Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [3420,2,Mod(449,3420)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3420, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([0, 3, 3, 5]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("3420.449");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 3420 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 19 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 3420.bc (of order \(6\), 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: | \(27.3088374913\) |
Analytic rank: | \(0\) |
Dimension: | \(48\) |
Relative dimension: | \(24\) over \(\Q(\zeta_{6})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{6}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
449.1 | 0 | 0 | 0 | −2.23604 | + | 0.0110334i | 0 | − | 2.24660i | 0 | 0 | 0 | |||||||||||||||
449.2 | 0 | 0 | 0 | −2.20846 | + | 0.350284i | 0 | − | 0.578724i | 0 | 0 | 0 | |||||||||||||||
449.3 | 0 | 0 | 0 | −2.14073 | − | 0.645964i | 0 | 3.56652i | 0 | 0 | 0 | ||||||||||||||||
449.4 | 0 | 0 | 0 | −2.10104 | + | 0.765261i | 0 | 4.46771i | 0 | 0 | 0 | ||||||||||||||||
449.5 | 0 | 0 | 0 | −1.71326 | + | 1.43692i | 0 | − | 4.46771i | 0 | 0 | 0 | |||||||||||||||
449.6 | 0 | 0 | 0 | −1.40759 | + | 1.73744i | 0 | 0.578724i | 0 | 0 | 0 | ||||||||||||||||
449.7 | 0 | 0 | 0 | −1.40503 | − | 1.73951i | 0 | − | 0.948149i | 0 | 0 | 0 | |||||||||||||||
449.8 | 0 | 0 | 0 | −1.31712 | − | 1.80699i | 0 | − | 2.24462i | 0 | 0 | 0 | |||||||||||||||
449.9 | 0 | 0 | 0 | −1.12758 | + | 1.93095i | 0 | 2.24660i | 0 | 0 | 0 | ||||||||||||||||
449.10 | 0 | 0 | 0 | −0.906340 | − | 2.04415i | 0 | 2.24462i | 0 | 0 | 0 | ||||||||||||||||
449.11 | 0 | 0 | 0 | −0.803945 | − | 2.08655i | 0 | 0.948149i | 0 | 0 | 0 | ||||||||||||||||
449.12 | 0 | 0 | 0 | −0.510944 | + | 2.17691i | 0 | − | 3.56652i | 0 | 0 | 0 | |||||||||||||||
449.13 | 0 | 0 | 0 | 0.510944 | − | 2.17691i | 0 | − | 3.56652i | 0 | 0 | 0 | |||||||||||||||
449.14 | 0 | 0 | 0 | 0.803945 | + | 2.08655i | 0 | 0.948149i | 0 | 0 | 0 | ||||||||||||||||
449.15 | 0 | 0 | 0 | 0.906340 | + | 2.04415i | 0 | 2.24462i | 0 | 0 | 0 | ||||||||||||||||
449.16 | 0 | 0 | 0 | 1.12758 | − | 1.93095i | 0 | 2.24660i | 0 | 0 | 0 | ||||||||||||||||
449.17 | 0 | 0 | 0 | 1.31712 | + | 1.80699i | 0 | − | 2.24462i | 0 | 0 | 0 | |||||||||||||||
449.18 | 0 | 0 | 0 | 1.40503 | + | 1.73951i | 0 | − | 0.948149i | 0 | 0 | 0 | |||||||||||||||
449.19 | 0 | 0 | 0 | 1.40759 | − | 1.73744i | 0 | 0.578724i | 0 | 0 | 0 | ||||||||||||||||
449.20 | 0 | 0 | 0 | 1.71326 | − | 1.43692i | 0 | − | 4.46771i | 0 | 0 | 0 | |||||||||||||||
See all 48 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
3.b | odd | 2 | 1 | inner |
5.b | even | 2 | 1 | inner |
15.d | odd | 2 | 1 | inner |
19.d | odd | 6 | 1 | inner |
57.f | even | 6 | 1 | inner |
95.h | odd | 6 | 1 | inner |
285.q | even | 6 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 3420.2.bc.d | ✓ | 48 |
3.b | odd | 2 | 1 | inner | 3420.2.bc.d | ✓ | 48 |
5.b | even | 2 | 1 | inner | 3420.2.bc.d | ✓ | 48 |
15.d | odd | 2 | 1 | inner | 3420.2.bc.d | ✓ | 48 |
19.d | odd | 6 | 1 | inner | 3420.2.bc.d | ✓ | 48 |
57.f | even | 6 | 1 | inner | 3420.2.bc.d | ✓ | 48 |
95.h | odd | 6 | 1 | inner | 3420.2.bc.d | ✓ | 48 |
285.q | even | 6 | 1 | inner | 3420.2.bc.d | ✓ | 48 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
3420.2.bc.d | ✓ | 48 | 1.a | even | 1 | 1 | trivial |
3420.2.bc.d | ✓ | 48 | 3.b | odd | 2 | 1 | inner |
3420.2.bc.d | ✓ | 48 | 5.b | even | 2 | 1 | inner |
3420.2.bc.d | ✓ | 48 | 15.d | odd | 2 | 1 | inner |
3420.2.bc.d | ✓ | 48 | 19.d | odd | 6 | 1 | inner |
3420.2.bc.d | ✓ | 48 | 57.f | even | 6 | 1 | inner |
3420.2.bc.d | ✓ | 48 | 95.h | odd | 6 | 1 | inner |
3420.2.bc.d | ✓ | 48 | 285.q | even | 6 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(3420, [\chi])\):
\( T_{7}^{12} + 44T_{7}^{10} + 662T_{7}^{8} + 4156T_{7}^{6} + 10825T_{7}^{4} + 8988T_{7}^{2} + 1944 \) |
\( T_{11}^{12} + 98T_{11}^{10} + 3401T_{11}^{8} + 47902T_{11}^{6} + 219868T_{11}^{4} + 229112T_{11}^{2} + 66564 \) |