Newspace parameters
| Level: | \( N \) | \(=\) | \( 1110 = 2 \cdot 3 \cdot 5 \cdot 37 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1110.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(65.4921201064\) |
| Analytic rank: | \(1\) |
| Dimension: | \(3\) |
| Coefficient field: | 3.3.243037.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{3} - x^{2} - 87x + 162 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| 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 \(1,\beta_1,\beta_2\) 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
\( x^{3} - x^{2} - 87x + 162 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{2} + 2\nu - 60 ) / 3 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( 3\beta_{2} - 2\beta _1 + 60 \)
|
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 |
|
−2.00000 | 3.00000 | 4.00000 | −5.00000 | −6.00000 | −26.5693 | −8.00000 | 9.00000 | 10.0000 | |||||||||||||||||||||||||||
| 1.2 | −2.00000 | 3.00000 | 4.00000 | −5.00000 | −6.00000 | −14.8613 | −8.00000 | 9.00000 | 10.0000 | ||||||||||||||||||||||||||||
| 1.3 | −2.00000 | 3.00000 | 4.00000 | −5.00000 | −6.00000 | 7.43060 | −8.00000 | 9.00000 | 10.0000 | ||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(1\) |
| \(3\) | \(-1\) |
| \(5\) | \(1\) |
| \(37\) | \(1\) |
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 | 1110.4.a.f | ✓ | 3 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 1110.4.a.f | ✓ | 3 | 1.a | even | 1 | 1 | trivial |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{7}^{3} + 34T_{7}^{2} + 87T_{7} - 2934 \)
acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1110))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T + 2)^{3} \)
|
| $3$ |
\( (T - 3)^{3} \)
|
| $5$ |
\( (T + 5)^{3} \)
|
| $7$ |
\( T^{3} + 34 T^{2} + \cdots - 2934 \)
|
| $11$ |
\( T^{3} - 51 T^{2} + \cdots + 23088 \)
|
| $13$ |
\( T^{3} + 73 T^{2} + \cdots - 7717 \)
|
| $17$ |
\( T^{3} - 41 T^{2} + \cdots + 35522 \)
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| $19$ |
\( T^{3} - 137 T^{2} + \cdots + 634910 \)
|
| $23$ |
\( T^{3} + 146 T^{2} + \cdots - 995670 \)
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| $29$ |
\( T^{3} - 78 T^{2} + \cdots + 941949 \)
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| $31$ |
\( T^{3} - 234 T^{2} + \cdots + 922410 \)
|
| $37$ |
\( (T + 37)^{3} \)
|
| $41$ |
\( T^{3} - 328 T^{2} + \cdots + 3744252 \)
|
| $43$ |
\( T^{3} + 60 T^{2} + \cdots - 24024403 \)
|
| $47$ |
\( T^{3} + 187 T^{2} + \cdots + 7155837 \)
|
| $53$ |
\( T^{3} + 465 T^{2} + \cdots - 124359216 \)
|
| $59$ |
\( T^{3} + 246 T^{2} + \cdots - 564213 \)
|
| $61$ |
\( T^{3} - 76 T^{2} + \cdots + 151014510 \)
|
| $67$ |
\( T^{3} + 1801 T^{2} + \cdots + 193171512 \)
|
| $71$ |
\( T^{3} + 151 T^{2} + \cdots - 24975000 \)
|
| $73$ |
\( T^{3} + 1449 T^{2} + \cdots - 223006554 \)
|
| $79$ |
\( T^{3} + 1775 T^{2} + \cdots + 172613740 \)
|
| $83$ |
\( T^{3} + 470 T^{2} + \cdots - 2116179 \)
|
| $89$ |
\( T^{3} - 425 T^{2} + \cdots - 18192523 \)
|
| $97$ |
\( T^{3} + 1058 T^{2} + \cdots - 652602150 \)
|
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