Newspace parameters
| Level: | \( N \) | \(=\) | \( 440 = 2^{3} \cdot 5 \cdot 11 \) |
| Weight: | \( k \) | \(=\) | \( 6 \) |
| Character orbit: | \([\chi]\) | \(=\) | 440.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(70.5688807177\) |
| Analytic rank: | \(1\) |
| Dimension: | \(5\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{5} - \cdots)\) |
|
|
|
| Defining polynomial: |
\( x^{5} - 2x^{4} - 637x^{3} + 4286x^{2} + 73104x - 585216 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{4} \) |
| 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
\( x^{5} - 2x^{4} - 637x^{3} + 4286x^{2} + 73104x - 585216 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -\nu^{4} + 58\nu^{3} + 1261\nu^{2} - 19726\nu - 110688 ) / 1452 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 7\nu^{4} + 78\nu^{3} - 3019\nu^{2} - 10506\nu + 223056 ) / 1452 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -7\nu^{4} - 78\nu^{3} + 3503\nu^{2} + 14862\nu - 348896 ) / 484 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{4} + 3\beta_{3} - 9\beta _1 + 260 \)
|
| \(\nu^{3}\) | \(=\) |
\( -12\beta_{4} - 33\beta_{3} + 21\beta_{2} + 415\beta _1 - 1980 \)
|
| \(\nu^{4}\) | \(=\) |
\( 565\beta_{4} + 1869\beta_{3} - 234\beta_{2} - 7005\beta _1 + 102332 \)
|
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 |
|
0 | −24.5255 | 0 | 25.0000 | 0 | 104.304 | 0 | 358.500 | 0 | |||||||||||||||||||||||||||||||||
| 1.2 | 0 | −12.1845 | 0 | 25.0000 | 0 | 39.3260 | 0 | −94.5390 | 0 | ||||||||||||||||||||||||||||||||||
| 1.3 | 0 | 8.93698 | 0 | 25.0000 | 0 | 189.026 | 0 | −163.130 | 0 | ||||||||||||||||||||||||||||||||||
| 1.4 | 0 | 13.3125 | 0 | 25.0000 | 0 | −84.4680 | 0 | −65.7781 | 0 | ||||||||||||||||||||||||||||||||||
| 1.5 | 0 | 16.4605 | 0 | 25.0000 | 0 | −114.187 | 0 | 27.9478 | 0 | ||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \( -1 \) |
| \(5\) | \( -1 \) |
| \(11\) | \( +1 \) |
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 | 440.6.a.c | ✓ | 5 |
| 4.b | odd | 2 | 1 | 880.6.a.q | 5 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 440.6.a.c | ✓ | 5 | 1.a | even | 1 | 1 | trivial |
| 880.6.a.q | 5 | 4.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{5} - 2T_{3}^{4} - 637T_{3}^{3} + 4286T_{3}^{2} + 73104T_{3} - 585216 \)
acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(440))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{5} \)
|
| $3$ |
\( T^{5} - 2 T^{4} + \cdots - 585216 \)
|
| $5$ |
\( (T - 25)^{5} \)
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| $7$ |
\( T^{5} + \cdots - 7478413056 \)
|
| $11$ |
\( (T + 121)^{5} \)
|
| $13$ |
\( T^{5} + \cdots + 50710246366400 \)
|
| $17$ |
\( T^{5} + \cdots - 374896840310784 \)
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| $19$ |
\( T^{5} + \cdots + 34\!\cdots\!44 \)
|
| $23$ |
\( T^{5} + \cdots + 10\!\cdots\!20 \)
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| $29$ |
\( T^{5} + \cdots - 18\!\cdots\!04 \)
|
| $31$ |
\( T^{5} + \cdots - 93\!\cdots\!48 \)
|
| $37$ |
\( T^{5} + \cdots + 10\!\cdots\!44 \)
|
| $41$ |
\( T^{5} + \cdots - 63\!\cdots\!00 \)
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| $43$ |
\( T^{5} + \cdots + 37\!\cdots\!48 \)
|
| $47$ |
\( T^{5} + \cdots + 40\!\cdots\!32 \)
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| $53$ |
\( T^{5} + \cdots + 27\!\cdots\!92 \)
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| $59$ |
\( T^{5} + \cdots + 74\!\cdots\!00 \)
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| $61$ |
\( T^{5} + \cdots - 33\!\cdots\!56 \)
|
| $67$ |
\( T^{5} + \cdots - 30\!\cdots\!00 \)
|
| $71$ |
\( T^{5} + \cdots + 27\!\cdots\!20 \)
|
| $73$ |
\( T^{5} + \cdots - 70\!\cdots\!16 \)
|
| $79$ |
\( T^{5} + \cdots + 16\!\cdots\!88 \)
|
| $83$ |
\( T^{5} + \cdots + 52\!\cdots\!08 \)
|
| $89$ |
\( T^{5} + \cdots + 12\!\cdots\!20 \)
|
| $97$ |
\( T^{5} + \cdots + 19\!\cdots\!32 \)
|
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