Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [7623,2,Mod(1,7623)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(7623, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0, 0]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("7623.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 7623 = 3^{2} \cdot 7 \cdot 11^{2} \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 7623.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: | \(60.8699614608\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | 4.4.725.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} - x^{3} - 3x^{2} + x + 1 \)
|
| Coefficient ring: | \(\Z[a_1, a_2]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 231) |
| 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 \(1,\beta_1,\beta_2,\beta_3\) 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
\( x^{4} - x^{3} - 3x^{2} + x + 1 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - \nu - 1 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{3} - \nu^{2} - 2\nu + 1 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + \beta _1 + 1 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{3} + \beta_{2} + 3\beta_1 \)
|
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1.1 |
|
−2.39026 | 0 | 3.71333 | 2.58993 | 0 | 1.00000 | −4.09529 | 0 | −6.19059 | ||||||||||||||||||||||||||||||
| 1.2 | 0.162147 | 0 | −1.97371 | 4.38705 | 0 | 1.00000 | −0.644326 | 0 | 0.711349 | |||||||||||||||||||||||||||||||
| 1.3 | 1.45589 | 0 | 0.119606 | −2.38705 | 0 | 1.00000 | −2.73764 | 0 | −3.47528 | |||||||||||||||||||||||||||||||
| 1.4 | 1.77222 | 0 | 1.14077 | −0.589926 | 0 | 1.00000 | −1.52274 | 0 | −1.04548 | |||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \(-1\) |
| \(7\) | \(-1\) |
| \(11\) | \(-1\) |
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 | 7623.2.a.cl | 4 | |
| 3.b | odd | 2 | 1 | 2541.2.a.bm | 4 | ||
| 11.b | odd | 2 | 1 | 7623.2.a.ci | 4 | ||
| 11.d | odd | 10 | 2 | 693.2.m.f | 8 | ||
| 33.d | even | 2 | 1 | 2541.2.a.bn | 4 | ||
| 33.f | even | 10 | 2 | 231.2.j.f | ✓ | 8 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 231.2.j.f | ✓ | 8 | 33.f | even | 10 | 2 | |
| 693.2.m.f | 8 | 11.d | odd | 10 | 2 | ||
| 2541.2.a.bm | 4 | 3.b | odd | 2 | 1 | ||
| 2541.2.a.bn | 4 | 33.d | even | 2 | 1 | ||
| 7623.2.a.ci | 4 | 11.b | odd | 2 | 1 | ||
| 7623.2.a.cl | 4 | 1.a | even | 1 | 1 | trivial | |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(7623))\):
|
\( T_{2}^{4} - T_{2}^{3} - 5T_{2}^{2} + 7T_{2} - 1 \)
|
|
\( T_{5}^{4} - 4T_{5}^{3} - 8T_{5}^{2} + 24T_{5} + 16 \)
|
|
\( T_{13}^{4} + 6T_{13}^{3} - 8T_{13}^{2} - 16T_{13} + 16 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} - T^{3} - 5 T^{2} + \cdots - 1 \)
|
| $3$ |
\( T^{4} \)
|
| $5$ |
\( T^{4} - 4 T^{3} + \cdots + 16 \)
|
| $7$ |
\( (T - 1)^{4} \)
|
| $11$ |
\( T^{4} \)
|
| $13$ |
\( T^{4} + 6 T^{3} + \cdots + 16 \)
|
| $17$ |
\( T^{4} - 8 T^{3} + \cdots + 304 \)
|
| $19$ |
\( T^{4} - 10 T^{3} + \cdots - 16 \)
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| $23$ |
\( T^{4} - 10 T^{3} + \cdots + 109 \)
|
| $29$ |
\( T^{4} - 79T^{2} + 29 \)
|
| $31$ |
\( T^{4} + 18 T^{3} + \cdots - 1744 \)
|
| $37$ |
\( T^{4} + 2 T^{3} + \cdots + 281 \)
|
| $41$ |
\( T^{4} - 10 T^{3} + \cdots - 4496 \)
|
| $43$ |
\( T^{4} - 4 T^{3} + \cdots + 1861 \)
|
| $47$ |
\( T^{4} + 4 T^{3} + \cdots - 976 \)
|
| $53$ |
\( T^{4} - 51 T^{2} + \cdots + 209 \)
|
| $59$ |
\( T^{4} - 16 T^{3} + \cdots + 16 \)
|
| $61$ |
\( T^{4} + 14 T^{3} + \cdots + 16 \)
|
| $67$ |
\( T^{4} + 28 T^{3} + \cdots + 1301 \)
|
| $71$ |
\( T^{4} - 18 T^{3} + \cdots - 179 \)
|
| $73$ |
\( T^{4} - 4 T^{3} + \cdots + 16 \)
|
| $79$ |
\( T^{4} - 20 T^{3} + \cdots + 149 \)
|
| $83$ |
\( T^{4} - 6 T^{3} + \cdots + 2864 \)
|
| $89$ |
\( T^{4} - 16 T^{3} + \cdots - 304 \)
|
| $97$ |
\( T^{4} - 32 T^{3} + \cdots - 8464 \)
|
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