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: | \(0\) |
Dimension: | \(85\) |
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 | −2.76864 | 3.12213 | 5.66539 | −1.61337 | −8.64407 | −3.34335 | −10.1482 | 6.74771 | 4.46684 | ||||||||||||||||||
1.2 | −2.68964 | −0.345637 | 5.23416 | 3.86855 | 0.929638 | −2.16399 | −8.69874 | −2.88054 | −10.4050 | ||||||||||||||||||
1.3 | −2.66672 | −1.32046 | 5.11137 | 0.282486 | 3.52128 | −2.11170 | −8.29715 | −1.25639 | −0.753309 | ||||||||||||||||||
1.4 | −2.62495 | 2.49737 | 4.89034 | 0.915854 | −6.55547 | 2.77253 | −7.58698 | 3.23688 | −2.40407 | ||||||||||||||||||
1.5 | −2.49324 | −1.70202 | 4.21624 | −1.58478 | 4.24353 | 2.99306 | −5.52560 | −0.103143 | 3.95124 | ||||||||||||||||||
1.6 | −2.42751 | 0.768452 | 3.89281 | −1.80408 | −1.86543 | −1.49541 | −4.59481 | −2.40948 | 4.37943 | ||||||||||||||||||
1.7 | −2.36110 | 2.47312 | 3.57478 | 1.50101 | −5.83928 | 4.30200 | −3.71820 | 3.11634 | −3.54403 | ||||||||||||||||||
1.8 | −2.31084 | 0.814794 | 3.33999 | −0.287417 | −1.88286 | −2.20640 | −3.09651 | −2.33611 | 0.664174 | ||||||||||||||||||
1.9 | −2.27621 | −2.60153 | 3.18111 | 2.30412 | 5.92161 | 1.78008 | −2.68845 | 3.76795 | −5.24464 | ||||||||||||||||||
1.10 | −2.26706 | −2.69699 | 3.13957 | −3.90758 | 6.11425 | −4.79951 | −2.58348 | 4.27377 | 8.85872 | ||||||||||||||||||
1.11 | −2.15921 | −0.135675 | 2.66221 | −3.14294 | 0.292951 | 0.215452 | −1.42985 | −2.98159 | 6.78628 | ||||||||||||||||||
1.12 | −2.11247 | 2.70421 | 2.46253 | 2.42834 | −5.71257 | 1.36575 | −0.977087 | 4.31276 | −5.12979 | ||||||||||||||||||
1.13 | −2.03156 | 3.25300 | 2.12723 | 1.52354 | −6.60865 | −4.26037 | −0.258468 | 7.58200 | −3.09515 | ||||||||||||||||||
1.14 | −1.99373 | −2.36120 | 1.97494 | −0.776635 | 4.70759 | −0.440394 | 0.0499623 | 2.57527 | 1.54840 | ||||||||||||||||||
1.15 | −1.96262 | −1.03474 | 1.85187 | 3.48986 | 2.03080 | −2.58926 | 0.290726 | −1.92931 | −6.84926 | ||||||||||||||||||
1.16 | −1.92648 | −3.23973 | 1.71134 | 1.34988 | 6.24129 | 3.96616 | 0.556094 | 7.49587 | −2.60052 | ||||||||||||||||||
1.17 | −1.88550 | 3.14155 | 1.55513 | −2.47668 | −5.92341 | 4.66231 | 0.838810 | 6.86935 | 4.66980 | ||||||||||||||||||
1.18 | −1.82466 | 0.0328358 | 1.32938 | −0.181740 | −0.0599140 | 4.41961 | 1.22366 | −2.99892 | 0.331613 | ||||||||||||||||||
1.19 | −1.67574 | 1.31893 | 0.808091 | −4.46158 | −2.21019 | 0.153470 | 1.99732 | −1.26041 | 7.47643 | ||||||||||||||||||
1.20 | −1.66354 | 1.93600 | 0.767368 | −2.27919 | −3.22061 | 0.280246 | 2.05053 | 0.748091 | 3.79153 | ||||||||||||||||||
See all 85 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.f | ✓ | 85 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
4033.2.a.f | ✓ | 85 | 1.a | even | 1 | 1 | trivial |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{85} - 11 T_{2}^{84} - 71 T_{2}^{83} + 1200 T_{2}^{82} + 1241 T_{2}^{81} - 61991 T_{2}^{80} + \cdots + 90944 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4033))\).