LMFDB - Embedded newform 810.2.e.b.271.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [810,2,Mod(271,810)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(810, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([2, 0]))

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

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

chi := DirichletCharacter("810.271");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 810 = 2 \cdot 3^{4} \cdot 5 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	810.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:	$6.46788256372$
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:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 30)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			271.1
Root			$0.500000 - 0.866025i$ of defining polynomial
Character	$\chi$	$=$	810.271
Dual form			810.2.e.b.541.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.500000 - 0.866025i) q^{2} +(-0.500000 + 0.866025i) q^{4} +(-0.500000 + 0.866025i) q^{5} +(2.00000 + 3.46410i) q^{7} +1.00000 q^{8} +O(q^{10})$	\(q+(-0.500000 - 0.866025i) q^{2} +(-0.500000 + 0.866025i) q^{4} +(-0.500000 + 0.866025i) q^{5} +(2.00000 + 3.46410i) q^{7} +1.00000 q^{8} +1.00000 q^{10} +(-1.00000 + 1.73205i) q^{13} +(2.00000 - 3.46410i) q^{14} +(-0.500000 - 0.866025i) q^{16} -6.00000 q^{17} -4.00000 q^{19} +(-0.500000 - 0.866025i) q^{20} +(-0.500000 - 0.866025i) q^{25} +2.00000 q^{26} -4.00000 q^{28} +(-3.00000 - 5.19615i) q^{29} +(-4.00000 + 6.92820i) q^{31} +(-0.500000 + 0.866025i) q^{32} +(3.00000 + 5.19615i) q^{34} -4.00000 q^{35} +2.00000 q^{37} +(2.00000 + 3.46410i) q^{38} +(-0.500000 + 0.866025i) q^{40} +(-3.00000 + 5.19615i) q^{41} +(2.00000 + 3.46410i) q^{43} +(-4.50000 + 7.79423i) q^{49} +(-0.500000 + 0.866025i) q^{50} +(-1.00000 - 1.73205i) q^{52} +6.00000 q^{53} +(2.00000 + 3.46410i) q^{56} +(-3.00000 + 5.19615i) q^{58} +(5.00000 + 8.66025i) q^{61} +8.00000 q^{62} +1.00000 q^{64} +(-1.00000 - 1.73205i) q^{65} +(2.00000 - 3.46410i) q^{67} +(3.00000 - 5.19615i) q^{68} +(2.00000 + 3.46410i) q^{70} +2.00000 q^{73} +(-1.00000 - 1.73205i) q^{74} +(2.00000 - 3.46410i) q^{76} +(-4.00000 - 6.92820i) q^{79} +1.00000 q^{80} +6.00000 q^{82} +(6.00000 + 10.3923i) q^{83} +(3.00000 - 5.19615i) q^{85} +(2.00000 - 3.46410i) q^{86} -18.0000 q^{89} -8.00000 q^{91} +(2.00000 - 3.46410i) q^{95} +(-1.00000 - 1.73205i) q^{97} +9.00000 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - q^{2} - q^{4} - q^{5} + 4 q^{7} + 2 q^{8}+O(q^{10}) $ 2 * q - q^2 - q^4 - q^5 + 4 * q^7 + 2 * q^8	$ 2 q - q^{2} - q^{4} - q^{5} + 4 q^{7} + 2 q^{8} + 2 q^{10} - 2 q^{13} + 4 q^{14} - q^{16} - 12 q^{17} - 8 q^{19} - q^{20} - q^{25} + 4 q^{26} - 8 q^{28} - 6 q^{29} - 8 q^{31} - q^{32} + 6 q^{34} - 8 q^{35} + 4 q^{37} + 4 q^{38} - q^{40} - 6 q^{41} + 4 q^{43} - 9 q^{49} - q^{50} - 2 q^{52} + 12 q^{53} + 4 q^{56} - 6 q^{58} + 10 q^{61} + 16 q^{62} + 2 q^{64} - 2 q^{65} + 4 q^{67} + 6 q^{68} + 4 q^{70} + 4 q^{73} - 2 q^{74} + 4 q^{76} - 8 q^{79} + 2 q^{80} + 12 q^{82} + 12 q^{83} + 6 q^{85} + 4 q^{86} - 36 q^{89} - 16 q^{91} + 4 q^{95} - 2 q^{97} + 18 q^{98}+O(q^{100}) $ 2 * q - q^2 - q^4 - q^5 + 4 * q^7 + 2 * q^8 + 2 * q^10 - 2 * q^13 + 4 * q^14 - q^16 - 12 * q^17 - 8 * q^19 - q^20 - q^25 + 4 * q^26 - 8 * q^28 - 6 * q^29 - 8 * q^31 - q^32 + 6 * q^34 - 8 * q^35 + 4 * q^37 + 4 * q^38 - q^40 - 6 * q^41 + 4 * q^43 - 9 * q^49 - q^50 - 2 * q^52 + 12 * q^53 + 4 * q^56 - 6 * q^58 + 10 * q^61 + 16 * q^62 + 2 * q^64 - 2 * q^65 + 4 * q^67 + 6 * q^68 + 4 * q^70 + 4 * q^73 - 2 * q^74 + 4 * q^76 - 8 * q^79 + 2 * q^80 + 12 * q^82 + 12 * q^83 + 6 * q^85 + 4 * q^86 - 36 * q^89 - 16 * q^91 + 4 * q^95 - 2 * q^97 + 18 * q^98

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/810\mathbb{Z}\right)^\times$.

$n$	$487$	$731$
$\chi(n)$	$1$	$e\left(\frac{1}{3}\right)$

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 $

$2$	−0.500000	−	0.866025i	−0.353553	−	0.612372i
$2$	−0.500000	−	0.866025i	−0.353553	−	0.612372i
$3$		0			0
$3$		0			0
$4$	−0.500000	+	0.866025i	−0.250000	+	0.433013i
$5$	−0.500000	+	0.866025i	−0.223607	+	0.387298i
$5$	−0.500000	+	0.866025i	−0.223607	+	0.387298i
$6$		0			0
$7$	2.00000	+	3.46410i	0.755929	+	1.30931i	0.944911	+	0.327327i	$0.106148\pi$
$7$	2.00000	+	3.46410i	0.755929	+	1.30931i	−0.188982	+	0.981981i	$0.560519\pi$
$8$	1.00000			0.353553
$9$		0			0
$10$	1.00000			0.316228
$11$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$11$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$12$		0			0
$13$	−1.00000	+	1.73205i	−0.277350	+	0.480384i	−0.970725	−	0.240192i	$-0.922790\pi$
$13$	−1.00000	+	1.73205i	−0.277350	+	0.480384i	0.693375	+	0.720577i	$0.256123\pi$
$14$	2.00000	−	3.46410i	0.534522	−	0.925820i
$15$		0			0
$16$	−0.500000	−	0.866025i	−0.125000	−	0.216506i
$17$	−6.00000			−1.45521			−0.727607	−	0.685994i	$-0.759367\pi$
$17$	−6.00000			−1.45521			−0.727607	+	0.685994i	$0.759367\pi$
$18$		0			0
$19$	−4.00000			−0.917663			−0.458831	−	0.888523i	$-0.651732\pi$
$19$	−4.00000			−0.917663			−0.458831	+	0.888523i	$0.651732\pi$
$20$	−0.500000	−	0.866025i	−0.111803	−	0.193649i
$21$		0			0
$22$		0			0
$23$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$23$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$24$		0			0
$25$	−0.500000	−	0.866025i	−0.100000	−	0.173205i
$26$	2.00000			0.392232
$27$		0			0
$28$	−4.00000			−0.755929
$29$	−3.00000	−	5.19615i	−0.557086	−	0.964901i	−0.997738	−	0.0672232i	$-0.978586\pi$
$29$	−3.00000	−	5.19615i	−0.557086	−	0.964901i	0.440652	−	0.897678i	$-0.354747\pi$
$30$		0			0
$31$	−4.00000	+	6.92820i	−0.718421	+	1.24434i	0.243204	+	0.969975i	$0.421802\pi$
$31$	−4.00000	+	6.92820i	−0.718421	+	1.24434i	−0.961625	+	0.274367i	$0.911532\pi$
$32$	−0.500000	+	0.866025i	−0.0883883	+	0.153093i
$33$		0			0
$34$	3.00000	+	5.19615i	0.514496	+	0.891133i
$35$	−4.00000			−0.676123
$36$		0			0
$37$	2.00000			0.328798			0.164399	−	0.986394i	$-0.447432\pi$
$37$	2.00000			0.328798			0.164399	+	0.986394i	$0.447432\pi$
$38$	2.00000	+	3.46410i	0.324443	+	0.561951i
$39$		0			0
$40$	−0.500000	+	0.866025i	−0.0790569	+	0.136931i
$41$	−3.00000	+	5.19615i	−0.468521	+	0.811503i	−0.999353	−	0.0359748i	$-0.988546\pi$
$41$	−3.00000	+	5.19615i	−0.468521	+	0.811503i	0.530831	+	0.847477i	$0.321880\pi$
$42$		0			0
$43$	2.00000	+	3.46410i	0.304997	+	0.528271i	0.977261	−	0.212041i	$-0.0680112\pi$
$43$	2.00000	+	3.46410i	0.304997	+	0.528271i	−0.672264	+	0.740312i	$0.734678\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$47$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$48$		0			0
$49$	−4.50000	+	7.79423i	−0.642857	+	1.11346i
$50$	−0.500000	+	0.866025i	−0.0707107	+	0.122474i
$51$		0			0
$52$	−1.00000	−	1.73205i	−0.138675	−	0.240192i
$53$	6.00000			0.824163			0.412082	−	0.911147i	$-0.364802\pi$
$53$	6.00000			0.824163			0.412082	+	0.911147i	$0.364802\pi$
$54$		0			0
$55$		0			0
$56$	2.00000	+	3.46410i	0.267261	+	0.462910i
$57$		0			0
$58$	−3.00000	+	5.19615i	−0.393919	+	0.682288i
$59$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$59$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$60$		0			0
$61$	5.00000	+	8.66025i	0.640184	+	1.10883i	0.985391	+	0.170305i	$0.0544754\pi$
$61$	5.00000	+	8.66025i	0.640184	+	1.10883i	−0.345207	+	0.938527i	$0.612191\pi$
$62$	8.00000			1.01600
$63$		0			0
$64$	1.00000			0.125000
$65$	−1.00000	−	1.73205i	−0.124035	−	0.214834i
$66$		0			0
$67$	2.00000	−	3.46410i	0.244339	−	0.423207i	−0.717607	−	0.696449i	$-0.754762\pi$
$67$	2.00000	−	3.46410i	0.244339	−	0.423207i	0.961946	+	0.273241i	$0.0880957\pi$
$68$	3.00000	−	5.19615i	0.363803	−	0.630126i
$69$		0			0
$70$	2.00000	+	3.46410i	0.239046	+	0.414039i
$71$		0			0			−	1.00000i	$-0.5\pi$
$71$		0			0				1.00000i	$0.5\pi$
$72$		0			0
$73$	2.00000			0.234082			0.117041	−	0.993127i	$-0.462659\pi$
$73$	2.00000			0.234082			0.117041	+	0.993127i	$0.462659\pi$
$74$	−1.00000	−	1.73205i	−0.116248	−	0.201347i
$75$		0			0
$76$	2.00000	−	3.46410i	0.229416	−	0.397360i
$77$		0			0
$78$		0			0
$79$	−4.00000	−	6.92820i	−0.450035	−	0.779484i	0.548352	−	0.836247i	$-0.315255\pi$
$79$	−4.00000	−	6.92820i	−0.450035	−	0.779484i	−0.998388	+	0.0567635i	$0.981922\pi$
$80$	1.00000			0.111803
$81$		0			0
$82$	6.00000			0.662589
$83$	6.00000	+	10.3923i	0.658586	+	1.14070i	0.980982	+	0.194099i	$0.0621783\pi$
$83$	6.00000	+	10.3923i	0.658586	+	1.14070i	−0.322396	+	0.946605i	$0.604488\pi$
$84$		0			0
$85$	3.00000	−	5.19615i	0.325396	−	0.563602i
$86$	2.00000	−	3.46410i	0.215666	−	0.373544i
$87$		0			0
$88$		0			0
$89$	−18.0000			−1.90800			−0.953998	−	0.299813i	$-0.903076\pi$
$89$	−18.0000			−1.90800			−0.953998	+	0.299813i	$0.903076\pi$
$90$		0			0
$91$	−8.00000			−0.838628
$92$		0			0
$93$		0			0
$94$		0			0
$95$	2.00000	−	3.46410i	0.205196	−	0.355409i
$96$		0			0
$97$	−1.00000	−	1.73205i	−0.101535	−	0.175863i	0.810782	−	0.585348i	$-0.199042\pi$
$97$	−1.00000	−	1.73205i	−0.101535	−	0.175863i	−0.912317	+	0.409484i	$0.865709\pi$
$98$	9.00000			0.909137
$99$		0			0
$100$	1.00000			0.100000
$101$	9.00000	+	15.5885i	0.895533	+	1.55111i	0.833143	+	0.553058i	$0.186539\pi$
$101$	9.00000	+	15.5885i	0.895533	+	1.55111i	0.0623905	+	0.998052i	$0.480128\pi$
$102$		0			0
$103$	2.00000	−	3.46410i	0.197066	−	0.341328i	−0.750510	−	0.660859i	$-0.770192\pi$
$103$	2.00000	−	3.46410i	0.197066	−	0.341328i	0.947576	+	0.319531i	$0.103525\pi$
$104$	−1.00000	+	1.73205i	−0.0980581	+	0.169842i
$105$		0			0
$106$	−3.00000	−	5.19615i	−0.291386	−	0.504695i
$107$	12.0000			1.16008			0.580042	−	0.814587i	$-0.303036\pi$
$107$	12.0000			1.16008			0.580042	+	0.814587i	$0.303036\pi$
$108$		0			0
$109$	−10.0000			−0.957826			−0.478913	−	0.877862i	$-0.658969\pi$
$109$	−10.0000			−0.957826			−0.478913	+	0.877862i	$0.658969\pi$
$110$		0			0
$111$		0			0
$112$	2.00000	−	3.46410i	0.188982	−	0.327327i
$113$	−9.00000	+	15.5885i	−0.846649	+	1.46644i	0.0375328	+	0.999295i	$0.488050\pi$
$113$	−9.00000	+	15.5885i	−0.846649	+	1.46644i	−0.884182	+	0.467143i	$0.845283\pi$
$114$		0			0
$115$		0			0
$116$	6.00000			0.557086
$117$		0			0
$118$		0			0
$119$	−12.0000	−	20.7846i	−1.10004	−	1.90532i
$120$		0			0
$121$	5.50000	−	9.52628i	0.500000	−	0.866025i
$122$	5.00000	−	8.66025i	0.452679	−	0.784063i
$123$		0			0
$124$	−4.00000	−	6.92820i	−0.359211	−	0.622171i
$125$	1.00000			0.0894427
$126$		0			0
$127$	20.0000			1.77471			0.887357	−	0.461084i	$-0.152539\pi$
$127$	20.0000			1.77471			0.887357	+	0.461084i	$0.152539\pi$
$128$	−0.500000	−	0.866025i	−0.0441942	−	0.0765466i
$129$		0			0
$130$	−1.00000	+	1.73205i	−0.0877058	+	0.151911i
$131$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$131$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$132$		0			0
$133$	−8.00000	−	13.8564i	−0.693688	−	1.20150i
$134$	−4.00000			−0.345547
$135$		0			0
$136$	−6.00000			−0.514496
$137$	3.00000	+	5.19615i	0.256307	+	0.443937i	0.965250	−	0.261329i	$-0.0841608\pi$
$137$	3.00000	+	5.19615i	0.256307	+	0.443937i	−0.708942	+	0.705266i	$0.750827\pi$
$138$		0			0
$139$	2.00000	−	3.46410i	0.169638	−	0.293821i	−0.768655	−	0.639664i	$-0.779074\pi$
$139$	2.00000	−	3.46410i	0.169638	−	0.293821i	0.938293	+	0.345843i	$0.112407\pi$
$140$	2.00000	−	3.46410i	0.169031	−	0.292770i
$141$		0			0
$142$		0			0
$143$		0			0
$144$		0			0
$145$	6.00000			0.498273
$146$	−1.00000	−	1.73205i	−0.0827606	−	0.143346i
$147$		0			0
$148$	−1.00000	+	1.73205i	−0.0821995	+	0.142374i
$149$	−3.00000	+	5.19615i	−0.245770	+	0.425685i	−0.962348	−	0.271821i	$-0.912374\pi$
$149$	−3.00000	+	5.19615i	−0.245770	+	0.425685i	0.716578	+	0.697507i	$0.245707\pi$
$150$		0			0
$151$	−4.00000	−	6.92820i	−0.325515	−	0.563809i	0.656101	−	0.754673i	$-0.272204\pi$
$151$	−4.00000	−	6.92820i	−0.325515	−	0.563809i	−0.981617	+	0.190864i	$0.938871\pi$
$152$	−4.00000			−0.324443
$153$		0			0
$154$		0			0
$155$	−4.00000	−	6.92820i	−0.321288	−	0.556487i
$156$		0			0
$157$	−1.00000	+	1.73205i	−0.0798087	+	0.138233i	−0.903167	−	0.429289i	$-0.858764\pi$
$157$	−1.00000	+	1.73205i	−0.0798087	+	0.138233i	0.823359	+	0.567521i	$0.192098\pi$
$158$	−4.00000	+	6.92820i	−0.318223	+	0.551178i
$159$		0			0
$160$	−0.500000	−	0.866025i	−0.0395285	−	0.0684653i
$161$		0			0
$162$		0			0
$163$	−4.00000			−0.313304			−0.156652	−	0.987654i	$-0.550070\pi$
$163$	−4.00000			−0.313304			−0.156652	+	0.987654i	$0.550070\pi$
$164$	−3.00000	−	5.19615i	−0.234261	−	0.405751i
$165$		0			0
$166$	6.00000	−	10.3923i	0.465690	−	0.806599i
$167$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$167$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$168$		0			0
$169$	4.50000	+	7.79423i	0.346154	+	0.599556i
$170$	−6.00000			−0.460179
$171$		0			0
$172$	−4.00000			−0.304997
$173$	9.00000	+	15.5885i	0.684257	+	1.18517i	0.973670	+	0.227964i	$0.0732068\pi$
$173$	9.00000	+	15.5885i	0.684257	+	1.18517i	−0.289412	+	0.957205i	$0.593460\pi$
$174$		0			0
$175$	2.00000	−	3.46410i	0.151186	−	0.261861i
$176$		0			0
$177$		0			0
$178$	9.00000	+	15.5885i	0.674579	+	1.16840i
$179$	−24.0000			−1.79384			−0.896922	−	0.442189i	$-0.854202\pi$
$179$	−24.0000			−1.79384			−0.896922	+	0.442189i	$0.854202\pi$
$180$		0			0
$181$	14.0000			1.04061			0.520306	−	0.853980i	$-0.325818\pi$
$181$	14.0000			1.04061			0.520306	+	0.853980i	$0.325818\pi$
$182$	4.00000	+	6.92820i	0.296500	+	0.513553i
$183$		0			0
$184$		0			0
$185$	−1.00000	+	1.73205i	−0.0735215	+	0.127343i
$186$		0			0
$187$		0			0
$188$		0			0
$189$		0			0
$190$	−4.00000			−0.290191
$191$	−12.0000	−	20.7846i	−0.868290	−	1.50392i	−0.863743	−	0.503932i	$-0.831886\pi$
$191$	−12.0000	−	20.7846i	−0.868290	−	1.50392i	−0.00454614	−	0.999990i	$-0.501447\pi$
$192$		0			0
$193$	11.0000	−	19.0526i	0.791797	−	1.37143i	−0.133056	−	0.991109i	$-0.542479\pi$
$193$	11.0000	−	19.0526i	0.791797	−	1.37143i	0.924853	−	0.380325i	$-0.124188\pi$
$194$	−1.00000	+	1.73205i	−0.0717958	+	0.124354i
$195$		0			0
$196$	−4.50000	−	7.79423i	−0.321429	−	0.556731i
$197$	6.00000			0.427482			0.213741	−	0.976890i	$-0.431435\pi$
$197$	6.00000			0.427482			0.213741	+	0.976890i	$0.431435\pi$
$198$		0			0
$199$	8.00000			0.567105			0.283552	−	0.958957i	$-0.408487\pi$
$199$	8.00000			0.567105			0.283552	+	0.958957i	$0.408487\pi$
$200$	−0.500000	−	0.866025i	−0.0353553	−	0.0612372i
$201$		0			0
$202$	9.00000	−	15.5885i	0.633238	−	1.09680i
$203$	12.0000	−	20.7846i	0.842235	−	1.45879i
$204$		0			0
$205$	−3.00000	−	5.19615i	−0.209529	−	0.362915i
$206$	−4.00000			−0.278693
$207$		0			0
$208$	2.00000			0.138675
$209$		0			0
$210$		0			0
$211$	−10.0000	+	17.3205i	−0.688428	+	1.19239i	0.283918	+	0.958849i	$0.408366\pi$
$211$	−10.0000	+	17.3205i	−0.688428	+	1.19239i	−0.972346	+	0.233544i	$0.924968\pi$
$212$	−3.00000	+	5.19615i	−0.206041	+	0.356873i
$213$		0			0
$214$	−6.00000	−	10.3923i	−0.410152	−	0.710403i
$215$	−4.00000			−0.272798
$216$		0			0
$217$	−32.0000			−2.17230
$218$	5.00000	+	8.66025i	0.338643	+	0.586546i
$219$		0			0
$220$		0			0
$221$	6.00000	−	10.3923i	0.403604	−	0.699062i
$222$		0			0
$223$	−10.0000	−	17.3205i	−0.669650	−	1.15987i	−0.978002	−	0.208595i	$-0.933111\pi$
$223$	−10.0000	−	17.3205i	−0.669650	−	1.15987i	0.308353	−	0.951272i	$-0.400222\pi$
$224$	−4.00000			−0.267261
$225$		0			0
$226$	18.0000			1.19734
$227$	−6.00000	−	10.3923i	−0.398234	−	0.689761i	0.595274	−	0.803523i	$-0.297043\pi$
$227$	−6.00000	−	10.3923i	−0.398234	−	0.689761i	−0.993508	+	0.113761i	$0.963710\pi$
$228$		0			0
$229$	5.00000	−	8.66025i	0.330409	−	0.572286i	−0.652183	−	0.758062i	$-0.726147\pi$
$229$	5.00000	−	8.66025i	0.330409	−	0.572286i	0.982592	+	0.185776i	$0.0594799\pi$
$230$		0			0
$231$		0			0
$232$	−3.00000	−	5.19615i	−0.196960	−	0.341144i
$233$	18.0000			1.17922			0.589610	−	0.807688i	$-0.299282\pi$
$233$	18.0000			1.17922			0.589610	+	0.807688i	$0.299282\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$		0			0
$238$	−12.0000	+	20.7846i	−0.777844	+	1.34727i
$239$	12.0000	−	20.7846i	0.776215	−	1.34444i	−0.157893	−	0.987456i	$-0.550470\pi$
$239$	12.0000	−	20.7846i	0.776215	−	1.34444i	0.934109	−	0.356988i	$-0.116196\pi$
$240$		0			0
$241$	−1.00000	−	1.73205i	−0.0644157	−	0.111571i	0.832019	−	0.554747i	$-0.187185\pi$
$241$	−1.00000	−	1.73205i	−0.0644157	−	0.111571i	−0.896435	+	0.443176i	$0.853852\pi$
$242$	−11.0000			−0.707107
$243$		0			0
$244$	−10.0000			−0.640184
$245$	−4.50000	−	7.79423i	−0.287494	−	0.497955i
$246$		0			0
$247$	4.00000	−	6.92820i	0.254514	−	0.440831i
$248$	−4.00000	+	6.92820i	−0.254000	+	0.439941i
$249$		0			0
$250$	−0.500000	−	0.866025i	−0.0316228	−	0.0547723i
$251$	24.0000			1.51487			0.757433	−	0.652913i	$-0.226453\pi$
$251$	24.0000			1.51487			0.757433	+	0.652913i	$0.226453\pi$
$252$		0			0
$253$		0			0
$254$	−10.0000	−	17.3205i	−0.627456	−	1.08679i
$255$		0			0
$256$	−0.500000	+	0.866025i	−0.0312500	+	0.0541266i
$257$	−9.00000	+	15.5885i	−0.561405	+	0.972381i	0.435970	+	0.899961i	$0.356405\pi$
$257$	−9.00000	+	15.5885i	−0.561405	+	0.972381i	−0.997374	+	0.0724199i	$0.976928\pi$
$258$		0			0
$259$	4.00000	+	6.92820i	0.248548	+	0.430498i
$260$	2.00000			0.124035
$261$		0			0
$262$		0			0
$263$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$263$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$264$		0			0
$265$	−3.00000	+	5.19615i	−0.184289	+	0.319197i
$266$	−8.00000	+	13.8564i	−0.490511	+	0.849591i
$267$		0			0
$268$	2.00000	+	3.46410i	0.122169	+	0.211604i
$269$	6.00000			0.365826			0.182913	−	0.983129i	$-0.441447\pi$
$269$	6.00000			0.365826			0.182913	+	0.983129i	$0.441447\pi$
$270$		0			0
$271$	−16.0000			−0.971931			−0.485965	−	0.873978i	$-0.661532\pi$
$271$	−16.0000			−0.971931			−0.485965	+	0.873978i	$0.661532\pi$
$272$	3.00000	+	5.19615i	0.181902	+	0.315063i
$273$		0			0
$274$	3.00000	−	5.19615i	0.181237	−	0.313911i
$275$		0			0
$276$		0			0
$277$	−1.00000	−	1.73205i	−0.0600842	−	0.104069i	0.834419	−	0.551131i	$-0.185804\pi$
$277$	−1.00000	−	1.73205i	−0.0600842	−	0.104069i	−0.894503	+	0.447062i	$0.852470\pi$
$278$	−4.00000			−0.239904
$279$		0			0
$280$	−4.00000			−0.239046
$281$	9.00000	+	15.5885i	0.536895	+	0.929929i	0.999069	+	0.0431402i	$0.0137362\pi$
$281$	9.00000	+	15.5885i	0.536895	+	0.929929i	−0.462174	+	0.886789i	$0.652930\pi$
$282$		0			0
$283$	14.0000	−	24.2487i	0.832214	−	1.44144i	−0.0640654	−	0.997946i	$-0.520407\pi$
$283$	14.0000	−	24.2487i	0.832214	−	1.44144i	0.896279	−	0.443491i	$-0.146260\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−24.0000			−1.41668
$288$		0			0
$289$	19.0000			1.11765
$290$	−3.00000	−	5.19615i	−0.176166	−	0.305129i
$291$		0			0
$292$	−1.00000	+	1.73205i	−0.0585206	+	0.101361i
$293$	−3.00000	+	5.19615i	−0.175262	+	0.303562i	−0.940252	−	0.340480i	$-0.889411\pi$
$293$	−3.00000	+	5.19615i	−0.175262	+	0.303562i	0.764990	+	0.644042i	$0.222744\pi$
$294$		0			0
$295$		0			0
$296$	2.00000			0.116248
$297$		0			0
$298$	6.00000			0.347571
$299$		0			0
$300$		0			0
$301$	−8.00000	+	13.8564i	−0.461112	+	0.798670i
$302$	−4.00000	+	6.92820i	−0.230174	+	0.398673i
$303$		0			0
$304$	2.00000	+	3.46410i	0.114708	+	0.198680i
$305$	−10.0000			−0.572598
$306$		0			0
$307$	20.0000			1.14146			0.570730	−	0.821138i	$-0.306660\pi$
$307$	20.0000			1.14146			0.570730	+	0.821138i	$0.306660\pi$
$308$		0			0
$309$		0			0
$310$	−4.00000	+	6.92820i	−0.227185	+	0.393496i
$311$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$311$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$312$		0			0
$313$	−1.00000	−	1.73205i	−0.0565233	−	0.0979013i	0.836379	−	0.548151i	$-0.184668\pi$
$313$	−1.00000	−	1.73205i	−0.0565233	−	0.0979013i	−0.892903	+	0.450250i	$0.851335\pi$
$314$	2.00000			0.112867
$315$		0			0
$316$	8.00000			0.450035
$317$	9.00000	+	15.5885i	0.505490	+	0.875535i	0.999980	+	0.00635137i	$0.00202172\pi$
$317$	9.00000	+	15.5885i	0.505490	+	0.875535i	−0.494489	+	0.869184i	$0.664645\pi$
$318$		0			0
$319$		0			0
$320$	−0.500000	+	0.866025i	−0.0279508	+	0.0484123i
$321$		0			0
$322$		0			0
$323$	24.0000			1.33540
$324$		0			0
$325$	2.00000			0.110940
$326$	2.00000	+	3.46410i	0.110770	+	0.191859i
$327$		0			0
$328$	−3.00000	+	5.19615i	−0.165647	+	0.286910i
$329$		0			0
$330$		0			0
$331$	14.0000	+	24.2487i	0.769510	+	1.33283i	0.937829	+	0.347097i	$0.112833\pi$
$331$	14.0000	+	24.2487i	0.769510	+	1.33283i	−0.168320	+	0.985732i	$0.553834\pi$
$332$	−12.0000			−0.658586
$333$		0			0
$334$		0			0
$335$	2.00000	+	3.46410i	0.109272	+	0.189264i
$336$		0			0
$337$	−13.0000	+	22.5167i	−0.708155	+	1.22656i	0.257386	+	0.966309i	$0.417139\pi$
$337$	−13.0000	+	22.5167i	−0.708155	+	1.22656i	−0.965541	+	0.260252i	$0.916194\pi$
$338$	4.50000	−	7.79423i	0.244768	−	0.423950i
$339$		0			0
$340$	3.00000	+	5.19615i	0.162698	+	0.281801i
$341$		0			0
$342$		0			0
$343$	−8.00000			−0.431959
$344$	2.00000	+	3.46410i	0.107833	+	0.186772i
$345$		0			0
$346$	9.00000	−	15.5885i	0.483843	−	0.838041i
$347$	6.00000	−	10.3923i	0.322097	−	0.557888i	−0.658824	−	0.752297i	$-0.728946\pi$
$347$	6.00000	−	10.3923i	0.322097	−	0.557888i	0.980921	+	0.194409i	$0.0622790\pi$
$348$		0			0
$349$	5.00000	+	8.66025i	0.267644	+	0.463573i	0.968253	−	0.249973i	$-0.0804216\pi$
$349$	5.00000	+	8.66025i	0.267644	+	0.463573i	−0.700609	+	0.713545i	$0.747088\pi$
$350$	−4.00000			−0.213809
$351$		0			0
$352$		0			0
$353$	3.00000	+	5.19615i	0.159674	+	0.276563i	0.934751	−	0.355303i	$-0.115622\pi$
$353$	3.00000	+	5.19615i	0.159674	+	0.276563i	−0.775077	+	0.631867i	$0.782289\pi$
$354$		0			0
$355$		0			0
$356$	9.00000	−	15.5885i	0.476999	−	0.826187i
$357$		0			0
$358$	12.0000	+	20.7846i	0.634220	+	1.09850i
$359$	24.0000			1.26667			0.633336	−	0.773877i	$-0.281685\pi$
$359$	24.0000			1.26667			0.633336	+	0.773877i	$0.281685\pi$
$360$		0			0
$361$	−3.00000			−0.157895
$362$	−7.00000	−	12.1244i	−0.367912	−	0.637242i
$363$		0			0
$364$	4.00000	−	6.92820i	0.209657	−	0.363137i
$365$	−1.00000	+	1.73205i	−0.0523424	+	0.0906597i
$366$		0			0
$367$	14.0000	+	24.2487i	0.730794	+	1.26577i	0.956544	+	0.291587i	$0.0941834\pi$
$367$	14.0000	+	24.2487i	0.730794	+	1.26577i	−0.225750	+	0.974185i	$0.572483\pi$
$368$		0			0
$369$		0			0
$370$	2.00000			0.103975
$371$	12.0000	+	20.7846i	0.623009	+	1.07908i
$372$		0			0
$373$	−13.0000	+	22.5167i	−0.673114	+	1.16587i	0.303902	+	0.952703i	$0.401711\pi$
$373$	−13.0000	+	22.5167i	−0.673114	+	1.16587i	−0.977016	+	0.213165i	$0.931623\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	12.0000			0.618031
$378$		0			0
$379$	−4.00000			−0.205466			−0.102733	−	0.994709i	$-0.532759\pi$
$379$	−4.00000			−0.205466			−0.102733	+	0.994709i	$0.532759\pi$
$380$	2.00000	+	3.46410i	0.102598	+	0.177705i
$381$		0			0
$382$	−12.0000	+	20.7846i	−0.613973	+	1.06343i
$383$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$383$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$384$		0			0
$385$		0			0
$386$	−22.0000			−1.11977
$387$		0			0
$388$	2.00000			0.101535
$389$	−3.00000	−	5.19615i	−0.152106	−	0.263455i	0.779895	−	0.625910i	$-0.215272\pi$
$389$	−3.00000	−	5.19615i	−0.152106	−	0.263455i	−0.932002	+	0.362454i	$0.881939\pi$
$390$		0			0
$391$		0			0
$392$	−4.50000	+	7.79423i	−0.227284	+	0.393668i
$393$		0			0
$394$	−3.00000	−	5.19615i	−0.151138	−	0.261778i
$395$	8.00000			0.402524
$396$		0			0
$397$	−22.0000			−1.10415			−0.552074	−	0.833795i	$-0.686163\pi$
$397$	−22.0000			−1.10415			−0.552074	+	0.833795i	$0.686163\pi$
$398$	−4.00000	−	6.92820i	−0.200502	−	0.347279i
$399$		0			0
$400$	−0.500000	+	0.866025i	−0.0250000	+	0.0433013i
$401$	−3.00000	+	5.19615i	−0.149813	+	0.259483i	−0.931158	−	0.364615i	$-0.881200\pi$
$401$	−3.00000	+	5.19615i	−0.149813	+	0.259483i	0.781345	+	0.624099i	$0.214534\pi$
$402$		0			0
$403$	−8.00000	−	13.8564i	−0.398508	−	0.690237i
$404$	−18.0000			−0.895533
$405$		0			0
$406$	−24.0000			−1.19110
$407$		0			0
$408$		0			0
$409$	−13.0000	+	22.5167i	−0.642809	+	1.11338i	0.341994	+	0.939702i	$0.388898\pi$
$409$	−13.0000	+	22.5167i	−0.642809	+	1.11338i	−0.984803	+	0.173675i	$0.944436\pi$
$410$	−3.00000	+	5.19615i	−0.148159	+	0.256620i
$411$		0			0
$412$	2.00000	+	3.46410i	0.0985329	+	0.170664i
$413$		0			0
$414$		0			0
$415$	−12.0000			−0.589057
$416$	−1.00000	−	1.73205i	−0.0490290	−	0.0849208i
$417$		0			0
$418$		0			0
$419$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$419$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$420$		0			0
$421$	5.00000	+	8.66025i	0.243685	+	0.422075i	0.961761	−	0.273890i	$-0.0883103\pi$
$421$	5.00000	+	8.66025i	0.243685	+	0.422075i	−0.718076	+	0.695965i	$0.754977\pi$
$422$	20.0000			0.973585
$423$		0			0
$424$	6.00000			0.291386
$425$	3.00000	+	5.19615i	0.145521	+	0.252050i
$426$		0			0
$427$	−20.0000	+	34.6410i	−0.967868	+	1.67640i
$428$	−6.00000	+	10.3923i	−0.290021	+	0.502331i
$429$		0			0
$430$	2.00000	+	3.46410i	0.0964486	+	0.167054i
$431$		0			0			−	1.00000i	$-0.5\pi$
$431$		0			0				1.00000i	$0.5\pi$
$432$		0			0
$433$	26.0000			1.24948			0.624740	−	0.780833i	$-0.285205\pi$
$433$	26.0000			1.24948			0.624740	+	0.780833i	$0.285205\pi$
$434$	16.0000	+	27.7128i	0.768025	+	1.33026i
$435$		0			0
$436$	5.00000	−	8.66025i	0.239457	−	0.414751i
$437$		0			0
$438$		0			0
$439$	−4.00000	−	6.92820i	−0.190910	−	0.330665i	0.754642	−	0.656136i	$-0.227810\pi$
$439$	−4.00000	−	6.92820i	−0.190910	−	0.330665i	−0.945552	+	0.325471i	$0.894477\pi$
$440$		0			0
$441$		0			0
$442$	−12.0000			−0.570782
$443$	6.00000	+	10.3923i	0.285069	+	0.493753i	0.972626	−	0.232377i	$-0.0746503\pi$
$443$	6.00000	+	10.3923i	0.285069	+	0.493753i	−0.687557	+	0.726130i	$0.741317\pi$
$444$		0			0
$445$	9.00000	−	15.5885i	0.426641	−	0.738964i
$446$	−10.0000	+	17.3205i	−0.473514	+	0.820150i
$447$		0			0
$448$	2.00000	+	3.46410i	0.0944911	+	0.163663i
$449$	6.00000			0.283158			0.141579	−	0.989927i	$-0.454782\pi$
$449$	6.00000			0.283158			0.141579	+	0.989927i	$0.454782\pi$
$450$		0			0
$451$		0			0
$452$	−9.00000	−	15.5885i	−0.423324	−	0.733219i
$453$		0			0
$454$	−6.00000	+	10.3923i	−0.281594	+	0.487735i
$455$	4.00000	−	6.92820i	0.187523	−	0.324799i
$456$		0			0
$457$	−13.0000	−	22.5167i	−0.608114	−	1.05328i	−0.991551	−	0.129718i	$-0.958593\pi$
$457$	−13.0000	−	22.5167i	−0.608114	−	1.05328i	0.383437	−	0.923567i	$-0.374740\pi$
$458$	−10.0000			−0.467269
$459$		0			0
$460$		0			0
$461$	−15.0000	−	25.9808i	−0.698620	−	1.21004i	−0.968945	−	0.247276i	$-0.920465\pi$
$461$	−15.0000	−	25.9808i	−0.698620	−	1.21004i	0.270326	−	0.962769i	$-0.412869\pi$
$462$		0			0
$463$	2.00000	−	3.46410i	0.0929479	−	0.160990i	−0.815802	−	0.578331i	$-0.803704\pi$
$463$	2.00000	−	3.46410i	0.0929479	−	0.160990i	0.908750	+	0.417340i	$0.137038\pi$
$464$	−3.00000	+	5.19615i	−0.139272	+	0.241225i
$465$		0			0
$466$	−9.00000	−	15.5885i	−0.416917	−	0.722121i
$467$	36.0000			1.66588			0.832941	−	0.553362i	$-0.186655\pi$
$467$	36.0000			1.66588			0.832941	+	0.553362i	$0.186655\pi$
$468$		0			0
$469$	16.0000			0.738811
$470$		0			0
$471$		0			0
$472$		0			0
$473$		0			0
$474$		0			0
$475$	2.00000	+	3.46410i	0.0917663	+	0.158944i
$476$	24.0000			1.10004
$477$		0			0
$478$	−24.0000			−1.09773
$479$	−12.0000	−	20.7846i	−0.548294	−	0.949673i	−0.998392	−	0.0566937i	$-0.981944\pi$
$479$	−12.0000	−	20.7846i	−0.548294	−	0.949673i	0.450098	−	0.892979i	$-0.351389\pi$
$480$		0			0
$481$	−2.00000	+	3.46410i	−0.0911922	+	0.157949i
$482$	−1.00000	+	1.73205i	−0.0455488	+	0.0788928i
$483$		0			0
$484$	5.50000	+	9.52628i	0.250000	+	0.433013i
$485$	2.00000			0.0908153
$486$		0			0
$487$	−28.0000			−1.26880			−0.634401	−	0.773004i	$-0.718753\pi$
$487$	−28.0000			−1.26880			−0.634401	+	0.773004i	$0.718753\pi$
$488$	5.00000	+	8.66025i	0.226339	+	0.392031i
$489$		0			0
$490$	−4.50000	+	7.79423i	−0.203289	+	0.352107i
$491$	12.0000	−	20.7846i	0.541552	−	0.937996i	−0.457263	−	0.889332i	$-0.651170\pi$
$491$	12.0000	−	20.7846i	0.541552	−	0.937996i	0.998815	−	0.0486647i	$-0.0154966\pi$
$492$		0			0
$493$	18.0000	+	31.1769i	0.810679	+	1.40414i
$494$	−8.00000			−0.359937
$495$		0			0
$496$	8.00000			0.359211
$497$		0			0
$498$		0			0
$499$	2.00000	−	3.46410i	0.0895323	−	0.155074i	−0.817781	−	0.575529i	$-0.804796\pi$
$499$	2.00000	−	3.46410i	0.0895323	−	0.155074i	0.907314	+	0.420455i	$0.138129\pi$
$500$	−0.500000	+	0.866025i	−0.0223607	+	0.0387298i
$501$		0			0
$502$	−12.0000	−	20.7846i	−0.535586	−	0.927663i
$503$	−24.0000			−1.07011			−0.535054	−	0.844818i	$-0.679709\pi$
$503$	−24.0000			−1.07011			−0.535054	+	0.844818i	$0.679709\pi$
$504$		0			0
$505$	−18.0000			−0.800989
$506$		0			0
$507$		0			0
$508$	−10.0000	+	17.3205i	−0.443678	+	0.768473i
$509$	−3.00000	+	5.19615i	−0.132973	+	0.230315i	−0.924821	−	0.380402i	$-0.875786\pi$
$509$	−3.00000	+	5.19615i	−0.132973	+	0.230315i	0.791849	+	0.610718i	$0.209119\pi$
$510$		0			0
$511$	4.00000	+	6.92820i	0.176950	+	0.306486i
$512$	1.00000			0.0441942
$513$		0			0
$514$	18.0000			0.793946
$515$	2.00000	+	3.46410i	0.0881305	+	0.152647i
$516$		0			0
$517$		0			0
$518$	4.00000	−	6.92820i	0.175750	−	0.304408i
$519$		0			0
$520$	−1.00000	−	1.73205i	−0.0438529	−	0.0759555i
$521$	−18.0000			−0.788594			−0.394297	−	0.918983i	$-0.629012\pi$
$521$	−18.0000			−0.788594			−0.394297	+	0.918983i	$0.629012\pi$
$522$		0			0
$523$	20.0000			0.874539			0.437269	−	0.899331i	$-0.355946\pi$
$523$	20.0000			0.874539			0.437269	+	0.899331i	$0.355946\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	24.0000	−	41.5692i	1.04546	−	1.81078i
$528$		0			0
$529$	11.5000	+	19.9186i	0.500000	+	0.866025i
$530$	6.00000			0.260623
$531$		0			0
$532$	16.0000			0.693688
$533$	−6.00000	−	10.3923i	−0.259889	−	0.450141i
$534$		0			0
$535$	−6.00000	+	10.3923i	−0.259403	+	0.449299i
$536$	2.00000	−	3.46410i	0.0863868	−	0.149626i
$537$		0			0
$538$	−3.00000	−	5.19615i	−0.129339	−	0.224022i
$539$		0			0
$540$		0			0
$541$	−10.0000			−0.429934			−0.214967	−	0.976621i	$-0.568964\pi$
$541$	−10.0000			−0.429934			−0.214967	+	0.976621i	$0.568964\pi$
$542$	8.00000	+	13.8564i	0.343629	+	0.595184i
$543$		0			0
$544$	3.00000	−	5.19615i	0.128624	−	0.222783i
$545$	5.00000	−	8.66025i	0.214176	−	0.370965i
$546$		0			0
$547$	14.0000	+	24.2487i	0.598597	+	1.03680i	0.993028	+	0.117875i	$0.0376081\pi$
$547$	14.0000	+	24.2487i	0.598597	+	1.03680i	−0.394432	+	0.918925i	$0.629059\pi$
$548$	−6.00000			−0.256307
$549$		0			0
$550$		0			0
$551$	12.0000	+	20.7846i	0.511217	+	0.885454i
$552$		0			0
$553$	16.0000	−	27.7128i	0.680389	−	1.17847i
$554$	−1.00000	+	1.73205i	−0.0424859	+	0.0735878i
$555$		0			0
$556$	2.00000	+	3.46410i	0.0848189	+	0.146911i
$557$	−18.0000			−0.762684			−0.381342	−	0.924434i	$-0.624538\pi$
$557$	−18.0000			−0.762684			−0.381342	+	0.924434i	$0.624538\pi$
$558$		0			0
$559$	−8.00000			−0.338364
$560$	2.00000	+	3.46410i	0.0845154	+	0.146385i
$561$		0			0
$562$	9.00000	−	15.5885i	0.379642	−	0.657559i
$563$	−6.00000	+	10.3923i	−0.252870	+	0.437983i	−0.964315	−	0.264758i	$-0.914708\pi$
$563$	−6.00000	+	10.3923i	−0.252870	+	0.437983i	0.711445	+	0.702742i	$0.248041\pi$
$564$		0			0
$565$	−9.00000	−	15.5885i	−0.378633	−	0.655811i
$566$	−28.0000			−1.17693
$567$		0			0
$568$		0			0
$569$	9.00000	+	15.5885i	0.377300	+	0.653502i	0.990668	−	0.136295i	$-0.0435194\pi$
$569$	9.00000	+	15.5885i	0.377300	+	0.653502i	−0.613369	+	0.789797i	$0.710186\pi$
$570$		0			0
$571$	−10.0000	+	17.3205i	−0.418487	+	0.724841i	−0.995788	−	0.0916910i	$-0.970773\pi$
$571$	−10.0000	+	17.3205i	−0.418487	+	0.724841i	0.577301	+	0.816532i	$0.304106\pi$
$572$		0			0
$573$		0			0
$574$	12.0000	+	20.7846i	0.500870	+	0.867533i
$575$		0			0
$576$		0			0
$577$	2.00000			0.0832611			0.0416305	−	0.999133i	$-0.486745\pi$
$577$	2.00000			0.0832611			0.0416305	+	0.999133i	$0.486745\pi$
$578$	−9.50000	−	16.4545i	−0.395148	−	0.684416i
$579$		0			0
$580$	−3.00000	+	5.19615i	−0.124568	+	0.215758i
$581$	−24.0000	+	41.5692i	−0.995688	+	1.72458i
$582$		0			0
$583$		0			0
$584$	2.00000			0.0827606
$585$		0			0
$586$	6.00000			0.247858
$587$	−6.00000	−	10.3923i	−0.247647	−	0.428936i	0.715226	−	0.698893i	$-0.246324\pi$
$587$	−6.00000	−	10.3923i	−0.247647	−	0.428936i	−0.962872	+	0.269957i	$0.912990\pi$
$588$		0			0
$589$	16.0000	−	27.7128i	0.659269	−	1.14189i
$590$		0			0
$591$		0			0
$592$	−1.00000	−	1.73205i	−0.0410997	−	0.0711868i
$593$	−30.0000			−1.23195			−0.615976	−	0.787765i	$-0.711238\pi$
$593$	−30.0000			−1.23195			−0.615976	+	0.787765i	$0.711238\pi$
$594$		0			0
$595$	24.0000			0.983904
$596$	−3.00000	−	5.19615i	−0.122885	−	0.212843i
$597$		0			0
$598$		0			0
$599$	−12.0000	+	20.7846i	−0.490307	+	0.849236i	−0.999938	−	0.0111569i	$-0.996449\pi$
$599$	−12.0000	+	20.7846i	−0.490307	+	0.849236i	0.509631	+	0.860393i	$0.329782\pi$
$600$		0			0
$601$	11.0000	+	19.0526i	0.448699	+	0.777170i	0.998302	−	0.0582563i	$-0.0185541\pi$
$601$	11.0000	+	19.0526i	0.448699	+	0.777170i	−0.549602	+	0.835426i	$0.685221\pi$
$602$	16.0000			0.652111
$603$		0			0
$604$	8.00000			0.325515
$605$	5.50000	+	9.52628i	0.223607	+	0.387298i
$606$		0			0
$607$	2.00000	−	3.46410i	0.0811775	−	0.140604i	−0.822578	−	0.568652i	$-0.807465\pi$
$607$	2.00000	−	3.46410i	0.0811775	−	0.140604i	0.903756	+	0.428048i	$0.140799\pi$
$608$	2.00000	−	3.46410i	0.0811107	−	0.140488i
$609$		0			0
$610$	5.00000	+	8.66025i	0.202444	+	0.350643i
$611$		0			0
$612$		0			0
$613$	2.00000			0.0807792			0.0403896	−	0.999184i	$-0.487140\pi$
$613$	2.00000			0.0807792			0.0403896	+	0.999184i	$0.487140\pi$
$614$	−10.0000	−	17.3205i	−0.403567	−	0.698999i
$615$		0			0
$616$		0			0
$617$	15.0000	−	25.9808i	0.603877	−	1.04595i	−0.388351	−	0.921512i	$-0.626955\pi$
$617$	15.0000	−	25.9808i	0.603877	−	1.04595i	0.992228	−	0.124434i	$-0.0397116\pi$
$618$		0			0
$619$	−22.0000	−	38.1051i	−0.884255	−	1.53157i	−0.846566	−	0.532284i	$-0.821334\pi$
$619$	−22.0000	−	38.1051i	−0.884255	−	1.53157i	−0.0376891	−	0.999290i	$-0.512000\pi$
$620$	8.00000			0.321288
$621$		0			0
$622$		0			0
$623$	−36.0000	−	62.3538i	−1.44231	−	2.49815i
$624$		0			0
$625$	−0.500000	+	0.866025i	−0.0200000	+	0.0346410i
$626$	−1.00000	+	1.73205i	−0.0399680	+	0.0692267i
$627$		0			0
$628$	−1.00000	−	1.73205i	−0.0399043	−	0.0691164i
$629$	−12.0000			−0.478471
$630$		0			0
$631$	32.0000			1.27390			0.636950	−	0.770905i	$-0.280196\pi$
$631$	32.0000			1.27390			0.636950	+	0.770905i	$0.280196\pi$
$632$	−4.00000	−	6.92820i	−0.159111	−	0.275589i
$633$		0			0
$634$	9.00000	−	15.5885i	0.357436	−	0.619097i
$635$	−10.0000	+	17.3205i	−0.396838	+	0.687343i
$636$		0			0
$637$	−9.00000	−	15.5885i	−0.356593	−	0.617637i
$638$		0			0
$639$		0			0
$640$	1.00000			0.0395285
$641$	−15.0000	−	25.9808i	−0.592464	−	1.02618i	−0.993899	−	0.110291i	$-0.964822\pi$
$641$	−15.0000	−	25.9808i	−0.592464	−	1.02618i	0.401435	−	0.915888i	$-0.368512\pi$
$642$		0			0
$643$	2.00000	−	3.46410i	0.0788723	−	0.136611i	−0.823891	−	0.566748i	$-0.808201\pi$
$643$	2.00000	−	3.46410i	0.0788723	−	0.136611i	0.902764	+	0.430137i	$0.141535\pi$
$644$		0			0
$645$		0			0
$646$	−12.0000	−	20.7846i	−0.472134	−	0.817760i
$647$	−24.0000			−0.943537			−0.471769	−	0.881722i	$-0.656384\pi$
$647$	−24.0000			−0.943537			−0.471769	+	0.881722i	$0.656384\pi$
$648$		0			0
$649$		0			0
$650$	−1.00000	−	1.73205i	−0.0392232	−	0.0679366i
$651$		0			0
$652$	2.00000	−	3.46410i	0.0783260	−	0.135665i
$653$	9.00000	−	15.5885i	0.352197	−	0.610023i	−0.634437	−	0.772975i	$-0.718768\pi$
$653$	9.00000	−	15.5885i	0.352197	−	0.610023i	0.986634	+	0.162951i	$0.0521013\pi$
$654$		0			0
$655$		0			0
$656$	6.00000			0.234261
$657$		0			0
$658$		0			0
$659$	24.0000	+	41.5692i	0.934907	+	1.61931i	0.774799	+	0.632207i	$0.217851\pi$
$659$	24.0000	+	41.5692i	0.934907	+	1.61931i	0.160108	+	0.987099i	$0.448816\pi$
$660$		0			0
$661$	−7.00000	+	12.1244i	−0.272268	+	0.471583i	−0.969442	−	0.245319i	$-0.921107\pi$
$661$	−7.00000	+	12.1244i	−0.272268	+	0.471583i	0.697174	+	0.716902i	$0.254441\pi$
$662$	14.0000	−	24.2487i	0.544125	−	0.942453i
$663$		0			0
$664$	6.00000	+	10.3923i	0.232845	+	0.403300i
$665$	16.0000			0.620453
$666$		0			0
$667$		0			0
$668$		0			0
$669$		0			0
$670$	2.00000	−	3.46410i	0.0772667	−	0.133830i
$671$		0			0
$672$		0			0
$673$	−13.0000	−	22.5167i	−0.501113	−	0.867953i	−0.999999	−	0.00128586i	$-0.999591\pi$
$673$	−13.0000	−	22.5167i	−0.501113	−	0.867953i	0.498886	−	0.866668i	$-0.333743\pi$
$674$	26.0000			1.00148
$675$		0			0
$676$	−9.00000			−0.346154
$677$	−3.00000	−	5.19615i	−0.115299	−	0.199704i	0.802600	−	0.596518i	$-0.203449\pi$
$677$	−3.00000	−	5.19615i	−0.115299	−	0.199704i	−0.917899	+	0.396813i	$0.870116\pi$
$678$		0			0
$679$	4.00000	−	6.92820i	0.153506	−	0.265880i
$680$	3.00000	−	5.19615i	0.115045	−	0.199263i
$681$		0			0
$682$		0			0
$683$	12.0000			0.459167			0.229584	−	0.973289i	$-0.426264\pi$
$683$	12.0000			0.459167			0.229584	+	0.973289i	$0.426264\pi$
$684$		0			0
$685$	−6.00000			−0.229248
$686$	4.00000	+	6.92820i	0.152721	+	0.264520i
$687$		0			0
$688$	2.00000	−	3.46410i	0.0762493	−	0.132068i
$689$	−6.00000	+	10.3923i	−0.228582	+	0.395915i
$690$		0			0
$691$	−22.0000	−	38.1051i	−0.836919	−	1.44959i	−0.892458	−	0.451130i	$-0.851021\pi$
$691$	−22.0000	−	38.1051i	−0.836919	−	1.44959i	0.0555386	−	0.998457i	$-0.482312\pi$
$692$	−18.0000			−0.684257
$693$		0			0
$694$	−12.0000			−0.455514
$695$	2.00000	+	3.46410i	0.0758643	+	0.131401i
$696$		0			0
$697$	18.0000	−	31.1769i	0.681799	−	1.18091i
$698$	5.00000	−	8.66025i	0.189253	−	0.327795i
$699$		0			0
$700$	2.00000	+	3.46410i	0.0755929	+	0.130931i
$701$	6.00000			0.226617			0.113308	−	0.993560i	$-0.463855\pi$
$701$	6.00000			0.226617			0.113308	+	0.993560i	$0.463855\pi$
$702$		0			0
$703$	−8.00000			−0.301726
$704$		0			0
$705$		0			0
$706$	3.00000	−	5.19615i	0.112906	−	0.195560i
$707$	−36.0000	+	62.3538i	−1.35392	+	2.34506i
$708$		0			0
$709$	−19.0000	−	32.9090i	−0.713560	−	1.23592i	−0.963512	−	0.267664i	$-0.913748\pi$
$709$	−19.0000	−	32.9090i	−0.713560	−	1.23592i	0.249952	−	0.968258i	$-0.419585\pi$
$710$		0			0
$711$		0			0
$712$	−18.0000			−0.674579
$713$		0			0
$714$		0			0
$715$		0			0
$716$	12.0000	−	20.7846i	0.448461	−	0.776757i
$717$		0			0
$718$	−12.0000	−	20.7846i	−0.447836	−	0.775675i
$719$		0			0			−	1.00000i	$-0.5\pi$
$719$		0			0				1.00000i	$0.5\pi$
$720$		0			0
$721$	16.0000			0.595871
$722$	1.50000	+	2.59808i	0.0558242	+	0.0966904i
$723$		0			0
$724$	−7.00000	+	12.1244i	−0.260153	+	0.450598i
$725$	−3.00000	+	5.19615i	−0.111417	+	0.192980i
$726$		0			0
$727$	14.0000	+	24.2487i	0.519231	+	0.899335i	0.999750	+	0.0223506i	$0.00711500\pi$
$727$	14.0000	+	24.2487i	0.519231	+	0.899335i	−0.480519	+	0.876984i	$0.659552\pi$
$728$	−8.00000			−0.296500
$729$		0			0
$730$	2.00000			0.0740233
$731$	−12.0000	−	20.7846i	−0.443836	−	0.768747i
$732$		0			0
$733$	11.0000	−	19.0526i	0.406294	−	0.703722i	−0.588177	−	0.808732i	$-0.700154\pi$
$733$	11.0000	−	19.0526i	0.406294	−	0.703722i	0.994471	+	0.105010i	$0.0334875\pi$
$734$	14.0000	−	24.2487i	0.516749	−	0.895036i
$735$		0			0
$736$		0			0
$737$		0			0
$738$		0			0
$739$	−52.0000			−1.91285			−0.956425	−	0.291977i	$-0.905687\pi$
$739$	−52.0000			−1.91285			−0.956425	+	0.291977i	$0.905687\pi$
$740$	−1.00000	−	1.73205i	−0.0367607	−	0.0636715i
$741$		0			0
$742$	12.0000	−	20.7846i	0.440534	−	0.763027i
$743$	−12.0000	+	20.7846i	−0.440237	+	0.762513i	−0.997707	−	0.0676840i	$-0.978439\pi$
$743$	−12.0000	+	20.7846i	−0.440237	+	0.762513i	0.557470	+	0.830197i	$0.311772\pi$
$744$		0			0
$745$	−3.00000	−	5.19615i	−0.109911	−	0.190372i
$746$	26.0000			0.951928
$747$		0			0
$748$		0			0
$749$	24.0000	+	41.5692i	0.876941	+	1.51891i
$750$		0			0
$751$	20.0000	−	34.6410i	0.729810	−	1.26407i	−0.227153	−	0.973859i	$-0.572942\pi$
$751$	20.0000	−	34.6410i	0.729810	−	1.26407i	0.956963	−	0.290209i	$-0.0937250\pi$
$752$		0			0
$753$		0			0
$754$	−6.00000	−	10.3923i	−0.218507	−	0.378465i
$755$	8.00000			0.291150
$756$		0			0
$757$	2.00000			0.0726912			0.0363456	−	0.999339i	$-0.488428\pi$
$757$	2.00000			0.0726912			0.0363456	+	0.999339i	$0.488428\pi$
$758$	2.00000	+	3.46410i	0.0726433	+	0.125822i
$759$		0			0
$760$	2.00000	−	3.46410i	0.0725476	−	0.125656i
$761$	9.00000	−	15.5885i	0.326250	−	0.565081i	−0.655515	−	0.755182i	$-0.727548\pi$
$761$	9.00000	−	15.5885i	0.326250	−	0.565081i	0.981764	+	0.190101i	$0.0608816\pi$
$762$		0			0
$763$	−20.0000	−	34.6410i	−0.724049	−	1.25409i
$764$	24.0000			0.868290
$765$		0			0
$766$		0			0
$767$		0			0
$768$		0			0
$769$	−1.00000	+	1.73205i	−0.0360609	+	0.0624593i	−0.883493	−	0.468445i	$-0.844814\pi$
$769$	−1.00000	+	1.73205i	−0.0360609	+	0.0624593i	0.847432	+	0.530904i	$0.178148\pi$
$770$		0			0
$771$		0			0
$772$	11.0000	+	19.0526i	0.395899	+	0.685717i
$773$	−42.0000			−1.51064			−0.755318	−	0.655359i	$-0.772517\pi$
$773$	−42.0000			−1.51064			−0.755318	+	0.655359i	$0.772517\pi$
$774$		0			0
$775$	8.00000			0.287368
$776$	−1.00000	−	1.73205i	−0.0358979	−	0.0621770i
$777$		0			0
$778$	−3.00000	+	5.19615i	−0.107555	+	0.186291i
$779$	12.0000	−	20.7846i	0.429945	−	0.744686i
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$	9.00000			0.321429
$785$	−1.00000	−	1.73205i	−0.0356915	−	0.0618195i
$786$		0			0
$787$	2.00000	−	3.46410i	0.0712923	−	0.123482i	−0.828176	−	0.560469i	$-0.810621\pi$
$787$	2.00000	−	3.46410i	0.0712923	−	0.123482i	0.899468	+	0.436987i	$0.143954\pi$
$788$	−3.00000	+	5.19615i	−0.106871	+	0.185105i
$789$		0			0
$790$	−4.00000	−	6.92820i	−0.142314	−	0.246494i
$791$	−72.0000			−2.56003
$792$		0			0
$793$	−20.0000			−0.710221
$794$	11.0000	+	19.0526i	0.390375	+	0.676150i
$795$		0			0
$796$	−4.00000	+	6.92820i	−0.141776	+	0.245564i
$797$	−15.0000	+	25.9808i	−0.531327	+	0.920286i	0.468004	+	0.883726i	$0.344973\pi$
$797$	−15.0000	+	25.9808i	−0.531327	+	0.920286i	−0.999331	+	0.0365596i	$0.988360\pi$
$798$		0			0
$799$		0			0
$800$	1.00000			0.0353553
$801$		0			0
$802$	6.00000			0.211867
$803$		0			0
$804$		0			0
$805$		0			0
$806$	−8.00000	+	13.8564i	−0.281788	+	0.488071i
$807$		0			0
$808$	9.00000	+	15.5885i	0.316619	+	0.548400i
$809$	54.0000			1.89854			0.949269	−	0.314464i	$-0.101825\pi$
$809$	54.0000			1.89854			0.949269	+	0.314464i	$0.101825\pi$
$810$		0			0
$811$	−4.00000			−0.140459			−0.0702295	−	0.997531i	$-0.522373\pi$
$811$	−4.00000			−0.140459			−0.0702295	+	0.997531i	$0.522373\pi$
$812$	12.0000	+	20.7846i	0.421117	+	0.729397i
$813$		0			0
$814$		0			0
$815$	2.00000	−	3.46410i	0.0700569	−	0.121342i
$816$		0			0
$817$	−8.00000	−	13.8564i	−0.279885	−	0.484774i
$818$	26.0000			0.909069
$819$		0			0
$820$	6.00000			0.209529
$821$	9.00000	+	15.5885i	0.314102	+	0.544041i	0.979246	−	0.202674i	$-0.0649632\pi$
$821$	9.00000	+	15.5885i	0.314102	+	0.544041i	−0.665144	+	0.746715i	$0.731630\pi$
$822$		0			0
$823$	−10.0000	+	17.3205i	−0.348578	+	0.603755i	−0.985997	−	0.166762i	$-0.946669\pi$
$823$	−10.0000	+	17.3205i	−0.348578	+	0.603755i	0.637419	+	0.770517i	$0.280002\pi$
$824$	2.00000	−	3.46410i	0.0696733	−	0.120678i
$825$		0			0
$826$		0			0
$827$	−12.0000			−0.417281			−0.208640	−	0.977992i	$-0.566904\pi$
$827$	−12.0000			−0.417281			−0.208640	+	0.977992i	$0.566904\pi$
$828$		0			0
$829$	38.0000			1.31979			0.659897	−	0.751356i	$-0.270600\pi$
$829$	38.0000			1.31979			0.659897	+	0.751356i	$0.270600\pi$
$830$	6.00000	+	10.3923i	0.208263	+	0.360722i
$831$		0			0
$832$	−1.00000	+	1.73205i	−0.0346688	+	0.0600481i
$833$	27.0000	−	46.7654i	0.935495	−	1.62032i
$834$		0			0
$835$		0			0
$836$		0			0
$837$		0			0
$838$		0			0
$839$	12.0000	+	20.7846i	0.414286	+	0.717564i	0.995353	−	0.0962912i	$-0.0306980\pi$
$839$	12.0000	+	20.7846i	0.414286	+	0.717564i	−0.581067	+	0.813856i	$0.697365\pi$
$840$		0			0
$841$	−3.50000	+	6.06218i	−0.120690	+	0.209041i
$842$	5.00000	−	8.66025i	0.172311	−	0.298452i
$843$		0			0
$844$	−10.0000	−	17.3205i	−0.344214	−	0.596196i
$845$	−9.00000			−0.309609
$846$		0			0
$847$	44.0000			1.51186
$848$	−3.00000	−	5.19615i	−0.103020	−	0.178437i
$849$		0			0
$850$	3.00000	−	5.19615i	0.102899	−	0.178227i
$851$		0			0
$852$		0			0
$853$	23.0000	+	39.8372i	0.787505	+	1.36400i	0.927491	+	0.373845i	$0.121961\pi$
$853$	23.0000	+	39.8372i	0.787505	+	1.36400i	−0.139986	+	0.990153i	$0.544706\pi$
$854$	40.0000			1.36877
$855$		0			0
$856$	12.0000			0.410152
$857$	−9.00000	−	15.5885i	−0.307434	−	0.532492i	0.670366	−	0.742030i	$-0.266137\pi$
$857$	−9.00000	−	15.5885i	−0.307434	−	0.532492i	−0.977800	+	0.209539i	$0.932804\pi$
$858$		0			0
$859$	2.00000	−	3.46410i	0.0682391	−	0.118194i	−0.829887	−	0.557931i	$-0.811595\pi$
$859$	2.00000	−	3.46410i	0.0682391	−	0.118194i	0.898126	+	0.439738i	$0.144929\pi$
$860$	2.00000	−	3.46410i	0.0681994	−	0.118125i
$861$		0			0
$862$		0			0
$863$	−24.0000			−0.816970			−0.408485	−	0.912765i	$-0.633943\pi$
$863$	−24.0000			−0.816970			−0.408485	+	0.912765i	$0.633943\pi$
$864$		0			0
$865$	−18.0000			−0.612018
$866$	−13.0000	−	22.5167i	−0.441758	−	0.765147i
$867$		0			0
$868$	16.0000	−	27.7128i	0.543075	−	0.940634i
$869$		0			0
$870$		0			0
$871$	4.00000	+	6.92820i	0.135535	+	0.234753i
$872$	−10.0000			−0.338643
$873$		0			0
$874$		0			0
$875$	2.00000	+	3.46410i	0.0676123	+	0.117108i
$876$		0			0
$877$	−1.00000	+	1.73205i	−0.0337676	+	0.0584872i	−0.882415	−	0.470471i	$-0.844084\pi$
$877$	−1.00000	+	1.73205i	−0.0337676	+	0.0584872i	0.848648	+	0.528958i	$0.177417\pi$
$878$	−4.00000	+	6.92820i	−0.134993	+	0.233816i
$879$		0			0
$880$		0			0
$881$	54.0000			1.81931			0.909653	−	0.415369i	$-0.136347\pi$
$881$	54.0000			1.81931			0.909653	+	0.415369i	$0.136347\pi$
$882$		0			0
$883$	−4.00000			−0.134611			−0.0673054	−	0.997732i	$-0.521440\pi$
$883$	−4.00000			−0.134611			−0.0673054	+	0.997732i	$0.521440\pi$
$884$	6.00000	+	10.3923i	0.201802	+	0.349531i
$885$		0			0
$886$	6.00000	−	10.3923i	0.201574	−	0.349136i
$887$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$887$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$888$		0			0
$889$	40.0000	+	69.2820i	1.34156	+	2.32364i
$890$	−18.0000			−0.603361
$891$		0			0
$892$	20.0000			0.669650
$893$		0			0
$894$		0			0
$895$	12.0000	−	20.7846i	0.401116	−	0.694753i
$896$	2.00000	−	3.46410i	0.0668153	−	0.115728i
$897$		0			0
$898$	−3.00000	−	5.19615i	−0.100111	−	0.173398i
$899$	48.0000			1.60089
$900$		0			0
$901$	−36.0000			−1.19933
$902$		0			0
$903$		0			0
$904$	−9.00000	+	15.5885i	−0.299336	+	0.518464i
$905$	−7.00000	+	12.1244i	−0.232688	+	0.403027i
$906$		0			0
$907$	−22.0000	−	38.1051i	−0.730498	−	1.26526i	−0.956671	−	0.291172i	$-0.905955\pi$
$907$	−22.0000	−	38.1051i	−0.730498	−	1.26526i	0.226173	−	0.974087i	$-0.427379\pi$
$908$	12.0000			0.398234
$909$		0			0
$910$	−8.00000			−0.265197
$911$	−24.0000	−	41.5692i	−0.795155	−	1.37725i	−0.922740	−	0.385422i	$-0.874056\pi$
$911$	−24.0000	−	41.5692i	−0.795155	−	1.37725i	0.127585	−	0.991828i	$-0.459277\pi$
$912$		0			0
$913$		0			0
$914$	−13.0000	+	22.5167i	−0.430002	+	0.744785i
$915$		0			0
$916$	5.00000	+	8.66025i	0.165205	+	0.286143i
$917$		0			0
$918$		0			0
$919$	−16.0000			−0.527791			−0.263896	−	0.964551i	$-0.585007\pi$
$919$	−16.0000			−0.527791			−0.263896	+	0.964551i	$0.585007\pi$
$920$		0			0
$921$		0			0
$922$	−15.0000	+	25.9808i	−0.493999	+	0.855631i
$923$		0			0
$924$		0			0
$925$	−1.00000	−	1.73205i	−0.0328798	−	0.0569495i
$926$	−4.00000			−0.131448
$927$		0			0
$928$	6.00000			0.196960
$929$	−3.00000	−	5.19615i	−0.0984268	−	0.170480i	0.812607	−	0.582812i	$-0.198048\pi$
$929$	−3.00000	−	5.19615i	−0.0984268	−	0.170480i	−0.911034	+	0.412332i	$0.864714\pi$
$930$		0			0
$931$	18.0000	−	31.1769i	0.589926	−	1.02178i
$932$	−9.00000	+	15.5885i	−0.294805	+	0.510617i
$933$		0			0
$934$	−18.0000	−	31.1769i	−0.588978	−	1.02014i
$935$		0			0
$936$		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	−	13.8564i	−0.261209	−	0.452428i
$939$		0			0
$940$		0			0
$941$	9.00000	−	15.5885i	0.293392	−	0.508169i	−0.681218	−	0.732081i	$-0.738549\pi$
$941$	9.00000	−	15.5885i	0.293392	−	0.508169i	0.974609	+	0.223912i	$0.0718827\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$	18.0000	+	31.1769i	0.584921	+	1.01311i	0.994885	+	0.101012i	$0.0322080\pi$
$947$	18.0000	+	31.1769i	0.584921	+	1.01311i	−0.409964	+	0.912102i	$0.634459\pi$
$948$		0			0
$949$	−2.00000	+	3.46410i	−0.0649227	+	0.112449i
$950$	2.00000	−	3.46410i	0.0648886	−	0.112390i
$951$		0			0
$952$	−12.0000	−	20.7846i	−0.388922	−	0.673633i
$953$	−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
$955$	24.0000			0.776622
$956$	12.0000	+	20.7846i	0.388108	+	0.672222i
$957$		0			0
$958$	−12.0000	+	20.7846i	−0.387702	+	0.671520i
$959$	−12.0000	+	20.7846i	−0.387500	+	0.671170i
$960$		0			0
$961$	−16.5000	−	28.5788i	−0.532258	−	0.921898i
$962$	4.00000			0.128965
$963$		0			0
$964$	2.00000			0.0644157
$965$	11.0000	+	19.0526i	0.354103	+	0.613324i
$966$		0			0
$967$	2.00000	−	3.46410i	0.0643157	−	0.111398i	−0.832075	−	0.554664i	$-0.812847\pi$
$967$	2.00000	−	3.46410i	0.0643157	−	0.111398i	0.896390	+	0.443266i	$0.146180\pi$
$968$	5.50000	−	9.52628i	0.176777	−	0.306186i
$969$		0			0
$970$	−1.00000	−	1.73205i	−0.0321081	−	0.0556128i
$971$	−24.0000			−0.770197			−0.385098	−	0.922876i	$-0.625832\pi$
$971$	−24.0000			−0.770197			−0.385098	+	0.922876i	$0.625832\pi$
$972$		0			0
$973$	16.0000			0.512936
$974$	14.0000	+	24.2487i	0.448589	+	0.776979i
$975$		0			0
$976$	5.00000	−	8.66025i	0.160046	−	0.277208i
$977$	−21.0000	+	36.3731i	−0.671850	+	1.16368i	0.305530	+	0.952183i	$0.401167\pi$
$977$	−21.0000	+	36.3731i	−0.671850	+	1.16368i	−0.977379	+	0.211495i	$0.932167\pi$
$978$		0			0
$979$		0			0
$980$	9.00000			0.287494
$981$		0			0
$982$	−24.0000			−0.765871
$983$	−12.0000	−	20.7846i	−0.382741	−	0.662926i	0.608712	−	0.793391i	$-0.291686\pi$
$983$	−12.0000	−	20.7846i	−0.382741	−	0.662926i	−0.991453	+	0.130465i	$0.958353\pi$
$984$		0			0
$985$	−3.00000	+	5.19615i	−0.0955879	+	0.165563i
$986$	18.0000	−	31.1769i	0.573237	−	0.992875i
$987$		0			0
$988$	4.00000	+	6.92820i	0.127257	+	0.220416i
$989$		0			0
$990$		0			0
$991$	−16.0000			−0.508257			−0.254128	−	0.967170i	$-0.581789\pi$
$991$	−16.0000			−0.508257			−0.254128	+	0.967170i	$0.581789\pi$
$992$	−4.00000	−	6.92820i	−0.127000	−	0.219971i
$993$		0			0
$994$		0			0
$995$	−4.00000	+	6.92820i	−0.126809	+	0.219639i
$996$		0			0
$997$	−13.0000	−	22.5167i	−0.411714	−	0.713110i	0.583363	−	0.812211i	$-0.301736\pi$
$997$	−13.0000	−	22.5167i	−0.411714	−	0.713110i	−0.995077	+	0.0991016i	$0.968403\pi$
$998$	−4.00000			−0.126618
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	810.2.e.b.271.1		2
3.2	odd	2		810.2.e.l.271.1		2
9.2	odd	6		810.2.e.l.541.1		2
9.4	even	3		90.2.a.c.1.1		1
9.5	odd	6		30.2.a.a.1.1	✓	1
9.7	even	3	inner	810.2.e.b.541.1		2
36.23	even	6		240.2.a.b.1.1		1
36.31	odd	6		720.2.a.j.1.1		1
45.4	even	6		450.2.a.d.1.1		1
45.13	odd	12		450.2.c.b.199.1		2
45.14	odd	6		150.2.a.b.1.1		1
45.22	odd	12		450.2.c.b.199.2		2
45.23	even	12		150.2.c.a.49.2		2
45.32	even	12		150.2.c.a.49.1		2
63.5	even	6		1470.2.i.q.361.1		2
63.13	odd	6		4410.2.a.z.1.1		1
63.23	odd	6		1470.2.i.o.361.1		2
63.32	odd	6		1470.2.i.o.961.1		2
63.41	even	6		1470.2.a.d.1.1		1
63.59	even	6		1470.2.i.q.961.1		2
72.5	odd	6		960.2.a.e.1.1		1
72.13	even	6		2880.2.a.a.1.1		1
72.59	even	6		960.2.a.p.1.1		1
72.67	odd	6		2880.2.a.q.1.1		1
99.32	even	6		3630.2.a.w.1.1		1
117.5	even	12		5070.2.b.k.1351.2		2
117.77	odd	6		5070.2.a.w.1.1		1
117.86	even	12		5070.2.b.k.1351.1		2
144.5	odd	12		3840.2.k.y.1921.1		2
144.59	even	12		3840.2.k.f.1921.2		2
144.77	odd	12		3840.2.k.y.1921.2		2
144.131	even	12		3840.2.k.f.1921.1		2
153.50	odd	6		8670.2.a.g.1.1		1
180.23	odd	12		1200.2.f.e.49.1		2
180.59	even	6		1200.2.a.k.1.1		1
180.67	even	12		3600.2.f.i.2449.2		2
180.103	even	12		3600.2.f.i.2449.1		2
180.139	odd	6		3600.2.a.f.1.1		1
180.167	odd	12		1200.2.f.e.49.2		2
315.104	even	6		7350.2.a.ct.1.1		1
360.59	even	6		4800.2.a.d.1.1		1
360.77	even	12		4800.2.f.p.3649.2		2
360.149	odd	6		4800.2.a.cq.1.1		1
360.203	odd	12		4800.2.f.w.3649.2		2
360.293	even	12		4800.2.f.p.3649.1		2
360.347	odd	12		4800.2.f.w.3649.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
30.2.a.a.1.1	✓	1	9.5	odd	6
90.2.a.c.1.1		1	9.4	even	3
150.2.a.b.1.1		1	45.14	odd	6
150.2.c.a.49.1		2	45.32	even	12
150.2.c.a.49.2		2	45.23	even	12
240.2.a.b.1.1		1	36.23	even	6
450.2.a.d.1.1		1	45.4	even	6
450.2.c.b.199.1		2	45.13	odd	12
450.2.c.b.199.2		2	45.22	odd	12
720.2.a.j.1.1		1	36.31	odd	6
810.2.e.b.271.1		2	1.1	even	1	trivial
810.2.e.b.541.1		2	9.7	even	3	inner
810.2.e.l.271.1		2	3.2	odd	2
810.2.e.l.541.1		2	9.2	odd	6
960.2.a.e.1.1		1	72.5	odd	6
960.2.a.p.1.1		1	72.59	even	6
1200.2.a.k.1.1		1	180.59	even	6
1200.2.f.e.49.1		2	180.23	odd	12
1200.2.f.e.49.2		2	180.167	odd	12
1470.2.a.d.1.1		1	63.41	even	6
1470.2.i.o.361.1		2	63.23	odd	6
1470.2.i.o.961.1		2	63.32	odd	6
1470.2.i.q.361.1		2	63.5	even	6
1470.2.i.q.961.1		2	63.59	even	6
2880.2.a.a.1.1		1	72.13	even	6
2880.2.a.q.1.1		1	72.67	odd	6
3600.2.a.f.1.1		1	180.139	odd	6
3600.2.f.i.2449.1		2	180.103	even	12
3600.2.f.i.2449.2		2	180.67	even	12
3630.2.a.w.1.1		1	99.32	even	6
3840.2.k.f.1921.1		2	144.131	even	12
3840.2.k.f.1921.2		2	144.59	even	12
3840.2.k.y.1921.1		2	144.5	odd	12
3840.2.k.y.1921.2		2	144.77	odd	12
4410.2.a.z.1.1		1	63.13	odd	6
4800.2.a.d.1.1		1	360.59	even	6
4800.2.a.cq.1.1		1	360.149	odd	6
4800.2.f.p.3649.1		2	360.293	even	12
4800.2.f.p.3649.2		2	360.77	even	12
4800.2.f.w.3649.1		2	360.347	odd	12
4800.2.f.w.3649.2		2	360.203	odd	12
5070.2.a.w.1.1		1	117.77	odd	6
5070.2.b.k.1351.1		2	117.86	even	12
5070.2.b.k.1351.2		2	117.5	even	12
7350.2.a.ct.1.1		1	315.104	even	6
8670.2.a.g.1.1		1	153.50	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:	\( N \)	\(=\)	\( 810 = 2 \cdot 3^{4} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	810.e (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-0.500000 - 0.866025i) q^{2} +(-0.500000 + 0.866025i) q^{4} +(-0.500000 + 0.866025i) q^{5} +(2.00000 + 3.46410i) q^{7} +1.00000 q^{8} +O(q^{10})\)	\(q+(-0.500000 - 0.866025i) q^{2} +(-0.500000 + 0.866025i) q^{4} +(-0.500000 + 0.866025i) q^{5} +(2.00000 + 3.46410i) q^{7} +1.00000 q^{8} +1.00000 q^{10} +(-1.00000 + 1.73205i) q^{13} +(2.00000 - 3.46410i) q^{14} +(-0.500000 - 0.866025i) q^{16} -6.00000 q^{17} -4.00000 q^{19} +(-0.500000 - 0.866025i) q^{20} +(-0.500000 - 0.866025i) q^{25} +2.00000 q^{26} -4.00000 q^{28} +(-3.00000 - 5.19615i) q^{29} +(-4.00000 + 6.92820i) q^{31} +(-0.500000 + 0.866025i) q^{32} +(3.00000 + 5.19615i) q^{34} -4.00000 q^{35} +2.00000 q^{37} +(2.00000 + 3.46410i) q^{38} +(-0.500000 + 0.866025i) q^{40} +(-3.00000 + 5.19615i) q^{41} +(2.00000 + 3.46410i) q^{43} +(-4.50000 + 7.79423i) q^{49} +(-0.500000 + 0.866025i) q^{50} +(-1.00000 - 1.73205i) q^{52} +6.00000 q^{53} +(2.00000 + 3.46410i) q^{56} +(-3.00000 + 5.19615i) q^{58} +(5.00000 + 8.66025i) q^{61} +8.00000 q^{62} +1.00000 q^{64} +(-1.00000 - 1.73205i) q^{65} +(2.00000 - 3.46410i) q^{67} +(3.00000 - 5.19615i) q^{68} +(2.00000 + 3.46410i) q^{70} +2.00000 q^{73} +(-1.00000 - 1.73205i) q^{74} +(2.00000 - 3.46410i) q^{76} +(-4.00000 - 6.92820i) q^{79} +1.00000 q^{80} +6.00000 q^{82} +(6.00000 + 10.3923i) q^{83} +(3.00000 - 5.19615i) q^{85} +(2.00000 - 3.46410i) q^{86} -18.0000 q^{89} -8.00000 q^{91} +(2.00000 - 3.46410i) q^{95} +(-1.00000 - 1.73205i) q^{97} +9.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - q^{2} - q^{4} - q^{5} + 4 q^{7} + 2 q^{8}+O(q^{10}) \) 2 * q - q^2 - q^4 - q^5 + 4 * q^7 + 2 * q^8	\( 2 q - q^{2} - q^{4} - q^{5} + 4 q^{7} + 2 q^{8} + 2 q^{10} - 2 q^{13} + 4 q^{14} - q^{16} - 12 q^{17} - 8 q^{19} - q^{20} - q^{25} + 4 q^{26} - 8 q^{28} - 6 q^{29} - 8 q^{31} - q^{32} + 6 q^{34} - 8 q^{35} + 4 q^{37} + 4 q^{38} - q^{40} - 6 q^{41} + 4 q^{43} - 9 q^{49} - q^{50} - 2 q^{52} + 12 q^{53} + 4 q^{56} - 6 q^{58} + 10 q^{61} + 16 q^{62} + 2 q^{64} - 2 q^{65} + 4 q^{67} + 6 q^{68} + 4 q^{70} + 4 q^{73} - 2 q^{74} + 4 q^{76} - 8 q^{79} + 2 q^{80} + 12 q^{82} + 12 q^{83} + 6 q^{85} + 4 q^{86} - 36 q^{89} - 16 q^{91} + 4 q^{95} - 2 q^{97} + 18 q^{98}+O(q^{100}) \) 2 * q - q^2 - q^4 - q^5 + 4 * q^7 + 2 * q^8 + 2 * q^10 - 2 * q^13 + 4 * q^14 - q^16 - 12 * q^17 - 8 * q^19 - q^20 - q^25 + 4 * q^26 - 8 * q^28 - 6 * q^29 - 8 * q^31 - q^32 + 6 * q^34 - 8 * q^35 + 4 * q^37 + 4 * q^38 - q^40 - 6 * q^41 + 4 * q^43 - 9 * q^49 - q^50 - 2 * q^52 + 12 * q^53 + 4 * q^56 - 6 * q^58 + 10 * q^61 + 16 * q^62 + 2 * q^64 - 2 * q^65 + 4 * q^67 + 6 * q^68 + 4 * q^70 + 4 * q^73 - 2 * q^74 + 4 * q^76 - 8 * q^79 + 2 * q^80 + 12 * q^82 + 12 * q^83 + 6 * q^85 + 4 * q^86 - 36 * q^89 - 16 * q^91 + 4 * q^95 - 2 * q^97 + 18 * q^98

Introduction

L-functions

Modular forms

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Database

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