Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [33,9,Mod(23,33)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(33, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([1, 0]))
N = Newforms(chi, 9, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("33.23");
S:= CuspForms(chi, 9);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 33 = 3 \cdot 11 \) |
Weight: | \( k \) | \(=\) | \( 9 \) |
Character orbit: | \([\chi]\) | \(=\) | 33.b (of order \(2\), degree \(1\), 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: | \(13.4434941320\) |
Analytic rank: | \(0\) |
Dimension: | \(26\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{2}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
23.1 | − | 30.7079i | 79.3894 | − | 16.0724i | −686.976 | − | 657.051i | −493.550 | − | 2437.88i | −1255.62 | 13234.4i | 6044.35 | − | 2551.96i | −20176.7 | ||||||||||
23.2 | − | 27.8469i | −68.6229 | + | 43.0336i | −519.450 | 436.996i | 1198.35 | + | 1910.94i | −547.160 | 7336.27i | 2857.21 | − | 5906.19i | 12169.0 | |||||||||||
23.3 | − | 27.3024i | −36.6798 | − | 72.2191i | −489.423 | 131.340i | −1971.76 | + | 1001.45i | −44.0163 | 6373.03i | −3870.19 | + | 5297.96i | 3585.90 | |||||||||||
23.4 | − | 23.5789i | −17.0295 | + | 79.1896i | −299.967 | − | 958.765i | 1867.21 | + | 401.538i | 3892.73 | 1036.68i | −5980.99 | − | 2697.12i | −22606.7 | ||||||||||
23.5 | − | 22.5549i | 76.8411 | − | 25.6211i | −252.723 | 1149.93i | −577.881 | − | 1733.14i | 3915.62 | − | 73.9026i | 5248.12 | − | 3937.51i | 25936.5 | ||||||||||
23.6 | − | 17.8320i | 18.7312 | − | 78.8045i | −61.9787 | − | 794.590i | −1405.24 | − | 334.014i | 2438.07 | − | 3459.78i | −5859.29 | − | 2952.20i | −14169.1 | |||||||||
23.7 | − | 16.2016i | −80.7607 | − | 6.22143i | −6.49112 | − | 809.269i | −100.797 | + | 1308.45i | −3530.96 | − | 4042.44i | 6483.59 | + | 1004.89i | −13111.4 | |||||||||
23.8 | − | 14.0184i | 36.8732 | − | 72.1205i | 59.4833 | 519.567i | −1011.02 | − | 516.905i | −3946.73 | − | 4422.58i | −3841.73 | − | 5318.63i | 7283.52 | ||||||||||
23.9 | − | 12.0893i | −77.6992 | − | 22.8875i | 109.849 | 288.931i | −276.694 | + | 939.329i | 3024.09 | − | 4422.86i | 5513.32 | + | 3556.68i | 3492.97 | ||||||||||
23.10 | − | 9.43364i | 80.9807 | + | 1.77018i | 167.006 | − | 654.316i | 16.6993 | − | 763.942i | −1003.87 | − | 3990.49i | 6554.73 | + | 286.701i | −6172.58 | |||||||||
23.11 | − | 8.17075i | −40.5699 | + | 70.1076i | 189.239 | 920.235i | 572.832 | + | 331.487i | −1093.89 | − | 3637.93i | −3269.16 | − | 5688.52i | 7519.00 | ||||||||||
23.12 | − | 4.28405i | 49.1853 | + | 64.3569i | 237.647 | 147.746i | 275.708 | − | 210.712i | 2560.99 | − | 2114.81i | −1722.62 | + | 6330.82i | 632.952 | ||||||||||
23.13 | − | 0.463681i | −38.1388 | + | 71.4593i | 255.785 | − | 649.991i | 33.1343 | + | 17.6842i | −831.270 | − | 237.305i | −3651.86 | − | 5450.75i | −301.389 | |||||||||
23.14 | 0.463681i | −38.1388 | − | 71.4593i | 255.785 | 649.991i | 33.1343 | − | 17.6842i | −831.270 | 237.305i | −3651.86 | + | 5450.75i | −301.389 | ||||||||||||
23.15 | 4.28405i | 49.1853 | − | 64.3569i | 237.647 | − | 147.746i | 275.708 | + | 210.712i | 2560.99 | 2114.81i | −1722.62 | − | 6330.82i | 632.952 | |||||||||||
23.16 | 8.17075i | −40.5699 | − | 70.1076i | 189.239 | − | 920.235i | 572.832 | − | 331.487i | −1093.89 | 3637.93i | −3269.16 | + | 5688.52i | 7519.00 | |||||||||||
23.17 | 9.43364i | 80.9807 | − | 1.77018i | 167.006 | 654.316i | 16.6993 | + | 763.942i | −1003.87 | 3990.49i | 6554.73 | − | 286.701i | −6172.58 | ||||||||||||
23.18 | 12.0893i | −77.6992 | + | 22.8875i | 109.849 | − | 288.931i | −276.694 | − | 939.329i | 3024.09 | 4422.86i | 5513.32 | − | 3556.68i | 3492.97 | |||||||||||
23.19 | 14.0184i | 36.8732 | + | 72.1205i | 59.4833 | − | 519.567i | −1011.02 | + | 516.905i | −3946.73 | 4422.58i | −3841.73 | + | 5318.63i | 7283.52 | |||||||||||
23.20 | 16.2016i | −80.7607 | + | 6.22143i | −6.49112 | 809.269i | −100.797 | − | 1308.45i | −3530.96 | 4042.44i | 6483.59 | − | 1004.89i | −13111.4 | ||||||||||||
See all 26 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
3.b | odd | 2 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 33.9.b.a | ✓ | 26 |
3.b | odd | 2 | 1 | inner | 33.9.b.a | ✓ | 26 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
33.9.b.a | ✓ | 26 | 1.a | even | 1 | 1 | trivial |
33.9.b.a | ✓ | 26 | 3.b | odd | 2 | 1 | inner |
Hecke kernels
This newform subspace is the entire newspace \(S_{9}^{\mathrm{new}}(33, [\chi])\).