Newform orbit 361.3.d.a

• Label: 361.3.d.a
• Level: $361$
• Weight: $3$
• Character orbit: 361.d
• Analytic conductor: $9.837$
• Analytic rank: $0$
• Dimension: $2$
• CM discriminant: -19
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 361 = 19^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	361.d (of order \(6\), degree \(2\), not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(9.83653754341\)
Analytic rank:	\(0\)
Dimension:	\(2\)
Coefficient field:	\(\Q(\sqrt{-3}) \)
Defining polynomial:	\( x^{2} - x + 1 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 19)
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.

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/361\mathbb{Z}\right)^\times\).

\(n\)	\(2\)
\(\chi(n)\)	\(\zeta_{6}\)

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  

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

69.1

0.500000	−	0.866025i
0.500000	+	0.866025i

0

0

−2.00000

+

3.46410i

4.50000

+

7.79423i

0

−5.00000

0

−4.50000

+

7.79423i

0

293.1

0

0

−2.00000

−

3.46410i

4.50000

−

7.79423i

0

−5.00000

0

−4.50000

−

7.79423i

0

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
19.b	odd	2	1	CM by \(\Q(\sqrt{-19}) \)
19.c	even	3	1	inner
19.d	odd	6	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	361.3.d.a		2
19.b	odd	2	1	CM	361.3.d.a		2
19.c	even	3	1		19.3.b.a	✓	1
19.c	even	3	1	inner	361.3.d.a		2
19.d	odd	6	1		19.3.b.a	✓	1
19.d	odd	6	1	inner	361.3.d.a		2
19.e	even	9	6		361.3.f.a		6
19.f	odd	18	6		361.3.f.a		6
57.f	even	6	1		171.3.c.a		1
57.h	odd	6	1		171.3.c.a		1
76.f	even	6	1		304.3.e.a		1
76.g	odd	6	1		304.3.e.a		1
95.h	odd	6	1		475.3.c.a		1
95.i	even	6	1		475.3.c.a		1
95.l	even	12	2		475.3.d.a		2
95.m	odd	12	2		475.3.d.a		2
152.k	odd	6	1		1216.3.e.b		1
152.l	odd	6	1		1216.3.e.a		1
152.o	even	6	1		1216.3.e.b		1
152.p	even	6	1		1216.3.e.a		1
228.m	even	6	1		2736.3.o.a		1
228.n	odd	6	1		2736.3.o.a		1

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
19.3.b.a	✓	1	19.c	even	3	1
19.3.b.a	✓	1	19.d	odd	6	1
171.3.c.a		1	57.f	even	6	1
171.3.c.a		1	57.h	odd	6	1
304.3.e.a		1	76.f	even	6	1
304.3.e.a		1	76.g	odd	6	1
361.3.d.a		2	1.a	even	1	1	trivial
361.3.d.a		2	19.b	odd	2	1	CM
361.3.d.a		2	19.c	even	3	1	inner
361.3.d.a		2	19.d	odd	6	1	inner
361.3.f.a		6	19.e	even	9	6
361.3.f.a		6	19.f	odd	18	6
475.3.c.a		1	95.h	odd	6	1
475.3.c.a		1	95.i	even	6	1
475.3.d.a		2	95.l	even	12	2
475.3.d.a		2	95.m	odd	12	2
1216.3.e.a		1	152.l	odd	6	1
1216.3.e.a		1	152.p	even	6	1
1216.3.e.b		1	152.k	odd	6	1
1216.3.e.b		1	152.o	even	6	1
2736.3.o.a		1	228.m	even	6	1
2736.3.o.a		1	228.n	odd	6	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2} \) acting on \(S_{3}^{\mathrm{new}}(361, [\chi])\).

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( T^{2} \)
$3$	\( T^{2} \)
$5$	\( T^{2} - 9T + 81 \)
$7$	\( (T + 5)^{2} \)
$11$	\( (T - 3)^{2} \)