Newform orbit 675.2.a.j

• Label: 675.2.a.j
• Level: $675$
• Weight: $2$
• Character orbit: 675.a
• Self dual: yes
• Analytic conductor: $5.390$
• Analytic rank: $1$
• Dimension: $2$
• CM discriminant: -15
• Inner twists: $2$

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:	\(1\)
Dimension:	\(2\)
Coefficient field:	\(\Q(\sqrt{5}) \)
Defining polynomial:	\( x^{2} - x - 1 \)
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:	$N(\mathrm{U}(1))$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \frac{1}{2}(1 + \sqrt{5})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\)	\(=\)	q + (-b - 1) * q^2 + 3*b * q^4 + (-4*b - 1) * q^8 + (3*b + 5) * q^16 + (2*b - 7) * q^17 + (-6*b + 1) * q^19 + (4*b - 5) * q^23 + (6*b - 7) * q^31 + (-3*b - 6) * q^32 + (3*b + 5) * q^34 + (11*b + 5) * q^38 + (-3*b + 1) * q^46 + (8*b - 4) * q^47 - 7 * q^49 + (-2*b - 11) * q^53 + (-12*b + 5) * q^61 + (-5*b + 1) * q^62 + (6*b - 1) * q^64 + (-15*b + 6) * q^68 + (-15*b - 18) * q^76 + (-6*b - 5) * q^79 + (-8*b + 1) * q^83 + (-3*b + 12) * q^92 + (-12*b - 4) * q^94 + (7*b + 7) * q^98
\(\operatorname{Tr}(f)(q)\)	\(=\)	2 * q - 3 * q^2 + 3 * q^4 - 6 * q^8 + 13 * q^16 - 12 * q^17 - 4 * q^19 - 6 * q^23 - 8 * q^31 - 15 * q^32 + 13 * q^34 + 21 * q^38 - q^46 - 14 * q^49 - 24 * q^53 - 2 * q^61 - 3 * q^62 + 4 * q^64 - 3 * q^68 - 51 * q^76 - 16 * q^79 - 6 * q^83 + 21 * q^92 - 20 * q^94 + 21 * q^98

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

1.61803
−0.618034

−2.61803

0

4.85410

0

0

0

−7.47214

0

0

1.2

−0.381966

0

−1.85410

0

0

0

1.47214

0

0

Atkin-Lehner signs

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

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
15.d	odd	2	1	CM by \(\Q(\sqrt{-15}) \)

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	675.2.a.j		2
3.b	odd	2	1		675.2.a.q		2
5.b	even	2	1		675.2.a.q		2
5.c	odd	4	2		135.2.b.a	✓	4
15.d	odd	2	1	CM	675.2.a.j		2
15.e	even	4	2		135.2.b.a	✓	4
20.e	even	4	2		2160.2.f.j		4
45.k	odd	12	4		405.2.j.h		8
45.l	even	12	4		405.2.j.h		8
60.l	odd	4	2		2160.2.f.j		4

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
135.2.b.a	✓	4	5.c	odd	4	2
135.2.b.a	✓	4	15.e	even	4	2
405.2.j.h		8	45.k	odd	12	4
405.2.j.h		8	45.l	even	12	4
675.2.a.j		2	1.a	even	1	1	trivial
675.2.a.j		2	15.d	odd	2	1	CM
675.2.a.q		2	3.b	odd	2	1
675.2.a.q		2	5.b	even	2	1
2160.2.f.j		4	20.e	even	4	2
2160.2.f.j		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 \(S_{2}^{\mathrm{new}}(\Gamma_0(675))\):

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

\( T_{7} \)

\( T_{11} \)

Hecke characteristic polynomials

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