LMFDB - Embedded newform 5472.2.p.a.3761.55

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5472,2,Mod(3761,5472)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5472, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("5472.3761");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 5472 = 2^{5} \cdot 3^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5472.p (of order $2$, degree $1$, not minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$43.6941399860$
Analytic rank:	$0$
Dimension:	$80$
Twist minimal:	no (minimal twist has level 1368)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			3761.55
Character	$\chi$	$=$	5472.3761
Dual form			5472.2.p.a.3761.54

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.64947 q^{5} +2.97899 q^{7} +O(q^{10})$	$q+1.64947 q^{5} +2.97899 q^{7} -2.43029 q^{11} +4.57864 q^{13} +0.746594i q^{17} +(1.01655 + 4.23871i) q^{19} -6.97482i q^{23} -2.27924 q^{25} -6.73090i q^{29} -1.79093i q^{31} +4.91376 q^{35} +10.0699 q^{37} +1.48683 q^{41} -7.95894i q^{43} +9.17522i q^{47} +1.87440 q^{49} -10.0354i q^{53} -4.00869 q^{55} -6.89777i q^{59} +12.1920i q^{61} +7.55233 q^{65} +2.52529 q^{67} +8.76160 q^{71} -1.97358 q^{73} -7.23981 q^{77} +7.02559i q^{79} +12.6320 q^{83} +1.23148i q^{85} +7.99416 q^{89} +13.6397 q^{91} +(1.67677 + 6.99162i) q^{95} -0.170738i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 80 q+O(q^{10}) $ 80 * q	$ 80 q + 80 q^{25} + 48 q^{49} + 32 q^{73}+O(q^{100}) $ 80 * q + 80 * q^25 + 48 * q^49 + 32 * q^73

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/5472\mathbb{Z}\right)^\times$.

$n$	$1217$	$2053$	$3745$	$4447$
$\chi(n)$	$-1$	$-1$	$-1$	$1$

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
$5$	1.64947			0.737666			0.368833	−	0.929496i	$-0.379757\pi$
$5$	1.64947			0.737666			0.368833	+	0.929496i	$0.379757\pi$
$6$		0			0
$7$	2.97899			1.12595			0.562977	−	0.826473i	$-0.309656\pi$
$7$	2.97899			1.12595			0.562977	+	0.826473i	$0.309656\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−2.43029			−0.732759			−0.366380	−	0.930465i	$-0.619403\pi$
$11$	−2.43029			−0.732759			−0.366380	+	0.930465i	$0.619403\pi$
$12$		0			0
$13$	4.57864			1.26989			0.634943	−	0.772559i	$-0.281024\pi$
$13$	4.57864			1.26989			0.634943	+	0.772559i	$0.281024\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$			0.746594i			0.181076i	0.995893	+	0.0905378i	$0.0288586\pi$
$17$			0.746594i			0.181076i	−0.995893	+	0.0905378i	$0.971141\pi$
$18$		0			0
$19$	1.01655	+	4.23871i	0.233212	+	0.972426i
$19$	1.01655	+	4.23871i	0.233212	+	0.972426i
$20$		0			0
$21$		0			0
$22$		0			0
$23$		−	6.97482i		−	1.45435i	−0.686451	−	0.727176i	$-0.740833\pi$
$23$		−	6.97482i		−	1.45435i	0.686451	−	0.727176i	$-0.259167\pi$
$24$		0			0
$25$	−2.27924			−0.455849
$26$		0			0
$27$		0			0
$28$		0			0
$29$		−	6.73090i		−	1.24990i	−0.780666	−	0.624948i	$-0.785120\pi$
$29$		−	6.73090i		−	1.24990i	0.780666	−	0.624948i	$-0.214880\pi$
$30$		0			0
$31$		−	1.79093i		−	0.321661i	−0.986982	−	0.160830i	$-0.948583\pi$
$31$		−	1.79093i		−	0.321661i	0.986982	−	0.160830i	$-0.0514172\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	4.91376			0.830577
$36$		0			0
$37$	10.0699			1.65548			0.827738	−	0.561114i	$-0.189627\pi$
$37$	10.0699			1.65548			0.827738	+	0.561114i	$0.189627\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	1.48683			0.232204			0.116102	−	0.993237i	$-0.462960\pi$
$41$	1.48683			0.232204			0.116102	+	0.993237i	$0.462960\pi$
$42$		0			0
$43$		−	7.95894i		−	1.21373i	−0.794806	−	0.606863i	$-0.792428\pi$
$43$		−	7.95894i		−	1.21373i	0.794806	−	0.606863i	$-0.207572\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$			9.17522i			1.33834i	0.743108	+	0.669172i	$0.233351\pi$
$47$			9.17522i			1.33834i	−0.743108	+	0.669172i	$0.766649\pi$
$48$		0			0
$49$	1.87440			0.267771
$50$		0			0
$51$		0			0
$52$		0			0
$53$		−	10.0354i		−	1.37846i	−0.724541	−	0.689232i	$-0.757948\pi$
$53$		−	10.0354i		−	1.37846i	0.724541	−	0.689232i	$-0.242052\pi$
$54$		0			0
$55$	−4.00869			−0.540532
$56$		0			0
$57$		0			0
$58$		0			0
$59$		−	6.89777i		−	0.898014i	−0.893528	−	0.449007i	$-0.851778\pi$
$59$		−	6.89777i		−	0.898014i	0.893528	−	0.449007i	$-0.148222\pi$
$60$		0			0
$61$			12.1920i			1.56102i	0.625141	+	0.780512i	$0.285041\pi$
$61$			12.1920i			1.56102i	−0.625141	+	0.780512i	$0.714959\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	7.55233			0.936751
$66$		0			0
$67$	2.52529			0.308513			0.154256	−	0.988031i	$-0.450702\pi$
$67$	2.52529			0.308513			0.154256	+	0.988031i	$0.450702\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	8.76160			1.03981			0.519905	−	0.854224i	$-0.325967\pi$
$71$	8.76160			1.03981			0.519905	+	0.854224i	$0.325967\pi$
$72$		0			0
$73$	−1.97358			−0.230990			−0.115495	−	0.993308i	$-0.536845\pi$
$73$	−1.97358			−0.230990			−0.115495	+	0.993308i	$0.536845\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−7.23981			−0.825053
$78$		0			0
$79$			7.02559i			0.790441i	0.918586	+	0.395220i	$0.129332\pi$
$79$			7.02559i			0.790441i	−0.918586	+	0.395220i	$0.870668\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	12.6320			1.38655			0.693274	−	0.720675i	$-0.256168\pi$
$83$	12.6320			1.38655			0.693274	+	0.720675i	$0.256168\pi$
$84$		0			0
$85$			1.23148i			0.133573i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	7.99416			0.847379			0.423690	−	0.905807i	$-0.360735\pi$
$89$	7.99416			0.847379			0.423690	+	0.905807i	$0.360735\pi$
$90$		0			0
$91$	13.6397			1.42983
$92$		0			0
$93$		0			0
$94$		0			0
$95$	1.67677	+	6.99162i	0.172033	+	0.717325i
$96$		0			0
$97$		−	0.170738i		−	0.0173358i	−0.999962	−	0.00866790i	$-0.997241\pi$
$97$		−	0.170738i		−	0.0173358i	0.999962	−	0.00866790i	$-0.00275911\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−5.91277			−0.588343			−0.294171	−	0.955753i	$-0.595044\pi$
$101$	−5.91277			−0.588343			−0.294171	+	0.955753i	$0.595044\pi$
$102$		0			0
$103$		−	16.8856i		−	1.66379i	−0.554933	−	0.831895i	$-0.687256\pi$
$103$		−	16.8856i		−	1.66379i	0.554933	−	0.831895i	$-0.312744\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$			11.8408i			1.14469i	0.820011	+	0.572347i	$0.193967\pi$
$107$			11.8408i			1.14469i	−0.820011	+	0.572347i	$0.806033\pi$
$108$		0			0
$109$	8.04657			0.770721			0.385361	−	0.922766i	$-0.374077\pi$
$109$	8.04657			0.770721			0.385361	+	0.922766i	$0.374077\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−17.3103			−1.62842			−0.814208	−	0.580574i	$-0.802828\pi$
$113$	−17.3103			−1.62842			−0.814208	+	0.580574i	$0.802828\pi$
$114$		0			0
$115$		−	11.5048i		−	1.07283i
$116$		0			0
$117$		0			0
$118$		0			0
$119$			2.22410i			0.203883i
$120$		0			0
$121$	−5.09370			−0.463064
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−12.0069			−1.07393
$126$		0			0
$127$		−	9.26771i		−	0.822376i	−0.911551	−	0.411188i	$-0.865114\pi$
$127$		−	9.26771i		−	0.822376i	0.911551	−	0.411188i	$-0.134886\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	17.5536			1.53367			0.766834	−	0.641845i	$-0.221831\pi$
$131$	17.5536			1.53367			0.766834	+	0.641845i	$0.221831\pi$
$132$		0			0
$133$	3.02829	+	12.6271i	0.262586	+	1.09491i
$134$		0			0
$135$		0			0
$136$		0			0
$137$			11.7575i			1.00451i	0.864720	+	0.502255i	$0.167496\pi$
$137$			11.7575i			1.00451i	−0.864720	+	0.502255i	$0.832504\pi$
$138$		0			0
$139$			3.19586i			0.271069i	0.990773	+	0.135535i	$0.0432752\pi$
$139$			3.19586i			0.271069i	−0.990773	+	0.135535i	$0.956725\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−11.1274			−0.930520
$144$		0			0
$145$		−	11.1024i		−	0.922006i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−7.47962			−0.612754			−0.306377	−	0.951910i	$-0.599117\pi$
$149$	−7.47962			−0.612754			−0.306377	+	0.951910i	$0.599117\pi$
$150$		0			0
$151$			22.1041i			1.79881i	0.437119	+	0.899404i	$0.355999\pi$
$151$			22.1041i			1.79881i	−0.437119	+	0.899404i	$0.644001\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		−	2.95409i		−	0.237278i
$156$		0			0
$157$			8.27782i			0.660642i	0.943869	+	0.330321i	$0.107157\pi$
$157$			8.27782i			0.660642i	−0.943869	+	0.330321i	$0.892843\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		−	20.7779i		−	1.63753i
$162$		0			0
$163$		−	10.6737i		−	0.836031i	−0.908440	−	0.418015i	$-0.862726\pi$
$163$		−	10.6737i		−	0.836031i	0.908440	−	0.418015i	$-0.137274\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−8.16879			−0.632120			−0.316060	−	0.948739i	$-0.602360\pi$
$167$	−8.16879			−0.632120			−0.316060	+	0.948739i	$0.602360\pi$
$168$		0			0
$169$	7.96392			0.612609
$170$		0			0
$171$		0			0
$172$		0			0
$173$		−	22.4235i		−	1.70483i	−0.522869	−	0.852413i	$-0.675138\pi$
$173$		−	22.4235i		−	1.70483i	0.522869	−	0.852413i	$-0.324862\pi$
$174$		0			0
$175$	−6.78985			−0.513265
$176$		0			0
$177$		0			0
$178$		0			0
$179$			1.45991i			0.109119i	0.998511	+	0.0545594i	$0.0173754\pi$
$179$			1.45991i			0.109119i	−0.998511	+	0.0545594i	$0.982625\pi$
$180$		0			0
$181$	2.13358			0.158588			0.0792940	−	0.996851i	$-0.474733\pi$
$181$	2.13358			0.158588			0.0792940	+	0.996851i	$0.474733\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	16.6100			1.22119
$186$		0			0
$187$		−	1.81444i		−	0.132685i
$188$		0			0
$189$		0			0
$190$		0			0
$191$		−	8.60268i		−	0.622468i	−0.950333	−	0.311234i	$-0.899258\pi$
$191$		−	8.60268i		−	0.622468i	0.950333	−	0.311234i	$-0.100742\pi$
$192$		0			0
$193$			19.1276i			1.37683i	0.725316	+	0.688416i	$0.241694\pi$
$193$			19.1276i			1.37683i	−0.725316	+	0.688416i	$0.758306\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	12.2365			0.871815			0.435908	−	0.899991i	$-0.356427\pi$
$197$	12.2365			0.871815			0.435908	+	0.899991i	$0.356427\pi$
$198$		0			0
$199$	13.4297			0.952009			0.476005	−	0.879443i	$-0.342085\pi$
$199$	13.4297			0.952009			0.476005	+	0.879443i	$0.342085\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$		−	20.0513i		−	1.40733i
$204$		0			0
$205$	2.45249			0.171289
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−2.47050	−	10.3013i	−0.170888	−	0.712554i
$210$		0			0
$211$	7.90829			0.544429			0.272214	−	0.962237i	$-0.412244\pi$
$211$	7.90829			0.544429			0.272214	+	0.962237i	$0.412244\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		−	13.1280i		−	0.895325i
$216$		0			0
$217$		−	5.33517i		−	0.362175i
$218$		0			0
$219$		0			0
$220$		0			0
$221$			3.41838i			0.229945i
$222$		0			0
$223$			1.85770i			0.124400i	0.998064	+	0.0622002i	$0.0198117\pi$
$223$			1.85770i			0.124400i	−0.998064	+	0.0622002i	$0.980188\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$		−	8.00287i		−	0.531169i	−0.964088	−	0.265585i	$-0.914435\pi$
$227$		−	8.00287i		−	0.531169i	0.964088	−	0.265585i	$-0.0855650\pi$
$228$		0			0
$229$		−	18.3440i		−	1.21221i	−0.795386	−	0.606103i	$-0.792732\pi$
$229$		−	18.3440i		−	1.21221i	0.795386	−	0.606103i	$-0.207268\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		−	0.594597i		−	0.0389534i	−0.999810	−	0.0194767i	$-0.993800\pi$
$233$		−	0.594597i		−	0.0389534i	0.999810	−	0.0194767i	$-0.00620001\pi$
$234$		0			0
$235$			15.1343i			0.987250i
$236$		0			0
$237$		0			0
$238$		0			0
$239$		−	3.89453i		−	0.251916i	−0.992036	−	0.125958i	$-0.959799\pi$
$239$		−	3.89453i		−	0.251916i	0.992036	−	0.125958i	$-0.0402005\pi$
$240$		0			0
$241$			23.9755i			1.54440i	0.635382	+	0.772198i	$0.280843\pi$
$241$			23.9755i			1.54440i	−0.635382	+	0.772198i	$0.719157\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	3.09176			0.197525
$246$		0			0
$247$	4.65441	+	19.4075i	0.296153	+	1.23487i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	1.81498			0.114560			0.0572802	−	0.998358i	$-0.481757\pi$
$251$	1.81498			0.114560			0.0572802	+	0.998358i	$0.481757\pi$
$252$		0			0
$253$			16.9508i			1.06569i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−24.5922			−1.53402			−0.767010	−	0.641635i	$-0.778256\pi$
$257$	−24.5922			−1.53402			−0.767010	+	0.641635i	$0.778256\pi$
$258$		0			0
$259$	29.9981			1.86399
$260$		0			0
$261$		0			0
$262$		0			0
$263$			14.7000i			0.906444i	0.891398	+	0.453222i	$0.149725\pi$
$263$			14.7000i			0.906444i	−0.891398	+	0.453222i	$0.850275\pi$
$264$		0			0
$265$		−	16.5530i		−	1.01685i
$266$		0			0
$267$		0			0
$268$		0			0
$269$			3.83007i			0.233523i	0.993160	+	0.116762i	$0.0372514\pi$
$269$			3.83007i			0.233523i	−0.993160	+	0.116762i	$0.962749\pi$
$270$		0			0
$271$	3.62738			0.220348			0.110174	−	0.993912i	$-0.464859\pi$
$271$	3.62738			0.220348			0.110174	+	0.993912i	$0.464859\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	5.53922			0.334028
$276$		0			0
$277$			0.168443i			0.0101208i	0.999987	+	0.00506038i	$0.00161078\pi$
$277$			0.168443i			0.0101208i	−0.999987	+	0.00506038i	$0.998389\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−19.9619			−1.19083			−0.595414	−	0.803419i	$-0.703012\pi$
$281$	−19.9619			−1.19083			−0.595414	+	0.803419i	$0.703012\pi$
$282$		0			0
$283$			21.2677i			1.26423i	0.774874	+	0.632116i	$0.217814\pi$
$283$			21.2677i			1.26423i	−0.774874	+	0.632116i	$0.782186\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	4.42926			0.261451
$288$		0			0
$289$	16.4426			0.967212
$290$		0			0
$291$		0			0
$292$		0			0
$293$		−	9.95018i		−	0.581296i	−0.956830	−	0.290648i	$-0.906129\pi$
$293$		−	9.95018i		−	0.581296i	0.956830	−	0.290648i	$-0.0938708\pi$
$294$		0			0
$295$		−	11.3777i		−	0.662434i
$296$		0			0
$297$		0			0
$298$		0			0
$299$		−	31.9352i		−	1.84686i
$300$		0			0
$301$		−	23.7096i		−	1.36660i
$302$		0			0
$303$		0			0
$304$		0			0
$305$			20.1103i			1.15151i
$306$		0			0
$307$	17.4039			0.993292			0.496646	−	0.867953i	$-0.334565\pi$
$307$	17.4039			0.993292			0.496646	+	0.867953i	$0.334565\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$			4.41009i			0.250073i	0.992152	+	0.125037i	$0.0399048\pi$
$311$			4.41009i			0.250073i	−0.992152	+	0.125037i	$0.960095\pi$
$312$		0			0
$313$	−24.8414			−1.40412			−0.702060	−	0.712117i	$-0.747736\pi$
$313$	−24.8414			−1.40412			−0.702060	+	0.712117i	$0.747736\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$			13.0987i			0.735696i	0.929886	+	0.367848i	$0.119905\pi$
$317$			13.0987i			0.735696i	−0.929886	+	0.367848i	$0.880095\pi$
$318$		0			0
$319$			16.3580i			0.915873i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−3.16459	+	0.758949i	−0.176083	+	0.0422290i
$324$		0			0
$325$	−10.4358			−0.578876
$326$		0			0
$327$		0			0
$328$		0			0
$329$			27.3329i			1.50691i
$330$		0			0
$331$	31.7178			1.74337			0.871684	−	0.490068i	$-0.163028\pi$
$331$	31.7178			1.74337			0.871684	+	0.490068i	$0.163028\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	4.16539			0.227579
$336$		0			0
$337$			14.5535i			0.792782i	0.918082	+	0.396391i	$0.129737\pi$
$337$			14.5535i			0.792782i	−0.918082	+	0.396391i	$0.870263\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$			4.35248i			0.235700i
$342$		0			0
$343$	−15.2691			−0.824456
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−16.5335			−0.887564			−0.443782	−	0.896135i	$-0.646364\pi$
$347$	−16.5335			−0.887564			−0.443782	+	0.896135i	$0.646364\pi$
$348$		0			0
$349$			23.5408i			1.26011i	0.776550	+	0.630056i	$0.216968\pi$
$349$			23.5408i			1.26011i	−0.776550	+	0.630056i	$0.783032\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$		−	2.47540i		−	0.131752i	−0.997828	−	0.0658761i	$-0.979016\pi$
$353$		−	2.47540i		−	0.131752i	0.997828	−	0.0658761i	$-0.0209842\pi$
$354$		0			0
$355$	14.4520			0.767033
$356$		0			0
$357$		0			0
$358$		0			0
$359$		−	11.7581i		−	0.620569i	−0.950644	−	0.310285i	$-0.899576\pi$
$359$		−	11.7581i		−	0.620569i	0.950644	−	0.310285i	$-0.100424\pi$
$360$		0			0
$361$	−16.9333	+	8.61770i	−0.891224	+	0.453563i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−3.25537			−0.170394
$366$		0			0
$367$	−21.5144			−1.12304			−0.561520	−	0.827463i	$-0.689783\pi$
$367$	−21.5144			−1.12304			−0.561520	+	0.827463i	$0.689783\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		−	29.8953i		−	1.55209i
$372$		0			0
$373$	34.6286			1.79300			0.896500	−	0.443043i	$-0.146101\pi$
$373$	34.6286			1.79300			0.896500	+	0.443043i	$0.146101\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		−	30.8184i		−	1.58723i
$378$		0			0
$379$	1.97134			0.101261			0.0506304	−	0.998717i	$-0.483877\pi$
$379$	1.97134			0.101261			0.0506304	+	0.998717i	$0.483877\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	8.85996			0.452723			0.226361	−	0.974043i	$-0.427317\pi$
$383$	8.85996			0.452723			0.226361	+	0.974043i	$0.427317\pi$
$384$		0			0
$385$	−11.9419			−0.608613
$386$		0			0
$387$		0			0
$388$		0			0
$389$	20.6015			1.04454			0.522269	−	0.852781i	$-0.325086\pi$
$389$	20.6015			1.04454			0.522269	+	0.852781i	$0.325086\pi$
$390$		0			0
$391$	5.20736			0.263347
$392$		0			0
$393$		0			0
$394$		0			0
$395$			11.5885i			0.583081i
$396$		0			0
$397$		−	20.6164i		−	1.03471i	−0.855771	−	0.517354i	$-0.826917\pi$
$397$		−	20.6164i		−	1.03471i	0.855771	−	0.517354i	$-0.173083\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	28.9010			1.44325			0.721624	−	0.692285i	$-0.243396\pi$
$401$	28.9010			1.44325			0.721624	+	0.692285i	$0.243396\pi$
$402$		0			0
$403$		−	8.20002i		−	0.408472i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−24.4727			−1.21307
$408$		0			0
$409$		−	36.1755i		−	1.78876i	−0.447305	−	0.894382i	$-0.647616\pi$
$409$		−	36.1755i		−	1.78876i	0.447305	−	0.894382i	$-0.352384\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		−	20.5484i		−	1.01112i
$414$		0			0
$415$	20.8362			1.02281
$416$		0			0
$417$		0			0
$418$		0			0
$419$	13.2900			0.649260			0.324630	−	0.945841i	$-0.394760\pi$
$419$	13.2900			0.649260			0.324630	+	0.945841i	$0.394760\pi$
$420$		0			0
$421$	22.6477			1.10378			0.551892	−	0.833916i	$-0.313906\pi$
$421$	22.6477			1.10378			0.551892	+	0.833916i	$0.313906\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		−	1.70167i		−	0.0825431i
$426$		0			0
$427$			36.3198i			1.75764i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−5.33265			−0.256865			−0.128432	−	0.991718i	$-0.540994\pi$
$431$	−5.33265			−0.256865			−0.128432	+	0.991718i	$0.540994\pi$
$432$		0			0
$433$		−	24.4969i		−	1.17725i	−0.808408	−	0.588623i	$-0.799670\pi$
$433$		−	24.4969i		−	1.17725i	0.808408	−	0.588623i	$-0.200330\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	29.5642	−	7.09025i	1.41425	−	0.339172i
$438$		0			0
$439$		−	17.8987i		−	0.854259i	−0.904190	−	0.427130i	$-0.859525\pi$
$439$		−	17.8987i		−	0.854259i	0.904190	−	0.427130i	$-0.140475\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	13.4274			0.637956			0.318978	−	0.947762i	$-0.396660\pi$
$443$	13.4274			0.637956			0.318978	+	0.947762i	$0.396660\pi$
$444$		0			0
$445$	13.1861			0.625083
$446$		0			0
$447$		0			0
$448$		0			0
$449$	12.1635			0.574030			0.287015	−	0.957926i	$-0.407337\pi$
$449$	12.1635			0.574030			0.287015	+	0.957926i	$0.407337\pi$
$450$		0			0
$451$	−3.61343			−0.170150
$452$		0			0
$453$		0			0
$454$		0			0
$455$	22.4983			1.05474
$456$		0			0
$457$	27.0173			1.26382			0.631908	−	0.775043i	$-0.282272\pi$
$457$	27.0173			1.26382			0.631908	+	0.775043i	$0.282272\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−10.0923			−0.470047			−0.235024	−	0.971990i	$-0.575517\pi$
$461$	−10.0923			−0.470047			−0.235024	+	0.971990i	$0.575517\pi$
$462$		0			0
$463$	−14.4435			−0.671248			−0.335624	−	0.941996i	$-0.608947\pi$
$463$	−14.4435			−0.671248			−0.335624	+	0.941996i	$0.608947\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−34.4659			−1.59489			−0.797445	−	0.603392i	$-0.793816\pi$
$467$	−34.4659			−1.59489			−0.797445	+	0.603392i	$0.793816\pi$
$468$		0			0
$469$	7.52281			0.347371
$470$		0			0
$471$		0			0
$472$		0			0
$473$			19.3425i			0.889369i
$474$		0			0
$475$	−2.31696	−	9.66105i	−0.106310	−	0.443279i
$476$		0			0
$477$		0			0
$478$		0			0
$479$		−	10.5450i		−	0.481815i	−0.970548	−	0.240908i	$-0.922555\pi$
$479$		−	10.5450i		−	0.481815i	0.970548	−	0.240908i	$-0.0774451\pi$
$480$		0			0
$481$	46.1063			2.10227
$482$		0			0
$483$		0			0
$484$		0			0
$485$		−	0.281627i		−	0.0127880i
$486$		0			0
$487$		−	28.3899i		−	1.28647i	−0.765669	−	0.643234i	$-0.777592\pi$
$487$		−	28.3899i		−	1.28647i	0.765669	−	0.643234i	$-0.222408\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−26.8461			−1.21155			−0.605773	−	0.795638i	$-0.707136\pi$
$491$	−26.8461			−1.21155			−0.605773	+	0.795638i	$0.707136\pi$
$492$		0			0
$493$	5.02525			0.226326
$494$		0			0
$495$		0			0
$496$		0			0
$497$	26.1007			1.17078
$498$		0			0
$499$			14.2028i			0.635804i	0.948124	+	0.317902i	$0.102978\pi$
$499$			14.2028i			0.635804i	−0.948124	+	0.317902i	$0.897022\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$			35.5971i			1.58720i	0.608442	+	0.793599i	$0.291795\pi$
$503$			35.5971i			1.58720i	−0.608442	+	0.793599i	$0.708205\pi$
$504$		0			0
$505$	−9.75294			−0.434000
$506$		0			0
$507$		0			0
$508$		0			0
$509$		−	16.4385i		−	0.728625i	−0.931277	−	0.364312i	$-0.881304\pi$
$509$		−	16.4385i		−	0.728625i	0.931277	−	0.364312i	$-0.118696\pi$
$510$		0			0
$511$	−5.87928			−0.260084
$512$		0			0
$513$		0			0
$514$		0			0
$515$		−	27.8523i		−	1.22732i
$516$		0			0
$517$		−	22.2984i		−	0.980683i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−21.2264			−0.929947			−0.464973	−	0.885325i	$-0.653936\pi$
$521$	−21.2264			−0.929947			−0.464973	+	0.885325i	$0.653936\pi$
$522$		0			0
$523$	−1.86129			−0.0813886			−0.0406943	−	0.999172i	$-0.512957\pi$
$523$	−1.86129			−0.0813886			−0.0406943	+	0.999172i	$0.512957\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	1.33710			0.0582449
$528$		0			0
$529$	−25.6482			−1.11514
$530$		0			0
$531$		0			0
$532$		0			0
$533$	6.80767			0.294873
$534$		0			0
$535$			19.5311i			0.844402i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−4.55532			−0.196212
$540$		0			0
$541$			18.1869i			0.781914i	0.920409	+	0.390957i	$0.127856\pi$
$541$			18.1869i			0.781914i	−0.920409	+	0.390957i	$0.872144\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	13.2726			0.568535
$546$		0			0
$547$	−40.6762			−1.73919			−0.869593	−	0.493769i	$-0.835619\pi$
$547$	−40.6762			−1.73919			−0.869593	+	0.493769i	$0.835619\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	28.5303	−	6.84229i	1.21543	−	0.291491i
$552$		0			0
$553$			20.9292i			0.889999i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	33.3325			1.41234			0.706171	−	0.708041i	$-0.250421\pi$
$557$	33.3325			1.41234			0.706171	+	0.708041i	$0.250421\pi$
$558$		0			0
$559$		−	36.4411i		−	1.54129i
$560$		0			0
$561$		0			0
$562$		0			0
$563$		−	3.33778i		−	0.140671i	−0.997523	−	0.0703353i	$-0.977593\pi$
$563$		−	3.33778i		−	0.140671i	0.997523	−	0.0703353i	$-0.0224069\pi$
$564$		0			0
$565$	−28.5528			−1.20123
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−9.69455			−0.406417			−0.203208	−	0.979136i	$-0.565137\pi$
$569$	−9.69455			−0.406417			−0.203208	+	0.979136i	$0.565137\pi$
$570$		0			0
$571$			21.6272i			0.905070i	0.891747	+	0.452535i	$0.149480\pi$
$571$			21.6272i			0.905070i	−0.891747	+	0.452535i	$0.850520\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$			15.8973i			0.662964i
$576$		0			0
$577$	29.6177			1.23300			0.616500	−	0.787355i	$-0.288550\pi$
$577$	29.6177			1.23300			0.616500	+	0.787355i	$0.288550\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	37.6308			1.56119
$582$		0			0
$583$			24.3888i			1.01008i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−19.0897			−0.787915			−0.393958	−	0.919129i	$-0.628894\pi$
$587$	−19.0897			−0.787915			−0.393958	+	0.919129i	$0.628894\pi$
$588$		0			0
$589$	7.59123	−	1.82057i	0.312791	−	0.0750152i
$590$		0			0
$591$		0			0
$592$		0			0
$593$			5.44767i			0.223709i	0.993725	+	0.111855i	$0.0356791\pi$
$593$			5.44767i			0.223709i	−0.993725	+	0.111855i	$0.964321\pi$
$594$		0			0
$595$			3.66858i			0.150397i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−10.2875			−0.420335			−0.210168	−	0.977665i	$-0.567401\pi$
$599$	−10.2875			−0.420335			−0.210168	+	0.977665i	$0.567401\pi$
$600$		0			0
$601$			32.4022i			1.32171i	0.750512	+	0.660856i	$0.229807\pi$
$601$			32.4022i			1.32171i	−0.750512	+	0.660856i	$0.770193\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−8.40192			−0.341586
$606$		0			0
$607$		−	10.5492i		−	0.428181i	−0.976814	−	0.214090i	$-0.931321\pi$
$607$		−	10.5492i		−	0.428181i	0.976814	−	0.214090i	$-0.0686787\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$			42.0100i			1.69954i
$612$		0			0
$613$		−	36.5980i		−	1.47818i	−0.673607	−	0.739090i	$-0.735256\pi$
$613$		−	36.5980i		−	1.47818i	0.673607	−	0.739090i	$-0.264744\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		−	15.3512i		−	0.618017i	−0.951059	−	0.309009i	$-0.900003\pi$
$617$		−	15.3512i		−	0.618017i	0.951059	−	0.309009i	$-0.0999972\pi$
$618$		0			0
$619$		−	27.2625i		−	1.09577i	−0.836553	−	0.547886i	$-0.815433\pi$
$619$		−	27.2625i		−	1.09577i	0.836553	−	0.547886i	$-0.184567\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	23.8145			0.954110
$624$		0			0
$625$	−8.40882			−0.336353
$626$		0			0
$627$		0			0
$628$		0			0
$629$			7.51810i			0.299766i
$630$		0			0
$631$	−12.7290			−0.506732			−0.253366	−	0.967371i	$-0.581538\pi$
$631$	−12.7290			−0.506732			−0.253366	+	0.967371i	$0.581538\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$		−	15.2868i		−	0.606639i
$636$		0			0
$637$	8.58218			0.340038
$638$		0			0
$639$		0			0
$640$		0			0
$641$	8.11642			0.320579			0.160290	−	0.987070i	$-0.448757\pi$
$641$	8.11642			0.320579			0.160290	+	0.987070i	$0.448757\pi$
$642$		0			0
$643$		−	14.2945i		−	0.563721i	−0.959455	−	0.281861i	$-0.909048\pi$
$643$		−	14.2945i		−	0.563721i	0.959455	−	0.281861i	$-0.0909516\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		−	22.7305i		−	0.893628i	−0.894627	−	0.446814i	$-0.852559\pi$
$647$		−	22.7305i		−	0.893628i	0.894627	−	0.446814i	$-0.147441\pi$
$648$		0			0
$649$			16.7636i			0.658028i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−25.2711			−0.988935			−0.494468	−	0.869196i	$-0.664637\pi$
$653$	−25.2711			−0.988935			−0.494468	+	0.869196i	$0.664637\pi$
$654$		0			0
$655$	28.9542			1.13134
$656$		0			0
$657$		0			0
$658$		0			0
$659$			44.1809i			1.72104i	0.509415	+	0.860521i	$0.329862\pi$
$659$			44.1809i			1.72104i	−0.509415	+	0.860521i	$0.670138\pi$
$660$		0			0
$661$	−11.1053			−0.431947			−0.215973	−	0.976399i	$-0.569292\pi$
$661$	−11.1053			−0.431947			−0.215973	+	0.976399i	$0.569292\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	4.99508	+	20.8280i	0.193701	+	0.807675i
$666$		0			0
$667$	−46.9468			−1.81779
$668$		0			0
$669$		0			0
$670$		0			0
$671$		−	29.6300i		−	1.14385i
$672$		0			0
$673$		−	18.3290i		−	0.706531i	−0.935523	−	0.353266i	$-0.885071\pi$
$673$		−	18.3290i		−	0.706531i	0.935523	−	0.353266i	$-0.114929\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$			35.6834i			1.37142i	0.727874	+	0.685711i	$0.240509\pi$
$677$			35.6834i			1.37142i	−0.727874	+	0.685711i	$0.759491\pi$
$678$		0			0
$679$		−	0.508627i		−	0.0195193i
$680$		0			0
$681$		0			0
$682$		0			0
$683$			20.9162i			0.800337i	0.916442	+	0.400169i	$0.131048\pi$
$683$			20.9162i			0.800337i	−0.916442	+	0.400169i	$0.868952\pi$
$684$		0			0
$685$			19.3936i			0.740993i
$686$		0			0
$687$		0			0
$688$		0			0
$689$		−	45.9483i		−	1.75049i
$690$		0			0
$691$			16.5019i			0.627763i	0.949462	+	0.313881i	$0.101629\pi$
$691$			16.5019i			0.627763i	−0.949462	+	0.313881i	$0.898371\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$			5.27148i			0.199958i
$696$		0			0
$697$			1.11006i			0.0420465i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−32.3479			−1.22176			−0.610881	−	0.791722i	$-0.709185\pi$
$701$	−32.3479			−1.22176			−0.610881	+	0.791722i	$0.709185\pi$
$702$		0			0
$703$	10.2365	+	42.6832i	0.386077	+	1.60983i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−17.6141			−0.662446
$708$		0			0
$709$			13.2883i			0.499052i	0.968368	+	0.249526i	$0.0802748\pi$
$709$			13.2883i			0.499052i	−0.968368	+	0.249526i	$0.919725\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−12.4914			−0.467808
$714$		0			0
$715$	−18.3543			−0.686413
$716$		0			0
$717$		0			0
$718$		0			0
$719$			32.9134i			1.22746i	0.789514	+	0.613732i	$0.210333\pi$
$719$			32.9134i			1.22746i	−0.789514	+	0.613732i	$0.789667\pi$
$720$		0			0
$721$		−	50.3021i		−	1.87335i
$722$		0			0
$723$		0			0
$724$		0			0
$725$			15.3414i			0.569764i
$726$		0			0
$727$	30.5725			1.13387			0.566936	−	0.823762i	$-0.308129\pi$
$727$	30.5725			1.13387			0.566936	+	0.823762i	$0.308129\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	5.94209			0.219776
$732$		0			0
$733$		−	0.0739768i		−	0.00273240i	−0.999999	−	0.00136620i	$-0.999565\pi$
$733$		−	0.0739768i		−	0.00273240i	0.999999	−	0.00136620i	$-0.000434875\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−6.13717			−0.226066
$738$		0			0
$739$		−	6.01384i		−	0.221223i	−0.993864	−	0.110611i	$-0.964719\pi$
$739$		−	6.01384i		−	0.221223i	0.993864	−	0.110611i	$-0.0352809\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−11.5289			−0.422956			−0.211478	−	0.977383i	$-0.567828\pi$
$743$	−11.5289			−0.422956			−0.211478	+	0.977383i	$0.567828\pi$
$744$		0			0
$745$	−12.3374			−0.452008
$746$		0			0
$747$		0			0
$748$		0			0
$749$			35.2737i			1.28887i
$750$		0			0
$751$		−	31.0952i		−	1.13468i	−0.823484	−	0.567339i	$-0.807973\pi$
$751$		−	31.0952i		−	1.13468i	0.823484	−	0.567339i	$-0.192027\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$			36.4601i			1.32692i
$756$		0			0
$757$		−	21.5505i		−	0.783265i	−0.920122	−	0.391633i	$-0.871910\pi$
$757$		−	21.5505i		−	0.783265i	0.920122	−	0.391633i	$-0.128090\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		−	12.5821i		−	0.456101i	−0.973649	−	0.228050i	$-0.926765\pi$
$761$		−	12.5821i		−	0.456101i	0.973649	−	0.228050i	$-0.0732351\pi$
$762$		0			0
$763$	23.9707			0.867796
$764$		0			0
$765$		0			0
$766$		0			0
$767$		−	31.5824i		−	1.14037i
$768$		0			0
$769$	−24.3454			−0.877916			−0.438958	−	0.898508i	$-0.644652\pi$
$769$	−24.3454			−0.877916			−0.438958	+	0.898508i	$0.644652\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$		−	4.76641i		−	0.171436i	−0.996319	−	0.0857180i	$-0.972682\pi$
$773$		−	4.76641i		−	0.171436i	0.996319	−	0.0857180i	$-0.0273184\pi$
$774$		0			0
$775$			4.08197i			0.146629i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	1.51144	+	6.30224i	0.0541528	+	0.225801i
$780$		0			0
$781$	−21.2932			−0.761931
$782$		0			0
$783$		0			0
$784$		0			0
$785$			13.6540i			0.487333i
$786$		0			0
$787$	−53.5279			−1.90806			−0.954031	−	0.299707i	$-0.903111\pi$
$787$	−53.5279			−1.90806			−0.954031	+	0.299707i	$0.903111\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−51.5672			−1.83352
$792$		0			0
$793$			55.8227i			1.98232i
$794$		0			0
$795$		0			0
$796$		0			0
$797$		−	24.7198i		−	0.875619i	−0.899068	−	0.437810i	$-0.855754\pi$
$797$		−	24.7198i		−	0.875619i	0.899068	−	0.437810i	$-0.144246\pi$
$798$		0			0
$799$	−6.85016			−0.242341
$800$		0			0
$801$		0			0
$802$		0			0
$803$	4.79637			0.169260
$804$		0			0
$805$		−	34.2726i		−	1.20795i
$806$		0			0
$807$		0			0
$808$		0			0
$809$			23.2380i			0.817003i	0.912758	+	0.408501i	$0.133949\pi$
$809$			23.2380i			0.817003i	−0.912758	+	0.408501i	$0.866051\pi$
$810$		0			0
$811$	−20.5201			−0.720557			−0.360278	−	0.932845i	$-0.617318\pi$
$811$	−20.5201			−0.720557			−0.360278	+	0.932845i	$0.617318\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		−	17.6060i		−	0.616711i
$816$		0			0
$817$	33.7356	−	8.09064i	1.18026	−	0.283056i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	25.3720			0.885488			0.442744	−	0.896648i	$-0.354005\pi$
$821$	25.3720			0.885488			0.442744	+	0.896648i	$0.354005\pi$
$822$		0			0
$823$	−18.2321			−0.635529			−0.317765	−	0.948170i	$-0.602932\pi$
$823$	−18.2321			−0.635529			−0.317765	+	0.948170i	$0.602932\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$			49.3209i			1.71506i	0.514437	+	0.857528i	$0.328001\pi$
$827$			49.3209i			1.71506i	−0.514437	+	0.857528i	$0.671999\pi$
$828$		0			0
$829$	−23.6567			−0.821631			−0.410815	−	0.911719i	$-0.634756\pi$
$829$	−23.6567			−0.821631			−0.410815	+	0.911719i	$0.634756\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$			1.39941i			0.0484868i
$834$		0			0
$835$	−13.4742			−0.466293
$836$		0			0
$837$		0			0
$838$		0			0
$839$	25.3598			0.875517			0.437758	−	0.899093i	$-0.355773\pi$
$839$	25.3598			0.875517			0.437758	+	0.899093i	$0.355773\pi$
$840$		0			0
$841$	−16.3050			−0.562242
$842$		0			0
$843$		0			0
$844$		0			0
$845$	13.1363			0.451901
$846$		0			0
$847$	−15.1741			−0.521388
$848$		0			0
$849$		0			0
$850$		0			0
$851$		−	70.2356i		−	2.40764i
$852$		0			0
$853$			28.4959i			0.975682i	0.872933	+	0.487841i	$0.162215\pi$
$853$			28.4959i			0.975682i	−0.872933	+	0.487841i	$0.837785\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−3.20130			−0.109354			−0.0546772	−	0.998504i	$-0.517413\pi$
$857$	−3.20130			−0.109354			−0.0546772	+	0.998504i	$0.517413\pi$
$858$		0			0
$859$		−	22.5083i		−	0.767974i	−0.923339	−	0.383987i	$-0.874551\pi$
$859$		−	22.5083i		−	0.767974i	0.923339	−	0.383987i	$-0.125449\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−49.8505			−1.69693			−0.848465	−	0.529252i	$-0.822473\pi$
$863$	−49.8505			−1.69693			−0.848465	+	0.529252i	$0.822473\pi$
$864$		0			0
$865$		−	36.9869i		−	1.25759i
$866$		0			0
$867$		0			0
$868$		0			0
$869$		−	17.0742i		−	0.579203i
$870$		0			0
$871$	11.5624			0.391776
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−35.7685			−1.20920
$876$		0			0
$877$	−9.97659			−0.336885			−0.168443	−	0.985711i	$-0.553874\pi$
$877$	−9.97659			−0.336885			−0.168443	+	0.985711i	$0.553874\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$			44.7298i			1.50698i	0.657457	+	0.753492i	$0.271632\pi$
$881$			44.7298i			1.50698i	−0.657457	+	0.753492i	$0.728368\pi$
$882$		0			0
$883$		−	2.15295i		−	0.0724525i	−0.999344	−	0.0362262i	$-0.988466\pi$
$883$		−	2.15295i		−	0.0724525i	0.999344	−	0.0362262i	$-0.0115337\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	25.9273			0.870553			0.435277	−	0.900297i	$-0.356651\pi$
$887$	25.9273			0.870553			0.435277	+	0.900297i	$0.356651\pi$
$888$		0			0
$889$		−	27.6084i		−	0.925957i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−38.8911	+	9.32705i	−1.30144	+	0.312118i
$894$		0			0
$895$			2.40808i			0.0804932i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−12.0546			−0.402043
$900$		0			0
$901$	7.49234			0.249606
$902$		0			0
$903$		0			0
$904$		0			0
$905$	3.51928			0.116985
$906$		0			0
$907$	−39.9517			−1.32657			−0.663287	−	0.748365i	$-0.730839\pi$
$907$	−39.9517			−1.32657			−0.663287	+	0.748365i	$0.730839\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−15.9066			−0.527010			−0.263505	−	0.964658i	$-0.584879\pi$
$911$	−15.9066			−0.527010			−0.263505	+	0.964658i	$0.584879\pi$
$912$		0			0
$913$	−30.6995			−1.01601
$914$		0			0
$915$		0			0
$916$		0			0
$917$	52.2922			1.72684
$918$		0			0
$919$	−24.6685			−0.813739			−0.406870	−	0.913486i	$-0.633380\pi$
$919$	−24.6685			−0.813739			−0.406870	+	0.913486i	$0.633380\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	40.1162			1.32044
$924$		0			0
$925$	−22.9517			−0.754647
$926$		0			0
$927$		0			0
$928$		0			0
$929$			7.99237i			0.262221i	0.991368	+	0.131111i	$0.0418543\pi$
$929$			7.99237i			0.262221i	−0.991368	+	0.131111i	$0.958146\pi$
$930$		0			0
$931$	1.90541	+	7.94501i	0.0624474	+	0.260387i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		−	2.99286i		−	0.0978771i
$936$		0			0
$937$	−47.0029			−1.53552			−0.767759	−	0.640738i	$-0.778628\pi$
$937$	−47.0029			−1.53552			−0.767759	+	0.640738i	$0.778628\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$		−	0.0556189i		−	0.00181312i	−1.00000		0.000906562i	$-0.999711\pi$
$941$		−	0.0556189i		−	0.00181312i	1.00000		0.000906562i	$-0.000288568\pi$
$942$		0			0
$943$		−	10.3704i		−	0.337706i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−16.4564			−0.534760			−0.267380	−	0.963591i	$-0.586158\pi$
$947$	−16.4564			−0.534760			−0.267380	+	0.963591i	$0.586158\pi$
$948$		0			0
$949$	−9.03632			−0.293331
$950$		0			0
$951$		0			0
$952$		0			0
$953$	0.871117			0.0282183			0.0141091	−	0.999900i	$-0.495509\pi$
$953$	0.871117			0.0282183			0.0141091	+	0.999900i	$0.495509\pi$
$954$		0			0
$955$		−	14.1899i		−	0.459173i
$956$		0			0
$957$		0			0
$958$		0			0
$959$			35.0255i			1.13103i
$960$		0			0
$961$	27.7926			0.896534
$962$		0			0
$963$		0			0
$964$		0			0
$965$			31.5504i			1.01564i
$966$		0			0
$967$	−13.5154			−0.434626			−0.217313	−	0.976102i	$-0.569729\pi$
$967$	−13.5154			−0.434626			−0.217313	+	0.976102i	$0.569729\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		−	1.13624i		−	0.0364637i	−0.999834	−	0.0182319i	$-0.994196\pi$
$971$		−	1.13624i		−	0.0364637i	0.999834	−	0.0182319i	$-0.00580370\pi$
$972$		0			0
$973$			9.52044i			0.305211i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−41.5063			−1.32790			−0.663952	−	0.747775i	$-0.731122\pi$
$977$	−41.5063			−1.32790			−0.663952	+	0.747775i	$0.731122\pi$
$978$		0			0
$979$	−19.4281			−0.620925
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−53.3934			−1.70298			−0.851492	−	0.524368i	$-0.824302\pi$
$983$	−53.3934			−1.70298			−0.851492	+	0.524368i	$0.824302\pi$
$984$		0			0
$985$	20.1838			0.643108
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−55.5122			−1.76518
$990$		0			0
$991$			45.2236i			1.43657i	0.695747	+	0.718287i	$0.255074\pi$
$991$			45.2236i			1.43657i	−0.695747	+	0.718287i	$0.744926\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	22.1520			0.702265
$996$		0			0
$997$		−	42.8940i		−	1.35847i	−0.733922	−	0.679233i	$-0.762312\pi$
$997$		−	42.8940i		−	1.35847i	0.733922	−	0.679233i	$-0.237688\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5472.2.p.a.3761.55		80
3.2	odd	2	inner	5472.2.p.a.3761.27		80
4.3	odd	2		1368.2.p.a.341.78	yes	80
8.3	odd	2		1368.2.p.a.341.79	yes	80
8.5	even	2	inner	5472.2.p.a.3761.25		80
12.11	even	2		1368.2.p.a.341.3	yes	80
19.18	odd	2	inner	5472.2.p.a.3761.56		80
24.5	odd	2	inner	5472.2.p.a.3761.53		80
24.11	even	2		1368.2.p.a.341.2	yes	80
57.56	even	2	inner	5472.2.p.a.3761.28		80
76.75	even	2		1368.2.p.a.341.4	yes	80
152.37	odd	2	inner	5472.2.p.a.3761.26		80
152.75	even	2		1368.2.p.a.341.1	✓	80
228.227	odd	2		1368.2.p.a.341.77	yes	80
456.227	odd	2		1368.2.p.a.341.80	yes	80
456.341	even	2	inner	5472.2.p.a.3761.54		80

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1368.2.p.a.341.1	✓	80	152.75	even	2
1368.2.p.a.341.2	yes	80	24.11	even	2
1368.2.p.a.341.3	yes	80	12.11	even	2
1368.2.p.a.341.4	yes	80	76.75	even	2
1368.2.p.a.341.77	yes	80	228.227	odd	2
1368.2.p.a.341.78	yes	80	4.3	odd	2
1368.2.p.a.341.79	yes	80	8.3	odd	2
1368.2.p.a.341.80	yes	80	456.227	odd	2
5472.2.p.a.3761.25		80	8.5	even	2	inner
5472.2.p.a.3761.26		80	152.37	odd	2	inner
5472.2.p.a.3761.27		80	3.2	odd	2	inner
5472.2.p.a.3761.28		80	57.56	even	2	inner
5472.2.p.a.3761.53		80	24.5	odd	2	inner
5472.2.p.a.3761.54		80	456.341	even	2	inner
5472.2.p.a.3761.55		80	1.1	even	1	trivial
5472.2.p.a.3761.56		80	19.18	odd	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 5472 = 2^{5} \cdot 3^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5472.p (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.64947 q^{5} +2.97899 q^{7} +O(q^{10})\)	\(q+1.64947 q^{5} +2.97899 q^{7} -2.43029 q^{11} +4.57864 q^{13} +0.746594i q^{17} +(1.01655 + 4.23871i) q^{19} -6.97482i q^{23} -2.27924 q^{25} -6.73090i q^{29} -1.79093i q^{31} +4.91376 q^{35} +10.0699 q^{37} +1.48683 q^{41} -7.95894i q^{43} +9.17522i q^{47} +1.87440 q^{49} -10.0354i q^{53} -4.00869 q^{55} -6.89777i q^{59} +12.1920i q^{61} +7.55233 q^{65} +2.52529 q^{67} +8.76160 q^{71} -1.97358 q^{73} -7.23981 q^{77} +7.02559i q^{79} +12.6320 q^{83} +1.23148i q^{85} +7.99416 q^{89} +13.6397 q^{91} +(1.67677 + 6.99162i) q^{95} -0.170738i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 80 q+O(q^{10}) \) 80 * q	\( 80 q + 80 q^{25} + 48 q^{49} + 32 q^{73}+O(q^{100}) \) 80 * q + 80 * q^25 + 48 * q^49 + 32 * q^73

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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