Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [69,7,Mod(47,69)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(69, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([1, 0]))
N = Newforms(chi, 7, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("69.47");
S:= CuspForms(chi, 7);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 69 = 3 \cdot 23 \) |
Weight: | \( k \) | \(=\) | \( 7 \) |
Character orbit: | \([\chi]\) | \(=\) | 69.b (of order \(2\), degree \(1\), 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: | \(15.8737317698\) |
Analytic rank: | \(0\) |
Dimension: | \(44\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{2}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
47.1 | − | 15.2045i | −26.4076 | + | 5.62468i | −167.177 | − | 4.50276i | 85.5205 | + | 401.515i | 443.657 | 1568.76i | 665.726 | − | 297.069i | −68.4623 | ||||||||||
47.2 | − | 14.6275i | 2.75073 | + | 26.8595i | −149.963 | − | 100.828i | 392.887 | − | 40.2362i | −263.111 | 1257.42i | −713.867 | + | 147.767i | −1474.86 | ||||||||||
47.3 | − | 14.4029i | 25.1960 | + | 9.70372i | −143.443 | 103.914i | 139.762 | − | 362.895i | 470.752 | 1144.21i | 540.676 | + | 488.990i | 1496.66 | |||||||||||
47.4 | − | 14.0722i | 4.93651 | − | 26.5449i | −134.028 | 199.248i | −373.546 | − | 69.4677i | 69.1270 | 985.449i | −680.262 | − | 262.078i | 2803.87 | |||||||||||
47.5 | − | 14.0000i | −13.7042 | − | 23.2636i | −132.001 | − | 135.214i | −325.691 | + | 191.860i | −169.403 | 952.014i | −353.387 | + | 637.619i | −1893.00 | ||||||||||
47.6 | − | 13.1440i | 26.0223 | − | 7.20015i | −108.766 | − | 47.7180i | −94.6390 | − | 342.037i | −438.649 | 588.401i | 625.316 | − | 374.728i | −627.206 | ||||||||||
47.7 | − | 12.3777i | −26.6682 | + | 4.21999i | −89.2071 | 195.553i | 52.2337 | + | 330.090i | −657.659 | 312.006i | 693.383 | − | 225.079i | 2420.50 | |||||||||||
47.8 | − | 11.0694i | −6.47589 | + | 26.2119i | −58.5309 | 94.9815i | 290.149 | + | 71.6840i | 96.5626 | − | 60.5391i | −645.126 | − | 339.491i | 1051.39 | ||||||||||
47.9 | − | 10.2941i | 12.6170 | − | 23.8707i | −41.9690 | − | 93.4381i | −245.728 | − | 129.881i | 330.565 | − | 226.790i | −410.623 | − | 602.354i | −961.863 | |||||||||
47.10 | − | 9.87745i | 22.8275 | + | 14.4190i | −33.5640 | − | 220.404i | 142.423 | − | 225.477i | 473.108 | − | 300.630i | 313.187 | + | 658.297i | −2177.03 | |||||||||
47.11 | − | 9.52809i | −24.3830 | + | 11.5961i | −26.7846 | − | 179.250i | 110.488 | + | 232.324i | −22.6651 | − | 354.592i | 460.063 | − | 565.494i | −1707.91 | |||||||||
47.12 | − | 8.50016i | −23.8542 | − | 12.6482i | −8.25265 | 65.5840i | −107.512 | + | 202.765i | 552.822 | − | 473.861i | 409.047 | + | 603.425i | 557.474 | ||||||||||
47.13 | − | 8.07729i | 18.6168 | + | 19.5554i | −1.24263 | 135.340i | 157.955 | − | 150.374i | −41.9690 | − | 506.910i | −35.8265 | + | 728.119i | 1093.18 | ||||||||||
47.14 | − | 7.89562i | −18.6193 | − | 19.5530i | 1.65917 | 10.5412i | −154.383 | + | 147.011i | −269.224 | − | 518.420i | −35.6409 | + | 728.128i | 83.2291 | ||||||||||
47.15 | − | 6.32520i | 24.9999 | − | 10.1983i | 23.9919 | 138.173i | −64.5060 | − | 158.129i | −112.869 | − | 556.566i | 520.991 | − | 509.911i | 873.969 | ||||||||||
47.16 | − | 5.20732i | 17.8105 | + | 20.2925i | 36.8838 | − | 125.797i | 105.670 | − | 92.7449i | −679.000 | − | 525.334i | −94.5737 | + | 722.839i | −655.063 | |||||||||
47.17 | − | 4.53332i | −0.406897 | − | 26.9969i | 43.4490 | 169.748i | −122.386 | + | 1.84460i | −134.238 | − | 487.101i | −728.669 | + | 21.9700i | 769.522 | ||||||||||
47.18 | − | 4.32001i | −14.6931 | + | 22.6520i | 45.3375 | − | 48.7369i | 97.8569 | + | 63.4743i | −61.9430 | − | 472.339i | −297.226 | − | 665.656i | −210.544 | |||||||||
47.19 | − | 2.95357i | 2.97092 | − | 26.8361i | 55.2764 | − | 175.802i | −79.2621 | − | 8.77482i | −435.351 | − | 352.291i | −711.347 | − | 159.456i | −519.242 | |||||||||
47.20 | − | 1.98217i | 25.5387 | − | 8.76199i | 60.0710 | − | 91.3581i | −17.3677 | − | 50.6220i | 169.692 | − | 245.929i | 575.455 | − | 447.541i | −181.087 | |||||||||
See all 44 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
3.b | odd | 2 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 69.7.b.a | ✓ | 44 |
3.b | odd | 2 | 1 | inner | 69.7.b.a | ✓ | 44 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
69.7.b.a | ✓ | 44 | 1.a | even | 1 | 1 | trivial |
69.7.b.a | ✓ | 44 | 3.b | odd | 2 | 1 | inner |
Hecke kernels
This newform subspace is the entire newspace \(S_{7}^{\mathrm{new}}(69, [\chi])\).