Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [48,4,Mod(11,48)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(48, base_ring=CyclotomicField(4))
chi = DirichletCharacter(H, H._module([2, 1, 2]))
N = Newforms(chi, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("48.11");
S:= CuspForms(chi, 4);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 48 = 2^{4} \cdot 3 \) |
Weight: | \( k \) | \(=\) | \( 4 \) |
Character orbit: | \([\chi]\) | \(=\) | 48.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: | \(2.83209168028\) |
Analytic rank: | \(0\) |
Dimension: | \(44\) |
Relative dimension: | \(22\) over \(\Q(i)\) |
Twist minimal: | yes |
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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
11.1 | −2.81978 | − | 0.221045i | −2.71261 | + | 4.43190i | 7.90228 | + | 1.24659i | 3.17566 | − | 3.17566i | 8.62861 | − | 11.8974i | −32.3513 | −22.0071 | − | 5.26187i | −12.2835 | − | 24.0440i | −9.65662 | + | 8.25269i | ||
11.2 | −2.60843 | + | 1.09367i | 4.68788 | + | 2.24138i | 5.60778 | − | 5.70550i | 2.69043 | − | 2.69043i | −14.6793 | − | 0.719504i | 10.6336 | −8.38758 | + | 21.0154i | 16.9524 | + | 21.0147i | −4.07535 | + | 9.96021i | ||
11.3 | −2.56930 | − | 1.18266i | −4.81196 | − | 1.96089i | 5.20263 | + | 6.07722i | −6.30133 | + | 6.30133i | 10.0443 | + | 10.7290i | 24.6728 | −6.17986 | − | 21.7672i | 19.3098 | + | 18.8714i | 23.6423 | − | 8.73769i | ||
11.4 | −2.53336 | − | 1.25780i | 3.76578 | − | 3.58035i | 4.83587 | + | 6.37294i | 4.71515 | − | 4.71515i | −14.0435 | + | 4.33371i | 4.67595 | −4.23510 | − | 22.2275i | 1.36225 | − | 26.9656i | −17.8759 | + | 6.01447i | ||
11.5 | −2.44754 | + | 1.41758i | 0.563550 | − | 5.16550i | 3.98092 | − | 6.93918i | −13.1633 | + | 13.1633i | 5.94321 | + | 13.4417i | −13.2717 | 0.0933841 | + | 22.6272i | −26.3648 | − | 5.82204i | 13.5576 | − | 50.8776i | ||
11.6 | −1.97996 | + | 2.01984i | −5.19561 | − | 0.0749974i | −0.159526 | − | 7.99841i | 5.37662 | − | 5.37662i | 10.4386 | − | 10.3458i | 14.8575 | 16.4714 | + | 15.5143i | 26.9888 | + | 0.779314i | 0.214439 | + | 21.5054i | ||
11.7 | −1.70353 | − | 2.25788i | 3.86039 | + | 3.47813i | −2.19600 | + | 7.69270i | −13.5794 | + | 13.5794i | 1.27693 | − | 14.6414i | −19.7355 | 21.1101 | − | 8.14640i | 2.80518 | + | 26.8539i | 53.7934 | + | 7.52774i | ||
11.8 | −1.13198 | − | 2.59203i | −1.13522 | + | 5.07063i | −5.43724 | + | 5.86826i | 11.2665 | − | 11.2665i | 14.4283 | − | 2.79732i | 30.2121 | 21.3655 | + | 7.45073i | −24.4225 | − | 11.5126i | −41.9567 | − | 16.4497i | ||
11.9 | −0.775594 | − | 2.72001i | −2.65327 | − | 4.46768i | −6.79691 | + | 4.21925i | 5.27809 | − | 5.27809i | −10.0943 | + | 10.6820i | −22.9284 | 16.7480 | + | 15.2152i | −12.9203 | + | 23.7079i | −18.4501 | − | 10.2628i | ||
11.10 | −0.763869 | + | 2.72333i | −0.151014 | + | 5.19396i | −6.83301 | − | 4.16053i | −4.66675 | + | 4.66675i | −14.0295 | − | 4.37877i | 0.405799 | 16.5500 | − | 15.4304i | −26.9544 | − | 1.56872i | −9.14429 | − | 16.2739i | ||
11.11 | −0.668281 | + | 2.74835i | 2.28317 | − | 4.66767i | −7.10680 | − | 3.67333i | 11.5146 | − | 11.5146i | 11.3026 | + | 9.39425i | 0.829117 | 14.8449 | − | 17.0771i | −16.5743 | − | 21.3142i | 23.9512 | + | 39.3412i | ||
11.12 | 0.668281 | − | 2.74835i | −4.66767 | + | 2.28317i | −7.10680 | − | 3.67333i | −11.5146 | + | 11.5146i | 3.15562 | + | 14.3542i | 0.829117 | −14.8449 | + | 17.0771i | 16.5743 | − | 21.3142i | 23.9512 | + | 39.3412i | ||
11.13 | 0.763869 | − | 2.72333i | 5.19396 | − | 0.151014i | −6.83301 | − | 4.16053i | 4.66675 | − | 4.66675i | 3.55625 | − | 14.2602i | 0.405799 | −16.5500 | + | 15.4304i | 26.9544 | − | 1.56872i | −9.14429 | − | 16.2739i | ||
11.14 | 0.775594 | + | 2.72001i | −4.46768 | − | 2.65327i | −6.79691 | + | 4.21925i | −5.27809 | + | 5.27809i | 3.75181 | − | 14.2100i | −22.9284 | −16.7480 | − | 15.2152i | 12.9203 | + | 23.7079i | −18.4501 | − | 10.2628i | ||
11.15 | 1.13198 | + | 2.59203i | 5.07063 | − | 1.13522i | −5.43724 | + | 5.86826i | −11.2665 | + | 11.2665i | 8.68239 | + | 11.8582i | 30.2121 | −21.3655 | − | 7.45073i | 24.4225 | − | 11.5126i | −41.9567 | − | 16.4497i | ||
11.16 | 1.70353 | + | 2.25788i | 3.47813 | + | 3.86039i | −2.19600 | + | 7.69270i | 13.5794 | − | 13.5794i | −2.79119 | + | 14.4295i | −19.7355 | −21.1101 | + | 8.14640i | −2.80518 | + | 26.8539i | 53.7934 | + | 7.52774i | ||
11.17 | 1.97996 | − | 2.01984i | −0.0749974 | − | 5.19561i | −0.159526 | − | 7.99841i | −5.37662 | + | 5.37662i | −10.6428 | − | 10.1356i | 14.8575 | −16.4714 | − | 15.5143i | −26.9888 | + | 0.779314i | 0.214439 | + | 21.5054i | ||
11.18 | 2.44754 | − | 1.41758i | −5.16550 | + | 0.563550i | 3.98092 | − | 6.93918i | 13.1633 | − | 13.1633i | −11.8439 | + | 8.70183i | −13.2717 | −0.0933841 | − | 22.6272i | 26.3648 | − | 5.82204i | 13.5576 | − | 50.8776i | ||
11.19 | 2.53336 | + | 1.25780i | −3.58035 | + | 3.76578i | 4.83587 | + | 6.37294i | −4.71515 | + | 4.71515i | −13.8069 | + | 5.03673i | 4.67595 | 4.23510 | + | 22.2275i | −1.36225 | − | 26.9656i | −17.8759 | + | 6.01447i | ||
11.20 | 2.56930 | + | 1.18266i | −1.96089 | − | 4.81196i | 5.20263 | + | 6.07722i | 6.30133 | − | 6.30133i | 0.652791 | − | 14.6824i | 24.6728 | 6.17986 | + | 21.7672i | −19.3098 | + | 18.8714i | 23.6423 | − | 8.73769i | ||
See all 44 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
3.b | odd | 2 | 1 | inner |
16.f | odd | 4 | 1 | inner |
48.k | even | 4 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 48.4.k.a | ✓ | 44 |
3.b | odd | 2 | 1 | inner | 48.4.k.a | ✓ | 44 |
4.b | odd | 2 | 1 | 192.4.k.a | 44 | ||
8.b | even | 2 | 1 | 384.4.k.b | 44 | ||
8.d | odd | 2 | 1 | 384.4.k.a | 44 | ||
12.b | even | 2 | 1 | 192.4.k.a | 44 | ||
16.e | even | 4 | 1 | 192.4.k.a | 44 | ||
16.e | even | 4 | 1 | 384.4.k.a | 44 | ||
16.f | odd | 4 | 1 | inner | 48.4.k.a | ✓ | 44 |
16.f | odd | 4 | 1 | 384.4.k.b | 44 | ||
24.f | even | 2 | 1 | 384.4.k.a | 44 | ||
24.h | odd | 2 | 1 | 384.4.k.b | 44 | ||
48.i | odd | 4 | 1 | 192.4.k.a | 44 | ||
48.i | odd | 4 | 1 | 384.4.k.a | 44 | ||
48.k | even | 4 | 1 | inner | 48.4.k.a | ✓ | 44 |
48.k | even | 4 | 1 | 384.4.k.b | 44 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
48.4.k.a | ✓ | 44 | 1.a | even | 1 | 1 | trivial |
48.4.k.a | ✓ | 44 | 3.b | odd | 2 | 1 | inner |
48.4.k.a | ✓ | 44 | 16.f | odd | 4 | 1 | inner |
48.4.k.a | ✓ | 44 | 48.k | even | 4 | 1 | inner |
192.4.k.a | 44 | 4.b | odd | 2 | 1 | ||
192.4.k.a | 44 | 12.b | even | 2 | 1 | ||
192.4.k.a | 44 | 16.e | even | 4 | 1 | ||
192.4.k.a | 44 | 48.i | odd | 4 | 1 | ||
384.4.k.a | 44 | 8.d | odd | 2 | 1 | ||
384.4.k.a | 44 | 16.e | even | 4 | 1 | ||
384.4.k.a | 44 | 24.f | even | 2 | 1 | ||
384.4.k.a | 44 | 48.i | odd | 4 | 1 | ||
384.4.k.b | 44 | 8.b | even | 2 | 1 | ||
384.4.k.b | 44 | 16.f | odd | 4 | 1 | ||
384.4.k.b | 44 | 24.h | odd | 2 | 1 | ||
384.4.k.b | 44 | 48.k | even | 4 | 1 |
Hecke kernels
This newform subspace is the entire newspace \(S_{4}^{\mathrm{new}}(48, [\chi])\).