Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [103,18,Mod(1,103)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(103, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0]))
N = Newforms(chi, 18, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("103.1");
S:= CuspForms(chi, 18);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 103 \) |
Weight: | \( k \) | \(=\) | \( 18 \) |
Character orbit: | \([\chi]\) | \(=\) | 103.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: | \(188.718749965\) |
Analytic rank: | \(0\) |
Dimension: | \(75\) |
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 | −684.133 | 15587.3 | 336966. | 1.49942e6 | −1.06638e7 | −1.70692e6 | −1.40859e8 | 1.13822e8 | −1.02580e9 | ||||||||||||||||||
1.2 | −683.258 | 6745.27 | 335769. | −704392. | −4.60876e6 | −1.06552e7 | −1.39861e8 | −8.36414e7 | 4.81281e8 | ||||||||||||||||||
1.3 | −681.827 | 15597.7 | 333816. | 239807. | −1.06349e7 | −2.04143e7 | −1.38236e8 | 1.14149e8 | −1.63507e8 | ||||||||||||||||||
1.4 | −660.506 | 899.492 | 305196. | 178972. | −594120. | −2.07107e7 | −1.15010e8 | −1.28331e8 | −1.18212e8 | ||||||||||||||||||
1.5 | −655.732 | −19150.8 | 298912. | −43957.1 | 1.25578e7 | −3.55696e6 | −1.10058e8 | 2.37613e8 | 2.88241e7 | ||||||||||||||||||
1.6 | −644.220 | 282.671 | 283947. | −178170. | −182102. | 2.20767e7 | −9.84850e7 | −1.29060e8 | 1.14781e8 | ||||||||||||||||||
1.7 | −619.709 | 12155.7 | 252967. | −668724. | −7.53303e6 | 1.48524e7 | −7.55395e7 | 1.86221e7 | 4.14414e8 | ||||||||||||||||||
1.8 | −607.179 | −15256.1 | 237594. | −1.13942e6 | 9.26320e6 | −2.76514e7 | −6.46780e7 | 1.03609e8 | 6.91832e8 | ||||||||||||||||||
1.9 | −565.256 | −19369.3 | 188442. | −244340. | 1.09486e7 | 2.06638e7 | −3.24290e7 | 2.46030e8 | 1.38115e8 | ||||||||||||||||||
1.10 | −560.848 | −13739.1 | 183478. | 964315. | 7.70553e6 | 1.95070e7 | −2.93921e7 | 5.96220e7 | −5.40834e8 | ||||||||||||||||||
1.11 | −524.816 | −21497.2 | 144360. | 1.49165e6 | 1.12821e7 | −1.60270e7 | −6.97353e6 | 3.32989e8 | −7.82843e8 | ||||||||||||||||||
1.12 | −505.425 | 12282.5 | 124383. | −1.08481e6 | −6.20786e6 | 1.56269e7 | 3.38103e6 | 2.17187e7 | 5.48290e8 | ||||||||||||||||||
1.13 | −484.615 | 17274.0 | 103780. | 1.05960e6 | −8.37122e6 | 34849.8 | 1.32263e7 | 1.69250e8 | −5.13500e8 | ||||||||||||||||||
1.14 | −472.360 | 8120.08 | 92052.4 | 863260. | −3.83561e6 | −1.83518e7 | 1.84313e7 | −6.32044e7 | −4.07770e8 | ||||||||||||||||||
1.15 | −457.450 | −9015.89 | 78188.7 | −815945. | 4.12432e6 | −2.21178e6 | 2.41915e7 | −4.78539e7 | 3.73254e8 | ||||||||||||||||||
1.16 | −429.129 | −9834.29 | 53079.9 | 480699. | 4.22018e6 | 2.34102e7 | 3.34687e7 | −3.24269e7 | −2.06282e8 | ||||||||||||||||||
1.17 | −418.493 | −3327.56 | 44064.0 | 777397. | 1.39256e6 | −1.89604e7 | 3.64122e7 | −1.18068e8 | −3.25335e8 | ||||||||||||||||||
1.18 | −383.136 | 4061.48 | 15721.5 | 623786. | −1.55610e6 | 6.34260e6 | 4.41950e7 | −1.12645e8 | −2.38995e8 | ||||||||||||||||||
1.19 | −381.945 | 21120.0 | 14809.7 | 834842. | −8.06668e6 | 2.86828e7 | 4.44058e7 | 3.16915e8 | −3.18863e8 | ||||||||||||||||||
1.20 | −379.273 | 2709.91 | 12775.7 | −1.61373e6 | −1.02779e6 | 2.70681e7 | 4.48665e7 | −1.21797e8 | 6.12042e8 | ||||||||||||||||||
See all 75 embeddings |
Atkin-Lehner signs
\( p \) | Sign |
---|---|
\(103\) | \(-1\) |
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 | 103.18.a.b | ✓ | 75 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
103.18.a.b | ✓ | 75 | 1.a | even | 1 | 1 | trivial |