Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [7378,2,Mod(1,7378)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(7378, 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("7378.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 7378 = 2 \cdot 7 \cdot 17 \cdot 31 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 7378.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: | \(58.9136266113\) |
| Analytic rank: | \(1\) |
| Dimension: | \(4\) |
| Coefficient field: | 4.4.7537.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} - x^{3} - 5x^{2} + 4x + 3 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | yes |
| 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
\( x^{4} - x^{3} - 5x^{2} + 4x + 3 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 3 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{3} - 4\nu \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{3} + 4\beta_1 \)
|
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 |
|
1.00000 | −3.04717 | 1.00000 | 2.58173 | −3.04717 | −1.00000 | 1.00000 | 6.28525 | 2.58173 | ||||||||||||||||||||||||||||||
| 1.2 | 1.00000 | −1.49192 | 1.00000 | −3.60665 | −1.49192 | −1.00000 | 1.00000 | −0.774179 | −3.60665 | |||||||||||||||||||||||||||||||
| 1.3 | 1.00000 | 0.379334 | 1.00000 | 2.79563 | 0.379334 | −1.00000 | 1.00000 | −2.85611 | 2.79563 | |||||||||||||||||||||||||||||||
| 1.4 | 1.00000 | 1.15976 | 1.00000 | 1.22929 | 1.15976 | −1.00000 | 1.00000 | −1.65497 | 1.22929 | |||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \( -1 \) |
| \(7\) | \( +1 \) |
| \(17\) | \( +1 \) |
| \(31\) | \( -1 \) |
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 | 7378.2.a.bd | ✓ | 4 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 7378.2.a.bd | ✓ | 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(7378))\):
|
\( T_{3}^{4} + 3T_{3}^{3} - 2T_{3}^{2} - 5T_{3} + 2 \)
|
|
\( T_{5}^{4} - 3T_{5}^{3} - 10T_{5}^{2} + 41T_{5} - 32 \)
|
|
\( T_{11}^{4} - 3T_{11}^{3} - 2T_{11}^{2} + 5T_{11} + 2 \)
|
|
\( T_{13}^{4} + 15T_{13}^{3} + 78T_{13}^{2} + 161T_{13} + 106 \)
|
|
\( T_{23}^{4} - 2T_{23}^{3} - 24T_{23}^{2} + 8T_{23} + 96 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T - 1)^{4} \)
|
| $3$ |
\( T^{4} + 3 T^{3} + \cdots + 2 \)
|
| $5$ |
\( T^{4} - 3 T^{3} + \cdots - 32 \)
|
| $7$ |
\( (T + 1)^{4} \)
|
| $11$ |
\( T^{4} - 3 T^{3} + \cdots + 2 \)
|
| $13$ |
\( T^{4} + 15 T^{3} + \cdots + 106 \)
|
| $17$ |
\( (T + 1)^{4} \)
|
| $19$ |
\( T^{4} - T^{3} + \cdots + 64 \)
|
| $23$ |
\( T^{4} - 2 T^{3} + \cdots + 96 \)
|
| $29$ |
\( T^{4} + 10 T^{3} + \cdots - 12 \)
|
| $31$ |
\( (T - 1)^{4} \)
|
| $37$ |
\( T^{4} - 12 T^{3} + \cdots - 192 \)
|
| $41$ |
\( T^{4} + 3 T^{3} + \cdots + 346 \)
|
| $43$ |
\( T^{4} - 6 T^{3} + \cdots + 472 \)
|
| $47$ |
\( T^{4} + 25 T^{3} + \cdots + 752 \)
|
| $53$ |
\( T^{4} + 6 T^{3} + \cdots + 458 \)
|
| $59$ |
\( T^{4} + 31 T^{3} + \cdots + 236 \)
|
| $61$ |
\( T^{4} + 6 T^{3} + \cdots + 92 \)
|
| $67$ |
\( T^{4} - 26 T^{3} + \cdots - 7016 \)
|
| $71$ |
\( T^{4} + 37 T^{3} + \cdots + 1068 \)
|
| $73$ |
\( T^{4} + 2 T^{3} + \cdots + 254 \)
|
| $79$ |
\( T^{4} - 20 T^{3} + \cdots - 21908 \)
|
| $83$ |
\( T^{4} + 7 T^{3} + \cdots - 2628 \)
|
| $89$ |
\( T^{4} + 9 T^{3} + \cdots + 94 \)
|
| $97$ |
\( T^{4} + 8 T^{3} + \cdots - 2 \)
|
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