Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [41,5,Mod(3,41)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(41, base_ring=CyclotomicField(8))
chi = DirichletCharacter(H, H._module([3]))
N = Newforms(chi, 5, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("41.3");
S:= CuspForms(chi, 5);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 41 \) |
Weight: | \( k \) | \(=\) | \( 5 \) |
Character orbit: | \([\chi]\) | \(=\) | 41.e (of order \(8\), degree \(4\), 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: | \(4.23816848644\) |
Analytic rank: | \(0\) |
Dimension: | \(52\) |
Relative dimension: | \(13\) over \(\Q(\zeta_{8})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{8}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
3.1 | −5.29291 | + | 5.29291i | −3.41101 | + | 8.23490i | − | 40.0297i | −15.5092 | − | 15.5092i | −25.5324 | − | 61.6407i | −7.15724 | + | 17.2791i | 127.187 | + | 127.187i | 1.09702 | + | 1.09702i | 164.177 | |||
3.2 | −4.22452 | + | 4.22452i | 2.48954 | − | 6.01029i | − | 19.6931i | 20.2159 | + | 20.2159i | 14.8735 | + | 35.9077i | −16.4789 | + | 39.7837i | 15.6016 | + | 15.6016i | 27.3499 | + | 27.3499i | −170.805 | |||
3.3 | −3.77357 | + | 3.77357i | 1.75756 | − | 4.24312i | − | 12.4797i | −6.75991 | − | 6.75991i | 9.37944 | + | 22.6440i | 28.3432 | − | 68.4265i | −13.2842 | − | 13.2842i | 42.3606 | + | 42.3606i | 51.0180 | |||
3.4 | −2.94669 | + | 2.94669i | −5.64860 | + | 13.6369i | − | 1.36599i | 29.0808 | + | 29.0808i | −23.5391 | − | 56.8285i | −4.25338 | + | 10.2686i | −43.1219 | − | 43.1219i | −96.7833 | − | 96.7833i | −171.384 | |||
3.5 | −2.30381 | + | 2.30381i | 5.64187 | − | 13.6207i | 5.38492i | −29.1694 | − | 29.1694i | 18.3817 | + | 44.3772i | −29.0115 | + | 70.0400i | −49.2668 | − | 49.2668i | −96.4165 | − | 96.4165i | 134.401 | ||||
3.6 | −1.51281 | + | 1.51281i | −3.63807 | + | 8.78307i | 11.4228i | −15.0929 | − | 15.0929i | −7.78341 | − | 18.7908i | 9.07206 | − | 21.9019i | −41.4855 | − | 41.4855i | −6.63110 | − | 6.63110i | 45.6653 | ||||
3.7 | 0.00440217 | − | 0.00440217i | 6.27012 | − | 15.1374i | 16.0000i | 21.5970 | + | 21.5970i | −0.0390353 | − | 0.0942396i | 32.8790 | − | 79.3770i | 0.140869 | + | 0.140869i | −132.551 | − | 132.551i | 0.190148 | ||||
3.8 | 0.289313 | − | 0.289313i | 0.442941 | − | 1.06935i | 15.8326i | 5.11819 | + | 5.11819i | −0.181230 | − | 0.437527i | −12.6159 | + | 30.4574i | 9.20959 | + | 9.20959i | 56.3283 | + | 56.3283i | 2.96152 | ||||
3.9 | 2.97521 | − | 2.97521i | −6.14728 | + | 14.8408i | − | 1.70370i | −22.5131 | − | 22.5131i | 25.8651 | + | 62.4440i | −31.7980 | + | 76.7671i | 42.5344 | + | 42.5344i | −125.186 | − | 125.186i | −133.962 | |||
3.10 | 2.98094 | − | 2.98094i | 1.87477 | − | 4.52610i | − | 1.77196i | −31.6311 | − | 31.6311i | −7.90344 | − | 19.0806i | 23.9981 | − | 57.9365i | 42.4129 | + | 42.4129i | 40.3048 | + | 40.3048i | −188.580 | |||
3.11 | 3.08327 | − | 3.08327i | −3.25836 | + | 7.86638i | − | 3.01309i | 21.5585 | + | 21.5585i | 14.2078 | + | 34.3006i | 13.4709 | − | 32.5217i | 40.0421 | + | 40.0421i | 6.01260 | + | 6.01260i | 132.941 | |||
3.12 | 4.13918 | − | 4.13918i | 4.26929 | − | 10.3070i | − | 18.2657i | 7.17275 | + | 7.17275i | −24.9911 | − | 60.3338i | −17.9434 | + | 43.3193i | −9.37801 | − | 9.37801i | −30.7311 | − | 30.7311i | 59.3787 | |||
3.13 | 5.58200 | − | 5.58200i | −2.10724 | + | 5.08733i | − | 46.3174i | −2.03816 | − | 2.03816i | 16.6349 | + | 40.1601i | 11.2022 | − | 27.0444i | −169.232 | − | 169.232i | 35.8352 | + | 35.8352i | −22.7540 | |||
14.1 | −5.29291 | − | 5.29291i | −3.41101 | − | 8.23490i | 40.0297i | −15.5092 | + | 15.5092i | −25.5324 | + | 61.6407i | −7.15724 | − | 17.2791i | 127.187 | − | 127.187i | 1.09702 | − | 1.09702i | 164.177 | ||||
14.2 | −4.22452 | − | 4.22452i | 2.48954 | + | 6.01029i | 19.6931i | 20.2159 | − | 20.2159i | 14.8735 | − | 35.9077i | −16.4789 | − | 39.7837i | 15.6016 | − | 15.6016i | 27.3499 | − | 27.3499i | −170.805 | ||||
14.3 | −3.77357 | − | 3.77357i | 1.75756 | + | 4.24312i | 12.4797i | −6.75991 | + | 6.75991i | 9.37944 | − | 22.6440i | 28.3432 | + | 68.4265i | −13.2842 | + | 13.2842i | 42.3606 | − | 42.3606i | 51.0180 | ||||
14.4 | −2.94669 | − | 2.94669i | −5.64860 | − | 13.6369i | 1.36599i | 29.0808 | − | 29.0808i | −23.5391 | + | 56.8285i | −4.25338 | − | 10.2686i | −43.1219 | + | 43.1219i | −96.7833 | + | 96.7833i | −171.384 | ||||
14.5 | −2.30381 | − | 2.30381i | 5.64187 | + | 13.6207i | − | 5.38492i | −29.1694 | + | 29.1694i | 18.3817 | − | 44.3772i | −29.0115 | − | 70.0400i | −49.2668 | + | 49.2668i | −96.4165 | + | 96.4165i | 134.401 | |||
14.6 | −1.51281 | − | 1.51281i | −3.63807 | − | 8.78307i | − | 11.4228i | −15.0929 | + | 15.0929i | −7.78341 | + | 18.7908i | 9.07206 | + | 21.9019i | −41.4855 | + | 41.4855i | −6.63110 | + | 6.63110i | 45.6653 | |||
14.7 | 0.00440217 | + | 0.00440217i | 6.27012 | + | 15.1374i | − | 16.0000i | 21.5970 | − | 21.5970i | −0.0390353 | + | 0.0942396i | 32.8790 | + | 79.3770i | 0.140869 | − | 0.140869i | −132.551 | + | 132.551i | 0.190148 | |||
See all 52 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
41.e | odd | 8 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 41.5.e.a | ✓ | 52 |
41.e | odd | 8 | 1 | inner | 41.5.e.a | ✓ | 52 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
41.5.e.a | ✓ | 52 | 1.a | even | 1 | 1 | trivial |
41.5.e.a | ✓ | 52 | 41.e | odd | 8 | 1 | inner |
Hecke kernels
This newform subspace is the entire newspace \(S_{5}^{\mathrm{new}}(41, [\chi])\).