Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [430,3,Mod(19,430)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(430, base_ring=CyclotomicField(42))
chi = DirichletCharacter(H, H._module([21, 19]))
N = Newforms(chi, 3, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("430.19");
S:= CuspForms(chi, 3);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 430 = 2 \cdot 5 \cdot 43 \) |
Weight: | \( k \) | \(=\) | \( 3 \) |
Character orbit: | \([\chi]\) | \(=\) | 430.v (of order \(42\), 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: | \(11.7166513675\) |
Analytic rank: | \(0\) |
Dimension: | \(528\) |
Relative dimension: | \(44\) over \(\Q(\zeta_{42})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{42}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
19.1 | −0.881748 | + | 1.10568i | −5.53717 | + | 0.834594i | −0.445042 | − | 1.94986i | 2.46053 | − | 4.35267i | 3.95960 | − | 6.85822i | −6.78387 | − | 11.7500i | 2.54832 | + | 1.22721i | 21.3636 | − | 6.58979i | 2.64308 | + | 6.55852i |
19.2 | −0.881748 | + | 1.10568i | −5.18599 | + | 0.781662i | −0.445042 | − | 1.94986i | −3.48247 | + | 3.58781i | 3.70847 | − | 6.42326i | −0.255468 | − | 0.442483i | 2.54832 | + | 1.22721i | 17.6833 | − | 5.45459i | −0.896301 | − | 7.01403i |
19.3 | −0.881748 | + | 1.10568i | −4.62376 | + | 0.696920i | −0.445042 | − | 1.94986i | −4.07322 | − | 2.89980i | 3.30642 | − | 5.72689i | 5.10344 | + | 8.83943i | 2.54832 | + | 1.22721i | 12.2933 | − | 3.79199i | 6.79780 | − | 1.94678i |
19.4 | −0.881748 | + | 1.10568i | −4.45936 | + | 0.672141i | −0.445042 | − | 1.94986i | 4.97131 | − | 0.534821i | 3.18886 | − | 5.52327i | 5.76015 | + | 9.97687i | 2.54832 | + | 1.22721i | 10.8340 | − | 3.34185i | −3.79211 | + | 5.96824i |
19.5 | −0.881748 | + | 1.10568i | −3.74963 | + | 0.565166i | −0.445042 | − | 1.94986i | 1.61914 | + | 4.73058i | 2.68134 | − | 4.64421i | −1.29937 | − | 2.25058i | 2.54832 | + | 1.22721i | 5.14017 | − | 1.58553i | −6.65816 | − | 2.38094i |
19.6 | −0.881748 | + | 1.10568i | −3.14566 | + | 0.474132i | −0.445042 | − | 1.94986i | −1.74550 | − | 4.68543i | 2.24944 | − | 3.89615i | 1.15128 | + | 1.99407i | 2.54832 | + | 1.22721i | 1.07023 | − | 0.330121i | 6.71966 | + | 2.20141i |
19.7 | −0.881748 | + | 1.10568i | −1.98509 | + | 0.299204i | −0.445042 | − | 1.94986i | −4.80167 | + | 1.39426i | 1.41953 | − | 2.45869i | −1.77310 | − | 3.07109i | 2.54832 | + | 1.22721i | −4.74910 | + | 1.46490i | 2.69227 | − | 6.53848i |
19.8 | −0.881748 | + | 1.10568i | −1.78481 | + | 0.269016i | −0.445042 | − | 1.94986i | 4.92419 | + | 0.867367i | 1.27630 | − | 2.21062i | −4.83720 | − | 8.37828i | 2.54832 | + | 1.22721i | −5.48699 | + | 1.69251i | −5.30092 | + | 4.67977i |
19.9 | −0.881748 | + | 1.10568i | −1.63988 | + | 0.247173i | −0.445042 | − | 1.94986i | 2.58854 | − | 4.27779i | 1.17267 | − | 2.03113i | 0.823044 | + | 1.42555i | 2.54832 | + | 1.22721i | −5.97203 | + | 1.84213i | 2.44741 | + | 6.63402i |
19.10 | −0.881748 | + | 1.10568i | −0.779641 | + | 0.117512i | −0.445042 | − | 1.94986i | 1.01477 | + | 4.89594i | 0.557517 | − | 0.965647i | 6.88523 | + | 11.9256i | 2.54832 | + | 1.22721i | −8.00612 | + | 2.46956i | −6.30810 | − | 3.19498i |
19.11 | −0.881748 | + | 1.10568i | 0.187544 | − | 0.0282677i | −0.445042 | − | 1.94986i | −4.96879 | + | 0.557772i | −0.134111 | + | 0.232288i | 3.01780 | + | 5.22699i | 2.54832 | + | 1.22721i | −8.56578 | + | 2.64219i | 3.76451 | − | 5.98569i |
19.12 | −0.881748 | + | 1.10568i | 0.848146 | − | 0.127837i | −0.445042 | − | 1.94986i | 4.27225 | + | 2.59767i | −0.606504 | + | 1.05050i | 3.17378 | + | 5.49714i | 2.54832 | + | 1.22721i | −7.89715 | + | 2.43595i | −6.63923 | + | 2.43323i |
19.13 | −0.881748 | + | 1.10568i | 1.49385 | − | 0.225161i | −0.445042 | − | 1.94986i | −1.94249 | + | 4.60725i | −1.06824 | + | 1.85025i | −5.73360 | − | 9.93089i | 2.54832 | + | 1.22721i | −6.41927 | + | 1.98008i | −3.38133 | − | 6.21020i |
19.14 | −0.881748 | + | 1.10568i | 1.56541 | − | 0.235948i | −0.445042 | − | 1.94986i | −3.98919 | − | 3.01435i | −1.11942 | + | 1.93889i | −4.02926 | − | 6.97889i | 2.54832 | + | 1.22721i | −6.20531 | + | 1.91408i | 6.85036 | − | 1.75286i |
19.15 | −0.881748 | + | 1.10568i | 1.92814 | − | 0.290620i | −0.445042 | − | 1.94986i | 4.97927 | + | 0.454829i | −1.37880 | + | 2.38815i | −2.45096 | − | 4.24519i | 2.54832 | + | 1.22721i | −4.96691 | + | 1.53209i | −4.89335 | + | 5.10442i |
19.16 | −0.881748 | + | 1.10568i | 2.53284 | − | 0.381764i | −0.445042 | − | 1.94986i | 0.994994 | − | 4.90000i | −1.81122 | + | 3.13712i | −3.83884 | − | 6.64907i | 2.54832 | + | 1.22721i | −2.33063 | + | 0.718904i | 4.54048 | + | 5.42070i |
19.17 | −0.881748 | + | 1.10568i | 2.54127 | − | 0.383034i | −0.445042 | − | 1.94986i | −1.54186 | + | 4.75633i | −1.81724 | + | 3.14756i | −0.460716 | − | 0.797984i | 2.54832 | + | 1.22721i | −2.28883 | + | 0.706011i | −3.89943 | − | 5.89868i |
19.18 | −0.881748 | + | 1.10568i | 2.92413 | − | 0.440742i | −0.445042 | − | 1.94986i | −4.00778 | − | 2.98959i | −2.09103 | + | 3.62176i | 4.77680 | + | 8.27366i | 2.54832 | + | 1.22721i | −0.243875 | + | 0.0752254i | 6.83938 | − | 1.79525i |
19.19 | −0.881748 | + | 1.10568i | 4.43770 | − | 0.668875i | −0.445042 | − | 1.94986i | 0.452191 | − | 4.97951i | −3.17337 | + | 5.49644i | 4.76870 | + | 8.25963i | 2.54832 | + | 1.22721i | 10.6456 | − | 3.28374i | 5.10701 | + | 4.89065i |
19.20 | −0.881748 | + | 1.10568i | 4.64965 | − | 0.700822i | −0.445042 | − | 1.94986i | 4.97525 | − | 0.496896i | −3.32494 | + | 5.75896i | 2.25382 | + | 3.90374i | 2.54832 | + | 1.22721i | 12.5279 | − | 3.86436i | −3.83751 | + | 5.93915i |
See next 80 embeddings (of 528 total) |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
5.b | even | 2 | 1 | inner |
43.h | odd | 42 | 1 | inner |
215.t | odd | 42 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 430.3.v.a | ✓ | 528 |
5.b | even | 2 | 1 | inner | 430.3.v.a | ✓ | 528 |
43.h | odd | 42 | 1 | inner | 430.3.v.a | ✓ | 528 |
215.t | odd | 42 | 1 | inner | 430.3.v.a | ✓ | 528 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
430.3.v.a | ✓ | 528 | 1.a | even | 1 | 1 | trivial |
430.3.v.a | ✓ | 528 | 5.b | even | 2 | 1 | inner |
430.3.v.a | ✓ | 528 | 43.h | odd | 42 | 1 | inner |
430.3.v.a | ✓ | 528 | 215.t | odd | 42 | 1 | inner |
Hecke kernels
This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(430, [\chi])\).