Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [6012,2,Mod(3005,6012)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6012, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 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("6012.3005");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 6012 = 2^{2} \cdot 3^{2} \cdot 167 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 6012.h (of order \(2\), degree \(1\), 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: | \(48.0060616952\) |
Analytic rank: | \(0\) |
Dimension: | \(56\) |
Twist minimal: | yes |
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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
3005.1 | 0 | 0 | 0 | −4.09297 | 0 | 2.29241 | 0 | 0 | 0 | ||||||||||||||||||
3005.2 | 0 | 0 | 0 | −4.09297 | 0 | 2.29241 | 0 | 0 | 0 | ||||||||||||||||||
3005.3 | 0 | 0 | 0 | −3.84494 | 0 | −2.14597 | 0 | 0 | 0 | ||||||||||||||||||
3005.4 | 0 | 0 | 0 | −3.84494 | 0 | −2.14597 | 0 | 0 | 0 | ||||||||||||||||||
3005.5 | 0 | 0 | 0 | −3.47368 | 0 | 3.93299 | 0 | 0 | 0 | ||||||||||||||||||
3005.6 | 0 | 0 | 0 | −3.47368 | 0 | 3.93299 | 0 | 0 | 0 | ||||||||||||||||||
3005.7 | 0 | 0 | 0 | −3.30258 | 0 | −4.47594 | 0 | 0 | 0 | ||||||||||||||||||
3005.8 | 0 | 0 | 0 | −3.30258 | 0 | −4.47594 | 0 | 0 | 0 | ||||||||||||||||||
3005.9 | 0 | 0 | 0 | −3.04368 | 0 | 0.233711 | 0 | 0 | 0 | ||||||||||||||||||
3005.10 | 0 | 0 | 0 | −3.04368 | 0 | 0.233711 | 0 | 0 | 0 | ||||||||||||||||||
3005.11 | 0 | 0 | 0 | −2.76473 | 0 | 0.618008 | 0 | 0 | 0 | ||||||||||||||||||
3005.12 | 0 | 0 | 0 | −2.76473 | 0 | 0.618008 | 0 | 0 | 0 | ||||||||||||||||||
3005.13 | 0 | 0 | 0 | −2.21423 | 0 | −1.10661 | 0 | 0 | 0 | ||||||||||||||||||
3005.14 | 0 | 0 | 0 | −2.21423 | 0 | −1.10661 | 0 | 0 | 0 | ||||||||||||||||||
3005.15 | 0 | 0 | 0 | −2.08738 | 0 | 3.97701 | 0 | 0 | 0 | ||||||||||||||||||
3005.16 | 0 | 0 | 0 | −2.08738 | 0 | 3.97701 | 0 | 0 | 0 | ||||||||||||||||||
3005.17 | 0 | 0 | 0 | −1.39469 | 0 | −4.57475 | 0 | 0 | 0 | ||||||||||||||||||
3005.18 | 0 | 0 | 0 | −1.39469 | 0 | −4.57475 | 0 | 0 | 0 | ||||||||||||||||||
3005.19 | 0 | 0 | 0 | −1.15018 | 0 | −0.123775 | 0 | 0 | 0 | ||||||||||||||||||
3005.20 | 0 | 0 | 0 | −1.15018 | 0 | −0.123775 | 0 | 0 | 0 | ||||||||||||||||||
See all 56 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
3.b | odd | 2 | 1 | inner |
167.b | odd | 2 | 1 | inner |
501.c | even | 2 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 6012.2.h.a | ✓ | 56 |
3.b | odd | 2 | 1 | inner | 6012.2.h.a | ✓ | 56 |
167.b | odd | 2 | 1 | inner | 6012.2.h.a | ✓ | 56 |
501.c | even | 2 | 1 | inner | 6012.2.h.a | ✓ | 56 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
6012.2.h.a | ✓ | 56 | 1.a | even | 1 | 1 | trivial |
6012.2.h.a | ✓ | 56 | 3.b | odd | 2 | 1 | inner |
6012.2.h.a | ✓ | 56 | 167.b | odd | 2 | 1 | inner |
6012.2.h.a | ✓ | 56 | 501.c | even | 2 | 1 | inner |
Hecke kernels
This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(6012, [\chi])\).