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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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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [432,4,Mod(145,432)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(432, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 2]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("432.145");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
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

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(25.4888251225\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{-11})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} - 2x^{2} - 3x + 9 \) Copy content Toggle raw display
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

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

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.

\(f(q)\) \(=\) \( q + (\beta_{3} + 7 \beta_1) q^{5} + (3 \beta_{3} - 3 \beta_{2} - 5 \beta_1 + 2) q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{3} + 7 \beta_1) q^{5} + (3 \beta_{3} - 3 \beta_{2} - 5 \beta_1 + 2) q^{7} + (8 \beta_{3} - 8 \beta_{2} + 29 \beta_1 - 37) q^{11} + ( - 15 \beta_{3} + 13 \beta_1) q^{13} + (9 \beta_{2} - 45) q^{17} + ( - 27 \beta_{2} + 25) q^{19} + ( - 19 \beta_{3} - 7 \beta_1) q^{23} + (15 \beta_{3} - 15 \beta_{2} - 68 \beta_1 + 53) q^{25} + (\beta_{3} - \beta_{2} + 25 \beta_1 - 26) q^{29} + ( - 3 \beta_{3} + 23 \beta_1) q^{31} + ( - 19 \beta_{2} - 8) q^{35} + ( - 54 \beta_{2} - 52) q^{37} + ( - 98 \beta_{3} + 115 \beta_1) q^{41} + ( - 6 \beta_{3} + 6 \beta_{2} - 41 \beta_1 + 47) q^{43} + ( - 91 \beta_{3} + 91 \beta_{2} + 245 \beta_1 - 154) q^{47} + (21 \beta_{3} + 246 \beta_1) q^{49} + ( - 162 \beta_{2} - 108) q^{53} + ( - 93 \beta_{2} - 360) q^{55} + ( - 136 \beta_{3} - 331 \beta_1) q^{59} + (105 \beta_{3} - 105 \beta_{2} + 167 \beta_1 - 272) q^{61} + ( - 107 \beta_{3} + 107 \beta_{2} - 29 \beta_1 + 136) q^{65} + ( - 66 \beta_{3} + 527 \beta_1) q^{67} + ( - 144 \beta_{2} + 612) q^{71} + ( - 243 \beta_{2} - 349) q^{73} + ( - 71 \beta_{3} - 47 \beta_1) q^{77} + (309 \beta_{3} - 309 \beta_{2} + 247 \beta_1 - 556) q^{79} + (107 \beta_{3} - 107 \beta_{2} + 353 \beta_1 - 460) q^{83} + (18 \beta_{3} - 306 \beta_1) q^{85} + (72 \beta_{2} + 234) q^{89} + ( - 69 \beta_{2} + 356) q^{91} + ( - 164 \beta_{3} + 148 \beta_1) q^{95} + ( - 102 \beta_{3} + 102 \beta_{2} + 419 \beta_1 - 317) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 15 q^{5} + 7 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 15 q^{5} + 7 q^{7} - 66 q^{11} + 11 q^{13} - 198 q^{17} + 154 q^{19} - 33 q^{23} + 121 q^{25} - 51 q^{29} + 43 q^{31} + 6 q^{35} - 100 q^{37} + 132 q^{41} + 88 q^{43} - 399 q^{47} + 513 q^{49} - 108 q^{53} - 1254 q^{55} - 798 q^{59} - 439 q^{61} + 165 q^{65} + 988 q^{67} + 2736 q^{71} - 910 q^{73} - 165 q^{77} - 803 q^{79} - 813 q^{83} - 594 q^{85} + 792 q^{89} + 1562 q^{91} + 132 q^{95} - 736 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

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

\(\beta_{1}\)\(=\) \( ( \nu^{3} + 2\nu^{2} - 2\nu - 3 ) / 6 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{3} + \nu^{2} + 5\nu ) / 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 2\nu^{3} + \nu^{2} + 2\nu - 9 ) / 3 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( ( \beta_{3} + \beta_{2} - 2\beta _1 + 2 ) / 3 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -\beta_{3} + 2\beta_{2} + 8\beta _1 + 1 ) / 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 4\beta_{3} - 2\beta_{2} - 2\beta _1 + 11 ) / 3 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
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 \( T_{5}^{4} - 15T_{5}^{3} + 177T_{5}^{2} - 720T_{5} + 2304 \) acting on \(S_{4}^{\mathrm{new}}(432, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( T^{4} - 15 T^{3} + 177 T^{2} + \cdots + 2304 \) Copy content Toggle raw display
$7$ \( T^{4} - 7 T^{3} + 111 T^{2} + \cdots + 3844 \) Copy content Toggle raw display
$11$ \( T^{4} + 66 T^{3} + 3795 T^{2} + \cdots + 314721 \) Copy content Toggle raw display
$13$ \( T^{4} - 11 T^{3} + 1947 T^{2} + \cdots + 3334276 \) Copy content Toggle raw display
$17$ \( (T^{2} + 99 T + 1782)^{2} \) Copy content Toggle raw display
$19$ \( (T^{2} - 77 T - 4532)^{2} \) Copy content Toggle raw display
$23$ \( T^{4} + 33 T^{3} + 3795 T^{2} + \cdots + 7322436 \) Copy content Toggle raw display
$29$ \( T^{4} + 51 T^{3} + 1959 T^{2} + \cdots + 412164 \) Copy content Toggle raw display
$31$ \( T^{4} - 43 T^{3} + 1461 T^{2} + \cdots + 150544 \) Copy content Toggle raw display
$37$ \( (T^{2} + 50 T - 23432)^{2} \) Copy content Toggle raw display
$41$ \( T^{4} - 132 T^{3} + \cdots + 5606565129 \) Copy content Toggle raw display
$43$ \( T^{4} - 88 T^{3} + 6105 T^{2} + \cdots + 2686321 \) Copy content Toggle raw display
$47$ \( T^{4} + 399 T^{3} + \cdots + 813276324 \) Copy content Toggle raw display
$53$ \( (T^{2} + 54 T - 215784)^{2} \) Copy content Toggle raw display
$59$ \( T^{4} + 798 T^{3} + \cdots + 43678881 \) Copy content Toggle raw display
$61$ \( T^{4} + 439 T^{3} + \cdots + 1829786176 \) Copy content Toggle raw display
$67$ \( T^{4} - 988 T^{3} + \cdots + 43305193801 \) Copy content Toggle raw display
$71$ \( (T^{2} - 1368 T + 296784)^{2} \) Copy content Toggle raw display
$73$ \( (T^{2} + 455 T - 435398)^{2} \) Copy content Toggle raw display
$79$ \( T^{4} + 803 T^{3} + \cdots + 392522298256 \) Copy content Toggle raw display
$83$ \( T^{4} + 813 T^{3} + \cdots + 5010940944 \) Copy content Toggle raw display
$89$ \( (T^{2} - 396 T - 3564)^{2} \) Copy content Toggle raw display
$97$ \( T^{4} + 736 T^{3} + \cdots + 2459267281 \) Copy content Toggle raw display
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