Newform orbit 4563.2.a.h

• Label: 4563.2.a.h
• Level: $4563$
• Weight: $2$
• Character orbit: 4563.a
• Self dual: yes
• Analytic conductor: $36.436$
• Analytic rank: $0$
• Dimension: $2$
• CM: no
• Inner twists: $1$

Newspace parameters

Level:	\( N \)	\(=\)	\( 4563 = 3^{3} \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4563.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	\(36.4357384423\)
Analytic rank:	\(0\)
Dimension:	\(2\)
Coefficient field:	\(\Q(\sqrt{5}) \)
Defining polynomial:	\( x^{2} - x - 1 \)
Coefficient ring:	\(\Z[a_1, a_2]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 351)
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 - b * q^2 + (b - 1) * q^4 + (-b - 1) * q^5 + (-2*b + 1) * q^7 + (2*b - 1) * q^8 + (2*b + 1) * q^10 + (3*b - 4) * q^11 + (b + 2) * q^14 - 3*b * q^16 + (b + 3) * q^17 + (5*b - 4) * q^19 - b * q^20 + (b - 3) * q^22 + (3*b + 2) * q^23 + (3*b - 3) * q^25 + (b - 3) * q^28 + (-6*b + 5) * q^29 + (4*b + 2) * q^31 + (-b + 5) * q^32 + (-4*b - 1) * q^34 + (3*b + 1) * q^35 + (-2*b - 3) * q^37 + (-b - 5) * q^38 + (-3*b - 1) * q^40 + (2*b - 3) * q^41 - 7*b * q^43 + (-4*b + 7) * q^44 + (-5*b - 3) * q^46 + (-5*b + 7) * q^47 - 2 * q^49 - 3 * q^50 + (-3*b + 7) * q^53 + (-2*b + 1) * q^55 - 5 * q^56 + (b + 6) * q^58 + (2*b + 3) * q^59 + (-10*b + 1) * q^61 + (-6*b - 4) * q^62 + (2*b + 1) * q^64 + (-5*b + 3) * q^67 + (3*b - 2) * q^68 + (-4*b - 3) * q^70 + (-6*b + 10) * q^71 + (b - 5) * q^73 + (5*b + 2) * q^74 + (-4*b + 9) * q^76 + (5*b - 10) * q^77 + (7*b - 8) * q^79 + (6*b + 3) * q^80 + (b - 2) * q^82 + (8*b + 1) * q^83 + (-5*b - 4) * q^85 + (7*b + 7) * q^86 + (-5*b + 10) * q^88 + (-2*b - 14) * q^89 + (2*b + 1) * q^92 + (-2*b + 5) * q^94 + (-6*b - 1) * q^95 + (-4*b + 4) * q^97 + 2*b * q^98
\(\operatorname{Tr}(f)(q)\)	\(=\)	2 * q - q^2 - q^4 - 3 * q^5 + 4 * q^10 - 5 * q^11 + 5 * q^14 - 3 * q^16 + 7 * q^17 - 3 * q^19 - q^20 - 5 * q^22 + 7 * q^23 - 3 * q^25 - 5 * q^28 + 4 * q^29 + 8 * q^31 + 9 * q^32 - 6 * q^34 + 5 * q^35 - 8 * q^37 - 11 * q^38 - 5 * q^40 - 4 * q^41 - 7 * q^43 + 10 * q^44 - 11 * q^46 + 9 * q^47 - 4 * q^49 - 6 * q^50 + 11 * q^53 - 10 * q^56 + 13 * q^58 + 8 * q^59 - 8 * q^61 - 14 * q^62 + 4 * q^64 + q^67 - q^68 - 10 * q^70 + 14 * q^71 - 9 * q^73 + 9 * q^74 + 14 * q^76 - 15 * q^77 - 9 * q^79 + 12 * q^80 - 3 * q^82 + 10 * q^83 - 13 * q^85 + 21 * q^86 + 15 * q^88 - 30 * q^89 + 4 * q^92 + 8 * q^94 - 8 * q^95 + 4 * q^97 + 2 * q^98

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

−1.61803

0

0.618034

−2.61803

0

−2.23607

2.23607

0

4.23607

1.2

0.618034

0

−1.61803

−0.381966

0

2.23607

−2.23607

0

−0.236068

Atkin-Lehner signs

\( p \)	Sign
\(3\)	\( -1 \)
\(13\)	\( +1 \)

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	4563.2.a.h		2
3.b	odd	2	1		4563.2.a.p		2
13.b	even	2	1		351.2.a.c	yes	2
39.d	odd	2	1		351.2.a.a	✓	2
52.b	odd	2	1		5616.2.a.bx		2
65.d	even	2	1		8775.2.a.u		2
117.n	odd	6	2		1053.2.e.l		4
117.t	even	6	2		1053.2.e.g		4
156.h	even	2	1		5616.2.a.bl		2
195.e	odd	2	1		8775.2.a.bb		2

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
351.2.a.a	✓	2	39.d	odd	2	1
351.2.a.c	yes	2	13.b	even	2	1
1053.2.e.g		4	117.t	even	6	2
1053.2.e.l		4	117.n	odd	6	2
4563.2.a.h		2	1.a	even	1	1	trivial
4563.2.a.p		2	3.b	odd	2	1
5616.2.a.bl		2	156.h	even	2	1
5616.2.a.bx		2	52.b	odd	2	1
8775.2.a.u		2	65.d	even	2	1
8775.2.a.bb		2	195.e	odd	2	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4563))\):

\( T_{2}^{2} + T_{2} - 1 \)

\( T_{5}^{2} + 3T_{5} + 1 \)

\( T_{7}^{2} - 5 \)

\( T_{17}^{2} - 7T_{17} + 11 \)

\( T_{19}^{2} + 3T_{19} - 29 \)

\( T_{23}^{2} - 7T_{23} + 1 \)

Hecke characteristic polynomials

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