LMFDB - Embedded newform 3248.1.dt.b.2323.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3248,1,Mod(27,3248)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3248, base_ring=CyclotomicField(28))

chi = DirichletCharacter(H, H._module([14, 7, 14, 15]))

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

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

chi := DirichletCharacter("3248.27");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 3248 = 2^{4} \cdot 7 \cdot 29 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	3248.dt (of order $28$, degree $12$, 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:	$1.62096316103$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	$\Q(\zeta_{28})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - x^{10} + x^{8} - x^{6} + x^{4} - x^{2} + 1 $ x^12 - x^10 + x^8 - x^6 + x^4 - x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{25}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Projective image:	$D_{28}$
Projective field:	Galois closure of $\mathbb{Q}[x]/(x^{28} + \cdots)$

Embedding invariants

Embedding label			2323.1
Root			$0.974928 - 0.222521i$ of defining polynomial
Character	$\chi$	$=$	3248.2323
Dual form			3248.1.dt.b.2939.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.222521 - 0.974928i) q^{2} +(-0.900969 - 0.433884i) q^{4} +(-0.900969 - 0.433884i) q^{7} +(-0.623490 + 0.781831i) q^{8} +(-0.623490 - 0.781831i) q^{9} +O(q^{10})$	\(q+(0.222521 - 0.974928i) q^{2} +(-0.900969 - 0.433884i) q^{4} +(-0.900969 - 0.433884i) q^{7} +(-0.623490 + 0.781831i) q^{8} +(-0.623490 - 0.781831i) q^{9} +(0.974928 - 1.22252i) q^{11} +(-0.623490 + 0.781831i) q^{14} +(0.623490 + 0.781831i) q^{16} +(-0.900969 + 0.433884i) q^{18} +(-0.974928 - 1.22252i) q^{22} +(-0.193096 + 0.846011i) q^{23} +(-0.433884 - 0.900969i) q^{25} +(0.623490 + 0.781831i) q^{28} +(-0.900969 + 0.433884i) q^{29} +(0.900969 - 0.433884i) q^{32} +(0.222521 + 0.974928i) q^{36} +(-1.21572 - 1.52446i) q^{37} +(-0.445042 + 1.94986i) q^{43} +(-1.40881 + 0.678448i) q^{44} +(0.781831 + 0.376510i) q^{46} +(0.623490 + 0.781831i) q^{49} +(-0.974928 + 0.222521i) q^{50} +(-1.68280 - 1.05737i) q^{53} +(0.900969 - 0.433884i) q^{56} +(0.222521 + 0.974928i) q^{58} +(0.222521 + 0.974928i) q^{63} +(-0.222521 - 0.974928i) q^{64} +(0.0739590 - 0.656405i) q^{67} +(-0.347948 - 0.277479i) q^{71} +1.00000 q^{72} +(-1.75676 + 0.846011i) q^{74} +(-1.40881 + 0.678448i) q^{77} +(-0.0250721 + 0.222521i) q^{79} +(-0.222521 + 0.974928i) q^{81} +(1.80194 + 0.867767i) q^{86} +(0.347948 + 1.52446i) q^{88} +(0.541044 - 0.678448i) q^{92} +(0.900969 - 0.433884i) q^{98} -1.56366 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q + 2 q^{2} - 2 q^{4} - 2 q^{7} + 2 q^{8} + 2 q^{9}+O(q^{10}) $ 12 * q + 2 * q^2 - 2 * q^4 - 2 * q^7 + 2 * q^8 + 2 * q^9	$ 12 q + 2 q^{2} - 2 q^{4} - 2 q^{7} + 2 q^{8} + 2 q^{9} + 2 q^{14} - 2 q^{16} - 2 q^{18} - 2 q^{28} - 2 q^{29} + 2 q^{32} + 2 q^{36} - 4 q^{43} - 2 q^{49} - 2 q^{53} + 2 q^{56} + 2 q^{58} + 2 q^{63} - 2 q^{64} - 2 q^{67} + 12 q^{72} - 12 q^{79} - 2 q^{81} + 4 q^{86} + 2 q^{98}+O(q^{100}) $ 12 * q + 2 * q^2 - 2 * q^4 - 2 * q^7 + 2 * q^8 + 2 * q^9 + 2 * q^14 - 2 * q^16 - 2 * q^18 - 2 * q^28 - 2 * q^29 + 2 * q^32 + 2 * q^36 - 4 * q^43 - 2 * q^49 - 2 * q^53 + 2 * q^56 + 2 * q^58 + 2 * q^63 - 2 * q^64 - 2 * q^67 + 12 * q^72 - 12 * q^79 - 2 * q^81 + 4 * q^86 + 2 * q^98

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3248\mathbb{Z}\right)^\times$.

$n$	$465$	$785$	$2031$	$2437$
$\chi(n)$	$-1$	$e\left(\frac{5}{28}\right)$	$-1$	$e\left(\frac{3}{4}\right)$

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 $

$2$	0.222521	−	0.974928i	0.222521	−	0.974928i
$2$	0.222521	−	0.974928i	0.222521	−	0.974928i
$3$		0			0		0.433884	−	0.900969i	$-0.357143\pi$
$3$		0			0		−0.433884	+	0.900969i	$0.642857\pi$
$4$	−0.900969	−	0.433884i	−0.900969	−	0.433884i
$5$		0			0		0.532032	−	0.846724i	$-0.321429\pi$
$5$		0			0		−0.532032	+	0.846724i	$0.678571\pi$
$6$		0			0
$7$	−0.900969	−	0.433884i	−0.900969	−	0.433884i
$7$	−0.900969	−	0.433884i	−0.900969	−	0.433884i
$8$	−0.623490	+	0.781831i	−0.623490	+	0.781831i
$9$	−0.623490	−	0.781831i	−0.623490	−	0.781831i
$10$		0			0
$11$	0.974928	−	1.22252i	0.974928	−	1.22252i		−	1.00000i	$-0.5\pi$
$11$	0.974928	−	1.22252i	0.974928	−	1.22252i	0.974928	−	0.222521i	$-0.0714286\pi$
$12$		0			0
$13$		0			0		−0.111964	−	0.993712i	$-0.535714\pi$
$13$		0			0		0.111964	+	0.993712i	$0.464286\pi$
$14$	−0.623490	+	0.781831i	−0.623490	+	0.781831i
$15$		0			0
$16$	0.623490	+	0.781831i	0.623490	+	0.781831i
$17$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$17$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$18$	−0.900969	+	0.433884i	−0.900969	+	0.433884i
$19$		0			0		−0.433884	−	0.900969i	$-0.642857\pi$
$19$		0			0		0.433884	+	0.900969i	$0.357143\pi$
$20$		0			0
$21$		0			0
$22$	−0.974928	−	1.22252i	−0.974928	−	1.22252i
$23$	−0.193096	+	0.846011i	−0.193096	+	0.846011i	0.781831	+	0.623490i	$0.214286\pi$
$23$	−0.193096	+	0.846011i	−0.193096	+	0.846011i	−0.974928	+	0.222521i	$0.928571\pi$
$24$		0			0
$25$	−0.433884	−	0.900969i	−0.433884	−	0.900969i
$26$		0			0
$27$		0			0
$28$	0.623490	+	0.781831i	0.623490	+	0.781831i
$29$	−0.900969	+	0.433884i	−0.900969	+	0.433884i
$29$	−0.900969	+	0.433884i	−0.900969	+	0.433884i
$30$		0			0
$31$		0			0		0.532032	−	0.846724i	$-0.321429\pi$
$31$		0			0		−0.532032	+	0.846724i	$0.678571\pi$
$32$	0.900969	−	0.433884i	0.900969	−	0.433884i
$33$		0			0
$34$		0			0
$35$		0			0
$36$	0.222521	+	0.974928i	0.222521	+	0.974928i
$37$	−1.21572	−	1.52446i	−1.21572	−	1.52446i	−0.781831	−	0.623490i	$-0.785714\pi$
$37$	−1.21572	−	1.52446i	−1.21572	−	1.52446i	−0.433884	−	0.900969i	$-0.642857\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$41$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$42$		0			0
$43$	−0.445042	+	1.94986i	−0.445042	+	1.94986i	−0.222521	+	0.974928i	$0.571429\pi$
$43$	−0.445042	+	1.94986i	−0.445042	+	1.94986i	−0.222521	+	0.974928i	$0.571429\pi$
$44$	−1.40881	+	0.678448i	−1.40881	+	0.678448i
$45$		0			0
$46$	0.781831	+	0.376510i	0.781831	+	0.376510i
$47$		0			0		−0.111964	−	0.993712i	$-0.535714\pi$
$47$		0			0		0.111964	+	0.993712i	$0.464286\pi$
$48$		0			0
$49$	0.623490	+	0.781831i	0.623490	+	0.781831i
$50$	−0.974928	+	0.222521i	−0.974928	+	0.222521i
$51$		0			0
$52$		0			0
$53$	−1.68280	−	1.05737i	−1.68280	−	1.05737i	−0.900969	−	0.433884i	$-0.857143\pi$
$53$	−1.68280	−	1.05737i	−1.68280	−	1.05737i	−0.781831	−	0.623490i	$-0.785714\pi$
$54$		0			0
$55$		0			0
$56$	0.900969	−	0.433884i	0.900969	−	0.433884i
$57$		0			0
$58$	0.222521	+	0.974928i	0.222521	+	0.974928i
$59$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$59$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$60$		0			0
$61$		0			0		−0.900969	−	0.433884i	$-0.857143\pi$
$61$		0			0		0.900969	+	0.433884i	$0.142857\pi$
$62$		0			0
$63$	0.222521	+	0.974928i	0.222521	+	0.974928i
$64$	−0.222521	−	0.974928i	−0.222521	−	0.974928i
$65$		0			0
$66$		0			0
$67$	0.0739590	−	0.656405i	0.0739590	−	0.656405i	−0.900969	−	0.433884i	$-0.857143\pi$
$67$	0.0739590	−	0.656405i	0.0739590	−	0.656405i	0.974928	−	0.222521i	$-0.0714286\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−0.347948	−	0.277479i	−0.347948	−	0.277479i	0.433884	−	0.900969i	$-0.357143\pi$
$71$	−0.347948	−	0.277479i	−0.347948	−	0.277479i	−0.781831	+	0.623490i	$0.785714\pi$
$72$	1.00000			1.00000
$73$		0			0		−0.532032	−	0.846724i	$-0.678571\pi$
$73$		0			0		0.532032	+	0.846724i	$0.321429\pi$
$74$	−1.75676	+	0.846011i	−1.75676	+	0.846011i
$75$		0			0
$76$		0			0
$77$	−1.40881	+	0.678448i	−1.40881	+	0.678448i
$78$		0			0
$79$	−0.0250721	+	0.222521i	−0.0250721	+	0.222521i	0.974928	+	0.222521i	$0.0714286\pi$
$79$	−0.0250721	+	0.222521i	−0.0250721	+	0.222521i	−1.00000			$\pi$
$80$		0			0
$81$	−0.222521	+	0.974928i	−0.222521	+	0.974928i
$82$		0			0
$83$		0			0		0.330279	−	0.943883i	$-0.392857\pi$
$83$		0			0		−0.330279	+	0.943883i	$0.607143\pi$
$84$		0			0
$85$		0			0
$86$	1.80194	+	0.867767i	1.80194	+	0.867767i
$87$		0			0
$88$	0.347948	+	1.52446i	0.347948	+	1.52446i
$89$		0			0		0.532032	−	0.846724i	$-0.321429\pi$
$89$		0			0		−0.532032	+	0.846724i	$0.678571\pi$
$90$		0			0
$91$		0			0
$92$	0.541044	−	0.678448i	0.541044	−	0.678448i
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$		0			0		−0.943883	−	0.330279i	$-0.892857\pi$
$97$		0			0		0.943883	+	0.330279i	$0.107143\pi$
$98$	0.900969	−	0.433884i	0.900969	−	0.433884i
$99$	−1.56366			−1.56366
$100$			1.00000i			1.00000i
$101$		0			0		−0.974928	−	0.222521i	$-0.928571\pi$
$101$		0			0		0.974928	+	0.222521i	$0.0714286\pi$
$102$		0			0
$103$		0			0		0.623490	−	0.781831i	$-0.285714\pi$
$103$		0			0		−0.623490	+	0.781831i	$0.714286\pi$
$104$		0			0
$105$		0			0
$106$	−1.40532	+	1.40532i	−1.40532	+	1.40532i
$107$	−0.189606	+	1.68280i	−0.189606	+	1.68280i	0.433884	+	0.900969i	$0.357143\pi$
$107$	−0.189606	+	1.68280i	−0.189606	+	1.68280i	−0.623490	+	0.781831i	$0.714286\pi$
$108$		0			0
$109$	1.59842	−	0.559311i	1.59842	−	0.559311i	0.623490	−	0.781831i	$-0.285714\pi$
$109$	1.59842	−	0.559311i	1.59842	−	0.559311i	0.974928	+	0.222521i	$0.0714286\pi$
$110$		0			0
$111$		0			0
$112$	−0.222521	−	0.974928i	−0.222521	−	0.974928i
$113$	−0.623490	−	1.78183i	−0.623490	−	1.78183i	−0.623490	−	0.781831i	$-0.714286\pi$
$113$	−0.623490	−	1.78183i	−0.623490	−	1.78183i		−	1.00000i	$-0.5\pi$
$114$		0			0
$115$		0			0
$116$	1.00000			1.00000
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−0.321552	−	1.40881i	−0.321552	−	1.40881i
$122$		0			0
$123$		0			0
$124$		0			0
$125$		0			0
$126$	1.00000			1.00000
$127$	−1.87590	+	0.211363i	−1.87590	+	0.211363i	−0.974928	−	0.222521i	$-0.928571\pi$
$127$	−1.87590	+	0.211363i	−1.87590	+	0.211363i	−0.900969	+	0.433884i	$0.857143\pi$
$128$	−1.00000			−1.00000
$129$		0			0
$130$		0			0
$131$		0			0		0.222521	−	0.974928i	$-0.428571\pi$
$131$		0			0		−0.222521	+	0.974928i	$0.571429\pi$
$132$		0			0
$133$		0			0
$134$	−0.623490	−	0.218169i	−0.623490	−	0.218169i
$135$		0			0
$136$		0			0
$137$	0.158342	−	1.40532i	0.158342	−	1.40532i	−0.623490	−	0.781831i	$-0.714286\pi$
$137$	0.158342	−	1.40532i	0.158342	−	1.40532i	0.781831	−	0.623490i	$-0.214286\pi$
$138$		0			0
$139$		0			0		0.846724	−	0.532032i	$-0.178571\pi$
$139$		0			0		−0.846724	+	0.532032i	$0.821429\pi$
$140$		0			0
$141$		0			0
$142$	−0.347948	+	0.277479i	−0.347948	+	0.277479i
$143$		0			0
$144$	0.222521	−	0.974928i	0.222521	−	0.974928i
$145$		0			0
$146$		0			0
$147$		0			0
$148$	0.433884	+	1.90097i	0.433884	+	1.90097i
$149$	1.78183	+	0.623490i	1.78183	+	0.623490i	1.00000			$0$
$149$	1.78183	+	0.623490i	1.78183	+	0.623490i	0.781831	+	0.623490i	$0.214286\pi$
$150$		0			0
$151$	−1.21572	−	0.277479i	−1.21572	−	0.277479i	−0.433884	−	0.900969i	$-0.642857\pi$
$151$	−1.21572	−	0.277479i	−1.21572	−	0.277479i	−0.781831	+	0.623490i	$0.785714\pi$
$152$		0			0
$153$		0			0
$154$	0.347948	+	1.52446i	0.347948	+	1.52446i
$155$		0			0
$156$		0			0
$157$		0			0			−	1.00000i	$-0.5\pi$
$157$		0			0				1.00000i	$0.5\pi$
$158$	0.211363	+	0.0739590i	0.211363	+	0.0739590i
$159$		0			0
$160$		0			0
$161$	0.541044	−	0.678448i	0.541044	−	0.678448i
$162$	0.900969	+	0.433884i	0.900969	+	0.433884i
$163$	1.21572	−	1.52446i	1.21572	−	1.52446i	0.433884	−	0.900969i	$-0.357143\pi$
$163$	1.21572	−	1.52446i	1.21572	−	1.52446i	0.781831	−	0.623490i	$-0.214286\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		0			0		0.433884	−	0.900969i	$-0.357143\pi$
$167$		0			0		−0.433884	+	0.900969i	$0.642857\pi$
$168$		0			0
$169$	−0.974928	+	0.222521i	−0.974928	+	0.222521i
$170$		0			0
$171$		0			0
$172$	1.24698	−	1.56366i	1.24698	−	1.56366i
$173$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$173$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$174$		0			0
$175$			1.00000i			1.00000i
$176$	1.56366			1.56366
$177$		0			0
$178$		0			0
$179$	1.05737	−	1.68280i	1.05737	−	1.68280i	0.433884	−	0.900969i	$-0.357143\pi$
$179$	1.05737	−	1.68280i	1.05737	−	1.68280i	0.623490	−	0.781831i	$-0.285714\pi$
$180$		0			0
$181$		0			0		0.943883	−	0.330279i	$-0.107143\pi$
$181$		0			0		−0.943883	+	0.330279i	$0.892857\pi$
$182$		0			0
$183$		0			0
$184$	−0.541044	−	0.678448i	−0.541044	−	0.678448i
$185$		0			0
$186$		0			0
$187$		0			0
$188$		0			0
$189$		0			0
$190$		0			0
$191$	0.752407	+	0.752407i	0.752407	+	0.752407i	0.974928	−	0.222521i	$-0.0714286\pi$
$191$	0.752407	+	0.752407i	0.752407	+	0.752407i	−0.222521	+	0.974928i	$0.571429\pi$
$192$		0			0
$193$	0.559311	−	1.59842i	0.559311	−	1.59842i	−0.222521	−	0.974928i	$-0.571429\pi$
$193$	0.559311	−	1.59842i	0.559311	−	1.59842i	0.781831	−	0.623490i	$-0.214286\pi$
$194$		0			0
$195$		0			0
$196$	−0.222521	−	0.974928i	−0.222521	−	0.974928i
$197$	0.566116	+	0.900969i	0.566116	+	0.900969i	1.00000			$0$
$197$	0.566116	+	0.900969i	0.566116	+	0.900969i	−0.433884	+	0.900969i	$0.642857\pi$
$198$	−0.347948	+	1.52446i	−0.347948	+	1.52446i
$199$		0			0		0.900969	−	0.433884i	$-0.142857\pi$
$199$		0			0		−0.900969	+	0.433884i	$0.857143\pi$
$200$	0.974928	+	0.222521i	0.974928	+	0.222521i
$201$		0			0
$202$		0			0
$203$	1.00000			1.00000
$204$		0			0
$205$		0			0
$206$		0			0
$207$	0.781831	−	0.376510i	0.781831	−	0.376510i
$208$		0			0
$209$		0			0
$210$		0			0
$211$	1.12349	+	1.40881i	1.12349	+	1.40881i	0.900969	+	0.433884i	$0.142857\pi$
$211$	1.12349	+	1.40881i	1.12349	+	1.40881i	0.222521	+	0.974928i	$0.428571\pi$
$212$	1.05737	+	1.68280i	1.05737	+	1.68280i
$213$		0			0
$214$	1.59842	+	0.559311i	1.59842	+	0.559311i
$215$		0			0
$216$		0			0
$217$		0			0
$218$	−0.189606	−	1.68280i	−0.189606	−	1.68280i
$219$		0			0
$220$		0			0
$221$		0			0
$222$		0			0
$223$		0			0		0.781831	−	0.623490i	$-0.214286\pi$
$223$		0			0		−0.781831	+	0.623490i	$0.785714\pi$
$224$	−1.00000			−1.00000
$225$	−0.433884	+	0.900969i	−0.433884	+	0.900969i
$226$	−1.87590	+	0.211363i	−1.87590	+	0.211363i
$227$		0			0		−0.846724	−	0.532032i	$-0.821429\pi$
$227$		0			0		0.846724	+	0.532032i	$0.178571\pi$
$228$		0			0
$229$		0			0		−0.900969	−	0.433884i	$-0.857143\pi$
$229$		0			0		0.900969	+	0.433884i	$0.142857\pi$
$230$		0			0
$231$		0			0
$232$	0.222521	−	0.974928i	0.222521	−	0.974928i
$233$		−	1.94986i		−	1.94986i	−0.222521	−	0.974928i	$-0.571429\pi$
$233$		−	1.94986i		−	1.94986i	0.222521	−	0.974928i	$-0.428571\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$	0.846011	−	1.75676i	0.846011	−	1.75676i	0.222521	−	0.974928i	$-0.428571\pi$
$239$	0.846011	−	1.75676i	0.846011	−	1.75676i	0.623490	−	0.781831i	$-0.285714\pi$
$240$		0			0
$241$		0			0		0.781831	−	0.623490i	$-0.214286\pi$
$241$		0			0		−0.781831	+	0.623490i	$0.785714\pi$
$242$	−1.44504			−1.44504
$243$		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		0.900969	−	0.433884i	$-0.142857\pi$
$251$		0			0		−0.900969	+	0.433884i	$0.857143\pi$
$252$	0.222521	−	0.974928i	0.222521	−	0.974928i
$253$	0.846011	+	1.06086i	0.846011	+	1.06086i
$254$	−0.211363	+	1.87590i	−0.211363	+	1.87590i
$255$		0			0
$256$	−0.222521	+	0.974928i	−0.222521	+	0.974928i
$257$		0			0		0.900969	−	0.433884i	$-0.142857\pi$
$257$		0			0		−0.900969	+	0.433884i	$0.857143\pi$
$258$		0			0
$259$	0.433884	+	1.90097i	0.433884	+	1.90097i
$260$		0			0
$261$	0.900969	+	0.433884i	0.900969	+	0.433884i
$262$		0			0
$263$	0.900969	+	0.566116i	0.900969	+	0.566116i	0.900969	−	0.433884i	$-0.142857\pi$
$263$	0.900969	+	0.566116i	0.900969	+	0.566116i			1.00000i	$0.5\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$	−0.351438	+	0.559311i	−0.351438	+	0.559311i
$269$		0			0		0.781831	−	0.623490i	$-0.214286\pi$
$269$		0			0		−0.781831	+	0.623490i	$0.785714\pi$
$270$		0			0
$271$		0			0		0.330279	−	0.943883i	$-0.392857\pi$
$271$		0			0		−0.330279	+	0.943883i	$0.607143\pi$
$272$		0			0
$273$		0			0
$274$	−1.33485	−	0.467085i	−1.33485	−	0.467085i
$275$	−1.52446	−	0.347948i	−1.52446	−	0.347948i
$276$		0			0
$277$	0.211363	+	1.87590i	0.211363	+	1.87590i	0.433884	+	0.900969i	$0.357143\pi$
$277$	0.211363	+	1.87590i	0.211363	+	1.87590i	−0.222521	+	0.974928i	$0.571429\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	1.40881	−	1.12349i	1.40881	−	1.12349i	0.433884	−	0.900969i	$-0.357143\pi$
$281$	1.40881	−	1.12349i	1.40881	−	1.12349i	0.974928	−	0.222521i	$-0.0714286\pi$
$282$		0			0
$283$		0			0		0.943883	−	0.330279i	$-0.107143\pi$
$283$		0			0		−0.943883	+	0.330279i	$0.892857\pi$
$284$	0.193096	+	0.400969i	0.193096	+	0.400969i
$285$		0			0
$286$		0			0
$287$		0			0
$288$	−0.900969	−	0.433884i	−0.900969	−	0.433884i
$289$		−	1.00000i		−	1.00000i
$290$		0			0
$291$		0			0
$292$		0			0
$293$		0			0		0.433884	−	0.900969i	$-0.357143\pi$
$293$		0			0		−0.433884	+	0.900969i	$0.642857\pi$
$294$		0			0
$295$		0			0
$296$	1.94986			1.94986
$297$		0			0
$298$	1.00435	−	1.59842i	1.00435	−	1.59842i
$299$		0			0
$300$		0			0
$301$	1.24698	−	1.56366i	1.24698	−	1.56366i
$302$	−0.541044	+	1.12349i	−0.541044	+	1.12349i
$303$		0			0
$304$		0			0
$305$		0			0
$306$		0			0
$307$		0			0			−	1.00000i	$-0.5\pi$
$307$		0			0				1.00000i	$0.5\pi$
$308$	1.56366			1.56366
$309$		0			0
$310$		0			0
$311$		0			0		−0.993712	−	0.111964i	$-0.964286\pi$
$311$		0			0		0.993712	+	0.111964i	$0.0357143\pi$
$312$		0			0
$313$		0			0		−0.974928	−	0.222521i	$-0.928571\pi$
$313$		0			0		0.974928	+	0.222521i	$0.0714286\pi$
$314$		0			0
$315$		0			0
$316$	0.119137	−	0.189606i	0.119137	−	0.189606i
$317$		0			0		−0.222521	−	0.974928i	$-0.571429\pi$
$317$		0			0		0.222521	+	0.974928i	$0.428571\pi$
$318$		0			0
$319$	−0.347948	+	1.52446i	−0.347948	+	1.52446i
$320$		0			0
$321$		0			0
$322$	−0.541044	−	0.678448i	−0.541044	−	0.678448i
$323$		0			0
$324$	0.623490	−	0.781831i	0.623490	−	0.781831i
$325$		0			0
$326$	−1.21572	−	1.52446i	−1.21572	−	1.52446i
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$			1.56366i			1.56366i	0.623490	+	0.781831i	$0.285714\pi$
$331$			1.56366i			1.56366i	−0.623490	+	0.781831i	$0.714286\pi$
$332$		0			0
$333$	−0.433884	+	1.90097i	−0.433884	+	1.90097i
$334$		0			0
$335$		0			0
$336$		0			0
$337$	−1.97493	+	0.222521i	−1.97493	+	0.222521i	−0.974928	+	0.222521i	$0.928571\pi$
$337$	−1.97493	+	0.222521i	−1.97493	+	0.222521i	−1.00000			$\pi$
$338$			1.00000i			1.00000i
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$	−0.222521	−	0.974928i	−0.222521	−	0.974928i
$344$	−1.24698	−	1.56366i	−1.24698	−	1.56366i
$345$		0			0
$346$		0			0
$347$	0.752407	−	0.752407i	0.752407	−	0.752407i	−0.222521	−	0.974928i	$-0.571429\pi$
$347$	0.752407	−	0.752407i	0.752407	−	0.752407i	0.974928	+	0.222521i	$0.0714286\pi$
$348$		0			0
$349$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$349$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$350$	0.974928	+	0.222521i	0.974928	+	0.222521i
$351$		0			0
$352$	0.347948	−	1.52446i	0.347948	−	1.52446i
$353$		0			0		0.974928	−	0.222521i	$-0.0714286\pi$
$353$		0			0		−0.974928	+	0.222521i	$0.928571\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$	−1.40532	−	1.40532i	−1.40532	−	1.40532i
$359$	−0.189606	−	1.68280i	−0.189606	−	1.68280i	−0.623490	−	0.781831i	$-0.714286\pi$
$359$	−0.189606	−	1.68280i	−0.189606	−	1.68280i	0.433884	−	0.900969i	$-0.357143\pi$
$360$		0			0
$361$	−0.623490	+	0.781831i	−0.623490	+	0.781831i
$362$		0			0
$363$		0			0
$364$		0			0
$365$		0			0
$366$		0			0
$367$		0			0		−0.943883	−	0.330279i	$-0.892857\pi$
$367$		0			0		0.943883	+	0.330279i	$0.107143\pi$
$368$	−0.781831	+	0.376510i	−0.781831	+	0.376510i
$369$		0			0
$370$		0			0
$371$	1.05737	+	1.68280i	1.05737	+	1.68280i
$372$		0			0
$373$	0.656405	−	1.87590i	0.656405	−	1.87590i	0.222521	−	0.974928i	$-0.428571\pi$
$373$	0.656405	−	1.87590i	0.656405	−	1.87590i	0.433884	−	0.900969i	$-0.357143\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$	−0.0990311	−	0.433884i	−0.0990311	−	0.433884i	0.900969	−	0.433884i	$-0.142857\pi$
$379$	−0.0990311	−	0.433884i	−0.0990311	−	0.433884i	−1.00000			$\pi$
$380$		0			0
$381$		0			0
$382$	0.900969	−	0.566116i	0.900969	−	0.566116i
$383$		0			0		0.222521	−	0.974928i	$-0.428571\pi$
$383$		0			0		−0.222521	+	0.974928i	$0.571429\pi$
$384$		0			0
$385$		0			0
$386$	−1.43388	−	0.900969i	−1.43388	−	0.900969i
$387$	1.80194	−	0.867767i	1.80194	−	0.867767i
$388$		0			0
$389$	−1.94986			−1.94986			−0.974928	−	0.222521i	$-0.928571\pi$
$389$	−1.94986			−1.94986			−0.974928	+	0.222521i	$0.928571\pi$
$390$		0			0
$391$		0			0
$392$	−1.00000			−1.00000
$393$		0			0
$394$	1.00435	−	0.351438i	1.00435	−	0.351438i
$395$		0			0
$396$	1.40881	+	0.678448i	1.40881	+	0.678448i
$397$		0			0		0.111964	−	0.993712i	$-0.464286\pi$
$397$		0			0		−0.111964	+	0.993712i	$0.535714\pi$
$398$		0			0
$399$		0			0
$400$	0.433884	−	0.900969i	0.433884	−	0.900969i
$401$	0.277479	+	1.21572i	0.277479	+	1.21572i	0.900969	+	0.433884i	$0.142857\pi$
$401$	0.277479	+	1.21572i	0.277479	+	1.21572i	−0.623490	+	0.781831i	$0.714286\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$		0			0
$406$	0.222521	−	0.974928i	0.222521	−	0.974928i
$407$	−3.04892			−3.04892
$408$		0			0
$409$		0			0		−0.330279	−	0.943883i	$-0.607143\pi$
$409$		0			0		0.330279	+	0.943883i	$0.392857\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		0			0
$414$	−0.193096	−	0.846011i	−0.193096	−	0.846011i
$415$		0			0
$416$		0			0
$417$		0			0
$418$		0			0
$419$		0			0		0.993712	−	0.111964i	$-0.0357143\pi$
$419$		0			0		−0.993712	+	0.111964i	$0.964286\pi$
$420$		0			0
$421$	0.193096	−	0.846011i	0.193096	−	0.846011i	−0.781831	−	0.623490i	$-0.785714\pi$
$421$	0.193096	−	0.846011i	0.193096	−	0.846011i	0.974928	−	0.222521i	$-0.0714286\pi$
$422$	1.62349	−	0.781831i	1.62349	−	0.781831i
$423$		0			0
$424$	1.87590	−	0.656405i	1.87590	−	0.656405i
$425$		0			0
$426$		0			0
$427$		0			0
$428$	0.900969	−	1.43388i	0.900969	−	1.43388i
$429$		0			0
$430$		0			0
$431$	−0.376510	−	0.781831i	−0.376510	−	0.781831i	0.623490	−	0.781831i	$-0.285714\pi$
$431$	−0.376510	−	0.781831i	−0.376510	−	0.781831i	−1.00000			$\pi$
$432$		0			0
$433$		0			0		0.532032	−	0.846724i	$-0.321429\pi$
$433$		0			0		−0.532032	+	0.846724i	$0.678571\pi$
$434$		0			0
$435$		0			0
$436$	−1.68280	−	0.189606i	−1.68280	−	0.189606i
$437$		0			0
$438$		0			0
$439$		0			0		−0.433884	−	0.900969i	$-0.642857\pi$
$439$		0			0		0.433884	+	0.900969i	$0.357143\pi$
$440$		0			0
$441$	0.222521	−	0.974928i	0.222521	−	0.974928i
$442$		0			0
$443$	0.974928	−	0.777479i	0.974928	−	0.777479i		−	1.00000i	$-0.5\pi$
$443$	0.974928	−	0.777479i	0.974928	−	0.777479i	0.974928	+	0.222521i	$0.0714286\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$	−0.222521	+	0.974928i	−0.222521	+	0.974928i
$449$	−0.119137	−	0.189606i	−0.119137	−	0.189606i	0.781831	−	0.623490i	$-0.214286\pi$
$449$	−0.119137	−	0.189606i	−0.119137	−	0.189606i	−0.900969	+	0.433884i	$0.857143\pi$
$450$	0.781831	+	0.623490i	0.781831	+	0.623490i
$451$		0			0
$452$	−0.211363	+	1.87590i	−0.211363	+	1.87590i
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−0.781831	−	0.376510i	−0.781831	−	0.376510i		−	1.00000i	$-0.5\pi$
$457$	−0.781831	−	0.376510i	−0.781831	−	0.376510i	−0.781831	+	0.623490i	$0.785714\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		0			0		0.433884	−	0.900969i	$-0.357143\pi$
$461$		0			0		−0.433884	+	0.900969i	$0.642857\pi$
$462$		0			0
$463$	1.56366			1.56366			0.781831	−	0.623490i	$-0.214286\pi$
$463$	1.56366			1.56366			0.781831	+	0.623490i	$0.214286\pi$
$464$	−0.900969	−	0.433884i	−0.900969	−	0.433884i
$465$		0			0
$466$	−1.90097	−	0.433884i	−1.90097	−	0.433884i
$467$		0			0		0.433884	−	0.900969i	$-0.357143\pi$
$467$		0			0		−0.433884	+	0.900969i	$0.642857\pi$
$468$		0			0
$469$	−0.351438	+	0.559311i	−0.351438	+	0.559311i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	1.94986	+	2.44504i	1.94986	+	2.44504i
$474$		0			0
$475$		0			0
$476$		0			0
$477$	0.222521	+	1.97493i	0.222521	+	1.97493i
$478$	−1.52446	−	1.21572i	−1.52446	−	1.21572i
$479$		0			0		−0.532032	−	0.846724i	$-0.678571\pi$
$479$		0			0		0.532032	+	0.846724i	$0.321429\pi$
$480$		0			0
$481$		0			0
$482$		0			0
$483$		0			0
$484$	−0.321552	+	1.40881i	−0.321552	+	1.40881i
$485$		0			0
$486$		0			0
$487$	−0.0990311	+	0.433884i	−0.0990311	+	0.433884i	0.900969	+	0.433884i	$0.142857\pi$
$487$	−0.0990311	+	0.433884i	−0.0990311	+	0.433884i	−1.00000			$\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	0.846011	−	0.193096i	0.846011	−	0.193096i	0.222521	−	0.974928i	$-0.428571\pi$
$491$	0.846011	−	0.193096i	0.846011	−	0.193096i	0.623490	+	0.781831i	$0.285714\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$		0			0
$497$	0.193096	+	0.400969i	0.193096	+	0.400969i
$498$		0			0
$499$	−1.43388	+	0.900969i	−1.43388	+	0.900969i	−0.433884	+	0.900969i	$0.642857\pi$
$499$	−1.43388	+	0.900969i	−1.43388	+	0.900969i	−1.00000			$1.00000\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		0			0		−0.943883	−	0.330279i	$-0.892857\pi$
$503$		0			0		0.943883	+	0.330279i	$0.107143\pi$
$504$	−0.900969	−	0.433884i	−0.900969	−	0.433884i
$505$		0			0
$506$	1.22252	−	0.588735i	1.22252	−	0.588735i
$507$		0			0
$508$	1.78183	+	0.623490i	1.78183	+	0.623490i
$509$		0			0		0.993712	−	0.111964i	$-0.0357143\pi$
$509$		0			0		−0.993712	+	0.111964i	$0.964286\pi$
$510$		0			0
$511$		0			0
$512$	0.900969	+	0.433884i	0.900969	+	0.433884i
$513$		0			0
$514$		0			0
$515$		0			0
$516$		0			0
$517$		0			0
$518$	1.94986			1.94986
$519$		0			0
$520$		0			0
$521$		0			0		1.00000			$0$
$521$		0			0		−1.00000			$\pi$
$522$	0.623490	−	0.781831i	0.623490	−	0.781831i
$523$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$523$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$524$		0			0
$525$		0			0
$526$	0.752407	−	0.752407i	0.752407	−	0.752407i
$527$		0			0
$528$		0			0
$529$	0.222521	+	0.107160i	0.222521	+	0.107160i
$530$		0			0
$531$		0			0
$532$		0			0
$533$		0			0
$534$		0			0
$535$		0			0
$536$	0.467085	+	0.467085i	0.467085	+	0.467085i
$537$		0			0
$538$		0			0
$539$	1.56366			1.56366
$540$		0			0
$541$	1.12349	−	0.541044i	1.12349	−	0.541044i	0.222521	−	0.974928i	$-0.428571\pi$
$541$	1.12349	−	0.541044i	1.12349	−	0.541044i	0.900969	+	0.433884i	$0.142857\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		0			0
$546$		0			0
$547$	−0.0739590	+	0.211363i	−0.0739590	+	0.211363i	−0.974928	−	0.222521i	$-0.928571\pi$
$547$	−0.0739590	+	0.211363i	−0.0739590	+	0.211363i	0.900969	+	0.433884i	$0.142857\pi$
$548$	−0.752407	+	1.19745i	−0.752407	+	1.19745i
$549$		0			0
$550$	−0.678448	+	1.40881i	−0.678448	+	1.40881i
$551$		0			0
$552$		0			0
$553$	0.119137	−	0.189606i	0.119137	−	0.189606i
$554$	1.87590	+	0.211363i	1.87590	+	0.211363i
$555$		0			0
$556$		0			0
$557$	−0.119137	−	0.189606i	−0.119137	−	0.189606i	0.781831	−	0.623490i	$-0.214286\pi$
$557$	−0.119137	−	0.189606i	−0.119137	−	0.189606i	−0.900969	+	0.433884i	$0.857143\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$		0			0
$562$	−0.781831	−	1.62349i	−0.781831	−	1.62349i
$563$		0			0		1.00000			$0$
$563$		0			0		−1.00000			$\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	0.623490	−	0.781831i	0.623490	−	0.781831i
$568$	0.433884	−	0.0990311i	0.433884	−	0.0990311i
$569$	−0.119137	−	1.05737i	−0.119137	−	1.05737i	−0.900969	−	0.433884i	$-0.857143\pi$
$569$	−0.119137	−	1.05737i	−0.119137	−	1.05737i	0.781831	−	0.623490i	$-0.214286\pi$
$570$		0			0
$571$	0.0250721	−	0.222521i	0.0250721	−	0.222521i	−0.974928	−	0.222521i	$-0.928571\pi$
$571$	0.0250721	−	0.222521i	0.0250721	−	0.222521i	1.00000			$0$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	0.846011	−	0.193096i	0.846011	−	0.193096i
$576$	−0.623490	+	0.781831i	−0.623490	+	0.781831i
$577$		0			0		−0.330279	−	0.943883i	$-0.607143\pi$
$577$		0			0		0.330279	+	0.943883i	$0.392857\pi$
$578$	−0.974928	−	0.222521i	−0.974928	−	0.222521i
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$	−2.93327	+	1.02640i	−2.93327	+	1.02640i
$584$		0			0
$585$		0			0
$586$		0			0
$587$		0			0		−0.330279	−	0.943883i	$-0.607143\pi$
$587$		0			0		0.330279	+	0.943883i	$0.392857\pi$
$588$		0			0
$589$		0			0
$590$		0			0
$591$		0			0
$592$	0.433884	−	1.90097i	0.433884	−	1.90097i
$593$		0			0		−0.781831	−	0.623490i	$-0.785714\pi$
$593$		0			0		0.781831	+	0.623490i	$0.214286\pi$
$594$		0			0
$595$		0			0
$596$	−1.33485	−	1.33485i	−1.33485	−	1.33485i
$597$		0			0
$598$		0			0
$599$	0.467085	−	1.33485i	0.467085	−	1.33485i	−0.433884	−	0.900969i	$-0.642857\pi$
$599$	0.467085	−	1.33485i	0.467085	−	1.33485i	0.900969	−	0.433884i	$-0.142857\pi$
$600$		0			0
$601$		0			0		0.111964	−	0.993712i	$-0.464286\pi$
$601$		0			0		−0.111964	+	0.993712i	$0.535714\pi$
$602$	−1.24698	−	1.56366i	−1.24698	−	1.56366i
$603$	−0.559311	+	0.351438i	−0.559311	+	0.351438i
$604$	0.974928	+	0.777479i	0.974928	+	0.777479i
$605$		0			0
$606$		0			0
$607$		0			0		−0.846724	−	0.532032i	$-0.821429\pi$
$607$		0			0		0.846724	+	0.532032i	$0.178571\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$	−1.59842	−	0.559311i	−1.59842	−	0.559311i	−0.623490	−	0.781831i	$-0.714286\pi$
$613$	−1.59842	−	0.559311i	−1.59842	−	0.559311i	−0.974928	+	0.222521i	$0.928571\pi$
$614$		0			0
$615$		0			0
$616$	0.347948	−	1.52446i	0.347948	−	1.52446i
$617$	−1.97493	−	0.222521i	−1.97493	−	0.222521i	−0.974928	−	0.222521i	$-0.928571\pi$
$617$	−1.97493	−	0.222521i	−1.97493	−	0.222521i	−1.00000			$\pi$
$618$		0			0
$619$		0			0		−0.433884	−	0.900969i	$-0.642857\pi$
$619$		0			0		0.433884	+	0.900969i	$0.357143\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	−0.623490	+	0.781831i	−0.623490	+	0.781831i
$626$		0			0
$627$		0			0
$628$		0			0
$629$		0			0
$630$		0			0
$631$	−0.678448	+	1.40881i	−0.678448	+	1.40881i	0.222521	+	0.974928i	$0.428571\pi$
$631$	−0.678448	+	1.40881i	−0.678448	+	1.40881i	−0.900969	+	0.433884i	$0.857143\pi$
$632$	−0.158342	−	0.158342i	−0.158342	−	0.158342i
$633$		0			0
$634$		0			0
$635$		0			0
$636$		0			0
$637$		0			0
$638$	1.40881	+	0.678448i	1.40881	+	0.678448i
$639$			0.445042i			0.445042i
$640$		0			0
$641$	1.00435	−	0.351438i	1.00435	−	0.351438i	0.222521	−	0.974928i	$-0.428571\pi$
$641$	1.00435	−	0.351438i	1.00435	−	0.351438i	0.781831	+	0.623490i	$0.214286\pi$
$642$		0			0
$643$		0			0		0.532032	−	0.846724i	$-0.321429\pi$
$643$		0			0		−0.532032	+	0.846724i	$0.678571\pi$
$644$	−0.781831	+	0.376510i	−0.781831	+	0.376510i
$645$		0			0
$646$		0			0
$647$		0			0		0.781831	−	0.623490i	$-0.214286\pi$
$647$		0			0		−0.781831	+	0.623490i	$0.785714\pi$
$648$	−0.623490	−	0.781831i	−0.623490	−	0.781831i
$649$		0			0
$650$		0			0
$651$		0			0
$652$	−1.75676	+	0.846011i	−1.75676	+	0.846011i
$653$	1.52446	+	0.347948i	1.52446	+	0.347948i	0.900969	−	0.433884i	$-0.142857\pi$
$653$	1.52446	+	0.347948i	1.52446	+	0.347948i	0.623490	+	0.781831i	$0.285714\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−0.347948	+	0.277479i	−0.347948	+	0.277479i	−0.781831	−	0.623490i	$-0.785714\pi$
$659$	−0.347948	+	0.277479i	−0.347948	+	0.277479i	0.433884	+	0.900969i	$0.357143\pi$
$660$		0			0
$661$		0			0		−0.532032	−	0.846724i	$-0.678571\pi$
$661$		0			0		0.532032	+	0.846724i	$0.321429\pi$
$662$	1.52446	+	0.347948i	1.52446	+	0.347948i
$663$		0			0
$664$		0			0
$665$		0			0
$666$	1.75676	+	0.846011i	1.75676	+	0.846011i
$667$	−0.193096	−	0.846011i	−0.193096	−	0.846011i
$668$		0			0
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$	−1.75676	−	0.400969i	−1.75676	−	0.400969i	−0.781831	−	0.623490i	$-0.785714\pi$
$673$	−1.75676	−	0.400969i	−1.75676	−	0.400969i	−0.974928	+	0.222521i	$0.928571\pi$
$674$	−0.222521	+	1.97493i	−0.222521	+	1.97493i
$675$		0			0
$676$	0.974928	+	0.222521i	0.974928	+	0.222521i
$677$		0			0		0.900969	−	0.433884i	$-0.142857\pi$
$677$		0			0		−0.900969	+	0.433884i	$0.857143\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$		0			0
$682$		0			0
$683$	0.222521	−	0.0250721i	0.222521	−	0.0250721i		−	1.00000i	$-0.5\pi$
$683$	0.222521	−	0.0250721i	0.222521	−	0.0250721i	0.222521	+	0.974928i	$0.428571\pi$
$684$		0			0
$685$		0			0
$686$	−1.00000			−1.00000
$687$		0			0
$688$	−1.80194	+	0.867767i	−1.80194	+	0.867767i
$689$		0			0
$690$		0			0
$691$		0			0		−0.846724	−	0.532032i	$-0.821429\pi$
$691$		0			0		0.846724	+	0.532032i	$0.178571\pi$
$692$		0			0
$693$	1.40881	+	0.678448i	1.40881	+	0.678448i
$694$	−0.566116	−	0.900969i	−0.566116	−	0.900969i
$695$		0			0
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$	0.433884	−	0.900969i	0.433884	−	0.900969i
$701$	−0.189606	−	0.119137i	−0.189606	−	0.119137i	0.433884	−	0.900969i	$-0.357143\pi$
$701$	−0.189606	−	0.119137i	−0.189606	−	0.119137i	−0.623490	+	0.781831i	$0.714286\pi$
$702$		0			0
$703$		0			0
$704$	−1.40881	−	0.678448i	−1.40881	−	0.678448i
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	−0.656405	+	0.0739590i	−0.656405	+	0.0739590i	−0.433884	−	0.900969i	$-0.642857\pi$
$709$	−0.656405	+	0.0739590i	−0.656405	+	0.0739590i	−0.222521	+	0.974928i	$0.571429\pi$
$710$		0			0
$711$	0.189606	−	0.119137i	0.189606	−	0.119137i
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$	−1.68280	+	1.05737i	−1.68280	+	1.05737i
$717$		0			0
$718$	−1.68280	−	0.189606i	−1.68280	−	0.189606i
$719$		0			0		−0.974928	−	0.222521i	$-0.928571\pi$
$719$		0			0		0.974928	+	0.222521i	$0.0714286\pi$
$720$		0			0
$721$		0			0
$722$	0.623490	+	0.781831i	0.623490	+	0.781831i
$723$		0			0
$724$		0			0
$725$	0.781831	+	0.623490i	0.781831	+	0.623490i
$726$		0			0
$727$		0			0		−0.846724	−	0.532032i	$-0.821429\pi$
$727$		0			0		0.846724	+	0.532032i	$0.178571\pi$
$728$		0			0
$729$	0.900969	−	0.433884i	0.900969	−	0.433884i
$730$		0			0
$731$		0			0
$732$		0			0
$733$		0			0		0.781831	−	0.623490i	$-0.214286\pi$
$733$		0			0		−0.781831	+	0.623490i	$0.785714\pi$
$734$		0			0
$735$		0			0
$736$	0.193096	+	0.846011i	0.193096	+	0.846011i
$737$	−0.730364	−	0.730364i	−0.730364	−	0.730364i
$738$		0			0
$739$	−0.846011	−	0.193096i	−0.846011	−	0.193096i	−0.222521	−	0.974928i	$-0.571429\pi$
$739$	−0.846011	−	0.193096i	−0.846011	−	0.193096i	−0.623490	+	0.781831i	$0.714286\pi$
$740$		0			0
$741$		0			0
$742$	1.87590	−	0.656405i	1.87590	−	0.656405i
$743$	0.656405	−	0.0739590i	0.656405	−	0.0739590i	0.222521	−	0.974928i	$-0.428571\pi$
$743$	0.656405	−	0.0739590i	0.656405	−	0.0739590i	0.433884	+	0.900969i	$0.357143\pi$
$744$		0			0
$745$		0			0
$746$	−1.68280	−	1.05737i	−1.68280	−	1.05737i
$747$		0			0
$748$		0			0
$749$	0.900969	−	1.43388i	0.900969	−	1.43388i
$750$		0			0
$751$	−0.623490	+	0.218169i	−0.623490	+	0.218169i	−0.623490	−	0.781831i	$-0.714286\pi$
$751$	−0.623490	+	0.218169i	−0.623490	+	0.218169i			1.00000i	$0.5\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$	−0.781831	+	1.62349i	−0.781831	+	1.62349i			1.00000i	$0.5\pi$
$757$	−0.781831	+	1.62349i	−0.781831	+	1.62349i	−0.781831	+	0.623490i	$0.785714\pi$
$758$	−0.445042			−0.445042
$759$		0			0
$760$		0			0
$761$		0			0		0.433884	−	0.900969i	$-0.357143\pi$
$761$		0			0		−0.433884	+	0.900969i	$0.642857\pi$
$762$		0			0
$763$	−1.68280	−	0.189606i	−1.68280	−	0.189606i
$764$	−0.351438	−	1.00435i	−0.351438	−	1.00435i
$765$		0			0
$766$		0			0
$767$		0			0
$768$		0			0
$769$		0			0		0.846724	−	0.532032i	$-0.178571\pi$
$769$		0			0		−0.846724	+	0.532032i	$0.821429\pi$
$770$		0			0
$771$		0			0
$772$	−1.19745	+	1.19745i	−1.19745	+	1.19745i
$773$		0			0		−0.433884	−	0.900969i	$-0.642857\pi$
$773$		0			0		0.433884	+	0.900969i	$0.357143\pi$
$774$	−0.445042	−	1.94986i	−0.445042	−	1.94986i
$775$		0			0
$776$		0			0
$777$		0			0
$778$	−0.433884	+	1.90097i	−0.433884	+	1.90097i
$779$		0			0
$780$		0			0
$781$	−0.678448	+	0.154851i	−0.678448	+	0.154851i
$782$		0			0
$783$		0			0
$784$	−0.222521	+	0.974928i	−0.222521	+	0.974928i
$785$		0			0
$786$		0			0
$787$		0			0		−0.943883	−	0.330279i	$-0.892857\pi$
$787$		0			0		0.943883	+	0.330279i	$0.107143\pi$
$788$	−0.119137	−	1.05737i	−0.119137	−	1.05737i
$789$		0			0
$790$		0			0
$791$	−0.211363	+	1.87590i	−0.211363	+	1.87590i
$792$	0.974928	−	1.22252i	0.974928	−	1.22252i
$793$		0			0
$794$		0			0
$795$		0			0
$796$		0			0
$797$		0			0		0.222521	−	0.974928i	$-0.428571\pi$
$797$		0			0		−0.222521	+	0.974928i	$0.571429\pi$
$798$		0			0
$799$		0			0
$800$	−0.781831	−	0.623490i	−0.781831	−	0.623490i
$801$		0			0
$802$	1.24698			1.24698
$803$		0			0
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$	1.33485	−	0.467085i	1.33485	−	0.467085i	0.433884	−	0.900969i	$-0.357143\pi$
$809$	1.33485	−	0.467085i	1.33485	−	0.467085i	0.900969	+	0.433884i	$0.142857\pi$
$810$		0			0
$811$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$811$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$812$	−0.900969	−	0.433884i	−0.900969	−	0.433884i
$813$		0			0
$814$	−0.678448	+	2.97247i	−0.678448	+	2.97247i
$815$		0			0
$816$		0			0
$817$		0			0
$818$		0			0
$819$		0			0
$820$		0			0
$821$	0.0739590	−	0.656405i	0.0739590	−	0.656405i	−0.900969	−	0.433884i	$-0.857143\pi$
$821$	0.0739590	−	0.656405i	0.0739590	−	0.656405i	0.974928	−	0.222521i	$-0.0714286\pi$
$822$		0			0
$823$	0.189606	+	1.68280i	0.189606	+	1.68280i	0.623490	+	0.781831i	$0.285714\pi$
$823$	0.189606	+	1.68280i	0.189606	+	1.68280i	−0.433884	+	0.900969i	$0.642857\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	1.75676	+	0.400969i	1.75676	+	0.400969i	0.974928	−	0.222521i	$-0.0714286\pi$
$827$	1.75676	+	0.400969i	1.75676	+	0.400969i	0.781831	+	0.623490i	$0.214286\pi$
$828$	−0.867767			−0.867767
$829$		0			0		1.00000			$0$
$829$		0			0		−1.00000			$\pi$
$830$		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		0.532032	−	0.846724i	$-0.321429\pi$
$839$		0			0		−0.532032	+	0.846724i	$0.678571\pi$
$840$		0			0
$841$	0.623490	−	0.781831i	0.623490	−	0.781831i
$842$	−0.781831	−	0.376510i	−0.781831	−	0.376510i
$843$		0			0
$844$	−0.400969	−	1.75676i	−0.400969	−	1.75676i
$845$		0			0
$846$		0			0
$847$	−0.321552	+	1.40881i	−0.321552	+	1.40881i
$848$	−0.222521	−	1.97493i	−0.222521	−	1.97493i
$849$		0			0
$850$		0			0
$851$	1.52446	−	0.734141i	1.52446	−	0.734141i
$852$		0			0
$853$		0			0		1.00000			$0$
$853$		0			0		−1.00000			$\pi$
$854$		0			0
$855$		0			0
$856$	−1.19745	−	1.19745i	−1.19745	−	1.19745i
$857$		0			0		−0.781831	−	0.623490i	$-0.785714\pi$
$857$		0			0		0.781831	+	0.623490i	$0.214286\pi$
$858$		0			0
$859$		0			0		−0.781831	−	0.623490i	$-0.785714\pi$
$859$		0			0		0.781831	+	0.623490i	$0.214286\pi$
$860$		0			0
$861$		0			0
$862$	−0.846011	+	0.193096i	−0.846011	+	0.193096i
$863$	−1.40881	−	0.678448i	−1.40881	−	0.678448i	−0.433884	−	0.900969i	$-0.642857\pi$
$863$	−1.40881	−	0.678448i	−1.40881	−	0.678448i	−0.974928	+	0.222521i	$0.928571\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$		0			0
$868$		0			0
$869$	0.247593	+	0.247593i	0.247593	+	0.247593i
$870$		0			0
$871$		0			0
$872$	−0.559311	+	1.59842i	−0.559311	+	1.59842i
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$	−0.218169	−	0.623490i	−0.218169	−	0.623490i	0.781831	−	0.623490i	$-0.214286\pi$
$877$	−0.218169	−	0.623490i	−0.218169	−	0.623490i	−1.00000			$\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		0			0		−0.111964	−	0.993712i	$-0.535714\pi$
$881$		0			0		0.111964	+	0.993712i	$0.464286\pi$
$882$	−0.900969	−	0.433884i	−0.900969	−	0.433884i
$883$	−1.05737	+	0.119137i	−1.05737	+	0.119137i	−0.623490	−	0.781831i	$-0.714286\pi$
$883$	−1.05737	+	0.119137i	−1.05737	+	0.119137i	−0.433884	+	0.900969i	$0.642857\pi$
$884$		0			0
$885$		0			0
$886$	−0.541044	−	1.12349i	−0.541044	−	1.12349i
$887$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$887$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$888$		0			0
$889$	1.78183	+	0.623490i	1.78183	+	0.623490i
$890$		0			0
$891$	0.974928	+	1.22252i	0.974928	+	1.22252i
$892$		0			0
$893$		0			0
$894$		0			0
$895$		0			0
$896$	0.900969	+	0.433884i	0.900969	+	0.433884i
$897$		0			0
$898$	−0.211363	+	0.0739590i	−0.211363	+	0.0739590i
$899$		0			0
$900$	0.781831	−	0.623490i	0.781831	−	0.623490i
$901$		0			0
$902$		0			0
$903$		0			0
$904$	1.78183	+	0.623490i	1.78183	+	0.623490i
$905$		0			0
$906$		0			0
$907$	−1.40881	+	1.12349i	−1.40881	+	1.12349i	−0.433884	+	0.900969i	$0.642857\pi$
$907$	−1.40881	+	1.12349i	−1.40881	+	1.12349i	−0.974928	+	0.222521i	$0.928571\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	1.00000	−	1.00000i	1.00000	−	1.00000i		−	1.00000i	$-0.5\pi$
$911$	1.00000	−	1.00000i	1.00000	−	1.00000i	1.00000			$0$
$912$		0			0
$913$		0			0
$914$	−0.541044	+	0.678448i	−0.541044	+	0.678448i
$915$		0			0
$916$		0			0
$917$		0			0
$918$		0			0
$919$	−0.541044	−	0.678448i	−0.541044	−	0.678448i	0.433884	−	0.900969i	$-0.357143\pi$
$919$	−0.541044	−	0.678448i	−0.541044	−	0.678448i	−0.974928	+	0.222521i	$0.928571\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		0			0
$924$		0			0
$925$	−0.846011	+	1.75676i	−0.846011	+	1.75676i
$926$	0.347948	−	1.52446i	0.347948	−	1.52446i
$927$		0			0
$928$	−0.623490	+	0.781831i	−0.623490	+	0.781831i
$929$		0			0		1.00000			$0$
$929$		0			0		−1.00000			$\pi$
$930$		0			0
$931$		0			0
$932$	−0.846011	+	1.75676i	−0.846011	+	1.75676i
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$		0			0		−0.623490	−	0.781831i	$-0.714286\pi$
$937$		0			0		0.623490	+	0.781831i	$0.285714\pi$
$938$	0.467085	+	0.467085i	0.467085	+	0.467085i
$939$		0			0
$940$		0			0
$941$		0			0		−0.111964	−	0.993712i	$-0.535714\pi$
$941$		0			0		0.111964	+	0.993712i	$0.464286\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$	2.81762	−	1.35690i	2.81762	−	1.35690i
$947$	0.376510	+	0.781831i	0.376510	+	0.781831i	1.00000			$0$
$947$	0.376510	+	0.781831i	0.376510	+	0.781831i	−0.623490	+	0.781831i	$0.714286\pi$
$948$		0			0
$949$		0			0
$950$		0			0
$951$		0			0
$952$		0			0
$953$	0.867767	+	1.80194i	0.867767	+	1.80194i	0.433884	+	0.900969i	$0.357143\pi$
$953$	0.867767	+	1.80194i	0.867767	+	1.80194i	0.433884	+	0.900969i	$0.357143\pi$
$954$	1.97493	+	0.222521i	1.97493	+	0.222521i
$955$		0			0
$956$	−1.52446	+	1.21572i	−1.52446	+	1.21572i
$957$		0			0
$958$		0			0
$959$	−0.752407	+	1.19745i	−0.752407	+	1.19745i
$960$		0			0
$961$	−0.433884	−	0.900969i	−0.433884	−	0.900969i
$962$		0			0
$963$	1.43388	−	0.900969i	1.43388	−	0.900969i
$964$		0			0
$965$		0			0
$966$		0			0
$967$	1.78183	+	0.623490i	1.78183	+	0.623490i	1.00000			$0$
$967$	1.78183	+	0.623490i	1.78183	+	0.623490i	0.781831	+	0.623490i	$0.214286\pi$
$968$	1.30194	+	0.626980i	1.30194	+	0.626980i
$969$		0			0
$970$		0			0
$971$		0			0		0.222521	−	0.974928i	$-0.428571\pi$
$971$		0			0		−0.222521	+	0.974928i	$0.571429\pi$
$972$		0			0
$973$		0			0
$974$	0.400969	+	0.193096i	0.400969	+	0.193096i
$975$		0			0
$976$		0			0
$977$	1.21572	+	1.52446i	1.21572	+	1.52446i	0.781831	+	0.623490i	$0.214286\pi$
$977$	1.21572	+	1.52446i	1.21572	+	1.52446i	0.433884	+	0.900969i	$0.357143\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$	−1.43388	−	0.900969i	−1.43388	−	0.900969i
$982$		−	0.867767i		−	0.867767i
$983$		0			0		−0.330279	−	0.943883i	$-0.607143\pi$
$983$		0			0		0.330279	+	0.943883i	$0.392857\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−1.56366	−	0.753020i	−1.56366	−	0.753020i
$990$		0			0
$991$	−0.400969	−	1.75676i	−0.400969	−	1.75676i	−0.623490	−	0.781831i	$-0.714286\pi$
$991$	−0.400969	−	1.75676i	−0.400969	−	1.75676i	0.222521	−	0.974928i	$-0.428571\pi$
$992$		0			0
$993$		0			0
$994$	0.433884	−	0.0990311i	0.433884	−	0.0990311i
$995$		0			0
$996$		0			0
$997$		0			0		−0.781831	−	0.623490i	$-0.785714\pi$
$997$		0			0		0.781831	+	0.623490i	$0.214286\pi$
$998$	0.559311	+	1.59842i	0.559311	+	1.59842i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3248.1.dt.b.2323.1	✓	12
7.6	odd	2	CM	3248.1.dt.b.2323.1	✓	12
16.11	odd	4		3248.1.ep.b.699.1	yes	12
29.10	odd	28		3248.1.ep.b.1315.1	yes	12
112.27	even	4		3248.1.ep.b.699.1	yes	12
203.97	even	28		3248.1.ep.b.1315.1	yes	12
464.155	even	28	inner	3248.1.dt.b.2939.1	yes	12
3248.2939	odd	28	inner	3248.1.dt.b.2939.1	yes	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3248.1.dt.b.2323.1	✓	12	1.1	even	1	trivial
3248.1.dt.b.2323.1	✓	12	7.6	odd	2	CM
3248.1.dt.b.2939.1	yes	12	464.155	even	28	inner
3248.1.dt.b.2939.1	yes	12	3248.2939	odd	28	inner
3248.1.ep.b.699.1	yes	12	16.11	odd	4
3248.1.ep.b.699.1	yes	12	112.27	even	4
3248.1.ep.b.1315.1	yes	12	29.10	odd	28
3248.1.ep.b.1315.1	yes	12	203.97	even	28

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:	\( N \)	\(=\)	\( 3248 = 2^{4} \cdot 7 \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	3248.dt (of order \(28\), degree \(12\), minimal)

\(f(q)\)	\(=\)	\(q+(0.222521 - 0.974928i) q^{2} +(-0.900969 - 0.433884i) q^{4} +(-0.900969 - 0.433884i) q^{7} +(-0.623490 + 0.781831i) q^{8} +(-0.623490 - 0.781831i) q^{9} +O(q^{10})\)	\(q+(0.222521 - 0.974928i) q^{2} +(-0.900969 - 0.433884i) q^{4} +(-0.900969 - 0.433884i) q^{7} +(-0.623490 + 0.781831i) q^{8} +(-0.623490 - 0.781831i) q^{9} +(0.974928 - 1.22252i) q^{11} +(-0.623490 + 0.781831i) q^{14} +(0.623490 + 0.781831i) q^{16} +(-0.900969 + 0.433884i) q^{18} +(-0.974928 - 1.22252i) q^{22} +(-0.193096 + 0.846011i) q^{23} +(-0.433884 - 0.900969i) q^{25} +(0.623490 + 0.781831i) q^{28} +(-0.900969 + 0.433884i) q^{29} +(0.900969 - 0.433884i) q^{32} +(0.222521 + 0.974928i) q^{36} +(-1.21572 - 1.52446i) q^{37} +(-0.445042 + 1.94986i) q^{43} +(-1.40881 + 0.678448i) q^{44} +(0.781831 + 0.376510i) q^{46} +(0.623490 + 0.781831i) q^{49} +(-0.974928 + 0.222521i) q^{50} +(-1.68280 - 1.05737i) q^{53} +(0.900969 - 0.433884i) q^{56} +(0.222521 + 0.974928i) q^{58} +(0.222521 + 0.974928i) q^{63} +(-0.222521 - 0.974928i) q^{64} +(0.0739590 - 0.656405i) q^{67} +(-0.347948 - 0.277479i) q^{71} +1.00000 q^{72} +(-1.75676 + 0.846011i) q^{74} +(-1.40881 + 0.678448i) q^{77} +(-0.0250721 + 0.222521i) q^{79} +(-0.222521 + 0.974928i) q^{81} +(1.80194 + 0.867767i) q^{86} +(0.347948 + 1.52446i) q^{88} +(0.541044 - 0.678448i) q^{92} +(0.900969 - 0.433884i) q^{98} -1.56366 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 2 q^{2} - 2 q^{4} - 2 q^{7} + 2 q^{8} + 2 q^{9}+O(q^{10}) \) 12 * q + 2 * q^2 - 2 * q^4 - 2 * q^7 + 2 * q^8 + 2 * q^9	\( 12 q + 2 q^{2} - 2 q^{4} - 2 q^{7} + 2 q^{8} + 2 q^{9} + 2 q^{14} - 2 q^{16} - 2 q^{18} - 2 q^{28} - 2 q^{29} + 2 q^{32} + 2 q^{36} - 4 q^{43} - 2 q^{49} - 2 q^{53} + 2 q^{56} + 2 q^{58} + 2 q^{63} - 2 q^{64} - 2 q^{67} + 12 q^{72} - 12 q^{79} - 2 q^{81} + 4 q^{86} + 2 q^{98}+O(q^{100}) \) 12 * q + 2 * q^2 - 2 * q^4 - 2 * q^7 + 2 * q^8 + 2 * q^9 + 2 * q^14 - 2 * q^16 - 2 * q^18 - 2 * q^28 - 2 * q^29 + 2 * q^32 + 2 * q^36 - 4 * q^43 - 2 * q^49 - 2 * q^53 + 2 * q^56 + 2 * q^58 + 2 * q^63 - 2 * q^64 - 2 * q^67 + 12 * q^72 - 12 * q^79 - 2 * q^81 + 4 * q^86 + 2 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more