Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [392,2,Mod(19,392)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(392, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([3, 3, 5]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("392.19");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 392 = 2^{3} \cdot 7^{2} \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 392.m (of order \(6\), degree \(2\), not 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: | \(3.13013575923\) |
| Analytic rank: | \(0\) |
| Dimension: | \(8\) |
| Relative dimension: | \(4\) over \(\Q(\zeta_{6})\) |
| 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: | \(\Z[a_1, \ldots, a_{9}]\) |
| Coefficient ring index: | \( 2^{2}\cdot 3^{2} \) |
| Twist minimal: | no (minimal twist has level 56) |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{6}]$ |
$q$-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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.
Basis of coefficient ring
| \(\beta_{1}\) | \(=\) |
\( \zeta_{24}^{2} \)
|
| \(\beta_{2}\) | \(=\) |
\( \zeta_{24}^{4} \)
|
| \(\beta_{3}\) | \(=\) |
\( \zeta_{24}^{6} \)
|
| \(\beta_{4}\) | \(=\) |
\( \zeta_{24}^{7} - \zeta_{24}^{5} + \zeta_{24}^{3} + 2\zeta_{24} \)
|
| \(\beta_{5}\) | \(=\) |
\( -\zeta_{24}^{7} + 2\zeta_{24}^{5} + 2\zeta_{24}^{3} - \zeta_{24} \)
|
| \(\beta_{6}\) | \(=\) |
\( -2\zeta_{24}^{7} + \zeta_{24}^{5} + \zeta_{24}^{3} + \zeta_{24} \)
|
| \(\beta_{7}\) | \(=\) |
\( 2\zeta_{24}^{7} + \zeta_{24}^{5} - \zeta_{24}^{3} + \zeta_{24} \)
|
| \(\zeta_{24}\) | \(=\) |
\( ( \beta_{7} + 2\beta_{6} - \beta_{5} + \beta_{4} ) / 6 \)
|
| \(\zeta_{24}^{2}\) | \(=\) |
\( \beta_1 \)
|
| \(\zeta_{24}^{3}\) | \(=\) |
\( ( -\beta_{7} - \beta_{6} + 2\beta_{5} + 2\beta_{4} ) / 6 \)
|
| \(\zeta_{24}^{4}\) | \(=\) |
\( \beta_{2} \)
|
| \(\zeta_{24}^{5}\) | \(=\) |
\( ( 2\beta_{7} + \beta_{6} + \beta_{5} - \beta_{4} ) / 6 \)
|
| \(\zeta_{24}^{6}\) | \(=\) |
\( \beta_{3} \)
|
| \(\zeta_{24}^{7}\) | \(=\) |
\( ( \beta_{7} - 2\beta_{6} + \beta_{5} + \beta_{4} ) / 6 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/392\mathbb{Z}\right)^\times\).
| \(n\) | \(197\) | \(295\) | \(297\) |
| \(\chi(n)\) | \(-1\) | \(-1\) | \(1 - \beta_{2}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 19.1 |
|
−0.366025 | − | 1.36603i | −2.12132 | + | 1.22474i | −1.73205 | + | 1.00000i | −1.22474 | + | 2.12132i | 2.44949 | + | 2.44949i | 0 | 2.00000 | + | 2.00000i | 1.50000 | − | 2.59808i | 3.34607 | + | 0.896575i | ||||||||||||||||||||||||||
| 19.2 | −0.366025 | − | 1.36603i | 2.12132 | − | 1.22474i | −1.73205 | + | 1.00000i | 1.22474 | − | 2.12132i | −2.44949 | − | 2.44949i | 0 | 2.00000 | + | 2.00000i | 1.50000 | − | 2.59808i | −3.34607 | − | 0.896575i | |||||||||||||||||||||||||||
| 19.3 | 1.36603 | − | 0.366025i | −2.12132 | + | 1.22474i | 1.73205 | − | 1.00000i | 1.22474 | − | 2.12132i | −2.44949 | + | 2.44949i | 0 | 2.00000 | − | 2.00000i | 1.50000 | − | 2.59808i | 0.896575 | − | 3.34607i | |||||||||||||||||||||||||||
| 19.4 | 1.36603 | − | 0.366025i | 2.12132 | − | 1.22474i | 1.73205 | − | 1.00000i | −1.22474 | + | 2.12132i | 2.44949 | − | 2.44949i | 0 | 2.00000 | − | 2.00000i | 1.50000 | − | 2.59808i | −0.896575 | + | 3.34607i | |||||||||||||||||||||||||||
| 227.1 | −0.366025 | + | 1.36603i | −2.12132 | − | 1.22474i | −1.73205 | − | 1.00000i | −1.22474 | − | 2.12132i | 2.44949 | − | 2.44949i | 0 | 2.00000 | − | 2.00000i | 1.50000 | + | 2.59808i | 3.34607 | − | 0.896575i | |||||||||||||||||||||||||||
| 227.2 | −0.366025 | + | 1.36603i | 2.12132 | + | 1.22474i | −1.73205 | − | 1.00000i | 1.22474 | + | 2.12132i | −2.44949 | + | 2.44949i | 0 | 2.00000 | − | 2.00000i | 1.50000 | + | 2.59808i | −3.34607 | + | 0.896575i | |||||||||||||||||||||||||||
| 227.3 | 1.36603 | + | 0.366025i | −2.12132 | − | 1.22474i | 1.73205 | + | 1.00000i | 1.22474 | + | 2.12132i | −2.44949 | − | 2.44949i | 0 | 2.00000 | + | 2.00000i | 1.50000 | + | 2.59808i | 0.896575 | + | 3.34607i | |||||||||||||||||||||||||||
| 227.4 | 1.36603 | + | 0.366025i | 2.12132 | + | 1.22474i | 1.73205 | + | 1.00000i | −1.22474 | − | 2.12132i | 2.44949 | + | 2.44949i | 0 | 2.00000 | + | 2.00000i | 1.50000 | + | 2.59808i | −0.896575 | − | 3.34607i | |||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 7.b | odd | 2 | 1 | inner |
| 7.c | even | 3 | 1 | inner |
| 7.d | odd | 6 | 1 | inner |
| 8.d | odd | 2 | 1 | inner |
| 56.e | even | 2 | 1 | inner |
| 56.k | odd | 6 | 1 | inner |
| 56.m | even | 6 | 1 | inner |
Twists
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(392, [\chi])\):
|
\( T_{3}^{4} - 6T_{3}^{2} + 36 \)
|
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\( T_{5}^{4} + 6T_{5}^{2} + 36 \)
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\( T_{23}^{4} - 16T_{23}^{2} + 256 \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{4} - 2 T^{3} + 2 T^{2} + \cdots + 4)^{2} \)
|
| $3$ |
\( (T^{4} - 6 T^{2} + 36)^{2} \)
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| $5$ |
\( (T^{4} + 6 T^{2} + 36)^{2} \)
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| $7$ |
\( T^{8} \)
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| $11$ |
\( (T^{2} + 2 T + 4)^{4} \)
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| $13$ |
\( (T^{2} - 6)^{4} \)
|
| $17$ |
\( (T^{4} - 24 T^{2} + 576)^{2} \)
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| $19$ |
\( (T^{4} - 6 T^{2} + 36)^{2} \)
|
| $23$ |
\( (T^{4} - 16 T^{2} + 256)^{2} \)
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| $29$ |
\( (T^{2} + 16)^{4} \)
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| $31$ |
\( (T^{4} + 24 T^{2} + 576)^{2} \)
|
| $37$ |
\( (T^{4} - 64 T^{2} + 4096)^{2} \)
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| $41$ |
\( T^{8} \)
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| $43$ |
\( (T + 6)^{8} \)
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| $47$ |
\( (T^{4} + 24 T^{2} + 576)^{2} \)
|
| $53$ |
\( (T^{4} - 16 T^{2} + 256)^{2} \)
|
| $59$ |
\( (T^{4} - 6 T^{2} + 36)^{2} \)
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| $61$ |
\( (T^{4} + 54 T^{2} + 2916)^{2} \)
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| $67$ |
\( (T^{2} - 2 T + 4)^{4} \)
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| $71$ |
\( (T^{2} + 100)^{4} \)
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| $73$ |
\( (T^{4} - 216 T^{2} + 46656)^{2} \)
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| $79$ |
\( (T^{4} - 36 T^{2} + 1296)^{2} \)
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| $83$ |
\( (T^{2} + 6)^{4} \)
|
| $89$ |
\( (T^{4} - 216 T^{2} + 46656)^{2} \)
|
| $97$ |
\( (T^{2} + 24)^{4} \)
|
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