Newform orbit 175.3.d.h

• Label: 175.3.d.h
• Level: $175$
• Weight: $3$
• Character orbit: 175.d
• Analytic conductor: $4.768$
• Analytic rank: $0$
• Dimension: $2$
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 175 = 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	175.d (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(4.76840462631\)
Analytic rank:	\(0\)
Dimension:	\(2\)
Coefficient field:	\(\Q(\sqrt{-10}) \)
Defining polynomial:	\( x^{2} + 10 \)
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 \(\beta = \sqrt{-10}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\)	\(=\)	q + 3 * q^2 + b * q^3 + 5 * q^4 + 3*b * q^6 + (2*b - 3) * q^7 + 3 * q^8 - q^9 + 14 * q^11 + 5*b * q^12 - b * q^13 + (6*b - 9) * q^14 - 11 * q^16 - 2*b * q^17 - 3 * q^18 - 9*b * q^19 + (-3*b - 20) * q^21 + 42 * q^22 + 12 * q^23 + 3*b * q^24 - 3*b * q^26 + 8*b * q^27 + (10*b - 15) * q^28 - 14 * q^29 - 12*b * q^31 - 45 * q^32 + 14*b * q^33 - 6*b * q^34 - 5 * q^36 + 18 * q^37 - 27*b * q^38 + 10 * q^39 + 6*b * q^41 + (-9*b - 60) * q^42 + 42 * q^43 + 70 * q^44 + 36 * q^46 - 14*b * q^47 - 11*b * q^48 + (-12*b - 31) * q^49 + 20 * q^51 - 5*b * q^52 - 54 * q^53 + 24*b * q^54 + (6*b - 9) * q^56 + 90 * q^57 - 42 * q^58 + 3*b * q^59 + 21*b * q^61 - 36*b * q^62 + (-2*b + 3) * q^63 - 91 * q^64 + 42*b * q^66 - 102 * q^67 - 10*b * q^68 + 12*b * q^69 - 16 * q^71 - 3 * q^72 + 20*b * q^73 + 54 * q^74 - 45*b * q^76 + (28*b - 42) * q^77 + 30 * q^78 + 76 * q^79 - 89 * q^81 + 18*b * q^82 + 23*b * q^83 + (-15*b - 100) * q^84 + 126 * q^86 - 14*b * q^87 + 42 * q^88 + 18*b * q^89 + (3*b + 20) * q^91 + 60 * q^92 + 120 * q^93 - 42*b * q^94 - 45*b * q^96 + 22*b * q^97 + (-36*b - 93) * q^98 - 14 * q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	2 * q + 6 * q^2 + 10 * q^4 - 6 * q^7 + 6 * q^8 - 2 * q^9 + 28 * q^11 - 18 * q^14 - 22 * q^16 - 6 * q^18 - 40 * q^21 + 84 * q^22 + 24 * q^23 - 30 * q^28 - 28 * q^29 - 90 * q^32 - 10 * q^36 + 36 * q^37 + 20 * q^39 - 120 * q^42 + 84 * q^43 + 140 * q^44 + 72 * q^46 - 62 * q^49 + 40 * q^51 - 108 * q^53 - 18 * q^56 + 180 * q^57 - 84 * q^58 + 6 * q^63 - 182 * q^64 - 204 * q^67 - 32 * q^71 - 6 * q^72 + 108 * q^74 - 84 * q^77 + 60 * q^78 + 152 * q^79 - 178 * q^81 - 200 * q^84 + 252 * q^86 + 84 * q^88 + 40 * q^91 + 120 * q^92 + 240 * q^93 - 186 * q^98 - 28 * q^99

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  

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

76.1

	−	3.16228i
		3.16228i

3.00000

−

3.16228i

5.00000

0

−

9.48683i

−3.00000

−

6.32456i

3.00000

−1.00000

0

76.2

3.00000

3.16228i

5.00000

0

9.48683i

−3.00000

+

6.32456i

3.00000

−1.00000

0

Inner twists

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

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	175.3.d.h		2
5.b	even	2	1		175.3.d.b		2
5.c	odd	4	2		35.3.c.c	✓	4
7.b	odd	2	1	inner	175.3.d.h		2
15.e	even	4	2		315.3.e.c		4
20.e	even	4	2		560.3.p.f		4
35.c	odd	2	1		175.3.d.b		2
35.f	even	4	2		35.3.c.c	✓	4
35.k	even	12	4		245.3.i.c		8
35.l	odd	12	4		245.3.i.c		8
105.k	odd	4	2		315.3.e.c		4
140.j	odd	4	2		560.3.p.f		4

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
35.3.c.c	✓	4	5.c	odd	4	2
35.3.c.c	✓	4	35.f	even	4	2
175.3.d.b		2	5.b	even	2	1
175.3.d.b		2	35.c	odd	2	1
175.3.d.h		2	1.a	even	1	1	trivial
175.3.d.h		2	7.b	odd	2	1	inner
245.3.i.c		8	35.k	even	12	4
245.3.i.c		8	35.l	odd	12	4
315.3.e.c		4	15.e	even	4	2
315.3.e.c		4	105.k	odd	4	2
560.3.p.f		4	20.e	even	4	2
560.3.p.f		4	140.j	odd	4	2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(175, [\chi])\):

\( T_{2} - 3 \)

\( T_{3}^{2} + 10 \)

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( (T - 3)^{2} \)
$3$	\( T^{2} + 10 \)
$5$	\( T^{2} \)
$7$	\( T^{2} + 6T + 49 \)
$11$	\( (T - 14)^{2} \)