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: | \(6\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{6} - \cdots)\) |
|
|
|
| Defining polynomial: |
\( x^{6} - x^{5} - 272724629 x^{4} - 165202445971 x^{3} + \cdots + 13\!\cdots\!00 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| Coefficient ring index: | \( 2^{48}\cdot 3^{8}\cdot 5^{5}\cdot 7^{2}\cdot 17 \) |
| 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 \(1,\beta_1,\ldots,\beta_{5}\) 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
\( x^{6} - x^{5} - 272724629 x^{4} - 165202445971 x^{3} + \cdots + 13\!\cdots\!00 \)
:
| \(\beta_{1}\) | \(=\) |
\( 12\nu - 2 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 20164153 \nu^{5} - 55325457126 \nu^{4} + \cdots + 19\!\cdots\!00 ) / 13\!\cdots\!00 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( - 20164153 \nu^{5} + 55325457126 \nu^{4} + \cdots - 64\!\cdots\!00 ) / 34\!\cdots\!00 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 2864534941 \nu^{5} + 15292996183422 \nu^{4} + \cdots + 10\!\cdots\!00 ) / 13\!\cdots\!00 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 6920440171 \nu^{5} + 32300329522482 \nu^{4} + \cdots - 32\!\cdots\!00 ) / 69\!\cdots\!00 \)
|
| \(\nu\) | \(=\) |
\( ( \beta _1 + 2 ) / 12 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{3} + 4\beta_{2} + 10921\beta _1 + 13090782216 ) / 144 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 1480\beta_{5} - 5301\beta_{4} - 44990\beta_{3} + 82863\beta_{2} + 24039774226\beta _1 + 142970547398688 ) / 1728 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( 1597202 \beta_{5} + 2372121 \beta_{4} + 610820429 \beta_{3} + 3876602781 \beta_{2} + \cdots + 65\!\cdots\!08 ) / 432 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( 391529827496 \beta_{5} - 1313544817179 \beta_{4} - 11957422056250 \beta_{3} + 46250854196481 \beta_{2} + \cdots - 39\!\cdots\!52 ) / 1728 \)
|
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 | −171908. | 0 | 9.76562e6 | 0 | 1.29638e9 | 0 | 1.90919e10 | 0 | ||||||||||||||||||||||||||||||||||||
| 1.2 | 0 | −128819. | 0 | 9.76562e6 | 0 | −8.24479e8 | 0 | 6.13398e9 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.3 | 0 | −2746.15 | 0 | 9.76562e6 | 0 | 1.17139e9 | 0 | −1.04528e10 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.4 | 0 | 45038.9 | 0 | 9.76562e6 | 0 | −7.62427e8 | 0 | −8.43185e9 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.5 | 0 | 53796.4 | 0 | 9.76562e6 | 0 | 8.92551e7 | 0 | −7.56630e9 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.6 | 0 | 166466. | 0 | 9.76562e6 | 0 | 4.05836e8 | 0 | 1.72505e10 | 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.l | 6 | |
| 4.b | odd | 2 | 1 | 40.22.a.d | ✓ | 6 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 40.22.a.d | ✓ | 6 | 4.b | odd | 2 | 1 | |
| 80.22.a.l | 6 | 1.a | even | 1 | 1 | trivial | |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{6} + 38172 T_{3}^{5} - 38665220976 T_{3}^{4} - 708468627183552 T_{3}^{3} + \cdots - 24\!\cdots\!00 \)
acting on \(S_{22}^{\mathrm{new}}(\Gamma_0(80))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} \)
|
| $3$ |
\( T^{6} + \cdots - 24\!\cdots\!00 \)
|
| $5$ |
\( (T - 9765625)^{6} \)
|
| $7$ |
\( T^{6} + \cdots + 34\!\cdots\!56 \)
|
| $11$ |
\( T^{6} + \cdots + 30\!\cdots\!16 \)
|
| $13$ |
\( T^{6} + \cdots - 16\!\cdots\!96 \)
|
| $17$ |
\( T^{6} + \cdots - 33\!\cdots\!00 \)
|
| $19$ |
\( T^{6} + \cdots - 21\!\cdots\!24 \)
|
| $23$ |
\( T^{6} + \cdots - 44\!\cdots\!84 \)
|
| $29$ |
\( T^{6} + \cdots + 80\!\cdots\!84 \)
|
| $31$ |
\( T^{6} + \cdots + 10\!\cdots\!00 \)
|
| $37$ |
\( T^{6} + \cdots + 99\!\cdots\!76 \)
|
| $41$ |
\( T^{6} + \cdots - 11\!\cdots\!64 \)
|
| $43$ |
\( T^{6} + \cdots - 60\!\cdots\!56 \)
|
| $47$ |
\( T^{6} + \cdots + 17\!\cdots\!56 \)
|
| $53$ |
\( T^{6} + \cdots - 14\!\cdots\!64 \)
|
| $59$ |
\( T^{6} + \cdots + 43\!\cdots\!96 \)
|
| $61$ |
\( T^{6} + \cdots + 51\!\cdots\!00 \)
|
| $67$ |
\( T^{6} + \cdots - 10\!\cdots\!96 \)
|
| $71$ |
\( T^{6} + \cdots - 10\!\cdots\!00 \)
|
| $73$ |
\( T^{6} + \cdots + 31\!\cdots\!56 \)
|
| $79$ |
\( T^{6} + \cdots - 55\!\cdots\!00 \)
|
| $83$ |
\( T^{6} + \cdots + 69\!\cdots\!36 \)
|
| $89$ |
\( T^{6} + \cdots + 11\!\cdots\!36 \)
|
| $97$ |
\( T^{6} + \cdots + 21\!\cdots\!96 \)
|
| show more | |
| show less |