Newform orbit 5247.2.a.e

• Label: 5247.2.a.e
• Level: $5247$
• Weight: $2$
• Character orbit: 5247.a
• Self dual: yes
• Analytic conductor: $41.898$
• Analytic rank: $0$
• Dimension: $2$
• CM: no
• Inner twists: $1$

Newspace parameters

Level:	\( N \)	\(=\)	\( 5247 = 3^{2} \cdot 11 \cdot 53 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5247.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	\(41.8975059406\)
Analytic rank:	\(0\)
Dimension:	\(2\)
Coefficient field:	\(\Q(\sqrt{2}) \)
Defining polynomial:	\( x^{2} - 2 \)
Coefficient ring:	\(\Z[a_1, a_2]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 583)
Fricke sign:	\(-1\)
Sato-Tate group:	$\mathrm{SU}(2)$

$q$-expansion

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

\(f(q)\)	\(=\)	q + b * q^2 + (-b + 3) * q^5 + (-b + 2) * q^7 - 2*b * q^8 + (3*b - 2) * q^10 + q^11 + b * q^13 + (2*b - 2) * q^14 - 4 * q^16 + (-4*b + 2) * q^17 + (2*b + 4) * q^19 + b * q^22 + (b + 3) * q^23 + (-6*b + 6) * q^25 + 2 * q^26 + (3*b - 4) * q^29 + (3*b - 1) * q^31 + (2*b - 8) * q^34 + (-5*b + 8) * q^35 + (-2*b - 9) * q^37 + (4*b + 4) * q^38 + (-6*b + 4) * q^40 + (3*b + 2) * q^41 + (b + 6) * q^43 + (3*b + 2) * q^46 + 10 * q^47 + (-4*b - 1) * q^49 + (6*b - 12) * q^50 + q^53 + (-b + 3) * q^55 + (-4*b + 4) * q^56 + (-4*b + 6) * q^58 + (2*b + 3) * q^59 + (b + 10) * q^61 + (-b + 6) * q^62 + 8 * q^64 + (3*b - 2) * q^65 + (3*b - 3) * q^67 + (8*b - 10) * q^70 + (-5*b + 1) * q^71 + (-4*b - 4) * q^73 + (-9*b - 4) * q^74 + (-b + 2) * q^77 + (2*b + 10) * q^79 + (4*b - 12) * q^80 + (2*b + 6) * q^82 + (6*b - 6) * q^83 + (-14*b + 14) * q^85 + (6*b + 2) * q^86 - 2*b * q^88 + (2*b + 7) * q^89 + (2*b - 2) * q^91 + 10*b * q^94 + (2*b + 8) * q^95 + (6*b - 7) * q^97 + (-b - 8) * q^98
\(\operatorname{Tr}(f)(q)\)	\(=\)	2 * q + 6 * q^5 + 4 * q^7 - 4 * q^10 + 2 * q^11 - 4 * q^14 - 8 * q^16 + 4 * q^17 + 8 * q^19 + 6 * q^23 + 12 * q^25 + 4 * q^26 - 8 * q^29 - 2 * q^31 - 16 * q^34 + 16 * q^35 - 18 * q^37 + 8 * q^38 + 8 * q^40 + 4 * q^41 + 12 * q^43 + 4 * q^46 + 20 * q^47 - 2 * q^49 - 24 * q^50 + 2 * q^53 + 6 * q^55 + 8 * q^56 + 12 * q^58 + 6 * q^59 + 20 * q^61 + 12 * q^62 + 16 * q^64 - 4 * q^65 - 6 * q^67 - 20 * q^70 + 2 * q^71 - 8 * q^73 - 8 * q^74 + 4 * q^77 + 20 * q^79 - 24 * q^80 + 12 * q^82 - 12 * q^83 + 28 * q^85 + 4 * q^86 + 14 * q^89 - 4 * q^91 + 16 * q^95 - 14 * q^97 - 16 * 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

−1.41421
1.41421

−1.41421

0

0

4.41421

0

3.41421

2.82843

0

−6.24264

1.2

1.41421

0

0

1.58579

0

0.585786

−2.82843

0

2.24264

Atkin-Lehner signs

\( p \)	Sign
\(3\)	\( -1 \)
\(11\)	\( -1 \)
\(53\)	\( -1 \)

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	5247.2.a.e		2
3.b	odd	2	1		583.2.a.e	✓	2
12.b	even	2	1		9328.2.a.q		2
33.d	even	2	1		6413.2.a.m		2

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
583.2.a.e	✓	2	3.b	odd	2	1
5247.2.a.e		2	1.a	even	1	1	trivial
6413.2.a.m		2	33.d	even	2	1
9328.2.a.q		2	12.b	even	2	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(5247))\):

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

\( T_{5}^{2} - 6T_{5} + 7 \)

Hecke characteristic polynomials

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