Newform orbit 2601.2.a.p

• Label: 2601.2.a.p
• Level: $2601$
• Weight: $2$
• Character orbit: 2601.a
• Self dual: yes
• Analytic conductor: $20.769$
• Analytic rank: $1$
• Dimension: $2$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 2601 = 3^{2} \cdot 17^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2601.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	\(20.7690895657\)
Analytic rank:	\(1\)
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 153)
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.

\(f(q)\)	\(=\)	\( q + \beta q^{2} - q^{5} + \beta q^{7} - 2 \beta q^{8} +O(q^{10}) \)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{5}+O(q^{10}) \)

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

−1.41421

2.82843

0

1.41421

1.2

1.41421

0

0

−1.00000

0

1.41421

−2.82843

0

−1.41421

Atkin-Lehner signs

\( p \)	Sign
\(3\)	\(1\)
\(17\)	\(1\)

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
51.c	odd	2	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	2601.2.a.p		2
3.b	odd	2	1		2601.2.a.q		2
17.b	even	2	1		2601.2.a.q		2
17.d	even	8	2		153.2.f.a	✓	4
51.c	odd	2	1	inner	2601.2.a.p		2
51.g	odd	8	2		153.2.f.a	✓	4
68.g	odd	8	2		2448.2.be.r		4
204.p	even	8	2		2448.2.be.r		4

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
153.2.f.a	✓	4	17.d	even	8	2
153.2.f.a	✓	4	51.g	odd	8	2
2448.2.be.r		4	68.g	odd	8	2
2448.2.be.r		4	204.p	even	8	2
2601.2.a.p		2	1.a	even	1	1	trivial
2601.2.a.p		2	51.c	odd	2	1	inner
2601.2.a.q		2	3.b	odd	2	1
2601.2.a.q		2	17.b	even	2	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(2601))\):

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

\( T_{5} + 1 \)

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

Hecke characteristic polynomials

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