Newform orbit 378.2.f.a

• Label: 378.2.f.a
• Level: $378$
• Weight: $2$
• Character orbit: 378.f
• Analytic conductor: $3.018$
• Analytic rank: $1$
• Dimension: $2$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 378 = 2 \cdot 3^{3} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	378.f (of order \(3\), degree \(2\), not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.01834519640\)
Analytic rank:	\(1\)
Dimension:	\(2\)
Coefficient field:	\(\Q(\sqrt{-3}) \)
Defining polynomial:	\( x^{2} - x + 1 \)
Coefficient ring:	\(\Z[a_1, a_2]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 126)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

$q$-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.

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/378\mathbb{Z}\right)^\times\).

\(n\)	\(29\)	\(325\)
\(\chi(n)\)	\(-\zeta_{6}\)	\(1\)

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

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

127.1

0.500000	+	0.866025i
0.500000	−	0.866025i

−0.500000

+

0.866025i

0

−0.500000

−

0.866025i

−1.50000

−

2.59808i

0

−0.500000

+

0.866025i

1.00000

0

3.00000

253.1

−0.500000

−

0.866025i

0

−0.500000

+

0.866025i

−1.50000

+

2.59808i

0

−0.500000

−

0.866025i

1.00000

0

3.00000

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
9.c	even	3	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	378.2.f.a		2
3.b	odd	2	1		126.2.f.a	✓	2
4.b	odd	2	1		3024.2.r.a		2
7.b	odd	2	1		2646.2.f.c		2
7.c	even	3	1		2646.2.e.f		2
7.c	even	3	1		2646.2.h.e		2
7.d	odd	6	1		2646.2.e.j		2
7.d	odd	6	1		2646.2.h.a		2
9.c	even	3	1	inner	378.2.f.a		2
9.c	even	3	1		1134.2.a.h		1
9.d	odd	6	1		126.2.f.a	✓	2
9.d	odd	6	1		1134.2.a.a		1
12.b	even	2	1		1008.2.r.d		2
21.c	even	2	1		882.2.f.h		2
21.g	even	6	1		882.2.e.d		2
21.g	even	6	1		882.2.h.f		2
21.h	odd	6	1		882.2.e.b		2
21.h	odd	6	1		882.2.h.j		2
36.f	odd	6	1		3024.2.r.a		2
36.f	odd	6	1		9072.2.a.w		1
36.h	even	6	1		1008.2.r.d		2
36.h	even	6	1		9072.2.a.c		1
63.g	even	3	1		2646.2.e.f		2
63.h	even	3	1		2646.2.h.e		2
63.i	even	6	1		882.2.h.f		2
63.j	odd	6	1		882.2.h.j		2
63.k	odd	6	1		2646.2.e.j		2
63.l	odd	6	1		2646.2.f.c		2
63.l	odd	6	1		7938.2.a.u		1
63.n	odd	6	1		882.2.e.b		2
63.o	even	6	1		882.2.f.h		2
63.o	even	6	1		7938.2.a.l		1
63.s	even	6	1		882.2.e.d		2
63.t	odd	6	1		2646.2.h.a		2

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
126.2.f.a	✓	2	3.b	odd	2	1
126.2.f.a	✓	2	9.d	odd	6	1
378.2.f.a		2	1.a	even	1	1	trivial
378.2.f.a		2	9.c	even	3	1	inner
882.2.e.b		2	21.h	odd	6	1
882.2.e.b		2	63.n	odd	6	1
882.2.e.d		2	21.g	even	6	1
882.2.e.d		2	63.s	even	6	1
882.2.f.h		2	21.c	even	2	1
882.2.f.h		2	63.o	even	6	1
882.2.h.f		2	21.g	even	6	1
882.2.h.f		2	63.i	even	6	1
882.2.h.j		2	21.h	odd	6	1
882.2.h.j		2	63.j	odd	6	1
1008.2.r.d		2	12.b	even	2	1
1008.2.r.d		2	36.h	even	6	1
1134.2.a.a		1	9.d	odd	6	1
1134.2.a.h		1	9.c	even	3	1
2646.2.e.f		2	7.c	even	3	1
2646.2.e.f		2	63.g	even	3	1
2646.2.e.j		2	7.d	odd	6	1
2646.2.e.j		2	63.k	odd	6	1
2646.2.f.c		2	7.b	odd	2	1
2646.2.f.c		2	63.l	odd	6	1
2646.2.h.a		2	7.d	odd	6	1
2646.2.h.a		2	63.t	odd	6	1
2646.2.h.e		2	7.c	even	3	1
2646.2.h.e		2	63.h	even	3	1
3024.2.r.a		2	4.b	odd	2	1
3024.2.r.a		2	36.f	odd	6	1
7938.2.a.l		1	63.o	even	6	1
7938.2.a.u		1	63.l	odd	6	1
9072.2.a.c		1	36.h	even	6	1
9072.2.a.w		1	36.f	odd	6	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{2} + 3T_{5} + 9 \) acting on \(S_{2}^{\mathrm{new}}(378, [\chi])\).

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( T^{2} + T + 1 \)
$3$	\( T^{2} \)
$5$	\( T^{2} + 3T + 9 \)
$7$	\( T^{2} + T + 1 \)
$11$	\( T^{2} + 6T + 36 \)