LMFDB - Embedded newform 1350.2.q.g.557.1

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 90)
Sato-Tate group:	$\mathrm{SU}(2)[C_{12}]$

Embedding invariants

Embedding label			557.1
Root			$0.258819 - 0.965926i$ of defining polynomial
Character	$\chi$	$=$	1350.557
Dual form			1350.2.q.g.143.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.965926 - 0.258819i) q^{2} +(0.866025 + 0.500000i) q^{4} +(-1.18034 + 4.40508i) q^{7} +(-0.707107 - 0.707107i) q^{8} +O(q^{10})$	\(q+(-0.965926 - 0.258819i) q^{2} +(0.866025 + 0.500000i) q^{4} +(-1.18034 + 4.40508i) q^{7} +(-0.707107 - 0.707107i) q^{8} +(0.550510 - 0.317837i) q^{11} +(-0.896575 - 3.34607i) q^{13} +(2.28024 - 3.94949i) q^{14} +(0.500000 + 0.866025i) q^{16} +(-0.317837 + 0.317837i) q^{17} +6.44949i q^{19} +(-0.614014 + 0.164525i) q^{22} +(-0.965926 + 0.258819i) q^{23} +3.46410i q^{26} +(-3.22474 + 3.22474i) q^{28} +(-0.158919 - 0.275255i) q^{29} +(-0.224745 + 0.389270i) q^{31} +(-0.258819 - 0.965926i) q^{32} +(0.389270 - 0.224745i) q^{34} +(3.00000 + 3.00000i) q^{37} +(1.66925 - 6.22973i) q^{38} +(-6.39898 - 3.69445i) q^{41} +(-3.34607 - 0.896575i) q^{43} +0.635674 q^{44} +1.00000 q^{46} +(-8.69333 - 2.32937i) q^{47} +(-11.9494 - 6.89898i) q^{49} +(0.896575 - 3.34607i) q^{52} +(-3.78194 - 3.78194i) q^{53} +(3.94949 - 2.28024i) q^{56} +(0.0822623 + 0.307007i) q^{58} +(-4.48905 + 7.77526i) q^{59} +(0.275255 + 0.476756i) q^{61} +(0.317837 - 0.317837i) q^{62} +1.00000i q^{64} +(-6.38512 + 1.71089i) q^{67} +(-0.434174 + 0.116337i) q^{68} +6.29253i q^{71} +(6.89898 - 6.89898i) q^{73} +(-2.12132 - 3.67423i) q^{74} +(-3.22474 + 5.58542i) q^{76} +(0.750311 + 2.80020i) q^{77} +(2.12132 - 1.22474i) q^{79} +(5.22474 + 5.22474i) q^{82} +(1.41043 - 5.26380i) q^{83} +(3.00000 + 1.73205i) q^{86} +(-0.614014 - 0.164525i) q^{88} -8.02458 q^{89} +15.7980 q^{91} +(-0.965926 - 0.258819i) q^{92} +(7.79423 + 4.50000i) q^{94} +(-0.695075 + 2.59405i) q^{97} +(9.75663 + 9.75663i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 8 q^{7}+O(q^{10}) $ 8 * q + 8 * q^7	$ 8 q + 8 q^{7} + 24 q^{11} + 4 q^{16} + 8 q^{22} - 16 q^{28} + 8 q^{31} + 24 q^{37} + 12 q^{38} - 12 q^{41} + 8 q^{46} + 12 q^{56} + 4 q^{58} + 12 q^{61} - 4 q^{67} - 12 q^{68} + 16 q^{73} - 16 q^{76} + 24 q^{77} + 32 q^{82} + 12 q^{83} + 24 q^{86} + 8 q^{88} + 48 q^{91} - 12 q^{97}+O(q^{100}) $ 8 * q + 8 * q^7 + 24 * q^11 + 4 * q^16 + 8 * q^22 - 16 * q^28 + 8 * q^31 + 24 * q^37 + 12 * q^38 - 12 * q^41 + 8 * q^46 + 12 * q^56 + 4 * q^58 + 12 * q^61 - 4 * q^67 - 12 * q^68 + 16 * q^73 - 16 * q^76 + 24 * q^77 + 32 * q^82 + 12 * q^83 + 24 * q^86 + 8 * q^88 + 48 * q^91 - 12 * q^97

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
$7$	−1.18034	+	4.40508i	−0.446126	+	1.66497i	0.266820	+	0.963746i	$0.414027\pi$
$7$	−1.18034	+	4.40508i	−0.446126	+	1.66497i	−0.712946	+	0.701219i	$0.752640\pi$
$8$	−0.707107	−	0.707107i	−0.250000	−	0.250000i
$9$		0			0
$10$		0			0
$11$	0.550510	−	0.317837i	0.165985	−	0.0958315i	−0.414706	−	0.909955i	$-0.636116\pi$
$11$	0.550510	−	0.317837i	0.165985	−	0.0958315i	0.580691	+	0.814124i	$0.302782\pi$
$12$		0			0
$13$	−0.896575	−	3.34607i	−0.248665	−	0.928032i	−0.971506	−	0.237016i	$-0.923830\pi$
$13$	−0.896575	−	3.34607i	−0.248665	−	0.928032i	0.722840	−	0.691015i	$-0.242836\pi$
$14$	2.28024	−	3.94949i	0.609419	−	1.05555i
$15$		0			0
$16$	0.500000	+	0.866025i	0.125000	+	0.216506i
$17$	−0.317837	+	0.317837i	−0.0770869	+	0.0770869i	−0.744599	−	0.667512i	$-0.767359\pi$
$17$	−0.317837	+	0.317837i	−0.0770869	+	0.0770869i	0.667512	+	0.744599i	$0.267359\pi$
$18$		0			0
$19$			6.44949i			1.47961i	0.672819	+	0.739807i	$0.265083\pi$
$19$			6.44949i			1.47961i	−0.672819	+	0.739807i	$0.734917\pi$
$20$		0			0
$21$		0			0
$22$	−0.614014	+	0.164525i	−0.130908	+	0.0350768i
$23$	−0.965926	+	0.258819i	−0.201409	+	0.0539675i	−0.358113	−	0.933678i	$-0.616580\pi$
$23$	−0.965926	+	0.258819i	−0.201409	+	0.0539675i	0.156704	+	0.987646i	$0.449913\pi$
$24$		0			0
$25$		0			0
$26$			3.46410i			0.679366i
$27$		0			0
$28$	−3.22474	+	3.22474i	−0.609419	+	0.609419i
$29$	−0.158919	−	0.275255i	−0.0295104	−	0.0511136i	0.850893	−	0.525339i	$-0.176061\pi$
$29$	−0.158919	−	0.275255i	−0.0295104	−	0.0511136i	−0.880403	+	0.474225i	$0.842728\pi$
$30$		0			0
$31$	−0.224745	+	0.389270i	−0.0403654	+	0.0699149i	−0.885502	−	0.464635i	$-0.846186\pi$
$31$	−0.224745	+	0.389270i	−0.0403654	+	0.0699149i	0.845137	+	0.534550i	$0.179519\pi$
$32$	−0.258819	−	0.965926i	−0.0457532	−	0.170753i
$33$		0			0
$34$	0.389270	−	0.224745i	0.0667592	−	0.0385434i
$35$		0			0
$36$		0			0
$37$	3.00000	+	3.00000i	0.493197	+	0.493197i	0.909312	−	0.416115i	$-0.136609\pi$
$37$	3.00000	+	3.00000i	0.493197	+	0.493197i	−0.416115	+	0.909312i	$0.636609\pi$
$38$	1.66925	−	6.22973i	0.270788	−	1.01060i
$39$		0			0
$40$		0			0
$41$	−6.39898	−	3.69445i	−0.999353	−	0.576977i	−0.0912960	−	0.995824i	$-0.529101\pi$
$41$	−6.39898	−	3.69445i	−0.999353	−	0.576977i	−0.908057	+	0.418847i	$0.862434\pi$
$42$		0			0
$43$	−3.34607	−	0.896575i	−0.510270	−	0.136726i	−0.00550783	−	0.999985i	$-0.501753\pi$
$43$	−3.34607	−	0.896575i	−0.510270	−	0.136726i	−0.504762	+	0.863258i	$0.668420\pi$
$44$	0.635674			0.0958315
$45$		0			0
$46$	1.00000			0.147442
$47$	−8.69333	−	2.32937i	−1.26805	−	0.339774i	−0.438768	−	0.898600i	$-0.644585\pi$
$47$	−8.69333	−	2.32937i	−1.26805	−	0.339774i	−0.829285	+	0.558827i	$0.811252\pi$
$48$		0			0
$49$	−11.9494	−	6.89898i	−1.70705	−	0.985568i
$50$		0			0
$51$		0			0
$52$	0.896575	−	3.34607i	0.124333	−	0.464016i
$53$	−3.78194	−	3.78194i	−0.519489	−	0.519489i	0.397928	−	0.917417i	$-0.369730\pi$
$53$	−3.78194	−	3.78194i	−0.519489	−	0.519489i	−0.917417	+	0.397928i	$0.869730\pi$
$54$		0			0
$55$		0			0
$56$	3.94949	−	2.28024i	0.527773	−	0.304710i
$57$		0			0
$58$	0.0822623	+	0.307007i	0.0108016	+	0.0403120i
$59$	−4.48905	+	7.77526i	−0.584424	+	1.01225i	0.410523	+	0.911850i	$0.365346\pi$
$59$	−4.48905	+	7.77526i	−0.584424	+	1.01225i	−0.994947	+	0.100402i	$0.967987\pi$
$60$		0			0
$61$	0.275255	+	0.476756i	0.0352428	+	0.0610423i	0.883109	−	0.469168i	$-0.155446\pi$
$61$	0.275255	+	0.476756i	0.0352428	+	0.0610423i	−0.847866	+	0.530211i	$0.822113\pi$
$62$	0.317837	−	0.317837i	0.0403654	−	0.0403654i
$63$		0			0
$64$			1.00000i			0.125000i
$65$		0			0
$66$		0			0
$67$	−6.38512	+	1.71089i	−0.780067	+	0.209018i	−0.626814	−	0.779169i	$-0.715642\pi$
$67$	−6.38512	+	1.71089i	−0.780067	+	0.209018i	−0.153253	+	0.988187i	$0.548975\pi$
$68$	−0.434174	+	0.116337i	−0.0526513	+	0.0141079i
$69$		0			0
$70$		0			0
$71$			6.29253i			0.746786i	0.927673	+	0.373393i	$0.121806\pi$
$71$			6.29253i			0.746786i	−0.927673	+	0.373393i	$0.878194\pi$
$72$		0			0
$73$	6.89898	−	6.89898i	0.807464	−	0.807464i	−0.176785	−	0.984249i	$-0.556570\pi$
$73$	6.89898	−	6.89898i	0.807464	−	0.807464i	0.984249	+	0.176785i	$0.0565697\pi$
$74$	−2.12132	−	3.67423i	−0.246598	−	0.427121i
$75$		0			0
$76$	−3.22474	+	5.58542i	−0.369904	+	0.640692i
$77$	0.750311	+	2.80020i	0.0855059	+	0.319112i
$78$		0			0
$79$	2.12132	−	1.22474i	0.238667	−	0.137795i	−0.375897	−	0.926662i	$-0.622665\pi$
$79$	2.12132	−	1.22474i	0.238667	−	0.137795i	0.614564	+	0.788867i	$0.289332\pi$
$80$		0			0
$81$		0			0
$82$	5.22474	+	5.22474i	0.576977	+	0.576977i
$83$	1.41043	−	5.26380i	0.154815	−	0.577777i	−0.844306	−	0.535861i	$-0.819987\pi$
$83$	1.41043	−	5.26380i	0.154815	−	0.577777i	0.999121	−	0.0419163i	$-0.0133463\pi$
$84$		0			0
$85$		0			0
$86$	3.00000	+	1.73205i	0.323498	+	0.186772i
$87$		0			0
$88$	−0.614014	−	0.164525i	−0.0654542	−	0.0175384i
$89$	−8.02458			−0.850604			−0.425302	−	0.905052i	$-0.639832\pi$
$89$	−8.02458			−0.850604			−0.425302	+	0.905052i	$0.639832\pi$
$90$		0			0
$91$	15.7980			1.65608
$92$	−0.965926	−	0.258819i	−0.100705	−	0.0269838i
$93$		0			0
$94$	7.79423	+	4.50000i	0.803913	+	0.464140i
$95$		0			0
$96$		0			0
$97$	−0.695075	+	2.59405i	−0.0705741	+	0.263386i	−0.992193	−	0.124709i	$-0.960200\pi$
$97$	−0.695075	+	2.59405i	−0.0705741	+	0.263386i	0.921619	+	0.388095i	$0.126867\pi$
$98$	9.75663	+	9.75663i	0.985568	+	0.985568i
$99$		0			0
$100$		0			0
$101$	−10.8990	+	6.29253i	−1.08449	+	0.626130i	−0.932104	−	0.362191i	$-0.882029\pi$
$101$	−10.8990	+	6.29253i	−1.08449	+	0.626130i	−0.152385	+	0.988321i	$0.548695\pi$
$102$		0			0
$103$	2.52520	+	9.42418i	0.248816	+	0.928592i	0.971427	+	0.237338i	$0.0762749\pi$
$103$	2.52520	+	9.42418i	0.248816	+	0.928592i	−0.722612	+	0.691254i	$0.757058\pi$
$104$	−1.73205	+	3.00000i	−0.169842	+	0.294174i
$105$		0			0
$106$	2.67423	+	4.63191i	0.259745	+	0.449891i
$107$	−13.6100	+	13.6100i	−1.31573	+	1.31573i	−0.398606	+	0.917122i	$0.630506\pi$
$107$	−13.6100	+	13.6100i	−1.31573	+	1.31573i	−0.917122	+	0.398606i	$0.869494\pi$
$108$		0			0
$109$		−	5.65153i		−	0.541318i	−0.962675	−	0.270659i	$-0.912758\pi$
$109$		−	5.65153i		−	0.541318i	0.962675	−	0.270659i	$-0.0872417\pi$
$110$		0			0
$111$		0			0
$112$	−4.40508	+	1.18034i	−0.416241	+	0.111532i
$113$	−5.60040	+	1.50062i	−0.526841	+	0.141167i	−0.512429	−	0.858729i	$-0.671254\pi$
$113$	−5.60040	+	1.50062i	−0.526841	+	0.141167i	−0.0144120	+	0.999896i	$0.504588\pi$
$114$		0			0
$115$		0			0
$116$		−	0.317837i		−	0.0295104i
$117$		0			0
$118$	6.34847	−	6.34847i	0.584424	−	0.584424i
$119$	−1.02494	−	1.77526i	−0.0939565	−	0.162737i
$120$		0			0
$121$	−5.29796	+	9.17633i	−0.481633	+	0.834212i
$122$	−0.142483	−	0.531752i	−0.0128998	−	0.0481426i
$123$		0			0
$124$	−0.389270	+	0.224745i	−0.0349574	+	0.0201827i
$125$		0			0
$126$		0			0
$127$	−1.87628	−	1.87628i	−0.166493	−	0.166493i	0.618943	−	0.785436i	$-0.287561\pi$
$127$	−1.87628	−	1.87628i	−0.166493	−	0.166493i	−0.785436	+	0.618943i	$0.787561\pi$
$128$	0.258819	−	0.965926i	0.0228766	−	0.0853766i
$129$		0			0
$130$		0			0
$131$	3.12372	+	1.80348i	0.272921	+	0.157571i	0.630214	−	0.776421i	$-0.282967\pi$
$131$	3.12372	+	1.80348i	0.272921	+	0.157571i	−0.357293	+	0.933992i	$0.616300\pi$
$132$		0			0
$133$	−28.4105	−	7.61258i	−2.46351	−	0.660095i
$134$	6.61037			0.571049
$135$		0			0
$136$	0.449490			0.0385434
$137$	21.0552	+	5.64173i	1.79887	+	0.482005i	0.993801	−	0.111178i	$-0.0354623\pi$
$137$	21.0552	+	5.64173i	1.79887	+	0.482005i	0.805068	+	0.593183i	$0.202129\pi$
$138$		0			0
$139$	−2.68556	−	1.55051i	−0.227786	−	0.131513i	0.381764	−	0.924260i	$-0.375317\pi$
$139$	−2.68556	−	1.55051i	−0.227786	−	0.131513i	−0.609550	+	0.792747i	$0.708650\pi$
$140$		0			0
$141$		0			0
$142$	1.62863	−	6.07812i	0.136671	−	0.510064i
$143$	−1.55708	−	1.55708i	−0.130209	−	0.130209i
$144$		0			0
$145$		0			0
$146$	−8.44949	+	4.87832i	−0.699285	+	0.403732i
$147$		0			0
$148$	1.09808	+	4.09808i	0.0902613	+	0.336860i
$149$	2.20881	−	3.82577i	0.180952	−	0.313419i	−0.761253	−	0.648455i	$-0.775415\pi$
$149$	2.20881	−	3.82577i	0.180952	−	0.313419i	0.942205	+	0.335036i	$0.108749\pi$
$150$		0			0
$151$	−8.79796	−	15.2385i	−0.715968	−	1.24009i	−0.962585	−	0.270980i	$-0.912652\pi$
$151$	−8.79796	−	15.2385i	−0.715968	−	1.24009i	0.246617	−	0.969113i	$-0.420681\pi$
$152$	4.56048	−	4.56048i	0.369904	−	0.369904i
$153$		0			0
$154$		−	2.89898i		−	0.233606i
$155$		0			0
$156$		0			0
$157$	−14.1363	+	3.78780i	−1.12820	+	0.302300i	−0.774196	−	0.632945i	$-0.781846\pi$
$157$	−14.1363	+	3.78780i	−1.12820	+	0.302300i	−0.354001	+	0.935245i	$0.615179\pi$
$158$	−2.36603	+	0.633975i	−0.188231	+	0.0504363i
$159$		0			0
$160$		0			0
$161$		−	4.56048i		−	0.359416i
$162$		0			0
$163$	−4.44949	+	4.44949i	−0.348511	+	0.348511i	−0.859555	−	0.511044i	$-0.829259\pi$
$163$	−4.44949	+	4.44949i	−0.348511	+	0.348511i	0.511044	+	0.859555i	$0.329259\pi$
$164$	−3.69445	−	6.39898i	−0.288488	−	0.499676i
$165$		0			0
$166$	−2.72474	+	4.71940i	−0.211481	+	0.366296i
$167$	−2.27708	−	8.49818i	−0.176206	−	0.657609i	−0.996343	−	0.0854420i	$-0.972770\pi$
$167$	−2.27708	−	8.49818i	−0.176206	−	0.657609i	0.820137	−	0.572167i	$-0.193897\pi$
$168$		0			0
$169$	0.866025	−	0.500000i	0.0666173	−	0.0384615i
$170$		0			0
$171$		0			0
$172$	−2.44949	−	2.44949i	−0.186772	−	0.186772i
$173$	−3.33850	+	12.4595i	−0.253822	+	0.947275i	0.714921	+	0.699206i	$0.246463\pi$
$173$	−3.33850	+	12.4595i	−0.253822	+	0.947275i	−0.968742	+	0.248069i	$0.920204\pi$
$174$		0			0
$175$		0			0
$176$	0.550510	+	0.317837i	0.0414963	+	0.0239579i
$177$		0			0
$178$	7.75115	+	2.07691i	0.580973	+	0.155671i
$179$	10.6780			0.798114			0.399057	−	0.916926i	$-0.369338\pi$
$179$	10.6780			0.798114			0.399057	+	0.916926i	$0.369338\pi$
$180$		0			0
$181$	−15.4495			−1.14835			−0.574176	−	0.818732i	$-0.694677\pi$
$181$	−15.4495			−1.14835			−0.574176	+	0.818732i	$0.694677\pi$
$182$	−15.2597	−	4.08881i	−1.13112	−	0.303083i
$183$		0			0
$184$	0.866025	+	0.500000i	0.0638442	+	0.0368605i
$185$		0			0
$186$		0			0
$187$	−0.0739521	+	0.275993i	−0.00540792	+	0.0201826i
$188$	−6.36396	−	6.36396i	−0.464140	−	0.464140i
$189$		0			0
$190$		0			0
$191$	15.1237	−	8.73169i	1.09431	−	0.631803i	0.159593	−	0.987183i	$-0.448982\pi$
$191$	15.1237	−	8.73169i	1.09431	−	0.631803i	0.934722	+	0.355380i	$0.115649\pi$
$192$		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	$0.702426\pi$
$194$	1.34278	−	2.32577i	0.0964061	−	0.166980i
$195$		0			0
$196$	−6.89898	−	11.9494i	−0.492784	−	0.853527i
$197$	−6.92820	+	6.92820i	−0.493614	+	0.493614i	−0.909443	−	0.415829i	$-0.863492\pi$
$197$	−6.92820	+	6.92820i	−0.493614	+	0.493614i	0.415829	+	0.909443i	$0.363492\pi$
$198$		0			0
$199$			8.44949i			0.598968i	0.954101	+	0.299484i	$0.0968146\pi$
$199$			8.44949i			0.598968i	−0.954101	+	0.299484i	$0.903185\pi$
$200$		0			0
$201$		0			0
$202$	12.1562	−	3.25725i	0.855310	−	0.229179i
$203$	1.40010	−	0.375156i	0.0982677	−	0.0263308i
$204$		0			0
$205$		0			0
$206$		−	9.75663i		−	0.679777i
$207$		0			0
$208$	2.44949	−	2.44949i	0.169842	−	0.169842i
$209$	2.04989	+	3.55051i	0.141794	+	0.245594i
$210$		0			0
$211$	−4.55051	+	7.88171i	−0.313270	+	0.542600i	−0.979068	−	0.203532i	$-0.934758\pi$
$211$	−4.55051	+	7.88171i	−0.313270	+	0.542600i	0.665798	+	0.746132i	$0.268091\pi$
$212$	−1.38429	−	5.16622i	−0.0950731	−	0.354818i
$213$		0			0
$214$	16.6688	−	9.62372i	1.13945	−	0.657864i
$215$		0			0
$216$		0			0
$217$	−1.44949	−	1.44949i	−0.0983978	−	0.0983978i
$218$	−1.46272	+	5.45896i	−0.0990682	+	0.369727i
$219$		0			0
$220$		0			0
$221$	1.34847	+	0.778539i	0.0907079	+	0.0523702i
$222$		0			0
$223$	24.7575	+	6.63374i	1.65788	+	0.444228i	0.961804	−	0.273738i	$-0.0882600\pi$
$223$	24.7575	+	6.63374i	1.65788	+	0.444228i	0.696078	+	0.717966i	$0.254927\pi$
$224$	4.56048			0.304710
$225$		0			0
$226$	5.79796			0.385674
$227$	24.0506	+	6.44433i	1.59629	+	0.427725i	0.943920	−	0.330174i	$-0.107107\pi$
$227$	24.0506	+	6.44433i	1.59629	+	0.427725i	0.652372	+	0.757899i	$0.273774\pi$
$228$		0			0
$229$	1.43027	+	0.825765i	0.0945147	+	0.0545681i	0.546512	−	0.837451i	$-0.315955\pi$
$229$	1.43027	+	0.825765i	0.0945147	+	0.0545681i	−0.451998	+	0.892019i	$0.649288\pi$
$230$		0			0
$231$		0			0
$232$	−0.0822623	+	0.307007i	−0.00540079	+	0.0201560i
$233$	14.4600	+	14.4600i	0.947304	+	0.947304i	0.998679	−	0.0513751i	$-0.0163604\pi$
$233$	14.4600	+	14.4600i	0.947304	+	0.947304i	−0.0513751	+	0.998679i	$0.516360\pi$
$234$		0			0
$235$		0			0
$236$	−7.77526	+	4.48905i	−0.506126	+	0.292212i
$237$		0			0
$238$	0.530550	+	1.98004i	0.0343905	+	0.128347i
$239$	−8.48528	+	14.6969i	−0.548867	+	0.950666i	0.449485	+	0.893288i	$0.351607\pi$
$239$	−8.48528	+	14.6969i	−0.548867	+	0.950666i	−0.998353	+	0.0573782i	$0.981726\pi$
$240$		0			0
$241$	−9.50000	−	16.4545i	−0.611949	−	1.05993i	−0.990912	−	0.134515i	$-0.957053\pi$
$241$	−9.50000	−	16.4545i	−0.611949	−	1.05993i	0.378963	−	0.925412i	$-0.376281\pi$
$242$	7.49245	−	7.49245i	0.481633	−	0.481633i
$243$		0			0
$244$			0.550510i			0.0352428i
$245$		0			0
$246$		0			0
$247$	21.5804	−	5.78245i	1.37313	−	0.367929i
$248$	0.434174	−	0.116337i	0.0275701	−	0.00738738i
$249$		0			0
$250$		0			0
$251$			2.68556i			0.169511i	0.996402	+	0.0847556i	$0.0270110\pi$
$251$			2.68556i			0.169511i	−0.996402	+	0.0847556i	$0.972989\pi$
$252$		0			0
$253$	−0.449490	+	0.449490i	−0.0282592	+	0.0282592i
$254$	1.32673	+	2.29796i	0.0832463	+	0.144187i
$255$		0			0
$256$	−0.500000	+	0.866025i	−0.0312500	+	0.0541266i
$257$	−4.67050	−	17.4305i	−0.291337	−	1.08729i	−0.944083	−	0.329709i	$-0.893049\pi$
$257$	−4.67050	−	17.4305i	−0.291337	−	1.08729i	0.652745	−	0.757578i	$-0.273617\pi$
$258$		0			0
$259$	−16.7563	+	9.67423i	−1.04118	+	0.601128i
$260$		0			0
$261$		0			0
$262$	−2.55051	−	2.55051i	−0.157571	−	0.157571i
$263$	4.32149	−	16.1280i	0.266474	−	0.994495i	−0.694868	−	0.719138i	$-0.744537\pi$
$263$	4.32149	−	16.1280i	0.266474	−	0.994495i	0.961342	−	0.275358i	$-0.0887963\pi$
$264$		0			0
$265$		0			0
$266$	25.4722	+	14.7064i	1.56180	+	0.901706i
$267$		0			0
$268$	−6.38512	−	1.71089i	−0.390033	−	0.104509i
$269$	−15.0956			−0.920398			−0.460199	−	0.887816i	$-0.652222\pi$
$269$	−15.0956			−0.920398			−0.460199	+	0.887816i	$0.652222\pi$
$270$		0			0
$271$	28.0454			1.70364			0.851819	−	0.523837i	$-0.175500\pi$
$271$	28.0454			1.70364			0.851819	+	0.523837i	$0.175500\pi$
$272$	−0.434174	−	0.116337i	−0.0263257	−	0.00705394i
$273$		0			0
$274$	−18.8776	−	10.8990i	−1.14044	−	0.658431i
$275$		0			0
$276$		0			0
$277$	7.28353	−	27.1825i	0.437625	−	1.63324i	−0.297080	−	0.954852i	$-0.596013\pi$
$277$	7.28353	−	27.1825i	0.437625	−	1.63324i	0.734705	−	0.678386i	$-0.237320\pi$
$278$	2.19275	+	2.19275i	0.131513	+	0.131513i
$279$		0			0
$280$		0			0
$281$	14.8485	−	8.57277i	0.885785	−	0.511408i	0.0132238	−	0.999913i	$-0.495791\pi$
$281$	14.8485	−	8.57277i	0.885785	−	0.511408i	0.872562	+	0.488504i	$0.162457\pi$
$282$		0			0
$283$	−6.26772	−	23.3914i	−0.372577	−	1.39048i	−0.856853	−	0.515561i	$-0.827583\pi$
$283$	−6.26772	−	23.3914i	−0.372577	−	1.39048i	0.484275	−	0.874916i	$-0.339083\pi$
$284$	−3.14626	+	5.44949i	−0.186696	+	0.323368i
$285$		0			0
$286$	1.10102	+	1.90702i	0.0651047	+	0.112765i
$287$	23.8273	−	23.8273i	1.40648	−	1.40648i
$288$		0			0
$289$			16.7980i			0.988115i
$290$		0			0
$291$		0			0
$292$	9.42418	−	2.52520i	0.551508	−	0.147776i
$293$	21.2942	−	5.70577i	1.24402	−	0.333335i	0.423998	−	0.905663i	$-0.360626\pi$
$293$	21.2942	−	5.70577i	1.24402	−	0.333335i	0.820024	+	0.572329i	$0.193960\pi$
$294$		0			0
$295$		0			0
$296$		−	4.24264i		−	0.246598i
$297$		0			0
$298$	−3.12372	+	3.12372i	−0.180952	+	0.180952i
$299$	1.73205	+	3.00000i	0.100167	+	0.173494i
$300$		0			0
$301$	7.89898	−	13.6814i	0.455290	−	0.788585i
$302$	4.55416	+	16.9964i	0.262062	+	0.978030i
$303$		0			0
$304$	−5.58542	+	3.22474i	−0.320346	+	0.184952i
$305$		0			0
$306$		0			0
$307$	6.67423	+	6.67423i	0.380919	+	0.380919i	0.871433	−	0.490514i	$-0.163191\pi$
$307$	6.67423	+	6.67423i	0.380919	+	0.380919i	−0.490514	+	0.871433i	$0.663191\pi$
$308$	−0.750311	+	2.80020i	−0.0427529	+	0.159556i
$309$		0			0
$310$		0			0
$311$	−23.8207	−	13.7529i	−1.35075	−	0.779853i	−0.362392	−	0.932026i	$-0.618040\pi$
$311$	−23.8207	−	13.7529i	−1.35075	−	0.779853i	−0.988354	+	0.152172i	$0.951373\pi$
$312$		0			0
$313$	11.5422	+	3.09273i	0.652405	+	0.174811i	0.569816	−	0.821772i	$-0.307015\pi$
$313$	11.5422	+	3.09273i	0.652405	+	0.174811i	0.0825888	+	0.996584i	$0.473681\pi$
$314$	14.6349			0.825898
$315$		0			0
$316$	2.44949			0.137795
$317$	−10.5276	−	2.82086i	−0.591289	−	0.158435i	−0.0492469	−	0.998787i	$-0.515682\pi$
$317$	−10.5276	−	2.82086i	−0.591289	−	0.158435i	−0.542042	+	0.840351i	$0.682349\pi$
$318$		0			0
$319$	−0.174973	−	0.101021i	−0.00979659	−	0.00565606i
$320$		0			0
$321$		0			0
$322$	−1.18034	+	4.40508i	−0.0657777	+	0.245486i
$323$	−2.04989	−	2.04989i	−0.114059	−	0.114059i
$324$		0			0
$325$		0			0
$326$	5.44949	−	3.14626i	0.301819	−	0.174255i
$327$		0			0
$328$	1.91239	+	7.13713i	0.105594	+	0.394082i
$329$	20.5222	−	35.5454i	1.13142	−	1.95968i
$330$		0			0
$331$	−0.224745	−	0.389270i	−0.0123531	−	0.0213962i	0.859783	−	0.510660i	$-0.170599\pi$
$331$	−0.224745	−	0.389270i	−0.0123531	−	0.0213962i	−0.872136	+	0.489264i	$0.837266\pi$
$332$	3.85337	−	3.85337i	0.211481	−	0.211481i
$333$		0			0
$334$			8.79796i			0.481403i
$335$		0			0
$336$		0			0
$337$	−3.00804	+	0.806003i	−0.163859	+	0.0439058i	−0.339816	−	0.940492i	$-0.610365\pi$
$337$	−3.00804	+	0.806003i	−0.163859	+	0.0439058i	0.175957	+	0.984398i	$0.443698\pi$
$338$	−0.965926	+	0.258819i	−0.0525394	+	0.0140779i
$339$		0			0
$340$		0			0
$341$			0.285729i			0.0154731i
$342$		0			0
$343$	21.9217	−	21.9217i	1.18366	−	1.18366i
$344$	1.73205	+	3.00000i	0.0933859	+	0.161749i
$345$		0			0
$346$	6.44949	−	11.1708i	0.346727	−	0.600548i
$347$	6.15937	+	22.9871i	0.330652	+	1.23401i	0.908507	+	0.417870i	$0.137223\pi$
$347$	6.15937	+	22.9871i	0.330652	+	1.23401i	−0.577855	+	0.816140i	$0.696110\pi$
$348$		0			0
$349$	25.1541	−	14.5227i	1.34647	−	0.777383i	0.358719	−	0.933446i	$-0.383214\pi$
$349$	25.1541	−	14.5227i	1.34647	−	0.777383i	0.987747	+	0.156063i	$0.0498803\pi$
$350$		0			0
$351$		0			0
$352$	−0.449490	−	0.449490i	−0.0239579	−	0.0239579i
$353$	−8.87564	+	33.1244i	−0.472403	+	1.76303i	0.158694	+	0.987328i	$0.449272\pi$
$353$	−8.87564	+	33.1244i	−0.472403	+	1.76303i	−0.631097	+	0.775704i	$0.717395\pi$
$354$		0			0
$355$		0			0
$356$	−6.94949	−	4.01229i	−0.368322	−	0.212651i
$357$		0			0
$358$	−10.3142	−	2.76368i	−0.545122	−	0.146065i
$359$	3.32124			0.175288			0.0876441	−	0.996152i	$-0.472066\pi$
$359$	3.32124			0.175288			0.0876441	+	0.996152i	$0.472066\pi$
$360$		0			0
$361$	−22.5959			−1.18926
$362$	14.9231	+	3.99862i	0.784339	+	0.210163i
$363$		0			0
$364$	13.6814	+	7.89898i	0.717102	+	0.414019i
$365$		0			0
$366$		0			0
$367$	1.06110	−	3.96008i	0.0553890	−	0.206714i	−0.932686	−	0.360690i	$-0.882541\pi$
$367$	1.06110	−	3.96008i	0.0553890	−	0.206714i	0.988075	+	0.153976i	$0.0492078\pi$
$368$	−0.707107	−	0.707107i	−0.0368605	−	0.0368605i
$369$		0			0
$370$		0			0
$371$	21.1237	−	12.1958i	1.09669	−	0.633174i
$372$		0			0
$373$	0.127549	+	0.476018i	0.00660422	+	0.0246473i	0.969149	−	0.246474i	$-0.0792721\pi$
$373$	0.127549	+	0.476018i	0.00660422	+	0.0246473i	−0.962545	+	0.271122i	$0.912605\pi$
$374$	0.142865	−	0.247449i	0.00738735	−	0.0127953i
$375$		0			0
$376$	4.50000	+	7.79423i	0.232070	+	0.401957i
$377$	−0.778539	+	0.778539i	−0.0400968	+	0.0400968i
$378$		0			0
$379$		−	21.3485i		−	1.09660i	−0.836283	−	0.548299i	$-0.815276\pi$
$379$		−	21.3485i		−	1.09660i	0.836283	−	0.548299i	$-0.184724\pi$
$380$		0			0
$381$		0			0
$382$	−16.8683	+	4.51985i	−0.863058	+	0.231256i
$383$	−7.92256	+	2.12284i	−0.404824	+	0.108472i	−0.455485	−	0.890244i	$-0.650534\pi$
$383$	−7.92256	+	2.12284i	−0.404824	+	0.108472i	0.0506606	+	0.998716i	$0.483867\pi$
$384$		0			0
$385$		0			0
$386$		−	17.3205i		−	0.881591i
$387$		0			0
$388$	−1.89898	+	1.89898i	−0.0964061	+	0.0964061i
$389$	18.4008	+	31.8712i	0.932959	+	1.61593i	0.778233	+	0.627975i	$0.216116\pi$
$389$	18.4008	+	31.8712i	0.932959	+	1.61593i	0.154726	+	0.987957i	$0.450551\pi$
$390$		0			0
$391$	0.224745	−	0.389270i	0.0113658	−	0.0196862i
$392$	3.57117	+	13.3278i	0.180372	+	0.673156i
$393$		0			0
$394$	8.48528	−	4.89898i	0.427482	−	0.246807i
$395$		0			0
$396$		0			0
$397$	−10.5505	−	10.5505i	−0.529515	−	0.529515i	0.390913	−	0.920428i	$-0.372159\pi$
$397$	−10.5505	−	10.5505i	−0.529515	−	0.529515i	−0.920428	+	0.390913i	$0.872159\pi$
$398$	2.18689	−	8.16158i	0.109619	−	0.409103i
$399$		0			0
$400$		0			0
$401$	−7.65153	−	4.41761i	−0.382099	−	0.220605i	0.296632	−	0.954992i	$-0.404137\pi$
$401$	−7.65153	−	4.41761i	−0.382099	−	0.220605i	−0.678731	+	0.734387i	$0.737470\pi$
$402$		0			0
$403$	1.50402	+	0.403001i	0.0749207	+	0.0200749i
$404$	−12.5851			−0.626130
$405$		0			0
$406$	−1.44949			−0.0719370
$407$	2.60504	+	0.698019i	0.129127	+	0.0345995i
$408$		0			0
$409$	25.0273	+	14.4495i	1.23752	+	0.714481i	0.968586	−	0.248678i	$-0.0799961\pi$
$409$	25.0273	+	14.4495i	1.23752	+	0.714481i	0.268932	+	0.963159i	$0.413329\pi$
$410$		0			0
$411$		0			0
$412$	−2.52520	+	9.42418i	−0.124408	+	0.464296i
$413$	−28.9521	−	28.9521i	−1.42464	−	1.42464i
$414$		0			0
$415$		0			0
$416$	−3.00000	+	1.73205i	−0.147087	+	0.0849208i
$417$		0			0
$418$	−1.06110	−	3.96008i	−0.0519001	−	0.193694i
$419$	2.51059	−	4.34847i	0.122650	−	0.212437i	−0.798162	−	0.602443i	$-0.794194\pi$
$419$	2.51059	−	4.34847i	0.122650	−	0.212437i	0.920812	+	0.390007i	$0.127527\pi$
$420$		0			0
$421$	2.55051	+	4.41761i	0.124304	+	0.215301i	0.921461	−	0.388471i	$-0.126997\pi$
$421$	2.55051	+	4.41761i	0.124304	+	0.215301i	−0.797157	+	0.603773i	$0.793663\pi$
$422$	6.43539	−	6.43539i	0.313270	−	0.313270i
$423$		0			0
$424$			5.34847i			0.259745i
$425$		0			0
$426$		0			0
$427$	−2.42504	+	0.649788i	−0.117356	+	0.0314455i
$428$	−18.5916	+	4.98161i	−0.898659	+	0.240795i
$429$		0			0
$430$		0			0
$431$		−	15.5563i		−	0.749323i	−0.927162	−	0.374661i	$-0.877759\pi$
$431$		−	15.5563i		−	0.749323i	0.927162	−	0.374661i	$-0.122241\pi$
$432$		0			0
$433$	−13.4495	+	13.4495i	−0.646341	+	0.646341i	−0.952107	−	0.305766i	$-0.901088\pi$
$433$	−13.4495	+	13.4495i	−0.646341	+	0.646341i	0.305766	+	0.952107i	$0.401088\pi$
$434$	1.02494	+	1.77526i	0.0491989	+	0.0852150i
$435$		0			0
$436$	2.82577	−	4.89437i	0.135330	−	0.234398i
$437$	−1.66925	−	6.22973i	−0.0798511	−	0.298008i
$438$		0			0
$439$	25.8058	−	14.8990i	1.23164	−	0.711089i	0.264271	−	0.964449i	$-0.414869\pi$
$439$	25.8058	−	14.8990i	1.23164	−	0.711089i	0.967372	+	0.253359i	$0.0815354\pi$
$440$		0			0
$441$		0			0
$442$	−1.10102	−	1.10102i	−0.0523702	−	0.0523702i
$443$	−1.41043	+	5.26380i	−0.0670116	+	0.250091i	−0.991303	−	0.131596i	$-0.957990\pi$
$443$	−1.41043	+	5.26380i	−0.0670116	+	0.250091i	0.924292	+	0.381687i	$0.124657\pi$
$444$		0			0
$445$		0			0
$446$	−22.1969	−	12.8154i	−1.05106	−	0.606827i
$447$		0			0
$448$	−4.40508	−	1.18034i	−0.208121	−	0.0557658i
$449$	0.921404			0.0434837			0.0217419	−	0.999764i	$-0.493079\pi$
$449$	0.921404			0.0434837			0.0217419	+	0.999764i	$0.493079\pi$
$450$		0			0
$451$	−4.69694			−0.221170
$452$	−5.60040	−	1.50062i	−0.263421	−	0.0705833i
$453$		0			0
$454$	−21.5631	−	12.4495i	−1.01201	−	0.584284i
$455$		0			0
$456$		0			0
$457$	3.78780	−	14.1363i	0.177186	−	0.661267i	−0.818983	−	0.573818i	$-0.805462\pi$
$457$	3.78780	−	14.1363i	0.177186	−	0.661267i	0.996169	−	0.0874492i	$-0.0278715\pi$
$458$	−1.16781	−	1.16781i	−0.0545681	−	0.0545681i
$459$		0			0
$460$		0			0
$461$	−28.6237	+	16.5259i	−1.33314	+	0.769689i	−0.985780	−	0.168043i	$-0.946255\pi$
$461$	−28.6237	+	16.5259i	−1.33314	+	0.769689i	−0.347360	+	0.937732i	$0.612922\pi$
$462$		0			0
$463$	3.16668	+	11.8182i	0.147168	+	0.549239i	0.999649	+	0.0264810i	$0.00843014\pi$
$463$	3.16668	+	11.8182i	0.147168	+	0.549239i	−0.852481	+	0.522758i	$0.824903\pi$
$464$	0.158919	−	0.275255i	0.00737761	−	0.0127784i
$465$		0			0
$466$	−10.2247	−	17.7098i	−0.473652	−	0.820390i
$467$	−2.82843	+	2.82843i	−0.130884	+	0.130884i	−0.769514	−	0.638630i	$-0.779501\pi$
$467$	−2.82843	+	2.82843i	−0.130884	+	0.130884i	0.638630	+	0.769514i	$0.279501\pi$
$468$		0			0
$469$		−	30.1464i		−	1.39203i
$470$		0			0
$471$		0			0
$472$	8.67217	−	2.32370i	0.399169	−	0.106957i
$473$	−2.12701	+	0.569930i	−0.0977999	+	0.0262054i
$474$		0			0
$475$		0			0
$476$		−	2.04989i		−	0.0939565i
$477$		0			0
$478$	12.0000	−	12.0000i	0.548867	−	0.548867i
$479$	−3.53553	−	6.12372i	−0.161543	−	0.279800i	0.773879	−	0.633333i	$-0.218314\pi$
$479$	−3.53553	−	6.12372i	−0.161543	−	0.279800i	−0.935422	+	0.353533i	$0.884980\pi$
$480$		0			0
$481$	7.34847	−	12.7279i	0.335061	−	0.580343i
$482$	4.91756	+	18.3526i	0.223989	+	0.835938i
$483$		0			0
$484$	−9.17633	+	5.29796i	−0.417106	+	0.240816i
$485$		0			0
$486$		0			0
$487$	12.0000	+	12.0000i	0.543772	+	0.543772i	0.924632	−	0.380861i	$-0.124372\pi$
$487$	12.0000	+	12.0000i	0.543772	+	0.543772i	−0.380861	+	0.924632i	$0.624372\pi$
$488$	0.142483	−	0.531752i	0.00644988	−	0.0240713i
$489$		0			0
$490$		0			0
$491$	24.2474	+	13.9993i	1.09427	+	0.631778i	0.934711	−	0.355410i	$-0.115659\pi$
$491$	24.2474	+	13.9993i	1.09427	+	0.631778i	0.159561	+	0.987188i	$0.448992\pi$
$492$		0			0
$493$	0.137997	+	0.0369761i	0.00621505	+	0.00166532i
$494$	−22.3417			−1.00520
$495$		0			0
$496$	−0.449490			−0.0201827
$497$	−27.7191	−	7.42731i	−1.24337	−	0.333161i
$498$		0			0
$499$	−0.778539	−	0.449490i	−0.0348522	−	0.0201219i	0.482473	−	0.875911i	$-0.339739\pi$
$499$	−0.778539	−	0.449490i	−0.0348522	−	0.0201219i	−0.517325	+	0.855789i	$0.673072\pi$
$500$		0			0
$501$		0			0
$502$	0.695075	−	2.59405i	0.0310227	−	0.115778i
$503$	−4.02834	−	4.02834i	−0.179615	−	0.179615i	0.611573	−	0.791188i	$-0.290537\pi$
$503$	−4.02834	−	4.02834i	−0.179615	−	0.179615i	−0.791188	+	0.611573i	$0.790537\pi$
$504$		0			0
$505$		0			0
$506$	0.550510	−	0.317837i	0.0244732	−	0.0141296i
$507$		0			0
$508$	−0.686765	−	2.56304i	−0.0304702	−	0.113717i
$509$	−4.22659	+	7.32066i	−0.187340	+	0.324483i	−0.944363	−	0.328906i	$-0.893320\pi$
$509$	−4.22659	+	7.32066i	−0.187340	+	0.324483i	0.757022	+	0.653389i	$0.226653\pi$
$510$		0			0
$511$	22.2474	+	38.5337i	0.984169	+	1.70463i
$512$	0.707107	−	0.707107i	0.0312500	−	0.0312500i
$513$		0			0
$514$			18.0454i			0.795949i
$515$		0			0
$516$		0			0
$517$	−5.52613	+	1.48072i	−0.243039	+	0.0651221i
$518$	18.6892	−	5.00775i	0.821156	−	0.220028i
$519$		0			0
$520$		0			0
$521$			29.4449i			1.29000i	0.764181	+	0.645001i	$0.223143\pi$
$521$			29.4449i			1.29000i	−0.764181	+	0.645001i	$0.776857\pi$
$522$		0			0
$523$	−4.22474	+	4.22474i	−0.184735	+	0.184735i	−0.793416	−	0.608680i	$-0.791699\pi$
$523$	−4.22474	+	4.22474i	−0.184735	+	0.184735i	0.608680	+	0.793416i	$0.291699\pi$
$524$	1.80348	+	3.12372i	0.0787855	+	0.136461i
$525$		0			0
$526$	−8.34847	+	14.4600i	−0.364011	+	0.630485i
$527$	−0.0522921	−	0.195157i	−0.00227788	−	0.00850116i
$528$		0			0
$529$	−19.0526	+	11.0000i	−0.828372	+	0.478261i
$530$		0			0
$531$		0			0
$532$	−20.7980	−	20.7980i	−0.901706	−	0.901706i
$533$	−6.62471	+	24.7238i	−0.286948	+	1.07090i
$534$		0			0
$535$		0			0
$536$	5.72474	+	3.30518i	0.247271	+	0.142762i
$537$		0			0
$538$	14.5813	+	3.90704i	0.628643	+	0.168444i
$539$	−8.77101			−0.377794
$540$		0			0
$541$	27.9444			1.20142			0.600712	−	0.799466i	$-0.294884\pi$
$541$	27.9444			1.20142			0.600712	+	0.799466i	$0.294884\pi$
$542$	−27.0898	−	7.25869i	−1.16361	−	0.311787i
$543$		0			0
$544$	0.389270	+	0.224745i	0.0166898	+	0.00963586i
$545$		0			0
$546$		0			0
$547$	1.05279	−	3.92907i	0.0450140	−	0.167995i	−0.939760	−	0.341836i	$-0.888951\pi$
$547$	1.05279	−	3.92907i	0.0450140	−	0.167995i	0.984774	+	0.173841i	$0.0556180\pi$
$548$	15.4135	+	15.4135i	0.658431	+	0.658431i
$549$		0			0
$550$		0			0
$551$	1.77526	−	1.02494i	0.0756284	−	0.0436641i
$552$		0			0
$553$	2.89123	+	10.7902i	0.122947	+	0.458846i
$554$	−14.0707	+	24.3712i	−0.597807	+	1.03543i
$555$		0			0
$556$	−1.55051	−	2.68556i	−0.0657563	−	0.113893i
$557$	7.88171	−	7.88171i	0.333959	−	0.333959i	−0.520129	−	0.854088i	$-0.674116\pi$
$557$	7.88171	−	7.88171i	0.333959	−	0.333959i	0.854088	+	0.520129i	$0.174116\pi$
$558$		0			0
$559$			12.0000i			0.507546i
$560$		0			0
$561$		0			0
$562$	−16.5613	+	4.43759i	−0.698597	+	0.187188i
$563$	18.9819	−	5.08619i	0.799993	−	0.214357i	0.164412	−	0.986392i	$-0.447427\pi$
$563$	18.9819	−	5.08619i	0.799993	−	0.214357i	0.635581	+	0.772034i	$0.280761\pi$
$564$		0			0
$565$		0			0
$566$			24.2166i			1.01790i
$567$		0			0
$568$	4.44949	−	4.44949i	0.186696	−	0.186696i
$569$	−9.58166	−	16.5959i	−0.401684	−	0.695737i	0.592245	−	0.805758i	$-0.298242\pi$
$569$	−9.58166	−	16.5959i	−0.401684	−	0.695737i	−0.993929	+	0.110021i	$0.964908\pi$
$570$		0			0
$571$	−18.4495	+	31.9555i	−0.772087	+	1.33729i	0.164330	+	0.986405i	$0.447454\pi$
$571$	−18.4495	+	31.9555i	−0.772087	+	1.33729i	−0.936417	+	0.350889i	$0.885880\pi$
$572$	−0.569930	−	2.12701i	−0.0238300	−	0.0889347i
$573$		0			0
$574$	−29.1824	+	16.8485i	−1.21805	+	0.703242i
$575$		0			0
$576$		0			0
$577$	17.0000	+	17.0000i	0.707719	+	0.707719i	0.966055	−	0.258336i	$-0.0831741\pi$
$577$	17.0000	+	17.0000i	0.707719	+	0.707719i	−0.258336	+	0.966055i	$0.583174\pi$
$578$	4.34763	−	16.2256i	0.180838	−	0.674895i
$579$		0			0
$580$		0			0
$581$	21.5227	+	12.4261i	0.892912	+	0.515523i
$582$		0			0
$583$	−3.28404	−	0.879955i	−0.136011	−	0.0364440i
$584$	−9.75663			−0.403732
$585$		0			0
$586$	−22.0454			−0.910687
$587$	12.6009	+	3.37640i	0.520095	+	0.139359i	0.509310	−	0.860583i	$-0.329901\pi$
$587$	12.6009	+	3.37640i	0.520095	+	0.139359i	0.0107843	+	0.999942i	$0.496567\pi$
$588$		0			0
$589$	−2.51059	−	1.44949i	−0.103447	−	0.0597252i
$590$		0			0
$591$		0			0
$592$	−1.09808	+	4.09808i	−0.0451307	+	0.168430i
$593$	7.24604	+	7.24604i	0.297559	+	0.297559i	0.840057	−	0.542498i	$-0.182521\pi$
$593$	7.24604	+	7.24604i	0.297559	+	0.297559i	−0.542498	+	0.840057i	$0.682521\pi$
$594$		0			0
$595$		0			0
$596$	3.82577	−	2.20881i	0.156709	−	0.0904762i
$597$		0			0
$598$	−0.896575	−	3.34607i	−0.0366637	−	0.136831i
$599$	9.97093	−	17.2702i	0.407401	−	0.705639i	−0.587197	−	0.809444i	$-0.699768\pi$
$599$	9.97093	−	17.2702i	0.407401	−	0.705639i	0.994598	+	0.103805i	$0.0331018\pi$
$600$		0			0
$601$	−2.65153	−	4.59259i	−0.108158	−	0.187335i	0.806866	−	0.590735i	$-0.201162\pi$
$601$	−2.65153	−	4.59259i	−0.108158	−	0.187335i	−0.915024	+	0.403399i	$0.867829\pi$
$602$	−11.1708	+	11.1708i	−0.455290	+	0.455290i
$603$		0			0
$604$		−	17.5959i		−	0.715968i
$605$		0			0
$606$		0			0
$607$	11.3732	−	3.04744i	0.461624	−	0.123692i	−0.0205092	−	0.999790i	$-0.506529\pi$
$607$	11.3732	−	3.04744i	0.461624	−	0.123692i	0.482133	+	0.876098i	$0.339862\pi$
$608$	6.22973	−	1.66925i	0.252649	−	0.0676971i
$609$		0			0
$610$		0			0
$611$			31.1769i			1.26128i
$612$		0			0
$613$	−6.79796	+	6.79796i	−0.274567	+	0.274567i	−0.830936	−	0.556369i	$-0.812194\pi$
$613$	−6.79796	+	6.79796i	−0.274567	+	0.274567i	0.556369	+	0.830936i	$0.312194\pi$
$614$	−4.71940	−	8.17423i	−0.190459	−	0.329885i
$615$		0			0
$616$	1.44949	−	2.51059i	0.0584016	−	0.101155i
$617$	−4.37378	−	16.3232i	−0.176082	−	0.657146i	−0.996365	−	0.0851882i	$-0.972851\pi$
$617$	−4.37378	−	16.3232i	−0.176082	−	0.657146i	0.820283	−	0.571958i	$-0.193816\pi$
$618$		0			0
$619$	−42.2121	+	24.3712i	−1.69665	+	0.979560i	−0.747748	+	0.663982i	$0.768865\pi$
$619$	−42.2121	+	24.3712i	−1.69665	+	0.979560i	−0.948900	+	0.315578i	$0.897802\pi$
$620$		0			0
$621$		0			0
$622$	19.4495	+	19.4495i	0.779853	+	0.779853i
$623$	9.47172	−	35.3489i	0.379476	−	1.41623i
$624$		0			0
$625$		0			0
$626$	−10.3485	−	5.97469i	−0.413608	−	0.238797i
$627$		0			0
$628$	−14.1363	−	3.78780i	−0.564099	−	0.151150i
$629$	−1.90702			−0.0760380
$630$		0			0
$631$	−3.10102			−0.123450			−0.0617248	−	0.998093i	$-0.519660\pi$
$631$	−3.10102			−0.123450			−0.0617248	+	0.998093i	$0.519660\pi$
$632$	−2.36603	−	0.633975i	−0.0941154	−	0.0252182i
$633$		0			0
$634$	9.43879	+	5.44949i	0.374862	+	0.216427i
$635$		0			0
$636$		0			0
$637$	−12.3709	+	46.1689i	−0.490153	+	1.82928i
$638$	0.142865	+	0.142865i	0.00565606	+	0.00565606i
$639$		0			0
$640$		0			0
$641$	−16.7474	+	9.66914i	−0.661484	+	0.381908i	−0.792842	−	0.609427i	$-0.791400\pi$
$641$	−16.7474	+	9.66914i	−0.661484	+	0.381908i	0.131358	+	0.991335i	$0.458066\pi$
$642$		0			0
$643$	−1.63694	−	6.10913i	−0.0645545	−	0.240921i	0.926108	−	0.377259i	$-0.123133\pi$
$643$	−1.63694	−	6.10913i	−0.0645545	−	0.240921i	−0.990662	+	0.136338i	$0.956467\pi$
$644$	2.28024	−	3.94949i	0.0898540	−	0.155632i
$645$		0			0
$646$	1.44949	+	2.51059i	0.0570294	+	0.0987778i
$647$	−23.5416	+	23.5416i	−0.925516	+	0.925516i	−0.997412	−	0.0718961i	$-0.977095\pi$
$647$	−23.5416	+	23.5416i	−0.925516	+	0.925516i	0.0718961	+	0.997412i	$0.477095\pi$
$648$		0			0
$649$			5.70714i			0.224025i
$650$		0			0
$651$		0			0
$652$	−6.07812	+	1.62863i	−0.238037	+	0.0637819i
$653$	25.5482	−	6.84563i	0.999780	−	0.267890i	0.278427	−	0.960457i	$-0.410187\pi$
$653$	25.5482	−	6.84563i	0.999780	−	0.267890i	0.721353	+	0.692567i	$0.243520\pi$
$654$		0			0
$655$		0			0
$656$		−	7.38891i		−	0.288488i
$657$		0			0
$658$	−29.0227	+	29.0227i	−1.13142	+	1.13142i
$659$	5.65685	+	9.79796i	0.220360	+	0.381674i	0.954917	−	0.296872i	$-0.0959435\pi$
$659$	5.65685	+	9.79796i	0.220360	+	0.381674i	−0.734557	+	0.678546i	$0.762610\pi$
$660$		0			0
$661$	0.651531	−	1.12848i	0.0253416	−	0.0438930i	−0.853076	−	0.521786i	$-0.825266\pi$
$661$	0.651531	−	1.12848i	0.0253416	−	0.0438930i	0.878418	+	0.477893i	$0.158599\pi$
$662$	0.116337	+	0.434174i	0.00452155	+	0.0168746i
$663$		0			0
$664$	−4.71940	+	2.72474i	−0.183148	+	0.105741i
$665$		0			0
$666$		0			0
$667$	0.224745	+	0.224745i	0.00870216	+	0.00870216i
$668$	2.27708	−	8.49818i	0.0881028	−	0.328804i
$669$		0			0
$670$		0			0
$671$	0.303062	+	0.174973i	0.0116996	+	0.00675474i
$672$		0			0
$673$	−22.4704	−	6.02093i	−0.866171	−	0.232090i	−0.201740	−	0.979439i	$-0.564660\pi$
$673$	−22.4704	−	6.02093i	−0.866171	−	0.232090i	−0.664431	+	0.747349i	$0.731326\pi$
$674$	3.11416			0.119953
$675$		0			0
$676$	1.00000			0.0384615
$677$	−44.0423	−	11.8011i	−1.69268	−	0.453553i	−0.721602	−	0.692308i	$-0.756594\pi$
$677$	−44.0423	−	11.8011i	−1.69268	−	0.453553i	−0.971080	+	0.238755i	$0.923261\pi$
$678$		0			0
$679$	−10.6066	−	6.12372i	−0.407044	−	0.235007i
$680$		0			0
$681$		0			0
$682$	0.0739521	−	0.275993i	0.00283177	−	0.0105683i
$683$	13.8564	+	13.8564i	0.530201	+	0.530201i	0.920632	−	0.390431i	$-0.127674\pi$
$683$	13.8564	+	13.8564i	0.530201	+	0.530201i	−0.390431	+	0.920632i	$0.627674\pi$
$684$		0			0
$685$		0			0
$686$	−26.8485	+	15.5010i	−1.02508	+	0.591830i
$687$		0			0
$688$	−0.896575	−	3.34607i	−0.0341816	−	0.127568i
$689$	−9.26382	+	16.0454i	−0.352923	+	0.611281i
$690$		0			0
$691$	10.4722	+	18.1384i	0.398381	+	0.690016i	0.993526	−	0.113602i	$-0.0362388\pi$
$691$	10.4722	+	18.1384i	0.398381	+	0.690016i	−0.595145	+	0.803618i	$0.702906\pi$
$692$	−9.12096	+	9.12096i	−0.346727	+	0.346727i
$693$		0			0
$694$		−	23.7980i		−	0.903358i
$695$		0			0
$696$		0			0
$697$	3.20807	−	0.859599i	0.121514	−	0.0325596i
$698$	−28.0557	+	7.51750i	−1.06192	+	0.284542i
$699$		0			0
$700$		0			0
$701$		−	21.1024i		−	0.797028i	−0.917162	−	0.398514i	$-0.869526\pi$
$701$		−	21.1024i		−	0.797028i	0.917162	−	0.398514i	$-0.130474\pi$
$702$		0			0
$703$	−19.3485	+	19.3485i	−0.729741	+	0.729741i
$704$	0.317837	+	0.550510i	0.0119789	+	0.0207481i
$705$		0			0
$706$	17.1464	−	29.6985i	0.645314	−	1.11772i
$707$	−14.8546	−	55.4382i	−0.558666	−	2.08497i
$708$		0			0
$709$	25.6790	−	14.8258i	0.964394	−	0.556793i	0.0668716	−	0.997762i	$-0.478698\pi$
$709$	25.6790	−	14.8258i	0.964394	−	0.556793i	0.897523	+	0.440968i	$0.145365\pi$
$710$		0			0
$711$		0			0
$712$	5.67423	+	5.67423i	0.212651	+	0.212651i
$713$	0.116337	−	0.434174i	0.00435684	−	0.0162599i
$714$		0			0
$715$		0			0
$716$	9.24745	+	5.33902i	0.345593	+	0.199528i
$717$		0			0
$718$	−3.20807	−	0.859599i	−0.119724	−	0.0320800i
$719$	32.5269			1.21305			0.606525	−	0.795065i	$-0.292563\pi$
$719$	32.5269			1.21305			0.606525	+	0.795065i	$0.292563\pi$
$720$		0			0
$721$	−44.4949			−1.65708
$722$	21.8260	+	5.84825i	0.812279	+	0.217649i
$723$		0			0
$724$	−13.3797	−	7.72474i	−0.497251	−	0.287088i
$725$		0			0
$726$		0			0
$727$	−12.5068	+	46.6759i	−0.463850	+	1.73111i	0.196822	+	0.980439i	$0.436938\pi$
$727$	−12.5068	+	46.6759i	−0.463850	+	1.73111i	−0.660673	+	0.750674i	$0.729729\pi$
$728$	−11.1708	−	11.1708i	−0.414019	−	0.414019i
$729$		0			0
$730$		0			0
$731$	1.34847	−	0.778539i	0.0498749	−	0.0287953i
$732$		0			0
$733$	−8.83821	−	32.9846i	−0.326447	−	1.21832i	−0.912850	−	0.408296i	$-0.866123\pi$
$733$	−8.83821	−	32.9846i	−0.326447	−	1.21832i	0.586403	−	0.810019i	$-0.300543\pi$
$734$	−2.04989	+	3.55051i	−0.0756627	+	0.131052i
$735$		0			0
$736$	0.500000	+	0.866025i	0.0184302	+	0.0319221i
$737$	−2.97129	+	2.97129i	−0.109449	+	0.109449i
$738$		0			0
$739$			28.9444i			1.06474i	0.846513	+	0.532368i	$0.178698\pi$
$739$			28.9444i			1.06474i	−0.846513	+	0.532368i	$0.821302\pi$
$740$		0			0
$741$		0			0
$742$	−23.5605	+	6.31300i	−0.864931	+	0.231758i
$743$	−9.56168	+	2.56204i	−0.350784	+	0.0939923i	−0.429909	−	0.902872i	$-0.641454\pi$
$743$	−9.56168	+	2.56204i	−0.350784	+	0.0939923i	0.0791245	+	0.996865i	$0.474788\pi$
$744$		0			0
$745$		0			0
$746$		−	0.492810i		−	0.0180431i
$747$		0			0
$748$	−0.202041	+	0.202041i	−0.00738735	+	0.00738735i
$749$	−43.8888	−	76.0176i	−1.60366	−	2.77762i
$750$		0			0
$751$	−10.3485	+	17.9241i	−0.377621	+	0.654059i	−0.990716	−	0.135951i	$-0.956591\pi$
$751$	−10.3485	+	17.9241i	−0.377621	+	0.654059i	0.613095	+	0.790010i	$0.289924\pi$
$752$	−2.32937	−	8.69333i	−0.0849434	−	0.317013i
$753$		0			0
$754$	0.953512	−	0.550510i	0.0347248	−	0.0200484i
$755$		0			0
$756$		0			0
$757$	−22.0454	−	22.0454i	−0.801254	−	0.801254i	0.182038	−	0.983292i	$-0.441731\pi$
$757$	−22.0454	−	22.0454i	−0.801254	−	0.801254i	−0.983292	+	0.182038i	$0.941731\pi$
$758$	−5.52539	+	20.6210i	−0.200691	+	0.748990i
$759$		0			0
$760$		0			0
$761$	5.60102	+	3.23375i	0.203037	+	0.117223i	0.598071	−	0.801443i	$-0.295934\pi$
$761$	5.60102	+	3.23375i	0.203037	+	0.117223i	−0.395034	+	0.918666i	$0.629267\pi$
$762$		0			0
$763$	24.8955	+	6.67072i	0.901276	+	0.241496i
$764$	17.4634			0.631803
$765$		0			0
$766$	8.20204			0.296352
$767$	30.0413	+	8.04954i	1.08473	+	0.290652i
$768$		0			0
$769$	8.39780	+	4.84847i	0.302832	+	0.174840i	0.643715	−	0.765266i	$-0.277392\pi$
$769$	8.39780	+	4.84847i	0.302832	+	0.174840i	−0.340882	+	0.940106i	$0.610726\pi$
$770$		0			0
$771$		0			0
$772$	−4.48288	+	16.7303i	−0.161342	+	0.602138i
$773$	−30.8270	−	30.8270i	−1.10877	−	1.10877i	−0.993313	−	0.115456i	$-0.963167\pi$
$773$	−30.8270	−	30.8270i	−1.10877	−	1.10877i	−0.115456	−	0.993313i	$-0.536833\pi$
$774$		0			0
$775$		0			0
$776$	2.32577	−	1.34278i	0.0834901	−	0.0482030i
$777$		0			0
$778$	−9.52497	−	35.5477i	−0.341487	−	1.27445i
$779$	23.8273	−	41.2702i	0.853703	−	1.47866i
$780$		0			0
$781$	2.00000	+	3.46410i	0.0715656	+	0.123955i
$782$	−0.317837	+	0.317837i	−0.0113658	+	0.0113658i
$783$		0			0
$784$		−	13.7980i		−	0.492784i
$785$		0			0
$786$		0			0
$787$	−9.42418	+	2.52520i	−0.335936	+	0.0900137i	−0.422844	−	0.906203i	$-0.638968\pi$
$787$	−9.42418	+	2.52520i	−0.335936	+	0.0900137i	0.0869079	+	0.996216i	$0.472301\pi$
$788$	−9.46410	+	2.53590i	−0.337145	+	0.0903376i
$789$		0			0
$790$		0			0
$791$		−	26.4415i		−	0.940150i
$792$		0			0
$793$	1.34847	−	1.34847i	0.0478855	−	0.0478855i
$794$	7.46034	+	12.9217i	0.264757	+	0.458573i
$795$		0			0
$796$	−4.22474	+	7.31747i	−0.149742	+	0.259361i
$797$	10.3005	+	38.4419i	0.364861	+	1.36168i	0.867609	+	0.497248i	$0.165656\pi$
$797$	10.3005	+	38.4419i	0.364861	+	1.36168i	−0.502747	+	0.864433i	$0.667677\pi$
$798$		0			0
$799$	3.50343	−	2.02270i	0.123942	−	0.0715581i
$800$		0			0
$801$		0			0
$802$	6.24745	+	6.24745i	0.220605	+	0.220605i
$803$	1.60521	−	5.99071i	0.0566465	−	0.211408i
$804$		0			0
$805$		0			0
$806$	−1.34847	−	0.778539i	−0.0474978	−	0.0274229i
$807$		0			0
$808$	12.1562	+	3.25725i	0.427655	+	0.114590i
$809$	−19.4490			−0.683792			−0.341896	−	0.939738i	$-0.611069\pi$
$809$	−19.4490			−0.683792			−0.341896	+	0.939738i	$0.611069\pi$
$810$		0			0
$811$	−39.6413			−1.39200			−0.695998	−	0.718044i	$-0.745038\pi$
$811$	−39.6413			−1.39200			−0.695998	+	0.718044i	$0.745038\pi$
$812$	1.40010	+	0.375156i	0.0491339	+	0.0131654i
$813$		0			0
$814$	−2.33562	−	1.34847i	−0.0818633	−	0.0472638i
$815$		0			0
$816$		0			0
$817$	5.78245	−	21.5804i	0.202302	−	0.755003i
$818$	−20.4347	−	20.4347i	−0.714481	−	0.714481i
$819$		0			0
$820$		0			0
$821$	−19.3207	+	11.1548i	−0.674296	+	0.389305i	−0.797702	−	0.603051i	$-0.793951\pi$
$821$	−19.3207	+	11.1548i	−0.674296	+	0.389305i	0.123407	+	0.992356i	$0.460618\pi$
$822$		0			0
$823$	−0.867910	−	3.23908i	−0.0302534	−	0.112907i	0.949148	−	0.314830i	$-0.101948\pi$
$823$	−0.867910	−	3.23908i	−0.0302534	−	0.112907i	−0.979401	+	0.201923i	$0.935281\pi$
$824$	4.87832	−	8.44949i	0.169944	−	0.294352i
$825$		0			0
$826$	20.4722	+	35.4589i	0.712319	+	1.23377i
$827$	−31.5662	+	31.5662i	−1.09766	+	1.09766i	−0.102980	+	0.994683i	$0.532838\pi$
$827$	−31.5662	+	31.5662i	−1.09766	+	1.09766i	−0.994683	+	0.102980i	$0.967162\pi$
$828$		0			0
$829$		−	10.5505i		−	0.366434i	−0.983072	−	0.183217i	$-0.941349\pi$
$829$		−	10.5505i		−	0.366434i	0.983072	−	0.183217i	$-0.0586512\pi$
$830$		0			0
$831$		0			0
$832$	3.34607	−	0.896575i	0.116004	−	0.0310832i
$833$	5.99071	−	1.60521i	0.207566	−	0.0556171i
$834$		0			0
$835$		0			0
$836$			4.09978i			0.141794i
$837$		0			0
$838$	−3.55051	+	3.55051i	−0.122650	+	0.122650i
$839$	0.246405	+	0.426786i	0.00850684	+	0.0147343i	0.870247	−	0.492615i	$-0.163959\pi$
$839$	0.246405	+	0.426786i	0.00850684	+	0.0147343i	−0.861741	+	0.507349i	$0.830625\pi$
$840$		0			0
$841$	14.4495	−	25.0273i	0.498258	−	0.863009i
$842$	−1.32024	−	4.92721i	−0.0454985	−	0.169803i
$843$		0			0
$844$	−7.88171	+	4.55051i	−0.271300	+	0.156635i
$845$		0			0
$846$		0			0
$847$	−34.1691	−	34.1691i	−1.17407	−	1.17407i
$848$	1.38429	−	5.16622i	0.0475366	−	0.177409i
$849$		0			0
$850$		0			0
$851$	−3.67423	−	2.12132i	−0.125951	−	0.0727179i
$852$		0			0
$853$	−2.39403	−	0.641478i	−0.0819700	−	0.0219638i	0.217601	−	0.976038i	$-0.430177\pi$
$853$	−2.39403	−	0.641478i	−0.0819700	−	0.0219638i	−0.299571	+	0.954074i	$0.596844\pi$
$854$	2.51059			0.0859106
$855$		0			0
$856$	19.2474			0.657864
$857$	15.2597	+	4.08881i	0.521260	+	0.139671i	0.509849	−	0.860264i	$-0.329701\pi$
$857$	15.2597	+	4.08881i	0.521260	+	0.139671i	0.0114106	+	0.999935i	$0.496368\pi$
$858$		0			0
$859$	40.2658	+	23.2474i	1.37385	+	0.793193i	0.991410	−	0.130788i	$-0.0417506\pi$
$859$	40.2658	+	23.2474i	1.37385	+	0.793193i	0.382440	+	0.923980i	$0.375084\pi$
$860$		0			0
$861$		0			0
$862$	−4.02628	+	15.0263i	−0.137136	+	0.511797i
$863$	20.7132	+	20.7132i	0.705085	+	0.705085i	0.965497	−	0.260413i	$-0.0838586\pi$
$863$	20.7132	+	20.7132i	0.705085	+	0.705085i	−0.260413	+	0.965497i	$0.583859\pi$
$864$		0			0
$865$		0			0
$866$	16.4722	−	9.51023i	0.559748	−	0.323171i
$867$		0			0
$868$	−0.530550	−	1.98004i	−0.0180080	−	0.0672069i
$869$	0.778539	−	1.34847i	0.0264101	−	0.0457437i
$870$		0			0
$871$	11.4495	+	19.8311i	0.387951	+	0.671951i
$872$	−3.99624	+	3.99624i	−0.135330	+	0.135330i
$873$		0			0
$874$			6.44949i			0.218157i
$875$		0			0
$876$		0			0
$877$	41.3188	−	11.0713i	1.39524	−	0.373852i	0.518604	−	0.855014i	$-0.326452\pi$
$877$	41.3188	−	11.0713i	1.39524	−	0.373852i	0.876632	+	0.481162i	$0.159785\pi$
$878$	−28.7826	+	7.71228i	−0.971366	+	0.260277i
$879$		0			0
$880$		0			0
$881$			54.8365i			1.84749i	0.383010	+	0.923744i	$0.374887\pi$
$881$			54.8365i			1.84749i	−0.383010	+	0.923744i	$0.625113\pi$
$882$		0			0
$883$	−6.27015	+	6.27015i	−0.211007	+	0.211007i	−0.804695	−	0.593688i	$-0.797671\pi$
$883$	−6.27015	+	6.27015i	−0.211007	+	0.211007i	0.593688	+	0.804695i	$0.297671\pi$
$884$	0.778539	+	1.34847i	0.0261851	+	0.0453539i
$885$		0			0
$886$	2.72474	−	4.71940i	0.0915396	−	0.158551i
$887$	−2.12284	−	7.92256i	−0.0712781	−	0.266014i	0.921085	−	0.389360i	$-0.127304\pi$
$887$	−2.12284	−	7.92256i	−0.0712781	−	0.266014i	−0.992364	+	0.123347i	$0.960637\pi$
$888$		0			0
$889$	10.4798	−	6.05051i	0.351481	−	0.202928i
$890$		0			0
$891$		0			0
$892$	18.1237	+	18.1237i	0.606827	+	0.606827i
$893$	15.0233	−	56.0676i	0.502734	−	1.87623i
$894$		0			0
$895$		0			0
$896$	3.94949	+	2.28024i	0.131943	+	0.0761774i
$897$		0			0
$898$	−0.890008	−	0.238477i	−0.0296999	−	0.00795807i
$899$	0.142865			0.00476480
$900$		0			0
$901$	2.40408			0.0800916
$902$	4.53689	+	1.21566i	0.151062	+	0.0404770i
$903$		0			0
$904$	5.02118	+	2.89898i	0.167002	+	0.0964186i
$905$		0			0
$906$		0			0
$907$	0.978838	−	3.65307i	0.0325018	−	0.121298i	−0.947769	−	0.318957i	$-0.896667\pi$
$907$	0.978838	−	3.65307i	0.0325018	−	0.121298i	0.980271	+	0.197659i	$0.0633339\pi$
$908$	17.6062	+	17.6062i	0.584284	+	0.584284i
$909$		0			0
$910$		0			0
$911$	6.12372	−	3.53553i	0.202888	−	0.117137i	−0.395114	−	0.918632i	$-0.629295\pi$
$911$	6.12372	−	3.53553i	0.202888	−	0.117137i	0.598002	+	0.801495i	$0.295962\pi$
$912$		0			0
$913$	−0.896575	−	3.34607i	−0.0296723	−	0.110739i
$914$	−7.31747	+	12.6742i	−0.242040	+	0.419226i
$915$		0			0
$916$	0.825765	+	1.43027i	0.0272841	+	0.0472574i
$917$	−11.6315	+	11.6315i	−0.384107	+	0.384107i
$918$		0			0
$919$			12.6515i			0.417335i	0.977987	+	0.208668i	$0.0669127\pi$
$919$			12.6515i			0.417335i	−0.977987	+	0.208668i	$0.933087\pi$
$920$		0			0
$921$		0			0
$922$	31.9256	−	8.55444i	1.05141	−	0.281726i
$923$	21.0552	−	5.64173i	0.693041	−	0.185700i
$924$		0			0
$925$		0			0
$926$		−	12.2351i		−	0.402071i
$927$		0			0
$928$	−0.224745	+	0.224745i	−0.00737761	+	0.00737761i
$929$	21.1024	+	36.5505i	0.692349	+	1.19918i	0.971066	+	0.238810i	$0.0767574\pi$
$929$	21.1024	+	36.5505i	0.692349	+	1.19918i	−0.278717	+	0.960373i	$0.589909\pi$
$930$		0			0
$931$	44.4949	−	77.0674i	1.45826	−	2.52578i
$932$	5.29272	+	19.7527i	0.173369	+	0.647021i
$933$		0			0
$934$	3.46410	−	2.00000i	0.113349	−	0.0654420i
$935$		0			0
$936$		0			0
$937$	3.10102	+	3.10102i	0.101306	+	0.101306i	0.755943	−	0.654637i	$-0.227179\pi$
$937$	3.10102	+	3.10102i	0.101306	+	0.101306i	−0.654637	+	0.755943i	$0.727179\pi$
$938$	−7.80247	+	29.1192i	−0.254760	+	0.950776i
$939$		0			0
$940$		0			0
$941$	−27.5227	−	15.8902i	−0.897215	−	0.518007i	−0.0209191	−	0.999781i	$-0.506659\pi$
$941$	−27.5227	−	15.8902i	−0.897215	−	0.518007i	−0.876295	+	0.481774i	$0.839993\pi$
$942$		0			0
$943$	7.13713	+	1.91239i	0.232417	+	0.0622760i
$944$	−8.97809			−0.292212
$945$		0			0
$946$	2.20204			0.0715945
$947$	2.94164	+	0.788210i	0.0955904	+	0.0256134i	0.306297	−	0.951936i	$-0.400910\pi$
$947$	2.94164	+	0.788210i	0.0955904	+	0.0256134i	−0.210707	+	0.977549i	$0.567577\pi$
$948$		0			0
$949$	−29.2699	−	16.8990i	−0.950141	−	0.548564i
$950$		0			0
$951$		0			0
$952$	−0.530550	+	1.98004i	−0.0171952	+	0.0641735i
$953$	5.79972	+	5.79972i	0.187871	+	0.187871i	0.794775	−	0.606904i	$-0.207589\pi$
$953$	5.79972	+	5.79972i	0.187871	+	0.187871i	−0.606904	+	0.794775i	$0.707589\pi$
$954$		0			0
$955$		0			0
$956$	−14.6969	+	8.48528i	−0.475333	+	0.274434i
$957$		0			0
$958$	1.83013	+	6.83013i	0.0591287	+	0.220671i
$959$	−49.7046	+	86.0908i	−1.60504	+	2.78002i
$960$		0			0
$961$	15.3990	+	26.6718i	0.496741	+	0.860381i
$962$	−10.3923	+	10.3923i	−0.335061	+	0.335061i
$963$		0			0
$964$		−	19.0000i		−	0.611949i
$965$		0			0
$966$		0			0
$967$	−38.6937	+	10.3679i	−1.24431	+	0.333411i	−0.820134	−	0.572171i	$-0.806101\pi$
$967$	−38.6937	+	10.3679i	−1.24431	+	0.333411i	−0.424172	+	0.905582i	$0.639435\pi$
$968$	10.2349	−	2.74243i	0.328961	−	0.0881449i
$969$		0			0
$970$		0			0
$971$		−	21.4989i		−	0.689934i	−0.938615	−	0.344967i	$-0.887890\pi$
$971$		−	21.4989i		−	0.689934i	0.938615	−	0.344967i	$-0.112110\pi$
$972$		0			0
$973$	10.0000	−	10.0000i	0.320585	−	0.320585i
$974$	−8.48528	−	14.6969i	−0.271886	−	0.470920i
$975$		0			0
$976$	−0.275255	+	0.476756i	−0.00881070	+	0.0152606i
$977$	1.68100	+	6.27359i	0.0537801	+	0.200710i	0.987589	−	0.157063i	$-0.0502027\pi$
$977$	1.68100	+	6.27359i	0.0537801	+	0.200710i	−0.933808	+	0.357773i	$0.883536\pi$
$978$		0			0
$979$	−4.41761	+	2.55051i	−0.141188	+	0.0815147i
$980$		0			0
$981$		0			0
$982$	−19.7980	−	19.7980i	−0.631778	−	0.631778i
$983$	−7.04041	+	26.2752i	−0.224554	+	0.838047i	0.758029	+	0.652221i	$0.226163\pi$
$983$	−7.04041	+	26.2752i	−0.224554	+	0.838047i	−0.982583	+	0.185826i	$0.940504\pi$
$984$		0			0
$985$		0			0
$986$	−0.123724	−	0.0714323i	−0.00394019	−	0.00227487i
$987$		0			0
$988$	21.5804	+	5.78245i	0.686564	+	0.183964i
$989$	3.46410			0.110152
$990$		0			0
$991$	16.7423			0.531838			0.265919	−	0.963995i	$-0.414325\pi$
$991$	16.7423			0.531838			0.265919	+	0.963995i	$0.414325\pi$
$992$	0.434174	+	0.116337i	0.0137850	+	0.00369369i
$993$		0			0
$994$	24.8523	+	14.3485i	0.788266	+	0.455106i
$995$		0			0
$996$		0			0
$997$	1.73955	−	6.49211i	0.0550922	−	0.205607i	−0.932893	−	0.360153i	$-0.882725\pi$
$997$	1.73955	−	6.49211i	0.0550922	−	0.205607i	0.987986	+	0.154545i	$0.0493913\pi$
$998$	0.635674	+	0.635674i	0.0201219	+	0.0201219i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1350.2.q.g.557.1		8
3.2	odd	2		450.2.p.a.257.2		8
5.2	odd	4		270.2.m.a.233.2		8
5.3	odd	4	inner	1350.2.q.g.1043.1		8
5.4	even	2		270.2.m.a.17.2		8
9.2	odd	6	inner	1350.2.q.g.1007.1		8
9.7	even	3		450.2.p.a.407.2		8
15.2	even	4		90.2.l.a.23.1	✓	8
15.8	even	4		450.2.p.a.293.2		8
15.14	odd	2		90.2.l.a.77.1	yes	8
45.2	even	12		270.2.m.a.143.2		8
45.4	even	6		810.2.f.b.647.2		8
45.7	odd	12		90.2.l.a.83.1	yes	8
45.14	odd	6		810.2.f.b.647.3		8
45.22	odd	12		810.2.f.b.323.4		8
45.29	odd	6		270.2.m.a.197.2		8
45.32	even	12		810.2.f.b.323.1		8
45.34	even	6		90.2.l.a.47.1	yes	8
45.38	even	12	inner	1350.2.q.g.143.1		8
45.43	odd	12		450.2.p.a.443.2		8
60.47	odd	4		720.2.cu.a.113.1		8
60.59	even	2		720.2.cu.a.257.1		8
180.7	even	12		720.2.cu.a.353.1		8
180.79	odd	6		720.2.cu.a.497.1		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
90.2.l.a.23.1	✓	8	15.2	even	4
90.2.l.a.47.1	yes	8	45.34	even	6
90.2.l.a.77.1	yes	8	15.14	odd	2
90.2.l.a.83.1	yes	8	45.7	odd	12
270.2.m.a.17.2		8	5.4	even	2
270.2.m.a.143.2		8	45.2	even	12
270.2.m.a.197.2		8	45.29	odd	6
270.2.m.a.233.2		8	5.2	odd	4
450.2.p.a.257.2		8	3.2	odd	2
450.2.p.a.293.2		8	15.8	even	4
450.2.p.a.407.2		8	9.7	even	3
450.2.p.a.443.2		8	45.43	odd	12
720.2.cu.a.113.1		8	60.47	odd	4
720.2.cu.a.257.1		8	60.59	even	2
720.2.cu.a.353.1		8	180.7	even	12
720.2.cu.a.497.1		8	180.79	odd	6
810.2.f.b.323.1		8	45.32	even	12
810.2.f.b.323.4		8	45.22	odd	12
810.2.f.b.647.2		8	45.4	even	6
810.2.f.b.647.3		8	45.14	odd	6
1350.2.q.g.143.1		8	45.38	even	12	inner
1350.2.q.g.557.1		8	1.1	even	1	trivial
1350.2.q.g.1007.1		8	9.2	odd	6	inner
1350.2.q.g.1043.1		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)

\(f(q)\)	\(=\)	\(q+(-0.965926 - 0.258819i) q^{2} +(0.866025 + 0.500000i) q^{4} +(-1.18034 + 4.40508i) q^{7} +(-0.707107 - 0.707107i) q^{8} +O(q^{10})\)	\(q+(-0.965926 - 0.258819i) q^{2} +(0.866025 + 0.500000i) q^{4} +(-1.18034 + 4.40508i) q^{7} +(-0.707107 - 0.707107i) q^{8} +(0.550510 - 0.317837i) q^{11} +(-0.896575 - 3.34607i) q^{13} +(2.28024 - 3.94949i) q^{14} +(0.500000 + 0.866025i) q^{16} +(-0.317837 + 0.317837i) q^{17} +6.44949i q^{19} +(-0.614014 + 0.164525i) q^{22} +(-0.965926 + 0.258819i) q^{23} +3.46410i q^{26} +(-3.22474 + 3.22474i) q^{28} +(-0.158919 - 0.275255i) q^{29} +(-0.224745 + 0.389270i) q^{31} +(-0.258819 - 0.965926i) q^{32} +(0.389270 - 0.224745i) q^{34} +(3.00000 + 3.00000i) q^{37} +(1.66925 - 6.22973i) q^{38} +(-6.39898 - 3.69445i) q^{41} +(-3.34607 - 0.896575i) q^{43} +0.635674 q^{44} +1.00000 q^{46} +(-8.69333 - 2.32937i) q^{47} +(-11.9494 - 6.89898i) q^{49} +(0.896575 - 3.34607i) q^{52} +(-3.78194 - 3.78194i) q^{53} +(3.94949 - 2.28024i) q^{56} +(0.0822623 + 0.307007i) q^{58} +(-4.48905 + 7.77526i) q^{59} +(0.275255 + 0.476756i) q^{61} +(0.317837 - 0.317837i) q^{62} +1.00000i q^{64} +(-6.38512 + 1.71089i) q^{67} +(-0.434174 + 0.116337i) q^{68} +6.29253i q^{71} +(6.89898 - 6.89898i) q^{73} +(-2.12132 - 3.67423i) q^{74} +(-3.22474 + 5.58542i) q^{76} +(0.750311 + 2.80020i) q^{77} +(2.12132 - 1.22474i) q^{79} +(5.22474 + 5.22474i) q^{82} +(1.41043 - 5.26380i) q^{83} +(3.00000 + 1.73205i) q^{86} +(-0.614014 - 0.164525i) q^{88} -8.02458 q^{89} +15.7980 q^{91} +(-0.965926 - 0.258819i) q^{92} +(7.79423 + 4.50000i) q^{94} +(-0.695075 + 2.59405i) q^{97} +(9.75663 + 9.75663i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 8 q^{7}+O(q^{10}) \) 8 * q + 8 * q^7	\( 8 q + 8 q^{7} + 24 q^{11} + 4 q^{16} + 8 q^{22} - 16 q^{28} + 8 q^{31} + 24 q^{37} + 12 q^{38} - 12 q^{41} + 8 q^{46} + 12 q^{56} + 4 q^{58} + 12 q^{61} - 4 q^{67} - 12 q^{68} + 16 q^{73} - 16 q^{76} + 24 q^{77} + 32 q^{82} + 12 q^{83} + 24 q^{86} + 8 q^{88} + 48 q^{91} - 12 q^{97}+O(q^{100}) \) 8 * q + 8 * q^7 + 24 * q^11 + 4 * q^16 + 8 * q^22 - 16 * q^28 + 8 * q^31 + 24 * q^37 + 12 * q^38 - 12 * q^41 + 8 * q^46 + 12 * q^56 + 4 * q^58 + 12 * q^61 - 4 * q^67 - 12 * q^68 + 16 * q^73 - 16 * q^76 + 24 * q^77 + 32 * q^82 + 12 * q^83 + 24 * q^86 + 8 * q^88 + 48 * q^91 - 12 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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