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Properties

Label 864.2.y.b
Level $864$
Weight $2$
Character orbit 864.y
Analytic conductor $6.899$
Analytic rank $0$
Dimension $54$
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] = [864,2,Mod(97,864)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(864, base_ring=CyclotomicField(18))
 
chi = DirichletCharacter(H, H._module([0, 0, 4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("864.97");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 864 = 2^{5} \cdot 3^{3} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 864.y (of order \(9\), degree \(6\), 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: \(6.89907473464\)
Analytic rank: \(0\)
Dimension: \(54\)
Relative dimension: \(9\) over \(\Q(\zeta_{9})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{9}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 54 q+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 54 q - 9 q^{11} + 12 q^{17} + 18 q^{19} + 12 q^{21} - 21 q^{27} + 6 q^{29} + 36 q^{31} - 9 q^{33} + 24 q^{39} + 3 q^{41} - 21 q^{43} + 42 q^{45} - 18 q^{49} + 24 q^{51} + 36 q^{53} - 72 q^{55} + 39 q^{57} + 18 q^{59} - 18 q^{61} - 30 q^{63} + 48 q^{65} - 27 q^{67} + 24 q^{69} - 84 q^{75} + 36 q^{77} + 72 q^{79} + 36 q^{81} + 6 q^{87} + 33 q^{89} + 36 q^{91} + 72 q^{93} + 36 q^{95} + 9 q^{97} + 120 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
97.1 0 −1.68626 − 0.395655i 0 −2.65901 − 0.967801i 0 −0.872194 + 4.94646i 0 2.68691 + 1.33435i 0
97.2 0 −1.59135 + 0.683824i 0 2.16034 + 0.786299i 0 0.181183 − 1.02754i 0 2.06477 − 2.17640i 0
97.3 0 −1.06169 − 1.36851i 0 0.217711 + 0.0792403i 0 0.399809 − 2.26743i 0 −0.745646 + 2.90586i 0
97.4 0 −1.00024 + 1.41404i 0 −3.61346 − 1.31519i 0 0.621731 − 3.52601i 0 −0.999043 − 2.82877i 0
97.5 0 −0.146186 − 1.72587i 0 2.31520 + 0.842664i 0 −0.690140 + 3.91398i 0 −2.95726 + 0.504595i 0
97.6 0 0.256084 + 1.71302i 0 −1.06964 − 0.389317i 0 −0.738542 + 4.18848i 0 −2.86884 + 0.877351i 0
97.7 0 1.11120 − 1.32862i 0 −0.779464 − 0.283702i 0 0.260701 − 1.47851i 0 −0.530450 − 2.95273i 0
97.8 0 1.53630 + 0.799856i 0 −1.92089 − 0.699146i 0 0.00661950 − 0.0375410i 0 1.72046 + 2.45764i 0
97.9 0 1.64243 + 0.549936i 0 3.46982 + 1.26291i 0 0.136240 − 0.772655i 0 2.39514 + 1.80646i 0
193.1 0 −1.72098 − 0.195544i 0 0.359864 − 2.04089i 0 2.05212 − 1.72193i 0 2.92353 + 0.673052i 0
193.2 0 −1.67597 + 0.437160i 0 −0.158606 + 0.899497i 0 −1.25116 + 1.04985i 0 2.61778 − 1.46534i 0
193.3 0 −0.661519 − 1.60075i 0 0.197119 − 1.11792i 0 −0.270013 + 0.226568i 0 −2.12478 + 2.11785i 0
193.4 0 −0.523926 + 1.65091i 0 −0.649143 + 3.68147i 0 1.60562 − 1.34727i 0 −2.45100 − 1.72991i 0
193.5 0 −0.0599758 + 1.73101i 0 0.392840 − 2.22791i 0 −3.55853 + 2.98596i 0 −2.99281 − 0.207638i 0
193.6 0 0.527750 − 1.64969i 0 −0.220114 + 1.24833i 0 −2.06659 + 1.73408i 0 −2.44296 − 1.74125i 0
193.7 0 1.17779 + 1.26996i 0 0.191385 − 1.08540i 0 2.62195 − 2.20007i 0 −0.225616 + 2.99150i 0
193.8 0 1.41286 − 1.00191i 0 0.727972 − 4.12853i 0 −0.397562 + 0.333594i 0 0.992364 − 2.83112i 0
193.9 0 1.69762 + 0.343652i 0 −0.494020 + 2.80173i 0 −1.80000 + 1.51038i 0 2.76381 + 1.16678i 0
385.1 0 −1.72098 + 0.195544i 0 0.359864 + 2.04089i 0 2.05212 + 1.72193i 0 2.92353 − 0.673052i 0
385.2 0 −1.67597 − 0.437160i 0 −0.158606 − 0.899497i 0 −1.25116 − 1.04985i 0 2.61778 + 1.46534i 0
See all 54 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 97.9
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
27.e even 9 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 864.2.y.b ✓ 54
4.b odd 2 1 864.2.y.c yes 54
27.e even 9 1 inner 864.2.y.b ✓ 54
108.j odd 18 1 864.2.y.c yes 54
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
864.2.y.b ✓ 54 1.a even 1 1 trivial
864.2.y.b ✓ 54 27.e even 9 1 inner
864.2.y.c yes 54 4.b odd 2 1
864.2.y.c yes 54 108.j odd 18 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(864, [\chi])\):

\( T_{5}^{54} + 26 T_{5}^{51} + 54 T_{5}^{49} + 4271 T_{5}^{48} + 1908 T_{5}^{47} - 4941 T_{5}^{46} + \cdots + 6135275584 \) Copy content Toggle raw display
\( T_{7}^{54} + 9 T_{7}^{52} - 106 T_{7}^{51} - 243 T_{7}^{50} - 84 T_{7}^{49} + 13673 T_{7}^{48} + \cdots + 1043846369344 \) Copy content Toggle raw display