Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [980,2,Mod(391,980)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(980, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([1, 0, 1]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("980.391");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 980 = 2^{2} \cdot 5 \cdot 7^{2} \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 980.g (of order \(2\), degree \(1\), 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: | \(7.82533939809\) |
Analytic rank: | \(0\) |
Dimension: | \(48\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{2}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
391.1 | −1.27833 | − | 0.604870i | −1.35351 | 1.26826 | + | 1.54645i | 1.00000i | 1.73023 | + | 0.818697i | 0 | −0.685859 | − | 2.74401i | −1.16802 | 0.604870 | − | 1.27833i | ||||||||
391.2 | −1.27833 | − | 0.604870i | 1.35351 | 1.26826 | + | 1.54645i | − | 1.00000i | −1.73023 | − | 0.818697i | 0 | −0.685859 | − | 2.74401i | −1.16802 | −0.604870 | + | 1.27833i | |||||||
391.3 | −1.27833 | + | 0.604870i | −1.35351 | 1.26826 | − | 1.54645i | − | 1.00000i | 1.73023 | − | 0.818697i | 0 | −0.685859 | + | 2.74401i | −1.16802 | 0.604870 | + | 1.27833i | |||||||
391.4 | −1.27833 | + | 0.604870i | 1.35351 | 1.26826 | − | 1.54645i | 1.00000i | −1.73023 | + | 0.818697i | 0 | −0.685859 | + | 2.74401i | −1.16802 | −0.604870 | − | 1.27833i | ||||||||
391.5 | −1.04820 | − | 0.949355i | −1.29812 | 0.197452 | + | 1.99023i | − | 1.00000i | 1.36069 | + | 1.23237i | 0 | 1.68246 | − | 2.27361i | −1.31490 | −0.949355 | + | 1.04820i | |||||||
391.6 | −1.04820 | − | 0.949355i | 1.29812 | 0.197452 | + | 1.99023i | 1.00000i | −1.36069 | − | 1.23237i | 0 | 1.68246 | − | 2.27361i | −1.31490 | 0.949355 | − | 1.04820i | ||||||||
391.7 | −1.04820 | + | 0.949355i | −1.29812 | 0.197452 | − | 1.99023i | 1.00000i | 1.36069 | − | 1.23237i | 0 | 1.68246 | + | 2.27361i | −1.31490 | −0.949355 | − | 1.04820i | ||||||||
391.8 | −1.04820 | + | 0.949355i | 1.29812 | 0.197452 | − | 1.99023i | − | 1.00000i | −1.36069 | + | 1.23237i | 0 | 1.68246 | + | 2.27361i | −1.31490 | 0.949355 | + | 1.04820i | |||||||
391.9 | −1.01402 | − | 0.985786i | −2.46214 | 0.0564536 | + | 1.99920i | − | 1.00000i | 2.49664 | + | 2.42714i | 0 | 1.91354 | − | 2.08287i | 3.06212 | −0.985786 | + | 1.01402i | |||||||
391.10 | −1.01402 | − | 0.985786i | 2.46214 | 0.0564536 | + | 1.99920i | 1.00000i | −2.49664 | − | 2.42714i | 0 | 1.91354 | − | 2.08287i | 3.06212 | 0.985786 | − | 1.01402i | ||||||||
391.11 | −1.01402 | + | 0.985786i | −2.46214 | 0.0564536 | − | 1.99920i | 1.00000i | 2.49664 | − | 2.42714i | 0 | 1.91354 | + | 2.08287i | 3.06212 | −0.985786 | − | 1.01402i | ||||||||
391.12 | −1.01402 | + | 0.985786i | 2.46214 | 0.0564536 | − | 1.99920i | − | 1.00000i | −2.49664 | + | 2.42714i | 0 | 1.91354 | + | 2.08287i | 3.06212 | 0.985786 | + | 1.01402i | |||||||
391.13 | −0.560755 | − | 1.29829i | −0.396131 | −1.37111 | + | 1.45604i | − | 1.00000i | 0.222133 | + | 0.514293i | 0 | 2.65922 | + | 0.963609i | −2.84308 | −1.29829 | + | 0.560755i | |||||||
391.14 | −0.560755 | − | 1.29829i | 0.396131 | −1.37111 | + | 1.45604i | 1.00000i | −0.222133 | − | 0.514293i | 0 | 2.65922 | + | 0.963609i | −2.84308 | 1.29829 | − | 0.560755i | ||||||||
391.15 | −0.560755 | + | 1.29829i | −0.396131 | −1.37111 | − | 1.45604i | 1.00000i | 0.222133 | − | 0.514293i | 0 | 2.65922 | − | 0.963609i | −2.84308 | −1.29829 | − | 0.560755i | ||||||||
391.16 | −0.560755 | + | 1.29829i | 0.396131 | −1.37111 | − | 1.45604i | − | 1.00000i | −0.222133 | + | 0.514293i | 0 | 2.65922 | − | 0.963609i | −2.84308 | 1.29829 | + | 0.560755i | |||||||
391.17 | −0.281497 | − | 1.38591i | −1.31255 | −1.84152 | + | 0.780262i | 1.00000i | 0.369479 | + | 1.81908i | 0 | 1.59976 | + | 2.33255i | −1.27722 | 1.38591 | − | 0.281497i | ||||||||
391.18 | −0.281497 | − | 1.38591i | 1.31255 | −1.84152 | + | 0.780262i | − | 1.00000i | −0.369479 | − | 1.81908i | 0 | 1.59976 | + | 2.33255i | −1.27722 | −1.38591 | + | 0.281497i | |||||||
391.19 | −0.281497 | + | 1.38591i | −1.31255 | −1.84152 | − | 0.780262i | − | 1.00000i | 0.369479 | − | 1.81908i | 0 | 1.59976 | − | 2.33255i | −1.27722 | 1.38591 | + | 0.281497i | |||||||
391.20 | −0.281497 | + | 1.38591i | 1.31255 | −1.84152 | − | 0.780262i | 1.00000i | −0.369479 | + | 1.81908i | 0 | 1.59976 | − | 2.33255i | −1.27722 | −1.38591 | − | 0.281497i | ||||||||
See all 48 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
4.b | odd | 2 | 1 | inner |
7.b | odd | 2 | 1 | inner |
28.d | even | 2 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 980.2.g.b | ✓ | 48 |
4.b | odd | 2 | 1 | inner | 980.2.g.b | ✓ | 48 |
7.b | odd | 2 | 1 | inner | 980.2.g.b | ✓ | 48 |
7.c | even | 3 | 2 | 980.2.o.g | 96 | ||
7.d | odd | 6 | 2 | 980.2.o.g | 96 | ||
28.d | even | 2 | 1 | inner | 980.2.g.b | ✓ | 48 |
28.f | even | 6 | 2 | 980.2.o.g | 96 | ||
28.g | odd | 6 | 2 | 980.2.o.g | 96 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
980.2.g.b | ✓ | 48 | 1.a | even | 1 | 1 | trivial |
980.2.g.b | ✓ | 48 | 4.b | odd | 2 | 1 | inner |
980.2.g.b | ✓ | 48 | 7.b | odd | 2 | 1 | inner |
980.2.g.b | ✓ | 48 | 28.d | even | 2 | 1 | inner |
980.2.o.g | 96 | 7.c | even | 3 | 2 | ||
980.2.o.g | 96 | 7.d | odd | 6 | 2 | ||
980.2.o.g | 96 | 28.f | even | 6 | 2 | ||
980.2.o.g | 96 | 28.g | odd | 6 | 2 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{24} - 48 T_{3}^{22} + 984 T_{3}^{20} - 11304 T_{3}^{18} + 80202 T_{3}^{16} - 365048 T_{3}^{14} + \cdots + 1348 \) acting on \(S_{2}^{\mathrm{new}}(980, [\chi])\).