Properties

Label 432.6.a.p
Level 432
Weight 6
Character orbit 432.a
Self dual yes
Analytic conductor 69.286
Analytic rank 1
Dimension 2
CM no
Inner twists 2

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Newspace parameters

Level: \( N \) \(=\) \( 432 = 2^{4} \cdot 3^{3} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 432.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(69.2858101592\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{6}) \)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{2}\cdot 3^{2} \)
Twist minimal: no (minimal twist has level 108)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = 36\sqrt{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta q^{5} -29 q^{7} +O(q^{10})\) \( q + \beta q^{5} -29 q^{7} -\beta q^{11} + 329 q^{13} -25 \beta q^{17} -1799 q^{19} -41 \beta q^{23} + 4651 q^{25} + 16 \beta q^{29} -5228 q^{31} -29 \beta q^{35} + 8783 q^{37} -176 \beta q^{41} -19976 q^{43} + 123 \beta q^{47} -15966 q^{49} + 334 \beta q^{53} -7776 q^{55} -65 \beta q^{59} -1069 q^{61} + 329 \beta q^{65} + 62077 q^{67} + 526 \beta q^{71} -48079 q^{73} + 29 \beta q^{77} -49979 q^{79} + 654 \beta q^{83} -194400 q^{85} -997 \beta q^{89} -9541 q^{91} -1799 \beta q^{95} + 12917 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 58q^{7} + O(q^{10}) \) \( 2q - 58q^{7} + 658q^{13} - 3598q^{19} + 9302q^{25} - 10456q^{31} + 17566q^{37} - 39952q^{43} - 31932q^{49} - 15552q^{55} - 2138q^{61} + 124154q^{67} - 96158q^{73} - 99958q^{79} - 388800q^{85} - 19082q^{91} + 25834q^{97} + O(q^{100}) \)

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
0 0 0 −88.1816 0 −29.0000 0 0 0
1.2 0 0 0 88.1816 0 −29.0000 0 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

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 432.6.a.p 2
3.b odd 2 1 inner 432.6.a.p 2
4.b odd 2 1 108.6.a.d 2
12.b even 2 1 108.6.a.d 2
36.f odd 6 2 324.6.e.e 4
36.h even 6 2 324.6.e.e 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
108.6.a.d 2 4.b odd 2 1
108.6.a.d 2 12.b even 2 1
324.6.e.e 4 36.f odd 6 2
324.6.e.e 4 36.h even 6 2
432.6.a.p 2 1.a even 1 1 trivial
432.6.a.p 2 3.b odd 2 1 inner

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(-1\)

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(432))\):

\( T_{5}^{2} - 7776 \)
\( T_{7} + 29 \)
\( T_{11}^{2} - 7776 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ 1
$5$ \( 1 - 1526 T^{2} + 9765625 T^{4} \)
$7$ \( ( 1 + 29 T + 16807 T^{2} )^{2} \)
$11$ \( 1 + 314326 T^{2} + 25937424601 T^{4} \)
$13$ \( ( 1 - 329 T + 371293 T^{2} )^{2} \)
$17$ \( 1 - 2020286 T^{2} + 2015993900449 T^{4} \)
$19$ \( ( 1 + 1799 T + 2476099 T^{2} )^{2} \)
$23$ \( 1 - 198770 T^{2} + 41426511213649 T^{4} \)
$29$ \( 1 + 39031642 T^{2} + 420707233300201 T^{4} \)
$31$ \( ( 1 + 5228 T + 28629151 T^{2} )^{2} \)
$37$ \( ( 1 - 8783 T + 69343957 T^{2} )^{2} \)
$41$ \( 1 - 9156974 T^{2} + 13422659310152401 T^{4} \)
$43$ \( ( 1 + 19976 T + 147008443 T^{2} )^{2} \)
$47$ \( 1 + 341046910 T^{2} + 52599132235830049 T^{4} \)
$53$ \( 1 - 31068470 T^{2} + 174887470365513049 T^{4} \)
$59$ \( 1 + 1396994998 T^{2} + 511116753300641401 T^{4} \)
$61$ \( ( 1 + 1069 T + 844596301 T^{2} )^{2} \)
$67$ \( ( 1 - 62077 T + 1350125107 T^{2} )^{2} \)
$71$ \( 1 + 1457026126 T^{2} + 3255243551009881201 T^{4} \)
$73$ \( ( 1 + 48079 T + 2073071593 T^{2} )^{2} \)
$79$ \( ( 1 + 49979 T + 3077056399 T^{2} )^{2} \)
$83$ \( 1 + 4552161670 T^{2} + 15516041187205853449 T^{4} \)
$89$ \( 1 + 3438704914 T^{2} + 31181719929966183601 T^{4} \)
$97$ \( ( 1 - 12917 T + 8587340257 T^{2} )^{2} \)
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