Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [224,3,Mod(13,224)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(224, base_ring=CyclotomicField(8))
chi = DirichletCharacter(H, H._module([0, 7, 4]))
N = Newforms(chi, 3, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("224.13");
S:= CuspForms(chi, 3);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 224 = 2^{5} \cdot 7 \) |
Weight: | \( k \) | \(=\) | \( 3 \) |
Character orbit: | \([\chi]\) | \(=\) | 224.v (of order \(8\), degree \(4\), 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: | \(6.10355792167\) |
Analytic rank: | \(0\) |
Dimension: | \(240\) |
Relative dimension: | \(60\) over \(\Q(\zeta_{8})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{8}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
13.1 | −1.96919 | + | 0.349710i | −0.313666 | − | 0.129925i | 3.75541 | − | 1.37729i | −3.56967 | − | 8.61795i | 0.663104 | + | 0.146154i | −6.10386 | + | 3.42679i | −6.91345 | + | 4.02544i | −6.28246 | − | 6.28246i | 10.0431 | + | 15.7220i |
13.2 | −1.96919 | + | 0.349710i | 0.313666 | + | 0.129925i | 3.75541 | − | 1.37729i | 3.56967 | + | 8.61795i | −0.663104 | − | 0.146154i | −3.42679 | + | 6.10386i | −6.91345 | + | 4.02544i | −6.28246 | − | 6.28246i | −10.0431 | − | 15.7220i |
13.3 | −1.91889 | − | 0.563790i | −4.49480 | − | 1.86181i | 3.36428 | + | 2.16370i | 0.432309 | + | 1.04369i | 7.57535 | + | 6.10672i | −6.93508 | − | 0.951145i | −5.23581 | − | 6.04866i | 10.3729 | + | 10.3729i | −0.241133 | − | 2.24645i |
13.4 | −1.91889 | − | 0.563790i | 4.49480 | + | 1.86181i | 3.36428 | + | 2.16370i | −0.432309 | − | 1.04369i | −7.57535 | − | 6.10672i | 0.951145 | + | 6.93508i | −5.23581 | − | 6.04866i | 10.3729 | + | 10.3729i | 0.241133 | + | 2.24645i |
13.5 | −1.91469 | − | 0.577879i | −2.09916 | − | 0.869500i | 3.33211 | + | 2.21292i | −0.289108 | − | 0.697967i | 3.51678 | + | 2.87788i | 4.51230 | + | 5.35156i | −5.10118 | − | 6.16263i | −2.71353 | − | 2.71353i | 0.150212 | + | 1.50346i |
13.6 | −1.91469 | − | 0.577879i | 2.09916 | + | 0.869500i | 3.33211 | + | 2.21292i | 0.289108 | + | 0.697967i | −3.51678 | − | 2.87788i | −5.35156 | − | 4.51230i | −5.10118 | − | 6.16263i | −2.71353 | − | 2.71353i | −0.150212 | − | 1.50346i |
13.7 | −1.89176 | + | 0.649047i | −5.08830 | − | 2.10764i | 3.15748 | − | 2.45567i | −0.976625 | − | 2.35778i | 10.9938 | + | 0.684602i | 4.79964 | + | 5.09544i | −4.37933 | + | 6.69489i | 15.0847 | + | 15.0847i | 3.37785 | + | 3.82647i |
13.8 | −1.89176 | + | 0.649047i | 5.08830 | + | 2.10764i | 3.15748 | − | 2.45567i | 0.976625 | + | 2.35778i | −10.9938 | − | 0.684602i | −5.09544 | − | 4.79964i | −4.37933 | + | 6.69489i | 15.0847 | + | 15.0847i | −3.37785 | − | 3.82647i |
13.9 | −1.84088 | + | 0.781766i | −3.58432 | − | 1.48467i | 2.77768 | − | 2.87828i | 3.04365 | + | 7.34802i | 7.75897 | − | 0.0689922i | −1.24702 | − | 6.88803i | −2.86325 | + | 7.47006i | 4.27913 | + | 4.27913i | −11.3474 | − | 11.1474i |
13.10 | −1.84088 | + | 0.781766i | 3.58432 | + | 1.48467i | 2.77768 | − | 2.87828i | −3.04365 | − | 7.34802i | −7.75897 | + | 0.0689922i | 6.88803 | + | 1.24702i | −2.86325 | + | 7.47006i | 4.27913 | + | 4.27913i | 11.3474 | + | 11.1474i |
13.11 | −1.62574 | − | 1.16489i | −3.33013 | − | 1.37939i | 1.28608 | + | 3.78761i | −3.52027 | − | 8.49868i | 3.80711 | + | 6.12175i | 1.37310 | − | 6.86401i | 2.32130 | − | 7.65582i | 2.82310 | + | 2.82310i | −4.17695 | + | 17.9174i |
13.12 | −1.62574 | − | 1.16489i | 3.33013 | + | 1.37939i | 1.28608 | + | 3.78761i | 3.52027 | + | 8.49868i | −3.80711 | − | 6.12175i | 6.86401 | − | 1.37310i | 2.32130 | − | 7.65582i | 2.82310 | + | 2.82310i | 4.17695 | − | 17.9174i |
13.13 | −1.46638 | + | 1.36005i | −1.65755 | − | 0.686579i | 0.300514 | − | 3.98870i | 0.205220 | + | 0.495445i | 3.36437 | − | 1.24757i | −0.193440 | + | 6.99733i | 4.98417 | + | 6.25764i | −4.08789 | − | 4.08789i | −0.974760 | − | 0.447398i |
13.14 | −1.46638 | + | 1.36005i | 1.65755 | + | 0.686579i | 0.300514 | − | 3.98870i | −0.205220 | − | 0.495445i | −3.36437 | + | 1.24757i | −6.99733 | + | 0.193440i | 4.98417 | + | 6.25764i | −4.08789 | − | 4.08789i | 0.974760 | + | 0.447398i |
13.15 | −1.41192 | + | 1.41651i | −2.79421 | − | 1.15740i | −0.0129803 | − | 3.99998i | −1.86072 | − | 4.49217i | 5.58465 | − | 2.32386i | 0.944293 | − | 6.93602i | 5.68432 | + | 5.62925i | 0.104065 | + | 0.104065i | 8.99036 | + | 3.70685i |
13.16 | −1.41192 | + | 1.41651i | 2.79421 | + | 1.15740i | −0.0129803 | − | 3.99998i | 1.86072 | + | 4.49217i | −5.58465 | + | 2.32386i | 6.93602 | − | 0.944293i | 5.68432 | + | 5.62925i | 0.104065 | + | 0.104065i | −8.99036 | − | 3.70685i |
13.17 | −1.33217 | − | 1.49175i | −2.62896 | − | 1.08895i | −0.450649 | + | 3.97453i | 2.38974 | + | 5.76934i | 1.87778 | + | 5.37243i | −3.62994 | + | 5.98528i | 6.52936 | − | 4.62250i | −0.638332 | − | 0.638332i | 5.42289 | − | 11.2506i |
13.18 | −1.33217 | − | 1.49175i | 2.62896 | + | 1.08895i | −0.450649 | + | 3.97453i | −2.38974 | − | 5.76934i | −1.87778 | − | 5.37243i | −5.98528 | + | 3.62994i | 6.52936 | − | 4.62250i | −0.638332 | − | 0.638332i | −5.42289 | + | 11.2506i |
13.19 | −1.28270 | − | 1.53450i | −0.416824 | − | 0.172654i | −0.709381 | + | 3.93659i | 1.21861 | + | 2.94198i | 0.269721 | + | 0.861080i | −3.48320 | − | 6.07185i | 6.95063 | − | 3.96091i | −6.22003 | − | 6.22003i | 2.95137 | − | 5.64363i |
13.20 | −1.28270 | − | 1.53450i | 0.416824 | + | 0.172654i | −0.709381 | + | 3.93659i | −1.21861 | − | 2.94198i | −0.269721 | − | 0.861080i | 6.07185 | + | 3.48320i | 6.95063 | − | 3.96091i | −6.22003 | − | 6.22003i | −2.95137 | + | 5.64363i |
See next 80 embeddings (of 240 total) |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
7.b | odd | 2 | 1 | inner |
32.g | even | 8 | 1 | inner |
224.v | odd | 8 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 224.3.v.b | ✓ | 240 |
7.b | odd | 2 | 1 | inner | 224.3.v.b | ✓ | 240 |
32.g | even | 8 | 1 | inner | 224.3.v.b | ✓ | 240 |
224.v | odd | 8 | 1 | inner | 224.3.v.b | ✓ | 240 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
224.3.v.b | ✓ | 240 | 1.a | even | 1 | 1 | trivial |
224.3.v.b | ✓ | 240 | 7.b | odd | 2 | 1 | inner |
224.3.v.b | ✓ | 240 | 32.g | even | 8 | 1 | inner |
224.3.v.b | ✓ | 240 | 224.v | odd | 8 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{240} + 4 T_{3}^{238} + 8 T_{3}^{236} - 1608 T_{3}^{234} + 4962748 T_{3}^{232} + 18548896 T_{3}^{230} + 35786432 T_{3}^{228} - 6084204608 T_{3}^{226} + 10440519986480 T_{3}^{224} + \cdots + 87\!\cdots\!36 \)
acting on \(S_{3}^{\mathrm{new}}(224, [\chi])\).