Newform orbit 6912.2.a.by

• Label: 6912.2.a.by
• Level: $6912$
• Weight: $2$
• Character orbit: 6912.a
• Self dual: yes
• Analytic conductor: $55.193$
• Analytic rank: $0$
• Dimension: $2$
• CM discriminant: -24
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 6912 = 2^{8} \cdot 3^{3} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6912.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	\(55.1925978771\)
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:	no (minimal twist has level 216)
Fricke sign:	\(-1\)
Sato-Tate group:	$N(\mathrm{U}(1))$

$q$-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.

\(f(q)\)	\(=\)	\( q + (\beta + 3) q^{5} + (3 \beta - 1) q^{7}+O(q^{10}) \)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 6 q^{5} - 2 q^{7}+O(q^{10}) \)

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  

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

1.1

−1.41421
1.41421

0

0

0

1.58579

0

−5.24264

0

0

0

1.2

0

0

0

4.41421

0

3.24264

0

0

0

Atkin-Lehner signs

\( p \)	Sign
\(2\)	\(-1\)
\(3\)	\(1\)

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
24.h	odd	2	1	CM by \(\Q(\sqrt{-6}) \)

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	6912.2.a.by		2
3.b	odd	2	1		6912.2.a.y		2
4.b	odd	2	1		6912.2.a.bz		2
8.b	even	2	1		6912.2.a.y		2
8.d	odd	2	1		6912.2.a.z		2
12.b	even	2	1		6912.2.a.z		2
16.e	even	4	2		864.2.d.b		4
16.f	odd	4	2		216.2.d.a	✓	4
24.f	even	2	1		6912.2.a.bz		2
24.h	odd	2	1	CM	6912.2.a.by		2
48.i	odd	4	2		864.2.d.b		4
48.k	even	4	2		216.2.d.a	✓	4
144.u	even	12	4		648.2.n.p		8
144.v	odd	12	4		648.2.n.p		8
144.w	odd	12	4		2592.2.r.o		8
144.x	even	12	4		2592.2.r.o		8

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
216.2.d.a	✓	4	16.f	odd	4	2
216.2.d.a	✓	4	48.k	even	4	2
648.2.n.p		8	144.u	even	12	4
648.2.n.p		8	144.v	odd	12	4
864.2.d.b		4	16.e	even	4	2
864.2.d.b		4	48.i	odd	4	2
2592.2.r.o		8	144.w	odd	12	4
2592.2.r.o		8	144.x	even	12	4
6912.2.a.y		2	3.b	odd	2	1
6912.2.a.y		2	8.b	even	2	1
6912.2.a.z		2	8.d	odd	2	1
6912.2.a.z		2	12.b	even	2	1
6912.2.a.by		2	1.a	even	1	1	trivial
6912.2.a.by		2	24.h	odd	2	1	CM
6912.2.a.bz		2	4.b	odd	2	1
6912.2.a.bz		2	24.f	even	2	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6912))\):

\( T_{5}^{2} - 6T_{5} + 7 \)

\( T_{7}^{2} + 2T_{7} - 17 \)

\( T_{11}^{2} + 6T_{11} + 1 \)

\( T_{13} \)

\( T_{17} \)

\( T_{19} \)

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( T^{2} \)
$3$	\( T^{2} \)
$5$	\( T^{2} - 6T + 7 \)
$7$	\( T^{2} + 2T - 17 \)
$11$	\( T^{2} + 6T + 1 \)