LMFDB - Embedded newform 6048.2.c.f.3025.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

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

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

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

chi = DirichletCharacter(H, H._module([0, 1, 0, 0]))

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

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

chi := DirichletCharacter("6048.3025");

S:= CuspForms(chi, 2);

N := Newforms(S);

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

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$48.2935231425$
Analytic rank:	$0$
Dimension:	$24$
Twist minimal:	no (minimal twist has level 1512)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			3025.8
Character	$\chi$	$=$	6048.3025
Dual form			6048.2.c.f.3025.17

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.82986i q^{5} -1.00000 q^{7} +O(q^{10})$	$q-1.82986i q^{5} -1.00000 q^{7} -3.75866i q^{11} +3.09942i q^{13} +3.61074 q^{17} -1.65297i q^{19} +7.47934 q^{23} +1.65160 q^{25} -2.38629i q^{29} -8.97301 q^{31} +1.82986i q^{35} -8.94392i q^{37} +3.04284 q^{41} -1.82597i q^{43} -6.21697 q^{47} +1.00000 q^{49} -3.21435i q^{53} -6.87784 q^{55} +12.4366i q^{59} +9.91962i q^{61} +5.67151 q^{65} -8.73156i q^{67} +12.4229 q^{71} +9.79655 q^{73} +3.75866i q^{77} +7.24388 q^{79} +8.99325i q^{83} -6.60717i q^{85} -1.54944 q^{89} -3.09942i q^{91} -3.02470 q^{95} -10.2583 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q - 24 q^{7}+O(q^{10}) $ 24 * q - 24 * q^7	$ 24 q - 24 q^{7} - 24 q^{25} - 8 q^{31} + 24 q^{49} - 16 q^{55} - 8 q^{79}+O(q^{100}) $ 24 * q - 24 * q^7 - 24 * q^25 - 8 * q^31 + 24 * q^49 - 16 * q^55 - 8 * q^79

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
$5$	− 1.82986i	− 0.818340i	−0.912458	−	0.409170i	$-0.865818\pi$
$5$	− 1.82986i	− 0.818340i	0.912458	−	0.409170i	$-0.134182\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	0	0
$10$	0	0
$11$	− 3.75866i	− 1.13328i	−0.823966	−	0.566639i	$-0.808243\pi$
$11$	− 3.75866i	− 1.13328i	0.823966	−	0.566639i	$-0.191757\pi$
$12$	0	0
$13$	3.09942i	0.859623i	0.902919	+	0.429812i	$0.141420\pi$
$13$	3.09942i	0.859623i	−0.902919	+	0.429812i	$0.858580\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	3.61074	0.875734	0.437867	−	0.899040i	$-0.355734\pi$
$17$	3.61074	0.875734	0.437867	+	0.899040i	$0.355734\pi$
$18$	0	0
$19$	− 1.65297i	− 0.379217i	−0.981860	−	0.189608i	$-0.939278\pi$
$19$	− 1.65297i	− 0.379217i	0.981860	−	0.189608i	$-0.0607218\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	7.47934	1.55955	0.779775	−	0.626060i	$-0.215334\pi$
$23$	7.47934	1.55955	0.779775	+	0.626060i	$0.215334\pi$
$24$	0	0
$25$	1.65160	0.330320
$26$	0	0
$27$	0	0
$28$	0	0
$29$	− 2.38629i	− 0.443123i	−0.975146	−	0.221561i	$-0.928885\pi$
$29$	− 2.38629i	− 0.443123i	0.975146	−	0.221561i	$-0.0711153\pi$
$30$	0	0
$31$	−8.97301	−1.61160	−0.805800	−	0.592188i	$-0.798264\pi$
$31$	−8.97301	−1.61160	−0.805800	+	0.592188i	$0.798264\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	1.82986i	0.309303i
$36$	0	0
$37$	− 8.94392i	− 1.47037i	−0.677865	−	0.735186i	$-0.737095\pi$
$37$	− 8.94392i	− 1.47037i	0.677865	−	0.735186i	$-0.262905\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	3.04284	0.475212	0.237606	−	0.971362i	$-0.423637\pi$
$41$	3.04284	0.475212	0.237606	+	0.971362i	$0.423637\pi$
$42$	0	0
$43$	− 1.82597i	− 0.278457i	−0.990260	−	0.139229i	$-0.955538\pi$
$43$	− 1.82597i	− 0.278457i	0.990260	−	0.139229i	$-0.0444623\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−6.21697	−0.906838	−0.453419	−	0.891297i	$-0.649796\pi$
$47$	−6.21697	−0.906838	−0.453419	+	0.891297i	$0.649796\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	0	0
$52$	0	0
$53$	− 3.21435i	− 0.441525i	−0.975328	−	0.220762i	$-0.929145\pi$
$53$	− 3.21435i	− 0.441525i	0.975328	−	0.220762i	$-0.0708546\pi$
$54$	0	0
$55$	−6.87784	−0.927407
$56$	0	0
$57$	0	0
$58$	0	0
$59$	12.4366i	1.61911i	0.587046	+	0.809554i	$0.300291\pi$
$59$	12.4366i	1.61911i	−0.587046	+	0.809554i	$0.699709\pi$
$60$	0	0
$61$	9.91962i	1.27008i	0.772480	+	0.635039i	$0.219016\pi$
$61$	9.91962i	1.27008i	−0.772480	+	0.635039i	$0.780984\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	5.67151	0.703464
$66$	0	0
$67$	− 8.73156i	− 1.06673i	−0.845885	−	0.533365i	$-0.820927\pi$
$67$	− 8.73156i	− 1.06673i	0.845885	−	0.533365i	$-0.179073\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	12.4229	1.47432	0.737161	−	0.675717i	$-0.236166\pi$
$71$	12.4229	1.47432	0.737161	+	0.675717i	$0.236166\pi$
$72$	0	0
$73$	9.79655	1.14660	0.573300	−	0.819346i	$-0.305663\pi$
$73$	9.79655	1.14660	0.573300	+	0.819346i	$0.305663\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	3.75866i	0.428339i
$78$	0	0
$79$	7.24388	0.815001	0.407500	−	0.913205i	$-0.366401\pi$
$79$	7.24388	0.815001	0.407500	+	0.913205i	$0.366401\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	8.99325i	0.987138i	0.869707	+	0.493569i	$0.164308\pi$
$83$	8.99325i	0.987138i	−0.869707	+	0.493569i	$0.835692\pi$
$84$	0	0
$85$	− 6.60717i	− 0.716648i
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−1.54944	−0.164240	−0.0821202	−	0.996622i	$-0.526169\pi$
$89$	−1.54944	−0.164240	−0.0821202	+	0.996622i	$0.526169\pi$
$90$	0	0
$91$	− 3.09942i	− 0.324907i
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−3.02470	−0.310328
$96$	0	0
$97$	−10.2583	−1.04158	−0.520788	−	0.853686i	$-0.674362\pi$
$97$	−10.2583	−1.04158	−0.520788	+	0.853686i	$0.674362\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	− 5.97880i	− 0.594913i	−0.954735	−	0.297456i	$-0.903862\pi$
$101$	− 5.97880i	− 0.594913i	0.954735	−	0.297456i	$-0.0961383\pi$
$102$	0	0
$103$	16.9060	1.66579	0.832896	−	0.553429i	$-0.186681\pi$
$103$	16.9060	1.66579	0.832896	+	0.553429i	$0.186681\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 2.14315i	− 0.207186i	−0.994620	−	0.103593i	$-0.966966\pi$
$107$	− 2.14315i	− 0.207186i	0.994620	−	0.103593i	$-0.0330339\pi$
$108$	0	0
$109$	− 8.34775i	− 0.799569i	−0.916609	−	0.399785i	$-0.869085\pi$
$109$	− 8.34775i	− 0.799569i	0.916609	−	0.399785i	$-0.130915\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−17.5901	−1.65474	−0.827369	−	0.561659i	$-0.810163\pi$
$113$	−17.5901	−1.65474	−0.827369	+	0.561659i	$0.810163\pi$
$114$	0	0
$115$	− 13.6862i	− 1.27624i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−3.61074	−0.330996
$120$	0	0
$121$	−3.12753	−0.284321
$122$	0	0
$123$	0	0
$124$	0	0
$125$	− 12.1715i	− 1.08865i
$126$	0	0
$127$	−9.73427	−0.863777	−0.431888	−	0.901927i	$-0.642153\pi$
$127$	−9.73427	−0.863777	−0.431888	+	0.901927i	$0.642153\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	− 16.7317i	− 1.46185i	−0.682456	−	0.730926i	$-0.739088\pi$
$131$	− 16.7317i	− 1.46185i	0.682456	−	0.730926i	$-0.260912\pi$
$132$	0	0
$133$	1.65297i	0.143330i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−15.1897	−1.29774	−0.648872	−	0.760897i	$-0.724759\pi$
$137$	−15.1897	−1.29774	−0.648872	+	0.760897i	$0.724759\pi$
$138$	0	0
$139$	− 9.32344i	− 0.790804i	−0.918508	−	0.395402i	$-0.870605\pi$
$139$	− 9.32344i	− 0.790804i	0.918508	−	0.395402i	$-0.129395\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	11.6497	0.974193
$144$	0	0
$145$	−4.36658	−0.362625
$146$	0	0
$147$	0	0
$148$	0	0
$149$	− 3.32322i	− 0.272248i	−0.990692	−	0.136124i	$-0.956535\pi$
$149$	− 3.32322i	− 0.272248i	0.990692	−	0.136124i	$-0.0434646\pi$
$150$	0	0
$151$	−12.3757	−1.00712	−0.503561	−	0.863960i	$-0.667977\pi$
$151$	−12.3757	−1.00712	−0.503561	+	0.863960i	$0.667977\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	16.4194i	1.31884i
$156$	0	0
$157$	− 12.9811i	− 1.03601i	−0.855378	−	0.518004i	$-0.826675\pi$
$157$	− 12.9811i	− 1.03601i	0.855378	−	0.518004i	$-0.173325\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−7.47934	−0.589454
$162$	0	0
$163$	− 6.17108i	− 0.483356i	−0.970356	−	0.241678i	$-0.922302\pi$
$163$	− 6.17108i	− 0.483356i	0.970356	−	0.241678i	$-0.0776978\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	5.75113	0.445036	0.222518	−	0.974929i	$-0.428572\pi$
$167$	5.75113	0.445036	0.222518	+	0.974929i	$0.428572\pi$
$168$	0	0
$169$	3.39362	0.261048
$170$	0	0
$171$	0	0
$172$	0	0
$173$	− 8.09698i	− 0.615602i	−0.951451	−	0.307801i	$-0.900407\pi$
$173$	− 8.09698i	− 0.615602i	0.951451	−	0.307801i	$-0.0995931\pi$
$174$	0	0
$175$	−1.65160	−0.124849
$176$	0	0
$177$	0	0
$178$	0	0
$179$	15.4128i	1.15201i	0.817447	+	0.576003i	$0.195389\pi$
$179$	15.4128i	1.15201i	−0.817447	+	0.576003i	$0.804611\pi$
$180$	0	0
$181$	− 6.06823i	− 0.451048i	−0.974238	−	0.225524i	$-0.927591\pi$
$181$	− 6.06823i	− 0.451048i	0.974238	−	0.225524i	$-0.0724095\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−16.3662	−1.20326
$186$	0	0
$187$	− 13.5716i	− 0.992450i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	2.76632	0.200164	0.100082	−	0.994979i	$-0.468090\pi$
$191$	2.76632	0.200164	0.100082	+	0.994979i	$0.468090\pi$
$192$	0	0
$193$	−17.9782	−1.29410	−0.647048	−	0.762449i	$-0.723997\pi$
$193$	−17.9782	−1.29410	−0.647048	+	0.762449i	$0.723997\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 10.5179i	− 0.749369i	−0.927152	−	0.374684i	$-0.877751\pi$
$197$	− 10.5179i	− 0.749369i	0.927152	−	0.374684i	$-0.122249\pi$
$198$	0	0
$199$	−4.09973	−0.290622	−0.145311	−	0.989386i	$-0.546418\pi$
$199$	−4.09973	−0.290622	−0.145311	+	0.989386i	$0.546418\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	2.38629i	0.167485i
$204$	0	0
$205$	− 5.56799i	− 0.388885i
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−6.21294	−0.429758
$210$	0	0
$211$	− 21.1789i	− 1.45802i	−0.684504	−	0.729009i	$-0.739981\pi$
$211$	− 21.1789i	− 1.45802i	0.684504	−	0.729009i	$-0.260019\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−3.34127	−0.227873
$216$	0	0
$217$	8.97301	0.609128
$218$	0	0
$219$	0	0
$220$	0	0
$221$	11.1912i	0.752801i
$222$	0	0
$223$	2.30718	0.154500	0.0772502	−	0.997012i	$-0.475386\pi$
$223$	2.30718	0.154500	0.0772502	+	0.997012i	$0.475386\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	2.69006i	0.178545i	0.996007	+	0.0892727i	$0.0284543\pi$
$227$	2.69006i	0.178545i	−0.996007	+	0.0892727i	$0.971546\pi$
$228$	0	0
$229$	− 15.3215i	− 1.01247i	−0.862394	−	0.506237i	$-0.831036\pi$
$229$	− 15.3215i	− 1.01247i	0.862394	−	0.506237i	$-0.168964\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	21.3885	1.40121	0.700605	−	0.713549i	$-0.252914\pi$
$233$	21.3885	1.40121	0.700605	+	0.713549i	$0.252914\pi$
$234$	0	0
$235$	11.3762i	0.742102i
$236$	0	0
$237$	0	0
$238$	0	0
$239$	7.04548	0.455734	0.227867	−	0.973692i	$-0.426825\pi$
$239$	7.04548	0.455734	0.227867	+	0.973692i	$0.426825\pi$
$240$	0	0
$241$	−0.343612	−0.0221340	−0.0110670	−	0.999939i	$-0.503523\pi$
$241$	−0.343612	−0.0221340	−0.0110670	+	0.999939i	$0.503523\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	− 1.82986i	− 0.116906i
$246$	0	0
$247$	5.12323	0.325983
$248$	0	0
$249$	0	0
$250$	0	0
$251$	7.65363i	0.483093i	0.970389	+	0.241546i	$0.0776546\pi$
$251$	7.65363i	0.483093i	−0.970389	+	0.241546i	$0.922345\pi$
$252$	0	0
$253$	− 28.1123i	− 1.76740i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	7.01134	0.437356	0.218678	−	0.975797i	$-0.429826\pi$
$257$	7.01134	0.437356	0.218678	+	0.975797i	$0.429826\pi$
$258$	0	0
$259$	8.94392i	0.555748i
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−24.0879	−1.48532	−0.742661	−	0.669667i	$-0.766437\pi$
$263$	−24.0879	−1.48532	−0.742661	+	0.669667i	$0.766437\pi$
$264$	0	0
$265$	−5.88182	−0.361317
$266$	0	0
$267$	0	0
$268$	0	0
$269$	− 14.4906i	− 0.883507i	−0.897136	−	0.441753i	$-0.854357\pi$
$269$	− 14.4906i	− 0.883507i	0.897136	−	0.441753i	$-0.145643\pi$
$270$	0	0
$271$	−28.3057	−1.71945	−0.859726	−	0.510756i	$-0.829365\pi$
$271$	−28.3057	−1.71945	−0.859726	+	0.510756i	$0.829365\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	− 6.20780i	− 0.374344i
$276$	0	0
$277$	− 9.70296i	− 0.582994i	−0.956572	−	0.291497i	$-0.905847\pi$
$277$	− 9.70296i	− 0.582994i	0.956572	−	0.291497i	$-0.0941534\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−16.0412	−0.956937	−0.478469	−	0.878105i	$-0.658808\pi$
$281$	−16.0412	−0.956937	−0.478469	+	0.878105i	$0.658808\pi$
$282$	0	0
$283$	− 11.0326i	− 0.655818i	−0.944709	−	0.327909i	$-0.893656\pi$
$283$	− 11.0326i	− 0.655818i	0.944709	−	0.327909i	$-0.106344\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−3.04284	−0.179613
$288$	0	0
$289$	−3.96254	−0.233091
$290$	0	0
$291$	0	0
$292$	0	0
$293$	10.9744i	0.641131i	0.947226	+	0.320566i	$0.103873\pi$
$293$	10.9744i	0.641131i	−0.947226	+	0.320566i	$0.896127\pi$
$294$	0	0
$295$	22.7573	1.32498
$296$	0	0
$297$	0	0
$298$	0	0
$299$	23.1816i	1.34062i
$300$	0	0
$301$	1.82597i	0.105247i
$302$	0	0
$303$	0	0
$304$	0	0
$305$	18.1516	1.03936
$306$	0	0
$307$	28.2184i	1.61051i	0.592931	+	0.805254i	$0.297971\pi$
$307$	28.2184i	1.61051i	−0.592931	+	0.805254i	$0.702029\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	24.3156	1.37881	0.689407	−	0.724375i	$-0.257871\pi$
$311$	24.3156	1.37881	0.689407	+	0.724375i	$0.257871\pi$
$312$	0	0
$313$	−8.29548	−0.468889	−0.234444	−	0.972130i	$-0.575327\pi$
$313$	−8.29548	−0.468889	−0.234444	+	0.972130i	$0.575327\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 0.906822i	− 0.0509322i	−0.999676	−	0.0254661i	$-0.991893\pi$
$317$	− 0.906822i	− 0.0509322i	0.999676	−	0.0254661i	$-0.00810699\pi$
$318$	0	0
$319$	−8.96925	−0.502182
$320$	0	0
$321$	0	0
$322$	0	0
$323$	− 5.96844i	− 0.332093i
$324$	0	0
$325$	5.11899i	0.283950i
$326$	0	0
$327$	0	0
$328$	0	0
$329$	6.21697	0.342753
$330$	0	0
$331$	31.7662i	1.74603i	0.487697	+	0.873013i	$0.337837\pi$
$331$	31.7662i	1.74603i	−0.487697	+	0.873013i	$0.662163\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−15.9776	−0.872948
$336$	0	0
$337$	−34.4231	−1.87515	−0.937573	−	0.347788i	$-0.886933\pi$
$337$	−34.4231	−1.87515	−0.937573	+	0.347788i	$0.886933\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	33.7265i	1.82639i
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 23.5611i	− 1.26483i	−0.774630	−	0.632414i	$-0.782064\pi$
$347$	− 23.5611i	− 1.26483i	0.774630	−	0.632414i	$-0.217936\pi$
$348$	0	0
$349$	− 28.6095i	− 1.53143i	−0.643179	−	0.765716i	$-0.722385\pi$
$349$	− 28.6095i	− 1.53143i	0.643179	−	0.765716i	$-0.277615\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−7.03843	−0.374618	−0.187309	−	0.982301i	$-0.559976\pi$
$353$	−7.03843	−0.374618	−0.187309	+	0.982301i	$0.559976\pi$
$354$	0	0
$355$	− 22.7321i	− 1.20650i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	13.2261	0.698046	0.349023	−	0.937114i	$-0.386513\pi$
$359$	13.2261	0.698046	0.349023	+	0.937114i	$0.386513\pi$
$360$	0	0
$361$	16.2677	0.856195
$362$	0	0
$363$	0	0
$364$	0	0
$365$	− 17.9263i	− 0.938308i
$366$	0	0
$367$	−11.6530	−0.608281	−0.304140	−	0.952627i	$-0.598369\pi$
$367$	−11.6530	−0.608281	−0.304140	+	0.952627i	$0.598369\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	3.21435i	0.166881i
$372$	0	0
$373$	− 9.85547i	− 0.510297i	−0.966902	−	0.255148i	$-0.917876\pi$
$373$	− 9.85547i	− 0.510297i	0.966902	−	0.255148i	$-0.0821243\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	7.39610	0.380919
$378$	0	0
$379$	− 34.5581i	− 1.77513i	−0.460684	−	0.887564i	$-0.652396\pi$
$379$	− 34.5581i	− 1.77513i	0.460684	−	0.887564i	$-0.347604\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−11.4073	−0.582884	−0.291442	−	0.956589i	$-0.594135\pi$
$383$	−11.4073	−0.582884	−0.291442	+	0.956589i	$0.594135\pi$
$384$	0	0
$385$	6.87784	0.350527
$386$	0	0
$387$	0	0
$388$	0	0
$389$	37.3093i	1.89166i	0.324667	+	0.945828i	$0.394748\pi$
$389$	37.3093i	1.89166i	−0.324667	+	0.945828i	$0.605252\pi$
$390$	0	0
$391$	27.0060	1.36575
$392$	0	0
$393$	0	0
$394$	0	0
$395$	− 13.2553i	− 0.666947i
$396$	0	0
$397$	10.8903i	0.546567i	0.961934	+	0.273283i	$0.0881097\pi$
$397$	10.8903i	0.546567i	−0.961934	+	0.273283i	$0.911890\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	20.6784	1.03263	0.516316	−	0.856398i	$-0.327303\pi$
$401$	20.6784	1.03263	0.516316	+	0.856398i	$0.327303\pi$
$402$	0	0
$403$	− 27.8111i	− 1.38537i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−33.6172	−1.66634
$408$	0	0
$409$	−16.1209	−0.797129	−0.398565	−	0.917140i	$-0.630492\pi$
$409$	−16.1209	−0.797129	−0.398565	+	0.917140i	$0.630492\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	− 12.4366i	− 0.611965i
$414$	0	0
$415$	16.4564	0.807814
$416$	0	0
$417$	0	0
$418$	0	0
$419$	− 26.5958i	− 1.29929i	−0.760237	−	0.649645i	$-0.774917\pi$
$419$	− 26.5958i	− 1.29929i	0.760237	−	0.649645i	$-0.225083\pi$
$420$	0	0
$421$	− 27.2198i	− 1.32661i	−0.748348	−	0.663307i	$-0.769153\pi$
$421$	− 27.2198i	− 1.32661i	0.748348	−	0.663307i	$-0.230847\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	5.96350	0.289272
$426$	0	0
$427$	− 9.91962i	− 0.480044i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−4.58135	−0.220676	−0.110338	−	0.993894i	$-0.535193\pi$
$431$	−4.58135	−0.220676	−0.110338	+	0.993894i	$0.535193\pi$
$432$	0	0
$433$	4.43085	0.212933	0.106466	−	0.994316i	$-0.466046\pi$
$433$	4.43085	0.212933	0.106466	+	0.994316i	$0.466046\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 12.3631i	− 0.591407i
$438$	0	0
$439$	11.2867	0.538685	0.269343	−	0.963044i	$-0.413194\pi$
$439$	11.2867	0.538685	0.269343	+	0.963044i	$0.413194\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	− 3.69129i	− 0.175378i	−0.996148	−	0.0876892i	$-0.972052\pi$
$443$	− 3.69129i	− 0.175378i	0.996148	−	0.0876892i	$-0.0279482\pi$
$444$	0	0
$445$	2.83526i	0.134404i
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−18.0692	−0.852737	−0.426368	−	0.904550i	$-0.640207\pi$
$449$	−18.0692	−0.852737	−0.426368	+	0.904550i	$0.640207\pi$
$450$	0	0
$451$	− 11.4370i	− 0.538548i
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−5.67151	−0.265884
$456$	0	0
$457$	−21.4880	−1.00517	−0.502583	−	0.864529i	$-0.667617\pi$
$457$	−21.4880	−1.00517	−0.502583	+	0.864529i	$0.667617\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	− 16.7622i	− 0.780695i	−0.920668	−	0.390348i	$-0.872355\pi$
$461$	− 16.7622i	− 0.780695i	0.920668	−	0.390348i	$-0.127645\pi$
$462$	0	0
$463$	9.74065	0.452686	0.226343	−	0.974048i	$-0.427323\pi$
$463$	9.74065	0.452686	0.226343	+	0.974048i	$0.427323\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 33.8683i	− 1.56724i	−0.621241	−	0.783620i	$-0.713371\pi$
$467$	− 33.8683i	− 1.56724i	0.621241	−	0.783620i	$-0.286629\pi$
$468$	0	0
$469$	8.73156i	0.403186i
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−6.86319	−0.315570
$474$	0	0
$475$	− 2.73004i	− 0.125263i
$476$	0	0
$477$	0	0
$478$	0	0
$479$	12.0611	0.551084	0.275542	−	0.961289i	$-0.411143\pi$
$479$	12.0611	0.551084	0.275542	+	0.961289i	$0.411143\pi$
$480$	0	0
$481$	27.7209	1.26397
$482$	0	0
$483$	0	0
$484$	0	0
$485$	18.7714i	0.852364i
$486$	0	0
$487$	11.5752	0.524522	0.262261	−	0.964997i	$-0.415532\pi$
$487$	11.5752	0.524522	0.262261	+	0.964997i	$0.415532\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	40.0965i	1.80953i	0.425911	+	0.904765i	$0.359954\pi$
$491$	40.0965i	1.80953i	−0.425911	+	0.904765i	$0.640046\pi$
$492$	0	0
$493$	− 8.61627i	− 0.388057i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−12.4229	−0.557241
$498$	0	0
$499$	33.2035i	1.48639i	0.669073	+	0.743197i	$0.266691\pi$
$499$	33.2035i	1.48639i	−0.669073	+	0.743197i	$0.733309\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−4.46295	−0.198993	−0.0994967	−	0.995038i	$-0.531723\pi$
$503$	−4.46295	−0.198993	−0.0994967	+	0.995038i	$0.531723\pi$
$504$	0	0
$505$	−10.9404	−0.486841
$506$	0	0
$507$	0	0
$508$	0	0
$509$	28.2457i	1.25197i	0.779837	+	0.625983i	$0.215302\pi$
$509$	28.2457i	1.25197i	−0.779837	+	0.625983i	$0.784698\pi$
$510$	0	0
$511$	−9.79655	−0.433374
$512$	0	0
$513$	0	0
$514$	0	0
$515$	− 30.9356i	− 1.36318i
$516$	0	0
$517$	23.3675i	1.02770i
$518$	0	0
$519$	0	0
$520$	0	0
$521$	30.4655	1.33472	0.667359	−	0.744737i	$-0.267425\pi$
$521$	30.4655	1.33472	0.667359	+	0.744737i	$0.267425\pi$
$522$	0	0
$523$	36.5909i	1.60001i	0.599994	+	0.800004i	$0.295169\pi$
$523$	36.5909i	1.60001i	−0.599994	+	0.800004i	$0.704831\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−32.3992	−1.41133
$528$	0	0
$529$	32.9405	1.43219
$530$	0	0
$531$	0	0
$532$	0	0
$533$	9.43104i	0.408504i
$534$	0	0
$535$	−3.92167	−0.169548
$536$	0	0
$537$	0	0
$538$	0	0
$539$	− 3.75866i	− 0.161897i
$540$	0	0
$541$	− 42.8045i	− 1.84031i	−0.391556	−	0.920154i	$-0.628063\pi$
$541$	− 42.8045i	− 1.84031i	0.391556	−	0.920154i	$-0.371937\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−15.2752	−0.654319
$546$	0	0
$547$	6.57128i	0.280968i	0.990083	+	0.140484i	$0.0448658\pi$
$547$	6.57128i	0.280968i	−0.990083	+	0.140484i	$0.955134\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−3.94446	−0.168040
$552$	0	0
$553$	−7.24388	−0.308041
$554$	0	0
$555$	0	0
$556$	0	0
$557$	31.2762i	1.32521i	0.748967	+	0.662607i	$0.230550\pi$
$557$	31.2762i	1.32521i	−0.748967	+	0.662607i	$0.769450\pi$
$558$	0	0
$559$	5.65943	0.239368
$560$	0	0
$561$	0	0
$562$	0	0
$563$	3.69436i	0.155699i	0.996965	+	0.0778493i	$0.0248053\pi$
$563$	3.69436i	0.155699i	−0.996965	+	0.0778493i	$0.975195\pi$
$564$	0	0
$565$	32.1875i	1.35414i
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−11.6082	−0.486641	−0.243320	−	0.969946i	$-0.578237\pi$
$569$	−11.6082	−0.486641	−0.243320	+	0.969946i	$0.578237\pi$
$570$	0	0
$571$	− 26.8739i	− 1.12464i	−0.826920	−	0.562319i	$-0.809909\pi$
$571$	− 26.8739i	− 1.12464i	0.826920	−	0.562319i	$-0.190091\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	12.3529	0.515150
$576$	0	0
$577$	10.2051	0.424843	0.212421	−	0.977178i	$-0.431865\pi$
$577$	10.2051	0.424843	0.212421	+	0.977178i	$0.431865\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	− 8.99325i	− 0.373103i
$582$	0	0
$583$	−12.0817	−0.500371
$584$	0	0
$585$	0	0
$586$	0	0
$587$	25.0075i	1.03217i	0.856538	+	0.516085i	$0.172611\pi$
$587$	25.0075i	1.03217i	−0.856538	+	0.516085i	$0.827389\pi$
$588$	0	0
$589$	14.8321i	0.611146i
$590$	0	0
$591$	0	0
$592$	0	0
$593$	43.4040	1.78239	0.891195	−	0.453620i	$-0.149868\pi$
$593$	43.4040	1.78239	0.891195	+	0.453620i	$0.149868\pi$
$594$	0	0
$595$	6.60717i	0.270867i
$596$	0	0
$597$	0	0
$598$	0	0
$599$	2.47057	0.100945	0.0504724	−	0.998725i	$-0.483927\pi$
$599$	2.47057	0.100945	0.0504724	+	0.998725i	$0.483927\pi$
$600$	0	0
$601$	−13.2142	−0.539018	−0.269509	−	0.962998i	$-0.586861\pi$
$601$	−13.2142	−0.539018	−0.269509	+	0.962998i	$0.586861\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	5.72296i	0.232671i
$606$	0	0
$607$	−29.4562	−1.19559	−0.597796	−	0.801649i	$-0.703957\pi$
$607$	−29.4562	−1.19559	−0.597796	+	0.801649i	$0.703957\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	− 19.2690i	− 0.779539i
$612$	0	0
$613$	16.9457i	0.684429i	0.939622	+	0.342215i	$0.111177\pi$
$613$	16.9457i	0.684429i	−0.939622	+	0.342215i	$0.888823\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	42.1917	1.69858	0.849288	−	0.527930i	$-0.177032\pi$
$617$	42.1917	1.69858	0.849288	+	0.527930i	$0.177032\pi$
$618$	0	0
$619$	− 47.7508i	− 1.91927i	−0.281256	−	0.959633i	$-0.590751\pi$
$619$	− 47.7508i	− 1.91927i	0.281256	−	0.959633i	$-0.409249\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	1.54944	0.0620770
$624$	0	0
$625$	−14.0142	−0.560569
$626$	0	0
$627$	0	0
$628$	0	0
$629$	− 32.2942i	− 1.28765i
$630$	0	0
$631$	−29.3539	−1.16856	−0.584279	−	0.811553i	$-0.698623\pi$
$631$	−29.3539	−1.16856	−0.584279	+	0.811553i	$0.698623\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	17.8124i	0.706863i
$636$	0	0
$637$	3.09942i	0.122803i
$638$	0	0
$639$	0	0
$640$	0	0
$641$	15.2796	0.603509	0.301754	−	0.953386i	$-0.402428\pi$
$641$	15.2796	0.603509	0.301754	+	0.953386i	$0.402428\pi$
$642$	0	0
$643$	26.8547i	1.05905i	0.848295	+	0.529524i	$0.177629\pi$
$643$	26.8547i	1.05905i	−0.848295	+	0.529524i	$0.822371\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	13.6808	0.537850	0.268925	−	0.963161i	$-0.413332\pi$
$647$	13.6808	0.537850	0.268925	+	0.963161i	$0.413332\pi$
$648$	0	0
$649$	46.7450	1.83490
$650$	0	0
$651$	0	0
$652$	0	0
$653$	− 5.65064i	− 0.221127i	−0.993869	−	0.110563i	$-0.964734\pi$
$653$	− 5.65064i	− 0.221127i	0.993869	−	0.110563i	$-0.0352655\pi$
$654$	0	0
$655$	−30.6167	−1.19629
$656$	0	0
$657$	0	0
$658$	0	0
$659$	− 24.7127i	− 0.962669i	−0.876537	−	0.481334i	$-0.840152\pi$
$659$	− 24.7127i	− 0.962669i	0.876537	−	0.481334i	$-0.159848\pi$
$660$	0	0
$661$	− 20.1467i	− 0.783615i	−0.920047	−	0.391807i	$-0.871850\pi$
$661$	− 20.1467i	− 0.783615i	0.920047	−	0.391807i	$-0.128150\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	3.02470	0.117293
$666$	0	0
$667$	− 17.8479i	− 0.691072i
$668$	0	0
$669$	0	0
$670$	0	0
$671$	37.2845	1.43935
$672$	0	0
$673$	−23.0926	−0.890155	−0.445077	−	0.895492i	$-0.646824\pi$
$673$	−23.0926	−0.890155	−0.445077	+	0.895492i	$0.646824\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 5.41515i	− 0.208121i	−0.994571	−	0.104061i	$-0.966816\pi$
$677$	− 5.41515i	− 0.208121i	0.994571	−	0.104061i	$-0.0331836\pi$
$678$	0	0
$679$	10.2583	0.393679
$680$	0	0
$681$	0	0
$682$	0	0
$683$	− 16.5403i	− 0.632898i	−0.948609	−	0.316449i	$-0.897509\pi$
$683$	− 16.5403i	− 0.632898i	0.948609	−	0.316449i	$-0.102491\pi$
$684$	0	0
$685$	27.7951i	1.06200i
$686$	0	0
$687$	0	0
$688$	0	0
$689$	9.96261	0.379545
$690$	0	0
$691$	17.8893i	0.680542i	0.940327	+	0.340271i	$0.110519\pi$
$691$	17.8893i	0.680542i	−0.940327	+	0.340271i	$0.889481\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−17.0606	−0.647147
$696$	0	0
$697$	10.9869	0.416160
$698$	0	0
$699$	0	0
$700$	0	0
$701$	28.6956i	1.08382i	0.840437	+	0.541909i	$0.182298\pi$
$701$	28.6956i	1.08382i	−0.840437	+	0.541909i	$0.817702\pi$
$702$	0	0
$703$	−14.7840	−0.557590
$704$	0	0
$705$	0	0
$706$	0	0
$707$	5.97880i	0.224856i
$708$	0	0
$709$	43.2083i	1.62272i	0.584546	+	0.811360i	$0.301272\pi$
$709$	43.2083i	1.62272i	−0.584546	+	0.811360i	$0.698728\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−67.1122	−2.51337
$714$	0	0
$715$	− 21.3173i	− 0.797221i
$716$	0	0
$717$	0	0
$718$	0	0
$719$	40.7179	1.51852	0.759261	−	0.650786i	$-0.225560\pi$
$719$	40.7179	1.51852	0.759261	+	0.650786i	$0.225560\pi$
$720$	0	0
$721$	−16.9060	−0.629611
$722$	0	0
$723$	0	0
$724$	0	0
$725$	− 3.94119i	− 0.146372i
$726$	0	0
$727$	1.96551	0.0728967	0.0364483	−	0.999336i	$-0.488396\pi$
$727$	1.96551	0.0728967	0.0364483	+	0.999336i	$0.488396\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	− 6.59310i	− 0.243854i
$732$	0	0
$733$	39.1211i	1.44497i	0.691386	+	0.722486i	$0.257001\pi$
$733$	39.1211i	1.44497i	−0.691386	+	0.722486i	$0.742999\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−32.8190	−1.20890
$738$	0	0
$739$	− 36.1851i	− 1.33109i	−0.746357	−	0.665546i	$-0.768199\pi$
$739$	− 36.1851i	− 1.33109i	0.746357	−	0.665546i	$-0.231801\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	20.1222	0.738210	0.369105	−	0.929388i	$-0.379664\pi$
$743$	20.1222	0.738210	0.369105	+	0.929388i	$0.379664\pi$
$744$	0	0
$745$	−6.08103	−0.222792
$746$	0	0
$747$	0	0
$748$	0	0
$749$	2.14315i	0.0783089i
$750$	0	0
$751$	36.2627	1.32324	0.661622	−	0.749838i	$-0.269869\pi$
$751$	36.2627	1.32324	0.661622	+	0.749838i	$0.269869\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	22.6459i	0.824167i
$756$	0	0
$757$	− 53.6865i	− 1.95127i	−0.219401	−	0.975635i	$-0.570410\pi$
$757$	− 53.6865i	− 1.95127i	0.219401	−	0.975635i	$-0.429590\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	37.3332	1.35333	0.676664	−	0.736292i	$-0.263425\pi$
$761$	37.3332	1.35333	0.676664	+	0.736292i	$0.263425\pi$
$762$	0	0
$763$	8.34775i	0.302209i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−38.5462	−1.39182
$768$	0	0
$769$	15.5423	0.560469	0.280235	−	0.959932i	$-0.409588\pi$
$769$	15.5423	0.560469	0.280235	+	0.959932i	$0.409588\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	− 39.0073i	− 1.40299i	−0.712672	−	0.701497i	$-0.752515\pi$
$773$	− 39.0073i	− 1.40299i	0.712672	−	0.701497i	$-0.247485\pi$
$774$	0	0
$775$	−14.8198	−0.532343
$776$	0	0
$777$	0	0
$778$	0	0
$779$	− 5.02972i	− 0.180209i
$780$	0	0
$781$	− 46.6933i	− 1.67082i
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−23.7537	−0.847807
$786$	0	0
$787$	9.50805i	0.338925i	0.985537	+	0.169463i	$0.0542032\pi$
$787$	9.50805i	0.338925i	−0.985537	+	0.169463i	$0.945797\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	17.5901	0.625432
$792$	0	0
$793$	−30.7450	−1.09179
$794$	0	0
$795$	0	0
$796$	0	0
$797$	40.4120i	1.43147i	0.698374	+	0.715733i	$0.253907\pi$
$797$	40.4120i	1.43147i	−0.698374	+	0.715733i	$0.746093\pi$
$798$	0	0
$799$	−22.4479	−0.794149
$800$	0	0
$801$	0	0
$802$	0	0
$803$	− 36.8219i	− 1.29942i
$804$	0	0
$805$	13.6862i	0.482374i
$806$	0	0
$807$	0	0
$808$	0	0
$809$	32.4907	1.14231	0.571157	−	0.820841i	$-0.306495\pi$
$809$	32.4907	1.14231	0.571157	+	0.820841i	$0.306495\pi$
$810$	0	0
$811$	− 8.34275i	− 0.292954i	−0.989214	−	0.146477i	$-0.953207\pi$
$811$	− 8.34275i	− 0.292954i	0.989214	−	0.146477i	$-0.0467934\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−11.2922	−0.395550
$816$	0	0
$817$	−3.01826	−0.105596
$818$	0	0
$819$	0	0
$820$	0	0
$821$	30.6627i	1.07014i	0.844809	+	0.535068i	$0.179714\pi$
$821$	30.6627i	1.07014i	−0.844809	+	0.535068i	$0.820286\pi$
$822$	0	0
$823$	25.5627	0.891059	0.445530	−	0.895267i	$-0.353015\pi$
$823$	25.5627	0.891059	0.445530	+	0.895267i	$0.353015\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 12.4138i	− 0.431670i	−0.976430	−	0.215835i	$-0.930753\pi$
$827$	− 12.4138i	− 0.431670i	0.976430	−	0.215835i	$-0.0692474\pi$
$828$	0	0
$829$	21.6360i	0.751449i	0.926731	+	0.375724i	$0.122606\pi$
$829$	21.6360i	0.751449i	−0.926731	+	0.375724i	$0.877394\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	3.61074	0.125105
$834$	0	0
$835$	− 10.5238i	− 0.364191i
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−1.76288	−0.0608612	−0.0304306	−	0.999537i	$-0.509688\pi$
$839$	−1.76288	−0.0608612	−0.0304306	+	0.999537i	$0.509688\pi$
$840$	0	0
$841$	23.3056	0.803642
$842$	0	0
$843$	0	0
$844$	0	0
$845$	− 6.20987i	− 0.213626i
$846$	0	0
$847$	3.12753	0.107463
$848$	0	0
$849$	0	0
$850$	0	0
$851$	− 66.8946i	− 2.29312i
$852$	0	0
$853$	0.627765i	0.0214943i	0.999942	+	0.0107471i	$0.00342098\pi$
$853$	0.627765i	0.0214943i	−0.999942	+	0.0107471i	$0.996579\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	35.3406	1.20721	0.603605	−	0.797283i	$-0.293730\pi$
$857$	35.3406	1.20721	0.603605	+	0.797283i	$0.293730\pi$
$858$	0	0
$859$	50.5205i	1.72374i	0.507133	+	0.861868i	$0.330705\pi$
$859$	50.5205i	1.72374i	−0.507133	+	0.861868i	$0.669295\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	35.7081	1.21552	0.607758	−	0.794122i	$-0.292069\pi$
$863$	35.7081	1.21552	0.607758	+	0.794122i	$0.292069\pi$
$864$	0	0
$865$	−14.8164	−0.503771
$866$	0	0
$867$	0	0
$868$	0	0
$869$	− 27.2273i	− 0.923623i
$870$	0	0
$871$	27.0627	0.916986
$872$	0	0
$873$	0	0
$874$	0	0
$875$	12.1715i	0.411472i
$876$	0	0
$877$	43.7066i	1.47587i	0.674873	+	0.737934i	$0.264198\pi$
$877$	43.7066i	1.47587i	−0.674873	+	0.737934i	$0.735802\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	39.3404	1.32541	0.662705	−	0.748880i	$-0.269408\pi$
$881$	39.3404	1.32541	0.662705	+	0.748880i	$0.269408\pi$
$882$	0	0
$883$	14.5600i	0.489981i	0.969525	+	0.244991i	$0.0787849\pi$
$883$	14.5600i	0.489981i	−0.969525	+	0.244991i	$0.921215\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−24.2186	−0.813182	−0.406591	−	0.913610i	$-0.633283\pi$
$887$	−24.2186	−0.813182	−0.406591	+	0.913610i	$0.633283\pi$
$888$	0	0
$889$	9.73427	0.326477
$890$	0	0
$891$	0	0
$892$	0	0
$893$	10.2764i	0.343888i
$894$	0	0
$895$	28.2033	0.942733
$896$	0	0
$897$	0	0
$898$	0	0
$899$	21.4122i	0.714137i
$900$	0	0
$901$	− 11.6062i	− 0.386658i
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−11.1040	−0.369111
$906$	0	0
$907$	32.8766i	1.09165i	0.837899	+	0.545825i	$0.183784\pi$
$907$	32.8766i	1.09165i	−0.837899	+	0.545825i	$0.816216\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−17.8694	−0.592039	−0.296020	−	0.955182i	$-0.595659\pi$
$911$	−17.8694	−0.592039	−0.296020	+	0.955182i	$0.595659\pi$
$912$	0	0
$913$	33.8026	1.11870
$914$	0	0
$915$	0	0
$916$	0	0
$917$	16.7317i	0.552528i
$918$	0	0
$919$	−25.3511	−0.836257	−0.418128	−	0.908388i	$-0.637314\pi$
$919$	−25.3511	−0.836257	−0.418128	+	0.908388i	$0.637314\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	38.5036i	1.26736i
$924$	0	0
$925$	− 14.7718i	− 0.485693i
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−39.3410	−1.29074	−0.645368	−	0.763872i	$-0.723296\pi$
$929$	−39.3410	−1.29074	−0.645368	+	0.763872i	$0.723296\pi$
$930$	0	0
$931$	− 1.65297i	− 0.0541738i
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−24.8341	−0.812162
$936$	0	0
$937$	44.9769	1.46933	0.734665	−	0.678430i	$-0.237339\pi$
$937$	44.9769	1.46933	0.734665	+	0.678430i	$0.237339\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	0.488014i	0.0159088i	0.999968	+	0.00795440i	$0.00253199\pi$
$941$	0.488014i	0.0159088i	−0.999968	+	0.00795440i	$0.997468\pi$
$942$	0	0
$943$	22.7585	0.741117
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 1.70377i	− 0.0553652i	−0.999617	−	0.0276826i	$-0.991187\pi$
$947$	− 1.70377i	− 0.0553652i	0.999617	−	0.0276826i	$-0.00881278\pi$
$948$	0	0
$949$	30.3636i	0.985643i
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−9.15979	−0.296715	−0.148357	−	0.988934i	$-0.547399\pi$
$953$	−9.15979	−0.296715	−0.148357	+	0.988934i	$0.547399\pi$
$954$	0	0
$955$	− 5.06198i	− 0.163802i
$956$	0	0
$957$	0	0
$958$	0	0
$959$	15.1897	0.490501
$960$	0	0
$961$	49.5149	1.59726
$962$	0	0
$963$	0	0
$964$	0	0
$965$	32.8976i	1.05901i
$966$	0	0
$967$	−36.1742	−1.16328	−0.581642	−	0.813445i	$-0.697589\pi$
$967$	−36.1742	−1.16328	−0.581642	+	0.813445i	$0.697589\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	− 7.10607i	− 0.228045i	−0.993478	−	0.114022i	$-0.963626\pi$
$971$	− 7.10607i	− 0.228045i	0.993478	−	0.114022i	$-0.0363735\pi$
$972$	0	0
$973$	9.32344i	0.298896i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	56.6938	1.81380	0.906898	−	0.421349i	$-0.138443\pi$
$977$	56.6938	1.81380	0.906898	+	0.421349i	$0.138443\pi$
$978$	0	0
$979$	5.82382i	0.186130i
$980$	0	0
$981$	0	0
$982$	0	0
$983$	21.6213	0.689614	0.344807	−	0.938674i	$-0.387944\pi$
$983$	21.6213	0.689614	0.344807	+	0.938674i	$0.387944\pi$
$984$	0	0
$985$	−19.2463	−0.613238
$986$	0	0
$987$	0	0
$988$	0	0
$989$	− 13.6570i	− 0.434268i
$990$	0	0
$991$	−22.3945	−0.711384	−0.355692	−	0.934603i	$-0.615755\pi$
$991$	−22.3945	−0.711384	−0.355692	+	0.934603i	$0.615755\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	7.50195i	0.237828i
$996$	0	0
$997$	11.3181i	0.358448i	0.983808	+	0.179224i	$0.0573587\pi$
$997$	11.3181i	0.358448i	−0.983808	+	0.179224i	$0.942641\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6048.2.c.f.3025.8		24
3.2	odd	2	inner	6048.2.c.f.3025.18		24
4.3	odd	2		1512.2.c.g.757.22	yes	24
8.3	odd	2		1512.2.c.g.757.21	yes	24
8.5	even	2	inner	6048.2.c.f.3025.17		24
12.11	even	2		1512.2.c.g.757.3	✓	24
24.5	odd	2	inner	6048.2.c.f.3025.7		24
24.11	even	2		1512.2.c.g.757.4	yes	24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1512.2.c.g.757.3	✓	24	12.11	even	2
1512.2.c.g.757.4	yes	24	24.11	even	2
1512.2.c.g.757.21	yes	24	8.3	odd	2
1512.2.c.g.757.22	yes	24	4.3	odd	2
6048.2.c.f.3025.7		24	24.5	odd	2	inner
6048.2.c.f.3025.8		24	1.1	even	1	trivial
6048.2.c.f.3025.17		24	8.5	even	2	inner
6048.2.c.f.3025.18		24	3.2	odd	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

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

\(f(q)\)	\(=\)	\(q-1.82986i q^{5} -1.00000 q^{7} +O(q^{10})\)	\(q-1.82986i q^{5} -1.00000 q^{7} -3.75866i q^{11} +3.09942i q^{13} +3.61074 q^{17} -1.65297i q^{19} +7.47934 q^{23} +1.65160 q^{25} -2.38629i q^{29} -8.97301 q^{31} +1.82986i q^{35} -8.94392i q^{37} +3.04284 q^{41} -1.82597i q^{43} -6.21697 q^{47} +1.00000 q^{49} -3.21435i q^{53} -6.87784 q^{55} +12.4366i q^{59} +9.91962i q^{61} +5.67151 q^{65} -8.73156i q^{67} +12.4229 q^{71} +9.79655 q^{73} +3.75866i q^{77} +7.24388 q^{79} +8.99325i q^{83} -6.60717i q^{85} -1.54944 q^{89} -3.09942i q^{91} -3.02470 q^{95} -10.2583 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q - 24 q^{7}+O(q^{10}) \) 24 * q - 24 * q^7	\( 24 q - 24 q^{7} - 24 q^{25} - 8 q^{31} + 24 q^{49} - 16 q^{55} - 8 q^{79}+O(q^{100}) \) 24 * q - 24 * q^7 - 24 * q^25 - 8 * q^31 + 24 * q^49 - 16 * q^55 - 8 * q^79

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