Newspace parameters
| Level: | \( N \) | \(=\) | \( 930 = 2 \cdot 3 \cdot 5 \cdot 31 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 930.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(54.8717763053\) |
| Analytic rank: | \(1\) |
| Dimension: | \(3\) |
| Coefficient field: | 3.3.4692.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{3} - x^{2} - 17x - 3 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| Coefficient ring index: | \( 2^{3} \) |
| 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
\( x^{3} - x^{2} - 17x - 3 \)
:
| \(\beta_{1}\) | \(=\) |
\( 4\nu - 1 \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 2\nu - 11 \)
|
| \(\nu\) | \(=\) |
\( ( \beta _1 + 1 ) / 4 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 2\beta_{2} + \beta _1 + 23 ) / 2 \)
|
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 |
|
2.00000 | −3.00000 | 4.00000 | −5.00000 | −6.00000 | −22.8214 | 8.00000 | 9.00000 | −10.0000 | |||||||||||||||||||||||||||
| 1.2 | 2.00000 | −3.00000 | 4.00000 | −5.00000 | −6.00000 | 3.49592 | 8.00000 | 9.00000 | −10.0000 | ||||||||||||||||||||||||||||
| 1.3 | 2.00000 | −3.00000 | 4.00000 | −5.00000 | −6.00000 | 9.32544 | 8.00000 | 9.00000 | −10.0000 | ||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(-1\) |
| \(3\) | \(1\) |
| \(5\) | \(1\) |
| \(31\) | \(1\) |
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 | 930.4.a.h | ✓ | 3 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 930.4.a.h | ✓ | 3 | 1.a | even | 1 | 1 | trivial |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{7}^{3} + 10T_{7}^{2} - 260T_{7} + 744 \)
acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(930))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T - 2)^{3} \)
|
| $3$ |
\( (T + 3)^{3} \)
|
| $5$ |
\( (T + 5)^{3} \)
|
| $7$ |
\( T^{3} + 10 T^{2} + \cdots + 744 \)
|
| $11$ |
\( T^{3} + 16 T^{2} + \cdots - 10240 \)
|
| $13$ |
\( T^{3} + 20 T^{2} + \cdots + 26928 \)
|
| $17$ |
\( T^{3} - 118 T^{2} + \cdots + 43320 \)
|
| $19$ |
\( T^{3} - 80 T^{2} + \cdots - 130944 \)
|
| $23$ |
\( T^{3} - 8 T^{2} + \cdots + 1142144 \)
|
| $29$ |
\( T^{3} + 244 T^{2} + \cdots + 269840 \)
|
| $31$ |
\( (T + 31)^{3} \)
|
| $37$ |
\( T^{3} + 364 T^{2} + \cdots - 3925648 \)
|
| $41$ |
\( T^{3} + 426 T^{2} + \cdots - 65044296 \)
|
| $43$ |
\( T^{3} + 488 T^{2} + \cdots - 18667904 \)
|
| $47$ |
\( T^{3} + 432 T^{2} + \cdots - 60690816 \)
|
| $53$ |
\( T^{3} + 126 T^{2} + \cdots + 10014824 \)
|
| $59$ |
\( T^{3} + 510 T^{2} + \cdots + 2842328 \)
|
| $61$ |
\( T^{3} + 650 T^{2} + \cdots - 186803016 \)
|
| $67$ |
\( T^{3} + 858 T^{2} + \cdots + 22347944 \)
|
| $71$ |
\( T^{3} + 990 T^{2} + \cdots - 97935336 \)
|
| $73$ |
\( T^{3} - 400 T^{2} + \cdots - 23190000 \)
|
| $79$ |
\( T^{3} - 728 T^{2} + \cdots + 769393536 \)
|
| $83$ |
\( T^{3} - 692 T^{2} + \cdots + 267988800 \)
|
| $89$ |
\( T^{3} + 2412 T^{2} + \cdots - 501239664 \)
|
| $97$ |
\( T^{3} + 1446 T^{2} + \cdots - 263690872 \)
|
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