Newspace parameters
| Level: | \( N \) | \(=\) | \( 9 = 3^{2} \) |
| Weight: | \( k \) | \(=\) | \( 44 \) |
| Character orbit: | \([\chi]\) | \(=\) | 9.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(105.399355811\) |
| Analytic rank: | \(1\) |
| Dimension: | \(4\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{4} - \cdots)\) |
|
|
|
| Defining polynomial: |
\( x^{4} - x^{3} - 886516819907x^{2} - 42308083143723387x + 94580276745082867224894 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2^{14}\cdot 3^{10}\cdot 5^{2}\cdot 7\cdot 11 \) |
| Twist minimal: | no (minimal twist has level 3) |
| 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
\( x^{4} - x^{3} - 886516819907x^{2} - 42308083143723387x + 94580276745082867224894 \)
:
| \(\beta_{1}\) | \(=\) |
\( 6\nu - 1 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 27\nu^{3} - 2660112\nu^{2} - 17984400403617\nu + 322364875892989014 ) / 43072 \)
|
| \(\beta_{3}\) | \(=\) |
\( 36\nu^{2} - 2577138\nu - 15957302114051 \)
|
| \(\nu\) | \(=\) |
\( ( \beta _1 + 1 ) / 6 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{3} + 429523\beta _1 + 15957302543574 ) / 36 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 147784\beta_{3} + 86144\beta_{2} + 6058276761571\beta _1 + 1713510242113696527 ) / 54 \)
|
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 |
|
−5.84038e6 | 0 | 2.53139e13 | −1.12222e15 | 0 | −1.55171e18 | −9.64704e19 | 0 | 6.55417e21 | ||||||||||||||||||||||||||||||
| 1.2 | −2.34437e6 | 0 | −3.30001e12 | 6.18875e14 | 0 | −6.01464e16 | 2.83578e19 | 0 | −1.45087e21 | |||||||||||||||||||||||||||||||
| 1.3 | 1.91590e6 | 0 | −5.12540e12 | −2.05854e15 | 0 | 1.37114e18 | −2.66723e19 | 0 | −3.94396e21 | |||||||||||||||||||||||||||||||
| 1.4 | 4.60883e6 | 0 | 1.24452e13 | 9.11231e14 | 0 | 3.55193e17 | 1.68183e19 | 0 | 4.19971e21 | |||||||||||||||||||||||||||||||
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 | 9.44.a.c | 4 | |
| 3.b | odd | 2 | 1 | 3.44.a.b | ✓ | 4 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 3.44.a.b | ✓ | 4 | 3.b | odd | 2 | 1 | |
| 9.44.a.c | 4 | 1.a | even | 1 | 1 | trivial | |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{4} + 1660014T_{2}^{3} - 30881238086592T_{2}^{2} - 17064803544699174912T_{2} + 120901670049507055201419264 \)
acting on \(S_{44}^{\mathrm{new}}(\Gamma_0(9))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} + \cdots + 12\!\cdots\!64 \)
|
| $3$ |
\( T^{4} \)
|
| $5$ |
\( T^{4} + \cdots + 13\!\cdots\!00 \)
|
| $7$ |
\( T^{4} + \cdots + 45\!\cdots\!00 \)
|
| $11$ |
\( T^{4} + \cdots + 16\!\cdots\!64 \)
|
| $13$ |
\( T^{4} + \cdots + 14\!\cdots\!84 \)
|
| $17$ |
\( T^{4} + \cdots + 53\!\cdots\!96 \)
|
| $19$ |
\( T^{4} + \cdots - 13\!\cdots\!00 \)
|
| $23$ |
\( T^{4} + \cdots - 36\!\cdots\!00 \)
|
| $29$ |
\( T^{4} + \cdots - 27\!\cdots\!00 \)
|
| $31$ |
\( T^{4} + \cdots - 22\!\cdots\!00 \)
|
| $37$ |
\( T^{4} + \cdots + 12\!\cdots\!00 \)
|
| $41$ |
\( T^{4} + \cdots + 18\!\cdots\!00 \)
|
| $43$ |
\( T^{4} + \cdots + 11\!\cdots\!36 \)
|
| $47$ |
\( T^{4} + \cdots + 24\!\cdots\!36 \)
|
| $53$ |
\( T^{4} + \cdots + 58\!\cdots\!00 \)
|
| $59$ |
\( T^{4} + \cdots + 67\!\cdots\!00 \)
|
| $61$ |
\( T^{4} + \cdots + 26\!\cdots\!76 \)
|
| $67$ |
\( T^{4} + \cdots + 28\!\cdots\!96 \)
|
| $71$ |
\( T^{4} + \cdots + 34\!\cdots\!76 \)
|
| $73$ |
\( T^{4} + \cdots + 56\!\cdots\!00 \)
|
| $79$ |
\( T^{4} + \cdots + 66\!\cdots\!00 \)
|
| $83$ |
\( T^{4} + \cdots - 23\!\cdots\!36 \)
|
| $89$ |
\( T^{4} + \cdots + 44\!\cdots\!00 \)
|
| $97$ |
\( T^{4} + \cdots - 42\!\cdots\!64 \)
|
| show more | |
| show less |