Properties

Label 7728.2.a.bq
Level $7728$
Weight $2$
Character orbit 7728.a
Self dual yes
Analytic conductor $61.708$
Analytic rank $0$
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] = [7728,2,Mod(1,7728)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(7728, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 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("7728.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 7728 = 2^{4} \cdot 3 \cdot 7 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 7728.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: \(61.7083906820\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.733.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 7x + 8 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 3864)
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.

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

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{3} - x^{2} - 7x + 8 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 5 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 5 \) 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
−2.69639
2.51820
1.17819
0 −1.00000 0 −3.27053 0 1.00000 0 1.00000 0
1.2 0 −1.00000 0 −2.34132 0 1.00000 0 1.00000 0
1.3 0 −1.00000 0 2.61186 0 1.00000 0 1.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(3\) \(1\)
\(7\) \(-1\)
\(23\) \(1\)

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 7728.2.a.bq 3
4.b odd 2 1 3864.2.a.o 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3864.2.a.o 3 4.b odd 2 1
7728.2.a.bq 3 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(7728))\):

\( T_{5}^{3} + 3T_{5}^{2} - 7T_{5} - 20 \) Copy content Toggle raw display
\( T_{11}^{3} - 2T_{11}^{2} - 9T_{11} + 14 \) Copy content Toggle raw display
\( T_{13}^{3} - 5T_{13}^{2} + T_{13} + 2 \) Copy content Toggle raw display
\( T_{17}^{3} + 4T_{17}^{2} - 19T_{17} - 50 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{3} \) Copy content Toggle raw display
$3$ \( (T + 1)^{3} \) Copy content Toggle raw display
$5$ \( T^{3} + 3 T^{2} - 7 T - 20 \) Copy content Toggle raw display
$7$ \( (T - 1)^{3} \) Copy content Toggle raw display
$11$ \( T^{3} - 2 T^{2} - 9 T + 14 \) Copy content Toggle raw display
$13$ \( T^{3} - 5T^{2} + T + 2 \) Copy content Toggle raw display
$17$ \( T^{3} + 4 T^{2} - 19 T - 50 \) Copy content Toggle raw display
$19$ \( T^{3} - 14 T^{2} + 55 T - 46 \) Copy content Toggle raw display
$23$ \( (T + 1)^{3} \) Copy content Toggle raw display
$29$ \( T^{3} + 4 T^{2} - 19 T - 50 \) Copy content Toggle raw display
$31$ \( T^{3} - 6 T^{2} - 43 T + 160 \) Copy content Toggle raw display
$37$ \( T^{3} + 6 T^{2} - 43 T - 44 \) Copy content Toggle raw display
$41$ \( T^{3} - 10 T^{2} + 23 T - 10 \) Copy content Toggle raw display
$43$ \( T^{3} - 21 T^{2} + 83 T + 302 \) Copy content Toggle raw display
$47$ \( T^{3} - 2 T^{2} - 28 T + 64 \) Copy content Toggle raw display
$53$ \( T^{3} + 13 T^{2} + 3 T - 16 \) Copy content Toggle raw display
$59$ \( T^{3} + 5 T^{2} - 91 T - 184 \) Copy content Toggle raw display
$61$ \( T^{3} + 17 T^{2} + 35 T - 196 \) Copy content Toggle raw display
$67$ \( T^{3} - 13 T^{2} - 43 T + 326 \) Copy content Toggle raw display
$71$ \( T^{3} + T^{2} - 7T - 8 \) Copy content Toggle raw display
$73$ \( T^{3} - 6 T^{2} - 85 T + 550 \) Copy content Toggle raw display
$79$ \( T^{3} - 16 T^{2} + 75 T - 104 \) Copy content Toggle raw display
$83$ \( T^{3} + 6 T^{2} - 81 T + 22 \) Copy content Toggle raw display
$89$ \( T^{3} + 11 T^{2} - 143 T - 1322 \) Copy content Toggle raw display
$97$ \( T^{3} - 8 T^{2} - 3 T + 26 \) Copy content Toggle raw display
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