Newspace parameters
| Level: | \( N \) | \(=\) | \( 4725 = 3^{3} \cdot 5^{2} \cdot 7 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 4725.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(37.7293149551\) |
| Analytic rank: | \(1\) |
| Dimension: | \(5\) |
| Coefficient field: | 5.5.2161212.1 |
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| Defining polynomial: |
\( x^{5} - 8x^{3} - x^{2} + 14x + 2 \)
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| Coefficient ring: | \(\Z[a_1, a_2]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 945) |
| 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
\( x^{5} - 8x^{3} - x^{2} + 14x + 2 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
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| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 3 \)
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| \(\beta_{3}\) | \(=\) |
\( \nu^{3} - 5\nu - 1 \)
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| \(\beta_{4}\) | \(=\) |
\( \nu^{4} - \nu^{3} - 6\nu^{2} + 4\nu + 5 \)
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| \(\nu\) | \(=\) |
\( \beta_1 \)
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| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 3 \)
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| \(\nu^{3}\) | \(=\) |
\( \beta_{3} + 5\beta _1 + 1 \)
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| \(\nu^{4}\) | \(=\) |
\( \beta_{4} + \beta_{3} + 6\beta_{2} + \beta _1 + 14 \)
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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 |
|
−2.42873 | 0 | 3.89875 | 0 | 0 | 1.00000 | −4.61157 | 0 | 0 | |||||||||||||||||||||||||||||||||
| 1.2 | −1.57529 | 0 | 0.481550 | 0 | 0 | 1.00000 | 2.39200 | 0 | 0 | ||||||||||||||||||||||||||||||||||
| 1.3 | 0.143064 | 0 | −1.97953 | 0 | 0 | 1.00000 | −0.569328 | 0 | 0 | ||||||||||||||||||||||||||||||||||
| 1.4 | 1.66057 | 0 | 0.757487 | 0 | 0 | 1.00000 | −2.06328 | 0 | 0 | ||||||||||||||||||||||||||||||||||
| 1.5 | 2.20040 | 0 | 2.84174 | 0 | 0 | 1.00000 | 1.85217 | 0 | 0 | ||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \( +1 \) |
| \(5\) | \( -1 \) |
| \(7\) | \( -1 \) |
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 | 4725.2.a.cc | 5 | |
| 3.b | odd | 2 | 1 | 4725.2.a.cd | 5 | ||
| 5.b | even | 2 | 1 | 4725.2.a.cb | 5 | ||
| 5.c | odd | 4 | 2 | 945.2.d.e | yes | 10 | |
| 15.d | odd | 2 | 1 | 4725.2.a.ca | 5 | ||
| 15.e | even | 4 | 2 | 945.2.d.d | ✓ | 10 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 945.2.d.d | ✓ | 10 | 15.e | even | 4 | 2 | |
| 945.2.d.e | yes | 10 | 5.c | odd | 4 | 2 | |
| 4725.2.a.ca | 5 | 15.d | odd | 2 | 1 | ||
| 4725.2.a.cb | 5 | 5.b | even | 2 | 1 | ||
| 4725.2.a.cc | 5 | 1.a | even | 1 | 1 | trivial | |
| 4725.2.a.cd | 5 | 3.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4725))\):
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\( T_{2}^{5} - 8T_{2}^{3} + T_{2}^{2} + 14T_{2} - 2 \)
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\( T_{11}^{5} + 9T_{11}^{4} - 142T_{11}^{2} - 144T_{11} + 384 \)
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\( T_{13}^{5} - 7T_{13}^{4} - 12T_{13}^{3} + 162T_{13}^{2} - 288T_{13} + 128 \)
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\( T_{37}^{5} - 95T_{37}^{3} + 388T_{37}^{2} - 544T_{37} + 256 \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{5} - 8 T^{3} + \cdots - 2 \)
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| $3$ |
\( T^{5} \)
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| $5$ |
\( T^{5} \)
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| $7$ |
\( (T - 1)^{5} \)
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| $11$ |
\( T^{5} + 9 T^{4} + \cdots + 384 \)
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| $13$ |
\( T^{5} - 7 T^{4} + \cdots + 128 \)
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| $17$ |
\( T^{5} + 14 T^{4} + \cdots - 556 \)
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| $19$ |
\( T^{5} + 3 T^{4} + \cdots - 6 \)
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| $23$ |
\( T^{5} + 2 T^{4} + \cdots + 768 \)
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| $29$ |
\( T^{5} + 12 T^{4} + \cdots - 4864 \)
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| $31$ |
\( T^{5} + 6 T^{4} + \cdots + 16 \)
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| $37$ |
\( T^{5} - 95 T^{3} + \cdots + 256 \)
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| $41$ |
\( T^{5} + 7 T^{4} + \cdots + 8896 \)
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| $43$ |
\( T^{5} - 7 T^{4} + \cdots - 8896 \)
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| $47$ |
\( T^{5} + 13 T^{4} + \cdots - 48 \)
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| $53$ |
\( T^{5} - 5 T^{4} + \cdots + 24372 \)
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| $59$ |
\( T^{5} + 26 T^{4} + \cdots - 128 \)
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| $61$ |
\( T^{5} - 2 T^{4} + \cdots + 556 \)
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| $67$ |
\( T^{5} + 21 T^{4} + \cdots - 2048 \)
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| $71$ |
\( (T + 8)^{5} \)
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| $73$ |
\( T^{5} + 9 T^{4} + \cdots + 768 \)
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| $79$ |
\( T^{5} - 8 T^{4} + \cdots - 4822 \)
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| $83$ |
\( T^{5} + 17 T^{4} + \cdots + 6661 \)
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| $89$ |
\( T^{5} + 11 T^{4} + \cdots + 97536 \)
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| $97$ |
\( T^{5} + 4 T^{4} + \cdots + 11008 \)
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