Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [348,2,Mod(347,348)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(348, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([1, 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("348.347");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 348 = 2^{2} \cdot 3 \cdot 29 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 348.b (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: | \(2.77879399034\) |
Analytic rank: | \(0\) |
Dimension: | \(24\) |
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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
347.1 | −1.37820 | − | 0.317122i | −1.04908 | − | 1.37820i | 1.79887 | + | 0.874114i | − | 3.30813i | 1.00878 | + | 2.23212i | − | 2.59774i | −2.20200 | − | 1.77516i | −0.798868 | + | 2.89168i | −1.04908 | + | 4.55926i | ||
347.2 | −1.37820 | − | 0.317122i | 1.04908 | − | 1.37820i | 1.79887 | + | 0.874114i | 3.30813i | −1.88290 | + | 1.56675i | − | 2.59774i | −2.20200 | − | 1.77516i | −0.798868 | − | 2.89168i | 1.04908 | − | 4.55926i | |||
347.3 | −1.37820 | + | 0.317122i | −1.04908 | + | 1.37820i | 1.79887 | − | 0.874114i | 3.30813i | 1.00878 | − | 2.23212i | 2.59774i | −2.20200 | + | 1.77516i | −0.798868 | − | 2.89168i | −1.04908 | − | 4.55926i | ||||
347.4 | −1.37820 | + | 0.317122i | 1.04908 | + | 1.37820i | 1.79887 | − | 0.874114i | − | 3.30813i | −1.88290 | − | 1.56675i | 2.59774i | −2.20200 | + | 1.77516i | −0.798868 | + | 2.89168i | 1.04908 | + | 4.55926i | |||
347.5 | −1.25694 | − | 0.648161i | −1.19168 | + | 1.25694i | 1.15978 | + | 1.62939i | − | 1.83856i | 2.31257 | − | 0.807485i | 1.31955i | −0.401655 | − | 2.79976i | −0.159775 | − | 2.99574i | −1.19168 | + | 2.31096i | |||
347.6 | −1.25694 | − | 0.648161i | 1.19168 | + | 1.25694i | 1.15978 | + | 1.62939i | 1.83856i | −0.683175 | − | 2.35229i | 1.31955i | −0.401655 | − | 2.79976i | −0.159775 | + | 2.99574i | 1.19168 | − | 2.31096i | ||||
347.7 | −1.25694 | + | 0.648161i | −1.19168 | − | 1.25694i | 1.15978 | − | 1.62939i | 1.83856i | 2.31257 | + | 0.807485i | − | 1.31955i | −0.401655 | + | 2.79976i | −0.159775 | + | 2.99574i | −1.19168 | − | 2.31096i | |||
347.8 | −1.25694 | + | 0.648161i | 1.19168 | − | 1.25694i | 1.15978 | − | 1.62939i | − | 1.83856i | −0.683175 | + | 2.35229i | − | 1.31955i | −0.401655 | + | 2.79976i | −0.159775 | − | 2.99574i | 1.19168 | + | 2.31096i | ||
347.9 | −0.721581 | − | 1.21627i | −1.57459 | − | 0.721581i | −0.958643 | + | 1.75528i | − | 1.29460i | 0.258551 | + | 2.43581i | 2.91729i | 2.82664 | − | 0.100603i | 1.95864 | + | 2.27238i | −1.57459 | + | 0.934157i | |||
347.10 | −0.721581 | − | 1.21627i | 1.57459 | − | 0.721581i | −0.958643 | + | 1.75528i | 1.29460i | −2.01383 | − | 1.39445i | 2.91729i | 2.82664 | − | 0.100603i | 1.95864 | − | 2.27238i | 1.57459 | − | 0.934157i | ||||
347.11 | −0.721581 | + | 1.21627i | −1.57459 | + | 0.721581i | −0.958643 | − | 1.75528i | 1.29460i | 0.258551 | − | 2.43581i | − | 2.91729i | 2.82664 | + | 0.100603i | 1.95864 | − | 2.27238i | −1.57459 | − | 0.934157i | |||
347.12 | −0.721581 | + | 1.21627i | 1.57459 | + | 0.721581i | −0.958643 | − | 1.75528i | − | 1.29460i | −2.01383 | + | 1.39445i | − | 2.91729i | 2.82664 | + | 0.100603i | 1.95864 | + | 2.27238i | 1.57459 | + | 0.934157i | ||
347.13 | 0.721581 | − | 1.21627i | −1.57459 | − | 0.721581i | −0.958643 | − | 1.75528i | − | 1.29460i | −2.01383 | + | 1.39445i | − | 2.91729i | −2.82664 | − | 0.100603i | 1.95864 | + | 2.27238i | −1.57459 | − | 0.934157i | ||
347.14 | 0.721581 | − | 1.21627i | 1.57459 | − | 0.721581i | −0.958643 | − | 1.75528i | 1.29460i | 0.258551 | − | 2.43581i | − | 2.91729i | −2.82664 | − | 0.100603i | 1.95864 | − | 2.27238i | 1.57459 | + | 0.934157i | |||
347.15 | 0.721581 | + | 1.21627i | −1.57459 | + | 0.721581i | −0.958643 | + | 1.75528i | 1.29460i | −2.01383 | − | 1.39445i | 2.91729i | −2.82664 | + | 0.100603i | 1.95864 | − | 2.27238i | −1.57459 | + | 0.934157i | ||||
347.16 | 0.721581 | + | 1.21627i | 1.57459 | + | 0.721581i | −0.958643 | + | 1.75528i | − | 1.29460i | 0.258551 | + | 2.43581i | 2.91729i | −2.82664 | + | 0.100603i | 1.95864 | + | 2.27238i | 1.57459 | − | 0.934157i | |||
347.17 | 1.25694 | − | 0.648161i | −1.19168 | + | 1.25694i | 1.15978 | − | 1.62939i | − | 1.83856i | −0.683175 | + | 2.35229i | − | 1.31955i | 0.401655 | − | 2.79976i | −0.159775 | − | 2.99574i | −1.19168 | − | 2.31096i | ||
347.18 | 1.25694 | − | 0.648161i | 1.19168 | + | 1.25694i | 1.15978 | − | 1.62939i | 1.83856i | 2.31257 | + | 0.807485i | − | 1.31955i | 0.401655 | − | 2.79976i | −0.159775 | + | 2.99574i | 1.19168 | + | 2.31096i | |||
347.19 | 1.25694 | + | 0.648161i | −1.19168 | − | 1.25694i | 1.15978 | + | 1.62939i | 1.83856i | −0.683175 | − | 2.35229i | 1.31955i | 0.401655 | + | 2.79976i | −0.159775 | + | 2.99574i | −1.19168 | + | 2.31096i | ||||
347.20 | 1.25694 | + | 0.648161i | 1.19168 | − | 1.25694i | 1.15978 | + | 1.62939i | − | 1.83856i | 2.31257 | − | 0.807485i | 1.31955i | 0.401655 | + | 2.79976i | −0.159775 | − | 2.99574i | 1.19168 | − | 2.31096i | |||
See all 24 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
3.b | odd | 2 | 1 | inner |
4.b | odd | 2 | 1 | inner |
12.b | even | 2 | 1 | inner |
29.b | even | 2 | 1 | inner |
87.d | odd | 2 | 1 | inner |
116.d | odd | 2 | 1 | inner |
348.b | even | 2 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 348.2.b.d | ✓ | 24 |
3.b | odd | 2 | 1 | inner | 348.2.b.d | ✓ | 24 |
4.b | odd | 2 | 1 | inner | 348.2.b.d | ✓ | 24 |
12.b | even | 2 | 1 | inner | 348.2.b.d | ✓ | 24 |
29.b | even | 2 | 1 | inner | 348.2.b.d | ✓ | 24 |
87.d | odd | 2 | 1 | inner | 348.2.b.d | ✓ | 24 |
116.d | odd | 2 | 1 | inner | 348.2.b.d | ✓ | 24 |
348.b | even | 2 | 1 | inner | 348.2.b.d | ✓ | 24 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
348.2.b.d | ✓ | 24 | 1.a | even | 1 | 1 | trivial |
348.2.b.d | ✓ | 24 | 3.b | odd | 2 | 1 | inner |
348.2.b.d | ✓ | 24 | 4.b | odd | 2 | 1 | inner |
348.2.b.d | ✓ | 24 | 12.b | even | 2 | 1 | inner |
348.2.b.d | ✓ | 24 | 29.b | even | 2 | 1 | inner |
348.2.b.d | ✓ | 24 | 87.d | odd | 2 | 1 | inner |
348.2.b.d | ✓ | 24 | 116.d | odd | 2 | 1 | inner |
348.2.b.d | ✓ | 24 | 348.b | even | 2 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{6} + 16T_{5}^{4} + 61T_{5}^{2} + 62 \) acting on \(S_{2}^{\mathrm{new}}(348, [\chi])\).