Properties

Label 7728.2.a.bd
Level $7728$
Weight $2$
Character orbit 7728.a
Self dual yes
Analytic conductor $61.708$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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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: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{5}) \)
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: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 966)
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 \(\beta = 2\sqrt{5}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - q^{3} + 2 q^{5} - q^{7} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - q^{3} + 2 q^{5} - q^{7} + q^{9} - \beta q^{13} - 2 q^{15} + \beta q^{17} + ( - \beta + 2) q^{19} + q^{21} + q^{23} - q^{25} - q^{27} - 2 q^{29} + (\beta - 2) q^{31} - 2 q^{35} + (2 \beta - 2) q^{37} + \beta q^{39} - 6 q^{41} + (2 \beta - 4) q^{43} + 2 q^{45} + (\beta - 2) q^{47} + q^{49} - \beta q^{51} + ( - 2 \beta - 2) q^{53} + (\beta - 2) q^{57} - 4 q^{59} - 6 q^{61} - q^{63} - 2 \beta q^{65} + ( - 2 \beta + 4) q^{67} - q^{69} + ( - 2 \beta + 4) q^{71} + (2 \beta + 6) q^{73} + q^{75} + (2 \beta - 4) q^{79} + q^{81} + ( - \beta + 2) q^{83} + 2 \beta q^{85} + 2 q^{87} + (3 \beta - 4) q^{89} + \beta q^{91} + ( - \beta + 2) q^{93} + ( - 2 \beta + 4) q^{95} + ( - \beta + 4) q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{3} + 4 q^{5} - 2 q^{7} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{3} + 4 q^{5} - 2 q^{7} + 2 q^{9} - 4 q^{15} + 4 q^{19} + 2 q^{21} + 2 q^{23} - 2 q^{25} - 2 q^{27} - 4 q^{29} - 4 q^{31} - 4 q^{35} - 4 q^{37} - 12 q^{41} - 8 q^{43} + 4 q^{45} - 4 q^{47} + 2 q^{49} - 4 q^{53} - 4 q^{57} - 8 q^{59} - 12 q^{61} - 2 q^{63} + 8 q^{67} - 2 q^{69} + 8 q^{71} + 12 q^{73} + 2 q^{75} - 8 q^{79} + 2 q^{81} + 4 q^{83} + 4 q^{87} - 8 q^{89} + 4 q^{93} + 8 q^{95} + 8 q^{97}+O(q^{100}) \) 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
1.61803
−0.618034
0 −1.00000 0 2.00000 0 −1.00000 0 1.00000 0
1.2 0 −1.00000 0 2.00000 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.bd 2
4.b odd 2 1 966.2.a.p 2
12.b even 2 1 2898.2.a.v 2
28.d even 2 1 6762.2.a.bz 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
966.2.a.p 2 4.b odd 2 1
2898.2.a.v 2 12.b even 2 1
6762.2.a.bz 2 28.d even 2 1
7728.2.a.bd 2 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(7728))\):

\( T_{5} - 2 \) Copy content Toggle raw display
\( T_{11} \) Copy content Toggle raw display
\( T_{13}^{2} - 20 \) Copy content Toggle raw display
\( T_{17}^{2} - 20 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( (T + 1)^{2} \) Copy content Toggle raw display
$5$ \( (T - 2)^{2} \) Copy content Toggle raw display
$7$ \( (T + 1)^{2} \) Copy content Toggle raw display
$11$ \( T^{2} \) Copy content Toggle raw display
$13$ \( T^{2} - 20 \) Copy content Toggle raw display
$17$ \( T^{2} - 20 \) Copy content Toggle raw display
$19$ \( T^{2} - 4T - 16 \) Copy content Toggle raw display
$23$ \( (T - 1)^{2} \) Copy content Toggle raw display
$29$ \( (T + 2)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} + 4T - 16 \) Copy content Toggle raw display
$37$ \( T^{2} + 4T - 76 \) Copy content Toggle raw display
$41$ \( (T + 6)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} + 8T - 64 \) Copy content Toggle raw display
$47$ \( T^{2} + 4T - 16 \) Copy content Toggle raw display
$53$ \( T^{2} + 4T - 76 \) Copy content Toggle raw display
$59$ \( (T + 4)^{2} \) Copy content Toggle raw display
$61$ \( (T + 6)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} - 8T - 64 \) Copy content Toggle raw display
$71$ \( T^{2} - 8T - 64 \) Copy content Toggle raw display
$73$ \( T^{2} - 12T - 44 \) Copy content Toggle raw display
$79$ \( T^{2} + 8T - 64 \) Copy content Toggle raw display
$83$ \( T^{2} - 4T - 16 \) Copy content Toggle raw display
$89$ \( T^{2} + 8T - 164 \) Copy content Toggle raw display
$97$ \( T^{2} - 8T - 4 \) Copy content Toggle raw display
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