LMFDB - Embedded newform 720.2.bd.a.523.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [720,2,Mod(307,720)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(720, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([2, 3, 0, 1]))

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

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

chi := DirichletCharacter("720.307");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 720 = 2^{4} \cdot 3^{2} \cdot 5 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	720.bd (of order $4$, degree $2$, 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:	$5.74922894553$
Analytic rank:	$1$
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 80)
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			523.1
Root			$-1.00000i$ of defining polynomial
Character	$\chi$	$=$	720.523
Dual form			720.2.bd.a.307.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+(-1.00000 - 1.00000i) q^{2} +2.00000i q^{4} +(-1.00000 + 2.00000i) q^{5} +(-3.00000 + 3.00000i) q^{7} +(2.00000 - 2.00000i) q^{8} +(3.00000 - 1.00000i) q^{10} +(1.00000 - 1.00000i) q^{11} -2.00000 q^{13} +6.00000 q^{14} -4.00000 q^{16} +(-1.00000 + 1.00000i) q^{17} +(3.00000 - 3.00000i) q^{19} +(-4.00000 - 2.00000i) q^{20} -2.00000 q^{22} +(1.00000 + 1.00000i) q^{23} +(-3.00000 - 4.00000i) q^{25} +(2.00000 + 2.00000i) q^{26} +(-6.00000 - 6.00000i) q^{28} +(-7.00000 - 7.00000i) q^{29} -2.00000i q^{31} +(4.00000 + 4.00000i) q^{32} +2.00000 q^{34} +(-3.00000 - 9.00000i) q^{35} -6.00000 q^{37} -6.00000 q^{38} +(2.00000 + 6.00000i) q^{40} -4.00000i q^{41} +4.00000 q^{43} +(2.00000 + 2.00000i) q^{44} -2.00000i q^{46} +(-7.00000 - 7.00000i) q^{47} -11.0000i q^{49} +(-1.00000 + 7.00000i) q^{50} -4.00000i q^{52} +8.00000i q^{53} +(1.00000 + 3.00000i) q^{55} +12.0000i q^{56} +14.0000i q^{58} +(3.00000 + 3.00000i) q^{59} +(-1.00000 + 1.00000i) q^{61} +(-2.00000 + 2.00000i) q^{62} -8.00000i q^{64} +(2.00000 - 4.00000i) q^{65} +4.00000 q^{67} +(-2.00000 - 2.00000i) q^{68} +(-6.00000 + 12.0000i) q^{70} +(-3.00000 + 3.00000i) q^{73} +(6.00000 + 6.00000i) q^{74} +(6.00000 + 6.00000i) q^{76} +6.00000i q^{77} -8.00000 q^{79} +(4.00000 - 8.00000i) q^{80} +(-4.00000 + 4.00000i) q^{82} +2.00000i q^{83} +(-1.00000 - 3.00000i) q^{85} +(-4.00000 - 4.00000i) q^{86} -4.00000i q^{88} -6.00000 q^{89} +(6.00000 - 6.00000i) q^{91} +(-2.00000 + 2.00000i) q^{92} +14.0000i q^{94} +(3.00000 + 9.00000i) q^{95} +(-11.0000 + 11.0000i) q^{97} +(-11.0000 + 11.0000i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{2} - 2 q^{5} - 6 q^{7} + 4 q^{8} + 6 q^{10} + 2 q^{11} - 4 q^{13} + 12 q^{14} - 8 q^{16} - 2 q^{17} + 6 q^{19} - 8 q^{20} - 4 q^{22} + 2 q^{23} - 6 q^{25} + 4 q^{26} - 12 q^{28} - 14 q^{29} + 8 q^{32}+ \cdots - 22 q^{98}+O(q^{100}) $ 2 * q - 2 * q^2 - 2 * q^5 - 6 * q^7 + 4 * q^8 + 6 * q^10 + 2 * q^11 - 4 * q^13 + 12 * q^14 - 8 * q^16 - 2 * q^17 + 6 * q^19 - 8 * q^20 - 4 * q^22 + 2 * q^23 - 6 * q^25 + 4 * q^26 - 12 * q^28 - 14 * q^29 + 8 * q^32 + 4 * q^34 - 6 * q^35 - 12 * q^37 - 12 * q^38 + 4 * q^40 + 8 * q^43 + 4 * q^44 - 14 * q^47 - 2 * q^50 + 2 * q^55 + 6 * q^59 - 2 * q^61 - 4 * q^62 + 4 * q^65 + 8 * q^67 - 4 * q^68 - 12 * q^70 - 6 * q^73 + 12 * q^74 + 12 * q^76 - 16 * q^79 + 8 * q^80 - 8 * q^82 - 2 * q^85 - 8 * q^86 - 12 * q^89 + 12 * q^91 - 4 * q^92 + 6 * q^95 - 22 * q^97 - 22 * q^98

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/720\mathbb{Z}\right)^\times$.

$n$	$181$	$271$	$577$	$641$
$\chi(n)$	$e\left(\frac{1}{4}\right)$	$-1$	$e\left(\frac{3}{4}\right)$	$1$

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 $

$2$	−1.00000	−	1.00000i	−0.707107	−	0.707107i
$2$	−1.00000	−	1.00000i	−0.707107	−	0.707107i
$3$		0			0
$3$		0			0
$4$			2.00000i			1.00000i
$5$	−1.00000	+	2.00000i	−0.447214	+	0.894427i
$5$	−1.00000	+	2.00000i	−0.447214	+	0.894427i
$6$		0			0
$7$	−3.00000	+	3.00000i	−1.13389	+	1.13389i	−0.144370	+	0.989524i	$0.546115\pi$
$7$	−3.00000	+	3.00000i	−1.13389	+	1.13389i	−0.989524	+	0.144370i	$0.953885\pi$
$8$	2.00000	−	2.00000i	0.707107	−	0.707107i
$9$		0			0
$10$	3.00000	−	1.00000i	0.948683	−	0.316228i
$11$	1.00000	−	1.00000i	0.301511	−	0.301511i	−0.540094	−	0.841605i	$-0.681611\pi$
$11$	1.00000	−	1.00000i	0.301511	−	0.301511i	0.841605	+	0.540094i	$0.181611\pi$
$12$		0			0
$13$	−2.00000			−0.554700			−0.277350	−	0.960769i	$-0.589456\pi$
$13$	−2.00000			−0.554700			−0.277350	+	0.960769i	$0.589456\pi$
$14$	6.00000			1.60357
$15$		0			0
$16$	−4.00000			−1.00000
$17$	−1.00000	+	1.00000i	−0.242536	+	0.242536i	−0.817898	−	0.575363i	$-0.804861\pi$
$17$	−1.00000	+	1.00000i	−0.242536	+	0.242536i	0.575363	+	0.817898i	$0.304861\pi$
$18$		0			0
$19$	3.00000	−	3.00000i	0.688247	−	0.688247i	−0.273597	−	0.961844i	$-0.588214\pi$
$19$	3.00000	−	3.00000i	0.688247	−	0.688247i	0.961844	+	0.273597i	$0.0882135\pi$
$20$	−4.00000	−	2.00000i	−0.894427	−	0.447214i
$21$		0			0
$22$	−2.00000			−0.426401
$23$	1.00000	+	1.00000i	0.208514	+	0.208514i	0.803636	−	0.595121i	$-0.202896\pi$
$23$	1.00000	+	1.00000i	0.208514	+	0.208514i	−0.595121	+	0.803636i	$0.702896\pi$
$24$		0			0
$25$	−3.00000	−	4.00000i	−0.600000	−	0.800000i
$26$	2.00000	+	2.00000i	0.392232	+	0.392232i
$27$		0			0
$28$	−6.00000	−	6.00000i	−1.13389	−	1.13389i
$29$	−7.00000	−	7.00000i	−1.29987	−	1.29987i	−0.928477	−	0.371391i	$-0.878881\pi$
$29$	−7.00000	−	7.00000i	−1.29987	−	1.29987i	−0.371391	−	0.928477i	$-0.621119\pi$
$30$		0			0
$31$		−	2.00000i		−	0.359211i	−0.983739	−	0.179605i	$-0.942518\pi$
$31$		−	2.00000i		−	0.359211i	0.983739	−	0.179605i	$-0.0574821\pi$
$32$	4.00000	+	4.00000i	0.707107	+	0.707107i
$33$		0			0
$34$	2.00000			0.342997
$35$	−3.00000	−	9.00000i	−0.507093	−	1.52128i
$36$		0			0
$37$	−6.00000			−0.986394			−0.493197	−	0.869918i	$-0.664172\pi$
$37$	−6.00000			−0.986394			−0.493197	+	0.869918i	$0.664172\pi$
$38$	−6.00000			−0.973329
$39$		0			0
$40$	2.00000	+	6.00000i	0.316228	+	0.948683i
$41$		−	4.00000i		−	0.624695i	−0.949968	−	0.312348i	$-0.898885\pi$
$41$		−	4.00000i		−	0.624695i	0.949968	−	0.312348i	$-0.101115\pi$
$42$		0			0
$43$	4.00000			0.609994			0.304997	−	0.952353i	$-0.401344\pi$
$43$	4.00000			0.609994			0.304997	+	0.952353i	$0.401344\pi$
$44$	2.00000	+	2.00000i	0.301511	+	0.301511i
$45$		0			0
$46$		−	2.00000i		−	0.294884i
$47$	−7.00000	−	7.00000i	−1.02105	−	1.02105i	−0.999774	−	0.0212814i	$-0.993225\pi$
$47$	−7.00000	−	7.00000i	−1.02105	−	1.02105i	−0.0212814	−	0.999774i	$-0.506775\pi$
$48$		0			0
$49$		−	11.0000i		−	1.57143i
$50$	−1.00000	+	7.00000i	−0.141421	+	0.989949i
$51$		0			0
$52$		−	4.00000i		−	0.554700i
$53$			8.00000i			1.09888i	0.835532	+	0.549442i	$0.185160\pi$
$53$			8.00000i			1.09888i	−0.835532	+	0.549442i	$0.814840\pi$
$54$		0			0
$55$	1.00000	+	3.00000i	0.134840	+	0.404520i
$56$			12.0000i			1.60357i
$57$		0			0
$58$			14.0000i			1.83829i
$59$	3.00000	+	3.00000i	0.390567	+	0.390567i	0.874889	−	0.484323i	$-0.160934\pi$
$59$	3.00000	+	3.00000i	0.390567	+	0.390567i	−0.484323	+	0.874889i	$0.660934\pi$
$60$		0			0
$61$	−1.00000	+	1.00000i	−0.128037	+	0.128037i	−0.768221	−	0.640184i	$-0.778858\pi$
$61$	−1.00000	+	1.00000i	−0.128037	+	0.128037i	0.640184	+	0.768221i	$0.278858\pi$
$62$	−2.00000	+	2.00000i	−0.254000	+	0.254000i
$63$		0			0
$64$		−	8.00000i		−	1.00000i
$65$	2.00000	−	4.00000i	0.248069	−	0.496139i
$66$		0			0
$67$	4.00000			0.488678			0.244339	−	0.969690i	$-0.421429\pi$
$67$	4.00000			0.488678			0.244339	+	0.969690i	$0.421429\pi$
$68$	−2.00000	−	2.00000i	−0.242536	−	0.242536i
$69$		0			0
$70$	−6.00000	+	12.0000i	−0.717137	+	1.43427i
$71$		0			0			−	1.00000i	$-0.5\pi$
$71$		0			0				1.00000i	$0.5\pi$
$72$		0			0
$73$	−3.00000	+	3.00000i	−0.351123	+	0.351123i	−0.860527	−	0.509404i	$-0.829866\pi$
$73$	−3.00000	+	3.00000i	−0.351123	+	0.351123i	0.509404	+	0.860527i	$0.329866\pi$
$74$	6.00000	+	6.00000i	0.697486	+	0.697486i
$75$		0			0
$76$	6.00000	+	6.00000i	0.688247	+	0.688247i
$77$			6.00000i			0.683763i
$78$		0			0
$79$	−8.00000			−0.900070			−0.450035	−	0.893011i	$-0.648589\pi$
$79$	−8.00000			−0.900070			−0.450035	+	0.893011i	$0.648589\pi$
$80$	4.00000	−	8.00000i	0.447214	−	0.894427i
$81$		0			0
$82$	−4.00000	+	4.00000i	−0.441726	+	0.441726i
$83$			2.00000i			0.219529i	0.993958	+	0.109764i	$0.0350096\pi$
$83$			2.00000i			0.219529i	−0.993958	+	0.109764i	$0.964990\pi$
$84$		0			0
$85$	−1.00000	−	3.00000i	−0.108465	−	0.325396i
$86$	−4.00000	−	4.00000i	−0.431331	−	0.431331i
$87$		0			0
$88$		−	4.00000i		−	0.426401i
$89$	−6.00000			−0.635999			−0.317999	−	0.948091i	$-0.603011\pi$
$89$	−6.00000			−0.635999			−0.317999	+	0.948091i	$0.603011\pi$
$90$		0			0
$91$	6.00000	−	6.00000i	0.628971	−	0.628971i
$92$	−2.00000	+	2.00000i	−0.208514	+	0.208514i
$93$		0			0
$94$			14.0000i			1.44399i
$95$	3.00000	+	9.00000i	0.307794	+	0.923381i
$96$		0			0
$97$	−11.0000	+	11.0000i	−1.11688	+	1.11688i	−0.124684	+	0.992196i	$0.539792\pi$
$97$	−11.0000	+	11.0000i	−1.11688	+	1.11688i	−0.992196	+	0.124684i	$0.960208\pi$
$98$	−11.0000	+	11.0000i	−1.11117	+	1.11117i
$99$		0			0
$100$	8.00000	−	6.00000i	0.800000	−	0.600000i
$101$	5.00000	+	5.00000i	0.497519	+	0.497519i	0.910665	−	0.413146i	$-0.135570\pi$
$101$	5.00000	+	5.00000i	0.497519	+	0.497519i	−0.413146	+	0.910665i	$0.635570\pi$
$102$		0			0
$103$	−5.00000	−	5.00000i	−0.492665	−	0.492665i	0.416480	−	0.909145i	$-0.363264\pi$
$103$	−5.00000	−	5.00000i	−0.492665	−	0.492665i	−0.909145	+	0.416480i	$0.863264\pi$
$104$	−4.00000	+	4.00000i	−0.392232	+	0.392232i
$105$		0			0
$106$	8.00000	−	8.00000i	0.777029	−	0.777029i
$107$		−	6.00000i		−	0.580042i	−0.957020	−	0.290021i	$-0.906338\pi$
$107$		−	6.00000i		−	0.580042i	0.957020	−	0.290021i	$-0.0936623\pi$
$108$		0			0
$109$	−5.00000	−	5.00000i	−0.478913	−	0.478913i	0.425871	−	0.904784i	$-0.359968\pi$
$109$	−5.00000	−	5.00000i	−0.478913	−	0.478913i	−0.904784	+	0.425871i	$0.859968\pi$
$110$	2.00000	−	4.00000i	0.190693	−	0.381385i
$111$		0			0
$112$	12.0000	−	12.0000i	1.13389	−	1.13389i
$113$	−13.0000	−	13.0000i	−1.22294	−	1.22294i	−0.966583	−	0.256354i	$-0.917479\pi$
$113$	−13.0000	−	13.0000i	−1.22294	−	1.22294i	−0.256354	−	0.966583i	$-0.582521\pi$
$114$		0			0
$115$	−3.00000	+	1.00000i	−0.279751	+	0.0932505i
$116$	14.0000	−	14.0000i	1.29987	−	1.29987i
$117$		0			0
$118$		−	6.00000i		−	0.552345i
$119$		−	6.00000i		−	0.550019i
$120$		0			0
$121$			9.00000i			0.818182i
$122$	2.00000			0.181071
$123$		0			0
$124$	4.00000			0.359211
$125$	11.0000	−	2.00000i	0.983870	−	0.178885i
$126$		0			0
$127$	7.00000	+	7.00000i	0.621150	+	0.621150i	0.945825	−	0.324676i	$-0.105255\pi$
$127$	7.00000	+	7.00000i	0.621150	+	0.621150i	−0.324676	+	0.945825i	$0.605255\pi$
$128$	−8.00000	+	8.00000i	−0.707107	+	0.707107i
$129$		0			0
$130$	−6.00000	+	2.00000i	−0.526235	+	0.175412i
$131$	7.00000	+	7.00000i	0.611593	+	0.611593i	0.943361	−	0.331768i	$-0.107645\pi$
$131$	7.00000	+	7.00000i	0.611593	+	0.611593i	−0.331768	+	0.943361i	$0.607645\pi$
$132$		0			0
$133$			18.0000i			1.56080i
$134$	−4.00000	−	4.00000i	−0.345547	−	0.345547i
$135$		0			0
$136$			4.00000i			0.342997i
$137$	−9.00000	−	9.00000i	−0.768922	−	0.768922i	0.208995	−	0.977917i	$-0.432981\pi$
$137$	−9.00000	−	9.00000i	−0.768922	−	0.768922i	−0.977917	+	0.208995i	$0.932981\pi$
$138$		0			0
$139$	9.00000	+	9.00000i	0.763370	+	0.763370i	0.976930	−	0.213560i	$-0.0685059\pi$
$139$	9.00000	+	9.00000i	0.763370	+	0.763370i	−0.213560	+	0.976930i	$0.568506\pi$
$140$	18.0000	−	6.00000i	1.52128	−	0.507093i
$141$		0			0
$142$		0			0
$143$	−2.00000	+	2.00000i	−0.167248	+	0.167248i
$144$		0			0
$145$	21.0000	−	7.00000i	1.74396	−	0.581318i
$146$	6.00000			0.496564
$147$		0			0
$148$		−	12.0000i		−	0.986394i
$149$	1.00000	−	1.00000i	0.0819232	−	0.0819232i	−0.664958	−	0.746881i	$-0.731550\pi$
$149$	1.00000	−	1.00000i	0.0819232	−	0.0819232i	0.746881	+	0.664958i	$0.231550\pi$
$150$		0			0
$151$	8.00000			0.651031			0.325515	−	0.945537i	$-0.394462\pi$
$151$	8.00000			0.651031			0.325515	+	0.945537i	$0.394462\pi$
$152$		−	12.0000i		−	0.973329i
$153$		0			0
$154$	6.00000	−	6.00000i	0.483494	−	0.483494i
$155$	4.00000	+	2.00000i	0.321288	+	0.160644i
$156$		0			0
$157$		−	20.0000i		−	1.59617i	−0.602542	−	0.798087i	$-0.705846\pi$
$157$		−	20.0000i		−	1.59617i	0.602542	−	0.798087i	$-0.294154\pi$
$158$	8.00000	+	8.00000i	0.636446	+	0.636446i
$159$		0			0
$160$	−12.0000	+	4.00000i	−0.948683	+	0.316228i
$161$	−6.00000			−0.472866
$162$		0			0
$163$			14.0000i			1.09656i	0.836293	+	0.548282i	$0.184718\pi$
$163$			14.0000i			1.09656i	−0.836293	+	0.548282i	$0.815282\pi$
$164$	8.00000			0.624695
$165$		0			0
$166$	2.00000	−	2.00000i	0.155230	−	0.155230i
$167$	3.00000	−	3.00000i	0.232147	−	0.232147i	−0.581441	−	0.813588i	$-0.697511\pi$
$167$	3.00000	−	3.00000i	0.232147	−	0.232147i	0.813588	+	0.581441i	$0.197511\pi$
$168$		0			0
$169$	−9.00000			−0.692308
$170$	−2.00000	+	4.00000i	−0.153393	+	0.306786i
$171$		0			0
$172$			8.00000i			0.609994i
$173$	6.00000			0.456172			0.228086	−	0.973641i	$-0.426753\pi$
$173$	6.00000			0.456172			0.228086	+	0.973641i	$0.426753\pi$
$174$		0			0
$175$	21.0000	+	3.00000i	1.58745	+	0.226779i
$176$	−4.00000	+	4.00000i	−0.301511	+	0.301511i
$177$		0			0
$178$	6.00000	+	6.00000i	0.449719	+	0.449719i
$179$	5.00000	−	5.00000i	0.373718	−	0.373718i	−0.495112	−	0.868829i	$-0.664873\pi$
$179$	5.00000	−	5.00000i	0.373718	−	0.373718i	0.868829	+	0.495112i	$0.164873\pi$
$180$		0			0
$181$	3.00000	+	3.00000i	0.222988	+	0.222988i	0.809756	−	0.586767i	$-0.199600\pi$
$181$	3.00000	+	3.00000i	0.222988	+	0.222988i	−0.586767	+	0.809756i	$0.699600\pi$
$182$	−12.0000			−0.889499
$183$		0			0
$184$	4.00000			0.294884
$185$	6.00000	−	12.0000i	0.441129	−	0.882258i
$186$		0			0
$187$			2.00000i			0.146254i
$188$	14.0000	−	14.0000i	1.02105	−	1.02105i
$189$		0			0
$190$	6.00000	−	12.0000i	0.435286	−	0.870572i
$191$			18.0000i			1.30243i	0.758891	+	0.651217i	$0.225741\pi$
$191$			18.0000i			1.30243i	−0.758891	+	0.651217i	$0.774259\pi$
$192$		0			0
$193$	−15.0000	−	15.0000i	−1.07972	−	1.07972i	−0.996534	−	0.0831899i	$-0.973489\pi$
$193$	−15.0000	−	15.0000i	−1.07972	−	1.07972i	−0.0831899	−	0.996534i	$-0.526511\pi$
$194$	22.0000			1.57951
$195$		0			0
$196$	22.0000			1.57143
$197$	−6.00000			−0.427482			−0.213741	−	0.976890i	$-0.568565\pi$
$197$	−6.00000			−0.427482			−0.213741	+	0.976890i	$0.568565\pi$
$198$		0			0
$199$		−	10.0000i		−	0.708881i	−0.935079	−	0.354441i	$-0.884671\pi$
$199$		−	10.0000i		−	0.708881i	0.935079	−	0.354441i	$-0.115329\pi$
$200$	−14.0000	−	2.00000i	−0.989949	−	0.141421i
$201$		0			0
$202$		−	10.0000i		−	0.703598i
$203$	42.0000			2.94782
$204$		0			0
$205$	8.00000	+	4.00000i	0.558744	+	0.279372i
$206$			10.0000i			0.696733i
$207$		0			0
$208$	8.00000			0.554700
$209$		−	6.00000i		−	0.415029i
$210$		0			0
$211$	−19.0000	−	19.0000i	−1.30801	−	1.30801i	−0.922847	−	0.385167i	$-0.874144\pi$
$211$	−19.0000	−	19.0000i	−1.30801	−	1.30801i	−0.385167	−	0.922847i	$-0.625856\pi$
$212$	−16.0000			−1.09888
$213$		0			0
$214$	−6.00000	+	6.00000i	−0.410152	+	0.410152i
$215$	−4.00000	+	8.00000i	−0.272798	+	0.545595i
$216$		0			0
$217$	6.00000	+	6.00000i	0.407307	+	0.407307i
$218$			10.0000i			0.677285i
$219$		0			0
$220$	−6.00000	+	2.00000i	−0.404520	+	0.134840i
$221$	2.00000	−	2.00000i	0.134535	−	0.134535i
$222$		0			0
$223$	9.00000	−	9.00000i	0.602685	−	0.602685i	−0.338340	−	0.941024i	$-0.609865\pi$
$223$	9.00000	−	9.00000i	0.602685	−	0.602685i	0.941024	+	0.338340i	$0.109865\pi$
$224$	−24.0000			−1.60357
$225$		0			0
$226$			26.0000i			1.72949i
$227$	−12.0000			−0.796468			−0.398234	−	0.917284i	$-0.630377\pi$
$227$	−12.0000			−0.796468			−0.398234	+	0.917284i	$0.630377\pi$
$228$		0			0
$229$	−1.00000	+	1.00000i	−0.0660819	+	0.0660819i	−0.739375	−	0.673293i	$-0.764879\pi$
$229$	−1.00000	+	1.00000i	−0.0660819	+	0.0660819i	0.673293	+	0.739375i	$0.264879\pi$
$230$	4.00000	+	2.00000i	0.263752	+	0.131876i
$231$		0			0
$232$	−28.0000			−1.83829
$233$	−9.00000	+	9.00000i	−0.589610	+	0.589610i	−0.937526	−	0.347916i	$-0.886889\pi$
$233$	−9.00000	+	9.00000i	−0.589610	+	0.589610i	0.347916	+	0.937526i	$0.386889\pi$
$234$		0			0
$235$	21.0000	−	7.00000i	1.36989	−	0.456630i
$236$	−6.00000	+	6.00000i	−0.390567	+	0.390567i
$237$		0			0
$238$	−6.00000	+	6.00000i	−0.388922	+	0.388922i
$239$		0			0			−	1.00000i	$-0.5\pi$
$239$		0			0				1.00000i	$0.5\pi$
$240$		0			0
$241$	−14.0000			−0.901819			−0.450910	−	0.892570i	$-0.648900\pi$
$241$	−14.0000			−0.901819			−0.450910	+	0.892570i	$0.648900\pi$
$242$	9.00000	−	9.00000i	0.578542	−	0.578542i
$243$		0			0
$244$	−2.00000	−	2.00000i	−0.128037	−	0.128037i
$245$	22.0000	+	11.0000i	1.40553	+	0.702764i
$246$		0			0
$247$	−6.00000	+	6.00000i	−0.381771	+	0.381771i
$248$	−4.00000	−	4.00000i	−0.254000	−	0.254000i
$249$		0			0
$250$	−13.0000	−	9.00000i	−0.822192	−	0.569210i
$251$	−11.0000	+	11.0000i	−0.694314	+	0.694314i	−0.963178	−	0.268864i	$-0.913352\pi$
$251$	−11.0000	+	11.0000i	−0.694314	+	0.694314i	0.268864	+	0.963178i	$0.413352\pi$
$252$		0			0
$253$	2.00000			0.125739
$254$		−	14.0000i		−	0.878438i
$255$		0			0
$256$	16.0000			1.00000
$257$	−13.0000	+	13.0000i	−0.810918	+	0.810918i	−0.984771	−	0.173854i	$-0.944378\pi$
$257$	−13.0000	+	13.0000i	−0.810918	+	0.810918i	0.173854	+	0.984771i	$0.444378\pi$
$258$		0			0
$259$	18.0000	−	18.0000i	1.11847	−	1.11847i
$260$	8.00000	+	4.00000i	0.496139	+	0.248069i
$261$		0			0
$262$		−	14.0000i		−	0.864923i
$263$	−7.00000	−	7.00000i	−0.431638	−	0.431638i	0.457547	−	0.889185i	$-0.348728\pi$
$263$	−7.00000	−	7.00000i	−0.431638	−	0.431638i	−0.889185	+	0.457547i	$0.848728\pi$
$264$		0			0
$265$	−16.0000	−	8.00000i	−0.982872	−	0.491436i
$266$	18.0000	−	18.0000i	1.10365	−	1.10365i
$267$		0			0
$268$			8.00000i			0.488678i
$269$	1.00000	+	1.00000i	0.0609711	+	0.0609711i	0.736935	−	0.675964i	$-0.236272\pi$
$269$	1.00000	+	1.00000i	0.0609711	+	0.0609711i	−0.675964	+	0.736935i	$0.736272\pi$
$270$		0			0
$271$			30.0000i			1.82237i	0.411997	+	0.911185i	$0.364831\pi$
$271$			30.0000i			1.82237i	−0.411997	+	0.911185i	$0.635169\pi$
$272$	4.00000	−	4.00000i	0.242536	−	0.242536i
$273$		0			0
$274$			18.0000i			1.08742i
$275$	−7.00000	−	1.00000i	−0.422116	−	0.0603023i
$276$		0			0
$277$	18.0000			1.08152			0.540758	−	0.841178i	$-0.318138\pi$
$277$	18.0000			1.08152			0.540758	+	0.841178i	$0.318138\pi$
$278$		−	18.0000i		−	1.07957i
$279$		0			0
$280$	−24.0000	−	12.0000i	−1.43427	−	0.717137i
$281$			16.0000i			0.954480i	0.878773	+	0.477240i	$0.158363\pi$
$281$			16.0000i			0.954480i	−0.878773	+	0.477240i	$0.841637\pi$
$282$		0			0
$283$	12.0000			0.713326			0.356663	−	0.934233i	$-0.383914\pi$
$283$	12.0000			0.713326			0.356663	+	0.934233i	$0.383914\pi$
$284$		0			0
$285$		0			0
$286$	4.00000			0.236525
$287$	12.0000	+	12.0000i	0.708338	+	0.708338i
$288$		0			0
$289$			15.0000i			0.882353i
$290$	−28.0000	−	14.0000i	−1.64422	−	0.822108i
$291$		0			0
$292$	−6.00000	−	6.00000i	−0.351123	−	0.351123i
$293$			12.0000i			0.701047i	0.936554	+	0.350524i	$0.113996\pi$
$293$			12.0000i			0.701047i	−0.936554	+	0.350524i	$0.886004\pi$
$294$		0			0
$295$	−9.00000	+	3.00000i	−0.524000	+	0.174667i
$296$	−12.0000	+	12.0000i	−0.697486	+	0.697486i
$297$		0			0
$298$	−2.00000			−0.115857
$299$	−2.00000	−	2.00000i	−0.115663	−	0.115663i
$300$		0			0
$301$	−12.0000	+	12.0000i	−0.691669	+	0.691669i
$302$	−8.00000	−	8.00000i	−0.460348	−	0.460348i
$303$		0			0
$304$	−12.0000	+	12.0000i	−0.688247	+	0.688247i
$305$	−1.00000	−	3.00000i	−0.0572598	−	0.171780i
$306$		0			0
$307$	−4.00000			−0.228292			−0.114146	−	0.993464i	$-0.536413\pi$
$307$	−4.00000			−0.228292			−0.114146	+	0.993464i	$0.536413\pi$
$308$	−12.0000			−0.683763
$309$		0			0
$310$	−2.00000	−	6.00000i	−0.113592	−	0.340777i
$311$	−16.0000			−0.907277			−0.453638	−	0.891186i	$-0.649874\pi$
$311$	−16.0000			−0.907277			−0.453638	+	0.891186i	$0.649874\pi$
$312$		0			0
$313$	13.0000	−	13.0000i	0.734803	−	0.734803i	−0.236764	−	0.971567i	$-0.576087\pi$
$313$	13.0000	−	13.0000i	0.734803	−	0.734803i	0.971567	+	0.236764i	$0.0760868\pi$
$314$	−20.0000	+	20.0000i	−1.12867	+	1.12867i
$315$		0			0
$316$		−	16.0000i		−	0.900070i
$317$			8.00000i			0.449325i	0.974437	+	0.224662i	$0.0721279\pi$
$317$			8.00000i			0.449325i	−0.974437	+	0.224662i	$0.927872\pi$
$318$		0			0
$319$	−14.0000			−0.783850
$320$	16.0000	+	8.00000i	0.894427	+	0.447214i
$321$		0			0
$322$	6.00000	+	6.00000i	0.334367	+	0.334367i
$323$			6.00000i			0.333849i
$324$		0			0
$325$	6.00000	+	8.00000i	0.332820	+	0.443760i
$326$	14.0000	−	14.0000i	0.775388	−	0.775388i
$327$		0			0
$328$	−8.00000	−	8.00000i	−0.441726	−	0.441726i
$329$	42.0000			2.31553
$330$		0			0
$331$	−21.0000	+	21.0000i	−1.15426	+	1.15426i	−0.168576	+	0.985689i	$0.553917\pi$
$331$	−21.0000	+	21.0000i	−1.15426	+	1.15426i	−0.985689	+	0.168576i	$0.946083\pi$
$332$	−4.00000			−0.219529
$333$		0			0
$334$	−6.00000			−0.328305
$335$	−4.00000	+	8.00000i	−0.218543	+	0.437087i
$336$		0			0
$337$	−11.0000	+	11.0000i	−0.599208	+	0.599208i	−0.940102	−	0.340894i	$-0.889270\pi$
$337$	−11.0000	+	11.0000i	−0.599208	+	0.599208i	0.340894	+	0.940102i	$0.389270\pi$
$338$	9.00000	+	9.00000i	0.489535	+	0.489535i
$339$		0			0
$340$	6.00000	−	2.00000i	0.325396	−	0.108465i
$341$	−2.00000	−	2.00000i	−0.108306	−	0.108306i
$342$		0			0
$343$	12.0000	+	12.0000i	0.647939	+	0.647939i
$344$	8.00000	−	8.00000i	0.431331	−	0.431331i
$345$		0			0
$346$	−6.00000	−	6.00000i	−0.322562	−	0.322562i
$347$			2.00000i			0.107366i	0.998558	+	0.0536828i	$0.0170960\pi$
$347$			2.00000i			0.107366i	−0.998558	+	0.0536828i	$0.982904\pi$
$348$		0			0
$349$	3.00000	+	3.00000i	0.160586	+	0.160586i	0.782826	−	0.622240i	$-0.213777\pi$
$349$	3.00000	+	3.00000i	0.160586	+	0.160586i	−0.622240	+	0.782826i	$0.713777\pi$
$350$	−18.0000	−	24.0000i	−0.962140	−	1.28285i
$351$		0			0
$352$	8.00000			0.426401
$353$	−13.0000	−	13.0000i	−0.691920	−	0.691920i	0.270734	−	0.962654i	$-0.412734\pi$
$353$	−13.0000	−	13.0000i	−0.691920	−	0.691920i	−0.962654	+	0.270734i	$0.912734\pi$
$354$		0			0
$355$		0			0
$356$		−	12.0000i		−	0.635999i
$357$		0			0
$358$	−10.0000			−0.528516
$359$		−	14.0000i		−	0.738892i	−0.929252	−	0.369446i	$-0.879548\pi$
$359$		−	14.0000i		−	0.738892i	0.929252	−	0.369446i	$-0.120452\pi$
$360$		0			0
$361$			1.00000i			0.0526316i
$362$		−	6.00000i		−	0.315353i
$363$		0			0
$364$	12.0000	+	12.0000i	0.628971	+	0.628971i
$365$	−3.00000	−	9.00000i	−0.157027	−	0.471082i
$366$		0			0
$367$	−21.0000	−	21.0000i	−1.09619	−	1.09619i	−0.994852	−	0.101339i	$-0.967687\pi$
$367$	−21.0000	−	21.0000i	−1.09619	−	1.09619i	−0.101339	−	0.994852i	$-0.532313\pi$
$368$	−4.00000	−	4.00000i	−0.208514	−	0.208514i
$369$		0			0
$370$	−18.0000	+	6.00000i	−0.935775	+	0.311925i
$371$	−24.0000	−	24.0000i	−1.24602	−	1.24602i
$372$		0			0
$373$		−	4.00000i		−	0.207112i	−0.994624	−	0.103556i	$-0.966978\pi$
$373$		−	4.00000i		−	0.207112i	0.994624	−	0.103556i	$-0.0330221\pi$
$374$	2.00000	−	2.00000i	0.103418	−	0.103418i
$375$		0			0
$376$	−28.0000			−1.44399
$377$	14.0000	+	14.0000i	0.721037	+	0.721037i
$378$		0			0
$379$	−15.0000	−	15.0000i	−0.770498	−	0.770498i	0.207695	−	0.978194i	$-0.433404\pi$
$379$	−15.0000	−	15.0000i	−0.770498	−	0.770498i	−0.978194	+	0.207695i	$0.933404\pi$
$380$	−18.0000	+	6.00000i	−0.923381	+	0.307794i
$381$		0			0
$382$	18.0000	−	18.0000i	0.920960	−	0.920960i
$383$	−5.00000	+	5.00000i	−0.255488	+	0.255488i	−0.823216	−	0.567728i	$-0.807823\pi$
$383$	−5.00000	+	5.00000i	−0.255488	+	0.255488i	0.567728	+	0.823216i	$0.307823\pi$
$384$		0			0
$385$	−12.0000	−	6.00000i	−0.611577	−	0.305788i
$386$			30.0000i			1.52696i
$387$		0			0
$388$	−22.0000	−	22.0000i	−1.11688	−	1.11688i
$389$	−23.0000	+	23.0000i	−1.16615	+	1.16615i	−0.183041	+	0.983105i	$0.558594\pi$
$389$	−23.0000	+	23.0000i	−1.16615	+	1.16615i	−0.983105	+	0.183041i	$0.941406\pi$
$390$		0			0
$391$	−2.00000			−0.101144
$392$	−22.0000	−	22.0000i	−1.11117	−	1.11117i
$393$		0			0
$394$	6.00000	+	6.00000i	0.302276	+	0.302276i
$395$	8.00000	−	16.0000i	0.402524	−	0.805047i
$396$		0			0
$397$			32.0000i			1.60603i	0.595956	+	0.803017i	$0.296773\pi$
$397$			32.0000i			1.60603i	−0.595956	+	0.803017i	$0.703227\pi$
$398$	−10.0000	+	10.0000i	−0.501255	+	0.501255i
$399$		0			0
$400$	12.0000	+	16.0000i	0.600000	+	0.800000i
$401$	2.00000			0.0998752			0.0499376	−	0.998752i	$-0.484098\pi$
$401$	2.00000			0.0998752			0.0499376	+	0.998752i	$0.484098\pi$
$402$		0			0
$403$			4.00000i			0.199254i
$404$	−10.0000	+	10.0000i	−0.497519	+	0.497519i
$405$		0			0
$406$	−42.0000	−	42.0000i	−2.08443	−	2.08443i
$407$	−6.00000	+	6.00000i	−0.297409	+	0.297409i
$408$		0			0
$409$	38.0000			1.87898			0.939490	−	0.342578i	$-0.111300\pi$
$409$	38.0000			1.87898			0.939490	+	0.342578i	$0.111300\pi$
$410$	−4.00000	−	12.0000i	−0.197546	−	0.592638i
$411$		0			0
$412$	10.0000	−	10.0000i	0.492665	−	0.492665i
$413$	−18.0000			−0.885722
$414$		0			0
$415$	−4.00000	−	2.00000i	−0.196352	−	0.0981761i
$416$	−8.00000	−	8.00000i	−0.392232	−	0.392232i
$417$		0			0
$418$	−6.00000	+	6.00000i	−0.293470	+	0.293470i
$419$	17.0000	−	17.0000i	0.830504	−	0.830504i	−0.157081	−	0.987586i	$-0.550208\pi$
$419$	17.0000	−	17.0000i	0.830504	−	0.830504i	0.987586	+	0.157081i	$0.0502085\pi$
$420$		0			0
$421$	−5.00000	−	5.00000i	−0.243685	−	0.243685i	0.574688	−	0.818373i	$-0.305124\pi$
$421$	−5.00000	−	5.00000i	−0.243685	−	0.243685i	−0.818373	+	0.574688i	$0.805124\pi$
$422$			38.0000i			1.84981i
$423$		0			0
$424$	16.0000	+	16.0000i	0.777029	+	0.777029i
$425$	7.00000	+	1.00000i	0.339550	+	0.0485071i
$426$		0			0
$427$		−	6.00000i		−	0.290360i
$428$	12.0000			0.580042
$429$		0			0
$430$	12.0000	−	4.00000i	0.578691	−	0.192897i
$431$			2.00000i			0.0963366i	0.998839	+	0.0481683i	$0.0153384\pi$
$431$			2.00000i			0.0963366i	−0.998839	+	0.0481683i	$0.984662\pi$
$432$		0			0
$433$	5.00000	+	5.00000i	0.240285	+	0.240285i	0.816968	−	0.576683i	$-0.195653\pi$
$433$	5.00000	+	5.00000i	0.240285	+	0.240285i	−0.576683	+	0.816968i	$0.695653\pi$
$434$		−	12.0000i		−	0.576018i
$435$		0			0
$436$	10.0000	−	10.0000i	0.478913	−	0.478913i
$437$	6.00000			0.287019
$438$		0			0
$439$		−	26.0000i		−	1.24091i	−0.784241	−	0.620456i	$-0.786947\pi$
$439$		−	26.0000i		−	1.24091i	0.784241	−	0.620456i	$-0.213053\pi$
$440$	8.00000	+	4.00000i	0.381385	+	0.190693i
$441$		0			0
$442$	−4.00000			−0.190261
$443$	4.00000			0.190046			0.0950229	−	0.995475i	$-0.469708\pi$
$443$	4.00000			0.190046			0.0950229	+	0.995475i	$0.469708\pi$
$444$		0			0
$445$	6.00000	−	12.0000i	0.284427	−	0.568855i
$446$	−18.0000			−0.852325
$447$		0			0
$448$	24.0000	+	24.0000i	1.13389	+	1.13389i
$449$			24.0000i			1.13263i	0.824189	+	0.566315i	$0.191631\pi$
$449$			24.0000i			1.13263i	−0.824189	+	0.566315i	$0.808369\pi$
$450$		0			0
$451$	−4.00000	−	4.00000i	−0.188353	−	0.188353i
$452$	26.0000	−	26.0000i	1.22294	−	1.22294i
$453$		0			0
$454$	12.0000	+	12.0000i	0.563188	+	0.563188i
$455$	6.00000	+	18.0000i	0.281284	+	0.843853i
$456$		0			0
$457$	−7.00000	−	7.00000i	−0.327446	−	0.327446i	0.524168	−	0.851615i	$-0.324376\pi$
$457$	−7.00000	−	7.00000i	−0.327446	−	0.327446i	−0.851615	+	0.524168i	$0.824376\pi$
$458$	2.00000			0.0934539
$459$		0			0
$460$	−2.00000	−	6.00000i	−0.0932505	−	0.279751i
$461$	21.0000	−	21.0000i	0.978068	−	0.978068i	−0.0216971	−	0.999765i	$-0.506907\pi$
$461$	21.0000	−	21.0000i	0.978068	−	0.978068i	0.999765	+	0.0216971i	$0.00690694\pi$
$462$		0			0
$463$	−19.0000	+	19.0000i	−0.883005	+	0.883005i	−0.993839	−	0.110834i	$-0.964648\pi$
$463$	−19.0000	+	19.0000i	−0.883005	+	0.883005i	0.110834	+	0.993839i	$0.464648\pi$
$464$	28.0000	+	28.0000i	1.29987	+	1.29987i
$465$		0			0
$466$	18.0000			0.833834
$467$	28.0000			1.29569			0.647843	−	0.761774i	$-0.275671\pi$
$467$	28.0000			1.29569			0.647843	+	0.761774i	$0.275671\pi$
$468$		0			0
$469$	−12.0000	+	12.0000i	−0.554109	+	0.554109i
$470$	−28.0000	−	14.0000i	−1.29154	−	0.645772i
$471$		0			0
$472$	12.0000			0.552345
$473$	4.00000	−	4.00000i	0.183920	−	0.183920i
$474$		0			0
$475$	−21.0000	−	3.00000i	−0.963546	−	0.137649i
$476$	12.0000			0.550019
$477$		0			0
$478$		0			0
$479$	−32.0000			−1.46212			−0.731059	−	0.682315i	$-0.760973\pi$
$479$	−32.0000			−1.46212			−0.731059	+	0.682315i	$0.760973\pi$
$480$		0			0
$481$	12.0000			0.547153
$482$	14.0000	+	14.0000i	0.637683	+	0.637683i
$483$		0			0
$484$	−18.0000			−0.818182
$485$	−11.0000	−	33.0000i	−0.499484	−	1.49845i
$486$		0			0
$487$	−15.0000	+	15.0000i	−0.679715	+	0.679715i	−0.959936	−	0.280221i	$-0.909592\pi$
$487$	−15.0000	+	15.0000i	−0.679715	+	0.679715i	0.280221	+	0.959936i	$0.409592\pi$
$488$			4.00000i			0.181071i
$489$		0			0
$490$	−11.0000	−	33.0000i	−0.496929	−	1.49079i
$491$	9.00000	−	9.00000i	0.406164	−	0.406164i	−0.474234	−	0.880399i	$-0.657275\pi$
$491$	9.00000	−	9.00000i	0.406164	−	0.406164i	0.880399	+	0.474234i	$0.157275\pi$
$492$		0			0
$493$	14.0000			0.630528
$494$	12.0000			0.539906
$495$		0			0
$496$			8.00000i			0.359211i
$497$		0			0
$498$		0			0
$499$	−29.0000	+	29.0000i	−1.29822	+	1.29822i	−0.368650	+	0.929568i	$0.620180\pi$
$499$	−29.0000	+	29.0000i	−1.29822	+	1.29822i	−0.929568	+	0.368650i	$0.879820\pi$
$500$	4.00000	+	22.0000i	0.178885	+	0.983870i
$501$		0			0
$502$	22.0000			0.981908
$503$	29.0000	+	29.0000i	1.29305	+	1.29305i	0.932893	+	0.360153i	$0.117275\pi$
$503$	29.0000	+	29.0000i	1.29305	+	1.29305i	0.360153	+	0.932893i	$0.382725\pi$
$504$		0			0
$505$	−15.0000	+	5.00000i	−0.667491	+	0.222497i
$506$	−2.00000	−	2.00000i	−0.0889108	−	0.0889108i
$507$		0			0
$508$	−14.0000	+	14.0000i	−0.621150	+	0.621150i
$509$	17.0000	+	17.0000i	0.753512	+	0.753512i	0.975133	−	0.221621i	$-0.0711348\pi$
$509$	17.0000	+	17.0000i	0.753512	+	0.753512i	−0.221621	+	0.975133i	$0.571135\pi$
$510$		0			0
$511$		−	18.0000i		−	0.796273i
$512$	−16.0000	−	16.0000i	−0.707107	−	0.707107i
$513$		0			0
$514$	26.0000			1.14681
$515$	15.0000	−	5.00000i	0.660979	−	0.220326i
$516$		0			0
$517$	−14.0000			−0.615719
$518$	−36.0000			−1.58175
$519$		0			0
$520$	−4.00000	−	12.0000i	−0.175412	−	0.526235i
$521$		−	16.0000i		−	0.700973i	−0.936568	−	0.350486i	$-0.886016\pi$
$521$		−	16.0000i		−	0.700973i	0.936568	−	0.350486i	$-0.113984\pi$
$522$		0			0
$523$	20.0000			0.874539			0.437269	−	0.899331i	$-0.355946\pi$
$523$	20.0000			0.874539			0.437269	+	0.899331i	$0.355946\pi$
$524$	−14.0000	+	14.0000i	−0.611593	+	0.611593i
$525$		0			0
$526$			14.0000i			0.610429i
$527$	2.00000	+	2.00000i	0.0871214	+	0.0871214i
$528$		0			0
$529$		−	21.0000i		−	0.913043i
$530$	8.00000	+	24.0000i	0.347498	+	1.04249i
$531$		0			0
$532$	−36.0000			−1.56080
$533$			8.00000i			0.346518i
$534$		0			0
$535$	12.0000	+	6.00000i	0.518805	+	0.259403i
$536$	8.00000	−	8.00000i	0.345547	−	0.345547i
$537$		0			0
$538$		−	2.00000i		−	0.0862261i
$539$	−11.0000	−	11.0000i	−0.473804	−	0.473804i
$540$		0			0
$541$	15.0000	−	15.0000i	0.644900	−	0.644900i	−0.306856	−	0.951756i	$-0.599277\pi$
$541$	15.0000	−	15.0000i	0.644900	−	0.644900i	0.951756	+	0.306856i	$0.0992769\pi$
$542$	30.0000	−	30.0000i	1.28861	−	1.28861i
$543$		0			0
$544$	−8.00000			−0.342997
$545$	15.0000	−	5.00000i	0.642529	−	0.214176i
$546$		0			0
$547$	28.0000			1.19719			0.598597	−	0.801050i	$-0.295725\pi$
$547$	28.0000			1.19719			0.598597	+	0.801050i	$0.295725\pi$
$548$	18.0000	−	18.0000i	0.768922	−	0.768922i
$549$		0			0
$550$	6.00000	+	8.00000i	0.255841	+	0.341121i
$551$	−42.0000			−1.78926
$552$		0			0
$553$	24.0000	−	24.0000i	1.02058	−	1.02058i
$554$	−18.0000	−	18.0000i	−0.764747	−	0.764747i
$555$		0			0
$556$	−18.0000	+	18.0000i	−0.763370	+	0.763370i
$557$			28.0000i			1.18640i	0.805056	+	0.593199i	$0.202135\pi$
$557$			28.0000i			1.18640i	−0.805056	+	0.593199i	$0.797865\pi$
$558$		0			0
$559$	−8.00000			−0.338364
$560$	12.0000	+	36.0000i	0.507093	+	1.52128i
$561$		0			0
$562$	16.0000	−	16.0000i	0.674919	−	0.674919i
$563$			18.0000i			0.758610i	0.925272	+	0.379305i	$0.123837\pi$
$563$			18.0000i			0.758610i	−0.925272	+	0.379305i	$0.876163\pi$
$564$		0			0
$565$	39.0000	−	13.0000i	1.64074	−	0.546914i
$566$	−12.0000	−	12.0000i	−0.504398	−	0.504398i
$567$		0			0
$568$		0			0
$569$	−6.00000			−0.251533			−0.125767	−	0.992060i	$-0.540139\pi$
$569$	−6.00000			−0.251533			−0.125767	+	0.992060i	$0.540139\pi$
$570$		0			0
$571$	−25.0000	+	25.0000i	−1.04622	+	1.04622i	−0.0473385	+	0.998879i	$0.515074\pi$
$571$	−25.0000	+	25.0000i	−1.04622	+	1.04622i	−0.998879	+	0.0473385i	$0.984926\pi$
$572$	−4.00000	−	4.00000i	−0.167248	−	0.167248i
$573$		0			0
$574$		−	24.0000i		−	1.00174i
$575$	1.00000	−	7.00000i	0.0417029	−	0.291920i
$576$		0			0
$577$	9.00000	−	9.00000i	0.374675	−	0.374675i	−0.494502	−	0.869177i	$-0.664649\pi$
$577$	9.00000	−	9.00000i	0.374675	−	0.374675i	0.869177	+	0.494502i	$0.164649\pi$
$578$	15.0000	−	15.0000i	0.623918	−	0.623918i
$579$		0			0
$580$	14.0000	+	42.0000i	0.581318	+	1.74396i
$581$	−6.00000	−	6.00000i	−0.248922	−	0.248922i
$582$		0			0
$583$	8.00000	+	8.00000i	0.331326	+	0.331326i
$584$			12.0000i			0.496564i
$585$		0			0
$586$	12.0000	−	12.0000i	0.495715	−	0.495715i
$587$			2.00000i			0.0825488i	0.999148	+	0.0412744i	$0.0131418\pi$
$587$			2.00000i			0.0825488i	−0.999148	+	0.0412744i	$0.986858\pi$
$588$		0			0
$589$	−6.00000	−	6.00000i	−0.247226	−	0.247226i
$590$	12.0000	+	6.00000i	0.494032	+	0.247016i
$591$		0			0
$592$	24.0000			0.986394
$593$	−17.0000	−	17.0000i	−0.698106	−	0.698106i	0.265896	−	0.964002i	$-0.414332\pi$
$593$	−17.0000	−	17.0000i	−0.698106	−	0.698106i	−0.964002	+	0.265896i	$0.914332\pi$
$594$		0			0
$595$	12.0000	+	6.00000i	0.491952	+	0.245976i
$596$	2.00000	+	2.00000i	0.0819232	+	0.0819232i
$597$		0			0
$598$			4.00000i			0.163572i
$599$		−	30.0000i		−	1.22577i	−0.790173	−	0.612883i	$-0.790010\pi$
$599$		−	30.0000i		−	1.22577i	0.790173	−	0.612883i	$-0.209990\pi$
$600$		0			0
$601$		−	16.0000i		−	0.652654i	−0.945257	−	0.326327i	$-0.894189\pi$
$601$		−	16.0000i		−	0.652654i	0.945257	−	0.326327i	$-0.105811\pi$
$602$	24.0000			0.978167
$603$		0			0
$604$			16.0000i			0.651031i
$605$	−18.0000	−	9.00000i	−0.731804	−	0.365902i
$606$		0			0
$607$	23.0000	+	23.0000i	0.933541	+	0.933541i	0.997925	−	0.0643840i	$-0.0205082\pi$
$607$	23.0000	+	23.0000i	0.933541	+	0.933541i	−0.0643840	+	0.997925i	$0.520508\pi$
$608$	24.0000			0.973329
$609$		0			0
$610$	−2.00000	+	4.00000i	−0.0809776	+	0.161955i
$611$	14.0000	+	14.0000i	0.566379	+	0.566379i
$612$		0			0
$613$			8.00000i			0.323117i	0.986863	+	0.161558i	$0.0516520\pi$
$613$			8.00000i			0.323117i	−0.986863	+	0.161558i	$0.948348\pi$
$614$	4.00000	+	4.00000i	0.161427	+	0.161427i
$615$		0			0
$616$	12.0000	+	12.0000i	0.483494	+	0.483494i
$617$	−25.0000	−	25.0000i	−1.00646	−	1.00646i	−0.999979	−	0.00648312i	$-0.997936\pi$
$617$	−25.0000	−	25.0000i	−1.00646	−	1.00646i	−0.00648312	−	0.999979i	$-0.502064\pi$
$618$		0			0
$619$	−7.00000	−	7.00000i	−0.281354	−	0.281354i	0.552295	−	0.833649i	$-0.313752\pi$
$619$	−7.00000	−	7.00000i	−0.281354	−	0.281354i	−0.833649	+	0.552295i	$0.813752\pi$
$620$	−4.00000	+	8.00000i	−0.160644	+	0.321288i
$621$		0			0
$622$	16.0000	+	16.0000i	0.641542	+	0.641542i
$623$	18.0000	−	18.0000i	0.721155	−	0.721155i
$624$		0			0
$625$	−7.00000	+	24.0000i	−0.280000	+	0.960000i
$626$	−26.0000			−1.03917
$627$		0			0
$628$	40.0000			1.59617
$629$	6.00000	−	6.00000i	0.239236	−	0.239236i
$630$		0			0
$631$	24.0000			0.955425			0.477712	−	0.878516i	$-0.341466\pi$
$631$	24.0000			0.955425			0.477712	+	0.878516i	$0.341466\pi$
$632$	−16.0000	+	16.0000i	−0.636446	+	0.636446i
$633$		0			0
$634$	8.00000	−	8.00000i	0.317721	−	0.317721i
$635$	−21.0000	+	7.00000i	−0.833360	+	0.277787i
$636$		0			0
$637$			22.0000i			0.871672i
$638$	14.0000	+	14.0000i	0.554265	+	0.554265i
$639$		0			0
$640$	−8.00000	−	24.0000i	−0.316228	−	0.948683i
$641$	30.0000			1.18493			0.592464	−	0.805597i	$-0.298155\pi$
$641$	30.0000			1.18493			0.592464	+	0.805597i	$0.298155\pi$
$642$		0			0
$643$		−	26.0000i		−	1.02534i	−0.858586	−	0.512670i	$-0.828656\pi$
$643$		−	26.0000i		−	1.02534i	0.858586	−	0.512670i	$-0.171344\pi$
$644$		−	12.0000i		−	0.472866i
$645$		0			0
$646$	6.00000	−	6.00000i	0.236067	−	0.236067i
$647$	15.0000	−	15.0000i	0.589711	−	0.589711i	−0.347842	−	0.937553i	$-0.613086\pi$
$647$	15.0000	−	15.0000i	0.589711	−	0.589711i	0.937553	+	0.347842i	$0.113086\pi$
$648$		0			0
$649$	6.00000			0.235521
$650$	2.00000	−	14.0000i	0.0784465	−	0.549125i
$651$		0			0
$652$	−28.0000			−1.09656
$653$	2.00000			0.0782660			0.0391330	−	0.999234i	$-0.487540\pi$
$653$	2.00000			0.0782660			0.0391330	+	0.999234i	$0.487540\pi$
$654$		0			0
$655$	−21.0000	+	7.00000i	−0.820538	+	0.273513i
$656$			16.0000i			0.624695i
$657$		0			0
$658$	−42.0000	−	42.0000i	−1.63733	−	1.63733i
$659$	−11.0000	+	11.0000i	−0.428499	+	0.428499i	−0.888117	−	0.459618i	$-0.847986\pi$
$659$	−11.0000	+	11.0000i	−0.428499	+	0.428499i	0.459618	+	0.888117i	$0.347986\pi$
$660$		0			0
$661$	−25.0000	−	25.0000i	−0.972387	−	0.972387i	0.0272416	−	0.999629i	$-0.491328\pi$
$661$	−25.0000	−	25.0000i	−0.972387	−	0.972387i	−0.999629	+	0.0272416i	$0.991328\pi$
$662$	42.0000			1.63238
$663$		0			0
$664$	4.00000	+	4.00000i	0.155230	+	0.155230i
$665$	−36.0000	−	18.0000i	−1.39602	−	0.698010i
$666$		0			0
$667$		−	14.0000i		−	0.542082i
$668$	6.00000	+	6.00000i	0.232147	+	0.232147i
$669$		0			0
$670$	12.0000	−	4.00000i	0.463600	−	0.154533i
$671$			2.00000i			0.0772091i
$672$		0			0
$673$	1.00000	+	1.00000i	0.0385472	+	0.0385472i	0.726118	−	0.687570i	$-0.241323\pi$
$673$	1.00000	+	1.00000i	0.0385472	+	0.0385472i	−0.687570	+	0.726118i	$0.741323\pi$
$674$	22.0000			0.847408
$675$		0			0
$676$		−	18.0000i		−	0.692308i
$677$	42.0000			1.61419			0.807096	−	0.590421i	$-0.201038\pi$
$677$	42.0000			1.61419			0.807096	+	0.590421i	$0.201038\pi$
$678$		0			0
$679$		−	66.0000i		−	2.53285i
$680$	−8.00000	−	4.00000i	−0.306786	−	0.153393i
$681$		0			0
$682$			4.00000i			0.153168i
$683$	4.00000			0.153056			0.0765279	−	0.997067i	$-0.475617\pi$
$683$	4.00000			0.153056			0.0765279	+	0.997067i	$0.475617\pi$
$684$		0			0
$685$	27.0000	−	9.00000i	1.03162	−	0.343872i
$686$		−	24.0000i		−	0.916324i
$687$		0			0
$688$	−16.0000			−0.609994
$689$		−	16.0000i		−	0.609551i
$690$		0			0
$691$	21.0000	+	21.0000i	0.798878	+	0.798878i	0.982919	−	0.184041i	$-0.0589179\pi$
$691$	21.0000	+	21.0000i	0.798878	+	0.798878i	−0.184041	+	0.982919i	$0.558918\pi$
$692$			12.0000i			0.456172i
$693$		0			0
$694$	2.00000	−	2.00000i	0.0759190	−	0.0759190i
$695$	−27.0000	+	9.00000i	−1.02417	+	0.341389i
$696$		0			0
$697$	4.00000	+	4.00000i	0.151511	+	0.151511i
$698$		−	6.00000i		−	0.227103i
$699$		0			0
$700$	−6.00000	+	42.0000i	−0.226779	+	1.58745i
$701$	13.0000	−	13.0000i	0.491003	−	0.491003i	−0.417619	−	0.908622i	$-0.637135\pi$
$701$	13.0000	−	13.0000i	0.491003	−	0.491003i	0.908622	+	0.417619i	$0.137135\pi$
$702$		0			0
$703$	−18.0000	+	18.0000i	−0.678883	+	0.678883i
$704$	−8.00000	−	8.00000i	−0.301511	−	0.301511i
$705$		0			0
$706$			26.0000i			0.978523i
$707$	−30.0000			−1.12827
$708$		0			0
$709$	−1.00000	+	1.00000i	−0.0375558	+	0.0375558i	−0.725635	−	0.688080i	$-0.758454\pi$
$709$	−1.00000	+	1.00000i	−0.0375558	+	0.0375558i	0.688080	+	0.725635i	$0.258454\pi$
$710$		0			0
$711$		0			0
$712$	−12.0000	+	12.0000i	−0.449719	+	0.449719i
$713$	2.00000	−	2.00000i	0.0749006	−	0.0749006i
$714$		0			0
$715$	−2.00000	−	6.00000i	−0.0747958	−	0.224387i
$716$	10.0000	+	10.0000i	0.373718	+	0.373718i
$717$		0			0
$718$	−14.0000	+	14.0000i	−0.522475	+	0.522475i
$719$	−32.0000			−1.19340			−0.596699	−	0.802465i	$-0.703521\pi$
$719$	−32.0000			−1.19340			−0.596699	+	0.802465i	$0.703521\pi$
$720$		0			0
$721$	30.0000			1.11726
$722$	1.00000	−	1.00000i	0.0372161	−	0.0372161i
$723$		0			0
$724$	−6.00000	+	6.00000i	−0.222988	+	0.222988i
$725$	−7.00000	+	49.0000i	−0.259973	+	1.81981i
$726$		0			0
$727$	−7.00000	+	7.00000i	−0.259616	+	0.259616i	−0.824898	−	0.565282i	$-0.808767\pi$
$727$	−7.00000	+	7.00000i	−0.259616	+	0.259616i	0.565282	+	0.824898i	$0.308767\pi$
$728$		−	24.0000i		−	0.889499i
$729$		0			0
$730$	−6.00000	+	12.0000i	−0.222070	+	0.444140i
$731$	−4.00000	+	4.00000i	−0.147945	+	0.147945i
$732$		0			0
$733$	−30.0000			−1.10808			−0.554038	−	0.832492i	$-0.686914\pi$
$733$	−30.0000			−1.10808			−0.554038	+	0.832492i	$0.686914\pi$
$734$			42.0000i			1.55025i
$735$		0			0
$736$			8.00000i			0.294884i
$737$	4.00000	−	4.00000i	0.147342	−	0.147342i
$738$		0			0
$739$	−21.0000	+	21.0000i	−0.772497	+	0.772497i	−0.978543	−	0.206045i	$-0.933941\pi$
$739$	−21.0000	+	21.0000i	−0.772497	+	0.772497i	0.206045	+	0.978543i	$0.433941\pi$
$740$	24.0000	+	12.0000i	0.882258	+	0.441129i
$741$		0			0
$742$			48.0000i			1.76214i
$743$	−31.0000	−	31.0000i	−1.13728	−	1.13728i	−0.988936	−	0.148344i	$-0.952606\pi$
$743$	−31.0000	−	31.0000i	−1.13728	−	1.13728i	−0.148344	−	0.988936i	$-0.547394\pi$
$744$		0			0
$745$	1.00000	+	3.00000i	0.0366372	+	0.109911i
$746$	−4.00000	+	4.00000i	−0.146450	+	0.146450i
$747$		0			0
$748$	−4.00000			−0.146254
$749$	18.0000	+	18.0000i	0.657706	+	0.657706i
$750$		0			0
$751$		−	50.0000i		−	1.82453i	−0.409605	−	0.912263i	$-0.634333\pi$
$751$		−	50.0000i		−	1.82453i	0.409605	−	0.912263i	$-0.365667\pi$
$752$	28.0000	+	28.0000i	1.02105	+	1.02105i
$753$		0			0
$754$		−	28.0000i		−	1.01970i
$755$	−8.00000	+	16.0000i	−0.291150	+	0.582300i
$756$		0			0
$757$	2.00000			0.0726912			0.0363456	−	0.999339i	$-0.488428\pi$
$757$	2.00000			0.0726912			0.0363456	+	0.999339i	$0.488428\pi$
$758$			30.0000i			1.08965i
$759$		0			0
$760$	24.0000	+	12.0000i	0.870572	+	0.435286i
$761$			40.0000i			1.45000i	0.688749	+	0.724999i	$0.258160\pi$
$761$			40.0000i			1.45000i	−0.688749	+	0.724999i	$0.741840\pi$
$762$		0			0
$763$	30.0000			1.08607
$764$	−36.0000			−1.30243
$765$		0			0
$766$	10.0000			0.361315
$767$	−6.00000	−	6.00000i	−0.216647	−	0.216647i
$768$		0			0
$769$		−	4.00000i		−	0.144244i	−0.997396	−	0.0721218i	$-0.977023\pi$
$769$		−	4.00000i		−	0.144244i	0.997396	−	0.0721218i	$-0.0229770\pi$
$770$	6.00000	+	18.0000i	0.216225	+	0.648675i
$771$		0			0
$772$	30.0000	−	30.0000i	1.07972	−	1.07972i
$773$		−	48.0000i		−	1.72644i	−0.504828	−	0.863220i	$-0.668444\pi$
$773$		−	48.0000i		−	1.72644i	0.504828	−	0.863220i	$-0.331556\pi$
$774$		0			0
$775$	−8.00000	+	6.00000i	−0.287368	+	0.215526i
$776$			44.0000i			1.57951i
$777$		0			0
$778$	46.0000			1.64918
$779$	−12.0000	−	12.0000i	−0.429945	−	0.429945i
$780$		0			0
$781$		0			0
$782$	2.00000	+	2.00000i	0.0715199	+	0.0715199i
$783$		0			0
$784$			44.0000i			1.57143i
$785$

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:	\( N \)	\(=\)	\( 720 = 2^{4} \cdot 3^{2} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	720.bd (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(-1.00000 - 1.00000i) q^{2} +2.00000i q^{4} +(-1.00000 + 2.00000i) q^{5} +(-3.00000 + 3.00000i) q^{7} +(2.00000 - 2.00000i) q^{8} +(3.00000 - 1.00000i) q^{10} +(1.00000 - 1.00000i) q^{11} -2.00000 q^{13} +6.00000 q^{14} -4.00000 q^{16} +(-1.00000 + 1.00000i) q^{17} +(3.00000 - 3.00000i) q^{19} +(-4.00000 - 2.00000i) q^{20} -2.00000 q^{22} +(1.00000 + 1.00000i) q^{23} +(-3.00000 - 4.00000i) q^{25} +(2.00000 + 2.00000i) q^{26} +(-6.00000 - 6.00000i) q^{28} +(-7.00000 - 7.00000i) q^{29} -2.00000i q^{31} +(4.00000 + 4.00000i) q^{32} +2.00000 q^{34} +(-3.00000 - 9.00000i) q^{35} -6.00000 q^{37} -6.00000 q^{38} +(2.00000 + 6.00000i) q^{40} -4.00000i q^{41} +4.00000 q^{43} +(2.00000 + 2.00000i) q^{44} -2.00000i q^{46} +(-7.00000 - 7.00000i) q^{47} -11.0000i q^{49} +(-1.00000 + 7.00000i) q^{50} -4.00000i q^{52} +8.00000i q^{53} +(1.00000 + 3.00000i) q^{55} +12.0000i q^{56} +14.0000i q^{58} +(3.00000 + 3.00000i) q^{59} +(-1.00000 + 1.00000i) q^{61} +(-2.00000 + 2.00000i) q^{62} -8.00000i q^{64} +(2.00000 - 4.00000i) q^{65} +4.00000 q^{67} +(-2.00000 - 2.00000i) q^{68} +(-6.00000 + 12.0000i) q^{70} +(-3.00000 + 3.00000i) q^{73} +(6.00000 + 6.00000i) q^{74} +(6.00000 + 6.00000i) q^{76} +6.00000i q^{77} -8.00000 q^{79} +(4.00000 - 8.00000i) q^{80} +(-4.00000 + 4.00000i) q^{82} +2.00000i q^{83} +(-1.00000 - 3.00000i) q^{85} +(-4.00000 - 4.00000i) q^{86} -4.00000i q^{88} -6.00000 q^{89} +(6.00000 - 6.00000i) q^{91} +(-2.00000 + 2.00000i) q^{92} +14.0000i q^{94} +(3.00000 + 9.00000i) q^{95} +(-11.0000 + 11.0000i) q^{97} +(-11.0000 + 11.0000i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{2} - 2 q^{5} - 6 q^{7} + 4 q^{8} + 6 q^{10} + 2 q^{11} - 4 q^{13} + 12 q^{14} - 8 q^{16} - 2 q^{17} + 6 q^{19} - 8 q^{20} - 4 q^{22} + 2 q^{23} - 6 q^{25} + 4 q^{26} - 12 q^{28} - 14 q^{29} + 8 q^{32}+ \cdots - 22 q^{98}+O(q^{100}) \) 2 * q - 2 * q^2 - 2 * q^5 - 6 * q^7 + 4 * q^8 + 6 * q^10 + 2 * q^11 - 4 * q^13 + 12 * q^14 - 8 * q^16 - 2 * q^17 + 6 * q^19 - 8 * q^20 - 4 * q^22 + 2 * q^23 - 6 * q^25 + 4 * q^26 - 12 * q^28 - 14 * q^29 + 8 * q^32 + 4 * q^34 - 6 * q^35 - 12 * q^37 - 12 * q^38 + 4 * q^40 + 8 * q^43 + 4 * q^44 - 14 * q^47 - 2 * q^50 + 2 * q^55 + 6 * q^59 - 2 * q^61 - 4 * q^62 + 4 * q^65 + 8 * q^67 - 4 * q^68 - 12 * q^70 - 6 * q^73 + 12 * q^74 + 12 * q^76 - 16 * q^79 + 8 * q^80 - 8 * q^82 - 2 * q^85 - 8 * q^86 - 12 * q^89 + 12 * q^91 - 4 * q^92 + 6 * q^95 - 22 * q^97 - 22 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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