LMFDB - Embedded newform 2790.2.d.g.559.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2790,2,Mod(559,2790)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2790, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1, 0]))

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

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

chi := DirichletCharacter("2790.559");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2790 = 2 \cdot 3^{2} \cdot 5 \cdot 31 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2790.d (of order $2$, degree $1$, 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:	$22.2782621639$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(i)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 1 $ x^2 + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 930)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			559.2
Root			$-1.00000i$ of defining polynomial
Character	$\chi$	$=$	2790.559
Dual form			2790.2.d.g.559.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{2} -1.00000 q^{4} +(1.00000 - 2.00000i) q^{5} -1.00000i q^{7} -1.00000i q^{8} +O(q^{10})$	\(q+1.00000i q^{2} -1.00000 q^{4} +(1.00000 - 2.00000i) q^{5} -1.00000i q^{7} -1.00000i q^{8} +(2.00000 + 1.00000i) q^{10} +5.00000 q^{11} -4.00000i q^{13} +1.00000 q^{14} +1.00000 q^{16} +5.00000 q^{19} +(-1.00000 + 2.00000i) q^{20} +5.00000i q^{22} +9.00000i q^{23} +(-3.00000 - 4.00000i) q^{25} +4.00000 q^{26} +1.00000i q^{28} -2.00000 q^{29} +1.00000 q^{31} +1.00000i q^{32} +(-2.00000 - 1.00000i) q^{35} -8.00000i q^{37} +5.00000i q^{38} +(-2.00000 - 1.00000i) q^{40} -6.00000 q^{41} +1.00000i q^{43} -5.00000 q^{44} -9.00000 q^{46} -12.0000i q^{47} +6.00000 q^{49} +(4.00000 - 3.00000i) q^{50} +4.00000i q^{52} +13.0000i q^{53} +(5.00000 - 10.0000i) q^{55} -1.00000 q^{56} -2.00000i q^{58} +10.0000 q^{59} -14.0000 q^{61} +1.00000i q^{62} -1.00000 q^{64} +(-8.00000 - 4.00000i) q^{65} -14.0000i q^{67} +(1.00000 - 2.00000i) q^{70} +9.00000 q^{71} -9.00000i q^{73} +8.00000 q^{74} -5.00000 q^{76} -5.00000i q^{77} -5.00000 q^{79} +(1.00000 - 2.00000i) q^{80} -6.00000i q^{82} -6.00000i q^{83} -1.00000 q^{86} -5.00000i q^{88} +3.00000 q^{89} -4.00000 q^{91} -9.00000i q^{92} +12.0000 q^{94} +(5.00000 - 10.0000i) q^{95} +18.0000i q^{97} +6.00000i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{4} + 2 q^{5}+O(q^{10}) $ 2 * q - 2 * q^4 + 2 * q^5	$ 2 q - 2 q^{4} + 2 q^{5} + 4 q^{10} + 10 q^{11} + 2 q^{14} + 2 q^{16} + 10 q^{19} - 2 q^{20} - 6 q^{25} + 8 q^{26} - 4 q^{29} + 2 q^{31} - 4 q^{35} - 4 q^{40} - 12 q^{41} - 10 q^{44} - 18 q^{46} + 12 q^{49} + 8 q^{50} + 10 q^{55} - 2 q^{56} + 20 q^{59} - 28 q^{61} - 2 q^{64} - 16 q^{65} + 2 q^{70} + 18 q^{71} + 16 q^{74} - 10 q^{76} - 10 q^{79} + 2 q^{80} - 2 q^{86} + 6 q^{89} - 8 q^{91} + 24 q^{94} + 10 q^{95}+O(q^{100}) $ 2 * q - 2 * q^4 + 2 * q^5 + 4 * q^10 + 10 * q^11 + 2 * q^14 + 2 * q^16 + 10 * q^19 - 2 * q^20 - 6 * q^25 + 8 * q^26 - 4 * q^29 + 2 * q^31 - 4 * q^35 - 4 * q^40 - 12 * q^41 - 10 * q^44 - 18 * q^46 + 12 * q^49 + 8 * q^50 + 10 * q^55 - 2 * q^56 + 20 * q^59 - 28 * q^61 - 2 * q^64 - 16 * q^65 + 2 * q^70 + 18 * q^71 + 16 * q^74 - 10 * q^76 - 10 * q^79 + 2 * q^80 - 2 * q^86 + 6 * q^89 - 8 * q^91 + 24 * q^94 + 10 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2790\mathbb{Z}\right)^\times$.

$n$	$1117$	$1801$	$2171$
$\chi(n)$	$-1$	$1$	$1$

Coefficient data

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

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

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

$2$			1.00000i			0.707107i
$2$			1.00000i			0.707107i
$3$		0			0
$3$		0			0
$4$	−1.00000			−0.500000
$5$	1.00000	−	2.00000i	0.447214	−	0.894427i
$5$	1.00000	−	2.00000i	0.447214	−	0.894427i
$6$		0			0
$7$		−	1.00000i		−	0.377964i	−0.981981	−	0.188982i	$-0.939481\pi$
$7$		−	1.00000i		−	0.377964i	0.981981	−	0.188982i	$-0.0605189\pi$
$8$		−	1.00000i		−	0.353553i
$9$		0			0
$10$	2.00000	+	1.00000i	0.632456	+	0.316228i
$11$	5.00000			1.50756			0.753778	−	0.657129i	$-0.228229\pi$
$11$	5.00000			1.50756			0.753778	+	0.657129i	$0.228229\pi$
$12$		0			0
$13$		−	4.00000i		−	1.10940i	−0.832050	−	0.554700i	$-0.812833\pi$
$13$		−	4.00000i		−	1.10940i	0.832050	−	0.554700i	$-0.187167\pi$
$14$	1.00000			0.267261
$15$		0			0
$16$	1.00000			0.250000
$17$		0			0		1.00000			$0$
$17$		0			0		−1.00000			$\pi$
$18$		0			0
$19$	5.00000			1.14708			0.573539	−	0.819178i	$-0.305570\pi$
$19$	5.00000			1.14708			0.573539	+	0.819178i	$0.305570\pi$
$20$	−1.00000	+	2.00000i	−0.223607	+	0.447214i
$21$		0			0
$22$			5.00000i			1.06600i
$23$			9.00000i			1.87663i	0.345782	+	0.938315i	$0.387614\pi$
$23$			9.00000i			1.87663i	−0.345782	+	0.938315i	$0.612386\pi$
$24$		0			0
$25$	−3.00000	−	4.00000i	−0.600000	−	0.800000i
$26$	4.00000			0.784465
$27$		0			0
$28$			1.00000i			0.188982i
$29$	−2.00000			−0.371391			−0.185695	−	0.982607i	$-0.559454\pi$
$29$	−2.00000			−0.371391			−0.185695	+	0.982607i	$0.559454\pi$
$30$		0			0
$31$	1.00000			0.179605
$31$	1.00000			0.179605
$32$			1.00000i			0.176777i
$33$		0			0
$34$		0			0
$35$	−2.00000	−	1.00000i	−0.338062	−	0.169031i
$36$		0			0
$37$		−	8.00000i		−	1.31519i	−0.753371	−	0.657596i	$-0.771573\pi$
$37$		−	8.00000i		−	1.31519i	0.753371	−	0.657596i	$-0.228427\pi$
$38$			5.00000i			0.811107i
$39$		0			0
$40$	−2.00000	−	1.00000i	−0.316228	−	0.158114i
$41$	−6.00000			−0.937043			−0.468521	−	0.883452i	$-0.655213\pi$
$41$	−6.00000			−0.937043			−0.468521	+	0.883452i	$0.655213\pi$
$42$		0			0
$43$			1.00000i			0.152499i	0.997089	+	0.0762493i	$0.0242945\pi$
$43$			1.00000i			0.152499i	−0.997089	+	0.0762493i	$0.975706\pi$
$44$	−5.00000			−0.753778
$45$		0			0
$46$	−9.00000			−1.32698
$47$		−	12.0000i		−	1.75038i	−0.483779	−	0.875190i	$-0.660736\pi$
$47$		−	12.0000i		−	1.75038i	0.483779	−	0.875190i	$-0.339264\pi$
$48$		0			0
$49$	6.00000			0.857143
$50$	4.00000	−	3.00000i	0.565685	−	0.424264i
$51$		0			0
$52$			4.00000i			0.554700i
$53$			13.0000i			1.78569i	0.450367	+	0.892844i	$0.351293\pi$
$53$			13.0000i			1.78569i	−0.450367	+	0.892844i	$0.648707\pi$
$54$		0			0
$55$	5.00000	−	10.0000i	0.674200	−	1.34840i
$56$	−1.00000			−0.133631
$57$		0			0
$58$		−	2.00000i		−	0.262613i
$59$	10.0000			1.30189			0.650945	−	0.759125i	$-0.274373\pi$
$59$	10.0000			1.30189			0.650945	+	0.759125i	$0.274373\pi$
$60$		0			0
$61$	−14.0000			−1.79252			−0.896258	−	0.443533i	$-0.853725\pi$
$61$	−14.0000			−1.79252			−0.896258	+	0.443533i	$0.853725\pi$
$62$			1.00000i			0.127000i
$63$		0			0
$64$	−1.00000			−0.125000
$65$	−8.00000	−	4.00000i	−0.992278	−	0.496139i
$66$		0			0
$67$		−	14.0000i		−	1.71037i	−0.518321	−	0.855186i	$-0.673443\pi$
$67$		−	14.0000i		−	1.71037i	0.518321	−	0.855186i	$-0.326557\pi$
$68$		0			0
$69$		0			0
$70$	1.00000	−	2.00000i	0.119523	−	0.239046i
$71$	9.00000			1.06810			0.534052	−	0.845452i	$-0.320669\pi$
$71$	9.00000			1.06810			0.534052	+	0.845452i	$0.320669\pi$
$72$		0			0
$73$		−	9.00000i		−	1.05337i	−0.850060	−	0.526685i	$-0.823435\pi$
$73$		−	9.00000i		−	1.05337i	0.850060	−	0.526685i	$-0.176565\pi$
$74$	8.00000			0.929981
$75$		0			0
$76$	−5.00000			−0.573539
$77$		−	5.00000i		−	0.569803i
$78$		0			0
$79$	−5.00000			−0.562544			−0.281272	−	0.959628i	$-0.590756\pi$
$79$	−5.00000			−0.562544			−0.281272	+	0.959628i	$0.590756\pi$
$80$	1.00000	−	2.00000i	0.111803	−	0.223607i
$81$		0			0
$82$		−	6.00000i		−	0.662589i
$83$		−	6.00000i		−	0.658586i	−0.944228	−	0.329293i	$-0.893190\pi$
$83$		−	6.00000i		−	0.658586i	0.944228	−	0.329293i	$-0.106810\pi$
$84$		0			0
$85$		0			0
$86$	−1.00000			−0.107833
$87$		0			0
$88$		−	5.00000i		−	0.533002i
$89$	3.00000			0.317999			0.159000	−	0.987279i	$-0.449173\pi$
$89$	3.00000			0.317999			0.159000	+	0.987279i	$0.449173\pi$
$90$		0			0
$91$	−4.00000			−0.419314
$92$		−	9.00000i		−	0.938315i
$93$		0			0
$94$	12.0000			1.23771
$95$	5.00000	−	10.0000i	0.512989	−	1.02598i
$96$		0			0
$97$			18.0000i			1.82762i	0.406138	+	0.913812i	$0.366875\pi$
$97$			18.0000i			1.82762i	−0.406138	+	0.913812i	$0.633125\pi$
$98$			6.00000i			0.606092i
$99$		0			0
$100$	3.00000	+	4.00000i	0.300000	+	0.400000i
$101$	7.00000			0.696526			0.348263	−	0.937397i	$-0.386772\pi$
$101$	7.00000			0.696526			0.348263	+	0.937397i	$0.386772\pi$
$102$		0			0
$103$			8.00000i			0.788263i	0.919054	+	0.394132i	$0.128955\pi$
$103$			8.00000i			0.788263i	−0.919054	+	0.394132i	$0.871045\pi$
$104$	−4.00000			−0.392232
$105$		0			0
$106$	−13.0000			−1.26267
$107$			11.0000i			1.06341i	0.846930	+	0.531705i	$0.178449\pi$
$107$			11.0000i			1.06341i	−0.846930	+	0.531705i	$0.821551\pi$
$108$		0			0
$109$	2.00000			0.191565			0.0957826	−	0.995402i	$-0.469465\pi$
$109$	2.00000			0.191565			0.0957826	+	0.995402i	$0.469465\pi$
$110$	10.0000	+	5.00000i	0.953463	+	0.476731i
$111$		0			0
$112$		−	1.00000i		−	0.0944911i
$113$		−	13.0000i		−	1.22294i	−0.791269	−	0.611469i	$-0.790579\pi$
$113$		−	13.0000i		−	1.22294i	0.791269	−	0.611469i	$-0.209421\pi$
$114$		0			0
$115$	18.0000	+	9.00000i	1.67851	+	0.839254i
$116$	2.00000			0.185695
$117$		0			0
$118$			10.0000i			0.920575i
$119$		0			0
$120$		0			0
$121$	14.0000			1.27273
$122$		−	14.0000i		−	1.26750i
$123$		0			0
$124$	−1.00000			−0.0898027
$125$	−11.0000	+	2.00000i	−0.983870	+	0.178885i
$126$		0			0
$127$		−	20.0000i		−	1.77471i	−0.461084	−	0.887357i	$-0.652539\pi$
$127$		−	20.0000i		−	1.77471i	0.461084	−	0.887357i	$-0.347461\pi$
$128$		−	1.00000i		−	0.0883883i
$129$		0			0
$130$	4.00000	−	8.00000i	0.350823	−	0.701646i
$131$	6.00000			0.524222			0.262111	−	0.965038i	$-0.415581\pi$
$131$	6.00000			0.524222			0.262111	+	0.965038i	$0.415581\pi$
$132$		0			0
$133$		−	5.00000i		−	0.433555i
$134$	14.0000			1.20942
$135$		0			0
$136$		0			0
$137$			2.00000i			0.170872i	0.996344	+	0.0854358i	$0.0272282\pi$
$137$			2.00000i			0.170872i	−0.996344	+	0.0854358i	$0.972772\pi$
$138$		0			0
$139$	−16.0000			−1.35710			−0.678551	−	0.734553i	$-0.737392\pi$
$139$	−16.0000			−1.35710			−0.678551	+	0.734553i	$0.737392\pi$
$140$	2.00000	+	1.00000i	0.169031	+	0.0845154i
$141$		0			0
$142$			9.00000i			0.755263i
$143$		−	20.0000i		−	1.67248i
$144$		0			0
$145$	−2.00000	+	4.00000i	−0.166091	+	0.332182i
$146$	9.00000			0.744845
$147$		0			0
$148$			8.00000i			0.657596i
$149$	7.00000			0.573462			0.286731	−	0.958011i	$-0.407431\pi$
$149$	7.00000			0.573462			0.286731	+	0.958011i	$0.407431\pi$
$150$		0			0
$151$		0			0			−	1.00000i	$-0.5\pi$
$151$		0			0				1.00000i	$0.5\pi$
$152$		−	5.00000i		−	0.405554i
$153$		0			0
$154$	5.00000			0.402911
$155$	1.00000	−	2.00000i	0.0803219	−	0.160644i
$156$		0			0
$157$		−	13.0000i		−	1.03751i	−0.854922	−	0.518756i	$-0.826395\pi$
$157$		−	13.0000i		−	1.03751i	0.854922	−	0.518756i	$-0.173605\pi$
$158$		−	5.00000i		−	0.397779i
$159$		0			0
$160$	2.00000	+	1.00000i	0.158114	+	0.0790569i
$161$	9.00000			0.709299
$162$		0			0
$163$			4.00000i			0.313304i	0.987654	+	0.156652i	$0.0500701\pi$
$163$			4.00000i			0.313304i	−0.987654	+	0.156652i	$0.949930\pi$
$164$	6.00000			0.468521
$165$		0			0
$166$	6.00000			0.465690
$167$		−	9.00000i		−	0.696441i	−0.937413	−	0.348220i	$-0.886786\pi$
$167$		−	9.00000i		−	0.696441i	0.937413	−	0.348220i	$-0.113214\pi$
$168$		0			0
$169$	−3.00000			−0.230769
$170$		0			0
$171$		0			0
$172$		−	1.00000i		−	0.0762493i
$173$			6.00000i			0.456172i	0.973641	+	0.228086i	$0.0732467\pi$
$173$			6.00000i			0.456172i	−0.973641	+	0.228086i	$0.926753\pi$
$174$		0			0
$175$	−4.00000	+	3.00000i	−0.302372	+	0.226779i
$176$	5.00000			0.376889
$177$		0			0
$178$			3.00000i			0.224860i
$179$	−16.0000			−1.19590			−0.597948	−	0.801535i	$-0.704017\pi$
$179$	−16.0000			−1.19590			−0.597948	+	0.801535i	$0.704017\pi$
$180$		0			0
$181$	17.0000			1.26360			0.631800	−	0.775131i	$-0.282316\pi$
$181$	17.0000			1.26360			0.631800	+	0.775131i	$0.282316\pi$
$182$		−	4.00000i		−	0.296500i
$183$		0			0
$184$	9.00000			0.663489
$185$	−16.0000	−	8.00000i	−1.17634	−	0.588172i
$186$		0			0
$187$		0			0
$188$			12.0000i			0.875190i
$189$		0			0
$190$	10.0000	+	5.00000i	0.725476	+	0.362738i
$191$	−8.00000			−0.578860			−0.289430	−	0.957199i	$-0.593466\pi$
$191$	−8.00000			−0.578860			−0.289430	+	0.957199i	$0.593466\pi$
$192$		0			0
$193$		−	22.0000i		−	1.58359i	−0.610784	−	0.791797i	$-0.709146\pi$
$193$		−	22.0000i		−	1.58359i	0.610784	−	0.791797i	$-0.290854\pi$
$194$	−18.0000			−1.29232
$195$		0			0
$196$	−6.00000			−0.428571
$197$		−	10.0000i		−	0.712470i	−0.934396	−	0.356235i	$-0.884060\pi$
$197$		−	10.0000i		−	0.712470i	0.934396	−	0.356235i	$-0.115940\pi$
$198$		0			0
$199$	11.0000			0.779769			0.389885	−	0.920864i	$-0.372515\pi$
$199$	11.0000			0.779769			0.389885	+	0.920864i	$0.372515\pi$
$200$	−4.00000	+	3.00000i	−0.282843	+	0.212132i
$201$		0			0
$202$			7.00000i			0.492518i
$203$			2.00000i			0.140372i
$204$		0			0
$205$	−6.00000	+	12.0000i	−0.419058	+	0.838116i
$206$	−8.00000			−0.557386
$207$		0			0
$208$		−	4.00000i		−	0.277350i
$209$	25.0000			1.72929
$210$		0			0
$211$	−13.0000			−0.894957			−0.447478	−	0.894295i	$-0.647678\pi$
$211$	−13.0000			−0.894957			−0.447478	+	0.894295i	$0.647678\pi$
$212$		−	13.0000i		−	0.892844i
$213$		0			0
$214$	−11.0000			−0.751945
$215$	2.00000	+	1.00000i	0.136399	+	0.0681994i
$216$		0			0
$217$		−	1.00000i		−	0.0678844i
$218$			2.00000i			0.135457i
$219$		0			0
$220$	−5.00000	+	10.0000i	−0.337100	+	0.674200i
$221$		0			0
$222$		0			0
$223$			16.0000i			1.07144i	0.844396	+	0.535720i	$0.179960\pi$
$223$			16.0000i			1.07144i	−0.844396	+	0.535720i	$0.820040\pi$
$224$	1.00000			0.0668153
$225$		0			0
$226$	13.0000			0.864747
$227$			15.0000i			0.995585i	0.867296	+	0.497792i	$0.165856\pi$
$227$			15.0000i			0.995585i	−0.867296	+	0.497792i	$0.834144\pi$
$228$		0			0
$229$	19.0000			1.25556			0.627778	−	0.778393i	$-0.283965\pi$
$229$	19.0000			1.25556			0.627778	+	0.778393i	$0.283965\pi$
$230$	−9.00000	+	18.0000i	−0.593442	+	1.18688i
$231$		0			0
$232$			2.00000i			0.131306i
$233$			1.00000i			0.0655122i	0.999463	+	0.0327561i	$0.0104285\pi$
$233$			1.00000i			0.0655122i	−0.999463	+	0.0327561i	$0.989572\pi$
$234$		0			0
$235$	−24.0000	−	12.0000i	−1.56559	−	0.782794i
$236$	−10.0000			−0.650945
$237$		0			0
$238$		0			0
$239$	−6.00000			−0.388108			−0.194054	−	0.980991i	$-0.562164\pi$
$239$	−6.00000			−0.388108			−0.194054	+	0.980991i	$0.562164\pi$
$240$		0			0
$241$	24.0000			1.54598			0.772988	−	0.634421i	$-0.218761\pi$
$241$	24.0000			1.54598			0.772988	+	0.634421i	$0.218761\pi$
$242$			14.0000i			0.899954i
$243$		0			0
$244$	14.0000			0.896258
$245$	6.00000	−	12.0000i	0.383326	−	0.766652i
$246$		0			0
$247$		−	20.0000i		−	1.27257i
$248$		−	1.00000i		−	0.0635001i
$249$		0			0
$250$	−2.00000	−	11.0000i	−0.126491	−	0.695701i
$251$	28.0000			1.76734			0.883672	−	0.468106i	$-0.155064\pi$
$251$	28.0000			1.76734			0.883672	+	0.468106i	$0.155064\pi$
$252$		0			0
$253$			45.0000i			2.82913i
$254$	20.0000			1.25491
$255$		0			0
$256$	1.00000			0.0625000
$257$		−	7.00000i		−	0.436648i	−0.975876	−	0.218324i	$-0.929941\pi$
$257$		−	7.00000i		−	0.436648i	0.975876	−	0.218324i	$-0.0700590\pi$
$258$		0			0
$259$	−8.00000			−0.497096
$260$	8.00000	+	4.00000i	0.496139	+	0.248069i
$261$		0			0
$262$			6.00000i			0.370681i
$263$			8.00000i			0.493301i	0.969104	+	0.246651i	$0.0793300\pi$
$263$			8.00000i			0.493301i	−0.969104	+	0.246651i	$0.920670\pi$
$264$		0			0
$265$	26.0000	+	13.0000i	1.59717	+	0.798584i
$266$	5.00000			0.306570
$267$		0			0
$268$			14.0000i			0.855186i
$269$	−22.0000			−1.34136			−0.670682	−	0.741745i	$-0.733998\pi$
$269$	−22.0000			−1.34136			−0.670682	+	0.741745i	$0.733998\pi$
$270$		0			0
$271$	−25.0000			−1.51864			−0.759321	−	0.650716i	$-0.774469\pi$
$271$	−25.0000			−1.51864			−0.759321	+	0.650716i	$0.774469\pi$
$272$		0			0
$273$		0			0
$274$	−2.00000			−0.120824
$275$	−15.0000	−	20.0000i	−0.904534	−	1.20605i
$276$		0			0
$277$			2.00000i			0.120168i	0.998193	+	0.0600842i	$0.0191369\pi$
$277$			2.00000i			0.120168i	−0.998193	+	0.0600842i	$0.980863\pi$
$278$		−	16.0000i		−	0.959616i
$279$		0			0
$280$	−1.00000	+	2.00000i	−0.0597614	+	0.119523i
$281$	−12.0000			−0.715860			−0.357930	−	0.933748i	$-0.616517\pi$
$281$	−12.0000			−0.715860			−0.357930	+	0.933748i	$0.616517\pi$
$282$		0			0
$283$			26.0000i			1.54554i	0.634686	+	0.772770i	$0.281129\pi$
$283$			26.0000i			1.54554i	−0.634686	+	0.772770i	$0.718871\pi$
$284$	−9.00000			−0.534052
$285$		0			0
$286$	20.0000			1.18262
$287$			6.00000i			0.354169i
$288$		0			0
$289$	17.0000			1.00000
$290$	−4.00000	−	2.00000i	−0.234888	−	0.117444i
$291$		0			0
$292$			9.00000i			0.526685i
$293$			14.0000i			0.817889i	0.912559	+	0.408944i	$0.134103\pi$
$293$			14.0000i			0.817889i	−0.912559	+	0.408944i	$0.865897\pi$
$294$		0			0
$295$	10.0000	−	20.0000i	0.582223	−	1.16445i
$296$	−8.00000			−0.464991
$297$		0			0
$298$			7.00000i			0.405499i
$299$	36.0000			2.08193
$300$		0			0
$301$	1.00000			0.0576390
$302$		0			0
$303$		0			0
$304$	5.00000			0.286770
$305$	−14.0000	+	28.0000i	−0.801638	+	1.60328i
$306$		0			0
$307$			8.00000i			0.456584i	0.973593	+	0.228292i	$0.0733141\pi$
$307$			8.00000i			0.456584i	−0.973593	+	0.228292i	$0.926686\pi$
$308$			5.00000i			0.284901i
$309$		0			0
$310$	2.00000	+	1.00000i	0.113592	+	0.0567962i
$311$	8.00000			0.453638			0.226819	−	0.973937i	$-0.427167\pi$
$311$	8.00000			0.453638			0.226819	+	0.973937i	$0.427167\pi$
$312$		0			0
$313$			6.00000i			0.339140i	0.985518	+	0.169570i	$0.0542379\pi$
$313$			6.00000i			0.339140i	−0.985518	+	0.169570i	$0.945762\pi$
$314$	13.0000			0.733632
$315$		0			0
$316$	5.00000			0.281272
$317$			34.0000i			1.90963i	0.297200	+	0.954815i	$0.403947\pi$
$317$			34.0000i			1.90963i	−0.297200	+	0.954815i	$0.596053\pi$
$318$		0			0
$319$	−10.0000			−0.559893
$320$	−1.00000	+	2.00000i	−0.0559017	+	0.111803i
$321$		0			0
$322$			9.00000i			0.501550i
$323$		0			0
$324$		0			0
$325$	−16.0000	+	12.0000i	−0.887520	+	0.665640i
$326$	−4.00000			−0.221540
$327$		0			0
$328$			6.00000i			0.331295i
$329$	−12.0000			−0.661581
$330$		0			0
$331$	−24.0000			−1.31916			−0.659580	−	0.751635i	$-0.729266\pi$
$331$	−24.0000			−1.31916			−0.659580	+	0.751635i	$0.729266\pi$
$332$			6.00000i			0.329293i
$333$		0			0
$334$	9.00000			0.492458
$335$	−28.0000	−	14.0000i	−1.52980	−	0.764902i
$336$		0			0
$337$		−	2.00000i		−	0.108947i	−0.998515	−	0.0544735i	$-0.982652\pi$
$337$		−	2.00000i		−	0.108947i	0.998515	−	0.0544735i	$-0.0173480\pi$
$338$		−	3.00000i		−	0.163178i
$339$		0			0
$340$		0			0
$341$	5.00000			0.270765
$342$		0			0
$343$		−	13.0000i		−	0.701934i
$344$	1.00000			0.0539164
$345$		0			0
$346$	−6.00000			−0.322562
$347$		−	18.0000i		−	0.966291i	−0.875540	−	0.483145i	$-0.839494\pi$
$347$		−	18.0000i		−	0.966291i	0.875540	−	0.483145i	$-0.160506\pi$
$348$		0			0
$349$	−4.00000			−0.214115			−0.107058	−	0.994253i	$-0.534143\pi$
$349$	−4.00000			−0.214115			−0.107058	+	0.994253i	$0.534143\pi$
$350$	−3.00000	−	4.00000i	−0.160357	−	0.213809i
$351$		0			0
$352$			5.00000i			0.266501i
$353$			4.00000i			0.212899i	0.994318	+	0.106449i	$0.0339482\pi$
$353$			4.00000i			0.212899i	−0.994318	+	0.106449i	$0.966052\pi$
$354$		0			0
$355$	9.00000	−	18.0000i	0.477670	−	0.955341i
$356$	−3.00000			−0.159000
$357$		0			0
$358$		−	16.0000i		−	0.845626i
$359$	−9.00000			−0.475002			−0.237501	−	0.971387i	$-0.576328\pi$
$359$	−9.00000			−0.475002			−0.237501	+	0.971387i	$0.576328\pi$
$360$		0			0
$361$	6.00000			0.315789
$362$			17.0000i			0.893500i
$363$		0			0
$364$	4.00000			0.209657
$365$	−18.0000	−	9.00000i	−0.942163	−	0.471082i
$366$		0			0
$367$		−	18.0000i		−	0.939592i	−0.882775	−	0.469796i	$-0.844327\pi$
$367$		−	18.0000i		−	0.939592i	0.882775	−	0.469796i	$-0.155673\pi$
$368$			9.00000i			0.469157i
$369$		0			0
$370$	8.00000	−	16.0000i	0.415900	−	0.831800i
$371$	13.0000			0.674926
$372$		0			0
$373$			1.00000i			0.0517780i	0.999665	+	0.0258890i	$0.00824165\pi$
$373$			1.00000i			0.0517780i	−0.999665	+	0.0258890i	$0.991758\pi$
$374$		0			0
$375$		0			0
$376$	−12.0000			−0.618853
$377$			8.00000i			0.412021i
$378$		0			0
$379$	11.0000			0.565032			0.282516	−	0.959263i	$-0.408831\pi$
$379$	11.0000			0.565032			0.282516	+	0.959263i	$0.408831\pi$
$380$	−5.00000	+	10.0000i	−0.256495	+	0.512989i
$381$		0			0
$382$		−	8.00000i		−	0.409316i
$383$		−	12.0000i		−	0.613171i	−0.951843	−	0.306586i	$-0.900813\pi$
$383$		−	12.0000i		−	0.613171i	0.951843	−	0.306586i	$-0.0991866\pi$
$384$		0			0
$385$	−10.0000	−	5.00000i	−0.509647	−	0.254824i
$386$	22.0000			1.11977
$387$		0			0
$388$		−	18.0000i		−	0.913812i
$389$	−16.0000			−0.811232			−0.405616	−	0.914044i	$-0.632943\pi$
$389$	−16.0000			−0.811232			−0.405616	+	0.914044i	$0.632943\pi$
$390$		0			0
$391$		0			0
$392$		−	6.00000i		−	0.303046i
$393$		0			0
$394$	10.0000			0.503793
$395$	−5.00000	+	10.0000i	−0.251577	+	0.503155i
$396$		0			0
$397$			3.00000i			0.150566i	0.997162	+	0.0752828i	$0.0239860\pi$
$397$			3.00000i			0.150566i	−0.997162	+	0.0752828i	$0.976014\pi$
$398$			11.0000i			0.551380i
$399$		0			0
$400$	−3.00000	−	4.00000i	−0.150000	−	0.200000i
$401$	25.0000			1.24844			0.624220	−	0.781248i	$-0.285417\pi$
$401$	25.0000			1.24844			0.624220	+	0.781248i	$0.285417\pi$
$402$		0			0
$403$		−	4.00000i		−	0.199254i
$404$	−7.00000			−0.348263
$405$		0			0
$406$	−2.00000			−0.0992583
$407$		−	40.0000i		−	1.98273i
$408$		0			0
$409$	28.0000			1.38451			0.692255	−	0.721653i	$-0.256617\pi$
$409$	28.0000			1.38451			0.692255	+	0.721653i	$0.256617\pi$
$410$	−12.0000	−	6.00000i	−0.592638	−	0.296319i
$411$		0			0
$412$		−	8.00000i		−	0.394132i
$413$		−	10.0000i		−	0.492068i
$414$		0			0
$415$	−12.0000	−	6.00000i	−0.589057	−	0.294528i
$416$	4.00000			0.196116
$417$		0			0
$418$			25.0000i			1.22279i
$419$	30.0000			1.46560			0.732798	−	0.680446i	$-0.238214\pi$
$419$	30.0000			1.46560			0.732798	+	0.680446i	$0.238214\pi$
$420$		0			0
$421$	−26.0000			−1.26716			−0.633581	−	0.773676i	$-0.718416\pi$
$421$	−26.0000			−1.26716			−0.633581	+	0.773676i	$0.718416\pi$
$422$		−	13.0000i		−	0.632830i
$423$		0			0
$424$	13.0000			0.631336
$425$		0			0
$426$		0			0
$427$			14.0000i			0.677507i
$428$		−	11.0000i		−	0.531705i
$429$		0			0
$430$	−1.00000	+	2.00000i	−0.0482243	+	0.0964486i
$431$	32.0000			1.54139			0.770693	−	0.637207i	$-0.219910\pi$
$431$	32.0000			1.54139			0.770693	+	0.637207i	$0.219910\pi$
$432$		0			0
$433$		−	11.0000i		−	0.528626i	−0.964437	−	0.264313i	$-0.914855\pi$
$433$		−	11.0000i		−	0.528626i	0.964437	−	0.264313i	$-0.0851452\pi$
$434$	1.00000			0.0480015
$435$		0			0
$436$	−2.00000			−0.0957826
$437$			45.0000i			2.15264i
$438$		0			0
$439$	−20.0000			−0.954548			−0.477274	−	0.878755i	$-0.658375\pi$
$439$	−20.0000			−0.954548			−0.477274	+	0.878755i	$0.658375\pi$
$440$	−10.0000	−	5.00000i	−0.476731	−	0.238366i
$441$		0			0
$442$		0			0
$443$		−	13.0000i		−	0.617649i	−0.951119	−	0.308824i	$-0.900064\pi$
$443$		−	13.0000i		−	0.617649i	0.951119	−	0.308824i	$-0.0999355\pi$
$444$		0			0
$445$	3.00000	−	6.00000i	0.142214	−	0.284427i
$446$	−16.0000			−0.757622
$447$		0			0
$448$			1.00000i			0.0472456i
$449$	6.00000			0.283158			0.141579	−	0.989927i	$-0.454782\pi$
$449$	6.00000			0.283158			0.141579	+	0.989927i	$0.454782\pi$
$450$		0			0
$451$	−30.0000			−1.41264
$452$			13.0000i			0.611469i
$453$		0			0
$454$	−15.0000			−0.703985
$455$	−4.00000	+	8.00000i	−0.187523	+	0.375046i
$456$		0			0
$457$			22.0000i			1.02912i	0.857455	+	0.514558i	$0.172044\pi$
$457$			22.0000i			1.02912i	−0.857455	+	0.514558i	$0.827956\pi$
$458$			19.0000i			0.887812i
$459$		0			0
$460$	−18.0000	−	9.00000i	−0.839254	−	0.419627i
$461$	−36.0000			−1.67669			−0.838344	−	0.545142i	$-0.816476\pi$
$461$	−36.0000			−1.67669			−0.838344	+	0.545142i	$0.816476\pi$
$462$		0			0
$463$			30.0000i			1.39422i	0.716965	+	0.697109i	$0.245531\pi$
$463$			30.0000i			1.39422i	−0.716965	+	0.697109i	$0.754469\pi$
$464$	−2.00000			−0.0928477
$465$		0			0
$466$	−1.00000			−0.0463241
$467$			20.0000i			0.925490i	0.886492	+	0.462745i	$0.153135\pi$
$467$			20.0000i			0.925490i	−0.886492	+	0.462745i	$0.846865\pi$
$468$		0			0
$469$	−14.0000			−0.646460
$470$	12.0000	−	24.0000i	0.553519	−	1.10704i
$471$		0			0
$472$		−	10.0000i		−	0.460287i
$473$			5.00000i			0.229900i
$474$		0			0
$475$	−15.0000	−	20.0000i	−0.688247	−	0.917663i
$476$		0			0
$477$		0			0
$478$		−	6.00000i		−	0.274434i
$479$	−21.0000			−0.959514			−0.479757	−	0.877401i	$-0.659275\pi$
$479$	−21.0000			−0.959514			−0.479757	+	0.877401i	$0.659275\pi$
$480$		0			0
$481$	−32.0000			−1.45907
$482$			24.0000i			1.09317i
$483$		0			0
$484$	−14.0000			−0.636364
$485$	36.0000	+	18.0000i	1.63468	+	0.817338i
$486$		0			0
$487$		−	24.0000i		−	1.08754i	−0.839233	−	0.543772i	$-0.816996\pi$
$487$		−	24.0000i		−	1.08754i	0.839233	−	0.543772i	$-0.183004\pi$
$488$			14.0000i			0.633750i
$489$		0			0
$490$	12.0000	+	6.00000i	0.542105	+	0.271052i
$491$	−15.0000			−0.676941			−0.338470	−	0.940977i	$-0.609909\pi$
$491$	−15.0000			−0.676941			−0.338470	+	0.940977i	$0.609909\pi$
$492$		0			0
$493$		0			0
$494$	20.0000			0.899843
$495$		0			0
$496$	1.00000			0.0449013
$497$		−	9.00000i		−	0.403705i
$498$		0			0
$499$	−6.00000			−0.268597			−0.134298	−	0.990941i	$-0.542878\pi$
$499$	−6.00000			−0.268597			−0.134298	+	0.990941i	$0.542878\pi$
$500$	11.0000	−	2.00000i	0.491935	−	0.0894427i
$501$		0			0
$502$			28.0000i			1.24970i
$503$		−	22.0000i		−	0.980932i	−0.871460	−	0.490466i	$-0.836827\pi$
$503$		−	22.0000i		−	0.980932i	0.871460	−	0.490466i	$-0.163173\pi$
$504$		0			0
$505$	7.00000	−	14.0000i	0.311496	−	0.622992i
$506$	−45.0000			−2.00049
$507$		0			0
$508$			20.0000i			0.887357i
$509$	−20.0000			−0.886484			−0.443242	−	0.896402i	$-0.646172\pi$
$509$	−20.0000			−0.886484			−0.443242	+	0.896402i	$0.646172\pi$
$510$		0			0
$511$	−9.00000			−0.398137
$512$			1.00000i			0.0441942i
$513$		0			0
$514$	7.00000			0.308757
$515$	16.0000	+	8.00000i	0.705044	+	0.352522i
$516$		0			0
$517$		−	60.0000i		−	2.63880i
$518$		−	8.00000i		−	0.351500i
$519$		0			0
$520$	−4.00000	+	8.00000i	−0.175412	+	0.350823i
$521$	36.0000			1.57719			0.788594	−	0.614914i	$-0.210809\pi$
$521$	36.0000			1.57719			0.788594	+	0.614914i	$0.210809\pi$
$522$		0			0
$523$			17.0000i			0.743358i	0.928361	+	0.371679i	$0.121218\pi$
$523$			17.0000i			0.743358i	−0.928361	+	0.371679i	$0.878782\pi$
$524$	−6.00000			−0.262111
$525$		0			0
$526$	−8.00000			−0.348817
$527$		0			0
$528$		0			0
$529$	−58.0000			−2.52174
$530$	−13.0000	+	26.0000i	−0.564684	+	1.12937i
$531$		0			0
$532$			5.00000i			0.216777i
$533$			24.0000i			1.03956i
$534$		0			0
$535$	22.0000	+	11.0000i	0.951143	+	0.475571i
$536$	−14.0000			−0.604708
$537$		0			0
$538$		−	22.0000i		−	0.948487i
$539$	30.0000			1.29219
$540$		0			0
$541$		0			0			−	1.00000i	$-0.5\pi$
$541$		0			0				1.00000i	$0.5\pi$
$542$		−	25.0000i		−	1.07384i
$543$		0			0
$544$		0			0
$545$	2.00000	−	4.00000i	0.0856706	−	0.171341i
$546$		0			0
$547$		−	16.0000i		−	0.684111i	−0.939680	−	0.342055i	$-0.888877\pi$
$547$		−	16.0000i		−	0.684111i	0.939680	−	0.342055i	$-0.111123\pi$
$548$		−	2.00000i		−	0.0854358i
$549$		0			0
$550$	20.0000	−	15.0000i	0.852803	−	0.639602i
$551$	−10.0000			−0.426014
$552$		0			0
$553$			5.00000i			0.212622i
$554$	−2.00000			−0.0849719
$555$		0			0
$556$	16.0000			0.678551
$557$			7.00000i			0.296600i	0.988942	+	0.148300i	$0.0473800\pi$
$557$			7.00000i			0.296600i	−0.988942	+	0.148300i	$0.952620\pi$
$558$		0			0
$559$	4.00000			0.169182
$560$	−2.00000	−	1.00000i	−0.0845154	−	0.0422577i
$561$		0			0
$562$		−	12.0000i		−	0.506189i
$563$			44.0000i			1.85438i	0.374593	+	0.927189i	$0.377783\pi$
$563$			44.0000i			1.85438i	−0.374593	+	0.927189i	$0.622217\pi$
$564$		0			0
$565$	−26.0000	−	13.0000i	−1.09383	−	0.546914i
$566$	−26.0000			−1.09286
$567$		0			0
$568$		−	9.00000i		−	0.377632i
$569$	−21.0000			−0.880366			−0.440183	−	0.897908i	$-0.645086\pi$
$569$	−21.0000			−0.880366			−0.440183	+	0.897908i	$0.645086\pi$
$570$		0			0
$571$	−10.0000			−0.418487			−0.209243	−	0.977864i	$-0.567100\pi$
$571$	−10.0000			−0.418487			−0.209243	+	0.977864i	$0.567100\pi$
$572$			20.0000i			0.836242i
$573$		0			0
$574$	−6.00000			−0.250435
$575$	36.0000	−	27.0000i	1.50130	−	1.12598i
$576$		0			0
$577$			2.00000i			0.0832611i	0.999133	+	0.0416305i	$0.0132552\pi$
$577$			2.00000i			0.0832611i	−0.999133	+	0.0416305i	$0.986745\pi$
$578$			17.0000i			0.707107i
$579$		0			0
$580$	2.00000	−	4.00000i	0.0830455	−	0.166091i
$581$	−6.00000			−0.248922
$582$		0			0
$583$			65.0000i			2.69202i
$584$	−9.00000			−0.372423
$585$		0			0
$586$	−14.0000			−0.578335
$587$		−	6.00000i		−	0.247647i	−0.992304	−	0.123823i	$-0.960484\pi$
$587$		−	6.00000i		−	0.247647i	0.992304	−	0.123823i	$-0.0395156\pi$
$588$		0			0
$589$	5.00000			0.206021
$590$	20.0000	+	10.0000i	0.823387	+	0.411693i
$591$		0			0
$592$		−	8.00000i		−	0.328798i
$593$			10.0000i			0.410651i	0.978694	+	0.205325i	$0.0658253\pi$
$593$			10.0000i			0.410651i	−0.978694	+	0.205325i	$0.934175\pi$
$594$		0			0
$595$		0			0
$596$	−7.00000			−0.286731
$597$		0			0
$598$			36.0000i			1.47215i
$599$	−39.0000			−1.59350			−0.796748	−	0.604311i	$-0.793448\pi$
$599$	−39.0000			−1.59350			−0.796748	+	0.604311i	$0.793448\pi$
$600$		0			0
$601$	−12.0000			−0.489490			−0.244745	−	0.969587i	$-0.578704\pi$
$601$	−12.0000			−0.489490			−0.244745	+	0.969587i	$0.578704\pi$
$602$			1.00000i			0.0407570i
$603$		0			0
$604$		0			0
$605$	14.0000	−	28.0000i	0.569181	−	1.13836i
$606$		0			0
$607$		−	13.0000i		−	0.527654i	−0.964570	−	0.263827i	$-0.915015\pi$
$607$		−	13.0000i		−	0.527654i	0.964570	−	0.263827i	$-0.0849848\pi$
$608$			5.00000i			0.202777i
$609$		0			0
$610$	−28.0000	−	14.0000i	−1.13369	−	0.566843i
$611$	−48.0000			−1.94187
$612$		0			0
$613$			4.00000i			0.161558i	0.996732	+	0.0807792i	$0.0257409\pi$
$613$			4.00000i			0.161558i	−0.996732	+	0.0807792i	$0.974259\pi$
$614$	−8.00000			−0.322854
$615$		0			0
$616$	−5.00000			−0.201456
$617$			9.00000i			0.362326i	0.983453	+	0.181163i	$0.0579862\pi$
$617$			9.00000i			0.362326i	−0.983453	+	0.181163i	$0.942014\pi$
$618$		0			0
$619$	−16.0000			−0.643094			−0.321547	−	0.946894i	$-0.604203\pi$
$619$	−16.0000			−0.643094			−0.321547	+	0.946894i	$0.604203\pi$
$620$	−1.00000	+	2.00000i	−0.0401610	+	0.0803219i
$621$		0			0
$622$			8.00000i			0.320771i
$623$		−	3.00000i		−	0.120192i
$624$		0			0
$625$	−7.00000	+	24.0000i	−0.280000	+	0.960000i
$626$	−6.00000			−0.239808
$627$		0			0
$628$			13.0000i			0.518756i
$629$		0			0
$630$		0			0
$631$	31.0000			1.23409			0.617045	−	0.786928i	$-0.288330\pi$
$631$	31.0000			1.23409			0.617045	+	0.786928i	$0.288330\pi$
$632$			5.00000i			0.198889i
$633$		0			0
$634$	−34.0000			−1.35031
$635$	−40.0000	−	20.0000i	−1.58735	−	0.793676i
$636$		0			0
$637$		−	24.0000i		−	0.950915i
$638$		−	10.0000i		−	0.395904i
$639$		0			0
$640$	−2.00000	−	1.00000i	−0.0790569	−	0.0395285i
$641$	2.00000			0.0789953			0.0394976	−	0.999220i	$-0.487424\pi$
$641$	2.00000			0.0789953			0.0394976	+	0.999220i	$0.487424\pi$
$642$		0			0
$643$		−	5.00000i		−	0.197181i	−0.995128	−	0.0985904i	$-0.968567\pi$
$643$		−	5.00000i		−	0.197181i	0.995128	−	0.0985904i	$-0.0314334\pi$
$644$	−9.00000			−0.354650
$645$		0			0
$646$		0			0
$647$			5.00000i			0.196570i	0.995158	+	0.0982851i	$0.0313357\pi$
$647$			5.00000i			0.196570i	−0.995158	+	0.0982851i	$0.968664\pi$
$648$		0			0
$649$	50.0000			1.96267
$650$	−12.0000	−	16.0000i	−0.470679	−	0.627572i
$651$		0			0
$652$		−	4.00000i		−	0.156652i
$653$		−	18.0000i		−	0.704394i	−0.935926	−	0.352197i	$-0.885435\pi$
$653$		−	18.0000i		−	0.704394i	0.935926	−	0.352197i	$-0.114565\pi$
$654$		0			0
$655$	6.00000	−	12.0000i	0.234439	−	0.468879i
$656$	−6.00000			−0.234261
$657$		0			0
$658$		−	12.0000i		−	0.467809i
$659$	30.0000			1.16863			0.584317	−	0.811525i	$-0.301362\pi$
$659$	30.0000			1.16863			0.584317	+	0.811525i	$0.301362\pi$
$660$		0			0
$661$	28.0000			1.08907			0.544537	−	0.838737i	$-0.316705\pi$
$661$	28.0000			1.08907			0.544537	+	0.838737i	$0.316705\pi$
$662$		−	24.0000i		−	0.932786i
$663$		0			0
$664$	−6.00000			−0.232845
$665$	−10.0000	−	5.00000i	−0.387783	−	0.193892i
$666$		0			0
$667$		−	18.0000i		−	0.696963i
$668$			9.00000i			0.348220i
$669$		0			0
$670$	14.0000	−	28.0000i	0.540867	−	1.08173i
$671$	−70.0000			−2.70232
$672$		0			0
$673$			38.0000i			1.46479i	0.680879	+	0.732396i	$0.261598\pi$
$673$			38.0000i			1.46479i	−0.680879	+	0.732396i	$0.738402\pi$
$674$	2.00000			0.0770371
$675$		0			0
$676$	3.00000			0.115385
$677$			51.0000i			1.96009i	0.198778	+	0.980045i	$0.436303\pi$
$677$			51.0000i			1.96009i	−0.198778	+	0.980045i	$0.563697\pi$
$678$		0			0
$679$	18.0000			0.690777
$680$		0			0
$681$		0			0
$682$			5.00000i			0.191460i
$683$		−	3.00000i		−	0.114792i	−0.998351	−	0.0573959i	$-0.981720\pi$
$683$		−	3.00000i		−	0.114792i	0.998351	−	0.0573959i	$-0.0182797\pi$
$684$		0			0
$685$	4.00000	+	2.00000i	0.152832	+	0.0764161i
$686$	13.0000			0.496342
$687$		0			0
$688$			1.00000i			0.0381246i
$689$	52.0000			1.98104
$690$		0			0
$691$	27.0000			1.02713			0.513564	−	0.858051i	$-0.328325\pi$
$691$	27.0000			1.02713			0.513564	+	0.858051i	$0.328325\pi$
$692$		−	6.00000i		−	0.228086i
$693$		0			0
$694$	18.0000			0.683271
$695$	−16.0000	+	32.0000i	−0.606915	+	1.21383i
$696$		0			0
$697$		0			0
$698$		−	4.00000i		−	0.151402i
$699$		0			0
$700$	4.00000	−	3.00000i	0.151186	−	0.113389i
$701$	−29.0000			−1.09531			−0.547657	−	0.836703i	$-0.684480\pi$
$701$	−29.0000			−1.09531			−0.547657	+	0.836703i	$0.684480\pi$
$702$		0			0
$703$		−	40.0000i		−	1.50863i
$704$	−5.00000			−0.188445
$705$		0			0
$706$	−4.00000			−0.150542
$707$		−	7.00000i		−	0.263262i
$708$		0			0
$709$	17.0000			0.638448			0.319224	−	0.947679i	$-0.396578\pi$
$709$	17.0000			0.638448			0.319224	+	0.947679i	$0.396578\pi$
$710$	18.0000	+	9.00000i	0.675528	+	0.337764i
$711$		0			0
$712$		−	3.00000i		−	0.112430i
$713$			9.00000i			0.337053i
$714$		0			0
$715$	−40.0000	−	20.0000i	−1.49592	−	0.747958i
$716$	16.0000			0.597948
$717$		0			0
$718$		−	9.00000i		−	0.335877i
$719$	−30.0000			−1.11881			−0.559406	−	0.828894i	$-0.688971\pi$
$719$	−30.0000			−1.11881			−0.559406	+	0.828894i	$0.688971\pi$
$720$		0			0
$721$	8.00000			0.297936
$722$			6.00000i			0.223297i
$723$		0			0
$724$	−17.0000			−0.631800
$725$	6.00000	+	8.00000i	0.222834	+	0.297113i
$726$		0			0
$727$		−	1.00000i		−	0.0370879i	−0.999828	−	0.0185440i	$-0.994097\pi$
$727$		−	1.00000i		−	0.0370879i	0.999828	−	0.0185440i	$-0.00590307\pi$
$728$			4.00000i			0.148250i
$729$		0			0
$730$	9.00000	−	18.0000i	0.333105	−	0.666210i
$731$		0			0
$732$		0			0
$733$		−	14.0000i		−	0.517102i	−0.965998	−	0.258551i	$-0.916755\pi$
$733$		−	14.0000i		−	0.517102i	0.965998	−	0.258551i	$-0.0832450\pi$
$734$	18.0000			0.664392
$735$		0			0
$736$	−9.00000			−0.331744
$737$		−	70.0000i		−	2.57848i
$738$		0			0
$739$	−10.0000			−0.367856			−0.183928	−	0.982940i	$-0.558881\pi$
$739$	−10.0000			−0.367856			−0.183928	+	0.982940i	$0.558881\pi$
$740$	16.0000	+	8.00000i	0.588172	+	0.294086i
$741$		0			0
$742$			13.0000i			0.477245i
$743$			43.0000i			1.57752i	0.614703	+	0.788759i	$0.289276\pi$
$743$			43.0000i			1.57752i	−0.614703	+	0.788759i	$0.710724\pi$
$744$		0			0
$745$	7.00000	−	14.0000i	0.256460	−	0.512920i
$746$	−1.00000			−0.0366126
$747$		0			0
$748$		0			0
$749$	11.0000			0.401931
$750$		0			0
$751$	18.0000			0.656829			0.328415	−	0.944534i	$-0.393486\pi$
$751$	18.0000			0.656829			0.328415	+	0.944534i	$0.393486\pi$
$752$		−	12.0000i		−	0.437595i
$753$		0			0
$754$	−8.00000			−0.291343
$755$		0			0
$756$		0			0
$757$			20.0000i			0.726912i	0.931611	+	0.363456i	$0.118403\pi$
$757$			20.0000i			0.726912i	−0.931611	+	0.363456i	$0.881597\pi$
$758$			11.0000i			0.399538i
$759$		0			0
$760$	−10.0000	−	5.00000i	−0.362738	−	0.181369i
$761$	−15.0000			−0.543750			−0.271875	−	0.962333i	$-0.587644\pi$
$761$	−15.0000			−0.543750			−0.271875	+	0.962333i	$0.587644\pi$
$762$		0			0
$763$		−	2.00000i		−	0.0724049i
$764$	8.00000			0.289430
$765$		0			0
$766$	12.0000			0.433578
$767$		−	40.0000i		−	1.44432i
$768$		0			0
$769$	35.0000			1.26213			0.631066	−	0.775729i	$-0.282618\pi$
$769$	35.0000			1.26213			0.631066	+	0.775729i	$0.282618\pi$
$770$	5.00000	−	10.0000i	0.180187	−	0.360375i
$771$		0			0
$772$			22.0000i			0.791797i
$773$			27.0000i			0.971123i	0.874203	+	0.485561i	$0.161385\pi$
$773$			27.0000i			0.971123i	−0.874203	+	0.485561i	$0.838615\pi$
$774$		0			0
$775$	−3.00000	−	4.00000i	−0.107763	−	0.143684i
$776$	18.0000			0.646162
$777$		0			0
$778$		−	16.0000i		−	0.573628i
$779$	−30.0000			−1.07486
$780$		0			0
$781$	45.0000			1.61023
$782$		0			0
$783$		0			0
$784$	6.00000			0.214286
$785$	−26.0000	−	13.0000i	−0.927980	−	0.463990i
$786$		0			0
$787$			23.0000i			0.819861i	0.912117	+	0.409931i	$0.134447\pi$
$787$			23.0000i			0.819861i	−0.912117	+	0.409931i	$0.865553\pi$
$788$			10.0000i			0.356235i
$789$		0			0
$790$	−10.0000	−	5.00000i	−0.355784	−	0.177892i
$791$	−13.0000			−0.462227
$792$		0			0

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:	\( N \)	\(=\)	\( 2790 = 2 \cdot 3^{2} \cdot 5 \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2790.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.00000i q^{2} -1.00000 q^{4} +(1.00000 - 2.00000i) q^{5} -1.00000i q^{7} -1.00000i q^{8} +O(q^{10})\)	\(q+1.00000i q^{2} -1.00000 q^{4} +(1.00000 - 2.00000i) q^{5} -1.00000i q^{7} -1.00000i q^{8} +(2.00000 + 1.00000i) q^{10} +5.00000 q^{11} -4.00000i q^{13} +1.00000 q^{14} +1.00000 q^{16} +5.00000 q^{19} +(-1.00000 + 2.00000i) q^{20} +5.00000i q^{22} +9.00000i q^{23} +(-3.00000 - 4.00000i) q^{25} +4.00000 q^{26} +1.00000i q^{28} -2.00000 q^{29} +1.00000 q^{31} +1.00000i q^{32} +(-2.00000 - 1.00000i) q^{35} -8.00000i q^{37} +5.00000i q^{38} +(-2.00000 - 1.00000i) q^{40} -6.00000 q^{41} +1.00000i q^{43} -5.00000 q^{44} -9.00000 q^{46} -12.0000i q^{47} +6.00000 q^{49} +(4.00000 - 3.00000i) q^{50} +4.00000i q^{52} +13.0000i q^{53} +(5.00000 - 10.0000i) q^{55} -1.00000 q^{56} -2.00000i q^{58} +10.0000 q^{59} -14.0000 q^{61} +1.00000i q^{62} -1.00000 q^{64} +(-8.00000 - 4.00000i) q^{65} -14.0000i q^{67} +(1.00000 - 2.00000i) q^{70} +9.00000 q^{71} -9.00000i q^{73} +8.00000 q^{74} -5.00000 q^{76} -5.00000i q^{77} -5.00000 q^{79} +(1.00000 - 2.00000i) q^{80} -6.00000i q^{82} -6.00000i q^{83} -1.00000 q^{86} -5.00000i q^{88} +3.00000 q^{89} -4.00000 q^{91} -9.00000i q^{92} +12.0000 q^{94} +(5.00000 - 10.0000i) q^{95} +18.0000i q^{97} +6.00000i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{4} + 2 q^{5}+O(q^{10}) \) 2 * q - 2 * q^4 + 2 * q^5	\( 2 q - 2 q^{4} + 2 q^{5} + 4 q^{10} + 10 q^{11} + 2 q^{14} + 2 q^{16} + 10 q^{19} - 2 q^{20} - 6 q^{25} + 8 q^{26} - 4 q^{29} + 2 q^{31} - 4 q^{35} - 4 q^{40} - 12 q^{41} - 10 q^{44} - 18 q^{46} + 12 q^{49} + 8 q^{50} + 10 q^{55} - 2 q^{56} + 20 q^{59} - 28 q^{61} - 2 q^{64} - 16 q^{65} + 2 q^{70} + 18 q^{71} + 16 q^{74} - 10 q^{76} - 10 q^{79} + 2 q^{80} - 2 q^{86} + 6 q^{89} - 8 q^{91} + 24 q^{94} + 10 q^{95}+O(q^{100}) \) 2 * q - 2 * q^4 + 2 * q^5 + 4 * q^10 + 10 * q^11 + 2 * q^14 + 2 * q^16 + 10 * q^19 - 2 * q^20 - 6 * q^25 + 8 * q^26 - 4 * q^29 + 2 * q^31 - 4 * q^35 - 4 * q^40 - 12 * q^41 - 10 * q^44 - 18 * q^46 + 12 * q^49 + 8 * q^50 + 10 * q^55 - 2 * q^56 + 20 * q^59 - 28 * q^61 - 2 * q^64 - 16 * q^65 + 2 * q^70 + 18 * q^71 + 16 * q^74 - 10 * q^76 - 10 * q^79 + 2 * q^80 - 2 * q^86 + 6 * q^89 - 8 * q^91 + 24 * q^94 + 10 * q^95

Introduction

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