Properties

Label 2160.2.f.k
Level $2160$
Weight $2$
Character orbit 2160.f
Analytic conductor $17.248$
Analytic rank $0$
Dimension $4$
Inner twists $4$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2160,2,Mod(1729,2160)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2160, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2160.1729");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2160 = 2^{4} \cdot 3^{3} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2160.f (of order \(2\), degree \(1\), 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: \(17.2476868366\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{8})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 3^{2} \)
Twist minimal: no (minimal twist has level 135)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_{3} q^{5} + \beta_1 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{3} q^{5} + \beta_1 q^{7} + (2 \beta_{3} - \beta_{2}) q^{11} + \beta_1 q^{13} - 2 \beta_{2} q^{17} + q^{19} + 5 \beta_{2} q^{23} + (\beta_1 + 4) q^{25} + ( - 2 \beta_{3} + \beta_{2}) q^{29} - 2 q^{31} + ( - \beta_{3} + 5 \beta_{2}) q^{35} - 3 \beta_1 q^{37} + (2 \beta_{3} - \beta_{2}) q^{41} - 2 \beta_1 q^{43} + 2 \beta_{2} q^{47} - 2 q^{49} + 7 \beta_{2} q^{53} + (\beta_1 + 9) q^{55} + (4 \beta_{3} - 2 \beta_{2}) q^{59} - 13 q^{61} + ( - \beta_{3} + 5 \beta_{2}) q^{65} - \beta_1 q^{67} + ( - 6 \beta_{3} + 3 \beta_{2}) q^{71} + 3 \beta_1 q^{73} + 9 \beta_{2} q^{77} - 5 q^{79} - \beta_{2} q^{83} + ( - 2 \beta_1 + 2) q^{85} - 9 q^{91} + \beta_{3} q^{95} - \beta_1 q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 4 q^{19} + 16 q^{25} - 8 q^{31} - 8 q^{49} + 36 q^{55} - 52 q^{61} - 20 q^{79} + 8 q^{85} - 36 q^{91}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring

\(\beta_{1}\)\(=\) \( 3\zeta_{8}^{2} \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \zeta_{8}^{3} + \zeta_{8} \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( -\zeta_{8}^{3} + 2\zeta_{8} \) Copy content Toggle raw display
\(\zeta_{8}\)\(=\) \( ( \beta_{3} + \beta_{2} ) / 3 \) Copy content Toggle raw display
\(\zeta_{8}^{2}\)\(=\) \( ( \beta_1 ) / 3 \) Copy content Toggle raw display
\(\zeta_{8}^{3}\)\(=\) \( ( -\beta_{3} + 2\beta_{2} ) / 3 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2160\mathbb{Z}\right)^\times\).

\(n\) \(271\) \(1297\) \(1621\) \(2081\)
\(\chi(n)\) \(1\) \(-1\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1729.1
−0.707107 0.707107i
−0.707107 + 0.707107i
0.707107 0.707107i
0.707107 + 0.707107i
0 0 0 −2.12132 0.707107i 0 3.00000i 0 0 0
1729.2 0 0 0 −2.12132 + 0.707107i 0 3.00000i 0 0 0
1729.3 0 0 0 2.12132 0.707107i 0 3.00000i 0 0 0
1729.4 0 0 0 2.12132 + 0.707107i 0 3.00000i 0 0 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
3.b odd 2 1 inner
5.b even 2 1 inner
15.d odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2160.2.f.k 4
3.b odd 2 1 inner 2160.2.f.k 4
4.b odd 2 1 135.2.b.b 4
5.b even 2 1 inner 2160.2.f.k 4
12.b even 2 1 135.2.b.b 4
15.d odd 2 1 inner 2160.2.f.k 4
20.d odd 2 1 135.2.b.b 4
20.e even 4 1 675.2.a.l 2
20.e even 4 1 675.2.a.m 2
36.f odd 6 2 405.2.j.f 8
36.h even 6 2 405.2.j.f 8
60.h even 2 1 135.2.b.b 4
60.l odd 4 1 675.2.a.l 2
60.l odd 4 1 675.2.a.m 2
180.n even 6 2 405.2.j.f 8
180.p odd 6 2 405.2.j.f 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
135.2.b.b 4 4.b odd 2 1
135.2.b.b 4 12.b even 2 1
135.2.b.b 4 20.d odd 2 1
135.2.b.b 4 60.h even 2 1
405.2.j.f 8 36.f odd 6 2
405.2.j.f 8 36.h even 6 2
405.2.j.f 8 180.n even 6 2
405.2.j.f 8 180.p odd 6 2
675.2.a.l 2 20.e even 4 1
675.2.a.l 2 60.l odd 4 1
675.2.a.m 2 20.e even 4 1
675.2.a.m 2 60.l odd 4 1
2160.2.f.k 4 1.a even 1 1 trivial
2160.2.f.k 4 3.b odd 2 1 inner
2160.2.f.k 4 5.b even 2 1 inner
2160.2.f.k 4 15.d odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(2160, [\chi])\):

\( T_{7}^{2} + 9 \) Copy content Toggle raw display
\( T_{11}^{2} - 18 \) Copy content Toggle raw display
\( T_{13}^{2} + 9 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( T^{4} - 8T^{2} + 25 \) Copy content Toggle raw display
$7$ \( (T^{2} + 9)^{2} \) Copy content Toggle raw display
$11$ \( (T^{2} - 18)^{2} \) Copy content Toggle raw display
$13$ \( (T^{2} + 9)^{2} \) Copy content Toggle raw display
$17$ \( (T^{2} + 8)^{2} \) Copy content Toggle raw display
$19$ \( (T - 1)^{4} \) Copy content Toggle raw display
$23$ \( (T^{2} + 50)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} - 18)^{2} \) Copy content Toggle raw display
$31$ \( (T + 2)^{4} \) Copy content Toggle raw display
$37$ \( (T^{2} + 81)^{2} \) Copy content Toggle raw display
$41$ \( (T^{2} - 18)^{2} \) Copy content Toggle raw display
$43$ \( (T^{2} + 36)^{2} \) Copy content Toggle raw display
$47$ \( (T^{2} + 8)^{2} \) Copy content Toggle raw display
$53$ \( (T^{2} + 98)^{2} \) Copy content Toggle raw display
$59$ \( (T^{2} - 72)^{2} \) Copy content Toggle raw display
$61$ \( (T + 13)^{4} \) Copy content Toggle raw display
$67$ \( (T^{2} + 9)^{2} \) Copy content Toggle raw display
$71$ \( (T^{2} - 162)^{2} \) Copy content Toggle raw display
$73$ \( (T^{2} + 81)^{2} \) Copy content Toggle raw display
$79$ \( (T + 5)^{4} \) Copy content Toggle raw display
$83$ \( (T^{2} + 2)^{2} \) Copy content Toggle raw display
$89$ \( T^{4} \) Copy content Toggle raw display
$97$ \( (T^{2} + 9)^{2} \) Copy content Toggle raw display
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