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: | \(3\) |
| Coefficient field: | 3.3.5333.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{3} - x^{2} - 11x + 8 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2 \) |
| 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 \(1,\beta_1,\beta_2\) 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
\( x^{3} - x^{2} - 11x + 8 \)
:
| \(\beta_{1}\) | \(=\) |
\( 2\nu - 1 \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - \nu - 7 \)
|
| \(\nu\) | \(=\) |
\( ( \beta _1 + 1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 2\beta_{2} + \beta _1 + 15 ) / 2 \)
|
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 |
|
−3.49429 | 0 | −19.7900 | 59.5955 | 0 | 96.8268 | 180.969 | 0 | −208.244 | |||||||||||||||||||||||||||
| 1.2 | 1.29009 | 0 | −30.3357 | −59.1123 | 0 | −213.331 | −80.4187 | 0 | −76.2602 | ||||||||||||||||||||||||||||
| 1.3 | 10.2042 | 0 | 72.1256 | 55.5168 | 0 | 2.50462 | 409.450 | 0 | 566.504 | ||||||||||||||||||||||||||||
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.c | 3 | |
| 3.b | odd | 2 | 1 | 69.6.a.b | ✓ | 3 | |
| 12.b | even | 2 | 1 | 1104.6.a.i | 3 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 69.6.a.b | ✓ | 3 | 3.b | odd | 2 | 1 | |
| 207.6.a.c | 3 | 1.a | even | 1 | 1 | trivial | |
| 1104.6.a.i | 3 | 12.b | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{3} - 8T_{2}^{2} - 27T_{2} + 46 \)
acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(207))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{3} - 8 T^{2} + \cdots + 46 \)
|
| $3$ |
\( T^{3} \)
|
| $5$ |
\( T^{3} - 56 T^{2} + \cdots + 195576 \)
|
| $7$ |
\( T^{3} + 114 T^{2} + \cdots + 51736 \)
|
| $11$ |
\( T^{3} - 376 T^{2} + \cdots - 21141352 \)
|
| $13$ |
\( T^{3} + 858 T^{2} + \cdots - 368282376 \)
|
| $17$ |
\( T^{3} - 2548 T^{2} + \cdots + 129112640 \)
|
| $19$ |
\( T^{3} + 2846 T^{2} + \cdots - 4313168 \)
|
| $23$ |
\( (T + 529)^{3} \)
|
| $29$ |
\( T^{3} + \cdots - 117835741080 \)
|
| $31$ |
\( T^{3} + \cdots + 16664141952 \)
|
| $37$ |
\( T^{3} + \cdots - 103469473312 \)
|
| $41$ |
\( T^{3} + \cdots - 213582513784 \)
|
| $43$ |
\( T^{3} + \cdots + 409945701888 \)
|
| $47$ |
\( T^{3} + \cdots - 524672802816 \)
|
| $53$ |
\( T^{3} + \cdots + 30187666172280 \)
|
| $59$ |
\( T^{3} + \cdots + 4070512924224 \)
|
| $61$ |
\( T^{3} + \cdots - 694379910768 \)
|
| $67$ |
\( T^{3} + \cdots + 128918418373088 \)
|
| $71$ |
\( T^{3} + \cdots - 26245560332032 \)
|
| $73$ |
\( T^{3} + \cdots - 119459239092680 \)
|
| $79$ |
\( T^{3} + \cdots - 93978622829240 \)
|
| $83$ |
\( T^{3} + \cdots + 694250483317800 \)
|
| $89$ |
\( T^{3} + \cdots - 102645237296960 \)
|
| $97$ |
\( T^{3} + \cdots - 82905816948920 \)
|
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