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.
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 | −1.41128 | − | 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) |
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
\( T_{5}^{128} - 512 T_{5}^{122} + 205312 T_{5}^{120} - 113408 T_{5}^{118} + 131072 T_{5}^{116} - 84445440 T_{5}^{114} + 14102444544 T_{5}^{112} - 20464036352 T_{5}^{110} + 22756098048 T_{5}^{108} + \cdots + 28\!\cdots\!96 \)
acting on \(S_{2}^{\mathrm{new}}(864, [\chi])\).