Newform orbit 175.2.a.f

• Label: 175.2.a.f
• Level: $175$
• Weight: $2$
• Character orbit: 175.a
• Self dual: yes
• Analytic conductor: $1.397$
• Analytic rank: $0$
• Dimension: $2$
• CM: no
• Inner twists: $1$

Newspace parameters

Level:	\( N \)	\(=\)	\( 175 = 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	175.a (trivial)

Newform invariants

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

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \frac{1}{2}(1 + \sqrt{17})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\)	\(=\)	\( q + \beta q^{2} + ( - \beta + 1) q^{3} + (\beta + 2) q^{4} - 4 q^{6} + q^{7} + (\beta + 4) q^{8} + ( - \beta + 2) q^{9}+O(q^{10}) \)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + q^{2} + q^{3} + 5 q^{4} - 8 q^{6} + 2 q^{7} + 9 q^{8} + 3 q^{9}+O(q^{10}) \)

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

−1.56155
2.56155

−1.56155

2.56155

0.438447

0

−4.00000

1.00000

2.43845

3.56155

0

1.2

2.56155

−1.56155

4.56155

0

−4.00000

1.00000

6.56155

−0.561553

0

Atkin-Lehner signs

\( p \)	Sign
\(5\)	\(1\)
\(7\)	\(-1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	175.2.a.f		2
3.b	odd	2	1		1575.2.a.p		2
4.b	odd	2	1		2800.2.a.bi		2
5.b	even	2	1		35.2.a.b	✓	2
5.c	odd	4	2		175.2.b.b		4
7.b	odd	2	1		1225.2.a.s		2
15.d	odd	2	1		315.2.a.e		2
15.e	even	4	2		1575.2.d.e		4
20.d	odd	2	1		560.2.a.i		2
20.e	even	4	2		2800.2.g.t		4
35.c	odd	2	1		245.2.a.d		2
35.f	even	4	2		1225.2.b.f		4
35.i	odd	6	2		245.2.e.h		4
35.j	even	6	2		245.2.e.i		4
40.e	odd	2	1		2240.2.a.bd		2
40.f	even	2	1		2240.2.a.bh		2
55.d	odd	2	1		4235.2.a.m		2
60.h	even	2	1		5040.2.a.bt		2
65.d	even	2	1		5915.2.a.l		2
105.g	even	2	1		2205.2.a.x		2
140.c	even	2	1		3920.2.a.bs		2

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
35.2.a.b	✓	2	5.b	even	2	1
175.2.a.f		2	1.a	even	1	1	trivial
175.2.b.b		4	5.c	odd	4	2
245.2.a.d		2	35.c	odd	2	1
245.2.e.h		4	35.i	odd	6	2
245.2.e.i		4	35.j	even	6	2
315.2.a.e		2	15.d	odd	2	1
560.2.a.i		2	20.d	odd	2	1
1225.2.a.s		2	7.b	odd	2	1
1225.2.b.f		4	35.f	even	4	2
1575.2.a.p		2	3.b	odd	2	1
1575.2.d.e		4	15.e	even	4	2
2205.2.a.x		2	105.g	even	2	1
2240.2.a.bd		2	40.e	odd	2	1
2240.2.a.bh		2	40.f	even	2	1
2800.2.a.bi		2	4.b	odd	2	1
2800.2.g.t		4	20.e	even	4	2
3920.2.a.bs		2	140.c	even	2	1
4235.2.a.m		2	55.d	odd	2	1
5040.2.a.bt		2	60.h	even	2	1
5915.2.a.l		2	65.d	even	2	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{2} - T_{2} - 4 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(175))\).

Hecke characteristic polynomials

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