Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [309,2,Mod(1,309)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(309, 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("309.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 309 = 3 \cdot 103 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 309.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: | \(2.46737742246\) |
| Analytic rank: | \(0\) |
| Dimension: | \(8\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{8} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{8} - x^{7} - 13x^{6} + 11x^{5} + 52x^{4} - 35x^{3} - 59x^{2} + 27x + 1 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| 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 \(1,\beta_1,\ldots,\beta_{7}\) 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
\( x^{8} - x^{7} - 13x^{6} + 11x^{5} + 52x^{4} - 35x^{3} - 59x^{2} + 27x + 1 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 3 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{3} - 5\nu \)
|
| \(\beta_{4}\) | \(=\) |
\( \nu^{5} - 8\nu^{3} + 13\nu - 2 \)
|
| \(\beta_{5}\) | \(=\) |
\( \nu^{6} - 8\nu^{4} - \nu^{3} + 13\nu^{2} + 3\nu - 1 \)
|
| \(\beta_{6}\) | \(=\) |
\( \nu^{5} - \nu^{4} - 7\nu^{3} + 6\nu^{2} + 8\nu - 5 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( \nu^{7} - 11\nu^{5} + 36\nu^{3} - 3\nu^{2} - 34\nu + 7 ) / 2 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{3} + 5\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( -\beta_{6} + \beta_{4} + \beta_{3} + 6\beta_{2} + 15 \)
|
| \(\nu^{5}\) | \(=\) |
\( \beta_{4} + 8\beta_{3} + 27\beta _1 + 2 \)
|
| \(\nu^{6}\) | \(=\) |
\( -8\beta_{6} + \beta_{5} + 8\beta_{4} + 9\beta_{3} + 35\beta_{2} + 2\beta _1 + 82 \)
|
| \(\nu^{7}\) | \(=\) |
\( 2\beta_{7} + 11\beta_{4} + 52\beta_{3} + 3\beta_{2} + 151\beta _1 + 24 \)
|
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 |
|
−2.49857 | 1.00000 | 4.24286 | 0.808843 | −2.49857 | 4.57320 | −5.60396 | 1.00000 | −2.02095 | ||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −2.43084 | 1.00000 | 3.90900 | −4.19036 | −2.43084 | −3.26427 | −4.64048 | 1.00000 | 10.1861 | |||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −1.53564 | 1.00000 | 0.358203 | 2.14118 | −1.53564 | 0.267492 | 2.52122 | 1.00000 | −3.28809 | |||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −0.458525 | 1.00000 | −1.78975 | −2.98439 | −0.458525 | 3.66803 | 1.73770 | 1.00000 | 1.36842 | |||||||||||||||||||||||||||||||||||||||||||
| 1.5 | 0.0344931 | 1.00000 | −1.99881 | 4.08386 | 0.0344931 | −0.0879825 | −0.137931 | 1.00000 | 0.140865 | |||||||||||||||||||||||||||||||||||||||||||
| 1.6 | 1.27306 | 1.00000 | −0.379326 | 1.25442 | 1.27306 | 2.55696 | −3.02902 | 1.00000 | 1.59695 | |||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 2.26366 | 1.00000 | 3.12415 | 0.128629 | 2.26366 | −4.08864 | 2.54469 | 1.00000 | 0.291172 | |||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 2.35238 | 1.00000 | 3.53367 | −2.24219 | 2.35238 | 2.37520 | 3.60778 | 1.00000 | −5.27447 | |||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \(-1\) |
| \(103\) | \(1\) |
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 | 309.2.a.d | ✓ | 8 |
| 3.b | odd | 2 | 1 | 927.2.a.g | 8 | ||
| 4.b | odd | 2 | 1 | 4944.2.a.bf | 8 | ||
| 5.b | even | 2 | 1 | 7725.2.a.z | 8 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 309.2.a.d | ✓ | 8 | 1.a | even | 1 | 1 | trivial |
| 927.2.a.g | 8 | 3.b | odd | 2 | 1 | ||
| 4944.2.a.bf | 8 | 4.b | odd | 2 | 1 | ||
| 7725.2.a.z | 8 | 5.b | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{8} + T_{2}^{7} - 13T_{2}^{6} - 11T_{2}^{5} + 52T_{2}^{4} + 35T_{2}^{3} - 59T_{2}^{2} - 27T_{2} + 1 \)
acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(309))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{8} + T^{7} - 13 T^{6} + \cdots + 1 \)
|
| $3$ |
\( (T - 1)^{8} \)
|
| $5$ |
\( T^{8} + T^{7} + \cdots - 32 \)
|
| $7$ |
\( T^{8} - 6 T^{7} + \cdots - 32 \)
|
| $11$ |
\( T^{8} - 6 T^{7} + \cdots - 256 \)
|
| $13$ |
\( T^{8} - 9 T^{7} + \cdots + 106 \)
|
| $17$ |
\( T^{8} + 4 T^{7} + \cdots + 32 \)
|
| $19$ |
\( T^{8} - 16 T^{7} + \cdots - 8000 \)
|
| $23$ |
\( T^{8} + 11 T^{7} + \cdots - 452240 \)
|
| $29$ |
\( T^{8} - 139 T^{6} + \cdots + 47800 \)
|
| $31$ |
\( T^{8} - 17 T^{7} + \cdots - 32000 \)
|
| $37$ |
\( T^{8} + 6 T^{7} + \cdots - 7040 \)
|
| $41$ |
\( T^{8} - 12 T^{7} + \cdots - 1960 \)
|
| $43$ |
\( T^{8} - 9 T^{7} + \cdots + 1088 \)
|
| $47$ |
\( T^{8} + 6 T^{7} + \cdots - 24991744 \)
|
| $53$ |
\( T^{8} + 16 T^{7} + \cdots + 835712 \)
|
| $59$ |
\( T^{8} - 11 T^{7} + \cdots - 2000020 \)
|
| $61$ |
\( T^{8} - 5 T^{7} + \cdots - 28310 \)
|
| $67$ |
\( T^{8} - 5 T^{7} + \cdots + 888256 \)
|
| $71$ |
\( T^{8} + 10 T^{7} + \cdots - 22528 \)
|
| $73$ |
\( T^{8} - 14 T^{7} + \cdots + 10240 \)
|
| $79$ |
\( T^{8} - 14 T^{7} + \cdots - 117632 \)
|
| $83$ |
\( T^{8} + 23 T^{7} + \cdots - 122188 \)
|
| $89$ |
\( T^{8} + 14 T^{7} + \cdots + 86144 \)
|
| $97$ |
\( T^{8} - 3 T^{7} + \cdots + 8331170 \)
|
| show more | |
| show less |