Newspace parameters
| Level: | \( N \) | \(=\) | \( 451 = 11 \cdot 41 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 451.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(3.60125313116\) |
| Analytic rank: | \(1\) |
| Dimension: | \(5\) |
| Coefficient field: | 5.5.24217.1 |
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|
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| Defining polynomial: |
\( x^{5} - 5x^{3} - x^{2} + 3x + 1 \)
|
| 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
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
\( x^{5} - 5x^{3} - x^{2} + 3x + 1 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu^{4} - 5\nu^{2} + 2 \)
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| \(\beta_{2}\) | \(=\) |
\( -\nu^{4} + 5\nu^{2} + \nu - 2 \)
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| \(\beta_{3}\) | \(=\) |
\( -\nu^{4} + \nu^{3} + 5\nu^{2} - 3\nu - 3 \)
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| \(\beta_{4}\) | \(=\) |
\( 2\nu^{4} - \nu^{3} - 9\nu^{2} + 3\nu + 3 \)
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| \(\nu\) | \(=\) |
\( \beta_{2} + \beta_1 \)
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| \(\nu^{2}\) | \(=\) |
\( \beta_{4} + \beta_{3} - \beta _1 + 2 \)
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| \(\nu^{3}\) | \(=\) |
\( \beta_{3} + 3\beta_{2} + 4\beta _1 + 1 \)
|
| \(\nu^{4}\) | \(=\) |
\( 5\beta_{4} + 5\beta_{3} - 4\beta _1 + 8 \)
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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 |
|
−2.44983 | −1.76009 | 4.00166 | −3.75177 | 4.31193 | 2.72012 | −4.90374 | 0.0979280 | 9.19119 | |||||||||||||||||||||||||||||||||
| 1.2 | −1.26073 | 2.57459 | −0.410549 | −2.02521 | −3.24587 | −4.45268 | 3.03906 | 3.62851 | 2.55325 | ||||||||||||||||||||||||||||||||||
| 1.3 | −0.339328 | −1.85294 | −1.88486 | 1.35431 | 0.628755 | 1.57575 | 1.31824 | 0.433386 | −0.459556 | ||||||||||||||||||||||||||||||||||
| 1.4 | 0.714533 | 0.131284 | −1.48944 | 0.818050 | 0.0938067 | −3.30571 | −2.49332 | −2.98276 | 0.584524 | ||||||||||||||||||||||||||||||||||
| 1.5 | 1.33536 | 0.907159 | −0.216816 | −4.39538 | 1.21138 | −1.53748 | −2.96025 | −2.17706 | −5.86941 | ||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(11\) | \( -1 \) |
| \(41\) | \( -1 \) |
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 | 451.2.a.c | ✓ | 5 |
| 3.b | odd | 2 | 1 | 4059.2.a.o | 5 | ||
| 4.b | odd | 2 | 1 | 7216.2.a.r | 5 | ||
| 11.b | odd | 2 | 1 | 4961.2.a.f | 5 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 451.2.a.c | ✓ | 5 | 1.a | even | 1 | 1 | trivial |
| 4059.2.a.o | 5 | 3.b | odd | 2 | 1 | ||
| 4961.2.a.f | 5 | 11.b | odd | 2 | 1 | ||
| 7216.2.a.r | 5 | 4.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{5} + 2T_{2}^{4} - 3T_{2}^{3} - 4T_{2}^{2} + 2T_{2} + 1 \)
acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(451))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{5} + 2 T^{4} + \cdots + 1 \)
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| $3$ |
\( T^{5} - 7 T^{3} + \cdots - 1 \)
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| $5$ |
\( T^{5} + 8 T^{4} + \cdots + 37 \)
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| $7$ |
\( T^{5} + 5 T^{4} + \cdots + 97 \)
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| $11$ |
\( (T - 1)^{5} \)
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| $13$ |
\( T^{5} + 5 T^{4} + \cdots + 1 \)
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| $17$ |
\( T^{5} - 3 T^{4} + \cdots + 493 \)
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| $19$ |
\( T^{5} + 11 T^{4} + \cdots - 835 \)
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| $23$ |
\( T^{5} + 7 T^{4} + \cdots - 59 \)
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| $29$ |
\( T^{5} + 24 T^{4} + \cdots + 365 \)
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| $31$ |
\( T^{5} + 13 T^{4} + \cdots - 391 \)
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| $37$ |
\( T^{5} - 2 T^{4} + \cdots - 1193 \)
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| $41$ |
\( (T - 1)^{5} \)
|
| $43$ |
\( T^{5} + 12 T^{4} + \cdots + 9547 \)
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| $47$ |
\( T^{5} + 6 T^{4} + \cdots + 1193 \)
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| $53$ |
\( T^{5} + 28 T^{4} + \cdots - 29087 \)
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| $59$ |
\( T^{5} + 5 T^{4} + \cdots - 1385 \)
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| $61$ |
\( T^{5} + 15 T^{4} + \cdots + 105361 \)
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| $67$ |
\( T^{5} - 30 T^{4} + \cdots + 13859 \)
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| $71$ |
\( T^{5} + T^{4} + \cdots + 17629 \)
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| $73$ |
\( T^{5} - T^{4} + \cdots - 661 \)
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| $79$ |
\( T^{5} + 4 T^{4} + \cdots + 17785 \)
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| $83$ |
\( T^{5} - 6 T^{4} + \cdots + 3209 \)
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| $89$ |
\( T^{5} + 6 T^{4} + \cdots + 12745 \)
|
| $97$ |
\( T^{5} - 23 T^{4} + \cdots - 13219 \)
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