Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [76,3,Mod(23,76)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(76, base_ring=CyclotomicField(18))
chi = DirichletCharacter(H, H._module([9, 2]))
N = Newforms(chi, 3, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("76.23");
S:= CuspForms(chi, 3);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 76 = 2^{2} \cdot 19 \) |
Weight: | \( k \) | \(=\) | \( 3 \) |
Character orbit: | \([\chi]\) | \(=\) | 76.l (of order \(18\), degree \(6\), 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: | \(2.07085000914\) |
Analytic rank: | \(0\) |
Dimension: | \(108\) |
Relative dimension: | \(18\) over \(\Q(\zeta_{18})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{18}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
23.1 | −1.96172 | − | 0.389448i | −1.94869 | + | 0.343607i | 3.69666 | + | 1.52797i | 1.01207 | − | 0.849231i | 3.95660 | + | 0.0848543i | −7.34996 | − | 4.24350i | −6.65673 | − | 4.43710i | −4.77790 | + | 1.73901i | −2.31613 | + | 1.27180i |
23.2 | −1.92177 | + | 0.553897i | 4.90961 | − | 0.865697i | 3.38640 | − | 2.12892i | 2.37531 | − | 1.99312i | −8.95563 | + | 4.38309i | −8.41114 | − | 4.85617i | −5.32867 | + | 5.96702i | 14.8976 | − | 5.42229i | −3.46081 | + | 5.14599i |
23.3 | −1.77891 | + | 0.914055i | 0.408304 | − | 0.0719951i | 2.32901 | − | 3.25203i | −5.90028 | + | 4.95092i | −0.660528 | + | 0.501285i | 6.78454 | + | 3.91705i | −1.17055 | + | 7.91390i | −8.29570 | + | 3.01939i | 5.97063 | − | 14.2004i |
23.4 | −1.75309 | − | 0.962633i | 1.94869 | − | 0.343607i | 2.14668 | + | 3.37517i | 1.01207 | − | 0.849231i | −3.74701 | − | 1.27350i | 7.34996 | + | 4.24350i | −0.514277 | − | 7.98345i | −4.77790 | + | 1.73901i | −2.59176 | + | 0.514526i |
23.5 | −1.57322 | + | 1.23490i | −4.66468 | + | 0.822509i | 0.950039 | − | 3.88554i | 2.08183 | − | 1.74687i | 6.32285 | − | 7.05441i | −0.551331 | − | 0.318311i | 3.30364 | + | 7.28601i | 12.6255 | − | 4.59531i | −1.11798 | + | 5.31906i |
23.6 | −1.11612 | − | 1.65960i | −4.90961 | + | 0.865697i | −1.50854 | + | 3.70463i | 2.37531 | − | 1.99312i | 6.91644 | + | 7.18176i | 8.41114 | + | 4.85617i | 7.83192 | − | 1.63125i | 14.8976 | − | 5.42229i | −5.95892 | − | 1.71749i |
23.7 | −0.831743 | + | 1.81885i | 1.75796 | − | 0.309976i | −2.61641 | − | 3.02563i | 6.16490 | − | 5.17297i | −0.898372 | + | 3.45528i | 6.99959 | + | 4.04121i | 7.67933 | − | 2.24230i | −5.46290 | + | 1.98833i | 4.28122 | + | 15.5156i |
23.8 | −0.775177 | − | 1.84366i | −0.408304 | + | 0.0719951i | −2.79820 | + | 2.85833i | −5.90028 | + | 4.95092i | 0.449243 | + | 0.696968i | −6.78454 | − | 3.91705i | 7.43891 | + | 2.94323i | −8.29570 | + | 3.01939i | 13.7016 | + | 7.04030i |
23.9 | −0.411377 | − | 1.95724i | 4.66468 | − | 0.822509i | −3.66154 | + | 1.61032i | 2.08183 | − | 1.74687i | −3.52879 | − | 8.79152i | 0.551331 | + | 0.318311i | 4.65805 | + | 6.50404i | 12.6255 | − | 4.59531i | −4.27545 | − | 3.35602i |
23.10 | 0.0616288 | + | 1.99905i | −2.53724 | + | 0.447383i | −3.99240 | + | 0.246398i | −1.56377 | + | 1.31216i | −1.05071 | − | 5.04449i | −3.64994 | − | 2.10730i | −0.738609 | − | 7.96583i | −2.21981 | + | 0.807946i | −2.71944 | − | 3.04518i |
23.11 | 0.156355 | + | 1.99388i | 5.33798 | − | 0.941229i | −3.95111 | + | 0.623505i | −5.57927 | + | 4.68156i | 2.71132 | + | 10.4961i | 3.83135 | + | 2.21203i | −1.86097 | − | 7.78054i | 19.1508 | − | 6.97034i | −10.2068 | − | 10.3924i |
23.12 | 0.531980 | − | 1.92795i | −1.75796 | + | 0.309976i | −3.43399 | − | 2.05126i | 6.16490 | − | 5.17297i | −0.337581 | + | 3.55416i | −6.99959 | − | 4.04121i | −5.78155 | + | 5.52934i | −5.46290 | + | 1.98833i | −6.69363 | − | 14.6375i |
23.13 | 1.32019 | + | 1.50237i | 2.09930 | − | 0.370163i | −0.514209 | + | 3.96681i | 3.93098 | − | 3.29849i | 3.32759 | + | 2.66523i | −3.79547 | − | 2.19131i | −6.63846 | + | 4.46440i | −4.18719 | + | 1.52401i | 10.1452 | + | 1.55116i |
23.14 | 1.33218 | − | 1.49175i | 2.53724 | − | 0.447383i | −0.450619 | − | 3.97454i | −1.56377 | + | 1.31216i | 2.71266 | − | 4.38091i | 3.64994 | + | 2.10730i | −6.52931 | − | 4.62257i | −2.21981 | + | 0.807946i | −0.125806 | + | 4.08077i |
23.15 | 1.40142 | − | 1.42690i | −5.33798 | + | 0.941229i | −0.0720694 | − | 3.99935i | −5.57927 | + | 4.68156i | −6.13769 | + | 8.93580i | −3.83135 | − | 2.21203i | −5.80766 | − | 5.50192i | 19.1508 | − | 6.97034i | −1.13877 | + | 14.5219i |
23.16 | 1.57762 | + | 1.22927i | −2.79584 | + | 0.492981i | 0.977798 | + | 3.87865i | −3.69543 | + | 3.10084i | −5.01679 | − | 2.65909i | 7.95020 | + | 4.59005i | −3.22530 | + | 7.32103i | −0.883563 | + | 0.321590i | −9.64176 | + | 0.349277i |
23.17 | 1.97702 | − | 0.302279i | −2.09930 | + | 0.370163i | 3.81725 | − | 1.19523i | 3.93098 | − | 3.29849i | −4.03848 | + | 1.36640i | 3.79547 | + | 2.19131i | 7.18552 | − | 3.51687i | −4.18719 | + | 1.52401i | 6.77459 | − | 7.70944i |
23.18 | 1.99869 | + | 0.0724033i | 2.79584 | − | 0.492981i | 3.98952 | + | 0.289423i | −3.69543 | + | 3.10084i | 5.62370 | − | 0.782889i | −7.95020 | − | 4.59005i | 7.95285 | + | 0.867321i | −0.883563 | + | 0.321590i | −7.61053 | + | 5.93004i |
35.1 | −1.97401 | − | 0.321352i | 1.80941 | + | 4.97132i | 3.79347 | + | 1.26871i | 0.0746739 | − | 0.423497i | −1.97426 | − | 10.3949i | −3.59587 | + | 2.07608i | −7.08065 | − | 3.72348i | −14.5457 | + | 12.2053i | −0.283499 | + | 0.811992i |
35.2 | −1.94561 | − | 0.463236i | −0.599610 | − | 1.64741i | 3.57083 | + | 1.80256i | −1.33437 | + | 7.56759i | 0.403468 | + | 3.48299i | 9.08670 | − | 5.24621i | −6.11244 | − | 5.16121i | 4.53996 | − | 3.80948i | 6.10175 | − | 14.1055i |
See next 80 embeddings (of 108 total) |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
4.b | odd | 2 | 1 | inner |
19.e | even | 9 | 1 | inner |
76.l | odd | 18 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 76.3.l.a | ✓ | 108 |
4.b | odd | 2 | 1 | inner | 76.3.l.a | ✓ | 108 |
19.e | even | 9 | 1 | inner | 76.3.l.a | ✓ | 108 |
76.l | odd | 18 | 1 | inner | 76.3.l.a | ✓ | 108 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
76.3.l.a | ✓ | 108 | 1.a | even | 1 | 1 | trivial |
76.3.l.a | ✓ | 108 | 4.b | odd | 2 | 1 | inner |
76.3.l.a | ✓ | 108 | 19.e | even | 9 | 1 | inner |
76.3.l.a | ✓ | 108 | 76.l | odd | 18 | 1 | inner |
Hecke kernels
This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(76, [\chi])\).