Newform orbit 243.4.a.g

• Label: 243.4.a.g
• 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:	\(\Q(\sqrt{21}) \)
Defining polynomial:	\( x^{2} - x - 5 \)
Coefficient ring:	\(\Z[a_1, a_2]\)
Coefficient ring index:	\( 2 \)
Twist minimal:	yes
Fricke sign:	\(-1\)
Sato-Tate group:	$\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{21}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\)	\(=\)	q - b * q^2 + 13 * q^4 + 4*b * q^5 - 25 * q^7 - 5*b * q^8 - 84 * q^10 + 2*b * q^11 - 61 * q^13 + 25*b * q^14 + q^16 - 18*b * q^17 + 89 * q^19 + 52*b * q^20 - 42 * q^22 + 10*b * q^23 + 211 * q^25 + 61*b * q^26 - 325 * q^28 + 2*b * q^29 - 79 * q^31 + 39*b * q^32 + 378 * q^34 - 100*b * q^35 - 307 * q^37 - 89*b * q^38 - 420 * q^40 - 68*b * q^41 - 223 * q^43 + 26*b * q^44 - 210 * q^46 - 106*b * q^47 + 282 * q^49 - 211*b * q^50 - 793 * q^52 + 78*b * q^53 + 168 * q^55 + 125*b * q^56 - 42 * q^58 + 112*b * q^59 - 490 * q^61 + 79*b * q^62 - 827 * q^64 - 244*b * q^65 - 292 * q^67 - 234*b * q^68 + 2100 * q^70 + 66*b * q^71 - 70 * q^73 + 307*b * q^74 + 1157 * q^76 - 50*b * q^77 - 421 * q^79 + 4*b * q^80 + 1428 * q^82 + 50*b * q^83 - 1512 * q^85 + 223*b * q^86 - 210 * q^88 - 168*b * q^89 + 1525 * q^91 + 130*b * q^92 + 2226 * q^94 + 356*b * q^95 + 65 * q^97 - 282*b * q^98
\(\operatorname{Tr}(f)(q)\)	\(=\)	2 * q + 26 * q^4 - 50 * q^7 - 168 * q^10 - 122 * q^13 + 2 * q^16 + 178 * q^19 - 84 * q^22 + 422 * q^25 - 650 * q^28 - 158 * q^31 + 756 * q^34 - 614 * q^37 - 840 * q^40 - 446 * q^43 - 420 * q^46 + 564 * q^49 - 1586 * q^52 + 336 * q^55 - 84 * q^58 - 980 * q^61 - 1654 * q^64 - 584 * q^67 + 4200 * q^70 - 140 * q^73 + 2314 * q^76 - 842 * q^79 + 2856 * q^82 - 3024 * q^85 - 420 * q^88 + 3050 * q^91 + 4452 * q^94 + 130 * 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

2.79129
−1.79129

−4.58258

0

13.0000

18.3303

0

−25.0000

−22.9129

0

−84.0000

1.2

4.58258

0

13.0000

−18.3303

0

−25.0000

22.9129

0

−84.0000

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.g	✓	2
3.b	odd	2	1	inner	243.4.a.g	✓	2
9.c	even	3	2		243.4.c.e		4
9.d	odd	6	2		243.4.c.e		4

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
243.4.a.g	✓	2	1.a	even	1	1	trivial
243.4.a.g	✓	2	3.b	odd	2	1	inner
243.4.c.e		4	9.c	even	3	2
243.4.c.e		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 \(S_{4}^{\mathrm{new}}(\Gamma_0(243))\):

\( T_{2}^{2} - 21 \)

\( T_{7} + 25 \)

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( T^{2} - 21 \)
$3$	\( T^{2} \)
$5$	\( T^{2} - 336 \)
$7$	\( (T + 25)^{2} \)
$11$	\( T^{2} - 84 \)