Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [91,10,Mod(19,91)]
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([10, 5]))
N = Newforms(chi, 10, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("91.19");
S:= CuspForms(chi, 10);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 91 = 7 \cdot 13 \) |
Weight: | \( k \) | \(=\) | \( 10 \) |
Character orbit: | \([\chi]\) | \(=\) | 91.w (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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
19.1 | −42.1214 | − | 11.2864i | − | 124.214i | 1203.42 | + | 694.796i | −2496.57 | + | 668.955i | −1401.93 | + | 5232.06i | −785.938 | + | 6303.64i | −27060.6 | − | 27060.6i | 4253.89 | 112709. | |||||
19.2 | −42.0539 | − | 11.2683i | 225.274i | 1198.15 | + | 691.751i | −405.575 | + | 108.674i | 2538.45 | − | 9473.63i | −4540.62 | − | 4442.57i | −26829.7 | − | 26829.7i | −31065.3 | 18280.6 | ||||||
19.3 | −41.9569 | − | 11.2423i | − | 46.2785i | 1190.59 | + | 687.385i | 222.605 | − | 59.6469i | −520.278 | + | 1941.70i | 2698.98 | − | 5750.57i | −26499.6 | − | 26499.6i | 17541.3 | −10010.4 | |||||
19.4 | −41.8918 | − | 11.2249i | 83.6686i | 1185.52 | + | 684.459i | 2006.85 | − | 537.734i | 939.169 | − | 3505.03i | −5369.22 | + | 3394.86i | −26279.0 | − | 26279.0i | 12682.6 | −90106.5 | ||||||
19.5 | −39.1361 | − | 10.4865i | − | 223.537i | 978.259 | + | 564.798i | 836.127 | − | 224.039i | −2344.12 | + | 8748.37i | 4429.68 | + | 4553.19i | −17693.9 | − | 17693.9i | −30286.0 | −35072.1 | |||||
19.6 | −37.6161 | − | 10.0792i | 168.334i | 869.976 | + | 502.281i | −1682.51 | + | 450.828i | 1696.67 | − | 6332.06i | −28.0274 | + | 6352.39i | −13563.6 | − | 13563.6i | −8653.25 | 67833.6 | ||||||
19.7 | −37.3346 | − | 10.0038i | − | 207.766i | 850.394 | + | 490.975i | 819.107 | − | 219.479i | −2078.45 | + | 7756.88i | −5236.89 | − | 3595.64i | −12844.1 | − | 12844.1i | −23483.9 | −32776.7 | |||||
19.8 | −36.7419 | − | 9.84496i | 65.3690i | 809.638 | + | 467.445i | −963.441 | + | 258.153i | 643.555 | − | 2401.78i | 5922.16 | − | 2298.17i | −11374.4 | − | 11374.4i | 15409.9 | 37940.1 | ||||||
19.9 | −36.3027 | − | 9.72728i | 195.464i | 779.863 | + | 450.254i | −142.800 | + | 38.2632i | 1901.33 | − | 7095.88i | 5520.27 | + | 3143.27i | −10324.8 | − | 10324.8i | −18523.2 | 5556.24 | ||||||
19.10 | −34.8437 | − | 9.33633i | − | 5.93488i | 683.508 | + | 394.624i | 2049.00 | − | 549.028i | −55.4100 | + | 206.793i | 4779.47 | + | 4184.53i | −7071.85 | − | 7071.85i | 19647.8 | −76520.5 | |||||
19.11 | −34.5640 | − | 9.26141i | 229.045i | 665.494 | + | 384.223i | 2307.82 | − | 618.377i | 2121.28 | − | 7916.72i | 5294.56 | − | 3510.16i | −6488.78 | − | 6488.78i | −32778.6 | −85494.5 | ||||||
19.12 | −34.1907 | − | 9.16138i | − | 11.3085i | 641.671 | + | 370.469i | 453.395 | − | 121.487i | −103.602 | + | 386.647i | −4887.48 | + | 4057.85i | −5730.18 | − | 5730.18i | 19555.1 | −16614.9 | |||||
19.13 | −34.0008 | − | 9.11049i | − | 115.784i | 629.650 | + | 363.528i | −1519.89 | + | 407.252i | −1054.85 | + | 3936.76i | −5885.54 | − | 2390.40i | −5352.84 | − | 5352.84i | 6277.01 | 55387.6 | |||||
19.14 | −31.8341 | − | 8.52993i | 99.4043i | 497.248 | + | 287.086i | −1864.94 | + | 499.711i | 847.912 | − | 3164.45i | −2436.99 | − | 5866.40i | −1448.88 | − | 1448.88i | 9801.78 | 63631.4 | ||||||
19.15 | −29.6101 | − | 7.93400i | − | 244.361i | 370.403 | + | 213.852i | −2005.83 | + | 537.459i | −1938.76 | + | 7235.53i | 5566.24 | − | 3061.14i | 1827.19 | + | 1827.19i | −40029.1 | 63656.8 | |||||
19.16 | −28.5420 | − | 7.64781i | − | 148.258i | 312.752 | + | 180.567i | −1354.51 | + | 362.939i | −1133.85 | + | 4231.59i | 5348.00 | − | 3428.20i | 3152.21 | + | 3152.21i | −2297.55 | 41436.0 | |||||
19.17 | −28.2074 | − | 7.55814i | 239.312i | 295.125 | + | 170.391i | 52.3963 | − | 14.0395i | 1808.76 | − | 6750.37i | −6075.99 | + | 1853.64i | 3535.54 | + | 3535.54i | −37587.4 | −1584.07 | ||||||
19.18 | −28.0223 | − | 7.50855i | 109.720i | 285.466 | + | 164.814i | 1623.73 | − | 435.078i | 823.836 | − | 3074.60i | −4554.16 | − | 4428.68i | 3741.14 | + | 3741.14i | 7644.59 | −48767.6 | ||||||
19.19 | −27.3496 | − | 7.32831i | − | 166.796i | 250.892 | + | 144.853i | 2135.35 | − | 572.165i | −1222.34 | + | 4561.82i | 3723.12 | − | 5147.04i | 4450.62 | + | 4450.62i | −8138.03 | −62593.9 | |||||
19.20 | −25.1179 | − | 6.73031i | − | 51.7685i | 142.205 | + | 82.1022i | 93.2990 | − | 24.9994i | −348.418 | + | 1300.32i | 3979.80 | + | 4951.24i | 6395.11 | + | 6395.11i | 17003.0 | −2511.73 | |||||
See next 80 embeddings (of 328 total) |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
91.w | 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.w.a | ✓ | 328 |
7.d | odd | 6 | 1 | 91.10.ba.a | yes | 328 | |
13.f | odd | 12 | 1 | 91.10.ba.a | yes | 328 | |
91.w | even | 12 | 1 | inner | 91.10.w.a | ✓ | 328 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
91.10.w.a | ✓ | 328 | 1.a | even | 1 | 1 | trivial |
91.10.w.a | ✓ | 328 | 91.w | even | 12 | 1 | inner |
91.10.ba.a | yes | 328 | 7.d | odd | 6 | 1 | |
91.10.ba.a | yes | 328 | 13.f | odd | 12 | 1 |
Hecke kernels
This newform subspace is the entire newspace \(S_{10}^{\mathrm{new}}(91, [\chi])\).