LMFDB - Embedded newform 2359.1.cv.a.797.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2359,1,Mod(6,2359)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2359, base_ring=CyclotomicField(56))

chi = DirichletCharacter(H, H._module([28, 47]))

N = Newforms(chi, 1, names="a")

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

chi := DirichletCharacter("2359.6");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 2359 = 7 \cdot 337 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	2359.cv (of order $56$, degree $24$, 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:	$1.17729436480$
Analytic rank:	$0$
Dimension:	$24$
Coefficient field:	$\Q(\zeta_{56})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{24} - x^{20} + x^{16} - x^{12} + x^{8} - x^{4} + 1 $ x^24 - x^20 + x^16 - x^12 + x^8 - x^4 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Projective image:	$D_{56}$
Projective field:	Galois closure of $\mathbb{Q}[x]/(x^{56} - \cdots)$

Embedding invariants

Embedding label			797.1
Root			$0.330279 - 0.943883i$ of defining polynomial
Character	$\chi$	$=$	2359.797
Dual form			2359.1.cv.a.811.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.201753 - 0.0971591i) q^{2} +(-0.592225 - 0.742627i) q^{4} +(0.330279 + 0.943883i) q^{7} +(0.0971591 + 0.425682i) q^{8} +(0.974928 + 0.222521i) q^{9} +O(q^{10})$	\(q+(-0.201753 - 0.0971591i) q^{2} +(-0.592225 - 0.742627i) q^{4} +(0.330279 + 0.943883i) q^{7} +(0.0971591 + 0.425682i) q^{8} +(0.974928 + 0.222521i) q^{9} +(0.249799 - 0.223234i) q^{11} +(0.0250721 - 0.222521i) q^{14} +(-0.189606 + 0.830718i) q^{16} +(-0.175075 - 0.139617i) q^{18} +(-0.0720870 + 0.0207679i) q^{22} +(-0.110556 + 0.0187843i) q^{23} +(-0.943883 + 0.330279i) q^{25} +(0.505354 - 0.804266i) q^{28} +(0.309511 + 1.82165i) q^{29} +(0.391199 - 0.490548i) q^{32} +(-0.412127 - 0.855791i) q^{36} +(0.678448 + 0.541044i) q^{37} +(0.656405 - 0.0739590i) q^{43} +(-0.313717 - 0.0533027i) q^{44} +(0.0241302 + 0.00695178i) q^{46} +(-0.781831 + 0.623490i) q^{49} +(0.222521 + 0.0250721i) q^{50} +(-0.330279 - 0.0561167i) q^{53} +(-0.369704 + 0.232301i) q^{56} +(0.114545 - 0.397596i) q^{58} +(0.111964 + 0.993712i) q^{63} +(0.641112 - 0.308743i) q^{64} +(1.01293 - 1.42760i) q^{67} +(0.993712 - 0.888036i) q^{71} +0.436629i q^{72} +(-0.0843115 - 0.175075i) q^{74} +(0.293211 + 0.162052i) q^{77} +(1.52574 + 0.734760i) q^{79} +(0.900969 + 0.433884i) q^{81} +(-0.139617 - 0.0488542i) q^{86} +(0.119297 + 0.0846458i) q^{88} +(0.0794241 + 0.0709777i) q^{92} +(0.218315 - 0.0498289i) q^{98} +(0.293211 - 0.162052i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q - 4 q^{4}+O(q^{10}) $ 24 * q - 4 * q^4	$ 24 q - 4 q^{4} + 24 q^{14} + 4 q^{16} + 24 q^{22} - 4 q^{23} - 4 q^{29} + 4 q^{43} - 4 q^{46} + 4 q^{50} - 24 q^{58} - 24 q^{64} + 4 q^{67} - 4 q^{77} + 4 q^{81} - 4 q^{92} - 4 q^{99}+O(q^{100}) $ 24 * q - 4 * q^4 + 24 * q^14 + 4 * q^16 + 24 * q^22 - 4 * q^23 - 4 * q^29 + 4 * q^43 - 4 * q^46 + 4 * q^50 - 24 * q^58 - 24 * q^64 + 4 * q^67 - 4 * q^77 + 4 * q^81 - 4 * q^92 - 4 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2359\mathbb{Z}\right)^\times$.

$n$	$675$	$1695$
$\chi(n)$	$-1$	$e\left(\frac{33}{56}\right)$

Coefficient data

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

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

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

$2$	−0.201753	−	0.0971591i	−0.201753	−	0.0971591i	0.330279	−	0.943883i	$-0.392857\pi$
$2$	−0.201753	−	0.0971591i	−0.201753	−	0.0971591i	−0.532032	+	0.846724i	$0.678571\pi$
$3$		0			0		−0.993712	−	0.111964i	$-0.964286\pi$
$3$		0			0		0.993712	+	0.111964i	$0.0357143\pi$
$4$	−0.592225	−	0.742627i	−0.592225	−	0.742627i
$5$		0			0		−0.167506	−	0.985871i	$-0.553571\pi$
$5$		0			0		0.167506	+	0.985871i	$0.446429\pi$
$6$		0			0
$7$	0.330279	+	0.943883i	0.330279	+	0.943883i
$7$	0.330279	+	0.943883i	0.330279	+	0.943883i
$8$	0.0971591	+	0.425682i	0.0971591	+	0.425682i
$9$	0.974928	+	0.222521i	0.974928	+	0.222521i
$10$		0			0
$11$	0.249799	−	0.223234i	0.249799	−	0.223234i	−0.532032	−	0.846724i	$-0.678571\pi$
$11$	0.249799	−	0.223234i	0.249799	−	0.223234i	0.781831	+	0.623490i	$0.214286\pi$
$12$		0			0
$13$		0			0		−0.222521	−	0.974928i	$-0.571429\pi$
$13$		0			0		0.222521	+	0.974928i	$0.428571\pi$
$14$	0.0250721	−	0.222521i	0.0250721	−	0.222521i
$15$		0			0
$16$	−0.189606	+	0.830718i	−0.189606	+	0.830718i
$17$		0			0		−0.578671	−	0.815561i	$-0.696429\pi$
$17$		0			0		0.578671	+	0.815561i	$0.303571\pi$
$18$	−0.175075	−	0.139617i	−0.175075	−	0.139617i
$19$		0			0		−0.745642	−	0.666347i	$-0.767857\pi$
$19$		0			0		0.745642	+	0.666347i	$0.232143\pi$
$20$		0			0
$21$		0			0
$22$	−0.0720870	+	0.0207679i	−0.0720870	+	0.0207679i
$23$	−0.110556	+	0.0187843i	−0.110556	+	0.0187843i	−0.222521	−	0.974928i	$-0.571429\pi$
$23$	−0.110556	+	0.0187843i	−0.110556	+	0.0187843i	0.111964	+	0.993712i	$0.464286\pi$
$24$		0			0
$25$	−0.943883	+	0.330279i	−0.943883	+	0.330279i
$26$		0			0
$27$		0			0
$28$	0.505354	−	0.804266i	0.505354	−	0.804266i
$29$	0.309511	+	1.82165i	0.309511	+	1.82165i	0.532032	+	0.846724i	$0.321429\pi$
$29$	0.309511	+	1.82165i	0.309511	+	1.82165i	−0.222521	+	0.974928i	$0.571429\pi$
$30$		0			0
$31$		0			0		0.0560704	−	0.998427i	$-0.482143\pi$
$31$		0			0		−0.0560704	+	0.998427i	$0.517857\pi$
$32$	0.391199	−	0.490548i	0.391199	−	0.490548i
$33$		0			0
$34$		0			0
$35$		0			0
$36$	−0.412127	−	0.855791i	−0.412127	−	0.855791i
$37$	0.678448	+	0.541044i	0.678448	+	0.541044i	0.900969	−	0.433884i	$-0.142857\pi$
$37$	0.678448	+	0.541044i	0.678448	+	0.541044i	−0.222521	+	0.974928i	$0.571429\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		0			0		0.433884	−	0.900969i	$-0.357143\pi$
$41$		0			0		−0.433884	+	0.900969i	$0.642857\pi$
$42$		0			0
$43$	0.656405	−	0.0739590i	0.656405	−	0.0739590i	0.222521	−	0.974928i	$-0.428571\pi$
$43$	0.656405	−	0.0739590i	0.656405	−	0.0739590i	0.433884	+	0.900969i	$0.357143\pi$
$44$	−0.313717	−	0.0533027i	−0.313717	−	0.0533027i
$45$		0			0
$46$	0.0241302	+	0.00695178i	0.0241302	+	0.00695178i
$47$		0			0		−0.993712	−	0.111964i	$-0.964286\pi$
$47$		0			0		0.993712	+	0.111964i	$0.0357143\pi$
$48$		0			0
$49$	−0.781831	+	0.623490i	−0.781831	+	0.623490i
$50$	0.222521	+	0.0250721i	0.222521	+	0.0250721i
$51$		0			0
$52$		0			0
$53$	−0.330279	−	0.0561167i	−0.330279	−	0.0561167i		−	1.00000i	$-0.5\pi$
$53$	−0.330279	−	0.0561167i	−0.330279	−	0.0561167i	−0.330279	+	0.943883i	$0.607143\pi$
$54$		0			0
$55$		0			0
$56$	−0.369704	+	0.232301i	−0.369704	+	0.232301i
$57$		0			0
$58$	0.114545	−	0.397596i	0.114545	−	0.397596i
$59$		0			0		−0.923880	−	0.382683i	$-0.875000\pi$
$59$		0			0		0.923880	+	0.382683i	$0.125000\pi$
$60$		0			0
$61$		0			0		−0.666347	−	0.745642i	$-0.732143\pi$
$61$		0			0		0.666347	+	0.745642i	$0.267857\pi$
$62$		0			0
$63$	0.111964	+	0.993712i	0.111964	+	0.993712i
$64$	0.641112	−	0.308743i	0.641112	−	0.308743i
$65$		0			0
$66$		0			0
$67$	1.01293	−	1.42760i	1.01293	−	1.42760i	0.111964	−	0.993712i	$-0.464286\pi$
$67$	1.01293	−	1.42760i	1.01293	−	1.42760i	0.900969	−	0.433884i	$-0.142857\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	0.993712	−	0.888036i	0.993712	−	0.888036i		−	1.00000i	$-0.5\pi$
$71$	0.993712	−	0.888036i	0.993712	−	0.888036i	0.993712	+	0.111964i	$0.0357143\pi$
$72$			0.436629i			0.436629i
$73$		0			0		−0.745642	−	0.666347i	$-0.767857\pi$
$73$		0			0		0.745642	+	0.666347i	$0.232143\pi$
$74$	−0.0843115	−	0.175075i	−0.0843115	−	0.175075i
$75$		0			0
$76$		0			0
$77$	0.293211	+	0.162052i	0.293211	+	0.162052i
$78$		0			0
$79$	1.52574	+	0.734760i	1.52574	+	0.734760i	0.993712	−	0.111964i	$-0.0357143\pi$
$79$	1.52574	+	0.734760i	1.52574	+	0.734760i	0.532032	+	0.846724i	$0.321429\pi$
$80$		0			0
$81$	0.900969	+	0.433884i	0.900969	+	0.433884i
$82$		0			0
$83$		0			0		−0.875223	−	0.483719i	$-0.839286\pi$
$83$		0			0		0.875223	+	0.483719i	$0.160714\pi$
$84$		0			0
$85$		0			0
$86$	−0.139617	−	0.0488542i	−0.139617	−	0.0488542i
$87$		0			0
$88$	0.119297	+	0.0846458i	0.119297	+	0.0846458i
$89$		0			0		0.985871	−	0.167506i	$-0.0535714\pi$
$89$		0			0		−0.985871	+	0.167506i	$0.946429\pi$
$90$		0			0
$91$		0			0
$92$	0.0794241	+	0.0709777i	0.0794241	+	0.0709777i
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$		0			0		−0.985871	−	0.167506i	$-0.946429\pi$
$97$		0			0		0.985871	+	0.167506i	$0.0535714\pi$
$98$	0.218315	−	0.0498289i	0.218315	−	0.0498289i
$99$	0.293211	−	0.162052i	0.293211	−	0.162052i
$100$	0.804266	+	0.505354i	0.804266	+	0.505354i
$101$		0			0		−0.276836	−	0.960917i	$-0.589286\pi$
$101$		0			0		0.276836	+	0.960917i	$0.410714\pi$
$102$		0			0
$103$		0			0		0.433884	−	0.900969i	$-0.357143\pi$
$103$		0			0		−0.433884	+	0.900969i	$0.642857\pi$
$104$		0			0
$105$		0			0
$106$	0.0611825	+	0.0434113i	0.0611825	+	0.0434113i
$107$	0.119137	−	1.05737i	0.119137	−	1.05737i	−0.781831	−	0.623490i	$-0.785714\pi$
$107$	0.119137	−	1.05737i	0.119137	−	1.05737i	0.900969	−	0.433884i	$-0.142857\pi$
$108$		0			0
$109$	0.249799	−	1.47021i	0.249799	−	1.47021i	−0.532032	−	0.846724i	$-0.678571\pi$
$109$	0.249799	−	1.47021i	0.249799	−	1.47021i	0.781831	−	0.623490i	$-0.214286\pi$
$110$		0			0
$111$		0			0
$112$	−0.846724	+	0.0954029i	−0.846724	+	0.0954029i
$113$	−1.87590	−	0.656405i	−1.87590	−	0.656405i	−0.974928	−	0.222521i	$-0.928571\pi$
$113$	−1.87590	−	0.656405i	−1.87590	−	0.656405i	−0.900969	−	0.433884i	$-0.857143\pi$
$114$		0			0
$115$		0			0
$116$	1.16951	−	1.30868i	1.16951	−	1.30868i
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−0.0993983	+	0.882185i	−0.0993983	+	0.882185i
$122$		0			0
$123$		0			0
$124$		0			0
$125$		0			0
$126$	0.0739590	−	0.211363i	0.0739590	−	0.211363i
$127$	1.33060	+	0.0747247i	1.33060	+	0.0747247i	0.707107	−	0.707107i	$-0.250000\pi$
$127$	1.33060	+	0.0747247i	1.33060	+	0.0747247i	0.623490	+	0.781831i	$0.285714\pi$
$128$	−0.786779			−0.786779
$129$		0			0
$130$		0			0
$131$		0			0		0.781831	−	0.623490i	$-0.214286\pi$
$131$		0			0		−0.781831	+	0.623490i	$0.785714\pi$
$132$		0			0
$133$		0			0
$134$	−0.343066	+	0.189606i	−0.343066	+	0.189606i
$135$		0			0
$136$		0			0
$137$	−0.958689	−	1.52574i	−0.958689	−	1.52574i	−0.846724	−	0.532032i	$-0.821429\pi$
$137$	−0.958689	−	1.52574i	−0.958689	−	1.52574i	−0.111964	−	0.993712i	$-0.535714\pi$
$138$		0			0
$139$		0			0		0.998427	−	0.0560704i	$-0.0178571\pi$
$139$		0			0		−0.998427	+	0.0560704i	$0.982143\pi$
$140$		0			0
$141$		0			0
$142$	−0.286765	+	0.0826156i	−0.286765	+	0.0826156i
$143$		0			0
$144$	−0.369704	+	0.767699i	−0.369704	+	0.767699i
$145$		0			0
$146$		0			0
$147$		0			0
$148$		−	0.824254i		−	0.824254i
$149$	0.900969	+	1.43388i	0.900969	+	1.43388i	0.900969	+	0.433884i	$0.142857\pi$
$149$	0.900969	+	1.43388i	0.900969	+	1.43388i			1.00000i	$0.5\pi$
$150$		0			0
$151$	−0.559828	+	0.789004i	−0.559828	+	0.789004i	−0.993712	−	0.111964i	$-0.964286\pi$
$151$	−0.559828	+	0.789004i	−0.559828	+	0.789004i	0.433884	+	0.900969i	$0.357143\pi$
$152$		0			0
$153$		0			0
$154$	−0.0434113	−	0.0611825i	−0.0434113	−	0.0611825i
$155$		0			0
$156$		0			0
$157$		0			0		0.815561	−	0.578671i	$-0.196429\pi$
$157$		0			0		−0.815561	+	0.578671i	$0.803571\pi$
$158$	−0.236435	−	0.296480i	−0.236435	−	0.296480i
$159$		0			0
$160$		0			0
$161$	−0.0542447	−	0.0981483i	−0.0542447	−	0.0981483i
$162$	−0.139617	−	0.175075i	−0.139617	−	0.175075i
$163$	−0.442896	+	1.06925i	−0.442896	+	1.06925i	0.532032	+	0.846724i	$0.321429\pi$
$163$	−0.442896	+	1.06925i	−0.442896	+	1.06925i	−0.974928	+	0.222521i	$0.928571\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		0			0		0.943883	−	0.330279i	$-0.107143\pi$
$167$		0			0		−0.943883	+	0.330279i	$0.892857\pi$
$168$		0			0
$169$	−0.900969	+	0.433884i	−0.900969	+	0.433884i
$170$		0			0
$171$		0			0
$172$	−0.443664	−	0.443664i	−0.443664	−	0.443664i
$173$		0			0		0.974928	−	0.222521i	$-0.0714286\pi$
$173$		0			0		−0.974928	+	0.222521i	$0.928571\pi$
$174$		0			0
$175$	−0.623490	−	0.781831i	−0.623490	−	0.781831i
$176$	0.138081	+	0.249840i	0.138081	+	0.249840i
$177$		0			0
$178$		0			0
$179$	−0.139617	−	0.175075i	−0.139617	−	0.175075i	0.707107	−	0.707107i	$-0.250000\pi$
$179$	−0.139617	−	0.175075i	−0.139617	−	0.175075i	−0.846724	+	0.532032i	$0.821429\pi$
$180$		0			0
$181$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$181$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$182$		0			0
$183$		0			0
$184$	−0.0187377	−	0.0452368i	−0.0187377	−	0.0452368i
$185$		0			0
$186$		0			0
$187$		0			0
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−0.442896	−	1.06925i	−0.442896	−	1.06925i	−0.974928	−	0.222521i	$-0.928571\pi$
$191$	−0.442896	−	1.06925i	−0.442896	−	1.06925i	0.532032	−	0.846724i	$-0.321429\pi$
$192$		0			0
$193$	−0.541044	+	1.12349i	−0.541044	+	1.12349i	0.433884	+	0.900969i	$0.357143\pi$
$193$	−0.541044	+	1.12349i	−0.541044	+	1.12349i	−0.974928	+	0.222521i	$0.928571\pi$
$194$		0			0
$195$		0			0
$196$	0.926041	+	0.211363i	0.926041	+	0.211363i
$197$	−1.70711	−	0.707107i	−1.70711	−	0.707107i	−0.707107	−	0.707107i	$-0.750000\pi$
$197$	−1.70711	−	0.707107i	−1.70711	−	0.707107i	−1.00000			$\pi$
$198$	−0.0749010	+	0.00420635i	−0.0749010	+	0.00420635i
$199$		0			0		−0.923880	−	0.382683i	$-0.875000\pi$
$199$		0			0		0.923880	+	0.382683i	$0.125000\pi$
$200$	−0.232301	−	0.369704i	−0.232301	−	0.369704i
$201$		0			0
$202$		0			0
$203$	−1.61720	+	0.893796i	−1.61720	+	0.893796i
$204$		0			0
$205$		0			0
$206$		0			0
$207$	−0.111964	−	0.00628779i	−0.111964	−	0.00628779i
$208$		0			0
$209$		0			0
$210$		0			0
$211$		0			0		−0.943883	−	0.330279i	$-0.892857\pi$
$211$		0			0		0.943883	+	0.330279i	$0.107143\pi$
$212$	0.153926	+	0.278508i	0.153926	+	0.278508i
$213$		0			0
$214$	−0.126770	+	0.201753i	−0.126770	+	0.201753i
$215$		0			0
$216$		0			0
$217$		0			0
$218$	−0.193242	+	0.272350i	−0.193242	+	0.272350i
$219$		0			0
$220$		0			0
$221$		0			0
$222$		0			0
$223$		0			0		−0.276836	−	0.960917i	$-0.589286\pi$
$223$		0			0		0.276836	+	0.960917i	$0.410714\pi$
$224$	0.592225	+	0.207229i	0.592225	+	0.207229i
$225$	−0.993712	+	0.111964i	−0.993712	+	0.111964i
$226$	0.314692	+	0.314692i	0.314692	+	0.314692i
$227$		0			0		−0.433884	−	0.900969i	$-0.642857\pi$
$227$		0			0		0.433884	+	0.900969i	$0.357143\pi$
$228$		0			0
$229$		0			0		−0.330279	−	0.943883i	$-0.607143\pi$
$229$		0			0		0.330279	+	0.943883i	$0.392857\pi$
$230$		0			0
$231$		0			0
$232$	−0.745373	+	0.308743i	−0.745373	+	0.308743i
$233$	−1.12349	+	0.541044i	−1.12349	+	0.541044i	−0.900969	−	0.433884i	$-0.857143\pi$
$233$	−1.12349	+	0.541044i	−1.12349	+	0.541044i	−0.222521	+	0.974928i	$0.571429\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−0.433884	+	0.0990311i	−0.433884	+	0.0990311i	−0.433884	−	0.900969i	$-0.642857\pi$
$239$	−0.433884	+	0.0990311i	−0.433884	+	0.0990311i			1.00000i	$0.5\pi$
$240$		0			0
$241$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$241$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$242$	0.105766	−	0.168326i	0.105766	−	0.168326i
$243$		0			0
$244$		0			0
$245$		0			0
$246$		0			0
$247$		0			0
$248$		0			0
$249$		0			0
$250$		0			0
$251$		0			0		−0.943883	−	0.330279i	$-0.892857\pi$
$251$		0			0		0.943883	+	0.330279i	$0.107143\pi$
$252$	0.671649	−	0.671649i	0.671649	−	0.671649i
$253$	−0.0234236	+	0.0293723i	−0.0234236	+	0.0293723i
$254$	−0.261192	−	0.144356i	−0.261192	−	0.144356i
$255$		0			0
$256$	−0.482377	−	0.232301i	−0.482377	−	0.232301i
$257$		0			0		−0.998427	−	0.0560704i	$-0.982143\pi$
$257$		0			0		0.998427	+	0.0560704i	$0.0178571\pi$
$258$		0			0
$259$	−0.286605	+	0.819071i	−0.286605	+	0.819071i
$260$		0			0
$261$	−0.103605	+	1.84485i	−0.103605	+	1.84485i
$262$		0			0
$263$	−0.376510	−	0.781831i	−0.376510	−	0.781831i	0.623490	−	0.781831i	$-0.285714\pi$
$263$	−0.376510	−	0.781831i	−0.376510	−	0.781831i	−1.00000			$\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$	−1.66006	+	0.0932268i	−1.66006	+	0.0932268i
$269$		0			0		0.276836	−	0.960917i	$-0.410714\pi$
$269$		0			0		−0.276836	+	0.960917i	$0.589286\pi$
$270$		0			0
$271$		0			0		0.985871	−	0.167506i	$-0.0535714\pi$
$271$		0			0		−0.985871	+	0.167506i	$0.946429\pi$
$272$		0			0
$273$		0			0
$274$	0.0451783	+	0.400969i	0.0451783	+	0.400969i
$275$	−0.162052	+	0.293211i	−0.162052	+	0.293211i
$276$		0			0
$277$	0.570690	−	0.510000i	0.570690	−	0.510000i	−0.330279	−	0.943883i	$-0.607143\pi$
$277$	0.570690	−	0.510000i	0.570690	−	0.510000i	0.900969	+	0.433884i	$0.142857\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	0.189606	−	0.119137i	0.189606	−	0.119137i	−0.433884	−	0.900969i	$-0.642857\pi$
$281$	0.189606	−	0.119137i	0.189606	−	0.119137i	0.623490	+	0.781831i	$0.285714\pi$
$282$		0			0
$283$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$283$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$284$	−1.24798	−	0.212040i	−1.24798	−	0.212040i
$285$		0			0
$286$		0			0
$287$		0			0
$288$	0.490548	−	0.391199i	0.490548	−	0.391199i
$289$	−0.330279	+	0.943883i	−0.330279	+	0.943883i
$290$		0			0
$291$		0			0
$292$		0			0
$293$		0			0		−0.985871	−	0.167506i	$-0.946429\pi$
$293$		0			0		0.985871	+	0.167506i	$0.0535714\pi$
$294$		0			0
$295$		0			0
$296$	−0.164395	+	0.341370i	−0.164395	+	0.341370i
$297$		0			0
$298$	−0.0424583	−	0.376828i	−0.0424583	−	0.376828i
$299$		0			0
$300$		0			0
$301$	0.286605	+	0.595142i	0.286605	+	0.595142i
$302$	0.189606	−	0.104792i	0.189606	−	0.104792i
$303$		0			0
$304$		0			0
$305$		0			0
$306$		0			0
$307$		0			0		0.382683	−	0.923880i	$-0.375000\pi$
$307$		0			0		−0.382683	+	0.923880i	$0.625000\pi$
$308$	−0.0533027	−	0.313717i	−0.0533027	−	0.313717i
$309$		0			0
$310$		0			0
$311$		0			0		0.222521	−	0.974928i	$-0.428571\pi$
$311$		0			0		−0.222521	+	0.974928i	$0.571429\pi$
$312$		0			0
$313$		0			0		−0.111964	−	0.993712i	$-0.535714\pi$
$313$		0			0		0.111964	+	0.993712i	$0.464286\pi$
$314$		0			0
$315$		0			0
$316$	−0.357932	−	1.56820i	−0.357932	−	1.56820i
$317$	0.223234	−	0.249799i	0.223234	−	0.249799i	−0.623490	−	0.781831i	$-0.714286\pi$
$317$	0.223234	−	0.249799i	0.223234	−	0.249799i	0.846724	+	0.532032i	$0.178571\pi$
$318$		0			0
$319$	0.483971	+	0.385954i	0.483971	+	0.385954i
$320$		0			0
$321$		0			0
$322$	0.00140802	+	0.0250721i	0.00140802	+	0.0250721i
$323$		0			0
$324$	−0.211363	−	0.926041i	−0.211363	−	0.926041i
$325$		0			0
$326$	0.193242	−	0.172692i	0.193242	−	0.172692i
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$	1.32399	+	0.831919i	1.32399	+	0.831919i	0.993712	−	0.111964i	$-0.0357143\pi$
$331$	1.32399	+	0.831919i	1.32399	+	0.831919i	0.330279	+	0.943883i	$0.392857\pi$
$332$		0			0
$333$	0.541044	+	0.678448i	0.541044	+	0.678448i
$334$		0			0
$335$		0			0
$336$		0			0
$337$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$337$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$338$	0.223929			0.223929
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$	−0.846724	−	0.532032i	−0.846724	−	0.532032i
$344$	0.0952587	+	0.272234i	0.0952587	+	0.272234i
$345$		0			0
$346$		0			0
$347$	0.545848	−	1.89468i	0.545848	−	1.89468i	0.111964	−	0.993712i	$-0.464286\pi$
$347$	0.545848	−	1.89468i	0.545848	−	1.89468i	0.433884	−	0.900969i	$-0.357143\pi$
$348$		0			0
$349$		0			0		−0.532032	−	0.846724i	$-0.678571\pi$
$349$		0			0		0.532032	+	0.846724i	$0.321429\pi$
$350$	0.0498289	+	0.218315i	0.0498289	+	0.218315i
$351$		0			0
$352$	−0.0117859	−	0.209868i	−0.0117859	−	0.209868i
$353$		0			0		0.222521	−	0.974928i	$-0.428571\pi$
$353$		0			0		−0.222521	+	0.974928i	$0.571429\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$	0.0111581	+	0.0488870i	0.0111581	+	0.0488870i
$359$	−0.532032	+	0.153276i	−0.532032	+	0.153276i	−0.532032	−	0.846724i	$-0.678571\pi$
$359$	−0.532032	+	0.153276i	−0.532032	+	0.153276i			1.00000i	$0.5\pi$
$360$		0			0
$361$	0.111964	+	0.993712i	0.111964	+	0.993712i
$362$		0			0
$363$		0			0
$364$		0			0
$365$		0			0
$366$		0			0
$367$		0			0		0.382683	−	0.923880i	$-0.375000\pi$
$367$		0			0		−0.382683	+	0.923880i	$0.625000\pi$
$368$	0.00535771	−	0.0954029i	0.00535771	−	0.0954029i
$369$		0			0
$370$		0			0
$371$	−0.0561167	−	0.330279i	−0.0561167	−	0.330279i
$372$		0			0
$373$	−0.613604	−	1.27416i	−0.613604	−	1.27416i	−0.943883	−	0.330279i	$-0.892857\pi$
$373$	−0.613604	−	1.27416i	−0.613604	−	1.27416i	0.330279	−	0.943883i	$-0.392857\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$	0.400969	−	1.75676i	0.400969	−	1.75676i	−0.222521	−	0.974928i	$-0.571429\pi$
$379$	0.400969	−	1.75676i	0.400969	−	1.75676i	0.623490	−	0.781831i	$-0.285714\pi$
$380$		0			0
$381$		0			0
$382$	−0.0145314	+	0.258755i	−0.0145314	+	0.258755i
$383$		0			0		−0.960917	−	0.276836i	$-0.910714\pi$
$383$		0			0		0.960917	+	0.276836i	$0.0892857\pi$
$384$		0			0
$385$		0			0
$386$	0.218315	−	0.174100i	0.218315	−	0.174100i
$387$	0.656405	+	0.0739590i	0.656405	+	0.0739590i
$388$		0			0
$389$	1.17700	−	1.47592i	1.17700	−	1.47592i	0.330279	−	0.943883i	$-0.392857\pi$
$389$	1.17700	−	1.47592i	1.17700	−	1.47592i	0.846724	−	0.532032i	$-0.178571\pi$
$390$		0			0
$391$		0			0
$392$	−0.341370	−	0.272234i	−0.341370	−	0.272234i
$393$		0			0
$394$	0.275712	+	0.308522i	0.275712	+	0.308522i
$395$		0			0
$396$	−0.293991	−	0.121775i	−0.293991	−	0.121775i
$397$		0			0		0.745642	−	0.666347i	$-0.232143\pi$
$397$		0			0		−0.745642	+	0.666347i	$0.767857\pi$
$398$		0			0
$399$		0			0
$400$	−0.0954029	−	0.846724i	−0.0954029	−	0.846724i
$401$	0.400969	−	0.193096i	0.400969	−	0.193096i	−0.222521	−	0.974928i	$-0.571429\pi$
$401$	0.400969	−	0.193096i	0.400969	−	0.193096i	0.623490	+	0.781831i	$0.285714\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$		0			0
$406$	0.413116	−	0.0232001i	0.413116	−	0.0232001i
$407$	0.290256	−	0.0163004i	0.290256	−	0.0163004i
$408$		0			0
$409$		0			0			−	1.00000i	$-0.5\pi$
$409$		0			0				1.00000i	$0.5\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		0			0
$414$	0.0219783	+	0.0121470i	0.0219783	+	0.0121470i
$415$		0			0
$416$		0			0
$417$		0			0
$418$		0			0
$419$		0			0		0.781831	−	0.623490i	$-0.214286\pi$
$419$		0			0		−0.781831	+	0.623490i	$0.785714\pi$
$420$		0			0
$421$	0.139617	−	0.175075i	0.139617	−	0.175075i	−0.707107	−	0.707107i	$-0.750000\pi$
$421$	0.139617	−	0.175075i	0.139617	−	0.175075i	0.846724	+	0.532032i	$0.178571\pi$
$422$		0			0
$423$		0			0
$424$	−0.00820177	−	0.146046i	−0.00820177	−	0.146046i
$425$		0			0
$426$		0			0
$427$		0			0
$428$	−0.855791	+	0.537729i	−0.855791	+	0.537729i
$429$		0			0
$430$		0			0
$431$	−1.32399	−	0.831919i	−1.32399	−	0.831919i	−0.330279	−	0.943883i	$-0.607143\pi$
$431$	−1.32399	−	0.831919i	−1.32399	−	0.831919i	−0.993712	+	0.111964i	$0.964286\pi$
$432$		0			0
$433$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$433$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$434$		0			0
$435$		0			0
$436$	−1.23976	+	0.685190i	−1.23976	+	0.685190i
$437$		0			0
$438$		0			0
$439$		0			0		−0.0560704	−	0.998427i	$-0.517857\pi$
$439$		0			0		0.0560704	+	0.998427i	$0.482143\pi$
$440$		0			0
$441$	−0.900969	+	0.433884i	−0.900969	+	0.433884i
$442$		0			0
$443$	−1.62856	−	1.15552i	−1.62856	−	1.15552i	−0.846724	−	0.532032i	$-0.821429\pi$
$443$	−1.62856	−	1.15552i	−1.62856	−	1.15552i	−0.781831	−	0.623490i	$-0.785714\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$	0.503164	+	0.503164i	0.503164	+	0.503164i
$449$	−1.93760	+	0.218315i	−1.93760	+	0.218315i	−0.993712	−	0.111964i	$-0.964286\pi$
$449$	−1.93760	+	0.218315i	−1.93760	+	0.218315i	−0.943883	+	0.330279i	$0.892857\pi$
$450$	0.211363	+	0.0739590i	0.211363	+	0.0739590i
$451$		0			0
$452$	0.623490	+	1.78183i	0.623490	+	1.78183i
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	1.11211	−	0.320394i	1.11211	−	0.320394i	0.330279	−	0.943883i	$-0.392857\pi$
$457$	1.11211	−	0.320394i	1.11211	−	0.320394i	0.781831	+	0.623490i	$0.214286\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		0			0		0.815561	−	0.578671i	$-0.196429\pi$
$461$		0			0		−0.815561	+	0.578671i	$0.803571\pi$
$462$		0			0
$463$		0			0		−0.943883	−	0.330279i	$-0.892857\pi$
$463$		0			0		0.943883	+	0.330279i	$0.107143\pi$
$464$	−1.57197	−	0.0882797i	−1.57197	−	0.0882797i
$465$		0			0
$466$	0.279235			0.279235
$467$		0			0		−0.998427	−	0.0560704i	$-0.982143\pi$
$467$		0			0		0.998427	+	0.0560704i	$0.0178571\pi$
$468$		0			0
$469$	1.68203	+	0.484586i	1.68203	+	0.484586i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	0.147459	−	0.165007i	0.147459	−	0.165007i
$474$		0			0
$475$		0			0
$476$		0			0
$477$	−0.309511	−	0.128204i	−0.309511	−	0.128204i
$478$	0.0971591	+	0.0221759i	0.0971591	+	0.0221759i
$479$		0			0		0.960917	−	0.276836i	$-0.0892857\pi$
$479$		0			0		−0.960917	+	0.276836i	$0.910714\pi$
$480$		0			0
$481$		0			0
$482$		0			0
$483$		0			0
$484$	0.714000	−	0.448636i	0.714000	−	0.448636i
$485$		0			0
$486$		0			0
$487$	1.84044	+	0.420068i	1.84044	+	0.420068i	0.993712	−	0.111964i	$-0.0357143\pi$
$487$	1.84044	+	0.420068i	1.84044	+	0.420068i	0.846724	+	0.532032i	$0.178571\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	0.320394	+	0.451552i	0.320394	+	0.451552i	0.943883	−	0.330279i	$-0.107143\pi$
$491$	0.320394	+	0.451552i	0.320394	+	0.451552i	−0.623490	+	0.781831i	$0.714286\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$		0			0
$497$	1.16640	+	0.644649i	1.16640	+	0.644649i
$498$		0			0
$499$	−0.974928	−	1.22252i	−0.974928	−	1.22252i	−0.974928	−	0.222521i	$-0.928571\pi$
$499$	−0.974928	−	1.22252i	−0.974928	−	1.22252i		−	1.00000i	$-0.5\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		0			0		−0.578671	−	0.815561i	$-0.696429\pi$
$503$		0			0		0.578671	+	0.815561i	$0.303571\pi$
$504$	−0.412127	+	0.144209i	−0.412127	+	0.144209i
$505$		0			0
$506$	0.00757958	−	0.00365013i	0.00757958	−	0.00365013i
$507$		0			0
$508$	−0.732521	−	1.03239i	−0.732521	−	1.03239i
$509$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$509$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$510$		0			0
$511$		0			0
$512$	0.565299	+	0.708863i	0.565299	+	0.708863i
$513$		0			0
$514$		0			0
$515$		0			0
$516$		0			0
$517$		0			0
$518$	0.137404	−	0.137404i	0.137404	−	0.137404i
$519$		0			0
$520$		0			0
$521$		0			0		−0.382683	−	0.923880i	$-0.625000\pi$
$521$		0			0		0.382683	+	0.923880i	$0.375000\pi$
$522$	0.200147	−	0.362138i	0.200147	−	0.362138i
$523$		0			0		0.578671	−	0.815561i	$-0.303571\pi$
$523$		0			0		−0.578671	+	0.815561i	$0.696429\pi$
$524$		0			0
$525$		0			0
$526$			0.194318i			0.194318i
$527$		0			0
$528$		0			0
$529$	−0.932013	+	0.326126i	−0.932013	+	0.326126i
$530$		0			0
$531$		0			0
$532$		0			0
$533$		0			0
$534$		0			0
$535$		0			0
$536$	0.706118	+	0.292483i	0.706118	+	0.292483i
$537$		0			0
$538$		0			0
$539$	−0.0561167	+	0.330279i	−0.0561167	+	0.330279i
$540$		0			0
$541$	0.292893	−	0.707107i	0.292893	−	0.707107i	−0.707107	−	0.707107i	$-0.750000\pi$
$541$	0.292893	−	0.707107i	0.292893	−	0.707107i	1.00000			$0$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		0			0
$546$		0			0
$547$	−0.965916	−	0.0542447i	−0.965916	−	0.0542447i	−0.433884	−	0.900969i	$-0.642857\pi$
$547$	−0.965916	−	0.0542447i	−0.965916	−	0.0542447i	−0.532032	+	0.846724i	$0.678571\pi$
$548$	−0.565299	+	1.61553i	−0.565299	+	1.61553i
$549$		0			0
$550$	0.0611825	−	0.0434113i	0.0611825	−	0.0434113i
$551$		0			0
$552$		0			0
$553$	−0.189606	+	1.68280i	−0.189606	+	1.68280i
$554$	−0.164690	+	0.0474462i	−0.164690	+	0.0474462i
$555$		0			0
$556$		0			0
$557$			1.41421i			1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$557$			1.41421i			1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$		0			0
$562$	−0.0498289	+	0.00561437i	−0.0498289	+	0.00561437i
$563$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$563$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	−0.111964	+	0.993712i	−0.111964	+	0.993712i
$568$	0.474569	+	0.336725i	0.474569	+	0.336725i
$569$	−0.511525	+	0.211881i	−0.511525	+	0.211881i	−0.623490	−	0.781831i	$-0.714286\pi$
$569$	−0.511525	+	0.211881i	−0.511525	+	0.211881i	0.111964	+	0.993712i	$0.464286\pi$
$570$		0			0
$571$	0.734760	−	1.52574i	0.734760	−	1.52574i	−0.111964	−	0.993712i	$-0.535714\pi$
$571$	0.734760	−	1.52574i	0.734760	−	1.52574i	0.846724	−	0.532032i	$-0.178571\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	0.0981483	−	0.0542447i	0.0981483	−	0.0542447i
$576$	0.693740	−	0.158342i	0.693740	−	0.158342i
$577$		0			0		−0.985871	−	0.167506i	$-0.946429\pi$
$577$		0			0		0.985871	+	0.167506i	$0.0535714\pi$
$578$	0.158342	−	0.158342i	0.158342	−	0.158342i
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$	−0.0950307	+	0.0597117i	−0.0950307	+	0.0597117i
$584$		0			0
$585$		0			0
$586$		0			0
$587$		0			0		−0.0560704	−	0.998427i	$-0.517857\pi$
$587$		0			0		0.0560704	+	0.998427i	$0.482143\pi$
$588$		0			0
$589$		0			0
$590$		0			0
$591$		0			0
$592$	−0.578093	+	0.461014i	−0.578093	+	0.461014i
$593$		0			0		−0.900969	−	0.433884i	$-0.857143\pi$
$593$		0			0		0.900969	+	0.433884i	$0.142857\pi$
$594$		0			0
$595$		0			0
$596$	0.531264	−	1.51827i	0.531264	−	1.51827i
$597$		0			0
$598$		0			0
$599$	−1.52446	+	0.347948i	−1.52446	+	0.347948i	−0.900969	−	0.433884i	$-0.857143\pi$
$599$	−1.52446	+	0.347948i	−1.52446	+	0.347948i	−0.623490	+	0.781831i	$0.714286\pi$
$600$		0			0
$601$		0			0		−0.745642	−	0.666347i	$-0.767857\pi$
$601$		0			0		0.745642	+	0.666347i	$0.232143\pi$
$602$		−	0.147918i		−	0.147918i
$603$	1.30521	−	1.16640i	1.30521	−	1.16640i
$604$	0.917481	−	0.0515246i	0.917481	−	0.0515246i
$605$		0			0
$606$		0			0
$607$		0			0		0.578671	−	0.815561i	$-0.303571\pi$
$607$		0			0		−0.578671	+	0.815561i	$0.696429\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$	0.993712	+	1.11196i	0.993712	+	1.11196i	0.993712	+	0.111964i	$0.0357143\pi$
$613$	0.993712	+	1.11196i	0.993712	+	1.11196i			1.00000i	$0.5\pi$
$614$		0			0
$615$		0			0
$616$	−0.0404944	+	0.140559i	−0.0404944	+	0.140559i
$617$	0.644649	+	0.721362i	0.644649	+	0.721362i	0.974928	−	0.222521i	$-0.0714286\pi$
$617$	0.644649	+	0.721362i	0.644649	+	0.721362i	−0.330279	+	0.943883i	$0.607143\pi$
$618$		0			0
$619$		0			0		−0.781831	−	0.623490i	$-0.785714\pi$
$619$		0			0		0.781831	+	0.623490i	$0.214286\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	0.781831	−	0.623490i	0.781831	−	0.623490i
$626$		0			0
$627$		0			0
$628$		0			0
$629$		0			0
$630$		0			0
$631$	−1.40532	+	0.158342i	−1.40532	+	0.158342i	−0.781831	−	0.623490i	$-0.785714\pi$
$631$	−1.40532	+	0.158342i	−1.40532	+	0.158342i	−0.623490	+	0.781831i	$0.714286\pi$
$632$	−0.164534	+	0.720870i	−0.164534	+	0.720870i
$633$		0			0
$634$	−0.0693085	+	0.0287085i	−0.0693085	+	0.0287085i
$635$		0			0
$636$		0			0
$637$		0			0
$638$	−0.0601437	−	0.124890i	−0.0601437	−	0.124890i
$639$	1.16640	−	0.644649i	1.16640	−	0.644649i
$640$		0			0
$641$	−1.60808	+	1.14099i	−1.60808	+	1.14099i	−0.707107	+	0.707107i	$0.750000\pi$
$641$	−1.60808	+	1.14099i	−1.60808	+	1.14099i	−0.900969	+	0.433884i	$0.857143\pi$
$642$		0			0
$643$		0			0		0.0560704	−	0.998427i	$-0.482143\pi$
$643$		0			0		−0.0560704	+	0.998427i	$0.517857\pi$
$644$	−0.0407626	+	0.0984095i	−0.0407626	+	0.0984095i
$645$		0			0
$646$		0			0
$647$		0			0		−0.943883	−	0.330279i	$-0.892857\pi$
$647$		0			0		0.943883	+	0.330279i	$0.107143\pi$
$648$	−0.0971591	+	0.425682i	−0.0971591	+	0.425682i
$649$		0			0
$650$		0			0
$651$		0			0
$652$	1.05634	−	0.304328i	1.05634	−	0.304328i
$653$	−0.442244	−	1.93760i	−0.442244	−	1.93760i	−0.330279	−	0.943883i	$-0.607143\pi$
$653$	−0.442244	−	1.93760i	−0.442244	−	1.93760i	−0.111964	−	0.993712i	$-0.535714\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$		0			0
$658$		0			0
$659$	0.0914577	+	1.62856i	0.0914577	+	1.62856i	0.623490	+	0.781831i	$0.285714\pi$
$659$	0.0914577	+	1.62856i	0.0914577	+	1.62856i	−0.532032	+	0.846724i	$0.678571\pi$
$660$		0			0
$661$		0			0		−0.222521	−	0.974928i	$-0.571429\pi$
$661$		0			0		0.222521	+	0.974928i	$0.428571\pi$
$662$	−0.186291	−	0.296480i	−0.186291	−	0.296480i
$663$		0			0
$664$		0			0
$665$		0			0
$666$	−0.0432399	−	0.189446i	−0.0432399	−	0.189446i
$667$	−0.0684369	−	0.195581i	−0.0684369	−	0.195581i
$668$		0			0
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$	−1.98742			−1.98742			−0.993712	−	0.111964i	$-0.964286\pi$
$673$	−1.98742			−1.98742			−0.993712	+	0.111964i	$0.964286\pi$
$674$	0.0739590	+	0.211363i	0.0739590	+	0.211363i
$675$		0			0
$676$	0.855791	+	0.412127i	0.855791	+	0.412127i
$677$		0			0		−0.993712	−	0.111964i	$-0.964286\pi$
$677$		0			0		0.993712	+	0.111964i	$0.0357143\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−1.37876	−	0.314692i	−1.37876	−	0.314692i	−0.532032	−	0.846724i	$-0.678571\pi$
$683$	−1.37876	−	0.314692i	−1.37876	−	0.314692i	−0.846724	+	0.532032i	$0.821429\pi$
$684$		0			0
$685$		0			0
$686$	0.119137	+	0.189606i	0.119137	+	0.189606i
$687$		0			0
$688$	−0.0630192	+	0.559311i	−0.0630192	+	0.559311i
$689$		0			0
$690$		0			0
$691$		0			0		−0.578671	−	0.815561i	$-0.696429\pi$
$691$		0			0		0.578671	+	0.815561i	$0.303571\pi$
$692$		0			0
$693$	0.249799	+	0.223234i	0.249799	+	0.223234i
$694$	−0.294212	+	0.329223i	−0.294212	+	0.329223i
$695$		0			0
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$	−0.211363	+	0.926041i	−0.211363	+	0.926041i
$701$	1.84044	+	0.643997i	1.84044	+	0.643997i	0.993712	+	0.111964i	$0.0357143\pi$
$701$	1.84044	+	0.643997i	1.84044	+	0.643997i	0.846724	+	0.532032i	$0.178571\pi$
$702$		0			0
$703$		0			0
$704$	0.0912273	−	0.220242i	0.0912273	−	0.220242i
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	1.53203	−	0.846724i	1.53203	−	0.846724i	0.532032	−	0.846724i	$-0.321429\pi$
$709$	1.53203	−	0.846724i	1.53203	−	0.846724i	1.00000			$0$
$710$		0			0
$711$	1.32399	+	1.05585i	1.32399	+	1.05585i
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$	−0.0473303	+	0.207367i	−0.0473303	+	0.207367i
$717$		0			0
$718$	0.122231	+	0.0207679i	0.122231	+	0.0207679i
$719$		0			0		0.0560704	−	0.998427i	$-0.482143\pi$
$719$		0			0		−0.0560704	+	0.998427i	$0.517857\pi$
$720$		0			0
$721$		0			0
$722$	0.0739590	−	0.211363i	0.0739590	−	0.211363i
$723$		0			0
$724$		0			0
$725$	−0.893796	−	1.61720i	−0.893796	−	1.61720i
$726$		0			0
$727$		0			0		−0.985871	−	0.167506i	$-0.946429\pi$
$727$		0			0		0.985871	+	0.167506i	$0.0535714\pi$
$728$		0			0
$729$	0.781831	+	0.623490i	0.781831	+	0.623490i
$730$		0			0
$731$		0			0
$732$		0			0
$733$		0			0		−0.923880	−	0.382683i	$-0.875000\pi$
$733$		0			0		0.923880	+	0.382683i	$0.125000\pi$
$734$		0			0
$735$		0			0
$736$	−0.0340350	+	0.0615817i	−0.0340350	+	0.0615817i
$737$	−0.0656584	−	0.582734i	−0.0656584	−	0.582734i
$738$		0			0
$739$	1.50696	−	0.624203i	1.50696	−	0.624203i	0.532032	−	0.846724i	$-0.321429\pi$
$739$	1.50696	−	0.624203i	1.50696	−	0.624203i	0.974928	+	0.222521i	$0.0714286\pi$
$740$		0			0
$741$		0			0
$742$	−0.0207679	+	0.0720870i	−0.0207679	+	0.0720870i
$743$	−1.48894	+	0.0836170i	−1.48894	+	0.0836170i	−0.781831	−	0.623490i	$-0.785714\pi$
$743$	−1.48894	+	0.0836170i	−1.48894	+	0.0836170i	−0.707107	+	0.707107i	$0.750000\pi$
$744$		0			0
$745$		0			0
$746$			0.316683i			0.316683i
$747$		0			0
$748$		0			0
$749$	1.03739	−	0.236777i	1.03739	−	0.236777i
$750$		0			0
$751$	1.61720	+	0.893796i	1.61720	+	0.893796i	0.993712	+	0.111964i	$0.0357143\pi$
$751$	1.61720	+	0.893796i	1.61720	+	0.893796i	0.623490	+	0.781831i	$0.285714\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$	0.0981483	+	0.0542447i	0.0981483	+	0.0542447i	0.532032	−	0.846724i	$-0.321429\pi$
$757$	0.0981483	+	0.0542447i	0.0981483	+	0.0542447i	−0.433884	+	0.900969i	$0.642857\pi$
$758$	−0.251582	+	0.315474i	−0.251582	+	0.315474i
$759$		0			0
$760$		0			0
$761$		0			0		−0.0560704	−	0.998427i	$-0.517857\pi$
$761$		0			0		0.0560704	+	0.998427i	$0.482143\pi$
$762$		0			0
$763$	1.47021	−	0.249799i	1.47021	−	0.249799i
$764$	−0.531756	+	0.962141i	−0.531756	+	0.962141i
$765$		0			0
$766$		0			0
$767$		0			0
$768$		0			0
$769$		0			0		0.532032	−	0.846724i	$-0.321429\pi$
$769$		0			0		−0.532032	+	0.846724i	$0.678571\pi$
$770$		0			0
$771$		0			0
$772$	1.15475	−	0.263565i	1.15475	−	0.263565i
$773$		0			0		0.875223	−	0.483719i	$-0.160714\pi$
$773$		0			0		−0.875223	+	0.483719i	$0.839286\pi$
$774$	−0.125246	−	0.0786972i	−0.125246	−	0.0786972i
$775$		0			0
$776$		0			0
$777$		0			0
$778$	−0.380863	+	0.183414i	−0.380863	+	0.183414i
$779$		0			0
$780$		0			0
$781$	0.0499886	−	0.443661i	0.0499886	−	0.443661i
$782$		0			0
$783$		0			0
$784$	−0.369704	−	0.767699i	−0.369704	−	0.767699i
$785$		0			0
$786$		0			0
$787$		0			0		−0.943883	−	0.330279i	$-0.892857\pi$
$787$		0			0		0.943883	+	0.330279i	$0.107143\pi$
$788$	0.485875	+	1.68651i	0.485875	+	1.68651i
$789$		0			0
$790$		0			0
$791$		−	1.98742i		−	1.98742i
$792$	0.0974707	+	0.109070i	0.0974707	+	0.109070i
$793$		0			0
$794$		0			0
$795$		0			0
$796$		0			0
$797$		0			0		0.532032	−	0.846724i	$-0.321429\pi$
$797$		0			0		−0.532032	+	0.846724i	$0.678571\pi$
$798$		0			0
$799$		0			0
$800$	−0.207229	+	0.592225i	−0.207229	+	0.592225i
$801$		0			0
$802$	−0.0996578			−0.0996578
$803$		0			0
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$	0.334485	−	1.96864i	0.334485	−	1.96864i	0.111964	−	0.993712i	$-0.464286\pi$
$809$	0.334485	−	1.96864i	0.334485	−	1.96864i	0.222521	−	0.974928i	$-0.428571\pi$
$810$		0			0
$811$		0			0		−0.532032	−	0.846724i	$-0.678571\pi$
$811$		0			0		0.532032	+	0.846724i	$0.321429\pi$
$812$	1.62151	+	0.671649i	1.62151	+	0.671649i
$813$		0			0
$814$	−0.0601437	−	0.0249123i	−0.0601437	−	0.0249123i
$815$		0			0
$816$		0			0
$817$		0			0
$818$		0			0
$819$		0			0
$820$		0			0
$821$	1.69345	−	1.06406i	1.69345	−	1.06406i	0.846724	−	0.532032i	$-0.178571\pi$
$821$	1.69345	−	1.06406i	1.69345	−	1.06406i	0.846724	−	0.532032i	$-0.178571\pi$
$822$		0			0
$823$	−0.461680	−	0.734760i	−0.461680	−	0.734760i	0.532032	−	0.846724i	$-0.321429\pi$
$823$	−0.461680	−	0.734760i	−0.461680	−	0.734760i	−0.993712	+	0.111964i	$0.964286\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	0.370222	+	0.893796i	0.370222	+	0.893796i	0.993712	+	0.111964i	$0.0357143\pi$
$827$	0.370222	+	0.893796i	0.370222	+	0.893796i	−0.623490	+	0.781831i	$0.714286\pi$
$828$	0.0616387	+	0.0868717i	0.0616387	+	0.0868717i
$829$		0			0		0.993712	−	0.111964i	$-0.0357143\pi$
$829$		0			0		−0.993712	+	0.111964i	$0.964286\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		0			0
$834$		0			0
$835$		0			0
$836$		0			0
$837$		0			0
$838$		0			0
$839$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$839$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$840$		0			0
$841$	−2.27874	+	0.797364i	−2.27874	+	0.797364i
$842$	−0.0451783	+	0.0217567i	−0.0451783	+	0.0217567i
$843$		0			0
$844$		0			0
$845$		0			0
$846$		0			0
$847$	−0.865508	+	0.197547i	−0.865508	+	0.197547i
$848$	0.109240	−	0.263729i	0.109240	−	0.263729i
$849$		0			0
$850$		0			0
$851$	−0.0851699	−	0.0470718i	−0.0851699	−	0.0470718i
$852$		0			0
$853$		0			0		−0.623490	−	0.781831i	$-0.714286\pi$
$853$		0			0		0.623490	+	0.781831i	$0.285714\pi$
$854$		0			0
$855$		0			0
$856$	0.461680	−	0.0520189i	0.461680	−	0.0520189i
$857$		0			0		−0.578671	−	0.815561i	$-0.696429\pi$
$857$		0			0		0.578671	+	0.815561i	$0.303571\pi$
$858$		0			0
$859$		0			0		0.483719	−	0.875223i	$-0.339286\pi$
$859$		0			0		−0.483719	+	0.875223i	$0.660714\pi$
$860$		0			0
$861$		0			0
$862$	0.186291	+	0.296480i	0.186291	+	0.296480i
$863$			0.445042i			0.445042i	0.974928	+	0.222521i	$0.0714286\pi$
$863$			0.445042i			0.445042i	−0.974928	+	0.222521i	$0.928571\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$		0			0
$868$		0			0
$869$	0.545154	−	0.157056i	0.545154	−	0.157056i
$870$		0			0
$871$		0			0
$872$	0.650114	−	0.0365096i	0.650114	−	0.0365096i
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$	−1.30521	+	0.721362i	−1.30521	+	0.721362i	−0.974928	−	0.222521i	$-0.928571\pi$
$877$	−1.30521	+	0.721362i	−1.30521	+	0.721362i	−0.330279	+	0.943883i	$0.607143\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		0			0		−0.998427	−	0.0560704i	$-0.982143\pi$
$881$		0			0		0.998427	+	0.0560704i	$0.0178571\pi$
$882$	0.223929			0.223929
$883$	0.445042			0.445042			0.222521	−	0.974928i	$-0.428571\pi$
$883$	0.445042			0.445042			0.222521	+	0.974928i	$0.428571\pi$
$884$		0			0
$885$		0			0
$886$	0.216297	+	0.391359i	0.216297	+	0.391359i
$887$		0			0		0.815561	−	0.578671i	$-0.196429\pi$
$887$		0			0		−0.815561	+	0.578671i	$0.803571\pi$
$888$		0			0
$889$	0.368937	+	1.28061i	0.368937	+	1.28061i
$890$		0			0
$891$	0.321919	−	0.0927433i	0.321919	−	0.0927433i
$892$		0			0
$893$		0			0
$894$		0			0
$895$		0			0
$896$	−0.259856	−	0.742627i	−0.259856	−	0.742627i
$897$		0			0
$898$	0.412127	+	0.144209i	0.412127	+	0.144209i
$899$		0			0
$900$	0.671649	+	0.671649i	0.671649	+	0.671649i
$901$		0			0
$902$		0			0
$903$		0			0
$904$	0.0971591	−	0.862311i	0.0971591	−	0.862311i
$905$		0			0
$906$		0			0
$907$	−0.958689	+	0.461680i	−0.958689	+	0.461680i	−0.846724	−	0.532032i	$-0.821429\pi$
$907$	−0.958689	+	0.461680i	−0.958689	+	0.461680i	−0.111964	+	0.993712i	$0.535714\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		0			0		0.532032	−	0.846724i	$-0.321429\pi$
$911$		0			0		−0.532032	+	0.846724i	$0.678571\pi$
$912$		0			0
$913$		0			0
$914$	−0.255501	−	0.0434113i	−0.255501	−	0.0434113i
$915$		0			0
$916$		0			0
$917$		0			0
$918$		0			0
$919$	−0.721362	−	0.644649i	−0.721362	−	0.644649i	0.222521	−	0.974928i	$-0.428571\pi$
$919$	−0.721362	−	0.644649i	−0.721362	−	0.644649i	−0.943883	+	0.330279i	$0.892857\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		0			0
$924$		0			0
$925$	−0.819071	−	0.286605i	−0.819071	−	0.286605i
$926$		0			0
$927$		0			0
$928$	1.01469	+	0.560799i	1.01469	+	0.560799i
$929$		0			0		0.781831	−	0.623490i	$-0.214286\pi$
$929$		0			0		−0.781831	+	0.623490i	$0.785714\pi$
$930$		0			0
$931$		0			0
$932$	1.06715	+	0.513914i	1.06715	+	0.513914i
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$		0			0		−0.433884	−	0.900969i	$-0.642857\pi$
$937$		0			0		0.433884	+	0.900969i	$0.357143\pi$
$938$	−0.292274	−	0.261192i	−0.292274	−	0.261192i
$939$		0			0
$940$		0			0
$941$		0			0		0.998427	−	0.0560704i	$-0.0178571\pi$
$941$		0			0		−0.998427	+	0.0560704i	$0.982143\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$	−0.0457823	+	0.0189636i	−0.0457823	+	0.0189636i
$947$	1.27416	−	0.613604i	1.27416	−	0.613604i	0.330279	−	0.943883i	$-0.392857\pi$
$947$	1.27416	−	0.613604i	1.27416	−	0.613604i	0.943883	+	0.330279i	$0.107143\pi$
$948$		0			0
$949$		0			0
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−0.0310446	+	0.107758i	−0.0310446	+	0.107758i	−0.974928	−	0.222521i	$-0.928571\pi$
$953$	−0.0310446	+	0.107758i	−0.0310446	+	0.107758i	0.943883	+	0.330279i	$0.107143\pi$
$954$	0.0499886	+	0.0559373i	0.0499886	+	0.0559373i
$955$		0			0
$956$	0.330500	+	0.263565i	0.330500	+	0.263565i
$957$		0			0
$958$		0			0
$959$	1.12349	−	1.40881i	1.12349	−	1.40881i
$960$		0			0
$961$	−0.993712	−	0.111964i	−0.993712	−	0.111964i
$962$		0			0
$963$	0.351438	−	1.00435i	0.351438	−	1.00435i
$964$		0			0
$965$		0			0
$966$		0			0
$967$	1.60808	+	0.273223i	1.60808	+	0.273223i	0.900969	−	0.433884i	$-0.142857\pi$
$967$	1.60808	+	0.273223i	1.60808	+	0.273223i	0.707107	+	0.707107i	$0.250000\pi$
$968$	−0.385188	+	0.0434002i	−0.385188	+	0.0434002i
$969$		0			0
$970$		0			0
$971$		0			0		0.923880	−	0.382683i	$-0.125000\pi$
$971$		0			0		−0.923880	+	0.382683i	$0.875000\pi$
$972$		0			0
$973$		0			0
$974$	−0.330500	−	0.263565i	−0.330500	−	0.263565i
$975$		0			0
$976$		0			0
$977$	0.162052	+	0.953769i	0.162052	+	0.953769i	0.943883	+	0.330279i	$0.107143\pi$
$977$	0.162052	+	0.953769i	0.162052	+	0.953769i	−0.781831	+	0.623490i	$0.785714\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$	0.570690	−	1.37777i	0.570690	−	1.37777i
$982$	−0.0207679	−	0.122231i	−0.0207679	−	0.122231i
$983$		0			0		0.532032	−	0.846724i	$-0.321429\pi$
$983$		0			0		−0.532032	+	0.846724i	$0.678571\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−0.0711805	+	0.0205068i	−0.0711805	+	0.0205068i
$990$		0			0
$991$	−0.771191	+	0.862963i	−0.771191	+	0.862963i	−0.993712	−	0.111964i	$-0.964286\pi$
$991$	−0.771191	+	0.862963i	−0.771191	+	0.862963i	0.222521	+	0.974928i	$0.428571\pi$
$992$		0			0
$993$		0			0
$994$	−0.172692	−	0.243387i	−0.172692	−	0.243387i
$995$		0			0
$996$		0			0
$997$		0			0		0.111964	−	0.993712i	$-0.464286\pi$
$997$		0			0		−0.111964	+	0.993712i	$0.535714\pi$
$998$	0.0779156	+	0.341370i	0.0779156	+	0.341370i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2359.1.cv.a.797.1	✓	24
7.6	odd	2	CM	2359.1.cv.a.797.1	✓	24
337.137	even	56	inner	2359.1.cv.a.811.1	yes	24
2359.811	odd	56	inner	2359.1.cv.a.811.1	yes	24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2359.1.cv.a.797.1	✓	24	1.1	even	1	trivial
2359.1.cv.a.797.1	✓	24	7.6	odd	2	CM
2359.1.cv.a.811.1	yes	24	337.137	even	56	inner
2359.1.cv.a.811.1	yes	24	2359.811	odd	56	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:	\( N \)	\(=\)	\( 2359 = 7 \cdot 337 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	2359.cv (of order \(56\), degree \(24\), minimal)

\(f(q)\)	\(=\)	\(q+(-0.201753 - 0.0971591i) q^{2} +(-0.592225 - 0.742627i) q^{4} +(0.330279 + 0.943883i) q^{7} +(0.0971591 + 0.425682i) q^{8} +(0.974928 + 0.222521i) q^{9} +O(q^{10})\)	\(q+(-0.201753 - 0.0971591i) q^{2} +(-0.592225 - 0.742627i) q^{4} +(0.330279 + 0.943883i) q^{7} +(0.0971591 + 0.425682i) q^{8} +(0.974928 + 0.222521i) q^{9} +(0.249799 - 0.223234i) q^{11} +(0.0250721 - 0.222521i) q^{14} +(-0.189606 + 0.830718i) q^{16} +(-0.175075 - 0.139617i) q^{18} +(-0.0720870 + 0.0207679i) q^{22} +(-0.110556 + 0.0187843i) q^{23} +(-0.943883 + 0.330279i) q^{25} +(0.505354 - 0.804266i) q^{28} +(0.309511 + 1.82165i) q^{29} +(0.391199 - 0.490548i) q^{32} +(-0.412127 - 0.855791i) q^{36} +(0.678448 + 0.541044i) q^{37} +(0.656405 - 0.0739590i) q^{43} +(-0.313717 - 0.0533027i) q^{44} +(0.0241302 + 0.00695178i) q^{46} +(-0.781831 + 0.623490i) q^{49} +(0.222521 + 0.0250721i) q^{50} +(-0.330279 - 0.0561167i) q^{53} +(-0.369704 + 0.232301i) q^{56} +(0.114545 - 0.397596i) q^{58} +(0.111964 + 0.993712i) q^{63} +(0.641112 - 0.308743i) q^{64} +(1.01293 - 1.42760i) q^{67} +(0.993712 - 0.888036i) q^{71} +0.436629i q^{72} +(-0.0843115 - 0.175075i) q^{74} +(0.293211 + 0.162052i) q^{77} +(1.52574 + 0.734760i) q^{79} +(0.900969 + 0.433884i) q^{81} +(-0.139617 - 0.0488542i) q^{86} +(0.119297 + 0.0846458i) q^{88} +(0.0794241 + 0.0709777i) q^{92} +(0.218315 - 0.0498289i) q^{98} +(0.293211 - 0.162052i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q - 4 q^{4}+O(q^{10}) \) 24 * q - 4 * q^4	\( 24 q - 4 q^{4} + 24 q^{14} + 4 q^{16} + 24 q^{22} - 4 q^{23} - 4 q^{29} + 4 q^{43} - 4 q^{46} + 4 q^{50} - 24 q^{58} - 24 q^{64} + 4 q^{67} - 4 q^{77} + 4 q^{81} - 4 q^{92} - 4 q^{99}+O(q^{100}) \) 24 * q - 4 * q^4 + 24 * q^14 + 4 * q^16 + 24 * q^22 - 4 * q^23 - 4 * q^29 + 4 * q^43 - 4 * q^46 + 4 * q^50 - 24 * q^58 - 24 * q^64 + 4 * q^67 - 4 * q^77 + 4 * q^81 - 4 * q^92 - 4 * q^99

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