Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [144,4,Mod(37,144)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(144, base_ring=CyclotomicField(4))
chi = DirichletCharacter(H, H._module([0, 1, 0]))
N = Newforms(chi, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("144.37");
S:= CuspForms(chi, 4);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 144 = 2^{4} \cdot 3^{2} \) |
Weight: | \( k \) | \(=\) | \( 4 \) |
Character orbit: | \([\chi]\) | \(=\) | 144.k (of order \(4\), degree \(2\), 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: | \(8.49627504083\) |
Analytic rank: | \(0\) |
Dimension: | \(24\) |
Relative dimension: | \(12\) over \(\Q(i)\) |
Twist minimal: | no (minimal twist has level 48) |
Sato-Tate group: | $\mathrm{SU}(2)[C_{4}]$ |
$q$-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 | −2.70307 | + | 0.832707i | 0 | 6.61320 | − | 4.50173i | −11.7911 | − | 11.7911i | 0 | − | 12.5754i | −14.1273 | + | 17.6754i | 0 | 41.6906 | + | 22.0536i | |||||||
37.2 | −2.59717 | − | 1.12013i | 0 | 5.49064 | + | 5.81833i | −0.706564 | − | 0.706564i | 0 | − | 4.44122i | −7.74288 | − | 21.2614i | 0 | 1.04363 | + | 2.62651i | |||||||
37.3 | −2.07099 | + | 1.92640i | 0 | 0.577966 | − | 7.97909i | −0.644922 | − | 0.644922i | 0 | 7.13926i | 14.1740 | + | 17.6380i | 0 | 2.57800 | + | 0.0932465i | ||||||||
37.4 | −1.94824 | − | 2.05046i | 0 | −0.408732 | + | 7.98955i | −2.24191 | − | 2.24191i | 0 | 9.00196i | 17.1785 | − | 14.7275i | 0 | −0.229160 | + | 8.96471i | ||||||||
37.5 | −0.954009 | + | 2.66268i | 0 | −6.17974 | − | 5.08044i | 8.83384 | + | 8.83384i | 0 | − | 29.4760i | 19.4231 | − | 11.6079i | 0 | −31.9493 | + | 15.0941i | |||||||
37.6 | 0.220074 | − | 2.81985i | 0 | −7.90313 | − | 1.24115i | −10.2951 | − | 10.2951i | 0 | 32.8369i | −5.23914 | + | 22.0125i | 0 | −31.2964 | + | 26.7650i | ||||||||
37.7 | 0.716137 | − | 2.73627i | 0 | −6.97430 | − | 3.91908i | 11.7719 | + | 11.7719i | 0 | 14.7089i | −15.7182 | + | 16.2769i | 0 | 40.6415 | − | 23.7808i | ||||||||
37.8 | 0.987020 | + | 2.65062i | 0 | −6.05158 | + | 5.23243i | 11.8955 | + | 11.8955i | 0 | − | 0.485059i | −19.8422 | − | 10.8759i | 0 | −19.7893 | + | 43.2714i | |||||||
37.9 | 1.40656 | + | 2.45389i | 0 | −4.04315 | + | 6.90311i | −3.22588 | − | 3.22588i | 0 | 24.6080i | −22.6264 | − | 0.211795i | 0 | 3.37855 | − | 12.4534i | ||||||||
37.10 | 1.92738 | − | 2.07008i | 0 | −0.570442 | − | 7.97964i | 7.29121 | + | 7.29121i | 0 | − | 22.1610i | −17.6179 | − | 14.1989i | 0 | 29.1463 | − | 1.04047i | |||||||
37.11 | 2.24080 | + | 1.72593i | 0 | 2.04234 | + | 7.73491i | −14.6111 | − | 14.6111i | 0 | − | 26.8889i | −8.77342 | + | 20.8573i | 0 | −7.52282 | − | 57.9584i | |||||||
37.12 | 2.77551 | + | 0.544550i | 0 | 7.40693 | + | 3.02281i | 3.72414 | + | 3.72414i | 0 | − | 20.2675i | 18.9120 | + | 12.4233i | 0 | 8.30842 | + | 12.3644i | |||||||
109.1 | −2.70307 | − | 0.832707i | 0 | 6.61320 | + | 4.50173i | −11.7911 | + | 11.7911i | 0 | 12.5754i | −14.1273 | − | 17.6754i | 0 | 41.6906 | − | 22.0536i | ||||||||
109.2 | −2.59717 | + | 1.12013i | 0 | 5.49064 | − | 5.81833i | −0.706564 | + | 0.706564i | 0 | 4.44122i | −7.74288 | + | 21.2614i | 0 | 1.04363 | − | 2.62651i | ||||||||
109.3 | −2.07099 | − | 1.92640i | 0 | 0.577966 | + | 7.97909i | −0.644922 | + | 0.644922i | 0 | − | 7.13926i | 14.1740 | − | 17.6380i | 0 | 2.57800 | − | 0.0932465i | |||||||
109.4 | −1.94824 | + | 2.05046i | 0 | −0.408732 | − | 7.98955i | −2.24191 | + | 2.24191i | 0 | − | 9.00196i | 17.1785 | + | 14.7275i | 0 | −0.229160 | − | 8.96471i | |||||||
109.5 | −0.954009 | − | 2.66268i | 0 | −6.17974 | + | 5.08044i | 8.83384 | − | 8.83384i | 0 | 29.4760i | 19.4231 | + | 11.6079i | 0 | −31.9493 | − | 15.0941i | ||||||||
109.6 | 0.220074 | + | 2.81985i | 0 | −7.90313 | + | 1.24115i | −10.2951 | + | 10.2951i | 0 | − | 32.8369i | −5.23914 | − | 22.0125i | 0 | −31.2964 | − | 26.7650i | |||||||
109.7 | 0.716137 | + | 2.73627i | 0 | −6.97430 | + | 3.91908i | 11.7719 | − | 11.7719i | 0 | − | 14.7089i | −15.7182 | − | 16.2769i | 0 | 40.6415 | + | 23.7808i | |||||||
109.8 | 0.987020 | − | 2.65062i | 0 | −6.05158 | − | 5.23243i | 11.8955 | − | 11.8955i | 0 | 0.485059i | −19.8422 | + | 10.8759i | 0 | −19.7893 | − | 43.2714i | ||||||||
See all 24 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
16.e | even | 4 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 144.4.k.b | 24 | |
3.b | odd | 2 | 1 | 48.4.j.a | ✓ | 24 | |
4.b | odd | 2 | 1 | 576.4.k.b | 24 | ||
12.b | even | 2 | 1 | 192.4.j.a | 24 | ||
16.e | even | 4 | 1 | inner | 144.4.k.b | 24 | |
16.f | odd | 4 | 1 | 576.4.k.b | 24 | ||
24.f | even | 2 | 1 | 384.4.j.a | 24 | ||
24.h | odd | 2 | 1 | 384.4.j.b | 24 | ||
48.i | odd | 4 | 1 | 48.4.j.a | ✓ | 24 | |
48.i | odd | 4 | 1 | 384.4.j.b | 24 | ||
48.k | even | 4 | 1 | 192.4.j.a | 24 | ||
48.k | even | 4 | 1 | 384.4.j.a | 24 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
48.4.j.a | ✓ | 24 | 3.b | odd | 2 | 1 | |
48.4.j.a | ✓ | 24 | 48.i | odd | 4 | 1 | |
144.4.k.b | 24 | 1.a | even | 1 | 1 | trivial | |
144.4.k.b | 24 | 16.e | even | 4 | 1 | inner | |
192.4.j.a | 24 | 12.b | even | 2 | 1 | ||
192.4.j.a | 24 | 48.k | even | 4 | 1 | ||
384.4.j.a | 24 | 24.f | even | 2 | 1 | ||
384.4.j.a | 24 | 48.k | even | 4 | 1 | ||
384.4.j.b | 24 | 24.h | odd | 2 | 1 | ||
384.4.j.b | 24 | 48.i | odd | 4 | 1 | ||
576.4.k.b | 24 | 4.b | odd | 2 | 1 | ||
576.4.k.b | 24 | 16.f | odd | 4 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{24} - 1936 T_{5}^{21} + 249216 T_{5}^{20} - 752832 T_{5}^{19} + 1874048 T_{5}^{18} - 217848704 T_{5}^{17} + 18944520384 T_{5}^{16} - 94398676992 T_{5}^{15} + \cdots + 15\!\cdots\!04 \)
acting on \(S_{4}^{\mathrm{new}}(144, [\chi])\).