LMFDB - Embedded newform 1350.2.q.g.1007.2

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			1007.2
Root			$-0.965926 + 0.258819i$ of defining polynomial
Character	$\chi$	$=$	1350.1007
Dual form			1350.2.q.g.1043.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.258819 + 0.965926i) q^{2} +(-0.866025 + 0.500000i) q^{4} +(1.05902 - 0.283763i) q^{7} +(-0.707107 - 0.707107i) q^{8} +O(q^{10})$	\(q+(0.258819 + 0.965926i) q^{2} +(-0.866025 + 0.500000i) q^{4} +(1.05902 - 0.283763i) q^{7} +(-0.707107 - 0.707107i) q^{8} +(5.44949 + 3.14626i) q^{11} +(-3.34607 - 0.896575i) q^{13} +(0.548188 + 0.949490i) q^{14} +(0.500000 - 0.866025i) q^{16} +(3.14626 - 3.14626i) q^{17} +1.55051i q^{19} +(-1.62863 + 6.07812i) q^{22} +(0.258819 - 0.965926i) q^{23} -3.46410i q^{26} +(-0.775255 + 0.775255i) q^{28} +(1.57313 - 2.72474i) q^{29} +(2.22474 + 3.85337i) q^{31} +(0.965926 + 0.258819i) q^{32} +(3.85337 + 2.22474i) q^{34} +(3.00000 + 3.00000i) q^{37} +(-1.49768 + 0.401302i) q^{38} +(3.39898 - 1.96240i) q^{41} +(-0.896575 - 3.34607i) q^{43} -6.29253 q^{44} +1.00000 q^{46} +(2.32937 + 8.69333i) q^{47} +(-5.02118 + 2.89898i) q^{49} +(3.34607 - 0.896575i) q^{52} +(6.61037 + 6.61037i) q^{53} +(-0.949490 - 0.548188i) q^{56} +(3.03906 + 0.814313i) q^{58} +(5.90326 + 10.2247i) q^{59} +(2.72474 - 4.71940i) q^{61} +(-3.14626 + 3.14626i) q^{62} +1.00000i q^{64} +(-0.978838 + 3.65307i) q^{67} +(-1.15161 + 4.29788i) q^{68} -0.635674i q^{71} +(-2.89898 + 2.89898i) q^{73} +(-2.12132 + 3.67423i) q^{74} +(-0.775255 - 1.34278i) q^{76} +(6.66390 + 1.78559i) q^{77} +(2.12132 + 1.22474i) q^{79} +(2.77526 + 2.77526i) q^{82} +(-0.531752 + 0.142483i) q^{83} +(3.00000 - 1.73205i) q^{86} +(-1.62863 - 6.07812i) q^{88} +2.36773 q^{89} -3.79796 q^{91} +(0.258819 + 0.965926i) q^{92} +(-7.79423 + 4.50000i) q^{94} +(-10.7902 + 2.89123i) q^{97} +(-4.09978 - 4.09978i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $\left(\mathbb{Z}/1350\mathbb{Z}\right)^\times$.

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

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 $

$2$	0.258819	+	0.965926i	0.183013	+	0.683013i
$2$	0.258819	+	0.965926i	0.183013	+	0.683013i
$3$		0			0
$3$		0			0
$4$	−0.866025	+	0.500000i	−0.433013	+	0.250000i
$5$		0			0
$5$		0			0
$6$		0			0
$7$	1.05902	−	0.283763i	0.400271	−	0.107252i	−0.0530669	−	0.998591i	$-0.516900\pi$
$7$	1.05902	−	0.283763i	0.400271	−	0.107252i	0.453338	+	0.891339i	$0.350233\pi$
$8$	−0.707107	−	0.707107i	−0.250000	−	0.250000i
$9$		0			0
$10$		0			0
$11$	5.44949	+	3.14626i	1.64308	+	0.948634i	0.979729	+	0.200329i	$0.0642011\pi$
$11$	5.44949	+	3.14626i	1.64308	+	0.948634i	0.663354	+	0.748305i	$0.269132\pi$
$12$		0			0
$13$	−3.34607	−	0.896575i	−0.928032	−	0.248665i	−0.237016	−	0.971506i	$-0.576170\pi$
$13$	−3.34607	−	0.896575i	−0.928032	−	0.248665i	−0.691015	+	0.722840i	$0.742836\pi$
$14$	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	−0.213797	−	0.976878i	$-0.568583\pi$
$17$	3.14626	−	3.14626i	0.763081	−	0.763081i	0.976878	+	0.213797i	$0.0685831\pi$
$18$		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
$22$	−1.62863	+	6.07812i	−0.347224	+	1.29586i
$23$	0.258819	−	0.965926i	0.0539675	−	0.201409i	−0.933678	−	0.358113i	$-0.883420\pi$
$23$	0.258819	−	0.965926i	0.0539675	−	0.201409i	0.987646	+	0.156704i	$0.0500868\pi$
$24$		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	$-0.738972\pi$
$29$	1.57313	−	2.72474i	0.292123	−	0.505972i	0.974312	+	0.225204i	$0.0723049\pi$
$30$		0			0
$31$	2.22474	+	3.85337i	0.399576	+	0.692086i	0.993674	−	0.112307i	$-0.0358240\pi$
$31$	2.22474	+	3.85337i	0.399576	+	0.692086i	−0.594098	+	0.804393i	$0.702491\pi$
$32$	0.965926	+	0.258819i	0.170753	+	0.0457532i
$33$		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	$0.636609\pi$
$38$	−1.49768	+	0.401302i	−0.242955	+	0.0650997i
$39$		0			0
$40$		0			0
$41$	3.39898	−	1.96240i	0.530831	−	0.306476i	−0.210524	−	0.977589i	$-0.567517\pi$
$41$	3.39898	−	1.96240i	0.530831	−	0.306476i	0.741355	+	0.671113i	$0.234184\pi$
$42$		0			0
$43$	−0.896575	−	3.34607i	−0.136726	−	0.510270i	−0.999985	−	0.00550783i	$-0.998247\pi$
$43$	−0.896575	−	3.34607i	−0.136726	−	0.510270i	0.863258	−	0.504762i	$-0.168420\pi$
$44$	−6.29253			−0.948634
$45$		0			0
$46$	1.00000			0.147442
$47$	2.32937	+	8.69333i	0.339774	+	1.26805i	0.898600	+	0.438768i	$0.144585\pi$
$47$	2.32937	+	8.69333i	0.339774	+	1.26805i	−0.558827	+	0.829285i	$0.688748\pi$
$48$		0			0
$49$	−5.02118	+	2.89898i	−0.717311	+	0.414140i
$50$		0			0
$51$		0			0
$52$	3.34607	−	0.896575i	0.464016	−	0.124333i
$53$	6.61037	+	6.61037i	0.908004	+	0.908004i	0.996111	−	0.0881074i	$-0.0280819\pi$
$53$	6.61037	+	6.61037i	0.908004	+	0.908004i	−0.0881074	+	0.996111i	$0.528082\pi$
$54$		0			0
$55$		0			0
$56$	−0.949490	−	0.548188i	−0.126881	−	0.0732547i
$57$		0			0
$58$	3.03906	+	0.814313i	0.399048	+	0.106925i
$59$	5.90326	+	10.2247i	0.768539	+	1.33115i	0.938355	+	0.345673i	$0.112349\pi$
$59$	5.90326	+	10.2247i	0.768539	+	1.33115i	−0.169816	+	0.985476i	$0.554317\pi$
$60$		0			0
$61$	2.72474	−	4.71940i	0.348868	−	0.604257i	−0.637181	−	0.770714i	$-0.719900\pi$
$61$	2.72474	−	4.71940i	0.348868	−	0.604257i	0.986049	+	0.166458i	$0.0532329\pi$
$62$	−3.14626	+	3.14626i	−0.399576	+	0.399576i
$63$		0			0
$64$			1.00000i			0.125000i
$65$		0			0
$66$		0			0
$67$	−0.978838	+	3.65307i	−0.119584	+	0.446294i	−0.999589	−	0.0286709i	$-0.990873\pi$
$67$	−0.978838	+	3.65307i	−0.119584	+	0.446294i	0.880005	+	0.474965i	$0.157539\pi$
$68$	−1.15161	+	4.29788i	−0.139654	+	0.521194i
$69$		0			0
$70$		0			0
$71$		−	0.635674i		−	0.0754407i	−0.999288	−	0.0377203i	$-0.987990\pi$
$71$		−	0.635674i		−	0.0754407i	0.999288	−	0.0377203i	$-0.0120096\pi$
$72$		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
$77$	6.66390	+	1.78559i	0.759422	+	0.203487i
$78$		0			0
$79$	2.12132	+	1.22474i	0.238667	+	0.137795i	0.614564	−	0.788867i	$-0.289332\pi$
$79$	2.12132	+	1.22474i	0.238667	+	0.137795i	−0.375897	+	0.926662i	$0.622665\pi$
$80$		0			0
$81$		0			0
$82$	2.77526	+	2.77526i	0.306476	+	0.306476i
$83$	−0.531752	+	0.142483i	−0.0583674	+	0.0156395i	−0.287885	−	0.957665i	$-0.592952\pi$
$83$	−0.531752	+	0.142483i	−0.0583674	+	0.0156395i	0.229517	+	0.973305i	$0.426285\pi$
$84$		0			0
$85$		0			0
$86$	3.00000	−	1.73205i	0.323498	−	0.186772i
$87$		0			0
$88$	−1.62863	−	6.07812i	−0.173612	−	0.647929i
$89$	2.36773			0.250978			0.125489	−	0.992095i	$-0.459950\pi$
$89$	2.36773			0.250978			0.125489	+	0.992095i	$0.459950\pi$
$90$		0			0
$91$	−3.79796			−0.398134
$92$	0.258819	+	0.965926i	0.0269838	+	0.100705i
$93$		0			0
$94$	−7.79423	+	4.50000i	−0.803913	+	0.464140i
$95$		0			0
$96$		0			0
$97$	−10.7902	+	2.89123i	−1.09558	+	0.293560i	−0.760963	−	0.648795i	$-0.775273\pi$
$97$	−10.7902	+	2.89123i	−1.09558	+	0.293560i	−0.334616	+	0.942355i	$0.608607\pi$
$98$	−4.09978	−	4.09978i	−0.414140	−	0.414140i
$99$		0			0
$100$		0			0
$101$	−1.10102	−	0.635674i	−0.109556	−	0.0632520i	0.444221	−	0.895917i	$-0.353481\pi$
$101$	−1.10102	−	0.635674i	−0.109556	−	0.0632520i	−0.553777	+	0.832665i	$0.686814\pi$
$102$		0			0
$103$	3.96008	+	1.06110i	0.390198	+	0.104553i	0.448584	−	0.893741i	$-0.351929\pi$
$103$	3.96008	+	1.06110i	0.390198	+	0.104553i	−0.0583855	+	0.998294i	$0.518595\pi$
$104$	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	−0.504628	−	0.863337i	$-0.668370\pi$
$107$	3.71051	−	3.71051i	0.358708	−	0.358708i	0.863337	+	0.504628i	$0.168370\pi$
$108$		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
$112$	0.283763	−	1.05902i	0.0268131	−	0.100068i
$113$	−3.57117	+	13.3278i	−0.335948	+	1.25377i	0.566890	+	0.823793i	$0.308146\pi$
$113$	−3.57117	+	13.3278i	−0.335948	+	1.25377i	−0.902838	+	0.429981i	$0.858520\pi$
$114$		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
$122$	5.26380	+	1.41043i	0.476562	+	0.127694i
$123$		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	$-0.596665\pi$
$128$	−0.965926	+	0.258819i	−0.0853766	+	0.0228766i
$129$		0			0
$130$		0			0
$131$	−9.12372	+	5.26758i	−0.797143	+	0.460231i	−0.842471	−	0.538741i	$-0.818900\pi$
$131$	−9.12372	+	5.26758i	−0.797143	+	0.460231i	0.0453278	+	0.998972i	$0.485567\pi$
$132$		0			0
$133$	0.439978	+	1.64202i	0.0381509	+	0.142381i
$134$	−3.78194			−0.326710
$135$		0			0
$136$	−4.44949			−0.381541
$137$	−0.569930	−	2.12701i	−0.0486924	−	0.181723i	0.937297	−	0.348533i	$-0.113320\pi$
$137$	−0.569930	−	2.12701i	−0.0486924	−	0.181723i	−0.985989	+	0.166810i	$0.946653\pi$
$138$		0			0
$139$	11.1708	−	6.44949i	0.947499	−	0.547039i	0.0551956	−	0.998476i	$-0.482422\pi$
$139$	11.1708	−	6.44949i	0.947499	−	0.547039i	0.892303	+	0.451437i	$0.149088\pi$
$140$		0			0
$141$		0			0
$142$	0.614014	−	0.164525i	0.0515269	−	0.0138066i
$143$	−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
$148$	−4.09808	−	1.09808i	−0.336860	−	0.0902613i
$149$	−6.45145	−	11.1742i	−0.528523	−	0.915429i	−0.999447	−	0.0332550i	$-0.989413\pi$
$149$	−6.45145	−	11.1742i	−0.528523	−	0.915429i	0.470924	−	0.882174i	$-0.343921\pi$
$150$		0			0
$151$	10.7980	−	18.7026i	0.878725	−	1.52200i	0.0259849	−	0.999662i	$-0.491728\pi$
$151$	10.7980	−	18.7026i	0.878725	−	1.52200i	0.852741	−	0.522335i	$-0.174939\pi$
$152$	1.09638	−	1.09638i	0.0889279	−	0.0889279i
$153$		0			0
$154$			6.89898i			0.555936i
$155$		0			0
$156$		0			0
$157$	−1.59165	+	5.94012i	−0.127028	+	0.474073i	−0.999904	−	0.0138684i	$-0.995585\pi$
$157$	−1.59165	+	5.94012i	−0.127028	+	0.474073i	0.872876	+	0.487942i	$0.162252\pi$
$158$	−0.633975	+	2.36603i	−0.0504363	+	0.188231i
$159$		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
$167$	−10.4300	−	2.79472i	−0.807100	−	0.216262i	−0.168401	−	0.985719i	$-0.553860\pi$
$167$	−10.4300	−	2.79472i	−0.807100	−	0.216262i	−0.638699	+	0.769457i	$0.720527\pi$
$168$		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
$173$	2.99536	−	0.802603i	0.227733	−	0.0610208i	−0.143148	−	0.989701i	$-0.545723\pi$
$173$	2.99536	−	0.802603i	0.227733	−	0.0610208i	0.370881	+	0.928680i	$0.379056\pi$
$174$		0			0
$175$		0			0
$176$	5.44949	−	3.14626i	0.410771	−	0.237159i
$177$		0			0
$178$	0.612812	+	2.28705i	0.0459322	+	0.171421i
$179$	17.6062			1.31595			0.657976	−	0.753039i	$-0.271413\pi$
$179$	17.6062			1.31595			0.657976	+	0.753039i	$0.271413\pi$
$180$		0			0
$181$	−10.5505			−0.784213			−0.392107	−	0.919920i	$-0.628254\pi$
$181$	−10.5505			−0.784213			−0.392107	+	0.919920i	$0.628254\pi$
$182$	−0.982984	−	3.66855i	−0.0728636	−	0.271931i
$183$		0			0
$184$	−0.866025	+	0.500000i	−0.0638442	+	0.0368605i
$185$		0			0
$186$		0			0
$187$	27.0445	−	7.24656i	1.97769	−	0.529921i
$188$	−6.36396	−	6.36396i	−0.464140	−	0.464140i
$189$		0			0
$190$		0			0
$191$	2.87628	+	1.66062i	0.208120	+	0.120158i	0.600437	−	0.799672i	$-0.294993\pi$
$191$	2.87628	+	1.66062i	0.208120	+	0.120158i	−0.392317	+	0.919830i	$0.628327\pi$
$192$		0			0
$193$	16.7303	+	4.48288i	1.20428	+	0.322685i	0.804513	−	0.593934i	$-0.202426\pi$
$193$	16.7303	+	4.48288i	1.20428	+	0.322685i	0.399762	+	0.916619i	$0.369093\pi$
$194$	−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	−0.415829	−	0.909443i	$-0.636508\pi$
$197$	6.92820	−	6.92820i	0.493614	−	0.493614i	0.909443	+	0.415829i	$0.136508\pi$
$198$		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
$202$	0.329049	−	1.22803i	0.0231518	−	0.0864038i
$203$	0.892794	−	3.33195i	0.0626618	−	0.233857i
$204$		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	−0.982995	−	0.183635i	$-0.941214\pi$
$211$	−9.44949	−	16.3670i	−0.650530	−	1.12675i	0.332465	−	0.943116i	$-0.392120\pi$
$212$	−9.02993	−	2.41956i	−0.620178	−	0.166176i
$213$		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
$218$	19.6551	−	5.26657i	1.33121	−	0.356697i
$219$		0			0
$220$		0			0
$221$	−13.3485	+	7.70674i	−0.897915	+	0.518412i
$222$		0			0
$223$	−2.15087	−	8.02714i	−0.144033	−	0.537537i	−0.999797	−	0.0201706i	$-0.993579\pi$
$223$	−2.15087	−	8.02714i	−0.144033	−	0.537537i	0.855764	−	0.517367i	$-0.173088\pi$
$224$	1.09638			0.0732547
$225$		0			0
$226$	−13.7980			−0.917827
$227$	−3.90843	−	14.5865i	−0.259412	−	0.968138i	−0.965583	−	0.260096i	$-0.916246\pi$
$227$	−3.90843	−	14.5865i	−0.259412	−	0.968138i	0.706171	−	0.708041i	$-0.250421\pi$
$228$		0			0
$229$	−14.1582	+	8.17423i	−0.935600	+	0.540169i	−0.888578	−	0.458725i	$-0.848306\pi$
$229$	−14.1582	+	8.17423i	−0.935600	+	0.540169i	−0.0470214	+	0.998894i	$0.514973\pi$
$230$		0			0
$231$		0			0
$232$	−3.03906	+	0.814313i	−0.199524	+	0.0534623i
$233$	10.9959	+	10.9959i	0.720363	+	0.720363i	0.968679	−	0.248316i	$-0.0798770\pi$
$233$	10.9959	+	10.9959i	0.720363	+	0.720363i	−0.248316	+	0.968679i	$0.579877\pi$
$234$		0			0
$235$		0			0
$236$	−10.2247	−	5.90326i	−0.665574	−	0.384269i
$237$		0			0
$238$	4.71209	+	1.26260i	0.305439	+	0.0818423i
$239$	−8.48528	−	14.6969i	−0.548867	−	0.950666i	−0.998353	−	0.0573782i	$-0.981726\pi$
$239$	−8.48528	−	14.6969i	−0.548867	−	0.950666i	0.449485	−	0.893288i	$-0.351607\pi$
$240$		0			0
$241$	−9.50000	+	16.4545i	−0.611949	+	1.05993i	0.378963	+	0.925412i	$0.376281\pi$
$241$	−9.50000	+	16.4545i	−0.611949	+	1.05993i	−0.990912	+	0.134515i	$0.957053\pi$
$242$	−20.2204	+	20.2204i	−1.29981	+	1.29981i
$243$		0			0
$244$			5.44949i			0.348868i
$245$		0			0
$246$		0			0
$247$	1.39015	−	5.18811i	0.0884531	−	0.330111i
$248$	1.15161	−	4.29788i	0.0731275	−	0.272915i
$249$		0			0
$250$		0			0
$251$		−	11.1708i		−	0.705097i	−0.935793	−	0.352549i	$-0.885315\pi$
$251$		−	11.1708i		−	0.705097i	0.935793	−	0.352549i	$-0.114685\pi$
$252$		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
$257$	−25.1579	−	6.74105i	−1.56931	−	0.420495i	−0.633713	−	0.773568i	$-0.718470\pi$
$257$	−25.1579	−	6.74105i	−1.56931	−	0.420495i	−0.935596	+	0.353073i	$0.885137\pi$
$258$		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
$263$	12.2643	−	3.28621i	0.756249	−	0.202636i	0.139961	−	0.990157i	$-0.455302\pi$
$263$	12.2643	−	3.28621i	0.756249	−	0.202636i	0.616288	+	0.787521i	$0.288636\pi$
$264$		0			0
$265$		0			0
$266$	−1.47219	+	0.849971i	−0.0902660	+	0.0521151i
$267$		0			0
$268$	−0.978838	−	3.65307i	−0.0597920	−	0.223147i
$269$	−4.70334			−0.286768			−0.143384	−	0.989667i	$-0.545798\pi$
$269$	−4.70334			−0.286768			−0.143384	+	0.989667i	$0.545798\pi$
$270$		0			0
$271$	−16.0454			−0.974689			−0.487345	−	0.873210i	$-0.662034\pi$
$271$	−16.0454			−0.974689			−0.487345	+	0.873210i	$0.662034\pi$
$272$	−1.15161	−	4.29788i	−0.0698268	−	0.260597i
$273$		0			0
$274$	1.90702	−	1.10102i	0.115208	−	0.0665151i
$275$		0			0
$276$		0			0
$277$	−13.7983	+	3.69723i	−0.829057	+	0.222145i	−0.648302	−	0.761383i	$-0.724521\pi$
$277$	−13.7983	+	3.69723i	−0.829057	+	0.222145i	−0.180754	+	0.983528i	$0.557854\pi$
$278$	9.12096	+	9.12096i	0.547039	+	0.547039i
$279$		0			0
$280$		0			0
$281$	0.151531	+	0.0874863i	0.00903957	+	0.00521900i	0.504513	−	0.863404i	$-0.331672\pi$
$281$	0.151531	+	0.0874863i	0.00903957	+	0.00521900i	−0.495473	+	0.868623i	$0.665005\pi$
$282$		0			0
$283$	6.66112	+	1.78484i	0.395962	+	0.106098i	0.451305	−	0.892370i	$-0.350959\pi$
$283$	6.66112	+	1.78484i	0.395962	+	0.106098i	−0.0553430	+	0.998467i	$0.517625\pi$
$284$	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
$292$	1.06110	−	3.96008i	0.0620962	−	0.231746i
$293$	5.70577	−	21.2942i	0.333335	−	1.24402i	−0.572329	−	0.820024i	$-0.693960\pi$
$293$	5.70577	−	21.2942i	0.333335	−	1.24402i	0.905663	−	0.423998i	$-0.139374\pi$
$294$		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
$302$	20.8601	+	5.58943i	1.20036	+	0.321636i
$303$		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	$0.758662\pi$
$308$	−6.66390	+	1.78559i	−0.379711	+	0.101743i
$309$		0			0
$310$		0			0
$311$	17.8207	−	10.2888i	1.01052	−	0.583422i	0.0991741	−	0.995070i	$-0.468380\pi$
$311$	17.8207	−	10.2888i	1.01052	−	0.583422i	0.911343	+	0.411648i	$0.135047\pi$
$312$		0			0
$313$	−1.29958	−	4.85009i	−0.0734564	−	0.274143i	0.919422	−	0.393271i	$-0.128657\pi$
$313$	−1.29958	−	4.85009i	−0.0734564	−	0.274143i	−0.992879	+	0.119128i	$0.961990\pi$
$314$	−6.14966			−0.347046
$315$		0			0
$316$	−2.44949			−0.137795
$317$	0.284965	+	1.06350i	0.0160052	+	0.0597323i	0.973467	−	0.228829i	$-0.0734897\pi$
$317$	0.284965	+	1.06350i	0.0160052	+	0.0597323i	−0.957461	+	0.288561i	$0.906823\pi$
$318$		0			0
$319$	17.1455	−	9.89898i	0.959966	−	0.554236i
$320$		0			0
$321$		0			0
$322$	1.05902	−	0.283763i	0.0590168	−	0.0158135i
$323$	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
$328$	−3.79107	−	1.01581i	−0.209327	−	0.0560889i
$329$	4.93369	+	8.54541i	0.272003	+	0.471124i
$330$		0			0
$331$	2.22474	−	3.85337i	0.122283	−	0.211800i	−0.798385	−	0.602148i	$-0.794312\pi$
$331$	2.22474	−	3.85337i	0.122283	−	0.211800i	0.920668	+	0.390347i	$0.127645\pi$
$332$	0.389270	−	0.389270i	0.0213639	−	0.0213639i
$333$		0			0
$334$		−	10.7980i		−	0.590838i
$335$		0			0
$336$		0			0
$337$	7.97861	−	29.7766i	0.434622	−	1.62203i	−0.307346	−	0.951598i	$-0.599441\pi$
$337$	7.97861	−	29.7766i	0.434622	−	1.62203i	0.741968	−	0.670435i	$-0.233892\pi$
$338$	0.258819	−	0.965926i	0.0140779	−	0.0525394i
$339$		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
$347$	−4.05886	−	1.08757i	−0.217891	−	0.0583837i	0.148222	−	0.988954i	$-0.452645\pi$
$347$	−4.05886	−	1.08757i	−0.217891	−	0.0583837i	−0.366113	+	0.930570i	$0.619312\pi$
$348$		0			0
$349$	13.0297	+	7.52270i	0.697464	+	0.402681i	0.806402	−	0.591367i	$-0.201412\pi$
$349$	13.0297	+	7.52270i	0.697464	+	0.402681i	−0.108938	+	0.994049i	$0.534745\pi$
$350$		0			0
$351$		0			0
$352$	4.44949	+	4.44949i	0.237159	+	0.237159i
$353$	−33.1244	+	8.87564i	−1.76303	+	0.472403i	−0.987328	−	0.158694i	$-0.949272\pi$
$353$	−33.1244	+	8.87564i	−1.76303	+	0.472403i	−0.775704	+	0.631097i	$0.782605\pi$
$354$		0			0
$355$		0			0
$356$	−2.05051	+	1.18386i	−0.108677	+	0.0627446i
$357$		0			0
$358$	4.55683	+	17.0063i	0.240836	+	0.898812i
$359$	−17.4634			−0.921682			−0.460841	−	0.887483i	$-0.652452\pi$
$359$	−17.4634			−0.921682			−0.460841	+	0.887483i	$0.652452\pi$
$360$		0			0
$361$	16.5959			0.873469
$362$	−2.73067	−	10.1910i	−0.143521	−	0.535628i
$363$		0			0
$364$	3.28913	−	1.89898i	0.172397	−	0.0995336i
$365$		0			0
$366$		0			0
$367$	9.42418	−	2.52520i	0.491938	−	0.131814i	−0.00431778	−	0.999991i	$-0.501374\pi$
$367$	9.42418	−	2.52520i	0.491938	−	0.131814i	0.496256	+	0.868176i	$0.334708\pi$
$368$	−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
$373$	19.6004	+	5.25190i	1.01487	+	0.271933i	0.727662	−	0.685935i	$-0.240607\pi$
$373$	19.6004	+	5.25190i	1.01487	+	0.271933i	0.287206	+	0.957869i	$0.407273\pi$
$374$	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
$382$	−0.859599	+	3.20807i	−0.0439809	+	0.164139i
$383$	7.19464	−	26.8508i	0.367629	−	1.37201i	−0.496193	−	0.868212i	$-0.665269\pi$
$383$	7.19464	−	26.8508i	0.367629	−	1.37201i	0.863822	−	0.503798i	$-0.168064\pi$
$384$		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	$-0.787789\pi$
$389$	2.81237	−	4.87117i	0.142593	−	0.246978i	0.928472	+	0.371402i	$0.121123\pi$
$390$		0			0
$391$	−2.22474	−	3.85337i	−0.112510	−	0.194873i
$392$	5.60040	+	1.50062i	0.282863	+	0.0757929i
$393$		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	$0.934718\pi$
$398$	−3.42953	+	0.918940i	−0.171907	+	0.0460623i
$399$		0			0
$400$		0			0
$401$	−22.3485	+	12.9029i	−1.11603	+	0.644340i	−0.940384	−	0.340114i	$-0.889534\pi$
$401$	−22.3485	+	12.9029i	−1.11603	+	0.644340i	−0.175645	+	0.984454i	$0.556201\pi$
$402$		0			0
$403$	−3.98930	−	14.8883i	−0.198721	−	0.741638i
$404$	1.27135			0.0632520
$405$		0			0
$406$	3.44949			0.171195
$407$	6.90968	+	25.7873i	0.342500	+	1.27823i
$408$		0			0
$409$	−16.5420	+	9.55051i	−0.817948	+	0.472242i	−0.849708	−	0.527253i	$-0.823222\pi$
$409$	−16.5420	+	9.55051i	−0.817948	+	0.472242i	0.0317605	+	0.999496i	$0.489889\pi$
$410$		0			0
$411$		0			0
$412$	−3.96008	+	1.06110i	−0.195099	+	0.0522767i
$413$	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
$418$	−9.42418	−	2.52520i	−0.460952	−	0.123512i
$419$	5.97469	+	10.3485i	0.291883	+	0.505556i	0.974255	−	0.225449i	$-0.0723850\pi$
$419$	5.97469	+	10.3485i	0.291883	+	0.505556i	−0.682372	+	0.731005i	$0.739052\pi$
$420$		0			0
$421$	7.44949	−	12.9029i	0.363066	−	0.628849i	−0.625398	−	0.780306i	$-0.715063\pi$
$421$	7.44949	−	12.9029i	0.363066	−	0.628849i	0.988464	+	0.151457i	$0.0483966\pi$
$422$	13.3636	−	13.3636i	0.650530	−	0.650530i
$423$		0			0
$424$		−	9.34847i		−	0.454002i
$425$		0			0
$426$		0			0
$427$	1.54636	−	5.77111i	0.0748338	−	0.279284i
$428$	−1.35814	+	5.06865i	−0.0656482	+	0.245002i
$429$		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
$433$	−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
$437$	1.49768	+	0.401302i	0.0716436	+	0.0191969i
$438$		0			0
$439$	−8.83523	−	5.10102i	−0.421682	−	0.243458i	0.274114	−	0.961697i	$-0.411615\pi$
$439$	−8.83523	−	5.10102i	−0.421682	−	0.243458i	−0.695797	+	0.718239i	$0.744949\pi$
$440$		0			0
$441$		0			0
$442$	−10.8990	−	10.8990i	−0.518412	−	0.518412i
$443$	0.531752	−	0.142483i	0.0252643	−	0.00676955i	−0.246165	−	0.969228i	$-0.579170\pi$
$443$	0.531752	−	0.142483i	0.0252643	−	0.00676955i	0.271429	+	0.962458i	$0.412504\pi$
$444$		0			0
$445$		0			0
$446$	7.19694	−	4.15515i	0.340785	−	0.196752i
$447$		0			0
$448$	0.283763	+	1.05902i	0.0134065	+	0.0500339i
$449$	21.7060			1.02437			0.512185	−	0.858875i	$-0.328836\pi$
$449$	21.7060			1.02437			0.512185	+	0.858875i	$0.328836\pi$
$450$		0			0
$451$	24.6969			1.16293
$452$	−3.57117	−	13.3278i	−0.167974	−	0.626887i
$453$		0			0
$454$	13.0779	−	7.55051i	0.613775	−	0.354363i
$455$		0			0
$456$		0			0
$457$	5.94012	−	1.59165i	0.277867	−	0.0744543i	−0.117194	−	0.993109i	$-0.537390\pi$
$457$	5.94012	−	1.59165i	0.277867	−	0.0744543i	0.395061	+	0.918655i	$0.370723\pi$
$458$	−11.5601	−	11.5601i	−0.540169	−	0.540169i
$459$		0			0
$460$		0			0
$461$	−16.3763	−	9.45485i	−0.762719	−	0.440356i	0.0675520	−	0.997716i	$-0.478481\pi$
$461$	−16.3763	−	9.45485i	−0.762719	−	0.440356i	−0.830271	+	0.557360i	$0.811814\pi$
$462$		0			0
$463$	−31.8946	−	8.54613i	−1.48227	−	0.397172i	−0.575150	−	0.818048i	$-0.695056\pi$
$463$	−31.8946	−	8.54613i	−1.48227	−	0.397172i	−0.907118	+	0.420876i	$0.861723\pi$
$464$	−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	−0.769514	−	0.638630i	$-0.779501\pi$
$467$	−2.82843	+	2.82843i	−0.130884	+	0.130884i	0.638630	+	0.769514i	$0.279501\pi$
$468$		0			0
$469$			4.14643i			0.191464i
$470$		0			0
$471$		0			0
$472$	3.05575	−	11.4042i	0.140652	−	0.524922i
$473$	5.64173	−	21.0552i	0.259407	−	0.968120i
$474$		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	$-0.884980\pi$
$479$	−3.53553	+	6.12372i	−0.161543	+	0.279800i	0.773879	+	0.633333i	$0.218314\pi$
$480$		0			0
$481$	−7.34847	−	12.7279i	−0.335061	−	0.580343i
$482$	−18.3526	−	4.91756i	−0.835938	−	0.223989i
$483$		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	$0.624372\pi$
$488$	−5.26380	+	1.41043i	−0.238281	+	0.0638472i
$489$		0			0
$490$		0			0
$491$	−0.247449	+	0.142865i	−0.0111672	+	0.00644739i	−0.505573	−	0.862784i	$-0.668719\pi$
$491$	−0.247449	+	0.142865i	−0.0111672	+	0.00644739i	0.494406	+	0.869231i	$0.335386\pi$
$492$		0			0
$493$	−3.62328	−	13.5223i	−0.163184	−	0.609012i
$494$	5.37113			0.241658
$495$		0			0
$496$	4.44949			0.199788
$497$	−0.180381	−	0.673191i	−0.00809119	−	0.0301967i
$498$		0			0
$499$	−7.70674	+	4.44949i	−0.345001	+	0.199187i	−0.662481	−	0.749078i	$-0.730497\pi$
$499$	−7.70674	+	4.44949i	−0.345001	+	0.199187i	0.317480	+	0.948265i	$0.397163\pi$
$500$		0			0
$501$		0			0
$502$	10.7902	−	2.89123i	0.481590	−	0.129042i
$503$	16.7563	+	16.7563i	0.747125	+	0.747125i	0.973938	−	0.226813i	$-0.0728307\pi$
$503$	16.7563	+	16.7563i	0.747125	+	0.747125i	−0.226813	+	0.973938i	$0.572831\pi$
$504$		0			0
$505$		0			0
$506$	5.44949	+	3.14626i	0.242259	+	0.139869i
$507$		0			0
$508$	19.2934	+	5.16964i	0.856005	+	0.229366i
$509$	−19.8150	−	34.3207i	−0.878286	−	1.52124i	−0.853220	−	0.521551i	$-0.825354\pi$
$509$	−19.8150	−	34.3207i	−0.878286	−	1.52124i	−0.0250662	−	0.999686i	$-0.507980\pi$
$510$		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
$517$	−14.6576	+	54.7030i	−0.644642	+	2.40584i
$518$	−1.20390	+	4.49303i	−0.0528965	+	0.197413i
$519$		0			0
$520$		0			0
$521$		−	29.4449i		−	1.29000i	−0.764181	−	0.645001i	$-0.776857\pi$
$521$		−	29.4449i		−	1.29000i	0.764181	−	0.645001i	$-0.223143\pi$
$522$		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
$527$	19.1234	+	5.12409i	0.833027	+	0.223209i
$528$		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
$533$	−13.1326	+	3.51888i	−0.568838	+	0.152420i
$534$		0			0
$535$		0			0
$536$	3.27526	−	1.89097i	0.141469	−	0.0816774i
$537$		0			0
$538$	−1.21731	−	4.54308i	−0.0524822	−	0.195866i
$539$	−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	$0.688323\pi$
$542$	−4.15286	−	15.4987i	−0.178380	−	0.665725i
$543$		0			0
$544$	3.85337	−	2.22474i	0.165212	−	0.0953851i
$545$		0			0
$546$		0			0
$547$	−20.6594	+	5.53567i	−0.883332	+	0.236688i	−0.671844	−	0.740693i	$-0.734498\pi$
$547$	−20.6594	+	5.53567i	−0.883332	+	0.236688i	−0.211488	+	0.977381i	$0.567831\pi$
$548$	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
$553$	2.59405	+	0.695075i	0.110310	+	0.0295576i
$554$	−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	−0.962999	−	0.269507i	$-0.913139\pi$
$557$	−16.3670	+	16.3670i	−0.693492	+	0.693492i	0.269507	+	0.962999i	$0.413139\pi$
$558$		0			0
$559$			12.0000i			0.507546i
$560$		0			0
$561$		0			0
$562$	−0.0452863	+	0.169011i	−0.00191029	+	0.00712928i
$563$	−8.89004	+	33.1781i	−0.374670	+	1.39829i	0.479155	+	0.877730i	$0.340943\pi$
$563$	−8.89004	+	33.1781i	−0.374670	+	1.39829i	−0.853826	+	0.520559i	$0.825724\pi$
$564$		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	0.451578	+	0.892232i	$0.350861\pi$
$569$	−13.0458	+	22.5959i	−0.546907	+	0.947270i	−0.998484	+	0.0550383i	$0.982472\pi$
$570$		0			0
$571$	−13.5505	−	23.4702i	−0.567071	−	0.982196i	−0.996854	−	0.0792637i	$-0.974743\pi$
$571$	−13.5505	−	23.4702i	−0.567071	−	0.982196i	0.429782	−	0.902932i	$-0.358590\pi$
$572$	21.0552	+	5.64173i	0.880363	+	0.235892i
$573$		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	$0.583174\pi$
$578$	2.70262	−	0.724165i	0.112414	−	0.0301213i
$579$		0			0
$580$		0			0
$581$	−0.522704	+	0.301783i	−0.0216854	+	0.0125201i
$582$		0			0
$583$	15.2252	+	56.8211i	0.630562	+	2.35329i
$584$	4.09978			0.169650
$585$		0			0
$586$	22.0454			0.910687
$587$	8.03514	+	29.9876i	0.331646	+	1.23772i	0.907460	+	0.420138i	$0.138019\pi$
$587$	8.03514	+	29.9876i	0.331646	+	1.23772i	−0.575814	+	0.817581i	$0.695315\pi$
$588$		0			0
$589$	−5.97469	+	3.44949i	−0.246183	+	0.142134i
$590$		0			0
$591$		0			0
$592$	4.09808	−	1.09808i	0.168430	−	0.0451307i
$593$	−10.0745	−	10.0745i	−0.413709	−	0.413709i	0.469320	−	0.883028i	$-0.344499\pi$
$593$	−10.0745	−	10.0745i	−0.413709	−	0.413709i	−0.883028	+	0.469320i	$0.844499\pi$
$594$		0			0
$595$		0			0
$596$	11.1742	+	6.45145i	0.457714	+	0.264262i
$597$		0			0
$598$	−3.34607	−	0.896575i	−0.136831	−	0.0366637i
$599$	16.8991	+	29.2702i	0.690480	+	1.19595i	0.971681	+	0.236297i	$0.0759339\pi$
$599$	16.8991	+	29.2702i	0.690480	+	1.19595i	−0.281201	+	0.959649i	$0.590733\pi$
$600$		0			0
$601$	−17.3485	+	30.0484i	−0.707659	+	1.22570i	0.258065	+	0.966128i	$0.416915\pi$
$601$	−17.3485	+	30.0484i	−0.707659	+	1.22570i	−0.965723	+	0.259573i	$0.916418\pi$
$602$	2.68556	−	2.68556i	0.109455	−	0.109455i
$603$		0			0
$604$			21.5959i			0.878725i
$605$		0			0
$606$		0			0
$607$	−5.73717	+	21.4114i	−0.232864	+	0.869062i	0.746235	+	0.665682i	$0.231859\pi$
$607$	−5.73717	+	21.4114i	−0.232864	+	0.869062i	−0.979100	+	0.203380i	$0.934807\pi$
$608$	−0.401302	+	1.49768i	−0.0162749	+	0.0607389i
$609$		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
$617$	6.85906	+	1.83788i	0.276135	+	0.0739902i	0.394229	−	0.919012i	$-0.371012\pi$
$617$	6.85906	+	1.83788i	0.276135	+	0.0739902i	−0.118094	+	0.993002i	$0.537678\pi$
$618$		0			0
$619$	−21.4275	−	12.3712i	−0.861244	−	0.497239i	0.00318471	−	0.999995i	$-0.498986\pi$
$619$	−21.4275	−	12.3712i	−0.861244	−	0.497239i	−0.864429	+	0.502756i	$0.832320\pi$
$620$		0			0
$621$		0			0
$622$	14.5505	+	14.5505i	0.583422	+	0.583422i
$623$	2.50746	−	0.671873i	0.100459	−	0.0269180i
$624$		0			0
$625$		0			0
$626$	4.34847	−	2.51059i	0.173800	−	0.100343i
$627$		0			0
$628$	−1.59165	−	5.94012i	−0.0635138	−	0.237037i
$629$	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	$0.582652\pi$
$632$	−0.633975	−	2.36603i	−0.0252182	−	0.0941154i
$633$		0			0
$634$	−0.953512	+	0.550510i	−0.0378688	+	0.0218636i
$635$		0			0
$636$		0			0
$637$	19.4003	−	5.19831i	0.768670	−	0.205964i
$638$	13.9993	+	13.9993i	0.554236	+	0.554236i
$639$		0			0
$640$		0			0
$641$	7.74745	+	4.47299i	0.306006	+	0.176673i	0.645138	−	0.764066i	$-0.276800\pi$
$641$	7.74745	+	4.47299i	0.306006	+	0.176673i	−0.339132	+	0.940739i	$0.610133\pi$
$642$		0			0
$643$	−30.6976	−	8.22539i	−1.21059	−	0.324378i	−0.403599	−	0.914936i	$-0.632241\pi$
$643$	−30.6976	−	8.22539i	−1.21059	−	0.324378i	−0.806995	+	0.590558i	$0.798908\pi$
$644$	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	−0.0187105	−	0.999825i	$-0.505956\pi$
$647$	24.9558	−	24.9558i	0.981114	−	0.981114i	0.999825	+	0.0187105i	$0.00595608\pi$
$648$		0			0
$649$			74.2929i			2.91625i
$650$		0			0
$651$		0			0
$652$	−0.164525	+	0.614014i	−0.00644328	+	0.0240467i
$653$	−5.57768	+	20.8162i	−0.218272	+	0.814601i	0.766718	+	0.641985i	$0.221889\pi$
$653$	−5.57768	+	20.8162i	−0.218272	+	0.814601i	−0.984989	+	0.172616i	$0.944778\pi$
$654$		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	$-0.762610\pi$
$659$	5.65685	−	9.79796i	0.220360	−	0.381674i	0.954917	+	0.296872i	$0.0959435\pi$
$660$		0			0
$661$	15.3485	+	26.5843i	0.596986	+	1.03401i	0.993263	+	0.115880i	$0.0369687\pi$
$661$	15.3485	+	26.5843i	0.596986	+	1.03401i	−0.396277	+	0.918131i	$0.629698\pi$
$662$	4.29788	+	1.15161i	0.167042	+	0.0447587i
$663$		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
$668$	10.4300	−	2.79472i	0.403550	−	0.108131i
$669$		0			0
$670$		0			0
$671$	29.6969	−	17.1455i	1.14644	−	0.661896i
$672$		0			0
$673$	4.22778	+	15.7783i	0.162969	+	0.608208i	0.998291	+	0.0584468i	$0.0186148\pi$
$673$	4.22778	+	15.7783i	0.162969	+	0.608208i	−0.835322	+	0.549762i	$0.814719\pi$
$674$	30.8270			1.18741
$675$		0			0
$676$	1.00000			0.0384615
$677$	1.65750	+	6.18587i	0.0637028	+	0.237742i	0.990435	−	0.137981i	$-0.0440612\pi$
$677$	1.65750	+	6.18587i	0.0637028	+	0.237742i	−0.926732	+	0.375723i	$0.877394\pi$
$678$		0			0
$679$	−10.6066	+	6.12372i	−0.407044	+	0.235007i
$680$		0			0
$681$		0			0
$682$	−27.0445	+	7.24656i	−1.03559	+	0.277485i
$683$	−13.8564	−	13.8564i	−0.530201	−	0.530201i	0.390431	−	0.920632i	$-0.372326\pi$
$683$	−13.8564	−	13.8564i	−0.530201	−	0.530201i	−0.920632	+	0.390431i	$0.872326\pi$
$684$		0			0
$685$		0			0
$686$	−12.1515	−	7.01569i	−0.463948	−	0.267860i
$687$		0			0
$688$	−3.34607	−	0.896575i	−0.127568	−	0.0341816i
$689$	−16.1920	−	28.0454i	−0.616867	−	1.06844i
$690$		0			0
$691$	−16.4722	+	28.5307i	−0.626632	+	1.08536i	0.361591	+	0.932337i	$0.382234\pi$
$691$	−16.4722	+	28.5307i	−0.626632	+	1.08536i	−0.988223	+	0.153021i	$0.951100\pi$
$692$	−2.19275	+	2.19275i	−0.0833559	+	0.0833559i
$693$		0			0
$694$		−	4.20204i		−	0.159507i
$695$		0			0
$696$		0			0
$697$	4.51985	−	16.8683i	0.171202	−	0.638933i
$698$	−3.89404	+	14.5327i	−0.147392	+	0.550073i
$699$		0			0
$700$		0			0
$701$			23.9309i			0.903857i	0.892054	+	0.451928i	$0.149264\pi$
$701$			23.9309i			0.903857i	−0.892054	+	0.451928i	$0.850736\pi$
$702$		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
$707$	−1.34638	−	0.360762i	−0.0506359	−	0.0135678i
$708$		0			0
$709$	−38.4069	−	22.1742i	−1.44240	−	0.832771i	−0.444392	−	0.895833i	$-0.646580\pi$
$709$	−38.4069	−	22.1742i	−1.44240	−	0.832771i	−0.998010	+	0.0630617i	$0.979914\pi$
$710$		0			0
$711$		0			0
$712$	−1.67423	−	1.67423i	−0.0627446	−	0.0627446i
$713$	4.29788	−	1.15161i	0.160957	−	0.0431282i
$714$		0			0
$715$		0			0
$716$	−15.2474	+	8.80312i	−0.569824	+	0.328988i
$717$		0			0
$718$	−4.51985	−	16.8683i	−0.168679	−	0.629520i
$719$	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
$721$	4.49490			0.167399
$722$	4.29534	+	16.0304i	0.159856	+	0.596591i
$723$		0			0
$724$	9.13701	−	5.27526i	0.339574	−	0.196053i
$725$		0			0
$726$		0			0
$727$	−30.2836	+	8.11447i	−1.12316	+	0.300949i	−0.772160	−	0.635428i	$-0.780824\pi$
$727$	−30.2836	+	8.11447i	−1.12316	+	0.300949i	−0.350996	+	0.936377i	$0.614157\pi$
$728$	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
$733$	−13.8603	−	3.71385i	−0.511941	−	0.137174i	−0.00640470	−	0.999979i	$-0.502039\pi$
$733$	−13.8603	−	3.71385i	−0.511941	−	0.137174i	−0.505536	+	0.862805i	$0.668705\pi$
$734$	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
$742$	−2.65275	+	9.90020i	−0.0973855	+	0.363448i
$743$	0.0261460	−	0.0975783i	0.000959205	−	0.00357980i	−0.965445	−	0.260609i	$-0.916077\pi$
$743$	0.0261460	−	0.0975783i	0.000959205	−	0.00357980i	0.966404	+	0.257029i	$0.0827435\pi$
$744$		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	0.934392	−	0.356246i	$-0.115944\pi$
$751$	4.34847	+	7.53177i	0.158678	+	0.274838i	−0.775714	+	0.631084i	$0.782610\pi$
$752$	8.69333	+	2.32937i	0.317013	+	0.0849434i
$753$		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	$0.558269\pi$
$758$	6.42489	−	1.72154i	0.233362	−	0.0625293i
$759$		0			0
$760$		0			0
$761$	15.3990	−	8.89060i	0.558213	−	0.322284i	−0.194215	−	0.980959i	$-0.562216\pi$
$761$	15.3990	−	8.89060i	0.558213	−	0.322284i	0.752428	+	0.658675i	$0.228883\pi$
$762$		0			0
$763$	−5.77414	−	21.5494i	−0.209038	−	0.780141i
$764$	−3.32124			−0.120158
$765$		0			0
$766$	27.7980			1.00438
$767$	−10.5854	−	39.5054i	−0.382218	−	1.42646i
$768$		0			0
$769$	17.0580	−	9.84847i	0.615129	−	0.355145i	−0.159841	−	0.987143i	$-0.551098\pi$
$769$	17.0580	−	9.84847i	0.615129	−	0.355145i	0.774970	+	0.631998i	$0.217765\pi$
$770$		0			0
$771$		0			0
$772$	−16.7303	+	4.48288i	−0.602138	+	0.161342i
$773$	−3.11416	−	3.11416i	−0.112008	−	0.112008i	0.648881	−	0.760890i	$-0.275237\pi$
$773$	−3.11416	−	3.11416i	−0.112008	−	0.112008i	−0.760890	+	0.648881i	$0.775237\pi$
$774$		0			0
$775$		0			0
$776$	9.67423	+	5.58542i	0.347285	+	0.200505i
$777$		0			0
$778$	5.43309	+	1.45579i	0.194786	+	0.0521927i
$779$	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
$787$	−1.06110	+	3.96008i	−0.0378241	+	0.141162i	−0.982256	−	0.187546i	$-0.939947\pi$
$787$	−1.06110	+	3.96008i	−0.0378241	+	0.141162i	0.944432	+	0.328708i	$0.106613\pi$
$788$	−2.53590	+	9.46410i	−0.0903376	+	0.337145i
$789$		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
$797$	−19.5137	−	5.22867i	−0.691210	−	0.185209i	−0.103920	−	0.994586i	$-0.533139\pi$
$797$	−19.5137	−	5.22867i	−0.691210	−	0.185209i	−0.587290	+	0.809377i	$0.699805\pi$
$798$		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
$803$	−24.9189	+	6.67700i	−0.879369	+	0.235626i
$804$		0			0
$805$		0			0
$806$	13.3485	−	7.70674i	0.470180	−	0.271458i
$807$		0			0
$808$	0.329049	+	1.22803i	0.0115759	+	0.0432019i
$809$	−54.0901			−1.90171			−0.950853	−	0.309644i	$-0.899790\pi$
$809$	−54.0901			−1.90171			−0.950853	+	0.309644i	$0.899790\pi$
$810$		0			0
$811$	43.6413			1.53245			0.766227	−	0.642570i	$-0.222132\pi$
$811$	43.6413			1.53245			0.766227	+	0.642570i	$0.222132\pi$
$812$	0.892794	+	3.33195i	0.0313309	+	0.116929i
$813$		0			0
$814$	−23.1202	+	13.3485i	−0.810364	+	0.467864i
$815$		0			0
$816$		0			0
$817$	5.18811	−	1.39015i	0.181509	−	0.0486352i
$818$	−13.5065	−	13.5065i	−0.472242	−	0.472242i
$819$		0			0
$820$		0			0
$821$	22.3207	+	12.8868i	0.778997	+	0.449754i	0.836075	−	0.548616i	$-0.184845\pi$
$821$	22.3207	+	12.8868i	0.778997	+	0.449754i	−0.0570780	+	0.998370i	$0.518178\pi$
$822$		0			0
$823$	−46.9519	−	12.5807i	−1.63664	−	0.438536i	−0.680811	−	0.732459i	$-0.738373\pi$
$823$	−46.9519	−	12.5807i	−1.63664	−	0.438536i	−0.955829	+	0.293923i	$0.905039\pi$
$824$	−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	−0.0486816	−	0.998814i	$-0.515502\pi$
$827$	27.3235	−	27.3235i	0.950133	−	0.950133i	0.998814	+	0.0486816i	$0.0155020\pi$
$828$		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
$832$	0.896575	−	3.34607i	0.0310832	−	0.116004i
$833$	−6.67700	+	24.9189i	−0.231344	+	0.863389i
$834$		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	$-0.947245\pi$
$839$	−10.1459	+	17.5732i	−0.350275	+	0.606695i	0.636022	+	0.771671i	$0.280579\pi$
$840$		0			0
$841$	9.55051	+	16.5420i	0.329328	+	0.570413i
$842$	14.3913	+	3.85614i	0.495957	+	0.132891i
$843$		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
$848$	9.02993	−	2.41956i	0.310089	−	0.0830881i
$849$		0			0
$850$		0			0
$851$	3.67423	−	2.12132i	0.125951	−	0.0727179i
$852$		0			0
$853$	9.60723	+	35.8547i	0.328945	+	1.22764i	0.910286	+	0.413980i	$0.135862\pi$
$853$	9.60723	+	35.8547i	0.328945	+	1.22764i	−0.581340	+	0.813660i	$0.697472\pi$
$854$	5.97469			0.204450
$855$		0			0
$856$	−5.24745			−0.179354
$857$	0.982984	+	3.66855i	0.0335781	+	0.125315i	0.980680	−	0.195616i	$-0.0626707\pi$
$857$	0.982984	+	3.66855i	0.0335781	+	0.125315i	−0.947102	+	0.320932i	$0.896004\pi$
$858$		0			0
$859$	2.16064	−	1.24745i	0.0737202	−	0.0425624i	−0.462687	−	0.886522i	$-0.653115\pi$
$859$	2.16064	−	1.24745i	0.0737202	−	0.0425624i	0.536407	+	0.843959i	$0.319781\pi$
$860$		0			0
$861$		0			0
$862$	15.0263	−	4.02628i	0.511797	−	0.137136i
$863$	−27.7842	−	27.7842i	−0.945787	−	0.945787i	0.0528175	−	0.998604i	$-0.483180\pi$
$863$	−27.7842	−	27.7842i	−0.945787	−	0.945787i	−0.998604	+	0.0528175i	$0.983180\pi$
$864$		0			0
$865$		0			0
$866$	−10.4722	−	6.04612i	−0.355860	−	0.205456i
$867$		0			0
$868$	−4.71209	−	1.26260i	−0.159939	−	0.0428555i
$869$	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
$877$	−2.10558	+	7.85813i	−0.0711004	+	0.265350i	−0.992321	−	0.123691i	$-0.960527\pi$
$877$	−2.10558	+	7.85813i	−0.0711004	+	0.265350i	0.921220	+	0.389041i	$0.127194\pi$
$878$	2.64048	−	9.85441i	0.0891120	−	0.332570i
$879$		0			0
$880$		0			0
$881$			58.3006i			1.96420i	0.188368	+	0.982098i	$0.439680\pi$
$881$			58.3006i			1.96420i	−0.188368	+	0.982098i	$0.560320\pi$
$882$		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
$887$	26.8508	+	7.19464i	0.901561	+	0.241572i	0.679686	−	0.733503i	$-0.262116\pi$
$887$	26.8508	+	7.19464i	0.901561	+	0.241572i	0.221874	+	0.975075i	$0.428783\pi$
$888$		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
$893$	−13.4791	+	3.61171i	−0.451061	+	0.120861i
$894$		0			0
$895$		0			0
$896$	−0.949490	+	0.548188i	−0.0317202	+	0.0183137i
$897$		0			0
$898$	5.61793	+	20.9664i	0.187473	+	0.699658i
$899$	13.9993			0.466902
$900$		0			0
$901$	41.5959			1.38576
$902$	6.39204	+	23.8554i	0.212832	+	0.794298i
$903$		0			0
$904$	11.9494	−	6.89898i	0.397431	−	0.229457i
$905$		0			0
$906$		0			0
$907$	6.38512	−	1.71089i	0.212015	−	0.0568091i	−0.151248	−	0.988496i	$-0.548329\pi$
$907$	6.38512	−	1.71089i	0.212015	−	0.0568091i	0.363263	+	0.931687i	$0.381663\pi$
$908$	10.6780	+	10.6780i	0.354363	+	0.354363i
$909$		0			0
$910$		0			0
$911$	−6.12372	−	3.53553i	−0.202888	−	0.117137i	0.395114	−	0.918632i	$-0.370705\pi$
$911$	−6.12372	−	3.53553i	−0.202888	−	0.117137i	−0.598002	+	0.801495i	$0.704038\pi$
$912$		0			0
$913$	−3.34607	−	0.896575i	−0.110739	−	0.0296723i
$914$	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
$922$	4.89419	−	18.2654i	0.161182	−	0.601538i
$923$	−0.569930	+	2.12701i	−0.0187595	+	0.0700113i
$924$		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	0.143765	+	0.989612i	$0.454079\pi$
$929$	−23.9309	+	41.4495i	−0.785147	+	1.35991i	−0.928912	+	0.370302i	$0.879254\pi$
$930$		0			0
$931$	−4.49490	−	7.78539i	−0.147314	−	0.255156i
$932$	−15.0206	−	4.02477i	−0.492017	−	0.131836i
$933$		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	$0.653691\pi$
$938$	−4.00514	+	1.07317i	−0.130773	+	0.0350404i
$939$		0			0
$940$		0			0
$941$	−5.47730	+	3.16232i	−0.178555	+	0.103089i	−0.586613	−	0.809867i	$-0.699539\pi$
$941$	−5.47730	+	3.16232i	−0.178555	+	0.103089i	0.408059	+	0.912956i	$0.366206\pi$
$942$		0			0
$943$	−1.01581	−	3.79107i	−0.0330795	−	0.123454i
$944$	11.8065			0.384269
$945$		0			0
$946$	21.7980			0.708713
$947$	10.6233	+	39.6468i	0.345212	+	1.28835i	0.892364	+	0.451316i	$0.149045\pi$
$947$	10.6233	+	39.6468i	0.345212	+	1.28835i	−0.547152	+	0.837033i	$0.684288\pi$
$948$		0			0
$949$	12.2993	−	7.10102i	0.399253	−	0.230509i
$950$		0			0
$951$		0			0
$952$	−4.71209	+	1.26260i	−0.152720	+	0.0409211i
$953$	19.6561	+	19.6561i	0.636724	+	0.636724i	0.949746	−	0.313022i	$-0.101341\pi$
$953$	19.6561	+	19.6561i	0.636724	+	0.636724i	−0.313022	+	0.949746i	$0.601341\pi$
$954$		0			0
$955$		0			0
$956$	14.6969	+	8.48528i	0.475333	+	0.274434i
$957$		0			0
$958$	−6.83013	−	1.83013i	−0.220671	−	0.0591287i
$959$	−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
$967$	13.0577	−	48.7319i	0.419907	−	1.56711i	−0.354894	−	0.934907i	$-0.615483\pi$
$967$	13.0577	−	48.7319i	0.419907	−	1.56711i	0.774800	−	0.632206i	$-0.217850\pi$
$968$	7.40117	−	27.6215i	0.237883	−	0.887790i
$969$		0			0
$970$		0			0
$971$		−	49.2117i		−	1.57928i	−0.613570	−	0.789640i	$-0.710267\pi$
$971$		−	49.2117i		−	1.57928i	0.613570	−	0.789640i	$-0.289733\pi$
$972$		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
$977$	41.0469	+	10.9985i	1.31321	+	0.351873i	0.846429	−	0.532502i	$-0.178748\pi$
$977$	41.0469	+	10.9985i	1.31321	+	0.351873i	0.466778	+	0.884374i	$0.345415\pi$
$978$		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
$983$	45.2034	−	12.1122i	1.44176	−	0.386319i	0.548612	−	0.836077i	$-0.315156\pi$
$983$	45.2034	−	12.1122i	1.44176	−	0.386319i	0.893151	+	0.449757i	$0.148490\pi$
$984$		0			0
$985$		0			0
$986$	12.1237	−	6.99964i	0.386098	−	0.222914i
$987$		0			0
$988$	1.39015	+	5.18811i	0.0442265	+	0.165056i
$989$	−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	$0.857342\pi$
$992$	1.15161	+	4.29788i	0.0365637	+	0.136458i
$993$		0			0
$994$	0.603566	−	0.348469i	0.0191440	−	0.0110528i
$995$		0			0
$996$		0			0
$997$	−39.9528	+	10.7053i	−1.26532	+	0.339041i	−0.828235	−	0.560381i	$-0.810655\pi$
$997$	−39.9528	+	10.7053i	−1.26532	+	0.339041i	−0.437082	+	0.899422i	$0.643988\pi$
$998$	−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.1007.2		8
3.2	odd	2		450.2.p.a.407.1		8
5.2	odd	4		270.2.m.a.143.1		8
5.3	odd	4	inner	1350.2.q.g.143.2		8
5.4	even	2		270.2.m.a.197.1		8
9.4	even	3		450.2.p.a.257.1		8
9.5	odd	6	inner	1350.2.q.g.557.2		8
15.2	even	4		90.2.l.a.83.2	yes	8
15.8	even	4		450.2.p.a.443.1		8
15.14	odd	2		90.2.l.a.47.2	yes	8
45.2	even	12		810.2.f.b.323.2		8
45.4	even	6		90.2.l.a.77.2	yes	8
45.7	odd	12		810.2.f.b.323.3		8
45.13	odd	12		450.2.p.a.293.1		8
45.14	odd	6		270.2.m.a.17.1		8
45.22	odd	12		90.2.l.a.23.2	✓	8
45.23	even	12	inner	1350.2.q.g.1043.2		8
45.29	odd	6		810.2.f.b.647.4		8
45.32	even	12		270.2.m.a.233.1		8
45.34	even	6		810.2.f.b.647.1		8
60.47	odd	4		720.2.cu.a.353.2		8
60.59	even	2		720.2.cu.a.497.2		8
180.67	even	12		720.2.cu.a.113.2		8
180.139	odd	6		720.2.cu.a.257.2		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
90.2.l.a.23.2	✓	8	45.22	odd	12
90.2.l.a.47.2	yes	8	15.14	odd	2
90.2.l.a.77.2	yes	8	45.4	even	6
90.2.l.a.83.2	yes	8	15.2	even	4
270.2.m.a.17.1		8	45.14	odd	6
270.2.m.a.143.1		8	5.2	odd	4
270.2.m.a.197.1		8	5.4	even	2
270.2.m.a.233.1		8	45.32	even	12
450.2.p.a.257.1		8	9.4	even	3
450.2.p.a.293.1		8	45.13	odd	12
450.2.p.a.407.1		8	3.2	odd	2
450.2.p.a.443.1		8	15.8	even	4
720.2.cu.a.113.2		8	180.67	even	12
720.2.cu.a.257.2		8	180.139	odd	6
720.2.cu.a.353.2		8	60.47	odd	4
720.2.cu.a.497.2		8	60.59	even	2
810.2.f.b.323.2		8	45.2	even	12
810.2.f.b.323.3		8	45.7	odd	12
810.2.f.b.647.1		8	45.34	even	6
810.2.f.b.647.4		8	45.29	odd	6
1350.2.q.g.143.2		8	5.3	odd	4	inner
1350.2.q.g.557.2		8	9.5	odd	6	inner
1350.2.q.g.1007.2		8	1.1	even	1	trivial
1350.2.q.g.1043.2		8	45.23	even	12	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)

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