Newform orbit 252.2.bf.a

• Label: 252.2.bf.a
• Level: $252$
• Weight: $2$
• Character orbit: 252.bf
• Analytic conductor: $2.012$
• Analytic rank: $0$
• Dimension: $4$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 252 = 2^{2} \cdot 3^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	252.bf (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(2.01223013094\)
Analytic rank:	\(0\)
Dimension:	\(4\)
Relative dimension:	\(2\) over \(\Q(\zeta_{6})\)
Coefficient field:	\(\Q(\sqrt{-3}, \sqrt{-7})\)
Defining polynomial:	\( x^{4} - x^{3} - x^{2} - 2x + 4 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

$q$-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.

\(f(q)\)	\(=\)	\( q - \beta_{3} q^{2} + ( - \beta_{2} + \beta_1 + 1) q^{4} + ( - \beta_{2} + 2) q^{5} + (\beta_{2} + 2 \beta_1 - 1) q^{7} + ( - \beta_{3} + \beta_{2} + \beta_1 - 3) q^{8}+O(q^{10}) \)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - q^{2} + 3 q^{4} + 6 q^{5} - 10 q^{8}+O(q^{10}) \)

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

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/252\mathbb{Z}\right)^\times\).

\(n\)	\(29\)	\(73\)	\(127\)
\(\chi(n)\)	\(1\)	\(\beta_{2}\)	\(-1\)

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  

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

19.1

1.39564	−	0.228425i
−0.895644	+	1.09445i
1.39564	+	0.228425i
−0.895644	−	1.09445i

−1.39564

−

0.228425i

0

1.89564

+

0.637600i

1.50000

+

0.866025i

0

2.29129

−

1.32288i

−2.50000

−

1.32288i

0

−1.89564

−

1.55130i

19.2

0.895644

+

1.09445i

0

−0.395644

+

1.96048i

1.50000

+

0.866025i

0

−2.29129

+

1.32288i

−2.50000

+

1.32288i

0

0.395644

+

2.41733i

199.1

−1.39564

+

0.228425i

0

1.89564

−

0.637600i

1.50000

−

0.866025i

0

2.29129

+

1.32288i

−2.50000

+

1.32288i

0

−1.89564

+

1.55130i

199.2

0.895644

−

1.09445i

0

−0.395644

−

1.96048i

1.50000

−

0.866025i

0

−2.29129

−

1.32288i

−2.50000

−

1.32288i

0

0.395644

−

2.41733i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
4.b	odd	2	1	inner
7.d	odd	6	1	inner
28.f	even	6	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	252.2.bf.a	✓	4
3.b	odd	2	1		252.2.bf.d	yes	4
4.b	odd	2	1	inner	252.2.bf.a	✓	4
7.c	even	3	1		1764.2.b.h		4
7.d	odd	6	1	inner	252.2.bf.a	✓	4
7.d	odd	6	1		1764.2.b.h		4
12.b	even	2	1		252.2.bf.d	yes	4
21.g	even	6	1		252.2.bf.d	yes	4
21.g	even	6	1		1764.2.b.b		4
21.h	odd	6	1		1764.2.b.b		4
28.f	even	6	1	inner	252.2.bf.a	✓	4
28.f	even	6	1		1764.2.b.h		4
28.g	odd	6	1		1764.2.b.h		4
84.j	odd	6	1		252.2.bf.d	yes	4
84.j	odd	6	1		1764.2.b.b		4
84.n	even	6	1		1764.2.b.b		4

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
252.2.bf.a	✓	4	1.a	even	1	1	trivial
252.2.bf.a	✓	4	4.b	odd	2	1	inner
252.2.bf.a	✓	4	7.d	odd	6	1	inner
252.2.bf.a	✓	4	28.f	even	6	1	inner
252.2.bf.d	yes	4	3.b	odd	2	1
252.2.bf.d	yes	4	12.b	even	2	1
252.2.bf.d	yes	4	21.g	even	6	1
252.2.bf.d	yes	4	84.j	odd	6	1
1764.2.b.b		4	21.g	even	6	1
1764.2.b.b		4	21.h	odd	6	1
1764.2.b.b		4	84.j	odd	6	1
1764.2.b.b		4	84.n	even	6	1
1764.2.b.h		4	7.c	even	3	1
1764.2.b.h		4	7.d	odd	6	1
1764.2.b.h		4	28.f	even	6	1
1764.2.b.h		4	28.g	odd	6	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(252, [\chi])\):

\( T_{5}^{2} - 3T_{5} + 3 \)

\( T_{11}^{4} - 7T_{11}^{2} + 49 \)

\( T_{19} \)

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	T^4 + T^3 - T^2 + 2*T + 4
$3$	\( T^{4} \)
$5$	\( (T^{2} - 3 T + 3)^{2} \)
$7$	\( T^{4} - 7T^{2} + 49 \)
$11$	\( T^{4} - 7T^{2} + 49 \)