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: | \(1\) |
Dimension: | \(109\) |
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 | −2.78763 | −0.692231 | 5.77089 | −0.511620 | 1.92969 | 1.79055 | −10.5119 | −2.52082 | 1.42621 | ||||||||||||||||||
1.2 | −2.71013 | 0.417619 | 5.34478 | −3.36649 | −1.13180 | −0.322957 | −9.06478 | −2.82559 | 9.12361 | ||||||||||||||||||
1.3 | −2.70989 | −1.09683 | 5.34348 | −2.68411 | 2.97228 | −1.74258 | −9.06044 | −1.79697 | 7.27363 | ||||||||||||||||||
1.4 | −2.69025 | 1.94949 | 5.23745 | 2.83503 | −5.24461 | −1.45115 | −8.70956 | 0.800495 | −7.62693 | ||||||||||||||||||
1.5 | −2.68695 | 0.942644 | 5.21970 | 3.35356 | −2.53284 | −4.89514 | −8.65118 | −2.11142 | −9.01084 | ||||||||||||||||||
1.6 | −2.61098 | 3.10227 | 4.81723 | −0.856662 | −8.09998 | −1.38221 | −7.35575 | 6.62408 | 2.23673 | ||||||||||||||||||
1.7 | −2.58083 | 0.172052 | 4.66069 | 0.569694 | −0.444038 | 4.52515 | −6.86679 | −2.97040 | −1.47028 | ||||||||||||||||||
1.8 | −2.49637 | −1.90525 | 4.23188 | 1.13152 | 4.75621 | −4.79924 | −5.57162 | 0.629973 | −2.82469 | ||||||||||||||||||
1.9 | −2.45790 | −2.51992 | 4.04129 | 3.33458 | 6.19373 | 1.77040 | −5.01729 | 3.35002 | −8.19607 | ||||||||||||||||||
1.10 | −2.45032 | −2.60360 | 4.00407 | −3.01282 | 6.37966 | 3.77378 | −4.91062 | 3.77875 | 7.38237 | ||||||||||||||||||
1.11 | −2.40987 | −2.85971 | 3.80748 | 1.79940 | 6.89154 | 2.51983 | −4.35580 | 5.17794 | −4.33633 | ||||||||||||||||||
1.12 | −2.39817 | 2.15006 | 3.75121 | −1.82413 | −5.15620 | 0.948079 | −4.19968 | 1.62275 | 4.37458 | ||||||||||||||||||
1.13 | −2.35175 | −1.69234 | 3.53073 | −3.31411 | 3.97997 | −2.65791 | −3.59990 | −0.135977 | 7.79397 | ||||||||||||||||||
1.14 | −2.30236 | 0.992330 | 3.30086 | 2.64832 | −2.28470 | −0.416453 | −2.99506 | −2.01528 | −6.09738 | ||||||||||||||||||
1.15 | −2.20321 | −0.260570 | 2.85413 | 0.532828 | 0.574090 | 4.62952 | −1.88183 | −2.93210 | −1.17393 | ||||||||||||||||||
1.16 | −2.18825 | 1.71398 | 2.78846 | 0.914186 | −3.75063 | 0.835020 | −1.72535 | −0.0622604 | −2.00047 | ||||||||||||||||||
1.17 | −2.14221 | 1.83370 | 2.58905 | −3.64172 | −3.92816 | 3.71202 | −1.26186 | 0.362447 | 7.80132 | ||||||||||||||||||
1.18 | −2.08654 | −0.221531 | 2.35365 | 3.05486 | 0.462234 | −0.0600719 | −0.737910 | −2.95092 | −6.37409 | ||||||||||||||||||
1.19 | −2.05458 | 0.484378 | 2.22130 | −1.81809 | −0.995193 | −4.27361 | −0.454677 | −2.76538 | 3.73540 | ||||||||||||||||||
1.20 | −2.05386 | −1.85254 | 2.21834 | −0.958510 | 3.80486 | 0.0563033 | −0.448438 | 0.431918 | 1.96864 | ||||||||||||||||||
See next 80 embeddings (of 109 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.b | ✓ | 109 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
6031.2.a.b | ✓ | 109 | 1.a | even | 1 | 1 | trivial |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{109} + 11 T_{2}^{108} - 98 T_{2}^{107} - 1498 T_{2}^{106} + 3459 T_{2}^{105} + 97915 T_{2}^{104} + \cdots - 1077199680 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6031))\).