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: | \(0\) |
| Dimension: | \(3\) |
| Coefficient field: | 3.3.1712869.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{3} - x^{2} - 157x + 130 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| Coefficient ring index: | \( 3 \) |
| 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
\( x^{3} - x^{2} - 157x + 130 \)
:
| \(\beta_{1}\) | \(=\) |
\( 3\nu - 1 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{2} - 3\nu - 103 ) / 3 \)
|
| \(\nu\) | \(=\) |
\( ( \beta _1 + 1 ) / 3 \)
|
| \(\nu^{2}\) | \(=\) |
\( 3\beta_{2} + \beta _1 + 104 \)
|
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 | −28.7804 | 8.00000 | 9.00000 | −10.0000 | |||||||||||||||||||||||||||
| 1.2 | 2.00000 | −3.00000 | 4.00000 | −5.00000 | −6.00000 | −5.15205 | 8.00000 | 9.00000 | −10.0000 | ||||||||||||||||||||||||||||
| 1.3 | 2.00000 | −3.00000 | 4.00000 | −5.00000 | −6.00000 | 35.9325 | 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.h | ✓ | 3 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 1110.4.a.h | ✓ | 3 | 1.a | even | 1 | 1 | trivial |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{7}^{3} - 2T_{7}^{2} - 1071T_{7} - 5328 \)
acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1110))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T - 2)^{3} \)
|
| $3$ |
\( (T + 3)^{3} \)
|
| $5$ |
\( (T + 5)^{3} \)
|
| $7$ |
\( T^{3} - 2 T^{2} + \cdots - 5328 \)
|
| $11$ |
\( T^{3} + 43 T^{2} + \cdots - 6024 \)
|
| $13$ |
\( T^{3} + 80 T^{2} + \cdots - 3590 \)
|
| $17$ |
\( T^{3} - 67 T^{2} + \cdots + 336150 \)
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| $19$ |
\( T^{3} + 77 T^{2} + \cdots - 148700 \)
|
| $23$ |
\( T^{3} - 152 T^{2} + \cdots + 234672 \)
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| $29$ |
\( T^{3} + 157 T^{2} + \cdots - 3071070 \)
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| $31$ |
\( T^{3} - 354 T^{2} + \cdots - 271580 \)
|
| $37$ |
\( (T - 37)^{3} \)
|
| $41$ |
\( T^{3} + 154 T^{2} + \cdots + 10170 \)
|
| $43$ |
\( T^{3} + 331 T^{2} + \cdots - 3612384 \)
|
| $47$ |
\( T^{3} - 552 T^{2} + \cdots + 64784772 \)
|
| $53$ |
\( T^{3} - 138 T^{2} + \cdots + 34082208 \)
|
| $59$ |
\( T^{3} - 269 T^{2} + \cdots + 80255220 \)
|
| $61$ |
\( T^{3} - 594 T^{2} + \cdots + 26809666 \)
|
| $67$ |
\( T^{3} + 501 T^{2} + \cdots - 63609404 \)
|
| $71$ |
\( T^{3} - 127 T^{2} + \cdots + 50727888 \)
|
| $73$ |
\( T^{3} - 713 T^{2} + \cdots - 10523358 \)
|
| $79$ |
\( T^{3} - 1651 T^{2} + \cdots - 22012916 \)
|
| $83$ |
\( T^{3} - 1537 T^{2} + \cdots - 115001988 \)
|
| $89$ |
\( T^{3} - 1544 T^{2} + \cdots + 12368826 \)
|
| $97$ |
\( T^{3} + 14 T^{2} + \cdots + 798115126 \)
|
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