Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [693,2,Mod(37,693)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(693, base_ring=CyclotomicField(30))
chi = DirichletCharacter(H, H._module([0, 10, 6]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("693.37");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 693 = 3^{2} \cdot 7 \cdot 11 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 693.by (of order \(15\), 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: | \(5.53363286007\) |
Analytic rank: | \(0\) |
Dimension: | \(64\) |
Relative dimension: | \(8\) over \(\Q(\zeta_{15})\) |
Twist minimal: | no (minimal twist has level 231) |
Sato-Tate group: | $\mathrm{SU}(2)[C_{15}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
37.1 | −1.83841 | + | 0.818513i | 0 | 1.37153 | − | 1.52323i | 0.0520867 | + | 0.495572i | 0 | 0.483130 | + | 2.60127i | −0.0309144 | + | 0.0951446i | 0 | −0.501389 | − | 0.868431i | ||||||
37.2 | −1.54620 | + | 0.688411i | 0 | 0.578555 | − | 0.642550i | 0.317169 | + | 3.01766i | 0 | −1.40565 | − | 2.24146i | 0.593816 | − | 1.82758i | 0 | −2.56780 | − | 4.44756i | ||||||
37.3 | −1.16334 | + | 0.517953i | 0 | −0.253175 | + | 0.281180i | −0.397932 | − | 3.78607i | 0 | 0.522748 | − | 2.59359i | 0.935917 | − | 2.88046i | 0 | 2.42393 | + | 4.19837i | ||||||
37.4 | −0.0592521 | + | 0.0263807i | 0 | −1.33545 | + | 1.48316i | −0.117657 | − | 1.11943i | 0 | −1.35463 | + | 2.27266i | 0.0800864 | − | 0.246481i | 0 | 0.0365029 | + | 0.0632249i | ||||||
37.5 | 0.122947 | − | 0.0547397i | 0 | −1.32614 | + | 1.47283i | 0.379677 | + | 3.61238i | 0 | 2.62556 | + | 0.326267i | −0.165600 | + | 0.509665i | 0 | 0.244421 | + | 0.423350i | ||||||
37.6 | 1.54210 | − | 0.686586i | 0 | 0.568401 | − | 0.631273i | 0.279932 | + | 2.66338i | 0 | −2.55686 | + | 0.680067i | −0.600157 | + | 1.84709i | 0 | 2.26032 | + | 3.91499i | ||||||
37.7 | 1.70284 | − | 0.758153i | 0 | 0.986607 | − | 1.09574i | −0.363031 | − | 3.45401i | 0 | 2.64131 | + | 0.153270i | −0.302713 | + | 0.931655i | 0 | −3.23685 | − | 5.60640i | ||||||
37.8 | 2.50180 | − | 1.11387i | 0 | 3.68004 | − | 4.08710i | −0.150245 | − | 1.42949i | 0 | −2.64282 | − | 0.124535i | 2.96170 | − | 9.11518i | 0 | −1.96815 | − | 3.40894i | ||||||
163.1 | −1.55128 | − | 1.72288i | 0 | −0.352761 | + | 3.35629i | −0.651377 | − | 0.138454i | 0 | −0.438868 | + | 2.60910i | 2.57853 | − | 1.87341i | 0 | 0.771931 | + | 1.33702i | ||||||
163.2 | −0.872627 | − | 0.969151i | 0 | 0.0312821 | − | 0.297629i | −0.151225 | − | 0.0321438i | 0 | 2.55326 | + | 0.693447i | −2.42586 | + | 1.76249i | 0 | 0.100810 | + | 0.174609i | ||||||
163.3 | −0.654921 | − | 0.727363i | 0 | 0.108921 | − | 1.03631i | 3.19676 | + | 0.679492i | 0 | −2.61318 | + | 0.413884i | −2.40878 | + | 1.75008i | 0 | −1.59938 | − | 2.77022i | ||||||
163.4 | 0.285304 | + | 0.316862i | 0 | 0.190054 | − | 1.80824i | −0.861829 | − | 0.183187i | 0 | −0.532095 | − | 2.59169i | 1.31708 | − | 0.956916i | 0 | −0.187838 | − | 0.325345i | ||||||
163.5 | 0.805605 | + | 0.894715i | 0 | 0.0575414 | − | 0.547470i | −3.19313 | − | 0.678721i | 0 | 2.33800 | + | 1.23845i | 2.48423 | − | 1.80490i | 0 | −1.96514 | − | 3.40372i | ||||||
163.6 | 1.18104 | + | 1.31168i | 0 | −0.116586 | + | 1.10924i | 0.872919 | + | 0.185545i | 0 | −0.544420 | + | 2.58913i | 1.26323 | − | 0.917790i | 0 | 0.787577 | + | 1.36412i | ||||||
163.7 | 1.40081 | + | 1.55575i | 0 | −0.249052 | + | 2.36957i | 3.68128 | + | 0.782479i | 0 | 0.128636 | − | 2.64262i | −0.648035 | + | 0.470825i | 0 | 3.93941 | + | 6.82326i | ||||||
163.8 | 1.82701 | + | 2.02911i | 0 | −0.570229 | + | 5.42537i | −2.89339 | − | 0.615009i | 0 | −2.43409 | − | 1.03693i | −7.63253 | + | 5.54536i | 0 | −4.03835 | − | 6.99462i | ||||||
235.1 | −0.286258 | + | 2.72356i | 0 | −5.37956 | − | 1.14346i | 1.31309 | + | 0.584627i | 0 | 2.06489 | − | 1.65416i | 2.96170 | − | 9.11518i | 0 | −1.96815 | + | 3.40894i | ||||||
235.2 | −0.194840 | + | 1.85378i | 0 | −1.44224 | − | 0.306558i | 3.17278 | + | 1.41261i | 0 | −2.04677 | + | 1.67652i | −0.302713 | + | 0.931655i | 0 | −3.23685 | + | 5.60640i | ||||||
235.3 | −0.176448 | + | 1.67879i | 0 | −0.830899 | − | 0.176613i | −2.44652 | − | 1.08926i | 0 | 2.46827 | − | 0.952696i | −0.600157 | + | 1.84709i | 0 | 2.26032 | − | 3.91499i | ||||||
235.4 | −0.0140677 | + | 0.133845i | 0 | 1.93858 | + | 0.412058i | −3.31825 | − | 1.47738i | 0 | −1.93235 | + | 1.80722i | −0.165600 | + | 0.509665i | 0 | 0.244421 | − | 0.423350i | ||||||
See all 64 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
7.c | even | 3 | 1 | inner |
11.c | even | 5 | 1 | inner |
77.m | even | 15 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 693.2.by.d | 64 | |
3.b | odd | 2 | 1 | 231.2.y.a | ✓ | 64 | |
7.c | even | 3 | 1 | inner | 693.2.by.d | 64 | |
11.c | even | 5 | 1 | inner | 693.2.by.d | 64 | |
21.h | odd | 6 | 1 | 231.2.y.a | ✓ | 64 | |
33.h | odd | 10 | 1 | 231.2.y.a | ✓ | 64 | |
77.m | even | 15 | 1 | inner | 693.2.by.d | 64 | |
231.z | odd | 30 | 1 | 231.2.y.a | ✓ | 64 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
231.2.y.a | ✓ | 64 | 3.b | odd | 2 | 1 | |
231.2.y.a | ✓ | 64 | 21.h | odd | 6 | 1 | |
231.2.y.a | ✓ | 64 | 33.h | odd | 10 | 1 | |
231.2.y.a | ✓ | 64 | 231.z | odd | 30 | 1 | |
693.2.by.d | 64 | 1.a | even | 1 | 1 | trivial | |
693.2.by.d | 64 | 7.c | even | 3 | 1 | inner | |
693.2.by.d | 64 | 11.c | even | 5 | 1 | inner | |
693.2.by.d | 64 | 77.m | even | 15 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{64} - 13 T_{2}^{62} + 16 T_{2}^{61} + 61 T_{2}^{60} - 96 T_{2}^{59} + 238 T_{2}^{58} - 614 T_{2}^{57} + \cdots + 625 \) acting on \(S_{2}^{\mathrm{new}}(693, [\chi])\).