Newform orbit 245.2.j.e

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

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:	\(\Q(\zeta_{12})\)
Defining polynomial:	\( x^{4} - x^{2} + 1 \)
Coefficient ring:	\(\Z[a_1, a_2, a_3]\)
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 primitive root of unity \(\zeta_{12}\). We also show the integral \(q\)-expansion of the trace form.

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

Character values

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

\(n\)	\(101\)	\(197\)
\(\chi(n)\)	\(-1 + \zeta_{12}^{2}\)	\(-1\)

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

−0.866025	−	0.500000i
0.866025	+	0.500000i
−0.866025	+	0.500000i
0.866025	−	0.500000i

−1.73205

−

1.00000i

−0.866025

+

0.500000i

1.00000

+

1.73205i

1.86603

−

1.23205i

2.00000

0

0

−1.00000

+

1.73205i

−4.46410

+

0.267949i

79.2

1.73205

+

1.00000i

0.866025

−

0.500000i

1.00000

+

1.73205i

0.133975

−

2.23205i

2.00000

0

0

−1.00000

+

1.73205i

2.46410

−

3.73205i

214.1

−1.73205

+

1.00000i

−0.866025

−

0.500000i

1.00000

−

1.73205i

1.86603

+

1.23205i

2.00000

0

0

−1.00000

−

1.73205i

−4.46410

−

0.267949i

214.2

1.73205

−

1.00000i

0.866025

+

0.500000i

1.00000

−

1.73205i

0.133975

+

2.23205i

2.00000

0

0

−1.00000

−

1.73205i

2.46410

+

3.73205i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
5.b	even	2	1	inner
7.c	even	3	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.e		4
5.b	even	2	1	inner	245.2.j.e		4
7.b	odd	2	1		245.2.j.d		4
7.c	even	3	1		35.2.b.a	✓	2
7.c	even	3	1	inner	245.2.j.e		4
7.d	odd	6	1		245.2.b.a		2
7.d	odd	6	1		245.2.j.d		4
21.g	even	6	1		2205.2.d.b		2
21.h	odd	6	1		315.2.d.a		2
28.g	odd	6	1		560.2.g.b		2
35.c	odd	2	1		245.2.j.d		4
35.i	odd	6	1		245.2.b.a		2
35.i	odd	6	1		245.2.j.d		4
35.j	even	6	1		35.2.b.a	✓	2
35.j	even	6	1	inner	245.2.j.e		4
35.k	even	12	1		1225.2.a.a		1
35.k	even	12	1		1225.2.a.i		1
35.l	odd	12	1		175.2.a.a		1
35.l	odd	12	1		175.2.a.c		1
56.k	odd	6	1		2240.2.g.g		2
56.p	even	6	1		2240.2.g.h		2
84.n	even	6	1		5040.2.t.p		2
105.o	odd	6	1		315.2.d.a		2
105.p	even	6	1		2205.2.d.b		2
105.x	even	12	1		1575.2.a.a		1
105.x	even	12	1		1575.2.a.k		1
140.p	odd	6	1		560.2.g.b		2
140.w	even	12	1		2800.2.a.l		1
140.w	even	12	1		2800.2.a.w		1
280.bf	even	6	1		2240.2.g.h		2
280.bi	odd	6	1		2240.2.g.g		2
420.ba	even	6	1		5040.2.t.p		2

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
35.2.b.a	✓	2	7.c	even	3	1
35.2.b.a	✓	2	35.j	even	6	1
175.2.a.a		1	35.l	odd	12	1
175.2.a.c		1	35.l	odd	12	1
245.2.b.a		2	7.d	odd	6	1
245.2.b.a		2	35.i	odd	6	1
245.2.j.d		4	7.b	odd	2	1
245.2.j.d		4	7.d	odd	6	1
245.2.j.d		4	35.c	odd	2	1
245.2.j.d		4	35.i	odd	6	1
245.2.j.e		4	1.a	even	1	1	trivial
245.2.j.e		4	5.b	even	2	1	inner
245.2.j.e		4	7.c	even	3	1	inner
245.2.j.e		4	35.j	even	6	1	inner
315.2.d.a		2	21.h	odd	6	1
315.2.d.a		2	105.o	odd	6	1
560.2.g.b		2	28.g	odd	6	1
560.2.g.b		2	140.p	odd	6	1
1225.2.a.a		1	35.k	even	12	1
1225.2.a.i		1	35.k	even	12	1
1575.2.a.a		1	105.x	even	12	1
1575.2.a.k		1	105.x	even	12	1
2205.2.d.b		2	21.g	even	6	1
2205.2.d.b		2	105.p	even	6	1
2240.2.g.g		2	56.k	odd	6	1
2240.2.g.g		2	280.bi	odd	6	1
2240.2.g.h		2	56.p	even	6	1
2240.2.g.h		2	280.bf	even	6	1
2800.2.a.l		1	140.w	even	12	1
2800.2.a.w		1	140.w	even	12	1
5040.2.t.p		2	84.n	even	6	1
5040.2.t.p		2	420.ba	even	6	1

Hecke kernels

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

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

\( T_{3}^{4} - T_{3}^{2} + 1 \)

\( T_{31}^{2} + 2T_{31} + 4 \)

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( T^{4} - 4T^{2} + 16 \)
$3$	\( T^{4} - T^{2} + 1 \)
$5$	T^4 - 4*T^3 + 11*T^2 - 20*T + 25
$7$	\( T^{4} \)
$11$	\( (T^{2} - 3 T + 9)^{2} \)