Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [91,10,Mod(30,91)]
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, 1]))
N = Newforms(chi, 10, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("91.30");
S:= CuspForms(chi, 10);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 91 = 7 \cdot 13 \) |
Weight: | \( k \) | \(=\) | \( 10 \) |
Character orbit: | \([\chi]\) | \(=\) | 91.u (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: | \(164\) |
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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
30.1 | −38.4423 | + | 22.1947i | −211.181 | 729.208 | − | 1263.02i | 1212.41 | + | 699.984i | 8118.27 | − | 4687.09i | −6075.53 | + | 1855.15i | 42010.8i | 24914.3 | −62143.7 | ||||||||
30.2 | −37.6617 | + | 21.7440i | 84.7460 | 689.605 | − | 1194.43i | 1069.03 | + | 617.207i | −3191.68 | + | 1842.72i | 664.570 | − | 6317.59i | 37713.2i | −12501.1 | −53682.2 | ||||||||
30.3 | −37.1580 | + | 21.4532i | 7.72989 | 664.476 | − | 1150.91i | −1575.27 | − | 909.485i | −287.227 | + | 165.831i | 1841.00 | + | 6079.83i | 35052.4i | −19623.2 | 78045.3 | ||||||||
30.4 | −35.5005 | + | 20.4962i | 132.067 | 584.192 | − | 1011.85i | 1239.03 | + | 715.356i | −4688.46 | + | 2706.88i | −3482.51 | + | 5312.79i | 26906.8i | −2241.26 | −58648.4 | ||||||||
30.5 | −35.1232 | + | 20.2784i | −146.367 | 566.427 | − | 981.080i | −436.798 | − | 252.185i | 5140.89 | − | 2968.10i | 5917.60 | + | 2309.88i | 25179.8i | 1740.41 | 20455.7 | ||||||||
30.6 | −35.0393 | + | 20.2299i | 269.124 | 562.501 | − | 974.280i | −63.4069 | − | 36.6080i | −9429.90 | + | 5444.36i | 5725.53 | + | 2751.71i | 24802.0i | 52744.6 | 2962.31 | ||||||||
30.7 | −34.8194 | + | 20.1030i | 195.742 | 552.260 | − | 956.543i | −1750.16 | − | 1010.46i | −6815.62 | + | 3935.00i | −6019.15 | − | 2030.62i | 23822.8i | 18631.9 | 81252.8 | ||||||||
30.8 | −32.7341 | + | 18.8991i | −212.836 | 458.349 | − | 793.883i | −1677.20 | − | 968.331i | 6967.00 | − | 4022.40i | 454.420 | − | 6336.17i | 15296.8i | 25616.1 | 73202.1 | ||||||||
30.9 | −32.3735 | + | 18.6908i | −146.497 | 442.694 | − | 766.768i | 1496.61 | + | 864.068i | 4742.60 | − | 2738.14i | 2994.62 | − | 5602.31i | 13957.8i | 1778.26 | −64600.5 | ||||||||
30.10 | −32.2435 | + | 18.6158i | −94.9975 | 437.095 | − | 757.071i | −1072.58 | − | 619.257i | 3063.05 | − | 1768.45i | −6352.31 | − | 41.5992i | 13484.9i | −10658.5 | 46111.8 | ||||||||
30.11 | −32.1895 | + | 18.5846i | 70.4411 | 434.776 | − | 753.054i | −714.561 | − | 412.552i | −2267.46 | + | 1309.12i | 5001.70 | − | 3916.20i | 13289.9i | −14721.1 | 30668.5 | ||||||||
30.12 | −29.1216 | + | 16.8134i | −81.0362 | 309.379 | − | 535.861i | 1658.21 | + | 957.369i | 2359.91 | − | 1362.49i | 4828.80 | + | 4127.50i | 3589.95i | −13116.1 | −64386.4 | ||||||||
30.13 | −28.4816 | + | 16.4439i | 94.5061 | 284.803 | − | 493.293i | 1116.08 | + | 644.368i | −2691.69 | + | 1554.05i | −1759.05 | + | 6104.04i | 1894.52i | −10751.6 | −42383.6 | ||||||||
30.14 | −28.2880 | + | 16.3321i | 21.2897 | 277.473 | − | 480.598i | 611.718 | + | 353.175i | −602.244 | + | 347.706i | −6100.90 | − | 1769.93i | 1402.82i | −19229.7 | −23072.3 | ||||||||
30.15 | −27.3854 | + | 15.8110i | 258.288 | 243.973 | − | 422.574i | 2120.68 | + | 1224.37i | −7073.31 | + | 4083.78i | −4845.78 | − | 4107.55i | − | 760.607i | 47029.5 | −77434.2 | |||||||
30.16 | −26.6838 | + | 15.4059i | −254.883 | 218.682 | − | 378.768i | 455.574 | + | 263.026i | 6801.24 | − | 3926.70i | −1765.12 | + | 6102.29i | − | 2299.67i | 45282.4 | −16208.6 | |||||||
30.17 | −26.3184 | + | 15.1950i | 183.597 | 205.773 | − | 356.410i | −1465.14 | − | 845.901i | −4831.98 | + | 2789.75i | −2061.39 | − | 6008.69i | − | 3052.77i | 14024.8 | 51413.7 | |||||||
30.18 | −23.8716 | + | 13.7823i | 180.122 | 123.902 | − | 214.605i | 97.1768 | + | 56.1050i | −4299.81 | + | 2482.50i | 6336.56 | − | 449.013i | − | 7282.43i | 12761.0 | −3093.02 | |||||||
30.19 | −23.7092 | + | 13.6885i | −252.262 | 118.751 | − | 205.683i | −1023.39 | − | 590.853i | 5980.94 | − | 3453.10i | 5001.22 | + | 3916.81i | − | 7514.94i | 43953.3 | 32351.6 | |||||||
30.20 | −22.2527 | + | 12.8476i | 176.414 | 74.1218 | − | 128.383i | −1048.20 | − | 605.176i | −3925.69 | + | 2266.50i | −1476.29 | + | 6178.53i | − | 9346.80i | 11439.0 | 31100.2 | |||||||
See next 80 embeddings (of 164 total) |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
91.u | 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.u.a | yes | 164 |
7.c | even | 3 | 1 | 91.10.k.a | ✓ | 164 | |
13.e | even | 6 | 1 | 91.10.k.a | ✓ | 164 | |
91.u | even | 6 | 1 | inner | 91.10.u.a | yes | 164 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
91.10.k.a | ✓ | 164 | 7.c | even | 3 | 1 | |
91.10.k.a | ✓ | 164 | 13.e | even | 6 | 1 | |
91.10.u.a | yes | 164 | 1.a | even | 1 | 1 | trivial |
91.10.u.a | yes | 164 | 91.u | even | 6 | 1 | inner |
Hecke kernels
This newform subspace is the entire newspace \(S_{10}^{\mathrm{new}}(91, [\chi])\).