Newspace parameters
| Level: | \( N \) | \(=\) | \( 9 = 3^{2} \) |
| Weight: | \( k \) | \(=\) | \( 70 \) |
| Character orbit: | \([\chi]\) | \(=\) | 9.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(271.363457963\) |
| Analytic rank: | \(0\) |
| Dimension: | \(5\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{5} - \cdots)\) |
|
|
|
| Defining polynomial: |
\( x^{5} - x^{4} + \cdots - 94\!\cdots\!36 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| Coefficient ring index: | \( 2^{43}\cdot 3^{24}\cdot 5^{5}\cdot 7^{2}\cdot 17\cdot 23 \) |
| Twist minimal: | no (minimal twist has level 1) |
| 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
\( x^{5} - x^{4} + \cdots - 94\!\cdots\!36 \)
:
| \(\beta_{1}\) | \(=\) |
\( 288\nu - 58 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 28625643 \nu^{4} - 275205114385317 \nu^{3} + \cdots + 12\!\cdots\!44 ) / 12\!\cdots\!28 \)
|
| \(\beta_{3}\) | \(=\) |
\( 82944\nu^{2} - 1539026111808\nu - 828972017711723645683 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 33342194997 \nu^{4} + \cdots - 39\!\cdots\!60 ) / 30\!\cdots\!32 \)
|
| \(\nu\) | \(=\) |
\( ( \beta _1 + 58 ) / 288 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{3} + 5343840666\beta _1 + 828972018021666404311 ) / 82944 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( - 279232 \beta_{4} + 447604165 \beta_{3} - 1300960512 \beta_{2} + \cdots + 13\!\cdots\!03 ) / 746496 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( - 24160668478272 \beta_{4} + \cdots + 11\!\cdots\!51 ) / 6718464 \)
|
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 |
|
−3.33859e10 | 0 | 5.24322e20 | −1.09328e24 | 0 | −1.48686e29 | 2.20259e30 | 0 | 3.65000e34 | |||||||||||||||||||||||||||||||||
| 1.2 | −2.12545e10 | 0 | −1.38540e20 | 2.01597e24 | 0 | 2.05778e29 | 1.54911e31 | 0 | −4.28486e34 | ||||||||||||||||||||||||||||||||||
| 1.3 | 6.72544e9 | 0 | −5.45064e20 | 6.57446e23 | 0 | 1.09182e29 | −7.63579e30 | 0 | 4.42161e33 | ||||||||||||||||||||||||||||||||||
| 1.4 | 1.83774e10 | 0 | −2.52566e20 | −1.12358e24 | 0 | −8.82897e28 | −1.54896e31 | 0 | −2.06484e34 | ||||||||||||||||||||||||||||||||||
| 1.5 | 4.75433e10 | 0 | 1.67007e21 | 1.40726e24 | 0 | −1.18429e27 | 5.13361e31 | 0 | 6.69057e34 | ||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \( -1 \) |
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 | 9.70.a.b | 5 | |
| 3.b | odd | 2 | 1 | 1.70.a.a | ✓ | 5 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 1.70.a.a | ✓ | 5 | 3.b | odd | 2 | 1 | |
| 9.70.a.b | 5 | 1.a | even | 1 | 1 | trivial | |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{5} - 18005734368 T_{2}^{4} + \cdots - 41\!\cdots\!68 \)
acting on \(S_{70}^{\mathrm{new}}(\Gamma_0(9))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{5} + \cdots - 41\!\cdots\!68 \)
|
| $3$ |
\( T^{5} \)
|
| $5$ |
\( T^{5} + \cdots - 22\!\cdots\!00 \)
|
| $7$ |
\( T^{5} + \cdots + 34\!\cdots\!68 \)
|
| $11$ |
\( T^{5} + \cdots + 77\!\cdots\!32 \)
|
| $13$ |
\( T^{5} + \cdots - 72\!\cdots\!76 \)
|
| $17$ |
\( T^{5} + \cdots + 31\!\cdots\!32 \)
|
| $19$ |
\( T^{5} + \cdots - 35\!\cdots\!00 \)
|
| $23$ |
\( T^{5} + \cdots - 72\!\cdots\!24 \)
|
| $29$ |
\( T^{5} + \cdots - 11\!\cdots\!00 \)
|
| $31$ |
\( T^{5} + \cdots + 40\!\cdots\!68 \)
|
| $37$ |
\( T^{5} + \cdots - 13\!\cdots\!32 \)
|
| $41$ |
\( T^{5} + \cdots - 12\!\cdots\!68 \)
|
| $43$ |
\( T^{5} + \cdots + 61\!\cdots\!24 \)
|
| $47$ |
\( T^{5} + \cdots - 32\!\cdots\!68 \)
|
| $53$ |
\( T^{5} + \cdots + 48\!\cdots\!76 \)
|
| $59$ |
\( T^{5} + \cdots - 12\!\cdots\!00 \)
|
| $61$ |
\( T^{5} + \cdots - 16\!\cdots\!32 \)
|
| $67$ |
\( T^{5} + \cdots + 50\!\cdots\!68 \)
|
| $71$ |
\( T^{5} + \cdots - 11\!\cdots\!68 \)
|
| $73$ |
\( T^{5} + \cdots - 37\!\cdots\!76 \)
|
| $79$ |
\( T^{5} + \cdots - 22\!\cdots\!00 \)
|
| $83$ |
\( T^{5} + \cdots - 16\!\cdots\!24 \)
|
| $89$ |
\( T^{5} + \cdots + 10\!\cdots\!00 \)
|
| $97$ |
\( T^{5} + \cdots - 93\!\cdots\!32 \)
|
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