Newspace parameters
| Level: | \( N \) | \(=\) | \( 801 = 3^{2} \cdot 89 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 801.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(6.39601720190\) |
| Analytic rank: | \(0\) |
| Dimension: | \(7\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{7} - \cdots)\) |
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| Defining polynomial: |
\( x^{7} - 3x^{6} - 5x^{5} + 15x^{4} + 10x^{3} - 20x^{2} - 9x + 3 \)
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| Coefficient ring: | \(\Z[a_1, a_2]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | yes |
| Fricke sign: | \(-1\) |
| Sato-Tate group: | $\mathrm{SU}(2)$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{6}\) 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
\( x^{7} - 3x^{6} - 5x^{5} + 15x^{4} + 10x^{3} - 20x^{2} - 9x + 3 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
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| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - \nu - 2 \)
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| \(\beta_{3}\) | \(=\) |
\( \nu^{3} - 2\nu^{2} - 2\nu + 2 \)
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| \(\beta_{4}\) | \(=\) |
\( \nu^{4} - 3\nu^{3} - 2\nu^{2} + 7\nu + 1 \)
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| \(\beta_{5}\) | \(=\) |
\( \nu^{5} - 3\nu^{4} - 2\nu^{3} + 8\nu^{2} - 3 \)
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| \(\beta_{6}\) | \(=\) |
\( \nu^{6} - 3\nu^{5} - 3\nu^{4} + 10\nu^{3} + 3\nu^{2} - 7\nu - 1 \)
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| \(\nu\) | \(=\) |
\( \beta_1 \)
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| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + \beta _1 + 2 \)
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| \(\nu^{3}\) | \(=\) |
\( \beta_{3} + 2\beta_{2} + 4\beta _1 + 2 \)
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| \(\nu^{4}\) | \(=\) |
\( \beta_{4} + 3\beta_{3} + 8\beta_{2} + 7\beta _1 + 9 \)
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| \(\nu^{5}\) | \(=\) |
\( \beta_{5} + 3\beta_{4} + 11\beta_{3} + 20\beta_{2} + 21\beta _1 + 18 \)
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| \(\nu^{6}\) | \(=\) |
\( \beta_{6} + 3\beta_{5} + 12\beta_{4} + 32\beta_{3} + 61\beta_{2} + 48\beta _1 + 56 \)
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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 |
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−1.76302 | 0 | 1.10824 | 3.07519 | 0 | 0.489655 | 1.57219 | 0 | −5.42162 | |||||||||||||||||||||||||||||||||||||||
| 1.2 | −1.05146 | 0 | −0.894441 | −2.39910 | 0 | −3.99775 | 3.04338 | 0 | 2.52254 | ||||||||||||||||||||||||||||||||||||||||
| 1.3 | −0.551339 | 0 | −1.69603 | 1.39554 | 0 | 2.30331 | 2.03776 | 0 | −0.769417 | ||||||||||||||||||||||||||||||||||||||||
| 1.4 | 0.767997 | 0 | −1.41018 | −1.60239 | 0 | −2.34822 | −2.61901 | 0 | −1.23063 | ||||||||||||||||||||||||||||||||||||||||
| 1.5 | 1.70701 | 0 | 0.913875 | 1.77946 | 0 | 1.81986 | −1.85402 | 0 | 3.03755 | ||||||||||||||||||||||||||||||||||||||||
| 1.6 | 2.34918 | 0 | 3.51863 | 3.06338 | 0 | −1.15148 | 3.56752 | 0 | 7.19642 | ||||||||||||||||||||||||||||||||||||||||
| 1.7 | 2.54163 | 0 | 4.45991 | −1.31208 | 0 | 2.88462 | 6.25219 | 0 | −3.33484 | ||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \( +1 \) |
| \(89\) | \( -1 \) |
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 | 801.2.a.k | yes | 7 |
| 3.b | odd | 2 | 1 | 801.2.a.j | ✓ | 7 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 801.2.a.j | ✓ | 7 | 3.b | odd | 2 | 1 | |
| 801.2.a.k | yes | 7 | 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(801))\):
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\( T_{2}^{7} - 4T_{2}^{6} - 2T_{2}^{5} + 20T_{2}^{4} - 5T_{2}^{3} - 26T_{2}^{2} + 5T_{2} + 8 \)
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\( T_{5}^{7} - 4T_{5}^{6} - 9T_{5}^{5} + 42T_{5}^{4} + 22T_{5}^{3} - 128T_{5}^{2} - 15T_{5} + 118 \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{7} - 4 T^{6} + \cdots + 8 \)
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| $3$ |
\( T^{7} \)
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| $5$ |
\( T^{7} - 4 T^{6} + \cdots + 118 \)
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| $7$ |
\( T^{7} - 20 T^{5} + \cdots + 64 \)
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| $11$ |
\( T^{7} - 16 T^{6} + \cdots - 262 \)
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| $13$ |
\( T^{7} + 2 T^{6} + \cdots - 1028 \)
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| $17$ |
\( T^{7} - 8 T^{6} + \cdots - 11874 \)
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| $19$ |
\( T^{7} + 6 T^{6} + \cdots - 116 \)
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| $23$ |
\( T^{7} - 17 T^{6} + \cdots - 79447 \)
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| $29$ |
\( T^{7} - 17 T^{6} + \cdots - 7509 \)
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| $31$ |
\( T^{7} + 2 T^{6} + \cdots + 7482 \)
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| $37$ |
\( T^{7} - 139 T^{5} + \cdots - 120446 \)
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| $41$ |
\( T^{7} - 13 T^{6} + \cdots + 133989 \)
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| $43$ |
\( T^{7} - 2 T^{6} + \cdots - 371822 \)
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| $47$ |
\( T^{7} - 20 T^{6} + \cdots + 34548 \)
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| $53$ |
\( T^{7} + 2 T^{6} + \cdots + 19932 \)
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| $59$ |
\( T^{7} - 19 T^{6} + \cdots - 1297 \)
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| $61$ |
\( T^{7} - 4 T^{6} + \cdots - 174522 \)
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| $67$ |
\( T^{7} + 11 T^{6} + \cdots + 321 \)
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| $71$ |
\( T^{7} - 36 T^{6} + \cdots + 383728 \)
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| $73$ |
\( T^{7} + 11 T^{6} + \cdots + 53003 \)
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| $79$ |
\( T^{7} + 15 T^{6} + \cdots + 59937 \)
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| $83$ |
\( T^{7} - 19 T^{6} + \cdots - 143991 \)
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| $89$ |
\( (T - 1)^{7} \)
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| $97$ |
\( T^{7} + 17 T^{6} + \cdots - 57747 \)
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