Properties

Label 2028.2.q.e
Level $2028$
Weight $2$
Character orbit 2028.q
Analytic conductor $16.194$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2028,2,Mod(361,2028)]
 
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, 5]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2028.361");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2028 = 2^{2} \cdot 3 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2028.q (of order \(6\), 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: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\zeta_{12})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 156)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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.

\(f(q)\) \(=\) \( q - \beta_{2} q^{3} + \beta_{3} q^{5} - 2 \beta_1 q^{7} + (\beta_{2} - 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{2} q^{3} + \beta_{3} q^{5} - 2 \beta_1 q^{7} + (\beta_{2} - 1) q^{9} + (3 \beta_{3} - 3 \beta_1) q^{11} + ( - \beta_{3} + \beta_1) q^{15} + ( - 2 \beta_{2} + 2) q^{17} + 2 \beta_{3} q^{21} + 8 \beta_{2} q^{23} + q^{25} + q^{27} - 2 \beta_{2} q^{29} - 4 \beta_{3} q^{31} + 3 \beta_1 q^{33} + ( - 8 \beta_{2} + 8) q^{35} + ( - 4 \beta_{3} + 4 \beta_1) q^{37} + (\beta_{3} - \beta_1) q^{41} + ( - 8 \beta_{2} + 8) q^{43} - \beta_1 q^{45} - 3 \beta_{3} q^{47} + 9 \beta_{2} q^{49} - 2 q^{51} + 6 q^{53} - 12 \beta_{2} q^{55} + \beta_1 q^{59} + (2 \beta_{2} - 2) q^{61} + ( - 2 \beta_{3} + 2 \beta_1) q^{63} + ( - 2 \beta_{3} + 2 \beta_1) q^{67} + ( - 8 \beta_{2} + 8) q^{69} - 3 \beta_1 q^{71} - 2 \beta_{3} q^{73} - \beta_{2} q^{75} + 24 q^{77} - \beta_{2} q^{81} + 7 \beta_{3} q^{83} + 2 \beta_1 q^{85} + (2 \beta_{2} - 2) q^{87} + ( - 3 \beta_{3} + 3 \beta_1) q^{89} + (4 \beta_{3} - 4 \beta_1) q^{93} + 6 \beta_1 q^{97} - 3 \beta_{3} q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 2 q^{3} - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 2 q^{3} - 2 q^{9} + 4 q^{17} + 16 q^{23} + 4 q^{25} + 4 q^{27} - 4 q^{29} + 16 q^{35} + 16 q^{43} + 18 q^{49} - 8 q^{51} + 24 q^{53} - 24 q^{55} - 4 q^{61} + 16 q^{69} - 2 q^{75} + 96 q^{77} - 2 q^{81} - 4 q^{87}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring

\(\beta_{1}\)\(=\) \( 2\zeta_{12} \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \zeta_{12}^{2} \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( 2\zeta_{12}^{3} \) Copy content Toggle raw display
\(\zeta_{12}\)\(=\) \( ( \beta_1 ) / 2 \) Copy content Toggle raw display
\(\zeta_{12}^{2}\)\(=\) \( \beta_{2} \) Copy content Toggle raw display
\(\zeta_{12}^{3}\)\(=\) \( ( \beta_{3} ) / 2 \) Copy content Toggle raw display

Character values

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

\(n\) \(677\) \(1015\) \(1861\)
\(\chi(n)\) \(1\) \(1\) \(1 - \beta_{2}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
361.1
−0.866025 0.500000i
0.866025 + 0.500000i
0.866025 0.500000i
−0.866025 + 0.500000i
0 −0.500000 0.866025i 0 2.00000i 0 3.46410 + 2.00000i 0 −0.500000 + 0.866025i 0
361.2 0 −0.500000 0.866025i 0 2.00000i 0 −3.46410 2.00000i 0 −0.500000 + 0.866025i 0
1837.1 0 −0.500000 + 0.866025i 0 2.00000i 0 −3.46410 + 2.00000i 0 −0.500000 0.866025i 0
1837.2 0 −0.500000 + 0.866025i 0 2.00000i 0 3.46410 2.00000i 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.b even 2 1 inner
13.c even 3 1 inner
13.e even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2028.2.q.e 4
13.b even 2 1 inner 2028.2.q.e 4
13.c even 3 1 156.2.b.b 2
13.c even 3 1 inner 2028.2.q.e 4
13.d odd 4 1 2028.2.i.a 2
13.d odd 4 1 2028.2.i.d 2
13.e even 6 1 156.2.b.b 2
13.e even 6 1 inner 2028.2.q.e 4
13.f odd 12 1 2028.2.a.d 1
13.f odd 12 1 2028.2.a.f 1
13.f odd 12 1 2028.2.i.a 2
13.f odd 12 1 2028.2.i.d 2
39.h odd 6 1 468.2.b.c 2
39.i odd 6 1 468.2.b.c 2
39.k even 12 1 6084.2.a.d 1
39.k even 12 1 6084.2.a.n 1
52.i odd 6 1 624.2.c.d 2
52.j odd 6 1 624.2.c.d 2
52.l even 12 1 8112.2.a.d 1
52.l even 12 1 8112.2.a.l 1
65.l even 6 1 3900.2.c.a 2
65.n even 6 1 3900.2.c.a 2
65.q odd 12 1 3900.2.j.b 2
65.q odd 12 1 3900.2.j.e 2
65.r odd 12 1 3900.2.j.b 2
65.r odd 12 1 3900.2.j.e 2
91.n odd 6 1 7644.2.e.b 2
91.t odd 6 1 7644.2.e.b 2
104.n odd 6 1 2496.2.c.i 2
104.p odd 6 1 2496.2.c.i 2
104.r even 6 1 2496.2.c.b 2
104.s even 6 1 2496.2.c.b 2
156.p even 6 1 1872.2.c.h 2
156.r even 6 1 1872.2.c.h 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
156.2.b.b 2 13.c even 3 1
156.2.b.b 2 13.e even 6 1
468.2.b.c 2 39.h odd 6 1
468.2.b.c 2 39.i odd 6 1
624.2.c.d 2 52.i odd 6 1
624.2.c.d 2 52.j odd 6 1
1872.2.c.h 2 156.p even 6 1
1872.2.c.h 2 156.r even 6 1
2028.2.a.d 1 13.f odd 12 1
2028.2.a.f 1 13.f odd 12 1
2028.2.i.a 2 13.d odd 4 1
2028.2.i.a 2 13.f odd 12 1
2028.2.i.d 2 13.d odd 4 1
2028.2.i.d 2 13.f odd 12 1
2028.2.q.e 4 1.a even 1 1 trivial
2028.2.q.e 4 13.b even 2 1 inner
2028.2.q.e 4 13.c even 3 1 inner
2028.2.q.e 4 13.e even 6 1 inner
2496.2.c.b 2 104.r even 6 1
2496.2.c.b 2 104.s even 6 1
2496.2.c.i 2 104.n odd 6 1
2496.2.c.i 2 104.p odd 6 1
3900.2.c.a 2 65.l even 6 1
3900.2.c.a 2 65.n even 6 1
3900.2.j.b 2 65.q odd 12 1
3900.2.j.b 2 65.r odd 12 1
3900.2.j.e 2 65.q odd 12 1
3900.2.j.e 2 65.r odd 12 1
6084.2.a.d 1 39.k even 12 1
6084.2.a.n 1 39.k even 12 1
7644.2.e.b 2 91.n odd 6 1
7644.2.e.b 2 91.t odd 6 1
8112.2.a.d 1 52.l even 12 1
8112.2.a.l 1 52.l even 12 1

Hecke kernels

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

\( T_{5}^{2} + 4 \) Copy content Toggle raw display
\( T_{7}^{4} - 16T_{7}^{2} + 256 \) Copy content Toggle raw display
\( T_{11}^{4} - 36T_{11}^{2} + 1296 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( (T^{2} + T + 1)^{2} \) Copy content Toggle raw display
$5$ \( (T^{2} + 4)^{2} \) Copy content Toggle raw display
$7$ \( T^{4} - 16T^{2} + 256 \) Copy content Toggle raw display
$11$ \( T^{4} - 36T^{2} + 1296 \) Copy content Toggle raw display
$13$ \( T^{4} \) Copy content Toggle raw display
$17$ \( (T^{2} - 2 T + 4)^{2} \) Copy content Toggle raw display
$19$ \( T^{4} \) Copy content Toggle raw display
$23$ \( (T^{2} - 8 T + 64)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} + 2 T + 4)^{2} \) Copy content Toggle raw display
$31$ \( (T^{2} + 64)^{2} \) Copy content Toggle raw display
$37$ \( T^{4} - 64T^{2} + 4096 \) Copy content Toggle raw display
$41$ \( T^{4} - 4T^{2} + 16 \) Copy content Toggle raw display
$43$ \( (T^{2} - 8 T + 64)^{2} \) Copy content Toggle raw display
$47$ \( (T^{2} + 36)^{2} \) Copy content Toggle raw display
$53$ \( (T - 6)^{4} \) Copy content Toggle raw display
$59$ \( T^{4} - 4T^{2} + 16 \) Copy content Toggle raw display
$61$ \( (T^{2} + 2 T + 4)^{2} \) Copy content Toggle raw display
$67$ \( T^{4} - 16T^{2} + 256 \) Copy content Toggle raw display
$71$ \( T^{4} - 36T^{2} + 1296 \) Copy content Toggle raw display
$73$ \( (T^{2} + 16)^{2} \) Copy content Toggle raw display
$79$ \( T^{4} \) Copy content Toggle raw display
$83$ \( (T^{2} + 196)^{2} \) Copy content Toggle raw display
$89$ \( T^{4} - 36T^{2} + 1296 \) Copy content Toggle raw display
$97$ \( T^{4} - 144 T^{2} + 20736 \) Copy content Toggle raw display
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