LMFDB - Embedded newform 378.2.e.b.37.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [378,2,Mod(37,378)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(378, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([2, 2]))

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

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

chi := DirichletCharacter("378.37");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 378 = 2 \cdot 3^{3} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	378.e (of order $3$, degree $2$, not minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$3.01834519640$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{6})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x + 1 $ x^2 - x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 126)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			37.1
Root			$0.500000 - 0.866025i$ of defining polynomial
Character	$\chi$	$=$	378.37
Dual form			378.2.e.b.235.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} +1.00000 q^{4} +(1.50000 + 2.59808i) q^{5} +(-2.00000 + 1.73205i) q^{7} +1.00000 q^{8} +O(q^{10})$	\(q+1.00000 q^{2} +1.00000 q^{4} +(1.50000 + 2.59808i) q^{5} +(-2.00000 + 1.73205i) q^{7} +1.00000 q^{8} +(1.50000 + 2.59808i) q^{10} +(-1.50000 + 2.59808i) q^{11} +(0.500000 - 0.866025i) q^{13} +(-2.00000 + 1.73205i) q^{14} +1.00000 q^{16} +(1.50000 + 2.59808i) q^{17} +(3.50000 - 6.06218i) q^{19} +(1.50000 + 2.59808i) q^{20} +(-1.50000 + 2.59808i) q^{22} +(-4.50000 - 7.79423i) q^{23} +(-2.00000 + 3.46410i) q^{25} +(0.500000 - 0.866025i) q^{26} +(-2.00000 + 1.73205i) q^{28} +(1.50000 + 2.59808i) q^{29} +8.00000 q^{31} +1.00000 q^{32} +(1.50000 + 2.59808i) q^{34} +(-7.50000 - 2.59808i) q^{35} +(0.500000 - 0.866025i) q^{37} +(3.50000 - 6.06218i) q^{38} +(1.50000 + 2.59808i) q^{40} +(1.50000 - 2.59808i) q^{41} +(0.500000 + 0.866025i) q^{43} +(-1.50000 + 2.59808i) q^{44} +(-4.50000 - 7.79423i) q^{46} +(1.00000 - 6.92820i) q^{49} +(-2.00000 + 3.46410i) q^{50} +(0.500000 - 0.866025i) q^{52} +(1.50000 + 2.59808i) q^{53} -9.00000 q^{55} +(-2.00000 + 1.73205i) q^{56} +(1.50000 + 2.59808i) q^{58} +2.00000 q^{61} +8.00000 q^{62} +1.00000 q^{64} +3.00000 q^{65} -4.00000 q^{67} +(1.50000 + 2.59808i) q^{68} +(-7.50000 - 2.59808i) q^{70} -12.0000 q^{71} +(-5.50000 - 9.52628i) q^{73} +(0.500000 - 0.866025i) q^{74} +(3.50000 - 6.06218i) q^{76} +(-1.50000 - 7.79423i) q^{77} -16.0000 q^{79} +(1.50000 + 2.59808i) q^{80} +(1.50000 - 2.59808i) q^{82} +(-4.50000 - 7.79423i) q^{83} +(-4.50000 + 7.79423i) q^{85} +(0.500000 + 0.866025i) q^{86} +(-1.50000 + 2.59808i) q^{88} +(1.50000 - 2.59808i) q^{89} +(0.500000 + 2.59808i) q^{91} +(-4.50000 - 7.79423i) q^{92} +21.0000 q^{95} +(0.500000 + 0.866025i) q^{97} +(1.00000 - 6.92820i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{2} + 2 q^{4} + 3 q^{5} - 4 q^{7} + 2 q^{8}+O(q^{10}) $ 2 * q + 2 * q^2 + 2 * q^4 + 3 * q^5 - 4 * q^7 + 2 * q^8	$ 2 q + 2 q^{2} + 2 q^{4} + 3 q^{5} - 4 q^{7} + 2 q^{8} + 3 q^{10} - 3 q^{11} + q^{13} - 4 q^{14} + 2 q^{16} + 3 q^{17} + 7 q^{19} + 3 q^{20} - 3 q^{22} - 9 q^{23} - 4 q^{25} + q^{26} - 4 q^{28} + 3 q^{29} + 16 q^{31} + 2 q^{32} + 3 q^{34} - 15 q^{35} + q^{37} + 7 q^{38} + 3 q^{40} + 3 q^{41} + q^{43} - 3 q^{44} - 9 q^{46} + 2 q^{49} - 4 q^{50} + q^{52} + 3 q^{53} - 18 q^{55} - 4 q^{56} + 3 q^{58} + 4 q^{61} + 16 q^{62} + 2 q^{64} + 6 q^{65} - 8 q^{67} + 3 q^{68} - 15 q^{70} - 24 q^{71} - 11 q^{73} + q^{74} + 7 q^{76} - 3 q^{77} - 32 q^{79} + 3 q^{80} + 3 q^{82} - 9 q^{83} - 9 q^{85} + q^{86} - 3 q^{88} + 3 q^{89} + q^{91} - 9 q^{92} + 42 q^{95} + q^{97} + 2 q^{98}+O(q^{100}) $ 2 * q + 2 * q^2 + 2 * q^4 + 3 * q^5 - 4 * q^7 + 2 * q^8 + 3 * q^10 - 3 * q^11 + q^13 - 4 * q^14 + 2 * q^16 + 3 * q^17 + 7 * q^19 + 3 * q^20 - 3 * q^22 - 9 * q^23 - 4 * q^25 + q^26 - 4 * q^28 + 3 * q^29 + 16 * q^31 + 2 * q^32 + 3 * q^34 - 15 * q^35 + q^37 + 7 * q^38 + 3 * q^40 + 3 * q^41 + q^43 - 3 * q^44 - 9 * q^46 + 2 * q^49 - 4 * q^50 + q^52 + 3 * q^53 - 18 * q^55 - 4 * q^56 + 3 * q^58 + 4 * q^61 + 16 * q^62 + 2 * q^64 + 6 * q^65 - 8 * q^67 + 3 * q^68 - 15 * q^70 - 24 * q^71 - 11 * q^73 + q^74 + 7 * q^76 - 3 * q^77 - 32 * q^79 + 3 * q^80 + 3 * q^82 - 9 * q^83 - 9 * q^85 + q^86 - 3 * q^88 + 3 * q^89 + q^91 - 9 * q^92 + 42 * q^95 + q^97 + 2 * q^98

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/378\mathbb{Z}\right)^\times$.

$n$	$29$	$325$
$\chi(n)$	$e\left(\frac{1}{3}\right)$	$e\left(\frac{1}{3}\right)$

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 $

$2$	1.00000			0.707107
$2$	1.00000			0.707107
$3$		0			0
$3$		0			0
$4$	1.00000			0.500000
$5$	1.50000	+	2.59808i	0.670820	+	1.16190i	0.977672	+	0.210138i	$0.0673912\pi$
$5$	1.50000	+	2.59808i	0.670820	+	1.16190i	−0.306851	+	0.951757i	$0.599275\pi$
$6$		0			0
$7$	−2.00000	+	1.73205i	−0.755929	+	0.654654i
$7$	−2.00000	+	1.73205i	−0.755929	+	0.654654i
$8$	1.00000			0.353553
$9$		0			0
$10$	1.50000	+	2.59808i	0.474342	+	0.821584i
$11$	−1.50000	+	2.59808i	−0.452267	+	0.783349i	−0.998526	−	0.0542666i	$-0.982718\pi$
$11$	−1.50000	+	2.59808i	−0.452267	+	0.783349i	0.546259	+	0.837616i	$0.316051\pi$
$12$		0			0
$13$	0.500000	−	0.866025i	0.138675	−	0.240192i	−0.788320	−	0.615265i	$-0.789049\pi$
$13$	0.500000	−	0.866025i	0.138675	−	0.240192i	0.926995	+	0.375073i	$0.122382\pi$
$14$	−2.00000	+	1.73205i	−0.534522	+	0.462910i
$15$		0			0
$16$	1.00000			0.250000
$17$	1.50000	+	2.59808i	0.363803	+	0.630126i	0.988583	−	0.150675i	$-0.0481447\pi$
$17$	1.50000	+	2.59808i	0.363803	+	0.630126i	−0.624780	+	0.780801i	$0.714811\pi$
$18$		0			0
$19$	3.50000	−	6.06218i	0.802955	−	1.39076i	−0.114708	−	0.993399i	$-0.536593\pi$
$19$	3.50000	−	6.06218i	0.802955	−	1.39076i	0.917663	−	0.397360i	$-0.130073\pi$
$20$	1.50000	+	2.59808i	0.335410	+	0.580948i
$21$		0			0
$22$	−1.50000	+	2.59808i	−0.319801	+	0.553912i
$23$	−4.50000	−	7.79423i	−0.938315	−	1.62521i	−0.768613	−	0.639713i	$-0.779053\pi$
$23$	−4.50000	−	7.79423i	−0.938315	−	1.62521i	−0.169701	−	0.985496i	$-0.554280\pi$
$24$		0			0
$25$	−2.00000	+	3.46410i	−0.400000	+	0.692820i
$26$	0.500000	−	0.866025i	0.0980581	−	0.169842i
$27$		0			0
$28$	−2.00000	+	1.73205i	−0.377964	+	0.327327i
$29$	1.50000	+	2.59808i	0.278543	+	0.482451i	0.971023	−	0.238987i	$-0.0768152\pi$
$29$	1.50000	+	2.59808i	0.278543	+	0.482451i	−0.692480	+	0.721437i	$0.743482\pi$
$30$		0			0
$31$	8.00000			1.43684			0.718421	−	0.695608i	$-0.244865\pi$
$31$	8.00000			1.43684			0.718421	+	0.695608i	$0.244865\pi$
$32$	1.00000			0.176777
$33$		0			0
$34$	1.50000	+	2.59808i	0.257248	+	0.445566i
$35$	−7.50000	−	2.59808i	−1.26773	−	0.439155i
$36$		0			0
$37$	0.500000	−	0.866025i	0.0821995	−	0.142374i	−0.821995	−	0.569495i	$-0.807139\pi$
$37$	0.500000	−	0.866025i	0.0821995	−	0.142374i	0.904194	+	0.427121i	$0.140472\pi$
$38$	3.50000	−	6.06218i	0.567775	−	0.983415i
$39$		0			0
$40$	1.50000	+	2.59808i	0.237171	+	0.410792i
$41$	1.50000	−	2.59808i	0.234261	−	0.405751i	−0.724797	−	0.688963i	$-0.758066\pi$
$41$	1.50000	−	2.59808i	0.234261	−	0.405751i	0.959058	+	0.283211i	$0.0913998\pi$
$42$		0			0
$43$	0.500000	+	0.866025i	0.0762493	+	0.132068i	0.901629	−	0.432511i	$-0.142372\pi$
$43$	0.500000	+	0.866025i	0.0762493	+	0.132068i	−0.825380	+	0.564578i	$0.809039\pi$
$44$	−1.50000	+	2.59808i	−0.226134	+	0.391675i
$45$		0			0
$46$	−4.50000	−	7.79423i	−0.663489	−	1.14920i
$47$		0			0			−	1.00000i	$-0.5\pi$
$47$		0			0				1.00000i	$0.5\pi$
$48$		0			0
$49$	1.00000	−	6.92820i	0.142857	−	0.989743i
$50$	−2.00000	+	3.46410i	−0.282843	+	0.489898i
$51$		0			0
$52$	0.500000	−	0.866025i	0.0693375	−	0.120096i
$53$	1.50000	+	2.59808i	0.206041	+	0.356873i	0.950464	−	0.310835i	$-0.100609\pi$
$53$	1.50000	+	2.59808i	0.206041	+	0.356873i	−0.744423	+	0.667708i	$0.767275\pi$
$54$		0			0
$55$	−9.00000			−1.21356
$56$	−2.00000	+	1.73205i	−0.267261	+	0.231455i
$57$		0			0
$58$	1.50000	+	2.59808i	0.196960	+	0.341144i
$59$		0			0			−	1.00000i	$-0.5\pi$
$59$		0			0				1.00000i	$0.5\pi$
$60$		0			0
$61$	2.00000			0.256074			0.128037	−	0.991769i	$-0.459132\pi$
$61$	2.00000			0.256074			0.128037	+	0.991769i	$0.459132\pi$
$62$	8.00000			1.01600
$63$		0			0
$64$	1.00000			0.125000
$65$	3.00000			0.372104
$66$		0			0
$67$	−4.00000			−0.488678			−0.244339	−	0.969690i	$-0.578571\pi$
$67$	−4.00000			−0.488678			−0.244339	+	0.969690i	$0.578571\pi$
$68$	1.50000	+	2.59808i	0.181902	+	0.315063i
$69$		0			0
$70$	−7.50000	−	2.59808i	−0.896421	−	0.310530i
$71$	−12.0000			−1.42414			−0.712069	−	0.702109i	$-0.752242\pi$
$71$	−12.0000			−1.42414			−0.712069	+	0.702109i	$0.752242\pi$
$72$		0			0
$73$	−5.50000	−	9.52628i	−0.643726	−	1.11497i	−0.984594	−	0.174855i	$-0.944054\pi$
$73$	−5.50000	−	9.52628i	−0.643726	−	1.11497i	0.340868	−	0.940111i	$-0.389279\pi$
$74$	0.500000	−	0.866025i	0.0581238	−	0.100673i
$75$		0			0
$76$	3.50000	−	6.06218i	0.401478	−	0.695379i
$77$	−1.50000	−	7.79423i	−0.170941	−	0.888235i
$78$		0			0
$79$	−16.0000			−1.80014			−0.900070	−	0.435745i	$-0.856485\pi$
$79$	−16.0000			−1.80014			−0.900070	+	0.435745i	$0.856485\pi$
$80$	1.50000	+	2.59808i	0.167705	+	0.290474i
$81$		0			0
$82$	1.50000	−	2.59808i	0.165647	−	0.286910i
$83$	−4.50000	−	7.79423i	−0.493939	−	0.855528i	0.506036	−	0.862512i	$-0.331110\pi$
$83$	−4.50000	−	7.79423i	−0.493939	−	0.855528i	−0.999976	+	0.00698436i	$0.997777\pi$
$84$		0			0
$85$	−4.50000	+	7.79423i	−0.488094	+	0.845403i
$86$	0.500000	+	0.866025i	0.0539164	+	0.0933859i
$87$		0			0
$88$	−1.50000	+	2.59808i	−0.159901	+	0.276956i
$89$	1.50000	−	2.59808i	0.159000	−	0.275396i	−0.775509	−	0.631337i	$-0.782506\pi$
$89$	1.50000	−	2.59808i	0.159000	−	0.275396i	0.934508	+	0.355942i	$0.115840\pi$
$90$		0			0
$91$	0.500000	+	2.59808i	0.0524142	+	0.272352i
$92$	−4.50000	−	7.79423i	−0.469157	−	0.812605i
$93$		0			0
$94$		0			0
$95$	21.0000			2.15455
$96$		0			0
$97$	0.500000	+	0.866025i	0.0507673	+	0.0879316i	0.890292	−	0.455389i	$-0.150500\pi$
$97$	0.500000	+	0.866025i	0.0507673	+	0.0879316i	−0.839525	+	0.543321i	$0.817167\pi$
$98$	1.00000	−	6.92820i	0.101015	−	0.699854i
$99$		0			0
$100$	−2.00000	+	3.46410i	−0.200000	+	0.346410i
$101$	1.50000	−	2.59808i	0.149256	−	0.258518i	−0.781697	−	0.623658i	$-0.785646\pi$
$101$	1.50000	−	2.59808i	0.149256	−	0.258518i	0.930953	+	0.365140i	$0.118979\pi$
$102$		0			0
$103$	6.50000	+	11.2583i	0.640464	+	1.10932i	0.985329	+	0.170664i	$0.0545913\pi$
$103$	6.50000	+	11.2583i	0.640464	+	1.10932i	−0.344865	+	0.938652i	$0.612075\pi$
$104$	0.500000	−	0.866025i	0.0490290	−	0.0849208i
$105$		0			0
$106$	1.50000	+	2.59808i	0.145693	+	0.252347i
$107$	4.50000	−	7.79423i	0.435031	−	0.753497i	−0.562267	−	0.826956i	$-0.690071\pi$
$107$	4.50000	−	7.79423i	0.435031	−	0.753497i	0.997298	+	0.0734594i	$0.0234039\pi$
$108$		0			0
$109$	6.50000	+	11.2583i	0.622587	+	1.07835i	0.989002	+	0.147901i	$0.0472517\pi$
$109$	6.50000	+	11.2583i	0.622587	+	1.07835i	−0.366415	+	0.930451i	$0.619415\pi$
$110$	−9.00000			−0.858116
$111$		0			0
$112$	−2.00000	+	1.73205i	−0.188982	+	0.163663i
$113$	−4.50000	+	7.79423i	−0.423324	+	0.733219i	−0.996262	−	0.0863794i	$-0.972470\pi$
$113$	−4.50000	+	7.79423i	−0.423324	+	0.733219i	0.572938	+	0.819599i	$0.305804\pi$
$114$		0			0
$115$	13.5000	−	23.3827i	1.25888	−	2.18045i
$116$	1.50000	+	2.59808i	0.139272	+	0.241225i
$117$		0			0
$118$		0			0
$119$	−7.50000	−	2.59808i	−0.687524	−	0.238165i
$120$		0			0
$121$	1.00000	+	1.73205i	0.0909091	+	0.157459i
$122$	2.00000			0.181071
$123$		0			0
$124$	8.00000			0.718421
$125$	3.00000			0.268328
$126$		0			0
$127$	−4.00000			−0.354943			−0.177471	−	0.984126i	$-0.556792\pi$
$127$	−4.00000			−0.354943			−0.177471	+	0.984126i	$0.556792\pi$
$128$	1.00000			0.0883883
$129$		0			0
$130$	3.00000			0.263117
$131$	7.50000	+	12.9904i	0.655278	+	1.13497i	0.981824	+	0.189794i	$0.0607819\pi$
$131$	7.50000	+	12.9904i	0.655278	+	1.13497i	−0.326546	+	0.945181i	$0.605885\pi$
$132$		0			0
$133$	3.50000	+	18.1865i	0.303488	+	1.57697i
$134$	−4.00000			−0.345547
$135$		0			0
$136$	1.50000	+	2.59808i	0.128624	+	0.222783i
$137$	−4.50000	+	7.79423i	−0.384461	+	0.665906i	−0.991694	−	0.128618i	$-0.958946\pi$
$137$	−4.50000	+	7.79423i	−0.384461	+	0.665906i	0.607233	+	0.794524i	$0.292279\pi$
$138$		0			0
$139$	3.50000	−	6.06218i	0.296866	−	0.514187i	−0.678551	−	0.734553i	$-0.737392\pi$
$139$	3.50000	−	6.06218i	0.296866	−	0.514187i	0.975417	+	0.220366i	$0.0707252\pi$
$140$	−7.50000	−	2.59808i	−0.633866	−	0.219578i
$141$		0			0
$142$	−12.0000			−1.00702
$143$	1.50000	+	2.59808i	0.125436	+	0.217262i
$144$		0			0
$145$	−4.50000	+	7.79423i	−0.373705	+	0.647275i
$146$	−5.50000	−	9.52628i	−0.455183	−	0.788400i
$147$		0			0
$148$	0.500000	−	0.866025i	0.0410997	−	0.0711868i
$149$	−4.50000	−	7.79423i	−0.368654	−	0.638528i	0.620701	−	0.784047i	$-0.286848\pi$
$149$	−4.50000	−	7.79423i	−0.368654	−	0.638528i	−0.989355	+	0.145519i	$0.953515\pi$
$150$		0			0
$151$	3.50000	−	6.06218i	0.284826	−	0.493333i	−0.687741	−	0.725956i	$-0.741398\pi$
$151$	3.50000	−	6.06218i	0.284826	−	0.493333i	0.972567	+	0.232623i	$0.0747309\pi$
$152$	3.50000	−	6.06218i	0.283887	−	0.491708i
$153$		0			0
$154$	−1.50000	−	7.79423i	−0.120873	−	0.628077i
$155$	12.0000	+	20.7846i	0.963863	+	1.66946i
$156$		0			0
$157$	−22.0000			−1.75579			−0.877896	−	0.478852i	$-0.841053\pi$
$157$	−22.0000			−1.75579			−0.877896	+	0.478852i	$0.841053\pi$
$158$	−16.0000			−1.27289
$159$		0			0
$160$	1.50000	+	2.59808i	0.118585	+	0.205396i
$161$	22.5000	+	7.79423i	1.77325	+	0.614271i
$162$		0			0
$163$	9.50000	−	16.4545i	0.744097	−	1.28881i	−0.206518	−	0.978443i	$-0.566213\pi$
$163$	9.50000	−	16.4545i	0.744097	−	1.28881i	0.950615	−	0.310372i	$-0.100454\pi$
$164$	1.50000	−	2.59808i	0.117130	−	0.202876i
$165$		0			0
$166$	−4.50000	−	7.79423i	−0.349268	−	0.604949i
$167$	−7.50000	+	12.9904i	−0.580367	+	1.00523i	0.415068	+	0.909790i	$0.363758\pi$
$167$	−7.50000	+	12.9904i	−0.580367	+	1.00523i	−0.995436	+	0.0954356i	$0.969576\pi$
$168$		0			0
$169$	6.00000	+	10.3923i	0.461538	+	0.799408i
$170$	−4.50000	+	7.79423i	−0.345134	+	0.597790i
$171$		0			0
$172$	0.500000	+	0.866025i	0.0381246	+	0.0660338i
$173$	6.00000			0.456172			0.228086	−	0.973641i	$-0.426753\pi$
$173$	6.00000			0.456172			0.228086	+	0.973641i	$0.426753\pi$
$174$		0			0
$175$	−2.00000	−	10.3923i	−0.151186	−	0.785584i
$176$	−1.50000	+	2.59808i	−0.113067	+	0.195837i
$177$		0			0
$178$	1.50000	−	2.59808i	0.112430	−	0.194734i
$179$	−10.5000	−	18.1865i	−0.784807	−	1.35933i	−0.929114	−	0.369792i	$-0.879429\pi$
$179$	−10.5000	−	18.1865i	−0.784807	−	1.35933i	0.144308	−	0.989533i	$-0.453905\pi$
$180$		0			0
$181$	2.00000			0.148659			0.0743294	−	0.997234i	$-0.476318\pi$
$181$	2.00000			0.148659			0.0743294	+	0.997234i	$0.476318\pi$
$182$	0.500000	+	2.59808i	0.0370625	+	0.192582i
$183$		0			0
$184$	−4.50000	−	7.79423i	−0.331744	−	0.574598i
$185$	3.00000			0.220564
$186$		0			0
$187$	−9.00000			−0.658145
$188$		0			0
$189$		0			0
$190$	21.0000			1.52350
$191$		0			0			−	1.00000i	$-0.5\pi$
$191$		0			0				1.00000i	$0.5\pi$
$192$		0			0
$193$	14.0000			1.00774			0.503871	−	0.863779i	$-0.331909\pi$
$193$	14.0000			1.00774			0.503871	+	0.863779i	$0.331909\pi$
$194$	0.500000	+	0.866025i	0.0358979	+	0.0621770i
$195$		0			0
$196$	1.00000	−	6.92820i	0.0714286	−	0.494872i
$197$	−18.0000			−1.28245			−0.641223	−	0.767354i	$-0.721573\pi$
$197$	−18.0000			−1.28245			−0.641223	+	0.767354i	$0.721573\pi$
$198$		0			0
$199$	12.5000	+	21.6506i	0.886102	+	1.53477i	0.844446	+	0.535641i	$0.179930\pi$
$199$	12.5000	+	21.6506i	0.886102	+	1.53477i	0.0416556	+	0.999132i	$0.486737\pi$
$200$	−2.00000	+	3.46410i	−0.141421	+	0.244949i
$201$		0			0
$202$	1.50000	−	2.59808i	0.105540	−	0.182800i
$203$	−7.50000	−	2.59808i	−0.526397	−	0.182349i
$204$		0			0
$205$	9.00000			0.628587
$206$	6.50000	+	11.2583i	0.452876	+	0.784405i
$207$		0			0
$208$	0.500000	−	0.866025i	0.0346688	−	0.0600481i
$209$	10.5000	+	18.1865i	0.726300	+	1.25799i
$210$		0			0
$211$	−2.50000	+	4.33013i	−0.172107	+	0.298098i	−0.939156	−	0.343490i	$-0.888391\pi$
$211$	−2.50000	+	4.33013i	−0.172107	+	0.298098i	0.767049	+	0.641588i	$0.221724\pi$
$212$	1.50000	+	2.59808i	0.103020	+	0.178437i
$213$		0			0
$214$	4.50000	−	7.79423i	0.307614	−	0.532803i
$215$	−1.50000	+	2.59808i	−0.102299	+	0.177187i
$216$		0			0
$217$	−16.0000	+	13.8564i	−1.08615	+	0.940634i
$218$	6.50000	+	11.2583i	0.440236	+	0.762510i
$219$		0			0
$220$	−9.00000			−0.606780
$221$	3.00000			0.201802
$222$		0			0
$223$	0.500000	+	0.866025i	0.0334825	+	0.0579934i	0.882281	−	0.470723i	$-0.156007\pi$
$223$	0.500000	+	0.866025i	0.0334825	+	0.0579934i	−0.848799	+	0.528716i	$0.822674\pi$
$224$	−2.00000	+	1.73205i	−0.133631	+	0.115728i
$225$		0			0
$226$	−4.50000	+	7.79423i	−0.299336	+	0.518464i
$227$	−1.50000	+	2.59808i	−0.0995585	+	0.172440i	−0.911502	−	0.411296i	$-0.865076\pi$
$227$	−1.50000	+	2.59808i	−0.0995585	+	0.172440i	0.811943	+	0.583736i	$0.198410\pi$
$228$		0			0
$229$	6.50000	+	11.2583i	0.429532	+	0.743971i	0.996832	−	0.0795401i	$-0.0253452\pi$
$229$	6.50000	+	11.2583i	0.429532	+	0.743971i	−0.567300	+	0.823511i	$0.692012\pi$
$230$	13.5000	−	23.3827i	0.890164	−	1.54181i
$231$		0			0
$232$	1.50000	+	2.59808i	0.0984798	+	0.170572i
$233$	1.50000	−	2.59808i	0.0982683	−	0.170206i	−0.812700	−	0.582683i	$-0.802003\pi$
$233$	1.50000	−	2.59808i	0.0982683	−	0.170206i	0.910968	+	0.412477i	$0.135336\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$		0			0
$238$	−7.50000	−	2.59808i	−0.486153	−	0.168408i
$239$	−1.50000	+	2.59808i	−0.0970269	+	0.168056i	−0.910453	−	0.413613i	$-0.864267\pi$
$239$	−1.50000	+	2.59808i	−0.0970269	+	0.168056i	0.813426	+	0.581669i	$0.197600\pi$
$240$		0			0
$241$	6.50000	−	11.2583i	0.418702	−	0.725213i	−0.577107	−	0.816668i	$-0.695819\pi$
$241$	6.50000	−	11.2583i	0.418702	−	0.725213i	0.995809	+	0.0914555i	$0.0291519\pi$
$242$	1.00000	+	1.73205i	0.0642824	+	0.111340i
$243$		0			0
$244$	2.00000			0.128037
$245$	19.5000	−	7.79423i	1.24581	−	0.497955i
$246$		0			0
$247$	−3.50000	−	6.06218i	−0.222700	−	0.385727i
$248$	8.00000			0.508001
$249$		0			0
$250$	3.00000			0.189737
$251$	−12.0000			−0.757433			−0.378717	−	0.925513i	$-0.623635\pi$
$251$	−12.0000			−0.757433			−0.378717	+	0.925513i	$0.623635\pi$
$252$		0			0
$253$	27.0000			1.69748
$254$	−4.00000			−0.250982
$255$		0			0
$256$	1.00000			0.0625000
$257$	−10.5000	−	18.1865i	−0.654972	−	1.13444i	−0.981901	−	0.189396i	$-0.939347\pi$
$257$	−10.5000	−	18.1865i	−0.654972	−	1.13444i	0.326929	−	0.945049i	$-0.393986\pi$
$258$		0			0
$259$	0.500000	+	2.59808i	0.0310685	+	0.161437i
$260$	3.00000			0.186052
$261$		0			0
$262$	7.50000	+	12.9904i	0.463352	+	0.802548i
$263$	4.50000	−	7.79423i	0.277482	−	0.480613i	−0.693276	−	0.720672i	$-0.743833\pi$
$263$	4.50000	−	7.79423i	0.277482	−	0.480613i	0.970758	+	0.240059i	$0.0771668\pi$
$264$		0			0
$265$	−4.50000	+	7.79423i	−0.276433	+	0.478796i
$266$	3.50000	+	18.1865i	0.214599	+	1.11509i
$267$		0			0
$268$	−4.00000			−0.244339
$269$	7.50000	+	12.9904i	0.457283	+	0.792038i	0.998816	−	0.0486418i	$-0.0154893\pi$
$269$	7.50000	+	12.9904i	0.457283	+	0.792038i	−0.541533	+	0.840679i	$0.682156\pi$
$270$		0			0
$271$	−2.50000	+	4.33013i	−0.151864	+	0.263036i	−0.931913	−	0.362682i	$-0.881861\pi$
$271$	−2.50000	+	4.33013i	−0.151864	+	0.263036i	0.780049	+	0.625719i	$0.215194\pi$
$272$	1.50000	+	2.59808i	0.0909509	+	0.157532i
$273$		0			0
$274$	−4.50000	+	7.79423i	−0.271855	+	0.470867i
$275$	−6.00000	−	10.3923i	−0.361814	−	0.626680i
$276$		0			0
$277$	0.500000	−	0.866025i	0.0300421	−	0.0520344i	−0.850613	−	0.525792i	$-0.823769\pi$
$277$	0.500000	−	0.866025i	0.0300421	−	0.0520344i	0.880656	+	0.473757i	$0.157103\pi$
$278$	3.50000	−	6.06218i	0.209916	−	0.363585i
$279$		0			0
$280$	−7.50000	−	2.59808i	−0.448211	−	0.155265i
$281$	−10.5000	−	18.1865i	−0.626377	−	1.08492i	−0.988273	−	0.152699i	$-0.951204\pi$
$281$	−10.5000	−	18.1865i	−0.626377	−	1.08492i	0.361895	−	0.932219i	$-0.382130\pi$
$282$		0			0
$283$	−4.00000			−0.237775			−0.118888	−	0.992908i	$-0.537933\pi$
$283$	−4.00000			−0.237775			−0.118888	+	0.992908i	$0.537933\pi$
$284$	−12.0000			−0.712069
$285$		0			0
$286$	1.50000	+	2.59808i	0.0886969	+	0.153627i
$287$	1.50000	+	7.79423i	0.0885422	+	0.460079i
$288$		0			0
$289$	4.00000	−	6.92820i	0.235294	−	0.407541i
$290$	−4.50000	+	7.79423i	−0.264249	+	0.457693i
$291$		0			0
$292$	−5.50000	−	9.52628i	−0.321863	−	0.557483i
$293$	−4.50000	+	7.79423i	−0.262893	+	0.455344i	−0.967009	−	0.254741i	$-0.918010\pi$
$293$	−4.50000	+	7.79423i	−0.262893	+	0.455344i	0.704117	+	0.710084i	$0.251343\pi$
$294$		0			0
$295$		0			0
$296$	0.500000	−	0.866025i	0.0290619	−	0.0503367i
$297$		0			0
$298$	−4.50000	−	7.79423i	−0.260678	−	0.451508i
$299$	−9.00000			−0.520483
$300$		0			0
$301$	−2.50000	−	0.866025i	−0.144098	−	0.0499169i
$302$	3.50000	−	6.06218i	0.201402	−	0.348839i
$303$		0			0
$304$	3.50000	−	6.06218i	0.200739	−	0.347690i
$305$	3.00000	+	5.19615i	0.171780	+	0.297531i
$306$		0			0
$307$	−28.0000			−1.59804			−0.799022	−	0.601302i	$-0.794649\pi$
$307$	−28.0000			−1.59804			−0.799022	+	0.601302i	$0.794649\pi$
$308$	−1.50000	−	7.79423i	−0.0854704	−	0.444117i
$309$		0			0
$310$	12.0000	+	20.7846i	0.681554	+	1.18049i
$311$	24.0000			1.36092			0.680458	−	0.732787i	$-0.261781\pi$
$311$	24.0000			1.36092			0.680458	+	0.732787i	$0.261781\pi$
$312$		0			0
$313$	−10.0000			−0.565233			−0.282617	−	0.959233i	$-0.591202\pi$
$313$	−10.0000			−0.565233			−0.282617	+	0.959233i	$0.591202\pi$
$314$	−22.0000			−1.24153
$315$		0			0
$316$	−16.0000			−0.900070
$317$	−18.0000			−1.01098			−0.505490	−	0.862832i	$-0.668688\pi$
$317$	−18.0000			−1.01098			−0.505490	+	0.862832i	$0.668688\pi$
$318$		0			0
$319$	−9.00000			−0.503903
$320$	1.50000	+	2.59808i	0.0838525	+	0.145237i
$321$		0			0
$322$	22.5000	+	7.79423i	1.25388	+	0.434355i
$323$	21.0000			1.16847
$324$		0			0
$325$	2.00000	+	3.46410i	0.110940	+	0.192154i
$326$	9.50000	−	16.4545i	0.526156	−	0.911330i
$327$		0			0
$328$	1.50000	−	2.59808i	0.0828236	−	0.143455i
$329$		0			0
$330$		0			0
$331$	8.00000			0.439720			0.219860	−	0.975531i	$-0.429440\pi$
$331$	8.00000			0.439720			0.219860	+	0.975531i	$0.429440\pi$
$332$	−4.50000	−	7.79423i	−0.246970	−	0.427764i
$333$		0			0
$334$	−7.50000	+	12.9904i	−0.410382	+	0.710802i
$335$	−6.00000	−	10.3923i	−0.327815	−	0.567792i
$336$		0			0
$337$	6.50000	−	11.2583i	0.354078	−	0.613280i	−0.632882	−	0.774248i	$-0.718128\pi$
$337$	6.50000	−	11.2583i	0.354078	−	0.613280i	0.986960	+	0.160968i	$0.0514616\pi$
$338$	6.00000	+	10.3923i	0.326357	+	0.565267i
$339$		0			0
$340$	−4.50000	+	7.79423i	−0.244047	+	0.422701i
$341$	−12.0000	+	20.7846i	−0.649836	+	1.12555i
$342$		0			0
$343$	10.0000	+	15.5885i	0.539949	+	0.841698i
$344$	0.500000	+	0.866025i	0.0269582	+	0.0466930i
$345$		0			0
$346$	6.00000			0.322562
$347$	−12.0000			−0.644194			−0.322097	−	0.946707i	$-0.604388\pi$
$347$	−12.0000			−0.644194			−0.322097	+	0.946707i	$0.604388\pi$
$348$		0			0
$349$	−11.5000	−	19.9186i	−0.615581	−	1.06622i	−0.990282	−	0.139072i	$-0.955588\pi$
$349$	−11.5000	−	19.9186i	−0.615581	−	1.06622i	0.374701	−	0.927146i	$-0.377745\pi$
$350$	−2.00000	−	10.3923i	−0.106904	−	0.555492i
$351$		0			0
$352$	−1.50000	+	2.59808i	−0.0799503	+	0.138478i
$353$	1.50000	−	2.59808i	0.0798369	−	0.138282i	−0.823343	−	0.567545i	$-0.807893\pi$
$353$	1.50000	−	2.59808i	0.0798369	−	0.138282i	0.903179	+	0.429263i	$0.141227\pi$
$354$		0			0
$355$	−18.0000	−	31.1769i	−0.955341	−	1.65470i
$356$	1.50000	−	2.59808i	0.0794998	−	0.137698i
$357$		0			0
$358$	−10.5000	−	18.1865i	−0.554942	−	0.961188i
$359$	4.50000	−	7.79423i	0.237501	−	0.411364i	−0.722496	−	0.691375i	$-0.757005\pi$
$359$	4.50000	−	7.79423i	0.237501	−	0.411364i	0.959997	+	0.280012i	$0.0903384\pi$
$360$		0			0
$361$	−15.0000	−	25.9808i	−0.789474	−	1.36741i
$362$	2.00000			0.105118
$363$		0			0
$364$	0.500000	+	2.59808i	0.0262071	+	0.136176i
$365$	16.5000	−	28.5788i	0.863649	−	1.49588i
$366$		0			0
$367$	−8.50000	+	14.7224i	−0.443696	+	0.768505i	−0.997960	−	0.0638362i	$-0.979666\pi$
$367$	−8.50000	+	14.7224i	−0.443696	+	0.768505i	0.554264	+	0.832341i	$0.313000\pi$
$368$	−4.50000	−	7.79423i	−0.234579	−	0.406302i
$369$		0			0
$370$	3.00000			0.155963
$371$	−7.50000	−	2.59808i	−0.389381	−	0.134885i
$372$		0			0
$373$	6.50000	+	11.2583i	0.336557	+	0.582934i	0.983783	−	0.179364i	$-0.0574041\pi$
$373$	6.50000	+	11.2583i	0.336557	+	0.582934i	−0.647225	+	0.762299i	$0.724071\pi$
$374$	−9.00000			−0.465379
$375$		0			0
$376$		0			0
$377$	3.00000			0.154508
$378$		0			0
$379$	−28.0000			−1.43826			−0.719132	−	0.694874i	$-0.755460\pi$
$379$	−28.0000			−1.43826			−0.719132	+	0.694874i	$0.755460\pi$
$380$	21.0000			1.07728
$381$		0			0
$382$		0			0
$383$	7.50000	+	12.9904i	0.383232	+	0.663777i	0.991522	−	0.129937i	$-0.0414776\pi$
$383$	7.50000	+	12.9904i	0.383232	+	0.663777i	−0.608290	+	0.793715i	$0.708144\pi$
$384$		0			0
$385$	18.0000	−	15.5885i	0.917365	−	0.794461i
$386$	14.0000			0.712581
$387$		0			0
$388$	0.500000	+	0.866025i	0.0253837	+	0.0439658i
$389$	13.5000	−	23.3827i	0.684477	−	1.18555i	−0.289124	−	0.957292i	$-0.593364\pi$
$389$	13.5000	−	23.3827i	0.684477	−	1.18555i	0.973601	−	0.228257i	$-0.0733028\pi$
$390$		0			0
$391$	13.5000	−	23.3827i	0.682724	−	1.18251i
$392$	1.00000	−	6.92820i	0.0505076	−	0.349927i
$393$		0			0
$394$	−18.0000			−0.906827
$395$	−24.0000	−	41.5692i	−1.20757	−	2.09157i
$396$		0			0
$397$	6.50000	−	11.2583i	0.326226	−	0.565039i	−0.655534	−	0.755166i	$-0.727556\pi$
$397$	6.50000	−	11.2583i	0.326226	−	0.565039i	0.981760	+	0.190126i	$0.0608897\pi$
$398$	12.5000	+	21.6506i	0.626568	+	1.08525i
$399$		0			0
$400$	−2.00000	+	3.46410i	−0.100000	+	0.173205i
$401$	13.5000	+	23.3827i	0.674158	+	1.16768i	0.976714	+	0.214544i	$0.0688266\pi$
$401$	13.5000	+	23.3827i	0.674158	+	1.16768i	−0.302556	+	0.953131i	$0.597840\pi$
$402$		0			0
$403$	4.00000	−	6.92820i	0.199254	−	0.345118i
$404$	1.50000	−	2.59808i	0.0746278	−	0.129259i
$405$		0			0
$406$	−7.50000	−	2.59808i	−0.372219	−	0.128940i
$407$	1.50000	+	2.59808i	0.0743522	+	0.128782i
$408$		0			0
$409$	−34.0000			−1.68119			−0.840596	−	0.541663i	$-0.817795\pi$
$409$	−34.0000			−1.68119			−0.840596	+	0.541663i	$0.817795\pi$
$410$	9.00000			0.444478
$411$		0			0
$412$	6.50000	+	11.2583i	0.320232	+	0.554658i
$413$		0			0
$414$		0			0
$415$	13.5000	−	23.3827i	0.662689	−	1.14781i
$416$	0.500000	−	0.866025i	0.0245145	−	0.0424604i
$417$		0			0
$418$	10.5000	+	18.1865i	0.513572	+	0.889532i
$419$	4.50000	−	7.79423i	0.219839	−	0.380773i	−0.734919	−	0.678155i	$-0.762780\pi$
$419$	4.50000	−	7.79423i	0.219839	−	0.380773i	0.954759	+	0.297382i	$0.0961133\pi$
$420$		0			0
$421$	−17.5000	−	30.3109i	−0.852898	−	1.47726i	−0.878582	−	0.477592i	$-0.841510\pi$
$421$	−17.5000	−	30.3109i	−0.852898	−	1.47726i	0.0256838	−	0.999670i	$-0.491824\pi$
$422$	−2.50000	+	4.33013i	−0.121698	+	0.210787i
$423$		0			0
$424$	1.50000	+	2.59808i	0.0728464	+	0.126174i
$425$	−12.0000			−0.582086
$426$		0			0
$427$	−4.00000	+	3.46410i	−0.193574	+	0.167640i
$428$	4.50000	−	7.79423i	0.217516	−	0.376748i
$429$		0			0
$430$	−1.50000	+	2.59808i	−0.0723364	+	0.125290i
$431$	13.5000	+	23.3827i	0.650272	+	1.12630i	0.983057	+	0.183301i	$0.0586785\pi$
$431$	13.5000	+	23.3827i	0.650272	+	1.12630i	−0.332785	+	0.943003i	$0.607988\pi$
$432$		0			0
$433$	2.00000			0.0961139			0.0480569	−	0.998845i	$-0.484697\pi$
$433$	2.00000			0.0961139			0.0480569	+	0.998845i	$0.484697\pi$
$434$	−16.0000	+	13.8564i	−0.768025	+	0.665129i
$435$		0			0
$436$	6.50000	+	11.2583i	0.311294	+	0.539176i
$437$	−63.0000			−3.01370
$438$		0			0
$439$	8.00000			0.381819			0.190910	−	0.981608i	$-0.438856\pi$
$439$	8.00000			0.381819			0.190910	+	0.981608i	$0.438856\pi$
$440$	−9.00000			−0.429058
$441$		0			0
$442$	3.00000			0.142695
$443$	−36.0000			−1.71041			−0.855206	−	0.518289i	$-0.826569\pi$
$443$	−36.0000			−1.71041			−0.855206	+	0.518289i	$0.826569\pi$
$444$		0			0
$445$	9.00000			0.426641
$446$	0.500000	+	0.866025i	0.0236757	+	0.0410075i
$447$		0			0
$448$	−2.00000	+	1.73205i	−0.0944911	+	0.0818317i
$449$	−6.00000			−0.283158			−0.141579	−	0.989927i	$-0.545218\pi$
$449$	−6.00000			−0.283158			−0.141579	+	0.989927i	$0.545218\pi$
$450$		0			0
$451$	4.50000	+	7.79423i	0.211897	+	0.367016i
$452$	−4.50000	+	7.79423i	−0.211662	+	0.366610i
$453$		0			0
$454$	−1.50000	+	2.59808i	−0.0703985	+	0.121934i
$455$	−6.00000	+	5.19615i	−0.281284	+	0.243599i
$456$		0			0
$457$	−10.0000			−0.467780			−0.233890	−	0.972263i	$-0.575146\pi$
$457$	−10.0000			−0.467780			−0.233890	+	0.972263i	$0.575146\pi$
$458$	6.50000	+	11.2583i	0.303725	+	0.526067i
$459$		0			0
$460$	13.5000	−	23.3827i	0.629441	−	1.09022i
$461$	−4.50000	−	7.79423i	−0.209586	−	0.363013i	0.741998	−	0.670402i	$-0.233878\pi$
$461$	−4.50000	−	7.79423i	−0.209586	−	0.363013i	−0.951584	+	0.307388i	$0.900545\pi$
$462$		0			0
$463$	−20.5000	+	35.5070i	−0.952716	+	1.65015i	−0.213205	+	0.977007i	$0.568390\pi$
$463$	−20.5000	+	35.5070i	−0.952716	+	1.65015i	−0.739511	+	0.673145i	$0.764943\pi$
$464$	1.50000	+	2.59808i	0.0696358	+	0.120613i
$465$		0			0
$466$	1.50000	−	2.59808i	0.0694862	−	0.120354i
$467$	−1.50000	+	2.59808i	−0.0694117	+	0.120225i	−0.898642	−	0.438682i	$-0.855446\pi$
$467$	−1.50000	+	2.59808i	−0.0694117	+	0.120225i	0.829231	+	0.558906i	$0.188779\pi$
$468$		0			0
$469$	8.00000	−	6.92820i	0.369406	−	0.319915i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−3.00000			−0.137940
$474$		0			0
$475$	14.0000	+	24.2487i	0.642364	+	1.11261i
$476$	−7.50000	−	2.59808i	−0.343762	−	0.119083i
$477$		0			0
$478$	−1.50000	+	2.59808i	−0.0686084	+	0.118833i
$479$	−1.50000	+	2.59808i	−0.0685367	+	0.118709i	−0.898257	−	0.439470i	$-0.855166\pi$
$479$	−1.50000	+	2.59808i	−0.0685367	+	0.118709i	0.829721	+	0.558179i	$0.188500\pi$
$480$		0			0
$481$	−0.500000	−	0.866025i	−0.0227980	−	0.0394874i
$482$	6.50000	−	11.2583i	0.296067	−	0.512803i
$483$		0			0
$484$	1.00000	+	1.73205i	0.0454545	+	0.0787296i
$485$	−1.50000	+	2.59808i	−0.0681115	+	0.117973i
$486$		0			0
$487$	12.5000	+	21.6506i	0.566429	+	0.981084i	0.996915	+	0.0784867i	$0.0250088\pi$
$487$	12.5000	+	21.6506i	0.566429	+	0.981084i	−0.430486	+	0.902597i	$0.641658\pi$
$488$	2.00000			0.0905357
$489$		0			0
$490$	19.5000	−	7.79423i	0.880920	−	0.352107i
$491$	10.5000	−	18.1865i	0.473858	−	0.820747i	−0.525694	−	0.850674i	$-0.676194\pi$
$491$	10.5000	−	18.1865i	0.473858	−	0.820747i	0.999552	+	0.0299272i	$0.00952753\pi$
$492$		0			0
$493$	−4.50000	+	7.79423i	−0.202670	+	0.351034i
$494$	−3.50000	−	6.06218i	−0.157472	−	0.272750i
$495$		0			0
$496$	8.00000			0.359211
$497$	24.0000	−	20.7846i	1.07655	−	0.932317i
$498$		0			0
$499$	12.5000	+	21.6506i	0.559577	+	0.969216i	0.997532	+	0.0702185i	$0.0223697\pi$
$499$	12.5000	+	21.6506i	0.559577	+	0.969216i	−0.437955	+	0.898997i	$0.644297\pi$
$500$	3.00000			0.134164
$501$		0			0
$502$	−12.0000			−0.535586
$503$	24.0000			1.07011			0.535054	−	0.844818i	$-0.320291\pi$
$503$	24.0000			1.07011			0.535054	+	0.844818i	$0.320291\pi$
$504$		0			0
$505$	9.00000			0.400495
$506$	27.0000			1.20030
$507$		0			0
$508$	−4.00000			−0.177471
$509$	−4.50000	−	7.79423i	−0.199459	−	0.345473i	0.748894	−	0.662690i	$-0.230585\pi$
$509$	−4.50000	−	7.79423i	−0.199459	−	0.345473i	−0.948353	+	0.317217i	$0.897252\pi$
$510$		0			0
$511$	27.5000	+	9.52628i	1.21653	+	0.421418i
$512$	1.00000			0.0441942
$513$		0			0
$514$	−10.5000	−	18.1865i	−0.463135	−	0.802174i
$515$	−19.5000	+	33.7750i	−0.859273	+	1.48830i
$516$		0			0
$517$		0			0
$518$	0.500000	+	2.59808i	0.0219687	+	0.114153i
$519$		0			0
$520$	3.00000			0.131559
$521$	1.50000	+	2.59808i	0.0657162	+	0.113824i	0.897011	−	0.442007i	$-0.145733\pi$
$521$	1.50000	+	2.59808i	0.0657162	+	0.113824i	−0.831295	+	0.555831i	$0.812400\pi$
$522$		0			0
$523$	3.50000	−	6.06218i	0.153044	−	0.265081i	−0.779301	−	0.626650i	$-0.784426\pi$
$523$	3.50000	−	6.06218i	0.153044	−	0.265081i	0.932345	+	0.361569i	$0.117759\pi$
$524$	7.50000	+	12.9904i	0.327639	+	0.567487i
$525$		0			0
$526$	4.50000	−	7.79423i	0.196209	−	0.339845i
$527$	12.0000	+	20.7846i	0.522728	+	0.905392i
$528$		0			0
$529$	−29.0000	+	50.2295i	−1.26087	+	2.18389i
$530$	−4.50000	+	7.79423i	−0.195468	+	0.338560i
$531$		0			0
$532$	3.50000	+	18.1865i	0.151744	+	0.788486i
$533$	−1.50000	−	2.59808i	−0.0649722	−	0.112535i
$534$		0			0
$535$	27.0000			1.16731
$536$	−4.00000			−0.172774
$537$		0			0
$538$	7.50000	+	12.9904i	0.323348	+	0.560055i
$539$	16.5000	+	12.9904i	0.710705	+	0.559535i
$540$		0			0
$541$	−5.50000	+	9.52628i	−0.236463	+	0.409567i	−0.959697	−	0.281037i	$-0.909322\pi$
$541$	−5.50000	+	9.52628i	−0.236463	+	0.409567i	0.723234	+	0.690604i	$0.242655\pi$
$542$	−2.50000	+	4.33013i	−0.107384	+	0.185995i
$543$		0			0
$544$	1.50000	+	2.59808i	0.0643120	+	0.111392i
$545$	−19.5000	+	33.7750i	−0.835288	+	1.44676i
$546$		0			0
$547$	−5.50000	−	9.52628i	−0.235163	−	0.407314i	0.724157	−	0.689635i	$-0.242229\pi$
$547$	−5.50000	−	9.52628i	−0.235163	−	0.407314i	−0.959320	+	0.282321i	$0.908896\pi$
$548$	−4.50000	+	7.79423i	−0.192230	+	0.332953i
$549$		0			0
$550$	−6.00000	−	10.3923i	−0.255841	−	0.443129i
$551$	21.0000			0.894630
$552$		0			0
$553$	32.0000	−	27.7128i	1.36078	−	1.17847i
$554$	0.500000	−	0.866025i	0.0212430	−	0.0367939i
$555$		0			0
$556$	3.50000	−	6.06218i	0.148433	−	0.257094i
$557$	−4.50000	−	7.79423i	−0.190671	−	0.330252i	0.754802	−	0.655953i	$-0.227733\pi$
$557$	−4.50000	−	7.79423i	−0.190671	−	0.330252i	−0.945473	+	0.325701i	$0.894400\pi$
$558$		0			0
$559$	1.00000			0.0422955
$560$	−7.50000	−	2.59808i	−0.316933	−	0.109789i
$561$		0			0
$562$	−10.5000	−	18.1865i	−0.442916	−	0.767153i
$563$	−12.0000			−0.505740			−0.252870	−	0.967500i	$-0.581374\pi$
$563$	−12.0000			−0.505740			−0.252870	+	0.967500i	$0.581374\pi$
$564$		0			0
$565$	−27.0000			−1.13590
$566$	−4.00000			−0.168133
$567$		0			0
$568$	−12.0000			−0.503509
$569$	18.0000			0.754599			0.377300	−	0.926091i	$-0.376853\pi$
$569$	18.0000			0.754599			0.377300	+	0.926091i	$0.376853\pi$
$570$		0			0
$571$	32.0000			1.33916			0.669579	−	0.742741i	$-0.266474\pi$
$571$	32.0000			1.33916			0.669579	+	0.742741i	$0.266474\pi$
$572$	1.50000	+	2.59808i	0.0627182	+	0.108631i
$573$		0			0
$574$	1.50000	+	7.79423i	0.0626088	+	0.325325i
$575$	36.0000			1.50130
$576$		0			0
$577$	12.5000	+	21.6506i	0.520382	+	0.901328i	0.999719	+	0.0236970i	$0.00754370\pi$
$577$	12.5000	+	21.6506i	0.520382	+	0.901328i	−0.479337	+	0.877631i	$0.659123\pi$
$578$	4.00000	−	6.92820i	0.166378	−	0.288175i
$579$		0			0
$580$	−4.50000	+	7.79423i	−0.186852	+	0.323638i
$581$	22.5000	+	7.79423i	0.933457	+	0.323359i
$582$		0			0
$583$	−9.00000			−0.372742
$584$	−5.50000	−	9.52628i	−0.227592	−	0.394200i
$585$		0			0
$586$	−4.50000	+	7.79423i	−0.185893	+	0.321977i
$587$	1.50000	+	2.59808i	0.0619116	+	0.107234i	0.895320	−	0.445424i	$-0.146947\pi$
$587$	1.50000	+	2.59808i	0.0619116	+	0.107234i	−0.833408	+	0.552658i	$0.813614\pi$
$588$		0			0
$589$	28.0000	−	48.4974i	1.15372	−	1.99830i
$590$		0			0
$591$		0			0
$592$	0.500000	−	0.866025i	0.0205499	−	0.0355934i
$593$	19.5000	−	33.7750i	0.800769	−	1.38697i	−0.118342	−	0.992973i	$-0.537758\pi$
$593$	19.5000	−	33.7750i	0.800769	−	1.38697i	0.919111	−	0.394000i	$-0.128909\pi$
$594$		0			0
$595$	−4.50000	−	23.3827i	−0.184482	−	0.958597i
$596$	−4.50000	−	7.79423i	−0.184327	−	0.319264i
$597$		0			0
$598$	−9.00000			−0.368037
$599$	24.0000			0.980613			0.490307	−	0.871550i	$-0.336885\pi$
$599$	24.0000			0.980613			0.490307	+	0.871550i	$0.336885\pi$
$600$		0			0
$601$	12.5000	+	21.6506i	0.509886	+	0.883148i	0.999934	+	0.0114528i	$0.00364562\pi$
$601$	12.5000	+	21.6506i	0.509886	+	0.883148i	−0.490049	+	0.871695i	$0.663021\pi$
$602$	−2.50000	−	0.866025i	−0.101892	−	0.0352966i
$603$		0			0
$604$	3.50000	−	6.06218i	0.142413	−	0.246667i
$605$	−3.00000	+	5.19615i	−0.121967	+	0.211254i
$606$		0			0
$607$	6.50000	+	11.2583i	0.263827	+	0.456962i	0.967256	−	0.253804i	$-0.0816819\pi$
$607$	6.50000	+	11.2583i	0.263827	+	0.456962i	−0.703429	+	0.710766i	$0.748349\pi$
$608$	3.50000	−	6.06218i	0.141944	−	0.245854i
$609$		0			0
$610$	3.00000	+	5.19615i	0.121466	+	0.210386i
$611$		0			0
$612$		0			0
$613$	−11.5000	−	19.9186i	−0.464481	−	0.804504i	0.534697	−	0.845044i	$-0.320426\pi$
$613$	−11.5000	−	19.9186i	−0.464481	−	0.804504i	−0.999178	+	0.0405396i	$0.987092\pi$
$614$	−28.0000			−1.12999
$615$		0			0
$616$	−1.50000	−	7.79423i	−0.0604367	−	0.314038i
$617$	−22.5000	+	38.9711i	−0.905816	+	1.56892i	−0.0859976	+	0.996295i	$0.527408\pi$
$617$	−22.5000	+	38.9711i	−0.905816	+	1.56892i	−0.819818	+	0.572624i	$0.805926\pi$
$618$		0			0
$619$	−8.50000	+	14.7224i	−0.341644	+	0.591744i	−0.984738	−	0.174042i	$-0.944317\pi$
$619$	−8.50000	+	14.7224i	−0.341644	+	0.591744i	0.643094	+	0.765787i	$0.277650\pi$
$620$	12.0000	+	20.7846i	0.481932	+	0.834730i
$621$		0			0
$622$	24.0000			0.962312
$623$	1.50000	+	7.79423i	0.0600962	+	0.312269i
$624$		0			0
$625$	14.5000	+	25.1147i	0.580000	+	1.00459i
$626$	−10.0000			−0.399680
$627$		0			0
$628$	−22.0000			−0.877896
$629$	3.00000			0.119618
$630$		0			0
$631$	8.00000			0.318475			0.159237	−	0.987240i	$-0.449096\pi$
$631$	8.00000			0.318475			0.159237	+	0.987240i	$0.449096\pi$
$632$	−16.0000			−0.636446
$633$		0			0
$634$	−18.0000			−0.714871
$635$	−6.00000	−	10.3923i	−0.238103	−	0.412406i
$636$		0			0
$637$	−5.50000	−	4.33013i	−0.217918	−	0.171566i
$638$	−9.00000			−0.356313
$639$		0			0
$640$	1.50000	+	2.59808i	0.0592927	+	0.102698i
$641$	−16.5000	+	28.5788i	−0.651711	+	1.12880i	0.330997	+	0.943632i	$0.392615\pi$
$641$	−16.5000	+	28.5788i	−0.651711	+	1.12880i	−0.982708	+	0.185164i	$0.940718\pi$
$642$		0			0
$643$	−14.5000	+	25.1147i	−0.571824	+	0.990429i	0.424555	+	0.905402i	$0.360431\pi$
$643$	−14.5000	+	25.1147i	−0.571824	+	0.990429i	−0.996379	+	0.0850262i	$0.972903\pi$
$644$	22.5000	+	7.79423i	0.886624	+	0.307136i
$645$		0			0
$646$	21.0000			0.826234
$647$	−10.5000	−	18.1865i	−0.412798	−	0.714986i	0.582397	−	0.812905i	$-0.302115\pi$
$647$	−10.5000	−	18.1865i	−0.412798	−	0.714986i	−0.995194	+	0.0979182i	$0.968782\pi$
$648$		0			0
$649$		0			0
$650$	2.00000	+	3.46410i	0.0784465	+	0.135873i
$651$		0			0
$652$	9.50000	−	16.4545i	0.372049	−	0.644407i
$653$	7.50000	+	12.9904i	0.293498	+	0.508353i	0.974634	−	0.223803i	$-0.0718474\pi$
$653$	7.50000	+	12.9904i	0.293498	+	0.508353i	−0.681137	+	0.732156i	$0.738514\pi$
$654$		0			0
$655$	−22.5000	+	38.9711i	−0.879148	+	1.52273i
$656$	1.50000	−	2.59808i	0.0585652	−	0.101438i
$657$		0			0
$658$		0			0
$659$	1.50000	+	2.59808i	0.0584317	+	0.101207i	0.893762	−	0.448542i	$-0.148057\pi$
$659$	1.50000	+	2.59808i	0.0584317	+	0.101207i	−0.835330	+	0.549749i	$0.814723\pi$
$660$		0			0
$661$	−22.0000			−0.855701			−0.427850	−	0.903850i	$-0.640729\pi$
$661$	−22.0000			−0.855701			−0.427850	+	0.903850i	$0.640729\pi$
$662$	8.00000			0.310929
$663$		0			0
$664$	−4.50000	−	7.79423i	−0.174634	−	0.302475i
$665$	−42.0000	+	36.3731i	−1.62869	+	1.41049i
$666$		0			0
$667$	13.5000	−	23.3827i	0.522722	−	0.905381i
$668$	−7.50000	+	12.9904i	−0.290184	+	0.502613i
$669$		0			0
$670$	−6.00000	−	10.3923i	−0.231800	−	0.401490i
$671$	−3.00000	+	5.19615i	−0.115814	+	0.200595i
$672$		0			0
$673$	−17.5000	−	30.3109i	−0.674575	−	1.16840i	−0.976593	−	0.215096i	$-0.930993\pi$
$673$	−17.5000	−	30.3109i	−0.674575	−	1.16840i	0.302017	−	0.953302i	$-0.402340\pi$
$674$	6.50000	−	11.2583i	0.250371	−	0.433655i
$675$		0			0
$676$	6.00000	+	10.3923i	0.230769	+	0.399704i
$677$	30.0000			1.15299			0.576497	−	0.817099i	$-0.304419\pi$
$677$	30.0000			1.15299			0.576497	+	0.817099i	$0.304419\pi$
$678$		0			0
$679$	−2.50000	−	0.866025i	−0.0959412	−	0.0332350i
$680$	−4.50000	+	7.79423i	−0.172567	+	0.298895i
$681$		0			0
$682$	−12.0000	+	20.7846i	−0.459504	+	0.795884i
$683$	−4.50000	−	7.79423i	−0.172188	−	0.298238i	0.766997	−	0.641651i	$-0.221750\pi$
$683$	−4.50000	−	7.79423i	−0.172188	−	0.298238i	−0.939184	+	0.343413i	$0.888417\pi$
$684$		0			0
$685$	−27.0000			−1.03162
$686$	10.0000	+	15.5885i	0.381802	+	0.595170i
$687$		0			0
$688$	0.500000	+	0.866025i	0.0190623	+	0.0330169i
$689$	3.00000			0.114291
$690$		0			0
$691$	44.0000			1.67384			0.836919	−	0.547326i	$-0.184354\pi$
$691$	44.0000			1.67384			0.836919	+	0.547326i	$0.184354\pi$
$692$	6.00000			0.228086
$693$		0			0
$694$	−12.0000			−0.455514
$695$	21.0000			0.796575
$696$		0			0
$697$	9.00000			0.340899
$698$	−11.5000	−	19.9186i	−0.435281	−	0.753930i
$699$		0			0
$700$	−2.00000	−	10.3923i	−0.0755929	−	0.392792i
$701$	18.0000			0.679851			0.339925	−	0.940452i	$-0.389598\pi$
$701$	18.0000			0.679851			0.339925	+	0.940452i	$0.389598\pi$
$702$		0			0
$703$	−3.50000	−	6.06218i	−0.132005	−	0.228639i
$704$	−1.50000	+	2.59808i	−0.0565334	+	0.0979187i
$705$		0			0
$706$	1.50000	−	2.59808i	0.0564532	−	0.0977799i
$707$	1.50000	+	7.79423i	0.0564133	+	0.293132i
$708$		0			0
$709$	26.0000			0.976450			0.488225	−	0.872718i	$-0.337644\pi$
$709$	26.0000			0.976450			0.488225	+	0.872718i	$0.337644\pi$
$710$	−18.0000	−	31.1769i	−0.675528	−	1.17005i
$711$		0			0
$712$	1.50000	−	2.59808i	0.0562149	−	0.0973670i
$713$	−36.0000	−	62.3538i	−1.34821	−	2.33517i
$714$		0			0
$715$	−4.50000	+	7.79423i	−0.168290	+	0.291488i
$716$	−10.5000	−	18.1865i	−0.392403	−	0.679663i
$717$		0			0
$718$	4.50000	−	7.79423i	0.167939	−	0.290878i
$719$	−7.50000	+	12.9904i	−0.279703	+	0.484459i	−0.971311	−	0.237814i	$-0.923569\pi$
$719$	−7.50000	+	12.9904i	−0.279703	+	0.484459i	0.691608	+	0.722273i	$0.256903\pi$
$720$		0			0
$721$	−32.5000	−	11.2583i	−1.21036	−	0.419282i
$722$	−15.0000	−	25.9808i	−0.558242	−	0.966904i
$723$		0			0
$724$	2.00000			0.0743294
$725$	−12.0000			−0.445669
$726$		0			0
$727$	6.50000	+	11.2583i	0.241072	+	0.417548i	0.961020	−	0.276479i	$-0.0891678\pi$
$727$	6.50000	+	11.2583i	0.241072	+	0.417548i	−0.719948	+	0.694028i	$0.755834\pi$
$728$	0.500000	+	2.59808i	0.0185312	+	0.0962911i
$729$		0			0
$730$	16.5000	−	28.5788i	0.610692	−	1.05775i
$731$	−1.50000	+	2.59808i	−0.0554795	+	0.0960933i
$732$		0			0
$733$	0.500000	+	0.866025i	0.0184679	+	0.0319874i	0.875112	−	0.483921i	$-0.160788\pi$
$733$	0.500000	+	0.866025i	0.0184679	+	0.0319874i	−0.856644	+	0.515908i	$0.827454\pi$
$734$	−8.50000	+	14.7224i	−0.313741	+	0.543415i
$735$		0			0
$736$	−4.50000	−	7.79423i	−0.165872	−	0.287299i
$737$	6.00000	−	10.3923i	0.221013	−	0.382805i
$738$		0			0
$739$	−11.5000	−	19.9186i	−0.423034	−	0.732717i	0.573200	−	0.819415i	$-0.305702\pi$
$739$	−11.5000	−	19.9186i	−0.423034	−	0.732717i	−0.996235	+	0.0866983i	$0.972368\pi$
$740$	3.00000			0.110282
$741$		0			0
$742$	−7.50000	−	2.59808i	−0.275334	−	0.0953784i
$743$	10.5000	−	18.1865i	0.385208	−	0.667199i	−0.606590	−	0.795015i	$-0.707463\pi$
$743$	10.5000	−	18.1865i	0.385208	−	0.667199i	0.991798	+	0.127815i	$0.0407965\pi$
$744$		0			0
$745$	13.5000	−	23.3827i	0.494602	−	0.856675i
$746$	6.50000	+	11.2583i	0.237982	+	0.412197i
$747$		0			0
$748$	−9.00000			−0.329073
$749$	4.50000	+	23.3827i	0.164426	+	0.854385i
$750$		0			0
$751$	6.50000	+	11.2583i	0.237188	+	0.410822i	0.959906	−	0.280321i	$-0.0904408\pi$
$751$	6.50000	+	11.2583i	0.237188	+	0.410822i	−0.722718	+	0.691143i	$0.757107\pi$
$752$		0			0
$753$		0			0
$754$	3.00000			0.109254
$755$	21.0000			0.764268
$756$		0			0
$757$	−22.0000			−0.799604			−0.399802	−	0.916602i	$-0.630921\pi$
$757$	−22.0000			−0.799604			−0.399802	+	0.916602i	$0.630921\pi$
$758$	−28.0000			−1.01701
$759$		0			0
$760$	21.0000			0.761750
$761$	−22.5000	−	38.9711i	−0.815624	−	1.41270i	−0.908879	−	0.417061i	$-0.863060\pi$
$761$	−22.5000	−	38.9711i	−0.815624	−	1.41270i	0.0932544	−	0.995642i	$-0.470273\pi$
$762$		0			0
$763$	−32.5000	−	11.2583i	−1.17658	−	0.407579i
$764$		0			0
$765$		0			0
$766$	7.50000	+	12.9904i	0.270986	+	0.469362i
$767$		0			0
$768$		0			0
$769$	−11.5000	+	19.9186i	−0.414701	+	0.718283i	−0.995397	−	0.0958377i	$-0.969447\pi$
$769$	−11.5000	+	19.9186i	−0.414701	+	0.718283i	0.580696	+	0.814120i	$0.302780\pi$
$770$	18.0000	−	15.5885i	0.648675	−	0.561769i
$771$		0			0
$772$	14.0000			0.503871
$773$	13.5000	+	23.3827i	0.485561	+	0.841017i	0.999862	−	0.0165929i	$-0.00528194\pi$
$773$	13.5000	+	23.3827i	0.485561	+	0.841017i	−0.514301	+	0.857610i	$0.671949\pi$
$774$		0			0
$775$	−16.0000	+	27.7128i	−0.574737	+	0.995474i
$776$	0.500000	+	0.866025i	0.0179490	+	0.0310885i
$777$		0			0
$778$	13.5000	−	23.3827i	0.483998	−	0.838310i
$779$	−10.5000	−	18.1865i	−0.376202	−	0.651600i
$780$		0			0
$781$	18.0000	−	31.1769i	0.644091	−	1.11560i
$782$	13.5000	−	23.3827i	0.482759	−	0.836163i
$783$		0			0
$784$	1.00000	−	6.92820i	0.0357143	−	0.247436i
$785$	−33.0000	−	57.1577i

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:	\( N \)	\(=\)	\( 378 = 2 \cdot 3^{3} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	378.e (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.00000 q^{4} +(1.50000 + 2.59808i) q^{5} +(-2.00000 + 1.73205i) q^{7} +1.00000 q^{8} +O(q^{10})\)	\(q+1.00000 q^{2} +1.00000 q^{4} +(1.50000 + 2.59808i) q^{5} +(-2.00000 + 1.73205i) q^{7} +1.00000 q^{8} +(1.50000 + 2.59808i) q^{10} +(-1.50000 + 2.59808i) q^{11} +(0.500000 - 0.866025i) q^{13} +(-2.00000 + 1.73205i) q^{14} +1.00000 q^{16} +(1.50000 + 2.59808i) q^{17} +(3.50000 - 6.06218i) q^{19} +(1.50000 + 2.59808i) q^{20} +(-1.50000 + 2.59808i) q^{22} +(-4.50000 - 7.79423i) q^{23} +(-2.00000 + 3.46410i) q^{25} +(0.500000 - 0.866025i) q^{26} +(-2.00000 + 1.73205i) q^{28} +(1.50000 + 2.59808i) q^{29} +8.00000 q^{31} +1.00000 q^{32} +(1.50000 + 2.59808i) q^{34} +(-7.50000 - 2.59808i) q^{35} +(0.500000 - 0.866025i) q^{37} +(3.50000 - 6.06218i) q^{38} +(1.50000 + 2.59808i) q^{40} +(1.50000 - 2.59808i) q^{41} +(0.500000 + 0.866025i) q^{43} +(-1.50000 + 2.59808i) q^{44} +(-4.50000 - 7.79423i) q^{46} +(1.00000 - 6.92820i) q^{49} +(-2.00000 + 3.46410i) q^{50} +(0.500000 - 0.866025i) q^{52} +(1.50000 + 2.59808i) q^{53} -9.00000 q^{55} +(-2.00000 + 1.73205i) q^{56} +(1.50000 + 2.59808i) q^{58} +2.00000 q^{61} +8.00000 q^{62} +1.00000 q^{64} +3.00000 q^{65} -4.00000 q^{67} +(1.50000 + 2.59808i) q^{68} +(-7.50000 - 2.59808i) q^{70} -12.0000 q^{71} +(-5.50000 - 9.52628i) q^{73} +(0.500000 - 0.866025i) q^{74} +(3.50000 - 6.06218i) q^{76} +(-1.50000 - 7.79423i) q^{77} -16.0000 q^{79} +(1.50000 + 2.59808i) q^{80} +(1.50000 - 2.59808i) q^{82} +(-4.50000 - 7.79423i) q^{83} +(-4.50000 + 7.79423i) q^{85} +(0.500000 + 0.866025i) q^{86} +(-1.50000 + 2.59808i) q^{88} +(1.50000 - 2.59808i) q^{89} +(0.500000 + 2.59808i) q^{91} +(-4.50000 - 7.79423i) q^{92} +21.0000 q^{95} +(0.500000 + 0.866025i) q^{97} +(1.00000 - 6.92820i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{2} + 2 q^{4} + 3 q^{5} - 4 q^{7} + 2 q^{8}+O(q^{10}) \) 2 * q + 2 * q^2 + 2 * q^4 + 3 * q^5 - 4 * q^7 + 2 * q^8	\( 2 q + 2 q^{2} + 2 q^{4} + 3 q^{5} - 4 q^{7} + 2 q^{8} + 3 q^{10} - 3 q^{11} + q^{13} - 4 q^{14} + 2 q^{16} + 3 q^{17} + 7 q^{19} + 3 q^{20} - 3 q^{22} - 9 q^{23} - 4 q^{25} + q^{26} - 4 q^{28} + 3 q^{29} + 16 q^{31} + 2 q^{32} + 3 q^{34} - 15 q^{35} + q^{37} + 7 q^{38} + 3 q^{40} + 3 q^{41} + q^{43} - 3 q^{44} - 9 q^{46} + 2 q^{49} - 4 q^{50} + q^{52} + 3 q^{53} - 18 q^{55} - 4 q^{56} + 3 q^{58} + 4 q^{61} + 16 q^{62} + 2 q^{64} + 6 q^{65} - 8 q^{67} + 3 q^{68} - 15 q^{70} - 24 q^{71} - 11 q^{73} + q^{74} + 7 q^{76} - 3 q^{77} - 32 q^{79} + 3 q^{80} + 3 q^{82} - 9 q^{83} - 9 q^{85} + q^{86} - 3 q^{88} + 3 q^{89} + q^{91} - 9 q^{92} + 42 q^{95} + q^{97} + 2 q^{98}+O(q^{100}) \) 2 * q + 2 * q^2 + 2 * q^4 + 3 * q^5 - 4 * q^7 + 2 * q^8 + 3 * q^10 - 3 * q^11 + q^13 - 4 * q^14 + 2 * q^16 + 3 * q^17 + 7 * q^19 + 3 * q^20 - 3 * q^22 - 9 * q^23 - 4 * q^25 + q^26 - 4 * q^28 + 3 * q^29 + 16 * q^31 + 2 * q^32 + 3 * q^34 - 15 * q^35 + q^37 + 7 * q^38 + 3 * q^40 + 3 * q^41 + q^43 - 3 * q^44 - 9 * q^46 + 2 * q^49 - 4 * q^50 + q^52 + 3 * q^53 - 18 * q^55 - 4 * q^56 + 3 * q^58 + 4 * q^61 + 16 * q^62 + 2 * q^64 + 6 * q^65 - 8 * q^67 + 3 * q^68 - 15 * q^70 - 24 * q^71 - 11 * q^73 + q^74 + 7 * q^76 - 3 * q^77 - 32 * q^79 + 3 * q^80 + 3 * q^82 - 9 * q^83 - 9 * q^85 + q^86 - 3 * q^88 + 3 * q^89 + q^91 - 9 * q^92 + 42 * q^95 + q^97 + 2 * q^98

Introduction

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