Newspace parameters
| Level: | \( N \) | \(=\) | \( 112 = 2^{4} \cdot 7 \) |
| Weight: | \( k \) | \(=\) | \( 5 \) |
| Character orbit: | \([\chi]\) | \(=\) | 112.l (of order \(4\), degree \(2\), minimal) |
Newform invariants
| Self dual: | no |
| Analytic conductor: | \(11.5774358654\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Relative dimension: | \(2\) over \(\Q(i)\) |
| Coefficient field: | \(\Q(i, \sqrt{7})\) |
|
|
|
| Defining polynomial: |
\( x^{4} - 3x^{2} + 4 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 3^{2} \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{U}(1)[D_{4}]$ |
$q$-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
\( x^{4} - 3x^{2} + 4 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( \nu^{3} - \nu ) / 2 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -\nu^{3} + 7\nu ) / 2 \)
|
| \(\beta_{3}\) | \(=\) |
\( 3\nu^{2} - 5 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{2} + \beta_1 ) / 3 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{3} + 5 ) / 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( \beta_{2} + 7\beta_1 ) / 3 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/112\mathbb{Z}\right)^\times\).
| \(n\) | \(15\) | \(17\) | \(85\) |
| \(\chi(n)\) | \(1\) | \(-1\) | \(\beta_{1}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 13.1 |
|
−3.96863 | − | 0.500000i | 0 | 15.5000 | + | 3.96863i | 0 | 0 | − | 49.0000i | −59.5294 | − | 23.5000i | 81.0000i | 0 | |||||||||||||||||||||||
| 13.2 | 3.96863 | − | 0.500000i | 0 | 15.5000 | − | 3.96863i | 0 | 0 | − | 49.0000i | 59.5294 | − | 23.5000i | 81.0000i | 0 | ||||||||||||||||||||||||
| 69.1 | −3.96863 | + | 0.500000i | 0 | 15.5000 | − | 3.96863i | 0 | 0 | 49.0000i | −59.5294 | + | 23.5000i | − | 81.0000i | 0 | ||||||||||||||||||||||||
| 69.2 | 3.96863 | + | 0.500000i | 0 | 15.5000 | + | 3.96863i | 0 | 0 | 49.0000i | 59.5294 | + | 23.5000i | − | 81.0000i | 0 | ||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 7.b | odd | 2 | 1 | CM by \(\Q(\sqrt{-7}) \) |
| 16.e | even | 4 | 1 | inner |
| 112.l | odd | 4 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 112.5.l.a | ✓ | 4 |
| 7.b | odd | 2 | 1 | CM | 112.5.l.a | ✓ | 4 |
| 16.e | even | 4 | 1 | inner | 112.5.l.a | ✓ | 4 |
| 112.l | odd | 4 | 1 | inner | 112.5.l.a | ✓ | 4 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 112.5.l.a | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
| 112.5.l.a | ✓ | 4 | 7.b | odd | 2 | 1 | CM |
| 112.5.l.a | ✓ | 4 | 16.e | even | 4 | 1 | inner |
| 112.5.l.a | ✓ | 4 | 112.l | odd | 4 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3} \)
acting on \(S_{5}^{\mathrm{new}}(112, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} - 31T^{2} + 256 \)
|
| $3$ |
\( T^{4} \)
|
| $5$ |
\( T^{4} \)
|
| $7$ |
\( (T^{2} + 2401)^{2} \)
|
| $11$ |
\( T^{4} - 412 T^{3} + \cdots + 173027716 \)
|
| $13$ |
\( T^{4} \)
|
| $17$ |
\( T^{4} \)
|
| $19$ |
\( T^{4} \)
|
| $23$ |
\( (T^{2} + 580608)^{2} \)
|
| $29$ |
\( T^{4} + \cdots + 11705941636 \)
|
| $31$ |
\( T^{4} \)
|
| $37$ |
\( T^{4} + \cdots + 4301003140996 \)
|
| $41$ |
\( T^{4} \)
|
| $43$ |
\( T^{4} + \cdots + 45239694794116 \)
|
| $47$ |
\( T^{4} \)
|
| $53$ |
\( T^{4} + \cdots + 236475564128644 \)
|
| $59$ |
\( T^{4} \)
|
| $61$ |
\( T^{4} \)
|
| $67$ |
\( T^{4} + \cdots + 250884248134276 \)
|
| $71$ |
\( (T^{2} + 8491396)^{2} \)
|
| $73$ |
\( T^{4} \)
|
| $79$ |
\( (T^{2} - 142507008)^{2} \)
|
| $83$ |
\( T^{4} \)
|
| $89$ |
\( T^{4} \)
|
| $97$ |
\( T^{4} \)
|
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