Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [124,5,Mod(67,124)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(124, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([3, 4]))
N = Newforms(chi, 5, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("124.67");
S:= CuspForms(chi, 5);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 124 = 2^{2} \cdot 31 \) |
Weight: | \( k \) | \(=\) | \( 5 \) |
Character orbit: | \([\chi]\) | \(=\) | 124.i (of order \(6\), 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: | \(12.8178754224\) |
Analytic rank: | \(0\) |
Dimension: | \(124\) |
Relative dimension: | \(62\) over \(\Q(\zeta_{6})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{6}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
67.1 | −3.99851 | − | 0.109072i | 4.29354 | + | 2.47888i | 15.9762 | + | 0.872250i | 12.1810 | + | 21.0980i | −16.8974 | − | 10.3801i | −61.1133 | − | 35.2838i | −63.7859 | − | 5.23026i | −28.2103 | − | 48.8617i | −46.4045 | − | 85.6893i |
67.2 | −3.99851 | + | 0.109072i | −4.29354 | − | 2.47888i | 15.9762 | − | 0.872250i | 12.1810 | + | 21.0980i | 17.4382 | + | 9.44352i | 61.1133 | + | 35.2838i | −63.7859 | + | 5.23026i | −28.2103 | − | 48.8617i | −51.0069 | − | 83.0321i |
67.3 | −3.96104 | − | 0.556897i | −13.1743 | − | 7.60620i | 15.3797 | + | 4.41179i | −9.10408 | − | 15.7687i | 47.9482 | + | 37.4653i | 16.7158 | + | 9.65089i | −58.4629 | − | 26.0402i | 75.2087 | + | 130.265i | 27.2801 | + | 67.5307i |
67.4 | −3.96104 | + | 0.556897i | 13.1743 | + | 7.60620i | 15.3797 | − | 4.41179i | −9.10408 | − | 15.7687i | −56.4200 | − | 22.7918i | −16.7158 | − | 9.65089i | −58.4629 | + | 26.0402i | 75.2087 | + | 130.265i | 44.8432 | + | 57.3906i |
67.5 | −3.82637 | − | 1.16570i | 4.04124 | + | 2.33321i | 13.2823 | + | 8.92080i | −15.1375 | − | 26.2189i | −12.7435 | − | 13.6386i | 48.4396 | + | 27.9666i | −40.4241 | − | 49.6175i | −29.6122 | − | 51.2899i | 27.3583 | + | 117.969i |
67.6 | −3.82637 | + | 1.16570i | −4.04124 | − | 2.33321i | 13.2823 | − | 8.92080i | −15.1375 | − | 26.2189i | 18.1831 | + | 4.21687i | −48.4396 | − | 27.9666i | −40.4241 | + | 49.6175i | −29.6122 | − | 51.2899i | 88.4850 | + | 82.6775i |
67.7 | −3.65497 | − | 1.62518i | 14.0734 | + | 8.12529i | 10.7176 | + | 11.8800i | 8.08713 | + | 14.0073i | −38.2328 | − | 52.5695i | 35.9919 | + | 20.7799i | −19.8652 | − | 60.8389i | 91.5405 | + | 158.553i | −6.79373 | − | 64.3393i |
67.8 | −3.65497 | + | 1.62518i | −14.0734 | − | 8.12529i | 10.7176 | − | 11.8800i | 8.08713 | + | 14.0073i | 64.6429 | + | 6.82578i | −35.9919 | − | 20.7799i | −19.8652 | + | 60.8389i | 91.5405 | + | 158.553i | −52.3226 | − | 38.0532i |
67.9 | −3.57848 | − | 1.78731i | −8.22563 | − | 4.74907i | 9.61106 | + | 12.7917i | −10.5069 | − | 18.1984i | 20.9472 | + | 31.6962i | −30.8214 | − | 17.7948i | −11.5303 | − | 62.9528i | 4.60732 | + | 7.98011i | 5.07244 | + | 83.9018i |
67.10 | −3.57848 | + | 1.78731i | 8.22563 | + | 4.74907i | 9.61106 | − | 12.7917i | −10.5069 | − | 18.1984i | −37.9233 | − | 2.29273i | 30.8214 | + | 17.7948i | −11.5303 | + | 62.9528i | 4.60732 | + | 7.98011i | 70.1249 | + | 46.3438i |
67.11 | −3.33266 | − | 2.21210i | −10.5786 | − | 6.10757i | 6.21325 | + | 14.7443i | 21.6007 | + | 37.4135i | 21.7444 | + | 43.7554i | −21.7292 | − | 12.5454i | 11.9093 | − | 62.8822i | 34.1049 | + | 59.0714i | 10.7745 | − | 172.469i |
67.12 | −3.33266 | + | 2.21210i | 10.5786 | + | 6.10757i | 6.21325 | − | 14.7443i | 21.6007 | + | 37.4135i | −48.7655 | + | 3.04649i | 21.7292 | + | 12.5454i | 11.9093 | + | 62.8822i | 34.1049 | + | 59.0714i | −154.750 | − | 76.9036i |
67.13 | −3.28486 | − | 2.28248i | 3.74080 | + | 2.15975i | 5.58060 | + | 14.9952i | 3.50661 | + | 6.07364i | −7.35842 | − | 15.6328i | −20.7012 | − | 11.9518i | 15.8948 | − | 61.9948i | −31.1709 | − | 53.9896i | 2.34420 | − | 27.9548i |
67.14 | −3.28486 | + | 2.28248i | −3.74080 | − | 2.15975i | 5.58060 | − | 14.9952i | 3.50661 | + | 6.07364i | 17.2176 | − | 1.44381i | 20.7012 | + | 11.9518i | 15.8948 | + | 61.9948i | −31.1709 | − | 53.9896i | −25.3817 | − | 11.9473i |
67.15 | −2.54741 | − | 3.08394i | 12.2602 | + | 7.07841i | −3.02140 | + | 15.7121i | −8.80527 | − | 15.2512i | −9.40226 | − | 55.8412i | −57.7873 | − | 33.3635i | 56.1521 | − | 30.7074i | 59.7077 | + | 103.417i | −24.6031 | + | 66.0060i |
67.16 | −2.54741 | + | 3.08394i | −12.2602 | − | 7.07841i | −3.02140 | − | 15.7121i | −8.80527 | − | 15.2512i | 53.0611 | − | 19.7780i | 57.7873 | + | 33.3635i | 56.1521 | + | 30.7074i | 59.7077 | + | 103.417i | 69.4644 | + | 11.6961i |
67.17 | −2.49213 | − | 3.12878i | 6.05257 | + | 3.49445i | −3.57854 | + | 15.5947i | 18.8621 | + | 32.6701i | −4.15044 | − | 27.6458i | 45.3324 | + | 26.1727i | 57.7105 | − | 27.6676i | −16.0776 | − | 27.8472i | 55.2108 | − | 140.434i |
67.18 | −2.49213 | + | 3.12878i | −6.05257 | − | 3.49445i | −3.57854 | − | 15.5947i | 18.8621 | + | 32.6701i | 26.0172 | − | 10.2285i | −45.3324 | − | 26.1727i | 57.7105 | + | 27.6676i | −16.0776 | − | 27.8472i | −149.225 | − | 22.4029i |
67.19 | −2.29228 | − | 3.27802i | −7.66968 | − | 4.42809i | −5.49089 | + | 15.0283i | 4.34301 | + | 7.52232i | 3.06567 | + | 35.2918i | 32.6813 | + | 18.8685i | 61.8498 | − | 16.4498i | −1.28398 | − | 2.22391i | 14.7029 | − | 31.4798i |
67.20 | −2.29228 | + | 3.27802i | 7.66968 | + | 4.42809i | −5.49089 | − | 15.0283i | 4.34301 | + | 7.52232i | −32.0965 | + | 14.9910i | −32.6813 | − | 18.8685i | 61.8498 | + | 16.4498i | −1.28398 | − | 2.22391i | −34.6138 | − | 3.00677i |
See next 80 embeddings (of 124 total) |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
4.b | odd | 2 | 1 | inner |
31.c | even | 3 | 1 | inner |
124.i | odd | 6 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 124.5.i.a | ✓ | 124 |
4.b | odd | 2 | 1 | inner | 124.5.i.a | ✓ | 124 |
31.c | even | 3 | 1 | inner | 124.5.i.a | ✓ | 124 |
124.i | odd | 6 | 1 | inner | 124.5.i.a | ✓ | 124 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
124.5.i.a | ✓ | 124 | 1.a | even | 1 | 1 | trivial |
124.5.i.a | ✓ | 124 | 4.b | odd | 2 | 1 | inner |
124.5.i.a | ✓ | 124 | 31.c | even | 3 | 1 | inner |
124.5.i.a | ✓ | 124 | 124.i | odd | 6 | 1 | inner |
Hecke kernels
This newform subspace is the entire newspace \(S_{5}^{\mathrm{new}}(124, [\chi])\).