Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [41,12,Mod(40,41)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(41, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([1]))
N = Newforms(chi, 12, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("41.40");
S:= CuspForms(chi, 12);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 41 \) |
Weight: | \( k \) | \(=\) | \( 12 \) |
Character orbit: | \([\chi]\) | \(=\) | 41.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: | \(31.5020704029\) |
Analytic rank: | \(0\) |
Dimension: | \(38\) |
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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
40.1 | −86.6525 | − | 468.692i | 5460.66 | 8950.34 | 40613.3i | − | 66695.3i | −295716. | −42524.8 | −775569. | ||||||||||||||||
40.2 | −86.6525 | 468.692i | 5460.66 | 8950.34 | − | 40613.3i | 66695.3i | −295716. | −42524.8 | −775569. | |||||||||||||||||
40.3 | −84.4624 | − | 636.943i | 5085.89 | −13046.4 | 53797.7i | 52159.6i | −256588. | −228549. | 1.10193e6 | |||||||||||||||||
40.4 | −84.4624 | 636.943i | 5085.89 | −13046.4 | − | 53797.7i | − | 52159.6i | −256588. | −228549. | 1.10193e6 | ||||||||||||||||
40.5 | −71.4234 | − | 66.9659i | 3053.30 | −1937.46 | 4782.93i | − | 230.167i | −71802.1 | 172663. | 138380. | ||||||||||||||||
40.6 | −71.4234 | 66.9659i | 3053.30 | −1937.46 | − | 4782.93i | 230.167i | −71802.1 | 172663. | 138380. | |||||||||||||||||
40.7 | −62.7044 | − | 600.500i | 1883.84 | 7906.03 | 37654.0i | 61105.2i | 10293.5 | −183453. | −495743. | |||||||||||||||||
40.8 | −62.7044 | 600.500i | 1883.84 | 7906.03 | − | 37654.0i | − | 61105.2i | 10293.5 | −183453. | −495743. | ||||||||||||||||
40.9 | −55.5144 | − | 693.355i | 1033.85 | −5551.71 | 38491.2i | − | 80006.7i | 56299.8 | −303594. | 308200. | ||||||||||||||||
40.10 | −55.5144 | 693.355i | 1033.85 | −5551.71 | − | 38491.2i | 80006.7i | 56299.8 | −303594. | 308200. | |||||||||||||||||
40.11 | −40.3637 | − | 515.574i | −418.774 | −1318.86 | 20810.4i | 11971.1i | 99568.1 | −88669.2 | 53234.1 | |||||||||||||||||
40.12 | −40.3637 | 515.574i | −418.774 | −1318.86 | − | 20810.4i | − | 11971.1i | 99568.1 | −88669.2 | 53234.1 | ||||||||||||||||
40.13 | −34.9176 | − | 275.392i | −828.758 | 10140.7 | 9616.05i | − | 26269.1i | 100450. | 101306. | −354088. | ||||||||||||||||
40.14 | −34.9176 | 275.392i | −828.758 | 10140.7 | − | 9616.05i | 26269.1i | 100450. | 101306. | −354088. | |||||||||||||||||
40.15 | −32.1724 | − | 38.7160i | −1012.94 | −10804.4 | 1245.58i | 61152.6i | 98477.7 | 175648. | 347603. | |||||||||||||||||
40.16 | −32.1724 | 38.7160i | −1012.94 | −10804.4 | − | 1245.58i | − | 61152.6i | 98477.7 | 175648. | 347603. | ||||||||||||||||
40.17 | −13.4608 | − | 693.996i | −1866.81 | −4832.15 | 9341.76i | 15652.4i | 52696.5 | −304484. | 65044.7 | |||||||||||||||||
40.18 | −13.4608 | 693.996i | −1866.81 | −4832.15 | − | 9341.76i | − | 15652.4i | 52696.5 | −304484. | 65044.7 | ||||||||||||||||
40.19 | 1.59232 | − | 160.353i | −2045.46 | 4151.31 | − | 255.333i | − | 65140.7i | −6518.11 | 151434. | 6610.22 | |||||||||||||||
40.20 | 1.59232 | 160.353i | −2045.46 | 4151.31 | 255.333i | 65140.7i | −6518.11 | 151434. | 6610.22 | ||||||||||||||||||
See all 38 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
41.b | even | 2 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 41.12.b.a | ✓ | 38 |
41.b | even | 2 | 1 | inner | 41.12.b.a | ✓ | 38 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
41.12.b.a | ✓ | 38 | 1.a | even | 1 | 1 | trivial |
41.12.b.a | ✓ | 38 | 41.b | even | 2 | 1 | inner |
Hecke kernels
This newform subspace is the entire newspace \(S_{12}^{\mathrm{new}}(41, [\chi])\).