Newform orbit 1344.2.bl.i

• Label: 1344.2.bl.i
• Level: $1344$
• Weight: $2$
• Character orbit: 1344.bl
• Analytic conductor: $10.732$
• Analytic rank: $0$
• Dimension: $8$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 1344 = 2^{6} \cdot 3 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1344.bl (of order \(6\), degree \(2\), not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(10.7318940317\)
Analytic rank:	\(0\)
Dimension:	\(8\)
Relative dimension:	\(4\) over \(\Q(\zeta_{6})\)
Coefficient field:	8.0.562828176.1
Defining polynomial:	\( x^{8} - 2x^{7} + x^{6} + 2x^{5} - 6x^{4} + 4x^{3} + 4x^{2} - 16x + 16 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index:	\( 2^{4} \)
Twist minimal:	no (minimal twist has level 84)
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.

\(f(q)\)	\(=\)	\( q + (\beta_1 - 1) q^{3} + ( - \beta_{5} - \beta_{2}) q^{5} + ( - \beta_{7} + \beta_{3} - \beta_1) q^{7} - \beta_1 q^{9}+O(q^{10}) \)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 4 q^{3} - 2 q^{7} - 4 q^{9}+O(q^{10}) \)

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

\(\beta_{1}\)	\(=\)	\( ( \nu^{7} + \nu^{5} + 4\nu^{4} - 6\nu^{3} + 4\nu - 8 ) / 16 \)
\(\beta_{2}\)	\(=\)	\( ( \nu^{7} - 4\nu^{6} + \nu^{5} + 8\nu^{4} - 6\nu^{3} + 16\nu^{2} + 4\nu - 24 ) / 16 \)
\(\beta_{3}\)	\(=\)	\( ( -3\nu^{7} + 4\nu^{6} - 3\nu^{5} + 18\nu^{3} - 12\nu + 40 ) / 16 \)
\(\beta_{4}\)	\(=\)	\( ( -5\nu^{7} + 4\nu^{6} - 5\nu^{5} - 8\nu^{4} + 14\nu^{3} - 16\nu^{2} - 20\nu + 56 ) / 16 \)
\(\beta_{5}\)	\(=\)	\( ( -\nu^{7} + \nu^{6} + \nu^{5} - 3\nu^{4} + 4\nu^{3} - 2\nu^{2} - 8\nu + 12 ) / 4 \)
\(\beta_{6}\)	\(=\)	\( ( -2\nu^{7} + \nu^{6} - 3\nu^{4} + 6\nu^{3} - 2\nu^{2} - 4\nu + 20 ) / 4 \)
\(\beta_{7}\)	\(=\)	\( ( -9\nu^{7} + 8\nu^{6} - \nu^{5} - 20\nu^{4} + 30\nu^{3} + 8\nu^{2} - 52\nu + 104 ) / 16 \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1344\mathbb{Z}\right)^\times\).

\(n\)	\(127\)	\(449\)	\(577\)	\(1093\)
\(\chi(n)\)	\(-1\)	\(1\)	\(\beta_{1}\)	\(1\)

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  

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

703.1

−1.33790	−	0.458297i
1.40376	+	0.171630i
0.0777157	+	1.41208i
0.856419	−	1.12541i
−1.33790	+	0.458297i
1.40376	−	0.171630i
0.0777157	−	1.41208i
0.856419	+	1.12541i

0

−0.500000

+

0.866025i

0

−2.12403

+

1.22631i

0

−2.63169

−

0.272415i

0

−0.500000

−

0.866025i

0

703.2

0

−0.500000

+

0.866025i

0

−0.834598

+

0.481855i

0

1.20103

+

2.35744i

0

−0.500000

−

0.866025i

0

703.3

0

−0.500000

+

0.866025i

0

−0.380152

+

0.219481i

0

2.02350

−

1.70453i

0

−0.500000

−

0.866025i

0

703.4

0

−0.500000

+

0.866025i

0

3.33878

−

1.92764i

0

−1.59285

−

2.11254i

0

−0.500000

−

0.866025i

0

1279.1

0

−0.500000

−

0.866025i

0

−2.12403

−

1.22631i

0

−2.63169

+

0.272415i

0

−0.500000

+

0.866025i

0

1279.2

0

−0.500000

−

0.866025i

0

−0.834598

−

0.481855i

0

1.20103

−

2.35744i

0

−0.500000

+

0.866025i

0

1279.3

0

−0.500000

−

0.866025i

0

−0.380152

−

0.219481i

0

2.02350

+

1.70453i

0

−0.500000

+

0.866025i

0

1279.4

0

−0.500000

−

0.866025i

0

3.33878

+

1.92764i

0

−1.59285

+

2.11254i

0

−0.500000

+

0.866025i

0

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
28.f	even	6	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	1344.2.bl.i		8
4.b	odd	2	1		1344.2.bl.j		8
7.d	odd	6	1		1344.2.bl.j		8
8.b	even	2	1		84.2.o.b	yes	8
8.d	odd	2	1		84.2.o.a	✓	8
24.f	even	2	1		252.2.bf.g		8
24.h	odd	2	1		252.2.bf.f		8
28.f	even	6	1	inner	1344.2.bl.i		8
56.e	even	2	1		588.2.o.d		8
56.h	odd	2	1		588.2.o.b		8
56.j	odd	6	1		84.2.o.a	✓	8
56.j	odd	6	1		588.2.b.b		8
56.k	odd	6	1		588.2.b.b		8
56.k	odd	6	1		588.2.o.b		8
56.m	even	6	1		84.2.o.b	yes	8
56.m	even	6	1		588.2.b.a		8
56.p	even	6	1		588.2.b.a		8
56.p	even	6	1		588.2.o.d		8
168.s	odd	6	1		1764.2.b.j		8
168.v	even	6	1		1764.2.b.i		8
168.ba	even	6	1		252.2.bf.g		8
168.ba	even	6	1		1764.2.b.i		8
168.be	odd	6	1		252.2.bf.f		8
168.be	odd	6	1		1764.2.b.j		8

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
84.2.o.a	✓	8	8.d	odd	2	1
84.2.o.a	✓	8	56.j	odd	6	1
84.2.o.b	yes	8	8.b	even	2	1
84.2.o.b	yes	8	56.m	even	6	1
252.2.bf.f		8	24.h	odd	2	1
252.2.bf.f		8	168.be	odd	6	1
252.2.bf.g		8	24.f	even	2	1
252.2.bf.g		8	168.ba	even	6	1
588.2.b.a		8	56.m	even	6	1
588.2.b.a		8	56.p	even	6	1
588.2.b.b		8	56.j	odd	6	1
588.2.b.b		8	56.k	odd	6	1
588.2.o.b		8	56.h	odd	2	1
588.2.o.b		8	56.k	odd	6	1
588.2.o.d		8	56.e	even	2	1
588.2.o.d		8	56.p	even	6	1
1344.2.bl.i		8	1.a	even	1	1	trivial
1344.2.bl.i		8	28.f	even	6	1	inner
1344.2.bl.j		8	4.b	odd	2	1
1344.2.bl.j		8	7.d	odd	6	1
1764.2.b.i		8	168.v	even	6	1
1764.2.b.i		8	168.ba	even	6	1
1764.2.b.j		8	168.s	odd	6	1
1764.2.b.j		8	168.be	odd	6	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1344, [\chi])\):

\( T_{5}^{8} - 11T_{5}^{6} + 125T_{5}^{4} + 264T_{5}^{3} + 236T_{5}^{2} + 96T_{5} + 16 \)

\( T_{11}^{8} - 6T_{11}^{7} - T_{11}^{6} + 78T_{11}^{5} + 125T_{11}^{4} - 156T_{11}^{3} - 212T_{11}^{2} + 240T_{11} + 400 \)

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( T^{8} \)
$3$	\( (T^{2} + T + 1)^{4} \)
$5$	T^8 - 11*T^6 + 125*T^4 + 264*T^3 + 236*T^2 + 96*T + 16
$7$	T^8 + 2*T^7 + 16*T^5 + 65*T^4 + 112*T^3 + 686*T + 2401
$11$	T^8 - 6*T^7 - T^6 + 78*T^5 + 125*T^4 - 156*T^3 - 212*T^2 + 240*T + 400