Newform orbit 675.2.a.p

• Label: 675.2.a.p
• Level: $675$
• Weight: $2$
• Character orbit: 675.a
• Self dual: yes
• Analytic conductor: $5.390$
• Analytic rank: $0$
• Dimension: $2$
• CM: no
• Inner twists: $1$

Newspace parameters

Level:	\( N \)	\(=\)	\( 675 = 3^{3} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	675.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	\(5.38990213644\)
Analytic rank:	\(0\)
Dimension:	\(2\)
Coefficient field:	\(\Q(\sqrt{13}) \)
Defining polynomial:	\( x^{2} - x - 3 \)
Coefficient ring:	\(\Z[a_1, a_2]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 135)
Fricke sign:	\(-1\)
Sato-Tate group:	$\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \frac{1}{2}(1 + \sqrt{13})\). We also show the integral \(q\)-expansion of the trace form.

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

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

−1.30278
2.30278

−1.30278

0

−0.302776

0

0

−4.60555

3.00000

0

0

1.2

2.30278

0

3.30278

0

0

2.60555

3.00000

0

0

Atkin-Lehner signs

\( p \)	Sign
\(3\)	\(-1\)
\(5\)	\(1\)

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	675.2.a.p		2
3.b	odd	2	1		675.2.a.k		2
5.b	even	2	1		135.2.a.c	✓	2
5.c	odd	4	2		675.2.b.i		4
15.d	odd	2	1		135.2.a.d	yes	2
15.e	even	4	2		675.2.b.h		4
20.d	odd	2	1		2160.2.a.ba		2
35.c	odd	2	1		6615.2.a.p		2
40.e	odd	2	1		8640.2.a.ck		2
40.f	even	2	1		8640.2.a.cr		2
45.h	odd	6	2		405.2.e.j		4
45.j	even	6	2		405.2.e.k		4
60.h	even	2	1		2160.2.a.y		2
105.g	even	2	1		6615.2.a.v		2
120.i	odd	2	1		8640.2.a.df		2
120.m	even	2	1		8640.2.a.cy		2

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
135.2.a.c	✓	2	5.b	even	2	1
135.2.a.d	yes	2	15.d	odd	2	1
405.2.e.j		4	45.h	odd	6	2
405.2.e.k		4	45.j	even	6	2
675.2.a.k		2	3.b	odd	2	1
675.2.a.p		2	1.a	even	1	1	trivial
675.2.b.h		4	15.e	even	4	2
675.2.b.i		4	5.c	odd	4	2
2160.2.a.y		2	60.h	even	2	1
2160.2.a.ba		2	20.d	odd	2	1
6615.2.a.p		2	35.c	odd	2	1
6615.2.a.v		2	105.g	even	2	1
8640.2.a.ck		2	40.e	odd	2	1
8640.2.a.cr		2	40.f	even	2	1
8640.2.a.cy		2	120.m	even	2	1
8640.2.a.df		2	120.i	odd	2	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(675))\):

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

\( T_{7}^{2} + 2T_{7} - 12 \)

\( T_{11}^{2} - 2T_{11} - 12 \)

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( T^{2} - T - 3 \)
$3$	\( T^{2} \)
$5$	\( T^{2} \)
$7$	\( T^{2} + 2T - 12 \)
$11$	\( T^{2} - 2T - 12 \)