Newspace parameters
| Level: | \( N \) | \(=\) | \( 1472 = 2^{6} \cdot 23 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1472.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(86.8508115285\) |
| Analytic rank: | \(1\) |
| Dimension: | \(3\) |
| Coefficient field: | 3.3.11032.1 |
|
|
|
| Defining polynomial: |
\( x^{3} - x^{2} - 14x + 10 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 736) |
| 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
\( x^{3} - x^{2} - 14x + 10 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 10 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 10 \)
|
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 |
|
0 | −2.92042 | 0 | 9.29008 | 0 | 10.3916 | 0 | −18.4712 | 0 | |||||||||||||||||||||||||||
| 1.2 | 0 | 0.296194 | 0 | −8.80085 | 0 | 15.6161 | 0 | −26.9123 | 0 | ||||||||||||||||||||||||||||
| 1.3 | 0 | 4.62422 | 0 | −0.489233 | 0 | −10.0077 | 0 | −5.61656 | 0 | ||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \( +1 \) |
| \(23\) | \( -1 \) |
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 | 1472.4.a.u | 3 | |
| 4.b | odd | 2 | 1 | 1472.4.a.r | 3 | ||
| 8.b | even | 2 | 1 | 736.4.a.a | ✓ | 3 | |
| 8.d | odd | 2 | 1 | 736.4.a.b | yes | 3 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 736.4.a.a | ✓ | 3 | 8.b | even | 2 | 1 | |
| 736.4.a.b | yes | 3 | 8.d | odd | 2 | 1 | |
| 1472.4.a.r | 3 | 4.b | odd | 2 | 1 | ||
| 1472.4.a.u | 3 | 1.a | even | 1 | 1 | trivial | |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{3} - 2T_{3}^{2} - 13T_{3} + 4 \)
acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1472))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{3} \)
|
| $3$ |
\( T^{3} - 2 T^{2} + \cdots + 4 \)
|
| $5$ |
\( T^{3} - 82T - 40 \)
|
| $7$ |
\( T^{3} - 16 T^{2} + \cdots + 1624 \)
|
| $11$ |
\( T^{3} - 6 T^{2} + \cdots + 24532 \)
|
| $13$ |
\( T^{3} + 36 T^{2} + \cdots - 69826 \)
|
| $17$ |
\( T^{3} - 142 T^{2} + \cdots + 535844 \)
|
| $19$ |
\( T^{3} - 94 T^{2} + \cdots - 166744 \)
|
| $23$ |
\( (T - 23)^{3} \)
|
| $29$ |
\( T^{3} - 168 T^{2} + \cdots + 2379442 \)
|
| $31$ |
\( T^{3} + 18 T^{2} + \cdots + 6577160 \)
|
| $37$ |
\( T^{3} - 504 T^{2} + \cdots + 3243368 \)
|
| $41$ |
\( T^{3} + 432 T^{2} + \cdots - 126214 \)
|
| $43$ |
\( T^{3} + 338 T^{2} + \cdots + 399776 \)
|
| $47$ |
\( T^{3} + 386 T^{2} + \cdots - 307136 \)
|
| $53$ |
\( T^{3} - 288 T^{2} + \cdots + 51713072 \)
|
| $59$ |
\( T^{3} + 1312 T^{2} + \cdots + 44115008 \)
|
| $61$ |
\( T^{3} - 84 T^{2} + \cdots - 15287648 \)
|
| $67$ |
\( T^{3} - 410 T^{2} + \cdots + 25079932 \)
|
| $71$ |
\( T^{3} + 1126 T^{2} + \cdots - 247560040 \)
|
| $73$ |
\( T^{3} + \cdots - 1521354982 \)
|
| $79$ |
\( T^{3} + 1656 T^{2} + \cdots + 8938384 \)
|
| $83$ |
\( T^{3} + 1226 T^{2} + \cdots - 235935260 \)
|
| $89$ |
\( T^{3} + \cdots - 1079992640 \)
|
| $97$ |
\( T^{3} + 1106 T^{2} + \cdots - 83924108 \)
|
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