Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [8037,2,Mod(1,8037)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8037, 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("8037.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 8037 = 3^{2} \cdot 19 \cdot 47 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 8037.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: | \(64.1757681045\) |
Analytic rank: | \(0\) |
Dimension: | \(23\) |
Twist minimal: | no (minimal twist has level 893) |
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 | −2.71530 | 0 | 5.37287 | −1.00168 | 0 | 2.38392 | −9.15838 | 0 | 2.71986 | ||||||||||||||||||
1.2 | −2.62457 | 0 | 4.88835 | −0.357317 | 0 | −2.31246 | −7.58066 | 0 | 0.937801 | ||||||||||||||||||
1.3 | −2.39494 | 0 | 3.73572 | 3.66484 | 0 | 3.11663 | −4.15695 | 0 | −8.77707 | ||||||||||||||||||
1.4 | −2.37437 | 0 | 3.63763 | 1.61117 | 0 | 0.108357 | −3.88834 | 0 | −3.82552 | ||||||||||||||||||
1.5 | −2.06457 | 0 | 2.26244 | −3.06045 | 0 | 3.81585 | −0.541831 | 0 | 6.31850 | ||||||||||||||||||
1.6 | −1.73522 | 0 | 1.01098 | −2.41866 | 0 | 1.62998 | 1.71616 | 0 | 4.19690 | ||||||||||||||||||
1.7 | −1.67375 | 0 | 0.801455 | −3.83185 | 0 | −3.94220 | 2.00607 | 0 | 6.41357 | ||||||||||||||||||
1.8 | −1.28005 | 0 | −0.361479 | 4.05982 | 0 | −2.17032 | 3.02280 | 0 | −5.19677 | ||||||||||||||||||
1.9 | −1.15471 | 0 | −0.666651 | −2.72924 | 0 | 1.99299 | 3.07920 | 0 | 3.15148 | ||||||||||||||||||
1.10 | −0.772110 | 0 | −1.40385 | 2.30868 | 0 | 3.13314 | 2.62814 | 0 | −1.78256 | ||||||||||||||||||
1.11 | −0.512932 | 0 | −1.73690 | 0.842347 | 0 | −1.17187 | 1.91678 | 0 | −0.432067 | ||||||||||||||||||
1.12 | 0.102666 | 0 | −1.98946 | −0.421030 | 0 | 4.07188 | −0.409581 | 0 | −0.0432253 | ||||||||||||||||||
1.13 | 0.154056 | 0 | −1.97627 | −1.39588 | 0 | −0.0231293 | −0.612568 | 0 | −0.215045 | ||||||||||||||||||
1.14 | 0.581855 | 0 | −1.66145 | 2.54642 | 0 | −4.55068 | −2.13043 | 0 | 1.48165 | ||||||||||||||||||
1.15 | 1.00631 | 0 | −0.987335 | −4.09447 | 0 | 3.87614 | −3.00619 | 0 | −4.12032 | ||||||||||||||||||
1.16 | 1.23080 | 0 | −0.485141 | 3.43966 | 0 | 5.19980 | −3.05870 | 0 | 4.23352 | ||||||||||||||||||
1.17 | 1.45599 | 0 | 0.119916 | −1.65894 | 0 | 2.91693 | −2.73739 | 0 | −2.41540 | ||||||||||||||||||
1.18 | 1.55024 | 0 | 0.403250 | 0.718559 | 0 | −2.40005 | −2.47535 | 0 | 1.11394 | ||||||||||||||||||
1.19 | 2.30976 | 0 | 3.33497 | 0.940736 | 0 | 4.70927 | 3.08346 | 0 | 2.17287 | ||||||||||||||||||
1.20 | 2.31786 | 0 | 3.37246 | −4.06165 | 0 | −2.19787 | 3.18116 | 0 | −9.41433 | ||||||||||||||||||
See all 23 embeddings |
Atkin-Lehner signs
\( p \) | Sign |
---|---|
\(3\) | \(-1\) |
\(19\) | \(-1\) |
\(47\) | \(-1\) |
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 | 8037.2.a.r | 23 | |
3.b | odd | 2 | 1 | 893.2.a.d | ✓ | 23 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
893.2.a.d | ✓ | 23 | 3.b | odd | 2 | 1 | |
8037.2.a.r | 23 | 1.a | even | 1 | 1 | trivial |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(8037))\):
\( T_{2}^{23} + T_{2}^{22} - 38 T_{2}^{21} - 37 T_{2}^{20} + 622 T_{2}^{19} + 586 T_{2}^{18} - 5746 T_{2}^{17} - 5195 T_{2}^{16} + 32986 T_{2}^{15} + 28295 T_{2}^{14} - 122210 T_{2}^{13} - 97857 T_{2}^{12} + 294219 T_{2}^{11} + \cdots + 315 \) |
\( T_{5}^{23} + T_{5}^{22} - 80 T_{5}^{21} - 78 T_{5}^{20} + 2715 T_{5}^{19} + 2569 T_{5}^{18} - 51111 T_{5}^{17} - 46571 T_{5}^{16} + 586779 T_{5}^{15} + 508457 T_{5}^{14} - 4262394 T_{5}^{13} - 3446926 T_{5}^{12} + \cdots - 2408448 \) |