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: | \(134\) |
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 | −2.78315 | −1.65667 | 5.74594 | 3.71472 | 4.61077 | 4.59606 | −10.4255 | −0.255436 | −10.3386 | ||||||||||||||||||
1.2 | −2.77108 | 1.62557 | 5.67890 | −0.123356 | −4.50458 | −1.34606 | −10.1945 | −0.357536 | 0.341829 | ||||||||||||||||||
1.3 | −2.71257 | −2.51910 | 5.35803 | −0.492593 | 6.83324 | −2.96732 | −9.10890 | 3.34588 | 1.33619 | ||||||||||||||||||
1.4 | −2.70867 | −0.769737 | 5.33687 | −2.82113 | 2.08496 | −1.98436 | −9.03847 | −2.40750 | 7.64150 | ||||||||||||||||||
1.5 | −2.62103 | −1.46003 | 4.86979 | 3.30025 | 3.82679 | −2.17387 | −7.52179 | −0.868300 | −8.65005 | ||||||||||||||||||
1.6 | −2.56681 | 3.22691 | 4.58852 | 1.32695 | −8.28287 | 4.31369 | −6.64424 | 7.41296 | −3.40604 | ||||||||||||||||||
1.7 | −2.54672 | −0.772659 | 4.48577 | 3.37038 | 1.96774 | 0.335508 | −6.33055 | −2.40300 | −8.58341 | ||||||||||||||||||
1.8 | −2.53107 | −0.695993 | 4.40632 | 1.03970 | 1.76161 | 0.140699 | −6.09057 | −2.51559 | −2.63157 | ||||||||||||||||||
1.9 | −2.49038 | 1.16097 | 4.20199 | 0.527322 | −2.89126 | 0.382261 | −5.48378 | −1.65215 | −1.31323 | ||||||||||||||||||
1.10 | −2.47900 | 1.64888 | 4.14545 | −0.897925 | −4.08758 | 2.76964 | −5.31858 | −0.281188 | 2.22596 | ||||||||||||||||||
1.11 | −2.47262 | 1.83929 | 4.11384 | −2.27118 | −4.54786 | −1.50415 | −5.22672 | 0.382983 | 5.61575 | ||||||||||||||||||
1.12 | −2.43815 | −2.94184 | 3.94457 | −1.35496 | 7.17265 | −0.834817 | −4.74114 | 5.65444 | 3.30360 | ||||||||||||||||||
1.13 | −2.41121 | 3.43364 | 3.81394 | −3.49586 | −8.27924 | 1.03348 | −4.37378 | 8.78990 | 8.42926 | ||||||||||||||||||
1.14 | −2.39957 | 0.225532 | 3.75794 | −1.91995 | −0.541181 | −3.36005 | −4.21829 | −2.94914 | 4.60705 | ||||||||||||||||||
1.15 | −2.39066 | −3.44347 | 3.71525 | −1.21628 | 8.23217 | −4.84430 | −4.10059 | 8.85751 | 2.90771 | ||||||||||||||||||
1.16 | −2.25933 | −1.97048 | 3.10458 | −1.05665 | 4.45197 | 1.22634 | −2.49561 | 0.882799 | 2.38733 | ||||||||||||||||||
1.17 | −2.23037 | 2.48760 | 2.97457 | 3.37158 | −5.54828 | −3.17676 | −2.17365 | 3.18815 | −7.51989 | ||||||||||||||||||
1.18 | −2.21168 | 1.70673 | 2.89152 | 4.27910 | −3.77473 | 1.99718 | −1.97176 | −0.0870880 | −9.46398 | ||||||||||||||||||
1.19 | −2.16741 | 3.17923 | 2.69765 | 1.60787 | −6.89069 | 1.52618 | −1.51209 | 7.10752 | −3.48491 | ||||||||||||||||||
1.20 | −2.07881 | 1.94163 | 2.32147 | −0.214355 | −4.03630 | −5.01421 | −0.668275 | 0.769946 | 0.445604 | ||||||||||||||||||
See next 80 embeddings (of 134 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.e | ✓ | 134 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
6031.2.a.e | ✓ | 134 | 1.a | even | 1 | 1 | trivial |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{134} - 9 T_{2}^{133} - 168 T_{2}^{132} + 1734 T_{2}^{131} + 13227 T_{2}^{130} + \cdots - 1624320801696 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6031))\).