LMFDB - Embedded newform 6048.2.h.i.2591.16

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			2591.16
Character	$\chi$	$=$	6048.2591
Dual form			6048.2.h.i.2591.15

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.29686i q^{5} -1.00000i q^{7} +O(q^{10})$	$q+1.29686i q^{5} -1.00000i q^{7} -2.21561 q^{11} -1.03730 q^{13} +2.24600i q^{17} +8.63898i q^{19} +8.96997 q^{23} +3.31815 q^{25} -3.26362i q^{29} +4.37958i q^{31} +1.29686 q^{35} -7.21713 q^{37} +7.58939i q^{41} -12.9813i q^{43} -5.24078 q^{47} -1.00000 q^{49} -11.1694i q^{53} -2.87334i q^{55} +2.46447 q^{59} -10.3300 q^{61} -1.34523i q^{65} -4.46833i q^{67} +13.5342 q^{71} -12.8567 q^{73} +2.21561i q^{77} +9.14521i q^{79} +9.96748 q^{83} -2.91275 q^{85} -3.48208i q^{89} +1.03730i q^{91} -11.2036 q^{95} -6.30706 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$	1.29686i	0.579975i	0.957030	+	0.289987i	$0.0936511\pi$
$5$	1.29686i	0.579975i	−0.957030	+	0.289987i	$0.906349\pi$
$6$	0	0
$7$	− 1.00000i	− 0.377964i
$7$	− 1.00000i	− 0.377964i
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−2.21561	−0.668031	−0.334016	−	0.942567i	$-0.608404\pi$
$11$	−2.21561	−0.668031	−0.334016	+	0.942567i	$0.608404\pi$
$12$	0	0
$13$	−1.03730	−0.287695	−0.143847	−	0.989600i	$-0.545947\pi$
$13$	−1.03730	−0.287695	−0.143847	+	0.989600i	$0.545947\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	2.24600i	0.544734i	0.962193	+	0.272367i	$0.0878065\pi$
$17$	2.24600i	0.544734i	−0.962193	+	0.272367i	$0.912193\pi$
$18$	0	0
$19$	8.63898i	1.98192i	0.134169	+	0.990959i	$0.457164\pi$
$19$	8.63898i	1.98192i	−0.134169	+	0.990959i	$0.542836\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	8.96997	1.87037	0.935184	−	0.354161i	$-0.115234\pi$
$23$	8.96997	1.87037	0.935184	+	0.354161i	$0.115234\pi$
$24$	0	0
$25$	3.31815	0.663629
$26$	0	0
$27$	0	0
$28$	0	0
$29$	− 3.26362i	− 0.606038i	−0.952984	−	0.303019i	$-0.902005\pi$
$29$	− 3.26362i	− 0.606038i	0.952984	−	0.303019i	$-0.0979946\pi$
$30$	0	0
$31$	4.37958i	0.786595i	0.919411	+	0.393298i	$0.128666\pi$
$31$	4.37958i	0.786595i	−0.919411	+	0.393298i	$0.871334\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	1.29686	0.219210
$36$	0	0
$37$	−7.21713	−1.18649	−0.593244	−	0.805023i	$-0.702153\pi$
$37$	−7.21713	−1.18649	−0.593244	+	0.805023i	$0.702153\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	7.58939i	1.18526i	0.805473	+	0.592632i	$0.201911\pi$
$41$	7.58939i	1.18526i	−0.805473	+	0.592632i	$0.798089\pi$
$42$	0	0
$43$	− 12.9813i	− 1.97962i	−0.142384	−	0.989811i	$-0.545477\pi$
$43$	− 12.9813i	− 1.97962i	0.142384	−	0.989811i	$-0.454523\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−5.24078	−0.764447	−0.382223	−	0.924070i	$-0.624842\pi$
$47$	−5.24078	−0.764447	−0.382223	+	0.924070i	$0.624842\pi$
$48$	0	0
$49$	−1.00000	−0.142857
$50$	0	0
$51$	0	0
$52$	0	0
$53$	− 11.1694i	− 1.53423i	−0.641511	−	0.767114i	$-0.721692\pi$
$53$	− 11.1694i	− 1.53423i	0.641511	−	0.767114i	$-0.278308\pi$
$54$	0	0
$55$	− 2.87334i	− 0.387441i
$56$	0	0
$57$	0	0
$58$	0	0
$59$	2.46447	0.320846	0.160423	−	0.987048i	$-0.448714\pi$
$59$	2.46447	0.320846	0.160423	+	0.987048i	$0.448714\pi$
$60$	0	0
$61$	−10.3300	−1.32262	−0.661311	−	0.750112i	$-0.730000\pi$
$61$	−10.3300	−1.32262	−0.661311	+	0.750112i	$0.730000\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	− 1.34523i	− 0.166856i
$66$	0	0
$67$	− 4.46833i	− 0.545894i	−0.962029	−	0.272947i	$-0.912002\pi$
$67$	− 4.46833i	− 0.545894i	0.962029	−	0.272947i	$-0.0879983\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	13.5342	1.60621	0.803107	−	0.595835i	$-0.203179\pi$
$71$	13.5342	1.60621	0.803107	+	0.595835i	$0.203179\pi$
$72$	0	0
$73$	−12.8567	−1.50476	−0.752382	−	0.658727i	$-0.771095\pi$
$73$	−12.8567	−1.50476	−0.752382	+	0.658727i	$0.771095\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	2.21561i	0.252492i
$78$	0	0
$79$	9.14521i	1.02892i	0.857515	+	0.514458i	$0.172007\pi$
$79$	9.14521i	1.02892i	−0.857515	+	0.514458i	$0.827993\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	9.96748	1.09407	0.547037	−	0.837109i	$-0.315756\pi$
$83$	9.96748	1.09407	0.547037	+	0.837109i	$0.315756\pi$
$84$	0	0
$85$	−2.91275	−0.315932
$86$	0	0
$87$	0	0
$88$	0	0
$89$	− 3.48208i	− 0.369100i	−0.982823	−	0.184550i	$-0.940917\pi$
$89$	− 3.48208i	− 0.369100i	0.982823	−	0.184550i	$-0.0590828\pi$
$90$	0	0
$91$	1.03730i	0.108738i
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−11.2036	−1.14946
$96$	0	0
$97$	−6.30706	−0.640385	−0.320192	−	0.947353i	$-0.603748\pi$
$97$	−6.30706	−0.640385	−0.320192	+	0.947353i	$0.603748\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	10.7845i	1.07310i	0.843868	+	0.536550i	$0.180273\pi$
$101$	10.7845i	1.07310i	−0.843868	+	0.536550i	$0.819727\pi$
$102$	0	0
$103$	7.98276i	0.786565i	0.919418	+	0.393282i	$0.128661\pi$
$103$	7.98276i	0.786565i	−0.919418	+	0.393282i	$0.871339\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−20.4082	−1.97293	−0.986465	−	0.163969i	$-0.947570\pi$
$107$	−20.4082	−1.97293	−0.986465	+	0.163969i	$0.947570\pi$
$108$	0	0
$109$	9.98623	0.956507	0.478254	−	0.878222i	$-0.341270\pi$
$109$	9.98623	0.956507	0.478254	+	0.878222i	$0.341270\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	9.60077i	0.903165i	0.892229	+	0.451582i	$0.149140\pi$
$113$	9.60077i	0.903165i	−0.892229	+	0.451582i	$0.850860\pi$
$114$	0	0
$115$	11.6328i	1.08477i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	2.24600	0.205890
$120$	0	0
$121$	−6.09107	−0.553734
$122$	0	0
$123$	0	0
$124$	0	0
$125$	10.7875i	0.964863i
$126$	0	0
$127$	9.84909i	0.873965i	0.899470	+	0.436983i	$0.143953\pi$
$127$	9.84909i	0.873965i	−0.899470	+	0.436983i	$0.856047\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−19.3232	−1.68827	−0.844137	−	0.536127i	$-0.819887\pi$
$131$	−19.3232	−1.68827	−0.844137	+	0.536127i	$0.819887\pi$
$132$	0	0
$133$	8.63898	0.749094
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 6.56515i	− 0.560899i	−0.959869	−	0.280449i	$-0.909517\pi$
$137$	− 6.56515i	− 0.560899i	0.959869	−	0.280449i	$-0.0904834\pi$
$138$	0	0
$139$	19.8276i	1.68176i	0.541225	+	0.840878i	$0.317961\pi$
$139$	19.8276i	1.68176i	−0.541225	+	0.840878i	$0.682039\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	2.29825	0.192189
$144$	0	0
$145$	4.23246	0.351487
$146$	0	0
$147$	0	0
$148$	0	0
$149$	8.58523i	0.703330i	0.936126	+	0.351665i	$0.114384\pi$
$149$	8.58523i	0.703330i	−0.936126	+	0.351665i	$0.885616\pi$
$150$	0	0
$151$	14.8203i	1.20606i	0.797719	+	0.603030i	$0.206040\pi$
$151$	14.8203i	1.20606i	−0.797719	+	0.603030i	$0.793960\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−5.67971	−0.456205
$156$	0	0
$157$	−19.6744	−1.57019	−0.785093	−	0.619378i	$-0.787385\pi$
$157$	−19.6744	−1.57019	−0.785093	+	0.619378i	$0.787385\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	− 8.96997i	− 0.706933i
$162$	0	0
$163$	6.75148i	0.528816i	0.964411	+	0.264408i	$0.0851766\pi$
$163$	6.75148i	0.528816i	−0.964411	+	0.264408i	$0.914823\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−6.10240	−0.472218	−0.236109	−	0.971727i	$-0.575872\pi$
$167$	−6.10240	−0.472218	−0.236109	+	0.971727i	$0.575872\pi$
$168$	0	0
$169$	−11.9240	−0.917232
$170$	0	0
$171$	0	0
$172$	0	0
$173$	− 2.60237i	− 0.197855i	−0.995095	−	0.0989274i	$-0.968459\pi$
$173$	− 2.60237i	− 0.197855i	0.995095	−	0.0989274i	$-0.0315412\pi$
$174$	0	0
$175$	− 3.31815i	− 0.250828i
$176$	0	0
$177$	0	0
$178$	0	0
$179$	16.0508	1.19969	0.599847	−	0.800114i	$-0.295228\pi$
$179$	16.0508	1.19969	0.599847	+	0.800114i	$0.295228\pi$
$180$	0	0
$181$	15.6876	1.16605	0.583026	−	0.812453i	$-0.301869\pi$
$181$	15.6876	1.16605	0.583026	+	0.812453i	$0.301869\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	− 9.35962i	− 0.688133i
$186$	0	0
$187$	− 4.97625i	− 0.363900i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	20.8131	1.50598	0.752991	−	0.658030i	$-0.228610\pi$
$191$	20.8131	1.50598	0.752991	+	0.658030i	$0.228610\pi$
$192$	0	0
$193$	−14.9102	−1.07326	−0.536631	−	0.843817i	$-0.680303\pi$
$193$	−14.9102	−1.07326	−0.536631	+	0.843817i	$0.680303\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	4.28412i	0.305231i	0.988286	+	0.152615i	$0.0487696\pi$
$197$	4.28412i	0.305231i	−0.988286	+	0.152615i	$0.951230\pi$
$198$	0	0
$199$	21.5110i	1.52488i	0.647061	+	0.762438i	$0.275998\pi$
$199$	21.5110i	1.52488i	−0.647061	+	0.762438i	$0.724002\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−3.26362	−0.229061
$204$	0	0
$205$	−9.84240	−0.687423
$206$	0	0
$207$	0	0
$208$	0	0
$209$	− 19.1406i	− 1.32398i
$210$	0	0
$211$	1.01467i	0.0698529i	0.999390	+	0.0349264i	$0.0111197\pi$
$211$	1.01467i	0.0698529i	−0.999390	+	0.0349264i	$0.988880\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	16.8349	1.14813
$216$	0	0
$217$	4.37958	0.297305
$218$	0	0
$219$	0	0
$220$	0	0
$221$	− 2.32977i	− 0.156717i
$222$	0	0
$223$	− 0.114187i	− 0.00764650i	−0.999993	−	0.00382325i	$-0.998783\pi$
$223$	− 0.114187i	− 0.00764650i	0.999993	−	0.00382325i	$-0.00121698\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−25.0865	−1.66505	−0.832526	−	0.553986i	$-0.813106\pi$
$227$	−25.0865	−1.66505	−0.832526	+	0.553986i	$0.813106\pi$
$228$	0	0
$229$	4.54175	0.300127	0.150064	−	0.988676i	$-0.452052\pi$
$229$	4.54175	0.300127	0.150064	+	0.988676i	$0.452052\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	23.4905i	1.53891i	0.638701	+	0.769455i	$0.279472\pi$
$233$	23.4905i	1.53891i	−0.638701	+	0.769455i	$0.720528\pi$
$234$	0	0
$235$	− 6.79658i	− 0.443360i
$236$	0	0
$237$	0	0
$238$	0	0
$239$	0.821010	0.0531067	0.0265534	−	0.999647i	$-0.491547\pi$
$239$	0.821010	0.0531067	0.0265534	+	0.999647i	$0.491547\pi$
$240$	0	0
$241$	−2.69884	−0.173848	−0.0869239	−	0.996215i	$-0.527704\pi$
$241$	−2.69884	−0.173848	−0.0869239	+	0.996215i	$0.527704\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	− 1.29686i	− 0.0828535i
$246$	0	0
$247$	− 8.96120i	− 0.570187i
$248$	0	0
$249$	0	0
$250$	0	0
$251$	14.7256	0.929475	0.464737	−	0.885449i	$-0.346149\pi$
$251$	14.7256	0.929475	0.464737	+	0.885449i	$0.346149\pi$
$252$	0	0
$253$	−19.8740	−1.24947
$254$	0	0
$255$	0	0
$256$	0	0
$257$	9.11220i	0.568404i	0.958764	+	0.284202i	$0.0917286\pi$
$257$	9.11220i	0.568404i	−0.958764	+	0.284202i	$0.908271\pi$
$258$	0	0
$259$	7.21713i	0.448450i
$260$	0	0
$261$	0	0
$262$	0	0
$263$	15.6206	0.963207	0.481604	−	0.876389i	$-0.340055\pi$
$263$	15.6206	0.963207	0.481604	+	0.876389i	$0.340055\pi$
$264$	0	0
$265$	14.4851	0.889814
$266$	0	0
$267$	0	0
$268$	0	0
$269$	− 28.3937i	− 1.73120i	−0.500739	−	0.865598i	$-0.666938\pi$
$269$	− 28.3937i	− 1.73120i	0.500739	−	0.865598i	$-0.333062\pi$
$270$	0	0
$271$	4.61964i	0.280623i	0.990107	+	0.140311i	$0.0448104\pi$
$271$	4.61964i	0.280623i	−0.990107	+	0.140311i	$0.955190\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−7.35172	−0.443325
$276$	0	0
$277$	−23.6330	−1.41997	−0.709985	−	0.704217i	$-0.751298\pi$
$277$	−23.6330	−1.41997	−0.709985	+	0.704217i	$0.751298\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	− 9.50949i	− 0.567288i	−0.958930	−	0.283644i	$-0.908457\pi$
$281$	− 9.50949i	− 0.567288i	0.958930	−	0.283644i	$-0.0915435\pi$
$282$	0	0
$283$	− 8.52211i	− 0.506587i	−0.967390	−	0.253293i	$-0.918486\pi$
$283$	− 8.52211i	− 0.506587i	0.967390	−	0.253293i	$-0.0815138\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	7.58939	0.447988
$288$	0	0
$289$	11.9555	0.703264
$290$	0	0
$291$	0	0
$292$	0	0
$293$	− 17.6710i	− 1.03235i	−0.856483	−	0.516175i	$-0.827355\pi$
$293$	− 17.6710i	− 1.03235i	0.856483	−	0.516175i	$-0.172645\pi$
$294$	0	0
$295$	3.19608i	0.186083i
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−9.30454	−0.538095
$300$	0	0
$301$	−12.9813	−0.748227
$302$	0	0
$303$	0	0
$304$	0	0
$305$	− 13.3966i	− 0.767088i
$306$	0	0
$307$	9.51533i	0.543068i	0.962429	+	0.271534i	$0.0875310\pi$
$307$	9.51533i	0.543068i	−0.962429	+	0.271534i	$0.912469\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−20.4716	−1.16084	−0.580419	−	0.814318i	$-0.697111\pi$
$311$	−20.4716	−1.16084	−0.580419	+	0.814318i	$0.697111\pi$
$312$	0	0
$313$	−9.49300	−0.536576	−0.268288	−	0.963339i	$-0.586458\pi$
$313$	−9.49300	−0.536576	−0.268288	+	0.963339i	$0.586458\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 16.5518i	− 0.929644i	−0.885404	−	0.464822i	$-0.846118\pi$
$317$	− 16.5518i	− 0.929644i	0.885404	−	0.464822i	$-0.153882\pi$
$318$	0	0
$319$	7.23090i	0.404853i
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−19.4031	−1.07962
$324$	0	0
$325$	−3.44191	−0.190923
$326$	0	0
$327$	0	0
$328$	0	0
$329$	5.24078i	0.288934i
$330$	0	0
$331$	30.8263i	1.69436i	0.531302	+	0.847182i	$0.321703\pi$
$331$	30.8263i	1.69436i	−0.531302	+	0.847182i	$0.678297\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	5.79481	0.316605
$336$	0	0
$337$	−15.4712	−0.842768	−0.421384	−	0.906882i	$-0.638456\pi$
$337$	−15.4712	−0.842768	−0.421384	+	0.906882i	$0.638456\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	− 9.70344i	− 0.525470i
$342$	0	0
$343$	1.00000i	0.0539949i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−13.5979	−0.729971	−0.364986	−	0.931013i	$-0.618926\pi$
$347$	−13.5979	−0.729971	−0.364986	+	0.931013i	$0.618926\pi$
$348$	0	0
$349$	−7.39184	−0.395676	−0.197838	−	0.980235i	$-0.563392\pi$
$349$	−7.39184	−0.395676	−0.197838	+	0.980235i	$0.563392\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	13.5532i	0.721365i	0.932689	+	0.360683i	$0.117456\pi$
$353$	13.5532i	0.721365i	−0.932689	+	0.360683i	$0.882544\pi$
$354$	0	0
$355$	17.5520i	0.931563i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−6.96702	−0.367705	−0.183853	−	0.982954i	$-0.558857\pi$
$359$	−6.96702	−0.367705	−0.183853	+	0.982954i	$0.558857\pi$
$360$	0	0
$361$	−55.6319	−2.92800
$362$	0	0
$363$	0	0
$364$	0	0
$365$	− 16.6734i	− 0.872725i
$366$	0	0
$367$	− 27.9129i	− 1.45704i	−0.685022	−	0.728522i	$-0.740208\pi$
$367$	− 27.9129i	− 1.45704i	0.685022	−	0.728522i	$-0.259792\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−11.1694	−0.579884
$372$	0	0
$373$	−3.28235	−0.169954	−0.0849769	−	0.996383i	$-0.527082\pi$
$373$	−3.28235	−0.169954	−0.0849769	+	0.996383i	$0.527082\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	3.38534i	0.174354i
$378$	0	0
$379$	2.59351i	0.133220i	0.997779	+	0.0666098i	$0.0212183\pi$
$379$	2.59351i	0.133220i	−0.997779	+	0.0666098i	$0.978782\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	12.5700	0.642295	0.321147	−	0.947029i	$-0.395931\pi$
$383$	12.5700	0.642295	0.321147	+	0.947029i	$0.395931\pi$
$384$	0	0
$385$	−2.87334	−0.146439
$386$	0	0
$387$	0	0
$388$	0	0
$389$	33.0483i	1.67562i	0.545964	+	0.837809i	$0.316164\pi$
$389$	33.0483i	1.67562i	−0.545964	+	0.837809i	$0.683836\pi$
$390$	0	0
$391$	20.1465i	1.01885i
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−11.8601	−0.596746
$396$	0	0
$397$	21.8925	1.09875	0.549377	−	0.835575i	$-0.314865\pi$
$397$	21.8925	1.09875	0.549377	+	0.835575i	$0.314865\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	9.11600i	0.455231i	0.973751	+	0.227616i	$0.0730929\pi$
$401$	9.11600i	0.455231i	−0.973751	+	0.227616i	$0.926907\pi$
$402$	0	0
$403$	− 4.54293i	− 0.226299i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	15.9903	0.792612
$408$	0	0
$409$	15.6721	0.774935	0.387467	−	0.921883i	$-0.373350\pi$
$409$	15.6721	0.774935	0.387467	+	0.921883i	$0.373350\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	− 2.46447i	− 0.121269i
$414$	0	0
$415$	12.9265i	0.634535i
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−21.9407	−1.07187	−0.535936	−	0.844259i	$-0.680041\pi$
$419$	−21.9407	−1.07187	−0.535936	+	0.844259i	$0.680041\pi$
$420$	0	0
$421$	−17.6611	−0.860752	−0.430376	−	0.902650i	$-0.641619\pi$
$421$	−17.6611	−0.860752	−0.430376	+	0.902650i	$0.641619\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	7.45255i	0.361502i
$426$	0	0
$427$	10.3300i	0.499904i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	22.2362	1.07108	0.535540	−	0.844510i	$-0.320108\pi$
$431$	22.2362	1.07108	0.535540	+	0.844510i	$0.320108\pi$
$432$	0	0
$433$	−36.2225	−1.74074	−0.870371	−	0.492396i	$-0.836121\pi$
$433$	−36.2225	−1.74074	−0.870371	+	0.492396i	$0.836121\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	77.4914i	3.70692i
$438$	0	0
$439$	− 4.36466i	− 0.208314i	−0.994561	−	0.104157i	$-0.966786\pi$
$439$	− 4.36466i	− 0.208314i	0.994561	−	0.104157i	$-0.0332145\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	23.0758	1.09636	0.548181	−	0.836359i	$-0.315320\pi$
$443$	23.0758	1.09636	0.548181	+	0.836359i	$0.315320\pi$
$444$	0	0
$445$	4.51579	0.214069
$446$	0	0
$447$	0	0
$448$	0	0
$449$	− 25.6147i	− 1.20883i	−0.796669	−	0.604416i	$-0.793407\pi$
$449$	− 25.6147i	− 1.20883i	0.796669	−	0.604416i	$-0.206593\pi$
$450$	0	0
$451$	− 16.8151i	− 0.791794i
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−1.34523	−0.0630655
$456$	0	0
$457$	19.0485	0.891052	0.445526	−	0.895269i	$-0.353017\pi$
$457$	19.0485	0.891052	0.445526	+	0.895269i	$0.353017\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	− 18.6533i	− 0.868773i	−0.900727	−	0.434386i	$-0.856965\pi$
$461$	− 18.6533i	− 0.868773i	0.900727	−	0.434386i	$-0.143035\pi$
$462$	0	0
$463$	− 26.5347i	− 1.23317i	−0.787288	−	0.616585i	$-0.788516\pi$
$463$	− 26.5347i	− 1.23317i	0.787288	−	0.616585i	$-0.211484\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	3.04573	0.140939	0.0704697	−	0.997514i	$-0.477550\pi$
$467$	3.04573	0.140939	0.0704697	+	0.997514i	$0.477550\pi$
$468$	0	0
$469$	−4.46833	−0.206328
$470$	0	0
$471$	0	0
$472$	0	0
$473$	28.7614i	1.32245i
$474$	0	0
$475$	28.6654i	1.31526i
$476$	0	0
$477$	0	0
$478$	0	0
$479$	14.4984	0.662450	0.331225	−	0.943552i	$-0.392538\pi$
$479$	14.4984	0.662450	0.331225	+	0.943552i	$0.392538\pi$
$480$	0	0
$481$	7.48631	0.341347
$482$	0	0
$483$	0	0
$484$	0	0
$485$	− 8.17939i	− 0.371407i
$486$	0	0
$487$	15.1551i	0.686743i	0.939200	+	0.343371i	$0.111569\pi$
$487$	15.1551i	0.686743i	−0.939200	+	0.343371i	$0.888431\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−5.74814	−0.259410	−0.129705	−	0.991553i	$-0.541403\pi$
$491$	−5.74814	−0.259410	−0.129705	+	0.991553i	$0.541403\pi$
$492$	0	0
$493$	7.33007	0.330130
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 13.5342i	− 0.607091i
$498$	0	0
$499$	− 10.9764i	− 0.491373i	−0.969349	−	0.245686i	$-0.920987\pi$
$499$	− 10.9764i	− 0.491373i	0.969349	−	0.245686i	$-0.0790134\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−4.71925	−0.210421	−0.105211	−	0.994450i	$-0.533552\pi$
$503$	−4.71925	−0.210421	−0.105211	+	0.994450i	$0.533552\pi$
$504$	0	0
$505$	−13.9860	−0.622371
$506$	0	0
$507$	0	0
$508$	0	0
$509$	28.5925i	1.26734i	0.773604	+	0.633669i	$0.218452\pi$
$509$	28.5925i	1.26734i	−0.773604	+	0.633669i	$0.781548\pi$
$510$	0	0
$511$	12.8567i	0.568747i
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−10.3525	−0.456188
$516$	0	0
$517$	11.6115	0.510674
$518$	0	0
$519$	0	0
$520$	0	0
$521$	− 29.2439i	− 1.28120i	−0.767876	−	0.640598i	$-0.778686\pi$
$521$	− 29.2439i	− 1.28120i	0.767876	−	0.640598i	$-0.221314\pi$
$522$	0	0
$523$	17.1008i	0.747765i	0.927476	+	0.373882i	$0.121974\pi$
$523$	17.1008i	0.747765i	−0.927476	+	0.373882i	$0.878026\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−9.83652	−0.428486
$528$	0	0
$529$	57.4604	2.49828
$530$	0	0
$531$	0	0
$532$	0	0
$533$	− 7.87246i	− 0.340994i
$534$	0	0
$535$	− 26.4666i	− 1.14425i
$536$	0	0
$537$	0	0
$538$	0	0
$539$	2.21561	0.0954331
$540$	0	0
$541$	27.8421	1.19702	0.598512	−	0.801114i	$-0.295759\pi$
$541$	27.8421	1.19702	0.598512	+	0.801114i	$0.295759\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	12.9508i	0.554750i
$546$	0	0
$547$	− 0.515933i	− 0.0220597i	−0.999939	−	0.0110299i	$-0.996489\pi$
$547$	− 0.515933i	− 0.0220597i	0.999939	−	0.0110299i	$-0.00351099\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	28.1943	1.20112
$552$	0	0
$553$	9.14521	0.388894
$554$	0	0
$555$	0	0
$556$	0	0
$557$	17.9248i	0.759498i	0.925090	+	0.379749i	$0.123990\pi$
$557$	17.9248i	0.759498i	−0.925090	+	0.379749i	$0.876010\pi$
$558$	0	0
$559$	13.4654i	0.569527i
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−32.4146	−1.36611	−0.683056	−	0.730366i	$-0.739350\pi$
$563$	−32.4146	−1.36611	−0.683056	+	0.730366i	$0.739350\pi$
$564$	0	0
$565$	−12.4509	−0.523813
$566$	0	0
$567$	0	0
$568$	0	0
$569$	0.202528i	0.00849044i	0.999991	+	0.00424522i	$0.00135130\pi$
$569$	0.202528i	0.00849044i	−0.999991	+	0.00424522i	$0.998649\pi$
$570$	0	0
$571$	26.2522i	1.09862i	0.835619	+	0.549310i	$0.185109\pi$
$571$	26.2522i	1.09862i	−0.835619	+	0.549310i	$0.814891\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	29.7637	1.24123
$576$	0	0
$577$	−6.79441	−0.282855	−0.141428	−	0.989949i	$-0.545169\pi$
$577$	−6.79441	−0.282855	−0.141428	+	0.989949i	$0.545169\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	− 9.96748i	− 0.413521i
$582$	0	0
$583$	24.7469i	1.02491i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−21.1087	−0.871248	−0.435624	−	0.900129i	$-0.643472\pi$
$587$	−21.1087	−0.871248	−0.435624	+	0.900129i	$0.643472\pi$
$588$	0	0
$589$	−37.8351	−1.55897
$590$	0	0
$591$	0	0
$592$	0	0
$593$	− 36.1673i	− 1.48521i	−0.669727	−	0.742607i	$-0.733589\pi$
$593$	− 36.1673i	− 1.48521i	0.669727	−	0.742607i	$-0.266411\pi$
$594$	0	0
$595$	2.91275i	0.119411i
$596$	0	0
$597$	0	0
$598$	0	0
$599$	31.5253	1.28809	0.644045	−	0.764988i	$-0.277255\pi$
$599$	31.5253	1.28809	0.644045	+	0.764988i	$0.277255\pi$
$600$	0	0
$601$	−20.2761	−0.827081	−0.413540	−	0.910486i	$-0.635708\pi$
$601$	−20.2761	−0.827081	−0.413540	+	0.910486i	$0.635708\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	− 7.89929i	− 0.321152i
$606$	0	0
$607$	32.7857i	1.33073i	0.746519	+	0.665364i	$0.231724\pi$
$607$	32.7857i	1.33073i	−0.746519	+	0.665364i	$0.768276\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	5.43626	0.219927
$612$	0	0
$613$	1.73331	0.0700077	0.0350039	−	0.999387i	$-0.488856\pi$
$613$	1.73331	0.0700077	0.0350039	+	0.999387i	$0.488856\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	26.2030i	1.05489i	0.849589	+	0.527446i	$0.176850\pi$
$617$	26.2030i	1.05489i	−0.849589	+	0.527446i	$0.823150\pi$
$618$	0	0
$619$	− 17.2543i	− 0.693509i	−0.937956	−	0.346754i	$-0.887284\pi$
$619$	− 17.2543i	− 0.693509i	0.937956	−	0.346754i	$-0.112716\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−3.48208	−0.139507
$624$	0	0
$625$	2.60083	0.104033
$626$	0	0
$627$	0	0
$628$	0	0
$629$	− 16.2096i	− 0.646321i
$630$	0	0
$631$	− 41.7871i	− 1.66352i	−0.555138	−	0.831758i	$-0.687334\pi$
$631$	− 41.7871i	− 1.66352i	0.555138	−	0.831758i	$-0.312666\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−12.7729	−0.506878
$636$	0	0
$637$	1.03730	0.0410993
$638$	0	0
$639$	0	0
$640$	0	0
$641$	− 33.9794i	− 1.34211i	−0.741410	−	0.671053i	$-0.765842\pi$
$641$	− 33.9794i	− 1.34211i	0.741410	−	0.671053i	$-0.234158\pi$
$642$	0	0
$643$	− 17.8079i	− 0.702275i	−0.936324	−	0.351138i	$-0.885795\pi$
$643$	− 17.8079i	− 0.702275i	0.936324	−	0.351138i	$-0.114205\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	24.6184	0.967849	0.483924	−	0.875110i	$-0.339211\pi$
$647$	24.6184	0.967849	0.483924	+	0.875110i	$0.339211\pi$
$648$	0	0
$649$	−5.46030	−0.214335
$650$	0	0
$651$	0	0
$652$	0	0
$653$	1.37615i	0.0538529i	0.999637	+	0.0269264i	$0.00857199\pi$
$653$	1.37615i	0.0538529i	−0.999637	+	0.0269264i	$0.991428\pi$
$654$	0	0
$655$	− 25.0595i	− 0.979157i
$656$	0	0
$657$	0	0
$658$	0	0
$659$	8.98020	0.349819	0.174909	−	0.984585i	$-0.444037\pi$
$659$	8.98020	0.349819	0.174909	+	0.984585i	$0.444037\pi$
$660$	0	0
$661$	−1.39794	−0.0543734	−0.0271867	−	0.999630i	$-0.508655\pi$
$661$	−1.39794	−0.0543734	−0.0271867	+	0.999630i	$0.508655\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	11.2036i	0.434456i
$666$	0	0
$667$	− 29.2745i	− 1.13351i
$668$	0	0
$669$	0	0
$670$	0	0
$671$	22.8873	0.883554
$672$	0	0
$673$	−22.8752	−0.881773	−0.440886	−	0.897563i	$-0.645336\pi$
$673$	−22.8752	−0.881773	−0.440886	+	0.897563i	$0.645336\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	14.3953i	0.553258i	0.960977	+	0.276629i	$0.0892173\pi$
$677$	14.3953i	0.553258i	−0.960977	+	0.276629i	$0.910783\pi$
$678$	0	0
$679$	6.30706i	0.242043i
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−0.558805	−0.0213821	−0.0106910	−	0.999943i	$-0.503403\pi$
$683$	−0.558805	−0.0213821	−0.0106910	+	0.999943i	$0.503403\pi$
$684$	0	0
$685$	8.51410	0.325307
$686$	0	0
$687$	0	0
$688$	0	0
$689$	11.5860i	0.441390i
$690$	0	0
$691$	− 0.0964465i	− 0.00366900i	−0.999998	−	0.00183450i	$-0.999416\pi$
$691$	− 0.0964465i	− 0.00366900i	0.999998	−	0.00183450i	$-0.000583940\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−25.7137	−0.975375
$696$	0	0
$697$	−17.0458	−0.645654
$698$	0	0
$699$	0	0
$700$	0	0
$701$	17.5851i	0.664179i	0.943248	+	0.332090i	$0.107754\pi$
$701$	17.5851i	0.664179i	−0.943248	+	0.332090i	$0.892246\pi$
$702$	0	0
$703$	− 62.3486i	− 2.35152i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	10.7845	0.405594
$708$	0	0
$709$	−22.4705	−0.843896	−0.421948	−	0.906620i	$-0.638654\pi$
$709$	−22.4705	−0.843896	−0.421948	+	0.906620i	$0.638654\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	39.2847i	1.47122i
$714$	0	0
$715$	2.98051i	0.111465i
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−3.41643	−0.127411	−0.0637057	−	0.997969i	$-0.520292\pi$
$719$	−3.41643	−0.127411	−0.0637057	+	0.997969i	$0.520292\pi$
$720$	0	0
$721$	7.98276	0.297294
$722$	0	0
$723$	0	0
$724$	0	0
$725$	− 10.8292i	− 0.402185i
$726$	0	0
$727$	6.76927i	0.251059i	0.992090	+	0.125529i	$0.0400629\pi$
$727$	6.76927i	0.251059i	−0.992090	+	0.125529i	$0.959937\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	29.1559	1.07837
$732$	0	0
$733$	−8.85387	−0.327025	−0.163513	−	0.986541i	$-0.552282\pi$
$733$	−8.85387	−0.327025	−0.163513	+	0.986541i	$0.552282\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	9.90008i	0.364674i
$738$	0	0
$739$	− 7.90087i	− 0.290638i	−0.989385	−	0.145319i	$-0.953579\pi$
$739$	− 7.90087i	− 0.290638i	0.989385	−	0.145319i	$-0.0464209\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−10.8356	−0.397521	−0.198761	−	0.980048i	$-0.563692\pi$
$743$	−10.8356	−0.397521	−0.198761	+	0.980048i	$0.563692\pi$
$744$	0	0
$745$	−11.1339	−0.407913
$746$	0	0
$747$	0	0
$748$	0	0
$749$	20.4082i	0.745698i
$750$	0	0
$751$	4.34951i	0.158716i	0.996846	+	0.0793580i	$0.0252870\pi$
$751$	4.34951i	0.158716i	−0.996846	+	0.0793580i	$0.974713\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−19.2199	−0.699484
$756$	0	0
$757$	9.63880	0.350328	0.175164	−	0.984539i	$-0.443954\pi$
$757$	9.63880	0.350328	0.175164	+	0.984539i	$0.443954\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	4.87867i	0.176852i	0.996083	+	0.0884259i	$0.0281836\pi$
$761$	4.87867i	0.176852i	−0.996083	+	0.0884259i	$0.971816\pi$
$762$	0	0
$763$	− 9.98623i	− 0.361526i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−2.55639	−0.0923058
$768$	0	0
$769$	19.2879	0.695540	0.347770	−	0.937580i	$-0.386939\pi$
$769$	19.2879	0.695540	0.347770	+	0.937580i	$0.386939\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	− 8.44127i	− 0.303611i	−0.988410	−	0.151806i	$-0.951491\pi$
$773$	− 8.44127i	− 0.303611i	0.988410	−	0.151806i	$-0.0485088\pi$
$774$	0	0
$775$	14.5321i	0.522008i
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−65.5646	−2.34909
$780$	0	0
$781$	−29.9865	−1.07300
$782$	0	0
$783$	0	0
$784$	0	0
$785$	− 25.5150i	− 0.910668i
$786$	0	0
$787$	32.2439i	1.14937i	0.818374	+	0.574686i	$0.194876\pi$
$787$	32.2439i	1.14937i	−0.818374	+	0.574686i	$0.805124\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	9.60077	0.341364
$792$	0	0
$793$	10.7153	0.380512
$794$	0	0
$795$	0	0
$796$	0	0
$797$	4.61145i	0.163346i	0.996659	+	0.0816730i	$0.0260263\pi$
$797$	4.61145i	0.163346i	−0.996659	+	0.0816730i	$0.973974\pi$
$798$	0	0
$799$	− 11.7708i	− 0.416420i
$800$	0	0
$801$	0	0
$802$	0	0
$803$	28.4854	1.00523
$804$	0	0
$805$	11.6328	0.410003
$806$	0	0
$807$	0	0
$808$	0	0
$809$	5.69680i	0.200289i	0.994973	+	0.100144i	$0.0319305\pi$
$809$	5.69680i	0.200289i	−0.994973	+	0.100144i	$0.968070\pi$
$810$	0	0
$811$	− 16.3693i	− 0.574803i	−0.957810	−	0.287401i	$-0.907209\pi$
$811$	− 16.3693i	− 0.574803i	0.957810	−	0.287401i	$-0.0927914\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−8.75574	−0.306700
$816$	0	0
$817$	112.145	3.92345
$818$	0	0
$819$	0	0
$820$	0	0
$821$	12.5960i	0.439602i	0.975545	+	0.219801i	$0.0705408\pi$
$821$	12.5960i	0.439602i	−0.975545	+	0.219801i	$0.929459\pi$
$822$	0	0
$823$	− 7.53241i	− 0.262563i	−0.991345	−	0.131282i	$-0.958091\pi$
$823$	− 7.53241i	− 0.262563i	0.991345	−	0.131282i	$-0.0419092\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	36.9398	1.28452	0.642262	−	0.766485i	$-0.277996\pi$
$827$	36.9398	1.28452	0.642262	+	0.766485i	$0.277996\pi$
$828$	0	0
$829$	−16.2971	−0.566022	−0.283011	−	0.959117i	$-0.591333\pi$
$829$	−16.2971	−0.566022	−0.283011	+	0.959117i	$0.591333\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	− 2.24600i	− 0.0778192i
$834$	0	0
$835$	− 7.91398i	− 0.273875i
$836$	0	0
$837$	0	0
$838$	0	0
$839$	29.0711	1.00365	0.501823	−	0.864970i	$-0.332663\pi$
$839$	29.0711	1.00365	0.501823	+	0.864970i	$0.332663\pi$
$840$	0	0
$841$	18.3488	0.632718
$842$	0	0
$843$	0	0
$844$	0	0
$845$	− 15.4638i	− 0.531971i
$846$	0	0
$847$	6.09107i	0.209292i
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−64.7374	−2.21917
$852$	0	0
$853$	45.3207	1.55175	0.775875	−	0.630887i	$-0.217309\pi$
$853$	45.3207	1.55175	0.775875	+	0.630887i	$0.217309\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 2.68393i	− 0.0916813i	−0.998949	−	0.0458407i	$-0.985403\pi$
$857$	− 2.68393i	− 0.0916813i	0.998949	−	0.0458407i	$-0.0145967\pi$
$858$	0	0
$859$	− 18.2862i	− 0.623917i	−0.950096	−	0.311959i	$-0.899015\pi$
$859$	− 18.2862i	− 0.623917i	0.950096	−	0.311959i	$-0.100985\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	13.0706	0.444927	0.222463	−	0.974941i	$-0.428590\pi$
$863$	13.0706	0.444927	0.222463	+	0.974941i	$0.428590\pi$
$864$	0	0
$865$	3.37492	0.114751
$866$	0	0
$867$	0	0
$868$	0	0
$869$	− 20.2622i	− 0.687349i
$870$	0	0
$871$	4.63499i	0.157051i
$872$	0	0
$873$	0	0
$874$	0	0
$875$	10.7875	0.364684
$876$	0	0
$877$	34.1346	1.15264	0.576322	−	0.817223i	$-0.304487\pi$
$877$	34.1346	1.15264	0.576322	+	0.817223i	$0.304487\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	− 6.07996i	− 0.204839i	−0.994741	−	0.102420i	$-0.967342\pi$
$881$	− 6.07996i	− 0.204839i	0.994741	−	0.102420i	$-0.0326584\pi$
$882$	0	0
$883$	8.63019i	0.290429i	0.989400	+	0.145214i	$0.0463872\pi$
$883$	8.63019i	0.290429i	−0.989400	+	0.145214i	$0.953613\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−43.4177	−1.45782	−0.728911	−	0.684608i	$-0.759973\pi$
$887$	−43.4177	−1.45782	−0.728911	+	0.684608i	$0.759973\pi$
$888$	0	0
$889$	9.84909	0.330328
$890$	0	0
$891$	0	0
$892$	0	0
$893$	− 45.2750i	− 1.51507i
$894$	0	0
$895$	20.8157i	0.695793i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	14.2933	0.476707
$900$	0	0
$901$	25.0863	0.835747
$902$	0	0
$903$	0	0
$904$	0	0
$905$	20.3447i	0.676281i
$906$	0	0
$907$	− 16.9055i	− 0.561337i	−0.959805	−	0.280669i	$-0.909444\pi$
$907$	− 16.9055i	− 0.561337i	0.959805	−	0.280669i	$-0.0905562\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−14.7515	−0.488737	−0.244369	−	0.969682i	$-0.578581\pi$
$911$	−14.7515	−0.488737	−0.244369	+	0.969682i	$0.578581\pi$
$912$	0	0
$913$	−22.0840	−0.730875
$914$	0	0
$915$	0	0
$916$	0	0
$917$	19.3232i	0.638108i
$918$	0	0
$919$	− 0.0621024i	− 0.00204857i	−0.999999	−	0.00102428i	$-0.999674\pi$
$919$	− 0.0621024i	− 0.00204857i	0.999999	−	0.00102428i	$-0.000326040\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−14.0390	−0.462099
$924$	0	0
$925$	−23.9475	−0.787388
$926$	0	0
$927$	0	0
$928$	0	0
$929$	7.38318i	0.242234i	0.992638	+	0.121117i	$0.0386476\pi$
$929$	7.38318i	0.242234i	−0.992638	+	0.121117i	$0.961352\pi$
$930$	0	0
$931$	− 8.63898i	− 0.283131i
$932$	0	0
$933$	0	0
$934$	0	0
$935$	6.45352	0.211053
$936$	0	0
$937$	33.9194	1.10810	0.554049	−	0.832484i	$-0.313082\pi$
$937$	33.9194	1.10810	0.554049	+	0.832484i	$0.313082\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	− 1.63805i	− 0.0533990i	−0.999644	−	0.0266995i	$-0.991500\pi$
$941$	− 1.63805i	− 0.0533990i	0.999644	−	0.0266995i	$-0.00849973\pi$
$942$	0	0
$943$	68.0766i	2.21688i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−7.85963	−0.255404	−0.127702	−	0.991813i	$-0.540760\pi$
$947$	−7.85963	−0.255404	−0.127702	+	0.991813i	$0.540760\pi$
$948$	0	0
$949$	13.3362	0.432913
$950$	0	0
$951$	0	0
$952$	0	0
$953$	36.0249i	1.16696i	0.812127	+	0.583480i	$0.198309\pi$
$953$	36.0249i	1.16696i	−0.812127	+	0.583480i	$0.801691\pi$
$954$	0	0
$955$	26.9917i	0.873432i
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−6.56515	−0.212000
$960$	0	0
$961$	11.8193	0.381268
$962$	0	0
$963$	0	0
$964$	0	0
$965$	− 19.3365i	− 0.622465i
$966$	0	0
$967$	− 60.0449i	− 1.93091i	−0.260562	−	0.965457i	$-0.583908\pi$
$967$	− 60.0449i	− 1.93091i	0.260562	−	0.965457i	$-0.416092\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	2.72865	0.0875665	0.0437832	−	0.999041i	$-0.486059\pi$
$971$	2.72865	0.0875665	0.0437832	+	0.999041i	$0.486059\pi$
$972$	0	0
$973$	19.8276	0.635644
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 53.0413i	− 1.69694i	−0.529243	−	0.848470i	$-0.677524\pi$
$977$	− 53.0413i	− 1.69694i	0.529243	−	0.848470i	$-0.322476\pi$
$978$	0	0
$979$	7.71494i	0.246571i
$980$	0	0
$981$	0	0
$982$	0	0
$983$	39.3167	1.25401	0.627004	−	0.779016i	$-0.284281\pi$
$983$	39.3167	1.25401	0.627004	+	0.779016i	$0.284281\pi$
$984$	0	0
$985$	−5.55591	−0.177026
$986$	0	0
$987$	0	0
$988$	0	0
$989$	− 116.442i	− 3.70262i
$990$	0	0
$991$	− 13.6496i	− 0.433595i	−0.976217	−	0.216798i	$-0.930439\pi$
$991$	− 13.6496i	− 0.433595i	0.976217	−	0.216798i	$-0.0695612\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−27.8968	−0.884389
$996$	0	0
$997$	30.6397	0.970370	0.485185	−	0.874412i	$-0.338752\pi$
$997$	30.6397	0.970370	0.485185	+	0.874412i	$0.338752\pi$
$998$	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.2591.16	yes	24
3.2	odd	2	inner	6048.2.h.i.2591.9	✓	24
4.3	odd	2	inner	6048.2.h.i.2591.10	yes	24
12.11	even	2	inner	6048.2.h.i.2591.15	yes	24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6048.2.h.i.2591.9	✓	24	3.2	odd	2	inner
6048.2.h.i.2591.10	yes	24	4.3	odd	2	inner
6048.2.h.i.2591.15	yes	24	12.11	even	2	inner
6048.2.h.i.2591.16	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)

\(f(q)\)	\(=\)	\(q+1.29686i q^{5} -1.00000i q^{7} +O(q^{10})\)	\(q+1.29686i q^{5} -1.00000i q^{7} -2.21561 q^{11} -1.03730 q^{13} +2.24600i q^{17} +8.63898i q^{19} +8.96997 q^{23} +3.31815 q^{25} -3.26362i q^{29} +4.37958i q^{31} +1.29686 q^{35} -7.21713 q^{37} +7.58939i q^{41} -12.9813i q^{43} -5.24078 q^{47} -1.00000 q^{49} -11.1694i q^{53} -2.87334i q^{55} +2.46447 q^{59} -10.3300 q^{61} -1.34523i q^{65} -4.46833i q^{67} +13.5342 q^{71} -12.8567 q^{73} +2.21561i q^{77} +9.14521i q^{79} +9.96748 q^{83} -2.91275 q^{85} -3.48208i q^{89} +1.03730i q^{91} -11.2036 q^{95} -6.30706 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q+O(q^{10}) \) 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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