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: | \(4\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{4} - \cdots)\) |
| Defining polynomial: |
\( x^{4} - 75x^{2} - 42x + 736 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2^{2} \) |
| 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 \(1,\beta_1,\beta_2,\beta_3\) 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
\( x^{4} - 75x^{2} - 42x + 736 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{3} - \nu^{2} - 54\nu + 4 ) / 4 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{2} - \nu - 38 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{3} + \beta _1 + 38 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{3} + 4\beta_{2} + 55\beta _1 + 34 \)
|
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 |
|
−8.50608 | 0 | 40.3535 | 86.6918 | 0 | −64.0051 | −71.0553 | 0 | −737.408 | ||||||||||||||||||||||||||||||
| 1.2 | −4.86863 | 0 | −8.29644 | −66.7344 | 0 | −185.299 | 196.188 | 0 | 324.905 | |||||||||||||||||||||||||||||||
| 1.3 | 2.04157 | 0 | −27.8320 | −42.3660 | 0 | 191.647 | −122.151 | 0 | −86.4934 | |||||||||||||||||||||||||||||||
| 1.4 | 7.33314 | 0 | 21.7749 | 0.408582 | 0 | −4.34307 | −74.9818 | 0 | 2.99619 | |||||||||||||||||||||||||||||||
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.e | 4 | |
| 3.b | odd | 2 | 1 | 69.6.a.d | ✓ | 4 | |
| 12.b | even | 2 | 1 | 1104.6.a.o | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 69.6.a.d | ✓ | 4 | 3.b | odd | 2 | 1 | |
| 207.6.a.e | 4 | 1.a | even | 1 | 1 | trivial | |
| 1104.6.a.o | 4 | 12.b | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{4} + 4T_{2}^{3} - 69T_{2}^{2} - 188T_{2} + 620 \)
acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(207))\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{4} + 4 T^{3} - 69 T^{2} - 188 T + 620 \)
$3$
\( T^{4} \)
$5$
\( T^{4} + 22 T^{3} - 6640 T^{2} + \cdots + 100144 \)
$7$
\( T^{4} + 62 T^{3} - 35668 T^{2} + \cdots - 9871616 \)
$11$
\( T^{4} - 1076 T^{3} + \cdots - 45159083072 \)
$13$
\( T^{4} + 396 T^{3} + \cdots + 16813724400 \)
$17$
\( T^{4} + 70 T^{3} + \cdots - 95458629376 \)
$19$
\( T^{4} + 6366 T^{3} + \cdots + 3908943190016 \)
$23$
\( (T - 529)^{4} \)
$29$
\( T^{4} + 3948 T^{3} + \cdots + 61820529282864 \)
$31$
\( T^{4} + \cdots + 378047008189440 \)
$37$
\( T^{4} + \cdots - 684323468629888 \)
$41$
\( T^{4} + 18680 T^{3} + \cdots - 12\!\cdots\!64 \)
$43$
\( T^{4} + 25846 T^{3} + \cdots + 13\!\cdots\!84 \)
$47$
\( T^{4} + 18392 T^{3} + \cdots + 12\!\cdots\!92 \)
$53$
\( T^{4} - 26518 T^{3} + \cdots + 40\!\cdots\!32 \)
$59$
\( T^{4} - 14520 T^{3} + \cdots + 23\!\cdots\!56 \)
$61$
\( T^{4} + 13688 T^{3} + \cdots + 60\!\cdots\!56 \)
$67$
\( T^{4} + 11098 T^{3} + \cdots - 73\!\cdots\!28 \)
$71$
\( T^{4} - 57496 T^{3} + \cdots + 12\!\cdots\!76 \)
$73$
\( T^{4} + 112272 T^{3} + \cdots - 14\!\cdots\!88 \)
$79$
\( T^{4} + 240754 T^{3} + \cdots + 75\!\cdots\!80 \)
$83$
\( T^{4} - 93268 T^{3} + \cdots + 12\!\cdots\!52 \)
$89$
\( T^{4} - 107582 T^{3} + \cdots - 75\!\cdots\!28 \)
$97$
\( T^{4} + 53076 T^{3} + \cdots + 20\!\cdots\!20 \)
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