Newform orbit 3626.2.a.a

• Label: 3626.2.a.a
• Level: $3626$
• Weight: $2$
• Character orbit: 3626.a
• Self dual: yes
• Analytic conductor: $28.954$
• Analytic rank: $1$
• Dimension: $2$
• CM: no
• Inner twists: $1$

Newspace parameters

Level:	\( N \)	\(=\)	\( 3626 = 2 \cdot 7^{2} \cdot 37 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3626.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	\(28.9537557729\)
Analytic rank:	\(1\)
Dimension:	\(2\)
Coefficient field:	\(\Q(\sqrt{13}) \)
Defining polynomial:	\( x^{2} - x - 3 \)
Coefficient ring:	\(\Z[a_1, a_2, a_3]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 74)
Fricke sign:	\(1\)
Sato-Tate group:	$\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \frac{1}{2}(1 + \sqrt{13})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\)	\(=\)	\( q - q^{2} + ( - \beta - 1) q^{3} + q^{4} + \beta q^{5} + (\beta + 1) q^{6} - q^{8} + (3 \beta + 1) q^{9}+O(q^{10}) \)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{2} - 3 q^{3} + 2 q^{4} + q^{5} + 3 q^{6} - 2 q^{8} + 5 q^{9}+O(q^{10}) \)

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

2.30278
−1.30278

−1.00000

−3.30278

1.00000

2.30278

3.30278

0

−1.00000

7.90833

−2.30278

1.2

−1.00000

0.302776

1.00000

−1.30278

−0.302776

0

−1.00000

−2.90833

1.30278

Atkin-Lehner signs

\( p \)	Sign
\(2\)	\(1\)
\(7\)	\(-1\)
\(37\)	\(-1\)

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	3626.2.a.a		2
7.b	odd	2	1		74.2.a.a	✓	2
21.c	even	2	1		666.2.a.j		2
28.d	even	2	1		592.2.a.f		2
35.c	odd	2	1		1850.2.a.u		2
35.f	even	4	2		1850.2.b.i		4
56.e	even	2	1		2368.2.a.ba		2
56.h	odd	2	1		2368.2.a.s		2
77.b	even	2	1		8954.2.a.p		2
84.h	odd	2	1		5328.2.a.bf		2
259.b	odd	2	1		2738.2.a.l		2

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
74.2.a.a	✓	2	7.b	odd	2	1
592.2.a.f		2	28.d	even	2	1
666.2.a.j		2	21.c	even	2	1
1850.2.a.u		2	35.c	odd	2	1
1850.2.b.i		4	35.f	even	4	2
2368.2.a.s		2	56.h	odd	2	1
2368.2.a.ba		2	56.e	even	2	1
2738.2.a.l		2	259.b	odd	2	1
3626.2.a.a		2	1.a	even	1	1	trivial
5328.2.a.bf		2	84.h	odd	2	1
8954.2.a.p		2	77.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 \(S_{2}^{\mathrm{new}}(\Gamma_0(3626))\):

\( T_{3}^{2} + 3T_{3} - 1 \)

\( T_{5}^{2} - T_{5} - 3 \)

\( T_{11}^{2} + T_{11} - 3 \)

Hecke characteristic polynomials

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