Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [38,9,Mod(27,38)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(38, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([1]))
N = Newforms(chi, 9, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("38.27");
S:= CuspForms(chi, 9);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 38 = 2 \cdot 19 \) |
Weight: | \( k \) | \(=\) | \( 9 \) |
Character orbit: | \([\chi]\) | \(=\) | 38.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: | \(15.4803871823\) |
Analytic rank: | \(0\) |
Dimension: | \(24\) |
Relative dimension: | \(12\) 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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
27.1 | −9.79796 | − | 5.65685i | −127.651 | − | 73.6994i | 64.0000 | + | 110.851i | 381.154 | − | 660.178i | 833.814 | + | 1444.21i | 2763.12 | − | 1448.15i | 7582.71 | + | 13133.6i | −7469.06 | + | 4312.26i | |||
27.2 | −9.79796 | − | 5.65685i | −57.0846 | − | 32.9578i | 64.0000 | + | 110.851i | −412.457 | + | 714.397i | 372.875 | + | 645.838i | 2843.86 | − | 1448.15i | −1108.07 | − | 1919.23i | 8082.48 | − | 4666.42i | |||
27.3 | −9.79796 | − | 5.65685i | −31.2989 | − | 18.0704i | 64.0000 | + | 110.851i | 117.295 | − | 203.161i | 204.444 | + | 354.107i | −867.592 | − | 1448.15i | −2627.42 | − | 4550.82i | −2298.50 | + | 1327.04i | |||
27.4 | −9.79796 | − | 5.65685i | 58.9458 | + | 34.0324i | 64.0000 | + | 110.851i | −530.398 | + | 918.676i | −385.033 | − | 666.896i | −2153.04 | − | 1448.15i | −964.092 | − | 1669.86i | 10393.6 | − | 6000.77i | |||
27.5 | −9.79796 | − | 5.65685i | 65.0070 | + | 37.5318i | 64.0000 | + | 110.851i | 393.618 | − | 681.767i | −424.624 | − | 735.470i | −1847.62 | − | 1448.15i | −463.230 | − | 802.337i | −7713.31 | + | 4453.28i | |||
27.6 | −9.79796 | − | 5.65685i | 99.7890 | + | 57.6132i | 64.0000 | + | 110.851i | −88.7120 | + | 153.654i | −651.819 | − | 1128.98i | 4081.69 | − | 1448.15i | 3358.06 | + | 5816.33i | 1738.39 | − | 1003.66i | |||
27.7 | 9.79796 | + | 5.65685i | −107.051 | − | 61.8057i | 64.0000 | + | 110.851i | 333.136 | − | 577.008i | −699.252 | − | 1211.14i | −377.866 | 1448.15i | 4359.40 | + | 7550.70i | 6528.10 | − | 3769.00i | ||||
27.8 | 9.79796 | + | 5.65685i | −36.3650 | − | 20.9954i | 64.0000 | + | 110.851i | 106.719 | − | 184.842i | −237.535 | − | 411.423i | 680.248 | 1448.15i | −2398.89 | − | 4155.00i | 2091.25 | − | 1207.38i | ||||
27.9 | 9.79796 | + | 5.65685i | −3.23455 | − | 1.86747i | 64.0000 | + | 110.851i | −499.152 | + | 864.556i | −21.1280 | − | 36.5948i | 3106.83 | 1448.15i | −3273.53 | − | 5669.91i | −9781.33 | + | 5647.26i | ||||
27.10 | 9.79796 | + | 5.65685i | 30.3133 | + | 17.5014i | 64.0000 | + | 110.851i | −189.461 | + | 328.157i | 198.006 | + | 342.956i | −4482.18 | 1448.15i | −2667.90 | − | 4620.94i | −3712.67 | + | 2143.51i | ||||
27.11 | 9.79796 | + | 5.65685i | 61.4665 | + | 35.4877i | 64.0000 | + | 110.851i | 219.968 | − | 380.996i | 401.497 | + | 695.414i | 2502.16 | 1448.15i | −761.746 | − | 1319.38i | 4310.48 | − | 2488.66i | ||||
27.12 | 9.79796 | + | 5.65685i | 131.163 | + | 75.7272i | 64.0000 | + | 110.851i | −110.709 | + | 191.754i | 856.755 | + | 1483.94i | −253.604 | 1448.15i | 8188.71 | + | 14183.3i | −2169.45 | + | 1252.53i | ||||
31.1 | −9.79796 | + | 5.65685i | −127.651 | + | 73.6994i | 64.0000 | − | 110.851i | 381.154 | + | 660.178i | 833.814 | − | 1444.21i | 2763.12 | 1448.15i | 7582.71 | − | 13133.6i | −7469.06 | − | 4312.26i | ||||
31.2 | −9.79796 | + | 5.65685i | −57.0846 | + | 32.9578i | 64.0000 | − | 110.851i | −412.457 | − | 714.397i | 372.875 | − | 645.838i | 2843.86 | 1448.15i | −1108.07 | + | 1919.23i | 8082.48 | + | 4666.42i | ||||
31.3 | −9.79796 | + | 5.65685i | −31.2989 | + | 18.0704i | 64.0000 | − | 110.851i | 117.295 | + | 203.161i | 204.444 | − | 354.107i | −867.592 | 1448.15i | −2627.42 | + | 4550.82i | −2298.50 | − | 1327.04i | ||||
31.4 | −9.79796 | + | 5.65685i | 58.9458 | − | 34.0324i | 64.0000 | − | 110.851i | −530.398 | − | 918.676i | −385.033 | + | 666.896i | −2153.04 | 1448.15i | −964.092 | + | 1669.86i | 10393.6 | + | 6000.77i | ||||
31.5 | −9.79796 | + | 5.65685i | 65.0070 | − | 37.5318i | 64.0000 | − | 110.851i | 393.618 | + | 681.767i | −424.624 | + | 735.470i | −1847.62 | 1448.15i | −463.230 | + | 802.337i | −7713.31 | − | 4453.28i | ||||
31.6 | −9.79796 | + | 5.65685i | 99.7890 | − | 57.6132i | 64.0000 | − | 110.851i | −88.7120 | − | 153.654i | −651.819 | + | 1128.98i | 4081.69 | 1448.15i | 3358.06 | − | 5816.33i | 1738.39 | + | 1003.66i | ||||
31.7 | 9.79796 | − | 5.65685i | −107.051 | + | 61.8057i | 64.0000 | − | 110.851i | 333.136 | + | 577.008i | −699.252 | + | 1211.14i | −377.866 | − | 1448.15i | 4359.40 | − | 7550.70i | 6528.10 | + | 3769.00i | |||
31.8 | 9.79796 | − | 5.65685i | −36.3650 | + | 20.9954i | 64.0000 | − | 110.851i | 106.719 | + | 184.842i | −237.535 | + | 411.423i | 680.248 | − | 1448.15i | −2398.89 | + | 4155.00i | 2091.25 | + | 1207.38i | |||
See all 24 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
19.d | odd | 6 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 38.9.d.a | ✓ | 24 |
19.d | odd | 6 | 1 | inner | 38.9.d.a | ✓ | 24 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
38.9.d.a | ✓ | 24 | 1.a | even | 1 | 1 | trivial |
38.9.d.a | ✓ | 24 | 19.d | odd | 6 | 1 | inner |
Hecke kernels
This newform subspace is the entire newspace \(S_{9}^{\mathrm{new}}(38, [\chi])\).