Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [72,8,Mod(11,72)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(72, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([3, 3, 1]))
N = Newforms(chi, 8, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("72.11");
S:= CuspForms(chi, 8);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 72 = 2^{3} \cdot 3^{2} \) |
| Weight: | \( k \) | \(=\) | \( 8 \) |
| Character orbit: | \([\chi]\) | \(=\) | 72.l (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: | \(22.4917218349\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Relative dimension: | \(2\) over \(\Q(\zeta_{6})\) |
| Coefficient field: | \(\Q(\sqrt{-2}, \sqrt{-3})\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} - 2x^{2} + 4 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{U}(1)[D_{6}]$ |
$q$-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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.
Basis of coefficient ring in terms of a root \(\nu\) of
\( x^{4} - 2x^{2} + 4 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{2} ) / 2 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{3} ) / 2 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( 2\beta_{2} \)
|
| \(\nu^{3}\) | \(=\) |
\( 2\beta_{3} \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/72\mathbb{Z}\right)^\times\).
| \(n\) | \(37\) | \(55\) | \(65\) |
| \(\chi(n)\) | \(-1\) | \(-1\) | \(1 - \beta_{2}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 11.1 |
|
−9.79796 | − | 5.65685i | 5.57832 | − | 46.4315i | 64.0000 | + | 110.851i | 0 | −317.312 | + | 423.378i | 0 | − | 1448.15i | −2124.76 | − | 518.019i | 0 | |||||||||||||||||||
| 11.2 | 9.79796 | + | 5.65685i | 37.4217 | − | 28.0467i | 64.0000 | + | 110.851i | 0 | 525.312 | − | 63.1114i | 0 | 1448.15i | 613.765 | − | 2099.11i | 0 | |||||||||||||||||||||
| 59.1 | −9.79796 | + | 5.65685i | 5.57832 | + | 46.4315i | 64.0000 | − | 110.851i | 0 | −317.312 | − | 423.378i | 0 | 1448.15i | −2124.76 | + | 518.019i | 0 | |||||||||||||||||||||
| 59.2 | 9.79796 | − | 5.65685i | 37.4217 | + | 28.0467i | 64.0000 | − | 110.851i | 0 | 525.312 | + | 63.1114i | 0 | − | 1448.15i | 613.765 | + | 2099.11i | 0 | ||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 8.d | odd | 2 | 1 | CM by \(\Q(\sqrt{-2}) \) |
| 9.d | odd | 6 | 1 | inner |
| 72.l | even | 6 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 72.8.l.a | ✓ | 4 |
| 8.d | odd | 2 | 1 | CM | 72.8.l.a | ✓ | 4 |
| 9.d | odd | 6 | 1 | inner | 72.8.l.a | ✓ | 4 |
| 72.l | even | 6 | 1 | inner | 72.8.l.a | ✓ | 4 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 72.8.l.a | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
| 72.8.l.a | ✓ | 4 | 8.d | odd | 2 | 1 | CM |
| 72.8.l.a | ✓ | 4 | 9.d | odd | 6 | 1 | inner |
| 72.8.l.a | ✓ | 4 | 72.l | even | 6 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5} \)
acting on \(S_{8}^{\mathrm{new}}(72, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} - 128 T^{2} + 16384 \)
|
| $3$ |
\( T^{4} - 86 T^{3} + \cdots + 4782969 \)
|
| $5$ |
\( T^{4} \)
|
| $7$ |
\( T^{4} \)
|
| $11$ |
\( T^{4} + \cdots + 33\!\cdots\!25 \)
|
| $13$ |
\( T^{4} \)
|
| $17$ |
\( T^{4} + \cdots + 66\!\cdots\!01 \)
|
| $19$ |
\( (T^{2} - 59722 T + 885102067)^{2} \)
|
| $23$ |
\( T^{4} \)
|
| $29$ |
\( T^{4} \)
|
| $31$ |
\( T^{4} \)
|
| $37$ |
\( T^{4} \)
|
| $41$ |
\( T^{4} + \cdots + 19\!\cdots\!25 \)
|
| $43$ |
\( T^{4} + \cdots + 58\!\cdots\!41 \)
|
| $47$ |
\( T^{4} \)
|
| $53$ |
\( T^{4} \)
|
| $59$ |
\( T^{4} + \cdots + 20\!\cdots\!49 \)
|
| $61$ |
\( T^{4} \)
|
| $67$ |
\( T^{4} + \cdots + 11\!\cdots\!25 \)
|
| $71$ |
\( T^{4} \)
|
| $73$ |
\( (T^{2} + \cdots - 9467995960295)^{2} \)
|
| $79$ |
\( T^{4} \)
|
| $83$ |
\( T^{4} + \cdots + 72\!\cdots\!04 \)
|
| $89$ |
\( (T^{2} + 126896341340192)^{2} \)
|
| $97$ |
\( T^{4} + \cdots + 20\!\cdots\!21 \)
|
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