LMFDB - Embedded newform 6048.2.h.d.2591.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6048,2,Mod(2591,6048)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

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

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

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

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

chi := DirichletCharacter("6048.2591");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6048 = 2^{5} \cdot 3^{3} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6048.h (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$48.2935231425$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\Q(\zeta_{24})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - x^{4} + 1 $ x^8 - x^4 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{6} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			2591.4
Root			$0.258819 - 0.965926i$ of defining polynomial
Character	$\chi$	$=$	6048.2591
Dual form			6048.2.h.d.2591.5

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.317837i q^{5} +1.00000i q^{7} +O(q^{10})$	$q-0.317837i q^{5} +1.00000i q^{7} -4.87832 q^{11} -2.44949 q^{13} -0.317837i q^{17} -0.449490i q^{19} +2.82843 q^{23} +4.89898 q^{25} +7.70674i q^{29} -4.44949i q^{31} +0.317837 q^{35} -7.00000 q^{37} +5.97469i q^{41} -10.3485i q^{43} +3.92480 q^{47} -1.00000 q^{49} -0.635674i q^{53} +1.55051i q^{55} -11.4887 q^{59} +9.34847 q^{61} +0.778539i q^{65} +8.00000i q^{67} +3.10102 q^{73} -4.87832i q^{77} -8.55051i q^{79} +10.2173 q^{83} -0.101021 q^{85} -14.1421i q^{89} -2.44949i q^{91} -0.142865 q^{95} +12.2474 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q+O(q^{10}) $ 8 * q	$ 8 q - 56 q^{37} - 8 q^{49} + 16 q^{61} + 64 q^{73} - 40 q^{85}+O(q^{100}) $ 8 * q - 56 * q^37 - 8 * q^49 + 16 * q^61 + 64 * q^73 - 40 * q^85

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
$5$	− 0.317837i	− 0.142141i	−0.997471	−	0.0710706i	$-0.977358\pi$
$5$	− 0.317837i	− 0.142141i	0.997471	−	0.0710706i	$-0.0226416\pi$
$6$	0	0
$7$	1.00000i	0.377964i
$7$	1.00000i	0.377964i
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−4.87832	−1.47087	−0.735434	−	0.677597i	$-0.763022\pi$
$11$	−4.87832	−1.47087	−0.735434	+	0.677597i	$0.763022\pi$
$12$	0	0
$13$	−2.44949	−0.679366	−0.339683	−	0.940540i	$-0.610320\pi$
$13$	−2.44949	−0.679366	−0.339683	+	0.940540i	$0.610320\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 0.317837i	− 0.0770869i	−0.999257	−	0.0385434i	$-0.987728\pi$
$17$	− 0.317837i	− 0.0770869i	0.999257	−	0.0385434i	$-0.0122718\pi$
$18$	0	0
$19$	− 0.449490i	− 0.103120i	−0.998670	−	0.0515600i	$-0.983581\pi$
$19$	− 0.449490i	− 0.103120i	0.998670	−	0.0515600i	$-0.0164193\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	2.82843	0.589768	0.294884	−	0.955533i	$-0.404719\pi$
$23$	2.82843	0.589768	0.294884	+	0.955533i	$0.404719\pi$
$24$	0	0
$25$	4.89898	0.979796
$26$	0	0
$27$	0	0
$28$	0	0
$29$	7.70674i	1.43111i	0.698558	+	0.715553i	$0.253825\pi$
$29$	7.70674i	1.43111i	−0.698558	+	0.715553i	$0.746175\pi$
$30$	0	0
$31$	− 4.44949i	− 0.799152i	−0.916700	−	0.399576i	$-0.869157\pi$
$31$	− 4.44949i	− 0.799152i	0.916700	−	0.399576i	$-0.130843\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	0.317837	0.0537243
$36$	0	0
$37$	−7.00000	−1.15079	−0.575396	−	0.817875i	$-0.695152\pi$
$37$	−7.00000	−1.15079	−0.575396	+	0.817875i	$0.695152\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	5.97469i	0.933090i	0.884497	+	0.466545i	$0.154501\pi$
$41$	5.97469i	0.933090i	−0.884497	+	0.466545i	$0.845499\pi$
$42$	0	0
$43$	− 10.3485i	− 1.57813i	−0.614312	−	0.789063i	$-0.710566\pi$
$43$	− 10.3485i	− 1.57813i	0.614312	−	0.789063i	$-0.289434\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	3.92480	0.572491	0.286246	−	0.958156i	$-0.407593\pi$
$47$	3.92480	0.572491	0.286246	+	0.958156i	$0.407593\pi$
$48$	0	0
$49$	−1.00000	−0.142857
$50$	0	0
$51$	0	0
$52$	0	0
$53$	− 0.635674i	− 0.0873166i	−0.999047	−	0.0436583i	$-0.986099\pi$
$53$	− 0.635674i	− 0.0873166i	0.999047	−	0.0436583i	$-0.0139013\pi$
$54$	0	0
$55$	1.55051i	0.209071i
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−11.4887	−1.49570	−0.747849	−	0.663868i	$-0.768914\pi$
$59$	−11.4887	−1.49570	−0.747849	+	0.663868i	$0.768914\pi$
$60$	0	0
$61$	9.34847	1.19695	0.598474	−	0.801142i	$-0.295774\pi$
$61$	9.34847	1.19695	0.598474	+	0.801142i	$0.295774\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	0.778539i	0.0965659i
$66$	0	0
$67$	8.00000i	0.977356i	0.872464	+	0.488678i	$0.162521\pi$
$67$	8.00000i	0.977356i	−0.872464	+	0.488678i	$0.837479\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	0	0		−	1.00000i	$-0.5\pi$
$71$	0	0			1.00000i	$0.5\pi$
$72$	0	0
$73$	3.10102	0.362947	0.181473	−	0.983396i	$-0.441913\pi$
$73$	3.10102	0.362947	0.181473	+	0.983396i	$0.441913\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 4.87832i	− 0.555936i
$78$	0	0
$79$	− 8.55051i	− 0.962008i	−0.876719	−	0.481004i	$-0.840272\pi$
$79$	− 8.55051i	− 0.962008i	0.876719	−	0.481004i	$-0.159728\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	10.2173	1.12150	0.560749	−	0.827986i	$-0.310513\pi$
$83$	10.2173	1.12150	0.560749	+	0.827986i	$0.310513\pi$
$84$	0	0
$85$	−0.101021	−0.0109572
$86$	0	0
$87$	0	0
$88$	0	0
$89$	− 14.1421i	− 1.49906i	−0.661968	−	0.749532i	$-0.730279\pi$
$89$	− 14.1421i	− 1.49906i	0.661968	−	0.749532i	$-0.269721\pi$
$90$	0	0
$91$	− 2.44949i	− 0.256776i
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−0.142865	−0.0146576
$96$	0	0
$97$	12.2474	1.24354	0.621770	−	0.783200i	$-0.286414\pi$
$97$	12.2474	1.24354	0.621770	+	0.783200i	$0.286414\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	− 4.38551i	− 0.436374i	−0.975907	−	0.218187i	$-0.929986\pi$
$101$	− 4.38551i	− 0.436374i	0.975907	−	0.218187i	$-0.0700143\pi$
$102$	0	0
$103$	− 3.10102i	− 0.305553i	−0.988261	−	0.152776i	$-0.951179\pi$
$103$	− 3.10102i	− 0.305553i	0.988261	−	0.152776i	$-0.0488214\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	14.1421	1.36717	0.683586	−	0.729870i	$-0.260419\pi$
$107$	14.1421	1.36717	0.683586	+	0.729870i	$0.260419\pi$
$108$	0	0
$109$	7.00000	0.670478	0.335239	−	0.942133i	$-0.391183\pi$
$109$	7.00000	0.670478	0.335239	+	0.942133i	$0.391183\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	− 2.68556i	− 0.252636i	−0.991990	−	0.126318i	$-0.959684\pi$
$113$	− 2.68556i	− 0.252636i	0.991990	−	0.126318i	$-0.0403160\pi$
$114$	0	0
$115$	− 0.898979i	− 0.0838303i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	0.317837	0.0291361
$120$	0	0
$121$	12.7980	1.16345
$122$	0	0
$123$	0	0
$124$	0	0
$125$	− 3.14626i	− 0.281410i
$126$	0	0
$127$	− 15.2474i	− 1.35299i	−0.736446	−	0.676496i	$-0.763498\pi$
$127$	− 15.2474i	− 1.35299i	0.736446	−	0.676496i	$-0.236502\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−16.0492	−1.40222	−0.701111	−	0.713052i	$-0.747312\pi$
$131$	−16.0492	−1.40222	−0.701111	+	0.713052i	$0.747312\pi$
$132$	0	0
$133$	0.449490	0.0389757
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 8.97809i	− 0.767050i	−0.923530	−	0.383525i	$-0.874710\pi$
$137$	− 8.97809i	− 0.767050i	0.923530	−	0.383525i	$-0.125290\pi$
$138$	0	0
$139$	− 12.2474i	− 1.03882i	−0.854527	−	0.519408i	$-0.826153\pi$
$139$	− 12.2474i	− 1.03882i	0.854527	−	0.519408i	$-0.173847\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	11.9494	0.999258
$144$	0	0
$145$	2.44949	0.203419
$146$	0	0
$147$	0	0
$148$	0	0
$149$	16.8277i	1.37858i	0.724486	+	0.689289i	$0.242077\pi$
$149$	16.8277i	1.37858i	−0.724486	+	0.689289i	$0.757923\pi$
$150$	0	0
$151$	− 17.2474i	− 1.40358i	−0.712385	−	0.701789i	$-0.752385\pi$
$151$	− 17.2474i	− 1.40358i	0.712385	−	0.701789i	$-0.247615\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−1.41421	−0.113592
$156$	0	0
$157$	15.3485	1.22494	0.612471	−	0.790493i	$-0.290176\pi$
$157$	15.3485	1.22494	0.612471	+	0.790493i	$0.290176\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	2.82843i	0.222911i
$162$	0	0
$163$	− 11.2474i	− 0.880968i	−0.897760	−	0.440484i	$-0.854807\pi$
$163$	− 11.2474i	− 0.880968i	0.897760	−	0.440484i	$-0.145193\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	9.93160	0.768531	0.384265	−	0.923223i	$-0.374455\pi$
$167$	9.93160	0.768531	0.384265	+	0.923223i	$0.374455\pi$
$168$	0	0
$169$	−7.00000	−0.538462
$170$	0	0
$171$	0	0
$172$	0	0
$173$	− 9.12096i	− 0.693453i	−0.937966	−	0.346727i	$-0.887293\pi$
$173$	− 9.12096i	− 0.693453i	0.937966	−	0.346727i	$-0.112707\pi$
$174$	0	0
$175$	4.89898i	0.370328i
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−4.87832	−0.364622	−0.182311	−	0.983241i	$-0.558358\pi$
$179$	−4.87832	−0.364622	−0.182311	+	0.983241i	$0.558358\pi$
$180$	0	0
$181$	24.6969	1.83571	0.917854	−	0.396917i	$-0.129920\pi$
$181$	24.6969	1.83571	0.917854	+	0.396917i	$0.129920\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	2.22486i	0.163575i
$186$	0	0
$187$	1.55051i	0.113385i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	16.0492	1.16128	0.580638	−	0.814162i	$-0.302803\pi$
$191$	16.0492	1.16128	0.580638	+	0.814162i	$0.302803\pi$
$192$	0	0
$193$	−11.0000	−0.791797	−0.395899	−	0.918294i	$-0.629567\pi$
$193$	−11.0000	−0.791797	−0.395899	+	0.918294i	$0.629567\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 0.778539i	− 0.0554686i	−0.999615	−	0.0277343i	$-0.991171\pi$
$197$	− 0.778539i	− 0.0554686i	0.999615	−	0.0277343i	$-0.00882924\pi$
$198$	0	0
$199$	10.0000i	0.708881i	0.935079	+	0.354441i	$0.115329\pi$
$199$	10.0000i	0.708881i	−0.935079	+	0.354441i	$0.884671\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−7.70674	−0.540907
$204$	0	0
$205$	1.89898	0.132630
$206$	0	0
$207$	0	0
$208$	0	0
$209$	2.19275i	0.151676i
$210$	0	0
$211$	6.00000i	0.413057i	0.978441	+	0.206529i	$0.0662166\pi$
$211$	6.00000i	0.413057i	−0.978441	+	0.206529i	$0.933783\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−3.28913	−0.224317
$216$	0	0
$217$	4.44949	0.302051
$218$	0	0
$219$	0	0
$220$	0	0
$221$	0.778539i	0.0523702i
$222$	0	0
$223$	18.2474i	1.22194i	0.791654	+	0.610970i	$0.209220\pi$
$223$	18.2474i	1.22194i	−0.791654	+	0.610970i	$0.790780\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−4.09978	−0.272112	−0.136056	−	0.990701i	$-0.543443\pi$
$227$	−4.09978	−0.272112	−0.136056	+	0.990701i	$0.543443\pi$
$228$	0	0
$229$	12.6969	0.839037	0.419519	−	0.907747i	$-0.362199\pi$
$229$	12.6969	0.839037	0.419519	+	0.907747i	$0.362199\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	18.7347i	1.22735i	0.789558	+	0.613676i	$0.210310\pi$
$233$	18.7347i	1.22735i	−0.789558	+	0.613676i	$0.789690\pi$
$234$	0	0
$235$	− 1.24745i	− 0.0813746i
$236$	0	0
$237$	0	0
$238$	0	0
$239$	3.32124	0.214833	0.107416	−	0.994214i	$-0.465742\pi$
$239$	3.32124	0.214833	0.107416	+	0.994214i	$0.465742\pi$
$240$	0	0
$241$	−3.75255	−0.241723	−0.120862	−	0.992669i	$-0.538566\pi$
$241$	−3.75255	−0.241723	−0.120862	+	0.992669i	$0.538566\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	0.317837i	0.0203059i
$246$	0	0
$247$	1.10102i	0.0700563i
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−18.4169	−1.16246	−0.581232	−	0.813738i	$-0.697429\pi$
$251$	−18.4169	−1.16246	−0.581232	+	0.813738i	$0.697429\pi$
$252$	0	0
$253$	−13.7980	−0.867470
$254$	0	0
$255$	0	0
$256$	0	0
$257$	11.0280i	0.687906i	0.938987	+	0.343953i	$0.111766\pi$
$257$	11.0280i	0.687906i	−0.938987	+	0.343953i	$0.888234\pi$
$258$	0	0
$259$	− 7.00000i	− 0.434959i
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−7.07107	−0.436021	−0.218010	−	0.975946i	$-0.569957\pi$
$263$	−7.07107	−0.436021	−0.218010	+	0.975946i	$0.569957\pi$
$264$	0	0
$265$	−0.202041	−0.0124113
$266$	0	0
$267$	0	0
$268$	0	0
$269$	− 27.0450i	− 1.64896i	−0.565888	−	0.824482i	$-0.691466\pi$
$269$	− 27.0450i	− 1.64896i	0.565888	−	0.824482i	$-0.308534\pi$
$270$	0	0
$271$	− 16.6969i	− 1.01427i	−0.861868	−	0.507133i	$-0.830705\pi$
$271$	− 16.6969i	− 1.01427i	0.861868	−	0.507133i	$-0.169295\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−23.8988	−1.44115
$276$	0	0
$277$	8.79796	0.528618	0.264309	−	0.964438i	$-0.414856\pi$
$277$	8.79796	0.528618	0.264309	+	0.964438i	$0.414856\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	8.48528i	0.506189i	0.967442	+	0.253095i	$0.0814484\pi$
$281$	8.48528i	0.506189i	−0.967442	+	0.253095i	$0.918552\pi$
$282$	0	0
$283$	− 32.2474i	− 1.91691i	−0.285241	−	0.958456i	$-0.592074\pi$
$283$	− 32.2474i	− 1.91691i	0.285241	−	0.958456i	$-0.407926\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−5.97469	−0.352675
$288$	0	0
$289$	16.8990	0.994058
$290$	0	0
$291$	0	0
$292$	0	0
$293$	20.7525i	1.21237i	0.795322	+	0.606187i	$0.207302\pi$
$293$	20.7525i	1.21237i	−0.795322	+	0.606187i	$0.792698\pi$
$294$	0	0
$295$	3.65153i	0.212600i
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−6.92820	−0.400668
$300$	0	0
$301$	10.3485	0.596476
$302$	0	0
$303$	0	0
$304$	0	0
$305$	− 2.97129i	− 0.170136i
$306$	0	0
$307$	4.89898i	0.279600i	0.990180	+	0.139800i	$0.0446459\pi$
$307$	4.89898i	0.279600i	−0.990180	+	0.139800i	$0.955354\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−9.58166	−0.543326	−0.271663	−	0.962392i	$-0.587574\pi$
$311$	−9.58166	−0.543326	−0.271663	+	0.962392i	$0.587574\pi$
$312$	0	0
$313$	6.89898	0.389953	0.194977	−	0.980808i	$-0.437537\pi$
$313$	6.89898	0.389953	0.194977	+	0.980808i	$0.437537\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 0.285729i	− 0.0160481i	−0.999968	−	0.00802407i	$-0.997446\pi$
$317$	− 0.285729i	− 0.0160481i	0.999968	−	0.00802407i	$-0.00255417\pi$
$318$	0	0
$319$	− 37.5959i	− 2.10497i
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−0.142865	−0.00794920
$324$	0	0
$325$	−12.0000	−0.665640
$326$	0	0
$327$	0	0
$328$	0	0
$329$	3.92480i	0.216381i
$330$	0	0
$331$	7.24745i	0.398356i	0.979963	+	0.199178i	$0.0638272\pi$
$331$	7.24745i	0.398356i	−0.979963	+	0.199178i	$0.936173\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	2.54270	0.138922
$336$	0	0
$337$	3.00000	0.163420	0.0817102	−	0.996656i	$-0.473962\pi$
$337$	3.00000	0.163420	0.0817102	+	0.996656i	$0.473962\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	21.7060i	1.17545i
$342$	0	0
$343$	− 1.00000i	− 0.0539949i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−1.69994	−0.0912577	−0.0456289	−	0.998958i	$-0.514529\pi$
$347$	−1.69994	−0.0912577	−0.0456289	+	0.998958i	$0.514529\pi$
$348$	0	0
$349$	14.2020	0.760218	0.380109	−	0.924942i	$-0.375887\pi$
$349$	14.2020	0.760218	0.380109	+	0.924942i	$0.375887\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	− 10.3602i	− 0.551418i	−0.961241	−	0.275709i	$-0.911087\pi$
$353$	− 10.3602i	− 0.551418i	0.961241	−	0.275709i	$-0.0889126\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	0	0
$358$	0	0
$359$	19.1633	1.01140	0.505701	−	0.862709i	$-0.331234\pi$
$359$	19.1633	1.01140	0.505701	+	0.862709i	$0.331234\pi$
$360$	0	0
$361$	18.7980	0.989366
$362$	0	0
$363$	0	0
$364$	0	0
$365$	− 0.985620i	− 0.0515897i
$366$	0	0
$367$	− 12.6969i	− 0.662775i	−0.943495	−	0.331387i	$-0.892483\pi$
$367$	− 12.6969i	− 0.662775i	0.943495	−	0.331387i	$-0.107517\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	0.635674	0.0330026
$372$	0	0
$373$	33.2929	1.72384	0.861919	−	0.507045i	$-0.169262\pi$
$373$	33.2929	1.72384	0.861919	+	0.507045i	$0.169262\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 18.8776i	− 0.972245i
$378$	0	0
$379$	− 33.0454i	− 1.69743i	−0.528852	−	0.848714i	$-0.677377\pi$
$379$	− 33.0454i	− 1.69743i	0.528852	−	0.848714i	$-0.322623\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	5.83183	0.297992	0.148996	−	0.988838i	$-0.452396\pi$
$383$	5.83183	0.297992	0.148996	+	0.988838i	$0.452396\pi$
$384$	0	0
$385$	−1.55051	−0.0790213
$386$	0	0
$387$	0	0
$388$	0	0
$389$	− 7.07107i	− 0.358517i	−0.983802	−	0.179259i	$-0.942630\pi$
$389$	− 7.07107i	− 0.358517i	0.983802	−	0.179259i	$-0.0573699\pi$
$390$	0	0
$391$	− 0.898979i	− 0.0454633i
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−2.71767	−0.136741
$396$	0	0
$397$	−15.1010	−0.757898	−0.378949	−	0.925417i	$-0.623715\pi$
$397$	−15.1010	−0.757898	−0.378949	+	0.925417i	$0.623715\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	− 11.6637i	− 0.582455i	−0.956654	−	0.291228i	$-0.905936\pi$
$401$	− 11.6637i	− 0.582455i	0.956654	−	0.291228i	$-0.0940637\pi$
$402$	0	0
$403$	10.8990i	0.542917i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	34.1482	1.69266
$408$	0	0
$409$	−5.79796	−0.286691	−0.143345	−	0.989673i	$-0.545786\pi$
$409$	−5.79796	−0.286691	−0.143345	+	0.989673i	$0.545786\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	− 11.4887i	− 0.565321i
$414$	0	0
$415$	− 3.24745i	− 0.159411i
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−20.6096	−1.00685	−0.503423	−	0.864040i	$-0.667926\pi$
$419$	−20.6096	−1.00685	−0.503423	+	0.864040i	$0.667926\pi$
$420$	0	0
$421$	7.10102	0.346083	0.173041	−	0.984915i	$-0.444641\pi$
$421$	7.10102	0.346083	0.173041	+	0.984915i	$0.444641\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	− 1.55708i	− 0.0755294i
$426$	0	0
$427$	9.34847i	0.452404i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	28.9842	1.39612	0.698059	−	0.716040i	$-0.254047\pi$
$431$	28.9842	1.39612	0.698059	+	0.716040i	$0.254047\pi$
$432$	0	0
$433$	25.8434	1.24195	0.620976	−	0.783829i	$-0.286736\pi$
$433$	25.8434	1.24195	0.620976	+	0.783829i	$0.286736\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 1.27135i	− 0.0608169i
$438$	0	0
$439$	2.00000i	0.0954548i	0.998860	+	0.0477274i	$0.0151979\pi$
$439$	2.00000i	0.0954548i	−0.998860	+	0.0477274i	$0.984802\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−10.8851	−0.517167	−0.258584	−	0.965989i	$-0.583256\pi$
$443$	−10.8851	−0.517167	−0.258584	+	0.965989i	$0.583256\pi$
$444$	0	0
$445$	−4.49490	−0.213079
$446$	0	0
$447$	0	0
$448$	0	0
$449$	− 31.6055i	− 1.49156i	−0.666194	−	0.745778i	$-0.732078\pi$
$449$	− 31.6055i	− 1.49156i	0.666194	−	0.745778i	$-0.267922\pi$
$450$	0	0
$451$	− 29.1464i	− 1.37245i
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−0.778539	−0.0364985
$456$	0	0
$457$	−21.3939	−1.00076	−0.500382	−	0.865805i	$-0.666807\pi$
$457$	−21.3939	−1.00076	−0.500382	+	0.865805i	$0.666807\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	9.43879i	0.439608i	0.975544	+	0.219804i	$0.0705419\pi$
$461$	9.43879i	0.439608i	−0.975544	+	0.219804i	$0.929458\pi$
$462$	0	0
$463$	− 28.8434i	− 1.34046i	−0.742151	−	0.670232i	$-0.766194\pi$
$463$	− 28.8434i	− 1.34046i	0.742151	−	0.670232i	$-0.233806\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	12.8708	0.595589	0.297794	−	0.954630i	$-0.403749\pi$
$467$	12.8708	0.595589	0.297794	+	0.954630i	$0.403749\pi$
$468$	0	0
$469$	−8.00000	−0.369406
$470$	0	0
$471$	0	0
$472$	0	0
$473$	50.4831i	2.32122i
$474$	0	0
$475$	− 2.20204i	− 0.101037i
$476$	0	0
$477$	0	0
$478$	0	0
$479$	5.48188	0.250474	0.125237	−	0.992127i	$-0.460031\pi$
$479$	5.48188	0.250474	0.125237	+	0.992127i	$0.460031\pi$
$480$	0	0
$481$	17.1464	0.781810
$482$	0	0
$483$	0	0
$484$	0	0
$485$	− 3.89270i	− 0.176758i
$486$	0	0
$487$	− 6.20204i	− 0.281041i	−0.990078	−	0.140521i	$-0.955122\pi$
$487$	− 6.20204i	− 0.281041i	0.990078	−	0.140521i	$-0.0448776\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−30.6841	−1.38475	−0.692377	−	0.721536i	$-0.743437\pi$
$491$	−30.6841	−1.38475	−0.692377	+	0.721536i	$0.743437\pi$
$492$	0	0
$493$	2.44949	0.110319
$494$	0	0
$495$	0	0
$496$	0	0
$497$	0	0
$498$	0	0
$499$	21.4495i	0.960211i	0.877211	+	0.480106i	$0.159402\pi$
$499$	21.4495i	0.960211i	−0.877211	+	0.480106i	$0.840598\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	34.1161	1.52116	0.760581	−	0.649243i	$-0.224914\pi$
$503$	34.1161	1.52116	0.760581	+	0.649243i	$0.224914\pi$
$504$	0	0
$505$	−1.39388	−0.0620267
$506$	0	0
$507$	0	0
$508$	0	0
$509$	2.79632i	0.123945i	0.998078	+	0.0619723i	$0.0197391\pi$
$509$	2.79632i	0.123945i	−0.998078	+	0.0619723i	$0.980261\pi$
$510$	0	0
$511$	3.10102i	0.137181i
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−0.985620	−0.0434316
$516$	0	0
$517$	−19.1464	−0.842059
$518$	0	0
$519$	0	0
$520$	0	0
$521$	33.0518i	1.44803i	0.689786	+	0.724013i	$0.257705\pi$
$521$	33.0518i	1.44803i	−0.689786	+	0.724013i	$0.742295\pi$
$522$	0	0
$523$	32.7423i	1.43172i	0.698242	+	0.715861i	$0.253966\pi$
$523$	32.7423i	1.43172i	−0.698242	+	0.715861i	$0.746034\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−1.41421	−0.0616041
$528$	0	0
$529$	−15.0000	−0.652174
$530$	0	0
$531$	0	0
$532$	0	0
$533$	− 14.6349i	− 0.633910i
$534$	0	0
$535$	− 4.49490i	− 0.194331i
$536$	0	0
$537$	0	0
$538$	0	0
$539$	4.87832	0.210124
$540$	0	0
$541$	2.59592	0.111607	0.0558036	−	0.998442i	$-0.482228\pi$
$541$	2.59592	0.111607	0.0558036	+	0.998442i	$0.482228\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	− 2.22486i	− 0.0953026i
$546$	0	0
$547$	− 41.0454i	− 1.75497i	−0.479600	−	0.877487i	$-0.659218\pi$
$547$	− 41.0454i	− 1.75497i	0.479600	−	0.877487i	$-0.340782\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	3.46410	0.147576
$552$	0	0
$553$	8.55051	0.363605
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 39.0265i	− 1.65361i	−0.562491	−	0.826803i	$-0.690157\pi$
$557$	− 39.0265i	− 1.65361i	0.562491	−	0.826803i	$-0.309843\pi$
$558$	0	0
$559$	25.3485i	1.07213i
$560$	0	0
$561$	0	0
$562$	0	0
$563$	8.77101	0.369654	0.184827	−	0.982771i	$-0.440828\pi$
$563$	8.77101	0.369654	0.184827	+	0.982771i	$0.440828\pi$
$564$	0	0
$565$	−0.853572	−0.0359100
$566$	0	0
$567$	0	0
$568$	0	0
$569$	− 41.8549i	− 1.75465i	−0.479896	−	0.877325i	$-0.659326\pi$
$569$	− 41.8549i	− 1.75465i	0.479896	−	0.877325i	$-0.340674\pi$
$570$	0	0
$571$	− 2.55051i	− 0.106736i	−0.998575	−	0.0533678i	$-0.983004\pi$
$571$	− 2.55051i	− 0.106736i	0.998575	−	0.0533678i	$-0.0169956\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	13.8564	0.577852
$576$	0	0
$577$	6.89898	0.287208	0.143604	−	0.989635i	$-0.454131\pi$
$577$	6.89898	0.287208	0.143604	+	0.989635i	$0.454131\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	10.2173i	0.423886i
$582$	0	0
$583$	3.10102i	0.128431i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−8.48528	−0.350225	−0.175113	−	0.984548i	$-0.556029\pi$
$587$	−8.48528	−0.350225	−0.175113	+	0.984548i	$0.556029\pi$
$588$	0	0
$589$	−2.00000	−0.0824086
$590$	0	0
$591$	0	0
$592$	0	0
$593$	− 16.0813i	− 0.660378i	−0.943915	−	0.330189i	$-0.892887\pi$
$593$	− 16.0813i	− 0.660378i	0.943915	−	0.330189i	$-0.107113\pi$
$594$	0	0
$595$	− 0.101021i	− 0.00414144i
$596$	0	0
$597$	0	0
$598$	0	0
$599$	6.43539	0.262943	0.131472	−	0.991320i	$-0.458030\pi$
$599$	6.43539	0.262943	0.131472	+	0.991320i	$0.458030\pi$
$600$	0	0
$601$	−11.7526	−0.479397	−0.239698	−	0.970847i	$-0.577049\pi$
$601$	−11.7526	−0.479397	−0.239698	+	0.970847i	$0.577049\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	− 4.06767i	− 0.165374i
$606$	0	0
$607$	39.8434i	1.61719i	0.588364	+	0.808596i	$0.299772\pi$
$607$	39.8434i	1.61719i	−0.588364	+	0.808596i	$0.700228\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−9.61377	−0.388931
$612$	0	0
$613$	−20.0000	−0.807792	−0.403896	−	0.914805i	$-0.632344\pi$
$613$	−20.0000	−0.807792	−0.403896	+	0.914805i	$0.632344\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	39.3123i	1.58265i	0.611395	+	0.791326i	$0.290609\pi$
$617$	39.3123i	1.58265i	−0.611395	+	0.791326i	$0.709391\pi$
$618$	0	0
$619$	− 3.14643i	− 0.126466i	−0.997999	−	0.0632328i	$-0.979859\pi$
$619$	− 3.14643i	− 0.126466i	0.997999	−	0.0632328i	$-0.0201411\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	14.1421	0.566593
$624$	0	0
$625$	23.4949	0.939796
$626$	0	0
$627$	0	0
$628$	0	0
$629$	2.22486i	0.0887110i
$630$	0	0
$631$	− 0.550510i	− 0.0219155i	−0.999940	−	0.0109577i	$-0.996512\pi$
$631$	− 0.550510i	− 0.0219155i	0.999940	−	0.0109577i	$-0.00348802\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−4.84621	−0.192316
$636$	0	0
$637$	2.44949	0.0970523
$638$	0	0
$639$	0	0
$640$	0	0
$641$	1.69994i	0.0671437i	0.999436	+	0.0335719i	$0.0106883\pi$
$641$	1.69994i	0.0671437i	−0.999436	+	0.0335719i	$0.989312\pi$
$642$	0	0
$643$	− 44.6969i	− 1.76268i	−0.472487	−	0.881338i	$-0.656644\pi$
$643$	− 44.6969i	− 1.76268i	0.472487	−	0.881338i	$-0.343356\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	24.1845	0.950791	0.475395	−	0.879772i	$-0.342305\pi$
$647$	24.1845	0.950791	0.475395	+	0.879772i	$0.342305\pi$
$648$	0	0
$649$	56.0454	2.19997
$650$	0	0
$651$	0	0
$652$	0	0
$653$	5.51399i	0.215779i	0.994163	+	0.107890i	$0.0344093\pi$
$653$	5.51399i	0.215779i	−0.994163	+	0.107890i	$0.965591\pi$
$654$	0	0
$655$	5.10102i	0.199313i
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−18.8776	−0.735366	−0.367683	−	0.929951i	$-0.619849\pi$
$659$	−18.8776	−0.735366	−0.367683	+	0.929951i	$0.619849\pi$
$660$	0	0
$661$	−46.7423	−1.81807	−0.909033	−	0.416724i	$-0.863178\pi$
$661$	−46.7423	−1.81807	−0.909033	+	0.416724i	$0.863178\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	− 0.142865i	− 0.00554005i
$666$	0	0
$667$	21.7980i	0.844020i
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−45.6048	−1.76055
$672$	0	0
$673$	−22.2929	−0.859326	−0.429663	−	0.902989i	$-0.641368\pi$
$673$	−22.2929	−0.859326	−0.429663	+	0.902989i	$0.641368\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 33.6554i	− 1.29348i	−0.762710	−	0.646741i	$-0.776131\pi$
$677$	− 33.6554i	− 1.29348i	0.762710	−	0.646741i	$-0.223869\pi$
$678$	0	0
$679$	12.2474i	0.470014i
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−29.1270	−1.11451	−0.557257	−	0.830340i	$-0.688146\pi$
$683$	−29.1270	−1.11451	−0.557257	+	0.830340i	$0.688146\pi$
$684$	0	0
$685$	−2.85357	−0.109029
$686$	0	0
$687$	0	0
$688$	0	0
$689$	1.55708i	0.0593200i
$690$	0	0
$691$	3.75255i	0.142754i	0.997449	+	0.0713769i	$0.0227393\pi$
$691$	3.75255i	0.142754i	−0.997449	+	0.0713769i	$0.977261\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−3.89270	−0.147658
$696$	0	0
$697$	1.89898	0.0719290
$698$	0	0
$699$	0	0
$700$	0	0
$701$	15.5563i	0.587555i	0.955874	+	0.293778i	$0.0949125\pi$
$701$	15.5563i	0.587555i	−0.955874	+	0.293778i	$0.905087\pi$
$702$	0	0
$703$	3.14643i	0.118670i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	4.38551	0.164934
$708$	0	0
$709$	35.0000	1.31445	0.657226	−	0.753693i	$-0.271730\pi$
$709$	35.0000	1.31445	0.657226	+	0.753693i	$0.271730\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	− 12.5851i	− 0.471314i
$714$	0	0
$715$	− 3.79796i	− 0.142036i
$716$	0	0
$717$	0	0
$718$	0	0
$719$	47.3368	1.76537	0.882683	−	0.469969i	$-0.155735\pi$
$719$	47.3368	1.76537	0.882683	+	0.469969i	$0.155735\pi$
$720$	0	0
$721$	3.10102	0.115488
$722$	0	0
$723$	0	0
$724$	0	0
$725$	37.7552i	1.40219i
$726$	0	0
$727$	− 17.7526i	− 0.658406i	−0.944259	−	0.329203i	$-0.893220\pi$
$727$	− 17.7526i	− 0.658406i	0.944259	−	0.329203i	$-0.106780\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−3.28913	−0.121653
$732$	0	0
$733$	33.1010	1.22261	0.611307	−	0.791394i	$-0.290644\pi$
$733$	33.1010	1.22261	0.611307	+	0.791394i	$0.290644\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 39.0265i	− 1.43756i
$738$	0	0
$739$	− 0.404082i	− 0.0148644i	−0.999972	−	0.00743220i	$-0.997634\pi$
$739$	− 0.404082i	− 0.0148644i	0.999972	−	0.00743220i	$-0.00236576\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−45.1120	−1.65500	−0.827499	−	0.561467i	$-0.810237\pi$
$743$	−45.1120	−1.65500	−0.827499	+	0.561467i	$0.810237\pi$
$744$	0	0
$745$	5.34847	0.195953
$746$	0	0
$747$	0	0
$748$	0	0
$749$	14.1421i	0.516742i
$750$	0	0
$751$	25.3939i	0.926636i	0.886192	+	0.463318i	$0.153341\pi$
$751$	25.3939i	0.926636i	−0.886192	+	0.463318i	$0.846659\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−5.48188	−0.199506
$756$	0	0
$757$	−53.6969	−1.95165	−0.975824	−	0.218557i	$-0.929865\pi$
$757$	−53.6969	−1.95165	−0.975824	+	0.218557i	$0.929865\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	33.9732i	1.23153i	0.787930	+	0.615764i	$0.211153\pi$
$761$	33.9732i	1.23153i	−0.787930	+	0.615764i	$0.788847\pi$
$762$	0	0
$763$	7.00000i	0.253417i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	28.1414	1.01613
$768$	0	0
$769$	−34.8990	−1.25849	−0.629245	−	0.777207i	$-0.716636\pi$
$769$	−34.8990	−1.25849	−0.629245	+	0.777207i	$0.716636\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	21.6739i	0.779556i	0.920909	+	0.389778i	$0.127448\pi$
$773$	21.6739i	0.779556i	−0.920909	+	0.389778i	$0.872552\pi$
$774$	0	0
$775$	− 21.7980i	− 0.783006i
$776$	0	0
$777$	0	0
$778$	0	0
$779$	2.68556	0.0962203
$780$	0	0
$781$	0	0
$782$	0	0
$783$	0	0
$784$	0	0
$785$	− 4.87832i	− 0.174115i
$786$	0	0
$787$	44.4949i	1.58607i	0.609175	+	0.793036i	$0.291501\pi$
$787$	44.4949i	1.58607i	−0.609175	+	0.793036i	$0.708499\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	2.68556	0.0954876
$792$	0	0
$793$	−22.8990	−0.813167
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 19.2275i	− 0.681074i	−0.940231	−	0.340537i	$-0.889391\pi$
$797$	− 19.2275i	− 0.681074i	0.940231	−	0.340537i	$-0.110609\pi$
$798$	0	0
$799$	− 1.24745i	− 0.0441316i
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−15.1278	−0.533847
$804$	0	0
$805$	0.898979	0.0316849
$806$	0	0
$807$	0	0
$808$	0	0
$809$	27.0771i	0.951981i	0.879450	+	0.475991i	$0.157910\pi$
$809$	27.0771i	0.951981i	−0.879450	+	0.475991i	$0.842090\pi$
$810$	0	0
$811$	36.7423i	1.29020i	0.764099	+	0.645099i	$0.223184\pi$
$811$	36.7423i	1.29020i	−0.764099	+	0.645099i	$0.776816\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−3.57486	−0.125222
$816$	0	0
$817$	−4.65153	−0.162736
$818$	0	0
$819$	0	0
$820$	0	0
$821$	33.3697i	1.16461i	0.812971	+	0.582305i	$0.197849\pi$
$821$	33.3697i	1.16461i	−0.812971	+	0.582305i	$0.802151\pi$
$822$	0	0
$823$	9.65153i	0.336431i	0.985750	+	0.168216i	$0.0538005\pi$
$823$	9.65153i	0.336431i	−0.985750	+	0.168216i	$0.946200\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−11.5208	−0.400617	−0.200309	−	0.979733i	$-0.564194\pi$
$827$	−11.5208	−0.400617	−0.200309	+	0.979733i	$0.564194\pi$
$828$	0	0
$829$	−6.04541	−0.209966	−0.104983	−	0.994474i	$-0.533479\pi$
$829$	−6.04541	−0.209966	−0.104983	+	0.994474i	$0.533479\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	0.317837i	0.0110124i
$834$	0	0
$835$	− 3.15663i	− 0.109240i
$836$	0	0
$837$	0	0
$838$	0	0
$839$	27.1879	0.938630	0.469315	−	0.883031i	$-0.344501\pi$
$839$	27.1879	0.938630	0.469315	+	0.883031i	$0.344501\pi$
$840$	0	0
$841$	−30.3939	−1.04806
$842$	0	0
$843$	0	0
$844$	0	0
$845$	2.22486i	0.0765375i
$846$	0	0
$847$	12.7980i	0.439743i
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−19.7990	−0.678701
$852$	0	0
$853$	46.8990	1.60579	0.802895	−	0.596120i	$-0.203292\pi$
$853$	46.8990	1.60579	0.802895	+	0.596120i	$0.203292\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	23.0095i	0.785989i	0.919541	+	0.392994i	$0.128561\pi$
$857$	23.0095i	0.785989i	−0.919541	+	0.392994i	$0.871439\pi$
$858$	0	0
$859$	26.4949i	0.903994i	0.892019	+	0.451997i	$0.149288\pi$
$859$	26.4949i	0.903994i	−0.892019	+	0.451997i	$0.850712\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−19.1633	−0.652327	−0.326163	−	0.945313i	$-0.605756\pi$
$863$	−19.1633	−0.652327	−0.326163	+	0.945313i	$0.605756\pi$
$864$	0	0
$865$	−2.89898	−0.0985683
$866$	0	0
$867$	0	0
$868$	0	0
$869$	41.7121i	1.41499i
$870$	0	0
$871$	− 19.5959i	− 0.663982i
$872$	0	0
$873$	0	0
$874$	0	0
$875$	3.14626	0.106363
$876$	0	0
$877$	−38.7980	−1.31011	−0.655057	−	0.755579i	$-0.727355\pi$
$877$	−38.7980	−1.31011	−0.655057	+	0.755579i	$0.727355\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	− 49.0689i	− 1.65317i	−0.562810	−	0.826586i	$-0.690280\pi$
$881$	− 49.0689i	− 1.65317i	0.562810	−	0.826586i	$-0.309720\pi$
$882$	0	0
$883$	− 7.04541i	− 0.237097i	−0.992948	−	0.118548i	$-0.962176\pi$
$883$	− 7.04541i	− 0.237097i	0.992948	−	0.118548i	$-0.0378241\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−9.58166	−0.321721	−0.160860	−	0.986977i	$-0.551427\pi$
$887$	−9.58166	−0.321721	−0.160860	+	0.986977i	$0.551427\pi$
$888$	0	0
$889$	15.2474	0.511383
$890$	0	0
$891$	0	0
$892$	0	0
$893$	− 1.76416i	− 0.0590353i
$894$	0	0
$895$	1.55051i	0.0518278i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	34.2911	1.14367
$900$	0	0
$901$	−0.202041	−0.00673096
$902$	0	0
$903$	0	0
$904$	0	0
$905$	− 7.84961i	− 0.260930i
$906$	0	0
$907$	2.05561i	0.0682555i	0.999417	+	0.0341278i	$0.0108653\pi$
$907$	2.05561i	0.0682555i	−0.999417	+	0.0341278i	$0.989135\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	0.778539	0.0257942	0.0128971	−	0.999917i	$-0.495895\pi$
$911$	0.778539	0.0257942	0.0128971	+	0.999917i	$0.495895\pi$
$912$	0	0
$913$	−49.8434	−1.64957
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 16.0492i	− 0.529990i
$918$	0	0
$919$	− 12.7526i	− 0.420668i	−0.977630	−	0.210334i	$-0.932545\pi$
$919$	− 12.7526i	− 0.420668i	0.977630	−	0.210334i	$-0.0674551\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	0	0
$924$	0	0
$925$	−34.2929	−1.12754
$926$	0	0
$927$	0	0
$928$	0	0
$929$	45.0012i	1.47644i	0.674559	+	0.738221i	$0.264334\pi$
$929$	45.0012i	1.47644i	−0.674559	+	0.738221i	$0.735666\pi$
$930$	0	0
$931$	0.449490i	0.0147314i
$932$	0	0
$933$	0	0
$934$	0	0
$935$	0.492810	0.0161166
$936$	0	0
$937$	15.3485	0.501413	0.250706	−	0.968063i	$-0.419337\pi$
$937$	15.3485	0.501413	0.250706	+	0.968063i	$0.419337\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	25.7737i	0.840198i	0.907478	+	0.420099i	$0.138005\pi$
$941$	25.7737i	0.840198i	−0.907478	+	0.420099i	$0.861995\pi$
$942$	0	0
$943$	16.8990i	0.550306i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−40.2979	−1.30950	−0.654752	−	0.755843i	$-0.727227\pi$
$947$	−40.2979	−1.30950	−0.654752	+	0.755843i	$0.727227\pi$
$948$	0	0
$949$	−7.59592	−0.246574
$950$	0	0
$951$	0	0
$952$	0	0
$953$	− 53.1687i	− 1.72230i	−0.508349	−	0.861151i	$-0.669744\pi$
$953$	− 53.1687i	− 1.72230i	0.508349	−	0.861151i	$-0.330256\pi$
$954$	0	0
$955$	− 5.10102i	− 0.165065i
$956$	0	0
$957$	0	0
$958$	0	0
$959$	8.97809	0.289918
$960$	0	0
$961$	11.2020	0.361356
$962$	0	0
$963$	0	0
$964$	0	0
$965$	3.49621i	0.112547i
$966$	0	0
$967$	4.89898i	0.157541i	0.996893	+	0.0787703i	$0.0250994\pi$
$967$	4.89898i	0.157541i	−0.996893	+	0.0787703i	$0.974901\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	17.7170	0.568565	0.284283	−	0.958741i	$-0.408245\pi$
$971$	17.7170	0.568565	0.284283	+	0.958741i	$0.408245\pi$
$972$	0	0
$973$	12.2474	0.392635
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 32.4483i	− 1.03811i	−0.854740	−	0.519056i	$-0.826284\pi$
$977$	− 32.4483i	− 1.03811i	0.854740	−	0.519056i	$-0.173716\pi$
$978$	0	0
$979$	68.9898i	2.20492i
$980$	0	0
$981$	0	0
$982$	0	0
$983$	16.8598	0.537744	0.268872	−	0.963176i	$-0.413349\pi$
$983$	16.8598	0.537744	0.268872	+	0.963176i	$0.413349\pi$
$984$	0	0
$985$	−0.247449	−0.00788437
$986$	0	0
$987$	0	0
$988$	0	0
$989$	− 29.2699i	− 0.930728i
$990$	0	0
$991$	19.4495i	0.617833i	0.951089	+	0.308917i	$0.0999664\pi$
$991$	19.4495i	0.617833i	−0.951089	+	0.308917i	$0.900034\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	3.17837	0.100761
$996$	0	0
$997$	7.75255	0.245526	0.122763	−	0.992436i	$-0.460825\pi$
$997$	7.75255	0.245526	0.122763	+	0.992436i	$0.460825\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6048.2.h.d.2591.4	yes	8
3.2	odd	2	inner	6048.2.h.d.2591.6	yes	8
4.3	odd	2	inner	6048.2.h.d.2591.3	✓	8
12.11	even	2	inner	6048.2.h.d.2591.5	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6048.2.h.d.2591.3	✓	8	4.3	odd	2	inner
6048.2.h.d.2591.4	yes	8	1.1	even	1	trivial
6048.2.h.d.2591.5	yes	8	12.11	even	2	inner
6048.2.h.d.2591.6	yes	8	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.h (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-0.317837i q^{5} +1.00000i q^{7} +O(q^{10})\)	\(q-0.317837i q^{5} +1.00000i q^{7} -4.87832 q^{11} -2.44949 q^{13} -0.317837i q^{17} -0.449490i q^{19} +2.82843 q^{23} +4.89898 q^{25} +7.70674i q^{29} -4.44949i q^{31} +0.317837 q^{35} -7.00000 q^{37} +5.97469i q^{41} -10.3485i q^{43} +3.92480 q^{47} -1.00000 q^{49} -0.635674i q^{53} +1.55051i q^{55} -11.4887 q^{59} +9.34847 q^{61} +0.778539i q^{65} +8.00000i q^{67} +3.10102 q^{73} -4.87832i q^{77} -8.55051i q^{79} +10.2173 q^{83} -0.101021 q^{85} -14.1421i q^{89} -2.44949i q^{91} -0.142865 q^{95} +12.2474 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q+O(q^{10}) \) 8 * q	\( 8 q - 56 q^{37} - 8 q^{49} + 16 q^{61} + 64 q^{73} - 40 q^{85}+O(q^{100}) \) 8 * q - 56 * q^37 - 8 * q^49 + 16 * q^61 + 64 * q^73 - 40 * q^85

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