Newspace parameters
| Level: | \( N \) | \(=\) | \( 1888 = 2^{5} \cdot 59 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1888.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(15.0757559016\) |
| Analytic rank: | \(1\) |
| Dimension: | \(4\) |
| Coefficient field: | 4.4.8468.1 |
|
|
|
| Defining polynomial: |
\( x^{4} - x^{3} - 5x^{2} + 3x + 4 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| Coefficient ring index: | \( 2^{2} \) |
| 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
\( x^{4} - x^{3} - 5x^{2} + 3x + 4 \)
:
| \(\beta_{1}\) | \(=\) |
\( 2\nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{3} - \nu^{2} - 3\nu + 2 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{3} + \nu^{2} - 5\nu - 3 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{3} - \beta_{2} + \beta _1 + 5 ) / 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( \beta_{3} + \beta_{2} + 4\beta _1 + 1 ) / 2 \)
|
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 |
|
0 | −3.26753 | 0 | 3.26753 | 0 | −1.00000 | 0 | 7.67678 | 0 | ||||||||||||||||||||||||||||||
| 1.2 | 0 | −1.80122 | 0 | 1.80122 | 0 | −1.00000 | 0 | 0.244393 | 0 | |||||||||||||||||||||||||||||||
| 1.3 | 0 | 1.40135 | 0 | −1.40135 | 0 | −1.00000 | 0 | −1.03621 | 0 | |||||||||||||||||||||||||||||||
| 1.4 | 0 | 2.66740 | 0 | −2.66740 | 0 | −1.00000 | 0 | 4.11504 | 0 | |||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \( +1 \) |
| \(59\) | \( +1 \) |
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 | 1888.2.a.h | ✓ | 4 |
| 4.b | odd | 2 | 1 | 1888.2.a.i | yes | 4 | |
| 8.b | even | 2 | 1 | 3776.2.a.bh | 4 | ||
| 8.d | odd | 2 | 1 | 3776.2.a.bf | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 1888.2.a.h | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
| 1888.2.a.i | yes | 4 | 4.b | odd | 2 | 1 | |
| 3776.2.a.bf | 4 | 8.d | odd | 2 | 1 | ||
| 3776.2.a.bh | 4 | 8.b | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{4} + T_{3}^{3} - 11T_{3}^{2} - 5T_{3} + 22 \)
acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(1888))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} \)
|
| $3$ |
\( T^{4} + T^{3} + \cdots + 22 \)
|
| $5$ |
\( T^{4} - T^{3} + \cdots + 22 \)
|
| $7$ |
\( (T + 1)^{4} \)
|
| $11$ |
\( T^{4} + T^{3} + \cdots - 32 \)
|
| $13$ |
\( T^{4} + 5 T^{3} + \cdots - 176 \)
|
| $17$ |
\( T^{4} + 11 T^{3} + \cdots - 8 \)
|
| $19$ |
\( T^{4} - 5 T^{3} + \cdots + 1384 \)
|
| $23$ |
\( T^{4} + 10 T^{3} + \cdots - 64 \)
|
| $29$ |
\( T^{4} + 7 T^{3} + \cdots - 4 \)
|
| $31$ |
\( T^{4} - 2 T^{3} + \cdots + 64 \)
|
| $37$ |
\( T^{4} + 3 T^{3} + \cdots - 184 \)
|
| $41$ |
\( T^{4} + 6 T^{3} + \cdots - 1117 \)
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| $43$ |
\( T^{4} + 5 T^{3} + \cdots + 8 \)
|
| $47$ |
\( T^{4} + 14 T^{3} + \cdots - 3872 \)
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| $53$ |
\( T^{4} - 3 T^{3} + \cdots + 1168 \)
|
| $59$ |
\( (T + 1)^{4} \)
|
| $61$ |
\( T^{4} - 24 T^{3} + \cdots - 7792 \)
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| $67$ |
\( T^{4} - 24 T^{3} + \cdots + 128 \)
|
| $71$ |
\( T^{4} + 13 T^{3} + \cdots - 128 \)
|
| $73$ |
\( T^{4} + 18 T^{3} + \cdots - 9376 \)
|
| $79$ |
\( (T + 7)^{4} \)
|
| $83$ |
\( T^{4} + 33 T^{3} + \cdots + 848 \)
|
| $89$ |
\( T^{4} - 64 T^{2} + \cdots + 16 \)
|
| $97$ |
\( T^{4} + 2 T^{3} + \cdots - 1888 \)
|
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