Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [880,2,Mod(17,880)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(880, base_ring=CyclotomicField(20))
chi = DirichletCharacter(H, H._module([0, 0, 5, 18]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("880.17");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 880 = 2^{4} \cdot 5 \cdot 11 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 880.cm (of order \(20\), degree \(8\), 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: | \(7.02683537787\) |
Analytic rank: | \(0\) |
Dimension: | \(48\) |
Relative dimension: | \(6\) over \(\Q(\zeta_{20})\) |
Twist minimal: | no (minimal twist has level 110) |
Sato-Tate group: | $\mathrm{SU}(2)[C_{20}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
17.1 | 0 | −2.03488 | + | 0.322293i | 0 | 0.667967 | + | 2.13397i | 0 | 0.0611439 | − | 0.386047i | 0 | 1.18368 | − | 0.384601i | 0 | ||||||||||
17.2 | 0 | −1.61854 | + | 0.256351i | 0 | −2.22860 | + | 0.182641i | 0 | −0.119289 | + | 0.753160i | 0 | −0.299227 | + | 0.0972249i | 0 | ||||||||||
17.3 | 0 | 0.216404 | − | 0.0342750i | 0 | 1.60893 | − | 1.55285i | 0 | 0.0165943 | − | 0.104772i | 0 | −2.80751 | + | 0.912216i | 0 | ||||||||||
17.4 | 0 | 1.23025 | − | 0.194853i | 0 | −1.87069 | + | 1.22496i | 0 | 0.466963 | − | 2.94829i | 0 | −1.37761 | + | 0.447613i | 0 | ||||||||||
17.5 | 0 | 2.20142 | − | 0.348671i | 0 | 2.18748 | − | 0.463629i | 0 | −0.620489 | + | 3.91761i | 0 | 1.87153 | − | 0.608097i | 0 | ||||||||||
17.6 | 0 | 2.79893 | − | 0.443308i | 0 | 0.537022 | + | 2.17062i | 0 | 0.516545 | − | 3.26134i | 0 | 4.78434 | − | 1.55453i | 0 | ||||||||||
193.1 | 0 | −0.322293 | − | 2.03488i | 0 | −1.79471 | + | 1.33380i | 0 | −0.386047 | − | 0.0611439i | 0 | −1.18368 | + | 0.384601i | 0 | ||||||||||
193.2 | 0 | −0.256351 | − | 1.61854i | 0 | 1.69562 | + | 1.45770i | 0 | 0.753160 | + | 0.119289i | 0 | 0.299227 | − | 0.0972249i | 0 | ||||||||||
193.3 | 0 | 0.0342750 | + | 0.216404i | 0 | −0.388914 | − | 2.20199i | 0 | −0.104772 | − | 0.0165943i | 0 | 2.80751 | − | 0.912216i | 0 | ||||||||||
193.4 | 0 | 0.194853 | + | 1.23025i | 0 | 0.793405 | + | 2.09058i | 0 | −2.94829 | − | 0.466963i | 0 | 1.37761 | − | 0.447613i | 0 | ||||||||||
193.5 | 0 | 0.348671 | + | 2.20142i | 0 | −1.49719 | − | 1.66085i | 0 | 3.91761 | + | 0.620489i | 0 | −1.87153 | + | 0.608097i | 0 | ||||||||||
193.6 | 0 | 0.443308 | + | 2.79893i | 0 | −1.71032 | + | 1.44042i | 0 | −3.26134 | − | 0.516545i | 0 | −4.78434 | + | 1.55453i | 0 | ||||||||||
337.1 | 0 | −1.18907 | − | 2.33368i | 0 | −1.05320 | − | 1.97251i | 0 | 2.88379 | + | 1.46937i | 0 | −2.26883 | + | 3.12278i | 0 | ||||||||||
337.2 | 0 | −0.992087 | − | 1.94708i | 0 | 0.244349 | + | 2.22268i | 0 | 3.05681 | + | 1.55752i | 0 | −1.04353 | + | 1.43630i | 0 | ||||||||||
337.3 | 0 | 0.176422 | + | 0.346247i | 0 | 0.0767418 | − | 2.23475i | 0 | 0.0136870 | + | 0.00697389i | 0 | 1.67459 | − | 2.30488i | 0 | ||||||||||
337.4 | 0 | 0.757609 | + | 1.48689i | 0 | −1.65111 | − | 1.50793i | 0 | −2.92930 | − | 1.49255i | 0 | 0.126480 | − | 0.174085i | 0 | ||||||||||
337.5 | 0 | 1.07350 | + | 2.10686i | 0 | 1.91408 | + | 1.15599i | 0 | 3.37294 | + | 1.71860i | 0 | −1.52312 | + | 2.09639i | 0 | ||||||||||
337.6 | 0 | 1.45771 | + | 2.86091i | 0 | −1.70643 | + | 1.44502i | 0 | 1.35828 | + | 0.692077i | 0 | −4.29653 | + | 5.91367i | 0 | ||||||||||
497.1 | 0 | −0.322293 | + | 2.03488i | 0 | −1.79471 | − | 1.33380i | 0 | −0.386047 | + | 0.0611439i | 0 | −1.18368 | − | 0.384601i | 0 | ||||||||||
497.2 | 0 | −0.256351 | + | 1.61854i | 0 | 1.69562 | − | 1.45770i | 0 | 0.753160 | − | 0.119289i | 0 | 0.299227 | + | 0.0972249i | 0 | ||||||||||
See all 48 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
5.c | odd | 4 | 1 | inner |
11.d | odd | 10 | 1 | inner |
55.l | even | 20 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 880.2.cm.c | 48 | |
4.b | odd | 2 | 1 | 110.2.k.a | ✓ | 48 | |
5.c | odd | 4 | 1 | inner | 880.2.cm.c | 48 | |
11.d | odd | 10 | 1 | inner | 880.2.cm.c | 48 | |
12.b | even | 2 | 1 | 990.2.bh.c | 48 | ||
20.d | odd | 2 | 1 | 550.2.bh.b | 48 | ||
20.e | even | 4 | 1 | 110.2.k.a | ✓ | 48 | |
20.e | even | 4 | 1 | 550.2.bh.b | 48 | ||
44.g | even | 10 | 1 | 110.2.k.a | ✓ | 48 | |
55.l | even | 20 | 1 | inner | 880.2.cm.c | 48 | |
60.l | odd | 4 | 1 | 990.2.bh.c | 48 | ||
132.n | odd | 10 | 1 | 990.2.bh.c | 48 | ||
220.o | even | 10 | 1 | 550.2.bh.b | 48 | ||
220.w | odd | 20 | 1 | 110.2.k.a | ✓ | 48 | |
220.w | odd | 20 | 1 | 550.2.bh.b | 48 | ||
660.bv | even | 20 | 1 | 990.2.bh.c | 48 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
110.2.k.a | ✓ | 48 | 4.b | odd | 2 | 1 | |
110.2.k.a | ✓ | 48 | 20.e | even | 4 | 1 | |
110.2.k.a | ✓ | 48 | 44.g | even | 10 | 1 | |
110.2.k.a | ✓ | 48 | 220.w | odd | 20 | 1 | |
550.2.bh.b | 48 | 20.d | odd | 2 | 1 | ||
550.2.bh.b | 48 | 20.e | even | 4 | 1 | ||
550.2.bh.b | 48 | 220.o | even | 10 | 1 | ||
550.2.bh.b | 48 | 220.w | odd | 20 | 1 | ||
880.2.cm.c | 48 | 1.a | even | 1 | 1 | trivial | |
880.2.cm.c | 48 | 5.c | odd | 4 | 1 | inner | |
880.2.cm.c | 48 | 11.d | odd | 10 | 1 | inner | |
880.2.cm.c | 48 | 55.l | even | 20 | 1 | inner | |
990.2.bh.c | 48 | 12.b | even | 2 | 1 | ||
990.2.bh.c | 48 | 60.l | odd | 4 | 1 | ||
990.2.bh.c | 48 | 132.n | odd | 10 | 1 | ||
990.2.bh.c | 48 | 660.bv | even | 20 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{48} - 4 T_{3}^{47} + 8 T_{3}^{46} - 60 T_{3}^{44} - 64 T_{3}^{43} + 736 T_{3}^{42} + \cdots + 723394816 \) acting on \(S_{2}^{\mathrm{new}}(880, [\chi])\).