Newform orbit 225.5.c.a

• Label: 225.5.c.a
• Level: $225$
• Weight: $5$
• Character orbit: 225.c
• Analytic conductor: $23.258$
• Analytic rank: $0$
• Dimension: $2$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 225 = 3^{2} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	225.c (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(23.2582416939\)
Analytic rank:	\(0\)
Dimension:	\(2\)
Coefficient field:	\(\Q(\sqrt{-2}) \)
Defining polynomial:	\( x^{2} + 2 \)
Coefficient ring:	\(\Z[a_1, a_2]\)
Coefficient ring index:	\( 3 \)
Twist minimal:	no (minimal twist has level 9)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = 3\sqrt{-2}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\)	\(=\)	\( q + \beta q^{2} - 2 q^{4} + 28 q^{7} + 14 \beta q^{8}+O(q^{10}) \)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 4 q^{4} + 56 q^{7}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/225\mathbb{Z}\right)^\times\).

\(n\)	\(101\)	\(127\)
\(\chi(n)\)	\(-1\)	\(1\)

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  

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

26.1

	−	1.41421i
		1.41421i

−

4.24264i

0

−2.00000

0

0

28.0000

−

59.3970i

0

0

26.2

4.24264i

0

−2.00000

0

0

28.0000

59.3970i

0

0

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	225.5.c.a		2
3.b	odd	2	1	inner	225.5.c.a		2
5.b	even	2	1		9.5.b.a	✓	2
5.c	odd	4	2		225.5.d.a		4
15.d	odd	2	1		9.5.b.a	✓	2
15.e	even	4	2		225.5.d.a		4
20.d	odd	2	1		144.5.e.c		2
35.c	odd	2	1		441.5.b.a		2
40.e	odd	2	1		576.5.e.g		2
40.f	even	2	1		576.5.e.d		2
45.h	odd	6	2		81.5.d.c		4
45.j	even	6	2		81.5.d.c		4
60.h	even	2	1		144.5.e.c		2
105.g	even	2	1		441.5.b.a		2
120.i	odd	2	1		576.5.e.d		2
120.m	even	2	1		576.5.e.g		2

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
9.5.b.a	✓	2	5.b	even	2	1
9.5.b.a	✓	2	15.d	odd	2	1
81.5.d.c		4	45.h	odd	6	2
81.5.d.c		4	45.j	even	6	2
144.5.e.c		2	20.d	odd	2	1
144.5.e.c		2	60.h	even	2	1
225.5.c.a		2	1.a	even	1	1	trivial
225.5.c.a		2	3.b	odd	2	1	inner
225.5.d.a		4	5.c	odd	4	2
225.5.d.a		4	15.e	even	4	2
441.5.b.a		2	35.c	odd	2	1
441.5.b.a		2	105.g	even	2	1
576.5.e.d		2	40.f	even	2	1
576.5.e.d		2	120.i	odd	2	1
576.5.e.g		2	40.e	odd	2	1
576.5.e.g		2	120.m	even	2	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{5}^{\mathrm{new}}(225, [\chi])\):

\( T_{2}^{2} + 18 \)

\( T_{7} - 28 \)

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( T^{2} + 18 \)
$3$	\( T^{2} \)
$5$	\( T^{2} \)
$7$	\( (T - 28)^{2} \)
$11$	\( T^{2} + 288 \)