Newspace parameters
| Level: | \( N \) | \(=\) | \( 29 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 29.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(1.71105539017\) |
| Analytic rank: | \(0\) |
| Dimension: | \(5\) |
| Coefficient field: | 5.5.13458092.1 |
| Defining polynomial: |
\( x^{5} - x^{4} - 14x^{3} + 18x^{2} + 20x - 8 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 2^{2} \) |
| 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 \(1,\beta_1,\beta_2,\beta_3,\beta_4\) 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
\( x^{5} - x^{4} - 14x^{3} + 18x^{2} + 20x - 8 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( \nu^{3} + \nu^{2} - 8\nu - 2 ) / 2 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -\nu^{3} - \nu^{2} + 12\nu + 2 ) / 2 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{2} - 6 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( \nu^{4} - 13\nu^{2} + 6\nu + 14 ) / 2 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{2} + \beta_1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{3} + 6 \)
|
| \(\nu^{3}\) | \(=\) |
\( -\beta_{3} + 4\beta_{2} + 6\beta _1 - 4 \)
|
| \(\nu^{4}\) | \(=\) |
\( 2\beta_{4} + 13\beta_{3} - 3\beta_{2} - 3\beta _1 + 64 \)
|
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 |
|
−5.49488 | −6.46343 | 22.1937 | 2.14270 | 35.5158 | 20.3573 | −77.9928 | 14.7760 | −11.7739 | |||||||||||||||||||||||||||||||||
| 1.2 | −2.84972 | 4.64574 | 0.120922 | 12.8729 | −13.2391 | 26.0540 | 22.4532 | −5.41713 | −36.6841 | ||||||||||||||||||||||||||||||||||
| 1.3 | 1.63099 | 9.87991 | −5.33986 | −16.8209 | 16.1141 | 5.21997 | −21.7572 | 70.6126 | −27.4348 | ||||||||||||||||||||||||||||||||||
| 1.4 | 2.24125 | 1.84328 | −2.97681 | 18.3339 | 4.13124 | −16.8583 | −24.6017 | −23.6023 | 41.0908 | ||||||||||||||||||||||||||||||||||
| 1.5 | 4.47236 | −1.90549 | 12.0020 | −6.52855 | −8.52204 | 5.22706 | 17.8986 | −23.3691 | −29.1981 | ||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(29\) | \(1\) |
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 | 29.4.a.b | ✓ | 5 |
| 3.b | odd | 2 | 1 | 261.4.a.f | 5 | ||
| 4.b | odd | 2 | 1 | 464.4.a.l | 5 | ||
| 5.b | even | 2 | 1 | 725.4.a.c | 5 | ||
| 7.b | odd | 2 | 1 | 1421.4.a.e | 5 | ||
| 8.b | even | 2 | 1 | 1856.4.a.y | 5 | ||
| 8.d | odd | 2 | 1 | 1856.4.a.bb | 5 | ||
| 29.b | even | 2 | 1 | 841.4.a.b | 5 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 29.4.a.b | ✓ | 5 | 1.a | even | 1 | 1 | trivial |
| 261.4.a.f | 5 | 3.b | odd | 2 | 1 | ||
| 464.4.a.l | 5 | 4.b | odd | 2 | 1 | ||
| 725.4.a.c | 5 | 5.b | even | 2 | 1 | ||
| 841.4.a.b | 5 | 29.b | even | 2 | 1 | ||
| 1421.4.a.e | 5 | 7.b | odd | 2 | 1 | ||
| 1856.4.a.y | 5 | 8.b | even | 2 | 1 | ||
| 1856.4.a.bb | 5 | 8.d | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{5} - 33T_{2}^{3} + 28T_{2}^{2} + 192T_{2} - 256 \)
acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(29))\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{5} - 33 T^{3} + 28 T^{2} + \cdots - 256 \)
$3$
\( T^{5} - 8 T^{4} - 52 T^{3} + \cdots - 1042 \)
$5$
\( T^{5} - 10 T^{4} - 366 T^{3} + \cdots - 55534 \)
$7$
\( T^{5} - 40 T^{4} + 84 T^{3} + \cdots + 243968 \)
$11$
\( T^{5} - 12 T^{4} - 4892 T^{3} + \cdots + 30997958 \)
$13$
\( T^{5} - 14 T^{4} - 7558 T^{3} + \cdots - 13078418 \)
$17$
\( T^{5} - 66 T^{4} - 2444 T^{3} + \cdots + 19935872 \)
$19$
\( T^{5} - 214 T^{4} + \cdots + 19441152 \)
$23$
\( T^{5} - 164 T^{4} + \cdots - 7938109184 \)
$29$
\( (T + 29)^{5} \)
$31$
\( T^{5} - 420 T^{4} + 45552 T^{3} + \cdots - 2094346 \)
$37$
\( T^{5} - 378 T^{4} + \cdots + 23564115968 \)
$41$
\( T^{5} + 1158 T^{4} + \cdots + 59613728000 \)
$43$
\( T^{5} + 204 T^{4} + \cdots + 198643410886 \)
$47$
\( T^{5} - 248 T^{4} + \cdots - 203435244846 \)
$53$
\( T^{5} + 554 T^{4} + \cdots - 786854101018 \)
$59$
\( T^{5} - 440 T^{4} + \cdots + 109032704000 \)
$61$
\( T^{5} - 618 T^{4} + \cdots + 2140697762176 \)
$67$
\( T^{5} - 1164 T^{4} + \cdots - 39308070146048 \)
$71$
\( T^{5} + 692 T^{4} + \cdots + 98341318953856 \)
$73$
\( T^{5} + 1950 T^{4} + \cdots + 7201878016 \)
$79$
\( T^{5} + \cdots - 240961986300538 \)
$83$
\( T^{5} - 512 T^{4} + \cdots + 6057622580224 \)
$89$
\( T^{5} - 866 T^{4} + \cdots + 21549994365568 \)
$97$
\( T^{5} - 1562 T^{4} + \cdots - 20480102175488 \)
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