Newform orbit 225.6.a.o

• Label: 225.6.a.o
• Level: $225$
• Weight: $6$
• Character orbit: 225.a
• Self dual: yes
• Analytic conductor: $36.086$
• Analytic rank: $1$
• Dimension: $2$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 225 = 3^{2} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	225.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	\(36.0863594579\)
Analytic rank:	\(1\)
Dimension:	\(2\)
Coefficient field:	\(\Q(\sqrt{70}) \)
Defining polynomial:	\( x^{2} - 70 \)
Coefficient ring:	\(\Z[a_1, a_2]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
Fricke sign:	\(1\)
Sato-Tate group:	$\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{70}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\)	\(=\)	\( q + \beta q^{2} + 38 q^{4} - 45 q^{7} + 6 \beta q^{8}+O(q^{10}) \)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 76 q^{4} - 90 q^{7}+O(q^{10}) \)

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

−8.36660
8.36660

−8.36660

0

38.0000

0

0

−45.0000

−50.1996

0

0

1.2

8.36660

0

38.0000

0

0

−45.0000

50.1996

0

0

Atkin-Lehner signs

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

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
3.b	odd	2	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	225.6.a.o	✓	2
3.b	odd	2	1	inner	225.6.a.o	✓	2
5.b	even	2	1		225.6.a.p	yes	2
5.c	odd	4	2		225.6.b.h		4
15.d	odd	2	1		225.6.a.p	yes	2
15.e	even	4	2		225.6.b.h		4

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
225.6.a.o	✓	2	1.a	even	1	1	trivial
225.6.a.o	✓	2	3.b	odd	2	1	inner
225.6.a.p	yes	2	5.b	even	2	1
225.6.a.p	yes	2	15.d	odd	2	1
225.6.b.h		4	5.c	odd	4	2
225.6.b.h		4	15.e	even	4	2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(225))\):

\( T_{2}^{2} - 70 \)

\( T_{7} + 45 \)

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( T^{2} - 70 \)
$3$	\( T^{2} \)
$5$	\( T^{2} \)
$7$	\( (T + 45)^{2} \)
$11$	\( T^{2} - 112000 \)