Newspace parameters
| Level: | \( N \) | \(=\) | \( 80 = 2^{4} \cdot 5 \) |
| Weight: | \( k \) | \(=\) | \( 22 \) |
| Character orbit: | \([\chi]\) | \(=\) | 80.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(223.581875430\) |
| Analytic rank: | \(1\) |
| Dimension: | \(5\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{5} - \cdots)\) |
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|
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| Defining polynomial: |
\( x^{5} - 2x^{4} - 135978135x^{3} - 15857319425x^{2} + 3799092634739500x - 601748185512337500 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{37}\cdot 3^{2}\cdot 5^{4}\cdot 7^{2} \) |
| Twist minimal: | no (minimal twist has level 40) |
| 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
\( x^{5} - 2x^{4} - 135978135x^{3} - 15857319425x^{2} + 3799092634739500x - 601748185512337500 \)
:
| \(\beta_{1}\) | \(=\) |
\( 16\nu - 6 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 1871872 \nu^{4} - 2513071104 \nu^{3} - 172749770640000 \nu^{2} + \cdots + 12\!\cdots\!00 ) / 2600304235425 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 217088 \nu^{4} + 10775030784 \nu^{3} - 30416179824000 \nu^{2} + \cdots + 60\!\cdots\!75 ) / 66674467575 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 1871872 \nu^{4} + 2513071104 \nu^{3} + 305885347493760 \nu^{2} + \cdots - 85\!\cdots\!15 ) / 520060847085 \)
|
| \(\nu\) | \(=\) |
\( ( \beta _1 + 6 ) / 16 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{4} + 5\beta_{2} + 2839\beta _1 + 13924160093 ) / 256 \)
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| \(\nu^{3}\) | \(=\) |
\( ( 7505\beta_{4} + 12339\beta_{3} - 18284\beta_{2} + 10252877235\beta _1 + 19815552822430 ) / 2048 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( 2993492640 \beta_{4} + 66262617 \beta_{3} + 26047646223 \beta_{2} + 26213873076580 \beta _1 + 35\!\cdots\!15 ) / 8192 \)
|
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 | −172242. | 0 | −9.76562e6 | 0 | 3.37861e8 | 0 | 1.92069e10 | 0 | |||||||||||||||||||||||||||||||||
| 1.2 | 0 | −108082. | 0 | −9.76562e6 | 0 | −1.05323e9 | 0 | 1.22131e9 | 0 | ||||||||||||||||||||||||||||||||||
| 1.3 | 0 | −14536.2 | 0 | −9.76562e6 | 0 | 4.47629e8 | 0 | −1.02491e10 | 0 | ||||||||||||||||||||||||||||||||||
| 1.4 | 0 | 92856.1 | 0 | −9.76562e6 | 0 | −9.95100e8 | 0 | −1.83810e9 | 0 | ||||||||||||||||||||||||||||||||||
| 1.5 | 0 | 141982. | 0 | −9.76562e6 | 0 | 1.05429e9 | 0 | 9.69849e9 | 0 | ||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \( +1 \) |
| \(5\) | \( +1 \) |
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 | 80.22.a.i | 5 | |
| 4.b | odd | 2 | 1 | 40.22.a.c | ✓ | 5 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 40.22.a.c | ✓ | 5 | 4.b | odd | 2 | 1 | |
| 80.22.a.i | 5 | 1.a | even | 1 | 1 | trivial | |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{5} + 60022 T_{3}^{4} - 33369346776 T_{3}^{3} + \cdots + 35\!\cdots\!60 \)
acting on \(S_{22}^{\mathrm{new}}(\Gamma_0(80))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{5} \)
|
| $3$ |
\( T^{5} + \cdots + 35\!\cdots\!60 \)
|
| $5$ |
\( (T + 9765625)^{5} \)
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| $7$ |
\( T^{5} + \cdots - 16\!\cdots\!36 \)
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| $11$ |
\( T^{5} + \cdots + 35\!\cdots\!04 \)
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| $13$ |
\( T^{5} + \cdots - 44\!\cdots\!52 \)
|
| $17$ |
\( T^{5} + \cdots + 82\!\cdots\!00 \)
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| $19$ |
\( T^{5} + \cdots - 80\!\cdots\!56 \)
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| $23$ |
\( T^{5} + \cdots + 58\!\cdots\!96 \)
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| $29$ |
\( T^{5} + \cdots - 29\!\cdots\!08 \)
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| $31$ |
\( T^{5} + \cdots + 83\!\cdots\!00 \)
|
| $37$ |
\( T^{5} + \cdots + 29\!\cdots\!96 \)
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| $41$ |
\( T^{5} + \cdots + 53\!\cdots\!56 \)
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| $43$ |
\( T^{5} + \cdots + 32\!\cdots\!68 \)
|
| $47$ |
\( T^{5} + \cdots - 15\!\cdots\!64 \)
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| $53$ |
\( T^{5} + \cdots - 44\!\cdots\!44 \)
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| $59$ |
\( T^{5} + \cdots + 27\!\cdots\!76 \)
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| $61$ |
\( T^{5} + \cdots - 12\!\cdots\!00 \)
|
| $67$ |
\( T^{5} + \cdots - 29\!\cdots\!88 \)
|
| $71$ |
\( T^{5} + \cdots - 34\!\cdots\!20 \)
|
| $73$ |
\( T^{5} + \cdots + 69\!\cdots\!24 \)
|
| $79$ |
\( T^{5} + \cdots - 59\!\cdots\!80 \)
|
| $83$ |
\( T^{5} + \cdots - 13\!\cdots\!04 \)
|
| $89$ |
\( T^{5} + \cdots + 34\!\cdots\!56 \)
|
| $97$ |
\( T^{5} + \cdots - 15\!\cdots\!64 \)
|
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