Newspace parameters
| Level: | \( N \) | \(=\) | \( 51 = 3 \cdot 17 \) |
| Weight: | \( k \) | \(=\) | \( 8 \) |
| Character orbit: | \([\chi]\) | \(=\) | 51.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(15.9316362997\) |
| Analytic rank: | \(1\) |
| Dimension: | \(3\) |
| Coefficient field: | 3.3.1514860.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{3} - x^{2} - 300x + 1512 \)
|
| Coefficient ring: | \(\Z[a_1, a_2]\) |
| Coefficient ring index: | \( 2\cdot 3 \) |
| Twist minimal: | yes |
| 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
\( x^{3} - x^{2} - 300x + 1512 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} + 6\nu - 202 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} - 6\beta _1 + 202 \)
|
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 |
|
−16.4988 | −27.0000 | 144.209 | −128.216 | 445.466 | −668.207 | −267.428 | 729.000 | 2115.41 | |||||||||||||||||||||||||||
| 1.2 | −7.49149 | −27.0000 | −71.8775 | 33.8265 | 202.270 | 970.589 | 1497.38 | 729.000 | −253.411 | ||||||||||||||||||||||||||||
| 1.3 | 16.9902 | −27.0000 | 160.669 | −345.610 | −458.737 | 505.618 | 555.047 | 729.000 | −5872.00 | ||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \(1\) |
| \(17\) | \(-1\) |
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 | 51.8.a.b | ✓ | 3 |
| 3.b | odd | 2 | 1 | 153.8.a.d | 3 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 51.8.a.b | ✓ | 3 | 1.a | even | 1 | 1 | trivial |
| 153.8.a.d | 3 | 3.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{3} + 7T_{2}^{2} - 284T_{2} - 2100 \)
acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(51))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{3} + 7 T^{2} + \cdots - 2100 \)
|
| $3$ |
\( (T + 27)^{3} \)
|
| $5$ |
\( T^{3} + 440 T^{2} + \cdots - 1498950 \)
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| $7$ |
\( T^{3} - 808 T^{2} + \cdots + 327920512 \)
|
| $11$ |
\( T^{3} + \cdots - 59443640748 \)
|
| $13$ |
\( T^{3} + \cdots - 125687205062 \)
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| $17$ |
\( (T - 4913)^{3} \)
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| $19$ |
\( T^{3} + \cdots + 4487084034916 \)
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| $23$ |
\( T^{3} + \cdots + 64362267945912 \)
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| $29$ |
\( T^{3} + \cdots - 410683043347704 \)
|
| $31$ |
\( T^{3} + \cdots + 430651093577152 \)
|
| $37$ |
\( T^{3} + \cdots + 79\!\cdots\!28 \)
|
| $41$ |
\( T^{3} + \cdots - 23\!\cdots\!82 \)
|
| $43$ |
\( T^{3} + \cdots - 80\!\cdots\!16 \)
|
| $47$ |
\( T^{3} + \cdots - 14\!\cdots\!32 \)
|
| $53$ |
\( T^{3} + \cdots - 23\!\cdots\!52 \)
|
| $59$ |
\( T^{3} + \cdots + 90\!\cdots\!60 \)
|
| $61$ |
\( T^{3} + \cdots - 10\!\cdots\!00 \)
|
| $67$ |
\( T^{3} + \cdots + 11\!\cdots\!80 \)
|
| $71$ |
\( T^{3} + \cdots + 96\!\cdots\!88 \)
|
| $73$ |
\( T^{3} + \cdots + 49\!\cdots\!76 \)
|
| $79$ |
\( T^{3} + \cdots + 34\!\cdots\!40 \)
|
| $83$ |
\( T^{3} + \cdots - 52\!\cdots\!68 \)
|
| $89$ |
\( T^{3} + \cdots - 18\!\cdots\!20 \)
|
| $97$ |
\( T^{3} + \cdots - 17\!\cdots\!20 \)
|
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