Newspace parameters
| Level: | \( N \) | \(=\) | \( 58 = 2 \cdot 29 \) |
| Weight: | \( k \) | \(=\) | \( 3 \) |
| Character orbit: | \([\chi]\) | \(=\) | 58.c (of order \(4\), degree \(2\), minimal) |
Newform invariants
| Self dual: | no |
| Analytic conductor: | \(1.58038553329\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Relative dimension: | \(2\) over \(\Q(i)\) |
| Coefficient field: | \(\Q(i, \sqrt{6})\) |
|
|
|
| Defining polynomial: |
\( x^{4} + 9 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{4}]$ |
$q$-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
\( x^{4} + 9 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{2} ) / 3 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{3} ) / 3 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( 3\beta_{2} \)
|
| \(\nu^{3}\) | \(=\) |
\( 3\beta_{3} \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/58\mathbb{Z}\right)^\times\).
| \(n\) | \(31\) |
| \(\chi(n)\) | \(-\beta_{2}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 17.1 |
|
1.00000 | − | 1.00000i | −0.224745 | + | 0.224745i | − | 2.00000i | − | 5.89898i | 0.449490i | 9.34847 | −2.00000 | − | 2.00000i | 8.89898i | −5.89898 | − | 5.89898i | ||||||||||||||||||||
| 17.2 | 1.00000 | − | 1.00000i | 2.22474 | − | 2.22474i | − | 2.00000i | 3.89898i | − | 4.44949i | −5.34847 | −2.00000 | − | 2.00000i | − | 0.898979i | 3.89898 | + | 3.89898i | ||||||||||||||||||||
| 41.1 | 1.00000 | + | 1.00000i | −0.224745 | − | 0.224745i | 2.00000i | 5.89898i | − | 0.449490i | 9.34847 | −2.00000 | + | 2.00000i | − | 8.89898i | −5.89898 | + | 5.89898i | |||||||||||||||||||||
| 41.2 | 1.00000 | + | 1.00000i | 2.22474 | + | 2.22474i | 2.00000i | − | 3.89898i | 4.44949i | −5.34847 | −2.00000 | + | 2.00000i | 0.898979i | 3.89898 | − | 3.89898i | ||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 29.c | odd | 4 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 58.3.c.a | ✓ | 4 |
| 3.b | odd | 2 | 1 | 522.3.f.a | 4 | ||
| 4.b | odd | 2 | 1 | 464.3.l.a | 4 | ||
| 29.c | odd | 4 | 1 | inner | 58.3.c.a | ✓ | 4 |
| 87.f | even | 4 | 1 | 522.3.f.a | 4 | ||
| 116.e | even | 4 | 1 | 464.3.l.a | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 58.3.c.a | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
| 58.3.c.a | ✓ | 4 | 29.c | odd | 4 | 1 | inner |
| 464.3.l.a | 4 | 4.b | odd | 2 | 1 | ||
| 464.3.l.a | 4 | 116.e | even | 4 | 1 | ||
| 522.3.f.a | 4 | 3.b | odd | 2 | 1 | ||
| 522.3.f.a | 4 | 87.f | even | 4 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{4} - 4T_{3}^{3} + 8T_{3}^{2} + 4T_{3} + 1 \)
acting on \(S_{3}^{\mathrm{new}}(58, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{2} - 2 T + 2)^{2} \)
|
| $3$ |
\( T^{4} - 4 T^{3} + \cdots + 1 \)
|
| $5$ |
\( T^{4} + 50T^{2} + 529 \)
|
| $7$ |
\( (T^{2} - 4 T - 50)^{2} \)
|
| $11$ |
\( T^{4} + 12 T^{3} + \cdots + 16641 \)
|
| $13$ |
\( T^{4} + 210T^{2} + 7569 \)
|
| $17$ |
\( T^{4} + 32 T^{3} + \cdots + 4096 \)
|
| $19$ |
\( T^{4} + 144 \)
|
| $23$ |
\( (T^{2} - 8 T - 1520)^{2} \)
|
| $29$ |
\( (T + 29)^{4} \)
|
| $31$ |
\( T^{4} - 76 T^{3} + \cdots + 128881 \)
|
| $37$ |
\( (T^{2} + 16 T + 128)^{2} \)
|
| $41$ |
\( T^{4} + 116 T^{3} + \cdots + 232324 \)
|
| $43$ |
\( T^{4} - 156 T^{3} + \cdots + 4730625 \)
|
| $47$ |
\( T^{4} + 28 T^{3} + \cdots + 4363921 \)
|
| $53$ |
\( (T^{2} + 70 T - 1175)^{2} \)
|
| $59$ |
\( (T^{2} + 32 T - 128)^{2} \)
|
| $61$ |
\( T^{4} + 16 T^{3} + \cdots + 1364224 \)
|
| $67$ |
\( T^{4} + 7040 T^{2} + 7573504 \)
|
| $71$ |
\( T^{4} + 500T^{2} + 2500 \)
|
| $73$ |
\( T^{4} - 48 T^{3} + \cdots + 110502144 \)
|
| $79$ |
\( T^{4} + 132 T^{3} + \cdots + 4626801 \)
|
| $83$ |
\( (T^{2} - 56 T - 5360)^{2} \)
|
| $89$ |
\( T^{4} - 204 T^{3} + \cdots + 16016004 \)
|
| $97$ |
\( T^{4} - 276 T^{3} + \cdots + 31741956 \)
|
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