Newspace parameters
| Level: | \( N \) | \(=\) | \( 7865 = 5 \cdot 11^{2} \cdot 13 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 7865.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(62.8023411897\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Coefficient field: | 6.6.3486377.1 |
|
|
|
| Defining polynomial: |
\( x^{6} - 6x^{4} - x^{3} + 7x^{2} + x - 1 \)
|
| Coefficient ring: | \(\Z[a_1, a_2]\) |
| Coefficient ring index: | \( 2 \) |
| Twist minimal: | no (minimal twist has level 715) |
| Fricke sign: | \(-1\) |
| Sato-Tate group: | $\mathrm{SU}(2)$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{5}\) 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
\( x^{6} - 6x^{4} - x^{3} + 7x^{2} + x - 1 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu^{5} - 5\nu^{3} - \nu^{2} + 2\nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{4} - 5\nu^{2} + 3 \)
|
| \(\beta_{3}\) | \(=\) |
\( -\nu^{5} + 5\nu^{3} + 2\nu^{2} - 3\nu - 2 \)
|
| \(\beta_{4}\) | \(=\) |
\( \nu^{5} - 6\nu^{3} - \nu^{2} + 6\nu + 1 \)
|
| \(\beta_{5}\) | \(=\) |
\( \nu^{5} - \nu^{4} - 5\nu^{3} + 4\nu^{2} + 4\nu - 2 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{5} + \beta_{2} - \beta _1 - 1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{5} + 2\beta_{3} + \beta_{2} + \beta _1 + 3 ) / 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( 2\beta_{5} - \beta_{4} + 2\beta_{2} - \beta _1 - 1 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( 5\beta_{5} + 10\beta_{3} + 7\beta_{2} + 5\beta _1 + 9 ) / 2 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( 19\beta_{5} - 10\beta_{4} + 2\beta_{3} + 19\beta_{2} - 5\beta _1 - 5 ) / 2 \)
|
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 |
|
−1.66974 | 0.417146 | 0.788028 | −1.00000 | −0.696526 | 5.01383 | 2.02368 | −2.82599 | 1.66974 | ||||||||||||||||||||||||||||||||||||
| 1.2 | −0.465241 | 0.943751 | −1.78355 | −1.00000 | −0.439072 | −0.235891 | 1.76026 | −2.10933 | 0.465241 | |||||||||||||||||||||||||||||||||||||
| 1.3 | 0.594807 | −3.26314 | −1.64620 | −1.00000 | −1.94094 | 0.486385 | −2.16879 | 7.64811 | −0.594807 | |||||||||||||||||||||||||||||||||||||
| 1.4 | 1.23213 | −0.651701 | −0.481844 | −1.00000 | −0.802983 | −1.09740 | −3.05797 | −2.57529 | −1.23213 | |||||||||||||||||||||||||||||||||||||
| 1.5 | 2.51998 | 2.48027 | 4.35029 | −1.00000 | 6.25024 | 5.08029 | 5.92267 | 3.15176 | −2.51998 | |||||||||||||||||||||||||||||||||||||
| 1.6 | 2.78806 | −1.92633 | 5.77328 | −1.00000 | −5.37071 | −1.24723 | 10.5201 | 0.710733 | −2.78806 | |||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(5\) | \( +1 \) |
| \(11\) | \( -1 \) |
| \(13\) | \( +1 \) |
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 | 7865.2.a.p | 6 | |
| 11.b | odd | 2 | 1 | 715.2.a.f | ✓ | 6 | |
| 33.d | even | 2 | 1 | 6435.2.a.bk | 6 | ||
| 55.d | odd | 2 | 1 | 3575.2.a.r | 6 | ||
| 143.d | odd | 2 | 1 | 9295.2.a.o | 6 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 715.2.a.f | ✓ | 6 | 11.b | odd | 2 | 1 | |
| 3575.2.a.r | 6 | 55.d | odd | 2 | 1 | ||
| 6435.2.a.bk | 6 | 33.d | even | 2 | 1 | ||
| 7865.2.a.p | 6 | 1.a | even | 1 | 1 | trivial | |
| 9295.2.a.o | 6 | 143.d | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(7865))\):
|
\( T_{2}^{6} - 5T_{2}^{5} + 3T_{2}^{4} + 15T_{2}^{3} - 17T_{2}^{2} - 2T_{2} + 4 \)
|
|
\( T_{3}^{6} + 2T_{3}^{5} - 9T_{3}^{4} - 12T_{3}^{3} + 15T_{3}^{2} + 6T_{3} - 4 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} - 5 T^{5} + \cdots + 4 \)
|
| $3$ |
\( T^{6} + 2 T^{5} + \cdots - 4 \)
|
| $5$ |
\( (T + 1)^{6} \)
|
| $7$ |
\( T^{6} - 8 T^{5} + \cdots - 4 \)
|
| $11$ |
\( T^{6} \)
|
| $13$ |
\( (T + 1)^{6} \)
|
| $17$ |
\( T^{6} - 16 T^{5} + \cdots - 592 \)
|
| $19$ |
\( T^{6} - 4 T^{5} + \cdots - 869 \)
|
| $23$ |
\( T^{6} + 2 T^{5} + \cdots - 6668 \)
|
| $29$ |
\( T^{6} - 2 T^{5} + \cdots + 14528 \)
|
| $31$ |
\( T^{6} - 4 T^{5} + \cdots + 5056 \)
|
| $37$ |
\( T^{6} + 14 T^{5} + \cdots - 31696 \)
|
| $41$ |
\( T^{6} - 18 T^{5} + \cdots + 29183 \)
|
| $43$ |
\( T^{6} - 12 T^{5} + \cdots + 23152 \)
|
| $47$ |
\( T^{6} + 24 T^{5} + \cdots + 592 \)
|
| $53$ |
\( T^{6} + 10 T^{5} + \cdots + 44864 \)
|
| $59$ |
\( T^{6} + 8 T^{5} + \cdots - 64576 \)
|
| $61$ |
\( T^{6} - 16 T^{5} + \cdots + 8128 \)
|
| $67$ |
\( T^{6} - 6 T^{5} + \cdots + 784 \)
|
| $71$ |
\( T^{6} - 10 T^{5} + \cdots + 379328 \)
|
| $73$ |
\( T^{6} - 32 T^{5} + \cdots + 45148 \)
|
| $79$ |
\( T^{6} + 16 T^{5} + \cdots + 36032 \)
|
| $83$ |
\( T^{6} - 22 T^{5} + \cdots + 1732 \)
|
| $89$ |
\( T^{6} + 26 T^{5} + \cdots + 97472 \)
|
| $97$ |
\( T^{6} + 4 T^{5} + \cdots + 2096 \)
|
| show more | |
| show less |