Newspace parameters
| Level: | \( N \) | \(=\) | \( 207 = 3^{2} \cdot 23 \) |
| Weight: | \( k \) | \(=\) | \( 6 \) |
| Character orbit: | \([\chi]\) | \(=\) | 207.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(33.1994507013\) |
| Analytic rank: | \(1\) |
| Dimension: | \(5\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{5} - \cdots)\) |
| Defining polynomial: |
\( x^{5} - 113x^{3} - 257x^{2} + 1404x + 2197 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2^{5} \) |
| Twist minimal: | no (minimal twist has level 69) |
| 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
\( x^{5} - 113x^{3} - 257x^{2} + 1404x + 2197 \)
:
| \(\beta_{1}\) | \(=\) |
\( 2\nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 7\nu^{4} - 13\nu^{3} - 700\nu^{2} - 499\nu + 5941 ) / 468 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -\nu^{4} + 13\nu^{3} - 56\nu^{2} - 419\nu + 4511 ) / 117 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 23\nu^{4} - 221\nu^{3} - 1676\nu^{2} + 11821\nu + 18005 ) / 468 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( -\beta_{4} - 4\beta_{3} + \beta_{2} + 6\beta _1 + 180 ) / 4 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( -7\beta_{4} - 7\beta_{3} + 19\beta_{2} + 86\beta _1 + 298 ) / 2 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( -63\beta_{4} - 213\beta_{3} + 219\beta_{2} + 531\beta _1 + 7856 ) / 2 \)
|
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 |
|
−9.34908 | 0 | 55.4053 | −70.5203 | 0 | −89.7132 | −218.818 | 0 | 659.300 | |||||||||||||||||||||||||||||||||
| 1.2 | −9.27315 | 0 | 53.9914 | 37.4928 | 0 | 154.850 | −203.929 | 0 | −347.676 | ||||||||||||||||||||||||||||||||||
| 1.3 | −2.24792 | 0 | −26.9469 | −53.3906 | 0 | 89.8688 | 132.508 | 0 | 120.018 | ||||||||||||||||||||||||||||||||||
| 1.4 | 3.54286 | 0 | −19.4482 | 92.1306 | 0 | −8.98894 | −182.273 | 0 | 326.406 | ||||||||||||||||||||||||||||||||||
| 1.5 | 9.32729 | 0 | 54.9984 | −99.7124 | 0 | 125.983 | 214.513 | 0 | −930.047 | ||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \(-1\) |
| \(23\) | \(-1\) |
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 | 207.6.a.f | 5 | |
| 3.b | odd | 2 | 1 | 69.6.a.e | ✓ | 5 | |
| 12.b | even | 2 | 1 | 1104.6.a.r | 5 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 69.6.a.e | ✓ | 5 | 3.b | odd | 2 | 1 | |
| 207.6.a.f | 5 | 1.a | even | 1 | 1 | trivial | |
| 1104.6.a.r | 5 | 12.b | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{5} + 8T_{2}^{4} - 107T_{2}^{3} - 770T_{2}^{2} + 1740T_{2} + 6440 \)
acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(207))\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{5} + 8 T^{4} - 107 T^{3} + \cdots + 6440 \)
$3$
\( T^{5} \)
$5$
\( T^{5} + 94 T^{4} + \cdots + 1296820224 \)
$7$
\( T^{5} - 272 T^{4} + \cdots - 1413831904 \)
$11$
\( T^{5} + 1100 T^{4} + \cdots - 5578745996288 \)
$13$
\( T^{5} + 978 T^{4} + \cdots - 2474410088160 \)
$17$
\( T^{5} + 2522 T^{4} + \cdots - 13327498013440 \)
$19$
\( T^{5} - 2060 T^{4} + \cdots + 30\!\cdots\!60 \)
$23$
\( (T - 529)^{5} \)
$29$
\( T^{5} + 1526 T^{4} + \cdots - 32\!\cdots\!40 \)
$31$
\( T^{5} + 7392 T^{4} + \cdots + 16\!\cdots\!48 \)
$37$
\( T^{5} + 8210 T^{4} + \cdots + 84\!\cdots\!80 \)
$41$
\( T^{5} + 21250 T^{4} + \cdots + 22\!\cdots\!12 \)
$43$
\( T^{5} + 4548 T^{4} + \cdots - 23\!\cdots\!72 \)
$47$
\( T^{5} + 536 T^{4} + \cdots - 94\!\cdots\!64 \)
$53$
\( T^{5} - 11482 T^{4} + \cdots - 77\!\cdots\!36 \)
$59$
\( T^{5} + 74676 T^{4} + \cdots + 10\!\cdots\!88 \)
$61$
\( T^{5} + 44618 T^{4} + \cdots + 11\!\cdots\!60 \)
$67$
\( T^{5} + 1412 T^{4} + \cdots - 11\!\cdots\!24 \)
$71$
\( T^{5} + 37912 T^{4} + \cdots + 42\!\cdots\!28 \)
$73$
\( T^{5} - 46546 T^{4} + \cdots + 27\!\cdots\!68 \)
$79$
\( T^{5} - 50544 T^{4} + \cdots - 39\!\cdots\!64 \)
$83$
\( T^{5} + 89588 T^{4} + \cdots + 29\!\cdots\!24 \)
$89$
\( T^{5} + 280410 T^{4} + \cdots - 28\!\cdots\!76 \)
$97$
\( T^{5} - 90074 T^{4} + \cdots + 55\!\cdots\!56 \)
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