Properties

Label 1728.4.a.k
Level $1728$
Weight $4$
Character orbit 1728.a
Self dual yes
Analytic conductor $101.955$
Analytic rank $2$
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] = [1728,4,Mod(1,1728)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
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")
 
//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

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: \(101.955300490\)
Analytic rank: \(2\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 54)
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)
 
\(f(q)\) \(=\) \( q - 3 q^{5} - 29 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q - 3 q^{5} - 29 q^{7} - 57 q^{11} - 20 q^{13} - 72 q^{17} - 106 q^{19} - 174 q^{23} - 116 q^{25} + 210 q^{29} - 47 q^{31} + 87 q^{35} - 2 q^{37} - 6 q^{41} + 218 q^{43} - 474 q^{47} + 498 q^{49} - 81 q^{53} + 171 q^{55} + 84 q^{59} - 56 q^{61} + 60 q^{65} - 142 q^{67} - 360 q^{71} - 1159 q^{73} + 1653 q^{77} + 160 q^{79} + 735 q^{83} + 216 q^{85} - 954 q^{89} + 580 q^{91} + 318 q^{95} + 191 q^{97}+O(q^{100}) \) 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
0
0 0 0 −3.00000 0 −29.0000 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

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 1728.4.a.k 1
3.b odd 2 1 1728.4.a.u 1
4.b odd 2 1 1728.4.a.l 1
8.b even 2 1 432.4.a.j 1
8.d odd 2 1 54.4.a.c yes 1
12.b even 2 1 1728.4.a.v 1
24.f even 2 1 54.4.a.b 1
24.h odd 2 1 432.4.a.e 1
40.e odd 2 1 1350.4.a.a 1
40.k even 4 2 1350.4.c.b 2
72.l even 6 2 162.4.c.g 2
72.p odd 6 2 162.4.c.b 2
120.m even 2 1 1350.4.a.o 1
120.q odd 4 2 1350.4.c.s 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
54.4.a.b 1 24.f even 2 1
54.4.a.c yes 1 8.d odd 2 1
162.4.c.b 2 72.p odd 6 2
162.4.c.g 2 72.l even 6 2
432.4.a.e 1 24.h odd 2 1
432.4.a.j 1 8.b even 2 1
1350.4.a.a 1 40.e odd 2 1
1350.4.a.o 1 120.m even 2 1
1350.4.c.b 2 40.k even 4 2
1350.4.c.s 2 120.q odd 4 2
1728.4.a.k 1 1.a even 1 1 trivial
1728.4.a.l 1 4.b odd 2 1
1728.4.a.u 1 3.b odd 2 1
1728.4.a.v 1 12.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 \(S_{4}^{\mathrm{new}}(\Gamma_0(1728))\):

\( T_{5} + 3 \) Copy content Toggle raw display
\( T_{7} + 29 \) Copy content Toggle raw display
\( T_{11} + 57 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \) Copy content Toggle raw display
$3$ \( T \) Copy content Toggle raw display
$5$ \( T + 3 \) Copy content Toggle raw display
$7$ \( T + 29 \) Copy content Toggle raw display
$11$ \( T + 57 \) Copy content Toggle raw display
$13$ \( T + 20 \) Copy content Toggle raw display
$17$ \( T + 72 \) Copy content Toggle raw display
$19$ \( T + 106 \) Copy content Toggle raw display
$23$ \( T + 174 \) Copy content Toggle raw display
$29$ \( T - 210 \) Copy content Toggle raw display
$31$ \( T + 47 \) Copy content Toggle raw display
$37$ \( T + 2 \) Copy content Toggle raw display
$41$ \( T + 6 \) Copy content Toggle raw display
$43$ \( T - 218 \) Copy content Toggle raw display
$47$ \( T + 474 \) Copy content Toggle raw display
$53$ \( T + 81 \) Copy content Toggle raw display
$59$ \( T - 84 \) Copy content Toggle raw display
$61$ \( T + 56 \) Copy content Toggle raw display
$67$ \( T + 142 \) Copy content Toggle raw display
$71$ \( T + 360 \) Copy content Toggle raw display
$73$ \( T + 1159 \) Copy content Toggle raw display
$79$ \( T - 160 \) Copy content Toggle raw display
$83$ \( T - 735 \) Copy content Toggle raw display
$89$ \( T + 954 \) Copy content Toggle raw display
$97$ \( T - 191 \) Copy content Toggle raw display
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