Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [29,11,Mod(2,29)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(29, base_ring=CyclotomicField(28))
chi = DirichletCharacter(H, H._module([1]))
N = Newforms(chi, 11, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("29.2");
S:= CuspForms(chi, 11);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 29 \) |
Weight: | \( k \) | \(=\) | \( 11 \) |
Character orbit: | \([\chi]\) | \(=\) | 29.f (of order \(28\), degree \(12\), 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: | \(18.4253603275\) |
Analytic rank: | \(0\) |
Dimension: | \(288\) |
Relative dimension: | \(24\) over \(\Q(\zeta_{28})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{28}]$ |
$q$-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 | −60.4165 | − | 6.80730i | 298.044 | + | 187.273i | 2605.48 | + | 594.685i | −931.971 | + | 743.222i | −16731.9 | − | 13343.3i | −2936.78 | − | 12866.9i | −94601.8 | − | 33102.6i | 28138.4 | + | 58429.9i | 61365.7 | − | 38558.6i |
2.2 | −54.5116 | − | 6.14198i | −196.863 | − | 123.697i | 1935.46 | + | 441.757i | 2084.33 | − | 1662.19i | 9971.57 | + | 7952.06i | 595.728 | + | 2610.06i | −49771.1 | − | 17415.7i | −2166.38 | − | 4498.53i | −123829. | + | 77807.0i |
2.3 | −53.3751 | − | 6.01393i | −211.856 | − | 133.118i | 1814.41 | + | 414.128i | −3826.59 | + | 3051.60i | 10507.3 | + | 8379.28i | −3164.18 | − | 13863.2i | −42438.5 | − | 14849.9i | 1542.20 | + | 3202.42i | 222597. | − | 139867.i |
2.4 | −47.2252 | − | 5.32100i | 125.327 | + | 78.7484i | 1203.58 | + | 274.709i | −564.100 | + | 449.855i | −5499.59 | − | 4385.77i | 7130.90 | + | 31242.5i | −9443.84 | − | 3304.54i | −16114.8 | − | 33462.7i | 29033.4 | − | 18242.9i |
2.5 | −43.8958 | − | 4.94587i | 168.989 | + | 106.183i | 904.055 | + | 206.345i | 4440.32 | − | 3541.03i | −6892.76 | − | 5496.79i | −2779.11 | − | 12176.1i | 4031.70 | + | 1410.75i | −8337.85 | − | 17313.7i | −212425. | + | 133475.i |
2.6 | −35.2938 | − | 3.97666i | 164.902 | + | 103.615i | 231.515 | + | 52.8418i | −2509.40 | + | 2001.18i | −5408.00 | − | 4312.74i | −2684.83 | − | 11763.0i | 26367.7 | + | 9226.46i | −9163.66 | − | 19028.5i | 96524.5 | − | 60650.4i |
2.7 | −25.6712 | − | 2.89245i | −357.922 | − | 224.897i | −347.681 | − | 79.3559i | 1823.02 | − | 1453.81i | 8537.79 | + | 6808.66i | −131.712 | − | 577.070i | 33665.0 | + | 11779.9i | 51908.9 | + | 107790.i | −51004.1 | + | 32048.0i |
2.8 | −24.2302 | − | 2.73009i | −222.660 | − | 139.906i | −418.676 | − | 95.5600i | −3285.43 | + | 2620.04i | 5013.14 | + | 3997.84i | 5541.19 | + | 24277.5i | 33451.3 | + | 11705.1i | 4383.12 | + | 9101.64i | 86759.7 | − | 54514.7i |
2.9 | −23.5047 | − | 2.64834i | −49.6686 | − | 31.2088i | −452.869 | − | 103.365i | 494.026 | − | 393.973i | 1084.79 | + | 865.093i | −3650.42 | − | 15993.5i | 33232.7 | + | 11628.6i | −24127.4 | − | 50101.1i | −12655.3 | + | 7951.86i |
2.10 | −19.3066 | − | 2.17533i | 391.127 | + | 245.761i | −630.313 | − | 143.865i | 593.623 | − | 473.399i | −7016.72 | − | 5595.65i | 1021.49 | + | 4475.44i | 30634.9 | + | 10719.6i | 66961.1 | + | 139046.i | −12490.7 | + | 7848.40i |
2.11 | −4.75908 | − | 0.536219i | 86.5441 | + | 54.3792i | −975.965 | − | 222.758i | 3659.28 | − | 2918.18i | −382.711 | − | 305.202i | 5076.39 | + | 22241.1i | 9154.17 | + | 3203.18i | −21087.6 | − | 43788.9i | −18979.6 | + | 11925.7i |
2.12 | 0.757424 | + | 0.0853412i | 157.893 | + | 99.2110i | −997.760 | − | 227.732i | −3734.49 | + | 2978.16i | 111.125 | + | 88.6196i | −313.352 | − | 1372.88i | −1473.00 | − | 515.426i | −10532.9 | − | 21871.8i | −3082.75 | + | 1937.02i |
2.13 | 7.04486 | + | 0.793765i | −103.202 | − | 64.8458i | −949.326 | − | 216.678i | 1386.51 | − | 1105.71i | −675.568 | − | 538.747i | −710.542 | − | 3113.09i | −13368.1 | − | 4677.69i | −19174.8 | − | 39816.9i | 10645.5 | − | 6688.99i |
2.14 | 12.3156 | + | 1.38763i | −317.080 | − | 199.234i | −848.579 | − | 193.683i | −3318.67 | + | 2646.55i | −3628.55 | − | 2893.67i | −7010.74 | − | 30716.1i | −22160.7 | − | 7754.37i | 35224.8 | + | 73145.1i | −44543.8 | + | 27988.7i |
2.15 | 19.8291 | + | 2.23421i | −232.445 | − | 146.055i | −610.123 | − | 139.257i | −1193.27 | + | 951.598i | −4282.87 | − | 3415.48i | 5593.25 | + | 24505.6i | −31073.9 | − | 10873.2i | 7078.35 | + | 14698.3i | −25787.5 | + | 16203.4i |
2.16 | 21.2097 | + | 2.38976i | 242.606 | + | 152.439i | −554.184 | − | 126.489i | −1573.54 | + | 1254.86i | 4781.31 | + | 3812.97i | 932.676 | + | 4086.32i | −32081.5 | − | 11225.8i | 9999.49 | + | 20764.1i | −36373.3 | + | 22854.9i |
2.17 | 23.5437 | + | 2.65273i | 243.217 | + | 152.824i | −451.060 | − | 102.951i | 3023.75 | − | 2411.36i | 5320.82 | + | 4243.21i | −7168.45 | − | 31407.0i | −33246.3 | − | 11633.4i | 10179.2 | + | 21137.3i | 77586.7 | − | 48751.0i |
2.18 | 34.3912 | + | 3.87495i | −318.343 | − | 200.028i | 169.411 | + | 38.6669i | 4257.16 | − | 3394.97i | −10173.1 | − | 8112.76i | 622.677 | + | 2728.12i | −27774.2 | − | 9718.63i | 35710.5 | + | 74153.6i | 159564. | − | 100261.i |
2.19 | 44.0608 | + | 4.96446i | 271.259 | + | 170.443i | 918.379 | + | 209.614i | 1792.39 | − | 1429.39i | 11105.7 | + | 8856.50i | 4523.66 | + | 19819.4i | −3431.97 | − | 1200.90i | 18910.0 | + | 39267.0i | 86070.4 | − | 54081.6i |
2.20 | 44.2102 | + | 4.98130i | −5.41800 | − | 3.40436i | 931.406 | + | 212.587i | −3133.77 | + | 2499.10i | −222.573 | − | 177.496i | 2520.68 | + | 11043.8i | −2882.49 | − | 1008.63i | −25602.6 | − | 53164.4i | −150993. | + | 94875.4i |
See next 80 embeddings (of 288 total) |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
29.f | odd | 28 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 29.11.f.a | ✓ | 288 |
29.f | odd | 28 | 1 | inner | 29.11.f.a | ✓ | 288 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
29.11.f.a | ✓ | 288 | 1.a | even | 1 | 1 | trivial |
29.11.f.a | ✓ | 288 | 29.f | odd | 28 | 1 | inner |
Hecke kernels
This newform subspace is the entire newspace \(S_{11}^{\mathrm{new}}(29, [\chi])\).