Newspace parameters
| Level: | \( N \) | \(=\) | \( 441 = 3^{2} \cdot 7^{2} \) |
| Weight: | \( k \) | \(=\) | \( 8 \) |
| Character orbit: | \([\chi]\) | \(=\) | 441.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(137.761796238\) |
| Analytic rank: | \(1\) |
| Dimension: | \(5\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{5} - \cdots)\) |
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| Defining polynomial: |
\( x^{5} - 412x^{3} - 48x^{2} + 36411x + 46368 \)
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| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2\cdot 3\cdot 7^{2} \) |
| Twist minimal: | no (minimal twist has level 21) |
| 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
\( x^{5} - 412x^{3} - 48x^{2} + 36411x + 46368 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
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| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 165 \)
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| \(\beta_{3}\) | \(=\) |
\( ( \nu^{4} + \nu^{3} - 299\nu^{2} - 11\nu + 10472 ) / 28 \)
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| \(\beta_{4}\) | \(=\) |
\( \nu^{3} + 3\nu^{2} - 236\nu - 523 \)
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| \(\nu\) | \(=\) |
\( \beta_1 \)
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| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 165 \)
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| \(\nu^{3}\) | \(=\) |
\( \beta_{4} - 3\beta_{2} + 236\beta _1 + 28 \)
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| \(\nu^{4}\) | \(=\) |
\( -\beta_{4} + 28\beta_{3} + 302\beta_{2} - 225\beta _1 + 38835 \)
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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 |
|
−20.1723 | 0 | 278.921 | 429.497 | 0 | 0 | −3044.41 | 0 | −8663.93 | |||||||||||||||||||||||||||||||||
| 1.2 | −13.2861 | 0 | 48.5217 | −348.341 | 0 | 0 | 1055.96 | 0 | 4628.11 | ||||||||||||||||||||||||||||||||||
| 1.3 | −4.29577 | 0 | −109.546 | 241.056 | 0 | 0 | 1020.44 | 0 | −1035.52 | ||||||||||||||||||||||||||||||||||
| 1.4 | 9.34884 | 0 | −40.5991 | −408.099 | 0 | 0 | −1576.21 | 0 | −3815.25 | ||||||||||||||||||||||||||||||||||
| 1.5 | 13.4053 | 0 | 51.7032 | 283.887 | 0 | 0 | −1022.78 | 0 | 3805.60 | ||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \( -1 \) |
| \(7\) | \( +1 \) |
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 | 441.8.a.x | 5 | |
| 3.b | odd | 2 | 1 | 147.8.a.j | 5 | ||
| 7.b | odd | 2 | 1 | 441.8.a.w | 5 | ||
| 7.c | even | 3 | 2 | 63.8.e.d | 10 | ||
| 21.c | even | 2 | 1 | 147.8.a.k | 5 | ||
| 21.g | even | 6 | 2 | 147.8.e.n | 10 | ||
| 21.h | odd | 6 | 2 | 21.8.e.b | ✓ | 10 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 21.8.e.b | ✓ | 10 | 21.h | odd | 6 | 2 | |
| 63.8.e.d | 10 | 7.c | even | 3 | 2 | ||
| 147.8.a.j | 5 | 3.b | odd | 2 | 1 | ||
| 147.8.a.k | 5 | 21.c | even | 2 | 1 | ||
| 147.8.e.n | 10 | 21.g | even | 6 | 2 | ||
| 441.8.a.w | 5 | 7.b | odd | 2 | 1 | ||
| 441.8.a.x | 5 | 1.a | even | 1 | 1 | trivial | |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(441))\):
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\( T_{2}^{5} + 15T_{2}^{4} - 322T_{2}^{3} - 3486T_{2}^{2} + 25404T_{2} + 144288 \)
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\( T_{5}^{5} - 198T_{5}^{4} - 285925T_{5}^{3} + 57240750T_{5}^{2} + 19546300500T_{5} - 4178247435000 \)
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\( T_{13}^{5} + 1273 T_{13}^{4} - 183213425 T_{13}^{3} - 831953102381 T_{13}^{2} + \cdots + 19\!\cdots\!84 \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{5} + 15 T^{4} + \cdots + 144288 \)
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| $3$ |
\( T^{5} \)
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| $5$ |
\( T^{5} + \cdots - 4178247435000 \)
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| $7$ |
\( T^{5} \)
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| $11$ |
\( T^{5} + \cdots - 40\!\cdots\!20 \)
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| $13$ |
\( T^{5} + \cdots + 19\!\cdots\!84 \)
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| $17$ |
\( T^{5} + \cdots - 27\!\cdots\!80 \)
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| $19$ |
\( T^{5} + \cdots + 16\!\cdots\!04 \)
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| $23$ |
\( T^{5} + \cdots - 12\!\cdots\!80 \)
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| $29$ |
\( T^{5} + \cdots + 51\!\cdots\!88 \)
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| $31$ |
\( T^{5} + \cdots + 24\!\cdots\!81 \)
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| $37$ |
\( T^{5} + \cdots + 56\!\cdots\!88 \)
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| $41$ |
\( T^{5} + \cdots - 10\!\cdots\!00 \)
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| $43$ |
\( T^{5} + \cdots - 21\!\cdots\!80 \)
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| $47$ |
\( T^{5} + \cdots - 15\!\cdots\!40 \)
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| $53$ |
\( T^{5} + \cdots + 70\!\cdots\!56 \)
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| $59$ |
\( T^{5} + \cdots - 80\!\cdots\!80 \)
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| $61$ |
\( T^{5} + \cdots - 58\!\cdots\!00 \)
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| $67$ |
\( T^{5} + \cdots - 26\!\cdots\!00 \)
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| $71$ |
\( T^{5} + \cdots + 37\!\cdots\!12 \)
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| $73$ |
\( T^{5} + \cdots + 15\!\cdots\!44 \)
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| $79$ |
\( T^{5} + \cdots - 73\!\cdots\!41 \)
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| $83$ |
\( T^{5} + \cdots + 83\!\cdots\!56 \)
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| $89$ |
\( T^{5} + \cdots + 34\!\cdots\!84 \)
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| $97$ |
\( T^{5} + \cdots - 10\!\cdots\!28 \)
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