Newform orbit 245.2.e.h

• Label: 245.2.e.h
• Level: $245$
• Weight: $2$
• Character orbit: 245.e
• Analytic conductor: $1.956$
• Analytic rank: $0$
• Dimension: $4$
• CM: no
• Inner twists: $2$

Newspace parameters

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

Newform invariants

Self dual:	no
Analytic conductor:	\(1.95633484952\)
Analytic rank:	\(0\)
Dimension:	\(4\)
Relative dimension:	\(2\) over \(\Q(\zeta_{3})\)
Coefficient field:	\(\Q(\sqrt{-3}, \sqrt{17})\)
Defining polynomial:	\( x^{4} - x^{3} + 5x^{2} + 4x + 16 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 35)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

$q$-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.

\(f(q)\)	\(=\)	\( q + \beta_1 q^{2} + ( - \beta_{3} + \beta_{2} - \beta_1 - 1) q^{3} + (\beta_{3} + 2 \beta_{2} + \beta_1 - 2) q^{4} + \beta_{2} q^{5} + 4 q^{6} + (\beta_{3} - 4) q^{8} + ( - 2 \beta_{2} + \beta_1) q^{9}+O(q^{10}) \)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + q^{2} - q^{3} - 5 q^{4} + 2 q^{5} + 16 q^{6} - 18 q^{8} - 3 q^{9}+O(q^{10}) \)

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

\(\beta_{1}\)	\(=\)	\( \nu \)
\(\beta_{2}\)	\(=\)	\( ( -\nu^{3} + 5\nu^{2} - 5\nu + 16 ) / 20 \)
\(\beta_{3}\)	\(=\)	\( ( \nu^{3} + 4 ) / 5 \)

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  

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

116.1

−0.780776	−	1.35234i
1.28078	+	2.21837i
−0.780776	+	1.35234i
1.28078	−	2.21837i

−0.780776

−

1.35234i

−1.28078

+

2.21837i

−0.219224

+

0.379706i

0.500000

+

0.866025i

4.00000

0

−2.43845

−1.78078

−

3.08440i

0.780776

−

1.35234i

116.2

1.28078

+

2.21837i

0.780776

−

1.35234i

−2.28078

+

3.95042i

0.500000

+

0.866025i

4.00000

0

−6.56155

0.280776

+

0.486319i

−1.28078

+

2.21837i

226.1

−0.780776

+

1.35234i

−1.28078

−

2.21837i

−0.219224

−

0.379706i

0.500000

−

0.866025i

4.00000

0

−2.43845

−1.78078

+

3.08440i

0.780776

+

1.35234i

226.2

1.28078

−

2.21837i

0.780776

+

1.35234i

−2.28078

−

3.95042i

0.500000

−

0.866025i

4.00000

0

−6.56155

0.280776

−

0.486319i

−1.28078

−

2.21837i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
7.c	even	3	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	245.2.e.h		4
7.b	odd	2	1		245.2.e.i		4
7.c	even	3	1		245.2.a.d		2
7.c	even	3	1	inner	245.2.e.h		4
7.d	odd	6	1		35.2.a.b	✓	2
7.d	odd	6	1		245.2.e.i		4
21.g	even	6	1		315.2.a.e		2
21.h	odd	6	1		2205.2.a.x		2
28.f	even	6	1		560.2.a.i		2
28.g	odd	6	1		3920.2.a.bs		2
35.i	odd	6	1		175.2.a.f		2
35.j	even	6	1		1225.2.a.s		2
35.k	even	12	2		175.2.b.b		4
35.l	odd	12	2		1225.2.b.f		4
56.j	odd	6	1		2240.2.a.bh		2
56.m	even	6	1		2240.2.a.bd		2
77.i	even	6	1		4235.2.a.m		2
84.j	odd	6	1		5040.2.a.bt		2
91.s	odd	6	1		5915.2.a.l		2
105.p	even	6	1		1575.2.a.p		2
105.w	odd	12	2		1575.2.d.e		4
140.s	even	6	1		2800.2.a.bi		2
140.x	odd	12	2		2800.2.g.t		4

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
35.2.a.b	✓	2	7.d	odd	6	1
175.2.a.f		2	35.i	odd	6	1
175.2.b.b		4	35.k	even	12	2
245.2.a.d		2	7.c	even	3	1
245.2.e.h		4	1.a	even	1	1	trivial
245.2.e.h		4	7.c	even	3	1	inner
245.2.e.i		4	7.b	odd	2	1
245.2.e.i		4	7.d	odd	6	1
315.2.a.e		2	21.g	even	6	1
560.2.a.i		2	28.f	even	6	1
1225.2.a.s		2	35.j	even	6	1
1225.2.b.f		4	35.l	odd	12	2
1575.2.a.p		2	105.p	even	6	1
1575.2.d.e		4	105.w	odd	12	2
2205.2.a.x		2	21.h	odd	6	1
2240.2.a.bd		2	56.m	even	6	1
2240.2.a.bh		2	56.j	odd	6	1
2800.2.a.bi		2	140.s	even	6	1
2800.2.g.t		4	140.x	odd	12	2
3920.2.a.bs		2	28.g	odd	6	1
4235.2.a.m		2	77.i	even	6	1
5040.2.a.bt		2	84.j	odd	6	1
5915.2.a.l		2	91.s	odd	6	1

Hecke kernels

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

\( T_{2}^{4} - T_{2}^{3} + 5T_{2}^{2} + 4T_{2} + 16 \)

\( T_{3}^{4} + T_{3}^{3} + 5T_{3}^{2} - 4T_{3} + 16 \)

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	T^4 - T^3 + 5*T^2 + 4*T + 16
$3$	T^4 + T^3 + 5*T^2 - 4*T + 16
$5$	\( (T^{2} - T + 1)^{2} \)
$7$	\( T^{4} \)
$11$	T^4 + T^3 + 5*T^2 - 4*T + 16