Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [17,14,Mod(2,17)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(17, base_ring=CyclotomicField(8))
chi = DirichletCharacter(H, H._module([7]))
N = Newforms(chi, 14, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("17.2");
S:= CuspForms(chi, 14);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 17 \) |
Weight: | \( k \) | \(=\) | \( 14 \) |
Character orbit: | \([\chi]\) | \(=\) | 17.d (of order \(8\), 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: | \(18.2292579218\) |
Analytic rank: | \(0\) |
Dimension: | \(76\) |
Relative dimension: | \(19\) over \(\Q(\zeta_{8})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{8}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
2.1 | −119.518 | − | 119.518i | 860.523 | − | 356.440i | 20377.2i | −19631.6 | − | 47394.9i | −145449. | − | 60247.1i | −71661.0 | + | 173005.i | 1.45636e6 | − | 1.45636e6i | −513906. | + | 513906.i | −3.31822e6 | + | 8.01089e6i | ||
2.2 | −112.431 | − | 112.431i | −344.690 | + | 142.775i | 17089.3i | 24752.8 | + | 59758.6i | 54806.0 | + | 22701.4i | −106446. | + | 256984.i | 1.00033e6 | − | 1.00033e6i | −1.02893e6 | + | 1.02893e6i | 3.93572e6 | − | 9.50168e6i | ||
2.3 | −105.777 | − | 105.777i | −2066.93 | + | 856.152i | 14185.4i | −10075.4 | − | 24324.3i | 309194. | + | 128073.i | 38475.1 | − | 92887.2i | 633964. | − | 633964.i | 2.41187e6 | − | 2.41187e6i | −1.50719e6 | + | 3.63869e6i | ||
2.4 | −85.9808 | − | 85.9808i | 1709.90 | − | 708.262i | 6593.38i | 10234.5 | + | 24708.2i | −207915. | − | 86121.2i | 131539. | − | 317563.i | −137450. | + | 137450.i | 1.29475e6 | − | 1.29475e6i | 1.24446e6 | − | 3.00439e6i | ||
2.5 | −82.3937 | − | 82.3937i | 15.7751 | − | 6.53428i | 5385.46i | −3541.82 | − | 8550.70i | −1838.16 | − | 761.390i | 149811. | − | 361675.i | −231242. | + | 231242.i | −1.12715e6 | + | 1.12715e6i | −412701. | + | 996348.i | ||
2.6 | −57.7630 | − | 57.7630i | 1786.66 | − | 740.060i | − | 1518.87i | −6597.89 | − | 15928.7i | −145951. | − | 60454.9i | −202896. | + | 489834.i | −560929. | + | 560929.i | 1.51712e6 | − | 1.51712e6i | −538977. | + | 1.30120e6i | |
2.7 | −53.0086 | − | 53.0086i | −1055.64 | + | 437.260i | − | 2572.17i | −1152.06 | − | 2781.33i | 79136.6 | + | 32779.4i | −89142.6 | + | 215209.i | −570594. | + | 570594.i | −204179. | + | 204179.i | −86365.1 | + | 208504.i | |
2.8 | −12.1255 | − | 12.1255i | −1763.94 | + | 730.648i | − | 7897.95i | 22183.3 | + | 53555.3i | 30248.1 | + | 12529.2i | 220223. | − | 531666.i | −195098. | + | 195098.i | 1.45029e6 | − | 1.45029e6i | 380400. | − | 918366.i | |
2.9 | −10.2774 | − | 10.2774i | 768.584 | − | 318.358i | − | 7980.75i | 18240.0 | + | 44035.3i | −11171.0 | − | 4627.18i | −83946.8 | + | 202666.i | −166215. | + | 166215.i | −637987. | + | 637987.i | 265109. | − | 640031.i | |
2.10 | −6.44357 | − | 6.44357i | 278.134 | − | 115.207i | − | 8108.96i | −21890.2 | − | 52847.5i | −2534.52 | − | 1049.83i | 63144.2 | − | 152443.i | −105036. | + | 105036.i | −1.06327e6 | + | 1.06327e6i | −199476. | + | 481577.i | |
2.11 | 19.3378 | + | 19.3378i | 1598.43 | − | 662.090i | − | 7444.10i | 3162.32 | + | 7634.51i | 43713.4 | + | 18106.7i | 140421. | − | 339006.i | 302368. | − | 302368.i | 989248. | − | 989248.i | −86482.3 | + | 208787.i | |
2.12 | 35.2841 | + | 35.2841i | −1889.98 | + | 782.855i | − | 5702.06i | −14202.2 | − | 34287.2i | −94308.6 | − | 39063.9i | −85800.7 | + | 207141.i | 490240. | − | 490240.i | 1.83181e6 | − | 1.83181e6i | 708680. | − | 1.71090e6i | |
2.13 | 43.7143 | + | 43.7143i | −624.013 | + | 258.475i | − | 4370.11i | 8305.42 | + | 20051.1i | −38577.4 | − | 15979.3i | −53765.5 | + | 129801.i | 549145. | − | 549145.i | −804773. | + | 804773.i | −513453. | + | 1.23958e6i | |
2.14 | 73.9767 | + | 73.9767i | 1820.07 | − | 753.899i | 2753.11i | −14191.6 | − | 34261.6i | 190414. | + | 78872.1i | −88983.0 | + | 214824.i | 402351. | − | 402351.i | 1.61695e6 | − | 1.61695e6i | 1.48471e6 | − | 3.58441e6i | ||
2.15 | 78.3701 | + | 78.3701i | 286.744 | − | 118.773i | 4091.76i | 8240.02 | + | 19893.2i | 31780.4 | + | 13163.9i | −71677.4 | + | 173045.i | 321337. | − | 321337.i | −1.05924e6 | + | 1.05924e6i | −913259. | + | 2.20480e6i | ||
2.16 | 90.1042 | + | 90.1042i | −914.334 | + | 378.730i | 8045.55i | −6954.34 | − | 16789.3i | −116511. | − | 48260.2i | 215725. | − | 520806.i | 13196.1 | − | 13196.1i | −434786. | + | 434786.i | 886168. | − | 2.13940e6i | ||
2.17 | 114.094 | + | 114.094i | −2071.82 | + | 858.176i | 17842.8i | 18497.9 | + | 44657.9i | −334294. | − | 138469.i | −142277. | + | 343487.i | −1.10109e6 | + | 1.10109e6i | 2.42861e6 | − | 2.42861e6i | −2.98469e6 | + | 7.20568e6i | ||
2.18 | 115.717 | + | 115.717i | 1642.47 | − | 680.335i | 18588.8i | 20808.7 | + | 50236.6i | 268789. | + | 111336.i | 141600. | − | 341853.i | −1.20309e6 | + | 1.20309e6i | 1.10751e6 | − | 1.10751e6i | −3.40531e6 | + | 8.22115e6i | ||
2.19 | 119.375 | + | 119.375i | 54.9833 | − | 22.7748i | 20308.8i | −19729.1 | − | 47630.4i | 9282.36 | + | 3844.88i | −102559. | + | 247601.i | −1.44644e6 | + | 1.44644e6i | −1.12485e6 | + | 1.12485e6i | 3.33071e6 | − | 8.04104e6i | ||
8.1 | −125.778 | + | 125.778i | −585.692 | + | 1413.99i | − | 23448.2i | −37851.1 | − | 15678.5i | −104181. | − | 251515.i | −47682.6 | + | 19750.8i | 1.91889e6 | + | 1.91889e6i | −528964. | − | 528964.i | 6.73284e6 | − | 2.78883e6i | |
See all 76 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
17.d | even | 8 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 17.14.d.a | ✓ | 76 |
17.d | even | 8 | 1 | inner | 17.14.d.a | ✓ | 76 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
17.14.d.a | ✓ | 76 | 1.a | even | 1 | 1 | trivial |
17.14.d.a | ✓ | 76 | 17.d | even | 8 | 1 | inner |
Hecke kernels
This newform subspace is the entire newspace \(S_{14}^{\mathrm{new}}(17, [\chi])\).