Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [2768,2,Mod(1729,2768)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2768, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 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("2768.1729");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 2768 = 2^{4} \cdot 173 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 2768.b (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: | \(22.1025912795\) |
Analytic rank: | \(0\) |
Dimension: | \(44\) |
Twist minimal: | no (minimal twist has level 1384) |
Sato-Tate group: | $\mathrm{SU}(2)[C_{2}]$ |
$q$-expansion
The algebraic \(q\)-expansion of this newform has not been computed, 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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
1729.1 | 0 | − | 3.34273i | 0 | 1.08216i | 0 | 1.18357i | 0 | −8.17385 | 0 | |||||||||||||||||
1729.2 | 0 | − | 2.98827i | 0 | − | 3.29747i | 0 | 2.01256i | 0 | −5.92978 | 0 | ||||||||||||||||
1729.3 | 0 | − | 2.91059i | 0 | − | 2.22251i | 0 | − | 4.78643i | 0 | −5.47151 | 0 | |||||||||||||||
1729.4 | 0 | − | 2.82601i | 0 | 2.66396i | 0 | − | 2.84218i | 0 | −4.98634 | 0 | ||||||||||||||||
1729.5 | 0 | − | 2.79123i | 0 | − | 2.03790i | 0 | 0.840900i | 0 | −4.79098 | 0 | ||||||||||||||||
1729.6 | 0 | − | 2.69600i | 0 | − | 0.439014i | 0 | 1.92795i | 0 | −4.26839 | 0 | ||||||||||||||||
1729.7 | 0 | − | 2.51796i | 0 | 3.83023i | 0 | − | 4.18232i | 0 | −3.34014 | 0 | ||||||||||||||||
1729.8 | 0 | − | 2.29475i | 0 | 1.43735i | 0 | 4.56720i | 0 | −2.26585 | 0 | |||||||||||||||||
1729.9 | 0 | − | 2.23667i | 0 | 3.04159i | 0 | 0.610946i | 0 | −2.00268 | 0 | |||||||||||||||||
1729.10 | 0 | − | 2.10812i | 0 | − | 4.35486i | 0 | − | 1.19177i | 0 | −1.44419 | 0 | |||||||||||||||
1729.11 | 0 | − | 2.01711i | 0 | 3.86214i | 0 | 4.99348i | 0 | −1.06874 | 0 | |||||||||||||||||
1729.12 | 0 | − | 1.77761i | 0 | − | 0.286868i | 0 | − | 3.24610i | 0 | −0.159893 | 0 | |||||||||||||||
1729.13 | 0 | − | 1.49730i | 0 | 1.79016i | 0 | − | 0.344616i | 0 | 0.758096 | 0 | ||||||||||||||||
1729.14 | 0 | − | 1.46130i | 0 | − | 0.951294i | 0 | 1.13488i | 0 | 0.864613 | 0 | ||||||||||||||||
1729.15 | 0 | − | 1.35762i | 0 | − | 2.59104i | 0 | 3.86607i | 0 | 1.15687 | 0 | ||||||||||||||||
1729.16 | 0 | − | 1.27837i | 0 | − | 1.46560i | 0 | − | 0.666662i | 0 | 1.36577 | 0 | |||||||||||||||
1729.17 | 0 | − | 0.728634i | 0 | − | 1.22522i | 0 | − | 4.33042i | 0 | 2.46909 | 0 | |||||||||||||||
1729.18 | 0 | − | 0.541837i | 0 | 2.70707i | 0 | − | 1.85652i | 0 | 2.70641 | 0 | ||||||||||||||||
1729.19 | 0 | − | 0.410769i | 0 | − | 4.25377i | 0 | − | 2.42984i | 0 | 2.83127 | 0 | |||||||||||||||
1729.20 | 0 | − | 0.397910i | 0 | 1.61278i | 0 | − | 3.47841i | 0 | 2.84167 | 0 | ||||||||||||||||
See all 44 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
173.b | even | 2 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 2768.2.b.e | 44 | |
4.b | odd | 2 | 1 | 1384.2.b.a | ✓ | 44 | |
173.b | even | 2 | 1 | inner | 2768.2.b.e | 44 | |
692.d | odd | 2 | 1 | 1384.2.b.a | ✓ | 44 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
1384.2.b.a | ✓ | 44 | 4.b | odd | 2 | 1 | |
1384.2.b.a | ✓ | 44 | 692.d | odd | 2 | 1 | |
2768.2.b.e | 44 | 1.a | even | 1 | 1 | trivial | |
2768.2.b.e | 44 | 173.b | even | 2 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{44} + 89 T_{3}^{42} + 3656 T_{3}^{40} + 92003 T_{3}^{38} + 1587449 T_{3}^{36} + 19916397 T_{3}^{34} + \cdots + 50176 \) acting on \(S_{2}^{\mathrm{new}}(2768, [\chi])\).