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Properties

Label 2028.2.i.b
Level $2028$
Weight $2$
Character orbit 2028.i
Analytic conductor $16.194$
Analytic rank $0$
Dimension $2$
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: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 156)
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 primitive root of unity \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\zeta_{6} - 1) q^{3} - 2 \zeta_{6} q^{7} - \zeta_{6} q^{9} + 6 \zeta_{6} q^{17} - 2 \zeta_{6} q^{19} + 2 q^{21} - 5 q^{25} + q^{27} + ( - 6 \zeta_{6} + 6) q^{29} + 2 q^{31} + (2 \zeta_{6} - 2) q^{37} + \cdots + 10 \zeta_{6} q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - q^{3} - 2 q^{7} - q^{9} + 6 q^{17} - 2 q^{19} + 4 q^{21} - 10 q^{25} + 2 q^{27} + 6 q^{29} + 4 q^{31} - 2 q^{37} + 12 q^{41} + 4 q^{43} + 3 q^{49} - 12 q^{51} + 12 q^{53} + 4 q^{57} - 12 q^{59}+ \cdots + 10 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(677\) \(1015\) \(1861\)
\(\chi(n)\) \(1\) \(1\) \(-\zeta_{6}\)

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
0.500000 + 0.866025i
0.500000 − 0.866025i
0 −0.500000 + 0.866025i 0 0 0 −1.00000 − 1.73205i 0 −0.500000 − 0.866025i 0
2005.1 0 −0.500000 − 0.866025i 0 0 0 −1.00000 + 1.73205i 0 −0.500000 + 0.866025i 0
\(n\): e.g. 2-40 or 990-1000
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.b 2
13.b even 2 1 2028.2.i.c 2
13.c even 3 1 156.2.a.b ✓ 1
13.c even 3 1 inner 2028.2.i.b 2
13.d odd 4 2 2028.2.q.d 4
13.e even 6 1 2028.2.a.e 1
13.e even 6 1 2028.2.i.c 2
13.f odd 12 2 2028.2.b.d 2
13.f odd 12 2 2028.2.q.d 4
39.h odd 6 1 6084.2.a.h 1
39.i odd 6 1 468.2.a.c 1
39.k even 12 2 6084.2.b.a 2
52.i odd 6 1 8112.2.a.i 1
52.j odd 6 1 624.2.a.b 1
65.n even 6 1 3900.2.a.a 1
65.q odd 12 2 3900.2.h.e 2
91.n odd 6 1 7644.2.a.a 1
104.n odd 6 1 2496.2.a.v 1
104.r even 6 1 2496.2.a.h 1
117.f even 3 1 4212.2.i.f 2
117.h even 3 1 4212.2.i.f 2
117.k odd 6 1 4212.2.i.g 2
117.u odd 6 1 4212.2.i.g 2
156.p even 6 1 1872.2.a.i 1
312.bh odd 6 1 7488.2.a.bf 1
312.bn even 6 1 7488.2.a.bb 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
156.2.a.b ✓ 1 13.c even 3 1
468.2.a.c 1 39.i odd 6 1
624.2.a.b 1 52.j odd 6 1
1872.2.a.i 1 156.p even 6 1
2028.2.a.e 1 13.e even 6 1
2028.2.b.d 2 13.f odd 12 2
2028.2.i.b 2 1.a even 1 1 trivial
2028.2.i.b 2 13.c even 3 1 inner
2028.2.i.c 2 13.b even 2 1
2028.2.i.c 2 13.e even 6 1
2028.2.q.d 4 13.d odd 4 2
2028.2.q.d 4 13.f odd 12 2
2496.2.a.h 1 104.r even 6 1
2496.2.a.v 1 104.n odd 6 1
3900.2.a.a 1 65.n even 6 1
3900.2.h.e 2 65.q odd 12 2
4212.2.i.f 2 117.f even 3 1
4212.2.i.f 2 117.h even 3 1
4212.2.i.g 2 117.k odd 6 1
4212.2.i.g 2 117.u odd 6 1
6084.2.a.h 1 39.h odd 6 1
6084.2.b.a 2 39.k even 12 2
7488.2.a.bb 1 312.bn even 6 1
7488.2.a.bf 1 312.bh odd 6 1
7644.2.a.a 1 91.n odd 6 1
8112.2.a.i 1 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 \(S_{2}^{\mathrm{new}}(2028, [\chi])\):

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} + T + 1 \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( T^{2} + 2T + 4 \) Copy content Toggle raw display
$11$ \( T^{2} \) Copy content Toggle raw display
$13$ \( T^{2} \) Copy content Toggle raw display
$17$ \( T^{2} - 6T + 36 \) Copy content Toggle raw display
$19$ \( T^{2} + 2T + 4 \) Copy content Toggle raw display
$23$ \( T^{2} \) Copy content Toggle raw display
$29$ \( T^{2} - 6T + 36 \) Copy content Toggle raw display
$31$ \( (T - 2)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} + 2T + 4 \) Copy content Toggle raw display
$41$ \( T^{2} - 12T + 144 \) Copy content Toggle raw display
$43$ \( T^{2} - 4T + 16 \) Copy content Toggle raw display
$47$ \( T^{2} \) Copy content Toggle raw display
$53$ \( (T - 6)^{2} \) Copy content Toggle raw display
$59$ \( T^{2} + 12T + 144 \) Copy content Toggle raw display
$61$ \( T^{2} + 2T + 4 \) Copy content Toggle raw display
$67$ \( T^{2} - 10T + 100 \) Copy content Toggle raw display
$71$ \( T^{2} + 12T + 144 \) Copy content Toggle raw display
$73$ \( (T - 14)^{2} \) Copy content Toggle raw display
$79$ \( (T - 8)^{2} \) Copy content Toggle raw display
$83$ \( (T - 12)^{2} \) Copy content Toggle raw display
$89$ \( T^{2} \) Copy content Toggle raw display
$97$ \( T^{2} - 10T + 100 \) Copy content Toggle raw display
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