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: | \(8\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{8} - \cdots)\) |
|
|
|
| Defining polynomial: |
\( x^{8} - x^{7} - 11x^{6} + 9x^{5} + 34x^{4} - 13x^{3} - 35x^{2} - 3x + 5 \)
|
| 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 \(1,\beta_1,\ldots,\beta_{7}\) 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
\( x^{8} - x^{7} - 11x^{6} + 9x^{5} + 34x^{4} - 13x^{3} - 35x^{2} - 3x + 5 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 3 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{7} - \nu^{6} - 9\nu^{5} + 10\nu^{4} + 18\nu^{3} - 18\nu^{2} - 10\nu + 4 \)
|
| \(\beta_{4}\) | \(=\) |
\( \nu^{7} - 2\nu^{6} - 10\nu^{5} + 18\nu^{4} + 24\nu^{3} - 31\nu^{2} - 17\nu + 7 \)
|
| \(\beta_{5}\) | \(=\) |
\( \nu^{7} - 2\nu^{6} - 10\nu^{5} + 19\nu^{4} + 24\nu^{3} - 38\nu^{2} - 16\nu + 12 \)
|
| \(\beta_{6}\) | \(=\) |
\( 2\nu^{7} - 3\nu^{6} - 19\nu^{5} + 28\nu^{4} + 41\nu^{3} - 49\nu^{2} - 22\nu + 11 \)
|
| \(\beta_{7}\) | \(=\) |
\( 2\nu^{7} - 3\nu^{6} - 20\nu^{5} + 28\nu^{4} + 49\nu^{3} - 50\nu^{2} - 33\nu + 10 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( -\beta_{6} + \beta_{4} + \beta_{3} + 5\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{5} - \beta_{4} + 7\beta_{2} - \beta _1 + 16 \)
|
| \(\nu^{5}\) | \(=\) |
\( -\beta_{7} - 7\beta_{6} + 8\beta_{4} + 8\beta_{3} - \beta_{2} + 29\beta _1 - 4 \)
|
| \(\nu^{6}\) | \(=\) |
\( \beta_{7} + \beta_{6} + 8\beta_{5} - 11\beta_{4} - \beta_{3} + 44\beta_{2} - 14\beta _1 + 96 \)
|
| \(\nu^{7}\) | \(=\) |
\( -8\beta_{7} - 44\beta_{6} - 2\beta_{5} + 53\beta_{4} + 54\beta_{3} - 17\beta_{2} + 177\beta _1 - 50 \)
|
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 |
|
−2.61443 | −1.48757 | 4.83523 | −1.00000 | 3.88914 | 3.12719 | −7.41251 | −0.787143 | 2.61443 | ||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −1.42417 | 2.07465 | 0.0282518 | −1.00000 | −2.95465 | −0.710933 | 2.80810 | 1.30416 | 1.42417 | |||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −0.769771 | −1.78335 | −1.40745 | −1.00000 | 1.37277 | 4.81089 | 2.62296 | 0.180351 | 0.769771 | |||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −0.652625 | 1.46424 | −1.57408 | −1.00000 | −0.955597 | 0.0215714 | 2.33253 | −0.856014 | 0.652625 | |||||||||||||||||||||||||||||||||||||||||||
| 1.5 | 0.336128 | −2.12071 | −1.88702 | −1.00000 | −0.712829 | −1.25721 | −1.30653 | 1.49741 | −0.336128 | |||||||||||||||||||||||||||||||||||||||||||
| 1.6 | 1.46613 | 0.200225 | 0.149532 | −1.00000 | 0.293556 | 3.52421 | −2.71302 | −2.95991 | −1.46613 | |||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 2.28679 | −0.0897034 | 3.22941 | −1.00000 | −0.205133 | −4.22248 | 2.81139 | −2.99195 | −2.28679 | |||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 2.37195 | −3.25777 | 3.62613 | −1.00000 | −7.72726 | 3.70677 | 3.85709 | 7.61309 | −2.37195 | |||||||||||||||||||||||||||||||||||||||||||
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.u | yes | 8 |
| 11.b | odd | 2 | 1 | 7865.2.a.t | ✓ | 8 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 7865.2.a.t | ✓ | 8 | 11.b | odd | 2 | 1 | |
| 7865.2.a.u | yes | 8 | 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(7865))\):
|
\( T_{2}^{8} - T_{2}^{7} - 11T_{2}^{6} + 9T_{2}^{5} + 34T_{2}^{4} - 13T_{2}^{3} - 35T_{2}^{2} - 3T_{2} + 5 \)
|
|
\( T_{3}^{8} + 5T_{3}^{7} - T_{3}^{6} - 33T_{3}^{5} - 26T_{3}^{4} + 51T_{3}^{3} + 51T_{3}^{2} - 7T_{3} - 1 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{8} - T^{7} - 11 T^{6} + \cdots + 5 \)
|
| $3$ |
\( T^{8} + 5 T^{7} + \cdots - 1 \)
|
| $5$ |
\( (T + 1)^{8} \)
|
| $7$ |
\( T^{8} - 9 T^{7} + \cdots - 16 \)
|
| $11$ |
\( T^{8} \)
|
| $13$ |
\( (T + 1)^{8} \)
|
| $17$ |
\( T^{8} - 4 T^{7} + \cdots - 269 \)
|
| $19$ |
\( T^{8} - 16 T^{7} + \cdots - 406400 \)
|
| $23$ |
\( T^{8} + 9 T^{7} + \cdots + 111535 \)
|
| $29$ |
\( T^{8} + 13 T^{7} + \cdots + 27 \)
|
| $31$ |
\( T^{8} - 224 T^{6} + \cdots + 880092 \)
|
| $37$ |
\( T^{8} + 5 T^{7} + \cdots + 4 \)
|
| $41$ |
\( T^{8} - 13 T^{7} + \cdots - 48796 \)
|
| $43$ |
\( T^{8} - 5 T^{7} + \cdots + 1189 \)
|
| $47$ |
\( T^{8} - 10 T^{7} + \cdots + 117508 \)
|
| $53$ |
\( T^{8} - 14 T^{7} + \cdots + 111563 \)
|
| $59$ |
\( T^{8} + 20 T^{7} + \cdots - 332236 \)
|
| $61$ |
\( T^{8} - 6 T^{7} + \cdots - 151121 \)
|
| $67$ |
\( T^{8} - 4 T^{7} + \cdots - 143540 \)
|
| $71$ |
\( T^{8} + 9 T^{7} + \cdots - 62596 \)
|
| $73$ |
\( T^{8} - 10 T^{7} + \cdots - 44806096 \)
|
| $79$ |
\( T^{8} - 13 T^{7} + \cdots + 734999 \)
|
| $83$ |
\( T^{8} - 26 T^{7} + \cdots + 7754860 \)
|
| $89$ |
\( T^{8} + 6 T^{7} + \cdots + 183680 \)
|
| $97$ |
\( T^{8} - 26 T^{7} + \cdots - 1408 \)
|
| show more | |
| show less |