Newspace parameters
| Level: | \( N \) | \(=\) | \( 2775 = 3 \cdot 5^{2} \cdot 37 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 2775.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(22.1584865609\) |
| Analytic rank: | \(0\) |
| Dimension: | \(5\) |
| Coefficient field: | 5.5.600268.1 |
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| Defining polynomial: |
\( x^{5} - x^{4} - 7x^{3} + 6x^{2} + 6x - 4 \)
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| Coefficient ring: | \(\Z[a_1, a_2]\) |
| Coefficient ring index: | \( 2 \) |
| Twist minimal: | no (minimal twist has level 555) |
| 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
\( x^{5} - x^{4} - 7x^{3} + 6x^{2} + 6x - 4 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( \nu^{4} - \nu^{3} - 5\nu^{2} + 4\nu ) / 2 \)
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| \(\beta_{2}\) | \(=\) |
\( ( -\nu^{4} - \nu^{3} + 7\nu^{2} + 8\nu - 6 ) / 2 \)
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| \(\beta_{3}\) | \(=\) |
\( ( \nu^{4} - \nu^{3} - 7\nu^{2} + 4\nu + 6 ) / 2 \)
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| \(\beta_{4}\) | \(=\) |
\( \nu^{4} - 7\nu^{2} + 5 \)
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| \(\nu\) | \(=\) |
\( ( \beta_{4} - \beta_{3} + \beta_{2} + 1 ) / 2 \)
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| \(\nu^{2}\) | \(=\) |
\( -\beta_{3} + \beta _1 + 3 \)
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| \(\nu^{3}\) | \(=\) |
\( 3\beta_{4} - 4\beta_{3} + 2\beta_{2} + 3 \)
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| \(\nu^{4}\) | \(=\) |
\( \beta_{4} - 7\beta_{3} + 7\beta _1 + 16 \)
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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 |
|
−1.97295 | 1.00000 | 1.89253 | 0 | −1.97295 | −2.73496 | 0.212042 | 1.00000 | 0 | |||||||||||||||||||||||||||||||||
| 1.2 | −0.568822 | 1.00000 | −1.67644 | 0 | −0.568822 | −3.69558 | 2.09124 | 1.00000 | 0 | ||||||||||||||||||||||||||||||||||
| 1.3 | 0.589830 | 1.00000 | −1.65210 | 0 | 0.589830 | 4.68977 | −2.15412 | 1.00000 | 0 | ||||||||||||||||||||||||||||||||||
| 1.4 | 2.18003 | 1.00000 | 2.75253 | 0 | 2.18003 | −2.45718 | 1.64054 | 1.00000 | 0 | ||||||||||||||||||||||||||||||||||
| 1.5 | 2.77191 | 1.00000 | 5.68349 | 0 | 2.77191 | 2.19795 | 10.2103 | 1.00000 | 0 | ||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \( -1 \) |
| \(5\) | \( +1 \) |
| \(37\) | \( +1 \) |
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 | 2775.2.a.bd | 5 | |
| 3.b | odd | 2 | 1 | 8325.2.a.bx | 5 | ||
| 5.b | even | 2 | 1 | 555.2.a.j | ✓ | 5 | |
| 15.d | odd | 2 | 1 | 1665.2.a.s | 5 | ||
| 20.d | odd | 2 | 1 | 8880.2.a.cm | 5 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 555.2.a.j | ✓ | 5 | 5.b | even | 2 | 1 | |
| 1665.2.a.s | 5 | 15.d | odd | 2 | 1 | ||
| 2775.2.a.bd | 5 | 1.a | even | 1 | 1 | trivial | |
| 8325.2.a.bx | 5 | 3.b | odd | 2 | 1 | ||
| 8880.2.a.cm | 5 | 20.d | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(2775))\):
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\( T_{2}^{5} - 3T_{2}^{4} - 4T_{2}^{3} + 13T_{2}^{2} + T_{2} - 4 \)
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\( T_{7}^{5} + 2T_{7}^{4} - 25T_{7}^{3} - 62T_{7}^{2} + 96T_{7} + 256 \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{5} - 3 T^{4} + \cdots - 4 \)
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| $3$ |
\( (T - 1)^{5} \)
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| $5$ |
\( T^{5} \)
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| $7$ |
\( T^{5} + 2 T^{4} + \cdots + 256 \)
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| $11$ |
\( T^{5} + T^{4} + \cdots + 128 \)
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| $13$ |
\( T^{5} + 7 T^{4} + \cdots + 4 \)
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| $17$ |
\( T^{5} + T^{4} - 41 T^{3} + \cdots - 8 \)
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| $19$ |
\( T^{5} - 13 T^{4} + \cdots + 1856 \)
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| $23$ |
\( T^{5} - 6 T^{4} + \cdots - 64 \)
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| $29$ |
\( T^{5} - 8 T^{4} + \cdots - 1244 \)
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| $31$ |
\( T^{5} - 8 T^{4} + \cdots - 3488 \)
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| $37$ |
\( (T + 1)^{5} \)
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| $41$ |
\( T^{5} - 4 T^{4} + \cdots + 1136 \)
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| $43$ |
\( T^{5} + 14 T^{4} + \cdots - 800 \)
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| $47$ |
\( T^{5} + T^{4} + \cdots - 128 \)
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| $53$ |
\( T^{5} + 17 T^{4} + \cdots - 5056 \)
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| $59$ |
\( T^{5} - 10 T^{4} + \cdots - 16 \)
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| $61$ |
\( T^{5} - 40 T^{4} + \cdots + 8152 \)
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| $67$ |
\( T^{5} - 13 T^{4} + \cdots - 64 \)
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| $71$ |
\( T^{5} + T^{4} + \cdots + 55808 \)
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| $73$ |
\( T^{5} + 3 T^{4} + \cdots - 18376 \)
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| $79$ |
\( T^{5} - 7 T^{4} + \cdots - 217984 \)
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| $83$ |
\( T^{5} - 6 T^{4} + \cdots - 40144 \)
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| $89$ |
\( T^{5} + 3 T^{4} + \cdots - 21452 \)
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| $97$ |
\( T^{5} - 71 T^{3} + \cdots - 584 \)
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