Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [2394,4,Mod(1,2394)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2394, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 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("2394.1");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 2394 = 2 \cdot 3^{2} \cdot 7 \cdot 19 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 2394.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: | \(141.250572554\) |
| Analytic rank: | \(1\) |
| Dimension: | \(5\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{5} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{5} - x^{4} - 47x^{3} - 11x^{2} + 348x - 236 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{13}]\) |
| Coefficient ring index: | \( 2^{4} \) |
| 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 \(1,\beta_1,\beta_2,\beta_3,\beta_4\) 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
\( x^{5} - x^{4} - 47x^{3} - 11x^{2} + 348x - 236 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( -\nu^{4} + 3\nu^{3} + 29\nu^{2} + \nu - 2 ) / 6 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{4} - 3\nu^{3} - 29\nu^{2} + 23\nu + 2 ) / 6 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{4} - 3\nu^{3} - 35\nu^{2} + 23\nu + 113 ) / 3 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -2\nu^{4} + 12\nu^{3} + 40\nu^{2} - 208\nu + 164 ) / 3 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{2} + \beta_1 ) / 4 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( -\beta_{3} + 2\beta_{2} + 37 ) / 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 2\beta_{4} - 6\beta_{3} + 47\beta_{2} + 27\beta _1 + 110 ) / 4 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( 3\beta_{4} - 38\beta_{3} + 129\beta_{2} + 29\beta _1 + 1234 ) / 2 \)
|
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 |
|
2.00000 | 0 | 4.00000 | −18.2558 | 0 | −7.00000 | 8.00000 | 0 | −36.5115 | |||||||||||||||||||||||||||||||||
| 1.2 | 2.00000 | 0 | 4.00000 | −0.466893 | 0 | −7.00000 | 8.00000 | 0 | −0.933785 | ||||||||||||||||||||||||||||||||||
| 1.3 | 2.00000 | 0 | 4.00000 | 1.68604 | 0 | −7.00000 | 8.00000 | 0 | 3.37207 | ||||||||||||||||||||||||||||||||||
| 1.4 | 2.00000 | 0 | 4.00000 | 6.76287 | 0 | −7.00000 | 8.00000 | 0 | 13.5257 | ||||||||||||||||||||||||||||||||||
| 1.5 | 2.00000 | 0 | 4.00000 | 18.2737 | 0 | −7.00000 | 8.00000 | 0 | 36.5475 | ||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(-1\) |
| \(3\) | \(1\) |
| \(7\) | \(1\) |
| \(19\) | \(1\) |
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 | 2394.4.a.bd | yes | 5 |
| 3.b | odd | 2 | 1 | 2394.4.a.bc | ✓ | 5 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 2394.4.a.bc | ✓ | 5 | 3.b | odd | 2 | 1 | |
| 2394.4.a.bd | yes | 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 \(S_{4}^{\mathrm{new}}(\Gamma_0(2394))\):
|
\( T_{5}^{5} - 8T_{5}^{4} - 326T_{5}^{3} + 2668T_{5}^{2} - 2488T_{5} - 1776 \)
|
|
\( T_{11}^{5} + 10T_{11}^{4} - 2680T_{11}^{3} + 11440T_{11}^{2} + 1863248T_{11} - 25523808 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T - 2)^{5} \)
|
| $3$ |
\( T^{5} \)
|
| $5$ |
\( T^{5} - 8 T^{4} + \cdots - 1776 \)
|
| $7$ |
\( (T + 7)^{5} \)
|
| $11$ |
\( T^{5} + 10 T^{4} + \cdots - 25523808 \)
|
| $13$ |
\( T^{5} + 12 T^{4} + \cdots + 17648128 \)
|
| $17$ |
\( T^{5} - 34 T^{4} + \cdots + 29045184 \)
|
| $19$ |
\( (T + 19)^{5} \)
|
| $23$ |
\( T^{5} + \cdots + 1052473216 \)
|
| $29$ |
\( T^{5} + \cdots - 1780532496 \)
|
| $31$ |
\( T^{5} + \cdots + 67210959008 \)
|
| $37$ |
\( T^{5} + \cdots + 783750296832 \)
|
| $41$ |
\( T^{5} + \cdots - 313304066208 \)
|
| $43$ |
\( T^{5} + \cdots - 78123329280 \)
|
| $47$ |
\( T^{5} + \cdots + 77011010816 \)
|
| $53$ |
\( T^{5} + \cdots - 2935999968688 \)
|
| $59$ |
\( T^{5} + \cdots - 1547804934144 \)
|
| $61$ |
\( T^{5} + \cdots - 68966802024032 \)
|
| $67$ |
\( T^{5} + \cdots - 6500762053632 \)
|
| $71$ |
\( T^{5} + \cdots - 41713546752 \)
|
| $73$ |
\( T^{5} + \cdots - 1935889839200 \)
|
| $79$ |
\( T^{5} + \cdots + 25388873556928 \)
|
| $83$ |
\( T^{5} + \cdots - 30151582804176 \)
|
| $89$ |
\( T^{5} + \cdots - 112024211616 \)
|
| $97$ |
\( T^{5} + \cdots + 514126232352 \)
|
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