LMFDB - Embedded newform 216.2.n.a.181.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [216,2,Mod(37,216)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(216, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 3, 2]))

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

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

chi := DirichletCharacter("216.37");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 216 = 2^{3} \cdot 3^{3} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	216.n (of order $6$, 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:	$1.72476868366$
Analytic rank:	$0$
Dimension:	$4$
Relative dimension:	$2$ over $\Q(\zeta_{6})$
Coefficient field:	$\Q(\zeta_{12})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{2} + 1 $ x^4 - x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 72)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			181.2
Root			$0.866025 - 0.500000i$ of defining polynomial
Character	$\chi$	$=$	216.181
Dual form			216.2.n.a.37.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.36603 - 0.366025i) q^{2} +(1.73205 - 1.00000i) q^{4} +(1.73205 - 1.00000i) q^{5} +(-2.00000 + 3.46410i) q^{7} +(2.00000 - 2.00000i) q^{8} +O(q^{10})$	\(q+(1.36603 - 0.366025i) q^{2} +(1.73205 - 1.00000i) q^{4} +(1.73205 - 1.00000i) q^{5} +(-2.00000 + 3.46410i) q^{7} +(2.00000 - 2.00000i) q^{8} +(2.00000 - 2.00000i) q^{10} +(-2.59808 - 1.50000i) q^{11} +(1.73205 - 1.00000i) q^{13} +(-1.46410 + 5.46410i) q^{14} +(2.00000 - 3.46410i) q^{16} -5.00000 q^{17} +1.00000i q^{19} +(2.00000 - 3.46410i) q^{20} +(-4.09808 - 1.09808i) q^{22} +(1.00000 + 1.73205i) q^{23} +(-0.500000 + 0.866025i) q^{25} +(2.00000 - 2.00000i) q^{26} +8.00000i q^{28} +(2.00000 + 3.46410i) q^{31} +(1.46410 - 5.46410i) q^{32} +(-6.83013 + 1.83013i) q^{34} +8.00000i q^{35} +2.00000i q^{37} +(0.366025 + 1.36603i) q^{38} +(1.46410 - 5.46410i) q^{40} +(-2.50000 - 4.33013i) q^{41} +(-9.52628 - 5.50000i) q^{43} -6.00000 q^{44} +(2.00000 + 2.00000i) q^{46} +(-3.00000 + 5.19615i) q^{47} +(-4.50000 - 7.79423i) q^{49} +(-0.366025 + 1.36603i) q^{50} +(2.00000 - 3.46410i) q^{52} -6.00000 q^{55} +(2.92820 + 10.9282i) q^{56} +(0.866025 - 0.500000i) q^{59} +(10.3923 + 6.00000i) q^{61} +(4.00000 + 4.00000i) q^{62} -8.00000i q^{64} +(2.00000 - 3.46410i) q^{65} +(2.59808 - 1.50000i) q^{67} +(-8.66025 + 5.00000i) q^{68} +(2.92820 + 10.9282i) q^{70} +6.00000 q^{71} +9.00000 q^{73} +(0.732051 + 2.73205i) q^{74} +(1.00000 + 1.73205i) q^{76} +(10.3923 - 6.00000i) q^{77} +(7.00000 - 12.1244i) q^{79} -8.00000i q^{80} +(-5.00000 - 5.00000i) q^{82} +(-3.46410 - 2.00000i) q^{83} +(-8.66025 + 5.00000i) q^{85} +(-15.0263 - 4.02628i) q^{86} +(-8.19615 + 2.19615i) q^{88} +14.0000 q^{89} +8.00000i q^{91} +(3.46410 + 2.00000i) q^{92} +(-2.19615 + 8.19615i) q^{94} +(1.00000 + 1.73205i) q^{95} +(-0.500000 + 0.866025i) q^{97} +(-9.00000 - 9.00000i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 2 q^{2} - 8 q^{7} + 8 q^{8}+O(q^{10}) $ 4 * q + 2 * q^2 - 8 * q^7 + 8 * q^8	$ 4 q + 2 q^{2} - 8 q^{7} + 8 q^{8} + 8 q^{10} + 8 q^{14} + 8 q^{16} - 20 q^{17} + 8 q^{20} - 6 q^{22} + 4 q^{23} - 2 q^{25} + 8 q^{26} + 8 q^{31} - 8 q^{32} - 10 q^{34} - 2 q^{38} - 8 q^{40} - 10 q^{41} - 24 q^{44} + 8 q^{46} - 12 q^{47} - 18 q^{49} + 2 q^{50} + 8 q^{52} - 24 q^{55} - 16 q^{56} + 16 q^{62} + 8 q^{65} - 16 q^{70} + 24 q^{71} + 36 q^{73} - 4 q^{74} + 4 q^{76} + 28 q^{79} - 20 q^{82} - 22 q^{86} - 12 q^{88} + 56 q^{89} + 12 q^{94} + 4 q^{95} - 2 q^{97} - 36 q^{98}+O(q^{100}) $ 4 * q + 2 * q^2 - 8 * q^7 + 8 * q^8 + 8 * q^10 + 8 * q^14 + 8 * q^16 - 20 * q^17 + 8 * q^20 - 6 * q^22 + 4 * q^23 - 2 * q^25 + 8 * q^26 + 8 * q^31 - 8 * q^32 - 10 * q^34 - 2 * q^38 - 8 * q^40 - 10 * q^41 - 24 * q^44 + 8 * q^46 - 12 * q^47 - 18 * q^49 + 2 * q^50 + 8 * q^52 - 24 * q^55 - 16 * q^56 + 16 * q^62 + 8 * q^65 - 16 * q^70 + 24 * q^71 + 36 * q^73 - 4 * q^74 + 4 * q^76 + 28 * q^79 - 20 * q^82 - 22 * q^86 - 12 * q^88 + 56 * q^89 + 12 * q^94 + 4 * q^95 - 2 * q^97 - 36 * q^98

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/216\mathbb{Z}\right)^\times$.

$n$	$55$	$109$	$137$
$\chi(n)$	$1$	$-1$	$e\left(\frac{2}{3}\right)$

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 $

$2$	1.36603	−	0.366025i	0.965926	−	0.258819i
$2$	1.36603	−	0.366025i	0.965926	−	0.258819i
$3$		0			0
$3$		0			0
$4$	1.73205	−	1.00000i	0.866025	−	0.500000i
$5$	1.73205	−	1.00000i	0.774597	−	0.447214i	−0.0599153	−	0.998203i	$-0.519083\pi$
$5$	1.73205	−	1.00000i	0.774597	−	0.447214i	0.834512	+	0.550990i	$0.185750\pi$
$6$		0			0
$7$	−2.00000	+	3.46410i	−0.755929	+	1.30931i	0.188982	+	0.981981i	$0.439481\pi$
$7$	−2.00000	+	3.46410i	−0.755929	+	1.30931i	−0.944911	+	0.327327i	$0.893852\pi$
$8$	2.00000	−	2.00000i	0.707107	−	0.707107i
$9$		0			0
$10$	2.00000	−	2.00000i	0.632456	−	0.632456i
$11$	−2.59808	−	1.50000i	−0.783349	−	0.452267i	0.0542666	−	0.998526i	$-0.482718\pi$
$11$	−2.59808	−	1.50000i	−0.783349	−	0.452267i	−0.837616	+	0.546259i	$0.816051\pi$
$12$		0			0
$13$	1.73205	−	1.00000i	0.480384	−	0.277350i	−0.240192	−	0.970725i	$-0.577210\pi$
$13$	1.73205	−	1.00000i	0.480384	−	0.277350i	0.720577	+	0.693375i	$0.243877\pi$
$14$	−1.46410	+	5.46410i	−0.391298	+	1.46034i
$15$		0			0
$16$	2.00000	−	3.46410i	0.500000	−	0.866025i
$17$	−5.00000			−1.21268			−0.606339	−	0.795206i	$-0.707363\pi$
$17$	−5.00000			−1.21268			−0.606339	+	0.795206i	$0.707363\pi$
$18$		0			0
$19$			1.00000i			0.229416i	0.993399	+	0.114708i	$0.0365932\pi$
$19$			1.00000i			0.229416i	−0.993399	+	0.114708i	$0.963407\pi$
$20$	2.00000	−	3.46410i	0.447214	−	0.774597i
$21$		0			0
$22$	−4.09808	−	1.09808i	−0.873713	−	0.234111i
$23$	1.00000	+	1.73205i	0.208514	+	0.361158i	0.951247	−	0.308431i	$-0.0998038\pi$
$23$	1.00000	+	1.73205i	0.208514	+	0.361158i	−0.742732	+	0.669588i	$0.766471\pi$
$24$		0			0
$25$	−0.500000	+	0.866025i	−0.100000	+	0.173205i
$26$	2.00000	−	2.00000i	0.392232	−	0.392232i
$27$		0			0
$28$			8.00000i			1.51186i
$29$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$29$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$30$		0			0
$31$	2.00000	+	3.46410i	0.359211	+	0.622171i	0.987829	−	0.155543i	$-0.0497126\pi$
$31$	2.00000	+	3.46410i	0.359211	+	0.622171i	−0.628619	+	0.777714i	$0.716379\pi$
$32$	1.46410	−	5.46410i	0.258819	−	0.965926i
$33$		0			0
$34$	−6.83013	+	1.83013i	−1.17136	+	0.313864i
$35$			8.00000i			1.35225i
$36$		0			0
$37$			2.00000i			0.328798i	0.986394	+	0.164399i	$0.0525685\pi$
$37$			2.00000i			0.328798i	−0.986394	+	0.164399i	$0.947432\pi$
$38$	0.366025	+	1.36603i	0.0593772	+	0.221599i
$39$		0			0
$40$	1.46410	−	5.46410i	0.231495	−	0.863950i
$41$	−2.50000	−	4.33013i	−0.390434	−	0.676252i	0.602072	−	0.798441i	$-0.294342\pi$
$41$	−2.50000	−	4.33013i	−0.390434	−	0.676252i	−0.992507	+	0.122189i	$0.961009\pi$
$42$		0			0
$43$	−9.52628	−	5.50000i	−1.45274	−	0.838742i	−0.454108	−	0.890947i	$-0.650042\pi$
$43$	−9.52628	−	5.50000i	−1.45274	−	0.838742i	−0.998636	+	0.0522047i	$0.983375\pi$
$44$	−6.00000			−0.904534
$45$		0			0
$46$	2.00000	+	2.00000i	0.294884	+	0.294884i
$47$	−3.00000	+	5.19615i	−0.437595	+	0.757937i	−0.997503	−	0.0706177i	$-0.977503\pi$
$47$	−3.00000	+	5.19615i	−0.437595	+	0.757937i	0.559908	+	0.828554i	$0.310836\pi$
$48$		0			0
$49$	−4.50000	−	7.79423i	−0.642857	−	1.11346i
$50$	−0.366025	+	1.36603i	−0.0517638	+	0.193185i
$51$		0			0
$52$	2.00000	−	3.46410i	0.277350	−	0.480384i
$53$		0			0		1.00000			$0$
$53$		0			0		−1.00000			$\pi$
$54$		0			0
$55$	−6.00000			−0.809040
$56$	2.92820	+	10.9282i	0.391298	+	1.46034i
$57$		0			0
$58$		0			0
$59$	0.866025	−	0.500000i	0.112747	−	0.0650945i	−0.442566	−	0.896736i	$-0.645932\pi$
$59$	0.866025	−	0.500000i	0.112747	−	0.0650945i	0.555313	+	0.831641i	$0.312598\pi$
$60$		0			0
$61$	10.3923	+	6.00000i	1.33060	+	0.768221i	0.985391	−	0.170305i	$-0.0544754\pi$
$61$	10.3923	+	6.00000i	1.33060	+	0.768221i	0.345207	+	0.938527i	$0.387809\pi$
$62$	4.00000	+	4.00000i	0.508001	+	0.508001i
$63$		0			0
$64$		−	8.00000i		−	1.00000i
$65$	2.00000	−	3.46410i	0.248069	−	0.429669i
$66$		0			0
$67$	2.59808	−	1.50000i	0.317406	−	0.183254i	−0.332830	−	0.942987i	$-0.608004\pi$
$67$	2.59808	−	1.50000i	0.317406	−	0.183254i	0.650236	+	0.759733i	$0.274670\pi$
$68$	−8.66025	+	5.00000i	−1.05021	+	0.606339i
$69$		0			0
$70$	2.92820	+	10.9282i	0.349987	+	1.30617i
$71$	6.00000			0.712069			0.356034	−	0.934473i	$-0.384129\pi$
$71$	6.00000			0.712069			0.356034	+	0.934473i	$0.384129\pi$
$72$		0			0
$73$	9.00000			1.05337			0.526685	−	0.850060i	$-0.323435\pi$
$73$	9.00000			1.05337			0.526685	+	0.850060i	$0.323435\pi$
$74$	0.732051	+	2.73205i	0.0850992	+	0.317594i
$75$		0			0
$76$	1.00000	+	1.73205i	0.114708	+	0.198680i
$77$	10.3923	−	6.00000i	1.18431	−	0.683763i
$78$		0			0
$79$	7.00000	−	12.1244i	0.787562	−	1.36410i	−0.139895	−	0.990166i	$-0.544677\pi$
$79$	7.00000	−	12.1244i	0.787562	−	1.36410i	0.927457	−	0.373930i	$-0.121990\pi$
$80$		−	8.00000i		−	0.894427i
$81$		0			0
$82$	−5.00000	−	5.00000i	−0.552158	−	0.552158i
$83$	−3.46410	−	2.00000i	−0.380235	−	0.219529i	0.297686	−	0.954664i	$-0.403785\pi$
$83$	−3.46410	−	2.00000i	−0.380235	−	0.219529i	−0.677920	+	0.735135i	$0.737119\pi$
$84$		0			0
$85$	−8.66025	+	5.00000i	−0.939336	+	0.542326i
$86$	−15.0263	−	4.02628i	−1.62033	−	0.434165i
$87$		0			0
$88$	−8.19615	+	2.19615i	−0.873713	+	0.234111i
$89$	14.0000			1.48400			0.741999	−	0.670402i	$-0.233878\pi$
$89$	14.0000			1.48400			0.741999	+	0.670402i	$0.233878\pi$
$90$		0			0
$91$			8.00000i			0.838628i
$92$	3.46410	+	2.00000i	0.361158	+	0.208514i
$93$		0			0
$94$	−2.19615	+	8.19615i	−0.226516	+	0.845369i
$95$	1.00000	+	1.73205i	0.102598	+	0.177705i
$96$		0			0
$97$	−0.500000	+	0.866025i	−0.0507673	+	0.0879316i	−0.890292	−	0.455389i	$-0.849500\pi$
$97$	−0.500000	+	0.866025i	−0.0507673	+	0.0879316i	0.839525	+	0.543321i	$0.182833\pi$
$98$	−9.00000	−	9.00000i	−0.909137	−	0.909137i
$99$		0			0
$100$			2.00000i			0.200000i
$101$	−12.1244	−	7.00000i	−1.20642	−	0.696526i	−0.244443	−	0.969664i	$-0.578605\pi$
$101$	−12.1244	−	7.00000i	−1.20642	−	0.696526i	−0.961975	+	0.273138i	$0.911939\pi$
$102$		0			0
$103$	3.00000	+	5.19615i	0.295599	+	0.511992i	0.975124	−	0.221660i	$-0.0711475\pi$
$103$	3.00000	+	5.19615i	0.295599	+	0.511992i	−0.679525	+	0.733652i	$0.737814\pi$
$104$	1.46410	−	5.46410i	0.143567	−	0.535799i
$105$		0			0
$106$		0			0
$107$		−	3.00000i		−	0.290021i	−0.989430	−	0.145010i	$-0.953678\pi$
$107$		−	3.00000i		−	0.290021i	0.989430	−	0.145010i	$-0.0463216\pi$
$108$		0			0
$109$		−	20.0000i		−	1.91565i	−0.287348	−	0.957826i	$-0.592774\pi$
$109$		−	20.0000i		−	1.91565i	0.287348	−	0.957826i	$-0.407226\pi$
$110$	−8.19615	+	2.19615i	−0.781472	+	0.209395i
$111$		0			0
$112$	8.00000	+	13.8564i	0.755929	+	1.30931i
$113$	3.00000	+	5.19615i	0.282216	+	0.488813i	0.971930	−	0.235269i	$-0.0755971\pi$
$113$	3.00000	+	5.19615i	0.282216	+	0.488813i	−0.689714	+	0.724082i	$0.742264\pi$
$114$		0			0
$115$	3.46410	+	2.00000i	0.323029	+	0.186501i
$116$		0			0
$117$		0			0
$118$	1.00000	−	1.00000i	0.0920575	−	0.0920575i
$119$	10.0000	−	17.3205i	0.916698	−	1.58777i
$120$		0			0
$121$	−1.00000	−	1.73205i	−0.0909091	−	0.157459i
$122$	16.3923	+	4.39230i	1.48409	+	0.397661i
$123$		0			0
$124$	6.92820	+	4.00000i	0.622171	+	0.359211i
$125$			12.0000i			1.07331i
$126$		0			0
$127$	−2.00000			−0.177471			−0.0887357	−	0.996055i	$-0.528283\pi$
$127$	−2.00000			−0.177471			−0.0887357	+	0.996055i	$0.528283\pi$
$128$	−2.92820	−	10.9282i	−0.258819	−	0.965926i
$129$		0			0
$130$	1.46410	−	5.46410i	0.128410	−	0.479233i
$131$	3.46410	−	2.00000i	0.302660	−	0.174741i	−0.340977	−	0.940072i	$-0.610758\pi$
$131$	3.46410	−	2.00000i	0.302660	−	0.174741i	0.643637	+	0.765331i	$0.277425\pi$
$132$		0			0
$133$	−3.46410	−	2.00000i	−0.300376	−	0.173422i
$134$	3.00000	−	3.00000i	0.259161	−	0.259161i
$135$		0			0
$136$	−10.0000	+	10.0000i	−0.857493	+	0.857493i
$137$	−4.50000	+	7.79423i	−0.384461	+	0.665906i	−0.991694	−	0.128618i	$-0.958946\pi$
$137$	−4.50000	+	7.79423i	−0.384461	+	0.665906i	0.607233	+	0.794524i	$0.292279\pi$
$138$		0			0
$139$	11.2583	−	6.50000i	0.954919	−	0.551323i	0.0603135	−	0.998179i	$-0.480790\pi$
$139$	11.2583	−	6.50000i	0.954919	−	0.551323i	0.894606	+	0.446857i	$0.147457\pi$
$140$	8.00000	+	13.8564i	0.676123	+	1.17108i
$141$		0			0
$142$	8.19615	−	2.19615i	0.687806	−	0.184297i
$143$	−6.00000			−0.501745
$144$		0			0
$145$		0			0
$146$	12.2942	−	3.29423i	1.01748	−	0.272632i
$147$		0			0
$148$	2.00000	+	3.46410i	0.164399	+	0.284747i
$149$	−15.5885	+	9.00000i	−1.27706	+	0.737309i	−0.976306	−	0.216394i	$-0.930570\pi$
$149$	−15.5885	+	9.00000i	−1.27706	+	0.737309i	−0.300750	+	0.953703i	$0.597237\pi$
$150$		0			0
$151$	3.00000	−	5.19615i	0.244137	−	0.422857i	−0.717752	−	0.696299i	$-0.754829\pi$
$151$	3.00000	−	5.19615i	0.244137	−	0.422857i	0.961888	+	0.273442i	$0.0881622\pi$
$152$	2.00000	+	2.00000i	0.162221	+	0.162221i
$153$		0			0
$154$	12.0000	−	12.0000i	0.966988	−	0.966988i
$155$	6.92820	+	4.00000i	0.556487	+	0.321288i
$156$		0			0
$157$	−3.46410	+	2.00000i	−0.276465	+	0.159617i	−0.631822	−	0.775113i	$-0.717693\pi$
$157$	−3.46410	+	2.00000i	−0.276465	+	0.159617i	0.355357	+	0.934731i	$0.384359\pi$
$158$	5.12436	−	19.1244i	0.407672	−	1.52145i
$159$		0			0
$160$	−2.92820	−	10.9282i	−0.231495	−	0.863950i
$161$	−8.00000			−0.630488
$162$		0			0
$163$		−	4.00000i		−	0.313304i	−0.987654	−	0.156652i	$-0.949930\pi$
$163$		−	4.00000i		−	0.313304i	0.987654	−	0.156652i	$-0.0500701\pi$
$164$	−8.66025	−	5.00000i	−0.676252	−	0.390434i
$165$		0			0
$166$	−5.46410	−	1.46410i	−0.424097	−	0.113636i
$167$	−1.00000	−	1.73205i	−0.0773823	−	0.134030i	0.824737	−	0.565516i	$-0.191323\pi$
$167$	−1.00000	−	1.73205i	−0.0773823	−	0.134030i	−0.902120	+	0.431486i	$0.857990\pi$
$168$		0			0
$169$	−4.50000	+	7.79423i	−0.346154	+	0.599556i
$170$	−10.0000	+	10.0000i	−0.766965	+	0.766965i
$171$		0			0
$172$	−22.0000			−1.67748
$173$	20.7846	+	12.0000i	1.58022	+	0.912343i	0.994826	+	0.101598i	$0.0323955\pi$
$173$	20.7846	+	12.0000i	1.58022	+	0.912343i	0.585399	+	0.810745i	$0.300938\pi$
$174$		0			0
$175$	−2.00000	−	3.46410i	−0.151186	−	0.261861i
$176$	−10.3923	+	6.00000i	−0.783349	+	0.452267i
$177$		0			0
$178$	19.1244	−	5.12436i	1.43343	−	0.384087i
$179$		−	20.0000i		−	1.49487i	−0.664335	−	0.747435i	$-0.731285\pi$
$179$		−	20.0000i		−	1.49487i	0.664335	−	0.747435i	$-0.268715\pi$
$180$		0			0
$181$			10.0000i			0.743294i	0.928374	+	0.371647i	$0.121207\pi$
$181$			10.0000i			0.743294i	−0.928374	+	0.371647i	$0.878793\pi$
$182$	2.92820	+	10.9282i	0.217053	+	0.810052i
$183$		0			0
$184$	5.46410	+	1.46410i	0.402819	+	0.107935i
$185$	2.00000	+	3.46410i	0.147043	+	0.254686i
$186$		0			0
$187$	12.9904	+	7.50000i	0.949951	+	0.548454i
$188$			12.0000i			0.875190i
$189$		0			0
$190$	2.00000	+	2.00000i	0.145095	+	0.145095i
$191$	−8.00000	+	13.8564i	−0.578860	+	1.00261i	0.416751	+	0.909021i	$0.363169\pi$
$191$	−8.00000	+	13.8564i	−0.578860	+	1.00261i	−0.995610	+	0.0935936i	$0.970165\pi$
$192$		0			0
$193$	7.50000	+	12.9904i	0.539862	+	0.935068i	0.998911	+	0.0466572i	$0.0148568\pi$
$193$	7.50000	+	12.9904i	0.539862	+	0.935068i	−0.459049	+	0.888411i	$0.651810\pi$
$194$	−0.366025	+	1.36603i	−0.0262791	+	0.0980749i
$195$		0			0
$196$	−15.5885	−	9.00000i	−1.11346	−	0.642857i
$197$		−	8.00000i		−	0.569976i	−0.958531	−	0.284988i	$-0.908010\pi$
$197$		−	8.00000i		−	0.569976i	0.958531	−	0.284988i	$-0.0919897\pi$
$198$		0			0
$199$	−8.00000			−0.567105			−0.283552	−	0.958957i	$-0.591513\pi$
$199$	−8.00000			−0.567105			−0.283552	+	0.958957i	$0.591513\pi$
$200$	0.732051	+	2.73205i	0.0517638	+	0.193185i
$201$		0			0
$202$	−19.1244	−	5.12436i	−1.34558	−	0.360548i
$203$		0			0
$204$		0			0
$205$	−8.66025	−	5.00000i	−0.604858	−	0.349215i
$206$	6.00000	+	6.00000i	0.418040	+	0.418040i
$207$		0			0
$208$		−	8.00000i		−	0.554700i
$209$	1.50000	−	2.59808i	0.103757	−	0.179713i
$210$		0			0
$211$	−13.8564	+	8.00000i	−0.953914	+	0.550743i	−0.894295	−	0.447478i	$-0.852322\pi$
$211$	−13.8564	+	8.00000i	−0.953914	+	0.550743i	−0.0596196	+	0.998221i	$0.518989\pi$
$212$		0			0
$213$		0			0
$214$	−1.09808	−	4.09808i	−0.0750629	−	0.280139i
$215$	−22.0000			−1.50039
$216$		0			0
$217$	−16.0000			−1.08615
$218$	−7.32051	−	27.3205i	−0.495807	−	1.85038i
$219$		0			0
$220$	−10.3923	+	6.00000i	−0.700649	+	0.404520i
$221$	−8.66025	+	5.00000i	−0.582552	+	0.336336i
$222$		0			0
$223$	−1.00000	+	1.73205i	−0.0669650	+	0.115987i	−0.897564	−	0.440884i	$-0.854665\pi$
$223$	−1.00000	+	1.73205i	−0.0669650	+	0.115987i	0.830599	+	0.556871i	$0.187998\pi$
$224$	16.0000	+	16.0000i	1.06904	+	1.06904i
$225$		0			0
$226$	6.00000	+	6.00000i	0.399114	+	0.399114i
$227$	6.06218	+	3.50000i	0.402361	+	0.232303i	0.687502	−	0.726182i	$-0.258707\pi$
$227$	6.06218	+	3.50000i	0.402361	+	0.232303i	−0.285141	+	0.958485i	$0.592041\pi$
$228$		0			0
$229$	−17.3205	+	10.0000i	−1.14457	+	0.660819i	−0.947559	−	0.319582i	$-0.896457\pi$
$229$	−17.3205	+	10.0000i	−1.14457	+	0.660819i	−0.197013	+	0.980401i	$0.563124\pi$
$230$	5.46410	+	1.46410i	0.360292	+	0.0965400i
$231$		0			0
$232$		0			0
$233$	−13.0000			−0.851658			−0.425829	−	0.904804i	$-0.640018\pi$
$233$	−13.0000			−0.851658			−0.425829	+	0.904804i	$0.640018\pi$
$234$		0			0
$235$			12.0000i			0.782794i
$236$	1.00000	−	1.73205i	0.0650945	−	0.112747i
$237$		0			0
$238$	7.32051	−	27.3205i	0.474518	−	1.77093i
$239$	−15.0000	−	25.9808i	−0.970269	−	1.68056i	−0.694737	−	0.719264i	$-0.744479\pi$
$239$	−15.0000	−	25.9808i	−0.970269	−	1.68056i	−0.275533	−	0.961292i	$-0.588854\pi$
$240$		0			0
$241$	−8.50000	+	14.7224i	−0.547533	+	0.948355i	0.450910	+	0.892570i	$0.351100\pi$
$241$	−8.50000	+	14.7224i	−0.547533	+	0.948355i	−0.998443	+	0.0557856i	$0.982234\pi$
$242$	−2.00000	−	2.00000i	−0.128565	−	0.128565i
$243$		0			0
$244$	24.0000			1.53644
$245$	−15.5885	−	9.00000i	−0.995910	−	0.574989i
$246$		0			0
$247$	1.00000	+	1.73205i	0.0636285	+	0.110208i
$248$	10.9282	+	2.92820i	0.693942	+	0.185941i
$249$		0			0
$250$	4.39230	+	16.3923i	0.277794	+	1.03674i
$251$			15.0000i			0.946792i	0.880850	+	0.473396i	$0.156972\pi$
$251$			15.0000i			0.946792i	−0.880850	+	0.473396i	$0.843028\pi$
$252$		0			0
$253$		−	6.00000i		−	0.377217i
$254$	−2.73205	+	0.732051i	−0.171424	+	0.0459330i
$255$		0			0
$256$	−8.00000	−	13.8564i	−0.500000	−	0.866025i
$257$	7.50000	+	12.9904i	0.467837	+	0.810318i	0.999325	−	0.0367485i	$-0.0117000\pi$
$257$	7.50000	+	12.9904i	0.467837	+	0.810318i	−0.531487	+	0.847066i	$0.678367\pi$
$258$		0			0
$259$	−6.92820	−	4.00000i	−0.430498	−	0.248548i
$260$		−	8.00000i		−	0.496139i
$261$		0			0
$262$	4.00000	−	4.00000i	0.247121	−	0.247121i
$263$	−4.00000	+	6.92820i	−0.246651	+	0.427211i	−0.962594	−	0.270947i	$-0.912663\pi$
$263$	−4.00000	+	6.92820i	−0.246651	+	0.427211i	0.715944	+	0.698158i	$0.245997\pi$
$264$		0			0
$265$		0			0
$266$	−5.46410	−	1.46410i	−0.335026	−	0.0897698i
$267$		0			0
$268$	3.00000	−	5.19615i	0.183254	−	0.317406i
$269$		−	12.0000i		−	0.731653i	−0.930683	−	0.365826i	$-0.880786\pi$
$269$		−	12.0000i		−	0.731653i	0.930683	−	0.365826i	$-0.119214\pi$
$270$		0			0
$271$	12.0000			0.728948			0.364474	−	0.931214i	$-0.381249\pi$
$271$	12.0000			0.728948			0.364474	+	0.931214i	$0.381249\pi$
$272$	−10.0000	+	17.3205i	−0.606339	+	1.05021i
$273$		0			0
$274$	−3.29423	+	12.2942i	−0.199012	+	0.742722i
$275$	2.59808	−	1.50000i	0.156670	−	0.0904534i
$276$		0			0
$277$	1.73205	+	1.00000i	0.104069	+	0.0600842i	0.551131	−	0.834419i	$-0.314196\pi$
$277$	1.73205	+	1.00000i	0.104069	+	0.0600842i	−0.447062	+	0.894503i	$0.647530\pi$
$278$	13.0000	−	13.0000i	0.779688	−	0.779688i
$279$		0			0
$280$	16.0000	+	16.0000i	0.956183	+	0.956183i
$281$	−3.00000	+	5.19615i	−0.178965	+	0.309976i	−0.941526	−	0.336939i	$-0.890608\pi$
$281$	−3.00000	+	5.19615i	−0.178965	+	0.309976i	0.762561	+	0.646916i	$0.223942\pi$
$282$		0			0
$283$	17.3205	−	10.0000i	1.02960	−	0.594438i	0.112728	−	0.993626i	$-0.464041\pi$
$283$	17.3205	−	10.0000i	1.02960	−	0.594438i	0.916869	+	0.399188i	$0.130708\pi$
$284$	10.3923	−	6.00000i	0.616670	−	0.356034i
$285$		0			0
$286$	−8.19615	+	2.19615i	−0.484649	+	0.129861i
$287$	20.0000			1.18056
$288$		0			0
$289$	8.00000			0.470588
$290$		0			0
$291$		0			0
$292$	15.5885	−	9.00000i	0.912245	−	0.526685i
$293$	−10.3923	+	6.00000i	−0.607125	+	0.350524i	−0.771839	−	0.635818i	$-0.780663\pi$
$293$	−10.3923	+	6.00000i	−0.607125	+	0.350524i	0.164714	+	0.986341i	$0.447330\pi$
$294$		0			0
$295$	1.00000	−	1.73205i	0.0582223	−	0.100844i
$296$	4.00000	+	4.00000i	0.232495	+	0.232495i
$297$		0			0
$298$	−18.0000	+	18.0000i	−1.04271	+	1.04271i
$299$	3.46410	+	2.00000i	0.200334	+	0.115663i
$300$		0			0
$301$	38.1051	−	22.0000i	2.19634	−	1.26806i
$302$	2.19615	−	8.19615i	0.126374	−	0.471636i
$303$		0			0
$304$	3.46410	+	2.00000i	0.198680	+	0.114708i
$305$	24.0000			1.37424
$306$		0			0
$307$		−	9.00000i		−	0.513657i	−0.966457	−	0.256829i	$-0.917322\pi$
$307$		−	9.00000i		−	0.513657i	0.966457	−	0.256829i	$-0.0826776\pi$
$308$	12.0000	−	20.7846i	0.683763	−	1.18431i
$309$		0			0
$310$	10.9282	+	2.92820i	0.620680	+	0.166311i
$311$	10.0000	+	17.3205i	0.567048	+	0.982156i	0.996856	+	0.0792356i	$0.0252479\pi$
$311$	10.0000	+	17.3205i	0.567048	+	0.982156i	−0.429808	+	0.902920i	$0.641419\pi$
$312$		0			0
$313$	0.500000	−	0.866025i	0.0282617	−	0.0489506i	−0.851549	−	0.524276i	$-0.824336\pi$
$313$	0.500000	−	0.866025i	0.0282617	−	0.0489506i	0.879810	+	0.475325i	$0.157669\pi$
$314$	−4.00000	+	4.00000i	−0.225733	+	0.225733i
$315$		0			0
$316$		−	28.0000i		−	1.57512i
$317$	19.0526	+	11.0000i	1.07010	+	0.617822i	0.928208	−	0.372061i	$-0.121349\pi$
$317$	19.0526	+	11.0000i	1.07010	+	0.617822i	0.141890	+	0.989882i	$0.454682\pi$
$318$		0			0
$319$		0			0
$320$	−8.00000	−	13.8564i	−0.447214	−	0.774597i
$321$		0			0
$322$	−10.9282	+	2.92820i	−0.609005	+	0.163182i
$323$		−	5.00000i		−	0.278207i
$324$		0			0
$325$			2.00000i			0.110940i
$326$	−1.46410	−	5.46410i	−0.0810891	−	0.302629i
$327$		0			0
$328$	−13.6603	−	3.66025i	−0.754261	−	0.202104i
$329$	−12.0000	−	20.7846i	−0.661581	−	1.14589i
$330$		0			0
$331$	−17.3205	−	10.0000i	−0.952021	−	0.549650i	−0.0583130	−	0.998298i	$-0.518572\pi$
$331$	−17.3205	−	10.0000i	−0.952021	−	0.549650i	−0.893708	+	0.448649i	$0.851905\pi$
$332$	−8.00000			−0.439057
$333$		0			0
$334$	−2.00000	−	2.00000i	−0.109435	−	0.109435i
$335$	3.00000	−	5.19615i	0.163908	−	0.283896i
$336$		0			0
$337$	−3.50000	−	6.06218i	−0.190657	−	0.330228i	0.754811	−	0.655942i	$-0.227729\pi$
$337$	−3.50000	−	6.06218i	−0.190657	−	0.330228i	−0.945468	+	0.325714i	$0.894395\pi$
$338$	−3.29423	+	12.2942i	−0.179182	+	0.668718i
$339$		0			0
$340$	−10.0000	+	17.3205i	−0.542326	+	0.939336i
$341$		−	12.0000i		−	0.649836i
$342$		0			0
$343$	8.00000			0.431959
$344$	−30.0526	+	8.05256i	−1.62033	+	0.434165i
$345$		0			0
$346$	32.7846	+	8.78461i	1.76251	+	0.472264i
$347$	−11.2583	+	6.50000i	−0.604379	+	0.348938i	−0.770762	−	0.637123i	$-0.780124\pi$
$347$	−11.2583	+	6.50000i	−0.604379	+	0.348938i	0.166383	+	0.986061i	$0.446791\pi$
$348$		0			0
$349$	13.8564	+	8.00000i	0.741716	+	0.428230i	0.822693	−	0.568486i	$-0.192471\pi$
$349$	13.8564	+	8.00000i	0.741716	+	0.428230i	−0.0809766	+	0.996716i	$0.525804\pi$
$350$	−4.00000	−	4.00000i	−0.213809	−	0.213809i
$351$		0			0
$352$	−12.0000	+	12.0000i	−0.639602	+	0.639602i
$353$	7.50000	−	12.9904i	0.399185	−	0.691408i	−0.594441	−	0.804139i	$-0.702627\pi$
$353$	7.50000	−	12.9904i	0.399185	−	0.691408i	0.993626	+	0.112731i	$0.0359599\pi$
$354$		0			0
$355$	10.3923	−	6.00000i	0.551566	−	0.318447i
$356$	24.2487	−	14.0000i	1.28518	−	0.741999i
$357$		0			0
$358$	−7.32051	−	27.3205i	−0.386901	−	1.44393i
$359$	4.00000			0.211112			0.105556	−	0.994413i	$-0.466338\pi$
$359$	4.00000			0.211112			0.105556	+	0.994413i	$0.466338\pi$
$360$		0			0
$361$	18.0000			0.947368
$362$	3.66025	+	13.6603i	0.192379	+	0.717967i
$363$		0			0
$364$	8.00000	+	13.8564i	0.419314	+	0.726273i
$365$	15.5885	−	9.00000i	0.815937	−	0.471082i
$366$		0			0
$367$	−9.00000	+	15.5885i	−0.469796	+	0.813711i	−0.999404	−	0.0345320i	$-0.989006\pi$
$367$	−9.00000	+	15.5885i	−0.469796	+	0.813711i	0.529607	+	0.848243i	$0.322339\pi$
$368$	8.00000			0.417029
$369$		0			0
$370$	4.00000	+	4.00000i	0.207950	+	0.207950i
$371$		0			0
$372$		0			0
$373$	−22.5167	+	13.0000i	−1.16587	+	0.673114i	−0.952703	−	0.303902i	$-0.901711\pi$
$373$	−22.5167	+	13.0000i	−1.16587	+	0.673114i	−0.213165	+	0.977016i	$0.568377\pi$
$374$	20.4904	+	5.49038i	1.05953	+	0.283901i
$375$		0			0
$376$	4.39230	+	16.3923i	0.226516	+	0.845369i
$377$		0			0
$378$		0			0
$379$		−	5.00000i		−	0.256833i	−0.991720	−	0.128416i	$-0.959011\pi$
$379$		−	5.00000i		−	0.256833i	0.991720	−	0.128416i	$-0.0409894\pi$
$380$	3.46410	+	2.00000i	0.177705	+	0.102598i
$381$		0			0
$382$	−5.85641	+	21.8564i	−0.299640	+	1.11827i
$383$	9.00000	+	15.5885i	0.459879	+	0.796533i	0.998954	−	0.0457244i	$-0.0145596\pi$
$383$	9.00000	+	15.5885i	0.459879	+	0.796533i	−0.539076	+	0.842257i	$0.681226\pi$
$384$		0			0
$385$	12.0000	−	20.7846i	0.611577	−	1.05928i
$386$	15.0000	+	15.0000i	0.763480	+	0.763480i
$387$		0			0
$388$			2.00000i			0.101535i
$389$	6.92820	+	4.00000i	0.351274	+	0.202808i	0.665246	−	0.746624i	$-0.268327\pi$
$389$	6.92820	+	4.00000i	0.351274	+	0.202808i	−0.313972	+	0.949432i	$0.601660\pi$
$390$		0			0
$391$	−5.00000	−	8.66025i	−0.252861	−	0.437968i
$392$	−24.5885	−	6.58846i	−1.24190	−	0.332767i
$393$		0			0
$394$	−2.92820	−	10.9282i	−0.147521	−	0.550555i
$395$		−	28.0000i		−	1.40883i
$396$		0			0
$397$			22.0000i			1.10415i	0.833795	+	0.552074i	$0.186163\pi$
$397$			22.0000i			1.10415i	−0.833795	+	0.552074i	$0.813837\pi$
$398$	−10.9282	+	2.92820i	−0.547781	+	0.146778i
$399$		0			0
$400$	2.00000	+	3.46410i	0.100000	+	0.173205i
$401$	−7.50000	−	12.9904i	−0.374532	−	0.648709i	0.615725	−	0.787961i	$-0.288863\pi$
$401$	−7.50000	−	12.9904i	−0.374532	−	0.648709i	−0.990257	+	0.139253i	$0.955530\pi$
$402$		0			0
$403$	6.92820	+	4.00000i	0.345118	+	0.199254i
$404$	−28.0000			−1.39305
$405$		0			0
$406$		0			0
$407$	3.00000	−	5.19615i	0.148704	−	0.257564i
$408$		0			0
$409$	−15.5000	−	26.8468i	−0.766426	−	1.32749i	−0.939490	−	0.342578i	$-0.888700\pi$
$409$	−15.5000	−	26.8468i	−0.766426	−	1.32749i	0.173064	−	0.984911i	$-0.444633\pi$
$410$	−13.6603	−	3.66025i	−0.674632	−	0.180767i
$411$		0			0
$412$	10.3923	+	6.00000i	0.511992	+	0.295599i
$413$			4.00000i			0.196827i
$414$		0			0
$415$	−8.00000			−0.392705
$416$	−2.92820	−	10.9282i	−0.143567	−	0.535799i
$417$		0			0
$418$	1.09808	−	4.09808i	0.0537087	−	0.200443i
$419$	−31.1769	+	18.0000i	−1.52309	+	0.879358i	−0.523465	+	0.852047i	$0.675361\pi$
$419$	−31.1769	+	18.0000i	−1.52309	+	0.879358i	−0.999627	+	0.0273103i	$0.991306\pi$
$420$		0			0
$421$	−19.0526	−	11.0000i	−0.928565	−	0.536107i	−0.0422075	−	0.999109i	$-0.513439\pi$
$421$	−19.0526	−	11.0000i	−0.928565	−	0.536107i	−0.886357	+	0.463002i	$0.846772\pi$
$422$	−16.0000	+	16.0000i	−0.778868	+	0.778868i
$423$		0			0
$424$		0			0
$425$	2.50000	−	4.33013i	0.121268	−	0.210042i
$426$		0			0
$427$	−41.5692	+	24.0000i	−2.01168	+	1.16144i
$428$	−3.00000	−	5.19615i	−0.145010	−	0.251166i
$429$		0			0
$430$	−30.0526	+	8.05256i	−1.44926	+	0.388329i
$431$	36.0000			1.73406			0.867029	−	0.498257i	$-0.166026\pi$
$431$	36.0000			1.73406			0.867029	+	0.498257i	$0.166026\pi$
$432$		0			0
$433$	5.00000			0.240285			0.120142	−	0.992757i	$-0.461665\pi$
$433$	5.00000			0.240285			0.120142	+	0.992757i	$0.461665\pi$
$434$	−21.8564	+	5.85641i	−1.04914	+	0.281117i
$435$		0			0
$436$	−20.0000	−	34.6410i	−0.957826	−	1.65900i
$437$	−1.73205	+	1.00000i	−0.0828552	+	0.0478365i
$438$		0			0
$439$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$439$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$440$	−12.0000	+	12.0000i	−0.572078	+	0.572078i
$441$		0			0
$442$	−10.0000	+	10.0000i	−0.475651	+	0.475651i
$443$	−7.79423	−	4.50000i	−0.370315	−	0.213801i	0.303281	−	0.952901i	$-0.401918\pi$
$443$	−7.79423	−	4.50000i	−0.370315	−	0.213801i	−0.673596	+	0.739100i	$0.735251\pi$
$444$		0			0
$445$	24.2487	−	14.0000i	1.14950	−	0.663664i
$446$	−0.732051	+	2.73205i	−0.0346636	+	0.129366i
$447$		0			0
$448$	27.7128	+	16.0000i	1.30931	+	0.755929i
$449$	−21.0000			−0.991051			−0.495526	−	0.868593i	$-0.665025\pi$
$449$	−21.0000			−0.991051			−0.495526	+	0.868593i	$0.665025\pi$
$450$		0			0
$451$			15.0000i			0.706322i
$452$	10.3923	+	6.00000i	0.488813	+	0.282216i
$453$		0			0
$454$	9.56218	+	2.56218i	0.448775	+	0.120249i
$455$	8.00000	+	13.8564i	0.375046	+	0.649598i
$456$		0			0
$457$	18.5000	−	32.0429i	0.865393	−	1.49891i	−0.00126243	−	0.999999i	$-0.500402\pi$
$457$	18.5000	−	32.0429i	0.865393	−	1.49891i	0.866656	−	0.498906i	$-0.166265\pi$
$458$	−20.0000	+	20.0000i	−0.934539	+	0.934539i
$459$		0			0
$460$	8.00000			0.373002
$461$	−5.19615	−	3.00000i	−0.242009	−	0.139724i	0.374091	−	0.927392i	$-0.377955\pi$
$461$	−5.19615	−	3.00000i	−0.242009	−	0.139724i	−0.616100	+	0.787668i	$0.711288\pi$
$462$		0			0
$463$	−5.00000	−	8.66025i	−0.232370	−	0.402476i	0.726135	−	0.687552i	$-0.241315\pi$
$463$	−5.00000	−	8.66025i	−0.232370	−	0.402476i	−0.958505	+	0.285076i	$0.907981\pi$
$464$		0			0
$465$		0			0
$466$	−17.7583	+	4.75833i	−0.822639	+	0.220425i
$467$			29.0000i			1.34196i	0.741475	+	0.670980i	$0.234126\pi$
$467$			29.0000i			1.34196i	−0.741475	+	0.670980i	$0.765874\pi$
$468$		0			0
$469$			12.0000i			0.554109i
$470$	4.39230	+	16.3923i	0.202602	+	0.756121i
$471$		0			0
$472$	0.732051	−	2.73205i	0.0336954	−	0.125753i
$473$	16.5000	+	28.5788i	0.758671	+	1.31406i
$474$		0			0
$475$	−0.866025	−	0.500000i	−0.0397360	−	0.0229416i
$476$		−	40.0000i		−	1.83340i
$477$		0			0
$478$	−30.0000	−	30.0000i	−1.37217	−	1.37217i
$479$	8.00000	−	13.8564i	0.365529	−	0.633115i	−0.623332	−	0.781958i	$-0.714221\pi$
$479$	8.00000	−	13.8564i	0.365529	−	0.633115i	0.988861	+	0.148842i	$0.0475547\pi$
$480$		0			0
$481$	2.00000	+	3.46410i	0.0911922	+	0.157949i
$482$	−6.22243	+	23.2224i	−0.283424	+	1.05775i
$483$		0			0
$484$	−3.46410	−	2.00000i	−0.157459	−	0.0909091i
$485$			2.00000i			0.0908153i
$486$		0			0
$487$	20.0000			0.906287			0.453143	−	0.891438i	$-0.350303\pi$
$487$	20.0000			0.906287			0.453143	+	0.891438i	$0.350303\pi$
$488$	32.7846	−	8.78461i	1.48409	−	0.397661i
$489$		0			0
$490$	−24.5885	−	6.58846i	−1.11079	−	0.297636i
$491$	16.4545	−	9.50000i	0.742580	−	0.428729i	−0.0804264	−	0.996761i	$-0.525628\pi$
$491$	16.4545	−	9.50000i	0.742580	−	0.428729i	0.823007	+	0.568032i	$0.192295\pi$
$492$		0			0
$493$		0			0
$494$	2.00000	+	2.00000i	0.0899843	+	0.0899843i
$495$		0			0
$496$	16.0000			0.718421
$497$	−12.0000	+	20.7846i	−0.538274	+	0.932317i
$498$		0			0
$499$	−12.9904	+	7.50000i	−0.581529	+	0.335746i	−0.761741	−	0.647882i	$-0.775655\pi$
$499$	−12.9904	+	7.50000i	−0.581529	+	0.335746i	0.180212	+	0.983628i	$0.442322\pi$
$500$	12.0000	+	20.7846i	0.536656	+	0.929516i
$501$		0			0
$502$	5.49038	+	20.4904i	0.245048	+	0.914530i
$503$	−36.0000			−1.60516			−0.802580	−	0.596544i	$-0.796540\pi$
$503$	−36.0000			−1.60516			−0.802580	+	0.596544i	$0.796540\pi$
$504$		0			0
$505$	−28.0000			−1.24598
$506$	−2.19615	−	8.19615i	−0.0976309	−	0.364363i
$507$		0			0
$508$	−3.46410	+	2.00000i	−0.153695	+	0.0887357i
$509$	31.1769	−	18.0000i	1.38189	−	0.797836i	0.389509	−	0.921023i	$-0.372645\pi$
$509$	31.1769	−	18.0000i	1.38189	−	0.797836i	0.992384	+	0.123187i	$0.0393114\pi$
$510$		0			0
$511$	−18.0000	+	31.1769i	−0.796273	+	1.37919i
$512$	−16.0000	−	16.0000i	−0.707107	−	0.707107i
$513$		0			0
$514$	15.0000	+	15.0000i	0.661622	+	0.661622i
$515$	10.3923	+	6.00000i	0.457940	+	0.264392i
$516$		0			0
$517$	15.5885	−	9.00000i	0.685580	−	0.395820i
$518$	−10.9282	−	2.92820i	−0.480158	−	0.128658i
$519$		0			0
$520$	−2.92820	−	10.9282i	−0.128410	−	0.479233i
$521$	−15.0000			−0.657162			−0.328581	−	0.944476i	$-0.606570\pi$
$521$	−15.0000			−0.657162			−0.328581	+	0.944476i	$0.606570\pi$
$522$		0			0
$523$		−	36.0000i		−	1.57417i	−0.616844	−	0.787085i	$-0.711589\pi$
$523$		−	36.0000i		−	1.57417i	0.616844	−	0.787085i	$-0.288411\pi$
$524$	4.00000	−	6.92820i	0.174741	−	0.302660i
$525$		0			0
$526$	−2.92820	+	10.9282i	−0.127676	+	0.476492i
$527$	−10.0000	−	17.3205i	−0.435607	−	0.754493i
$528$		0			0
$529$	9.50000	−	16.4545i	0.413043	−	0.715412i
$530$		0			0
$531$		0			0
$532$	−8.00000			−0.346844
$533$	−8.66025	−	5.00000i	−0.375117	−	0.216574i
$534$		0			0
$535$	−3.00000	−	5.19615i	−0.129701	−	0.224649i
$536$	2.19615	−	8.19615i	0.0948593	−	0.354020i
$537$		0			0
$538$	−4.39230	−	16.3923i	−0.189366	−	0.706722i
$539$			27.0000i			1.16297i
$540$		0			0
$541$		−	22.0000i		−	0.945854i	−0.881102	−	0.472927i	$-0.843197\pi$
$541$		−	22.0000i		−	0.945854i	0.881102	−	0.472927i	$-0.156803\pi$
$542$	16.3923	−	4.39230i	0.704110	−	0.188666i
$543$		0			0
$544$	−7.32051	+	27.3205i	−0.313864	+	1.17136i
$545$	−20.0000	−	34.6410i	−0.856706	−	1.48386i
$546$		0			0
$547$	30.3109	+	17.5000i	1.29600	+	0.748246i	0.979711	−	0.200417i	$-0.0642296\pi$
$547$	30.3109	+	17.5000i	1.29600	+	0.748246i	0.316289	+	0.948663i	$0.397563\pi$
$548$			18.0000i			0.768922i
$549$		0			0
$550$	3.00000	−	3.00000i	0.127920	−	0.127920i
$551$		0			0
$552$		0			0
$553$	28.0000	+	48.4974i	1.19068	+	2.06232i
$554$	2.73205	+	0.732051i	0.116074	+	0.0311019i
$555$		0			0
$556$	13.0000	−	22.5167i	0.551323	−	0.954919i
$557$			24.0000i			1.01691i	0.861088	+	0.508456i	$0.169784\pi$
$557$			24.0000i			1.01691i	−0.861088	+	0.508456i	$0.830216\pi$
$558$		0			0
$559$	−22.0000			−0.930501
$560$	27.7128	+	16.0000i	1.17108	+	0.676123i
$561$		0			0
$562$	−2.19615	+	8.19615i	−0.0926391	+	0.345734i
$563$	7.79423	−	4.50000i	0.328488	−	0.189652i	−0.326682	−	0.945134i	$-0.605931\pi$
$563$	7.79423	−	4.50000i	0.328488	−	0.189652i	0.655169	+	0.755482i	$0.272597\pi$
$564$		0			0
$565$	10.3923	+	6.00000i	0.437208	+	0.252422i
$566$	20.0000	−	20.0000i	0.840663	−	0.840663i
$567$		0			0
$568$	12.0000	−	12.0000i	0.503509	−	0.503509i
$569$	19.5000	−	33.7750i	0.817483	−	1.41592i	−0.0900490	−	0.995937i	$-0.528702\pi$
$569$	19.5000	−	33.7750i	0.817483	−	1.41592i	0.907532	−	0.419984i	$-0.137964\pi$
$570$		0			0
$571$	−4.33013	+	2.50000i	−0.181210	+	0.104622i	−0.587861	−	0.808962i	$-0.700030\pi$
$571$	−4.33013	+	2.50000i	−0.181210	+	0.104622i	0.406651	+	0.913584i	$0.366697\pi$
$572$	−10.3923	+	6.00000i	−0.434524	+	0.250873i
$573$		0			0
$574$	27.3205	−	7.32051i	1.14034	−	0.305552i
$575$	−2.00000			−0.0834058
$576$		0			0
$577$	−33.0000			−1.37381			−0.686904	−	0.726748i	$-0.741031\pi$
$577$	−33.0000			−1.37381			−0.686904	+	0.726748i	$0.741031\pi$
$578$	10.9282	−	2.92820i	0.454553	−	0.121797i
$579$		0			0
$580$		0			0
$581$	13.8564	−	8.00000i	0.574861	−	0.331896i
$582$		0			0
$583$		0			0
$584$	18.0000	−	18.0000i	0.744845	−	0.744845i
$585$		0			0
$586$	−12.0000	+	12.0000i	−0.495715	+	0.495715i
$587$	−32.0429	−	18.5000i	−1.32255	−	0.763577i	−0.338418	−	0.940996i	$-0.609892\pi$
$587$	−32.0429	−	18.5000i	−1.32255	−	0.763577i	−0.984135	+	0.177419i	$0.943225\pi$
$588$		0			0
$589$	−3.46410	+	2.00000i	−0.142736	+	0.0824086i
$590$	0.732051	−	2.73205i	0.0301381	−	0.112477i
$591$		0			0
$592$	6.92820	+	4.00000i	0.284747	+	0.164399i
$593$	30.0000			1.23195			0.615976	−	0.787765i	$-0.288762\pi$
$593$	30.0000			1.23195			0.615976	+	0.787765i	$0.288762\pi$
$594$		0			0
$595$		−	40.0000i		−	1.63984i
$596$	−18.0000	+	31.1769i	−0.737309	+	1.27706i
$597$		0			0
$598$	5.46410	+	1.46410i	0.223444	+	0.0598716i
$599$	19.0000	+	32.9090i	0.776319	+	1.34462i	0.934050	+	0.357142i	$0.116249\pi$
$599$	19.0000	+	32.9090i	0.776319	+	1.34462i	−0.157731	+	0.987482i	$0.550418\pi$
$600$		0			0
$601$	−2.50000	+	4.33013i	−0.101977	+	0.176630i	−0.912499	−	0.409079i	$-0.865850\pi$
$601$	−2.50000	+	4.33013i	−0.101977	+	0.176630i	0.810522	+	0.585708i	$0.199184\pi$
$602$	44.0000	−	44.0000i	1.79331	−	1.79331i
$603$		0			0
$604$		−	12.0000i		−	0.488273i
$605$	−3.46410	−	2.00000i	−0.140836	−	0.0813116i
$606$		0			0
$607$	10.0000	+	17.3205i	0.405887	+	0.703018i	0.994424	−	0.105453i	$-0.0336291\pi$
$607$	10.0000	+	17.3205i	0.405887	+	0.703018i	−0.588537	+	0.808470i	$0.700296\pi$
$608$	5.46410	+	1.46410i	0.221599	+	0.0593772i
$609$		0			0
$610$	32.7846	−	8.78461i	1.32741	−	0.355678i
$611$			12.0000i			0.485468i
$612$		0			0
$613$			34.0000i			1.37325i	0.727013	+	0.686624i	$0.240908\pi$
$613$			34.0000i			1.37325i	−0.727013	+	0.686624i	$0.759092\pi$
$614$	−3.29423	−	12.2942i	−0.132944	−	0.496155i
$615$		0			0
$616$	8.78461	−	32.7846i	0.353942	−	1.32093i
$617$	2.50000	+	4.33013i	0.100646	+	0.174324i	0.911951	−	0.410299i	$-0.134576\pi$
$617$	2.50000	+	4.33013i	0.100646	+	0.174324i	−0.811305	+	0.584623i	$0.801242\pi$
$618$		0			0
$619$	−2.59808	−	1.50000i	−0.104425	−	0.0602901i	0.446878	−	0.894595i	$-0.352536\pi$
$619$	−2.59808	−	1.50000i	−0.104425	−	0.0602901i	−0.551303	+	0.834305i	$0.685869\pi$
$620$	16.0000			0.642575
$621$		0			0
$622$	20.0000	+	20.0000i	0.801927	+	0.801927i
$623$	−28.0000	+	48.4974i	−1.12180	+	1.94301i
$624$		0			0
$625$	9.50000	+	16.4545i	0.380000	+	0.658179i
$626$	0.366025	−	1.36603i	0.0146293	−	0.0545974i
$627$		0			0
$628$	−4.00000	+	6.92820i	−0.159617	+	0.276465i
$629$		−	10.0000i		−	0.398726i
$630$		0			0
$631$	34.0000			1.35352			0.676759	−	0.736204i	$-0.263384\pi$
$631$	34.0000			1.35352			0.676759	+	0.736204i	$0.263384\pi$
$632$	−10.2487	−	38.2487i	−0.407672	−	1.52145i
$633$		0			0
$634$	30.0526	+	8.05256i	1.19354	+	0.319808i
$635$	−3.46410	+	2.00000i	−0.137469	+	0.0793676i
$636$		0			0
$637$	−15.5885	−	9.00000i	−0.617637	−	0.356593i
$638$		0			0
$639$		0			0
$640$	−16.0000	−	16.0000i	−0.632456	−	0.632456i
$641$	−15.5000	+	26.8468i	−0.612213	+	1.06038i	0.378653	+	0.925539i	$0.376387\pi$
$641$	−15.5000	+	26.8468i	−0.612213	+	1.06038i	−0.990867	+	0.134846i	$0.956946\pi$
$642$		0			0
$643$	4.33013	−	2.50000i	0.170764	−	0.0985904i	−0.412182	−	0.911101i	$-0.635233\pi$
$643$	4.33013	−	2.50000i	0.170764	−	0.0985904i	0.582946	+	0.812511i	$0.301900\pi$
$644$	−13.8564	+	8.00000i	−0.546019	+	0.315244i
$645$		0			0
$646$	−1.83013	−	6.83013i	−0.0720054	−	0.268728i
$647$	−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
$649$	−3.00000			−0.117760
$650$	0.732051	+	2.73205i	0.0287134	+	0.107160i
$651$		0			0
$652$	−4.00000	−	6.92820i	−0.156652	−	0.271329i
$653$	−31.1769	+	18.0000i	−1.22005	+	0.704394i	−0.964928	−	0.262515i	$-0.915448\pi$
$653$	−31.1769	+	18.0000i	−1.22005	+	0.704394i	−0.255119	+	0.966910i	$0.582115\pi$
$654$		0			0
$655$	4.00000	−	6.92820i	0.156293	−	0.270707i
$656$	−20.0000			−0.780869
$657$		0			0
$658$	−24.0000	−	24.0000i	−0.935617	−	0.935617i
$659$	−10.3923	−	6.00000i	−0.404827	−	0.233727i	0.283738	−	0.958902i	$-0.408425\pi$
$659$	−10.3923	−	6.00000i	−0.404827	−	0.233727i	−0.688565	+	0.725175i	$0.741759\pi$
$660$		0			0
$661$	12.1244	−	7.00000i	0.471583	−	0.272268i	−0.245319	−	0.969442i	$-0.578893\pi$
$661$	12.1244	−	7.00000i	0.471583	−	0.272268i	0.716902	+	0.697174i	$0.245559\pi$
$662$	−27.3205	−	7.32051i	−1.06184	−	0.284520i
$663$		0			0
$664$	−10.9282	+	2.92820i	−0.424097	+	0.113636i
$665$	−8.00000			−0.310227
$666$		0			0
$667$		0			0
$668$	−3.46410	−	2.00000i	−0.134030	−	0.0773823i
$669$		0			0
$670$	2.19615	−	8.19615i	0.0848448	−	0.316645i
$671$	−18.0000	−	31.1769i	−0.694882	−	1.20357i
$672$		0			0
$673$	−13.0000	+	22.5167i	−0.501113	+	0.867953i	0.498886	+	0.866668i	$0.333743\pi$
$673$	−13.0000	+	22.5167i	−0.501113	+	0.867953i	−0.999999	+	0.00128586i	$0.999591\pi$
$674$	−7.00000	−	7.00000i	−0.269630	−	0.269630i
$675$		0			0
$676$			18.0000i			0.692308i
$677$	−41.5692	−	24.0000i	−1.59763	−	0.922395i	−0.991941	−	0.126697i	$-0.959562\pi$
$677$	−41.5692	−	24.0000i	−1.59763	−	0.922395i	−0.605693	−	0.795698i	$-0.707104\pi$
$678$		0			0
$679$	−2.00000	−	3.46410i	−0.0767530	−	0.132940i
$680$	−7.32051	+	27.3205i	−0.280729	+	1.04769i
$681$		0			0
$682$	−4.39230	−	16.3923i	−0.168190	−	0.627694i
$683$		−	35.0000i		−	1.33924i	−0.742705	−	0.669619i	$-0.766457\pi$
$683$		−	35.0000i		−	1.33924i	0.742705	−	0.669619i	$-0.233543\pi$
$684$		0			0
$685$			18.0000i			0.687745i
$686$	10.9282	−	2.92820i	0.417241	−	0.111799i
$687$		0			0
$688$	−38.1051	+	22.0000i	−1.45274	+	0.838742i
$689$		0			0
$690$		0			0
$691$	−24.2487	−	14.0000i	−0.922464	−	0.532585i	−0.0380440	−	0.999276i	$-0.512113\pi$
$691$	−24.2487	−	14.0000i	−0.922464	−	0.532585i	−0.884420	+	0.466691i	$0.845446\pi$
$692$	48.0000			1.82469
$693$		0			0
$694$	−13.0000	+	13.0000i	−0.493473	+	0.493473i
$695$	13.0000	−	22.5167i	0.493118	−	0.854106i
$696$		0			0
$697$	12.5000	+	21.6506i	0.473471	+	0.820076i
$698$	21.8564	+	5.85641i	0.827277	+	0.221668i
$699$		0			0
$700$	−6.92820	−	4.00000i	−0.261861	−	0.151186i
$701$		−	10.0000i		−	0.377695i	−0.982006	−	0.188847i	$-0.939525\pi$
$701$		−	10.0000i		−	0.377695i	0.982006	−	0.188847i	$-0.0604752\pi$
$702$		0			0
$703$	−2.00000			−0.0754314
$704$	−12.0000	+	20.7846i	−0.452267	+	0.783349i
$705$		0			0
$706$	5.49038	−	20.4904i	0.206633	−	0.771166i
$707$	48.4974	−	28.0000i	1.82393	−	1.05305i
$708$		0			0
$709$	6.92820	+	4.00000i	0.260194	+	0.150223i	0.624423	−	0.781086i	$-0.285334\pi$
$709$	6.92820	+	4.00000i	0.260194	+	0.150223i	−0.364229	+	0.931309i	$0.618667\pi$
$710$	12.0000	−	12.0000i	0.450352	−	0.450352i
$711$		0			0
$712$	28.0000	−	28.0000i	1.04934	−	1.04934i
$713$	−4.00000	+	6.92820i	−0.149801	+	0.259463i
$714$		0			0
$715$	−10.3923	+	6.00000i	−0.388650	+	0.224387i
$716$	−20.0000	−	34.6410i	−0.747435	−	1.29460i
$717$		0			0
$718$	5.46410	−	1.46410i	0.203918	−	0.0546398i
$719$	34.0000			1.26799			0.633993	−	0.773339i	$-0.281415\pi$
$719$	34.0000			1.26799			0.633993	+	0.773339i	$0.281415\pi$
$720$		0			0
$721$	−24.0000			−0.893807
$722$	24.5885	−	6.58846i	0.915088	−	0.245197i
$723$		0			0
$724$	10.0000	+	17.3205i	0.371647	+	0.643712i
$725$		0			0
$726$		0			0
$727$	−3.00000	+	5.19615i	−0.111264	+	0.192715i	−0.916280	−	0.400538i	$-0.868823\pi$
$727$	−3.00000	+	5.19615i	−0.111264	+	0.192715i	0.805016	+	0.593253i	$0.202157\pi$
$728$	16.0000	+	16.0000i	0.592999	+	0.592999i
$729$		0			0
$730$	18.0000	−	18.0000i	0.666210	−	0.666210i
$731$	47.6314	+	27.5000i	1.76171	+	1.01712i
$732$		0			0
$733$	−24.2487	+	14.0000i	−0.895647	+	0.517102i	−0.875785	−	0.482701i	$-0.839656\pi$
$733$	−24.2487	+	14.0000i	−0.895647	+	0.517102i	−0.0198613	+	0.999803i	$0.506322\pi$
$734$	−6.58846	+	24.5885i	−0.243184	+	0.907577i
$735$		0			0
$736$	10.9282	−	2.92820i	0.402819	−	0.107935i
$737$	−9.00000			−0.331519
$738$		0			0
$739$			41.0000i			1.50821i	0.656754	+	0.754105i	$0.271929\pi$
$739$			41.0000i			1.50821i	−0.656754	+	0.754105i	$0.728071\pi$
$740$	6.92820	+	4.00000i	0.254686	+	0.147043i
$741$		0			0
$742$		0			0
$743$	−22.0000	−	38.1051i	−0.807102	−	1.39794i	−0.914863	−	0.403764i	$-0.867702\pi$
$743$	−22.0000	−	38.1051i	−0.807102	−	1.39794i	0.107761	−	0.994177i	$-0.465632\pi$
$744$		0			0
$745$	−18.0000	+	31.1769i	−0.659469	+	1.14223i
$746$	−26.0000	+	26.0000i	−0.951928	+	0.951928i
$747$		0			0
$748$	30.0000			1.09691
$749$	10.3923	+	6.00000i	0.379727	+	0.219235i
$750$		0			0
$751$	−1.00000	−	1.73205i	−0.0364905	−	0.0632034i	0.847203	−	0.531269i	$-0.178285\pi$
$751$	−1.00000	−	1.73205i	−0.0364905	−	0.0632034i	−0.883694	+	0.468065i	$0.844951\pi$
$752$	12.0000	+	20.7846i	0.437595	+	0.757937i
$753$		0			0
$754$		0			0
$755$		−	12.0000i		−	0.436725i
$756$		0			0
$757$		−	36.0000i		−	1.30844i	−0.756303	−	0.654221i	$-0.772997\pi$
$757$		−	36.0000i		−	1.30844i	0.756303	−	0.654221i	$-0.227003\pi$
$758$	−1.83013	−	6.83013i	−0.0664732	−	0.248081i
$759$		0			0
$760$	5.46410	+	1.46410i	0.198204	+	0.0531085i
$761$	7.00000	+	12.1244i	0.253750	+	0.439508i	0.964555	−	0.263881i	$-0.0850027\pi$
$761$	7.00000	+	12.1244i	0.253750	+	0.439508i	−0.710805	+	0.703389i	$0.751669\pi$
$762$		0			0
$763$	69.2820	+	40.0000i	2.50818	+	1.44810i
$764$			32.0000i			1.15772i
$765$		0			0
$766$	18.0000	+	18.0000i	0.650366	+	0.650366i
$767$	1.00000	−	1.73205i	0.0361079	−	0.0625407i
$768$		0			0
$769$	−17.0000	−	29.4449i	−0.613036	−	1.06181i	−0.990726	−	0.135877i	$-0.956615\pi$
$769$	−17.0000	−	29.4449i	−0.613036	−	1.06181i	0.377690	−	0.925932i	$-0.376718\pi$
$770$	8.78461	−	32.7846i	0.316575	−	1.18148i
$771$		0			0
$772$	25.9808	+	15.0000i	0.935068	+	0.539862i
$773$			38.0000i			1.36677i	0.730061	+	0.683383i	$0.239492\pi$
$773$			38.0000i			1.36677i	−0.730061	+	0.683383i	$0.760508\pi$
$774$		0			0
$775$	−4.00000			−0.143684
$776$	0.732051	+	2.73205i	0.0262791	+	0.0980749i
$777$		0			0
$778$	10.9282	+	2.92820i	0.391795	+	0.104981i
$779$	4.33013	−	2.50000i	0.155143	−	0.0895718i
$780$		0			0
$781$	−15.5885	−	9.00000i	−0.557799	−	0.322045i
$782$	−10.0000	−	10.0000i	−0.357599	−	0.357599i
$783$		0			0
$784$	−36.0000			−1.28571
$785$	−4.00000	+	6.92820i	−0.142766	+	0.247278i
$786$		0			0
$787$	10.3923	−	6.00000i	0.370446	−	0.213877i	−0.303207	−	0.952925i	$-0.598058\pi$
$787$	10.3923	−	6.00000i	0.370446	−	0.213877i	0.673653	+	0.739048i	$0.264724\pi$
$788$	−8.00000	−	13.8564i	−0.284988	−	0.493614i
$789$		0			0
$790$	−10.2487	−	38.2487i	−0.364633	−	1.36083i
$791$	−24.0000			−0.853342
$792$		0			0
$793$	24.0000			0.852265
$794$	8.05256	+	30.0526i	0.285775	+	1.06653i
$795$		0			0
$796$	−13.8564	+	8.00000i	−0.491127	+	0.283552i
$797$	19.0526	−	11.0000i	0.674876	−	0.389640i	−0.123045	−	0.992401i	$-0.539266\pi$
$797$	19.0526	−	11.0000i	0.674876	−	0.389640i	0.797922	+	0.602761i	$0.205933\pi$
$798$		0			0
$799$	15.0000	−	25.9808i	0.530662	−	0.919133i
$800$	4.00000	+	4.00000i	0.141421	+	0.141421i
$801$		0			0
$802$	−15.0000	−	15.0000i	−0.529668	−	0.529668i
$803$	−23.3827	−	13.5000i	−0.825157	−	0.476405i
$804$		0			0
$805$	−13.8564	+	8.00000i	−0.488374	+	0.281963i
$806$	10.9282	+	2.92820i	0.384930	+	0.103142i
$807$		0			0
$808$	−38.2487	+	10.2487i	−1.34558	+	0.360548i
$809$	9.00000			0.316423			0.158212	−	0.987405i	$-0.449427\pi$
$809$	9.00000			0.316423			0.158212	+	0.987405i	$0.449427\pi$
$810$		0			0
$811$		−	7.00000i		−	0.245803i	−0.992419	−	0.122902i	$-0.960780\pi$
$811$		−	7.00000i		−	0.245803i	0.992419	−	0.122902i	$-0.0392200\pi$
$812$		0			0
$813$		0			0
$814$	2.19615	−	8.19615i	0.0769751	−	0.287275i
$815$	−4.00000	−	6.92820i	−0.140114	−	0.242684i
$816$		0			0
$817$	5.50000	−	9.52628i	0.192421	−	0.333282i
$818$	−31.0000	−	31.0000i	−1.08389	−	1.08389i
$819$		0			0
$820$	−20.0000			−0.698430
$821$	24.2487	+	14.0000i	0.846286	+	0.488603i	0.859396	−	0.511311i	$-0.170840\pi$
$821$	24.2487	+	14.0000i	0.846286	+	0.488603i	−0.0131101	+	0.999914i	$0.504173\pi$
$822$		0			0
$823$	−13.0000	−	22.5167i	−0.453152	−	0.784881i	0.545428	−	0.838157i	$-0.316367\pi$
$823$	−13.0000	−	22.5167i	−0.453152	−	0.784881i	−0.998580	+	0.0532760i	$0.983034\pi$
$824$	16.3923	+	4.39230i	0.571053	+	0.153013i
$825$		0			0
$826$	1.46410	+	5.46410i	0.0509426	+	0.190120i
$827$			48.0000i			1.66912i	0.550914	+	0.834562i	$0.314279\pi$
$827$			48.0000i			1.66912i	−0.550914	+	0.834562i	$0.685721\pi$
$828$		0			0
$829$		−	32.0000i		−	1.11141i	−0.831381	−	0.555703i	$-0.812449\pi$
$829$		−	32.0000i		−	1.11141i	0.831381	−	0.555703i	$-0.187551\pi$
$830$	−10.9282	+	2.92820i	−0.379323	+	0.101639i
$831$		0			0
$832$	−8.00000	−	13.8564i	−0.277350	−	0.480384i
$833$	22.5000	+	38.9711i	0.779579	+	1.35027i
$834$		0			0
$835$	−3.46410	−	2.00000i	−0.119880	−	0.0692129i
$836$		−	6.00000i		−	0.207514i
$837$		0			0
$838$	−36.0000	+	36.0000i	−1.24360	+	1.24360i
$839$	2.00000	−	3.46410i	0.0690477	−	0.119594i	−0.829435	−	0.558604i	$-0.811337\pi$
$839$	2.00000	−	3.46410i	0.0690477	−	0.119594i	0.898482	+	0.439010i	$0.144671\pi$
$840$		0			0
$841$	−14.5000	−	25.1147i	−0.500000	−	0.866025i
$842$	−30.0526	−	8.05256i	−1.03568	−	0.277510i
$843$		0			0
$844$	−16.0000	+	27.7128i	−0.550743	+	0.953914i
$845$			18.0000i			0.619219i
$846$		0			0
$847$	8.00000			0.274883
$848$		0			0
$849$		0			0
$850$	1.83013	−	6.83013i	0.0627728	−	0.234271i
$851$	−3.46410	+	2.00000i	−0.118748	+	0.0685591i
$852$		0			0
$853$	−36.3731	−	21.0000i	−1.24539	−	0.719026i	−0.275204	−	0.961386i	$-0.588745\pi$
$853$	−36.3731	−	21.0000i	−1.24539	−	0.719026i	−0.970186	+	0.242360i	$0.922079\pi$
$854$	−48.0000	+	48.0000i	−1.64253	+	1.64253i
$855$		0			0
$856$	−6.00000	−	6.00000i	−0.205076	−	0.205076i
$857$	3.00000	−	5.19615i	0.102478	−	0.177497i	−0.810227	−	0.586116i	$-0.800656\pi$
$857$	3.00000	−	5.19615i	0.102478	−	0.177497i	0.912705	+	0.408619i	$0.133990\pi$
$858$		0			0
$859$	30.3109	−	17.5000i	1.03419	−	0.597092i	0.116011	−	0.993248i	$-0.462989\pi$
$859$	30.3109	−	17.5000i	1.03419	−	0.597092i	0.918183	+	0.396156i	$0.129656\pi$
$860$	−38.1051	+	22.0000i	−1.29937	+	0.750194i
$861$		0			0
$862$	49.1769	−	13.1769i	1.67497	−	0.448807i
$863$	−6.00000			−0.204242			−0.102121	−	0.994772i	$-0.532563\pi$
$863$	−6.00000			−0.204242			−0.102121	+	0.994772i	$0.532563\pi$
$864$		0			0
$865$	48.0000			1.63205
$866$	6.83013	−	1.83013i	0.232097	−	0.0621902i
$867$		0			0
$868$	−27.7128	+	16.0000i	−0.940634	+	0.543075i
$869$	−36.3731	+	21.0000i	−1.23387	+	0.712376i
$870$		0			0
$871$	3.00000	−	5.19615i	0.101651	−	0.176065i
$872$	−40.0000	−	40.0000i	−1.35457	−	1.35457i
$873$		0			0
$874$	−2.00000	+	2.00000i	−0.0676510	+	0.0676510i
$875$	−41.5692	−	24.0000i	−1.40530	−	0.811348i
$876$		0			0
$877$	−6.92820	+	4.00000i	−0.233949	+	0.135070i	−0.612392	−	0.790554i	$-0.709793\pi$
$877$	−6.92820	+	4.00000i	−0.233949	+	0.135070i	0.378444	+	0.925624i	$0.376459\pi$
$878$		0			0
$879$		0			0
$880$	−12.0000	+	20.7846i	−0.404520	+	0.700649i
$881$	2.00000			0.0673817			0.0336909	−	0.999432i	$-0.489274\pi$
$881$	2.00000			0.0673817			0.0336909	+	0.999432i	$0.489274\pi$
$882$		0			0
$883$			23.0000i			0.774012i	0.922077	+	0.387006i	$0.126491\pi$
$883$			23.0000i			0.774012i	−0.922077	+	0.387006i	$0.873509\pi$
$884$	−10.0000	+	17.3205i	−0.336336	+	0.582552i
$885$		0			0
$886$	−12.2942	−	3.29423i	−0.413033	−	0.110672i
$887$	15.0000	+	25.9808i	0.503651	+	0.872349i	0.999991	+	0.00422062i	$0.00134347\pi$
$887$	15.0000	+	25.9808i	0.503651	+	0.872349i	−0.496340	+	0.868128i	$0.665323\pi$
$888$		0			0
$889$	4.00000	−	6.92820i	0.134156	−	0.232364i
$890$	28.0000	−	28.0000i	0.938562	−	0.938562i
$891$		0			0
$892$			4.00000i			0.133930i
$893$	−5.19615	−	3.00000i	−0.173883	−	0.100391i
$894$		0			0
$895$	−20.0000	−	34.6410i	−0.668526	−	1.15792i
$896$	43.7128	+	11.7128i	1.46034	+	0.391298i
$897$		0			0
$898$	−28.6865	+	7.68653i	−0.957282	+	0.256503i
$899$		0			0
$900$		0			0
$901$		0			0
$902$	5.49038	+	20.4904i	0.182810	+	0.682255i
$903$		0			0
$904$	16.3923	+	4.39230i	0.545200	+	0.146086i
$905$	10.0000	+	17.3205i	0.332411	+	0.575753i
$906$		0			0
$907$	−21.6506	−	12.5000i	−0.718898	−	0.415056i	0.0954492	−	0.995434i	$-0.469571\pi$
$907$	−21.6506	−	12.5000i	−0.718898	−	0.415056i	−0.814347	+	0.580379i	$0.802905\pi$
$908$	14.0000			0.464606
$909$		0			0
$910$	16.0000	+	16.0000i	0.530395	+	0.530395i
$911$	6.00000	−	10.3923i	0.198789	−	0.344312i	−0.749347	−	0.662177i	$-0.769633\pi$
$911$	6.00000	−	10.3923i	0.198789	−	0.344312i	0.948136	+	0.317865i	$0.102966\pi$
$912$		0			0
$913$	6.00000	+	10.3923i	0.198571	+	0.343935i
$914$	13.5429	−	50.5429i	0.447961	−	1.67181i
$915$		0			0
$916$	−20.0000	+	34.6410i	−0.660819	+	1.14457i
$917$			16.0000i			0.528367i
$918$		0			0
$919$	−42.0000			−1.38545			−0.692726	−	0.721201i	$-0.743591\pi$
$919$	−42.0000			−1.38545			−0.692726	+	0.721201i	$0.743591\pi$
$920$	10.9282	−	2.92820i	0.360292	−	0.0965400i
$921$		0			0
$922$	−8.19615	−	2.19615i	−0.269926	−	0.0723264i
$923$	10.3923	−	6.00000i	0.342067	−	0.197492i
$924$		0			0
$925$	−1.73205	−	1.00000i	−0.0569495	−	0.0328798i
$926$	−10.0000	−	10.0000i	−0.328620	−	0.328620i
$927$		0			0
$928$		0			0
$929$	7.00000	−	12.1244i	0.229663	−	0.397787i	−0.728046	−	0.685529i	$-0.759571\pi$
$929$	7.00000	−	12.1244i	0.229663	−	0.397787i	0.957708	+	0.287742i	$0.0929044\pi$
$930$		0			0
$931$	7.79423	−	4.50000i	0.255446	−	0.147482i
$932$	−22.5167	+	13.0000i	−0.737558	+	0.425829i
$933$		0			0
$934$	10.6147	+	39.6147i	0.347325	+	1.29623i
$935$	30.0000			0.981105
$936$		0			0
$937$	2.00000			0.0653372			0.0326686	−	0.999466i	$-0.489599\pi$
$937$	2.00000			0.0653372			0.0326686	+	0.999466i	$0.489599\pi$
$938$	4.39230	+	16.3923i	0.143414	+	0.535228i
$939$		0			0
$940$	12.0000	+	20.7846i	0.391397	+	0.677919i
$941$	−20.7846	+	12.0000i	−0.677559	+	0.391189i	−0.798935	−	0.601418i	$-0.794603\pi$
$941$	−20.7846	+	12.0000i	−0.677559	+	0.391189i	0.121376	+	0.992607i	$0.461269\pi$
$942$		0			0
$943$	5.00000	−	8.66025i	0.162822	−	0.282017i
$944$		−	4.00000i		−	0.130189i
$945$		0			0
$946$	33.0000	+	33.0000i	1.07292	+	1.07292i
$947$	−2.59808	−	1.50000i	−0.0844261	−	0.0487435i	0.457193	−	0.889368i	$-0.348855\pi$
$947$	−2.59808	−	1.50000i	−0.0844261	−	0.0487435i	−0.541619	+	0.840624i	$0.682188\pi$
$948$		0			0
$949$	15.5885	−	9.00000i	0.506023	−	0.292152i
$950$	−1.36603	−	0.366025i	−0.0443197	−	0.0118754i
$951$		0			0
$952$	−14.6410	−	54.6410i	−0.474518	−	1.77093i
$953$	21.0000			0.680257			0.340128	−	0.940379i	$-0.389529\pi$
$953$	21.0000			0.680257			0.340128	+	0.940379i	$0.389529\pi$
$954$		0			0
$955$			32.0000i			1.03550i
$956$	−51.9615	−	30.0000i	−1.68056	−	0.970269i
$957$		0			0
$958$	5.85641	−	21.8564i	0.189212	−	0.706148i
$959$	−18.0000	−	31.1769i	−0.581250	−	1.00676i
$960$		0			0
$961$	7.50000	−	12.9904i	0.241935	−	0.419045i
$962$	4.00000	+	4.00000i	0.128965	+	0.128965i
$963$		0			0
$964$			34.0000i			1.09507i
$965$	25.9808	+	15.0000i	0.836350	+	0.482867i
$966$		0			0
$967$	2.00000	+	3.46410i	0.0643157	+	0.111398i	0.896390	−	0.443266i	$-0.146180\pi$
$967$	2.00000	+	3.46410i	0.0643157	+	0.111398i	−0.832075	+	0.554664i	$0.812847\pi$
$968$	−5.46410	−	1.46410i	−0.175623	−	0.0470580i
$969$		0			0
$970$	0.732051	+	2.73205i	0.0235047	+	0.0877209i
$971$		−	12.0000i		−	0.385098i	−0.981287	−	0.192549i	$-0.938325\pi$
$971$		−	12.0000i		−	0.385098i	0.981287	−	0.192549i	$-0.0616755\pi$
$972$		0			0
$973$			52.0000i			1.66704i
$974$	27.3205	−	7.32051i	0.875406	−	0.234564i
$975$		0			0
$976$	41.5692	−	24.0000i	1.33060	−	0.768221i
$977$	−15.5000	−	26.8468i	−0.495889	−	0.858905i	0.504100	−	0.863645i	$-0.331824\pi$
$977$	−15.5000	−	26.8468i	−0.495889	−	0.858905i	−0.999989	+	0.00474056i	$0.998491\pi$
$978$		0			0
$979$	−36.3731	−	21.0000i	−1.16249	−	0.671163i
$980$	−36.0000			−1.14998
$981$		0			0
$982$	19.0000	−	19.0000i	0.606314	−	0.606314i
$983$	23.0000	−	39.8372i	0.733586	−	1.27061i	−0.221755	−	0.975102i	$-0.571178\pi$
$983$	23.0000	−	39.8372i	0.733586	−	1.27061i	0.955341	−	0.295506i	$-0.0954882\pi$
$984$		0			0
$985$	−8.00000	−	13.8564i	−0.254901	−	0.441502i
$986$		0			0
$987$		0			0
$988$	3.46410	+	2.00000i	0.110208	+	0.0636285i
$989$		−	22.0000i		−	0.699559i
$990$		0			0
$991$	56.0000			1.77890			0.889449	−	0.457034i	$-0.151088\pi$
$991$	56.0000			1.77890			0.889449	+	0.457034i	$0.151088\pi$
$992$	21.8564	−	5.85641i	0.693942	−	0.185941i
$993$		0			0
$994$	−8.78461	+	32.7846i	−0.278631	+	1.03986i
$995$	−13.8564	+	8.00000i	−0.439278	+	0.253617i
$996$		0			0
$997$	10.3923	+	6.00000i	0.329128	+	0.190022i	0.655454	−	0.755235i	$-0.272477\pi$
$997$	10.3923	+	6.00000i	0.329128	+	0.190022i	−0.326326	+	0.945257i	$0.605811\pi$
$998$	−15.0000	+	15.0000i	−0.474817	+	0.474817i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	216.2.n.a.181.2		4
3.2	odd	2		72.2.n.a.61.1	yes	4
4.3	odd	2		864.2.r.a.721.2		4
8.3	odd	2		864.2.r.a.721.1		4
8.5	even	2	inner	216.2.n.a.181.1		4
9.2	odd	6		648.2.d.d.325.2		2
9.4	even	3	inner	216.2.n.a.37.1		4
9.5	odd	6		72.2.n.a.13.2	yes	4
9.7	even	3		648.2.d.a.325.1		2
12.11	even	2		288.2.r.a.241.2		4
24.5	odd	2		72.2.n.a.61.2	yes	4
24.11	even	2		288.2.r.a.241.1		4
36.7	odd	6		2592.2.d.a.1297.2		2
36.11	even	6		2592.2.d.b.1297.1		2
36.23	even	6		288.2.r.a.49.1		4
36.31	odd	6		864.2.r.a.145.1		4
72.5	odd	6		72.2.n.a.13.1	✓	4
72.11	even	6		2592.2.d.b.1297.2		2
72.13	even	6	inner	216.2.n.a.37.2		4
72.29	odd	6		648.2.d.d.325.1		2
72.43	odd	6		2592.2.d.a.1297.1		2
72.59	even	6		288.2.r.a.49.2		4
72.61	even	6		648.2.d.a.325.2		2
72.67	odd	6		864.2.r.a.145.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
72.2.n.a.13.1	✓	4	72.5	odd	6
72.2.n.a.13.2	yes	4	9.5	odd	6
72.2.n.a.61.1	yes	4	3.2	odd	2
72.2.n.a.61.2	yes	4	24.5	odd	2
216.2.n.a.37.1		4	9.4	even	3	inner
216.2.n.a.37.2		4	72.13	even	6	inner
216.2.n.a.181.1		4	8.5	even	2	inner
216.2.n.a.181.2		4	1.1	even	1	trivial
288.2.r.a.49.1		4	36.23	even	6
288.2.r.a.49.2		4	72.59	even	6
288.2.r.a.241.1		4	24.11	even	2
288.2.r.a.241.2		4	12.11	even	2
648.2.d.a.325.1		2	9.7	even	3
648.2.d.a.325.2		2	72.61	even	6
648.2.d.d.325.1		2	72.29	odd	6
648.2.d.d.325.2		2	9.2	odd	6
864.2.r.a.145.1		4	36.31	odd	6
864.2.r.a.145.2		4	72.67	odd	6
864.2.r.a.721.1		4	8.3	odd	2
864.2.r.a.721.2		4	4.3	odd	2
2592.2.d.a.1297.1		2	72.43	odd	6
2592.2.d.a.1297.2		2	36.7	odd	6
2592.2.d.b.1297.1		2	36.11	even	6
2592.2.d.b.1297.2		2	72.11	even	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:	\( N \)	\(=\)	\( 216 = 2^{3} \cdot 3^{3} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	216.n (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(1.36603 - 0.366025i) q^{2} +(1.73205 - 1.00000i) q^{4} +(1.73205 - 1.00000i) q^{5} +(-2.00000 + 3.46410i) q^{7} +(2.00000 - 2.00000i) q^{8} +O(q^{10})\)	\(q+(1.36603 - 0.366025i) q^{2} +(1.73205 - 1.00000i) q^{4} +(1.73205 - 1.00000i) q^{5} +(-2.00000 + 3.46410i) q^{7} +(2.00000 - 2.00000i) q^{8} +(2.00000 - 2.00000i) q^{10} +(-2.59808 - 1.50000i) q^{11} +(1.73205 - 1.00000i) q^{13} +(-1.46410 + 5.46410i) q^{14} +(2.00000 - 3.46410i) q^{16} -5.00000 q^{17} +1.00000i q^{19} +(2.00000 - 3.46410i) q^{20} +(-4.09808 - 1.09808i) q^{22} +(1.00000 + 1.73205i) q^{23} +(-0.500000 + 0.866025i) q^{25} +(2.00000 - 2.00000i) q^{26} +8.00000i q^{28} +(2.00000 + 3.46410i) q^{31} +(1.46410 - 5.46410i) q^{32} +(-6.83013 + 1.83013i) q^{34} +8.00000i q^{35} +2.00000i q^{37} +(0.366025 + 1.36603i) q^{38} +(1.46410 - 5.46410i) q^{40} +(-2.50000 - 4.33013i) q^{41} +(-9.52628 - 5.50000i) q^{43} -6.00000 q^{44} +(2.00000 + 2.00000i) q^{46} +(-3.00000 + 5.19615i) q^{47} +(-4.50000 - 7.79423i) q^{49} +(-0.366025 + 1.36603i) q^{50} +(2.00000 - 3.46410i) q^{52} -6.00000 q^{55} +(2.92820 + 10.9282i) q^{56} +(0.866025 - 0.500000i) q^{59} +(10.3923 + 6.00000i) q^{61} +(4.00000 + 4.00000i) q^{62} -8.00000i q^{64} +(2.00000 - 3.46410i) q^{65} +(2.59808 - 1.50000i) q^{67} +(-8.66025 + 5.00000i) q^{68} +(2.92820 + 10.9282i) q^{70} +6.00000 q^{71} +9.00000 q^{73} +(0.732051 + 2.73205i) q^{74} +(1.00000 + 1.73205i) q^{76} +(10.3923 - 6.00000i) q^{77} +(7.00000 - 12.1244i) q^{79} -8.00000i q^{80} +(-5.00000 - 5.00000i) q^{82} +(-3.46410 - 2.00000i) q^{83} +(-8.66025 + 5.00000i) q^{85} +(-15.0263 - 4.02628i) q^{86} +(-8.19615 + 2.19615i) q^{88} +14.0000 q^{89} +8.00000i q^{91} +(3.46410 + 2.00000i) q^{92} +(-2.19615 + 8.19615i) q^{94} +(1.00000 + 1.73205i) q^{95} +(-0.500000 + 0.866025i) q^{97} +(-9.00000 - 9.00000i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 2 q^{2} - 8 q^{7} + 8 q^{8}+O(q^{10}) \) 4 * q + 2 * q^2 - 8 * q^7 + 8 * q^8	\( 4 q + 2 q^{2} - 8 q^{7} + 8 q^{8} + 8 q^{10} + 8 q^{14} + 8 q^{16} - 20 q^{17} + 8 q^{20} - 6 q^{22} + 4 q^{23} - 2 q^{25} + 8 q^{26} + 8 q^{31} - 8 q^{32} - 10 q^{34} - 2 q^{38} - 8 q^{40} - 10 q^{41} - 24 q^{44} + 8 q^{46} - 12 q^{47} - 18 q^{49} + 2 q^{50} + 8 q^{52} - 24 q^{55} - 16 q^{56} + 16 q^{62} + 8 q^{65} - 16 q^{70} + 24 q^{71} + 36 q^{73} - 4 q^{74} + 4 q^{76} + 28 q^{79} - 20 q^{82} - 22 q^{86} - 12 q^{88} + 56 q^{89} + 12 q^{94} + 4 q^{95} - 2 q^{97} - 36 q^{98}+O(q^{100}) \) 4 * q + 2 * q^2 - 8 * q^7 + 8 * q^8 + 8 * q^10 + 8 * q^14 + 8 * q^16 - 20 * q^17 + 8 * q^20 - 6 * q^22 + 4 * q^23 - 2 * q^25 + 8 * q^26 + 8 * q^31 - 8 * q^32 - 10 * q^34 - 2 * q^38 - 8 * q^40 - 10 * q^41 - 24 * q^44 + 8 * q^46 - 12 * q^47 - 18 * q^49 + 2 * q^50 + 8 * q^52 - 24 * q^55 - 16 * q^56 + 16 * q^62 + 8 * q^65 - 16 * q^70 + 24 * q^71 + 36 * q^73 - 4 * q^74 + 4 * q^76 + 28 * q^79 - 20 * q^82 - 22 * q^86 - 12 * q^88 + 56 * q^89 + 12 * q^94 + 4 * q^95 - 2 * q^97 - 36 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more