Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [429,2,Mod(73,429)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(429, base_ring=CyclotomicField(20))
chi = DirichletCharacter(H, H._module([0, 14, 5]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("429.73");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 429 = 3 \cdot 11 \cdot 13 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 429.bj (of order \(20\), degree \(8\), 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: | \(3.42558224671\) |
Analytic rank: | \(0\) |
Dimension: | \(112\) |
Relative dimension: | \(14\) over \(\Q(\zeta_{20})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{20}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
73.1 | −2.64667 | + | 0.419191i | 0.309017 | − | 0.951057i | 4.92702 | − | 1.60089i | −2.31578 | − | 0.366783i | −0.419191 | + | 2.64667i | −2.21285 | + | 4.34297i | −7.59393 | + | 3.86930i | −0.809017 | − | 0.587785i | 6.28285 | ||
73.2 | −2.45153 | + | 0.388283i | 0.309017 | − | 0.951057i | 3.95710 | − | 1.28574i | −1.53570 | − | 0.243230i | −0.388283 | + | 2.45153i | 2.15383 | − | 4.22713i | −4.77859 | + | 2.43481i | −0.809017 | − | 0.587785i | 3.85924 | ||
73.3 | −1.67413 | + | 0.265156i | 0.309017 | − | 0.951057i | 0.830297 | − | 0.269780i | 1.80923 | + | 0.286554i | −0.265156 | + | 1.67413i | 0.0352382 | − | 0.0691588i | 1.70202 | − | 0.867223i | −0.809017 | − | 0.587785i | −3.10487 | ||
73.4 | −1.65467 | + | 0.262074i | 0.309017 | − | 0.951057i | 0.767143 | − | 0.249260i | −1.95063 | − | 0.308949i | −0.262074 | + | 1.65467i | 1.17205 | − | 2.30028i | 1.78136 | − | 0.907647i | −0.809017 | − | 0.587785i | 3.30862 | ||
73.5 | −0.962531 | + | 0.152450i | 0.309017 | − | 0.951057i | −0.998888 | + | 0.324558i | −3.13516 | − | 0.496561i | −0.152450 | + | 0.962531i | −0.833499 | + | 1.63583i | 2.64861 | − | 1.34953i | −0.809017 | − | 0.587785i | 3.09339 | ||
73.6 | −0.904052 | + | 0.143188i | 0.309017 | − | 0.951057i | −1.10531 | + | 0.359135i | 1.60359 | + | 0.253983i | −0.143188 | + | 0.904052i | −2.17730 | + | 4.27319i | 2.57894 | − | 1.31404i | −0.809017 | − | 0.587785i | −1.48609 | ||
73.7 | −0.447102 | + | 0.0708140i | 0.309017 | − | 0.951057i | −1.70723 | + | 0.554712i | 4.30790 | + | 0.682304i | −0.0708140 | + | 0.447102i | 1.76348 | − | 3.46103i | 1.53070 | − | 0.779929i | −0.809017 | − | 0.587785i | −1.97439 | ||
73.8 | 0.502609 | − | 0.0796054i | 0.309017 | − | 0.951057i | −1.65583 | + | 0.538013i | 0.387478 | + | 0.0613705i | 0.0796054 | − | 0.502609i | −1.68058 | + | 3.29833i | −1.69623 | + | 0.864271i | −0.809017 | − | 0.587785i | 0.199635 | ||
73.9 | 0.523380 | − | 0.0828953i | 0.309017 | − | 0.951057i | −1.63506 | + | 0.531262i | 1.16191 | + | 0.184028i | 0.0828953 | − | 0.523380i | 0.580635 | − | 1.13956i | −1.75601 | + | 0.894734i | −0.809017 | − | 0.587785i | 0.623374 | ||
73.10 | 0.724135 | − | 0.114692i | 0.309017 | − | 0.951057i | −1.39090 | + | 0.451929i | −1.84470 | − | 0.292172i | 0.114692 | − | 0.724135i | 1.08006 | − | 2.11974i | −2.26187 | + | 1.15248i | −0.809017 | − | 0.587785i | −1.36932 | ||
73.11 | 1.92680 | − | 0.305175i | 0.309017 | − | 0.951057i | 1.71732 | − | 0.557991i | 3.27451 | + | 0.518632i | 0.305175 | − | 1.92680i | −0.858865 | + | 1.68562i | −0.337739 | + | 0.172087i | −0.809017 | − | 0.587785i | 6.46761 | ||
73.12 | 2.08536 | − | 0.330289i | 0.309017 | − | 0.951057i | 2.33753 | − | 0.759508i | −3.67776 | − | 0.582500i | 0.330289 | − | 2.08536i | 1.72106 | − | 3.37776i | 0.861266 | − | 0.438837i | −0.809017 | − | 0.587785i | −7.86185 | ||
73.13 | 2.19389 | − | 0.347478i | 0.309017 | − | 0.951057i | 2.79029 | − | 0.906622i | 1.48353 | + | 0.234969i | 0.347478 | − | 2.19389i | 0.326561 | − | 0.640911i | 1.84829 | − | 0.941752i | −0.809017 | − | 0.587785i | 3.33636 | ||
73.14 | 2.78451 | − | 0.441023i | 0.309017 | − | 0.951057i | 5.65687 | − | 1.83803i | −1.82850 | − | 0.289605i | 0.441023 | − | 2.78451i | −1.06982 | + | 2.09963i | 9.91711 | − | 5.05302i | −0.809017 | − | 0.587785i | −5.21918 | ||
112.1 | −1.13354 | − | 2.22470i | −0.809017 | + | 0.587785i | −2.48880 | + | 3.42554i | −0.0990937 | + | 0.194482i | 2.22470 | + | 1.13354i | −0.728003 | − | 4.59643i | 5.50974 | + | 0.872657i | 0.309017 | − | 0.951057i | 0.544991 | ||
112.2 | −1.01823 | − | 1.99839i | −0.809017 | + | 0.587785i | −1.78119 | + | 2.45160i | −0.716768 | + | 1.40674i | 1.99839 | + | 1.01823i | 0.347529 | + | 2.19421i | 2.28246 | + | 0.361506i | 0.309017 | − | 0.951057i | 3.54104 | ||
112.3 | −0.753510 | − | 1.47885i | −0.809017 | + | 0.587785i | −0.443641 | + | 0.610619i | 1.42413 | − | 2.79501i | 1.47885 | + | 0.753510i | 0.104512 | + | 0.659862i | −2.04133 | − | 0.323315i | 0.309017 | − | 0.951057i | −5.20649 | ||
112.4 | −0.661175 | − | 1.29763i | −0.809017 | + | 0.587785i | −0.0711199 | + | 0.0978882i | −1.28004 | + | 2.51223i | 1.29763 | + | 0.661175i | −0.219997 | − | 1.38901i | −2.70282 | − | 0.428085i | 0.309017 | − | 0.951057i | 4.10627 | ||
112.5 | −0.501725 | − | 0.984691i | −0.809017 | + | 0.587785i | 0.457683 | − | 0.629946i | 1.47255 | − | 2.89004i | 0.984691 | + | 0.501725i | −0.482591 | − | 3.04696i | −3.03301 | − | 0.480381i | 0.309017 | − | 0.951057i | −3.58461 | ||
112.6 | −0.379529 | − | 0.744868i | −0.809017 | + | 0.587785i | 0.764785 | − | 1.05264i | −0.448929 | + | 0.881074i | 0.744868 | + | 0.379529i | 0.542880 | + | 3.42761i | −2.72572 | − | 0.431711i | 0.309017 | − | 0.951057i | 0.826665 | ||
See next 80 embeddings (of 112 total) |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
11.d | odd | 10 | 1 | inner |
13.d | odd | 4 | 1 | inner |
143.s | even | 20 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 429.2.bj.a | ✓ | 112 |
11.d | odd | 10 | 1 | inner | 429.2.bj.a | ✓ | 112 |
13.d | odd | 4 | 1 | inner | 429.2.bj.a | ✓ | 112 |
143.s | even | 20 | 1 | inner | 429.2.bj.a | ✓ | 112 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
429.2.bj.a | ✓ | 112 | 1.a | even | 1 | 1 | trivial |
429.2.bj.a | ✓ | 112 | 11.d | odd | 10 | 1 | inner |
429.2.bj.a | ✓ | 112 | 13.d | odd | 4 | 1 | inner |
429.2.bj.a | ✓ | 112 | 143.s | even | 20 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{112} - 130 T_{2}^{108} + 9399 T_{2}^{104} + 1780 T_{2}^{103} - 12220 T_{2}^{101} - 506840 T_{2}^{100} - 210830 T_{2}^{99} + 1588600 T_{2}^{97} + 26948093 T_{2}^{96} + 15995580 T_{2}^{95} + \cdots + 214358881 \)
acting on \(S_{2}^{\mathrm{new}}(429, [\chi])\).