Newspace parameters
| Level: | \( N \) | \(=\) | \( 83 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 83.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(0.662758336777\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Coefficient field: | 6.6.9059636.1 |
| Defining polynomial: |
\( x^{6} - x^{5} - 8x^{4} + 11x^{3} + 4x^{2} - 4x - 1 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | yes |
| Fricke sign: | \(-1\) |
| Sato-Tate group: | $\mathrm{SU}(2)$ |
$q$-expansion
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
\( x^{6} - x^{5} - 8x^{4} + 11x^{3} + 4x^{2} - 4x - 1 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{4} - 7\nu^{2} + 4\nu + 2 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{5} - \nu^{4} - 7\nu^{3} + 12\nu^{2} - 2\nu - 4 \)
|
| \(\beta_{4}\) | \(=\) |
\( -\nu^{5} + \nu^{4} + 8\nu^{3} - 11\nu^{2} - 4\nu + 3 \)
|
| \(\beta_{5}\) | \(=\) |
\( -2\nu^{5} + 2\nu^{4} + 15\nu^{3} - 22\nu^{2} - \nu + 4 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{5} - \beta_{4} + \beta_{3} - \beta _1 + 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( -\beta_{5} + 2\beta_{4} + 7\beta _1 - 2 \)
|
| \(\nu^{4}\) | \(=\) |
\( 7\beta_{5} - 7\beta_{4} + 7\beta_{3} + \beta_{2} - 11\beta _1 + 19 \)
|
| \(\nu^{5}\) | \(=\) |
\( -12\beta_{5} + 19\beta_{4} - 4\beta_{3} + \beta_{2} + 52\beta _1 - 27 \)
|
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.
| Label | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1.1 |
|
−2.29018 | 1.96807 | 3.24494 | −1.62860 | −4.50724 | 3.12266 | −2.85114 | 0.873293 | 3.72978 | ||||||||||||||||||||||||||||||||||||
| 1.2 | −1.66658 | −2.04941 | 0.777479 | 2.79494 | 3.41549 | 3.61008 | 2.03743 | 1.20006 | −4.65798 | |||||||||||||||||||||||||||||||||||||
| 1.3 | −0.429349 | 2.76735 | −1.81566 | 1.71413 | −1.18816 | −3.46533 | 1.63825 | 4.65821 | −0.735961 | |||||||||||||||||||||||||||||||||||||
| 1.4 | 1.16417 | 1.13227 | −0.644717 | −1.45742 | 1.31815 | 3.35950 | −3.07889 | −1.71797 | −1.69668 | |||||||||||||||||||||||||||||||||||||
| 1.5 | 1.59835 | −1.32239 | 0.554733 | 4.05060 | −2.11364 | −2.22837 | −2.31005 | −1.25129 | 6.47429 | |||||||||||||||||||||||||||||||||||||
| 1.6 | 2.62359 | −1.49589 | 4.88322 | −3.47366 | −3.92460 | −1.39854 | 7.56440 | −0.762314 | −9.11346 | |||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(83\) | \(-1\) |
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 | 83.2.a.b | ✓ | 6 |
| 3.b | odd | 2 | 1 | 747.2.a.j | 6 | ||
| 4.b | odd | 2 | 1 | 1328.2.a.l | 6 | ||
| 5.b | even | 2 | 1 | 2075.2.a.g | 6 | ||
| 7.b | odd | 2 | 1 | 4067.2.a.d | 6 | ||
| 8.b | even | 2 | 1 | 5312.2.a.bn | 6 | ||
| 8.d | odd | 2 | 1 | 5312.2.a.bo | 6 | ||
| 83.b | odd | 2 | 1 | 6889.2.a.e | 6 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 83.2.a.b | ✓ | 6 | 1.a | even | 1 | 1 | trivial |
| 747.2.a.j | 6 | 3.b | odd | 2 | 1 | ||
| 1328.2.a.l | 6 | 4.b | odd | 2 | 1 | ||
| 2075.2.a.g | 6 | 5.b | even | 2 | 1 | ||
| 4067.2.a.d | 6 | 7.b | odd | 2 | 1 | ||
| 5312.2.a.bn | 6 | 8.b | even | 2 | 1 | ||
| 5312.2.a.bo | 6 | 8.d | odd | 2 | 1 | ||
| 6889.2.a.e | 6 | 83.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{6} - T_{2}^{5} - 9T_{2}^{4} + 7T_{2}^{3} + 20T_{2}^{2} - 12T_{2} - 8 \)
acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(83))\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{6} - T^{5} - 9 T^{4} + 7 T^{3} + 20 T^{2} + \cdots - 8 \)
$3$
\( T^{6} - T^{5} - 10 T^{4} + 5 T^{3} + \cdots - 25 \)
$5$
\( T^{6} - 2 T^{5} - 20 T^{4} + 28 T^{3} + \cdots - 160 \)
$7$
\( T^{6} - 3 T^{5} - 22 T^{4} + 55 T^{3} + \cdots - 409 \)
$11$
\( T^{6} + 3 T^{5} - 26 T^{4} - 83 T^{3} + \cdots - 113 \)
$13$
\( T^{6} - 14 T^{5} + 44 T^{4} + \cdots + 992 \)
$17$
\( T^{6} + 5 T^{5} - 20 T^{4} - 77 T^{3} + \cdots - 275 \)
$19$
\( T^{6} + 4 T^{5} - 68 T^{4} + \cdots + 6176 \)
$23$
\( T^{6} + 5 T^{5} - 61 T^{4} + \cdots + 10912 \)
$29$
\( T^{6} + T^{5} - 88 T^{4} - 181 T^{3} + \cdots - 55 \)
$31$
\( T^{6} - 3 T^{5} - 66 T^{4} - 93 T^{3} + \cdots - 313 \)
$37$
\( T^{6} - 39 T^{5} + 576 T^{4} + \cdots - 91499 \)
$41$
\( T^{6} + T^{5} - 47 T^{4} - T^{3} + \cdots - 248 \)
$43$
\( T^{6} + 8 T^{5} - 44 T^{4} + \cdots + 6400 \)
$47$
\( T^{6} + 12 T^{5} - 96 T^{4} + \cdots + 25952 \)
$53$
\( T^{6} - 14 T^{5} - 64 T^{4} + 1064 T^{3} + \cdots - 64 \)
$59$
\( T^{6} + 17 T^{5} + 10 T^{4} + \cdots + 3527 \)
$61$
\( T^{6} + 5 T^{5} - 208 T^{4} + \cdots - 47347 \)
$67$
\( T^{6} - 16 T^{5} - 128 T^{4} + \cdots + 264256 \)
$71$
\( T^{6} + 26 T^{5} + 168 T^{4} + \cdots + 7232 \)
$73$
\( T^{6} + 6 T^{5} - 268 T^{4} + \cdots - 39136 \)
$79$
\( T^{6} + 12 T^{5} - 12 T^{4} + \cdots - 160 \)
$83$
\( (T - 1)^{6} \)
$89$
\( T^{6} + 22 T^{5} - 28 T^{4} + \cdots + 144896 \)
$97$
\( T^{6} - 6 T^{5} - 300 T^{4} + \cdots - 101120 \)
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