LMFDB - Embedded newform 456.2.u.a.11.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [456,2,Mod(11,456)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(456, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([3, 3, 3, 4]))

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

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

chi := DirichletCharacter("456.11");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 456 = 2^{3} \cdot 3 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	456.u (of order $6$, degree $2$, 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:	$3.64117833217$
Analytic rank:	$0$
Dimension:	$4$
Relative dimension:	$2$ over $\Q(\zeta_{6})$
Coefficient field:	$\Q(\sqrt{-2}, \sqrt{-3})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{2} + 4 $ x^4 - 2*x^2 + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{6}]$

Embedding invariants

Embedding label			11.2
Root			$1.22474 + 0.707107i$ of defining polynomial
Character	$\chi$	$=$	456.11
Dual form			456.2.u.a.83.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.22474 + 0.707107i) q^{2} +(-1.00000 + 1.41421i) q^{3} +(1.00000 + 1.73205i) q^{4} +(-2.22474 + 1.02494i) q^{6} +2.82843i q^{8} +(-1.00000 - 2.82843i) q^{9} +O(q^{10})$	\(q+(1.22474 + 0.707107i) q^{2} +(-1.00000 + 1.41421i) q^{3} +(1.00000 + 1.73205i) q^{4} +(-2.22474 + 1.02494i) q^{6} +2.82843i q^{8} +(-1.00000 - 2.82843i) q^{9} +6.61037i q^{11} +(-3.44949 - 0.317837i) q^{12} +(-2.00000 + 3.46410i) q^{16} +(-4.89898 - 2.82843i) q^{17} +(0.775255 - 4.17121i) q^{18} +(3.17423 - 2.98735i) q^{19} +(-4.67423 + 8.09601i) q^{22} +(-4.00000 - 2.82843i) q^{24} +(2.50000 + 4.33013i) q^{25} +(5.00000 + 1.41421i) q^{27} +(-4.89898 + 2.82843i) q^{32} +(-9.34847 - 6.61037i) q^{33} +(-4.00000 - 6.92820i) q^{34} +(3.89898 - 4.56048i) q^{36} +(6.00000 - 1.41421i) q^{38} +(-0.398979 - 0.230351i) q^{41} +(5.00000 - 8.66025i) q^{43} +(-11.4495 + 6.61037i) q^{44} +(-2.89898 - 6.29253i) q^{48} +7.00000 q^{49} +7.07107i q^{50} +(8.89898 - 4.09978i) q^{51} +(5.12372 + 5.26758i) q^{54} +(1.05051 + 7.47639i) q^{57} +(10.6237 + 6.13361i) q^{59} -8.00000 q^{64} +(-6.77526 - 14.7064i) q^{66} +(7.17423 + 12.4261i) q^{67} -11.3137i q^{68} +(8.00000 - 2.82843i) q^{72} +(7.84847 - 13.5939i) q^{73} +(-8.62372 - 0.794593i) q^{75} +(8.34847 + 2.51059i) q^{76} +(-7.00000 + 5.65685i) q^{81} +(-0.325765 - 0.564242i) q^{82} +17.0027i q^{83} +(12.2474 - 7.07107i) q^{86} -18.6969 q^{88} +(4.89898 - 2.82843i) q^{89} +(0.898979 - 9.75663i) q^{96} +(4.84847 - 8.39780i) q^{97} +(8.57321 + 4.94975i) q^{98} +(18.6969 - 6.61037i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{3} + 4 q^{4} - 4 q^{6} - 4 q^{9}+O(q^{10}) $ 4 * q - 4 * q^3 + 4 * q^4 - 4 * q^6 - 4 * q^9	$ 4 q - 4 q^{3} + 4 q^{4} - 4 q^{6} - 4 q^{9} - 4 q^{12} - 8 q^{16} + 8 q^{18} - 2 q^{19} - 4 q^{22} - 16 q^{24} + 10 q^{25} + 20 q^{27} - 8 q^{33} - 16 q^{34} - 4 q^{36} + 24 q^{38} + 18 q^{41} + 20 q^{43} - 36 q^{44} + 8 q^{48} + 28 q^{49} + 16 q^{51} - 4 q^{54} + 14 q^{57} + 18 q^{59} - 32 q^{64} - 32 q^{66} + 14 q^{67} + 32 q^{72} + 2 q^{73} - 10 q^{75} + 4 q^{76} - 28 q^{81} - 16 q^{82} - 16 q^{88} - 16 q^{96} - 10 q^{97} + 16 q^{99}+O(q^{100}) $ 4 * q - 4 * q^3 + 4 * q^4 - 4 * q^6 - 4 * q^9 - 4 * q^12 - 8 * q^16 + 8 * q^18 - 2 * q^19 - 4 * q^22 - 16 * q^24 + 10 * q^25 + 20 * q^27 - 8 * q^33 - 16 * q^34 - 4 * q^36 + 24 * q^38 + 18 * q^41 + 20 * q^43 - 36 * q^44 + 8 * q^48 + 28 * q^49 + 16 * q^51 - 4 * q^54 + 14 * q^57 + 18 * q^59 - 32 * q^64 - 32 * q^66 + 14 * q^67 + 32 * q^72 + 2 * q^73 - 10 * q^75 + 4 * q^76 - 28 * q^81 - 16 * q^82 - 16 * q^88 - 16 * q^96 - 10 * q^97 + 16 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/456\mathbb{Z}\right)^\times$.

$n$	$97$	$229$	$305$	$343$
$\chi(n)$	$e\left(\frac{2}{3}\right)$	$-1$	$-1$	$-1$

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 $

$2$	1.22474	+	0.707107i	0.866025	+	0.500000i
$2$	1.22474	+	0.707107i	0.866025	+	0.500000i
$3$	−1.00000	+	1.41421i	−0.577350	+	0.816497i
$3$	−1.00000	+	1.41421i	−0.577350	+	0.816497i
$4$	1.00000	+	1.73205i	0.500000	+	0.866025i
$5$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$5$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$6$	−2.22474	+	1.02494i	−0.908248	+	0.418432i
$7$		0			0		1.00000			$0$
$7$		0			0		−1.00000			$\pi$
$8$			2.82843i			1.00000i
$9$	−1.00000	−	2.82843i	−0.333333	−	0.942809i
$10$		0			0
$11$			6.61037i			1.99310i	0.0829925	+	0.996550i	$0.473552\pi$
$11$			6.61037i			1.99310i	−0.0829925	+	0.996550i	$0.526448\pi$
$12$	−3.44949	−	0.317837i	−0.995782	−	0.0917517i
$13$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$13$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$14$		0			0
$15$		0			0
$16$	−2.00000	+	3.46410i	−0.500000	+	0.866025i
$17$	−4.89898	−	2.82843i	−1.18818	−	0.685994i	−0.230285	−	0.973123i	$-0.573966\pi$
$17$	−4.89898	−	2.82843i	−1.18818	−	0.685994i	−0.957892	+	0.287129i	$0.907299\pi$
$18$	0.775255	−	4.17121i	0.182729	−	0.983163i
$19$	3.17423	−	2.98735i	0.728219	−	0.685344i
$19$	3.17423	−	2.98735i	0.728219	−	0.685344i
$20$		0			0
$21$		0			0
$22$	−4.67423	+	8.09601i	−0.996550	+	1.72608i
$23$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$23$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$24$	−4.00000	−	2.82843i	−0.816497	−	0.577350i
$25$	2.50000	+	4.33013i	0.500000	+	0.866025i
$26$		0			0
$27$	5.00000	+	1.41421i	0.962250	+	0.272166i
$28$		0			0
$29$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$29$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$30$		0			0
$31$		0			0		1.00000			$0$
$31$		0			0		−1.00000			$\pi$
$32$	−4.89898	+	2.82843i	−0.866025	+	0.500000i
$33$	−9.34847	−	6.61037i	−1.62736	−	1.15072i
$34$	−4.00000	−	6.92820i	−0.685994	−	1.18818i
$35$		0			0
$36$	3.89898	−	4.56048i	0.649830	−	0.760080i
$37$		0			0		1.00000			$0$
$37$		0			0		−1.00000			$\pi$
$38$	6.00000	−	1.41421i	0.973329	−	0.229416i
$39$		0			0
$40$		0			0
$41$	−0.398979	−	0.230351i	−0.0623101	−	0.0359748i	0.468521	−	0.883452i	$-0.344787\pi$
$41$	−0.398979	−	0.230351i	−0.0623101	−	0.0359748i	−0.530831	+	0.847477i	$0.678120\pi$
$42$		0			0
$43$	5.00000	−	8.66025i	0.762493	−	1.32068i	−0.179069	−	0.983836i	$-0.557309\pi$
$43$	5.00000	−	8.66025i	0.762493	−	1.32068i	0.941562	−	0.336840i	$-0.109358\pi$
$44$	−11.4495	+	6.61037i	−1.72608	+	0.996550i
$45$		0			0
$46$		0			0
$47$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$47$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$48$	−2.89898	−	6.29253i	−0.418432	−	0.908248i
$49$	7.00000			1.00000
$50$			7.07107i			1.00000i
$51$	8.89898	−	4.09978i	1.24611	−	0.574083i
$52$		0			0
$53$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$53$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$54$	5.12372	+	5.26758i	0.697251	+	0.716827i
$55$		0			0
$56$		0			0
$57$	1.05051	+	7.47639i	0.139143	+	0.990272i
$58$		0			0
$59$	10.6237	+	6.13361i	1.38309	+	0.798528i	0.992524	−	0.122047i	$-0.0389457\pi$
$59$	10.6237	+	6.13361i	1.38309	+	0.798528i	0.390567	+	0.920575i	$0.372279\pi$
$60$		0			0
$61$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$61$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$62$		0			0
$63$		0			0
$64$	−8.00000			−1.00000
$65$		0			0
$66$	−6.77526	−	14.7064i	−0.833976	−	1.81023i
$67$	7.17423	+	12.4261i	0.876472	+	1.51809i	0.855186	+	0.518321i	$0.173443\pi$
$67$	7.17423	+	12.4261i	0.876472	+	1.51809i	0.0212861	+	0.999773i	$0.493224\pi$
$68$		−	11.3137i		−	1.37199i
$69$		0			0
$70$		0			0
$71$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$71$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$72$	8.00000	−	2.82843i	0.942809	−	0.333333i
$73$	7.84847	−	13.5939i	0.918594	−	1.59105i	0.117041	−	0.993127i	$-0.462659\pi$
$73$	7.84847	−	13.5939i	0.918594	−	1.59105i	0.801553	−	0.597924i	$-0.204008\pi$
$74$		0			0
$75$	−8.62372	−	0.794593i	−0.995782	−	0.0917517i
$76$	8.34847	+	2.51059i	0.957635	+	0.287984i
$77$		0			0
$78$		0			0
$79$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$79$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$80$		0			0
$81$	−7.00000	+	5.65685i	−0.777778	+	0.628539i
$82$	−0.325765	−	0.564242i	−0.0359748	−	0.0623101i
$83$			17.0027i			1.86629i	0.359506	+	0.933143i	$0.382945\pi$
$83$			17.0027i			1.86629i	−0.359506	+	0.933143i	$0.617055\pi$
$84$		0			0
$85$		0			0
$86$	12.2474	−	7.07107i	1.32068	−	0.762493i
$87$		0			0
$88$	−18.6969			−1.99310
$89$	4.89898	−	2.82843i	0.519291	−	0.299813i	−0.217354	−	0.976093i	$-0.569742\pi$
$89$	4.89898	−	2.82843i	0.519291	−	0.299813i	0.736644	+	0.676280i	$0.236409\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$	0.898979	−	9.75663i	0.0917517	−	0.995782i
$97$	4.84847	−	8.39780i	0.492287	−	0.852667i	−0.507673	−	0.861550i	$-0.669494\pi$
$97$	4.84847	−	8.39780i	0.492287	−	0.852667i	0.999961	+	0.00888289i	$0.00282755\pi$
$98$	8.57321	+	4.94975i	0.866025	+	0.500000i
$99$	18.6969	−	6.61037i	1.87911	−	0.664367i
$100$	−5.00000	+	8.66025i	−0.500000	+	0.866025i
$101$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$101$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$102$	13.7980	+	1.27135i	1.36620	+	0.125882i
$103$		0			0		1.00000			$0$
$103$		0			0		−1.00000			$\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		−	19.7990i		−	1.91404i	−0.290021	−	0.957020i	$-0.593662\pi$
$107$		−	19.7990i		−	1.91404i	0.290021	−	0.957020i	$-0.406338\pi$
$108$	2.55051	+	10.0745i	0.245423	+	0.969416i
$109$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$109$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$		−	9.93160i		−	0.934287i	−0.884182	−	0.467143i	$-0.845283\pi$
$113$		−	9.93160i		−	0.934287i	0.884182	−	0.467143i	$-0.154717\pi$
$114$	−4.00000	+	9.89949i	−0.374634	+	0.927173i
$115$		0			0
$116$		0			0
$117$		0			0
$118$	8.67423	+	15.0242i	0.798528	+	1.38309i
$119$		0			0
$120$		0			0
$121$	−32.6969			−2.97245
$122$		0			0
$123$	0.724745	−	0.333891i	0.0653480	−	0.0301060i
$124$		0			0
$125$		0			0
$126$		0			0
$127$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$127$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$128$	−9.79796	−	5.65685i	−0.866025	−	0.500000i
$129$	7.24745	+	15.7313i	0.638102	+	1.38507i
$130$		0			0
$131$	−7.37628	−	4.25869i	−0.644468	−	0.372084i	0.141865	−	0.989886i	$-0.454690\pi$
$131$	−7.37628	−	4.25869i	−0.644468	−	0.372084i	−0.786334	+	0.617802i	$0.788023\pi$
$132$	2.10102	−	22.8024i	0.182870	−	1.98469i
$133$		0			0
$134$			20.2918i			1.75294i
$135$		0			0
$136$	8.00000	−	13.8564i	0.685994	−	1.18818i
$137$	−5.29796	+	3.05878i	−0.452635	+	0.261329i	−0.708942	−	0.705266i	$-0.750827\pi$
$137$	−5.29796	+	3.05878i	−0.452635	+	0.261329i	0.256307	+	0.966595i	$0.417494\pi$
$138$		0			0
$139$	−9.17423	−	15.8902i	−0.778148	−	1.34779i	−0.933008	−	0.359856i	$-0.882826\pi$
$139$	−9.17423	−	15.8902i	−0.778148	−	1.34779i	0.154859	−	0.987937i	$-0.450508\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$	11.7980	+	2.19275i	0.983163	+	0.182729i
$145$		0			0
$146$	19.2247	−	11.0994i	1.59105	−	0.918594i
$147$	−7.00000	+	9.89949i	−0.577350	+	0.816497i
$148$		0			0
$149$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$149$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$150$	−10.0000	−	7.07107i	−0.816497	−	0.577350i
$151$		0			0		1.00000			$0$
$151$		0			0		−1.00000			$\pi$
$152$	8.44949	+	8.97809i	0.685344	+	0.728219i
$153$	−3.10102	+	16.6848i	−0.250703	+	1.34889i
$154$		0			0
$155$		0			0
$156$		0			0
$157$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$157$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$	−12.5732	+	1.97846i	−0.987845	+	0.155442i
$163$	−23.0454			−1.80506			−0.902528	−	0.430632i	$-0.858291\pi$
$163$	−23.0454			−1.80506			−0.902528	+	0.430632i	$0.858291\pi$
$164$		−	0.921404i		−	0.0719495i
$165$		0			0
$166$	−12.0227	+	20.8239i	−0.933143	+	1.61625i
$167$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$167$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$168$		0			0
$169$	−6.50000	+	11.2583i	−0.500000	+	0.866025i
$170$		0			0
$171$	−11.6237	−	5.99075i	−0.888888	−	0.458124i
$172$	20.0000			1.52499
$173$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$173$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$174$		0			0
$175$		0			0
$176$	−22.8990	−	13.2207i	−1.72608	−	0.996550i
$177$	−19.2980	+	8.89060i	−1.45052	+	0.668259i
$178$	8.00000			0.599625
$179$			25.4880i			1.90506i	0.304446	+	0.952529i	$0.401529\pi$
$179$			25.4880i			1.90506i	−0.304446	+	0.952529i	$0.598471\pi$
$180$		0			0
$181$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$181$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$		0			0
$186$		0			0
$187$	18.6969	−	32.3840i	1.36726	−	2.36816i
$188$		0			0
$189$		0			0
$190$		0			0
$191$		0			0			−	1.00000i	$-0.5\pi$
$191$		0			0				1.00000i	$0.5\pi$
$192$	8.00000	−	11.3137i	0.577350	−	0.816497i
$193$	11.0000	−	19.0526i	0.791797	−	1.37143i	−0.133056	−	0.991109i	$-0.542479\pi$
$193$	11.0000	−	19.0526i	0.791797	−	1.37143i	0.924853	−	0.380325i	$-0.124188\pi$
$194$	11.8763	−	6.85677i	0.852667	−	0.492287i
$195$		0			0
$196$	7.00000	+	12.1244i	0.500000	+	0.866025i
$197$		0			0			−	1.00000i	$-0.5\pi$
$197$		0			0				1.00000i	$0.5\pi$
$198$	27.5732	+	5.12472i	1.95954	+	0.364198i
$199$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$199$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$200$	−12.2474	+	7.07107i	−0.866025	+	0.500000i
$201$	−24.7474	−	2.28024i	−1.74555	−	0.160836i
$202$		0			0
$203$		0			0
$204$	16.0000	+	11.3137i	1.12022	+	0.792118i
$205$		0			0
$206$		0			0
$207$		0			0
$208$		0			0
$209$	19.7474	+	20.9829i	1.36596	+	1.45141i
$210$		0			0
$211$	−7.00000	+	12.1244i	−0.481900	+	0.834675i	−0.999784	−	0.0207756i	$-0.993386\pi$
$211$	−7.00000	+	12.1244i	−0.481900	+	0.834675i	0.517884	+	0.855451i	$0.326720\pi$
$212$		0			0
$213$		0			0
$214$	14.0000	−	24.2487i	0.957020	−	1.65761i
$215$		0			0
$216$	−4.00000	+	14.1421i	−0.272166	+	0.962250i
$217$		0			0
$218$		0			0
$219$	11.3763	+	24.6934i	0.768737	+	1.66862i
$220$		0			0
$221$		0			0
$222$		0			0
$223$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$223$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$224$		0			0
$225$	9.74745	−	11.4012i	0.649830	−	0.760080i
$226$	7.02270	−	12.1637i	0.467143	−	0.809116i
$227$		−	24.5665i		−	1.63054i	−0.579082	−	0.815270i	$-0.696589\pi$
$227$		−	24.5665i		−	1.63054i	0.579082	−	0.815270i	$-0.303411\pi$
$228$	−11.8990	+	9.29593i	−0.788029	+	0.615638i
$229$		0			0		1.00000			$0$
$229$		0			0		−1.00000			$\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−20.0505	−	11.5762i	−1.31355	−	0.758380i	−0.330870	−	0.943676i	$-0.607342\pi$
$233$	−20.0505	−	11.5762i	−1.31355	−	0.758380i	−0.982683	+	0.185296i	$0.940675\pi$
$234$		0			0
$235$		0			0
$236$			24.5344i			1.59706i
$237$		0			0
$238$		0			0
$239$		0			0			−	1.00000i	$-0.5\pi$
$239$		0			0				1.00000i	$0.5\pi$
$240$		0			0
$241$	−0.848469	−	1.46959i	−0.0546547	−	0.0946647i	0.837404	−	0.546585i	$-0.184072\pi$
$241$	−0.848469	−	1.46959i	−0.0546547	−	0.0946647i	−0.892058	+	0.451920i	$0.850739\pi$
$242$	−40.0454	−	23.1202i	−2.57422	−	1.48622i
$243$	−1.00000	−	15.5563i	−0.0641500	−	0.997940i
$244$		0			0
$245$		0			0
$246$	1.12372	+	0.103540i	0.0716460	+	0.00660149i
$247$		0			0
$248$		0			0
$249$	−24.0454	−	17.0027i	−1.52382	−	1.07750i
$250$		0			0
$251$	−17.9722	+	10.3763i	−1.13439	+	0.654943i	−0.945036	−	0.326965i	$-0.893974\pi$
$251$	−17.9722	+	10.3763i	−1.13439	+	0.654943i	−0.189358	+	0.981908i	$0.560641\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$		0			0
$256$	−8.00000	−	13.8564i	−0.500000	−	0.866025i
$257$	27.3990	−	15.8188i	1.70910	−	0.986750i	0.773427	−	0.633885i	$-0.218541\pi$
$257$	27.3990	−	15.8188i	1.70910	−	0.986750i	0.935674	−	0.352865i	$-0.114792\pi$
$258$	−2.24745	+	24.3916i	−0.139920	+	1.51855i
$259$		0			0
$260$		0			0
$261$		0			0
$262$	−6.02270	−	10.4316i	−0.372084	−	0.644468i
$263$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$263$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$264$	18.6969	−	26.4415i	1.15072	−	1.62736i
$265$		0			0
$266$		0			0
$267$	−0.898979	+	9.75663i	−0.0550167	+	0.597096i
$268$	−14.3485	+	24.8523i	−0.876472	+	1.51809i
$269$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$269$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$270$		0			0
$271$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$271$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$272$	19.5959	−	11.3137i	1.18818	−	0.685994i
$273$		0			0
$274$	−8.65153			−0.522658
$275$	−28.6237	+	16.5259i	−1.72608	+	0.996550i
$276$		0			0
$277$		0			0		1.00000			$0$
$277$		0			0		−1.00000			$\pi$
$278$		−	25.9487i		−	1.55630i
$279$		0			0
$280$		0			0
$281$	25.7474	−	14.8653i	1.53596	−	0.886789i	0.536895	−	0.843649i	$-0.319597\pi$
$281$	25.7474	−	14.8653i	1.53596	−	0.886789i	0.999069	−	0.0431402i	$-0.0137362\pi$
$282$		0			0
$283$	−16.5227	+	28.6182i	−0.982173	+	1.70117i	−0.328291	+	0.944577i	$0.606473\pi$
$283$	−16.5227	+	28.6182i	−0.982173	+	1.70117i	−0.653882	+	0.756596i	$0.726861\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$	12.8990	+	11.0280i	0.760080	+	0.649830i
$289$	7.50000	+	12.9904i	0.441176	+	0.764140i
$290$		0			0
$291$	7.02781	+	15.2546i	0.411977	+	0.894238i
$292$	31.3939			1.83719
$293$		0			0			−	1.00000i	$-0.5\pi$
$293$		0			0				1.00000i	$0.5\pi$
$294$	−15.5732	+	7.17461i	−0.908248	+	0.418432i
$295$		0			0
$296$		0			0
$297$	−9.34847	+	33.0518i	−0.542453	+	1.91786i
$298$		0			0
$299$		0			0
$300$	−7.24745	−	15.7313i	−0.418432	−	0.908248i
$301$		0			0
$302$		0			0
$303$		0			0
$304$	4.00000	+	16.9706i	0.229416	+	0.973329i
$305$		0			0
$306$	−15.5959	+	18.2419i	−0.891559	+	1.04282i
$307$	−4.82577	+	8.35847i	−0.275421	+	0.477043i	−0.970241	−	0.242140i	$-0.922151\pi$
$307$	−4.82577	+	8.35847i	−0.275421	+	0.477043i	0.694820	+	0.719183i	$0.255484\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		0			0			−	1.00000i	$-0.5\pi$
$311$		0			0				1.00000i	$0.5\pi$
$312$		0			0
$313$	12.1969	+	21.1257i	0.689412	+	1.19410i	0.972028	+	0.234863i	$0.0754642\pi$
$313$	12.1969	+	21.1257i	0.689412	+	1.19410i	−0.282617	+	0.959233i	$0.591202\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$317$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$	28.0000	+	19.7990i	1.56281	+	1.10507i
$322$		0			0
$323$	−24.0000	+	5.65685i	−1.33540	+	0.314756i
$324$	−16.7980	−	6.46750i	−0.933220	−	0.359306i
$325$		0			0
$326$	−28.2247	−	16.2956i	−1.56322	−	0.902528i
$327$		0			0
$328$	0.651531	−	1.12848i	0.0359748	−	0.0623101i
$329$		0			0
$330$		0			0
$331$	9.04541			0.497181			0.248590	−	0.968609i	$-0.420033\pi$
$331$	9.04541			0.497181			0.248590	+	0.968609i	$0.420033\pi$
$332$	−29.4495	+	17.0027i	−1.61625	+	0.933143i
$333$		0			0
$334$		0			0
$335$		0			0
$336$		0			0
$337$	−11.1969	+	19.3937i	−0.609936	+	1.05644i	0.381314	+	0.924445i	$0.375472\pi$
$337$	−11.1969	+	19.3937i	−0.609936	+	1.05644i	−0.991250	+	0.131995i	$0.957862\pi$
$338$	−15.9217	+	9.19239i	−0.866025	+	0.500000i
$339$	14.0454	+	9.93160i	0.762842	+	0.539411i
$340$		0			0
$341$		0			0
$342$	−10.0000	−	15.5563i	−0.540738	−	0.841191i
$343$		0			0
$344$	24.4949	+	14.1421i	1.32068	+	0.762493i
$345$		0			0
$346$		0			0
$347$	−20.4217	−	11.7905i	−1.09629	−	0.632945i	−0.161048	−	0.986947i	$-0.551488\pi$
$347$	−20.4217	−	11.7905i	−1.09629	−	0.632945i	−0.935245	+	0.354001i	$0.884821\pi$
$348$		0			0
$349$		0			0		1.00000			$0$
$349$		0			0		−1.00000			$\pi$
$350$		0			0
$351$		0			0
$352$	−18.6969	−	32.3840i	−0.996550	−	1.72608i
$353$		−	37.2945i		−	1.98498i	−0.122308	−	0.992492i	$-0.539030\pi$
$353$		−	37.2945i		−	1.98498i	0.122308	−	0.992492i	$-0.460970\pi$
$354$	−29.9217	−	2.75699i	−1.59032	−	0.146533i
$355$		0			0
$356$	9.79796	+	5.65685i	0.519291	+	0.299813i
$357$		0			0
$358$	−18.0227	+	31.2162i	−0.952529	+	1.64983i
$359$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$359$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$360$		0			0
$361$	1.15153	−	18.9651i	0.0606069	−	0.998162i
$362$		0			0
$363$	32.6969	−	46.2405i	1.71614	−	2.42699i
$364$		0			0
$365$		0			0
$366$		0			0
$367$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$367$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$368$		0			0
$369$	−0.252551	+	1.35884i	−0.0131473	+	0.0707381i
$370$		0			0
$371$		0			0
$372$		0			0
$373$		0			0		1.00000			$0$
$373$		0			0		−1.00000			$\pi$
$374$	45.7980	−	26.4415i	2.36816	−	1.36726i
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$	38.0000			1.95193			0.975964	−	0.217930i	$-0.0699304\pi$
$379$	38.0000			1.95193			0.975964	+	0.217930i	$0.0699304\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$383$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$384$	17.7980	−	8.19955i	0.908248	−	0.418432i
$385$		0			0
$386$	26.9444	−	15.5563i	1.37143	−	0.791797i
$387$	−29.4949	−	5.48188i	−1.49931	−	0.278660i
$388$	19.3939			0.984575
$389$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$389$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$390$		0			0
$391$		0			0
$392$			19.7990i			1.00000i
$393$	13.3990	−	6.17293i	0.675889	−	0.311383i
$394$		0			0
$395$		0			0
$396$	30.1464	+	25.7737i	1.51492	+	1.29518i
$397$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$397$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$398$		0			0
$399$		0			0
$400$	−20.0000			−1.00000
$401$	12.6464	+	7.30142i	0.631532	+	0.364615i	0.781345	−	0.624099i	$-0.214534\pi$
$401$	12.6464	+	7.30142i	0.631532	+	0.364615i	−0.149813	+	0.988714i	$0.547867\pi$
$402$	−28.6969	−	20.2918i	−1.43127	−	1.01206i
$403$		0			0
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$	11.5959	+	25.1701i	0.574083	+	1.24611i
$409$	9.19694	+	15.9296i	0.454759	+	0.787666i	0.998674	−	0.0514740i	$-0.0163919\pi$
$409$	9.19694	+	15.9296i	0.454759	+	0.787666i	−0.543915	+	0.839140i	$0.683059\pi$
$410$		0			0
$411$	0.972194	−	10.5512i	0.0479548	−	0.520453i
$412$		0			0
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$	31.6464	+	2.91591i	1.54973	+	0.142793i
$418$	9.34847	+	39.6622i	0.457249	+	1.93994i
$419$		−	36.7696i		−	1.79631i	−0.439679	−	0.898155i	$-0.644908\pi$
$419$		−	36.7696i		−	1.79631i	0.439679	−	0.898155i	$-0.355092\pi$
$420$		0			0
$421$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$421$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$422$	−17.1464	+	9.89949i	−0.834675	+	0.481900i
$423$		0			0
$424$		0			0
$425$		−	28.2843i		−	1.37199i
$426$		0			0
$427$		0			0
$428$	34.2929	−	19.7990i	1.65761	−	0.957020i
$429$		0			0
$430$		0			0
$431$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$431$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$432$	−14.8990	+	14.4921i	−0.716827	+	0.697251i
$433$	−19.0000	−	32.9090i	−0.913082	−	1.58150i	−0.809686	−	0.586864i	$-0.800362\pi$
$433$	−19.0000	−	32.9090i	−0.913082	−	1.58150i	−0.103396	−	0.994640i	$-0.532971\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		0			0
$438$	−3.52781	+	38.2873i	−0.168565	+	1.82944i
$439$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$439$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$440$		0			0
$441$	−7.00000	−	19.7990i	−0.333333	−	0.942809i
$442$		0			0
$443$	32.7247	−	18.8936i	1.55480	−	0.897664i	0.557059	−	0.830473i	$-0.311930\pi$
$443$	32.7247	−	18.8936i	1.55480	−	0.897664i	0.997740	−	0.0671913i	$-0.0214038\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$		0			0
$449$			33.5446i			1.58307i	0.611124	+	0.791535i	$0.290718\pi$
$449$			33.5446i			1.58307i	−0.611124	+	0.791535i	$0.709282\pi$
$450$	20.0000	−	7.07107i	0.942809	−	0.333333i
$451$	1.52270	−	2.63740i	0.0717013	−	0.124190i
$452$	17.2020	−	9.93160i	0.809116	−	0.467143i
$453$		0			0
$454$	17.3712	−	30.0878i	0.815270	−	1.41209i
$455$		0			0
$456$	−21.1464	+	2.97129i	−0.990272	+	0.139143i
$457$	16.3939			0.766873			0.383437	−	0.923567i	$-0.374740\pi$
$457$	16.3939			0.766873			0.383437	+	0.923567i	$0.374740\pi$
$458$		0			0
$459$	−20.4949	−	21.0703i	−0.956620	−	0.983479i
$460$		0			0
$461$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$461$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$462$		0			0
$463$		0			0		1.00000			$0$
$463$		0			0		−1.00000			$\pi$
$464$		0			0
$465$		0			0
$466$	−16.3712	−	28.3557i	−0.758380	−	1.31355i
$467$			10.4244i			0.482384i	0.970477	+	0.241192i	$0.0775384\pi$
$467$			10.4244i			0.482384i	−0.970477	+	0.241192i	$0.922462\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$		0			0
$472$	−17.3485	+	30.0484i	−0.798528	+	1.38309i
$473$	57.2474	+	33.0518i	2.63224	+	1.51972i
$474$		0			0
$475$	20.8712	+	6.27647i	0.957635	+	0.287984i
$476$		0			0
$477$		0			0
$478$		0			0
$479$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$479$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$480$		0			0
$481$		0			0
$482$		−	2.39983i		−	0.109309i
$483$		0			0
$484$	−32.6969	−	56.6328i	−1.48622	−	2.57422i
$485$		0			0
$486$	9.77526	−	19.7597i	0.443415	−	0.896317i
$487$		0			0		1.00000			$0$
$487$		0			0		−1.00000			$\pi$
$488$		0			0
$489$	23.0454	−	32.5911i	1.04215	−	1.47382i
$490$		0			0
$491$	−12.2474	−	7.07107i	−0.552720	−	0.319113i	0.197499	−	0.980303i	$-0.436718\pi$
$491$	−12.2474	−	7.07107i	−0.552720	−	0.319113i	−0.750218	+	0.661190i	$0.770052\pi$
$492$	1.30306	+	0.921404i	0.0587466	+	0.0415401i
$493$		0			0
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$	−17.4268	−	37.8266i	−0.780913	−	1.69505i
$499$	−14.8712	+	25.7576i	−0.665725	+	1.15307i	0.313363	+	0.949633i	$0.398544\pi$
$499$	−14.8712	+	25.7576i	−0.665725	+	1.15307i	−0.979088	+	0.203436i	$0.934789\pi$
$500$		0			0
$501$		0			0
$502$	−29.3485			−1.30989
$503$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$503$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	−9.42168	−	20.4507i	−0.418432	−	0.908248i
$508$		0			0
$509$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$509$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$510$		0			0
$511$		0			0
$512$		−	22.6274i		−	1.00000i
$513$	20.0959	−	10.4477i	0.887256	−	0.461276i
$514$	44.7423			1.97350
$515$		0			0
$516$	−20.0000	+	28.2843i	−0.880451	+	1.24515i
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$			27.8236i			1.21897i	0.792797	+	0.609486i	$0.208624\pi$
$521$			27.8236i			1.21897i	−0.792797	+	0.609486i	$0.791376\pi$
$522$		0			0
$523$	−19.0000	−	32.9090i	−0.830812	−	1.43901i	−0.897395	−	0.441228i	$-0.854543\pi$
$523$	−19.0000	−	32.9090i	−0.830812	−	1.43901i	0.0665832	−	0.997781i	$-0.478790\pi$
$524$		−	17.0348i		−	0.744168i
$525$		0			0
$526$		0			0
$527$		0			0
$528$	41.5959	−	19.1633i	1.81023	−	0.833976i
$529$	11.5000	−	19.9186i	0.500000	−	0.866025i
$530$		0			0
$531$	6.72474	−	36.1820i	0.291829	−	1.57017i
$532$		0			0
$533$		0			0
$534$	−8.00000	+	11.3137i	−0.346194	+	0.489592i
$535$		0			0
$536$	−35.1464	+	20.2918i	−1.51809	+	0.876472i
$537$	−36.0454	−	25.4880i	−1.55547	−	1.09989i
$538$		0			0
$539$			46.2726i			1.99310i
$540$		0			0
$541$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$541$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$542$		0			0
$543$		0			0
$544$	32.0000			1.37199
$545$		0			0
$546$		0			0
$547$	23.0000	+	39.8372i	0.983409	+	1.70331i	0.648803	+	0.760956i	$0.275270\pi$
$547$	23.0000	+	39.8372i	0.983409	+	1.70331i	0.334606	+	0.942358i	$0.391397\pi$
$548$	−10.5959	−	6.11756i	−0.452635	−	0.261329i
$549$		0			0
$550$	−46.7423			−1.99310
$551$		0			0
$552$		0			0
$553$		0			0
$554$		0			0
$555$		0			0
$556$	18.3485	−	31.7805i	0.778148	−	1.34779i
$557$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$557$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$	27.1010	+	58.8255i	1.14421	+	2.48362i
$562$	42.0454			1.77358
$563$		−	7.59599i		−	0.320133i	−0.987106	−	0.160066i	$-0.948829\pi$
$563$		−	7.59599i		−	0.320133i	0.987106	−	0.160066i	$-0.0511708\pi$
$564$		0			0
$565$		0			0
$566$	−40.4722	+	23.3666i	−1.70117	+	0.982173i
$567$		0			0
$568$		0			0
$569$			22.6274i			0.948591i	0.880366	+	0.474295i	$0.157297\pi$
$569$			22.6274i			0.948591i	−0.880366	+	0.474295i	$0.842703\pi$
$570$		0			0
$571$	−25.7423			−1.07728			−0.538642	−	0.842535i	$-0.681062\pi$
$571$	−25.7423			−1.07728			−0.538642	+	0.842535i	$0.681062\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$	8.00000	+	22.6274i	0.333333	+	0.942809i
$577$	−12.3939			−0.515964			−0.257982	−	0.966150i	$-0.583058\pi$
$577$	−12.3939			−0.515964			−0.257982	+	0.966150i	$0.583058\pi$
$578$			21.2132i			0.882353i
$579$	15.9444	+	34.6089i	0.662626	+	1.43830i
$580$		0			0
$581$		0			0
$582$	−2.17934	+	23.6524i	−0.0903364	+	0.980422i
$583$		0			0
$584$	38.4495	+	22.1988i	1.59105	+	0.918594i
$585$		0			0
$586$		0			0
$587$	−41.6413	−	24.0416i	−1.71872	−	0.992304i	−0.921272	−	0.388918i	$-0.872849\pi$
$587$	−41.6413	−	24.0416i	−1.71872	−	0.992304i	−0.797449	−	0.603386i	$-0.793818\pi$
$588$	−24.1464	−	2.22486i	−0.995782	−	0.0917517i
$589$		0			0
$590$		0			0
$591$		0			0
$592$		0			0
$593$	6.09592	−	3.51948i	0.250329	−	0.144528i	−0.369586	−	0.929197i	$-0.620500\pi$
$593$	6.09592	−	3.51948i	0.250329	−	0.144528i	0.619915	+	0.784669i	$0.287167\pi$
$594$	−34.8207	+	33.8697i	−1.42871	+	1.38969i
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$599$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$600$	2.24745	−	24.3916i	0.0917517	−	0.995782i
$601$	37.6969			1.53769			0.768845	−	0.639435i	$-0.220832\pi$
$601$	37.6969			1.53769			0.768845	+	0.639435i	$0.220832\pi$
$602$		0			0
$603$	27.9722	−	32.7179i	1.13912	−	1.33238i
$604$		0			0
$605$		0			0
$606$		0			0
$607$		0			0		1.00000			$0$
$607$		0			0		−1.00000			$\pi$
$608$	−7.10102	+	23.6130i	−0.287984	+	0.957635i
$609$		0			0
$610$		0			0
$611$		0			0
$612$	−32.0000	+	11.3137i	−1.29352	+	0.457330i
$613$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$613$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$614$	−11.8207	+	6.82466i	−0.477043	+	0.275421i
$615$		0			0
$616$		0			0
$617$	5.35357	−	3.09089i	0.215527	−	0.124434i	−0.388351	−	0.921512i	$-0.626955\pi$
$617$	5.35357	−	3.09089i	0.215527	−	0.124434i	0.603877	+	0.797077i	$0.293622\pi$
$618$		0			0
$619$	26.0000			1.04503			0.522514	−	0.852631i	$-0.324994\pi$
$619$	26.0000			1.04503			0.522514	+	0.852631i	$0.324994\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	−12.5000	+	21.6506i	−0.500000	+	0.866025i
$626$			34.4982i			1.37882i
$627$	−49.4217	+	6.94426i	−1.97371	+	0.277327i
$628$		0			0
$629$		0			0
$630$		0			0
$631$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$631$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$632$		0			0
$633$	−10.1464	−	22.0239i	−0.403284	−	0.875369i
$634$		0			0
$635$		0			0
$636$		0			0
$637$		0			0
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−43.7474	−	25.2576i	−1.72792	−	0.997615i	−0.898470	−	0.439034i	$-0.855321\pi$
$641$	−43.7474	−	25.2576i	−1.72792	−	0.997615i	−0.829450	−	0.558581i	$-0.811346\pi$
$642$	20.2929	+	44.0477i	0.800895	+	1.73842i
$643$	8.82577	−	15.2867i	0.348054	−	0.602848i	−0.637850	−	0.770161i	$-0.720176\pi$
$643$	8.82577	−	15.2867i	0.348054	−	0.602848i	0.985904	+	0.167313i	$0.0535092\pi$
$644$		0			0
$645$		0			0
$646$	−33.3939	−	10.0424i	−1.31386	−	0.395111i
$647$		0			0			−	1.00000i	$-0.5\pi$
$647$		0			0				1.00000i	$0.5\pi$
$648$	−16.0000	−	19.7990i	−0.628539	−	0.777778i
$649$	−40.5454	+	70.2267i	−1.59155	+	2.75664i
$650$		0			0
$651$		0			0
$652$	−23.0454	−	39.9158i	−0.902528	−	1.56322i
$653$		0			0			−	1.00000i	$-0.5\pi$
$653$		0			0				1.00000i	$0.5\pi$
$654$		0			0
$655$		0			0
$656$	1.59592	−	0.921404i	0.0623101	−	0.0359748i
$657$	−46.2980	−	8.60488i	−1.80626	−	0.335708i
$658$		0			0
$659$	41.6413	−	24.0416i	1.62212	−	0.936529i	0.635763	−	0.771885i	$-0.280686\pi$
$659$	41.6413	−	24.0416i	1.62212	−	0.936529i	0.986353	−	0.164644i	$-0.0526477\pi$
$660$		0			0
$661$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$661$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$662$	11.0783	+	6.39607i	0.430571	+	0.248590i
$663$		0			0
$664$	−48.0908			−1.86629
$665$		0			0
$666$		0			0
$667$		0			0
$668$		0			0
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$	−10.0000			−0.385472			−0.192736	−	0.981251i	$-0.561736\pi$
$673$	−10.0000			−0.385472			−0.192736	+	0.981251i	$0.561736\pi$
$674$	−27.4268	+	15.8349i	−1.05644	+	0.609936i
$675$	6.37628	+	25.1862i	0.245423	+	0.969416i
$676$	−26.0000			−1.00000
$677$		0			0			−	1.00000i	$-0.5\pi$
$677$		0			0				1.00000i	$0.5\pi$
$678$	10.1793	+	22.0953i	0.390935	+	0.848564i
$679$		0			0
$680$		0			0
$681$	34.7423	+	24.5665i	1.33133	+	0.941392i
$682$		0			0
$683$			31.1127i			1.19049i	0.803543	+	0.595247i	$0.202946\pi$
$683$			31.1127i			1.19049i	−0.803543	+	0.595247i	$0.797054\pi$
$684$	−1.24745	−	26.1236i	−0.0476974	−	0.998862i
$685$		0			0
$686$		0			0
$687$		0			0
$688$	20.0000	+	34.6410i	0.762493	+	1.32068i
$689$		0			0
$690$		0			0
$691$	−46.0000			−1.74992			−0.874961	−	0.484193i	$-0.839113\pi$
$691$	−46.0000			−1.74992			−0.874961	+	0.484193i	$0.839113\pi$
$692$		0			0
$693$		0			0
$694$	−16.6742	−	28.8806i	−0.632945	−	1.09629i
$695$		0			0
$696$		0			0
$697$	1.30306	+	2.25697i	0.0493570	+	0.0854888i
$698$		0			0
$699$	36.4217	−	16.7795i	1.37759	−	0.634660i
$700$		0			0
$701$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$701$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$702$		0			0
$703$		0			0
$704$		−	52.8829i		−	1.99310i
$705$		0			0
$706$	26.3712	−	45.6762i	0.992492	−	1.71905i
$707$		0			0
$708$	−34.6969	−	24.5344i	−1.30399	−	0.922061i
$709$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$709$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$710$		0			0
$711$		0			0
$712$	8.00000	+	13.8564i	0.299813	+	0.519291i
$713$		0			0
$714$		0			0
$715$		0			0
$716$	−44.1464	+	25.4880i	−1.64983	+	0.952529i
$717$		0			0
$718$		0			0
$719$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$719$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$720$		0			0
$721$		0			0
$722$	14.8207	−	22.4131i	0.551568	−	0.834130i
$723$	2.92679	+	0.269675i	0.108848	+	0.0100293i
$724$		0			0
$725$		0			0
$726$	72.7423	−	33.5125i	2.69972	−	1.24377i
$727$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$727$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$728$		0			0
$729$	23.0000	+	14.1421i	0.851852	+	0.523783i
$730$		0			0
$731$	−48.9898	+	28.2843i	−1.81195	+	1.04613i
$732$		0			0
$733$		0			0		1.00000			$0$
$733$		0			0		−1.00000			$\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−82.1413	+	47.4243i	−3.02571	+	1.74690i
$738$	−1.27015	+	1.48565i	−0.0467550	+	0.0546874i
$739$	−26.8712	+	46.5422i	−0.988472	+	1.71208i	−0.363117	+	0.931744i	$0.618287\pi$
$739$	−26.8712	+	46.5422i	−0.988472	+	1.71208i	−0.625355	+	0.780340i	$0.715046\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$743$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$	48.0908	−	17.0027i	1.75955	−	0.622095i
$748$	74.7878			2.73451
$749$		0			0
$750$		0			0
$751$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$751$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$752$		0			0
$753$	3.29796	−	35.7928i	0.120184	−	1.30436i
$754$		0			0
$755$		0			0
$756$		0			0
$757$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$757$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$758$	46.5403	+	26.8701i	1.69042	+	0.975964i
$759$		0			0
$760$		0			0
$761$			52.4222i			1.90030i	0.311787	+	0.950152i	$0.399073\pi$
$761$			52.4222i			1.90030i	−0.311787	+	0.950152i	$0.600927\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$	27.5959	+	2.54270i	0.995782	+	0.0917517i
$769$	11.0000	+	19.0526i	0.396670	+	0.687053i	0.993313	−	0.115454i	$-0.0368323\pi$
$769$	11.0000	+	19.0526i	0.396670	+	0.687053i	−0.596643	+	0.802507i	$0.703499\pi$
$770$		0			0
$771$	−5.02781	+	54.5668i	−0.181072	+	1.96518i
$772$	44.0000			1.58359
$773$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$773$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$774$	−32.2474	−	27.5699i	−1.15911	−	0.990981i
$775$		0			0
$776$	23.7526	+	13.7135i	0.852667	+	0.492287i
$777$		0			0
$778$		0			0
$779$	−1.95459	+	0.460702i	−0.0700305	+	0.0165064i
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$	−14.0000	+	24.2487i	−0.500000	+	0.866025i
$785$		0			0
$786$	20.7753	+	1.91424i	0.741029	+	0.0682787i
$787$	−47.0454			−1.67699			−0.838494	−	0.544911i	$-0.816563\pi$
$787$	−47.0454			−1.67699			−0.838494	+	0.544911i	$0.816563\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$	18.6969	+	52.8829i	0.664367	+	1.87911i
$793$		0			0
$794$		0			0
$795$		0			0
$796$		0			0
$797$		0			0			−	1.00000i	$-0.5\pi$
$797$		0			0				1.00000i	$0.5\pi$
$798$		0			0
$799$		0			0
$800$	−24.4949	−	14.1421i	−0.866025	−	0.500000i
$801$	−12.8990	−	11.0280i	−0.455763	−	0.389654i
$802$	10.3258	+	17.8848i	0.364615	+	0.631532i
$803$	89.8610	+	51.8813i	3.17112	+	1.83085i
$804$	−20.7980	−	45.1441i	−0.733487	−	1.59211i
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$		−	33.4804i		−	1.17711i	−0.808458	−	0.588555i	$-0.799697\pi$
$809$		−	33.4804i		−	1.17711i	0.808458	−	0.588555i	$-0.200303\pi$
$810$		0			0
$811$	−19.0000	−	32.9090i	−0.667180	−	1.15559i	−0.978689	−	0.205347i	$-0.934168\pi$
$811$	−19.0000	−	32.9090i	−0.667180	−	1.15559i	0.311509	−	0.950243i	$-0.399166\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		0			0
$816$	−3.59592	+	39.0265i	−0.125882	+	1.36620i
$817$	−10.0000	−	42.4264i	−0.349856	−	1.48431i
$818$			26.0129i			0.909519i
$819$		0			0
$820$		0			0
$821$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$821$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$822$	8.65153	−	12.2351i	0.301757	−	0.426749i
$823$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$823$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$824$		0			0
$825$	5.25255	−	57.0060i	0.182870	−	1.98469i
$826$		0			0
$827$	49.0732	−	28.3324i	1.70644	−	0.985215i	0.767561	−	0.640976i	$-0.221470\pi$
$827$	49.0732	−	28.3324i	1.70644	−	0.985215i	0.938882	−	0.344239i	$-0.111863\pi$
$828$		0			0
$829$		0			0		1.00000			$0$
$829$		0			0		−1.00000			$\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−34.2929	−	19.7990i	−1.18818	−	0.685994i
$834$	36.6969	+	25.9487i	1.27071	+	0.898528i
$835$		0			0
$836$	−16.5959	+	55.1864i	−0.573982	+	1.90866i
$837$		0			0
$838$	26.0000	−	45.0333i	0.898155	−	1.55565i
$839$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$839$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$840$		0			0
$841$	14.5000	−	25.1147i	0.500000	−	0.866025i
$842$		0			0
$843$	−4.72474	+	51.2777i	−0.162729	+	1.76610i
$844$	−28.0000			−0.963800
$845$		0			0
$846$		0			0
$847$		0			0
$848$		0			0
$849$	−23.9495	−	51.9848i	−0.821944	−	1.78411i
$850$	20.0000	−	34.6410i	0.685994	−	1.18818i
$851$		0			0
$852$		0			0
$853$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$853$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$854$		0			0
$855$		0			0
$856$	56.0000			1.91404
$857$	−30.7020	−	17.7258i	−1.04876	−	0.605503i	−0.126459	−	0.991972i	$-0.540361\pi$
$857$	−30.7020	−	17.7258i	−1.04876	−	0.605503i	−0.922302	+	0.386469i	$0.873695\pi$
$858$		0			0
$859$	−18.1742	−	31.4787i	−0.620097	−	1.07404i	−0.989467	−	0.144757i	$-0.953760\pi$
$859$	−18.1742	−	31.4787i	−0.620097	−	1.07404i	0.369370	−	0.929282i	$-0.379573\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		0			0			−	1.00000i	$-0.5\pi$
$863$		0			0				1.00000i	$0.5\pi$
$864$	−28.4949	+	7.21393i	−0.969416	+	0.245423i
$865$		0			0
$866$		−	53.7401i		−	1.82616i
$867$	−25.8712	−	2.38378i	−0.878631	−	0.0809574i
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$		0			0
$873$	−28.6010	−	5.31575i	−0.967998	−	0.179911i
$874$		0			0
$875$		0			0
$876$	−31.3939	+	44.3976i	−1.06070	+	1.50006i
$877$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$877$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		−	43.8727i		−	1.47811i	−0.673645	−	0.739055i	$-0.735272\pi$
$881$		−	43.8727i		−	1.47811i	0.673645	−	0.739055i	$-0.264728\pi$
$882$	5.42679	−	29.1985i	0.182729	−	0.983163i
$883$	26.2196	+	45.4138i	0.882361	+	1.52829i	0.848709	+	0.528861i	$0.177381\pi$
$883$	26.2196	+	45.4138i	0.882361	+	1.52829i	0.0336527	+	0.999434i	$0.489286\pi$
$884$		0			0
$885$		0			0
$886$	53.4393			1.79533
$887$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$887$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$	−37.3939	−	46.2726i	−1.25274	−	1.55019i
$892$		0			0
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$	−23.7196	+	41.0836i	−0.791535	+	1.37098i
$899$		0			0
$900$	29.4949	+	5.48188i	0.983163	+	0.182729i
$901$		0			0
$902$	3.72985	−	2.15343i	0.124190	−	0.0717013i
$903$		0			0
$904$	28.0908			0.934287
$905$		0			0
$906$		0			0
$907$	23.2196	−	40.2176i	0.770996	−	1.33540i	−0.166022	−	0.986122i	$-0.553092\pi$
$907$	23.2196	−	40.2176i	0.770996	−	1.33540i	0.937018	−	0.349281i	$-0.113574\pi$
$908$	42.5505	−	24.5665i	1.41209	−	0.815270i
$909$		0			0
$910$		0			0
$911$		0			0			−	1.00000i	$-0.5\pi$
$911$		0			0				1.00000i	$0.5\pi$
$912$	−28.0000	−	11.3137i	−0.927173	−	0.374634i
$913$	−112.394			−3.71969
$914$	20.0783	+	11.5922i	0.664132	+	0.383437i
$915$		0			0
$916$		0			0
$917$		0			0
$918$	−10.2020	−	40.2979i	−0.336718	−	1.33003i
$919$		0			0		1.00000			$0$
$919$		0			0		−1.00000			$\pi$
$920$		0			0
$921$	−6.99490	−	15.1831i	−0.230490	−	0.500301i
$922$		0			0
$923$		0			0
$924$		0			0
$925$		0			0
$926$		0			0
$927$		0			0
$928$		0			0
$929$	28.2526	+	16.3116i	0.926936	+	0.535167i	0.885841	−	0.463988i	$-0.153582\pi$
$929$	28.2526	+	16.3116i	0.926936	+	0.535167i	0.0410949	+	0.999155i	$0.486915\pi$
$930$		0			0
$931$	22.2196	−	20.9114i	0.728219	−	0.685344i
$932$		−	46.3047i		−	1.51676i
$933$		0			0
$934$	−7.37117	+	12.7672i	−0.241192	+	0.417757i
$935$		0			0
$936$		0			0
$937$	13.5454	+	23.4613i	0.442509	+	0.766448i	0.997875	−	0.0651578i	$-0.0207551\pi$
$937$	13.5454	+	23.4613i	0.442509	+	0.766448i	−0.555366	+	0.831606i	$0.687422\pi$
$938$		0			0
$939$	−42.0732	−	3.87664i	−1.37301	−	0.126509i
$940$		0			0
$941$		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
$944$	−42.4949	+	24.5344i	−1.38309	+	0.798528i
$945$		0			0
$946$	46.7423	+	80.9601i	1.51972	+	2.63224i
$947$	46.5403	+	26.8701i	1.51236	+	0.873160i	0.999896	+	0.0144491i	$0.00459946\pi$
$947$	46.5403	+	26.8701i	1.51236	+	0.873160i	0.512461	+	0.858710i	$0.328734\pi$
$948$		0			0
$949$		0			0
$950$	21.1237	+	22.4452i	0.685344	+	0.728219i
$951$		0			0
$952$		0			0
$953$	−51.0959	−	29.5002i	−1.65516	−	0.955607i	−0.974902	−	0.222633i	$-0.928535\pi$
$953$	−51.0959	−	29.5002i	−1.65516	−	0.955607i	−0.680257	−	0.732974i	$-0.738132\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	31.0000			1.00000
$962$		0			0
$963$	−56.0000	+	19.7990i	−1.80457	+	0.638014i
$964$	1.69694	−	2.93918i	0.0546547	−	0.0946647i
$965$		0			0
$966$		0			0
$967$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$967$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$968$		−	92.4809i		−	2.97245i
$969$	16.0000	−	39.5980i	0.513994	−	1.27207i
$970$		0			0
$971$	53.9722	+	31.1609i	1.73205	+	1.00000i	0.866471	+	0.499227i	$0.166383\pi$
$971$	53.9722	+	31.1609i	1.73205	+	1.00000i	0.865579	+	0.500773i	$0.166951\pi$
$972$	25.9444	−	17.2884i	0.832167	−	0.554526i
$973$		0			0
$974$		0			0
$975$		0			0
$976$		0			0
$977$			25.9165i			0.829144i	0.910017	+	0.414572i	$0.136069\pi$
$977$			25.9165i			0.829144i	−0.910017	+	0.414572i	$0.863931\pi$
$978$	51.2702	−	23.6203i	1.63944	−	0.755292i
$979$	18.6969	+	32.3840i	0.597557	+	1.03500i
$980$		0			0
$981$		0			0
$982$	−10.0000	−	17.3205i	−0.319113	−	0.552720i
$983$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$983$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$984$	0.944387	+	2.04989i	0.0301060	+	0.0653480i
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$991$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$992$		0			0
$993$	−9.04541	+	12.7921i	−0.287047	+	0.405946i
$994$		0			0
$995$		0			0
$996$	5.40408	−	58.6505i	0.171235	−	1.85841i
$997$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$997$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$998$	−36.4268	+	21.0310i	−1.15307	+	0.665725i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	456.2.u.a.11.2	✓	4
3.2	odd	2		456.2.u.b.11.1	yes	4
8.3	odd	2	CM	456.2.u.a.11.2	✓	4
19.7	even	3		456.2.u.b.83.1	yes	4
24.11	even	2		456.2.u.b.11.1	yes	4
57.26	odd	6	inner	456.2.u.a.83.2	yes	4
152.83	odd	6		456.2.u.b.83.1	yes	4
456.83	even	6	inner	456.2.u.a.83.2	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
456.2.u.a.11.2	✓	4	1.1	even	1	trivial
456.2.u.a.11.2	✓	4	8.3	odd	2	CM
456.2.u.a.83.2	yes	4	57.26	odd	6	inner
456.2.u.a.83.2	yes	4	456.83	even	6	inner
456.2.u.b.11.1	yes	4	3.2	odd	2
456.2.u.b.11.1	yes	4	24.11	even	2
456.2.u.b.83.1	yes	4	19.7	even	3
456.2.u.b.83.1	yes	4	152.83	odd	6

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 456 = 2^{3} \cdot 3 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	456.u (of order \(6\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(1.22474 + 0.707107i) q^{2} +(-1.00000 + 1.41421i) q^{3} +(1.00000 + 1.73205i) q^{4} +(-2.22474 + 1.02494i) q^{6} +2.82843i q^{8} +(-1.00000 - 2.82843i) q^{9} +O(q^{10})\)	\(q+(1.22474 + 0.707107i) q^{2} +(-1.00000 + 1.41421i) q^{3} +(1.00000 + 1.73205i) q^{4} +(-2.22474 + 1.02494i) q^{6} +2.82843i q^{8} +(-1.00000 - 2.82843i) q^{9} +6.61037i q^{11} +(-3.44949 - 0.317837i) q^{12} +(-2.00000 + 3.46410i) q^{16} +(-4.89898 - 2.82843i) q^{17} +(0.775255 - 4.17121i) q^{18} +(3.17423 - 2.98735i) q^{19} +(-4.67423 + 8.09601i) q^{22} +(-4.00000 - 2.82843i) q^{24} +(2.50000 + 4.33013i) q^{25} +(5.00000 + 1.41421i) q^{27} +(-4.89898 + 2.82843i) q^{32} +(-9.34847 - 6.61037i) q^{33} +(-4.00000 - 6.92820i) q^{34} +(3.89898 - 4.56048i) q^{36} +(6.00000 - 1.41421i) q^{38} +(-0.398979 - 0.230351i) q^{41} +(5.00000 - 8.66025i) q^{43} +(-11.4495 + 6.61037i) q^{44} +(-2.89898 - 6.29253i) q^{48} +7.00000 q^{49} +7.07107i q^{50} +(8.89898 - 4.09978i) q^{51} +(5.12372 + 5.26758i) q^{54} +(1.05051 + 7.47639i) q^{57} +(10.6237 + 6.13361i) q^{59} -8.00000 q^{64} +(-6.77526 - 14.7064i) q^{66} +(7.17423 + 12.4261i) q^{67} -11.3137i q^{68} +(8.00000 - 2.82843i) q^{72} +(7.84847 - 13.5939i) q^{73} +(-8.62372 - 0.794593i) q^{75} +(8.34847 + 2.51059i) q^{76} +(-7.00000 + 5.65685i) q^{81} +(-0.325765 - 0.564242i) q^{82} +17.0027i q^{83} +(12.2474 - 7.07107i) q^{86} -18.6969 q^{88} +(4.89898 - 2.82843i) q^{89} +(0.898979 - 9.75663i) q^{96} +(4.84847 - 8.39780i) q^{97} +(8.57321 + 4.94975i) q^{98} +(18.6969 - 6.61037i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{3} + 4 q^{4} - 4 q^{6} - 4 q^{9}+O(q^{10}) \) 4 * q - 4 * q^3 + 4 * q^4 - 4 * q^6 - 4 * q^9	\( 4 q - 4 q^{3} + 4 q^{4} - 4 q^{6} - 4 q^{9} - 4 q^{12} - 8 q^{16} + 8 q^{18} - 2 q^{19} - 4 q^{22} - 16 q^{24} + 10 q^{25} + 20 q^{27} - 8 q^{33} - 16 q^{34} - 4 q^{36} + 24 q^{38} + 18 q^{41} + 20 q^{43} - 36 q^{44} + 8 q^{48} + 28 q^{49} + 16 q^{51} - 4 q^{54} + 14 q^{57} + 18 q^{59} - 32 q^{64} - 32 q^{66} + 14 q^{67} + 32 q^{72} + 2 q^{73} - 10 q^{75} + 4 q^{76} - 28 q^{81} - 16 q^{82} - 16 q^{88} - 16 q^{96} - 10 q^{97} + 16 q^{99}+O(q^{100}) \) 4 * q - 4 * q^3 + 4 * q^4 - 4 * q^6 - 4 * q^9 - 4 * q^12 - 8 * q^16 + 8 * q^18 - 2 * q^19 - 4 * q^22 - 16 * q^24 + 10 * q^25 + 20 * q^27 - 8 * q^33 - 16 * q^34 - 4 * q^36 + 24 * q^38 + 18 * q^41 + 20 * q^43 - 36 * q^44 + 8 * q^48 + 28 * q^49 + 16 * q^51 - 4 * q^54 + 14 * q^57 + 18 * q^59 - 32 * q^64 - 32 * q^66 + 14 * q^67 + 32 * q^72 + 2 * q^73 - 10 * q^75 + 4 * q^76 - 28 * q^81 - 16 * q^82 - 16 * q^88 - 16 * q^96 - 10 * q^97 + 16 * q^99

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