Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [960,2,Mod(49,960)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(960, base_ring=CyclotomicField(4))
chi = DirichletCharacter(H, H._module([0, 1, 0, 2]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("960.49");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 960 = 2^{6} \cdot 3 \cdot 5 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 960.bl (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: | \(7.66563859404\) |
Analytic rank: | \(0\) |
Dimension: | \(48\) |
Relative dimension: | \(24\) over \(\Q(i)\) |
Twist minimal: | no (minimal twist has level 240) |
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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
49.1 | 0 | −0.707107 | − | 0.707107i | 0 | −2.09919 | + | 0.770325i | 0 | −3.05002 | 0 | 1.00000i | 0 | ||||||||||||||
49.2 | 0 | −0.707107 | − | 0.707107i | 0 | −1.98421 | − | 1.03097i | 0 | 3.91927 | 0 | 1.00000i | 0 | ||||||||||||||
49.3 | 0 | −0.707107 | − | 0.707107i | 0 | −1.86022 | − | 1.24079i | 0 | −1.58988 | 0 | 1.00000i | 0 | ||||||||||||||
49.4 | 0 | −0.707107 | − | 0.707107i | 0 | −1.50085 | + | 1.65754i | 0 | 2.58977 | 0 | 1.00000i | 0 | ||||||||||||||
49.5 | 0 | −0.707107 | − | 0.707107i | 0 | −0.466917 | + | 2.18678i | 0 | 1.00010 | 0 | 1.00000i | 0 | ||||||||||||||
49.6 | 0 | −0.707107 | − | 0.707107i | 0 | 0.162008 | − | 2.23019i | 0 | −2.93661 | 0 | 1.00000i | 0 | ||||||||||||||
49.7 | 0 | −0.707107 | − | 0.707107i | 0 | 0.404088 | + | 2.19925i | 0 | −1.81567 | 0 | 1.00000i | 0 | ||||||||||||||
49.8 | 0 | −0.707107 | − | 0.707107i | 0 | 0.607542 | − | 2.15195i | 0 | 2.25286 | 0 | 1.00000i | 0 | ||||||||||||||
49.9 | 0 | −0.707107 | − | 0.707107i | 0 | 0.860885 | − | 2.06370i | 0 | 0.707398 | 0 | 1.00000i | 0 | ||||||||||||||
49.10 | 0 | −0.707107 | − | 0.707107i | 0 | 1.75308 | + | 1.38805i | 0 | 4.66030 | 0 | 1.00000i | 0 | ||||||||||||||
49.11 | 0 | −0.707107 | − | 0.707107i | 0 | 1.95942 | + | 1.07735i | 0 | −1.22137 | 0 | 1.00000i | 0 | ||||||||||||||
49.12 | 0 | −0.707107 | − | 0.707107i | 0 | 2.16437 | − | 0.561697i | 0 | −4.51614 | 0 | 1.00000i | 0 | ||||||||||||||
49.13 | 0 | 0.707107 | + | 0.707107i | 0 | −2.23019 | + | 0.162008i | 0 | 2.93661 | 0 | 1.00000i | 0 | ||||||||||||||
49.14 | 0 | 0.707107 | + | 0.707107i | 0 | −2.15195 | + | 0.607542i | 0 | −2.25286 | 0 | 1.00000i | 0 | ||||||||||||||
49.15 | 0 | 0.707107 | + | 0.707107i | 0 | −2.06370 | + | 0.860885i | 0 | −0.707398 | 0 | 1.00000i | 0 | ||||||||||||||
49.16 | 0 | 0.707107 | + | 0.707107i | 0 | −1.24079 | − | 1.86022i | 0 | 1.58988 | 0 | 1.00000i | 0 | ||||||||||||||
49.17 | 0 | 0.707107 | + | 0.707107i | 0 | −1.03097 | − | 1.98421i | 0 | −3.91927 | 0 | 1.00000i | 0 | ||||||||||||||
49.18 | 0 | 0.707107 | + | 0.707107i | 0 | −0.561697 | + | 2.16437i | 0 | 4.51614 | 0 | 1.00000i | 0 | ||||||||||||||
49.19 | 0 | 0.707107 | + | 0.707107i | 0 | 0.770325 | − | 2.09919i | 0 | 3.05002 | 0 | 1.00000i | 0 | ||||||||||||||
49.20 | 0 | 0.707107 | + | 0.707107i | 0 | 1.07735 | + | 1.95942i | 0 | 1.22137 | 0 | 1.00000i | 0 | ||||||||||||||
See all 48 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
5.b | even | 2 | 1 | inner |
16.e | even | 4 | 1 | inner |
80.q | even | 4 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 960.2.bl.a | 48 | |
4.b | odd | 2 | 1 | 240.2.bl.a | ✓ | 48 | |
5.b | even | 2 | 1 | inner | 960.2.bl.a | 48 | |
8.b | even | 2 | 1 | 1920.2.bl.b | 48 | ||
8.d | odd | 2 | 1 | 1920.2.bl.a | 48 | ||
12.b | even | 2 | 1 | 720.2.bm.h | 48 | ||
16.e | even | 4 | 1 | inner | 960.2.bl.a | 48 | |
16.e | even | 4 | 1 | 1920.2.bl.b | 48 | ||
16.f | odd | 4 | 1 | 240.2.bl.a | ✓ | 48 | |
16.f | odd | 4 | 1 | 1920.2.bl.a | 48 | ||
20.d | odd | 2 | 1 | 240.2.bl.a | ✓ | 48 | |
40.e | odd | 2 | 1 | 1920.2.bl.a | 48 | ||
40.f | even | 2 | 1 | 1920.2.bl.b | 48 | ||
48.k | even | 4 | 1 | 720.2.bm.h | 48 | ||
60.h | even | 2 | 1 | 720.2.bm.h | 48 | ||
80.k | odd | 4 | 1 | 240.2.bl.a | ✓ | 48 | |
80.k | odd | 4 | 1 | 1920.2.bl.a | 48 | ||
80.q | even | 4 | 1 | inner | 960.2.bl.a | 48 | |
80.q | even | 4 | 1 | 1920.2.bl.b | 48 | ||
240.t | even | 4 | 1 | 720.2.bm.h | 48 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
240.2.bl.a | ✓ | 48 | 4.b | odd | 2 | 1 | |
240.2.bl.a | ✓ | 48 | 16.f | odd | 4 | 1 | |
240.2.bl.a | ✓ | 48 | 20.d | odd | 2 | 1 | |
240.2.bl.a | ✓ | 48 | 80.k | odd | 4 | 1 | |
720.2.bm.h | 48 | 12.b | even | 2 | 1 | ||
720.2.bm.h | 48 | 48.k | even | 4 | 1 | ||
720.2.bm.h | 48 | 60.h | even | 2 | 1 | ||
720.2.bm.h | 48 | 240.t | even | 4 | 1 | ||
960.2.bl.a | 48 | 1.a | even | 1 | 1 | trivial | |
960.2.bl.a | 48 | 5.b | even | 2 | 1 | inner | |
960.2.bl.a | 48 | 16.e | even | 4 | 1 | inner | |
960.2.bl.a | 48 | 80.q | even | 4 | 1 | inner | |
1920.2.bl.a | 48 | 8.d | odd | 2 | 1 | ||
1920.2.bl.a | 48 | 16.f | odd | 4 | 1 | ||
1920.2.bl.a | 48 | 40.e | odd | 2 | 1 | ||
1920.2.bl.a | 48 | 80.k | odd | 4 | 1 | ||
1920.2.bl.b | 48 | 8.b | even | 2 | 1 | ||
1920.2.bl.b | 48 | 16.e | even | 4 | 1 | ||
1920.2.bl.b | 48 | 40.f | even | 2 | 1 | ||
1920.2.bl.b | 48 | 80.q | even | 4 | 1 |
Hecke kernels
This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(960, [\chi])\).