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Properties

Label 2028.2.a.m
Level $2028$
Weight $2$
Character orbit 2028.a
Self dual yes
Analytic conductor $16.194$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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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: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{3}, \sqrt{43})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 23x^{2} + 100 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
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)
 

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + q^{3} + \beta_1 q^{5} + ( - \beta_{2} + \beta_1) q^{7} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{3} + \beta_1 q^{5} + ( - \beta_{2} + \beta_1) q^{7} + q^{9} - 2 \beta_{2} q^{11} + \beta_1 q^{15} - \beta_{3} q^{17} + 2 \beta_{2} q^{19} + ( - \beta_{2} + \beta_1) q^{21} + 2 q^{23} + (\beta_{3} + 7) q^{25} + q^{27} + (\beta_{3} - 2) q^{29} + (3 \beta_{2} - \beta_1) q^{31} - 2 \beta_{2} q^{33} + 10 q^{35} + ( - 2 \beta_{2} - \beta_1) q^{37} + (2 \beta_{2} - \beta_1) q^{41} + (\beta_{3} + 7) q^{43} + \beta_1 q^{45} + 6 \beta_{2} q^{47} + ( - \beta_{3} + 4) q^{49} - \beta_{3} q^{51} + (\beta_{3} - 4) q^{53} + ( - 2 \beta_{3} - 4) q^{55} + 2 \beta_{2} q^{57} + ( - 2 \beta_{2} - 2 \beta_1) q^{59} + 5 q^{61} + ( - \beta_{2} + \beta_1) q^{63} + (\beta_{2} - \beta_1) q^{67} + 2 q^{69} + 2 \beta_{2} q^{71} + (\beta_{2} - 4 \beta_1) q^{73} + (\beta_{3} + 7) q^{75} + ( - 2 \beta_{3} + 2) q^{77} + ( - \beta_{3} + 7) q^{79} + q^{81} + (4 \beta_{2} - 2 \beta_1) q^{83} + ( - 10 \beta_{2} - \beta_1) q^{85} + (\beta_{3} - 2) q^{87} - 4 \beta_{2} q^{89} + (3 \beta_{2} - \beta_1) q^{93} + (2 \beta_{3} + 4) q^{95} + ( - 3 \beta_{2} - 3 \beta_1) q^{97} - 2 \beta_{2} q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 4 q^{3} + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 4 q^{3} + 4 q^{9} + 2 q^{17} + 8 q^{23} + 26 q^{25} + 4 q^{27} - 10 q^{29} + 40 q^{35} + 26 q^{43} + 18 q^{49} + 2 q^{51} - 18 q^{53} - 12 q^{55} + 20 q^{61} + 8 q^{69} + 26 q^{75} + 12 q^{77} + 30 q^{79} + 4 q^{81} - 10 q^{87} + 12 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - 23x^{2} + 100 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{3} - 13\nu ) / 10 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{2} - 12 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{3} + 12 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 10\beta_{2} + 13\beta_1 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

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
−4.14474
−2.41269
2.41269
4.14474
0 1.00000 0 −4.14474 0 −2.41269 0 1.00000 0
1.2 0 1.00000 0 −2.41269 0 −4.14474 0 1.00000 0
1.3 0 1.00000 0 2.41269 0 4.14474 0 1.00000 0
1.4 0 1.00000 0 4.14474 0 2.41269 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

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2028.2.a.m 4
3.b odd 2 1 6084.2.a.bd 4
4.b odd 2 1 8112.2.a.cr 4
13.b even 2 1 inner 2028.2.a.m 4
13.c even 3 2 2028.2.i.n 8
13.d odd 4 2 2028.2.b.e 4
13.e even 6 2 2028.2.i.n 8
13.f odd 12 2 156.2.q.b ✓ 4
13.f odd 12 2 2028.2.q.f 4
39.d odd 2 1 6084.2.a.bd 4
39.f even 4 2 6084.2.b.o 4
39.k even 12 2 468.2.t.d 4
52.b odd 2 1 8112.2.a.cr 4
52.l even 12 2 624.2.bv.f 4
65.o even 12 2 3900.2.bw.j 8
65.s odd 12 2 3900.2.cd.i 4
65.t even 12 2 3900.2.bw.j 8
156.v odd 12 2 1872.2.by.j 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
156.2.q.b ✓ 4 13.f odd 12 2
468.2.t.d 4 39.k even 12 2
624.2.bv.f 4 52.l even 12 2
1872.2.by.j 4 156.v odd 12 2
2028.2.a.m 4 1.a even 1 1 trivial
2028.2.a.m 4 13.b even 2 1 inner
2028.2.b.e 4 13.d odd 4 2
2028.2.i.n 8 13.c even 3 2
2028.2.i.n 8 13.e even 6 2
2028.2.q.f 4 13.f odd 12 2
3900.2.bw.j 8 65.o even 12 2
3900.2.bw.j 8 65.t even 12 2
3900.2.cd.i 4 65.s odd 12 2
6084.2.a.bd 4 3.b odd 2 1
6084.2.a.bd 4 39.d odd 2 1
6084.2.b.o 4 39.f even 4 2
8112.2.a.cr 4 4.b odd 2 1
8112.2.a.cr 4 52.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 \(S_{2}^{\mathrm{new}}(\Gamma_0(2028))\):

\( T_{5}^{4} - 23T_{5}^{2} + 100 \) Copy content Toggle raw display
\( T_{7}^{4} - 23T_{7}^{2} + 100 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( (T - 1)^{4} \) Copy content Toggle raw display
$5$ \( T^{4} - 23T^{2} + 100 \) Copy content Toggle raw display
$7$ \( T^{4} - 23T^{2} + 100 \) Copy content Toggle raw display
$11$ \( (T^{2} - 12)^{2} \) Copy content Toggle raw display
$13$ \( T^{4} \) Copy content Toggle raw display
$17$ \( (T^{2} - T - 32)^{2} \) Copy content Toggle raw display
$19$ \( (T^{2} - 12)^{2} \) Copy content Toggle raw display
$23$ \( (T - 2)^{4} \) Copy content Toggle raw display
$29$ \( (T^{2} + 5 T - 26)^{2} \) Copy content Toggle raw display
$31$ \( T^{4} - 59T^{2} + 64 \) Copy content Toggle raw display
$37$ \( T^{4} - 59T^{2} + 64 \) Copy content Toggle raw display
$41$ \( T^{4} - 35T^{2} + 16 \) Copy content Toggle raw display
$43$ \( (T^{2} - 13 T + 10)^{2} \) Copy content Toggle raw display
$47$ \( (T^{2} - 108)^{2} \) Copy content Toggle raw display
$53$ \( (T^{2} + 9 T - 12)^{2} \) Copy content Toggle raw display
$59$ \( T^{4} - 140T^{2} + 256 \) Copy content Toggle raw display
$61$ \( (T - 5)^{4} \) Copy content Toggle raw display
$67$ \( T^{4} - 23T^{2} + 100 \) Copy content Toggle raw display
$71$ \( (T^{2} - 12)^{2} \) Copy content Toggle raw display
$73$ \( T^{4} - 350 T^{2} + 28561 \) Copy content Toggle raw display
$79$ \( (T^{2} - 15 T + 24)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} - 140T^{2} + 256 \) Copy content Toggle raw display
$89$ \( (T^{2} - 48)^{2} \) Copy content Toggle raw display
$97$ \( T^{4} - 315T^{2} + 1296 \) Copy content Toggle raw display
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