Newform orbit 925.2.m.b

• Label: 925.2.m.b
• Level: $925$
• Weight: $2$
• Character orbit: 925.m
• Analytic conductor: $7.386$
• Analytic rank: $0$
• Dimension: $4$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 925 = 5^{2} \cdot 37 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	925.m (of order \(6\), degree \(2\), not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(7.38616218697\)
Analytic rank:	\(0\)
Dimension:	\(4\)
Relative dimension:	\(2\) over \(\Q(\zeta_{6})\)
Coefficient field:	\(\Q(\zeta_{12})\)
Defining polynomial:	\( x^{4} - x^{2} + 1 \)
Coefficient ring:	\(\Z[a_1, a_2, a_3]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 37)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{12}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\)	\(=\)	q + (-z^2 + 1) * q^2 + (-z^3 - z^2 + z + 2) * q^3 + z^2 * q^4 + (-z^3 - 2*z^2 + 1) * q^6 + (-2*z^2 + 4) * q^7 + 3 * q^8 + (-4*z^3 - z^2 + 2*z + 1) * q^9
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 2 q^{2} + 6 q^{3} + 2 q^{4} + 12 q^{7} + 12 q^{8} + 2 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/925\mathbb{Z}\right)^\times\).

\(n\)	\(76\)	\(852\)
\(\chi(n)\)	\(\zeta_{12}^{2}\)	\(-1\)

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  

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

249.1

−0.866025	−	0.500000i
0.866025	+	0.500000i
−0.866025	+	0.500000i
0.866025	−	0.500000i

0.500000

−

0.866025i

0.633975

−

0.366025i

0.500000

+

0.866025i

0

−

0.732051i

3.00000

−

1.73205i

3.00000

−1.23205

+

2.13397i

0

249.2

0.500000

−

0.866025i

2.36603

−

1.36603i

0.500000

+

0.866025i

0

−

2.73205i

3.00000

−

1.73205i

3.00000

2.23205

−

3.86603i

0

899.1

0.500000

+

0.866025i

0.633975

+

0.366025i

0.500000

−

0.866025i

0

0.732051i

3.00000

+

1.73205i

3.00000

−1.23205

−

2.13397i

0

899.2

0.500000

+

0.866025i

2.36603

+

1.36603i

0.500000

−

0.866025i

0

2.73205i

3.00000

+

1.73205i

3.00000

2.23205

+

3.86603i

0

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
185.l	even	6	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	925.2.m.b		4
5.b	even	2	1		925.2.m.a		4
5.c	odd	4	1		37.2.e.a	✓	4
5.c	odd	4	1		925.2.n.a		4
15.e	even	4	1		333.2.s.b		4
20.e	even	4	1		592.2.w.c		4
37.e	even	6	1		925.2.m.a		4
185.l	even	6	1	inner	925.2.m.b		4
185.p	even	12	1		1369.2.a.g		2
185.r	odd	12	1		37.2.e.a	✓	4
185.r	odd	12	1		925.2.n.a		4
185.r	odd	12	1		1369.2.b.d		4
185.s	odd	12	1		1369.2.b.d		4
185.u	even	12	1		1369.2.a.h		2
555.bh	even	12	1		333.2.s.b		4
740.bh	even	12	1		592.2.w.c		4

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
37.2.e.a	✓	4	5.c	odd	4	1
37.2.e.a	✓	4	185.r	odd	12	1
333.2.s.b		4	15.e	even	4	1
333.2.s.b		4	555.bh	even	12	1
592.2.w.c		4	20.e	even	4	1
592.2.w.c		4	740.bh	even	12	1
925.2.m.a		4	5.b	even	2	1
925.2.m.a		4	37.e	even	6	1
925.2.m.b		4	1.a	even	1	1	trivial
925.2.m.b		4	185.l	even	6	1	inner
925.2.n.a		4	5.c	odd	4	1
925.2.n.a		4	185.r	odd	12	1
1369.2.a.g		2	185.p	even	12	1
1369.2.a.h		2	185.u	even	12	1
1369.2.b.d		4	185.r	odd	12	1
1369.2.b.d		4	185.s	odd	12	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{2} - T_{2} + 1 \) acting on \(S_{2}^{\mathrm{new}}(925, [\chi])\).

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( (T^{2} - T + 1)^{2} \)
$3$	T^4 - 6*T^3 + 14*T^2 - 12*T + 4
$5$	\( T^{4} \)
$7$	\( (T^{2} - 6 T + 12)^{2} \)
$11$	\( (T^{2} + 6 T + 6)^{2} \)