Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [819,2,Mod(50,819)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(819, base_ring=CyclotomicField(12))
chi = DirichletCharacter(H, H._module([10, 0, 7]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("819.50");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 819 = 3^{2} \cdot 7 \cdot 13 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 819.fy (of order \(12\), 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: | \(6.53974792554\) |
Analytic rank: | \(0\) |
Dimension: | \(336\) |
Relative dimension: | \(84\) over \(\Q(\zeta_{12})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{12}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
50.1 | −0.727236 | − | 2.71408i | 1.68275 | + | 0.410318i | −5.10531 | + | 2.94755i | −0.0876862 | − | 0.327249i | −0.110119 | − | 4.86551i | −0.707107 | + | 0.707107i | 7.73897 | + | 7.73897i | 2.66328 | + | 1.38092i | −0.824413 | + | 0.475975i |
50.2 | −0.721163 | − | 2.69142i | −1.27046 | + | 1.17726i | −4.99160 | + | 2.88190i | −0.822532 | − | 3.06973i | 4.08470 | + | 2.57033i | 0.707107 | − | 0.707107i | 7.41566 | + | 7.41566i | 0.228118 | − | 2.99131i | −7.66875 | + | 4.42756i |
50.3 | −0.708020 | − | 2.64237i | 0.160527 | + | 1.72460i | −4.74876 | + | 2.74170i | 0.977651 | + | 3.64864i | 4.44336 | − | 1.64522i | 0.707107 | − | 0.707107i | 6.73810 | + | 6.73810i | −2.94846 | + | 0.553689i | 8.94886 | − | 5.16663i |
50.4 | −0.693142 | − | 2.58684i | −1.33631 | − | 1.10194i | −4.47926 | + | 2.58610i | 0.766927 | + | 2.86221i | −1.92428 | + | 4.22063i | −0.707107 | + | 0.707107i | 6.00720 | + | 6.00720i | 0.571467 | + | 2.94507i | 6.87250 | − | 3.96784i |
50.5 | −0.676409 | − | 2.52439i | 1.29284 | − | 1.15263i | −4.18297 | + | 2.41504i | −0.237212 | − | 0.885285i | −3.78419 | − | 2.48398i | 0.707107 | − | 0.707107i | 5.22994 | + | 5.22994i | 0.342872 | − | 2.98034i | −2.07435 | + | 1.19763i |
50.6 | −0.663488 | − | 2.47617i | −0.692898 | − | 1.58742i | −3.95915 | + | 2.28582i | −0.816615 | − | 3.04765i | −3.47098 | + | 2.76896i | −0.707107 | + | 0.707107i | 4.66155 | + | 4.66155i | −2.03979 | + | 2.19984i | −7.00468 | + | 4.04416i |
50.7 | −0.634582 | − | 2.36829i | −1.61523 | − | 0.625330i | −3.47406 | + | 2.00575i | 0.0143461 | + | 0.0535404i | −0.455971 | + | 4.22216i | 0.707107 | − | 0.707107i | 3.48736 | + | 3.48736i | 2.21792 | + | 2.02010i | 0.117696 | − | 0.0679516i |
50.8 | −0.629766 | − | 2.35032i | −0.786534 | + | 1.54317i | −3.39535 | + | 1.96031i | −0.0503052 | − | 0.187742i | 4.12227 | + | 0.876773i | −0.707107 | + | 0.707107i | 3.30451 | + | 3.30451i | −1.76273 | − | 2.42751i | −0.409572 | + | 0.236467i |
50.9 | −0.606137 | − | 2.26213i | 1.35078 | − | 1.08415i | −3.01779 | + | 1.74232i | 0.760590 | + | 2.83856i | −3.27125 | − | 2.39851i | −0.707107 | + | 0.707107i | 2.45857 | + | 2.45857i | 0.649233 | − | 2.92891i | 5.96018 | − | 3.44111i |
50.10 | −0.603210 | − | 2.25121i | 0.907564 | + | 1.47524i | −2.97204 | + | 1.71591i | −1.06637 | − | 3.97976i | 2.77362 | − | 2.93300i | 0.707107 | − | 0.707107i | 2.35963 | + | 2.35963i | −1.35266 | + | 2.67775i | −8.31604 | + | 4.80127i |
50.11 | −0.585079 | − | 2.18354i | 0.305712 | + | 1.70486i | −2.69350 | + | 1.55509i | 0.0226908 | + | 0.0846831i | 3.54377 | − | 1.66501i | −0.707107 | + | 0.707107i | 1.77459 | + | 1.77459i | −2.81308 | + | 1.04239i | 0.171633 | − | 0.0990926i |
50.12 | −0.580718 | − | 2.16727i | 0.116460 | − | 1.72813i | −2.62776 | + | 1.51714i | 0.952172 | + | 3.55355i | −3.81295 | + | 0.751156i | 0.707107 | − | 0.707107i | 1.64094 | + | 1.64094i | −2.97287 | − | 0.402516i | 7.14856 | − | 4.12722i |
50.13 | −0.572772 | − | 2.13762i | 1.16473 | + | 1.28195i | −2.50928 | + | 1.44873i | 0.0776352 | + | 0.289738i | 2.07319 | − | 3.22402i | 0.707107 | − | 0.707107i | 1.40439 | + | 1.40439i | −0.286797 | + | 2.98626i | 0.574882 | − | 0.331908i |
50.14 | −0.563544 | − | 2.10318i | −1.71440 | + | 0.246659i | −2.37372 | + | 1.37047i | −0.775066 | − | 2.89259i | 1.48491 | + | 3.46668i | −0.707107 | + | 0.707107i | 1.14076 | + | 1.14076i | 2.87832 | − | 0.845744i | −5.64684 | + | 3.26020i |
50.15 | −0.548397 | − | 2.04664i | −1.71570 | + | 0.237418i | −2.15597 | + | 1.24475i | 0.530257 | + | 1.97895i | 1.42680 | + | 3.38123i | 0.707107 | − | 0.707107i | 0.733384 | + | 0.733384i | 2.88726 | − | 0.814679i | 3.75941 | − | 2.17049i |
50.16 | −0.541434 | − | 2.02066i | 1.72494 | + | 0.156795i | −2.05786 | + | 1.18811i | −1.05055 | − | 3.92071i | −0.617111 | − | 3.57041i | −0.707107 | + | 0.707107i | 0.556508 | + | 0.556508i | 2.95083 | + | 0.540924i | −7.35361 | + | 4.24561i |
50.17 | −0.503059 | − | 1.87744i | 1.45910 | + | 0.933282i | −1.53967 | + | 0.888931i | 0.750845 | + | 2.80219i | 1.01817 | − | 3.20888i | −0.707107 | + | 0.707107i | −0.305303 | − | 0.305303i | 1.25797 | + | 2.72351i | 4.88324 | − | 2.81934i |
50.18 | −0.459010 | − | 1.71305i | 1.54753 | − | 0.777908i | −0.991796 | + | 0.572614i | −0.538351 | − | 2.00915i | −2.04293 | − | 2.29393i | 0.707107 | − | 0.707107i | −1.07192 | − | 1.07192i | 1.78972 | − | 2.40768i | −3.19467 | + | 1.84444i |
50.19 | −0.452193 | − | 1.68761i | −1.18434 | + | 1.26385i | −0.911485 | + | 0.526246i | 0.371355 | + | 1.38591i | 2.66844 | + | 1.42720i | 0.707107 | − | 0.707107i | −1.17056 | − | 1.17056i | −0.194657 | − | 2.99368i | 2.17095 | − | 1.25340i |
50.20 | −0.445095 | − | 1.66112i | −0.117775 | − | 1.72804i | −0.829146 | + | 0.478708i | 0.109473 | + | 0.408558i | −2.81806 | + | 0.964781i | −0.707107 | + | 0.707107i | −1.26780 | − | 1.26780i | −2.97226 | + | 0.407041i | 0.629936 | − | 0.363694i |
See next 80 embeddings (of 336 total) |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
117.bc | even | 12 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 819.2.fy.a | yes | 336 |
9.d | odd | 6 | 1 | 819.2.ew.a | ✓ | 336 | |
13.f | odd | 12 | 1 | 819.2.ew.a | ✓ | 336 | |
117.bc | even | 12 | 1 | inner | 819.2.fy.a | yes | 336 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
819.2.ew.a | ✓ | 336 | 9.d | odd | 6 | 1 | |
819.2.ew.a | ✓ | 336 | 13.f | odd | 12 | 1 | |
819.2.fy.a | yes | 336 | 1.a | even | 1 | 1 | trivial |
819.2.fy.a | yes | 336 | 117.bc | even | 12 | 1 | inner |
Hecke kernels
This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(819, [\chi])\).