Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [144,4,Mod(49,144)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(144, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([0, 0, 2]))
N = Newforms(chi, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("144.49");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 144 = 2^{4} \cdot 3^{2} \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 144.i (of order \(3\), degree \(2\), not 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: | \(8.49627504083\) |
| Analytic rank: | \(0\) |
| Dimension: | \(2\) |
| Coefficient field: | \(\Q(\sqrt{-3}) \) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{2} - x + 1 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 18) |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{3}]$ |
$q$-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 \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/144\mathbb{Z}\right)^\times\).
| \(n\) | \(37\) | \(65\) | \(127\) |
| \(\chi(n)\) | \(1\) | \(-\zeta_{6}\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 49.1 |
|
0 | 5.19615i | 0 | 4.50000 | − | 7.79423i | 0 | −15.5000 | − | 26.8468i | 0 | −27.0000 | 0 | ||||||||||||||||||||
| 97.1 | 0 | − | 5.19615i | 0 | 4.50000 | + | 7.79423i | 0 | −15.5000 | + | 26.8468i | 0 | −27.0000 | 0 | ||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 9.c | even | 3 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 144.4.i.a | 2 | |
| 3.b | odd | 2 | 1 | 432.4.i.a | 2 | ||
| 4.b | odd | 2 | 1 | 18.4.c.a | ✓ | 2 | |
| 9.c | even | 3 | 1 | inner | 144.4.i.a | 2 | |
| 9.c | even | 3 | 1 | 1296.4.a.b | 1 | ||
| 9.d | odd | 6 | 1 | 432.4.i.a | 2 | ||
| 9.d | odd | 6 | 1 | 1296.4.a.g | 1 | ||
| 12.b | even | 2 | 1 | 54.4.c.a | 2 | ||
| 36.f | odd | 6 | 1 | 18.4.c.a | ✓ | 2 | |
| 36.f | odd | 6 | 1 | 162.4.a.d | 1 | ||
| 36.h | even | 6 | 1 | 54.4.c.a | 2 | ||
| 36.h | even | 6 | 1 | 162.4.a.a | 1 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 18.4.c.a | ✓ | 2 | 4.b | odd | 2 | 1 | |
| 18.4.c.a | ✓ | 2 | 36.f | odd | 6 | 1 | |
| 54.4.c.a | 2 | 12.b | even | 2 | 1 | ||
| 54.4.c.a | 2 | 36.h | even | 6 | 1 | ||
| 144.4.i.a | 2 | 1.a | even | 1 | 1 | trivial | |
| 144.4.i.a | 2 | 9.c | even | 3 | 1 | inner | |
| 162.4.a.a | 1 | 36.h | even | 6 | 1 | ||
| 162.4.a.d | 1 | 36.f | odd | 6 | 1 | ||
| 432.4.i.a | 2 | 3.b | odd | 2 | 1 | ||
| 432.4.i.a | 2 | 9.d | odd | 6 | 1 | ||
| 1296.4.a.b | 1 | 9.c | even | 3 | 1 | ||
| 1296.4.a.g | 1 | 9.d | odd | 6 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{2} - 9T_{5} + 81 \)
acting on \(S_{4}^{\mathrm{new}}(144, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{2} \)
|
| $3$ |
\( T^{2} + 27 \)
|
| $5$ |
\( T^{2} - 9T + 81 \)
|
| $7$ |
\( T^{2} + 31T + 961 \)
|
| $11$ |
\( T^{2} + 15T + 225 \)
|
| $13$ |
\( T^{2} - 37T + 1369 \)
|
| $17$ |
\( (T + 42)^{2} \)
|
| $19$ |
\( (T - 28)^{2} \)
|
| $23$ |
\( T^{2} - 195T + 38025 \)
|
| $29$ |
\( T^{2} + 111T + 12321 \)
|
| $31$ |
\( T^{2} + 205T + 42025 \)
|
| $37$ |
\( (T + 166)^{2} \)
|
| $41$ |
\( T^{2} - 261T + 68121 \)
|
| $43$ |
\( T^{2} + 43T + 1849 \)
|
| $47$ |
\( T^{2} - 177T + 31329 \)
|
| $53$ |
\( (T - 114)^{2} \)
|
| $59$ |
\( T^{2} - 159T + 25281 \)
|
| $61$ |
\( T^{2} + 191T + 36481 \)
|
| $67$ |
\( T^{2} + 421T + 177241 \)
|
| $71$ |
\( (T + 156)^{2} \)
|
| $73$ |
\( (T - 182)^{2} \)
|
| $79$ |
\( T^{2} - 1133 T + 1283689 \)
|
| $83$ |
\( T^{2} + 1083 T + 1172889 \)
|
| $89$ |
\( (T + 1050)^{2} \)
|
| $97$ |
\( T^{2} - 901T + 811801 \)
|
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