Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [91,10,Mod(45,91)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(91, base_ring=CyclotomicField(12))
chi = DirichletCharacter(H, H._module([2, 5]))
N = Newforms(chi, 10, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("91.45");
S:= CuspForms(chi, 10);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 91 = 7 \cdot 13 \) |
Weight: | \( k \) | \(=\) | \( 10 \) |
Character orbit: | \([\chi]\) | \(=\) | 91.ba (of order \(12\), degree \(4\), 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: | \(328\) |
Relative dimension: | \(82\) over \(\Q(\zeta_{12})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{12}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
45.1 | −31.9799 | + | 31.9799i | 138.161 | − | 79.7670i | − | 1533.43i | 257.967 | − | 962.747i | −1867.42 | + | 6969.31i | −5442.14 | − | 3276.69i | 32665.3 | + | 32665.3i | 2884.06 | − | 4995.34i | 22538.8 | + | 39038.4i | |
45.2 | −30.5163 | + | 30.5163i | −150.738 | + | 87.0288i | − | 1350.49i | −264.030 | + | 985.373i | 1944.18 | − | 7255.78i | 6319.73 | + | 643.947i | 25587.8 | + | 25587.8i | 5306.51 | − | 9191.15i | −22012.8 | − | 38127.2i | |
45.3 | −29.5517 | + | 29.5517i | 84.7315 | − | 48.9197i | − | 1234.60i | −686.606 | + | 2562.45i | −1058.30 | + | 3949.62i | 2936.94 | − | 5632.76i | 21354.2 | + | 21354.2i | −5055.22 | + | 8755.90i | −55434.3 | − | 96015.0i | |
45.4 | −29.1246 | + | 29.1246i | −83.6873 | + | 48.3169i | − | 1184.49i | −174.709 | + | 652.025i | 1030.15 | − | 3844.57i | −2227.44 | + | 5949.13i | 19585.9 | + | 19585.9i | −5172.46 | + | 8958.97i | −13901.6 | − | 24078.3i | |
45.5 | −28.8523 | + | 28.8523i | 62.1682 | − | 35.8928i | − | 1152.91i | 446.104 | − | 1664.88i | −758.104 | + | 2829.28i | 4603.08 | + | 4377.82i | 18491.6 | + | 18491.6i | −7264.91 | + | 12583.2i | 35164.6 | + | 60906.8i | |
45.6 | −28.6005 | + | 28.6005i | −224.468 | + | 129.597i | − | 1123.98i | 572.201 | − | 2135.48i | 2713.37 | − | 10126.4i | −6293.02 | + | 866.889i | 17502.9 | + | 17502.9i | 23749.0 | − | 41134.5i | 44710.7 | + | 77441.2i | |
45.7 | −27.6091 | + | 27.6091i | −116.526 | + | 67.2766i | − | 1012.52i | 499.218 | − | 1863.11i | 1359.74 | − | 5074.63i | 4051.21 | − | 4892.99i | 13818.9 | + | 13818.9i | −789.230 | + | 1366.99i | 37655.7 | + | 65221.6i | |
45.8 | −27.5677 | + | 27.5677i | −75.6898 | + | 43.6996i | − | 1007.96i | −177.770 | + | 663.448i | 881.899 | − | 3291.29i | −5466.40 | − | 3236.06i | 13672.4 | + | 13672.4i | −6022.20 | + | 10430.8i | −13389.0 | − | 23190.5i | |
45.9 | −26.6211 | + | 26.6211i | 223.790 | − | 129.205i | − | 905.364i | 59.1663 | − | 220.812i | −2517.95 | + | 9397.13i | 6258.70 | + | 1087.35i | 10471.8 | + | 10471.8i | 23546.6 | − | 40783.9i | 4303.18 | + | 7453.32i | |
45.10 | −26.2285 | + | 26.2285i | 199.647 | − | 115.267i | − | 863.870i | −451.287 | + | 1684.23i | −2213.19 | + | 8259.73i | −4590.26 | + | 4391.26i | 9229.03 | + | 9229.03i | 16731.2 | − | 28979.4i | −32338.2 | − | 56011.3i | |
45.11 | −25.3891 | + | 25.3891i | 77.9998 | − | 45.0332i | − | 777.216i | −209.760 | + | 782.836i | −836.993 | + | 3123.70i | −1005.97 | + | 6272.29i | 6733.61 | + | 6733.61i | −5785.52 | + | 10020.8i | −14549.9 | − | 25201.2i | |
45.12 | −23.9096 | + | 23.9096i | 69.0080 | − | 39.8418i | − | 631.341i | −145.288 | + | 542.221i | −697.354 | + | 2602.56i | 4387.03 | − | 4594.29i | 2853.40 | + | 2853.40i | −6666.76 | + | 11547.2i | −9490.53 | − | 16438.1i | |
45.13 | −22.8366 | + | 22.8366i | −55.4636 | + | 32.0219i | − | 531.022i | 243.133 | − | 907.386i | 535.329 | − | 1997.87i | −1493.80 | − | 6174.31i | 434.405 | + | 434.405i | −7790.69 | + | 13493.9i | 15169.3 | + | 26274.0i | |
45.14 | −22.7343 | + | 22.7343i | 181.092 | − | 104.554i | − | 521.696i | 382.464 | − | 1427.38i | −1740.05 | + | 6493.95i | −705.954 | − | 6313.10i | 220.439 | + | 220.439i | 12021.4 | − | 20821.7i | 23755.3 | + | 41145.5i | |
45.15 | −22.1382 | + | 22.1382i | −202.243 | + | 116.765i | − | 468.204i | −28.7375 | + | 107.250i | 1892.34 | − | 7062.29i | 271.382 | + | 6346.65i | −969.563 | − | 969.563i | 17426.8 | − | 30184.0i | −1738.13 | − | 3010.52i | |
45.16 | −22.0837 | + | 22.0837i | −223.252 | + | 128.895i | − | 463.380i | −332.592 | + | 1241.25i | 2083.76 | − | 7776.70i | 3576.33 | − | 5250.09i | −1073.72 | − | 1073.72i | 23386.1 | − | 40506.0i | −20066.5 | − | 34756.3i | |
45.17 | −21.8909 | + | 21.8909i | 34.2183 | − | 19.7559i | − | 446.422i | 529.063 | − | 1974.49i | −316.594 | + | 1181.54i | −5769.80 | + | 2657.64i | −1435.57 | − | 1435.57i | −9060.91 | + | 15693.9i | 31641.7 | + | 54805.0i | |
45.18 | −21.1278 | + | 21.1278i | −148.902 | + | 85.9683i | − | 380.769i | −684.658 | + | 2555.18i | 1329.64 | − | 4962.29i | −6234.92 | − | 1216.32i | −2772.63 | − | 2772.63i | 4939.61 | − | 8555.66i | −39520.0 | − | 68450.7i | |
45.19 | −19.6515 | + | 19.6515i | 67.4394 | − | 38.9362i | − | 260.365i | −185.704 | + | 693.057i | −560.133 | + | 2090.44i | 6132.49 | + | 1657.16i | −4945.01 | − | 4945.01i | −6809.45 | + | 11794.3i | −9970.26 | − | 17269.0i | |
45.20 | −17.8765 | + | 17.8765i | −109.368 | + | 63.1438i | − | 127.139i | 538.246 | − | 2008.76i | 826.332 | − | 3083.91i | 5367.83 | + | 3397.06i | −6879.97 | − | 6879.97i | −1867.23 | + | 3234.14i | 26287.7 | + | 45531.6i | |
See next 80 embeddings (of 328 total) |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
91.ba | even | 12 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 91.10.ba.a | yes | 328 |
7.d | odd | 6 | 1 | 91.10.w.a | ✓ | 328 | |
13.f | odd | 12 | 1 | 91.10.w.a | ✓ | 328 | |
91.ba | even | 12 | 1 | inner | 91.10.ba.a | yes | 328 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
91.10.w.a | ✓ | 328 | 7.d | odd | 6 | 1 | |
91.10.w.a | ✓ | 328 | 13.f | odd | 12 | 1 | |
91.10.ba.a | yes | 328 | 1.a | even | 1 | 1 | trivial |
91.10.ba.a | yes | 328 | 91.ba | even | 12 | 1 | inner |
Hecke kernels
This newform subspace is the entire newspace \(S_{10}^{\mathrm{new}}(91, [\chi])\).