Newform orbit 1215.1.d.b

• Label: 1215.1.d.b
• Level: $1215$
• Weight: $1$
• Character orbit: 1215.d
• Self dual: yes
• Analytic conductor: $0.606$
• Analytic rank: $0$
• Dimension: $3$
• Projective image: $D_{9}$
• CM discriminant: -15
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 1215 = 3^{5} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	1215.d (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual:	yes
Analytic conductor:	\(0.606363990349\)
Analytic rank:	\(0\)
Dimension:	\(3\)
Coefficient field:	\(\Q(\zeta_{18})^+\)
Defining polynomial:	\( x^{3} - 3x - 1 \)
Coefficient ring:	\(\Z[a_1, a_2]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
Projective image:	\(D_{9}\)
Projective field:	Galois closure of 9.1.242137805625.3
Artin image:	$D_9$
Artin field:	Galois closure of 9.1.242137805625.3

$q$-expansion

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

\(f(q)\)	\(=\)	\( q - \beta_1 q^{2} + (\beta_{2} + 1) q^{4} + q^{5} + ( - \beta_1 - 1) q^{8}+O(q^{10}) \)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 3 q^{4} + 3 q^{5} - 3 q^{8}+O(q^{10}) \)

Basis of coefficient ring in terms of \(\nu = \zeta_{18} + \zeta_{18}^{-1}\):

\(\beta_{1}\)	\(=\)	\( \nu \)
\(\beta_{2}\)	\(=\)	\( \nu^{2} - 2 \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1215\mathbb{Z}\right)^\times\).

\(n\)	\(487\)	\(731\)
\(\chi(n)\)	\(-1\)	\(-1\)

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

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

1214.1

1.87939
−0.347296
−1.53209

−1.87939

0

2.53209

1.00000

0

0

−2.87939

0

−1.87939

1214.2

0.347296

0

−0.879385

1.00000

0

0

−0.652704

0

0.347296

1214.3

1.53209

0

1.34730

1.00000

0

0

0.532089

0

1.53209

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	1215.1.d.b	yes	3
3.b	odd	2	1		1215.1.d.a	✓	3
5.b	even	2	1		1215.1.d.a	✓	3
9.c	even	3	2		1215.1.h.a		6
9.d	odd	6	2		1215.1.h.b		6
15.d	odd	2	1	CM	1215.1.d.b	yes	3
27.e	even	9	2		3645.1.n.b		6
27.e	even	9	2		3645.1.n.c		6
27.e	even	9	2		3645.1.n.h		6
27.f	odd	18	2		3645.1.n.a		6
27.f	odd	18	2		3645.1.n.f		6
27.f	odd	18	2		3645.1.n.g		6
45.h	odd	6	2		1215.1.h.a		6
45.j	even	6	2		1215.1.h.b		6
135.n	odd	18	2		3645.1.n.b		6
135.n	odd	18	2		3645.1.n.c		6
135.n	odd	18	2		3645.1.n.h		6
135.p	even	18	2		3645.1.n.a		6
135.p	even	18	2		3645.1.n.f		6
135.p	even	18	2		3645.1.n.g		6

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
1215.1.d.a	✓	3	3.b	odd	2	1
1215.1.d.a	✓	3	5.b	even	2	1
1215.1.d.b	yes	3	1.a	even	1	1	trivial
1215.1.d.b	yes	3	15.d	odd	2	1	CM
1215.1.h.a		6	9.c	even	3	2
1215.1.h.a		6	45.h	odd	6	2
1215.1.h.b		6	9.d	odd	6	2
1215.1.h.b		6	45.j	even	6	2
3645.1.n.a		6	27.f	odd	18	2
3645.1.n.a		6	135.p	even	18	2
3645.1.n.b		6	27.e	even	9	2
3645.1.n.b		6	135.n	odd	18	2
3645.1.n.c		6	27.e	even	9	2
3645.1.n.c		6	135.n	odd	18	2
3645.1.n.f		6	27.f	odd	18	2
3645.1.n.f		6	135.p	even	18	2
3645.1.n.g		6	27.f	odd	18	2
3645.1.n.g		6	135.p	even	18	2
3645.1.n.h		6	27.e	even	9	2
3645.1.n.h		6	135.n	odd	18	2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{3} - 3T_{2} + 1 \) acting on \(S_{1}^{\mathrm{new}}(1215, [\chi])\).

Hecke characteristic polynomials

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