Properties

Label 2028.2.a.e
Level $2028$
Weight $2$
Character orbit 2028.a
Self dual yes
Analytic conductor $16.194$
Analytic rank $1$
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] = [2028,2,Mod(1,2028)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2028, 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("2028.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2028 = 2^{2} \cdot 3 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2028.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: \(16.1936615299\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 156)
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)
 
\(f(q)\) \(=\) \( q + q^{3} - 2 q^{7} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{3} - 2 q^{7} + q^{9} - 6 q^{17} - 2 q^{19} - 2 q^{21} - 5 q^{25} + q^{27} - 6 q^{29} - 2 q^{31} - 2 q^{37} + 12 q^{41} - 4 q^{43} - 3 q^{49} - 6 q^{51} + 6 q^{53} - 2 q^{57} - 12 q^{59} + 2 q^{61} - 2 q^{63} + 10 q^{67} - 12 q^{71} - 14 q^{73} - 5 q^{75} + 8 q^{79} + q^{81} - 12 q^{83} - 6 q^{87} - 2 q^{93} + 10 q^{97}+O(q^{100}) \) 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 1.00000 0 0 0 −2.00000 0 1.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(-1\)
\(13\) \(1\)

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 2028.2.a.e 1
3.b odd 2 1 6084.2.a.h 1
4.b odd 2 1 8112.2.a.i 1
13.b even 2 1 156.2.a.b 1
13.c even 3 2 2028.2.i.c 2
13.d odd 4 2 2028.2.b.d 2
13.e even 6 2 2028.2.i.b 2
13.f odd 12 4 2028.2.q.d 4
39.d odd 2 1 468.2.a.c 1
39.f even 4 2 6084.2.b.a 2
52.b odd 2 1 624.2.a.b 1
65.d even 2 1 3900.2.a.a 1
65.h odd 4 2 3900.2.h.e 2
91.b odd 2 1 7644.2.a.a 1
104.e even 2 1 2496.2.a.h 1
104.h odd 2 1 2496.2.a.v 1
117.n odd 6 2 4212.2.i.g 2
117.t even 6 2 4212.2.i.f 2
156.h even 2 1 1872.2.a.i 1
312.b odd 2 1 7488.2.a.bf 1
312.h even 2 1 7488.2.a.bb 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
156.2.a.b 1 13.b even 2 1
468.2.a.c 1 39.d odd 2 1
624.2.a.b 1 52.b odd 2 1
1872.2.a.i 1 156.h even 2 1
2028.2.a.e 1 1.a even 1 1 trivial
2028.2.b.d 2 13.d odd 4 2
2028.2.i.b 2 13.e even 6 2
2028.2.i.c 2 13.c even 3 2
2028.2.q.d 4 13.f odd 12 4
2496.2.a.h 1 104.e even 2 1
2496.2.a.v 1 104.h odd 2 1
3900.2.a.a 1 65.d even 2 1
3900.2.h.e 2 65.h odd 4 2
4212.2.i.f 2 117.t even 6 2
4212.2.i.g 2 117.n odd 6 2
6084.2.a.h 1 3.b odd 2 1
6084.2.b.a 2 39.f even 4 2
7488.2.a.bb 1 312.h even 2 1
7488.2.a.bf 1 312.b odd 2 1
7644.2.a.a 1 91.b odd 2 1
8112.2.a.i 1 4.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(2028))\):

\( T_{5} \) Copy content Toggle raw display
\( T_{7} + 2 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \) Copy content Toggle raw display
$3$ \( T - 1 \) Copy content Toggle raw display
$5$ \( T \) Copy content Toggle raw display
$7$ \( T + 2 \) Copy content Toggle raw display
$11$ \( T \) Copy content Toggle raw display
$13$ \( T \) Copy content Toggle raw display
$17$ \( T + 6 \) Copy content Toggle raw display
$19$ \( T + 2 \) Copy content Toggle raw display
$23$ \( T \) Copy content Toggle raw display
$29$ \( T + 6 \) Copy content Toggle raw display
$31$ \( T + 2 \) Copy content Toggle raw display
$37$ \( T + 2 \) Copy content Toggle raw display
$41$ \( T - 12 \) Copy content Toggle raw display
$43$ \( T + 4 \) Copy content Toggle raw display
$47$ \( T \) Copy content Toggle raw display
$53$ \( T - 6 \) Copy content Toggle raw display
$59$ \( T + 12 \) Copy content Toggle raw display
$61$ \( T - 2 \) Copy content Toggle raw display
$67$ \( T - 10 \) Copy content Toggle raw display
$71$ \( T + 12 \) Copy content Toggle raw display
$73$ \( T + 14 \) Copy content Toggle raw display
$79$ \( T - 8 \) Copy content Toggle raw display
$83$ \( T + 12 \) Copy content Toggle raw display
$89$ \( T \) Copy content Toggle raw display
$97$ \( T - 10 \) Copy content Toggle raw display
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