Properties

Label 54.4.a.c
Level $54$
Weight $4$
Character orbit 54.a
Self dual yes
Analytic conductor $3.186$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [54,4,Mod(1,54)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(54, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("54.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 54 = 2 \cdot 3^{3} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 54.a (trivial)

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: yes
Analytic conductor: \(3.18610314031\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \( q + 2 q^{2} + 4 q^{4} + 3 q^{5} + 29 q^{7} + 8 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + 2 q^{2} + 4 q^{4} + 3 q^{5} + 29 q^{7} + 8 q^{8} + 6 q^{10} - 57 q^{11} + 20 q^{13} + 58 q^{14} + 16 q^{16} - 72 q^{17} - 106 q^{19} + 12 q^{20} - 114 q^{22} + 174 q^{23} - 116 q^{25} + 40 q^{26} + 116 q^{28} - 210 q^{29} + 47 q^{31} + 32 q^{32} - 144 q^{34} + 87 q^{35} + 2 q^{37} - 212 q^{38} + 24 q^{40} - 6 q^{41} + 218 q^{43} - 228 q^{44} + 348 q^{46} + 474 q^{47} + 498 q^{49} - 232 q^{50} + 80 q^{52} + 81 q^{53} - 171 q^{55} + 232 q^{56} - 420 q^{58} + 84 q^{59} + 56 q^{61} + 94 q^{62} + 64 q^{64} + 60 q^{65} - 142 q^{67} - 288 q^{68} + 174 q^{70} + 360 q^{71} - 1159 q^{73} + 4 q^{74} - 424 q^{76} - 1653 q^{77} - 160 q^{79} + 48 q^{80} - 12 q^{82} + 735 q^{83} - 216 q^{85} + 436 q^{86} - 456 q^{88} - 954 q^{89} + 580 q^{91} + 696 q^{92} + 948 q^{94} - 318 q^{95} + 191 q^{97} + 996 q^{98}+O(q^{100}) \) Copy content Toggle raw display

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
0
2.00000 0 4.00000 3.00000 0 29.0000 8.00000 0 6.00000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

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

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 54.4.a.c yes 1
3.b odd 2 1 54.4.a.b 1
4.b odd 2 1 432.4.a.j 1
5.b even 2 1 1350.4.a.a 1
5.c odd 4 2 1350.4.c.b 2
8.b even 2 1 1728.4.a.l 1
8.d odd 2 1 1728.4.a.k 1
9.c even 3 2 162.4.c.b 2
9.d odd 6 2 162.4.c.g 2
12.b even 2 1 432.4.a.e 1
15.d odd 2 1 1350.4.a.o 1
15.e even 4 2 1350.4.c.s 2
24.f even 2 1 1728.4.a.u 1
24.h odd 2 1 1728.4.a.v 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
54.4.a.b 1 3.b odd 2 1
54.4.a.c yes 1 1.a even 1 1 trivial
162.4.c.b 2 9.c even 3 2
162.4.c.g 2 9.d odd 6 2
432.4.a.e 1 12.b even 2 1
432.4.a.j 1 4.b odd 2 1
1350.4.a.a 1 5.b even 2 1
1350.4.a.o 1 15.d odd 2 1
1350.4.c.b 2 5.c odd 4 2
1350.4.c.s 2 15.e even 4 2
1728.4.a.k 1 8.d odd 2 1
1728.4.a.l 1 8.b even 2 1
1728.4.a.u 1 24.f even 2 1
1728.4.a.v 1 24.h odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5} - 3 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(54))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T - 2 \) Copy content Toggle raw display
$3$ \( T \) Copy content Toggle raw display
$5$ \( T - 3 \) Copy content Toggle raw display
$7$ \( T - 29 \) Copy content Toggle raw display
$11$ \( T + 57 \) Copy content Toggle raw display
$13$ \( T - 20 \) Copy content Toggle raw display
$17$ \( T + 72 \) Copy content Toggle raw display
$19$ \( T + 106 \) Copy content Toggle raw display
$23$ \( T - 174 \) Copy content Toggle raw display
$29$ \( T + 210 \) Copy content Toggle raw display
$31$ \( T - 47 \) Copy content Toggle raw display
$37$ \( T - 2 \) Copy content Toggle raw display
$41$ \( T + 6 \) Copy content Toggle raw display
$43$ \( T - 218 \) Copy content Toggle raw display
$47$ \( T - 474 \) Copy content Toggle raw display
$53$ \( T - 81 \) Copy content Toggle raw display
$59$ \( T - 84 \) Copy content Toggle raw display
$61$ \( T - 56 \) Copy content Toggle raw display
$67$ \( T + 142 \) Copy content Toggle raw display
$71$ \( T - 360 \) Copy content Toggle raw display
$73$ \( T + 1159 \) Copy content Toggle raw display
$79$ \( T + 160 \) Copy content Toggle raw display
$83$ \( T - 735 \) Copy content Toggle raw display
$89$ \( T + 954 \) Copy content Toggle raw display
$97$ \( T - 191 \) Copy content Toggle raw display
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