Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [416,2,Mod(15,416)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(416, base_ring=CyclotomicField(12))
chi = DirichletCharacter(H, H._module([6, 6, 1]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("416.15");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 416 = 2^{5} \cdot 13 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 416.bk (of order \(12\), degree \(4\), not 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: | \(3.32177672409\) |
Analytic rank: | \(0\) |
Dimension: | \(48\) |
Relative dimension: | \(12\) over \(\Q(\zeta_{12})\) |
Twist minimal: | no (minimal twist has level 104) |
Sato-Tate group: | $\mathrm{SU}(2)[C_{12}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
15.1 | 0 | −1.49630 | − | 2.59167i | 0 | −1.40823 | − | 1.40823i | 0 | −1.02175 | − | 3.81323i | 0 | −2.97784 | + | 5.15776i | 0 | ||||||||||
15.2 | 0 | −1.49630 | − | 2.59167i | 0 | 1.40823 | + | 1.40823i | 0 | 1.02175 | + | 3.81323i | 0 | −2.97784 | + | 5.15776i | 0 | ||||||||||
15.3 | 0 | −0.793616 | − | 1.37458i | 0 | −0.203095 | − | 0.203095i | 0 | −0.207385 | − | 0.773970i | 0 | 0.240346 | − | 0.416291i | 0 | ||||||||||
15.4 | 0 | −0.793616 | − | 1.37458i | 0 | 0.203095 | + | 0.203095i | 0 | 0.207385 | + | 0.773970i | 0 | 0.240346 | − | 0.416291i | 0 | ||||||||||
15.5 | 0 | 0.217361 | + | 0.376480i | 0 | −1.34711 | − | 1.34711i | 0 | 1.10888 | + | 4.13839i | 0 | 1.40551 | − | 2.43441i | 0 | ||||||||||
15.6 | 0 | 0.217361 | + | 0.376480i | 0 | 1.34711 | + | 1.34711i | 0 | −1.10888 | − | 4.13839i | 0 | 1.40551 | − | 2.43441i | 0 | ||||||||||
15.7 | 0 | 0.253720 | + | 0.439456i | 0 | −2.25762 | − | 2.25762i | 0 | 0.0840763 | + | 0.313777i | 0 | 1.37125 | − | 2.37508i | 0 | ||||||||||
15.8 | 0 | 0.253720 | + | 0.439456i | 0 | 2.25762 | + | 2.25762i | 0 | −0.0840763 | − | 0.313777i | 0 | 1.37125 | − | 2.37508i | 0 | ||||||||||
15.9 | 0 | 0.958009 | + | 1.65932i | 0 | −2.08788 | − | 2.08788i | 0 | −0.981174 | − | 3.66179i | 0 | −0.335561 | + | 0.581208i | 0 | ||||||||||
15.10 | 0 | 0.958009 | + | 1.65932i | 0 | 2.08788 | + | 2.08788i | 0 | 0.981174 | + | 3.66179i | 0 | −0.335561 | + | 0.581208i | 0 | ||||||||||
15.11 | 0 | 1.36083 | + | 2.35702i | 0 | −1.35602 | − | 1.35602i | 0 | 0.327458 | + | 1.22209i | 0 | −2.20371 | + | 3.81694i | 0 | ||||||||||
15.12 | 0 | 1.36083 | + | 2.35702i | 0 | 1.35602 | + | 1.35602i | 0 | −0.327458 | − | 1.22209i | 0 | −2.20371 | + | 3.81694i | 0 | ||||||||||
111.1 | 0 | −1.49630 | + | 2.59167i | 0 | −1.40823 | + | 1.40823i | 0 | −1.02175 | + | 3.81323i | 0 | −2.97784 | − | 5.15776i | 0 | ||||||||||
111.2 | 0 | −1.49630 | + | 2.59167i | 0 | 1.40823 | − | 1.40823i | 0 | 1.02175 | − | 3.81323i | 0 | −2.97784 | − | 5.15776i | 0 | ||||||||||
111.3 | 0 | −0.793616 | + | 1.37458i | 0 | −0.203095 | + | 0.203095i | 0 | −0.207385 | + | 0.773970i | 0 | 0.240346 | + | 0.416291i | 0 | ||||||||||
111.4 | 0 | −0.793616 | + | 1.37458i | 0 | 0.203095 | − | 0.203095i | 0 | 0.207385 | − | 0.773970i | 0 | 0.240346 | + | 0.416291i | 0 | ||||||||||
111.5 | 0 | 0.217361 | − | 0.376480i | 0 | −1.34711 | + | 1.34711i | 0 | 1.10888 | − | 4.13839i | 0 | 1.40551 | + | 2.43441i | 0 | ||||||||||
111.6 | 0 | 0.217361 | − | 0.376480i | 0 | 1.34711 | − | 1.34711i | 0 | −1.10888 | + | 4.13839i | 0 | 1.40551 | + | 2.43441i | 0 | ||||||||||
111.7 | 0 | 0.253720 | − | 0.439456i | 0 | −2.25762 | + | 2.25762i | 0 | 0.0840763 | − | 0.313777i | 0 | 1.37125 | + | 2.37508i | 0 | ||||||||||
111.8 | 0 | 0.253720 | − | 0.439456i | 0 | 2.25762 | − | 2.25762i | 0 | −0.0840763 | + | 0.313777i | 0 | 1.37125 | + | 2.37508i | 0 | ||||||||||
See all 48 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
8.d | odd | 2 | 1 | inner |
13.f | odd | 12 | 1 | inner |
104.u | even | 12 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 416.2.bk.a | 48 | |
4.b | odd | 2 | 1 | 104.2.u.a | ✓ | 48 | |
8.b | even | 2 | 1 | 104.2.u.a | ✓ | 48 | |
8.d | odd | 2 | 1 | inner | 416.2.bk.a | 48 | |
12.b | even | 2 | 1 | 936.2.ed.d | 48 | ||
13.f | odd | 12 | 1 | inner | 416.2.bk.a | 48 | |
24.h | odd | 2 | 1 | 936.2.ed.d | 48 | ||
52.l | even | 12 | 1 | 104.2.u.a | ✓ | 48 | |
104.u | even | 12 | 1 | inner | 416.2.bk.a | 48 | |
104.x | odd | 12 | 1 | 104.2.u.a | ✓ | 48 | |
156.v | odd | 12 | 1 | 936.2.ed.d | 48 | ||
312.bo | even | 12 | 1 | 936.2.ed.d | 48 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
104.2.u.a | ✓ | 48 | 4.b | odd | 2 | 1 | |
104.2.u.a | ✓ | 48 | 8.b | even | 2 | 1 | |
104.2.u.a | ✓ | 48 | 52.l | even | 12 | 1 | |
104.2.u.a | ✓ | 48 | 104.x | odd | 12 | 1 | |
416.2.bk.a | 48 | 1.a | even | 1 | 1 | trivial | |
416.2.bk.a | 48 | 8.d | odd | 2 | 1 | inner | |
416.2.bk.a | 48 | 13.f | odd | 12 | 1 | inner | |
416.2.bk.a | 48 | 104.u | even | 12 | 1 | inner | |
936.2.ed.d | 48 | 12.b | even | 2 | 1 | ||
936.2.ed.d | 48 | 24.h | odd | 2 | 1 | ||
936.2.ed.d | 48 | 156.v | odd | 12 | 1 | ||
936.2.ed.d | 48 | 312.bo | even | 12 | 1 |
Hecke kernels
This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(416, [\chi])\).