LMFDB - Embedded newform 810.2.c.g.649.4

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [810,2,Mod(649,810)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(810, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 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("810.649"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 810 = 2 \cdot 3^{4} \cdot 5 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	810.c (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [8,0,0,-8,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)

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

Self dual:	no
Analytic conductor:	$6.46788256372$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.2702336256.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 9x^{6} + 56x^{4} + 225x^{2} + 625 $ x^8 + 9x^6 + 56x^4 + 225*x^2 + 625
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			649.4
Root			$0.656712 - 2.13746i$ of defining polynomial
Character	$\chi$	$=$	810.649
Dual form			810.2.c.g.649.8

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-1.00000i q^{2} -1.00000 q^{4} +(1.52274 + 1.63746i) q^{5} +1.31342i q^{7} +1.00000i q^{8} +(1.63746 - 1.52274i) q^{10} -2.62685 q^{11} +1.73205i q^{13} +1.31342 q^{14} +1.00000 q^{16} +1.27492i q^{17} +2.27492 q^{19} +(-1.52274 - 1.63746i) q^{20} +2.62685i q^{22} +6.27492i q^{23} +(-0.362541 + 4.98684i) q^{25} +1.73205 q^{26} -1.31342i q^{28} -9.13642 q^{29} +8.54983 q^{31} -1.00000i q^{32} +1.27492 q^{34} +(-2.15068 + 2.00000i) q^{35} +9.97368i q^{37} -2.27492i q^{38} +(-1.63746 + 1.52274i) q^{40} -5.61478 q^{41} +2.62685i q^{43} +2.62685 q^{44} +6.27492 q^{46} -10.2749i q^{47} +5.27492 q^{49} +(4.98684 + 0.362541i) q^{50} -1.73205i q^{52} +8.27492i q^{53} +(-4.00000 - 4.30136i) q^{55} -1.31342 q^{56} +9.13642i q^{58} +8.24163 q^{59} +7.27492 q^{61} -8.54983i q^{62} -1.00000 q^{64} +(-2.83616 + 2.63746i) q^{65} -12.1819i q^{67} -1.27492i q^{68} +(2.00000 + 2.15068i) q^{70} +6.92820 q^{71} +4.83507i q^{73} +9.97368 q^{74} -2.27492 q^{76} -3.45017i q^{77} +(1.52274 + 1.63746i) q^{80} +5.61478i q^{82} +4.00000i q^{83} +(-2.08762 + 1.94136i) q^{85} +2.62685 q^{86} -2.62685i q^{88} -3.04547 q^{89} -2.27492 q^{91} -6.27492i q^{92} -10.2749 q^{94} +(3.46410 + 3.72508i) q^{95} -19.1101i q^{97} -5.27492i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 8 q^{4} - 2 q^{10} + 8 q^{16} - 12 q^{19} - 18 q^{25} + 8 q^{31} - 20 q^{34} + 2 q^{40} + 20 q^{46} + 12 q^{49} - 32 q^{55} + 28 q^{61} - 8 q^{64} + 16 q^{70} + 12 q^{76} - 62 q^{85} + 12 q^{91} - 52 q^{94}+O(q^{100}) $ 8 * q - 8 * q^4 - 2 * q^10 + 8 * q^16 - 12 * q^19 - 18 * q^25 + 8 * q^31 - 20 * q^34 + 2 * q^40 + 20 * q^46 + 12 * q^49 - 32 * q^55 + 28 * q^61 - 8 * q^64 + 16 * q^70 + 12 * q^76 - 62 * q^85 + 12 * q^91 - 52 * q^94

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/810\mathbb{Z}\right)^\times$.

$n$	$487$	$731$
$\chi(n)$	$-1$	$1$

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 $

$2$		−	1.00000i		−	0.707107i
$2$		−	1.00000i		−	0.707107i
$3$		0			0
$3$		0			0
$4$	−1.00000			−0.500000
$5$	1.52274	+	1.63746i	0.680989	+	0.732294i
$5$	1.52274	+	1.63746i	0.680989	+	0.732294i
$6$		0			0
$7$			1.31342i			0.496428i	0.968705	+	0.248214i	$0.0798436\pi$
$7$			1.31342i			0.496428i	−0.968705	+	0.248214i	$0.920156\pi$
$8$			1.00000i			0.353553i
$9$		0			0
$10$	1.63746	−	1.52274i	0.517810	−	0.481532i
$11$	−2.62685			−0.792025			−0.396012	−	0.918245i	$-0.629606\pi$
$11$	−2.62685			−0.792025			−0.396012	+	0.918245i	$0.629606\pi$
$12$		0			0
$13$			1.73205i			0.480384i	0.970725	+	0.240192i	$0.0772105\pi$
$13$			1.73205i			0.480384i	−0.970725	+	0.240192i	$0.922790\pi$
$14$	1.31342			0.351027
$15$		0			0
$16$	1.00000			0.250000
$17$			1.27492i			0.309213i	0.987976	+	0.154606i	$0.0494109\pi$
$17$			1.27492i			0.309213i	−0.987976	+	0.154606i	$0.950589\pi$
$18$		0			0
$19$	2.27492			0.521902			0.260951	−	0.965352i	$-0.415964\pi$
$19$	2.27492			0.521902			0.260951	+	0.965352i	$0.415964\pi$
$20$	−1.52274	−	1.63746i	−0.340494	−	0.366147i
$21$		0			0
$22$			2.62685i			0.560046i
$23$			6.27492i			1.30841i	0.756317	+	0.654205i	$0.226997\pi$
$23$			6.27492i			1.30841i	−0.756317	+	0.654205i	$0.773003\pi$
$24$		0			0
$25$	−0.362541	+	4.98684i	−0.0725083	+	0.997368i
$26$	1.73205			0.339683
$27$		0			0
$28$		−	1.31342i		−	0.248214i
$29$	−9.13642			−1.69659			−0.848296	−	0.529523i	$-0.822371\pi$
$29$	−9.13642			−1.69659			−0.848296	+	0.529523i	$0.822371\pi$
$30$		0			0
$31$	8.54983			1.53560			0.767798	−	0.640692i	$-0.221353\pi$
$31$	8.54983			1.53560			0.767798	+	0.640692i	$0.221353\pi$
$32$		−	1.00000i		−	0.176777i
$33$		0			0
$34$	1.27492			0.218646
$35$	−2.15068	+	2.00000i	−0.363531	+	0.338062i
$36$		0			0
$37$			9.97368i			1.63966i	0.572605	+	0.819831i	$0.305933\pi$
$37$			9.97368i			1.63966i	−0.572605	+	0.819831i	$0.694067\pi$
$38$		−	2.27492i		−	0.369040i
$39$		0			0
$40$	−1.63746	+	1.52274i	−0.258905	+	0.240766i
$41$	−5.61478			−0.876881			−0.438441	−	0.898760i	$-0.644469\pi$
$41$	−5.61478			−0.876881			−0.438441	+	0.898760i	$0.644469\pi$
$42$		0			0
$43$			2.62685i			0.400591i	0.979736	+	0.200295i	$0.0641902\pi$
$43$			2.62685i			0.400591i	−0.979736	+	0.200295i	$0.935810\pi$
$44$	2.62685			0.396012
$45$		0			0
$46$	6.27492			0.925186
$47$		−	10.2749i		−	1.49875i	−0.662145	−	0.749375i	$-0.730354\pi$
$47$		−	10.2749i		−	1.49875i	0.662145	−	0.749375i	$-0.269646\pi$
$48$		0			0
$49$	5.27492			0.753560
$50$	4.98684	+	0.362541i	0.705246	+	0.0512711i
$51$		0			0
$52$		−	1.73205i		−	0.240192i
$53$			8.27492i			1.13665i	0.822805	+	0.568324i	$0.192408\pi$
$53$			8.27492i			1.13665i	−0.822805	+	0.568324i	$0.807592\pi$
$54$		0			0
$55$	−4.00000	−	4.30136i	−0.539360	−	0.579995i
$56$	−1.31342			−0.175514
$57$		0			0
$58$			9.13642i			1.19967i
$59$	8.24163			1.07297			0.536484	−	0.843910i	$-0.319752\pi$
$59$	8.24163			1.07297			0.536484	+	0.843910i	$0.319752\pi$
$60$		0			0
$61$	7.27492			0.931458			0.465729	−	0.884927i	$-0.345792\pi$
$61$	7.27492			0.931458			0.465729	+	0.884927i	$0.345792\pi$
$62$		−	8.54983i		−	1.08583i
$63$		0			0
$64$	−1.00000			−0.125000
$65$	−2.83616	+	2.63746i	−0.351783	+	0.327136i
$66$		0			0
$67$		−	12.1819i		−	1.48826i	−0.668037	−	0.744128i	$-0.732865\pi$
$67$		−	12.1819i		−	1.48826i	0.668037	−	0.744128i	$-0.267135\pi$
$68$		−	1.27492i		−	0.154606i
$69$		0			0
$70$	2.00000	+	2.15068i	0.239046	+	0.257055i
$71$	6.92820			0.822226			0.411113	−	0.911584i	$-0.365140\pi$
$71$	6.92820			0.822226			0.411113	+	0.911584i	$0.365140\pi$
$72$		0			0
$73$			4.83507i			0.565902i	0.959134	+	0.282951i	$0.0913134\pi$
$73$			4.83507i			0.565902i	−0.959134	+	0.282951i	$0.908687\pi$
$74$	9.97368			1.15942
$75$		0			0
$76$	−2.27492			−0.260951
$77$		−	3.45017i		−	0.393183i
$78$		0			0
$79$		0			0			−	1.00000i	$-0.5\pi$
$79$		0			0				1.00000i	$0.5\pi$
$80$	1.52274	+	1.63746i	0.170247	+	0.183073i
$81$		0			0
$82$			5.61478i			0.620049i
$83$			4.00000i			0.439057i	0.975606	+	0.219529i	$0.0704519\pi$
$83$			4.00000i			0.439057i	−0.975606	+	0.219529i	$0.929548\pi$
$84$		0			0
$85$	−2.08762	+	1.94136i	−0.226435	+	0.210571i
$86$	2.62685			0.283260
$87$		0			0
$88$		−	2.62685i		−	0.280023i
$89$	−3.04547			−0.322820			−0.161410	−	0.986887i	$-0.551604\pi$
$89$	−3.04547			−0.322820			−0.161410	+	0.986887i	$0.551604\pi$
$90$		0			0
$91$	−2.27492			−0.238476
$92$		−	6.27492i		−	0.654205i
$93$		0			0
$94$	−10.2749			−1.05978
$95$	3.46410	+	3.72508i	0.355409	+	0.382185i
$96$		0			0
$97$		−	19.1101i		−	1.94034i	−0.242432	−	0.970168i	$-0.577945\pi$
$97$		−	19.1101i		−	1.94034i	0.242432	−	0.970168i	$-0.422055\pi$
$98$		−	5.27492i		−	0.532847i
$99$		0			0
$100$	0.362541	−	4.98684i	0.0362541	−	0.498684i
$101$	9.55505			0.950763			0.475382	−	0.879780i	$-0.342310\pi$
$101$	9.55505			0.950763			0.475382	+	0.879780i	$0.342310\pi$
$102$		0			0
$103$		−	6.56712i		−	0.647078i	−0.946215	−	0.323539i	$-0.895127\pi$
$103$		−	6.56712i		−	0.647078i	0.946215	−	0.323539i	$-0.104873\pi$
$104$	−1.73205			−0.169842
$105$		0			0
$106$	8.27492			0.803731
$107$		−	12.0000i		−	1.16008i	−0.814587	−	0.580042i	$-0.803036\pi$
$107$		−	12.0000i		−	1.16008i	0.814587	−	0.580042i	$-0.196964\pi$
$108$		0			0
$109$	−7.82475			−0.749475			−0.374738	−	0.927131i	$-0.622267\pi$
$109$	−7.82475			−0.749475			−0.374738	+	0.927131i	$0.622267\pi$
$110$	−4.30136	+	4.00000i	−0.410118	+	0.381385i
$111$		0			0
$112$			1.31342i			0.124107i
$113$			0.725083i			0.0682101i	0.999418	+	0.0341050i	$0.0108581\pi$
$113$			0.725083i			0.0682101i	−0.999418	+	0.0341050i	$0.989142\pi$
$114$		0			0
$115$	−10.2749	+	9.55505i	−0.958141	+	0.891013i
$116$	9.13642			0.848296
$117$		0			0
$118$		−	8.24163i		−	0.758703i
$119$	−1.67451			−0.153502
$120$		0			0
$121$	−4.09967			−0.372697
$122$		−	7.27492i		−	0.658640i
$123$		0			0
$124$	−8.54983			−0.767798
$125$	−8.71780	+	7.00000i	−0.779744	+	0.626099i
$126$		0			0
$127$			22.0980i			1.96088i	0.196810	+	0.980442i	$0.436942\pi$
$127$			22.0980i			1.96088i	−0.196810	+	0.980442i	$0.563058\pi$
$128$			1.00000i			0.0883883i
$129$		0			0
$130$	2.63746	+	2.83616i	0.231320	+	0.248748i
$131$	−17.7967			−1.55490			−0.777452	−	0.628943i	$-0.783488\pi$
$131$	−17.7967			−1.55490			−0.777452	+	0.628943i	$0.783488\pi$
$132$		0			0
$133$			2.98793i			0.259086i
$134$	−12.1819			−1.05236
$135$		0			0
$136$	−1.27492			−0.109323
$137$			3.82475i			0.326771i	0.986562	+	0.163385i	$0.0522414\pi$
$137$			3.82475i			0.326771i	−0.986562	+	0.163385i	$0.947759\pi$
$138$		0			0
$139$	10.2749			0.871507			0.435754	−	0.900066i	$-0.356482\pi$
$139$	10.2749			0.871507			0.435754	+	0.900066i	$0.356482\pi$
$140$	2.15068	−	2.00000i	0.181765	−	0.169031i
$141$		0			0
$142$		−	6.92820i		−	0.581402i
$143$		−	4.54983i		−	0.380476i
$144$		0			0
$145$	−13.9124	−	14.9605i	−1.15536	−	1.24240i
$146$	4.83507			0.400153
$147$		0			0
$148$		−	9.97368i		−	0.819831i
$149$	−1.37097			−0.112314			−0.0561570	−	0.998422i	$-0.517885\pi$
$149$	−1.37097			−0.112314			−0.0561570	+	0.998422i	$0.517885\pi$
$150$		0			0
$151$	−17.0997			−1.39155			−0.695776	−	0.718259i	$-0.744939\pi$
$151$	−17.0997			−1.39155			−0.695776	+	0.718259i	$0.744939\pi$
$152$			2.27492i			0.184520i
$153$		0			0
$154$	−3.45017			−0.278022
$155$	13.0192	+	14.0000i	1.04572	+	1.12451i
$156$		0			0
$157$		−	6.03341i		−	0.481518i	−0.970585	−	0.240759i	$-0.922604\pi$
$157$		−	6.03341i		−	0.481518i	0.970585	−	0.240759i	$-0.0773964\pi$
$158$		0			0
$159$		0			0
$160$	1.63746	−	1.52274i	0.129452	−	0.120383i
$161$	−8.24163			−0.649531
$162$		0			0
$163$		−	24.3638i		−	1.90832i	−0.299297	−	0.954160i	$-0.596752\pi$
$163$		−	24.3638i		−	1.90832i	0.299297	−	0.954160i	$-0.403248\pi$
$164$	5.61478			0.438441
$165$		0			0
$166$	4.00000			0.310460
$167$		−	21.0997i		−	1.63274i	−0.577528	−	0.816371i	$-0.695983\pi$
$167$		−	21.0997i		−	1.63274i	0.577528	−	0.816371i	$-0.304017\pi$
$168$		0			0
$169$	10.0000			0.769231
$170$	1.94136	+	2.08762i	0.148896	+	0.160113i
$171$		0			0
$172$		−	2.62685i		−	0.200295i
$173$			20.0997i			1.52815i	0.645128	+	0.764075i	$0.276804\pi$
$173$			20.0997i			1.52815i	−0.645128	+	0.764075i	$0.723196\pi$
$174$		0			0
$175$	−6.54983	−	0.476171i	−0.495121	−	0.0359951i
$176$	−2.62685			−0.198006
$177$		0			0
$178$			3.04547i			0.228268i
$179$	1.31342			0.0981699			0.0490850	−	0.998795i	$-0.484369\pi$
$179$	1.31342			0.0981699			0.0490850	+	0.998795i	$0.484369\pi$
$180$		0			0
$181$	6.54983			0.486845			0.243423	−	0.969920i	$-0.421730\pi$
$181$	6.54983			0.486845			0.243423	+	0.969920i	$0.421730\pi$
$182$			2.27492i			0.168628i
$183$		0			0
$184$	−6.27492			−0.462593
$185$	−16.3315	+	15.1873i	−1.20071	+	1.11659i
$186$		0			0
$187$		−	3.34901i		−	0.244904i
$188$			10.2749i			0.749375i
$189$		0			0
$190$	3.72508	−	3.46410i	0.270246	−	0.251312i
$191$	14.8087			1.07152			0.535762	−	0.844369i	$-0.320025\pi$
$191$	14.8087			1.07152			0.535762	+	0.844369i	$0.320025\pi$
$192$		0			0
$193$			8.29917i			0.597387i	0.954349	+	0.298694i	$0.0965509\pi$
$193$			8.29917i			0.597387i	−0.954349	+	0.298694i	$0.903449\pi$
$194$	−19.1101			−1.37203
$195$		0			0
$196$	−5.27492			−0.376780
$197$		−	11.0000i		−	0.783718i	−0.920025	−	0.391859i	$-0.871832\pi$
$197$		−	11.0000i		−	0.783718i	0.920025	−	0.391859i	$-0.128168\pi$
$198$		0			0
$199$	7.45017			0.528128			0.264064	−	0.964505i	$-0.414937\pi$
$199$	7.45017			0.528128			0.264064	+	0.964505i	$0.414937\pi$
$200$	−4.98684	−	0.362541i	−0.352623	−	0.0256355i
$201$		0			0
$202$		−	9.55505i		−	0.672291i
$203$		−	12.0000i		−	0.842235i
$204$		0			0
$205$	−8.54983	−	9.19397i	−0.597146	−	0.642135i
$206$	−6.56712			−0.457553
$207$		0			0
$208$			1.73205i			0.120096i
$209$	−5.97586			−0.413359
$210$		0			0
$211$	−9.72508			−0.669502			−0.334751	−	0.942307i	$-0.608652\pi$
$211$	−9.72508			−0.669502			−0.334751	+	0.942307i	$0.608652\pi$
$212$		−	8.27492i		−	0.568324i
$213$		0			0
$214$	−12.0000			−0.820303
$215$	−4.30136	+	4.00000i	−0.293350	+	0.272798i
$216$		0			0
$217$			11.2296i			0.762312i
$218$			7.82475i			0.529959i
$219$		0			0
$220$	4.00000	+	4.30136i	0.269680	+	0.289997i
$221$	−2.20822			−0.148541
$222$		0			0
$223$			16.4833i			1.10380i	0.833910	+	0.551900i	$0.186097\pi$
$223$			16.4833i			1.10380i	−0.833910	+	0.551900i	$0.813903\pi$
$224$	1.31342			0.0877568
$225$		0			0
$226$	0.725083			0.0482318
$227$			0.549834i			0.0364938i	0.999834	+	0.0182469i	$0.00580849\pi$
$227$			0.549834i			0.0364938i	−0.999834	+	0.0182469i	$0.994192\pi$
$228$		0			0
$229$	−22.3746			−1.47855			−0.739277	−	0.673401i	$-0.764833\pi$
$229$	−22.3746			−1.47855			−0.739277	+	0.673401i	$0.764833\pi$
$230$	9.55505	+	10.2749i	0.630041	+	0.677508i
$231$		0			0
$232$		−	9.13642i		−	0.599836i
$233$		−	0.175248i		−	0.0114809i	−0.999984	−	0.00574045i	$-0.998173\pi$
$233$		−	0.175248i		−	0.0114809i	0.999984	−	0.00574045i	$-0.00182725\pi$
$234$		0			0
$235$	16.8248	−	15.6460i	1.09753	−	1.02063i
$236$	−8.24163			−0.536484
$237$		0			0
$238$			1.67451i			0.108542i
$239$	9.55505			0.618065			0.309032	−	0.951051i	$-0.399995\pi$
$239$	9.55505			0.618065			0.309032	+	0.951051i	$0.399995\pi$
$240$		0			0
$241$	13.8248			0.890531			0.445265	−	0.895399i	$-0.353109\pi$
$241$	13.8248			0.890531			0.445265	+	0.895399i	$0.353109\pi$
$242$			4.09967i			0.263537i
$243$		0			0
$244$	−7.27492			−0.465729
$245$	8.03231	+	8.63746i	0.513166	+	0.551827i
$246$		0			0
$247$			3.94027i			0.250714i
$248$			8.54983i			0.542915i
$249$		0			0
$250$	7.00000	+	8.71780i	0.442719	+	0.551362i
$251$	2.98793			0.188597			0.0942983	−	0.995544i	$-0.469939\pi$
$251$	2.98793			0.188597			0.0942983	+	0.995544i	$0.469939\pi$
$252$		0			0
$253$		−	16.4833i		−	1.03629i
$254$	22.0980			1.38655
$255$		0			0
$256$	1.00000			0.0625000
$257$		−	0.725083i		−	0.0452294i	−0.999744	−	0.0226147i	$-0.992801\pi$
$257$		−	0.725083i		−	0.0452294i	0.999744	−	0.0226147i	$-0.00719910\pi$
$258$		0			0
$259$	−13.0997			−0.813974
$260$	2.83616	−	2.63746i	0.175891	−	0.163568i
$261$		0			0
$262$			17.7967i			1.09948i
$263$		−	19.3746i		−	1.19469i	−0.801985	−	0.597344i	$-0.796223\pi$
$263$		−	19.3746i		−	1.19469i	0.801985	−	0.597344i	$-0.203777\pi$
$264$		0			0
$265$	−13.5498	+	12.6005i	−0.832360	+	0.774044i
$266$	2.98793			0.183202
$267$		0			0
$268$			12.1819i			0.744128i
$269$	11.6482			0.710202			0.355101	−	0.934828i	$-0.384446\pi$
$269$	11.6482			0.710202			0.355101	+	0.934828i	$0.384446\pi$
$270$		0			0
$271$	−12.5498			−0.762348			−0.381174	−	0.924503i	$-0.624480\pi$
$271$	−12.5498			−0.762348			−0.381174	+	0.924503i	$0.624480\pi$
$272$			1.27492i			0.0773032i
$273$		0			0
$274$	3.82475			0.231062
$275$	0.952341	−	13.0997i	0.0574283	−	0.789940i
$276$		0			0
$277$		−	20.4235i		−	1.22713i	−0.789644	−	0.613565i	$-0.789735\pi$
$277$		−	20.4235i		−	1.22713i	0.789644	−	0.613565i	$-0.210265\pi$
$278$		−	10.2749i		−	0.616249i
$279$		0			0
$280$	−2.00000	−	2.15068i	−0.119523	−	0.128528i
$281$	−2.68439			−0.160137			−0.0800687	−	0.996789i	$-0.525514\pi$
$281$	−2.68439			−0.160137			−0.0800687	+	0.996789i	$0.525514\pi$
$282$		0			0
$283$		−	9.55505i		−	0.567989i	−0.958826	−	0.283994i	$-0.908340\pi$
$283$		−	9.55505i		−	0.567989i	0.958826	−	0.283994i	$-0.0916597\pi$
$284$	−6.92820			−0.411113
$285$		0			0
$286$	−4.54983			−0.269037
$287$		−	7.37459i		−	0.435308i
$288$		0			0
$289$	15.3746			0.904387
$290$	−14.9605	+	13.9124i	−0.878512	+	0.816963i
$291$		0			0
$292$		−	4.83507i		−	0.282951i
$293$			15.0000i			0.876309i	0.898900	+	0.438155i	$0.144368\pi$
$293$			15.0000i			0.876309i	−0.898900	+	0.438155i	$0.855632\pi$
$294$		0			0
$295$	12.5498	+	13.4953i	0.730680	+	0.785728i
$296$	−9.97368			−0.579708
$297$		0			0
$298$			1.37097i			0.0794180i
$299$	−10.8685			−0.628540
$300$		0			0
$301$	−3.45017			−0.198864
$302$			17.0997i			0.983975i
$303$		0			0
$304$	2.27492			0.130475
$305$	11.0778	+	11.9124i	0.634312	+	0.682101i
$306$		0			0
$307$		−	0.952341i		−	0.0543530i	−0.999631	−	0.0271765i	$-0.991348\pi$
$307$		−	0.952341i		−	0.0543530i	0.999631	−	0.0271765i	$-0.00865161\pi$
$308$			3.45017i			0.196591i
$309$		0			0
$310$	14.0000	−	13.0192i	0.795147	−	0.739438i
$311$	21.7370			1.23259			0.616295	−	0.787516i	$-0.288633\pi$
$311$	21.7370			1.23259			0.616295	+	0.787516i	$0.288633\pi$
$312$		0			0
$313$		−	17.8542i		−	1.00918i	−0.863359	−	0.504590i	$-0.831644\pi$
$313$		−	17.8542i		−	1.00918i	0.863359	−	0.504590i	$-0.168356\pi$
$314$	−6.03341			−0.340485
$315$		0			0
$316$		0			0
$317$		−	22.0997i		−	1.24124i	−0.784111	−	0.620621i	$-0.786881\pi$
$317$		−	22.0997i		−	1.24124i	0.784111	−	0.620621i	$-0.213119\pi$
$318$		0			0
$319$	24.0000			1.34374
$320$	−1.52274	−	1.63746i	−0.0851236	−	0.0915367i
$321$		0			0
$322$			8.24163i			0.459288i
$323$			2.90033i			0.161379i
$324$		0			0
$325$	−8.63746	−	0.627940i	−0.479120	−	0.0348319i
$326$	−24.3638			−1.34939
$327$		0			0
$328$		−	5.61478i		−	0.310024i
$329$	13.4953			0.744021
$330$		0			0
$331$	−20.0000			−1.09930			−0.549650	−	0.835395i	$-0.685239\pi$
$331$	−20.0000			−1.09930			−0.549650	+	0.835395i	$0.685239\pi$
$332$		−	4.00000i		−	0.219529i
$333$		0			0
$334$	−21.0997			−1.15452
$335$	19.9474	−	18.5498i	1.08984	−	1.01349i
$336$		0			0
$337$			5.25370i			0.286187i	0.989709	+	0.143094i	$0.0457050\pi$
$337$			5.25370i			0.286187i	−0.989709	+	0.143094i	$0.954295\pi$
$338$		−	10.0000i		−	0.543928i
$339$		0			0
$340$	2.08762	−	1.94136i	0.113217	−	0.105285i
$341$	−22.4591			−1.21623
$342$		0			0
$343$			16.1222i			0.870515i
$344$	−2.62685			−0.141630
$345$		0			0
$346$	20.0997			1.08056
$347$		−	7.45017i		−	0.399946i	−0.979801	−	0.199973i	$-0.935915\pi$
$347$		−	7.45017i		−	0.399946i	0.979801	−	0.199973i	$-0.0640854\pi$
$348$		0			0
$349$	32.1993			1.72359			0.861796	−	0.507256i	$-0.169340\pi$
$349$	32.1993			1.72359			0.861796	+	0.507256i	$0.169340\pi$
$350$	−0.476171	+	6.54983i	−0.0254524	+	0.350103i
$351$		0			0
$352$			2.62685i			0.140011i
$353$		−	14.5498i		−	0.774410i	−0.921994	−	0.387205i	$-0.873441\pi$
$353$		−	14.5498i		−	0.774410i	0.921994	−	0.387205i	$-0.126559\pi$
$354$		0			0
$355$	10.5498	+	11.3446i	0.559927	+	0.602111i
$356$	3.04547			0.161410
$357$		0			0
$358$		−	1.31342i		−	0.0694166i
$359$	21.7370			1.14723			0.573616	−	0.819124i	$-0.305540\pi$
$359$	21.7370			1.14723			0.573616	+	0.819124i	$0.305540\pi$
$360$		0			0
$361$	−13.8248			−0.727619
$362$		−	6.54983i		−	0.344252i
$363$		0			0
$364$	2.27492			0.119238
$365$	−7.91723	+	7.36254i	−0.414407	+	0.385373i
$366$		0			0
$367$		−	2.62685i		−	0.137120i	−0.997647	−	0.0685602i	$-0.978159\pi$
$367$		−	2.62685i		−	0.137120i	0.997647	−	0.0685602i	$-0.0218405\pi$
$368$			6.27492i			0.327103i
$369$		0			0
$370$	15.1873	+	16.3315i	0.789550	+	0.849033i
$371$	−10.8685			−0.564263
$372$		0			0
$373$			22.6893i			1.17481i	0.809294	+	0.587404i	$0.199850\pi$
$373$			22.6893i			1.17481i	−0.809294	+	0.587404i	$0.800150\pi$
$374$	−3.34901			−0.173173
$375$		0			0
$376$	10.2749			0.529888
$377$		−	15.8248i		−	0.815016i
$378$		0			0
$379$	27.3746			1.40614			0.703069	−	0.711122i	$-0.251812\pi$
$379$	27.3746			1.40614			0.703069	+	0.711122i	$0.251812\pi$
$380$	−3.46410	−	3.72508i	−0.177705	−	0.191093i
$381$		0			0
$382$		−	14.8087i		−	0.757681i
$383$		−	3.37459i		−	0.172433i	−0.996276	−	0.0862166i	$-0.972522\pi$
$383$		−	3.37459i		−	0.172433i	0.996276	−	0.0862166i	$-0.0274777\pi$
$384$		0			0
$385$	5.64950	−	5.25370i	0.287925	−	0.267753i
$386$	8.29917			0.422417
$387$		0			0
$388$			19.1101i			0.970168i
$389$	−33.9189			−1.71975			−0.859877	−	0.510501i	$-0.829460\pi$
$389$	−33.9189			−1.71975			−0.859877	+	0.510501i	$0.829460\pi$
$390$		0			0
$391$	−8.00000			−0.404577
$392$			5.27492i			0.266424i
$393$		0			0
$394$	−11.0000			−0.554172
$395$		0			0
$396$		0			0
$397$			14.3901i			0.722219i	0.932523	+	0.361110i	$0.117602\pi$
$397$			14.3901i			0.722219i	−0.932523	+	0.361110i	$0.882398\pi$
$398$		−	7.45017i		−	0.373443i
$399$		0			0
$400$	−0.362541	+	4.98684i	−0.0181271	+	0.249342i
$401$	21.6794			1.08262			0.541309	−	0.840824i	$-0.317929\pi$
$401$	21.6794			1.08262			0.541309	+	0.840824i	$0.317929\pi$
$402$		0			0
$403$			14.8087i			0.737676i
$404$	−9.55505			−0.475382
$405$		0			0
$406$	−12.0000			−0.595550
$407$		−	26.1993i		−	1.29865i
$408$		0			0
$409$	16.0997			0.796077			0.398039	−	0.917369i	$-0.369691\pi$
$409$	16.0997			0.796077			0.398039	+	0.917369i	$0.369691\pi$
$410$	−9.19397	+	8.54983i	−0.454058	+	0.422246i
$411$		0			0
$412$			6.56712i			0.323539i
$413$			10.8248i			0.532651i
$414$		0			0
$415$	−6.54983	+	6.09095i	−0.321519	+	0.298993i
$416$	1.73205			0.0849208
$417$		0			0
$418$			5.97586i			0.292289i
$419$	16.4833			0.805260			0.402630	−	0.915363i	$-0.368096\pi$
$419$	16.4833			0.805260			0.402630	+	0.915363i	$0.368096\pi$
$420$		0			0
$421$	9.27492			0.452032			0.226016	−	0.974124i	$-0.427430\pi$
$421$	9.27492			0.452032			0.226016	+	0.974124i	$0.427430\pi$
$422$			9.72508i			0.473410i
$423$		0			0
$424$	−8.27492			−0.401866
$425$	−6.35781	−	0.462210i	−0.308399	−	0.0224205i
$426$		0			0
$427$			9.55505i			0.462401i
$428$			12.0000i			0.580042i
$429$		0			0
$430$	4.00000	+	4.30136i	0.192897	+	0.207430i
$431$	33.9189			1.63381			0.816907	−	0.576770i	$-0.195687\pi$
$431$	33.9189			1.63381			0.816907	+	0.576770i	$0.195687\pi$
$432$		0			0
$433$			10.8109i			0.519540i	0.965670	+	0.259770i	$0.0836468\pi$
$433$			10.8109i			0.519540i	−0.965670	+	0.259770i	$0.916353\pi$
$434$	11.2296			0.539036
$435$		0			0
$436$	7.82475			0.374738
$437$			14.2749i			0.682862i
$438$		0			0
$439$	36.0000			1.71819			0.859093	−	0.511819i	$-0.171028\pi$
$439$	36.0000			1.71819			0.859093	+	0.511819i	$0.171028\pi$
$440$	4.30136	−	4.00000i	0.205059	−	0.190693i
$441$		0			0
$442$			2.20822i			0.105034i
$443$			7.45017i			0.353968i	0.984214	+	0.176984i	$0.0566341\pi$
$443$			7.45017i			0.353968i	−0.984214	+	0.176984i	$0.943366\pi$
$444$		0			0
$445$	−4.63746	−	4.98684i	−0.219837	−	0.236399i
$446$	16.4833			0.780505
$447$		0			0
$448$		−	1.31342i		−	0.0620535i
$449$	10.8685			0.512915			0.256458	−	0.966555i	$-0.417445\pi$
$449$	10.8685			0.512915			0.256458	+	0.966555i	$0.417445\pi$
$450$		0			0
$451$	14.7492			0.694511
$452$		−	0.725083i		−	0.0341050i
$453$		0			0
$454$	0.549834			0.0258050
$455$	−3.46410	−	3.72508i	−0.162400	−	0.174635i
$456$		0			0
$457$			15.9495i			0.746088i	0.927814	+	0.373044i	$0.121686\pi$
$457$			15.9495i			0.746088i	−0.927814	+	0.373044i	$0.878314\pi$
$458$			22.3746i			1.04550i
$459$		0			0
$460$	10.2749	−	9.55505i	0.479070	−	0.445507i
$461$	6.92820			0.322679			0.161339	−	0.986899i	$-0.448419\pi$
$461$	6.92820			0.322679			0.161339	+	0.986899i	$0.448419\pi$
$462$		0			0
$463$			16.8443i			0.782823i	0.920216	+	0.391411i	$0.128013\pi$
$463$			16.8443i			0.782823i	−0.920216	+	0.391411i	$0.871987\pi$
$464$	−9.13642			−0.424148
$465$		0			0
$466$	−0.175248			−0.00811822
$467$			13.6495i			0.631624i	0.948822	+	0.315812i	$0.102277\pi$
$467$			13.6495i			0.631624i	−0.948822	+	0.315812i	$0.897723\pi$
$468$		0			0
$469$	16.0000			0.738811
$470$	−15.6460	−	16.8248i	−0.721696	−	0.776068i
$471$		0			0
$472$			8.24163i			0.379352i
$473$		−	6.90033i		−	0.317278i
$474$		0			0
$475$	−0.824752	+	11.3446i	−0.0378422	+	0.520528i
$476$	1.67451			0.0767509
$477$		0			0
$478$		−	9.55505i		−	0.437038i
$479$	1.67451			0.0765102			0.0382551	−	0.999268i	$-0.487820\pi$
$479$	1.67451			0.0765102			0.0382551	+	0.999268i	$0.487820\pi$
$480$		0			0
$481$	−17.2749			−0.787668
$482$		−	13.8248i		−	0.629700i
$483$		0			0
$484$	4.09967			0.186349
$485$	31.2920	−	29.0997i	1.42090	−	1.32135i
$486$		0			0
$487$		−	9.91613i		−	0.449343i	−0.974435	−	0.224671i	$-0.927869\pi$
$487$		−	9.91613i		−	0.449343i	0.974435	−	0.224671i	$-0.0721309\pi$
$488$			7.27492i			0.329320i
$489$		0			0
$490$	8.63746	−	8.03231i	0.390201	−	0.362863i
$491$	18.7490			0.846131			0.423066	−	0.906099i	$-0.360954\pi$
$491$	18.7490			0.846131			0.423066	+	0.906099i	$0.360954\pi$
$492$		0			0
$493$		−	11.6482i		−	0.524608i
$494$	3.94027			0.177281
$495$		0			0
$496$	8.54983			0.383899
$497$			9.09967i			0.408176i
$498$		0			0
$499$	−14.2749			−0.639033			−0.319517	−	0.947581i	$-0.603520\pi$
$499$	−14.2749			−0.639033			−0.319517	+	0.947581i	$0.603520\pi$
$500$	8.71780	−	7.00000i	0.389872	−	0.313050i
$501$		0			0
$502$		−	2.98793i		−	0.133358i
$503$		−	33.0997i		−	1.47584i	−0.674887	−	0.737921i	$-0.735808\pi$
$503$		−	33.0997i		−	1.47584i	0.674887	−	0.737921i	$-0.264192\pi$
$504$		0			0
$505$	14.5498	+	15.6460i	0.647459	+	0.696238i
$506$	−16.4833			−0.732770
$507$		0			0
$508$		−	22.0980i		−	0.980442i
$509$	−6.92820			−0.307087			−0.153544	−	0.988142i	$-0.549069\pi$
$509$	−6.92820			−0.307087			−0.153544	+	0.988142i	$0.549069\pi$
$510$		0			0
$511$	−6.35050			−0.280929
$512$		−	1.00000i		−	0.0441942i
$513$		0			0
$514$	−0.725083			−0.0319820
$515$	10.7534	−	10.0000i	0.473851	−	0.440653i
$516$		0			0
$517$			26.9906i			1.18705i
$518$			13.0997i			0.575566i
$519$		0			0
$520$	−2.63746	−	2.83616i	−0.115660	−	0.124374i
$521$	−43.8350			−1.92045			−0.960223	−	0.279235i	$-0.909919\pi$
$521$	−43.8350			−1.92045			−0.960223	+	0.279235i	$0.909919\pi$
$522$		0			0
$523$		−	10.5074i		−	0.459456i	−0.973255	−	0.229728i	$-0.926216\pi$
$523$		−	10.5074i		−	0.459456i	0.973255	−	0.229728i	$-0.0737837\pi$
$524$	17.7967			0.777452
$525$		0			0
$526$	−19.3746			−0.844772
$527$			10.9003i			0.474826i
$528$		0			0
$529$	−16.3746			−0.711939
$530$	12.6005	+	13.5498i	0.547332	+	0.588567i
$531$		0			0
$532$		−	2.98793i		−	0.129543i
$533$		−	9.72508i		−	0.421240i
$534$		0			0
$535$	19.6495	−	18.2728i	0.849522	−	0.790004i
$536$	12.1819			0.526178
$537$		0			0
$538$		−	11.6482i		−	0.502189i
$539$	−13.8564			−0.596838
$540$		0			0
$541$	10.7251			0.461107			0.230554	−	0.973060i	$-0.425946\pi$
$541$	10.7251			0.461107			0.230554	+	0.973060i	$0.425946\pi$
$542$			12.5498i			0.539062i
$543$		0			0
$544$	1.27492			0.0546616
$545$	−11.9150	−	12.8127i	−0.510384	−	0.548836i
$546$		0			0
$547$			33.9189i			1.45027i	0.688609	+	0.725133i	$0.258222\pi$
$547$			33.9189i			1.45027i	−0.688609	+	0.725133i	$0.741778\pi$
$548$		−	3.82475i		−	0.163385i
$549$		0			0
$550$	−13.0997	−	0.952341i	−0.558572	−	0.0406080i
$551$	−20.7846			−0.885454
$552$		0			0
$553$		0			0
$554$	−20.4235			−0.867713
$555$		0			0
$556$	−10.2749			−0.435754
$557$			45.1993i			1.91516i	0.288172	+	0.957579i	$0.406953\pi$
$557$			45.1993i			1.91516i	−0.288172	+	0.957579i	$0.593047\pi$
$558$		0			0
$559$	−4.54983			−0.192437
$560$	−2.15068	+	2.00000i	−0.0908827	+	0.0845154i
$561$		0			0
$562$			2.68439i			0.113234i
$563$			29.0997i			1.22640i	0.789926	+	0.613202i	$0.210119\pi$
$563$			29.0997i			1.22640i	−0.789926	+	0.613202i	$0.789881\pi$
$564$		0			0
$565$	−1.18729	+	1.10411i	−0.0499498	+	0.0464503i
$566$	−9.55505			−0.401629
$567$		0			0
$568$			6.92820i			0.290701i
$569$	−12.9616			−0.543379			−0.271689	−	0.962385i	$-0.587582\pi$
$569$	−12.9616			−0.543379			−0.271689	+	0.962385i	$0.587582\pi$
$570$		0			0
$571$	−17.0997			−0.715599			−0.357799	−	0.933798i	$-0.616473\pi$
$571$	−17.0997			−0.715599			−0.357799	+	0.933798i	$0.616473\pi$
$572$			4.54983i			0.190238i
$573$		0			0
$574$	−7.37459			−0.307809
$575$	−31.2920	−	2.27492i	−1.30497	−	0.0948706i
$576$		0			0
$577$		−	29.0838i		−	1.21077i	−0.795931	−	0.605387i	$-0.793018\pi$
$577$		−	29.0838i		−	1.21077i	0.795931	−	0.605387i	$-0.206982\pi$
$578$		−	15.3746i		−	0.639498i
$579$		0			0
$580$	13.9124	+	14.9605i	0.577680	+	0.621202i
$581$	−5.25370			−0.217960
$582$		0			0
$583$		−	21.7370i		−	0.900253i
$584$	−4.83507			−0.200077
$585$		0			0
$586$	15.0000			0.619644
$587$			33.0997i			1.36617i	0.730339	+	0.683085i	$0.239362\pi$
$587$			33.0997i			1.36617i	−0.730339	+	0.683085i	$0.760638\pi$
$588$		0			0
$589$	19.4502			0.801430
$590$	13.4953	−	12.5498i	0.555594	−	0.516669i
$591$		0			0
$592$			9.97368i			0.409916i
$593$			20.1752i			0.828498i	0.910164	+	0.414249i	$0.135956\pi$
$593$			20.1752i			0.828498i	−0.910164	+	0.414249i	$0.864044\pi$
$594$		0			0
$595$	−2.54983	−	2.74194i	−0.104533	−	0.112408i
$596$	1.37097			0.0561570
$597$		0			0
$598$			10.8685i			0.444445i
$599$	6.92820			0.283079			0.141539	−	0.989933i	$-0.454795\pi$
$599$	6.92820			0.283079			0.141539	+	0.989933i	$0.454795\pi$
$600$		0			0
$601$	0.450166			0.0183626			0.00918132	−	0.999958i	$-0.497077\pi$
$601$	0.450166			0.0183626			0.00918132	+	0.999958i	$0.497077\pi$
$602$			3.45017i			0.140618i
$603$		0			0
$604$	17.0997			0.695776
$605$	−6.24272	−	6.71304i	−0.253803	−	0.272924i
$606$		0			0
$607$		−	33.9189i		−	1.37672i	−0.725367	−	0.688362i	$-0.758330\pi$
$607$		−	33.9189i		−	1.37672i	0.725367	−	0.688362i	$-0.241670\pi$
$608$		−	2.27492i		−	0.0922601i
$609$		0			0
$610$	11.9124	−	11.0778i	0.482318	−	0.448527i
$611$	17.7967			0.719977
$612$		0			0
$613$			12.5430i			0.506606i	0.967387	+	0.253303i	$0.0815170\pi$
$613$			12.5430i			0.506606i	−0.967387	+	0.253303i	$0.918483\pi$
$614$	−0.952341			−0.0384334
$615$		0			0
$616$	3.45017			0.139011
$617$		−	7.82475i		−	0.315013i	−0.987518	−	0.157506i	$-0.949655\pi$
$617$		−	7.82475i		−	0.315013i	0.987518	−	0.157506i	$-0.0503455\pi$
$618$		0			0
$619$	−22.2749			−0.895305			−0.447652	−	0.894208i	$-0.647740\pi$
$619$	−22.2749			−0.895305			−0.447652	+	0.894208i	$0.647740\pi$
$620$	−13.0192	−	14.0000i	−0.522862	−	0.562254i
$621$		0			0
$622$		−	21.7370i		−	0.871572i
$623$		−	4.00000i		−	0.160257i
$624$		0			0
$625$	−24.7371	−	3.61587i	−0.989485	−	0.144635i
$626$	−17.8542			−0.713598
$627$		0			0
$628$			6.03341i			0.240759i
$629$	−12.7156			−0.507005
$630$		0			0
$631$	−40.0000			−1.59237			−0.796187	−	0.605050i	$-0.793153\pi$
$631$	−40.0000			−1.59237			−0.796187	+	0.605050i	$0.793153\pi$
$632$		0			0
$633$		0			0
$634$	−22.0997			−0.877690
$635$	−36.1846	+	33.6495i	−1.43594	+	1.33534i
$636$		0			0
$637$			9.13642i			0.361998i
$638$		−	24.0000i		−	0.950169i
$639$		0			0
$640$	−1.63746	+	1.52274i	−0.0647262	+	0.0601915i
$641$	30.7583			1.21488			0.607440	−	0.794366i	$-0.292197\pi$
$641$	30.7583			1.21488			0.607440	+	0.794366i	$0.292197\pi$
$642$		0			0
$643$			0.952341i			0.0375567i	0.999824	+	0.0187783i	$0.00597768\pi$
$643$			0.952341i			0.0375567i	−0.999824	+	0.0187783i	$0.994022\pi$
$644$	8.24163			0.324766
$645$		0			0
$646$	2.90033			0.114112
$647$		−	21.0997i		−	0.829514i	−0.909932	−	0.414757i	$-0.863867\pi$
$647$		−	21.0997i		−	0.829514i	0.909932	−	0.414757i	$-0.136133\pi$
$648$		0			0
$649$	−21.6495			−0.849817
$650$	−0.627940	+	8.63746i	−0.0246298	+	0.338789i
$651$		0			0
$652$			24.3638i			0.954160i
$653$		−	14.0000i		−	0.547862i	−0.961749	−	0.273931i	$-0.911676\pi$
$653$		−	14.0000i		−	0.547862i	0.961749	−	0.273931i	$-0.0883240\pi$
$654$		0			0
$655$	−27.0997	−	29.1413i	−1.05887	−	1.13865i
$656$	−5.61478			−0.219220
$657$		0			0
$658$		−	13.4953i		−	0.526102i
$659$	−4.66244			−0.181623			−0.0908114	−	0.995868i	$-0.528946\pi$
$659$	−4.66244			−0.181623			−0.0908114	+	0.995868i	$0.528946\pi$
$660$		0			0
$661$	31.2749			1.21645			0.608227	−	0.793763i	$-0.291881\pi$
$661$	31.2749			1.21645			0.608227	+	0.793763i	$0.291881\pi$
$662$			20.0000i			0.777322i
$663$		0			0
$664$	−4.00000			−0.155230
$665$	−4.89261	+	4.54983i	−0.189727	+	0.176435i
$666$		0			0
$667$		−	57.3303i		−	2.21984i
$668$			21.0997i			0.816371i
$669$		0			0
$670$	−18.5498	−	19.9474i	−0.716643	−	0.770634i
$671$	−19.1101			−0.737737
$672$		0			0
$673$			5.67232i			0.218652i	0.994006	+	0.109326i	$0.0348692\pi$
$673$			5.67232i			0.218652i	−0.994006	+	0.109326i	$0.965131\pi$
$674$	5.25370			0.202365
$675$		0			0
$676$	−10.0000			−0.384615
$677$			23.7251i			0.911829i	0.890024	+	0.455915i	$0.150688\pi$
$677$			23.7251i			0.911829i	−0.890024	+	0.455915i	$0.849312\pi$
$678$		0			0
$679$	25.0997			0.963237
$680$	−1.94136	−	2.08762i	−0.0744479	−	0.0800567i
$681$		0			0
$682$			22.4591i			0.860004i
$683$			26.7492i			1.02353i	0.859126	+	0.511764i	$0.171008\pi$
$683$			26.7492i			1.02353i	−0.859126	+	0.511764i	$0.828992\pi$
$684$		0			0
$685$	−6.26287	+	5.82409i	−0.239292	+	0.222527i
$686$	16.1222			0.615547
$687$		0			0
$688$			2.62685i			0.100148i
$689$	−14.3326			−0.546028
$690$		0			0
$691$	−23.9244			−0.910128			−0.455064	−	0.890459i	$-0.650384\pi$
$691$	−23.9244			−0.910128			−0.455064	+	0.890459i	$0.650384\pi$
$692$		−	20.0997i		−	0.764075i
$693$		0			0
$694$	−7.45017			−0.282804
$695$	15.6460	+	16.8248i	0.593487	+	0.638199i
$696$		0			0
$697$		−	7.15838i		−	0.271143i
$698$		−	32.1993i		−	1.21876i
$699$		0			0
$700$	6.54983	+	0.476171i	0.247560	+	0.0179976i
$701$	−18.6915			−0.705967			−0.352984	−	0.935629i	$-0.614833\pi$
$701$	−18.6915			−0.705967			−0.352984	+	0.935629i	$0.614833\pi$
$702$		0			0
$703$			22.6893i			0.855743i
$704$	2.62685			0.0990031
$705$		0			0
$706$	−14.5498			−0.547590
$707$			12.5498i			0.471985i
$708$		0			0
$709$	3.82475			0.143642			0.0718208	−	0.997418i	$-0.477119\pi$
$709$	3.82475			0.143642			0.0718208	+	0.997418i	$0.477119\pi$
$710$	11.3446	−	10.5498i	0.425757	−	0.395928i
$711$		0			0
$712$		−	3.04547i		−	0.114134i
$713$			53.6495i			2.00919i
$714$		0			0
$715$	7.45017	−	6.92820i	0.278620	−	0.259100i
$716$	−1.31342			−0.0490850
$717$		0			0
$718$		−	21.7370i		−	0.811216i
$719$	8.60271			0.320827			0.160413	−	0.987050i	$-0.448717\pi$
$719$	8.60271			0.320827			0.160413	+	0.987050i	$0.448717\pi$
$720$		0			0
$721$	8.62541			0.321227
$722$			13.8248i			0.514504i
$723$		0			0
$724$	−6.54983			−0.243423
$725$	3.31233	−	45.5619i	0.123017	−	1.69213i
$726$		0			0
$727$		−	11.8208i		−	0.438410i	−0.975679	−	0.219205i	$-0.929654\pi$
$727$		−	11.8208i		−	0.438410i	0.975679	−	0.219205i	$-0.0703463\pi$
$728$		−	2.27492i		−	0.0843140i
$729$		0			0
$730$	7.36254	+	7.91723i	0.272500	+	0.293030i
$731$	−3.34901			−0.123868
$732$		0			0
$733$		−	47.0531i		−	1.73795i	−0.494860	−	0.868973i	$-0.664781\pi$
$733$		−	47.0531i		−	1.73795i	0.494860	−	0.868973i	$-0.335219\pi$
$734$	−2.62685			−0.0969587
$735$		0			0
$736$	6.27492			0.231297
$737$			32.0000i			1.17874i
$738$		0			0
$739$	10.9003			0.400975			0.200488	−	0.979696i	$-0.435747\pi$
$739$	10.9003			0.400975			0.200488	+	0.979696i	$0.435747\pi$
$740$	16.3315	−	15.1873i	0.600357	−	0.558296i
$741$		0			0
$742$			10.8685i			0.398994i
$743$		−	45.0997i		−	1.65455i	−0.561800	−	0.827273i	$-0.689891\pi$
$743$		−	45.0997i		−	1.65455i	0.561800	−	0.827273i	$-0.310109\pi$
$744$		0			0
$745$	−2.08762	−	2.24490i	−0.0764846	−	0.0822469i
$746$	22.6893			0.830714
$747$		0			0
$748$			3.34901i			0.122452i
$749$	15.7611			0.575898
$750$		0			0
$751$	−32.5498			−1.18776			−0.593880	−	0.804554i	$-0.702405\pi$
$751$	−32.5498			−1.18776			−0.593880	+	0.804554i	$0.702405\pi$
$752$		−	10.2749i		−	0.374688i
$753$		0			0
$754$	−15.8248			−0.576303
$755$	−26.0383	−	28.0000i	−0.947631	−	1.01902i
$756$		0			0
$757$		−	26.3994i		−	0.959502i	−0.877405	−	0.479751i	$-0.840727\pi$
$757$		−	26.3994i		−	0.959502i	0.877405	−	0.479751i	$-0.159273\pi$
$758$		−	27.3746i		−	0.994290i
$759$		0			0
$760$	−3.72508	+	3.46410i	−0.135123	+	0.125656i
$761$	−17.4931			−0.634126			−0.317063	−	0.948405i	$-0.602697\pi$
$761$	−17.4931			−0.634126			−0.317063	+	0.948405i	$0.602697\pi$
$762$		0			0
$763$		−	10.2772i		−	0.372060i
$764$	−14.8087			−0.535762
$765$		0			0
$766$	−3.37459			−0.121929
$767$			14.2749i			0.515437i
$768$		0			0
$769$	−35.2749			−1.27205			−0.636023	−	0.771670i	$-0.719422\pi$
$769$	−35.2749			−1.27205			−0.636023	+	0.771670i	$0.719422\pi$
$770$	−5.25370	−	5.64950i	−0.189330	−	0.203594i
$771$		0			0
$772$		−	8.29917i		−	0.298694i
$773$		−	45.8248i		−	1.64820i	−0.566443	−	0.824101i	$-0.691681\pi$
$773$		−	45.8248i		−	1.64820i	0.566443	−	0.824101i	$-0.308319\pi$
$774$		0			0
$775$	−3.09967	+	42.6366i	−0.111343	+	1.53155i
$776$	19.1101			0.686013
$777$		0			0
$778$			33.9189i			1.21605i
$779$	−12.7732			−0.457646
$780$		0			0
$781$	−18.1993			−0.651224
$782$			8.00000i			0.286079i
$783$		0			0
$784$	5.27492			0.188390
$785$	9.87945	−	9.18729i	0.352613	−	0.327909i
$786$		0			0
$787$			3.34901i			0.119379i	0.998217	+	0.0596897i	$0.0190111\pi$
$787$			3.34901i			0.119379i	−0.998217	+	0.0596897i	$0.980989\pi$
$788$			11.0000i			0.391859i
$789$		0			0
$790$		0			0
$791$	−0.952341			−0.0338614
$792$		0			0
$793$			12.6005i			0.447458i
$794$	14.3901			0.510686
$795$		0			0
$796$	−7.45017			−0.264064
$797$			28.3746i			1.00508i	0.864554	+	0.502540i	$0.167601\pi$
$797$			28.3746i			1.00508i	−0.864554	+	0.502540i	$0.832399\pi$
$798$		0			0
$799$	13.0997			0.463433
$800$	4.98684	+	0.362541i	0.176311	+	0.0128178i
$801$		0			0
$802$		−	21.6794i		−	0.765526i
$803$		−	12.7010i		−	0.448208i
$804$		0			0
$805$	−12.5498	−	13.4953i	−0.442324	−	0.475648i
$806$	14.8087			0.521616
$807$		0			0
$808$			9.55505i			0.336146i
$809$	49.6224			1.74463			0.872315	−	0.488944i	$-0.162618\pi$
$809$	49.6224			1.74463			0.872315	+	0.488944i	$0.162618\pi$
$810$		0			0
$811$	6.90033			0.242303			0.121152	−	0.992634i	$-0.461341\pi$
$811$	6.90033			0.242303			0.121152	+	0.992634i	$0.461341\pi$
$812$			12.0000i			0.421117i
$813$		0			0
$814$	−26.1993			−0.918286
$815$	39.8947	−	37.0997i	1.39745	−	1.29954i
$816$		0			0
$817$			5.97586i			0.209069i
$818$		−	16.0997i		−	0.562912i
$819$		0			0
$820$	8.54983	+	9.19397i	0.298573	+	0.321067i
$821$	21.2032			0.739998			0.369999	−	0.929032i	$-0.379358\pi$
$821$	21.2032			0.739998			0.369999	+	0.929032i	$0.379358\pi$
$822$		0			0
$823$		−	42.5216i		−	1.48221i	−0.671390	−	0.741104i	$-0.734302\pi$
$823$		−	42.5216i		−	1.48221i	0.671390	−	0.741104i	$-0.265698\pi$
$824$	6.56712			0.228776
$825$		0			0
$826$	10.8248			0.376641
$827$			27.4502i			0.954536i	0.878758	+	0.477268i	$0.158373\pi$
$827$			27.4502i			0.954536i	−0.878758	+	0.477268i	$0.841627\pi$
$828$		0			0
$829$	−46.5498			−1.61674			−0.808371	−	0.588673i	$-0.799651\pi$
$829$	−46.5498			−1.61674			−0.808371	+	0.588673i	$0.799651\pi$
$830$	6.09095	+	6.54983i	0.211420	+	0.227348i
$831$		0			0
$832$		−	1.73205i		−	0.0600481i
$833$			6.72508i			0.233010i
$834$		0			0
$835$	34.5498	−	32.1293i	1.19565	−	1.11188i
$836$	5.97586			0.206680
$837$		0			0
$838$		−	16.4833i		−	0.569405i
$839$	48.7276			1.68226			0.841132	−	0.540830i	$-0.181890\pi$
$839$	48.7276			1.68226			0.841132	+	0.540830i	$0.181890\pi$
$840$		0			0
$841$	54.4743			1.87842
$842$		−	9.27492i		−	0.319635i
$843$		0			0
$844$	9.72508			0.334751
$845$	15.2274	+	16.3746i	0.523838	+	0.563303i
$846$		0			0
$847$		−	5.38460i		−	0.185017i
$848$			8.27492i			0.284162i
$849$		0			0
$850$	−0.462210	+	6.35781i	−0.0158537	+	0.218071i
$851$	−62.5840			−2.14535
$852$		0			0
$853$			10.2772i			0.351885i	0.984400	+	0.175943i	$0.0562973\pi$
$853$			10.2772i			0.351885i	−0.984400	+	0.175943i	$0.943703\pi$
$854$	9.55505			0.326967
$855$		0			0
$856$	12.0000			0.410152
$857$			33.8248i			1.15543i	0.816238	+	0.577716i	$0.196056\pi$
$857$			33.8248i			1.15543i	−0.816238	+	0.577716i	$0.803944\pi$
$858$		0			0
$859$	4.00000			0.136478			0.0682391	−	0.997669i	$-0.478262\pi$
$859$	4.00000			0.136478			0.0682391	+	0.997669i	$0.478262\pi$
$860$	4.30136	−	4.00000i	0.146675	−	0.136399i
$861$		0			0
$862$		−	33.9189i		−	1.15528i
$863$			9.72508i			0.331046i	0.986206	+	0.165523i	$0.0529312\pi$
$863$			9.72508i			0.331046i	−0.986206	+	0.165523i	$0.947069\pi$
$864$		0			0
$865$	−32.9124	+	30.6065i	−1.11905	+	1.04065i
$866$	10.8109			0.367370
$867$		0			0
$868$		−	11.2296i		−	0.381156i
$869$		0			0
$870$		0			0
$871$	21.0997			0.714935
$872$		−	7.82475i		−	0.264980i
$873$		0			0
$874$	14.2749			0.482856
$875$	−9.19397	−	11.4502i	−0.310813	−	0.387086i
$876$		0			0
$877$			10.3348i			0.348980i	0.984659	+	0.174490i	$0.0558277\pi$
$877$			10.3348i			0.348980i	−0.984659	+	0.174490i	$0.944172\pi$
$878$		−	36.0000i		−	1.21494i
$879$		0			0
$880$	−4.00000	−	4.30136i	−0.134840	−	0.144999i
$881$	−15.7611			−0.531005			−0.265502	−	0.964110i	$-0.585538\pi$
$881$	−15.7611			−0.531005			−0.265502	+	0.964110i	$0.585538\pi$
$882$		0			0
$883$		−	0.952341i		−	0.0320488i	−0.999872	−	0.0160244i	$-0.994899\pi$
$883$		−	0.952341i		−	0.0320488i	0.999872	−	0.0160244i	$-0.00510095\pi$
$884$	2.20822			0.0742705
$885$		0			0
$886$	7.45017			0.250293
$887$			18.8248i			0.632073i	0.948747	+	0.316037i	$0.102352\pi$
$887$			18.8248i			0.632073i	−0.948747	+	0.316037i	$0.897648\pi$
$888$		0			0
$889$	−29.0241			−0.973437
$890$	−4.98684	+	4.63746i	−0.167159	+	0.155448i
$891$		0			0
$892$		−	16.4833i		−	0.551900i
$893$		−	23.3746i		−	0.782201i
$894$		0			0
$895$	2.00000	+	2.15068i	0.0668526	+	0.0718892i
$896$	−1.31342			−0.0438784
$897$		0			0
$898$		−	10.8685i		−	0.362686i
$899$	−78.1149			−2.60528
$900$		0			0
$901$	−10.5498			−0.351466
$902$		−	14.7492i		−	0.491094i
$903$		0			0
$904$	−0.725083			−0.0241159
$905$	9.97368	+	10.7251i	0.331536	+	0.356514i
$906$		0			0
$907$			41.5692i			1.38028i	0.723674	+	0.690142i	$0.242452\pi$
$907$			41.5692i			1.38028i	−0.723674	+	0.690142i	$0.757548\pi$
$908$		−	0.549834i		−	0.0182469i
$909$		0			0
$910$	−3.72508	+	3.46410i	−0.123485	+	0.114834i
$911$	−5.97586			−0.197989			−0.0989946	−	0.995088i	$-0.531563\pi$
$911$	−5.97586			−0.197989			−0.0989946	+	0.995088i	$0.531563\pi$
$912$		0			0
$913$		−	10.5074i		−	0.347744i
$914$	15.9495			0.527564
$915$		0			0
$916$	22.3746			0.739277
$917$		−	23.3746i		−	0.771897i
$918$		0			0
$919$	−33.0997			−1.09186			−0.545929	−	0.837832i	$-0.683823\pi$
$919$	−33.0997			−1.09186			−0.545929	+	0.837832i	$0.683823\pi$
$920$	−9.55505	−	10.2749i	−0.315021	−	0.338754i
$921$		0			0
$922$		−	6.92820i		−	0.228168i
$923$			12.0000i			0.394985i
$924$		0			0
$925$	−49.7371	−	3.61587i	−1.63535	−	0.118889i
$926$	16.8443			0.553539
$927$		0			0
$928$			9.13642i			0.299918i
$929$	−13.5529			−0.444655			−0.222328	−	0.974972i	$-0.571365\pi$
$929$	−13.5529			−0.444655			−0.222328	+	0.974972i	$0.571365\pi$
$930$		0			0
$931$	12.0000			0.393284
$932$			0.175248i			0.00574045i
$933$		0			0
$934$	13.6495			0.446625
$935$	5.48387	−	5.09967i	0.179342	−	0.166777i
$936$		0			0
$937$		−	5.55724i		−	0.181547i	−0.995872	−	0.0907735i	$-0.971066\pi$
$937$		−	5.55724i		−	0.181547i	0.995872	−	0.0907735i	$-0.0289339\pi$
$938$		−	16.0000i		−	0.522419i
$939$		0			0
$940$	−16.8248	+	15.6460i	−0.548763	+	0.510316i
$941$	10.9260			0.356178			0.178089	−	0.984014i	$-0.443008\pi$
$941$	10.9260			0.356178			0.178089	+	0.984014i	$0.443008\pi$
$942$		0			0
$943$		−	35.2323i		−	1.14732i
$944$	8.24163			0.268242
$945$		0			0
$946$	−6.90033			−0.224349
$947$			16.5498i			0.537797i	0.963168	+	0.268899i	$0.0866597\pi$
$947$			16.5498i			0.537797i	−0.963168	+	0.268899i	$0.913340\pi$
$948$		0			0
$949$	−8.37459			−0.271851
$950$	11.3446	+	0.824752i	0.368069	+	0.0267585i
$951$		0			0
$952$		−	1.67451i		−	0.0542711i
$953$		−	39.2749i		−	1.27224i	−0.771590	−	0.636120i	$-0.780538\pi$
$953$		−	39.2749i		−	1.27224i	0.771590	−	0.636120i	$-0.219462\pi$
$954$		0			0
$955$	22.5498	+	24.2487i	0.729696	+	0.784670i
$956$	−9.55505			−0.309032
$957$		0			0
$958$		−	1.67451i		−	0.0541009i
$959$	−5.02352			−0.162218
$960$		0			0
$961$	42.0997			1.35805
$962$			17.2749i			0.556966i
$963$		0			0
$964$	−13.8248			−0.445265
$965$	−13.5895	+	12.6375i	−0.437463	+	0.406814i
$966$		0			0
$967$			9.55505i			0.307270i	0.988128	+	0.153635i	$0.0490980\pi$
$967$			9.55505i			0.307270i	−0.988128	+	0.153635i	$0.950902\pi$
$968$		−	4.09967i		−	0.131768i
$969$		0			0
$970$	−29.0997	−	31.2920i	−0.934334	−	1.00473i
$971$	37.6289			1.20757			0.603785	−	0.797147i	$-0.293658\pi$
$971$	37.6289			1.20757			0.603785	+	0.797147i	$0.293658\pi$
$972$		0			0
$973$			13.4953i			0.432640i
$974$	−9.91613			−0.317733
$975$		0			0
$976$	7.27492			0.232864
$977$		−	0.199338i		−	0.00637738i	−0.999995	−	0.00318869i	$-0.998985\pi$
$977$		−	0.199338i		−	0.00637738i	0.999995	−	0.00318869i	$-0.00101499\pi$
$978$		0			0
$979$	8.00000			0.255681
$980$	−8.03231	−	8.63746i	−0.256583	−	0.275913i
$981$		0			0
$982$		−	18.7490i		−	0.598305i
$983$			32.0000i			1.02064i	0.859984	+	0.510321i	$0.170473\pi$
$983$			32.0000i			1.02064i	−0.859984	+	0.510321i	$0.829527\pi$
$984$		0			0
$985$	18.0120	−	16.7501i	0.573911	−	0.533703i
$986$	−11.6482			−0.370954
$987$		0			0
$988$		−	3.94027i		−	0.125357i
$989$	−16.4833			−0.524137
$990$		0			0
$991$	−47.2990			−1.50250			−0.751251	−	0.660016i	$-0.770549\pi$
$991$	−47.2990			−1.50250			−0.751251	+	0.660016i	$0.770549\pi$
$992$		−	8.54983i		−	0.271458i
$993$		0			0
$994$	9.09967			0.288624
$995$	11.3446	+	12.1993i	0.359649	+	0.386745i
$996$		0			0
$997$			46.1583i			1.46185i	0.682459	+	0.730924i	$0.260911\pi$
$997$			46.1583i			1.46185i	−0.682459	+	0.730924i	$0.739089\pi$
$998$			14.2749i			0.451865i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	810.2.c.g.649.4	yes	8
3.2	odd	2	inner	810.2.c.g.649.5	yes	8
5.2	odd	4		4050.2.a.cb.1.2		4
5.3	odd	4		4050.2.a.ca.1.3		4
5.4	even	2	inner	810.2.c.g.649.8	yes	8
9.2	odd	6		810.2.i.g.109.4		8
9.4	even	3		810.2.i.i.379.4		8
9.5	odd	6		810.2.i.i.379.1		8
9.7	even	3		810.2.i.g.109.1		8
15.2	even	4		4050.2.a.ca.1.2		4
15.8	even	4		4050.2.a.cb.1.3		4
15.14	odd	2	inner	810.2.c.g.649.1	✓	8
45.4	even	6		810.2.i.g.379.1		8
45.14	odd	6		810.2.i.g.379.4		8
45.29	odd	6		810.2.i.i.109.1		8
45.34	even	6		810.2.i.i.109.4		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
810.2.c.g.649.1	✓	8	15.14	odd	2	inner
810.2.c.g.649.4	yes	8	1.1	even	1	trivial
810.2.c.g.649.5	yes	8	3.2	odd	2	inner
810.2.c.g.649.8	yes	8	5.4	even	2	inner
810.2.i.g.109.1		8	9.7	even	3
810.2.i.g.109.4		8	9.2	odd	6
810.2.i.g.379.1		8	45.4	even	6
810.2.i.g.379.4		8	45.14	odd	6
810.2.i.i.109.1		8	45.29	odd	6
810.2.i.i.109.4		8	45.34	even	6
810.2.i.i.379.1		8	9.5	odd	6
810.2.i.i.379.4		8	9.4	even	3
4050.2.a.ca.1.2		4	15.2	even	4
4050.2.a.ca.1.3		4	5.3	odd	4
4050.2.a.cb.1.2		4	5.2	odd	4
4050.2.a.cb.1.3		4	15.8	even	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 810 = 2 \cdot 3^{4} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	810.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.00000i q^{2} -1.00000 q^{4} +(1.52274 + 1.63746i) q^{5} +1.31342i q^{7} +1.00000i q^{8} +(1.63746 - 1.52274i) q^{10} -2.62685 q^{11} +1.73205i q^{13} +1.31342 q^{14} +1.00000 q^{16} +1.27492i q^{17} +2.27492 q^{19} +(-1.52274 - 1.63746i) q^{20} +2.62685i q^{22} +6.27492i q^{23} +(-0.362541 + 4.98684i) q^{25} +1.73205 q^{26} -1.31342i q^{28} -9.13642 q^{29} +8.54983 q^{31} -1.00000i q^{32} +1.27492 q^{34} +(-2.15068 + 2.00000i) q^{35} +9.97368i q^{37} -2.27492i q^{38} +(-1.63746 + 1.52274i) q^{40} -5.61478 q^{41} +2.62685i q^{43} +2.62685 q^{44} +6.27492 q^{46} -10.2749i q^{47} +5.27492 q^{49} +(4.98684 + 0.362541i) q^{50} -1.73205i q^{52} +8.27492i q^{53} +(-4.00000 - 4.30136i) q^{55} -1.31342 q^{56} +9.13642i q^{58} +8.24163 q^{59} +7.27492 q^{61} -8.54983i q^{62} -1.00000 q^{64} +(-2.83616 + 2.63746i) q^{65} -12.1819i q^{67} -1.27492i q^{68} +(2.00000 + 2.15068i) q^{70} +6.92820 q^{71} +4.83507i q^{73} +9.97368 q^{74} -2.27492 q^{76} -3.45017i q^{77} +(1.52274 + 1.63746i) q^{80} +5.61478i q^{82} +4.00000i q^{83} +(-2.08762 + 1.94136i) q^{85} +2.62685 q^{86} -2.62685i q^{88} -3.04547 q^{89} -2.27492 q^{91} -6.27492i q^{92} -10.2749 q^{94} +(3.46410 + 3.72508i) q^{95} -19.1101i q^{97} -5.27492i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 8 q^{4} - 2 q^{10} + 8 q^{16} - 12 q^{19} - 18 q^{25} + 8 q^{31} - 20 q^{34} + 2 q^{40} + 20 q^{46} + 12 q^{49} - 32 q^{55} + 28 q^{61} - 8 q^{64} + 16 q^{70} + 12 q^{76} - 62 q^{85} + 12 q^{91} - 52 q^{94}+O(q^{100}) \) 8 * q - 8 * q^4 - 2 * q^10 + 8 * q^16 - 12 * q^19 - 18 * q^25 + 8 * q^31 - 20 * q^34 + 2 * q^40 + 20 * q^46 + 12 * q^49 - 32 * q^55 + 28 * q^61 - 8 * q^64 + 16 * q^70 + 12 * q^76 - 62 * q^85 + 12 * q^91 - 52 * q^94

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