Newform orbit 3675.2.a.be

• Label: 3675.2.a.be
• Level: $3675$
• Weight: $2$
• Character orbit: 3675.a
• Self dual: yes
• Analytic conductor: $29.345$
• Analytic rank: $0$
• Dimension: $2$
• CM: no
• Inner twists: $1$

Newspace parameters

Level:	\( N \)	\(=\)	\( 3675 = 3 \cdot 5^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3675.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	\(29.3450227428\)
Analytic rank:	\(0\)
Dimension:	\(2\)
Coefficient field:	\(\Q(\sqrt{3}) \)
Defining polynomial:	\( x^{2} - 3 \)
Coefficient ring:	\(\Z[a_1, a_2]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 105)
Fricke sign:	\(-1\)
Sato-Tate group:	$\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{3}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\)	\(=\)	\( q + (\beta + 1) q^{2} - q^{3} + (2 \beta + 2) q^{4} + ( - \beta - 1) q^{6} + (2 \beta + 6) q^{8} + q^{9}+O(q^{10}) \)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{2} - 2 q^{3} + 4 q^{4} - 2 q^{6} + 12 q^{8} + 2 q^{9}+O(q^{10}) \)

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

−1.73205
1.73205

−0.732051

−1.00000

−1.46410

0

0.732051

0

2.53590

1.00000

0

1.2

2.73205

−1.00000

5.46410

0

−2.73205

0

9.46410

1.00000

0

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	3675.2.a.be		2
5.b	even	2	1		735.2.a.h		2
7.b	odd	2	1		3675.2.a.bg		2
7.d	odd	6	2		525.2.i.f		4
15.d	odd	2	1		2205.2.a.ba		2
35.c	odd	2	1		735.2.a.g		2
35.i	odd	6	2		105.2.i.d	✓	4
35.j	even	6	2		735.2.i.l		4
35.k	even	12	2		525.2.r.a		4
35.k	even	12	2		525.2.r.f		4
105.g	even	2	1		2205.2.a.z		2
105.p	even	6	2		315.2.j.c		4
140.s	even	6	2		1680.2.bg.o		4

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
105.2.i.d	✓	4	35.i	odd	6	2
315.2.j.c		4	105.p	even	6	2
525.2.i.f		4	7.d	odd	6	2
525.2.r.a		4	35.k	even	12	2
525.2.r.f		4	35.k	even	12	2
735.2.a.g		2	35.c	odd	2	1
735.2.a.h		2	5.b	even	2	1
735.2.i.l		4	35.j	even	6	2
1680.2.bg.o		4	140.s	even	6	2
2205.2.a.z		2	105.g	even	2	1
2205.2.a.ba		2	15.d	odd	2	1
3675.2.a.be		2	1.a	even	1	1	trivial
3675.2.a.bg		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 \(S_{2}^{\mathrm{new}}(\Gamma_0(3675))\):

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

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

\( T_{13}^{2} - 8T_{13} + 13 \)

Hecke characteristic polynomials

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