Newform orbit 243.4.a.e

• Label: 243.4.a.e
• Level: $243$
• Weight: $4$
• Character orbit: 243.a
• Self dual: yes
• Analytic conductor: $14.337$
• Analytic rank: $1$
• Dimension: $2$
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 243 = 3^{5} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	243.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	\(14.3374641314\)
Analytic rank:	\(1\)
Dimension:	\(2\)
Coefficient field:	\(\Q(\sqrt{6}) \)
Defining polynomial:	\( x^{2} - 6 \)
Coefficient ring:	\(\Z[a_1, a_2]\)
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 \(\beta = \sqrt{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\)	\(=\)	q + b * q^2 - 2 * q^4 + 2*b * q^5 - 4 * q^7 - 10*b * q^8 + 12 * q^10 - 26*b * q^11 - 31 * q^13 - 4*b * q^14 - 44 * q^16 + 6*b * q^17 + 35 * q^19 - 4*b * q^20 - 156 * q^22 + 38*b * q^23 - 101 * q^25 - 31*b * q^26 + 8 * q^28 - 56*b * q^29 - 265 * q^31 + 36*b * q^32 + 36 * q^34 - 8*b * q^35 + 134 * q^37 + 35*b * q^38 - 120 * q^40 - 148*b * q^41 - 463 * q^43 + 52*b * q^44 + 228 * q^46 + 220*b * q^47 - 327 * q^49 - 101*b * q^50 + 62 * q^52 + 174*b * q^53 - 312 * q^55 + 40*b * q^56 - 336 * q^58 + 218*b * q^59 + 71 * q^61 - 265*b * q^62 + 568 * q^64 - 62*b * q^65 + 809 * q^67 - 12*b * q^68 - 48 * q^70 + 354*b * q^71 - 223 * q^73 + 134*b * q^74 - 70 * q^76 + 104*b * q^77 + 23 * q^79 - 88*b * q^80 - 888 * q^82 - 86*b * q^83 + 72 * q^85 - 463*b * q^86 + 1560 * q^88 + 36*b * q^89 + 124 * q^91 - 76*b * q^92 + 1320 * q^94 + 70*b * q^95 - 1075 * q^97 - 327*b * q^98
\(\operatorname{Tr}(f)(q)\)	\(=\)	2 * q - 4 * q^4 - 8 * q^7 + 24 * q^10 - 62 * q^13 - 88 * q^16 + 70 * q^19 - 312 * q^22 - 202 * q^25 + 16 * q^28 - 530 * q^31 + 72 * q^34 + 268 * q^37 - 240 * q^40 - 926 * q^43 + 456 * q^46 - 654 * q^49 + 124 * q^52 - 624 * q^55 - 672 * q^58 + 142 * q^61 + 1136 * q^64 + 1618 * q^67 - 96 * q^70 - 446 * q^73 - 140 * q^76 + 46 * q^79 - 1776 * q^82 + 144 * q^85 + 3120 * q^88 + 248 * q^91 + 2640 * q^94 - 2150 * q^97

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

−2.44949
2.44949

−2.44949

0

−2.00000

−4.89898

0

−4.00000

24.4949

0

12.0000

1.2

2.44949

0

−2.00000

4.89898

0

−4.00000

−24.4949

0

12.0000

Atkin-Lehner signs

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

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
3.b	odd	2	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	243.4.a.e	✓	2
3.b	odd	2	1	inner	243.4.a.e	✓	2
9.c	even	3	2		243.4.c.g		4
9.d	odd	6	2		243.4.c.g		4

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
243.4.a.e	✓	2	1.a	even	1	1	trivial
243.4.a.e	✓	2	3.b	odd	2	1	inner
243.4.c.g		4	9.c	even	3	2
243.4.c.g		4	9.d	odd	6	2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(243))\):

\( T_{2}^{2} - 6 \)

\( T_{7} + 4 \)

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( T^{2} - 6 \)
$3$	\( T^{2} \)
$5$	\( T^{2} - 24 \)
$7$	\( (T + 4)^{2} \)
$11$	\( T^{2} - 4056 \)