Newspace parameters
| Level: | \( N \) | \(=\) | \( 2277 = 3^{2} \cdot 11 \cdot 23 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 2277.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(18.1819365402\) |
| Analytic rank: | \(0\) |
| Dimension: | \(5\) |
| Coefficient field: | 5.5.170701.1 |
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| Defining polynomial: |
\( x^{5} - x^{4} - 6x^{3} + 4x + 1 \)
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| Coefficient ring: | \(\Z[a_1, a_2]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 253) |
| 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
\( x^{5} - x^{4} - 6x^{3} + 4x + 1 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
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| \(\beta_{2}\) | \(=\) |
\( \nu^{4} - \nu^{3} - 6\nu^{2} + 3 \)
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| \(\beta_{3}\) | \(=\) |
\( -\nu^{4} + 2\nu^{3} + 4\nu^{2} - 4\nu - 1 \)
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| \(\beta_{4}\) | \(=\) |
\( -\nu^{4} + 2\nu^{3} + 5\nu^{2} - 5\nu - 3 \)
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| \(\nu\) | \(=\) |
\( \beta_1 \)
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| \(\nu^{2}\) | \(=\) |
\( \beta_{4} - \beta_{3} + \beta _1 + 2 \)
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| \(\nu^{3}\) | \(=\) |
\( 2\beta_{4} - \beta_{3} + \beta_{2} + 6\beta _1 + 2 \)
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| \(\nu^{4}\) | \(=\) |
\( 8\beta_{4} - 7\beta_{3} + 2\beta_{2} + 12\beta _1 + 11 \)
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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 |
|
−1.89398 | 0 | 1.58716 | 4.08349 | 0 | 0.994750 | 0.781916 | 0 | −7.73404 | |||||||||||||||||||||||||||||||||
| 1.2 | 0.0846188 | 0 | −1.99284 | −0.462928 | 0 | −4.03328 | −0.337869 | 0 | −0.0391724 | ||||||||||||||||||||||||||||||||||
| 1.3 | 1.28142 | 0 | −0.357954 | −3.80110 | 0 | −1.85610 | −3.02154 | 0 | −4.87082 | ||||||||||||||||||||||||||||||||||
| 1.4 | 1.75774 | 0 | 1.08966 | 2.14004 | 0 | 4.58759 | −1.60015 | 0 | 3.76164 | ||||||||||||||||||||||||||||||||||
| 1.5 | 2.77019 | 0 | 5.67398 | 1.04050 | 0 | −2.69296 | 10.1776 | 0 | 2.88239 | ||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \( -1 \) |
| \(11\) | \( -1 \) |
| \(23\) | \( -1 \) |
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 | 2277.2.a.l | 5 | |
| 3.b | odd | 2 | 1 | 253.2.a.c | ✓ | 5 | |
| 12.b | even | 2 | 1 | 4048.2.a.bb | 5 | ||
| 15.d | odd | 2 | 1 | 6325.2.a.l | 5 | ||
| 33.d | even | 2 | 1 | 2783.2.a.g | 5 | ||
| 69.c | even | 2 | 1 | 5819.2.a.d | 5 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 253.2.a.c | ✓ | 5 | 3.b | odd | 2 | 1 | |
| 2277.2.a.l | 5 | 1.a | even | 1 | 1 | trivial | |
| 2783.2.a.g | 5 | 33.d | even | 2 | 1 | ||
| 4048.2.a.bb | 5 | 12.b | even | 2 | 1 | ||
| 5819.2.a.d | 5 | 69.c | even | 2 | 1 | ||
| 6325.2.a.l | 5 | 15.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 \(S_{2}^{\mathrm{new}}(\Gamma_0(2277))\):
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\( T_{2}^{5} - 4T_{2}^{4} + 14T_{2}^{2} - 13T_{2} + 1 \)
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\( T_{5}^{5} - 3T_{5}^{4} - 14T_{5}^{3} + 43T_{5}^{2} - 12T_{5} - 16 \)
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\( T_{17}^{5} - 9T_{17}^{4} - 16T_{17}^{3} + 257T_{17}^{2} - 72T_{17} - 1492 \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{5} - 4 T^{4} + \cdots + 1 \)
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| $3$ |
\( T^{5} \)
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| $5$ |
\( T^{5} - 3 T^{4} + \cdots - 16 \)
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| $7$ |
\( T^{5} + 3 T^{4} + \cdots + 92 \)
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| $11$ |
\( (T - 1)^{5} \)
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| $13$ |
\( T^{5} + 15 T^{4} + \cdots + 89 \)
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| $17$ |
\( T^{5} - 9 T^{4} + \cdots - 1492 \)
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| $19$ |
\( T^{5} + 5 T^{4} + \cdots + 4 \)
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| $23$ |
\( (T - 1)^{5} \)
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| $29$ |
\( T^{5} - 8 T^{4} + \cdots - 2011 \)
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| $31$ |
\( T^{5} + 4 T^{4} + \cdots - 161 \)
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| $37$ |
\( T^{5} - 8 T^{4} + \cdots - 1948 \)
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| $41$ |
\( T^{5} - 6 T^{4} + \cdots - 10459 \)
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| $43$ |
\( T^{5} + 8 T^{4} + \cdots + 3988 \)
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| $47$ |
\( T^{5} - 34 T^{4} + \cdots - 5272 \)
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| $53$ |
\( T^{5} + 2 T^{4} + \cdots - 4 \)
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| $59$ |
\( T^{5} - 13 T^{4} + \cdots - 8368 \)
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| $61$ |
\( T^{5} - 18 T^{4} + \cdots - 4 \)
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| $67$ |
\( T^{5} + 4 T^{4} + \cdots + 1568 \)
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| $71$ |
\( T^{5} + 10 T^{4} + \cdots + 73601 \)
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| $73$ |
\( T^{5} + 31 T^{4} + \cdots - 3659 \)
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| $79$ |
\( T^{5} + 3 T^{4} + \cdots + 2476 \)
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| $83$ |
\( T^{5} - 43 T^{4} + \cdots + 71188 \)
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| $89$ |
\( T^{5} + 15 T^{4} + \cdots + 4804 \)
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| $97$ |
\( T^{5} - 17 T^{4} + \cdots - 3716 \)
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