Properties

Label 432.4.i.c
Level $432$
Weight $4$
Character orbit 432.i
Analytic conductor $25.489$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Newspace parameters

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

Newform invariants

Self dual: no
Analytic conductor: \(25.4888251225\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{-11})\)
Defining polynomial: \(x^{4} - x^{3} - 2 x^{2} - 3 x + 9\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 3 \)
Twist minimal: no (minimal twist has level 9)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( 7 \beta_{1} + \beta_{3} ) q^{5} + ( 2 - 5 \beta_{1} - 3 \beta_{2} + 3 \beta_{3} ) q^{7} +O(q^{10})\) \( q + ( 7 \beta_{1} + \beta_{3} ) q^{5} + ( 2 - 5 \beta_{1} - 3 \beta_{2} + 3 \beta_{3} ) q^{7} + ( -37 + 29 \beta_{1} - 8 \beta_{2} + 8 \beta_{3} ) q^{11} + ( 13 \beta_{1} - 15 \beta_{3} ) q^{13} + ( -45 + 9 \beta_{2} ) q^{17} + ( 25 - 27 \beta_{2} ) q^{19} + ( -7 \beta_{1} - 19 \beta_{3} ) q^{23} + ( 53 - 68 \beta_{1} - 15 \beta_{2} + 15 \beta_{3} ) q^{25} + ( -26 + 25 \beta_{1} - \beta_{2} + \beta_{3} ) q^{29} + ( 23 \beta_{1} - 3 \beta_{3} ) q^{31} + ( -8 - 19 \beta_{2} ) q^{35} + ( -52 - 54 \beta_{2} ) q^{37} + ( 115 \beta_{1} - 98 \beta_{3} ) q^{41} + ( 47 - 41 \beta_{1} + 6 \beta_{2} - 6 \beta_{3} ) q^{43} + ( -154 + 245 \beta_{1} + 91 \beta_{2} - 91 \beta_{3} ) q^{47} + ( 246 \beta_{1} + 21 \beta_{3} ) q^{49} + ( -108 - 162 \beta_{2} ) q^{53} + ( -360 - 93 \beta_{2} ) q^{55} + ( -331 \beta_{1} - 136 \beta_{3} ) q^{59} + ( -272 + 167 \beta_{1} - 105 \beta_{2} + 105 \beta_{3} ) q^{61} + ( 136 - 29 \beta_{1} + 107 \beta_{2} - 107 \beta_{3} ) q^{65} + ( 527 \beta_{1} - 66 \beta_{3} ) q^{67} + ( 612 - 144 \beta_{2} ) q^{71} + ( -349 - 243 \beta_{2} ) q^{73} + ( -47 \beta_{1} - 71 \beta_{3} ) q^{77} + ( -556 + 247 \beta_{1} - 309 \beta_{2} + 309 \beta_{3} ) q^{79} + ( -460 + 353 \beta_{1} - 107 \beta_{2} + 107 \beta_{3} ) q^{83} + ( -306 \beta_{1} + 18 \beta_{3} ) q^{85} + ( 234 + 72 \beta_{2} ) q^{89} + ( 356 - 69 \beta_{2} ) q^{91} + ( 148 \beta_{1} - 164 \beta_{3} ) q^{95} + ( -317 + 419 \beta_{1} + 102 \beta_{2} - 102 \beta_{3} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 15q^{5} + 7q^{7} + O(q^{10}) \) \( 4q + 15q^{5} + 7q^{7} - 66q^{11} + 11q^{13} - 198q^{17} + 154q^{19} - 33q^{23} + 121q^{25} - 51q^{29} + 43q^{31} + 6q^{35} - 100q^{37} + 132q^{41} + 88q^{43} - 399q^{47} + 513q^{49} - 108q^{53} - 1254q^{55} - 798q^{59} - 439q^{61} + 165q^{65} + 988q^{67} + 2736q^{71} - 910q^{73} - 165q^{77} - 803q^{79} - 813q^{83} - 594q^{85} + 792q^{89} + 1562q^{91} + 132q^{95} - 736q^{97} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} - x^{3} - 2 x^{2} - 3 x + 9\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( \nu^{3} + 2 \nu^{2} - 2 \nu - 3 \)\()/6\)
\(\beta_{2}\)\(=\)\((\)\( -\nu^{3} + \nu^{2} + 5 \nu \)\()/3\)
\(\beta_{3}\)\(=\)\((\)\( 2 \nu^{3} + \nu^{2} + 2 \nu - 9 \)\()/3\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{3} + \beta_{2} - 2 \beta_{1} + 2\)\()/3\)
\(\nu^{2}\)\(=\)\((\)\(-\beta_{3} + 2 \beta_{2} + 8 \beta_{1} + 1\)\()/3\)
\(\nu^{3}\)\(=\)\((\)\(4 \beta_{3} - 2 \beta_{2} - 2 \beta_{1} + 11\)\()/3\)

Character values

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

\(n\) \(271\) \(325\) \(353\)
\(\chi(n)\) \(1\) \(1\) \(-\beta_{1}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
145.1
−1.18614 + 1.26217i
1.68614 0.396143i
−1.18614 1.26217i
1.68614 + 0.396143i
0 0 0 2.31386 4.00772i 0 6.05842 + 10.4935i 0 0 0
145.2 0 0 0 5.18614 8.98266i 0 −2.55842 4.43132i 0 0 0
289.1 0 0 0 2.31386 + 4.00772i 0 6.05842 10.4935i 0 0 0
289.2 0 0 0 5.18614 + 8.98266i 0 −2.55842 + 4.43132i 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
9.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 432.4.i.c 4
3.b odd 2 1 144.4.i.c 4
4.b odd 2 1 27.4.c.a 4
9.c even 3 1 inner 432.4.i.c 4
9.c even 3 1 1296.4.a.i 2
9.d odd 6 1 144.4.i.c 4
9.d odd 6 1 1296.4.a.u 2
12.b even 2 1 9.4.c.a 4
36.f odd 6 1 27.4.c.a 4
36.f odd 6 1 81.4.a.a 2
36.h even 6 1 9.4.c.a 4
36.h even 6 1 81.4.a.d 2
60.h even 2 1 225.4.e.b 4
60.l odd 4 2 225.4.k.b 8
180.n even 6 1 225.4.e.b 4
180.n even 6 1 2025.4.a.g 2
180.p odd 6 1 2025.4.a.n 2
180.v odd 12 2 225.4.k.b 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
9.4.c.a 4 12.b even 2 1
9.4.c.a 4 36.h even 6 1
27.4.c.a 4 4.b odd 2 1
27.4.c.a 4 36.f odd 6 1
81.4.a.a 2 36.f odd 6 1
81.4.a.d 2 36.h even 6 1
144.4.i.c 4 3.b odd 2 1
144.4.i.c 4 9.d odd 6 1
225.4.e.b 4 60.h even 2 1
225.4.e.b 4 180.n even 6 1
225.4.k.b 8 60.l odd 4 2
225.4.k.b 8 180.v odd 12 2
432.4.i.c 4 1.a even 1 1 trivial
432.4.i.c 4 9.c even 3 1 inner
1296.4.a.i 2 9.c even 3 1
1296.4.a.u 2 9.d odd 6 1
2025.4.a.g 2 180.n even 6 1
2025.4.a.n 2 180.p odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{4} - 15 T_{5}^{3} + 177 T_{5}^{2} - 720 T_{5} + 2304 \) acting on \(S_{4}^{\mathrm{new}}(432, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( T^{4} \)
$5$ \( 2304 - 720 T + 177 T^{2} - 15 T^{3} + T^{4} \)
$7$ \( 3844 + 434 T + 111 T^{2} - 7 T^{3} + T^{4} \)
$11$ \( 314721 + 37026 T + 3795 T^{2} + 66 T^{3} + T^{4} \)
$13$ \( 3334276 + 20086 T + 1947 T^{2} - 11 T^{3} + T^{4} \)
$17$ \( ( 1782 + 99 T + T^{2} )^{2} \)
$19$ \( ( -4532 - 77 T + T^{2} )^{2} \)
$23$ \( 7322436 - 89298 T + 3795 T^{2} + 33 T^{3} + T^{4} \)
$29$ \( 412164 + 32742 T + 1959 T^{2} + 51 T^{3} + T^{4} \)
$31$ \( 150544 - 16684 T + 1461 T^{2} - 43 T^{3} + T^{4} \)
$37$ \( ( -23432 + 50 T + T^{2} )^{2} \)
$41$ \( 5606565129 + 9883764 T + 92301 T^{2} - 132 T^{3} + T^{4} \)
$43$ \( 2686321 - 144232 T + 6105 T^{2} - 88 T^{3} + T^{4} \)
$47$ \( 813276324 - 11378682 T + 187719 T^{2} + 399 T^{3} + T^{4} \)
$53$ \( ( -215784 + 54 T + T^{2} )^{2} \)
$59$ \( 43678881 + 5273982 T + 630195 T^{2} + 798 T^{3} + T^{4} \)
$61$ \( 1829786176 - 18778664 T + 235497 T^{2} + 439 T^{3} + T^{4} \)
$67$ \( 43305193801 - 205601812 T + 768045 T^{2} - 988 T^{3} + T^{4} \)
$71$ \( ( 296784 - 1368 T + T^{2} )^{2} \)
$73$ \( ( -435398 + 455 T + T^{2} )^{2} \)
$79$ \( 392522298256 - 503092348 T + 1271325 T^{2} + 803 T^{3} + T^{4} \)
$83$ \( 5010940944 + 57550644 T + 590181 T^{2} + 813 T^{3} + T^{4} \)
$89$ \( ( -3564 - 396 T + T^{2} )^{2} \)
$97$ \( 2459267281 + 36498976 T + 492105 T^{2} + 736 T^{3} + T^{4} \)
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