Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [600,2,Mod(43,600)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(600, base_ring=CyclotomicField(4))
chi = DirichletCharacter(H, H._module([2, 2, 0, 3]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("600.43");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 600 = 2^{3} \cdot 3 \cdot 5^{2} \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 600.v (of order \(4\), degree \(2\), 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: | \(4.79102412128\) |
Analytic rank: | \(0\) |
Dimension: | \(24\) |
Relative dimension: | \(12\) over \(\Q(i)\) |
Twist minimal: | no (minimal twist has level 120) |
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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
43.1 | −1.31581 | + | 0.518298i | −0.707107 | − | 0.707107i | 1.46273 | − | 1.36397i | 0 | 1.29691 | + | 0.563929i | −0.645414 | − | 0.645414i | −1.21775 | + | 2.55286i | 1.00000i | 0 | ||||||
43.2 | −1.25738 | + | 0.647304i | 0.707107 | + | 0.707107i | 1.16200 | − | 1.62781i | 0 | −1.34681 | − | 0.431387i | 1.45533 | + | 1.45533i | −0.407381 | + | 2.79894i | 1.00000i | 0 | ||||||
43.3 | −1.08304 | − | 0.909406i | −0.707107 | − | 0.707107i | 0.345961 | + | 1.96985i | 0 | 0.122779 | + | 1.40887i | 2.10796 | + | 2.10796i | 1.41670 | − | 2.44805i | 1.00000i | 0 | ||||||
43.4 | −0.647304 | + | 1.25738i | 0.707107 | + | 0.707107i | −1.16200 | − | 1.62781i | 0 | −1.34681 | + | 0.431387i | −1.45533 | − | 1.45533i | 2.79894 | − | 0.407381i | 1.00000i | 0 | ||||||
43.5 | −0.624608 | − | 1.26880i | 0.707107 | + | 0.707107i | −1.21973 | + | 1.58501i | 0 | 0.455516 | − | 1.33884i | −1.93078 | − | 1.93078i | 2.77292 | + | 0.557590i | 1.00000i | 0 | ||||||
43.6 | −0.518298 | + | 1.31581i | −0.707107 | − | 0.707107i | −1.46273 | − | 1.36397i | 0 | 1.29691 | − | 0.563929i | 0.645414 | + | 0.645414i | 2.55286 | − | 1.21775i | 1.00000i | 0 | ||||||
43.7 | −0.109339 | − | 1.40998i | −0.707107 | − | 0.707107i | −1.97609 | + | 0.308331i | 0 | −0.919693 | + | 1.07432i | 1.21782 | + | 1.21782i | 0.650804 | + | 2.75254i | 1.00000i | 0 | ||||||
43.8 | 0.804501 | − | 1.16309i | 0.707107 | + | 0.707107i | −0.705556 | − | 1.87141i | 0 | 1.39130 | − | 0.253561i | 3.43671 | + | 3.43671i | −2.74424 | − | 0.684930i | 1.00000i | 0 | ||||||
43.9 | 0.909406 | + | 1.08304i | −0.707107 | − | 0.707107i | −0.345961 | + | 1.96985i | 0 | 0.122779 | − | 1.40887i | −2.10796 | − | 2.10796i | −2.44805 | + | 1.41670i | 1.00000i | 0 | ||||||
43.10 | 1.16309 | − | 0.804501i | 0.707107 | + | 0.707107i | 0.705556 | − | 1.87141i | 0 | 1.39130 | + | 0.253561i | −3.43671 | − | 3.43671i | −0.684930 | − | 2.74424i | 1.00000i | 0 | ||||||
43.11 | 1.26880 | + | 0.624608i | 0.707107 | + | 0.707107i | 1.21973 | + | 1.58501i | 0 | 0.455516 | + | 1.33884i | 1.93078 | + | 1.93078i | 0.557590 | + | 2.77292i | 1.00000i | 0 | ||||||
43.12 | 1.40998 | + | 0.109339i | −0.707107 | − | 0.707107i | 1.97609 | + | 0.308331i | 0 | −0.919693 | − | 1.07432i | −1.21782 | − | 1.21782i | 2.75254 | + | 0.650804i | 1.00000i | 0 | ||||||
307.1 | −1.31581 | − | 0.518298i | −0.707107 | + | 0.707107i | 1.46273 | + | 1.36397i | 0 | 1.29691 | − | 0.563929i | −0.645414 | + | 0.645414i | −1.21775 | − | 2.55286i | − | 1.00000i | 0 | |||||
307.2 | −1.25738 | − | 0.647304i | 0.707107 | − | 0.707107i | 1.16200 | + | 1.62781i | 0 | −1.34681 | + | 0.431387i | 1.45533 | − | 1.45533i | −0.407381 | − | 2.79894i | − | 1.00000i | 0 | |||||
307.3 | −1.08304 | + | 0.909406i | −0.707107 | + | 0.707107i | 0.345961 | − | 1.96985i | 0 | 0.122779 | − | 1.40887i | 2.10796 | − | 2.10796i | 1.41670 | + | 2.44805i | − | 1.00000i | 0 | |||||
307.4 | −0.647304 | − | 1.25738i | 0.707107 | − | 0.707107i | −1.16200 | + | 1.62781i | 0 | −1.34681 | − | 0.431387i | −1.45533 | + | 1.45533i | 2.79894 | + | 0.407381i | − | 1.00000i | 0 | |||||
307.5 | −0.624608 | + | 1.26880i | 0.707107 | − | 0.707107i | −1.21973 | − | 1.58501i | 0 | 0.455516 | + | 1.33884i | −1.93078 | + | 1.93078i | 2.77292 | − | 0.557590i | − | 1.00000i | 0 | |||||
307.6 | −0.518298 | − | 1.31581i | −0.707107 | + | 0.707107i | −1.46273 | + | 1.36397i | 0 | 1.29691 | + | 0.563929i | 0.645414 | − | 0.645414i | 2.55286 | + | 1.21775i | − | 1.00000i | 0 | |||||
307.7 | −0.109339 | + | 1.40998i | −0.707107 | + | 0.707107i | −1.97609 | − | 0.308331i | 0 | −0.919693 | − | 1.07432i | 1.21782 | − | 1.21782i | 0.650804 | − | 2.75254i | − | 1.00000i | 0 | |||||
307.8 | 0.804501 | + | 1.16309i | 0.707107 | − | 0.707107i | −0.705556 | + | 1.87141i | 0 | 1.39130 | + | 0.253561i | 3.43671 | − | 3.43671i | −2.74424 | + | 0.684930i | − | 1.00000i | 0 | |||||
See all 24 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
5.c | odd | 4 | 1 | inner |
8.d | odd | 2 | 1 | inner |
40.k | even | 4 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 600.2.v.b | 24 | |
4.b | odd | 2 | 1 | 2400.2.bh.b | 24 | ||
5.b | even | 2 | 1 | 120.2.v.a | ✓ | 24 | |
5.c | odd | 4 | 1 | 120.2.v.a | ✓ | 24 | |
5.c | odd | 4 | 1 | inner | 600.2.v.b | 24 | |
8.b | even | 2 | 1 | 2400.2.bh.b | 24 | ||
8.d | odd | 2 | 1 | inner | 600.2.v.b | 24 | |
15.d | odd | 2 | 1 | 360.2.w.e | 24 | ||
15.e | even | 4 | 1 | 360.2.w.e | 24 | ||
20.d | odd | 2 | 1 | 480.2.bh.a | 24 | ||
20.e | even | 4 | 1 | 480.2.bh.a | 24 | ||
20.e | even | 4 | 1 | 2400.2.bh.b | 24 | ||
40.e | odd | 2 | 1 | 120.2.v.a | ✓ | 24 | |
40.f | even | 2 | 1 | 480.2.bh.a | 24 | ||
40.i | odd | 4 | 1 | 480.2.bh.a | 24 | ||
40.i | odd | 4 | 1 | 2400.2.bh.b | 24 | ||
40.k | even | 4 | 1 | 120.2.v.a | ✓ | 24 | |
40.k | even | 4 | 1 | inner | 600.2.v.b | 24 | |
60.h | even | 2 | 1 | 1440.2.bi.e | 24 | ||
60.l | odd | 4 | 1 | 1440.2.bi.e | 24 | ||
120.i | odd | 2 | 1 | 1440.2.bi.e | 24 | ||
120.m | even | 2 | 1 | 360.2.w.e | 24 | ||
120.q | odd | 4 | 1 | 360.2.w.e | 24 | ||
120.w | even | 4 | 1 | 1440.2.bi.e | 24 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
120.2.v.a | ✓ | 24 | 5.b | even | 2 | 1 | |
120.2.v.a | ✓ | 24 | 5.c | odd | 4 | 1 | |
120.2.v.a | ✓ | 24 | 40.e | odd | 2 | 1 | |
120.2.v.a | ✓ | 24 | 40.k | even | 4 | 1 | |
360.2.w.e | 24 | 15.d | odd | 2 | 1 | ||
360.2.w.e | 24 | 15.e | even | 4 | 1 | ||
360.2.w.e | 24 | 120.m | even | 2 | 1 | ||
360.2.w.e | 24 | 120.q | odd | 4 | 1 | ||
480.2.bh.a | 24 | 20.d | odd | 2 | 1 | ||
480.2.bh.a | 24 | 20.e | even | 4 | 1 | ||
480.2.bh.a | 24 | 40.f | even | 2 | 1 | ||
480.2.bh.a | 24 | 40.i | odd | 4 | 1 | ||
600.2.v.b | 24 | 1.a | even | 1 | 1 | trivial | |
600.2.v.b | 24 | 5.c | odd | 4 | 1 | inner | |
600.2.v.b | 24 | 8.d | odd | 2 | 1 | inner | |
600.2.v.b | 24 | 40.k | even | 4 | 1 | inner | |
1440.2.bi.e | 24 | 60.h | even | 2 | 1 | ||
1440.2.bi.e | 24 | 60.l | odd | 4 | 1 | ||
1440.2.bi.e | 24 | 120.i | odd | 2 | 1 | ||
1440.2.bi.e | 24 | 120.w | even | 4 | 1 | ||
2400.2.bh.b | 24 | 4.b | odd | 2 | 1 | ||
2400.2.bh.b | 24 | 8.b | even | 2 | 1 | ||
2400.2.bh.b | 24 | 20.e | even | 4 | 1 | ||
2400.2.bh.b | 24 | 40.i | odd | 4 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{24} + 720T_{7}^{20} + 98656T_{7}^{16} + 4752640T_{7}^{12} + 81309952T_{7}^{8} + 440926208T_{7}^{4} + 268435456 \) acting on \(S_{2}^{\mathrm{new}}(600, [\chi])\).