Newform orbit 1875.2.a.d

• Label: 1875.2.a.d
• Level: $1875$
• Weight: $2$
• Character orbit: 1875.a
• Self dual: yes
• Analytic conductor: $14.972$
• Analytic rank: $0$
• Dimension: $2$
• CM: no
• Inner twists: $1$

Newspace parameters

Level:	\( N \)	\(=\)	\( 1875 = 3 \cdot 5^{4} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1875.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	\(14.9719503790\)
Analytic rank:	\(0\)
Dimension:	\(2\)
Coefficient field:	\(\Q(\sqrt{5}) \)
Defining polynomial:	\( x^{2} - x - 1 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 75)
Fricke sign:	\(-1\)
Sato-Tate group:	$\mathrm{SU}(2)$

$q$-expansion

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

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

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

−0.618034
1.61803

1.00000

1.00000

−1.00000

0

1.00000

−4.47214

−3.00000

1.00000

0

1.2

1.00000

1.00000

−1.00000

0

1.00000

4.47214

−3.00000

1.00000

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	1875.2.a.d		2
3.b	odd	2	1		5625.2.a.a		2
5.b	even	2	1		1875.2.a.a		2
5.c	odd	4	2		1875.2.b.b		4
15.d	odd	2	1		5625.2.a.h		2
25.d	even	5	2		75.2.g.a	✓	4
25.e	even	10	2		375.2.g.a		4
25.f	odd	20	4		375.2.i.a		8
75.j	odd	10	2		225.2.h.a		4

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
75.2.g.a	✓	4	25.d	even	5	2
225.2.h.a		4	75.j	odd	10	2
375.2.g.a		4	25.e	even	10	2
375.2.i.a		8	25.f	odd	20	4
1875.2.a.a		2	5.b	even	2	1
1875.2.a.d		2	1.a	even	1	1	trivial
1875.2.b.b		4	5.c	odd	4	2
5625.2.a.a		2	3.b	odd	2	1
5625.2.a.h		2	15.d	odd	2	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2} - 1 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(1875))\).

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( (T - 1)^{2} \)
$3$	\( (T - 1)^{2} \)
$5$	\( T^{2} \)
$7$	\( T^{2} - 20 \)
$11$	\( T^{2} - 2T - 4 \)