Newform orbit 294.3.g.b

• Label: 294.3.g.b
• Level: $294$
• Weight: $3$
• Character orbit: 294.g
• Analytic conductor: $8.011$
• Analytic rank: $0$
• Dimension: $4$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 294 = 2 \cdot 3 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	294.g (of order \(6\), degree \(2\), not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(8.01091977219\)
Analytic rank:	\(0\)
Dimension:	\(4\)
Relative dimension:	\(2\) over \(\Q(\zeta_{6})\)
Coefficient field:	\(\Q(\sqrt{2}, \sqrt{-3})\)
Defining polynomial:	\(x^{4} + 2 x^{2} + 4\)
Coefficient ring:	\(\Z[a_1, a_2, a_3]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 42)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\)	\(=\)	\( q + \beta_{1} q^{2} + ( -2 - \beta_{2} ) q^{3} + 2 \beta_{2} q^{4} + ( 2 - \beta_{1} - 2 \beta_{2} - 2 \beta_{3} ) q^{5} + ( -2 \beta_{1} - \beta_{3} ) q^{6} + 2 \beta_{3} q^{8} + ( 3 + 3 \beta_{2} ) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4q - 6q^{3} - 4q^{4} + 12q^{5} + 6q^{9} + O(q^{10}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} + 2 x^{2} + 4\):

\(\beta_{0}\)	\(=\)	\( 1 \)
\(\beta_{1}\)	\(=\)	\( \nu \)
\(\beta_{2}\)	\(=\)	\( \nu^{2} \)\(/2\)
\(\beta_{3}\)	\(=\)	\( \nu^{3} \)\(/2\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/294\mathbb{Z}\right)^\times\).

\(n\)	\(197\)	\(199\)
\(\chi(n)\)	\(1\)	\(1 + \beta_{2}\)

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

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

19.1

−0.707107	+	1.22474i
0.707107	−	1.22474i
−0.707107	−	1.22474i
0.707107	+	1.22474i

−0.707107

+

1.22474i

−1.50000

+

0.866025i

−1.00000

−

1.73205i

0.878680

+

0.507306i

−

2.44949i

0

2.82843

1.50000

−

2.59808i

−1.24264

+

0.717439i

19.2

0.707107

−

1.22474i

−1.50000

+

0.866025i

−1.00000

−

1.73205i

5.12132

+

2.95680i

2.44949i

0

−2.82843

1.50000

−

2.59808i

7.24264

−

4.18154i

31.1

−0.707107

−

1.22474i

−1.50000

−

0.866025i

−1.00000

+

1.73205i

0.878680

−

0.507306i

2.44949i

0

2.82843

1.50000

+

2.59808i

−1.24264

−

0.717439i

31.2

0.707107

+

1.22474i

−1.50000

−

0.866025i

−1.00000

+

1.73205i

5.12132

−

2.95680i

−

2.44949i

0

−2.82843

1.50000

+

2.59808i

7.24264

+

4.18154i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
7.d	odd	6	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	294.3.g.b		4
3.b	odd	2	1		882.3.n.a		4
7.b	odd	2	1		294.3.g.c		4
7.c	even	3	1		42.3.c.a	✓	4
7.c	even	3	1		294.3.g.c		4
7.d	odd	6	1		42.3.c.a	✓	4
7.d	odd	6	1	inner	294.3.g.b		4
21.c	even	2	1		882.3.n.d		4
21.g	even	6	1		126.3.c.b		4
21.g	even	6	1		882.3.n.a		4
21.h	odd	6	1		126.3.c.b		4
21.h	odd	6	1		882.3.n.d		4
28.f	even	6	1		336.3.f.c		4
28.g	odd	6	1		336.3.f.c		4
35.i	odd	6	1		1050.3.f.a		4
35.j	even	6	1		1050.3.f.a		4
35.k	even	12	2		1050.3.h.a		8
35.l	odd	12	2		1050.3.h.a		8
56.j	odd	6	1		1344.3.f.f		4
56.k	odd	6	1		1344.3.f.e		4
56.m	even	6	1		1344.3.f.e		4
56.p	even	6	1		1344.3.f.f		4
84.j	odd	6	1		1008.3.f.g		4
84.n	even	6	1		1008.3.f.g		4

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
42.3.c.a	✓	4	7.c	even	3	1
42.3.c.a	✓	4	7.d	odd	6	1
126.3.c.b		4	21.g	even	6	1
126.3.c.b		4	21.h	odd	6	1
294.3.g.b		4	1.a	even	1	1	trivial
294.3.g.b		4	7.d	odd	6	1	inner
294.3.g.c		4	7.b	odd	2	1
294.3.g.c		4	7.c	even	3	1
336.3.f.c		4	28.f	even	6	1
336.3.f.c		4	28.g	odd	6	1
882.3.n.a		4	3.b	odd	2	1
882.3.n.a		4	21.g	even	6	1
882.3.n.d		4	21.c	even	2	1
882.3.n.d		4	21.h	odd	6	1
1008.3.f.g		4	84.j	odd	6	1
1008.3.f.g		4	84.n	even	6	1
1050.3.f.a		4	35.i	odd	6	1
1050.3.f.a		4	35.j	even	6	1
1050.3.h.a		8	35.k	even	12	2
1050.3.h.a		8	35.l	odd	12	2
1344.3.f.e		4	56.k	odd	6	1
1344.3.f.e		4	56.m	even	6	1
1344.3.f.f		4	56.j	odd	6	1
1344.3.f.f		4	56.p	even	6	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{4} - 12 T_{5}^{3} + 54 T_{5}^{2} - 72 T_{5} + 36 \) acting on \(S_{3}^{\mathrm{new}}(294, [\chi])\).

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( 4 + 2 T^{2} + T^{4} \)
$3$	\( ( 3 + 3 T + T^{2} )^{2} \)
$5$	\( 36 - 72 T + 54 T^{2} - 12 T^{3} + T^{4} \)
$7$	\( T^{4} \)
$11$	\( 324 - 216 T + 126 T^{2} - 12 T^{3} + T^{4} \)