Newform orbit 245.2.e.b

• Label: 245.2.e.b
• Level: $245$
• Weight: $2$
• Character orbit: 245.e
• Analytic conductor: $1.956$
• Analytic rank: $0$
• Dimension: $2$
• 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:	\(2\)
Coefficient field:	\(\Q(\sqrt{-3}) \)
Defining polynomial:	\( x^{2} - x + 1 \)
Coefficient ring:	\(\Z[a_1, a_2, a_3]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 35)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

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

Character values

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

\(n\)	\(101\)	\(197\)
\(\chi(n)\)	\(-\zeta_{6}\)	\(1\)

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

0.500000	+	0.866025i
0.500000	−	0.866025i

0

0.500000

−

0.866025i

1.00000

−

1.73205i

−0.500000

−

0.866025i

0

0

0

1.00000

+

1.73205i

0

226.1

0

0.500000

+

0.866025i

1.00000

+

1.73205i

−0.500000

+

0.866025i

0

0

0

1.00000

−

1.73205i

0

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.b		2
7.b	odd	2	1		245.2.e.a		2
7.c	even	3	1		245.2.a.c		1
7.c	even	3	1	inner	245.2.e.b		2
7.d	odd	6	1		35.2.a.a	✓	1
7.d	odd	6	1		245.2.e.a		2
21.g	even	6	1		315.2.a.b		1
21.h	odd	6	1		2205.2.a.e		1
28.f	even	6	1		560.2.a.b		1
28.g	odd	6	1		3920.2.a.ba		1
35.i	odd	6	1		175.2.a.b		1
35.j	even	6	1		1225.2.a.e		1
35.k	even	12	2		175.2.b.a		2
35.l	odd	12	2		1225.2.b.d		2
56.j	odd	6	1		2240.2.a.k		1
56.m	even	6	1		2240.2.a.u		1
77.i	even	6	1		4235.2.a.c		1
84.j	odd	6	1		5040.2.a.v		1
91.s	odd	6	1		5915.2.a.f		1
105.p	even	6	1		1575.2.a.f		1
105.w	odd	12	2		1575.2.d.c		2
140.s	even	6	1		2800.2.a.z		1
140.x	odd	12	2		2800.2.g.l		2

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
35.2.a.a	✓	1	7.d	odd	6	1
175.2.a.b		1	35.i	odd	6	1
175.2.b.a		2	35.k	even	12	2
245.2.a.c		1	7.c	even	3	1
245.2.e.a		2	7.b	odd	2	1
245.2.e.a		2	7.d	odd	6	1
245.2.e.b		2	1.a	even	1	1	trivial
245.2.e.b		2	7.c	even	3	1	inner
315.2.a.b		1	21.g	even	6	1
560.2.a.b		1	28.f	even	6	1
1225.2.a.e		1	35.j	even	6	1
1225.2.b.d		2	35.l	odd	12	2
1575.2.a.f		1	105.p	even	6	1
1575.2.d.c		2	105.w	odd	12	2
2205.2.a.e		1	21.h	odd	6	1
2240.2.a.k		1	56.j	odd	6	1
2240.2.a.u		1	56.m	even	6	1
2800.2.a.z		1	140.s	even	6	1
2800.2.g.l		2	140.x	odd	12	2
3920.2.a.ba		1	28.g	odd	6	1
4235.2.a.c		1	77.i	even	6	1
5040.2.a.v		1	84.j	odd	6	1
5915.2.a.f		1	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 \(S_{2}^{\mathrm{new}}(245, [\chi])\):

\( T_{2} \)

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

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( T^{2} \)
$3$	\( T^{2} - T + 1 \)
$5$	\( T^{2} + T + 1 \)
$7$	\( T^{2} \)
$11$	\( T^{2} - 3T + 9 \)