Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [24,8,Mod(11,24)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(24, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([1, 1, 1]))
N = Newforms(chi, 8, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("24.11");
S:= CuspForms(chi, 8);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 24 = 2^{3} \cdot 3 \) |
| Weight: | \( k \) | \(=\) | \( 8 \) |
| Character orbit: | \([\chi]\) | \(=\) | 24.f (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: | \(7.49724061162\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | \(\Q(\sqrt{6}, \sqrt{-26})\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} + 10x^{2} + 64 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 2^{5} \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{2}]$ |
$q$-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
\( x^{4} + 10x^{2} + 64 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( \nu^{3} + 18\nu ) / 8 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -\nu^{3} - 2\nu ) / 4 \)
|
| \(\beta_{3}\) | \(=\) |
\( 4\nu^{2} + 20 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{2} + 2\beta_1 ) / 4 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{3} - 20 ) / 4 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( -9\beta_{2} - 2\beta_1 ) / 2 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/24\mathbb{Z}\right)^\times\).
| \(n\) | \(7\) | \(13\) | \(17\) |
| \(\chi(n)\) | \(-1\) | \(-1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 11.1 |
|
−4.89898 | − | 10.1980i | −9.00000 | + | 45.8912i | −80.0000 | + | 99.9200i | 97.9796 | 512.091 | − | 133.038i | − | 1548.76i | 1410.91 | + | 326.337i | −2025.00 | − | 826.041i | −480.000 | − | 999.200i | |||||||||||||||
| 11.2 | −4.89898 | + | 10.1980i | −9.00000 | − | 45.8912i | −80.0000 | − | 99.9200i | 97.9796 | 512.091 | + | 133.038i | 1548.76i | 1410.91 | − | 326.337i | −2025.00 | + | 826.041i | −480.000 | + | 999.200i | |||||||||||||||||
| 11.3 | 4.89898 | − | 10.1980i | −9.00000 | + | 45.8912i | −80.0000 | − | 99.9200i | −97.9796 | 423.909 | + | 316.602i | 1548.76i | −1410.91 | + | 326.337i | −2025.00 | − | 826.041i | −480.000 | + | 999.200i | |||||||||||||||||
| 11.4 | 4.89898 | + | 10.1980i | −9.00000 | − | 45.8912i | −80.0000 | + | 99.9200i | −97.9796 | 423.909 | − | 316.602i | − | 1548.76i | −1410.91 | − | 326.337i | −2025.00 | + | 826.041i | −480.000 | − | 999.200i | ||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 3.b | odd | 2 | 1 | inner |
| 8.d | odd | 2 | 1 | inner |
| 24.f | even | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 24.8.f.b | ✓ | 4 |
| 3.b | odd | 2 | 1 | inner | 24.8.f.b | ✓ | 4 |
| 4.b | odd | 2 | 1 | 96.8.f.b | 4 | ||
| 8.b | even | 2 | 1 | 96.8.f.b | 4 | ||
| 8.d | odd | 2 | 1 | inner | 24.8.f.b | ✓ | 4 |
| 12.b | even | 2 | 1 | 96.8.f.b | 4 | ||
| 24.f | even | 2 | 1 | inner | 24.8.f.b | ✓ | 4 |
| 24.h | odd | 2 | 1 | 96.8.f.b | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 24.8.f.b | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
| 24.8.f.b | ✓ | 4 | 3.b | odd | 2 | 1 | inner |
| 24.8.f.b | ✓ | 4 | 8.d | odd | 2 | 1 | inner |
| 24.8.f.b | ✓ | 4 | 24.f | even | 2 | 1 | inner |
| 96.8.f.b | 4 | 4.b | odd | 2 | 1 | ||
| 96.8.f.b | 4 | 8.b | even | 2 | 1 | ||
| 96.8.f.b | 4 | 12.b | even | 2 | 1 | ||
| 96.8.f.b | 4 | 24.h | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{2} - 9600 \)
acting on \(S_{8}^{\mathrm{new}}(24, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} + 160 T^{2} + 16384 \)
|
| $3$ |
\( (T^{2} + 18 T + 2187)^{2} \)
|
| $5$ |
\( (T^{2} - 9600)^{2} \)
|
| $7$ |
\( (T^{2} + 2398656)^{2} \)
|
| $11$ |
\( (T^{2} + 3185000)^{2} \)
|
| $13$ |
\( (T^{2} + 102857664)^{2} \)
|
| $17$ |
\( (T^{2} + 759283616)^{2} \)
|
| $19$ |
\( (T - 11570)^{4} \)
|
| $23$ |
\( (T^{2} - 3084117504)^{2} \)
|
| $29$ |
\( (T^{2} - 2999817600)^{2} \)
|
| $31$ |
\( (T^{2} + 5139825600)^{2} \)
|
| $37$ |
\( (T^{2} + 79989114816)^{2} \)
|
| $41$ |
\( (T^{2} + 162923945600)^{2} \)
|
| $43$ |
\( (T - 495062)^{4} \)
|
| $47$ |
\( (T^{2} - 1290350985216)^{2} \)
|
| $53$ |
\( (T^{2} - 327660488064)^{2} \)
|
| $59$ |
\( (T^{2} + 2075050640936)^{2} \)
|
| $61$ |
\( (T^{2} + 3190936382400)^{2} \)
|
| $67$ |
\( (T - 1400126)^{4} \)
|
| $71$ |
\( (T^{2} - 12741150610944)^{2} \)
|
| $73$ |
\( (T + 2223598)^{4} \)
|
| $79$ |
\( (T^{2} + 31254497167296)^{2} \)
|
| $83$ |
\( (T^{2} + 9080138221544)^{2} \)
|
| $89$ |
\( (T^{2} + 34167910932896)^{2} \)
|
| $97$ |
\( (T - 6867926)^{4} \)
|
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