\pi$
$198$	0	0
$199$	13.7258i	0.972996i	0.873682	+	0.486498i	$0.161726\pi$
$199$	13.7258i	0.972996i	−0.873682	+	0.486498i	$0.838274\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−0.263176	−0.0184713
$204$	0	0
$205$	−45.5077	−3.17840
$206$	0	0
$207$	0	0
$208$	0	0
$209$	− 23.1699i	− 1.60270i
$210$	0	0
$211$	− 2.67280i	− 0.184003i	−0.995759	−	0.0920017i	$-0.970673\pi$
$211$	− 2.67280i	− 0.184003i	0.995759	−	0.0920017i	$-0.0293265\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−18.2384	−1.24385
$216$	0	0
$217$	−0.304131	−0.0206458
$218$	0	0
$219$	0	0
$220$	0	0
$221$	− 0.218226i	− 0.0146795i
$222$	0	0
$223$	13.8944i	0.930435i	0.885196	+	0.465217i	$0.154024\pi$
$223$	13.8944i	0.930435i	−0.885196	+	0.465217i	$0.845976\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−16.7141	−1.10935	−0.554677	−	0.832065i	$-0.687158\pi$
$227$	−16.7141	−1.10935	−0.554677	+	0.832065i	$0.687158\pi$
$228$	0	0
$229$	−9.78011	−0.646288	−0.323144	−	0.946350i	$-0.604740\pi$
$229$	−9.78011	−0.646288	−0.323144	+	0.946350i	$0.604740\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	− 7.69348i	− 0.504016i	−0.967725	−	0.252008i	$-0.918909\pi$
$233$	− 7.69348i	− 0.504016i	0.967725	−	0.252008i	$-0.0810910\pi$
$234$	0	0
$235$	42.3641i	2.76353i
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−22.5536	−1.45887	−0.729436	−	0.684049i	$-0.760217\pi$
$239$	−22.5536	−1.45887	−0.729436	+	0.684049i	$0.760217\pi$
$240$	0	0
$241$	4.08026	0.262833	0.131416	−	0.991327i	$-0.458047\pi$
$241$	4.08026	0.262833	0.131416	+	0.991327i	$0.458047\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	− 4.29730i	− 0.274545i
$246$	0	0
$247$	0.406221i	0.0258472i
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−9.67785	−0.610861	−0.305430	−	0.952214i	$-0.598800\pi$
$251$	−9.67785	−0.610861	−0.305430	+	0.952214i	$0.598800\pi$
$252$	0	0
$253$	−22.8077	−1.43391
$254$	0	0
$255$	0	0
$256$	0	0
$257$	3.83710i	0.239352i	0.992813	+	0.119676i	$0.0381856\pi$
$257$	3.83710i	0.239352i	−0.992813	+	0.119676i	$0.961814\pi$
$258$	0	0
$259$	− 9.56782i	− 0.594516i
$260$	0	0
$261$	0	0
$262$	0	0
$263$	9.29994	0.573459	0.286729	−	0.958012i	$-0.407432\pi$
$263$	9.29994	0.573459	0.286729	+	0.958012i	$0.407432\pi$
$264$	0	0
$265$	−51.1472	−3.14194
$266$	0	0
$267$	0	0
$268$	0	0
$269$	18.7782i	1.14493i	0.819931	+	0.572463i	$0.194012\pi$
$269$	18.7782i	1.14493i	−0.819931	+	0.572463i	$0.805988\pi$
$270$	0	0
$271$	30.7720i	1.86927i	0.355614	+	0.934633i	$0.384272\pi$
$271$	30.7720i	1.86927i	−0.355614	+	0.934633i	$0.615728\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−64.2507	−3.87446
$276$	0	0
$277$	16.7278	1.00508	0.502538	−	0.864555i	$-0.332400\pi$
$277$	16.7278	1.00508	0.502538	+	0.864555i	$0.332400\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	− 16.3993i	− 0.978298i	−0.872200	−	0.489149i	$-0.837307\pi$
$281$	− 16.3993i	− 0.978298i	0.872200	−	0.489149i	$-0.162693\pi$
$282$	0	0
$283$	11.0392i	0.656215i	0.944640	+	0.328108i	$0.106411\pi$
$283$	11.0392i	0.656215i	−0.944640	+	0.328108i	$0.893589\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	10.5898	0.625098
$288$	0	0
$289$	10.1937	0.599630
$290$	0	0
$291$	0	0
$292$	0	0
$293$	12.1045i	0.707152i	0.935406	+	0.353576i	$0.115034\pi$
$293$	12.1045i	0.707152i	−0.935406	+	0.353576i	$0.884966\pi$
$294$	0	0
$295$	− 23.6694i	− 1.37808i
$296$	0	0
$297$	0	0
$298$	0	0
$299$	0.399870	0.0231251
$300$	0	0
$301$	4.24414	0.244628
$302$	0	0
$303$	0	0
$304$	0	0
$305$	− 37.0539i	− 2.12170i
$306$	0	0
$307$	− 6.71383i	− 0.383179i	−0.981475	−	0.191589i	$-0.938636\pi$
$307$	− 6.71383i	− 0.383179i	0.981475	−	0.191589i	$-0.0613642\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−20.7974	−1.17931	−0.589656	−	0.807655i	$-0.700737\pi$
$311$	−20.7974	−1.17931	−0.589656	+	0.807655i	$0.700737\pi$
$312$	0	0
$313$	32.0502	1.81159	0.905793	−	0.423721i	$-0.139276\pi$
$313$	32.0502	1.81159	0.905793	+	0.423721i	$0.139276\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	8.53433i	0.479336i	0.970855	+	0.239668i	$0.0770386\pi$
$317$	8.53433i	0.479336i	−0.970855	+	0.239668i	$0.922961\pi$
$318$	0	0
$319$	− 1.25563i	− 0.0703015i
$320$	0	0
$321$	0	0
$322$	0	0
$323$	12.6697	0.704961
$324$	0	0
$325$	1.12646	0.0624848
$326$	0	0
$327$	0	0
$328$	0	0
$329$	− 9.85829i	− 0.543505i
$330$	0	0
$331$	− 4.90328i	− 0.269509i	−0.990879	−	0.134754i	$-0.956975\pi$
$331$	− 4.90328i	− 0.269509i	0.990879	−	0.134754i	$-0.0430245\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	7.76637	0.424322
$336$	0	0
$337$	14.6839	0.799883	0.399942	−	0.916541i	$-0.369030\pi$
$337$	14.6839	0.799883	0.399942	+	0.916541i	$0.369030\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	− 1.45102i	− 0.0785773i
$342$	0	0
$343$	1.00000i	0.0539949i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−23.1623	−1.24342	−0.621708	−	0.783249i	$-0.713561\pi$
$347$	−23.1623	−1.24342	−0.621708	+	0.783249i	$0.713561\pi$
$348$	0	0
$349$	−0.685492	−0.0366936	−0.0183468	−	0.999832i	$-0.505840\pi$
$349$	−0.685492	−0.0366936	−0.0183468	+	0.999832i	$0.505840\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	− 4.88650i	− 0.260082i	−0.991509	−	0.130041i	$-0.958489\pi$
$353$	− 4.88650i	− 0.260082i	0.991509	−	0.130041i	$-0.0415109\pi$
$354$	0	0
$355$	46.8262i	2.48527i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	16.5643	0.874228	0.437114	−	0.899406i	$-0.356001\pi$
$359$	16.5643	0.874228	0.437114	+	0.899406i	$0.356001\pi$
$360$	0	0
$361$	−4.58426	−0.241277
$362$	0	0
$363$	0	0
$364$	0	0
$365$	− 20.9854i	− 1.09843i
$366$	0	0
$367$	3.56155i	0.185912i	0.995670	+	0.0929558i	$0.0296315\pi$
$367$	3.56155i	0.185912i	−0.995670	+	0.0929558i	$0.970368\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	11.9022	0.617929
$372$	0	0
$373$	2.22788	0.115355	0.0576775	−	0.998335i	$-0.481630\pi$
$373$	2.22788	0.115355	0.0576775	+	0.998335i	$0.481630\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	0.0220139i	0.00113378i
$378$	0	0
$379$	− 31.3323i	− 1.60943i	−0.593660	−	0.804716i	$-0.702318\pi$
$379$	− 31.3323i	− 1.60943i	0.593660	−	0.804716i	$-0.297682\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	3.47176	0.177399	0.0886994	−	0.996058i	$-0.471729\pi$
$383$	3.47176	0.177399	0.0886994	+	0.996058i	$0.471729\pi$
$384$	0	0
$385$	20.5026	1.04491
$386$	0	0
$387$	0	0
$388$	0	0
$389$	− 14.8062i	− 0.750705i	−0.926882	−	0.375353i	$-0.877522\pi$
$389$	− 14.8062i	− 0.750705i	0.926882	−	0.375353i	$-0.122478\pi$
$390$	0	0
$391$	− 12.4716i	− 0.630717i
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−61.6324	−3.10106
$396$	0	0
$397$	2.64446	0.132722	0.0663609	−	0.997796i	$-0.478861\pi$
$397$	2.64446	0.132722	0.0663609	+	0.997796i	$0.478861\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	32.2306i	1.60952i	0.593600	+	0.804760i	$0.297706\pi$
$401$	32.2306i	1.60952i	−0.593600	+	0.804760i	$0.702294\pi$
$402$	0	0
$403$	0.0254397i	0.00126724i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	45.6485	2.26271
$408$	0	0
$409$	−29.1725	−1.44249	−0.721244	−	0.692681i	$-0.756430\pi$
$409$	−29.1725	−1.44249	−0.721244	+	0.692681i	$0.756430\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	5.50796i	0.271029i
$414$	0	0
$415$	− 44.7207i	− 2.19525i
$416$	0	0
$417$	0	0
$418$	0	0
$419$	1.33707	0.0653201	0.0326601	−	0.999467i	$-0.489602\pi$
$419$	1.33707	0.0653201	0.0326601	+	0.999467i	$0.489602\pi$
$420$	0	0
$421$	−28.9359	−1.41025	−0.705124	−	0.709084i	$-0.749109\pi$
$421$	−28.9359	−1.41025	−0.705124	+	0.709084i	$0.749109\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	− 35.1334i	− 1.70422i
$426$	0	0
$427$	8.62260i	0.417277i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	24.9335	1.20101	0.600503	−	0.799623i	$-0.294967\pi$
$431$	24.9335	1.20101	0.600503	+	0.799623i	$0.294967\pi$
$432$	0	0
$433$	−15.2671	−0.733688	−0.366844	−	0.930282i	$-0.619562\pi$
$433$	−15.2671	−0.733688	−0.366844	+	0.930282i	$0.619562\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	23.2155i	1.11055i
$438$	0	0
$439$	− 21.0126i	− 1.00288i	−0.865193	−	0.501438i	$-0.832804\pi$
$439$	− 21.0126i	− 1.00288i	0.865193	−	0.501438i	$-0.167196\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	3.73048	0.177240	0.0886202	−	0.996065i	$-0.471754\pi$
$443$	3.73048	0.177240	0.0886202	+	0.996065i	$0.471754\pi$
$444$	0	0
$445$	−33.7496	−1.59989
$446$	0	0
$447$	0	0
$448$	0	0
$449$	22.1203i	1.04392i	0.852969	+	0.521961i	$0.174800\pi$
$449$	22.1203i	1.04392i	−0.852969	+	0.521961i	$0.825200\pi$
$450$	0	0
$451$	50.5246i	2.37911i
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−0.359457	−0.0168516
$456$	0	0
$457$	−28.5181	−1.33402	−0.667011	−	0.745048i	$-0.732427\pi$
$457$	−28.5181	−1.33402	−0.667011	+	0.745048i	$0.732427\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	1.01774i	0.0474010i	0.999719	+	0.0237005i	$0.00754482\pi$
$461$	1.01774i	0.0474010i	−0.999719	+	0.0237005i	$0.992455\pi$
$462$	0	0
$463$	− 6.24844i	− 0.290389i	−0.989403	−	0.145195i	$-0.953619\pi$
$463$	− 6.24844i	− 0.290389i	0.989403	−	0.145195i	$-0.0463809\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	21.0484	0.974002	0.487001	−	0.873401i	$-0.338091\pi$
$467$	21.0484	0.974002	0.487001	+	0.873401i	$0.338091\pi$
$468$	0	0
$469$	−1.80727	−0.0834518
$470$	0	0
$471$	0	0
$472$	0	0
$473$	20.2490i	0.931049i
$474$	0	0
$475$	65.3997i	3.00074i
$476$	0	0
$477$	0	0
$478$	0	0
$479$	18.3215	0.837132	0.418566	−	0.908186i	$-0.362533\pi$
$479$	18.3215	0.837132	0.418566	+	0.908186i	$0.362533\pi$
$480$	0	0
$481$	−0.800322	−0.0364915
$482$	0	0
$483$	0	0
$484$	0	0
$485$	− 5.57894i	− 0.253326i
$486$	0	0
$487$	− 36.0381i	− 1.63304i	−0.577317	−	0.816520i	$-0.695900\pi$
$487$	− 36.0381i	− 1.63304i	0.577317	−	0.816520i	$-0.304100\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	19.2411	0.868341	0.434170	−	0.900831i	$-0.357042\pi$
$491$	19.2411	0.868341	0.434170	+	0.900831i	$0.357042\pi$
$492$	0	0
$493$	0.686597	0.0309228
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 10.8966i	− 0.488781i
$498$	0	0
$499$	14.6319i	0.655015i	0.944849	+	0.327508i	$0.106209\pi$
$499$	14.6319i	0.655015i	−0.944849	+	0.327508i	$0.893791\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	42.3558	1.88855	0.944275	−	0.329157i	$-0.106765\pi$
$503$	42.3558	1.88855	0.944275	+	0.329157i	$0.106765\pi$
$504$	0	0
$505$	−49.4632	−2.20109
$506$	0	0
$507$	0	0
$508$	0	0
$509$	− 1.90879i	− 0.0846056i	−0.999105	−	0.0423028i	$-0.986531\pi$
$509$	− 1.90879i	− 0.0846056i	0.999105	−	0.0423028i	$-0.0134694\pi$
$510$	0	0
$511$	4.88339i	0.216028i
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−4.63435	−0.204214
$516$	0	0
$517$	47.0344	2.06857
$518$	0	0
$519$	0	0
$520$	0	0
$521$	− 0.934351i	− 0.0409347i	−0.999791	−	0.0204673i	$-0.993485\pi$
$521$	− 0.934351i	− 0.0409347i	0.999791	−	0.0204673i	$-0.00651541\pi$
$522$	0	0
$523$	− 13.3676i	− 0.584525i	−0.956338	−	0.292262i	$-0.905592\pi$
$523$	− 13.3676i	− 0.584525i	0.956338	−	0.292262i	$-0.0944081\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	0.793444	0.0345630
$528$	0	0
$529$	−0.147435	−0.00641022
$530$	0	0
$531$	0	0
$532$	0	0
$533$	− 0.885810i	− 0.0383687i
$534$	0	0
$535$	16.9839i	0.734280i
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−4.77104	−0.205503
$540$	0	0
$541$	−45.2077	−1.94363	−0.971816	−	0.235740i	$-0.924249\pi$
$541$	−45.2077	−1.94363	−0.971816	+	0.235740i	$0.924249\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	7.25300i	0.310684i
$546$	0	0
$547$	− 6.59374i	− 0.281928i	−0.990015	−	0.140964i	$-0.954980\pi$
$547$	− 6.59374i	− 0.281928i	0.990015	−	0.140964i	$-0.0450202\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−1.27808	−0.0544480
$552$	0	0
$553$	14.3421	0.609889
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 35.9511i	− 1.52330i	−0.647991	−	0.761648i	$-0.724391\pi$
$557$	− 35.9511i	− 1.52330i	0.647991	−	0.761648i	$-0.275609\pi$
$558$	0	0
$559$	− 0.355011i	− 0.0150153i
$560$	0	0
$561$	0	0
$562$	0	0
$563$	7.93375	0.334368	0.167184	−	0.985926i	$-0.446533\pi$
$563$	7.93375	0.334368	0.167184	+	0.985926i	$0.446533\pi$
$564$	0	0
$565$	36.5217	1.53648
$566$	0	0
$567$	0	0
$568$	0	0
$569$	45.2629i	1.89752i	0.315999	+	0.948759i	$0.397660\pi$
$569$	45.2629i	1.89752i	−0.315999	+	0.948759i	$0.602340\pi$
$570$	0	0
$571$	14.7251i	0.616227i	0.951350	+	0.308114i	$0.0996977\pi$
$571$	14.7251i	0.616227i	−0.951350	+	0.308114i	$0.900302\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	64.3772	2.68471
$576$	0	0
$577$	−38.0893	−1.58568	−0.792840	−	0.609430i	$-0.791398\pi$
$577$	−38.0893	−1.58568	−0.792840	+	0.609430i	$0.791398\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	10.4067i	0.431742i
$582$	0	0
$583$	56.7857i	2.35182i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	17.3877	0.717668	0.358834	−	0.933401i	$-0.383174\pi$
$587$	17.3877	0.717668	0.358834	+	0.933401i	$0.383174\pi$
$588$	0	0
$589$	−1.47697	−0.0608575
$590$	0	0
$591$	0	0
$592$	0	0
$593$	− 0.551386i	− 0.0226427i	−0.999936	−	0.0113214i	$-0.996396\pi$
$593$	− 0.551386i	− 0.0226427i	0.999936	−	0.0113214i	$-0.00360378\pi$
$594$	0	0
$595$	11.2112i	0.459613i
$596$	0	0
$597$	0	0
$598$	0	0
$599$	4.43049	0.181025	0.0905124	−	0.995895i	$-0.471150\pi$
$599$	4.43049	0.181025	0.0905124	+	0.995895i	$0.471150\pi$
$600$	0	0
$601$	−29.9890	−1.22328	−0.611638	−	0.791138i	$-0.709489\pi$
$601$	−29.9890	−1.22328	−0.611638	+	0.791138i	$0.709489\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	50.5486i	2.05509i
$606$	0	0
$607$	− 0.0521455i	− 0.00211652i	−0.999999	−	0.00105826i	$-0.999663\pi$
$607$	− 0.0521455i	− 0.00211652i	0.999999	−	0.00105826i	$-0.000336854\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−0.824619	−0.0333605
$612$	0	0
$613$	21.4946	0.868157	0.434078	−	0.900875i	$-0.357074\pi$
$613$	21.4946	0.868157	0.434078	+	0.900875i	$0.357074\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	3.81413i	0.153551i	0.997048	+	0.0767756i	$0.0244625\pi$
$617$	3.81413i	0.153551i	−0.997048	+	0.0767756i	$0.975537\pi$
$618$	0	0
$619$	− 38.3553i	− 1.54163i	−0.637060	−	0.770814i	$-0.719850\pi$
$619$	− 38.3553i	− 1.54163i	0.637060	−	0.770814i	$-0.280150\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	7.85368	0.314651
$624$	0	0
$625$	89.0208	3.56083
$626$	0	0
$627$	0	0
$628$	0	0
$629$	24.9614i	0.995275i
$630$	0	0
$631$	− 13.7903i	− 0.548982i	−0.961590	−	0.274491i	$-0.911491\pi$
$631$	− 13.7903i	− 0.548982i	0.961590	−	0.274491i	$-0.0885094\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−52.6755	−2.09036
$636$	0	0
$637$	0.0836472	0.00331422
$638$	0	0
$639$	0	0
$640$	0	0
$641$	39.7307i	1.56927i	0.619960	+	0.784634i	$0.287149\pi$
$641$	39.7307i	1.56927i	−0.619960	+	0.784634i	$0.712851\pi$
$642$	0	0
$643$	− 35.6618i	− 1.40637i	−0.711009	−	0.703183i	$-0.751761\pi$
$643$	− 35.6618i	− 1.40637i	0.711009	−	0.703183i	$-0.248239\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−25.5540	−1.00463	−0.502315	−	0.864685i	$-0.667518\pi$
$647$	−25.5540	−1.00463	−0.502315	+	0.864685i	$0.667518\pi$
$648$	0	0
$649$	−26.2787	−1.03153
$650$	0	0
$651$	0	0
$652$	0	0
$653$	3.13082i	0.122518i	0.998122	+	0.0612591i	$0.0195116\pi$
$653$	3.13082i	0.122518i	−0.998122	+	0.0612591i	$0.980488\pi$
$654$	0	0
$655$	− 43.9399i	− 1.71687i
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−17.9580	−0.699543	−0.349771	−	0.936835i	$-0.613741\pi$
$659$	−17.9580	−0.699543	−0.349771	+	0.936835i	$0.613741\pi$
$660$	0	0
$661$	−18.3081	−0.712103	−0.356051	−	0.934466i	$-0.615877\pi$
$661$	−18.3081	−0.712103	−0.356051	+	0.934466i	$0.615877\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	− 20.8693i	− 0.809275i
$666$	0	0
$667$	1.25810i	0.0487137i
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−41.1388	−1.58815
$672$	0	0
$673$	33.2016	1.27983	0.639914	−	0.768447i	$-0.278970\pi$
$673$	33.2016	1.27983	0.639914	+	0.768447i	$0.278970\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	22.3678i	0.859663i	0.902909	+	0.429832i	$0.141427\pi$
$677$	22.3678i	0.859663i	−0.902909	+	0.429832i	$0.858573\pi$
$678$	0	0
$679$	1.29824i	0.0498220i
$680$	0	0
$681$	0	0
$682$	0	0
$683$	44.5984	1.70651	0.853256	−	0.521493i	$-0.174625\pi$
$683$	44.5984	1.70651	0.853256	+	0.521493i	$0.174625\pi$
$684$	0	0
$685$	95.6150	3.65326
$686$	0	0
$687$	0	0
$688$	0	0
$689$	− 0.995582i	− 0.0379286i
$690$	0	0
$691$	25.9307i	0.986450i	0.869902	+	0.493225i	$0.164182\pi$
$691$	25.9307i	0.986450i	−0.869902	+	0.493225i	$0.835818\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	43.5213	1.65086
$696$	0	0
$697$	−27.6277	−1.04647
$698$	0	0
$699$	0	0
$700$	0	0
$701$	− 24.3122i	− 0.918258i	−0.888370	−	0.459129i	$-0.848162\pi$
$701$	− 24.3122i	− 0.918258i	0.888370	−	0.459129i	$-0.151838\pi$
$702$	0	0
$703$	− 46.4648i	− 1.75245i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	11.5103	0.432890
$708$	0	0
$709$	−12.6409	−0.474740	−0.237370	−	0.971419i	$-0.576285\pi$
$709$	−12.6409	−0.474740	−0.237370	+	0.971419i	$0.576285\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	1.45388i	0.0544482i
$714$	0	0
$715$	− 1.71499i	− 0.0641369i
$716$	0	0
$717$	0	0
$718$	0	0
$719$	44.9044	1.67465	0.837325	−	0.546705i	$-0.184118\pi$
$719$	44.9044	1.67465	0.837325	+	0.546705i	$0.184118\pi$
$720$	0	0
$721$	1.07843	0.0401629
$722$	0	0
$723$	0	0
$724$	0	0
$725$	3.54414i	0.131626i
$726$	0	0
$727$	19.8025i	0.734434i	0.930135	+	0.367217i	$0.119689\pi$
$727$	19.8025i	0.734434i	−0.930135	+	0.367217i	$0.880311\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−11.0725	−0.409531
$732$	0	0
$733$	−28.3746	−1.04804	−0.524019	−	0.851707i	$-0.675568\pi$
$733$	−28.3746	−1.04804	−0.524019	+	0.851707i	$0.675568\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 8.62255i	− 0.317616i
$738$	0	0
$739$	− 13.8535i	− 0.509608i	−0.966993	−	0.254804i	$-0.917989\pi$
$739$	− 13.8535i	− 0.509608i	0.966993	−	0.254804i	$-0.0820109\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	9.90143	0.363248	0.181624	−	0.983368i	$-0.441865\pi$
$743$	9.90143	0.363248	0.181624	+	0.983368i	$0.441865\pi$
$744$	0	0
$745$	37.9521	1.39046
$746$	0	0
$747$	0	0
$748$	0	0
$749$	− 3.95223i	− 0.144411i
$750$	0	0
$751$	− 5.39410i	− 0.196833i	−0.995145	−	0.0984167i	$-0.968622\pi$
$751$	− 5.39410i	− 0.196833i	0.995145	−	0.0984167i	$-0.0313778\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	17.9853	0.654553
$756$	0	0
$757$	−8.40323	−0.305421	−0.152710	−	0.988271i	$-0.548800\pi$
$757$	−8.40323	−0.305421	−0.152710	+	0.988271i	$0.548800\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	34.7139i	1.25838i	0.777252	+	0.629190i	$0.216613\pi$
$761$	34.7139i	1.25838i	−0.777252	+	0.629190i	$0.783387\pi$
$762$	0	0
$763$	− 1.68780i	− 0.0611026i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	0.460726	0.0166358
$768$	0	0
$769$	45.7418	1.64949	0.824746	−	0.565504i	$-0.191318\pi$
$769$	45.7418	1.64949	0.824746	+	0.565504i	$0.191318\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	8.53248i	0.306892i	0.988157	+	0.153446i	$0.0490371\pi$
$773$	8.53248i	0.306892i	−0.988157	+	0.153446i	$0.950963\pi$
$774$	0	0
$775$	4.09567i	0.147121i
$776$	0	0
$777$	0	0
$778$	0	0
$779$	51.4281	1.84260
$780$	0	0
$781$	51.9884	1.86029
$782$	0	0
$783$	0	0
$784$	0	0
$785$	− 51.5869i	− 1.84122i
$786$	0	0
$787$	− 30.5916i	− 1.09047i	−0.838282	−	0.545237i	$-0.816440\pi$
$787$	− 30.5916i	− 1.09047i	0.838282	−	0.545237i	$-0.183560\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−8.49874	−0.302180
$792$	0	0
$793$	0.721256	0.0256126
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 6.66452i	− 0.236070i	−0.993009	−	0.118035i	$-0.962341\pi$
$797$	− 6.66452i	− 0.236070i	0.993009	−	0.118035i	$-0.0376594\pi$
$798$	0	0
$799$	25.7192i	0.909879i
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−23.2989	−0.822199
$804$	0	0
$805$	−20.5430	−0.724045
$806$	0	0
$807$	0	0
$808$	0	0
$809$	32.5914i	1.14585i	0.819607	+	0.572926i	$0.194192\pi$
$809$	32.5914i	1.14585i	−0.819607	+	0.572926i	$0.805808\pi$
$810$	0	0
$811$	− 1.99304i	− 0.0699851i	−0.999388	−	0.0349926i	$-0.988859\pi$
$811$	− 1.99304i	− 0.0699851i	0.999388	−	0.0349926i	$-0.0111408\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−54.5850	−1.91203
$816$	0	0
$817$	20.6111	0.721091
$818$	0	0
$819$	0	0
$820$	0	0
$821$	9.77861i	0.341276i	0.985334	+	0.170638i	$0.0545829\pi$
$821$	9.77861i	0.341276i	−0.985334	+	0.170638i	$0.945417\pi$
$822$	0	0
$823$	2.33642i	0.0814425i	0.999171	+	0.0407213i	$0.0129656\pi$
$823$	2.33642i	0.0814425i	−0.999171	+	0.0407213i	$0.987034\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−15.1072	−0.525328	−0.262664	−	0.964887i	$-0.584601\pi$
$827$	−15.1072	−0.525328	−0.262664	+	0.964887i	$0.584601\pi$
$828$	0	0
$829$	6.56565	0.228034	0.114017	−	0.993479i	$-0.463628\pi$
$829$	6.56565	0.228034	0.114017	+	0.993479i	$0.463628\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	− 2.60889i	− 0.0903926i
$834$	0	0
$835$	87.1177i	3.01483i
$836$	0	0
$837$	0	0
$838$	0	0
$839$	31.9747	1.10389	0.551944	−	0.833881i	$-0.313886\pi$
$839$	31.9747	1.10389	0.551944	+	0.833881i	$0.313886\pi$
$840$	0	0
$841$	28.9307	0.997612
$842$	0	0
$843$	0	0
$844$	0	0
$845$	− 55.8349i	− 1.92078i
$846$	0	0
$847$	− 11.7629i	− 0.404177i
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−45.7384	−1.56789
$852$	0	0
$853$	17.2085	0.589209	0.294604	−	0.955619i	$-0.404812\pi$
$853$	17.2085	0.589209	0.294604	+	0.955619i	$0.404812\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	10.2007i	0.348450i	0.984706	+	0.174225i	$0.0557419\pi$
$857$	10.2007i	0.348450i	−0.984706	+	0.174225i	$0.944258\pi$
$858$	0	0
$859$	36.5760i	1.24796i	0.781441	+	0.623979i	$0.214485\pi$
$859$	36.5760i	1.24796i	−0.781441	+	0.623979i	$0.785515\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−18.4332	−0.627472	−0.313736	−	0.949510i	$-0.601581\pi$
$863$	−18.4332	−0.627472	−0.313736	+	0.949510i	$0.601581\pi$
$864$	0	0
$865$	−88.8482	−3.02093
$866$	0	0
$867$	0	0
$868$	0	0
$869$	68.4269i	2.32122i
$870$	0	0
$871$	0.151173i	0.00512229i
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−36.3844	−1.23002
$876$	0	0
$877$	56.2219	1.89848	0.949240	−	0.314552i	$-0.101854\pi$
$877$	56.2219	1.89848	0.949240	+	0.314552i	$0.101854\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	4.61624i	0.155525i	0.996972	+	0.0777625i	$0.0247776\pi$
$881$	4.61624i	0.155525i	−0.996972	+	0.0777625i	$0.975222\pi$
$882$	0	0
$883$	− 52.5589i	− 1.76875i	−0.466778	−	0.884374i	$-0.654585\pi$
$883$	− 52.5589i	− 1.76875i	0.466778	−	0.884374i	$-0.345415\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	12.4298	0.417351	0.208675	−	0.977985i	$-0.433085\pi$
$887$	12.4298	0.417351	0.208675	+	0.977985i	$0.433085\pi$
$888$	0	0
$889$	12.2578	0.411114
$890$	0	0
$891$	0	0
$892$	0	0
$893$	− 47.8754i	− 1.60209i
$894$	0	0
$895$	20.1026i	0.671957i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−0.0800401	−0.00266949
$900$	0	0
$901$	−31.0514	−1.03447
$902$	0	0
$903$	0	0
$904$	0	0
$905$	102.850i	3.41884i
$906$	0	0
$907$	− 27.4245i	− 0.910617i	−0.890334	−	0.455308i	$-0.849529\pi$
$907$	− 27.4245i	− 0.910617i	0.890334	−	0.455308i	$-0.150471\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−49.0785	−1.62604	−0.813022	−	0.582233i	$-0.802179\pi$
$911$	−49.0785	−1.62604	−0.813022	+	0.582233i	$0.802179\pi$
$912$	0	0
$913$	−49.6508	−1.64320
$914$	0	0
$915$	0	0
$916$	0	0
$917$	10.2250i	0.337659i
$918$	0	0
$919$	− 5.08992i	− 0.167901i	−0.996470	−	0.0839505i	$-0.973246\pi$
$919$	− 5.08992i	− 0.167901i	0.996470	−	0.0839505i	$-0.0267538\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−0.911474	−0.0300015
$924$	0	0
$925$	−128.848	−4.23650
$926$	0	0
$927$	0	0
$928$	0	0
$929$	− 28.5127i	− 0.935473i	−0.883868	−	0.467736i	$-0.845070\pi$
$929$	− 28.5127i	− 0.935473i	0.883868	−	0.467736i	$-0.154930\pi$
$930$	0	0
$931$	4.85636i	0.159161i
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−53.4890	−1.74928
$936$	0	0
$937$	−16.9605	−0.554075	−0.277037	−	0.960859i	$-0.589353\pi$
$937$	−16.9605	−0.554075	−0.277037	+	0.960859i	$0.589353\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	0.636605i	0.0207527i	0.999946	+	0.0103764i	$0.00330296\pi$
$941$	0.636605i	0.0207527i	−0.999946	+	0.0103764i	$0.996697\pi$
$942$	0	0
$943$	− 50.6240i	− 1.64854i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	3.81589	0.124000	0.0619999	−	0.998076i	$-0.480252\pi$
$947$	3.81589	0.124000	0.0619999	+	0.998076i	$0.480252\pi$
$948$	0	0
$949$	0.408482	0.0132599
$950$	0	0
$951$	0	0
$952$	0	0
$953$	4.41783i	0.143108i	0.997437	+	0.0715538i	$0.0227958\pi$
$953$	4.41783i	0.143108i	−0.997437	+	0.0715538i	$0.977204\pi$
$954$	0	0
$955$	− 103.497i	− 3.34908i
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−22.2500	−0.718490
$960$	0	0
$961$	30.9075	0.997016
$962$	0	0
$963$	0	0
$964$	0	0
$965$	− 33.0067i	− 1.06252i
$966$	0	0
$967$	− 53.4421i	− 1.71858i	−0.511487	−	0.859291i	$-0.670905\pi$
$967$	− 53.4421i	− 1.71858i	0.511487	−	0.859291i	$-0.329095\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−52.7188	−1.69183	−0.845913	−	0.533321i	$-0.820944\pi$
$971$	−52.7188	−1.69183	−0.845913	+	0.533321i	$0.820944\pi$
$972$	0	0
$973$	−10.1276	−0.324675
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 42.0146i	− 1.34417i	−0.740476	−	0.672083i	$-0.765400\pi$
$977$	− 42.0146i	− 1.34417i	0.740476	−	0.672083i	$-0.234600\pi$
$978$	0	0
$979$	37.4702i	1.19755i
$980$	0	0
$981$	0	0
$982$	0	0
$983$	34.9736	1.11549	0.557743	−	0.830014i	$-0.311668\pi$
$983$	34.9736	1.11549	0.557743	+	0.830014i	$0.311668\pi$
$984$	0	0
$985$	96.7478	3.08264
$986$	0	0
$987$	0	0
$988$	0	0
$989$	− 20.2888i	− 0.645148i
$990$	0	0
$991$	42.1138i	1.33779i	0.743357	+	0.668894i	$0.233232\pi$
$991$	42.1138i	1.33779i	−0.743357	+	0.668894i	$0.766768\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−58.9839	−1.86991
$996$	0	0
$997$	9.13760	0.289391	0.144695	−	0.989476i	$-0.453780\pi$
$997$	9.13760	0.289391	0.144695	+	0.989476i	$0.453780\pi$
$998$	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.2591.2	yes	24
3.2	odd	2	inner	6048.2.h.i.2591.23	yes	24
4.3	odd	2	inner	6048.2.h.i.2591.24	yes	24
12.11	even	2	inner	6048.2.h.i.2591.1	✓	24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6048.2.h.i.2591.1	✓	24	12.11	even	2	inner
6048.2.h.i.2591.2	yes	24	1.1	even	1	trivial
6048.2.h.i.2591.23	yes	24	3.2	odd	2	inner
6048.2.h.i.2591.24	yes	24	4.3	odd	2	inner

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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)

\(f(q)\)	\(=\)	\(q+4.29730i q^{5} -1.00000i q^{7} +O(q^{10})\)	\(q+4.29730i q^{5} -1.00000i q^{7} +4.77104 q^{11} -0.0836472 q^{13} +2.60889i q^{17} -4.85636i q^{19} -4.78044 q^{23} -13.4668 q^{25} -0.263176i q^{29} -0.304131i q^{31} +4.29730 q^{35} +9.56782 q^{37} +10.5898i q^{41} +4.24414i q^{43} +9.85829 q^{47} -1.00000 q^{49} +11.9022i q^{53} +20.5026i q^{55} -5.50796 q^{59} -8.62260 q^{61} -0.359457i q^{65} -1.80727i q^{67} +10.8966 q^{71} -4.88339 q^{73} -4.77104i q^{77} +14.3421i q^{79} -10.4067 q^{83} -11.2112 q^{85} +7.85368i q^{89} +0.0836472i q^{91} +20.8693 q^{95} -1.29824 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q+O(q^{10}) \) 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

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