Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [4024,2,Mod(1,4024)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4024, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0, 0]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("4024.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 4024 = 2^{3} \cdot 503 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 4024.a (trivial) |
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: | yes |
Analytic conductor: | \(32.1318017734\) |
Analytic rank: | \(1\) |
Dimension: | \(29\) |
Twist minimal: | yes |
Fricke sign: | \(1\) |
Sato-Tate group: | $\mathrm{SU}(2)$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
1.1 | 0 | −3.34526 | 0 | −0.499712 | 0 | 1.77499 | 0 | 8.19075 | 0 | ||||||||||||||||||
1.2 | 0 | −3.05929 | 0 | 0.773308 | 0 | −3.93634 | 0 | 6.35923 | 0 | ||||||||||||||||||
1.3 | 0 | −2.89251 | 0 | 3.16304 | 0 | −0.0867661 | 0 | 5.36660 | 0 | ||||||||||||||||||
1.4 | 0 | −2.70977 | 0 | 2.99505 | 0 | −2.81033 | 0 | 4.34283 | 0 | ||||||||||||||||||
1.5 | 0 | −2.46401 | 0 | −4.08257 | 0 | −4.62698 | 0 | 3.07134 | 0 | ||||||||||||||||||
1.6 | 0 | −2.34298 | 0 | −1.32410 | 0 | 0.439798 | 0 | 2.48955 | 0 | ||||||||||||||||||
1.7 | 0 | −2.20984 | 0 | −0.810761 | 0 | 4.21793 | 0 | 1.88340 | 0 | ||||||||||||||||||
1.8 | 0 | −1.85667 | 0 | 2.88879 | 0 | 1.11301 | 0 | 0.447227 | 0 | ||||||||||||||||||
1.9 | 0 | −1.79909 | 0 | 0.0213618 | 0 | −3.47965 | 0 | 0.236734 | 0 | ||||||||||||||||||
1.10 | 0 | −1.15550 | 0 | −3.26045 | 0 | −1.92982 | 0 | −1.66482 | 0 | ||||||||||||||||||
1.11 | 0 | −1.10456 | 0 | −2.70098 | 0 | 3.17910 | 0 | −1.77994 | 0 | ||||||||||||||||||
1.12 | 0 | −0.795174 | 0 | 3.83818 | 0 | 0.297725 | 0 | −2.36770 | 0 | ||||||||||||||||||
1.13 | 0 | −0.782310 | 0 | 2.74195 | 0 | −3.13445 | 0 | −2.38799 | 0 | ||||||||||||||||||
1.14 | 0 | −0.582422 | 0 | −3.18976 | 0 | −3.00819 | 0 | −2.66078 | 0 | ||||||||||||||||||
1.15 | 0 | −0.502218 | 0 | −3.38571 | 0 | 3.39706 | 0 | −2.74778 | 0 | ||||||||||||||||||
1.16 | 0 | −0.278789 | 0 | −0.533193 | 0 | 1.93823 | 0 | −2.92228 | 0 | ||||||||||||||||||
1.17 | 0 | −0.137144 | 0 | 1.57958 | 0 | 4.44075 | 0 | −2.98119 | 0 | ||||||||||||||||||
1.18 | 0 | 0.454325 | 0 | 0.622756 | 0 | −4.14589 | 0 | −2.79359 | 0 | ||||||||||||||||||
1.19 | 0 | 0.936010 | 0 | 1.25557 | 0 | 0.290471 | 0 | −2.12388 | 0 | ||||||||||||||||||
1.20 | 0 | 1.28849 | 0 | −1.09309 | 0 | 0.799248 | 0 | −1.33979 | 0 | ||||||||||||||||||
See all 29 embeddings |
Atkin-Lehner signs
\( p \) | Sign |
---|---|
\(2\) | \(1\) |
\(503\) | \(1\) |
Inner twists
This newform does not admit any (nontrivial) inner twists.
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 4024.2.a.e | ✓ | 29 |
4.b | odd | 2 | 1 | 8048.2.a.w | 29 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
4024.2.a.e | ✓ | 29 | 1.a | even | 1 | 1 | trivial |
8048.2.a.w | 29 | 4.b | odd | 2 | 1 |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4024))\):
\( T_{3}^{29} + 7 T_{3}^{28} - 29 T_{3}^{27} - 295 T_{3}^{26} + 203 T_{3}^{25} + 5370 T_{3}^{24} + 2583 T_{3}^{23} - 55568 T_{3}^{22} - 60915 T_{3}^{21} + 361198 T_{3}^{20} + 547243 T_{3}^{19} - 1532820 T_{3}^{18} - 2880795 T_{3}^{17} + \cdots + 6975 \) |
\( T_{5}^{29} + 4 T_{5}^{28} - 75 T_{5}^{27} - 297 T_{5}^{26} + 2432 T_{5}^{25} + 9505 T_{5}^{24} - 44755 T_{5}^{23} - 171998 T_{5}^{22} + 515671 T_{5}^{21} + 1941171 T_{5}^{20} - 3870424 T_{5}^{19} - 14229831 T_{5}^{18} + \cdots + 16384 \) |
\( T_{7}^{29} + 13 T_{7}^{28} - 24 T_{7}^{27} - 969 T_{7}^{26} - 1832 T_{7}^{25} + 29380 T_{7}^{24} + 104502 T_{7}^{23} - 465151 T_{7}^{22} - 2376115 T_{7}^{21} + 4047703 T_{7}^{20} + 30762437 T_{7}^{19} + \cdots - 64747 \) |