Newform orbit 361.2.a.e

• Label: 361.2.a.e
• Level: $361$
• Weight: $2$
• Character orbit: 361.a
• Self dual: yes
• Analytic conductor: $2.883$
• Analytic rank: $0$
• Dimension: $2$
• CM: no
• Inner twists: $1$

Newspace parameters

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

Newform invariants

Self dual:	yes
Analytic conductor:	\(2.88259951297\)
Analytic rank:	\(0\)
Dimension:	\(2\)
Coefficient field:	\(\Q(\sqrt{5}) \)
Defining polynomial:	\( x^{2} - x - 1 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
Fricke sign:	\(-1\)
Sato-Tate group:	$\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \frac{1}{2}(1 + \sqrt{5})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\)	\(=\)	q + (-2*b + 1) * q^2 + 2 * q^3 + 3 * q^4 + b * q^5 + (-4*b + 2) * q^6 + (2*b - 2) * q^7 + (-2*b + 1) * q^8 + q^9 + (-b - 2) * q^10 + (2*b + 2) * q^11 + 6 * q^12 + (-3*b + 3) * q^13 + (2*b - 6) * q^14 + 2*b * q^15 - q^16 + (b - 1) * q^17 + (-2*b + 1) * q^18 + 3*b * q^20 + (4*b - 4) * q^21 + (-6*b - 2) * q^22 + (-4*b + 2) * q^23 + (-4*b + 2) * q^24 + (b - 4) * q^25 + (-3*b + 9) * q^26 - 4 * q^27 + (6*b - 6) * q^28 + (5*b - 1) * q^29 + (-2*b - 4) * q^30 - 6 * q^31 + (6*b - 3) * q^32 + (4*b + 4) * q^33 + (b - 3) * q^34 + 2 * q^35 + 3 * q^36 + (3*b + 4) * q^37 + (-6*b + 6) * q^39 + (-b - 2) * q^40 + (5*b - 4) * q^41 + (4*b - 12) * q^42 + (-4*b + 6) * q^43 + (6*b + 6) * q^44 + b * q^45 + 10 * q^46 + (-2*b + 8) * q^47 - 2 * q^48 + (-4*b + 1) * q^49 + (7*b - 6) * q^50 + (2*b - 2) * q^51 + (-9*b + 9) * q^52 + (-5*b + 8) * q^53 + (8*b - 4) * q^54 + (4*b + 2) * q^55 + (2*b - 6) * q^56 + (-3*b - 11) * q^58 + 2*b * q^59 + 6*b * q^60 + (3*b + 5) * q^61 + (12*b - 6) * q^62 + (2*b - 2) * q^63 - 13 * q^64 - 3 * q^65 + (-12*b - 4) * q^66 + (-4*b + 6) * q^67 + (3*b - 3) * q^68 + (-8*b + 4) * q^69 + (-4*b + 2) * q^70 - 2*b * q^71 + (-2*b + 1) * q^72 - 9*b * q^73 + (-11*b - 2) * q^74 + (2*b - 8) * q^75 + 4*b * q^77 + (-6*b + 18) * q^78 + 2 * q^79 - b * q^80 - 11 * q^81 + (3*b - 14) * q^82 - 6*b * q^83 + (12*b - 12) * q^84 + q^85 + (-8*b + 14) * q^86 + (10*b - 2) * q^87 + (-6*b - 2) * q^88 + (-7*b + 4) * q^89 + (-b - 2) * q^90 + (6*b - 12) * q^91 + (-12*b + 6) * q^92 - 12 * q^93 + (-14*b + 12) * q^94 + (12*b - 6) * q^96 + (-5*b - 3) * q^97 + (2*b + 9) * q^98 + (2*b + 2) * q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	2 * q + 4 * q^3 + 6 * q^4 + q^5 - 2 * q^7 + 2 * q^9 - 5 * q^10 + 6 * q^11 + 12 * q^12 + 3 * q^13 - 10 * q^14 + 2 * q^15 - 2 * q^16 - q^17 + 3 * q^20 - 4 * q^21 - 10 * q^22 - 7 * q^25 + 15 * q^26 - 8 * q^27 - 6 * q^28 + 3 * q^29 - 10 * q^30 - 12 * q^31 + 12 * q^33 - 5 * q^34 + 4 * q^35 + 6 * q^36 + 11 * q^37 + 6 * q^39 - 5 * q^40 - 3 * q^41 - 20 * q^42 + 8 * q^43 + 18 * q^44 + q^45 + 20 * q^46 + 14 * q^47 - 4 * q^48 - 2 * q^49 - 5 * q^50 - 2 * q^51 + 9 * q^52 + 11 * q^53 + 8 * q^55 - 10 * q^56 - 25 * q^58 + 2 * q^59 + 6 * q^60 + 13 * q^61 - 2 * q^63 - 26 * q^64 - 6 * q^65 - 20 * q^66 + 8 * q^67 - 3 * q^68 - 2 * q^71 - 9 * q^73 - 15 * q^74 - 14 * q^75 + 4 * q^77 + 30 * q^78 + 4 * q^79 - q^80 - 22 * q^81 - 25 * q^82 - 6 * q^83 - 12 * q^84 + 2 * q^85 + 20 * q^86 + 6 * q^87 - 10 * q^88 + q^89 - 5 * q^90 - 18 * q^91 - 24 * q^93 + 10 * q^94 - 11 * q^97 + 20 * q^98 + 6 * q^99

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

1.61803
−0.618034

−2.23607

2.00000

3.00000

1.61803

−4.47214

1.23607

−2.23607

1.00000

−3.61803

1.2

2.23607

2.00000

3.00000

−0.618034

4.47214

−3.23607

2.23607

1.00000

−1.38197

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	361.2.a.e	yes	2
3.b	odd	2	1		3249.2.a.m		2
4.b	odd	2	1		5776.2.a.r		2
5.b	even	2	1		9025.2.a.o		2
19.b	odd	2	1		361.2.a.d	✓	2
19.c	even	3	2		361.2.c.e		4
19.d	odd	6	2		361.2.c.f		4
19.e	even	9	6		361.2.e.k		12
19.f	odd	18	6		361.2.e.l		12
57.d	even	2	1		3249.2.a.n		2
76.d	even	2	1		5776.2.a.bh		2
95.d	odd	2	1		9025.2.a.r		2

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
361.2.a.d	✓	2	19.b	odd	2	1
361.2.a.e	yes	2	1.a	even	1	1	trivial
361.2.c.e		4	19.c	even	3	2
361.2.c.f		4	19.d	odd	6	2
361.2.e.k		12	19.e	even	9	6
361.2.e.l		12	19.f	odd	18	6
3249.2.a.m		2	3.b	odd	2	1
3249.2.a.n		2	57.d	even	2	1
5776.2.a.r		2	4.b	odd	2	1
5776.2.a.bh		2	76.d	even	2	1
9025.2.a.o		2	5.b	even	2	1
9025.2.a.r		2	95.d	odd	2	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(361))\):

\( T_{2}^{2} - 5 \)

\( T_{3} - 2 \)

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( T^{2} - 5 \)
$3$	\( (T - 2)^{2} \)
$5$	\( T^{2} - T - 1 \)
$7$	\( T^{2} + 2T - 4 \)
$11$	\( T^{2} - 6T + 4 \)