Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [319,2,Mod(1,319)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(319, 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("319.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 319 = 11 \cdot 29 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 319.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: | \(2.54722782448\) |
| Analytic rank: | \(0\) |
| Dimension: | \(7\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{7} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{7} - 3x^{6} - 4x^{5} + 15x^{4} + x^{3} - 14x^{2} + 1 \)
|
| 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 \(1,\beta_1,\ldots,\beta_{6}\) 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
\( x^{7} - 3x^{6} - 4x^{5} + 15x^{4} + x^{3} - 14x^{2} + 1 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - \nu - 2 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{3} - \nu^{2} - 3\nu + 1 \)
|
| \(\beta_{4}\) | \(=\) |
\( \nu^{5} - 2\nu^{4} - 5\nu^{3} + 8\nu^{2} + 5\nu - 3 \)
|
| \(\beta_{5}\) | \(=\) |
\( \nu^{6} - 2\nu^{5} - 5\nu^{4} + 9\nu^{3} + 4\nu^{2} - 6\nu \)
|
| \(\beta_{6}\) | \(=\) |
\( \nu^{6} - 3\nu^{5} - 4\nu^{4} + 15\nu^{3} + \nu^{2} - 14\nu \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + \beta _1 + 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{3} + \beta_{2} + 4\beta _1 + 1 \)
|
| \(\nu^{4}\) | \(=\) |
\( -\beta_{6} + \beta_{5} - \beta_{4} + \beta_{3} + 6\beta_{2} + 6\beta _1 + 8 \)
|
| \(\nu^{5}\) | \(=\) |
\( -2\beta_{6} + 2\beta_{5} - \beta_{4} + 7\beta_{3} + 9\beta_{2} + 19\beta _1 + 8 \)
|
| \(\nu^{6}\) | \(=\) |
\( -9\beta_{6} + 10\beta_{5} - 7\beta_{4} + 10\beta_{3} + 35\beta_{2} + 34\beta _1 + 39 \)
|
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 |
|
−1.94021 | 0.168176 | 1.76441 | 2.63360 | −0.326296 | 4.76292 | 0.457103 | −2.97172 | −5.10973 | |||||||||||||||||||||||||||||||||||||||
| 1.2 | −0.954260 | 2.71767 | −1.08939 | −0.444107 | −2.59336 | −1.03527 | 2.94808 | 4.38571 | 0.423793 | ||||||||||||||||||||||||||||||||||||||||
| 1.3 | −0.276682 | −1.81423 | −1.92345 | −2.51096 | 0.501965 | 1.88195 | 1.08555 | 0.291435 | 0.694737 | ||||||||||||||||||||||||||||||||||||||||
| 1.4 | 0.281296 | −3.43225 | −1.92087 | 1.14753 | −0.965479 | −3.65422 | −1.10293 | 8.78037 | 0.322796 | ||||||||||||||||||||||||||||||||||||||||
| 1.5 | 1.36486 | 1.29197 | −0.137158 | 2.01641 | 1.76336 | 1.68223 | −2.91692 | −1.33082 | 2.75211 | ||||||||||||||||||||||||||||||||||||||||
| 1.6 | 2.07719 | 2.68771 | 2.31473 | −2.92190 | 5.58288 | 0.102367 | 0.653749 | 4.22376 | −6.06936 | ||||||||||||||||||||||||||||||||||||||||
| 1.7 | 2.44780 | −1.61903 | 3.99173 | 4.07943 | −3.96307 | −2.73997 | 4.87537 | −0.378738 | 9.98564 | ||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(11\) | \(-1\) |
| \(29\) | \(1\) |
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 | 319.2.a.d | ✓ | 7 |
| 3.b | odd | 2 | 1 | 2871.2.a.m | 7 | ||
| 4.b | odd | 2 | 1 | 5104.2.a.x | 7 | ||
| 5.b | even | 2 | 1 | 7975.2.a.j | 7 | ||
| 11.b | odd | 2 | 1 | 3509.2.a.m | 7 | ||
| 29.b | even | 2 | 1 | 9251.2.a.m | 7 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 319.2.a.d | ✓ | 7 | 1.a | even | 1 | 1 | trivial |
| 2871.2.a.m | 7 | 3.b | odd | 2 | 1 | ||
| 3509.2.a.m | 7 | 11.b | odd | 2 | 1 | ||
| 5104.2.a.x | 7 | 4.b | odd | 2 | 1 | ||
| 7975.2.a.j | 7 | 5.b | even | 2 | 1 | ||
| 9251.2.a.m | 7 | 29.b | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{7} - 3T_{2}^{6} - 4T_{2}^{5} + 15T_{2}^{4} + T_{2}^{3} - 14T_{2}^{2} + 1 \)
acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(319))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{7} - 3 T^{6} + \cdots + 1 \)
|
| $3$ |
\( T^{7} - 17 T^{5} + \cdots + 16 \)
|
| $5$ |
\( T^{7} - 4 T^{6} + \cdots + 81 \)
|
| $7$ |
\( T^{7} - T^{6} + \cdots + 16 \)
|
| $11$ |
\( (T - 1)^{7} \)
|
| $13$ |
\( T^{7} - 51 T^{5} + \cdots + 464 \)
|
| $17$ |
\( T^{7} - 18 T^{6} + \cdots + 9 \)
|
| $19$ |
\( T^{7} - 10 T^{6} + \cdots - 11805 \)
|
| $23$ |
\( T^{7} - 4 T^{6} + \cdots - 64 \)
|
| $29$ |
\( (T + 1)^{7} \)
|
| $31$ |
\( T^{7} + 13 T^{6} + \cdots - 67248 \)
|
| $37$ |
\( T^{7} + 5 T^{6} + \cdots + 9168 \)
|
| $41$ |
\( T^{7} - 31 T^{6} + \cdots + 16141 \)
|
| $43$ |
\( T^{7} - 9 T^{6} + \cdots + 27 \)
|
| $47$ |
\( T^{7} + 7 T^{6} + \cdots - 37904 \)
|
| $53$ |
\( T^{7} - 12 T^{6} + \cdots - 17253 \)
|
| $59$ |
\( T^{7} - 12 T^{6} + \cdots + 14604175 \)
|
| $61$ |
\( T^{7} - 15 T^{6} + \cdots - 53325 \)
|
| $67$ |
\( T^{7} + 25 T^{6} + \cdots + 71 \)
|
| $71$ |
\( T^{7} - 4 T^{6} + \cdots - 92187 \)
|
| $73$ |
\( T^{7} - 7 T^{6} + \cdots + 529136 \)
|
| $79$ |
\( T^{7} - 11 T^{6} + \cdots + 25 \)
|
| $83$ |
\( T^{7} - 36 T^{6} + \cdots - 608688 \)
|
| $89$ |
\( T^{7} - 26 T^{6} + \cdots - 174960 \)
|
| $97$ |
\( T^{7} - 14 T^{6} + \cdots - 19408 \)
|
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