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Properties

Label 2160.2.by.f
Level $2160$
Weight $2$
Character orbit 2160.by
Analytic conductor $17.248$
Analytic rank $0$
Dimension $32$
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] = [2160,2,Mod(289,2160)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2160, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 4, 3]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2160.289");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2160 = 2^{4} \cdot 3^{3} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2160.by (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: \(17.2476868366\)
Analytic rank: \(0\)
Dimension: \(32\)
Relative dimension: \(16\) over \(\Q(\zeta_{6})\)
Twist minimal: no (minimal twist has level 360)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 32 q + 2 q^{5}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 32 q + 2 q^{5} + 16 q^{11} - 8 q^{19} - 6 q^{25} - 20 q^{29} + 12 q^{31} + 4 q^{35} + 8 q^{41} + 36 q^{49} - 20 q^{55} - 20 q^{61} - 10 q^{65} + 16 q^{71} - 4 q^{79} + 36 q^{85} + 96 q^{89} + 8 q^{91} - 32 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
289.1 0 0 0 −2.23514 − 0.0643013i 0 1.19102 + 0.687633i 0 0 0
289.2 0 0 0 −1.61222 + 1.54943i 0 −1.98012 − 1.14322i 0 0 0
289.3 0 0 0 −1.59892 − 1.56315i 0 −0.608912 − 0.351555i 0 0 0
289.4 0 0 0 −1.52089 + 1.63917i 0 2.51643 + 1.45286i 0 0 0
289.5 0 0 0 −1.16287 − 1.90990i 0 −3.55262 − 2.05111i 0 0 0
289.6 0 0 0 −1.07259 − 1.96203i 0 3.55262 + 2.05111i 0 0 0
289.7 0 0 0 −0.668629 + 2.13376i 0 −3.90215 − 2.25291i 0 0 0
289.8 0 0 0 −0.554267 − 2.16628i 0 0.608912 + 0.351555i 0 0 0
289.9 0 0 0 −0.307984 + 2.21476i 0 3.28422 + 1.89614i 0 0 0
289.10 0 0 0 0.355011 + 2.20771i 0 −2.19623 − 1.26799i 0 0 0
289.11 0 0 0 1.06189 − 1.96784i 0 −1.19102 − 0.687633i 0 0 0
289.12 0 0 0 1.73442 + 1.41130i 0 2.19623 + 1.26799i 0 0 0
289.13 0 0 0 2.07203 + 0.840657i 0 −3.28422 − 1.89614i 0 0 0
289.14 0 0 0 2.14796 − 0.621508i 0 1.98012 + 1.14322i 0 0 0
289.15 0 0 0 2.18001 − 0.497547i 0 −2.51643 − 1.45286i 0 0 0
289.16 0 0 0 2.18221 + 0.487830i 0 3.90215 + 2.25291i 0 0 0
1009.1 0 0 0 −2.23514 + 0.0643013i 0 1.19102 − 0.687633i 0 0 0
1009.2 0 0 0 −1.61222 − 1.54943i 0 −1.98012 + 1.14322i 0 0 0
1009.3 0 0 0 −1.59892 + 1.56315i 0 −0.608912 + 0.351555i 0 0 0
1009.4 0 0 0 −1.52089 − 1.63917i 0 2.51643 − 1.45286i 0 0 0
See all 32 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 289.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
9.c even 3 1 inner
45.j even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2160.2.by.f 32
3.b odd 2 1 720.2.by.f 32
4.b odd 2 1 1080.2.bi.b 32
5.b even 2 1 inner 2160.2.by.f 32
9.c even 3 1 inner 2160.2.by.f 32
9.d odd 6 1 720.2.by.f 32
12.b even 2 1 360.2.bi.b ✓ 32
15.d odd 2 1 720.2.by.f 32
20.d odd 2 1 1080.2.bi.b 32
36.f odd 6 1 1080.2.bi.b 32
36.f odd 6 1 3240.2.f.i 16
36.h even 6 1 360.2.bi.b ✓ 32
36.h even 6 1 3240.2.f.k 16
45.h odd 6 1 720.2.by.f 32
45.j even 6 1 inner 2160.2.by.f 32
60.h even 2 1 360.2.bi.b ✓ 32
180.n even 6 1 360.2.bi.b ✓ 32
180.n even 6 1 3240.2.f.k 16
180.p odd 6 1 1080.2.bi.b 32
180.p odd 6 1 3240.2.f.i 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
360.2.bi.b ✓ 32 12.b even 2 1
360.2.bi.b ✓ 32 36.h even 6 1
360.2.bi.b ✓ 32 60.h even 2 1
360.2.bi.b ✓ 32 180.n even 6 1
720.2.by.f 32 3.b odd 2 1
720.2.by.f 32 9.d odd 6 1
720.2.by.f 32 15.d odd 2 1
720.2.by.f 32 45.h odd 6 1
1080.2.bi.b 32 4.b odd 2 1
1080.2.bi.b 32 20.d odd 2 1
1080.2.bi.b 32 36.f odd 6 1
1080.2.bi.b 32 180.p odd 6 1
2160.2.by.f 32 1.a even 1 1 trivial
2160.2.by.f 32 5.b even 2 1 inner
2160.2.by.f 32 9.c even 3 1 inner
2160.2.by.f 32 45.j even 6 1 inner
3240.2.f.i 16 36.f odd 6 1
3240.2.f.i 16 180.p odd 6 1
3240.2.f.k 16 36.h even 6 1
3240.2.f.k 16 180.n even 6 1

Hecke kernels

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

\( T_{7}^{32} - 74 T_{7}^{30} + 3261 T_{7}^{28} - 94822 T_{7}^{26} + 2048402 T_{7}^{24} + \cdots + 1700843738896 \) Copy content Toggle raw display
\( T_{11}^{16} - 8 T_{11}^{15} + 84 T_{11}^{14} - 272 T_{11}^{13} + 2101 T_{11}^{12} - 5148 T_{11}^{11} + \cdots + 16 \) Copy content Toggle raw display