Newform orbit 245.2.j.b

• Label: 245.2.j.b
• Level: $245$
• Weight: $2$
• Character orbit: 245.j
• Analytic conductor: $1.956$
• Analytic rank: $0$
• Dimension: $4$
• CM discriminant: -35
• Inner twists: $8$

Newspace parameters

Level:	\( N \)	\(=\)	\( 245 = 5 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	245.j (of order \(6\), degree \(2\), not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(1.95633484952\)
Analytic rank:	\(0\)
Dimension:	\(4\)
Relative dimension:	\(2\) over \(\Q(\zeta_{6})\)
Coefficient field:	\(\Q(\sqrt{-3}, \sqrt{-5})\)
Defining polynomial:	\( x^{4} - 5x^{2} + 25 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{6}]$

$q$-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.

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

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - 5x^{2} + 25 \) :

\(\beta_{1}\)	\(=\)	\( \nu \)
\(\beta_{2}\)	\(=\)	\( ( \nu^{2} ) / 5 \)
\(\beta_{3}\)	\(=\)	\( ( \nu^{3} ) / 5 \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/245\mathbb{Z}\right)^\times\).

\(n\)	\(101\)	\(197\)
\(\chi(n)\)	\(-\beta_{2}\)	\(-1\)

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  

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

79.1

−1.93649	+	1.11803i
1.93649	−	1.11803i
−1.93649	−	1.11803i
1.93649	+	1.11803i

0

−1.93649

+

1.11803i

−1.00000

−

1.73205i

1.93649

+

1.11803i

0

0

0

1.00000

−

1.73205i

0

79.2

0

1.93649

−

1.11803i

−1.00000

−

1.73205i

−1.93649

−

1.11803i

0

0

0

1.00000

−

1.73205i

0

214.1

0

−1.93649

−

1.11803i

−1.00000

+

1.73205i

1.93649

−

1.11803i

0

0

0

1.00000

+

1.73205i

0

214.2

0

1.93649

+

1.11803i

−1.00000

+

1.73205i

−1.93649

+

1.11803i

0

0

0

1.00000

+

1.73205i

0

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
35.c	odd	2	1	CM by \(\Q(\sqrt{-35}) \)
5.b	even	2	1	inner
7.b	odd	2	1	inner
7.c	even	3	1	inner
7.d	odd	6	1	inner
35.i	odd	6	1	inner
35.j	even	6	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	245.2.j.b		4
5.b	even	2	1	inner	245.2.j.b		4
7.b	odd	2	1	inner	245.2.j.b		4
7.c	even	3	1		245.2.b.d	✓	2
7.c	even	3	1	inner	245.2.j.b		4
7.d	odd	6	1		245.2.b.d	✓	2
7.d	odd	6	1	inner	245.2.j.b		4
21.g	even	6	1		2205.2.d.h		2
21.h	odd	6	1		2205.2.d.h		2
35.c	odd	2	1	CM	245.2.j.b		4
35.i	odd	6	1		245.2.b.d	✓	2
35.i	odd	6	1	inner	245.2.j.b		4
35.j	even	6	1		245.2.b.d	✓	2
35.j	even	6	1	inner	245.2.j.b		4
35.k	even	12	2		1225.2.a.q		2
35.l	odd	12	2		1225.2.a.q		2
105.o	odd	6	1		2205.2.d.h		2
105.p	even	6	1		2205.2.d.h		2

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
245.2.b.d	✓	2	7.c	even	3	1
245.2.b.d	✓	2	7.d	odd	6	1
245.2.b.d	✓	2	35.i	odd	6	1
245.2.b.d	✓	2	35.j	even	6	1
245.2.j.b		4	1.a	even	1	1	trivial
245.2.j.b		4	5.b	even	2	1	inner
245.2.j.b		4	7.b	odd	2	1	inner
245.2.j.b		4	7.c	even	3	1	inner
245.2.j.b		4	7.d	odd	6	1	inner
245.2.j.b		4	35.c	odd	2	1	CM
245.2.j.b		4	35.i	odd	6	1	inner
245.2.j.b		4	35.j	even	6	1	inner
1225.2.a.q		2	35.k	even	12	2
1225.2.a.q		2	35.l	odd	12	2
2205.2.d.h		2	21.g	even	6	1
2205.2.d.h		2	21.h	odd	6	1
2205.2.d.h		2	105.o	odd	6	1
2205.2.d.h		2	105.p	even	6	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(245, [\chi])\):

\( T_{2} \)

\( T_{3}^{4} - 5T_{3}^{2} + 25 \)

\( T_{31} \)

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( T^{4} \)
$3$	\( T^{4} - 5T^{2} + 25 \)
$5$	\( T^{4} - 5T^{2} + 25 \)
$7$	\( T^{4} \)
$11$	\( (T^{2} - 3 T + 9)^{2} \)