Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [43,9,Mod(7,43)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(43, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([5]))
N = Newforms(chi, 9, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("43.7");
S:= CuspForms(chi, 9);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 43 \) |
Weight: | \( k \) | \(=\) | \( 9 \) |
Character orbit: | \([\chi]\) | \(=\) | 43.d (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: | \(17.5172802326\) |
Analytic rank: | \(0\) |
Dimension: | \(56\) |
Relative dimension: | \(28\) 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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
7.1 | − | 31.1706i | −22.2576 | + | 12.8504i | −715.604 | 466.420 | − | 269.288i | 400.555 | + | 693.781i | 2878.38 | + | 1661.83i | 14326.1i | −2950.23 | + | 5109.95i | −8393.86 | − | 14538.6i | |||||
7.2 | − | 29.0831i | −96.3585 | + | 55.6326i | −589.829 | −695.509 | + | 401.552i | 1617.97 | + | 2802.41i | −1511.66 | − | 872.755i | 9708.78i | 2909.48 | − | 5039.36i | 11678.4 | + | 20227.6i | |||||
7.3 | − | 26.9022i | 74.3127 | − | 42.9044i | −467.727 | −575.075 | + | 332.020i | −1154.22 | − | 1999.17i | −1517.67 | − | 876.227i | 5695.91i | 401.081 | − | 694.692i | 8932.05 | + | 15470.8i | |||||
7.4 | − | 25.0164i | 105.358 | − | 60.8284i | −369.821 | 539.881 | − | 311.700i | −1521.71 | − | 2635.68i | 1531.35 | + | 884.128i | 2847.41i | 4119.69 | − | 7135.51i | −7797.62 | − | 13505.9i | |||||
7.5 | − | 23.8148i | −40.9385 | + | 23.6358i | −311.143 | 728.687 | − | 420.708i | 562.882 | + | 974.940i | −3233.30 | − | 1866.75i | 1313.23i | −2163.19 | + | 3746.76i | −10019.1 | − | 17353.5i | |||||
7.6 | − | 21.4690i | −45.9397 | + | 26.5233i | −204.918 | −326.905 | + | 188.739i | 569.429 | + | 986.280i | 2218.06 | + | 1280.60i | − | 1096.68i | −1873.53 | + | 3245.04i | 4052.03 | + | 7018.32i | ||||
7.7 | − | 18.7653i | −129.819 | + | 74.9511i | −96.1366 | 343.803 | − | 198.495i | 1406.48 | + | 2436.10i | 1031.87 | + | 595.751i | − | 2999.89i | 7954.84 | − | 13778.2i | −3724.82 | − | 6451.57i | ||||
7.8 | − | 15.0304i | 10.2856 | − | 5.93838i | 30.0883 | −349.334 | + | 201.688i | −89.2560 | − | 154.596i | 764.498 | + | 441.383i | − | 4300.01i | −3209.97 | + | 5559.83i | 3031.45 | + | 5250.62i | ||||
7.9 | − | 13.9154i | 49.0781 | − | 28.3352i | 62.3606 | 389.929 | − | 225.126i | −394.297 | − | 682.943i | −1302.81 | − | 752.179i | − | 4430.13i | −1674.73 | + | 2900.72i | −3132.72 | − | 5426.03i | ||||
7.10 | − | 9.44319i | −73.7010 | + | 42.5513i | 166.826 | −1017.04 | + | 587.188i | 401.820 | + | 695.972i | −3125.69 | − | 1804.62i | − | 3992.83i | 340.724 | − | 590.151i | 5544.92 | + | 9604.09i | ||||
7.11 | − | 8.98674i | 117.071 | − | 67.5912i | 175.239 | −869.912 | + | 502.244i | −607.424 | − | 1052.09i | 3700.56 | + | 2136.52i | − | 3875.43i | 5856.64 | − | 10144.0i | 4513.54 | + | 7817.67i | ||||
7.12 | − | 7.02236i | 126.337 | − | 72.9406i | 206.686 | 353.829 | − | 204.284i | −512.215 | − | 887.182i | −2295.79 | − | 1325.47i | − | 3249.15i | 7360.15 | − | 12748.2i | −1434.55 | − | 2484.72i | ||||
7.13 | − | 5.01466i | 14.2615 | − | 8.23388i | 230.853 | 929.674 | − | 536.748i | −41.2901 | − | 71.5166i | 3070.94 | + | 1773.01i | − | 2441.40i | −3144.91 | + | 5447.14i | −2691.61 | − | 4662.00i | ||||
7.14 | − | 4.04858i | −89.5347 | + | 51.6929i | 239.609 | −50.9246 | + | 29.4013i | 209.283 | + | 362.488i | 839.413 | + | 484.635i | − | 2006.51i | 2063.81 | − | 3574.63i | 119.034 | + | 206.172i | ||||
7.15 | 0.0978549i | 50.1238 | − | 28.9390i | 255.990 | −516.114 | + | 297.978i | 2.83182 | + | 4.90486i | −3014.43 | − | 1740.38i | 50.1008i | −1605.57 | + | 2780.93i | −29.1587 | − | 50.5043i | ||||||
7.16 | 0.360163i | −80.5202 | + | 46.4884i | 255.870 | 539.177 | − | 311.294i | −16.7434 | − | 29.0004i | −1661.31 | − | 959.156i | 184.357i | 1041.83 | − | 1804.51i | 112.117 | + | 194.192i | ||||||
7.17 | 7.28626i | 47.2038 | − | 27.2531i | 202.910 | −237.272 | + | 136.989i | 198.573 | + | 343.939i | 742.931 | + | 428.932i | 3343.74i | −1795.04 | + | 3109.09i | −998.137 | − | 1728.82i | ||||||
7.18 | 9.49614i | −65.9694 | + | 38.0875i | 165.823 | −622.087 | + | 359.162i | −361.684 | − | 626.455i | 2996.66 | + | 1730.12i | 4005.69i | −379.191 | + | 656.777i | −3410.65 | − | 5907.42i | ||||||
7.19 | 13.2055i | 103.620 | − | 59.8253i | 81.6138 | 541.475 | − | 312.621i | 790.025 | + | 1368.36i | 754.120 | + | 435.391i | 4458.37i | 3877.63 | − | 6716.26i | 4128.32 | + | 7150.47i | ||||||
7.20 | 13.4706i | −22.2334 | + | 12.8365i | 74.5432 | 377.208 | − | 217.781i | −172.915 | − | 299.497i | −2801.83 | − | 1617.64i | 4452.61i | −2950.95 | + | 5111.20i | 2933.64 | + | 5081.22i | ||||||
See all 56 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
43.d | odd | 6 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 43.9.d.a | ✓ | 56 |
43.d | odd | 6 | 1 | inner | 43.9.d.a | ✓ | 56 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
43.9.d.a | ✓ | 56 | 1.a | even | 1 | 1 | trivial |
43.9.d.a | ✓ | 56 | 43.d | odd | 6 | 1 | inner |
Hecke kernels
This newform subspace is the entire newspace \(S_{9}^{\mathrm{new}}(43, [\chi])\).