LMFDB - Embedded newform 6336.2.f.c.3169.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [6336,2,Mod(3169,6336)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("6336.3169"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(6336, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 6336 = 2^{6} \cdot 3^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6336.f (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-4,0,0,0,0,0, 0,0,0,0,0,0,0,0,0,0,-24] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(41)] == traces)

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

Self dual:	no
Analytic conductor:	$50.5932147207$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(i, \sqrt{6})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 9 $ x^4 + 9
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 2112)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			3169.1
Root			$1.22474 - 1.22474i$ of defining polynomial
Character	$\chi$	$=$	6336.3169
Dual form			6336.2.f.c.3169.3

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-2.44949i q^{5} -2.44949 q^{7} +1.00000i q^{11} -2.44949i q^{13} -8.00000i q^{19} +7.34847 q^{23} -1.00000 q^{25} +4.89898i q^{29} +9.79796 q^{31} +6.00000i q^{35} -6.00000 q^{41} -4.00000i q^{43} +7.34847 q^{47} -1.00000 q^{49} -7.34847i q^{53} +2.44949 q^{55} +7.34847i q^{61} -6.00000 q^{65} -10.0000i q^{67} +2.44949 q^{71} +2.00000 q^{73} -2.44949i q^{77} +12.2474 q^{79} +6.00000i q^{83} -18.0000 q^{89} +6.00000i q^{91} -19.5959 q^{95} -8.00000 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{25} - 24 q^{41} - 4 q^{49} - 24 q^{65} + 8 q^{73} - 72 q^{89} - 32 q^{97}+O(q^{100}) $ 4 * q - 4 * q^25 - 24 * q^41 - 4 * q^49 - 24 * q^65 + 8 * q^73 - 72 * q^89 - 32 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/6336\mathbb{Z}\right)^\times$.

$n$	$1729$	$3521$	$4159$	$4357$
$\chi(n)$	$1$	$1$	$1$	$-1$

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
$5$	− 2.44949i	− 1.09545i	−0.836660	−	0.547723i	$-0.815495\pi$
$5$	− 2.44949i	− 1.09545i	0.836660	−	0.547723i	$-0.184505\pi$
$6$	0	0
$7$	−2.44949	−0.925820	−0.462910	−	0.886405i	$-0.653195\pi$
$7$	−2.44949	−0.925820	−0.462910	+	0.886405i	$0.653195\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	1.00000i	0.301511i
$11$	1.00000i	0.301511i
$12$	0	0
$13$	− 2.44949i	− 0.679366i	−0.940540	−	0.339683i	$-0.889680\pi$
$13$	− 2.44949i	− 0.679366i	0.940540	−	0.339683i	$-0.110320\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	0	0		−	1.00000i	$-0.5\pi$
$17$	0	0			1.00000i	$0.5\pi$
$18$	0	0
$19$	− 8.00000i	− 1.83533i	−0.397360	−	0.917663i	$-0.630073\pi$
$19$	− 8.00000i	− 1.83533i	0.397360	−	0.917663i	$-0.369927\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	7.34847	1.53226	0.766131	−	0.642685i	$-0.222179\pi$
$23$	7.34847	1.53226	0.766131	+	0.642685i	$0.222179\pi$
$24$	0	0
$25$	−1.00000	−0.200000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	4.89898i	0.909718i	0.890564	+	0.454859i	$0.150310\pi$
$29$	4.89898i	0.909718i	−0.890564	+	0.454859i	$0.849690\pi$
$30$	0	0
$31$	9.79796	1.75977	0.879883	−	0.475191i	$-0.157621\pi$
$31$	9.79796	1.75977	0.879883	+	0.475191i	$0.157621\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	6.00000i	1.01419i
$36$	0	0
$37$	0	0	1.00000			$0$
$37$	0	0	−1.00000			$\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−6.00000	−0.937043	−0.468521	−	0.883452i	$-0.655213\pi$
$41$	−6.00000	−0.937043	−0.468521	+	0.883452i	$0.655213\pi$
$42$	0	0
$43$	− 4.00000i	− 0.609994i	−0.952353	−	0.304997i	$-0.901344\pi$
$43$	− 4.00000i	− 0.609994i	0.952353	−	0.304997i	$-0.0986555\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	7.34847	1.07188	0.535942	−	0.844255i	$-0.319956\pi$
$47$	7.34847	1.07188	0.535942	+	0.844255i	$0.319956\pi$
$48$	0	0
$49$	−1.00000	−0.142857
$50$	0	0
$51$	0	0
$52$	0	0
$53$	− 7.34847i	− 1.00939i	−0.863298	−	0.504695i	$-0.831605\pi$
$53$	− 7.34847i	− 1.00939i	0.863298	−	0.504695i	$-0.168395\pi$
$54$	0	0
$55$	2.44949	0.330289
$56$	0	0
$57$	0	0
$58$	0	0
$59$	0	0	1.00000			$0$
$59$	0	0	−1.00000			$\pi$
$60$	0	0
$61$	7.34847i	0.940875i	0.882433	+	0.470438i	$0.155904\pi$
$61$	7.34847i	0.940875i	−0.882433	+	0.470438i	$0.844096\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−6.00000	−0.744208
$66$	0	0
$67$	− 10.0000i	− 1.22169i	−0.791748	−	0.610847i	$-0.790829\pi$
$67$	− 10.0000i	− 1.22169i	0.791748	−	0.610847i	$-0.209171\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	2.44949	0.290701	0.145350	−	0.989380i	$-0.453569\pi$
$71$	2.44949	0.290701	0.145350	+	0.989380i	$0.453569\pi$
$72$	0	0
$73$	2.00000	0.234082	0.117041	−	0.993127i	$-0.462659\pi$
$73$	2.00000	0.234082	0.117041	+	0.993127i	$0.462659\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 2.44949i	− 0.279145i
$78$	0	0
$79$	12.2474	1.37795	0.688973	−	0.724787i	$-0.258062\pi$
$79$	12.2474	1.37795	0.688973	+	0.724787i	$0.258062\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	6.00000i	0.658586i	0.944228	+	0.329293i	$0.106810\pi$
$83$	6.00000i	0.658586i	−0.944228	+	0.329293i	$0.893190\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−18.0000	−1.90800	−0.953998	−	0.299813i	$-0.903076\pi$
$89$	−18.0000	−1.90800	−0.953998	+	0.299813i	$0.903076\pi$
$90$	0	0
$91$	6.00000i	0.628971i
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−19.5959	−2.01050
$96$	0	0
$97$	−8.00000	−0.812277	−0.406138	−	0.913812i	$-0.633125\pi$
$97$	−8.00000	−0.812277	−0.406138	+	0.913812i	$0.633125\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	4.89898i	0.487467i	0.969842	+	0.243733i	$0.0783722\pi$
$101$	4.89898i	0.487467i	−0.969842	+	0.243733i	$0.921628\pi$
$102$	0	0
$103$	−4.89898	−0.482711	−0.241355	−	0.970437i	$-0.577592\pi$
$103$	−4.89898	−0.482711	−0.241355	+	0.970437i	$0.577592\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	12.0000i	1.16008i	0.814587	+	0.580042i	$0.196964\pi$
$107$	12.0000i	1.16008i	−0.814587	+	0.580042i	$0.803036\pi$
$108$	0	0
$109$	− 12.2474i	− 1.17309i	−0.809916	−	0.586546i	$-0.800487\pi$
$109$	− 12.2474i	− 1.17309i	0.809916	−	0.586546i	$-0.199513\pi$
$110$	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
$115$	− 18.0000i	− 1.67851i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−1.00000	−0.0909091
$122$	0	0
$123$	0	0
$124$	0	0
$125$	− 9.79796i	− 0.876356i
$126$	0	0
$127$	−17.1464	−1.52150	−0.760750	−	0.649045i	$-0.775169\pi$
$127$	−17.1464	−1.52150	−0.760750	+	0.649045i	$0.775169\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	− 18.0000i	− 1.57267i	−0.617802	−	0.786334i	$-0.711977\pi$
$131$	− 18.0000i	− 1.57267i	0.617802	−	0.786334i	$-0.288023\pi$
$132$	0	0
$133$	19.5959i	1.69918i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−6.00000	−0.512615	−0.256307	−	0.966595i	$-0.582506\pi$
$137$	−6.00000	−0.512615	−0.256307	+	0.966595i	$0.582506\pi$
$138$	0	0
$139$	− 8.00000i	− 0.678551i	−0.940687	−	0.339276i	$-0.889818\pi$
$139$	− 8.00000i	− 0.678551i	0.940687	−	0.339276i	$-0.110182\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	2.44949	0.204837
$144$	0	0
$145$	12.0000	0.996546
$146$	0	0
$147$	0	0
$148$	0	0
$149$	− 19.5959i	− 1.60536i	−0.596410	−	0.802680i	$-0.703407\pi$
$149$	− 19.5959i	− 1.60536i	0.596410	−	0.802680i	$-0.296593\pi$
$150$	0	0
$151$	−17.1464	−1.39536	−0.697678	−	0.716411i	$-0.745783\pi$
$151$	−17.1464	−1.39536	−0.697678	+	0.716411i	$0.745783\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	− 24.0000i	− 1.92773i
$156$	0	0
$157$	− 4.89898i	− 0.390981i	−0.980706	−	0.195491i	$-0.937370\pi$
$157$	− 4.89898i	− 0.390981i	0.980706	−	0.195491i	$-0.0626299\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−18.0000	−1.41860
$162$	0	0
$163$	− 20.0000i	− 1.56652i	−0.621694	−	0.783260i	$-0.713555\pi$
$163$	− 20.0000i	− 1.56652i	0.621694	−	0.783260i	$-0.286445\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−4.89898	−0.379094	−0.189547	−	0.981872i	$-0.560702\pi$
$167$	−4.89898	−0.379094	−0.189547	+	0.981872i	$0.560702\pi$
$168$	0	0
$169$	7.00000	0.538462
$170$	0	0
$171$	0	0
$172$	0	0
$173$	24.4949i	1.86231i	0.364620	+	0.931156i	$0.381199\pi$
$173$	24.4949i	1.86231i	−0.364620	+	0.931156i	$0.618801\pi$
$174$	0	0
$175$	2.44949	0.185164
$176$	0	0
$177$	0	0
$178$	0	0
$179$	12.0000i	0.896922i	0.893802	+	0.448461i	$0.148028\pi$
$179$	12.0000i	0.896922i	−0.893802	+	0.448461i	$0.851972\pi$
$180$	0	0
$181$	− 24.4949i	− 1.82069i	−0.413849	−	0.910346i	$-0.635816\pi$
$181$	− 24.4949i	− 1.82069i	0.413849	−	0.910346i	$-0.364184\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	0	0
$186$	0	0
$187$	0	0
$188$	0	0
$189$	0	0
$190$	0	0
$191$	17.1464	1.24067	0.620336	−	0.784336i	$-0.286996\pi$
$191$	17.1464	1.24067	0.620336	+	0.784336i	$0.286996\pi$
$192$	0	0
$193$	2.00000	0.143963	0.0719816	−	0.997406i	$-0.477068\pi$
$193$	2.00000	0.143963	0.0719816	+	0.997406i	$0.477068\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	4.89898i	0.349038i	0.984654	+	0.174519i	$0.0558370\pi$
$197$	4.89898i	0.349038i	−0.984654	+	0.174519i	$0.944163\pi$
$198$	0	0
$199$	14.6969	1.04184	0.520919	−	0.853606i	$-0.325589\pi$
$199$	14.6969	1.04184	0.520919	+	0.853606i	$0.325589\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	− 12.0000i	− 0.842235i
$204$	0	0
$205$	14.6969i	1.02648i
$206$	0	0
$207$	0	0
$208$	0	0
$209$	8.00000	0.553372
$210$	0	0
$211$	− 4.00000i	− 0.275371i	−0.990476	−	0.137686i	$-0.956034\pi$
$211$	− 4.00000i	− 0.275371i	0.990476	−	0.137686i	$-0.0439664\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−9.79796	−0.668215
$216$	0	0
$217$	−24.0000	−1.62923
$218$	0	0
$219$	0	0
$220$	0	0
$221$	0	0
$222$	0	0
$223$	−9.79796	−0.656120	−0.328060	−	0.944657i	$-0.606395\pi$
$223$	−9.79796	−0.656120	−0.328060	+	0.944657i	$0.606395\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	12.0000i	0.796468i	0.917284	+	0.398234i	$0.130377\pi$
$227$	12.0000i	0.796468i	−0.917284	+	0.398234i	$0.869623\pi$
$228$	0	0
$229$	− 14.6969i	− 0.971201i	−0.874181	−	0.485601i	$-0.838601\pi$
$229$	− 14.6969i	− 0.971201i	0.874181	−	0.485601i	$-0.161399\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	6.00000	0.393073	0.196537	−	0.980497i	$-0.437031\pi$
$233$	6.00000	0.393073	0.196537	+	0.980497i	$0.437031\pi$
$234$	0	0
$235$	− 18.0000i	− 1.17419i
$236$	0	0
$237$	0	0
$238$	0	0
$239$	4.89898	0.316889	0.158444	−	0.987368i	$-0.449352\pi$
$239$	4.89898	0.316889	0.158444	+	0.987368i	$0.449352\pi$
$240$	0	0
$241$	−10.0000	−0.644157	−0.322078	−	0.946713i	$-0.604381\pi$
$241$	−10.0000	−0.644157	−0.322078	+	0.946713i	$0.604381\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	2.44949i	0.156492i
$246$	0	0
$247$	−19.5959	−1.24686
$248$	0	0
$249$	0	0
$250$	0	0
$251$	0	0	1.00000			$0$
$251$	0	0	−1.00000			$\pi$
$252$	0	0
$253$	7.34847i	0.461994i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−18.0000	−1.12281	−0.561405	−	0.827541i	$-0.689739\pi$
$257$	−18.0000	−1.12281	−0.561405	+	0.827541i	$0.689739\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−19.5959	−1.20834	−0.604168	−	0.796857i	$-0.706494\pi$
$263$	−19.5959	−1.20834	−0.604168	+	0.796857i	$0.706494\pi$
$264$	0	0
$265$	−18.0000	−1.10573
$266$	0	0
$267$	0	0
$268$	0	0
$269$	12.2474i	0.746740i	0.927682	+	0.373370i	$0.121798\pi$
$269$	12.2474i	0.746740i	−0.927682	+	0.373370i	$0.878202\pi$
$270$	0	0
$271$	12.2474	0.743980	0.371990	−	0.928237i	$-0.378676\pi$
$271$	12.2474	0.743980	0.371990	+	0.928237i	$0.378676\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	− 1.00000i	− 0.0603023i
$276$	0	0
$277$	− 17.1464i	− 1.03023i	−0.857121	−	0.515115i	$-0.827749\pi$
$277$	− 17.1464i	− 1.03023i	0.857121	−	0.515115i	$-0.172251\pi$
$278$	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
$283$	− 8.00000i	− 0.475551i	−0.971320	−	0.237775i	$-0.923582\pi$
$283$	− 8.00000i	− 0.475551i	0.971320	−	0.237775i	$-0.0764182\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	14.6969	0.867533
$288$	0	0
$289$	−17.0000	−1.00000
$290$	0	0
$291$	0	0
$292$	0	0
$293$	29.3939i	1.71721i	0.512639	+	0.858604i	$0.328668\pi$
$293$	29.3939i	1.71721i	−0.512639	+	0.858604i	$0.671332\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	0	0
$298$	0	0
$299$	− 18.0000i	− 1.04097i
$300$	0	0
$301$	9.79796i	0.564745i
$302$	0	0
$303$	0	0
$304$	0	0
$305$	18.0000	1.03068
$306$	0	0
$307$	20.0000i	1.14146i	0.821138	+	0.570730i	$0.193340\pi$
$307$	20.0000i	1.14146i	−0.821138	+	0.570730i	$0.806660\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−17.1464	−0.972285	−0.486142	−	0.873880i	$-0.661596\pi$
$311$	−17.1464	−0.972285	−0.486142	+	0.873880i	$0.661596\pi$
$312$	0	0
$313$	−26.0000	−1.46961	−0.734803	−	0.678280i	$-0.762726\pi$
$313$	−26.0000	−1.46961	−0.734803	+	0.678280i	$0.762726\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 2.44949i	− 0.137577i	−0.997631	−	0.0687885i	$-0.978087\pi$
$317$	− 2.44949i	− 0.137577i	0.997631	−	0.0687885i	$-0.0219134\pi$
$318$	0	0
$319$	−4.89898	−0.274290
$320$	0	0
$321$	0	0
$322$	0	0
$323$	0	0
$324$	0	0
$325$	2.44949i	0.135873i
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−18.0000	−0.992372
$330$	0	0
$331$	28.0000i	1.53902i	0.638635	+	0.769510i	$0.279499\pi$
$331$	28.0000i	1.53902i	−0.638635	+	0.769510i	$0.720501\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−24.4949	−1.33830
$336$	0	0
$337$	−2.00000	−0.108947	−0.0544735	−	0.998515i	$-0.517348\pi$
$337$	−2.00000	−0.108947	−0.0544735	+	0.998515i	$0.517348\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	9.79796i	0.530589i
$342$	0	0
$343$	19.5959	1.05808
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 18.0000i	− 0.966291i	−0.875540	−	0.483145i	$-0.839494\pi$
$347$	− 18.0000i	− 0.966291i	0.875540	−	0.483145i	$-0.160506\pi$
$348$	0	0
$349$	7.34847i	0.393355i	0.980468	+	0.196677i	$0.0630151\pi$
$349$	7.34847i	0.393355i	−0.980468	+	0.196677i	$0.936985\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	6.00000	0.319348	0.159674	−	0.987170i	$-0.448956\pi$
$353$	6.00000	0.319348	0.159674	+	0.987170i	$0.448956\pi$
$354$	0	0
$355$	− 6.00000i	− 0.318447i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−4.89898	−0.258558	−0.129279	−	0.991608i	$-0.541266\pi$
$359$	−4.89898	−0.258558	−0.129279	+	0.991608i	$0.541266\pi$
$360$	0	0
$361$	−45.0000	−2.36842
$362$	0	0
$363$	0	0
$364$	0	0
$365$	− 4.89898i	− 0.256424i
$366$	0	0
$367$	−9.79796	−0.511449	−0.255725	−	0.966750i	$-0.582314\pi$
$367$	−9.79796	−0.511449	−0.255725	+	0.966750i	$0.582314\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	18.0000i	0.934513i
$372$	0	0
$373$	2.44949i	0.126830i	0.997987	+	0.0634149i	$0.0201991\pi$
$373$	2.44949i	0.126830i	−0.997987	+	0.0634149i	$0.979801\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	12.0000	0.618031
$378$	0	0
$379$	− 4.00000i	− 0.205466i	−0.994709	−	0.102733i	$-0.967241\pi$
$379$	− 4.00000i	− 0.205466i	0.994709	−	0.102733i	$-0.0327588\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−12.2474	−0.625815	−0.312908	−	0.949784i	$-0.601303\pi$
$383$	−12.2474	−0.625815	−0.312908	+	0.949784i	$0.601303\pi$
$384$	0	0
$385$	−6.00000	−0.305788
$386$	0	0
$387$	0	0
$388$	0	0
$389$	− 26.9444i	− 1.36613i	−0.730355	−	0.683067i	$-0.760646\pi$
$389$	− 26.9444i	− 1.36613i	0.730355	−	0.683067i	$-0.239354\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	0	0
$394$	0	0
$395$	− 30.0000i	− 1.50946i
$396$	0	0
$397$	− 4.89898i	− 0.245873i	−0.992415	−	0.122936i	$-0.960769\pi$
$397$	− 4.89898i	− 0.245873i	0.992415	−	0.122936i	$-0.0392311\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	30.0000	1.49813	0.749064	−	0.662497i	$-0.230503\pi$
$401$	30.0000	1.49813	0.749064	+	0.662497i	$0.230503\pi$
$402$	0	0
$403$	− 24.0000i	− 1.19553i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	0	0
$408$	0	0
$409$	2.00000	0.0988936	0.0494468	−	0.998777i	$-0.484254\pi$
$409$	2.00000	0.0988936	0.0494468	+	0.998777i	$0.484254\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	0	0
$414$	0	0
$415$	14.6969	0.721444
$416$	0	0
$417$	0	0
$418$	0	0
$419$	0	0	1.00000			$0$
$419$	0	0	−1.00000			$\pi$
$420$	0	0
$421$	19.5959i	0.955047i	0.878619	+	0.477523i	$0.158465\pi$
$421$	19.5959i	0.955047i	−0.878619	+	0.477523i	$0.841535\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	0	0
$426$	0	0
$427$	− 18.0000i	− 0.871081i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	29.3939	1.41585	0.707927	−	0.706286i	$-0.249631\pi$
$431$	29.3939	1.41585	0.707927	+	0.706286i	$0.249631\pi$
$432$	0	0
$433$	−14.0000	−0.672797	−0.336399	−	0.941720i	$-0.609209\pi$
$433$	−14.0000	−0.672797	−0.336399	+	0.941720i	$0.609209\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 58.7878i	− 2.81220i
$438$	0	0
$439$	−31.8434	−1.51980	−0.759900	−	0.650039i	$-0.774752\pi$
$439$	−31.8434	−1.51980	−0.759900	+	0.650039i	$0.774752\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	− 24.0000i	− 1.14027i	−0.821549	−	0.570137i	$-0.806890\pi$
$443$	− 24.0000i	− 1.14027i	0.821549	−	0.570137i	$-0.193110\pi$
$444$	0	0
$445$	44.0908i	2.09011i
$446$	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
$451$	− 6.00000i	− 0.282529i
$452$	0	0
$453$	0	0
$454$	0	0
$455$	14.6969	0.689003
$456$	0	0
$457$	−10.0000	−0.467780	−0.233890	−	0.972263i	$-0.575146\pi$
$457$	−10.0000	−0.467780	−0.233890	+	0.972263i	$0.575146\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	9.79796i	0.456336i	0.973622	+	0.228168i	$0.0732736\pi$
$461$	9.79796i	0.456336i	−0.973622	+	0.228168i	$0.926726\pi$
$462$	0	0
$463$	24.4949	1.13837	0.569187	−	0.822208i	$-0.307258\pi$
$463$	24.4949	1.13837	0.569187	+	0.822208i	$0.307258\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	0	0	1.00000			$0$
$467$	0	0	−1.00000			$\pi$
$468$	0	0
$469$	24.4949i	1.13107i
$470$	0	0
$471$	0	0
$472$	0	0
$473$	4.00000	0.183920
$474$	0	0
$475$	8.00000i	0.367065i
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−9.79796	−0.447680	−0.223840	−	0.974626i	$-0.571859\pi$
$479$	−9.79796	−0.447680	−0.223840	+	0.974626i	$0.571859\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	0	0
$484$	0	0
$485$	19.5959i	0.889805i
$486$	0	0
$487$	24.4949	1.10997	0.554985	−	0.831860i	$-0.312724\pi$
$487$	24.4949	1.10997	0.554985	+	0.831860i	$0.312724\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	− 12.0000i	− 0.541552i	−0.962642	−	0.270776i	$-0.912720\pi$
$491$	− 12.0000i	− 0.541552i	0.962642	−	0.270776i	$-0.0872803\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−6.00000	−0.269137
$498$	0	0
$499$	− 4.00000i	− 0.179065i	−0.995984	−	0.0895323i	$-0.971463\pi$
$499$	− 4.00000i	− 0.179065i	0.995984	−	0.0895323i	$-0.0285372\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	4.89898	0.218435	0.109217	−	0.994018i	$-0.465166\pi$
$503$	4.89898	0.218435	0.109217	+	0.994018i	$0.465166\pi$
$504$	0	0
$505$	12.0000	0.533993
$506$	0	0
$507$	0	0
$508$	0	0
$509$	26.9444i	1.19429i	0.802134	+	0.597144i	$0.203698\pi$
$509$	26.9444i	1.19429i	−0.802134	+	0.597144i	$0.796302\pi$
$510$	0	0
$511$	−4.89898	−0.216718
$512$	0	0
$513$	0	0
$514$	0	0
$515$	12.0000i	0.528783i
$516$	0	0
$517$	7.34847i	0.323185i
$518$	0	0
$519$	0	0
$520$	0	0
$521$	30.0000	1.31432	0.657162	−	0.753749i	$-0.271757\pi$
$521$	30.0000	1.31432	0.657162	+	0.753749i	$0.271757\pi$
$522$	0	0
$523$	20.0000i	0.874539i	0.899331	+	0.437269i	$0.144054\pi$
$523$	20.0000i	0.874539i	−0.899331	+	0.437269i	$0.855946\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	0	0
$528$	0	0
$529$	31.0000	1.34783
$530$	0	0
$531$	0	0
$532$	0	0
$533$	14.6969i	0.636595i
$534$	0	0
$535$	29.3939	1.27081
$536$	0	0
$537$	0	0
$538$	0	0
$539$	− 1.00000i	− 0.0430730i
$540$	0	0
$541$	− 2.44949i	− 0.105312i	−0.998613	−	0.0526559i	$-0.983231\pi$
$541$	− 2.44949i	− 0.105312i	0.998613	−	0.0526559i	$-0.0167686\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−30.0000	−1.28506
$546$	0	0
$547$	− 20.0000i	− 0.855138i	−0.903983	−	0.427569i	$-0.859370\pi$
$547$	− 20.0000i	− 0.855138i	0.903983	−	0.427569i	$-0.140630\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	39.1918	1.66963
$552$	0	0
$553$	−30.0000	−1.27573
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 9.79796i	− 0.415153i	−0.978219	−	0.207576i	$-0.933442\pi$
$557$	− 9.79796i	− 0.415153i	0.978219	−	0.207576i	$-0.0665576\pi$
$558$	0	0
$559$	−9.79796	−0.414410
$560$	0	0
$561$	0	0
$562$	0	0
$563$	− 36.0000i	− 1.51722i	−0.651546	−	0.758610i	$-0.725879\pi$
$563$	− 36.0000i	− 1.51722i	0.651546	−	0.758610i	$-0.274121\pi$
$564$	0	0
$565$	− 14.6969i	− 0.618305i
$566$	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
$571$	40.0000i	1.67395i	0.547243	+	0.836974i	$0.315677\pi$
$571$	40.0000i	1.67395i	−0.547243	+	0.836974i	$0.684323\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−7.34847	−0.306452
$576$	0	0
$577$	28.0000	1.16566	0.582828	−	0.812596i	$-0.301946\pi$
$577$	28.0000	1.16566	0.582828	+	0.812596i	$0.301946\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	− 14.6969i	− 0.609732i
$582$	0	0
$583$	7.34847	0.304342
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 36.0000i	− 1.48588i	−0.669359	−	0.742940i	$-0.733431\pi$
$587$	− 36.0000i	− 1.48588i	0.669359	−	0.742940i	$-0.266569\pi$
$588$	0	0
$589$	− 78.3837i	− 3.22974i
$590$	0	0
$591$	0	0
$592$	0	0
$593$	12.0000	0.492781	0.246390	−	0.969171i	$-0.420755\pi$
$593$	12.0000	0.492781	0.246390	+	0.969171i	$0.420755\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−7.34847	−0.300250	−0.150125	−	0.988667i	$-0.547968\pi$
$599$	−7.34847	−0.300250	−0.150125	+	0.988667i	$0.547968\pi$
$600$	0	0
$601$	−46.0000	−1.87638	−0.938190	−	0.346122i	$-0.887498\pi$
$601$	−46.0000	−1.87638	−0.938190	+	0.346122i	$0.887498\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	2.44949i	0.0995859i
$606$	0	0
$607$	−41.6413	−1.69017	−0.845085	−	0.534633i	$-0.820450\pi$
$607$	−41.6413	−1.69017	−0.845085	+	0.534633i	$0.820450\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	− 18.0000i	− 0.728202i
$612$	0	0
$613$	− 46.5403i	− 1.87975i	−0.341525	−	0.939873i	$-0.610943\pi$
$613$	− 46.5403i	− 1.87975i	0.341525	−	0.939873i	$-0.389057\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−30.0000	−1.20775	−0.603877	−	0.797077i	$-0.706378\pi$
$617$	−30.0000	−1.20775	−0.603877	+	0.797077i	$0.706378\pi$
$618$	0	0
$619$	10.0000i	0.401934i	0.979598	+	0.200967i	$0.0644084\pi$
$619$	10.0000i	0.401934i	−0.979598	+	0.200967i	$0.935592\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	44.0908	1.76646
$624$	0	0
$625$	−29.0000	−1.16000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	0	0
$630$	0	0
$631$	19.5959	0.780101	0.390051	−	0.920793i	$-0.372458\pi$
$631$	19.5959	0.780101	0.390051	+	0.920793i	$0.372458\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	42.0000i	1.66672i
$636$	0	0
$637$	2.44949i	0.0970523i
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−18.0000	−0.710957	−0.355479	−	0.934684i	$-0.615682\pi$
$641$	−18.0000	−0.710957	−0.355479	+	0.934684i	$0.615682\pi$
$642$	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
$647$	31.8434	1.25189	0.625946	−	0.779866i	$-0.284713\pi$
$647$	31.8434	1.25189	0.625946	+	0.779866i	$0.284713\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	0	0
$652$	0	0
$653$	− 2.44949i	− 0.0958559i	−0.998851	−	0.0479280i	$-0.984738\pi$
$653$	− 2.44949i	− 0.0958559i	0.998851	−	0.0479280i	$-0.0152618\pi$
$654$	0	0
$655$	−44.0908	−1.72277
$656$	0	0
$657$	0	0
$658$	0	0
$659$	− 12.0000i	− 0.467454i	−0.972302	−	0.233727i	$-0.924908\pi$
$659$	− 12.0000i	− 0.467454i	0.972302	−	0.233727i	$-0.0750921\pi$
$660$	0	0
$661$	34.2929i	1.33384i	0.745130	+	0.666919i	$0.232387\pi$
$661$	34.2929i	1.33384i	−0.745130	+	0.666919i	$0.767613\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	48.0000	1.86136
$666$	0	0
$667$	36.0000i	1.39393i
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−7.34847	−0.283685
$672$	0	0
$673$	46.0000	1.77317	0.886585	−	0.462566i	$-0.153071\pi$
$673$	46.0000	1.77317	0.886585	+	0.462566i	$0.153071\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	24.4949i	0.941415i	0.882289	+	0.470708i	$0.156001\pi$
$677$	24.4949i	0.941415i	−0.882289	+	0.470708i	$0.843999\pi$
$678$	0	0
$679$	19.5959	0.752022
$680$	0	0
$681$	0	0
$682$	0	0
$683$	12.0000i	0.459167i	0.973289	+	0.229584i	$0.0737364\pi$
$683$	12.0000i	0.459167i	−0.973289	+	0.229584i	$0.926264\pi$
$684$	0	0
$685$	14.6969i	0.561541i
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−18.0000	−0.685745
$690$	0	0
$691$	− 2.00000i	− 0.0760836i	−0.999276	−	0.0380418i	$-0.987888\pi$
$691$	− 2.00000i	− 0.0760836i	0.999276	−	0.0380418i	$-0.0121120\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−19.5959	−0.743316
$696$	0	0
$697$	0	0
$698$	0	0
$699$	0	0
$700$	0	0
$701$	9.79796i	0.370064i	0.982733	+	0.185032i	$0.0592388\pi$
$701$	9.79796i	0.370064i	−0.982733	+	0.185032i	$0.940761\pi$
$702$	0	0
$703$	0	0
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 12.0000i	− 0.451306i
$708$	0	0
$709$	− 24.4949i	− 0.919925i	−0.887938	−	0.459963i	$-0.847863\pi$
$709$	− 24.4949i	− 0.919925i	0.887938	−	0.459963i	$-0.152137\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	72.0000	2.69642
$714$	0	0
$715$	− 6.00000i	− 0.224387i
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−31.8434	−1.18756	−0.593779	−	0.804628i	$-0.702364\pi$
$719$	−31.8434	−1.18756	−0.593779	+	0.804628i	$0.702364\pi$
$720$	0	0
$721$	12.0000	0.446903
$722$	0	0
$723$	0	0
$724$	0	0
$725$	− 4.89898i	− 0.181944i
$726$	0	0
$727$	14.6969	0.545079	0.272540	−	0.962145i	$-0.412136\pi$
$727$	14.6969	0.545079	0.272540	+	0.962145i	$0.412136\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	0	0
$732$	0	0
$733$	− 22.0454i	− 0.814266i	−0.913369	−	0.407133i	$-0.866529\pi$
$733$	− 22.0454i	− 0.814266i	0.913369	−	0.407133i	$-0.133471\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	10.0000	0.368355
$738$	0	0
$739$	− 16.0000i	− 0.588570i	−0.955718	−	0.294285i	$-0.904919\pi$
$739$	− 16.0000i	− 0.588570i	0.955718	−	0.294285i	$-0.0950814\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	44.0908	1.61754	0.808768	−	0.588128i	$-0.200135\pi$
$743$	44.0908	1.61754	0.808768	+	0.588128i	$0.200135\pi$
$744$	0	0
$745$	−48.0000	−1.75858
$746$	0	0
$747$	0	0
$748$	0	0
$749$	− 29.3939i	− 1.07403i
$750$	0	0
$751$	48.9898	1.78766	0.893832	−	0.448403i	$-0.148007\pi$
$751$	48.9898	1.78766	0.893832	+	0.448403i	$0.148007\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	42.0000i	1.52854i
$756$	0	0
$757$	− 19.5959i	− 0.712226i	−0.934443	−	0.356113i	$-0.884102\pi$
$757$	− 19.5959i	− 0.712226i	0.934443	−	0.356113i	$-0.115898\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−48.0000	−1.74000	−0.869999	−	0.493053i	$-0.835881\pi$
$761$	−48.0000	−1.74000	−0.869999	+	0.493053i	$0.835881\pi$
$762$	0	0
$763$	30.0000i	1.08607i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	0	0
$768$	0	0
$769$	50.0000	1.80305	0.901523	−	0.432731i	$-0.142450\pi$
$769$	50.0000	1.80305	0.901523	+	0.432731i	$0.142450\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	− 41.6413i	− 1.49773i	−0.662720	−	0.748867i	$-0.730598\pi$
$773$	− 41.6413i	− 1.49773i	0.662720	−	0.748867i	$-0.269402\pi$
$774$	0	0
$775$	−9.79796	−0.351953
$776$	0	0
$777$	0	0
$778$	0	0
$779$	48.0000i	1.71978i
$780$	0	0
$781$	2.44949i	0.0876496i
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−12.0000	−0.428298
$786$	0	0
$787$	40.0000i	1.42585i	0.701242	+	0.712923i	$0.252629\pi$
$787$	40.0000i	1.42585i	−0.701242	+	0.712923i	$0.747371\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−14.6969	−0.522563
$792$	0	0
$793$	18.0000	0.639199
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 31.8434i	− 1.12795i	−0.825792	−	0.563975i	$-0.809271\pi$
$797$	− 31.8434i	− 1.12795i	0.825792	−	0.563975i	$-0.190729\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	0	0
$802$	0	0
$803$	2.00000i	0.0705785i
$804$	0	0
$805$	44.0908i	1.55400i
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−42.0000	−1.47664	−0.738321	−	0.674450i	$-0.764381\pi$
$809$	−42.0000	−1.47664	−0.738321	+	0.674450i	$0.764381\pi$
$810$	0	0
$811$	− 16.0000i	− 0.561836i	−0.959732	−	0.280918i	$-0.909361\pi$
$811$	− 16.0000i	− 0.561836i	0.959732	−	0.280918i	$-0.0906389\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−48.9898	−1.71604
$816$	0	0
$817$	−32.0000	−1.11954
$818$	0	0
$819$	0	0
$820$	0	0
$821$	4.89898i	0.170976i	0.996339	+	0.0854878i	$0.0272449\pi$
$821$	4.89898i	0.170976i	−0.996339	+	0.0854878i	$0.972755\pi$
$822$	0	0
$823$	−29.3939	−1.02461	−0.512303	−	0.858805i	$-0.671208\pi$
$823$	−29.3939	−1.02461	−0.512303	+	0.858805i	$0.671208\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 54.0000i	− 1.87776i	−0.344239	−	0.938882i	$-0.611863\pi$
$827$	− 54.0000i	− 1.87776i	0.344239	−	0.938882i	$-0.388137\pi$
$828$	0	0
$829$	0	0	1.00000			$0$
$829$	0	0	−1.00000			$\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	0	0
$834$	0	0
$835$	12.0000i	0.415277i
$836$	0	0
$837$	0	0
$838$	0	0
$839$	31.8434	1.09936	0.549678	−	0.835377i	$-0.314751\pi$
$839$	31.8434	1.09936	0.549678	+	0.835377i	$0.314751\pi$
$840$	0	0
$841$	5.00000	0.172414
$842$	0	0
$843$	0	0
$844$	0	0
$845$	− 17.1464i	− 0.589855i
$846$	0	0
$847$	2.44949	0.0841655
$848$	0	0
$849$	0	0
$850$	0	0
$851$	0	0
$852$	0	0
$853$	2.44949i	0.0838689i	0.999120	+	0.0419345i	$0.0133521\pi$
$853$	2.44949i	0.0838689i	−0.999120	+	0.0419345i	$0.986648\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−42.0000	−1.43469	−0.717346	−	0.696717i	$-0.754643\pi$
$857$	−42.0000	−1.43469	−0.717346	+	0.696717i	$0.754643\pi$
$858$	0	0
$859$	20.0000i	0.682391i	0.939992	+	0.341196i	$0.110832\pi$
$859$	20.0000i	0.682391i	−0.939992	+	0.341196i	$0.889168\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−31.8434	−1.08396	−0.541980	−	0.840391i	$-0.682325\pi$
$863$	−31.8434	−1.08396	−0.541980	+	0.840391i	$0.682325\pi$
$864$	0	0
$865$	60.0000	2.04006
$866$	0	0
$867$	0	0
$868$	0	0
$869$	12.2474i	0.415466i
$870$	0	0
$871$	−24.4949	−0.829978
$872$	0	0
$873$	0	0
$874$	0	0
$875$	24.0000i	0.811348i
$876$	0	0
$877$	− 36.7423i	− 1.24070i	−0.784325	−	0.620350i	$-0.786990\pi$
$877$	− 36.7423i	− 1.24070i	0.784325	−	0.620350i	$-0.213010\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	42.0000	1.41502	0.707508	−	0.706705i	$-0.249819\pi$
$881$	42.0000	1.41502	0.707508	+	0.706705i	$0.249819\pi$
$882$	0	0
$883$	− 2.00000i	− 0.0673054i	−0.999434	−	0.0336527i	$-0.989286\pi$
$883$	− 2.00000i	− 0.0673054i	0.999434	−	0.0336527i	$-0.0107140\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−4.89898	−0.164492	−0.0822458	−	0.996612i	$-0.526209\pi$
$887$	−4.89898	−0.164492	−0.0822458	+	0.996612i	$0.526209\pi$
$888$	0	0
$889$	42.0000	1.40863
$890$	0	0
$891$	0	0
$892$	0	0
$893$	− 58.7878i	− 1.96726i
$894$	0	0
$895$	29.3939	0.982529
$896$	0	0
$897$	0	0
$898$	0	0
$899$	48.0000i	1.60089i
$900$	0	0
$901$	0	0
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−60.0000	−1.99447
$906$	0	0
$907$	− 58.0000i	− 1.92586i	−0.269754	−	0.962929i	$-0.586942\pi$
$907$	− 58.0000i	− 1.92586i	0.269754	−	0.962929i	$-0.413058\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	51.4393	1.70426	0.852130	−	0.523331i	$-0.175311\pi$
$911$	51.4393	1.70426	0.852130	+	0.523331i	$0.175311\pi$
$912$	0	0
$913$	−6.00000	−0.198571
$914$	0	0
$915$	0	0
$916$	0	0
$917$	44.0908i	1.45601i
$918$	0	0
$919$	26.9444	0.888813	0.444407	−	0.895825i	$-0.353415\pi$
$919$	26.9444	0.888813	0.444407	+	0.895825i	$0.353415\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	− 6.00000i	− 0.197492i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	0	0
$928$	0	0
$929$	30.0000	0.984268	0.492134	−	0.870519i	$-0.336217\pi$
$929$	30.0000	0.984268	0.492134	+	0.870519i	$0.336217\pi$
$930$	0	0
$931$	8.00000i	0.262189i
$932$	0	0
$933$	0	0
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−38.0000	−1.24141	−0.620703	−	0.784046i	$-0.713153\pi$
$937$	−38.0000	−1.24141	−0.620703	+	0.784046i	$0.713153\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	− 14.6969i	− 0.479107i	−0.970883	−	0.239553i	$-0.922999\pi$
$941$	− 14.6969i	− 0.479107i	0.970883	−	0.239553i	$-0.0770010\pi$
$942$	0	0
$943$	−44.0908	−1.43579
$944$	0	0
$945$	0	0
$946$	0	0
$947$	24.0000i	0.779895i	0.920837	+	0.389948i	$0.127507\pi$
$947$	24.0000i	0.779895i	−0.920837	+	0.389948i	$0.872493\pi$
$948$	0	0
$949$	− 4.89898i	− 0.159028i
$950$	0	0
$951$	0	0
$952$	0	0
$953$	30.0000	0.971795	0.485898	−	0.874016i	$-0.338493\pi$
$953$	30.0000	0.971795	0.485898	+	0.874016i	$0.338493\pi$
$954$	0	0
$955$	− 42.0000i	− 1.35909i
$956$	0	0
$957$	0	0
$958$	0	0
$959$	14.6969	0.474589
$960$	0	0
$961$	65.0000	2.09677
$962$	0	0
$963$	0	0
$964$	0	0
$965$	− 4.89898i	− 0.157704i
$966$	0	0
$967$	−22.0454	−0.708933	−0.354466	−	0.935069i	$-0.615337\pi$
$967$	−22.0454	−0.708933	−0.354466	+	0.935069i	$0.615337\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	60.0000i	1.92549i	0.270408	+	0.962746i	$0.412841\pi$
$971$	60.0000i	1.92549i	−0.270408	+	0.962746i	$0.587159\pi$
$972$	0	0
$973$	19.5959i	0.628216i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	18.0000	0.575871	0.287936	−	0.957650i	$-0.407031\pi$
$977$	18.0000	0.575871	0.287936	+	0.957650i	$0.407031\pi$
$978$	0	0
$979$	− 18.0000i	− 0.575282i
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−26.9444	−0.859392	−0.429696	−	0.902973i	$-0.641379\pi$
$983$	−26.9444	−0.859392	−0.429696	+	0.902973i	$0.641379\pi$
$984$	0	0
$985$	12.0000	0.382352
$986$	0	0
$987$	0	0
$988$	0	0
$989$	− 29.3939i	− 0.934671i
$990$	0	0
$991$	19.5959	0.622485	0.311242	−	0.950331i	$-0.399255\pi$
$991$	19.5959	0.622485	0.311242	+	0.950331i	$0.399255\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	− 36.0000i	− 1.14128i
$996$	0	0
$997$	56.3383i	1.78425i	0.451788	+	0.892125i	$0.350786\pi$
$997$	56.3383i	1.78425i	−0.451788	+	0.892125i	$0.649214\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6336.2.f.c.3169.1		4
3.2	odd	2		2112.2.f.d.1057.4	yes	4
4.3	odd	2	inner	6336.2.f.c.3169.2		4
8.3	odd	2	inner	6336.2.f.c.3169.4		4
8.5	even	2	inner	6336.2.f.c.3169.3		4
12.11	even	2		2112.2.f.d.1057.2	yes	4
24.5	odd	2		2112.2.f.d.1057.1	✓	4
24.11	even	2		2112.2.f.d.1057.3	yes	4
48.5	odd	4		8448.2.a.by.1.1		2
48.11	even	4		8448.2.a.be.1.1		2
48.29	odd	4		8448.2.a.be.1.2		2
48.35	even	4		8448.2.a.by.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2112.2.f.d.1057.1	✓	4	24.5	odd	2
2112.2.f.d.1057.2	yes	4	12.11	even	2
2112.2.f.d.1057.3	yes	4	24.11	even	2
2112.2.f.d.1057.4	yes	4	3.2	odd	2
6336.2.f.c.3169.1		4	1.1	even	1	trivial
6336.2.f.c.3169.2		4	4.3	odd	2	inner
6336.2.f.c.3169.3		4	8.5	even	2	inner
6336.2.f.c.3169.4		4	8.3	odd	2	inner
8448.2.a.be.1.1		2	48.11	even	4
8448.2.a.be.1.2		2	48.29	odd	4
8448.2.a.by.1.1		2	48.5	odd	4
8448.2.a.by.1.2		2	48.35	even	4

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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 6336 = 2^{6} \cdot 3^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6336.f (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-2.44949i q^{5} -2.44949 q^{7} +1.00000i q^{11} -2.44949i q^{13} -8.00000i q^{19} +7.34847 q^{23} -1.00000 q^{25} +4.89898i q^{29} +9.79796 q^{31} +6.00000i q^{35} -6.00000 q^{41} -4.00000i q^{43} +7.34847 q^{47} -1.00000 q^{49} -7.34847i q^{53} +2.44949 q^{55} +7.34847i q^{61} -6.00000 q^{65} -10.0000i q^{67} +2.44949 q^{71} +2.00000 q^{73} -2.44949i q^{77} +12.2474 q^{79} +6.00000i q^{83} -18.0000 q^{89} +6.00000i q^{91} -19.5959 q^{95} -8.00000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{25} - 24 q^{41} - 4 q^{49} - 24 q^{65} + 8 q^{73} - 72 q^{89} - 32 q^{97}+O(q^{100}) \) 4 * q - 4 * q^25 - 24 * q^41 - 4 * q^49 - 24 * q^65 + 8 * q^73 - 72 * q^89 - 32 * q^97

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