LMFDB - Embedded newform 378.2.e.b.235.1

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			235.1
Root			$0.500000 + 0.866025i$ of defining polynomial
Character	$\chi$	$=$	378.235
Dual form			378.2.e.b.37.1

$q$-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 $\left(\mathbb{Z}/378\mathbb{Z}\right)^\times$.

$n$	$29$	$325$
$\chi(n)$	$e\left(\frac{2}{3}\right)$	$e\left(\frac{2}{3}\right)$

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	−0.306851	−	0.951757i	$-0.599275\pi$
$5$	1.50000	−	2.59808i	0.670820	−	1.16190i	0.977672	−	0.210138i	$-0.0673912\pi$
$6$		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	0.546259	−	0.837616i	$-0.316051\pi$
$11$	−1.50000	−	2.59808i	−0.452267	−	0.783349i	−0.998526	+	0.0542666i	$0.982718\pi$
$12$		0			0
$13$	0.500000	+	0.866025i	0.138675	+	0.240192i	0.926995	−	0.375073i	$-0.122382\pi$
$13$	0.500000	+	0.866025i	0.138675	+	0.240192i	−0.788320	+	0.615265i	$0.789049\pi$
$14$	−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	−0.624780	−	0.780801i	$-0.714811\pi$
$17$	1.50000	−	2.59808i	0.363803	−	0.630126i	0.988583	+	0.150675i	$0.0481447\pi$
$18$		0			0
$19$	3.50000	+	6.06218i	0.802955	+	1.39076i	0.917663	+	0.397360i	$0.130073\pi$
$19$	3.50000	+	6.06218i	0.802955	+	1.39076i	−0.114708	+	0.993399i	$0.536593\pi$
$20$	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	−0.169701	+	0.985496i	$0.554280\pi$
$23$	−4.50000	+	7.79423i	−0.938315	+	1.62521i	−0.768613	+	0.639713i	$0.779053\pi$
$24$		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	−0.692480	−	0.721437i	$-0.743482\pi$
$29$	1.50000	−	2.59808i	0.278543	−	0.482451i	0.971023	+	0.238987i	$0.0768152\pi$
$30$		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	0.904194	−	0.427121i	$-0.140472\pi$
$37$	0.500000	+	0.866025i	0.0821995	+	0.142374i	−0.821995	+	0.569495i	$0.807139\pi$
$38$	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	0.959058	−	0.283211i	$-0.0913998\pi$
$41$	1.50000	+	2.59808i	0.234261	+	0.405751i	−0.724797	+	0.688963i	$0.758066\pi$
$42$		0			0
$43$	0.500000	−	0.866025i	0.0762493	−	0.132068i	−0.825380	−	0.564578i	$-0.809039\pi$
$43$	0.500000	−	0.866025i	0.0762493	−	0.132068i	0.901629	+	0.432511i	$0.142372\pi$
$44$	−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	−0.744423	−	0.667708i	$-0.767275\pi$
$53$	1.50000	−	2.59808i	0.206041	−	0.356873i	0.950464	+	0.310835i	$0.100609\pi$
$54$		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	0.340868	+	0.940111i	$0.389279\pi$
$73$	−5.50000	+	9.52628i	−0.643726	+	1.11497i	−0.984594	+	0.174855i	$0.944054\pi$
$74$	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	−0.999976	−	0.00698436i	$-0.997777\pi$
$83$	−4.50000	+	7.79423i	−0.493939	+	0.855528i	0.506036	+	0.862512i	$0.331110\pi$
$84$		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	0.934508	−	0.355942i	$-0.115840\pi$
$89$	1.50000	+	2.59808i	0.159000	+	0.275396i	−0.775509	+	0.631337i	$0.782506\pi$
$90$		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	−0.839525	−	0.543321i	$-0.817167\pi$
$97$	0.500000	−	0.866025i	0.0507673	−	0.0879316i	0.890292	+	0.455389i	$0.150500\pi$
$98$	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	0.930953	−	0.365140i	$-0.118979\pi$
$101$	1.50000	+	2.59808i	0.149256	+	0.258518i	−0.781697	+	0.623658i	$0.785646\pi$
$102$		0			0
$103$	6.50000	−	11.2583i	0.640464	−	1.10932i	−0.344865	−	0.938652i	$-0.612075\pi$
$103$	6.50000	−	11.2583i	0.640464	−	1.10932i	0.985329	−	0.170664i	$-0.0545913\pi$
$104$	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	0.997298	−	0.0734594i	$-0.0234039\pi$
$107$	4.50000	+	7.79423i	0.435031	+	0.753497i	−0.562267	+	0.826956i	$0.690071\pi$
$108$		0			0
$109$	6.50000	−	11.2583i	0.622587	−	1.07835i	−0.366415	−	0.930451i	$-0.619415\pi$
$109$	6.50000	−	11.2583i	0.622587	−	1.07835i	0.989002	−	0.147901i	$-0.0472517\pi$
$110$	−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	0.572938	−	0.819599i	$-0.305804\pi$
$113$	−4.50000	−	7.79423i	−0.423324	−	0.733219i	−0.996262	+	0.0863794i	$0.972470\pi$
$114$		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	−0.326546	−	0.945181i	$-0.605885\pi$
$131$	7.50000	−	12.9904i	0.655278	−	1.13497i	0.981824	−	0.189794i	$-0.0607819\pi$
$132$		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	0.607233	−	0.794524i	$-0.292279\pi$
$137$	−4.50000	−	7.79423i	−0.384461	−	0.665906i	−0.991694	+	0.128618i	$0.958946\pi$
$138$		0			0
$139$	3.50000	+	6.06218i	0.296866	+	0.514187i	0.975417	−	0.220366i	$-0.0707252\pi$
$139$	3.50000	+	6.06218i	0.296866	+	0.514187i	−0.678551	+	0.734553i	$0.737392\pi$
$140$	−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	−0.989355	−	0.145519i	$-0.953515\pi$
$149$	−4.50000	+	7.79423i	−0.368654	+	0.638528i	0.620701	+	0.784047i	$0.286848\pi$
$150$		0			0
$151$	3.50000	+	6.06218i	0.284826	+	0.493333i	0.972567	−	0.232623i	$-0.0747309\pi$
$151$	3.50000	+	6.06218i	0.284826	+	0.493333i	−0.687741	+	0.725956i	$0.741398\pi$
$152$	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	0.950615	+	0.310372i	$0.100454\pi$
$163$	9.50000	+	16.4545i	0.744097	+	1.28881i	−0.206518	+	0.978443i	$0.566213\pi$
$164$	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	−0.995436	−	0.0954356i	$-0.969576\pi$
$167$	−7.50000	−	12.9904i	−0.580367	−	1.00523i	0.415068	−	0.909790i	$-0.363758\pi$
$168$		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	0.144308	+	0.989533i	$0.453905\pi$
$179$	−10.5000	+	18.1865i	−0.784807	+	1.35933i	−0.929114	+	0.369792i	$0.879429\pi$
$180$		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	0.0416556	−	0.999132i	$-0.486737\pi$
$199$	12.5000	−	21.6506i	0.886102	−	1.53477i	0.844446	−	0.535641i	$-0.179930\pi$
$200$	−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	0.767049	−	0.641588i	$-0.221724\pi$
$211$	−2.50000	−	4.33013i	−0.172107	−	0.298098i	−0.939156	+	0.343490i	$0.888391\pi$
$212$	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	−0.848799	−	0.528716i	$-0.822674\pi$
$223$	0.500000	−	0.866025i	0.0334825	−	0.0579934i	0.882281	+	0.470723i	$0.156007\pi$
$224$	−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	0.811943	−	0.583736i	$-0.198410\pi$
$227$	−1.50000	−	2.59808i	−0.0995585	−	0.172440i	−0.911502	+	0.411296i	$0.865076\pi$
$228$		0			0
$229$	6.50000	−	11.2583i	0.429532	−	0.743971i	−0.567300	−	0.823511i	$-0.692012\pi$
$229$	6.50000	−	11.2583i	0.429532	−	0.743971i	0.996832	+	0.0795401i	$0.0253452\pi$
$230$	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	0.910968	−	0.412477i	$-0.135336\pi$
$233$	1.50000	+	2.59808i	0.0982683	+	0.170206i	−0.812700	+	0.582683i	$0.802003\pi$
$234$		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	0.813426	−	0.581669i	$-0.197600\pi$
$239$	−1.50000	−	2.59808i	−0.0970269	−	0.168056i	−0.910453	+	0.413613i	$0.864267\pi$
$240$		0			0
$241$	6.50000	+	11.2583i	0.418702	+	0.725213i	0.995809	−	0.0914555i	$-0.0291519\pi$
$241$	6.50000	+	11.2583i	0.418702	+	0.725213i	−0.577107	+	0.816668i	$0.695819\pi$
$242$	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	0.326929	+	0.945049i	$0.393986\pi$
$257$	−10.5000	+	18.1865i	−0.654972	+	1.13444i	−0.981901	+	0.189396i	$0.939347\pi$
$258$		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	0.970758	−	0.240059i	$-0.0771668\pi$
$263$	4.50000	+	7.79423i	0.277482	+	0.480613i	−0.693276	+	0.720672i	$0.743833\pi$
$264$		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	−0.541533	−	0.840679i	$-0.682156\pi$
$269$	7.50000	−	12.9904i	0.457283	−	0.792038i	0.998816	+	0.0486418i	$0.0154893\pi$
$270$		0			0
$271$	−2.50000	−	4.33013i	−0.151864	−	0.263036i	0.780049	−	0.625719i	$-0.215194\pi$
$271$	−2.50000	−	4.33013i	−0.151864	−	0.263036i	−0.931913	+	0.362682i	$0.881861\pi$
$272$	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	0.880656	−	0.473757i	$-0.157103\pi$
$277$	0.500000	+	0.866025i	0.0300421	+	0.0520344i	−0.850613	+	0.525792i	$0.823769\pi$
$278$	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	0.361895	+	0.932219i	$0.382130\pi$
$281$	−10.5000	+	18.1865i	−0.626377	+	1.08492i	−0.988273	+	0.152699i	$0.951204\pi$
$282$		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	0.704117	−	0.710084i	$-0.251343\pi$
$293$	−4.50000	−	7.79423i	−0.262893	−	0.455344i	−0.967009	+	0.254741i	$0.918010\pi$
$294$		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	0.986960	−	0.160968i	$-0.0514616\pi$
$337$	6.50000	+	11.2583i	0.354078	+	0.613280i	−0.632882	+	0.774248i	$0.718128\pi$
$338$	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	0.374701	+	0.927146i	$0.377745\pi$
$349$	−11.5000	+	19.9186i	−0.615581	+	1.06622i	−0.990282	+	0.139072i	$0.955588\pi$
$350$	−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	0.903179	−	0.429263i	$-0.141227\pi$
$353$	1.50000	+	2.59808i	0.0798369	+	0.138282i	−0.823343	+	0.567545i	$0.807893\pi$
$354$		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	0.959997	−	0.280012i	$-0.0903384\pi$
$359$	4.50000	+	7.79423i	0.237501	+	0.411364i	−0.722496	+	0.691375i	$0.757005\pi$
$360$		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	0.554264	−	0.832341i	$-0.313000\pi$
$367$	−8.50000	−	14.7224i	−0.443696	−	0.768505i	−0.997960	+	0.0638362i	$0.979666\pi$
$368$	−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	−0.647225	−	0.762299i	$-0.724071\pi$
$373$	6.50000	−	11.2583i	0.336557	−	0.582934i	0.983783	+	0.179364i	$0.0574041\pi$
$374$	−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	−0.608290	−	0.793715i	$-0.708144\pi$
$383$	7.50000	−	12.9904i	0.383232	−	0.663777i	0.991522	+	0.129937i	$0.0414776\pi$
$384$		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	0.973601	+	0.228257i	$0.0733028\pi$
$389$	13.5000	+	23.3827i	0.684477	+	1.18555i	−0.289124	+	0.957292i	$0.593364\pi$
$390$		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	0.981760	−	0.190126i	$-0.0608897\pi$
$397$	6.50000	+	11.2583i	0.326226	+	0.565039i	−0.655534	+	0.755166i	$0.727556\pi$
$398$	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	−0.302556	−	0.953131i	$-0.597840\pi$
$401$	13.5000	−	23.3827i	0.674158	−	1.16768i	0.976714	−	0.214544i	$-0.0688266\pi$
$402$		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	0.954759	−	0.297382i	$-0.0961133\pi$
$419$	4.50000	+	7.79423i	0.219839	+	0.380773i	−0.734919	+	0.678155i	$0.762780\pi$
$420$		0			0
$421$	−17.5000	+	30.3109i	−0.852898	+	1.47726i	0.0256838	+	0.999670i	$0.491824\pi$
$421$	−17.5000	+	30.3109i	−0.852898	+	1.47726i	−0.878582	+	0.477592i	$0.841510\pi$
$422$	−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	−0.332785	−	0.943003i	$-0.607988\pi$
$431$	13.5000	−	23.3827i	0.650272	−	1.12630i	0.983057	−	0.183301i	$-0.0586785\pi$
$432$		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	−0.951584	−	0.307388i	$-0.900545\pi$
$461$	−4.50000	+	7.79423i	−0.209586	+	0.363013i	0.741998	+	0.670402i	$0.233878\pi$
$462$		0			0
$463$	−20.5000	−	35.5070i	−0.952716	−	1.65015i	−0.739511	−	0.673145i	$-0.764943\pi$
$463$	−20.5000	−	35.5070i	−0.952716	−	1.65015i	−0.213205	−	0.977007i	$-0.568390\pi$
$464$	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	0.829231	−	0.558906i	$-0.188779\pi$
$467$	−1.50000	−	2.59808i	−0.0694117	−	0.120225i	−0.898642	+	0.438682i	$0.855446\pi$
$468$		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	0.829721	−	0.558179i	$-0.188500\pi$
$479$	−1.50000	−	2.59808i	−0.0685367	−	0.118709i	−0.898257	+	0.439470i	$0.855166\pi$
$480$		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	−0.430486	−	0.902597i	$-0.641658\pi$
$487$	12.5000	−	21.6506i	0.566429	−	0.981084i	0.996915	−	0.0784867i	$-0.0250088\pi$
$488$	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	0.999552	−	0.0299272i	$-0.00952753\pi$
$491$	10.5000	+	18.1865i	0.473858	+	0.820747i	−0.525694	+	0.850674i	$0.676194\pi$
$492$		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	−0.437955	−	0.898997i	$-0.644297\pi$
$499$	12.5000	−	21.6506i	0.559577	−	0.969216i	0.997532	−	0.0702185i	$-0.0223697\pi$
$500$	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	−0.948353	−	0.317217i	$-0.897252\pi$
$509$	−4.50000	+	7.79423i	−0.199459	+	0.345473i	0.748894	+	0.662690i	$0.230585\pi$
$510$		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	−0.831295	−	0.555831i	$-0.812400\pi$
$521$	1.50000	−	2.59808i	0.0657162	−	0.113824i	0.897011	+	0.442007i	$0.145733\pi$
$522$		0			0
$523$	3.50000	+	6.06218i	0.153044	+	0.265081i	0.932345	−	0.361569i	$-0.117759\pi$
$523$	3.50000	+	6.06218i	0.153044	+	0.265081i	−0.779301	+	0.626650i	$0.784426\pi$
$524$	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	0.723234	−	0.690604i	$-0.242655\pi$
$541$	−5.50000	−	9.52628i	−0.236463	−	0.409567i	−0.959697	+	0.281037i	$0.909322\pi$
$542$	−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	−0.959320	−	0.282321i	$-0.908896\pi$
$547$	−5.50000	+	9.52628i	−0.235163	+	0.407314i	0.724157	+	0.689635i	$0.242229\pi$
$548$	−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	−0.945473	−	0.325701i	$-0.894400\pi$
$557$	−4.50000	+	7.79423i	−0.190671	+	0.330252i	0.754802	+	0.655953i	$0.227733\pi$
$558$		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	−0.479337	−	0.877631i	$-0.659123\pi$
$577$	12.5000	−	21.6506i	0.520382	−	0.901328i	0.999719	−	0.0236970i	$-0.00754370\pi$
$578$	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	−0.833408	−	0.552658i	$-0.813614\pi$
$587$	1.50000	−	2.59808i	0.0619116	−	0.107234i	0.895320	+	0.445424i	$0.146947\pi$
$588$		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	0.919111	+	0.394000i	$0.128909\pi$
$593$	19.5000	+	33.7750i	0.800769	+	1.38697i	−0.118342	+	0.992973i	$0.537758\pi$
$594$		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	−0.490049	−	0.871695i	$-0.663021\pi$
$601$	12.5000	−	21.6506i	0.509886	−	0.883148i	0.999934	−	0.0114528i	$-0.00364562\pi$
$602$	−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	−0.703429	−	0.710766i	$-0.748349\pi$
$607$	6.50000	−	11.2583i	0.263827	−	0.456962i	0.967256	+	0.253804i	$0.0816819\pi$
$608$	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	−0.999178	−	0.0405396i	$-0.987092\pi$
$613$	−11.5000	+	19.9186i	−0.464481	+	0.804504i	0.534697	+	0.845044i	$0.320426\pi$
$614$	−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	−0.819818	−	0.572624i	$-0.805926\pi$
$617$	−22.5000	−	38.9711i	−0.905816	−	1.56892i	−0.0859976	−	0.996295i	$-0.527408\pi$
$618$		0			0
$619$	−8.50000	−	14.7224i	−0.341644	−	0.591744i	0.643094	−	0.765787i	$-0.277650\pi$
$619$	−8.50000	−	14.7224i	−0.341644	−	0.591744i	−0.984738	+	0.174042i	$0.944317\pi$
$620$	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	−0.982708	−	0.185164i	$-0.940718\pi$
$641$	−16.5000	−	28.5788i	−0.651711	−	1.12880i	0.330997	−	0.943632i	$-0.392615\pi$
$642$		0			0
$643$	−14.5000	−	25.1147i	−0.571824	−	0.990429i	−0.996379	−	0.0850262i	$-0.972903\pi$
$643$	−14.5000	−	25.1147i	−0.571824	−	0.990429i	0.424555	−	0.905402i	$-0.360431\pi$
$644$	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	−0.995194	−	0.0979182i	$-0.968782\pi$
$647$	−10.5000	+	18.1865i	−0.412798	+	0.714986i	0.582397	+	0.812905i	$0.302115\pi$
$648$		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	−0.681137	−	0.732156i	$-0.738514\pi$
$653$	7.50000	−	12.9904i	0.293498	−	0.508353i	0.974634	+	0.223803i	$0.0718474\pi$
$654$		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	−0.835330	−	0.549749i	$-0.814723\pi$
$659$	1.50000	−	2.59808i	0.0584317	−	0.101207i	0.893762	+	0.448542i	$0.148057\pi$
$660$		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	0.302017	+	0.953302i	$0.402340\pi$
$673$	−17.5000	+	30.3109i	−0.674575	+	1.16840i	−0.976593	+	0.215096i	$0.930993\pi$
$674$	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	−0.939184	−	0.343413i	$-0.888417\pi$
$683$	−4.50000	+	7.79423i	−0.172188	+	0.298238i	0.766997	+	0.641651i	$0.221750\pi$
$684$		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	0.691608	−	0.722273i	$-0.256903\pi$
$719$	−7.50000	−	12.9904i	−0.279703	−	0.484459i	−0.971311	+	0.237814i	$0.923569\pi$
$720$		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	−0.719948	−	0.694028i	$-0.755834\pi$
$727$	6.50000	−	11.2583i	0.241072	−	0.417548i	0.961020	+	0.276479i	$0.0891678\pi$
$728$	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	−0.856644	−	0.515908i	$-0.827454\pi$
$733$	0.500000	−	0.866025i	0.0184679	−	0.0319874i	0.875112	+	0.483921i	$0.160788\pi$
$734$	−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	−0.996235	−	0.0866983i	$-0.972368\pi$
$739$	−11.5000	+	19.9186i	−0.423034	+	0.732717i	0.573200	+	0.819415i	$0.305702\pi$
$740$	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	0.991798	−	0.127815i	$-0.0407965\pi$
$743$	10.5000	+	18.1865i	0.385208	+	0.667199i	−0.606590	+	0.795015i	$0.707463\pi$
$744$		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	−0.722718	−	0.691143i	$-0.757107\pi$
$751$	6.50000	−	11.2583i	0.237188	−	0.410822i	0.959906	+	0.280321i	$0.0904408\pi$
$752$		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	0.0932544	+	0.995642i	$0.470273\pi$
$761$	−22.5000	+	38.9711i	−0.815624	+	1.41270i	−0.908879	+	0.417061i	$0.863060\pi$
$762$		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	0.580696	−	0.814120i	$-0.302780\pi$
$769$	−11.5000	−	19.9186i	−0.414701	−	0.718283i	−0.995397	+	0.0958377i	$0.969447\pi$
$770$	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	−0.514301	−	0.857610i	$-0.671949\pi$
$773$	13.5000	−	23.3827i	0.485561	−	0.841017i	0.999862	+	0.0165929i	$0.00528194\pi$
$774$		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	−1.17782	+	2.04004i
$786$		0			0
$787$	−28.0000			−0.998092			−0.499046	−	0.866575i	$-0.666316\pi$
$787$	−28.0000			−0.998092			−0.499046	+	0.866575i	$0.666316\pi$
$788$	−18.0000			−0.641223
$789$		0			0
$790$	−24.0000	+	41.5692i	−0.853882	+	1.47897i
$791$	−4.50000	+	23.3827i	−0.160002	+	0.831393i
$792$		0			0
$793$	1.00000	+	1.73205i	0.0355110	+	0.0615069i
$794$	6.50000	+	11.2583i	0.230676	+	0.399543i
$795$		0			0
$796$	12.5000	−	21.6506i	0.443051	−	0.767386i
$797$	−10.5000	−	18.1865i	−0.371929	−	0.644200i	0.617933	−	0.786231i	$-0.287970\pi$
$797$	−10.5000	−	18.1865i	−0.371929	−	0.644200i	−0.989862	+	0.142031i	$0.954637\pi$
$798$		0			0
$799$		0			0
$800$	−2.00000	−	3.46410i	−0.0707107	−	0.122474i
$801$		0			0
$802$	13.5000	−	23.3827i	0.476702	−	0.825671i
$803$	33.0000			1.16454
$804$		0			0
$805$	13.5000	−	70.1481i	0.475812	−	2.47239i
$806$	4.00000	+	6.92820i	0.140894	+	0.244036i
$807$		0			0
$808$	1.50000	+	2.59808i	0.0527698	+	0.0914000i
$809$	−16.5000	+	28.5788i	−0.580109	+	1.00478i	0.415357	+	0.909659i	$0.363657\pi$
$809$	−16.5000	+	28.5788i	−0.580109	+	1.00478i	−0.995466	+	0.0951198i	$0.969677\pi$
$810$		0			0
$811$	20.0000			0.702295			0.351147	−	0.936320i	$-0.385792\pi$
$811$	20.0000			0.702295			0.351147	+	0.936320i	$0.385792\pi$
$812$	−7.50000	+	2.59808i	−0.263198	+	0.0911746i
$813$		0			0
$814$	1.50000	−	2.59808i	0.0525750	−	0.0910625i
$815$	57.0000			1.99662
$816$		0			0
$817$	7.00000			0.244899
$818$	−34.0000			−1.18878
$819$		0			0
$820$	9.00000			0.314294
$821$	42.0000			1.46581			0.732905	−	0.680331i	$-0.238164\pi$
$821$	42.0000			1.46581			0.732905	+	0.680331i	$0.238164\pi$
$822$		0			0
$823$	−40.0000			−1.39431			−0.697156	−	0.716919i	$-0.745552\pi$
$823$	−40.0000			−1.39431			−0.697156	+	0.716919i	$0.745552\pi$
$824$	6.50000	−	11.2583i	0.226438	−	0.392203i
$825$		0			0
$826$		0			0
$827$	−36.0000			−1.25184			−0.625921	−	0.779886i	$-0.715277\pi$
$827$	−36.0000			−1.25184			−0.625921	+	0.779886i	$0.715277\pi$
$828$		0			0
$829$	−5.50000	+	9.52628i	−0.191023	+	0.330861i	−0.945589	−	0.325362i	$-0.894514\pi$
$829$	−5.50000	+	9.52628i	−0.191023	+	0.330861i	0.754567	+	0.656223i	$0.227847\pi$
$830$	13.5000	+	23.3827i	0.468592	+	0.811625i
$831$		0			0
$832$	0.500000	+	0.866025i	0.0173344	+	0.0300240i
$833$	19.5000	+	7.79423i	0.675635	+	0.270054i
$834$		0			0
$835$	−45.0000			−1.55729
$836$	10.5000	−	18.1865i	0.363150	−	0.628994i
$837$		0			0
$838$	4.50000	+	7.79423i	0.155450	+	0.269247i
$839$	7.50000	−	12.9904i	0.258929	−	0.448478i	−0.707026	−	0.707187i	$-0.749964\pi$
$839$	7.50000	−	12.9904i	0.258929	−	0.448478i	0.965955	+	0.258709i	$0.0832972\pi$
$840$		0			0
$841$	10.0000	+	17.3205i	0.344828	+	0.597259i
$842$	−17.5000	+	30.3109i	−0.603090	+	1.04458i
$843$		0			0
$844$	−2.50000	−	4.33013i	−0.0860535	−	0.149049i
$845$	−18.0000	−	31.1769i	−0.619219	−	1.07252i
$846$		0			0
$847$	−5.00000	+	1.73205i	−0.171802	+	0.0595140i
$848$	1.50000	−	2.59808i	0.0515102	−	0.0892183i
$849$		0			0
$850$	−12.0000			−0.411597
$851$	−9.00000			−0.308516
$852$		0			0
$853$	0.500000	−	0.866025i	0.0171197	−	0.0296521i	−0.857339	−	0.514753i	$-0.827884\pi$
$853$	0.500000	−	0.866025i	0.0171197	−	0.0296521i	0.874458	+	0.485101i	$0.161217\pi$
$854$	−4.00000	−	3.46410i	−0.136877	−	0.118539i
$855$		0			0
$856$	4.50000	+	7.79423i	0.153807	+	0.266401i
$857$	1.50000	+	2.59808i	0.0512390	+	0.0887486i	0.890507	−	0.454969i	$-0.150350\pi$
$857$	1.50000	+	2.59808i	0.0512390	+	0.0887486i	−0.839268	+	0.543718i	$0.817016\pi$
$858$		0			0
$859$	12.5000	−	21.6506i	0.426494	−	0.738710i	−0.570064	−	0.821600i	$-0.693082\pi$
$859$	12.5000	−	21.6506i	0.426494	−	0.738710i	0.996559	+	0.0828900i	$0.0264150\pi$
$860$	−1.50000	−	2.59808i	−0.0511496	−	0.0885937i
$861$		0			0
$862$	13.5000	−	23.3827i	0.459812	−	0.796417i
$863$	−25.5000	−	44.1673i	−0.868030	−	1.50347i	−0.864007	−	0.503480i	$-0.832053\pi$
$863$	−25.5000	−	44.1673i	−0.868030	−	1.50347i	−0.00402340	−	0.999992i	$-0.501281\pi$
$864$		0			0
$865$	9.00000	−	15.5885i	0.306009	−	0.530023i
$866$	2.00000			0.0679628
$867$		0			0
$868$	−16.0000	−	13.8564i	−0.543075	−	0.470317i
$869$	24.0000	+	41.5692i	0.814144	+	1.41014i
$870$		0			0
$871$	−2.00000	−	3.46410i	−0.0677674	−	0.117377i
$872$	6.50000	−	11.2583i	0.220118	−	0.381255i
$873$		0			0
$874$	−63.0000			−2.13101
$875$	−6.00000	−	5.19615i	−0.202837	−	0.175662i
$876$		0			0
$877$	−23.5000	+	40.7032i	−0.793539	+	1.37445i	0.130224	+	0.991485i	$0.458430\pi$
$877$	−23.5000	+	40.7032i	−0.793539	+	1.37445i	−0.923763	+	0.382965i	$0.874903\pi$
$878$	8.00000			0.269987
$879$		0			0
$880$	−9.00000			−0.303390
$881$	18.0000			0.606435			0.303218	−	0.952921i	$-0.401939\pi$
$881$	18.0000			0.606435			0.303218	+	0.952921i	$0.401939\pi$
$882$		0			0
$883$	20.0000			0.673054			0.336527	−	0.941674i	$-0.390748\pi$
$883$	20.0000			0.673054			0.336527	+	0.941674i	$0.390748\pi$
$884$	3.00000			0.100901
$885$		0			0
$886$	−36.0000			−1.20944
$887$	−16.5000	+	28.5788i	−0.554016	+	0.959583i	0.443964	+	0.896045i	$0.353572\pi$
$887$	−16.5000	+	28.5788i	−0.554016	+	0.959583i	−0.997979	+	0.0635387i	$0.979761\pi$
$888$		0			0
$889$	8.00000	+	6.92820i	0.268311	+	0.232364i
$890$	9.00000			0.301681
$891$		0			0
$892$	0.500000	−	0.866025i	0.0167412	−	0.0289967i
$893$		0			0
$894$		0			0
$895$	31.5000	+	54.5596i	1.05293	+	1.82373i
$896$	−2.00000	−	1.73205i	−0.0668153	−	0.0578638i
$897$		0			0
$898$	−6.00000			−0.200223
$899$	12.0000	−	20.7846i	0.400222	−	0.693206i
$900$		0			0
$901$	−4.50000	−	7.79423i	−0.149917	−	0.259663i
$902$	4.50000	−	7.79423i	0.149834	−	0.259519i
$903$		0			0
$904$	−4.50000	−	7.79423i	−0.149668	−	0.259232i
$905$	3.00000	−	5.19615i	0.0997234	−	0.172726i
$906$		0			0
$907$	21.5000	+	37.2391i	0.713896	+	1.23650i	0.963384	+	0.268126i	$0.0864043\pi$
$907$	21.5000	+	37.2391i	0.713896	+	1.23650i	−0.249488	+	0.968378i	$0.580262\pi$
$908$	−1.50000	−	2.59808i	−0.0497792	−	0.0862202i
$909$		0			0
$910$	−6.00000	−	5.19615i	−0.198898	−	0.172251i
$911$	19.5000	−	33.7750i	0.646064	−	1.11902i	−0.337991	−	0.941149i	$-0.609747\pi$
$911$	19.5000	−	33.7750i	0.646064	−	1.11902i	0.984055	−	0.177866i	$-0.0569194\pi$
$912$		0			0
$913$	27.0000			0.893570
$914$	−10.0000			−0.330771
$915$		0			0
$916$	6.50000	−	11.2583i	0.214766	−	0.371986i
$917$	−37.5000	+	12.9904i	−1.23836	+	0.428980i
$918$		0			0
$919$	−26.5000	−	45.8993i	−0.874154	−	1.51408i	−0.857661	−	0.514216i	$-0.828083\pi$
$919$	−26.5000	−	45.8993i	−0.874154	−	1.51408i	−0.0164935	−	0.999864i	$-0.505250\pi$
$920$	13.5000	+	23.3827i	0.445082	+	0.770904i
$921$		0			0
$922$	−4.50000	+	7.79423i	−0.148200	+	0.256689i
$923$	−6.00000	−	10.3923i	−0.197492	−	0.342067i
$924$		0			0
$925$	2.00000	−	3.46410i	0.0657596	−	0.113899i
$926$	−20.5000	−	35.5070i	−0.673672	−	1.16683i
$927$		0			0
$928$	1.50000	−	2.59808i	0.0492399	−	0.0852860i
$929$	−18.0000			−0.590561			−0.295280	−	0.955411i	$-0.595413\pi$
$929$	−18.0000			−0.590561			−0.295280	+	0.955411i	$0.595413\pi$
$930$		0			0
$931$	−38.5000	+	30.3109i	−1.26179	+	0.993399i
$932$	1.50000	+	2.59808i	0.0491341	+	0.0851028i
$933$		0			0
$934$	−1.50000	−	2.59808i	−0.0490815	−	0.0850117i
$935$	−13.5000	+	23.3827i	−0.441497	+	0.764696i
$936$		0			0
$937$	26.0000			0.849383			0.424691	−	0.905338i	$-0.360383\pi$
$937$	26.0000			0.849383			0.424691	+	0.905338i	$0.360383\pi$
$938$	8.00000	+	6.92820i	0.261209	+	0.226214i
$939$		0			0
$940$		0			0
$941$	6.00000			0.195594			0.0977972	−	0.995206i	$-0.468820\pi$
$941$	6.00000			0.195594			0.0977972	+	0.995206i	$0.468820\pi$
$942$		0			0
$943$	−27.0000			−0.879241
$944$		0			0
$945$		0			0
$946$	−3.00000			−0.0975384
$947$	−12.0000			−0.389948			−0.194974	−	0.980808i	$-0.562462\pi$
$947$	−12.0000			−0.389948			−0.194974	+	0.980808i	$0.562462\pi$
$948$		0			0
$949$	−11.0000			−0.357075
$950$	14.0000	−	24.2487i	0.454220	−	0.786732i
$951$		0			0
$952$	−7.50000	+	2.59808i	−0.243076	+	0.0842041i
$953$	−6.00000			−0.194359			−0.0971795	−	0.995267i	$-0.530982\pi$
$953$	−6.00000			−0.194359			−0.0971795	+	0.995267i	$0.530982\pi$
$954$		0			0
$955$		0			0
$956$	−1.50000	−	2.59808i	−0.0485135	−	0.0840278i
$957$		0			0
$958$	−1.50000	−	2.59808i	−0.0484628	−	0.0839400i
$959$	−4.50000	+	23.3827i	−0.145313	+	0.755066i
$960$		0			0
$961$	33.0000			1.06452
$962$	−0.500000	+	0.866025i	−0.0161206	+	0.0279218i
$963$		0			0
$964$	6.50000	+	11.2583i	0.209351	+	0.362606i
$965$	21.0000	−	36.3731i	0.676014	−	1.17089i
$966$		0			0
$967$	−20.5000	−	35.5070i	−0.659236	−	1.14183i	−0.980814	−	0.194946i	$-0.937547\pi$
$967$	−20.5000	−	35.5070i	−0.659236	−	1.14183i	0.321578	−	0.946883i	$-0.395787\pi$
$968$	1.00000	−	1.73205i	0.0321412	−	0.0556702i
$969$		0			0
$970$	−1.50000	−	2.59808i	−0.0481621	−	0.0834192i
$971$	16.5000	+	28.5788i	0.529510	+	0.917139i	0.999408	+	0.0344175i	$0.0109576\pi$
$971$	16.5000	+	28.5788i	0.529510	+	0.917139i	−0.469897	+	0.882721i	$0.655709\pi$
$972$		0			0
$973$	3.50000	−	18.1865i	0.112205	−	0.583033i
$974$	12.5000	−	21.6506i	0.400526	−	0.693731i
$975$		0			0
$976$	2.00000			0.0640184
$977$	30.0000			0.959785			0.479893	−	0.877327i	$-0.340676\pi$
$977$	30.0000			0.959785			0.479893	+	0.877327i	$0.340676\pi$
$978$		0			0
$979$	4.50000	−	7.79423i	0.143821	−	0.249105i
$980$	19.5000	+	7.79423i	0.622905	+	0.248978i
$981$		0			0
$982$	10.5000	+	18.1865i	0.335068	+	0.580356i
$983$	−7.50000	−	12.9904i	−0.239213	−	0.414329i	0.721276	−	0.692648i	$-0.243556\pi$
$983$	−7.50000	−	12.9904i	−0.239213	−	0.414329i	−0.960489	+	0.278319i	$0.910223\pi$
$984$		0			0
$985$	−27.0000	+	46.7654i	−0.860292	+	1.49007i
$986$	−4.50000	−	7.79423i	−0.143309	−	0.248219i
$987$		0			0
$988$	−3.50000	+	6.06218i	−0.111350	+	0.192864i
$989$	4.50000	+	7.79423i	0.143092	+	0.247842i
$990$		0			0
$991$	12.5000	−	21.6506i	0.397076	−	0.687755i	−0.596288	−	0.802771i	$-0.703358\pi$
$991$	12.5000	−	21.6506i	0.397076	−	0.687755i	0.993364	+	0.115015i	$0.0366917\pi$
$992$	8.00000			0.254000
$993$		0			0
$994$	24.0000	+	20.7846i	0.761234	+	0.659248i
$995$	−37.5000	−	64.9519i	−1.18883	−	2.05911i
$996$		0			0
$997$	6.50000	+	11.2583i	0.205857	+	0.356555i	0.950405	−	0.311014i	$-0.100668\pi$
$997$	6.50000	+	11.2583i	0.205857	+	0.356555i	−0.744548	+	0.667568i	$0.767335\pi$
$998$	12.5000	−	21.6506i	0.395681	−	0.685339i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	378.2.e.b.235.1		2
3.2	odd	2		126.2.e.a.25.1	✓	2
4.3	odd	2		3024.2.q.f.2881.1		2
7.2	even	3		378.2.h.a.289.1		2
7.3	odd	6		2646.2.f.a.883.1		2
7.4	even	3		2646.2.f.d.883.1		2
7.5	odd	6		2646.2.h.d.667.1		2
7.6	odd	2		2646.2.e.g.2125.1		2
9.2	odd	6		1134.2.g.e.487.1		2
9.4	even	3		378.2.h.a.361.1		2
9.5	odd	6		126.2.h.b.67.1	yes	2
9.7	even	3		1134.2.g.c.487.1		2
12.11	even	2		1008.2.q.a.529.1		2
21.2	odd	6		126.2.h.b.79.1	yes	2
21.5	even	6		882.2.h.i.79.1		2
21.11	odd	6		882.2.f.i.295.1		2
21.17	even	6		882.2.f.g.295.1		2
21.20	even	2		882.2.e.c.655.1		2
28.23	odd	6		3024.2.t.a.289.1		2
36.23	even	6		1008.2.t.f.193.1		2
36.31	odd	6		3024.2.t.a.1873.1		2
63.2	odd	6		1134.2.g.e.163.1		2
63.4	even	3		2646.2.f.d.1765.1		2
63.5	even	6		882.2.e.c.373.1		2
63.11	odd	6		7938.2.a.m.1.1		1
63.13	odd	6		2646.2.h.d.361.1		2
63.16	even	3		1134.2.g.c.163.1		2
63.23	odd	6		126.2.e.a.121.1	yes	2
63.25	even	3		7938.2.a.t.1.1		1
63.31	odd	6		2646.2.f.a.1765.1		2
63.32	odd	6		882.2.f.i.589.1		2
63.38	even	6		7938.2.a.b.1.1		1
63.40	odd	6		2646.2.e.g.1549.1		2
63.41	even	6		882.2.h.i.67.1		2
63.52	odd	6		7938.2.a.be.1.1		1
63.58	even	3	inner	378.2.e.b.37.1		2
63.59	even	6		882.2.f.g.589.1		2
84.23	even	6		1008.2.t.f.961.1		2
252.23	even	6		1008.2.q.a.625.1		2
252.247	odd	6		3024.2.q.f.2305.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
126.2.e.a.25.1	✓	2	3.2	odd	2
126.2.e.a.121.1	yes	2	63.23	odd	6
126.2.h.b.67.1	yes	2	9.5	odd	6
126.2.h.b.79.1	yes	2	21.2	odd	6
378.2.e.b.37.1		2	63.58	even	3	inner
378.2.e.b.235.1		2	1.1	even	1	trivial
378.2.h.a.289.1		2	7.2	even	3
378.2.h.a.361.1		2	9.4	even	3
882.2.e.c.373.1		2	63.5	even	6
882.2.e.c.655.1		2	21.20	even	2
882.2.f.g.295.1		2	21.17	even	6
882.2.f.g.589.1		2	63.59	even	6
882.2.f.i.295.1		2	21.11	odd	6
882.2.f.i.589.1		2	63.32	odd	6
882.2.h.i.67.1		2	63.41	even	6
882.2.h.i.79.1		2	21.5	even	6
1008.2.q.a.529.1		2	12.11	even	2
1008.2.q.a.625.1		2	252.23	even	6
1008.2.t.f.193.1		2	36.23	even	6
1008.2.t.f.961.1		2	84.23	even	6
1134.2.g.c.163.1		2	63.16	even	3
1134.2.g.c.487.1		2	9.7	even	3
1134.2.g.e.163.1		2	63.2	odd	6
1134.2.g.e.487.1		2	9.2	odd	6
2646.2.e.g.1549.1		2	63.40	odd	6
2646.2.e.g.2125.1		2	7.6	odd	2
2646.2.f.a.883.1		2	7.3	odd	6
2646.2.f.a.1765.1		2	63.31	odd	6
2646.2.f.d.883.1		2	7.4	even	3
2646.2.f.d.1765.1		2	63.4	even	3
2646.2.h.d.361.1		2	63.13	odd	6
2646.2.h.d.667.1		2	7.5	odd	6
3024.2.q.f.2305.1		2	252.247	odd	6
3024.2.q.f.2881.1		2	4.3	odd	2
3024.2.t.a.289.1		2	28.23	odd	6
3024.2.t.a.1873.1		2	36.31	odd	6
7938.2.a.b.1.1		1	63.38	even	6
7938.2.a.m.1.1		1	63.11	odd	6
7938.2.a.t.1.1		1	63.25	even	3
7938.2.a.be.1.1		1	63.52	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:	\( N \)	\(=\)	\( 378 = 2 \cdot 3^{3} \cdot 7 \)
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

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