Properties

Label 432.5.e.d
Level 432
Weight 5
Character orbit 432.e
Analytic conductor 44.656
Analytic rank 0
Dimension 2
CM no
Inner twists 2

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Newspace parameters

Level: \( N \) \(=\) \( 432 = 2^{4} \cdot 3^{3} \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 432.e (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(44.6558240522\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 3^{2} \)
Twist minimal: no (minimal twist has level 108)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(i = \sqrt{-1}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 9 i q^{5} -5 q^{7} +O(q^{10})\) \( q + 9 i q^{5} -5 q^{7} -117 i q^{11} -34 q^{13} + 450 i q^{17} + 64 q^{19} -612 i q^{23} + 544 q^{25} + 1062 i q^{29} + 697 q^{31} -45 i q^{35} -748 q^{37} + 684 i q^{41} -2618 q^{43} + 2646 i q^{47} -2376 q^{49} -1071 i q^{53} + 1053 q^{55} + 5814 i q^{59} + 6404 q^{61} -306 i q^{65} + 5218 q^{67} + 6570 i q^{71} -4519 q^{73} + 585 i q^{77} -7502 q^{79} + 5481 i q^{83} -4050 q^{85} + 8874 i q^{89} + 170 q^{91} + 576 i q^{95} + 10571 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 10q^{7} + O(q^{10}) \) \( 2q - 10q^{7} - 68q^{13} + 128q^{19} + 1088q^{25} + 1394q^{31} - 1496q^{37} - 5236q^{43} - 4752q^{49} + 2106q^{55} + 12808q^{61} + 10436q^{67} - 9038q^{73} - 15004q^{79} - 8100q^{85} + 340q^{91} + 21142q^{97} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/432\mathbb{Z}\right)^\times\).

\(n\) \(271\) \(325\) \(353\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
161.1
1.00000i
1.00000i
0 0 0 9.00000i 0 −5.00000 0 0 0
161.2 0 0 0 9.00000i 0 −5.00000 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 432.5.e.d 2
3.b odd 2 1 inner 432.5.e.d 2
4.b odd 2 1 108.5.c.c 2
12.b even 2 1 108.5.c.c 2
36.f odd 6 2 324.5.g.d 4
36.h even 6 2 324.5.g.d 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
108.5.c.c 2 4.b odd 2 1
108.5.c.c 2 12.b even 2 1
324.5.g.d 4 36.f odd 6 2
324.5.g.d 4 36.h even 6 2
432.5.e.d 2 1.a even 1 1 trivial
432.5.e.d 2 3.b odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{5}^{\mathrm{new}}(432, [\chi])\):

\( T_{5}^{2} + 81 \)
\( T_{7} + 5 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ 1
$5$ \( 1 - 1169 T^{2} + 390625 T^{4} \)
$7$ \( ( 1 + 5 T + 2401 T^{2} )^{2} \)
$11$ \( 1 - 15593 T^{2} + 214358881 T^{4} \)
$13$ \( ( 1 + 34 T + 28561 T^{2} )^{2} \)
$17$ \( 1 + 35458 T^{2} + 6975757441 T^{4} \)
$19$ \( ( 1 - 64 T + 130321 T^{2} )^{2} \)
$23$ \( 1 - 185138 T^{2} + 78310985281 T^{4} \)
$29$ \( 1 - 286718 T^{2} + 500246412961 T^{4} \)
$31$ \( ( 1 - 697 T + 923521 T^{2} )^{2} \)
$37$ \( ( 1 + 748 T + 1874161 T^{2} )^{2} \)
$41$ \( 1 - 5183666 T^{2} + 7984925229121 T^{4} \)
$43$ \( ( 1 + 2618 T + 3418801 T^{2} )^{2} \)
$47$ \( 1 - 2758046 T^{2} + 23811286661761 T^{4} \)
$53$ \( 1 - 14633921 T^{2} + 62259690411361 T^{4} \)
$59$ \( 1 + 9567874 T^{2} + 146830437604321 T^{4} \)
$61$ \( ( 1 - 6404 T + 13845841 T^{2} )^{2} \)
$67$ \( ( 1 - 5218 T + 20151121 T^{2} )^{2} \)
$71$ \( 1 - 7658462 T^{2} + 645753531245761 T^{4} \)
$73$ \( ( 1 + 4519 T + 28398241 T^{2} )^{2} \)
$79$ \( ( 1 + 7502 T + 38950081 T^{2} )^{2} \)
$83$ \( 1 - 64875281 T^{2} + 2252292232139041 T^{4} \)
$89$ \( 1 - 46736606 T^{2} + 3936588805702081 T^{4} \)
$97$ \( ( 1 - 10571 T + 88529281 T^{2} )^{2} \)
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