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Properties

Label 864.2.v.b
Level $864$
Weight $2$
Character orbit 864.v
Analytic conductor $6.899$
Analytic rank $0$
Dimension $128$
Inner twists $4$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [864,2,Mod(109,864)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://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);
 
Copy content sage: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")
 
Level: \( N \) \(=\) \( 864 = 2^{5} \cdot 3^{3} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 864.v (of order \(8\), degree \(4\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [128,0,0,0,0,0,0,0,0,16] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(10)] == traces)
 
Copy content 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 algebraic \(q\)-expansion of this newform has not been computed, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 128 q + 16 q^{10} - 32 q^{16} - 16 q^{22} - 32 q^{40} - 32 q^{46} - 80 q^{52} + 32 q^{55} - 32 q^{58} + 64 q^{61} + 48 q^{64} + 64 q^{67} - 96 q^{70} + 32 q^{76} - 80 q^{82} - 80 q^{88} + 96 q^{91} - 48 q^{94}+O(q^{100}) \) Copy content Toggle raw display

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.

Copy content comment:embeddings in the coefficient field
 
Copy content gp:mfembed(f)
 
Label   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
109.1 −1.41128 − 0.0910249i 0 1.98343 + 0.256923i 2.84116 − 1.17684i 0 −0.237847 + 0.237847i −2.77579 − 0.543133i 0 −4.11679 + 1.40224i
109.2 −1.39831 + 0.211463i 0 1.91057 − 0.591382i −1.77214 + 0.734044i 0 −0.0355868 + 0.0355868i −2.54652 + 1.23095i 0 2.32278 − 1.40116i
109.3 −1.33723 − 0.460244i 0 1.57635 + 1.23090i −2.37116 + 0.982165i 0 2.29537 − 2.29537i −1.54142 − 2.37150i 0 3.62281 − 0.222067i
109.4 −1.23997 + 0.680049i 0 1.07507 − 1.68649i 1.02237 − 0.423478i 0 0.485592 − 0.485592i −0.186159 + 2.82229i 0 −0.979722 + 1.22036i
109.5 −1.19327 + 0.759019i 0 0.847781 − 1.81143i 2.08348 − 0.863006i 0 −2.85037 + 2.85037i 0.363276 + 2.80500i 0 −1.83111 + 2.61120i
109.6 −1.14607 − 0.828563i 0 0.626968 + 1.89919i −2.89399 + 1.19873i 0 −3.03933 + 3.03933i 0.855044 − 2.69609i 0 4.30995 + 1.02402i
109.7 −1.12159 − 0.861417i 0 0.515922 + 1.93231i 1.04543 − 0.433032i 0 2.00753 − 2.00753i 1.08587 − 2.61168i 0 −1.54557 − 0.414869i
109.8 −1.11341 + 0.871966i 0 0.479352 − 1.94171i −2.70909 + 1.12214i 0 0.103960 − 0.103960i 1.15939 + 2.57989i 0 2.03785 − 3.61163i
109.9 −0.933793 − 1.06209i 0 −0.256062 + 1.98354i 1.30670 − 0.541253i 0 −1.80429 + 1.80429i 2.34580 − 1.58026i 0 −1.79505 − 0.882413i
109.10 −0.877362 + 1.10916i 0 −0.460473 − 1.94627i 3.16129 − 1.30945i 0 3.16733 − 3.16733i 2.56273 + 1.19684i 0 −1.32120 + 4.65523i
109.11 −0.679990 + 1.24001i 0 −1.07523 − 1.68638i −3.54476 + 1.46829i 0 1.60611 − 1.60611i 2.82227 − 0.186565i 0 0.589716 − 5.39394i
109.12 −0.479175 + 1.33056i 0 −1.54078 − 1.27514i −0.855212 + 0.354240i 0 −3.04155 + 3.04155i 2.43496 − 1.43909i 0 −0.0615424 − 1.30765i
109.13 −0.463875 − 1.33597i 0 −1.56964 + 1.23945i −0.137547 + 0.0569738i 0 0.385875 − 0.385875i 2.38398 + 1.52205i 0 0.139920 + 0.157330i
109.14 −0.322202 − 1.37702i 0 −1.79237 + 0.887359i 1.57161 − 0.650984i 0 3.07534 − 3.07534i 1.79942 + 2.18222i 0 −1.40280 − 1.95440i
109.15 −0.274404 − 1.38734i 0 −1.84940 + 0.761382i −3.91016 + 1.61964i 0 −0.876571 + 0.876571i 1.56378 + 2.35682i 0 3.31995 + 4.98027i
109.16 −0.239074 + 1.39386i 0 −1.88569 − 0.666470i 0.369546 − 0.153071i 0 −1.24156 + 1.24156i 1.37978 − 2.46905i 0 0.125011 + 0.551690i
109.17 0.239074 − 1.39386i 0 −1.88569 − 0.666470i −0.369546 + 0.153071i 0 −1.24156 + 1.24156i −1.37978 + 2.46905i 0 0.125011 + 0.551690i
109.18 0.274404 + 1.38734i 0 −1.84940 + 0.761382i 3.91016 − 1.61964i 0 −0.876571 + 0.876571i −1.56378 − 2.35682i 0 3.31995 + 4.98027i
109.19 0.322202 + 1.37702i 0 −1.79237 + 0.887359i −1.57161 + 0.650984i 0 3.07534 − 3.07534i −1.79942 − 2.18222i 0 −1.40280 − 1.95440i
109.20 0.463875 + 1.33597i 0 −1.56964 + 1.23945i 0.137547 − 0.0569738i 0 0.385875 − 0.385875i −2.38398 − 1.52205i 0 0.139920 + 0.157330i
See next 80 embeddings (of 128 total)
\(n\): e.g. 2-40 or 80-90
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.b ✓ 128
3.b odd 2 1 inner 864.2.v.b ✓ 128
32.g even 8 1 inner 864.2.v.b ✓ 128
96.p odd 8 1 inner 864.2.v.b ✓ 128
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
864.2.v.b ✓ 128 1.a even 1 1 trivial
864.2.v.b ✓ 128 3.b odd 2 1 inner
864.2.v.b ✓ 128 32.g even 8 1 inner
864.2.v.b ✓ 128 96.p odd 8 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{128} - 512 T_{5}^{122} + 205312 T_{5}^{120} - 113408 T_{5}^{118} + 131072 T_{5}^{116} + \cdots + 28\!\cdots\!96 \) acting on \(S_{2}^{\mathrm{new}}(864, [\chi])\). Copy content Toggle raw display