Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [89,10,Mod(11,89)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(89, base_ring=CyclotomicField(22))
chi = DirichletCharacter(H, H._module([21]))
N = Newforms(chi, 10, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("89.11");
S:= CuspForms(chi, 10);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 89 \) |
Weight: | \( k \) | \(=\) | \( 10 \) |
Character orbit: | \([\chi]\) | \(=\) | 89.f (of order \(22\), degree \(10\), 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: | \(45.8381894186\) |
Analytic rank: | \(0\) |
Dimension: | \(660\) |
Relative dimension: | \(66\) over \(\Q(\zeta_{22})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{22}]$ |
$q$-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 | −29.3718 | − | 33.8968i | 100.152 | − | 14.3997i | −213.429 | + | 1484.43i | −1324.01 | + | 850.889i | −3429.76 | − | 2971.90i | −2456.23 | − | 3821.96i | 37267.6 | − | 23950.5i | −9062.56 | + | 2661.01i | 67731.0 | + | 19887.6i |
11.2 | −27.8962 | − | 32.1940i | −111.443 | + | 16.0230i | −185.387 | + | 1289.39i | −383.758 | + | 246.626i | 3624.67 | + | 3140.80i | 3200.31 | + | 4979.78i | 28334.0 | − | 18209.2i | −6722.97 | + | 1974.04i | 18645.3 | + | 5474.75i |
11.3 | −27.4737 | − | 31.7064i | −251.974 | + | 36.2284i | −177.623 | + | 1235.40i | 988.921 | − | 635.541i | 8071.35 | + | 6993.87i | −4048.34 | − | 6299.35i | 25979.7 | − | 16696.1i | 43292.9 | − | 12711.9i | −47320.1 | − | 13894.4i |
11.4 | −26.8447 | − | 30.9805i | 276.126 | − | 39.7009i | −166.285 | + | 1156.54i | 514.520 | − | 330.662i | −8642.48 | − | 7488.75i | −609.321 | − | 948.122i | 22637.3 | − | 14548.1i | 55783.7 | − | 16379.6i | −24056.2 | − | 7063.54i |
11.5 | −26.5347 | − | 30.6227i | 57.2379 | − | 8.22957i | −160.793 | + | 1118.34i | 1789.65 | − | 1150.14i | −1770.80 | − | 1534.41i | −680.903 | − | 1059.51i | 21060.6 | − | 13534.8i | −15677.2 | + | 4603.26i | −82708.0 | − | 24285.3i |
11.6 | −24.4588 | − | 28.2269i | 162.887 | − | 23.4196i | −125.662 | + | 874.001i | 299.561 | − | 192.516i | −4645.08 | − | 4024.98i | 3266.28 | + | 5082.43i | 11656.6 | − | 7491.26i | 7098.00 | − | 2084.16i | −12761.0 | − | 3746.97i |
11.7 | −24.0656 | − | 27.7732i | −110.529 | + | 15.8917i | −119.332 | + | 829.971i | 1590.10 | − | 1021.89i | 3101.32 | + | 2687.31i | 2917.13 | + | 4539.15i | 10094.1 | − | 6487.05i | −6921.55 | + | 2032.35i | −66647.9 | − | 19569.6i |
11.8 | −23.8701 | − | 27.5475i | −96.7895 | + | 13.9162i | −116.221 | + | 808.333i | −236.144 | + | 151.761i | 2693.73 | + | 2334.13i | −6039.03 | − | 9396.92i | 9341.70 | − | 6003.54i | −9711.15 | + | 2851.45i | 9817.41 | + | 2882.65i |
11.9 | −23.6242 | − | 27.2637i | −245.101 | + | 35.2402i | −112.345 | + | 781.376i | −2304.70 | + | 1481.14i | 6751.07 | + | 5849.84i | 2623.57 | + | 4082.35i | 8418.97 | − | 5410.54i | 39946.8 | − | 11729.4i | 94828.0 | + | 27844.0i |
11.10 | −23.1036 | − | 26.6629i | −44.4319 | + | 6.38834i | −104.272 | + | 725.230i | −1564.09 | + | 1005.18i | 1196.87 | + | 1037.09i | −1347.71 | − | 2097.08i | 6549.91 | − | 4209.37i | −16952.3 | + | 4977.65i | 62937.2 | + | 18480.0i |
11.11 | −22.5505 | − | 26.0247i | 122.825 | − | 17.6596i | −95.8928 | + | 666.949i | −945.067 | + | 607.358i | −3229.35 | − | 2798.25i | 5568.18 | + | 8664.25i | 4687.40 | − | 3012.41i | −4111.55 | + | 1207.26i | 37118.0 | + | 10898.8i |
11.12 | −22.2001 | − | 25.6203i | 121.444 | − | 17.4610i | −90.6890 | + | 630.756i | 992.362 | − | 637.752i | −3143.42 | − | 2723.79i | −5968.88 | − | 9287.77i | 3571.77 | − | 2295.44i | −4441.94 | + | 1304.27i | −38369.9 | − | 11266.4i |
11.13 | −20.5229 | − | 23.6847i | 180.388 | − | 25.9359i | −66.9096 | + | 465.366i | −1796.71 | + | 1154.67i | −4316.36 | − | 3740.15i | −2806.51 | − | 4367.02i | −1103.30 | + | 709.049i | 12981.4 | − | 3811.68i | 64221.6 | + | 18857.2i |
11.14 | −18.9979 | − | 21.9247i | −176.954 | + | 25.4421i | −46.9086 | + | 326.256i | −667.049 | + | 428.686i | 3919.55 | + | 3396.31i | −1569.31 | − | 2441.89i | −4451.24 | + | 2860.64i | 11779.6 | − | 3458.79i | 22071.3 | + | 6480.72i |
11.15 | −18.6156 | − | 21.4835i | −248.677 | + | 35.7544i | −42.1371 | + | 293.070i | 754.308 | − | 484.765i | 5397.41 | + | 4676.88i | 5204.71 | + | 8098.69i | −5163.46 | + | 3318.35i | 41676.3 | − | 12237.3i | −24456.4 | − | 7181.04i |
11.16 | −17.6929 | − | 20.4187i | −78.9795 | + | 11.3555i | −31.0189 | + | 215.741i | 256.751 | − | 165.004i | 1629.24 | + | 1411.74i | 3112.86 | + | 4843.70i | −6683.18 | + | 4295.02i | −12776.9 | + | 3751.63i | −7911.82 | − | 2323.12i |
11.17 | −15.6407 | − | 18.0504i | 57.7859 | − | 8.30835i | −8.31807 | + | 57.8534i | 915.893 | − | 588.609i | −1053.78 | − | 913.107i | −1849.17 | − | 2877.36i | −9113.01 | + | 5856.58i | −15615.5 | + | 4585.13i | −24949.8 | − | 7325.93i |
11.18 | −15.6053 | − | 18.0095i | −169.737 | + | 24.4045i | −7.95116 | + | 55.3015i | 2179.61 | − | 1400.75i | 3088.32 | + | 2676.05i | −1004.03 | − | 1562.30i | −9144.08 | + | 5876.54i | 9329.46 | − | 2739.38i | −59240.4 | − | 17394.5i |
11.19 | −15.4169 | − | 17.7921i | 217.920 | − | 31.3321i | −6.01132 | + | 41.8096i | 614.570 | − | 394.960i | −3917.12 | − | 3394.20i | −2069.29 | − | 3219.88i | −9303.62 | + | 5979.08i | 27621.6 | − | 8110.45i | −16501.9 | − | 4845.41i |
11.20 | −13.7990 | − | 15.9249i | 202.658 | − | 29.1378i | 9.67508 | − | 67.2917i | 2267.64 | − | 1457.33i | −3260.50 | − | 2825.24i | 6234.30 | + | 9700.76i | −10281.2 | + | 6607.30i | 21335.5 | − | 6264.68i | −54499.1 | − | 16002.4i |
See next 80 embeddings (of 660 total) |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
89.f | even | 22 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 89.10.f.a | ✓ | 660 |
89.f | even | 22 | 1 | inner | 89.10.f.a | ✓ | 660 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
89.10.f.a | ✓ | 660 | 1.a | even | 1 | 1 | trivial |
89.10.f.a | ✓ | 660 | 89.f | even | 22 | 1 | inner |
Hecke kernels
This newform subspace is the entire newspace \(S_{10}^{\mathrm{new}}(89, [\chi])\).