Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [91,10,Mod(16,91)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(91, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([2, 2]))
N = Newforms(chi, 10, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("91.16");
S:= CuspForms(chi, 10);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 91 = 7 \cdot 13 \) |
Weight: | \( k \) | \(=\) | \( 10 \) |
Character orbit: | \([\chi]\) | \(=\) | 91.h (of order \(3\), 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: | \(46.8682610909\) |
Analytic rank: | \(0\) |
Dimension: | \(164\) |
Relative dimension: | \(82\) over \(\Q(\zeta_{3})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{3}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
16.1 | −44.9267 | −48.7021 | + | 84.3546i | 1506.41 | −218.727 | + | 378.846i | 2188.03 | − | 3789.78i | −3419.91 | + | 5353.30i | −44675.7 | 5097.70 | + | 8829.48i | 9826.67 | − | 17020.3i | ||||||
16.2 | −42.8048 | 136.894 | − | 237.108i | 1320.25 | 98.6739 | − | 170.908i | −5859.74 | + | 10149.4i | 4801.69 | + | 4159.02i | −34597.0 | −27638.7 | − | 47871.6i | −4223.72 | + | 7315.69i | ||||||
16.3 | −41.4944 | 79.2639 | − | 137.289i | 1209.79 | −958.800 | + | 1660.69i | −3289.01 | + | 5696.74i | −3727.25 | − | 5144.04i | −28954.3 | −2724.04 | − | 4718.18i | 39784.9 | − | 68909.4i | ||||||
16.4 | −41.2731 | 64.2311 | − | 111.252i | 1191.47 | 877.691 | − | 1520.21i | −2651.02 | + | 4591.70i | −5795.09 | − | 2602.03i | −28044.0 | 1590.23 | + | 2754.36i | −36225.1 | + | 62743.7i | ||||||
16.5 | −41.2552 | −8.92868 | + | 15.4649i | 1189.99 | −842.207 | + | 1458.75i | 368.355 | − | 638.009i | 6320.66 | + | 634.750i | −27970.8 | 9682.06 | + | 16769.8i | 34745.4 | − | 60180.9i | ||||||
16.6 | −40.0257 | −36.1738 | + | 62.6549i | 1090.06 | 319.890 | − | 554.065i | 1447.88 | − | 2507.80i | 2743.86 | − | 5729.30i | −23137.1 | 7224.41 | + | 12513.0i | −12803.8 | + | 22176.8i | ||||||
16.7 | −39.7623 | 42.2099 | − | 73.1097i | 1069.04 | 915.875 | − | 1586.34i | −1678.36 | + | 2907.01i | 6254.10 | − | 1113.51i | −22149.3 | 6278.15 | + | 10874.1i | −36417.3 | + | 63076.7i | ||||||
16.8 | −39.6408 | −110.934 | + | 192.143i | 1059.40 | 854.067 | − | 1479.29i | 4397.50 | − | 7616.70i | −6023.08 | − | 2018.95i | −21699.3 | −14771.1 | − | 25584.2i | −33855.9 | + | 58640.2i | ||||||
16.9 | −39.3517 | −134.908 | + | 233.667i | 1036.55 | −1292.21 | + | 2238.17i | 5308.84 | − | 9195.18i | 797.607 | − | 6302.18i | −20642.1 | −26558.7 | − | 46000.9i | 50850.6 | − | 88075.8i | ||||||
16.10 | −36.4163 | −91.6449 | + | 158.734i | 814.145 | 939.046 | − | 1626.48i | 3337.36 | − | 5780.49i | 6277.96 | + | 969.960i | −11003.0 | −6956.07 | − | 12048.3i | −34196.6 | + | 59230.2i | ||||||
16.11 | −35.7216 | 56.1841 | − | 97.3137i | 764.034 | 568.907 | − | 985.376i | −2006.99 | + | 3476.20i | −1487.09 | + | 6175.94i | −9003.08 | 3528.20 | + | 6111.01i | −20322.3 | + | 35199.2i | ||||||
16.12 | −35.3278 | −104.968 | + | 181.810i | 736.050 | −141.598 | + | 245.255i | 3708.29 | − | 6422.95i | 2929.99 | + | 5636.38i | −7915.19 | −12195.2 | − | 21122.6i | 5002.34 | − | 8664.31i | ||||||
16.13 | −33.6954 | −7.73692 | + | 13.4007i | 623.381 | −406.981 | + | 704.911i | 260.699 | − | 451.543i | −6248.30 | + | 1145.56i | −3753.02 | 9721.78 | + | 16838.6i | 13713.4 | − | 23752.3i | ||||||
16.14 | −33.5094 | 52.0738 | − | 90.1944i | 610.880 | −1097.77 | + | 1901.40i | −1744.96 | + | 3022.36i | 2263.61 | + | 5935.46i | −3313.40 | 4418.14 | + | 7652.45i | 36785.7 | − | 63714.7i | ||||||
16.15 | −32.2619 | 79.9269 | − | 138.437i | 528.833 | −412.767 | + | 714.934i | −2578.60 | + | 4466.26i | 2253.08 | − | 5939.46i | −543.049 | −2935.12 | − | 5083.77i | 13316.7 | − | 23065.2i | ||||||
16.16 | −29.7445 | −60.8004 | + | 105.309i | 372.734 | −912.935 | + | 1581.25i | 1808.48 | − | 3132.37i | −6325.30 | + | 586.682i | 4142.40 | 2448.13 | + | 4240.28i | 27154.8 | − | 47033.4i | ||||||
16.17 | −29.7072 | −48.0812 | + | 83.2791i | 370.519 | −95.8890 | + | 166.085i | 1428.36 | − | 2473.99i | −1720.94 | − | 6114.90i | 4203.00 | 5217.89 | + | 9037.66i | 2848.60 | − | 4933.91i | ||||||
16.18 | −29.4349 | 122.315 | − | 211.855i | 354.414 | −310.503 | + | 537.807i | −3600.32 | + | 6235.94i | −5780.64 | + | 2633.98i | 4638.52 | −20080.3 | − | 34780.0i | 9139.63 | − | 15830.3i | ||||||
16.19 | −26.7467 | 112.136 | − | 194.225i | 203.387 | 1156.03 | − | 2002.30i | −2999.27 | + | 5194.89i | −2358.40 | − | 5898.44i | 8254.38 | −15307.5 | − | 26513.3i | −30920.0 | + | 53555.0i | ||||||
16.20 | −26.4661 | 2.88206 | − | 4.99188i | 188.453 | 778.057 | − | 1347.63i | −76.2769 | + | 132.115i | 442.842 | + | 6336.99i | 8563.02 | 9824.89 | + | 17017.2i | −20592.1 | + | 35666.6i | ||||||
See next 80 embeddings (of 164 total) |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
91.h | even | 3 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 91.10.h.a | yes | 164 |
7.c | even | 3 | 1 | 91.10.g.a | ✓ | 164 | |
13.c | even | 3 | 1 | 91.10.g.a | ✓ | 164 | |
91.h | even | 3 | 1 | inner | 91.10.h.a | yes | 164 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
91.10.g.a | ✓ | 164 | 7.c | even | 3 | 1 | |
91.10.g.a | ✓ | 164 | 13.c | even | 3 | 1 | |
91.10.h.a | yes | 164 | 1.a | even | 1 | 1 | trivial |
91.10.h.a | yes | 164 | 91.h | even | 3 | 1 | inner |
Hecke kernels
This newform subspace is the entire newspace \(S_{10}^{\mathrm{new}}(91, [\chi])\).