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: | \(0\) |
| Dimension: | \(6\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{6} - \cdots)\) |
| Defining polynomial: |
\( x^{6} - 2x^{5} - 149x^{4} + 215x^{3} + 6182x^{2} - 4625x - 79150 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{13}]\) |
| Coefficient ring index: | \( 2\cdot 3^{2}\cdot 5 \) |
| Twist minimal: | no (minimal twist has level 23) |
| 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
\( x^{6} - 2x^{5} - 149x^{4} + 215x^{3} + 6182x^{2} - 4625x - 79150 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -3\nu^{5} + 45\nu^{4} + 274\nu^{3} - 5031\nu^{2} - 2171\nu + 101838 ) / 412 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 3\nu^{5} - 45\nu^{4} - 274\nu^{3} + 5443\nu^{2} + 2171\nu - 122850 ) / 412 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 27\nu^{5} + 7\nu^{4} - 3290\nu^{3} - 2513\nu^{2} + 75571\nu + 106866 ) / 412 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 2\nu^{5} - 30\nu^{4} - 217\nu^{3} + 3457\nu^{2} + 3782\nu - 72424 ) / 103 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{3} + \beta_{2} + 51 \)
|
| \(\nu^{3}\) | \(=\) |
\( -3\beta_{5} + 3\beta_{3} - 5\beta_{2} + 68\beta _1 + 21 \)
|
| \(\nu^{4}\) | \(=\) |
\( -6\beta_{5} + \beta_{4} + 122\beta_{3} + 115\beta_{2} + 3474 \)
|
| \(\nu^{5}\) | \(=\) |
\( -364\beta_{5} + 15\beta_{4} + 427\beta_{3} - 546\beta_{2} + 5487\beta _1 + 2447 \)
|
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 |
|
−10.3365 | 0 | 74.8423 | 32.5264 | 0 | 43.6495 | −442.838 | 0 | −336.208 | ||||||||||||||||||||||||||||||||||||
| 1.2 | −6.29078 | 0 | 7.57397 | 45.2266 | 0 | −206.967 | 153.659 | 0 | −284.511 | |||||||||||||||||||||||||||||||||||||
| 1.3 | −5.81709 | 0 | 1.83858 | −83.3054 | 0 | 84.5782 | 175.452 | 0 | 484.596 | |||||||||||||||||||||||||||||||||||||
| 1.4 | 4.64307 | 0 | −10.4419 | −9.65155 | 0 | 199.091 | −197.061 | 0 | −44.8128 | |||||||||||||||||||||||||||||||||||||
| 1.5 | 5.03655 | 0 | −6.63314 | −108.846 | 0 | −33.8609 | −194.578 | 0 | −548.208 | |||||||||||||||||||||||||||||||||||||
| 1.6 | 8.76471 | 0 | 44.8202 | 82.0499 | 0 | 213.509 | 112.365 | 0 | 719.144 | |||||||||||||||||||||||||||||||||||||
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.g | 6 | |
| 3.b | odd | 2 | 1 | 23.6.a.b | ✓ | 6 | |
| 12.b | even | 2 | 1 | 368.6.a.h | 6 | ||
| 15.d | odd | 2 | 1 | 575.6.a.c | 6 | ||
| 69.c | even | 2 | 1 | 529.6.a.c | 6 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 23.6.a.b | ✓ | 6 | 3.b | odd | 2 | 1 | |
| 207.6.a.g | 6 | 1.a | even | 1 | 1 | trivial | |
| 368.6.a.h | 6 | 12.b | even | 2 | 1 | ||
| 529.6.a.c | 6 | 69.c | even | 2 | 1 | ||
| 575.6.a.c | 6 | 15.d | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{6} + 4T_{2}^{5} - 144T_{2}^{4} - 381T_{2}^{3} + 5928T_{2}^{2} + 7784T_{2} - 77528 \)
acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(207))\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{6} + 4 T^{5} - 144 T^{4} + \cdots - 77528 \)
$3$
\( T^{6} \)
$5$
\( T^{6} + 42 T^{5} + \cdots - 10563094144 \)
$7$
\( T^{6} - 300 T^{5} + \cdots + 1099779388928 \)
$11$
\( T^{6} - 58 T^{5} + \cdots + 97362234604672 \)
$13$
\( T^{6} - 792 T^{5} + \cdots - 83\!\cdots\!88 \)
$17$
\( T^{6} - 400 T^{5} + \cdots - 22\!\cdots\!32 \)
$19$
\( T^{6} - 2738 T^{5} + \cdots - 18\!\cdots\!48 \)
$23$
\( (T + 529)^{6} \)
$29$
\( T^{6} + 11244 T^{5} + \cdots + 15\!\cdots\!00 \)
$31$
\( T^{6} - 13748 T^{5} + \cdots + 76\!\cdots\!36 \)
$37$
\( T^{6} - 25426 T^{5} + \cdots - 24\!\cdots\!28 \)
$41$
\( T^{6} - 14268 T^{5} + \cdots - 63\!\cdots\!16 \)
$43$
\( T^{6} + 18082 T^{5} + \cdots - 11\!\cdots\!28 \)
$47$
\( T^{6} - 23084 T^{5} + \cdots - 16\!\cdots\!00 \)
$53$
\( T^{6} + 17522 T^{5} + \cdots - 60\!\cdots\!96 \)
$59$
\( T^{6} - 36392 T^{5} + \cdots - 35\!\cdots\!00 \)
$61$
\( T^{6} - 27062 T^{5} + \cdots - 14\!\cdots\!32 \)
$67$
\( T^{6} - 37138 T^{5} + \cdots - 47\!\cdots\!72 \)
$71$
\( T^{6} - 158556 T^{5} + \cdots + 63\!\cdots\!32 \)
$73$
\( T^{6} - 112228 T^{5} + \cdots + 30\!\cdots\!00 \)
$79$
\( T^{6} - 36844 T^{5} + \cdots + 19\!\cdots\!92 \)
$83$
\( T^{6} - 76350 T^{5} + \cdots + 75\!\cdots\!36 \)
$89$
\( T^{6} + 16100 T^{5} + \cdots - 71\!\cdots\!00 \)
$97$
\( T^{6} - 259432 T^{5} + \cdots + 29\!\cdots\!00 \)
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