Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [775,2,Mod(156,775)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(775, base_ring=CyclotomicField(10))
chi = DirichletCharacter(H, H._module([4, 0]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("775.156");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 775 = 5^{2} \cdot 31 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 775.j (of order \(5\), 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: | \(6.18840615665\) |
Analytic rank: | \(0\) |
Dimension: | \(160\) |
Relative dimension: | \(40\) over \(\Q(\zeta_{5})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{5}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
156.1 | −0.848625 | − | 2.61180i | 2.44944 | − | 1.77962i | −4.48329 | + | 3.25730i | 2.19781 | + | 0.411887i | −6.72666 | − | 4.88720i | 1.14447 | 7.86859 | + | 5.71687i | 1.90564 | − | 5.86496i | −0.789347 | − | 6.08976i | ||
156.2 | −0.823075 | − | 2.53316i | −1.25286 | + | 0.910254i | −4.12144 | + | 2.99440i | −2.23433 | + | 0.0880920i | 3.33702 | + | 2.42449i | −0.296470 | 6.66788 | + | 4.84450i | −0.185963 | + | 0.572335i | 2.06217 | + | 5.58742i | ||
156.3 | −0.801339 | − | 2.46627i | 0.547645 | − | 0.397887i | −3.82229 | + | 2.77706i | 0.533421 | − | 2.17151i | −1.42015 | − | 1.03180i | −3.54137 | 5.71605 | + | 4.15295i | −0.785450 | + | 2.41737i | −5.78298 | + | 0.424558i | ||
156.4 | −0.749608 | − | 2.30706i | −0.693590 | + | 0.503923i | −3.14256 | + | 2.28320i | −1.28236 | − | 1.83182i | 1.68250 | + | 1.22241i | 2.66727 | 3.69817 | + | 2.68688i | −0.699922 | + | 2.15414i | −3.26485 | + | 4.33161i | ||
156.5 | −0.745963 | − | 2.29584i | 1.42406 | − | 1.03464i | −3.09638 | + | 2.24965i | −1.72323 | + | 1.42495i | −3.43767 | − | 2.49762i | 3.97778 | 3.56871 | + | 2.59282i | 0.0304220 | − | 0.0936293i | 4.55692 | + | 2.89330i | ||
156.6 | −0.743746 | − | 2.28902i | −2.09240 | + | 1.52022i | −3.06840 | + | 2.22932i | 0.487900 | + | 2.18219i | 5.03601 | + | 3.65888i | 4.93338 | 3.49076 | + | 2.53618i | 1.14002 | − | 3.50863i | 4.63219 | − | 2.73981i | ||
156.7 | −0.683199 | − | 2.10267i | 2.53065 | − | 1.83863i | −2.33643 | + | 1.69751i | −1.64075 | − | 1.51919i | −5.59496 | − | 4.06498i | −0.194679 | 1.58828 | + | 1.15395i | 2.09660 | − | 6.45267i | −2.07339 | + | 4.48787i | ||
156.8 | −0.609561 | − | 1.87604i | −1.75820 | + | 1.27741i | −1.52991 | + | 1.11155i | 1.46351 | + | 1.69060i | 3.46820 | + | 2.51980i | −0.863422 | −0.173831 | − | 0.126295i | 0.532455 | − | 1.63873i | 2.27953 | − | 3.77612i | ||
156.9 | −0.555168 | − | 1.70863i | −1.92643 | + | 1.39963i | −0.993180 | + | 0.721587i | −1.73604 | + | 1.40931i | 3.46096 | + | 2.51453i | −4.33155 | −1.12259 | − | 0.815609i | 0.825112 | − | 2.53943i | 3.37179 | + | 2.18385i | ||
156.10 | −0.524831 | − | 1.61527i | 1.13868 | − | 0.827302i | −0.715599 | + | 0.519913i | 2.09379 | + | 0.784886i | −1.93393 | − | 1.40508i | −4.31341 | −1.53269 | − | 1.11356i | −0.314879 | + | 0.969099i | 0.168912 | − | 3.79396i | ||
156.11 | −0.459923 | − | 1.41550i | 0.828343 | − | 0.601826i | −0.174068 | + | 0.126468i | 2.03639 | − | 0.923634i | −1.23286 | − | 0.895723i | 3.39926 | −2.14911 | − | 1.56142i | −0.603094 | + | 1.85613i | −2.24398 | − | 2.45771i | ||
156.12 | −0.381577 | − | 1.17437i | 0.756089 | − | 0.549331i | 0.384480 | − | 0.279341i | 0.0519802 | + | 2.23546i | −0.933627 | − | 0.678320i | −0.771186 | −2.47272 | − | 1.79654i | −0.657145 | + | 2.02248i | 2.60544 | − | 0.914047i | ||
156.13 | −0.379931 | − | 1.16931i | −2.41947 | + | 1.75785i | 0.395100 | − | 0.287057i | 1.77514 | − | 1.35973i | 2.97470 | + | 2.16124i | 4.68592 | −2.47511 | − | 1.79828i | 1.83675 | − | 5.65293i | −2.26438 | − | 1.55908i | ||
156.14 | −0.369570 | − | 1.13742i | 0.759672 | − | 0.551934i | 0.460895 | − | 0.334860i | −1.65224 | − | 1.50669i | −0.908532 | − | 0.660087i | −1.94021 | −2.48630 | − | 1.80640i | −0.654580 | + | 2.01459i | −1.10312 | + | 2.43611i | ||
156.15 | −0.322877 | − | 0.993713i | −2.20206 | + | 1.59989i | 0.734818 | − | 0.533877i | −0.985462 | − | 2.00720i | 2.30083 | + | 1.67165i | −3.45698 | −2.45838 | − | 1.78612i | 1.36237 | − | 4.19296i | −1.67640 | + | 1.62735i | ||
156.16 | −0.218896 | − | 0.673692i | 2.00537 | − | 1.45699i | 1.21209 | − | 0.880634i | 2.00345 | − | 0.993066i | −1.42053 | − | 1.03207i | −0.368794 | −2.00475 | − | 1.45654i | 0.971653 | − | 2.99044i | −1.10757 | − | 1.13233i | ||
156.17 | −0.200194 | − | 0.616134i | 1.52398 | − | 1.10724i | 1.27849 | − | 0.928878i | −1.99743 | + | 1.00513i | −0.987297 | − | 0.717313i | 3.29259 | −1.87649 | − | 1.36335i | 0.169491 | − | 0.521639i | 1.01917 | + | 1.02946i | ||
156.18 | −0.145907 | − | 0.449056i | −1.13168 | + | 0.822213i | 1.43767 | − | 1.04453i | −2.14491 | + | 0.631944i | 0.534339 | + | 0.388220i | 0.465253 | −1.44280 | − | 1.04825i | −0.322388 | + | 0.992209i | 0.596736 | + | 0.870980i | ||
156.19 | −0.0759340 | − | 0.233701i | 2.63647 | − | 1.91551i | 1.56918 | − | 1.14008i | −2.22553 | + | 0.216824i | −0.647853 | − | 0.470693i | −4.97983 | −0.783187 | − | 0.569019i | 2.35475 | − | 7.24717i | 0.219665 | + | 0.503644i | ||
156.20 | −0.0232911 | − | 0.0716827i | 0.686947 | − | 0.499096i | 1.61344 | − | 1.17223i | 1.74588 | − | 1.39711i | −0.0517764 | − | 0.0376177i | −0.320307 | −0.243561 | − | 0.176958i | −0.704252 | + | 2.16746i | −0.140812 | − | 0.0926090i | ||
See next 80 embeddings (of 160 total) |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
25.d | even | 5 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 775.2.j.c | ✓ | 160 |
25.d | even | 5 | 1 | inner | 775.2.j.c | ✓ | 160 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
775.2.j.c | ✓ | 160 | 1.a | even | 1 | 1 | trivial |
775.2.j.c | ✓ | 160 | 25.d | even | 5 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{160} - T_{2}^{159} + 62 T_{2}^{158} - 65 T_{2}^{157} + 2081 T_{2}^{156} - 2267 T_{2}^{155} + 50180 T_{2}^{154} - 56474 T_{2}^{153} + 974476 T_{2}^{152} - 1127114 T_{2}^{151} + 16126729 T_{2}^{150} + \cdots + 7307701132176 \)
acting on \(S_{2}^{\mathrm{new}}(775, [\chi])\).