Newspace parameters
| Level: | \( N \) | \(=\) | \( 39 = 3 \cdot 13 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 39.k (of order \(12\), degree \(4\), minimal) |
Newform invariants
| Self dual: | no |
| Analytic conductor: | \(0.311416567883\) |
| Analytic rank: | \(0\) |
| Dimension: | \(8\) |
| Relative dimension: | \(2\) over \(\Q(\zeta_{12})\) |
| Coefficient field: | 8.0.56070144.2 |
| Defining polynomial: |
\( x^{8} - 4x^{7} + 16x^{6} - 34x^{5} + 63x^{4} - 74x^{3} + 70x^{2} - 38x + 13 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{12}]$ |
$q$-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
\( x^{8} - 4x^{7} + 16x^{6} - 34x^{5} + 63x^{4} - 74x^{3} + 70x^{2} - 38x + 13 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -\nu^{7} - 15\nu^{6} + 32\nu^{5} - 172\nu^{4} + 221\nu^{3} - 426\nu^{2} + 235\nu - 159 ) / 37 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -3\nu^{7} - 8\nu^{6} + 22\nu^{5} - 146\nu^{4} + 256\nu^{3} - 390\nu^{2} + 298\nu - 70 ) / 37 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -3\nu^{7} - 8\nu^{6} + 22\nu^{5} - 146\nu^{4} + 256\nu^{3} - 427\nu^{2} + 335\nu - 181 ) / 37 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 3\nu^{7} - 29\nu^{6} + 89\nu^{5} - 261\nu^{4} + 373\nu^{3} - 498\nu^{2} + 294\nu - 152 ) / 37 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( -8\nu^{7} + 28\nu^{6} - 114\nu^{5} + 215\nu^{4} - 378\nu^{3} + 366\nu^{2} - 266\nu + 97 ) / 37 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 17\nu^{7} - 41\nu^{6} + 159\nu^{5} - 184\nu^{4} + 276\nu^{3} - 84\nu^{2} + 38\nu + 39 ) / 37 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( -\beta_{4} + \beta_{3} + \beta _1 - 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{7} + \beta_{6} - \beta_{5} + 2\beta_{3} - 2\beta _1 - 4 \)
|
| \(\nu^{4}\) | \(=\) |
\( 2\beta_{7} + 3\beta_{6} + 6\beta_{4} - 2\beta_{3} - 2\beta_{2} - 6\beta _1 + 7 \)
|
| \(\nu^{5}\) | \(=\) |
\( -4\beta_{7} - 3\beta_{6} + 7\beta_{5} + 6\beta_{4} - 12\beta_{3} - 5\beta_{2} + \beta _1 + 26 \)
|
| \(\nu^{6}\) | \(=\) |
\( -17\beta_{7} - 25\beta_{6} + 3\beta_{5} - 24\beta_{4} - 5\beta_{3} + 7\beta_{2} + 27\beta _1 - 1 \)
|
| \(\nu^{7}\) | \(=\) |
\( 4\beta_{7} - 16\beta_{6} - 42\beta_{5} - 54\beta_{4} + 51\beta_{3} + 42\beta_{2} + 26\beta _1 - 122 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/39\mathbb{Z}\right)^\times\).
| \(n\) | \(14\) | \(28\) |
| \(\chi(n)\) | \(-1\) | \(\beta_{5}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 2.1 |
|
−0.619657 | + | 2.31259i | 1.64914 | − | 0.529480i | −3.23205 | − | 1.86603i | −1.69293 | − | 1.69293i | 0.202571 | + | 4.14187i | −1.36603 | + | 0.366025i | 2.93225 | − | 2.93225i | 2.43930 | − | 1.74637i | 4.96410 | − | 2.86603i | ||||||||||||||||||||||||
| 2.2 | 0.619657 | − | 2.31259i | −1.28311 | + | 1.16345i | −3.23205 | − | 1.86603i | 1.69293 | + | 1.69293i | 1.89551 | + | 3.68825i | −1.36603 | + | 0.366025i | −2.93225 | + | 2.93225i | 0.292748 | − | 2.98568i | 4.96410 | − | 2.86603i | |||||||||||||||||||||||||
| 11.1 | −1.45466 | − | 0.389774i | 0.239203 | − | 1.71545i | 0.232051 | + | 0.133975i | 1.06488 | − | 1.06488i | −1.01660 | + | 2.40216i | 0.366025 | + | 1.36603i | 1.84443 | + | 1.84443i | −2.88556 | − | 0.820682i | −1.96410 | + | 1.13397i | |||||||||||||||||||||||||
| 11.2 | 1.45466 | + | 0.389774i | −1.60523 | − | 0.650571i | 0.232051 | + | 0.133975i | −1.06488 | + | 1.06488i | −2.08148 | − | 1.57203i | 0.366025 | + | 1.36603i | −1.84443 | − | 1.84443i | 2.15351 | + | 2.08863i | −1.96410 | + | 1.13397i | |||||||||||||||||||||||||
| 20.1 | −0.619657 | − | 2.31259i | 1.64914 | + | 0.529480i | −3.23205 | + | 1.86603i | −1.69293 | + | 1.69293i | 0.202571 | − | 4.14187i | −1.36603 | − | 0.366025i | 2.93225 | + | 2.93225i | 2.43930 | + | 1.74637i | 4.96410 | + | 2.86603i | |||||||||||||||||||||||||
| 20.2 | 0.619657 | + | 2.31259i | −1.28311 | − | 1.16345i | −3.23205 | + | 1.86603i | 1.69293 | − | 1.69293i | 1.89551 | − | 3.68825i | −1.36603 | − | 0.366025i | −2.93225 | − | 2.93225i | 0.292748 | + | 2.98568i | 4.96410 | + | 2.86603i | |||||||||||||||||||||||||
| 32.1 | −1.45466 | + | 0.389774i | 0.239203 | + | 1.71545i | 0.232051 | − | 0.133975i | 1.06488 | + | 1.06488i | −1.01660 | − | 2.40216i | 0.366025 | − | 1.36603i | 1.84443 | − | 1.84443i | −2.88556 | + | 0.820682i | −1.96410 | − | 1.13397i | |||||||||||||||||||||||||
| 32.2 | 1.45466 | − | 0.389774i | −1.60523 | + | 0.650571i | 0.232051 | − | 0.133975i | −1.06488 | − | 1.06488i | −2.08148 | + | 1.57203i | 0.366025 | − | 1.36603i | −1.84443 | + | 1.84443i | 2.15351 | − | 2.08863i | −1.96410 | − | 1.13397i | |||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 3.b | odd | 2 | 1 | inner |
| 13.f | odd | 12 | 1 | inner |
| 39.k | even | 12 | 1 | inner |
Twists
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{8} + 6T_{2}^{6} - T_{2}^{4} - 78T_{2}^{2} + 169 \)
acting on \(S_{2}^{\mathrm{new}}(39, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{8} + 6 T^{6} - T^{4} - 78 T^{2} + \cdots + 169 \)
$3$
\( T^{8} + 2 T^{7} - 4 T^{5} - 5 T^{4} + \cdots + 81 \)
$5$
\( T^{8} + 38T^{4} + 169 \)
$7$
\( (T^{4} + 2 T^{3} + 2 T^{2} + 4 T + 4)^{2} \)
$11$
\( T^{8} + 24 T^{6} + 140 T^{4} + \cdots + 2704 \)
$13$
\( (T^{4} - 4 T^{3} + 3 T^{2} - 52 T + 169)^{2} \)
$17$
\( T^{8} + 30 T^{6} + 783 T^{4} + \cdots + 13689 \)
$19$
\( (T^{4} + 8 T^{3} + 20 T^{2} + 16 T + 16)^{2} \)
$23$
\( T^{8} \)
$29$
\( T^{8} - 82 T^{6} + 5151 T^{4} + \cdots + 2474329 \)
$31$
\( (T^{4} - 4 T^{3} + 8 T^{2} + 88 T + 484)^{2} \)
$37$
\( (T^{4} + 14 T^{3} + 113 T^{2} + 592 T + 1369)^{2} \)
$41$
\( T^{8} - 54 T^{6} + 959 T^{4} + \cdots + 169 \)
$43$
\( (T^{4} - 18 T^{3} + 126 T^{2} - 324 T + 324)^{2} \)
$47$
\( T^{8} + 9728 T^{4} + \cdots + 11075584 \)
$53$
\( (T^{4} + 22 T^{2} + 13)^{2} \)
$59$
\( T^{8} + 24 T^{6} - 16 T^{4} + \cdots + 43264 \)
$61$
\( (T^{2} - 7 T + 49)^{4} \)
$67$
\( (T^{4} + 20 T^{3} + 164 T^{2} + 832 T + 2704)^{2} \)
$71$
\( T^{8} - 24 T^{6} - 16 T^{4} + \cdots + 43264 \)
$73$
\( (T^{4} + 14 T^{3} + 98 T^{2} + 154 T + 121)^{2} \)
$79$
\( (T - 2)^{8} \)
$83$
\( T^{8} + 296T^{4} + 2704 \)
$89$
\( T^{8} - 24 T^{6} - 8596 T^{4} + \cdots + 77228944 \)
$97$
\( (T^{4} - 10 T^{3} + 194 T^{2} - 572 T + 484)^{2} \)
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