Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [2645,2,Mod(1,2645)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2645, 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("2645.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 2645 = 5 \cdot 23^{2} \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 2645.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: | \(21.1204313346\) |
| Analytic rank: | \(1\) |
| Dimension: | \(5\) |
| Coefficient field: | \(\Q(\zeta_{22})^+\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{5} - x^{4} - 4x^{3} + 3x^{2} + 3x - 1 \)
|
| Coefficient ring: | \(\Z[a_1, a_2]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 115) |
| 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 \(\nu = \zeta_{22} + \zeta_{22}^{-1}\):
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 2 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{3} - 3\nu \)
|
| \(\beta_{4}\) | \(=\) |
\( \nu^{4} - 4\nu^{2} + 2 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{3} + 3\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{4} + 4\beta_{2} + 6 \)
|
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 |
|
−2.39788 | −1.00000 | 3.74982 | 1.00000 | 2.39788 | 1.37279 | −4.19584 | −2.00000 | −2.39788 | |||||||||||||||||||||||||||||||||
| 1.2 | −1.54620 | −1.00000 | 0.390736 | 1.00000 | 1.54620 | 2.39788 | 2.48825 | −2.00000 | −1.54620 | ||||||||||||||||||||||||||||||||||
| 1.3 | −1.37279 | −1.00000 | −0.115460 | 1.00000 | 1.37279 | −2.22871 | 2.90407 | −2.00000 | −1.37279 | ||||||||||||||||||||||||||||||||||
| 1.4 | 0.0881559 | −1.00000 | −1.99223 | 1.00000 | −0.0881559 | 1.54620 | −0.351939 | −2.00000 | 0.0881559 | ||||||||||||||||||||||||||||||||||
| 1.5 | 2.22871 | −1.00000 | 2.96714 | 1.00000 | −2.22871 | −0.0881559 | 2.15546 | −2.00000 | 2.22871 | ||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(5\) | \(-1\) |
| \(23\) | \(-1\) |
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 | 2645.2.a.o | 5 | |
| 23.b | odd | 2 | 1 | 2645.2.a.n | 5 | ||
| 23.d | odd | 22 | 2 | 115.2.g.a | ✓ | 10 | |
| 115.i | odd | 22 | 2 | 575.2.k.a | 10 | ||
| 115.l | even | 44 | 4 | 575.2.p.a | 20 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 115.2.g.a | ✓ | 10 | 23.d | odd | 22 | 2 | |
| 575.2.k.a | 10 | 115.i | odd | 22 | 2 | ||
| 575.2.p.a | 20 | 115.l | even | 44 | 4 | ||
| 2645.2.a.n | 5 | 23.b | odd | 2 | 1 | ||
| 2645.2.a.o | 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(2645))\):
|
\( T_{2}^{5} + 3T_{2}^{4} - 3T_{2}^{3} - 15T_{2}^{2} - 10T_{2} + 1 \)
|
|
\( T_{7}^{5} - 3T_{7}^{4} - 3T_{7}^{3} + 15T_{7}^{2} - 10T_{7} - 1 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{5} + 3 T^{4} + \cdots + 1 \)
|
| $3$ |
\( (T + 1)^{5} \)
|
| $5$ |
\( (T - 1)^{5} \)
|
| $7$ |
\( T^{5} - 3 T^{4} + \cdots - 1 \)
|
| $11$ |
\( T^{5} + 4 T^{4} + \cdots + 89 \)
|
| $13$ |
\( T^{5} - 11 T^{3} + \cdots - 11 \)
|
| $17$ |
\( T^{5} + 5 T^{4} + \cdots + 683 \)
|
| $19$ |
\( T^{5} - T^{4} + \cdots + 109 \)
|
| $23$ |
\( T^{5} \)
|
| $29$ |
\( T^{5} - T^{4} + \cdots - 11309 \)
|
| $31$ |
\( T^{5} + 10 T^{4} + \cdots - 89 \)
|
| $37$ |
\( T^{5} + T^{4} + \cdots + 3389 \)
|
| $41$ |
\( T^{5} + 8 T^{4} + \cdots + 13903 \)
|
| $43$ |
\( T^{5} + 24 T^{4} + \cdots - 2069 \)
|
| $47$ |
\( T^{5} + 26 T^{4} + \cdots - 9811 \)
|
| $53$ |
\( T^{5} + 3 T^{4} + \cdots - 1187 \)
|
| $59$ |
\( T^{5} - 41 T^{4} + \cdots - 20921 \)
|
| $61$ |
\( T^{5} + 4 T^{4} + \cdots + 661 \)
|
| $67$ |
\( T^{5} - 23 T^{4} + \cdots + 5059 \)
|
| $71$ |
\( T^{5} - 121 T^{3} + \cdots - 1199 \)
|
| $73$ |
\( T^{5} + 17 T^{4} + \cdots - 8867 \)
|
| $79$ |
\( T^{5} - 13 T^{4} + \cdots + 15709 \)
|
| $83$ |
\( T^{5} + 19 T^{4} + \cdots + 3167 \)
|
| $89$ |
\( T^{5} - 32 T^{4} + \cdots - 6577 \)
|
| $97$ |
\( T^{5} + 15 T^{4} + \cdots + 7877 \)
|
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