LMFDB - Embedded newform 1620.2.r.h.109.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1620,2,Mod(109,1620)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1620, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 2, 3]))

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

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

chi := DirichletCharacter("1620.109");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1620 = 2^{2} \cdot 3^{4} \cdot 5 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1620.r (of order $6$, degree $2$, 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:	$12.9357651274$
Analytic rank:	$0$
Dimension:	$16$
Relative dimension:	$8$ over $\Q(\zeta_{6})$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} + 3x^{14} - 11x^{12} - 90x^{10} - 450x^{8} - 2250x^{6} - 6875x^{4} + 46875x^{2} + 390625 $ x^16 + 3x^14 - 11x^12 - 90x^10 - 450x^8 - 2250x^6 - 6875x^4 + 46875*x^2 + 390625
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{2}\cdot 3^{4} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			109.2
Root			$1.76348 + 1.37482i$ of defining polynomial
Character	$\chi$	$=$	1620.109
Dual form			1620.2.r.h.1189.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.76348 + 1.37482i) q^{5} +(-4.27415 + 2.46768i) q^{7} +O(q^{10})$	$q+(-1.76348 + 1.37482i) q^{5} +(-4.27415 + 2.46768i) q^{7} +(-1.20635 - 2.08945i) q^{11} +(-2.51942 - 1.45459i) q^{13} +6.86869i q^{17} +4.17891 q^{19} +(2.90917 + 1.67961i) q^{23} +(1.21972 - 4.84895i) q^{25} +(-2.59808 - 4.50000i) q^{29} +(3.08945 - 5.35109i) q^{31} +(4.14474 - 10.2279i) q^{35} -7.84453i q^{37} +(-2.93840 + 5.08945i) q^{41} +(4.27415 - 2.46768i) q^{43} +(-10.3122 + 5.95376i) q^{47} +(8.67891 - 15.0323i) q^{49} -8.54830i q^{53} +(5.00000 + 2.02619i) q^{55} +(0.525704 - 0.910546i) q^{59} +(-4.58945 - 7.94917i) q^{61} +(6.44274 - 0.898624i) q^{65} +(3.50947 + 2.02619i) q^{67} +14.1663 q^{71} +2.02619i q^{73} +(10.3122 + 5.95376i) q^{77} +(3.00000 + 5.19615i) q^{79} +(4.49387 - 2.59454i) q^{83} +(-9.44324 - 12.1128i) q^{85} +3.09334 q^{89} +14.3578 q^{91} +(-7.36942 + 5.74527i) q^{95} +(-0.764681 + 0.441489i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q+O(q^{10}) $ 16 * q	$ 16 q - 24 q^{19} - 6 q^{25} + 4 q^{31} + 48 q^{49} + 80 q^{55} - 28 q^{61} + 48 q^{79} - 22 q^{85} + 48 q^{91}+O(q^{100}) $ 16 * q - 24 * q^19 - 6 * q^25 + 4 * q^31 + 48 * q^49 + 80 * q^55 - 28 * q^61 + 48 * q^79 - 22 * q^85 + 48 * q^91

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1620\mathbb{Z}\right)^\times$.

$n$	$811$	$1297$	$1541$
$\chi(n)$	$1$	$-1$	$e\left(\frac{1}{3}\right)$

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
$5$	−1.76348	+	1.37482i	−0.788652	+	0.614840i
$5$	−1.76348	+	1.37482i	−0.788652	+	0.614840i
$6$		0			0
$7$	−4.27415	+	2.46768i	−1.61548	+	0.932696i	−0.627407	+	0.778692i	$0.715884\pi$
$7$	−4.27415	+	2.46768i	−1.61548	+	0.932696i	−0.988070	+	0.154005i	$0.950783\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−1.20635	−	2.08945i	−0.363727	−	0.629994i	0.624844	−	0.780750i	$-0.285163\pi$
$11$	−1.20635	−	2.08945i	−0.363727	−	0.629994i	−0.988571	+	0.150756i	$0.951829\pi$
$12$		0			0
$13$	−2.51942	−	1.45459i	−0.698760	−	0.403429i	0.108125	−	0.994137i	$-0.465515\pi$
$13$	−2.51942	−	1.45459i	−0.698760	−	0.403429i	−0.806886	+	0.590708i	$0.798849\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$			6.86869i			1.66590i	0.553347	+	0.832951i	$0.313350\pi$
$17$			6.86869i			1.66590i	−0.553347	+	0.832951i	$0.686650\pi$
$18$		0			0
$19$	4.17891			0.958707			0.479354	−	0.877622i	$-0.340871\pi$
$19$	4.17891			0.958707			0.479354	+	0.877622i	$0.340871\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	2.90917	+	1.67961i	0.606604	+	0.350223i	0.771635	−	0.636065i	$-0.219439\pi$
$23$	2.90917	+	1.67961i	0.606604	+	0.350223i	−0.165031	+	0.986288i	$0.552772\pi$
$24$		0			0
$25$	1.21972	−	4.84895i	0.243943	−	0.969790i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−2.59808	−	4.50000i	−0.482451	−	0.835629i	0.517346	−	0.855776i	$-0.326920\pi$
$29$	−2.59808	−	4.50000i	−0.482451	−	0.835629i	−0.999797	+	0.0201471i	$0.993587\pi$
$30$		0			0
$31$	3.08945	−	5.35109i	0.554882	−	0.961084i	−0.443030	−	0.896507i	$-0.646097\pi$
$31$	3.08945	−	5.35109i	0.554882	−	0.961084i	0.997913	−	0.0645778i	$-0.0205701\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	4.14474	−	10.2279i	0.700590	−	1.72883i
$36$		0			0
$37$		−	7.84453i		−	1.28963i	−0.764337	−	0.644817i	$-0.776934\pi$
$37$		−	7.84453i		−	1.28963i	0.764337	−	0.644817i	$-0.223066\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−2.93840	+	5.08945i	−0.458901	+	0.794839i	−0.998903	−	0.0468242i	$-0.985090\pi$
$41$	−2.93840	+	5.08945i	−0.458901	+	0.794839i	0.540003	+	0.841663i	$0.318423\pi$
$42$		0			0
$43$	4.27415	−	2.46768i	0.651802	−	0.376318i	−0.137344	−	0.990523i	$-0.543857\pi$
$43$	4.27415	−	2.46768i	0.651802	−	0.376318i	0.789146	+	0.614205i	$0.210523\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−10.3122	+	5.95376i	−1.50419	+	0.868445i	−0.504203	+	0.863585i	$0.668214\pi$
$47$	−10.3122	+	5.95376i	−1.50419	+	0.868445i	−0.999988	+	0.00486027i	$0.998453\pi$
$48$		0			0
$49$	8.67891	−	15.0323i	1.23984	−	2.14747i
$50$		0			0
$51$		0			0
$52$		0			0
$53$		−	8.54830i		−	1.17420i	−0.809515	−	0.587100i	$-0.800270\pi$
$53$		−	8.54830i		−	1.17420i	0.809515	−	0.587100i	$-0.199730\pi$
$54$		0			0
$55$	5.00000	+	2.02619i	0.674200	+	0.273212i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	0.525704	−	0.910546i	0.0684408	−	0.118543i	−0.829774	−	0.558099i	$-0.811531\pi$
$59$	0.525704	−	0.910546i	0.0684408	−	0.118543i	0.898215	+	0.439556i	$0.144864\pi$
$60$		0			0
$61$	−4.58945	−	7.94917i	−0.587619	−	1.01779i	−0.994543	−	0.104325i	$-0.966732\pi$
$61$	−4.58945	−	7.94917i	−0.587619	−	1.01779i	0.406924	−	0.913462i	$-0.366601\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	6.44274	−	0.898624i	0.799123	−	0.111461i
$66$		0			0
$67$	3.50947	+	2.02619i	0.428750	+	0.247539i	0.698814	−	0.715303i	$-0.253712\pi$
$67$	3.50947	+	2.02619i	0.428750	+	0.247539i	−0.270064	+	0.962842i	$0.587045\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	14.1663			1.68123			0.840614	−	0.541634i	$-0.182194\pi$
$71$	14.1663			1.68123			0.840614	+	0.541634i	$0.182194\pi$
$72$		0			0
$73$			2.02619i			0.237148i	0.992945	+	0.118574i	$0.0378323\pi$
$73$			2.02619i			0.237148i	−0.992945	+	0.118574i	$0.962168\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	10.3122	+	5.95376i	1.17519	+	0.678494i
$78$		0			0
$79$	3.00000	+	5.19615i	0.337526	+	0.584613i	0.983967	−	0.178352i	$-0.0570765\pi$
$79$	3.00000	+	5.19615i	0.337526	+	0.584613i	−0.646440	+	0.762964i	$0.723743\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	4.49387	−	2.59454i	0.493267	−	0.284788i	−0.232662	−	0.972558i	$-0.574744\pi$
$83$	4.49387	−	2.59454i	0.493267	−	0.284788i	0.725929	+	0.687770i	$0.241410\pi$
$84$		0			0
$85$	−9.44324	−	12.1128i	−1.02426	−	1.31382i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	3.09334			0.327893			0.163947	−	0.986469i	$-0.447578\pi$
$89$	3.09334			0.327893			0.163947	+	0.986469i	$0.447578\pi$
$90$		0			0
$91$	14.3578			1.50511
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−7.36942	+	5.74527i	−0.756086	+	0.589452i
$96$		0			0
$97$	−0.764681	+	0.441489i	−0.0776416	+	0.0448264i	−0.538318	−	0.842742i	$-0.680940\pi$
$97$	−0.764681	+	0.441489i	−0.0776416	+	0.0448264i	0.460677	+	0.887568i	$0.347607\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	2.93840	+	5.08945i	0.292382	+	0.506420i	0.974372	−	0.224941i	$-0.0722190\pi$
$101$	2.93840	+	5.08945i	0.292382	+	0.506420i	−0.681991	+	0.731361i	$0.738886\pi$
$102$		0			0
$103$	−8.54830	−	4.93536i	−0.842289	−	0.486296i	0.0157525	−	0.999876i	$-0.494986\pi$
$103$	−8.54830	−	4.93536i	−0.842289	−	0.486296i	−0.858042	+	0.513580i	$0.828319\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		0			0		1.00000			$0$
$107$		0			0		−1.00000			$\pi$
$108$		0			0
$109$	−7.00000			−0.670478			−0.335239	−	0.942133i	$-0.608817\pi$
$109$	−7.00000			−0.670478			−0.335239	+	0.942133i	$0.608817\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−11.7668	−	6.79357i	−1.10693	−	0.639085i	−0.168896	−	0.985634i	$-0.554020\pi$
$113$	−11.7668	−	6.79357i	−1.10693	−	0.639085i	−0.938032	+	0.346549i	$0.887353\pi$
$114$		0			0
$115$	−7.43943	+	1.03764i	−0.693731	+	0.0967606i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−16.9497	−	29.3578i	−1.55378	−	2.69123i
$120$		0			0
$121$	2.58945	−	4.48507i	0.235405	−	0.407733i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	4.51551	+	10.2279i	0.403879	+	0.914812i
$126$		0			0
$127$		−	4.93536i		−	0.437943i	−0.975731	−	0.218971i	$-0.929730\pi$
$127$		−	4.93536i		−	0.437943i	0.975731	−	0.218971i	$-0.0702701\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−7.08314	+	12.2684i	−0.618857	+	1.07189i	0.370838	+	0.928698i	$0.379071\pi$
$131$	−7.08314	+	12.2684i	−0.618857	+	1.07189i	−0.989695	+	0.143194i	$0.954263\pi$
$132$		0			0
$133$	−17.8613	+	10.3122i	−1.54877	+	0.894183i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	4.36376	−	2.51942i	0.372821	−	0.215248i	−0.301869	−	0.953349i	$-0.597611\pi$
$137$	4.36376	−	2.51942i	0.372821	−	0.215248i	0.674690	+	0.738101i	$0.264277\pi$
$138$		0			0
$139$	2.91055	−	5.04121i	0.246869	−	0.427590i	−0.715786	−	0.698319i	$-0.753932\pi$
$139$	2.91055	−	5.04121i	0.246869	−	0.427590i	0.962656	+	0.270729i	$0.0872648\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$			7.01894i			0.586953i
$144$		0			0
$145$	10.7684	+	4.36376i	0.894264	+	0.362390i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	6.58788	−	11.4105i	0.539700	−	0.934788i	−0.459220	−	0.888323i	$-0.651871\pi$
$149$	6.58788	−	11.4105i	0.539700	−	0.934788i	0.998920	−	0.0464656i	$-0.0147958\pi$
$150$		0			0
$151$	3.08945	+	5.35109i	0.251416	+	0.435466i	0.963916	−	0.266207i	$-0.0857704\pi$
$151$	3.08945	+	5.35109i	0.251416	+	0.435466i	−0.712500	+	0.701672i	$0.752437\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	1.90862	+	13.6840i	0.153304	+	1.09912i
$156$		0			0
$157$	−6.79357	−	3.92227i	−0.542186	−	0.313031i	0.203779	−	0.979017i	$-0.434678\pi$
$157$	−6.79357	−	3.92227i	−0.542186	−	0.313031i	−0.745964	+	0.665986i	$0.768011\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−16.5790			−1.30661
$162$		0			0
$163$		−	10.7537i		−	0.842295i	−0.906992	−	0.421148i	$-0.861627\pi$
$163$		−	10.7537i		−	0.842295i	0.906992	−	0.421148i	$-0.138373\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−7.40305	−	4.27415i	−0.572865	−	0.330744i	0.185428	−	0.982658i	$-0.440633\pi$
$167$	−7.40305	−	4.27415i	−0.572865	−	0.330744i	−0.758293	+	0.651914i	$0.773966\pi$
$168$		0			0
$169$	−2.26836	−	3.92892i	−0.174489	−	0.302225i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−16.2607	+	9.38811i	−1.23628	+	0.713765i	−0.968331	−	0.249670i	$-0.919678\pi$
$173$	−16.2607	+	9.38811i	−1.23628	+	0.713765i	−0.267945	+	0.963434i	$0.586345\pi$
$174$		0			0
$175$	6.75241	+	23.7350i	0.510434	+	1.79420i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−7.97961			−0.596424			−0.298212	−	0.954500i	$-0.596390\pi$
$179$	−7.97961			−0.596424			−0.298212	+	0.954500i	$0.596390\pi$
$180$		0			0
$181$	3.82109			0.284020			0.142010	−	0.989865i	$-0.454644\pi$
$181$	3.82109			0.284020			0.142010	+	0.989865i	$0.454644\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	10.7849	+	13.8337i	0.792919	+	1.01707i
$186$		0			0
$187$	14.3518	−	8.28602i	1.04951	−	0.605934i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	3.30916	+	5.73164i	0.239443	+	0.414727i	0.960554	−	0.278092i	$-0.0897020\pi$
$191$	3.30916	+	5.73164i	0.239443	+	0.414727i	−0.721112	+	0.692819i	$0.756369\pi$
$192$		0			0
$193$	18.8513	+	10.8838i	1.35695	+	0.783435i	0.989211	−	0.146495i	$-0.0467992\pi$
$193$	18.8513	+	10.8838i	1.35695	+	0.783435i	0.367737	+	0.929930i	$0.380133\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		−	15.4170i		−	1.09842i	−0.835686	−	0.549208i	$-0.814930\pi$
$197$		−	15.4170i		−	1.09842i	0.835686	−	0.549208i	$-0.185070\pi$
$198$		0			0
$199$	−20.3578			−1.44313			−0.721564	−	0.692348i	$-0.756576\pi$
$199$	−20.3578			−1.44313			−0.721564	+	0.692348i	$0.756576\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	22.2091	+	12.8225i	1.55878	+	0.899960i
$204$		0			0
$205$	−1.81530	−	13.0149i	−0.126786	−	0.909002i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−5.04121	−	8.73164i	−0.348708	−	0.603980i
$210$		0			0
$211$	−8.08945	+	14.0113i	−0.556901	+	0.964581i	0.440852	+	0.897580i	$0.354676\pi$
$211$	−8.08945	+	14.0113i	−0.556901	+	0.964581i	−0.997753	+	0.0670010i	$0.978657\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−4.14474	+	10.2279i	−0.282669	+	0.697538i
$216$		0			0
$217$			30.4952i			2.07015i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	9.99110	−	17.3051i	0.672074	−	1.16407i
$222$		0			0
$223$	22.1354	−	12.7799i	1.48230	−	0.855805i	0.482500	−	0.875896i	$-0.339729\pi$
$223$	22.1354	−	12.7799i	1.48230	−	0.855805i	0.999798	+	0.0200906i	$0.00639547\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	16.3908	−	9.46323i	1.08790	−	0.628097i	0.154880	−	0.987933i	$-0.450501\pi$
$227$	16.3908	−	9.46323i	1.08790	−	0.628097i	0.933015	+	0.359837i	$0.117168\pi$
$228$		0			0
$229$	3.41055	−	5.90724i	0.225375	−	0.390361i	−0.731057	−	0.682317i	$-0.760973\pi$
$229$	3.41055	−	5.90724i	0.225375	−	0.390361i	0.956432	+	0.291955i	$0.0943059\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		−	0.150248i		−	0.00984309i	−0.999988	−	0.00492154i	$-0.998433\pi$
$233$		−	0.150248i		−	0.00984309i	0.999988	−	0.00492154i	$-0.00156658\pi$
$234$		0			0
$235$	10.0000	−	24.6768i	0.652328	−	1.60974i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	4.51551	−	7.82109i	0.292084	−	0.505904i	−0.682218	−	0.731149i	$-0.738985\pi$
$239$	4.51551	−	7.82109i	0.292084	−	0.505904i	0.974302	+	0.225244i	$0.0723180\pi$
$240$		0			0
$241$	−9.50000	−	16.4545i	−0.611949	−	1.05993i	−0.990912	−	0.134515i	$-0.957053\pi$
$241$	−9.50000	−	16.4545i	−0.611949	−	1.05993i	0.378963	−	0.925412i	$-0.376281\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	5.36172	+	38.4411i	0.342547	+	2.45591i
$246$		0			0
$247$	−10.5284	−	6.07858i	−0.669907	−	0.386771i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	17.3205			1.09326			0.546630	−	0.837374i	$-0.315910\pi$
$251$	17.3205			1.09326			0.546630	+	0.837374i	$0.315910\pi$
$252$		0			0
$253$		−	8.10477i		−	0.509543i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−19.1698	−	11.0677i	−1.19578	−	0.690385i	−0.236170	−	0.971712i	$-0.575892\pi$
$257$	−19.1698	−	11.0677i	−1.19578	−	0.690385i	−0.959612	+	0.281327i	$0.909226\pi$
$258$		0			0
$259$	19.3578	+	33.5287i	1.20284	+	2.08337i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	5.81834	−	3.35922i	0.358774	−	0.207138i	−0.309769	−	0.950812i	$-0.600252\pi$
$263$	5.81834	−	3.35922i	0.358774	−	0.207138i	0.668543	+	0.743674i	$0.266918\pi$
$264$		0			0
$265$	11.7524	+	15.0747i	0.721945	+	0.926034i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−30.0646			−1.83307			−0.916536	−	0.399952i	$-0.869027\pi$
$269$	−30.0646			−1.83307			−0.916536	+	0.399952i	$0.869027\pi$
$270$		0			0
$271$	−22.3578			−1.35814			−0.679070	−	0.734073i	$-0.737617\pi$
$271$	−22.3578			−1.35814			−0.679070	+	0.734073i	$0.737617\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−11.6031	+	3.30097i	−0.699690	+	0.199056i
$276$		0			0
$277$	−9.31298	+	5.37685i	−0.559563	+	0.323064i	−0.752970	−	0.658055i	$-0.771380\pi$
$277$	−9.31298	+	5.37685i	−0.559563	+	0.323064i	0.193407	+	0.981119i	$0.438046\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	6.21712	+	10.7684i	0.370882	+	0.642387i	0.989701	−	0.143147i	$-0.0457221\pi$
$281$	6.21712	+	10.7684i	0.370882	+	0.642387i	−0.618819	+	0.785533i	$0.712389\pi$
$282$		0			0
$283$	1.52936	+	0.882978i	0.0909112	+	0.0524876i	0.544766	−	0.838588i	$-0.316618\pi$
$283$	1.52936	+	0.882978i	0.0909112	+	0.0524876i	−0.453855	+	0.891075i	$0.649952\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		−	29.0041i		−	1.71206i
$288$		0			0
$289$	−30.1789			−1.77523
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−17.8454	−	10.3030i	−1.04254	−	0.601910i	−0.121987	−	0.992532i	$-0.538927\pi$
$293$	−17.8454	−	10.3030i	−1.04254	−	0.601910i	−0.920551	+	0.390622i	$0.872260\pi$
$294$		0			0
$295$	0.324773	+	2.32848i	0.0189090	+	0.135569i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−4.88627	−	8.46327i	−0.282581	−	0.489444i
$300$		0			0
$301$	−12.1789	+	21.0945i	−0.701981	+	1.21587i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	19.0221	+	7.70850i	1.08920	+	0.441387i
$306$		0			0
$307$			8.98775i			0.512958i	0.966550	+	0.256479i	$0.0825624\pi$
$307$			8.98775i			0.512958i	−0.966550	+	0.256479i	$0.917438\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−5.35109	+	9.26836i	−0.303433	+	0.525561i	−0.976911	−	0.213646i	$-0.931466\pi$
$311$	−5.35109	+	9.26836i	−0.303433	+	0.525561i	0.673479	+	0.739207i	$0.264799\pi$
$312$		0			0
$313$	−6.02889	+	3.48078i	−0.340773	+	0.196745i	−0.660614	−	0.750726i	$-0.729704\pi$
$313$	−6.02889	+	3.48078i	−0.340773	+	0.196745i	0.319841	+	0.947471i	$0.396371\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	29.4821	−	17.0215i	1.65588	−	0.956021i	0.681290	−	0.732013i	$-0.261419\pi$
$317$	29.4821	−	17.0215i	1.65588	−	0.956021i	0.974587	−	0.224008i	$-0.0719143\pi$
$318$		0			0
$319$	−6.26836	+	10.8571i	−0.350961	+	0.607882i
$320$		0			0
$321$		0			0
$322$		0			0
$323$			28.7036i			1.59711i
$324$		0			0
$325$	−10.1262	+	10.4423i	−0.561699	+	0.579237i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	29.3840	−	50.8945i	1.61999	−	2.80591i
$330$		0			0
$331$	16.2684	+	28.1776i	0.894190	+	1.54878i	0.834804	+	0.550548i	$0.185581\pi$
$331$	16.2684	+	28.1776i	0.894190	+	1.54878i	0.0593863	+	0.998235i	$0.481086\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−8.97453	+	1.25176i	−0.490331	+	0.0683907i
$336$		0			0
$337$	−8.54830	−	4.93536i	−0.465656	−	0.268846i	0.248764	−	0.968564i	$-0.419976\pi$
$337$	−8.54830	−	4.93536i	−0.465656	−	0.268846i	−0.714419	+	0.699718i	$0.753309\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−14.9078			−0.807303
$342$		0			0
$343$			51.1196i			2.76020i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−7.40305	−	4.27415i	−0.397416	−	0.229448i	0.287952	−	0.957645i	$-0.407026\pi$
$347$	−7.40305	−	4.27415i	−0.397416	−	0.229448i	−0.685369	+	0.728196i	$0.740359\pi$
$348$		0			0
$349$	10.0895	+	17.4754i	0.540076	+	0.935439i	0.998899	+	0.0469115i	$0.0149379\pi$
$349$	10.0895	+	17.4754i	0.540076	+	0.935439i	−0.458823	+	0.888528i	$0.651729\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	19.0397	−	10.9926i	1.01338	−	0.585077i	0.101202	−	0.994866i	$-0.467731\pi$
$353$	19.0397	−	10.9926i	1.01338	−	0.585077i	0.912180	+	0.409789i	$0.134398\pi$
$354$		0			0
$355$	−24.9819	+	19.4762i	−1.32590	+	1.03369i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	0.309878			0.0163548			0.00817738	−	0.999967i	$-0.497397\pi$
$359$	0.309878			0.0163548			0.00817738	+	0.999967i	$0.497397\pi$
$360$		0			0
$361$	−1.53673			−0.0808803
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−2.78566	−	3.57315i	−0.145808	−	0.187027i
$366$		0			0
$367$	8.54830	−	4.93536i	0.446218	−	0.257624i	−0.260014	−	0.965605i	$-0.583727\pi$
$367$	8.54830	−	4.93536i	0.446218	−	0.257624i	0.706231	+	0.707981i	$0.250394\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	21.0945	+	36.5367i	1.09517	+	1.89689i
$372$		0			0
$373$	−7.78362	−	4.49387i	−0.403021	−	0.232684i	0.284766	−	0.958597i	$-0.408084\pi$
$373$	−7.78362	−	4.49387i	−0.403021	−	0.232684i	−0.687786	+	0.725913i	$0.741417\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$			15.1165i			0.778539i
$378$		0			0
$379$	0.357817			0.0183798			0.00918990	−	0.999958i	$-0.497075\pi$
$379$	0.357817			0.0183798			0.00918990	+	0.999958i	$0.497075\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	11.8969	+	6.86869i	0.607904	+	0.350974i	0.772145	−	0.635447i	$-0.219184\pi$
$383$	11.8969	+	6.86869i	0.607904	+	0.350974i	−0.164241	+	0.986420i	$0.552517\pi$
$384$		0			0
$385$	−26.3708	+	3.67816i	−1.34398	+	0.187456i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−12.8050	−	22.1789i	−0.649239	−	1.12452i	−0.983305	−	0.181966i	$-0.941754\pi$
$389$	−12.8050	−	22.1789i	−0.649239	−	1.12452i	0.334066	−	0.942550i	$-0.391579\pi$
$390$		0			0
$391$	−11.5367	+	19.9822i	−0.583437	+	1.01054i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−12.4342	−	5.03883i	−0.625634	−	0.253531i
$396$		0			0
$397$		−	11.0139i		−	0.552774i	−0.961046	−	0.276387i	$-0.910863\pi$
$397$		−	11.0139i		−	0.552774i	0.961046	−	0.276387i	$-0.0891371\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−10.3619	+	17.9473i	−0.517447	+	0.896244i	0.482348	+	0.875980i	$0.339784\pi$
$401$	−10.3619	+	17.9473i	−0.517447	+	0.896244i	−0.999795	+	0.0202642i	$0.993549\pi$
$402$		0			0
$403$	−15.5672	+	8.98775i	−0.775459	+	0.447712i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−16.3908	+	9.46323i	−0.812462	+	0.469075i
$408$		0			0
$409$	3.41055	−	5.90724i	0.168641	−	0.292094i	−0.769302	−	0.638886i	$-0.779396\pi$
$409$	3.41055	−	5.90724i	0.168641	−	0.292094i	0.937942	+	0.346792i	$0.112729\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$			5.18908i			0.255338i
$414$		0			0
$415$	−4.35782	+	10.7537i	−0.213917	+	0.527879i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	8.28949	−	14.3578i	0.404968	−	0.701425i	−0.589350	−	0.807878i	$-0.700616\pi$
$419$	8.28949	−	14.3578i	0.404968	−	0.701425i	0.994318	+	0.106453i	$0.0339493\pi$
$420$		0			0
$421$	−14.8578	−	25.7345i	−0.724126	−	1.25422i	−0.959333	−	0.282277i	$-0.908910\pi$
$421$	−14.8578	−	25.7345i	−0.724126	−	1.25422i	0.235207	−	0.971945i	$-0.424423\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	33.3059	+	8.37785i	1.61557	+	0.406385i
$426$		0			0
$427$	39.2320	+	22.6506i	1.89857	+	1.09614i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−1.67116			−0.0804972			−0.0402486	−	0.999190i	$-0.512815\pi$
$431$	−1.67116			−0.0804972			−0.0402486	+	0.999190i	$0.512815\pi$
$432$		0			0
$433$		−	36.5737i		−	1.75762i	−0.477170	−	0.878811i	$-0.658337\pi$
$433$		−	36.5737i		−	1.75762i	0.477170	−	0.878811i	$-0.341663\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	12.1572	+	7.01894i	0.581556	+	0.335761i
$438$		0			0
$439$	9.26836	+	16.0533i	0.442355	+	0.766181i	0.997864	−	0.0653296i	$-0.0208099\pi$
$439$	9.26836	+	16.0533i	0.442355	+	0.766181i	−0.555509	+	0.831511i	$0.687477\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−16.1306	+	9.31298i	−0.766386	+	0.442473i	−0.831584	−	0.555399i	$-0.812565\pi$
$443$	−16.1306	+	9.31298i	−0.766386	+	0.442473i	0.0651979	+	0.997872i	$0.479232\pi$
$444$		0			0
$445$	−5.45503	+	4.25280i	−0.258593	+	0.201602i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−25.1783			−1.18824			−0.594120	−	0.804377i	$-0.702500\pi$
$449$	−25.1783			−1.18824			−0.594120	+	0.804377i	$0.702500\pi$
$450$		0			0
$451$	14.1789			0.667659
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−25.3197	+	19.7395i	−1.18701	+	0.925401i
$456$		0			0
$457$	23.8902	−	13.7930i	1.11753	−	0.645209i	0.176764	−	0.984253i	$-0.443437\pi$
$457$	23.8902	−	13.7930i	1.11753	−	0.645209i	0.940770	+	0.339044i	$0.110104\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	5.35109	+	9.26836i	0.249225	+	0.431671i	0.963311	−	0.268387i	$-0.0864907\pi$
$461$	5.35109	+	9.26836i	0.249225	+	0.431671i	−0.714086	+	0.700058i	$0.753157\pi$
$462$		0			0
$463$	25.6449	+	14.8061i	1.19182	+	0.688097i	0.958719	−	0.284355i	$-0.0917795\pi$
$463$	25.6449	+	14.8061i	1.19182	+	0.688097i	0.233101	+	0.972453i	$0.425113\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$			15.5672i			0.720366i	0.932882	+	0.360183i	$0.117286\pi$
$467$			15.5672i			0.720366i	−0.932882	+	0.360183i	$0.882714\pi$
$468$		0			0
$469$	−20.0000			−0.923514
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−10.3122	−	5.95376i	−0.474156	−	0.273754i
$474$		0			0
$475$	5.09708	−	20.2633i	0.233870	−	0.929744i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−12.6501	−	21.9105i	−0.577996	−	1.00112i	−0.995709	−	0.0925394i	$-0.970502\pi$
$479$	−12.6501	−	21.9105i	−0.577996	−	1.00112i	0.417713	−	0.908579i	$-0.362832\pi$
$480$		0			0
$481$	−11.4105	+	19.7636i	−0.520276	+	0.901145i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	0.741529	−	1.82986i	0.0336711	−	0.0830896i
$486$		0			0
$487$		−	17.9755i		−	0.814548i	−0.913306	−	0.407274i	$-0.866479\pi$
$487$		−	17.9755i		−	0.814548i	0.913306	−	0.407274i	$-0.133521\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	17.7853	−	30.8051i	0.802641	−	1.39021i	−0.115232	−	0.993339i	$-0.536761\pi$
$491$	17.7853	−	30.8051i	0.802641	−	1.39021i	0.917872	−	0.396876i	$-0.129906\pi$
$492$		0			0
$493$	30.9091	−	17.8454i	1.39208	−	0.803716i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−60.5488	+	34.9579i	−2.71599	+	1.56808i
$498$		0			0
$499$	2.08945	−	3.61904i	0.0935368	−	0.162011i	−0.815460	−	0.578813i	$-0.803516\pi$
$499$	2.08945	−	3.61904i	0.0935368	−	0.162011i	0.908997	+	0.416803i	$0.136849\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		−	37.2519i		−	1.66098i	−0.557032	−	0.830491i	$-0.688060\pi$
$503$		−	37.2519i		−	1.66098i	0.557032	−	0.830491i	$-0.311940\pi$
$504$		0			0
$505$	−12.1789	−	4.93536i	−0.541954	−	0.219621i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−3.77398	+	6.53673i	−0.167279	+	0.289735i	−0.937462	−	0.348087i	$-0.886831\pi$
$509$	−3.77398	+	6.53673i	−0.167279	+	0.289735i	0.770183	+	0.637822i	$0.220165\pi$
$510$		0			0
$511$	−5.00000	−	8.66025i	−0.221187	−	0.383107i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	21.8600	−	3.04900i	0.963267	−	0.134355i
$516$		0			0
$517$	24.8802	+	14.3646i	1.09423	+	0.631755i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−9.03102			−0.395656			−0.197828	−	0.980237i	$-0.563389\pi$
$521$	−9.03102			−0.395656			−0.197828	+	0.980237i	$0.563389\pi$
$522$		0			0
$523$		−	7.58430i		−	0.331638i	−0.986156	−	0.165819i	$-0.946973\pi$
$523$		−	7.58430i		−	0.331638i	0.986156	−	0.165819i	$-0.0530268\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	36.7550	+	21.2205i	1.60107	+	0.924380i
$528$		0			0
$529$	−5.85782	−	10.1460i	−0.254688	−	0.441132i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	14.8061	−	8.54830i	0.641323	−	0.370268i
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−41.8791			−1.80386
$540$		0			0
$541$	3.35782			0.144364			0.0721819	−	0.997391i	$-0.477004\pi$
$541$	3.35782			0.144364			0.0721819	+	0.997391i	$0.477004\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	12.3444	−	9.62377i	0.528774	−	0.412237i
$546$		0			0
$547$	−17.0966	+	9.87073i	−0.730998	+	0.422042i	−0.818787	−	0.574097i	$-0.805353\pi$
$547$	−17.0966	+	9.87073i	−0.730998	+	0.422042i	0.0877892	+	0.996139i	$0.472020\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−10.8571	−	18.8051i	−0.462529	−	0.801124i
$552$		0			0
$553$	−25.6449	−	14.8061i	−1.09053	−	0.629619i
$554$		0			0
$555$		0			0
$556$		0			0
$557$			39.2320i			1.66231i	0.556037	+	0.831157i	$0.312321\pi$
$557$			39.2320i			1.66231i	−0.556037	+	0.831157i	$0.687679\pi$
$558$		0			0
$559$	−14.3578			−0.607271
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−10.3122	−	5.95376i	−0.434608	−	0.250921i	0.266700	−	0.963780i	$-0.414067\pi$
$563$	−10.3122	−	5.95376i	−0.434608	−	0.250921i	−0.701308	+	0.712859i	$0.747400\pi$
$564$		0			0
$565$	30.0905	−	4.19698i	1.26592	−	0.176568i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	2.22731	+	3.85782i	0.0933738	+	0.161728i	0.908929	−	0.416952i	$-0.136902\pi$
$569$	2.22731	+	3.85782i	0.0933738	+	0.161728i	−0.815555	+	0.578680i	$0.803568\pi$
$570$		0			0
$571$	9.08945	−	15.7434i	0.380382	−	0.658841i	−0.610735	−	0.791835i	$-0.709126\pi$
$571$	9.08945	−	15.7434i	0.380382	−	0.658841i	0.991117	+	0.132994i	$0.0424592\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	11.6927	−	12.0578i	0.487619	−	0.502844i
$576$		0			0
$577$		−	7.84453i		−	0.326572i	−0.986579	−	0.163286i	$-0.947791\pi$
$577$		−	7.84453i		−	0.326572i	0.986579	−	0.163286i	$-0.0522094\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−12.8050	+	22.1789i	−0.531241	+	0.920136i
$582$		0			0
$583$	−17.8613	+	10.3122i	−0.739739	+	0.427088i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	10.0520	−	5.80351i	0.414890	−	0.239537i	−0.277999	−	0.960581i	$-0.589671\pi$
$587$	10.0520	−	5.80351i	0.414890	−	0.239537i	0.692888	+	0.721045i	$0.256338\pi$
$588$		0			0
$589$	12.9105	−	22.3617i	0.531970	−	0.921399i
$590$		0			0
$591$		0			0
$592$		0			0
$593$		−	5.03883i		−	0.206920i	−0.994634	−	0.103460i	$-0.967009\pi$
$593$		−	5.03883i		−	0.206920i	0.994634	−	0.103460i	$-0.0329914\pi$
$594$		0			0
$595$	70.2524	+	28.4690i	2.88007	+	1.16711i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	8.50531	−	14.7316i	0.347518	−	0.601918i	−0.638290	−	0.769796i	$-0.720358\pi$
$599$	8.50531	−	14.7316i	0.347518	−	0.601918i	0.985808	+	0.167878i	$0.0536913\pi$
$600$		0			0
$601$	1.67891	+	2.90795i	0.0684841	+	0.118618i	0.898234	−	0.439517i	$-0.144850\pi$
$601$	1.67891	+	2.90795i	0.0684841	+	0.118618i	−0.829750	+	0.558135i	$0.811517\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	1.59973	+	11.4694i	0.0650383	+	0.466296i
$606$		0			0
$607$	−4.27415	−	2.46768i	−0.173482	−	0.100160i	0.410744	−	0.911751i	$-0.365269\pi$
$607$	−4.27415	−	2.46768i	−0.173482	−	0.100160i	−0.584227	+	0.811590i	$0.698602\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	34.6410			1.40143
$612$		0			0
$613$			37.7170i			1.52337i	0.647945	+	0.761687i	$0.275628\pi$
$613$			37.7170i			1.52337i	−0.647945	+	0.761687i	$0.724372\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−4.62399	−	2.66966i	−0.186155	−	0.107477i	0.404026	−	0.914747i	$-0.367610\pi$
$617$	−4.62399	−	2.66966i	−0.186155	−	0.107477i	−0.590181	+	0.807271i	$0.700944\pi$
$618$		0			0
$619$	5.82109	+	10.0824i	0.233969	+	0.405247i	0.958973	−	0.283499i	$-0.0914951\pi$
$619$	5.82109	+	10.0824i	0.233969	+	0.405247i	−0.725003	+	0.688745i	$0.758162\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−13.2214	+	7.63337i	−0.529704	+	0.305825i
$624$		0			0
$625$	−22.0246	−	11.8287i	−0.880984	−	0.473147i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	53.8817			2.14840
$630$		0			0
$631$	−22.5367			−0.897173			−0.448586	−	0.893739i	$-0.648072\pi$
$631$	−22.5367			−0.897173			−0.448586	+	0.893739i	$0.648072\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	6.78526	+	8.70341i	0.269265	+	0.345384i
$636$		0			0
$637$	−43.7316	+	25.2484i	−1.73271	+	1.00038i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−9.15551	−	15.8578i	−0.361621	−	0.626346i	0.626607	−	0.779336i	$-0.284443\pi$
$641$	−9.15551	−	15.8578i	−0.361621	−	0.626346i	−0.988228	+	0.152990i	$0.951110\pi$
$642$		0			0
$643$	−17.8613	−	10.3122i	−0.704380	−	0.406674i	0.104597	−	0.994515i	$-0.466645\pi$
$643$	−17.8613	−	10.3122i	−0.704380	−	0.406674i	−0.808977	+	0.587841i	$0.799978\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$			44.2709i			1.74047i	0.492639	+	0.870234i	$0.336032\pi$
$647$			44.2709i			1.74047i	−0.492639	+	0.870234i	$0.663968\pi$
$648$		0			0
$649$	−2.53673			−0.0995752
$650$		0			0
$651$		0			0
$652$		0			0
$653$	0.260237	+	0.150248i	0.0101839	+	0.00587967i	0.505083	−	0.863071i	$-0.331462\pi$
$653$	0.260237	+	0.150248i	0.0101839	+	0.00587967i	−0.494899	+	0.868950i	$0.664795\pi$
$654$		0			0
$655$	−4.37587	−	31.3731i	−0.170979	−	1.22585i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−20.7237	−	35.8945i	−0.807282	−	1.39825i	−0.914740	−	0.404043i	$-0.867605\pi$
$659$	−20.7237	−	35.8945i	−0.807282	−	1.39825i	0.107458	−	0.994210i	$-0.465729\pi$
$660$		0			0
$661$	23.2156	−	40.2107i	0.902983	−	1.56401i	0.0793821	−	0.996844i	$-0.474705\pi$
$661$	23.2156	−	40.2107i	0.902983	−	1.56401i	0.823601	−	0.567169i	$-0.191961\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	17.3205	−	42.7415i	0.671660	−	1.65744i
$666$		0			0
$667$		−	17.4550i		−	0.675861i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−11.0729	+	19.1789i	−0.427466	+	0.740394i
$672$		0			0
$673$	−12.5971	+	7.27293i	−0.485582	+	0.280351i	−0.722740	−	0.691120i	$-0.757117\pi$
$673$	−12.5971	+	7.27293i	−0.485582	+	0.280351i	0.237158	+	0.971471i	$0.423784\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	29.8724	−	17.2469i	1.14809	−	0.662850i	0.199669	−	0.979863i	$-0.436013\pi$
$677$	29.8724	−	17.2469i	1.14809	−	0.662850i	0.948421	+	0.317013i	$0.102680\pi$
$678$		0			0
$679$	2.17891	−	3.77398i	0.0836188	−	0.144832i
$680$		0			0
$681$		0			0
$682$		0			0
$683$		−	44.5714i		−	1.70548i	−0.522339	−	0.852738i	$-0.674940\pi$
$683$		−	44.5714i		−	1.70548i	0.522339	−	0.852738i	$-0.325060\pi$
$684$		0			0
$685$	−4.23164	+	10.4423i	−0.161683	+	0.398981i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−12.4342	+	21.5367i	−0.473707	+	0.820484i
$690$		0			0
$691$	15.3578	+	26.6005i	0.584239	+	1.01193i	0.994970	+	0.100175i	$0.0319402\pi$
$691$	15.3578	+	26.6005i	0.584239	+	1.01193i	−0.410731	+	0.911757i	$0.634726\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	1.79810	+	12.8916i	0.0682057	+	0.489005i
$696$		0			0
$697$	−34.9579	−	20.1829i	−1.32412	−	0.764484i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−27.3420			−1.03269			−0.516347	−	0.856379i	$-0.672709\pi$
$701$	−27.3420			−1.03269			−0.516347	+	0.856379i	$0.672709\pi$
$702$		0			0
$703$		−	32.7816i		−	1.23638i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−25.1183	−	14.5021i	−0.944671	−	0.545406i
$708$		0			0
$709$	7.76836	+	13.4552i	0.291747	+	0.505321i	0.974223	−	0.225587i	$-0.0724301\pi$
$709$	7.76836	+	13.4552i	0.291747	+	0.505321i	−0.682476	+	0.730908i	$0.739097\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	17.9755	−	10.3782i	0.673188	−	0.388665i
$714$		0			0
$715$	−9.64981	−	12.3777i	−0.360882	−	0.462902i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	40.3960			1.50652			0.753259	−	0.657724i	$-0.228481\pi$
$719$	40.3960			1.50652			0.753259	+	0.657724i	$0.228481\pi$
$720$		0			0
$721$	48.7156			1.81426
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−24.9892	+	7.10922i	−0.928075	+	0.264030i
$726$		0			0
$727$	−40.7614	+	23.5336i	−1.51176	+	0.872813i	−0.511851	+	0.859074i	$0.671040\pi$
$727$	−40.7614	+	23.5336i	−1.51176	+	0.872813i	−0.999906	+	0.0137388i	$0.995627\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	16.9497	+	29.3578i	0.626909	+	1.08584i
$732$		0			0
$733$	17.0966	+	9.87073i	0.631477	+	0.364584i	0.781324	−	0.624126i	$-0.214545\pi$
$733$	17.0966	+	9.87073i	0.631477	+	0.364584i	−0.149847	+	0.988709i	$0.547878\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		−	9.77717i		−	0.360147i
$738$		0			0
$739$	22.8945			0.842189			0.421095	−	0.907017i	$-0.361646\pi$
$739$	22.8945			0.842189			0.421095	+	0.907017i	$0.361646\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−43.0938	−	24.8802i	−1.58096	−	0.912767i	−0.994720	−	0.102628i	$-0.967275\pi$
$743$	−43.0938	−	24.8802i	−1.58096	−	0.912767i	−0.586239	−	0.810138i	$-0.699392\pi$
$744$		0			0
$745$	4.06991	+	29.1794i	0.149110	+	1.06905i
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	3.17891	−	5.50603i	0.116000	−	0.200918i	−0.802179	−	0.597084i	$-0.796326\pi$
$751$	3.17891	−	5.50603i	0.116000	−	0.200918i	0.918179	+	0.396166i	$0.129659\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−12.8050	−	5.18908i	−0.466022	−	0.188850i
$756$		0			0
$757$		−	1.76596i		−	0.0641847i	−0.999485	−	0.0320924i	$-0.989783\pi$
$757$		−	1.76596i		−	0.0641847i	0.999485	−	0.0320924i	$-0.0102171\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	0.866025	−	1.50000i	0.0313934	−	0.0543750i	−0.849902	−	0.526941i	$-0.823339\pi$
$761$	0.866025	−	1.50000i	0.0313934	−	0.0543750i	0.881295	+	0.472566i	$0.156672\pi$
$762$		0			0
$763$	29.9191	−	17.2738i	1.08314	−	0.625353i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−2.64893	+	1.52936i	−0.0956474	+	0.0552221i
$768$		0			0
$769$	12.8578	−	22.2704i	0.463665	−	0.803091i	−0.535475	−	0.844551i	$-0.679868\pi$
$769$	12.8578	−	22.2704i	0.463665	−	0.803091i	0.999140	+	0.0414599i	$0.0132009\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$			30.6837i			1.10362i	0.833971	+	0.551809i	$0.186062\pi$
$773$			30.6837i			1.10362i	−0.833971	+	0.551809i	$0.813938\pi$
$774$		0			0
$775$	−22.1789	−	21.5074i	−0.796690	−	0.772569i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−12.2793	+	21.2684i	−0.439951	+	0.762018i
$780$		0			0
$781$	−17.0895	−	29.5998i	−0.611509	−	1.05916i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	17.3727	−	2.42313i	0.620060	−	0.0864851i
$786$		0			0
$787$	−41.5261	−	23.9751i	−1.48024	−	0.854620i	−0.480495	−	0.876997i	$-0.659543\pi$
$787$	−41.5261	−	23.9751i	−1.48024	−	0.854620i	−0.999750	+	0.0223775i	$0.992876\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	67.0574			2.38429
$792$		0			0
$793$			26.7030i			0.948252i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	10.1821	+	5.87864i	0.360668	+	0.208232i	0.669374	−	0.742926i	$-0.266562\pi$
$797$	10.1821	+	5.87864i	0.360668	+	0.208232i	−0.308706	+	0.951158i	$0.599896\pi$
$798$		0			0
$799$	−40.8945	−	70.8314i	−1.44674	−	2.50584i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	4.23364	−	2.44429i	0.149402	−	0.0862572i
$804$		0			0
$805$	29.2367	−	22.7932i	1.03046	−	0.803354i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	5.19615			0.182687			0.0913435	−	0.995819i	$-0.470884\pi$
$809$	5.19615			0.182687			0.0913435	+	0.995819i	$0.470884\pi$
$810$		0			0
$811$	6.89454			0.242100			0.121050	−	0.992646i	$-0.461374\pi$
$811$	6.89454			0.242100			0.121050	+	0.992646i	$0.461374\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	14.7845	+	18.9639i	0.517877	+	0.664278i
$816$		0			0
$817$	17.8613	−	10.3122i	0.624887	−	0.360779i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−26.1053	−	45.2156i	−0.911080	−	1.57804i	−0.812542	−	0.582903i	$-0.801917\pi$
$821$	−26.1053	−	45.2156i	−0.911080	−	1.57804i	−0.0985382	−	0.995133i	$-0.531417\pi$
$822$		0			0
$823$	−10.0777	−	5.81834i	−0.351285	−	0.202815i	0.313966	−	0.949434i	$-0.398342\pi$
$823$	−10.0777	−	5.81834i	−0.351285	−	0.202815i	−0.665251	+	0.746620i	$0.731676\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		−	13.7374i		−	0.477696i	−0.971057	−	0.238848i	$-0.923230\pi$
$827$		−	13.7374i		−	0.477696i	0.971057	−	0.238848i	$-0.0767697\pi$
$828$		0			0
$829$	4.17891			0.145139			0.0725697	−	0.997363i	$-0.476880\pi$
$829$	4.17891			0.145139			0.0725697	+	0.997363i	$0.476880\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	103.252	+	59.6127i	3.57748	+	2.06546i
$834$		0			0
$835$	18.9313	−	2.64051i	0.655145	−	0.0913788i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−11.9085	−	20.6262i	−0.411128	−	0.712095i	0.583885	−	0.811836i	$-0.301532\pi$
$839$	−11.9085	−	20.6262i	−0.411128	−	0.712095i	−0.995013	+	0.0997414i	$0.968198\pi$
$840$		0			0
$841$	1.00000	−	1.73205i	0.0344828	−	0.0597259i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	9.40178	+	3.80997i	0.323431	+	0.131067i
$846$		0			0
$847$			25.5598i			0.878245i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	13.1758	−	22.8211i	0.451659	−	0.782297i
$852$		0			0
$853$	10.8423	−	6.25983i	0.371235	−	0.214333i	−0.302763	−	0.953066i	$-0.597909\pi$
$853$	10.8423	−	6.25983i	0.371235	−	0.214333i	0.673998	+	0.738733i	$0.264576\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−48.7820	+	28.1643i	−1.66636	+	0.962075i	−0.696788	+	0.717277i	$0.745388\pi$
$857$	−48.7820	+	28.1643i	−1.66636	+	0.962075i	−0.969574	+	0.244798i	$0.921278\pi$
$858$		0			0
$859$	19.2684	−	33.3738i	0.657428	−	1.13870i	−0.323851	−	0.946108i	$-0.604978\pi$
$859$	19.2684	−	33.3738i	0.657428	−	1.13870i	0.981279	−	0.192591i	$-0.0616890\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$			8.54830i			0.290988i	0.989359	+	0.145494i	$0.0464771\pi$
$863$			8.54830i			0.290988i	−0.989359	+	0.145494i	$0.953523\pi$
$864$		0			0
$865$	15.7684	−	38.9113i	0.536140	−	1.32302i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	7.23808	−	12.5367i	0.245535	−	0.425279i
$870$		0			0
$871$	−5.89454	−	10.2096i	−0.199729	−	0.345941i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−44.5392	−	32.5728i	−1.50570	−	1.10116i
$876$		0			0
$877$	41.7515	+	24.1052i	1.40985	+	0.813975i	0.995373	−	0.0960863i	$-0.0306325\pi$
$877$	41.7515	+	24.1052i	1.40985	+	0.813975i	0.414473	+	0.910061i	$0.363966\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−23.9388			−0.806520			−0.403260	−	0.915085i	$-0.632123\pi$
$881$	−23.9388			−0.806520			−0.403260	+	0.915085i	$0.632123\pi$
$882$		0			0
$883$		−	50.2366i		−	1.69060i	−0.534295	−	0.845298i	$-0.679423\pi$
$883$		−	50.2366i		−	1.69060i	0.534295	−	0.845298i	$-0.320577\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	22.2091	+	12.8225i	0.745710	+	0.430536i	0.824142	−	0.566384i	$-0.191658\pi$
$887$	22.2091	+	12.8225i	0.745710	+	0.430536i	−0.0784318	+	0.996919i	$0.524991\pi$
$888$		0			0
$889$	12.1789	+	21.0945i	0.408467	+	0.707486i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−43.0938	+	24.8802i	−1.44208	+	0.832585i
$894$		0			0
$895$	14.0719	−	10.9706i	0.470371	−	0.366706i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−32.1065			−1.07081
$900$		0			0
$901$	58.7156			1.95610
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−6.73841	+	5.25333i	−0.223992	+	0.174627i
$906$		0			0
$907$	51.2898	−	29.6122i	1.70305	−	0.983256i	0.760411	−	0.649442i	$-0.224997\pi$
$907$	51.2898	−	29.6122i	1.70305	−	0.983256i	0.942639	−	0.333815i	$-0.108336\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	17.7853	+	30.8051i	0.589254	+	1.02062i	0.994330	+	0.106335i	$0.0339116\pi$
$911$	17.7853	+	30.8051i	0.589254	+	1.02062i	−0.405076	+	0.914283i	$0.632755\pi$
$912$		0			0
$913$	−10.8423	−	6.25983i	−0.358829	−	0.207170i
$914$		0			0
$915$		0			0
$916$		0			0
$917$		−	69.9158i		−	2.30882i
$918$		0			0
$919$	19.8211			0.653837			0.326919	−	0.945052i	$-0.393990\pi$
$919$	19.8211			0.653837			0.326919	+	0.945052i	$0.393990\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−35.6908	−	20.6061i	−1.17478	−	0.678257i
$924$		0			0
$925$	−38.0377	−	9.56810i	−1.25067	−	0.314597i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−6.37206	−	11.0367i	−0.209060	−	0.362103i	0.742358	−	0.670003i	$-0.233707\pi$
$929$	−6.37206	−	11.0367i	−0.209060	−	0.362103i	−0.951419	+	0.307900i	$0.900374\pi$
$930$		0			0
$931$	36.2684	−	62.8186i	1.18865	−	2.05880i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−13.9173	+	34.3435i	−0.455144	+	1.12315i
$936$		0			0
$937$		−	12.7799i		−	0.417501i	−0.977969	−	0.208751i	$-0.933060\pi$
$937$		−	12.7799i		−	0.417501i	0.977969	−	0.208751i	$-0.0669397\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−26.9408	+	46.6629i	−0.878246	+	1.52117i	−0.0249824	+	0.999688i	$0.507953\pi$
$941$	−26.9408	+	46.6629i	−0.878246	+	1.52117i	−0.853264	+	0.521479i	$0.825380\pi$
$942$		0			0
$943$	−17.0966	+	9.87073i	−0.556742	+	0.321435i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−6.07858	+	3.50947i	−0.197527	+	0.114042i	−0.595502	−	0.803354i	$-0.703047\pi$
$947$	−6.07858	+	3.50947i	−0.197527	+	0.114042i	0.397974	+	0.917397i	$0.369713\pi$
$948$		0			0
$949$	2.94727	−	5.10482i	0.0956725	−	0.165710i
$950$		0			0
$951$		0			0
$952$		0			0
$953$		−	11.7573i		−	0.380855i	−0.981701	−	0.190428i	$-0.939013\pi$
$953$		−	11.7573i		−	0.380855i	0.981701	−	0.190428i	$-0.0609875\pi$
$954$		0			0
$955$	−13.7156	−	5.55810i	−0.443827	−	0.179856i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−12.4342	+	21.5367i	−0.401522	+	0.695457i
$960$		0			0
$961$	−3.58945	−	6.21712i	−0.115789	−	0.200552i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−48.2073	+	6.72388i	−1.55185	+	0.216449i
$966$		0			0
$967$	11.2931	+	6.52007i	0.363161	+	0.209671i	0.670467	−	0.741940i	$-0.266094\pi$
$967$	11.2931	+	6.52007i	0.363161	+	0.209671i	−0.307305	+	0.951611i	$0.599427\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−21.7142			−0.696843			−0.348422	−	0.937338i	$-0.613282\pi$
$971$	−21.7142			−0.696843			−0.348422	+	0.937338i	$0.613282\pi$
$972$		0			0
$973$			28.7292i			0.921016i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−23.5336	−	13.5871i	−0.752907	−	0.434691i	0.0738365	−	0.997270i	$-0.476476\pi$
$977$	−23.5336	−	13.5871i	−0.752907	−	0.434691i	−0.826743	+	0.562579i	$0.809809\pi$
$978$		0			0
$979$	−3.73164	−	6.46339i	−0.119264	−	0.206571i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	22.4694	−	12.9727i	0.716662	−	0.413765i	−0.0968610	−	0.995298i	$-0.530880\pi$
$983$	22.4694	−	12.9727i	0.716662	−	0.413765i	0.813523	+	0.581533i	$0.197547\pi$
$984$		0			0
$985$	21.1957	+	27.1875i	0.675350	+	0.866267i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	16.5790			0.527181
$990$		0			0
$991$	−32.5367			−1.03356			−0.516782	−	0.856117i	$-0.672870\pi$
$991$	−32.5367			−1.03356			−0.516782	+	0.856117i	$0.672870\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	35.9006	−	27.9884i	1.13812	−	0.887293i
$996$		0			0
$997$	25.8703	−	14.9362i	0.819320	−	0.473035i	−0.0308620	−	0.999524i	$-0.509825\pi$
$997$	25.8703	−	14.9362i	0.819320	−	0.473035i	0.850182	+	0.526489i	$0.176492\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1620.2.r.h.109.2		16
3.2	odd	2	inner	1620.2.r.h.109.7		16
5.4	even	2	inner	1620.2.r.h.109.4		16
9.2	odd	6	inner	1620.2.r.h.1189.5		16
9.4	even	3		1620.2.d.e.649.8	yes	8
9.5	odd	6		1620.2.d.e.649.1	✓	8
9.7	even	3	inner	1620.2.r.h.1189.4		16
15.14	odd	2	inner	1620.2.r.h.109.5		16
45.4	even	6		1620.2.d.e.649.7	yes	8
45.13	odd	12		8100.2.a.be.1.2		8
45.14	odd	6		1620.2.d.e.649.2	yes	8
45.22	odd	12		8100.2.a.be.1.8		8
45.23	even	12		8100.2.a.be.1.1		8
45.29	odd	6	inner	1620.2.r.h.1189.7		16
45.32	even	12		8100.2.a.be.1.7		8
45.34	even	6	inner	1620.2.r.h.1189.2		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1620.2.d.e.649.1	✓	8	9.5	odd	6
1620.2.d.e.649.2	yes	8	45.14	odd	6
1620.2.d.e.649.7	yes	8	45.4	even	6
1620.2.d.e.649.8	yes	8	9.4	even	3
1620.2.r.h.109.2		16	1.1	even	1	trivial
1620.2.r.h.109.4		16	5.4	even	2	inner
1620.2.r.h.109.5		16	15.14	odd	2	inner
1620.2.r.h.109.7		16	3.2	odd	2	inner
1620.2.r.h.1189.2		16	45.34	even	6	inner
1620.2.r.h.1189.4		16	9.7	even	3	inner
1620.2.r.h.1189.5		16	9.2	odd	6	inner
1620.2.r.h.1189.7		16	45.29	odd	6	inner
8100.2.a.be.1.1		8	45.23	even	12
8100.2.a.be.1.2		8	45.13	odd	12
8100.2.a.be.1.7		8	45.32	even	12
8100.2.a.be.1.8		8	45.22	odd	12

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:	\( N \)	\(=\)	\( 1620 = 2^{2} \cdot 3^{4} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1620.r (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-1.76348 + 1.37482i) q^{5} +(-4.27415 + 2.46768i) q^{7} +O(q^{10})\)	\(q+(-1.76348 + 1.37482i) q^{5} +(-4.27415 + 2.46768i) q^{7} +(-1.20635 - 2.08945i) q^{11} +(-2.51942 - 1.45459i) q^{13} +6.86869i q^{17} +4.17891 q^{19} +(2.90917 + 1.67961i) q^{23} +(1.21972 - 4.84895i) q^{25} +(-2.59808 - 4.50000i) q^{29} +(3.08945 - 5.35109i) q^{31} +(4.14474 - 10.2279i) q^{35} -7.84453i q^{37} +(-2.93840 + 5.08945i) q^{41} +(4.27415 - 2.46768i) q^{43} +(-10.3122 + 5.95376i) q^{47} +(8.67891 - 15.0323i) q^{49} -8.54830i q^{53} +(5.00000 + 2.02619i) q^{55} +(0.525704 - 0.910546i) q^{59} +(-4.58945 - 7.94917i) q^{61} +(6.44274 - 0.898624i) q^{65} +(3.50947 + 2.02619i) q^{67} +14.1663 q^{71} +2.02619i q^{73} +(10.3122 + 5.95376i) q^{77} +(3.00000 + 5.19615i) q^{79} +(4.49387 - 2.59454i) q^{83} +(-9.44324 - 12.1128i) q^{85} +3.09334 q^{89} +14.3578 q^{91} +(-7.36942 + 5.74527i) q^{95} +(-0.764681 + 0.441489i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q+O(q^{10}) \) 16 * q	\( 16 q - 24 q^{19} - 6 q^{25} + 4 q^{31} + 48 q^{49} + 80 q^{55} - 28 q^{61} + 48 q^{79} - 22 q^{85} + 48 q^{91}+O(q^{100}) \) 16 * q - 24 * q^19 - 6 * q^25 + 4 * q^31 + 48 * q^49 + 80 * q^55 - 28 * q^61 + 48 * q^79 - 22 * q^85 + 48 * q^91

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