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: | \(6\) |
| Coefficient field: | 6.6.27387072.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} - 9x^{4} + 18x^{2} - 6x - 3 \)
|
| Coefficient ring: | \(\Z[a_1, a_2]\) |
| 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 \(1,\beta_1,\ldots,\beta_{5}\) 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
\( x^{6} - 9x^{4} + 18x^{2} - 6x - 3 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{3} - \nu^{2} - 5\nu + 3 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{4} - \nu^{3} - 5\nu^{2} + 3\nu \)
|
| \(\beta_{4}\) | \(=\) |
\( \nu^{4} - \nu^{3} - 6\nu^{2} + 3\nu + 3 \)
|
| \(\beta_{5}\) | \(=\) |
\( \nu^{5} - 7\nu^{3} - 2\nu^{2} + 7\nu + 1 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( -\beta_{4} + \beta_{3} + 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( -\beta_{4} + \beta_{3} + \beta_{2} + 5\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( -6\beta_{4} + 7\beta_{3} + \beta_{2} + 2\beta _1 + 15 \)
|
| \(\nu^{5}\) | \(=\) |
\( \beta_{5} - 9\beta_{4} + 9\beta_{3} + 7\beta_{2} + 28\beta _1 + 5 \)
|
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 |
|
−2.59569 | −2.10536 | 4.73759 | 1.00000 | 5.46486 | −4.49586 | −7.10591 | 1.43255 | −2.59569 | ||||||||||||||||||||||||||||||||||||
| 1.2 | −1.20427 | 2.16337 | −0.549727 | 1.00000 | −2.60528 | −2.08586 | 3.07057 | 1.68016 | −1.20427 | |||||||||||||||||||||||||||||||||||||
| 1.3 | −0.756215 | −3.36993 | −1.42814 | 1.00000 | 2.54839 | 1.30980 | 2.59241 | 8.35644 | −0.756215 | |||||||||||||||||||||||||||||||||||||
| 1.4 | 0.277618 | −0.0506556 | −1.92293 | 1.00000 | −0.0140629 | −0.480849 | −1.08908 | −2.99743 | 0.277618 | |||||||||||||||||||||||||||||||||||||
| 1.5 | 2.06791 | −3.05801 | 2.27624 | 1.00000 | −6.32367 | 3.58172 | 0.571245 | 6.35140 | 2.06791 | |||||||||||||||||||||||||||||||||||||
| 1.6 | 2.21065 | 0.420587 | 2.88696 | 1.00000 | 0.929770 | −3.82895 | 1.96077 | −2.82311 | 2.21065 | |||||||||||||||||||||||||||||||||||||
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.q | yes | 6 |
| 23.b | odd | 2 | 1 | 2645.2.a.p | ✓ | 6 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 2645.2.a.p | ✓ | 6 | 23.b | odd | 2 | 1 | |
| 2645.2.a.q | yes | 6 | 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(2645))\):
|
\( T_{2}^{6} - 9T_{2}^{4} + 18T_{2}^{2} + 6T_{2} - 3 \)
|
|
\( T_{7}^{6} + 6T_{7}^{5} - 9T_{7}^{4} - 90T_{7}^{3} - 54T_{7}^{2} + 162T_{7} + 81 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} - 9 T^{4} + \cdots - 3 \)
|
| $3$ |
\( T^{6} + 6 T^{5} + \cdots + 1 \)
|
| $5$ |
\( (T - 1)^{6} \)
|
| $7$ |
\( T^{6} + 6 T^{5} + \cdots + 81 \)
|
| $11$ |
\( T^{6} + 12 T^{5} + \cdots + 2997 \)
|
| $13$ |
\( T^{6} + 12 T^{5} + \cdots - 23 \)
|
| $17$ |
\( T^{6} - 45 T^{4} + \cdots + 81 \)
|
| $19$ |
\( T^{6} + 12 T^{5} + \cdots + 1917 \)
|
| $23$ |
\( T^{6} \)
|
| $29$ |
\( T^{6} - 18 T^{5} + \cdots + 789 \)
|
| $31$ |
\( T^{6} - 24 T^{5} + \cdots + 517 \)
|
| $37$ |
\( T^{6} + 18 T^{5} + \cdots + 18441 \)
|
| $41$ |
\( T^{6} - 6 T^{5} + \cdots + 177 \)
|
| $43$ |
\( T^{6} + 24 T^{5} + \cdots - 15471 \)
|
| $47$ |
\( T^{6} + 6 T^{5} + \cdots + 9573 \)
|
| $53$ |
\( T^{6} + 30 T^{5} + \cdots - 891 \)
|
| $59$ |
\( T^{6} - 90 T^{4} + \cdots + 5157 \)
|
| $61$ |
\( T^{6} + 12 T^{5} + \cdots - 129951 \)
|
| $67$ |
\( T^{6} + 6 T^{5} + \cdots + 1269 \)
|
| $71$ |
\( T^{6} - 6 T^{5} + \cdots + 116349 \)
|
| $73$ |
\( T^{6} - 24 T^{5} + \cdots - 26807 \)
|
| $79$ |
\( T^{6} + 12 T^{5} + \cdots - 15471 \)
|
| $83$ |
\( T^{6} + 6 T^{5} + \cdots + 12717 \)
|
| $89$ |
\( T^{6} - 12 T^{5} + \cdots - 1863 \)
|
| $97$ |
\( T^{6} + 6 T^{5} + \cdots - 1971 \)
|
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