Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [29,8,Mod(4,29)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(29, base_ring=CyclotomicField(14))
chi = DirichletCharacter(H, H._module([1]))
N = Newforms(chi, 8, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("29.4");
S:= CuspForms(chi, 8);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 29 \) |
Weight: | \( k \) | \(=\) | \( 8 \) |
Character orbit: | \([\chi]\) | \(=\) | 29.e (of order \(14\), degree \(6\), 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: | \(9.05916573904\) |
Analytic rank: | \(0\) |
Dimension: | \(96\) |
Relative dimension: | \(16\) over \(\Q(\zeta_{14})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{14}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
4.1 | −19.9614 | − | 4.55605i | 16.7140 | + | 34.7070i | 262.375 | + | 126.353i | 46.2709 | − | 202.726i | −175.507 | − | 768.948i | −856.134 | + | 412.292i | −2612.69 | − | 2083.55i | 438.356 | − | 549.681i | −1847.26 | + | 3835.88i |
4.2 | −17.1849 | − | 3.92235i | −28.6166 | − | 59.4230i | 164.613 | + | 79.2734i | −53.9771 | + | 236.489i | 258.697 | + | 1133.42i | −694.864 | + | 334.629i | −753.925 | − | 601.235i | −1348.61 | + | 1691.11i | 1855.19 | − | 3852.33i |
4.3 | −16.3184 | − | 3.72456i | 24.7323 | + | 51.3572i | 137.092 | + | 66.0202i | −116.379 | + | 509.891i | −212.308 | − | 930.182i | 1113.60 | − | 536.279i | −316.180 | − | 252.145i | −662.302 | + | 830.500i | 3798.24 | − | 7887.12i |
4.4 | −16.0299 | − | 3.65873i | −13.6888 | − | 28.4250i | 128.249 | + | 61.7614i | 57.5812 | − | 252.280i | 115.431 | + | 505.735i | 1251.46 | − | 602.670i | −184.410 | − | 147.062i | 742.974 | − | 931.660i | −1846.05 | + | 3833.35i |
4.5 | −9.35936 | − | 2.13621i | 15.2659 | + | 31.7000i | −32.2898 | − | 15.5499i | −15.7357 | + | 68.9427i | −75.1613 | − | 329.303i | −676.409 | + | 325.741i | 1229.71 | + | 980.664i | 591.731 | − | 742.007i | 294.553 | − | 611.645i |
4.6 | −6.62931 | − | 1.51310i | 34.2559 | + | 71.1332i | −73.6657 | − | 35.4756i | 94.1794 | − | 412.627i | −119.462 | − | 523.396i | 849.727 | − | 409.207i | 1115.16 | + | 889.311i | −2522.89 | + | 3163.60i | −1248.69 | + | 2592.93i |
4.7 | −5.36405 | − | 1.22431i | −9.60762 | − | 19.9504i | −88.0499 | − | 42.4026i | 41.6869 | − | 182.642i | 27.1103 | + | 118.778i | −510.272 | + | 245.734i | 971.000 | + | 774.347i | 1057.86 | − | 1326.51i | −447.222 | + | 928.666i |
4.8 | −3.46579 | − | 0.791044i | −14.7361 | − | 30.5998i | −103.938 | − | 50.0539i | −91.2666 | + | 399.865i | 26.8664 | + | 117.709i | 200.782 | − | 96.6916i | 676.389 | + | 539.402i | 644.377 | − | 808.023i | 632.622 | − | 1313.65i |
4.9 | −1.04671 | − | 0.238905i | −39.6898 | − | 82.4167i | −114.285 | − | 55.0370i | 73.7799 | − | 323.251i | 21.8540 | + | 95.7484i | 216.705 | − | 104.360i | 213.918 | + | 170.594i | −3853.65 | + | 4832.33i | −154.452 | + | 320.724i |
4.10 | 5.20280 | + | 1.18750i | 29.1385 | + | 60.5068i | −89.6651 | − | 43.1804i | −44.8992 | + | 196.716i | 79.7498 | + | 349.407i | −458.849 | + | 220.970i | −949.289 | − | 757.033i | −1448.45 | + | 1816.30i | −467.202 | + | 970.156i |
4.11 | 6.40796 | + | 1.46258i | 1.99118 | + | 4.13473i | −76.4012 | − | 36.7929i | −7.61716 | + | 33.3729i | 6.71206 | + | 29.4074i | 1563.59 | − | 752.987i | −1093.53 | − | 872.060i | 1350.44 | − | 1693.40i | −97.6209 | + | 202.712i |
4.12 | 9.92390 | + | 2.26507i | 1.31301 | + | 2.72650i | −21.9707 | − | 10.5805i | 100.667 | − | 441.052i | 6.85452 | + | 30.0316i | −925.876 | + | 445.878i | −1212.74 | − | 967.127i | 1357.86 | − | 1702.71i | 1998.02 | − | 4148.94i |
4.13 | 10.9235 | + | 2.49321i | −26.6467 | − | 55.3324i | −2.21759 | − | 1.06794i | −50.3658 | + | 220.667i | −153.119 | − | 670.858i | −425.854 | + | 205.081i | −1142.84 | − | 911.381i | −988.053 | + | 1238.98i | −1100.34 | + | 2284.88i |
4.14 | 17.9262 | + | 4.09153i | 6.16655 | + | 12.8050i | 189.283 | + | 91.1538i | −94.6542 | + | 414.707i | 58.1506 | + | 254.775i | −762.411 | + | 367.158i | 1180.07 | + | 941.076i | 1237.63 | − | 1551.94i | −3393.57 | + | 7046.83i |
4.15 | 17.9566 | + | 4.09849i | 26.3233 | + | 54.6609i | 190.320 | + | 91.6531i | 30.2961 | − | 132.736i | 248.651 | + | 1089.41i | 459.975 | − | 221.512i | 1198.65 | + | 955.891i | −931.324 | + | 1167.84i | 1088.03 | − | 2259.33i |
4.16 | 19.1958 | + | 4.38132i | −23.2918 | − | 48.3658i | 233.959 | + | 112.669i | 36.5819 | − | 160.276i | −235.198 | − | 1030.47i | 541.526 | − | 260.785i | 2026.99 | + | 1616.47i | −433.175 | + | 543.185i | 1404.44 | − | 2916.35i |
5.1 | −16.8298 | + | 13.4213i | −72.7388 | + | 16.6021i | 74.6272 | − | 326.963i | 216.453 | + | 271.423i | 1001.35 | − | 1255.66i | −204.467 | − | 895.829i | 1936.81 | + | 4021.83i | 3044.88 | − | 1466.34i | −7285.70 | − | 1662.91i |
5.2 | −15.7422 | + | 12.5540i | 39.5051 | − | 9.01678i | 61.7319 | − | 270.465i | −50.2510 | − | 63.0128i | −508.702 | + | 637.892i | 190.445 | + | 834.393i | 1305.38 | + | 2710.65i | −491.067 | + | 236.485i | 1582.13 | + | 361.110i |
5.3 | −11.8009 | + | 9.41091i | 31.2296 | − | 7.12795i | 22.2135 | − | 97.3238i | 62.6921 | + | 78.6134i | −301.457 | + | 378.015i | −274.053 | − | 1200.70i | −184.508 | − | 383.134i | −1045.94 | + | 503.698i | −1479.65 | − | 337.720i |
5.4 | −10.9983 | + | 8.77088i | −44.4854 | + | 10.1535i | 15.5523 | − | 68.1393i | −208.135 | − | 260.993i | 400.210 | − | 501.847i | −9.74527 | − | 42.6968i | −354.671 | − | 736.482i | −94.5645 | + | 45.5399i | 4578.28 | + | 1044.96i |
See all 96 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
29.e | even | 14 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 29.8.e.a | ✓ | 96 |
29.e | even | 14 | 1 | inner | 29.8.e.a | ✓ | 96 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
29.8.e.a | ✓ | 96 | 1.a | even | 1 | 1 | trivial |
29.8.e.a | ✓ | 96 | 29.e | even | 14 | 1 | inner |
Hecke kernels
This newform subspace is the entire newspace \(S_{8}^{\mathrm{new}}(29, [\chi])\).