Properties

Label 2028.2.i.a
Level $2028$
Weight $2$
Character orbit 2028.i
Analytic conductor $16.194$
Analytic rank $1$
Dimension $2$
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: \(1\)
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.

\(f(q)\) \(=\) \( q + (\zeta_{6} - 1) q^{3} - 2 q^{5} - 4 \zeta_{6} q^{7} - \zeta_{6} q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + (\zeta_{6} - 1) q^{3} - 2 q^{5} - 4 \zeta_{6} q^{7} - \zeta_{6} q^{9} + ( - 6 \zeta_{6} + 6) q^{11} + ( - 2 \zeta_{6} + 2) q^{15} - 2 \zeta_{6} q^{17} + 4 q^{21} + (8 \zeta_{6} - 8) q^{23} - q^{25} + q^{27} + (2 \zeta_{6} - 2) q^{29} + 8 q^{31} + 6 \zeta_{6} q^{33} + 8 \zeta_{6} q^{35} + (8 \zeta_{6} - 8) q^{37} + (2 \zeta_{6} - 2) q^{41} - 8 \zeta_{6} q^{43} + 2 \zeta_{6} q^{45} - 6 q^{47} + (9 \zeta_{6} - 9) q^{49} + 2 q^{51} + 6 q^{53} + (12 \zeta_{6} - 12) q^{55} + 2 \zeta_{6} q^{59} - 2 \zeta_{6} q^{61} + (4 \zeta_{6} - 4) q^{63} + ( - 4 \zeta_{6} + 4) q^{67} - 8 \zeta_{6} q^{69} + 6 \zeta_{6} q^{71} - 4 q^{73} + ( - \zeta_{6} + 1) q^{75} - 24 q^{77} + (\zeta_{6} - 1) q^{81} - 14 q^{83} + 4 \zeta_{6} q^{85} - 2 \zeta_{6} q^{87} + (6 \zeta_{6} - 6) q^{89} + (8 \zeta_{6} - 8) q^{93} - 12 \zeta_{6} q^{97} - 6 q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - q^{3} - 4 q^{5} - 4 q^{7} - q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - q^{3} - 4 q^{5} - 4 q^{7} - q^{9} + 6 q^{11} + 2 q^{15} - 2 q^{17} + 8 q^{21} - 8 q^{23} - 2 q^{25} + 2 q^{27} - 2 q^{29} + 16 q^{31} + 6 q^{33} + 8 q^{35} - 8 q^{37} - 2 q^{41} - 8 q^{43} + 2 q^{45} - 12 q^{47} - 9 q^{49} + 4 q^{51} + 12 q^{53} - 12 q^{55} + 2 q^{59} - 2 q^{61} - 4 q^{63} + 4 q^{67} - 8 q^{69} + 6 q^{71} - 8 q^{73} + q^{75} - 48 q^{77} - q^{81} - 28 q^{83} + 4 q^{85} - 2 q^{87} - 6 q^{89} - 8 q^{93} - 12 q^{97} - 12 q^{99}+O(q^{100}) \) 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 −2.00000 0 −2.00000 3.46410i 0 −0.500000 0.866025i 0
2005.1 0 −0.500000 0.866025i 0 −2.00000 0 −2.00000 + 3.46410i 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.a 2
13.b even 2 1 2028.2.i.d 2
13.c even 3 1 2028.2.a.d 1
13.c even 3 1 inner 2028.2.i.a 2
13.d odd 4 2 2028.2.q.e 4
13.e even 6 1 2028.2.a.f 1
13.e even 6 1 2028.2.i.d 2
13.f odd 12 2 156.2.b.b 2
13.f odd 12 2 2028.2.q.e 4
39.h odd 6 1 6084.2.a.d 1
39.i odd 6 1 6084.2.a.n 1
39.k even 12 2 468.2.b.c 2
52.i odd 6 1 8112.2.a.l 1
52.j odd 6 1 8112.2.a.d 1
52.l even 12 2 624.2.c.d 2
65.o even 12 2 3900.2.j.b 2
65.s odd 12 2 3900.2.c.a 2
65.t even 12 2 3900.2.j.e 2
91.bc even 12 2 7644.2.e.b 2
104.u even 12 2 2496.2.c.i 2
104.x odd 12 2 2496.2.c.b 2
156.v odd 12 2 1872.2.c.h 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
156.2.b.b 2 13.f odd 12 2
468.2.b.c 2 39.k even 12 2
624.2.c.d 2 52.l even 12 2
1872.2.c.h 2 156.v odd 12 2
2028.2.a.d 1 13.c even 3 1
2028.2.a.f 1 13.e even 6 1
2028.2.i.a 2 1.a even 1 1 trivial
2028.2.i.a 2 13.c even 3 1 inner
2028.2.i.d 2 13.b even 2 1
2028.2.i.d 2 13.e even 6 1
2028.2.q.e 4 13.d odd 4 2
2028.2.q.e 4 13.f odd 12 2
2496.2.c.b 2 104.x odd 12 2
2496.2.c.i 2 104.u even 12 2
3900.2.c.a 2 65.s odd 12 2
3900.2.j.b 2 65.o even 12 2
3900.2.j.e 2 65.t even 12 2
6084.2.a.d 1 39.h odd 6 1
6084.2.a.n 1 39.i odd 6 1
7644.2.e.b 2 91.bc even 12 2
8112.2.a.d 1 52.j odd 6 1
8112.2.a.l 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 \(S_{2}^{\mathrm{new}}(2028, [\chi])\):

\( T_{5} + 2 \) Copy content Toggle raw display
\( T_{7}^{2} + 4T_{7} + 16 \) 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
$5$ \( (T + 2)^{2} \) Copy content Toggle raw display
$7$ \( T^{2} + 4T + 16 \) Copy content Toggle raw display
$11$ \( T^{2} - 6T + 36 \) Copy content Toggle raw display
$13$ \( T^{2} \) Copy content Toggle raw display
$17$ \( T^{2} + 2T + 4 \) Copy content Toggle raw display
$19$ \( T^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 8T + 64 \) Copy content Toggle raw display
$29$ \( T^{2} + 2T + 4 \) Copy content Toggle raw display
$31$ \( (T - 8)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} + 8T + 64 \) Copy content Toggle raw display
$41$ \( T^{2} + 2T + 4 \) Copy content Toggle raw display
$43$ \( T^{2} + 8T + 64 \) Copy content Toggle raw display
$47$ \( (T + 6)^{2} \) Copy content Toggle raw display
$53$ \( (T - 6)^{2} \) Copy content Toggle raw display
$59$ \( T^{2} - 2T + 4 \) Copy content Toggle raw display
$61$ \( T^{2} + 2T + 4 \) Copy content Toggle raw display
$67$ \( T^{2} - 4T + 16 \) Copy content Toggle raw display
$71$ \( T^{2} - 6T + 36 \) Copy content Toggle raw display
$73$ \( (T + 4)^{2} \) Copy content Toggle raw display
$79$ \( T^{2} \) Copy content Toggle raw display
$83$ \( (T + 14)^{2} \) Copy content Toggle raw display
$89$ \( T^{2} + 6T + 36 \) Copy content Toggle raw display
$97$ \( T^{2} + 12T + 144 \) Copy content Toggle raw display
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