Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [156,2,Mod(155,156)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(156, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([1, 1, 1]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("156.155");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 156 = 2^{2} \cdot 3 \cdot 13 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 156.h (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: | \(1.24566627153\) |
| Analytic rank: | \(0\) |
| Dimension: | \(8\) |
| Coefficient field: | 8.0.592240896.6 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{8} - 3x^{6} + 7x^{4} - 12x^{2} + 16 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2^{5} \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{U}(1)[D_{2}]$ |
$q$-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 \(1,\beta_1,\ldots,\beta_{7}\) 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
\( x^{8} - 3x^{6} + 7x^{4} - 12x^{2} + 16 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( \nu^{7} + \nu^{5} + 11\nu^{3} + 32\nu ) / 24 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -\nu^{7} - \nu^{5} - 11\nu^{3} + 16\nu ) / 24 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{6} + \nu^{4} - \nu^{2} + 2 ) / 6 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -\nu^{6} + 5\nu^{4} - 5\nu^{2} + 10 ) / 6 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( \nu^{6} - 2\nu^{4} + 8\nu^{2} - 7 ) / 3 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( -\nu^{7} - \nu^{5} + 5\nu^{3} ) / 8 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( -5\nu^{7} + 19\nu^{5} - 31\nu^{3} + 56\nu ) / 24 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{2} + \beta_1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{5} + \beta_{4} - \beta_{3} + 1 ) / 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( \beta_{6} - 2\beta_{2} + \beta_1 ) / 2 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( \beta_{5} + 3\beta_{4} + \beta_{3} - 3 ) / 2 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( 2\beta_{7} - \beta_{6} - 7\beta_{2} ) / 2 \)
|
| \(\nu^{6}\) | \(=\) |
\( -\beta_{4} + 5\beta_{3} \)
|
| \(\nu^{7}\) | \(=\) |
\( ( -2\beta_{7} - 10\beta_{6} - 3\beta_{2} + 5\beta_1 ) / 2 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/156\mathbb{Z}\right)^\times\).
| \(n\) | \(53\) | \(79\) | \(145\) |
| \(\chi(n)\) | \(-1\) | \(-1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 155.1 |
|
−1.37891 | − | 0.314026i | 1.73205i | 1.80278 | + | 0.866025i | 1.67000 | 0.543909 | − | 2.38834i | 0 | −2.21391 | − | 1.76029i | −3.00000 | −2.30278 | − | 0.524423i | ||||||||||||||||||||||||||||||||
| 155.2 | −1.37891 | + | 0.314026i | − | 1.73205i | 1.80278 | − | 0.866025i | 1.67000 | 0.543909 | + | 2.38834i | 0 | −2.21391 | + | 1.76029i | −3.00000 | −2.30278 | + | 0.524423i | ||||||||||||||||||||||||||||||||
| 155.3 | −0.314026 | − | 1.37891i | 1.73205i | −1.80278 | + | 0.866025i | −4.14863 | 2.38834 | − | 0.543909i | 0 | 1.76029 | + | 2.21391i | −3.00000 | 1.30278 | + | 5.72058i | |||||||||||||||||||||||||||||||||
| 155.4 | −0.314026 | + | 1.37891i | − | 1.73205i | −1.80278 | − | 0.866025i | −4.14863 | 2.38834 | + | 0.543909i | 0 | 1.76029 | − | 2.21391i | −3.00000 | 1.30278 | − | 5.72058i | ||||||||||||||||||||||||||||||||
| 155.5 | 0.314026 | − | 1.37891i | − | 1.73205i | −1.80278 | − | 0.866025i | 4.14863 | −2.38834 | − | 0.543909i | 0 | −1.76029 | + | 2.21391i | −3.00000 | 1.30278 | − | 5.72058i | ||||||||||||||||||||||||||||||||
| 155.6 | 0.314026 | + | 1.37891i | 1.73205i | −1.80278 | + | 0.866025i | 4.14863 | −2.38834 | + | 0.543909i | 0 | −1.76029 | − | 2.21391i | −3.00000 | 1.30278 | + | 5.72058i | |||||||||||||||||||||||||||||||||
| 155.7 | 1.37891 | − | 0.314026i | − | 1.73205i | 1.80278 | − | 0.866025i | −1.67000 | −0.543909 | − | 2.38834i | 0 | 2.21391 | − | 1.76029i | −3.00000 | −2.30278 | + | 0.524423i | ||||||||||||||||||||||||||||||||
| 155.8 | 1.37891 | + | 0.314026i | 1.73205i | 1.80278 | + | 0.866025i | −1.67000 | −0.543909 | + | 2.38834i | 0 | 2.21391 | + | 1.76029i | −3.00000 | −2.30278 | − | 0.524423i | |||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 39.d | odd | 2 | 1 | CM by \(\Q(\sqrt{-39}) \) |
| 3.b | odd | 2 | 1 | inner |
| 4.b | odd | 2 | 1 | inner |
| 12.b | even | 2 | 1 | inner |
| 13.b | even | 2 | 1 | inner |
| 52.b | odd | 2 | 1 | inner |
| 156.h | even | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 156.2.h.a | ✓ | 8 |
| 3.b | odd | 2 | 1 | inner | 156.2.h.a | ✓ | 8 |
| 4.b | odd | 2 | 1 | inner | 156.2.h.a | ✓ | 8 |
| 12.b | even | 2 | 1 | inner | 156.2.h.a | ✓ | 8 |
| 13.b | even | 2 | 1 | inner | 156.2.h.a | ✓ | 8 |
| 39.d | odd | 2 | 1 | CM | 156.2.h.a | ✓ | 8 |
| 52.b | odd | 2 | 1 | inner | 156.2.h.a | ✓ | 8 |
| 156.h | even | 2 | 1 | inner | 156.2.h.a | ✓ | 8 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 156.2.h.a | ✓ | 8 | 1.a | even | 1 | 1 | trivial |
| 156.2.h.a | ✓ | 8 | 3.b | odd | 2 | 1 | inner |
| 156.2.h.a | ✓ | 8 | 4.b | odd | 2 | 1 | inner |
| 156.2.h.a | ✓ | 8 | 12.b | even | 2 | 1 | inner |
| 156.2.h.a | ✓ | 8 | 13.b | even | 2 | 1 | inner |
| 156.2.h.a | ✓ | 8 | 39.d | odd | 2 | 1 | CM |
| 156.2.h.a | ✓ | 8 | 52.b | odd | 2 | 1 | inner |
| 156.2.h.a | ✓ | 8 | 156.h | even | 2 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{4} - 20T_{5}^{2} + 48 \)
acting on \(S_{2}^{\mathrm{new}}(156, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{8} - 5T^{4} + 16 \)
|
| $3$ |
\( (T^{2} + 3)^{4} \)
|
| $5$ |
\( (T^{4} - 20 T^{2} + 48)^{2} \)
|
| $7$ |
\( T^{8} \)
|
| $11$ |
\( (T^{4} + 44 T^{2} + 432)^{2} \)
|
| $13$ |
\( (T^{2} - 13)^{4} \)
|
| $17$ |
\( T^{8} \)
|
| $19$ |
\( T^{8} \)
|
| $23$ |
\( T^{8} \)
|
| $29$ |
\( T^{8} \)
|
| $31$ |
\( T^{8} \)
|
| $37$ |
\( T^{8} \)
|
| $41$ |
\( (T^{4} - 164 T^{2} + 432)^{2} \)
|
| $43$ |
\( (T^{2} + 156)^{4} \)
|
| $47$ |
\( (T^{4} + 188 T^{2} + 48)^{2} \)
|
| $53$ |
\( T^{8} \)
|
| $59$ |
\( (T^{4} + 236 T^{2} + 13872)^{2} \)
|
| $61$ |
\( (T^{2} - 52)^{4} \)
|
| $67$ |
\( T^{8} \)
|
| $71$ |
\( (T^{4} + 284 T^{2} + 13872)^{2} \)
|
| $73$ |
\( T^{8} \)
|
| $79$ |
\( (T^{2} + 108)^{4} \)
|
| $83$ |
\( (T^{4} + 332 T^{2} + 48)^{2} \)
|
| $89$ |
\( (T^{4} - 356 T^{2} + 25392)^{2} \)
|
| $97$ |
\( T^{8} \)
|
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