LMFDB - Embedded newform 1350.2.q.g.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 90)
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.g.143.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.965926 + 0.258819i) q^{2} +(0.866025 + 0.500000i) q^{4} +(-0.283763 + 1.05902i) q^{7} +(0.707107 + 0.707107i) q^{8} +O(q^{10})$	\(q+(0.965926 + 0.258819i) q^{2} +(0.866025 + 0.500000i) q^{4} +(-0.283763 + 1.05902i) q^{7} +(0.707107 + 0.707107i) q^{8} +(5.44949 - 3.14626i) q^{11} +(0.896575 + 3.34607i) q^{13} +(-0.548188 + 0.949490i) q^{14} +(0.500000 + 0.866025i) q^{16} +(-3.14626 + 3.14626i) q^{17} +1.55051i q^{19} +(6.07812 - 1.62863i) q^{22} +(0.965926 - 0.258819i) q^{23} +3.46410i q^{26} +(-0.775255 + 0.775255i) q^{28} +(-1.57313 - 2.72474i) q^{29} +(2.22474 - 3.85337i) q^{31} +(0.258819 + 0.965926i) q^{32} +(-3.85337 + 2.22474i) q^{34} +(3.00000 + 3.00000i) q^{37} +(-0.401302 + 1.49768i) q^{38} +(3.39898 + 1.96240i) q^{41} +(3.34607 + 0.896575i) q^{43} +6.29253 q^{44} +1.00000 q^{46} +(8.69333 + 2.32937i) q^{47} +(5.02118 + 2.89898i) q^{49} +(-0.896575 + 3.34607i) q^{52} +(-6.61037 - 6.61037i) q^{53} +(-0.949490 + 0.548188i) q^{56} +(-0.814313 - 3.03906i) q^{58} +(-5.90326 + 10.2247i) q^{59} +(2.72474 + 4.71940i) q^{61} +(3.14626 - 3.14626i) q^{62} +1.00000i q^{64} +(3.65307 - 0.978838i) q^{67} +(-4.29788 + 1.15161i) q^{68} +0.635674i q^{71} +(-2.89898 + 2.89898i) q^{73} +(2.12132 + 3.67423i) q^{74} +(-0.775255 + 1.34278i) q^{76} +(1.78559 + 6.66390i) q^{77} +(-2.12132 + 1.22474i) q^{79} +(2.77526 + 2.77526i) q^{82} +(-0.142483 + 0.531752i) q^{83} +(3.00000 + 1.73205i) q^{86} +(6.07812 + 1.62863i) q^{88} -2.36773 q^{89} -3.79796 q^{91} +(0.965926 + 0.258819i) q^{92} +(7.79423 + 4.50000i) q^{94} +(2.89123 - 10.7902i) q^{97} +(4.09978 + 4.09978i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$7$	−0.283763	+	1.05902i	−0.107252	+	0.400271i	−0.998591	−	0.0530669i	$-0.983100\pi$
$7$	−0.283763	+	1.05902i	−0.107252	+	0.400271i	0.891339	+	0.453338i	$0.149767\pi$
$8$	0.707107	+	0.707107i	0.250000	+	0.250000i
$9$		0			0
$10$		0			0
$11$	5.44949	−	3.14626i	1.64308	−	0.948634i	0.663354	−	0.748305i	$-0.269132\pi$
$11$	5.44949	−	3.14626i	1.64308	−	0.948634i	0.979729	−	0.200329i	$-0.0642011\pi$
$12$		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	$0.757164\pi$
$14$	−0.548188	+	0.949490i	−0.146509	+	0.253762i
$15$		0			0
$16$	0.500000	+	0.866025i	0.125000	+	0.216506i
$17$	−3.14626	+	3.14626i	−0.763081	+	0.763081i	−0.976878	−	0.213797i	$-0.931417\pi$
$17$	−3.14626	+	3.14626i	−0.763081	+	0.763081i	0.213797	+	0.976878i	$0.431417\pi$
$18$		0			0
$19$			1.55051i			0.355711i	0.984057	+	0.177856i	$0.0569160\pi$
$19$			1.55051i			0.355711i	−0.984057	+	0.177856i	$0.943084\pi$
$20$		0			0
$21$		0			0
$22$	6.07812	−	1.62863i	1.29586	−	0.347224i
$23$	0.965926	−	0.258819i	0.201409	−	0.0539675i	−0.156704	−	0.987646i	$-0.550087\pi$
$23$	0.965926	−	0.258819i	0.201409	−	0.0539675i	0.358113	+	0.933678i	$0.383420\pi$
$24$		0			0
$25$		0			0
$26$			3.46410i			0.679366i
$27$		0			0
$28$	−0.775255	+	0.775255i	−0.146509	+	0.146509i
$29$	−1.57313	−	2.72474i	−0.292123	−	0.505972i	0.682188	−	0.731177i	$-0.261028\pi$
$29$	−1.57313	−	2.72474i	−0.292123	−	0.505972i	−0.974312	+	0.225204i	$0.927695\pi$
$30$		0			0
$31$	2.22474	−	3.85337i	0.399576	−	0.692086i	−0.594098	−	0.804393i	$-0.702491\pi$
$31$	2.22474	−	3.85337i	0.399576	−	0.692086i	0.993674	+	0.112307i	$0.0358240\pi$
$32$	0.258819	+	0.965926i	0.0457532	+	0.170753i
$33$		0			0
$34$	−3.85337	+	2.22474i	−0.660848	+	0.381541i
$35$		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	$0.636609\pi$
$38$	−0.401302	+	1.49768i	−0.0650997	+	0.242955i
$39$		0			0
$40$		0			0
$41$	3.39898	+	1.96240i	0.530831	+	0.306476i	0.741355	−	0.671113i	$-0.234184\pi$
$41$	3.39898	+	1.96240i	0.530831	+	0.306476i	−0.210524	+	0.977589i	$0.567517\pi$
$42$		0			0
$43$	3.34607	+	0.896575i	0.510270	+	0.136726i	0.504762	−	0.863258i	$-0.331580\pi$
$43$	3.34607	+	0.896575i	0.510270	+	0.136726i	0.00550783	+	0.999985i	$0.498247\pi$
$44$	6.29253			0.948634
$45$		0			0
$46$	1.00000			0.147442
$47$	8.69333	+	2.32937i	1.26805	+	0.339774i	0.829285	−	0.558827i	$-0.188748\pi$
$47$	8.69333	+	2.32937i	1.26805	+	0.339774i	0.438768	+	0.898600i	$0.355415\pi$
$48$		0			0
$49$	5.02118	+	2.89898i	0.717311	+	0.414140i
$50$		0			0
$51$		0			0
$52$	−0.896575	+	3.34607i	−0.124333	+	0.464016i
$53$	−6.61037	−	6.61037i	−0.908004	−	0.908004i	0.0881074	−	0.996111i	$-0.471918\pi$
$53$	−6.61037	−	6.61037i	−0.908004	−	0.908004i	−0.996111	+	0.0881074i	$0.971918\pi$
$54$		0			0
$55$		0			0
$56$	−0.949490	+	0.548188i	−0.126881	+	0.0732547i
$57$		0			0
$58$	−0.814313	−	3.03906i	−0.106925	−	0.399048i
$59$	−5.90326	+	10.2247i	−0.768539	+	1.33115i	0.169816	+	0.985476i	$0.445683\pi$
$59$	−5.90326	+	10.2247i	−0.768539	+	1.33115i	−0.938355	+	0.345673i	$0.887651\pi$
$60$		0			0
$61$	2.72474	+	4.71940i	0.348868	+	0.604257i	0.986049	−	0.166458i	$-0.0532329\pi$
$61$	2.72474	+	4.71940i	0.348868	+	0.604257i	−0.637181	+	0.770714i	$0.719900\pi$
$62$	3.14626	−	3.14626i	0.399576	−	0.399576i
$63$		0			0
$64$			1.00000i			0.125000i
$65$		0			0
$66$		0			0
$67$	3.65307	−	0.978838i	0.446294	−	0.119584i	−0.0286709	−	0.999589i	$-0.509127\pi$
$67$	3.65307	−	0.978838i	0.446294	−	0.119584i	0.474965	+	0.880005i	$0.342461\pi$
$68$	−4.29788	+	1.15161i	−0.521194	+	0.139654i
$69$		0			0
$70$		0			0
$71$			0.635674i			0.0754407i	0.999288	+	0.0377203i	$0.0120096\pi$
$71$			0.635674i			0.0754407i	−0.999288	+	0.0377203i	$0.987990\pi$
$72$		0			0
$73$	−2.89898	+	2.89898i	−0.339300	+	0.339300i	−0.856104	−	0.516804i	$-0.827122\pi$
$73$	−2.89898	+	2.89898i	−0.339300	+	0.339300i	0.516804	+	0.856104i	$0.327122\pi$
$74$	2.12132	+	3.67423i	0.246598	+	0.427121i
$75$		0			0
$76$	−0.775255	+	1.34278i	−0.0889279	+	0.154028i
$77$	1.78559	+	6.66390i	0.203487	+	0.759422i
$78$		0			0
$79$	−2.12132	+	1.22474i	−0.238667	+	0.137795i	−0.614564	−	0.788867i	$-0.710668\pi$
$79$	−2.12132	+	1.22474i	−0.238667	+	0.137795i	0.375897	+	0.926662i	$0.377335\pi$
$80$		0			0
$81$		0			0
$82$	2.77526	+	2.77526i	0.306476	+	0.306476i
$83$	−0.142483	+	0.531752i	−0.0156395	+	0.0583674i	−0.973305	−	0.229517i	$-0.926285\pi$
$83$	−0.142483	+	0.531752i	−0.0156395	+	0.0583674i	0.957665	+	0.287885i	$0.0929519\pi$
$84$		0			0
$85$		0			0
$86$	3.00000	+	1.73205i	0.323498	+	0.186772i
$87$		0			0
$88$	6.07812	+	1.62863i	0.647929	+	0.173612i
$89$	−2.36773			−0.250978			−0.125489	−	0.992095i	$-0.540050\pi$
$89$	−2.36773			−0.250978			−0.125489	+	0.992095i	$0.540050\pi$
$90$		0			0
$91$	−3.79796			−0.398134
$92$	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
$97$	2.89123	−	10.7902i	0.293560	−	1.09558i	−0.648795	−	0.760963i	$-0.724727\pi$
$97$	2.89123	−	10.7902i	0.293560	−	1.09558i	0.942355	−	0.334616i	$-0.108607\pi$
$98$	4.09978	+	4.09978i	0.414140	+	0.414140i
$99$		0			0
$100$		0			0
$101$	−1.10102	+	0.635674i	−0.109556	+	0.0632520i	−0.553777	−	0.832665i	$-0.686814\pi$
$101$	−1.10102	+	0.635674i	−0.109556	+	0.0632520i	0.444221	+	0.895917i	$0.353481\pi$
$102$		0			0
$103$	−1.06110	−	3.96008i	−0.104553	−	0.390198i	0.893741	−	0.448584i	$-0.148071\pi$
$103$	−1.06110	−	3.96008i	−0.104553	−	0.390198i	−0.998294	+	0.0583855i	$0.981405\pi$
$104$	−1.73205	+	3.00000i	−0.169842	+	0.294174i
$105$		0			0
$106$	−4.67423	−	8.09601i	−0.454002	−	0.786354i
$107$	−3.71051	+	3.71051i	−0.358708	+	0.358708i	−0.863337	−	0.504628i	$-0.831630\pi$
$107$	−3.71051	+	3.71051i	−0.358708	+	0.358708i	0.504628	+	0.863337i	$0.331630\pi$
$108$		0			0
$109$		−	20.3485i		−	1.94903i	−0.224323	−	0.974515i	$-0.572017\pi$
$109$		−	20.3485i		−	1.94903i	0.224323	−	0.974515i	$-0.427983\pi$
$110$		0			0
$111$		0			0
$112$	−1.05902	+	0.283763i	−0.100068	+	0.0268131i
$113$	−13.3278	+	3.57117i	−1.25377	+	0.335948i	−0.823793	−	0.566890i	$-0.808146\pi$
$113$	−13.3278	+	3.57117i	−1.25377	+	0.335948i	−0.429981	+	0.902838i	$0.641480\pi$
$114$		0			0
$115$		0			0
$116$		−	3.14626i		−	0.292123i
$117$		0			0
$118$	−8.34847	+	8.34847i	−0.768539	+	0.768539i
$119$	−2.43916	−	4.22474i	−0.223597	−	0.387282i
$120$		0			0
$121$	14.2980	−	24.7648i	1.29981	−	2.25134i
$122$	1.41043	+	5.26380i	0.127694	+	0.476562i
$123$		0			0
$124$	3.85337	−	2.22474i	0.346043	−	0.199788i
$125$		0			0
$126$		0			0
$127$	−14.1237	−	14.1237i	−1.25328	−	1.25328i	−0.954242	−	0.299036i	$-0.903335\pi$
$127$	−14.1237	−	14.1237i	−1.25328	−	1.25328i	−0.299036	−	0.954242i	$-0.596665\pi$
$128$	−0.258819	+	0.965926i	−0.0228766	+	0.0853766i
$129$		0			0
$130$		0			0
$131$	−9.12372	−	5.26758i	−0.797143	−	0.460231i	0.0453278	−	0.998972i	$-0.485567\pi$
$131$	−9.12372	−	5.26758i	−0.797143	−	0.460231i	−0.842471	+	0.538741i	$0.818900\pi$
$132$		0			0
$133$	−1.64202	−	0.439978i	−0.142381	−	0.0381509i
$134$	3.78194			0.326710
$135$		0			0
$136$	−4.44949			−0.381541
$137$	−2.12701	−	0.569930i	−0.181723	−	0.0486924i	0.166810	−	0.985989i	$-0.446653\pi$
$137$	−2.12701	−	0.569930i	−0.181723	−	0.0486924i	−0.348533	+	0.937297i	$0.613320\pi$
$138$		0			0
$139$	−11.1708	−	6.44949i	−0.947499	−	0.547039i	−0.0551956	−	0.998476i	$-0.517578\pi$
$139$	−11.1708	−	6.44949i	−0.947499	−	0.547039i	−0.892303	+	0.451437i	$0.850912\pi$
$140$		0			0
$141$		0			0
$142$	−0.164525	+	0.614014i	−0.0138066	+	0.0515269i
$143$	15.4135	+	15.4135i	1.28894	+	1.28894i
$144$		0			0
$145$		0			0
$146$	−3.55051	+	2.04989i	−0.293842	+	0.169650i
$147$		0			0
$148$	1.09808	+	4.09808i	0.0902613	+	0.336860i
$149$	6.45145	−	11.1742i	0.528523	−	0.915429i	−0.470924	−	0.882174i	$-0.656079\pi$
$149$	6.45145	−	11.1742i	0.528523	−	0.915429i	0.999447	−	0.0332550i	$-0.0105874\pi$
$150$		0			0
$151$	10.7980	+	18.7026i	0.878725	+	1.52200i	0.852741	+	0.522335i	$0.174939\pi$
$151$	10.7980	+	18.7026i	0.878725	+	1.52200i	0.0259849	+	0.999662i	$0.491728\pi$
$152$	−1.09638	+	1.09638i	−0.0889279	+	0.0889279i
$153$		0			0
$154$			6.89898i			0.555936i
$155$		0			0
$156$		0			0
$157$	5.94012	−	1.59165i	0.474073	−	0.127028i	−0.0138684	−	0.999904i	$-0.504415\pi$
$157$	5.94012	−	1.59165i	0.474073	−	0.127028i	0.487942	+	0.872876i	$0.337748\pi$
$158$	−2.36603	+	0.633975i	−0.188231	+	0.0504363i
$159$		0			0
$160$		0			0
$161$			1.09638i			0.0864066i
$162$		0			0
$163$	0.449490	−	0.449490i	0.0352068	−	0.0352068i	−0.689284	−	0.724491i	$-0.742075\pi$
$163$	0.449490	−	0.449490i	0.0352068	−	0.0352068i	0.724491	+	0.689284i	$0.242075\pi$
$164$	1.96240	+	3.39898i	0.153238	+	0.265416i
$165$		0			0
$166$	−0.275255	+	0.476756i	−0.0213639	+	0.0370034i
$167$	−2.79472	−	10.4300i	−0.216262	−	0.807100i	−0.985719	−	0.168401i	$-0.946140\pi$
$167$	−2.79472	−	10.4300i	−0.216262	−	0.807100i	0.769457	−	0.638699i	$-0.220527\pi$
$168$		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
$173$	0.802603	−	2.99536i	0.0610208	−	0.227733i	−0.928680	−	0.370881i	$-0.879056\pi$
$173$	0.802603	−	2.99536i	0.0610208	−	0.227733i	0.989701	+	0.143148i	$0.0457225\pi$
$174$		0			0
$175$		0			0
$176$	5.44949	+	3.14626i	0.410771	+	0.237159i
$177$		0			0
$178$	−2.28705	−	0.612812i	−0.171421	−	0.0459322i
$179$	−17.6062			−1.31595			−0.657976	−	0.753039i	$-0.728587\pi$
$179$	−17.6062			−1.31595			−0.657976	+	0.753039i	$0.728587\pi$
$180$		0			0
$181$	−10.5505			−0.784213			−0.392107	−	0.919920i	$-0.628254\pi$
$181$	−10.5505			−0.784213			−0.392107	+	0.919920i	$0.628254\pi$
$182$	−3.66855	−	0.982984i	−0.271931	−	0.0728636i
$183$		0			0
$184$	0.866025	+	0.500000i	0.0638442	+	0.0368605i
$185$		0			0
$186$		0			0
$187$	−7.24656	+	27.0445i	−0.529921	+	1.97769i
$188$	6.36396	+	6.36396i	0.464140	+	0.464140i
$189$		0			0
$190$		0			0
$191$	2.87628	−	1.66062i	0.208120	−	0.120158i	−0.392317	−	0.919830i	$-0.628327\pi$
$191$	2.87628	−	1.66062i	0.208120	−	0.120158i	0.600437	+	0.799672i	$0.294993\pi$
$192$		0			0
$193$	−4.48288	−	16.7303i	−0.322685	−	1.20428i	−0.916619	−	0.399762i	$-0.869093\pi$
$193$	−4.48288	−	16.7303i	−0.322685	−	1.20428i	0.593934	−	0.804513i	$-0.297574\pi$
$194$	5.58542	−	9.67423i	0.401010	−	0.694570i
$195$		0			0
$196$	2.89898	+	5.02118i	0.207070	+	0.358656i
$197$	−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
$199$			3.55051i			0.251689i	0.992050	+	0.125844i	$0.0401640\pi$
$199$			3.55051i			0.251689i	−0.992050	+	0.125844i	$0.959836\pi$
$200$		0			0
$201$		0			0
$202$	−1.22803	+	0.329049i	−0.0864038	+	0.0231518i
$203$	3.33195	−	0.892794i	0.233857	−	0.0626618i
$204$		0			0
$205$		0			0
$206$		−	4.09978i		−	0.285645i
$207$		0			0
$208$	−2.44949	+	2.44949i	−0.169842	+	0.169842i
$209$	4.87832	+	8.44949i	0.337440	+	0.584463i
$210$		0			0
$211$	−9.44949	+	16.3670i	−0.650530	+	1.12675i	0.332465	+	0.943116i	$0.392120\pi$
$211$	−9.44949	+	16.3670i	−0.650530	+	1.12675i	−0.982995	+	0.183635i	$0.941214\pi$
$212$	−2.41956	−	9.02993i	−0.166176	−	0.620178i
$213$		0			0
$214$	−4.54442	+	2.62372i	−0.310650	+	0.179354i
$215$		0			0
$216$		0			0
$217$	3.44949	+	3.44949i	0.234167	+	0.234167i
$218$	5.26657	−	19.6551i	0.356697	−	1.33121i
$219$		0			0
$220$		0			0
$221$	−13.3485	−	7.70674i	−0.897915	−	0.518412i
$222$		0			0
$223$	8.02714	+	2.15087i	0.537537	+	0.144033i	0.517367	−	0.855764i	$-0.326912\pi$
$223$	8.02714	+	2.15087i	0.537537	+	0.144033i	0.0201706	+	0.999797i	$0.493579\pi$
$224$	−1.09638			−0.0732547
$225$		0			0
$226$	−13.7980			−0.917827
$227$	−14.5865	−	3.90843i	−0.968138	−	0.259412i	−0.260096	−	0.965583i	$-0.583754\pi$
$227$	−14.5865	−	3.90843i	−0.968138	−	0.259412i	−0.708041	+	0.706171i	$0.750421\pi$
$228$		0			0
$229$	14.1582	+	8.17423i	0.935600	+	0.540169i	0.888578	−	0.458725i	$-0.151694\pi$
$229$	14.1582	+	8.17423i	0.935600	+	0.540169i	0.0470214	+	0.998894i	$0.485027\pi$
$230$		0			0
$231$		0			0
$232$	0.814313	−	3.03906i	0.0534623	−	0.199524i
$233$	−10.9959	−	10.9959i	−0.720363	−	0.720363i	0.248316	−	0.968679i	$-0.420123\pi$
$233$	−10.9959	−	10.9959i	−0.720363	−	0.720363i	−0.968679	+	0.248316i	$0.920123\pi$
$234$		0			0
$235$		0			0
$236$	−10.2247	+	5.90326i	−0.665574	+	0.384269i
$237$		0			0
$238$	−1.26260	−	4.71209i	−0.0818423	−	0.305439i
$239$	8.48528	−	14.6969i	0.548867	−	0.950666i	−0.449485	−	0.893288i	$-0.648393\pi$
$239$	8.48528	−	14.6969i	0.548867	−	0.950666i	0.998353	−	0.0573782i	$-0.0182741\pi$
$240$		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	$-0.376281\pi$
$242$	20.2204	−	20.2204i	1.29981	−	1.29981i
$243$		0			0
$244$			5.44949i			0.348868i
$245$		0			0
$246$		0			0
$247$	−5.18811	+	1.39015i	−0.330111	+	0.0884531i
$248$	4.29788	−	1.15161i	0.272915	−	0.0731275i
$249$		0			0
$250$		0			0
$251$			11.1708i			0.705097i	0.935793	+	0.352549i	$0.114685\pi$
$251$			11.1708i			0.705097i	−0.935793	+	0.352549i	$0.885315\pi$
$252$		0			0
$253$	4.44949	−	4.44949i	0.279737	−	0.279737i
$254$	−9.98698	−	17.2980i	−0.626639	−	1.08537i
$255$		0			0
$256$	−0.500000	+	0.866025i	−0.0312500	+	0.0541266i
$257$	−6.74105	−	25.1579i	−0.420495	−	1.56931i	−0.773568	−	0.633713i	$-0.781530\pi$
$257$	−6.74105	−	25.1579i	−0.420495	−	1.56931i	0.353073	−	0.935596i	$-0.385137\pi$
$258$		0			0
$259$	−4.02834	+	2.32577i	−0.250309	+	0.144516i
$260$		0			0
$261$		0			0
$262$	−7.44949	−	7.44949i	−0.460231	−	0.460231i
$263$	3.28621	−	12.2643i	0.202636	−	0.756249i	−0.787521	−	0.616288i	$-0.788636\pi$
$263$	3.28621	−	12.2643i	0.202636	−	0.756249i	0.990157	−	0.139961i	$-0.0446977\pi$
$264$		0			0
$265$		0			0
$266$	−1.47219	−	0.849971i	−0.0902660	−	0.0521151i
$267$		0			0
$268$	3.65307	+	0.978838i	0.223147	+	0.0597920i
$269$	4.70334			0.286768			0.143384	−	0.989667i	$-0.454202\pi$
$269$	4.70334			0.286768			0.143384	+	0.989667i	$0.454202\pi$
$270$		0			0
$271$	−16.0454			−0.974689			−0.487345	−	0.873210i	$-0.662034\pi$
$271$	−16.0454			−0.974689			−0.487345	+	0.873210i	$0.662034\pi$
$272$	−4.29788	−	1.15161i	−0.260597	−	0.0698268i
$273$		0			0
$274$	−1.90702	−	1.10102i	−0.115208	−	0.0665151i
$275$		0			0
$276$		0			0
$277$	3.69723	−	13.7983i	0.222145	−	0.829057i	−0.761383	−	0.648302i	$-0.775479\pi$
$277$	3.69723	−	13.7983i	0.222145	−	0.829057i	0.983528	−	0.180754i	$-0.0578539\pi$
$278$	−9.12096	−	9.12096i	−0.547039	−	0.547039i
$279$		0			0
$280$		0			0
$281$	0.151531	−	0.0874863i	0.00903957	−	0.00521900i	−0.495473	−	0.868623i	$-0.665005\pi$
$281$	0.151531	−	0.0874863i	0.00903957	−	0.00521900i	0.504513	+	0.863404i	$0.331672\pi$
$282$		0			0
$283$	−1.78484	−	6.66112i	−0.106098	−	0.395962i	0.892370	−	0.451305i	$-0.149041\pi$
$283$	−1.78484	−	6.66112i	−0.106098	−	0.395962i	−0.998467	+	0.0553430i	$0.982375\pi$
$284$	−0.317837	+	0.550510i	−0.0188602	+	0.0326668i
$285$		0			0
$286$	10.8990	+	18.8776i	0.644470	+	1.11626i
$287$	−3.04272	+	3.04272i	−0.179606	+	0.179606i
$288$		0			0
$289$		−	2.79796i		−	0.164586i
$290$		0			0
$291$		0			0
$292$	−3.96008	+	1.06110i	−0.231746	+	0.0620962i
$293$	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
$298$	9.12372	−	9.12372i	0.528523	−	0.528523i
$299$	1.73205	+	3.00000i	0.100167	+	0.173494i
$300$		0			0
$301$	−1.89898	+	3.28913i	−0.109455	+	0.189582i
$302$	5.58943	+	20.8601i	0.321636	+	1.20036i
$303$		0			0
$304$	−1.34278	+	0.775255i	−0.0770138	+	0.0444639i
$305$		0			0
$306$		0			0
$307$	−0.674235	−	0.674235i	−0.0384806	−	0.0384806i	0.687605	−	0.726085i	$-0.258662\pi$
$307$	−0.674235	−	0.674235i	−0.0384806	−	0.0384806i	−0.726085	+	0.687605i	$0.758662\pi$
$308$	−1.78559	+	6.66390i	−0.101743	+	0.379711i
$309$		0			0
$310$		0			0
$311$	17.8207	+	10.2888i	1.01052	+	0.583422i	0.911343	−	0.411648i	$-0.135047\pi$
$311$	17.8207	+	10.2888i	1.01052	+	0.583422i	0.0991741	+	0.995070i	$0.468380\pi$
$312$		0			0
$313$	4.85009	+	1.29958i	0.274143	+	0.0734564i	0.393271	−	0.919422i	$-0.371343\pi$
$313$	4.85009	+	1.29958i	0.274143	+	0.0734564i	−0.119128	+	0.992879i	$0.538010\pi$
$314$	6.14966			0.347046
$315$		0			0
$316$	−2.44949			−0.137795
$317$	1.06350	+	0.284965i	0.0597323	+	0.0160052i	0.288561	−	0.957461i	$-0.406823\pi$
$317$	1.06350	+	0.284965i	0.0597323	+	0.0160052i	−0.228829	+	0.973467i	$0.573490\pi$
$318$		0			0
$319$	−17.1455	−	9.89898i	−0.959966	−	0.554236i
$320$		0			0
$321$		0			0
$322$	−0.283763	+	1.05902i	−0.0158135	+	0.0590168i
$323$	−4.87832	−	4.87832i	−0.271437	−	0.271437i
$324$		0			0
$325$		0			0
$326$	0.550510	−	0.317837i	0.0304899	−	0.0176034i
$327$		0			0
$328$	1.01581	+	3.79107i	0.0560889	+	0.209327i
$329$	−4.93369	+	8.54541i	−0.272003	+	0.471124i
$330$		0			0
$331$	2.22474	+	3.85337i	0.122283	+	0.211800i	0.920668	−	0.390347i	$-0.127645\pi$
$331$	2.22474	+	3.85337i	0.122283	+	0.211800i	−0.798385	+	0.602148i	$0.794312\pi$
$332$	−0.389270	+	0.389270i	−0.0213639	+	0.0213639i
$333$		0			0
$334$		−	10.7980i		−	0.590838i
$335$		0			0
$336$		0			0
$337$	−29.7766	+	7.97861i	−1.62203	+	0.434622i	−0.951598	−	0.307346i	$-0.900559\pi$
$337$	−29.7766	+	7.97861i	−1.62203	+	0.434622i	−0.670435	+	0.741968i	$0.733892\pi$
$338$	0.965926	−	0.258819i	0.0525394	−	0.0140779i
$339$		0			0
$340$		0			0
$341$		−	27.9985i		−	1.51621i
$342$		0			0
$343$	−9.92168	+	9.92168i	−0.535721	+	0.535721i
$344$	1.73205	+	3.00000i	0.0933859	+	0.161749i
$345$		0			0
$346$	1.55051	−	2.68556i	0.0833559	−	0.144377i
$347$	−1.08757	−	4.05886i	−0.0583837	−	0.217891i	0.930570	−	0.366113i	$-0.119312\pi$
$347$	−1.08757	−	4.05886i	−0.0583837	−	0.217891i	−0.988954	+	0.148222i	$0.952645\pi$
$348$		0			0
$349$	−13.0297	+	7.52270i	−0.697464	+	0.402681i	−0.806402	−	0.591367i	$-0.798588\pi$
$349$	−13.0297	+	7.52270i	−0.697464	+	0.402681i	0.108938	+	0.994049i	$0.465255\pi$
$350$		0			0
$351$		0			0
$352$	4.44949	+	4.44949i	0.237159	+	0.237159i
$353$	−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
$356$	−2.05051	−	1.18386i	−0.108677	−	0.0627446i
$357$		0			0
$358$	−17.0063	−	4.55683i	−0.898812	−	0.240836i
$359$	17.4634			0.921682			0.460841	−	0.887483i	$-0.347548\pi$
$359$	17.4634			0.921682			0.460841	+	0.887483i	$0.347548\pi$
$360$		0			0
$361$	16.5959			0.873469
$362$	−10.1910	−	2.73067i	−0.535628	−	0.143521i
$363$		0			0
$364$	−3.28913	−	1.89898i	−0.172397	−	0.0995336i
$365$		0			0
$366$		0			0
$367$	−2.52520	+	9.42418i	−0.131814	+	0.491938i	−0.999991	−	0.00431778i	$-0.998626\pi$
$367$	−2.52520	+	9.42418i	−0.131814	+	0.491938i	0.868176	+	0.496256i	$0.165292\pi$
$368$	0.707107	+	0.707107i	0.0368605	+	0.0368605i
$369$		0			0
$370$		0			0
$371$	8.87628	−	5.12472i	0.460833	−	0.266062i
$372$		0			0
$373$	−5.25190	−	19.6004i	−0.271933	−	1.01487i	−0.957869	−	0.287206i	$-0.907273\pi$
$373$	−5.25190	−	19.6004i	−0.271933	−	1.01487i	0.685935	−	0.727662i	$-0.259393\pi$
$374$	−13.9993	+	24.2474i	−0.723885	+	1.25381i
$375$		0			0
$376$	4.50000	+	7.79423i	0.232070	+	0.401957i
$377$	7.70674	−	7.70674i	0.396917	−	0.396917i
$378$		0			0
$379$		−	6.65153i		−	0.341666i	−0.985300	−	0.170833i	$-0.945354\pi$
$379$		−	6.65153i		−	0.341666i	0.985300	−	0.170833i	$-0.0546459\pi$
$380$		0			0
$381$		0			0
$382$	3.20807	−	0.859599i	0.164139	−	0.0439809i
$383$	26.8508	−	7.19464i	1.37201	−	0.367629i	0.503798	−	0.863822i	$-0.331936\pi$
$383$	26.8508	−	7.19464i	1.37201	−	0.367629i	0.868212	+	0.496193i	$0.165269\pi$
$384$		0			0
$385$		0			0
$386$		−	17.3205i		−	0.881591i
$387$		0			0
$388$	7.89898	−	7.89898i	0.401010	−	0.401010i
$389$	−2.81237	−	4.87117i	−0.142593	−	0.246978i	0.785879	−	0.618380i	$-0.212211\pi$
$389$	−2.81237	−	4.87117i	−0.142593	−	0.246978i	−0.928472	+	0.371402i	$0.878877\pi$
$390$		0			0
$391$	−2.22474	+	3.85337i	−0.112510	+	0.194873i
$392$	1.50062	+	5.60040i	0.0757929	+	0.282863i
$393$		0			0
$394$	−8.48528	+	4.89898i	−0.427482	+	0.246807i
$395$		0			0
$396$		0			0
$397$	−15.4495	−	15.4495i	−0.775388	−	0.775388i	0.203655	−	0.979043i	$-0.434718\pi$
$397$	−15.4495	−	15.4495i	−0.775388	−	0.775388i	−0.979043	+	0.203655i	$0.934718\pi$
$398$	−0.918940	+	3.42953i	−0.0460623	+	0.171907i
$399$		0			0
$400$		0			0
$401$	−22.3485	−	12.9029i	−1.11603	−	0.644340i	−0.175645	−	0.984454i	$-0.556201\pi$
$401$	−22.3485	−	12.9029i	−1.11603	−	0.644340i	−0.940384	+	0.340114i	$0.889534\pi$
$402$		0			0
$403$	14.8883	+	3.98930i	0.741638	+	0.198721i
$404$	−1.27135			−0.0632520
$405$		0			0
$406$	3.44949			0.171195
$407$	25.7873	+	6.90968i	1.27823	+	0.342500i
$408$		0			0
$409$	16.5420	+	9.55051i	0.817948	+	0.472242i	0.849708	−	0.527253i	$-0.176778\pi$
$409$	16.5420	+	9.55051i	0.817948	+	0.472242i	−0.0317605	+	0.999496i	$0.510111\pi$
$410$		0			0
$411$		0			0
$412$	1.06110	−	3.96008i	0.0522767	−	0.195099i
$413$	−9.15306	−	9.15306i	−0.450393	−	0.450393i
$414$		0			0
$415$		0			0
$416$	−3.00000	+	1.73205i	−0.147087	+	0.0849208i
$417$		0			0
$418$	2.52520	+	9.42418i	0.123512	+	0.460952i
$419$	−5.97469	+	10.3485i	−0.291883	+	0.505556i	−0.974255	−	0.225449i	$-0.927615\pi$
$419$	−5.97469	+	10.3485i	−0.291883	+	0.505556i	0.682372	+	0.731005i	$0.260948\pi$
$420$		0			0
$421$	7.44949	+	12.9029i	0.363066	+	0.628849i	0.988464	−	0.151457i	$-0.0483966\pi$
$421$	7.44949	+	12.9029i	0.363066	+	0.628849i	−0.625398	+	0.780306i	$0.715063\pi$
$422$	−13.3636	+	13.3636i	−0.650530	+	0.650530i
$423$		0			0
$424$		−	9.34847i		−	0.454002i
$425$		0			0
$426$		0			0
$427$	−5.77111	+	1.54636i	−0.279284	+	0.0748338i
$428$	−5.06865	+	1.35814i	−0.245002	+	0.0656482i
$429$		0			0
$430$		0			0
$431$			15.5563i			0.749323i	0.927162	+	0.374661i	$0.122241\pi$
$431$			15.5563i			0.749323i	−0.927162	+	0.374661i	$0.877759\pi$
$432$		0			0
$433$	−8.55051	+	8.55051i	−0.410911	+	0.410911i	−0.882056	−	0.471145i	$-0.843841\pi$
$433$	−8.55051	+	8.55051i	−0.410911	+	0.410911i	0.471145	+	0.882056i	$0.343841\pi$
$434$	2.43916	+	4.22474i	0.117083	+	0.202794i
$435$		0			0
$436$	10.1742	−	17.6223i	0.487257	−	0.843955i
$437$	0.401302	+	1.49768i	0.0191969	+	0.0716436i
$438$		0			0
$439$	8.83523	−	5.10102i	0.421682	−	0.243458i	−0.274114	−	0.961697i	$-0.588385\pi$
$439$	8.83523	−	5.10102i	0.421682	−	0.243458i	0.695797	+	0.718239i	$0.255051\pi$
$440$		0			0
$441$		0			0
$442$	−10.8990	−	10.8990i	−0.518412	−	0.518412i
$443$	0.142483	−	0.531752i	0.00676955	−	0.0252643i	−0.962458	−	0.271429i	$-0.912504\pi$
$443$	0.142483	−	0.531752i	0.00676955	−	0.0252643i	0.969228	+	0.246165i	$0.0791704\pi$
$444$		0			0
$445$		0			0
$446$	7.19694	+	4.15515i	0.340785	+	0.196752i
$447$		0			0
$448$	−1.05902	−	0.283763i	−0.0500339	−	0.0134065i
$449$	−21.7060			−1.02437			−0.512185	−	0.858875i	$-0.671164\pi$
$449$	−21.7060			−1.02437			−0.512185	+	0.858875i	$0.671164\pi$
$450$		0			0
$451$	24.6969			1.16293
$452$	−13.3278	−	3.57117i	−0.626887	−	0.167974i
$453$		0			0
$454$	−13.0779	−	7.55051i	−0.613775	−	0.354363i
$455$		0			0
$456$		0			0
$457$	−1.59165	+	5.94012i	−0.0744543	+	0.277867i	−0.993109	−	0.117194i	$-0.962610\pi$
$457$	−1.59165	+	5.94012i	−0.0744543	+	0.277867i	0.918655	+	0.395061i	$0.129277\pi$
$458$	11.5601	+	11.5601i	0.540169	+	0.540169i
$459$		0			0
$460$		0			0
$461$	−16.3763	+	9.45485i	−0.762719	+	0.440356i	−0.830271	−	0.557360i	$-0.811814\pi$
$461$	−16.3763	+	9.45485i	−0.762719	+	0.440356i	0.0675520	+	0.997716i	$0.478481\pi$
$462$		0			0
$463$	8.54613	+	31.8946i	0.397172	+	1.48227i	0.818048	+	0.575150i	$0.195056\pi$
$463$	8.54613	+	31.8946i	0.397172	+	1.48227i	−0.420876	+	0.907118i	$0.638277\pi$
$464$	1.57313	−	2.72474i	0.0730308	−	0.126493i
$465$		0			0
$466$	−7.77526	−	13.4671i	−0.360182	−	0.623853i
$467$	2.82843	−	2.82843i	0.130884	−	0.130884i	−0.638630	−	0.769514i	$-0.720499\pi$
$467$	2.82843	−	2.82843i	0.130884	−	0.130884i	0.769514	+	0.638630i	$0.220499\pi$
$468$		0			0
$469$			4.14643i			0.191464i
$470$		0			0
$471$		0			0
$472$	−11.4042	+	3.05575i	−0.524922	+	0.140652i
$473$	21.0552	−	5.64173i	0.968120	−	0.259407i
$474$		0			0
$475$		0			0
$476$		−	4.87832i		−	0.223597i
$477$		0			0
$478$	12.0000	−	12.0000i	0.548867	−	0.548867i
$479$	3.53553	+	6.12372i	0.161543	+	0.279800i	0.935422	−	0.353533i	$-0.115020\pi$
$479$	3.53553	+	6.12372i	0.161543	+	0.279800i	−0.773879	+	0.633333i	$0.781686\pi$
$480$		0			0
$481$	−7.34847	+	12.7279i	−0.335061	+	0.580343i
$482$	−4.91756	−	18.3526i	−0.223989	−	0.835938i
$483$		0			0
$484$	24.7648	−	14.2980i	1.12567	−	0.649907i
$485$		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	$0.624372\pi$
$488$	−1.41043	+	5.26380i	−0.0638472	+	0.238281i
$489$		0			0
$490$		0			0
$491$	−0.247449	−	0.142865i	−0.0111672	−	0.00644739i	0.494406	−	0.869231i	$-0.335386\pi$
$491$	−0.247449	−	0.142865i	−0.0111672	−	0.00644739i	−0.505573	+	0.862784i	$0.668719\pi$
$492$		0			0
$493$	13.5223	+	3.62328i	0.609012	+	0.163184i
$494$	−5.37113			−0.241658
$495$		0			0
$496$	4.44949			0.199788
$497$	−0.673191	−	0.180381i	−0.0301967	−	0.00809119i
$498$		0			0
$499$	7.70674	+	4.44949i	0.345001	+	0.199187i	0.662481	−	0.749078i	$-0.269503\pi$
$499$	7.70674	+	4.44949i	0.345001	+	0.199187i	−0.317480	+	0.948265i	$0.602837\pi$
$500$		0			0
$501$		0			0
$502$	−2.89123	+	10.7902i	−0.129042	+	0.481590i
$503$	−16.7563	−	16.7563i	−0.747125	−	0.747125i	0.226813	−	0.973938i	$-0.427169\pi$
$503$	−16.7563	−	16.7563i	−0.747125	−	0.747125i	−0.973938	+	0.226813i	$0.927169\pi$
$504$		0			0
$505$		0			0
$506$	5.44949	−	3.14626i	0.242259	−	0.139869i
$507$		0			0
$508$	−5.16964	−	19.2934i	−0.229366	−	0.856005i
$509$	19.8150	−	34.3207i	0.878286	−	1.52124i	0.0250662	−	0.999686i	$-0.492020\pi$
$509$	19.8150	−	34.3207i	0.878286	−	1.52124i	0.853220	−	0.521551i	$-0.174646\pi$
$510$		0			0
$511$	−2.24745	−	3.89270i	−0.0994213	−	0.172203i
$512$	−0.707107	+	0.707107i	−0.0312500	+	0.0312500i
$513$		0			0
$514$		−	26.0454i		−	1.14881i
$515$		0			0
$516$		0			0
$517$	54.7030	−	14.6576i	2.40584	−	0.644642i
$518$	−4.49303	+	1.20390i	−0.197413	+	0.0528965i
$519$		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
$523$	−1.77526	+	1.77526i	−0.0776265	+	0.0776265i	−0.744854	−	0.667228i	$-0.767481\pi$
$523$	−1.77526	+	1.77526i	−0.0776265	+	0.0776265i	0.667228	+	0.744854i	$0.267481\pi$
$524$	−5.26758	−	9.12372i	−0.230116	−	0.398572i
$525$		0			0
$526$	6.34847	−	10.9959i	0.276806	−	0.479443i
$527$	5.12409	+	19.1234i	0.223209	+	0.833027i
$528$		0			0
$529$	−19.0526	+	11.0000i	−0.828372	+	0.478261i
$530$		0			0
$531$		0			0
$532$	−1.20204	−	1.20204i	−0.0521151	−	0.0521151i
$533$	−3.51888	+	13.1326i	−0.152420	+	0.568838i
$534$		0			0
$535$		0			0
$536$	3.27526	+	1.89097i	0.141469	+	0.0816774i
$537$		0			0
$538$	4.54308	+	1.21731i	0.195866	+	0.0524822i
$539$	36.4838			1.57147
$540$		0			0
$541$	−25.9444			−1.11544			−0.557718	−	0.830030i	$-0.688323\pi$
$541$	−25.9444			−1.11544			−0.557718	+	0.830030i	$0.688323\pi$
$542$	−15.4987	−	4.15286i	−0.665725	−	0.178380i
$543$		0			0
$544$	−3.85337	−	2.22474i	−0.165212	−	0.0953851i
$545$		0			0
$546$		0			0
$547$	5.53567	−	20.6594i	0.236688	−	0.883332i	−0.740693	−	0.671844i	$-0.765502\pi$
$547$	5.53567	−	20.6594i	0.236688	−	0.883332i	0.977381	−	0.211488i	$-0.0678308\pi$
$548$	−1.55708	−	1.55708i	−0.0665151	−	0.0665151i
$549$		0			0
$550$		0			0
$551$	4.22474	−	2.43916i	0.179980	−	0.103912i
$552$		0			0
$553$	−0.695075	−	2.59405i	−0.0295576	−	0.110310i
$554$	7.14250	−	12.3712i	0.303456	−	0.525601i
$555$		0			0
$556$	−6.44949	−	11.1708i	−0.273519	−	0.473749i
$557$	16.3670	−	16.3670i	0.693492	−	0.693492i	−0.269507	−	0.962999i	$-0.586861\pi$
$557$	16.3670	−	16.3670i	0.693492	−	0.693492i	0.962999	+	0.269507i	$0.0868606\pi$
$558$		0			0
$559$			12.0000i			0.507546i
$560$		0			0
$561$		0			0
$562$	0.169011	−	0.0452863i	0.00712928	−	0.00191029i
$563$	−33.1781	+	8.89004i	−1.39829	+	0.374670i	−0.877730	−	0.479155i	$-0.840943\pi$
$563$	−33.1781	+	8.89004i	−1.39829	+	0.374670i	−0.520559	+	0.853826i	$0.674276\pi$
$564$		0			0
$565$		0			0
$566$		−	6.89610i		−	0.289864i
$567$		0			0
$568$	−0.449490	+	0.449490i	−0.0188602	+	0.0188602i
$569$	13.0458	+	22.5959i	0.546907	+	0.947270i	0.998484	+	0.0550383i	$0.0175281\pi$
$569$	13.0458	+	22.5959i	0.546907	+	0.947270i	−0.451578	+	0.892232i	$0.649139\pi$
$570$		0			0
$571$	−13.5505	+	23.4702i	−0.567071	+	0.982196i	0.429782	+	0.902932i	$0.358590\pi$
$571$	−13.5505	+	23.4702i	−0.567071	+	0.982196i	−0.996854	+	0.0792637i	$0.974743\pi$
$572$	5.64173	+	21.0552i	0.235892	+	0.880363i
$573$		0			0
$574$	−3.72656	+	2.15153i	−0.155544	+	0.0898032i
$575$		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	$0.583174\pi$
$578$	0.724165	−	2.70262i	0.0301213	−	0.112414i
$579$		0			0
$580$		0			0
$581$	−0.522704	−	0.301783i	−0.0216854	−	0.0125201i
$582$		0			0
$583$	−56.8211	−	15.2252i	−2.35329	−	0.630562i
$584$	−4.09978			−0.169650
$585$		0			0
$586$	22.0454			0.910687
$587$	29.9876	+	8.03514i	1.23772	+	0.331646i	0.817581	−	0.575814i	$-0.195315\pi$
$587$	29.9876	+	8.03514i	1.23772	+	0.331646i	0.420138	+	0.907460i	$0.361981\pi$
$588$		0			0
$589$	5.97469	+	3.44949i	0.246183	+	0.142134i
$590$		0			0
$591$		0			0
$592$	−1.09808	+	4.09808i	−0.0451307	+	0.168430i
$593$	10.0745	+	10.0745i	0.413709	+	0.413709i	0.883028	−	0.469320i	$-0.155501\pi$
$593$	10.0745	+	10.0745i	0.413709	+	0.413709i	−0.469320	+	0.883028i	$0.655501\pi$
$594$		0			0
$595$		0			0
$596$	11.1742	−	6.45145i	0.457714	−	0.264262i
$597$		0			0
$598$	0.896575	+	3.34607i	0.0366637	+	0.136831i
$599$	−16.8991	+	29.2702i	−0.690480	+	1.19595i	0.281201	+	0.959649i	$0.409267\pi$
$599$	−16.8991	+	29.2702i	−0.690480	+	1.19595i	−0.971681	+	0.236297i	$0.924066\pi$
$600$		0			0
$601$	−17.3485	−	30.0484i	−0.707659	−	1.22570i	−0.965723	−	0.259573i	$-0.916418\pi$
$601$	−17.3485	−	30.0484i	−0.707659	−	1.22570i	0.258065	−	0.966128i	$-0.416915\pi$
$602$	−2.68556	+	2.68556i	−0.109455	+	0.109455i
$603$		0			0
$604$			21.5959i			0.878725i
$605$		0			0
$606$		0			0
$607$	21.4114	−	5.73717i	0.869062	−	0.232864i	0.203380	−	0.979100i	$-0.434807\pi$
$607$	21.4114	−	5.73717i	0.869062	−	0.232864i	0.665682	+	0.746235i	$0.268141\pi$
$608$	−1.49768	+	0.401302i	−0.0607389	+	0.0162749i
$609$		0			0
$610$		0			0
$611$			31.1769i			1.26128i
$612$		0			0
$613$	12.7980	−	12.7980i	0.516905	−	0.516905i	−0.399729	−	0.916633i	$-0.630896\pi$
$613$	12.7980	−	12.7980i	0.516905	−	0.516905i	0.916633	+	0.399729i	$0.130896\pi$
$614$	−0.476756	−	0.825765i	−0.0192403	−	0.0333252i
$615$		0			0
$616$	−3.44949	+	5.97469i	−0.138984	+	0.240727i
$617$	1.83788	+	6.85906i	0.0739902	+	0.276135i	0.993002	−	0.118094i	$-0.0376783\pi$
$617$	1.83788	+	6.85906i	0.0739902	+	0.276135i	−0.919012	+	0.394229i	$0.871012\pi$
$618$		0			0
$619$	21.4275	−	12.3712i	0.861244	−	0.497239i	−0.00318471	−	0.999995i	$-0.501014\pi$
$619$	21.4275	−	12.3712i	0.861244	−	0.497239i	0.864429	+	0.502756i	$0.167680\pi$
$620$		0			0
$621$		0			0
$622$	14.5505	+	14.5505i	0.583422	+	0.583422i
$623$	0.671873	−	2.50746i	0.0269180	−	0.100459i
$624$		0			0
$625$		0			0
$626$	4.34847	+	2.51059i	0.173800	+	0.100343i
$627$		0			0
$628$	5.94012	+	1.59165i	0.237037	+	0.0635138i
$629$	−18.8776			−0.752699
$630$		0			0
$631$	−12.8990			−0.513500			−0.256750	−	0.966478i	$-0.582652\pi$
$631$	−12.8990			−0.513500			−0.256750	+	0.966478i	$0.582652\pi$
$632$	−2.36603	−	0.633975i	−0.0941154	−	0.0252182i
$633$		0			0
$634$	0.953512	+	0.550510i	0.0378688	+	0.0218636i
$635$		0			0
$636$		0			0
$637$	−5.19831	+	19.4003i	−0.205964	+	0.768670i
$638$	−13.9993	−	13.9993i	−0.554236	−	0.554236i
$639$		0			0
$640$		0			0
$641$	7.74745	−	4.47299i	0.306006	−	0.176673i	−0.339132	−	0.940739i	$-0.610133\pi$
$641$	7.74745	−	4.47299i	0.306006	−	0.176673i	0.645138	+	0.764066i	$0.276800\pi$
$642$		0			0
$643$	8.22539	+	30.6976i	0.324378	+	1.21059i	0.914936	+	0.403599i	$0.132241\pi$
$643$	8.22539	+	30.6976i	0.324378	+	1.21059i	−0.590558	+	0.806995i	$0.701092\pi$
$644$	−0.548188	+	0.949490i	−0.0216016	+	0.0374151i
$645$		0			0
$646$	−3.44949	−	5.97469i	−0.135718	−	0.235071i
$647$	−24.9558	+	24.9558i	−0.981114	+	0.981114i	−0.999825	−	0.0187105i	$-0.994044\pi$
$647$	−24.9558	+	24.9558i	−0.981114	+	0.981114i	0.0187105	+	0.999825i	$0.494044\pi$
$648$		0			0
$649$			74.2929i			2.91625i
$650$		0			0
$651$		0			0
$652$	0.614014	−	0.164525i	0.0240467	−	0.00644328i
$653$	−20.8162	+	5.57768i	−0.814601	+	0.218272i	−0.641985	−	0.766718i	$-0.721889\pi$
$653$	−20.8162	+	5.57768i	−0.814601	+	0.218272i	−0.172616	+	0.984989i	$0.555222\pi$
$654$		0			0
$655$		0			0
$656$			3.92480i			0.153238i
$657$		0			0
$658$	−6.97730	+	6.97730i	−0.272003	+	0.272003i
$659$	−5.65685	−	9.79796i	−0.220360	−	0.381674i	0.734557	−	0.678546i	$-0.237390\pi$
$659$	−5.65685	−	9.79796i	−0.220360	−	0.381674i	−0.954917	+	0.296872i	$0.904056\pi$
$660$		0			0
$661$	15.3485	−	26.5843i	0.596986	−	1.03401i	−0.396277	−	0.918131i	$-0.629698\pi$
$661$	15.3485	−	26.5843i	0.596986	−	1.03401i	0.993263	−	0.115880i	$-0.0369687\pi$
$662$	1.15161	+	4.29788i	0.0447587	+	0.167042i
$663$		0			0
$664$	−0.476756	+	0.275255i	−0.0185017	+	0.0106820i
$665$		0			0
$666$		0			0
$667$	−2.22474	−	2.22474i	−0.0861425	−	0.0861425i
$668$	2.79472	−	10.4300i	0.108131	−	0.403550i
$669$		0			0
$670$		0			0
$671$	29.6969	+	17.1455i	1.14644	+	0.661896i
$672$		0			0
$673$	−15.7783	−	4.22778i	−0.608208	−	0.162969i	−0.0584468	−	0.998291i	$-0.518615\pi$
$673$	−15.7783	−	4.22778i	−0.608208	−	0.162969i	−0.549762	+	0.835322i	$0.685281\pi$
$674$	−30.8270			−1.18741
$675$		0			0
$676$	1.00000			0.0384615
$677$	6.18587	+	1.65750i	0.237742	+	0.0637028i	0.375723	−	0.926732i	$-0.377394\pi$
$677$	6.18587	+	1.65750i	0.237742	+	0.0637028i	−0.137981	+	0.990435i	$0.544061\pi$
$678$		0			0
$679$	10.6066	+	6.12372i	0.407044	+	0.235007i
$680$		0			0
$681$		0			0
$682$	7.24656	−	27.0445i	0.277485	−	1.03559i
$683$	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
$686$	−12.1515	+	7.01569i	−0.463948	+	0.267860i
$687$		0			0
$688$	0.896575	+	3.34607i	0.0341816	+	0.127568i
$689$	16.1920	−	28.0454i	0.616867	−	1.06844i
$690$		0			0
$691$	−16.4722	−	28.5307i	−0.626632	−	1.08536i	−0.988223	−	0.153021i	$-0.951100\pi$
$691$	−16.4722	−	28.5307i	−0.626632	−	1.08536i	0.361591	−	0.932337i	$-0.382234\pi$
$692$	2.19275	−	2.19275i	0.0833559	−	0.0833559i
$693$		0			0
$694$		−	4.20204i		−	0.159507i
$695$		0			0
$696$		0			0
$697$	−16.8683	+	4.51985i	−0.638933	+	0.171202i
$698$	−14.5327	+	3.89404i	−0.550073	+	0.147392i
$699$		0			0
$700$		0			0
$701$		−	23.9309i		−	0.903857i	−0.892054	−	0.451928i	$-0.850736\pi$
$701$		−	23.9309i		−	0.903857i	0.892054	−	0.451928i	$-0.149264\pi$
$702$		0			0
$703$	−4.65153	+	4.65153i	−0.175436	+	0.175436i
$704$	3.14626	+	5.44949i	0.118579	+	0.205385i
$705$		0			0
$706$	−17.1464	+	29.6985i	−0.645314	+	1.11772i
$707$	−0.360762	−	1.34638i	−0.0135678	−	0.0506359i
$708$		0			0
$709$	38.4069	−	22.1742i	1.44240	−	0.832771i	0.444392	−	0.895833i	$-0.353420\pi$
$709$	38.4069	−	22.1742i	1.44240	−	0.832771i	0.998010	+	0.0630617i	$0.0200865\pi$
$710$		0			0
$711$		0			0
$712$	−1.67423	−	1.67423i	−0.0627446	−	0.0627446i
$713$	1.15161	−	4.29788i	0.0431282	−	0.160957i
$714$		0			0
$715$		0			0
$716$	−15.2474	−	8.80312i	−0.569824	−	0.328988i
$717$		0			0
$718$	16.8683	+	4.51985i	0.629520	+	0.168679i
$719$	−32.5269			−1.21305			−0.606525	−	0.795065i	$-0.707437\pi$
$719$	−32.5269			−1.21305			−0.606525	+	0.795065i	$0.707437\pi$
$720$		0			0
$721$	4.49490			0.167399
$722$	16.0304	+	4.29534i	0.596591	+	0.159856i
$723$		0			0
$724$	−9.13701	−	5.27526i	−0.339574	−	0.196053i
$725$		0			0
$726$		0			0
$727$	8.11447	−	30.2836i	0.300949	−	1.12316i	−0.635428	−	0.772160i	$-0.719176\pi$
$727$	8.11447	−	30.2836i	0.300949	−	1.12316i	0.936377	−	0.350996i	$-0.114157\pi$
$728$	−2.68556	−	2.68556i	−0.0995336	−	0.0995336i
$729$		0			0
$730$		0			0
$731$	−13.3485	+	7.70674i	−0.493711	+	0.285044i
$732$		0			0
$733$	3.71385	+	13.8603i	0.137174	+	0.511941i	0.999979	+	0.00640470i	$0.00203869\pi$
$733$	3.71385	+	13.8603i	0.137174	+	0.511941i	−0.862805	+	0.505536i	$0.831295\pi$
$734$	−4.87832	+	8.44949i	−0.180062	+	0.311876i
$735$		0			0
$736$	0.500000	+	0.866025i	0.0184302	+	0.0319221i
$737$	16.8277	−	16.8277i	0.619856	−	0.619856i
$738$		0			0
$739$		−	24.9444i		−	0.917594i	−0.888541	−	0.458797i	$-0.848280\pi$
$739$		−	24.9444i		−	0.917594i	0.888541	−	0.458797i	$-0.151720\pi$
$740$		0			0
$741$		0			0
$742$	9.90020	−	2.65275i	0.363448	−	0.0973855i
$743$	0.0975783	−	0.0261460i	0.00357980	−	0.000959205i	−0.257029	−	0.966404i	$-0.582743\pi$
$743$	0.0975783	−	0.0261460i	0.00357980	−	0.000959205i	0.260609	+	0.965445i	$0.416077\pi$
$744$		0			0
$745$		0			0
$746$		−	20.2918i		−	0.742936i
$747$		0			0
$748$	−19.7980	+	19.7980i	−0.723885	+	0.723885i
$749$	−2.87659	−	4.98240i	−0.105108	−	0.182053i
$750$		0			0
$751$	4.34847	−	7.53177i	0.158678	−	0.274838i	−0.775714	−	0.631084i	$-0.782610\pi$
$751$	4.34847	−	7.53177i	0.158678	−	0.274838i	0.934392	+	0.356246i	$0.115944\pi$
$752$	2.32937	+	8.69333i	0.0849434	+	0.317013i
$753$		0			0
$754$	9.43879	−	5.44949i	0.343741	−	0.198459i
$755$		0			0
$756$		0			0
$757$	22.0454	+	22.0454i	0.801254	+	0.801254i	0.983292	−	0.182038i	$-0.0582693\pi$
$757$	22.0454	+	22.0454i	0.801254	+	0.801254i	−0.182038	+	0.983292i	$0.558269\pi$
$758$	1.72154	−	6.42489i	0.0625293	−	0.233362i
$759$		0			0
$760$		0			0
$761$	15.3990	+	8.89060i	0.558213	+	0.322284i	0.752428	−	0.658675i	$-0.228883\pi$
$761$	15.3990	+	8.89060i	0.558213	+	0.322284i	−0.194215	+	0.980959i	$0.562216\pi$
$762$		0			0
$763$	21.5494	+	5.77414i	0.780141	+	0.209038i
$764$	3.32124			0.120158
$765$		0			0
$766$	27.7980			1.00438
$767$	−39.5054	−	10.5854i	−1.42646	−	0.382218i
$768$		0			0
$769$	−17.0580	−	9.84847i	−0.615129	−	0.355145i	0.159841	−	0.987143i	$-0.448902\pi$
$769$	−17.0580	−	9.84847i	−0.615129	−	0.355145i	−0.774970	+	0.631998i	$0.782235\pi$
$770$		0			0
$771$		0			0
$772$	4.48288	−	16.7303i	0.161342	−	0.602138i
$773$	3.11416	+	3.11416i	0.112008	+	0.112008i	0.760890	−	0.648881i	$-0.224763\pi$
$773$	3.11416	+	3.11416i	0.112008	+	0.112008i	−0.648881	+	0.760890i	$0.724763\pi$
$774$		0			0
$775$		0			0
$776$	9.67423	−	5.58542i	0.347285	−	0.200505i
$777$		0			0
$778$	−1.45579	−	5.43309i	−0.0521927	−	0.194786i
$779$	−3.04272	+	5.27015i	−0.109017	+	0.188823i
$780$		0			0
$781$	2.00000	+	3.46410i	0.0715656	+	0.123955i
$782$	−3.14626	+	3.14626i	−0.112510	+	0.112510i
$783$		0			0
$784$			5.79796i			0.207070i
$785$		0			0
$786$		0			0
$787$	3.96008	−	1.06110i	0.141162	−	0.0378241i	−0.187546	−	0.982256i	$-0.560053\pi$
$787$	3.96008	−	1.06110i	0.141162	−	0.0378241i	0.328708	+	0.944432i	$0.393387\pi$
$788$	−9.46410	+	2.53590i	−0.337145	+	0.0903376i
$789$		0			0
$790$		0			0
$791$		−	15.1278i		−	0.537881i
$792$		0			0
$793$	−13.3485	+	13.3485i	−0.474018	+	0.474018i
$794$	−10.9244	−	18.9217i	−0.387694	−	0.671505i
$795$		0			0
$796$	−1.77526	+	3.07483i	−0.0629222	+	0.108985i
$797$	−5.22867	−	19.5137i	−0.185209	−	0.691210i	−0.994586	−	0.103920i	$-0.966861\pi$
$797$	−5.22867	−	19.5137i	−0.185209	−	0.691210i	0.809377	−	0.587290i	$-0.199805\pi$
$798$		0			0
$799$	−34.6803	+	20.0227i	−1.22690	+	0.708352i
$800$		0			0
$801$		0			0
$802$	−18.2474	−	18.2474i	−0.644340	−	0.644340i
$803$	−6.67700	+	24.9189i	−0.235626	+	0.879369i
$804$		0			0
$805$		0			0
$806$	13.3485	+	7.70674i	0.470180	+	0.271458i
$807$		0			0
$808$	−1.22803	−	0.329049i	−0.0432019	−	0.0115759i
$809$	54.0901			1.90171			0.950853	−	0.309644i	$-0.100210\pi$
$809$	54.0901			1.90171			0.950853	+	0.309644i	$0.100210\pi$
$810$		0			0
$811$	43.6413			1.53245			0.766227	−	0.642570i	$-0.222132\pi$
$811$	43.6413			1.53245			0.766227	+	0.642570i	$0.222132\pi$
$812$	3.33195	+	0.892794i	0.116929	+	0.0313309i
$813$		0			0
$814$	23.1202	+	13.3485i	0.810364	+	0.467864i
$815$		0			0
$816$		0			0
$817$	−1.39015	+	5.18811i	−0.0486352	+	0.181509i
$818$	13.5065	+	13.5065i	0.472242	+	0.472242i
$819$		0			0
$820$		0			0
$821$	22.3207	−	12.8868i	0.778997	−	0.449754i	−0.0570780	−	0.998370i	$-0.518178\pi$
$821$	22.3207	−	12.8868i	0.778997	−	0.449754i	0.836075	+	0.548616i	$0.184845\pi$
$822$		0			0
$823$	12.5807	+	46.9519i	0.438536	+	1.63664i	0.732459	+	0.680811i	$0.238373\pi$
$823$	12.5807	+	46.9519i	0.438536	+	1.63664i	−0.293923	+	0.955829i	$0.594961\pi$
$824$	2.04989	−	3.55051i	0.0714112	−	0.123688i
$825$		0			0
$826$	−6.47219	−	11.2102i	−0.225196	−	0.390052i
$827$	−27.3235	+	27.3235i	−0.950133	+	0.950133i	−0.998814	−	0.0486816i	$-0.984498\pi$
$827$	−27.3235	+	27.3235i	−0.950133	+	0.950133i	0.0486816	+	0.998814i	$0.484498\pi$
$828$		0			0
$829$		−	15.4495i		−	0.536583i	−0.963338	−	0.268291i	$-0.913541\pi$
$829$		−	15.4495i		−	0.536583i	0.963338	−	0.268291i	$-0.0864590\pi$
$830$		0			0
$831$		0			0
$832$	−3.34607	+	0.896575i	−0.116004	+	0.0310832i
$833$	−24.9189	+	6.67700i	−0.863389	+	0.231344i
$834$		0			0
$835$		0			0
$836$			9.75663i			0.337440i
$837$		0			0
$838$	−8.44949	+	8.44949i	−0.291883	+	0.291883i
$839$	10.1459	+	17.5732i	0.350275	+	0.606695i	0.986298	−	0.164976i	$-0.0527547\pi$
$839$	10.1459	+	17.5732i	0.350275	+	0.606695i	−0.636022	+	0.771671i	$0.719421\pi$
$840$		0			0
$841$	9.55051	−	16.5420i	0.329328	−	0.570413i
$842$	3.85614	+	14.3913i	0.132891	+	0.495957i
$843$		0			0
$844$	−16.3670	+	9.44949i	−0.563375	+	0.325265i
$845$		0			0
$846$		0			0
$847$	22.1691	+	22.1691i	0.761740	+	0.761740i
$848$	2.41956	−	9.02993i	0.0830881	−	0.310089i
$849$		0			0
$850$		0			0
$851$	3.67423	+	2.12132i	0.125951	+	0.0727179i
$852$		0			0
$853$	−35.8547	−	9.60723i	−1.22764	−	0.328945i	−0.413980	−	0.910286i	$-0.635862\pi$
$853$	−35.8547	−	9.60723i	−1.22764	−	0.328945i	−0.813660	+	0.581340i	$0.802528\pi$
$854$	−5.97469			−0.204450
$855$		0			0
$856$	−5.24745			−0.179354
$857$	3.66855	+	0.982984i	0.125315	+	0.0335781i	0.320932	−	0.947102i	$-0.396004\pi$
$857$	3.66855	+	0.982984i	0.125315	+	0.0335781i	−0.195616	+	0.980680i	$0.562671\pi$
$858$		0			0
$859$	−2.16064	−	1.24745i	−0.0737202	−	0.0425624i	0.462687	−	0.886522i	$-0.346885\pi$
$859$	−2.16064	−	1.24745i	−0.0737202	−	0.0425624i	−0.536407	+	0.843959i	$0.680219\pi$
$860$		0			0
$861$		0			0
$862$	−4.02628	+	15.0263i	−0.137136	+	0.511797i
$863$	27.7842	+	27.7842i	0.945787	+	0.945787i	0.998604	−	0.0528175i	$-0.0168202\pi$
$863$	27.7842	+	27.7842i	0.945787	+	0.945787i	−0.0528175	+	0.998604i	$0.516820\pi$
$864$		0			0
$865$		0			0
$866$	−10.4722	+	6.04612i	−0.355860	+	0.205456i
$867$		0			0
$868$	1.26260	+	4.71209i	0.0428555	+	0.159939i
$869$	−7.70674	+	13.3485i	−0.261433	+	0.452816i
$870$		0			0
$871$	6.55051	+	11.3458i	0.221956	+	0.384438i
$872$	14.3885	−	14.3885i	0.487257	−	0.487257i
$873$		0			0
$874$			1.55051i			0.0524468i
$875$		0			0
$876$		0			0
$877$	7.85813	−	2.10558i	0.265350	−	0.0711004i	−0.123691	−	0.992321i	$-0.539473\pi$
$877$	7.85813	−	2.10558i	0.265350	−	0.0711004i	0.389041	+	0.921220i	$0.372806\pi$
$878$	9.85441	−	2.64048i	0.332570	−	0.0891120i
$879$		0			0
$880$		0			0
$881$		−	58.3006i		−	1.96420i	−0.188368	−	0.982098i	$-0.560320\pi$
$881$		−	58.3006i		−	1.96420i	0.188368	−	0.982098i	$-0.439680\pi$
$882$		0			0
$883$	40.2702	−	40.2702i	1.35520	−	1.35520i	0.475463	−	0.879736i	$-0.342281\pi$
$883$	40.2702	−	40.2702i	1.35520	−	1.35520i	0.879736	−	0.475463i	$-0.157719\pi$
$884$	−7.70674	−	13.3485i	−0.259206	−	0.448958i
$885$		0			0
$886$	0.275255	−	0.476756i	0.00924738	−	0.0160169i
$887$	7.19464	+	26.8508i	0.241572	+	0.901561i	0.975075	+	0.221874i	$0.0712174\pi$
$887$	7.19464	+	26.8508i	0.241572	+	0.901561i	−0.733503	+	0.679686i	$0.762116\pi$
$888$		0			0
$889$	18.9651	−	10.9495i	0.636068	−	0.367234i
$890$		0			0
$891$		0			0
$892$	5.87628	+	5.87628i	0.196752	+	0.196752i
$893$	−3.61171	+	13.4791i	−0.120861	+	0.451061i
$894$		0			0
$895$		0			0
$896$	−0.949490	−	0.548188i	−0.0317202	−	0.0183137i
$897$		0			0
$898$	−20.9664	−	5.61793i	−0.699658	−	0.187473i
$899$	−13.9993			−0.466902
$900$		0			0
$901$	41.5959			1.38576
$902$	23.8554	+	6.39204i	0.794298	+	0.212832i
$903$		0			0
$904$	−11.9494	−	6.89898i	−0.397431	−	0.229457i
$905$		0			0
$906$		0			0
$907$	−1.71089	+	6.38512i	−0.0568091	+	0.212015i	−0.988496	−	0.151248i	$-0.951671\pi$
$907$	−1.71089	+	6.38512i	−0.0568091	+	0.212015i	0.931687	+	0.363263i	$0.118337\pi$
$908$	−10.6780	−	10.6780i	−0.354363	−	0.354363i
$909$		0			0
$910$		0			0
$911$	−6.12372	+	3.53553i	−0.202888	+	0.117137i	−0.598002	−	0.801495i	$-0.704038\pi$
$911$	−6.12372	+	3.53553i	−0.202888	+	0.117137i	0.395114	+	0.918632i	$0.370705\pi$
$912$		0			0
$913$	0.896575	+	3.34607i	0.0296723	+	0.110739i
$914$	−3.07483	+	5.32577i	−0.101706	+	0.176161i
$915$		0			0
$916$	8.17423	+	14.1582i	0.270084	+	0.467800i
$917$	8.16744	−	8.16744i	0.269713	−	0.269713i
$918$		0			0
$919$			27.3485i			0.902143i	0.892488	+	0.451071i	$0.148958\pi$
$919$			27.3485i			0.902143i	−0.892488	+	0.451071i	$0.851042\pi$
$920$		0			0
$921$		0			0
$922$	−18.2654	+	4.89419i	−0.601538	+	0.161182i
$923$	−2.12701	+	0.569930i	−0.0700113	+	0.0187595i
$924$		0			0
$925$		0			0
$926$			33.0197i			1.08510i
$927$		0			0
$928$	2.22474	−	2.22474i	0.0730308	−	0.0730308i
$929$	23.9309	+	41.4495i	0.785147	+	1.35991i	0.928912	+	0.370302i	$0.120746\pi$
$929$	23.9309	+	41.4495i	0.785147	+	1.35991i	−0.143765	+	0.989612i	$0.545921\pi$
$930$		0			0
$931$	−4.49490	+	7.78539i	−0.147314	+	0.255156i
$932$	−4.02477	−	15.0206i	−0.131836	−	0.492017i
$933$		0			0
$934$	3.46410	−	2.00000i	0.113349	−	0.0654420i
$935$		0			0
$936$		0			0
$937$	12.8990	+	12.8990i	0.421391	+	0.421391i	0.885683	−	0.464291i	$-0.153691\pi$
$937$	12.8990	+	12.8990i	0.421391	+	0.421391i	−0.464291	+	0.885683i	$0.653691\pi$
$938$	−1.07317	+	4.00514i	−0.0350404	+	0.130773i
$939$		0			0
$940$		0			0
$941$	−5.47730	−	3.16232i	−0.178555	−	0.103089i	0.408059	−	0.912956i	$-0.366206\pi$
$941$	−5.47730	−	3.16232i	−0.178555	−	0.103089i	−0.586613	+	0.809867i	$0.699539\pi$
$942$		0			0
$943$	3.79107	+	1.01581i	0.123454	+	0.0330795i
$944$	−11.8065			−0.384269
$945$		0			0
$946$	21.7980			0.708713
$947$	39.6468	+	10.6233i	1.28835	+	0.345212i	0.837033	−	0.547152i	$-0.184288\pi$
$947$	39.6468	+	10.6233i	1.28835	+	0.345212i	0.451316	+	0.892364i	$0.350955\pi$
$948$		0			0
$949$	−12.2993	−	7.10102i	−0.399253	−	0.230509i
$950$		0			0
$951$		0			0
$952$	1.26260	−	4.71209i	0.0409211	−	0.152720i
$953$	−19.6561	−	19.6561i	−0.636724	−	0.636724i	0.313022	−	0.949746i	$-0.398659\pi$
$953$	−19.6561	−	19.6561i	−0.636724	−	0.636724i	−0.949746	+	0.313022i	$0.898659\pi$
$954$		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
$959$	1.20713	−	2.09082i	0.0389804	−	0.0675159i
$960$		0			0
$961$	5.60102	+	9.70125i	0.180678	+	0.312944i
$962$	−10.3923	+	10.3923i	−0.335061	+	0.335061i
$963$		0			0
$964$		−	19.0000i		−	0.611949i
$965$		0			0
$966$		0			0
$967$	−48.7319	+	13.0577i	−1.56711	+	0.419907i	−0.934907	−	0.354894i	$-0.884517\pi$
$967$	−48.7319	+	13.0577i	−1.56711	+	0.419907i	−0.632206	+	0.774800i	$0.717850\pi$
$968$	27.6215	−	7.40117i	0.887790	−	0.237883i
$969$		0			0
$970$		0			0
$971$			49.2117i			1.57928i	0.613570	+	0.789640i	$0.289733\pi$
$971$			49.2117i			1.57928i	−0.613570	+	0.789640i	$0.710267\pi$
$972$		0			0
$973$	10.0000	−	10.0000i	0.320585	−	0.320585i
$974$	8.48528	+	14.6969i	0.271886	+	0.470920i
$975$		0			0
$976$	−2.72474	+	4.71940i	−0.0872170	+	0.151064i
$977$	10.9985	+	41.0469i	0.351873	+	1.31321i	0.884374	+	0.466778i	$0.154585\pi$
$977$	10.9985	+	41.0469i	0.351873	+	1.31321i	−0.532502	+	0.846429i	$0.678748\pi$
$978$		0			0
$979$	−12.9029	+	7.44949i	−0.412378	+	0.238087i
$980$		0			0
$981$		0			0
$982$	−0.202041	−	0.202041i	−0.00644739	−	0.00644739i
$983$	12.1122	−	45.2034i	0.386319	−	1.44176i	−0.449757	−	0.893151i	$-0.648490\pi$
$983$	12.1122	−	45.2034i	0.386319	−	1.44176i	0.836077	−	0.548612i	$-0.184844\pi$
$984$		0			0
$985$		0			0
$986$	12.1237	+	6.99964i	0.386098	+	0.222914i
$987$		0			0
$988$	−5.18811	−	1.39015i	−0.165056	−	0.0442265i
$989$	3.46410			0.110152
$990$		0			0
$991$	−56.7423			−1.80248			−0.901240	−	0.433320i	$-0.857342\pi$
$991$	−56.7423			−1.80248			−0.901240	+	0.433320i	$0.857342\pi$
$992$	4.29788	+	1.15161i	0.136458	+	0.0365637i
$993$		0			0
$994$	−0.603566	−	0.348469i	−0.0191440	−	0.0110528i
$995$		0			0
$996$		0			0
$997$	10.7053	−	39.9528i	0.339041	−	1.26532i	−0.560381	−	0.828235i	$-0.689345\pi$
$997$	10.7053	−	39.9528i	0.339041	−	1.26532i	0.899422	−	0.437082i	$-0.143988\pi$
$998$	6.29253	+	6.29253i	0.199187	+	0.199187i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1350.2.q.g.557.2		8
3.2	odd	2		450.2.p.a.257.1		8
5.2	odd	4		270.2.m.a.233.1		8
5.3	odd	4	inner	1350.2.q.g.1043.2		8
5.4	even	2		270.2.m.a.17.1		8
9.2	odd	6	inner	1350.2.q.g.1007.2		8
9.7	even	3		450.2.p.a.407.1		8
15.2	even	4		90.2.l.a.23.2	✓	8
15.8	even	4		450.2.p.a.293.1		8
15.14	odd	2		90.2.l.a.77.2	yes	8
45.2	even	12		270.2.m.a.143.1		8
45.4	even	6		810.2.f.b.647.4		8
45.7	odd	12		90.2.l.a.83.2	yes	8
45.14	odd	6		810.2.f.b.647.1		8
45.22	odd	12		810.2.f.b.323.2		8
45.29	odd	6		270.2.m.a.197.1		8
45.32	even	12		810.2.f.b.323.3		8
45.34	even	6		90.2.l.a.47.2	yes	8
45.38	even	12	inner	1350.2.q.g.143.2		8
45.43	odd	12		450.2.p.a.443.1		8
60.47	odd	4		720.2.cu.a.113.2		8
60.59	even	2		720.2.cu.a.257.2		8
180.7	even	12		720.2.cu.a.353.2		8
180.79	odd	6		720.2.cu.a.497.2		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
90.2.l.a.23.2	✓	8	15.2	even	4
90.2.l.a.47.2	yes	8	45.34	even	6
90.2.l.a.77.2	yes	8	15.14	odd	2
90.2.l.a.83.2	yes	8	45.7	odd	12
270.2.m.a.17.1		8	5.4	even	2
270.2.m.a.143.1		8	45.2	even	12
270.2.m.a.197.1		8	45.29	odd	6
270.2.m.a.233.1		8	5.2	odd	4
450.2.p.a.257.1		8	3.2	odd	2
450.2.p.a.293.1		8	15.8	even	4
450.2.p.a.407.1		8	9.7	even	3
450.2.p.a.443.1		8	45.43	odd	12
720.2.cu.a.113.2		8	60.47	odd	4
720.2.cu.a.257.2		8	60.59	even	2
720.2.cu.a.353.2		8	180.7	even	12
720.2.cu.a.497.2		8	180.79	odd	6
810.2.f.b.323.2		8	45.22	odd	12
810.2.f.b.323.3		8	45.32	even	12
810.2.f.b.647.1		8	45.14	odd	6
810.2.f.b.647.4		8	45.4	even	6
1350.2.q.g.143.2		8	45.38	even	12	inner
1350.2.q.g.557.2		8	1.1	even	1	trivial
1350.2.q.g.1007.2		8	9.2	odd	6	inner
1350.2.q.g.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)

\(f(q)\)	\(=\)	\(q+(0.965926 + 0.258819i) q^{2} +(0.866025 + 0.500000i) q^{4} +(-0.283763 + 1.05902i) q^{7} +(0.707107 + 0.707107i) q^{8} +O(q^{10})\)	\(q+(0.965926 + 0.258819i) q^{2} +(0.866025 + 0.500000i) q^{4} +(-0.283763 + 1.05902i) q^{7} +(0.707107 + 0.707107i) q^{8} +(5.44949 - 3.14626i) q^{11} +(0.896575 + 3.34607i) q^{13} +(-0.548188 + 0.949490i) q^{14} +(0.500000 + 0.866025i) q^{16} +(-3.14626 + 3.14626i) q^{17} +1.55051i q^{19} +(6.07812 - 1.62863i) q^{22} +(0.965926 - 0.258819i) q^{23} +3.46410i q^{26} +(-0.775255 + 0.775255i) q^{28} +(-1.57313 - 2.72474i) q^{29} +(2.22474 - 3.85337i) q^{31} +(0.258819 + 0.965926i) q^{32} +(-3.85337 + 2.22474i) q^{34} +(3.00000 + 3.00000i) q^{37} +(-0.401302 + 1.49768i) q^{38} +(3.39898 + 1.96240i) q^{41} +(3.34607 + 0.896575i) q^{43} +6.29253 q^{44} +1.00000 q^{46} +(8.69333 + 2.32937i) q^{47} +(5.02118 + 2.89898i) q^{49} +(-0.896575 + 3.34607i) q^{52} +(-6.61037 - 6.61037i) q^{53} +(-0.949490 + 0.548188i) q^{56} +(-0.814313 - 3.03906i) q^{58} +(-5.90326 + 10.2247i) q^{59} +(2.72474 + 4.71940i) q^{61} +(3.14626 - 3.14626i) q^{62} +1.00000i q^{64} +(3.65307 - 0.978838i) q^{67} +(-4.29788 + 1.15161i) q^{68} +0.635674i q^{71} +(-2.89898 + 2.89898i) q^{73} +(2.12132 + 3.67423i) q^{74} +(-0.775255 + 1.34278i) q^{76} +(1.78559 + 6.66390i) q^{77} +(-2.12132 + 1.22474i) q^{79} +(2.77526 + 2.77526i) q^{82} +(-0.142483 + 0.531752i) q^{83} +(3.00000 + 1.73205i) q^{86} +(6.07812 + 1.62863i) q^{88} -2.36773 q^{89} -3.79796 q^{91} +(0.965926 + 0.258819i) q^{92} +(7.79423 + 4.50000i) q^{94} +(2.89123 - 10.7902i) q^{97} +(4.09978 + 4.09978i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 8 q^{7}+O(q^{10}) \) 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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