Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [22,9,Mod(7,22)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(22, base_ring=CyclotomicField(10))
chi = DirichletCharacter(H, H._module([7]))
N = Newforms(chi, 9, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("22.7");
S:= CuspForms(chi, 9);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 22 = 2 \cdot 11 \) |
Weight: | \( k \) | \(=\) | \( 9 \) |
Character orbit: | \([\chi]\) | \(=\) | 22.d (of order \(10\), degree \(4\), 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: | \(8.96232942134\) |
Analytic rank: | \(0\) |
Dimension: | \(32\) |
Relative dimension: | \(8\) over \(\Q(\zeta_{10})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{10}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
7.1 | −6.65003 | + | 9.15298i | −24.3639 | + | 74.9844i | −39.5542 | − | 121.735i | −467.457 | + | 339.628i | −524.310 | − | 721.651i | −1196.71 | + | 388.833i | 1377.28 | + | 447.504i | 278.905 | + | 202.637i | − | 6537.16i | |
7.2 | −6.65003 | + | 9.15298i | −20.1856 | + | 62.1248i | −39.5542 | − | 121.735i | 927.050 | − | 673.541i | −434.393 | − | 597.890i | 320.984 | − | 104.294i | 1377.28 | + | 447.504i | 1855.92 | + | 1348.41i | 12964.3i | ||
7.3 | −6.65003 | + | 9.15298i | 15.7928 | − | 48.6053i | −39.5542 | − | 121.735i | −394.920 | + | 286.926i | 339.861 | + | 467.778i | 1393.15 | − | 452.662i | 1377.28 | + | 447.504i | 3194.90 | + | 2321.23i | − | 5522.77i | |
7.4 | −6.65003 | + | 9.15298i | 39.9571 | − | 122.975i | −39.5542 | − | 121.735i | 594.183 | − | 431.699i | 859.874 | + | 1183.52i | −4362.77 | + | 1417.55i | 1377.28 | + | 447.504i | −8218.38 | − | 5971.00i | 8309.36i | ||
7.5 | 6.65003 | − | 9.15298i | −43.1911 | + | 132.929i | −39.5542 | − | 121.735i | 371.274 | − | 269.746i | 929.471 | + | 1279.31i | −2980.04 | + | 968.273i | −1377.28 | − | 447.504i | −10496.6 | − | 7626.22i | − | 5192.08i | |
7.6 | 6.65003 | − | 9.15298i | −17.2507 | + | 53.0922i | −39.5542 | − | 121.735i | −95.9356 | + | 69.7013i | 371.234 | + | 510.960i | 3856.66 | − | 1253.10i | −1377.28 | − | 447.504i | 2786.76 | + | 2024.70i | 1341.61i | ||
7.7 | 6.65003 | − | 9.15298i | 10.6908 | − | 32.9028i | −39.5542 | − | 121.735i | −864.151 | + | 627.842i | −230.065 | − | 316.657i | −1443.95 | + | 469.167i | −1377.28 | − | 447.504i | 4339.66 | + | 3152.95i | 12084.7i | ||
7.8 | 6.65003 | − | 9.15298i | 32.1819 | − | 99.0456i | −39.5542 | − | 121.735i | 446.808 | − | 324.625i | −692.552 | − | 953.217i | 361.475 | − | 117.450i | −1377.28 | − | 447.504i | −3466.40 | − | 2518.49i | − | 6248.39i | |
13.1 | −10.7600 | − | 3.49613i | −118.443 | + | 86.0541i | 103.554 | + | 75.2365i | 263.736 | + | 811.695i | 1575.30 | − | 511.847i | −2234.95 | + | 3076.14i | −851.204 | − | 1171.58i | 4596.05 | − | 14145.2i | − | 9655.88i | |
13.2 | −10.7600 | − | 3.49613i | −42.1296 | + | 30.6089i | 103.554 | + | 75.2365i | −302.083 | − | 929.717i | 560.326 | − | 182.061i | −997.523 | + | 1372.97i | −851.204 | − | 1171.58i | −1189.47 | + | 3660.80i | 11059.9i | ||
13.3 | −10.7600 | − | 3.49613i | −9.77853 | + | 7.10452i | 103.554 | + | 75.2365i | 24.4502 | + | 75.2499i | 130.055 | − | 42.2574i | 1395.71 | − | 1921.02i | −851.204 | − | 1171.58i | −1982.32 | + | 6100.94i | − | 895.168i | |
13.4 | −10.7600 | − | 3.49613i | 67.1008 | − | 48.7516i | 103.554 | + | 75.2365i | 202.501 | + | 623.233i | −892.445 | + | 289.973i | −159.403 | + | 219.399i | −851.204 | − | 1171.58i | 98.3408 | − | 302.662i | − | 7413.94i | |
13.5 | 10.7600 | + | 3.49613i | −124.474 | + | 90.4358i | 103.554 | + | 75.2365i | −172.270 | − | 530.193i | −1655.51 | + | 537.909i | −92.0512 | + | 126.698i | 851.204 | + | 1171.58i | 5287.73 | − | 16274.0i | − | 6307.14i | |
13.6 | 10.7600 | + | 3.49613i | −23.2078 | + | 16.8615i | 103.554 | + | 75.2365i | 88.9820 | + | 273.858i | −308.665 | + | 100.291i | −550.185 | + | 757.265i | 851.204 | + | 1171.58i | −1773.17 | + | 5457.25i | 3257.80i | ||
13.7 | 10.7600 | + | 3.49613i | 43.1191 | − | 31.3278i | 103.554 | + | 75.2365i | −227.453 | − | 700.028i | 573.486 | − | 186.337i | 1739.62 | − | 2394.38i | 851.204 | + | 1171.58i | −1149.64 | + | 3538.23i | − | 8327.49i | |
13.8 | 10.7600 | + | 3.49613i | 123.182 | − | 89.4972i | 103.554 | + | 75.2365i | 310.287 | + | 954.964i | 1638.33 | − | 532.327i | −525.022 | + | 722.631i | 851.204 | + | 1171.58i | 5136.68 | − | 15809.1i | 11360.2i | ||
17.1 | −10.7600 | + | 3.49613i | −118.443 | − | 86.0541i | 103.554 | − | 75.2365i | 263.736 | − | 811.695i | 1575.30 | + | 511.847i | −2234.95 | − | 3076.14i | −851.204 | + | 1171.58i | 4596.05 | + | 14145.2i | 9655.88i | ||
17.2 | −10.7600 | + | 3.49613i | −42.1296 | − | 30.6089i | 103.554 | − | 75.2365i | −302.083 | + | 929.717i | 560.326 | + | 182.061i | −997.523 | − | 1372.97i | −851.204 | + | 1171.58i | −1189.47 | − | 3660.80i | − | 11059.9i | |
17.3 | −10.7600 | + | 3.49613i | −9.77853 | − | 7.10452i | 103.554 | − | 75.2365i | 24.4502 | − | 75.2499i | 130.055 | + | 42.2574i | 1395.71 | + | 1921.02i | −851.204 | + | 1171.58i | −1982.32 | − | 6100.94i | 895.168i | ||
17.4 | −10.7600 | + | 3.49613i | 67.1008 | + | 48.7516i | 103.554 | − | 75.2365i | 202.501 | − | 623.233i | −892.445 | − | 289.973i | −159.403 | − | 219.399i | −851.204 | + | 1171.58i | 98.3408 | + | 302.662i | 7413.94i | ||
See all 32 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
11.d | odd | 10 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 22.9.d.a | ✓ | 32 |
11.c | even | 5 | 1 | 242.9.b.e | 32 | ||
11.d | odd | 10 | 1 | inner | 22.9.d.a | ✓ | 32 |
11.d | odd | 10 | 1 | 242.9.b.e | 32 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
22.9.d.a | ✓ | 32 | 1.a | even | 1 | 1 | trivial |
22.9.d.a | ✓ | 32 | 11.d | odd | 10 | 1 | inner |
242.9.b.e | 32 | 11.c | even | 5 | 1 | ||
242.9.b.e | 32 | 11.d | odd | 10 | 1 |
Hecke kernels
This newform subspace is the entire newspace \(S_{9}^{\mathrm{new}}(22, [\chi])\).