Newform orbit 2205.2.a.t

• Label: 2205.2.a.t
• Level: $2205$
• Weight: $2$
• Character orbit: 2205.a
• Self dual: yes
• Analytic conductor: $17.607$
• Analytic rank: $0$
• Dimension: $2$
• CM: no
• Inner twists: $1$

Newspace parameters

Level:	\( N \)	\(=\)	\( 2205 = 3^{2} \cdot 5 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2205.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	\(17.6070136457\)
Analytic rank:	\(0\)
Dimension:	\(2\)
Coefficient field:	\(\Q(\sqrt{2}) \)
Defining polynomial:	\( x^{2} - 2 \)
Coefficient ring:	\(\Z[a_1, a_2]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 245)
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.

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

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

−1.41421

0

0

−1.00000

0

0

2.82843

0

1.41421

1.2

1.41421

0

0

−1.00000

0

0

−2.82843

0

−1.41421

Atkin-Lehner signs

\( p \)	Sign
\(3\)	\(-1\)
\(5\)	\(1\)
\(7\)	\(1\)

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	2205.2.a.t		2
3.b	odd	2	1		245.2.a.f	yes	2
7.b	odd	2	1		2205.2.a.v		2
12.b	even	2	1		3920.2.a.br		2
15.d	odd	2	1		1225.2.a.p		2
15.e	even	4	2		1225.2.b.j		4
21.c	even	2	1		245.2.a.e	✓	2
21.g	even	6	2		245.2.e.g		4
21.h	odd	6	2		245.2.e.f		4
84.h	odd	2	1		3920.2.a.bw		2
105.g	even	2	1		1225.2.a.r		2
105.k	odd	4	2		1225.2.b.i		4

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
245.2.a.e	✓	2	21.c	even	2	1
245.2.a.f	yes	2	3.b	odd	2	1
245.2.e.f		4	21.h	odd	6	2
245.2.e.g		4	21.g	even	6	2
1225.2.a.p		2	15.d	odd	2	1
1225.2.a.r		2	105.g	even	2	1
1225.2.b.i		4	105.k	odd	4	2
1225.2.b.j		4	15.e	even	4	2
2205.2.a.t		2	1.a	even	1	1	trivial
2205.2.a.v		2	7.b	odd	2	1
3920.2.a.br		2	12.b	even	2	1
3920.2.a.bw		2	84.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 \(S_{2}^{\mathrm{new}}(\Gamma_0(2205))\):

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

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

\( T_{13}^{2} - 6T_{13} + 7 \)

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

Hecke characteristic polynomials

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