Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [4004,2,Mod(3849,4004)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4004, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 1, 1, 0]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("4004.3849");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 4004 = 2^{2} \cdot 7 \cdot 11 \cdot 13 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 4004.e (of order \(2\), degree \(1\), 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: | \(31.9721009693\) |
Analytic rank: | \(0\) |
Dimension: | \(48\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{2}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
3849.1 | 0 | − | 3.23317i | 0 | 0.717019i | 0 | −0.284426 | + | 2.63042i | 0 | −7.45341 | 0 | |||||||||||||||
3849.2 | 0 | − | 3.15005i | 0 | − | 2.09918i | 0 | 1.59876 | − | 2.10807i | 0 | −6.92283 | 0 | ||||||||||||||
3849.3 | 0 | − | 3.04894i | 0 | 4.30537i | 0 | −2.63608 | − | 0.225969i | 0 | −6.29604 | 0 | |||||||||||||||
3849.4 | 0 | − | 2.99006i | 0 | 1.25381i | 0 | −2.36377 | − | 1.18853i | 0 | −5.94047 | 0 | |||||||||||||||
3849.5 | 0 | − | 2.95250i | 0 | 1.47372i | 0 | 0.700125 | + | 2.55144i | 0 | −5.71727 | 0 | |||||||||||||||
3849.6 | 0 | − | 2.73693i | 0 | − | 3.14756i | 0 | −1.11363 | − | 2.39996i | 0 | −4.49077 | 0 | ||||||||||||||
3849.7 | 0 | − | 2.42199i | 0 | − | 0.343428i | 0 | 2.23070 | − | 1.42266i | 0 | −2.86603 | 0 | ||||||||||||||
3849.8 | 0 | − | 2.41288i | 0 | 2.25637i | 0 | 2.16066 | − | 1.52694i | 0 | −2.82198 | 0 | |||||||||||||||
3849.9 | 0 | − | 2.39724i | 0 | − | 2.83291i | 0 | 2.55736 | + | 0.678178i | 0 | −2.74675 | 0 | ||||||||||||||
3849.10 | 0 | − | 2.30818i | 0 | 0.586386i | 0 | −2.63094 | − | 0.279585i | 0 | −2.32771 | 0 | |||||||||||||||
3849.11 | 0 | − | 1.96820i | 0 | − | 3.88595i | 0 | −0.0756292 | + | 2.64467i | 0 | −0.873803 | 0 | ||||||||||||||
3849.12 | 0 | − | 1.77770i | 0 | 3.16364i | 0 | 0.435138 | + | 2.60972i | 0 | −0.160220 | 0 | |||||||||||||||
3849.13 | 0 | − | 1.76248i | 0 | − | 1.09765i | 0 | −1.82169 | + | 1.91871i | 0 | −0.106338 | 0 | ||||||||||||||
3849.14 | 0 | − | 1.33626i | 0 | 2.57665i | 0 | 2.48110 | + | 0.918779i | 0 | 1.21441 | 0 | |||||||||||||||
3849.15 | 0 | − | 1.12498i | 0 | − | 0.542613i | 0 | 1.59076 | − | 2.11411i | 0 | 1.73443 | 0 | ||||||||||||||
3849.16 | 0 | − | 1.04079i | 0 | − | 4.22530i | 0 | −2.02914 | + | 1.69782i | 0 | 1.91676 | 0 | ||||||||||||||
3849.17 | 0 | − | 1.00653i | 0 | 1.31275i | 0 | −1.40205 | − | 2.24371i | 0 | 1.98690 | 0 | |||||||||||||||
3849.18 | 0 | − | 0.968020i | 0 | − | 0.815183i | 0 | −2.53164 | + | 0.768635i | 0 | 2.06294 | 0 | ||||||||||||||
3849.19 | 0 | − | 0.950260i | 0 | 3.52969i | 0 | −2.60374 | + | 0.469588i | 0 | 2.09701 | 0 | |||||||||||||||
3849.20 | 0 | − | 0.867297i | 0 | − | 0.687824i | 0 | 2.28648 | + | 1.33116i | 0 | 2.24780 | 0 | ||||||||||||||
See all 48 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
77.b | even | 2 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 4004.2.e.a | ✓ | 48 |
7.b | odd | 2 | 1 | 4004.2.e.b | yes | 48 | |
11.b | odd | 2 | 1 | 4004.2.e.b | yes | 48 | |
77.b | even | 2 | 1 | inner | 4004.2.e.a | ✓ | 48 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
4004.2.e.a | ✓ | 48 | 1.a | even | 1 | 1 | trivial |
4004.2.e.a | ✓ | 48 | 77.b | even | 2 | 1 | inner |
4004.2.e.b | yes | 48 | 7.b | odd | 2 | 1 | |
4004.2.e.b | yes | 48 | 11.b | odd | 2 | 1 |