Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [211,2,Mod(34,211)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(211, base_ring=CyclotomicField(42))
chi = DirichletCharacter(H, H._module([40]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("211.34");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 211 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 211.j (of order \(21\), degree \(12\), 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: | \(1.68484348265\) |
Analytic rank: | \(0\) |
Dimension: | \(204\) |
Relative dimension: | \(17\) over \(\Q(\zeta_{21})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{21}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
34.1 | −2.53452 | + | 0.382018i | 0.970920 | − | 0.146343i | 4.36672 | − | 1.34696i | −1.83349 | + | 2.29912i | −2.40491 | + | 0.741818i | −0.0538484 | − | 0.137204i | −5.93436 | + | 2.85784i | −1.94545 | + | 0.600092i | 3.76871 | − | 6.52760i |
34.2 | −2.28371 | + | 0.344214i | 0.351495 | − | 0.0529794i | 3.18569 | − | 0.982657i | 2.26526 | − | 2.84054i | −0.784477 | + | 0.241979i | −0.258865 | − | 0.659578i | −2.77537 | + | 1.33655i | −2.74598 | + | 0.847021i | −4.19543 | + | 7.26671i |
34.3 | −2.26358 | + | 0.341180i | −2.77231 | + | 0.417859i | 3.09625 | − | 0.955067i | 0.0173325 | − | 0.0217343i | 6.13279 | − | 1.89172i | −1.25488 | − | 3.19737i | −2.55786 | + | 1.23180i | 4.64440 | − | 1.43261i | −0.0318182 | + | 0.0551108i |
34.4 | −1.58818 | + | 0.239379i | −2.04144 | + | 0.307697i | 0.553853 | − | 0.170841i | −1.61165 | + | 2.02095i | 3.16850 | − | 0.977353i | 1.59575 | + | 4.06591i | 2.05540 | − | 0.989828i | 1.20607 | − | 0.372022i | 2.07581 | − | 3.59541i |
34.5 | −1.41407 | + | 0.213137i | 3.02510 | − | 0.455960i | 0.0430174 | − | 0.0132691i | −0.0500366 | + | 0.0627439i | −4.18052 | + | 1.28952i | −1.79686 | − | 4.57832i | 2.51884 | − | 1.21301i | 6.07661 | − | 1.87439i | 0.0573821 | − | 0.0993888i |
34.6 | −1.18456 | + | 0.178543i | −0.125810 | + | 0.0189628i | −0.539852 | + | 0.166522i | 0.502029 | − | 0.629525i | 0.145643 | − | 0.0449250i | 0.509781 | + | 1.29890i | 2.76836 | − | 1.33317i | −2.85125 | + | 0.879494i | −0.482284 | + | 0.835341i |
34.7 | −0.333125 | + | 0.0502106i | −0.575392 | + | 0.0867265i | −1.80269 | + | 0.556058i | 0.597837 | − | 0.749664i | 0.187323 | − | 0.0577816i | −1.00675 | − | 2.56516i | 1.17965 | − | 0.568092i | −2.54316 | + | 0.784462i | −0.161514 | + | 0.279750i |
34.8 | −0.215343 | + | 0.0324577i | 2.32503 | − | 0.350442i | −1.86583 | + | 0.575531i | 1.67617 | − | 2.10185i | −0.489304 | + | 0.150930i | 1.51530 | + | 3.86091i | 0.775529 | − | 0.373475i | 2.41625 | − | 0.745316i | −0.292730 | + | 0.507023i |
34.9 | −0.152293 | + | 0.0229544i | −3.00837 | + | 0.453439i | −1.88848 | + | 0.582519i | 1.29176 | − | 1.61982i | 0.447745 | − | 0.138111i | 0.193898 | + | 0.494044i | 0.551752 | − | 0.265710i | 5.97798 | − | 1.84396i | −0.159544 | + | 0.276339i |
34.10 | −0.0842224 | + | 0.0126945i | 1.95243 | − | 0.294281i | −1.90421 | + | 0.587372i | −2.63127 | + | 3.29951i | −0.160703 | + | 0.0495702i | 0.702130 | + | 1.78900i | 0.306399 | − | 0.147554i | 0.858659 | − | 0.264861i | 0.179726 | − | 0.311295i |
34.11 | 0.557194 | − | 0.0839835i | −0.740325 | + | 0.111586i | −1.60773 | + | 0.495920i | −1.49600 | + | 1.87592i | −0.403133 | + | 0.124350i | −1.04909 | − | 2.67303i | −1.86954 | + | 0.900323i | −2.33109 | + | 0.719046i | −0.676015 | + | 1.17089i |
34.12 | 1.16660 | − | 0.175837i | 2.31080 | − | 0.348297i | −0.581100 | + | 0.179246i | 0.657712 | − | 0.824745i | 2.63455 | − | 0.812650i | −0.422359 | − | 1.07615i | −2.77229 | + | 1.33506i | 2.35177 | − | 0.725425i | 0.622269 | − | 1.07780i |
34.13 | 1.43489 | − | 0.216275i | −2.07678 | + | 0.313024i | 0.100993 | − | 0.0311523i | −1.41511 | + | 1.77449i | −2.91226 | + | 0.898312i | 0.743002 | + | 1.89314i | −2.47661 | + | 1.19267i | 1.34832 | − | 0.415900i | −1.64675 | + | 2.85226i |
34.14 | 1.80504 | − | 0.272066i | −0.728178 | + | 0.109755i | 1.27301 | − | 0.392671i | 2.53820 | − | 3.18281i | −1.28453 | + | 0.396225i | −0.692092 | − | 1.76342i | −1.09831 | + | 0.528920i | −2.34852 | + | 0.724423i | 3.71563 | − | 6.43566i |
34.15 | 2.05505 | − | 0.309749i | 0.856257 | − | 0.129060i | 2.21614 | − | 0.683590i | −0.0498308 | + | 0.0624859i | 1.71967 | − | 0.530449i | 0.632720 | + | 1.61214i | 0.597644 | − | 0.287810i | −2.15020 | + | 0.663249i | −0.0830499 | + | 0.143847i |
34.16 | 2.61212 | − | 0.393713i | 0.261941 | − | 0.0394813i | 4.75699 | − | 1.46734i | −2.10245 | + | 2.63639i | 0.668677 | − | 0.206259i | −1.70049 | − | 4.33278i | 7.08807 | − | 3.41344i | −2.79966 | + | 0.863582i | −4.45387 | + | 7.71433i |
34.17 | 2.68901 | − | 0.405303i | −2.73392 | + | 0.412072i | 5.15535 | − | 1.59021i | 0.866052 | − | 1.08599i | −7.18452 | + | 2.21613i | 0.979677 | + | 2.49618i | 8.31810 | − | 4.00579i | 4.43780 | − | 1.36888i | 1.88866 | − | 3.27126i |
43.1 | −0.980584 | − | 2.49849i | 0.808541 | + | 2.06013i | −3.81479 | + | 3.53960i | −1.10500 | − | 1.38563i | 4.35436 | − | 4.04026i | 2.88602 | − | 0.434998i | 7.74793 | + | 3.73120i | −1.39124 | + | 1.29088i | −2.37843 | + | 4.11956i |
43.2 | −0.944044 | − | 2.40539i | −0.289907 | − | 0.738670i | −3.42856 | + | 3.18123i | 0.134583 | + | 0.168761i | −1.50310 | + | 1.39467i | −4.48382 | + | 0.675827i | 6.23258 | + | 3.00145i | 1.73757 | − | 1.61223i | 0.278884 | − | 0.483042i |
43.3 | −0.761572 | − | 1.94045i | 0.0593356 | + | 0.151185i | −1.71926 | + | 1.59524i | 1.79982 | + | 2.25690i | 0.248179 | − | 0.230276i | 0.409818 | − | 0.0617701i | 0.648605 | + | 0.312352i | 2.17982 | − | 2.02258i | 3.00872 | − | 5.21125i |
See next 80 embeddings (of 204 total) |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
211.j | even | 21 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 211.2.j.a | ✓ | 204 |
211.j | even | 21 | 1 | inner | 211.2.j.a | ✓ | 204 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
211.2.j.a | ✓ | 204 | 1.a | even | 1 | 1 | trivial |
211.2.j.a | ✓ | 204 | 211.j | even | 21 | 1 | inner |
Hecke kernels
This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(211, [\chi])\).