Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [75,9,Mod(11,75)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(75, base_ring=CyclotomicField(10))
chi = DirichletCharacter(H, H._module([5, 8]))
N = Newforms(chi, 9, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("75.11");
S:= CuspForms(chi, 9);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 75 = 3 \cdot 5^{2} \) |
Weight: | \( k \) | \(=\) | \( 9 \) |
Character orbit: | \([\chi]\) | \(=\) | 75.j (of order \(10\), 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: | \(30.5533957546\) |
Analytic rank: | \(0\) |
Dimension: | \(312\) |
Relative dimension: | \(78\) over \(\Q(\zeta_{10})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{10}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
11.1 | −18.2305 | − | 25.0921i | −80.9595 | − | 2.56149i | −218.155 | + | 671.412i | −34.5911 | + | 624.042i | 1411.66 | + | 2078.14i | −3717.22 | 13272.8 | − | 4312.60i | 6547.88 | + | 414.754i | 16289.1 | − | 10508.6i | ||
11.2 | −17.9702 | − | 24.7339i | −3.85988 | + | 80.9080i | −209.728 | + | 645.475i | 609.095 | + | 140.101i | 2070.53 | − | 1358.46i | 3815.59 | 12290.4 | − | 3993.39i | −6531.20 | − | 624.590i | −7480.33 | − | 17582.9i | ||
11.3 | −17.6326 | − | 24.2691i | 12.6048 | + | 80.0132i | −198.976 | + | 612.384i | −559.573 | − | 278.393i | 1719.60 | − | 1716.75i | −2015.68 | 11066.8 | − | 3595.82i | −6243.24 | + | 2017.11i | 3110.35 | + | 18489.2i | ||
11.4 | −17.3034 | − | 23.8161i | 79.9396 | + | 13.0637i | −188.690 | + | 580.728i | 518.937 | − | 348.324i | −1072.10 | − | 2129.89i | −3638.55 | 9928.28 | − | 3225.89i | 6219.68 | + | 2088.62i | −17275.1 | − | 6331.84i | ||
11.5 | −17.0444 | − | 23.4596i | −29.5485 | − | 75.4181i | −180.733 | + | 556.239i | −601.366 | − | 170.248i | −1265.64 | + | 1978.65i | 1578.44 | 9069.54 | − | 2946.87i | −4814.77 | + | 4456.98i | 6255.98 | + | 17009.6i | ||
11.6 | −16.9218 | − | 23.2908i | 40.8105 | − | 69.9679i | −177.008 | + | 544.774i | 356.232 | + | 513.541i | −2320.20 | + | 233.469i | 1515.19 | 8674.23 | − | 2818.43i | −3230.00 | − | 5710.85i | 5932.71 | − | 16987.0i | ||
11.7 | −16.6151 | − | 22.8688i | 76.4502 | − | 26.7649i | −167.810 | + | 516.465i | −553.809 | + | 289.691i | −1882.31 | − | 1303.62i | −219.465 | 7716.82 | − | 2507.35i | 5128.28 | − | 4092.37i | 15826.5 | + | 7851.68i | ||
11.8 | −16.3948 | − | 22.5655i | −80.8710 | − | 4.56896i | −161.305 | + | 496.445i | 295.172 | − | 550.907i | 1222.76 | + | 1899.80i | 1770.00 | 7056.09 | − | 2292.66i | 6519.25 | + | 738.994i | −17270.8 | + | 2371.32i | ||
11.9 | −15.8525 | − | 21.8191i | −5.79551 | − | 80.7924i | −145.663 | + | 448.304i | 426.206 | − | 457.136i | −1670.94 | + | 1407.21i | −1651.43 | 5524.34 | − | 1794.97i | −6493.82 | + | 936.466i | −16730.7 | − | 2052.68i | ||
11.10 | −14.9990 | − | 20.6444i | 74.9382 | + | 30.7451i | −122.112 | + | 375.821i | −233.811 | − | 579.619i | −489.288 | − | 2008.20i | 4241.10 | 3377.31 | − | 1097.36i | 4670.48 | + | 4607.96i | −8458.94 | + | 13520.6i | ||
11.11 | −14.0772 | − | 19.3756i | −55.8311 | + | 58.6847i | −98.1381 | + | 302.038i | −216.170 | − | 586.426i | 1923.00 | + | 255.644i | −1564.18 | 1402.67 | − | 455.754i | −326.786 | − | 6552.86i | −8319.29 | + | 12443.7i | ||
11.12 | −13.8207 | − | 19.0226i | 34.2136 | + | 73.4196i | −91.7383 | + | 282.341i | −323.695 | + | 534.646i | 923.774 | − | 1665.54i | −716.299 | 913.979 | − | 296.970i | −4219.86 | + | 5023.89i | 14644.1 | − | 1231.69i | ||
11.13 | −13.7113 | − | 18.8720i | −65.6121 | + | 47.4980i | −89.0441 | + | 274.050i | −427.105 | + | 456.297i | 1796.01 | + | 586.972i | 3153.01 | 713.322 | − | 231.773i | 2048.89 | − | 6232.88i | 14467.4 | + | 1803.89i | ||
11.14 | −13.0609 | − | 17.9768i | −58.5411 | − | 55.9816i | −73.4700 | + | 226.118i | 333.636 | + | 528.500i | −241.770 | + | 1783.55i | 2630.47 | −385.592 | + | 125.287i | 293.121 | + | 6554.45i | 5143.16 | − | 12900.4i | ||
11.15 | −13.0487 | − | 17.9600i | 76.2368 | + | 27.3669i | −73.1849 | + | 225.240i | 326.265 | + | 533.081i | −503.283 | − | 1726.32i | 791.787 | −404.710 | + | 131.498i | 5063.11 | + | 4172.73i | 5316.81 | − | 12815.8i | ||
11.16 | −12.6564 | − | 17.4201i | −41.7102 | + | 69.4353i | −64.1661 | + | 197.483i | 571.801 | + | 252.327i | 1737.47 | − | 152.208i | −2610.02 | −990.225 | + | 321.744i | −3081.52 | − | 5792.32i | −2841.41 | − | 13154.4i | ||
11.17 | −11.9719 | − | 16.4779i | 55.6718 | − | 58.8358i | −49.0862 | + | 151.072i | −314.756 | − | 539.957i | −1635.98 | − | 212.979i | −1258.15 | −1881.96 | + | 611.485i | −362.291 | − | 6550.99i | −5129.12 | + | 11650.8i | ||
11.18 | −11.3839 | − | 15.6686i | −62.4373 | − | 51.6002i | −36.8032 | + | 113.269i | 619.683 | + | 81.3505i | −97.7219 | + | 1565.72i | −1394.12 | −2521.68 | + | 819.342i | 1235.84 | + | 6443.56i | −5779.76 | − | 10635.6i | ||
11.19 | −11.3177 | − | 15.5774i | −6.75329 | − | 80.7180i | −35.4584 | + | 109.130i | −344.132 | + | 521.726i | −1180.95 | + | 1018.74i | −4430.09 | −2586.70 | + | 840.469i | −6469.79 | + | 1090.22i | 12021.9 | − | 544.023i | ||
11.20 | −10.5050 | − | 14.4588i | −77.0617 | − | 24.9499i | −19.5955 | + | 60.3088i | −616.818 | − | 100.797i | 448.784 | + | 1376.32i | −818.953 | −3273.49 | + | 1063.62i | 5316.01 | + | 3845.36i | 5022.24 | + | 9977.35i | ||
See next 80 embeddings (of 312 total) |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
3.b | odd | 2 | 1 | inner |
25.d | even | 5 | 1 | inner |
75.j | odd | 10 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 75.9.j.a | ✓ | 312 |
3.b | odd | 2 | 1 | inner | 75.9.j.a | ✓ | 312 |
25.d | even | 5 | 1 | inner | 75.9.j.a | ✓ | 312 |
75.j | odd | 10 | 1 | inner | 75.9.j.a | ✓ | 312 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
75.9.j.a | ✓ | 312 | 1.a | even | 1 | 1 | trivial |
75.9.j.a | ✓ | 312 | 3.b | odd | 2 | 1 | inner |
75.9.j.a | ✓ | 312 | 25.d | even | 5 | 1 | inner |
75.9.j.a | ✓ | 312 | 75.j | odd | 10 | 1 | inner |
Hecke kernels
This newform subspace is the entire newspace \(S_{9}^{\mathrm{new}}(75, [\chi])\).