Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [240,3,Mod(47,240)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(240, base_ring=CyclotomicField(4))
chi = DirichletCharacter(H, H._module([2, 0, 2, 1]))
N = Newforms(chi, 3, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("240.47");
S:= CuspForms(chi, 3);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 240 = 2^{4} \cdot 3 \cdot 5 \) |
| Weight: | \( k \) | \(=\) | \( 3 \) |
| Character orbit: | \([\chi]\) | \(=\) | 240.bj (of order \(4\), 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: | \(6.53952634465\) |
| Analytic rank: | \(0\) |
| Dimension: | \(8\) |
| Relative dimension: | \(4\) over \(\Q(i)\) |
| Coefficient field: | \(\Q(\zeta_{24})\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{8} - x^{4} + 1 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{4} \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{4}]$ |
$q$-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 basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.
Basis of coefficient ring
| \(\beta_{1}\) | \(=\) |
\( \zeta_{24}^{3} \)
|
| \(\beta_{2}\) | \(=\) |
\( \zeta_{24}^{5} + \zeta_{24} \)
|
| \(\beta_{3}\) | \(=\) |
\( \zeta_{24}^{6} \)
|
| \(\beta_{4}\) | \(=\) |
\( 2\zeta_{24}^{4} - 1 \)
|
| \(\beta_{5}\) | \(=\) |
\( -\zeta_{24}^{5} + \zeta_{24} \)
|
| \(\beta_{6}\) | \(=\) |
\( -\zeta_{24}^{6} + 2\zeta_{24}^{2} \)
|
| \(\beta_{7}\) | \(=\) |
\( 2\zeta_{24}^{7} - \zeta_{24}^{3} \)
|
| \(\zeta_{24}\) | \(=\) |
\( ( \beta_{5} + \beta_{2} ) / 2 \)
|
| \(\zeta_{24}^{2}\) | \(=\) |
\( ( \beta_{6} + \beta_{3} ) / 2 \)
|
| \(\zeta_{24}^{3}\) | \(=\) |
\( \beta_1 \)
|
| \(\zeta_{24}^{4}\) | \(=\) |
\( ( \beta_{4} + 1 ) / 2 \)
|
| \(\zeta_{24}^{5}\) | \(=\) |
\( ( -\beta_{5} + \beta_{2} ) / 2 \)
|
| \(\zeta_{24}^{6}\) | \(=\) |
\( \beta_{3} \)
|
| \(\zeta_{24}^{7}\) | \(=\) |
\( ( \beta_{7} + \beta_1 ) / 2 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/240\mathbb{Z}\right)^\times\).
| \(n\) | \(31\) | \(97\) | \(161\) | \(181\) |
| \(\chi(n)\) | \(-1\) | \(\beta_{3}\) | \(-1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 47.1 |
|
0 | −2.95680 | − | 0.507306i | 0 | −3.53553 | − | 3.53553i | 0 | −8.66025 | + | 8.66025i | 0 | 8.48528 | + | 3.00000i | 0 | ||||||||||||||||||||||||||||||||||
| 47.2 | 0 | −0.507306 | − | 2.95680i | 0 | 3.53553 | + | 3.53553i | 0 | −8.66025 | + | 8.66025i | 0 | −8.48528 | + | 3.00000i | 0 | |||||||||||||||||||||||||||||||||||
| 47.3 | 0 | 0.507306 | + | 2.95680i | 0 | 3.53553 | + | 3.53553i | 0 | 8.66025 | − | 8.66025i | 0 | −8.48528 | + | 3.00000i | 0 | |||||||||||||||||||||||||||||||||||
| 47.4 | 0 | 2.95680 | + | 0.507306i | 0 | −3.53553 | − | 3.53553i | 0 | 8.66025 | − | 8.66025i | 0 | 8.48528 | + | 3.00000i | 0 | |||||||||||||||||||||||||||||||||||
| 143.1 | 0 | −2.95680 | + | 0.507306i | 0 | −3.53553 | + | 3.53553i | 0 | −8.66025 | − | 8.66025i | 0 | 8.48528 | − | 3.00000i | 0 | |||||||||||||||||||||||||||||||||||
| 143.2 | 0 | −0.507306 | + | 2.95680i | 0 | 3.53553 | − | 3.53553i | 0 | −8.66025 | − | 8.66025i | 0 | −8.48528 | − | 3.00000i | 0 | |||||||||||||||||||||||||||||||||||
| 143.3 | 0 | 0.507306 | − | 2.95680i | 0 | 3.53553 | − | 3.53553i | 0 | 8.66025 | + | 8.66025i | 0 | −8.48528 | − | 3.00000i | 0 | |||||||||||||||||||||||||||||||||||
| 143.4 | 0 | 2.95680 | − | 0.507306i | 0 | −3.53553 | + | 3.53553i | 0 | 8.66025 | + | 8.66025i | 0 | 8.48528 | − | 3.00000i | 0 | |||||||||||||||||||||||||||||||||||
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 |
| 5.c | odd | 4 | 1 | inner |
| 12.b | even | 2 | 1 | inner |
| 15.e | even | 4 | 1 | inner |
| 20.e | even | 4 | 1 | inner |
| 60.l | odd | 4 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 240.3.bj.a | ✓ | 8 |
| 3.b | odd | 2 | 1 | inner | 240.3.bj.a | ✓ | 8 |
| 4.b | odd | 2 | 1 | inner | 240.3.bj.a | ✓ | 8 |
| 5.c | odd | 4 | 1 | inner | 240.3.bj.a | ✓ | 8 |
| 12.b | even | 2 | 1 | inner | 240.3.bj.a | ✓ | 8 |
| 15.e | even | 4 | 1 | inner | 240.3.bj.a | ✓ | 8 |
| 20.e | even | 4 | 1 | inner | 240.3.bj.a | ✓ | 8 |
| 60.l | odd | 4 | 1 | inner | 240.3.bj.a | ✓ | 8 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 240.3.bj.a | ✓ | 8 | 1.a | even | 1 | 1 | trivial |
| 240.3.bj.a | ✓ | 8 | 3.b | odd | 2 | 1 | inner |
| 240.3.bj.a | ✓ | 8 | 4.b | odd | 2 | 1 | inner |
| 240.3.bj.a | ✓ | 8 | 5.c | odd | 4 | 1 | inner |
| 240.3.bj.a | ✓ | 8 | 12.b | even | 2 | 1 | inner |
| 240.3.bj.a | ✓ | 8 | 15.e | even | 4 | 1 | inner |
| 240.3.bj.a | ✓ | 8 | 20.e | even | 4 | 1 | inner |
| 240.3.bj.a | ✓ | 8 | 60.l | odd | 4 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{7}^{4} + 22500 \)
acting on \(S_{3}^{\mathrm{new}}(240, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{8} \)
|
| $3$ |
\( T^{8} - 126T^{4} + 6561 \)
|
| $5$ |
\( (T^{4} + 625)^{2} \)
|
| $7$ |
\( (T^{4} + 22500)^{2} \)
|
| $11$ |
\( (T^{2} - 294)^{4} \)
|
| $13$ |
\( (T^{2} - 8 T + 32)^{4} \)
|
| $17$ |
\( (T^{4} + 234256)^{2} \)
|
| $19$ |
\( (T^{2} - 48)^{4} \)
|
| $23$ |
\( (T^{4} + 2304)^{2} \)
|
| $29$ |
\( (T^{2} - 722)^{4} \)
|
| $31$ |
\( (T^{2} + 12)^{4} \)
|
| $37$ |
\( (T^{2} - 76 T + 2888)^{4} \)
|
| $41$ |
\( (T^{2} + 3872)^{4} \)
|
| $43$ |
\( (T^{4} + 1382976)^{2} \)
|
| $47$ |
\( (T^{4} + 36864)^{2} \)
|
| $53$ |
\( (T^{4} + 4096)^{2} \)
|
| $59$ |
\( (T^{2} + 2166)^{4} \)
|
| $61$ |
\( (T + 42)^{8} \)
|
| $67$ |
\( (T^{4} + 746496)^{2} \)
|
| $71$ |
\( (T^{2} - 216)^{4} \)
|
| $73$ |
\( (T^{2} + 74 T + 2738)^{4} \)
|
| $79$ |
\( (T^{2} - 5292)^{4} \)
|
| $83$ |
\( (T^{4} + 300259584)^{2} \)
|
| $89$ |
\( (T^{2} - 7688)^{4} \)
|
| $97$ |
\( (T^{2} - 70 T + 2450)^{4} \)
|
| show more | |
| show less |