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*z + 1) * q^3 + (-2*z + 2) * q^4 + (-z - 2) * q^7 - 3 * q^9 + (-2*z - 2) * q^12 + (-3*z + 6) * q^13 - 4*z * q^16 + (2*z - 5) * q^19 + (5*z - 4) * q^21 + (5*z - 5) * q^25 + (6*z - 3) * q^27 + (4*z - 6) * q^28 + (-6*z + 12) * q^31 + (6*z - 6) * q^36 + (-3*z - 3) * q^37 - 9*z * q^39 + (-13*z + 13) * q^43 + (4*z - 8) * q^48 + (5*z + 3) * q^49 + (-12*z + 6) * q^52 + (8*z - 1) * q^57 + 13 * q^61 + (3*z + 6) * q^63 - 8 * q^64 + (-2*z - 2) * q^67 + 17 * q^73 + (5*z + 5) * q^75 + (10*z - 6) * q^76 + (3*z - 6) * q^79 + 9 * q^81 + (8*z + 2) * q^84 + (3*z - 15) * q^91 - 18*z * q^93 + (8*z + 8) * q^97
$\operatorname{Tr}(f)(q)$	$=$	2 * q + 2 * q^4 - 5 * q^7 - 6 * q^9 - 6 * q^12 + 9 * q^13 - 4 * q^16 - 8 * q^19 - 3 * q^21 - 5 * q^25 - 8 * q^28 + 18 * q^31 - 6 * q^36 - 9 * q^37 - 9 * q^39 + 13 * q^43 - 12 * q^48 + 11 * q^49 + 6 * q^57 + 26 * q^61 + 15 * q^63 - 16 * q^64 - 6 * q^67 + 34 * q^73 + 15 * q^75 - 2 * q^76 - 9 * q^79 + 18 * q^81 + 12 * q^84 - 27 * q^91 - 18 * q^93 + 24 * q^97

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

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