LMFDB - Embedded newform 2359.1.cv.a.986.1

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			986.1
Root			$0.111964 - 0.993712i$ of defining polynomial
Character	$\chi$	$=$	2359.986
Dual form			2359.1.cv.a.1658.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.05585 - 1.32399i) q^{2} +(-0.415617 - 1.82094i) q^{4} +(0.111964 + 0.993712i) q^{7} +(-1.32399 - 0.637601i) q^{8} +(-0.433884 - 0.900969i) q^{9} +O(q^{10})$	\(q+(1.05585 - 1.32399i) q^{2} +(-0.415617 - 1.82094i) q^{4} +(0.111964 + 0.993712i) q^{7} +(-1.32399 - 0.637601i) q^{8} +(-0.433884 - 0.900969i) q^{9} +(1.91881 + 0.552800i) q^{11} +(1.43388 + 0.900969i) q^{14} +(-0.559311 + 0.269350i) q^{16} +(-1.65099 - 0.376828i) q^{18} +(2.75788 - 1.95682i) q^{22} +(-0.0542447 - 0.965916i) q^{23} +(0.993712 - 0.111964i) q^{25} +(1.76295 - 0.616884i) q^{28} +(-1.84485 + 0.103605i) q^{29} +(0.0930688 - 0.407761i) q^{32} +(-1.46028 + 1.16453i) q^{36} +(-1.52446 - 0.347948i) q^{37} +(0.119137 + 0.189606i) q^{43} +(0.209124 - 3.72379i) q^{44} +(-1.33614 - 0.948041i) q^{46} +(-0.974928 + 0.222521i) q^{49} +(0.900969 - 1.43388i) q^{50} +(-0.111964 + 1.99371i) q^{53} +(0.485352 - 1.38705i) q^{56} +(-1.81071 + 2.55196i) q^{58} +(0.846724 - 0.532032i) q^{63} +(-0.828660 - 1.03911i) q^{64} +(0.223234 - 0.249799i) q^{67} +(0.532032 + 0.153276i) q^{71} +1.46952i q^{72} +(-2.07028 + 1.65099i) q^{74} +(-0.334485 + 1.96864i) q^{77} +(-0.411851 + 0.516445i) q^{79} +(-0.623490 + 0.781831i) q^{81} +(0.376828 + 0.0424583i) q^{86} +(-2.18802 - 1.95534i) q^{88} +(-1.73633 + 0.500228i) q^{92} +(-0.734760 + 1.52574i) q^{98} +(-0.334485 - 1.96864i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $\left(\mathbb{Z}/2359\mathbb{Z}\right)^\times$.

$n$	$675$	$1695$
$\chi(n)$	$-1$	$e\left(\frac{39}{56}\right)$

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 $

$2$	1.05585	−	1.32399i	1.05585	−	1.32399i	0.111964	−	0.993712i	$-0.464286\pi$
$2$	1.05585	−	1.32399i	1.05585	−	1.32399i	0.943883	−	0.330279i	$-0.107143\pi$
$3$		0			0		0.532032	−	0.846724i	$-0.321429\pi$
$3$		0			0		−0.532032	+	0.846724i	$0.678571\pi$
$4$	−0.415617	−	1.82094i	−0.415617	−	1.82094i
$5$		0			0		0.998427	−	0.0560704i	$-0.0178571\pi$
$5$		0			0		−0.998427	+	0.0560704i	$0.982143\pi$
$6$		0			0
$7$	0.111964	+	0.993712i	0.111964	+	0.993712i
$7$	0.111964	+	0.993712i	0.111964	+	0.993712i
$8$	−1.32399	−	0.637601i	−1.32399	−	0.637601i
$9$	−0.433884	−	0.900969i	−0.433884	−	0.900969i
$10$		0			0
$11$	1.91881	+	0.552800i	1.91881	+	0.552800i	0.974928	+	0.222521i	$0.0714286\pi$
$11$	1.91881	+	0.552800i	1.91881	+	0.552800i	0.943883	+	0.330279i	$0.107143\pi$
$12$		0			0
$13$		0			0		−0.900969	−	0.433884i	$-0.857143\pi$
$13$		0			0		0.900969	+	0.433884i	$0.142857\pi$
$14$	1.43388	+	0.900969i	1.43388	+	0.900969i
$15$		0			0
$16$	−0.559311	+	0.269350i	−0.559311	+	0.269350i
$17$		0			0		−0.666347	−	0.745642i	$-0.732143\pi$
$17$		0			0		0.666347	+	0.745642i	$0.267857\pi$
$18$	−1.65099	−	0.376828i	−1.65099	−	0.376828i
$19$		0			0		0.960917	−	0.276836i	$-0.0892857\pi$
$19$		0			0		−0.960917	+	0.276836i	$0.910714\pi$
$20$		0			0
$21$		0			0
$22$	2.75788	−	1.95682i	2.75788	−	1.95682i
$23$	−0.0542447	−	0.965916i	−0.0542447	−	0.965916i	−0.900969	−	0.433884i	$-0.857143\pi$
$23$	−0.0542447	−	0.965916i	−0.0542447	−	0.965916i	0.846724	−	0.532032i	$-0.178571\pi$
$24$		0			0
$25$	0.993712	−	0.111964i	0.993712	−	0.111964i
$26$		0			0
$27$		0			0
$28$	1.76295	−	0.616884i	1.76295	−	0.616884i
$29$	−1.84485	+	0.103605i	−1.84485	+	0.103605i	−0.943883	−	0.330279i	$-0.892857\pi$
$29$	−1.84485	+	0.103605i	−1.84485	+	0.103605i	−0.900969	+	0.433884i	$0.857143\pi$
$30$		0			0
$31$		0			0		−0.483719	−	0.875223i	$-0.660714\pi$
$31$		0			0		0.483719	+	0.875223i	$0.339286\pi$
$32$	0.0930688	−	0.407761i	0.0930688	−	0.407761i
$33$		0			0
$34$		0			0
$35$		0			0
$36$	−1.46028	+	1.16453i	−1.46028	+	1.16453i
$37$	−1.52446	−	0.347948i	−1.52446	−	0.347948i	−0.623490	−	0.781831i	$-0.714286\pi$
$37$	−1.52446	−	0.347948i	−1.52446	−	0.347948i	−0.900969	+	0.433884i	$0.857143\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		0			0		−0.781831	−	0.623490i	$-0.785714\pi$
$41$		0			0		0.781831	+	0.623490i	$0.214286\pi$
$42$		0			0
$43$	0.119137	+	0.189606i	0.119137	+	0.189606i	0.900969	−	0.433884i	$-0.142857\pi$
$43$	0.119137	+	0.189606i	0.119137	+	0.189606i	−0.781831	+	0.623490i	$0.785714\pi$
$44$	0.209124	−	3.72379i	0.209124	−	3.72379i
$45$		0			0
$46$	−1.33614	−	0.948041i	−1.33614	−	0.948041i
$47$		0			0		0.532032	−	0.846724i	$-0.321429\pi$
$47$		0			0		−0.532032	+	0.846724i	$0.678571\pi$
$48$		0			0
$49$	−0.974928	+	0.222521i	−0.974928	+	0.222521i
$50$	0.900969	−	1.43388i	0.900969	−	1.43388i
$51$		0			0
$52$		0			0
$53$	−0.111964	+	1.99371i	−0.111964	+	1.99371i			1.00000i	$0.5\pi$
$53$	−0.111964	+	1.99371i	−0.111964	+	1.99371i	−0.111964	+	0.993712i	$0.535714\pi$
$54$		0			0
$55$		0			0
$56$	0.485352	−	1.38705i	0.485352	−	1.38705i
$57$		0			0
$58$	−1.81071	+	2.55196i	−1.81071	+	2.55196i
$59$		0			0		0.923880	−	0.382683i	$-0.125000\pi$
$59$		0			0		−0.923880	+	0.382683i	$0.875000\pi$
$60$		0			0
$61$		0			0		0.276836	−	0.960917i	$-0.410714\pi$
$61$		0			0		−0.276836	+	0.960917i	$0.589286\pi$
$62$		0			0
$63$	0.846724	−	0.532032i	0.846724	−	0.532032i
$64$	−0.828660	−	1.03911i	−0.828660	−	1.03911i
$65$		0			0
$66$		0			0
$67$	0.223234	−	0.249799i	0.223234	−	0.249799i	−0.623490	−	0.781831i	$-0.714286\pi$
$67$	0.223234	−	0.249799i	0.223234	−	0.249799i	0.846724	+	0.532032i	$0.178571\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	0.532032	+	0.153276i	0.532032	+	0.153276i	0.532032	−	0.846724i	$-0.321429\pi$
$71$	0.532032	+	0.153276i	0.532032	+	0.153276i			1.00000i	$0.5\pi$
$72$			1.46952i			1.46952i
$73$		0			0		0.960917	−	0.276836i	$-0.0892857\pi$
$73$		0			0		−0.960917	+	0.276836i	$0.910714\pi$
$74$	−2.07028	+	1.65099i	−2.07028	+	1.65099i
$75$		0			0
$76$		0			0
$77$	−0.334485	+	1.96864i	−0.334485	+	1.96864i
$78$		0			0
$79$	−0.411851	+	0.516445i	−0.411851	+	0.516445i	−0.943883	−	0.330279i	$-0.892857\pi$
$79$	−0.411851	+	0.516445i	−0.411851	+	0.516445i	0.532032	+	0.846724i	$0.321429\pi$
$80$		0			0
$81$	−0.623490	+	0.781831i	−0.623490	+	0.781831i
$82$		0			0
$83$		0			0		0.167506	−	0.985871i	$-0.446429\pi$
$83$		0			0		−0.167506	+	0.985871i	$0.553571\pi$
$84$		0			0
$85$		0			0
$86$	0.376828	+	0.0424583i	0.376828	+	0.0424583i
$87$		0			0
$88$	−2.18802	−	1.95534i	−2.18802	−	1.95534i
$89$		0			0		−0.0560704	−	0.998427i	$-0.517857\pi$
$89$		0			0		0.0560704	+	0.998427i	$0.482143\pi$
$90$		0			0
$91$		0			0
$92$	−1.73633	+	0.500228i	−1.73633	+	0.500228i
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$		0			0		0.0560704	−	0.998427i	$-0.482143\pi$
$97$		0			0		−0.0560704	+	0.998427i	$0.517857\pi$
$98$	−0.734760	+	1.52574i	−0.734760	+	1.52574i
$99$	−0.334485	−	1.96864i	−0.334485	−	1.96864i
$100$	−0.616884	−	1.76295i	−0.616884	−	1.76295i
$101$		0			0		−0.578671	−	0.815561i	$-0.696429\pi$
$101$		0			0		0.578671	+	0.815561i	$0.303571\pi$
$102$		0			0
$103$		0			0		−0.781831	−	0.623490i	$-0.785714\pi$
$103$		0			0		0.781831	+	0.623490i	$0.214286\pi$
$104$		0			0
$105$		0			0
$106$	2.52144	+	2.25330i	2.52144	+	2.25330i
$107$	−1.59842	−	1.00435i	−1.59842	−	1.00435i	−0.974928	−	0.222521i	$-0.928571\pi$
$107$	−1.59842	−	1.00435i	−1.59842	−	1.00435i	−0.623490	−	0.781831i	$-0.714286\pi$
$108$		0			0
$109$	1.91881	+	0.107758i	1.91881	+	0.107758i	0.974928	−	0.222521i	$-0.0714286\pi$
$109$	1.91881	+	0.107758i	1.91881	+	0.107758i	0.943883	+	0.330279i	$0.107143\pi$
$110$		0			0
$111$		0			0
$112$	−0.330279	−	0.525636i	−0.330279	−	0.525636i
$113$	1.05737	+	0.119137i	1.05737	+	0.119137i	0.623490	−	0.781831i	$-0.285714\pi$
$113$	1.05737	+	0.119137i	1.05737	+	0.119137i	0.433884	+	0.900969i	$0.357143\pi$
$114$		0			0
$115$		0			0
$116$	0.955410	+	3.31630i	0.955410	+	3.31630i
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$	2.52952	+	1.58941i	2.52952	+	1.58941i
$122$		0			0
$123$		0			0
$124$		0			0
$125$		0			0
$126$	0.189606	−	1.68280i	0.189606	−	1.68280i
$127$	0.484586	−	0.267821i	0.484586	−	0.267821i	−0.222521	−	0.974928i	$-0.571429\pi$
$127$	0.484586	−	0.267821i	0.484586	−	0.267821i	0.707107	+	0.707107i	$0.250000\pi$
$128$	−1.83246			−1.83246
$129$		0			0
$130$		0			0
$131$		0			0		0.974928	−	0.222521i	$-0.0714286\pi$
$131$		0			0		−0.974928	+	0.222521i	$0.928571\pi$
$132$		0			0
$133$		0			0
$134$	−0.0950307	−	0.559311i	−0.0950307	−	0.559311i
$135$		0			0
$136$		0			0
$137$	−1.17700	−	0.411851i	−1.17700	−	0.411851i	−0.330279	−	0.943883i	$-0.607143\pi$
$137$	−1.17700	−	0.411851i	−1.17700	−	0.411851i	−0.846724	+	0.532032i	$0.821429\pi$
$138$		0			0
$139$		0			0		−0.875223	−	0.483719i	$-0.839286\pi$
$139$		0			0		0.875223	+	0.483719i	$0.160714\pi$
$140$		0			0
$141$		0			0
$142$	0.764681	−	0.542570i	0.764681	−	0.542570i
$143$		0			0
$144$	0.485352	+	0.387055i	0.485352	+	0.387055i
$145$		0			0
$146$		0			0
$147$		0			0
$148$			2.92056i			2.92056i
$149$	−0.623490	−	0.218169i	−0.623490	−	0.218169i		−	1.00000i	$-0.5\pi$
$149$	−0.623490	−	0.218169i	−0.623490	−	0.218169i	−0.623490	+	0.781831i	$0.714286\pi$
$150$		0			0
$151$	−1.31386	+	1.47021i	−1.31386	+	1.47021i	−0.532032	+	0.846724i	$0.678571\pi$
$151$	−1.31386	+	1.47021i	−1.31386	+	1.47021i	−0.781831	+	0.623490i	$0.785714\pi$
$152$		0			0
$153$		0			0
$154$	2.25330	+	2.52144i	2.25330	+	2.52144i
$155$		0			0
$156$		0			0
$157$		0			0		0.745642	−	0.666347i	$-0.232143\pi$
$157$		0			0		−0.745642	+	0.666347i	$0.767857\pi$
$158$	0.248917	+	1.09057i	0.248917	+	1.09057i
$159$		0			0
$160$		0			0
$161$	0.953769	−	0.162052i	0.953769	−	0.162052i
$162$	0.376828	+	1.65099i	0.376828	+	1.65099i
$163$	−0.510000	−	1.23125i	−0.510000	−	1.23125i	−0.943883	−	0.330279i	$-0.892857\pi$
$163$	−0.510000	−	1.23125i	−0.510000	−	1.23125i	0.433884	−	0.900969i	$-0.357143\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		0			0		0.993712	−	0.111964i	$-0.0357143\pi$
$167$		0			0		−0.993712	+	0.111964i	$0.964286\pi$
$168$		0			0
$169$	0.623490	+	0.781831i	0.623490	+	0.781831i
$170$		0			0
$171$		0			0
$172$	0.295745	−	0.295745i	0.295745	−	0.295745i
$173$		0			0		0.433884	−	0.900969i	$-0.357143\pi$
$173$		0			0		−0.433884	+	0.900969i	$0.642857\pi$
$174$		0			0
$175$	0.222521	+	0.974928i	0.222521	+	0.974928i
$176$	−1.22211	+	0.207644i	−1.22211	+	0.207644i
$177$		0			0
$178$		0			0
$179$	0.376828	+	1.65099i	0.376828	+	1.65099i	0.707107	+	0.707107i	$0.250000\pi$
$179$	0.376828	+	1.65099i	0.376828	+	1.65099i	−0.330279	+	0.943883i	$0.607143\pi$
$180$		0			0
$181$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$181$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$182$		0			0
$183$		0			0
$184$	−0.544049	+	1.31345i	−0.544049	+	1.31345i
$185$		0			0
$186$		0			0
$187$		0			0
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−0.510000	+	1.23125i	−0.510000	+	1.23125i	0.433884	+	0.900969i	$0.357143\pi$
$191$	−0.510000	+	1.23125i	−0.510000	+	1.23125i	−0.943883	+	0.330279i	$0.892857\pi$
$192$		0			0
$193$	−0.347948	−	0.277479i	−0.347948	−	0.277479i	0.433884	−	0.900969i	$-0.357143\pi$
$193$	−0.347948	−	0.277479i	−0.347948	−	0.277479i	−0.781831	+	0.623490i	$0.785714\pi$
$194$		0			0
$195$		0			0
$196$	0.810394	+	1.68280i	0.810394	+	1.68280i
$197$	−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			$\pi$
$198$	−2.95963	−	1.63573i	−2.95963	−	1.63573i
$199$		0			0		0.923880	−	0.382683i	$-0.125000\pi$
$199$		0			0		−0.923880	+	0.382683i	$0.875000\pi$
$200$	−1.38705	−	0.485352i	−1.38705	−	0.485352i
$201$		0			0
$202$		0			0
$203$	−0.309511	−	1.82165i	−0.309511	−	1.82165i
$204$		0			0
$205$		0			0
$206$		0			0
$207$	−0.846724	+	0.467968i	−0.846724	+	0.467968i
$208$		0			0
$209$		0			0
$210$		0			0
$211$		0			0		−0.993712	−	0.111964i	$-0.964286\pi$
$211$		0			0		0.993712	+	0.111964i	$0.0357143\pi$
$212$	3.67696	−	0.624741i	3.67696	−	0.624741i
$213$		0			0
$214$	−3.01744	+	1.05585i	−3.01744	+	1.05585i
$215$		0			0
$216$		0			0
$217$		0			0
$218$	2.16864	−	2.42671i	2.16864	−	2.42671i
$219$		0			0
$220$		0			0
$221$		0			0
$222$		0			0
$223$		0			0		−0.578671	−	0.815561i	$-0.696429\pi$
$223$		0			0		0.578671	+	0.815561i	$0.303571\pi$
$224$	0.415617	+	0.0468288i	0.415617	+	0.0468288i
$225$	−0.532032	−	0.846724i	−0.532032	−	0.846724i
$226$	1.27416	−	1.27416i	1.27416	−	1.27416i
$227$		0			0		0.781831	−	0.623490i	$-0.214286\pi$
$227$		0			0		−0.781831	+	0.623490i	$0.785714\pi$
$228$		0			0
$229$		0			0		−0.111964	−	0.993712i	$-0.535714\pi$
$229$		0			0		0.111964	+	0.993712i	$0.464286\pi$
$230$		0			0
$231$		0			0
$232$	2.50863	+	1.03911i	2.50863	+	1.03911i
$233$	−0.277479	−	0.347948i	−0.277479	−	0.347948i	0.623490	−	0.781831i	$-0.285714\pi$
$233$	−0.277479	−	0.347948i	−0.277479	−	0.347948i	−0.900969	+	0.433884i	$0.857143\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$	0.781831	−	1.62349i	0.781831	−	1.62349i		−	1.00000i	$-0.5\pi$
$239$	0.781831	−	1.62349i	0.781831	−	1.62349i	0.781831	−	0.623490i	$-0.214286\pi$
$240$		0			0
$241$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$241$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$242$	4.77515	−	1.67090i	4.77515	−	1.67090i
$243$		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		−0.993712	−	0.111964i	$-0.964286\pi$
$251$		0			0		0.993712	+	0.111964i	$0.0357143\pi$
$252$	−1.32071	−	1.32071i	−1.32071	−	1.32071i
$253$	0.429873	−	1.88340i	0.429873	−	1.88340i
$254$	0.157056	−	0.924366i	0.157056	−	0.924366i
$255$		0			0
$256$	−1.10614	+	1.38705i	−1.10614	+	1.38705i
$257$		0			0		0.875223	−	0.483719i	$-0.160714\pi$
$257$		0			0		−0.875223	+	0.483719i	$0.839286\pi$
$258$		0			0
$259$	0.175075	−	1.55383i	0.175075	−	1.55383i
$260$		0			0
$261$	0.893796	+	1.61720i	0.893796	+	1.61720i
$262$		0			0
$263$	−1.22252	+	0.974928i	−1.22252	+	0.974928i	−0.222521	+	0.974928i	$0.571429\pi$
$263$	−1.22252	+	0.974928i	−1.22252	+	0.974928i	−1.00000			$\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$	−0.547649	−	0.302675i	−0.547649	−	0.302675i
$269$		0			0		0.578671	−	0.815561i	$-0.303571\pi$
$269$		0			0		−0.578671	+	0.815561i	$0.696429\pi$
$270$		0			0
$271$		0			0		−0.0560704	−	0.998427i	$-0.517857\pi$
$271$		0			0		0.0560704	+	0.998427i	$0.482143\pi$
$272$		0			0
$273$		0			0
$274$	−1.78802	+	1.12349i	−1.78802	+	1.12349i
$275$	1.96864	+	0.334485i	1.96864	+	0.334485i
$276$		0			0
$277$	−0.735454	−	0.211881i	−0.735454	−	0.211881i	−0.111964	−	0.993712i	$-0.535714\pi$
$277$	−0.735454	−	0.211881i	−0.735454	−	0.211881i	−0.623490	+	0.781831i	$0.714286\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	0.559311	−	1.59842i	0.559311	−	1.59842i	−0.222521	−	0.974928i	$-0.571429\pi$
$281$	0.559311	−	1.59842i	0.559311	−	1.59842i	0.781831	−	0.623490i	$-0.214286\pi$
$282$		0			0
$283$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$283$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$284$	0.0579841	−	1.03250i	0.0579841	−	1.03250i
$285$		0			0
$286$		0			0
$287$		0			0
$288$	−0.407761	+	0.0930688i	−0.407761	+	0.0930688i
$289$	−0.111964	+	0.993712i	−0.111964	+	0.993712i
$290$		0			0
$291$		0			0
$292$		0			0
$293$		0			0		0.0560704	−	0.998427i	$-0.482143\pi$
$293$		0			0		−0.0560704	+	0.998427i	$0.517857\pi$
$294$		0			0
$295$		0			0
$296$	1.79652	+	1.43268i	1.79652	+	1.43268i
$297$		0			0
$298$	−0.947164	+	0.595142i	−0.947164	+	0.595142i
$299$		0			0
$300$		0			0
$301$	−0.175075	+	0.139617i	−0.175075	+	0.139617i
$302$	0.559311	+	3.29187i	0.559311	+	3.29187i
$303$		0			0
$304$		0			0
$305$		0			0
$306$		0			0
$307$		0			0		−0.382683	−	0.923880i	$-0.625000\pi$
$307$		0			0		0.382683	+	0.923880i	$0.375000\pi$
$308$	3.72379	−	0.209124i	3.72379	−	0.209124i
$309$		0			0
$310$		0			0
$311$		0			0		0.900969	−	0.433884i	$-0.142857\pi$
$311$		0			0		−0.900969	+	0.433884i	$0.857143\pi$
$312$		0			0
$313$		0			0		0.846724	−	0.532032i	$-0.178571\pi$
$313$		0			0		−0.846724	+	0.532032i	$0.821429\pi$
$314$		0			0
$315$		0			0
$316$	1.11159	+	0.535312i	1.11159	+	0.535312i
$317$	0.552800	+	1.91881i	0.552800	+	1.91881i	0.330279	+	0.943883i	$0.392857\pi$
$317$	0.552800	+	1.91881i	0.552800	+	1.91881i	0.222521	+	0.974928i	$0.428571\pi$
$318$		0			0
$319$	−3.59720	−	0.821036i	−3.59720	−	0.821036i
$320$		0			0
$321$		0			0
$322$	0.792480	−	1.43388i	0.792480	−	1.43388i
$323$		0			0
$324$	1.68280	+	0.810394i	1.68280	+	0.810394i
$325$		0			0
$326$	−2.16864	−	0.624775i	−2.16864	−	0.624775i
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$	0.643997	+	1.84044i	0.643997	+	1.84044i	0.532032	+	0.846724i	$0.321429\pi$
$331$	0.643997	+	1.84044i	0.643997	+	1.84044i	0.111964	+	0.993712i	$0.464286\pi$
$332$		0			0
$333$	0.347948	+	1.52446i	0.347948	+	1.52446i
$334$		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
$338$	1.69345			1.69345
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$	−0.330279	−	0.943883i	−0.330279	−	0.943883i
$344$	−0.0368439	−	0.326999i	−0.0368439	−	0.326999i
$345$		0			0
$346$		0			0
$347$	0.0648927	−	0.0914577i	0.0648927	−	0.0914577i	−0.781831	−	0.623490i	$-0.785714\pi$
$347$	0.0648927	−	0.0914577i	0.0648927	−	0.0914577i	0.846724	+	0.532032i	$0.178571\pi$
$348$		0			0
$349$		0			0		−0.943883	−	0.330279i	$-0.892857\pi$
$349$		0			0		0.943883	+	0.330279i	$0.107143\pi$
$350$	1.52574	+	0.734760i	1.52574	+	0.734760i
$351$		0			0
$352$	0.403992	−	0.730968i	0.403992	−	0.730968i
$353$		0			0		0.900969	−	0.433884i	$-0.142857\pi$
$353$		0			0		−0.900969	+	0.433884i	$0.857143\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$	2.58377	+	1.24428i	2.58377	+	1.24428i
$359$	0.943883	−	0.669721i	0.943883	−	0.669721i		−	1.00000i	$-0.5\pi$
$359$	0.943883	−	0.669721i	0.943883	−	0.669721i	0.943883	+	0.330279i	$0.107143\pi$
$360$		0			0
$361$	0.846724	−	0.532032i	0.846724	−	0.532032i
$362$		0			0
$363$		0			0
$364$		0			0
$365$		0			0
$366$		0			0
$367$		0			0		−0.382683	−	0.923880i	$-0.625000\pi$
$367$		0			0		0.382683	+	0.923880i	$0.375000\pi$
$368$	0.290509	+	0.525636i	0.290509	+	0.525636i
$369$		0			0
$370$		0			0
$371$	−1.99371	+	0.111964i	−1.99371	+	0.111964i
$372$		0			0
$373$	1.10568	−	0.881748i	1.10568	−	0.881748i	0.111964	−	0.993712i	$-0.464286\pi$
$373$	1.10568	−	0.881748i	1.10568	−	0.881748i	0.993712	+	0.111964i	$0.0357143\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$	−1.12349	+	0.541044i	−1.12349	+	0.541044i	−0.900969	−	0.433884i	$-0.857143\pi$
$379$	−1.12349	+	0.541044i	−1.12349	+	0.541044i	−0.222521	+	0.974928i	$0.571429\pi$
$380$		0			0
$381$		0			0
$382$	1.09168	+	1.97525i	1.09168	+	1.97525i
$383$		0			0		−0.815561	−	0.578671i	$-0.803571\pi$
$383$		0			0		0.815561	+	0.578671i	$0.196429\pi$
$384$		0			0
$385$		0			0
$386$	−0.734760	+	0.167704i	−0.734760	+	0.167704i
$387$	0.119137	−	0.189606i	0.119137	−	0.189606i
$388$		0			0
$389$	0.442244	−	1.93760i	0.442244	−	1.93760i	0.111964	−	0.993712i	$-0.464286\pi$
$389$	0.442244	−	1.93760i	0.442244	−	1.93760i	0.330279	−	0.943883i	$-0.392857\pi$
$390$		0			0
$391$		0			0
$392$	1.43268	+	0.326999i	1.43268	+	0.326999i
$393$		0			0
$394$	−0.866242	+	3.00679i	−0.866242	+	3.00679i
$395$		0			0
$396$	−3.44575	+	1.42728i	−3.44575	+	1.42728i
$397$		0			0		−0.960917	−	0.276836i	$-0.910714\pi$
$397$		0			0		0.960917	+	0.276836i	$0.0892857\pi$
$398$		0			0
$399$		0			0
$400$	−0.525636	+	0.330279i	−0.525636	+	0.330279i
$401$	−1.12349	−	1.40881i	−1.12349	−	1.40881i	−0.900969	−	0.433884i	$-0.857143\pi$
$401$	−1.12349	−	1.40881i	−1.12349	−	1.40881i	−0.222521	−	0.974928i	$-0.571429\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$		0			0
$406$	−2.73865	−	1.51360i	−2.73865	−	1.51360i
$407$	−2.73280	−	1.51037i	−2.73280	−	1.51037i
$408$		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
$414$	−0.274426	+	1.61516i	−0.274426	+	1.61516i
$415$		0			0
$416$		0			0
$417$		0			0
$418$		0			0
$419$		0			0		0.974928	−	0.222521i	$-0.0714286\pi$
$419$		0			0		−0.974928	+	0.222521i	$0.928571\pi$
$420$		0			0
$421$	−0.376828	+	1.65099i	−0.376828	+	1.65099i	0.330279	+	0.943883i	$0.392857\pi$
$421$	−0.376828	+	1.65099i	−0.376828	+	1.65099i	−0.707107	+	0.707107i	$0.750000\pi$
$422$		0			0
$423$		0			0
$424$	1.41943	−	2.56827i	1.41943	−	2.56827i
$425$		0			0
$426$		0			0
$427$		0			0
$428$	−1.16453	+	3.32805i	−1.16453	+	3.32805i
$429$		0			0
$430$		0			0
$431$	−0.643997	−	1.84044i	−0.643997	−	1.84044i	−0.532032	−	0.846724i	$-0.678571\pi$
$431$	−0.643997	−	1.84044i	−0.643997	−	1.84044i	−0.111964	−	0.993712i	$-0.535714\pi$
$432$		0			0
$433$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$433$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$434$		0			0
$435$		0			0
$436$	−0.601270	−	3.53882i	−0.601270	−	3.53882i
$437$		0			0
$438$		0			0
$439$		0			0		0.483719	−	0.875223i	$-0.339286\pi$
$439$		0			0		−0.483719	+	0.875223i	$0.660714\pi$
$440$		0			0
$441$	0.623490	+	0.781831i	0.623490	+	0.781831i
$442$		0			0
$443$	−1.30521	−	1.16640i	−1.30521	−	1.16640i	−0.974928	−	0.222521i	$-0.928571\pi$
$443$	−1.30521	−	1.16640i	−1.30521	−	1.16640i	−0.330279	−	0.943883i	$-0.607143\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$	0.939793	−	0.939793i	0.939793	−	0.939793i
$449$	0.461680	+	0.734760i	0.461680	+	0.734760i	0.993712	−	0.111964i	$-0.0357143\pi$
$449$	0.461680	+	0.734760i	0.461680	+	0.734760i	−0.532032	+	0.846724i	$0.678571\pi$
$450$	−1.68280	−	0.189606i	−1.68280	−	0.189606i
$451$		0			0
$452$	−0.222521	−	1.97493i	−0.222521	−	1.97493i
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	1.08689	−	0.771191i	1.08689	−	0.771191i	0.111964	−	0.993712i	$-0.464286\pi$
$457$	1.08689	−	0.771191i	1.08689	−	0.771191i	0.974928	+	0.222521i	$0.0714286\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		0			0		0.745642	−	0.666347i	$-0.232143\pi$
$461$		0			0		−0.745642	+	0.666347i	$0.767857\pi$
$462$		0			0
$463$		0			0		−0.993712	−	0.111964i	$-0.964286\pi$
$463$		0			0		0.993712	+	0.111964i	$0.0357143\pi$
$464$	1.00394	−	0.554858i	1.00394	−	0.554858i
$465$		0			0
$466$	−0.753655			−0.753655
$467$		0			0		0.875223	−	0.483719i	$-0.160714\pi$
$467$		0			0		−0.875223	+	0.483719i	$0.839286\pi$
$468$		0			0
$469$	0.273223	+	0.193862i	0.273223	+	0.193862i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	0.123788	+	0.429677i	0.123788	+	0.429677i
$474$		0			0
$475$		0			0
$476$		0			0
$477$	1.84485	−	0.764163i	1.84485	−	0.764163i
$478$	−1.32399	−	2.74930i	−1.32399	−	2.74930i
$479$		0			0		0.815561	−	0.578671i	$-0.196429\pi$
$479$		0			0		−0.815561	+	0.578671i	$0.803571\pi$
$480$		0			0
$481$		0			0
$482$		0			0
$483$		0			0
$484$	1.84290	−	5.26669i	1.84290	−	5.26669i
$485$		0			0
$486$		0			0
$487$	0.862311	+	1.79061i	0.862311	+	1.79061i	0.532032	+	0.846724i	$0.321429\pi$
$487$	0.862311	+	1.79061i	0.862311	+	1.79061i	0.330279	+	0.943883i	$0.392857\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−0.771191	−	0.862963i	−0.771191	−	0.862963i	0.222521	−	0.974928i	$-0.428571\pi$
$491$	−0.771191	−	0.862963i	−0.771191	−	0.862963i	−0.993712	+	0.111964i	$0.964286\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−0.0927433	+	0.545848i	−0.0927433	+	0.545848i
$498$		0			0
$499$	0.433884	+	1.90097i	0.433884	+	1.90097i	0.433884	+	0.900969i	$0.357143\pi$
$499$	0.433884	+	1.90097i	0.433884	+	1.90097i			1.00000i	$0.500000\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		0			0		−0.666347	−	0.745642i	$-0.732143\pi$
$503$		0			0		0.666347	+	0.745642i	$0.267857\pi$
$504$	−1.46028	+	0.164534i	−1.46028	+	0.164534i
$505$		0			0
$506$	−2.03972	−	2.55773i	−2.03972	−	2.55773i
$507$		0			0
$508$	−0.689088	−	0.771090i	−0.689088	−	0.771090i
$509$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$509$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$510$		0			0
$511$		0			0
$512$	0.260773	+	1.14252i	0.260773	+	1.14252i
$513$		0			0
$514$		0			0
$515$		0			0
$516$		0			0
$517$		0			0
$518$	−1.87241	−	1.87241i	−1.87241	−	1.87241i
$519$		0			0
$520$		0			0
$521$		0			0		0.382683	−	0.923880i	$-0.375000\pi$
$521$		0			0		−0.382683	+	0.923880i	$0.625000\pi$
$522$	3.08487	+	0.524141i	3.08487	+	0.524141i
$523$		0			0		0.666347	−	0.745642i	$-0.267857\pi$
$523$		0			0		−0.666347	+	0.745642i	$0.732143\pi$
$524$		0			0
$525$		0			0
$526$			2.64798i			2.64798i
$527$		0			0
$528$		0			0
$529$	0.0636613	−	0.00717291i	0.0636613	−	0.00717291i
$530$		0			0
$531$		0			0
$532$		0			0
$533$		0			0
$534$		0			0
$535$		0			0
$536$	−0.454833	+	0.188398i	−0.454833	+	0.188398i
$537$		0			0
$538$		0			0
$539$	−1.99371	−	0.111964i	−1.99371	−	0.111964i
$540$		0			0
$541$	0.292893	+	0.707107i	0.292893	+	0.707107i	1.00000			$0$
$541$	0.292893	+	0.707107i	0.292893	+	0.707107i	−0.707107	+	0.707107i	$0.750000\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		0			0
$546$		0			0
$547$	1.72571	−	0.953769i	1.72571	−	0.953769i	0.781831	−	0.623490i	$-0.214286\pi$
$547$	1.72571	−	0.953769i	1.72571	−	0.953769i	0.943883	−	0.330279i	$-0.107143\pi$
$548$	−0.260773	+	2.31442i	−0.260773	+	2.31442i
$549$		0			0
$550$	2.52144	−	2.25330i	2.52144	−	2.25330i
$551$		0			0
$552$		0			0
$553$	−0.559311	−	0.351438i	−0.559311	−	0.351438i
$554$	−1.05706	+	0.750021i	−1.05706	+	0.750021i
$555$		0			0
$556$		0			0
$557$		−	1.41421i		−	1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$557$		−	1.41421i		−	1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$		0			0
$562$	−1.52574	−	2.42821i	−1.52574	−	2.42821i
$563$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$563$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	−0.846724	−	0.532032i	−0.846724	−	0.532032i
$568$	−0.606677	−	0.542160i	−0.606677	−	0.542160i
$569$	1.06925	+	0.442896i	1.06925	+	0.442896i	0.846724	−	0.532032i	$-0.178571\pi$
$569$	1.06925	+	0.442896i	1.06925	+	0.442896i	0.222521	+	0.974928i	$0.428571\pi$
$570$		0			0
$571$	−0.516445	−	0.411851i	−0.516445	−	0.411851i	0.330279	−	0.943883i	$-0.392857\pi$
$571$	−0.516445	−	0.411851i	−0.516445	−	0.411851i	−0.846724	+	0.532032i	$0.821429\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−0.162052	−	0.953769i	−0.162052	−	0.953769i
$576$	−0.576661	+	1.19745i	−0.576661	+	1.19745i
$577$		0			0		0.0560704	−	0.998427i	$-0.482143\pi$
$577$		0			0		−0.0560704	+	0.998427i	$0.517857\pi$
$578$	1.19745	+	1.19745i	1.19745	+	1.19745i
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$	−1.31696	+	3.76366i	−1.31696	+	3.76366i
$584$		0			0
$585$		0			0
$586$		0			0
$587$		0			0		0.483719	−	0.875223i	$-0.339286\pi$
$587$		0			0		−0.483719	+	0.875223i	$0.660714\pi$
$588$		0			0
$589$		0			0
$590$		0			0
$591$		0			0
$592$	0.946365	−	0.216002i	0.946365	−	0.216002i
$593$		0			0		0.623490	−	0.781831i	$-0.285714\pi$
$593$		0			0		−0.623490	+	0.781831i	$0.714286\pi$
$594$		0			0
$595$		0			0
$596$	−0.138138	+	1.22601i	−0.138138	+	1.22601i
$597$		0			0
$598$		0			0
$599$	0.846011	−	1.75676i	0.846011	−	1.75676i	0.222521	−	0.974928i	$-0.428571\pi$
$599$	0.846011	−	1.75676i	0.846011	−	1.75676i	0.623490	−	0.781831i	$-0.285714\pi$
$600$		0			0
$601$		0			0		0.960917	−	0.276836i	$-0.0892857\pi$
$601$		0			0		−0.960917	+	0.276836i	$0.910714\pi$
$602$			0.379212i			0.379212i
$603$	−0.321919	−	0.0927433i	−0.321919	−	0.0927433i
$604$	3.22323	+	1.78142i	3.22323	+	1.78142i
$605$		0			0
$606$		0			0
$607$		0			0		0.666347	−	0.745642i	$-0.267857\pi$
$607$		0			0		−0.666347	+	0.745642i	$0.732143\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$	0.532032	−	1.84672i	0.532032	−	1.84672i		−	1.00000i	$-0.5\pi$
$613$	0.532032	−	1.84672i	0.532032	−	1.84672i	0.532032	−	0.846724i	$-0.321429\pi$
$614$		0			0
$615$		0			0
$616$	1.69806	−	2.39319i	1.69806	−	2.39319i
$617$	−0.545848	+	1.89468i	−0.545848	+	1.89468i	−0.111964	+	0.993712i	$0.535714\pi$
$617$	−0.545848	+	1.89468i	−0.545848	+	1.89468i	−0.433884	+	0.900969i	$0.642857\pi$
$618$		0			0
$619$		0			0		−0.974928	−	0.222521i	$-0.928571\pi$
$619$		0			0		0.974928	+	0.222521i	$0.0714286\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	0.974928	−	0.222521i	0.974928	−	0.222521i
$626$		0			0
$627$		0			0
$628$		0			0
$629$		0			0
$630$		0			0
$631$	−0.752407	−	1.19745i	−0.752407	−	1.19745i	−0.974928	−	0.222521i	$-0.928571\pi$
$631$	−0.752407	−	1.19745i	−0.752407	−	1.19745i	0.222521	−	0.974928i	$-0.428571\pi$
$632$	0.874573	−	0.421172i	0.874573	−	0.421172i
$633$		0			0
$634$	3.12416	+	1.29407i	3.12416	+	1.29407i
$635$		0			0
$636$		0			0
$637$		0			0
$638$	−4.88514	+	3.89577i	−4.88514	+	3.89577i
$639$	−0.0927433	−	0.545848i	−0.0927433	−	0.545848i
$640$		0			0
$641$	−0.0836170	+	0.0747247i	−0.0836170	+	0.0747247i	−0.707107	−	0.707107i	$-0.750000\pi$
$641$	−0.0836170	+	0.0747247i	−0.0836170	+	0.0747247i	0.623490	+	0.781831i	$0.285714\pi$
$642$		0			0
$643$		0			0		−0.483719	−	0.875223i	$-0.660714\pi$
$643$		0			0		0.483719	+	0.875223i	$0.339286\pi$
$644$	−0.691489	−	1.66940i	−0.691489	−	1.66940i
$645$		0			0
$646$		0			0
$647$		0			0		−0.993712	−	0.111964i	$-0.964286\pi$
$647$		0			0		0.993712	+	0.111964i	$0.0357143\pi$
$648$	1.32399	−	0.637601i	1.32399	−	0.637601i
$649$		0			0
$650$		0			0
$651$		0			0
$652$	−2.03006	+	1.44041i	−2.03006	+	1.44041i
$653$	−0.958689	−	0.461680i	−0.958689	−	0.461680i	−0.111964	−	0.993712i	$-0.535714\pi$
$653$	−0.958689	−	0.461680i	−0.958689	−	0.461680i	−0.846724	+	0.532032i	$0.821429\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$		0			0
$658$		0			0
$659$	0.721362	−	1.30521i	0.721362	−	1.30521i	−0.222521	−	0.974928i	$-0.571429\pi$
$659$	0.721362	−	1.30521i	0.721362	−	1.30521i	0.943883	−	0.330279i	$-0.107143\pi$
$660$		0			0
$661$		0			0		−0.900969	−	0.433884i	$-0.857143\pi$
$661$		0			0		0.900969	+	0.433884i	$0.142857\pi$
$662$	3.11668	+	1.09057i	3.11668	+	1.09057i
$663$		0			0
$664$		0			0
$665$		0			0
$666$	2.38575	+	1.14892i	2.38575	+	1.14892i
$667$	0.200147	+	1.77635i	0.200147	+	1.77635i
$668$		0			0
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$	−1.06406			−1.06406			−0.532032	−	0.846724i	$-0.678571\pi$
$673$	−1.06406			−1.06406			−0.532032	+	0.846724i	$0.678571\pi$
$674$	0.189606	+	1.68280i	0.189606	+	1.68280i
$675$		0			0
$676$	1.16453	−	1.46028i	1.16453	−	1.46028i
$677$		0			0		0.532032	−	0.846724i	$-0.321429\pi$
$677$		0			0		−0.532032	+	0.846724i	$0.678571\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$		0			0
$682$		0			0
$683$	0.613604	+	1.27416i	0.613604	+	1.27416i	0.943883	+	0.330279i	$0.107143\pi$
$683$	0.613604	+	1.27416i	0.613604	+	1.27416i	−0.330279	+	0.943883i	$0.607143\pi$
$684$		0			0
$685$		0			0
$686$	−1.59842	−	0.559311i	−1.59842	−	0.559311i
$687$		0			0
$688$	−0.117705	−	0.0739590i	−0.117705	−	0.0739590i
$689$		0			0
$690$		0			0
$691$		0			0		−0.666347	−	0.745642i	$-0.732143\pi$
$691$		0			0		0.666347	+	0.745642i	$0.267857\pi$
$692$		0			0
$693$	1.91881	−	0.552800i	1.91881	−	0.552800i
$694$	−0.0525724	−	0.182483i	−0.0525724	−	0.182483i
$695$		0			0
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$	1.68280	−	0.810394i	1.68280	−	0.810394i
$701$	0.862311	+	0.0971591i	0.862311	+	0.0971591i	0.532032	−	0.846724i	$-0.321429\pi$
$701$	0.862311	+	0.0971591i	0.862311	+	0.0971591i	0.330279	+	0.943883i	$0.392857\pi$
$702$		0			0
$703$		0			0
$704$	−1.01562	−	2.45193i	−1.01562	−	2.45193i
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	0.0561167	+	0.330279i	0.0561167	+	0.330279i	1.00000			$0$
$709$	0.0561167	+	0.330279i	0.0561167	+	0.330279i	−0.943883	+	0.330279i	$0.892857\pi$
$710$		0			0
$711$	0.643997	+	0.146988i	0.643997	+	0.146988i
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$	2.84974	−	1.37236i	2.84974	−	1.37236i
$717$		0			0
$718$	0.109892	−	1.95682i	0.109892	−	1.95682i
$719$		0			0		−0.483719	−	0.875223i	$-0.660714\pi$
$719$		0			0		0.483719	+	0.875223i	$0.339286\pi$
$720$		0			0
$721$		0			0
$722$	0.189606	−	1.68280i	0.189606	−	1.68280i
$723$		0			0
$724$		0			0
$725$	−1.82165	+	0.309511i	−1.82165	+	0.309511i
$726$		0			0
$727$		0			0		0.0560704	−	0.998427i	$-0.482143\pi$
$727$		0			0		−0.0560704	+	0.998427i	$0.517857\pi$
$728$		0			0
$729$	0.974928	+	0.222521i	0.974928	+	0.222521i
$730$		0			0
$731$		0			0
$732$		0			0
$733$		0			0		0.923880	−	0.382683i	$-0.125000\pi$
$733$		0			0		−0.923880	+	0.382683i	$0.875000\pi$
$734$		0			0
$735$		0			0
$736$	−0.398911	−	0.0677777i	−0.398911	−	0.0677777i
$737$	0.566434	−	0.355914i	0.566434	−	0.355914i
$738$		0			0
$739$	−1.37777	−	0.570690i	−1.37777	−	0.570690i	−0.433884	−	0.900969i	$-0.642857\pi$
$739$	−1.37777	−	0.570690i	−1.37777	−	0.570690i	−0.943883	+	0.330279i	$0.892857\pi$
$740$		0			0
$741$		0			0
$742$	−1.95682	+	2.75788i	−1.95682	+	2.75788i
$743$	−1.68203	−	0.929628i	−1.68203	−	0.929628i	−0.974928	−	0.222521i	$-0.928571\pi$
$743$	−1.68203	−	0.929628i	−1.68203	−	0.929628i	−0.707107	−	0.707107i	$-0.750000\pi$
$744$		0			0
$745$		0			0
$746$		−	2.39490i		−	2.39490i
$747$		0			0
$748$		0			0
$749$	0.819071	−	1.70082i	0.819071	−	1.70082i
$750$		0			0
$751$	0.309511	−	1.82165i	0.309511	−	1.82165i	−0.222521	−	0.974928i	$-0.571429\pi$
$751$	0.309511	−	1.82165i	0.309511	−	1.82165i	0.532032	−	0.846724i	$-0.321429\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$	−0.162052	+	0.953769i	−0.162052	+	0.953769i	0.781831	+	0.623490i	$0.214286\pi$
$757$	−0.162052	+	0.953769i	−0.162052	+	0.953769i	−0.943883	+	0.330279i	$0.892857\pi$
$758$	−0.469896	+	2.05875i	−0.469896	+	2.05875i
$759$		0			0
$760$		0			0
$761$		0			0		0.483719	−	0.875223i	$-0.339286\pi$
$761$		0			0		−0.483719	+	0.875223i	$0.660714\pi$
$762$		0			0
$763$	0.107758	+	1.91881i	0.107758	+	1.91881i
$764$	2.45399	+	0.416950i	2.45399	+	0.416950i
$765$		0			0
$766$		0			0
$767$		0			0
$768$		0			0
$769$		0			0		0.943883	−	0.330279i	$-0.107143\pi$
$769$		0			0		−0.943883	+	0.330279i	$0.892857\pi$
$770$		0			0
$771$		0			0
$772$	−0.360659	+	0.748917i	−0.360659	+	0.748917i
$773$		0			0		−0.167506	−	0.985871i	$-0.553571\pi$
$773$		0			0		0.167506	+	0.985871i	$0.446429\pi$
$774$	−0.125246	−	0.357932i	−0.125246	−	0.357932i
$775$		0			0
$776$		0			0
$777$		0			0
$778$	−2.09842	−	2.63133i	−2.09842	−	2.63133i
$779$		0			0
$780$		0			0
$781$	0.936138	+	0.588215i	0.936138	+	0.588215i
$782$		0			0
$783$		0			0
$784$	0.485352	−	0.387055i	0.485352	−	0.387055i
$785$		0			0
$786$		0			0
$787$		0			0		−0.993712	−	0.111964i	$-0.964286\pi$
$787$		0			0		0.993712	+	0.111964i	$0.0357143\pi$
$788$	1.99710	+	2.81465i	1.99710	+	2.81465i
$789$		0			0
$790$		0			0
$791$			1.06406i			1.06406i
$792$	−0.812350	+	2.81973i	−0.812350	+	2.81973i
$793$		0			0
$794$		0			0
$795$		0			0
$796$		0			0
$797$		0			0		0.943883	−	0.330279i	$-0.107143\pi$
$797$		0			0		−0.943883	+	0.330279i	$0.892857\pi$
$798$		0			0
$799$		0			0
$800$	0.0468288	−	0.415617i	0.0468288	−	0.415617i
$801$		0			0
$802$	−3.05149			−3.05149
$803$		0			0
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$	1.74769	+	0.0981483i	1.74769	+	0.0981483i	0.900969	−	0.433884i	$-0.142857\pi$
$809$	1.74769	+	0.0981483i	1.74769	+	0.0981483i	0.846724	+	0.532032i	$0.178571\pi$
$810$		0			0
$811$		0			0		−0.943883	−	0.330279i	$-0.892857\pi$
$811$		0			0		0.943883	+	0.330279i	$0.107143\pi$
$812$	−3.18848	+	1.32071i	−3.18848	+	1.32071i
$813$		0			0
$814$	−4.88514	+	2.02349i	−4.88514	+	2.02349i
$815$		0			0
$816$		0			0
$817$		0			0
$818$		0			0
$819$		0			0
$820$		0			0
$821$	0.660558	−	1.88777i	0.660558	−	1.88777i	0.330279	−	0.943883i	$-0.392857\pi$
$821$	0.660558	−	1.88777i	0.660558	−	1.88777i	0.330279	−	0.943883i	$-0.392857\pi$
$822$		0			0
$823$	−1.47592	−	0.516445i	−1.47592	−	0.516445i	−0.532032	−	0.846724i	$-0.678571\pi$
$823$	−1.47592	−	0.516445i	−1.47592	−	0.516445i	−0.943883	+	0.330279i	$0.892857\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	0.754553	−	1.82165i	0.754553	−	1.82165i	0.222521	−	0.974928i	$-0.428571\pi$
$827$	0.754553	−	1.82165i	0.754553	−	1.82165i	0.532032	−	0.846724i	$-0.321429\pi$
$828$	1.20405	+	1.34734i	1.20405	+	1.34734i
$829$		0			0		−0.532032	−	0.846724i	$-0.678571\pi$
$829$		0			0		0.532032	+	0.846724i	$0.321429\pi$
$830$		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	$-0.250000\pi$
$839$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$840$		0			0
$841$	2.39903	−	0.270306i	2.39903	−	0.270306i
$842$	1.78802	+	2.24211i	1.78802	+	2.24211i
$843$		0			0
$844$		0			0
$845$		0			0
$846$		0			0
$847$	−1.29619	+	2.69158i	−1.29619	+	2.69158i
$848$	−0.474383	−	1.14526i	−0.474383	−	1.14526i
$849$		0			0
$850$		0			0
$851$	−0.253394	+	1.49137i	−0.253394	+	1.49137i
$852$		0			0
$853$		0			0		−0.222521	−	0.974928i	$-0.571429\pi$
$853$		0			0		0.222521	+	0.974928i	$0.428571\pi$
$854$		0			0
$855$		0			0
$856$	1.47592	+	2.34891i	1.47592	+	2.34891i
$857$		0			0		−0.666347	−	0.745642i	$-0.732143\pi$
$857$		0			0		0.666347	+	0.745642i	$0.267857\pi$
$858$		0			0
$859$		0			0		−0.985871	−	0.167506i	$-0.946429\pi$
$859$		0			0		0.985871	+	0.167506i	$0.0535714\pi$
$860$		0			0
$861$		0			0
$862$	−3.11668	−	1.09057i	−3.11668	−	1.09057i
$863$		−	1.80194i		−	1.80194i	−0.433884	−	0.900969i	$-0.642857\pi$
$863$		−	1.80194i		−	1.80194i	0.433884	−	0.900969i	$-0.357143\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−1.07576	+	0.763289i	−1.07576	+	0.763289i
$870$		0			0
$871$		0			0
$872$	−2.47178	−	1.36611i	−2.47178	−	1.36611i
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$	0.321919	+	1.89468i	0.321919	+	1.89468i	0.433884	+	0.900969i	$0.357143\pi$
$877$	0.321919	+	1.89468i	0.321919	+	1.89468i	−0.111964	+	0.993712i	$0.535714\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		0			0		0.875223	−	0.483719i	$-0.160714\pi$
$881$		0			0		−0.875223	+	0.483719i	$0.839286\pi$
$882$	1.69345			1.69345
$883$	1.80194			1.80194			0.900969	−	0.433884i	$-0.142857\pi$
$883$	1.80194			1.80194			0.900969	+	0.433884i	$0.142857\pi$
$884$		0			0
$885$		0			0
$886$	−2.92241	+	0.496537i	−2.92241	+	0.496537i
$887$		0			0		0.745642	−	0.666347i	$-0.232143\pi$
$887$		0			0		−0.745642	+	0.666347i	$0.767857\pi$
$888$		0			0
$889$	0.320394	+	0.451552i	0.320394	+	0.451552i
$890$		0			0
$891$	−1.62856	+	1.15552i	−1.62856	+	1.15552i
$892$		0			0
$893$		0			0
$894$		0			0
$895$		0			0
$896$	−0.205171	−	1.82094i	−0.205171	−	1.82094i
$897$		0			0
$898$	1.46028	+	0.164534i	1.46028	+	0.164534i
$899$		0			0
$900$	−1.32071	+	1.32071i	−1.32071	+	1.32071i
$901$		0			0
$902$		0			0
$903$		0			0
$904$	−1.32399	−	0.831919i	−1.32399	−	0.831919i
$905$		0			0
$906$		0			0
$907$	−1.17700	−	1.47592i	−1.17700	−	1.47592i	−0.846724	−	0.532032i	$-0.821429\pi$
$907$	−1.17700	−	1.47592i	−1.17700	−	1.47592i	−0.330279	−	0.943883i	$-0.607143\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		0			0		0.943883	−	0.330279i	$-0.107143\pi$
$911$		0			0		−0.943883	+	0.330279i	$0.892857\pi$
$912$		0			0
$913$		0			0
$914$	0.126542	−	2.25330i	0.126542	−	2.25330i
$915$		0			0
$916$		0			0
$917$		0			0
$918$		0			0
$919$	1.89468	−	0.545848i	1.89468	−	0.545848i	0.900969	−	0.433884i	$-0.142857\pi$
$919$	1.89468	−	0.545848i	1.89468	−	0.545848i	0.993712	−	0.111964i	$-0.0357143\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		0			0
$924$		0			0
$925$	−1.55383	−	0.175075i	−1.55383	−	0.175075i
$926$		0			0
$927$		0			0
$928$	−0.129452	+	0.761901i	−0.129452	+	0.761901i
$929$		0			0		0.974928	−	0.222521i	$-0.0714286\pi$
$929$		0			0		−0.974928	+	0.222521i	$0.928571\pi$
$930$		0			0
$931$		0			0
$932$	−0.518266	+	0.649885i	−0.518266	+	0.649885i
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$		0			0		0.781831	−	0.623490i	$-0.214286\pi$
$937$		0			0		−0.781831	+	0.623490i	$0.785714\pi$
$938$	0.545154	−	0.157056i	0.545154	−	0.157056i
$939$		0			0
$940$		0			0
$941$		0			0		−0.875223	−	0.483719i	$-0.839286\pi$
$941$		0			0		0.875223	+	0.483719i	$0.160714\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$	0.699590	+	0.289780i	0.699590	+	0.289780i
$947$	−0.881748	−	1.10568i	−0.881748	−	1.10568i	−0.993712	−	0.111964i	$-0.964286\pi$
$947$	−0.881748	−	1.10568i	−0.881748	−	1.10568i	0.111964	−	0.993712i	$-0.464286\pi$
$948$		0			0
$949$		0			0
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−0.559828	+	0.789004i	−0.559828	+	0.789004i	−0.993712	−	0.111964i	$-0.964286\pi$
$953$	−0.559828	+	0.789004i	−0.559828	+	0.789004i	0.433884	+	0.900969i	$0.357143\pi$
$954$	0.936138	−	3.24941i	0.936138	−	3.24941i
$955$		0			0
$956$	−3.28122	−	0.748917i	−3.28122	−	0.748917i
$957$		0			0
$958$		0			0
$959$	0.277479	−	1.21572i	0.277479	−	1.21572i
$960$		0			0
$961$	−0.532032	+	0.846724i	−0.532032	+	0.846724i
$962$		0			0
$963$	−0.211363	+	1.87590i	−0.211363	+	1.87590i
$964$		0			0
$965$		0			0
$966$		0			0
$967$	0.0836170	−	1.48894i	0.0836170	−	1.48894i	−0.623490	−	0.781831i	$-0.714286\pi$
$967$	0.0836170	−	1.48894i	0.0836170	−	1.48894i	0.707107	−	0.707107i	$-0.250000\pi$
$968$	−2.33566	−	3.71719i	−2.33566	−	3.71719i
$969$		0			0
$970$		0			0
$971$		0			0		−0.923880	−	0.382683i	$-0.875000\pi$
$971$		0			0		0.923880	+	0.382683i	$0.125000\pi$
$972$		0			0
$973$		0			0
$974$	3.28122	+	0.748917i	3.28122	+	0.748917i
$975$		0			0
$976$		0			0
$977$	−1.96864	+	0.110556i	−1.96864	+	0.110556i	−0.993712	−	0.111964i	$-0.964286\pi$
$977$	−1.96864	+	0.110556i	−1.96864	+	0.110556i	−0.974928	+	0.222521i	$0.928571\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$	−0.735454	−	1.77554i	−0.735454	−	1.77554i
$982$	−1.95682	+	0.109892i	−1.95682	+	0.109892i
$983$		0			0		0.943883	−	0.330279i	$-0.107143\pi$
$983$		0			0		−0.943883	+	0.330279i	$0.892857\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$	0.176681	−	0.125362i	0.176681	−	0.125362i
$990$		0			0
$991$	0.368937	+	1.28061i	0.368937	+	1.28061i	0.900969	+	0.433884i	$0.142857\pi$
$991$	0.368937	+	1.28061i	0.368937	+	1.28061i	−0.532032	+	0.846724i	$0.678571\pi$
$992$		0			0
$993$		0			0
$994$	0.624775	+	0.699124i	0.624775	+	0.699124i
$995$		0			0
$996$		0			0
$997$		0			0		−0.846724	−	0.532032i	$-0.821429\pi$
$997$		0			0		0.846724	+	0.532032i	$0.178571\pi$
$998$	2.97498	+	1.43268i	2.97498	+	1.43268i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2359.1.cv.a.986.1	✓	24
7.6	odd	2	CM	2359.1.cv.a.986.1	✓	24
337.310	even	56	inner	2359.1.cv.a.1658.1	yes	24
2359.1658	odd	56	inner	2359.1.cv.a.1658.1	yes	24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2359.1.cv.a.986.1	✓	24	1.1	even	1	trivial
2359.1.cv.a.986.1	✓	24	7.6	odd	2	CM
2359.1.cv.a.1658.1	yes	24	337.310	even	56	inner
2359.1.cv.a.1658.1	yes	24	2359.1658	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)

\(f(q)\)	\(=\)	\(q+(1.05585 - 1.32399i) q^{2} +(-0.415617 - 1.82094i) q^{4} +(0.111964 + 0.993712i) q^{7} +(-1.32399 - 0.637601i) q^{8} +(-0.433884 - 0.900969i) q^{9} +O(q^{10})\)	\(q+(1.05585 - 1.32399i) q^{2} +(-0.415617 - 1.82094i) q^{4} +(0.111964 + 0.993712i) q^{7} +(-1.32399 - 0.637601i) q^{8} +(-0.433884 - 0.900969i) q^{9} +(1.91881 + 0.552800i) q^{11} +(1.43388 + 0.900969i) q^{14} +(-0.559311 + 0.269350i) q^{16} +(-1.65099 - 0.376828i) q^{18} +(2.75788 - 1.95682i) q^{22} +(-0.0542447 - 0.965916i) q^{23} +(0.993712 - 0.111964i) q^{25} +(1.76295 - 0.616884i) q^{28} +(-1.84485 + 0.103605i) q^{29} +(0.0930688 - 0.407761i) q^{32} +(-1.46028 + 1.16453i) q^{36} +(-1.52446 - 0.347948i) q^{37} +(0.119137 + 0.189606i) q^{43} +(0.209124 - 3.72379i) q^{44} +(-1.33614 - 0.948041i) q^{46} +(-0.974928 + 0.222521i) q^{49} +(0.900969 - 1.43388i) q^{50} +(-0.111964 + 1.99371i) q^{53} +(0.485352 - 1.38705i) q^{56} +(-1.81071 + 2.55196i) q^{58} +(0.846724 - 0.532032i) q^{63} +(-0.828660 - 1.03911i) q^{64} +(0.223234 - 0.249799i) q^{67} +(0.532032 + 0.153276i) q^{71} +1.46952i q^{72} +(-2.07028 + 1.65099i) q^{74} +(-0.334485 + 1.96864i) q^{77} +(-0.411851 + 0.516445i) q^{79} +(-0.623490 + 0.781831i) q^{81} +(0.376828 + 0.0424583i) q^{86} +(-2.18802 - 1.95534i) q^{88} +(-1.73633 + 0.500228i) q^{92} +(-0.734760 + 1.52574i) q^{98} +(-0.334485 - 1.96864i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q - 4 q^{4}+O(q^{10}) \) 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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