LMFDB - Embedded newform 1350.2.e.g.451.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1350,2,Mod(451,1350)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1350, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([4, 0]))

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

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

chi := DirichletCharacter("1350.451");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1350 = 2 \cdot 3^{3} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1350.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:	$10.7798042729$
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 90)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			451.1
Root			$0.500000 + 0.866025i$ of defining polynomial
Character	$\chi$	$=$	1350.451
Dual form			1350.2.e.g.901.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.500000 - 0.866025i) q^{2} +(-0.500000 - 0.866025i) q^{4} +(-2.00000 + 3.46410i) q^{7} -1.00000 q^{8} +O(q^{10})$	\(q+(0.500000 - 0.866025i) q^{2} +(-0.500000 - 0.866025i) q^{4} +(-2.00000 + 3.46410i) q^{7} -1.00000 q^{8} +(1.50000 - 2.59808i) q^{11} +(-2.00000 - 3.46410i) q^{13} +(2.00000 + 3.46410i) q^{14} +(-0.500000 + 0.866025i) q^{16} +3.00000 q^{17} +5.00000 q^{19} +(-1.50000 - 2.59808i) q^{22} +(-3.00000 - 5.19615i) q^{23} -4.00000 q^{26} +4.00000 q^{28} +(3.00000 - 5.19615i) q^{29} +(-1.00000 - 1.73205i) q^{31} +(0.500000 + 0.866025i) q^{32} +(1.50000 - 2.59808i) q^{34} +4.00000 q^{37} +(2.50000 - 4.33013i) q^{38} +(-1.50000 - 2.59808i) q^{41} +(5.50000 - 9.52628i) q^{43} -3.00000 q^{44} -6.00000 q^{46} +(-4.50000 - 7.79423i) q^{49} +(-2.00000 + 3.46410i) q^{52} +6.00000 q^{53} +(2.00000 - 3.46410i) q^{56} +(-3.00000 - 5.19615i) q^{58} +(-1.50000 - 2.59808i) q^{59} +(5.00000 - 8.66025i) q^{61} -2.00000 q^{62} +1.00000 q^{64} +(2.50000 + 4.33013i) q^{67} +(-1.50000 - 2.59808i) q^{68} -6.00000 q^{71} +7.00000 q^{73} +(2.00000 - 3.46410i) q^{74} +(-2.50000 - 4.33013i) q^{76} +(6.00000 + 10.3923i) q^{77} +(-7.00000 + 12.1244i) q^{79} -3.00000 q^{82} +(-6.00000 + 10.3923i) q^{83} +(-5.50000 - 9.52628i) q^{86} +(-1.50000 + 2.59808i) q^{88} -6.00000 q^{89} +16.0000 q^{91} +(-3.00000 + 5.19615i) q^{92} +(5.50000 - 9.52628i) q^{97} -9.00000 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + q^{2} - q^{4} - 4 q^{7} - 2 q^{8}+O(q^{10}) $ 2 * q + q^2 - q^4 - 4 * q^7 - 2 * q^8	$ 2 q + q^{2} - q^{4} - 4 q^{7} - 2 q^{8} + 3 q^{11} - 4 q^{13} + 4 q^{14} - q^{16} + 6 q^{17} + 10 q^{19} - 3 q^{22} - 6 q^{23} - 8 q^{26} + 8 q^{28} + 6 q^{29} - 2 q^{31} + q^{32} + 3 q^{34} + 8 q^{37} + 5 q^{38} - 3 q^{41} + 11 q^{43} - 6 q^{44} - 12 q^{46} - 9 q^{49} - 4 q^{52} + 12 q^{53} + 4 q^{56} - 6 q^{58} - 3 q^{59} + 10 q^{61} - 4 q^{62} + 2 q^{64} + 5 q^{67} - 3 q^{68} - 12 q^{71} + 14 q^{73} + 4 q^{74} - 5 q^{76} + 12 q^{77} - 14 q^{79} - 6 q^{82} - 12 q^{83} - 11 q^{86} - 3 q^{88} - 12 q^{89} + 32 q^{91} - 6 q^{92} + 11 q^{97} - 18 q^{98}+O(q^{100}) $ 2 * q + q^2 - q^4 - 4 * q^7 - 2 * q^8 + 3 * q^11 - 4 * q^13 + 4 * q^14 - q^16 + 6 * q^17 + 10 * q^19 - 3 * q^22 - 6 * q^23 - 8 * q^26 + 8 * q^28 + 6 * q^29 - 2 * q^31 + q^32 + 3 * q^34 + 8 * q^37 + 5 * q^38 - 3 * q^41 + 11 * q^43 - 6 * q^44 - 12 * q^46 - 9 * q^49 - 4 * q^52 + 12 * q^53 + 4 * q^56 - 6 * q^58 - 3 * q^59 + 10 * q^61 - 4 * q^62 + 2 * q^64 + 5 * q^67 - 3 * q^68 - 12 * q^71 + 14 * q^73 + 4 * q^74 - 5 * q^76 + 12 * q^77 - 14 * q^79 - 6 * q^82 - 12 * q^83 - 11 * q^86 - 3 * q^88 - 12 * q^89 + 32 * q^91 - 6 * q^92 + 11 * q^97 - 18 * q^98

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1350\mathbb{Z}\right)^\times$.

$n$	$1001$	$1027$
$\chi(n)$	$e\left(\frac{2}{3}\right)$	$1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$	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			0
$5$		0			0
$6$		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	$0.893852\pi$
$8$	−1.00000			−0.353553
$9$		0			0
$10$		0			0
$11$	1.50000	−	2.59808i	0.452267	−	0.783349i	−0.546259	−	0.837616i	$-0.683949\pi$
$11$	1.50000	−	2.59808i	0.452267	−	0.783349i	0.998526	+	0.0542666i	$0.0172821\pi$
$12$		0			0
$13$	−2.00000	−	3.46410i	−0.554700	−	0.960769i	−0.997927	−	0.0643593i	$-0.979500\pi$
$13$	−2.00000	−	3.46410i	−0.554700	−	0.960769i	0.443227	−	0.896410i	$-0.353834\pi$
$14$	2.00000	+	3.46410i	0.534522	+	0.925820i
$15$		0			0
$16$	−0.500000	+	0.866025i	−0.125000	+	0.216506i
$17$	3.00000			0.727607			0.363803	−	0.931476i	$-0.381478\pi$
$17$	3.00000			0.727607			0.363803	+	0.931476i	$0.381478\pi$
$18$		0			0
$19$	5.00000			1.14708			0.573539	−	0.819178i	$-0.305570\pi$
$19$	5.00000			1.14708			0.573539	+	0.819178i	$0.305570\pi$
$20$		0			0
$21$		0			0
$22$	−1.50000	−	2.59808i	−0.319801	−	0.553912i
$23$	−3.00000	−	5.19615i	−0.625543	−	1.08347i	−0.988436	−	0.151642i	$-0.951544\pi$
$23$	−3.00000	−	5.19615i	−0.625543	−	1.08347i	0.362892	−	0.931831i	$-0.381789\pi$
$24$		0			0
$25$		0			0
$26$	−4.00000			−0.784465
$27$		0			0
$28$	4.00000			0.755929
$29$	3.00000	−	5.19615i	0.557086	−	0.964901i	−0.440652	−	0.897678i	$-0.645253\pi$
$29$	3.00000	−	5.19615i	0.557086	−	0.964901i	0.997738	−	0.0672232i	$-0.0214140\pi$
$30$		0			0
$31$	−1.00000	−	1.73205i	−0.179605	−	0.311086i	0.762140	−	0.647412i	$-0.224149\pi$
$31$	−1.00000	−	1.73205i	−0.179605	−	0.311086i	−0.941745	+	0.336327i	$0.890815\pi$
$32$	0.500000	+	0.866025i	0.0883883	+	0.153093i
$33$		0			0
$34$	1.50000	−	2.59808i	0.257248	−	0.445566i
$35$		0			0
$36$		0			0
$37$	4.00000			0.657596			0.328798	−	0.944400i	$-0.393356\pi$
$37$	4.00000			0.657596			0.328798	+	0.944400i	$0.393356\pi$
$38$	2.50000	−	4.33013i	0.405554	−	0.702439i
$39$		0			0
$40$		0			0
$41$	−1.50000	−	2.59808i	−0.234261	−	0.405751i	0.724797	−	0.688963i	$-0.241934\pi$
$41$	−1.50000	−	2.59808i	−0.234261	−	0.405751i	−0.959058	+	0.283211i	$0.908600\pi$
$42$		0			0
$43$	5.50000	−	9.52628i	0.838742	−	1.45274i	−0.0522047	−	0.998636i	$-0.516625\pi$
$43$	5.50000	−	9.52628i	0.838742	−	1.45274i	0.890947	−	0.454108i	$-0.150042\pi$
$44$	−3.00000			−0.452267
$45$		0			0
$46$	−6.00000			−0.884652
$47$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$47$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$48$		0			0
$49$	−4.50000	−	7.79423i	−0.642857	−	1.11346i
$50$		0			0
$51$		0			0
$52$	−2.00000	+	3.46410i	−0.277350	+	0.480384i
$53$	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
$59$	−1.50000	−	2.59808i	−0.195283	−	0.338241i	0.751710	−	0.659494i	$-0.229229\pi$
$59$	−1.50000	−	2.59808i	−0.195283	−	0.338241i	−0.946993	+	0.321253i	$0.895896\pi$
$60$		0			0
$61$	5.00000	−	8.66025i	0.640184	−	1.10883i	−0.345207	−	0.938527i	$-0.612191\pi$
$61$	5.00000	−	8.66025i	0.640184	−	1.10883i	0.985391	−	0.170305i	$-0.0544754\pi$
$62$	−2.00000			−0.254000
$63$		0			0
$64$	1.00000			0.125000
$65$		0			0
$66$		0			0
$67$	2.50000	+	4.33013i	0.305424	+	0.529009i	0.977356	−	0.211604i	$-0.0678686\pi$
$67$	2.50000	+	4.33013i	0.305424	+	0.529009i	−0.671932	+	0.740613i	$0.734535\pi$
$68$	−1.50000	−	2.59808i	−0.181902	−	0.315063i
$69$		0			0
$70$		0			0
$71$	−6.00000			−0.712069			−0.356034	−	0.934473i	$-0.615871\pi$
$71$	−6.00000			−0.712069			−0.356034	+	0.934473i	$0.615871\pi$
$72$		0			0
$73$	7.00000			0.819288			0.409644	−	0.912245i	$-0.365653\pi$
$73$	7.00000			0.819288			0.409644	+	0.912245i	$0.365653\pi$
$74$	2.00000	−	3.46410i	0.232495	−	0.402694i
$75$		0			0
$76$	−2.50000	−	4.33013i	−0.286770	−	0.496700i
$77$	6.00000	+	10.3923i	0.683763	+	1.18431i
$78$		0			0
$79$	−7.00000	+	12.1244i	−0.787562	+	1.36410i	0.139895	+	0.990166i	$0.455323\pi$
$79$	−7.00000	+	12.1244i	−0.787562	+	1.36410i	−0.927457	+	0.373930i	$0.878010\pi$
$80$		0			0
$81$		0			0
$82$	−3.00000			−0.331295
$83$	−6.00000	+	10.3923i	−0.658586	+	1.14070i	0.322396	+	0.946605i	$0.395512\pi$
$83$	−6.00000	+	10.3923i	−0.658586	+	1.14070i	−0.980982	+	0.194099i	$0.937822\pi$
$84$		0			0
$85$		0			0
$86$	−5.50000	−	9.52628i	−0.593080	−	1.02725i
$87$		0			0
$88$	−1.50000	+	2.59808i	−0.159901	+	0.276956i
$89$	−6.00000			−0.635999			−0.317999	−	0.948091i	$-0.603011\pi$
$89$	−6.00000			−0.635999			−0.317999	+	0.948091i	$0.603011\pi$
$90$		0			0
$91$	16.0000			1.67726
$92$	−3.00000	+	5.19615i	−0.312772	+	0.541736i
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$	5.50000	−	9.52628i	0.558440	−	0.967247i	−0.439187	−	0.898396i	$-0.644733\pi$
$97$	5.50000	−	9.52628i	0.558440	−	0.967247i	0.997627	−	0.0688512i	$-0.0219334\pi$
$98$	−9.00000			−0.909137
$99$		0			0
$100$		0			0
$101$	−6.00000	+	10.3923i	−0.597022	+	1.03407i	0.396236	+	0.918149i	$0.370316\pi$
$101$	−6.00000	+	10.3923i	−0.597022	+	1.03407i	−0.993258	+	0.115924i	$0.963017\pi$
$102$		0			0
$103$	−2.00000	−	3.46410i	−0.197066	−	0.341328i	0.750510	−	0.660859i	$-0.229808\pi$
$103$	−2.00000	−	3.46410i	−0.197066	−	0.341328i	−0.947576	+	0.319531i	$0.896475\pi$
$104$	2.00000	+	3.46410i	0.196116	+	0.339683i
$105$		0			0
$106$	3.00000	−	5.19615i	0.291386	−	0.504695i
$107$	−9.00000			−0.870063			−0.435031	−	0.900415i	$-0.643263\pi$
$107$	−9.00000			−0.870063			−0.435031	+	0.900415i	$0.643263\pi$
$108$		0			0
$109$	−4.00000			−0.383131			−0.191565	−	0.981480i	$-0.561356\pi$
$109$	−4.00000			−0.383131			−0.191565	+	0.981480i	$0.561356\pi$
$110$		0			0
$111$		0			0
$112$	−2.00000	−	3.46410i	−0.188982	−	0.327327i
$113$	−9.00000	−	15.5885i	−0.846649	−	1.46644i	−0.884182	−	0.467143i	$-0.845283\pi$
$113$	−9.00000	−	15.5885i	−0.846649	−	1.46644i	0.0375328	−	0.999295i	$-0.488050\pi$
$114$		0			0
$115$		0			0
$116$	−6.00000			−0.557086
$117$		0			0
$118$	−3.00000			−0.276172
$119$	−6.00000	+	10.3923i	−0.550019	+	0.952661i
$120$		0			0
$121$	1.00000	+	1.73205i	0.0909091	+	0.157459i
$122$	−5.00000	−	8.66025i	−0.452679	−	0.784063i
$123$		0			0
$124$	−1.00000	+	1.73205i	−0.0898027	+	0.155543i
$125$		0			0
$126$		0			0
$127$	−2.00000			−0.177471			−0.0887357	−	0.996055i	$-0.528283\pi$
$127$	−2.00000			−0.177471			−0.0887357	+	0.996055i	$0.528283\pi$
$128$	0.500000	−	0.866025i	0.0441942	−	0.0765466i
$129$		0			0
$130$		0			0
$131$	6.00000	+	10.3923i	0.524222	+	0.907980i	0.999602	+	0.0281993i	$0.00897729\pi$
$131$	6.00000	+	10.3923i	0.524222	+	0.907980i	−0.475380	+	0.879781i	$0.657689\pi$
$132$		0			0
$133$	−10.0000	+	17.3205i	−0.867110	+	1.50188i
$134$	5.00000			0.431934
$135$		0			0
$136$	−3.00000			−0.257248
$137$	4.50000	−	7.79423i	0.384461	−	0.665906i	−0.607233	−	0.794524i	$-0.707721\pi$
$137$	4.50000	−	7.79423i	0.384461	−	0.665906i	0.991694	+	0.128618i	$0.0410540\pi$
$138$		0			0
$139$	0.500000	+	0.866025i	0.0424094	+	0.0734553i	0.886451	−	0.462822i	$-0.153163\pi$
$139$	0.500000	+	0.866025i	0.0424094	+	0.0734553i	−0.844042	+	0.536278i	$0.819830\pi$
$140$		0			0
$141$		0			0
$142$	−3.00000	+	5.19615i	−0.251754	+	0.436051i
$143$	−12.0000			−1.00349
$144$		0			0
$145$		0			0
$146$	3.50000	−	6.06218i	0.289662	−	0.501709i
$147$		0			0
$148$	−2.00000	−	3.46410i	−0.164399	−	0.284747i
$149$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$149$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$150$		0			0
$151$	5.00000	−	8.66025i	0.406894	−	0.704761i	−0.587646	−	0.809118i	$-0.699945\pi$
$151$	5.00000	−	8.66025i	0.406894	−	0.704761i	0.994540	+	0.104357i	$0.0332784\pi$
$152$	−5.00000			−0.405554
$153$		0			0
$154$	12.0000			0.966988
$155$		0			0
$156$		0			0
$157$	4.00000	+	6.92820i	0.319235	+	0.552931i	0.980329	−	0.197372i	$-0.0632408\pi$
$157$	4.00000	+	6.92820i	0.319235	+	0.552931i	−0.661094	+	0.750303i	$0.729907\pi$
$158$	7.00000	+	12.1244i	0.556890	+	0.964562i
$159$		0			0
$160$		0			0
$161$	24.0000			1.89146
$162$		0			0
$163$	16.0000			1.25322			0.626608	−	0.779334i	$-0.284443\pi$
$163$	16.0000			1.25322			0.626608	+	0.779334i	$0.284443\pi$
$164$	−1.50000	+	2.59808i	−0.117130	+	0.202876i
$165$		0			0
$166$	6.00000	+	10.3923i	0.465690	+	0.806599i
$167$	9.00000	+	15.5885i	0.696441	+	1.20627i	0.969693	+	0.244328i	$0.0785675\pi$
$167$	9.00000	+	15.5885i	0.696441	+	1.20627i	−0.273252	+	0.961943i	$0.588099\pi$
$168$		0			0
$169$	−1.50000	+	2.59808i	−0.115385	+	0.199852i
$170$		0			0
$171$		0			0
$172$	−11.0000			−0.838742
$173$	9.00000	−	15.5885i	0.684257	−	1.18517i	−0.289412	−	0.957205i	$-0.593460\pi$
$173$	9.00000	−	15.5885i	0.684257	−	1.18517i	0.973670	−	0.227964i	$-0.0732068\pi$
$174$		0			0
$175$		0			0
$176$	1.50000	+	2.59808i	0.113067	+	0.195837i
$177$		0			0
$178$	−3.00000	+	5.19615i	−0.224860	+	0.389468i
$179$		0			0			−	1.00000i	$-0.5\pi$
$179$		0			0				1.00000i	$0.5\pi$
$180$		0			0
$181$	−16.0000			−1.18927			−0.594635	−	0.803996i	$-0.702704\pi$
$181$	−16.0000			−1.18927			−0.594635	+	0.803996i	$0.702704\pi$
$182$	8.00000	−	13.8564i	0.592999	−	1.02711i
$183$		0			0
$184$	3.00000	+	5.19615i	0.221163	+	0.383065i
$185$		0			0
$186$		0			0
$187$	4.50000	−	7.79423i	0.329073	−	0.569970i
$188$		0			0
$189$		0			0
$190$		0			0
$191$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$191$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$192$		0			0
$193$	−6.50000	−	11.2583i	−0.467880	−	0.810392i	0.531446	−	0.847092i	$-0.321649\pi$
$193$	−6.50000	−	11.2583i	−0.467880	−	0.810392i	−0.999326	+	0.0366998i	$0.988315\pi$
$194$	−5.50000	−	9.52628i	−0.394877	−	0.683947i
$195$		0			0
$196$	−4.50000	+	7.79423i	−0.321429	+	0.556731i
$197$		0			0			−	1.00000i	$-0.5\pi$
$197$		0			0				1.00000i	$0.5\pi$
$198$		0			0
$199$	20.0000			1.41776			0.708881	−	0.705328i	$-0.249200\pi$
$199$	20.0000			1.41776			0.708881	+	0.705328i	$0.249200\pi$
$200$		0			0
$201$		0			0
$202$	6.00000	+	10.3923i	0.422159	+	0.731200i
$203$	12.0000	+	20.7846i	0.842235	+	1.45879i
$204$		0			0
$205$		0			0
$206$	−4.00000			−0.278693
$207$		0			0
$208$	4.00000			0.277350
$209$	7.50000	−	12.9904i	0.518786	−	0.898563i
$210$		0			0
$211$	2.00000	+	3.46410i	0.137686	+	0.238479i	0.926620	−	0.375999i	$-0.122700\pi$
$211$	2.00000	+	3.46410i	0.137686	+	0.238479i	−0.788935	+	0.614477i	$0.789367\pi$
$212$	−3.00000	−	5.19615i	−0.206041	−	0.356873i
$213$		0			0
$214$	−4.50000	+	7.79423i	−0.307614	+	0.532803i
$215$		0			0
$216$		0			0
$217$	8.00000			0.543075
$218$	−2.00000	+	3.46410i	−0.135457	+	0.234619i
$219$		0			0
$220$		0			0
$221$	−6.00000	−	10.3923i	−0.403604	−	0.699062i
$222$		0			0
$223$	−11.0000	+	19.0526i	−0.736614	+	1.27585i	0.217397	+	0.976083i	$0.430243\pi$
$223$	−11.0000	+	19.0526i	−0.736614	+	1.27585i	−0.954011	+	0.299770i	$0.903090\pi$
$224$	−4.00000			−0.267261
$225$		0			0
$226$	−18.0000			−1.19734
$227$	1.50000	−	2.59808i	0.0995585	−	0.172440i	−0.811943	−	0.583736i	$-0.801590\pi$
$227$	1.50000	−	2.59808i	0.0995585	−	0.172440i	0.911502	+	0.411296i	$0.134924\pi$
$228$		0			0
$229$	−10.0000	−	17.3205i	−0.660819	−	1.14457i	−0.980401	−	0.197013i	$-0.936876\pi$
$229$	−10.0000	−	17.3205i	−0.660819	−	1.14457i	0.319582	−	0.947559i	$-0.396457\pi$
$230$		0			0
$231$		0			0
$232$	−3.00000	+	5.19615i	−0.196960	+	0.341144i
$233$	21.0000			1.37576			0.687878	−	0.725826i	$-0.258542\pi$
$233$	21.0000			1.37576			0.687878	+	0.725826i	$0.258542\pi$
$234$		0			0
$235$		0			0
$236$	−1.50000	+	2.59808i	−0.0976417	+	0.169120i
$237$		0			0
$238$	6.00000	+	10.3923i	0.388922	+	0.673633i
$239$	3.00000	+	5.19615i	0.194054	+	0.336111i	0.946590	−	0.322440i	$-0.104503\pi$
$239$	3.00000	+	5.19615i	0.194054	+	0.336111i	−0.752536	+	0.658551i	$0.771170\pi$
$240$		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			0.128565
$243$		0			0
$244$	−10.0000			−0.640184
$245$		0			0
$246$		0			0
$247$	−10.0000	−	17.3205i	−0.636285	−	1.10208i
$248$	1.00000	+	1.73205i	0.0635001	+	0.109985i
$249$		0			0
$250$		0			0
$251$	21.0000			1.32551			0.662754	−	0.748837i	$-0.269387\pi$
$251$	21.0000			1.32551			0.662754	+	0.748837i	$0.269387\pi$
$252$		0			0
$253$	−18.0000			−1.13165
$254$	−1.00000	+	1.73205i	−0.0627456	+	0.108679i
$255$		0			0
$256$	−0.500000	−	0.866025i	−0.0312500	−	0.0541266i
$257$	−4.50000	−	7.79423i	−0.280702	−	0.486191i	0.690856	−	0.722993i	$-0.257234\pi$
$257$	−4.50000	−	7.79423i	−0.280702	−	0.486191i	−0.971558	+	0.236802i	$0.923901\pi$
$258$		0			0
$259$	−8.00000	+	13.8564i	−0.497096	+	0.860995i
$260$		0			0
$261$		0			0
$262$	12.0000			0.741362
$263$	−12.0000	+	20.7846i	−0.739952	+	1.28163i	0.212565	+	0.977147i	$0.431818\pi$
$263$	−12.0000	+	20.7846i	−0.739952	+	1.28163i	−0.952517	+	0.304487i	$0.901515\pi$
$264$		0			0
$265$		0			0
$266$	10.0000	+	17.3205i	0.613139	+	1.06199i
$267$		0			0
$268$	2.50000	−	4.33013i	0.152712	−	0.264505i
$269$	−6.00000			−0.365826			−0.182913	−	0.983129i	$-0.558553\pi$
$269$	−6.00000			−0.365826			−0.182913	+	0.983129i	$0.558553\pi$
$270$		0			0
$271$	−16.0000			−0.971931			−0.485965	−	0.873978i	$-0.661532\pi$
$271$	−16.0000			−0.971931			−0.485965	+	0.873978i	$0.661532\pi$
$272$	−1.50000	+	2.59808i	−0.0909509	+	0.157532i
$273$		0			0
$274$	−4.50000	−	7.79423i	−0.271855	−	0.470867i
$275$		0			0
$276$		0			0
$277$	−11.0000	+	19.0526i	−0.660926	+	1.14476i	0.319447	+	0.947604i	$0.396503\pi$
$277$	−11.0000	+	19.0526i	−0.660926	+	1.14476i	−0.980373	+	0.197153i	$0.936830\pi$
$278$	1.00000			0.0599760
$279$		0			0
$280$		0			0
$281$	−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
$283$	10.0000	+	17.3205i	0.594438	+	1.02960i	0.993626	+	0.112728i	$0.0359589\pi$
$283$	10.0000	+	17.3205i	0.594438	+	1.02960i	−0.399188	+	0.916869i	$0.630708\pi$
$284$	3.00000	+	5.19615i	0.178017	+	0.308335i
$285$		0			0
$286$	−6.00000	+	10.3923i	−0.354787	+	0.614510i
$287$	12.0000			0.708338
$288$		0			0
$289$	−8.00000			−0.470588
$290$		0			0
$291$		0			0
$292$	−3.50000	−	6.06218i	−0.204822	−	0.354762i
$293$	9.00000	+	15.5885i	0.525786	+	0.910687i	0.999549	+	0.0300351i	$0.00956192\pi$
$293$	9.00000	+	15.5885i	0.525786	+	0.910687i	−0.473763	+	0.880652i	$0.657105\pi$
$294$		0			0
$295$		0			0
$296$	−4.00000			−0.232495
$297$		0			0
$298$		0			0
$299$	−12.0000	+	20.7846i	−0.693978	+	1.20201i
$300$		0			0
$301$	22.0000	+	38.1051i	1.26806	+	2.19634i
$302$	−5.00000	−	8.66025i	−0.287718	−	0.498342i
$303$		0			0
$304$	−2.50000	+	4.33013i	−0.143385	+	0.248350i
$305$		0			0
$306$		0			0
$307$	7.00000			0.399511			0.199756	−	0.979846i	$-0.435985\pi$
$307$	7.00000			0.399511			0.199756	+	0.979846i	$0.435985\pi$
$308$	6.00000	−	10.3923i	0.341882	−	0.592157i
$309$		0			0
$310$		0			0
$311$	−3.00000	−	5.19615i	−0.170114	−	0.294647i	0.768345	−	0.640036i	$-0.221080\pi$
$311$	−3.00000	−	5.19615i	−0.170114	−	0.294647i	−0.938460	+	0.345389i	$0.887747\pi$
$312$		0			0
$313$	−0.500000	+	0.866025i	−0.0282617	+	0.0489506i	−0.879810	−	0.475325i	$-0.842331\pi$
$313$	−0.500000	+	0.866025i	−0.0282617	+	0.0489506i	0.851549	+	0.524276i	$0.175664\pi$
$314$	8.00000			0.451466
$315$		0			0
$316$	14.0000			0.787562
$317$	−12.0000	+	20.7846i	−0.673987	+	1.16738i	0.302777	+	0.953062i	$0.402086\pi$
$317$	−12.0000	+	20.7846i	−0.673987	+	1.16738i	−0.976764	+	0.214318i	$0.931247\pi$
$318$		0			0
$319$	−9.00000	−	15.5885i	−0.503903	−	0.872786i
$320$		0			0
$321$		0			0
$322$	12.0000	−	20.7846i	0.668734	−	1.15828i
$323$	15.0000			0.834622
$324$		0			0
$325$		0			0
$326$	8.00000	−	13.8564i	0.443079	−	0.767435i
$327$		0			0
$328$	1.50000	+	2.59808i	0.0828236	+	0.143455i
$329$		0			0
$330$		0			0
$331$	2.00000	−	3.46410i	0.109930	−	0.190404i	−0.805812	−	0.592172i	$-0.798271\pi$
$331$	2.00000	−	3.46410i	0.109930	−	0.190404i	0.915742	+	0.401768i	$0.131604\pi$
$332$	12.0000			0.658586
$333$		0			0
$334$	18.0000			0.984916
$335$		0			0
$336$		0			0
$337$	−15.5000	−	26.8468i	−0.844339	−	1.46244i	−0.886194	−	0.463314i	$-0.846660\pi$
$337$	−15.5000	−	26.8468i	−0.844339	−	1.46244i	0.0418554	−	0.999124i	$-0.486673\pi$
$338$	1.50000	+	2.59808i	0.0815892	+	0.141317i
$339$		0			0
$340$		0			0
$341$	−6.00000			−0.324918
$342$		0			0
$343$	8.00000			0.431959
$344$	−5.50000	+	9.52628i	−0.296540	+	0.513623i
$345$		0			0
$346$	−9.00000	−	15.5885i	−0.483843	−	0.838041i
$347$	−10.5000	−	18.1865i	−0.563670	−	0.976304i	−0.997172	−	0.0751519i	$-0.976056\pi$
$347$	−10.5000	−	18.1865i	−0.563670	−	0.976304i	0.433503	−	0.901152i	$-0.357278\pi$
$348$		0			0
$349$	8.00000	−	13.8564i	0.428230	−	0.741716i	−0.568486	−	0.822693i	$-0.692471\pi$
$349$	8.00000	−	13.8564i	0.428230	−	0.741716i	0.996716	+	0.0809766i	$0.0258039\pi$
$350$		0			0
$351$		0			0
$352$	3.00000			0.159901
$353$	−4.50000	+	7.79423i	−0.239511	+	0.414845i	−0.960574	−	0.278024i	$-0.910320\pi$
$353$	−4.50000	+	7.79423i	−0.239511	+	0.414845i	0.721063	+	0.692869i	$0.243654\pi$
$354$		0			0
$355$		0			0
$356$	3.00000	+	5.19615i	0.159000	+	0.275396i
$357$		0			0
$358$		0			0
$359$	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
$361$	6.00000			0.315789
$362$	−8.00000	+	13.8564i	−0.420471	+	0.728277i
$363$		0			0
$364$	−8.00000	−	13.8564i	−0.419314	−	0.726273i
$365$		0			0
$366$		0			0
$367$	4.00000	−	6.92820i	0.208798	−	0.361649i	−0.742538	−	0.669804i	$-0.766378\pi$
$367$	4.00000	−	6.92820i	0.208798	−	0.361649i	0.951336	+	0.308155i	$0.0997115\pi$
$368$	6.00000			0.312772
$369$		0			0
$370$		0			0
$371$	−12.0000	+	20.7846i	−0.623009	+	1.07908i
$372$		0			0
$373$	−5.00000	−	8.66025i	−0.258890	−	0.448411i	0.707055	−	0.707159i	$-0.250023\pi$
$373$	−5.00000	−	8.66025i	−0.258890	−	0.448411i	−0.965945	+	0.258748i	$0.916690\pi$
$374$	−4.50000	−	7.79423i	−0.232689	−	0.403030i
$375$		0			0
$376$		0			0
$377$	−24.0000			−1.23606
$378$		0			0
$379$	29.0000			1.48963			0.744815	−	0.667271i	$-0.232538\pi$
$379$	29.0000			1.48963			0.744815	+	0.667271i	$0.232538\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−6.00000	−	10.3923i	−0.306586	−	0.531022i	0.671027	−	0.741433i	$-0.265853\pi$
$383$	−6.00000	−	10.3923i	−0.306586	−	0.531022i	−0.977613	+	0.210411i	$0.932520\pi$
$384$		0			0
$385$		0			0
$386$	−13.0000			−0.661683
$387$		0			0
$388$	−11.0000			−0.558440
$389$	18.0000	−	31.1769i	0.912636	−	1.58073i	0.102311	−	0.994753i	$-0.467376\pi$
$389$	18.0000	−	31.1769i	0.912636	−	1.58073i	0.810326	−	0.585980i	$-0.199290\pi$
$390$		0			0
$391$	−9.00000	−	15.5885i	−0.455150	−	0.788342i
$392$	4.50000	+	7.79423i	0.227284	+	0.393668i
$393$		0			0
$394$		0			0
$395$		0			0
$396$		0			0
$397$	−8.00000			−0.401508			−0.200754	−	0.979642i	$-0.564339\pi$
$397$	−8.00000			−0.401508			−0.200754	+	0.979642i	$0.564339\pi$
$398$	10.0000	−	17.3205i	0.501255	−	0.868199i
$399$		0			0
$400$		0			0
$401$	16.5000	+	28.5788i	0.823971	+	1.42716i	0.902703	+	0.430263i	$0.141579\pi$
$401$	16.5000	+	28.5788i	0.823971	+	1.42716i	−0.0787327	+	0.996896i	$0.525087\pi$
$402$		0			0
$403$	−4.00000	+	6.92820i	−0.199254	+	0.345118i
$404$	12.0000			0.597022
$405$		0			0
$406$	24.0000			1.19110
$407$	6.00000	−	10.3923i	0.297409	−	0.515127i
$408$		0			0
$409$	15.5000	+	26.8468i	0.766426	+	1.32749i	0.939490	+	0.342578i	$0.111300\pi$
$409$	15.5000	+	26.8468i	0.766426	+	1.32749i	−0.173064	+	0.984911i	$0.555367\pi$
$410$		0			0
$411$		0			0
$412$	−2.00000	+	3.46410i	−0.0985329	+	0.170664i
$413$	12.0000			0.590481
$414$		0			0
$415$		0			0
$416$	2.00000	−	3.46410i	0.0980581	−	0.169842i
$417$		0			0
$418$	−7.50000	−	12.9904i	−0.366837	−	0.635380i
$419$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$419$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$420$		0			0
$421$	−1.00000	+	1.73205i	−0.0487370	+	0.0844150i	−0.889365	−	0.457198i	$-0.848853\pi$
$421$	−1.00000	+	1.73205i	−0.0487370	+	0.0844150i	0.840628	+	0.541613i	$0.182186\pi$
$422$	4.00000			0.194717
$423$		0			0
$424$	−6.00000			−0.291386
$425$		0			0
$426$		0			0
$427$	20.0000	+	34.6410i	0.967868	+	1.67640i
$428$	4.50000	+	7.79423i	0.217516	+	0.376748i
$429$		0			0
$430$		0			0
$431$	−24.0000			−1.15604			−0.578020	−	0.816023i	$-0.696174\pi$
$431$	−24.0000			−1.15604			−0.578020	+	0.816023i	$0.696174\pi$
$432$		0			0
$433$	13.0000			0.624740			0.312370	−	0.949960i	$-0.398877\pi$
$433$	13.0000			0.624740			0.312370	+	0.949960i	$0.398877\pi$
$434$	4.00000	−	6.92820i	0.192006	−	0.332564i
$435$		0			0
$436$	2.00000	+	3.46410i	0.0957826	+	0.165900i
$437$	−15.0000	−	25.9808i	−0.717547	−	1.24283i
$438$		0			0
$439$	5.00000	−	8.66025i	0.238637	−	0.413331i	−0.721686	−	0.692220i	$-0.756633\pi$
$439$	5.00000	−	8.66025i	0.238637	−	0.413331i	0.960323	+	0.278889i	$0.0899661\pi$
$440$		0			0
$441$		0			0
$442$	−12.0000			−0.570782
$443$	−1.50000	+	2.59808i	−0.0712672	+	0.123438i	−0.899457	−	0.437009i	$-0.856038\pi$
$443$	−1.50000	+	2.59808i	−0.0712672	+	0.123438i	0.828190	+	0.560448i	$0.189371\pi$
$444$		0			0
$445$		0			0
$446$	11.0000	+	19.0526i	0.520865	+	0.902165i
$447$		0			0
$448$	−2.00000	+	3.46410i	−0.0944911	+	0.163663i
$449$	15.0000			0.707894			0.353947	−	0.935266i	$-0.384839\pi$
$449$	15.0000			0.707894			0.353947	+	0.935266i	$0.384839\pi$
$450$		0			0
$451$	−9.00000			−0.423793
$452$	−9.00000	+	15.5885i	−0.423324	+	0.733219i
$453$		0			0
$454$	−1.50000	−	2.59808i	−0.0703985	−	0.121934i
$455$		0			0
$456$		0			0
$457$	−0.500000	+	0.866025i	−0.0233890	+	0.0405110i	−0.877483	−	0.479608i	$-0.840779\pi$
$457$	−0.500000	+	0.866025i	−0.0233890	+	0.0405110i	0.854094	+	0.520119i	$0.174112\pi$
$458$	−20.0000			−0.934539
$459$		0			0
$460$		0			0
$461$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$461$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$462$		0			0
$463$	10.0000	+	17.3205i	0.464739	+	0.804952i	0.999190	−	0.0402476i	$-0.0128147\pi$
$463$	10.0000	+	17.3205i	0.464739	+	0.804952i	−0.534450	+	0.845200i	$0.679481\pi$
$464$	3.00000	+	5.19615i	0.139272	+	0.241225i
$465$		0			0
$466$	10.5000	−	18.1865i	0.486403	−	0.842475i
$467$	21.0000			0.971764			0.485882	−	0.874024i	$-0.338498\pi$
$467$	21.0000			0.971764			0.485882	+	0.874024i	$0.338498\pi$
$468$		0			0
$469$	−20.0000			−0.923514
$470$		0			0
$471$		0			0
$472$	1.50000	+	2.59808i	0.0690431	+	0.119586i
$473$	−16.5000	−	28.5788i	−0.758671	−	1.31406i
$474$		0			0
$475$		0			0
$476$	12.0000			0.550019
$477$		0			0
$478$	6.00000			0.274434
$479$	−3.00000	+	5.19615i	−0.137073	+	0.237418i	−0.926388	−	0.376571i	$-0.877103\pi$
$479$	−3.00000	+	5.19615i	−0.137073	+	0.237418i	0.789314	+	0.613990i	$0.210436\pi$
$480$		0			0
$481$	−8.00000	−	13.8564i	−0.364769	−	0.631798i
$482$	8.50000	+	14.7224i	0.387164	+	0.670588i
$483$		0			0
$484$	1.00000	−	1.73205i	0.0454545	−	0.0787296i
$485$		0			0
$486$		0			0
$487$	−2.00000			−0.0906287			−0.0453143	−	0.998973i	$-0.514429\pi$
$487$	−2.00000			−0.0906287			−0.0453143	+	0.998973i	$0.514429\pi$
$488$	−5.00000	+	8.66025i	−0.226339	+	0.392031i
$489$		0			0
$490$		0			0
$491$	−16.5000	−	28.5788i	−0.744635	−	1.28974i	−0.950365	−	0.311136i	$-0.899290\pi$
$491$	−16.5000	−	28.5788i	−0.744635	−	1.28974i	0.205731	−	0.978609i	$-0.434043\pi$
$492$		0			0
$493$	9.00000	−	15.5885i	0.405340	−	0.702069i
$494$	−20.0000			−0.899843
$495$		0			0
$496$	2.00000			0.0898027
$497$	12.0000	−	20.7846i	0.538274	−	0.932317i
$498$		0			0
$499$	15.5000	+	26.8468i	0.693875	+	1.20183i	0.970558	+	0.240866i	$0.0774314\pi$
$499$	15.5000	+	26.8468i	0.693875	+	1.20183i	−0.276683	+	0.960961i	$0.589235\pi$
$500$		0			0
$501$		0			0
$502$	10.5000	−	18.1865i	0.468638	−	0.811705i
$503$	6.00000			0.267527			0.133763	−	0.991013i	$-0.457294\pi$
$503$	6.00000			0.267527			0.133763	+	0.991013i	$0.457294\pi$
$504$		0			0
$505$		0			0
$506$	−9.00000	+	15.5885i	−0.400099	+	0.692991i
$507$		0			0
$508$	1.00000	+	1.73205i	0.0443678	+	0.0768473i
$509$	−6.00000	−	10.3923i	−0.265945	−	0.460631i	0.701866	−	0.712309i	$-0.252351\pi$
$509$	−6.00000	−	10.3923i	−0.265945	−	0.460631i	−0.967811	+	0.251679i	$0.919017\pi$
$510$		0			0
$511$	−14.0000	+	24.2487i	−0.619324	+	1.07270i
$512$	−1.00000			−0.0441942
$513$		0			0
$514$	−9.00000			−0.396973
$515$		0			0
$516$		0			0
$517$		0			0
$518$	8.00000	+	13.8564i	0.351500	+	0.608816i
$519$		0			0
$520$		0			0
$521$	3.00000			0.131432			0.0657162	−	0.997838i	$-0.479067\pi$
$521$	3.00000			0.131432			0.0657162	+	0.997838i	$0.479067\pi$
$522$		0			0
$523$	−20.0000			−0.874539			−0.437269	−	0.899331i	$-0.644054\pi$
$523$	−20.0000			−0.874539			−0.437269	+	0.899331i	$0.644054\pi$
$524$	6.00000	−	10.3923i	0.262111	−	0.453990i
$525$		0			0
$526$	12.0000	+	20.7846i	0.523225	+	0.906252i
$527$	−3.00000	−	5.19615i	−0.130682	−	0.226348i
$528$		0			0
$529$	−6.50000	+	11.2583i	−0.282609	+	0.489493i
$530$		0			0
$531$		0			0
$532$	20.0000			0.867110
$533$	−6.00000	+	10.3923i	−0.259889	+	0.450141i
$534$		0			0
$535$		0			0
$536$	−2.50000	−	4.33013i	−0.107984	−	0.187033i
$537$		0			0
$538$	−3.00000	+	5.19615i	−0.129339	+	0.224022i
$539$	−27.0000			−1.16297
$540$		0			0
$541$	8.00000			0.343947			0.171973	−	0.985102i	$-0.444986\pi$
$541$	8.00000			0.343947			0.171973	+	0.985102i	$0.444986\pi$
$542$	−8.00000	+	13.8564i	−0.343629	+	0.595184i
$543$		0			0
$544$	1.50000	+	2.59808i	0.0643120	+	0.111392i
$545$		0			0
$546$		0			0
$547$	−0.500000	+	0.866025i	−0.0213785	+	0.0370286i	−0.876517	−	0.481371i	$-0.840139\pi$
$547$	−0.500000	+	0.866025i	−0.0213785	+	0.0370286i	0.855138	+	0.518400i	$0.173472\pi$
$548$	−9.00000			−0.384461
$549$		0			0
$550$		0			0
$551$	15.0000	−	25.9808i	0.639021	−	1.10682i
$552$		0			0
$553$	−28.0000	−	48.4974i	−1.19068	−	2.06232i
$554$	11.0000	+	19.0526i	0.467345	+	0.809466i
$555$		0			0
$556$	0.500000	−	0.866025i	0.0212047	−	0.0367277i
$557$	−12.0000			−0.508456			−0.254228	−	0.967144i	$-0.581821\pi$
$557$	−12.0000			−0.508456			−0.254228	+	0.967144i	$0.581821\pi$
$558$		0			0
$559$	−44.0000			−1.86100
$560$		0			0
$561$		0			0
$562$	3.00000	+	5.19615i	0.126547	+	0.219186i
$563$	1.50000	+	2.59808i	0.0632175	+	0.109496i	0.895902	−	0.444252i	$-0.146530\pi$
$563$	1.50000	+	2.59808i	0.0632175	+	0.109496i	−0.832684	+	0.553748i	$0.813197\pi$
$564$		0			0
$565$		0			0
$566$	20.0000			0.840663
$567$		0			0
$568$	6.00000			0.251754
$569$	−19.5000	+	33.7750i	−0.817483	+	1.41592i	0.0900490	+	0.995937i	$0.471298\pi$
$569$	−19.5000	+	33.7750i	−0.817483	+	1.41592i	−0.907532	+	0.419984i	$0.862036\pi$
$570$		0			0
$571$	−14.5000	−	25.1147i	−0.606806	−	1.05102i	−0.991763	−	0.128085i	$-0.959117\pi$
$571$	−14.5000	−	25.1147i	−0.606806	−	1.05102i	0.384957	−	0.922934i	$-0.374216\pi$
$572$	6.00000	+	10.3923i	0.250873	+	0.434524i
$573$		0			0
$574$	6.00000	−	10.3923i	0.250435	−	0.433766i
$575$		0			0
$576$		0			0
$577$	7.00000			0.291414			0.145707	−	0.989328i	$-0.453454\pi$
$577$	7.00000			0.291414			0.145707	+	0.989328i	$0.453454\pi$
$578$	−4.00000	+	6.92820i	−0.166378	+	0.288175i
$579$		0			0
$580$		0			0
$581$	−24.0000	−	41.5692i	−0.995688	−	1.72458i
$582$		0			0
$583$	9.00000	−	15.5885i	0.372742	−	0.645608i
$584$	−7.00000			−0.289662
$585$		0			0
$586$	18.0000			0.743573
$587$	−19.5000	+	33.7750i	−0.804851	+	1.39404i	0.111540	+	0.993760i	$0.464422\pi$
$587$	−19.5000	+	33.7750i	−0.804851	+	1.39404i	−0.916392	+	0.400283i	$0.868912\pi$
$588$		0			0
$589$	−5.00000	−	8.66025i	−0.206021	−	0.356840i
$590$		0			0
$591$		0			0
$592$	−2.00000	+	3.46410i	−0.0821995	+	0.142374i
$593$	−18.0000			−0.739171			−0.369586	−	0.929197i	$-0.620500\pi$
$593$	−18.0000			−0.739171			−0.369586	+	0.929197i	$0.620500\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$	12.0000	+	20.7846i	0.490716	+	0.849946i
$599$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$599$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$600$		0			0
$601$	12.5000	−	21.6506i	0.509886	−	0.883148i	−0.490049	−	0.871695i	$-0.663021\pi$
$601$	12.5000	−	21.6506i	0.509886	−	0.883148i	0.999934	−	0.0114528i	$-0.00364562\pi$
$602$	44.0000			1.79331
$603$		0			0
$604$	−10.0000			−0.406894
$605$		0			0
$606$		0			0
$607$	4.00000	+	6.92820i	0.162355	+	0.281207i	0.935713	−	0.352763i	$-0.114758\pi$
$607$	4.00000	+	6.92820i	0.162355	+	0.281207i	−0.773358	+	0.633970i	$0.781424\pi$
$608$	2.50000	+	4.33013i	0.101388	+	0.175610i
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$	−2.00000			−0.0807792			−0.0403896	−	0.999184i	$-0.512860\pi$
$613$	−2.00000			−0.0807792			−0.0403896	+	0.999184i	$0.512860\pi$
$614$	3.50000	−	6.06218i	0.141249	−	0.244650i
$615$		0			0
$616$	−6.00000	−	10.3923i	−0.241747	−	0.418718i
$617$	19.5000	+	33.7750i	0.785040	+	1.35973i	0.928975	+	0.370143i	$0.120691\pi$
$617$	19.5000	+	33.7750i	0.785040	+	1.35973i	−0.143934	+	0.989587i	$0.545975\pi$
$618$		0			0
$619$	9.50000	−	16.4545i	0.381837	−	0.661361i	−0.609488	−	0.792796i	$-0.708625\pi$
$619$	9.50000	−	16.4545i	0.381837	−	0.661361i	0.991325	+	0.131434i	$0.0419582\pi$
$620$		0			0
$621$		0			0
$622$	−6.00000			−0.240578
$623$	12.0000	−	20.7846i	0.480770	−	0.832718i
$624$		0			0
$625$		0			0
$626$	0.500000	+	0.866025i	0.0199840	+	0.0346133i
$627$		0			0
$628$	4.00000	−	6.92820i	0.159617	−	0.276465i
$629$	12.0000			0.478471
$630$		0			0
$631$	−34.0000			−1.35352			−0.676759	−	0.736204i	$-0.736616\pi$
$631$	−34.0000			−1.35352			−0.676759	+	0.736204i	$0.736616\pi$
$632$	7.00000	−	12.1244i	0.278445	−	0.482281i
$633$		0			0
$634$	12.0000	+	20.7846i	0.476581	+	0.825462i
$635$		0			0
$636$		0			0
$637$	−18.0000	+	31.1769i	−0.713186	+	1.23527i
$638$	−18.0000			−0.712627
$639$		0			0
$640$		0			0
$641$	4.50000	−	7.79423i	0.177739	−	0.307854i	−0.763367	−	0.645966i	$-0.776455\pi$
$641$	4.50000	−	7.79423i	0.177739	−	0.307854i	0.941106	+	0.338112i	$0.109788\pi$
$642$		0			0
$643$	11.5000	+	19.9186i	0.453516	+	0.785512i	0.998602	−	0.0528680i	$-0.0168363\pi$
$643$	11.5000	+	19.9186i	0.453516	+	0.785512i	−0.545086	+	0.838380i	$0.683503\pi$
$644$	−12.0000	−	20.7846i	−0.472866	−	0.819028i
$645$		0			0
$646$	7.50000	−	12.9904i	0.295084	−	0.511100i
$647$	6.00000			0.235884			0.117942	−	0.993020i	$-0.462370\pi$
$647$	6.00000			0.235884			0.117942	+	0.993020i	$0.462370\pi$
$648$		0			0
$649$	−9.00000			−0.353281
$650$		0			0
$651$		0			0
$652$	−8.00000	−	13.8564i	−0.313304	−	0.542659i
$653$	−21.0000	−	36.3731i	−0.821794	−	1.42339i	−0.904345	−	0.426801i	$-0.859640\pi$
$653$	−21.0000	−	36.3731i	−0.821794	−	1.42339i	0.0825519	−	0.996587i	$-0.473693\pi$
$654$		0			0
$655$		0			0
$656$	3.00000			0.117130
$657$		0			0
$658$		0			0
$659$	−18.0000	+	31.1769i	−0.701180	+	1.21448i	0.266872	+	0.963732i	$0.414010\pi$
$659$	−18.0000	+	31.1769i	−0.701180	+	1.21448i	−0.968052	+	0.250748i	$0.919323\pi$
$660$		0			0
$661$	−16.0000	−	27.7128i	−0.622328	−	1.07790i	−0.989051	−	0.147573i	$-0.952854\pi$
$661$	−16.0000	−	27.7128i	−0.622328	−	1.07790i	0.366723	−	0.930330i	$-0.380480\pi$
$662$	−2.00000	−	3.46410i	−0.0777322	−	0.134636i
$663$		0			0
$664$	6.00000	−	10.3923i	0.232845	−	0.403300i
$665$		0			0
$666$		0			0
$667$	−36.0000			−1.39393
$668$	9.00000	−	15.5885i	0.348220	−	0.603136i
$669$		0			0
$670$		0			0
$671$	−15.0000	−	25.9808i	−0.579069	−	1.00298i
$672$		0			0
$673$	7.00000	−	12.1244i	0.269830	−	0.467360i	−0.698988	−	0.715134i	$-0.746366\pi$
$673$	7.00000	−	12.1244i	0.269830	−	0.467360i	0.968818	+	0.247774i	$0.0796991\pi$
$674$	−31.0000			−1.19408
$675$		0			0
$676$	3.00000			0.115385
$677$	18.0000	−	31.1769i	0.691796	−	1.19823i	−0.279453	−	0.960159i	$-0.590153\pi$
$677$	18.0000	−	31.1769i	0.691796	−	1.19823i	0.971249	−	0.238067i	$-0.0765137\pi$
$678$		0			0
$679$	22.0000	+	38.1051i	0.844283	+	1.46234i
$680$		0			0
$681$		0			0
$682$	−3.00000	+	5.19615i	−0.114876	+	0.198971i
$683$	−21.0000			−0.803543			−0.401771	−	0.915740i	$-0.631605\pi$
$683$	−21.0000			−0.803543			−0.401771	+	0.915740i	$0.631605\pi$
$684$		0			0
$685$		0			0
$686$	4.00000	−	6.92820i	0.152721	−	0.264520i
$687$		0			0
$688$	5.50000	+	9.52628i	0.209686	+	0.363186i
$689$	−12.0000	−	20.7846i	−0.457164	−	0.791831i
$690$		0			0
$691$	−4.00000	+	6.92820i	−0.152167	+	0.263561i	−0.932024	−	0.362397i	$-0.881959\pi$
$691$	−4.00000	+	6.92820i	−0.152167	+	0.263561i	0.779857	+	0.625958i	$0.215292\pi$
$692$	−18.0000			−0.684257
$693$		0			0
$694$	−21.0000			−0.797149
$695$		0			0
$696$		0			0
$697$	−4.50000	−	7.79423i	−0.170450	−	0.295227i
$698$	−8.00000	−	13.8564i	−0.302804	−	0.524473i
$699$		0			0
$700$		0			0
$701$	−42.0000			−1.58632			−0.793159	−	0.609015i	$-0.791565\pi$
$701$	−42.0000			−1.58632			−0.793159	+	0.609015i	$0.791565\pi$
$702$		0			0
$703$	20.0000			0.754314
$704$	1.50000	−	2.59808i	0.0565334	−	0.0979187i
$705$		0			0
$706$	4.50000	+	7.79423i	0.169360	+	0.293340i
$707$	−24.0000	−	41.5692i	−0.902613	−	1.56337i
$708$		0			0
$709$	17.0000	−	29.4449i	0.638448	−	1.10583i	−0.347325	−	0.937745i	$-0.612910\pi$
$709$	17.0000	−	29.4449i	0.638448	−	1.10583i	0.985773	−	0.168080i	$-0.0537568\pi$
$710$		0			0
$711$		0			0
$712$	6.00000			0.224860
$713$	−6.00000	+	10.3923i	−0.224702	+	0.389195i
$714$		0			0
$715$		0			0
$716$		0			0
$717$		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
$722$	3.00000	−	5.19615i	0.111648	−	0.193381i
$723$		0			0
$724$	8.00000	+	13.8564i	0.297318	+	0.514969i
$725$		0			0
$726$		0			0
$727$	−14.0000	+	24.2487i	−0.519231	+	0.899335i	0.480519	+	0.876984i	$0.340448\pi$
$727$	−14.0000	+	24.2487i	−0.519231	+	0.899335i	−0.999750	+	0.0223506i	$0.992885\pi$
$728$	−16.0000			−0.592999
$729$		0			0
$730$		0			0
$731$	16.5000	−	28.5788i	0.610275	−	1.05703i
$732$		0			0
$733$	16.0000	+	27.7128i	0.590973	+	1.02360i	0.994102	+	0.108453i	$0.0345896\pi$
$733$	16.0000	+	27.7128i	0.590973	+	1.02360i	−0.403128	+	0.915144i	$0.632077\pi$
$734$	−4.00000	−	6.92820i	−0.147643	−	0.255725i
$735$		0			0
$736$	3.00000	−	5.19615i	0.110581	−	0.191533i
$737$	15.0000			0.552532
$738$		0			0
$739$	29.0000			1.06678			0.533391	−	0.845869i	$-0.320917\pi$
$739$	29.0000			1.06678			0.533391	+	0.845869i	$0.320917\pi$
$740$		0			0
$741$		0			0
$742$	12.0000	+	20.7846i	0.440534	+	0.763027i
$743$	3.00000	+	5.19615i	0.110059	+	0.190628i	0.915794	−	0.401648i	$-0.131563\pi$
$743$	3.00000	+	5.19615i	0.110059	+	0.190628i	−0.805735	+	0.592277i	$0.798229\pi$
$744$		0			0
$745$		0			0
$746$	−10.0000			−0.366126
$747$		0			0
$748$	−9.00000			−0.329073
$749$	18.0000	−	31.1769i	0.657706	−	1.13918i
$750$		0			0
$751$	14.0000	+	24.2487i	0.510867	+	0.884848i	0.999921	+	0.0125942i	$0.00400897\pi$
$751$	14.0000	+	24.2487i	0.510867	+	0.884848i	−0.489053	+	0.872254i	$0.662658\pi$
$752$		0			0
$753$		0			0
$754$	−12.0000	+	20.7846i	−0.437014	+	0.756931i
$755$		0			0
$756$		0			0
$757$	−38.0000			−1.38113			−0.690567	−	0.723269i	$-0.742639\pi$
$757$	−38.0000			−1.38113			−0.690567	+	0.723269i	$0.742639\pi$
$758$	14.5000	−	25.1147i	0.526664	−	0.912208i
$759$		0			0
$760$		0			0
$761$	9.00000	+	15.5885i	0.326250	+	0.565081i	0.981764	−	0.190101i	$-0.0608816\pi$
$761$	9.00000	+	15.5885i	0.326250	+	0.565081i	−0.655515	+	0.755182i	$0.727548\pi$
$762$		0			0
$763$	8.00000	−	13.8564i	0.289619	−	0.501636i
$764$		0			0
$765$		0			0
$766$	−12.0000			−0.433578
$767$	−6.00000	+	10.3923i	−0.216647	+	0.375244i
$768$		0			0
$769$	−25.0000	−	43.3013i	−0.901523	−	1.56148i	−0.825518	−	0.564376i	$-0.809117\pi$
$769$	−25.0000	−	43.3013i	−0.901523	−	1.56148i	−0.0760054	−	0.997107i	$-0.524217\pi$
$770$		0			0
$771$		0			0
$772$	−6.50000	+	11.2583i	−0.233940	+	0.405196i
$773$	−30.0000			−1.07903			−0.539513	−	0.841978i	$-0.681391\pi$
$773$	−30.0000			−1.07903			−0.539513	+	0.841978i	$0.681391\pi$
$774$		0			0
$775$		0			0
$776$	−5.50000	+	9.52628i	−0.197438	+	0.341974i
$777$		0			0
$778$	−18.0000	−	31.1769i	−0.645331	−	1.11775i
$779$	−7.50000	−	12.9904i	−0.268715	−	0.465429i
$780$		0			0
$781$	−9.00000	+	15.5885i	−0.322045	+	0.557799i
$782$	−18.0000			−0.643679
$783$		0			0
$784$	9.00000			0.321429
$785$		0			0
$786$		0			0
$787$	−2.00000	−	3.46410i	−0.0712923	−	0.123482i	0.828176	−	0.560469i	$-0.189379\pi$
$787$	−2.00000	−	3.46410i	−0.0712923	−	0.123482i	−0.899468	+	0.436987i	$0.856046\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	72.0000			2.56003
$792$		0			0
$793$	−40.0000			−1.42044
$794$	−4.00000	+	6.92820i	−0.141955	+	0.245873i
$795$		0			0
$796$	−10.0000	−	17.3205i	−0.354441	−	0.613909i
$797$	24.0000	+	41.5692i	0.850124	+	1.47246i	0.881096	+	0.472937i	$0.156806\pi$
$797$	24.0000	+	41.5692i	0.850124	+	1.47246i	−0.0309726	+	0.999520i	$0.509860\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$		0			0
$802$	33.0000			1.16527
$803$	10.5000	−	18.1865i	0.370537	−	0.641789i
$804$		0			0
$805$		0			0
$806$	4.00000	+	6.92820i	0.140894	+	0.244036i
$807$		0			0
$808$	6.00000	−	10.3923i	0.211079	−	0.365600i
$809$	39.0000			1.37117			0.685583	−	0.727994i	$-0.259547\pi$
$809$	39.0000			1.37117			0.685583	+	0.727994i	$0.259547\pi$
$810$		0			0
$811$	35.0000			1.22902			0.614508	−	0.788911i	$-0.289355\pi$
$811$	35.0000			1.22902			0.614508	+	0.788911i	$0.289355\pi$
$812$	12.0000	−	20.7846i	0.421117	−	0.729397i
$813$		0			0
$814$	−6.00000	−	10.3923i	−0.210300	−	0.364250i
$815$		0			0
$816$		0			0
$817$	27.5000	−	47.6314i	0.962103	−	1.66641i
$818$	31.0000			1.08389
$819$		0			0
$820$		0			0
$821$	−24.0000	+	41.5692i	−0.837606	+	1.45078i	0.0542853	+	0.998525i	$0.482712\pi$
$821$	−24.0000	+	41.5692i	−0.837606	+	1.45078i	−0.891891	+	0.452250i	$0.850621\pi$
$822$		0			0
$823$	−2.00000	−	3.46410i	−0.0697156	−	0.120751i	0.829060	−	0.559159i	$-0.188876\pi$
$823$	−2.00000	−	3.46410i	−0.0697156	−	0.120751i	−0.898776	+	0.438408i	$0.855543\pi$
$824$	2.00000	+	3.46410i	0.0696733	+	0.120678i
$825$		0			0
$826$	6.00000	−	10.3923i	0.208767	−	0.361595i
$827$	−36.0000			−1.25184			−0.625921	−	0.779886i	$-0.715277\pi$
$827$	−36.0000			−1.25184			−0.625921	+	0.779886i	$0.715277\pi$
$828$		0			0
$829$	44.0000			1.52818			0.764092	−	0.645108i	$-0.223188\pi$
$829$	44.0000			1.52818			0.764092	+	0.645108i	$0.223188\pi$
$830$		0			0
$831$		0			0
$832$	−2.00000	−	3.46410i	−0.0693375	−	0.120096i
$833$	−13.5000	−	23.3827i	−0.467747	−	0.810162i
$834$		0			0
$835$		0			0
$836$	−15.0000			−0.518786
$837$		0			0
$838$		0			0
$839$	−15.0000	+	25.9808i	−0.517858	+	0.896956i	0.481927	+	0.876211i	$0.339937\pi$
$839$	−15.0000	+	25.9808i	−0.517858	+	0.896956i	−0.999785	+	0.0207443i	$0.993396\pi$
$840$		0			0
$841$	−3.50000	−	6.06218i	−0.120690	−	0.209041i
$842$	1.00000	+	1.73205i	0.0344623	+	0.0596904i
$843$		0			0
$844$	2.00000	−	3.46410i	0.0688428	−	0.119239i
$845$		0			0
$846$		0			0
$847$	−8.00000			−0.274883
$848$	−3.00000	+	5.19615i	−0.103020	+	0.178437i
$849$		0			0
$850$		0			0
$851$	−12.0000	−	20.7846i	−0.411355	−	0.712487i
$852$		0			0
$853$	22.0000	−	38.1051i	0.753266	−	1.30469i	−0.192966	−	0.981205i	$-0.561811\pi$
$853$	22.0000	−	38.1051i	0.753266	−	1.30469i	0.946232	−	0.323489i	$-0.104856\pi$
$854$	40.0000			1.36877
$855$		0			0
$856$	9.00000			0.307614
$857$	9.00000	−	15.5885i	0.307434	−	0.532492i	−0.670366	−	0.742030i	$-0.733863\pi$
$857$	9.00000	−	15.5885i	0.307434	−	0.532492i	0.977800	+	0.209539i	$0.0671963\pi$
$858$		0			0
$859$	9.50000	+	16.4545i	0.324136	+	0.561420i	0.981337	−	0.192295i	$-0.0615932\pi$
$859$	9.50000	+	16.4545i	0.324136	+	0.561420i	−0.657201	+	0.753715i	$0.728260\pi$
$860$		0			0
$861$		0			0
$862$	−12.0000	+	20.7846i	−0.408722	+	0.707927i
$863$	−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
$865$		0			0
$866$	6.50000	−	11.2583i	0.220879	−	0.382574i
$867$		0			0
$868$	−4.00000	−	6.92820i	−0.135769	−	0.235159i
$869$	21.0000	+	36.3731i	0.712376	+	1.23387i
$870$		0			0
$871$	10.0000	−	17.3205i	0.338837	−	0.586883i
$872$	4.00000			0.135457
$873$		0			0
$874$	−30.0000			−1.01477
$875$		0			0
$876$		0			0
$877$	1.00000	+	1.73205i	0.0337676	+	0.0584872i	0.882415	−	0.470471i	$-0.155916\pi$
$877$	1.00000	+	1.73205i	0.0337676	+	0.0584872i	−0.848648	+	0.528958i	$0.822583\pi$
$878$	−5.00000	−	8.66025i	−0.168742	−	0.292269i
$879$		0			0
$880$		0			0
$881$	−30.0000			−1.01073			−0.505363	−	0.862907i	$-0.668641\pi$
$881$	−30.0000			−1.01073			−0.505363	+	0.862907i	$0.668641\pi$
$882$		0			0
$883$	−41.0000			−1.37976			−0.689880	−	0.723924i	$-0.742337\pi$
$883$	−41.0000			−1.37976			−0.689880	+	0.723924i	$0.742337\pi$
$884$	−6.00000	+	10.3923i	−0.201802	+	0.349531i
$885$		0			0
$886$	1.50000	+	2.59808i	0.0503935	+	0.0872841i
$887$	21.0000	+	36.3731i	0.705111	+	1.22129i	0.966651	+	0.256096i	$0.0824362\pi$
$887$	21.0000	+	36.3731i	0.705111	+	1.22129i	−0.261540	+	0.965193i	$0.584230\pi$
$888$		0			0
$889$	4.00000	−	6.92820i	0.134156	−	0.232364i
$890$		0			0
$891$		0			0
$892$	22.0000			0.736614
$893$		0			0
$894$		0			0
$895$		0			0
$896$	2.00000	+	3.46410i	0.0668153	+	0.115728i
$897$		0			0
$898$	7.50000	−	12.9904i	0.250278	−	0.433495i
$899$	−12.0000			−0.400222
$900$		0			0
$901$	18.0000			0.599667
$902$	−4.50000	+	7.79423i	−0.149834	+	0.259519i
$903$		0			0
$904$	9.00000	+	15.5885i	0.299336	+	0.518464i
$905$		0			0
$906$		0			0
$907$	2.50000	−	4.33013i	0.0830111	−	0.143780i	−0.821531	−	0.570164i	$-0.806880\pi$
$907$	2.50000	−	4.33013i	0.0830111	−	0.143780i	0.904542	+	0.426385i	$0.140213\pi$
$908$	−3.00000			−0.0995585
$909$		0			0
$910$		0			0
$911$	−15.0000	+	25.9808i	−0.496972	+	0.860781i	−0.999994	−	0.00349271i	$-0.998888\pi$
$911$	−15.0000	+	25.9808i	−0.496972	+	0.860781i	0.503022	+	0.864274i	$0.332222\pi$
$912$		0			0
$913$	18.0000	+	31.1769i	0.595713	+	1.03181i
$914$	0.500000	+	0.866025i	0.0165385	+	0.0286456i
$915$		0			0
$916$	−10.0000	+	17.3205i	−0.330409	+	0.572286i
$917$	−48.0000			−1.58510
$918$		0			0
$919$	−34.0000			−1.12156			−0.560778	−	0.827966i	$-0.689498\pi$
$919$	−34.0000			−1.12156			−0.560778	+	0.827966i	$0.689498\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	12.0000	+	20.7846i	0.394985	+	0.684134i
$924$		0			0
$925$		0			0
$926$	20.0000			0.657241
$927$		0			0
$928$	6.00000			0.196960
$929$	15.0000	−	25.9808i	0.492134	−	0.852401i	−0.507825	−	0.861460i	$-0.669550\pi$
$929$	15.0000	−	25.9808i	0.492134	−	0.852401i	0.999959	+	0.00905914i	$0.00288365\pi$
$930$		0			0
$931$	−22.5000	−	38.9711i	−0.737408	−	1.27723i
$932$	−10.5000	−	18.1865i	−0.343939	−	0.595720i
$933$		0			0
$934$	10.5000	−	18.1865i	0.343570	−	0.595082i
$935$		0			0
$936$		0			0
$937$	10.0000			0.326686			0.163343	−	0.986569i	$-0.447772\pi$
$937$	10.0000			0.326686			0.163343	+	0.986569i	$0.447772\pi$
$938$	−10.0000	+	17.3205i	−0.326512	+	0.565535i
$939$		0			0
$940$		0			0
$941$	12.0000	+	20.7846i	0.391189	+	0.677559i	0.992607	−	0.121376i	$-0.0387306\pi$
$941$	12.0000	+	20.7846i	0.391189	+	0.677559i	−0.601418	+	0.798935i	$0.705397\pi$
$942$		0			0
$943$	−9.00000	+	15.5885i	−0.293080	+	0.507630i
$944$	3.00000			0.0976417
$945$		0			0
$946$	−33.0000			−1.07292
$947$	−13.5000	+	23.3827i	−0.438691	+	0.759835i	−0.997589	−	0.0694014i	$-0.977891\pi$
$947$	−13.5000	+	23.3827i	−0.438691	+	0.759835i	0.558898	+	0.829237i	$0.311224\pi$
$948$		0			0
$949$	−14.0000	−	24.2487i	−0.454459	−	0.787146i
$950$		0			0
$951$		0			0
$952$	6.00000	−	10.3923i	0.194461	−	0.336817i
$953$	51.0000			1.65205			0.826026	−	0.563632i	$-0.190596\pi$
$953$	51.0000			1.65205			0.826026	+	0.563632i	$0.190596\pi$
$954$		0			0
$955$		0			0
$956$	3.00000	−	5.19615i	0.0970269	−	0.168056i
$957$		0			0
$958$	3.00000	+	5.19615i	0.0969256	+	0.167880i
$959$	18.0000	+	31.1769i	0.581250	+	1.00676i
$960$		0			0
$961$	13.5000	−	23.3827i	0.435484	−	0.754280i
$962$	−16.0000			−0.515861
$963$		0			0
$964$	17.0000			0.547533
$965$		0			0
$966$		0			0
$967$	−11.0000	−	19.0526i	−0.353736	−	0.612689i	0.633165	−	0.774017i	$-0.281756\pi$
$967$	−11.0000	−	19.0526i	−0.353736	−	0.612689i	−0.986901	+	0.161328i	$0.948422\pi$
$968$	−1.00000	−	1.73205i	−0.0321412	−	0.0556702i
$969$		0			0
$970$		0			0
$971$	−60.0000			−1.92549			−0.962746	−	0.270408i	$-0.912841\pi$
$971$	−60.0000			−1.92549			−0.962746	+	0.270408i	$0.912841\pi$
$972$		0			0
$973$	−4.00000			−0.128234
$974$	−1.00000	+	1.73205i	−0.0320421	+	0.0554985i
$975$		0			0
$976$	5.00000	+	8.66025i	0.160046	+	0.277208i
$977$	4.50000	+	7.79423i	0.143968	+	0.249359i	0.928987	−	0.370111i	$-0.120681\pi$
$977$	4.50000	+	7.79423i	0.143968	+	0.249359i	−0.785020	+	0.619471i	$0.787347\pi$
$978$		0			0
$979$	−9.00000	+	15.5885i	−0.287641	+	0.498209i
$980$		0			0
$981$		0			0
$982$	−33.0000			−1.05307
$983$	−18.0000	+	31.1769i	−0.574111	+	0.994389i	0.422027	+	0.906583i	$0.361319\pi$
$983$	−18.0000	+	31.1769i	−0.574111	+	0.994389i	−0.996138	+	0.0878058i	$0.972015\pi$
$984$		0			0
$985$		0			0
$986$	−9.00000	−	15.5885i	−0.286618	−	0.496438i
$987$		0			0
$988$	−10.0000	+	17.3205i	−0.318142	+	0.551039i
$989$	−66.0000			−2.09868
$990$		0			0
$991$	20.0000			0.635321			0.317660	−	0.948205i	$-0.397103\pi$
$991$	20.0000			0.635321			0.317660	+	0.948205i	$0.397103\pi$
$992$	1.00000	−	1.73205i	0.0317500	−	0.0549927i
$993$		0			0
$994$	−12.0000	−	20.7846i	−0.380617	−	0.659248i
$995$		0			0
$996$		0			0
$997$	13.0000	−	22.5167i	0.411714	−	0.713110i	−0.583363	−	0.812211i	$-0.698264\pi$
$997$	13.0000	−	22.5167i	0.411714	−	0.713110i	0.995077	+	0.0991016i	$0.0315969\pi$
$998$	31.0000			0.981288
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1350.2.e.g.451.1		2
3.2	odd	2		450.2.e.d.151.1		2
5.2	odd	4		1350.2.j.c.1099.2		4
5.3	odd	4		1350.2.j.c.1099.1		4
5.4	even	2		270.2.e.a.181.1		2
9.2	odd	6		4050.2.a.bi.1.1		1
9.4	even	3	inner	1350.2.e.g.901.1		2
9.5	odd	6		450.2.e.d.301.1		2
9.7	even	3		4050.2.a.q.1.1		1
15.2	even	4		450.2.j.a.349.1		4
15.8	even	4		450.2.j.a.349.2		4
15.14	odd	2		90.2.e.b.61.1	yes	2
20.19	odd	2		2160.2.q.d.721.1		2
45.2	even	12		4050.2.c.p.649.2		2
45.4	even	6		270.2.e.a.91.1		2
45.7	odd	12		4050.2.c.d.649.1		2
45.13	odd	12		1350.2.j.c.199.2		4
45.14	odd	6		90.2.e.b.31.1	✓	2
45.22	odd	12		1350.2.j.c.199.1		4
45.23	even	12		450.2.j.a.49.1		4
45.29	odd	6		810.2.a.a.1.1		1
45.32	even	12		450.2.j.a.49.2		4
45.34	even	6		810.2.a.e.1.1		1
45.38	even	12		4050.2.c.p.649.1		2
45.43	odd	12		4050.2.c.d.649.2		2
60.59	even	2		720.2.q.c.241.1		2
180.59	even	6		720.2.q.c.481.1		2
180.79	odd	6		6480.2.a.l.1.1		1
180.119	even	6		6480.2.a.z.1.1		1
180.139	odd	6		2160.2.q.d.1441.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
90.2.e.b.31.1	✓	2	45.14	odd	6
90.2.e.b.61.1	yes	2	15.14	odd	2
270.2.e.a.91.1		2	45.4	even	6
270.2.e.a.181.1		2	5.4	even	2
450.2.e.d.151.1		2	3.2	odd	2
450.2.e.d.301.1		2	9.5	odd	6
450.2.j.a.49.1		4	45.23	even	12
450.2.j.a.49.2		4	45.32	even	12
450.2.j.a.349.1		4	15.2	even	4
450.2.j.a.349.2		4	15.8	even	4
720.2.q.c.241.1		2	60.59	even	2
720.2.q.c.481.1		2	180.59	even	6
810.2.a.a.1.1		1	45.29	odd	6
810.2.a.e.1.1		1	45.34	even	6
1350.2.e.g.451.1		2	1.1	even	1	trivial
1350.2.e.g.901.1		2	9.4	even	3	inner
1350.2.j.c.199.1		4	45.22	odd	12
1350.2.j.c.199.2		4	45.13	odd	12
1350.2.j.c.1099.1		4	5.3	odd	4
1350.2.j.c.1099.2		4	5.2	odd	4
2160.2.q.d.721.1		2	20.19	odd	2
2160.2.q.d.1441.1		2	180.139	odd	6
4050.2.a.q.1.1		1	9.7	even	3
4050.2.a.bi.1.1		1	9.2	odd	6
4050.2.c.d.649.1		2	45.7	odd	12
4050.2.c.d.649.2		2	45.43	odd	12
4050.2.c.p.649.1		2	45.38	even	12
4050.2.c.p.649.2		2	45.2	even	12
6480.2.a.l.1.1		1	180.79	odd	6
6480.2.a.z.1.1		1	180.119	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:	\( N \)	\(=\)	\( 1350 = 2 \cdot 3^{3} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1350.e (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(0.500000 - 0.866025i) q^{2} +(-0.500000 - 0.866025i) q^{4} +(-2.00000 + 3.46410i) q^{7} -1.00000 q^{8} +O(q^{10})\)	\(q+(0.500000 - 0.866025i) q^{2} +(-0.500000 - 0.866025i) q^{4} +(-2.00000 + 3.46410i) q^{7} -1.00000 q^{8} +(1.50000 - 2.59808i) q^{11} +(-2.00000 - 3.46410i) q^{13} +(2.00000 + 3.46410i) q^{14} +(-0.500000 + 0.866025i) q^{16} +3.00000 q^{17} +5.00000 q^{19} +(-1.50000 - 2.59808i) q^{22} +(-3.00000 - 5.19615i) q^{23} -4.00000 q^{26} +4.00000 q^{28} +(3.00000 - 5.19615i) q^{29} +(-1.00000 - 1.73205i) q^{31} +(0.500000 + 0.866025i) q^{32} +(1.50000 - 2.59808i) q^{34} +4.00000 q^{37} +(2.50000 - 4.33013i) q^{38} +(-1.50000 - 2.59808i) q^{41} +(5.50000 - 9.52628i) q^{43} -3.00000 q^{44} -6.00000 q^{46} +(-4.50000 - 7.79423i) q^{49} +(-2.00000 + 3.46410i) q^{52} +6.00000 q^{53} +(2.00000 - 3.46410i) q^{56} +(-3.00000 - 5.19615i) q^{58} +(-1.50000 - 2.59808i) q^{59} +(5.00000 - 8.66025i) q^{61} -2.00000 q^{62} +1.00000 q^{64} +(2.50000 + 4.33013i) q^{67} +(-1.50000 - 2.59808i) q^{68} -6.00000 q^{71} +7.00000 q^{73} +(2.00000 - 3.46410i) q^{74} +(-2.50000 - 4.33013i) q^{76} +(6.00000 + 10.3923i) q^{77} +(-7.00000 + 12.1244i) q^{79} -3.00000 q^{82} +(-6.00000 + 10.3923i) q^{83} +(-5.50000 - 9.52628i) q^{86} +(-1.50000 + 2.59808i) q^{88} -6.00000 q^{89} +16.0000 q^{91} +(-3.00000 + 5.19615i) q^{92} +(5.50000 - 9.52628i) q^{97} -9.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + q^{2} - q^{4} - 4 q^{7} - 2 q^{8}+O(q^{10}) \) 2 * q + q^2 - q^4 - 4 * q^7 - 2 * q^8	\( 2 q + q^{2} - q^{4} - 4 q^{7} - 2 q^{8} + 3 q^{11} - 4 q^{13} + 4 q^{14} - q^{16} + 6 q^{17} + 10 q^{19} - 3 q^{22} - 6 q^{23} - 8 q^{26} + 8 q^{28} + 6 q^{29} - 2 q^{31} + q^{32} + 3 q^{34} + 8 q^{37} + 5 q^{38} - 3 q^{41} + 11 q^{43} - 6 q^{44} - 12 q^{46} - 9 q^{49} - 4 q^{52} + 12 q^{53} + 4 q^{56} - 6 q^{58} - 3 q^{59} + 10 q^{61} - 4 q^{62} + 2 q^{64} + 5 q^{67} - 3 q^{68} - 12 q^{71} + 14 q^{73} + 4 q^{74} - 5 q^{76} + 12 q^{77} - 14 q^{79} - 6 q^{82} - 12 q^{83} - 11 q^{86} - 3 q^{88} - 12 q^{89} + 32 q^{91} - 6 q^{92} + 11 q^{97} - 18 q^{98}+O(q^{100}) \) 2 * q + q^2 - q^4 - 4 * q^7 - 2 * q^8 + 3 * q^11 - 4 * q^13 + 4 * q^14 - q^16 + 6 * q^17 + 10 * q^19 - 3 * q^22 - 6 * q^23 - 8 * q^26 + 8 * q^28 + 6 * q^29 - 2 * q^31 + q^32 + 3 * q^34 + 8 * q^37 + 5 * q^38 - 3 * q^41 + 11 * q^43 - 6 * q^44 - 12 * q^46 - 9 * q^49 - 4 * q^52 + 12 * q^53 + 4 * q^56 - 6 * q^58 - 3 * q^59 + 10 * q^61 - 4 * q^62 + 2 * q^64 + 5 * q^67 - 3 * q^68 - 12 * q^71 + 14 * q^73 + 4 * q^74 - 5 * q^76 + 12 * q^77 - 14 * q^79 - 6 * q^82 - 12 * q^83 - 11 * q^86 - 3 * q^88 - 12 * q^89 + 32 * q^91 - 6 * q^92 + 11 * q^97 - 18 * q^98

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