Properties

Label 4864.2.a.bc
Level $4864$
Weight $2$
Character orbit 4864.a
Self dual yes
Analytic conductor $38.839$
Analytic rank $1$
Dimension $3$
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] = [4864,2,Mod(1,4864)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4864, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("4864.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4864 = 2^{8} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4864.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: \(38.8392355432\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.892.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 8x + 10 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 2432)
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)
 

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

\(f(q)\) \(=\) \( q - \beta_1 q^{3} + (\beta_1 - 1) q^{5} + ( - \beta_{2} - 1) q^{7} + (\beta_{2} - \beta_1 + 3) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{3} + (\beta_1 - 1) q^{5} + ( - \beta_{2} - 1) q^{7} + (\beta_{2} - \beta_1 + 3) q^{9} + ( - \beta_{2} + \beta_1 - 1) q^{11} + (\beta_{2} - \beta_1) q^{13} + ( - \beta_{2} + 2 \beta_1 - 6) q^{15} + 3 q^{17} + q^{19} + (2 \beta_{2} + \beta_1 + 2) q^{21} + (\beta_{2} + \beta_1 - 4) q^{23} + (\beta_{2} - 3 \beta_1 + 2) q^{25} + ( - \beta_{2} - \beta_1 + 4) q^{27} + ( - 2 \beta_{2} - \beta_1 + 2) q^{29} + (\beta_{2} - 4) q^{31} + (\beta_{2} + 2 \beta_1 - 4) q^{33} + ( - \beta_{2} - \beta_1 - 1) q^{35} + ( - \beta_{2} + 2 \beta_1 + 2) q^{37} + ( - \beta_{2} - \beta_1 + 4) q^{39} + (\beta_{2} + 2 \beta_1 - 4) q^{41} + (3 \beta_{2} - \beta_1 + 3) q^{43} + (5 \beta_1 - 7) q^{45} + ( - \beta_1 - 5) q^{47} + (2 \beta_{2} + 2 \beta_1) q^{49} - 3 \beta_1 q^{51} + (\beta_1 + 4) q^{53} + ( - 3 \beta_1 + 5) q^{55} - \beta_1 q^{57} + (\beta_{2} - \beta_1 + 6) q^{59} + (2 \beta_{2} - \beta_1 + 5) q^{61} + ( - 2 \beta_{2} - \beta_1 - 7) q^{63} + (2 \beta_1 - 4) q^{65} + (\beta_1 - 10) q^{67} + ( - 3 \beta_{2} + 5 \beta_1 - 8) q^{69} + ( - 2 \beta_{2} + 4 \beta_1 - 2) q^{71} + ( - 6 \beta_{2} + 4 \beta_1 - 7) q^{73} + (\beta_{2} - 5 \beta_1 + 16) q^{75} + (\beta_1 + 5) q^{77} + (\beta_{2} - 4) q^{79} + ( - 2 \beta_1 - 1) q^{81} + (4 \beta_{2} + 2 \beta_1 - 6) q^{83} + (3 \beta_1 - 3) q^{85} + (5 \beta_{2} - 3 \beta_1 + 10) q^{87} + ( - \beta_{2} - 2 \beta_1 - 6) q^{89} + (\beta_{2} - \beta_1 - 4) q^{91} + ( - 2 \beta_{2} + 4 \beta_1 - 2) q^{93} + (\beta_1 - 1) q^{95} + ( - 4 \beta_{2} - 2 \beta_1 + 8) q^{97} + ( - \beta_{2} + 3 \beta_1 - 11) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - q^{3} - 2 q^{5} - 3 q^{7} + 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q - q^{3} - 2 q^{5} - 3 q^{7} + 8 q^{9} - 2 q^{11} - q^{13} - 16 q^{15} + 9 q^{17} + 3 q^{19} + 7 q^{21} - 11 q^{23} + 3 q^{25} + 11 q^{27} + 5 q^{29} - 12 q^{31} - 10 q^{33} - 4 q^{35} + 8 q^{37} + 11 q^{39} - 10 q^{41} + 8 q^{43} - 16 q^{45} - 16 q^{47} + 2 q^{49} - 3 q^{51} + 13 q^{53} + 12 q^{55} - q^{57} + 17 q^{59} + 14 q^{61} - 22 q^{63} - 10 q^{65} - 29 q^{67} - 19 q^{69} - 2 q^{71} - 17 q^{73} + 43 q^{75} + 16 q^{77} - 12 q^{79} - 5 q^{81} - 16 q^{83} - 6 q^{85} + 27 q^{87} - 20 q^{89} - 13 q^{91} - 2 q^{93} - 2 q^{95} + 22 q^{97} - 30 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{3} - x^{2} - 8x + 10 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} + \nu - 6 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} - \beta _1 + 6 \) 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
2.59774
1.31955
−2.91729
0 −2.59774 0 1.59774 0 −4.34596 0 3.74823 0
1.2 0 −1.31955 0 0.319551 0 1.93923 0 −1.25879 0
1.3 0 2.91729 0 −3.91729 0 −0.593272 0 5.51056 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(19\) \(-1\)

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 4864.2.a.bc 3
4.b odd 2 1 4864.2.a.be 3
8.b even 2 1 4864.2.a.bf 3
8.d odd 2 1 4864.2.a.bd 3
16.e even 4 2 2432.2.c.g yes 6
16.f odd 4 2 2432.2.c.f 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2432.2.c.f 6 16.f odd 4 2
2432.2.c.g yes 6 16.e even 4 2
4864.2.a.bc 3 1.a even 1 1 trivial
4864.2.a.bd 3 8.d odd 2 1
4864.2.a.be 3 4.b odd 2 1
4864.2.a.bf 3 8.b even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4864))\):

\( T_{3}^{3} + T_{3}^{2} - 8T_{3} - 10 \) Copy content Toggle raw display
\( T_{5}^{3} + 2T_{5}^{2} - 7T_{5} + 2 \) Copy content Toggle raw display
\( T_{7}^{3} + 3T_{7}^{2} - 7T_{7} - 5 \) Copy content Toggle raw display
\( T_{11}^{3} + 2T_{11}^{2} - 11T_{11} - 20 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{3} \) Copy content Toggle raw display
$3$ \( T^{3} + T^{2} - 8T - 10 \) Copy content Toggle raw display
$5$ \( T^{3} + 2 T^{2} - 7 T + 2 \) Copy content Toggle raw display
$7$ \( T^{3} + 3 T^{2} - 7 T - 5 \) Copy content Toggle raw display
$11$ \( T^{3} + 2 T^{2} - 11 T - 20 \) Copy content Toggle raw display
$13$ \( T^{3} + T^{2} - 12T + 8 \) Copy content Toggle raw display
$17$ \( (T - 3)^{3} \) Copy content Toggle raw display
$19$ \( (T - 1)^{3} \) Copy content Toggle raw display
$23$ \( T^{3} + 11 T^{2} + 16 T - 80 \) Copy content Toggle raw display
$29$ \( T^{3} - 5 T^{2} - 52 T + 274 \) Copy content Toggle raw display
$31$ \( T^{3} + 12 T^{2} + 38 T + 20 \) Copy content Toggle raw display
$37$ \( T^{3} - 8 T^{2} - 10 T + 100 \) Copy content Toggle raw display
$41$ \( T^{3} + 10 T^{2} - 22 T - 200 \) Copy content Toggle raw display
$43$ \( T^{3} - 8 T^{2} - 59 T + 350 \) Copy content Toggle raw display
$47$ \( T^{3} + 16 T^{2} + 77 T + 100 \) Copy content Toggle raw display
$53$ \( T^{3} - 13 T^{2} + 48 T - 38 \) Copy content Toggle raw display
$59$ \( T^{3} - 17 T^{2} + 84 T - 100 \) Copy content Toggle raw display
$61$ \( T^{3} - 14 T^{2} + 29 T + 142 \) Copy content Toggle raw display
$67$ \( T^{3} + 29 T^{2} + 272 T + 830 \) Copy content Toggle raw display
$71$ \( T^{3} + 2 T^{2} - 124 T + 200 \) Copy content Toggle raw display
$73$ \( T^{3} + 17 T^{2} - 253 T - 4309 \) Copy content Toggle raw display
$79$ \( T^{3} + 12 T^{2} + 38 T + 20 \) Copy content Toggle raw display
$83$ \( T^{3} + 16 T^{2} - 156 T - 2560 \) Copy content Toggle raw display
$89$ \( T^{3} + 20 T^{2} + 78 T - 20 \) Copy content Toggle raw display
$97$ \( T^{3} - 22 T^{2} - 80 T + 2800 \) Copy content Toggle raw display
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