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: | \(110\) |
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 | −2.78685 | −0.498757 | 5.76655 | 1.84399 | 1.38996 | −1.25608 | −10.4968 | −2.75124 | −5.13894 | ||||||||||||||||||
1.2 | −2.77072 | 1.69880 | 5.67692 | −3.04963 | −4.70691 | 3.28914 | −10.1877 | −0.114069 | 8.44968 | ||||||||||||||||||
1.3 | −2.69148 | −2.62726 | 5.24405 | 0.394241 | 7.07121 | −0.479189 | −8.73128 | 3.90249 | −1.06109 | ||||||||||||||||||
1.4 | −2.67187 | 1.06249 | 5.13890 | 3.00541 | −2.83884 | 2.91292 | −8.38673 | −1.87111 | −8.03007 | ||||||||||||||||||
1.5 | −2.64482 | −1.59347 | 4.99505 | −2.42826 | 4.21443 | 4.09769 | −7.92136 | −0.460856 | 6.42231 | ||||||||||||||||||
1.6 | −2.64313 | 2.88413 | 4.98613 | −1.94351 | −7.62312 | −0.878417 | −7.89274 | 5.31819 | 5.13695 | ||||||||||||||||||
1.7 | −2.55998 | 2.69028 | 4.55348 | −2.87781 | −6.88705 | 3.57411 | −6.53686 | 4.23759 | 7.36714 | ||||||||||||||||||
1.8 | −2.51631 | 1.27091 | 4.33182 | 1.33976 | −3.19800 | 0.944830 | −5.86759 | −1.38479 | −3.37125 | ||||||||||||||||||
1.9 | −2.48517 | 3.11043 | 4.17608 | 3.13519 | −7.72995 | −2.09644 | −5.40793 | 6.67478 | −7.79149 | ||||||||||||||||||
1.10 | −2.45577 | −2.50707 | 4.03080 | −3.71982 | 6.15679 | −2.33961 | −4.98718 | 3.28541 | 9.13502 | ||||||||||||||||||
1.11 | −2.41585 | 0.542785 | 3.83634 | −2.85515 | −1.31129 | −2.66009 | −4.43632 | −2.70538 | 6.89763 | ||||||||||||||||||
1.12 | −2.35892 | −1.34501 | 3.56450 | 0.355993 | 3.17276 | −3.31316 | −3.69052 | −1.19096 | −0.839759 | ||||||||||||||||||
1.13 | −2.33551 | 0.518713 | 3.45460 | −1.17306 | −1.21146 | −2.50606 | −3.39724 | −2.73094 | 2.73970 | ||||||||||||||||||
1.14 | −2.29692 | −1.19284 | 3.27585 | 2.94374 | 2.73987 | 2.63830 | −2.93052 | −1.57712 | −6.76154 | ||||||||||||||||||
1.15 | −2.27127 | −0.567339 | 3.15866 | −3.83448 | 1.28858 | 2.52768 | −2.63164 | −2.67813 | 8.70914 | ||||||||||||||||||
1.16 | −2.17795 | 2.23759 | 2.74348 | 1.66421 | −4.87337 | −3.06154 | −1.61926 | 2.00681 | −3.62457 | ||||||||||||||||||
1.17 | −2.13628 | −0.861354 | 2.56370 | 1.50403 | 1.84010 | −3.07386 | −1.20423 | −2.25807 | −3.21303 | ||||||||||||||||||
1.18 | −2.12748 | −2.55539 | 2.52618 | 2.02951 | 5.43655 | −1.20737 | −1.11945 | 3.53003 | −4.31775 | ||||||||||||||||||
1.19 | −2.05303 | −2.17913 | 2.21492 | 2.99569 | 4.47382 | 4.05870 | −0.441239 | 1.74862 | −6.15023 | ||||||||||||||||||
1.20 | −1.98962 | 1.97988 | 1.95861 | 0.230568 | −3.93921 | −0.245933 | 0.0823553 | 0.919909 | −0.458744 | ||||||||||||||||||
See next 80 embeddings (of 110 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.c | ✓ | 110 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
6031.2.a.c | ✓ | 110 | 1.a | even | 1 | 1 | trivial |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{110} + 9 T_{2}^{109} - 118 T_{2}^{108} - 1284 T_{2}^{107} + 6219 T_{2}^{106} + 88433 T_{2}^{105} + \cdots - 1023360 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6031))\).