Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [9,15,Mod(2,9)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(9, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([1]))
N = Newforms(chi, 15, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("9.2");
S:= CuspForms(chi, 15);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 9 = 3^{2} \) |
Weight: | \( k \) | \(=\) | \( 15 \) |
Character orbit: | \([\chi]\) | \(=\) | 9.d (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: | \(11.1896071337\) |
Analytic rank: | \(0\) |
Dimension: | \(26\) |
Relative dimension: | \(13\) 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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
2.1 | −193.294 | − | 111.598i | 1404.39 | + | 1676.50i | 16716.4 | + | 28953.6i | −20060.8 | + | 11582.1i | −84364.2 | − | 480785.i | 361557. | − | 626235.i | − | 3.80522e6i | −838368. | + | 4.70892e6i | 5.17016e6 | |||
2.2 | −165.573 | − | 95.5937i | 919.037 | − | 1984.52i | 10084.3 | + | 17466.5i | 5072.07 | − | 2928.36i | −341876. | + | 240730.i | −596946. | + | 1.03394e6i | − | 723570.i | −3.09371e6 | − | 3.64771e6i | −1.11973e6 | |||
2.3 | −164.198 | − | 94.7998i | −2133.26 | − | 481.824i | 9782.02 | + | 16943.0i | 93735.9 | − | 54118.4i | 304601. | + | 281348.i | 584546. | − | 1.01246e6i | − | 602934.i | 4.31866e6 | + | 2.05572e6i | −2.05217e7 | |||
2.4 | −122.622 | − | 70.7960i | −1808.59 | + | 1229.62i | 1832.14 | + | 3173.36i | −101737. | + | 58738.1i | 308826. | − | 22738.4i | −525106. | + | 909510.i | 1.80101e6i | 1.75902e6 | − | 4.44777e6i | 1.66337e7 | ||||
2.5 | −47.7787 | − | 27.5850i | 2180.12 | − | 173.279i | −6670.13 | − | 11553.0i | 20922.0 | − | 12079.3i | −108943. | − | 51859.8i | 105744. | − | 183153.i | 1.63989e6i | 4.72292e6 | − | 755540.i | −1.33284e6 | ||||
2.6 | −40.5755 | − | 23.4263i | −594.691 | − | 2104.59i | −7094.42 | − | 12287.9i | −113204. | + | 65358.5i | −25172.9 | + | 99326.3i | 742609. | − | 1.28624e6i | 1.43242e6i | −4.07565e6 | + | 2.50316e6i | 6.12443e6 | ||||
2.7 | −36.6279 | − | 21.1471i | 13.1197 | + | 2186.96i | −7297.60 | − | 12639.8i | 53737.7 | − | 31025.5i | 45767.4 | − | 80381.2i | −26704.6 | + | 46253.7i | 1.31024e6i | −4.78262e6 | + | 57384.4i | −2.62440e6 | ||||
2.8 | 25.4810 | + | 14.7114i | −1531.84 | − | 1560.91i | −7759.15 | − | 13439.2i | 68095.1 | − | 39314.7i | −16069.4 | − | 62309.0i | −576494. | + | 998517.i | − | 938657.i | −89920.6 | + | 4.78212e6i | 2.31351e6 | |||
2.9 | 104.538 | + | 60.3552i | −1916.47 | + | 1053.62i | −906.505 | − | 1570.11i | −10010.1 | + | 5779.31i | −263936. | − | 5524.99i | 308433. | − | 534222.i | − | 2.19657e6i | 2.56273e6 | − | 4.03847e6i | −1.39525e6 | |||
2.10 | 104.562 | + | 60.3690i | 1696.28 | + | 1380.43i | −903.169 | − | 1564.33i | −116056. | + | 67005.0i | 94031.7 | + | 246744.i | −391422. | + | 677963.i | − | 2.19626e6i | 971781. | + | 4.68321e6i | −1.61801e7 | |||
2.11 | 129.548 | + | 74.7948i | 1398.48 | − | 1681.44i | 2996.54 | + | 5190.15i | 31950.1 | − | 18446.4i | 306934. | − | 113228.i | 153772. | − | 266342.i | − | 1.55438e6i | −871480. | − | 4.70290e6i | 5.51879e6 | |||
2.12 | 195.306 | + | 112.760i | 840.821 | + | 2018.91i | 17237.5 | + | 29856.3i | 103308. | − | 59645.1i | −63434.5 | + | 489115.i | 163246. | − | 282751.i | 4.07989e6i | −3.36901e6 | + | 3.39508e6i | 2.69023e7 | ||||
2.13 | 209.734 | + | 121.090i | −1566.90 | − | 1525.71i | 21133.6 | + | 36604.4i | −69749.9 | + | 40270.1i | −143885. | − | 509729.i | −376400. | + | 651944.i | 6.26837e6i | 127396. | + | 4.78127e6i | −1.95052e7 | ||||
5.1 | −193.294 | + | 111.598i | 1404.39 | − | 1676.50i | 16716.4 | − | 28953.6i | −20060.8 | − | 11582.1i | −84364.2 | + | 480785.i | 361557. | + | 626235.i | 3.80522e6i | −838368. | − | 4.70892e6i | 5.17016e6 | ||||
5.2 | −165.573 | + | 95.5937i | 919.037 | + | 1984.52i | 10084.3 | − | 17466.5i | 5072.07 | + | 2928.36i | −341876. | − | 240730.i | −596946. | − | 1.03394e6i | 723570.i | −3.09371e6 | + | 3.64771e6i | −1.11973e6 | ||||
5.3 | −164.198 | + | 94.7998i | −2133.26 | + | 481.824i | 9782.02 | − | 16943.0i | 93735.9 | + | 54118.4i | 304601. | − | 281348.i | 584546. | + | 1.01246e6i | 602934.i | 4.31866e6 | − | 2.05572e6i | −2.05217e7 | ||||
5.4 | −122.622 | + | 70.7960i | −1808.59 | − | 1229.62i | 1832.14 | − | 3173.36i | −101737. | − | 58738.1i | 308826. | + | 22738.4i | −525106. | − | 909510.i | − | 1.80101e6i | 1.75902e6 | + | 4.44777e6i | 1.66337e7 | |||
5.5 | −47.7787 | + | 27.5850i | 2180.12 | + | 173.279i | −6670.13 | + | 11553.0i | 20922.0 | + | 12079.3i | −108943. | + | 51859.8i | 105744. | + | 183153.i | − | 1.63989e6i | 4.72292e6 | + | 755540.i | −1.33284e6 | |||
5.6 | −40.5755 | + | 23.4263i | −594.691 | + | 2104.59i | −7094.42 | + | 12287.9i | −113204. | − | 65358.5i | −25172.9 | − | 99326.3i | 742609. | + | 1.28624e6i | − | 1.43242e6i | −4.07565e6 | − | 2.50316e6i | 6.12443e6 | |||
5.7 | −36.6279 | + | 21.1471i | 13.1197 | − | 2186.96i | −7297.60 | + | 12639.8i | 53737.7 | + | 31025.5i | 45767.4 | + | 80381.2i | −26704.6 | − | 46253.7i | − | 1.31024e6i | −4.78262e6 | − | 57384.4i | −2.62440e6 | |||
See all 26 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
9.d | odd | 6 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 9.15.d.a | ✓ | 26 |
3.b | odd | 2 | 1 | 27.15.d.a | 26 | ||
9.c | even | 3 | 1 | 27.15.d.a | 26 | ||
9.c | even | 3 | 1 | 81.15.b.a | 26 | ||
9.d | odd | 6 | 1 | inner | 9.15.d.a | ✓ | 26 |
9.d | odd | 6 | 1 | 81.15.b.a | 26 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
9.15.d.a | ✓ | 26 | 1.a | even | 1 | 1 | trivial |
9.15.d.a | ✓ | 26 | 9.d | odd | 6 | 1 | inner |
27.15.d.a | 26 | 3.b | odd | 2 | 1 | ||
27.15.d.a | 26 | 9.c | even | 3 | 1 | ||
81.15.b.a | 26 | 9.c | even | 3 | 1 | ||
81.15.b.a | 26 | 9.d | odd | 6 | 1 |
Hecke kernels
This newform subspace is the entire newspace \(S_{15}^{\mathrm{new}}(9, [\chi])\).