Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [6045,2,Mod(1,6045)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6045, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 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("6045.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 6045 = 3 \cdot 5 \cdot 13 \cdot 31 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 6045.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: | \(48.2695680219\) |
| Analytic rank: | \(1\) |
| Dimension: | \(9\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{9} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{9} - 3x^{8} - 7x^{7} + 20x^{6} + 20x^{5} - 38x^{4} - 27x^{3} + 13x^{2} + 6x - 1 \)
|
| 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 \(1,\beta_1,\ldots,\beta_{8}\) 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
\( x^{9} - 3x^{8} - 7x^{7} + 20x^{6} + 20x^{5} - 38x^{4} - 27x^{3} + 13x^{2} + 6x - 1 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - \nu - 2 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{3} - \nu^{2} - 4\nu + 1 \)
|
| \(\beta_{4}\) | \(=\) |
\( \nu^{5} - 3\nu^{4} - 3\nu^{3} + 10\nu^{2} + 2\nu - 3 \)
|
| \(\beta_{5}\) | \(=\) |
\( \nu^{5} - 2\nu^{4} - 5\nu^{3} + 7\nu^{2} + 6\nu - 2 \)
|
| \(\beta_{6}\) | \(=\) |
\( \nu^{6} - 3\nu^{5} - 3\nu^{4} + 11\nu^{3} + \nu^{2} - 6\nu + 1 \)
|
| \(\beta_{7}\) | \(=\) |
\( -\nu^{8} + 4\nu^{7} + 3\nu^{6} - 23\nu^{5} + 2\nu^{4} + 38\nu^{3} - 7\nu^{2} - 12\nu + 3 \)
|
| \(\beta_{8}\) | \(=\) |
\( \nu^{8} - 3\nu^{7} - 7\nu^{6} + 20\nu^{5} + 20\nu^{4} - 38\nu^{3} - 26\nu^{2} + 12\nu + 3 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + \beta _1 + 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{3} + \beta_{2} + 5\beta _1 + 1 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{5} - \beta_{4} + 2\beta_{3} + 5\beta_{2} + 9\beta _1 + 7 \)
|
| \(\nu^{5}\) | \(=\) |
\( 3\beta_{5} - 2\beta_{4} + 9\beta_{3} + 8\beta_{2} + 30\beta _1 + 7 \)
|
| \(\nu^{6}\) | \(=\) |
\( \beta_{6} + 12\beta_{5} - 9\beta_{4} + 22\beta_{3} + 27\beta_{2} + 67\beta _1 + 28 \)
|
| \(\nu^{7}\) | \(=\) |
\( \beta_{8} + \beta_{7} + 4\beta_{6} + 35\beta_{5} - 20\beta_{4} + 71\beta_{3} + 55\beta_{2} + 193\beta _1 + 39 \)
|
| \(\nu^{8}\) | \(=\) |
\( 4\beta_{8} + 3\beta_{7} + 19\beta_{6} + 109\beta_{5} - 63\beta_{4} + 185\beta_{3} + 158\beta_{2} + 472\beta _1 + 120 \)
|
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 |
|
−2.67506 | 1.00000 | 5.15596 | 1.00000 | −2.67506 | 1.28643 | −8.44238 | 1.00000 | −2.67506 | |||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −2.37659 | 1.00000 | 3.64818 | 1.00000 | −2.37659 | −1.73113 | −3.91706 | 1.00000 | −2.37659 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −1.89284 | 1.00000 | 1.58283 | 1.00000 | −1.89284 | −4.37246 | 0.789632 | 1.00000 | −1.89284 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −0.529693 | 1.00000 | −1.71943 | 1.00000 | −0.529693 | −2.25073 | 1.97015 | 1.00000 | −0.529693 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | −0.139045 | 1.00000 | −1.98067 | 1.00000 | −0.139045 | −0.572827 | 0.553494 | 1.00000 | −0.139045 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | 0.503427 | 1.00000 | −1.74656 | 1.00000 | 0.503427 | 1.63268 | −1.88612 | 1.00000 | 0.503427 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 0.839112 | 1.00000 | −1.29589 | 1.00000 | 0.839112 | 2.06151 | −2.76562 | 1.00000 | 0.839112 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 1.57676 | 1.00000 | 0.486159 | 1.00000 | 1.57676 | −0.0992003 | −2.38696 | 1.00000 | 1.57676 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | 1.69393 | 1.00000 | 0.869413 | 1.00000 | 1.69393 | −0.954273 | −1.91514 | 1.00000 | 1.69393 | ||||||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \(-1\) |
| \(5\) | \(-1\) |
| \(13\) | \(1\) |
| \(31\) | \(1\) |
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 | 6045.2.a.u | ✓ | 9 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 6045.2.a.u | ✓ | 9 | 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(6045))\):
|
\( T_{2}^{9} + 3T_{2}^{8} - 7T_{2}^{7} - 20T_{2}^{6} + 20T_{2}^{5} + 38T_{2}^{4} - 27T_{2}^{3} - 13T_{2}^{2} + 6T_{2} + 1 \)
|
|
\( T_{7}^{9} + 5T_{7}^{8} - 6T_{7}^{7} - 43T_{7}^{6} + 2T_{7}^{5} + 112T_{7}^{4} + 36T_{7}^{3} - 85T_{7}^{2} - 49T_{7} - 4 \)
|
|
\( T_{11}^{9} - 55T_{11}^{7} - 70T_{11}^{6} + 882T_{11}^{5} + 1972T_{11}^{4} - 2898T_{11}^{3} - 9452T_{11}^{2} - 2295T_{11} + 4516 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{9} + 3 T^{8} + \cdots + 1 \)
|
| $3$ |
\( (T - 1)^{9} \)
|
| $5$ |
\( (T - 1)^{9} \)
|
| $7$ |
\( T^{9} + 5 T^{8} + \cdots - 4 \)
|
| $11$ |
\( T^{9} - 55 T^{7} + \cdots + 4516 \)
|
| $13$ |
\( (T + 1)^{9} \)
|
| $17$ |
\( T^{9} + 5 T^{8} + \cdots - 2098 \)
|
| $19$ |
\( T^{9} + 12 T^{8} + \cdots + 235708 \)
|
| $23$ |
\( T^{9} + 21 T^{8} + \cdots + 1376 \)
|
| $29$ |
\( T^{9} + 15 T^{8} + \cdots - 5469706 \)
|
| $31$ |
\( (T + 1)^{9} \)
|
| $37$ |
\( T^{9} + 13 T^{8} + \cdots - 294482 \)
|
| $41$ |
\( T^{9} + 11 T^{8} + \cdots + 15660058 \)
|
| $43$ |
\( T^{9} + 26 T^{8} + \cdots - 7596044 \)
|
| $47$ |
\( T^{9} + 21 T^{8} + \cdots + 896 \)
|
| $53$ |
\( T^{9} - 18 T^{8} + \cdots - 897086 \)
|
| $59$ |
\( T^{9} + 24 T^{8} + \cdots - 32552 \)
|
| $61$ |
\( T^{9} + 4 T^{8} + \cdots - 27695434 \)
|
| $67$ |
\( T^{9} + 20 T^{8} + \cdots - 27655256 \)
|
| $71$ |
\( T^{9} + 13 T^{8} + \cdots + 6198692 \)
|
| $73$ |
\( T^{9} - 15 T^{8} + \cdots + 11507282 \)
|
| $79$ |
\( T^{9} + 15 T^{8} + \cdots + 685112 \)
|
| $83$ |
\( T^{9} + 5 T^{8} + \cdots - 12258892 \)
|
| $89$ |
\( T^{9} - 31 T^{8} + \cdots - 293238022 \)
|
| $97$ |
\( T^{9} + 11 T^{8} + \cdots - 145375154 \)
|
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