Properties

Label 1728.2.a.n
Level $1728$
Weight $2$
Character orbit 1728.a
Self dual yes
Analytic conductor $13.798$
Analytic rank $1$
Dimension $1$
CM discriminant -3
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1728,2,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, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1728.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1728 = 2^{6} \cdot 3^{3} \)
Weight: \( k \) \(=\) \( 2 \)
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: \(13.7981494693\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 27)
Fricke sign: \(1\)
Sato-Tate group: $N(\mathrm{U}(1))$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \( q - q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q - q^{7} - 5 q^{13} + 7 q^{19} - 5 q^{25} - 4 q^{31} - 11 q^{37} - 8 q^{43} - 6 q^{49} + q^{61} - 5 q^{67} - 7 q^{73} + 17 q^{79} + 5 q^{91} - 19 q^{97}+O(q^{100}) \) 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 0 0 −1.00000 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
3.b odd 2 1 CM by \(\Q(\sqrt{-3}) \)

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1728.2.a.n 1
3.b odd 2 1 CM 1728.2.a.n 1
4.b odd 2 1 1728.2.a.o 1
8.b even 2 1 27.2.a.a 1
8.d odd 2 1 432.2.a.e 1
12.b even 2 1 1728.2.a.o 1
24.f even 2 1 432.2.a.e 1
24.h odd 2 1 27.2.a.a 1
40.f even 2 1 675.2.a.e 1
40.i odd 4 2 675.2.b.f 2
56.h odd 2 1 1323.2.a.i 1
72.j odd 6 2 81.2.c.a 2
72.l even 6 2 1296.2.i.i 2
72.n even 6 2 81.2.c.a 2
72.p odd 6 2 1296.2.i.i 2
88.b odd 2 1 3267.2.a.f 1
104.e even 2 1 4563.2.a.e 1
120.i odd 2 1 675.2.a.e 1
120.w even 4 2 675.2.b.f 2
136.h even 2 1 7803.2.a.k 1
152.g odd 2 1 9747.2.a.f 1
168.i even 2 1 1323.2.a.i 1
216.t even 18 6 729.2.e.f 6
216.x odd 18 6 729.2.e.f 6
264.m even 2 1 3267.2.a.f 1
312.b odd 2 1 4563.2.a.e 1
408.b odd 2 1 7803.2.a.k 1
456.p even 2 1 9747.2.a.f 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
27.2.a.a 1 8.b even 2 1
27.2.a.a 1 24.h odd 2 1
81.2.c.a 2 72.j odd 6 2
81.2.c.a 2 72.n even 6 2
432.2.a.e 1 8.d odd 2 1
432.2.a.e 1 24.f even 2 1
675.2.a.e 1 40.f even 2 1
675.2.a.e 1 120.i odd 2 1
675.2.b.f 2 40.i odd 4 2
675.2.b.f 2 120.w even 4 2
729.2.e.f 6 216.t even 18 6
729.2.e.f 6 216.x odd 18 6
1296.2.i.i 2 72.l even 6 2
1296.2.i.i 2 72.p odd 6 2
1323.2.a.i 1 56.h odd 2 1
1323.2.a.i 1 168.i even 2 1
1728.2.a.n 1 1.a even 1 1 trivial
1728.2.a.n 1 3.b odd 2 1 CM
1728.2.a.o 1 4.b odd 2 1
1728.2.a.o 1 12.b even 2 1
3267.2.a.f 1 88.b odd 2 1
3267.2.a.f 1 264.m even 2 1
4563.2.a.e 1 104.e even 2 1
4563.2.a.e 1 312.b odd 2 1
7803.2.a.k 1 136.h even 2 1
7803.2.a.k 1 408.b odd 2 1
9747.2.a.f 1 152.g odd 2 1
9747.2.a.f 1 456.p even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(1728))\):

\( T_{5} \) Copy content Toggle raw display
\( T_{7} + 1 \) Copy content Toggle raw display
\( T_{11} \) 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
$5$ \( T \) Copy content Toggle raw display
$7$ \( T + 1 \) Copy content Toggle raw display
$11$ \( T \) Copy content Toggle raw display
$13$ \( T + 5 \) Copy content Toggle raw display
$17$ \( T \) Copy content Toggle raw display
$19$ \( T - 7 \) Copy content Toggle raw display
$23$ \( T \) Copy content Toggle raw display
$29$ \( T \) Copy content Toggle raw display
$31$ \( T + 4 \) Copy content Toggle raw display
$37$ \( T + 11 \) Copy content Toggle raw display
$41$ \( T \) Copy content Toggle raw display
$43$ \( T + 8 \) Copy content Toggle raw display
$47$ \( T \) Copy content Toggle raw display
$53$ \( T \) Copy content Toggle raw display
$59$ \( T \) Copy content Toggle raw display
$61$ \( T - 1 \) Copy content Toggle raw display
$67$ \( T + 5 \) Copy content Toggle raw display
$71$ \( T \) Copy content Toggle raw display
$73$ \( T + 7 \) Copy content Toggle raw display
$79$ \( T - 17 \) Copy content Toggle raw display
$83$ \( T \) Copy content Toggle raw display
$89$ \( T \) Copy content Toggle raw display
$97$ \( T + 19 \) Copy content Toggle raw display
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