Newspace parameters
| Level: | \( N \) | \(=\) | \( 2499 = 3 \cdot 7^{2} \cdot 17 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 2499.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(19.9546154651\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | 4.4.14013.1 |
|
|
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| Defining polynomial: |
\( x^{4} - x^{3} - 6x^{2} + 6x + 3 \)
|
| Coefficient ring: | \(\Z[a_1, a_2]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 357) |
| Fricke sign: | \(-1\) |
| Sato-Tate group: | $\mathrm{SU}(2)$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) 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
\( x^{4} - x^{3} - 6x^{2} + 6x + 3 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 3 \)
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| \(\beta_{3}\) | \(=\) |
\( \nu^{3} - 5\nu + 1 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 3 \)
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| \(\nu^{3}\) | \(=\) |
\( \beta_{3} + 5\beta _1 - 1 \)
|
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 |
|
−2.20800 | 1.00000 | 2.87525 | 0.275456 | −2.20800 | 0 | −1.93254 | 1.00000 | −0.608206 | ||||||||||||||||||||||||||||||
| 1.2 | −1.53652 | 1.00000 | 0.360904 | 4.05503 | −1.53652 | 0 | 2.51851 | 1.00000 | −6.23065 | |||||||||||||||||||||||||||||||
| 1.3 | 0.372845 | 1.00000 | −1.86099 | −1.81240 | 0.372845 | 0 | −1.43955 | 1.00000 | −0.675744 | |||||||||||||||||||||||||||||||
| 1.4 | 2.37167 | 1.00000 | 3.62484 | 1.48191 | 2.37167 | 0 | 3.85358 | 1.00000 | 3.51460 | |||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \( -1 \) |
| \(7\) | \( +1 \) |
| \(17\) | \( +1 \) |
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 | 2499.2.a.w | 4 | |
| 3.b | odd | 2 | 1 | 7497.2.a.bm | 4 | ||
| 7.b | odd | 2 | 1 | 2499.2.a.v | 4 | ||
| 7.c | even | 3 | 2 | 357.2.i.d | ✓ | 8 | |
| 21.c | even | 2 | 1 | 7497.2.a.bn | 4 | ||
| 21.h | odd | 6 | 2 | 1071.2.i.f | 8 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 357.2.i.d | ✓ | 8 | 7.c | even | 3 | 2 | |
| 1071.2.i.f | 8 | 21.h | odd | 6 | 2 | ||
| 2499.2.a.v | 4 | 7.b | odd | 2 | 1 | ||
| 2499.2.a.w | 4 | 1.a | even | 1 | 1 | trivial | |
| 7497.2.a.bm | 4 | 3.b | odd | 2 | 1 | ||
| 7497.2.a.bn | 4 | 21.c | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(2499))\):
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\( T_{2}^{4} + T_{2}^{3} - 6T_{2}^{2} - 6T_{2} + 3 \)
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\( T_{5}^{4} - 4T_{5}^{3} - 3T_{5}^{2} + 12T_{5} - 3 \)
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\( T_{11}^{4} - T_{11}^{3} - 6T_{11}^{2} + 6T_{11} + 3 \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} + T^{3} - 6 T^{2} + \cdots + 3 \)
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| $3$ |
\( (T - 1)^{4} \)
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| $5$ |
\( T^{4} - 4 T^{3} + \cdots - 3 \)
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| $7$ |
\( T^{4} \)
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| $11$ |
\( T^{4} - T^{3} - 6 T^{2} + \cdots + 3 \)
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| $13$ |
\( T^{4} - 10 T^{3} + \cdots - 103 \)
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| $17$ |
\( (T + 1)^{4} \)
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| $19$ |
\( T^{4} - 13 T^{3} + \cdots - 367 \)
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| $23$ |
\( T^{4} - 3 T^{3} + \cdots + 27 \)
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| $29$ |
\( T^{4} - 7 T^{3} + \cdots + 147 \)
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| $31$ |
\( T^{4} + 6 T^{3} + \cdots - 63 \)
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| $37$ |
\( T^{4} + 6 T^{3} + \cdots - 21 \)
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| $41$ |
\( T^{4} - 26 T^{3} + \cdots - 4527 \)
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| $43$ |
\( T^{4} + 4 T^{3} + \cdots - 929 \)
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| $47$ |
\( T^{4} + 6 T^{3} + \cdots - 189 \)
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| $53$ |
\( T^{4} - 15 T^{3} + \cdots - 1827 \)
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| $59$ |
\( T^{4} - 15 T^{3} + \cdots + 2457 \)
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| $61$ |
\( T^{4} + 11 T^{3} + \cdots - 1477 \)
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| $67$ |
\( T^{4} + 8 T^{3} + \cdots + 16349 \)
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| $71$ |
\( T^{4} - 15 T^{3} + \cdots - 531 \)
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| $73$ |
\( T^{4} - 3 T^{3} + \cdots + 6951 \)
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| $79$ |
\( T^{4} - 5 T^{3} + \cdots + 2557 \)
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| $83$ |
\( T^{4} + 12 T^{3} + \cdots - 783 \)
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| $89$ |
\( T^{4} - 16 T^{3} + \cdots + 12537 \)
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| $97$ |
\( T^{4} - 3 T^{3} + \cdots + 15567 \)
|
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