LMFDB - Embedded newform 735.2.p.d.374.8

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [735,2,Mod(374,735)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("735.374"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(735, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([3, 3, 1])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 735 = 3 \cdot 5 \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	735.p (of order $6$, degree $2$, not minimal)

Newform invariants

comment:select newform

sage:traces = [16,0,-24] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)

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

Self dual:	no
Analytic conductor:	$5.86900454856$
Analytic rank:	$0$
Dimension:	$16$
Relative dimension:	$8$ over $\Q(\zeta_{6})$
Coefficient field:	16.0.721389578983833600000000.5
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} + 8x^{14} + 44x^{12} + 128x^{10} + 223x^{8} - 464x^{6} - 724x^{4} + 784x^{2} + 2401 $ x^16 + 8x^14 + 44x^12 + 128x^10 + 223x^8 - 464x^6 - 724x^4 + 784*x^2 + 2401
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 7^{6} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{6}]$

Embedding invariants

Embedding label			374.8
Root			$-1.22796 - 0.279124i$ of defining polynomial
Character	$\chi$	$=$	735.374
Dual form			735.2.p.d.509.8

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+(1.36434 + 2.36311i) q^{2} +(-1.50000 - 0.866025i) q^{3} +(-2.72286 + 4.71613i) q^{4} +(-1.93649 + 1.11803i) q^{5} -4.72622i q^{6} -9.40228 q^{8} +(1.50000 + 2.59808i) q^{9} +(-5.28407 - 3.05076i) q^{10} +(8.16858 - 4.71613i) q^{12} +3.87298 q^{15} +(-7.38221 - 12.7864i) q^{16} +(-1.50403 - 0.868351i) q^{17} +(-4.09303 + 7.08933i) q^{18} +(-0.633294 + 0.365633i) q^{19} -12.1770i q^{20} +(-3.31162 - 5.73589i) q^{23} +(14.1034 + 8.14261i) q^{24} +(2.50000 - 4.33013i) q^{25} -5.19615i q^{27} +(5.28407 + 9.15229i) q^{30} +(3.54626 + 2.04744i) q^{31} +(10.7414 - 18.6047i) q^{32} -4.73891i q^{34} -16.3372 q^{36} +(-1.72806 - 0.997696i) q^{38} +(18.2074 - 10.5121i) q^{40} +(-5.80948 - 3.35410i) q^{45} +(9.03636 - 15.6514i) q^{46} +(-11.8412 + 6.83651i) q^{47} +25.5727i q^{48} +13.6434 q^{50} +(1.50403 + 2.60505i) q^{51} +(-7.25653 + 12.5687i) q^{53} +(12.2791 - 7.08933i) q^{54} +1.26659 q^{57} +(-10.5456 + 18.2655i) q^{60} +(-11.7323 + 6.77366i) q^{61} +11.1736i q^{62} +29.0912 q^{64} +(8.19051 - 4.72880i) q^{68} +11.4718i q^{69} +(-14.1034 - 24.4278i) q^{72} +(-7.50000 + 4.33013i) q^{75} -3.98226i q^{76} +(-8.39547 - 14.5414i) q^{79} +(28.5912 + 16.5071i) q^{80} +(-4.50000 + 7.79423i) q^{81} -10.1996i q^{83} +3.88338 q^{85} -18.3046i q^{90} +36.0683 q^{92} +(-3.54626 - 6.14231i) q^{93} +(-32.3109 - 18.6547i) q^{94} +(0.817579 - 1.41609i) q^{95} +(-32.2243 + 18.6047i) q^{96} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 24 q^{3} - 16 q^{4} + 24 q^{9} + 48 q^{12} - 32 q^{16} + 40 q^{25} - 96 q^{36} + 128 q^{64} - 72 q^{68} - 120 q^{75} + 120 q^{80} - 72 q^{81}+O(q^{100}) $ 16 * q - 24 * q^3 - 16 * q^4 + 24 * q^9 + 48 * q^12 - 32 * q^16 + 40 * q^25 - 96 * q^36 + 128 * q^64 - 72 * q^68 - 120 * q^75 + 120 * q^80 - 72 * q^81

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/735\mathbb{Z}\right)^\times$.

$n$	$346$	$442$	$491$
$\chi(n)$	$e\left(\frac{1}{6}\right)$	$-1$	$-1$

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 $

$2$	1.36434	+	2.36311i	0.964736	+	1.67097i	0.710324	+	0.703875i	$0.248549\pi$
$2$	1.36434	+	2.36311i	0.964736	+	1.67097i	0.254412	+	0.967096i	$0.418118\pi$
$3$	−1.50000	−	0.866025i	−0.866025	−	0.500000i
$3$	−1.50000	−	0.866025i	−0.866025	−	0.500000i
$4$	−2.72286	+	4.71613i	−1.36143	+	2.35807i
$5$	−1.93649	+	1.11803i	−0.866025	+	0.500000i
$5$	−1.93649	+	1.11803i	−0.866025	+	0.500000i
$6$		−	4.72622i		−	1.92947i
$7$		0			0
$7$		0			0
$8$	−9.40228			−3.32421
$9$	1.50000	+	2.59808i	0.500000	+	0.866025i
$10$	−5.28407	−	3.05076i	−1.67097	−	0.964736i
$11$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$11$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$12$	8.16858	−	4.71613i	2.35807	−	1.36143i
$13$		0			0			−	1.00000i	$-0.5\pi$
$13$		0			0				1.00000i	$0.5\pi$
$14$		0			0
$15$	3.87298			1.00000
$16$	−7.38221	−	12.7864i	−1.84555	−	3.19659i
$17$	−1.50403	−	0.868351i	−0.364780	−	0.210606i	0.306395	−	0.951904i	$-0.400877\pi$
$17$	−1.50403	−	0.868351i	−0.364780	−	0.210606i	−0.671176	+	0.741298i	$0.734210\pi$
$18$	−4.09303	+	7.08933i	−0.964736	+	1.67097i
$19$	−0.633294	+	0.365633i	−0.145288	+	0.0838818i	−0.570882	−	0.821032i	$-0.693398\pi$
$19$	−0.633294	+	0.365633i	−0.145288	+	0.0838818i	0.425594	+	0.904914i	$0.360065\pi$
$20$		−	12.1770i		−	2.72286i
$21$		0			0
$22$		0			0
$23$	−3.31162	−	5.73589i	−0.690520	−	1.19602i	−0.971668	−	0.236352i	$-0.924048\pi$
$23$	−3.31162	−	5.73589i	−0.690520	−	1.19602i	0.281147	−	0.959665i	$-0.409285\pi$
$24$	14.1034	+	8.14261i	2.87885	+	1.66210i
$25$	2.50000	−	4.33013i	0.500000	−	0.866025i
$26$		0			0
$27$		−	5.19615i		−	1.00000i
$28$		0			0
$29$		0			0		1.00000			$0$
$29$		0			0		−1.00000			$\pi$
$30$	5.28407	+	9.15229i	0.964736	+	1.67097i
$31$	3.54626	+	2.04744i	0.636928	+	0.367730i	0.783430	−	0.621480i	$-0.213468\pi$
$31$	3.54626	+	2.04744i	0.636928	+	0.367730i	−0.146502	+	0.989210i	$0.546802\pi$
$32$	10.7414	−	18.6047i	1.89884	−	3.28888i
$33$		0			0
$34$		−	4.73891i		−	0.812717i
$35$		0			0
$36$	−16.3372			−2.72286
$37$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$37$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$38$	−1.72806	−	0.997696i	−0.280328	−	0.161848i
$39$		0			0
$40$	18.2074	−	10.5121i	2.87885	−	1.66210i
$41$		0			0			−	1.00000i	$-0.5\pi$
$41$		0			0				1.00000i	$0.5\pi$
$42$		0			0
$43$		0			0		1.00000			$0$
$43$		0			0		−1.00000			$\pi$
$44$		0			0
$45$	−5.80948	−	3.35410i	−0.866025	−	0.500000i
$46$	9.03636	−	15.6514i	1.33234	−	2.30768i
$47$	−11.8412	+	6.83651i	−1.72721	+	0.997208i	−0.826269	+	0.563276i	$0.809541\pi$
$47$	−11.8412	+	6.83651i	−1.72721	+	0.997208i	−0.900946	+	0.433932i	$0.857126\pi$
$48$			25.5727i			3.69111i
$49$		0			0
$50$	13.6434			1.92947
$51$	1.50403	+	2.60505i	0.210606	+	0.364780i
$52$		0			0
$53$	−7.25653	+	12.5687i	−0.996761	+	1.72644i	−0.428737	+	0.903430i	$0.641041\pi$
$53$	−7.25653	+	12.5687i	−0.996761	+	1.72644i	−0.568025	+	0.823012i	$0.692292\pi$
$54$	12.2791	−	7.08933i	1.67097	−	0.964736i
$55$		0			0
$56$		0			0
$57$	1.26659			0.167764
$58$		0			0
$59$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$59$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$60$	−10.5456	+	18.2655i	−1.36143	+	2.35807i
$61$	−11.7323	+	6.77366i	−1.50217	+	0.867278i	−0.502172	+	0.864767i	$0.667466\pi$
$61$	−11.7323	+	6.77366i	−1.50217	+	0.867278i	−0.999997	+	0.00251038i	$0.999201\pi$
$62$			11.1736i			1.41905i
$63$		0			0
$64$	29.0912			3.63640
$65$		0			0
$66$		0			0
$67$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$67$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$68$	8.19051	−	4.72880i	0.993246	−	0.573451i
$69$			11.4718i			1.38104i
$70$		0			0
$71$		0			0		1.00000			$0$
$71$		0			0		−1.00000			$\pi$
$72$	−14.1034	−	24.4278i	−1.66210	−	2.87885i
$73$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$73$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$74$		0			0
$75$	−7.50000	+	4.33013i	−0.866025	+	0.500000i
$76$		−	3.98226i		−	0.456797i
$77$		0			0
$78$		0			0
$79$	−8.39547	−	14.5414i	−0.944564	−	1.63603i	−0.756622	−	0.653852i	$-0.773152\pi$
$79$	−8.39547	−	14.5414i	−0.944564	−	1.63603i	−0.187942	−	0.982180i	$-0.560182\pi$
$80$	28.5912	+	16.5071i	3.19659	+	1.84555i
$81$	−4.50000	+	7.79423i	−0.500000	+	0.866025i
$82$		0			0
$83$		−	10.1996i		−	1.11955i	−0.828643	−	0.559777i	$-0.810887\pi$
$83$		−	10.1996i		−	1.11955i	0.828643	−	0.559777i	$-0.189113\pi$
$84$		0			0
$85$	3.88338			0.421212
$86$		0			0
$87$		0			0
$88$		0			0
$89$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$89$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$90$		−	18.3046i		−	1.92947i
$91$		0			0
$92$	36.0683			3.76038
$93$	−3.54626	−	6.14231i	−0.367730	−	0.636928i
$94$	−32.3109	−	18.6547i	−3.33261	−	1.92408i
$95$	0.817579	−	1.41609i	0.0838818	−	0.145288i
$96$	−32.2243	+	18.6047i	−3.28888	+	1.89884i
$97$		0			0			−	1.00000i	$-0.5\pi$
$97$		0			0				1.00000i	$0.5\pi$
$98$		0			0
$99$		0			0
$100$	13.6143	+	23.5807i	1.36143	+	2.35807i
$101$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$101$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$102$	−4.10402	+	7.10837i	−0.406358	+	0.703833i
$103$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$103$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$104$		0			0
$105$		0			0
$106$	−39.6016			−3.84644
$107$	6.27495	+	10.8685i	0.606622	+	1.05070i	0.991793	+	0.127855i	$0.0408092\pi$
$107$	6.27495	+	10.8685i	0.606622	+	1.05070i	−0.385171	+	0.922845i	$0.625857\pi$
$108$	24.5057	+	14.1484i	2.35807	+	1.36143i
$109$	−7.74597	+	13.4164i	−0.741929	+	1.28506i	0.209687	+	0.977769i	$0.432756\pi$
$109$	−7.74597	+	13.4164i	−0.741929	+	1.28506i	−0.951616	+	0.307290i	$0.900578\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	3.82221			0.359563			0.179782	−	0.983707i	$-0.442461\pi$
$113$	3.82221			0.359563			0.179782	+	0.983707i	$0.442461\pi$
$114$	1.72806	+	2.99309i	0.161848	+	0.280328i
$115$	12.8258	+	7.40500i	1.19602	+	0.690520i
$116$		0			0
$117$		0			0
$118$		0			0
$119$		0			0
$120$	−36.4149			−3.32421
$121$	−5.50000	−	9.52628i	−0.500000	−	0.866025i
$122$	−32.0138	−	18.4832i	−2.89839	−	1.67339i
$123$		0			0
$124$	−19.3120	+	11.1498i	−1.73427	+	1.00128i
$125$			11.1803i			1.00000i
$126$		0			0
$127$		0			0		1.00000			$0$
$127$		0			0		−1.00000			$\pi$
$128$	18.2074	+	31.5362i	1.60933	+	2.78743i
$129$		0			0
$130$		0			0
$131$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$131$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$	5.80948	+	10.0623i	0.500000	+	0.866025i
$136$	14.1413	+	8.16448i	1.21261	+	0.700098i
$137$	−9.00363	+	15.5947i	−0.769232	+	1.33235i	0.168747	+	0.985659i	$0.446028\pi$
$137$	−9.00363	+	15.5947i	−0.769232	+	1.33235i	−0.937980	+	0.346690i	$0.887306\pi$
$138$	−27.1091	+	15.6514i	−2.30768	+	1.33234i
$139$			22.9998i			1.95081i	0.220412	+	0.975407i	$0.429260\pi$
$139$			22.9998i			1.95081i	−0.220412	+	0.975407i	$0.570740\pi$
$140$		0			0
$141$	23.6824			1.99442
$142$		0			0
$143$		0			0
$144$	22.1466	−	38.3591i	1.84555	−	3.19659i
$145$		0			0
$146$		0			0
$147$		0			0
$148$		0			0
$149$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$149$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$150$	−20.4651	−	11.8156i	−1.67097	−	0.964736i
$151$	5.38741	−	9.33127i	0.438421	−	0.759368i	−0.559147	−	0.829069i	$-0.688871\pi$
$151$	5.38741	−	9.33127i	0.438421	−	0.759368i	0.997568	+	0.0697008i	$0.0222045\pi$
$152$	5.95441	−	3.43778i	0.482966	−	0.278841i
$153$		−	5.21011i		−	0.421212i
$154$		0			0
$155$	−9.15641			−0.735461
$156$		0			0
$157$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$157$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$158$	22.9086	−	39.6788i	1.82251	−	3.15668i
$159$	21.7696	−	12.5687i	1.72644	−	0.996761i
$160$			48.0372i			3.79767i
$161$		0			0
$162$	−24.5582			−1.92947
$163$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$163$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$164$		0			0
$165$		0			0
$166$	24.1028	−	13.9158i	1.87074	−	1.08007i
$167$		−	8.94427i		−	0.692129i	−0.938211	−	0.346064i	$-0.887518\pi$
$167$		−	8.94427i		−	0.692129i	0.938211	−	0.346064i	$-0.112482\pi$
$168$		0			0
$169$	−13.0000			−1.00000
$170$	5.29826	+	9.17686i	0.406358	+	0.703833i
$171$	−1.89988	−	1.09690i	−0.145288	−	0.0838818i
$172$		0			0
$173$	22.1783	−	12.8047i	1.68619	−	0.973521i	0.728796	−	0.684731i	$-0.240080\pi$
$173$	22.1783	−	12.8047i	1.68619	−	0.973521i	0.957392	−	0.288790i	$-0.0932531\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		0			0
$178$		0			0
$179$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$179$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$180$	31.6368	−	18.2655i	2.35807	−	1.36143i
$181$		−	14.3968i		−	1.07011i	−0.844818	−	0.535054i	$-0.820291\pi$
$181$		−	14.3968i		−	1.07011i	0.844818	−	0.535054i	$-0.179709\pi$
$182$		0			0
$183$	23.4646			1.73456
$184$	31.1368	+	53.9305i	2.29543	+	3.97581i
$185$		0			0
$186$	9.67664	−	16.7604i	0.709525	−	1.22893i
$187$		0			0
$188$		−	74.4595i		−	5.43051i
$189$		0			0
$190$	4.46183			0.323695
$191$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$191$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$192$	−43.6368	−	25.1937i	−3.14921	−	1.81820i
$193$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$193$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−14.7369			−1.04996			−0.524982	−	0.851113i	$-0.675928\pi$
$197$	−14.7369			−1.04996			−0.524982	+	0.851113i	$0.675928\pi$
$198$		0			0
$199$	20.5030	+	11.8374i	1.45342	+	0.839132i	0.998674	−	0.0514890i	$-0.0163967\pi$
$199$	20.5030	+	11.8374i	1.45342	+	0.839132i	0.454746	+	0.890621i	$0.349730\pi$
$200$	−23.5057	+	40.7131i	−1.66210	+	2.87885i
$201$		0			0
$202$		0			0
$203$		0			0
$204$	−16.3810			−1.14690
$205$		0			0
$206$		0			0
$207$	9.93486	−	17.2077i	0.690520	−	1.19602i
$208$		0			0
$209$		0			0
$210$		0			0
$211$	7.74597			0.533254			0.266627	−	0.963800i	$-0.414091\pi$
$211$	7.74597			0.533254			0.266627	+	0.963800i	$0.414091\pi$
$212$	−39.5170	−	68.4455i	−2.71404	−	4.70086i
$213$		0			0
$214$	−17.1224	+	29.6568i	−1.17046	+	2.02730i
$215$		0			0
$216$			48.8557i			3.32421i
$217$		0			0
$218$	−42.2726			−2.86306
$219$		0			0
$220$		0			0
$221$		0			0
$222$		0			0
$223$		0			0			−	1.00000i	$-0.5\pi$
$223$		0			0				1.00000i	$0.5\pi$
$224$		0			0
$225$	15.0000			1.00000
$226$	5.21480	+	9.03231i	0.346884	+	0.600820i
$227$	15.4919	+	8.94427i	1.02824	+	0.593652i	0.916479	−	0.400083i	$-0.131019\pi$
$227$	15.4919	+	8.94427i	1.02824	+	0.593652i	0.111757	+	0.993736i	$0.464352\pi$
$228$	−3.44874	+	5.97340i	−0.228399	+	0.395598i
$229$	−21.0119	+	12.1312i	−1.38851	+	0.801654i	−0.993147	−	0.116873i	$-0.962713\pi$
$229$	−21.0119	+	12.1312i	−1.38851	+	0.801654i	−0.395358	+	0.918527i	$0.629380\pi$
$230$			40.4118i			2.66468i
$231$		0			0
$232$		0			0
$233$	13.8798	+	24.0405i	0.909294	+	1.57494i	0.815047	+	0.579394i	$0.196711\pi$
$233$	13.8798	+	24.0405i	0.909294	+	1.57494i	0.0942465	+	0.995549i	$0.469956\pi$
$234$		0			0
$235$	15.2869	−	26.4777i	0.997208	−	1.72721i
$236$		0			0
$237$			29.0828i			1.88913i
$238$		0			0
$239$		0			0		1.00000			$0$
$239$		0			0		−1.00000			$\pi$
$240$	−28.5912	−	49.5214i	−1.84555	−	3.19659i
$241$	−8.66827	−	5.00463i	−0.558372	−	0.322376i	0.194120	−	0.980978i	$-0.437815\pi$
$241$	−8.66827	−	5.00463i	−0.558372	−	0.322376i	−0.752492	+	0.658602i	$0.771148\pi$
$242$	15.0078	−	25.9942i	0.964736	−	1.67097i
$243$	13.5000	−	7.79423i	0.866025	−	0.500000i
$244$		−	73.7749i		−	4.72295i
$245$		0			0
$246$		0			0
$247$		0			0
$248$	−33.3430	−	19.2506i	−2.11728	−	1.22241i
$249$	−8.83313	+	15.2994i	−0.559777	+	0.969562i
$250$	−26.4204	+	15.2538i	−1.67097	+	0.964736i
$251$		0			0			−	1.00000i	$-0.5\pi$
$251$		0			0				1.00000i	$0.5\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$	−5.82508	−	3.36311i	−0.364780	−	0.210606i
$256$	−20.5912	+	35.6650i	−1.28695	+	2.22906i
$257$	6.00000	−	3.46410i	0.374270	−	0.216085i	−0.301052	−	0.953608i	$-0.597338\pi$
$257$	6.00000	−	3.46410i	0.374270	−	0.216085i	0.675322	+	0.737523i	$0.264005\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$		0			0
$262$		0			0
$263$	10.0972	−	17.4888i	0.622618	−	1.07841i	−0.366379	−	0.930466i	$-0.619403\pi$
$263$	10.0972	−	17.4888i	0.622618	−	1.07841i	0.988996	−	0.147939i	$-0.0472641\pi$
$264$		0			0
$265$		−	32.4522i		−	1.99352i
$266$		0			0
$267$		0			0
$268$		0			0
$269$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$269$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$270$	−15.8522	+	27.4569i	−0.964736	+	1.67097i
$271$	−23.0362	+	13.2999i	−1.39935	+	0.807914i	−0.994324	−	0.106392i	$-0.966070\pi$
$271$	−23.0362	+	13.2999i	−1.39935	+	0.807914i	−0.405024	+	0.914306i	$0.632737\pi$
$272$			25.6414i			1.55474i
$273$		0			0
$274$	−49.1361			−2.96842
$275$		0			0
$276$	−54.1024	−	31.2361i	−3.25658	−	1.88019i
$277$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$277$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$278$	−54.3510	+	31.3795i	−3.25975	+	1.88202i
$279$			12.2846i			0.735461i
$280$		0			0
$281$		0			0		1.00000			$0$
$281$		0			0		−1.00000			$\pi$
$282$	32.3109	+	55.9641i	1.92408	+	3.33261i
$283$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$283$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$284$		0			0
$285$	−2.45274	+	1.41609i	−0.145288	+	0.0838818i
$286$		0			0
$287$		0			0
$288$	64.4487			3.79767
$289$	−6.99193	−	12.1104i	−0.411290	−	0.712376i
$290$		0			0
$291$		0			0
$292$		0			0
$293$			13.8564i			0.809500i	0.914427	+	0.404750i	$0.132641\pi$
$293$			13.8564i			0.809500i	−0.914427	+	0.404750i	$0.867359\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$		0			0
$300$		−	47.1613i		−	2.72286i
$301$		0			0
$302$	29.4011			1.69184
$303$		0			0
$304$	9.35022	+	5.39835i	0.536272	+	0.309617i
$305$	15.1464	−	26.2343i	0.867278	−	1.50217i
$306$	12.3121	−	7.10837i	0.703833	−	0.406358i
$307$		0			0			−	1.00000i	$-0.5\pi$
$307$		0			0				1.00000i	$0.5\pi$
$308$		0			0
$309$		0			0
$310$	−12.4925	−	21.6376i	−0.709525	−	1.22893i
$311$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$311$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$312$		0			0
$313$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$313$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$314$		0			0
$315$		0			0
$316$	91.4387			5.14383
$317$	−8.52312	−	14.7625i	−0.478706	−	0.829143i	0.520996	−	0.853559i	$-0.325561\pi$
$317$	−8.52312	−	14.7625i	−0.478706	−	0.829143i	−0.999702	+	0.0244160i	$0.992227\pi$
$318$	59.4024	+	34.2960i	3.33112	+	1.92322i
$319$		0			0
$320$	−56.3348	+	32.5249i	−3.14921	+	1.81820i
$321$		−	21.7371i		−	1.21324i
$322$		0			0
$323$	1.26999			0.0706641
$324$	−24.5057	−	42.4452i	−1.36143	−	2.35807i
$325$		0			0
$326$		0			0
$327$	23.2379	−	13.4164i	1.28506	−	0.741929i
$328$		0			0
$329$		0			0
$330$		0			0
$331$	11.6190	+	20.1246i	0.638635	+	1.10615i	0.985732	+	0.168320i	$0.0538340\pi$
$331$	11.6190	+	20.1246i	0.638635	+	1.10615i	−0.347097	+	0.937829i	$0.612833\pi$
$332$	48.1028	+	27.7721i	2.63998	+	1.52419i
$333$		0			0
$334$	21.1363	−	12.2030i	1.15653	−	0.667721i
$335$		0			0
$336$		0			0
$337$		0			0		1.00000			$0$
$337$		0			0		−1.00000			$\pi$
$338$	−17.7364	−	30.7204i	−0.964736	−	1.67097i
$339$	−5.73332	−	3.31013i	−0.311391	−	0.179782i
$340$	−10.5739	+	18.3145i	−0.573451	+	0.993246i
$341$		0			0
$342$		−	5.98617i		−	0.323695i
$343$		0			0
$344$		0			0
$345$	−12.8258	−	22.2150i	−0.690520	−	1.19602i
$346$	60.5177	+	34.9399i	3.25345	+	1.87838i
$347$	−2.04503	+	3.54210i	−0.109783	+	0.190150i	−0.915682	−	0.401903i	$-0.868349\pi$
$347$	−2.04503	+	3.54210i	−0.109783	+	0.190150i	0.805899	+	0.592053i	$0.201682\pi$
$348$		0			0
$349$			37.3404i			1.99879i	0.0348323	+	0.999393i	$0.488910\pi$
$349$			37.3404i			1.99879i	−0.0348323	+	0.999393i	$0.511090\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	18.0000	+	10.3923i	0.958043	+	0.553127i	0.895570	−	0.444920i	$-0.146768\pi$
$353$	18.0000	+	10.3923i	0.958043	+	0.553127i	0.0624731	+	0.998047i	$0.480101\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$359$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$360$	54.6223	+	31.5362i	2.87885	+	1.66210i
$361$	−9.23263	+	15.9914i	−0.485928	+	0.841651i
$362$	34.0213	−	19.6422i	1.78812	−	1.03237i
$363$			19.0526i			1.00000i
$364$		0			0
$365$		0			0
$366$	32.0138	+	55.4495i	1.67339	+	2.89839i
$367$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$367$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$368$	−48.8941	+	84.6871i	−2.54878	+	4.41462i
$369$		0			0
$370$		0			0
$371$		0			0
$372$	38.6239			2.00256
$373$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$373$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$374$		0			0
$375$	9.68246	−	16.7705i	0.500000	−	0.866025i
$376$	111.334	−	64.2788i	5.74162	−	3.31493i
$377$		0			0
$378$		0			0
$379$	−24.5577			−1.26144			−0.630722	−	0.776009i	$-0.717241\pi$
$379$	−24.5577			−1.26144			−0.630722	+	0.776009i	$0.717241\pi$
$380$	4.45231	+	7.71162i	0.228399	+	0.395598i
$381$		0			0
$382$		0			0
$383$	17.8573	−	10.3099i	0.912465	−	0.526812i	0.0312418	−	0.999512i	$-0.490054\pi$
$383$	17.8573	−	10.3099i	0.912465	−	0.526812i	0.881224	+	0.472700i	$0.156720\pi$
$384$		−	63.0724i		−	3.21865i
$385$		0			0
$386$		0			0
$387$		0			0
$388$		0			0
$389$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$389$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$390$		0			0
$391$			11.5026i			0.581711i
$392$		0			0
$393$		0			0
$394$	−20.1062	−	34.8250i	−1.01294	−	1.75446i
$395$	32.5155	+	18.7728i	1.63603	+	0.944564i
$396$		0			0
$397$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$397$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$398$			64.6011i			3.23816i
$399$		0			0
$400$	−73.8221			−3.69111
$401$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$401$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$		−	20.1246i		−	1.00000i
$406$		0			0
$407$		0			0
$408$	−14.1413	−	24.4934i	−0.700098	−	1.21261i
$409$	−31.0712	−	17.9389i	−1.53637	−	0.887024i	−0.999047	−	0.0436468i	$-0.986102\pi$
$409$	−31.0712	−	17.9389i	−1.53637	−	0.887024i	−0.537323	−	0.843377i	$-0.680564\pi$
$410$		0			0
$411$	27.0109	−	15.5947i	1.33235	−	0.769232i
$412$		0			0
$413$		0			0
$414$	54.2182			2.66468
$415$	11.4035	+	19.7515i	0.559777	+	0.969562i
$416$		0			0
$417$	19.9184	−	34.4996i	0.975407	−	1.68945i
$418$		0			0
$419$		0			0			−	1.00000i	$-0.5\pi$
$419$		0			0				1.00000i	$0.5\pi$
$420$		0			0
$421$	−15.9156			−0.775679			−0.387840	−	0.921727i	$-0.626779\pi$
$421$	−15.9156			−0.775679			−0.387840	+	0.921727i	$0.626779\pi$
$422$	10.5681	+	18.3046i	0.514449	+	0.891053i
$423$	−35.5236	−	20.5095i	−1.72721	−	0.997208i
$424$	68.2280	−	118.174i	3.31344	−	5.73905i
$425$	−7.52014	+	4.34175i	−0.364780	+	0.210606i
$426$		0			0
$427$		0			0
$428$	−68.3432			−3.30349
$429$		0			0
$430$		0			0
$431$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$431$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$432$	−66.4399	+	38.3591i	−3.19659	+	1.84555i
$433$		0			0			−	1.00000i	$-0.5\pi$
$433$		0			0				1.00000i	$0.5\pi$
$434$		0			0
$435$		0			0
$436$	−42.1824	−	73.0620i	−2.02017	−	3.49903i
$437$	4.19446	+	2.42167i	0.200648	+	0.115844i
$438$		0			0
$439$	36.2905	−	20.9523i	1.73205	−	1.00000i	0.865557	−	0.500810i	$-0.166964\pi$
$439$	36.2905	−	20.9523i	1.73205	−	1.00000i	0.866493	−	0.499190i	$-0.166369\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	15.5545	+	26.9412i	0.739018	+	1.28002i	0.952938	+	0.303165i	$0.0980435\pi$
$443$	15.5545	+	26.9412i	0.739018	+	1.28002i	−0.213920	+	0.976851i	$0.568623\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$		0			0
$449$		0			0		1.00000			$0$
$449$		0			0		−1.00000			$\pi$
$450$	20.4651	+	35.4467i	0.964736	+	1.67097i
$451$		0			0
$452$	−10.4073	+	18.0260i	−0.489520	+	0.847874i
$453$	−16.1622	+	9.33127i	−0.759368	+	0.438421i
$454$			48.8122i			2.29087i
$455$		0			0
$456$	−11.9088			−0.557682
$457$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$457$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$458$	−57.3348	−	33.1023i	−2.67908	−	1.54677i
$459$	−4.51208	+	7.81516i	−0.210606	+	0.364780i
$460$	−69.8459	+	40.3256i	−3.25658	+	1.88019i
$461$		0			0			−	1.00000i	$-0.5\pi$
$461$		0			0				1.00000i	$0.5\pi$
$462$		0			0
$463$		0			0		1.00000			$0$
$463$		0			0		−1.00000			$\pi$
$464$		0			0
$465$	13.7346	+	7.92969i	0.636928	+	0.367730i
$466$	−37.8735	+	65.5988i	−1.75446	+	3.03881i
$467$	−30.9839	+	17.8885i	−1.43376	+	0.827783i	−0.997405	−	0.0719905i	$-0.977065\pi$
$467$	−30.9839	+	17.8885i	−1.43376	+	0.827783i	−0.436357	+	0.899774i	$0.643732\pi$
$468$		0			0
$469$		0			0
$470$	83.4263			3.84817
$471$		0			0
$472$		0			0
$473$		0			0
$474$	−68.7257	+	39.6788i	−3.15668	+	1.82251i
$475$			3.65633i			0.167764i
$476$		0			0
$477$	−43.5392			−1.99352
$478$		0			0
$479$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$479$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$480$	41.6014	−	72.0558i	1.89884	−	3.28888i
$481$		0			0
$482$		−	27.3121i		−	1.24403i
$483$		0			0
$484$	59.9029			2.72286
$485$		0			0
$486$	36.8372	+	21.2680i	1.67097	+	0.964736i
$487$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$487$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$488$	110.311	−	63.6878i	4.99352	−	2.88301i
$489$		0			0
$490$		0			0
$491$		0			0		1.00000			$0$
$491$		0			0		−1.00000			$\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$		−	60.4584i		−	2.71466i
$497$		0			0
$498$	−48.2057			−2.16015
$499$	−18.2950	−	31.6878i	−0.818995	−	1.41854i	−0.906423	−	0.422370i	$-0.861198\pi$
$499$	−18.2950	−	31.6878i	−0.818995	−	1.41854i	0.0874285	−	0.996171i	$-0.472135\pi$
$500$	−52.7280	−	30.4425i	−2.35807	−	1.36143i
$501$	−7.74597	+	13.4164i	−0.346064	+	0.599401i
$502$		0			0
$503$			34.0723i			1.51921i	0.650386	+	0.759604i	$0.274607\pi$
$503$			34.0723i			1.51921i	−0.650386	+	0.759604i	$0.725393\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	19.5000	+	11.2583i	0.866025	+	0.500000i
$508$		0			0
$509$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$509$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$510$		−	18.3537i		−	0.812717i
$511$		0			0
$512$	−39.5439			−1.74761
$513$	1.89988	+	3.29069i	0.0838818	+	0.145288i
$514$	16.3721	+	9.45244i	0.722143	+	0.416929i
$515$		0			0
$516$		0			0
$517$		0			0
$518$		0			0
$519$	−44.3567			−1.94704
$520$		0			0
$521$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$521$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$522$		0			0
$523$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$523$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$524$		0			0
$525$		0			0
$526$	55.1039			2.40265
$527$	−3.55579	−	6.15880i	−0.154892	−	0.268282i
$528$		0			0
$529$	−10.4336	+	18.0716i	−0.453636	+	0.785721i
$530$	76.6881	−	44.2759i	3.33112	−	1.92322i
$531$		0			0
$532$		0			0
$533$		0			0
$534$		0			0
$535$	−24.3028	−	14.0312i	−1.05070	−	0.606622i
$536$		0			0
$537$		0			0
$538$		0			0
$539$		0			0
$540$	−63.2735			−2.72286
$541$	15.7246	+	27.2358i	0.676052	+	1.17096i	0.976160	+	0.217051i	$0.0696439\pi$
$541$	15.7246	+	27.2358i	0.676052	+	1.17096i	−0.300108	+	0.953905i	$0.597023\pi$
$542$	−62.8585	−	36.2914i	−2.70000	−	1.55885i
$543$	−12.4680	+	21.5953i	−0.535054	+	0.926742i
$544$	−32.3109	+	18.6547i	−1.38532	+	0.799813i
$545$		−	34.6410i		−	1.48386i
$546$		0			0
$547$		0			0		1.00000			$0$
$547$		0			0		−1.00000			$\pi$
$548$	−49.0313	−	84.9246i	−2.09451	−	3.62780i
$549$	−35.1970	−	20.3210i	−1.50217	−	0.867278i
$550$		0			0
$551$		0			0
$552$		−	107.861i		−	4.59087i
$553$		0			0
$554$		0			0
$555$		0			0
$556$	−108.470	−	62.6251i	−4.60015	−	2.65590i
$557$	−18.2832	+	31.6675i	−0.774685	+	1.34179i	0.160287	+	0.987070i	$0.448758\pi$
$557$	−18.2832	+	31.6675i	−0.774685	+	1.34179i	−0.934972	+	0.354723i	$0.884575\pi$
$558$	−29.0299	+	16.7604i	−1.22893	+	0.709525i
$559$		0			0
$560$		0			0
$561$		0			0
$562$		0			0
$563$	2.81702	+	1.62641i	0.118723	+	0.0685449i	0.558186	−	0.829716i	$-0.311498\pi$
$563$	2.81702	+	1.62641i	0.118723	+	0.0685449i	−0.439462	+	0.898261i	$0.644831\pi$
$564$	−64.4838	+	111.689i	−2.71526	+	4.70296i
$565$	−7.40168	+	4.27336i	−0.311391	+	0.179782i
$566$		0			0
$567$		0			0
$568$		0			0
$569$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$569$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$570$	−6.69275	−	3.86406i	−0.280328	−	0.161848i
$571$	19.3649	−	33.5410i	0.810397	−	1.40365i	−0.102190	−	0.994765i	$-0.532585\pi$
$571$	19.3649	−	33.5410i	0.810397	−	1.40365i	0.912587	−	0.408883i	$-0.134082\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−33.1162			−1.38104
$576$	43.6368	+	75.5811i	1.81820	+	3.14921i
$577$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$577$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$578$	19.0788	−	33.0454i	0.793573	−	1.37451i
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$		0			0
$584$		0			0
$585$		0			0
$586$	−32.7442	+	18.9049i	−1.35265	+	0.780953i
$587$		−	44.4925i		−	1.83640i	−0.396116	−	0.918201i	$-0.629642\pi$
$587$		−	44.4925i		−	1.83640i	0.396116	−	0.918201i	$-0.370358\pi$
$588$		0			0
$589$	−2.99444			−0.123384
$590$		0			0
$591$	22.1054	+	12.7626i	0.909296	+	0.524982i
$592$		0			0
$593$	−3.87298	+	2.23607i	−0.159044	+	0.0918243i	−0.577410	−	0.816454i	$-0.695936\pi$
$593$	−3.87298	+	2.23607i	−0.159044	+	0.0918243i	0.418365	+	0.908279i	$0.362603\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	−20.5030	−	35.5122i	−0.839132	−	1.45342i
$598$		0			0
$599$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$599$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$600$	70.5171	−	40.7131i	2.87885	−	1.66210i
$601$		−	43.1673i		−	1.76083i	−0.474202	−	0.880416i	$-0.657263\pi$
$601$		−	43.1673i		−	1.76083i	0.474202	−	0.880416i	$-0.342737\pi$
$602$		0			0
$603$		0			0
$604$	29.3383	+	50.8155i	1.19376	+	2.06765i
$605$	21.3014	+	12.2984i	0.866025	+	0.500000i
$606$		0			0
$607$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$607$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$608$			15.7097i			0.637112i
$609$		0			0
$610$	82.6593			3.34678
$611$		0			0
$612$	24.5715	+	14.1864i	0.993246	+	0.573451i
$613$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$613$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	25.2264			1.01558			0.507788	−	0.861482i	$-0.330464\pi$
$617$	25.2264			1.01558			0.507788	+	0.861482i	$0.330464\pi$
$618$		0			0
$619$	42.9059	+	24.7717i	1.72453	+	0.995660i	0.908800	+	0.417232i	$0.137000\pi$
$619$	42.9059	+	24.7717i	1.72453	+	0.995660i	0.815733	+	0.578428i	$0.196334\pi$
$620$	24.9316	−	43.1829i	1.00128	−	1.73427i
$621$	−29.8046	+	17.2077i	−1.19602	+	0.690520i
$622$		0			0
$623$		0			0
$624$		0			0
$625$	−12.5000	−	21.6506i	−0.500000	−	0.866025i
$626$		0			0
$627$		0			0
$628$		0			0
$629$		0			0
$630$		0			0
$631$	−4.75871			−0.189441			−0.0947206	−	0.995504i	$-0.530196\pi$
$631$	−4.75871			−0.189441			−0.0947206	+	0.995504i	$0.530196\pi$
$632$	78.9365	+	136.722i	3.13993	+	5.43851i
$633$	−11.6190	−	6.70820i	−0.461812	−	0.266627i
$634$	23.2569	−	40.2821i	0.923650	−	1.59981i
$635$		0			0
$636$			136.891i			5.42808i
$637$		0			0
$638$		0			0
$639$		0			0
$640$	−70.5171	−	40.7131i	−2.78743	−	1.60933i
$641$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$641$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$642$	51.3671	−	29.6568i	2.02730	−	1.17046i
$643$		0			0			−	1.00000i	$-0.5\pi$
$643$		0			0				1.00000i	$0.5\pi$
$644$		0			0
$645$		0			0
$646$	1.73270	+	3.00112i	0.0681722	+	0.118078i
$647$	−38.7298	−	22.3607i	−1.52263	−	0.879089i	−0.999642	−	0.0267469i	$-0.991485\pi$
$647$	−38.7298	−	22.3607i	−1.52263	−	0.879089i	−0.522985	−	0.852342i	$-0.675181\pi$
$648$	42.3103	−	73.2835i	1.66210	−	2.87885i
$649$		0			0
$650$		0			0
$651$		0			0
$652$		0			0
$653$	25.3757	+	43.9521i	0.993029	+	1.71998i	0.598590	+	0.801056i	$0.295728\pi$
$653$	25.3757	+	43.9521i	0.993029	+	1.71998i	0.394439	+	0.918922i	$0.370939\pi$
$654$	63.4089	+	36.6091i	2.47948	+	1.43153i
$655$		0			0
$656$		0			0
$657$		0			0
$658$		0			0
$659$		0			0		1.00000			$0$
$659$		0			0		−1.00000			$\pi$
$660$		0			0
$661$	−44.4765	−	25.6785i	−1.72994	−	0.998779i	−0.889665	−	0.456613i	$-0.849062\pi$
$661$	−44.4765	−	25.6785i	−1.72994	−	0.998779i	−0.840271	−	0.542166i	$-0.817604\pi$
$662$	−31.7044	+	54.9137i	−1.23223	+	2.13428i
$663$		0			0
$664$			95.8997i			3.72163i
$665$		0			0
$666$		0			0
$667$		0			0
$668$	42.1824	+	24.3540i	1.63208	+	0.942284i
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$		0			0		1.00000			$0$
$673$		0			0		−1.00000			$\pi$
$674$		0			0
$675$	−22.5000	−	12.9904i	−0.866025	−	0.500000i
$676$	35.3972	−	61.3097i	1.36143	−	2.35807i
$677$	−36.0000	+	20.7846i	−1.38359	+	0.798817i	−0.992583	−	0.121569i	$-0.961207\pi$
$677$	−36.0000	+	20.7846i	−1.38359	+	0.798817i	−0.391009	+	0.920387i	$0.627874\pi$
$678$		−	18.0646i		−	0.693767i
$679$		0			0
$680$	−36.5127			−1.40020
$681$	−15.4919	−	26.8328i	−0.593652	−	1.02824i
$682$		0			0
$683$	−24.4479	+	42.3450i	−0.935474	+	1.62029i	−0.161686	+	0.986842i	$0.551693\pi$
$683$	−24.4479	+	42.3450i	−0.935474	+	1.62029i	−0.773787	+	0.633446i	$0.781640\pi$
$684$	10.3462	−	5.97340i	0.395598	−	0.228399i
$685$		−	40.2655i		−	1.53846i
$686$		0			0
$687$	42.0238			1.60331
$688$		0			0
$689$		0			0
$690$	34.9977	−	60.6178i	1.33234	−	2.30768i
$691$	44.1725	−	25.5030i	1.68040	−	0.970180i	0.719007	−	0.695003i	$-0.244597\pi$
$691$	44.1725	−	25.5030i	1.68040	−	0.970180i	0.961394	−	0.275176i	$-0.0887363\pi$
$692$			139.461i			5.30152i
$693$		0			0
$694$	−11.1605			−0.423646
$695$	−25.7145	−	44.5388i	−0.975407	−	1.68945i
$696$		0			0
$697$		0			0
$698$	−88.2395	+	50.9451i	−3.33991	+	1.92830i
$699$		−	48.0809i		−	1.81859i
$700$		0			0
$701$		0			0		1.00000			$0$
$701$		0			0		−1.00000			$\pi$
$702$		0			0
$703$		0			0
$704$		0			0
$705$	−45.8607	+	26.4777i	−1.72721	+	0.997208i
$706$			56.7146i			2.13448i
$707$		0			0
$708$		0			0
$709$	25.6241	+	44.3822i	0.962332	+	1.66681i	0.716619	+	0.697465i	$0.245689\pi$
$709$	25.6241	+	44.3822i	0.962332	+	1.66681i	0.245713	+	0.969343i	$0.420978\pi$
$710$		0			0
$711$	25.1864	−	43.6241i	0.944564	−	1.63603i
$712$		0			0
$713$		−	27.1213i		−	1.01570i
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$719$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$720$			99.0428i			3.69111i
$721$		0			0
$722$	−50.3858			−1.87517
$723$	8.66827	+	15.0139i	0.322376	+	0.558372i
$724$	67.8974	+	39.2006i	2.52339	+	1.45688i
$725$		0			0
$726$	−45.0233	+	25.9942i	−1.67097	+	0.964736i
$727$		0			0			−	1.00000i	$-0.5\pi$
$727$		0			0				1.00000i	$0.5\pi$
$728$		0			0
$729$	−27.0000			−1.00000
$730$		0			0
$731$		0			0
$732$	−63.8909	+	110.662i	−2.36148	+	4.09020i
$733$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$733$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$734$		0			0
$735$		0			0
$736$	−142.286			−5.24474
$737$		0			0
$738$		0			0
$739$	−2.00000	+	3.46410i	−0.0735712	+	0.127429i	−0.900464	−	0.434930i	$-0.856773\pi$
$739$	−2.00000	+	3.46410i	−0.0735712	+	0.127429i	0.826893	+	0.562360i	$0.190106\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	11.6896			0.428849			0.214425	−	0.976741i	$-0.431212\pi$
$743$	11.6896			0.428849			0.214425	+	0.976741i	$0.431212\pi$
$744$	33.3430	+	57.7517i	1.22241	+	2.11728i
$745$		0			0
$746$		0			0
$747$	26.4994	−	15.2994i	0.969562	−	0.559777i
$748$		0			0
$749$		0			0
$750$	52.8407			1.92947
$751$	−4.00000	−	6.92820i	−0.145962	−	0.252814i	0.783769	−	0.621052i	$-0.213294\pi$
$751$	−4.00000	−	6.92820i	−0.145962	−	0.252814i	−0.929731	+	0.368238i	$0.879961\pi$
$752$	174.828	+	100.937i	6.37533	+	3.68080i
$753$		0			0
$754$		0			0
$755$			24.0932i			0.876843i
$756$		0			0
$757$		0			0		1.00000			$0$
$757$		0			0		−1.00000			$\pi$
$758$	−33.5051	−	58.0326i	−1.21696	−	2.10784i
$759$		0			0
$760$	−7.68711	+	13.3145i	−0.278841	+	0.482966i
$761$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$761$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$	5.82508	+	10.0893i	0.210606	+	0.364780i
$766$	48.7269	+	28.1325i	1.76058	+	1.01647i
$767$		0			0
$768$	61.7735	−	35.6650i	2.22906	−	1.28695i
$769$		−	11.0219i		−	0.397461i	−0.980054	−	0.198730i	$-0.936318\pi$
$769$		−	11.0219i		−	0.397461i	0.980054	−	0.198730i	$-0.0636818\pi$
$770$		0			0
$771$	−12.0000			−0.432169
$772$		0			0
$773$	−31.2025	−	18.0148i	−1.12228	−	0.647947i	−0.180295	−	0.983613i	$-0.557705\pi$
$773$	−31.2025	−	18.0148i	−1.12228	−	0.647947i	−0.941981	+	0.335666i	$0.891039\pi$
$774$		0			0
$775$	17.7313	−	10.2372i	0.636928	−	0.367730i
$776$		0			0
$777$		0			0
$778$		0			0
$779$		0			0
$780$		0			0
$781$		0			0
$782$	−27.1819	+	15.6935i	−0.972022	+	0.561197i
$783$		0			0
$784$		0			0
$785$		0			0
$786$		0			0
$787$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$787$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$788$	40.1266	−	69.5014i	1.42945	−	2.47588i
$789$	−30.2915	+	17.4888i	−1.07841	+	0.622618i
$790$			102.450i			3.64502i
$791$		0			0
$792$		0			0
$793$		0			0
$794$		0			0
$795$	−28.1044	+	48.6783i	−0.996761	+	1.72644i
$796$	−111.654	+	64.4632i	−3.95746	+	2.28484i
$797$		−	15.1891i		−	0.538027i	−0.963136	−	0.269013i	$-0.913302\pi$
$797$		−	15.1891i		−	0.538027i	0.963136	−	0.269013i	$-0.0866976\pi$
$798$		0			0
$799$	23.7460			0.840072
$800$	−53.7072	−	93.0236i	−1.89884	−	3.28888i
$801$		0			0
$802$		0			0
$803$		0			0
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$809$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$810$	47.5567	−	27.4569i	1.67097	−	0.964736i
$811$			1.56948i			0.0551120i	0.999620	+	0.0275560i	$0.00877245\pi$
$811$			1.56948i			0.0551120i	−0.999620	+	0.0275560i	$0.991228\pi$
$812$		0			0
$813$	46.0724			1.61583
$814$		0			0
$815$		0			0
$816$	22.2061	−	38.4621i	0.777369	−	1.34644i
$817$		0			0
$818$		−	97.8994i		−	3.42297i
$819$		0			0
$820$		0			0
$821$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$821$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$822$	73.7042	+	42.5532i	2.57073	+	1.48421i
$823$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$823$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−15.8202			−0.550123			−0.275061	−	0.961427i	$-0.588698\pi$
$827$	−15.8202			−0.550123			−0.275061	+	0.961427i	$0.588698\pi$
$828$	54.1024	+	93.7082i	1.88019	+	3.25658i
$829$	−25.9174	−	14.9634i	−0.900148	−	0.519700i	−0.0228995	−	0.999738i	$-0.507290\pi$
$829$	−25.9174	−	14.9634i	−0.900148	−	0.519700i	−0.877248	+	0.480037i	$0.840623\pi$
$830$	−31.1166	+	53.8956i	−1.08007	+	1.87074i
$831$		0			0
$832$		0			0
$833$		0			0
$834$	108.702			3.76404
$835$	10.0000	+	17.3205i	0.346064	+	0.599401i
$836$		0			0
$837$	10.6388	−	18.4269i	0.367730	−	0.636928i
$838$		0			0
$839$		0			0			−	1.00000i	$-0.5\pi$
$839$		0			0				1.00000i	$0.5\pi$
$840$		0			0
$841$	29.0000			1.00000
$842$	−21.7143	−	37.6103i	−0.748325	−	1.29614i
$843$		0			0
$844$	−21.0912	+	36.5310i	−0.725988	+	1.25745i
$845$	25.1744	−	14.5344i	0.866025	−	0.500000i
$846$		−	111.928i		−	3.84817i
$847$		0			0
$848$	214.277			7.35830
$849$		0			0
$850$	−20.5201	−	11.8473i	−0.703833	−	0.406358i
$851$		0			0
$852$		0			0
$853$		0			0			−	1.00000i	$-0.5\pi$
$853$		0			0				1.00000i	$0.5\pi$
$854$		0			0
$855$	4.90547			0.167764
$856$	−58.9988	−	102.189i	−2.01654	−	3.49275i
$857$	−39.8446	−	23.0043i	−1.36107	−	0.785812i	−0.371300	−	0.928513i	$-0.621088\pi$
$857$	−39.8446	−	23.0043i	−1.36107	−	0.785812i	−0.989766	+	0.142701i	$0.954421\pi$
$858$		0			0
$859$	45.5701	−	26.3099i	1.55483	−	0.897682i	0.557093	−	0.830450i	$-0.311917\pi$
$859$	45.5701	−	26.3099i	1.55483	−	0.897682i	0.997737	−	0.0672316i	$-0.0214166\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−13.3675	−	23.1531i	−0.455034	−	0.788142i	0.543656	−	0.839308i	$-0.317040\pi$
$863$	−13.3675	−	23.1531i	−0.455034	−	0.788142i	−0.998690	+	0.0511658i	$0.983706\pi$
$864$	−96.6730	−	55.8142i	−3.28888	−	1.89884i
$865$	−28.6321	+	49.5923i	−0.973521	+	1.68619i
$866$		0			0
$867$			24.2208i			0.822580i
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$	72.8298	−	126.145i	2.46633	−	4.27180i
$873$		0			0
$874$			13.2160i			0.447036i
$875$		0			0
$876$		0			0
$877$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$877$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$878$	99.0253	+	57.1723i	3.34194	+	1.92947i
$879$	12.0000	−	20.7846i	0.404750	−	0.701047i
$880$		0			0
$881$		0			0			−	1.00000i	$-0.5\pi$
$881$		0			0				1.00000i	$0.5\pi$
$882$		0			0
$883$		0			0		1.00000			$0$
$883$		0			0		−1.00000			$\pi$
$884$		0			0
$885$		0			0
$886$	−42.4434	+	73.5141i	−1.42591	+	2.46976i
$887$	−41.5397	+	23.9829i	−1.39477	+	0.805268i	−0.993838	−	0.110841i	$-0.964645\pi$
$887$	−41.5397	+	23.9829i	−1.39477	+	0.805268i	−0.400928	+	0.916110i	$0.631312\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$		0			0
$892$		0			0
$893$	4.99930	−	8.65905i	0.167295	−	0.289764i
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$		0			0
$899$		0			0
$900$	−40.8429	+	70.7420i	−1.36143	+	2.35807i
$901$	21.8281	−	12.6024i	0.727198	−	0.419848i
$902$		0			0
$903$		0			0
$904$	−35.9375			−1.19526
$905$	16.0962	+	27.8794i	0.535054	+	0.926742i
$906$	−44.1016	−	25.4621i	−1.46518	−	0.845921i
$907$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$907$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$908$	−84.3647	+	48.7080i	−2.79974	+	1.61643i
$909$		0			0
$910$		0			0
$911$		0			0		1.00000			$0$
$911$		0			0		−1.00000			$\pi$
$912$	−9.35022	−	16.1951i	−0.309617	−	0.536272i
$913$		0			0
$914$		0			0
$915$	−45.4391	+	26.2343i	−1.50217	+	0.867278i
$916$		−	132.126i		−	4.36558i
$917$		0			0
$918$	−24.6241			−0.812717
$919$	11.6190	+	20.1246i	0.383274	+	0.663850i	0.991528	−	0.129893i	$-0.0414632\pi$
$919$	11.6190	+	20.1246i	0.383274	+	0.663850i	−0.608254	+	0.793742i	$0.708130\pi$
$920$	−120.592	−	69.6239i	−3.97581	−	2.29543i
$921$		0			0
$922$		0			0
$923$		0			0
$924$		0			0
$925$		0			0
$926$		0			0
$927$		0			0
$928$		0			0
$929$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$929$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$930$			43.2752i			1.41905i
$931$		0			0
$932$	−151.171			−4.95176
$933$		0			0
$934$	−84.5452	−	48.8122i	−2.76640	−	1.59718i
$935$		0			0
$936$		0			0
$937$		0			0			−	1.00000i	$-0.5\pi$
$937$		0			0				1.00000i	$0.5\pi$
$938$		0			0
$939$		0			0
$940$	83.2482	+	144.190i	2.71526	+	4.70296i
$941$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$941$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−3.00463	−	5.20417i	−0.0976374	−	0.169113i	0.813069	−	0.582168i	$-0.197795\pi$
$947$	−3.00463	−	5.20417i	−0.0976374	−	0.169113i	−0.910706	+	0.413055i	$0.864462\pi$
$948$	−137.158	−	79.1883i	−4.45469	−	2.57191i
$949$		0			0
$950$	−8.64030	+	4.98848i	−0.280328	+	0.161848i
$951$			29.5250i			0.957412i
$952$		0			0
$953$	61.6662			1.99756			0.998782	−	0.0493329i	$-0.0157095\pi$
$953$	61.6662			1.99756			0.998782	+	0.0493329i	$0.0157095\pi$
$954$	−59.4024	−	102.888i	−1.92322	−	3.33112i
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$	112.670			3.63640
$961$	−7.11601	−	12.3253i	−0.229549	−	0.397590i
$962$		0			0
$963$	−18.8248	+	32.6056i	−0.606622	+	1.05070i
$964$	47.2050	−	27.2538i	1.52037	−	0.877786i
$965$		0			0
$966$		0			0
$967$		0			0		1.00000			$0$
$967$		0			0		−1.00000			$\pi$
$968$	51.7126	+	89.5688i	1.66210	+	2.87885i
$969$	−1.90498	−	1.09984i	−0.0611969	−	0.0353320i
$970$		0			0
$971$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$971$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$972$			84.8904i			2.72286i
$973$		0			0
$974$		0			0
$975$		0			0
$976$	173.221	+	100.009i	5.54467	+	3.20121i
$977$	−29.6594	+	51.3716i	−0.948889	+	1.64352i	−0.201118	+	0.979567i	$0.564457\pi$
$977$	−29.6594	+	51.3716i	−0.948889	+	1.64352i	−0.747771	+	0.663957i	$0.768876\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$	−46.4758			−1.48386
$982$		0			0
$983$	−3.00000	−	1.73205i	−0.0956851	−	0.0552438i	0.451394	−	0.892325i	$-0.350927\pi$
$983$	−3.00000	−	1.73205i	−0.0956851	−	0.0552438i	−0.547079	+	0.837081i	$0.684260\pi$
$984$		0			0
$985$	28.5380	−	16.4764i	0.909296	−	0.524982i
$986$		0			0
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$	−16.0000	+	27.7128i	−0.508257	+	0.880327i	0.491698	+	0.870766i	$0.336377\pi$
$991$	−16.0000	+	27.7128i	−0.508257	+	0.880327i	−0.999954	+	0.00956046i	$0.996957\pi$
$992$	76.1840	−	43.9848i	2.41884	−	1.39652i
$993$		−	40.2492i		−	1.27727i
$994$		0			0
$995$	−52.9385			−1.67826
$996$	−48.1028	−	83.3164i	−1.52419	−	2.63998i
$997$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$997$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$998$	49.9212	−	86.4660i	1.58023	−	2.73703i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	735.2.p.d.374.8		16
3.2	odd	2		735.2.p.e.374.1		16
5.4	even	2		735.2.p.e.374.1		16
7.2	even	3		735.2.p.e.509.8		16
7.3	odd	6		735.2.g.a.734.1	✓	16
7.4	even	3		735.2.g.a.734.2	yes	16
7.5	odd	6	inner	735.2.p.d.509.8		16
7.6	odd	2		735.2.p.e.374.8		16
15.14	odd	2	CM	735.2.p.d.374.8		16
21.2	odd	6	inner	735.2.p.d.509.1		16
21.5	even	6		735.2.p.e.509.1		16
21.11	odd	6		735.2.g.a.734.15	yes	16
21.17	even	6		735.2.g.a.734.16	yes	16
21.20	even	2	inner	735.2.p.d.374.1		16
35.4	even	6		735.2.g.a.734.15	yes	16
35.9	even	6	inner	735.2.p.d.509.1		16
35.19	odd	6		735.2.p.e.509.1		16
35.24	odd	6		735.2.g.a.734.16	yes	16
35.34	odd	2	inner	735.2.p.d.374.1		16
105.44	odd	6		735.2.p.e.509.8		16
105.59	even	6		735.2.g.a.734.1	✓	16
105.74	odd	6		735.2.g.a.734.2	yes	16
105.89	even	6	inner	735.2.p.d.509.8		16
105.104	even	2		735.2.p.e.374.8		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
735.2.g.a.734.1	✓	16	7.3	odd	6
735.2.g.a.734.1	✓	16	105.59	even	6
735.2.g.a.734.2	yes	16	7.4	even	3
735.2.g.a.734.2	yes	16	105.74	odd	6
735.2.g.a.734.15	yes	16	21.11	odd	6
735.2.g.a.734.15	yes	16	35.4	even	6
735.2.g.a.734.16	yes	16	21.17	even	6
735.2.g.a.734.16	yes	16	35.24	odd	6
735.2.p.d.374.1		16	21.20	even	2	inner
735.2.p.d.374.1		16	35.34	odd	2	inner
735.2.p.d.374.8		16	1.1	even	1	trivial
735.2.p.d.374.8		16	15.14	odd	2	CM
735.2.p.d.509.1		16	21.2	odd	6	inner
735.2.p.d.509.1		16	35.9	even	6	inner
735.2.p.d.509.8		16	7.5	odd	6	inner
735.2.p.d.509.8		16	105.89	even	6	inner
735.2.p.e.374.1		16	3.2	odd	2
735.2.p.e.374.1		16	5.4	even	2
735.2.p.e.374.8		16	7.6	odd	2
735.2.p.e.374.8		16	105.104	even	2
735.2.p.e.509.1		16	21.5	even	6
735.2.p.e.509.1		16	35.19	odd	6
735.2.p.e.509.8		16	7.2	even	3
735.2.p.e.509.8		16	105.44	odd	6

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:	\( N \)	\(=\)	\( 735 = 3 \cdot 5 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	735.p (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(1.36434 + 2.36311i) q^{2} +(-1.50000 - 0.866025i) q^{3} +(-2.72286 + 4.71613i) q^{4} +(-1.93649 + 1.11803i) q^{5} -4.72622i q^{6} -9.40228 q^{8} +(1.50000 + 2.59808i) q^{9} +(-5.28407 - 3.05076i) q^{10} +(8.16858 - 4.71613i) q^{12} +3.87298 q^{15} +(-7.38221 - 12.7864i) q^{16} +(-1.50403 - 0.868351i) q^{17} +(-4.09303 + 7.08933i) q^{18} +(-0.633294 + 0.365633i) q^{19} -12.1770i q^{20} +(-3.31162 - 5.73589i) q^{23} +(14.1034 + 8.14261i) q^{24} +(2.50000 - 4.33013i) q^{25} -5.19615i q^{27} +(5.28407 + 9.15229i) q^{30} +(3.54626 + 2.04744i) q^{31} +(10.7414 - 18.6047i) q^{32} -4.73891i q^{34} -16.3372 q^{36} +(-1.72806 - 0.997696i) q^{38} +(18.2074 - 10.5121i) q^{40} +(-5.80948 - 3.35410i) q^{45} +(9.03636 - 15.6514i) q^{46} +(-11.8412 + 6.83651i) q^{47} +25.5727i q^{48} +13.6434 q^{50} +(1.50403 + 2.60505i) q^{51} +(-7.25653 + 12.5687i) q^{53} +(12.2791 - 7.08933i) q^{54} +1.26659 q^{57} +(-10.5456 + 18.2655i) q^{60} +(-11.7323 + 6.77366i) q^{61} +11.1736i q^{62} +29.0912 q^{64} +(8.19051 - 4.72880i) q^{68} +11.4718i q^{69} +(-14.1034 - 24.4278i) q^{72} +(-7.50000 + 4.33013i) q^{75} -3.98226i q^{76} +(-8.39547 - 14.5414i) q^{79} +(28.5912 + 16.5071i) q^{80} +(-4.50000 + 7.79423i) q^{81} -10.1996i q^{83} +3.88338 q^{85} -18.3046i q^{90} +36.0683 q^{92} +(-3.54626 - 6.14231i) q^{93} +(-32.3109 - 18.6547i) q^{94} +(0.817579 - 1.41609i) q^{95} +(-32.2243 + 18.6047i) q^{96} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 24 q^{3} - 16 q^{4} + 24 q^{9} + 48 q^{12} - 32 q^{16} + 40 q^{25} - 96 q^{36} + 128 q^{64} - 72 q^{68} - 120 q^{75} + 120 q^{80} - 72 q^{81}+O(q^{100}) \) 16 * q - 24 * q^3 - 16 * q^4 + 24 * q^9 + 48 * q^12 - 32 * q^16 + 40 * q^25 - 96 * q^36 + 128 * q^64 - 72 * q^68 - 120 * q^75 + 120 * q^80 - 72 * q^81

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