Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [121,4,Mod(3,121)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(121, base_ring=CyclotomicField(10))
chi = DirichletCharacter(H, H._module([8]))
N = Newforms(chi, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("121.3");
S:= CuspForms(chi, 4);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 121 = 11^{2} \) |
Weight: | \( k \) | \(=\) | \( 4 \) |
Character orbit: | \([\chi]\) | \(=\) | 121.c (of order \(5\), degree \(4\), not 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: | \(7.13923111069\) |
Analytic rank: | \(0\) |
Dimension: | \(24\) |
Relative dimension: | \(6\) over \(\Q(\zeta_{5})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{5}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
3.1 | −3.62629 | + | 2.63465i | 0.719300 | + | 2.21378i | 3.73643 | − | 11.4995i | −15.2540 | − | 11.0827i | −8.44091 | − | 6.13268i | −5.80624 | + | 17.8698i | 5.66698 | + | 17.4412i | 17.4600 | − | 12.6855i | 84.5143 | ||
3.2 | −2.93750 | + | 2.13422i | 2.92726 | + | 9.00917i | 1.60188 | − | 4.93009i | 1.70313 | + | 1.23740i | −27.8264 | − | 20.2170i | −3.03234 | + | 9.33259i | −3.15984 | − | 9.72499i | −50.7528 | + | 36.8741i | −7.64382 | ||
3.3 | −0.688785 | + | 0.500431i | −2.41049 | − | 7.41872i | −2.24814 | + | 6.91907i | 7.88774 | + | 5.73078i | 5.37287 | + | 3.90362i | 7.93077 | − | 24.4084i | −4.01877 | − | 12.3685i | −27.3835 | + | 19.8953i | −8.30082 | ||
3.4 | 0.688785 | − | 0.500431i | −2.41049 | − | 7.41872i | −2.24814 | + | 6.91907i | 7.88774 | + | 5.73078i | −5.37287 | − | 3.90362i | −7.93077 | + | 24.4084i | 4.01877 | + | 12.3685i | −27.3835 | + | 19.8953i | 8.30082 | ||
3.5 | 2.93750 | − | 2.13422i | 2.92726 | + | 9.00917i | 1.60188 | − | 4.93009i | 1.70313 | + | 1.23740i | 27.8264 | + | 20.2170i | 3.03234 | − | 9.33259i | 3.15984 | + | 9.72499i | −50.7528 | + | 36.8741i | 7.64382 | ||
3.6 | 3.62629 | − | 2.63465i | 0.719300 | + | 2.21378i | 3.73643 | − | 11.4995i | −15.2540 | − | 11.0827i | 8.44091 | + | 6.13268i | 5.80624 | − | 17.8698i | −5.66698 | − | 17.4412i | 17.4600 | − | 12.6855i | −84.5143 | ||
9.1 | −1.38512 | + | 4.26295i | −1.88315 | + | 1.36819i | −9.78210 | − | 7.10711i | 5.82651 | + | 17.9321i | −3.22414 | − | 9.92289i | −15.2009 | − | 11.0441i | 14.8364 | − | 10.7792i | −6.66914 | + | 20.5255i | −84.5143 | ||
9.2 | −1.12203 | + | 3.45324i | −7.66366 | + | 5.56797i | −4.19379 | − | 3.04697i | −0.650538 | − | 2.00215i | −10.6287 | − | 32.7119i | −7.93878 | − | 5.76786i | −8.27257 | + | 6.01037i | 19.3859 | − | 59.6635i | 7.64382 | ||
9.3 | −0.263092 | + | 0.809715i | 6.31074 | − | 4.58502i | 5.88572 | + | 4.27622i | −3.01285 | − | 9.27260i | 2.05225 | + | 6.31618i | 20.7630 | + | 15.0852i | −10.5213 | + | 7.64416i | 10.4596 | − | 32.1912i | 8.30082 | ||
9.4 | 0.263092 | − | 0.809715i | 6.31074 | − | 4.58502i | 5.88572 | + | 4.27622i | −3.01285 | − | 9.27260i | −2.05225 | − | 6.31618i | −20.7630 | − | 15.0852i | 10.5213 | − | 7.64416i | 10.4596 | − | 32.1912i | −8.30082 | ||
9.5 | 1.12203 | − | 3.45324i | −7.66366 | + | 5.56797i | −4.19379 | − | 3.04697i | −0.650538 | − | 2.00215i | 10.6287 | + | 32.7119i | 7.93878 | + | 5.76786i | 8.27257 | − | 6.01037i | 19.3859 | − | 59.6635i | −7.64382 | ||
9.6 | 1.38512 | − | 4.26295i | −1.88315 | + | 1.36819i | −9.78210 | − | 7.10711i | 5.82651 | + | 17.9321i | 3.22414 | + | 9.92289i | 15.2009 | + | 11.0441i | −14.8364 | + | 10.7792i | −6.66914 | + | 20.5255i | 84.5143 | ||
27.1 | −1.38512 | − | 4.26295i | −1.88315 | − | 1.36819i | −9.78210 | + | 7.10711i | 5.82651 | − | 17.9321i | −3.22414 | + | 9.92289i | −15.2009 | + | 11.0441i | 14.8364 | + | 10.7792i | −6.66914 | − | 20.5255i | −84.5143 | ||
27.2 | −1.12203 | − | 3.45324i | −7.66366 | − | 5.56797i | −4.19379 | + | 3.04697i | −0.650538 | + | 2.00215i | −10.6287 | + | 32.7119i | −7.93878 | + | 5.76786i | −8.27257 | − | 6.01037i | 19.3859 | + | 59.6635i | 7.64382 | ||
27.3 | −0.263092 | − | 0.809715i | 6.31074 | + | 4.58502i | 5.88572 | − | 4.27622i | −3.01285 | + | 9.27260i | 2.05225 | − | 6.31618i | 20.7630 | − | 15.0852i | −10.5213 | − | 7.64416i | 10.4596 | + | 32.1912i | 8.30082 | ||
27.4 | 0.263092 | + | 0.809715i | 6.31074 | + | 4.58502i | 5.88572 | − | 4.27622i | −3.01285 | + | 9.27260i | −2.05225 | + | 6.31618i | −20.7630 | + | 15.0852i | 10.5213 | + | 7.64416i | 10.4596 | + | 32.1912i | −8.30082 | ||
27.5 | 1.12203 | + | 3.45324i | −7.66366 | − | 5.56797i | −4.19379 | + | 3.04697i | −0.650538 | + | 2.00215i | 10.6287 | − | 32.7119i | 7.93878 | − | 5.76786i | 8.27257 | + | 6.01037i | 19.3859 | + | 59.6635i | −7.64382 | ||
27.6 | 1.38512 | + | 4.26295i | −1.88315 | − | 1.36819i | −9.78210 | + | 7.10711i | 5.82651 | − | 17.9321i | 3.22414 | − | 9.92289i | 15.2009 | − | 11.0441i | −14.8364 | − | 10.7792i | −6.66914 | − | 20.5255i | 84.5143 | ||
81.1 | −3.62629 | − | 2.63465i | 0.719300 | − | 2.21378i | 3.73643 | + | 11.4995i | −15.2540 | + | 11.0827i | −8.44091 | + | 6.13268i | −5.80624 | − | 17.8698i | 5.66698 | − | 17.4412i | 17.4600 | + | 12.6855i | 84.5143 | ||
81.2 | −2.93750 | − | 2.13422i | 2.92726 | − | 9.00917i | 1.60188 | + | 4.93009i | 1.70313 | − | 1.23740i | −27.8264 | + | 20.2170i | −3.03234 | − | 9.33259i | −3.15984 | + | 9.72499i | −50.7528 | − | 36.8741i | −7.64382 | ||
See all 24 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
11.b | odd | 2 | 1 | inner |
11.c | even | 5 | 3 | inner |
11.d | odd | 10 | 3 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 121.4.c.j | 24 | |
11.b | odd | 2 | 1 | inner | 121.4.c.j | 24 | |
11.c | even | 5 | 1 | 121.4.a.h | ✓ | 6 | |
11.c | even | 5 | 3 | inner | 121.4.c.j | 24 | |
11.d | odd | 10 | 1 | 121.4.a.h | ✓ | 6 | |
11.d | odd | 10 | 3 | inner | 121.4.c.j | 24 | |
33.f | even | 10 | 1 | 1089.4.a.bj | 6 | ||
33.h | odd | 10 | 1 | 1089.4.a.bj | 6 | ||
44.g | even | 10 | 1 | 1936.4.a.bq | 6 | ||
44.h | odd | 10 | 1 | 1936.4.a.bq | 6 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
121.4.a.h | ✓ | 6 | 11.c | even | 5 | 1 | |
121.4.a.h | ✓ | 6 | 11.d | odd | 10 | 1 | |
121.4.c.j | 24 | 1.a | even | 1 | 1 | trivial | |
121.4.c.j | 24 | 11.b | odd | 2 | 1 | inner | |
121.4.c.j | 24 | 11.c | even | 5 | 3 | inner | |
121.4.c.j | 24 | 11.d | odd | 10 | 3 | inner | |
1089.4.a.bj | 6 | 33.f | even | 10 | 1 | ||
1089.4.a.bj | 6 | 33.h | odd | 10 | 1 | ||
1936.4.a.bq | 6 | 44.g | even | 10 | 1 | ||
1936.4.a.bq | 6 | 44.h | odd | 10 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{24} + 34 T_{2}^{22} + 867 T_{2}^{20} + 19844 T_{2}^{18} + 430661 T_{2}^{16} + 5401988 T_{2}^{14} + \cdots + 1358954496 \) acting on \(S_{4}^{\mathrm{new}}(121, [\chi])\).