Newspace parameters
| Level: | \( N \) | \(=\) | \( 3724 = 2^{2} \cdot 7^{2} \cdot 19 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 3724.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(29.7362897127\) |
| Analytic rank: | \(0\) |
| Dimension: | \(5\) |
| Coefficient field: | 5.5.11350832.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{5} - 2x^{4} - 9x^{3} + 8x^{2} + 23x + 4 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| 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 \(1,\beta_1,\beta_2,\beta_3,\beta_4\) 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
\( x^{5} - 2x^{4} - 9x^{3} + 8x^{2} + 23x + 4 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - \nu - 4 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{3} - 2\nu^{2} - 4\nu + 3 \)
|
| \(\beta_{4}\) | \(=\) |
\( \nu^{4} - 3\nu^{3} - 5\nu^{2} + 11\nu + 7 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + \beta _1 + 4 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{3} + 2\beta_{2} + 6\beta _1 + 5 \)
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| \(\nu^{4}\) | \(=\) |
\( \beta_{4} + 3\beta_{3} + 11\beta_{2} + 12\beta _1 + 28 \)
|
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 | −1.90555 | 0 | −1.53668 | 0 | 0 | 0 | 0.631128 | 0 | |||||||||||||||||||||||||||||||||
| 1.2 | 0 | −1.48377 | 0 | 0.314641 | 0 | 0 | 0 | −0.798415 | 0 | ||||||||||||||||||||||||||||||||||
| 1.3 | 0 | −0.188829 | 0 | 3.77551 | 0 | 0 | 0 | −2.96434 | 0 | ||||||||||||||||||||||||||||||||||
| 1.4 | 0 | 2.25348 | 0 | 1.17530 | 0 | 0 | 0 | 2.07818 | 0 | ||||||||||||||||||||||||||||||||||
| 1.5 | 0 | 3.32467 | 0 | −3.72878 | 0 | 0 | 0 | 8.05345 | 0 | ||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(-1\) |
| \(7\) | \(-1\) |
| \(19\) | \(-1\) |
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 | 3724.2.a.l | yes | 5 |
| 7.b | odd | 2 | 1 | 3724.2.a.k | ✓ | 5 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 3724.2.a.k | ✓ | 5 | 7.b | odd | 2 | 1 | |
| 3724.2.a.l | yes | 5 | 1.a | even | 1 | 1 | trivial |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{5} - 2T_{3}^{4} - 9T_{3}^{3} + 8T_{3}^{2} + 23T_{3} + 4 \)
acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(3724))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{5} \)
|
| $3$ |
\( T^{5} - 2 T^{4} + \cdots + 4 \)
|
| $5$ |
\( T^{5} - 16 T^{3} + \cdots - 8 \)
|
| $7$ |
\( T^{5} \)
|
| $11$ |
\( T^{5} - 2 T^{4} + \cdots + 20 \)
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| $13$ |
\( T^{5} - 4 T^{4} + \cdots + 62 \)
|
| $17$ |
\( T^{5} - 4 T^{4} + \cdots - 500 \)
|
| $19$ |
\( (T - 1)^{5} \)
|
| $23$ |
\( T^{5} - 54 T^{3} + \cdots - 100 \)
|
| $29$ |
\( T^{5} + 8 T^{4} + \cdots + 3106 \)
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| $31$ |
\( T^{5} - 147 T^{3} + \cdots - 3664 \)
|
| $37$ |
\( T^{5} + 8 T^{4} + \cdots + 1706 \)
|
| $41$ |
\( T^{5} - 18 T^{4} + \cdots + 250 \)
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| $43$ |
\( T^{5} - 4 T^{4} + \cdots + 20480 \)
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| $47$ |
\( T^{5} - 20 T^{4} + \cdots - 100 \)
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| $53$ |
\( T^{5} + 2 T^{4} + \cdots + 27198 \)
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| $59$ |
\( T^{5} - 14 T^{4} + \cdots - 52740 \)
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| $61$ |
\( T^{5} - 12 T^{4} + \cdots - 18496 \)
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| $67$ |
\( T^{5} - 12 T^{4} + \cdots + 3166 \)
|
| $71$ |
\( T^{5} - 12 T^{4} + \cdots - 61650 \)
|
| $73$ |
\( T^{5} + 36 T^{4} + \cdots - 51076 \)
|
| $79$ |
\( T^{5} - 6 T^{4} + \cdots - 800 \)
|
| $83$ |
\( T^{5} - 12 T^{4} + \cdots + 4904 \)
|
| $89$ |
\( T^{5} - 50 T^{4} + \cdots - 40064 \)
|
| $97$ |
\( T^{5} - 32 T^{4} + \cdots + 23922 \)
|
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