Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1344,2,Mod(337,1344)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1344, base_ring=CyclotomicField(4))
chi = DirichletCharacter(H, H._module([0, 3, 0, 0]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("1344.337");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 1344 = 2^{6} \cdot 3 \cdot 7 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 1344.w (of order \(4\), degree \(2\), 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: | \(10.7318940317\) |
Analytic rank: | \(0\) |
Dimension: | \(28\) |
Relative dimension: | \(14\) over \(\Q(i)\) |
Twist minimal: | no (minimal twist has level 336) |
Sato-Tate group: | $\mathrm{SU}(2)[C_{4}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
337.1 | 0 | −0.707107 | − | 0.707107i | 0 | −1.77267 | + | 1.77267i | 0 | − | 1.00000i | 0 | 1.00000i | 0 | |||||||||||||
337.2 | 0 | −0.707107 | − | 0.707107i | 0 | −1.84737 | + | 1.84737i | 0 | − | 1.00000i | 0 | 1.00000i | 0 | |||||||||||||
337.3 | 0 | −0.707107 | − | 0.707107i | 0 | −0.646579 | + | 0.646579i | 0 | − | 1.00000i | 0 | 1.00000i | 0 | |||||||||||||
337.4 | 0 | −0.707107 | − | 0.707107i | 0 | −0.539395 | + | 0.539395i | 0 | − | 1.00000i | 0 | 1.00000i | 0 | |||||||||||||
337.5 | 0 | −0.707107 | − | 0.707107i | 0 | 1.25389 | − | 1.25389i | 0 | − | 1.00000i | 0 | 1.00000i | 0 | |||||||||||||
337.6 | 0 | −0.707107 | − | 0.707107i | 0 | 2.44528 | − | 2.44528i | 0 | − | 1.00000i | 0 | 1.00000i | 0 | |||||||||||||
337.7 | 0 | −0.707107 | − | 0.707107i | 0 | 2.52107 | − | 2.52107i | 0 | − | 1.00000i | 0 | 1.00000i | 0 | |||||||||||||
337.8 | 0 | 0.707107 | + | 0.707107i | 0 | −3.03597 | + | 3.03597i | 0 | − | 1.00000i | 0 | 1.00000i | 0 | |||||||||||||
337.9 | 0 | 0.707107 | + | 0.707107i | 0 | −0.805240 | + | 0.805240i | 0 | − | 1.00000i | 0 | 1.00000i | 0 | |||||||||||||
337.10 | 0 | 0.707107 | + | 0.707107i | 0 | −0.979857 | + | 0.979857i | 0 | − | 1.00000i | 0 | 1.00000i | 0 | |||||||||||||
337.11 | 0 | 0.707107 | + | 0.707107i | 0 | 0.116928 | − | 0.116928i | 0 | − | 1.00000i | 0 | 1.00000i | 0 | |||||||||||||
337.12 | 0 | 0.707107 | + | 0.707107i | 0 | 2.39875 | − | 2.39875i | 0 | − | 1.00000i | 0 | 1.00000i | 0 | |||||||||||||
337.13 | 0 | 0.707107 | + | 0.707107i | 0 | 2.66347 | − | 2.66347i | 0 | − | 1.00000i | 0 | 1.00000i | 0 | |||||||||||||
337.14 | 0 | 0.707107 | + | 0.707107i | 0 | −1.77230 | + | 1.77230i | 0 | − | 1.00000i | 0 | 1.00000i | 0 | |||||||||||||
1009.1 | 0 | −0.707107 | + | 0.707107i | 0 | −1.77267 | − | 1.77267i | 0 | 1.00000i | 0 | − | 1.00000i | 0 | |||||||||||||
1009.2 | 0 | −0.707107 | + | 0.707107i | 0 | −1.84737 | − | 1.84737i | 0 | 1.00000i | 0 | − | 1.00000i | 0 | |||||||||||||
1009.3 | 0 | −0.707107 | + | 0.707107i | 0 | −0.646579 | − | 0.646579i | 0 | 1.00000i | 0 | − | 1.00000i | 0 | |||||||||||||
1009.4 | 0 | −0.707107 | + | 0.707107i | 0 | −0.539395 | − | 0.539395i | 0 | 1.00000i | 0 | − | 1.00000i | 0 | |||||||||||||
1009.5 | 0 | −0.707107 | + | 0.707107i | 0 | 1.25389 | + | 1.25389i | 0 | 1.00000i | 0 | − | 1.00000i | 0 | |||||||||||||
1009.6 | 0 | −0.707107 | + | 0.707107i | 0 | 2.44528 | + | 2.44528i | 0 | 1.00000i | 0 | − | 1.00000i | 0 | |||||||||||||
See all 28 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
16.e | even | 4 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 1344.2.w.b | 28 | |
4.b | odd | 2 | 1 | 336.2.w.b | ✓ | 28 | |
8.b | even | 2 | 1 | 2688.2.w.c | 28 | ||
8.d | odd | 2 | 1 | 2688.2.w.d | 28 | ||
16.e | even | 4 | 1 | inner | 1344.2.w.b | 28 | |
16.e | even | 4 | 1 | 2688.2.w.c | 28 | ||
16.f | odd | 4 | 1 | 336.2.w.b | ✓ | 28 | |
16.f | odd | 4 | 1 | 2688.2.w.d | 28 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
336.2.w.b | ✓ | 28 | 4.b | odd | 2 | 1 | |
336.2.w.b | ✓ | 28 | 16.f | odd | 4 | 1 | |
1344.2.w.b | 28 | 1.a | even | 1 | 1 | trivial | |
1344.2.w.b | 28 | 16.e | even | 4 | 1 | inner | |
2688.2.w.c | 28 | 8.b | even | 2 | 1 | ||
2688.2.w.c | 28 | 16.e | even | 4 | 1 | ||
2688.2.w.d | 28 | 8.d | odd | 2 | 1 | ||
2688.2.w.d | 28 | 16.f | odd | 4 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{28} + 24 T_{5}^{25} + 560 T_{5}^{24} + 144 T_{5}^{23} + 288 T_{5}^{22} + 13392 T_{5}^{21} + 103880 T_{5}^{20} + 103872 T_{5}^{19} + 170496 T_{5}^{18} + 1865824 T_{5}^{17} + 9658432 T_{5}^{16} + \cdots + 12845056 \)
acting on \(S_{2}^{\mathrm{new}}(1344, [\chi])\).