Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [211,4,Mod(4,211)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(211, base_ring=CyclotomicField(210))
chi = DirichletCharacter(H, H._module([2]))
N = Newforms(chi, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("211.4");
S:= CuspForms(chi, 4);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 211 \) |
Weight: | \( k \) | \(=\) | \( 4 \) |
Character orbit: | \([\chi]\) | \(=\) | 211.o (of order \(105\), degree \(48\), 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: | \(12.4494030112\) |
Analytic rank: | \(0\) |
Dimension: | \(2496\) |
Relative dimension: | \(52\) over \(\Q(\zeta_{105})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{105}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
4.1 | −5.49394 | − | 0.164427i | 0.452282 | + | 1.54811i | 22.1706 | + | 1.32827i | 10.4149 | + | 10.8932i | −2.23025 | − | 8.57961i | −6.86029 | − | 11.1016i | −77.7915 | − | 7.00136i | 20.5614 | − | 13.1351i | −55.4279 | − | 61.5589i |
4.2 | −5.44300 | − | 0.162903i | 1.03697 | + | 3.54946i | 21.6140 | + | 1.29493i | −4.76118 | − | 4.97980i | −5.06603 | − | 19.4886i | 14.8661 | + | 24.0568i | −74.0460 | − | 6.66426i | 11.2301 | − | 7.17406i | 25.1039 | + | 27.8807i |
4.3 | −5.14700 | − | 0.154044i | −1.94870 | − | 6.67019i | 18.4822 | + | 1.10729i | −13.3869 | − | 14.0016i | 9.00244 | + | 34.6317i | −7.85085 | − | 12.7045i | −53.9288 | − | 4.85368i | −17.9406 | + | 11.4609i | 66.7457 | + | 74.1286i |
4.4 | −5.06083 | − | 0.151465i | −1.94159 | − | 6.64588i | 17.6034 | + | 1.05464i | 4.64734 | + | 4.86074i | 8.81945 | + | 33.9277i | −8.50593 | − | 13.7646i | −48.5863 | − | 4.37284i | −17.6444 | + | 11.2717i | −22.7832 | − | 25.3033i |
4.5 | −5.03399 | − | 0.150661i | 2.08944 | + | 7.15196i | 17.3326 | + | 1.03842i | −11.5818 | − | 12.1136i | −9.44070 | − | 36.3177i | −16.9640 | − | 27.4517i | −46.9680 | − | 4.22720i | −24.0313 | + | 15.3518i | 56.4775 | + | 62.7246i |
4.6 | −4.67229 | − | 0.139836i | −2.20351 | − | 7.54241i | 13.8251 | + | 0.828278i | −1.55134 | − | 1.62257i | 9.24076 | + | 35.5485i | 18.1831 | + | 29.4245i | −27.2343 | − | 2.45113i | −29.2791 | + | 18.7042i | 7.02140 | + | 7.79805i |
4.7 | −4.55004 | − | 0.136178i | 2.50504 | + | 8.57450i | 12.6987 | + | 0.760795i | 8.65994 | + | 9.05758i | −10.2304 | − | 39.3555i | −1.26238 | − | 2.04283i | −21.4058 | − | 1.92656i | −44.4933 | + | 28.4234i | −38.1696 | − | 42.3917i |
4.8 | −4.49221 | − | 0.134447i | −0.513553 | − | 1.75784i | 12.1762 | + | 0.729491i | 2.09910 | + | 2.19548i | 2.07065 | + | 7.96563i | −2.08593 | − | 3.37553i | −18.7907 | − | 1.69120i | 19.9272 | − | 12.7300i | −9.13440 | − | 10.1448i |
4.9 | −4.23488 | − | 0.126745i | −0.0687177 | − | 0.235214i | 9.93245 | + | 0.595067i | −7.85520 | − | 8.21590i | 0.261199 | + | 1.00481i | −0.602483 | − | 0.974960i | −8.22956 | − | 0.740674i | 22.7029 | − | 14.5031i | 32.2245 | + | 35.7889i |
4.10 | −3.94947 | − | 0.118203i | 1.95596 | + | 6.69506i | 7.59863 | + | 0.455245i | −4.72732 | − | 4.94439i | −6.93362 | − | 26.6731i | 5.59437 | + | 9.05301i | 1.52589 | + | 0.137332i | −18.2446 | + | 11.6551i | 18.0860 | + | 20.0865i |
4.11 | −3.81768 | − | 0.114259i | −0.420949 | − | 1.44087i | 6.57597 | + | 0.393975i | 14.1094 | + | 14.7573i | 1.44242 | + | 5.54886i | 18.1428 | + | 29.3593i | 5.37220 | + | 0.483506i | 20.8546 | − | 13.3224i | −52.1791 | − | 57.9507i |
4.12 | −3.78977 | − | 0.113423i | −2.23757 | − | 7.65898i | 6.36378 | + | 0.381263i | 12.0320 | + | 12.5845i | 7.61116 | + | 29.2795i | −5.70166 | − | 9.22662i | 6.13560 | + | 0.552214i | −30.8998 | + | 19.7395i | −44.1713 | − | 49.0572i |
4.13 | −2.97205 | − | 0.0889500i | 1.11700 | + | 3.82337i | 0.839479 | + | 0.0502943i | 9.58614 | + | 10.0263i | −2.97968 | − | 11.4626i | −5.32177 | − | 8.61188i | 21.2008 | + | 1.90810i | 9.38297 | − | 5.99407i | −27.5986 | − | 30.6514i |
4.14 | −2.79080 | − | 0.0835254i | −2.62448 | − | 8.98332i | −0.204111 | − | 0.0122286i | −0.127435 | − | 0.133286i | 6.57404 | + | 25.2898i | −4.87602 | − | 7.89055i | 22.8151 | + | 2.05339i | −51.0587 | + | 32.6176i | 0.344512 | + | 0.382620i |
4.15 | −2.75774 | − | 0.0825360i | −0.0747413 | − | 0.255832i | −0.387370 | − | 0.0232079i | −11.3656 | − | 11.8875i | 0.185002 | + | 0.711687i | 6.87859 | + | 11.1312i | 23.0493 | + | 2.07447i | 22.6936 | − | 14.4972i | 30.3623 | + | 33.7207i |
4.16 | −2.73023 | − | 0.0817127i | 0.951897 | + | 3.25825i | −0.538206 | − | 0.0322446i | 2.42673 | + | 2.53816i | −2.33266 | − | 8.97355i | −13.2353 | − | 21.4178i | 23.2304 | + | 2.09078i | 13.0434 | − | 8.33243i | −6.41813 | − | 7.12805i |
4.17 | −2.52390 | − | 0.0755376i | −0.964932 | − | 3.30287i | −1.62130 | − | 0.0971343i | −6.03664 | − | 6.31383i | 2.18590 | + | 8.40900i | −18.9702 | − | 30.6983i | 24.2036 | + | 2.17836i | 12.7756 | − | 8.16139i | 14.7590 | + | 16.3915i |
4.18 | −2.50543 | − | 0.0749848i | −1.81833 | − | 6.22396i | −1.71410 | − | 0.102694i | −9.86413 | − | 10.3171i | 4.08900 | + | 15.7301i | 12.8470 | + | 20.7894i | 24.2586 | + | 2.18331i | −12.6779 | + | 8.09894i | 23.9403 | + | 26.5884i |
4.19 | −1.89843 | − | 0.0568179i | 2.03911 | + | 6.97967i | −4.38486 | − | 0.262703i | −13.7334 | − | 14.3640i | −3.47454 | − | 13.3663i | −7.73264 | − | 12.5132i | 23.4425 | + | 2.10987i | −21.8044 | + | 13.9292i | 25.2558 | + | 28.0494i |
4.20 | −1.85867 | − | 0.0556278i | 2.40174 | + | 8.22092i | −4.53413 | − | 0.271646i | −1.51949 | − | 1.58927i | −4.00673 | − | 15.4136i | 16.1974 | + | 26.2112i | 23.2284 | + | 2.09060i | −39.0617 | + | 24.9535i | 2.73583 | + | 3.03845i |
See next 80 embeddings (of 2496 total) |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
211.o | even | 105 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 211.4.o.a | ✓ | 2496 |
211.o | even | 105 | 1 | inner | 211.4.o.a | ✓ | 2496 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
211.4.o.a | ✓ | 2496 | 1.a | even | 1 | 1 | trivial |
211.4.o.a | ✓ | 2496 | 211.o | even | 105 | 1 | inner |
Hecke kernels
This newform subspace is the entire newspace \(S_{4}^{\mathrm{new}}(211, [\chi])\).