Newspace parameters
| Level: | \( N \) | \(=\) | \( 9801 = 3^{4} \cdot 11^{2} \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 9801.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(78.2613790211\) |
| Analytic rank: | \(0\) |
| Dimension: | \(3\) |
| Coefficient field: | 3.3.837.1 |
|
|
|
| Defining polynomial: |
\( x^{3} - 6x - 1 \)
|
| Coefficient ring: | \(\Z[a_1, a_2]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 891) |
| 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
\( x^{3} - 6x - 1 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 4 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
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| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 4 \)
|
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 |
|
−2.52892 | 0 | 4.39543 | −0.528918 | 0 | −0.133492 | −6.05784 | 0 | 1.33759 | |||||||||||||||||||||||||||
| 1.2 | 0.167449 | 0 | −1.97196 | 2.16745 | 0 | −3.80451 | −0.665102 | 0 | 0.362938 | ||||||||||||||||||||||||||||
| 1.3 | 2.36147 | 0 | 3.57653 | 4.36147 | 0 | 3.93800 | 3.72294 | 0 | 10.2995 | ||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \( +1 \) |
| \(11\) | \( -1 \) |
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 | 9801.2.a.bg | 3 | |
| 3.b | odd | 2 | 1 | 9801.2.a.bf | 3 | ||
| 11.b | odd | 2 | 1 | 891.2.a.n | yes | 3 | |
| 33.d | even | 2 | 1 | 891.2.a.m | ✓ | 3 | |
| 99.g | even | 6 | 2 | 891.2.e.s | 6 | ||
| 99.h | odd | 6 | 2 | 891.2.e.r | 6 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 891.2.a.m | ✓ | 3 | 33.d | even | 2 | 1 | |
| 891.2.a.n | yes | 3 | 11.b | odd | 2 | 1 | |
| 891.2.e.r | 6 | 99.h | odd | 6 | 2 | ||
| 891.2.e.s | 6 | 99.g | even | 6 | 2 | ||
| 9801.2.a.bf | 3 | 3.b | odd | 2 | 1 | ||
| 9801.2.a.bg | 3 | 1.a | even | 1 | 1 | trivial | |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(9801))\):
|
\( T_{2}^{3} - 6T_{2} + 1 \)
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\( T_{5}^{3} - 6T_{5}^{2} + 6T_{5} + 5 \)
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\( T_{7}^{3} - 15T_{7} - 2 \)
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\( T_{17}^{3} + 9T_{17}^{2} + 6T_{17} - 20 \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{3} - 6T + 1 \)
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| $3$ |
\( T^{3} \)
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| $5$ |
\( T^{3} - 6 T^{2} + \cdots + 5 \)
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| $7$ |
\( T^{3} - 15T - 2 \)
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| $11$ |
\( T^{3} \)
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| $13$ |
\( T^{3} - 6T + 1 \)
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| $17$ |
\( T^{3} + 9 T^{2} + \cdots - 20 \)
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| $19$ |
\( T^{3} - 3 T^{2} + \cdots + 90 \)
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| $23$ |
\( T^{3} - 21T + 16 \)
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| $29$ |
\( T^{3} + 18 T^{2} + \cdots + 159 \)
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| $31$ |
\( T^{3} + 15 T^{2} + \cdots + 50 \)
|
| $37$ |
\( T^{3} - 9 T^{2} + \cdots + 237 \)
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| $41$ |
\( T^{3} + 3 T^{2} + \cdots - 26 \)
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| $43$ |
\( T^{3} + 6 T^{2} + \cdots - 24 \)
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| $47$ |
\( T^{3} - 9 T^{2} + \cdots + 124 \)
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| $53$ |
\( T^{3} + 3 T^{2} + \cdots - 150 \)
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| $59$ |
\( T^{3} - 6 T^{2} + \cdots + 802 \)
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| $61$ |
\( T^{3} + 3 T^{2} + \cdots - 12 \)
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| $67$ |
\( T^{3} + 21 T^{2} + \cdots - 458 \)
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| $71$ |
\( T^{3} + 3 T^{2} + \cdots + 120 \)
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| $73$ |
\( T^{3} - 3 T^{2} + \cdots + 967 \)
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| $79$ |
\( T^{3} - 3 T^{2} + \cdots - 292 \)
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| $83$ |
\( (T + 6)^{3} \)
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| $89$ |
\( T^{3} - 36 T^{2} + \cdots - 1569 \)
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| $97$ |
\( T^{3} + 6 T^{2} + \cdots - 32 \)
|
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