Newform orbit 405.2.e.j

• Label: 405.2.e.j
• Level: $405$
• Weight: $2$
• Character orbit: 405.e
• Analytic conductor: $3.234$
• Analytic rank: $0$
• Dimension: $4$
• CM: no
• Inner twists: $2$

Newspace parameters

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

Newform invariants

Self dual:	no
Analytic conductor:	\(3.23394128186\)
Analytic rank:	\(0\)
Dimension:	\(4\)
Relative dimension:	\(2\) over \(\Q(\zeta_{3})\)
Coefficient field:	\(\Q(\sqrt{-3}, \sqrt{13})\)
Defining polynomial:	\( x^{4} - x^{3} + 4x^{2} + 3x + 9 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 135)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

$q$-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.

\(f(q)\)	\(=\)	\( q - \beta_1 q^{2} + (\beta_{3} + \beta_{2} + \beta_1 - 1) q^{4} - \beta_{2} q^{5} + ( - 2 \beta_{2} + 2 \beta_1 - 2) q^{7} + 3 q^{8}+O(q^{10}) \)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - q^{2} - 3 q^{4} + 2 q^{5} - 2 q^{7} + 12 q^{8}+O(q^{10}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - x^{3} + 4x^{2} + 3x + 9 \) :

\(\beta_{1}\)	\(=\)	\( \nu \)
\(\beta_{2}\)	\(=\)	\( ( -\nu^{3} + 4\nu^{2} - 4\nu - 3 ) / 12 \)
\(\beta_{3}\)	\(=\)	\( ( \nu^{3} + 7 ) / 4 \)

Character values

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

\(n\)	\(82\)	\(326\)
\(\chi(n)\)	\(1\)	\(\beta_{2}\)

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  

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

136.1

1.15139	−	1.99426i
−0.651388	+	1.12824i
1.15139	+	1.99426i
−0.651388	−	1.12824i

−1.15139

+

1.99426i

0

−1.65139

−

2.86029i

0.500000

+

0.866025i

0

1.30278

−

2.25647i

3.00000

0

−2.30278

136.2

0.651388

−

1.12824i

0

0.151388

+

0.262211i

0.500000

+

0.866025i

0

−2.30278

+

3.98852i

3.00000

0

1.30278

271.1

−1.15139

−

1.99426i

0

−1.65139

+

2.86029i

0.500000

−

0.866025i

0

1.30278

+

2.25647i

3.00000

0

−2.30278

271.2

0.651388

+

1.12824i

0

0.151388

−

0.262211i

0.500000

−

0.866025i

0

−2.30278

−

3.98852i

3.00000

0

1.30278

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
9.c	even	3	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	405.2.e.j		4
3.b	odd	2	1		405.2.e.k		4
9.c	even	3	1		135.2.a.d	yes	2
9.c	even	3	1	inner	405.2.e.j		4
9.d	odd	6	1		135.2.a.c	✓	2
9.d	odd	6	1		405.2.e.k		4
36.f	odd	6	1		2160.2.a.y		2
36.h	even	6	1		2160.2.a.ba		2
45.h	odd	6	1		675.2.a.p		2
45.j	even	6	1		675.2.a.k		2
45.k	odd	12	2		675.2.b.h		4
45.l	even	12	2		675.2.b.i		4
63.l	odd	6	1		6615.2.a.v		2
63.o	even	6	1		6615.2.a.p		2
72.j	odd	6	1		8640.2.a.cr		2
72.l	even	6	1		8640.2.a.ck		2
72.n	even	6	1		8640.2.a.df		2
72.p	odd	6	1		8640.2.a.cy		2

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
135.2.a.c	✓	2	9.d	odd	6	1
135.2.a.d	yes	2	9.c	even	3	1
405.2.e.j		4	1.a	even	1	1	trivial
405.2.e.j		4	9.c	even	3	1	inner
405.2.e.k		4	3.b	odd	2	1
405.2.e.k		4	9.d	odd	6	1
675.2.a.k		2	45.j	even	6	1
675.2.a.p		2	45.h	odd	6	1
675.2.b.h		4	45.k	odd	12	2
675.2.b.i		4	45.l	even	12	2
2160.2.a.y		2	36.f	odd	6	1
2160.2.a.ba		2	36.h	even	6	1
6615.2.a.p		2	63.o	even	6	1
6615.2.a.v		2	63.l	odd	6	1
8640.2.a.ck		2	72.l	even	6	1
8640.2.a.cr		2	72.j	odd	6	1
8640.2.a.cy		2	72.p	odd	6	1
8640.2.a.df		2	72.n	even	6	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(405, [\chi])\):

\( T_{2}^{4} + T_{2}^{3} + 4T_{2}^{2} - 3T_{2} + 9 \)

\( T_{7}^{4} + 2T_{7}^{3} + 16T_{7}^{2} - 24T_{7} + 144 \)

\( T_{11}^{4} - 2T_{11}^{3} + 16T_{11}^{2} + 24T_{11} + 144 \)

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	T^4 + T^3 + 4*T^2 - 3*T + 9
$3$	\( T^{4} \)
$5$	\( (T^{2} - T + 1)^{2} \)
$7$	T^4 + 2*T^3 + 16*T^2 - 24*T + 144
$11$	T^4 - 2*T^3 + 16*T^2 + 24*T + 144