Newform orbit 405.1.h.a

• Label: 405.1.h.a
• Level: $405$
• Weight: $1$
• Character orbit: 405.h
• Analytic conductor: $0.202$
• Analytic rank: $0$
• Dimension: $2$
• Projective image: $D_{3}$
• CM discriminant: -15
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 405 = 3^{4} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	405.h (of order \(6\), degree \(2\), not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(0.202121330116\)
Analytic rank:	\(0\)
Dimension:	\(2\)
Coefficient field:	\(\Q(\zeta_{6})\)
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)
Projective image:	\(D_{3}\)
Projective field:	Galois closure of 3.1.135.1
Artin image:	$C_6\times S_3$
Artin field:	Galois closure of 12.0.6053445140625.2

$q$-expansion

The \(q\)-expansion and trace form are shown below.

\(f(q)\)	\(=\)	\( q - \zeta_{6} q^{2} - \zeta_{6}^{2} q^{5} - q^{8} +O(q^{10}) \)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - q^{2} + q^{5} - 2 q^{8}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/405\mathbb{Z}\right)^\times\).

\(n\)	\(82\)	\(326\)
\(\chi(n)\)	\(-1\)	\(-\zeta_{6}^{2}\)

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

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

134.1

0.500000	+	0.866025i
0.500000	−	0.866025i

−0.500000

−

0.866025i

0

0

0.500000

−

0.866025i

0

0

−1.00000

0

−1.00000

269.1

−0.500000

+

0.866025i

0

0

0.500000

+

0.866025i

0

0

−1.00000

0

−1.00000

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
15.d	odd	2	1	CM by \(\Q(\sqrt{-15}) \)
9.c	even	3	1	inner
45.h	odd	6	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	405.1.h.a		2
3.b	odd	2	1		405.1.h.b		2
5.b	even	2	1		405.1.h.b		2
5.c	odd	4	2		2025.1.j.c		4
9.c	even	3	1		135.1.d.b	yes	1
9.c	even	3	1	inner	405.1.h.a		2
9.d	odd	6	1		135.1.d.a	✓	1
9.d	odd	6	1		405.1.h.b		2
15.d	odd	2	1	CM	405.1.h.a		2
15.e	even	4	2		2025.1.j.c		4
27.e	even	9	6		3645.1.n.e		6
27.f	odd	18	6		3645.1.n.d		6
36.f	odd	6	1		2160.1.c.a		1
36.h	even	6	1		2160.1.c.b		1
45.h	odd	6	1		135.1.d.b	yes	1
45.h	odd	6	1	inner	405.1.h.a		2
45.j	even	6	1		135.1.d.a	✓	1
45.j	even	6	1		405.1.h.b		2
45.k	odd	12	2		675.1.c.c		2
45.k	odd	12	2		2025.1.j.c		4
45.l	even	12	2		675.1.c.c		2
45.l	even	12	2		2025.1.j.c		4
135.n	odd	18	6		3645.1.n.e		6
135.p	even	18	6		3645.1.n.d		6
180.n	even	6	1		2160.1.c.a		1
180.p	odd	6	1		2160.1.c.b		1

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
135.1.d.a	✓	1	9.d	odd	6	1
135.1.d.a	✓	1	45.j	even	6	1
135.1.d.b	yes	1	9.c	even	3	1
135.1.d.b	yes	1	45.h	odd	6	1
405.1.h.a		2	1.a	even	1	1	trivial
405.1.h.a		2	9.c	even	3	1	inner
405.1.h.a		2	15.d	odd	2	1	CM
405.1.h.a		2	45.h	odd	6	1	inner
405.1.h.b		2	3.b	odd	2	1
405.1.h.b		2	5.b	even	2	1
405.1.h.b		2	9.d	odd	6	1
405.1.h.b		2	45.j	even	6	1
675.1.c.c		2	45.k	odd	12	2
675.1.c.c		2	45.l	even	12	2
2025.1.j.c		4	5.c	odd	4	2
2025.1.j.c		4	15.e	even	4	2
2025.1.j.c		4	45.k	odd	12	2
2025.1.j.c		4	45.l	even	12	2
2160.1.c.a		1	36.f	odd	6	1
2160.1.c.a		1	180.n	even	6	1
2160.1.c.b		1	36.h	even	6	1
2160.1.c.b		1	180.p	odd	6	1
3645.1.n.d		6	27.f	odd	18	6
3645.1.n.d		6	135.p	even	18	6
3645.1.n.e		6	27.e	even	9	6
3645.1.n.e		6	135.n	odd	18	6

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{2} + T_{2} + 1 \) acting on \(S_{1}^{\mathrm{new}}(405, [\chi])\).

Hecke characteristic polynomials

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