Newspace parameters
| Level: | \( N \) | \(=\) | \( 765 = 3^{2} \cdot 5 \cdot 17 \) |
| Weight: | \( k \) | \(=\) | \( 6 \) |
| Character orbit: | \([\chi]\) | \(=\) | 765.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(122.693622157\) |
| Analytic rank: | \(0\) |
| Dimension: | \(5\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{5} - \cdots)\) |
|
|
|
| Defining polynomial: |
\( x^{5} - 2x^{4} - 95x^{3} + 220x^{2} + 1668x - 4640 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{13}]\) |
| Coefficient ring index: | \( 2 \) |
| Twist minimal: | no (minimal twist has level 85) |
| 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
\( x^{5} - 2x^{4} - 95x^{3} + 220x^{2} + 1668x - 4640 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{4} - 75\nu^{2} + 10\nu + 672 ) / 16 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{4} - 91\nu^{2} - 6\nu + 1296 ) / 16 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( \nu^{4} + 2\nu^{3} - 89\nu^{2} - 122\nu + 1304 ) / 8 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( -\beta_{3} + \beta_{2} - \beta _1 + 39 \)
|
| \(\nu^{3}\) | \(=\) |
\( 4\beta_{4} - 7\beta_{3} - \beta_{2} + 59\beta _1 - 43 \)
|
| \(\nu^{4}\) | \(=\) |
\( -75\beta_{3} + 91\beta_{2} - 85\beta _1 + 2253 \)
|
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 |
|
−7.57023 | 0 | 25.3084 | −25.0000 | 0 | 57.9243 | 50.6569 | 0 | 189.256 | |||||||||||||||||||||||||||||||||
| 1.2 | −3.99434 | 0 | −16.0453 | −25.0000 | 0 | −173.307 | 191.909 | 0 | 99.8584 | ||||||||||||||||||||||||||||||||||
| 1.3 | 4.29890 | 0 | −13.5195 | −25.0000 | 0 | 45.8012 | −195.684 | 0 | −107.472 | ||||||||||||||||||||||||||||||||||
| 1.4 | 4.95319 | 0 | −7.46587 | −25.0000 | 0 | 13.5050 | −195.482 | 0 | −123.830 | ||||||||||||||||||||||||||||||||||
| 1.5 | 9.31248 | 0 | 54.7222 | −25.0000 | 0 | −147.923 | 211.600 | 0 | −232.812 | ||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \( -1 \) |
| \(5\) | \( +1 \) |
| \(17\) | \( +1 \) |
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 | 765.6.a.g | 5 | |
| 3.b | odd | 2 | 1 | 85.6.a.a | ✓ | 5 | |
| 15.d | odd | 2 | 1 | 425.6.a.d | 5 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 85.6.a.a | ✓ | 5 | 3.b | odd | 2 | 1 | |
| 425.6.a.d | 5 | 15.d | odd | 2 | 1 | ||
| 765.6.a.g | 5 | 1.a | even | 1 | 1 | trivial | |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{5} - 7T_{2}^{4} - 77T_{2}^{3} + 483T_{2}^{2} + 956T_{2} - 5996 \)
acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(765))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{5} - 7 T^{4} + \cdots - 5996 \)
|
| $3$ |
\( T^{5} \)
|
| $5$ |
\( (T + 25)^{5} \)
|
| $7$ |
\( T^{5} + 204 T^{4} + \cdots - 918510464 \)
|
| $11$ |
\( T^{5} + \cdots + 6830846460160 \)
|
| $13$ |
\( T^{5} + \cdots - 5736558121912 \)
|
| $17$ |
\( (T + 289)^{5} \)
|
| $19$ |
\( T^{5} + \cdots + 500794756887808 \)
|
| $23$ |
\( T^{5} + \cdots + 17\!\cdots\!60 \)
|
| $29$ |
\( T^{5} + \cdots + 50\!\cdots\!80 \)
|
| $31$ |
\( T^{5} + \cdots - 98\!\cdots\!56 \)
|
| $37$ |
\( T^{5} + \cdots - 42\!\cdots\!68 \)
|
| $41$ |
\( T^{5} + \cdots + 25\!\cdots\!68 \)
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| $43$ |
\( T^{5} + \cdots + 87\!\cdots\!56 \)
|
| $47$ |
\( T^{5} + \cdots - 11\!\cdots\!40 \)
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| $53$ |
\( T^{5} + \cdots - 12\!\cdots\!36 \)
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| $59$ |
\( T^{5} + \cdots - 75\!\cdots\!72 \)
|
| $61$ |
\( T^{5} + \cdots - 28\!\cdots\!00 \)
|
| $67$ |
\( T^{5} + \cdots - 25\!\cdots\!88 \)
|
| $71$ |
\( T^{5} + \cdots + 16\!\cdots\!52 \)
|
| $73$ |
\( T^{5} + \cdots + 17\!\cdots\!64 \)
|
| $79$ |
\( T^{5} + \cdots - 12\!\cdots\!48 \)
|
| $83$ |
\( T^{5} + \cdots + 61\!\cdots\!96 \)
|
| $89$ |
\( T^{5} + \cdots - 93\!\cdots\!40 \)
|
| $97$ |
\( T^{5} + \cdots + 90\!\cdots\!00 \)
|
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