Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [72,8,Mod(13,72)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(72, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([0, 3, 2]))
N = Newforms(chi, 8, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("72.13");
S:= CuspForms(chi, 8);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 72 = 2^{3} \cdot 3^{2} \) |
Weight: | \( k \) | \(=\) | \( 8 \) |
Character orbit: | \([\chi]\) | \(=\) | 72.n (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: | \(22.4917218349\) |
Analytic rank: | \(0\) |
Dimension: | \(164\) |
Relative dimension: | \(82\) 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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
13.1 | −11.3038 | + | 0.473330i | −23.1239 | − | 40.6483i | 127.552 | − | 10.7009i | 156.431 | + | 90.3155i | 280.628 | + | 448.536i | 183.338 | + | 317.552i | −1436.76 | + | 181.334i | −1117.57 | + | 1879.89i | −1811.01 | − | 946.865i |
13.2 | −11.2976 | + | 0.602606i | −46.7246 | − | 1.95340i | 127.274 | − | 13.6161i | −29.6157 | − | 17.0986i | 529.055 | − | 6.08770i | −438.175 | − | 758.942i | −1429.69 | + | 230.525i | 2179.37 | + | 182.543i | 344.891 | + | 175.328i |
13.3 | −11.2966 | − | 0.622684i | 19.3085 | + | 42.5932i | 127.225 | + | 14.0684i | −458.263 | − | 264.578i | −191.598 | − | 493.180i | −430.048 | − | 744.865i | −1428.44 | − | 238.145i | −1441.36 | + | 1644.82i | 5012.04 | + | 3274.18i |
13.4 | −11.2910 | − | 0.716509i | 41.0005 | − | 22.4936i | 126.973 | + | 16.1802i | −73.9107 | − | 42.6724i | −479.053 | + | 224.598i | 208.656 | + | 361.403i | −1422.06 | − | 273.668i | 1175.08 | − | 1844.50i | 803.951 | + | 534.772i |
13.5 | −11.1884 | + | 1.67919i | −34.9028 | + | 31.1255i | 122.361 | − | 37.5749i | −271.359 | − | 156.669i | 338.240 | − | 406.853i | 704.673 | + | 1220.53i | −1305.92 | + | 625.870i | 249.406 | − | 2172.73i | 3299.15 | + | 1297.21i |
13.6 | −11.1359 | + | 1.99766i | 18.1379 | + | 43.1047i | 120.019 | − | 44.4916i | 205.108 | + | 118.419i | −288.092 | − | 443.778i | 841.491 | + | 1457.51i | −1247.64 | + | 735.213i | −1529.03 | + | 1563.66i | −2520.63 | − | 908.972i |
13.7 | −10.9054 | + | 3.01190i | 34.0751 | + | 32.0294i | 109.857 | − | 65.6921i | 318.063 | + | 183.634i | −468.074 | − | 246.664i | −768.035 | − | 1330.28i | −1000.18 | + | 1047.28i | 135.231 | + | 2182.82i | −4021.70 | − | 1044.63i |
13.8 | −10.8000 | + | 3.37058i | 5.68683 | − | 46.4183i | 105.278 | − | 72.8043i | −392.053 | − | 226.352i | 95.0392 | + | 520.484i | −563.528 | − | 976.059i | −891.609 | + | 1141.13i | −2122.32 | − | 527.946i | 4997.09 | + | 1123.14i |
13.9 | −10.6787 | − | 3.73693i | 27.7884 | − | 37.6139i | 100.071 | + | 79.8114i | 390.655 | + | 225.545i | −437.305 | + | 297.825i | −517.710 | − | 896.700i | −770.378 | − | 1226.24i | −642.611 | − | 2090.46i | −3328.86 | − | 3868.38i |
13.10 | −10.6617 | − | 3.78537i | 45.3121 | + | 11.5679i | 99.3419 | + | 80.7167i | −121.295 | − | 70.0297i | −439.313 | − | 294.856i | 248.188 | + | 429.875i | −753.607 | − | 1236.62i | 1919.37 | + | 1048.33i | 1028.12 | + | 1205.78i |
13.11 | −10.5554 | − | 4.07240i | −14.9325 | + | 44.3173i | 94.8311 | + | 85.9713i | 159.215 | + | 91.9229i | 338.096 | − | 406.973i | −195.310 | − | 338.288i | −650.867 | − | 1293.65i | −1741.04 | − | 1323.54i | −1306.23 | − | 1618.67i |
13.12 | −9.94286 | + | 5.39810i | −42.4698 | + | 19.5784i | 69.7211 | − | 107.345i | 395.210 | + | 228.175i | 316.586 | − | 423.921i | −20.8306 | − | 36.0796i | −113.767 | + | 1443.68i | 1420.38 | − | 1662.98i | −5161.23 | − | 135.326i |
13.13 | −9.91032 | + | 5.45762i | 46.3951 | − | 5.87285i | 68.4287 | − | 108.174i | −56.1042 | − | 32.3918i | −427.739 | + | 311.409i | −152.197 | − | 263.613i | −87.7794 | + | 1445.49i | 2118.02 | − | 544.944i | 732.793 | + | 14.8171i |
13.14 | −9.87518 | − | 5.52094i | 5.24322 | − | 46.4705i | 67.0384 | + | 109.041i | −315.728 | − | 182.286i | −308.339 | + | 429.957i | 611.374 | + | 1058.93i | −60.0094 | − | 1446.91i | −2132.02 | − | 487.310i | 2111.48 | + | 3543.22i |
13.15 | −9.70577 | + | 5.81361i | 18.5086 | − | 42.9468i | 60.4040 | − | 112.851i | 223.068 | + | 128.788i | 70.0355 | + | 524.434i | 512.210 | + | 887.174i | 69.8041 | + | 1446.47i | −1501.86 | − | 1589.77i | −2913.77 | + | 46.8387i |
13.16 | −9.44441 | − | 6.22921i | −46.3527 | + | 6.19919i | 50.3939 | + | 117.662i | −277.107 | − | 159.988i | 476.390 | + | 230.193i | 41.7818 | + | 72.3683i | 257.003 | − | 1425.17i | 2110.14 | − | 574.698i | 1620.52 | + | 3237.15i |
13.17 | −9.38659 | − | 6.31601i | −26.9410 | − | 38.2254i | 48.2160 | + | 118.572i | −76.9557 | − | 44.4304i | 11.4521 | + | 528.966i | −589.934 | − | 1021.79i | 296.316 | − | 1417.52i | −735.363 | + | 2059.66i | 441.728 | + | 903.103i |
13.18 | −9.24600 | − | 6.52008i | −45.8311 | − | 9.30116i | 42.9772 | + | 120.569i | 410.301 | + | 236.888i | 363.110 | + | 384.821i | 806.324 | + | 1396.59i | 388.755 | − | 1395.00i | 2013.98 | + | 852.565i | −2249.12 | − | 4865.46i |
13.19 | −8.85286 | + | 7.04464i | −37.3311 | − | 28.1671i | 28.7461 | − | 124.730i | −339.926 | − | 196.256i | 528.914 | − | 13.6246i | 470.927 | + | 815.670i | 624.195 | + | 1306.73i | 600.225 | + | 2103.02i | 4391.87 | − | 657.226i |
13.20 | −8.70945 | + | 7.22118i | −18.9377 | + | 42.7594i | 23.7091 | − | 125.785i | −134.181 | − | 77.4696i | −143.836 | − | 509.163i | −357.842 | − | 619.801i | 701.823 | + | 1266.73i | −1469.72 | − | 1619.53i | 1728.07 | − | 294.229i |
See next 80 embeddings (of 164 total) |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
8.b | even | 2 | 1 | inner |
9.c | even | 3 | 1 | inner |
72.n | even | 6 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 72.8.n.a | ✓ | 164 |
8.b | even | 2 | 1 | inner | 72.8.n.a | ✓ | 164 |
9.c | even | 3 | 1 | inner | 72.8.n.a | ✓ | 164 |
72.n | even | 6 | 1 | inner | 72.8.n.a | ✓ | 164 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
72.8.n.a | ✓ | 164 | 1.a | even | 1 | 1 | trivial |
72.8.n.a | ✓ | 164 | 8.b | even | 2 | 1 | inner |
72.8.n.a | ✓ | 164 | 9.c | even | 3 | 1 | inner |
72.8.n.a | ✓ | 164 | 72.n | even | 6 | 1 | inner |
Hecke kernels
This newform subspace is the entire newspace \(S_{8}^{\mathrm{new}}(72, [\chi])\).