Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [4033,2,Mod(1,4033)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4033, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([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("4033.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 4033 = 37 \cdot 109 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 4033.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: | \(32.2036671352\) |
Analytic rank: | \(1\) |
Dimension: | \(77\) |
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 | −2.77942 | −0.824598 | 5.72520 | 1.36321 | 2.29191 | 0.958419 | −10.3539 | −2.32004 | −3.78894 | ||||||||||||||||||
1.2 | −2.66405 | 1.68420 | 5.09718 | 0.183970 | −4.48681 | 1.02981 | −8.25104 | −0.163454 | −0.490107 | ||||||||||||||||||
1.3 | −2.64480 | 0.673502 | 4.99498 | −3.52022 | −1.78128 | −3.10314 | −7.92112 | −2.54639 | 9.31030 | ||||||||||||||||||
1.4 | −2.64408 | −3.27443 | 4.99114 | −3.17028 | 8.65784 | 2.13489 | −7.90879 | 7.72189 | 8.38246 | ||||||||||||||||||
1.5 | −2.57358 | −2.79713 | 4.62329 | 0.767586 | 7.19864 | −3.79460 | −6.75123 | 4.82396 | −1.97544 | ||||||||||||||||||
1.6 | −2.57274 | −3.14788 | 4.61898 | −2.11075 | 8.09868 | −0.361357 | −6.73794 | 6.90917 | 5.43041 | ||||||||||||||||||
1.7 | −2.51099 | 0.0536632 | 4.30505 | −1.45353 | −0.134748 | 3.51496 | −5.78795 | −2.99712 | 3.64980 | ||||||||||||||||||
1.8 | −2.50194 | −1.32701 | 4.25970 | 0.884945 | 3.32011 | 3.50232 | −5.65363 | −1.23903 | −2.21408 | ||||||||||||||||||
1.9 | −2.37004 | 2.31466 | 3.61710 | 2.16488 | −5.48585 | −1.68976 | −3.83259 | 2.35766 | −5.13086 | ||||||||||||||||||
1.10 | −2.36112 | 2.69902 | 3.57489 | −1.67448 | −6.37271 | 0.928256 | −3.71850 | 4.28471 | 3.95366 | ||||||||||||||||||
1.11 | −2.25607 | 0.265455 | 3.08984 | 2.86324 | −0.598885 | −5.00198 | −2.45875 | −2.92953 | −6.45966 | ||||||||||||||||||
1.12 | −2.25323 | −3.09773 | 3.07703 | 4.21432 | 6.97988 | −2.71387 | −2.42679 | 6.59591 | −9.49583 | ||||||||||||||||||
1.13 | −2.16010 | −1.27573 | 2.66602 | −3.85876 | 2.75571 | 4.13321 | −1.43868 | −1.37250 | 8.33530 | ||||||||||||||||||
1.14 | −2.03838 | 1.12284 | 2.15500 | 3.13166 | −2.28878 | 0.892972 | −0.315945 | −1.73922 | −6.38351 | ||||||||||||||||||
1.15 | −1.98064 | −1.34264 | 1.92294 | −0.618382 | 2.65929 | −3.38656 | 0.152622 | −1.19732 | 1.22479 | ||||||||||||||||||
1.16 | −1.93065 | −1.88783 | 1.72740 | 2.91327 | 3.64473 | 1.86456 | 0.526302 | 0.563889 | −5.62450 | ||||||||||||||||||
1.17 | −1.80263 | 2.09678 | 1.24948 | −2.41088 | −3.77972 | 2.66635 | 1.35291 | 1.39648 | 4.34594 | ||||||||||||||||||
1.18 | −1.78823 | 2.78240 | 1.19778 | −2.10202 | −4.97558 | −2.70114 | 1.43455 | 4.74173 | 3.75890 | ||||||||||||||||||
1.19 | −1.77580 | −1.89592 | 1.15346 | 0.136329 | 3.36677 | −1.80819 | 1.50328 | 0.594515 | −0.242093 | ||||||||||||||||||
1.20 | −1.67758 | 0.712654 | 0.814260 | 2.97956 | −1.19553 | 2.13168 | 1.98917 | −2.49212 | −4.99844 | ||||||||||||||||||
See all 77 embeddings |
Atkin-Lehner signs
\( p \) | Sign |
---|---|
\(37\) | \(-1\) |
\(109\) | \(-1\) |
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 | 4033.2.a.c | ✓ | 77 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
4033.2.a.c | ✓ | 77 | 1.a | even | 1 | 1 | trivial |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{77} + 9 T_{2}^{76} - 73 T_{2}^{75} - 880 T_{2}^{74} + 2021 T_{2}^{73} + 40805 T_{2}^{72} + \cdots + 5180112 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4033))\).