Newform orbit 175.2.b.b

• Label: 175.2.b.b
• Level: $175$
• Weight: $2$
• Character orbit: 175.b
• Analytic conductor: $1.397$
• Analytic rank: $0$
• Dimension: $4$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 175 = 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	175.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(1.39738203537\)
Analytic rank:	\(0\)
Dimension:	\(4\)
Coefficient field:	\(\Q(i, \sqrt{17})\)
Defining polynomial:	\( x^{4} + 9x^{2} + 16 \)
Coefficient ring:	\(\Z[a_1, a_2, a_3]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 35)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

$q$-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.

\(f(q)\)	\(=\)	\( q + \beta_1 q^{2} + (\beta_{2} + \beta_1) q^{3} + (\beta_{3} - 3) q^{4} - 4 q^{6} - \beta_{2} q^{7} + (4 \beta_{2} - \beta_1) q^{8} + ( - \beta_{3} - 1) q^{9}+O(q^{10}) \)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 10 q^{4} - 16 q^{6} - 6 q^{9}+O(q^{10}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} + 9x^{2} + 16 \) :

\(\beta_{1}\)	\(=\)	\( \nu \)
\(\beta_{2}\)	\(=\)	\( ( \nu^{3} + 5\nu ) / 4 \)
\(\beta_{3}\)	\(=\)	\( \nu^{2} + 5 \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/175\mathbb{Z}\right)^\times\).

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

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

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

99.1

	−	2.56155i
	−	1.56155i
		1.56155i
		2.56155i

−

2.56155i

−

1.56155i

−4.56155

0

−4.00000

−

1.00000i

6.56155i

0.561553

0

99.2

−

1.56155i

−

2.56155i

−0.438447

0

−4.00000

1.00000i

−

2.43845i

−3.56155

0

99.3

1.56155i

2.56155i

−0.438447

0

−4.00000

−

1.00000i

2.43845i

−3.56155

0

99.4

2.56155i

1.56155i

−4.56155

0

−4.00000

1.00000i

−

6.56155i

0.561553

0

Inner twists

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

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	175.2.b.b		4
3.b	odd	2	1		1575.2.d.e		4
4.b	odd	2	1		2800.2.g.t		4
5.b	even	2	1	inner	175.2.b.b		4
5.c	odd	4	1		35.2.a.b	✓	2
5.c	odd	4	1		175.2.a.f		2
7.b	odd	2	1		1225.2.b.f		4
15.d	odd	2	1		1575.2.d.e		4
15.e	even	4	1		315.2.a.e		2
15.e	even	4	1		1575.2.a.p		2
20.d	odd	2	1		2800.2.g.t		4
20.e	even	4	1		560.2.a.i		2
20.e	even	4	1		2800.2.a.bi		2
35.c	odd	2	1		1225.2.b.f		4
35.f	even	4	1		245.2.a.d		2
35.f	even	4	1		1225.2.a.s		2
35.k	even	12	2		245.2.e.h		4
35.l	odd	12	2		245.2.e.i		4
40.i	odd	4	1		2240.2.a.bh		2
40.k	even	4	1		2240.2.a.bd		2
55.e	even	4	1		4235.2.a.m		2
60.l	odd	4	1		5040.2.a.bt		2
65.h	odd	4	1		5915.2.a.l		2
105.k	odd	4	1		2205.2.a.x		2
140.j	odd	4	1		3920.2.a.bs		2

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
35.2.a.b	✓	2	5.c	odd	4	1
175.2.a.f		2	5.c	odd	4	1
175.2.b.b		4	1.a	even	1	1	trivial
175.2.b.b		4	5.b	even	2	1	inner
245.2.a.d		2	35.f	even	4	1
245.2.e.h		4	35.k	even	12	2
245.2.e.i		4	35.l	odd	12	2
315.2.a.e		2	15.e	even	4	1
560.2.a.i		2	20.e	even	4	1
1225.2.a.s		2	35.f	even	4	1
1225.2.b.f		4	7.b	odd	2	1
1225.2.b.f		4	35.c	odd	2	1
1575.2.a.p		2	15.e	even	4	1
1575.2.d.e		4	3.b	odd	2	1
1575.2.d.e		4	15.d	odd	2	1
2205.2.a.x		2	105.k	odd	4	1
2240.2.a.bd		2	40.k	even	4	1
2240.2.a.bh		2	40.i	odd	4	1
2800.2.a.bi		2	20.e	even	4	1
2800.2.g.t		4	4.b	odd	2	1
2800.2.g.t		4	20.d	odd	2	1
3920.2.a.bs		2	140.j	odd	4	1
4235.2.a.m		2	55.e	even	4	1
5040.2.a.bt		2	60.l	odd	4	1
5915.2.a.l		2	65.h	odd	4	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{4} + 9T_{2}^{2} + 16 \) acting on \(S_{2}^{\mathrm{new}}(175, [\chi])\).

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( T^{4} + 9T^{2} + 16 \)
$3$	\( T^{4} + 9T^{2} + 16 \)
$5$	\( T^{4} \)
$7$	\( (T^{2} + 1)^{2} \)
$11$	\( (T^{2} - T - 4)^{2} \)