Newform orbit 1175.1.b.b

• Label: 1175.1.b.b
• Level: $1175$
• Weight: $1$
• Character orbit: 1175.b
• Analytic conductor: $0.586$
• Analytic rank: $0$
• Dimension: $4$
• Projective image: $D_{5}$
• CM discriminant: -47
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 1175 = 5^{2} \cdot 47 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	1175.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(0.586401389844\)
Analytic rank:	\(0\)
Dimension:	\(4\)
Coefficient field:	\(\Q(i, \sqrt{5})\)
Defining polynomial:	\( x^{4} + 3x^{2} + 1 \)
Coefficient ring:	\(\Z[a_1, a_2]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 47)
Projective image:	\(D_{5}\)
Projective field:	Galois closure of 5.1.2209.1
Artin image:	$C_4\times D_5$
Artin field:	Galois closure of \(\mathbb{Q}[x]/(x^{20} - \cdots)\)

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\)	\(=\)	\( q - \beta_1 q^{2} + ( - \beta_{3} - \beta_1) q^{3} + \beta_{2} q^{4} - q^{6} - \beta_1 q^{7} - \beta_{3} q^{8} + ( - \beta_{2} - 1) q^{9}+O(q^{10}) \)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 2 q^{4} - 4 q^{6} - 2 q^{9}+O(q^{10}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} + 3x^{2} + 1 \) :

\(\beta_{1}\)	\(=\)	\( \nu \)
\(\beta_{2}\)	\(=\)	\( \nu^{2} + 1 \)
\(\beta_{3}\)	\(=\)	\( \nu^{3} + 2\nu \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1175\mathbb{Z}\right)^\times\).

\(n\)	\(377\)	\(851\)
\(\chi(n)\)	\(-1\)	\(-1\)

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  

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

1174.1

		1.61803i
		0.618034i
	−	0.618034i
	−	1.61803i

−

1.61803i

−

0.618034i

−1.61803

0

−1.00000

−

1.61803i

1.00000i

0.618034

0

1174.2

−

0.618034i

−

1.61803i

0.618034

0

−1.00000

−

0.618034i

−

1.00000i

−1.61803

0

1174.3

0.618034i

1.61803i

0.618034

0

−1.00000

0.618034i

1.00000i

−1.61803

0

1174.4

1.61803i

0.618034i

−1.61803

0

−1.00000

1.61803i

−

1.00000i

0.618034

0

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
47.b	odd	2	1	CM by \(\Q(\sqrt{-47}) \)
5.b	even	2	1	inner
235.b	odd	2	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	1175.1.b.b		4
5.b	even	2	1	inner	1175.1.b.b		4
5.c	odd	4	1		47.1.b.a	✓	2
5.c	odd	4	1		1175.1.d.c		2
15.e	even	4	1		423.1.d.a		2
20.e	even	4	1		752.1.g.a		2
35.f	even	4	1		2303.1.d.c		2
35.k	even	12	2		2303.1.f.b		4
35.l	odd	12	2		2303.1.f.c		4
40.i	odd	4	1		3008.1.g.b		2
40.k	even	4	1		3008.1.g.a		2
45.k	odd	12	2		3807.1.f.b		4
45.l	even	12	2		3807.1.f.a		4
47.b	odd	2	1	CM	1175.1.b.b		4
235.b	odd	2	1	inner	1175.1.b.b		4
235.e	even	4	1		47.1.b.a	✓	2
235.e	even	4	1		1175.1.d.c		2
235.k	odd	92	22		2209.1.d.a		44
235.l	even	92	22		2209.1.d.a		44
705.l	odd	4	1		423.1.d.a		2
940.l	odd	4	1		752.1.g.a		2
1645.j	odd	4	1		2303.1.d.c		2
1645.v	odd	12	2		2303.1.f.b		4
1645.x	even	12	2		2303.1.f.c		4
1880.q	odd	4	1		3008.1.g.a		2
1880.u	even	4	1		3008.1.g.b		2
2115.u	odd	12	2		3807.1.f.a		4
2115.x	even	12	2		3807.1.f.b		4

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
47.1.b.a	✓	2	5.c	odd	4	1
47.1.b.a	✓	2	235.e	even	4	1
423.1.d.a		2	15.e	even	4	1
423.1.d.a		2	705.l	odd	4	1
752.1.g.a		2	20.e	even	4	1
752.1.g.a		2	940.l	odd	4	1
1175.1.b.b		4	1.a	even	1	1	trivial
1175.1.b.b		4	5.b	even	2	1	inner
1175.1.b.b		4	47.b	odd	2	1	CM
1175.1.b.b		4	235.b	odd	2	1	inner
1175.1.d.c		2	5.c	odd	4	1
1175.1.d.c		2	235.e	even	4	1
2209.1.d.a		44	235.k	odd	92	22
2209.1.d.a		44	235.l	even	92	22
2303.1.d.c		2	35.f	even	4	1
2303.1.d.c		2	1645.j	odd	4	1
2303.1.f.b		4	35.k	even	12	2
2303.1.f.b		4	1645.v	odd	12	2
2303.1.f.c		4	35.l	odd	12	2
2303.1.f.c		4	1645.x	even	12	2
3008.1.g.a		2	40.k	even	4	1
3008.1.g.a		2	1880.q	odd	4	1
3008.1.g.b		2	40.i	odd	4	1
3008.1.g.b		2	1880.u	even	4	1
3807.1.f.a		4	45.l	even	12	2
3807.1.f.a		4	2115.u	odd	12	2
3807.1.f.b		4	45.k	odd	12	2
3807.1.f.b		4	2115.x	even	12	2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{4} + 3T_{2}^{2} + 1 \) acting on \(S_{1}^{\mathrm{new}}(1175, [\chi])\).

Hecke characteristic polynomials

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