Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [7381,2,Mod(1,7381)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(7381, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([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("7381.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 7381 = 11^{2} \cdot 61 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 7381.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: | \(58.9375817319\) |
| Analytic rank: | \(0\) |
| Dimension: | \(5\) |
| Coefficient field: | 5.5.24217.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{5} - 5x^{3} - x^{2} + 3x + 1 \)
|
| Coefficient ring: | \(\Z[a_1, a_2]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 671) |
| 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 \(1,\beta_1,\beta_2,\beta_3,\beta_4\) 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
\( x^{5} - 5x^{3} - x^{2} + 3x + 1 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu^{4} - 5\nu^{2} + 2 \)
|
| \(\beta_{2}\) | \(=\) |
\( -\nu^{4} + 5\nu^{2} + \nu - 2 \)
|
| \(\beta_{3}\) | \(=\) |
\( -\nu^{4} + \nu^{3} + 5\nu^{2} - 3\nu - 3 \)
|
| \(\beta_{4}\) | \(=\) |
\( 2\nu^{4} - \nu^{3} - 9\nu^{2} + 3\nu + 3 \)
|
| \(\nu\) | \(=\) |
\( \beta_{2} + \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{4} + \beta_{3} - \beta _1 + 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{3} + 3\beta_{2} + 4\beta _1 + 1 \)
|
| \(\nu^{4}\) | \(=\) |
\( 5\beta_{4} + 5\beta_{3} - 4\beta _1 + 8 \)
|
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 |
|
−1.33536 | −0.586497 | −0.216816 | 1.70504 | 0.783184 | 3.82358 | 2.96025 | −2.65602 | −2.27684 | |||||||||||||||||||||||||||||||||
| 1.2 | −0.714533 | 0.684982 | −1.48944 | −1.45989 | −0.489443 | −1.23480 | 2.49332 | −2.53080 | 1.04314 | ||||||||||||||||||||||||||||||||||
| 1.3 | 0.339328 | −2.60767 | −1.88486 | 0.383484 | −0.884856 | −0.840700 | −1.31824 | 3.79994 | 0.130127 | ||||||||||||||||||||||||||||||||||
| 1.4 | 1.26073 | 0.467546 | −0.410549 | −2.13883 | 0.589451 | −2.81011 | −3.03906 | −2.78140 | −2.69649 | ||||||||||||||||||||||||||||||||||
| 1.5 | 2.44983 | 2.04164 | 4.00166 | −0.489803 | 5.00166 | 2.06203 | 4.90374 | 1.16828 | −1.19993 | ||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(11\) | \(-1\) |
| \(61\) | \(1\) |
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 | 7381.2.a.g | 5 | |
| 11.b | odd | 2 | 1 | 671.2.a.a | ✓ | 5 | |
| 33.d | even | 2 | 1 | 6039.2.a.a | 5 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 671.2.a.a | ✓ | 5 | 11.b | odd | 2 | 1 | |
| 6039.2.a.a | 5 | 33.d | even | 2 | 1 | ||
| 7381.2.a.g | 5 | 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(7381))\):
|
\( T_{2}^{5} - 2T_{2}^{4} - 3T_{2}^{3} + 4T_{2}^{2} + 2T_{2} - 1 \)
|
|
\( T_{7}^{5} - T_{7}^{4} - 14T_{7}^{3} + T_{7}^{2} + 37T_{7} + 23 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{5} - 2 T^{4} + \cdots - 1 \)
|
| $3$ |
\( T^{5} - 6 T^{3} + \cdots - 1 \)
|
| $5$ |
\( T^{5} + 2 T^{4} + \cdots + 1 \)
|
| $7$ |
\( T^{5} - T^{4} + \cdots + 23 \)
|
| $11$ |
\( T^{5} \)
|
| $13$ |
\( T^{5} - 10 T^{4} + \cdots + 23 \)
|
| $17$ |
\( T^{5} - 3 T^{4} + \cdots - 1 \)
|
| $19$ |
\( T^{5} - 13 T^{4} + \cdots + 5 \)
|
| $23$ |
\( T^{5} - 35 T^{3} + \cdots - 59 \)
|
| $29$ |
\( T^{5} - 7 T^{4} + \cdots + 2885 \)
|
| $31$ |
\( T^{5} + 13 T^{4} + \cdots + 313 \)
|
| $37$ |
\( T^{5} + 6 T^{4} + \cdots + 97 \)
|
| $41$ |
\( T^{5} - 9 T^{4} + \cdots + 43 \)
|
| $43$ |
\( T^{5} + 2 T^{4} + \cdots + 1037 \)
|
| $47$ |
\( T^{5} + 3 T^{4} + \cdots - 17 \)
|
| $53$ |
\( T^{5} - 3 T^{4} + \cdots + 2347 \)
|
| $59$ |
\( T^{5} + 14 T^{4} + \cdots + 655 \)
|
| $61$ |
\( (T + 1)^{5} \)
|
| $67$ |
\( T^{5} + 5 T^{4} + \cdots + 1249 \)
|
| $71$ |
\( T^{5} - 3 T^{4} + \cdots - 149 \)
|
| $73$ |
\( T^{5} - 4 T^{4} + \cdots - 431 \)
|
| $79$ |
\( T^{5} - 27 T^{4} + \cdots - 1285 \)
|
| $83$ |
\( T^{5} - 3 T^{4} + \cdots - 3457 \)
|
| $89$ |
\( T^{5} + 12 T^{4} + \cdots + 3995 \)
|
| $97$ |
\( T^{5} + 5 T^{4} + \cdots + 62147 \)
|
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