Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [6031,2,Mod(1,6031)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6031, 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("6031.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 6031 = 37 \cdot 163 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 6031.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: | \(48.1577774590\) |
Analytic rank: | \(0\) |
Dimension: | \(133\) |
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 | −2.76277 | 2.80088 | 5.63291 | −0.925460 | −7.73820 | −3.46221 | −10.0369 | 4.84493 | 2.55683 | ||||||||||||||||||
1.2 | −2.74334 | −3.12442 | 5.52594 | 0.187024 | 8.57137 | 1.31323 | −9.67287 | 6.76201 | −0.513072 | ||||||||||||||||||
1.3 | −2.65451 | 1.31557 | 5.04644 | 0.400622 | −3.49219 | 1.95447 | −8.08682 | −1.26929 | −1.06346 | ||||||||||||||||||
1.4 | −2.64085 | 0.791428 | 4.97411 | −0.110767 | −2.09005 | −4.52560 | −7.85420 | −2.37364 | 0.292520 | ||||||||||||||||||
1.5 | −2.62634 | −3.19656 | 4.89764 | 4.04178 | 8.39525 | −2.68408 | −7.61017 | 7.21801 | −10.6151 | ||||||||||||||||||
1.6 | −2.59785 | −1.74392 | 4.74884 | 1.33654 | 4.53044 | −2.87132 | −7.14109 | 0.0412485 | −3.47214 | ||||||||||||||||||
1.7 | −2.58445 | 2.91776 | 4.67939 | 3.62109 | −7.54082 | 1.31660 | −6.92475 | 5.51335 | −9.35852 | ||||||||||||||||||
1.8 | −2.55998 | 1.31640 | 4.55348 | 3.87746 | −3.36994 | 3.54726 | −6.53684 | −1.26710 | −9.92621 | ||||||||||||||||||
1.9 | −2.55877 | −0.511653 | 4.54728 | −3.89293 | 1.30920 | 1.47164 | −6.51789 | −2.73821 | 9.96108 | ||||||||||||||||||
1.10 | −2.45235 | −2.50002 | 4.01402 | −3.16357 | 6.13092 | −0.556288 | −4.93908 | 3.25009 | 7.75817 | ||||||||||||||||||
1.11 | −2.39714 | 0.221428 | 3.74630 | −0.354828 | −0.530796 | 3.04567 | −4.18612 | −2.95097 | 0.850573 | ||||||||||||||||||
1.12 | −2.38561 | −1.27766 | 3.69114 | 1.05833 | 3.04800 | 3.63452 | −4.03440 | −1.36759 | −2.52477 | ||||||||||||||||||
1.13 | −2.36309 | −0.0862351 | 3.58421 | −0.328445 | 0.203782 | −0.467187 | −3.74364 | −2.99256 | 0.776147 | ||||||||||||||||||
1.14 | −2.30996 | −1.13672 | 3.33593 | 4.36294 | 2.62577 | −2.53540 | −3.08596 | −1.70788 | −10.0782 | ||||||||||||||||||
1.15 | −2.29540 | 2.78712 | 3.26888 | −0.0528483 | −6.39757 | 2.45561 | −2.91258 | 4.76805 | 0.121308 | ||||||||||||||||||
1.16 | −2.27893 | 2.16909 | 3.19353 | 2.05388 | −4.94321 | −4.20533 | −2.71998 | 1.70495 | −4.68065 | ||||||||||||||||||
1.17 | −2.27645 | −1.27799 | 3.18222 | 2.39479 | 2.90928 | 2.25162 | −2.69127 | −1.36674 | −5.45162 | ||||||||||||||||||
1.18 | −2.24334 | −3.44407 | 3.03258 | −0.143577 | 7.72622 | 5.03410 | −2.31642 | 8.86161 | 0.322092 | ||||||||||||||||||
1.19 | −2.20408 | 0.816725 | 2.85795 | −2.28441 | −1.80012 | 0.445697 | −1.89098 | −2.33296 | 5.03501 | ||||||||||||||||||
1.20 | −2.20055 | −1.32438 | 2.84242 | −2.94026 | 2.91437 | −4.27239 | −1.85380 | −1.24601 | 6.47020 | ||||||||||||||||||
See next 80 embeddings (of 133 total) |
Atkin-Lehner signs
\( p \) | Sign |
---|---|
\(37\) | \(1\) |
\(163\) | \(-1\) |
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 | 6031.2.a.d | ✓ | 133 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
6031.2.a.d | ✓ | 133 | 1.a | even | 1 | 1 | trivial |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{133} - 14 T_{2}^{132} - 106 T_{2}^{131} + 2366 T_{2}^{130} + 2353 T_{2}^{129} - 191882 T_{2}^{128} + \cdots + 5092864 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6031))\).