Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [75,9,Mod(13,75)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(75, base_ring=CyclotomicField(20))
chi = DirichletCharacter(H, H._module([0, 19]))
N = Newforms(chi, 9, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("75.13");
S:= CuspForms(chi, 9);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 75 = 3 \cdot 5^{2} \) |
Weight: | \( k \) | \(=\) | \( 9 \) |
Character orbit: | \([\chi]\) | \(=\) | 75.k (of order \(20\), degree \(8\), 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: | \(30.5533957546\) |
Analytic rank: | \(0\) |
Dimension: | \(320\) |
Relative dimension: | \(40\) over \(\Q(\zeta_{20})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{20}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
13.1 | −31.1680 | + | 4.93653i | −21.2310 | + | 41.6683i | 703.605 | − | 228.615i | 559.718 | + | 278.101i | 456.033 | − | 1403.52i | 1699.82 | − | 1699.82i | −13603.4 | + | 6931.29i | −1285.49 | − | 1769.32i | −18818.2 | − | 5904.80i |
13.2 | −29.6660 | + | 4.69863i | 21.2310 | − | 41.6683i | 614.522 | − | 199.670i | −583.809 | − | 223.142i | −434.056 | + | 1335.89i | 2556.62 | − | 2556.62i | −10441.1 | + | 5320.03i | −1285.49 | − | 1769.32i | 18367.7 | + | 3876.61i |
13.3 | −28.6989 | + | 4.54547i | −21.2310 | + | 41.6683i | 559.498 | − | 181.792i | −53.0876 | − | 622.741i | 419.907 | − | 1292.34i | −1163.90 | + | 1163.90i | −8602.90 | + | 4383.40i | −1285.49 | − | 1769.32i | 4354.21 | + | 17630.7i |
13.4 | −27.5831 | + | 4.36873i | 21.2310 | − | 41.6683i | 498.269 | − | 161.897i | 111.872 | + | 614.906i | −403.580 | + | 1242.09i | −1146.98 | + | 1146.98i | −6666.44 | + | 3396.72i | −1285.49 | − | 1769.32i | −5772.14 | − | 16472.3i |
13.5 | −26.6845 | + | 4.22641i | 21.2310 | − | 41.6683i | 450.730 | − | 146.451i | 365.591 | − | 506.921i | −390.433 | + | 1201.63i | 516.394 | − | 516.394i | −5246.00 | + | 2672.97i | −1285.49 | − | 1769.32i | −7613.15 | + | 15072.1i |
13.6 | −24.6881 | + | 3.91021i | −21.2310 | + | 41.6683i | 350.742 | − | 113.963i | −624.780 | − | 16.5691i | 361.222 | − | 1111.73i | 1512.89 | − | 1512.89i | −2512.05 | + | 1279.95i | −1285.49 | − | 1769.32i | 15489.4 | − | 2033.96i |
13.7 | −22.6671 | + | 3.59011i | −21.2310 | + | 41.6683i | 257.437 | − | 83.6462i | −191.814 | + | 594.838i | 331.652 | − | 1020.72i | −1074.42 | + | 1074.42i | −300.281 | + | 153.001i | −1285.49 | − | 1769.32i | 2212.32 | − | 14171.9i |
13.8 | −21.3788 | + | 3.38607i | −21.2310 | + | 41.6683i | 202.117 | − | 65.6718i | 611.912 | + | 127.235i | 312.802 | − | 962.707i | −2654.08 | + | 2654.08i | 838.588 | − | 427.282i | −1285.49 | − | 1769.32i | −13512.8 | − | 648.155i |
13.9 | −20.1080 | + | 3.18480i | 21.2310 | − | 41.6683i | 150.719 | − | 48.9717i | −314.176 | − | 540.295i | −294.209 | + | 905.483i | −3324.52 | + | 3324.52i | 1769.06 | − | 901.383i | −1285.49 | − | 1769.32i | 8038.19 | + | 9863.67i |
13.10 | −19.1290 | + | 3.02974i | 21.2310 | − | 41.6683i | 113.269 | − | 36.8032i | −568.921 | + | 258.756i | −279.885 | + | 861.396i | 339.914 | − | 339.914i | 2362.46 | − | 1203.73i | −1285.49 | − | 1769.32i | 10098.9 | − | 6673.41i |
13.11 | −16.6143 | + | 2.63144i | 21.2310 | − | 41.6683i | 25.6393 | − | 8.33071i | 450.950 | + | 432.746i | −243.091 | + | 748.156i | 2407.36 | − | 2407.36i | 3432.86 | − | 1749.13i | −1285.49 | − | 1769.32i | −8630.96 | − | 6003.11i |
13.12 | −14.4958 | + | 2.29591i | −21.2310 | + | 41.6683i | −38.6136 | + | 12.5463i | 555.364 | − | 286.698i | 212.094 | − | 652.759i | 2517.93 | − | 2517.93i | 3878.60 | − | 1976.25i | −1285.49 | − | 1769.32i | −7392.21 | + | 5430.98i |
13.13 | −13.4052 | + | 2.12318i | 21.2310 | − | 41.6683i | −68.2788 | + | 22.1851i | 619.380 | − | 83.6279i | −196.137 | + | 603.649i | −1056.12 | + | 1056.12i | 3964.00 | − | 2019.76i | −1285.49 | − | 1769.32i | −8125.36 | + | 2436.10i |
13.14 | −11.6504 | + | 1.84524i | −21.2310 | + | 41.6683i | −111.144 | + | 36.1127i | −559.413 | − | 278.715i | 170.462 | − | 524.628i | 285.116 | − | 285.116i | 3918.78 | − | 1996.72i | −1285.49 | − | 1769.32i | 7031.68 | + | 2214.88i |
13.15 | −9.28548 | + | 1.47068i | −21.2310 | + | 41.6683i | −159.413 | + | 51.7965i | −136.341 | − | 609.948i | 135.860 | − | 418.134i | −1847.56 | + | 1847.56i | 3548.45 | − | 1808.03i | −1285.49 | − | 1769.32i | 2163.02 | + | 5463.14i |
13.16 | −9.06505 | + | 1.43576i | 21.2310 | − | 41.6683i | −163.357 | + | 53.0778i | −123.101 | − | 612.757i | −132.635 | + | 408.208i | 2061.95 | − | 2061.95i | 3498.12 | − | 1782.38i | −1285.49 | − | 1769.32i | 1995.69 | + | 5377.93i |
13.17 | −6.59120 | + | 1.04394i | −21.2310 | + | 41.6683i | −201.116 | + | 65.3467i | −166.950 | + | 602.290i | 96.4387 | − | 296.808i | 872.910 | − | 872.910i | 2779.56 | − | 1416.26i | −1285.49 | − | 1769.32i | 471.642 | − | 4144.10i |
13.18 | −5.04321 | + | 0.798766i | −21.2310 | + | 41.6683i | −218.674 | + | 71.0517i | 357.372 | + | 512.748i | 73.7894 | − | 227.101i | 267.664 | − | 267.664i | 2210.75 | − | 1126.43i | −1285.49 | − | 1769.32i | −2211.87 | − | 2300.44i |
13.19 | −2.33258 | + | 0.369444i | 21.2310 | − | 41.6683i | −238.166 | + | 77.3848i | 379.135 | − | 496.872i | −34.1290 | + | 105.038i | −309.158 | + | 309.158i | 1065.64 | − | 542.970i | −1285.49 | − | 1769.32i | −700.795 | + | 1299.06i |
13.20 | −0.0204195 | + | 0.00323412i | 21.2310 | − | 41.6683i | −243.470 | + | 79.1082i | 217.278 | + | 586.017i | −0.298766 | + | 0.919507i | −2757.81 | + | 2757.81i | 9.43137 | − | 4.80552i | −1285.49 | − | 1769.32i | −6.33194 | − | 11.2634i |
See next 80 embeddings (of 320 total) |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
25.f | odd | 20 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 75.9.k.a | ✓ | 320 |
25.f | odd | 20 | 1 | inner | 75.9.k.a | ✓ | 320 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
75.9.k.a | ✓ | 320 | 1.a | even | 1 | 1 | trivial |
75.9.k.a | ✓ | 320 | 25.f | odd | 20 | 1 | inner |
Hecke kernels
This newform subspace is the entire newspace \(S_{9}^{\mathrm{new}}(75, [\chi])\).