Newform orbit 350.2.a.g

• Label: 350.2.a.g
• Level: $350$
• Weight: $2$
• Character orbit: 350.a
• Self dual: yes
• Analytic conductor: $2.795$
• Analytic rank: $0$
• Dimension: $2$
• CM: no
• Inner twists: $1$

Newspace parameters

Level:	\( N \)	\(=\)	\( 350 = 2 \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	350.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	\(2.79476407074\)
Analytic rank:	\(0\)
Dimension:	\(2\)
Coefficient field:	\(\Q(\sqrt{6}) \)
Defining polynomial:	\( x^{2} - 6 \)
Coefficient ring:	\(\Z[a_1, a_2, a_3]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 70)
Fricke sign:	\(-1\)
Sato-Tate group:	$\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\)	\(=\)	\( q - q^{2} + \beta q^{3} + q^{4} - \beta q^{6} - q^{7} - q^{8} + 3 q^{9} +O(q^{10}) \)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{2} + 2 q^{4} - 2 q^{7} - 2 q^{8} + 6 q^{9}+O(q^{10}) \)

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  

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

1.1

−2.44949
2.44949

−1.00000

−2.44949

1.00000

0

2.44949

−1.00000

−1.00000

3.00000

0

1.2

−1.00000

2.44949

1.00000

0

−2.44949

−1.00000

−1.00000

3.00000

0

Atkin-Lehner signs

\( p \)	Sign
\(2\)	\(1\)
\(5\)	\(-1\)
\(7\)	\(1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	350.2.a.g		2
3.b	odd	2	1		3150.2.a.bt		2
4.b	odd	2	1		2800.2.a.bm		2
5.b	even	2	1		350.2.a.h		2
5.c	odd	4	2		70.2.c.a	✓	4
7.b	odd	2	1		2450.2.a.bl		2
15.d	odd	2	1		3150.2.a.bs		2
15.e	even	4	2		630.2.g.g		4
20.d	odd	2	1		2800.2.a.bl		2
20.e	even	4	2		560.2.g.e		4
35.c	odd	2	1		2450.2.a.bq		2
35.f	even	4	2		490.2.c.e		4
35.k	even	12	4		490.2.i.f		8
35.l	odd	12	4		490.2.i.c		8
40.i	odd	4	2		2240.2.g.j		4
40.k	even	4	2		2240.2.g.i		4
60.l	odd	4	2		5040.2.t.t		4

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
70.2.c.a	✓	4	5.c	odd	4	2
350.2.a.g		2	1.a	even	1	1	trivial
350.2.a.h		2	5.b	even	2	1
490.2.c.e		4	35.f	even	4	2
490.2.i.c		8	35.l	odd	12	4
490.2.i.f		8	35.k	even	12	4
560.2.g.e		4	20.e	even	4	2
630.2.g.g		4	15.e	even	4	2
2240.2.g.i		4	40.k	even	4	2
2240.2.g.j		4	40.i	odd	4	2
2450.2.a.bl		2	7.b	odd	2	1
2450.2.a.bq		2	35.c	odd	2	1
2800.2.a.bl		2	20.d	odd	2	1
2800.2.a.bm		2	4.b	odd	2	1
3150.2.a.bs		2	15.d	odd	2	1
3150.2.a.bt		2	3.b	odd	2	1
5040.2.t.t		4	60.l	odd	4	2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(350))\):

\( T_{3}^{2} - 6 \)

\( T_{13}^{2} - 4T_{13} - 2 \)

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( (T + 1)^{2} \)
$3$	\( T^{2} - 6 \)
$5$	\( T^{2} \)
$7$	\( (T + 1)^{2} \)
$11$	\( T^{2} - 24 \)