Newform orbit 675.1.c.a

• Label: 675.1.c.a
• Level: $675$
• Weight: $1$
• Character orbit: 675.c
• Self dual: yes
• Analytic conductor: $0.337$
• Analytic rank: $0$
• Dimension: $1$
• Projective image: $D_{3}$
• CM discriminant: -3
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 675 = 3^{3} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	675.c (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual:	yes
Analytic conductor:	\(0.336868883527\)
Analytic rank:	\(0\)
Dimension:	\(1\)
Coefficient field:	\(\mathbb{Q}\)
Coefficient ring:	\(\mathbb{Z}\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
Projective image:	\(D_{3}\)
Projective field:	Galois closure of 3.1.675.1
Artin image:	$S_3$
Artin field:	Galois closure of 3.1.675.1
Stark unit:	Root of $x^{3} - 123x^{2} + 3x - 1$

$q$-expansion

\(f(q)\)

\(=\)

\( q + q^{4} - q^{7} + 2 q^{13} + q^{16} - q^{19} - q^{28} - q^{31} - q^{37} - q^{43} + 2 q^{52} - q^{61} + q^{64} + 2 q^{67} - q^{73} - q^{76} - q^{79} - 2 q^{91} - q^{97}+O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/675\mathbb{Z}\right)^\times\).

\(n\)	\(326\)	\(352\)
\(\chi(n)\)	\(1\)	\(0\)

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  

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

26.1

0

0

0

1.00000

0

0

−1.00000

0

0

0

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
3.b	odd	2	1	CM by \(\Q(\sqrt{-3}) \)

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	675.1.c.a	✓	1
3.b	odd	2	1	CM	675.1.c.a	✓	1
5.b	even	2	1		675.1.c.b	yes	1
5.c	odd	4	2		675.1.d.a		2
9.c	even	3	2		2025.1.j.b		2
9.d	odd	6	2		2025.1.j.b		2
15.d	odd	2	1		675.1.c.b	yes	1
15.e	even	4	2		675.1.d.a		2
45.h	odd	6	2		2025.1.j.a		2
45.j	even	6	2		2025.1.j.a		2
45.k	odd	12	4		2025.1.i.a		4
45.l	even	12	4		2025.1.i.a		4

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
675.1.c.a	✓	1	1.a	even	1	1	trivial
675.1.c.a	✓	1	3.b	odd	2	1	CM
675.1.c.b	yes	1	5.b	even	2	1
675.1.c.b	yes	1	15.d	odd	2	1
675.1.d.a		2	5.c	odd	4	2
675.1.d.a		2	15.e	even	4	2
2025.1.i.a		4	45.k	odd	12	4
2025.1.i.a		4	45.l	even	12	4
2025.1.j.a		2	45.h	odd	6	2
2025.1.j.a		2	45.j	even	6	2
2025.1.j.b		2	9.c	even	3	2
2025.1.j.b		2	9.d	odd	6	2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{1}^{\mathrm{new}}(675, [\chi])\):

\( T_{2} \)

\( T_{7} + 1 \)

Hecke characteristic polynomials

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