Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [12,8,Mod(11,12)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(12, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([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("12.11");
S:= CuspForms(chi, 8);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 12 = 2^{2} \cdot 3 \) |
| Weight: | \( k \) | \(=\) | \( 8 \) |
| Character orbit: | \([\chi]\) | \(=\) | 12.b (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: | \(3.74862030581\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | \(\Q(\sqrt{3}, \sqrt{-5})\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} + x^{2} + 4 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 2^{5}\cdot 3^{2} \) |
| 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
\( x^{4} + x^{2} + 4 \)
:
| \(\beta_{1}\) | \(=\) |
\( -2\nu^{3} - 6\nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{3} + 6\nu^{2} + 3\nu + 3 \)
|
| \(\beta_{3}\) | \(=\) |
\( -2\nu^{3} + 6\nu \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{3} - \beta_1 ) / 12 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 2\beta_{2} + \beta _1 - 6 ) / 12 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( -\beta_{3} - \beta_1 ) / 4 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/12\mathbb{Z}\right)^\times\).
| \(n\) | \(5\) | \(7\) |
| \(\chi(n)\) | \(-1\) | \(-1\) |
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 |
|
−10.3923 | − | 4.47214i | −31.1769 | − | 34.8569i | 88.0000 | + | 92.9516i | 205.718i | 168.115 | + | 501.670i | 999.230i | −498.831 | − | 1359.53i | −243.000 | + | 2173.46i | 920.000 | − | 2137.89i | ||||||||||||||||
| 11.2 | −10.3923 | + | 4.47214i | −31.1769 | + | 34.8569i | 88.0000 | − | 92.9516i | − | 205.718i | 168.115 | − | 501.670i | − | 999.230i | −498.831 | + | 1359.53i | −243.000 | − | 2173.46i | 920.000 | + | 2137.89i | |||||||||||||||
| 11.3 | 10.3923 | − | 4.47214i | 31.1769 | + | 34.8569i | 88.0000 | − | 92.9516i | 205.718i | 479.885 | + | 222.816i | − | 999.230i | 498.831 | − | 1359.53i | −243.000 | + | 2173.46i | 920.000 | + | 2137.89i | ||||||||||||||||
| 11.4 | 10.3923 | + | 4.47214i | 31.1769 | − | 34.8569i | 88.0000 | + | 92.9516i | − | 205.718i | 479.885 | − | 222.816i | 999.230i | 498.831 | + | 1359.53i | −243.000 | − | 2173.46i | 920.000 | − | 2137.89i | ||||||||||||||||
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 |
| 12.b | even | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 12.8.b.a | ✓ | 4 |
| 3.b | odd | 2 | 1 | inner | 12.8.b.a | ✓ | 4 |
| 4.b | odd | 2 | 1 | inner | 12.8.b.a | ✓ | 4 |
| 8.b | even | 2 | 1 | 192.8.c.b | 4 | ||
| 8.d | odd | 2 | 1 | 192.8.c.b | 4 | ||
| 12.b | even | 2 | 1 | inner | 12.8.b.a | ✓ | 4 |
| 24.f | even | 2 | 1 | 192.8.c.b | 4 | ||
| 24.h | odd | 2 | 1 | 192.8.c.b | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 12.8.b.a | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
| 12.8.b.a | ✓ | 4 | 3.b | odd | 2 | 1 | inner |
| 12.8.b.a | ✓ | 4 | 4.b | odd | 2 | 1 | inner |
| 12.8.b.a | ✓ | 4 | 12.b | even | 2 | 1 | inner |
| 192.8.c.b | 4 | 8.b | even | 2 | 1 | ||
| 192.8.c.b | 4 | 8.d | odd | 2 | 1 | ||
| 192.8.c.b | 4 | 24.f | even | 2 | 1 | ||
| 192.8.c.b | 4 | 24.h | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{2} + 42320 \)
acting on \(S_{8}^{\mathrm{new}}(12, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} - 176 T^{2} + 16384 \)
|
| $3$ |
\( T^{4} + 486 T^{2} + 4782969 \)
|
| $5$ |
\( (T^{2} + 42320)^{2} \)
|
| $7$ |
\( (T^{2} + 998460)^{2} \)
|
| $11$ |
\( (T^{2} - 3898800)^{2} \)
|
| $13$ |
\( (T + 12730)^{4} \)
|
| $17$ |
\( (T^{2} + 355493120)^{2} \)
|
| $19$ |
\( (T^{2} + 571774140)^{2} \)
|
| $23$ |
\( (T^{2} - 5237707968)^{2} \)
|
| $29$ |
\( (T^{2} + 11258461520)^{2} \)
|
| $31$ |
\( (T^{2} + 8167133340)^{2} \)
|
| $37$ |
\( (T - 43310)^{4} \)
|
| $41$ |
\( (T^{2} + 617466025280)^{2} \)
|
| $43$ |
\( (T^{2} + 5013472860)^{2} \)
|
| $47$ |
\( (T^{2} - 317827314432)^{2} \)
|
| $53$ |
\( (T^{2} + 121761133520)^{2} \)
|
| $59$ |
\( (T^{2} - 574743870000)^{2} \)
|
| $61$ |
\( (T - 314198)^{4} \)
|
| $67$ |
\( (T^{2} + 658290901500)^{2} \)
|
| $71$ |
\( (T^{2} - 70227000000)^{2} \)
|
| $73$ |
\( (T + 259270)^{4} \)
|
| $79$ |
\( (T^{2} + 27403392658140)^{2} \)
|
| $83$ |
\( (T^{2} - 102043810492848)^{2} \)
|
| $89$ |
\( (T^{2} + 15089736130880)^{2} \)
|
| $97$ |
\( (T - 7243010)^{4} \)
|
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