Newform orbit 75.2.g.b

• Label: 75.2.g.b
• Level: $75$
• Weight: $2$
• Character orbit: 75.g
• Analytic conductor: $0.599$
• Analytic rank: $0$
• Dimension: $8$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 75 = 3 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	75.g (of order \(5\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(0.598878015160\)
Analytic rank:	\(0\)
Dimension:	\(8\)
Relative dimension:	\(2\) over \(\Q(\zeta_{5})\)
Coefficient field:	8.0.26265625.1
Defining polynomial:	\( x^{8} - 3x^{7} + 2x^{6} + x^{4} + 8x^{2} - 24x + 16 \)
Coefficient ring:	\(\Z[a_1, a_2, a_3]\)
Coefficient ring index:	\( 5 \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{5}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\)	\(=\)	q + (-b7 - b3) * q^2 + (b6 + b3 + b1 - 1) * q^3 + (b6 - b4 + 2*b3 + 2*b1 - 1) * q^4 + (-b6 - b4 - b1) * q^5 + (b5 - b1) * q^6 + (b7 - 2*b6 - b3 - b2 + 1) * q^7 + (b5 - b3 - b2 - 5*b1 + 1) * q^8 - b6 * q^9
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - q^{2} - 2 q^{3} + q^{4} - 5 q^{5} - q^{6} + 4 q^{7} - 2 q^{9}+O(q^{10}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 3x^{7} + 2x^{6} + x^{4} + 8x^{2} - 24x + 16 \) :

\(\beta_{1}\)	\(=\)	\( ( \nu^{7} - \nu^{6} + \nu^{3} + 2\nu^{2} + 4\nu - 8 ) / 8 \)
\(\beta_{2}\)	\(=\)	\( ( \nu^{7} - \nu^{6} + \nu^{3} - 2\nu^{2} + 12\nu - 12 ) / 4 \)
\(\beta_{3}\)	\(=\)	\( ( -7\nu^{7} + 9\nu^{6} + 2\nu^{5} + 4\nu^{4} + \nu^{3} - 4\nu^{2} - 60\nu + 64 ) / 8 \)
\(\beta_{4}\)	\(=\)	\( ( 11\nu^{7} - 15\nu^{6} - 4\nu^{5} - 8\nu^{4} + 3\nu^{3} + 2\nu^{2} + 92\nu - 96 ) / 8 \)
\(\beta_{5}\)	\(=\)	\( ( 11\nu^{7} - 13\nu^{6} - 6\nu^{5} - 8\nu^{4} + 3\nu^{3} + 4\nu^{2} + 96\nu - 88 ) / 8 \)
\(\beta_{6}\)	\(=\)	\( ( 13\nu^{7} - 21\nu^{6} - 4\nu^{5} - 4\nu^{4} + 5\nu^{3} + 10\nu^{2} + 120\nu - 144 ) / 8 \)
\(\beta_{7}\)	\(=\)	\( ( 19\nu^{7} - 31\nu^{6} - 4\nu^{5} - 8\nu^{4} + 11\nu^{3} + 18\nu^{2} + 180\nu - 224 ) / 8 \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/75\mathbb{Z}\right)^\times\).

\(n\)	\(26\)	\(52\)
\(\chi(n)\)	\(1\)	\(-1 + \beta_{1} + \beta_{3} + \beta_{6}\)

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  

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

16.1

1.33631	+	0.462894i
−0.0272949	−	1.41395i
1.40799	+	0.132563i
−1.21700	−	0.720348i
1.40799	−	0.132563i
−1.21700	+	0.720348i
1.33631	−	0.462894i
−0.0272949	+	1.41395i

−2.18949

−

1.59076i

0.309017

+

0.951057i

1.64533

+

5.06380i

0.336312

+

2.21063i

0.836312

−

2.57390i

0.470294

2.78023

−

8.55667i

−0.809017

+

0.587785i

2.78023

−

5.37515i

16.2

1.38048

+

1.00297i

0.309017

+

0.951057i

0.281722

+

0.867051i

−1.02729

−

1.98612i

−0.527295

+

1.62285i

−3.94243

0.573870

−

1.76619i

−0.809017

+

0.587785i

0.573870

−

3.77214i

31.1

−0.346820

−

1.06740i

−0.809017

+

0.587785i

0.598970

−

0.435177i

0.407987

−

2.19853i

0.907987

+

0.659691i

1.11373

−2.48822

−

1.80780i

0.309017

−

0.951057i

−2.48822

+

0.327009i

31.2

0.655837

+

2.01846i

−0.809017

+

0.587785i

−2.02602

+

1.47199i

−2.21700

−

0.291365i

−1.71700

−

1.24748i

4.35840

−0.865884

−

0.629102i

0.309017

−

0.951057i

−0.865884

−

4.66602i

46.1

−0.346820

+

1.06740i

−0.809017

−

0.587785i

0.598970

+

0.435177i

0.407987

+

2.19853i

0.907987

−

0.659691i

1.11373

−2.48822

+

1.80780i

0.309017

+

0.951057i

−2.48822

−

0.327009i

46.2

0.655837

−

2.01846i

−0.809017

−

0.587785i

−2.02602

−

1.47199i

−2.21700

+

0.291365i

−1.71700

+

1.24748i

4.35840

−0.865884

+

0.629102i

0.309017

+

0.951057i

−0.865884

+

4.66602i

61.1

−2.18949

+

1.59076i

0.309017

−

0.951057i

1.64533

−

5.06380i

0.336312

−

2.21063i

0.836312

+

2.57390i

0.470294

2.78023

+

8.55667i

−0.809017

−

0.587785i

2.78023

+

5.37515i

61.2

1.38048

−

1.00297i

0.309017

−

0.951057i

0.281722

−

0.867051i

−1.02729

+

1.98612i

−0.527295

−

1.62285i

−3.94243

0.573870

+

1.76619i

−0.809017

−

0.587785i

0.573870

+

3.77214i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
25.d	even	5	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	75.2.g.b	✓	8
3.b	odd	2	1		225.2.h.c		8
5.b	even	2	1		375.2.g.b		8
5.c	odd	4	2		375.2.i.b		16
25.d	even	5	1	inner	75.2.g.b	✓	8
25.d	even	5	1		1875.2.a.h		4
25.e	even	10	1		375.2.g.b		8
25.e	even	10	1		1875.2.a.e		4
25.f	odd	20	2		375.2.i.b		16
25.f	odd	20	2		1875.2.b.c		8
75.h	odd	10	1		5625.2.a.n		4
75.j	odd	10	1		225.2.h.c		8
75.j	odd	10	1		5625.2.a.i		4

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
75.2.g.b	✓	8	1.a	even	1	1	trivial
75.2.g.b	✓	8	25.d	even	5	1	inner
225.2.h.c		8	3.b	odd	2	1
225.2.h.c		8	75.j	odd	10	1
375.2.g.b		8	5.b	even	2	1
375.2.g.b		8	25.e	even	10	1
375.2.i.b		16	5.c	odd	4	2
375.2.i.b		16	25.f	odd	20	2
1875.2.a.e		4	25.e	even	10	1
1875.2.a.h		4	25.d	even	5	1
1875.2.b.c		8	25.f	odd	20	2
5625.2.a.i		4	75.j	odd	10	1
5625.2.a.n		4	75.h	odd	10	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{8} + T_{2}^{7} + 2T_{2}^{6} + 3T_{2}^{5} + 25T_{2}^{4} - 43T_{2}^{3} + 82T_{2}^{2} - 11T_{2} + 121 \) acting on \(S_{2}^{\mathrm{new}}(75, [\chi])\).

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	T^8 + T^7 + 2*T^6 + 3*T^5 + 25*T^4 - 43*T^3 + 82*T^2 - 11*T + 121
$3$	(T^4 + T^3 + T^2 + T + 1)^2
$5$	T^8 + 5*T^7 + 20*T^6 + 65*T^5 + 155*T^4 + 325*T^3 + 500*T^2 + 625*T + 625
$7$	(T^4 - 2*T^3 - 16*T^2 + 27*T - 9)^2
$11$	T^8 - 16*T^7 + 127*T^6 - 593*T^5 + 1930*T^4 - 4047*T^3 + 5957*T^2 - 6649*T + 11881