LMFDB - Embedded newform 6048.2.c.b.3025.2

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:	$2$
Coefficient field:	$\Q(i)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 1 $ x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1512)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			3025.2
Root			$1.00000i$ of defining polynomial
Character	$\chi$	$=$	6048.3025
Dual form			6048.2.c.b.3025.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.00000i q^{5} +1.00000 q^{7} +O(q^{10})$	$q+2.00000i q^{5} +1.00000 q^{7} +5.00000i q^{13} +1.00000 q^{17} -4.00000i q^{19} +5.00000 q^{23} +1.00000 q^{25} -9.00000i q^{29} +7.00000 q^{31} +2.00000i q^{35} +2.00000i q^{37} +2.00000 q^{41} -5.00000i q^{43} +1.00000 q^{49} -9.00000i q^{53} -1.00000i q^{59} +6.00000i q^{61} -10.0000 q^{65} +9.00000i q^{67} +15.0000 q^{71} +14.0000 q^{79} -4.00000i q^{83} +2.00000i q^{85} -13.0000 q^{89} +5.00000i q^{91} +8.00000 q^{95} -14.0000 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{7}+O(q^{10}) $ 2 * q + 2 * q^7	$ 2 q + 2 q^{7} + 2 q^{17} + 10 q^{23} + 2 q^{25} + 14 q^{31} + 4 q^{41} + 2 q^{49} - 20 q^{65} + 30 q^{71} + 28 q^{79} - 26 q^{89} + 16 q^{95} - 28 q^{97}+O(q^{100}) $ 2 * q + 2 * q^7 + 2 * q^17 + 10 * q^23 + 2 * q^25 + 14 * q^31 + 4 * q^41 + 2 * q^49 - 20 * q^65 + 30 * q^71 + 28 * q^79 - 26 * q^89 + 16 * q^95 - 28 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/6048\mathbb{Z}\right)^\times$.

$n$	$2593$	$3781$	$3809$	$4159$
$\chi(n)$	$1$	$-1$	$1$	$1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	0	0
$3$	0	0
$4$	0	0
$5$	2.00000i	0.894427i	0.894427	+	0.447214i	$0.147584\pi$
$5$	2.00000i	0.894427i	−0.894427	+	0.447214i	$0.852416\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0	0
$9$	0	0
$10$	0	0
$11$	0	0	1.00000			$0$
$11$	0	0	−1.00000			$\pi$
$12$	0	0
$13$	5.00000i	1.38675i	0.720577	+	0.693375i	$0.243877\pi$
$13$	5.00000i	1.38675i	−0.720577	+	0.693375i	$0.756123\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	1.00000	0.242536	0.121268	−	0.992620i	$-0.461304\pi$
$17$	1.00000	0.242536	0.121268	+	0.992620i	$0.461304\pi$
$18$	0	0
$19$	− 4.00000i	− 0.917663i	−0.888523	−	0.458831i	$-0.848268\pi$
$19$	− 4.00000i	− 0.917663i	0.888523	−	0.458831i	$-0.151732\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	5.00000	1.04257	0.521286	−	0.853382i	$-0.325452\pi$
$23$	5.00000	1.04257	0.521286	+	0.853382i	$0.325452\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	− 9.00000i	− 1.67126i	−0.549294	−	0.835629i	$-0.685103\pi$
$29$	− 9.00000i	− 1.67126i	0.549294	−	0.835629i	$-0.314897\pi$
$30$	0	0
$31$	7.00000	1.25724	0.628619	−	0.777714i	$-0.283621\pi$
$31$	7.00000	1.25724	0.628619	+	0.777714i	$0.283621\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	2.00000i	0.338062i
$36$	0	0
$37$	2.00000i	0.328798i	0.986394	+	0.164399i	$0.0525685\pi$
$37$	2.00000i	0.328798i	−0.986394	+	0.164399i	$0.947432\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	2.00000	0.312348	0.156174	−	0.987730i	$-0.450084\pi$
$41$	2.00000	0.312348	0.156174	+	0.987730i	$0.450084\pi$
$42$	0	0
$43$	− 5.00000i	− 0.762493i	−0.924473	−	0.381246i	$-0.875495\pi$
$43$	− 5.00000i	− 0.762493i	0.924473	−	0.381246i	$-0.124505\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	0	0		−	1.00000i	$-0.5\pi$
$47$	0	0			1.00000i	$0.5\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	0	0
$52$	0	0
$53$	− 9.00000i	− 1.23625i	−0.786082	−	0.618123i	$-0.787894\pi$
$53$	− 9.00000i	− 1.23625i	0.786082	−	0.618123i	$-0.212106\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	0	0
$58$	0	0
$59$	− 1.00000i	− 0.130189i	−0.997879	−	0.0650945i	$-0.979265\pi$
$59$	− 1.00000i	− 0.130189i	0.997879	−	0.0650945i	$-0.0207349\pi$
$60$	0	0
$61$	6.00000i	0.768221i	0.923287	+	0.384111i	$0.125492\pi$
$61$	6.00000i	0.768221i	−0.923287	+	0.384111i	$0.874508\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−10.0000	−1.24035
$66$	0	0
$67$	9.00000i	1.09952i	0.835321	+	0.549762i	$0.185282\pi$
$67$	9.00000i	1.09952i	−0.835321	+	0.549762i	$0.814718\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	15.0000	1.78017	0.890086	−	0.455792i	$-0.150644\pi$
$71$	15.0000	1.78017	0.890086	+	0.455792i	$0.150644\pi$
$72$	0	0
$73$	0	0		−	1.00000i	$-0.5\pi$
$73$	0	0			1.00000i	$0.5\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	0	0
$78$	0	0
$79$	14.0000	1.57512	0.787562	−	0.616236i	$-0.211343\pi$
$79$	14.0000	1.57512	0.787562	+	0.616236i	$0.211343\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	− 4.00000i	− 0.439057i	−0.975606	−	0.219529i	$-0.929548\pi$
$83$	− 4.00000i	− 0.439057i	0.975606	−	0.219529i	$-0.0704519\pi$
$84$	0	0
$85$	2.00000i	0.216930i
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−13.0000	−1.37800	−0.688999	−	0.724763i	$-0.741949\pi$
$89$	−13.0000	−1.37800	−0.688999	+	0.724763i	$0.741949\pi$
$90$	0	0
$91$	5.00000i	0.524142i
$92$	0	0
$93$	0	0
$94$	0	0
$95$	8.00000	0.820783
$96$	0	0
$97$	−14.0000	−1.42148	−0.710742	−	0.703452i	$-0.751641\pi$
$97$	−14.0000	−1.42148	−0.710742	+	0.703452i	$0.751641\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	8.00000i	0.796030i	0.917379	+	0.398015i	$0.130301\pi$
$101$	8.00000i	0.796030i	−0.917379	+	0.398015i	$0.869699\pi$
$102$	0	0
$103$	3.00000	0.295599	0.147799	−	0.989017i	$-0.452781\pi$
$103$	3.00000	0.295599	0.147799	+	0.989017i	$0.452781\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	0	0	1.00000			$0$
$107$	0	0	−1.00000			$\pi$
$108$	0	0
$109$	16.0000i	1.53252i	0.642529	+	0.766261i	$0.277885\pi$
$109$	16.0000i	1.53252i	−0.642529	+	0.766261i	$0.722115\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	6.00000	0.564433	0.282216	−	0.959351i	$-0.408930\pi$
$113$	6.00000	0.564433	0.282216	+	0.959351i	$0.408930\pi$
$114$	0	0
$115$	10.0000i	0.932505i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	1.00000	0.0916698
$120$	0	0
$121$	11.0000	1.00000
$122$	0	0
$123$	0	0
$124$	0	0
$125$	12.0000i	1.07331i
$126$	0	0
$127$	8.00000	0.709885	0.354943	−	0.934888i	$-0.384500\pi$
$127$	8.00000	0.709885	0.354943	+	0.934888i	$0.384500\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	− 13.0000i	− 1.13582i	−0.823092	−	0.567908i	$-0.807753\pi$
$131$	− 13.0000i	− 1.13582i	0.823092	−	0.567908i	$-0.192247\pi$
$132$	0	0
$133$	− 4.00000i	− 0.346844i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−12.0000	−1.02523	−0.512615	−	0.858619i	$-0.671323\pi$
$137$	−12.0000	−1.02523	−0.512615	+	0.858619i	$0.671323\pi$
$138$	0	0
$139$	20.0000i	1.69638i	0.529694	+	0.848189i	$0.322307\pi$
$139$	20.0000i	1.69638i	−0.529694	+	0.848189i	$0.677693\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	0	0
$144$	0	0
$145$	18.0000	1.49482
$146$	0	0
$147$	0	0
$148$	0	0
$149$	9.00000i	0.737309i	0.929567	+	0.368654i	$0.120181\pi$
$149$	9.00000i	0.737309i	−0.929567	+	0.368654i	$0.879819\pi$
$150$	0	0
$151$	−12.0000	−0.976546	−0.488273	−	0.872691i	$-0.662373\pi$
$151$	−12.0000	−0.976546	−0.488273	+	0.872691i	$0.662373\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	14.0000i	1.12451i
$156$	0	0
$157$	− 13.0000i	− 1.03751i	−0.854922	−	0.518756i	$-0.826395\pi$
$157$	− 13.0000i	− 1.03751i	0.854922	−	0.518756i	$-0.173605\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	5.00000	0.394055
$162$	0	0
$163$	− 11.0000i	− 0.861586i	−0.902451	−	0.430793i	$-0.858234\pi$
$163$	− 11.0000i	− 0.861586i	0.902451	−	0.430793i	$-0.141766\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	4.00000	0.309529	0.154765	−	0.987951i	$-0.450538\pi$
$167$	4.00000	0.309529	0.154765	+	0.987951i	$0.450538\pi$
$168$	0	0
$169$	−12.0000	−0.923077
$170$	0	0
$171$	0	0
$172$	0	0
$173$	6.00000i	0.456172i	0.973641	+	0.228086i	$0.0732467\pi$
$173$	6.00000i	0.456172i	−0.973641	+	0.228086i	$0.926753\pi$
$174$	0	0
$175$	1.00000	0.0755929
$176$	0	0
$177$	0	0
$178$	0	0
$179$	− 10.0000i	− 0.747435i	−0.927543	−	0.373718i	$-0.878083\pi$
$179$	− 10.0000i	− 0.747435i	0.927543	−	0.373718i	$-0.121917\pi$
$180$	0	0
$181$	19.0000i	1.41226i	0.708083	+	0.706129i	$0.249560\pi$
$181$	19.0000i	1.41226i	−0.708083	+	0.706129i	$0.750440\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−4.00000	−0.294086
$186$	0	0
$187$	0	0
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−4.00000	−0.289430	−0.144715	−	0.989473i	$-0.546227\pi$
$191$	−4.00000	−0.289430	−0.144715	+	0.989473i	$0.546227\pi$
$192$	0	0
$193$	9.00000	0.647834	0.323917	−	0.946085i	$-0.395000\pi$
$193$	9.00000	0.647834	0.323917	+	0.946085i	$0.395000\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 14.0000i	− 0.997459i	−0.866758	−	0.498729i	$-0.833800\pi$
$197$	− 14.0000i	− 0.997459i	0.866758	−	0.498729i	$-0.166200\pi$
$198$	0	0
$199$	−13.0000	−0.921546	−0.460773	−	0.887518i	$-0.652428\pi$
$199$	−13.0000	−0.921546	−0.460773	+	0.887518i	$0.652428\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	− 9.00000i	− 0.631676i
$204$	0	0
$205$	4.00000i	0.279372i
$206$	0	0
$207$	0	0
$208$	0	0
$209$	0	0
$210$	0	0
$211$	1.00000i	0.0688428i	0.999407	+	0.0344214i	$0.0109588\pi$
$211$	1.00000i	0.0688428i	−0.999407	+	0.0344214i	$0.989041\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	10.0000	0.681994
$216$	0	0
$217$	7.00000	0.475191
$218$	0	0
$219$	0	0
$220$	0	0
$221$	5.00000i	0.336336i
$222$	0	0
$223$	16.0000	1.07144	0.535720	−	0.844396i	$-0.320040\pi$
$223$	16.0000	1.07144	0.535720	+	0.844396i	$0.320040\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	25.0000i	1.65931i	0.558278	+	0.829654i	$0.311462\pi$
$227$	25.0000i	1.65931i	−0.558278	+	0.829654i	$0.688538\pi$
$228$	0	0
$229$	10.0000i	0.660819i	0.943838	+	0.330409i	$0.107187\pi$
$229$	10.0000i	0.660819i	−0.943838	+	0.330409i	$0.892813\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−10.0000	−0.655122	−0.327561	−	0.944830i	$-0.606227\pi$
$233$	−10.0000	−0.655122	−0.327561	+	0.944830i	$0.606227\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−24.0000	−1.55243	−0.776215	−	0.630468i	$-0.782863\pi$
$239$	−24.0000	−1.55243	−0.776215	+	0.630468i	$0.782863\pi$
$240$	0	0
$241$	26.0000	1.67481	0.837404	−	0.546585i	$-0.184072\pi$
$241$	26.0000	1.67481	0.837404	+	0.546585i	$0.184072\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	2.00000i	0.127775i
$246$	0	0
$247$	20.0000	1.27257
$248$	0	0
$249$	0	0
$250$	0	0
$251$	− 12.0000i	− 0.757433i	−0.925513	−	0.378717i	$-0.876365\pi$
$251$	− 12.0000i	− 0.757433i	0.925513	−	0.378717i	$-0.123635\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−6.00000	−0.374270	−0.187135	−	0.982334i	$-0.559920\pi$
$257$	−6.00000	−0.374270	−0.187135	+	0.982334i	$0.559920\pi$
$258$	0	0
$259$	2.00000i	0.124274i
$260$	0	0
$261$	0	0
$262$	0	0
$263$	31.0000	1.91154	0.955771	−	0.294112i	$-0.0950239\pi$
$263$	31.0000	1.91154	0.955771	+	0.294112i	$0.0950239\pi$
$264$	0	0
$265$	18.0000	1.10573
$266$	0	0
$267$	0	0
$268$	0	0
$269$	0	0	1.00000			$0$
$269$	0	0	−1.00000			$\pi$
$270$	0	0
$271$	15.0000	0.911185	0.455593	−	0.890188i	$-0.349427\pi$
$271$	15.0000	0.911185	0.455593	+	0.890188i	$0.349427\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	0	0
$276$	0	0
$277$	− 26.0000i	− 1.56219i	−0.624413	−	0.781094i	$-0.714662\pi$
$277$	− 26.0000i	− 1.56219i	0.624413	−	0.781094i	$-0.285338\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−12.0000	−0.715860	−0.357930	−	0.933748i	$-0.616517\pi$
$281$	−12.0000	−0.715860	−0.357930	+	0.933748i	$0.616517\pi$
$282$	0	0
$283$	16.0000i	0.951101i	0.879688	+	0.475551i	$0.157751\pi$
$283$	16.0000i	0.951101i	−0.879688	+	0.475551i	$0.842249\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	2.00000	0.118056
$288$	0	0
$289$	−16.0000	−0.941176
$290$	0	0
$291$	0	0
$292$	0	0
$293$	− 12.0000i	− 0.701047i	−0.936554	−	0.350524i	$-0.886004\pi$
$293$	− 12.0000i	− 0.701047i	0.936554	−	0.350524i	$-0.113996\pi$
$294$	0	0
$295$	2.00000	0.116445
$296$	0	0
$297$	0	0
$298$	0	0
$299$	25.0000i	1.44579i
$300$	0	0
$301$	− 5.00000i	− 0.288195i
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−12.0000	−0.687118
$306$	0	0
$307$	− 6.00000i	− 0.342438i	−0.985233	−	0.171219i	$-0.945229\pi$
$307$	− 6.00000i	− 0.342438i	0.985233	−	0.171219i	$-0.0547706\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−34.0000	−1.92796	−0.963982	−	0.265969i	$-0.914308\pi$
$311$	−34.0000	−1.92796	−0.963982	+	0.265969i	$0.914308\pi$
$312$	0	0
$313$	32.0000	1.80875	0.904373	−	0.426742i	$-0.140339\pi$
$313$	32.0000	1.80875	0.904373	+	0.426742i	$0.140339\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	14.0000i	0.786318i	0.919470	+	0.393159i	$0.128618\pi$
$317$	14.0000i	0.786318i	−0.919470	+	0.393159i	$0.871382\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	0	0
$322$	0	0
$323$	− 4.00000i	− 0.222566i
$324$	0	0
$325$	5.00000i	0.277350i
$326$	0	0
$327$	0	0
$328$	0	0
$329$	0	0
$330$	0	0
$331$	13.0000i	0.714545i	0.934000	+	0.357272i	$0.116293\pi$
$331$	13.0000i	0.714545i	−0.934000	+	0.357272i	$0.883707\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−18.0000	−0.983445
$336$	0	0
$337$	19.0000	1.03500	0.517498	−	0.855684i	$-0.326864\pi$
$337$	19.0000	1.03500	0.517498	+	0.855684i	$0.326864\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	0	0
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	0	0
$346$	0	0
$347$	16.0000i	0.858925i	0.903085	+	0.429463i	$0.141297\pi$
$347$	16.0000i	0.858925i	−0.903085	+	0.429463i	$0.858703\pi$
$348$	0	0
$349$	23.0000i	1.23116i	0.788074	+	0.615581i	$0.211079\pi$
$349$	23.0000i	1.23116i	−0.788074	+	0.615581i	$0.788921\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−33.0000	−1.75641	−0.878206	−	0.478282i	$-0.841260\pi$
$353$	−33.0000	−1.75641	−0.878206	+	0.478282i	$0.841260\pi$
$354$	0	0
$355$	30.0000i	1.59223i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−1.00000	−0.0527780	−0.0263890	−	0.999652i	$-0.508401\pi$
$359$	−1.00000	−0.0527780	−0.0263890	+	0.999652i	$0.508401\pi$
$360$	0	0
$361$	3.00000	0.157895
$362$	0	0
$363$	0	0
$364$	0	0
$365$	0	0
$366$	0	0
$367$	−21.0000	−1.09619	−0.548096	−	0.836416i	$-0.684647\pi$
$367$	−21.0000	−1.09619	−0.548096	+	0.836416i	$0.684647\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	− 9.00000i	− 0.467257i
$372$	0	0
$373$	22.0000i	1.13912i	0.821951	+	0.569558i	$0.192886\pi$
$373$	22.0000i	1.13912i	−0.821951	+	0.569558i	$0.807114\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	45.0000	2.31762
$378$	0	0
$379$	− 16.0000i	− 0.821865i	−0.911666	−	0.410932i	$-0.865203\pi$
$379$	− 16.0000i	− 0.821865i	0.911666	−	0.410932i	$-0.134797\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	24.0000	1.22634	0.613171	−	0.789950i	$-0.289894\pi$
$383$	24.0000	1.22634	0.613171	+	0.789950i	$0.289894\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	0	0
$388$	0	0
$389$	10.0000i	0.507020i	0.967333	+	0.253510i	$0.0815851\pi$
$389$	10.0000i	0.507020i	−0.967333	+	0.253510i	$0.918415\pi$
$390$	0	0
$391$	5.00000	0.252861
$392$	0	0
$393$	0	0
$394$	0	0
$395$	28.0000i	1.40883i
$396$	0	0
$397$	10.0000i	0.501886i	0.968002	+	0.250943i	$0.0807406\pi$
$397$	10.0000i	0.501886i	−0.968002	+	0.250943i	$0.919259\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	24.0000	1.19850	0.599251	−	0.800561i	$-0.295465\pi$
$401$	24.0000	1.19850	0.599251	+	0.800561i	$0.295465\pi$
$402$	0	0
$403$	35.0000i	1.74347i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	0	0
$408$	0	0
$409$	−32.0000	−1.58230	−0.791149	−	0.611623i	$-0.790517\pi$
$409$	−32.0000	−1.58230	−0.791149	+	0.611623i	$0.790517\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	− 1.00000i	− 0.0492068i
$414$	0	0
$415$	8.00000	0.392705
$416$	0	0
$417$	0	0
$418$	0	0
$419$	− 15.0000i	− 0.732798i	−0.930458	−	0.366399i	$-0.880591\pi$
$419$	− 15.0000i	− 0.732798i	0.930458	−	0.366399i	$-0.119409\pi$
$420$	0	0
$421$	4.00000i	0.194948i	0.995238	+	0.0974740i	$0.0310763\pi$
$421$	4.00000i	0.194948i	−0.995238	+	0.0974740i	$0.968924\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	1.00000	0.0485071
$426$	0	0
$427$	6.00000i	0.290360i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	12.0000	0.578020	0.289010	−	0.957326i	$-0.406674\pi$
$431$	12.0000	0.578020	0.289010	+	0.957326i	$0.406674\pi$
$432$	0	0
$433$	8.00000	0.384455	0.192228	−	0.981350i	$-0.438429\pi$
$433$	8.00000	0.384455	0.192228	+	0.981350i	$0.438429\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 20.0000i	− 0.956730i
$438$	0	0
$439$	−9.00000	−0.429547	−0.214773	−	0.976664i	$-0.568901\pi$
$439$	−9.00000	−0.429547	−0.214773	+	0.976664i	$0.568901\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	6.00000i	0.285069i	0.989790	+	0.142534i	$0.0455251\pi$
$443$	6.00000i	0.285069i	−0.989790	+	0.142534i	$0.954475\pi$
$444$	0	0
$445$	− 26.0000i	− 1.23252i
$446$	0	0
$447$	0	0
$448$	0	0
$449$	30.0000	1.41579	0.707894	−	0.706319i	$-0.249646\pi$
$449$	30.0000	1.41579	0.707894	+	0.706319i	$0.249646\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−10.0000	−0.468807
$456$	0	0
$457$	5.00000	0.233890	0.116945	−	0.993138i	$-0.462690\pi$
$457$	5.00000	0.233890	0.116945	+	0.993138i	$0.462690\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	− 12.0000i	− 0.558896i	−0.960161	−	0.279448i	$-0.909849\pi$
$461$	− 12.0000i	− 0.558896i	0.960161	−	0.279448i	$-0.0901514\pi$
$462$	0	0
$463$	−16.0000	−0.743583	−0.371792	−	0.928316i	$-0.621256\pi$
$463$	−16.0000	−0.743583	−0.371792	+	0.928316i	$0.621256\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 32.0000i	− 1.48078i	−0.672176	−	0.740392i	$-0.734640\pi$
$467$	− 32.0000i	− 1.48078i	0.672176	−	0.740392i	$-0.265360\pi$
$468$	0	0
$469$	9.00000i	0.415581i
$470$	0	0
$471$	0	0
$472$	0	0
$473$	0	0
$474$	0	0
$475$	− 4.00000i	− 0.183533i
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−26.0000	−1.18797	−0.593985	−	0.804476i	$-0.702446\pi$
$479$	−26.0000	−1.18797	−0.593985	+	0.804476i	$0.702446\pi$
$480$	0	0
$481$	−10.0000	−0.455961
$482$	0	0
$483$	0	0
$484$	0	0
$485$	− 28.0000i	− 1.27141i
$486$	0	0
$487$	16.0000	0.725029	0.362515	−	0.931978i	$-0.381918\pi$
$487$	16.0000	0.725029	0.362515	+	0.931978i	$0.381918\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	− 4.00000i	− 0.180517i	−0.995918	−	0.0902587i	$-0.971231\pi$
$491$	− 4.00000i	− 0.180517i	0.995918	−	0.0902587i	$-0.0287694\pi$
$492$	0	0
$493$	− 9.00000i	− 0.405340i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	15.0000	0.672842
$498$	0	0
$499$	24.0000i	1.07439i	0.843459	+	0.537194i	$0.180516\pi$
$499$	24.0000i	1.07439i	−0.843459	+	0.537194i	$0.819484\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−6.00000	−0.267527	−0.133763	−	0.991013i	$-0.542706\pi$
$503$	−6.00000	−0.267527	−0.133763	+	0.991013i	$0.542706\pi$
$504$	0	0
$505$	−16.0000	−0.711991
$506$	0	0
$507$	0	0
$508$	0	0
$509$	18.0000i	0.797836i	0.916987	+	0.398918i	$0.130614\pi$
$509$	18.0000i	0.797836i	−0.916987	+	0.398918i	$0.869386\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	0	0
$514$	0	0
$515$	6.00000i	0.264392i
$516$	0	0
$517$	0	0
$518$	0	0
$519$	0	0
$520$	0	0
$521$	39.0000	1.70862	0.854311	−	0.519763i	$-0.173980\pi$
$521$	39.0000	1.70862	0.854311	+	0.519763i	$0.173980\pi$
$522$	0	0
$523$	36.0000i	1.57417i	0.616844	+	0.787085i	$0.288411\pi$
$523$	36.0000i	1.57417i	−0.616844	+	0.787085i	$0.711589\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	7.00000	0.304925
$528$	0	0
$529$	2.00000	0.0869565
$530$	0	0
$531$	0	0
$532$	0	0
$533$	10.0000i	0.433148i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	0	0
$538$	0	0
$539$	0	0
$540$	0	0
$541$	38.0000i	1.63375i	0.576816	+	0.816874i	$0.304295\pi$
$541$	38.0000i	1.63375i	−0.576816	+	0.816874i	$0.695705\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−32.0000	−1.37073
$546$	0	0
$547$	− 28.0000i	− 1.19719i	−0.801050	−	0.598597i	$-0.795725\pi$
$547$	− 28.0000i	− 1.19719i	0.801050	−	0.598597i	$-0.204275\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−36.0000	−1.53365
$552$	0	0
$553$	14.0000	0.595341
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 33.0000i	− 1.39825i	−0.714997	−	0.699127i	$-0.753572\pi$
$557$	− 33.0000i	− 1.39825i	0.714997	−	0.699127i	$-0.246428\pi$
$558$	0	0
$559$	25.0000	1.05739
$560$	0	0
$561$	0	0
$562$	0	0
$563$	− 15.0000i	− 0.632175i	−0.948730	−	0.316087i	$-0.897631\pi$
$563$	− 15.0000i	− 0.632175i	0.948730	−	0.316087i	$-0.102369\pi$
$564$	0	0
$565$	12.0000i	0.504844i
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−24.0000	−1.00613	−0.503066	−	0.864248i	$-0.667795\pi$
$569$	−24.0000	−1.00613	−0.503066	+	0.864248i	$0.667795\pi$
$570$	0	0
$571$	− 19.0000i	− 0.795125i	−0.917575	−	0.397563i	$-0.869856\pi$
$571$	− 19.0000i	− 0.795125i	0.917575	−	0.397563i	$-0.130144\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	5.00000	0.208514
$576$	0	0
$577$	30.0000	1.24892	0.624458	−	0.781058i	$-0.285320\pi$
$577$	30.0000	1.24892	0.624458	+	0.781058i	$0.285320\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	− 4.00000i	− 0.165948i
$582$	0	0
$583$	0	0
$584$	0	0
$585$	0	0
$586$	0	0
$587$	23.0000i	0.949312i	0.880172	+	0.474656i	$0.157427\pi$
$587$	23.0000i	0.949312i	−0.880172	+	0.474656i	$0.842573\pi$
$588$	0	0
$589$	− 28.0000i	− 1.15372i
$590$	0	0
$591$	0	0
$592$	0	0
$593$	6.00000	0.246390	0.123195	−	0.992382i	$-0.460686\pi$
$593$	6.00000	0.246390	0.123195	+	0.992382i	$0.460686\pi$
$594$	0	0
$595$	2.00000i	0.0819920i
$596$	0	0
$597$	0	0
$598$	0	0
$599$	35.0000	1.43006	0.715031	−	0.699093i	$-0.246413\pi$
$599$	35.0000	1.43006	0.715031	+	0.699093i	$0.246413\pi$
$600$	0	0
$601$	38.0000	1.55005	0.775026	−	0.631929i	$-0.217737\pi$
$601$	38.0000	1.55005	0.775026	+	0.631929i	$0.217737\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	22.0000i	0.894427i
$606$	0	0
$607$	−1.00000	−0.0405887	−0.0202944	−	0.999794i	$-0.506460\pi$
$607$	−1.00000	−0.0405887	−0.0202944	+	0.999794i	$0.506460\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	0	0
$612$	0	0
$613$	16.0000i	0.646234i	0.946359	+	0.323117i	$0.104731\pi$
$613$	16.0000i	0.646234i	−0.946359	+	0.323117i	$0.895269\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	46.0000	1.85189	0.925945	−	0.377658i	$-0.123271\pi$
$617$	46.0000	1.85189	0.925945	+	0.377658i	$0.123271\pi$
$618$	0	0
$619$	0	0	1.00000			$0$
$619$	0	0	−1.00000			$\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−13.0000	−0.520834
$624$	0	0
$625$	−19.0000	−0.760000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	2.00000i	0.0797452i
$630$	0	0
$631$	−34.0000	−1.35352	−0.676759	−	0.736204i	$-0.736616\pi$
$631$	−34.0000	−1.35352	−0.676759	+	0.736204i	$0.736616\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	16.0000i	0.634941i
$636$	0	0
$637$	5.00000i	0.198107i
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−26.0000	−1.02694	−0.513469	−	0.858108i	$-0.671640\pi$
$641$	−26.0000	−1.02694	−0.513469	+	0.858108i	$0.671640\pi$
$642$	0	0
$643$	− 14.0000i	− 0.552106i	−0.961142	−	0.276053i	$-0.910973\pi$
$643$	− 14.0000i	− 0.552106i	0.961142	−	0.276053i	$-0.0890266\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−18.0000	−0.707653	−0.353827	−	0.935311i	$-0.615120\pi$
$647$	−18.0000	−0.707653	−0.353827	+	0.935311i	$0.615120\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	0	0
$652$	0	0
$653$	21.0000i	0.821794i	0.911682	+	0.410897i	$0.134784\pi$
$653$	21.0000i	0.821794i	−0.911682	+	0.410897i	$0.865216\pi$
$654$	0	0
$655$	26.0000	1.01590
$656$	0	0
$657$	0	0
$658$	0	0
$659$	− 36.0000i	− 1.40236i	−0.712984	−	0.701180i	$-0.752657\pi$
$659$	− 36.0000i	− 1.40236i	0.712984	−	0.701180i	$-0.247343\pi$
$660$	0	0
$661$	− 22.0000i	− 0.855701i	−0.903850	−	0.427850i	$-0.859271\pi$
$661$	− 22.0000i	− 0.855701i	0.903850	−	0.427850i	$-0.140729\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	8.00000	0.310227
$666$	0	0
$667$	− 45.0000i	− 1.74241i
$668$	0	0
$669$	0	0
$670$	0	0
$671$	0	0
$672$	0	0
$673$	−37.0000	−1.42625	−0.713123	−	0.701039i	$-0.752720\pi$
$673$	−37.0000	−1.42625	−0.713123	+	0.701039i	$0.752720\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	12.0000i	0.461197i	0.973049	+	0.230599i	$0.0740685\pi$
$677$	12.0000i	0.461197i	−0.973049	+	0.230599i	$0.925932\pi$
$678$	0	0
$679$	−14.0000	−0.537271
$680$	0	0
$681$	0	0
$682$	0	0
$683$	− 46.0000i	− 1.76014i	−0.474843	−	0.880071i	$-0.657495\pi$
$683$	− 46.0000i	− 1.76014i	0.474843	−	0.880071i	$-0.342505\pi$
$684$	0	0
$685$	− 24.0000i	− 0.916993i
$686$	0	0
$687$	0	0
$688$	0	0
$689$	45.0000	1.71436
$690$	0	0
$691$	8.00000i	0.304334i	0.988355	+	0.152167i	$0.0486252\pi$
$691$	8.00000i	0.304334i	−0.988355	+	0.152167i	$0.951375\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−40.0000	−1.51729
$696$	0	0
$697$	2.00000	0.0757554
$698$	0	0
$699$	0	0
$700$	0	0
$701$	− 22.0000i	− 0.830929i	−0.909610	−	0.415464i	$-0.863619\pi$
$701$	− 22.0000i	− 0.830929i	0.909610	−	0.415464i	$-0.136381\pi$
$702$	0	0
$703$	8.00000	0.301726
$704$	0	0
$705$	0	0
$706$	0	0
$707$	8.00000i	0.300871i
$708$	0	0
$709$	16.0000i	0.600893i	0.953799	+	0.300446i	$0.0971356\pi$
$709$	16.0000i	0.600893i	−0.953799	+	0.300446i	$0.902864\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	35.0000	1.31076
$714$	0	0
$715$	0	0
$716$	0	0
$717$	0	0
$718$	0	0
$719$	14.0000	0.522112	0.261056	−	0.965324i	$-0.415929\pi$
$719$	14.0000	0.522112	0.261056	+	0.965324i	$0.415929\pi$
$720$	0	0
$721$	3.00000	0.111726
$722$	0	0
$723$	0	0
$724$	0	0
$725$	− 9.00000i	− 0.334252i
$726$	0	0
$727$	33.0000	1.22390	0.611951	−	0.790896i	$-0.290385\pi$
$727$	33.0000	1.22390	0.611951	+	0.790896i	$0.290385\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	− 5.00000i	− 0.184932i
$732$	0	0
$733$	− 49.0000i	− 1.80986i	−0.425564	−	0.904928i	$-0.639924\pi$
$733$	− 49.0000i	− 1.80986i	0.425564	−	0.904928i	$-0.360076\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	0	0
$738$	0	0
$739$	− 32.0000i	− 1.17714i	−0.808447	−	0.588570i	$-0.799691\pi$
$739$	− 32.0000i	− 1.17714i	0.808447	−	0.588570i	$-0.200309\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−23.0000	−0.843788	−0.421894	−	0.906645i	$-0.638635\pi$
$743$	−23.0000	−0.843788	−0.421894	+	0.906645i	$0.638635\pi$
$744$	0	0
$745$	−18.0000	−0.659469
$746$	0	0
$747$	0	0
$748$	0	0
$749$	0	0
$750$	0	0
$751$	46.0000	1.67856	0.839282	−	0.543696i	$-0.182976\pi$
$751$	46.0000	1.67856	0.839282	+	0.543696i	$0.182976\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	− 24.0000i	− 0.873449i
$756$	0	0
$757$	− 24.0000i	− 0.872295i	−0.899875	−	0.436147i	$-0.856343\pi$
$757$	− 24.0000i	− 0.872295i	0.899875	−	0.436147i	$-0.143657\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−11.0000	−0.398750	−0.199375	−	0.979923i	$-0.563891\pi$
$761$	−11.0000	−0.398750	−0.199375	+	0.979923i	$0.563891\pi$
$762$	0	0
$763$	16.0000i	0.579239i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	5.00000	0.180540
$768$	0	0
$769$	−2.00000	−0.0721218	−0.0360609	−	0.999350i	$-0.511481\pi$
$769$	−2.00000	−0.0721218	−0.0360609	+	0.999350i	$0.511481\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	32.0000i	1.15096i	0.817816	+	0.575480i	$0.195185\pi$
$773$	32.0000i	1.15096i	−0.817816	+	0.575480i	$0.804815\pi$
$774$	0	0
$775$	7.00000	0.251447
$776$	0	0
$777$	0	0
$778$	0	0
$779$	− 8.00000i	− 0.286630i
$780$	0	0
$781$	0	0
$782$	0	0
$783$	0	0
$784$	0	0
$785$	26.0000	0.927980
$786$	0	0
$787$	− 48.0000i	− 1.71102i	−0.517790	−	0.855508i	$-0.673245\pi$
$787$	− 48.0000i	− 1.71102i	0.517790	−	0.855508i	$-0.326755\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	6.00000	0.213335
$792$	0	0
$793$	−30.0000	−1.06533
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 2.00000i	− 0.0708436i	−0.999372	−	0.0354218i	$-0.988723\pi$
$797$	− 2.00000i	− 0.0708436i	0.999372	−	0.0354218i	$-0.0112775\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	0	0
$802$	0	0
$803$	0	0
$804$	0	0
$805$	10.0000i	0.352454i
$806$	0	0
$807$	0	0
$808$	0	0
$809$	6.00000	0.210949	0.105474	−	0.994422i	$-0.466364\pi$
$809$	6.00000	0.210949	0.105474	+	0.994422i	$0.466364\pi$
$810$	0	0
$811$	− 14.0000i	− 0.491606i	−0.969320	−	0.245803i	$-0.920948\pi$
$811$	− 14.0000i	− 0.491606i	0.969320	−	0.245803i	$-0.0790517\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	22.0000	0.770626
$816$	0	0
$817$	−20.0000	−0.699711
$818$	0	0
$819$	0	0
$820$	0	0
$821$	35.0000i	1.22151i	0.791820	+	0.610754i	$0.209134\pi$
$821$	35.0000i	1.22151i	−0.791820	+	0.610754i	$0.790866\pi$
$822$	0	0
$823$	−44.0000	−1.53374	−0.766872	−	0.641800i	$-0.778188\pi$
$823$	−44.0000	−1.53374	−0.766872	+	0.641800i	$0.778188\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	12.0000i	0.417281i	0.977992	+	0.208640i	$0.0669038\pi$
$827$	12.0000i	0.417281i	−0.977992	+	0.208640i	$0.933096\pi$
$828$	0	0
$829$	34.0000i	1.18087i	0.807086	+	0.590434i	$0.201044\pi$
$829$	34.0000i	1.18087i	−0.807086	+	0.590434i	$0.798956\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	1.00000	0.0346479
$834$	0	0
$835$	8.00000i	0.276851i
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−44.0000	−1.51905	−0.759524	−	0.650479i	$-0.774568\pi$
$839$	−44.0000	−1.51905	−0.759524	+	0.650479i	$0.774568\pi$
$840$	0	0
$841$	−52.0000	−1.79310
$842$	0	0
$843$	0	0
$844$	0	0
$845$	− 24.0000i	− 0.825625i
$846$	0	0
$847$	11.0000	0.377964
$848$	0	0
$849$	0	0
$850$	0	0
$851$	10.0000i	0.342796i
$852$	0	0
$853$	9.00000i	0.308154i	0.988059	+	0.154077i	$0.0492404\pi$
$853$	9.00000i	0.308154i	−0.988059	+	0.154077i	$0.950760\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	9.00000	0.307434	0.153717	−	0.988115i	$-0.450876\pi$
$857$	9.00000	0.307434	0.153717	+	0.988115i	$0.450876\pi$
$858$	0	0
$859$	− 2.00000i	− 0.0682391i	−0.999418	−	0.0341196i	$-0.989137\pi$
$859$	− 2.00000i	− 0.0682391i	0.999418	−	0.0341196i	$-0.0108627\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	45.0000	1.53182	0.765909	−	0.642949i	$-0.222289\pi$
$863$	45.0000	1.53182	0.765909	+	0.642949i	$0.222289\pi$
$864$	0	0
$865$	−12.0000	−0.408012
$866$	0	0
$867$	0	0
$868$	0	0
$869$	0	0
$870$	0	0
$871$	−45.0000	−1.52477
$872$	0	0
$873$	0	0
$874$	0	0
$875$	12.0000i	0.405674i
$876$	0	0
$877$	34.0000i	1.14810i	0.818821	+	0.574049i	$0.194628\pi$
$877$	34.0000i	1.14810i	−0.818821	+	0.574049i	$0.805372\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	47.0000	1.58347	0.791735	−	0.610865i	$-0.209178\pi$
$881$	47.0000	1.58347	0.791735	+	0.610865i	$0.209178\pi$
$882$	0	0
$883$	− 29.0000i	− 0.975928i	−0.872864	−	0.487964i	$-0.837740\pi$
$883$	− 29.0000i	− 0.975928i	0.872864	−	0.487964i	$-0.162260\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	12.0000	0.402921	0.201460	−	0.979497i	$-0.435431\pi$
$887$	12.0000	0.402921	0.201460	+	0.979497i	$0.435431\pi$
$888$	0	0
$889$	8.00000	0.268311
$890$	0	0
$891$	0	0
$892$	0	0
$893$	0	0
$894$	0	0
$895$	20.0000	0.668526
$896$	0	0
$897$	0	0
$898$	0	0
$899$	− 63.0000i	− 2.10117i
$900$	0	0
$901$	− 9.00000i	− 0.299833i
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−38.0000	−1.26316
$906$	0	0
$907$	− 28.0000i	− 0.929725i	−0.885383	−	0.464862i	$-0.846104\pi$
$907$	− 28.0000i	− 0.929725i	0.885383	−	0.464862i	$-0.153896\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	0	0		−	1.00000i	$-0.5\pi$
$911$	0	0			1.00000i	$0.5\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 13.0000i	− 0.429298i
$918$	0	0
$919$	−18.0000	−0.593765	−0.296883	−	0.954914i	$-0.595947\pi$
$919$	−18.0000	−0.593765	−0.296883	+	0.954914i	$0.595947\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	75.0000i	2.46866i
$924$	0	0
$925$	2.00000i	0.0657596i
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−14.0000	−0.459325	−0.229663	−	0.973270i	$-0.573762\pi$
$929$	−14.0000	−0.459325	−0.229663	+	0.973270i	$0.573762\pi$
$930$	0	0
$931$	− 4.00000i	− 0.131095i
$932$	0	0
$933$	0	0
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−22.0000	−0.718709	−0.359354	−	0.933201i	$-0.617003\pi$
$937$	−22.0000	−0.718709	−0.359354	+	0.933201i	$0.617003\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	− 24.0000i	− 0.782378i	−0.920310	−	0.391189i	$-0.872064\pi$
$941$	− 24.0000i	− 0.782378i	0.920310	−	0.391189i	$-0.127936\pi$
$942$	0	0
$943$	10.0000	0.325645
$944$	0	0
$945$	0	0
$946$	0	0
$947$	12.0000i	0.389948i	0.980808	+	0.194974i	$0.0624622\pi$
$947$	12.0000i	0.389948i	−0.980808	+	0.194974i	$0.937538\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−36.0000	−1.16615	−0.583077	−	0.812417i	$-0.698151\pi$
$953$	−36.0000	−1.16615	−0.583077	+	0.812417i	$0.698151\pi$
$954$	0	0
$955$	− 8.00000i	− 0.258874i
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−12.0000	−0.387500
$960$	0	0
$961$	18.0000	0.580645
$962$	0	0
$963$	0	0
$964$	0	0
$965$	18.0000i	0.579441i
$966$	0	0
$967$	−50.0000	−1.60789	−0.803946	−	0.594703i	$-0.797270\pi$
$967$	−50.0000	−1.60789	−0.803946	+	0.594703i	$0.797270\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	33.0000i	1.05902i	0.848304	+	0.529510i	$0.177624\pi$
$971$	33.0000i	1.05902i	−0.848304	+	0.529510i	$0.822376\pi$
$972$	0	0
$973$	20.0000i	0.641171i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−56.0000	−1.79160	−0.895799	−	0.444459i	$-0.853396\pi$
$977$	−56.0000	−1.79160	−0.895799	+	0.444459i	$0.853396\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−62.0000	−1.97749	−0.988746	−	0.149601i	$-0.952201\pi$
$983$	−62.0000	−1.97749	−0.988746	+	0.149601i	$0.952201\pi$
$984$	0	0
$985$	28.0000	0.892154
$986$	0	0
$987$	0	0
$988$	0	0
$989$	− 25.0000i	− 0.794954i
$990$	0	0
$991$	−20.0000	−0.635321	−0.317660	−	0.948205i	$-0.602897\pi$
$991$	−20.0000	−0.635321	−0.317660	+	0.948205i	$0.602897\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	− 26.0000i	− 0.824255i
$996$	0	0
$997$	− 57.0000i	− 1.80521i	−0.430472	−	0.902604i	$-0.641653\pi$
$997$	− 57.0000i	− 1.80521i	0.430472	−	0.902604i	$-0.358347\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6048.2.c.b.3025.2		2
3.2	odd	2		6048.2.c.a.3025.1		2
4.3	odd	2		1512.2.c.a.757.1	✓	2
8.3	odd	2		1512.2.c.a.757.2	yes	2
8.5	even	2	inner	6048.2.c.b.3025.1		2
12.11	even	2		1512.2.c.b.757.2	yes	2
24.5	odd	2		6048.2.c.a.3025.2		2
24.11	even	2		1512.2.c.b.757.1	yes	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1512.2.c.a.757.1	✓	2	4.3	odd	2
1512.2.c.a.757.2	yes	2	8.3	odd	2
1512.2.c.b.757.1	yes	2	24.11	even	2
1512.2.c.b.757.2	yes	2	12.11	even	2
6048.2.c.a.3025.1		2	3.2	odd	2
6048.2.c.a.3025.2		2	24.5	odd	2
6048.2.c.b.3025.1		2	8.5	even	2	inner
6048.2.c.b.3025.2		2	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)

\(f(q)\)	\(=\)	\(q+2.00000i q^{5} +1.00000 q^{7} +O(q^{10})\)	\(q+2.00000i q^{5} +1.00000 q^{7} +5.00000i q^{13} +1.00000 q^{17} -4.00000i q^{19} +5.00000 q^{23} +1.00000 q^{25} -9.00000i q^{29} +7.00000 q^{31} +2.00000i q^{35} +2.00000i q^{37} +2.00000 q^{41} -5.00000i q^{43} +1.00000 q^{49} -9.00000i q^{53} -1.00000i q^{59} +6.00000i q^{61} -10.0000 q^{65} +9.00000i q^{67} +15.0000 q^{71} +14.0000 q^{79} -4.00000i q^{83} +2.00000i q^{85} -13.0000 q^{89} +5.00000i q^{91} +8.00000 q^{95} -14.0000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{7}+O(q^{10}) \) 2 * q + 2 * q^7	\( 2 q + 2 q^{7} + 2 q^{17} + 10 q^{23} + 2 q^{25} + 14 q^{31} + 4 q^{41} + 2 q^{49} - 20 q^{65} + 30 q^{71} + 28 q^{79} - 26 q^{89} + 16 q^{95} - 28 q^{97}+O(q^{100}) \) 2 * q + 2 * q^7 + 2 * q^17 + 10 * q^23 + 2 * q^25 + 14 * q^31 + 4 * q^41 + 2 * q^49 - 20 * q^65 + 30 * q^71 + 28 * q^79 - 26 * q^89 + 16 * q^95 - 28 * q^97

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