LMFDB - Embedded newform 6048.2.c.g.3025.16

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			3025.16
Character	$\chi$	$=$	6048.3025
Dual form			6048.2.c.g.3025.9

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.53368i q^{5} -1.00000 q^{7} +O(q^{10})$	$q+1.53368i q^{5} -1.00000 q^{7} +2.28335i q^{11} +7.10447i q^{13} +6.81307 q^{17} +1.60676i q^{19} -1.16757 q^{23} +2.64784 q^{25} +4.07059i q^{29} +7.90568 q^{31} -1.53368i q^{35} -7.04836i q^{37} +10.1710 q^{41} -0.344719i q^{43} -3.10859 q^{47} +1.00000 q^{49} -5.66835i q^{53} -3.50191 q^{55} +1.38165i q^{59} +5.50516i q^{61} -10.8960 q^{65} -8.35907i q^{67} -12.2819 q^{71} -5.95132 q^{73} -2.28335i q^{77} +15.0027 q^{79} +14.7419i q^{83} +10.4490i q^{85} -11.8923 q^{89} -7.10447i q^{91} -2.46425 q^{95} -2.60256 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q - 24 q^{7}+O(q^{10}) $ 24 * q - 24 * q^7	$ 24 q - 24 q^{7} - 24 q^{25} + 16 q^{31} + 24 q^{49} + 8 q^{55} - 32 q^{79}+O(q^{100}) $ 24 * q - 24 * q^7 - 24 * q^25 + 16 * q^31 + 24 * q^49 + 8 * q^55 - 32 * 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
$5$	1.53368i	0.685881i	0.939357	+	0.342940i	$0.111423\pi$
$5$	1.53368i	0.685881i	−0.939357	+	0.342940i	$0.888577\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	0	0
$10$	0	0
$11$	2.28335i	0.688455i	0.938886	+	0.344228i	$0.111859\pi$
$11$	2.28335i	0.688455i	−0.938886	+	0.344228i	$0.888141\pi$
$12$	0	0
$13$	7.10447i	1.97043i	0.171333	+	0.985213i	$0.445193\pi$
$13$	7.10447i	1.97043i	−0.171333	+	0.985213i	$0.554807\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	6.81307	1.65241	0.826206	−	0.563368i	$-0.190495\pi$
$17$	6.81307	1.65241	0.826206	+	0.563368i	$0.190495\pi$
$18$	0	0
$19$	1.60676i	0.368616i	0.982869	+	0.184308i	$0.0590044\pi$
$19$	1.60676i	0.368616i	−0.982869	+	0.184308i	$0.940996\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−1.16757	−0.243455	−0.121727	−	0.992564i	$-0.538843\pi$
$23$	−1.16757	−0.243455	−0.121727	+	0.992564i	$0.538843\pi$
$24$	0	0
$25$	2.64784	0.529568
$26$	0	0
$27$	0	0
$28$	0	0
$29$	4.07059i	0.755889i	0.925828	+	0.377944i	$0.123369\pi$
$29$	4.07059i	0.755889i	−0.925828	+	0.377944i	$0.876631\pi$
$30$	0	0
$31$	7.90568	1.41990	0.709951	−	0.704251i	$-0.248717\pi$
$31$	7.90568	1.41990	0.709951	+	0.704251i	$0.248717\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	− 1.53368i	− 0.259238i
$36$	0	0
$37$	− 7.04836i	− 1.15874i	−0.815063	−	0.579372i	$-0.803298\pi$
$37$	− 7.04836i	− 1.15874i	0.815063	−	0.579372i	$-0.196702\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	10.1710	1.58844	0.794218	−	0.607633i	$-0.207881\pi$
$41$	10.1710	1.58844	0.794218	+	0.607633i	$0.207881\pi$
$42$	0	0
$43$	− 0.344719i	− 0.0525691i	−0.999655	−	0.0262846i	$-0.991632\pi$
$43$	− 0.344719i	− 0.0525691i	0.999655	−	0.0262846i	$-0.00836760\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−3.10859	−0.453434	−0.226717	−	0.973961i	$-0.572799\pi$
$47$	−3.10859	−0.453434	−0.226717	+	0.973961i	$0.572799\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	0	0
$52$	0	0
$53$	− 5.66835i	− 0.778608i	−0.921109	−	0.389304i	$-0.872716\pi$
$53$	− 5.66835i	− 0.778608i	0.921109	−	0.389304i	$-0.127284\pi$
$54$	0	0
$55$	−3.50191	−0.472198
$56$	0	0
$57$	0	0
$58$	0	0
$59$	1.38165i	0.179875i	0.995947	+	0.0899374i	$0.0286667\pi$
$59$	1.38165i	0.179875i	−0.995947	+	0.0899374i	$0.971333\pi$
$60$	0	0
$61$	5.50516i	0.704863i	0.935838	+	0.352431i	$0.114645\pi$
$61$	5.50516i	0.704863i	−0.935838	+	0.352431i	$0.885355\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−10.8960	−1.35148
$66$	0	0
$67$	− 8.35907i	− 1.02122i	−0.859811	−	0.510612i	$-0.829419\pi$
$67$	− 8.35907i	− 1.02122i	0.859811	−	0.510612i	$-0.170581\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−12.2819	−1.45759	−0.728795	−	0.684732i	$-0.759919\pi$
$71$	−12.2819	−1.45759	−0.728795	+	0.684732i	$0.759919\pi$
$72$	0	0
$73$	−5.95132	−0.696549	−0.348275	−	0.937393i	$-0.613232\pi$
$73$	−5.95132	−0.696549	−0.348275	+	0.937393i	$0.613232\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 2.28335i	− 0.260212i
$78$	0	0
$79$	15.0027	1.68794	0.843968	−	0.536393i	$-0.180213\pi$
$79$	15.0027	1.68794	0.843968	+	0.536393i	$0.180213\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	14.7419i	1.61814i	0.587715	+	0.809068i	$0.300027\pi$
$83$	14.7419i	1.61814i	−0.587715	+	0.809068i	$0.699973\pi$
$84$	0	0
$85$	10.4490i	1.13336i
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−11.8923	−1.26058	−0.630290	−	0.776360i	$-0.717064\pi$
$89$	−11.8923	−1.26058	−0.630290	+	0.776360i	$0.717064\pi$
$90$	0	0
$91$	− 7.10447i	− 0.744751i
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−2.46425	−0.252827
$96$	0	0
$97$	−2.60256	−0.264250	−0.132125	−	0.991233i	$-0.542180\pi$
$97$	−2.60256	−0.264250	−0.132125	+	0.991233i	$0.542180\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	9.58182i	0.953427i	0.879059	+	0.476713i	$0.158172\pi$
$101$	9.58182i	0.953427i	−0.879059	+	0.476713i	$0.841828\pi$
$102$	0	0
$103$	−9.44175	−0.930323	−0.465162	−	0.885226i	$-0.654004\pi$
$103$	−9.44175	−0.930323	−0.465162	+	0.885226i	$0.654004\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 5.04240i	− 0.487467i	−0.969842	−	0.243733i	$-0.921628\pi$
$107$	− 5.04240i	− 0.487467i	0.969842	−	0.243733i	$-0.0783722\pi$
$108$	0	0
$109$	− 2.35923i	− 0.225973i	−0.993597	−	0.112987i	$-0.963958\pi$
$109$	− 2.35923i	− 0.225973i	0.993597	−	0.112987i	$-0.0360417\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−14.0317	−1.31999	−0.659994	−	0.751271i	$-0.729441\pi$
$113$	−14.0317	−1.31999	−0.659994	+	0.751271i	$0.729441\pi$
$114$	0	0
$115$	− 1.79067i	− 0.166981i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−6.81307	−0.624553
$120$	0	0
$121$	5.78632	0.526029
$122$	0	0
$123$	0	0
$124$	0	0
$125$	11.7293i	1.04910i
$126$	0	0
$127$	17.0218	1.51044	0.755220	−	0.655472i	$-0.227530\pi$
$127$	17.0218	1.51044	0.755220	+	0.655472i	$0.227530\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	17.8861i	1.56272i	0.624082	+	0.781359i	$0.285473\pi$
$131$	17.8861i	1.56272i	−0.624082	+	0.781359i	$0.714527\pi$
$132$	0	0
$133$	− 1.60676i	− 0.139324i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	9.18009	0.784308	0.392154	−	0.919900i	$-0.371730\pi$
$137$	9.18009	0.784308	0.392154	+	0.919900i	$0.371730\pi$
$138$	0	0
$139$	− 12.0518i	− 1.02222i	−0.859516	−	0.511108i	$-0.829235\pi$
$139$	− 12.0518i	− 1.02222i	0.859516	−	0.511108i	$-0.170765\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−16.2220	−1.35655
$144$	0	0
$145$	−6.24296	−0.518449
$146$	0	0
$147$	0	0
$148$	0	0
$149$	7.71091i	0.631702i	0.948809	+	0.315851i	$0.102290\pi$
$149$	7.71091i	0.631702i	−0.948809	+	0.315851i	$0.897710\pi$
$150$	0	0
$151$	−5.91312	−0.481203	−0.240601	−	0.970624i	$-0.577345\pi$
$151$	−5.91312	−0.481203	−0.240601	+	0.970624i	$0.577345\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	12.1247i	0.973883i
$156$	0	0
$157$	0.906198i	0.0723225i	0.999346	+	0.0361612i	$0.0115130\pi$
$157$	0.906198i	0.0723225i	−0.999346	+	0.0361612i	$0.988487\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	1.16757	0.0920172
$162$	0	0
$163$	4.46083i	0.349400i	0.984622	+	0.174700i	$0.0558955\pi$
$163$	4.46083i	0.349400i	−0.984622	+	0.174700i	$0.944105\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	22.6885	1.75569	0.877845	−	0.478944i	$-0.158980\pi$
$167$	22.6885	1.75569	0.877845	+	0.478944i	$0.158980\pi$
$168$	0	0
$169$	−37.4735	−2.88258
$170$	0	0
$171$	0	0
$172$	0	0
$173$	− 24.2316i	− 1.84229i	−0.389218	−	0.921145i	$-0.627255\pi$
$173$	− 24.2316i	− 1.84229i	0.389218	−	0.921145i	$-0.372745\pi$
$174$	0	0
$175$	−2.64784	−0.200158
$176$	0	0
$177$	0	0
$178$	0	0
$179$	8.82249i	0.659424i	0.944082	+	0.329712i	$0.106952\pi$
$179$	8.82249i	0.659424i	−0.944082	+	0.329712i	$0.893048\pi$
$180$	0	0
$181$	4.29583i	0.319306i	0.987173	+	0.159653i	$0.0510376\pi$
$181$	4.29583i	0.319306i	−0.987173	+	0.159653i	$0.948962\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	10.8099	0.794760
$186$	0	0
$187$	15.5566i	1.13761i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	0.224849	0.0162695	0.00813476	−	0.999967i	$-0.497411\pi$
$191$	0.224849	0.0162695	0.00813476	+	0.999967i	$0.497411\pi$
$192$	0	0
$193$	−11.7447	−0.845404	−0.422702	−	0.906269i	$-0.638918\pi$
$193$	−11.7447	−0.845404	−0.422702	+	0.906269i	$0.638918\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	14.7998i	1.05444i	0.849729	+	0.527220i	$0.176766\pi$
$197$	14.7998i	1.05444i	−0.849729	+	0.527220i	$0.823234\pi$
$198$	0	0
$199$	19.6242	1.39112	0.695561	−	0.718467i	$-0.255156\pi$
$199$	19.6242	1.39112	0.695561	+	0.718467i	$0.255156\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	− 4.07059i	− 0.285699i
$204$	0	0
$205$	15.5989i	1.08948i
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−3.66879	−0.253776
$210$	0	0
$211$	− 0.913482i	− 0.0628867i	−0.999506	−	0.0314434i	$-0.989990\pi$
$211$	− 0.913482i	− 0.0628867i	0.999506	−	0.0314434i	$-0.0100104\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	0.528687	0.0360561
$216$	0	0
$217$	−7.90568	−0.536673
$218$	0	0
$219$	0	0
$220$	0	0
$221$	48.4033i	3.25596i
$222$	0	0
$223$	11.8646	0.794512	0.397256	−	0.917708i	$-0.369963\pi$
$223$	11.8646	0.794512	0.397256	+	0.917708i	$0.369963\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	19.2403i	1.27702i	0.769613	+	0.638510i	$0.220449\pi$
$227$	19.2403i	1.27702i	−0.769613	+	0.638510i	$0.779551\pi$
$228$	0	0
$229$	7.11176i	0.469958i	0.972000	+	0.234979i	$0.0755022\pi$
$229$	7.11176i	0.469958i	−0.972000	+	0.234979i	$0.924498\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−18.5527	−1.21542	−0.607712	−	0.794157i	$-0.707913\pi$
$233$	−18.5527	−1.21542	−0.607712	+	0.794157i	$0.707913\pi$
$234$	0	0
$235$	− 4.76757i	− 0.311002i
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−3.86610	−0.250077	−0.125039	−	0.992152i	$-0.539905\pi$
$239$	−3.86610	−0.250077	−0.125039	+	0.992152i	$0.539905\pi$
$240$	0	0
$241$	−11.5687	−0.745205	−0.372603	−	0.927991i	$-0.621535\pi$
$241$	−11.5687	−0.745205	−0.372603	+	0.927991i	$0.621535\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	1.53368i	0.0979829i
$246$	0	0
$247$	−11.4152	−0.726331
$248$	0	0
$249$	0	0
$250$	0	0
$251$	− 13.5653i	− 0.856235i	−0.903723	−	0.428117i	$-0.859177\pi$
$251$	− 13.5653i	− 0.856235i	0.903723	−	0.428117i	$-0.140823\pi$
$252$	0	0
$253$	− 2.66596i	− 0.167608i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	11.1567	0.695938	0.347969	−	0.937506i	$-0.386871\pi$
$257$	11.1567	0.695938	0.347969	+	0.937506i	$0.386871\pi$
$258$	0	0
$259$	7.04836i	0.437964i
$260$	0	0
$261$	0	0
$262$	0	0
$263$	13.9609	0.860864	0.430432	−	0.902623i	$-0.358361\pi$
$263$	13.9609	0.860864	0.430432	+	0.902623i	$0.358361\pi$
$264$	0	0
$265$	8.69341	0.534032
$266$	0	0
$267$	0	0
$268$	0	0
$269$	16.1185i	0.982761i	0.870945	+	0.491380i	$0.163507\pi$
$269$	16.1185i	0.982761i	−0.870945	+	0.491380i	$0.836493\pi$
$270$	0	0
$271$	−8.91716	−0.541679	−0.270840	−	0.962624i	$-0.587301\pi$
$271$	−8.91716	−0.541679	−0.270840	+	0.962624i	$0.587301\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	6.04594i	0.364584i
$276$	0	0
$277$	− 26.8189i	− 1.61139i	−0.592328	−	0.805697i	$-0.701791\pi$
$277$	− 26.8189i	− 1.61139i	0.592328	−	0.805697i	$-0.298209\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−11.0688	−0.660306	−0.330153	−	0.943927i	$-0.607100\pi$
$281$	−11.0688	−0.660306	−0.330153	+	0.943927i	$0.607100\pi$
$282$	0	0
$283$	− 21.7405i	− 1.29234i	−0.763194	−	0.646170i	$-0.776370\pi$
$283$	− 21.7405i	− 1.29234i	0.763194	−	0.646170i	$-0.223630\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−10.1710	−0.600373
$288$	0	0
$289$	29.4179	1.73046
$290$	0	0
$291$	0	0
$292$	0	0
$293$	8.80042i	0.514126i	0.966395	+	0.257063i	$0.0827548\pi$
$293$	8.80042i	0.514126i	−0.966395	+	0.257063i	$0.917245\pi$
$294$	0	0
$295$	−2.11900	−0.123373
$296$	0	0
$297$	0	0
$298$	0	0
$299$	− 8.29496i	− 0.479710i
$300$	0	0
$301$	0.344719i	0.0198693i
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−8.44312	−0.483452
$306$	0	0
$307$	− 3.11170i	− 0.177594i	−0.996050	−	0.0887971i	$-0.971698\pi$
$307$	− 3.11170i	− 0.177594i	0.996050	−	0.0887971i	$-0.0283023\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−0.443815	−0.0251664	−0.0125832	−	0.999921i	$-0.504005\pi$
$311$	−0.443815	−0.0251664	−0.0125832	+	0.999921i	$0.504005\pi$
$312$	0	0
$313$	19.1601	1.08299	0.541497	−	0.840703i	$-0.317858\pi$
$313$	19.1601	1.08299	0.541497	+	0.840703i	$0.317858\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	11.2142i	0.629854i	0.949116	+	0.314927i	$0.101980\pi$
$317$	11.2142i	0.629854i	−0.949116	+	0.314927i	$0.898020\pi$
$318$	0	0
$319$	−9.29456	−0.520396
$320$	0	0
$321$	0	0
$322$	0	0
$323$	10.9470i	0.609105i
$324$	0	0
$325$	18.8115i	1.04347i
$326$	0	0
$327$	0	0
$328$	0	0
$329$	3.10859	0.171382
$330$	0	0
$331$	− 17.8007i	− 0.978414i	−0.872168	−	0.489207i	$-0.837286\pi$
$331$	− 17.8007i	− 0.978414i	0.872168	−	0.489207i	$-0.162714\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	12.8201	0.700437
$336$	0	0
$337$	−3.46393	−0.188692	−0.0943461	−	0.995539i	$-0.530076\pi$
$337$	−3.46393	−0.188692	−0.0943461	+	0.995539i	$0.530076\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	18.0514i	0.977539i
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	0	0
$346$	0	0
$347$	27.3962i	1.47070i	0.677686	+	0.735352i	$0.262983\pi$
$347$	27.3962i	1.47070i	−0.677686	+	0.735352i	$0.737017\pi$
$348$	0	0
$349$	− 1.62149i	− 0.0867961i	−0.999058	−	0.0433981i	$-0.986182\pi$
$349$	− 1.62149i	− 0.0867961i	0.999058	−	0.0433981i	$-0.0138184\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−16.5882	−0.882903	−0.441452	−	0.897285i	$-0.645536\pi$
$353$	−16.5882	−0.882903	−0.441452	+	0.897285i	$0.645536\pi$
$354$	0	0
$355$	− 18.8364i	− 0.999732i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	7.84388	0.413984	0.206992	−	0.978343i	$-0.433633\pi$
$359$	7.84388	0.413984	0.206992	+	0.978343i	$0.433633\pi$
$360$	0	0
$361$	16.4183	0.864122
$362$	0	0
$363$	0	0
$364$	0	0
$365$	− 9.12740i	− 0.477750i
$366$	0	0
$367$	−0.761448	−0.0397472	−0.0198736	−	0.999803i	$-0.506326\pi$
$367$	−0.761448	−0.0397472	−0.0198736	+	0.999803i	$0.506326\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	5.66835i	0.294286i
$372$	0	0
$373$	13.3240i	0.689888i	0.938623	+	0.344944i	$0.112102\pi$
$373$	13.3240i	0.689888i	−0.938623	+	0.344944i	$0.887898\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−28.9194	−1.48942
$378$	0	0
$379$	− 13.6595i	− 0.701640i	−0.936443	−	0.350820i	$-0.885903\pi$
$379$	− 13.6595i	− 0.701640i	0.936443	−	0.350820i	$-0.114097\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−9.70332	−0.495816	−0.247908	−	0.968784i	$-0.579743\pi$
$383$	−9.70332	−0.495816	−0.247908	+	0.968784i	$0.579743\pi$
$384$	0	0
$385$	3.50191	0.178474
$386$	0	0
$387$	0	0
$388$	0	0
$389$	− 24.7223i	− 1.25347i	−0.779233	−	0.626734i	$-0.784391\pi$
$389$	− 24.7223i	− 1.25347i	0.779233	−	0.626734i	$-0.215609\pi$
$390$	0	0
$391$	−7.95472	−0.402287
$392$	0	0
$393$	0	0
$394$	0	0
$395$	23.0093i	1.15772i
$396$	0	0
$397$	− 29.1070i	− 1.46084i	−0.682999	−	0.730420i	$-0.739325\pi$
$397$	− 29.1070i	− 1.46084i	0.682999	−	0.730420i	$-0.260675\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−13.7247	−0.685380	−0.342690	−	0.939449i	$-0.611338\pi$
$401$	−13.7247	−0.685380	−0.342690	+	0.939449i	$0.611338\pi$
$402$	0	0
$403$	56.1657i	2.79781i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	16.0939	0.797743
$408$	0	0
$409$	18.1707	0.898486	0.449243	−	0.893410i	$-0.351694\pi$
$409$	18.1707	0.898486	0.449243	+	0.893410i	$0.351694\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	− 1.38165i	− 0.0679863i
$414$	0	0
$415$	−22.6093	−1.10985
$416$	0	0
$417$	0	0
$418$	0	0
$419$	− 32.9489i	− 1.60966i	−0.593507	−	0.804829i	$-0.702257\pi$
$419$	− 32.9489i	− 1.60966i	0.593507	−	0.804829i	$-0.297743\pi$
$420$	0	0
$421$	− 27.5222i	− 1.34135i	−0.741751	−	0.670675i	$-0.766004\pi$
$421$	− 27.5222i	− 1.34135i	0.741751	−	0.670675i	$-0.233996\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	18.0399	0.875064
$426$	0	0
$427$	− 5.50516i	− 0.266413i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−40.3524	−1.94371	−0.971855	−	0.235582i	$-0.924300\pi$
$431$	−40.3524	−1.94371	−0.971855	+	0.235582i	$0.924300\pi$
$432$	0	0
$433$	−38.5928	−1.85465	−0.927327	−	0.374253i	$-0.877899\pi$
$433$	−38.5928	−1.85465	−0.927327	+	0.374253i	$0.877899\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 1.87600i	− 0.0897413i
$438$	0	0
$439$	1.68908	0.0806152	0.0403076	−	0.999187i	$-0.487166\pi$
$439$	1.68908	0.0806152	0.0403076	+	0.999187i	$0.487166\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	− 18.3190i	− 0.870361i	−0.900343	−	0.435180i	$-0.856685\pi$
$443$	− 18.3190i	− 0.870361i	0.900343	−	0.435180i	$-0.143315\pi$
$444$	0	0
$445$	− 18.2389i	− 0.864607i
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−34.2644	−1.61704	−0.808518	−	0.588471i	$-0.799730\pi$
$449$	−34.2644	−1.61704	−0.808518	+	0.588471i	$0.799730\pi$
$450$	0	0
$451$	23.2238i	1.09357i
$452$	0	0
$453$	0	0
$454$	0	0
$455$	10.8960	0.510810
$456$	0	0
$457$	3.08541	0.144330	0.0721648	−	0.997393i	$-0.477009\pi$
$457$	3.08541	0.144330	0.0721648	+	0.997393i	$0.477009\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	− 8.62369i	− 0.401645i	−0.979628	−	0.200823i	$-0.935639\pi$
$461$	− 8.62369i	− 0.401645i	0.979628	−	0.200823i	$-0.0643615\pi$
$462$	0	0
$463$	−3.29178	−0.152982	−0.0764910	−	0.997070i	$-0.524372\pi$
$463$	−3.29178	−0.152982	−0.0764910	+	0.997070i	$0.524372\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 31.3473i	− 1.45058i	−0.688443	−	0.725290i	$-0.741706\pi$
$467$	− 31.3473i	− 1.45058i	0.688443	−	0.725290i	$-0.258294\pi$
$468$	0	0
$469$	8.35907i	0.385986i
$470$	0	0
$471$	0	0
$472$	0	0
$473$	0.787113	0.0361915
$474$	0	0
$475$	4.25444i	0.195207i
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−23.7139	−1.08351	−0.541757	−	0.840535i	$-0.682241\pi$
$479$	−23.7139	−1.08351	−0.541757	+	0.840535i	$0.682241\pi$
$480$	0	0
$481$	50.0749	2.28322
$482$	0	0
$483$	0	0
$484$	0	0
$485$	− 3.99148i	− 0.181244i
$486$	0	0
$487$	−35.7317	−1.61916	−0.809580	−	0.587010i	$-0.800305\pi$
$487$	−35.7317	−1.61916	−0.809580	+	0.587010i	$0.800305\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	40.3543i	1.82116i	0.413330	+	0.910581i	$0.364366\pi$
$491$	40.3543i	1.82116i	−0.413330	+	0.910581i	$0.635634\pi$
$492$	0	0
$493$	27.7332i	1.24904i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	12.2819	0.550917
$498$	0	0
$499$	41.0646i	1.83831i	0.393902	+	0.919153i	$0.371125\pi$
$499$	41.0646i	1.83831i	−0.393902	+	0.919153i	$0.628875\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−13.5744	−0.605252	−0.302626	−	0.953109i	$-0.597863\pi$
$503$	−13.5744	−0.605252	−0.302626	+	0.953109i	$0.597863\pi$
$504$	0	0
$505$	−14.6954	−0.653937
$506$	0	0
$507$	0	0
$508$	0	0
$509$	− 13.3591i	− 0.592132i	−0.955167	−	0.296066i	$-0.904325\pi$
$509$	− 13.3591i	− 0.592132i	0.955167	−	0.296066i	$-0.0956749\pi$
$510$	0	0
$511$	5.95132	0.263271
$512$	0	0
$513$	0	0
$514$	0	0
$515$	− 14.4806i	− 0.638091i
$516$	0	0
$517$	− 7.09799i	− 0.312169i
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−20.5338	−0.899600	−0.449800	−	0.893129i	$-0.648505\pi$
$521$	−20.5338	−0.899600	−0.449800	+	0.893129i	$0.648505\pi$
$522$	0	0
$523$	4.37298i	0.191217i	0.995419	+	0.0956086i	$0.0304797\pi$
$523$	4.37298i	0.191217i	−0.995419	+	0.0956086i	$0.969520\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	53.8619	2.34626
$528$	0	0
$529$	−21.6368	−0.940730
$530$	0	0
$531$	0	0
$532$	0	0
$533$	72.2593i	3.12990i
$534$	0	0
$535$	7.73340	0.334344
$536$	0	0
$537$	0	0
$538$	0	0
$539$	2.28335i	0.0983507i
$540$	0	0
$541$	− 11.8762i	− 0.510598i	−0.966862	−	0.255299i	$-0.917826\pi$
$541$	− 11.8762i	− 0.510598i	0.966862	−	0.255299i	$-0.0821740\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	3.61829	0.154991
$546$	0	0
$547$	− 5.50044i	− 0.235182i	−0.993062	−	0.117591i	$-0.962483\pi$
$547$	− 5.50044i	− 0.235182i	0.993062	−	0.117591i	$-0.0375171\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−6.54045	−0.278633
$552$	0	0
$553$	−15.0027	−0.637980
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 8.19596i	− 0.347274i	−0.984810	−	0.173637i	$-0.944448\pi$
$557$	− 8.19596i	− 0.347274i	0.984810	−	0.173637i	$-0.0555519\pi$
$558$	0	0
$559$	2.44905	0.103584
$560$	0	0
$561$	0	0
$562$	0	0
$563$	0.145215i	0.00612008i	0.999995	+	0.00306004i	$0.000974043\pi$
$563$	0.145215i	0.00612008i	−0.999995	+	0.00306004i	$0.999026\pi$
$564$	0	0
$565$	− 21.5200i	− 0.905354i
$566$	0	0
$567$	0	0
$568$	0	0
$569$	5.09548	0.213613	0.106807	−	0.994280i	$-0.465937\pi$
$569$	5.09548	0.213613	0.106807	+	0.994280i	$0.465937\pi$
$570$	0	0
$571$	25.3061i	1.05903i	0.848301	+	0.529514i	$0.177626\pi$
$571$	25.3061i	1.05903i	−0.848301	+	0.529514i	$0.822374\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−3.09153	−0.128926
$576$	0	0
$577$	19.5004	0.811814	0.405907	−	0.913914i	$-0.366956\pi$
$577$	19.5004	0.811814	0.405907	+	0.913914i	$0.366956\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	− 14.7419i	− 0.611598i
$582$	0	0
$583$	12.9428	0.536037
$584$	0	0
$585$	0	0
$586$	0	0
$587$	36.0783i	1.48911i	0.667561	+	0.744555i	$0.267338\pi$
$587$	36.0783i	1.48911i	−0.667561	+	0.744555i	$0.732662\pi$
$588$	0	0
$589$	12.7025i	0.523399i
$590$	0	0
$591$	0	0
$592$	0	0
$593$	16.7669	0.688535	0.344268	−	0.938872i	$-0.388127\pi$
$593$	16.7669	0.688535	0.344268	+	0.938872i	$0.388127\pi$
$594$	0	0
$595$	− 10.4490i	− 0.428369i
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−31.1008	−1.27074	−0.635372	−	0.772206i	$-0.719153\pi$
$599$	−31.1008	−1.27074	−0.635372	+	0.772206i	$0.719153\pi$
$600$	0	0
$601$	17.1858	0.701022	0.350511	−	0.936559i	$-0.386008\pi$
$601$	17.1858	0.701022	0.350511	+	0.936559i	$0.386008\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	8.87434i	0.360793i
$606$	0	0
$607$	−12.9692	−0.526404	−0.263202	−	0.964741i	$-0.584779\pi$
$607$	−12.9692	−0.526404	−0.263202	+	0.964741i	$0.584779\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	− 22.0849i	− 0.893459i
$612$	0	0
$613$	26.5265i	1.07139i	0.844410	+	0.535697i	$0.179951\pi$
$613$	26.5265i	1.07139i	−0.844410	+	0.535697i	$0.820049\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	37.3263	1.50270	0.751350	−	0.659904i	$-0.229403\pi$
$617$	37.3263	1.50270	0.751350	+	0.659904i	$0.229403\pi$
$618$	0	0
$619$	− 10.8774i	− 0.437200i	−0.975815	−	0.218600i	$-0.929851\pi$
$619$	− 10.8774i	− 0.437200i	0.975815	−	0.218600i	$-0.0701489\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	11.8923	0.476454
$624$	0	0
$625$	−4.74975	−0.189990
$626$	0	0
$627$	0	0
$628$	0	0
$629$	− 48.0210i	− 1.91472i
$630$	0	0
$631$	24.2943	0.967140	0.483570	−	0.875306i	$-0.339340\pi$
$631$	24.2943	0.967140	0.483570	+	0.875306i	$0.339340\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	26.1059i	1.03598i
$636$	0	0
$637$	7.10447i	0.281489i
$638$	0	0
$639$	0	0
$640$	0	0
$641$	16.5765	0.654731	0.327365	−	0.944898i	$-0.393839\pi$
$641$	16.5765	0.654731	0.327365	+	0.944898i	$0.393839\pi$
$642$	0	0
$643$	− 7.15319i	− 0.282094i	−0.990003	−	0.141047i	$-0.954953\pi$
$643$	− 7.15319i	− 0.282094i	0.990003	−	0.141047i	$-0.0450469\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−7.94968	−0.312534	−0.156267	−	0.987715i	$-0.549946\pi$
$647$	−7.94968	−0.312534	−0.156267	+	0.987715i	$0.549946\pi$
$648$	0	0
$649$	−3.15478	−0.123836
$650$	0	0
$651$	0	0
$652$	0	0
$653$	7.45217i	0.291626i	0.989312	+	0.145813i	$0.0465798\pi$
$653$	7.45217i	0.291626i	−0.989312	+	0.145813i	$0.953420\pi$
$654$	0	0
$655$	−27.4315	−1.07184
$656$	0	0
$657$	0	0
$658$	0	0
$659$	26.2741i	1.02349i	0.859136	+	0.511747i	$0.171001\pi$
$659$	26.2741i	1.02349i	−0.859136	+	0.511747i	$0.828999\pi$
$660$	0	0
$661$	− 3.98821i	− 0.155123i	−0.996988	−	0.0775617i	$-0.975287\pi$
$661$	− 3.98821i	− 0.155123i	0.996988	−	0.0775617i	$-0.0247135\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	2.46425	0.0955594
$666$	0	0
$667$	− 4.75268i	− 0.184025i
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−12.5702	−0.485267
$672$	0	0
$673$	30.5709	1.17842	0.589211	−	0.807979i	$-0.299439\pi$
$673$	30.5709	1.17842	0.589211	+	0.807979i	$0.299439\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	21.9032i	0.841808i	0.907105	+	0.420904i	$0.138287\pi$
$677$	21.9032i	0.841808i	−0.907105	+	0.420904i	$0.861713\pi$
$678$	0	0
$679$	2.60256	0.0998770
$680$	0	0
$681$	0	0
$682$	0	0
$683$	− 13.0423i	− 0.499051i	−0.968368	−	0.249525i	$-0.919725\pi$
$683$	− 13.0423i	− 0.499051i	0.968368	−	0.249525i	$-0.0802746\pi$
$684$	0	0
$685$	14.0793i	0.537942i
$686$	0	0
$687$	0	0
$688$	0	0
$689$	40.2707	1.53419
$690$	0	0
$691$	− 3.25399i	− 0.123788i	−0.998083	−	0.0618938i	$-0.980286\pi$
$691$	− 3.25399i	− 0.123788i	0.998083	−	0.0618938i	$-0.0197140\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	18.4835	0.701118
$696$	0	0
$697$	69.2954	2.62475
$698$	0	0
$699$	0	0
$700$	0	0
$701$	2.70404i	0.102130i	0.998695	+	0.0510650i	$0.0162616\pi$
$701$	2.70404i	0.102130i	−0.998695	+	0.0510650i	$0.983738\pi$
$702$	0	0
$703$	11.3250	0.427132
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 9.58182i	− 0.360361i
$708$	0	0
$709$	− 49.0447i	− 1.84191i	−0.389668	−	0.920955i	$-0.627410\pi$
$709$	− 49.0447i	− 1.84191i	0.389668	−	0.920955i	$-0.372590\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−9.23042	−0.345682
$714$	0	0
$715$	− 24.8793i	− 0.930431i
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−21.3291	−0.795442	−0.397721	−	0.917506i	$-0.630199\pi$
$719$	−21.3291	−0.795442	−0.397721	+	0.917506i	$0.630199\pi$
$720$	0	0
$721$	9.44175	0.351629
$722$	0	0
$723$	0	0
$724$	0	0
$725$	10.7783i	0.400294i
$726$	0	0
$727$	23.5713	0.874210	0.437105	−	0.899411i	$-0.356004\pi$
$727$	23.5713	0.874210	0.437105	+	0.899411i	$0.356004\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	− 2.34859i	− 0.0868658i
$732$	0	0
$733$	− 2.28803i	− 0.0845105i	−0.999107	−	0.0422552i	$-0.986546\pi$
$733$	− 2.28803i	− 0.0845105i	0.999107	−	0.0422552i	$-0.0134543\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	19.0867	0.703067
$738$	0	0
$739$	33.6953i	1.23950i	0.784798	+	0.619751i	$0.212766\pi$
$739$	33.6953i	1.23950i	−0.784798	+	0.619751i	$0.787234\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	35.7898	1.31300	0.656501	−	0.754326i	$-0.272036\pi$
$743$	35.7898	1.31300	0.656501	+	0.754326i	$0.272036\pi$
$744$	0	0
$745$	−11.8260	−0.433272
$746$	0	0
$747$	0	0
$748$	0	0
$749$	5.04240i	0.184245i
$750$	0	0
$751$	−7.35858	−0.268518	−0.134259	−	0.990946i	$-0.542865\pi$
$751$	−7.35858	−0.268518	−0.134259	+	0.990946i	$0.542865\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	− 9.06881i	− 0.330048i
$756$	0	0
$757$	23.2086i	0.843530i	0.906705	+	0.421765i	$0.138589\pi$
$757$	23.2086i	0.843530i	−0.906705	+	0.421765i	$0.861411\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	22.1491	0.802902	0.401451	−	0.915880i	$-0.368506\pi$
$761$	22.1491	0.802902	0.401451	+	0.915880i	$0.368506\pi$
$762$	0	0
$763$	2.35923i	0.0854099i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−9.81586	−0.354430
$768$	0	0
$769$	−27.4818	−0.991020	−0.495510	−	0.868602i	$-0.665019\pi$
$769$	−27.4818	−0.991020	−0.495510	+	0.868602i	$0.665019\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	− 39.9693i	− 1.43760i	−0.695219	−	0.718798i	$-0.744692\pi$
$773$	− 39.9693i	− 1.43760i	0.695219	−	0.718798i	$-0.255308\pi$
$774$	0	0
$775$	20.9330	0.751935
$776$	0	0
$777$	0	0
$778$	0	0
$779$	16.3423i	0.585523i
$780$	0	0
$781$	− 28.0438i	− 1.00348i
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−1.38981	−0.0496046
$786$	0	0
$787$	34.5651i	1.23211i	0.787702	+	0.616057i	$0.211271\pi$
$787$	34.5651i	1.23211i	−0.787702	+	0.616057i	$0.788729\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	14.0317	0.498908
$792$	0	0
$793$	−39.1112	−1.38888
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 16.9968i	− 0.602059i	−0.953615	−	0.301029i	$-0.902670\pi$
$797$	− 16.9968i	− 0.602059i	0.953615	−	0.301029i	$-0.0973302\pi$
$798$	0	0
$799$	−21.1790	−0.749260
$800$	0	0
$801$	0	0
$802$	0	0
$803$	− 13.5889i	− 0.479543i
$804$	0	0
$805$	1.79067i	0.0631128i
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−2.33018	−0.0819249	−0.0409624	−	0.999161i	$-0.513042\pi$
$809$	−2.33018	−0.0819249	−0.0409624	+	0.999161i	$0.513042\pi$
$810$	0	0
$811$	− 8.79524i	− 0.308843i	−0.988005	−	0.154421i	$-0.950649\pi$
$811$	− 8.79524i	− 0.308843i	0.988005	−	0.154421i	$-0.0493513\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−6.84147	−0.239646
$816$	0	0
$817$	0.553880	0.0193778
$818$	0	0
$819$	0	0
$820$	0	0
$821$	− 14.1689i	− 0.494497i	−0.968952	−	0.247249i	$-0.920474\pi$
$821$	− 14.1689i	− 0.494497i	0.968952	−	0.247249i	$-0.0795265\pi$
$822$	0	0
$823$	−5.53482	−0.192932	−0.0964659	−	0.995336i	$-0.530754\pi$
$823$	−5.53482	−0.192932	−0.0964659	+	0.995336i	$0.530754\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	7.32030i	0.254552i	0.991867	+	0.127276i	$0.0406234\pi$
$827$	7.32030i	0.254552i	−0.991867	+	0.127276i	$0.959377\pi$
$828$	0	0
$829$	− 32.7046i	− 1.13588i	−0.823071	−	0.567939i	$-0.807741\pi$
$829$	− 32.7046i	− 1.13588i	0.823071	−	0.567939i	$-0.192259\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	6.81307	0.236059
$834$	0	0
$835$	34.7968i	1.20419i
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−19.2836	−0.665744	−0.332872	−	0.942972i	$-0.608018\pi$
$839$	−19.2836	−0.665744	−0.332872	+	0.942972i	$0.608018\pi$
$840$	0	0
$841$	12.4303	0.428632
$842$	0	0
$843$	0	0
$844$	0	0
$845$	− 57.4723i	− 1.97711i
$846$	0	0
$847$	−5.78632	−0.198820
$848$	0	0
$849$	0	0
$850$	0	0
$851$	8.22944i	0.282102i
$852$	0	0
$853$	− 24.5003i	− 0.838874i	−0.907784	−	0.419437i	$-0.862228\pi$
$853$	− 24.5003i	− 0.838874i	0.907784	−	0.419437i	$-0.137772\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−24.0113	−0.820211	−0.410105	−	0.912038i	$-0.634508\pi$
$857$	−24.0113	−0.820211	−0.410105	+	0.912038i	$0.634508\pi$
$858$	0	0
$859$	− 2.15328i	− 0.0734689i	−0.999325	−	0.0367344i	$-0.988304\pi$
$859$	− 2.15328i	− 0.0734689i	0.999325	−	0.0367344i	$-0.0116956\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	6.94727	0.236488	0.118244	−	0.992985i	$-0.462274\pi$
$863$	6.94727	0.236488	0.118244	+	0.992985i	$0.462274\pi$
$864$	0	0
$865$	37.1633	1.26359
$866$	0	0
$867$	0	0
$868$	0	0
$869$	34.2564i	1.16207i
$870$	0	0
$871$	59.3868	2.01225
$872$	0	0
$873$	0	0
$874$	0	0
$875$	− 11.7293i	− 0.396523i
$876$	0	0
$877$	24.2307i	0.818211i	0.912487	+	0.409106i	$0.134159\pi$
$877$	24.2307i	0.818211i	−0.912487	+	0.409106i	$0.865841\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−42.4462	−1.43005	−0.715024	−	0.699100i	$-0.753584\pi$
$881$	−42.4462	−1.43005	−0.715024	+	0.699100i	$0.753584\pi$
$882$	0	0
$883$	0.397509i	0.0133773i	0.999978	+	0.00668863i	$0.00212907\pi$
$883$	0.397509i	0.0133773i	−0.999978	+	0.00668863i	$0.997871\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	36.3698	1.22118	0.610590	−	0.791947i	$-0.290932\pi$
$887$	36.3698	1.22118	0.610590	+	0.791947i	$0.290932\pi$
$888$	0	0
$889$	−17.0218	−0.570892
$890$	0	0
$891$	0	0
$892$	0	0
$893$	− 4.99476i	− 0.167143i
$894$	0	0
$895$	−13.5308	−0.452286
$896$	0	0
$897$	0	0
$898$	0	0
$899$	32.1807i	1.07329i
$900$	0	0
$901$	− 38.6189i	− 1.28658i
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−6.58840	−0.219006
$906$	0	0
$907$	− 52.1212i	− 1.73066i	−0.501205	−	0.865329i	$-0.667110\pi$
$907$	− 52.1212i	− 1.73066i	0.501205	−	0.865329i	$-0.332890\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−25.8160	−0.855322	−0.427661	−	0.903939i	$-0.640662\pi$
$911$	−25.8160	−0.855322	−0.427661	+	0.903939i	$0.640662\pi$
$912$	0	0
$913$	−33.6609	−1.11401
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 17.8861i	− 0.590652i
$918$	0	0
$919$	4.71609	0.155569	0.0777847	−	0.996970i	$-0.475215\pi$
$919$	4.71609	0.155569	0.0777847	+	0.996970i	$0.475215\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	− 87.2562i	− 2.87207i
$924$	0	0
$925$	− 18.6629i	− 0.613634i
$926$	0	0
$927$	0	0
$928$	0	0
$929$	48.4774	1.59049	0.795246	−	0.606287i	$-0.207342\pi$
$929$	48.4774	1.59049	0.795246	+	0.606287i	$0.207342\pi$
$930$	0	0
$931$	1.60676i	0.0526594i
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−23.8588	−0.780266
$936$	0	0
$937$	11.5193	0.376319	0.188159	−	0.982138i	$-0.439748\pi$
$937$	11.5193	0.376319	0.188159	+	0.982138i	$0.439748\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	32.3390i	1.05422i	0.849796	+	0.527111i	$0.176725\pi$
$941$	32.3390i	1.05422i	−0.849796	+	0.527111i	$0.823275\pi$
$942$	0	0
$943$	−11.8753	−0.386712
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 37.4382i	− 1.21658i	−0.793715	−	0.608289i	$-0.791856\pi$
$947$	− 37.4382i	− 1.21658i	0.793715	−	0.608289i	$-0.208144\pi$
$948$	0	0
$949$	− 42.2810i	− 1.37250i
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−18.4974	−0.599190	−0.299595	−	0.954066i	$-0.596852\pi$
$953$	−18.4974	−0.599190	−0.299595	+	0.954066i	$0.596852\pi$
$954$	0	0
$955$	0.344846i	0.0111590i
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−9.18009	−0.296441
$960$	0	0
$961$	31.4998	1.01612
$962$	0	0
$963$	0	0
$964$	0	0
$965$	− 18.0126i	− 0.579846i
$966$	0	0
$967$	−44.4221	−1.42852	−0.714259	−	0.699882i	$-0.753236\pi$
$967$	−44.4221	−1.42852	−0.714259	+	0.699882i	$0.753236\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	− 37.5434i	− 1.20482i	−0.798185	−	0.602412i	$-0.794206\pi$
$971$	− 37.5434i	− 1.20482i	0.798185	−	0.602412i	$-0.205794\pi$
$972$	0	0
$973$	12.0518i	0.386361i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−6.89949	−0.220734	−0.110367	−	0.993891i	$-0.535203\pi$
$977$	−6.89949	−0.220734	−0.110367	+	0.993891i	$0.535203\pi$
$978$	0	0
$979$	− 27.1542i	− 0.867853i
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−28.9252	−0.922571	−0.461285	−	0.887252i	$-0.652612\pi$
$983$	−28.9252	−0.922571	−0.461285	+	0.887252i	$0.652612\pi$
$984$	0	0
$985$	−22.6980	−0.723220
$986$	0	0
$987$	0	0
$988$	0	0
$989$	0.402483i	0.0127982i
$990$	0	0
$991$	−37.7278	−1.19846	−0.599232	−	0.800575i	$-0.704527\pi$
$991$	−37.7278	−1.19846	−0.599232	+	0.800575i	$0.704527\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	30.0971i	0.954144i
$996$	0	0
$997$	53.0914i	1.68142i	0.541484	+	0.840711i	$0.317863\pi$
$997$	53.0914i	1.68142i	−0.541484	+	0.840711i	$0.682137\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6048.2.c.g.3025.16		24
3.2	odd	2	inner	6048.2.c.g.3025.10		24
4.3	odd	2		1512.2.c.f.757.21	yes	24
8.3	odd	2		1512.2.c.f.757.22	yes	24
8.5	even	2	inner	6048.2.c.g.3025.9		24
12.11	even	2		1512.2.c.f.757.4	yes	24
24.5	odd	2	inner	6048.2.c.g.3025.15		24
24.11	even	2		1512.2.c.f.757.3	✓	24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1512.2.c.f.757.3	✓	24	24.11	even	2
1512.2.c.f.757.4	yes	24	12.11	even	2
1512.2.c.f.757.21	yes	24	4.3	odd	2
1512.2.c.f.757.22	yes	24	8.3	odd	2
6048.2.c.g.3025.9		24	8.5	even	2	inner
6048.2.c.g.3025.10		24	3.2	odd	2	inner
6048.2.c.g.3025.15		24	24.5	odd	2	inner
6048.2.c.g.3025.16		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.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.53368i q^{5} -1.00000 q^{7} +O(q^{10})\)	\(q+1.53368i q^{5} -1.00000 q^{7} +2.28335i q^{11} +7.10447i q^{13} +6.81307 q^{17} +1.60676i q^{19} -1.16757 q^{23} +2.64784 q^{25} +4.07059i q^{29} +7.90568 q^{31} -1.53368i q^{35} -7.04836i q^{37} +10.1710 q^{41} -0.344719i q^{43} -3.10859 q^{47} +1.00000 q^{49} -5.66835i q^{53} -3.50191 q^{55} +1.38165i q^{59} +5.50516i q^{61} -10.8960 q^{65} -8.35907i q^{67} -12.2819 q^{71} -5.95132 q^{73} -2.28335i q^{77} +15.0027 q^{79} +14.7419i q^{83} +10.4490i q^{85} -11.8923 q^{89} -7.10447i q^{91} -2.46425 q^{95} -2.60256 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q - 24 q^{7}+O(q^{10}) \) 24 * q - 24 * q^7	\( 24 q - 24 q^{7} - 24 q^{25} + 16 q^{31} + 24 q^{49} + 8 q^{55} - 32 q^{79}+O(q^{100}) \) 24 * q - 24 * q^7 - 24 * q^25 + 16 * q^31 + 24 * q^49 + 8 * q^55 - 32 * q^79

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