Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [91,10,Mod(36,91)]
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([0, 5]))
N = Newforms(chi, 10, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("91.36");
S:= CuspForms(chi, 10);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 91 = 7 \cdot 13 \) |
Weight: | \( k \) | \(=\) | \( 10 \) |
Character orbit: | \([\chi]\) | \(=\) | 91.q (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: | \(46.8682610909\) |
Analytic rank: | \(0\) |
Dimension: | \(124\) |
Relative dimension: | \(62\) 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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
36.1 | −38.9200 | + | 22.4705i | −55.4073 | − | 95.9683i | 753.843 | − | 1305.69i | 740.456i | 4312.90 | + | 2490.06i | −2079.33 | − | 1200.50i | 44747.0i | 3701.56 | − | 6411.29i | −16638.4 | − | 28818.5i | ||||
36.2 | −38.5034 | + | 22.2300i | −91.4804 | − | 158.449i | 732.343 | − | 1268.46i | − | 2102.73i | 7044.62 | + | 4067.21i | 2079.33 | + | 1200.50i | 42356.4i | −6895.84 | + | 11943.9i | 46743.6 | + | 80962.4i | |||
36.3 | −35.6614 | + | 20.5891i | 116.392 | + | 201.597i | 591.821 | − | 1025.06i | − | 877.811i | −8301.40 | − | 4792.82i | −2079.33 | − | 1200.50i | 27657.0i | −17252.8 | + | 29882.7i | 18073.3 | + | 31303.9i | |||
36.4 | −35.5029 | + | 20.4976i | 54.8657 | + | 95.0301i | 584.302 | − | 1012.04i | 455.216i | −3895.78 | − | 2249.23i | 2079.33 | + | 1200.50i | 26917.6i | 3821.02 | − | 6618.20i | −9330.84 | − | 16161.5i | ||||
36.5 | −35.0192 | + | 20.2183i | 52.5461 | + | 91.0125i | 561.562 | − | 972.654i | 1911.46i | −3680.24 | − | 2124.79i | −2079.33 | − | 1200.50i | 24711.8i | 4319.32 | − | 7481.28i | −38646.5 | − | 66937.7i | ||||
36.6 | −32.9663 | + | 19.0331i | −108.110 | − | 187.253i | 468.516 | − | 811.494i | 2130.68i | 7127.99 | + | 4115.35i | 2079.33 | + | 1200.50i | 16179.3i | −13534.2 | + | 23442.0i | −40553.5 | − | 70240.7i | ||||
36.7 | −32.0069 | + | 18.4792i | −34.1506 | − | 59.1506i | 426.960 | − | 739.517i | 1335.08i | 2186.11 | + | 1262.15i | 2079.33 | + | 1200.50i | 12636.8i | 7508.97 | − | 13005.9i | −24671.1 | − | 42731.6i | ||||
36.8 | −31.2009 | + | 18.0138i | 134.462 | + | 232.896i | 392.997 | − | 680.690i | 2021.81i | −8390.69 | − | 4844.37i | 2079.33 | + | 1200.50i | 9871.33i | −26318.8 | + | 45585.5i | −36420.6 | − | 63082.3i | ||||
36.9 | −30.7856 | + | 17.7740i | −47.1433 | − | 81.6546i | 375.833 | − | 650.963i | − | 1486.53i | 2902.67 | + | 1675.85i | −2079.33 | − | 1200.50i | 8519.70i | 5396.52 | − | 9347.04i | 26421.7 | + | 45763.7i | |||
36.10 | −30.3753 | + | 17.5372i | 13.0681 | + | 22.6346i | 359.107 | − | 621.991i | − | 1205.75i | −793.895 | − | 458.355i | 2079.33 | + | 1200.50i | 7232.81i | 9499.95 | − | 16454.4i | 21145.4 | + | 36625.0i | |||
36.11 | −29.7427 | + | 17.1720i | −63.0297 | − | 109.171i | 333.753 | − | 578.077i | 894.218i | 3749.35 | + | 2164.69i | −2079.33 | − | 1200.50i | 5340.70i | 1896.02 | − | 3284.00i | −15355.5 | − | 26596.5i | ||||
36.12 | −26.9451 | + | 15.5567i | 20.0545 | + | 34.7353i | 228.024 | − | 394.949i | − | 2230.19i | −1080.74 | − | 623.963i | −2079.33 | − | 1200.50i | − | 1740.87i | 9037.14 | − | 15652.8i | 34694.4 | + | 60092.5i | ||
36.13 | −25.2553 | + | 14.5812i | −95.5625 | − | 165.519i | 169.221 | − | 293.099i | − | 1073.46i | 4826.92 | + | 2786.83i | 2079.33 | + | 1200.50i | − | 5061.36i | −8422.88 | + | 14588.9i | 15652.4 | + | 27110.7i | ||
36.14 | −23.8670 | + | 13.7796i | 110.756 | + | 191.835i | 123.755 | − | 214.349i | − | 2578.04i | −5286.82 | − | 3052.35i | 2079.33 | + | 1200.50i | − | 7289.15i | −14692.3 | + | 25447.8i | 35524.4 | + | 61530.0i | ||
36.15 | −22.8497 | + | 13.1923i | 61.9983 | + | 107.384i | 92.0727 | − | 159.475i | 1099.16i | −2833.29 | − | 1635.80i | −2079.33 | − | 1200.50i | − | 8650.30i | 2153.91 | − | 3730.69i | −14500.4 | − | 25115.4i | |||
36.16 | −21.5135 | + | 12.4208i | 94.2787 | + | 163.295i | 52.5525 | − | 91.0236i | 179.352i | −4056.52 | − | 2342.03i | 2079.33 | + | 1200.50i | − | 10107.9i | −7935.43 | + | 13744.6i | −2227.69 | − | 3858.47i | |||
36.17 | −21.0351 | + | 12.1446i | 102.233 | + | 177.072i | 38.9839 | − | 67.5221i | − | 705.619i | −4300.95 | − | 2483.16i | −2079.33 | − | 1200.50i | − | 10542.3i | −11061.6 | + | 19159.2i | 8569.48 | + | 14842.8i | ||
36.18 | −18.9950 | + | 10.9668i | −12.3077 | − | 21.3175i | −15.4603 | + | 26.7780i | 364.739i | 467.569 | + | 269.951i | 2079.33 | + | 1200.50i | − | 11908.2i | 9538.54 | − | 16521.2i | −4000.01 | − | 6928.22i | |||
36.19 | −17.4096 | + | 10.0514i | −79.7414 | − | 138.116i | −53.9368 | + | 93.4213i | − | 2021.81i | 2776.53 | + | 1603.03i | 2079.33 | + | 1200.50i | − | 12461.3i | −2875.88 | + | 4981.18i | 20322.1 | + | 35198.9i | ||
36.20 | −14.7416 | + | 8.51109i | −92.7073 | − | 160.574i | −111.123 | + | 192.470i | 2393.29i | 2733.32 | + | 1578.08i | −2079.33 | − | 1200.50i | − | 12498.5i | −7347.78 | + | 12726.7i | −20369.6 | − | 35281.1i | |||
See next 80 embeddings (of 124 total) |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
13.e | even | 6 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 91.10.q.a | ✓ | 124 |
13.e | even | 6 | 1 | inner | 91.10.q.a | ✓ | 124 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
91.10.q.a | ✓ | 124 | 1.a | even | 1 | 1 | trivial |
91.10.q.a | ✓ | 124 | 13.e | even | 6 | 1 | inner |
Hecke kernels
This newform subspace is the entire newspace \(S_{10}^{\mathrm{new}}(91, [\chi])\).