Newform orbit 75.2.e.b

• Label: 75.2.e.b
• Level: $75$
• Weight: $2$
• Character orbit: 75.e
• Analytic conductor: $0.599$
• Analytic rank: $0$
• Dimension: $4$
• CM discriminant: -3
• Inner twists: $8$

Newspace parameters

Level:	\( N \)	\(=\)	\( 75 = 3 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	75.e (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(0.598878015160\)
Analytic rank:	\(0\)
Dimension:	\(4\)
Relative dimension:	\(2\) over \(\Q(i)\)
Coefficient field:	\(\Q(i, \sqrt{6})\)
Defining polynomial:	\( x^{4} + 9 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{4}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\)	\(=\)	\( q + \beta_1 q^{3} - 2 \beta_{2} q^{4} + \beta_{3} q^{7} + 3 \beta_{2} q^{9}+O(q^{10}) \)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q+O(q^{10}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} + 9 \) :

\(\beta_{1}\)	\(=\)	\( \nu \)
\(\beta_{2}\)	\(=\)	\( ( \nu^{2} ) / 3 \)
\(\beta_{3}\)	\(=\)	\( ( \nu^{3} ) / 3 \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/75\mathbb{Z}\right)^\times\).

\(n\)	\(26\)	\(52\)
\(\chi(n)\)	\(-1\)	\(-\beta_{2}\)

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

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

32.1

−1.22474	+	1.22474i
1.22474	−	1.22474i
−1.22474	−	1.22474i
1.22474	+	1.22474i

0

−1.22474

+

1.22474i

2.00000i

0

0

1.22474

+

1.22474i

0

−

3.00000i

0

32.2

0

1.22474

−

1.22474i

2.00000i

0

0

−1.22474

−

1.22474i

0

−

3.00000i

0

68.1

0

−1.22474

−

1.22474i

−

2.00000i

0

0

1.22474

−

1.22474i

0

3.00000i

0

68.2

0

1.22474

+

1.22474i

−

2.00000i

0

0

−1.22474

+

1.22474i

0

3.00000i

0

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
3.b	odd	2	1	CM by \(\Q(\sqrt{-3}) \)
5.b	even	2	1	inner
5.c	odd	4	2	inner
15.d	odd	2	1	inner
15.e	even	4	2	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	75.2.e.b	✓	4
3.b	odd	2	1	CM	75.2.e.b	✓	4
4.b	odd	2	1		1200.2.v.f		4
5.b	even	2	1	inner	75.2.e.b	✓	4
5.c	odd	4	2	inner	75.2.e.b	✓	4
12.b	even	2	1		1200.2.v.f		4
15.d	odd	2	1	inner	75.2.e.b	✓	4
15.e	even	4	2	inner	75.2.e.b	✓	4
20.d	odd	2	1		1200.2.v.f		4
20.e	even	4	2		1200.2.v.f		4
60.h	even	2	1		1200.2.v.f		4
60.l	odd	4	2		1200.2.v.f		4

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
75.2.e.b	✓	4	1.a	even	1	1	trivial
75.2.e.b	✓	4	3.b	odd	2	1	CM
75.2.e.b	✓	4	5.b	even	2	1	inner
75.2.e.b	✓	4	5.c	odd	4	2	inner
75.2.e.b	✓	4	15.d	odd	2	1	inner
75.2.e.b	✓	4	15.e	even	4	2	inner
1200.2.v.f		4	4.b	odd	2	1
1200.2.v.f		4	12.b	even	2	1
1200.2.v.f		4	20.d	odd	2	1
1200.2.v.f		4	20.e	even	4	2
1200.2.v.f		4	60.h	even	2	1
1200.2.v.f		4	60.l	odd	4	2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2} \) acting on \(S_{2}^{\mathrm{new}}(75, [\chi])\).

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( T^{4} \)
$3$	\( T^{4} + 9 \)
$5$	\( T^{4} \)
$7$	\( T^{4} + 9 \)
$11$	\( T^{4} \)