Newform orbit 342.2.j.e

• Label: 342.2.j.e
• Level: $342$
• Weight: $2$
• Character orbit: 342.j
• Analytic conductor: $2.731$
• Analytic rank: $0$
• Dimension: $8$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 342 = 2 \cdot 3^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	342.j (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(2.73088374913\)
Analytic rank:	\(0\)
Dimension:	\(8\)
Relative dimension:	\(4\) over \(\Q(\zeta_{6})\)
Coefficient field:	8.0.152695449.1
Defining polynomial:	\( x^{8} - 2x^{7} + 3x^{5} - 5x^{4} + 6x^{3} - 16x + 16 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index:	\( 3 \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\)	\(=\)	q + (b6 + 1) * q^2 + (b6 + b5 + b4 - b1 + 1) * q^3 + b6 * q^4 + (3*b6 + b5 + b4 - b3 - 2*b1 + 1) * q^5 + (2*b6 + b5 + b4 - b3 - b1 + 1) * q^6 + (-b7 - 2*b6 + b3 - b2 + b1) * q^7 - q^8 + (b7 + b6 + 2*b5 + b4 - b3 + b2 - 2*b1 - 1) * q^9
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 4 q^{2} + 5 q^{3} - 4 q^{4} + q^{6} + 6 q^{7} - 8 q^{8} - 7 q^{9}+O(q^{10}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 2x^{7} + 3x^{5} - 5x^{4} + 6x^{3} - 16x + 16 \) :

\(\beta_{1}\)	\(=\)	\( ( -\nu^{6} - 2\nu^{5} + \nu^{3} - 3\nu^{2} + 10\nu - 8 ) / 12 \)
\(\beta_{2}\)	\(=\)	\( ( \nu^{7} - 4\nu^{6} + 11\nu^{4} - 3\nu^{3} - 4\nu^{2} - 16\nu - 24 ) / 24 \)
\(\beta_{3}\)	\(=\)	\( ( -\nu^{6} + 4\nu^{5} + \nu^{3} + 3\nu^{2} - 8\nu + 4 ) / 12 \)
\(\beta_{4}\)	\(=\)	\( ( \nu^{7} - \nu^{6} - \nu^{4} - 6\nu^{3} - \nu^{2} + 8\nu - 12 ) / 12 \)
\(\beta_{5}\)	\(=\)	\( ( -\nu^{7} + \nu^{6} + \nu^{4} + 6\nu^{3} - 11\nu^{2} + 4\nu + 12 ) / 12 \)
\(\beta_{6}\)	\(=\)	\( ( 5\nu^{7} - 4\nu^{6} - 4\nu^{5} + 7\nu^{4} - 7\nu^{3} + 16\nu^{2} + 12\nu - 64 ) / 24 \)
\(\beta_{7}\)	\(=\)	\( ( 4\nu^{7} - 3\nu^{6} - 4\nu^{5} + 8\nu^{4} - 13\nu^{3} + 17\nu^{2} + 28\nu - 52 ) / 12 \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/342\mathbb{Z}\right)^\times\).

\(n\)	\(191\)	\(325\)
\(\chi(n)\)	\(1 + \beta_{6}\)	\(1 + \beta_{6}\)

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

Label  

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

65.1

−1.37475	+	0.331768i
1.05924	+	0.937022i
1.40702	+	0.142460i
−0.0915132	−	1.41125i
−1.37475	−	0.331768i
1.05924	−	0.937022i
1.40702	−	0.142460i
−0.0915132	+	1.41125i

0.500000

+

0.866025i

−0.474693

+

1.66573i

−0.500000

+

0.866025i

0.843540i

−1.67991

+

0.421770i

−0.874747

+

1.51511i

−1.00000

−2.54933

−

1.58142i

−0.730527

+

0.421770i

65.2

0.500000

+

0.866025i

0.218136

−

1.71826i

−0.500000

+

0.866025i

−

1.34044i

1.59712

−

0.670219i

1.55924

−

2.70068i

−1.00000

−2.90483

−

0.749627i

1.16085

−

0.670219i

65.3

0.500000

+

0.866025i

1.08014

+

1.35400i

−0.500000

+

0.866025i

3.22485i

−0.632527

+

1.61242i

1.90702

−

3.30306i

−1.00000

−0.666611

+

2.92500i

−2.79280

+

1.61242i

65.4

0.500000

+

0.866025i

1.67642

−

0.435444i

−0.500000

+

0.866025i

2.46820i

1.21532

+

1.23410i

0.408487

−

0.707520i

−1.00000

2.62078

−

1.45997i

−2.13753

+

1.23410i

221.1

0.500000

−

0.866025i

−0.474693

−

1.66573i

−0.500000

−

0.866025i

−

0.843540i

−1.67991

−

0.421770i

−0.874747

−

1.51511i

−1.00000

−2.54933

+

1.58142i

−0.730527

−

0.421770i

221.2

0.500000

−

0.866025i

0.218136

+

1.71826i

−0.500000

−

0.866025i

1.34044i

1.59712

+

0.670219i

1.55924

+

2.70068i

−1.00000

−2.90483

+

0.749627i

1.16085

+

0.670219i

221.3

0.500000

−

0.866025i

1.08014

−

1.35400i

−0.500000

−

0.866025i

−

3.22485i

−0.632527

−

1.61242i

1.90702

+

3.30306i

−1.00000

−0.666611

−

2.92500i

−2.79280

−

1.61242i

221.4

0.500000

−

0.866025i

1.67642

+

0.435444i

−0.500000

−

0.866025i

−

2.46820i

1.21532

−

1.23410i

0.408487

+

0.707520i

−1.00000

2.62078

+

1.45997i

−2.13753

−

1.23410i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
171.k	even	6	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	342.2.j.e	✓	8
3.b	odd	2	1		1026.2.j.e		8
9.c	even	3	1		1026.2.n.e		8
9.d	odd	6	1		342.2.n.e	yes	8
19.d	odd	6	1		342.2.n.e	yes	8
57.f	even	6	1		1026.2.n.e		8
171.k	even	6	1	inner	342.2.j.e	✓	8
171.s	odd	6	1		1026.2.j.e		8

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
342.2.j.e	✓	8	1.a	even	1	1	trivial
342.2.j.e	✓	8	171.k	even	6	1	inner
342.2.n.e	yes	8	9.d	odd	6	1
342.2.n.e	yes	8	19.d	odd	6	1
1026.2.j.e		8	3.b	odd	2	1
1026.2.j.e		8	171.s	odd	6	1
1026.2.n.e		8	9.c	even	3	1
1026.2.n.e		8	57.f	even	6	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(342, [\chi])\):

\( T_{5}^{8} + 19T_{5}^{6} + 106T_{5}^{4} + 180T_{5}^{2} + 81 \)

\( T_{7}^{8} - 6T_{7}^{7} + 32T_{7}^{6} - 66T_{7}^{5} + 159T_{7}^{4} - 120T_{7}^{3} + 509T_{7}^{2} - 357T_{7} + 289 \)

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( (T^{2} - T + 1)^{4} \)
$3$	T^8 - 5*T^7 + 16*T^6 - 39*T^5 + 75*T^4 - 117*T^3 + 144*T^2 - 135*T + 81
$5$	T^8 + 19*T^6 + 106*T^4 + 180*T^2 + 81
$7$	T^8 - 6*T^7 + 32*T^6 - 66*T^5 + 159*T^4 - 120*T^3 + 509*T^2 - 357*T + 289
$11$	T^8 - 20*T^6 + 343*T^4 - 300*T^3 - 1065*T^2 + 855*T + 3249