Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [192,7,Mod(17,192)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(192, base_ring=CyclotomicField(4))
chi = DirichletCharacter(H, H._module([0, 3, 2]))
N = Newforms(chi, 7, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("192.17");
S:= CuspForms(chi, 7);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 192 = 2^{6} \cdot 3 \) |
Weight: | \( k \) | \(=\) | \( 7 \) |
Character orbit: | \([\chi]\) | \(=\) | 192.i (of order \(4\), degree \(2\), not 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: | \(44.1703840550\) |
Analytic rank: | \(0\) |
Dimension: | \(92\) |
Relative dimension: | \(46\) over \(\Q(i)\) |
Twist minimal: | no (minimal twist has level 48) |
Sato-Tate group: | $\mathrm{SU}(2)[C_{4}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
17.1 | 0 | −26.8720 | − | 2.62588i | 0 | 141.081 | + | 141.081i | 0 | 308.364i | 0 | 715.209 | + | 141.126i | 0 | ||||||||||||
17.2 | 0 | −26.7985 | − | 3.29233i | 0 | 59.2583 | + | 59.2583i | 0 | − | 2.09133i | 0 | 707.321 | + | 176.459i | 0 | |||||||||||
17.3 | 0 | −26.4520 | + | 5.41236i | 0 | 54.1549 | + | 54.1549i | 0 | − | 639.122i | 0 | 670.413 | − | 286.335i | 0 | |||||||||||
17.4 | 0 | −26.4071 | + | 5.62718i | 0 | −81.2421 | − | 81.2421i | 0 | 51.0073i | 0 | 665.670 | − | 297.195i | 0 | ||||||||||||
17.5 | 0 | −26.3911 | − | 5.70179i | 0 | −89.3741 | − | 89.3741i | 0 | − | 331.991i | 0 | 663.979 | + | 300.953i | 0 | |||||||||||
17.6 | 0 | −26.3319 | + | 5.96897i | 0 | −73.2326 | − | 73.2326i | 0 | 386.845i | 0 | 657.743 | − | 314.349i | 0 | ||||||||||||
17.7 | 0 | −23.2271 | + | 13.7659i | 0 | 133.976 | + | 133.976i | 0 | − | 25.6967i | 0 | 350.001 | − | 639.485i | 0 | |||||||||||
17.8 | 0 | −22.8925 | − | 14.3156i | 0 | −4.45378 | − | 4.45378i | 0 | 601.125i | 0 | 319.129 | + | 655.437i | 0 | ||||||||||||
17.9 | 0 | −20.6842 | − | 17.3541i | 0 | −77.8915 | − | 77.8915i | 0 | − | 310.530i | 0 | 126.674 | + | 717.910i | 0 | |||||||||||
17.10 | 0 | −20.4285 | − | 17.6543i | 0 | −132.844 | − | 132.844i | 0 | − | 91.8613i | 0 | 105.651 | + | 721.304i | 0 | |||||||||||
17.11 | 0 | −19.9194 | + | 18.2268i | 0 | 14.1816 | + | 14.1816i | 0 | 191.419i | 0 | 64.5655 | − | 726.135i | 0 | ||||||||||||
17.12 | 0 | −18.9466 | + | 19.2360i | 0 | −148.490 | − | 148.490i | 0 | 263.428i | 0 | −11.0511 | − | 728.916i | 0 | ||||||||||||
17.13 | 0 | −17.6543 | − | 20.4285i | 0 | 132.844 | + | 132.844i | 0 | − | 91.8613i | 0 | −105.651 | + | 721.304i | 0 | |||||||||||
17.14 | 0 | −17.3541 | − | 20.6842i | 0 | 77.8915 | + | 77.8915i | 0 | − | 310.530i | 0 | −126.674 | + | 717.910i | 0 | |||||||||||
17.15 | 0 | −15.0602 | + | 22.4096i | 0 | 13.2524 | + | 13.2524i | 0 | − | 437.375i | 0 | −275.382 | − | 674.986i | 0 | |||||||||||
17.16 | 0 | −14.3156 | − | 22.8925i | 0 | 4.45378 | + | 4.45378i | 0 | 601.125i | 0 | −319.129 | + | 655.437i | 0 | ||||||||||||
17.17 | 0 | −12.2138 | + | 24.0795i | 0 | −131.879 | − | 131.879i | 0 | − | 543.561i | 0 | −430.647 | − | 588.204i | 0 | |||||||||||
17.18 | 0 | −9.91420 | + | 25.1139i | 0 | 41.0809 | + | 41.0809i | 0 | − | 200.806i | 0 | −532.417 | − | 497.969i | 0 | |||||||||||
17.19 | 0 | −9.03029 | + | 25.4451i | 0 | 65.5814 | + | 65.5814i | 0 | 515.866i | 0 | −565.908 | − | 459.554i | 0 | ||||||||||||
17.20 | 0 | −5.70179 | − | 26.3911i | 0 | 89.3741 | + | 89.3741i | 0 | − | 331.991i | 0 | −663.979 | + | 300.953i | 0 | |||||||||||
See all 92 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
3.b | odd | 2 | 1 | inner |
16.e | even | 4 | 1 | inner |
48.i | odd | 4 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 192.7.i.a | 92 | |
3.b | odd | 2 | 1 | inner | 192.7.i.a | 92 | |
4.b | odd | 2 | 1 | 48.7.i.a | ✓ | 92 | |
12.b | even | 2 | 1 | 48.7.i.a | ✓ | 92 | |
16.e | even | 4 | 1 | inner | 192.7.i.a | 92 | |
16.f | odd | 4 | 1 | 48.7.i.a | ✓ | 92 | |
48.i | odd | 4 | 1 | inner | 192.7.i.a | 92 | |
48.k | even | 4 | 1 | 48.7.i.a | ✓ | 92 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
48.7.i.a | ✓ | 92 | 4.b | odd | 2 | 1 | |
48.7.i.a | ✓ | 92 | 12.b | even | 2 | 1 | |
48.7.i.a | ✓ | 92 | 16.f | odd | 4 | 1 | |
48.7.i.a | ✓ | 92 | 48.k | even | 4 | 1 | |
192.7.i.a | 92 | 1.a | even | 1 | 1 | trivial | |
192.7.i.a | 92 | 3.b | odd | 2 | 1 | inner | |
192.7.i.a | 92 | 16.e | even | 4 | 1 | inner | |
192.7.i.a | 92 | 48.i | odd | 4 | 1 | inner |
Hecke kernels
This newform subspace is the entire newspace \(S_{7}^{\mathrm{new}}(192, [\chi])\).