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: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{4} - \cdots)\) |
|
|
|
| Defining polynomial: |
\( x^{4} - x^{3} - 39396895x^{2} + 10754000272x + 339264711383190 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| Coefficient ring index: | \( 2^{24}\cdot 3^{2}\cdot 5^{2}\cdot 7 \) |
| Twist minimal: | no (minimal twist has level 20) |
| Fricke sign: | \(-1\) |
| Sato-Tate group: | $\mathrm{SU}(2)$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) 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
\( x^{4} - x^{3} - 39396895x^{2} + 10754000272x + 339264711383190 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( 3152\nu^{3} + 18028576\nu^{2} - 56806176576\nu - 329791438416320 ) / 386602825 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -1800368\nu^{3} - 5273737184\nu^{2} + 36096552309184\nu + 89407740439146880 ) / 386602825 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -24125712\nu^{3} - 107849213856\nu^{2} + 539834093729856\nu + 1930454069052793920 ) / 386602825 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{3} - 6\beta_{2} + 4227\beta _1 + 53760 ) / 215040 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( -1453\beta_{3} + 41814\beta_{2} + 12762033\beta _1 + 8471908408320 ) / 430080 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 3168235\beta_{3} - 32530842\beta_{2} + 9436826457\beta _1 - 246864461798400 ) / 30720 \)
|
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 | −142328. | 0 | −9.76562e6 | 0 | −7.69825e8 | 0 | 9.79697e9 | 0 | ||||||||||||||||||||||||||||||
| 1.2 | 0 | −5365.82 | 0 | −9.76562e6 | 0 | 1.42026e9 | 0 | −1.04316e10 | 0 | |||||||||||||||||||||||||||||||
| 1.3 | 0 | 31390.6 | 0 | −9.76562e6 | 0 | −2.16499e8 | 0 | −9.47498e9 | 0 | |||||||||||||||||||||||||||||||
| 1.4 | 0 | 199543. | 0 | −9.76562e6 | 0 | −4.14672e8 | 0 | 2.93572e10 | 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.h | 4 | |
| 4.b | odd | 2 | 1 | 20.22.a.b | ✓ | 4 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 20.22.a.b | ✓ | 4 | 4.b | odd | 2 | 1 | |
| 80.22.a.h | 4 | 1.a | even | 1 | 1 | trivial | |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{4} - 83240T_{3}^{3} - 27080079528T_{3}^{2} + 748758571198560T_{3} + 4783704534323137296 \)
acting on \(S_{22}^{\mathrm{new}}(\Gamma_0(80))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} \)
|
| $3$ |
\( T^{4} + \cdots + 47\!\cdots\!96 \)
|
| $5$ |
\( (T + 9765625)^{4} \)
|
| $7$ |
\( T^{4} + \cdots - 98\!\cdots\!24 \)
|
| $11$ |
\( T^{4} + \cdots - 10\!\cdots\!00 \)
|
| $13$ |
\( T^{4} + \cdots - 18\!\cdots\!64 \)
|
| $17$ |
\( T^{4} + \cdots + 62\!\cdots\!76 \)
|
| $19$ |
\( T^{4} + \cdots + 60\!\cdots\!16 \)
|
| $23$ |
\( T^{4} + \cdots + 13\!\cdots\!36 \)
|
| $29$ |
\( T^{4} + \cdots - 26\!\cdots\!84 \)
|
| $31$ |
\( T^{4} + \cdots + 25\!\cdots\!56 \)
|
| $37$ |
\( T^{4} + \cdots + 12\!\cdots\!56 \)
|
| $41$ |
\( T^{4} + \cdots - 33\!\cdots\!24 \)
|
| $43$ |
\( T^{4} + \cdots + 44\!\cdots\!00 \)
|
| $47$ |
\( T^{4} + \cdots - 57\!\cdots\!04 \)
|
| $53$ |
\( T^{4} + \cdots + 55\!\cdots\!16 \)
|
| $59$ |
\( T^{4} + \cdots + 73\!\cdots\!96 \)
|
| $61$ |
\( T^{4} + \cdots - 17\!\cdots\!64 \)
|
| $67$ |
\( T^{4} + \cdots + 49\!\cdots\!56 \)
|
| $71$ |
\( T^{4} + \cdots - 24\!\cdots\!84 \)
|
| $73$ |
\( T^{4} + \cdots + 47\!\cdots\!36 \)
|
| $79$ |
\( T^{4} + \cdots - 54\!\cdots\!64 \)
|
| $83$ |
\( T^{4} + \cdots + 44\!\cdots\!96 \)
|
| $89$ |
\( T^{4} + \cdots - 35\!\cdots\!64 \)
|
| $97$ |
\( T^{4} + \cdots - 10\!\cdots\!64 \)
|
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