Newform orbit 3456.1.e.c

• Label: 3456.1.e.c
• Level: $3456$
• Weight: $1$
• Character orbit: 3456.e
• Analytic conductor: $1.725$
• Analytic rank: $0$
• Dimension: $2$
• Projective image: $S_{4}$
• CM/RM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 3456 = 2^{7} \cdot 3^{3} \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	3456.e (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(1.72476868366\)
Analytic rank:	\(0\)
Dimension:	\(2\)
Coefficient field:	\(\Q(\sqrt{-2}) \)
Defining polynomial:	\( x^{2} + 2 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
Projective image:	\(S_{4}\)
Projective field:	Galois closure of 4.2.6912.2
Artin image:	$\GL(2,3)$
Artin field:	Galois closure of 8.2.9172942848.6

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{-2}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\)	\(=\)	q - b * q^5 + q^7 - b * q^11 - q^13 + b * q^17 + q^19 - b * q^23 - q^25 - b * q^35 + q^37 - b * q^47 - 2 * q^55 + b * q^59 - q^61 + b * q^65 - q^67 - q^73 - b * q^77 - q^79 + 2 * q^85 + b * q^89 - q^91 - b * q^95 + q^97
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{7} - 2 q^{13} + 2 q^{19} - 2 q^{25} + 2 q^{37} - 4 q^{55} - 2 q^{61} - 2 q^{67} - 2 q^{73} - 2 q^{79} + 4 q^{85} - 2 q^{91} + 2 q^{97}+O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3456\mathbb{Z}\right)^\times\).

\(n\)	\(2053\)	\(2431\)	\(2945\)
\(\chi(n)\)	\(1\)	\(1\)	\(-1\)

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  

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

1025.1

		1.41421i
	−	1.41421i

0

0

0

−

1.41421i

0

1.00000

0

0

0

1025.2

0

0

0

1.41421i

0

1.00000

0

0

0

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
3.b	odd	2	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	3456.1.e.c	yes	2
3.b	odd	2	1	inner	3456.1.e.c	yes	2
4.b	odd	2	1		3456.1.e.a	✓	2
8.b	even	2	1		3456.1.e.d	yes	2
8.d	odd	2	1		3456.1.e.b	yes	2
12.b	even	2	1		3456.1.e.a	✓	2
24.f	even	2	1		3456.1.e.b	yes	2
24.h	odd	2	1		3456.1.e.d	yes	2

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
3456.1.e.a	✓	2	4.b	odd	2	1
3456.1.e.a	✓	2	12.b	even	2	1
3456.1.e.b	yes	2	8.d	odd	2	1
3456.1.e.b	yes	2	24.f	even	2	1
3456.1.e.c	yes	2	1.a	even	1	1	trivial
3456.1.e.c	yes	2	3.b	odd	2	1	inner
3456.1.e.d	yes	2	8.b	even	2	1
3456.1.e.d	yes	2	24.h	odd	2	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{1}^{\mathrm{new}}(3456, [\chi])\):

\( T_{7} - 1 \)

\( T_{13} + 1 \)

Hecke characteristic polynomials

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