Newform orbit 756.2.n.a

• Label: 756.2.n.a
• Level: $756$
• Weight: $2$
• Character orbit: 756.n
• Analytic conductor: $6.037$
• Analytic rank: $0$
• Dimension: $4$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 756 = 2^{2} \cdot 3^{3} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	756.n (of order \(6\), degree \(2\), not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(6.03669039281\)
Analytic rank:	\(0\)
Dimension:	\(4\)
Relative dimension:	\(2\) over \(\Q(\zeta_{6})\)
Coefficient field:	\(\Q(\zeta_{12})\)
Defining polynomial:	\( x^{4} - x^{2} + 1 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 252)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{12}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\)	\(=\)	\( q + (\zeta_{12}^{2} + \zeta_{12} - 1) q^{2} + (2 \zeta_{12}^{3} - 2 \zeta_{12}) q^{4} + ( - 4 \zeta_{12}^{2} + 2) q^{5} + (3 \zeta_{12}^{3} - 2 \zeta_{12}) q^{7} + ( - 2 \zeta_{12}^{3} - 2) q^{8}+O(q^{10}) \)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 2 q^{2} - 8 q^{8}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/756\mathbb{Z}\right)^\times\).

\(n\)	\(29\)	\(325\)	\(379\)
\(\chi(n)\)	\(-1 + \zeta_{12}^{2}\)	\(\zeta_{12}^{2}\)	\(-1\)

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

−0.866025	+	0.500000i
0.866025	−	0.500000i
−0.866025	−	0.500000i
0.866025	+	0.500000i

−1.36603

−

0.366025i

0

1.73205

+

1.00000i

3.46410i

0

1.73205

+

2.00000i

−2.00000

−

2.00000i

0

1.26795

−

4.73205i

19.2

0.366025

−

1.36603i

0

−1.73205

−

1.00000i

3.46410i

0

−1.73205

−

2.00000i

−2.00000

+

2.00000i

0

4.73205

+

1.26795i

199.1

−1.36603

+

0.366025i

0

1.73205

−

1.00000i

−

3.46410i

0

1.73205

−

2.00000i

−2.00000

+

2.00000i

0

1.26795

+

4.73205i

199.2

0.366025

+

1.36603i

0

−1.73205

+

1.00000i

−

3.46410i

0

−1.73205

+

2.00000i

−2.00000

−

2.00000i

0

4.73205

−

1.26795i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
4.b	odd	2	1	inner
63.k	odd	6	1	inner
252.n	even	6	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	756.2.n.a		4
3.b	odd	2	1		252.2.n.a	✓	4
4.b	odd	2	1	inner	756.2.n.a		4
7.d	odd	6	1		756.2.bj.a		4
9.c	even	3	1		756.2.bj.a		4
9.d	odd	6	1		252.2.bj.a	yes	4
12.b	even	2	1		252.2.n.a	✓	4
21.g	even	6	1		252.2.bj.a	yes	4
28.f	even	6	1		756.2.bj.a		4
36.f	odd	6	1		756.2.bj.a		4
36.h	even	6	1		252.2.bj.a	yes	4
63.k	odd	6	1	inner	756.2.n.a		4
63.s	even	6	1		252.2.n.a	✓	4
84.j	odd	6	1		252.2.bj.a	yes	4
252.n	even	6	1	inner	756.2.n.a		4
252.bn	odd	6	1		252.2.n.a	✓	4

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
252.2.n.a	✓	4	3.b	odd	2	1
252.2.n.a	✓	4	12.b	even	2	1
252.2.n.a	✓	4	63.s	even	6	1
252.2.n.a	✓	4	252.bn	odd	6	1
252.2.bj.a	yes	4	9.d	odd	6	1
252.2.bj.a	yes	4	21.g	even	6	1
252.2.bj.a	yes	4	36.h	even	6	1
252.2.bj.a	yes	4	84.j	odd	6	1
756.2.n.a		4	1.a	even	1	1	trivial
756.2.n.a		4	4.b	odd	2	1	inner
756.2.n.a		4	63.k	odd	6	1	inner
756.2.n.a		4	252.n	even	6	1	inner
756.2.bj.a		4	7.d	odd	6	1
756.2.bj.a		4	9.c	even	3	1
756.2.bj.a		4	28.f	even	6	1
756.2.bj.a		4	36.f	odd	6	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{2} + 12 \) acting on \(S_{2}^{\mathrm{new}}(756, [\chi])\).

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	T^4 + 2*T^3 + 2*T^2 + 4*T + 4
$3$	\( T^{4} \)
$5$	\( (T^{2} + 12)^{2} \)
$7$	\( T^{4} + 2T^{2} + 49 \)
$11$	\( (T^{2} + 4)^{2} \)