LMFDB - Embedded newform 1350.2.q.b.557.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1350,2,Mod(143,1350)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1350, base_ring=CyclotomicField(12))

chi = DirichletCharacter(H, H._module([2, 9]))

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

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

chi := DirichletCharacter("1350.143");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1350 = 2 \cdot 3^{3} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1350.q (of order $12$, degree $4$, 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:	$8$
Relative dimension:	$2$ over $\Q(\zeta_{12})$
Coefficient field:	$\Q(\zeta_{24})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - x^{4} + 1 $ x^8 - x^4 + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 450)
Sato-Tate group:	$\mathrm{SU}(2)[C_{12}]$

Embedding invariants

Embedding label			557.2
Root			$0.258819 + 0.965926i$ of defining polynomial
Character	$\chi$	$=$	1350.557
Dual form			1350.2.q.b.143.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.965926 + 0.258819i) q^{2} +(0.866025 + 0.500000i) q^{4} +(-0.448288 + 1.67303i) q^{7} +(0.707107 + 0.707107i) q^{8} +O(q^{10})$	\(q+(0.965926 + 0.258819i) q^{2} +(0.866025 + 0.500000i) q^{4} +(-0.448288 + 1.67303i) q^{7} +(0.707107 + 0.707107i) q^{8} +(-3.00000 + 1.73205i) q^{11} +(0.896575 + 3.34607i) q^{13} +(-0.866025 + 1.50000i) q^{14} +(0.500000 + 0.866025i) q^{16} +(4.24264 - 4.24264i) q^{17} +2.00000i q^{19} +(-3.34607 + 0.896575i) q^{22} +(-2.89778 + 0.776457i) q^{23} +3.46410i q^{26} +(-1.22474 + 1.22474i) q^{28} +(4.33013 + 7.50000i) q^{29} +(-5.00000 + 8.66025i) q^{31} +(0.258819 + 0.965926i) q^{32} +(5.19615 - 3.00000i) q^{34} +(2.44949 + 2.44949i) q^{37} +(-0.517638 + 1.93185i) q^{38} +(-7.50000 - 4.33013i) q^{41} +(-10.0382 - 2.68973i) q^{43} -3.46410 q^{44} -3.00000 q^{46} +(2.89778 + 0.776457i) q^{47} +(3.46410 + 2.00000i) q^{49} +(-0.896575 + 3.34607i) q^{52} +(-1.50000 + 0.866025i) q^{56} +(2.24144 + 8.36516i) q^{58} +(3.46410 - 6.00000i) q^{59} +(6.50000 + 11.2583i) q^{61} +(-7.07107 + 7.07107i) q^{62} +1.00000i q^{64} +(11.7112 - 3.13801i) q^{67} +(5.79555 - 1.55291i) q^{68} +3.46410i q^{71} +(9.79796 - 9.79796i) q^{73} +(1.73205 + 3.00000i) q^{74} +(-1.00000 + 1.73205i) q^{76} +(-1.55291 - 5.79555i) q^{77} +(3.46410 - 2.00000i) q^{79} +(-6.12372 - 6.12372i) q^{82} +(-0.776457 + 2.89778i) q^{83} +(-9.00000 - 5.19615i) q^{86} +(-3.34607 - 0.896575i) q^{88} -1.73205 q^{89} -6.00000 q^{91} +(-2.89778 - 0.776457i) q^{92} +(2.59808 + 1.50000i) q^{94} +(1.79315 - 6.69213i) q^{97} +(2.82843 + 2.82843i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q+O(q^{10}) $ 8 * q	$ 8 q - 24 q^{11} + 4 q^{16} - 40 q^{31} - 60 q^{41} - 24 q^{46} - 12 q^{56} + 52 q^{61} - 8 q^{76} - 72 q^{86} - 48 q^{91}+O(q^{100}) $ 8 * q - 24 * q^11 + 4 * q^16 - 40 * q^31 - 60 * q^41 - 24 * q^46 - 12 * q^56 + 52 * q^61 - 8 * q^76 - 72 * q^86 - 48 * q^91

Display 100 coefficients 10 coefficients

Character values

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

$n$	$1001$	$1027$
$\chi(n)$	$e\left(\frac{5}{6}\right)$	$e\left(\frac{1}{4}\right)$

Coefficient data

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

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

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

$2$	0.965926	+	0.258819i	0.683013	+	0.183013i
$2$	0.965926	+	0.258819i	0.683013	+	0.183013i
$3$		0			0
$3$		0			0
$4$	0.866025	+	0.500000i	0.433013	+	0.250000i
$5$		0			0
$5$		0			0
$6$		0			0
$7$	−0.448288	+	1.67303i	−0.169437	+	0.632347i	0.827996	+	0.560734i	$0.189481\pi$
$7$	−0.448288	+	1.67303i	−0.169437	+	0.632347i	−0.997433	+	0.0716124i	$0.977186\pi$
$8$	0.707107	+	0.707107i	0.250000	+	0.250000i
$9$		0			0
$10$		0			0
$11$	−3.00000	+	1.73205i	−0.904534	+	0.522233i	−0.878668	−	0.477432i	$-0.841568\pi$
$11$	−3.00000	+	1.73205i	−0.904534	+	0.522233i	−0.0258656	+	0.999665i	$0.508234\pi$
$12$		0			0
$13$	0.896575	+	3.34607i	0.248665	+	0.928032i	0.971506	+	0.237016i	$0.0761695\pi$
$13$	0.896575	+	3.34607i	0.248665	+	0.928032i	−0.722840	+	0.691015i	$0.757164\pi$
$14$	−0.866025	+	1.50000i	−0.231455	+	0.400892i
$15$		0			0
$16$	0.500000	+	0.866025i	0.125000	+	0.216506i
$17$	4.24264	−	4.24264i	1.02899	−	1.02899i	0.0294245	−	0.999567i	$-0.490633\pi$
$17$	4.24264	−	4.24264i	1.02899	−	1.02899i	0.999567	−	0.0294245i	$-0.00936746\pi$
$18$		0			0
$19$			2.00000i			0.458831i	0.973329	+	0.229416i	$0.0736815\pi$
$19$			2.00000i			0.458831i	−0.973329	+	0.229416i	$0.926318\pi$
$20$		0			0
$21$		0			0
$22$	−3.34607	+	0.896575i	−0.713384	+	0.191151i
$23$	−2.89778	+	0.776457i	−0.604228	+	0.161903i	−0.547948	−	0.836512i	$-0.684591\pi$
$23$	−2.89778	+	0.776457i	−0.604228	+	0.161903i	−0.0562805	+	0.998415i	$0.517924\pi$
$24$		0			0
$25$		0			0
$26$			3.46410i			0.679366i
$27$		0			0
$28$	−1.22474	+	1.22474i	−0.231455	+	0.231455i
$29$	4.33013	+	7.50000i	0.804084	+	1.39272i	0.916907	+	0.399100i	$0.130677\pi$
$29$	4.33013	+	7.50000i	0.804084	+	1.39272i	−0.112823	+	0.993615i	$0.535989\pi$
$30$		0			0
$31$	−5.00000	+	8.66025i	−0.898027	+	1.55543i	−0.0680129	+	0.997684i	$0.521666\pi$
$31$	−5.00000	+	8.66025i	−0.898027	+	1.55543i	−0.830014	+	0.557743i	$0.811667\pi$
$32$	0.258819	+	0.965926i	0.0457532	+	0.170753i
$33$		0			0
$34$	5.19615	−	3.00000i	0.891133	−	0.514496i
$35$		0			0
$36$		0			0
$37$	2.44949	+	2.44949i	0.402694	+	0.402694i	0.879181	−	0.476488i	$-0.158090\pi$
$37$	2.44949	+	2.44949i	0.402694	+	0.402694i	−0.476488	+	0.879181i	$0.658090\pi$
$38$	−0.517638	+	1.93185i	−0.0839720	+	0.313388i
$39$		0			0
$40$		0			0
$41$	−7.50000	−	4.33013i	−1.17130	−	0.676252i	−0.217317	−	0.976101i	$-0.569730\pi$
$41$	−7.50000	−	4.33013i	−1.17130	−	0.676252i	−0.953987	+	0.299849i	$0.903064\pi$
$42$		0			0
$43$	−10.0382	−	2.68973i	−1.53081	−	0.410179i	−0.607527	−	0.794299i	$-0.707838\pi$
$43$	−10.0382	−	2.68973i	−1.53081	−	0.410179i	−0.923283	+	0.384120i	$0.874505\pi$
$44$	−3.46410			−0.522233
$45$		0			0
$46$	−3.00000			−0.442326
$47$	2.89778	+	0.776457i	0.422684	+	0.113258i	0.463890	−	0.885893i	$-0.346453\pi$
$47$	2.89778	+	0.776457i	0.422684	+	0.113258i	−0.0412058	+	0.999151i	$0.513120\pi$
$48$		0			0
$49$	3.46410	+	2.00000i	0.494872	+	0.285714i
$50$		0			0
$51$		0			0
$52$	−0.896575	+	3.34607i	−0.124333	+	0.464016i
$53$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$53$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$54$		0			0
$55$		0			0
$56$	−1.50000	+	0.866025i	−0.200446	+	0.115728i
$57$		0			0
$58$	2.24144	+	8.36516i	0.294315	+	1.09840i
$59$	3.46410	−	6.00000i	0.450988	−	0.781133i	−0.547460	−	0.836832i	$-0.684405\pi$
$59$	3.46410	−	6.00000i	0.450988	−	0.781133i	0.998448	+	0.0556984i	$0.0177385\pi$
$60$		0			0
$61$	6.50000	+	11.2583i	0.832240	+	1.44148i	0.896258	+	0.443533i	$0.146275\pi$
$61$	6.50000	+	11.2583i	0.832240	+	1.44148i	−0.0640184	+	0.997949i	$0.520392\pi$
$62$	−7.07107	+	7.07107i	−0.898027	+	0.898027i
$63$		0			0
$64$			1.00000i			0.125000i
$65$		0			0
$66$		0			0
$67$	11.7112	−	3.13801i	1.43075	−	0.383369i	0.541468	−	0.840721i	$-0.317869\pi$
$67$	11.7112	−	3.13801i	1.43075	−	0.383369i	0.889286	+	0.457352i	$0.151202\pi$
$68$	5.79555	−	1.55291i	0.702814	−	0.188319i
$69$		0			0
$70$		0			0
$71$			3.46410i			0.411113i	0.978645	+	0.205557i	$0.0659005\pi$
$71$			3.46410i			0.411113i	−0.978645	+	0.205557i	$0.934100\pi$
$72$		0			0
$73$	9.79796	−	9.79796i	1.14676	−	1.14676i	0.159579	−	0.987185i	$-0.448986\pi$
$73$	9.79796	−	9.79796i	1.14676	−	1.14676i	0.987185	−	0.159579i	$-0.0510137\pi$
$74$	1.73205	+	3.00000i	0.201347	+	0.348743i
$75$		0			0
$76$	−1.00000	+	1.73205i	−0.114708	+	0.198680i
$77$	−1.55291	−	5.79555i	−0.176971	−	0.660465i
$78$		0			0
$79$	3.46410	−	2.00000i	0.389742	−	0.225018i	−0.292306	−	0.956325i	$-0.594423\pi$
$79$	3.46410	−	2.00000i	0.389742	−	0.225018i	0.682048	+	0.731307i	$0.261089\pi$
$80$		0			0
$81$		0			0
$82$	−6.12372	−	6.12372i	−0.676252	−	0.676252i
$83$	−0.776457	+	2.89778i	−0.0852272	+	0.318072i	−0.995357	−	0.0962507i	$-0.969315\pi$
$83$	−0.776457	+	2.89778i	−0.0852272	+	0.318072i	0.910130	+	0.414323i	$0.135982\pi$
$84$		0			0
$85$		0			0
$86$	−9.00000	−	5.19615i	−0.970495	−	0.560316i
$87$		0			0
$88$	−3.34607	−	0.896575i	−0.356692	−	0.0955753i
$89$	−1.73205			−0.183597			−0.0917985	−	0.995778i	$-0.529262\pi$
$89$	−1.73205			−0.183597			−0.0917985	+	0.995778i	$0.529262\pi$
$90$		0			0
$91$	−6.00000			−0.628971
$92$	−2.89778	−	0.776457i	−0.302114	−	0.0809513i
$93$		0			0
$94$	2.59808	+	1.50000i	0.267971	+	0.154713i
$95$		0			0
$96$		0			0
$97$	1.79315	−	6.69213i	0.182067	−	0.679483i	−0.813173	−	0.582023i	$-0.802261\pi$
$97$	1.79315	−	6.69213i	0.182067	−	0.679483i	0.995239	−	0.0974602i	$-0.0310719\pi$
$98$	2.82843	+	2.82843i	0.285714	+	0.285714i
$99$		0			0
$100$		0			0
$101$	−12.0000	+	6.92820i	−1.19404	+	0.689382i	−0.959221	−	0.282656i	$-0.908784\pi$
$101$	−12.0000	+	6.92820i	−1.19404	+	0.689382i	−0.234823	+	0.972038i	$0.575451\pi$
$102$		0			0
$103$	−0.896575	−	3.34607i	−0.0883422	−	0.329698i	0.907584	−	0.419871i	$-0.137925\pi$
$103$	−0.896575	−	3.34607i	−0.0883422	−	0.329698i	−0.995926	+	0.0901732i	$0.971258\pi$
$104$	−1.73205	+	3.00000i	−0.169842	+	0.294174i
$105$		0			0
$106$		0			0
$107$	−2.12132	+	2.12132i	−0.205076	+	0.205076i	−0.802171	−	0.597095i	$-0.796322\pi$
$107$	−2.12132	+	2.12132i	−0.205076	+	0.205076i	0.597095	+	0.802171i	$0.296322\pi$
$108$		0			0
$109$		−	5.00000i		−	0.478913i	−0.970907	−	0.239457i	$-0.923031\pi$
$109$		−	5.00000i		−	0.478913i	0.970907	−	0.239457i	$-0.0769693\pi$
$110$		0			0
$111$		0			0
$112$	−1.67303	+	0.448288i	−0.158087	+	0.0423592i
$113$		0			0		−0.258819	−	0.965926i	$-0.583333\pi$
$113$		0			0		0.258819	+	0.965926i	$0.416667\pi$
$114$		0			0
$115$		0			0
$116$			8.66025i			0.804084i
$117$		0			0
$118$	4.89898	−	4.89898i	0.450988	−	0.450988i
$119$	5.19615	+	9.00000i	0.476331	+	0.825029i
$120$		0			0
$121$	0.500000	−	0.866025i	0.0454545	−	0.0787296i
$122$	3.36465	+	12.5570i	0.304621	+	1.13686i
$123$		0			0
$124$	−8.66025	+	5.00000i	−0.777714	+	0.449013i
$125$		0			0
$126$		0			0
$127$	−3.67423	−	3.67423i	−0.326036	−	0.326036i	0.525041	−	0.851077i	$-0.324050\pi$
$127$	−3.67423	−	3.67423i	−0.326036	−	0.326036i	−0.851077	+	0.525041i	$0.824050\pi$
$128$	−0.258819	+	0.965926i	−0.0228766	+	0.0853766i
$129$		0			0
$130$		0			0
$131$	6.00000	+	3.46410i	0.524222	+	0.302660i	0.738661	−	0.674078i	$-0.235459\pi$
$131$	6.00000	+	3.46410i	0.524222	+	0.302660i	−0.214438	+	0.976738i	$0.568792\pi$
$132$		0			0
$133$	−3.34607	−	0.896575i	−0.290141	−	0.0777430i
$134$	12.1244			1.04738
$135$		0			0
$136$	6.00000			0.514496
$137$	−11.5911	−	3.10583i	−0.990295	−	0.265349i	−0.272921	−	0.962037i	$-0.587990\pi$
$137$	−11.5911	−	3.10583i	−0.990295	−	0.265349i	−0.717375	+	0.696688i	$0.754656\pi$
$138$		0			0
$139$	3.46410	+	2.00000i	0.293821	+	0.169638i	0.639664	−	0.768655i	$-0.279074\pi$
$139$	3.46410	+	2.00000i	0.293821	+	0.169638i	−0.345843	+	0.938293i	$0.612407\pi$
$140$		0			0
$141$		0			0
$142$	−0.896575	+	3.34607i	−0.0752389	+	0.280796i
$143$	−8.48528	−	8.48528i	−0.709575	−	0.709575i
$144$		0			0
$145$		0			0
$146$	12.0000	−	6.92820i	0.993127	−	0.573382i
$147$		0			0
$148$	0.896575	+	3.34607i	0.0736980	+	0.275045i
$149$	−2.59808	+	4.50000i	−0.212843	+	0.368654i	−0.952603	−	0.304216i	$-0.901606\pi$
$149$	−2.59808	+	4.50000i	−0.212843	+	0.368654i	0.739760	+	0.672870i	$0.234939\pi$
$150$		0			0
$151$	−5.00000	−	8.66025i	−0.406894	−	0.704761i	0.587646	−	0.809118i	$-0.300055\pi$
$151$	−5.00000	−	8.66025i	−0.406894	−	0.704761i	−0.994540	+	0.104357i	$0.966722\pi$
$152$	−1.41421	+	1.41421i	−0.114708	+	0.114708i
$153$		0			0
$154$		−	6.00000i		−	0.483494i
$155$		0			0
$156$		0			0
$157$	20.0764	−	5.37945i	1.60227	−	0.429327i	0.656543	−	0.754288i	$-0.272018\pi$
$157$	20.0764	−	5.37945i	1.60227	−	0.429327i	0.945727	+	0.324961i	$0.105351\pi$
$158$	3.86370	−	1.03528i	0.307380	−	0.0823622i
$159$		0			0
$160$		0			0
$161$		−	5.19615i		−	0.409514i
$162$		0			0
$163$	12.2474	−	12.2474i	0.959294	−	0.959294i	−0.0399091	−	0.999203i	$-0.512707\pi$
$163$	12.2474	−	12.2474i	0.959294	−	0.959294i	0.999203	+	0.0399091i	$0.0127068\pi$
$164$	−4.33013	−	7.50000i	−0.338126	−	0.585652i
$165$		0			0
$166$	−1.50000	+	2.59808i	−0.116423	+	0.201650i
$167$	−2.32937	−	8.69333i	−0.180252	−	0.672710i	−0.995597	−	0.0937349i	$-0.970119\pi$
$167$	−2.32937	−	8.69333i	−0.180252	−	0.672710i	0.815345	−	0.578975i	$-0.196547\pi$
$168$		0			0
$169$	0.866025	−	0.500000i	0.0666173	−	0.0384615i
$170$		0			0
$171$		0			0
$172$	−7.34847	−	7.34847i	−0.560316	−	0.560316i
$173$	6.21166	−	23.1822i	0.472264	−	1.76251i	−0.159344	−	0.987223i	$-0.550938\pi$
$173$	6.21166	−	23.1822i	0.472264	−	1.76251i	0.631607	−	0.775288i	$-0.282395\pi$
$174$		0			0
$175$		0			0
$176$	−3.00000	−	1.73205i	−0.226134	−	0.130558i
$177$		0			0
$178$	−1.67303	−	0.448288i	−0.125399	−	0.0336006i
$179$	−24.2487			−1.81243			−0.906217	−	0.422813i	$-0.861043\pi$
$179$	−24.2487			−1.81243			−0.906217	+	0.422813i	$0.861043\pi$
$180$		0			0
$181$	7.00000			0.520306			0.260153	−	0.965567i	$-0.416227\pi$
$181$	7.00000			0.520306			0.260153	+	0.965567i	$0.416227\pi$
$182$	−5.79555	−	1.55291i	−0.429595	−	0.115110i
$183$		0			0
$184$	−2.59808	−	1.50000i	−0.191533	−	0.110581i
$185$		0			0
$186$		0			0
$187$	−5.37945	+	20.0764i	−0.393385	+	1.46813i
$188$	2.12132	+	2.12132i	0.154713	+	0.154713i
$189$		0			0
$190$		0			0
$191$	15.0000	−	8.66025i	1.08536	−	0.626634i	0.153024	−	0.988222i	$-0.451099\pi$
$191$	15.0000	−	8.66025i	1.08536	−	0.626634i	0.932338	+	0.361588i	$0.117765\pi$
$192$		0			0
$193$	4.48288	+	16.7303i	0.322685	+	1.20428i	0.916619	+	0.399762i	$0.130907\pi$
$193$	4.48288	+	16.7303i	0.322685	+	1.20428i	−0.593934	+	0.804513i	$0.702426\pi$
$194$	3.46410	−	6.00000i	0.248708	−	0.430775i
$195$		0			0
$196$	2.00000	+	3.46410i	0.142857	+	0.247436i
$197$	8.48528	−	8.48528i	0.604551	−	0.604551i	−0.336966	−	0.941517i	$-0.609401\pi$
$197$	8.48528	−	8.48528i	0.604551	−	0.604551i	0.941517	+	0.336966i	$0.109401\pi$
$198$		0			0
$199$			10.0000i			0.708881i	0.935079	+	0.354441i	$0.115329\pi$
$199$			10.0000i			0.708881i	−0.935079	+	0.354441i	$0.884671\pi$
$200$		0			0
$201$		0			0
$202$	−13.3843	+	3.58630i	−0.941713	+	0.252331i
$203$	−14.4889	+	3.88229i	−1.01692	+	0.272483i
$204$		0			0
$205$		0			0
$206$		−	3.46410i		−	0.241355i
$207$		0			0
$208$	−2.44949	+	2.44949i	−0.169842	+	0.169842i
$209$	−3.46410	−	6.00000i	−0.239617	−	0.415029i
$210$		0			0
$211$	11.0000	−	19.0526i	0.757271	−	1.31163i	−0.186966	−	0.982366i	$-0.559865\pi$
$211$	11.0000	−	19.0526i	0.757271	−	1.31163i	0.944237	−	0.329266i	$-0.106801\pi$
$212$		0			0
$213$		0			0
$214$	−2.59808	+	1.50000i	−0.177601	+	0.102538i
$215$		0			0
$216$		0			0
$217$	−12.2474	−	12.2474i	−0.831411	−	0.831411i
$218$	1.29410	−	4.82963i	0.0876472	−	0.327104i
$219$		0			0
$220$		0			0
$221$	18.0000	+	10.3923i	1.21081	+	0.699062i
$222$		0			0
$223$	−15.0573	−	4.03459i	−1.00831	−	0.270176i	−0.283387	−	0.959005i	$-0.591458\pi$
$223$	−15.0573	−	4.03459i	−1.00831	−	0.270176i	−0.724923	+	0.688829i	$0.758125\pi$
$224$	−1.73205			−0.115728
$225$		0			0
$226$		0			0
$227$	23.1822	+	6.21166i	1.53866	+	0.412282i	0.925832	−	0.377936i	$-0.123366\pi$
$227$	23.1822	+	6.21166i	1.53866	+	0.412282i	0.612826	+	0.790218i	$0.290033\pi$
$228$		0			0
$229$	−4.33013	−	2.50000i	−0.286143	−	0.165205i	0.350058	−	0.936728i	$-0.386162\pi$
$229$	−4.33013	−	2.50000i	−0.286143	−	0.165205i	−0.636201	+	0.771523i	$0.719495\pi$
$230$		0			0
$231$		0			0
$232$	−2.24144	+	8.36516i	−0.147158	+	0.549200i
$233$	−4.24264	−	4.24264i	−0.277945	−	0.277945i	0.554343	−	0.832288i	$-0.312969\pi$
$233$	−4.24264	−	4.24264i	−0.277945	−	0.277945i	−0.832288	+	0.554343i	$0.812969\pi$
$234$		0			0
$235$		0			0
$236$	6.00000	−	3.46410i	0.390567	−	0.225494i
$237$		0			0
$238$	2.68973	+	10.0382i	0.174349	+	0.650680i
$239$	10.3923	−	18.0000i	0.672222	−	1.16432i	−0.305050	−	0.952336i	$-0.598673\pi$
$239$	10.3923	−	18.0000i	0.672222	−	1.16432i	0.977273	−	0.211987i	$-0.0679934\pi$
$240$		0			0
$241$	−8.50000	−	14.7224i	−0.547533	−	0.948355i	−0.998443	−	0.0557856i	$-0.982234\pi$
$241$	−8.50000	−	14.7224i	−0.547533	−	0.948355i	0.450910	−	0.892570i	$-0.351100\pi$
$242$	0.707107	−	0.707107i	0.0454545	−	0.0454545i
$243$		0			0
$244$			13.0000i			0.832240i
$245$		0			0
$246$		0			0
$247$	−6.69213	+	1.79315i	−0.425810	+	0.114095i
$248$	−9.65926	+	2.58819i	−0.613364	+	0.164350i
$249$		0			0
$250$		0			0
$251$		−	10.3923i		−	0.655956i	−0.944685	−	0.327978i	$-0.893633\pi$
$251$		−	10.3923i		−	0.655956i	0.944685	−	0.327978i	$-0.106367\pi$
$252$		0			0
$253$	7.34847	−	7.34847i	0.461994	−	0.461994i
$254$	−2.59808	−	4.50000i	−0.163018	−	0.282355i
$255$		0			0
$256$	−0.500000	+	0.866025i	−0.0312500	+	0.0541266i
$257$		0			0		0.965926	−	0.258819i	$-0.0833333\pi$
$257$		0			0		−0.965926	+	0.258819i	$0.916667\pi$
$258$		0			0
$259$	−5.19615	+	3.00000i	−0.322873	+	0.186411i
$260$		0			0
$261$		0			0
$262$	4.89898	+	4.89898i	0.302660	+	0.302660i
$263$	−6.21166	+	23.1822i	−0.383027	+	1.42948i	0.458226	+	0.888836i	$0.348485\pi$
$263$	−6.21166	+	23.1822i	−0.383027	+	1.42948i	−0.841253	+	0.540641i	$0.818182\pi$
$264$		0			0
$265$		0			0
$266$	−3.00000	−	1.73205i	−0.183942	−	0.106199i
$267$		0			0
$268$	11.7112	+	3.13801i	0.715377	+	0.191685i
$269$	−1.73205			−0.105605			−0.0528025	−	0.998605i	$-0.516815\pi$
$269$	−1.73205			−0.105605			−0.0528025	+	0.998605i	$0.516815\pi$
$270$		0			0
$271$	−10.0000			−0.607457			−0.303728	−	0.952759i	$-0.598232\pi$
$271$	−10.0000			−0.607457			−0.303728	+	0.952759i	$0.598232\pi$
$272$	5.79555	+	1.55291i	0.351407	+	0.0941593i
$273$		0			0
$274$	−10.3923	−	6.00000i	−0.627822	−	0.362473i
$275$		0			0
$276$		0			0
$277$	−2.68973	+	10.0382i	−0.161610	+	0.603137i	0.836838	+	0.547450i	$0.184401\pi$
$277$	−2.68973	+	10.0382i	−0.161610	+	0.603137i	−0.998448	+	0.0556866i	$0.982265\pi$
$278$	2.82843	+	2.82843i	0.169638	+	0.169638i
$279$		0			0
$280$		0			0
$281$	−4.50000	+	2.59808i	−0.268447	+	0.154988i	−0.628182	−	0.778067i	$-0.716201\pi$
$281$	−4.50000	+	2.59808i	−0.268447	+	0.154988i	0.359734	+	0.933055i	$0.382867\pi$
$282$		0			0
$283$	1.34486	+	5.01910i	0.0799438	+	0.298354i	0.994309	−	0.106537i	$-0.0339761\pi$
$283$	1.34486	+	5.01910i	0.0799438	+	0.298354i	−0.914365	+	0.404891i	$0.867309\pi$
$284$	−1.73205	+	3.00000i	−0.102778	+	0.178017i
$285$		0			0
$286$	−6.00000	−	10.3923i	−0.354787	−	0.614510i
$287$	10.6066	−	10.6066i	0.626088	−	0.626088i
$288$		0			0
$289$		−	19.0000i		−	1.11765i
$290$		0			0
$291$		0			0
$292$	13.3843	−	3.58630i	0.783255	−	0.209872i
$293$	28.9778	−	7.76457i	1.69290	−	0.453611i	0.721764	−	0.692139i	$-0.243332\pi$
$293$	28.9778	−	7.76457i	1.69290	−	0.453611i	0.971136	+	0.238528i	$0.0766649\pi$
$294$		0			0
$295$		0			0
$296$			3.46410i			0.201347i
$297$		0			0
$298$	−3.67423	+	3.67423i	−0.212843	+	0.212843i
$299$	−5.19615	−	9.00000i	−0.300501	−	0.520483i
$300$		0			0
$301$	9.00000	−	15.5885i	0.518751	−	0.898504i
$302$	−2.58819	−	9.65926i	−0.148934	−	0.555828i
$303$		0			0
$304$	−1.73205	+	1.00000i	−0.0993399	+	0.0573539i
$305$		0			0
$306$		0			0
$307$	3.67423	+	3.67423i	0.209700	+	0.209700i	0.804140	−	0.594440i	$-0.202626\pi$
$307$	3.67423	+	3.67423i	0.209700	+	0.209700i	−0.594440	+	0.804140i	$0.702626\pi$
$308$	1.55291	−	5.79555i	0.0884855	−	0.330232i
$309$		0			0
$310$		0			0
$311$	21.0000	+	12.1244i	1.19080	+	0.687509i	0.958488	−	0.285132i	$-0.0920375\pi$
$311$	21.0000	+	12.1244i	1.19080	+	0.687509i	0.232313	+	0.972641i	$0.425371\pi$
$312$		0			0
$313$	3.34607	+	0.896575i	0.189131	+	0.0506774i	0.352141	−	0.935947i	$-0.385454\pi$
$313$	3.34607	+	0.896575i	0.189131	+	0.0506774i	−0.163010	+	0.986624i	$0.552120\pi$
$314$	20.7846			1.17294
$315$		0			0
$316$	4.00000			0.225018
$317$	−5.79555	−	1.55291i	−0.325511	−	0.0872204i	0.0923631	−	0.995725i	$-0.470558\pi$
$317$	−5.79555	−	1.55291i	−0.325511	−	0.0872204i	−0.417874	+	0.908505i	$0.637225\pi$
$318$		0			0
$319$	−25.9808	−	15.0000i	−1.45464	−	0.839839i
$320$		0			0
$321$		0			0
$322$	1.34486	−	5.01910i	0.0749463	−	0.279703i
$323$	8.48528	+	8.48528i	0.472134	+	0.472134i
$324$		0			0
$325$		0			0
$326$	15.0000	−	8.66025i	0.830773	−	0.479647i
$327$		0			0
$328$	−2.24144	−	8.36516i	−0.123763	−	0.461889i
$329$	−2.59808	+	4.50000i	−0.143237	+	0.248093i
$330$		0			0
$331$	−4.00000	−	6.92820i	−0.219860	−	0.380808i	0.734905	−	0.678170i	$-0.237227\pi$
$331$	−4.00000	−	6.92820i	−0.219860	−	0.380808i	−0.954765	+	0.297361i	$0.903893\pi$
$332$	−2.12132	+	2.12132i	−0.116423	+	0.116423i
$333$		0			0
$334$		−	9.00000i		−	0.492458i
$335$		0			0
$336$		0			0
$337$	13.3843	−	3.58630i	0.729087	−	0.195358i	0.124864	−	0.992174i	$-0.460150\pi$
$337$	13.3843	−	3.58630i	0.729087	−	0.195358i	0.604223	+	0.796815i	$0.293484\pi$
$338$	0.965926	−	0.258819i	0.0525394	−	0.0140779i
$339$		0			0
$340$		0			0
$341$		−	34.6410i		−	1.87592i
$342$		0			0
$343$	−13.4722	+	13.4722i	−0.727430	+	0.727430i
$344$	−5.19615	−	9.00000i	−0.280158	−	0.485247i
$345$		0			0
$346$	12.0000	−	20.7846i	0.645124	−	1.11739i
$347$	3.10583	+	11.5911i	0.166730	+	0.622243i	0.997813	+	0.0660960i	$0.0210543\pi$
$347$	3.10583	+	11.5911i	0.166730	+	0.622243i	−0.831084	+	0.556147i	$0.812279\pi$
$348$		0			0
$349$	−0.866025	+	0.500000i	−0.0463573	+	0.0267644i	−0.523000	−	0.852333i	$-0.675187\pi$
$349$	−0.866025	+	0.500000i	−0.0463573	+	0.0267644i	0.476642	+	0.879097i	$0.341854\pi$
$350$		0			0
$351$		0			0
$352$	−2.44949	−	2.44949i	−0.130558	−	0.130558i
$353$	−6.21166	+	23.1822i	−0.330613	+	1.23387i	0.577934	+	0.816083i	$0.303859\pi$
$353$	−6.21166	+	23.1822i	−0.330613	+	1.23387i	−0.908547	+	0.417782i	$0.862808\pi$
$354$		0			0
$355$		0			0
$356$	−1.50000	−	0.866025i	−0.0794998	−	0.0458993i
$357$		0			0
$358$	−23.4225	−	6.27603i	−1.23792	−	0.331698i
$359$	17.3205			0.914141			0.457071	−	0.889430i	$-0.348899\pi$
$359$	17.3205			0.914141			0.457071	+	0.889430i	$0.348899\pi$
$360$		0			0
$361$	15.0000			0.789474
$362$	6.76148	+	1.81173i	0.355376	+	0.0952226i
$363$		0			0
$364$	−5.19615	−	3.00000i	−0.272352	−	0.157243i
$365$		0			0
$366$		0			0
$367$	−0.896575	+	3.34607i	−0.0468009	+	0.174663i	−0.985370	−	0.170427i	$-0.945485\pi$
$367$	−0.896575	+	3.34607i	−0.0468009	+	0.174663i	0.938569	+	0.345091i	$0.112152\pi$
$368$	−2.12132	−	2.12132i	−0.110581	−	0.110581i
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$	3.58630	+	13.3843i	0.185692	+	0.693011i	0.994481	+	0.104913i	$0.0334564\pi$
$373$	3.58630	+	13.3843i	0.185692	+	0.693011i	−0.808790	+	0.588098i	$0.799877\pi$
$374$	−10.3923	+	18.0000i	−0.537373	+	0.930758i
$375$		0			0
$376$	1.50000	+	2.59808i	0.0773566	+	0.133986i
$377$	−21.2132	+	21.2132i	−1.09254	+	1.09254i
$378$		0			0
$379$			8.00000i			0.410932i	0.978664	+	0.205466i	$0.0658711\pi$
$379$			8.00000i			0.410932i	−0.978664	+	0.205466i	$0.934129\pi$
$380$		0			0
$381$		0			0
$382$	16.7303	−	4.48288i	0.855998	−	0.229364i
$383$	−23.1822	+	6.21166i	−1.18456	+	0.317401i	−0.796732	−	0.604333i	$-0.793440\pi$
$383$	−23.1822	+	6.21166i	−1.18456	+	0.317401i	−0.387824	+	0.921733i	$0.626773\pi$
$384$		0			0
$385$		0			0
$386$			17.3205i			0.881591i
$387$		0			0
$388$	4.89898	−	4.89898i	0.248708	−	0.248708i
$389$	−12.9904	−	22.5000i	−0.658638	−	1.14080i	−0.980968	−	0.194168i	$-0.937799\pi$
$389$	−12.9904	−	22.5000i	−0.658638	−	1.14080i	0.322330	−	0.946627i	$-0.395534\pi$
$390$		0			0
$391$	−9.00000	+	15.5885i	−0.455150	+	0.788342i
$392$	1.03528	+	3.86370i	0.0522893	+	0.195146i
$393$		0			0
$394$	10.3923	−	6.00000i	0.523557	−	0.302276i
$395$		0			0
$396$		0			0
$397$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$397$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$398$	−2.58819	+	9.65926i	−0.129734	+	0.484175i
$399$		0			0
$400$		0			0
$401$	18.0000	+	10.3923i	0.898877	+	0.518967i	0.876836	−	0.480790i	$-0.159650\pi$
$401$	18.0000	+	10.3923i	0.898877	+	0.518967i	0.0220414	+	0.999757i	$0.492983\pi$
$402$		0			0
$403$	−33.4607	−	8.96575i	−1.66679	−	0.446616i
$404$	−13.8564			−0.689382
$405$		0			0
$406$	−15.0000			−0.744438
$407$	−11.5911	−	3.10583i	−0.574550	−	0.153950i
$408$		0			0
$409$	−1.73205	−	1.00000i	−0.0856444	−	0.0494468i	0.456566	−	0.889689i	$-0.349079\pi$
$409$	−1.73205	−	1.00000i	−0.0856444	−	0.0494468i	−0.542211	+	0.840243i	$0.682412\pi$
$410$		0			0
$411$		0			0
$412$	0.896575	−	3.34607i	0.0441711	−	0.164849i
$413$	8.48528	+	8.48528i	0.417533	+	0.417533i
$414$		0			0
$415$		0			0
$416$	−3.00000	+	1.73205i	−0.147087	+	0.0849208i
$417$		0			0
$418$	−1.79315	−	6.69213i	−0.0877059	−	0.327323i
$419$	5.19615	−	9.00000i	0.253849	−	0.439679i	−0.710734	−	0.703461i	$-0.751637\pi$
$419$	5.19615	−	9.00000i	0.253849	−	0.439679i	0.964582	+	0.263783i	$0.0849701\pi$
$420$		0			0
$421$	−5.00000	−	8.66025i	−0.243685	−	0.422075i	0.718076	−	0.695965i	$-0.245023\pi$
$421$	−5.00000	−	8.66025i	−0.243685	−	0.422075i	−0.961761	+	0.273890i	$0.911690\pi$
$422$	15.5563	−	15.5563i	0.757271	−	0.757271i
$423$		0			0
$424$		0			0
$425$		0			0
$426$		0			0
$427$	−21.7494	+	5.82774i	−1.05253	+	0.282024i
$428$	−2.89778	+	0.776457i	−0.140069	+	0.0375315i
$429$		0			0
$430$		0			0
$431$		−	6.92820i		−	0.333720i	−0.985981	−	0.166860i	$-0.946637\pi$
$431$		−	6.92820i		−	0.333720i	0.985981	−	0.166860i	$-0.0533628\pi$
$432$		0			0
$433$	−7.34847	+	7.34847i	−0.353145	+	0.353145i	−0.861278	−	0.508133i	$-0.830336\pi$
$433$	−7.34847	+	7.34847i	−0.353145	+	0.353145i	0.508133	+	0.861278i	$0.330336\pi$
$434$	−8.66025	−	15.0000i	−0.415705	−	0.720023i
$435$		0			0
$436$	2.50000	−	4.33013i	0.119728	−	0.207375i
$437$	−1.55291	−	5.79555i	−0.0742860	−	0.277239i
$438$		0			0
$439$	−17.3205	+	10.0000i	−0.826663	+	0.477274i	−0.852709	−	0.522387i	$-0.825042\pi$
$439$	−17.3205	+	10.0000i	−0.826663	+	0.477274i	0.0260459	+	0.999661i	$0.491708\pi$
$440$		0			0
$441$		0			0
$442$	14.6969	+	14.6969i	0.699062	+	0.699062i
$443$	2.32937	−	8.69333i	0.110672	−	0.413033i	−0.888255	−	0.459351i	$-0.848082\pi$
$443$	2.32937	−	8.69333i	0.110672	−	0.413033i	0.998927	+	0.0463181i	$0.0147488\pi$
$444$		0			0
$445$		0			0
$446$	−13.5000	−	7.79423i	−0.639244	−	0.369067i
$447$		0			0
$448$	−1.67303	−	0.448288i	−0.0790434	−	0.0211796i
$449$	13.8564			0.653924			0.326962	−	0.945037i	$-0.393975\pi$
$449$	13.8564			0.653924			0.326962	+	0.945037i	$0.393975\pi$
$450$		0			0
$451$	30.0000			1.41264
$452$		0			0
$453$		0			0
$454$	20.7846	+	12.0000i	0.975470	+	0.563188i
$455$		0			0
$456$		0			0
$457$	8.06918	−	30.1146i	0.377460	−	1.40870i	−0.472256	−	0.881461i	$-0.656560\pi$
$457$	8.06918	−	30.1146i	0.377460	−	1.40870i	0.849717	−	0.527240i	$-0.176773\pi$
$458$	−3.53553	−	3.53553i	−0.165205	−	0.165205i
$459$		0			0
$460$		0			0
$461$	−10.5000	+	6.06218i	−0.489034	+	0.282344i	−0.724174	−	0.689618i	$-0.757779\pi$
$461$	−10.5000	+	6.06218i	−0.489034	+	0.282344i	0.235140	+	0.971962i	$0.424445\pi$
$462$		0			0
$463$	−4.48288	−	16.7303i	−0.208337	−	0.777524i	−0.988406	−	0.151831i	$-0.951483\pi$
$463$	−4.48288	−	16.7303i	−0.208337	−	0.777524i	0.780069	−	0.625693i	$-0.215184\pi$
$464$	−4.33013	+	7.50000i	−0.201021	+	0.348179i
$465$		0			0
$466$	−3.00000	−	5.19615i	−0.138972	−	0.240707i
$467$	−25.4558	+	25.4558i	−1.17796	+	1.17796i	−0.197692	+	0.980264i	$0.563345\pi$
$467$	−25.4558	+	25.4558i	−1.17796	+	1.17796i	−0.980264	+	0.197692i	$0.936655\pi$
$468$		0			0
$469$			21.0000i			0.969690i
$470$		0			0
$471$		0			0
$472$	6.69213	−	1.79315i	0.308030	−	0.0825365i
$473$	34.7733	−	9.31749i	1.59888	−	0.428418i
$474$		0			0
$475$		0			0
$476$			10.3923i			0.476331i
$477$		0			0
$478$	14.6969	−	14.6969i	0.672222	−	0.672222i
$479$	13.8564	+	24.0000i	0.633115	+	1.09659i	0.986911	+	0.161265i	$0.0515575\pi$
$479$	13.8564	+	24.0000i	0.633115	+	1.09659i	−0.353796	+	0.935323i	$0.615109\pi$
$480$		0			0
$481$	−6.00000	+	10.3923i	−0.273576	+	0.473848i
$482$	−4.39992	−	16.4207i	−0.200411	−	0.747944i
$483$		0			0
$484$	0.866025	−	0.500000i	0.0393648	−	0.0227273i
$485$		0			0
$486$		0			0
$487$	17.1464	+	17.1464i	0.776979	+	0.776979i	0.979316	−	0.202337i	$-0.0648537\pi$
$487$	17.1464	+	17.1464i	0.776979	+	0.776979i	−0.202337	+	0.979316i	$0.564854\pi$
$488$	−3.36465	+	12.5570i	−0.152310	+	0.568430i
$489$		0			0
$490$		0			0
$491$	−12.0000	−	6.92820i	−0.541552	−	0.312665i	0.204155	−	0.978938i	$-0.434555\pi$
$491$	−12.0000	−	6.92820i	−0.541552	−	0.312665i	−0.745708	+	0.666273i	$0.767889\pi$
$492$		0			0
$493$	50.1910	+	13.4486i	2.26049	+	0.605696i
$494$	−6.92820			−0.311715
$495$		0			0
$496$	−10.0000			−0.449013
$497$	−5.79555	−	1.55291i	−0.259966	−	0.0696577i
$498$		0			0
$499$	38.1051	+	22.0000i	1.70582	+	0.984855i	0.939599	+	0.342277i	$0.111198\pi$
$499$	38.1051	+	22.0000i	1.70582	+	0.984855i	0.766220	+	0.642578i	$0.222135\pi$
$500$		0			0
$501$		0			0
$502$	2.68973	−	10.0382i	0.120048	−	0.448027i
$503$	6.36396	+	6.36396i	0.283755	+	0.283755i	0.834605	−	0.550850i	$-0.185696\pi$
$503$	6.36396	+	6.36396i	0.283755	+	0.283755i	−0.550850	+	0.834605i	$0.685696\pi$
$504$		0			0
$505$		0			0
$506$	9.00000	−	5.19615i	0.400099	−	0.230997i
$507$		0			0
$508$	−1.34486	−	5.01910i	−0.0596687	−	0.222686i
$509$	−2.59808	+	4.50000i	−0.115158	+	0.199459i	−0.917843	−	0.396944i	$-0.870071\pi$
$509$	−2.59808	+	4.50000i	−0.115158	+	0.199459i	0.802685	+	0.596403i	$0.203404\pi$
$510$		0			0
$511$	12.0000	+	20.7846i	0.530849	+	0.919457i
$512$	−0.707107	+	0.707107i	−0.0312500	+	0.0312500i
$513$		0			0
$514$		0			0
$515$		0			0
$516$		0			0
$517$	−10.0382	+	2.68973i	−0.441479	+	0.118294i
$518$	−5.79555	+	1.55291i	−0.254642	+	0.0682311i
$519$		0			0
$520$		0			0
$521$		−	5.19615i		−	0.227648i	−0.993501	−	0.113824i	$-0.963690\pi$
$521$		−	5.19615i		−	0.227648i	0.993501	−	0.113824i	$-0.0363099\pi$
$522$		0			0
$523$	−8.57321	+	8.57321i	−0.374880	+	0.374880i	−0.869251	−	0.494371i	$-0.835399\pi$
$523$	−8.57321	+	8.57321i	−0.374880	+	0.374880i	0.494371	+	0.869251i	$0.335399\pi$
$524$	3.46410	+	6.00000i	0.151330	+	0.262111i
$525$		0			0
$526$	−12.0000	+	20.7846i	−0.523225	+	0.906252i
$527$	15.5291	+	57.9555i	0.676460	+	2.52458i
$528$		0			0
$529$	−12.1244	+	7.00000i	−0.527146	+	0.304348i
$530$		0			0
$531$		0			0
$532$	−2.44949	−	2.44949i	−0.106199	−	0.106199i
$533$	7.76457	−	28.9778i	0.336321	−	1.25517i
$534$		0			0
$535$		0			0
$536$	10.5000	+	6.06218i	0.453531	+	0.261846i
$537$		0			0
$538$	−1.67303	−	0.448288i	−0.0721296	−	0.0193271i
$539$	−13.8564			−0.596838
$540$		0			0
$541$	−43.0000			−1.84871			−0.924357	−	0.381528i	$-0.875398\pi$
$541$	−43.0000			−1.84871			−0.924357	+	0.381528i	$0.875398\pi$
$542$	−9.65926	−	2.58819i	−0.414901	−	0.111172i
$543$		0			0
$544$	5.19615	+	3.00000i	0.222783	+	0.128624i
$545$		0			0
$546$		0			0
$547$	0.448288	−	1.67303i	0.0191674	−	0.0715337i	−0.955680	−	0.294408i	$-0.904877\pi$
$547$	0.448288	−	1.67303i	0.0191674	−	0.0715337i	0.974847	+	0.222875i	$0.0715441\pi$
$548$	−8.48528	−	8.48528i	−0.362473	−	0.362473i
$549$		0			0
$550$		0			0
$551$	−15.0000	+	8.66025i	−0.639021	+	0.368939i
$552$		0			0
$553$	1.79315	+	6.69213i	0.0762525	+	0.284578i
$554$	−5.19615	+	9.00000i	−0.220763	+	0.382373i
$555$		0			0
$556$	2.00000	+	3.46410i	0.0848189	+	0.146911i
$557$	25.4558	−	25.4558i	1.07860	−	1.07860i	0.0819634	−	0.996635i	$-0.473881\pi$
$557$	25.4558	−	25.4558i	1.07860	−	1.07860i	0.996635	−	0.0819634i	$-0.0261191\pi$
$558$		0			0
$559$		−	36.0000i		−	1.52264i
$560$		0			0
$561$		0			0
$562$	−5.01910	+	1.34486i	−0.211718	+	0.0567296i
$563$	−20.2844	+	5.43520i	−0.854887	+	0.229066i	−0.659542	−	0.751668i	$-0.729250\pi$
$563$	−20.2844	+	5.43520i	−0.854887	+	0.229066i	−0.195346	+	0.980734i	$0.562583\pi$
$564$		0			0
$565$		0			0
$566$			5.19615i			0.218411i
$567$		0			0
$568$	−2.44949	+	2.44949i	−0.102778	+	0.102778i
$569$	3.46410	+	6.00000i	0.145223	+	0.251533i	0.929456	−	0.368933i	$-0.120277\pi$
$569$	3.46410	+	6.00000i	0.145223	+	0.251533i	−0.784233	+	0.620466i	$0.786943\pi$
$570$		0			0
$571$	2.00000	−	3.46410i	0.0836974	−	0.144968i	−0.821138	−	0.570730i	$-0.806660\pi$
$571$	2.00000	−	3.46410i	0.0836974	−	0.144968i	0.904835	+	0.425762i	$0.139994\pi$
$572$	−3.10583	−	11.5911i	−0.129861	−	0.484649i
$573$		0			0
$574$	12.9904	−	7.50000i	0.542208	−	0.313044i
$575$		0			0
$576$		0			0
$577$	−24.4949	−	24.4949i	−1.01974	−	1.01974i	−0.999801	−	0.0199346i	$-0.993654\pi$
$577$	−24.4949	−	24.4949i	−1.01974	−	1.01974i	−0.0199346	−	0.999801i	$-0.506346\pi$
$578$	4.91756	−	18.3526i	0.204544	−	0.763367i
$579$		0			0
$580$		0			0
$581$	−4.50000	−	2.59808i	−0.186691	−	0.107786i
$582$		0			0
$583$		0			0
$584$	13.8564			0.573382
$585$		0			0
$586$	30.0000			1.23929
$587$	−2.89778	−	0.776457i	−0.119604	−	0.0320478i	0.198520	−	0.980097i	$-0.436386\pi$
$587$	−2.89778	−	0.776457i	−0.119604	−	0.0320478i	−0.318125	+	0.948049i	$0.603053\pi$
$588$		0			0
$589$	−17.3205	−	10.0000i	−0.713679	−	0.412043i
$590$		0			0
$591$		0			0
$592$	−0.896575	+	3.34607i	−0.0368490	+	0.137522i
$593$	12.7279	+	12.7279i	0.522673	+	0.522673i	0.918378	−	0.395705i	$-0.129500\pi$
$593$	12.7279	+	12.7279i	0.522673	+	0.522673i	−0.395705	+	0.918378i	$0.629500\pi$
$594$		0			0
$595$		0			0
$596$	−4.50000	+	2.59808i	−0.184327	+	0.106421i
$597$		0			0
$598$	−2.68973	−	10.0382i	−0.109991	−	0.410492i
$599$	−13.8564	+	24.0000i	−0.566157	+	0.980613i	0.430784	+	0.902455i	$0.358237\pi$
$599$	−13.8564	+	24.0000i	−0.566157	+	0.980613i	−0.996941	+	0.0781581i	$0.975096\pi$
$600$		0			0
$601$	19.0000	+	32.9090i	0.775026	+	1.34238i	0.934780	+	0.355228i	$0.115597\pi$
$601$	19.0000	+	32.9090i	0.775026	+	1.34238i	−0.159754	+	0.987157i	$0.551070\pi$
$602$	12.7279	−	12.7279i	0.518751	−	0.518751i
$603$		0			0
$604$		−	10.0000i		−	0.406894i
$605$		0			0
$606$		0			0
$607$	−38.4797	+	10.3106i	−1.56184	+	0.418495i	−0.933247	−	0.359235i	$-0.883038\pi$
$607$	−38.4797	+	10.3106i	−1.56184	+	0.418495i	−0.628598	+	0.777730i	$0.716371\pi$
$608$	−1.93185	+	0.517638i	−0.0783469	+	0.0209930i
$609$		0			0
$610$		0			0
$611$			10.3923i			0.420428i
$612$		0			0
$613$	12.2474	−	12.2474i	0.494670	−	0.494670i	−0.415104	−	0.909774i	$-0.636255\pi$
$613$	12.2474	−	12.2474i	0.494670	−	0.494670i	0.909774	+	0.415104i	$0.136255\pi$
$614$	2.59808	+	4.50000i	0.104850	+	0.181605i
$615$		0			0
$616$	3.00000	−	5.19615i	0.120873	−	0.209359i
$617$	−9.31749	−	34.7733i	−0.375108	−	1.39992i	−0.853187	−	0.521605i	$-0.825334\pi$
$617$	−9.31749	−	34.7733i	−0.375108	−	1.39992i	0.478079	−	0.878317i	$-0.341333\pi$
$618$		0			0
$619$	19.0526	−	11.0000i	0.765787	−	0.442127i	−0.0655827	−	0.997847i	$-0.520891\pi$
$619$	19.0526	−	11.0000i	0.765787	−	0.442127i	0.831370	+	0.555720i	$0.187557\pi$
$620$		0			0
$621$		0			0
$622$	17.1464	+	17.1464i	0.687509	+	0.687509i
$623$	0.776457	−	2.89778i	0.0311081	−	0.116097i
$624$		0			0
$625$		0			0
$626$	3.00000	+	1.73205i	0.119904	+	0.0692267i
$627$		0			0
$628$	20.0764	+	5.37945i	0.801135	+	0.214664i
$629$	20.7846			0.828737
$630$		0			0
$631$	−16.0000			−0.636950			−0.318475	−	0.947931i	$-0.603171\pi$
$631$	−16.0000			−0.636950			−0.318475	+	0.947931i	$0.603171\pi$
$632$	3.86370	+	1.03528i	0.153690	+	0.0411811i
$633$		0			0
$634$	−5.19615	−	3.00000i	−0.206366	−	0.119145i
$635$		0			0
$636$		0			0
$637$	−3.58630	+	13.3843i	−0.142094	+	0.530304i
$638$	−21.2132	−	21.2132i	−0.839839	−	0.839839i
$639$		0			0
$640$		0			0
$641$	−28.5000	+	16.4545i	−1.12568	+	0.649913i	−0.942845	−	0.333230i	$-0.891861\pi$
$641$	−28.5000	+	16.4545i	−1.12568	+	0.649913i	−0.182837	+	0.983143i	$0.558528\pi$
$642$		0			0
$643$	−1.34486	−	5.01910i	−0.0530362	−	0.197934i	0.934324	−	0.356424i	$-0.116004\pi$
$643$	−1.34486	−	5.01910i	−0.0530362	−	0.197934i	−0.987361	+	0.158490i	$0.949337\pi$
$644$	2.59808	−	4.50000i	0.102379	−	0.177325i
$645$		0			0
$646$	6.00000	+	10.3923i	0.236067	+	0.408880i
$647$	−14.8492	+	14.8492i	−0.583784	+	0.583784i	−0.935941	−	0.352157i	$-0.885448\pi$
$647$	−14.8492	+	14.8492i	−0.583784	+	0.583784i	0.352157	+	0.935941i	$0.385448\pi$
$648$		0			0
$649$			24.0000i			0.942082i
$650$		0			0
$651$		0			0
$652$	16.7303	−	4.48288i	0.655210	−	0.175563i
$653$	−17.3867	+	4.65874i	−0.680393	+	0.182311i	−0.582432	−	0.812880i	$-0.697899\pi$
$653$	−17.3867	+	4.65874i	−0.680393	+	0.182311i	−0.0979610	+	0.995190i	$0.531232\pi$
$654$		0			0
$655$		0			0
$656$		−	8.66025i		−	0.338126i
$657$		0			0
$658$	−3.67423	+	3.67423i	−0.143237	+	0.143237i
$659$	10.3923	+	18.0000i	0.404827	+	0.701180i	0.994301	−	0.106606i	$-0.0339985\pi$
$659$	10.3923	+	18.0000i	0.404827	+	0.701180i	−0.589475	+	0.807787i	$0.700665\pi$
$660$		0			0
$661$	−11.0000	+	19.0526i	−0.427850	+	0.741059i	−0.996682	−	0.0813955i	$-0.974062\pi$
$661$	−11.0000	+	19.0526i	−0.427850	+	0.741059i	0.568831	+	0.822454i	$0.307396\pi$
$662$	−2.07055	−	7.72741i	−0.0804743	−	0.300334i
$663$		0			0
$664$	−2.59808	+	1.50000i	−0.100825	+	0.0582113i
$665$		0			0
$666$		0			0
$667$	−18.3712	−	18.3712i	−0.711335	−	0.711335i
$668$	2.32937	−	8.69333i	0.0901261	−	0.336355i
$669$		0			0
$670$		0			0
$671$	−39.0000	−	22.5167i	−1.50558	−	0.869246i
$672$		0			0
$673$	36.8067	+	9.86233i	1.41879	+	0.380165i	0.885056	−	0.465485i	$-0.154120\pi$
$673$	36.8067	+	9.86233i	1.41879	+	0.380165i	0.533739	+	0.845649i	$0.320787\pi$
$674$	13.8564			0.533729
$675$		0			0
$676$	1.00000			0.0384615
$677$	5.79555	+	1.55291i	0.222741	+	0.0596833i	0.368464	−	0.929642i	$-0.379884\pi$
$677$	5.79555	+	1.55291i	0.222741	+	0.0596833i	−0.145722	+	0.989326i	$0.546551\pi$
$678$		0			0
$679$	10.3923	+	6.00000i	0.398820	+	0.230259i
$680$		0			0
$681$		0			0
$682$	8.96575	−	33.4607i	0.343316	−	1.28127i
$683$	−8.48528	−	8.48528i	−0.324680	−	0.324680i	0.525879	−	0.850559i	$-0.323736\pi$
$683$	−8.48528	−	8.48528i	−0.324680	−	0.324680i	−0.850559	+	0.525879i	$0.823736\pi$
$684$		0			0
$685$		0			0
$686$	−16.5000	+	9.52628i	−0.629973	+	0.363715i
$687$		0			0
$688$	−2.68973	−	10.0382i	−0.102545	−	0.382703i
$689$		0			0
$690$		0			0
$691$	4.00000	+	6.92820i	0.152167	+	0.263561i	0.932024	−	0.362397i	$-0.118041\pi$
$691$	4.00000	+	6.92820i	0.152167	+	0.263561i	−0.779857	+	0.625958i	$0.784708\pi$
$692$	16.9706	−	16.9706i	0.645124	−	0.645124i
$693$		0			0
$694$			12.0000i			0.455514i
$695$		0			0
$696$		0			0
$697$	−50.1910	+	13.4486i	−1.90112	+	0.509403i
$698$	−0.965926	+	0.258819i	−0.0365608	+	0.00979645i
$699$		0			0
$700$		0			0
$701$			12.1244i			0.457931i	0.973435	+	0.228965i	$0.0735342\pi$
$701$			12.1244i			0.457931i	−0.973435	+	0.228965i	$0.926466\pi$
$702$		0			0
$703$	−4.89898	+	4.89898i	−0.184769	+	0.184769i
$704$	−1.73205	−	3.00000i	−0.0652791	−	0.113067i
$705$		0			0
$706$	−12.0000	+	20.7846i	−0.451626	+	0.782239i
$707$	−6.21166	−	23.1822i	−0.233613	−	0.871857i
$708$		0			0
$709$	35.5070	−	20.5000i	1.33349	−	0.769894i	0.347661	−	0.937620i	$-0.386976\pi$
$709$	35.5070	−	20.5000i	1.33349	−	0.769894i	0.985834	+	0.167727i	$0.0536426\pi$
$710$		0			0
$711$		0			0
$712$	−1.22474	−	1.22474i	−0.0458993	−	0.0458993i
$713$	7.76457	−	28.9778i	0.290785	−	1.08523i
$714$		0			0
$715$		0			0
$716$	−21.0000	−	12.1244i	−0.784807	−	0.453108i
$717$		0			0
$718$	16.7303	+	4.48288i	0.624370	+	0.167299i
$719$	−6.92820			−0.258378			−0.129189	−	0.991620i	$-0.541237\pi$
$719$	−6.92820			−0.258378			−0.129189	+	0.991620i	$0.541237\pi$
$720$		0			0
$721$	6.00000			0.223452
$722$	14.4889	+	3.88229i	0.539221	+	0.144484i
$723$		0			0
$724$	6.06218	+	3.50000i	0.225299	+	0.130076i
$725$		0			0
$726$		0			0
$727$	−1.34486	+	5.01910i	−0.0498782	+	0.186148i	−0.986370	−	0.164541i	$-0.947386\pi$
$727$	−1.34486	+	5.01910i	−0.0498782	+	0.186148i	0.936492	+	0.350689i	$0.114052\pi$
$728$	−4.24264	−	4.24264i	−0.157243	−	0.157243i
$729$		0			0
$730$		0			0
$731$	−54.0000	+	31.1769i	−1.99726	+	1.15312i
$732$		0			0
$733$		0			0		0.965926	−	0.258819i	$-0.0833333\pi$
$733$		0			0		−0.965926	+	0.258819i	$0.916667\pi$
$734$	−1.73205	+	3.00000i	−0.0639312	+	0.110732i
$735$		0			0
$736$	−1.50000	−	2.59808i	−0.0552907	−	0.0957664i
$737$	−29.6985	+	29.6985i	−1.09396	+	1.09396i
$738$		0			0
$739$		−	4.00000i		−	0.147142i	−0.997290	−	0.0735712i	$-0.976560\pi$
$739$		−	4.00000i		−	0.147142i	0.997290	−	0.0735712i	$-0.0234396\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	26.0800	−	6.98811i	0.956782	−	0.256369i	0.253544	−	0.967324i	$-0.418404\pi$
$743$	26.0800	−	6.98811i	0.956782	−	0.256369i	0.703238	+	0.710955i	$0.251737\pi$
$744$		0			0
$745$		0			0
$746$			13.8564i			0.507319i
$747$		0			0
$748$	−14.6969	+	14.6969i	−0.537373	+	0.537373i
$749$	−2.59808	−	4.50000i	−0.0949316	−	0.164426i
$750$		0			0
$751$	1.00000	−	1.73205i	0.0364905	−	0.0632034i	−0.847203	−	0.531269i	$-0.821715\pi$
$751$	1.00000	−	1.73205i	0.0364905	−	0.0632034i	0.883694	+	0.468065i	$0.155049\pi$
$752$	0.776457	+	2.89778i	0.0283145	+	0.105671i
$753$		0			0
$754$	−25.9808	+	15.0000i	−0.946164	+	0.546268i
$755$		0			0
$756$		0			0
$757$	31.8434	+	31.8434i	1.15737	+	1.15737i	0.985040	+	0.172327i	$0.0551286\pi$
$757$	31.8434	+	31.8434i	1.15737	+	1.15737i	0.172327	+	0.985040i	$0.444871\pi$
$758$	−2.07055	+	7.72741i	−0.0752058	+	0.280672i
$759$		0			0
$760$		0			0
$761$	34.5000	+	19.9186i	1.25062	+	0.722048i	0.971233	−	0.238129i	$-0.0765342\pi$
$761$	34.5000	+	19.9186i	1.25062	+	0.722048i	0.279391	+	0.960178i	$0.409868\pi$
$762$		0			0
$763$	8.36516	+	2.24144i	0.302839	+	0.0811455i
$764$	17.3205			0.626634
$765$		0			0
$766$	−24.0000			−0.867155
$767$	23.1822	+	6.21166i	0.837061	+	0.224290i
$768$		0			0
$769$	19.9186	+	11.5000i	0.718283	+	0.414701i	0.814120	−	0.580696i	$-0.197220\pi$
$769$	19.9186	+	11.5000i	0.718283	+	0.414701i	−0.0958377	+	0.995397i	$0.530553\pi$
$770$		0			0
$771$		0			0
$772$	−4.48288	+	16.7303i	−0.161342	+	0.602138i
$773$	−25.4558	−	25.4558i	−0.915583	−	0.915583i	0.0811212	−	0.996704i	$-0.474150\pi$
$773$	−25.4558	−	25.4558i	−0.915583	−	0.915583i	−0.996704	+	0.0811212i	$0.974150\pi$
$774$		0			0
$775$		0			0
$776$	6.00000	−	3.46410i	0.215387	−	0.124354i
$777$		0			0
$778$	−6.72432	−	25.0955i	−0.241078	−	0.899717i
$779$	8.66025	−	15.0000i	0.310286	−	0.537431i
$780$		0			0
$781$	−6.00000	−	10.3923i	−0.214697	−	0.371866i
$782$	−12.7279	+	12.7279i	−0.455150	+	0.455150i
$783$		0			0
$784$			4.00000i			0.142857i
$785$		0			0
$786$		0			0
$787$	−3.34607	+	0.896575i	−0.119274	+	0.0319595i	−0.317962	−	0.948103i	$-0.602999\pi$
$787$	−3.34607	+	0.896575i	−0.119274	+	0.0319595i	0.198688	+	0.980063i	$0.436332\pi$
$788$	11.5911	−	3.10583i	0.412916	−	0.110641i
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	−31.8434	+	31.8434i	−1.13079	+	1.13079i
$794$		0			0
$795$		0			0
$796$	−5.00000	+	8.66025i	−0.177220	+	0.306955i
$797$	10.8704	+	40.5689i	0.385049	+	1.43702i	0.838089	+	0.545533i	$0.183673\pi$
$797$	10.8704	+	40.5689i	0.385049	+	1.43702i	−0.453040	+	0.891490i	$0.649660\pi$
$798$		0			0
$799$	15.5885	−	9.00000i	0.551480	−	0.318397i
$800$		0			0
$801$		0			0
$802$	14.6969	+	14.6969i	0.518967	+	0.518967i
$803$	−12.4233	+	46.3644i	−0.438409	+	1.63617i
$804$		0			0
$805$		0			0
$806$	−30.0000	−	17.3205i	−1.05670	−	0.610089i
$807$		0			0
$808$	−13.3843	−	3.58630i	−0.470857	−	0.126166i
$809$	6.92820			0.243583			0.121791	−	0.992556i	$-0.461136\pi$
$809$	6.92820			0.243583			0.121791	+	0.992556i	$0.461136\pi$
$810$		0			0
$811$	4.00000			0.140459			0.0702295	−	0.997531i	$-0.477627\pi$
$811$	4.00000			0.140459			0.0702295	+	0.997531i	$0.477627\pi$
$812$	−14.4889	−	3.88229i	−0.508460	−	0.136242i
$813$		0			0
$814$	−10.3923	−	6.00000i	−0.364250	−	0.210300i
$815$		0			0
$816$		0			0
$817$	5.37945	−	20.0764i	0.188203	−	0.702384i
$818$	−1.41421	−	1.41421i	−0.0494468	−	0.0494468i
$819$		0			0
$820$		0			0
$821$	16.5000	−	9.52628i	0.575854	−	0.332469i	−0.183630	−	0.982995i	$-0.558785\pi$
$821$	16.5000	−	9.52628i	0.575854	−	0.332469i	0.759484	+	0.650526i	$0.225452\pi$
$822$		0			0
$823$	−7.62089	−	28.4416i	−0.265648	−	0.991410i	−0.961853	−	0.273567i	$-0.911796\pi$
$823$	−7.62089	−	28.4416i	−0.265648	−	0.991410i	0.696205	−	0.717843i	$-0.254870\pi$
$824$	1.73205	−	3.00000i	0.0603388	−	0.104510i
$825$		0			0
$826$	6.00000	+	10.3923i	0.208767	+	0.361595i
$827$	31.8198	−	31.8198i	1.10648	−	1.10648i	0.112874	−	0.993609i	$-0.463994\pi$
$827$	31.8198	−	31.8198i	1.10648	−	1.10648i	0.993609	−	0.112874i	$-0.0360055\pi$
$828$		0			0
$829$			29.0000i			1.00721i	0.863934	+	0.503606i	$0.167994\pi$
$829$			29.0000i			1.00721i	−0.863934	+	0.503606i	$0.832006\pi$
$830$		0			0
$831$		0			0
$832$	−3.34607	+	0.896575i	−0.116004	+	0.0310832i
$833$	23.1822	−	6.21166i	0.803216	−	0.215221i
$834$		0			0
$835$		0			0
$836$		−	6.92820i		−	0.239617i
$837$		0			0
$838$	7.34847	−	7.34847i	0.253849	−	0.253849i
$839$	19.0526	+	33.0000i	0.657767	+	1.13929i	0.981192	+	0.193033i	$0.0618323\pi$
$839$	19.0526	+	33.0000i	0.657767	+	1.13929i	−0.323425	+	0.946254i	$0.604834\pi$
$840$		0			0
$841$	−23.0000	+	39.8372i	−0.793103	+	1.37370i
$842$	−2.58819	−	9.65926i	−0.0891949	−	0.332880i
$843$		0			0
$844$	19.0526	−	11.0000i	0.655816	−	0.378636i
$845$		0			0
$846$		0			0
$847$	1.22474	+	1.22474i	0.0420827	+	0.0420827i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−9.00000	−	5.19615i	−0.308516	−	0.178122i
$852$		0			0
$853$	16.7303	+	4.48288i	0.572835	+	0.153491i	0.533597	−	0.845739i	$-0.320840\pi$
$853$	16.7303	+	4.48288i	0.572835	+	0.153491i	0.0392388	+	0.999230i	$0.487507\pi$
$854$	−22.5167			−0.770504
$855$		0			0
$856$	−3.00000			−0.102538
$857$	−40.5689	−	10.8704i	−1.38581	−	0.371326i	−0.512580	−	0.858640i	$-0.671310\pi$
$857$	−40.5689	−	10.8704i	−1.38581	−	0.371326i	−0.873227	+	0.487314i	$0.837977\pi$
$858$		0			0
$859$	−19.0526	−	11.0000i	−0.650065	−	0.375315i	0.138416	−	0.990374i	$-0.455799\pi$
$859$	−19.0526	−	11.0000i	−0.650065	−	0.375315i	−0.788481	+	0.615059i	$0.789132\pi$
$860$		0			0
$861$		0			0
$862$	1.79315	−	6.69213i	0.0610750	−	0.227935i
$863$	10.6066	+	10.6066i	0.361053	+	0.361053i	0.864201	−	0.503148i	$-0.167825\pi$
$863$	10.6066	+	10.6066i	0.361053	+	0.361053i	−0.503148	+	0.864201i	$0.667825\pi$
$864$		0			0
$865$		0			0
$866$	−9.00000	+	5.19615i	−0.305832	+	0.176572i
$867$		0			0
$868$	−4.48288	−	16.7303i	−0.152159	−	0.567864i
$869$	−6.92820	+	12.0000i	−0.235023	+	0.407072i
$870$		0			0
$871$	21.0000	+	36.3731i	0.711558	+	1.23245i
$872$	3.53553	−	3.53553i	0.119728	−	0.119728i
$873$		0			0
$874$		−	6.00000i		−	0.202953i
$875$		0			0
$876$		0			0
$877$	16.7303	−	4.48288i	0.564943	−	0.151376i	0.0349667	−	0.999388i	$-0.488867\pi$
$877$	16.7303	−	4.48288i	0.564943	−	0.151376i	0.529976	+	0.848012i	$0.322201\pi$
$878$	−19.3185	+	5.17638i	−0.651968	+	0.174694i
$879$		0			0
$880$		0			0
$881$			1.73205i			0.0583543i	0.999574	+	0.0291771i	$0.00928869\pi$
$881$			1.73205i			0.0583543i	−0.999574	+	0.0291771i	$0.990711\pi$
$882$		0			0
$883$	−30.6186	+	30.6186i	−1.03040	+	1.03040i	−0.0308754	+	0.999523i	$0.509830\pi$
$883$	−30.6186	+	30.6186i	−1.03040	+	1.03040i	−0.999523	+	0.0308754i	$0.990170\pi$
$884$	10.3923	+	18.0000i	0.349531	+	0.605406i
$885$		0			0
$886$	4.50000	−	7.79423i	0.151180	−	0.261852i
$887$	6.21166	+	23.1822i	0.208567	+	0.778383i	0.988333	+	0.152311i	$0.0486716\pi$
$887$	6.21166	+	23.1822i	0.208567	+	0.778383i	−0.779766	+	0.626072i	$0.784662\pi$
$888$		0			0
$889$	7.79423	−	4.50000i	0.261410	−	0.150925i
$890$		0			0
$891$		0			0
$892$	−11.0227	−	11.0227i	−0.369067	−	0.369067i
$893$	−1.55291	+	5.79555i	−0.0519663	+	0.193941i
$894$		0			0
$895$		0			0
$896$	−1.50000	−	0.866025i	−0.0501115	−	0.0289319i
$897$		0			0
$898$	13.3843	+	3.58630i	0.446639	+	0.119676i
$899$	−86.6025			−2.88836
$900$		0			0
$901$		0			0
$902$	28.9778	+	7.76457i	0.964854	+	0.258532i
$903$		0			0
$904$		0			0
$905$		0			0
$906$		0			0
$907$	2.24144	−	8.36516i	0.0744257	−	0.277761i	−0.918677	−	0.395010i	$-0.870741\pi$
$907$	2.24144	−	8.36516i	0.0744257	−	0.277761i	0.993102	+	0.117250i	$0.0374077\pi$
$908$	16.9706	+	16.9706i	0.563188	+	0.563188i
$909$		0			0
$910$		0			0
$911$	48.0000	−	27.7128i	1.59031	−	0.918166i	0.597058	−	0.802198i	$-0.296336\pi$
$911$	48.0000	−	27.7128i	1.59031	−	0.918166i	0.993253	−	0.115968i	$-0.0369971\pi$
$912$		0			0
$913$	−2.68973	−	10.0382i	−0.0890170	−	0.332216i
$914$	15.5885	−	27.0000i	0.515620	−	0.893081i
$915$		0			0
$916$	−2.50000	−	4.33013i	−0.0826023	−	0.143071i
$917$	−8.48528	+	8.48528i	−0.280209	+	0.280209i
$918$		0			0
$919$		−	40.0000i		−	1.31948i	−0.751495	−	0.659739i	$-0.770667\pi$
$919$		−	40.0000i		−	1.31948i	0.751495	−	0.659739i	$-0.229333\pi$
$920$		0			0
$921$		0			0
$922$	−11.7112	+	3.13801i	−0.385689	+	0.103345i
$923$	−11.5911	+	3.10583i	−0.381526	+	0.102230i
$924$		0			0
$925$		0			0
$926$		−	17.3205i		−	0.569187i
$927$		0			0
$928$	−6.12372	+	6.12372i	−0.201021	+	0.201021i
$929$	−3.46410	−	6.00000i	−0.113653	−	0.196854i	0.803587	−	0.595187i	$-0.202922\pi$
$929$	−3.46410	−	6.00000i	−0.113653	−	0.196854i	−0.917241	+	0.398333i	$0.869589\pi$
$930$		0			0
$931$	−4.00000	+	6.92820i	−0.131095	+	0.227063i
$932$	−1.55291	−	5.79555i	−0.0508674	−	0.189840i
$933$		0			0
$934$	−31.1769	+	18.0000i	−1.02014	+	0.588978i
$935$		0			0
$936$		0			0
$937$	9.79796	+	9.79796i	0.320085	+	0.320085i	0.848800	−	0.528714i	$-0.177326\pi$
$937$	9.79796	+	9.79796i	0.320085	+	0.320085i	−0.528714	+	0.848800i	$0.677326\pi$
$938$	−5.43520	+	20.2844i	−0.177466	+	0.662311i
$939$		0			0
$940$		0			0
$941$	4.50000	+	2.59808i	0.146696	+	0.0846949i	0.571551	−	0.820566i	$-0.306342\pi$
$941$	4.50000	+	2.59808i	0.146696	+	0.0846949i	−0.424856	+	0.905261i	$0.639675\pi$
$942$		0			0
$943$	25.0955	+	6.72432i	0.817222	+	0.218974i
$944$	6.92820			0.225494
$945$		0			0
$946$	36.0000			1.17046
$947$	−26.0800	−	6.98811i	−0.847486	−	0.227083i	−0.191158	−	0.981559i	$-0.561224\pi$
$947$	−26.0800	−	6.98811i	−0.847486	−	0.227083i	−0.656328	+	0.754476i	$0.727891\pi$
$948$		0			0
$949$	41.5692	+	24.0000i	1.34939	+	0.779073i
$950$		0			0
$951$		0			0
$952$	−2.68973	+	10.0382i	−0.0871745	+	0.325340i
$953$	29.6985	+	29.6985i	0.962028	+	0.962028i	0.999305	−	0.0372767i	$-0.0118683\pi$
$953$	29.6985	+	29.6985i	0.962028	+	0.962028i	−0.0372767	+	0.999305i	$0.511868\pi$
$954$		0			0
$955$		0			0
$956$	18.0000	−	10.3923i	0.582162	−	0.336111i
$957$		0			0
$958$	7.17260	+	26.7685i	0.231736	+	0.864852i
$959$	10.3923	−	18.0000i	0.335585	−	0.581250i
$960$		0			0
$961$	−34.5000	−	59.7558i	−1.11290	−	1.92760i
$962$	−8.48528	+	8.48528i	−0.273576	+	0.273576i
$963$		0			0
$964$		−	17.0000i		−	0.547533i
$965$		0			0
$966$		0			0
$967$	51.8640	−	13.8969i	1.66783	−	0.446895i	0.703309	−	0.710885i	$-0.251705\pi$
$967$	51.8640	−	13.8969i	1.66783	−	0.446895i	0.964525	+	0.263990i	$0.0850385\pi$
$968$	0.965926	−	0.258819i	0.0310460	−	0.00831876i
$969$		0			0
$970$		0			0
$971$			38.1051i			1.22285i	0.791302	+	0.611426i	$0.209404\pi$
$971$			38.1051i			1.22285i	−0.791302	+	0.611426i	$0.790596\pi$
$972$		0			0
$973$	−4.89898	+	4.89898i	−0.157054	+	0.157054i
$974$	12.1244	+	21.0000i	0.388489	+	0.672883i
$975$		0			0
$976$	−6.50000	+	11.2583i	−0.208060	+	0.360370i
$977$	−10.8704	−	40.5689i	−0.347775	−	1.29791i	−0.889337	−	0.457253i	$-0.848833\pi$
$977$	−10.8704	−	40.5689i	−0.347775	−	1.29791i	0.541562	−	0.840661i	$-0.317833\pi$
$978$		0			0
$979$	5.19615	−	3.00000i	0.166070	−	0.0958804i
$980$		0			0
$981$		0			0
$982$	−9.79796	−	9.79796i	−0.312665	−	0.312665i
$983$	−2.32937	+	8.69333i	−0.0742954	+	0.277274i	−0.993073	−	0.117502i	$-0.962511\pi$
$983$	−2.32937	+	8.69333i	−0.0742954	+	0.277274i	0.918777	+	0.394776i	$0.129178\pi$
$984$		0			0
$985$		0			0
$986$	45.0000	+	25.9808i	1.43309	+	0.827396i
$987$		0			0
$988$	−6.69213	−	1.79315i	−0.212905	−	0.0570477i
$989$	31.1769			0.991368
$990$		0			0
$991$	20.0000			0.635321			0.317660	−	0.948205i	$-0.397103\pi$
$991$	20.0000			0.635321			0.317660	+	0.948205i	$0.397103\pi$
$992$	−9.65926	−	2.58819i	−0.306682	−	0.0821751i
$993$		0			0
$994$	−5.19615	−	3.00000i	−0.164812	−	0.0951542i
$995$		0			0
$996$		0			0
$997$	10.7589	−	40.1528i	0.340738	−	1.27165i	−0.556775	−	0.830663i	$-0.687961\pi$
$997$	10.7589	−	40.1528i	0.340738	−	1.27165i	0.897513	−	0.440988i	$-0.145372\pi$
$998$	31.1127	+	31.1127i	0.984855	+	0.984855i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1350.2.q.b.557.2		8
3.2	odd	2		450.2.p.d.257.1	✓	8
5.2	odd	4	inner	1350.2.q.b.1043.1		8
5.3	odd	4	inner	1350.2.q.b.1043.2		8
5.4	even	2	inner	1350.2.q.b.557.1		8
9.2	odd	6	inner	1350.2.q.b.1007.2		8
9.7	even	3		450.2.p.d.407.1	yes	8
15.2	even	4		450.2.p.d.293.2	yes	8
15.8	even	4		450.2.p.d.293.1	yes	8
15.14	odd	2		450.2.p.d.257.2	yes	8
45.2	even	12	inner	1350.2.q.b.143.1		8
45.7	odd	12		450.2.p.d.443.2	yes	8
45.29	odd	6	inner	1350.2.q.b.1007.1		8
45.34	even	6		450.2.p.d.407.2	yes	8
45.38	even	12	inner	1350.2.q.b.143.2		8
45.43	odd	12		450.2.p.d.443.1	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
450.2.p.d.257.1	✓	8	3.2	odd	2
450.2.p.d.257.2	yes	8	15.14	odd	2
450.2.p.d.293.1	yes	8	15.8	even	4
450.2.p.d.293.2	yes	8	15.2	even	4
450.2.p.d.407.1	yes	8	9.7	even	3
450.2.p.d.407.2	yes	8	45.34	even	6
450.2.p.d.443.1	yes	8	45.43	odd	12
450.2.p.d.443.2	yes	8	45.7	odd	12
1350.2.q.b.143.1		8	45.2	even	12	inner
1350.2.q.b.143.2		8	45.38	even	12	inner
1350.2.q.b.557.1		8	5.4	even	2	inner
1350.2.q.b.557.2		8	1.1	even	1	trivial
1350.2.q.b.1007.1		8	45.29	odd	6	inner
1350.2.q.b.1007.2		8	9.2	odd	6	inner
1350.2.q.b.1043.1		8	5.2	odd	4	inner
1350.2.q.b.1043.2		8	5.3	odd	4	inner

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.q (of order \(12\), degree \(4\), not minimal)

\(f(q)\)	\(=\)	\(q+(0.965926 + 0.258819i) q^{2} +(0.866025 + 0.500000i) q^{4} +(-0.448288 + 1.67303i) q^{7} +(0.707107 + 0.707107i) q^{8} +O(q^{10})\)	\(q+(0.965926 + 0.258819i) q^{2} +(0.866025 + 0.500000i) q^{4} +(-0.448288 + 1.67303i) q^{7} +(0.707107 + 0.707107i) q^{8} +(-3.00000 + 1.73205i) q^{11} +(0.896575 + 3.34607i) q^{13} +(-0.866025 + 1.50000i) q^{14} +(0.500000 + 0.866025i) q^{16} +(4.24264 - 4.24264i) q^{17} +2.00000i q^{19} +(-3.34607 + 0.896575i) q^{22} +(-2.89778 + 0.776457i) q^{23} +3.46410i q^{26} +(-1.22474 + 1.22474i) q^{28} +(4.33013 + 7.50000i) q^{29} +(-5.00000 + 8.66025i) q^{31} +(0.258819 + 0.965926i) q^{32} +(5.19615 - 3.00000i) q^{34} +(2.44949 + 2.44949i) q^{37} +(-0.517638 + 1.93185i) q^{38} +(-7.50000 - 4.33013i) q^{41} +(-10.0382 - 2.68973i) q^{43} -3.46410 q^{44} -3.00000 q^{46} +(2.89778 + 0.776457i) q^{47} +(3.46410 + 2.00000i) q^{49} +(-0.896575 + 3.34607i) q^{52} +(-1.50000 + 0.866025i) q^{56} +(2.24144 + 8.36516i) q^{58} +(3.46410 - 6.00000i) q^{59} +(6.50000 + 11.2583i) q^{61} +(-7.07107 + 7.07107i) q^{62} +1.00000i q^{64} +(11.7112 - 3.13801i) q^{67} +(5.79555 - 1.55291i) q^{68} +3.46410i q^{71} +(9.79796 - 9.79796i) q^{73} +(1.73205 + 3.00000i) q^{74} +(-1.00000 + 1.73205i) q^{76} +(-1.55291 - 5.79555i) q^{77} +(3.46410 - 2.00000i) q^{79} +(-6.12372 - 6.12372i) q^{82} +(-0.776457 + 2.89778i) q^{83} +(-9.00000 - 5.19615i) q^{86} +(-3.34607 - 0.896575i) q^{88} -1.73205 q^{89} -6.00000 q^{91} +(-2.89778 - 0.776457i) q^{92} +(2.59808 + 1.50000i) q^{94} +(1.79315 - 6.69213i) q^{97} +(2.82843 + 2.82843i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q+O(q^{10}) \) 8 * q	\( 8 q - 24 q^{11} + 4 q^{16} - 40 q^{31} - 60 q^{41} - 24 q^{46} - 12 q^{56} + 52 q^{61} - 8 q^{76} - 72 q^{86} - 48 q^{91}+O(q^{100}) \) 8 * q - 24 * q^11 + 4 * q^16 - 40 * q^31 - 60 * q^41 - 24 * q^46 - 12 * q^56 + 52 * q^61 - 8 * q^76 - 72 * q^86 - 48 * q^91

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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