Newform orbit 361.4.a.g

• Label: 361.4.a.g
• Level: $361$
• Weight: $4$
• Character orbit: 361.a
• Self dual: yes
• Analytic conductor: $21.300$
• Analytic rank: $0$
• Dimension: $2$
• CM: no
• Inner twists: $1$

Newspace parameters

Level:	\( N \)	\(=\)	\( 361 = 19^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	361.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	\(21.2996895121\)
Analytic rank:	\(0\)
Dimension:	\(2\)
Coefficient field:	\(\Q(\sqrt{55}) \)
Defining polynomial:	\( x^{2} - 55 \)
Coefficient ring:	\(\Z[a_1, a_2, a_3]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 19)
Fricke sign:	\(1\)
Sato-Tate group:	$\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{55}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\)	\(=\)	\( q + q^{2} + \beta q^{3} - 7 q^{4} + ( - \beta - 7) q^{5} + \beta q^{6} + (\beta - 7) q^{7} - 15 q^{8} + 28 q^{9}+O(q^{10}) \)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{2} - 14 q^{4} - 14 q^{5} - 14 q^{7} - 30 q^{8} + 56 q^{9}+O(q^{10}) \)

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  

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

1.1

−7.41620
7.41620

1.00000

−7.41620

−7.00000

0.416198

−7.41620

−14.4162

−15.0000

28.0000

0.416198

1.2

1.00000

7.41620

−7.00000

−14.4162

7.41620

0.416198

−15.0000

28.0000

−14.4162

Atkin-Lehner signs

\( p \)	Sign
\(19\)	\(1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	361.4.a.g		2
19.b	odd	2	1		361.4.a.d		2
19.c	even	3	2		19.4.c.a	✓	4
57.h	odd	6	2		171.4.f.e		4
76.g	odd	6	2		304.4.i.c		4

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
19.4.c.a	✓	4	19.c	even	3	2
171.4.f.e		4	57.h	odd	6	2
304.4.i.c		4	76.g	odd	6	2
361.4.a.d		2	19.b	odd	2	1
361.4.a.g		2	1.a	even	1	1	trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2} - 1 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(361))\).

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( (T - 1)^{2} \)
$3$	\( T^{2} - 55 \)
$5$	\( T^{2} + 14T - 6 \)
$7$	\( T^{2} + 14T - 6 \)
$11$	\( T^{2} - 28T - 2499 \)