Newform orbit 399.2.x.a

• Label: 399.2.x.a
• Level: $399$
• Weight: $2$
• Character orbit: 399.x
• Analytic conductor: $3.186$
• Analytic rank: $0$
• Dimension: $2$
• CM discriminant: -3
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 399 = 3 \cdot 7 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	399.x (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.18603104065\)
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:	yes
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.

\(f(q)\)	\(=\)	\( q + ( - 2 \zeta_{6} + 1) q^{3} + ( - 2 \zeta_{6} + 2) q^{4} + ( - \zeta_{6} - 2) q^{7} - 3 q^{9}+O(q^{10}) \)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{4} - 5 q^{7} - 6 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/399\mathbb{Z}\right)^\times\).

\(n\)	\(115\)	\(134\)	\(211\)
\(\chi(n)\)	\(-1 + \zeta_{6}\)	\(-1\)	\(\zeta_{6}\)

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

−

1.73205i

1.00000

−

1.73205i

0

0

−2.50000

−

0.866025i

0

−3.00000

0

221.1

0

1.73205i

1.00000

+

1.73205i

0

0

−2.50000

+

0.866025i

0

−3.00000

0

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
3.b	odd	2	1	CM by \(\Q(\sqrt{-3}) \)
133.n	odd	6	1	inner
399.x	even	6	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	399.2.x.a	✓	2
3.b	odd	2	1	CM	399.2.x.a	✓	2
7.c	even	3	1		399.2.bm.a	yes	2
19.d	odd	6	1		399.2.bm.a	yes	2
21.h	odd	6	1		399.2.bm.a	yes	2
57.f	even	6	1		399.2.bm.a	yes	2
133.n	odd	6	1	inner	399.2.x.a	✓	2
399.x	even	6	1	inner	399.2.x.a	✓	2

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
399.2.x.a	✓	2	1.a	even	1	1	trivial
399.2.x.a	✓	2	3.b	odd	2	1	CM
399.2.x.a	✓	2	133.n	odd	6	1	inner
399.2.x.a	✓	2	399.x	even	6	1	inner
399.2.bm.a	yes	2	7.c	even	3	1
399.2.bm.a	yes	2	19.d	odd	6	1
399.2.bm.a	yes	2	21.h	odd	6	1
399.2.bm.a	yes	2	57.f	even	6	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(399, [\chi])\):

\( T_{2} \)

\( T_{13}^{2} - 9T_{13} + 27 \)

Hecke characteristic polynomials

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