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: | \(3\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{3} - \cdots)\) |
|
|
|
| Defining polynomial: |
\( x^{3} - x^{2} - 54559x - 2833496 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{12}\cdot 3\cdot 5 \) |
| Twist minimal: | no (minimal twist has level 5) |
| Fricke sign: | \(-1\) |
| Sato-Tate group: | $\mathrm{SU}(2)$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2\) 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
\( x^{3} - x^{2} - 54559x - 2833496 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( -64\nu^{2} - 5184\nu + 2329633 ) / 99 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -48\nu^{2} + 6672\nu + 1743680 ) / 11 \)
|
| \(\nu\) | \(=\) |
\( ( 4\beta_{2} - 27\beta _1 + 1289 ) / 3840 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( -108\beta_{2} - 1251\beta _1 + 46557857 ) / 1280 \)
|
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 | −156099. | 0 | −9.76562e6 | 0 | 2.53688e8 | 0 | 1.39065e10 | 0 | |||||||||||||||||||||||||||
| 1.2 | 0 | 9397.92 | 0 | −9.76562e6 | 0 | 1.90393e8 | 0 | −1.03720e10 | 0 | ||||||||||||||||||||||||||||
| 1.3 | 0 | 94506.8 | 0 | −9.76562e6 | 0 | −1.12850e9 | 0 | −1.52881e9 | 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.e | 3 | |
| 4.b | odd | 2 | 1 | 5.22.a.a | ✓ | 3 | |
| 12.b | even | 2 | 1 | 45.22.a.d | 3 | ||
| 20.d | odd | 2 | 1 | 25.22.a.b | 3 | ||
| 20.e | even | 4 | 2 | 25.22.b.b | 6 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 5.22.a.a | ✓ | 3 | 4.b | odd | 2 | 1 | |
| 25.22.a.b | 3 | 20.d | odd | 2 | 1 | ||
| 25.22.b.b | 6 | 20.e | even | 4 | 2 | ||
| 45.22.a.d | 3 | 12.b | even | 2 | 1 | ||
| 80.22.a.e | 3 | 1.a | even | 1 | 1 | trivial | |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{3} + 52194T_{3}^{2} - 15331235796T_{3} + 138641899095576 \)
acting on \(S_{22}^{\mathrm{new}}(\Gamma_0(80))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{3} \)
|
| $3$ |
\( T^{3} + \cdots + 138641899095576 \)
|
| $5$ |
\( (T + 9765625)^{3} \)
|
| $7$ |
\( T^{3} + \cdots + 54\!\cdots\!88 \)
|
| $11$ |
\( T^{3} + \cdots - 36\!\cdots\!72 \)
|
| $13$ |
\( T^{3} + \cdots - 17\!\cdots\!36 \)
|
| $17$ |
\( T^{3} + \cdots - 18\!\cdots\!88 \)
|
| $19$ |
\( T^{3} + \cdots - 30\!\cdots\!00 \)
|
| $23$ |
\( T^{3} + \cdots - 19\!\cdots\!04 \)
|
| $29$ |
\( T^{3} + \cdots + 85\!\cdots\!00 \)
|
| $31$ |
\( T^{3} + \cdots + 17\!\cdots\!68 \)
|
| $37$ |
\( T^{3} + \cdots - 23\!\cdots\!88 \)
|
| $41$ |
\( T^{3} + \cdots - 96\!\cdots\!88 \)
|
| $43$ |
\( T^{3} + \cdots - 11\!\cdots\!84 \)
|
| $47$ |
\( T^{3} + \cdots - 88\!\cdots\!12 \)
|
| $53$ |
\( T^{3} + \cdots - 87\!\cdots\!76 \)
|
| $59$ |
\( T^{3} + \cdots + 25\!\cdots\!00 \)
|
| $61$ |
\( T^{3} + \cdots + 39\!\cdots\!72 \)
|
| $67$ |
\( T^{3} + \cdots + 33\!\cdots\!88 \)
|
| $71$ |
\( T^{3} + \cdots + 55\!\cdots\!48 \)
|
| $73$ |
\( T^{3} + \cdots + 45\!\cdots\!04 \)
|
| $79$ |
\( T^{3} + \cdots - 60\!\cdots\!00 \)
|
| $83$ |
\( T^{3} + \cdots - 41\!\cdots\!44 \)
|
| $89$ |
\( T^{3} + \cdots + 77\!\cdots\!00 \)
|
| $97$ |
\( T^{3} + \cdots + 27\!\cdots\!12 \)
|
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