Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [61,9,Mod(11,61)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(61, base_ring=CyclotomicField(4))
chi = DirichletCharacter(H, H._module([1]))
N = Newforms(chi, 9, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("61.11");
S:= CuspForms(chi, 9);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 61 \) |
Weight: | \( k \) | \(=\) | \( 9 \) |
Character orbit: | \([\chi]\) | \(=\) | 61.d (of order \(4\), 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: | \(24.8500952137\) |
Analytic rank: | \(0\) |
Dimension: | \(82\) |
Relative dimension: | \(41\) over \(\Q(i)\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{4}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
11.1 | −21.2881 | − | 21.2881i | 12.3702i | 650.367i | 793.137i | 263.338 | − | 263.338i | −2412.03 | − | 2412.03i | 8395.32 | − | 8395.32i | 6407.98 | 16884.4 | − | 16884.4i | ||||||||
11.2 | −20.8664 | − | 20.8664i | − | 38.5690i | 614.817i | 69.4108i | −804.797 | + | 804.797i | 1997.03 | + | 1997.03i | 7487.23 | − | 7487.23i | 5073.43 | 1448.36 | − | 1448.36i | |||||||
11.3 | −20.5638 | − | 20.5638i | − | 121.719i | 589.743i | − | 1083.18i | −2503.01 | + | 2503.01i | −1781.46 | − | 1781.46i | 6863.04 | − | 6863.04i | −8254.53 | −22274.3 | + | 22274.3i | ||||||
11.4 | −20.1232 | − | 20.1232i | 127.522i | 553.885i | − | 671.220i | 2566.14 | − | 2566.14i | 536.994 | + | 536.994i | 5994.39 | − | 5994.39i | −9700.77 | −13507.1 | + | 13507.1i | |||||||
11.5 | −18.6699 | − | 18.6699i | 93.9195i | 441.127i | 616.035i | 1753.46 | − | 1753.46i | 1452.34 | + | 1452.34i | 3456.30 | − | 3456.30i | −2259.87 | 11501.3 | − | 11501.3i | ||||||||
11.6 | −18.5138 | − | 18.5138i | 9.00826i | 429.525i | − | 775.655i | 166.778 | − | 166.778i | −581.583 | − | 581.583i | 3212.61 | − | 3212.61i | 6479.85 | −14360.4 | + | 14360.4i | |||||||
11.7 | −17.4953 | − | 17.4953i | − | 150.121i | 356.170i | 862.462i | −2626.41 | + | 2626.41i | −1749.89 | − | 1749.89i | 1752.51 | − | 1752.51i | −15975.3 | 15089.0 | − | 15089.0i | |||||||
11.8 | −16.0682 | − | 16.0682i | − | 98.3603i | 260.374i | 156.143i | −1580.47 | + | 1580.47i | 1884.54 | + | 1884.54i | 70.2833 | − | 70.2833i | −3113.75 | 2508.94 | − | 2508.94i | |||||||
11.9 | −13.9825 | − | 13.9825i | 129.045i | 135.022i | 482.619i | 1804.38 | − | 1804.38i | −1709.24 | − | 1709.24i | −1691.58 | + | 1691.58i | −10091.7 | 6748.23 | − | 6748.23i | ||||||||
11.10 | −13.2958 | − | 13.2958i | − | 37.3980i | 97.5592i | 131.019i | −497.239 | + | 497.239i | −1655.82 | − | 1655.82i | −2106.60 | + | 2106.60i | 5162.39 | 1742.01 | − | 1742.01i | |||||||
11.11 | −13.0804 | − | 13.0804i | 48.4247i | 86.1913i | − | 475.622i | 633.412 | − | 633.412i | −1359.87 | − | 1359.87i | −2221.16 | + | 2221.16i | 4216.05 | −6221.30 | + | 6221.30i | |||||||
11.12 | −10.9363 | − | 10.9363i | 25.8158i | − | 16.7929i | 1164.26i | 282.331 | − | 282.331i | 1657.56 | + | 1657.56i | −2983.36 | + | 2983.36i | 5894.54 | 12732.8 | − | 12732.8i | |||||||
11.13 | −10.9355 | − | 10.9355i | 40.9506i | − | 16.8293i | − | 742.032i | 447.815 | − | 447.815i | 2481.46 | + | 2481.46i | −2983.53 | + | 2983.53i | 4884.05 | −8114.50 | + | 8114.50i | ||||||
11.14 | −8.83447 | − | 8.83447i | − | 119.831i | − | 99.9043i | − | 948.313i | −1058.64 | + | 1058.64i | 1107.39 | + | 1107.39i | −3144.23 | + | 3144.23i | −7798.38 | −8377.84 | + | 8377.84i | |||||
11.15 | −6.85491 | − | 6.85491i | 130.247i | − | 162.020i | − | 18.3560i | 892.832 | − | 892.832i | 2331.62 | + | 2331.62i | −2865.49 | + | 2865.49i | −10403.3 | −125.829 | + | 125.829i | ||||||
11.16 | −6.15613 | − | 6.15613i | − | 88.4223i | − | 180.204i | − | 287.135i | −544.339 | + | 544.339i | −2555.37 | − | 2555.37i | −2685.33 | + | 2685.33i | −1257.50 | −1767.64 | + | 1767.64i | |||||
11.17 | −6.14056 | − | 6.14056i | − | 66.5883i | − | 180.587i | 857.795i | −408.890 | + | 408.890i | −216.769 | − | 216.769i | −2680.89 | + | 2680.89i | 2127.00 | 5267.34 | − | 5267.34i | ||||||
11.18 | −3.70176 | − | 3.70176i | 117.318i | − | 228.594i | − | 1215.69i | 434.282 | − | 434.282i | −2288.13 | − | 2288.13i | −1793.85 | + | 1793.85i | −7202.43 | −4500.21 | + | 4500.21i | ||||||
11.19 | −2.43956 | − | 2.43956i | 26.2435i | − | 244.097i | − | 56.8987i | 64.0225 | − | 64.0225i | 1007.09 | + | 1007.09i | −1220.02 | + | 1220.02i | 5872.28 | −138.808 | + | 138.808i | ||||||
11.20 | −1.16699 | − | 1.16699i | 110.695i | − | 253.276i | 625.680i | 129.181 | − | 129.181i | −1738.72 | − | 1738.72i | −594.320 | + | 594.320i | −5692.48 | 730.163 | − | 730.163i | |||||||
See all 82 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
61.d | odd | 4 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 61.9.d.a | ✓ | 82 |
61.d | odd | 4 | 1 | inner | 61.9.d.a | ✓ | 82 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
61.9.d.a | ✓ | 82 | 1.a | even | 1 | 1 | trivial |
61.9.d.a | ✓ | 82 | 61.d | odd | 4 | 1 | inner |
Hecke kernels
This newform subspace is the entire newspace \(S_{9}^{\mathrm{new}}(61, [\chi])\).