Newform orbit 325.2.a.i

• Label: 325.2.a.i
• Level: $325$
• Weight: $2$
• Character orbit: 325.a
• Self dual: yes
• Analytic conductor: $2.595$
• Analytic rank: $0$
• Dimension: $2$
• CM: no
• Inner twists: $1$

Newspace parameters

Level:	\( N \)	\(=\)	\( 325 = 5^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	325.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	\(2.59513806569\)
Analytic rank:	\(0\)
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.

\(f(q)\)	\(=\)	\( q + (\beta + 1) q^{2} + \beta q^{3} + (2 \beta + 1) q^{4} + (\beta + 2) q^{6} + ( - 2 \beta - 2) q^{7} + (\beta + 3) q^{8} - q^{9}+O(q^{10}) \)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{2} + 2 q^{4} + 4 q^{6} - 4 q^{7} + 6 q^{8} - 2 q^{9}+O(q^{10}) \)

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

−0.414214

−1.41421

−1.82843

0

0.585786

0.828427

1.58579

−1.00000

0

1.2

2.41421

1.41421

3.82843

0

3.41421

−4.82843

4.41421

−1.00000

0

Atkin-Lehner signs

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

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	325.2.a.i		2
3.b	odd	2	1		2925.2.a.u		2
4.b	odd	2	1		5200.2.a.bu		2
5.b	even	2	1		65.2.a.b	✓	2
5.c	odd	4	2		325.2.b.f		4
13.b	even	2	1		4225.2.a.r		2
15.d	odd	2	1		585.2.a.m		2
15.e	even	4	2		2925.2.c.r		4
20.d	odd	2	1		1040.2.a.j		2
35.c	odd	2	1		3185.2.a.j		2
40.e	odd	2	1		4160.2.a.z		2
40.f	even	2	1		4160.2.a.bf		2
55.d	odd	2	1		7865.2.a.j		2
60.h	even	2	1		9360.2.a.cd		2
65.d	even	2	1		845.2.a.g		2
65.g	odd	4	2		845.2.c.b		4
65.l	even	6	2		845.2.e.c		4
65.n	even	6	2		845.2.e.h		4
65.s	odd	12	4		845.2.m.f		8
195.e	odd	2	1		7605.2.a.x		2

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
65.2.a.b	✓	2	5.b	even	2	1
325.2.a.i		2	1.a	even	1	1	trivial
325.2.b.f		4	5.c	odd	4	2
585.2.a.m		2	15.d	odd	2	1
845.2.a.g		2	65.d	even	2	1
845.2.c.b		4	65.g	odd	4	2
845.2.e.c		4	65.l	even	6	2
845.2.e.h		4	65.n	even	6	2
845.2.m.f		8	65.s	odd	12	4
1040.2.a.j		2	20.d	odd	2	1
2925.2.a.u		2	3.b	odd	2	1
2925.2.c.r		4	15.e	even	4	2
3185.2.a.j		2	35.c	odd	2	1
4160.2.a.z		2	40.e	odd	2	1
4160.2.a.bf		2	40.f	even	2	1
4225.2.a.r		2	13.b	even	2	1
5200.2.a.bu		2	4.b	odd	2	1
7605.2.a.x		2	195.e	odd	2	1
7865.2.a.j		2	55.d	odd	2	1
9360.2.a.cd		2	60.h	even	2	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(325))\):

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

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

Hecke characteristic polynomials

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