Newform orbit 3150.3.c.c

• Label: 3150.3.c.c
• Level: $3150$
• Weight: $3$
• Character orbit: 3150.c
• Analytic conductor: $85.831$
• Analytic rank: $0$
• Dimension: $8$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 3150 = 2 \cdot 3^{2} \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	3150.c (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(85.8312832735\)
Analytic rank:	\(0\)
Dimension:	\(8\)
Coefficient field:	8.0.157351936.1
Defining polynomial:	\( x^{8} + x^{4} + 16 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index:	\( 2^{4} \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

$q$-expansion

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

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

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + x^{4} + 16 \) :

\(\beta_{1}\)	\(=\)	\( ( 2\nu^{4} + 1 ) / 3 \)
\(\beta_{2}\)	\(=\)	\( ( \nu^{6} + 5\nu^{2} ) / 12 \)
\(\beta_{3}\)	\(=\)	\( ( -\nu^{7} + 4\nu^{5} + 7\nu^{3} - 4\nu ) / 24 \)
\(\beta_{4}\)	\(=\)	\( ( -\nu^{6} + 3\nu^{2} ) / 4 \)
\(\beta_{5}\)	\(=\)	\( ( -\nu^{7} + 2\nu^{5} - 5\nu^{3} + 10\nu ) / 12 \)
\(\beta_{6}\)	\(=\)	\( ( -\nu^{7} - 4\nu^{5} + 7\nu^{3} + 4\nu ) / 24 \)
\(\beta_{7}\)	\(=\)	\( ( \nu^{7} + 2\nu^{5} + 5\nu^{3} + 10\nu ) / 12 \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3150\mathbb{Z}\right)^\times\).

\(n\)	\(127\)	\(451\)	\(2801\)
\(\chi(n)\)	\(-1\)	\(1\)	\(-1\)

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

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

449.1

0.581861	−	1.28897i
−1.28897	−	0.581861i
−1.28897	+	0.581861i
0.581861	+	1.28897i
1.28897	+	0.581861i
−0.581861	+	1.28897i
−0.581861	−	1.28897i
1.28897	−	0.581861i

−1.41421

0

2.00000

0

0

−

2.64575i

−2.82843

0

0

449.2

−1.41421

0

2.00000

0

0

−

2.64575i

−2.82843

0

0

449.3

−1.41421

0

2.00000

0

0

2.64575i

−2.82843

0

0

449.4

−1.41421

0

2.00000

0

0

2.64575i

−2.82843

0

0

449.5

1.41421

0

2.00000

0

0

−

2.64575i

2.82843

0

0

449.6

1.41421

0

2.00000

0

0

−

2.64575i

2.82843

0

0

449.7

1.41421

0

2.00000

0

0

2.64575i

2.82843

0

0

449.8

1.41421

0

2.00000

0

0

2.64575i

2.82843

0

0

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
3.b	odd	2	1	inner
5.b	even	2	1	inner
15.d	odd	2	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	3150.3.c.c		8
3.b	odd	2	1	inner	3150.3.c.c		8
5.b	even	2	1	inner	3150.3.c.c		8
5.c	odd	4	1		3150.3.e.a	✓	4
5.c	odd	4	1		3150.3.e.d	yes	4
15.d	odd	2	1	inner	3150.3.c.c		8
15.e	even	4	1		3150.3.e.a	✓	4
15.e	even	4	1		3150.3.e.d	yes	4

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
3150.3.c.c		8	1.a	even	1	1	trivial
3150.3.c.c		8	3.b	odd	2	1	inner
3150.3.c.c		8	5.b	even	2	1	inner
3150.3.c.c		8	15.d	odd	2	1	inner
3150.3.e.a	✓	4	5.c	odd	4	1
3150.3.e.a	✓	4	15.e	even	4	1
3150.3.e.d	yes	4	5.c	odd	4	1
3150.3.e.d	yes	4	15.e	even	4	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{11}^{4} + 464T_{11}^{2} + 12321 \) acting on \(S_{3}^{\mathrm{new}}(3150, [\chi])\).

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( (T^{2} - 2)^{4} \)
$3$	\( T^{8} \)
$5$	\( T^{8} \)
$7$	\( (T^{2} + 7)^{4} \)
$11$	\( (T^{4} + 464 T^{2} + 12321)^{2} \)