LMFDB - Embedded newform 6336.2.b.a.2177.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [6336,2,Mod(2177,6336)] 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, 0, 1, 1])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("6336.2177"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 6336 = 2^{6} \cdot 3^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6336.b (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [2,0,0,0,0,0,0,0,0,0,-6,0,0,0,0,0,-12,0,0,0,0,0,0,0,-6,0,0,0, 0,0,4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(31)] == traces)

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

Self dual:	no
Analytic conductor:	$50.5932147207$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{-2}) $
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 2 $ x^2 + 2
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 198)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			2177.1
Root			$-1.41421i$ of defining polynomial
Character	$\chi$	$=$	6336.2177
Dual form			6336.2.b.a.2177.2

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-2.82843i q^{5} +(-3.00000 - 1.41421i) q^{11} +4.24264i q^{13} -6.00000 q^{17} +4.24264i q^{19} -1.41421i q^{23} -3.00000 q^{25} +2.00000 q^{31} +10.0000 q^{37} -6.00000 q^{41} -12.7279i q^{43} -9.89949i q^{47} +7.00000 q^{49} +5.65685i q^{53} +(-4.00000 + 8.48528i) q^{55} +5.65685i q^{59} +12.7279i q^{61} +12.0000 q^{65} -8.00000 q^{67} +15.5563i q^{71} +8.48528i q^{73} +8.48528i q^{79} -6.00000 q^{83} +16.9706i q^{85} +7.07107i q^{89} +12.0000 q^{95} -4.00000 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 6 q^{11} - 12 q^{17} - 6 q^{25} + 4 q^{31} + 20 q^{37} - 12 q^{41} + 14 q^{49} - 8 q^{55} + 24 q^{65} - 16 q^{67} - 12 q^{83} + 24 q^{95} - 8 q^{97}+O(q^{100}) $ 2 * q - 6 * q^11 - 12 * q^17 - 6 * q^25 + 4 * q^31 + 20 * q^37 - 12 * q^41 + 14 * q^49 - 8 * q^55 + 24 * q^65 - 16 * q^67 - 12 * q^83 + 24 * q^95 - 8 * 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
$5$		−	2.82843i		−	1.26491i	−0.774597	−	0.632456i	$-0.782047\pi$
$5$		−	2.82843i		−	1.26491i	0.774597	−	0.632456i	$-0.217953\pi$
$6$		0			0
$7$		0			0		1.00000			$0$
$7$		0			0		−1.00000			$\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−3.00000	−	1.41421i	−0.904534	−	0.426401i
$11$	−3.00000	−	1.41421i	−0.904534	−	0.426401i
$12$		0			0
$13$			4.24264i			1.17670i	0.808608	+	0.588348i	$0.200222\pi$
$13$			4.24264i			1.17670i	−0.808608	+	0.588348i	$0.799778\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−6.00000			−1.45521			−0.727607	−	0.685994i	$-0.759367\pi$
$17$	−6.00000			−1.45521			−0.727607	+	0.685994i	$0.759367\pi$
$18$		0			0
$19$			4.24264i			0.973329i	0.873589	+	0.486664i	$0.161786\pi$
$19$			4.24264i			0.973329i	−0.873589	+	0.486664i	$0.838214\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$		−	1.41421i		−	0.294884i	−0.989071	−	0.147442i	$-0.952896\pi$
$23$		−	1.41421i		−	0.294884i	0.989071	−	0.147442i	$-0.0471040\pi$
$24$		0			0
$25$	−3.00000			−0.600000
$26$		0			0
$27$		0			0
$28$		0			0
$29$		0			0			−	1.00000i	$-0.5\pi$
$29$		0			0				1.00000i	$0.5\pi$
$30$		0			0
$31$	2.00000			0.359211			0.179605	−	0.983739i	$-0.442518\pi$
$31$	2.00000			0.359211			0.179605	+	0.983739i	$0.442518\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$		0			0
$37$	10.0000			1.64399			0.821995	−	0.569495i	$-0.192861\pi$
$37$	10.0000			1.64399			0.821995	+	0.569495i	$0.192861\pi$
$38$		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
$43$		−	12.7279i		−	1.94099i	−0.241121	−	0.970495i	$-0.577515\pi$
$43$		−	12.7279i		−	1.94099i	0.241121	−	0.970495i	$-0.422485\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		−	9.89949i		−	1.44399i	−0.691898	−	0.721995i	$-0.743225\pi$
$47$		−	9.89949i		−	1.44399i	0.691898	−	0.721995i	$-0.256775\pi$
$48$		0			0
$49$	7.00000			1.00000
$50$		0			0
$51$		0			0
$52$		0			0
$53$			5.65685i			0.777029i	0.921443	+	0.388514i	$0.127012\pi$
$53$			5.65685i			0.777029i	−0.921443	+	0.388514i	$0.872988\pi$
$54$		0			0
$55$	−4.00000	+	8.48528i	−0.539360	+	1.14416i
$56$		0			0
$57$		0			0
$58$		0			0
$59$			5.65685i			0.736460i	0.929735	+	0.368230i	$0.120036\pi$
$59$			5.65685i			0.736460i	−0.929735	+	0.368230i	$0.879964\pi$
$60$		0			0
$61$			12.7279i			1.62964i	0.579712	+	0.814822i	$0.303165\pi$
$61$			12.7279i			1.62964i	−0.579712	+	0.814822i	$0.696835\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	12.0000			1.48842
$66$		0			0
$67$	−8.00000			−0.977356			−0.488678	−	0.872464i	$-0.662521\pi$
$67$	−8.00000			−0.977356			−0.488678	+	0.872464i	$0.662521\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$			15.5563i			1.84620i	0.384561	+	0.923099i	$0.374353\pi$
$71$			15.5563i			1.84620i	−0.384561	+	0.923099i	$0.625647\pi$
$72$		0			0
$73$			8.48528i			0.993127i	0.868000	+	0.496564i	$0.165405\pi$
$73$			8.48528i			0.993127i	−0.868000	+	0.496564i	$0.834595\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$		0			0
$78$		0			0
$79$			8.48528i			0.954669i	0.878722	+	0.477334i	$0.158397\pi$
$79$			8.48528i			0.954669i	−0.878722	+	0.477334i	$0.841603\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−6.00000			−0.658586			−0.329293	−	0.944228i	$-0.606810\pi$
$83$	−6.00000			−0.658586			−0.329293	+	0.944228i	$0.606810\pi$
$84$		0			0
$85$			16.9706i			1.84072i
$86$		0			0
$87$		0			0
$88$		0			0
$89$			7.07107i			0.749532i	0.927119	+	0.374766i	$0.122277\pi$
$89$			7.07107i			0.749532i	−0.927119	+	0.374766i	$0.877723\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$		0			0
$94$		0			0
$95$	12.0000			1.23117
$96$		0			0
$97$	−4.00000			−0.406138			−0.203069	−	0.979164i	$-0.565092\pi$
$97$	−4.00000			−0.406138			−0.203069	+	0.979164i	$0.565092\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	6.00000			0.597022			0.298511	−	0.954406i	$-0.403510\pi$
$101$	6.00000			0.597022			0.298511	+	0.954406i	$0.403510\pi$
$102$		0			0
$103$	14.0000			1.37946			0.689730	−	0.724066i	$-0.257729\pi$
$103$	14.0000			1.37946			0.689730	+	0.724066i	$0.257729\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	12.0000			1.16008			0.580042	−	0.814587i	$-0.303036\pi$
$107$	12.0000			1.16008			0.580042	+	0.814587i	$0.303036\pi$
$108$		0			0
$109$		−	4.24264i		−	0.406371i	−0.979140	−	0.203186i	$-0.934871\pi$
$109$		−	4.24264i		−	0.406371i	0.979140	−	0.203186i	$-0.0651295\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$			7.07107i			0.665190i	0.943070	+	0.332595i	$0.107924\pi$
$113$			7.07107i			0.665190i	−0.943070	+	0.332595i	$0.892076\pi$
$114$		0			0
$115$	−4.00000			−0.373002
$116$		0			0
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$	7.00000	+	8.48528i	0.636364	+	0.771389i
$122$		0			0
$123$		0			0
$124$		0			0
$125$		−	5.65685i		−	0.505964i
$126$		0			0
$127$		−	8.48528i		−	0.752947i	−0.926427	−	0.376473i	$-0.877137\pi$
$127$		−	8.48528i		−	0.752947i	0.926427	−	0.376473i	$-0.122863\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−12.0000			−1.04844			−0.524222	−	0.851581i	$-0.675644\pi$
$131$	−12.0000			−1.04844			−0.524222	+	0.851581i	$0.675644\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		0			0
$136$		0			0
$137$		−	1.41421i		−	0.120824i	−0.998174	−	0.0604122i	$-0.980758\pi$
$137$		−	1.41421i		−	0.120824i	0.998174	−	0.0604122i	$-0.0192415\pi$
$138$		0			0
$139$		−	12.7279i		−	1.07957i	−0.841803	−	0.539784i	$-0.818506\pi$
$139$		−	12.7279i		−	1.07957i	0.841803	−	0.539784i	$-0.181494\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	6.00000	−	12.7279i	0.501745	−	1.06436i
$144$		0			0
$145$		0			0
$146$		0			0
$147$		0			0
$148$		0			0
$149$	6.00000			0.491539			0.245770	−	0.969328i	$-0.420959\pi$
$149$	6.00000			0.491539			0.245770	+	0.969328i	$0.420959\pi$
$150$		0			0
$151$		−	8.48528i		−	0.690522i	−0.938507	−	0.345261i	$-0.887790\pi$
$151$		−	8.48528i		−	0.690522i	0.938507	−	0.345261i	$-0.112210\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		−	5.65685i		−	0.454369i
$156$		0			0
$157$	10.0000			0.798087			0.399043	−	0.916932i	$-0.369342\pi$
$157$	10.0000			0.798087			0.399043	+	0.916932i	$0.369342\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$	16.0000			1.25322			0.626608	−	0.779334i	$-0.284443\pi$
$163$	16.0000			1.25322			0.626608	+	0.779334i	$0.284443\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		0			0			−	1.00000i	$-0.5\pi$
$167$		0			0				1.00000i	$0.5\pi$
$168$		0			0
$169$	−5.00000			−0.384615
$170$		0			0
$171$		0			0
$172$		0			0
$173$	12.0000			0.912343			0.456172	−	0.889892i	$-0.349220\pi$
$173$	12.0000			0.912343			0.456172	+	0.889892i	$0.349220\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		0			0
$178$		0			0
$179$			14.1421i			1.05703i	0.848923	+	0.528516i	$0.177252\pi$
$179$			14.1421i			1.05703i	−0.848923	+	0.528516i	$0.822748\pi$
$180$		0			0
$181$	−14.0000			−1.04061			−0.520306	−	0.853980i	$-0.674182\pi$
$181$	−14.0000			−1.04061			−0.520306	+	0.853980i	$0.674182\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$		−	28.2843i		−	2.07950i
$186$		0			0
$187$	18.0000	+	8.48528i	1.31629	+	0.620505i
$188$		0			0
$189$		0			0
$190$		0			0
$191$			7.07107i			0.511645i	0.966724	+	0.255822i	$0.0823462\pi$
$191$			7.07107i			0.511645i	−0.966724	+	0.255822i	$0.917654\pi$
$192$		0			0
$193$			8.48528i			0.610784i	0.952227	+	0.305392i	$0.0987875\pi$
$193$			8.48528i			0.610784i	−0.952227	+	0.305392i	$0.901213\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	18.0000			1.28245			0.641223	−	0.767354i	$-0.278427\pi$
$197$	18.0000			1.28245			0.641223	+	0.767354i	$0.278427\pi$
$198$		0			0
$199$	−16.0000			−1.13421			−0.567105	−	0.823646i	$-0.691937\pi$
$199$	−16.0000			−1.13421			−0.567105	+	0.823646i	$0.691937\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$		0			0
$204$		0			0
$205$			16.9706i			1.18528i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	6.00000	−	12.7279i	0.415029	−	0.880409i
$210$		0			0
$211$			21.2132i			1.46038i	0.683246	+	0.730189i	$0.260568\pi$
$211$			21.2132i			1.46038i	−0.683246	+	0.730189i	$0.739432\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−36.0000			−2.45518
$216$		0			0
$217$		0			0
$218$		0			0
$219$		0			0
$220$		0			0
$221$		−	25.4558i		−	1.71235i
$222$		0			0
$223$	8.00000			0.535720			0.267860	−	0.963458i	$-0.413684\pi$
$223$	8.00000			0.535720			0.267860	+	0.963458i	$0.413684\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	12.0000			0.796468			0.398234	−	0.917284i	$-0.369623\pi$
$227$	12.0000			0.796468			0.398234	+	0.917284i	$0.369623\pi$
$228$		0			0
$229$	22.0000			1.45380			0.726900	−	0.686743i	$-0.240960\pi$
$229$	22.0000			1.45380			0.726900	+	0.686743i	$0.240960\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	18.0000			1.17922			0.589610	−	0.807688i	$-0.299282\pi$
$233$	18.0000			1.17922			0.589610	+	0.807688i	$0.299282\pi$
$234$		0			0
$235$	−28.0000			−1.82652
$236$		0			0
$237$		0			0
$238$		0			0
$239$	12.0000			0.776215			0.388108	−	0.921614i	$-0.373129\pi$
$239$	12.0000			0.776215			0.388108	+	0.921614i	$0.373129\pi$
$240$		0			0
$241$			16.9706i			1.09317i	0.837404	+	0.546585i	$0.184072\pi$
$241$			16.9706i			1.09317i	−0.837404	+	0.546585i	$0.815928\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$		−	19.7990i		−	1.26491i
$246$		0			0
$247$	−18.0000			−1.14531
$248$		0			0
$249$		0			0
$250$		0			0
$251$		−	19.7990i		−	1.24970i	−0.780744	−	0.624851i	$-0.785160\pi$
$251$		−	19.7990i		−	1.24970i	0.780744	−	0.624851i	$-0.214840\pi$
$252$		0			0
$253$	−2.00000	+	4.24264i	−0.125739	+	0.266733i
$254$		0			0
$255$		0			0
$256$		0			0
$257$		−	1.41421i		−	0.0882162i	−0.999027	−	0.0441081i	$-0.985955\pi$
$257$		−	1.41421i		−	0.0882162i	0.999027	−	0.0441081i	$-0.0140446\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$		0			0
$262$		0			0
$263$		0			0			−	1.00000i	$-0.5\pi$
$263$		0			0				1.00000i	$0.5\pi$
$264$		0			0
$265$	16.0000			0.982872
$266$		0			0
$267$		0			0
$268$		0			0
$269$			5.65685i			0.344904i	0.985018	+	0.172452i	$0.0551690\pi$
$269$			5.65685i			0.344904i	−0.985018	+	0.172452i	$0.944831\pi$
$270$		0			0
$271$		−	8.48528i		−	0.515444i	−0.966219	−	0.257722i	$-0.917028\pi$
$271$		−	8.48528i		−	0.515444i	0.966219	−	0.257722i	$-0.0829719\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	9.00000	+	4.24264i	0.542720	+	0.255841i
$276$		0			0
$277$		−	4.24264i		−	0.254916i	−0.991844	−	0.127458i	$-0.959318\pi$
$277$		−	4.24264i		−	0.254916i	0.991844	−	0.127458i	$-0.0406817\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	30.0000			1.78965			0.894825	−	0.446417i	$-0.147300\pi$
$281$	30.0000			1.78965			0.894825	+	0.446417i	$0.147300\pi$
$282$		0			0
$283$		−	12.7279i		−	0.756596i	−0.925684	−	0.378298i	$-0.876509\pi$
$283$		−	12.7279i		−	0.756596i	0.925684	−	0.378298i	$-0.123491\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	19.0000			1.11765
$290$		0			0
$291$		0			0
$292$		0			0
$293$		0			0			−	1.00000i	$-0.5\pi$
$293$		0			0				1.00000i	$0.5\pi$
$294$		0			0
$295$	16.0000			0.931556
$296$		0			0
$297$		0			0
$298$		0			0
$299$	6.00000			0.346989
$300$		0			0
$301$		0			0
$302$		0			0
$303$		0			0
$304$		0			0
$305$	36.0000			2.06135
$306$		0			0
$307$			21.2132i			1.21070i	0.795959	+	0.605351i	$0.206967\pi$
$307$			21.2132i			1.21070i	−0.795959	+	0.605351i	$0.793033\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		−	1.41421i		−	0.0801927i	−0.999196	−	0.0400963i	$-0.987234\pi$
$311$		−	1.41421i		−	0.0801927i	0.999196	−	0.0400963i	$-0.0127665\pi$
$312$		0			0
$313$	8.00000			0.452187			0.226093	−	0.974106i	$-0.427405\pi$
$313$	8.00000			0.452187			0.226093	+	0.974106i	$0.427405\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$			22.6274i			1.27088i	0.772149	+	0.635441i	$0.219182\pi$
$317$			22.6274i			1.27088i	−0.772149	+	0.635441i	$0.780818\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$		0			0
$322$		0			0
$323$		−	25.4558i		−	1.41640i
$324$		0			0
$325$		−	12.7279i		−	0.706018i
$326$		0			0
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$	−20.0000			−1.09930			−0.549650	−	0.835395i	$-0.685239\pi$
$331$	−20.0000			−1.09930			−0.549650	+	0.835395i	$0.685239\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$			22.6274i			1.23627i
$336$		0			0
$337$			8.48528i			0.462223i	0.972927	+	0.231111i	$0.0742362\pi$
$337$			8.48528i			0.462223i	−0.972927	+	0.231111i	$0.925764\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−6.00000	−	2.82843i	−0.324918	−	0.153168i
$342$		0			0
$343$		0			0
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−30.0000			−1.61048			−0.805242	−	0.592946i	$-0.797965\pi$
$347$	−30.0000			−1.61048			−0.805242	+	0.592946i	$0.797965\pi$
$348$		0			0
$349$		−	29.6985i		−	1.58972i	−0.606791	−	0.794862i	$-0.707543\pi$
$349$		−	29.6985i		−	1.58972i	0.606791	−	0.794862i	$-0.292457\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$		−	9.89949i		−	0.526897i	−0.964673	−	0.263448i	$-0.915140\pi$
$353$		−	9.89949i		−	0.526897i	0.964673	−	0.263448i	$-0.0848599\pi$
$354$		0			0
$355$	44.0000			2.33528
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−12.0000			−0.633336			−0.316668	−	0.948536i	$-0.602564\pi$
$359$	−12.0000			−0.633336			−0.316668	+	0.948536i	$0.602564\pi$
$360$		0			0
$361$	1.00000			0.0526316
$362$		0			0
$363$		0			0
$364$		0			0
$365$	24.0000			1.25622
$366$		0			0
$367$	−16.0000			−0.835193			−0.417597	−	0.908633i	$-0.637127\pi$
$367$	−16.0000			−0.835193			−0.417597	+	0.908633i	$0.637127\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$			21.2132i			1.09838i	0.835698	+	0.549189i	$0.185063\pi$
$373$			21.2132i			1.09838i	−0.835698	+	0.549189i	$0.814937\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$	−20.0000			−1.02733			−0.513665	−	0.857991i	$-0.671713\pi$
$379$	−20.0000			−1.02733			−0.513665	+	0.857991i	$0.671713\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		−	9.89949i		−	0.505841i	−0.967487	−	0.252920i	$-0.918609\pi$
$383$		−	9.89949i		−	0.505841i	0.967487	−	0.252920i	$-0.0813910\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$		0			0
$389$			14.1421i			0.717035i	0.933523	+	0.358517i	$0.116718\pi$
$389$			14.1421i			0.717035i	−0.933523	+	0.358517i	$0.883282\pi$
$390$		0			0
$391$			8.48528i			0.429119i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	24.0000			1.20757
$396$		0			0
$397$	−2.00000			−0.100377			−0.0501886	−	0.998740i	$-0.515982\pi$
$397$	−2.00000			−0.100377			−0.0501886	+	0.998740i	$0.515982\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$		−	1.41421i		−	0.0706225i	−0.999376	−	0.0353112i	$-0.988758\pi$
$401$		−	1.41421i		−	0.0706225i	0.999376	−	0.0353112i	$-0.0112422\pi$
$402$		0			0
$403$			8.48528i			0.422682i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−30.0000	−	14.1421i	−1.48704	−	0.701000i
$408$		0			0
$409$			33.9411i			1.67828i	0.543915	+	0.839140i	$0.316941\pi$
$409$			33.9411i			1.67828i	−0.543915	+	0.839140i	$0.683059\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		0			0
$414$		0			0
$415$			16.9706i			0.833052i
$416$		0			0
$417$		0			0
$418$		0			0
$419$			14.1421i			0.690889i	0.938439	+	0.345444i	$0.112272\pi$
$419$			14.1421i			0.690889i	−0.938439	+	0.345444i	$0.887728\pi$
$420$		0			0
$421$	−26.0000			−1.26716			−0.633581	−	0.773676i	$-0.718416\pi$
$421$	−26.0000			−1.26716			−0.633581	+	0.773676i	$0.718416\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	18.0000			0.873128
$426$		0			0
$427$		0			0
$428$		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
$433$	−16.0000			−0.768911			−0.384455	−	0.923144i	$-0.625611\pi$
$433$	−16.0000			−0.768911			−0.384455	+	0.923144i	$0.625611\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	6.00000			0.287019
$438$		0			0
$439$		−	33.9411i		−	1.61992i	−0.586484	−	0.809961i	$-0.699488\pi$
$439$		−	33.9411i		−	1.61992i	0.586484	−	0.809961i	$-0.300512\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$			39.5980i			1.88136i	0.339300	+	0.940678i	$0.389810\pi$
$443$			39.5980i			1.88136i	−0.339300	+	0.940678i	$0.610190\pi$
$444$		0			0
$445$	20.0000			0.948091
$446$		0			0
$447$		0			0
$448$		0			0
$449$		−	9.89949i		−	0.467186i	−0.972334	−	0.233593i	$-0.924952\pi$
$449$		−	9.89949i		−	0.467186i	0.972334	−	0.233593i	$-0.0750483\pi$
$450$		0			0
$451$	18.0000	+	8.48528i	0.847587	+	0.399556i
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$			16.9706i			0.793849i	0.917851	+	0.396925i	$0.129923\pi$
$457$			16.9706i			0.793849i	−0.917851	+	0.396925i	$0.870077\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−12.0000			−0.558896			−0.279448	−	0.960161i	$-0.590151\pi$
$461$	−12.0000			−0.558896			−0.279448	+	0.960161i	$0.590151\pi$
$462$		0			0
$463$	−22.0000			−1.02243			−0.511213	−	0.859454i	$-0.670804\pi$
$463$	−22.0000			−1.02243			−0.511213	+	0.859454i	$0.670804\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$			14.1421i			0.654420i	0.944952	+	0.327210i	$0.106108\pi$
$467$			14.1421i			0.654420i	−0.944952	+	0.327210i	$0.893892\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−18.0000	+	38.1838i	−0.827641	+	1.75569i
$474$		0			0
$475$		−	12.7279i		−	0.583997i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	36.0000			1.64488			0.822441	−	0.568850i	$-0.192612\pi$
$479$	36.0000			1.64488			0.822441	+	0.568850i	$0.192612\pi$
$480$		0			0
$481$			42.4264i			1.93448i
$482$		0			0
$483$		0			0
$484$		0			0
$485$			11.3137i			0.513729i
$486$		0			0
$487$	8.00000			0.362515			0.181257	−	0.983436i	$-0.441983\pi$
$487$	8.00000			0.362515			0.181257	+	0.983436i	$0.441983\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	12.0000			0.541552			0.270776	−	0.962642i	$-0.412720\pi$
$491$	12.0000			0.541552			0.270776	+	0.962642i	$0.412720\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$	4.00000			0.179065			0.0895323	−	0.995984i	$-0.471463\pi$
$499$	4.00000			0.179065			0.0895323	+	0.995984i	$0.471463\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	12.0000			0.535054			0.267527	−	0.963550i	$-0.413794\pi$
$503$	12.0000			0.535054			0.267527	+	0.963550i	$0.413794\pi$
$504$		0			0
$505$		−	16.9706i		−	0.755180i
$506$		0			0
$507$		0			0
$508$		0			0
$509$		−	11.3137i		−	0.501471i	−0.968056	−	0.250736i	$-0.919328\pi$
$509$		−	11.3137i		−	0.501471i	0.968056	−	0.250736i	$-0.0806725\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$		0			0
$514$		0			0
$515$		−	39.5980i		−	1.74490i
$516$		0			0
$517$	−14.0000	+	29.6985i	−0.615719	+	1.30614i
$518$		0			0
$519$		0			0
$520$		0			0
$521$		−	35.3553i		−	1.54895i	−0.632607	−	0.774473i	$-0.718015\pi$
$521$		−	35.3553i		−	1.54895i	0.632607	−	0.774473i	$-0.281985\pi$
$522$		0			0
$523$		−	12.7279i		−	0.556553i	−0.960501	−	0.278277i	$-0.910237\pi$
$523$		−	12.7279i		−	0.556553i	0.960501	−	0.278277i	$-0.0897632\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−12.0000			−0.522728
$528$		0			0
$529$	21.0000			0.913043
$530$		0			0
$531$		0			0
$532$		0			0
$533$		−	25.4558i		−	1.10262i
$534$		0			0
$535$		−	33.9411i		−	1.46740i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−21.0000	−	9.89949i	−0.904534	−	0.426401i
$540$		0			0
$541$			21.2132i			0.912027i	0.889973	+	0.456013i	$0.150723\pi$
$541$			21.2132i			0.912027i	−0.889973	+	0.456013i	$0.849277\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−12.0000			−0.514024
$546$		0			0
$547$			12.7279i			0.544207i	0.962268	+	0.272103i	$0.0877193\pi$
$547$			12.7279i			0.544207i	−0.962268	+	0.272103i	$0.912281\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		0			0
$552$		0			0
$553$		0			0
$554$		0			0
$555$		0			0
$556$		0			0
$557$	12.0000			0.508456			0.254228	−	0.967144i	$-0.418179\pi$
$557$	12.0000			0.508456			0.254228	+	0.967144i	$0.418179\pi$
$558$		0			0
$559$	54.0000			2.28396
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−6.00000			−0.252870			−0.126435	−	0.991975i	$-0.540353\pi$
$563$	−6.00000			−0.252870			−0.126435	+	0.991975i	$0.540353\pi$
$564$		0			0
$565$	20.0000			0.841406
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−6.00000			−0.251533			−0.125767	−	0.992060i	$-0.540139\pi$
$569$	−6.00000			−0.251533			−0.125767	+	0.992060i	$0.540139\pi$
$570$		0			0
$571$		−	4.24264i		−	0.177549i	−0.996052	−	0.0887745i	$-0.971705\pi$
$571$		−	4.24264i		−	0.177549i	0.996052	−	0.0887745i	$-0.0282950\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$			4.24264i			0.176930i
$576$		0			0
$577$	−4.00000			−0.166522			−0.0832611	−	0.996528i	$-0.526534\pi$
$577$	−4.00000			−0.166522			−0.0832611	+	0.996528i	$0.526534\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$	8.00000	−	16.9706i	0.331326	−	0.702849i
$584$		0			0
$585$		0			0
$586$		0			0
$587$		−	28.2843i		−	1.16742i	−0.811963	−	0.583708i	$-0.801601\pi$
$587$		−	28.2843i		−	1.16742i	0.811963	−	0.583708i	$-0.198399\pi$
$588$		0			0
$589$			8.48528i			0.349630i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−30.0000			−1.23195			−0.615976	−	0.787765i	$-0.711238\pi$
$593$	−30.0000			−1.23195			−0.615976	+	0.787765i	$0.711238\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$			24.0416i			0.982314i	0.871071	+	0.491157i	$0.163426\pi$
$599$			24.0416i			0.982314i	−0.871071	+	0.491157i	$0.836574\pi$
$600$		0			0
$601$		0			0		1.00000			$0$
$601$		0			0		−1.00000			$\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	24.0000	−	19.7990i	0.975739	−	0.804943i
$606$		0			0
$607$			8.48528i			0.344407i	0.985061	+	0.172203i	$0.0550887\pi$
$607$			8.48528i			0.344407i	−0.985061	+	0.172203i	$0.944911\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	42.0000			1.69914
$612$		0			0
$613$			12.7279i			0.514076i	0.966401	+	0.257038i	$0.0827465\pi$
$613$			12.7279i			0.514076i	−0.966401	+	0.257038i	$0.917253\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$			24.0416i			0.967880i	0.875101	+	0.483940i	$0.160795\pi$
$617$			24.0416i			0.967880i	−0.875101	+	0.483940i	$0.839205\pi$
$618$		0			0
$619$	40.0000			1.60774			0.803868	−	0.594808i	$-0.202772\pi$
$619$	40.0000			1.60774			0.803868	+	0.594808i	$0.202772\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	−31.0000			−1.24000
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−60.0000			−2.39236
$630$		0			0
$631$	−22.0000			−0.875806			−0.437903	−	0.899022i	$-0.644279\pi$
$631$	−22.0000			−0.875806			−0.437903	+	0.899022i	$0.644279\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−24.0000			−0.952411
$636$		0			0
$637$			29.6985i			1.17670i
$638$		0			0
$639$		0			0
$640$		0			0
$641$			15.5563i			0.614439i	0.951639	+	0.307219i	$0.0993986\pi$
$641$			15.5563i			0.614439i	−0.951639	+	0.307219i	$0.900601\pi$
$642$		0			0
$643$	−32.0000			−1.26196			−0.630978	−	0.775800i	$-0.717346\pi$
$643$	−32.0000			−1.26196			−0.630978	+	0.775800i	$0.717346\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$			32.5269i			1.27876i	0.768889	+	0.639382i	$0.220810\pi$
$647$			32.5269i			1.27876i	−0.768889	+	0.639382i	$0.779190\pi$
$648$		0			0
$649$	8.00000	−	16.9706i	0.314027	−	0.666153i
$650$		0			0
$651$		0			0
$652$		0			0
$653$			31.1127i			1.21753i	0.793349	+	0.608767i	$0.208336\pi$
$653$			31.1127i			1.21753i	−0.793349	+	0.608767i	$0.791664\pi$
$654$		0			0
$655$			33.9411i			1.32619i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	18.0000			0.701180			0.350590	−	0.936529i	$-0.385981\pi$
$659$	18.0000			0.701180			0.350590	+	0.936529i	$0.385981\pi$
$660$		0			0
$661$	−2.00000			−0.0777910			−0.0388955	−	0.999243i	$-0.512384\pi$
$661$	−2.00000			−0.0777910			−0.0388955	+	0.999243i	$0.512384\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$		0			0
$668$		0			0
$669$		0			0
$670$		0			0
$671$	18.0000	−	38.1838i	0.694882	−	1.47407i
$672$		0			0
$673$		−	42.4264i		−	1.63542i	−0.575632	−	0.817709i	$-0.695244\pi$
$673$		−	42.4264i		−	1.63542i	0.575632	−	0.817709i	$-0.304756\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	6.00000			0.230599			0.115299	−	0.993331i	$-0.463217\pi$
$677$	6.00000			0.230599			0.115299	+	0.993331i	$0.463217\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$		0			0
$682$		0			0
$683$			31.1127i			1.19049i	0.803543	+	0.595247i	$0.202946\pi$
$683$			31.1127i			1.19049i	−0.803543	+	0.595247i	$0.797054\pi$
$684$		0			0
$685$	−4.00000			−0.152832
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−24.0000			−0.914327
$690$		0			0
$691$	−20.0000			−0.760836			−0.380418	−	0.924815i	$-0.624220\pi$
$691$	−20.0000			−0.760836			−0.380418	+	0.924815i	$0.624220\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−36.0000			−1.36556
$696$		0			0
$697$	36.0000			1.36360
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−48.0000			−1.81293			−0.906467	−	0.422276i	$-0.861231\pi$
$701$	−48.0000			−1.81293			−0.906467	+	0.422276i	$0.861231\pi$
$702$		0			0
$703$			42.4264i			1.60014i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	34.0000			1.27690			0.638448	−	0.769665i	$-0.279577\pi$
$709$	34.0000			1.27690			0.638448	+	0.769665i	$0.279577\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$		−	2.82843i		−	0.105925i
$714$		0			0
$715$	−36.0000	−	16.9706i	−1.34632	−	0.634663i
$716$		0			0
$717$		0			0
$718$		0			0
$719$		−	1.41421i		−	0.0527413i	−0.999652	−	0.0263706i	$-0.991605\pi$
$719$		−	1.41421i		−	0.0527413i	0.999652	−	0.0263706i	$-0.00839501\pi$
$720$		0			0
$721$		0			0
$722$		0			0
$723$		0			0
$724$		0			0
$725$		0			0
$726$		0			0
$727$	−40.0000			−1.48352			−0.741759	−	0.670667i	$-0.766008\pi$
$727$	−40.0000			−1.48352			−0.741759	+	0.670667i	$0.766008\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$			76.3675i			2.82456i
$732$		0			0
$733$		−	4.24264i		−	0.156706i	−0.996926	−	0.0783528i	$-0.975034\pi$
$733$		−	4.24264i		−	0.156706i	0.996926	−	0.0783528i	$-0.0249660\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	24.0000	+	11.3137i	0.884051	+	0.416746i
$738$		0			0
$739$		−	21.2132i		−	0.780340i	−0.920743	−	0.390170i	$-0.872416\pi$
$739$		−	21.2132i		−	0.780340i	0.920743	−	0.390170i	$-0.127584\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	36.0000			1.32071			0.660356	−	0.750953i	$-0.270405\pi$
$743$	36.0000			1.32071			0.660356	+	0.750953i	$0.270405\pi$
$744$		0			0
$745$		−	16.9706i		−	0.621753i
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	2.00000			0.0729810			0.0364905	−	0.999334i	$-0.488382\pi$
$751$	2.00000			0.0729810			0.0364905	+	0.999334i	$0.488382\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−24.0000			−0.873449
$756$		0			0
$757$	34.0000			1.23575			0.617876	−	0.786276i	$-0.287994\pi$
$757$	34.0000			1.23575			0.617876	+	0.786276i	$0.287994\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−42.0000			−1.52250			−0.761249	−	0.648459i	$-0.775414\pi$
$761$	−42.0000			−1.52250			−0.761249	+	0.648459i	$0.775414\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−24.0000			−0.866590
$768$		0			0
$769$			8.48528i			0.305987i	0.988227	+	0.152994i	$0.0488914\pi$
$769$			8.48528i			0.305987i	−0.988227	+	0.152994i	$0.951109\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$			5.65685i			0.203463i	0.994812	+	0.101731i	$0.0324382\pi$
$773$			5.65685i			0.203463i	−0.994812	+	0.101731i	$0.967562\pi$
$774$		0			0
$775$	−6.00000			−0.215526
$776$		0			0
$777$		0			0
$778$		0			0
$779$		−	25.4558i		−	0.912050i
$780$		0			0
$781$	22.0000	−	46.6690i	0.787222	−	1.66995i
$782$		0			0
$783$		0			0
$784$		0			0
$785$		−	28.2843i		−	1.00951i
$786$		0			0
$787$			21.2132i			0.756169i	0.925771	+	0.378085i	$0.123417\pi$
$787$			21.2132i			0.756169i	−0.925771	+	0.378085i	$0.876583\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	−54.0000			−1.91760
$794$		0			0
$795$		0			0
$796$		0			0
$797$			31.1127i			1.10207i	0.834483	+	0.551034i	$0.185767\pi$
$797$			31.1127i			1.10207i	−0.834483	+	0.551034i	$0.814233\pi$
$798$		0			0
$799$			59.3970i			2.10131i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	12.0000	−	25.4558i	0.423471	−	0.898317i
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−18.0000			−0.632846			−0.316423	−	0.948618i	$-0.602482\pi$
$809$	−18.0000			−0.632846			−0.316423	+	0.948618i	$0.602482\pi$
$810$		0			0
$811$		−	46.6690i		−	1.63877i	−0.573242	−	0.819386i	$-0.694315\pi$
$811$		−	46.6690i		−	1.63877i	0.573242	−	0.819386i	$-0.305685\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		−	45.2548i		−	1.58521i
$816$		0			0
$817$	54.0000			1.88922
$818$		0			0
$819$		0			0
$820$		0			0
$821$	18.0000			0.628204			0.314102	−	0.949389i	$-0.398297\pi$
$821$	18.0000			0.628204			0.314102	+	0.949389i	$0.398297\pi$
$822$		0			0
$823$	32.0000			1.11545			0.557725	−	0.830026i	$-0.311674\pi$
$823$	32.0000			1.11545			0.557725	+	0.830026i	$0.311674\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−42.0000			−1.46048			−0.730242	−	0.683189i	$-0.760592\pi$
$827$	−42.0000			−1.46048			−0.730242	+	0.683189i	$0.760592\pi$
$828$		0			0
$829$	−14.0000			−0.486240			−0.243120	−	0.969996i	$-0.578171\pi$
$829$	−14.0000			−0.486240			−0.243120	+	0.969996i	$0.578171\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−42.0000			−1.45521
$834$		0			0
$835$		0			0
$836$		0			0
$837$		0			0
$838$		0			0
$839$		−	9.89949i		−	0.341769i	−0.985291	−	0.170884i	$-0.945338\pi$
$839$		−	9.89949i		−	0.341769i	0.985291	−	0.170884i	$-0.0546624\pi$
$840$		0			0
$841$	−29.0000			−1.00000
$842$		0			0
$843$		0			0
$844$		0			0
$845$			14.1421i			0.486504i
$846$		0			0
$847$		0			0
$848$		0			0
$849$		0			0
$850$		0			0
$851$		−	14.1421i		−	0.484786i
$852$		0			0
$853$			4.24264i			0.145265i	0.997359	+	0.0726326i	$0.0231401\pi$
$853$			4.24264i			0.145265i	−0.997359	+	0.0726326i	$0.976860\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	30.0000			1.02478			0.512390	−	0.858753i	$-0.328760\pi$
$857$	30.0000			1.02478			0.512390	+	0.858753i	$0.328760\pi$
$858$		0			0
$859$	−8.00000			−0.272956			−0.136478	−	0.990643i	$-0.543578\pi$
$859$	−8.00000			−0.272956			−0.136478	+	0.990643i	$0.543578\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$			7.07107i			0.240702i	0.992731	+	0.120351i	$0.0384020\pi$
$863$			7.07107i			0.240702i	−0.992731	+	0.120351i	$0.961598\pi$
$864$		0			0
$865$		−	33.9411i		−	1.15403i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	12.0000	−	25.4558i	0.407072	−	0.863530i
$870$		0			0
$871$		−	33.9411i		−	1.15005i
$872$		0			0
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$			38.1838i			1.28937i	0.764447	+	0.644687i	$0.223012\pi$
$877$			38.1838i			1.28937i	−0.764447	+	0.644687i	$0.776988\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$			24.0416i			0.809983i	0.914320	+	0.404992i	$0.132726\pi$
$881$			24.0416i			0.809983i	−0.914320	+	0.404992i	$0.867274\pi$
$882$		0			0
$883$	16.0000			0.538443			0.269221	−	0.963078i	$-0.413234\pi$
$883$	16.0000			0.538443			0.269221	+	0.963078i	$0.413234\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	36.0000			1.20876			0.604381	−	0.796696i	$-0.293421\pi$
$887$	36.0000			1.20876			0.604381	+	0.796696i	$0.293421\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$		0			0
$892$		0			0
$893$	42.0000			1.40548
$894$		0			0
$895$	40.0000			1.33705
$896$		0			0
$897$		0			0
$898$		0			0
$899$		0			0
$900$		0			0
$901$		−	33.9411i		−	1.13074i
$902$		0			0
$903$		0			0
$904$		0			0
$905$			39.5980i			1.31628i
$906$		0			0
$907$	−8.00000			−0.265636			−0.132818	−	0.991140i	$-0.542403\pi$
$907$	−8.00000			−0.265636			−0.132818	+	0.991140i	$0.542403\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$			15.5563i			0.515405i	0.966224	+	0.257702i	$0.0829654\pi$
$911$			15.5563i			0.515405i	−0.966224	+	0.257702i	$0.917035\pi$
$912$		0			0
$913$	18.0000	+	8.48528i	0.595713	+	0.280822i
$914$		0			0
$915$		0			0
$916$		0			0
$917$		0			0
$918$		0			0
$919$		−	25.4558i		−	0.839711i	−0.907591	−	0.419855i	$-0.862081\pi$
$919$		−	25.4558i		−	0.839711i	0.907591	−	0.419855i	$-0.137919\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−66.0000			−2.17242
$924$		0			0
$925$	−30.0000			−0.986394
$926$		0			0
$927$		0			0
$928$		0			0
$929$		−	9.89949i		−	0.324792i	−0.986726	−	0.162396i	$-0.948078\pi$
$929$		−	9.89949i		−	0.324792i	0.986726	−	0.162396i	$-0.0519222\pi$
$930$		0			0
$931$			29.6985i			0.973329i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	24.0000	−	50.9117i	0.784884	−	1.66499i
$936$		0			0
$937$			33.9411i			1.10881i	0.832248	+	0.554404i	$0.187054\pi$
$937$			33.9411i			1.10881i	−0.832248	+	0.554404i	$0.812946\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−30.0000			−0.977972			−0.488986	−	0.872292i	$-0.662633\pi$
$941$	−30.0000			−0.977972			−0.488986	+	0.872292i	$0.662633\pi$
$942$		0			0
$943$			8.48528i			0.276319i
$944$		0			0
$945$		0			0
$946$		0			0
$947$		−	45.2548i		−	1.47058i	−0.677750	−	0.735292i	$-0.737045\pi$
$947$		−	45.2548i		−	1.47058i	0.677750	−	0.735292i	$-0.262955\pi$
$948$		0			0
$949$	−36.0000			−1.16861
$950$		0			0
$951$		0			0
$952$		0			0
$953$	6.00000			0.194359			0.0971795	−	0.995267i	$-0.469018\pi$
$953$	6.00000			0.194359			0.0971795	+	0.995267i	$0.469018\pi$
$954$		0			0
$955$	20.0000			0.647185
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	−27.0000			−0.870968
$962$		0			0
$963$		0			0
$964$		0			0
$965$	24.0000			0.772587
$966$		0			0
$967$			25.4558i			0.818605i	0.912399	+	0.409302i	$0.134228\pi$
$967$			25.4558i			0.818605i	−0.912399	+	0.409302i	$0.865772\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$			5.65685i			0.181537i	0.995872	+	0.0907685i	$0.0289323\pi$
$971$			5.65685i			0.181537i	−0.995872	+	0.0907685i	$0.971068\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$		0			0
$976$		0			0
$977$			24.0416i			0.769160i	0.923092	+	0.384580i	$0.125654\pi$
$977$			24.0416i			0.769160i	−0.923092	+	0.384580i	$0.874346\pi$
$978$		0			0
$979$	10.0000	−	21.2132i	0.319601	−	0.677977i
$980$		0			0
$981$		0			0
$982$		0			0
$983$			49.4975i			1.57872i	0.613928	+	0.789362i	$0.289589\pi$
$983$			49.4975i			1.57872i	−0.613928	+	0.789362i	$0.710411\pi$
$984$		0			0
$985$		−	50.9117i		−	1.62218i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−18.0000			−0.572367
$990$		0			0
$991$	−34.0000			−1.08005			−0.540023	−	0.841650i	$-0.681584\pi$
$991$	−34.0000			−1.08005			−0.540023	+	0.841650i	$0.681584\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$			45.2548i			1.43467i
$996$		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.b.a.2177.1		2
3.2	odd	2		6336.2.b.n.2177.2		2
4.3	odd	2		6336.2.b.i.2177.1		2
8.3	odd	2		1584.2.b.a.593.2		2
8.5	even	2		198.2.b.b.197.2	yes	2
11.10	odd	2		6336.2.b.n.2177.1		2
12.11	even	2		6336.2.b.f.2177.2		2
24.5	odd	2		198.2.b.a.197.1	✓	2
24.11	even	2		1584.2.b.d.593.1		2
33.32	even	2	inner	6336.2.b.a.2177.2		2
40.13	odd	4		4950.2.f.b.4949.2		4
40.29	even	2		4950.2.d.b.4751.2		2
40.37	odd	4		4950.2.f.b.4949.4		4
44.43	even	2		6336.2.b.f.2177.1		2
72.5	odd	6		1782.2.i.g.593.1		4
72.13	even	6		1782.2.i.b.593.2		4
72.29	odd	6		1782.2.i.g.1187.2		4
72.61	even	6		1782.2.i.b.1187.1		4
88.21	odd	2		198.2.b.a.197.2	yes	2
88.43	even	2		1584.2.b.d.593.2		2
120.29	odd	2		4950.2.d.e.4751.1		2
120.53	even	4		4950.2.f.a.4949.3		4
120.77	even	4		4950.2.f.a.4949.1		4
132.131	odd	2		6336.2.b.i.2177.2		2
264.131	odd	2		1584.2.b.a.593.1		2
264.197	even	2		198.2.b.b.197.1	yes	2
440.109	odd	2		4950.2.d.e.4751.2		2
440.197	even	4		4950.2.f.a.4949.2		4
440.373	even	4		4950.2.f.a.4949.4		4
792.373	odd	6		1782.2.i.g.593.2		4
792.461	even	6		1782.2.i.b.1187.2		4
792.637	odd	6		1782.2.i.g.1187.1		4
792.725	even	6		1782.2.i.b.593.1		4
1320.197	odd	4		4950.2.f.b.4949.3		4
1320.989	even	2		4950.2.d.b.4751.1		2
1320.1253	odd	4		4950.2.f.b.4949.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
198.2.b.a.197.1	✓	2	24.5	odd	2
198.2.b.a.197.2	yes	2	88.21	odd	2
198.2.b.b.197.1	yes	2	264.197	even	2
198.2.b.b.197.2	yes	2	8.5	even	2
1584.2.b.a.593.1		2	264.131	odd	2
1584.2.b.a.593.2		2	8.3	odd	2
1584.2.b.d.593.1		2	24.11	even	2
1584.2.b.d.593.2		2	88.43	even	2
1782.2.i.b.593.1		4	792.725	even	6
1782.2.i.b.593.2		4	72.13	even	6
1782.2.i.b.1187.1		4	72.61	even	6
1782.2.i.b.1187.2		4	792.461	even	6
1782.2.i.g.593.1		4	72.5	odd	6
1782.2.i.g.593.2		4	792.373	odd	6
1782.2.i.g.1187.1		4	792.637	odd	6
1782.2.i.g.1187.2		4	72.29	odd	6
4950.2.d.b.4751.1		2	1320.989	even	2
4950.2.d.b.4751.2		2	40.29	even	2
4950.2.d.e.4751.1		2	120.29	odd	2
4950.2.d.e.4751.2		2	440.109	odd	2
4950.2.f.a.4949.1		4	120.77	even	4
4950.2.f.a.4949.2		4	440.197	even	4
4950.2.f.a.4949.3		4	120.53	even	4
4950.2.f.a.4949.4		4	440.373	even	4
4950.2.f.b.4949.1		4	1320.1253	odd	4
4950.2.f.b.4949.2		4	40.13	odd	4
4950.2.f.b.4949.3		4	1320.197	odd	4
4950.2.f.b.4949.4		4	40.37	odd	4
6336.2.b.a.2177.1		2	1.1	even	1	trivial
6336.2.b.a.2177.2		2	33.32	even	2	inner
6336.2.b.f.2177.1		2	44.43	even	2
6336.2.b.f.2177.2		2	12.11	even	2
6336.2.b.i.2177.1		2	4.3	odd	2
6336.2.b.i.2177.2		2	132.131	odd	2
6336.2.b.n.2177.1		2	11.10	odd	2
6336.2.b.n.2177.2		2	3.2	odd	2

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.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-2.82843i q^{5} +(-3.00000 - 1.41421i) q^{11} +4.24264i q^{13} -6.00000 q^{17} +4.24264i q^{19} -1.41421i q^{23} -3.00000 q^{25} +2.00000 q^{31} +10.0000 q^{37} -6.00000 q^{41} -12.7279i q^{43} -9.89949i q^{47} +7.00000 q^{49} +5.65685i q^{53} +(-4.00000 + 8.48528i) q^{55} +5.65685i q^{59} +12.7279i q^{61} +12.0000 q^{65} -8.00000 q^{67} +15.5563i q^{71} +8.48528i q^{73} +8.48528i q^{79} -6.00000 q^{83} +16.9706i q^{85} +7.07107i q^{89} +12.0000 q^{95} -4.00000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 6 q^{11} - 12 q^{17} - 6 q^{25} + 4 q^{31} + 20 q^{37} - 12 q^{41} + 14 q^{49} - 8 q^{55} + 24 q^{65} - 16 q^{67} - 12 q^{83} + 24 q^{95} - 8 q^{97}+O(q^{100}) \) 2 * q - 6 * q^11 - 12 * q^17 - 6 * q^25 + 4 * q^31 + 20 * q^37 - 12 * q^41 + 14 * q^49 - 8 * q^55 + 24 * q^65 - 16 * q^67 - 12 * q^83 + 24 * q^95 - 8 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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