LMFDB - Embedded newform 735.2.p.d.509.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [735,2,Mod(374,735)]

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

lf = mfeigenbasis(mf)

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")

//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);

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

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			509.2
Root			$-1.36434 + 1.59774i$ of defining polynomial
Character	$\chi$	$=$	735.509
Dual form			735.2.p.d.374.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+(-1.22796 + 2.12688i) q^{2} +(-1.50000 + 0.866025i) q^{3} +(-2.01575 - 3.49139i) q^{4} +(1.93649 + 1.11803i) q^{5} -4.25377i q^{6} +4.98920 q^{8} +(1.50000 - 2.59808i) q^{9} +(-4.75585 + 2.74579i) q^{10} +(6.04726 + 3.49139i) q^{12} -3.87298 q^{15} +(-2.09501 + 3.62867i) q^{16} +(6.98125 - 4.03063i) q^{17} +(3.68387 + 6.38065i) q^{18} +(4.87192 + 2.81280i) q^{19} -9.01472i q^{20} +(-4.79454 + 8.30439i) q^{23} +(-7.48380 + 4.32077i) q^{24} +(2.50000 + 4.33013i) q^{25} +5.19615i q^{27} +(4.75585 - 8.23738i) q^{30} +(3.83371 - 2.21339i) q^{31} +(-0.155971 - 0.270150i) q^{32} +19.7977i q^{34} -12.0945 q^{36} +(-11.9650 + 6.90800i) q^{38} +(9.66155 + 5.57810i) q^{40} +(5.80948 - 3.35410i) q^{45} +(-11.7750 - 20.3949i) q^{46} +(0.886735 + 0.511957i) q^{47} -7.25734i q^{48} -12.2796 q^{50} +(-6.98125 + 12.0919i) q^{51} +(-4.71717 - 8.17037i) q^{53} +(-11.0516 - 6.38065i) q^{54} -9.74384 q^{57} +(7.80698 + 13.5221i) q^{60} +(3.53403 + 2.04037i) q^{61} +10.8718i q^{62} -7.61396 q^{64} +(-28.1450 - 16.2495i) q^{68} -16.6088i q^{69} +(7.48380 - 12.9623i) q^{72} +(-7.50000 - 4.33013i) q^{75} -22.6797i q^{76} +(2.91824 - 5.05454i) q^{79} +(-8.11396 + 4.68459i) q^{80} +(-4.50000 - 7.79423i) q^{81} +15.0986i q^{83} +18.0255 q^{85} +16.4748i q^{90} +38.6584 q^{92} +(-3.83371 + 6.64017i) q^{93} +(-2.17774 + 1.25732i) q^{94} +(6.28962 + 10.8939i) q^{95} +(0.467913 + 0.270150i) q^{96} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $\left(\mathbb{Z}/735\mathbb{Z}\right)^\times$.

$n$	$346$	$442$	$491$
$\chi(n)$	$e\left(\frac{5}{6}\right)$	$-1$	$-1$

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 $

$2$	−1.22796	+	2.12688i	−0.868296	+	1.50393i	−0.00455955	+	0.999990i	$0.501451\pi$
$2$	−1.22796	+	2.12688i	−0.868296	+	1.50393i	−0.863737	+	0.503943i	$0.831882\pi$
$3$	−1.50000	+	0.866025i	−0.866025	+	0.500000i
$3$	−1.50000	+	0.866025i	−0.866025	+	0.500000i
$4$	−2.01575	−	3.49139i	−1.00788	−	1.74569i
$5$	1.93649	+	1.11803i	0.866025	+	0.500000i
$5$	1.93649	+	1.11803i	0.866025	+	0.500000i
$6$		−	4.25377i		−	1.73659i
$7$		0			0
$7$		0			0
$8$	4.98920			1.76395
$9$	1.50000	−	2.59808i	0.500000	−	0.866025i
$10$	−4.75585	+	2.74579i	−1.50393	+	0.868296i
$11$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$11$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$12$	6.04726	+	3.49139i	1.74569	+	1.00788i
$13$		0			0			−	1.00000i	$-0.5\pi$
$13$		0			0				1.00000i	$0.5\pi$
$14$		0			0
$15$	−3.87298			−1.00000
$16$	−2.09501	+	3.62867i	−0.523754	+	0.907168i
$17$	6.98125	−	4.03063i	1.69320	−	0.977571i	0.741298	−	0.671176i	$-0.234210\pi$
$17$	6.98125	−	4.03063i	1.69320	−	0.977571i	0.951904	−	0.306395i	$-0.0991229\pi$
$18$	3.68387	+	6.38065i	0.868296	+	1.50393i
$19$	4.87192	+	2.81280i	1.11769	+	0.645301i	0.940811	−	0.338931i	$-0.110065\pi$
$19$	4.87192	+	2.81280i	1.11769	+	0.645301i	0.176883	+	0.984232i	$0.443398\pi$
$20$		−	9.01472i		−	2.01575i
$21$		0			0
$22$		0			0
$23$	−4.79454	+	8.30439i	−0.999731	+	1.73159i	−0.479784	+	0.877387i	$0.659285\pi$
$23$	−4.79454	+	8.30439i	−0.999731	+	1.73159i	−0.519947	+	0.854199i	$0.674048\pi$
$24$	−7.48380	+	4.32077i	−1.52762	+	0.881974i
$25$	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			$\pi$
$30$	4.75585	−	8.23738i	0.868296	−	1.50393i
$31$	3.83371	−	2.21339i	0.688554	−	0.397537i	−0.114516	−	0.993421i	$-0.536532\pi$
$31$	3.83371	−	2.21339i	0.688554	−	0.397537i	0.803070	+	0.595885i	$0.203198\pi$
$32$	−0.155971	−	0.270150i	−0.0275720	−	0.0477561i
$33$		0			0
$34$			19.7977i			3.39528i
$35$		0			0
$36$	−12.0945			−2.01575
$37$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$37$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$38$	−11.9650	+	6.90800i	−1.94098	+	1.12063i
$39$		0			0
$40$	9.66155	+	5.57810i	1.52762	+	0.881974i
$41$		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
$46$	−11.7750	−	20.3949i	−1.73613	−	3.00706i
$47$	0.886735	+	0.511957i	0.129344	+	0.0746766i	0.563276	−	0.826269i	$-0.309541\pi$
$47$	0.886735	+	0.511957i	0.129344	+	0.0746766i	−0.433932	+	0.900946i	$0.642874\pi$
$48$		−	7.25734i		−	1.04751i
$49$		0			0
$50$	−12.2796			−1.73659
$51$	−6.98125	+	12.0919i	−0.977571	+	1.69320i
$52$		0			0
$53$	−4.71717	−	8.17037i	−0.647953	−	1.12229i	−0.983611	−	0.180303i	$-0.942292\pi$
$53$	−4.71717	−	8.17037i	−0.647953	−	1.12229i	0.335659	−	0.941984i	$-0.391041\pi$
$54$	−11.0516	−	6.38065i	−1.50393	−	0.868296i
$55$		0			0
$56$		0			0
$57$	−9.74384			−1.29060
$58$		0			0
$59$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$59$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$60$	7.80698	+	13.5221i	1.00788	+	1.74569i
$61$	3.53403	+	2.04037i	0.452486	+	0.261243i	0.708880	−	0.705329i	$-0.249201\pi$
$61$	3.53403	+	2.04037i	0.452486	+	0.261243i	−0.256393	+	0.966573i	$0.582534\pi$
$62$			10.8718i			1.38072i
$63$		0			0
$64$	−7.61396			−0.951744
$65$		0			0
$66$		0			0
$67$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$67$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$68$	−28.1450	−	16.2495i	−3.41308	−	1.97054i
$69$		−	16.6088i		−	1.99946i
$70$		0			0
$71$		0			0		1.00000			$0$
$71$		0			0		−1.00000			$\pi$
$72$	7.48380	−	12.9623i	0.881974	−	1.52762i
$73$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$73$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$74$		0			0
$75$	−7.50000	−	4.33013i	−0.866025	−	0.500000i
$76$		−	22.6797i		−	2.60154i
$77$		0			0
$78$		0			0
$79$	2.91824	−	5.05454i	0.328328	−	0.568680i	−0.653852	−	0.756622i	$-0.726848\pi$
$79$	2.91824	−	5.05454i	0.328328	−	0.568680i	0.982180	+	0.187942i	$0.0601816\pi$
$80$	−8.11396	+	4.68459i	−0.907168	+	0.523754i
$81$	−4.50000	−	7.79423i	−0.500000	−	0.866025i
$82$		0			0
$83$			15.0986i			1.65729i	0.559777	+	0.828643i	$0.310887\pi$
$83$			15.0986i			1.65729i	−0.559777	+	0.828643i	$0.689113\pi$
$84$		0			0
$85$	18.0255			1.95514
$86$		0			0
$87$		0			0
$88$		0			0
$89$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$89$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$90$			16.4748i			1.73659i
$91$		0			0
$92$	38.6584			4.03042
$93$	−3.83371	+	6.64017i	−0.397537	+	0.688554i
$94$	−2.17774	+	1.25732i	−0.224617	+	0.129683i
$95$	6.28962	+	10.8939i	0.645301	+	1.11769i
$96$	0.467913	+	0.270150i	0.0477561	+	0.0275720i
$97$		0			0			−	1.00000i	$-0.5\pi$
$97$		0			0				1.00000i	$0.5\pi$
$98$		0			0
$99$		0			0
$100$	10.0788	−	17.4569i	1.00788	−	1.74569i
$101$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$101$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$102$	−17.1453	−	29.6966i	−1.69764	−	2.94040i
$103$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$103$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$104$		0			0
$105$		0			0
$106$	23.1699			2.25046
$107$	1.37779	−	2.38641i	0.133196	−	0.230703i	−0.791711	−	0.610896i	$-0.790809\pi$
$107$	1.37779	−	2.38641i	0.133196	−	0.230703i	0.924907	+	0.380193i	$0.124143\pi$
$108$	18.1418	−	10.4742i	1.74569	−	1.00788i
$109$	7.74597	+	13.4164i	0.741929	+	1.28506i	0.951616	+	0.307290i	$0.0994222\pi$
$109$	7.74597	+	13.4164i	0.741929	+	1.28506i	−0.209687	+	0.977769i	$0.567244\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−17.4911			−1.64542			−0.822710	−	0.568461i	$-0.807539\pi$
$113$	−17.4911			−1.64542			−0.822710	+	0.568461i	$0.807539\pi$
$114$	11.9650	−	20.7240i	1.12063	−	1.94098i
$115$	−18.5692	+	10.7209i	−1.73159	+	0.999731i
$116$		0			0
$117$		0			0
$118$		0			0
$119$		0			0
$120$	−19.3231			−1.76395
$121$	−5.50000	+	9.52628i	−0.500000	+	0.866025i
$122$	−8.67927	+	5.01098i	−0.785784	+	0.453673i
$123$		0			0
$124$	−15.4556	−	8.92330i	−1.38795	−	0.801336i
$125$			11.1803i			1.00000i
$126$		0			0
$127$		0			0		1.00000			$0$
$127$		0			0		−1.00000			$\pi$
$128$	9.66155	−	16.7343i	0.853968	−	1.47912i
$129$		0			0
$130$		0			0
$131$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$131$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$	−5.80948	+	10.0623i	−0.500000	+	0.866025i
$136$	34.8309	−	20.1096i	2.98672	−	1.72439i
$137$	1.07812	+	1.86736i	0.0921099	+	0.159539i	0.908399	−	0.418105i	$-0.137306\pi$
$137$	1.07812	+	1.86736i	0.0921099	+	0.159539i	−0.816289	+	0.577644i	$0.803972\pi$
$138$	35.3249	+	20.3949i	3.00706	+	1.73613i
$139$			12.5883i			1.06772i	0.845572	+	0.533862i	$0.179260\pi$
$139$			12.5883i			1.06772i	−0.845572	+	0.533862i	$0.820740\pi$
$140$		0			0
$141$	−1.77347			−0.149353
$142$		0			0
$143$		0			0
$144$	6.28504	+	10.8860i	0.523754	+	0.907168i
$145$		0			0
$146$		0			0
$147$		0			0
$148$		0			0
$149$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$149$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$150$	18.4193	−	10.6344i	1.50393	−	0.868296i
$151$	11.0443	+	19.1292i	0.898770	+	1.55671i	0.829069	+	0.559147i	$0.188871\pi$
$151$	11.0443	+	19.1292i	0.898770	+	1.55671i	0.0697008	+	0.997568i	$0.477796\pi$
$152$	24.3070	+	14.0336i	1.97156	+	1.13828i
$153$		−	24.1838i		−	1.95514i
$154$		0			0
$155$	9.89859			0.795074
$156$		0			0
$157$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$157$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$158$	7.16695	+	12.4135i	0.570172	+	0.987566i
$159$	14.1515	+	8.17037i	1.12229	+	0.647953i
$160$		−	0.697523i		−	0.0551440i
$161$		0			0
$162$	22.1032			1.73659
$163$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$163$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$164$		0			0
$165$		0			0
$166$	−32.1130	−	18.5404i	−2.49245	−	1.43902i
$167$		−	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
$170$	−22.1345	+	38.3382i	−1.69764	+	2.94040i
$171$	14.6158	−	8.43841i	1.11769	−	0.645301i
$172$		0			0
$173$	5.20778	+	3.00671i	0.395940	+	0.228596i	0.684731	−	0.728796i	$-0.259920\pi$
$173$	5.20778	+	3.00671i	0.395940	+	0.228596i	−0.288790	+	0.957392i	$0.593253\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		0			0
$178$		0			0
$179$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$179$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$180$	−23.4209	−	13.5221i	−1.74569	−	1.00788i
$181$		−	5.89364i		−	0.438071i	−0.975717	−	0.219036i	$-0.929709\pi$
$181$		−	5.89364i		−	0.438071i	0.975717	−	0.219036i	$-0.0702911\pi$
$182$		0			0
$183$	−7.06806			−0.522486
$184$	−23.9209	+	41.4323i	−1.76347	+	3.05443i
$185$		0			0
$186$	−9.41525	−	16.3077i	−0.690359	−	1.19574i
$187$		0			0
$188$		−	4.12792i		−	0.301059i
$189$		0			0
$190$	−30.8935			−2.24125
$191$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$191$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$192$	11.4209	−	6.59388i	0.824235	−	0.475872i
$193$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$193$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	27.3147			1.94609			0.973046	−	0.230610i	$-0.0740720\pi$
$197$	27.3147			1.94609			0.973046	+	0.230610i	$0.0740720\pi$
$198$		0			0
$199$	23.8953	−	13.7960i	1.69390	−	0.977971i	0.742577	−	0.669761i	$-0.233603\pi$
$199$	23.8953	−	13.7960i	1.69390	−	0.977971i	0.951318	−	0.308210i	$-0.0997300\pi$
$200$	12.4730	+	21.6039i	0.881974	+	1.52762i
$201$		0			0
$202$		0			0
$203$		0			0
$204$	56.2899			3.94108
$205$		0			0
$206$		0			0
$207$	14.3836	+	24.9132i	0.999731	+	1.73159i
$208$		0			0
$209$		0			0
$210$		0			0
$211$	−7.74597			−0.533254			−0.266627	−	0.963800i	$-0.585909\pi$
$211$	−7.74597			−0.533254			−0.266627	+	0.963800i	$0.585909\pi$
$212$	−19.0173	+	32.9389i	−1.30611	+	2.26225i
$213$		0			0
$214$	3.38374	+	5.86081i	0.231308	+	0.400637i
$215$		0			0
$216$			25.9246i			1.76395i
$217$		0			0
$218$	−38.0468			−2.57686
$219$		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
$226$	21.4783	−	37.2014i	1.42871	−	2.47460i
$227$	−15.4919	+	8.94427i	−1.02824	+	0.593652i	−0.916479	−	0.400083i	$-0.868981\pi$
$227$	−15.4919	+	8.94427i	−1.02824	+	0.593652i	−0.111757	+	0.993736i	$0.535648\pi$
$228$	19.6412	+	34.0195i	1.30077	+	2.25300i
$229$	25.9369	+	14.9747i	1.71396	+	0.989555i	0.929057	+	0.369936i	$0.120620\pi$
$229$	25.9369	+	14.9747i	1.71396	+	0.989555i	0.784903	+	0.619619i	$0.212713\pi$
$230$		−	52.6593i		−	3.47225i
$231$		0			0
$232$		0			0
$233$	14.3063	−	24.7792i	0.937234	−	1.62334i	0.166632	−	0.986019i	$-0.446711\pi$
$233$	14.3063	−	24.7792i	0.937234	−	1.62334i	0.770602	−	0.637317i	$-0.219956\pi$
$234$		0			0
$235$	1.14477	+	1.98280i	0.0746766	+	0.129344i
$236$		0			0
$237$			10.1091i			0.656656i
$238$		0			0
$239$		0			0		1.00000			$0$
$239$		0			0		−1.00000			$\pi$
$240$	8.11396	−	14.0538i	0.523754	−	0.907168i
$241$	−24.1275	+	13.9300i	−1.55419	+	0.897310i	−0.556393	+	0.830919i	$0.687815\pi$
$241$	−24.1275	+	13.9300i	−1.55419	+	0.897310i	−0.997794	+	0.0663905i	$0.978852\pi$
$242$	−13.5075	−	23.3957i	−0.868296	−	1.50393i
$243$	13.5000	+	7.79423i	0.866025	+	0.500000i
$244$		−	16.4516i		−	1.05320i
$245$		0			0
$246$		0			0
$247$		0			0
$248$	19.1271	−	11.0431i	1.21457	−	0.701235i
$249$	−13.0758	−	22.6479i	−0.828643	−	1.43525i
$250$	−23.7793	−	13.7290i	−1.50393	−	0.868296i
$251$		0			0			−	1.00000i	$-0.5\pi$
$251$		0			0				1.00000i	$0.5\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$	−27.0383	+	15.6106i	−1.69320	+	0.977571i
$256$	16.1140	+	27.9102i	1.00712	+	1.74439i
$257$	6.00000	+	3.46410i	0.374270	+	0.216085i	0.675322	−	0.737523i	$-0.264005\pi$
$257$	6.00000	+	3.46410i	0.374270	+	0.216085i	−0.301052	+	0.953608i	$0.597338\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−16.1133	−	27.9090i	−0.993587	−	1.72094i	−0.594717	−	0.803935i	$-0.702736\pi$
$263$	−16.1133	−	27.9090i	−0.993587	−	1.72094i	−0.398870	−	0.917008i	$-0.630597\pi$
$264$		0			0
$265$		−	21.0958i		−	1.29591i
$266$		0			0
$267$		0			0
$268$		0			0
$269$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$269$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$270$	−14.2676	−	24.7121i	−0.868296	−	1.50393i
$271$	−4.40766	−	2.54476i	−0.267746	−	0.154583i	0.360117	−	0.932907i	$-0.382737\pi$
$271$	−4.40766	−	2.54476i	−0.267746	−	0.154583i	−0.627863	+	0.778324i	$0.716070\pi$
$272$			33.7769i			2.04803i
$273$		0			0
$274$	−5.29553			−0.319915
$275$		0			0
$276$	−57.9877	+	33.4792i	−3.49045	+	2.01521i
$277$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$277$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$278$	−26.7738	−	15.4579i	−1.60579	−	0.927100i
$279$		−	13.2803i		−	0.795074i
$280$		0			0
$281$		0			0		1.00000			$0$
$281$		0			0		−1.00000			$\pi$
$282$	2.17774	−	3.77196i	0.129683	−	0.224617i
$283$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$283$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$284$		0			0
$285$	−18.8689	−	10.8939i	−1.11769	−	0.645301i
$286$		0			0
$287$		0			0
$288$	−0.935825			−0.0551440
$289$	23.9919	−	41.5552i	1.41129	−	2.44443i
$290$		0			0
$291$		0			0
$292$		0			0
$293$		−	13.8564i		−	0.809500i	−0.914427	−	0.404750i	$-0.867359\pi$
$293$		−	13.8564i		−	0.809500i	0.914427	−	0.404750i	$-0.132641\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$		0			0
$300$			34.9139i			2.01575i
$301$		0			0
$302$	−54.2475			−3.12159
$303$		0			0
$304$	−20.4135	+	11.7857i	−1.17079	+	0.675958i
$305$	4.56241	+	7.90233i	0.261243	+	0.452486i
$306$	51.4360	+	29.6966i	2.94040	+	1.69764i
$307$		0			0			−	1.00000i	$-0.5\pi$
$307$		0			0				1.00000i	$0.5\pi$
$308$		0			0
$309$		0			0
$310$	−12.1550	+	21.0531i	−0.690359	+	1.19574i
$311$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$311$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$312$		0			0
$313$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$313$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$314$		0			0
$315$		0			0
$316$	−23.5298			−1.32366
$317$	5.02667	−	8.70645i	0.282326	−	0.489003i	−0.689631	−	0.724161i	$-0.742227\pi$
$317$	5.02667	−	8.70645i	0.282326	−	0.489003i	0.971957	+	0.235158i	$0.0755607\pi$
$318$	−34.7548	+	20.0657i	−1.94895	+	1.12523i
$319$		0			0
$320$	−14.7444	−	8.51266i	−0.824235	−	0.475872i
$321$			4.77282i			0.266393i
$322$		0			0
$323$	45.3495			2.52331
$324$	−18.1418	+	31.4225i	−1.00788	+	1.74569i
$325$		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	0.347097	+	0.937829i	$0.387167\pi$
$331$	−11.6190	+	20.1246i	−0.638635	+	1.10615i	−0.985732	+	0.168320i	$0.946166\pi$
$332$	52.7151	−	30.4350i	2.89311	−	1.67034i
$333$		0			0
$334$	19.0234	+	10.9832i	1.04092	+	0.600973i
$335$		0			0
$336$		0			0
$337$		0			0		1.00000			$0$
$337$		0			0		−1.00000			$\pi$
$338$	15.9634	−	27.6495i	0.868296	−	1.50393i
$339$	26.2366	−	15.1477i	1.42498	−	0.822710i
$340$	−36.3350	−	62.9341i	−1.97054	−	3.41308i
$341$		0			0
$342$			41.4480i			2.24125i
$343$		0			0
$344$		0			0
$345$	18.5692	−	32.1628i	0.999731	−	1.73159i
$346$	−12.7899	+	7.38423i	−0.687587	+	0.396979i
$347$	−14.5384	−	25.1812i	−0.780461	−	1.35180i	−0.931673	−	0.363297i	$-0.881651\pi$
$347$	−14.5384	−	25.1812i	−0.780461	−	1.35180i	0.151212	−	0.988501i	$-0.451682\pi$
$348$		0			0
$349$		−	27.3239i		−	1.46262i	−0.682048	−	0.731308i	$-0.738910\pi$
$349$		−	27.3239i		−	1.46262i	0.682048	−	0.731308i	$-0.261090\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	18.0000	−	10.3923i	0.958043	−	0.553127i	0.0624731	−	0.998047i	$-0.480101\pi$
$353$	18.0000	−	10.3923i	0.958043	−	0.553127i	0.895570	+	0.444920i	$0.146768\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$359$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$360$	28.9846	−	16.7343i	1.52762	−	0.881974i
$361$	6.32372	+	10.9530i	0.332828	+	0.576474i
$362$	12.5351	+	7.23714i	0.658830	+	0.380376i
$363$		−	19.0526i		−	1.00000i
$364$		0			0
$365$		0			0
$366$	8.67927	−	15.0329i	0.453673	−	0.785784i
$367$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$367$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$368$	−20.0893	−	34.7956i	−1.04723	−	1.81385i
$369$		0			0
$370$		0			0
$371$		0			0
$372$	30.9112			1.60267
$373$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$373$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$374$		0			0
$375$	−9.68246	−	16.7705i	−0.500000	−	0.866025i
$376$	4.42410	+	2.55426i	0.228156	+	0.131726i
$377$		0			0
$378$		0			0
$379$	−30.2146			−1.55202			−0.776009	−	0.630722i	$-0.782759\pi$
$379$	−30.2146			−1.55202			−0.776009	+	0.630722i	$0.782759\pi$
$380$	25.3566	−	43.9190i	1.30077	−	2.25300i
$381$		0			0
$382$		0			0
$383$	−28.8117	−	16.6345i	−1.47221	−	0.849982i	−0.472700	−	0.881224i	$-0.656720\pi$
$383$	−28.8117	−	16.6345i	−1.47221	−	0.849982i	−0.999512	+	0.0312418i	$0.990054\pi$
$384$			33.4686i			1.70794i
$385$		0			0
$386$		0			0
$387$		0			0
$388$		0			0
$389$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$389$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$390$		0			0
$391$			77.3001i			3.90923i
$392$		0			0
$393$		0			0
$394$	−33.5413	+	58.0952i	−1.68978	+	2.92679i
$395$	11.3023	−	6.52539i	0.568680	−	0.328328i
$396$		0			0
$397$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$397$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$398$			67.7634i			3.39667i
$399$		0			0
$400$	−20.9501			−1.04751
$401$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$401$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$		−	20.1246i		−	1.00000i
$406$		0			0
$407$		0			0
$408$	−34.8309	+	60.3288i	−1.72439	+	2.98672i
$409$	−33.4070	+	19.2876i	−1.65187	+	0.953709i	−0.675570	+	0.737296i	$0.736102\pi$
$409$	−33.4070	+	19.2876i	−1.65187	+	0.953709i	−0.976302	+	0.216413i	$0.930564\pi$
$410$		0			0
$411$	−3.23435	−	1.86736i	−0.159539	−	0.0921099i
$412$		0			0
$413$		0			0
$414$	−70.6499			−3.47225
$415$	−16.8807	+	29.2383i	−0.828643	+	1.43525i
$416$		0			0
$417$	−10.9018	−	18.8824i	−0.533862	−	0.924676i
$418$		0			0
$419$		0			0			−	1.00000i	$-0.5\pi$
$419$		0			0				1.00000i	$0.5\pi$
$420$		0			0
$421$	37.8245			1.84345			0.921727	−	0.387840i	$-0.126779\pi$
$421$	37.8245			1.84345			0.921727	+	0.387840i	$0.126779\pi$
$422$	9.51171	−	16.4748i	0.463023	−	0.801979i
$423$	2.66021	−	1.53587i	0.129344	−	0.0746766i
$424$	−23.5349	−	40.7636i	−1.14296	−	1.97966i
$425$	34.9063	+	20.1531i	1.69320	+	0.977571i
$426$		0			0
$427$		0			0
$428$	−11.1092			−0.536982
$429$		0			0
$430$		0			0
$431$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$431$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$432$	−18.8551	−	10.8860i	−0.907168	−	0.523754i
$433$		0			0			−	1.00000i	$-0.5\pi$
$433$		0			0				1.00000i	$0.5\pi$
$434$		0			0
$435$		0			0
$436$	31.2279	−	54.0883i	1.49555	−	2.59036i
$437$	−46.7172	+	26.9722i	−2.23479	+	1.29026i
$438$		0			0
$439$	−25.6372	−	14.8017i	−1.22360	−	0.706445i	−0.257916	−	0.966167i	$-0.583036\pi$
$439$	−25.6372	−	14.8017i	−1.22360	−	0.706445i	−0.965683	+	0.259722i	$0.916369\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−21.0251	+	36.4165i	−0.998932	+	1.73020i	−0.459459	+	0.888199i	$0.651957\pi$
$443$	−21.0251	+	36.4165i	−0.998932	+	1.73020i	−0.539474	+	0.842002i	$0.681377\pi$
$444$		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			$\pi$
$450$	−18.4193	+	31.9032i	−0.868296	+	1.50393i
$451$		0			0
$452$	35.2577	+	61.0681i	1.65838	+	2.87240i
$453$	−33.1328	−	19.1292i	−1.55671	−	0.898770i
$454$		−	43.9327i		−	2.06186i
$455$		0			0
$456$	−48.6140			−2.27656
$457$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$457$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$458$	−63.6988	+	36.7765i	−2.97645	+	1.71845i
$459$	20.9438	+	36.2757i	0.977571	+	1.69320i
$460$	74.8618	+	43.2215i	3.49045	+	2.01521i
$461$		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
$465$	−14.8479	+	8.57243i	−0.688554	+	0.397537i
$466$	35.1349	+	60.8554i	1.62759	+	2.81907i
$467$	30.9839	+	17.8885i	1.43376	+	0.827783i	0.997405	−	0.0719905i	$-0.0229351\pi$
$467$	30.9839	+	17.8885i	1.43376	+	0.827783i	0.436357	+	0.899774i	$0.356268\pi$
$468$		0			0
$469$		0			0
$470$	−5.62291			−0.259366
$471$		0			0
$472$		0			0
$473$		0			0
$474$	−21.5008	−	12.4135i	−0.987566	−	0.570172i
$475$			28.1280i			1.29060i
$476$		0			0
$477$	−28.3030			−1.29591
$478$		0			0
$479$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$479$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$480$	0.604073	+	1.04628i	0.0275720	+	0.0477561i
$481$		0			0
$482$		−	68.4217i		−	3.11652i
$483$		0			0
$484$	44.3466			2.01575
$485$		0			0
$486$	−33.1548	+	19.1419i	−1.50393	+	0.868296i
$487$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$487$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$488$	17.6320	+	10.1798i	0.798163	+	0.460819i
$489$		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
$496$			18.5483i			0.832845i
$497$		0			0
$498$	64.2259			2.87803
$499$	12.8177	−	22.2010i	0.573801	−	0.993852i	−0.422370	−	0.906423i	$-0.638802\pi$
$499$	12.8177	−	22.2010i	0.573801	−	0.993852i	0.996171	−	0.0874285i	$-0.0278649\pi$
$500$	39.0349	−	22.5368i	1.74569	−	1.00788i
$501$	7.74597	+	13.4164i	0.346064	+	0.599401i
$502$		0			0
$503$		−	29.1733i		−	1.30077i	−0.759604	−	0.650386i	$-0.774607\pi$
$503$		−	29.1733i		−	1.30077i	0.759604	−	0.650386i	$-0.225393\pi$
$504$		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	$-0.833333\pi$
$509$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$510$		−	76.6763i		−	3.39528i
$511$		0			0
$512$	−40.5027			−1.78999
$513$	−14.6158	+	25.3152i	−0.645301	+	1.11769i
$514$	−14.7355	+	8.50753i	−0.649954	+	0.375251i
$515$		0			0
$516$		0			0
$517$		0			0
$518$		0			0
$519$	−10.4156			−0.457193
$520$		0			0
$521$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$521$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$522$		0			0
$523$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$523$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$524$		0			0
$525$		0			0
$526$	79.1456			3.45091
$527$	17.8427	−	30.9045i	0.777241	−	1.34622i
$528$		0			0
$529$	−34.4753	−	59.7129i	−1.49892	−	2.59621i
$530$	44.8683	+	25.9047i	1.94895	+	1.12523i
$531$		0			0
$532$		0			0
$533$		0			0
$534$		0			0
$535$	5.33617	−	3.08084i	0.230703	−	0.133196i
$536$		0			0
$537$		0			0
$538$		0			0
$539$		0			0
$540$	46.8419			2.01575
$541$	17.1388	−	29.6852i	0.736854	−	1.27627i	−0.217051	−	0.976160i	$-0.569644\pi$
$541$	17.1388	−	29.6852i	0.736854	−	1.27627i	0.953905	−	0.300108i	$-0.0970228\pi$
$542$	10.8248	−	6.24972i	0.464966	−	0.268448i
$543$	5.10405	+	8.84047i	0.219036	+	0.379381i
$544$	−2.17774	−	1.25732i	−0.0933700	−	0.0539072i
$545$			34.6410i			1.48386i
$546$		0			0
$547$		0			0		1.00000			$0$
$547$		0			0		−1.00000			$\pi$
$548$	4.34644	−	7.52826i	0.185671	−	0.321591i
$549$	10.6021	−	6.12112i	0.452486	−	0.261243i
$550$		0			0
$551$		0			0
$552$		−	82.8645i		−	3.52695i
$553$		0			0
$554$		0			0
$555$		0			0
$556$	43.9505	−	25.3749i	1.86392	−	1.07613i
$557$	23.4810	+	40.6703i	0.994922	+	1.72326i	0.584624	+	0.811304i	$0.301242\pi$
$557$	23.4810	+	40.6703i	0.994922	+	1.72326i	0.410298	+	0.911952i	$0.365425\pi$
$558$	28.2457	+	16.3077i	1.19574	+	0.690359i
$559$		0			0
$560$		0			0
$561$		0			0
$562$		0			0
$563$	41.0008	−	23.6718i	1.72798	−	0.997648i	0.829716	−	0.558186i	$-0.188502\pi$
$563$	41.0008	−	23.6718i	1.72798	−	0.997648i	0.898261	−	0.439462i	$-0.144831\pi$
$564$	3.57488	+	6.19187i	0.150530	+	0.260725i
$565$	−33.8713	−	19.5556i	−1.42498	−	0.822710i
$566$		0			0
$567$		0			0
$568$		0			0
$569$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$569$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$570$	46.3403	−	26.7546i	1.94098	−	1.12063i
$571$	−19.3649	−	33.5410i	−0.810397	−	1.40365i	−0.912587	−	0.408883i	$-0.865918\pi$
$571$	−19.3649	−	33.5410i	−0.810397	−	1.40365i	0.102190	−	0.994765i	$-0.467415\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−47.9454			−1.99946
$576$	−11.4209	+	19.7816i	−0.475872	+	0.824235i
$577$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$577$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$578$	58.9221	+	102.056i	2.45084	+	4.24497i
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$		0			0
$584$		0			0
$585$		0			0
$586$	29.4709	+	17.0151i	1.21743	+	0.702886i
$587$		−	19.1943i		−	0.792232i	−0.918201	−	0.396116i	$-0.870358\pi$
$587$		−	19.1943i		−	0.792232i	0.918201	−	0.396116i	$-0.129642\pi$
$588$		0			0
$589$	24.9033			1.02612
$590$		0			0
$591$	−40.9721	+	23.6552i	−1.68537	+	0.973046i
$592$		0			0
$593$	3.87298	+	2.23607i	0.159044	+	0.0918243i	0.577410	−	0.816454i	$-0.304064\pi$
$593$	3.87298	+	2.23607i	0.159044	+	0.0918243i	−0.418365	+	0.908279i	$0.637397\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	−23.8953	+	41.3879i	−0.977971	+	1.69390i
$598$		0			0
$599$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$599$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$600$	−37.4190	−	21.6039i	−1.52762	−	0.881974i
$601$		−	46.9644i		−	1.91572i	−0.287237	−	0.957860i	$-0.592737\pi$
$601$		−	46.9644i		−	1.91572i	0.287237	−	0.957860i	$-0.407263\pi$
$602$		0			0
$603$		0			0
$604$	44.5250	−	77.1196i	1.81170	−	3.13795i
$605$	−21.3014	+	12.2984i	−0.866025	+	0.500000i
$606$		0			0
$607$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$607$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$608$		−	1.75486i		−	0.0711690i
$609$		0			0
$610$	−22.4098			−0.907345
$611$		0			0
$612$	−84.4349	+	48.7485i	−3.41308	+	1.97054i
$613$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$613$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	48.1002			1.93644			0.968220	−	0.250100i	$-0.0804635\pi$
$617$	48.1002			1.93644			0.968220	+	0.250100i	$0.0804635\pi$
$618$		0			0
$619$	33.1749	−	19.1535i	1.33341	−	0.769846i	0.347591	−	0.937646i	$-0.387000\pi$
$619$	33.1749	−	19.1535i	1.33341	−	0.769846i	0.985821	+	0.167800i	$0.0536663\pi$
$620$	−19.9531	−	34.5598i	−0.801336	−	1.38795i
$621$	−43.1509	−	24.9132i	−1.73159	−	0.999731i
$622$		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
$631$	−50.0135			−1.99101			−0.995504	−	0.0947206i	$-0.969804\pi$
$631$	−50.0135			−1.99101			−0.995504	+	0.0947206i	$0.969804\pi$
$632$	14.5597	−	25.2181i	0.579153	−	1.00312i
$633$	11.6190	−	6.70820i	0.461812	−	0.266627i
$634$	12.3451	+	21.3823i	0.490285	+	0.849199i
$635$		0			0
$636$		−	65.8778i		−	2.61222i
$637$		0			0
$638$		0			0
$639$		0			0
$640$	37.4190	−	21.6039i	1.47912	−	0.853968i
$641$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$641$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$642$	−10.1512	−	5.86081i	−0.400637	−	0.231308i
$643$		0			0			−	1.00000i	$-0.5\pi$
$643$		0			0				1.00000i	$0.5\pi$
$644$		0			0
$645$		0			0
$646$	−55.6872	+	96.4530i	−2.19098	+	3.79489i
$647$	38.7298	−	22.3607i	1.52263	−	0.879089i	0.522985	−	0.852342i	$-0.324819\pi$
$647$	38.7298	−	22.3607i	1.52263	−	0.879089i	0.999642	−	0.0267469i	$-0.00851482\pi$
$648$	−22.4514	−	38.8870i	−0.881974	−	1.52762i
$649$		0			0
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−15.8136	+	27.3899i	−0.618834	+	1.07185i	0.370865	+	0.928687i	$0.379061\pi$
$653$	−15.8136	+	27.3899i	−0.618834	+	1.07185i	−0.989699	+	0.143165i	$0.954272\pi$
$654$	57.0702	−	32.9495i	2.23162	−	1.28843i
$655$		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
$661$	33.0050	−	19.0554i	1.28374	−	0.741170i	0.306214	−	0.951963i	$-0.400938\pi$
$661$	33.0050	−	19.0554i	1.28374	−	0.741170i	0.977531	+	0.210792i	$0.0676043\pi$
$662$	−28.5351	−	49.4243i	−1.10905	−	1.92093i
$663$		0			0
$664$			75.3300i			2.92337i
$665$		0			0
$666$		0			0
$667$		0			0
$668$	−31.2279	+	18.0294i	−1.20824	+	0.697580i
$669$		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
$676$	26.2048	+	45.3880i	1.00788	+	1.74569i
$677$	−36.0000	−	20.7846i	−1.38359	−	0.798817i	−0.391009	−	0.920387i	$-0.627874\pi$
$677$	−36.0000	−	20.7846i	−1.38359	−	0.798817i	−0.992583	+	0.121569i	$0.961207\pi$
$678$			74.4029i			2.85743i
$679$		0			0
$680$	89.9329			3.44877
$681$	15.4919	−	26.8328i	0.593652	−	1.02824i
$682$		0			0
$683$	−23.8180	−	41.2539i	−0.911369	−	1.57854i	−0.812132	−	0.583473i	$-0.801693\pi$
$683$	−23.8180	−	41.2539i	−0.911369	−	1.57854i	−0.0992365	−	0.995064i	$-0.531640\pi$
$684$	−58.9235	−	34.0195i	−2.25300	−	1.30077i
$685$			4.82149i			0.184220i
$686$		0			0
$687$	−51.8738			−1.97911
$688$		0			0
$689$		0			0
$690$	45.6043	+	78.9889i	1.73613	+	3.00706i
$691$	23.4311	+	13.5279i	0.891360	+	0.514627i	0.874387	−	0.485229i	$-0.161264\pi$
$691$	23.4311	+	13.5279i	0.891360	+	0.514627i	0.0169730	+	0.999856i	$0.494597\pi$
$692$		−	24.2432i		−	0.921587i
$693$		0			0
$694$	71.4100			2.71069
$695$	−14.0741	+	24.3771i	−0.533862	+	0.924676i
$696$		0			0
$697$		0			0
$698$	58.1148	+	33.5526i	2.19968	+	1.26998i
$699$			49.5583i			1.87447i
$700$		0			0
$701$		0			0		1.00000			$0$
$701$		0			0		−1.00000			$\pi$
$702$		0			0
$703$		0			0
$704$		0			0
$705$	−3.43431	−	1.98280i	−0.129344	−	0.0746766i
$706$			51.0452i			1.92111i
$707$		0			0
$708$		0			0
$709$	7.23929	−	12.5388i	0.271877	−	0.470905i	−0.697465	−	0.716619i	$-0.745689\pi$
$709$	7.23929	−	12.5388i	0.271877	−	0.470905i	0.969343	+	0.245713i	$0.0790222\pi$
$710$		0			0
$711$	−8.75472	−	15.1636i	−0.328328	−	0.568680i
$712$		0			0
$713$			42.4488i			1.58972i
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$719$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$720$			28.1076i			1.04751i
$721$		0			0
$722$	−31.0610			−1.15597
$723$	24.1275	−	41.7900i	0.897310	−	1.55419i
$724$	−20.5770	+	11.8801i	−0.764738	+	0.441522i
$725$		0			0
$726$	40.5226	+	23.3957i	1.50393	+	0.868296i
$727$		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
$732$	14.2475	+	24.6773i	0.526601	+	0.912101i
$733$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$733$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$734$		0			0
$735$		0			0
$736$	2.99124			0.110258
$737$		0			0
$738$		0			0
$739$	−2.00000	−	3.46410i	−0.0735712	−	0.127429i	0.826893	−	0.562360i	$-0.190106\pi$
$739$	−2.00000	−	3.46410i	−0.0735712	−	0.127429i	−0.900464	+	0.434930i	$0.856773\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−29.3863			−1.07808			−0.539039	−	0.842281i	$-0.681212\pi$
$743$	−29.3863			−1.07808			−0.539039	+	0.842281i	$0.681212\pi$
$744$	−19.1271	+	33.1292i	−0.701235	+	1.21457i
$745$		0			0
$746$		0			0
$747$	39.2273	+	22.6479i	1.43525	+	0.828643i
$748$		0			0
$749$		0			0
$750$	47.5585			1.73659
$751$	−4.00000	+	6.92820i	−0.145962	+	0.252814i	−0.929731	−	0.368238i	$-0.879961\pi$
$751$	−4.00000	+	6.92820i	−0.145962	+	0.252814i	0.783769	+	0.621052i	$0.213294\pi$
$752$	−3.71545	+	2.14511i	−0.135488	+	0.0782243i
$753$		0			0
$754$		0			0
$755$			49.3915i			1.79754i
$756$		0			0
$757$		0			0		1.00000			$0$
$757$		0			0		−1.00000			$\pi$
$758$	37.1022	−	64.2628i	1.34761	−	2.33413i
$759$		0			0
$760$	31.3802	+	54.3521i	1.13828	+	1.97156i
$761$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$761$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$	27.0383	−	46.8317i	0.977571	−	1.69320i
$766$	70.7591	−	40.8528i	2.55663	−	1.47607i
$767$		0			0
$768$	−48.3419	−	27.9102i	−1.74439	−	1.00712i
$769$			30.6414i			1.10496i	0.833527	+	0.552479i	$0.186318\pi$
$769$			30.6414i			1.10496i	−0.833527	+	0.552479i	$0.813682\pi$
$770$		0			0
$771$	−12.0000			−0.432169
$772$		0			0
$773$	36.6797	−	21.1771i	1.31928	−	0.761686i	0.335666	−	0.941981i	$-0.391039\pi$
$773$	36.6797	−	21.1771i	1.31928	−	0.761686i	0.983613	+	0.180295i	$0.0577053\pi$
$774$		0			0
$775$	19.1685	+	11.0670i	0.688554	+	0.397537i
$776$		0			0
$777$		0			0
$778$		0			0
$779$		0			0
$780$		0			0
$781$		0			0
$782$	−164.408	−	94.9211i	−5.87922	−	3.39437i
$783$		0			0
$784$		0			0
$785$		0			0
$786$		0			0
$787$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$787$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$788$	−55.0597	−	95.3662i	−1.96142	−	3.39728i
$789$	48.3398	+	27.9090i	1.72094	+	0.993587i
$790$			32.0516i			1.14034i
$791$		0			0
$792$		0			0
$793$		0			0
$794$		0			0
$795$	18.2695	+	31.6437i	0.647953	+	1.12229i
$796$	−96.3342	−	55.6186i	−3.41447	−	1.97135i
$797$			54.3810i			1.92627i	0.269013	+	0.963136i	$0.413302\pi$
$797$			54.3810i			1.92627i	−0.269013	+	0.963136i	$0.586698\pi$
$798$		0			0
$799$	8.25403			0.292007
$800$	0.779854	−	1.35075i	0.0275720	−	0.0477561i
$801$		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	$-0.666667\pi$
$809$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$810$	42.8027	+	24.7121i	1.50393	+	0.868296i
$811$		−	39.1490i		−	1.37471i	−0.726323	−	0.687353i	$-0.758772\pi$
$811$		−	39.1490i		−	1.37471i	0.726323	−	0.687353i	$-0.241228\pi$
$812$		0			0
$813$	8.81532			0.309167
$814$		0			0
$815$		0			0
$816$	−29.2517	−	50.6653i	−1.02401	−	1.77364i
$817$		0			0
$818$		−	94.7371i		−	3.31241i
$819$		0			0
$820$		0			0
$821$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$821$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$822$	7.94329	−	4.58606i	0.277054	−	0.159957i
$823$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$823$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−27.9141			−0.970667			−0.485334	−	0.874329i	$-0.661302\pi$
$827$	−27.9141			−0.970667			−0.485334	+	0.874329i	$0.661302\pi$
$828$	57.9877	−	100.438i	2.01521	−	3.49045i
$829$	−11.8008	+	6.81319i	−0.409859	+	0.236632i	−0.690729	−	0.723114i	$-0.742710\pi$
$829$	−11.8008	+	6.81319i	−0.409859	+	0.236632i	0.280870	+	0.959746i	$0.409377\pi$
$830$	−41.4576	−	71.8067i	−1.43902	−	2.49245i
$831$		0			0
$832$		0			0
$833$		0			0
$834$	53.5476			1.85420
$835$	10.0000	−	17.3205i	0.346064	−	0.599401i
$836$		0			0
$837$	11.5011	+	19.9205i	0.397537	+	0.688554i
$838$		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
$842$	−46.4468	+	80.4483i	−1.60066	+	2.77243i
$843$		0			0
$844$	15.6140	+	27.0442i	0.537455	+	0.930899i
$845$	−25.1744	−	14.5344i	−0.866025	−	0.500000i
$846$			7.54393i			0.259366i
$847$		0			0
$848$	39.5301			1.35747
$849$		0			0
$850$	−85.7267	+	49.4944i	−2.94040	+	1.69764i
$851$		0			0
$852$		0			0
$853$		0			0			−	1.00000i	$-0.5\pi$
$853$		0			0				1.00000i	$0.5\pi$
$854$		0			0
$855$	37.7377			1.29060
$856$	6.87409	−	11.9063i	0.234952	−	0.406948i
$857$	−31.3593	+	18.1053i	−1.07121	+	0.618466i	−0.928513	−	0.371300i	$-0.878912\pi$
$857$	−31.3593	+	18.1053i	−1.07121	+	0.618466i	−0.142701	+	0.989766i	$0.545579\pi$
$858$		0			0
$859$	−48.0401	−	27.7360i	−1.63911	−	0.946340i	−0.981141	−	0.193293i	$-0.938083\pi$
$859$	−48.0401	−	27.7360i	−1.63911	−	0.946340i	−0.657967	−	0.753047i	$-0.728583\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−9.04521	+	15.6668i	−0.307902	+	0.533303i	−0.977903	−	0.209058i	$-0.932960\pi$
$863$	−9.04521	+	15.6668i	−0.307902	+	0.533303i	0.670001	+	0.742360i	$0.266294\pi$
$864$	1.40374	−	0.810449i	0.0477561	−	0.0275720i
$865$	6.72322	+	11.6450i	0.228596	+	0.395940i
$866$		0			0
$867$			83.1105i			2.82258i
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$	38.6462	+	66.9372i	1.30872	+	2.26678i
$873$		0			0
$874$		−	132.483i		−	4.48130i
$875$		0			0
$876$		0			0
$877$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$877$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$878$	62.9628	−	36.3516i	2.12489	−	1.22681i
$879$	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
$886$	−51.6358	−	89.4358i	−1.73474	−	3.00465i
$887$	30.5852	+	17.6584i	1.02695	+	0.592911i	0.916110	−	0.400928i	$-0.131312\pi$
$887$	30.5852	+	17.6584i	1.02695	+	0.592911i	0.110841	+	0.993838i	$0.464645\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$		0			0
$892$		0			0
$893$	2.88007	+	4.98842i	0.0963778	+	0.166931i
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$		0			0
$899$		0			0
$900$	−30.2363	−	52.3708i	−1.00788	−	1.74569i
$901$	−65.8635	−	38.0263i	−2.19423	−	1.26684i
$902$		0			0
$903$		0			0
$904$	−87.2664			−2.90244
$905$	6.58930	−	11.4130i	0.219036	−	0.379381i
$906$	81.3712	−	46.9797i	2.70338	−	1.56080i
$907$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$907$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$908$	62.4558	+	36.0589i	2.07267	+	1.19666i
$909$		0			0
$910$		0			0
$911$		0			0		1.00000			$0$
$911$		0			0		−1.00000			$\pi$
$912$	20.4135	−	35.3572i	0.675958	−	1.17079i
$913$		0			0
$914$		0			0
$915$	−13.6872	−	7.90233i	−0.452486	−	0.261243i
$916$		−	120.741i		−	3.98940i
$917$		0			0
$918$	−102.872			−3.39528
$919$	−11.6190	+	20.1246i	−0.383274	+	0.663850i	−0.991528	−	0.129893i	$-0.958537\pi$
$919$	−11.6190	+	20.1246i	−0.383274	+	0.663850i	0.608254	+	0.793742i	$0.291870\pi$
$920$	−92.6454	+	53.4888i	−3.05443	+	1.76347i
$921$		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	$-0.833333\pi$
$929$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$930$		−	42.1063i		−	1.38072i
$931$		0			0
$932$	−115.351			−3.77846
$933$		0			0
$934$	−76.0937	+	43.9327i	−2.48986	+	1.43752i
$935$		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
$940$	4.61515	−	7.99367i	0.150530	−	0.260725i
$941$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$941$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$	23.7807	−	41.1893i	0.772768	−	1.33847i	−0.163272	−	0.986581i	$-0.552205\pi$
$947$	23.7807	−	41.1893i	0.772768	−	1.33847i	0.936040	−	0.351893i	$-0.114462\pi$
$948$	35.2947	−	20.3774i	1.14632	−	0.661828i
$949$		0			0
$950$	−59.8250	−	34.5400i	−1.94098	−	1.12063i
$951$			17.4129i			0.564652i
$952$		0			0
$953$	−41.4508			−1.34272			−0.671362	−	0.741129i	$-0.734290\pi$
$953$	−41.4508			−1.34272			−0.671362	+	0.741129i	$0.734290\pi$
$954$	34.7548	−	60.1971i	1.12523	−	1.94895i
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$	29.4887			0.951744
$961$	−5.70180	+	9.87580i	−0.183929	+	0.318574i
$962$		0			0
$963$	−4.13338	−	7.15923i	−0.133196	−	0.230703i
$964$	97.2700	+	56.1589i	3.13286	+	1.80876i
$965$		0			0
$966$		0			0
$967$		0			0		1.00000			$0$
$967$		0			0		−1.00000			$\pi$
$968$	−27.4406	+	47.5285i	−0.881974	+	1.52762i
$969$	−68.0242	+	39.2738i	−2.18525	+	1.26166i
$970$		0			0
$971$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$971$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$972$		−	62.8450i		−	2.01575i
$973$		0			0
$974$		0			0
$975$		0			0
$976$	−14.8077	+	8.54922i	−0.473983	+	0.273654i
$977$	−13.9967	−	24.2431i	−0.447796	−	0.775605i	0.550447	−	0.834870i	$-0.314457\pi$
$977$	−13.9967	−	24.2431i	−0.447796	−	0.775605i	−0.998242	+	0.0592656i	$0.981124\pi$
$978$		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	−0.547079	−	0.837081i	$-0.684260\pi$
$983$	−3.00000	+	1.73205i	−0.0956851	+	0.0552438i	0.451394	+	0.892325i	$0.350927\pi$
$984$		0			0
$985$	52.8947	+	30.5388i	1.68537	+	0.973046i
$986$		0			0
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$	−16.0000	−	27.7128i	−0.508257	−	0.880327i	−0.999954	−	0.00956046i	$-0.996957\pi$
$991$	−16.0000	−	27.7128i	−0.508257	−	0.880327i	0.491698	−	0.870766i	$-0.336377\pi$
$992$	−1.19589	−	0.690449i	−0.0379696	−	0.0219218i
$993$		−	40.2492i		−	1.27727i
$994$		0			0
$995$	61.6975			1.95594
$996$	−52.7151	+	91.3051i	−1.67034	+	2.89311i
$997$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$997$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$998$	31.4792	+	54.5236i	0.996458	+	1.72592i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	735.2.p.d.509.2		16
3.2	odd	2		735.2.p.e.509.7		16
5.4	even	2		735.2.p.e.509.7		16
7.2	even	3		735.2.g.a.734.13	yes	16
7.3	odd	6	inner	735.2.p.d.374.2		16
7.4	even	3		735.2.p.e.374.2		16
7.5	odd	6		735.2.g.a.734.14	yes	16
7.6	odd	2		735.2.p.e.509.2		16
15.14	odd	2	CM	735.2.p.d.509.2		16
21.2	odd	6		735.2.g.a.734.4	yes	16
21.5	even	6		735.2.g.a.734.3	✓	16
21.11	odd	6	inner	735.2.p.d.374.7		16
21.17	even	6		735.2.p.e.374.7		16
21.20	even	2	inner	735.2.p.d.509.7		16
35.4	even	6	inner	735.2.p.d.374.7		16
35.9	even	6		735.2.g.a.734.4	yes	16
35.19	odd	6		735.2.g.a.734.3	✓	16
35.24	odd	6		735.2.p.e.374.7		16
35.34	odd	2	inner	735.2.p.d.509.7		16
105.44	odd	6		735.2.g.a.734.13	yes	16
105.59	even	6	inner	735.2.p.d.374.2		16
105.74	odd	6		735.2.p.e.374.2		16
105.89	even	6		735.2.g.a.734.14	yes	16
105.104	even	2		735.2.p.e.509.2		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
735.2.g.a.734.3	✓	16	21.5	even	6
735.2.g.a.734.3	✓	16	35.19	odd	6
735.2.g.a.734.4	yes	16	21.2	odd	6
735.2.g.a.734.4	yes	16	35.9	even	6
735.2.g.a.734.13	yes	16	7.2	even	3
735.2.g.a.734.13	yes	16	105.44	odd	6
735.2.g.a.734.14	yes	16	7.5	odd	6
735.2.g.a.734.14	yes	16	105.89	even	6
735.2.p.d.374.2		16	7.3	odd	6	inner
735.2.p.d.374.2		16	105.59	even	6	inner
735.2.p.d.374.7		16	21.11	odd	6	inner
735.2.p.d.374.7		16	35.4	even	6	inner
735.2.p.d.509.2		16	1.1	even	1	trivial
735.2.p.d.509.2		16	15.14	odd	2	CM
735.2.p.d.509.7		16	21.20	even	2	inner
735.2.p.d.509.7		16	35.34	odd	2	inner
735.2.p.e.374.2		16	7.4	even	3
735.2.p.e.374.2		16	105.74	odd	6
735.2.p.e.374.7		16	21.17	even	6
735.2.p.e.374.7		16	35.24	odd	6
735.2.p.e.509.2		16	7.6	odd	2
735.2.p.e.509.2		16	105.104	even	2
735.2.p.e.509.7		16	3.2	odd	2
735.2.p.e.509.7		16	5.4	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q+(-1.22796 + 2.12688i) q^{2} +(-1.50000 + 0.866025i) q^{3} +(-2.01575 - 3.49139i) q^{4} +(1.93649 + 1.11803i) q^{5} -4.25377i q^{6} +4.98920 q^{8} +(1.50000 - 2.59808i) q^{9} +(-4.75585 + 2.74579i) q^{10} +(6.04726 + 3.49139i) q^{12} -3.87298 q^{15} +(-2.09501 + 3.62867i) q^{16} +(6.98125 - 4.03063i) q^{17} +(3.68387 + 6.38065i) q^{18} +(4.87192 + 2.81280i) q^{19} -9.01472i q^{20} +(-4.79454 + 8.30439i) q^{23} +(-7.48380 + 4.32077i) q^{24} +(2.50000 + 4.33013i) q^{25} +5.19615i q^{27} +(4.75585 - 8.23738i) q^{30} +(3.83371 - 2.21339i) q^{31} +(-0.155971 - 0.270150i) q^{32} +19.7977i q^{34} -12.0945 q^{36} +(-11.9650 + 6.90800i) q^{38} +(9.66155 + 5.57810i) q^{40} +(5.80948 - 3.35410i) q^{45} +(-11.7750 - 20.3949i) q^{46} +(0.886735 + 0.511957i) q^{47} -7.25734i q^{48} -12.2796 q^{50} +(-6.98125 + 12.0919i) q^{51} +(-4.71717 - 8.17037i) q^{53} +(-11.0516 - 6.38065i) q^{54} -9.74384 q^{57} +(7.80698 + 13.5221i) q^{60} +(3.53403 + 2.04037i) q^{61} +10.8718i q^{62} -7.61396 q^{64} +(-28.1450 - 16.2495i) q^{68} -16.6088i q^{69} +(7.48380 - 12.9623i) q^{72} +(-7.50000 - 4.33013i) q^{75} -22.6797i q^{76} +(2.91824 - 5.05454i) q^{79} +(-8.11396 + 4.68459i) q^{80} +(-4.50000 - 7.79423i) q^{81} +15.0986i q^{83} +18.0255 q^{85} +16.4748i q^{90} +38.6584 q^{92} +(-3.83371 + 6.64017i) q^{93} +(-2.17774 + 1.25732i) q^{94} +(6.28962 + 10.8939i) q^{95} +(0.467913 + 0.270150i) q^{96} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 24 q^{3} - 16 q^{4} + 24 q^{9} + 48 q^{12} - 32 q^{16} + 40 q^{25} - 96 q^{36} + 128 q^{64} - 72 q^{68} - 120 q^{75} + 120 q^{80} - 72 q^{81}+O(q^{100}) \) 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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