Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [864,2,Mod(37,864)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(864, base_ring=CyclotomicField(24))
chi = DirichletCharacter(H, H._module([0, 3, 8]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("864.37");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 864 = 2^{5} \cdot 3^{3} \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 864.bk (of order \(24\), degree \(8\), not 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: | \(368\) |
Relative dimension: | \(46\) over \(\Q(\zeta_{24})\) |
Twist minimal: | no (minimal twist has level 288) |
Sato-Tate group: | $\mathrm{SU}(2)[C_{24}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
37.1 | −1.41330 | + | 0.0509169i | 0 | 1.99481 | − | 0.143921i | −2.15664 | + | 1.65485i | 0 | −0.251218 | + | 0.937559i | −2.81194 | + | 0.304973i | 0 | 2.96372 | − | 2.44861i | ||||||
37.2 | −1.39820 | − | 0.212242i | 0 | 1.90991 | + | 0.593513i | 0.306163 | − | 0.234927i | 0 | 0.593754 | − | 2.21592i | −2.54446 | − | 1.23521i | 0 | −0.477938 | + | 0.263494i | ||||||
37.3 | −1.39517 | + | 0.231288i | 0 | 1.89301 | − | 0.645374i | 2.21674 | − | 1.70096i | 0 | −0.400471 | + | 1.49458i | −2.49181 | + | 1.33824i | 0 | −2.69932 | + | 2.88584i | ||||||
37.4 | −1.34055 | + | 0.450480i | 0 | 1.59413 | − | 1.20778i | −1.37652 | + | 1.05624i | 0 | 1.04771 | − | 3.91012i | −1.59293 | + | 2.33721i | 0 | 1.36948 | − | 2.03604i | ||||||
37.5 | −1.32629 | − | 0.490865i | 0 | 1.51810 | + | 1.30206i | −0.489423 | + | 0.375547i | 0 | −0.584477 | + | 2.18130i | −1.37431 | − | 2.47210i | 0 | 0.833461 | − | 0.257845i | ||||||
37.6 | −1.32317 | − | 0.499220i | 0 | 1.50156 | + | 1.32111i | 1.16198 | − | 0.891617i | 0 | 0.778386 | − | 2.90498i | −1.32729 | − | 2.49766i | 0 | −1.98261 | + | 0.599678i | ||||||
37.7 | −1.31129 | + | 0.529637i | 0 | 1.43897 | − | 1.38902i | 0.931920 | − | 0.715087i | 0 | −0.366881 | + | 1.36922i | −1.15123 | + | 2.58354i | 0 | −0.843281 | + | 1.43127i | ||||||
37.8 | −1.16868 | − | 0.796364i | 0 | 0.731610 | + | 1.86138i | 3.18925 | − | 2.44720i | 0 | −0.890421 | + | 3.32310i | 0.627322 | − | 2.75798i | 0 | −5.67606 | + | 0.320180i | ||||||
37.9 | −1.15525 | + | 0.815721i | 0 | 0.669198 | − | 1.88472i | −1.28782 | + | 0.988180i | 0 | −0.0841003 | + | 0.313867i | 0.764318 | + | 2.72320i | 0 | 0.681674 | − | 2.19210i | ||||||
37.10 | −1.04770 | + | 0.949902i | 0 | 0.195371 | − | 1.99043i | 2.93870 | − | 2.25494i | 0 | 1.16359 | − | 4.34257i | 1.68603 | + | 2.27097i | 0 | −0.936914 | + | 5.15400i | ||||||
37.11 | −0.930084 | − | 1.06534i | 0 | −0.269888 | + | 1.98171i | −2.36155 | + | 1.81208i | 0 | 0.338610 | − | 1.26371i | 2.36221 | − | 1.55563i | 0 | 4.12692 | + | 0.830461i | ||||||
37.12 | −0.923453 | + | 1.07109i | 0 | −0.294470 | − | 1.97820i | −2.37671 | + | 1.82371i | 0 | −0.989396 | + | 3.69248i | 2.39076 | + | 1.51137i | 0 | 0.241417 | − | 4.22979i | ||||||
37.13 | −0.917969 | + | 1.07579i | 0 | −0.314664 | − | 1.97509i | 2.56852 | − | 1.97090i | 0 | −0.846806 | + | 3.16032i | 2.41364 | + | 1.47456i | 0 | −0.237547 | + | 4.57243i | ||||||
37.14 | −0.874211 | − | 1.11165i | 0 | −0.471511 | + | 1.94362i | −1.36482 | + | 1.04726i | 0 | 1.09705 | − | 4.09423i | 2.57282 | − | 1.17498i | 0 | 2.35732 | + | 0.601666i | ||||||
37.15 | −0.830777 | − | 1.14447i | 0 | −0.619620 | + | 1.90160i | 1.83769 | − | 1.41011i | 0 | 0.454899 | − | 1.69771i | 2.69109 | − | 0.870667i | 0 | −3.14054 | − | 0.931695i | ||||||
37.16 | −0.685362 | − | 1.23704i | 0 | −1.06056 | + | 1.69565i | −2.94194 | + | 2.25743i | 0 | −1.22999 | + | 4.59037i | 2.82446 | + | 0.149828i | 0 | 4.80884 | + | 2.09216i | ||||||
37.17 | −0.683019 | + | 1.23834i | 0 | −1.06697 | − | 1.69162i | 0.526137 | − | 0.403719i | 0 | 0.911495 | − | 3.40174i | 2.82356 | − | 0.165864i | 0 | 0.140580 | + | 0.927284i | ||||||
37.18 | −0.624817 | − | 1.26870i | 0 | −1.21921 | + | 1.58541i | 1.33786 | − | 1.02658i | 0 | −0.437818 | + | 1.63396i | 2.77320 | + | 0.556221i | 0 | −2.13834 | − | 1.05592i | ||||||
37.19 | −0.532138 | + | 1.31028i | 0 | −1.43366 | − | 1.39450i | −0.794880 | + | 0.609933i | 0 | 0.0316510 | − | 0.118123i | 2.59008 | − | 1.13642i | 0 | −0.376196 | − | 1.36608i | ||||||
37.20 | −0.218918 | + | 1.39717i | 0 | −1.90415 | − | 0.611729i | 0.538426 | − | 0.413148i | 0 | −0.988129 | + | 3.68775i | 1.27154 | − | 2.52650i | 0 | 0.459366 | + | 0.842716i | ||||||
See next 80 embeddings (of 368 total) |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
9.c | even | 3 | 1 | inner |
32.g | even | 8 | 1 | inner |
288.bc | even | 24 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 864.2.bk.a | 368 | |
3.b | odd | 2 | 1 | 288.2.bc.a | ✓ | 368 | |
9.c | even | 3 | 1 | inner | 864.2.bk.a | 368 | |
9.d | odd | 6 | 1 | 288.2.bc.a | ✓ | 368 | |
32.g | even | 8 | 1 | inner | 864.2.bk.a | 368 | |
96.p | odd | 8 | 1 | 288.2.bc.a | ✓ | 368 | |
288.bc | even | 24 | 1 | inner | 864.2.bk.a | 368 | |
288.be | odd | 24 | 1 | 288.2.bc.a | ✓ | 368 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
288.2.bc.a | ✓ | 368 | 3.b | odd | 2 | 1 | |
288.2.bc.a | ✓ | 368 | 9.d | odd | 6 | 1 | |
288.2.bc.a | ✓ | 368 | 96.p | odd | 8 | 1 | |
288.2.bc.a | ✓ | 368 | 288.be | odd | 24 | 1 | |
864.2.bk.a | 368 | 1.a | even | 1 | 1 | trivial | |
864.2.bk.a | 368 | 9.c | even | 3 | 1 | inner | |
864.2.bk.a | 368 | 32.g | even | 8 | 1 | inner | |
864.2.bk.a | 368 | 288.bc | even | 24 | 1 | inner |
Hecke kernels
This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(864, [\chi])\).