Newspace parameters
| Level: | \( N \) | \(=\) | \( 51 = 3 \cdot 17 \) |
| Weight: | \( k \) | \(=\) | \( 6 \) |
| Character orbit: | \([\chi]\) | \(=\) | 51.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(8.17957481046\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{4} - \cdots)\) |
|
|
|
| Defining polynomial: |
\( x^{4} - x^{3} - 117x^{2} + 299x + 1026 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 1 \) |
| 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 \(1,\beta_1,\beta_2,\beta_3\) 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
\( x^{4} - x^{3} - 117x^{2} + 299x + 1026 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{3} + 10\nu^{2} - 75\nu - 424 ) / 34 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -2\nu^{3} - 3\nu^{2} + 184\nu - 138 ) / 17 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{3} + 4\beta_{2} - 2\beta _1 + 58 \)
|
| \(\nu^{3}\) | \(=\) |
\( -10\beta_{3} - 6\beta_{2} + 95\beta _1 - 156 \)
|
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 |
|
−10.1538 | 9.00000 | 71.0987 | 87.4534 | −91.3838 | −148.200 | −396.998 | 81.0000 | −887.980 | ||||||||||||||||||||||||||||||
| 1.2 | −0.977811 | 9.00000 | −31.0439 | −8.49105 | −8.80030 | 164.461 | 61.6450 | 81.0000 | 8.30264 | |||||||||||||||||||||||||||||||
| 1.3 | 6.21760 | 9.00000 | 6.65857 | 86.8235 | 55.9584 | 10.7167 | −157.563 | 81.0000 | 539.834 | |||||||||||||||||||||||||||||||
| 1.4 | 9.91396 | 9.00000 | 66.2866 | −19.7859 | 89.2257 | 33.0231 | 339.916 | 81.0000 | −196.156 | |||||||||||||||||||||||||||||||
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 | 51.6.a.c | ✓ | 4 |
| 3.b | odd | 2 | 1 | 153.6.a.e | 4 | ||
| 4.b | odd | 2 | 1 | 816.6.a.o | 4 | ||
| 17.b | even | 2 | 1 | 867.6.a.f | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 51.6.a.c | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
| 153.6.a.e | 4 | 3.b | odd | 2 | 1 | ||
| 816.6.a.o | 4 | 4.b | odd | 2 | 1 | ||
| 867.6.a.f | 4 | 17.b | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{4} - 5T_{2}^{3} - 108T_{2}^{2} + 526T_{2} + 612 \)
acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(51))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} - 5 T^{3} + \cdots + 612 \)
|
| $3$ |
\( (T - 9)^{4} \)
|
| $5$ |
\( T^{4} - 146 T^{3} + \cdots + 1275648 \)
|
| $7$ |
\( T^{4} - 60 T^{3} + \cdots - 8625600 \)
|
| $11$ |
\( T^{4} + \cdots - 3437913600 \)
|
| $13$ |
\( T^{4} + \cdots + 29011393972 \)
|
| $17$ |
\( (T + 289)^{4} \)
|
| $19$ |
\( T^{4} + \cdots - 3576219089616 \)
|
| $23$ |
\( T^{4} + \cdots + 5662798513668 \)
|
| $29$ |
\( T^{4} + \cdots - 133725174557184 \)
|
| $31$ |
\( T^{4} + \cdots - 15\!\cdots\!76 \)
|
| $37$ |
\( T^{4} + \cdots + 10\!\cdots\!36 \)
|
| $41$ |
\( T^{4} + \cdots + 47677475772324 \)
|
| $43$ |
\( T^{4} + \cdots - 40\!\cdots\!92 \)
|
| $47$ |
\( T^{4} + \cdots - 36\!\cdots\!44 \)
|
| $53$ |
\( T^{4} + \cdots - 28\!\cdots\!84 \)
|
| $59$ |
\( T^{4} + \cdots - 60\!\cdots\!64 \)
|
| $61$ |
\( T^{4} + \cdots + 56\!\cdots\!68 \)
|
| $67$ |
\( T^{4} + \cdots - 43\!\cdots\!52 \)
|
| $71$ |
\( T^{4} + \cdots + 39\!\cdots\!24 \)
|
| $73$ |
\( T^{4} + \cdots + 48\!\cdots\!52 \)
|
| $79$ |
\( T^{4} + \cdots - 52\!\cdots\!96 \)
|
| $83$ |
\( T^{4} + \cdots + 48\!\cdots\!72 \)
|
| $89$ |
\( T^{4} + \cdots - 56\!\cdots\!16 \)
|
| $97$ |
\( T^{4} + \cdots + 11\!\cdots\!68 \)
|
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