Newform orbit 245.3.h.b

• Label: 245.3.h.b
• Level: $245$
• Weight: $3$
• Character orbit: 245.h
• Analytic conductor: $6.676$
• Analytic rank: $0$
• Dimension: $4$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 245 = 5 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	245.h (of order \(6\), degree \(2\), not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(6.67576647683\)
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:	\(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 35)
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.

\(f(q)\)	\(=\)	\( q + \beta_{2} q^{2} - 2 \beta_1 q^{3} + ( - 3 \beta_{2} + 3) q^{4} + ( - \beta_{3} + \beta_1) q^{5} - 2 \beta_{3} q^{6} + 7 q^{8} + 11 \beta_{2} q^{9}+O(q^{10}) \)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 2 q^{2} + 6 q^{4} + 28 q^{8} + 22 q^{9}+O(q^{10}) \)

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

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

31.1

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

0.500000

+

0.866025i

−3.87298

−

2.23607i

1.50000

−

2.59808i

1.93649

−

1.11803i

−

4.47214i

0

7.00000

5.50000

+

9.52628i

1.93649

+

1.11803i

31.2

0.500000

+

0.866025i

3.87298

+

2.23607i

1.50000

−

2.59808i

−1.93649

+

1.11803i

4.47214i

0

7.00000

5.50000

+

9.52628i

−1.93649

−

1.11803i

166.1

0.500000

−

0.866025i

−3.87298

+

2.23607i

1.50000

+

2.59808i

1.93649

+

1.11803i

4.47214i

0

7.00000

5.50000

−

9.52628i

1.93649

−

1.11803i

166.2

0.500000

−

0.866025i

3.87298

−

2.23607i

1.50000

+

2.59808i

−1.93649

−

1.11803i

−

4.47214i

0

7.00000

5.50000

−

9.52628i

−1.93649

+

1.11803i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
7.b	odd	2	1	inner
7.c	even	3	1	inner
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	245.3.h.b		4
7.b	odd	2	1	inner	245.3.h.b		4
7.c	even	3	1		35.3.d.a	✓	2
7.c	even	3	1	inner	245.3.h.b		4
7.d	odd	6	1		35.3.d.a	✓	2
7.d	odd	6	1	inner	245.3.h.b		4
21.g	even	6	1		315.3.h.b		2
21.h	odd	6	1		315.3.h.b		2
28.f	even	6	1		560.3.f.a		2
28.g	odd	6	1		560.3.f.a		2
35.i	odd	6	1		175.3.d.f		2
35.j	even	6	1		175.3.d.f		2
35.k	even	12	2		175.3.c.d		4
35.l	odd	12	2		175.3.c.d		4

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
35.3.d.a	✓	2	7.c	even	3	1
35.3.d.a	✓	2	7.d	odd	6	1
175.3.c.d		4	35.k	even	12	2
175.3.c.d		4	35.l	odd	12	2
175.3.d.f		2	35.i	odd	6	1
175.3.d.f		2	35.j	even	6	1
245.3.h.b		4	1.a	even	1	1	trivial
245.3.h.b		4	7.b	odd	2	1	inner
245.3.h.b		4	7.c	even	3	1	inner
245.3.h.b		4	7.d	odd	6	1	inner
315.3.h.b		2	21.g	even	6	1
315.3.h.b		2	21.h	odd	6	1
560.3.f.a		2	28.f	even	6	1
560.3.f.a		2	28.g	odd	6	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{2} - T_{2} + 1 \) acting on \(S_{3}^{\mathrm{new}}(245, [\chi])\).

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( (T^{2} - T + 1)^{2} \)
$3$	\( T^{4} - 20T^{2} + 400 \)
$5$	\( T^{4} - 5T^{2} + 25 \)
$7$	\( T^{4} \)
$11$	\( (T^{2} + 2 T + 4)^{2} \)