↑

Properties

Label 1728.4.a.bn
Level $1728$
Weight $4$
Character orbit 1728.a
Self dual yes
Analytic conductor $101.955$
Analytic rank $0$
Dimension $2$
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [1728,4,Mod(1,1728)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1728, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 4, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1728.1"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Level: \( N \) \(=\) \( 1728 = 2^{6} \cdot 3^{3} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1728.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,0,0,0,0,0,0,0,0,2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(101.955300490\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{73}) \)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 18 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 864)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content gp:mfcoefs(f, 20)
 

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{73}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta q^{5} - \beta q^{7} + q^{11} + 8 q^{13} - 12 \beta q^{17} - 14 \beta q^{19} + 18 q^{23} - 52 q^{25} + 14 \beta q^{29} - 11 \beta q^{31} + 73 q^{35} + 146 q^{37} + 10 \beta q^{41} + 26 \beta q^{43} + \cdots - 1241 q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{11} + 16 q^{13} + 36 q^{23} - 104 q^{25} + 146 q^{35} + 292 q^{37} + 212 q^{47} - 540 q^{49} + 40 q^{59} + 816 q^{61} - 40 q^{71} - 1182 q^{73} + 690 q^{83} + 1752 q^{85} + 2044 q^{95} - 2482 q^{97}+O(q^{100}) \) 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.

Copy content comment:embeddings in the coefficient field
 
Copy content 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
4.77200
−3.77200
0 0 0 −8.54400 0 −8.54400 0 0 0
1.2 0 0 0 8.54400 0 8.54400 0 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

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

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
12.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1728.4.a.bn 2
3.b odd 2 1 1728.4.a.bm 2
4.b odd 2 1 1728.4.a.bm 2
8.b even 2 1 864.4.a.f ✓ 2
8.d odd 2 1 864.4.a.g yes 2
12.b even 2 1 inner 1728.4.a.bn 2
24.f even 2 1 864.4.a.f ✓ 2
24.h odd 2 1 864.4.a.g yes 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
864.4.a.f ✓ 2 8.b even 2 1
864.4.a.f ✓ 2 24.f even 2 1
864.4.a.g yes 2 8.d odd 2 1
864.4.a.g yes 2 24.h odd 2 1
1728.4.a.bm 2 3.b odd 2 1
1728.4.a.bm 2 4.b odd 2 1
1728.4.a.bn 2 1.a even 1 1 trivial
1728.4.a.bn 2 12.b even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1728))\):

\( T_{5}^{2} - 73 \) Copy content Toggle raw display
\( T_{7}^{2} - 73 \) Copy content Toggle raw display
\( T_{11} - 1 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} - 73 \) Copy content Toggle raw display
$7$ \( T^{2} - 73 \) Copy content Toggle raw display
$11$ \( (T - 1)^{2} \) Copy content Toggle raw display
$13$ \( (T - 8)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} - 10512 \) Copy content Toggle raw display
$19$ \( T^{2} - 14308 \) Copy content Toggle raw display
$23$ \( (T - 18)^{2} \) Copy content Toggle raw display
$29$ \( T^{2} - 14308 \) Copy content Toggle raw display
$31$ \( T^{2} - 8833 \) Copy content Toggle raw display
$37$ \( (T - 146)^{2} \) Copy content Toggle raw display
$41$ \( T^{2} - 7300 \) Copy content Toggle raw display
$43$ \( T^{2} - 49348 \) Copy content Toggle raw display
$47$ \( (T - 106)^{2} \) Copy content Toggle raw display
$53$ \( T^{2} - 89425 \) Copy content Toggle raw display
$59$ \( (T - 20)^{2} \) Copy content Toggle raw display
$61$ \( (T - 408)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} - 357700 \) Copy content Toggle raw display
$71$ \( (T + 20)^{2} \) Copy content Toggle raw display
$73$ \( (T + 591)^{2} \) Copy content Toggle raw display
$79$ \( T^{2} - 467200 \) Copy content Toggle raw display
$83$ \( (T - 345)^{2} \) Copy content Toggle raw display
$89$ \( T^{2} - 49348 \) Copy content Toggle raw display
$97$ \( (T + 1241)^{2} \) Copy content Toggle raw display
show more
show less