LMFDB - Embedded newform 465.2.t.a.254.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [465,2,Mod(119,465)] 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("465.119"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(465, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([3, 3, 1])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 465 = 3 \cdot 5 \cdot 31 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	465.t (of order $6$, degree $2$, minimal)

Newform invariants

comment:select newform

sage:traces = [4,-6] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)

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

Self dual:	no
Analytic conductor:	$3.71304369399$
Analytic rank:	$0$
Dimension:	$4$
Relative dimension:	$2$ over $\Q(\zeta_{6})$
Coefficient field:	$\Q(\sqrt{-3}, \sqrt{5})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} + 2x^{2} + x + 1 $ x^4 - x^3 + 2*x^2 + x + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{6}]$

Embedding invariants

Embedding label			254.1
Root			$-0.309017 + 0.535233i$ of defining polynomial
Character	$\chi$	$=$	465.254
Dual form			465.2.t.a.119.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-2.61803 q^{2} +(1.50000 - 0.866025i) q^{3} +4.85410 q^{4} +(-1.11803 + 1.93649i) q^{5} +(-3.92705 + 2.26728i) q^{6} -7.47214 q^{8} +(1.50000 - 2.59808i) q^{9} +(2.92705 - 5.06980i) q^{10} +(7.28115 - 4.20378i) q^{12} +3.87298i q^{15} +9.85410 q^{16} +(6.35410 - 3.66854i) q^{17} +(-3.92705 + 6.80185i) q^{18} +(2.35410 + 4.07742i) q^{19} +(-5.42705 + 9.39993i) q^{20} +3.46410i q^{23} +(-11.2082 + 6.47106i) q^{24} +(-2.50000 - 4.33013i) q^{25} -5.19615i q^{27} -10.1396i q^{30} +(4.00000 + 3.87298i) q^{31} -10.8541 q^{32} +(-16.6353 + 9.60437i) q^{34} +(7.28115 - 12.6113i) q^{36} +(-6.16312 - 10.6748i) q^{38} +(8.35410 - 14.4697i) q^{40} +(3.35410 + 5.80948i) q^{45} -9.06914i q^{46} +13.4721 q^{47} +(14.7812 - 8.53390i) q^{48} +(-3.50000 + 6.06218i) q^{49} +(6.54508 + 11.3364i) q^{50} +(6.35410 - 11.0056i) q^{51} +(12.0000 + 6.92820i) q^{53} +13.6037i q^{54} +(7.06231 + 4.07742i) q^{57} +18.7999i q^{60} -9.47802i q^{61} +(-10.4721 - 10.1396i) q^{62} +8.70820 q^{64} +(30.8435 - 17.8075i) q^{68} +(3.00000 + 5.19615i) q^{69} +(-11.2082 + 19.4132i) q^{72} +(-7.50000 - 4.33013i) q^{75} +(11.4271 + 19.7922i) q^{76} +(15.3541 - 8.86469i) q^{79} +(-11.0172 + 19.0824i) q^{80} +(-4.50000 - 7.79423i) q^{81} +(-14.9164 - 8.61199i) q^{83} +16.4062i q^{85} +(-8.78115 - 15.2094i) q^{90} +16.8151i q^{92} +(9.35410 + 2.34537i) q^{93} -35.2705 q^{94} -10.5279 q^{95} +(-16.2812 + 9.39993i) q^{96} +(9.16312 - 15.8710i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 6 q^{2} + 6 q^{3} + 6 q^{4} - 9 q^{6} - 12 q^{8} + 6 q^{9} + 5 q^{10} + 9 q^{12} + 26 q^{16} + 12 q^{17} - 9 q^{18} - 4 q^{19} - 15 q^{20} - 18 q^{24} - 10 q^{25} + 16 q^{31} - 30 q^{32} - 33 q^{34}+ \cdots + 21 q^{98}+O(q^{100}) $ 4 * q - 6 * q^2 + 6 * q^3 + 6 * q^4 - 9 * q^6 - 12 * q^8 + 6 * q^9 + 5 * q^10 + 9 * q^12 + 26 * q^16 + 12 * q^17 - 9 * q^18 - 4 * q^19 - 15 * q^20 - 18 * q^24 - 10 * q^25 + 16 * q^31 - 30 * q^32 - 33 * q^34 + 9 * q^36 - 9 * q^38 + 20 * q^40 + 36 * q^47 + 39 * q^48 - 14 * q^49 + 15 * q^50 + 12 * q^51 + 48 * q^53 - 12 * q^57 - 24 * q^62 + 8 * q^64 + 63 * q^68 + 12 * q^69 - 18 * q^72 - 30 * q^75 + 39 * q^76 + 48 * q^79 - 15 * q^80 - 18 * q^81 - 6 * q^83 - 15 * q^90 + 24 * q^93 - 74 * q^94 - 60 * q^95 - 45 * q^96 + 21 * q^98

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/465\mathbb{Z}\right)^\times$.

$n$	$187$	$311$	$406$
$\chi(n)$	$-1$	$-1$	$e\left(\frac{5}{6}\right)$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$	−2.61803			−1.85123			−0.925615	−	0.378467i	$-0.876451\pi$
$2$	−2.61803			−1.85123			−0.925615	+	0.378467i	$0.876451\pi$
$3$	1.50000	−	0.866025i	0.866025	−	0.500000i
$3$	1.50000	−	0.866025i	0.866025	−	0.500000i
$4$	4.85410			2.42705
$5$	−1.11803	+	1.93649i	−0.500000	+	0.866025i
$5$	−1.11803	+	1.93649i	−0.500000	+	0.866025i
$6$	−3.92705	+	2.26728i	−1.60321	+	0.925615i
$7$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$7$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$8$	−7.47214			−2.64180
$9$	1.50000	−	2.59808i	0.500000	−	0.866025i
$10$	2.92705	−	5.06980i	0.925615	−	1.60321i
$11$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$11$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$12$	7.28115	−	4.20378i	2.10189	−	1.21353i
$13$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$13$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$14$		0			0
$15$			3.87298i			1.00000i
$16$	9.85410			2.46353
$17$	6.35410	−	3.66854i	1.54110	−	0.889752i	0.542326	−	0.840168i	$-0.317544\pi$
$17$	6.35410	−	3.66854i	1.54110	−	0.889752i	0.998770	−	0.0495842i	$-0.0157896\pi$
$18$	−3.92705	+	6.80185i	−0.925615	+	1.60321i
$19$	2.35410	+	4.07742i	0.540068	+	0.935425i	0.998899	+	0.0469020i	$0.0149348\pi$
$19$	2.35410	+	4.07742i	0.540068	+	0.935425i	−0.458831	+	0.888523i	$0.651732\pi$
$20$	−5.42705	+	9.39993i	−1.21353	+	2.10189i
$21$		0			0
$22$		0			0
$23$			3.46410i			0.722315i	0.932505	+	0.361158i	$0.117618\pi$
$23$			3.46410i			0.722315i	−0.932505	+	0.361158i	$0.882382\pi$
$24$	−11.2082	+	6.47106i	−2.28787	+	1.32090i
$25$	−2.50000	−	4.33013i	−0.500000	−	0.866025i
$26$		0			0
$27$		−	5.19615i		−	1.00000i
$28$		0			0
$29$		0			0			−	1.00000i	$-0.5\pi$
$29$		0			0				1.00000i	$0.5\pi$
$30$		−	10.1396i		−	1.85123i
$31$	4.00000	+	3.87298i	0.718421	+	0.695608i
$31$	4.00000	+	3.87298i	0.718421	+	0.695608i
$32$	−10.8541			−1.91875
$33$		0			0
$34$	−16.6353	+	9.60437i	−2.85292	+	1.64714i
$35$		0			0
$36$	7.28115	−	12.6113i	1.21353	−	2.10189i
$37$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$37$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$38$	−6.16312	−	10.6748i	−0.999790	−	1.73169i
$39$		0			0
$40$	8.35410	−	14.4697i	1.32090	−	2.28787i
$41$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$41$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$42$		0			0
$43$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$43$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$44$		0			0
$45$	3.35410	+	5.80948i	0.500000	+	0.866025i
$46$		−	9.06914i		−	1.33717i
$47$	13.4721			1.96511			0.982556	−	0.185964i	$-0.0595409\pi$
$47$	13.4721			1.96511			0.982556	+	0.185964i	$0.0595409\pi$
$48$	14.7812	−	8.53390i	2.13348	−	1.23176i
$49$	−3.50000	+	6.06218i	−0.500000	+	0.866025i
$50$	6.54508	+	11.3364i	0.925615	+	1.60321i
$51$	6.35410	−	11.0056i	0.889752	−	1.54110i
$52$		0			0
$53$	12.0000	+	6.92820i	1.64833	+	0.951662i	0.977737	+	0.209833i	$0.0672922\pi$
$53$	12.0000	+	6.92820i	1.64833	+	0.951662i	0.670590	+	0.741829i	$0.266041\pi$
$54$			13.6037i			1.85123i
$55$		0			0
$56$		0			0
$57$	7.06231	+	4.07742i	0.935425	+	0.540068i
$58$		0			0
$59$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$59$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$60$			18.7999i			2.42705i
$61$		−	9.47802i		−	1.21354i	−0.794879	−	0.606768i	$-0.792466\pi$
$61$		−	9.47802i		−	1.21354i	0.794879	−	0.606768i	$-0.207534\pi$
$62$	−10.4721	−	10.1396i	−1.32996	−	1.28773i
$63$		0			0
$64$	8.70820			1.08853
$65$		0			0
$66$		0			0
$67$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$67$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$68$	30.8435	−	17.8075i	3.74032	−	2.15947i
$69$	3.00000	+	5.19615i	0.361158	+	0.625543i
$70$		0			0
$71$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$71$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$72$	−11.2082	+	19.4132i	−1.32090	+	2.28787i
$73$		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
$76$	11.4271	+	19.7922i	1.31077	+	2.27032i
$77$		0			0
$78$		0			0
$79$	15.3541	−	8.86469i	1.72747	−	0.997356i	0.827401	−	0.561611i	$-0.189818\pi$
$79$	15.3541	−	8.86469i	1.72747	−	0.997356i	0.900070	−	0.435745i	$-0.143515\pi$
$80$	−11.0172	+	19.0824i	−1.23176	+	2.13348i
$81$	−4.50000	−	7.79423i	−0.500000	−	0.866025i
$82$		0			0
$83$	−14.9164	−	8.61199i	−1.63729	−	0.945289i	−0.981761	−	0.190117i	$-0.939113\pi$
$83$	−14.9164	−	8.61199i	−1.63729	−	0.945289i	−0.655527	−	0.755172i	$-0.727553\pi$
$84$		0			0
$85$			16.4062i			1.77950i
$86$		0			0
$87$		0			0
$88$		0			0
$89$		0			0			−	1.00000i	$-0.5\pi$
$89$		0			0				1.00000i	$0.5\pi$
$90$	−8.78115	−	15.2094i	−0.925615	−	1.60321i
$91$		0			0
$92$			16.8151i			1.75310i
$93$	9.35410	+	2.34537i	0.969975	+	0.243204i
$94$	−35.2705			−3.63788
$95$	−10.5279			−1.08014
$96$	−16.2812	+	9.39993i	−1.66169	+	0.959376i
$97$		0			0		1.00000			$0$
$97$		0			0		−1.00000			$\pi$
$98$	9.16312	−	15.8710i	0.925615	−	1.60321i
$99$		0			0
$100$	−12.1353	−	21.0189i	−1.21353	−	2.10189i
$101$		0			0		1.00000			$0$
$101$		0			0		−1.00000			$\pi$
$102$	−16.6353	+	28.8131i	−1.64714	+	2.85292i
$103$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$103$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$104$		0			0
$105$		0			0
$106$	−31.4164	−	18.1383i	−3.05143	−	1.76174i
$107$	−8.94427	−	15.4919i	−0.864675	−	1.49766i	−0.867369	−	0.497665i	$-0.834191\pi$
$107$	−8.94427	−	15.4919i	−0.864675	−	1.49766i	0.00269372	−	0.999996i	$-0.499143\pi$
$108$		−	25.2227i		−	2.42705i
$109$	−14.0000			−1.34096			−0.670478	−	0.741929i	$-0.733911\pi$
$109$	−14.0000			−1.34096			−0.670478	+	0.741929i	$0.733911\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−10.1180	+	17.5249i	−0.951825	+	1.64861i	−0.210352	+	0.977626i	$0.567461\pi$
$113$	−10.1180	+	17.5249i	−0.951825	+	1.64861i	−0.741473	+	0.670983i	$0.765872\pi$
$114$	−18.4894	−	10.6748i	−1.73169	−	0.999790i
$115$	−6.70820	−	3.87298i	−0.625543	−	0.361158i
$116$		0			0
$117$		0			0
$118$		0			0
$119$		0			0
$120$		−	28.9395i		−	2.64180i
$121$	5.50000	+	9.52628i	0.500000	+	0.866025i
$122$			24.8138i			2.24653i
$123$		0			0
$124$	19.4164	+	18.7999i	1.74364	+	1.68828i
$125$	11.1803			1.00000
$126$		0			0
$127$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$127$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$128$	−1.09017			−0.0963583
$129$		0			0
$130$		0			0
$131$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$131$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$	10.0623	+	5.80948i	0.866025	+	0.500000i
$136$	−47.4787	+	27.4118i	−4.07127	+	2.35055i
$137$	−13.7705	−	7.95041i	−1.17649	−	0.679249i	−0.221293	−	0.975207i	$-0.571028\pi$
$137$	−13.7705	−	7.95041i	−1.17649	−	0.679249i	−0.955201	+	0.295958i	$0.904361\pi$
$138$	−7.85410	−	13.6037i	−0.668586	−	1.15802i
$139$		−	8.15485i		−	0.691685i	−0.938293	−	0.345843i	$-0.887593\pi$
$139$		−	8.15485i		−	0.691685i	0.938293	−	0.345843i	$-0.112407\pi$
$140$		0			0
$141$	20.2082	−	11.6672i	1.70184	−	0.982556i
$142$		0			0
$143$		0			0
$144$	14.7812	−	25.6017i	1.23176	−	2.13348i
$145$		0			0
$146$		0			0
$147$			12.1244i			1.00000i
$148$		0			0
$149$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$149$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$150$	19.6353	+	11.3364i	1.60321	+	0.925615i
$151$			23.2379i			1.89107i	0.325515	+	0.945537i	$0.394462\pi$
$151$			23.2379i			1.89107i	−0.325515	+	0.945537i	$0.605538\pi$
$152$	−17.5902	−	30.4671i	−1.42675	−	2.47121i
$153$		−	22.0113i		−	1.77950i
$154$		0			0
$155$	−11.9721	+	3.41584i	−0.961625	+	0.274367i
$156$		0			0
$157$		0			0		1.00000			$0$
$157$		0			0		−1.00000			$\pi$
$158$	−40.1976	+	23.2081i	−3.19795	+	1.84634i
$159$	24.0000			1.90332
$160$	12.1353	−	21.0189i	0.959376	−	1.66169i
$161$		0			0
$162$	11.7812	+	20.4056i	0.925615	+	1.60321i
$163$		0			0		1.00000			$0$
$163$		0			0		−1.00000			$\pi$
$164$		0			0
$165$		0			0
$166$	39.0517	+	22.5465i	3.03100	+	1.74995i
$167$	−3.79180	+	2.18919i	−0.293418	+	0.169405i	−0.639482	−	0.768806i	$-0.720851\pi$
$167$	−3.79180	+	2.18919i	−0.293418	+	0.169405i	0.346064	+	0.938211i	$0.387518\pi$
$168$		0			0
$169$	6.50000	+	11.2583i	0.500000	+	0.866025i
$170$		−	42.9520i		−	3.29427i
$171$	14.1246			1.08014
$172$		0			0
$173$	−11.1803	+	19.3649i	−0.850026	+	1.47229i	0.0311588	+	0.999514i	$0.490080\pi$
$173$	−11.1803	+	19.3649i	−0.850026	+	1.47229i	−0.881184	+	0.472773i	$0.843253\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		0			0
$178$		0			0
$179$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$179$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$180$	16.2812	+	28.1998i	1.21353	+	2.10189i
$181$	−13.4164	−	7.74597i	−0.997234	−	0.575753i	−0.0898051	−	0.995959i	$-0.528624\pi$
$181$	−13.4164	−	7.74597i	−0.997234	−	0.575753i	−0.907429	+	0.420206i	$0.861958\pi$
$182$		0			0
$183$	−8.20820	−	14.2170i	−0.606768	−	1.05095i
$184$		−	25.8842i		−	1.90821i
$185$		0			0
$186$	−24.4894	−	6.14027i	−1.79565	−	0.450226i
$187$		0			0
$188$	65.3951			4.76943
$189$		0			0
$190$	27.5623			1.99958
$191$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$191$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$192$	13.0623	−	7.54153i	0.942691	−	0.544263i
$193$		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
$196$	−16.9894	+	29.4264i	−1.21353	+	2.10189i
$197$	−15.3541	−	8.86469i	−1.09393	−	0.631583i	−0.159313	−	0.987228i	$-0.550928\pi$
$197$	−15.3541	−	8.86469i	−1.09393	−	0.631583i	−0.934621	+	0.355645i	$0.884261\pi$
$198$		0			0
$199$	−1.93769	−	1.11873i	−0.137359	−	0.0793045i	0.429745	−	0.902950i	$-0.358603\pi$
$199$	−1.93769	−	1.11873i	−0.137359	−	0.0793045i	−0.567105	+	0.823646i	$0.691937\pi$
$200$	18.6803	+	32.3553i	1.32090	+	2.28787i
$201$		0			0
$202$		0			0
$203$		0			0
$204$	30.8435	−	53.4224i	2.15947	−	3.74032i
$205$		0			0
$206$		0			0
$207$	9.00000	+	5.19615i	0.625543	+	0.361158i
$208$		0			0
$209$		0			0
$210$		0			0
$211$	−14.0000	−	24.2487i	−0.963800	−	1.66935i	−0.712806	−	0.701361i	$-0.752576\pi$
$211$	−14.0000	−	24.2487i	−0.963800	−	1.66935i	−0.250994	−	0.967989i	$-0.580757\pi$
$212$	58.2492	+	33.6302i	4.00057	+	2.30973i
$213$		0			0
$214$	23.4164	+	40.5584i	1.60071	+	2.77252i
$215$		0			0
$216$			38.8264i			2.64180i
$217$		0			0
$218$	36.6525			2.48242
$219$		0			0
$220$		0			0
$221$		0			0
$222$		0			0
$223$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$223$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$224$		0			0
$225$	−15.0000			−1.00000
$226$	26.4894	−	45.8809i	1.76205	−	3.05195i
$227$	6.02786	−	10.4406i	0.400083	−	0.692965i	−0.593652	−	0.804722i	$-0.702314\pi$
$227$	6.02786	−	10.4406i	0.400083	−	0.692965i	0.993736	+	0.111757i	$0.0356478\pi$
$228$	34.2812	+	19.7922i	2.27032	+	1.31077i
$229$	−13.4164	+	7.74597i	−0.886581	+	0.511868i	−0.872823	−	0.488037i	$-0.837713\pi$
$229$	−13.4164	+	7.74597i	−0.886581	+	0.511868i	−0.0137585	+	0.999905i	$0.504380\pi$
$230$	17.5623	+	10.1396i	1.15802	+	0.668586i
$231$		0			0
$232$		0			0
$233$	−29.1803			−1.91167			−0.955834	−	0.293908i	$-0.905044\pi$
$233$	−29.1803			−1.91167			−0.955834	+	0.293908i	$0.905044\pi$
$234$		0			0
$235$	−15.0623	+	26.0887i	−0.982556	+	1.70184i
$236$		0			0
$237$	15.3541	−	26.5941i	0.997356	−	1.72747i
$238$		0			0
$239$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$239$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$240$			38.1648i			2.46353i
$241$	14.9164	−	8.61199i	0.960850	−	0.554747i	0.0644157	−	0.997923i	$-0.479482\pi$
$241$	14.9164	−	8.61199i	0.960850	−	0.554747i	0.896435	+	0.443176i	$0.146148\pi$
$242$	−14.3992	−	24.9401i	−0.925615	−	1.60321i
$243$	−13.5000	−	7.79423i	−0.866025	−	0.500000i
$244$		−	46.0073i		−	2.94531i
$245$	−7.82624	−	13.5554i	−0.500000	−	0.866025i
$246$		0			0
$247$		0			0
$248$	−29.8885	−	28.9395i	−1.89792	−	1.83766i
$249$	−29.8328			−1.89058
$250$	−29.2705			−1.85123
$251$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$251$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$	14.2082	+	24.6093i	0.889752	+	1.54110i
$256$	−14.5623			−0.910144
$257$	−4.82624	+	8.35929i	−0.301052	+	0.521438i	−0.976375	−	0.216085i	$-0.930671\pi$
$257$	−4.82624	+	8.35929i	−0.301052	+	0.521438i	0.675322	+	0.737523i	$0.264005\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$		0			0
$262$		0			0
$263$			23.3344i			1.43886i	0.694564	+	0.719431i	$0.255597\pi$
$263$			23.3344i			1.43886i	−0.694564	+	0.719431i	$0.744403\pi$
$264$		0			0
$265$	−26.8328	+	15.4919i	−1.64833	+	0.951662i
$266$		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	$0.166667\pi$
$270$	−26.3435	−	15.2094i	−1.60321	−	0.925615i
$271$		−	31.5858i		−	1.91870i	−0.282218	−	0.959350i	$-0.591070\pi$
$271$		−	31.5858i		−	1.91870i	0.282218	−	0.959350i	$-0.408930\pi$
$272$	62.6140	−	36.1502i	3.79653	−	2.19193i
$273$		0			0
$274$	36.0517	+	20.8144i	2.17796	+	1.25745i
$275$		0			0
$276$	14.5623	+	25.2227i	0.876548	+	1.51823i
$277$		0			0			−	1.00000i	$-0.5\pi$
$277$		0			0				1.00000i	$0.5\pi$
$278$			21.3497i			1.28047i
$279$	16.0623	−	4.58283i	0.961625	−	0.274367i
$280$		0			0
$281$		0			0		1.00000			$0$
$281$		0			0		−1.00000			$\pi$
$282$	−52.9058	+	30.5452i	−3.15049	+	1.81894i
$283$		0			0		1.00000			$0$
$283$		0			0		−1.00000			$\pi$
$284$		0			0
$285$	−15.7918	+	9.11740i	−0.935425	+	0.540068i
$286$		0			0
$287$		0			0
$288$	−16.2812	+	28.1998i	−0.959376	+	1.66169i
$289$	18.4164	−	31.8982i	1.08332	−	1.87636i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	1.82624	+	3.16314i	0.106690	+	0.184792i	0.914427	−	0.404750i	$-0.132641\pi$
$293$	1.82624	+	3.16314i	0.106690	+	0.184792i	−0.807737	+	0.589542i	$0.799308\pi$
$294$		−	31.7420i		−	1.85123i
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$		0			0
$300$	−36.4058	−	21.0189i	−2.10189	−	1.21353i
$301$		0			0
$302$		−	60.8376i		−	3.50081i
$303$		0			0
$304$	23.1976	+	40.1794i	1.33047	+	2.30444i
$305$	18.3541	+	10.5967i	1.05095	+	0.606768i
$306$			57.6262i			3.29427i
$307$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$307$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$308$		0			0
$309$		0			0
$310$	31.3435	−	8.94278i	1.78019	−	0.507916i
$311$		0			0		1.00000			$0$
$311$		0			0		−1.00000			$\pi$
$312$		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
$316$	74.5304	−	43.0301i	4.19266	−	2.42063i
$317$	6.40983	+	11.1022i	0.360012	+	0.623559i	0.987962	−	0.154694i	$-0.0494393\pi$
$317$	6.40983	+	11.1022i	0.360012	+	0.623559i	−0.627950	+	0.778253i	$0.716106\pi$
$318$	−62.8328			−3.52349
$319$		0			0
$320$	−9.73607	+	16.8634i	−0.544263	+	0.942691i
$321$	−26.8328	−	15.4919i	−1.49766	−	0.864675i
$322$		0			0
$323$	29.9164	+	17.2722i	1.66459	+	0.961053i
$324$	−21.8435	−	37.8340i	−1.21353	−	2.10189i
$325$		0			0
$326$		0			0
$327$	−21.0000	+	12.1244i	−1.16130	+	0.670478i
$328$		0			0
$329$		0			0
$330$		0			0
$331$	10.9377	+	6.31488i	0.601190	+	0.347097i	0.769510	−	0.638635i	$-0.220501\pi$
$331$	10.9377	+	6.31488i	0.601190	+	0.347097i	−0.168320	+	0.985732i	$0.553834\pi$
$332$	−72.4058	−	41.8035i	−3.97378	−	2.29426i
$333$		0			0
$334$	9.92705	−	5.73139i	0.543184	−	0.313607i
$335$		0			0
$336$		0			0
$337$		0			0			−	1.00000i	$-0.5\pi$
$337$		0			0				1.00000i	$0.5\pi$
$338$	−17.0172	−	29.4747i	−0.925615	−	1.60321i
$339$			35.0499i			1.90365i
$340$			79.6375i			4.31895i
$341$		0			0
$342$	−36.9787			−1.99958
$343$		0			0
$344$		0			0
$345$	−13.4164			−0.722315
$346$	29.2705	−	50.6980i	1.57359	−	2.72554i
$347$	31.3328	−	18.0900i	1.68203	−	0.971123i	0.721726	−	0.692179i	$-0.243349\pi$
$347$	31.3328	−	18.0900i	1.68203	−	0.971123i	0.960307	−	0.278944i	$-0.0899844\pi$
$348$		0			0
$349$	−30.4164			−1.62815			−0.814076	−	0.580758i	$-0.802756\pi$
$349$	−30.4164			−1.62815			−0.814076	+	0.580758i	$0.802756\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	14.4787	−	8.35929i	0.770624	−	0.444920i	−0.0624731	−	0.998047i	$-0.519899\pi$
$353$	14.4787	−	8.35929i	0.770624	−	0.444920i	0.833097	+	0.553127i	$0.186565\pi$
$354$		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	$0.666667\pi$
$360$	−25.0623	−	43.4092i	−1.32090	−	2.28787i
$361$	−1.58359	+	2.74286i	−0.0833470	+	0.144361i
$362$	35.1246	+	20.2792i	1.84611	+	1.06585i
$363$	16.5000	+	9.52628i	0.866025	+	0.500000i
$364$		0			0
$365$		0			0
$366$	21.4894	+	37.2207i	1.12327	+	1.94555i
$367$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$367$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$368$			34.1356i			1.77944i
$369$		0			0
$370$		0			0
$371$		0			0
$372$	45.4058	+	11.3847i	2.35418	+	0.590268i
$373$		0			0		1.00000			$0$
$373$		0			0		−1.00000			$\pi$
$374$		0			0
$375$	16.7705	−	9.68246i	0.866025	−	0.500000i
$376$	−100.666			−5.19143
$377$		0			0
$378$		0			0
$379$	2.00000	+	3.46410i	0.102733	+	0.177939i	0.912810	−	0.408385i	$-0.133908\pi$
$379$	2.00000	+	3.46410i	0.102733	+	0.177939i	−0.810077	+	0.586324i	$0.800575\pi$
$380$	−51.1033			−2.62155
$381$		0			0
$382$		0			0
$383$	−9.79180	−	5.65330i	−0.500337	−	0.288870i	0.228515	−	0.973540i	$-0.426613\pi$
$383$	−9.79180	−	5.65330i	−0.500337	−	0.288870i	−0.728853	+	0.684670i	$0.759946\pi$
$384$	−1.63525	+	0.944115i	−0.0834488	+	0.0481792i
$385$		0			0
$386$		0			0
$387$		0			0
$388$		0			0
$389$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$389$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$390$		0			0
$391$	12.7082	+	22.0113i	0.642681	+	1.11316i
$392$	26.1525	−	45.2974i	1.32090	−	2.28787i
$393$		0			0
$394$	40.1976	+	23.2081i	2.02512	+	1.16921i
$395$			39.6441i			1.99471i
$396$		0			0
$397$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$397$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$398$	5.07295	+	2.92887i	0.254284	+	0.146811i
$399$		0			0
$400$	−24.6353	−	42.6695i	−1.23176	−	2.13348i
$401$		0			0			−	1.00000i	$-0.5\pi$
$401$		0			0				1.00000i	$0.5\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$	20.1246			1.00000
$406$		0			0
$407$		0			0
$408$	−47.4787	+	82.2355i	−2.35055	+	4.07127i
$409$	−6.08359	+	3.51236i	−0.300814	+	0.173675i	−0.642809	−	0.766027i	$-0.722231\pi$
$409$	−6.08359	+	3.51236i	−0.300814	+	0.173675i	0.341994	+	0.939702i	$0.388898\pi$
$410$		0			0
$411$	−27.5410			−1.35850
$412$		0			0
$413$		0			0
$414$	−23.5623	−	13.6037i	−1.15802	−	0.668586i
$415$	33.3541	−	19.2570i	1.63729	−	0.945289i
$416$		0			0
$417$	−7.06231	−	12.2323i	−0.345843	−	0.599017i
$418$		0			0
$419$		0			0		1.00000			$0$
$419$		0			0		−1.00000			$\pi$
$420$		0			0
$421$	−2.79180	+	4.83553i	−0.136064	+	0.235669i	−0.926003	−	0.377515i	$-0.876779\pi$
$421$	−2.79180	+	4.83553i	−0.136064	+	0.235669i	0.789940	+	0.613185i	$0.210112\pi$
$422$	36.6525	+	63.4840i	1.78421	+	3.09035i
$423$	20.2082	−	35.0016i	0.982556	−	1.70184i
$424$	−89.6656	−	51.7685i	−4.35455	−	2.51410i
$425$	−31.7705	−	18.3427i	−1.54110	−	0.889752i
$426$		0			0
$427$		0			0
$428$	−43.4164	−	75.1994i	−2.09861	−	3.63490i
$429$		0			0
$430$		0			0
$431$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$431$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$432$		−	51.2034i		−	2.46353i
$433$		0			0			−	1.00000i	$-0.5\pi$
$433$		0			0				1.00000i	$0.5\pi$
$434$		0			0
$435$		0			0
$436$	−67.9574			−3.25457
$437$	−14.1246	+	8.15485i	−0.675672	+	0.390099i
$438$		0			0
$439$	−8.00000	+	13.8564i	−0.381819	+	0.661330i	−0.991322	−	0.131453i	$-0.958036\pi$
$439$	−8.00000	+	13.8564i	−0.381819	+	0.661330i	0.609503	+	0.792784i	$0.291369\pi$
$440$		0			0
$441$	10.5000	+	18.1865i	0.500000	+	0.866025i
$442$		0			0
$443$	20.9721	−	36.3248i	0.996416	−	1.72584i	0.424955	−	0.905214i	$-0.360290\pi$
$443$	20.9721	−	36.3248i	0.996416	−	1.72584i	0.571461	−	0.820629i	$-0.306377\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$		0			0
$449$		0			0			−	1.00000i	$-0.5\pi$
$449$		0			0				1.00000i	$0.5\pi$
$450$	39.2705			1.85123
$451$		0			0
$452$	−49.1140	+	85.0679i	−2.31013	+	4.00126i
$453$	20.1246	+	34.8569i	0.945537	+	1.63772i
$454$	−15.7812	+	27.3338i	−0.740646	+	1.28284i
$455$		0			0
$456$	−52.7705	−	30.4671i	−2.47121	−	1.42675i
$457$		0			0			−	1.00000i	$-0.5\pi$
$457$		0			0				1.00000i	$0.5\pi$
$458$	35.1246	−	20.2792i	1.64127	−	0.947585i
$459$	−19.0623	−	33.0169i	−0.889752	−	1.54110i
$460$	−32.5623	−	18.7999i	−1.51823	−	0.876548i
$461$		0			0			−	1.00000i	$-0.5\pi$
$461$		0			0				1.00000i	$0.5\pi$
$462$		0			0
$463$		0			0			−	1.00000i	$-0.5\pi$
$463$		0			0				1.00000i	$0.5\pi$
$464$		0			0
$465$	−15.0000	+	15.4919i	−0.695608	+	0.718421i
$466$	76.3951			3.53894
$467$	−35.7771			−1.65557			−0.827783	−	0.561048i	$-0.810398\pi$
$467$	−35.7771			−1.65557			−0.827783	+	0.561048i	$0.810398\pi$
$468$		0			0
$469$		0			0
$470$	39.4336	−	68.3010i	1.81894	−	3.15049i
$471$		0			0
$472$		0			0
$473$		0			0
$474$	−40.1976	+	69.6242i	−1.84634	+	3.19795i
$475$	11.7705	−	20.3871i	0.540068	−	0.935425i
$476$		0			0
$477$	36.0000	−	20.7846i	1.64833	−	0.951662i
$478$		0			0
$479$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$479$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$480$		−	42.0378i		−	1.91875i
$481$		0			0
$482$	−39.0517	+	22.5465i	−1.77875	+	1.02696i
$483$		0			0
$484$	26.6976	+	46.2415i	1.21353	+	2.10189i
$485$		0			0
$486$	35.3435	+	20.4056i	1.60321	+	0.925615i
$487$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$487$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$488$			70.8210i			3.20592i
$489$		0			0
$490$	20.4894	+	35.4886i	0.925615	+	1.60321i
$491$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$491$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$	39.4164	+	38.1648i	1.76985	+	1.71365i
$497$		0			0
$498$	78.1033			3.49989
$499$	−6.70820	+	3.87298i	−0.300300	+	0.173379i	−0.642578	−	0.766220i	$-0.722135\pi$
$499$	−6.70820	+	3.87298i	−0.300300	+	0.173379i	0.342277	+	0.939599i	$0.388802\pi$
$500$	54.2705			2.42705
$501$	−3.79180	+	6.56758i	−0.169405	+	0.293418i
$502$		0			0
$503$	22.3607	+	38.7298i	0.997013	+	1.72688i	0.565388	+	0.824825i	$0.308726\pi$
$503$	22.3607	+	38.7298i	0.997013	+	1.72688i	0.431625	+	0.902053i	$0.357940\pi$
$504$		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	$0.166667\pi$
$510$	−37.1976	−	64.4281i	−1.64714	−	2.85292i
$511$		0			0
$512$	40.3050			1.78124
$513$	21.1869	−	12.2323i	0.935425	−	0.540068i
$514$	12.6353	−	21.8849i	0.557317	−	0.965302i
$515$		0			0
$516$		0			0
$517$		0			0
$518$		0			0
$519$			38.7298i			1.70005i
$520$		0			0
$521$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$521$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$522$		0			0
$523$		0			0			−	1.00000i	$-0.5\pi$
$523$		0			0				1.00000i	$0.5\pi$
$524$		0			0
$525$		0			0
$526$		−	61.0903i		−	2.66366i
$527$	39.6246	+	9.93516i	1.72608	+	0.432782i
$528$		0			0
$529$	11.0000			0.478261
$530$	70.2492	−	40.5584i	3.05143	−	1.76174i
$531$		0			0
$532$		0			0
$533$		0			0
$534$		0			0
$535$	40.0000			1.72935
$536$		0			0
$537$		0			0
$538$		0			0
$539$		0			0
$540$	48.8435	+	28.1998i	2.10189	+	1.21353i
$541$	−19.6246	−	33.9908i	−0.843728	−	1.46138i	−0.886721	−	0.462304i	$-0.847023\pi$
$541$	−19.6246	−	33.9908i	−0.843728	−	1.46138i	0.0429934	−	0.999075i	$-0.486311\pi$
$542$			82.6927i			3.55196i
$543$	−26.8328			−1.15151
$544$	−68.9681	+	39.8187i	−2.95698	+	1.70721i
$545$	15.6525	−	27.1109i	0.670478	−	1.16130i
$546$		0			0
$547$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$547$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$548$	−66.8435	−	38.5921i	−2.85541	−	1.64857i
$549$	−24.6246	−	14.2170i	−1.05095	−	0.606768i
$550$		0			0
$551$		0			0
$552$	−22.4164	−	38.8264i	−0.954106	−	1.65256i
$553$		0			0
$554$		0			0
$555$		0			0
$556$		−	39.5845i		−	1.67876i
$557$			1.41969i			0.0601543i	0.999548	+	0.0300772i	$0.00957530\pi$
$557$			1.41969i			0.0601543i	−0.999548	+	0.0300772i	$0.990425\pi$
$558$	−42.0517	+	11.9980i	−1.78019	+	0.507916i
$559$		0			0
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−4.55573	+	7.89075i	−0.192001	+	0.332556i	−0.945913	−	0.324420i	$-0.894831\pi$
$563$	−4.55573	+	7.89075i	−0.192001	+	0.332556i	0.753912	+	0.656975i	$0.228164\pi$
$564$	98.0927	−	56.6338i	4.13045	−	2.38471i
$565$	−22.6246	−	39.1870i	−0.951825	−	1.64861i
$566$		0			0
$567$		0			0
$568$		0			0
$569$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$569$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$570$	41.3435	−	23.8697i	1.73169	−	0.999790i
$571$	33.5410	+	19.3649i	1.40365	+	0.810397i	0.994765	−	0.102190i	$-0.0325850\pi$
$571$	33.5410	+	19.3649i	1.40365	+	0.810397i	0.408883	+	0.912587i	$0.365918\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	15.0000	−	8.66025i	0.625543	−	0.361158i
$576$	13.0623	−	22.6246i	0.544263	−	0.942691i
$577$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$577$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$578$	−48.2148	+	83.5105i	−2.00547	+	3.47358i
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$		0			0
$584$		0			0
$585$		0			0
$586$	−4.78115	−	8.28120i	−0.197508	−	0.342093i
$587$		−	7.02473i		−	0.289942i	−0.989436	−	0.144971i	$-0.953691\pi$
$587$		−	7.02473i		−	0.289942i	0.989436	−	0.144971i	$-0.0463088\pi$
$588$			58.8529i			2.42705i
$589$	−6.37539	+	25.4271i	−0.262693	+	1.04771i
$590$		0			0
$591$	−30.7082			−1.26317
$592$		0			0
$593$	−39.7639			−1.63291			−0.816454	−	0.577410i	$-0.804064\pi$
$593$	−39.7639			−1.63291			−0.816454	+	0.577410i	$0.804064\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	−3.87539			−0.158609
$598$		0			0
$599$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$599$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$600$	56.0410	+	32.3553i	2.28787	+	1.32090i
$601$	15.0836	−	8.70852i	0.615273	−	0.355228i	−0.159754	−	0.987157i	$-0.551070\pi$
$601$	15.0836	−	8.70852i	0.615273	−	0.355228i	0.775026	+	0.631929i	$0.217737\pi$
$602$		0			0
$603$		0			0
$604$			112.799i			4.58973i
$605$	−24.5967			−1.00000
$606$		0			0
$607$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$607$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$608$	−25.5517	−	44.2568i	−1.03626	−	1.79485i
$609$		0			0
$610$	−48.0517	−	27.7426i	−1.94555	−	1.12327i
$611$		0			0
$612$		−	106.845i		−	4.31895i
$613$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$613$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−24.5967	−	42.6028i	−0.990228	−	1.71512i	−0.615889	−	0.787833i	$-0.711203\pi$
$617$	−24.5967	−	42.6028i	−0.990228	−	1.71512i	−0.374338	−	0.927292i	$-0.622130\pi$
$618$		0			0
$619$		−	26.4862i		−	1.06457i	−0.846566	−	0.532284i	$-0.821334\pi$
$619$		−	26.4862i		−	1.06457i	0.846566	−	0.532284i	$-0.178666\pi$
$620$	−58.1140	+	16.5808i	−2.33391	+	0.665902i
$621$	18.0000			0.722315
$622$		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
$631$	−7.22949	−	4.17395i	−0.287801	−	0.166162i	0.349148	−	0.937067i	$-0.386471\pi$
$631$	−7.22949	−	4.17395i	−0.287801	−	0.166162i	−0.636950	+	0.770905i	$0.719804\pi$
$632$	−114.728	+	66.2382i	−4.56363	+	2.63481i
$633$	−42.0000	−	24.2487i	−1.66935	−	0.963800i
$634$	−16.7812	−	29.0658i	−0.666465	−	1.15435i
$635$		0			0
$636$	116.498			4.61946
$637$		0			0
$638$		0			0
$639$		0			0
$640$	1.21885	−	2.11111i	0.0481792	−	0.0834488i
$641$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$641$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$642$	70.2492	+	40.5584i	2.77252	+	1.60071i
$643$		0			0			−	1.00000i	$-0.5\pi$
$643$		0			0				1.00000i	$0.5\pi$
$644$		0			0
$645$		0			0
$646$	−78.3222	−	45.2193i	−3.08154	−	1.77913i
$647$		−	50.8542i		−	1.99928i	−0.0267469	−	0.999642i	$-0.508515\pi$
$647$		−	50.8542i		−	1.99928i	0.0267469	−	0.999642i	$-0.491485\pi$
$648$	33.6246	+	58.2395i	1.32090	+	2.28787i
$649$		0			0
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−36.5967			−1.43214			−0.716071	−	0.698028i	$-0.754061\pi$
$653$	−36.5967			−1.43214			−0.716071	+	0.698028i	$0.754061\pi$
$654$	54.9787	−	31.7420i	2.14984	−	1.24121i
$655$		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
$661$	−14.6246	+	25.3306i	−0.568831	+	0.985245i	0.427850	+	0.903850i	$0.359271\pi$
$661$	−14.6246	+	25.3306i	−0.568831	+	0.985245i	−0.996682	+	0.0813955i	$0.974062\pi$
$662$	−28.6353	−	16.5326i	−1.11294	−	0.642557i
$663$		0			0
$664$	111.457	+	64.3500i	4.32539	+	2.49726i
$665$		0			0
$666$		0			0
$667$		0			0
$668$	−18.4058	+	10.6266i	−0.712140	+	0.411154i
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$673$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$674$		0			0
$675$	−22.5000	+	12.9904i	−0.866025	+	0.500000i
$676$	31.5517	+	54.6491i	1.21353	+	2.10189i
$677$	36.0000	+	20.7846i	1.38359	+	0.798817i	0.992583	−	0.121569i	$-0.0387926\pi$
$677$	36.0000	+	20.7846i	1.38359	+	0.798817i	0.391009	+	0.920387i	$0.372126\pi$
$678$		−	91.7618i		−	3.52409i
$679$		0			0
$680$		−	122.590i		−	4.70109i
$681$		−	20.8811i		−	0.800167i
$682$		0			0
$683$	50.8885			1.94720			0.973598	−	0.228269i	$-0.0733067\pi$
$683$	50.8885			1.94720			0.973598	+	0.228269i	$0.0733067\pi$
$684$	68.5623			2.62155
$685$	30.7918	−	17.7777i	1.17649	−	0.679249i
$686$		0			0
$687$	−13.4164	+	23.2379i	−0.511868	+	0.886581i
$688$		0			0
$689$		0			0
$690$	35.1246			1.33717
$691$	−9.64590	+	16.7072i	−0.366947	+	0.635571i	−0.989087	−	0.147335i	$-0.952930\pi$
$691$	−9.64590	+	16.7072i	−0.366947	+	0.635571i	0.622139	+	0.782907i	$0.286264\pi$
$692$	−54.2705	+	93.9993i	−2.06306	+	3.57332i
$693$		0			0
$694$	−82.0304	+	47.3603i	−3.11383	+	1.79777i
$695$	15.7918	+	9.11740i	0.599017	+	0.345843i
$696$		0			0
$697$		0			0
$698$	79.6312			3.01408
$699$	−43.7705	+	25.2709i	−1.65555	+	0.955834i
$700$		0			0
$701$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$701$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$702$		0			0
$703$		0			0
$704$		0			0
$705$			52.1774i			1.96511i
$706$	−37.9058	+	21.8849i	−1.42660	+	0.823649i
$707$		0			0
$708$		0			0
$709$			0.721240i			0.0270867i	0.999908	+	0.0135434i	$0.00431112\pi$
$709$			0.721240i			0.0270867i	−0.999908	+	0.0135434i	$0.995689\pi$
$710$		0			0
$711$		−	53.1882i		−	1.99471i
$712$		0			0
$713$	−13.4164	+	13.8564i	−0.502448	+	0.518927i
$714$		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	$0.833333\pi$
$720$	33.0517	+	57.2472i	1.23176	+	2.13348i
$721$		0			0
$722$	4.14590	−	7.18091i	0.154294	−	0.267246i
$723$	14.9164	−	25.8360i	0.554747	−	0.960850i
$724$	−65.1246	−	37.5997i	−2.42034	−	1.39738i
$725$		0			0
$726$	−43.1976	−	24.9401i	−1.60321	−	0.925615i
$727$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$727$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$728$		0			0
$729$	−27.0000			−1.00000
$730$		0			0
$731$		0			0
$732$	−39.8435	−	69.0109i	−1.47266	−	2.55072i
$733$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$733$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$734$		0			0
$735$	−23.4787	−	13.5554i	−0.866025	−	0.500000i
$736$		−	37.5997i		−	1.38594i
$737$		0			0
$738$		0			0
$739$	−26.4787	−	15.2875i	−0.974035	−	0.562360i	−0.0735712	−	0.997290i	$-0.523440\pi$
$739$	−26.4787	−	15.2875i	−0.974035	−	0.562360i	−0.900464	+	0.434930i	$0.856773\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		−	31.1769i		−	1.14377i	−0.820334	−	0.571885i	$-0.806212\pi$
$743$		−	31.1769i		−	1.14377i	0.820334	−	0.571885i	$-0.193788\pi$
$744$	−69.8951	−	17.5249i	−2.56248	−	0.642496i
$745$		0			0
$746$		0			0
$747$	−44.7492	+	25.8360i	−1.63729	+	0.945289i
$748$		0			0
$749$		0			0
$750$	−43.9058	+	25.3490i	−1.60321	+	0.925615i
$751$	25.4787	+	44.1304i	0.929731	+	1.61034i	0.783769	+	0.621052i	$0.213294\pi$
$751$	25.4787	+	44.1304i	0.929731	+	1.61034i	0.145962	+	0.989290i	$0.453372\pi$
$752$	132.756			4.84111
$753$		0			0
$754$		0			0
$755$	−45.0000	−	25.9808i	−1.63772	−	0.945537i
$756$		0			0
$757$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$757$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$758$	−5.23607	−	9.06914i	−0.190183	−	0.329406i
$759$		0			0
$760$	78.6656			2.85350
$761$		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
$765$	42.6246	+	24.6093i	1.54110	+	0.889752i
$766$	25.6353	+	14.8005i	0.926239	+	0.534765i
$767$		0			0
$768$	−21.8435	+	12.6113i	−0.788208	+	0.455072i
$769$	24.9164	+	43.1565i	0.898509	+	1.55626i	0.829401	+	0.558653i	$0.188682\pi$
$769$	24.9164	+	43.1565i	0.898509	+	1.55626i	0.0691074	+	0.997609i	$0.477985\pi$
$770$		0			0
$771$			16.7186i			0.602105i
$772$		0			0
$773$			55.4256i			1.99352i	0.0804258	+	0.996761i	$0.474372\pi$
$773$			55.4256i			1.99352i	−0.0804258	+	0.996761i	$0.525628\pi$
$774$		0			0
$775$	6.77051	−	27.0030i	0.243204	−	0.969975i
$776$		0			0
$777$		0			0
$778$		0			0
$779$		0			0
$780$		0			0
$781$		0			0
$782$	−33.2705	−	57.6262i	−1.18975	−	2.06071i
$783$		0			0
$784$	−34.4894	+	59.7373i	−1.23176	+	2.13348i
$785$		0			0
$786$		0			0
$787$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$787$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$788$	−74.5304	−	43.0301i	−2.65503	−	1.53289i
$789$	20.2082	+	35.0016i	0.719431	+	1.24609i
$790$		−	103.790i		−	3.69267i
$791$		0			0
$792$		0			0
$793$		0			0
$794$		0			0
$795$	−26.8328	+	46.4758i	−0.951662	+	1.64833i
$796$	−9.40576	−	5.43042i	−0.333378	−	0.192476i
$797$	48.8951	+	28.2296i	1.73195	+	0.999944i	0.871262	+	0.490819i	$0.163302\pi$
$797$	48.8951	+	28.2296i	1.73195	+	0.999944i	0.860693	+	0.509125i	$0.170031\pi$
$798$		0			0
$799$	85.6033	−	49.4231i	3.02843	−	1.74846i
$800$	27.1353	+	46.9996i	0.959376	+	1.66169i
$801$		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		0.866025	−	0.500000i	$-0.166667\pi$
$809$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$810$	−52.6869			−1.85123
$811$	−26.0000	+	45.0333i	−0.912983	+	1.58133i	−0.103156	+	0.994665i	$0.532894\pi$
$811$	−26.0000	+	45.0333i	−0.912983	+	1.58133i	−0.809827	+	0.586669i	$0.800439\pi$
$812$		0			0
$813$	−27.3541	−	47.3787i	−0.959350	−	1.66164i
$814$		0			0
$815$		0			0
$816$	62.6140	−	108.451i	2.19193	−	3.79653i
$817$		0			0
$818$	15.9271	−	9.19549i	0.556876	−	0.321513i
$819$		0			0
$820$		0			0
$821$		0			0			−	1.00000i	$-0.5\pi$
$821$		0			0				1.00000i	$0.5\pi$
$822$	72.1033			2.51489
$823$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$823$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	7.33282	+	4.23360i	0.254987	+	0.147217i	0.622046	−	0.782981i	$-0.286302\pi$
$827$	7.33282	+	4.23360i	0.254987	+	0.147217i	−0.367059	+	0.930198i	$0.619635\pi$
$828$	43.6869	+	25.2227i	1.51823	+	0.876548i
$829$		−	52.6828i		−	1.82975i	−0.403739	−	0.914874i	$-0.632290\pi$
$829$		−	52.6828i		−	1.82975i	0.403739	−	0.914874i	$-0.367710\pi$
$830$	−87.3222	+	50.4155i	−3.03100	+	1.74995i
$831$		0			0
$832$		0			0
$833$			51.3596i			1.77950i
$834$	18.4894	+	32.0245i	0.640234	+	1.10892i
$835$		−	9.79038i		−	0.338810i
$836$		0			0
$837$	20.1246	−	20.7846i	0.695608	−	0.718421i
$838$		0			0
$839$		0			0		1.00000			$0$
$839$		0			0		−1.00000			$\pi$
$840$		0			0
$841$	−29.0000			−1.00000
$842$	7.30902	−	12.6596i	0.251885	−	0.436278i
$843$		0			0
$844$	−67.9574	−	117.706i	−2.33919	−	4.05160i
$845$	−29.0689			−1.00000
$846$	−52.9058	+	91.6355i	−1.81894	+	3.15049i
$847$		0			0
$848$	118.249	+	68.2712i	4.06070	+	2.34444i
$849$		0			0
$850$	83.1763	+	48.0218i	2.85292	+	1.64714i
$851$		0			0
$852$		0			0
$853$		0			0		1.00000			$0$
$853$		0			0		−1.00000			$\pi$
$854$		0			0
$855$	−15.7918	+	27.3522i	−0.540068	+	0.935425i
$856$	66.8328	+	115.758i	2.28430	+	3.95652i
$857$	29.0689	−	50.3488i	0.992974	−	1.71988i	0.394009	−	0.919107i	$-0.371088\pi$
$857$	29.0689	−	50.3488i	0.992974	−	1.71988i	0.598965	−	0.800775i	$-0.295579\pi$
$858$		0			0
$859$	16.2295	+	9.37010i	0.553743	+	0.319704i	0.750630	−	0.660722i	$-0.229750\pi$
$859$	16.2295	+	9.37010i	0.553743	+	0.319704i	−0.196887	+	0.980426i	$0.563083\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−17.0410	−	9.83864i	−0.580083	−	0.334911i	0.181083	−	0.983468i	$-0.442040\pi$
$863$	−17.0410	−	9.83864i	−0.580083	−	0.334911i	−0.761166	+	0.648557i	$0.775373\pi$
$864$			56.3996i			1.91875i
$865$	−25.0000	−	43.3013i	−0.850026	−	1.47229i
$866$		0			0
$867$		−	63.7963i		−	2.16664i
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$	104.610			3.54254
$873$		0			0
$874$	36.9787	−	21.3497i	1.25082	−	0.722163i
$875$		0			0
$876$		0			0
$877$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$877$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$878$	20.9443	−	36.2765i	0.706835	−	1.22427i
$879$	5.47871	+	3.16314i	0.184792	+	0.106690i
$880$		0			0
$881$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$881$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$882$	−27.4894	−	47.6130i	−0.925615	−	1.60321i
$883$		0			0			−	1.00000i	$-0.5\pi$
$883$		0			0				1.00000i	$0.5\pi$
$884$		0			0
$885$		0			0
$886$	−54.9058	+	95.0996i	−1.84460	+	3.19493i
$887$	−4.47214	−	7.74597i	−0.150160	−	0.260084i	0.781126	−	0.624373i	$-0.214645\pi$
$887$	−4.47214	−	7.74597i	−0.150160	−	0.260084i	−0.931286	+	0.364289i	$0.881312\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$		0			0
$892$		0			0
$893$	31.7148	+	54.9316i	1.06129	+	1.83822i
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$		0			0
$899$		0			0
$900$	−72.8115			−2.42705
$901$	101.666			3.38697
$902$		0			0
$903$		0			0
$904$	75.6033	−	130.949i	2.51453	−	4.35529i
$905$	30.0000	−	17.3205i	0.997234	−	0.575753i
$906$	−52.6869	−	91.2564i	−1.75041	−	3.03179i
$907$		0			0		1.00000			$0$
$907$		0			0		−1.00000			$\pi$
$908$	29.2599	−	50.6796i	0.971023	−	1.68186i
$909$		0			0
$910$		0			0
$911$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$911$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$912$	69.5927	+	40.1794i	2.30444	+	1.33047i
$913$		0			0
$914$		0			0
$915$	36.7082			1.21354
$916$	−65.1246	+	37.5997i	−2.15178	+	1.24233i
$917$		0			0
$918$	49.9058	+	86.4393i	1.64714	+	2.85292i
$919$	24.0623	−	41.6771i	0.793742	−	1.37480i	−0.129893	−	0.991528i	$-0.541463\pi$
$919$	24.0623	−	41.6771i	0.793742	−	1.37480i	0.923635	−	0.383274i	$-0.125203\pi$
$920$	50.1246	+	28.9395i	1.65256	+	0.954106i
$921$		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			−	1.00000i	$-0.5\pi$
$929$		0			0				1.00000i	$0.5\pi$
$930$	39.2705	−	40.5584i	1.28773	−	1.32996i
$931$	−32.9574			−1.08014
$932$	−141.644			−4.63971
$933$		0			0
$934$	93.6656			3.06483
$935$		0			0
$936$		0			0
$937$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$937$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$938$		0			0
$939$		0			0
$940$	−73.1140	+	126.637i	−2.38471	+	4.13045i
$941$		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
$947$	38.9164	−	22.4684i	1.26461	−	0.730125i	0.290650	−	0.956830i	$-0.406129\pi$
$947$	38.9164	−	22.4684i	1.26461	−	0.730125i	0.973964	+	0.226705i	$0.0727952\pi$
$948$	74.5304	−	129.090i	2.42063	−	4.19266i
$949$		0			0
$950$	−30.8156	+	53.3742i	−0.999790	+	1.73169i
$951$	19.2295	+	11.1022i	0.623559	+	0.360012i
$952$		0			0
$953$			60.7411i			1.96760i	0.179278	+	0.983798i	$0.442624\pi$
$953$			60.7411i			1.96760i	−0.179278	+	0.983798i	$0.557376\pi$
$954$	−94.2492	+	54.4148i	−3.05143	+	1.76174i
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$			33.7267i			1.08853i
$961$	1.00000	+	30.9839i	0.0322581	+	0.999480i
$962$		0			0
$963$	−53.6656			−1.72935
$964$	72.4058	−	41.8035i	2.33203	−	1.34640i
$965$		0			0
$966$		0			0
$967$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$967$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$968$	−41.0967	−	71.1817i	−1.32090	−	2.28787i
$969$	59.8328			1.92211
$970$		0			0
$971$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$971$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$972$	−65.5304	−	37.8340i	−2.10189	−	1.21353i
$973$		0			0
$974$		0			0
$975$		0			0
$976$		−	93.3974i		−	2.98958i
$977$	−51.7639			−1.65607			−0.828037	−	0.560673i	$-0.810543\pi$
$977$	−51.7639			−1.65607			−0.828037	+	0.560673i	$0.810543\pi$
$978$		0			0
$979$		0			0
$980$	−37.9894	−	65.7995i	−1.21353	−	2.10189i
$981$	−21.0000	+	36.3731i	−0.670478	+	1.16130i
$982$		0			0
$983$	−48.4574	−	27.9769i	−1.54555	−	0.892325i	−0.998473	−	0.0552438i	$-0.982406\pi$
$983$	−48.4574	−	27.9769i	−1.54555	−	0.892325i	−0.547079	−	0.837081i	$-0.684260\pi$
$984$		0			0
$985$	34.3328	−	19.8221i	1.09393	−	0.631583i
$986$		0			0
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$			54.8237i			1.74153i	0.491698	+	0.870766i	$0.336377\pi$
$991$			54.8237i			1.74153i	−0.491698	+	0.870766i	$0.663623\pi$
$992$	−43.4164	−	42.0378i	−1.37847	−	1.33470i
$993$	21.8754			0.694194
$994$		0			0
$995$	4.33282	−	2.50155i	0.137359	−	0.0793045i
$996$	−144.812			−4.58853
$997$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$997$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$998$	17.5623	−	10.1396i	0.555925	−	0.320963i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	465.2.t.a.254.1	yes	4
3.2	odd	2		465.2.t.b.254.2	yes	4
5.4	even	2		465.2.t.b.254.2	yes	4
15.14	odd	2	CM	465.2.t.a.254.1	yes	4
31.26	odd	6	inner	465.2.t.a.119.1	✓	4
93.26	even	6		465.2.t.b.119.2	yes	4
155.119	odd	6		465.2.t.b.119.2	yes	4
465.119	even	6	inner	465.2.t.a.119.1	✓	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
465.2.t.a.119.1	✓	4	31.26	odd	6	inner
465.2.t.a.119.1	✓	4	465.119	even	6	inner
465.2.t.a.254.1	yes	4	1.1	even	1	trivial
465.2.t.a.254.1	yes	4	15.14	odd	2	CM
465.2.t.b.119.2	yes	4	93.26	even	6
465.2.t.b.119.2	yes	4	155.119	odd	6
465.2.t.b.254.2	yes	4	3.2	odd	2
465.2.t.b.254.2	yes	4	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 465 = 3 \cdot 5 \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	465.t (of order \(6\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q-2.61803 q^{2} +(1.50000 - 0.866025i) q^{3} +4.85410 q^{4} +(-1.11803 + 1.93649i) q^{5} +(-3.92705 + 2.26728i) q^{6} -7.47214 q^{8} +(1.50000 - 2.59808i) q^{9} +(2.92705 - 5.06980i) q^{10} +(7.28115 - 4.20378i) q^{12} +3.87298i q^{15} +9.85410 q^{16} +(6.35410 - 3.66854i) q^{17} +(-3.92705 + 6.80185i) q^{18} +(2.35410 + 4.07742i) q^{19} +(-5.42705 + 9.39993i) q^{20} +3.46410i q^{23} +(-11.2082 + 6.47106i) q^{24} +(-2.50000 - 4.33013i) q^{25} -5.19615i q^{27} -10.1396i q^{30} +(4.00000 + 3.87298i) q^{31} -10.8541 q^{32} +(-16.6353 + 9.60437i) q^{34} +(7.28115 - 12.6113i) q^{36} +(-6.16312 - 10.6748i) q^{38} +(8.35410 - 14.4697i) q^{40} +(3.35410 + 5.80948i) q^{45} -9.06914i q^{46} +13.4721 q^{47} +(14.7812 - 8.53390i) q^{48} +(-3.50000 + 6.06218i) q^{49} +(6.54508 + 11.3364i) q^{50} +(6.35410 - 11.0056i) q^{51} +(12.0000 + 6.92820i) q^{53} +13.6037i q^{54} +(7.06231 + 4.07742i) q^{57} +18.7999i q^{60} -9.47802i q^{61} +(-10.4721 - 10.1396i) q^{62} +8.70820 q^{64} +(30.8435 - 17.8075i) q^{68} +(3.00000 + 5.19615i) q^{69} +(-11.2082 + 19.4132i) q^{72} +(-7.50000 - 4.33013i) q^{75} +(11.4271 + 19.7922i) q^{76} +(15.3541 - 8.86469i) q^{79} +(-11.0172 + 19.0824i) q^{80} +(-4.50000 - 7.79423i) q^{81} +(-14.9164 - 8.61199i) q^{83} +16.4062i q^{85} +(-8.78115 - 15.2094i) q^{90} +16.8151i q^{92} +(9.35410 + 2.34537i) q^{93} -35.2705 q^{94} -10.5279 q^{95} +(-16.2812 + 9.39993i) q^{96} +(9.16312 - 15.8710i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 6 q^{2} + 6 q^{3} + 6 q^{4} - 9 q^{6} - 12 q^{8} + 6 q^{9} + 5 q^{10} + 9 q^{12} + 26 q^{16} + 12 q^{17} - 9 q^{18} - 4 q^{19} - 15 q^{20} - 18 q^{24} - 10 q^{25} + 16 q^{31} - 30 q^{32} - 33 q^{34}+ \cdots + 21 q^{98}+O(q^{100}) \) 4 * q - 6 * q^2 + 6 * q^3 + 6 * q^4 - 9 * q^6 - 12 * q^8 + 6 * q^9 + 5 * q^10 + 9 * q^12 + 26 * q^16 + 12 * q^17 - 9 * q^18 - 4 * q^19 - 15 * q^20 - 18 * q^24 - 10 * q^25 + 16 * q^31 - 30 * q^32 - 33 * q^34 + 9 * q^36 - 9 * q^38 + 20 * q^40 + 36 * q^47 + 39 * q^48 - 14 * q^49 + 15 * q^50 + 12 * q^51 + 48 * q^53 - 12 * q^57 - 24 * q^62 + 8 * q^64 + 63 * q^68 + 12 * q^69 - 18 * q^72 - 30 * q^75 + 39 * q^76 + 48 * q^79 - 15 * q^80 - 18 * q^81 - 6 * q^83 - 15 * q^90 + 24 * q^93 - 74 * q^94 - 60 * q^95 - 45 * q^96 + 21 * q^98

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