Newform orbit 3822.2.a.bp

• Label: 3822.2.a.bp
• Level: $3822$
• Weight: $2$
• Character orbit: 3822.a
• Self dual: yes
• Analytic conductor: $30.519$
• Analytic rank: $1$
• Dimension: $2$
• CM: no
• Inner twists: $1$

Newspace parameters

Level:	\( N \)	\(=\)	\( 3822 = 2 \cdot 3 \cdot 7^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3822.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	\(30.5188236525\)
Analytic rank:	\(1\)
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 546)
Fricke sign:	\(1\)
Sato-Tate group:	$\mathrm{SU}(2)$

$q$-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.

\(f(q)\)	\(=\)	\( q + q^{2} - q^{3} + q^{4} + \beta q^{5} - q^{6} + q^{8} + q^{9}+O(q^{10}) \)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{2} - 2 q^{3} + 2 q^{4} - 2 q^{6} + 2 q^{8} + 2 q^{9}+O(q^{10}) \)

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

1.00000

−1.00000

1.00000

−1.41421

−1.00000

0

1.00000

1.00000

−1.41421

1.2

1.00000

−1.00000

1.00000

1.41421

−1.00000

0

1.00000

1.00000

1.41421

Atkin-Lehner signs

\( p \)	Sign
\(2\)	\(-1\)
\(3\)	\(1\)
\(7\)	\(-1\)
\(13\)	\(1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	3822.2.a.bp		2
7.b	odd	2	1		3822.2.a.bs		2
7.d	odd	6	2		546.2.i.h	✓	4
21.g	even	6	2		1638.2.j.n		4

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
546.2.i.h	✓	4	7.d	odd	6	2
1638.2.j.n		4	21.g	even	6	2
3822.2.a.bp		2	1.a	even	1	1	trivial
3822.2.a.bs		2	7.b	odd	2	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(3822))\):

\( T_{5}^{2} - 2 \)

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

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

\( T_{29} - 5 \)

Hecke characteristic polynomials

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