Newform orbit 3185.2.a.j

• Label: 3185.2.a.j
• Level: $3185$
• Weight: $2$
• Character orbit: 3185.a
• Self dual: yes
• Analytic conductor: $25.432$
• Analytic rank: $1$
• Dimension: $2$
• CM: no
• Inner twists: $1$

Newspace parameters

Level:	\( N \)	\(=\)	\( 3185 = 5 \cdot 7^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3185.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	\(25.4323530438\)
Analytic rank:	\(1\)
Dimension:	\(2\)
Coefficient field:	\(\Q(\sqrt{2}) \)
Defining polynomial:	\( x^{2} - 2 \)
Coefficient ring:	\(\Z[a_1, a_2]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 65)
Fricke sign:	\(1\)
Sato-Tate group:	$\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{2}\). We also show the integral \(q\)-expansion of the trace form.

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

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

−1.41421
1.41421

−2.41421

1.41421

3.82843

−1.00000

−3.41421

0

−4.41421

−1.00000

2.41421

1.2

0.414214

−1.41421

−1.82843

−1.00000

−0.585786

0

−1.58579

−1.00000

−0.414214

Atkin-Lehner signs

\( p \)	Sign
\(5\)	\(1\)
\(7\)	\(-1\)
\(13\)	\(-1\)

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	3185.2.a.j		2
7.b	odd	2	1		65.2.a.b	✓	2
21.c	even	2	1		585.2.a.m		2
28.d	even	2	1		1040.2.a.j		2
35.c	odd	2	1		325.2.a.i		2
35.f	even	4	2		325.2.b.f		4
56.e	even	2	1		4160.2.a.z		2
56.h	odd	2	1		4160.2.a.bf		2
77.b	even	2	1		7865.2.a.j		2
84.h	odd	2	1		9360.2.a.cd		2
91.b	odd	2	1		845.2.a.g		2
91.i	even	4	2		845.2.c.b		4
91.n	odd	6	2		845.2.e.h		4
91.t	odd	6	2		845.2.e.c		4
91.bc	even	12	4		845.2.m.f		8
105.g	even	2	1		2925.2.a.u		2
105.k	odd	4	2		2925.2.c.r		4
140.c	even	2	1		5200.2.a.bu		2
273.g	even	2	1		7605.2.a.x		2
455.h	odd	2	1		4225.2.a.r		2

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
65.2.a.b	✓	2	7.b	odd	2	1
325.2.a.i		2	35.c	odd	2	1
325.2.b.f		4	35.f	even	4	2
585.2.a.m		2	21.c	even	2	1
845.2.a.g		2	91.b	odd	2	1
845.2.c.b		4	91.i	even	4	2
845.2.e.c		4	91.t	odd	6	2
845.2.e.h		4	91.n	odd	6	2
845.2.m.f		8	91.bc	even	12	4
1040.2.a.j		2	28.d	even	2	1
2925.2.a.u		2	105.g	even	2	1
2925.2.c.r		4	105.k	odd	4	2
3185.2.a.j		2	1.a	even	1	1	trivial
4160.2.a.z		2	56.e	even	2	1
4160.2.a.bf		2	56.h	odd	2	1
4225.2.a.r		2	455.h	odd	2	1
5200.2.a.bu		2	140.c	even	2	1
7605.2.a.x		2	273.g	even	2	1
7865.2.a.j		2	77.b	even	2	1
9360.2.a.cd		2	84.h	odd	2	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(3185))\):

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

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

\( T_{11}^{2} - 4T_{11} + 2 \)

Hecke characteristic polynomials

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