Properties

Label 2160.2.f.o
Level $2160$
Weight $2$
Character orbit 2160.f
Analytic conductor $17.248$
Analytic rank $0$
Dimension $8$
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] = [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: \(8\)
Coefficient field: 8.0.2702336256.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 9x^{6} + 56x^{4} + 225x^{2} + 625 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{4} \)
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{7} q^{5} - \beta_1 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{7} q^{5} - \beta_1 q^{7} + (2 \beta_{7} - \beta_{6} - 2 \beta_{3}) q^{11} + (\beta_{5} - \beta_{2} - \beta_1) q^{13} + (\beta_{7} + \beta_{4} + \beta_{3}) q^{17} + ( - \beta_{5} - \beta_{2} + 1) q^{19} + ( - \beta_{7} + \beta_{4} - \beta_{3}) q^{23} + (\beta_{5} - 2 \beta_{2} + \beta_1 - 2) q^{25} + ( - \beta_{7} - \beta_{6} + \beta_{3}) q^{29} - 3 q^{31} + (2 \beta_{6} - \beta_{4} - \beta_{3}) q^{35} + ( - 2 \beta_{5} + 2 \beta_{2}) q^{37} + ( - 3 \beta_{7} - \beta_{6} + 3 \beta_{3}) q^{41} + (3 \beta_{5} - 3 \beta_{2} + 3 \beta_1) q^{43} + ( - 2 \beta_{7} + 4 \beta_{4} - 2 \beta_{3}) q^{47} + ( - 3 \beta_{5} - 3 \beta_{2} - 3) q^{49} + (2 \beta_{7} + \beta_{4} + 2 \beta_{3}) q^{53} + (2 \beta_{5} - 5 \beta_{2} + 5) q^{55} + (\beta_{7} - \beta_{6} - \beta_{3}) q^{59} + ( - \beta_{5} - \beta_{2} + 7) q^{61} + ( - \beta_{7} + 3 \beta_{6} - 4 \beta_{4} + \beta_{3}) q^{65} + ( - 2 \beta_{5} + 2 \beta_{2}) q^{67} + ( - \beta_{7} - \beta_{6} + \beta_{3}) q^{71} + (2 \beta_{5} - 2 \beta_{2} + \beta_1) q^{73} + (\beta_{7} + 6 \beta_{4} + \beta_{3}) q^{77} + ( - \beta_{5} - \beta_{2} + 3) q^{79} + ( - 2 \beta_{7} - 7 \beta_{4} - 2 \beta_{3}) q^{83} + ( - 2 \beta_{2} + \beta_1 - 7) q^{85} + ( - 3 \beta_{7} + 3 \beta_{6} + 3 \beta_{3}) q^{89} + ( - 4 \beta_{5} - 4 \beta_{2} - 12) q^{91} + (2 \beta_{7} - \beta_{6} - 7 \beta_{4} - 2 \beta_{3}) q^{95} + (5 \beta_{5} - 5 \beta_{2} - 2 \beta_1) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 4 q^{19} - 18 q^{25} - 24 q^{31} - 36 q^{49} + 34 q^{55} + 52 q^{61} + 20 q^{79} - 60 q^{85} - 112 q^{91}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + 9x^{6} + 56x^{4} + 225x^{2} + 625 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -\nu^{6} - 84\nu^{4} - 756\nu^{2} - 2325 ) / 700 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -11\nu^{6} - 224\nu^{4} - 616\nu^{2} - 2475 ) / 1400 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{7} + 279\nu ) / 280 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -9\nu^{7} - 56\nu^{5} - 154\nu^{3} - 625\nu ) / 1750 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -9\nu^{6} - 56\nu^{4} - 224\nu^{2} - 625 ) / 280 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 3\nu^{7} + 52\nu^{5} + 268\nu^{3} + 575\nu ) / 500 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 9\nu^{7} + 56\nu^{5} + 504\nu^{3} + 2025\nu ) / 1400 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{7} - \beta_{6} - 2\beta_{4} + 3\beta_{3} ) / 4 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -\beta_{5} + 5\beta_{2} - 5\beta _1 - 10 ) / 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 3\beta_{7} + \beta_{6} + 7\beta_{4} - 3\beta_{3} \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 11\beta_{5} - 47\beta_{2} + 11\beta _1 - 22 ) / 4 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -89\beta_{7} + 45\beta_{6} - 90\beta_{4} + 45\beta_{3} ) / 4 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -42\beta_{5} + 42\beta_{2} + 14\beta _1 + 27 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 279\beta_{7} - 279\beta_{6} - 558\beta_{4} - 283\beta_{3} ) / 4 \) 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
0.656712 2.13746i
0.656712 + 2.13746i
1.52274 1.63746i
1.52274 + 1.63746i
−1.52274 1.63746i
−1.52274 + 1.63746i
−0.656712 2.13746i
−0.656712 + 2.13746i
0 0 0 −1.52274 1.63746i 0 0.418627i 0 0 0
1729.2 0 0 0 −1.52274 + 1.63746i 0 0.418627i 0 0 0
1729.3 0 0 0 −0.656712 2.13746i 0 4.77753i 0 0 0
1729.4 0 0 0 −0.656712 + 2.13746i 0 4.77753i 0 0 0
1729.5 0 0 0 0.656712 2.13746i 0 4.77753i 0 0 0
1729.6 0 0 0 0.656712 + 2.13746i 0 4.77753i 0 0 0
1729.7 0 0 0 1.52274 1.63746i 0 0.418627i 0 0 0
1729.8 0 0 0 1.52274 + 1.63746i 0 0.418627i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1729.8
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.o 8
3.b odd 2 1 inner 2160.2.f.o 8
4.b odd 2 1 1080.2.f.g 8
5.b even 2 1 inner 2160.2.f.o 8
12.b even 2 1 1080.2.f.g 8
15.d odd 2 1 inner 2160.2.f.o 8
20.d odd 2 1 1080.2.f.g 8
20.e even 4 1 5400.2.a.cg 4
20.e even 4 1 5400.2.a.ch 4
60.h even 2 1 1080.2.f.g 8
60.l odd 4 1 5400.2.a.cg 4
60.l odd 4 1 5400.2.a.ch 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1080.2.f.g 8 4.b odd 2 1
1080.2.f.g 8 12.b even 2 1
1080.2.f.g 8 20.d odd 2 1
1080.2.f.g 8 60.h even 2 1
2160.2.f.o 8 1.a even 1 1 trivial
2160.2.f.o 8 3.b odd 2 1 inner
2160.2.f.o 8 5.b even 2 1 inner
2160.2.f.o 8 15.d odd 2 1 inner
5400.2.a.cg 4 20.e even 4 1
5400.2.a.cg 4 60.l odd 4 1
5400.2.a.ch 4 20.e even 4 1
5400.2.a.ch 4 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 \(S_{2}^{\mathrm{new}}(2160, [\chi])\):

\( T_{7}^{4} + 23T_{7}^{2} + 4 \) Copy content Toggle raw display
\( T_{11}^{4} - 47T_{11}^{2} + 196 \) Copy content Toggle raw display
\( T_{13}^{4} + 44T_{13}^{2} + 256 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( T^{8} \) Copy content Toggle raw display
$5$ \( T^{8} + 9 T^{6} + 56 T^{4} + 225 T^{2} + \cdots + 625 \) Copy content Toggle raw display
$7$ \( (T^{4} + 23 T^{2} + 4)^{2} \) Copy content Toggle raw display
$11$ \( (T^{4} - 47 T^{2} + 196)^{2} \) Copy content Toggle raw display
$13$ \( (T^{4} + 44 T^{2} + 256)^{2} \) Copy content Toggle raw display
$17$ \( (T^{4} + 33 T^{2} + 144)^{2} \) Copy content Toggle raw display
$19$ \( (T^{2} - T - 14)^{4} \) Copy content Toggle raw display
$23$ \( (T^{4} + 29 T^{2} + 196)^{2} \) Copy content Toggle raw display
$29$ \( (T^{4} - 44 T^{2} + 256)^{2} \) Copy content Toggle raw display
$31$ \( (T + 3)^{8} \) Copy content Toggle raw display
$37$ \( (T^{4} + 44 T^{2} + 256)^{2} \) Copy content Toggle raw display
$41$ \( (T^{2} - 76)^{4} \) Copy content Toggle raw display
$43$ \( (T^{2} + 108)^{4} \) Copy content Toggle raw display
$47$ \( (T^{4} + 132 T^{2} + 2304)^{2} \) Copy content Toggle raw display
$53$ \( (T^{4} + 122 T^{2} + 2809)^{2} \) Copy content Toggle raw display
$59$ \( (T^{2} - 12)^{4} \) Copy content Toggle raw display
$61$ \( (T^{2} - 13 T + 28)^{4} \) Copy content Toggle raw display
$67$ \( (T^{4} + 44 T^{2} + 256)^{2} \) Copy content Toggle raw display
$71$ \( (T^{4} - 44 T^{2} + 256)^{2} \) Copy content Toggle raw display
$73$ \( (T^{4} + 47 T^{2} + 196)^{2} \) Copy content Toggle raw display
$79$ \( (T^{2} - 5 T - 8)^{4} \) Copy content Toggle raw display
$83$ \( (T^{4} + 242 T^{2} + 49)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} - 108)^{4} \) Copy content Toggle raw display
$97$ \( (T^{4} + 467 T^{2} + 53824)^{2} \) Copy content Toggle raw display
show more
show less