Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [91,8,Mod(16,91)]
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, 2]))
N = Newforms(chi, 8, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("91.16");
S:= CuspForms(chi, 8);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 91 = 7 \cdot 13 \) |
Weight: | \( k \) | \(=\) | \( 8 \) |
Character orbit: | \([\chi]\) | \(=\) | 91.h (of order \(3\), 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: | \(28.4270373191\) |
Analytic rank: | \(0\) |
Dimension: | \(126\) |
Relative dimension: | \(63\) over \(\Q(\zeta_{3})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{3}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
16.1 | −21.8777 | −23.9756 | + | 41.5269i | 350.632 | −21.7436 | + | 37.6610i | 524.529 | − | 908.511i | −34.8107 | − | 906.825i | −4870.67 | −56.1551 | − | 97.2635i | 475.699 | − | 823.934i | ||||||
16.2 | −21.6162 | −8.26223 | + | 14.3106i | 339.261 | 269.902 | − | 467.485i | 178.598 | − | 309.341i | 280.582 | + | 863.028i | −4566.66 | 956.971 | + | 1657.52i | −5834.27 | + | 10105.3i | ||||||
16.3 | −21.0975 | 26.1888 | − | 45.3604i | 317.105 | −66.6229 | + | 115.394i | −552.519 | + | 956.990i | 658.312 | − | 624.634i | −3989.64 | −278.208 | − | 481.870i | 1405.58 | − | 2434.53i | ||||||
16.4 | −20.9336 | 26.1734 | − | 45.3336i | 310.215 | −90.5560 | + | 156.848i | −547.902 | + | 948.995i | −623.340 | + | 659.538i | −3814.40 | −276.592 | − | 479.071i | 1895.66 | − | 3283.38i | ||||||
16.5 | −18.7319 | −26.1441 | + | 45.2829i | 222.884 | −97.2612 | + | 168.461i | 489.729 | − | 848.235i | 880.970 | + | 217.796i | −1777.37 | −273.527 | − | 473.763i | 1821.89 | − | 3155.60i | ||||||
16.6 | −18.3767 | −24.0823 | + | 41.7118i | 209.703 | −251.617 | + | 435.813i | 442.554 | − | 766.525i | −798.848 | + | 430.564i | −1501.43 | −66.4166 | − | 115.037i | 4623.89 | − | 8008.81i | ||||||
16.7 | −18.3303 | −41.3019 | + | 71.5369i | 208.000 | 68.8190 | − | 119.198i | 757.075 | − | 1311.29i | −737.556 | + | 528.729i | −1466.42 | −2318.19 | − | 4015.22i | −1261.47 | + | 2184.93i | ||||||
16.8 | −18.1736 | 6.23452 | − | 10.7985i | 202.279 | −132.496 | + | 229.490i | −113.303 | + | 196.247i | 592.599 | + | 687.291i | −1349.91 | 1015.76 | + | 1759.35i | 2407.93 | − | 4170.66i | ||||||
16.9 | −18.0124 | 2.20243 | − | 3.81472i | 196.447 | 111.005 | − | 192.267i | −39.6710 | + | 68.7122i | −798.622 | − | 430.983i | −1232.89 | 1083.80 | + | 1877.19i | −1999.47 | + | 3463.19i | ||||||
16.10 | −17.9775 | 33.7545 | − | 58.4645i | 195.192 | 176.342 | − | 305.434i | −606.823 | + | 1051.05i | 544.696 | − | 725.844i | −1207.95 | −1185.23 | − | 2052.88i | −3170.20 | + | 5490.95i | ||||||
16.11 | −15.9608 | 38.5973 | − | 66.8525i | 126.746 | 172.468 | − | 298.723i | −616.042 | + | 1067.02i | −37.4390 | + | 906.720i | 20.0123 | −1886.00 | − | 3266.65i | −2752.72 | + | 4767.84i | ||||||
16.12 | −14.8162 | −42.5519 | + | 73.7021i | 91.5204 | 187.452 | − | 324.677i | 630.458 | − | 1091.99i | 581.575 | − | 696.644i | 540.490 | −2527.83 | − | 4378.33i | −2777.33 | + | 4810.48i | ||||||
16.13 | −14.8112 | 4.92210 | − | 8.52533i | 91.3725 | 67.2870 | − | 116.545i | −72.9024 | + | 126.271i | −792.466 | − | 442.199i | 542.499 | 1045.05 | + | 1810.07i | −996.603 | + | 1726.17i | ||||||
16.14 | −14.2960 | −8.02646 | + | 13.9022i | 76.3766 | 107.975 | − | 187.018i | 114.747 | − | 198.747i | 868.950 | + | 261.665i | 738.010 | 964.652 | + | 1670.83i | −1543.61 | + | 2673.61i | ||||||
16.15 | −14.2658 | 42.8483 | − | 74.2155i | 75.5130 | −151.993 | + | 263.260i | −611.265 | + | 1058.74i | −754.932 | − | 503.607i | 748.769 | −2578.46 | − | 4466.02i | 2168.30 | − | 3755.61i | ||||||
16.16 | −13.3993 | 9.37130 | − | 16.2316i | 51.5416 | −257.461 | + | 445.935i | −125.569 | + | 217.492i | 237.569 | − | 875.845i | 1024.49 | 917.858 | + | 1589.78i | 3449.80 | − | 5975.23i | ||||||
16.17 | −13.0238 | −26.2436 | + | 45.4552i | 41.6200 | −63.3839 | + | 109.784i | 341.792 | − | 592.001i | 185.431 | − | 888.346i | 1125.00 | −283.950 | − | 491.816i | 825.501 | − | 1429.81i | ||||||
16.18 | −11.4816 | 28.8130 | − | 49.9056i | 3.82665 | −60.6077 | + | 104.976i | −330.819 | + | 572.995i | 720.487 | + | 551.762i | 1425.71 | −566.877 | − | 981.860i | 695.873 | − | 1205.29i | ||||||
16.19 | −11.4327 | −25.3886 | + | 43.9743i | 2.70595 | 116.445 | − | 201.688i | 290.259 | − | 502.743i | −337.374 | + | 842.450i | 1432.45 | −195.658 | − | 338.890i | −1331.27 | + | 2305.83i | ||||||
16.20 | −9.50049 | 9.63027 | − | 16.6801i | −37.7406 | −115.203 | + | 199.537i | −91.4923 | + | 158.469i | −595.376 | + | 684.888i | 1574.62 | 908.016 | + | 1572.73i | 1094.48 | − | 1895.70i | ||||||
See next 80 embeddings (of 126 total) |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
91.h | even | 3 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 91.8.h.a | yes | 126 |
7.c | even | 3 | 1 | 91.8.g.a | ✓ | 126 | |
13.c | even | 3 | 1 | 91.8.g.a | ✓ | 126 | |
91.h | even | 3 | 1 | inner | 91.8.h.a | yes | 126 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
91.8.g.a | ✓ | 126 | 7.c | even | 3 | 1 | |
91.8.g.a | ✓ | 126 | 13.c | even | 3 | 1 | |
91.8.h.a | yes | 126 | 1.a | even | 1 | 1 | trivial |
91.8.h.a | yes | 126 | 91.h | even | 3 | 1 | inner |
Hecke kernels
This newform subspace is the entire newspace \(S_{8}^{\mathrm{new}}(91, [\chi])\).