Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [399,2,Mod(5,399)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(399, base_ring=CyclotomicField(18))
chi = DirichletCharacter(H, H._module([9, 15, 16]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("399.5");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 399 = 3 \cdot 7 \cdot 19 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 399.cb (of order \(18\), degree \(6\), 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: | \(3.18603104065\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Coefficient field: | \(\Q(\zeta_{18})\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} - x^{3} + 1 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{4}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{U}(1)[D_{18}]$ |
$q$-expansion
comment: q-expansion
sage: f.q_expansion() # note that sage often uses an isomorphic number field
gp: mfcoefs(f, 20)
Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{18}\). We also show the integral \(q\)-expansion of the trace form.
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/399\mathbb{Z}\right)^\times\).
| \(n\) | \(115\) | \(134\) | \(211\) |
| \(\chi(n)\) | \(\zeta_{18}^{3}\) | \(-1\) | \(\zeta_{18}^{2}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 5.1 |
|
0 | −0.592396 | − | 1.62760i | −1.87939 | − | 0.684040i | 0 | 0 | −1.35844 | + | 2.27038i | 0 | −2.29813 | + | 1.92836i | 0 | ||||||||||||||||||||||||||||
| 80.1 | 0 | −0.592396 | + | 1.62760i | −1.87939 | + | 0.684040i | 0 | 0 | −1.35844 | − | 2.27038i | 0 | −2.29813 | − | 1.92836i | 0 | |||||||||||||||||||||||||||||
| 101.1 | 0 | −1.11334 | − | 1.32683i | 1.53209 | + | 1.28558i | 0 | 0 | −1.28699 | + | 2.31164i | 0 | −0.520945 | + | 2.95442i | 0 | |||||||||||||||||||||||||||||
| 131.1 | 0 | 1.70574 | + | 0.300767i | 0.347296 | + | 1.96962i | 0 | 0 | 2.64543 | + | 0.0412527i | 0 | 2.81908 | + | 1.02606i | 0 | |||||||||||||||||||||||||||||
| 320.1 | 0 | −1.11334 | + | 1.32683i | 1.53209 | − | 1.28558i | 0 | 0 | −1.28699 | − | 2.31164i | 0 | −0.520945 | − | 2.95442i | 0 | |||||||||||||||||||||||||||||
| 332.1 | 0 | 1.70574 | − | 0.300767i | 0.347296 | − | 1.96962i | 0 | 0 | 2.64543 | − | 0.0412527i | 0 | 2.81908 | − | 1.02606i | 0 | |||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 3.b | odd | 2 | 1 | CM by \(\Q(\sqrt{-3}) \) |
| 133.z | odd | 18 | 1 | inner |
| 399.cb | even | 18 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 399.2.cb.a | ✓ | 6 |
| 3.b | odd | 2 | 1 | CM | 399.2.cb.a | ✓ | 6 |
| 7.d | odd | 6 | 1 | 399.2.ci.a | yes | 6 | |
| 19.e | even | 9 | 1 | 399.2.ci.a | yes | 6 | |
| 21.g | even | 6 | 1 | 399.2.ci.a | yes | 6 | |
| 57.l | odd | 18 | 1 | 399.2.ci.a | yes | 6 | |
| 133.z | odd | 18 | 1 | inner | 399.2.cb.a | ✓ | 6 |
| 399.cb | even | 18 | 1 | inner | 399.2.cb.a | ✓ | 6 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 399.2.cb.a | ✓ | 6 | 1.a | even | 1 | 1 | trivial |
| 399.2.cb.a | ✓ | 6 | 3.b | odd | 2 | 1 | CM |
| 399.2.cb.a | ✓ | 6 | 133.z | odd | 18 | 1 | inner |
| 399.2.cb.a | ✓ | 6 | 399.cb | even | 18 | 1 | inner |
| 399.2.ci.a | yes | 6 | 7.d | odd | 6 | 1 | |
| 399.2.ci.a | yes | 6 | 19.e | even | 9 | 1 | |
| 399.2.ci.a | yes | 6 | 21.g | even | 6 | 1 | |
| 399.2.ci.a | yes | 6 | 57.l | odd | 18 | 1 | |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2} \)
acting on \(S_{2}^{\mathrm{new}}(399, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} \)
|
| $3$ |
\( T^{6} - 9T^{3} + 27 \)
|
| $5$ |
\( T^{6} \)
|
| $7$ |
\( T^{6} - 37T^{3} + 343 \)
|
| $11$ |
\( T^{6} \)
|
| $13$ |
\( T^{6} + 6 T^{5} + \cdots + 867 \)
|
| $17$ |
\( T^{6} \)
|
| $19$ |
\( (T^{2} + T + 19)^{3} \)
|
| $23$ |
\( T^{6} \)
|
| $29$ |
\( T^{6} \)
|
| $31$ |
\( T^{6} - 93 T^{4} + \cdots + 35643 \)
|
| $37$ |
\( T^{6} + 111 T^{4} + \cdots + 187489 \)
|
| $41$ |
\( T^{6} \)
|
| $43$ |
\( T^{6} + 24 T^{5} + \cdots + 201601 \)
|
| $47$ |
\( T^{6} \)
|
| $53$ |
\( T^{6} \)
|
| $59$ |
\( T^{6} \)
|
| $61$ |
\( T^{6} + 39 T^{5} + \cdots + 96123 \)
|
| $67$ |
\( T^{6} - 15 T^{5} + \cdots + 16129 \)
|
| $71$ |
\( T^{6} \)
|
| $73$ |
\( T^{6} + 21 T^{5} + \cdots + 711507 \)
|
| $79$ |
\( T^{6} + 39 T^{5} + \cdots + 253009 \)
|
| $83$ |
\( T^{6} \)
|
| $89$ |
\( T^{6} \)
|
| $97$ |
\( T^{6} + 11979 T^{3} + 47832147 \)
|
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