Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [89,8,Mod(2,89)]
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([4]))
N = Newforms(chi, 8, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("89.2");
S:= CuspForms(chi, 8);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 89 \) |
Weight: | \( k \) | \(=\) | \( 8 \) |
Character orbit: | \([\chi]\) | \(=\) | 89.e (of order \(11\), 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: | \(27.8022672681\) |
Analytic rank: | \(0\) |
Dimension: | \(520\) |
Relative dimension: | \(52\) over \(\Q(\zeta_{11})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{11}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
2.1 | −21.2001 | + | 6.22490i | −33.4095 | − | 21.4710i | 303.013 | − | 194.734i | 263.036 | − | 303.560i | 841.939 | + | 247.216i | −0.516662 | + | 0.596260i | −3359.63 | + | 3877.21i | −253.319 | − | 554.692i | −3686.75 | + | 8072.86i |
2.2 | −20.5565 | + | 6.03594i | 11.9643 | + | 7.68901i | 278.457 | − | 178.954i | −48.2949 | + | 55.7353i | −292.355 | − | 85.8433i | 928.914 | − | 1072.02i | −2848.13 | + | 3286.91i | −824.488 | − | 1805.38i | 656.360 | − | 1437.23i |
2.3 | −20.5363 | + | 6.03001i | −70.3873 | − | 45.2351i | 277.699 | − | 178.467i | −294.237 | + | 339.568i | 1718.26 | + | 504.528i | −185.636 | + | 214.236i | −2832.70 | + | 3269.11i | 1999.64 | + | 4378.59i | 3994.95 | − | 8747.72i |
2.4 | −20.1719 | + | 5.92300i | 47.9334 | + | 30.8049i | 264.142 | − | 169.754i | −212.353 | + | 245.068i | −1149.36 | − | 337.484i | −863.515 | + | 996.550i | −2560.56 | + | 2955.04i | 440.157 | + | 963.809i | 2832.01 | − | 6201.24i |
2.5 | −18.2426 | + | 5.35651i | 38.0824 | + | 24.4741i | 196.420 | − | 126.232i | 40.0015 | − | 46.1642i | −825.818 | − | 242.482i | 69.3538 | − | 80.0385i | −1313.36 | + | 1515.70i | −57.2230 | − | 125.301i | −482.453 | + | 1056.42i |
2.6 | −18.0262 | + | 5.29297i | 67.4808 | + | 43.3672i | 189.247 | − | 121.622i | 344.724 | − | 397.833i | −1445.96 | − | 424.573i | 152.883 | − | 176.436i | −1192.89 | + | 1376.66i | 1764.42 | + | 3863.54i | −4108.35 | + | 8996.02i |
2.7 | −17.3159 | + | 5.08442i | −17.8251 | − | 11.4555i | 166.310 | − | 106.881i | −102.105 | + | 117.835i | 366.903 | + | 107.733i | −550.699 | + | 635.540i | −823.654 | + | 950.548i | −722.007 | − | 1580.97i | 1168.92 | − | 2559.58i |
2.8 | −16.8471 | + | 4.94675i | −5.05763 | − | 3.25034i | 151.673 | − | 97.4745i | 264.971 | − | 305.792i | 101.285 | + | 29.7399i | −756.962 | + | 873.581i | −601.294 | + | 693.931i | −893.498 | − | 1956.49i | −2951.30 | + | 6462.45i |
2.9 | −16.0704 | + | 4.71869i | −37.9848 | − | 24.4114i | 128.310 | − | 82.4600i | −78.3375 | + | 90.4063i | 725.620 | + | 213.061i | 850.478 | − | 981.503i | −268.968 | + | 310.406i | −61.5796 | − | 134.841i | 832.314 | − | 1822.51i |
2.10 | −15.9834 | + | 4.69316i | −61.8700 | − | 39.7614i | 125.764 | − | 80.8235i | 133.709 | − | 154.308i | 1175.50 | + | 345.158i | −363.499 | + | 419.500i | −234.492 | + | 270.619i | 1338.41 | + | 2930.71i | −1412.93 | + | 3093.89i |
2.11 | −14.9883 | + | 4.40096i | 69.5461 | + | 44.6946i | 97.5996 | − | 62.7235i | −310.556 | + | 358.400i | −1239.08 | − | 363.825i | 1121.77 | − | 1294.59i | 122.583 | − | 141.468i | 1930.55 | + | 4227.31i | 3077.39 | − | 6738.54i |
2.12 | −13.5119 | + | 3.96747i | 36.9806 | + | 23.7660i | 59.1515 | − | 38.0144i | 77.9143 | − | 89.9179i | −593.971 | − | 174.406i | 275.511 | − | 317.957i | 531.984 | − | 613.943i | −105.768 | − | 231.601i | −696.028 | + | 1524.09i |
2.13 | −12.8781 | + | 3.78134i | −4.79284 | − | 3.08017i | 43.8653 | − | 28.1905i | −332.998 | + | 384.300i | 73.3696 | + | 21.5432i | −1.23373 | + | 1.42380i | 666.737 | − | 769.455i | −895.029 | − | 1959.84i | 2835.20 | − | 6208.22i |
2.14 | −12.5833 | + | 3.69480i | −64.2506 | − | 41.2914i | 37.0083 | − | 23.7838i | 194.842 | − | 224.859i | 961.051 | + | 282.190i | 538.997 | − | 622.035i | 721.480 | − | 832.632i | 1514.65 | + | 3316.63i | −1620.95 | + | 3549.38i |
2.15 | −11.2460 | + | 3.30213i | 73.6479 | + | 47.3307i | 7.88855 | − | 5.06967i | −40.5184 | + | 46.7607i | −984.539 | − | 289.087i | −886.851 | + | 1023.48i | 910.489 | − | 1050.76i | 2275.32 | + | 4982.25i | 301.261 | − | 659.669i |
2.16 | −9.60772 | + | 2.82108i | −4.29476 | − | 2.76008i | −23.3306 | + | 14.9937i | 257.829 | − | 297.550i | 49.0493 | + | 14.4022i | 682.330 | − | 787.451i | 1021.20 | − | 1178.52i | −897.686 | − | 1965.66i | −1637.73 | + | 3586.14i |
2.17 | −9.38620 | + | 2.75604i | 40.4729 | + | 26.0104i | −27.1754 | + | 17.4646i | 25.6024 | − | 29.5467i | −451.573 | − | 132.594i | −106.327 | + | 122.708i | 1026.93 | − | 1185.14i | 53.0067 | + | 116.068i | −158.877 | + | 347.893i |
2.18 | −8.44733 | + | 2.48036i | −56.0085 | − | 35.9945i | −42.4752 | + | 27.2972i | −115.185 | + | 132.931i | 562.402 | + | 165.136i | −800.208 | + | 923.489i | 1029.06 | − | 1187.60i | 932.838 | + | 2042.63i | 643.292 | − | 1408.61i |
2.19 | −8.20805 | + | 2.41010i | 20.3231 | + | 13.0608i | −46.1169 | + | 29.6375i | −196.806 | + | 227.126i | −198.291 | − | 58.2234i | −232.155 | + | 267.922i | 1024.16 | − | 1181.95i | −666.071 | − | 1458.49i | 1067.99 | − | 2338.58i |
2.20 | −8.19033 | + | 2.40490i | 2.16744 | + | 1.39293i | −46.3825 | + | 29.8082i | 228.112 | − | 263.255i | −21.1019 | − | 6.19607i | −1145.57 | + | 1322.06i | 1023.72 | − | 1181.43i | −905.755 | − | 1983.33i | −1235.21 | + | 2704.73i |
See next 80 embeddings (of 520 total) |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
89.e | even | 11 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 89.8.e.a | ✓ | 520 |
89.e | even | 11 | 1 | inner | 89.8.e.a | ✓ | 520 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
89.8.e.a | ✓ | 520 | 1.a | even | 1 | 1 | trivial |
89.8.e.a | ✓ | 520 | 89.e | even | 11 | 1 | inner |
Hecke kernels
This newform subspace is the entire newspace \(S_{8}^{\mathrm{new}}(89, [\chi])\).