Newform orbit 336.3.bn.a

• Label: 336.3.bn.a
• Level: $336$
• Weight: $3$
• Character orbit: 336.bn
• Analytic conductor: $9.155$
• Analytic rank: $0$
• Dimension: $2$
• CM discriminant: -3
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 336 = 2^{4} \cdot 3 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	336.bn (of order \(6\), degree \(2\), not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(9.15533688251\)
Analytic rank:	\(0\)
Dimension:	\(2\)
Coefficient field:	\(\Q(\sqrt{-3}) \)
Defining polynomial:	\( x^{2} - x + 1 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 84)
Sato-Tate group:	$\mathrm{U}(1)[D_{6}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\)	\(=\)	\( q - 3 \zeta_{6} q^{3} + ( - 5 \zeta_{6} - 3) q^{7} + (9 \zeta_{6} - 9) q^{9} +O(q^{10}) \)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 3 q^{3} - 11 q^{7} - 9 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/336\mathbb{Z}\right)^\times\).

\(n\)	\(85\)	\(113\)	\(127\)	\(241\)
\(\chi(n)\)	\(1\)	\(-1\)	\(1\)	\(-1 + \zeta_{6}\)

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  

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

65.1

0.500000	+	0.866025i
0.500000	−	0.866025i

0

−1.50000

−

2.59808i

0

0

0

−5.50000

−

4.33013i

0

−4.50000

+

7.79423i

0

305.1

0

−1.50000

+

2.59808i

0

0

0

−5.50000

+

4.33013i

0

−4.50000

−

7.79423i

0

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
3.b	odd	2	1	CM by \(\Q(\sqrt{-3}) \)
7.c	even	3	1	inner
21.h	odd	6	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	336.3.bn.a		2
3.b	odd	2	1	CM	336.3.bn.a		2
4.b	odd	2	1		84.3.p.a	✓	2
7.c	even	3	1	inner	336.3.bn.a		2
12.b	even	2	1		84.3.p.a	✓	2
21.h	odd	6	1	inner	336.3.bn.a		2
28.d	even	2	1		588.3.p.a		2
28.f	even	6	1		588.3.c.b		1
28.f	even	6	1		588.3.p.a		2
28.g	odd	6	1		84.3.p.a	✓	2
28.g	odd	6	1		588.3.c.a		1
84.h	odd	2	1		588.3.p.a		2
84.j	odd	6	1		588.3.c.b		1
84.j	odd	6	1		588.3.p.a		2
84.n	even	6	1		84.3.p.a	✓	2
84.n	even	6	1		588.3.c.a		1

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
84.3.p.a	✓	2	4.b	odd	2	1
84.3.p.a	✓	2	12.b	even	2	1
84.3.p.a	✓	2	28.g	odd	6	1
84.3.p.a	✓	2	84.n	even	6	1
336.3.bn.a		2	1.a	even	1	1	trivial
336.3.bn.a		2	3.b	odd	2	1	CM
336.3.bn.a		2	7.c	even	3	1	inner
336.3.bn.a		2	21.h	odd	6	1	inner
588.3.c.a		1	28.g	odd	6	1
588.3.c.a		1	84.n	even	6	1
588.3.c.b		1	28.f	even	6	1
588.3.c.b		1	84.j	odd	6	1
588.3.p.a		2	28.d	even	2	1
588.3.p.a		2	28.f	even	6	1
588.3.p.a		2	84.h	odd	2	1
588.3.p.a		2	84.j	odd	6	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(336, [\chi])\):

\( T_{5} \)

\( T_{13} + 1 \)

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( T^{2} \)
$3$	\( T^{2} + 3T + 9 \)
$5$	\( T^{2} \)
$7$	\( T^{2} + 11T + 49 \)
$11$	\( T^{2} \)