↑

Properties

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

Related objects

Downloads

Learn more

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: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
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 primitive root of unity \(\zeta_{12}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \zeta_{12}^{2} q^{3} + 2 \zeta_{12}^{3} q^{5} + \zeta_{12} q^{7} + (\zeta_{12}^{2} - 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \zeta_{12}^{2} q^{3} + 2 \zeta_{12}^{3} q^{5} + \zeta_{12} q^{7} + (\zeta_{12}^{2} - 1) q^{9} + (2 \zeta_{12}^{3} - 2 \zeta_{12}) q^{11} + (2 \zeta_{12}^{3} - 2 \zeta_{12}) q^{15} + (4 \zeta_{12}^{2} - 4) q^{17} - 4 \zeta_{12} q^{19} + \zeta_{12}^{3} q^{21} + 6 \zeta_{12}^{2} q^{23} + q^{25} - q^{27} - 6 \zeta_{12}^{2} q^{29} - \zeta_{12}^{3} q^{31} - 2 \zeta_{12} q^{33} + (2 \zeta_{12}^{2} - 2) q^{35} + (10 \zeta_{12}^{3} - 10 \zeta_{12}) q^{37} + (4 \zeta_{12}^{3} - 4 \zeta_{12}) q^{41} + ( - \zeta_{12}^{2} + 1) q^{43} - 2 \zeta_{12} q^{45} + 10 \zeta_{12}^{3} q^{47} - 6 \zeta_{12}^{2} q^{49} - 4 q^{51} + 8 q^{53} - 4 \zeta_{12}^{2} q^{55} - 4 \zeta_{12}^{3} q^{57} - 2 \zeta_{12} q^{59} + ( - 5 \zeta_{12}^{2} + 5) q^{61} + (\zeta_{12}^{3} - \zeta_{12}) q^{63} + (7 \zeta_{12}^{3} - 7 \zeta_{12}) q^{67} + (6 \zeta_{12}^{2} - 6) q^{69} - 10 \zeta_{12} q^{71} + 7 \zeta_{12}^{3} q^{73} + \zeta_{12}^{2} q^{75} - 2 q^{77} + 17 q^{79} - \zeta_{12}^{2} q^{81} + 12 \zeta_{12}^{3} q^{83} - 8 \zeta_{12} q^{85} + ( - 6 \zeta_{12}^{2} + 6) q^{87} + ( - 16 \zeta_{12}^{3} + 16 \zeta_{12}) q^{89} + ( - \zeta_{12}^{3} + \zeta_{12}) q^{93} + ( - 8 \zeta_{12}^{2} + 8) q^{95} - 13 \zeta_{12} q^{97} - 2 \zeta_{12}^{3} q^{99} +O(q^{100}) \) 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 \( 4 q + 2 q^{3} - 2 q^{9} - 8 q^{17} + 12 q^{23} + 4 q^{25} - 4 q^{27} - 12 q^{29} - 4 q^{35} + 2 q^{43} - 12 q^{49} - 16 q^{51} + 32 q^{53} - 8 q^{55} + 10 q^{61} - 12 q^{69} + 2 q^{75} - 8 q^{77} + 68 q^{79} - 2 q^{81} + 12 q^{87} + 16 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Character values

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

\(n\) \(677\) \(1015\) \(1861\)
\(\chi(n)\) \(1\) \(1\) \(1 - \zeta_{12}^{2}\)

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 −0.866025 − 0.500000i 0 −0.500000 + 0.866025i 0
361.2 0 0.500000 + 0.866025i 0 2.00000i 0 0.866025 + 0.500000i 0 −0.500000 + 0.866025i 0
1837.1 0 0.500000 − 0.866025i 0 − 2.00000i 0 0.866025 − 0.500000i 0 −0.500000 − 0.866025i 0
1837.2 0 0.500000 − 0.866025i 0 2.00000i 0 −0.866025 + 0.500000i 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.g 4
13.b even 2 1 inner 2028.2.q.g 4
13.c even 3 1 2028.2.b.c 2
13.c even 3 1 inner 2028.2.q.g 4
13.d odd 4 1 156.2.i.a ✓ 2
13.d odd 4 1 2028.2.i.f 2
13.e even 6 1 2028.2.b.c 2
13.e even 6 1 inner 2028.2.q.g 4
13.f odd 12 1 156.2.i.a ✓ 2
13.f odd 12 1 2028.2.a.a 1
13.f odd 12 1 2028.2.a.b 1
13.f odd 12 1 2028.2.i.f 2
39.f even 4 1 468.2.l.b 2
39.h odd 6 1 6084.2.b.b 2
39.i odd 6 1 6084.2.b.b 2
39.k even 12 1 468.2.l.b 2
39.k even 12 1 6084.2.a.e 1
39.k even 12 1 6084.2.a.l 1
52.f even 4 1 624.2.q.d 2
52.l even 12 1 624.2.q.d 2
52.l even 12 1 8112.2.a.u 1
52.l even 12 1 8112.2.a.bd 1
65.f even 4 1 3900.2.by.c 4
65.g odd 4 1 3900.2.q.e 2
65.k even 4 1 3900.2.by.c 4
65.o even 12 1 3900.2.by.c 4
65.s odd 12 1 3900.2.q.e 2
65.t even 12 1 3900.2.by.c 4
156.l odd 4 1 1872.2.t.e 2
156.v odd 12 1 1872.2.t.e 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
156.2.i.a ✓ 2 13.d odd 4 1
156.2.i.a ✓ 2 13.f odd 12 1
468.2.l.b 2 39.f even 4 1
468.2.l.b 2 39.k even 12 1
624.2.q.d 2 52.f even 4 1
624.2.q.d 2 52.l even 12 1
1872.2.t.e 2 156.l odd 4 1
1872.2.t.e 2 156.v odd 12 1
2028.2.a.a 1 13.f odd 12 1
2028.2.a.b 1 13.f odd 12 1
2028.2.b.c 2 13.c even 3 1
2028.2.b.c 2 13.e even 6 1
2028.2.i.f 2 13.d odd 4 1
2028.2.i.f 2 13.f odd 12 1
2028.2.q.g 4 1.a even 1 1 trivial
2028.2.q.g 4 13.b even 2 1 inner
2028.2.q.g 4 13.c even 3 1 inner
2028.2.q.g 4 13.e even 6 1 inner
3900.2.q.e 2 65.g odd 4 1
3900.2.q.e 2 65.s odd 12 1
3900.2.by.c 4 65.f even 4 1
3900.2.by.c 4 65.k even 4 1
3900.2.by.c 4 65.o even 12 1
3900.2.by.c 4 65.t even 12 1
6084.2.a.e 1 39.k even 12 1
6084.2.a.l 1 39.k even 12 1
6084.2.b.b 2 39.h odd 6 1
6084.2.b.b 2 39.i odd 6 1
8112.2.a.u 1 52.l even 12 1
8112.2.a.bd 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
\( T_{7}^{4} - T_{7}^{2} + 1 \) Copy content Toggle raw display
\( T_{11}^{4} - 4T_{11}^{2} + 16 \) 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
$7$ \( T^{4} - T^{2} + 1 \) Copy content Toggle raw display
$11$ \( T^{4} - 4T^{2} + 16 \) Copy content Toggle raw display
$13$ \( T^{4} \) Copy content Toggle raw display
$17$ \( (T^{2} + 4 T + 16)^{2} \) Copy content Toggle raw display
$19$ \( T^{4} - 16T^{2} + 256 \) Copy content Toggle raw display
$23$ \( (T^{2} - 6 T + 36)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} + 6 T + 36)^{2} \) Copy content Toggle raw display
$31$ \( (T^{2} + 1)^{2} \) Copy content Toggle raw display
$37$ \( T^{4} - 100 T^{2} + 10000 \) Copy content Toggle raw display
$41$ \( T^{4} - 16T^{2} + 256 \) Copy content Toggle raw display
$43$ \( (T^{2} - T + 1)^{2} \) Copy content Toggle raw display
$47$ \( (T^{2} + 100)^{2} \) Copy content Toggle raw display
$53$ \( (T - 8)^{4} \) Copy content Toggle raw display
$59$ \( T^{4} - 4T^{2} + 16 \) Copy content Toggle raw display
$61$ \( (T^{2} - 5 T + 25)^{2} \) Copy content Toggle raw display
$67$ \( T^{4} - 49T^{2} + 2401 \) Copy content Toggle raw display
$71$ \( T^{4} - 100 T^{2} + 10000 \) Copy content Toggle raw display
$73$ \( (T^{2} + 49)^{2} \) Copy content Toggle raw display
$79$ \( (T - 17)^{4} \) Copy content Toggle raw display
$83$ \( (T^{2} + 144)^{2} \) Copy content Toggle raw display
$89$ \( T^{4} - 256 T^{2} + 65536 \) Copy content Toggle raw display
$97$ \( T^{4} - 169 T^{2} + 28561 \) Copy content Toggle raw display
show more
show less