Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [21,30,Mod(4,21)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(21, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([0, 4]))
N = Newforms(chi, 30, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("21.4");
S:= CuspForms(chi, 30);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 21 = 3 \cdot 7 \) |
Weight: | \( k \) | \(=\) | \( 30 \) |
Character orbit: | \([\chi]\) | \(=\) | 21.e (of order \(3\), degree \(2\), minimal) |
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: | no |
Analytic conductor: | \(111.883889004\) |
Analytic rank: | \(0\) |
Dimension: | \(38\) |
Relative dimension: | \(19\) over \(\Q(\zeta_{3})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{3}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
4.1 | −21682.7 | − | 37555.5i | 2.39148e6 | − | 4.14217e6i | −6.71844e8 | + | 1.16367e9i | −3.79032e9 | − | 6.56503e9i | −2.07415e11 | −1.42775e12 | + | 1.08693e12i | 3.49879e13 | −1.14384e13 | − | 1.98119e13i | −1.64369e14 | + | 2.84695e14i | ||||
4.2 | −19318.8 | − | 33461.1i | 2.39148e6 | − | 4.14217e6i | −4.77996e8 | + | 8.27914e8i | −7.79808e8 | − | 1.35067e9i | −1.84802e11 | −1.66873e11 | − | 1.78663e12i | 1.61939e13 | −1.14384e13 | − | 1.98119e13i | −3.01299e13 | + | 5.21865e13i | ||||
4.3 | −19095.2 | − | 33073.8i | 2.39148e6 | − | 4.14217e6i | −4.60816e8 | + | 7.98156e8i | 9.98170e9 | + | 1.72888e10i | −1.82663e11 | 1.32001e12 | + | 1.21552e12i | 1.46941e13 | −1.14384e13 | − | 1.98119e13i | 3.81205e14 | − | 6.60266e14i | ||||
4.4 | −13945.7 | − | 24154.6i | 2.39148e6 | − | 4.14217e6i | −1.20528e8 | + | 2.08761e8i | −1.12432e10 | − | 1.94737e10i | −1.33403e11 | 1.74806e12 | + | 4.05197e11i | −8.25066e12 | −1.14384e13 | − | 1.98119e13i | −3.13587e14 | + | 5.43149e14i | ||||
4.5 | −13747.8 | − | 23811.9i | 2.39148e6 | − | 4.14217e6i | −1.09569e8 | + | 1.89779e8i | −5.45988e9 | − | 9.45678e9i | −1.31511e11 | −1.44405e12 | + | 1.06518e12i | −8.73628e12 | −1.14384e13 | − | 1.98119e13i | −1.50123e14 | + | 2.60020e14i | ||||
4.6 | −10085.3 | − | 17468.3i | 2.39148e6 | − | 4.14217e6i | 6.50070e7 | − | 1.12595e8i | 4.83495e9 | + | 8.37438e9i | −9.64758e10 | −7.77849e11 | + | 1.61705e12i | −1.34515e13 | −1.14384e13 | − | 1.98119e13i | 9.75244e13 | − | 1.68917e14i | ||||
4.7 | −9506.59 | − | 16465.9i | 2.39148e6 | − | 4.14217e6i | 8.76848e7 | − | 1.51874e8i | 4.01268e9 | + | 6.95016e9i | −9.09395e10 | 1.51909e12 | − | 9.55123e11i | −1.35420e13 | −1.14384e13 | − | 1.98119e13i | 7.62938e13 | − | 1.32145e14i | ||||
4.8 | −5312.73 | − | 9201.91i | 2.39148e6 | − | 4.14217e6i | 2.11985e8 | − | 3.67169e8i | 6.21129e8 | + | 1.07583e9i | −5.08212e10 | −1.19318e12 | − | 1.34023e12i | −1.02094e13 | −1.14384e13 | − | 1.98119e13i | 6.59978e12 | − | 1.14312e13i | ||||
4.9 | −803.934 | − | 1392.45i | 2.39148e6 | − | 4.14217e6i | 2.67143e8 | − | 4.62705e8i | −1.04266e10 | − | 1.80594e10i | −7.69038e9 | −3.52235e11 | − | 1.75950e12i | −1.72228e12 | −1.14384e13 | − | 1.98119e13i | −1.67646e13 | + | 2.90371e13i | ||||
4.10 | 1314.29 | + | 2276.42i | 2.39148e6 | − | 4.14217e6i | 2.64981e8 | − | 4.58960e8i | 1.11285e10 | + | 1.92751e10i | 1.25724e10 | −1.79440e12 | + | 6.33421e9i | 2.80425e12 | −1.14384e13 | − | 1.98119e13i | −2.92521e13 | + | 5.06661e13i | ||||
4.11 | 3171.13 | + | 5492.56i | 2.39148e6 | − | 4.14217e6i | 2.48323e8 | − | 4.30109e8i | −5.04465e9 | − | 8.73759e9i | 3.03349e10 | 7.64524e11 | + | 1.62339e12i | 6.55484e12 | −1.14384e13 | − | 1.98119e13i | 3.19945e13 | − | 5.54161e13i | ||||
4.12 | 4961.84 | + | 8594.16i | 2.39148e6 | − | 4.14217e6i | 2.19196e8 | − | 3.79658e8i | 8.80219e9 | + | 1.52458e10i | 4.74646e10 | 1.78765e12 | + | 1.55597e11i | 9.67819e12 | −1.14384e13 | − | 1.98119e13i | −8.73501e13 | + | 1.51295e14i | ||||
4.13 | 10661.0 | + | 18465.4i | 2.39148e6 | − | 4.14217e6i | 4.11213e7 | − | 7.12241e7i | −5.16991e9 | − | 8.95455e9i | 1.01983e11 | −1.66601e12 | + | 6.66564e11i | 1.32007e13 | −1.14384e13 | − | 1.98119e13i | 1.10233e14 | − | 1.90929e14i | ||||
4.14 | 11213.9 | + | 19423.1i | 2.39148e6 | − | 4.14217e6i | 1.69302e7 | − | 2.93240e7i | 4.19413e9 | + | 7.26445e9i | 1.07272e11 | 4.56481e11 | − | 1.73538e12i | 1.28003e13 | −1.14384e13 | − | 1.98119e13i | −9.40656e13 | + | 1.62926e14i | ||||
4.15 | 14667.6 | + | 25405.0i | 2.39148e6 | − | 4.14217e6i | −1.61841e8 | + | 2.80316e8i | −8.78577e9 | − | 1.52174e10i | 1.40309e11 | 1.75086e12 | − | 3.92932e11i | 6.25395e12 | −1.14384e13 | − | 1.98119e13i | 2.57732e14 | − | 4.46405e14i | ||||
4.16 | 17773.2 | + | 30784.0i | 2.39148e6 | − | 4.14217e6i | −3.63334e8 | + | 6.29314e8i | 6.21359e9 | + | 1.07623e10i | 1.70017e11 | −3.33772e10 | + | 1.79410e12i | −6.74661e12 | −1.14384e13 | − | 1.98119e13i | −2.20870e14 | + | 3.82558e14i | ||||
4.17 | 20203.6 | + | 34993.6i | 2.39148e6 | − | 4.14217e6i | −5.47933e8 | + | 9.49048e8i | 5.22782e7 | + | 9.05485e7i | 1.93266e11 | 1.78686e12 | + | 1.64388e11i | −2.25874e13 | −1.14384e13 | − | 1.98119e13i | −2.11241e12 | + | 3.65880e12i | ||||
4.18 | 20330.6 | + | 35213.6i | 2.39148e6 | − | 4.14217e6i | −5.58229e8 | + | 9.66881e8i | 9.23100e9 | + | 1.59886e10i | 1.94481e11 | −1.57660e12 | − | 8.56870e11i | −2.35667e13 | −1.14384e13 | − | 1.98119e13i | −3.75343e14 | + | 6.50113e14i | ||||
4.19 | 21714.1 | + | 37610.0i | 2.39148e6 | − | 4.14217e6i | −6.74573e8 | + | 1.16839e9i | −9.42284e9 | − | 1.63208e10i | 2.07716e11 | −1.27489e12 | − | 1.26276e12i | −3.52757e13 | −1.14384e13 | − | 1.98119e13i | 4.09218e14 | − | 7.08786e14i | ||||
16.1 | −21682.7 | + | 37555.5i | 2.39148e6 | + | 4.14217e6i | −6.71844e8 | − | 1.16367e9i | −3.79032e9 | + | 6.56503e9i | −2.07415e11 | −1.42775e12 | − | 1.08693e12i | 3.49879e13 | −1.14384e13 | + | 1.98119e13i | −1.64369e14 | − | 2.84695e14i | ||||
See all 38 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
7.c | even | 3 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 21.30.e.a | ✓ | 38 |
7.c | even | 3 | 1 | inner | 21.30.e.a | ✓ | 38 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
21.30.e.a | ✓ | 38 | 1.a | even | 1 | 1 | trivial |
21.30.e.a | ✓ | 38 | 7.c | even | 3 | 1 | inner |