LMFDB - Embedded newform 6336.2.f.b.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)

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")

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);

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,-16,0,0,0,0,0,0,0,12,0,0,0,0, 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:	$\Q(\zeta_{8})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 1 $ x^4 + 1
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			$-0.707107 - 0.707107i$ of defining polynomial
Character	$\chi$	$=$	6336.3169
Dual form			6336.2.f.b.3169.3

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-1.41421i q^{5} -4.24264 q^{7} -1.00000i q^{11} -1.41421i q^{13} -4.00000 q^{17} +1.41421 q^{23} +3.00000 q^{25} +2.82843i q^{29} -5.65685 q^{31} +6.00000i q^{35} -6.00000 q^{41} +4.00000i q^{43} -9.89949 q^{47} +11.0000 q^{49} +1.41421i q^{53} -1.41421 q^{55} +8.00000i q^{59} -12.7279i q^{61} -2.00000 q^{65} -10.0000i q^{67} -12.7279 q^{71} +10.0000 q^{73} +4.24264i q^{77} +4.24264 q^{79} +6.00000i q^{83} +5.65685i q^{85} +6.00000 q^{89} +6.00000i q^{91} +12.0000 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 16 q^{17} + 12 q^{25} - 24 q^{41} + 44 q^{49} - 8 q^{65} + 40 q^{73} + 24 q^{89} + 48 q^{97}+O(q^{100}) $ 4 * q - 16 * q^17 + 12 * q^25 - 24 * q^41 + 44 * q^49 - 8 * q^65 + 40 * q^73 + 24 * q^89 + 48 * 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
$5$	− 1.41421i	− 0.632456i	−0.948683	−	0.316228i	$-0.897584\pi$
$5$	− 1.41421i	− 0.632456i	0.948683	−	0.316228i	$-0.102416\pi$
$6$	0	0
$7$	−4.24264	−1.60357	−0.801784	−	0.597614i	$-0.796115\pi$
$7$	−4.24264	−1.60357	−0.801784	+	0.597614i	$0.796115\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	− 1.00000i	− 0.301511i
$11$	− 1.00000i	− 0.301511i
$12$	0	0
$13$	− 1.41421i	− 0.392232i	−0.980581	−	0.196116i	$-0.937167\pi$
$13$	− 1.41421i	− 0.392232i	0.980581	−	0.196116i	$-0.0628330\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−4.00000	−0.970143	−0.485071	−	0.874475i	$-0.661206\pi$
$17$	−4.00000	−0.970143	−0.485071	+	0.874475i	$0.661206\pi$
$18$	0	0
$19$	0	0	1.00000			$0$
$19$	0	0	−1.00000			$\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	1.41421	0.294884	0.147442	−	0.989071i	$-0.452896\pi$
$23$	1.41421	0.294884	0.147442	+	0.989071i	$0.452896\pi$
$24$	0	0
$25$	3.00000	0.600000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	2.82843i	0.525226i	0.964901	+	0.262613i	$0.0845842\pi$
$29$	2.82843i	0.525226i	−0.964901	+	0.262613i	$0.915416\pi$
$30$	0	0
$31$	−5.65685	−1.01600	−0.508001	−	0.861357i	$-0.669615\pi$
$31$	−5.65685	−1.01600	−0.508001	+	0.861357i	$0.669615\pi$
$32$	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	$0.0986555\pi$
$43$	4.00000i	0.609994i	−0.952353	+	0.304997i	$0.901344\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−9.89949	−1.44399	−0.721995	−	0.691898i	$-0.756775\pi$
$47$	−9.89949	−1.44399	−0.721995	+	0.691898i	$0.756775\pi$
$48$	0	0
$49$	11.0000	1.57143
$50$	0	0
$51$	0	0
$52$	0	0
$53$	1.41421i	0.194257i	0.995272	+	0.0971286i	$0.0309658\pi$
$53$	1.41421i	0.194257i	−0.995272	+	0.0971286i	$0.969034\pi$
$54$	0	0
$55$	−1.41421	−0.190693
$56$	0	0
$57$	0	0
$58$	0	0
$59$	8.00000i	1.04151i	0.853706	+	0.520756i	$0.174350\pi$
$59$	8.00000i	1.04151i	−0.853706	+	0.520756i	$0.825650\pi$
$60$	0	0
$61$	− 12.7279i	− 1.62964i	−0.579712	−	0.814822i	$-0.696835\pi$
$61$	− 12.7279i	− 1.62964i	0.579712	−	0.814822i	$-0.303165\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−2.00000	−0.248069
$66$	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
$71$	−12.7279	−1.51053	−0.755263	−	0.655422i	$-0.772491\pi$
$71$	−12.7279	−1.51053	−0.755263	+	0.655422i	$0.772491\pi$
$72$	0	0
$73$	10.0000	1.17041	0.585206	−	0.810885i	$-0.301014\pi$
$73$	10.0000	1.17041	0.585206	+	0.810885i	$0.301014\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	4.24264i	0.483494i
$78$	0	0
$79$	4.24264	0.477334	0.238667	−	0.971101i	$-0.423290\pi$
$79$	4.24264	0.477334	0.238667	+	0.971101i	$0.423290\pi$
$80$	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
$85$	5.65685i	0.613572i
$86$	0	0
$87$	0	0
$88$	0	0
$89$	6.00000	0.635999	0.317999	−	0.948091i	$-0.396989\pi$
$89$	6.00000	0.635999	0.317999	+	0.948091i	$0.396989\pi$
$90$	0	0
$91$	6.00000i	0.628971i
$92$	0	0
$93$	0	0
$94$	0	0
$95$	0	0
$96$	0	0
$97$	12.0000	1.21842	0.609208	−	0.793011i	$-0.291488\pi$
$97$	12.0000	1.21842	0.609208	+	0.793011i	$0.291488\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	− 8.48528i	− 0.844317i	−0.906522	−	0.422159i	$-0.861273\pi$
$101$	− 8.48528i	− 0.844317i	0.906522	−	0.422159i	$-0.138727\pi$
$102$	0	0
$103$	14.1421	1.39347	0.696733	−	0.717331i	$-0.254636\pi$
$103$	14.1421	1.39347	0.696733	+	0.717331i	$0.254636\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	4.00000i	0.386695i	0.981130	+	0.193347i	$0.0619344\pi$
$107$	4.00000i	0.386695i	−0.981130	+	0.193347i	$0.938066\pi$
$108$	0	0
$109$	4.24264i	0.406371i	0.979140	+	0.203186i	$0.0651295\pi$
$109$	4.24264i	0.406371i	−0.979140	+	0.203186i	$0.934871\pi$
$110$	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
$115$	− 2.00000i	− 0.186501i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	16.9706	1.55569
$120$	0	0
$121$	−1.00000	−0.0909091
$122$	0	0
$123$	0	0
$124$	0	0
$125$	− 11.3137i	− 1.01193i
$126$	0	0
$127$	−12.7279	−1.12942	−0.564710	−	0.825289i	$-0.691012\pi$
$127$	−12.7279	−1.12942	−0.564710	+	0.825289i	$0.691012\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	14.0000i	1.22319i	0.791173	+	0.611593i	$0.209471\pi$
$131$	14.0000i	1.22319i	−0.791173	+	0.611593i	$0.790529\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	0	0
$136$	0	0
$137$	2.00000	0.170872	0.0854358	−	0.996344i	$-0.472772\pi$
$137$	2.00000	0.170872	0.0854358	+	0.996344i	$0.472772\pi$
$138$	0	0
$139$	0	0	1.00000			$0$
$139$	0	0	−1.00000			$\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−1.41421	−0.118262
$144$	0	0
$145$	4.00000	0.332182
$146$	0	0
$147$	0	0
$148$	0	0
$149$	22.6274i	1.85371i	0.375419	+	0.926855i	$0.377499\pi$
$149$	22.6274i	1.85371i	−0.375419	+	0.926855i	$0.622501\pi$
$150$	0	0
$151$	4.24264	0.345261	0.172631	−	0.984987i	$-0.444773\pi$
$151$	4.24264	0.345261	0.172631	+	0.984987i	$0.444773\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	8.00000i	0.642575i
$156$	0	0
$157$	19.7990i	1.58013i	0.613022	+	0.790066i	$0.289954\pi$
$157$	19.7990i	1.58013i	−0.613022	+	0.790066i	$0.710046\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−6.00000	−0.472866
$162$	0	0
$163$	12.0000i	0.939913i	0.882690	+	0.469956i	$0.155730\pi$
$163$	12.0000i	0.939913i	−0.882690	+	0.469956i	$0.844270\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	25.4558	1.96983	0.984916	−	0.173032i	$-0.0553564\pi$
$167$	25.4558	1.96983	0.984916	+	0.173032i	$0.0553564\pi$
$168$	0	0
$169$	11.0000	0.846154
$170$	0	0
$171$	0	0
$172$	0	0
$173$	− 8.48528i	− 0.645124i	−0.946548	−	0.322562i	$-0.895456\pi$
$173$	− 8.48528i	− 0.645124i	0.946548	−	0.322562i	$-0.104544\pi$
$174$	0	0
$175$	−12.7279	−0.962140
$176$	0	0
$177$	0	0
$178$	0	0
$179$	4.00000i	0.298974i	0.988764	+	0.149487i	$0.0477622\pi$
$179$	4.00000i	0.298974i	−0.988764	+	0.149487i	$0.952238\pi$
$180$	0	0
$181$	8.48528i	0.630706i	0.948974	+	0.315353i	$0.102123\pi$
$181$	8.48528i	0.630706i	−0.948974	+	0.315353i	$0.897877\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	0	0
$186$	0	0
$187$	4.00000i	0.292509i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−4.24264	−0.306987	−0.153493	−	0.988150i	$-0.549052\pi$
$191$	−4.24264	−0.306987	−0.153493	+	0.988150i	$0.549052\pi$
$192$	0	0
$193$	10.0000	0.719816	0.359908	−	0.932988i	$-0.382808\pi$
$193$	10.0000	0.719816	0.359908	+	0.932988i	$0.382808\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	2.82843i	0.201517i	0.994911	+	0.100759i	$0.0321270\pi$
$197$	2.82843i	0.201517i	−0.994911	+	0.100759i	$0.967873\pi$
$198$	0	0
$199$	25.4558	1.80452	0.902258	−	0.431196i	$-0.141908\pi$
$199$	25.4558	1.80452	0.902258	+	0.431196i	$0.141908\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	− 12.0000i	− 0.842235i
$204$	0	0
$205$	8.48528i	0.592638i
$206$	0	0
$207$	0	0
$208$	0	0
$209$	0	0
$210$	0	0
$211$	− 12.0000i	− 0.826114i	−0.910705	−	0.413057i	$-0.864461\pi$
$211$	− 12.0000i	− 0.826114i	0.910705	−	0.413057i	$-0.135539\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	5.65685	0.385794
$216$	0	0
$217$	24.0000	1.62923
$218$	0	0
$219$	0	0
$220$	0	0
$221$	5.65685i	0.380521i
$222$	0	0
$223$	16.9706	1.13643	0.568216	−	0.822879i	$-0.307634\pi$
$223$	16.9706	1.13643	0.568216	+	0.822879i	$0.307634\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	28.0000i	1.85843i	0.369546	+	0.929213i	$0.379513\pi$
$227$	28.0000i	1.85843i	−0.369546	+	0.929213i	$0.620487\pi$
$228$	0	0
$229$	− 19.7990i	− 1.30835i	−0.756341	−	0.654177i	$-0.773015\pi$
$229$	− 19.7990i	− 1.30835i	0.756341	−	0.654177i	$-0.226985\pi$
$230$	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
$235$	14.0000i	0.913259i
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−2.82843	−0.182956	−0.0914779	−	0.995807i	$-0.529159\pi$
$239$	−2.82843	−0.182956	−0.0914779	+	0.995807i	$0.529159\pi$
$240$	0	0
$241$	−26.0000	−1.67481	−0.837404	−	0.546585i	$-0.815928\pi$
$241$	−26.0000	−1.67481	−0.837404	+	0.546585i	$0.815928\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	− 15.5563i	− 0.993859i
$246$	0	0
$247$	0	0
$248$	0	0
$249$	0	0
$250$	0	0
$251$	− 24.0000i	− 1.51487i	−0.652913	−	0.757433i	$-0.726453\pi$
$251$	− 24.0000i	− 1.51487i	0.652913	−	0.757433i	$-0.273547\pi$
$252$	0	0
$253$	− 1.41421i	− 0.0889108i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−2.00000	−0.124757	−0.0623783	−	0.998053i	$-0.519869\pi$
$257$	−2.00000	−0.124757	−0.0623783	+	0.998053i	$0.519869\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−22.6274	−1.39527	−0.697633	−	0.716455i	$-0.745763\pi$
$263$	−22.6274	−1.39527	−0.697633	+	0.716455i	$0.745763\pi$
$264$	0	0
$265$	2.00000	0.122859
$266$	0	0
$267$	0	0
$268$	0	0
$269$	18.3848i	1.12094i	0.828175	+	0.560470i	$0.189379\pi$
$269$	18.3848i	1.12094i	−0.828175	+	0.560470i	$0.810621\pi$
$270$	0	0
$271$	−12.7279	−0.773166	−0.386583	−	0.922255i	$-0.626345\pi$
$271$	−12.7279	−0.773166	−0.386583	+	0.922255i	$0.626345\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	− 3.00000i	− 0.180907i
$276$	0	0
$277$	7.07107i	0.424859i	0.977176	+	0.212430i	$0.0681376\pi$
$277$	7.07107i	0.424859i	−0.977176	+	0.212430i	$0.931862\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	24.0000	1.43172	0.715860	−	0.698244i	$-0.246035\pi$
$281$	24.0000	1.43172	0.715860	+	0.698244i	$0.246035\pi$
$282$	0	0
$283$	32.0000i	1.90220i	0.308879	+	0.951101i	$0.400046\pi$
$283$	32.0000i	1.90220i	−0.308879	+	0.951101i	$0.599954\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	25.4558	1.50261
$288$	0	0
$289$	−1.00000	−0.0588235
$290$	0	0
$291$	0	0
$292$	0	0
$293$	28.2843i	1.65238i	0.563388	+	0.826192i	$0.309498\pi$
$293$	28.2843i	1.65238i	−0.563388	+	0.826192i	$0.690502\pi$
$294$	0	0
$295$	11.3137	0.658710
$296$	0	0
$297$	0	0
$298$	0	0
$299$	− 2.00000i	− 0.115663i
$300$	0	0
$301$	− 16.9706i	− 0.978167i
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−18.0000	−1.03068
$306$	0	0
$307$	12.0000i	0.684876i	0.939540	+	0.342438i	$0.111253\pi$
$307$	12.0000i	0.684876i	−0.939540	+	0.342438i	$0.888747\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	9.89949	0.561349	0.280674	−	0.959803i	$-0.409442\pi$
$311$	9.89949	0.561349	0.280674	+	0.959803i	$0.409442\pi$
$312$	0	0
$313$	22.0000	1.24351	0.621757	−	0.783210i	$-0.286419\pi$
$313$	22.0000	1.24351	0.621757	+	0.783210i	$0.286419\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 35.3553i	− 1.98575i	−0.119145	−	0.992877i	$-0.538015\pi$
$317$	− 35.3553i	− 1.98575i	0.119145	−	0.992877i	$-0.461985\pi$
$318$	0	0
$319$	2.82843	0.158362
$320$	0	0
$321$	0	0
$322$	0	0
$323$	0	0
$324$	0	0
$325$	− 4.24264i	− 0.235339i
$326$	0	0
$327$	0	0
$328$	0	0
$329$	42.0000	2.31553
$330$	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
$335$	−14.1421	−0.772667
$336$	0	0
$337$	−10.0000	−0.544735	−0.272367	−	0.962193i	$-0.587807\pi$
$337$	−10.0000	−0.544735	−0.272367	+	0.962193i	$0.587807\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	5.65685i	0.306336i
$342$	0	0
$343$	−16.9706	−0.916324
$344$	0	0
$345$	0	0
$346$	0	0
$347$	14.0000i	0.751559i	0.926709	+	0.375780i	$0.122625\pi$
$347$	14.0000i	0.751559i	−0.926709	+	0.375780i	$0.877375\pi$
$348$	0	0
$349$	− 7.07107i	− 0.378506i	−0.981928	−	0.189253i	$-0.939393\pi$
$349$	− 7.07107i	− 0.378506i	0.981928	−	0.189253i	$-0.0606066\pi$
$350$	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
$355$	18.0000i	0.955341i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−31.1127	−1.64207	−0.821033	−	0.570881i	$-0.806602\pi$
$359$	−31.1127	−1.64207	−0.821033	+	0.570881i	$0.806602\pi$
$360$	0	0
$361$	19.0000	1.00000
$362$	0	0
$363$	0	0
$364$	0	0
$365$	− 14.1421i	− 0.740233i
$366$	0	0
$367$	5.65685	0.295285	0.147643	−	0.989041i	$-0.452831\pi$
$367$	5.65685	0.295285	0.147643	+	0.989041i	$0.452831\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	− 6.00000i	− 0.311504i
$372$	0	0
$373$	1.41421i	0.0732252i	0.999330	+	0.0366126i	$0.0116568\pi$
$373$	1.41421i	0.0732252i	−0.999330	+	0.0366126i	$0.988343\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	4.00000	0.206010
$378$	0	0
$379$	4.00000i	0.205466i	0.994709	+	0.102733i	$0.0327588\pi$
$379$	4.00000i	0.205466i	−0.994709	+	0.102733i	$0.967241\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	29.6985	1.51752	0.758761	−	0.651369i	$-0.225805\pi$
$383$	29.6985	1.51752	0.758761	+	0.651369i	$0.225805\pi$
$384$	0	0
$385$	6.00000	0.305788
$386$	0	0
$387$	0	0
$388$	0	0
$389$	− 9.89949i	− 0.501924i	−0.967997	−	0.250962i	$-0.919253\pi$
$389$	− 9.89949i	− 0.501924i	0.967997	−	0.250962i	$-0.0807470\pi$
$390$	0	0
$391$	−5.65685	−0.286079
$392$	0	0
$393$	0	0
$394$	0	0
$395$	− 6.00000i	− 0.301893i
$396$	0	0
$397$	8.48528i	0.425864i	0.977067	+	0.212932i	$0.0683013\pi$
$397$	8.48528i	0.425864i	−0.977067	+	0.212932i	$0.931699\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	22.0000	1.09863	0.549314	−	0.835616i	$-0.314889\pi$
$401$	22.0000	1.09863	0.549314	+	0.835616i	$0.314889\pi$
$402$	0	0
$403$	8.00000i	0.398508i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	0	0
$408$	0	0
$409$	18.0000	0.890043	0.445021	−	0.895520i	$-0.353196\pi$
$409$	18.0000	0.890043	0.445021	+	0.895520i	$0.353196\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	− 33.9411i	− 1.67013i
$414$	0	0
$415$	8.48528	0.416526
$416$	0	0
$417$	0	0
$418$	0	0
$419$	24.0000i	1.17248i	0.810139	+	0.586238i	$0.199392\pi$
$419$	24.0000i	1.17248i	−0.810139	+	0.586238i	$0.800608\pi$
$420$	0	0
$421$	− 22.6274i	− 1.10279i	−0.834243	−	0.551396i	$-0.814095\pi$
$421$	− 22.6274i	− 1.10279i	0.834243	−	0.551396i	$-0.185905\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−12.0000	−0.582086
$426$	0	0
$427$	54.0000i	2.61324i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−5.65685	−0.272481	−0.136241	−	0.990676i	$-0.543502\pi$
$431$	−5.65685	−0.272481	−0.136241	+	0.990676i	$0.543502\pi$
$432$	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
$437$	0	0
$438$	0	0
$439$	−4.24264	−0.202490	−0.101245	−	0.994862i	$-0.532283\pi$
$439$	−4.24264	−0.202490	−0.101245	+	0.994862i	$0.532283\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	− 32.0000i	− 1.52037i	−0.649709	−	0.760183i	$-0.725109\pi$
$443$	− 32.0000i	− 1.52037i	0.649709	−	0.760183i	$-0.274891\pi$
$444$	0	0
$445$	− 8.48528i	− 0.402241i
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−26.0000	−1.22702	−0.613508	−	0.789689i	$-0.710242\pi$
$449$	−26.0000	−1.22702	−0.613508	+	0.789689i	$0.710242\pi$
$450$	0	0
$451$	6.00000i	0.282529i
$452$	0	0
$453$	0	0
$454$	0	0
$455$	8.48528	0.397796
$456$	0	0
$457$	−18.0000	−0.842004	−0.421002	−	0.907060i	$-0.638322\pi$
$457$	−18.0000	−0.842004	−0.421002	+	0.907060i	$0.638322\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	5.65685i	0.263466i	0.991285	+	0.131733i	$0.0420541\pi$
$461$	5.65685i	0.263466i	−0.991285	+	0.131733i	$0.957946\pi$
$462$	0	0
$463$	−14.1421	−0.657241	−0.328620	−	0.944462i	$-0.606584\pi$
$463$	−14.1421	−0.657241	−0.328620	+	0.944462i	$0.606584\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 24.0000i	− 1.11059i	−0.831654	−	0.555294i	$-0.812606\pi$
$467$	− 24.0000i	− 1.11059i	0.831654	−	0.555294i	$-0.187394\pi$
$468$	0	0
$469$	42.4264i	1.95907i
$470$	0	0
$471$	0	0
$472$	0	0
$473$	4.00000	0.183920
$474$	0	0
$475$	0	0
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−39.5980	−1.80928	−0.904639	−	0.426179i	$-0.859859\pi$
$479$	−39.5980	−1.80928	−0.904639	+	0.426179i	$0.859859\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	0	0
$484$	0	0
$485$	− 16.9706i	− 0.770594i
$486$	0	0
$487$	−25.4558	−1.15351	−0.576757	−	0.816916i	$-0.695682\pi$
$487$	−25.4558	−1.15351	−0.576757	+	0.816916i	$0.695682\pi$
$488$	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
$493$	− 11.3137i	− 0.509544i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	54.0000	2.42223
$498$	0	0
$499$	− 28.0000i	− 1.25345i	−0.779240	−	0.626726i	$-0.784395\pi$
$499$	− 28.0000i	− 1.25345i	0.779240	−	0.626726i	$-0.215605\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	8.48528	0.378340	0.189170	−	0.981944i	$-0.439420\pi$
$503$	8.48528	0.378340	0.189170	+	0.981944i	$0.439420\pi$
$504$	0	0
$505$	−12.0000	−0.533993
$506$	0	0
$507$	0	0
$508$	0	0
$509$	21.2132i	0.940259i	0.882598	+	0.470129i	$0.155793\pi$
$509$	21.2132i	0.940259i	−0.882598	+	0.470129i	$0.844207\pi$
$510$	0	0
$511$	−42.4264	−1.87683
$512$	0	0
$513$	0	0
$514$	0	0
$515$	− 20.0000i	− 0.881305i
$516$	0	0
$517$	9.89949i	0.435379i
$518$	0	0
$519$	0	0
$520$	0	0
$521$	14.0000	0.613351	0.306676	−	0.951814i	$-0.400783\pi$
$521$	14.0000	0.613351	0.306676	+	0.951814i	$0.400783\pi$
$522$	0	0
$523$	− 4.00000i	− 0.174908i	−0.996169	−	0.0874539i	$-0.972127\pi$
$523$	− 4.00000i	− 0.174908i	0.996169	−	0.0874539i	$-0.0278730\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	22.6274	0.985666
$528$	0	0
$529$	−21.0000	−0.913043
$530$	0	0
$531$	0	0
$532$	0	0
$533$	8.48528i	0.367538i
$534$	0	0
$535$	5.65685	0.244567
$536$	0	0
$537$	0	0
$538$	0	0
$539$	− 11.0000i	− 0.473804i
$540$	0	0
$541$	− 24.0416i	− 1.03363i	−0.856097	−	0.516815i	$-0.827117\pi$
$541$	− 24.0416i	− 1.03363i	0.856097	−	0.516815i	$-0.172883\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	6.00000	0.257012
$546$	0	0
$547$	20.0000i	0.855138i	0.903983	+	0.427569i	$0.140630\pi$
$547$	20.0000i	0.855138i	−0.903983	+	0.427569i	$0.859370\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	0	0
$552$	0	0
$553$	−18.0000	−0.765438
$554$	0	0
$555$	0	0
$556$	0	0
$557$	28.2843i	1.19844i	0.800583	+	0.599222i	$0.204523\pi$
$557$	28.2843i	1.19844i	−0.800583	+	0.599222i	$0.795477\pi$
$558$	0	0
$559$	5.65685	0.239259
$560$	0	0
$561$	0	0
$562$	0	0
$563$	12.0000i	0.505740i	0.967500	+	0.252870i	$0.0813744\pi$
$563$	12.0000i	0.505740i	−0.967500	+	0.252870i	$0.918626\pi$
$564$	0	0
$565$	− 8.48528i	− 0.356978i
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−36.0000	−1.50920	−0.754599	−	0.656186i	$-0.772169\pi$
$569$	−36.0000	−1.50920	−0.754599	+	0.656186i	$0.772169\pi$
$570$	0	0
$571$	− 32.0000i	− 1.33916i	−0.742741	−	0.669579i	$-0.766474\pi$
$571$	− 32.0000i	− 1.33916i	0.742741	−	0.669579i	$-0.233526\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	4.24264	0.176930
$576$	0	0
$577$	0	0		−	1.00000i	$-0.5\pi$
$577$	0	0			1.00000i	$0.5\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	− 25.4558i	− 1.05609i
$582$	0	0
$583$	1.41421	0.0585707
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 12.0000i	− 0.495293i	−0.968850	−	0.247647i	$-0.920343\pi$
$587$	− 12.0000i	− 0.495293i	0.968850	−	0.247647i	$-0.0796572\pi$
$588$	0	0
$589$	0	0
$590$	0	0
$591$	0	0
$592$	0	0
$593$	24.0000	0.985562	0.492781	−	0.870153i	$-0.335980\pi$
$593$	24.0000	0.985562	0.492781	+	0.870153i	$0.335980\pi$
$594$	0	0
$595$	− 24.0000i	− 0.983904i
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−18.3848	−0.751182	−0.375591	−	0.926786i	$-0.622560\pi$
$599$	−18.3848	−0.751182	−0.375591	+	0.926786i	$0.622560\pi$
$600$	0	0
$601$	26.0000	1.06056	0.530281	−	0.847822i	$-0.322086\pi$
$601$	26.0000	1.06056	0.530281	+	0.847822i	$0.322086\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	1.41421i	0.0574960i
$606$	0	0
$607$	41.0122	1.66463	0.832317	−	0.554300i	$-0.187014\pi$
$607$	41.0122	1.66463	0.832317	+	0.554300i	$0.187014\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	14.0000i	0.566379i
$612$	0	0
$613$	− 43.8406i	− 1.77071i	−0.464919	−	0.885353i	$-0.653917\pi$
$613$	− 43.8406i	− 1.77071i	0.464919	−	0.885353i	$-0.346083\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	34.0000	1.36879	0.684394	−	0.729112i	$-0.260067\pi$
$617$	34.0000	1.36879	0.684394	+	0.729112i	$0.260067\pi$
$618$	0	0
$619$	26.0000i	1.04503i	0.852631	+	0.522514i	$0.175006\pi$
$619$	26.0000i	1.04503i	−0.852631	+	0.522514i	$0.824994\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−25.4558	−1.01987
$624$	0	0
$625$	−1.00000	−0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	0	0
$630$	0	0
$631$	−22.6274	−0.900783	−0.450392	−	0.892831i	$-0.648716\pi$
$631$	−22.6274	−0.900783	−0.450392	+	0.892831i	$0.648716\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	18.0000i	0.714308i
$636$	0	0
$637$	− 15.5563i	− 0.616365i
$638$	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
$643$	34.0000i	1.34083i	0.741987	+	0.670415i	$0.233884\pi$
$643$	34.0000i	1.34083i	−0.741987	+	0.670415i	$0.766116\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−29.6985	−1.16757	−0.583784	−	0.811909i	$-0.698428\pi$
$647$	−29.6985	−1.16757	−0.583784	+	0.811909i	$0.698428\pi$
$648$	0	0
$649$	8.00000	0.314027
$650$	0	0
$651$	0	0
$652$	0	0
$653$	32.5269i	1.27288i	0.771328	+	0.636438i	$0.219593\pi$
$653$	32.5269i	1.27288i	−0.771328	+	0.636438i	$0.780407\pi$
$654$	0	0
$655$	19.7990	0.773611
$656$	0	0
$657$	0	0
$658$	0	0
$659$	28.0000i	1.09073i	0.838200	+	0.545363i	$0.183608\pi$
$659$	28.0000i	1.09073i	−0.838200	+	0.545363i	$0.816392\pi$
$660$	0	0
$661$	8.48528i	0.330039i	0.986290	+	0.165020i	$0.0527687\pi$
$661$	8.48528i	0.330039i	−0.986290	+	0.165020i	$0.947231\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	4.00000i	0.154881i
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−12.7279	−0.491356
$672$	0	0
$673$	−34.0000	−1.31060	−0.655302	−	0.755367i	$-0.727459\pi$
$673$	−34.0000	−1.31060	−0.655302	+	0.755367i	$0.727459\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	2.82843i	0.108705i	0.998522	+	0.0543526i	$0.0173095\pi$
$677$	2.82843i	0.108705i	−0.998522	+	0.0543526i	$0.982690\pi$
$678$	0	0
$679$	−50.9117	−1.95381
$680$	0	0
$681$	0	0
$682$	0	0
$683$	− 28.0000i	− 1.07139i	−0.844411	−	0.535695i	$-0.820050\pi$
$683$	− 28.0000i	− 1.07139i	0.844411	−	0.535695i	$-0.179950\pi$
$684$	0	0
$685$	− 2.82843i	− 0.108069i
$686$	0	0
$687$	0	0
$688$	0	0
$689$	2.00000	0.0761939
$690$	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
$695$	0	0
$696$	0	0
$697$	24.0000	0.909065
$698$	0	0
$699$	0	0
$700$	0	0
$701$	28.2843i	1.06828i	0.845395	+	0.534141i	$0.179365\pi$
$701$	28.2843i	1.06828i	−0.845395	+	0.534141i	$0.820635\pi$
$702$	0	0
$703$	0	0
$704$	0	0
$705$	0	0
$706$	0	0
$707$	36.0000i	1.35392i
$708$	0	0
$709$	8.48528i	0.318671i	0.987224	+	0.159336i	$0.0509352\pi$
$709$	8.48528i	0.318671i	−0.987224	+	0.159336i	$0.949065\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−8.00000	−0.299602
$714$	0	0
$715$	2.00000i	0.0747958i
$716$	0	0
$717$	0	0
$718$	0	0
$719$	41.0122	1.52950	0.764748	−	0.644329i	$-0.222863\pi$
$719$	41.0122	1.52950	0.764748	+	0.644329i	$0.222863\pi$
$720$	0	0
$721$	−60.0000	−2.23452
$722$	0	0
$723$	0	0
$724$	0	0
$725$	8.48528i	0.315135i
$726$	0	0
$727$	36.7696	1.36371	0.681854	−	0.731489i	$-0.261174\pi$
$727$	36.7696	1.36371	0.681854	+	0.731489i	$0.261174\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	− 16.0000i	− 0.591781i
$732$	0	0
$733$	21.2132i	0.783528i	0.920066	+	0.391764i	$0.128135\pi$
$733$	21.2132i	0.783528i	−0.920066	+	0.391764i	$0.871865\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−10.0000	−0.368355
$738$	0	0
$739$	− 40.0000i	− 1.47142i	−0.677295	−	0.735712i	$-0.736848\pi$
$739$	− 40.0000i	− 1.47142i	0.677295	−	0.735712i	$-0.263152\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−2.82843	−0.103765	−0.0518825	−	0.998653i	$-0.516522\pi$
$743$	−2.82843	−0.103765	−0.0518825	+	0.998653i	$0.516522\pi$
$744$	0	0
$745$	32.0000	1.17239
$746$	0	0
$747$	0	0
$748$	0	0
$749$	− 16.9706i	− 0.620091i
$750$	0	0
$751$	16.9706	0.619265	0.309632	−	0.950856i	$-0.399794\pi$
$751$	16.9706	0.619265	0.309632	+	0.950856i	$0.399794\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	− 6.00000i	− 0.218362i
$756$	0	0
$757$	45.2548i	1.64481i	0.568899	+	0.822407i	$0.307370\pi$
$757$	45.2548i	1.64481i	−0.568899	+	0.822407i	$0.692630\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	20.0000	0.724999	0.362500	−	0.931984i	$-0.381923\pi$
$761$	20.0000	0.724999	0.362500	+	0.931984i	$0.381923\pi$
$762$	0	0
$763$	− 18.0000i	− 0.651644i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	11.3137	0.408514
$768$	0	0
$769$	−6.00000	−0.216366	−0.108183	−	0.994131i	$-0.534503\pi$
$769$	−6.00000	−0.216366	−0.108183	+	0.994131i	$0.534503\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	9.89949i	0.356060i	0.984025	+	0.178030i	$0.0569724\pi$
$773$	9.89949i	0.356060i	−0.984025	+	0.178030i	$0.943028\pi$
$774$	0	0
$775$	−16.9706	−0.609601
$776$	0	0
$777$	0	0
$778$	0	0
$779$	0	0
$780$	0	0
$781$	12.7279i	0.455441i
$782$	0	0
$783$	0	0
$784$	0	0
$785$	28.0000	0.999363
$786$	0	0
$787$	0	0	1.00000			$0$
$787$	0	0	−1.00000			$\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−25.4558	−0.905106
$792$	0	0
$793$	−18.0000	−0.639199
$794$	0	0
$795$	0	0
$796$	0	0
$797$	43.8406i	1.55291i	0.630170	+	0.776457i	$0.282985\pi$
$797$	43.8406i	1.55291i	−0.630170	+	0.776457i	$0.717015\pi$
$798$	0	0
$799$	39.5980	1.40088
$800$	0	0
$801$	0	0
$802$	0	0
$803$	− 10.0000i	− 0.352892i
$804$	0	0
$805$	8.48528i	0.299067i
$806$	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
$811$	24.0000i	0.842754i	0.906886	+	0.421377i	$0.138453\pi$
$811$	24.0000i	0.842754i	−0.906886	+	0.421377i	$0.861547\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	16.9706	0.594453
$816$	0	0
$817$	0	0
$818$	0	0
$819$	0	0
$820$	0	0
$821$	14.1421i	0.493564i	0.969071	+	0.246782i	$0.0793731\pi$
$821$	14.1421i	0.493564i	−0.969071	+	0.246782i	$0.920627\pi$
$822$	0	0
$823$	28.2843	0.985928	0.492964	−	0.870050i	$-0.335913\pi$
$823$	28.2843	0.985928	0.492964	+	0.870050i	$0.335913\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 6.00000i	− 0.208640i	−0.994544	−	0.104320i	$-0.966733\pi$
$827$	− 6.00000i	− 0.208640i	0.994544	−	0.104320i	$-0.0332667\pi$
$828$	0	0
$829$	33.9411i	1.17882i	0.807833	+	0.589412i	$0.200641\pi$
$829$	33.9411i	1.17882i	−0.807833	+	0.589412i	$0.799359\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−44.0000	−1.52451
$834$	0	0
$835$	− 36.0000i	− 1.24583i
$836$	0	0
$837$	0	0
$838$	0	0
$839$	43.8406	1.51355	0.756773	−	0.653678i	$-0.226775\pi$
$839$	43.8406	1.51355	0.756773	+	0.653678i	$0.226775\pi$
$840$	0	0
$841$	21.0000	0.724138
$842$	0	0
$843$	0	0
$844$	0	0
$845$	− 15.5563i	− 0.535155i
$846$	0	0
$847$	4.24264	0.145779
$848$	0	0
$849$	0	0
$850$	0	0
$851$	0	0
$852$	0	0
$853$	29.6985i	1.01686i	0.861104	+	0.508428i	$0.169773\pi$
$853$	29.6985i	1.01686i	−0.861104	+	0.508428i	$0.830227\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	22.0000	0.751506	0.375753	−	0.926720i	$-0.377384\pi$
$857$	22.0000	0.751506	0.375753	+	0.926720i	$0.377384\pi$
$858$	0	0
$859$	− 36.0000i	− 1.22830i	−0.789188	−	0.614152i	$-0.789498\pi$
$859$	− 36.0000i	− 1.22830i	0.789188	−	0.614152i	$-0.210502\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	1.41421	0.0481404	0.0240702	−	0.999710i	$-0.492337\pi$
$863$	1.41421	0.0481404	0.0240702	+	0.999710i	$0.492337\pi$
$864$	0	0
$865$	−12.0000	−0.408012
$866$	0	0
$867$	0	0
$868$	0	0
$869$	− 4.24264i	− 0.143922i
$870$	0	0
$871$	−14.1421	−0.479188
$872$	0	0
$873$	0	0
$874$	0	0
$875$	48.0000i	1.62270i
$876$	0	0
$877$	52.3259i	1.76692i	0.468506	+	0.883460i	$0.344792\pi$
$877$	52.3259i	1.76692i	−0.468506	+	0.883460i	$0.655208\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	2.00000	0.0673817	0.0336909	−	0.999432i	$-0.489274\pi$
$881$	2.00000	0.0673817	0.0336909	+	0.999432i	$0.489274\pi$
$882$	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
$887$	2.82843	0.0949693	0.0474846	−	0.998872i	$-0.484879\pi$
$887$	2.82843	0.0949693	0.0474846	+	0.998872i	$0.484879\pi$
$888$	0	0
$889$	54.0000	1.81110
$890$	0	0
$891$	0	0
$892$	0	0
$893$	0	0
$894$	0	0
$895$	5.65685	0.189088
$896$	0	0
$897$	0	0
$898$	0	0
$899$	− 16.0000i	− 0.533630i
$900$	0	0
$901$	− 5.65685i	− 0.188457i
$902$	0	0
$903$	0	0
$904$	0	0
$905$	12.0000	0.398893
$906$	0	0
$907$	− 26.0000i	− 0.863316i	−0.902037	−	0.431658i	$-0.857929\pi$
$907$	− 26.0000i	− 0.863316i	0.902037	−	0.431658i	$-0.142071\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−1.41421	−0.0468550	−0.0234275	−	0.999726i	$-0.507458\pi$
$911$	−1.41421	−0.0468550	−0.0234275	+	0.999726i	$0.507458\pi$
$912$	0	0
$913$	6.00000	0.198571
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 59.3970i	− 1.96146i
$918$	0	0
$919$	1.41421	0.0466506	0.0233253	−	0.999728i	$-0.492575\pi$
$919$	1.41421	0.0466506	0.0233253	+	0.999728i	$0.492575\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	18.0000i	0.592477i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−42.0000	−1.37798	−0.688988	−	0.724773i	$-0.741945\pi$
$929$	−42.0000	−1.37798	−0.688988	+	0.724773i	$0.741945\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	0	0
$934$	0	0
$935$	5.65685	0.184999
$936$	0	0
$937$	2.00000	0.0653372	0.0326686	−	0.999466i	$-0.489599\pi$
$937$	2.00000	0.0653372	0.0326686	+	0.999466i	$0.489599\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	− 53.7401i	− 1.75188i	−0.482422	−	0.875939i	$-0.660243\pi$
$941$	− 53.7401i	− 1.75188i	0.482422	−	0.875939i	$-0.339757\pi$
$942$	0	0
$943$	−8.48528	−0.276319
$944$	0	0
$945$	0	0
$946$	0	0
$947$	48.0000i	1.55979i	0.625910	+	0.779895i	$0.284728\pi$
$947$	48.0000i	1.55979i	−0.625910	+	0.779895i	$0.715272\pi$
$948$	0	0
$949$	− 14.1421i	− 0.459073i
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−50.0000	−1.61966	−0.809829	−	0.586665i	$-0.800440\pi$
$953$	−50.0000	−1.61966	−0.809829	+	0.586665i	$0.800440\pi$
$954$	0	0
$955$	6.00000i	0.194155i
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−8.48528	−0.274004
$960$	0	0
$961$	1.00000	0.0322581
$962$	0	0
$963$	0	0
$964$	0	0
$965$	− 14.1421i	− 0.455251i
$966$	0	0
$967$	−9.89949	−0.318346	−0.159173	−	0.987251i	$-0.550883\pi$
$967$	−9.89949	−0.318346	−0.159173	+	0.987251i	$0.550883\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	36.0000i	1.15529i	0.816286	+	0.577647i	$0.196029\pi$
$971$	36.0000i	1.15529i	−0.816286	+	0.577647i	$0.803971\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−46.0000	−1.47167	−0.735835	−	0.677161i	$-0.763210\pi$
$977$	−46.0000	−1.47167	−0.735835	+	0.677161i	$0.763210\pi$
$978$	0	0
$979$	− 6.00000i	− 0.191761i
$980$	0	0
$981$	0	0
$982$	0	0
$983$	26.8701	0.857022	0.428511	−	0.903537i	$-0.359038\pi$
$983$	26.8701	0.857022	0.428511	+	0.903537i	$0.359038\pi$
$984$	0	0
$985$	4.00000	0.127451
$986$	0	0
$987$	0	0
$988$	0	0
$989$	5.65685i	0.179878i
$990$	0	0
$991$	22.6274	0.718784	0.359392	−	0.933187i	$-0.382984\pi$
$991$	22.6274	0.718784	0.359392	+	0.933187i	$0.382984\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	− 36.0000i	− 1.14128i
$996$	0	0
$997$	− 29.6985i	− 0.940560i	−0.882517	−	0.470280i	$-0.844153\pi$
$997$	− 29.6985i	− 0.940560i	0.882517	−	0.470280i	$-0.155847\pi$
$998$	0	0
$999$	0	0

Twists

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

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

\(f(q)\)	\(=\)	\(q-1.41421i q^{5} -4.24264 q^{7} -1.00000i q^{11} -1.41421i q^{13} -4.00000 q^{17} +1.41421 q^{23} +3.00000 q^{25} +2.82843i q^{29} -5.65685 q^{31} +6.00000i q^{35} -6.00000 q^{41} +4.00000i q^{43} -9.89949 q^{47} +11.0000 q^{49} +1.41421i q^{53} -1.41421 q^{55} +8.00000i q^{59} -12.7279i q^{61} -2.00000 q^{65} -10.0000i q^{67} -12.7279 q^{71} +10.0000 q^{73} +4.24264i q^{77} +4.24264 q^{79} +6.00000i q^{83} +5.65685i q^{85} +6.00000 q^{89} +6.00000i q^{91} +12.0000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 16 q^{17} + 12 q^{25} - 24 q^{41} + 44 q^{49} - 8 q^{65} + 40 q^{73} + 24 q^{89} + 48 q^{97}+O(q^{100}) \) 4 * q - 16 * q^17 + 12 * q^25 - 24 * q^41 + 44 * q^49 - 8 * q^65 + 40 * q^73 + 24 * q^89 + 48 * q^97

Introduction

L-functions

Modular forms

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Database

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