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Properties

Label 289.3.e.r
Level $289$
Weight $3$
Character orbit 289.e
Analytic conductor $7.875$
Analytic rank $0$
Dimension $96$
CM no
Inner twists $16$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [289,3,Mod(40,289)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(289, base_ring=CyclotomicField(16))
 
chi = DirichletCharacter(H, H._module([15]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("289.40");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 289 = 17^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 289.e (of order \(16\), degree \(8\), 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: \(7.87467964001\)
Analytic rank: \(0\)
Dimension: \(96\)
Relative dimension: \(12\) over \(\Q(\zeta_{16})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{16}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 96 q+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 96 q + 288 q^{18} + 384 q^{35} + 1920 q^{52} - 1152 q^{69} - 6240 q^{86}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
40.1 −3.45680 − 1.43186i −2.43709 + 0.484768i 7.07086 + 7.07086i 0.635929 − 0.951734i 9.11867 + 1.81382i −3.04050 − 4.55043i −8.59071 − 20.7398i −2.61050 + 1.08130i −3.56103 + 2.37940i
40.2 −3.45680 − 1.43186i 2.43709 − 0.484768i 7.07086 + 7.07086i −0.635929 + 0.951734i −9.11867 − 1.81382i 3.04050 + 4.55043i −8.59071 − 20.7398i −2.61050 + 1.08130i 3.56103 − 2.37940i
40.3 −2.10235 − 0.870820i −1.65357 + 0.328916i 0.833100 + 0.833100i −2.49895 + 3.73994i 3.76280 + 0.748468i −5.13709 − 7.68820i 2.45730 + 5.93244i −5.68881 + 2.35638i 8.51046 − 5.68651i
40.4 −2.10235 − 0.870820i 1.65357 − 0.328916i 0.833100 + 0.833100i 2.49895 − 3.73994i −3.76280 − 0.748468i 5.13709 + 7.68820i 2.45730 + 5.93244i −5.68881 + 2.35638i −8.51046 + 5.68651i
40.5 −0.577354 − 0.239148i −4.01529 + 0.798690i −2.55228 − 2.55228i 2.45675 − 3.67678i 2.50925 + 0.499120i −5.69243 − 8.51932i 1.81979 + 4.39336i 7.16969 − 2.96978i −2.29771 + 1.53528i
40.6 −0.577354 − 0.239148i 4.01529 − 0.798690i −2.55228 − 2.55228i −2.45675 + 3.67678i −2.50925 − 0.499120i 5.69243 + 8.51932i 1.81979 + 4.39336i 7.16969 − 2.96978i 2.29771 − 1.53528i
40.7 −0.0158468 − 0.00656394i −2.90309 + 0.577461i −2.82822 − 2.82822i −4.90560 + 7.34175i 0.0497950 + 0.00990485i −3.56844 − 5.34054i 0.0525096 + 0.126769i −0.220434 + 0.0913066i 0.125929 − 0.0841428i
40.8 −0.0158468 − 0.00656394i 2.90309 − 0.577461i −2.82822 − 2.82822i 4.90560 − 7.34175i −0.0497950 − 0.00990485i 3.56844 + 5.34054i 0.0525096 + 0.126769i −0.220434 + 0.0913066i −0.125929 + 0.0841428i
40.9 2.74407 + 1.13663i −4.26572 + 0.848504i 3.40954 + 3.40954i 3.75511 − 5.61992i −12.6698 − 2.52019i −1.56892 − 2.34806i 0.934102 + 2.25512i 9.16146 − 3.79480i 16.6920 − 11.1533i
40.10 2.74407 + 1.13663i 4.26572 − 0.848504i 3.40954 + 3.40954i −3.75511 + 5.61992i 12.6698 + 2.52019i 1.56892 + 2.34806i 0.934102 + 2.25512i 9.16146 − 3.79480i −16.6920 + 11.1533i
40.11 3.40829 + 1.41176i −0.724048 + 0.144022i 6.79492 + 6.79492i −3.48078 + 5.20935i −2.67109 − 0.531312i 0.323193 + 0.483692i 7.91921 + 19.1187i −7.81141 + 3.23559i −19.2178 + 12.8409i
40.12 3.40829 + 1.41176i 0.724048 − 0.144022i 6.79492 + 6.79492i 3.48078 − 5.20935i 2.67109 + 0.531312i −0.323193 − 0.483692i 7.91921 + 19.1187i −7.81141 + 3.23559i 19.2178 − 12.8409i
65.1 −3.40829 + 1.41176i −0.144022 + 0.724048i 6.79492 − 6.79492i −5.20935 + 3.48078i −0.531312 − 2.67109i 0.483692 + 0.323193i −7.91921 + 19.1187i 7.81141 + 3.23559i 12.8409 − 19.2178i
65.2 −3.40829 + 1.41176i 0.144022 − 0.724048i 6.79492 − 6.79492i 5.20935 − 3.48078i 0.531312 + 2.67109i −0.483692 − 0.323193i −7.91921 + 19.1187i 7.81141 + 3.23559i −12.8409 + 19.2178i
65.3 −2.74407 + 1.13663i −0.848504 + 4.26572i 3.40954 − 3.40954i 5.61992 − 3.75511i −2.52019 − 12.6698i −2.34806 − 1.56892i −0.934102 + 2.25512i −9.16146 − 3.79480i −11.1533 + 16.6920i
65.4 −2.74407 + 1.13663i 0.848504 − 4.26572i 3.40954 − 3.40954i −5.61992 + 3.75511i 2.52019 + 12.6698i 2.34806 + 1.56892i −0.934102 + 2.25512i −9.16146 − 3.79480i 11.1533 − 16.6920i
65.5 0.0158468 − 0.00656394i −0.577461 + 2.90309i −2.82822 + 2.82822i −7.34175 + 4.90560i 0.00990485 + 0.0497950i −5.34054 − 3.56844i −0.0525096 + 0.126769i 0.220434 + 0.0913066i −0.0841428 + 0.125929i
65.6 0.0158468 − 0.00656394i 0.577461 − 2.90309i −2.82822 + 2.82822i 7.34175 − 4.90560i −0.00990485 − 0.0497950i 5.34054 + 3.56844i −0.0525096 + 0.126769i 0.220434 + 0.0913066i 0.0841428 − 0.125929i
65.7 0.577354 − 0.239148i −0.798690 + 4.01529i −2.55228 + 2.55228i 3.67678 − 2.45675i 0.499120 + 2.50925i −8.51932 − 5.69243i −1.81979 + 4.39336i −7.16969 − 2.96978i 1.53528 − 2.29771i
65.8 0.577354 − 0.239148i 0.798690 − 4.01529i −2.55228 + 2.55228i −3.67678 + 2.45675i −0.499120 − 2.50925i 8.51932 + 5.69243i −1.81979 + 4.39336i −7.16969 − 2.96978i −1.53528 + 2.29771i
See all 96 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 40.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
17.b even 2 1 inner
17.c even 4 2 inner
17.d even 8 4 inner
17.e odd 16 8 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 289.3.e.r ✓ 96
17.b even 2 1 inner 289.3.e.r ✓ 96
17.c even 4 2 inner 289.3.e.r ✓ 96
17.d even 8 4 inner 289.3.e.r ✓ 96
17.e odd 16 8 inner 289.3.e.r ✓ 96
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
289.3.e.r ✓ 96 1.a even 1 1 trivial
289.3.e.r ✓ 96 17.b even 2 1 inner
289.3.e.r ✓ 96 17.c even 4 2 inner
289.3.e.r ✓ 96 17.d even 8 4 inner
289.3.e.r ✓ 96 17.e odd 16 8 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(289, [\chi])\):

\( T_{2}^{48} + 79494 T_{2}^{40} + 1814840847 T_{2}^{32} + 9245419445012 T_{2}^{24} + \cdots + 1 \) Copy content Toggle raw display
\( T_{3}^{96} + 22657938090 T_{3}^{80} + \cdots + 24\!\cdots\!41 \) Copy content Toggle raw display