Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [126,10,Mod(17,126)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(126, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([3, 1]))
N = Newforms(chi, 10, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("126.17");
S:= CuspForms(chi, 10);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 126 = 2 \cdot 3^{2} \cdot 7 \) |
Weight: | \( k \) | \(=\) | \( 10 \) |
Character orbit: | \([\chi]\) | \(=\) | 126.k (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: | \(64.8945153566\) |
Analytic rank: | \(0\) |
Dimension: | \(48\) |
Relative dimension: | \(24\) 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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
17.1 | −13.8564 | + | 8.00000i | 0 | 128.000 | − | 221.703i | −1351.83 | − | 2341.43i | 0 | −3938.59 | + | 4984.09i | 4096.00i | 0 | 37463.0 | + | 21629.2i | ||||||||
17.2 | −13.8564 | + | 8.00000i | 0 | 128.000 | − | 221.703i | 940.085 | + | 1628.28i | 0 | −5173.04 | + | 3686.91i | 4096.00i | 0 | −26052.4 | − | 15041.4i | ||||||||
17.3 | −13.8564 | + | 8.00000i | 0 | 128.000 | − | 221.703i | 788.973 | + | 1366.54i | 0 | −3939.49 | − | 4983.38i | 4096.00i | 0 | −21864.7 | − | 12623.6i | ||||||||
17.4 | −13.8564 | + | 8.00000i | 0 | 128.000 | − | 221.703i | 718.886 | + | 1245.15i | 0 | 6327.61 | − | 561.268i | 4096.00i | 0 | −19922.3 | − | 11502.2i | ||||||||
17.5 | −13.8564 | + | 8.00000i | 0 | 128.000 | − | 221.703i | −356.977 | − | 618.302i | 0 | 3246.94 | + | 5459.94i | 4096.00i | 0 | 9892.83 | + | 5711.63i | ||||||||
17.6 | −13.8564 | + | 8.00000i | 0 | 128.000 | − | 221.703i | −349.092 | − | 604.645i | 0 | 4657.86 | − | 4319.48i | 4096.00i | 0 | 9674.32 | + | 5585.47i | ||||||||
17.7 | −13.8564 | + | 8.00000i | 0 | 128.000 | − | 221.703i | 262.462 | + | 454.598i | 0 | −5311.53 | − | 3484.43i | 4096.00i | 0 | −7273.57 | − | 4199.40i | ||||||||
17.8 | −13.8564 | + | 8.00000i | 0 | 128.000 | − | 221.703i | −74.4501 | − | 128.951i | 0 | −4177.48 | + | 4785.63i | 4096.00i | 0 | 2063.22 | + | 1191.20i | ||||||||
17.9 | −13.8564 | + | 8.00000i | 0 | 128.000 | − | 221.703i | 554.620 | + | 960.629i | 0 | 1350.61 | + | 6207.21i | 4096.00i | 0 | −15370.1 | − | 8873.91i | ||||||||
17.10 | −13.8564 | + | 8.00000i | 0 | 128.000 | − | 221.703i | −558.488 | − | 967.329i | 0 | 3171.64 | − | 5504.03i | 4096.00i | 0 | 15477.3 | + | 8935.81i | ||||||||
17.11 | −13.8564 | + | 8.00000i | 0 | 128.000 | − | 221.703i | −999.354 | − | 1730.93i | 0 | −5636.61 | − | 2929.55i | 4096.00i | 0 | 27694.9 | + | 15989.7i | ||||||||
17.12 | −13.8564 | + | 8.00000i | 0 | 128.000 | − | 221.703i | −962.211 | − | 1666.60i | 0 | 6242.08 | + | 1179.01i | 4096.00i | 0 | 26665.6 | + | 15395.4i | ||||||||
17.13 | 13.8564 | − | 8.00000i | 0 | 128.000 | − | 221.703i | 962.211 | + | 1666.60i | 0 | 6242.08 | + | 1179.01i | − | 4096.00i | 0 | 26665.6 | + | 15395.4i | |||||||
17.14 | 13.8564 | − | 8.00000i | 0 | 128.000 | − | 221.703i | 999.354 | + | 1730.93i | 0 | −5636.61 | − | 2929.55i | − | 4096.00i | 0 | 27694.9 | + | 15989.7i | |||||||
17.15 | 13.8564 | − | 8.00000i | 0 | 128.000 | − | 221.703i | 558.488 | + | 967.329i | 0 | 3171.64 | − | 5504.03i | − | 4096.00i | 0 | 15477.3 | + | 8935.81i | |||||||
17.16 | 13.8564 | − | 8.00000i | 0 | 128.000 | − | 221.703i | −554.620 | − | 960.629i | 0 | 1350.61 | + | 6207.21i | − | 4096.00i | 0 | −15370.1 | − | 8873.91i | |||||||
17.17 | 13.8564 | − | 8.00000i | 0 | 128.000 | − | 221.703i | 74.4501 | + | 128.951i | 0 | −4177.48 | + | 4785.63i | − | 4096.00i | 0 | 2063.22 | + | 1191.20i | |||||||
17.18 | 13.8564 | − | 8.00000i | 0 | 128.000 | − | 221.703i | −262.462 | − | 454.598i | 0 | −5311.53 | − | 3484.43i | − | 4096.00i | 0 | −7273.57 | − | 4199.40i | |||||||
17.19 | 13.8564 | − | 8.00000i | 0 | 128.000 | − | 221.703i | 349.092 | + | 604.645i | 0 | 4657.86 | − | 4319.48i | − | 4096.00i | 0 | 9674.32 | + | 5585.47i | |||||||
17.20 | 13.8564 | − | 8.00000i | 0 | 128.000 | − | 221.703i | 356.977 | + | 618.302i | 0 | 3246.94 | + | 5459.94i | − | 4096.00i | 0 | 9892.83 | + | 5711.63i | |||||||
See all 48 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
3.b | odd | 2 | 1 | inner |
7.d | odd | 6 | 1 | inner |
21.g | even | 6 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 126.10.k.a | ✓ | 48 |
3.b | odd | 2 | 1 | inner | 126.10.k.a | ✓ | 48 |
7.d | odd | 6 | 1 | inner | 126.10.k.a | ✓ | 48 |
21.g | even | 6 | 1 | inner | 126.10.k.a | ✓ | 48 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
126.10.k.a | ✓ | 48 | 1.a | even | 1 | 1 | trivial |
126.10.k.a | ✓ | 48 | 3.b | odd | 2 | 1 | inner |
126.10.k.a | ✓ | 48 | 7.d | odd | 6 | 1 | inner |
126.10.k.a | ✓ | 48 | 21.g | even | 6 | 1 | inner |
Hecke kernels
This newform subspace is the entire newspace \(S_{10}^{\mathrm{new}}(126, [\chi])\).