LMFDB - Embedded newform 4560.2.d.l.2431.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4560,2,Mod(2431,4560)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4560, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 0, 0, 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("4560.2431");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4560 = 2^{4} \cdot 3 \cdot 5 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4560.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:	$36.4117833217$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} + 35x^{10} + 202x^{8} + 362x^{6} + 245x^{4} + 63x^{2} + 4 $ x^12 + 35x^10 + 202x^8 + 362x^6 + 245x^4 + 63*x^2 + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{11}\cdot 3^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			2431.1
Root			$-0.303940i$ of defining polynomial
Character	$\chi$	$=$	4560.2431
Dual form			4560.2.d.l.2431.12

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} +1.00000 q^{5} -4.54425i q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} +1.00000 q^{5} -4.54425i q^{7} +1.00000 q^{9} +2.47405i q^{11} -2.88758i q^{13} +1.00000 q^{15} +6.99322 q^{17} +(-3.60766 - 2.44639i) q^{19} -4.54425i q^{21} -2.99525i q^{23} +1.00000 q^{25} +1.00000 q^{27} +0.990054i q^{29} -3.62961 q^{31} +2.47405i q^{33} -4.54425i q^{35} -9.46589i q^{37} -2.88758i q^{39} +1.54900i q^{41} +7.98190i q^{43} +1.00000 q^{45} -0.413532i q^{47} -13.6502 q^{49} +6.99322 q^{51} -6.45935i q^{53} +2.47405i q^{55} +(-3.60766 - 2.44639i) q^{57} +13.0206 q^{59} +6.65022 q^{61} -4.54425i q^{63} -2.88758i q^{65} -7.16283 q^{67} -2.99525i q^{69} -8.62284 q^{71} +0.592475 q^{73} +1.00000 q^{75} +11.2427 q^{77} -14.9687 q^{79} +1.00000 q^{81} -3.02411i q^{83} +6.99322 q^{85} +0.990054i q^{87} +18.4643i q^{89} -13.1219 q^{91} -3.62961 q^{93} +(-3.60766 - 2.44639i) q^{95} -11.4460i q^{97} +2.47405i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q + 12 q^{3} + 12 q^{5} + 12 q^{9}+O(q^{10}) $ 12 * q + 12 * q^3 + 12 * q^5 + 12 * q^9	$ 12 q + 12 q^{3} + 12 q^{5} + 12 q^{9} + 12 q^{15} + 4 q^{17} + 12 q^{25} + 12 q^{27} - 12 q^{31} + 12 q^{45} - 28 q^{49} + 4 q^{51} + 52 q^{59} - 56 q^{61} - 32 q^{67} + 8 q^{71} + 32 q^{73} + 12 q^{75} + 24 q^{77} - 28 q^{79} + 12 q^{81} + 4 q^{85} + 32 q^{91} - 12 q^{93}+O(q^{100}) $ 12 * q + 12 * q^3 + 12 * q^5 + 12 * q^9 + 12 * q^15 + 4 * q^17 + 12 * q^25 + 12 * q^27 - 12 * q^31 + 12 * q^45 - 28 * q^49 + 4 * q^51 + 52 * q^59 - 56 * q^61 - 32 * q^67 + 8 * q^71 + 32 * q^73 + 12 * q^75 + 24 * q^77 - 28 * q^79 + 12 * q^81 + 4 * q^85 + 32 * q^91 - 12 * q^93

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/4560\mathbb{Z}\right)^\times$.

$n$	$1141$	$1711$	$1921$	$2737$	$3041$
$\chi(n)$	$1$	$-1$	$-1$	$1$	$1$

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 $

$2$		0			0
$2$		0			0
$3$	1.00000			0.577350
$3$	1.00000			0.577350
$4$		0			0
$5$	1.00000			0.447214
$5$	1.00000			0.447214
$6$		0			0
$7$		−	4.54425i		−	1.71757i	−0.512340	−	0.858783i	$-0.671221\pi$
$7$		−	4.54425i		−	1.71757i	0.512340	−	0.858783i	$-0.328779\pi$
$8$		0			0
$9$	1.00000			0.333333
$10$		0			0
$11$			2.47405i			0.745954i	0.927841	+	0.372977i	$0.121663\pi$
$11$			2.47405i			0.745954i	−0.927841	+	0.372977i	$0.878337\pi$
$12$		0			0
$13$		−	2.88758i		−	0.800871i	−0.916325	−	0.400435i	$-0.868859\pi$
$13$		−	2.88758i		−	0.800871i	0.916325	−	0.400435i	$-0.131141\pi$
$14$		0			0
$15$	1.00000			0.258199
$16$		0			0
$17$	6.99322			1.69611			0.848053	−	0.529912i	$-0.177775\pi$
$17$	6.99322			1.69611			0.848053	+	0.529912i	$0.177775\pi$
$18$		0			0
$19$	−3.60766	−	2.44639i	−0.827653	−	0.561240i
$19$	−3.60766	−	2.44639i	−0.827653	−	0.561240i
$20$		0			0
$21$		−	4.54425i		−	0.991637i
$22$		0			0
$23$		−	2.99525i		−	0.624553i	−0.949991	−	0.312277i	$-0.898908\pi$
$23$		−	2.99525i		−	0.624553i	0.949991	−	0.312277i	$-0.101092\pi$
$24$		0			0
$25$	1.00000			0.200000
$26$		0			0
$27$	1.00000			0.192450
$28$		0			0
$29$			0.990054i			0.183848i	0.995766	+	0.0919242i	$0.0293017\pi$
$29$			0.990054i			0.183848i	−0.995766	+	0.0919242i	$0.970698\pi$
$30$		0			0
$31$	−3.62961			−0.651898			−0.325949	−	0.945387i	$-0.605684\pi$
$31$	−3.62961			−0.651898			−0.325949	+	0.945387i	$0.605684\pi$
$32$		0			0
$33$			2.47405i			0.430676i
$34$		0			0
$35$		−	4.54425i		−	0.768119i
$36$		0			0
$37$		−	9.46589i		−	1.55618i	−0.628151	−	0.778092i	$-0.716188\pi$
$37$		−	9.46589i		−	1.55618i	0.628151	−	0.778092i	$-0.283812\pi$
$38$		0			0
$39$		−	2.88758i		−	0.462383i
$40$		0			0
$41$			1.54900i			0.241913i	0.992658	+	0.120957i	$0.0385962\pi$
$41$			1.54900i			0.241913i	−0.992658	+	0.120957i	$0.961404\pi$
$42$		0			0
$43$			7.98190i			1.21723i	0.793466	+	0.608614i	$0.208274\pi$
$43$			7.98190i			1.21723i	−0.793466	+	0.608614i	$0.791726\pi$
$44$		0			0
$45$	1.00000			0.149071
$46$		0			0
$47$		−	0.413532i		−	0.0603199i	−0.999545	−	0.0301599i	$-0.990398\pi$
$47$		−	0.413532i		−	0.0603199i	0.999545	−	0.0301599i	$-0.00960166\pi$
$48$		0			0
$49$	−13.6502			−1.95003
$50$		0			0
$51$	6.99322			0.979247
$52$		0			0
$53$		−	6.45935i		−	0.887260i	−0.896210	−	0.443630i	$-0.853690\pi$
$53$		−	6.45935i		−	0.887260i	0.896210	−	0.443630i	$-0.146310\pi$
$54$		0			0
$55$			2.47405i			0.333601i
$56$		0			0
$57$	−3.60766	−	2.44639i	−0.477846	−	0.324032i
$58$		0			0
$59$	13.0206			1.69514			0.847569	−	0.530685i	$-0.178065\pi$
$59$	13.0206			1.69514			0.847569	+	0.530685i	$0.178065\pi$
$60$		0			0
$61$	6.65022			0.851474			0.425737	−	0.904847i	$-0.360015\pi$
$61$	6.65022			0.851474			0.425737	+	0.904847i	$0.360015\pi$
$62$		0			0
$63$		−	4.54425i		−	0.572522i
$64$		0			0
$65$		−	2.88758i		−	0.358160i
$66$		0			0
$67$	−7.16283			−0.875079			−0.437539	−	0.899199i	$-0.644150\pi$
$67$	−7.16283			−0.875079			−0.437539	+	0.899199i	$0.644150\pi$
$68$		0			0
$69$		−	2.99525i		−	0.360586i
$70$		0			0
$71$	−8.62284			−1.02334			−0.511671	−	0.859181i	$-0.670973\pi$
$71$	−8.62284			−1.02334			−0.511671	+	0.859181i	$0.670973\pi$
$72$		0			0
$73$	0.592475			0.0693439			0.0346720	−	0.999399i	$-0.488961\pi$
$73$	0.592475			0.0693439			0.0346720	+	0.999399i	$0.488961\pi$
$74$		0			0
$75$	1.00000			0.115470
$76$		0			0
$77$	11.2427			1.28122
$78$		0			0
$79$	−14.9687			−1.68411			−0.842056	−	0.539391i	$-0.818655\pi$
$79$	−14.9687			−1.68411			−0.842056	+	0.539391i	$0.818655\pi$
$80$		0			0
$81$	1.00000			0.111111
$82$		0			0
$83$		−	3.02411i		−	0.331940i	−0.986131	−	0.165970i	$-0.946925\pi$
$83$		−	3.02411i		−	0.331940i	0.986131	−	0.165970i	$-0.0530754\pi$
$84$		0			0
$85$	6.99322			0.758521
$86$		0			0
$87$			0.990054i			0.106145i
$88$		0			0
$89$			18.4643i			1.95721i	0.205744	+	0.978606i	$0.434039\pi$
$89$			18.4643i			1.95721i	−0.205744	+	0.978606i	$0.565961\pi$
$90$		0			0
$91$	−13.1219			−1.37555
$92$		0			0
$93$	−3.62961			−0.376373
$94$		0			0
$95$	−3.60766	−	2.44639i	−0.370138	−	0.250994i
$96$		0			0
$97$		−	11.4460i		−	1.16217i	−0.813845	−	0.581083i	$-0.802629\pi$
$97$		−	11.4460i		−	1.16217i	0.813845	−	0.581083i	$-0.197371\pi$
$98$		0			0
$99$			2.47405i			0.248651i
$100$		0			0
$101$	0.688875			0.0685456			0.0342728	−	0.999413i	$-0.489088\pi$
$101$	0.688875			0.0685456			0.0342728	+	0.999413i	$0.489088\pi$
$102$		0			0
$103$	7.16283			0.705774			0.352887	−	0.935666i	$-0.385200\pi$
$103$	7.16283			0.705774			0.352887	+	0.935666i	$0.385200\pi$
$104$		0			0
$105$		−	4.54425i		−	0.443474i
$106$		0			0
$107$	9.46001			0.914534			0.457267	−	0.889330i	$-0.348828\pi$
$107$	9.46001			0.914534			0.457267	+	0.889330i	$0.348828\pi$
$108$		0			0
$109$		−	4.91923i		−	0.471177i	−0.971853	−	0.235589i	$-0.924298\pi$
$109$		−	4.91923i		−	0.471177i	0.971853	−	0.235589i	$-0.0757018\pi$
$110$		0			0
$111$		−	9.46589i		−	0.898463i
$112$		0			0
$113$		−	13.6978i		−	1.28858i	−0.764783	−	0.644288i	$-0.777154\pi$
$113$		−	13.6978i		−	1.28858i	0.764783	−	0.644288i	$-0.222846\pi$
$114$		0			0
$115$		−	2.99525i		−	0.279309i
$116$		0			0
$117$		−	2.88758i		−	0.266957i
$118$		0			0
$119$		−	31.7790i		−	2.91317i
$120$		0			0
$121$	4.87909			0.443553
$122$		0			0
$123$			1.54900i			0.139669i
$124$		0			0
$125$	1.00000			0.0894427
$126$		0			0
$127$	5.75530			0.510701			0.255350	−	0.966849i	$-0.417809\pi$
$127$	5.75530			0.510701			0.255350	+	0.966849i	$0.417809\pi$
$128$		0			0
$129$			7.98190i			0.702767i
$130$		0			0
$131$		−	2.40252i		−	0.209909i	−0.994477	−	0.104955i	$-0.966530\pi$
$131$		−	2.40252i		−	0.209909i	0.994477	−	0.104955i	$-0.0334697\pi$
$132$		0			0
$133$	−11.1170	+	16.3941i	−0.963967	+	1.42155i
$134$		0			0
$135$	1.00000			0.0860663
$136$		0			0
$137$	−5.46191			−0.466643			−0.233321	−	0.972400i	$-0.574959\pi$
$137$	−5.46191			−0.466643			−0.233321	+	0.972400i	$0.574959\pi$
$138$		0			0
$139$			5.09885i			0.432479i	0.976340	+	0.216240i	$0.0693792\pi$
$139$			5.09885i			0.432479i	−0.976340	+	0.216240i	$0.930621\pi$
$140$		0			0
$141$		−	0.413532i		−	0.0348257i
$142$		0			0
$143$	7.14401			0.597412
$144$		0			0
$145$			0.990054i			0.0822195i
$146$		0			0
$147$	−13.6502			−1.12585
$148$		0			0
$149$	−17.8821			−1.46496			−0.732478	−	0.680791i	$-0.761636\pi$
$149$	−17.8821			−1.46496			−0.732478	+	0.680791i	$0.761636\pi$
$150$		0			0
$151$	−10.8997			−0.887004			−0.443502	−	0.896273i	$-0.646264\pi$
$151$	−10.8997			−0.887004			−0.443502	+	0.896273i	$0.646264\pi$
$152$		0			0
$153$	6.99322			0.565369
$154$		0			0
$155$	−3.62961			−0.291538
$156$		0			0
$157$	5.65471			0.451295			0.225648	−	0.974209i	$-0.427550\pi$
$157$	5.65471			0.451295			0.225648	+	0.974209i	$0.427550\pi$
$158$		0			0
$159$		−	6.45935i		−	0.512260i
$160$		0			0
$161$	−13.6112			−1.07271
$162$		0			0
$163$		−	22.3684i		−	1.75203i	−0.482286	−	0.876014i	$-0.660193\pi$
$163$		−	22.3684i		−	1.75203i	0.482286	−	0.876014i	$-0.339807\pi$
$164$		0			0
$165$			2.47405i			0.192604i
$166$		0			0
$167$	−10.4032			−0.805027			−0.402513	−	0.915414i	$-0.631863\pi$
$167$	−10.4032			−0.805027			−0.402513	+	0.915414i	$0.631863\pi$
$168$		0			0
$169$	4.66188			0.358606
$170$		0			0
$171$	−3.60766	−	2.44639i	−0.275884	−	0.187080i
$172$		0			0
$173$			6.97563i			0.530348i	0.964201	+	0.265174i	$0.0854293\pi$
$173$			6.97563i			0.530348i	−0.964201	+	0.265174i	$0.914571\pi$
$174$		0			0
$175$		−	4.54425i		−	0.343513i
$176$		0			0
$177$	13.0206			0.978689
$178$		0			0
$179$	−24.9552			−1.86524			−0.932618	−	0.360865i	$-0.882481\pi$
$179$	−24.9552			−1.86524			−0.932618	+	0.360865i	$0.882481\pi$
$180$		0			0
$181$			21.6147i			1.60660i	0.595572	+	0.803302i	$0.296926\pi$
$181$			21.6147i			1.60660i	−0.595572	+	0.803302i	$0.703074\pi$
$182$		0			0
$183$	6.65022			0.491598
$184$		0			0
$185$		−	9.46589i		−	0.695946i
$186$		0			0
$187$			17.3016i			1.26522i
$188$		0			0
$189$		−	4.54425i		−	0.330546i
$190$		0			0
$191$		−	11.6341i		−	0.841812i	−0.907104	−	0.420906i	$-0.861712\pi$
$191$		−	11.6341i		−	0.841812i	0.907104	−	0.420906i	$-0.138288\pi$
$192$		0			0
$193$		−	3.69073i		−	0.265665i	−0.991139	−	0.132832i	$-0.957593\pi$
$193$		−	3.69073i		−	0.265665i	0.991139	−	0.132832i	$-0.0424072\pi$
$194$		0			0
$195$		−	2.88758i		−	0.206784i
$196$		0			0
$197$	24.9736			1.77929			0.889647	−	0.456648i	$-0.150950\pi$
$197$	24.9736			1.77929			0.889647	+	0.456648i	$0.150950\pi$
$198$		0			0
$199$		−	19.0594i		−	1.35108i	−0.737321	−	0.675542i	$-0.763910\pi$
$199$		−	19.0594i		−	1.35108i	0.737321	−	0.675542i	$-0.236090\pi$
$200$		0			0
$201$	−7.16283			−0.505227
$202$		0			0
$203$	4.49905			0.315772
$204$		0			0
$205$			1.54900i			0.108187i
$206$		0			0
$207$		−	2.99525i		−	0.208184i
$208$		0			0
$209$	6.05248	−	8.92551i	0.418659	−	0.617391i
$210$		0			0
$211$	19.8679			1.36776			0.683882	−	0.729593i	$-0.260290\pi$
$211$	19.8679			1.36776			0.683882	+	0.729593i	$0.260290\pi$
$212$		0			0
$213$	−8.62284			−0.590827
$214$		0			0
$215$			7.98190i			0.544361i
$216$		0			0
$217$			16.4939i			1.11968i
$218$		0			0
$219$	0.592475			0.0400357
$220$		0			0
$221$		−	20.1935i		−	1.35836i
$222$		0			0
$223$	−18.9046			−1.26594			−0.632972	−	0.774175i	$-0.718165\pi$
$223$	−18.9046			−1.26594			−0.632972	+	0.774175i	$0.718165\pi$
$224$		0			0
$225$	1.00000			0.0666667
$226$		0			0
$227$	6.84243			0.454148			0.227074	−	0.973877i	$-0.427084\pi$
$227$	6.84243			0.454148			0.227074	+	0.973877i	$0.427084\pi$
$228$		0			0
$229$	−4.17568			−0.275937			−0.137968	−	0.990437i	$-0.544057\pi$
$229$	−4.17568			−0.275937			−0.137968	+	0.990437i	$0.544057\pi$
$230$		0			0
$231$	11.2427			0.739715
$232$		0			0
$233$	−7.50870			−0.491911			−0.245956	−	0.969281i	$-0.579102\pi$
$233$	−7.50870			−0.491911			−0.245956	+	0.969281i	$0.579102\pi$
$234$		0			0
$235$		−	0.413532i		−	0.0269759i
$236$		0			0
$237$	−14.9687			−0.972322
$238$		0			0
$239$		−	23.3282i		−	1.50898i	−0.656313	−	0.754488i	$-0.727885\pi$
$239$		−	23.3282i		−	1.50898i	0.656313	−	0.754488i	$-0.272115\pi$
$240$		0			0
$241$			16.6377i			1.07173i	0.844304	+	0.535864i	$0.180014\pi$
$241$			16.6377i			1.07173i	−0.844304	+	0.535864i	$0.819986\pi$
$242$		0			0
$243$	1.00000			0.0641500
$244$		0			0
$245$	−13.6502			−0.872081
$246$		0			0
$247$	−7.06414	+	10.4174i	−0.449481	+	0.662843i
$248$		0			0
$249$		−	3.02411i		−	0.191645i
$250$		0			0
$251$			21.0241i			1.32703i	0.748163	+	0.663515i	$0.230936\pi$
$251$			21.0241i			1.32703i	−0.748163	+	0.663515i	$0.769064\pi$
$252$		0			0
$253$	7.41040			0.465888
$254$		0			0
$255$	6.99322			0.437933
$256$		0			0
$257$		−	1.89753i		−	0.118364i	−0.998247	−	0.0591822i	$-0.981151\pi$
$257$		−	1.89753i		−	0.118364i	0.998247	−	0.0591822i	$-0.0188493\pi$
$258$		0			0
$259$	−43.0154			−2.67285
$260$		0			0
$261$			0.990054i			0.0612828i
$262$		0			0
$263$			9.22640i			0.568924i	0.958687	+	0.284462i	$0.0918150\pi$
$263$			9.22640i			0.568924i	−0.958687	+	0.284462i	$0.908185\pi$
$264$		0			0
$265$		−	6.45935i		−	0.396795i
$266$		0			0
$267$			18.4643i			1.13000i
$268$		0			0
$269$		−	19.7203i		−	1.20237i	−0.799111	−	0.601184i	$-0.794696\pi$
$269$		−	19.7203i		−	1.20237i	0.799111	−	0.601184i	$-0.205304\pi$
$270$		0			0
$271$			4.37592i			0.265818i	0.991128	+	0.132909i	$0.0424318\pi$
$271$			4.37592i			0.265818i	−0.991128	+	0.132909i	$0.957568\pi$
$272$		0			0
$273$	−13.1219			−0.794173
$274$		0			0
$275$			2.47405i			0.149191i
$276$		0			0
$277$	24.1315			1.44992			0.724962	−	0.688789i	$-0.241857\pi$
$277$	24.1315			1.44992			0.724962	+	0.688789i	$0.241857\pi$
$278$		0			0
$279$	−3.62961			−0.217299
$280$		0			0
$281$		−	15.2039i		−	0.906987i	−0.891260	−	0.453493i	$-0.850178\pi$
$281$		−	15.2039i		−	0.906987i	0.891260	−	0.453493i	$-0.149822\pi$
$282$		0			0
$283$		−	11.2469i		−	0.668557i	−0.942474	−	0.334278i	$-0.891507\pi$
$283$		−	11.2469i		−	0.668557i	0.942474	−	0.334278i	$-0.108493\pi$
$284$		0			0
$285$	−3.60766	−	2.44639i	−0.213699	−	0.144912i
$286$		0			0
$287$	7.03904			0.415502
$288$		0			0
$289$	31.9052			1.87677
$290$		0			0
$291$		−	11.4460i		−	0.670976i
$292$		0			0
$293$			25.2957i			1.47779i	0.673820	+	0.738895i	$0.264652\pi$
$293$			25.2957i			1.47779i	−0.673820	+	0.738895i	$0.735348\pi$
$294$		0			0
$295$	13.0206			0.758089
$296$		0			0
$297$			2.47405i			0.143559i
$298$		0			0
$299$	−8.64903			−0.500186
$300$		0			0
$301$	36.2718			2.09067
$302$		0			0
$303$	0.688875			0.0395748
$304$		0			0
$305$	6.65022			0.380791
$306$		0			0
$307$	29.2126			1.66725			0.833626	−	0.552330i	$-0.186261\pi$
$307$	29.2126			1.66725			0.833626	+	0.552330i	$0.186261\pi$
$308$		0			0
$309$	7.16283			0.407479
$310$		0			0
$311$			16.4577i			0.933233i	0.884460	+	0.466616i	$0.154527\pi$
$311$			16.4577i			0.933233i	−0.884460	+	0.466616i	$0.845473\pi$
$312$		0			0
$313$	−6.34778			−0.358798			−0.179399	−	0.983776i	$-0.557415\pi$
$313$	−6.34778			−0.358798			−0.179399	+	0.983776i	$0.557415\pi$
$314$		0			0
$315$		−	4.54425i		−	0.256040i
$316$		0			0
$317$			17.8188i			1.00080i	0.865793	+	0.500402i	$0.166814\pi$
$317$			17.8188i			1.00080i	−0.865793	+	0.500402i	$0.833186\pi$
$318$		0			0
$319$	−2.44944			−0.137142
$320$		0			0
$321$	9.46001			0.528006
$322$		0			0
$323$	−25.2291	−	17.1081i	−1.40379	−	0.951923i
$324$		0			0
$325$		−	2.88758i		−	0.160174i
$326$		0			0
$327$		−	4.91923i		−	0.272034i
$328$		0			0
$329$	−1.87919			−0.103603
$330$		0			0
$331$	4.16770			0.229078			0.114539	−	0.993419i	$-0.463461\pi$
$331$	4.16770			0.229078			0.114539	+	0.993419i	$0.463461\pi$
$332$		0			0
$333$		−	9.46589i		−	0.518728i
$334$		0			0
$335$	−7.16283			−0.391347
$336$		0			0
$337$			6.09029i			0.331759i	0.986146	+	0.165880i	$0.0530463\pi$
$337$			6.09029i			0.331759i	−0.986146	+	0.165880i	$0.946954\pi$
$338$		0			0
$339$		−	13.6978i		−	0.743960i
$340$		0			0
$341$		−	8.97984i		−	0.486286i
$342$		0			0
$343$			30.2203i			1.63174i
$344$		0			0
$345$		−	2.99525i		−	0.161259i
$346$		0			0
$347$		−	10.8641i		−	0.583217i	−0.956538	−	0.291609i	$-0.905810\pi$
$347$		−	10.8641i		−	0.583217i	0.956538	−	0.291609i	$-0.0941905\pi$
$348$		0			0
$349$	35.2075			1.88461			0.942307	−	0.334750i	$-0.108652\pi$
$349$	35.2075			1.88461			0.942307	+	0.334750i	$0.108652\pi$
$350$		0			0
$351$		−	2.88758i		−	0.154128i
$352$		0			0
$353$	9.68677			0.515575			0.257787	−	0.966202i	$-0.417007\pi$
$353$	9.68677			0.515575			0.257787	+	0.966202i	$0.417007\pi$
$354$		0			0
$355$	−8.62284			−0.457653
$356$		0			0
$357$		−	31.7790i		−	1.68192i
$358$		0			0
$359$			16.4968i			0.870670i	0.900269	+	0.435335i	$0.143370\pi$
$359$			16.4968i			0.870670i	−0.900269	+	0.435335i	$0.856630\pi$
$360$		0			0
$361$	7.03036	+	17.6515i	0.370019	+	0.929024i
$362$		0			0
$363$	4.87909			0.256086
$364$		0			0
$365$	0.592475			0.0310116
$366$		0			0
$367$		−	13.1027i		−	0.683954i	−0.939708	−	0.341977i	$-0.888904\pi$
$367$		−	13.1027i		−	0.683954i	0.939708	−	0.341977i	$-0.111096\pi$
$368$		0			0
$369$			1.54900i			0.0806377i
$370$		0			0
$371$	−29.3529			−1.52393
$372$		0			0
$373$		−	20.1846i		−	1.04512i	−0.852603	−	0.522560i	$-0.824977\pi$
$373$		−	20.1846i		−	1.04512i	0.852603	−	0.522560i	$-0.175023\pi$
$374$		0			0
$375$	1.00000			0.0516398
$376$		0			0
$377$	2.85886			0.147239
$378$		0			0
$379$	−2.40552			−0.123563			−0.0617817	−	0.998090i	$-0.519678\pi$
$379$	−2.40552			−0.123563			−0.0617817	+	0.998090i	$0.519678\pi$
$380$		0			0
$381$	5.75530			0.294853
$382$		0			0
$383$	23.2046			1.18570			0.592851	−	0.805313i	$-0.298002\pi$
$383$	23.2046			1.18570			0.592851	+	0.805313i	$0.298002\pi$
$384$		0			0
$385$	11.2427			0.572981
$386$		0			0
$387$			7.98190i			0.405743i
$388$		0			0
$389$	30.4171			1.54221			0.771103	−	0.636710i	$-0.219705\pi$
$389$	30.4171			1.54221			0.771103	+	0.636710i	$0.219705\pi$
$390$		0			0
$391$		−	20.9465i		−	1.05931i
$392$		0			0
$393$		−	2.40252i		−	0.121191i
$394$		0			0
$395$	−14.9687			−0.753158
$396$		0			0
$397$	21.3001			1.06902			0.534510	−	0.845162i	$-0.320496\pi$
$397$	21.3001			1.06902			0.534510	+	0.845162i	$0.320496\pi$
$398$		0			0
$399$	−11.1170	+	16.3941i	−0.556546	+	0.820731i
$400$		0			0
$401$			7.24706i			0.361901i	0.983492	+	0.180950i	$0.0579174\pi$
$401$			7.24706i			0.361901i	−0.983492	+	0.180950i	$0.942083\pi$
$402$		0			0
$403$			10.4808i			0.522086i
$404$		0			0
$405$	1.00000			0.0496904
$406$		0			0
$407$	23.4191			1.16084
$408$		0			0
$409$		−	28.2264i		−	1.39570i	−0.716242	−	0.697852i	$-0.754139\pi$
$409$		−	28.2264i		−	1.39570i	0.716242	−	0.697852i	$-0.245861\pi$
$410$		0			0
$411$	−5.46191			−0.269416
$412$		0			0
$413$		−	59.1689i		−	2.91151i
$414$		0			0
$415$		−	3.02411i		−	0.148448i
$416$		0			0
$417$			5.09885i			0.249692i
$418$		0			0
$419$		−	4.34352i		−	0.212195i	−0.994356	−	0.106097i	$-0.966164\pi$
$419$		−	4.34352i		−	0.212195i	0.994356	−	0.106097i	$-0.0338355\pi$
$420$		0			0
$421$			37.1151i			1.80888i	0.426603	+	0.904439i	$0.359710\pi$
$421$			37.1151i			1.80888i	−0.426603	+	0.904439i	$0.640290\pi$
$422$		0			0
$423$		−	0.413532i		−	0.0201066i
$424$		0			0
$425$	6.99322			0.339221
$426$		0			0
$427$		−	30.2203i		−	1.46246i
$428$		0			0
$429$	7.14401			0.344916
$430$		0			0
$431$	8.92670			0.429984			0.214992	−	0.976616i	$-0.431027\pi$
$431$	8.92670			0.429984			0.214992	+	0.976616i	$0.431027\pi$
$432$		0			0
$433$		−	5.56613i		−	0.267491i	−0.991016	−	0.133746i	$-0.957299\pi$
$433$		−	5.56613i		−	0.267491i	0.991016	−	0.133746i	$-0.0427005\pi$
$434$		0			0
$435$			0.990054i			0.0474694i
$436$		0			0
$437$	−7.32755	+	10.8058i	−0.350524	+	0.516913i
$438$		0			0
$439$	13.7889			0.658107			0.329053	−	0.944311i	$-0.393270\pi$
$439$	13.7889			0.658107			0.329053	+	0.944311i	$0.393270\pi$
$440$		0			0
$441$	−13.6502			−0.650011
$442$		0			0
$443$		−	1.74379i		−	0.0828499i	−0.999142	−	0.0414250i	$-0.986810\pi$
$443$		−	1.74379i		−	0.0828499i	0.999142	−	0.0414250i	$-0.0131898\pi$
$444$		0			0
$445$			18.4643i			0.875292i
$446$		0			0
$447$	−17.8821			−0.845793
$448$		0			0
$449$			12.3829i			0.584385i	0.956360	+	0.292192i	$0.0943848\pi$
$449$			12.3829i			0.584385i	−0.956360	+	0.292192i	$0.905615\pi$
$450$		0			0
$451$	−3.83230			−0.180456
$452$		0			0
$453$	−10.8997			−0.512112
$454$		0			0
$455$	−13.1219			−0.615164
$456$		0			0
$457$	−37.8765			−1.77179			−0.885894	−	0.463888i	$-0.846454\pi$
$457$	−37.8765			−1.77179			−0.885894	+	0.463888i	$0.846454\pi$
$458$		0			0
$459$	6.99322			0.326416
$460$		0			0
$461$	6.43004			0.299477			0.149738	−	0.988726i	$-0.452157\pi$
$461$	6.43004			0.299477			0.149738	+	0.988726i	$0.452157\pi$
$462$		0			0
$463$			17.7248i			0.823741i	0.911242	+	0.411870i	$0.135124\pi$
$463$			17.7248i			0.823741i	−0.911242	+	0.411870i	$0.864876\pi$
$464$		0			0
$465$	−3.62961			−0.168319
$466$		0			0
$467$			23.7477i			1.09891i	0.835523	+	0.549456i	$0.185165\pi$
$467$			23.7477i			1.09891i	−0.835523	+	0.549456i	$0.814835\pi$
$468$		0			0
$469$			32.5497i			1.50301i
$470$		0			0
$471$	5.65471			0.260555
$472$		0			0
$473$	−19.7476			−0.907995
$474$		0			0
$475$	−3.60766	−	2.44639i	−0.165531	−	0.112248i
$476$		0			0
$477$		−	6.45935i		−	0.295753i
$478$		0			0
$479$		−	33.4817i		−	1.52982i	−0.644138	−	0.764909i	$-0.722784\pi$
$479$		−	33.4817i		−	1.52982i	0.644138	−	0.764909i	$-0.277216\pi$
$480$		0			0
$481$	−27.3335			−1.24630
$482$		0			0
$483$	−13.6112			−0.619330
$484$		0			0
$485$		−	11.4460i		−	0.519736i
$486$		0			0
$487$	−27.2262			−1.23374			−0.616868	−	0.787067i	$-0.711599\pi$
$487$	−27.2262			−1.23374			−0.616868	+	0.787067i	$0.711599\pi$
$488$		0			0
$489$		−	22.3684i		−	1.01153i
$490$		0			0
$491$			14.6076i			0.659234i	0.944115	+	0.329617i	$0.106920\pi$
$491$			14.6076i			0.659234i	−0.944115	+	0.329617i	$0.893080\pi$
$492$		0			0
$493$			6.92367i			0.311826i
$494$		0			0
$495$			2.47405i			0.111200i
$496$		0			0
$497$			39.1843i			1.75766i
$498$		0			0
$499$		−	22.1925i		−	0.993474i	−0.867901	−	0.496737i	$-0.834531\pi$
$499$		−	22.1925i		−	0.993474i	0.867901	−	0.496737i	$-0.165469\pi$
$500$		0			0
$501$	−10.4032			−0.464782
$502$		0			0
$503$			32.7200i			1.45891i	0.684027	+	0.729457i	$0.260227\pi$
$503$			32.7200i			1.45891i	−0.684027	+	0.729457i	$0.739773\pi$
$504$		0			0
$505$	0.688875			0.0306545
$506$		0			0
$507$	4.66188			0.207041
$508$		0			0
$509$		−	9.44237i		−	0.418526i	−0.977859	−	0.209263i	$-0.932894\pi$
$509$		−	9.44237i		−	0.418526i	0.977859	−	0.209263i	$-0.0671065\pi$
$510$		0			0
$511$		−	2.69236i		−	0.119103i
$512$		0			0
$513$	−3.60766	−	2.44639i	−0.159282	−	0.108011i
$514$		0			0
$515$	7.16283			0.315632
$516$		0			0
$517$	1.02310			0.0449958
$518$		0			0
$519$			6.97563i			0.306196i
$520$		0			0
$521$		−	20.9506i		−	0.917861i	−0.888472	−	0.458931i	$-0.848233\pi$
$521$		−	20.9506i		−	0.917861i	0.888472	−	0.458931i	$-0.151767\pi$
$522$		0			0
$523$	20.8910			0.913501			0.456750	−	0.889595i	$-0.349013\pi$
$523$	20.8910			0.913501			0.456750	+	0.889595i	$0.349013\pi$
$524$		0			0
$525$		−	4.54425i		−	0.198327i
$526$		0			0
$527$	−25.3827			−1.10569
$528$		0			0
$529$	14.0285			0.609933
$530$		0			0
$531$	13.0206			0.565046
$532$		0			0
$533$	4.47286			0.193741
$534$		0			0
$535$	9.46001			0.408992
$536$		0			0
$537$	−24.9552			−1.07689
$538$		0			0
$539$		−	33.7713i		−	1.45463i
$540$		0			0
$541$	−3.88148			−0.166878			−0.0834389	−	0.996513i	$-0.526590\pi$
$541$	−3.88148			−0.166878			−0.0834389	+	0.996513i	$0.526590\pi$
$542$		0			0
$543$			21.6147i			0.927573i
$544$		0			0
$545$		−	4.91923i		−	0.210717i
$546$		0			0
$547$	−11.4025			−0.487537			−0.243769	−	0.969833i	$-0.578384\pi$
$547$	−11.4025			−0.487537			−0.243769	+	0.969833i	$0.578384\pi$
$548$		0			0
$549$	6.65022			0.283825
$550$		0			0
$551$	2.42206	−	3.57177i	0.103183	−	0.152163i
$552$		0			0
$553$			68.0216i			2.89257i
$554$		0			0
$555$		−	9.46589i		−	0.401805i
$556$		0			0
$557$	−25.5891			−1.08424			−0.542122	−	0.840300i	$-0.682379\pi$
$557$	−25.5891			−1.08424			−0.542122	+	0.840300i	$0.682379\pi$
$558$		0			0
$559$	23.0484			0.974842
$560$		0			0
$561$			17.3016i			0.730473i
$562$		0			0
$563$	−2.25566			−0.0950645			−0.0475323	−	0.998870i	$-0.515136\pi$
$563$	−2.25566			−0.0950645			−0.0475323	+	0.998870i	$0.515136\pi$
$564$		0			0
$565$		−	13.6978i		−	0.576269i
$566$		0			0
$567$		−	4.54425i		−	0.190841i
$568$		0			0
$569$		−	4.05465i		−	0.169980i	−0.996382	−	0.0849900i	$-0.972914\pi$
$569$		−	4.05465i		−	0.169980i	0.996382	−	0.0849900i	$-0.0270858\pi$
$570$		0			0
$571$		−	7.56936i		−	0.316768i	−0.987378	−	0.158384i	$-0.949372\pi$
$571$		−	7.56936i		−	0.316768i	0.987378	−	0.158384i	$-0.0506284\pi$
$572$		0			0
$573$		−	11.6341i		−	0.486021i
$574$		0			0
$575$		−	2.99525i		−	0.124911i
$576$		0			0
$577$	−33.5963			−1.39863			−0.699317	−	0.714812i	$-0.746512\pi$
$577$	−33.5963			−1.39863			−0.699317	+	0.714812i	$0.746512\pi$
$578$		0			0
$579$		−	3.69073i		−	0.153382i
$580$		0			0
$581$	−13.7423			−0.570128
$582$		0			0
$583$	15.9807			0.661855
$584$		0			0
$585$		−	2.88758i		−	0.119387i
$586$		0			0
$587$		−	33.7652i		−	1.39364i	−0.717246	−	0.696820i	$-0.754598\pi$
$587$		−	33.7652i		−	1.39364i	0.717246	−	0.696820i	$-0.245402\pi$
$588$		0			0
$589$	13.0944	+	8.87945i	0.539545	+	0.365871i
$590$		0			0
$591$	24.9736			1.02728
$592$		0			0
$593$	−40.1130			−1.64724			−0.823622	−	0.567140i	$-0.808050\pi$
$593$	−40.1130			−1.64724			−0.823622	+	0.567140i	$0.808050\pi$
$594$		0			0
$595$		−	31.7790i		−	1.30281i
$596$		0			0
$597$		−	19.0594i		−	0.780049i
$598$		0			0
$599$	14.0937			0.575853			0.287926	−	0.957653i	$-0.407034\pi$
$599$	14.0937			0.575853			0.287926	+	0.957653i	$0.407034\pi$
$600$		0			0
$601$			20.5785i			0.839416i	0.907659	+	0.419708i	$0.137867\pi$
$601$			20.5785i			0.839416i	−0.907659	+	0.419708i	$0.862133\pi$
$602$		0			0
$603$	−7.16283			−0.291693
$604$		0			0
$605$	4.87909			0.198363
$606$		0			0
$607$	−33.7948			−1.37169			−0.685845	−	0.727748i	$-0.740567\pi$
$607$	−33.7948			−1.37169			−0.685845	+	0.727748i	$0.740567\pi$
$608$		0			0
$609$	4.49905			0.182311
$610$		0			0
$611$	−1.19411			−0.0483084
$612$		0			0
$613$	−15.1256			−0.610916			−0.305458	−	0.952206i	$-0.598810\pi$
$613$	−15.1256			−0.610916			−0.305458	+	0.952206i	$0.598810\pi$
$614$		0			0
$615$			1.54900i			0.0624617i
$616$		0			0
$617$	−15.4922			−0.623691			−0.311846	−	0.950133i	$-0.600947\pi$
$617$	−15.4922			−0.623691			−0.311846	+	0.950133i	$0.600947\pi$
$618$		0			0
$619$			32.5614i			1.30875i	0.756169	+	0.654376i	$0.227069\pi$
$619$			32.5614i			1.30875i	−0.756169	+	0.654376i	$0.772931\pi$
$620$		0			0
$621$		−	2.99525i		−	0.120195i
$622$		0			0
$623$	83.9064			3.36164
$624$		0			0
$625$	1.00000			0.0400000
$626$		0			0
$627$	6.05248	−	8.92551i	0.241713	−	0.356451i
$628$		0			0
$629$		−	66.1971i		−	2.63945i
$630$		0			0
$631$			46.4010i			1.84719i	0.383366	+	0.923597i	$0.374765\pi$
$631$			46.4010i			1.84719i	−0.383366	+	0.923597i	$0.625235\pi$
$632$		0			0
$633$	19.8679			0.789679
$634$		0			0
$635$	5.75530			0.228392
$636$		0			0
$637$			39.4161i			1.56172i
$638$		0			0
$639$	−8.62284			−0.341114
$640$		0			0
$641$			24.9081i			0.983809i	0.870649	+	0.491905i	$0.163699\pi$
$641$			24.9081i			0.983809i	−0.870649	+	0.491905i	$0.836301\pi$
$642$		0			0
$643$			7.92417i			0.312499i	0.987718	+	0.156249i	$0.0499404\pi$
$643$			7.92417i			0.312499i	−0.987718	+	0.156249i	$0.950060\pi$
$644$		0			0
$645$			7.98190i			0.314287i
$646$		0			0
$647$		−	12.8731i		−	0.506094i	−0.967454	−	0.253047i	$-0.918567\pi$
$647$		−	12.8731i		−	0.506094i	0.967454	−	0.253047i	$-0.0814327\pi$
$648$		0			0
$649$			32.2136i			1.26449i
$650$		0			0
$651$			16.4939i			0.646446i
$652$		0			0
$653$	9.45703			0.370082			0.185041	−	0.982731i	$-0.440758\pi$
$653$	9.45703			0.370082			0.185041	+	0.982731i	$0.440758\pi$
$654$		0			0
$655$		−	2.40252i		−	0.0938742i
$656$		0			0
$657$	0.592475			0.0231146
$658$		0			0
$659$	48.0599			1.87215			0.936074	−	0.351802i	$-0.114431\pi$
$659$	48.0599			1.87215			0.936074	+	0.351802i	$0.114431\pi$
$660$		0			0
$661$			9.48844i			0.369058i	0.982827	+	0.184529i	$0.0590759\pi$
$661$			9.48844i			0.369058i	−0.982827	+	0.184529i	$0.940924\pi$
$662$		0			0
$663$		−	20.1935i		−	0.784250i
$664$		0			0
$665$	−11.1170	+	16.3941i	−0.431099	+	0.635736i
$666$		0			0
$667$	2.96546			0.114823
$668$		0			0
$669$	−18.9046			−0.730893
$670$		0			0
$671$			16.4530i			0.635160i
$672$		0			0
$673$			21.2448i			0.818926i	0.912327	+	0.409463i	$0.134284\pi$
$673$			21.2448i			0.818926i	−0.912327	+	0.409463i	$0.865716\pi$
$674$		0			0
$675$	1.00000			0.0384900
$676$		0			0
$677$			29.8682i			1.14793i	0.818880	+	0.573964i	$0.194595\pi$
$677$			29.8682i			1.14793i	−0.818880	+	0.573964i	$0.805405\pi$
$678$		0			0
$679$	−52.0135			−1.99609
$680$		0			0
$681$	6.84243			0.262203
$682$		0			0
$683$	−10.0080			−0.382945			−0.191472	−	0.981498i	$-0.561326\pi$
$683$	−10.0080			−0.382945			−0.191472	+	0.981498i	$0.561326\pi$
$684$		0			0
$685$	−5.46191			−0.208689
$686$		0			0
$687$	−4.17568			−0.159312
$688$		0			0
$689$	−18.6519			−0.710581
$690$		0			0
$691$			13.3219i			0.506791i	0.967363	+	0.253395i	$0.0815473\pi$
$691$			13.3219i			0.506791i	−0.967363	+	0.253395i	$0.918453\pi$
$692$		0			0
$693$	11.2427			0.427075
$694$		0			0
$695$			5.09885i			0.193411i
$696$		0			0
$697$			10.8325i			0.410310i
$698$		0			0
$699$	−7.50870			−0.284005
$700$		0			0
$701$	−35.8354			−1.35349			−0.676743	−	0.736219i	$-0.736609\pi$
$701$	−35.8354			−1.35349			−0.676743	+	0.736219i	$0.736609\pi$
$702$		0			0
$703$	−23.1573	+	34.1497i	−0.873392	+	1.28798i
$704$		0			0
$705$		−	0.413532i		−	0.0155745i
$706$		0			0
$707$		−	3.13042i		−	0.117732i
$708$		0			0
$709$	46.3265			1.73983			0.869914	−	0.493203i	$-0.164174\pi$
$709$	46.3265			1.73983			0.869914	+	0.493203i	$0.164174\pi$
$710$		0			0
$711$	−14.9687			−0.561370
$712$		0			0
$713$			10.8716i			0.407145i
$714$		0			0
$715$	7.14401			0.267171
$716$		0			0
$717$		−	23.3282i		−	0.871208i
$718$		0			0
$719$		−	2.09034i		−	0.0779567i	−0.999240	−	0.0389783i	$-0.987590\pi$
$719$		−	2.09034i		−	0.0779567i	0.999240	−	0.0389783i	$-0.0124103\pi$
$720$		0			0
$721$		−	32.5497i		−	1.21221i
$722$		0			0
$723$			16.6377i			0.618762i
$724$		0			0
$725$			0.990054i			0.0367697i
$726$		0			0
$727$			24.3884i			0.904517i	0.891887	+	0.452258i	$0.149382\pi$
$727$			24.3884i			0.904517i	−0.891887	+	0.452258i	$0.850618\pi$
$728$		0			0
$729$	1.00000			0.0370370
$730$		0			0
$731$			55.8192i			2.06455i
$732$		0			0
$733$	−11.3598			−0.419584			−0.209792	−	0.977746i	$-0.567279\pi$
$733$	−11.3598			−0.419584			−0.209792	+	0.977746i	$0.567279\pi$
$734$		0			0
$735$	−13.6502			−0.503496
$736$		0			0
$737$		−	17.7212i		−	0.652768i
$738$		0			0
$739$		−	26.4284i		−	0.972184i	−0.873908	−	0.486092i	$-0.838422\pi$
$739$		−	26.4284i		−	0.972184i	0.873908	−	0.486092i	$-0.161578\pi$
$740$		0			0
$741$	−7.06414	+	10.4174i	−0.259508	+	0.382693i
$742$		0			0
$743$	34.9080			1.28065			0.640326	−	0.768103i	$-0.278799\pi$
$743$	34.9080			1.28065			0.640326	+	0.768103i	$0.278799\pi$
$744$		0			0
$745$	−17.8821			−0.655148
$746$		0			0
$747$		−	3.02411i		−	0.110647i
$748$		0			0
$749$		−	42.9887i		−	1.57077i
$750$		0			0
$751$	10.9328			0.398942			0.199471	−	0.979904i	$-0.436078\pi$
$751$	10.9328			0.398942			0.199471	+	0.979904i	$0.436078\pi$
$752$		0			0
$753$			21.0241i			0.766161i
$754$		0			0
$755$	−10.8997			−0.396680
$756$		0			0
$757$	24.2411			0.881057			0.440528	−	0.897739i	$-0.354791\pi$
$757$	24.2411			0.881057			0.440528	+	0.897739i	$0.354791\pi$
$758$		0			0
$759$	7.41040			0.268980
$760$		0			0
$761$	−28.3127			−1.02633			−0.513167	−	0.858289i	$-0.671528\pi$
$761$	−28.3127			−1.02633			−0.513167	+	0.858289i	$0.671528\pi$
$762$		0			0
$763$	−22.3542			−0.809278
$764$		0			0
$765$	6.99322			0.252840
$766$		0			0
$767$		−	37.5980i		−	1.35759i
$768$		0			0
$769$	−35.7118			−1.28780			−0.643900	−	0.765110i	$-0.722685\pi$
$769$	−35.7118			−1.28780			−0.643900	+	0.765110i	$0.722685\pi$
$770$		0			0
$771$		−	1.89753i		−	0.0683377i
$772$		0			0
$773$		−	12.2162i		−	0.439385i	−0.975569	−	0.219693i	$-0.929495\pi$
$773$		−	12.2162i		−	0.439385i	0.975569	−	0.219693i	$-0.0705054\pi$
$774$		0			0
$775$	−3.62961			−0.130380
$776$		0			0
$777$	−43.0154			−1.54317
$778$		0			0
$779$	3.78946	−	5.58826i	0.135771	−	0.200220i
$780$		0			0
$781$		−	21.3333i		−	0.763366i
$782$		0			0
$783$			0.990054i			0.0353816i
$784$		0			0
$785$	5.65471			0.201825
$786$		0			0
$787$	−26.7570			−0.953784			−0.476892	−	0.878962i	$-0.658237\pi$
$787$	−26.7570			−0.953784			−0.476892	+	0.878962i	$0.658237\pi$
$788$		0			0
$789$			9.22640i			0.328469i
$790$		0			0
$791$	−62.2461			−2.21322
$792$		0			0
$793$		−	19.2030i		−	0.681920i
$794$		0			0
$795$		−	6.45935i		−	0.229090i
$796$		0			0
$797$		−	28.0106i		−	0.992185i	−0.868270	−	0.496093i	$-0.834768\pi$
$797$		−	28.0106i		−	0.992185i	0.868270	−	0.496093i	$-0.165232\pi$
$798$		0			0
$799$		−	2.89192i		−	0.102309i
$800$		0			0
$801$			18.4643i			0.652404i
$802$		0			0
$803$			1.46581i			0.0517274i
$804$		0			0
$805$	−13.6112			−0.479731
$806$		0			0
$807$		−	19.7203i		−	0.694187i
$808$		0			0
$809$	34.0383			1.19672			0.598362	−	0.801226i	$-0.295818\pi$
$809$	34.0383			1.19672			0.598362	+	0.801226i	$0.295818\pi$
$810$		0			0
$811$	54.1178			1.90033			0.950166	−	0.311744i	$-0.100913\pi$
$811$	54.1178			1.90033			0.950166	+	0.311744i	$0.100913\pi$
$812$		0			0
$813$			4.37592i			0.153470i
$814$		0			0
$815$		−	22.3684i		−	0.783530i
$816$		0			0
$817$	19.5268	−	28.7959i	0.683157	−	1.00744i
$818$		0			0
$819$	−13.1219			−0.458516
$820$		0			0
$821$	23.5316			0.821258			0.410629	−	0.911803i	$-0.365309\pi$
$821$	23.5316			0.821258			0.410629	+	0.911803i	$0.365309\pi$
$822$		0			0
$823$			54.5957i			1.90309i	0.307517	+	0.951543i	$0.400502\pi$
$823$			54.5957i			1.90309i	−0.307517	+	0.951543i	$0.599498\pi$
$824$		0			0
$825$			2.47405i			0.0861353i
$826$		0			0
$827$	−9.21255			−0.320352			−0.160176	−	0.987088i	$-0.551206\pi$
$827$	−9.21255			−0.320352			−0.160176	+	0.987088i	$0.551206\pi$
$828$		0			0
$829$			13.4990i			0.468840i	0.972135	+	0.234420i	$0.0753190\pi$
$829$			13.4990i			0.468840i	−0.972135	+	0.234420i	$0.924681\pi$
$830$		0			0
$831$	24.1315			0.837114
$832$		0			0
$833$	−95.4590			−3.30746
$834$		0			0
$835$	−10.4032			−0.360019
$836$		0			0
$837$	−3.62961			−0.125458
$838$		0			0
$839$	26.0044			0.897771			0.448886	−	0.893589i	$-0.351821\pi$
$839$	26.0044			0.897771			0.448886	+	0.893589i	$0.351821\pi$
$840$		0			0
$841$	28.0198			0.966200
$842$		0			0
$843$		−	15.2039i		−	0.523649i
$844$		0			0
$845$	4.66188			0.160374
$846$		0			0
$847$		−	22.1718i		−	0.761832i
$848$		0			0
$849$		−	11.2469i		−	0.385991i
$850$		0			0
$851$	−28.3527			−0.971919
$852$		0			0
$853$	24.1315			0.826248			0.413124	−	0.910675i	$-0.364438\pi$
$853$	24.1315			0.826248			0.413124	+	0.910675i	$0.364438\pi$
$854$		0			0
$855$	−3.60766	−	2.44639i	−0.123379	−	0.0836647i
$856$		0			0
$857$			16.3145i			0.557292i	0.960394	+	0.278646i	$0.0898855\pi$
$857$			16.3145i			0.557292i	−0.960394	+	0.278646i	$0.910114\pi$
$858$		0			0
$859$			50.9676i			1.73899i	0.493940	+	0.869496i	$0.335556\pi$
$859$			50.9676i			1.73899i	−0.493940	+	0.869496i	$0.664444\pi$
$860$		0			0
$861$	7.03904			0.239890
$862$		0			0
$863$	51.9884			1.76971			0.884853	−	0.465871i	$-0.154259\pi$
$863$	51.9884			1.76971			0.884853	+	0.465871i	$0.154259\pi$
$864$		0			0
$865$			6.97563i			0.237179i
$866$		0			0
$867$	31.9052			1.08356
$868$		0			0
$869$		−	37.0333i		−	1.25627i
$870$		0			0
$871$			20.6832i			0.700825i
$872$		0			0
$873$		−	11.4460i		−	0.387388i
$874$		0			0
$875$		−	4.54425i		−	0.153624i
$876$		0			0
$877$			21.1707i			0.714883i	0.933935	+	0.357442i	$0.116351\pi$
$877$			21.1707i			0.714883i	−0.933935	+	0.357442i	$0.883649\pi$
$878$		0			0
$879$			25.2957i			0.853203i
$880$		0			0
$881$	38.9612			1.31264			0.656319	−	0.754484i	$-0.272113\pi$
$881$	38.9612			1.31264			0.656319	+	0.754484i	$0.272113\pi$
$882$		0			0
$883$			23.5665i			0.793077i	0.918018	+	0.396538i	$0.129789\pi$
$883$			23.5665i			0.793077i	−0.918018	+	0.396538i	$0.870211\pi$
$884$		0			0
$885$	13.0206			0.437683
$886$		0			0
$887$	37.6400			1.26383			0.631914	−	0.775038i	$-0.282269\pi$
$887$	37.6400			1.26383			0.631914	+	0.775038i	$0.282269\pi$
$888$		0			0
$889$		−	26.1535i		−	0.877162i
$890$		0			0
$891$			2.47405i			0.0828837i
$892$		0			0
$893$	−1.01166	+	1.49188i	−0.0338539	+	0.0499239i
$894$		0			0
$895$	−24.9552			−0.834159
$896$		0			0
$897$	−8.64903			−0.288783
$898$		0			0
$899$		−	3.59351i		−	0.119850i
$900$		0			0
$901$		−	45.1717i		−	1.50489i
$902$		0			0
$903$	36.2718			1.20705
$904$		0			0
$905$			21.6147i			0.718495i
$906$		0			0
$907$	34.2140			1.13606			0.568029	−	0.823009i	$-0.307706\pi$
$907$	34.2140			1.13606			0.568029	+	0.823009i	$0.307706\pi$
$908$		0			0
$909$	0.688875			0.0228485
$910$		0			0
$911$	27.2485			0.902785			0.451392	−	0.892326i	$-0.350928\pi$
$911$	27.2485			0.902785			0.451392	+	0.892326i	$0.350928\pi$
$912$		0			0
$913$	7.48180			0.247612
$914$		0			0
$915$	6.65022			0.219850
$916$		0			0
$917$	−10.9177			−0.360533
$918$		0			0
$919$		−	0.549024i		−	0.0181106i	−0.999959	−	0.00905531i	$-0.997118\pi$
$919$		−	0.549024i		−	0.0181106i	0.999959	−	0.00905531i	$-0.00288243\pi$
$920$		0			0
$921$	29.2126			0.962588
$922$		0			0
$923$			24.8991i			0.819565i
$924$		0			0
$925$		−	9.46589i		−	0.311237i
$926$		0			0
$927$	7.16283			0.235258
$928$		0			0
$929$	−13.5467			−0.444452			−0.222226	−	0.974995i	$-0.571332\pi$
$929$	−13.5467			−0.444452			−0.222226	+	0.974995i	$0.571332\pi$
$930$		0			0
$931$	49.2453	+	33.3937i	1.61395	+	1.09444i
$932$		0			0
$933$			16.4577i			0.538802i
$934$		0			0
$935$			17.3016i			0.565822i
$936$		0			0
$937$	−26.8777			−0.878055			−0.439028	−	0.898474i	$-0.644677\pi$
$937$	−26.8777			−0.878055			−0.439028	+	0.898474i	$0.644677\pi$
$938$		0			0
$939$	−6.34778			−0.207152
$940$		0			0
$941$			16.4891i			0.537531i	0.963206	+	0.268765i	$0.0866156\pi$
$941$			16.4891i			0.537531i	−0.963206	+	0.268765i	$0.913384\pi$
$942$		0			0
$943$	4.63964			0.151088
$944$		0			0
$945$		−	4.54425i		−	0.147825i
$946$		0			0
$947$			25.9187i			0.842244i	0.907004	+	0.421122i	$0.138364\pi$
$947$			25.9187i			0.842244i	−0.907004	+	0.421122i	$0.861636\pi$
$948$		0			0
$949$		−	1.71082i		−	0.0555355i
$950$		0			0
$951$			17.8188i			0.577814i
$952$		0			0
$953$			4.47821i			0.145063i	0.997366	+	0.0725317i	$0.0231079\pi$
$953$			4.47821i			0.145063i	−0.997366	+	0.0725317i	$0.976892\pi$
$954$		0			0
$955$		−	11.6341i		−	0.376470i
$956$		0			0
$957$	−2.44944			−0.0791792
$958$		0			0
$959$			24.8203i			0.801490i
$960$		0			0
$961$	−17.8259			−0.575029
$962$		0			0
$963$	9.46001			0.304845
$964$		0			0
$965$		−	3.69073i		−	0.118809i
$966$		0			0
$967$			49.4779i			1.59110i	0.605887	+	0.795550i	$0.292818\pi$
$967$			49.4779i			1.59110i	−0.605887	+	0.795550i	$0.707182\pi$
$968$		0			0
$969$	−25.2291	−	17.1081i	−0.810477	−	0.549593i
$970$		0			0
$971$	51.9726			1.66788			0.833939	−	0.551856i	$-0.186080\pi$
$971$	51.9726			1.66788			0.833939	+	0.551856i	$0.186080\pi$
$972$		0			0
$973$	23.1705			0.742811
$974$		0			0
$975$		−	2.88758i		−	0.0924766i
$976$		0			0
$977$		−	6.61194i		−	0.211535i	−0.994391	−	0.105767i	$-0.966270\pi$
$977$		−	6.61194i		−	0.211535i	0.994391	−	0.105767i	$-0.0337299\pi$
$978$		0			0
$979$	−45.6816			−1.45999
$980$		0			0
$981$		−	4.91923i		−	0.157059i
$982$		0			0
$983$	21.5537			0.687456			0.343728	−	0.939069i	$-0.388310\pi$
$983$	21.5537			0.687456			0.343728	+	0.939069i	$0.388310\pi$
$984$		0			0
$985$	24.9736			0.795725
$986$		0			0
$987$	−1.87919			−0.0598154
$988$		0			0
$989$	23.9078			0.760224
$990$		0			0
$991$	2.27566			0.0722889			0.0361444	−	0.999347i	$-0.488492\pi$
$991$	2.27566			0.0722889			0.0361444	+	0.999347i	$0.488492\pi$
$992$		0			0
$993$	4.16770			0.132258
$994$		0			0
$995$		−	19.0594i		−	0.604223i
$996$		0			0
$997$	−53.1823			−1.68430			−0.842150	−	0.539243i	$-0.818710\pi$
$997$	−53.1823			−1.68430			−0.842150	+	0.539243i	$0.818710\pi$
$998$		0			0
$999$		−	9.46589i		−	0.299488i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4560.2.d.l.2431.1	yes	12
4.3	odd	2		4560.2.d.j.2431.12	yes	12
19.18	odd	2		4560.2.d.j.2431.1	✓	12
76.75	even	2	inner	4560.2.d.l.2431.12	yes	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4560.2.d.j.2431.1	✓	12	19.18	odd	2
4560.2.d.j.2431.12	yes	12	4.3	odd	2
4560.2.d.l.2431.1	yes	12	1.1	even	1	trivial
4560.2.d.l.2431.12	yes	12	76.75	even	2	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:	\( N \)	\(=\)	\( 4560 = 2^{4} \cdot 3 \cdot 5 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4560.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} +1.00000 q^{5} -4.54425i q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} +1.00000 q^{5} -4.54425i q^{7} +1.00000 q^{9} +2.47405i q^{11} -2.88758i q^{13} +1.00000 q^{15} +6.99322 q^{17} +(-3.60766 - 2.44639i) q^{19} -4.54425i q^{21} -2.99525i q^{23} +1.00000 q^{25} +1.00000 q^{27} +0.990054i q^{29} -3.62961 q^{31} +2.47405i q^{33} -4.54425i q^{35} -9.46589i q^{37} -2.88758i q^{39} +1.54900i q^{41} +7.98190i q^{43} +1.00000 q^{45} -0.413532i q^{47} -13.6502 q^{49} +6.99322 q^{51} -6.45935i q^{53} +2.47405i q^{55} +(-3.60766 - 2.44639i) q^{57} +13.0206 q^{59} +6.65022 q^{61} -4.54425i q^{63} -2.88758i q^{65} -7.16283 q^{67} -2.99525i q^{69} -8.62284 q^{71} +0.592475 q^{73} +1.00000 q^{75} +11.2427 q^{77} -14.9687 q^{79} +1.00000 q^{81} -3.02411i q^{83} +6.99322 q^{85} +0.990054i q^{87} +18.4643i q^{89} -13.1219 q^{91} -3.62961 q^{93} +(-3.60766 - 2.44639i) q^{95} -11.4460i q^{97} +2.47405i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 12 q^{3} + 12 q^{5} + 12 q^{9}+O(q^{10}) \) 12 * q + 12 * q^3 + 12 * q^5 + 12 * q^9	\( 12 q + 12 q^{3} + 12 q^{5} + 12 q^{9} + 12 q^{15} + 4 q^{17} + 12 q^{25} + 12 q^{27} - 12 q^{31} + 12 q^{45} - 28 q^{49} + 4 q^{51} + 52 q^{59} - 56 q^{61} - 32 q^{67} + 8 q^{71} + 32 q^{73} + 12 q^{75} + 24 q^{77} - 28 q^{79} + 12 q^{81} + 4 q^{85} + 32 q^{91} - 12 q^{93}+O(q^{100}) \) 12 * q + 12 * q^3 + 12 * q^5 + 12 * q^9 + 12 * q^15 + 4 * q^17 + 12 * q^25 + 12 * q^27 - 12 * q^31 + 12 * q^45 - 28 * q^49 + 4 * q^51 + 52 * q^59 - 56 * q^61 - 32 * q^67 + 8 * q^71 + 32 * q^73 + 12 * q^75 + 24 * q^77 - 28 * q^79 + 12 * q^81 + 4 * q^85 + 32 * q^91 - 12 * q^93

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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