Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [95,4,Mod(6,95)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(95, base_ring=CyclotomicField(18))
chi = DirichletCharacter(H, H._module([0, 14]))
N = Newforms(chi, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("95.6");
S:= CuspForms(chi, 4);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 95 = 5 \cdot 19 \) |
Weight: | \( k \) | \(=\) | \( 4 \) |
Character orbit: | \([\chi]\) | \(=\) | 95.k (of order \(9\), degree \(6\), 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: | \(5.60518145055\) |
Analytic rank: | \(0\) |
Dimension: | \(60\) |
Relative dimension: | \(10\) over \(\Q(\zeta_{9})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{9}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
6.1 | −4.11693 | + | 3.45451i | −4.83870 | − | 1.76114i | 3.62625 | − | 20.5655i | −0.868241 | − | 4.92404i | 26.0045 | − | 9.46486i | −0.675317 | + | 1.16968i | 34.6177 | + | 59.9597i | −0.371804 | − | 0.311981i | 20.5846 | + | 17.2726i |
6.2 | −3.16499 | + | 2.65575i | 3.89076 | + | 1.41612i | 1.57502 | − | 8.93238i | −0.868241 | − | 4.92404i | −16.0751 | + | 5.85086i | 9.34491 | − | 16.1859i | 2.21078 | + | 3.82918i | −7.55056 | − | 6.33568i | 15.8250 | + | 13.2787i |
6.3 | −2.84678 | + | 2.38873i | 6.40608 | + | 2.33162i | 1.00893 | − | 5.72194i | −0.868241 | − | 4.92404i | −23.8063 | + | 8.66480i | −13.6371 | + | 23.6201i | −4.06886 | − | 7.04748i | 14.9182 | + | 12.5179i | 14.2339 | + | 11.9437i |
6.4 | −1.97601 | + | 1.65807i | −3.68433 | − | 1.34099i | −0.233758 | + | 1.32571i | −0.868241 | − | 4.92404i | 9.50376 | − | 3.45908i | −0.488995 | + | 0.846964i | −12.0542 | − | 20.8785i | −8.90713 | − | 7.47397i | 9.88007 | + | 8.29037i |
6.5 | −0.218503 | + | 0.183346i | −1.99430 | − | 0.725867i | −1.37506 | + | 7.79834i | −0.868241 | − | 4.92404i | 0.568845 | − | 0.207043i | −1.41581 | + | 2.45225i | −2.27028 | − | 3.93223i | −17.2328 | − | 14.4601i | 1.09251 | + | 0.916728i |
6.6 | 0.478693 | − | 0.401671i | 8.62374 | + | 3.13878i | −1.32138 | + | 7.49391i | −0.868241 | − | 4.92404i | 5.38888 | − | 1.96139i | 7.60280 | − | 13.1684i | 4.87711 | + | 8.44740i | 43.8337 | + | 36.7808i | −2.39347 | − | 2.00836i |
6.7 | 1.29856 | − | 1.08962i | −8.85206 | − | 3.22189i | −0.890203 | + | 5.04859i | −0.868241 | − | 4.92404i | −15.0055 | + | 5.46157i | 3.60688 | − | 6.24730i | 11.1257 | + | 19.2702i | 47.2952 | + | 39.6854i | −6.49279 | − | 5.44810i |
6.8 | 2.39352 | − | 2.00841i | −0.310724 | − | 0.113094i | 0.306081 | − | 1.73587i | −0.868241 | − | 4.92404i | −0.970865 | + | 0.353366i | 18.0464 | − | 31.2572i | 9.74437 | + | 16.8777i | −20.5994 | − | 17.2850i | −11.9676 | − | 10.0420i |
6.9 | 3.58404 | − | 3.00736i | 5.73918 | + | 2.08889i | 2.41190 | − | 13.6786i | −0.868241 | − | 4.92404i | 26.8515 | − | 9.77314i | −4.62023 | + | 8.00248i | −13.7775 | − | 23.8634i | 7.89151 | + | 6.62176i | −17.9202 | − | 15.0368i |
6.10 | 3.77028 | − | 3.16364i | −6.41934 | − | 2.33645i | 2.81719 | − | 15.9771i | −0.868241 | − | 4.92404i | −31.5943 | + | 11.4994i | −7.43684 | + | 12.8810i | −20.2372 | − | 35.0518i | 15.0657 | + | 12.6416i | −18.8514 | − | 15.8182i |
16.1 | −4.11693 | − | 3.45451i | −4.83870 | + | 1.76114i | 3.62625 | + | 20.5655i | −0.868241 | + | 4.92404i | 26.0045 | + | 9.46486i | −0.675317 | − | 1.16968i | 34.6177 | − | 59.9597i | −0.371804 | + | 0.311981i | 20.5846 | − | 17.2726i |
16.2 | −3.16499 | − | 2.65575i | 3.89076 | − | 1.41612i | 1.57502 | + | 8.93238i | −0.868241 | + | 4.92404i | −16.0751 | − | 5.85086i | 9.34491 | + | 16.1859i | 2.21078 | − | 3.82918i | −7.55056 | + | 6.33568i | 15.8250 | − | 13.2787i |
16.3 | −2.84678 | − | 2.38873i | 6.40608 | − | 2.33162i | 1.00893 | + | 5.72194i | −0.868241 | + | 4.92404i | −23.8063 | − | 8.66480i | −13.6371 | − | 23.6201i | −4.06886 | + | 7.04748i | 14.9182 | − | 12.5179i | 14.2339 | − | 11.9437i |
16.4 | −1.97601 | − | 1.65807i | −3.68433 | + | 1.34099i | −0.233758 | − | 1.32571i | −0.868241 | + | 4.92404i | 9.50376 | + | 3.45908i | −0.488995 | − | 0.846964i | −12.0542 | + | 20.8785i | −8.90713 | + | 7.47397i | 9.88007 | − | 8.29037i |
16.5 | −0.218503 | − | 0.183346i | −1.99430 | + | 0.725867i | −1.37506 | − | 7.79834i | −0.868241 | + | 4.92404i | 0.568845 | + | 0.207043i | −1.41581 | − | 2.45225i | −2.27028 | + | 3.93223i | −17.2328 | + | 14.4601i | 1.09251 | − | 0.916728i |
16.6 | 0.478693 | + | 0.401671i | 8.62374 | − | 3.13878i | −1.32138 | − | 7.49391i | −0.868241 | + | 4.92404i | 5.38888 | + | 1.96139i | 7.60280 | + | 13.1684i | 4.87711 | − | 8.44740i | 43.8337 | − | 36.7808i | −2.39347 | + | 2.00836i |
16.7 | 1.29856 | + | 1.08962i | −8.85206 | + | 3.22189i | −0.890203 | − | 5.04859i | −0.868241 | + | 4.92404i | −15.0055 | − | 5.46157i | 3.60688 | + | 6.24730i | 11.1257 | − | 19.2702i | 47.2952 | − | 39.6854i | −6.49279 | + | 5.44810i |
16.8 | 2.39352 | + | 2.00841i | −0.310724 | + | 0.113094i | 0.306081 | + | 1.73587i | −0.868241 | + | 4.92404i | −0.970865 | − | 0.353366i | 18.0464 | + | 31.2572i | 9.74437 | − | 16.8777i | −20.5994 | + | 17.2850i | −11.9676 | + | 10.0420i |
16.9 | 3.58404 | + | 3.00736i | 5.73918 | − | 2.08889i | 2.41190 | + | 13.6786i | −0.868241 | + | 4.92404i | 26.8515 | + | 9.77314i | −4.62023 | − | 8.00248i | −13.7775 | + | 23.8634i | 7.89151 | − | 6.62176i | −17.9202 | + | 15.0368i |
16.10 | 3.77028 | + | 3.16364i | −6.41934 | + | 2.33645i | 2.81719 | + | 15.9771i | −0.868241 | + | 4.92404i | −31.5943 | − | 11.4994i | −7.43684 | − | 12.8810i | −20.2372 | + | 35.0518i | 15.0657 | − | 12.6416i | −18.8514 | + | 15.8182i |
See all 60 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
19.e | even | 9 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 95.4.k.b | ✓ | 60 |
19.e | even | 9 | 1 | inner | 95.4.k.b | ✓ | 60 |
19.e | even | 9 | 1 | 1805.4.a.x | 30 | ||
19.f | odd | 18 | 1 | 1805.4.a.y | 30 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
95.4.k.b | ✓ | 60 | 1.a | even | 1 | 1 | trivial |
95.4.k.b | ✓ | 60 | 19.e | even | 9 | 1 | inner |
1805.4.a.x | 30 | 19.e | even | 9 | 1 | ||
1805.4.a.y | 30 | 19.f | odd | 18 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{60} - 9 T_{2}^{59} + 36 T_{2}^{58} - 93 T_{2}^{57} + 411 T_{2}^{56} - 3450 T_{2}^{55} + \cdots + 11\!\cdots\!04 \) acting on \(S_{4}^{\mathrm{new}}(95, [\chi])\).