Newform orbit 735.2.a.k

• Label: 735.2.a.k
• Level: $735$
• Weight: $2$
• Character orbit: 735.a
• Self dual: yes
• Analytic conductor: $5.869$
• Analytic rank: $0$
• Dimension: $2$
• CM: no
• Inner twists: $1$

Newspace parameters

Level:	\( N \)	\(=\)	\( 735 = 3 \cdot 5 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	735.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	\(5.86900454856\)
Analytic rank:	\(0\)
Dimension:	\(2\)
Coefficient field:	\(\Q(\sqrt{5}) \)
Defining polynomial:	\( x^{2} - x - 1 \)
Coefficient ring:	\(\Z[a_1, a_2]\)
Coefficient ring index:	\( 2 \)
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 \(\beta = \sqrt{5}\). We also show the integral \(q\)-expansion of the trace form.

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

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

1.61803
−0.618034

−2.23607

1.00000

3.00000

1.00000

−2.23607

0

−2.23607

1.00000

−2.23607

1.2

2.23607

1.00000

3.00000

1.00000

2.23607

0

2.23607

1.00000

2.23607

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	735.2.a.k		2
3.b	odd	2	1		2205.2.a.w		2
5.b	even	2	1		3675.2.a.y		2
7.b	odd	2	1		105.2.a.b	✓	2
7.c	even	3	2		735.2.i.i		4
7.d	odd	6	2		735.2.i.k		4
21.c	even	2	1		315.2.a.d		2
28.d	even	2	1		1680.2.a.v		2
35.c	odd	2	1		525.2.a.g		2
35.f	even	4	2		525.2.d.c		4
56.e	even	2	1		6720.2.a.cs		2
56.h	odd	2	1		6720.2.a.cx		2
84.h	odd	2	1		5040.2.a.bw		2
105.g	even	2	1		1575.2.a.r		2
105.k	odd	4	2		1575.2.d.d		4
140.c	even	2	1		8400.2.a.cx		2

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
105.2.a.b	✓	2	7.b	odd	2	1
315.2.a.d		2	21.c	even	2	1
525.2.a.g		2	35.c	odd	2	1
525.2.d.c		4	35.f	even	4	2
735.2.a.k		2	1.a	even	1	1	trivial
735.2.i.i		4	7.c	even	3	2
735.2.i.k		4	7.d	odd	6	2
1575.2.a.r		2	105.g	even	2	1
1575.2.d.d		4	105.k	odd	4	2
1680.2.a.v		2	28.d	even	2	1
2205.2.a.w		2	3.b	odd	2	1
3675.2.a.y		2	5.b	even	2	1
5040.2.a.bw		2	84.h	odd	2	1
6720.2.a.cs		2	56.e	even	2	1
6720.2.a.cx		2	56.h	odd	2	1
8400.2.a.cx		2	140.c	even	2	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(735))\):

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

\( T_{13}^{2} - 20 \)

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( T^{2} - 5 \)
$3$	\( (T - 1)^{2} \)
$5$	\( (T - 1)^{2} \)
$7$	\( T^{2} \)
$11$	\( T^{2} - 4T - 16 \)