Newspace parameters
| Level: | \( N \) | \(=\) | \( 99 = 3^{2} \cdot 11 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 99.e (of order \(3\), degree \(2\), minimal) |
Newform invariants
| Self dual: | no |
| Analytic conductor: | \(0.790518980011\) |
| Analytic rank: | \(0\) |
| Dimension: | \(8\) |
| Relative dimension: | \(4\) over \(\Q(\zeta_{3})\) |
| Coefficient field: | 8.0.508277025.1 |
| Defining polynomial: |
\( x^{8} - 3x^{7} + 5x^{6} - 15x^{5} + 21x^{4} + 3x^{3} - 22x^{2} + 3x + 19 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{3}]$ |
$q$-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
\( x^{8} - 3x^{7} + 5x^{6} - 15x^{5} + 21x^{4} + 3x^{3} - 22x^{2} + 3x + 19 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -4\nu^{7} + 79\nu^{6} - 177\nu^{5} + 459\nu^{4} - 1008\nu^{3} + 1011\nu^{2} - 752\nu - 478 ) / 933 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 35\nu^{7} + 164\nu^{6} - 395\nu^{5} + 260\nu^{4} - 2687\nu^{3} + 2894\nu^{2} + 1604\nu - 2193 ) / 1866 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 217\nu^{7} + 146\nu^{6} + 39\nu^{5} - 876\nu^{4} - 4095\nu^{3} + 6498\nu^{2} + 1610\nu - 2525 ) / 5598 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 241\nu^{7} - 328\nu^{6} + 1101\nu^{5} - 3630\nu^{4} + 1953\nu^{3} - 5166\nu^{2} + 6122\nu + 343 ) / 5598 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( -145\nu^{7} + 298\nu^{6} - 585\nu^{5} + 1944\nu^{4} - 2019\nu^{3} - 438\nu^{2} + 730\nu + 1799 ) / 1866 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 701\nu^{7} - 1016\nu^{6} + 1863\nu^{5} - 6966\nu^{4} + 2181\nu^{3} + 10122\nu^{2} - 6296\nu - 6265 ) / 5598 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( -\beta_{5} + \beta_{4} - \beta_{2} \)
|
| \(\nu^{3}\) | \(=\) |
\( -3\beta_{7} - 5\beta_{6} - \beta_{5} + \beta_{4} + 2\beta_{2} + \beta _1 + 3 \)
|
| \(\nu^{4}\) | \(=\) |
\( -\beta_{6} + \beta_{5} - \beta_{4} - 3\beta_{3} + 6\beta_{2} + 7\beta_1 \)
|
| \(\nu^{5}\) | \(=\) |
\( 5\beta_{7} + 12\beta_{6} - \beta_{5} + 12\beta_{4} - 8\beta_{3} - 8\beta_{2} - \beta _1 - 14 \)
|
| \(\nu^{6}\) | \(=\) |
\( -29\beta_{7} - 35\beta_{6} - 7\beta_{5} + 32\beta_{4} - \beta_{3} + 2\beta_{2} - 18\beta _1 + 16 \)
|
| \(\nu^{7}\) | \(=\) |
\( -38\beta_{7} - 77\beta_{6} + 20\beta_{5} - 13\beta_{4} - 10\beta_{3} + 92\beta_{2} + 52\beta _1 + 60 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/99\mathbb{Z}\right)^\times\).
| \(n\) | \(46\) | \(56\) |
| \(\chi(n)\) | \(1\) | \(-1 + \beta_{6}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 34.1 |
|
−1.23467 | + | 2.13851i | 1.66933 | − | 0.461883i | −2.04881 | − | 3.54864i | 1.21814 | + | 2.10988i | −1.07333 | + | 4.14015i | −1.16933 | + | 2.02534i | 5.17972 | 2.57333 | − | 1.54207i | −6.01598 | ||||||||||||||||||||||||||||
| 34.2 | −1.07781 | + | 1.86682i | −0.635299 | − | 1.61133i | −1.32333 | − | 2.29208i | −1.81197 | − | 3.13842i | 3.69279 | + | 0.550720i | 1.13530 | − | 1.96640i | 1.39396 | −2.19279 | + | 2.04736i | 7.81179 | |||||||||||||||||||||||||||||
| 34.3 | 0.447217 | − | 0.774602i | 1.22553 | + | 1.22396i | 0.599994 | + | 1.03922i | −1.87447 | − | 3.24667i | 1.49616 | − | 0.401921i | −0.725528 | + | 1.25665i | 2.86218 | 0.00384004 | + | 3.00000i | −3.35317 | |||||||||||||||||||||||||||||
| 34.4 | 1.36526 | − | 2.36469i | 0.240440 | + | 1.71528i | −2.72785 | − | 4.72478i | 0.468293 | + | 0.811107i | 4.38438 | + | 1.77323i | 0.259560 | − | 0.449571i | −9.43585 | −2.88438 | + | 0.824844i | 2.55736 | |||||||||||||||||||||||||||||
| 67.1 | −1.23467 | − | 2.13851i | 1.66933 | + | 0.461883i | −2.04881 | + | 3.54864i | 1.21814 | − | 2.10988i | −1.07333 | − | 4.14015i | −1.16933 | − | 2.02534i | 5.17972 | 2.57333 | + | 1.54207i | −6.01598 | |||||||||||||||||||||||||||||
| 67.2 | −1.07781 | − | 1.86682i | −0.635299 | + | 1.61133i | −1.32333 | + | 2.29208i | −1.81197 | + | 3.13842i | 3.69279 | − | 0.550720i | 1.13530 | + | 1.96640i | 1.39396 | −2.19279 | − | 2.04736i | 7.81179 | |||||||||||||||||||||||||||||
| 67.3 | 0.447217 | + | 0.774602i | 1.22553 | − | 1.22396i | 0.599994 | − | 1.03922i | −1.87447 | + | 3.24667i | 1.49616 | + | 0.401921i | −0.725528 | − | 1.25665i | 2.86218 | 0.00384004 | − | 3.00000i | −3.35317 | |||||||||||||||||||||||||||||
| 67.4 | 1.36526 | + | 2.36469i | 0.240440 | − | 1.71528i | −2.72785 | + | 4.72478i | 0.468293 | − | 0.811107i | 4.38438 | − | 1.77323i | 0.259560 | + | 0.449571i | −9.43585 | −2.88438 | − | 0.824844i | 2.55736 | |||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 9.c | even | 3 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 99.2.e.e | ✓ | 8 |
| 3.b | odd | 2 | 1 | 297.2.e.e | 8 | ||
| 9.c | even | 3 | 1 | inner | 99.2.e.e | ✓ | 8 |
| 9.c | even | 3 | 1 | 891.2.a.q | 4 | ||
| 9.d | odd | 6 | 1 | 297.2.e.e | 8 | ||
| 9.d | odd | 6 | 1 | 891.2.a.p | 4 | ||
| 11.b | odd | 2 | 1 | 1089.2.e.i | 8 | ||
| 99.g | even | 6 | 1 | 9801.2.a.bl | 4 | ||
| 99.h | odd | 6 | 1 | 1089.2.e.i | 8 | ||
| 99.h | odd | 6 | 1 | 9801.2.a.bi | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 99.2.e.e | ✓ | 8 | 1.a | even | 1 | 1 | trivial |
| 99.2.e.e | ✓ | 8 | 9.c | even | 3 | 1 | inner |
| 297.2.e.e | 8 | 3.b | odd | 2 | 1 | ||
| 297.2.e.e | 8 | 9.d | odd | 6 | 1 | ||
| 891.2.a.p | 4 | 9.d | odd | 6 | 1 | ||
| 891.2.a.q | 4 | 9.c | even | 3 | 1 | ||
| 1089.2.e.i | 8 | 11.b | odd | 2 | 1 | ||
| 1089.2.e.i | 8 | 99.h | odd | 6 | 1 | ||
| 9801.2.a.bi | 4 | 99.h | odd | 6 | 1 | ||
| 9801.2.a.bl | 4 | 99.g | even | 6 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{8} + T_{2}^{7} + 10T_{2}^{6} + 7T_{2}^{5} + 76T_{2}^{4} + 46T_{2}^{3} + 181T_{2}^{2} - 104T_{2} + 169 \)
acting on \(S_{2}^{\mathrm{new}}(99, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{8} + T^{7} + 10 T^{6} + 7 T^{5} + \cdots + 169 \)
$3$
\( T^{8} - 5 T^{7} + 15 T^{6} - 35 T^{5} + \cdots + 81 \)
$5$
\( T^{8} + 4 T^{7} + 25 T^{6} + 22 T^{5} + \cdots + 961 \)
$7$
\( T^{8} + T^{7} + 7 T^{6} + 4 T^{5} + \cdots + 16 \)
$11$
\( (T^{2} + T + 1)^{4} \)
$13$
\( T^{8} + 7 T^{7} + 64 T^{6} + \cdots + 24964 \)
$17$
\( (T^{4} + 5 T^{3} - 24 T^{2} - 169 T - 236)^{2} \)
$19$
\( (T^{4} - 9 T^{3} + 81 T - 54)^{2} \)
$23$
\( T^{8} + 14 T^{7} + 145 T^{6} + \cdots + 35344 \)
$29$
\( T^{8} - 6 T^{7} + 45 T^{6} + \cdots + 22500 \)
$31$
\( T^{8} - 2 T^{7} + 25 T^{6} + \cdots + 3481 \)
$37$
\( (T^{4} - 3 T^{3} - 81 T^{2} + 144 T + 57)^{2} \)
$41$
\( T^{8} - 2 T^{7} + 67 T^{6} + 448 T^{5} + \cdots + 676 \)
$43$
\( T^{8} - 21 T^{7} + 285 T^{6} + \cdots + 248004 \)
$47$
\( T^{8} - 7 T^{7} + 64 T^{6} + 101 T^{5} + \cdots + 1 \)
$53$
\( (T^{4} + 6 T^{3} - 45 T^{2} - 165 T - 123)^{2} \)
$59$
\( T^{8} + 2 T^{7} + 25 T^{6} + 56 T^{5} + \cdots + 169 \)
$61$
\( T^{8} + 15 T^{7} + 147 T^{6} + \cdots + 14400 \)
$67$
\( T^{8} + 14 T^{7} + 217 T^{6} + \cdots + 4713241 \)
$71$
\( (T^{4} + 3 T^{3} - 87 T^{2} + 270 T - 195)^{2} \)
$73$
\( (T^{4} - 22 T^{3} + 129 T^{2} + 11 T - 1028)^{2} \)
$79$
\( T^{8} + 11 T^{7} + 145 T^{6} + \cdots + 1600 \)
$83$
\( T^{8} + 18 T^{7} + 327 T^{6} + \cdots + 2396304 \)
$89$
\( (T^{4} + 6 T^{3} - 132 T^{2} - 48 T + 2064)^{2} \)
$97$
\( T^{8} + 26 T^{7} + 520 T^{6} + \cdots + 1510441 \)
show more
show less