Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [672,4,Mod(193,672)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(672, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([0, 0, 0, 4]))
N = Newforms(chi, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("672.193");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 672 = 2^{5} \cdot 3 \cdot 7 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 672.q (of order \(3\), 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: | \(39.6492835239\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Relative dimension: | \(2\) over \(\Q(\zeta_{3})\) |
| Coefficient field: | \(\Q(\sqrt{-3}, \sqrt{37})\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} - x^{3} + 10x^{2} + 9x + 81 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2^{2} \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{3}]$ |
$q$-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
\( x^{4} - x^{3} + 10x^{2} + 9x + 81 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( -\nu^{3} + 10\nu^{2} - 10\nu + 81 ) / 90 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{3} - 10\nu^{2} + 190\nu - 81 ) / 90 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{3} + 14 ) / 5 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{2} + \beta_1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{3} + \beta_{2} + 19\beta _1 - 19 ) / 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( 5\beta_{3} - 14 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/672\mathbb{Z}\right)^\times\).
| \(n\) | \(127\) | \(421\) | \(449\) | \(577\) |
| \(\chi(n)\) | \(1\) | \(1\) | \(1\) | \(-1 + \beta_{1}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 193.1 |
|
0 | 1.50000 | − | 2.59808i | 0 | −0.0413813 | − | 0.0716745i | 0 | 5.95862 | − | 17.5355i | 0 | −4.50000 | − | 7.79423i | 0 | ||||||||||||||||||||||
| 193.2 | 0 | 1.50000 | − | 2.59808i | 0 | 6.04138 | + | 10.4640i | 0 | 12.0414 | + | 14.0714i | 0 | −4.50000 | − | 7.79423i | 0 | |||||||||||||||||||||||
| 289.1 | 0 | 1.50000 | + | 2.59808i | 0 | −0.0413813 | + | 0.0716745i | 0 | 5.95862 | + | 17.5355i | 0 | −4.50000 | + | 7.79423i | 0 | |||||||||||||||||||||||
| 289.2 | 0 | 1.50000 | + | 2.59808i | 0 | 6.04138 | − | 10.4640i | 0 | 12.0414 | − | 14.0714i | 0 | −4.50000 | + | 7.79423i | 0 | |||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 7.c | even | 3 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 672.4.q.d | yes | 4 |
| 4.b | odd | 2 | 1 | 672.4.q.c | ✓ | 4 | |
| 7.c | even | 3 | 1 | inner | 672.4.q.d | yes | 4 |
| 28.g | odd | 6 | 1 | 672.4.q.c | ✓ | 4 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 672.4.q.c | ✓ | 4 | 4.b | odd | 2 | 1 | |
| 672.4.q.c | ✓ | 4 | 28.g | odd | 6 | 1 | |
| 672.4.q.d | yes | 4 | 1.a | even | 1 | 1 | trivial |
| 672.4.q.d | yes | 4 | 7.c | even | 3 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(672, [\chi])\):
|
\( T_{5}^{4} - 12T_{5}^{3} + 145T_{5}^{2} + 12T_{5} + 1 \)
|
|
\( T_{11}^{4} + 46T_{11}^{3} + 2179T_{11}^{2} - 2898T_{11} + 3969 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} \)
|
| $3$ |
\( (T^{2} - 3 T + 9)^{2} \)
|
| $5$ |
\( T^{4} - 12 T^{3} + \cdots + 1 \)
|
| $7$ |
\( T^{4} - 36 T^{3} + \cdots + 117649 \)
|
| $11$ |
\( T^{4} + 46 T^{3} + \cdots + 3969 \)
|
| $13$ |
\( (T^{2} - 44 T + 336)^{2} \)
|
| $17$ |
\( T^{4} + 12 T^{3} + \cdots + 13424896 \)
|
| $19$ |
\( T^{4} + 84 T^{3} + \cdots + 2611456 \)
|
| $23$ |
\( T^{4} + 148 T^{3} + \cdots + 154554624 \)
|
| $29$ |
\( (T^{2} + 76 T - 21681)^{2} \)
|
| $31$ |
\( T^{4} + 140 T^{3} + \cdots + 9529569 \)
|
| $37$ |
\( T^{4} + \cdots + 14113440000 \)
|
| $41$ |
\( (T^{2} + 304 T - 36096)^{2} \)
|
| $43$ |
\( (T^{2} + 348 T + 29684)^{2} \)
|
| $47$ |
\( T^{4} + \cdots + 10047255696 \)
|
| $53$ |
\( T^{4} + 244 T^{3} + \cdots + 170851041 \)
|
| $59$ |
\( T^{4} + \cdots + 67778477649 \)
|
| $61$ |
\( T^{4} + \cdots + 4176778384 \)
|
| $67$ |
\( T^{4} + \cdots + 119436595216 \)
|
| $71$ |
\( (T^{2} + 392 T - 86052)^{2} \)
|
| $73$ |
\( T^{4} + \cdots + 397671494544 \)
|
| $79$ |
\( T^{4} + \cdots + 53276487489 \)
|
| $83$ |
\( (T^{2} - 78 T - 190287)^{2} \)
|
| $89$ |
\( T^{4} + \cdots + 1039396562064 \)
|
| $97$ |
\( (T^{2} - 86 T - 1419543)^{2} \)
|
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