Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [423,4,Mod(1,423)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(423, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0]))
N = Newforms(chi, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("423.1");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 423 = 3^{2} \cdot 47 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 423.a (trivial) |
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: | yes |
| Analytic conductor: | \(24.9578079324\) |
| Analytic rank: | \(1\) |
| Dimension: | \(3\) |
| Coefficient field: | 3.3.1101.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{3} - x^{2} - 9x + 12 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 47) |
| Fricke sign: | \(-1\) |
| Sato-Tate group: | $\mathrm{SU}(2)$ |
$q$-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 \(1,\beta_1,\beta_2\) 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
\( x^{3} - x^{2} - 9x + 12 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} + \nu - 7 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} - \beta _1 + 7 \)
|
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | |||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1.1 |
|
−1.51882 | 0 | −5.69320 | 6.17438 | 0 | −3.35636 | 20.7975 | 0 | −9.37775 | |||||||||||||||||||||||||||
| 1.2 | 1.60930 | 0 | −5.41015 | 9.01945 | 0 | −11.3182 | −21.5810 | 0 | 14.5150 | ||||||||||||||||||||||||||||
| 1.3 | 4.90952 | 0 | 16.1033 | −9.19383 | 0 | −30.3255 | 39.7835 | 0 | −45.1373 | ||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \(-1\) |
| \(47\) | \(1\) |
Inner twists
This newform does not admit any (nontrivial) inner twists.
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 423.4.a.b | 3 | |
| 3.b | odd | 2 | 1 | 47.4.a.a | ✓ | 3 | |
| 12.b | even | 2 | 1 | 752.4.a.c | 3 | ||
| 15.d | odd | 2 | 1 | 1175.4.a.a | 3 | ||
| 21.c | even | 2 | 1 | 2303.4.a.a | 3 | ||
| 141.c | even | 2 | 1 | 2209.4.a.a | 3 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 47.4.a.a | ✓ | 3 | 3.b | odd | 2 | 1 | |
| 423.4.a.b | 3 | 1.a | even | 1 | 1 | trivial | |
| 752.4.a.c | 3 | 12.b | even | 2 | 1 | ||
| 1175.4.a.a | 3 | 15.d | odd | 2 | 1 | ||
| 2209.4.a.a | 3 | 141.c | even | 2 | 1 | ||
| 2303.4.a.a | 3 | 21.c | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{3} - 5T_{2}^{2} - 2T_{2} + 12 \)
acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(423))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{3} - 5 T^{2} + \cdots + 12 \)
|
| $3$ |
\( T^{3} \)
|
| $5$ |
\( T^{3} - 6 T^{2} + \cdots + 512 \)
|
| $7$ |
\( T^{3} + 45 T^{2} + \cdots + 1152 \)
|
| $11$ |
\( T^{3} + 2 T^{2} + \cdots - 18432 \)
|
| $13$ |
\( T^{3} + 80 T^{2} + \cdots + 4288 \)
|
| $17$ |
\( T^{3} - 39 T^{2} + \cdots + 114146 \)
|
| $19$ |
\( T^{3} + 24 T^{2} + \cdots - 16776 \)
|
| $23$ |
\( T^{3} + 120 T^{2} + \cdots - 997488 \)
|
| $29$ |
\( T^{3} - 184 T^{2} + \cdots + 5017536 \)
|
| $31$ |
\( T^{3} + 4 T^{2} + \cdots + 8556992 \)
|
| $37$ |
\( T^{3} + 589 T^{2} + \cdots + 7475042 \)
|
| $41$ |
\( T^{3} - 92 T^{2} + \cdots + 4686008 \)
|
| $43$ |
\( T^{3} + 250 T^{2} + \cdots + 2995448 \)
|
| $47$ |
\( (T + 47)^{3} \)
|
| $53$ |
\( T^{3} + 459 T^{2} + \cdots - 72682394 \)
|
| $59$ |
\( T^{3} + 579 T^{2} + \cdots - 143703316 \)
|
| $61$ |
\( T^{3} - 267 T^{2} + \cdots - 9210494 \)
|
| $67$ |
\( T^{3} + 540 T^{2} + \cdots - 467662264 \)
|
| $71$ |
\( T^{3} + 749 T^{2} + \cdots - 335163288 \)
|
| $73$ |
\( T^{3} + 1924 T^{2} + \cdots + 63904184 \)
|
| $79$ |
\( T^{3} - 805 T^{2} + \cdots + 122645808 \)
|
| $83$ |
\( T^{3} + 712 T^{2} + \cdots - 211373952 \)
|
| $89$ |
\( T^{3} + 835 T^{2} + \cdots - 112057878 \)
|
| $97$ |
\( T^{3} + 2243 T^{2} + \cdots + 137445082 \)
|
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