Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [177,6,Mod(176,177)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(177, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([1, 1]))
N = Newforms(chi, 6, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("177.176");
S:= CuspForms(chi, 6);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 177 = 3 \cdot 59 \) |
Weight: | \( k \) | \(=\) | \( 6 \) |
Character orbit: | \([\chi]\) | \(=\) | 177.d (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: | \(28.3879361069\) |
Analytic rank: | \(0\) |
Dimension: | \(92\) |
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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
176.1 | −10.9379 | 4.58836 | + | 14.8979i | 87.6366 | − | 82.7497i | −50.1868 | − | 162.951i | 211.698 | −608.545 | −200.894 | + | 136.714i | 905.104i | |||||||||||
176.2 | −10.9379 | 4.58836 | − | 14.8979i | 87.6366 | 82.7497i | −50.1868 | + | 162.951i | 211.698 | −608.545 | −200.894 | − | 136.714i | − | 905.104i | |||||||||||
176.3 | −10.7490 | 9.69295 | + | 12.2085i | 83.5412 | 33.1454i | −104.190 | − | 131.229i | −174.539 | −554.017 | −55.0934 | + | 236.672i | − | 356.281i | |||||||||||
176.4 | −10.7490 | 9.69295 | − | 12.2085i | 83.5412 | − | 33.1454i | −104.190 | + | 131.229i | −174.539 | −554.017 | −55.0934 | − | 236.672i | 356.281i | |||||||||||
176.5 | −10.3821 | −12.6016 | − | 9.17604i | 75.7873 | − | 9.35226i | 130.831 | + | 95.2663i | 67.1723 | −454.603 | 74.6006 | + | 231.266i | 97.0958i | |||||||||||
176.6 | −10.3821 | −12.6016 | + | 9.17604i | 75.7873 | 9.35226i | 130.831 | − | 95.2663i | 67.1723 | −454.603 | 74.6006 | − | 231.266i | − | 97.0958i | |||||||||||
176.7 | −9.95283 | −15.5523 | − | 1.06177i | 67.0588 | − | 97.9414i | 154.789 | + | 10.5677i | −155.750 | −348.934 | 240.745 | + | 33.0259i | 974.794i | |||||||||||
176.8 | −9.95283 | −15.5523 | + | 1.06177i | 67.0588 | 97.9414i | 154.789 | − | 10.5677i | −155.750 | −348.934 | 240.745 | − | 33.0259i | − | 974.794i | |||||||||||
176.9 | −9.63720 | 15.3454 | − | 2.74211i | 60.8757 | − | 48.0995i | −147.887 | + | 26.4263i | 55.2643 | −278.281 | 227.962 | − | 84.1575i | 463.545i | |||||||||||
176.10 | −9.63720 | 15.3454 | + | 2.74211i | 60.8757 | 48.0995i | −147.887 | − | 26.4263i | 55.2643 | −278.281 | 227.962 | + | 84.1575i | − | 463.545i | |||||||||||
176.11 | −9.16184 | −4.65820 | + | 14.8762i | 51.9393 | − | 80.4460i | 42.6777 | − | 136.293i | −176.825 | −182.680 | −199.602 | − | 138.593i | 737.033i | |||||||||||
176.12 | −9.16184 | −4.65820 | − | 14.8762i | 51.9393 | 80.4460i | 42.6777 | + | 136.293i | −176.825 | −182.680 | −199.602 | + | 138.593i | − | 737.033i | |||||||||||
176.13 | −8.73168 | 1.48415 | − | 15.5176i | 44.2422 | − | 95.2512i | −12.9591 | + | 135.495i | 144.410 | −106.895 | −238.595 | − | 46.0609i | 831.703i | |||||||||||
176.14 | −8.73168 | 1.48415 | + | 15.5176i | 44.2422 | 95.2512i | −12.9591 | − | 135.495i | 144.410 | −106.895 | −238.595 | + | 46.0609i | − | 831.703i | |||||||||||
176.15 | −8.62331 | −4.57233 | + | 14.9028i | 42.3615 | 15.6511i | 39.4286 | − | 128.512i | −96.1896 | −89.3504 | −201.188 | − | 136.281i | − | 134.964i | |||||||||||
176.16 | −8.62331 | −4.57233 | − | 14.9028i | 42.3615 | − | 15.6511i | 39.4286 | + | 128.512i | −96.1896 | −89.3504 | −201.188 | + | 136.281i | 134.964i | |||||||||||
176.17 | −7.56324 | 11.2364 | + | 10.8048i | 25.2027 | − | 51.3528i | −84.9838 | − | 81.7190i | 90.7178 | 51.4099 | 9.51431 | + | 242.814i | 388.394i | |||||||||||
176.18 | −7.56324 | 11.2364 | − | 10.8048i | 25.2027 | 51.3528i | −84.9838 | + | 81.7190i | 90.7178 | 51.4099 | 9.51431 | − | 242.814i | − | 388.394i | |||||||||||
176.19 | −7.43533 | −12.1962 | − | 9.70834i | 23.2842 | 26.1059i | 90.6831 | + | 72.1847i | 171.796 | 64.8050 | 54.4964 | + | 236.810i | − | 194.106i | |||||||||||
176.20 | −7.43533 | −12.1962 | + | 9.70834i | 23.2842 | − | 26.1059i | 90.6831 | − | 72.1847i | 171.796 | 64.8050 | 54.4964 | − | 236.810i | 194.106i | |||||||||||
See all 92 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
3.b | odd | 2 | 1 | inner |
59.b | odd | 2 | 1 | inner |
177.d | even | 2 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 177.6.d.b | ✓ | 92 |
3.b | odd | 2 | 1 | inner | 177.6.d.b | ✓ | 92 |
59.b | odd | 2 | 1 | inner | 177.6.d.b | ✓ | 92 |
177.d | even | 2 | 1 | inner | 177.6.d.b | ✓ | 92 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
177.6.d.b | ✓ | 92 | 1.a | even | 1 | 1 | trivial |
177.6.d.b | ✓ | 92 | 3.b | odd | 2 | 1 | inner |
177.6.d.b | ✓ | 92 | 59.b | odd | 2 | 1 | inner |
177.6.d.b | ✓ | 92 | 177.d | even | 2 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{46} - 1167 T_{2}^{44} + 634864 T_{2}^{42} - 213934715 T_{2}^{40} + 50049221944 T_{2}^{38} + \cdots - 21\!\cdots\!32 \) acting on \(S_{6}^{\mathrm{new}}(177, [\chi])\).