Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [63,6,Mod(47,63)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(63, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([1, 5]))
N = Newforms(chi, 6, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("63.47");
S:= CuspForms(chi, 6);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 63 = 3^{2} \cdot 7 \) |
Weight: | \( k \) | \(=\) | \( 6 \) |
Character orbit: | \([\chi]\) | \(=\) | 63.s (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: | \(10.1041806482\) |
Analytic rank: | \(0\) |
Dimension: | \(76\) |
Relative dimension: | \(38\) 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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
47.1 | −9.58962 | + | 5.53657i | −0.877361 | + | 15.5637i | 45.3072 | − | 78.4743i | 21.5696 | −77.7562 | − | 154.108i | 61.1717 | + | 114.302i | 649.045i | −241.460 | − | 27.3101i | −206.844 | + | 119.422i | ||||
47.2 | −9.06532 | + | 5.23386i | 7.40328 | − | 13.7183i | 38.7866 | − | 67.1804i | 85.8214 | 4.68654 | + | 163.108i | 39.9810 | − | 123.323i | 477.048i | −133.383 | − | 203.121i | −777.998 | + | 449.178i | ||||
47.3 | −8.69257 | + | 5.01866i | −15.4559 | − | 2.02881i | 34.3739 | − | 59.5373i | −49.6832 | 144.533 | − | 59.9322i | 56.4374 | − | 116.713i | 368.849i | 234.768 | + | 62.7140i | 431.875 | − | 249.343i | ||||
47.4 | −8.37859 | + | 4.83738i | 9.11067 | − | 12.6489i | 30.8005 | − | 53.3480i | −104.481 | −15.1468 | + | 150.052i | −115.003 | + | 59.8436i | 286.382i | −76.9914 | − | 230.481i | 875.403 | − | 505.414i | ||||
47.5 | −7.50494 | + | 4.33298i | 14.2230 | + | 6.38014i | 21.5494 | − | 37.3246i | 15.7794 | −134.388 | + | 13.7454i | −117.294 | − | 55.2192i | 96.1815i | 161.588 | + | 181.489i | −118.424 | + | 68.3718i | ||||
47.6 | −7.40857 | + | 4.27734i | −9.98088 | − | 11.9742i | 20.5913 | − | 35.6651i | 38.9005 | 125.162 | + | 46.0202i | −32.6607 | + | 125.460i | 78.5539i | −43.7639 | + | 239.027i | −288.197 | + | 166.391i | ||||
47.7 | −7.18813 | + | 4.15007i | 15.2650 | + | 3.15899i | 18.4461 | − | 31.9497i | −13.4718 | −122.837 | + | 40.6437i | 121.599 | + | 44.9527i | 40.6067i | 223.042 | + | 96.4440i | 96.8369 | − | 55.9088i | ||||
47.8 | −6.85527 | + | 3.95789i | −11.6841 | + | 10.3190i | 15.3298 | − | 26.5520i | −25.8967 | 39.2562 | − | 116.984i | −120.641 | + | 47.4627i | − | 10.6099i | 30.0365 | − | 241.136i | 177.529 | − | 102.497i | |||
47.9 | −5.95180 | + | 3.43627i | −2.44206 | + | 15.3960i | 7.61597 | − | 13.1912i | 82.4113 | −38.3702 | − | 100.025i | −27.2714 | − | 126.741i | − | 115.239i | −231.073 | − | 75.1959i | −490.496 | + | 283.188i | |||
47.10 | −5.22444 | + | 3.01633i | 1.37473 | + | 15.5277i | 2.19649 | − | 3.80443i | −106.554 | −54.0189 | − | 76.9769i | 99.6671 | − | 82.9064i | − | 166.544i | −239.220 | + | 42.6928i | 556.683 | − | 321.401i | |||
47.11 | −4.52842 | + | 2.61449i | −15.1040 | + | 3.85599i | −2.32894 | + | 4.03383i | 57.3397 | 58.3159 | − | 56.9508i | 128.059 | + | 20.1980i | − | 191.683i | 213.263 | − | 116.482i | −259.658 | + | 149.914i | |||
47.12 | −4.44864 | + | 2.56843i | 6.98225 | − | 13.9373i | −2.80638 | + | 4.86080i | −22.3974 | 4.73533 | + | 79.9354i | 125.358 | + | 33.0507i | − | 193.211i | −145.496 | − | 194.627i | 99.6378 | − | 57.5259i | |||
47.13 | −4.43807 | + | 2.56232i | −5.73757 | − | 14.4941i | −2.86900 | + | 4.96925i | −12.0147 | 62.6024 | + | 49.6246i | −44.5801 | − | 121.736i | − | 193.394i | −177.161 | + | 166.322i | 53.3223 | − | 30.7856i | |||
47.14 | −3.42420 | + | 1.97696i | 13.4136 | − | 7.94196i | −8.18323 | + | 14.1738i | 96.1527 | −30.2300 | + | 53.7131i | −67.3980 | + | 110.745i | − | 191.237i | 116.850 | − | 213.061i | −329.246 | + | 190.090i | |||
47.15 | −2.69688 | + | 1.55704i | 8.07896 | + | 13.3316i | −11.1512 | + | 19.3145i | −14.7675 | −42.5458 | − | 23.3743i | −76.3428 | + | 104.780i | − | 169.103i | −112.461 | + | 215.410i | 39.8260 | − | 22.9936i | |||
47.16 | −1.69409 | + | 0.978081i | −13.0902 | − | 8.46437i | −14.0867 | + | 24.3989i | −99.5152 | 30.4548 | + | 1.53606i | 54.4830 | + | 117.638i | − | 117.709i | 99.7087 | + | 221.601i | 168.587 | − | 97.3339i | |||
47.17 | −1.45110 | + | 0.837793i | 14.6627 | − | 5.29211i | −14.5962 | + | 25.2814i | −41.1571 | −16.8433 | + | 19.9637i | −56.9736 | − | 116.452i | − | 102.533i | 186.987 | − | 155.193i | 59.7231 | − | 34.4811i | |||
47.18 | −1.22697 | + | 0.708392i | −14.9068 | + | 4.55938i | −14.9964 | + | 25.9745i | −39.2064 | 15.0604 | − | 16.1541i | −128.843 | − | 14.3719i | − | 87.8303i | 201.424 | − | 135.931i | 48.1052 | − | 27.7735i | |||
47.19 | 0.0316775 | − | 0.0182890i | 12.3725 | + | 9.48272i | −15.9993 | + | 27.7117i | 62.5776 | 0.565358 | + | 0.0741087i | 87.2452 | − | 95.8920i | 2.34094i | 63.1559 | + | 234.649i | 1.98230 | − | 1.14448i | ||||
47.20 | 0.0675663 | − | 0.0390094i | −14.2360 | − | 6.35113i | −15.9970 | + | 27.7075i | 92.3091 | −1.20963 | + | 0.126215i | −119.454 | − | 50.3751i | 4.99273i | 162.326 | + | 180.829i | 6.23698 | − | 3.60092i | ||||
See all 76 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
63.s | even | 6 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 63.6.s.a | yes | 76 |
3.b | odd | 2 | 1 | 189.6.s.a | 76 | ||
7.d | odd | 6 | 1 | 63.6.i.a | ✓ | 76 | |
9.c | even | 3 | 1 | 189.6.i.a | 76 | ||
9.d | odd | 6 | 1 | 63.6.i.a | ✓ | 76 | |
21.g | even | 6 | 1 | 189.6.i.a | 76 | ||
63.k | odd | 6 | 1 | 189.6.s.a | 76 | ||
63.s | even | 6 | 1 | inner | 63.6.s.a | yes | 76 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
63.6.i.a | ✓ | 76 | 7.d | odd | 6 | 1 | |
63.6.i.a | ✓ | 76 | 9.d | odd | 6 | 1 | |
63.6.s.a | yes | 76 | 1.a | even | 1 | 1 | trivial |
63.6.s.a | yes | 76 | 63.s | even | 6 | 1 | inner |
189.6.i.a | 76 | 9.c | even | 3 | 1 | ||
189.6.i.a | 76 | 21.g | even | 6 | 1 | ||
189.6.s.a | 76 | 3.b | odd | 2 | 1 | ||
189.6.s.a | 76 | 63.k | odd | 6 | 1 |
Hecke kernels
This newform subspace is the entire newspace \(S_{6}^{\mathrm{new}}(63, [\chi])\).