LMFDB - Embedded newform 968.1.l.a.3.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [968,1,Mod(3,968)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(968, base_ring=CyclotomicField(10))

chi = DirichletCharacter(H, H._module([5, 5, 8]))

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

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

chi := DirichletCharacter("968.3");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$N$	$=$	$968 = 2^{3} \cdot 11^{2}$
Weight:	$k$	$=$	$1$
Character orbit:	$[\chi]$	$=$	968.l (of order $10$ , degree $4$ , not 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:	$0.483094932229$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{10})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$x^{4} - x^{3} + x^{2} - x + 1$ x^4 - x^3 + x^2 - x + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$1$
Twist minimal:	no (minimal twist has level 88)
Projective image:	$D_{5}$
Projective field:	Galois closure of 5.1.937024.1

Embedding invariants

Embedding label			3.1
Root			$0.809017 - 0.587785i$ of defining polynomial
Character	$\chi$	$=$	968.3
Dual form			968.1.l.a.323.1

$q$ -expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.809017 + 0.587785i) q^{2} +(0.190983 + 0.587785i) q^{3} +(0.309017 - 0.951057i) q^{4} +(-0.500000 - 0.363271i) q^{6} +(0.309017 + 0.951057i) q^{8} +(0.500000 - 0.363271i) q^{9} +0.618034 q^{12} +(-0.809017 - 0.587785i) q^{16} +(1.30902 + 0.951057i) q^{17} +(-0.190983 + 0.587785i) q^{18} +(-0.500000 - 1.53884i) q^{19} +(-0.500000 + 0.363271i) q^{24} +(0.309017 + 0.951057i) q^{25} +(0.809017 + 0.587785i) q^{27} +1.00000 q^{32} -1.61803 q^{34} +(-0.190983 - 0.587785i) q^{36} +(1.30902 + 0.951057i) q^{38} +(0.190983 + 0.587785i) q^{41} +0.618034 q^{43} +(0.190983 - 0.587785i) q^{48} +(-0.809017 - 0.587785i) q^{49} +(-0.809017 - 0.587785i) q^{50} +(-0.309017 + 0.951057i) q^{51} -1.00000 q^{54} +(0.809017 - 0.587785i) q^{57} +(-0.500000 + 1.53884i) q^{59} +(-0.809017 + 0.587785i) q^{64} -1.61803 q^{67} +(1.30902 - 0.951057i) q^{68} +(0.500000 + 0.363271i) q^{72} +(0.190983 - 0.587785i) q^{73} +(-0.500000 + 0.363271i) q^{75} -1.61803 q^{76} +(-0.500000 - 0.363271i) q^{82} +(-0.500000 - 0.363271i) q^{83} +(-0.500000 + 0.363271i) q^{86} +0.618034 q^{89} +(0.190983 + 0.587785i) q^{96} +(-0.500000 + 0.363271i) q^{97} +1.00000 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$4 q - q^{2} + 3 q^{3} - q^{4} - 2 q^{6} - q^{8} + 2 q^{9} - 2 q^{12} - q^{16} + 3 q^{17} - 3 q^{18} - 2 q^{19} - 2 q^{24} - q^{25} + q^{27} + 4 q^{32} - 2 q^{34} - 3 q^{36} + 3 q^{38} + 3 q^{41} - 2 q^{43}+ \cdots + 4 q^{98}+O(q^{100})$ 4 * q - q^2 + 3 * q^3 - q^4 - 2 * q^6 - q^8 + 2 * q^9 - 2 * q^12 - q^16 + 3 * q^17 - 3 * q^18 - 2 * q^19 - 2 * q^24 - q^25 + q^27 + 4 * q^32 - 2 * q^34 - 3 * q^36 + 3 * q^38 + 3 * q^41 - 2 * q^43 + 3 * q^48 - q^49 - q^50 + q^51 - 4 * q^54 + q^57 - 2 * q^59 - q^64 - 2 * q^67 + 3 * q^68 + 2 * q^72 + 3 * q^73 - 2 * q^75 - 2 * q^76 - 2 * q^82 - 2 * q^83 - 2 * q^86 - 2 * q^89 + 3 * q^96 - 2 * q^97 + 4 * q^98

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/968\mathbb{Z}\right)^\times$ .

$n$	$485$	$727$	$849$
$\chi(n)$	$-1$	$-1$	$e\left(\frac{4}{5}\right)$

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

)

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

$2$	−0.809017	+	0.587785i	−0.809017	+	0.587785i
$2$	−0.809017	+	0.587785i	−0.809017	+	0.587785i
$3$	0.190983	+	0.587785i	0.190983	+	0.587785i	1.00000			$0$
$3$	0.190983	+	0.587785i	0.190983	+	0.587785i	−0.809017	+	0.587785i	$0.800000\pi$
$4$	0.309017	−	0.951057i	0.309017	−	0.951057i
$5$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$5$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$6$	−0.500000	−	0.363271i	−0.500000	−	0.363271i
$7$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$7$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$8$	0.309017	+	0.951057i	0.309017	+	0.951057i
$9$	0.500000	−	0.363271i	0.500000	−	0.363271i
$10$		0			0
$11$		0			0
$11$		0			0
$12$	0.618034			0.618034
$13$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$13$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$14$		0			0
$15$		0			0
$16$	−0.809017	−	0.587785i	−0.809017	−	0.587785i
$17$	1.30902	+	0.951057i	1.30902	+	0.951057i	1.00000			$0$
$17$	1.30902	+	0.951057i	1.30902	+	0.951057i	0.309017	+	0.951057i	$0.400000\pi$
$18$	−0.190983	+	0.587785i	−0.190983	+	0.587785i
$19$	−0.500000	−	1.53884i	−0.500000	−	1.53884i	−0.809017	−	0.587785i	$-0.800000\pi$
$19$	−0.500000	−	1.53884i	−0.500000	−	1.53884i	0.309017	−	0.951057i	$-0.400000\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$		0			0		1.00000			$0$
$23$		0			0		−1.00000			$\pi$
$24$	−0.500000	+	0.363271i	−0.500000	+	0.363271i
$25$	0.309017	+	0.951057i	0.309017	+	0.951057i
$26$		0			0
$27$	0.809017	+	0.587785i	0.809017	+	0.587785i
$28$		0			0
$29$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$29$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$30$		0			0
$31$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$31$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$32$	1.00000			1.00000
$33$		0			0
$34$	−1.61803			−1.61803
$35$		0			0
$36$	−0.190983	−	0.587785i	−0.190983	−	0.587785i
$37$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$37$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$38$	1.30902	+	0.951057i	1.30902	+	0.951057i
$39$		0			0
$40$		0			0
$41$	0.190983	+	0.587785i	0.190983	+	0.587785i	1.00000			$0$
$41$	0.190983	+	0.587785i	0.190983	+	0.587785i	−0.809017	+	0.587785i	$0.800000\pi$
$42$		0			0
$43$	0.618034			0.618034			0.309017	−	0.951057i	$-0.400000\pi$
$43$	0.618034			0.618034			0.309017	+	0.951057i	$0.400000\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$47$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$48$	0.190983	−	0.587785i	0.190983	−	0.587785i
$49$	−0.809017	−	0.587785i	−0.809017	−	0.587785i
$50$	−0.809017	−	0.587785i	−0.809017	−	0.587785i
$51$	−0.309017	+	0.951057i	−0.309017	+	0.951057i
$52$		0			0
$53$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$53$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$54$	−1.00000			−1.00000
$55$		0			0
$56$		0			0
$57$	0.809017	−	0.587785i	0.809017	−	0.587785i
$58$		0			0
$59$	−0.500000	+	1.53884i	−0.500000	+	1.53884i	0.309017	+	0.951057i	$0.400000\pi$
$59$	−0.500000	+	1.53884i	−0.500000	+	1.53884i	−0.809017	+	0.587785i	$0.800000\pi$
$60$		0			0
$61$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$61$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$62$		0			0
$63$		0			0
$64$	−0.809017	+	0.587785i	−0.809017	+	0.587785i
$65$		0			0
$66$		0			0
$67$	−1.61803			−1.61803			−0.809017	−	0.587785i	$-0.800000\pi$
$67$	−1.61803			−1.61803			−0.809017	+	0.587785i	$0.800000\pi$
$68$	1.30902	−	0.951057i	1.30902	−	0.951057i
$69$		0			0
$70$		0			0
$71$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$71$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$72$	0.500000	+	0.363271i	0.500000	+	0.363271i
$73$	0.190983	−	0.587785i	0.190983	−	0.587785i	−0.809017	−	0.587785i	$-0.800000\pi$
$73$	0.190983	−	0.587785i	0.190983	−	0.587785i	1.00000			$0$
$74$		0			0
$75$	−0.500000	+	0.363271i	−0.500000	+	0.363271i
$76$	−1.61803			−1.61803
$77$		0			0
$78$		0			0
$79$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$79$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$80$		0			0
$81$		0			0
$82$	−0.500000	−	0.363271i	−0.500000	−	0.363271i
$83$	−0.500000	−	0.363271i	−0.500000	−	0.363271i	0.309017	−	0.951057i	$-0.400000\pi$
$83$	−0.500000	−	0.363271i	−0.500000	−	0.363271i	−0.809017	+	0.587785i	$0.800000\pi$
$84$		0			0
$85$		0			0
$86$	−0.500000	+	0.363271i	−0.500000	+	0.363271i
$87$		0			0
$88$		0			0
$89$	0.618034			0.618034			0.309017	−	0.951057i	$-0.400000\pi$
$89$	0.618034			0.618034			0.309017	+	0.951057i	$0.400000\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$	0.190983	+	0.587785i	0.190983	+	0.587785i
$97$	−0.500000	+	0.363271i	−0.500000	+	0.363271i	−0.809017	−	0.587785i	$-0.800000\pi$
$97$	−0.500000	+	0.363271i	−0.500000	+	0.363271i	0.309017	+	0.951057i	$0.400000\pi$
$98$	1.00000			1.00000
$99$		0			0
$100$	1.00000			1.00000
$101$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$101$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$102$	−0.309017	−	0.951057i	−0.309017	−	0.951057i
$103$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$103$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	0.190983	+	0.587785i	0.190983	+	0.587785i	1.00000			$0$
$107$	0.190983	+	0.587785i	0.190983	+	0.587785i	−0.809017	+	0.587785i	$0.800000\pi$
$108$	0.809017	−	0.587785i	0.809017	−	0.587785i
$109$		0			0		1.00000			$0$
$109$		0			0		−1.00000			$\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−0.500000	−	1.53884i	−0.500000	−	1.53884i	−0.809017	−	0.587785i	$-0.800000\pi$
$113$	−0.500000	−	1.53884i	−0.500000	−	1.53884i	0.309017	−	0.951057i	$-0.400000\pi$
$114$	−0.309017	+	0.951057i	−0.309017	+	0.951057i
$115$		0			0
$116$		0			0
$117$		0			0
$118$	−0.500000	−	1.53884i	−0.500000	−	1.53884i
$119$		0			0
$120$		0			0
$121$		0			0
$122$		0			0
$123$	−0.309017	+	0.224514i	−0.309017	+	0.224514i
$124$		0			0
$125$		0			0
$126$		0			0
$127$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$127$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$128$	0.309017	−	0.951057i	0.309017	−	0.951057i
$129$	0.118034	+	0.363271i	0.118034	+	0.363271i
$130$		0			0
$131$	−1.61803			−1.61803			−0.809017	−	0.587785i	$-0.800000\pi$
$131$	−1.61803			−1.61803			−0.809017	+	0.587785i	$0.800000\pi$
$132$		0			0
$133$		0			0
$134$	1.30902	−	0.951057i	1.30902	−	0.951057i
$135$		0			0
$136$	−0.500000	+	1.53884i	−0.500000	+	1.53884i
$137$	−0.500000	−	0.363271i	−0.500000	−	0.363271i	0.309017	−	0.951057i	$-0.400000\pi$
$137$	−0.500000	−	0.363271i	−0.500000	−	0.363271i	−0.809017	+	0.587785i	$0.800000\pi$
$138$		0			0
$139$	0.618034	−	1.90211i	0.618034	−	1.90211i	0.309017	−	0.951057i	$-0.400000\pi$
$139$	0.618034	−	1.90211i	0.618034	−	1.90211i	0.309017	−	0.951057i	$-0.400000\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$	−0.618034			−0.618034
$145$		0			0
$146$	0.190983	+	0.587785i	0.190983	+	0.587785i
$147$	0.190983	−	0.587785i	0.190983	−	0.587785i
$148$		0			0
$149$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$149$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$150$	0.190983	−	0.587785i	0.190983	−	0.587785i
$151$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$151$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$152$	1.30902	−	0.951057i	1.30902	−	0.951057i
$153$	1.00000			1.00000
$154$		0			0
$155$		0			0
$156$		0			0
$157$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$157$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$	1.30902	−	0.951057i	1.30902	−	0.951057i	0.309017	−	0.951057i	$-0.400000\pi$
$163$	1.30902	−	0.951057i	1.30902	−	0.951057i	1.00000			$0$
$164$	0.618034			0.618034
$165$		0			0
$166$	0.618034			0.618034
$167$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$167$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$168$		0			0
$169$	0.309017	−	0.951057i	0.309017	−	0.951057i
$170$		0			0
$171$	−0.809017	−	0.587785i	−0.809017	−	0.587785i
$172$	0.190983	−	0.587785i	0.190983	−	0.587785i
$173$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$173$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$	−1.00000			−1.00000
$178$	−0.500000	+	0.363271i	−0.500000	+	0.363271i
$179$	−0.500000	−	1.53884i	−0.500000	−	1.53884i	−0.809017	−	0.587785i	$-0.800000\pi$
$179$	−0.500000	−	1.53884i	−0.500000	−	1.53884i	0.309017	−	0.951057i	$-0.400000\pi$
$180$		0			0
$181$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$181$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$		0			0
$186$		0			0
$187$		0			0
$188$		0			0
$189$		0			0
$190$		0			0
$191$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$191$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$192$	−0.500000	−	0.363271i	−0.500000	−	0.363271i
$193$	−1.61803	−	1.17557i	−1.61803	−	1.17557i	−0.809017	−	0.587785i	$-0.800000\pi$
$193$	−1.61803	−	1.17557i	−1.61803	−	1.17557i	−0.809017	−	0.587785i	$-0.800000\pi$
$194$	0.190983	−	0.587785i	0.190983	−	0.587785i
$195$		0			0
$196$	−0.809017	+	0.587785i	−0.809017	+	0.587785i
$197$		0			0		1.00000			$0$
$197$		0			0		−1.00000			$\pi$
$198$		0			0
$199$		0			0		1.00000			$0$
$199$		0			0		−1.00000			$\pi$
$200$	−0.809017	+	0.587785i	−0.809017	+	0.587785i
$201$	−0.309017	−	0.951057i	−0.309017	−	0.951057i
$202$		0			0
$203$		0			0
$204$	0.809017	+	0.587785i	0.809017	+	0.587785i
$205$		0			0
$206$		0			0
$207$		0			0
$208$		0			0
$209$		0			0
$210$		0			0
$211$	−0.500000	+	0.363271i	−0.500000	+	0.363271i	−0.809017	−	0.587785i	$-0.800000\pi$
$211$	−0.500000	+	0.363271i	−0.500000	+	0.363271i	0.309017	+	0.951057i	$0.400000\pi$
$212$		0			0
$213$		0			0
$214$	−0.500000	−	0.363271i	−0.500000	−	0.363271i
$215$		0			0
$216$	−0.309017	+	0.951057i	−0.309017	+	0.951057i
$217$		0			0
$218$		0			0
$219$	0.381966			0.381966
$220$		0			0
$221$		0			0
$222$		0			0
$223$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$223$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$224$		0			0
$225$	0.500000	+	0.363271i	0.500000	+	0.363271i
$226$	1.30902	+	0.951057i	1.30902	+	0.951057i
$227$	−0.500000	+	1.53884i	−0.500000	+	1.53884i	0.309017	+	0.951057i	$0.400000\pi$
$227$	−0.500000	+	1.53884i	−0.500000	+	1.53884i	−0.809017	+	0.587785i	$0.800000\pi$
$228$	−0.309017	−	0.951057i	−0.309017	−	0.951057i
$229$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$229$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−0.500000	+	0.363271i	−0.500000	+	0.363271i	−0.809017	−	0.587785i	$-0.800000\pi$
$233$	−0.500000	+	0.363271i	−0.500000	+	0.363271i	0.309017	+	0.951057i	$0.400000\pi$
$234$		0			0
$235$		0			0
$236$	1.30902	+	0.951057i	1.30902	+	0.951057i
$237$		0			0
$238$		0			0
$239$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$239$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$240$		0			0
$241$	−1.61803			−1.61803			−0.809017	−	0.587785i	$-0.800000\pi$
$241$	−1.61803			−1.61803			−0.809017	+	0.587785i	$0.800000\pi$
$242$		0			0
$243$	1.00000			1.00000
$244$		0			0
$245$		0			0
$246$	0.118034	−	0.363271i	0.118034	−	0.363271i
$247$		0			0
$248$		0			0
$249$	0.118034	−	0.363271i	0.118034	−	0.363271i
$250$		0			0
$251$	−1.61803	+	1.17557i	−1.61803	+	1.17557i	−0.809017	+	0.587785i	$0.800000\pi$
$251$	−1.61803	+	1.17557i	−1.61803	+	1.17557i	−0.809017	+	0.587785i	$0.800000\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$		0			0
$256$	0.309017	+	0.951057i	0.309017	+	0.951057i
$257$	0.190983	−	0.587785i	0.190983	−	0.587785i	−0.809017	−	0.587785i	$-0.800000\pi$
$257$	0.190983	−	0.587785i	0.190983	−	0.587785i	1.00000			$0$
$258$	−0.309017	−	0.224514i	−0.309017	−	0.224514i
$259$		0			0
$260$		0			0
$261$		0			0
$262$	1.30902	−	0.951057i	1.30902	−	0.951057i
$263$		0			0		1.00000			$0$
$263$		0			0		−1.00000			$\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$	0.118034	+	0.363271i	0.118034	+	0.363271i
$268$	−0.500000	+	1.53884i	−0.500000	+	1.53884i
$269$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$269$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$270$		0			0
$271$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$271$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$272$	−0.500000	−	1.53884i	−0.500000	−	1.53884i
$273$		0			0
$274$	0.618034			0.618034
$275$		0			0
$276$		0			0
$277$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$277$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$278$	0.618034	+	1.90211i	0.618034	+	1.90211i
$279$		0			0
$280$		0			0
$281$	−0.500000	−	0.363271i	−0.500000	−	0.363271i	0.309017	−	0.951057i	$-0.400000\pi$
$281$	−0.500000	−	0.363271i	−0.500000	−	0.363271i	−0.809017	+	0.587785i	$0.800000\pi$
$282$		0			0
$283$	0.618034	+	1.90211i	0.618034	+	1.90211i	0.309017	+	0.951057i	$0.400000\pi$
$283$	0.618034	+	1.90211i	0.618034	+	1.90211i	0.309017	+	0.951057i	$0.400000\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$	0.500000	−	0.363271i	0.500000	−	0.363271i
$289$	0.500000	+	1.53884i	0.500000	+	1.53884i
$290$		0			0
$291$	−0.309017	−	0.224514i	−0.309017	−	0.224514i
$292$	−0.500000	−	0.363271i	−0.500000	−	0.363271i
$293$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$293$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$294$	0.190983	+	0.587785i	0.190983	+	0.587785i
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$		0			0
$300$	0.190983	+	0.587785i	0.190983	+	0.587785i
$301$		0			0
$302$		0			0
$303$		0			0
$304$	−0.500000	+	1.53884i	−0.500000	+	1.53884i
$305$		0			0
$306$	−0.809017	+	0.587785i	−0.809017	+	0.587785i
$307$	0.618034			0.618034			0.309017	−	0.951057i	$-0.400000\pi$
$307$	0.618034			0.618034			0.309017	+	0.951057i	$0.400000\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$311$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$312$		0			0
$313$	1.30902	+	0.951057i	1.30902	+	0.951057i	1.00000			$0$
$313$	1.30902	+	0.951057i	1.30902	+	0.951057i	0.309017	+	0.951057i	$0.400000\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$317$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$	−0.309017	+	0.224514i	−0.309017	+	0.224514i
$322$		0			0
$323$	0.809017	−	2.48990i	0.809017	−	2.48990i
$324$		0			0
$325$		0			0
$326$	−0.500000	+	1.53884i	−0.500000	+	1.53884i
$327$		0			0
$328$	−0.500000	+	0.363271i	−0.500000	+	0.363271i
$329$		0			0
$330$		0			0
$331$	0.618034			0.618034			0.309017	−	0.951057i	$-0.400000\pi$
$331$	0.618034			0.618034			0.309017	+	0.951057i	$0.400000\pi$
$332$	−0.500000	+	0.363271i	−0.500000	+	0.363271i
$333$		0			0
$334$		0			0
$335$		0			0
$336$		0			0
$337$	−0.500000	+	1.53884i	−0.500000	+	1.53884i	0.309017	+	0.951057i	$0.400000\pi$
$337$	−0.500000	+	1.53884i	−0.500000	+	1.53884i	−0.809017	+	0.587785i	$0.800000\pi$
$338$	0.309017	+	0.951057i	0.309017	+	0.951057i
$339$	0.809017	−	0.587785i	0.809017	−	0.587785i
$340$		0			0
$341$		0			0
$342$	1.00000			1.00000
$343$		0			0
$344$	0.190983	+	0.587785i	0.190983	+	0.587785i
$345$		0			0
$346$		0			0
$347$	1.30902	+	0.951057i	1.30902	+	0.951057i	1.00000			$0$
$347$	1.30902	+	0.951057i	1.30902	+	0.951057i	0.309017	+	0.951057i	$0.400000\pi$
$348$		0			0
$349$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$349$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−1.61803			−1.61803			−0.809017	−	0.587785i	$-0.800000\pi$
$353$	−1.61803			−1.61803			−0.809017	+	0.587785i	$0.800000\pi$
$354$	0.809017	−	0.587785i	0.809017	−	0.587785i
$355$		0			0
$356$	0.190983	−	0.587785i	0.190983	−	0.587785i
$357$		0			0
$358$	1.30902	+	0.951057i	1.30902	+	0.951057i
$359$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$359$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$360$		0			0
$361$	−1.30902	+	0.951057i	−1.30902	+	0.951057i
$362$		0			0
$363$		0			0
$364$		0			0
$365$		0			0
$366$		0			0
$367$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$367$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$368$		0			0
$369$	0.309017	+	0.224514i	0.309017	+	0.224514i
$370$		0			0
$371$		0			0
$372$		0			0
$373$		0			0		1.00000			$0$
$373$		0			0		−1.00000			$\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$	−0.500000	−	0.363271i	−0.500000	−	0.363271i	0.309017	−	0.951057i	$-0.400000\pi$
$379$	−0.500000	−	0.363271i	−0.500000	−	0.363271i	−0.809017	+	0.587785i	$0.800000\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$383$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$384$	0.618034			0.618034
$385$		0			0
$386$	2.00000			2.00000
$387$	0.309017	−	0.224514i	0.309017	−	0.224514i
$388$	0.190983	+	0.587785i	0.190983	+	0.587785i
$389$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$389$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$390$		0			0
$391$		0			0
$392$	0.309017	−	0.951057i	0.309017	−	0.951057i
$393$	−0.309017	−	0.951057i	−0.309017	−	0.951057i
$394$		0			0
$395$		0			0
$396$		0			0
$397$		0			0		1.00000			$0$
$397$		0			0		−1.00000			$\pi$
$398$		0			0
$399$		0			0
$400$	0.309017	−	0.951057i	0.309017	−	0.951057i
$401$	−0.500000	−	0.363271i	−0.500000	−	0.363271i	0.309017	−	0.951057i	$-0.400000\pi$
$401$	−0.500000	−	0.363271i	−0.500000	−	0.363271i	−0.809017	+	0.587785i	$0.800000\pi$
$402$	0.809017	+	0.587785i	0.809017	+	0.587785i
$403$		0			0
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$	−1.00000			−1.00000
$409$	−1.61803	+	1.17557i	−1.61803	+	1.17557i	−0.809017	+	0.587785i	$0.800000\pi$
$409$	−1.61803	+	1.17557i	−1.61803	+	1.17557i	−0.809017	+	0.587785i	$0.800000\pi$
$410$		0			0
$411$	0.118034	−	0.363271i	0.118034	−	0.363271i
$412$		0			0
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$	1.23607			1.23607
$418$		0			0
$419$	0.618034			0.618034			0.309017	−	0.951057i	$-0.400000\pi$
$419$	0.618034			0.618034			0.309017	+	0.951057i	$0.400000\pi$
$420$		0			0
$421$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$421$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$422$	0.190983	−	0.587785i	0.190983	−	0.587785i
$423$		0			0
$424$		0			0
$425$	−0.500000	+	1.53884i	−0.500000	+	1.53884i
$426$		0			0
$427$		0			0
$428$	0.618034			0.618034
$429$		0			0
$430$		0			0
$431$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$431$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$432$	−0.309017	−	0.951057i	−0.309017	−	0.951057i
$433$	−0.500000	+	1.53884i	−0.500000	+	1.53884i	0.309017	+	0.951057i	$0.400000\pi$
$433$	−0.500000	+	1.53884i	−0.500000	+	1.53884i	−0.809017	+	0.587785i	$0.800000\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		0			0
$438$	−0.309017	+	0.224514i	−0.309017	+	0.224514i
$439$		0			0		1.00000			$0$
$439$		0			0		−1.00000			$\pi$
$440$		0			0
$441$	−0.618034			−0.618034
$442$		0			0
$443$	0.190983	+	0.587785i	0.190983	+	0.587785i	1.00000			$0$
$443$	0.190983	+	0.587785i	0.190983	+	0.587785i	−0.809017	+	0.587785i	$0.800000\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$		0			0
$449$	1.30902	−	0.951057i	1.30902	−	0.951057i	0.309017	−	0.951057i	$-0.400000\pi$
$449$	1.30902	−	0.951057i	1.30902	−	0.951057i	1.00000			$0$
$450$	−0.618034			−0.618034
$451$		0			0
$452$	−1.61803			−1.61803
$453$		0			0
$454$	−0.500000	−	1.53884i	−0.500000	−	1.53884i
$455$		0			0
$456$	0.809017	+	0.587785i	0.809017	+	0.587785i
$457$	−0.500000	−	0.363271i	−0.500000	−	0.363271i	0.309017	−	0.951057i	$-0.400000\pi$
$457$	−0.500000	−	0.363271i	−0.500000	−	0.363271i	−0.809017	+	0.587785i	$0.800000\pi$
$458$		0			0
$459$	0.500000	+	1.53884i	0.500000	+	1.53884i
$460$		0			0
$461$		0			0		1.00000			$0$
$461$		0			0		−1.00000			$\pi$
$462$		0			0
$463$		0			0		1.00000			$0$
$463$		0			0		−1.00000			$\pi$
$464$		0			0
$465$		0			0
$466$	0.190983	−	0.587785i	0.190983	−	0.587785i
$467$	−1.61803	−	1.17557i	−1.61803	−	1.17557i	−0.809017	−	0.587785i	$-0.800000\pi$
$467$	−1.61803	−	1.17557i	−1.61803	−	1.17557i	−0.809017	−	0.587785i	$-0.800000\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$		0			0
$472$	−1.61803			−1.61803
$473$		0			0
$474$		0			0
$475$	1.30902	−	0.951057i	1.30902	−	0.951057i
$476$		0			0
$477$		0			0
$478$		0			0
$479$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$479$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$480$		0			0
$481$		0			0
$482$	1.30902	−	0.951057i	1.30902	−	0.951057i
$483$		0			0
$484$		0			0
$485$		0			0
$486$	−0.809017	+	0.587785i	−0.809017	+	0.587785i
$487$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$487$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$488$		0			0
$489$	0.809017	+	0.587785i	0.809017	+	0.587785i
$490$		0			0
$491$	0.190983	−	0.587785i	0.190983	−	0.587785i	−0.809017	−	0.587785i	$-0.800000\pi$
$491$	0.190983	−	0.587785i	0.190983	−	0.587785i	1.00000			$0$
$492$	0.118034	+	0.363271i	0.118034	+	0.363271i
$493$		0			0
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$	0.118034	+	0.363271i	0.118034	+	0.363271i
$499$	0.190983	−	0.587785i	0.190983	−	0.587785i	−0.809017	−	0.587785i	$-0.800000\pi$
$499$	0.190983	−	0.587785i	0.190983	−	0.587785i	1.00000			$0$
$500$		0			0
$501$		0			0
$502$	0.618034	−	1.90211i	0.618034	−	1.90211i
$503$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$503$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	0.618034			0.618034
$508$		0			0
$509$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$509$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$510$		0			0
$511$		0			0
$512$	−0.809017	−	0.587785i	−0.809017	−	0.587785i
$513$	0.500000	−	1.53884i	0.500000	−	1.53884i
$514$	0.190983	+	0.587785i	0.190983	+	0.587785i
$515$		0			0
$516$	0.381966			0.381966
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−0.500000	+	1.53884i	−0.500000	+	1.53884i	0.309017	+	0.951057i	$0.400000\pi$
$521$	−0.500000	+	1.53884i	−0.500000	+	1.53884i	−0.809017	+	0.587785i	$0.800000\pi$
$522$		0			0
$523$	1.30902	+	0.951057i	1.30902	+	0.951057i	1.00000			$0$
$523$	1.30902	+	0.951057i	1.30902	+	0.951057i	0.309017	+	0.951057i	$0.400000\pi$
$524$	−0.500000	+	1.53884i	−0.500000	+	1.53884i
$525$		0			0
$526$		0			0
$527$		0			0
$528$		0			0
$529$	1.00000			1.00000
$530$		0			0
$531$	0.309017	+	0.951057i	0.309017	+	0.951057i
$532$		0			0
$533$		0			0
$534$	−0.309017	−	0.224514i	−0.309017	−	0.224514i
$535$		0			0
$536$	−0.500000	−	1.53884i	−0.500000	−	1.53884i
$537$	0.809017	−	0.587785i	0.809017	−	0.587785i
$538$		0			0
$539$		0			0
$540$		0			0
$541$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$541$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$542$		0			0
$543$		0			0
$544$	1.30902	+	0.951057i	1.30902	+	0.951057i
$545$		0			0
$546$		0			0
$547$	−0.500000	−	1.53884i	−0.500000	−	1.53884i	−0.809017	−	0.587785i	$-0.800000\pi$
$547$	−0.500000	−	1.53884i	−0.500000	−	1.53884i	0.309017	−	0.951057i	$-0.400000\pi$
$548$	−0.500000	+	0.363271i	−0.500000	+	0.363271i
$549$		0			0
$550$		0			0
$551$		0			0
$552$		0			0
$553$		0			0
$554$		0			0
$555$		0			0
$556$	−1.61803	−	1.17557i	−1.61803	−	1.17557i
$557$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$557$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$		0			0
$562$	0.618034			0.618034
$563$	−0.500000	+	0.363271i	−0.500000	+	0.363271i	−0.809017	−	0.587785i	$-0.800000\pi$
$563$	−0.500000	+	0.363271i	−0.500000	+	0.363271i	0.309017	+	0.951057i	$0.400000\pi$
$564$		0			0
$565$		0			0
$566$	−1.61803	−	1.17557i	−1.61803	−	1.17557i
$567$		0			0
$568$		0			0
$569$	−0.500000	−	1.53884i	−0.500000	−	1.53884i	−0.809017	−	0.587785i	$-0.800000\pi$
$569$	−0.500000	−	1.53884i	−0.500000	−	1.53884i	0.309017	−	0.951057i	$-0.400000\pi$
$570$		0			0
$571$	2.00000			2.00000			1.00000			$0$
$571$	2.00000			2.00000			1.00000			$0$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$	−0.190983	+	0.587785i	−0.190983	+	0.587785i
$577$	1.30902	+	0.951057i	1.30902	+	0.951057i	1.00000			$0$
$577$	1.30902	+	0.951057i	1.30902	+	0.951057i	0.309017	+	0.951057i	$0.400000\pi$
$578$	−1.30902	−	0.951057i	−1.30902	−	0.951057i
$579$	0.381966	−	1.17557i	0.381966	−	1.17557i
$580$		0			0
$581$		0			0
$582$	0.381966			0.381966
$583$		0			0
$584$	0.618034			0.618034
$585$		0			0
$586$		0			0
$587$	−0.500000	+	1.53884i	−0.500000	+	1.53884i	0.309017	+	0.951057i	$0.400000\pi$
$587$	−0.500000	+	1.53884i	−0.500000	+	1.53884i	−0.809017	+	0.587785i	$0.800000\pi$
$588$	−0.500000	−	0.363271i	−0.500000	−	0.363271i
$589$		0			0
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−1.61803			−1.61803			−0.809017	−	0.587785i	$-0.800000\pi$
$593$	−1.61803			−1.61803			−0.809017	+	0.587785i	$0.800000\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$599$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$600$	−0.500000	−	0.363271i	−0.500000	−	0.363271i
$601$	0.190983	−	0.587785i	0.190983	−	0.587785i	−0.809017	−	0.587785i	$-0.800000\pi$
$601$	0.190983	−	0.587785i	0.190983	−	0.587785i	1.00000			$0$
$602$		0			0
$603$	−0.809017	+	0.587785i	−0.809017	+	0.587785i
$604$		0			0
$605$		0			0
$606$		0			0
$607$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$607$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$608$	−0.500000	−	1.53884i	−0.500000	−	1.53884i
$609$		0			0
$610$		0			0
$611$		0			0
$612$	0.309017	−	0.951057i	0.309017	−	0.951057i
$613$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$613$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$614$	−0.500000	+	0.363271i	−0.500000	+	0.363271i
$615$		0			0
$616$		0			0
$617$	−1.61803			−1.61803			−0.809017	−	0.587785i	$-0.800000\pi$
$617$	−1.61803			−1.61803			−0.809017	+	0.587785i	$0.800000\pi$
$618$		0			0
$619$	−0.500000	−	1.53884i	−0.500000	−	1.53884i	−0.809017	−	0.587785i	$-0.800000\pi$
$619$	−0.500000	−	1.53884i	−0.500000	−	1.53884i	0.309017	−	0.951057i	$-0.400000\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	−0.809017	+	0.587785i	−0.809017	+	0.587785i
$626$	−1.61803			−1.61803
$627$		0			0
$628$		0			0
$629$		0			0
$630$		0			0
$631$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$631$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$632$		0			0
$633$	−0.309017	−	0.224514i	−0.309017	−	0.224514i
$634$		0			0
$635$		0			0
$636$		0			0
$637$		0			0
$638$		0			0
$639$		0			0
$640$		0			0
$641$	0.190983	+	0.587785i	0.190983	+	0.587785i	1.00000			$0$
$641$	0.190983	+	0.587785i	0.190983	+	0.587785i	−0.809017	+	0.587785i	$0.800000\pi$
$642$	0.118034	−	0.363271i	0.118034	−	0.363271i
$643$	−0.500000	−	0.363271i	−0.500000	−	0.363271i	0.309017	−	0.951057i	$-0.400000\pi$
$643$	−0.500000	−	0.363271i	−0.500000	−	0.363271i	−0.809017	+	0.587785i	$0.800000\pi$
$644$		0			0
$645$		0			0
$646$	0.809017	+	2.48990i	0.809017	+	2.48990i
$647$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$647$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$		0			0
$652$	−0.500000	−	1.53884i	−0.500000	−	1.53884i
$653$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$653$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$654$		0			0
$655$		0			0
$656$	0.190983	−	0.587785i	0.190983	−	0.587785i
$657$	−0.118034	−	0.363271i	−0.118034	−	0.363271i
$658$		0			0
$659$	−1.61803			−1.61803			−0.809017	−	0.587785i	$-0.800000\pi$
$659$	−1.61803			−1.61803			−0.809017	+	0.587785i	$0.800000\pi$
$660$		0			0
$661$		0			0		1.00000			$0$
$661$		0			0		−1.00000			$\pi$
$662$	−0.500000	+	0.363271i	−0.500000	+	0.363271i
$663$		0			0
$664$	0.190983	−	0.587785i	0.190983	−	0.587785i
$665$		0			0
$666$		0			0
$667$		0			0
$668$		0			0
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$	1.30902	−	0.951057i	1.30902	−	0.951057i	0.309017	−	0.951057i	$-0.400000\pi$
$673$	1.30902	−	0.951057i	1.30902	−	0.951057i	1.00000			$0$
$674$	−0.500000	−	1.53884i	−0.500000	−	1.53884i
$675$	−0.309017	+	0.951057i	−0.309017	+	0.951057i
$676$	−0.809017	−	0.587785i	−0.809017	−	0.587785i
$677$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$677$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$678$	−0.309017	+	0.951057i	−0.309017	+	0.951057i
$679$		0			0
$680$		0			0
$681$	−1.00000			−1.00000
$682$		0			0
$683$	2.00000			2.00000			1.00000			$0$
$683$	2.00000			2.00000			1.00000			$0$
$684$	−0.809017	+	0.587785i	−0.809017	+	0.587785i
$685$		0			0
$686$		0			0
$687$		0			0
$688$	−0.500000	−	0.363271i	−0.500000	−	0.363271i
$689$		0			0
$690$		0			0
$691$	1.30902	−	0.951057i	1.30902	−	0.951057i	0.309017	−	0.951057i	$-0.400000\pi$
$691$	1.30902	−	0.951057i	1.30902	−	0.951057i	1.00000			$0$
$692$		0			0
$693$		0			0
$694$	−1.61803			−1.61803
$695$		0			0
$696$		0			0
$697$	−0.309017	+	0.951057i	−0.309017	+	0.951057i
$698$		0			0
$699$	−0.309017	−	0.224514i	−0.309017	−	0.224514i
$700$		0			0
$701$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$701$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$702$		0			0
$703$		0			0
$704$		0			0
$705$		0			0
$706$	1.30902	−	0.951057i	1.30902	−	0.951057i
$707$		0			0
$708$	−0.309017	+	0.951057i	−0.309017	+	0.951057i
$709$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$709$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$710$		0			0
$711$		0			0
$712$	0.190983	+	0.587785i	0.190983	+	0.587785i
$713$		0			0
$714$		0			0
$715$		0			0
$716$	−1.61803			−1.61803
$717$		0			0
$718$		0			0
$719$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$719$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$720$		0			0
$721$		0			0
$722$	0.500000	−	1.53884i	0.500000	−	1.53884i
$723$	−0.309017	−	0.951057i	−0.309017	−	0.951057i
$724$		0			0
$725$		0			0
$726$		0			0
$727$		0			0		1.00000			$0$
$727$		0			0		−1.00000			$\pi$
$728$		0			0
$729$	0.190983	+	0.587785i	0.190983	+	0.587785i
$730$		0			0
$731$	0.809017	+	0.587785i	0.809017	+	0.587785i
$732$		0			0
$733$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$733$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$	−0.381966			−0.381966
$739$	1.30902	−	0.951057i	1.30902	−	0.951057i	0.309017	−	0.951057i	$-0.400000\pi$
$739$	1.30902	−	0.951057i	1.30902	−	0.951057i	1.00000			$0$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$743$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$	−0.381966			−0.381966
$748$		0			0
$749$		0			0
$750$		0			0
$751$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$751$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$752$		0			0
$753$	−1.00000	−	0.726543i	−1.00000	−	0.726543i
$754$		0			0
$755$		0			0
$756$		0			0
$757$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$757$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$758$	0.618034			0.618034
$759$		0			0
$760$		0			0
$761$	1.30902	−	0.951057i	1.30902	−	0.951057i	0.309017	−	0.951057i	$-0.400000\pi$
$761$	1.30902	−	0.951057i	1.30902	−	0.951057i	1.00000			$0$
$762$		0			0
$763$		0			0
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$	−0.500000	+	0.363271i	−0.500000	+	0.363271i
$769$	2.00000			2.00000			1.00000			$0$
$769$	2.00000			2.00000			1.00000			$0$
$770$		0			0
$771$	0.381966			0.381966
$772$	−1.61803	+	1.17557i	−1.61803	+	1.17557i
$773$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$773$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$774$	−0.118034	+	0.363271i	−0.118034	+	0.363271i
$775$		0			0
$776$	−0.500000	−	0.363271i	−0.500000	−	0.363271i
$777$		0			0
$778$		0			0
$779$	0.809017	−	0.587785i	0.809017	−	0.587785i
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$	0.309017	+	0.951057i	0.309017	+	0.951057i
$785$		0			0
$786$	0.809017	+	0.587785i	0.809017	+	0.587785i
$787$	1.30902	+	0.951057i	1.30902	+	0.951057i	1.00000			$0$
$787$	1.30902	+	0.951057i	1.30902	+	0.951057i	0.309017	+	0.951057i	$0.400000\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$		0			0
$794$		0			0
$795$		0			0
$796$		0			0
$797$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$797$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$798$		0			0
$799$		0			0
$800$	0.309017	+	0.951057i	0.309017	+	0.951057i
$801$	0.309017	−	0.224514i	0.309017	−	0.224514i
$802$	0.618034			0.618034
$803$		0			0
$804$	−1.00000			−1.00000
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$	1.30902	+	0.951057i	1.30902	+	0.951057i	1.00000			$0$
$809$	1.30902	+	0.951057i	1.30902	+	0.951057i	0.309017	+	0.951057i	$0.400000\pi$
$810$		0			0
$811$	0.190983	+	0.587785i	0.190983	+	0.587785i	1.00000			$0$
$811$	0.190983	+	0.587785i	0.190983	+	0.587785i	−0.809017	+	0.587785i	$0.800000\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		0			0
$816$	0.809017	−	0.587785i	0.809017	−	0.587785i
$817$	−0.309017	−	0.951057i	−0.309017	−	0.951057i
$818$	0.618034	−	1.90211i	0.618034	−	1.90211i
$819$		0			0
$820$		0			0
$821$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$821$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$822$	0.118034	+	0.363271i	0.118034	+	0.363271i
$823$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$823$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	1.30902	−	0.951057i	1.30902	−	0.951057i	0.309017	−	0.951057i	$-0.400000\pi$
$827$	1.30902	−	0.951057i	1.30902	−	0.951057i	1.00000			$0$
$828$		0			0
$829$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$829$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−0.500000	−	1.53884i	−0.500000	−	1.53884i
$834$	−1.00000	+	0.726543i	−1.00000	+	0.726543i
$835$		0			0
$836$		0			0
$837$		0			0
$838$	−0.500000	+	0.363271i	−0.500000	+	0.363271i
$839$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$839$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$840$		0			0
$841$	−0.809017	−	0.587785i	−0.809017	−	0.587785i
$842$		0			0
$843$	0.118034	−	0.363271i	0.118034	−	0.363271i
$844$	0.190983	+	0.587785i	0.190983	+	0.587785i
$845$		0			0
$846$		0			0
$847$		0			0
$848$		0			0
$849$	−1.00000	+	0.726543i	−1.00000	+	0.726543i
$850$	−0.500000	−	1.53884i	−0.500000	−	1.53884i
$851$		0			0
$852$		0			0
$853$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$853$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$854$		0			0
$855$		0			0
$856$	−0.500000	+	0.363271i	−0.500000	+	0.363271i
$857$	0.618034			0.618034			0.309017	−	0.951057i	$-0.400000\pi$
$857$	0.618034			0.618034			0.309017	+	0.951057i	$0.400000\pi$
$858$		0			0
$859$	−1.61803			−1.61803			−0.809017	−	0.587785i	$-0.800000\pi$
$859$	−1.61803			−1.61803			−0.809017	+	0.587785i	$0.800000\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$863$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$864$	0.809017	+	0.587785i	0.809017	+	0.587785i
$865$		0			0
$866$	−0.500000	−	1.53884i	−0.500000	−	1.53884i
$867$	−0.809017	+	0.587785i	−0.809017	+	0.587785i
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$		0			0
$873$	−0.118034	+	0.363271i	−0.118034	+	0.363271i
$874$		0			0
$875$		0			0
$876$	0.118034	−	0.363271i	0.118034	−	0.363271i
$877$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$877$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	0.618034			0.618034			0.309017	−	0.951057i	$-0.400000\pi$
$881$	0.618034			0.618034			0.309017	+	0.951057i	$0.400000\pi$
$882$	0.500000	−	0.363271i	0.500000	−	0.363271i
$883$	0.190983	+	0.587785i	0.190983	+	0.587785i	1.00000			$0$
$883$	0.190983	+	0.587785i	0.190983	+	0.587785i	−0.809017	+	0.587785i	$0.800000\pi$
$884$		0			0
$885$		0			0
$886$	−0.500000	−	0.363271i	−0.500000	−	0.363271i
$887$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$887$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$		0			0
$892$		0			0
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$	−0.500000	+	1.53884i	−0.500000	+	1.53884i
$899$		0			0
$900$	0.500000	−	0.363271i	0.500000	−	0.363271i
$901$		0			0
$902$		0			0
$903$		0			0
$904$	1.30902	−	0.951057i	1.30902	−	0.951057i
$905$		0			0
$906$		0			0
$907$	1.30902	+	0.951057i	1.30902	+	0.951057i	1.00000			$0$
$907$	1.30902	+	0.951057i	1.30902	+	0.951057i	0.309017	+	0.951057i	$0.400000\pi$
$908$	1.30902	+	0.951057i	1.30902	+	0.951057i
$909$		0			0
$910$		0			0
$911$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$911$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$912$	−1.00000			−1.00000
$913$		0			0
$914$	0.618034			0.618034
$915$		0			0
$916$		0			0
$917$		0			0
$918$	−1.30902	−	0.951057i	−1.30902	−	0.951057i
$919$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$919$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$920$		0			0
$921$	0.118034	+	0.363271i	0.118034	+	0.363271i
$922$		0			0
$923$		0			0
$924$		0			0
$925$		0			0
$926$		0			0
$927$		0			0
$928$		0			0
$929$	1.30902	+	0.951057i	1.30902	+	0.951057i	1.00000			$0$
$929$	1.30902	+	0.951057i	1.30902	+	0.951057i	0.309017	+	0.951057i	$0.400000\pi$
$930$		0			0
$931$	−0.500000	+	1.53884i	−0.500000	+	1.53884i
$932$	0.190983	+	0.587785i	0.190983	+	0.587785i
$933$		0			0
$934$	2.00000			2.00000
$935$		0			0
$936$		0			0
$937$	1.30902	−	0.951057i	1.30902	−	0.951057i	0.309017	−	0.951057i	$-0.400000\pi$
$937$	1.30902	−	0.951057i	1.30902	−	0.951057i	1.00000			$0$
$938$		0			0
$939$	−0.309017	+	0.951057i	−0.309017	+	0.951057i
$940$		0			0
$941$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$941$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$942$		0			0
$943$		0			0
$944$	1.30902	−	0.951057i	1.30902	−	0.951057i
$945$		0			0
$946$		0			0
$947$	−1.61803			−1.61803			−0.809017	−	0.587785i	$-0.800000\pi$
$947$	−1.61803			−1.61803			−0.809017	+	0.587785i	$0.800000\pi$
$948$		0			0
$949$		0			0
$950$	−0.500000	+	1.53884i	−0.500000	+	1.53884i
$951$		0			0
$952$		0			0
$953$	0.190983	−	0.587785i	0.190983	−	0.587785i	−0.809017	−	0.587785i	$-0.800000\pi$
$953$	0.190983	−	0.587785i	0.190983	−	0.587785i	1.00000			$0$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	0.309017	−	0.951057i	0.309017	−	0.951057i
$962$		0			0
$963$	0.309017	+	0.224514i	0.309017	+	0.224514i
$964$	−0.500000	+	1.53884i	−0.500000	+	1.53884i
$965$		0			0
$966$		0			0
$967$		0			0		1.00000			$0$
$967$		0			0		−1.00000			$\pi$
$968$		0			0
$969$	1.61803			1.61803
$970$		0			0
$971$	0.618034	+	1.90211i	0.618034	+	1.90211i	0.309017	+	0.951057i	$0.400000\pi$
$971$	0.618034	+	1.90211i	0.618034	+	1.90211i	0.309017	+	0.951057i	$0.400000\pi$
$972$	0.309017	−	0.951057i	0.309017	−	0.951057i
$973$		0			0
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−1.61803	+	1.17557i	−1.61803	+	1.17557i	−0.809017	+	0.587785i	$0.800000\pi$
$977$	−1.61803	+	1.17557i	−1.61803	+	1.17557i	−0.809017	+	0.587785i	$0.800000\pi$
$978$	−1.00000			−1.00000
$979$		0			0
$980$		0			0
$981$		0			0
$982$	0.190983	+	0.587785i	0.190983	+	0.587785i
$983$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$983$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$984$	−0.309017	−	0.224514i	−0.309017	−	0.224514i
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$		0			0		1.00000			$0$
$991$		0			0		−1.00000			$\pi$
$992$		0			0
$993$	0.118034	+	0.363271i	0.118034	+	0.363271i
$994$		0			0
$995$		0			0
$996$	−0.309017	−	0.224514i	−0.309017	−	0.224514i
$997$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$997$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$998$	0.190983	+	0.587785i	0.190983	+	0.587785i
$999$		0			0

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

)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	968.1.l.a.3.1		4
4.3	odd	2		3872.1.t.b.1455.1		4
8.3	odd	2	CM	968.1.l.a.3.1		4
8.5	even	2		3872.1.t.b.1455.1		4
11.2	odd	10		968.1.f.a.243.2		2
11.3	even	5		88.1.l.a.27.1	✓	4
11.4	even	5	inner	968.1.l.a.323.1		4
11.5	even	5		88.1.l.a.75.1	yes	4
11.6	odd	10		968.1.l.b.251.1		4
11.7	odd	10		968.1.l.c.323.1		4
11.8	odd	10		968.1.l.b.27.1		4
11.9	even	5		968.1.f.b.243.2		2
11.10	odd	2		968.1.l.c.3.1		4
33.5	odd	10		792.1.bu.a.163.1		4
33.14	odd	10		792.1.bu.a.379.1		4
44.3	odd	10		352.1.t.a.335.1		4
44.7	even	10		3872.1.t.a.1775.1		4
44.15	odd	10		3872.1.t.b.1775.1		4
44.19	even	10		3872.1.t.c.2447.1		4
44.27	odd	10		352.1.t.a.207.1		4
44.31	odd	10		3872.1.f.a.3631.1		2
44.35	even	10		3872.1.f.b.3631.1		2
44.39	even	10		3872.1.t.c.2671.1		4
44.43	even	2		3872.1.t.a.1455.1		4
55.3	odd	20		2200.1.dd.a.1699.2		8
55.14	even	10		2200.1.cl.a.2051.1		4
55.27	odd	20		2200.1.dd.a.2099.2		8
55.38	odd	20		2200.1.dd.a.2099.1		8
55.47	odd	20		2200.1.dd.a.1699.1		8
55.49	even	10		2200.1.cl.a.251.1		4
88.3	odd	10		88.1.l.a.27.1	✓	4
88.5	even	10		352.1.t.a.207.1		4
88.13	odd	10		3872.1.f.b.3631.1		2
88.19	even	10		968.1.l.b.27.1		4
88.21	odd	2		3872.1.t.a.1455.1		4
88.27	odd	10		88.1.l.a.75.1	yes	4
88.29	odd	10		3872.1.t.a.1775.1		4
88.35	even	10		968.1.f.a.243.2		2
88.37	even	10		3872.1.t.b.1775.1		4
88.43	even	2		968.1.l.c.3.1		4
88.51	even	10		968.1.l.c.323.1		4
88.53	even	10		3872.1.f.a.3631.1		2
88.59	odd	10	inner	968.1.l.a.323.1		4
88.61	odd	10		3872.1.t.c.2671.1		4
88.69	even	10		352.1.t.a.335.1		4
88.75	odd	10		968.1.f.b.243.2		2
88.83	even	10		968.1.l.b.251.1		4
88.85	odd	10		3872.1.t.c.2447.1		4
132.47	even	10		3168.1.ck.a.3151.1		4
132.71	even	10		3168.1.ck.a.559.1		4
176.3	odd	20		2816.1.v.c.511.1		8
176.5	even	20		2816.1.v.c.1791.2		8
176.27	odd	20		2816.1.v.c.1791.1		8
176.69	even	20		2816.1.v.c.511.1		8
176.91	odd	20		2816.1.v.c.511.2		8
176.93	even	20		2816.1.v.c.1791.1		8
176.115	odd	20		2816.1.v.c.1791.2		8
176.157	even	20		2816.1.v.c.511.2		8
264.5	odd	10		3168.1.ck.a.559.1		4
264.179	even	10		792.1.bu.a.379.1		4
264.203	even	10		792.1.bu.a.163.1		4
264.245	odd	10		3168.1.ck.a.3151.1		4
440.3	even	20		2200.1.dd.a.1699.2		8
440.27	even	20		2200.1.dd.a.2099.2		8
440.179	odd	10		2200.1.cl.a.2051.1		4
440.203	even	20		2200.1.dd.a.2099.1		8
440.267	even	20		2200.1.dd.a.1699.1		8
440.379	odd	10		2200.1.cl.a.251.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
88.1.l.a.27.1	✓	4	11.3	even	5
88.1.l.a.27.1	✓	4	88.3	odd	10
88.1.l.a.75.1	yes	4	11.5	even	5
88.1.l.a.75.1	yes	4	88.27	odd	10
352.1.t.a.207.1		4	44.27	odd	10
352.1.t.a.207.1		4	88.5	even	10
352.1.t.a.335.1		4	44.3	odd	10
352.1.t.a.335.1		4	88.69	even	10
792.1.bu.a.163.1		4	33.5	odd	10
792.1.bu.a.163.1		4	264.203	even	10
792.1.bu.a.379.1		4	33.14	odd	10
792.1.bu.a.379.1		4	264.179	even	10
968.1.f.a.243.2		2	11.2	odd	10
968.1.f.a.243.2		2	88.35	even	10
968.1.f.b.243.2		2	11.9	even	5
968.1.f.b.243.2		2	88.75	odd	10
968.1.l.a.3.1		4	1.1	even	1	trivial
968.1.l.a.3.1		4	8.3	odd	2	CM
968.1.l.a.323.1		4	11.4	even	5	inner
968.1.l.a.323.1		4	88.59	odd	10	inner
968.1.l.b.27.1		4	11.8	odd	10
968.1.l.b.27.1		4	88.19	even	10
968.1.l.b.251.1		4	11.6	odd	10
968.1.l.b.251.1		4	88.83	even	10
968.1.l.c.3.1		4	11.10	odd	2
968.1.l.c.3.1		4	88.43	even	2
968.1.l.c.323.1		4	11.7	odd	10
968.1.l.c.323.1		4	88.51	even	10
2200.1.cl.a.251.1		4	55.49	even	10
2200.1.cl.a.251.1		4	440.379	odd	10
2200.1.cl.a.2051.1		4	55.14	even	10
2200.1.cl.a.2051.1		4	440.179	odd	10
2200.1.dd.a.1699.1		8	55.47	odd	20
2200.1.dd.a.1699.1		8	440.267	even	20
2200.1.dd.a.1699.2		8	55.3	odd	20
2200.1.dd.a.1699.2		8	440.3	even	20
2200.1.dd.a.2099.1		8	55.38	odd	20
2200.1.dd.a.2099.1		8	440.203	even	20
2200.1.dd.a.2099.2		8	55.27	odd	20
2200.1.dd.a.2099.2		8	440.27	even	20
2816.1.v.c.511.1		8	176.3	odd	20
2816.1.v.c.511.1		8	176.69	even	20
2816.1.v.c.511.2		8	176.91	odd	20
2816.1.v.c.511.2		8	176.157	even	20
2816.1.v.c.1791.1		8	176.27	odd	20
2816.1.v.c.1791.1		8	176.93	even	20
2816.1.v.c.1791.2		8	176.5	even	20
2816.1.v.c.1791.2		8	176.115	odd	20
3168.1.ck.a.559.1		4	132.71	even	10
3168.1.ck.a.559.1		4	264.5	odd	10
3168.1.ck.a.3151.1		4	132.47	even	10
3168.1.ck.a.3151.1		4	264.245	odd	10
3872.1.f.a.3631.1		2	44.31	odd	10
3872.1.f.a.3631.1		2	88.53	even	10
3872.1.f.b.3631.1		2	44.35	even	10
3872.1.f.b.3631.1		2	88.13	odd	10
3872.1.t.a.1455.1		4	44.43	even	2
3872.1.t.a.1455.1		4	88.21	odd	2
3872.1.t.a.1775.1		4	44.7	even	10
3872.1.t.a.1775.1		4	88.29	odd	10
3872.1.t.b.1455.1		4	4.3	odd	2
3872.1.t.b.1455.1		4	8.5	even	2
3872.1.t.b.1775.1		4	44.15	odd	10
3872.1.t.b.1775.1		4	88.37	even	10
3872.1.t.c.2447.1		4	44.19	even	10
3872.1.t.c.2447.1		4	88.85	odd	10
3872.1.t.c.2671.1		4	44.39	even	10
3872.1.t.c.2671.1		4	88.61	odd	10

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 $\F_{q}$

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more

Newspace parameters

Newform invariants

Embedding invariants

$q$ -expansion

Character values

Coefficient data

Twists

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	968.1.l.a.3.1

Level	$968$
Weight	$1$
Character	968.3
Analytic conductor	$0.483$
Analytic rank	$0$
Dimension	$4$
Projective image	$D_{5}$
CM discriminant	-8
Inner twists	$4$