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Properties

Label 2028.2.i.j
Level $2028$
Weight $2$
Character orbit 2028.i
Analytic conductor $16.194$
Analytic rank $0$
Dimension $6$
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(529,2028)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2028, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2028.529");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2028 = 2^{2} \cdot 3 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2028.i (of order \(3\), degree \(2\), not minimal)

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: no
Analytic conductor: \(16.1936615299\)
Analytic rank: \(0\)
Dimension: \(6\)
Relative dimension: \(3\) over \(\Q(\zeta_{3})\)
Coefficient field: 6.0.64827.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - x^{5} + 3x^{4} + 5x^{2} - 2x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_{5} - 1) q^{3} + (\beta_{3} - 2 \beta_{2} + 1) q^{5} + ( - \beta_{5} + \beta_{4} + \cdots + 2 \beta_1) q^{7}+ \cdots - \beta_{5} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{5} - 1) q^{3} + (\beta_{3} - 2 \beta_{2} + 1) q^{5} + ( - \beta_{5} + \beta_{4} + \cdots + 2 \beta_1) q^{7}+ \cdots + (\beta_{2} + 2) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 3 q^{3} - 6 q^{7} - 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 3 q^{3} - 6 q^{7} - 3 q^{9} - 7 q^{11} - 3 q^{17} - 13 q^{19} + 12 q^{21} + 14 q^{23} - 2 q^{25} + 6 q^{27} - 10 q^{29} + 12 q^{31} - 7 q^{33} + 14 q^{35} - 6 q^{37} - 13 q^{41} + 5 q^{43} + 14 q^{47} - 5 q^{49} + 6 q^{51} - 6 q^{53} + 7 q^{55} + 26 q^{57} + 6 q^{59} + q^{61} - 6 q^{63} - 33 q^{67} + 14 q^{69} + 6 q^{71} + 28 q^{73} + q^{75} + 42 q^{77} - 22 q^{79} - 3 q^{81} + 16 q^{83} - 7 q^{85} - 10 q^{87} + 7 q^{89} - 6 q^{93} + 7 q^{95} - 19 q^{97} + 14 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - x^{5} + 3x^{4} + 5x^{2} - 2x + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{5} + 3\nu^{4} - 9\nu^{3} + 5\nu^{2} - 2\nu + 6 ) / 13 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -3\nu^{5} + 9\nu^{4} - 14\nu^{3} + 15\nu^{2} - 6\nu + 18 ) / 13 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -4\nu^{5} - \nu^{4} - 10\nu^{3} - 6\nu^{2} - 34\nu - 2 ) / 13 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -6\nu^{5} + 5\nu^{4} - 15\nu^{3} - 9\nu^{2} - 25\nu + 10 ) / 13 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{5} + \beta_{4} + \beta_{3} - \beta_{2} + \beta_1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} - 3\beta_{2} \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 2\beta_{5} - 3\beta_{4} - 4\beta _1 - 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( \beta_{5} - 4\beta_{4} - 4\beta_{3} + 9\beta_{2} - 9\beta_1 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2028\mathbb{Z}\right)^\times\).

\(n\) \(677\) \(1015\) \(1861\)
\(\chi(n)\) \(1\) \(1\) \(-\beta_{5}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
529.1
0.900969 − 1.56052i
0.222521 − 0.385418i
−0.623490 + 1.07992i
0.900969 + 1.56052i
0.222521 + 0.385418i
−0.623490 − 1.07992i
0 −0.500000 + 0.866025i 0 −3.04892 0 −2.52446 − 4.37249i 0 −0.500000 − 0.866025i 0
529.2 0 −0.500000 + 0.866025i 0 1.35690 0 −0.321552 − 0.556945i 0 −0.500000 − 0.866025i 0
529.3 0 −0.500000 + 0.866025i 0 1.69202 0 −0.153989 − 0.266717i 0 −0.500000 − 0.866025i 0
2005.1 0 −0.500000 − 0.866025i 0 −3.04892 0 −2.52446 + 4.37249i 0 −0.500000 + 0.866025i 0
2005.2 0 −0.500000 − 0.866025i 0 1.35690 0 −0.321552 + 0.556945i 0 −0.500000 + 0.866025i 0
2005.3 0 −0.500000 − 0.866025i 0 1.69202 0 −0.153989 + 0.266717i 0 −0.500000 + 0.866025i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 529.3
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2028.2.i.j 6
13.b even 2 1 2028.2.i.k 6
13.c even 3 1 2028.2.a.l yes 3
13.c even 3 1 inner 2028.2.i.j 6
13.d odd 4 2 2028.2.q.i 12
13.e even 6 1 2028.2.a.k ✓ 3
13.e even 6 1 2028.2.i.k 6
13.f odd 12 2 2028.2.b.g 6
13.f odd 12 2 2028.2.q.i 12
39.h odd 6 1 6084.2.a.z 3
39.i odd 6 1 6084.2.a.ba 3
39.k even 12 2 6084.2.b.q 6
52.i odd 6 1 8112.2.a.cd 3
52.j odd 6 1 8112.2.a.ca 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2028.2.a.k ✓ 3 13.e even 6 1
2028.2.a.l yes 3 13.c even 3 1
2028.2.b.g 6 13.f odd 12 2
2028.2.i.j 6 1.a even 1 1 trivial
2028.2.i.j 6 13.c even 3 1 inner
2028.2.i.k 6 13.b even 2 1
2028.2.i.k 6 13.e even 6 1
2028.2.q.i 12 13.d odd 4 2
2028.2.q.i 12 13.f odd 12 2
6084.2.a.z 3 39.h odd 6 1
6084.2.a.ba 3 39.i odd 6 1
6084.2.b.q 6 39.k even 12 2
8112.2.a.ca 3 52.j odd 6 1
8112.2.a.cd 3 52.i odd 6 1

Hecke kernels

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

\( T_{5}^{3} - 7T_{5} + 7 \) Copy content Toggle raw display
\( T_{7}^{6} + 6T_{7}^{5} + 31T_{7}^{4} + 28T_{7}^{3} + 19T_{7}^{2} + 5T_{7} + 1 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} \) Copy content Toggle raw display
$3$ \( (T^{2} + T + 1)^{3} \) Copy content Toggle raw display
$5$ \( (T^{3} - 7 T + 7)^{2} \) Copy content Toggle raw display
$7$ \( T^{6} + 6 T^{5} + \cdots + 1 \) Copy content Toggle raw display
$11$ \( T^{6} + 7 T^{5} + \cdots + 49 \) Copy content Toggle raw display
$13$ \( T^{6} \) Copy content Toggle raw display
$17$ \( T^{6} + 3 T^{5} + \cdots + 1 \) Copy content Toggle raw display
$19$ \( T^{6} + 13 T^{5} + \cdots + 169 \) Copy content Toggle raw display
$23$ \( T^{6} - 14 T^{5} + \cdots + 49 \) Copy content Toggle raw display
$29$ \( T^{6} + 10 T^{5} + \cdots + 169 \) Copy content Toggle raw display
$31$ \( (T^{3} - 6 T^{2} + \cdots + 181)^{2} \) Copy content Toggle raw display
$37$ \( T^{6} + 6 T^{5} + \cdots + 1 \) Copy content Toggle raw display
$41$ \( T^{6} + 13 T^{5} + \cdots + 6889 \) Copy content Toggle raw display
$43$ \( T^{6} - 5 T^{5} + \cdots + 1681 \) Copy content Toggle raw display
$47$ \( (T^{3} - 7 T^{2} + \cdots + 301)^{2} \) Copy content Toggle raw display
$53$ \( (T^{3} + 3 T^{2} + \cdots - 643)^{2} \) Copy content Toggle raw display
$59$ \( T^{6} - 6 T^{5} + \cdots + 1236544 \) Copy content Toggle raw display
$61$ \( T^{6} - T^{5} + \cdots + 1681 \) Copy content Toggle raw display
$67$ \( T^{6} + 33 T^{5} + \cdots + 1555009 \) Copy content Toggle raw display
$71$ \( T^{6} - 6 T^{5} + \cdots + 5041 \) Copy content Toggle raw display
$73$ \( (T^{3} - 14 T^{2} + \cdots + 91)^{2} \) Copy content Toggle raw display
$79$ \( (T^{3} + 11 T^{2} + \cdots + 13)^{2} \) Copy content Toggle raw display
$83$ \( (T^{3} - 8 T^{2} - 149 T + 29)^{2} \) Copy content Toggle raw display
$89$ \( T^{6} - 7 T^{5} + \cdots + 946729 \) Copy content Toggle raw display
$97$ \( T^{6} + 19 T^{5} + \cdots + 841 \) Copy content Toggle raw display
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