Properties

Label 432.4.i.a
Level $432$
Weight $4$
Character orbit 432.i
Analytic conductor $25.489$
Analytic rank $0$
Dimension $2$
Inner twists $2$

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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: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 18)
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 primitive root of unity \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - 9 \zeta_{6} q^{5} + (31 \zeta_{6} - 31) q^{7} +O(q^{10}) \) Copy content Toggle raw display \( q - 9 \zeta_{6} q^{5} + (31 \zeta_{6} - 31) q^{7} + ( - 15 \zeta_{6} + 15) q^{11} + 37 \zeta_{6} q^{13} + 42 q^{17} + 28 q^{19} - 195 \zeta_{6} q^{23} + ( - 44 \zeta_{6} + 44) q^{25} + ( - 111 \zeta_{6} + 111) q^{29} - 205 \zeta_{6} q^{31} + 279 q^{35} - 166 q^{37} - 261 \zeta_{6} q^{41} + (43 \zeta_{6} - 43) q^{43} + (177 \zeta_{6} - 177) q^{47} - 618 \zeta_{6} q^{49} - 114 q^{53} - 135 q^{55} - 159 \zeta_{6} q^{59} + (191 \zeta_{6} - 191) q^{61} + ( - 333 \zeta_{6} + 333) q^{65} - 421 \zeta_{6} q^{67} + 156 q^{71} + 182 q^{73} + 465 \zeta_{6} q^{77} + ( - 1133 \zeta_{6} + 1133) q^{79} + ( - 1083 \zeta_{6} + 1083) q^{83} - 378 \zeta_{6} q^{85} + 1050 q^{89} - 1147 q^{91} - 252 \zeta_{6} q^{95} + ( - 901 \zeta_{6} + 901) q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 9 q^{5} - 31 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 9 q^{5} - 31 q^{7} + 15 q^{11} + 37 q^{13} + 84 q^{17} + 56 q^{19} - 195 q^{23} + 44 q^{25} + 111 q^{29} - 205 q^{31} + 558 q^{35} - 332 q^{37} - 261 q^{41} - 43 q^{43} - 177 q^{47} - 618 q^{49} - 228 q^{53} - 270 q^{55} - 159 q^{59} - 191 q^{61} + 333 q^{65} - 421 q^{67} + 312 q^{71} + 364 q^{73} + 465 q^{77} + 1133 q^{79} + 1083 q^{83} - 378 q^{85} + 2100 q^{89} - 2294 q^{91} - 252 q^{95} + 901 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(271\) \(325\) \(353\)
\(\chi(n)\) \(1\) \(1\) \(-\zeta_{6}\)

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
0.500000 0.866025i
0.500000 + 0.866025i
0 0 0 −4.50000 + 7.79423i 0 −15.5000 26.8468i 0 0 0
289.1 0 0 0 −4.50000 7.79423i 0 −15.5000 + 26.8468i 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.a 2
3.b odd 2 1 144.4.i.a 2
4.b odd 2 1 54.4.c.a 2
9.c even 3 1 inner 432.4.i.a 2
9.c even 3 1 1296.4.a.g 1
9.d odd 6 1 144.4.i.a 2
9.d odd 6 1 1296.4.a.b 1
12.b even 2 1 18.4.c.a 2
36.f odd 6 1 54.4.c.a 2
36.f odd 6 1 162.4.a.a 1
36.h even 6 1 18.4.c.a 2
36.h even 6 1 162.4.a.d 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
18.4.c.a 2 12.b even 2 1
18.4.c.a 2 36.h even 6 1
54.4.c.a 2 4.b odd 2 1
54.4.c.a 2 36.f odd 6 1
144.4.i.a 2 3.b odd 2 1
144.4.i.a 2 9.d odd 6 1
162.4.a.a 1 36.f odd 6 1
162.4.a.d 1 36.h even 6 1
432.4.i.a 2 1.a even 1 1 trivial
432.4.i.a 2 9.c even 3 1 inner
1296.4.a.b 1 9.d odd 6 1
1296.4.a.g 1 9.c even 3 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{2} + 9T_{5} + 81 \) acting on \(S_{4}^{\mathrm{new}}(432, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} + 9T + 81 \) Copy content Toggle raw display
$7$ \( T^{2} + 31T + 961 \) Copy content Toggle raw display
$11$ \( T^{2} - 15T + 225 \) Copy content Toggle raw display
$13$ \( T^{2} - 37T + 1369 \) Copy content Toggle raw display
$17$ \( (T - 42)^{2} \) Copy content Toggle raw display
$19$ \( (T - 28)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 195T + 38025 \) Copy content Toggle raw display
$29$ \( T^{2} - 111T + 12321 \) Copy content Toggle raw display
$31$ \( T^{2} + 205T + 42025 \) Copy content Toggle raw display
$37$ \( (T + 166)^{2} \) Copy content Toggle raw display
$41$ \( T^{2} + 261T + 68121 \) Copy content Toggle raw display
$43$ \( T^{2} + 43T + 1849 \) Copy content Toggle raw display
$47$ \( T^{2} + 177T + 31329 \) Copy content Toggle raw display
$53$ \( (T + 114)^{2} \) Copy content Toggle raw display
$59$ \( T^{2} + 159T + 25281 \) Copy content Toggle raw display
$61$ \( T^{2} + 191T + 36481 \) Copy content Toggle raw display
$67$ \( T^{2} + 421T + 177241 \) Copy content Toggle raw display
$71$ \( (T - 156)^{2} \) Copy content Toggle raw display
$73$ \( (T - 182)^{2} \) Copy content Toggle raw display
$79$ \( T^{2} - 1133 T + 1283689 \) Copy content Toggle raw display
$83$ \( T^{2} - 1083 T + 1172889 \) Copy content Toggle raw display
$89$ \( (T - 1050)^{2} \) Copy content Toggle raw display
$97$ \( T^{2} - 901T + 811801 \) Copy content Toggle raw display
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