LMFDB - Embedded newform 6048.2.c.g.3025.22

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			3025.22
Character	$\chi$	$=$	6048.3025
Dual form			6048.2.c.g.3025.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+3.11390i q^{5} -1.00000 q^{7} +O(q^{10})$	$q+3.11390i q^{5} -1.00000 q^{7} -2.37402i q^{11} +1.09044i q^{13} -3.69300 q^{17} -1.08263i q^{19} +4.87366 q^{23} -4.69640 q^{25} +1.59607i q^{29} -7.45516 q^{31} -3.11390i q^{35} -4.61410i q^{37} +0.0380616 q^{41} +11.0608i q^{43} +0.337175 q^{47} +1.00000 q^{49} -8.14672i q^{53} +7.39246 q^{55} -15.0761i q^{59} +5.53745i q^{61} -3.39553 q^{65} -7.70421i q^{67} -13.5472 q^{71} -14.6727 q^{73} +2.37402i q^{77} -5.18122 q^{79} +0.138321i q^{83} -11.4996i q^{85} +11.9531 q^{89} -1.09044i q^{91} +3.37121 q^{95} -5.30202 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$	3.11390i	1.39258i	0.717760	+	0.696290i	$0.245167\pi$
$5$	3.11390i	1.39258i	−0.717760	+	0.696290i	$0.754833\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	0	0
$10$	0	0
$11$	− 2.37402i	− 0.715793i	−0.933761	−	0.357897i	$-0.883494\pi$
$11$	− 2.37402i	− 0.715793i	0.933761	−	0.357897i	$-0.116506\pi$
$12$	0	0
$13$	1.09044i	0.302434i	0.988501	+	0.151217i	$0.0483192\pi$
$13$	1.09044i	0.302434i	−0.988501	+	0.151217i	$0.951681\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−3.69300	−0.895684	−0.447842	−	0.894113i	$-0.647807\pi$
$17$	−3.69300	−0.895684	−0.447842	+	0.894113i	$0.647807\pi$
$18$	0	0
$19$	− 1.08263i	− 0.248372i	−0.992259	−	0.124186i	$-0.960368\pi$
$19$	− 1.08263i	− 0.248372i	0.992259	−	0.124186i	$-0.0396320\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	4.87366	1.01623	0.508114	−	0.861290i	$-0.330343\pi$
$23$	4.87366	1.01623	0.508114	+	0.861290i	$0.330343\pi$
$24$	0	0
$25$	−4.69640	−0.939280
$26$	0	0
$27$	0	0
$28$	0	0
$29$	1.59607i	0.296382i	0.988959	+	0.148191i	$0.0473451\pi$
$29$	1.59607i	0.296382i	−0.988959	+	0.148191i	$0.952655\pi$
$30$	0	0
$31$	−7.45516	−1.33899	−0.669493	−	0.742818i	$-0.733489\pi$
$31$	−7.45516	−1.33899	−0.669493	+	0.742818i	$0.733489\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	− 3.11390i	− 0.526346i
$36$	0	0
$37$	− 4.61410i	− 0.758554i	−0.925283	−	0.379277i	$-0.876173\pi$
$37$	− 4.61410i	− 0.758554i	0.925283	−	0.379277i	$-0.123827\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	0.0380616	0.00594422	0.00297211	−	0.999996i	$-0.499054\pi$
$41$	0.0380616	0.00594422	0.00297211	+	0.999996i	$0.499054\pi$
$42$	0	0
$43$	11.0608i	1.68675i	0.537323	+	0.843377i	$0.319436\pi$
$43$	11.0608i	1.68675i	−0.537323	+	0.843377i	$0.680564\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	0.337175	0.0491821	0.0245910	−	0.999698i	$-0.492172\pi$
$47$	0.337175	0.0491821	0.0245910	+	0.999698i	$0.492172\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	0	0
$52$	0	0
$53$	− 8.14672i	− 1.11904i	−0.828818	−	0.559519i	$-0.810986\pi$
$53$	− 8.14672i	− 1.11904i	0.828818	−	0.559519i	$-0.189014\pi$
$54$	0	0
$55$	7.39246	0.996799
$56$	0	0
$57$	0	0
$58$	0	0
$59$	− 15.0761i	− 1.96274i	−0.192129	−	0.981370i	$-0.561539\pi$
$59$	− 15.0761i	− 1.96274i	0.192129	−	0.981370i	$-0.438461\pi$
$60$	0	0
$61$	5.53745i	0.708998i	0.935056	+	0.354499i	$0.115349\pi$
$61$	5.53745i	0.708998i	−0.935056	+	0.354499i	$0.884651\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−3.39553	−0.421163
$66$	0	0
$67$	− 7.70421i	− 0.941219i	−0.882342	−	0.470609i	$-0.844034\pi$
$67$	− 7.70421i	− 0.941219i	0.882342	−	0.470609i	$-0.155966\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−13.5472	−1.60776	−0.803878	−	0.594794i	$-0.797234\pi$
$71$	−13.5472	−1.60776	−0.803878	+	0.594794i	$0.797234\pi$
$72$	0	0
$73$	−14.6727	−1.71731	−0.858656	−	0.512553i	$-0.828700\pi$
$73$	−14.6727	−1.71731	−0.858656	+	0.512553i	$0.828700\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	2.37402i	0.270544i
$78$	0	0
$79$	−5.18122	−0.582932	−0.291466	−	0.956581i	$-0.594143\pi$
$79$	−5.18122	−0.582932	−0.291466	+	0.956581i	$0.594143\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	0.138321i	0.0151827i	0.999971	+	0.00759136i	$0.00241643\pi$
$83$	0.138321i	0.0151827i	−0.999971	+	0.00759136i	$0.997584\pi$
$84$	0	0
$85$	− 11.4996i	− 1.24731i
$86$	0	0
$87$	0	0
$88$	0	0
$89$	11.9531	1.26703	0.633514	−	0.773731i	$-0.281612\pi$
$89$	11.9531	1.26703	0.633514	+	0.773731i	$0.281612\pi$
$90$	0	0
$91$	− 1.09044i	− 0.114309i
$92$	0	0
$93$	0	0
$94$	0	0
$95$	3.37121	0.345879
$96$	0	0
$97$	−5.30202	−0.538339	−0.269169	−	0.963093i	$-0.586749\pi$
$97$	−5.30202	−0.538339	−0.269169	+	0.963093i	$0.586749\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	0.0753369i	0.00749630i	0.999993	+	0.00374815i	$0.00119308\pi$
$101$	0.0753369i	0.00749630i	−0.999993	+	0.00374815i	$0.998807\pi$
$102$	0	0
$103$	−15.3347	−1.51097	−0.755487	−	0.655163i	$-0.772600\pi$
$103$	−15.3347	−1.51097	−0.755487	+	0.655163i	$0.772600\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	4.03726i	0.390297i	0.980774	+	0.195148i	$0.0625189\pi$
$107$	4.03726i	0.390297i	−0.980774	+	0.195148i	$0.937481\pi$
$108$	0	0
$109$	− 12.2335i	− 1.17176i	−0.810399	−	0.585879i	$-0.800749\pi$
$109$	− 12.2335i	− 1.17176i	0.810399	−	0.585879i	$-0.199251\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	11.1934	1.05299	0.526495	−	0.850178i	$-0.323506\pi$
$113$	11.1934	1.05299	0.526495	+	0.850178i	$0.323506\pi$
$114$	0	0
$115$	15.1761i	1.41518i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	3.69300	0.338537
$120$	0	0
$121$	5.36404	0.487640
$122$	0	0
$123$	0	0
$124$	0	0
$125$	0.945386i	0.0845579i
$126$	0	0
$127$	2.43139	0.215751	0.107876	−	0.994164i	$-0.465595\pi$
$127$	2.43139	0.215751	0.107876	+	0.994164i	$0.465595\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	− 8.83723i	− 0.772112i	−0.922475	−	0.386056i	$-0.873837\pi$
$131$	− 8.83723i	− 0.772112i	0.922475	−	0.386056i	$-0.126163\pi$
$132$	0	0
$133$	1.08263i	0.0938760i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−11.1478	−0.952422	−0.476211	−	0.879331i	$-0.657990\pi$
$137$	−11.1478	−0.952422	−0.476211	+	0.879331i	$0.657990\pi$
$138$	0	0
$139$	17.2724i	1.46502i	0.680755	+	0.732511i	$0.261652\pi$
$139$	17.2724i	1.46502i	−0.680755	+	0.732511i	$0.738348\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	2.58872	0.216480
$144$	0	0
$145$	−4.97000	−0.412736
$146$	0	0
$147$	0	0
$148$	0	0
$149$	− 13.7236i	− 1.12428i	−0.827042	−	0.562140i	$-0.809978\pi$
$149$	− 13.7236i	− 1.12428i	0.827042	−	0.562140i	$-0.190022\pi$
$150$	0	0
$151$	12.8195	1.04324	0.521620	−	0.853178i	$-0.325328\pi$
$151$	12.8195	1.04324	0.521620	+	0.853178i	$0.325328\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	− 23.2146i	− 1.86465i
$156$	0	0
$157$	− 21.1418i	− 1.68730i	−0.536892	−	0.843651i	$-0.680402\pi$
$157$	− 21.1418i	− 1.68730i	0.536892	−	0.843651i	$-0.319598\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−4.87366	−0.384098
$162$	0	0
$163$	− 0.386568i	− 0.0302784i	−0.999885	−	0.0151392i	$-0.995181\pi$
$163$	− 0.386568i	− 0.0302784i	0.999885	−	0.0151392i	$-0.00481914\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−14.4336	−1.11690	−0.558451	−	0.829537i	$-0.688604\pi$
$167$	−14.4336	−1.11690	−0.558451	+	0.829537i	$0.688604\pi$
$168$	0	0
$169$	11.8109	0.908534
$170$	0	0
$171$	0	0
$172$	0	0
$173$	− 13.9493i	− 1.06055i	−0.847826	−	0.530274i	$-0.822089\pi$
$173$	− 13.9493i	− 1.06055i	0.847826	−	0.530274i	$-0.177911\pi$
$174$	0	0
$175$	4.69640	0.355014
$176$	0	0
$177$	0	0
$178$	0	0
$179$	− 10.6569i	− 0.796537i	−0.917269	−	0.398269i	$-0.869611\pi$
$179$	− 10.6569i	− 0.796537i	0.917269	−	0.398269i	$-0.130389\pi$
$180$	0	0
$181$	− 6.63866i	− 0.493448i	−0.969086	−	0.246724i	$-0.920646\pi$
$181$	− 6.63866i	− 0.493448i	0.969086	−	0.246724i	$-0.0793540\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	14.3679	1.05635
$186$	0	0
$187$	8.76724i	0.641124i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	9.81148	0.709934	0.354967	−	0.934879i	$-0.384492\pi$
$191$	9.81148	0.709934	0.354967	+	0.934879i	$0.384492\pi$
$192$	0	0
$193$	17.4539	1.25636	0.628180	−	0.778068i	$-0.283800\pi$
$193$	17.4539	1.25636	0.628180	+	0.778068i	$0.283800\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 27.6871i	− 1.97262i	−0.164888	−	0.986312i	$-0.552726\pi$
$197$	− 27.6871i	− 1.97262i	0.164888	−	0.986312i	$-0.447274\pi$
$198$	0	0
$199$	−9.29804	−0.659121	−0.329560	−	0.944134i	$-0.606901\pi$
$199$	−9.29804	−0.659121	−0.329560	+	0.944134i	$0.606901\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	− 1.59607i	− 0.112022i
$204$	0	0
$205$	0.118520i	0.00827780i
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−2.57018	−0.177783
$210$	0	0
$211$	16.3068i	1.12260i	0.827611	+	0.561302i	$0.189699\pi$
$211$	16.3068i	1.12260i	−0.827611	+	0.561302i	$0.810301\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−34.4422	−2.34894
$216$	0	0
$217$	7.45516	0.506089
$218$	0	0
$219$	0	0
$220$	0	0
$221$	− 4.02699i	− 0.270885i
$222$	0	0
$223$	18.8846	1.26461	0.632304	−	0.774720i	$-0.282109\pi$
$223$	18.8846	1.26461	0.632304	+	0.774720i	$0.282109\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 12.9709i	− 0.860907i	−0.902613	−	0.430454i	$-0.858354\pi$
$227$	− 12.9709i	− 0.860907i	0.902613	−	0.430454i	$-0.141646\pi$
$228$	0	0
$229$	5.92552i	0.391569i	0.980647	+	0.195785i	$0.0627253\pi$
$229$	5.92552i	0.391569i	−0.980647	+	0.195785i	$0.937275\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−12.2093	−0.799861	−0.399930	−	0.916546i	$-0.630966\pi$
$233$	−12.2093	−0.799861	−0.399930	+	0.916546i	$0.630966\pi$
$234$	0	0
$235$	1.04993i	0.0684900i
$236$	0	0
$237$	0	0
$238$	0	0
$239$	22.5273	1.45717	0.728585	−	0.684956i	$-0.240178\pi$
$239$	22.5273	1.45717	0.728585	+	0.684956i	$0.240178\pi$
$240$	0	0
$241$	−16.1826	−1.04241	−0.521206	−	0.853431i	$-0.674518\pi$
$241$	−16.1826	−1.04241	−0.521206	+	0.853431i	$0.674518\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	3.11390i	0.198940i
$246$	0	0
$247$	1.18054	0.0751162
$248$	0	0
$249$	0	0
$250$	0	0
$251$	− 6.88454i	− 0.434548i	−0.976111	−	0.217274i	$-0.930283\pi$
$251$	− 6.88454i	− 0.434548i	0.976111	−	0.217274i	$-0.0697166\pi$
$252$	0	0
$253$	− 11.5702i	− 0.727409i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	2.70139	0.168508	0.0842541	−	0.996444i	$-0.473149\pi$
$257$	2.70139	0.168508	0.0842541	+	0.996444i	$0.473149\pi$
$258$	0	0
$259$	4.61410i	0.286707i
$260$	0	0
$261$	0	0
$262$	0	0
$263$	13.5500	0.835527	0.417763	−	0.908556i	$-0.362814\pi$
$263$	13.5500	0.835527	0.417763	+	0.908556i	$0.362814\pi$
$264$	0	0
$265$	25.3681	1.55835
$266$	0	0
$267$	0	0
$268$	0	0
$269$	− 6.41522i	− 0.391143i	−0.980689	−	0.195571i	$-0.937344\pi$
$269$	− 6.41522i	− 0.391143i	0.980689	−	0.195571i	$-0.0626562\pi$
$270$	0	0
$271$	14.5096	0.881397	0.440698	−	0.897655i	$-0.354731\pi$
$271$	14.5096	0.881397	0.440698	+	0.897655i	$0.354731\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	11.1493i	0.672330i
$276$	0	0
$277$	25.3175i	1.52118i	0.649231	+	0.760591i	$0.275091\pi$
$277$	25.3175i	1.52118i	−0.649231	+	0.760591i	$0.724909\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	0.719979	0.0429504	0.0214752	−	0.999769i	$-0.493164\pi$
$281$	0.719979	0.0429504	0.0214752	+	0.999769i	$0.493164\pi$
$282$	0	0
$283$	11.3262i	0.673275i	0.941634	+	0.336637i	$0.109290\pi$
$283$	11.3262i	0.673275i	−0.941634	+	0.336637i	$0.890710\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−0.0380616	−0.00224670
$288$	0	0
$289$	−3.36176	−0.197751
$290$	0	0
$291$	0	0
$292$	0	0
$293$	− 15.2567i	− 0.891304i	−0.895206	−	0.445652i	$-0.852972\pi$
$293$	− 15.2567i	− 0.891304i	0.895206	−	0.445652i	$-0.147028\pi$
$294$	0	0
$295$	46.9455	2.73327
$296$	0	0
$297$	0	0
$298$	0	0
$299$	5.31443i	0.307342i
$300$	0	0
$301$	− 11.0608i	− 0.637533i
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−17.2431	−0.987337
$306$	0	0
$307$	− 25.5497i	− 1.45820i	−0.684409	−	0.729098i	$-0.739940\pi$
$307$	− 25.5497i	− 1.45820i	0.684409	−	0.729098i	$-0.260060\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	3.00328	0.170300	0.0851501	−	0.996368i	$-0.472863\pi$
$311$	3.00328	0.170300	0.0851501	+	0.996368i	$0.472863\pi$
$312$	0	0
$313$	−5.53962	−0.313118	−0.156559	−	0.987669i	$-0.550040\pi$
$313$	−5.53962	−0.313118	−0.156559	+	0.987669i	$0.550040\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 32.1705i	− 1.80688i	−0.428718	−	0.903439i	$-0.641034\pi$
$317$	− 32.1705i	− 1.80688i	0.428718	−	0.903439i	$-0.358966\pi$
$318$	0	0
$319$	3.78909	0.212148
$320$	0	0
$321$	0	0
$322$	0	0
$323$	3.99815i	0.222463i
$324$	0	0
$325$	− 5.12114i	− 0.284070i
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−0.337175	−0.0185891
$330$	0	0
$331$	− 24.9617i	− 1.37202i	−0.727592	−	0.686010i	$-0.759361\pi$
$331$	− 24.9617i	− 1.37202i	0.727592	−	0.686010i	$-0.240639\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	23.9902	1.31072
$336$	0	0
$337$	17.7899	0.969076	0.484538	−	0.874770i	$-0.338988\pi$
$337$	17.7899	0.969076	0.484538	+	0.874770i	$0.338988\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	17.6987i	0.958437i
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 2.08108i	− 0.111718i	−0.998439	−	0.0558591i	$-0.982210\pi$
$347$	− 2.08108i	− 0.111718i	0.998439	−	0.0558591i	$-0.0177898\pi$
$348$	0	0
$349$	− 7.11683i	− 0.380955i	−0.981692	−	0.190477i	$-0.938996\pi$
$349$	− 7.11683i	− 0.380955i	0.981692	−	0.190477i	$-0.0610036\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−28.9449	−1.54058	−0.770290	−	0.637694i	$-0.779889\pi$
$353$	−28.9449	−1.54058	−0.770290	+	0.637694i	$0.779889\pi$
$354$	0	0
$355$	− 42.1847i	− 2.23893i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−10.0190	−0.528780	−0.264390	−	0.964416i	$-0.585171\pi$
$359$	−10.0190	−0.528780	−0.264390	+	0.964416i	$0.585171\pi$
$360$	0	0
$361$	17.8279	0.938311
$362$	0	0
$363$	0	0
$364$	0	0
$365$	− 45.6894i	− 2.39149i
$366$	0	0
$367$	−31.4662	−1.64252	−0.821261	−	0.570552i	$-0.806729\pi$
$367$	−31.4662	−1.64252	−0.821261	+	0.570552i	$0.806729\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	8.14672i	0.422956i
$372$	0	0
$373$	31.8121i	1.64717i	0.567193	+	0.823585i	$0.308030\pi$
$373$	31.8121i	1.64717i	−0.567193	+	0.823585i	$0.691970\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−1.74042	−0.0896360
$378$	0	0
$379$	15.2806i	0.784911i	0.919771	+	0.392456i	$0.128374\pi$
$379$	15.2806i	0.784911i	−0.919771	+	0.392456i	$0.871626\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	19.0555	0.973691	0.486845	−	0.873488i	$-0.338147\pi$
$383$	19.0555	0.973691	0.486845	+	0.873488i	$0.338147\pi$
$384$	0	0
$385$	−7.39246	−0.376755
$386$	0	0
$387$	0	0
$388$	0	0
$389$	3.69055i	0.187118i	0.995614	+	0.0935591i	$0.0298244\pi$
$389$	3.69055i	0.187118i	−0.995614	+	0.0935591i	$0.970176\pi$
$390$	0	0
$391$	−17.9984	−0.910219
$392$	0	0
$393$	0	0
$394$	0	0
$395$	− 16.1338i	− 0.811780i
$396$	0	0
$397$	14.8184i	0.743715i	0.928290	+	0.371857i	$0.121279\pi$
$397$	14.8184i	0.743715i	−0.928290	+	0.371857i	$0.878721\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	1.80006	0.0898908	0.0449454	−	0.998989i	$-0.485689\pi$
$401$	1.80006	0.0898908	0.0449454	+	0.998989i	$0.485689\pi$
$402$	0	0
$403$	− 8.12940i	− 0.404954i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−10.9540	−0.542968
$408$	0	0
$409$	−25.6731	−1.26945	−0.634727	−	0.772736i	$-0.718888\pi$
$409$	−25.6731	−1.26945	−0.634727	+	0.772736i	$0.718888\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	15.0761i	0.741846i
$414$	0	0
$415$	−0.430719	−0.0211431
$416$	0	0
$417$	0	0
$418$	0	0
$419$	17.3661i	0.848392i	0.905571	+	0.424196i	$0.139443\pi$
$419$	17.3661i	0.848392i	−0.905571	+	0.424196i	$0.860557\pi$
$420$	0	0
$421$	− 30.4837i	− 1.48568i	−0.669468	−	0.742841i	$-0.733478\pi$
$421$	− 30.4837i	− 1.48568i	0.669468	−	0.742841i	$-0.266522\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	17.3438	0.841298
$426$	0	0
$427$	− 5.53745i	− 0.267976i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−9.49698	−0.457453	−0.228727	−	0.973491i	$-0.573456\pi$
$431$	−9.49698	−0.457453	−0.228727	+	0.973491i	$0.573456\pi$
$432$	0	0
$433$	25.6935	1.23475	0.617376	−	0.786668i	$-0.288196\pi$
$433$	25.6935	1.23475	0.617376	+	0.786668i	$0.288196\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 5.27637i	− 0.252403i
$438$	0	0
$439$	−11.0047	−0.525227	−0.262614	−	0.964901i	$-0.584584\pi$
$439$	−11.0047	−0.525227	−0.262614	+	0.964901i	$0.584584\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	32.0650i	1.52346i	0.647897	+	0.761728i	$0.275649\pi$
$443$	32.0650i	1.52346i	−0.647897	+	0.761728i	$0.724351\pi$
$444$	0	0
$445$	37.2208i	1.76444i
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−16.5474	−0.780922	−0.390461	−	0.920619i	$-0.627684\pi$
$449$	−16.5474	−0.780922	−0.390461	+	0.920619i	$0.627684\pi$
$450$	0	0
$451$	− 0.0903588i	− 0.00425483i
$452$	0	0
$453$	0	0
$454$	0	0
$455$	3.39553	0.159185
$456$	0	0
$457$	−26.9067	−1.25864	−0.629321	−	0.777145i	$-0.716667\pi$
$457$	−26.9067	−1.25864	−0.629321	+	0.777145i	$0.716667\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	− 28.9351i	− 1.34764i	−0.738895	−	0.673821i	$-0.764652\pi$
$461$	− 28.9351i	− 1.34764i	0.738895	−	0.673821i	$-0.235348\pi$
$462$	0	0
$463$	−10.3287	−0.480017	−0.240009	−	0.970771i	$-0.577150\pi$
$463$	−10.3287	−0.480017	−0.240009	+	0.970771i	$0.577150\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 22.1113i	− 1.02319i	−0.859227	−	0.511595i	$-0.829055\pi$
$467$	− 22.1113i	− 1.02319i	0.859227	−	0.511595i	$-0.170945\pi$
$468$	0	0
$469$	7.70421i	0.355747i
$470$	0	0
$471$	0	0
$472$	0	0
$473$	26.2585	1.20737
$474$	0	0
$475$	5.08446i	0.233291i
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−5.77499	−0.263866	−0.131933	−	0.991259i	$-0.542118\pi$
$479$	−5.77499	−0.263866	−0.131933	+	0.991259i	$0.542118\pi$
$480$	0	0
$481$	5.03140	0.229412
$482$	0	0
$483$	0	0
$484$	0	0
$485$	− 16.5100i	− 0.749680i
$486$	0	0
$487$	−12.5566	−0.568993	−0.284497	−	0.958677i	$-0.591826\pi$
$487$	−12.5566	−0.568993	−0.284497	+	0.958677i	$0.591826\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	− 20.9548i	− 0.945677i	−0.881149	−	0.472839i	$-0.843229\pi$
$491$	− 20.9548i	− 0.945677i	0.881149	−	0.472839i	$-0.156771\pi$
$492$	0	0
$493$	− 5.89428i	− 0.265465i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	13.5472	0.607675
$498$	0	0
$499$	13.0152i	0.582641i	0.956626	+	0.291320i	$0.0940946\pi$
$499$	13.0152i	0.582641i	−0.956626	+	0.291320i	$0.905905\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−22.9100	−1.02150	−0.510752	−	0.859728i	$-0.670633\pi$
$503$	−22.9100	−1.02150	−0.510752	+	0.859728i	$0.670633\pi$
$504$	0	0
$505$	−0.234592	−0.0104392
$506$	0	0
$507$	0	0
$508$	0	0
$509$	24.5703i	1.08906i	0.838741	+	0.544530i	$0.183292\pi$
$509$	24.5703i	1.08906i	−0.838741	+	0.544530i	$0.816708\pi$
$510$	0	0
$511$	14.6727	0.649083
$512$	0	0
$513$	0	0
$514$	0	0
$515$	− 47.7508i	− 2.10415i
$516$	0	0
$517$	− 0.800460i	− 0.0352042i
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−1.51316	−0.0662929	−0.0331465	−	0.999451i	$-0.510553\pi$
$521$	−1.51316	−0.0662929	−0.0331465	+	0.999451i	$0.510553\pi$
$522$	0	0
$523$	− 0.202190i	− 0.00884117i	−0.999990	−	0.00442059i	$-0.998593\pi$
$523$	− 0.202190i	− 0.00884117i	0.999990	−	0.00442059i	$-0.00140712\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	27.5319	1.19931
$528$	0	0
$529$	0.752559	0.0327199
$530$	0	0
$531$	0	0
$532$	0	0
$533$	0.0415038i	0.00179773i
$534$	0	0
$535$	−12.5716	−0.543520
$536$	0	0
$537$	0	0
$538$	0	0
$539$	− 2.37402i	− 0.102256i
$540$	0	0
$541$	28.5045i	1.22550i	0.790275	+	0.612752i	$0.209938\pi$
$541$	28.5045i	1.22550i	−0.790275	+	0.612752i	$0.790062\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	38.0940	1.63177
$546$	0	0
$547$	− 37.9150i	− 1.62113i	−0.585648	−	0.810565i	$-0.699160\pi$
$547$	− 37.9150i	− 1.62113i	0.585648	−	0.810565i	$-0.300840\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	1.72795	0.0736132
$552$	0	0
$553$	5.18122	0.220328
$554$	0	0
$555$	0	0
$556$	0	0
$557$	16.6109i	0.703828i	0.936032	+	0.351914i	$0.114469\pi$
$557$	16.6109i	0.703828i	−0.936032	+	0.351914i	$0.885531\pi$
$558$	0	0
$559$	−12.0611	−0.510131
$560$	0	0
$561$	0	0
$562$	0	0
$563$	− 6.84219i	− 0.288364i	−0.989551	−	0.144182i	$-0.953945\pi$
$563$	− 6.84219i	− 0.288364i	0.989551	−	0.144182i	$-0.0460551\pi$
$564$	0	0
$565$	34.8553i	1.46637i
$566$	0	0
$567$	0	0
$568$	0	0
$569$	15.3184	0.642180	0.321090	−	0.947049i	$-0.395951\pi$
$569$	15.3184	0.642180	0.321090	+	0.947049i	$0.395951\pi$
$570$	0	0
$571$	− 36.6853i	− 1.53523i	−0.640911	−	0.767616i	$-0.721443\pi$
$571$	− 36.6853i	− 1.53523i	0.640911	−	0.767616i	$-0.278557\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−22.8886	−0.954523
$576$	0	0
$577$	−23.9150	−0.995596	−0.497798	−	0.867293i	$-0.665858\pi$
$577$	−23.9150	−0.995596	−0.497798	+	0.867293i	$0.665858\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	− 0.138321i	− 0.00573852i
$582$	0	0
$583$	−19.3404	−0.800999
$584$	0	0
$585$	0	0
$586$	0	0
$587$	45.0433i	1.85914i	0.368651	+	0.929568i	$0.379820\pi$
$587$	45.0433i	1.85914i	−0.368651	+	0.929568i	$0.620180\pi$
$588$	0	0
$589$	8.07118i	0.332567i
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−13.4524	−0.552425	−0.276212	−	0.961097i	$-0.589079\pi$
$593$	−13.4524	−0.552425	−0.276212	+	0.961097i	$0.589079\pi$
$594$	0	0
$595$	11.4996i	0.471439i
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−21.8377	−0.892263	−0.446132	−	0.894967i	$-0.647199\pi$
$599$	−21.8377	−0.892263	−0.446132	+	0.894967i	$0.647199\pi$
$600$	0	0
$601$	−16.3704	−0.667764	−0.333882	−	0.942615i	$-0.608359\pi$
$601$	−16.3704	−0.667764	−0.333882	+	0.942615i	$0.608359\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	16.7031i	0.679078i
$606$	0	0
$607$	−28.8256	−1.17000	−0.584998	−	0.811034i	$-0.698905\pi$
$607$	−28.8256	−1.17000	−0.584998	+	0.811034i	$0.698905\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	0.367669i	0.0148743i
$612$	0	0
$613$	34.6397i	1.39908i	0.714592	+	0.699541i	$0.246612\pi$
$613$	34.6397i	1.39908i	−0.714592	+	0.699541i	$0.753388\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−38.6359	−1.55542	−0.777711	−	0.628622i	$-0.783619\pi$
$617$	−38.6359	−1.55542	−0.777711	+	0.628622i	$0.783619\pi$
$618$	0	0
$619$	25.9642i	1.04359i	0.853071	+	0.521794i	$0.174737\pi$
$619$	25.9642i	1.04359i	−0.853071	+	0.521794i	$0.825263\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−11.9531	−0.478891
$624$	0	0
$625$	−26.4258	−1.05703
$626$	0	0
$627$	0	0
$628$	0	0
$629$	17.0399i	0.679425i
$630$	0	0
$631$	−5.94731	−0.236759	−0.118379	−	0.992968i	$-0.537770\pi$
$631$	−5.94731	−0.236759	−0.118379	+	0.992968i	$0.537770\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	7.57112i	0.300451i
$636$	0	0
$637$	1.09044i	0.0432048i
$638$	0	0
$639$	0	0
$640$	0	0
$641$	41.7290	1.64820	0.824098	−	0.566447i	$-0.191682\pi$
$641$	41.7290	1.64820	0.824098	+	0.566447i	$0.191682\pi$
$642$	0	0
$643$	36.9825i	1.45845i	0.684275	+	0.729224i	$0.260119\pi$
$643$	36.9825i	1.45845i	−0.684275	+	0.729224i	$0.739881\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−2.39044	−0.0939778	−0.0469889	−	0.998895i	$-0.514963\pi$
$647$	−2.39044	−0.0939778	−0.0469889	+	0.998895i	$0.514963\pi$
$648$	0	0
$649$	−35.7909	−1.40492
$650$	0	0
$651$	0	0
$652$	0	0
$653$	27.1588i	1.06280i	0.847120	+	0.531402i	$0.178335\pi$
$653$	27.1588i	1.06280i	−0.847120	+	0.531402i	$0.821665\pi$
$654$	0	0
$655$	27.5183	1.07523
$656$	0	0
$657$	0	0
$658$	0	0
$659$	− 41.3343i	− 1.61016i	−0.593169	−	0.805078i	$-0.702123\pi$
$659$	− 41.3343i	− 1.61016i	0.593169	−	0.805078i	$-0.297877\pi$
$660$	0	0
$661$	− 6.44768i	− 0.250786i	−0.992107	−	0.125393i	$-0.959981\pi$
$661$	− 6.44768i	− 0.250786i	0.992107	−	0.125393i	$-0.0400192\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−3.37121	−0.130730
$666$	0	0
$667$	7.77869i	0.301192i
$668$	0	0
$669$	0	0
$670$	0	0
$671$	13.1460	0.507496
$672$	0	0
$673$	−36.1564	−1.39373	−0.696863	−	0.717204i	$-0.745421\pi$
$673$	−36.1564	−1.39373	−0.696863	+	0.717204i	$0.745421\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	47.4088i	1.82207i	0.412330	+	0.911035i	$0.364715\pi$
$677$	47.4088i	1.82207i	−0.412330	+	0.911035i	$0.635285\pi$
$678$	0	0
$679$	5.30202	0.203473
$680$	0	0
$681$	0	0
$682$	0	0
$683$	− 3.55646i	− 0.136084i	−0.997682	−	0.0680420i	$-0.978325\pi$
$683$	− 3.55646i	− 0.136084i	0.997682	−	0.0680420i	$-0.0216752\pi$
$684$	0	0
$685$	− 34.7132i	− 1.32632i
$686$	0	0
$687$	0	0
$688$	0	0
$689$	8.88350	0.338435
$690$	0	0
$691$	− 41.6181i	− 1.58323i	−0.611023	−	0.791613i	$-0.709242\pi$
$691$	− 41.6181i	− 1.58323i	0.611023	−	0.791613i	$-0.290758\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−53.7845	−2.04016
$696$	0	0
$697$	−0.140561	−0.00532414
$698$	0	0
$699$	0	0
$700$	0	0
$701$	− 34.8945i	− 1.31795i	−0.752166	−	0.658973i	$-0.770991\pi$
$701$	− 34.8945i	− 1.31795i	0.752166	−	0.658973i	$-0.229009\pi$
$702$	0	0
$703$	−4.99537	−0.188404
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 0.0753369i	− 0.00283334i
$708$	0	0
$709$	− 49.2296i	− 1.84886i	−0.381357	−	0.924428i	$-0.624543\pi$
$709$	− 49.2296i	− 1.84886i	0.381357	−	0.924428i	$-0.375457\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−36.3339	−1.36072
$714$	0	0
$715$	8.06103i	0.301466i
$716$	0	0
$717$	0	0
$718$	0	0
$719$	10.9969	0.410117	0.205058	−	0.978750i	$-0.434262\pi$
$719$	10.9969	0.410117	0.205058	+	0.978750i	$0.434262\pi$
$720$	0	0
$721$	15.3347	0.571095
$722$	0	0
$723$	0	0
$724$	0	0
$725$	− 7.49577i	− 0.278386i
$726$	0	0
$727$	−9.03008	−0.334907	−0.167454	−	0.985880i	$-0.553554\pi$
$727$	−9.03008	−0.334907	−0.167454	+	0.985880i	$0.553554\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	− 40.8474i	− 1.51080i
$732$	0	0
$733$	− 41.6840i	− 1.53963i	−0.638265	−	0.769817i	$-0.720347\pi$
$733$	− 41.6840i	− 1.53963i	0.638265	−	0.769817i	$-0.279653\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−18.2899	−0.673718
$738$	0	0
$739$	− 34.7320i	− 1.27764i	−0.769358	−	0.638818i	$-0.779424\pi$
$739$	− 34.7320i	− 1.27764i	0.769358	−	0.638818i	$-0.220576\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	15.9173	0.583951	0.291975	−	0.956426i	$-0.405687\pi$
$743$	15.9173	0.583951	0.291975	+	0.956426i	$0.405687\pi$
$744$	0	0
$745$	42.7339	1.56565
$746$	0	0
$747$	0	0
$748$	0	0
$749$	− 4.03726i	− 0.147518i
$750$	0	0
$751$	−13.7001	−0.499925	−0.249963	−	0.968255i	$-0.580418\pi$
$751$	−13.7001	−0.499925	−0.249963	+	0.968255i	$0.580418\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	39.9188i	1.45279i
$756$	0	0
$757$	27.3072i	0.992496i	0.868181	+	0.496248i	$0.165289\pi$
$757$	27.3072i	0.992496i	−0.868181	+	0.496248i	$0.834711\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−18.3078	−0.663657	−0.331829	−	0.943340i	$-0.607666\pi$
$761$	−18.3078	−0.663657	−0.331829	+	0.943340i	$0.607666\pi$
$762$	0	0
$763$	12.2335i	0.442883i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	16.4396	0.593598
$768$	0	0
$769$	−13.3631	−0.481884	−0.240942	−	0.970540i	$-0.577456\pi$
$769$	−13.3631	−0.481884	−0.240942	+	0.970540i	$0.577456\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	− 39.3918i	− 1.41682i	−0.705799	−	0.708412i	$-0.749412\pi$
$773$	− 39.3918i	− 1.41682i	0.705799	−	0.708412i	$-0.250588\pi$
$774$	0	0
$775$	35.0124	1.25768
$776$	0	0
$777$	0	0
$778$	0	0
$779$	− 0.0412066i	− 0.00147638i
$780$	0	0
$781$	32.1613i	1.15082i
$782$	0	0
$783$	0	0
$784$	0	0
$785$	65.8336	2.34970
$786$	0	0
$787$	− 36.3862i	− 1.29703i	−0.761203	−	0.648514i	$-0.775391\pi$
$787$	− 36.3862i	− 1.29703i	0.761203	−	0.648514i	$-0.224609\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−11.1934	−0.397993
$792$	0	0
$793$	−6.03826	−0.214425
$794$	0	0
$795$	0	0
$796$	0	0
$797$	33.1598i	1.17458i	0.809377	+	0.587290i	$0.199805\pi$
$797$	33.1598i	1.17458i	−0.809377	+	0.587290i	$0.800195\pi$
$798$	0	0
$799$	−1.24519	−0.0440516
$800$	0	0
$801$	0	0
$802$	0	0
$803$	34.8333i	1.22924i
$804$	0	0
$805$	− 15.1761i	− 0.534888i
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−44.9173	−1.57921	−0.789604	−	0.613616i	$-0.789714\pi$
$809$	−44.9173	−1.57921	−0.789604	+	0.613616i	$0.789714\pi$
$810$	0	0
$811$	37.5222i	1.31758i	0.752325	+	0.658792i	$0.228932\pi$
$811$	37.5222i	1.31758i	−0.752325	+	0.658792i	$0.771068\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	1.20374	0.0421650
$816$	0	0
$817$	11.9747	0.418943
$818$	0	0
$819$	0	0
$820$	0	0
$821$	0.214362i	0.00748129i	0.999993	+	0.00374065i	$0.00119069\pi$
$821$	0.214362i	0.00748129i	−0.999993	+	0.00374065i	$0.998809\pi$
$822$	0	0
$823$	4.84153	0.168765	0.0843825	−	0.996433i	$-0.473108\pi$
$823$	4.84153	0.168765	0.0843825	+	0.996433i	$0.473108\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	9.13088i	0.317512i	0.987318	+	0.158756i	$0.0507483\pi$
$827$	9.13088i	0.317512i	−0.987318	+	0.158756i	$0.949252\pi$
$828$	0	0
$829$	22.8115i	0.792275i	0.918191	+	0.396138i	$0.129650\pi$
$829$	22.8115i	0.792275i	−0.918191	+	0.396138i	$0.870350\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−3.69300	−0.127955
$834$	0	0
$835$	− 44.9447i	− 1.55538i
$836$	0	0
$837$	0	0
$838$	0	0
$839$	37.9537	1.31031	0.655154	−	0.755496i	$-0.272604\pi$
$839$	37.9537	1.31031	0.655154	+	0.755496i	$0.272604\pi$
$840$	0	0
$841$	26.4526	0.912157
$842$	0	0
$843$	0	0
$844$	0	0
$845$	36.7781i	1.26521i
$846$	0	0
$847$	−5.36404	−0.184311
$848$	0	0
$849$	0	0
$850$	0	0
$851$	− 22.4876i	− 0.770864i
$852$	0	0
$853$	− 35.9465i	− 1.23078i	−0.788221	−	0.615392i	$-0.788998\pi$
$853$	− 35.9465i	− 1.23078i	0.788221	−	0.615392i	$-0.211002\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	36.9904	1.26357	0.631783	−	0.775145i	$-0.282323\pi$
$857$	36.9904	1.26357	0.631783	+	0.775145i	$0.282323\pi$
$858$	0	0
$859$	0.924367i	0.0315390i	0.999876	+	0.0157695i	$0.00501979\pi$
$859$	0.924367i	0.0315390i	−0.999876	+	0.0157695i	$0.994980\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	11.2057	0.381445	0.190723	−	0.981644i	$-0.438917\pi$
$863$	11.2057	0.381445	0.190723	+	0.981644i	$0.438917\pi$
$864$	0	0
$865$	43.4369	1.47690
$866$	0	0
$867$	0	0
$868$	0	0
$869$	12.3003i	0.417259i
$870$	0	0
$871$	8.40098	0.284656
$872$	0	0
$873$	0	0
$874$	0	0
$875$	− 0.945386i	− 0.0319599i
$876$	0	0
$877$	38.6354i	1.30463i	0.757950	+	0.652313i	$0.226201\pi$
$877$	38.6354i	1.30463i	−0.757950	+	0.652313i	$0.773799\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−47.0860	−1.58637	−0.793184	−	0.608982i	$-0.791578\pi$
$881$	−47.0860	−1.58637	−0.793184	+	0.608982i	$0.791578\pi$
$882$	0	0
$883$	2.23864i	0.0753362i	0.999290	+	0.0376681i	$0.0119930\pi$
$883$	2.23864i	0.0753362i	−0.999290	+	0.0376681i	$0.988007\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	56.5911	1.90014	0.950071	−	0.312032i	$-0.101010\pi$
$887$	56.5911	1.90014	0.950071	+	0.312032i	$0.101010\pi$
$888$	0	0
$889$	−2.43139	−0.0815463
$890$	0	0
$891$	0	0
$892$	0	0
$893$	− 0.365036i	− 0.0122155i
$894$	0	0
$895$	33.1847	1.10924
$896$	0	0
$897$	0	0
$898$	0	0
$899$	− 11.8989i	− 0.396852i
$900$	0	0
$901$	30.0858i	1.00230i
$902$	0	0
$903$	0	0
$904$	0	0
$905$	20.6721	0.687165
$906$	0	0
$907$	− 37.0996i	− 1.23187i	−0.787797	−	0.615935i	$-0.788778\pi$
$907$	− 37.0996i	− 1.23187i	0.787797	−	0.615935i	$-0.211222\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	45.3701	1.50318	0.751589	−	0.659632i	$-0.229288\pi$
$911$	45.3701	1.50318	0.751589	+	0.659632i	$0.229288\pi$
$912$	0	0
$913$	0.328377	0.0108677
$914$	0	0
$915$	0	0
$916$	0	0
$917$	8.83723i	0.291831i
$918$	0	0
$919$	−28.8038	−0.950148	−0.475074	−	0.879946i	$-0.657579\pi$
$919$	−28.8038	−0.950148	−0.475074	+	0.879946i	$0.657579\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	− 14.7724i	− 0.486240i
$924$	0	0
$925$	21.6697i	0.712495i
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−45.5518	−1.49451	−0.747254	−	0.664539i	$-0.768628\pi$
$929$	−45.5518	−1.49451	−0.747254	+	0.664539i	$0.768628\pi$
$930$	0	0
$931$	− 1.08263i	− 0.0354818i
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−27.3003	−0.892817
$936$	0	0
$937$	−43.3973	−1.41773	−0.708864	−	0.705346i	$-0.750792\pi$
$937$	−43.3973	−1.41773	−0.708864	+	0.705346i	$0.750792\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	− 17.5629i	− 0.572534i	−0.958150	−	0.286267i	$-0.907586\pi$
$941$	− 17.5629i	− 0.572534i	0.958150	−	0.286267i	$-0.0924145\pi$
$942$	0	0
$943$	0.185499	0.00604068
$944$	0	0
$945$	0	0
$946$	0	0
$947$	33.8968i	1.10150i	0.834671	+	0.550749i	$0.185658\pi$
$947$	33.8968i	1.10150i	−0.834671	+	0.550749i	$0.814342\pi$
$948$	0	0
$949$	− 15.9997i	− 0.519373i
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−42.7021	−1.38326	−0.691628	−	0.722254i	$-0.743106\pi$
$953$	−42.7021	−1.38326	−0.691628	+	0.722254i	$0.743106\pi$
$954$	0	0
$955$	30.5520i	0.988639i
$956$	0	0
$957$	0	0
$958$	0	0
$959$	11.1478	0.359982
$960$	0	0
$961$	24.5794	0.792883
$962$	0	0
$963$	0	0
$964$	0	0
$965$	54.3498i	1.74958i
$966$	0	0
$967$	−23.9796	−0.771132	−0.385566	−	0.922680i	$-0.625994\pi$
$967$	−23.9796	−0.771132	−0.385566	+	0.922680i	$0.625994\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	13.2383i	0.424837i	0.977179	+	0.212418i	$0.0681340\pi$
$971$	13.2383i	0.424837i	−0.977179	+	0.212418i	$0.931866\pi$
$972$	0	0
$973$	− 17.2724i	− 0.553726i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	49.7224	1.59076	0.795380	−	0.606110i	$-0.207271\pi$
$977$	49.7224	1.59076	0.795380	+	0.606110i	$0.207271\pi$
$978$	0	0
$979$	− 28.3769i	− 0.906929i
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−33.8135	−1.07848	−0.539241	−	0.842151i	$-0.681289\pi$
$983$	−33.8135	−1.07848	−0.539241	+	0.842151i	$0.681289\pi$
$984$	0	0
$985$	86.2150	2.74704
$986$	0	0
$987$	0	0
$988$	0	0
$989$	53.9065i	1.71413i
$990$	0	0
$991$	−36.2781	−1.15241	−0.576206	−	0.817304i	$-0.695468\pi$
$991$	−36.2781	−1.15241	−0.576206	+	0.817304i	$0.695468\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	− 28.9532i	− 0.917879i
$996$	0	0
$997$	22.7054i	0.719088i	0.933128	+	0.359544i	$0.117068\pi$
$997$	22.7054i	0.719088i	−0.933128	+	0.359544i	$0.882932\pi$
$998$	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.3025.22		24
3.2	odd	2	inner	6048.2.c.g.3025.4		24
4.3	odd	2		1512.2.c.f.757.24	yes	24
8.3	odd	2		1512.2.c.f.757.23	yes	24
8.5	even	2	inner	6048.2.c.g.3025.3		24
12.11	even	2		1512.2.c.f.757.1	✓	24
24.5	odd	2	inner	6048.2.c.g.3025.21		24
24.11	even	2		1512.2.c.f.757.2	yes	24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1512.2.c.f.757.1	✓	24	12.11	even	2
1512.2.c.f.757.2	yes	24	24.11	even	2
1512.2.c.f.757.23	yes	24	8.3	odd	2
1512.2.c.f.757.24	yes	24	4.3	odd	2
6048.2.c.g.3025.3		24	8.5	even	2	inner
6048.2.c.g.3025.4		24	3.2	odd	2	inner
6048.2.c.g.3025.21		24	24.5	odd	2	inner
6048.2.c.g.3025.22		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)

\(f(q)\)	\(=\)	\(q+3.11390i q^{5} -1.00000 q^{7} +O(q^{10})\)	\(q+3.11390i q^{5} -1.00000 q^{7} -2.37402i q^{11} +1.09044i q^{13} -3.69300 q^{17} -1.08263i q^{19} +4.87366 q^{23} -4.69640 q^{25} +1.59607i q^{29} -7.45516 q^{31} -3.11390i q^{35} -4.61410i q^{37} +0.0380616 q^{41} +11.0608i q^{43} +0.337175 q^{47} +1.00000 q^{49} -8.14672i q^{53} +7.39246 q^{55} -15.0761i q^{59} +5.53745i q^{61} -3.39553 q^{65} -7.70421i q^{67} -13.5472 q^{71} -14.6727 q^{73} +2.37402i q^{77} -5.18122 q^{79} +0.138321i q^{83} -11.4996i q^{85} +11.9531 q^{89} -1.09044i q^{91} +3.37121 q^{95} -5.30202 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q - 24 q^{7}+O(q^{10}) \) 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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