Properties

Label 54.8.a.e
Level $54$
Weight $8$
Character orbit 54.a
Self dual yes
Analytic conductor $16.869$
Analytic rank $1$
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] = [54,8,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, 8, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("54.1");
 
S:= CuspForms(chi, 8);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 54 = 2 \cdot 3^{3} \)
Weight: \( k \) \(=\) \( 8 \)
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: \(16.8687913761\)
Analytic rank: \(1\)
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)
 
\(f(q)\) \(=\) \( q + 8 q^{2} + 64 q^{4} - 105 q^{5} - 937 q^{7} + 512 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + 8 q^{2} + 64 q^{4} - 105 q^{5} - 937 q^{7} + 512 q^{8} - 840 q^{10} - 5943 q^{11} + 68 q^{13} - 7496 q^{14} + 4096 q^{16} + 5400 q^{17} - 48382 q^{19} - 6720 q^{20} - 47544 q^{22} + 642 q^{23} - 67100 q^{25} + 544 q^{26} - 59968 q^{28} + 125934 q^{29} - 161275 q^{31} + 32768 q^{32} + 43200 q^{34} + 98385 q^{35} - 414286 q^{37} - 387056 q^{38} - 53760 q^{40} + 627474 q^{41} + 570590 q^{43} - 380352 q^{44} + 5136 q^{46} - 538698 q^{47} + 54426 q^{49} - 536800 q^{50} + 4352 q^{52} - 356283 q^{53} + 624015 q^{55} - 479744 q^{56} + 1007472 q^{58} + 2910828 q^{59} + 2684168 q^{61} - 1290200 q^{62} + 262144 q^{64} - 7140 q^{65} + 2681078 q^{67} + 345600 q^{68} + 787080 q^{70} + 3705480 q^{71} - 153151 q^{73} - 3314288 q^{74} - 3096448 q^{76} + 5568591 q^{77} - 7579288 q^{79} - 430080 q^{80} + 5019792 q^{82} - 9345999 q^{83} - 567000 q^{85} + 4564720 q^{86} - 3042816 q^{88} - 4033602 q^{89} - 63716 q^{91} + 41088 q^{92} - 4309584 q^{94} + 5080110 q^{95} - 5754097 q^{97} + 435408 q^{98}+O(q^{100}) \) 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
8.00000 0 64.0000 −105.000 0 −937.000 512.000 0 −840.000
\(n\): 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 54.8.a.e yes 1
3.b odd 2 1 54.8.a.b 1
4.b odd 2 1 432.8.a.c 1
9.c even 3 2 162.8.c.c 2
9.d odd 6 2 162.8.c.j 2
12.b even 2 1 432.8.a.f 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
54.8.a.b 1 3.b odd 2 1
54.8.a.e yes 1 1.a even 1 1 trivial
162.8.c.c 2 9.c even 3 2
162.8.c.j 2 9.d odd 6 2
432.8.a.c 1 4.b odd 2 1
432.8.a.f 1 12.b even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5} + 105 \) acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(54))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T - 8 \) Copy content Toggle raw display
$3$ \( T \) Copy content Toggle raw display
$5$ \( T + 105 \) Copy content Toggle raw display
$7$ \( T + 937 \) Copy content Toggle raw display
$11$ \( T + 5943 \) Copy content Toggle raw display
$13$ \( T - 68 \) Copy content Toggle raw display
$17$ \( T - 5400 \) Copy content Toggle raw display
$19$ \( T + 48382 \) Copy content Toggle raw display
$23$ \( T - 642 \) Copy content Toggle raw display
$29$ \( T - 125934 \) Copy content Toggle raw display
$31$ \( T + 161275 \) Copy content Toggle raw display
$37$ \( T + 414286 \) Copy content Toggle raw display
$41$ \( T - 627474 \) Copy content Toggle raw display
$43$ \( T - 570590 \) Copy content Toggle raw display
$47$ \( T + 538698 \) Copy content Toggle raw display
$53$ \( T + 356283 \) Copy content Toggle raw display
$59$ \( T - 2910828 \) Copy content Toggle raw display
$61$ \( T - 2684168 \) Copy content Toggle raw display
$67$ \( T - 2681078 \) Copy content Toggle raw display
$71$ \( T - 3705480 \) Copy content Toggle raw display
$73$ \( T + 153151 \) Copy content Toggle raw display
$79$ \( T + 7579288 \) Copy content Toggle raw display
$83$ \( T + 9345999 \) Copy content Toggle raw display
$89$ \( T + 4033602 \) Copy content Toggle raw display
$97$ \( T + 5754097 \) Copy content Toggle raw display
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