Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [400,2,Mod(29,400)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(400, base_ring=CyclotomicField(20))
chi = DirichletCharacter(H, H._module([0, 15, 2]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("400.29");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 400 = 2^{4} \cdot 5^{2} \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 400.bl (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: | \(3.19401608085\) |
Analytic rank: | \(0\) |
Dimension: | \(464\) |
Relative dimension: | \(58\) over \(\Q(\zeta_{20})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{20}]$ |
$q$-expansion
The algebraic \(q\)-expansion of this newform has not been computed, 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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
29.1 | −1.41418 | − | 0.0103712i | 1.06269 | − | 0.168314i | 1.99978 | + | 0.0293334i | −2.11616 | + | 0.722412i | −1.50458 | + | 0.227004i | −0.232144 | −2.82774 | − | 0.0622227i | −1.75218 | + | 0.569319i | 3.00011 | − | 0.999671i | ||
29.2 | −1.41356 | − | 0.0431277i | −3.09356 | + | 0.489972i | 1.99628 | + | 0.121927i | −2.23084 | + | 0.152807i | 4.39405 | − | 0.559185i | 2.16271 | −2.81659 | − | 0.258445i | 6.47689 | − | 2.10447i | 3.16001 | − | 0.119790i | ||
29.3 | −1.41316 | − | 0.0546657i | 1.83259 | − | 0.290254i | 1.99402 | + | 0.154502i | 0.995469 | + | 2.00226i | −2.60560 | + | 0.309994i | −0.288575 | −2.80942 | − | 0.327341i | 0.420967 | − | 0.136781i | −1.29730 | − | 2.88392i | ||
29.4 | −1.39795 | + | 0.213865i | 3.11943 | − | 0.494069i | 1.90852 | − | 0.597945i | 1.43675 | − | 1.71340i | −4.25514 | + | 1.35782i | −3.50228 | −2.54014 | + | 1.24406i | 6.63355 | − | 2.15537i | −1.64207 | + | 2.70252i | ||
29.5 | −1.38613 | − | 0.280454i | −0.894740 | + | 0.141713i | 1.84269 | + | 0.777490i | 0.571921 | − | 2.16169i | 1.27997 | + | 0.0545018i | 4.11279 | −2.33615 | − | 1.59449i | −2.07269 | + | 0.673458i | −1.39901 | + | 2.83598i | ||
29.6 | −1.37753 | + | 0.320006i | −0.593985 | + | 0.0940780i | 1.79519 | − | 0.881639i | −0.779471 | − | 2.09581i | 0.788128 | − | 0.319675i | −1.40656 | −2.19081 | + | 1.78896i | −2.50920 | + | 0.815289i | 1.74442 | + | 2.63761i | ||
29.7 | −1.35280 | − | 0.412237i | −1.98405 | + | 0.314242i | 1.66012 | + | 1.11535i | −0.149735 | + | 2.23105i | 2.81356 | + | 0.392793i | −4.59495 | −1.78602 | − | 2.19320i | 0.984533 | − | 0.319894i | 1.12228 | − | 2.95643i | ||
29.8 | −1.29143 | + | 0.576366i | −0.752396 | + | 0.119168i | 1.33560 | − | 1.48868i | 0.231849 | + | 2.22402i | 0.902985 | − | 0.587553i | 2.56554 | −0.866820 | + | 2.69233i | −2.30127 | + | 0.747728i | −1.58127 | − | 2.73854i | ||
29.9 | −1.27798 | + | 0.605618i | −2.41193 | + | 0.382013i | 1.26645 | − | 1.54793i | 2.20655 | − | 0.362152i | 2.85104 | − | 1.94891i | −1.42770 | −0.681042 | + | 2.74521i | 2.81831 | − | 0.915726i | −2.60059 | + | 1.79915i | ||
29.10 | −1.26619 | − | 0.629895i | 2.62418 | − | 0.415629i | 1.20646 | + | 1.59513i | −0.886089 | − | 2.05301i | −3.58451 | − | 1.12669i | 3.34726 | −0.522846 | − | 2.77968i | 3.86039 | − | 1.25432i | −0.171225 | + | 3.15764i | ||
29.11 | −1.21289 | − | 0.727254i | −0.266037 | + | 0.0421361i | 0.942204 | + | 1.76416i | 1.58641 | − | 1.57585i | 0.353317 | + | 0.142370i | −4.44042 | 0.140201 | − | 2.82495i | −2.78417 | + | 0.904631i | −3.07018 | + | 0.757610i | ||
29.12 | −1.09755 | + | 0.891847i | 1.78947 | − | 0.283424i | 0.409216 | − | 1.95769i | 2.13879 | − | 0.652354i | −1.71125 | + | 1.90700i | 4.96308 | 1.29682 | + | 2.51361i | 0.268690 | − | 0.0873027i | −1.76562 | + | 2.62347i | ||
29.13 | −1.09639 | + | 0.893266i | 2.76199 | − | 0.437457i | 0.404150 | − | 1.95874i | −1.79261 | + | 1.33662i | −2.63746 | + | 2.94682i | 0.291269 | 1.30657 | + | 2.50856i | 4.58407 | − | 1.48945i | 0.771442 | − | 3.06674i | ||
29.14 | −1.08196 | − | 0.910696i | 1.78850 | − | 0.283271i | 0.341265 | + | 1.97067i | 2.01441 | + | 0.970637i | −2.19306 | − | 1.32229i | 1.01538 | 1.42545 | − | 2.44297i | 0.265328 | − | 0.0862102i | −1.29556 | − | 2.88471i | ||
29.15 | −1.01647 | − | 0.983253i | −0.224268 | + | 0.0355205i | 0.0664253 | + | 1.99890i | −1.20871 | + | 1.88123i | 0.262887 | + | 0.184406i | 2.48678 | 1.89790 | − | 2.09713i | −2.80414 | + | 0.911119i | 3.07834 | − | 0.723748i | ||
29.16 | −1.01050 | + | 0.989384i | 0.847034 | − | 0.134157i | 0.0422379 | − | 1.99955i | −1.16539 | − | 1.90836i | −0.723199 | + | 0.973608i | −1.50111 | 1.93565 | + | 2.06235i | −2.15370 | + | 0.699780i | 3.06574 | + | 0.775388i | ||
29.17 | −0.919307 | − | 1.07465i | −1.25267 | + | 0.198404i | −0.309751 | + | 1.97587i | −2.17568 | − | 0.516161i | 1.36480 | + | 1.16379i | −0.199333 | 2.40813 | − | 1.48355i | −1.32335 | + | 0.429981i | 1.44542 | + | 2.81261i | ||
29.18 | −0.919044 | − | 1.07488i | −3.33287 | + | 0.527874i | −0.310718 | + | 1.97572i | 0.778461 | − | 2.09619i | 3.63045 | + | 3.09728i | −0.463968 | 2.40921 | − | 1.48179i | 7.97618 | − | 2.59162i | −2.96858 | + | 1.08974i | ||
29.19 | −0.842684 | + | 1.13573i | −2.41612 | + | 0.382676i | −0.579767 | − | 1.91412i | −2.22343 | + | 0.237412i | 1.60141 | − | 3.06654i | −1.53357 | 2.66249 | + | 0.954542i | 2.83804 | − | 0.922135i | 1.60401 | − | 2.72528i | ||
29.20 | −0.667125 | + | 1.24697i | 0.284389 | − | 0.0450428i | −1.10989 | − | 1.66378i | 2.23360 | − | 0.104931i | −0.133556 | + | 0.384675i | −1.82171 | 2.81512 | − | 0.274055i | −2.77432 | + | 0.901432i | −1.35925 | + | 2.85525i | ||
See next 80 embeddings (of 464 total) |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
16.e | even | 4 | 1 | inner |
25.e | even | 10 | 1 | inner |
400.bl | even | 20 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 400.2.bl.a | ✓ | 464 |
16.e | even | 4 | 1 | inner | 400.2.bl.a | ✓ | 464 |
25.e | even | 10 | 1 | inner | 400.2.bl.a | ✓ | 464 |
400.bl | even | 20 | 1 | inner | 400.2.bl.a | ✓ | 464 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
400.2.bl.a | ✓ | 464 | 1.a | even | 1 | 1 | trivial |
400.2.bl.a | ✓ | 464 | 16.e | even | 4 | 1 | inner |
400.2.bl.a | ✓ | 464 | 25.e | even | 10 | 1 | inner |
400.2.bl.a | ✓ | 464 | 400.bl | even | 20 | 1 | inner |
Hecke kernels
This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(400, [\chi])\).