Properties

Label 2160.2.f.n
Level $2160$
Weight $2$
Character orbit 2160.f
Analytic conductor $17.248$
Analytic rank $0$
Dimension $4$
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] = [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: \(\Q(i, \sqrt{6})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 1080)
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.

\(f(q)\) \(=\) \( q + (\beta_{3} + \beta_{2} + 1) q^{5} + \beta_{2} q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{3} + \beta_{2} + 1) q^{5} + \beta_{2} q^{7} + ( - \beta_{3} + \beta_1 + 2) q^{11} + ( - 2 \beta_{3} - \beta_{2} - 2 \beta_1) q^{13} + ( - 2 \beta_{3} - 2 \beta_1) q^{17} + ( - 2 \beta_{3} + 2 \beta_1 - 1) q^{19} + ( - \beta_{3} + 2 \beta_{2} - \beta_1) q^{23} + (2 \beta_{3} - \beta_{2} - 2 \beta_1) q^{25} + ( - \beta_{3} + \beta_1 + 2) q^{29} - 6 q^{31} + (\beta_{2} - \beta_1 - 1) q^{35} + ( - 2 \beta_{3} - 5 \beta_{2} - 2 \beta_1) q^{37} + ( - 3 \beta_{3} + 3 \beta_1 - 2) q^{41} + 6 \beta_{2} q^{43} + ( - 2 \beta_{3} - 2 \beta_1) q^{47} + 6 q^{49} + ( - \beta_{3} + 2 \beta_{2} - \beta_1) q^{53} + (2 \beta_{3} + 5 \beta_{2} + 2 \beta_1 - 1) q^{55} + ( - 2 \beta_{3} + 2 \beta_1) q^{59} + (4 \beta_{3} - 4 \beta_1 - 1) q^{61} + ( - 4 \beta_{3} + 5 \beta_{2} + \beta_1 + 7) q^{65} + (4 \beta_{3} - 5 \beta_{2} + 4 \beta_1) q^{67} + ( - \beta_{3} + \beta_1 + 14) q^{71} + (2 \beta_{3} + \beta_{2} + 2 \beta_1) q^{73} + (\beta_{3} + 2 \beta_{2} + \beta_1) q^{77} + ( - 2 \beta_{3} + 2 \beta_1 + 1) q^{79} + (\beta_{3} + 10 \beta_{2} + \beta_1) q^{83} + ( - 4 \beta_{3} + 6 \beta_{2} + 6) q^{85} + 12 q^{89} + ( - 2 \beta_{3} + 2 \beta_1 + 1) q^{91} + ( - \beta_{3} + 5 \beta_{2} + 4 \beta_1 - 7) q^{95} + (2 \beta_{3} - 11 \beta_{2} + 2 \beta_1) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 4 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 4 q^{5} + 8 q^{11} - 4 q^{19} + 8 q^{29} - 24 q^{31} - 4 q^{35} - 8 q^{41} + 24 q^{49} - 4 q^{55} - 4 q^{61} + 28 q^{65} + 56 q^{71} + 4 q^{79} + 24 q^{85} + 48 q^{89} + 4 q^{91} - 28 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} + 9 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{2} ) / 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{3} ) / 3 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( 3\beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 3\beta_{3} \) 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
1.22474 1.22474i
1.22474 + 1.22474i
−1.22474 1.22474i
−1.22474 + 1.22474i
0 0 0 −0.224745 2.22474i 0 1.00000i 0 0 0
1729.2 0 0 0 −0.224745 + 2.22474i 0 1.00000i 0 0 0
1729.3 0 0 0 2.22474 0.224745i 0 1.00000i 0 0 0
1729.4 0 0 0 2.22474 + 0.224745i 0 1.00000i 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
5.b even 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.n 4
3.b odd 2 1 2160.2.f.i 4
4.b odd 2 1 1080.2.f.f yes 4
5.b even 2 1 inner 2160.2.f.n 4
12.b even 2 1 1080.2.f.e 4
15.d odd 2 1 2160.2.f.i 4
20.d odd 2 1 1080.2.f.f yes 4
20.e even 4 1 5400.2.a.bw 2
20.e even 4 1 5400.2.a.cc 2
60.h even 2 1 1080.2.f.e 4
60.l odd 4 1 5400.2.a.bz 2
60.l odd 4 1 5400.2.a.cf 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1080.2.f.e 4 12.b even 2 1
1080.2.f.e 4 60.h even 2 1
1080.2.f.f yes 4 4.b odd 2 1
1080.2.f.f yes 4 20.d odd 2 1
2160.2.f.i 4 3.b odd 2 1
2160.2.f.i 4 15.d odd 2 1
2160.2.f.n 4 1.a even 1 1 trivial
2160.2.f.n 4 5.b even 2 1 inner
5400.2.a.bw 2 20.e even 4 1
5400.2.a.bz 2 60.l odd 4 1
5400.2.a.cc 2 20.e even 4 1
5400.2.a.cf 2 60.l odd 4 1

Hecke kernels

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

\( T_{7}^{2} + 1 \) Copy content Toggle raw display
\( T_{11}^{2} - 4T_{11} - 2 \) Copy content Toggle raw display
\( T_{13}^{4} + 50T_{13}^{2} + 529 \) 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
$5$ \( T^{4} - 4 T^{3} + 8 T^{2} - 20 T + 25 \) Copy content Toggle raw display
$7$ \( (T^{2} + 1)^{2} \) Copy content Toggle raw display
$11$ \( (T^{2} - 4 T - 2)^{2} \) Copy content Toggle raw display
$13$ \( T^{4} + 50T^{2} + 529 \) Copy content Toggle raw display
$17$ \( (T^{2} + 24)^{2} \) Copy content Toggle raw display
$19$ \( (T^{2} + 2 T - 23)^{2} \) Copy content Toggle raw display
$23$ \( T^{4} + 20T^{2} + 4 \) Copy content Toggle raw display
$29$ \( (T^{2} - 4 T - 2)^{2} \) Copy content Toggle raw display
$31$ \( (T + 6)^{4} \) Copy content Toggle raw display
$37$ \( T^{4} + 98T^{2} + 1 \) Copy content Toggle raw display
$41$ \( (T^{2} + 4 T - 50)^{2} \) Copy content Toggle raw display
$43$ \( (T^{2} + 36)^{2} \) Copy content Toggle raw display
$47$ \( (T^{2} + 24)^{2} \) Copy content Toggle raw display
$53$ \( T^{4} + 20T^{2} + 4 \) Copy content Toggle raw display
$59$ \( (T^{2} - 24)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} + 2 T - 95)^{2} \) Copy content Toggle raw display
$67$ \( T^{4} + 242T^{2} + 5041 \) Copy content Toggle raw display
$71$ \( (T^{2} - 28 T + 190)^{2} \) Copy content Toggle raw display
$73$ \( T^{4} + 50T^{2} + 529 \) Copy content Toggle raw display
$79$ \( (T^{2} - 2 T - 23)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} + 212T^{2} + 8836 \) Copy content Toggle raw display
$89$ \( (T - 12)^{4} \) Copy content Toggle raw display
$97$ \( T^{4} + 290T^{2} + 9409 \) Copy content Toggle raw display
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