Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [4020,2,Mod(1609,4020)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4020, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0, 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("4020.1609");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 4020 = 2^{2} \cdot 3 \cdot 5 \cdot 67 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 4020.g (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: | \(32.0998616126\) |
Analytic rank: | \(0\) |
Dimension: | \(38\) |
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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
1609.1 | 0 | − | 1.00000i | 0 | −2.23543 | + | 0.0533841i | 0 | − | 4.50874i | 0 | −1.00000 | 0 | ||||||||||||||
1609.2 | 0 | − | 1.00000i | 0 | −2.22744 | + | 0.196195i | 0 | 0.128273i | 0 | −1.00000 | 0 | |||||||||||||||
1609.3 | 0 | − | 1.00000i | 0 | −2.06455 | + | 0.858848i | 0 | 3.58783i | 0 | −1.00000 | 0 | |||||||||||||||
1609.4 | 0 | − | 1.00000i | 0 | −2.02556 | − | 0.947153i | 0 | 2.95752i | 0 | −1.00000 | 0 | |||||||||||||||
1609.5 | 0 | − | 1.00000i | 0 | −1.48830 | + | 1.66882i | 0 | − | 2.27974i | 0 | −1.00000 | 0 | ||||||||||||||
1609.6 | 0 | − | 1.00000i | 0 | −0.953985 | − | 2.02235i | 0 | 5.11769i | 0 | −1.00000 | 0 | |||||||||||||||
1609.7 | 0 | − | 1.00000i | 0 | −0.847850 | + | 2.06909i | 0 | 3.67313i | 0 | −1.00000 | 0 | |||||||||||||||
1609.8 | 0 | − | 1.00000i | 0 | −0.676225 | − | 2.13137i | 0 | − | 4.23231i | 0 | −1.00000 | 0 | ||||||||||||||
1609.9 | 0 | − | 1.00000i | 0 | −0.599682 | − | 2.15415i | 0 | 0.0778518i | 0 | −1.00000 | 0 | |||||||||||||||
1609.10 | 0 | − | 1.00000i | 0 | −0.504884 | − | 2.17832i | 0 | − | 3.91807i | 0 | −1.00000 | 0 | ||||||||||||||
1609.11 | 0 | − | 1.00000i | 0 | −0.471887 | + | 2.18571i | 0 | − | 1.05400i | 0 | −1.00000 | 0 | ||||||||||||||
1609.12 | 0 | − | 1.00000i | 0 | 0.145309 | + | 2.23134i | 0 | − | 1.12842i | 0 | −1.00000 | 0 | ||||||||||||||
1609.13 | 0 | − | 1.00000i | 0 | 1.17292 | − | 1.90375i | 0 | 2.37828i | 0 | −1.00000 | 0 | |||||||||||||||
1609.14 | 0 | − | 1.00000i | 0 | 1.34274 | + | 1.78803i | 0 | 0.521655i | 0 | −1.00000 | 0 | |||||||||||||||
1609.15 | 0 | − | 1.00000i | 0 | 1.96436 | + | 1.06831i | 0 | − | 3.03191i | 0 | −1.00000 | 0 | ||||||||||||||
1609.16 | 0 | − | 1.00000i | 0 | 1.96624 | + | 1.06486i | 0 | 4.32230i | 0 | −1.00000 | 0 | |||||||||||||||
1609.17 | 0 | − | 1.00000i | 0 | 2.08125 | − | 0.817560i | 0 | 3.41849i | 0 | −1.00000 | 0 | |||||||||||||||
1609.18 | 0 | − | 1.00000i | 0 | 2.20926 | − | 0.345224i | 0 | 0.0620642i | 0 | −1.00000 | 0 | |||||||||||||||
1609.19 | 0 | − | 1.00000i | 0 | 2.21373 | + | 0.315287i | 0 | − | 0.0918930i | 0 | −1.00000 | 0 | ||||||||||||||
1609.20 | 0 | 1.00000i | 0 | −2.23543 | − | 0.0533841i | 0 | 4.50874i | 0 | −1.00000 | 0 | ||||||||||||||||
See all 38 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
5.b | even | 2 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 4020.2.g.c | ✓ | 38 |
5.b | even | 2 | 1 | inner | 4020.2.g.c | ✓ | 38 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
4020.2.g.c | ✓ | 38 | 1.a | even | 1 | 1 | trivial |
4020.2.g.c | ✓ | 38 | 5.b | even | 2 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{7}^{38} + 168 T_{7}^{36} + 12756 T_{7}^{34} + 579200 T_{7}^{32} + 17532664 T_{7}^{30} + 373287946 T_{7}^{28} + 5745701054 T_{7}^{26} + 64680970638 T_{7}^{24} + 532307001632 T_{7}^{22} + \cdots + 16384 \)
acting on \(S_{2}^{\mathrm{new}}(4020, [\chi])\).