Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1680,2,Mod(209,1680)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1680, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0, 1, 1, 1]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("1680.209");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 1680 = 2^{4} \cdot 3 \cdot 5 \cdot 7 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 1680.k (of order \(2\), degree \(1\), 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: | \(13.4148675396\) |
Analytic rank: | \(0\) |
Dimension: | \(24\) |
Twist minimal: | no (minimal twist has level 840) |
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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
209.1 | 0 | −1.70556 | − | 0.301745i | 0 | 2.04579 | + | 0.902636i | 0 | 1.30232 | − | 2.30304i | 0 | 2.81790 | + | 1.02929i | 0 | ||||||||||
209.2 | 0 | −1.70556 | + | 0.301745i | 0 | 2.04579 | − | 0.902636i | 0 | 1.30232 | + | 2.30304i | 0 | 2.81790 | − | 1.02929i | 0 | ||||||||||
209.3 | 0 | −1.64602 | − | 0.539084i | 0 | −1.30213 | − | 1.81782i | 0 | −2.19974 | + | 1.47008i | 0 | 2.41878 | + | 1.77469i | 0 | ||||||||||
209.4 | 0 | −1.64602 | + | 0.539084i | 0 | −1.30213 | + | 1.81782i | 0 | −2.19974 | − | 1.47008i | 0 | 2.41878 | − | 1.77469i | 0 | ||||||||||
209.5 | 0 | −1.37873 | − | 1.04838i | 0 | −1.84099 | + | 1.26915i | 0 | 2.63201 | − | 0.269297i | 0 | 0.801780 | + | 2.89087i | 0 | ||||||||||
209.6 | 0 | −1.37873 | + | 1.04838i | 0 | −1.84099 | − | 1.26915i | 0 | 2.63201 | + | 0.269297i | 0 | 0.801780 | − | 2.89087i | 0 | ||||||||||
209.7 | 0 | −0.773053 | − | 1.54996i | 0 | 0.194052 | + | 2.22763i | 0 | −0.942148 | + | 2.47232i | 0 | −1.80478 | + | 2.39641i | 0 | ||||||||||
209.8 | 0 | −0.773053 | + | 1.54996i | 0 | 0.194052 | − | 2.22763i | 0 | −0.942148 | − | 2.47232i | 0 | −1.80478 | − | 2.39641i | 0 | ||||||||||
209.9 | 0 | −0.726113 | − | 1.57250i | 0 | 2.20987 | − | 0.341309i | 0 | −2.06855 | − | 1.64958i | 0 | −1.94552 | + | 2.28363i | 0 | ||||||||||
209.10 | 0 | −0.726113 | + | 1.57250i | 0 | 2.20987 | + | 0.341309i | 0 | −2.06855 | + | 1.64958i | 0 | −1.94552 | − | 2.28363i | 0 | ||||||||||
209.11 | 0 | −0.325454 | − | 1.70120i | 0 | 1.34492 | − | 1.78639i | 0 | 1.53984 | + | 2.15149i | 0 | −2.78816 | + | 1.10732i | 0 | ||||||||||
209.12 | 0 | −0.325454 | + | 1.70120i | 0 | 1.34492 | + | 1.78639i | 0 | 1.53984 | − | 2.15149i | 0 | −2.78816 | − | 1.10732i | 0 | ||||||||||
209.13 | 0 | 0.325454 | − | 1.70120i | 0 | −1.34492 | − | 1.78639i | 0 | −1.53984 | − | 2.15149i | 0 | −2.78816 | − | 1.10732i | 0 | ||||||||||
209.14 | 0 | 0.325454 | + | 1.70120i | 0 | −1.34492 | + | 1.78639i | 0 | −1.53984 | + | 2.15149i | 0 | −2.78816 | + | 1.10732i | 0 | ||||||||||
209.15 | 0 | 0.726113 | − | 1.57250i | 0 | −2.20987 | − | 0.341309i | 0 | 2.06855 | + | 1.64958i | 0 | −1.94552 | − | 2.28363i | 0 | ||||||||||
209.16 | 0 | 0.726113 | + | 1.57250i | 0 | −2.20987 | + | 0.341309i | 0 | 2.06855 | − | 1.64958i | 0 | −1.94552 | + | 2.28363i | 0 | ||||||||||
209.17 | 0 | 0.773053 | − | 1.54996i | 0 | −0.194052 | + | 2.22763i | 0 | 0.942148 | − | 2.47232i | 0 | −1.80478 | − | 2.39641i | 0 | ||||||||||
209.18 | 0 | 0.773053 | + | 1.54996i | 0 | −0.194052 | − | 2.22763i | 0 | 0.942148 | + | 2.47232i | 0 | −1.80478 | + | 2.39641i | 0 | ||||||||||
209.19 | 0 | 1.37873 | − | 1.04838i | 0 | 1.84099 | + | 1.26915i | 0 | −2.63201 | + | 0.269297i | 0 | 0.801780 | − | 2.89087i | 0 | ||||||||||
209.20 | 0 | 1.37873 | + | 1.04838i | 0 | 1.84099 | − | 1.26915i | 0 | −2.63201 | − | 0.269297i | 0 | 0.801780 | + | 2.89087i | 0 | ||||||||||
See all 24 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
7.b | odd | 2 | 1 | inner |
15.d | odd | 2 | 1 | inner |
105.g | even | 2 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 1680.2.k.h | 24 | |
3.b | odd | 2 | 1 | 1680.2.k.i | 24 | ||
4.b | odd | 2 | 1 | 840.2.k.b | yes | 24 | |
5.b | even | 2 | 1 | 1680.2.k.i | 24 | ||
7.b | odd | 2 | 1 | inner | 1680.2.k.h | 24 | |
12.b | even | 2 | 1 | 840.2.k.a | ✓ | 24 | |
15.d | odd | 2 | 1 | inner | 1680.2.k.h | 24 | |
20.d | odd | 2 | 1 | 840.2.k.a | ✓ | 24 | |
21.c | even | 2 | 1 | 1680.2.k.i | 24 | ||
28.d | even | 2 | 1 | 840.2.k.b | yes | 24 | |
35.c | odd | 2 | 1 | 1680.2.k.i | 24 | ||
60.h | even | 2 | 1 | 840.2.k.b | yes | 24 | |
84.h | odd | 2 | 1 | 840.2.k.a | ✓ | 24 | |
105.g | even | 2 | 1 | inner | 1680.2.k.h | 24 | |
140.c | even | 2 | 1 | 840.2.k.a | ✓ | 24 | |
420.o | odd | 2 | 1 | 840.2.k.b | yes | 24 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
840.2.k.a | ✓ | 24 | 12.b | even | 2 | 1 | |
840.2.k.a | ✓ | 24 | 20.d | odd | 2 | 1 | |
840.2.k.a | ✓ | 24 | 84.h | odd | 2 | 1 | |
840.2.k.a | ✓ | 24 | 140.c | even | 2 | 1 | |
840.2.k.b | yes | 24 | 4.b | odd | 2 | 1 | |
840.2.k.b | yes | 24 | 28.d | even | 2 | 1 | |
840.2.k.b | yes | 24 | 60.h | even | 2 | 1 | |
840.2.k.b | yes | 24 | 420.o | odd | 2 | 1 | |
1680.2.k.h | 24 | 1.a | even | 1 | 1 | trivial | |
1680.2.k.h | 24 | 7.b | odd | 2 | 1 | inner | |
1680.2.k.h | 24 | 15.d | odd | 2 | 1 | inner | |
1680.2.k.h | 24 | 105.g | even | 2 | 1 | inner | |
1680.2.k.i | 24 | 3.b | odd | 2 | 1 | ||
1680.2.k.i | 24 | 5.b | even | 2 | 1 | ||
1680.2.k.i | 24 | 21.c | even | 2 | 1 | ||
1680.2.k.i | 24 | 35.c | odd | 2 | 1 |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1680, [\chi])\):
\( T_{11}^{12} + 71T_{11}^{10} + 1628T_{11}^{8} + 13408T_{11}^{6} + 38400T_{11}^{4} + 43008T_{11}^{2} + 16384 \) |
\( T_{13}^{12} - 79T_{13}^{10} + 2368T_{13}^{8} - 32776T_{13}^{6} + 197088T_{13}^{4} - 335872T_{13}^{2} + 8192 \) |
\( T_{23}^{6} - 4T_{23}^{5} - 68T_{23}^{4} + 64T_{23}^{3} + 864T_{23}^{2} + 128T_{23} - 2048 \) |