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Properties

Label 864.2.v.a
Level $864$
Weight $2$
Character orbit 864.v
Analytic conductor $6.899$
Analytic rank $0$
Dimension $128$
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] = [864,2,Mod(109,864)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(864, base_ring=CyclotomicField(8))
 
chi = DirichletCharacter(H, H._module([0, 7, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("864.109");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 864 = 2^{5} \cdot 3^{3} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 864.v (of order \(8\), degree \(4\), 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: \(128\)
Relative dimension: \(32\) over \(\Q(\zeta_{8})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{8}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 128 q+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 128 q - 8 q^{10} - 32 q^{16} + 32 q^{22} + 64 q^{40} + 64 q^{46} + 88 q^{52} - 64 q^{55} + 64 q^{58} - 32 q^{61} - 96 q^{64} + 64 q^{67} + 48 q^{70} + 32 q^{76} + 40 q^{82} + 40 q^{88} - 48 q^{91} + 24 q^{94}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
109.1 −1.39805 − 0.213220i 0 1.90907 + 0.596184i 0.0181233 − 0.00750691i 0 −1.64798 + 1.64798i −2.54186 − 1.24055i 0 −0.0269378 + 0.00663076i
109.2 −1.39716 + 0.218946i 0 1.90413 − 0.611807i −2.13352 + 0.883735i 0 0.426205 − 0.426205i −2.52642 + 1.27169i 0 2.78739 − 1.70185i
109.3 −1.38295 + 0.295740i 0 1.82508 − 0.817983i 3.70533 − 1.53480i 0 1.70420 − 1.70420i −2.28207 + 1.67097i 0 −4.67037 + 3.21836i
109.4 −1.33208 − 0.474937i 0 1.54887 + 1.26531i −0.0485815 + 0.0201231i 0 2.80669 − 2.80669i −1.46228 − 2.42110i 0 0.0742716 − 0.00373246i
109.5 −1.29834 − 0.560644i 0 1.37136 + 1.45581i 1.49059 − 0.617423i 0 −3.02902 + 3.02902i −0.964291 − 2.65897i 0 −2.28144 − 0.0340682i
109.6 −1.18499 + 0.771885i 0 0.808388 − 1.82935i −0.796479 + 0.329912i 0 3.41414 − 3.41414i 0.454114 + 2.79173i 0 0.689163 − 1.00573i
109.7 −1.17722 + 0.783679i 0 0.771696 − 1.84512i −3.84162 + 1.59125i 0 −2.86540 + 2.86540i 0.537529 + 2.77688i 0 3.27540 − 4.88384i
109.8 −1.01339 − 0.986432i 0 0.0539048 + 1.99927i −2.55155 + 1.05689i 0 1.06497 − 1.06497i 1.91752 − 2.07921i 0 3.62825 + 1.44589i
109.9 −0.957543 − 1.04073i 0 −0.166221 + 1.99308i 3.57149 − 1.47936i 0 1.49427 − 1.49427i 2.23341 − 1.73547i 0 −4.95946 − 2.30039i
109.10 −0.934820 + 1.06118i 0 −0.252222 − 1.98403i 0.930451 − 0.385405i 0 −0.938500 + 0.938500i 2.34121 + 1.58706i 0 −0.460818 + 1.34766i
109.11 −0.791979 + 1.17165i 0 −0.745540 − 1.85585i 2.17386 − 0.900442i 0 −1.13840 + 1.13840i 2.76486 + 0.596278i 0 −0.666644 + 3.26014i
109.12 −0.694736 − 1.23180i 0 −1.03468 + 1.71156i −1.00178 + 0.414949i 0 −1.78507 + 1.78507i 2.82714 + 0.0854454i 0 1.20711 + 0.945711i
109.13 −0.567462 − 1.29537i 0 −1.35597 + 1.47015i 2.16011 − 0.894745i 0 1.46314 − 1.46314i 2.67385 + 0.922239i 0 −2.38480 − 2.29041i
109.14 −0.139754 + 1.40729i 0 −1.96094 − 0.393350i −0.345772 + 0.143223i 0 0.299071 − 0.299071i 0.827608 − 2.70464i 0 −0.153234 − 0.506617i
109.15 −0.114947 + 1.40953i 0 −1.97357 − 0.324044i −2.86147 + 1.18526i 0 −2.81522 + 2.81522i 0.683608 − 2.74457i 0 −1.34175 − 4.16959i
109.16 −0.0434074 + 1.41355i 0 −1.99623 − 0.122717i 2.68483 − 1.11209i 0 1.54691 − 1.54691i 0.260117 − 2.81644i 0 1.45545 + 3.84340i
109.17 0.0434074 − 1.41355i 0 −1.99623 − 0.122717i −2.68483 + 1.11209i 0 1.54691 − 1.54691i −0.260117 + 2.81644i 0 1.45545 + 3.84340i
109.18 0.114947 − 1.40953i 0 −1.97357 − 0.324044i 2.86147 − 1.18526i 0 −2.81522 + 2.81522i −0.683608 + 2.74457i 0 −1.34175 − 4.16959i
109.19 0.139754 − 1.40729i 0 −1.96094 − 0.393350i 0.345772 − 0.143223i 0 0.299071 − 0.299071i −0.827608 + 2.70464i 0 −0.153234 − 0.506617i
109.20 0.567462 + 1.29537i 0 −1.35597 + 1.47015i −2.16011 + 0.894745i 0 1.46314 − 1.46314i −2.67385 − 0.922239i 0 −2.38480 − 2.29041i
See next 80 embeddings (of 128 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 109.32
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
32.g even 8 1 inner
96.p odd 8 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 864.2.v.a ✓ 128
3.b odd 2 1 inner 864.2.v.a ✓ 128
32.g even 8 1 inner 864.2.v.a ✓ 128
96.p odd 8 1 inner 864.2.v.a ✓ 128
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
864.2.v.a ✓ 128 1.a even 1 1 trivial
864.2.v.a ✓ 128 3.b odd 2 1 inner
864.2.v.a ✓ 128 32.g even 8 1 inner
864.2.v.a ✓ 128 96.p odd 8 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{128} + 1024 T_{5}^{122} + 204544 T_{5}^{120} + 220672 T_{5}^{118} + 524288 T_{5}^{116} + \cdots + 31\!\cdots\!01 \) acting on \(S_{2}^{\mathrm{new}}(864, [\chi])\). Copy content Toggle raw display