Newspace parameters
| Level: | \( N \) | \(=\) | \( 2240 = 2^{6} \cdot 5 \cdot 7 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 2240.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(132.164278413\) |
| Analytic rank: | \(0\) |
| Dimension: | \(3\) |
| Coefficient field: | 3.3.6053.1 |
|
|
|
| Defining polynomial: |
\( x^{3} - 20x - 31 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| Coefficient ring index: | \( 2^{2} \) |
| Twist minimal: | no (minimal twist has level 280) |
| Fricke sign: | \(+1\) |
| Sato-Tate group: | $\mathrm{SU}(2)$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2\) 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
\( x^{3} - 20x - 31 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu^{2} - \nu - 13 \)
|
| \(\beta_{2}\) | \(=\) |
\( -2\nu^{2} + 6\nu + 27 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{2} + 2\beta _1 - 1 ) / 4 \)
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| \(\nu^{2}\) | \(=\) |
\( ( \beta_{2} + 6\beta _1 + 51 ) / 4 \)
|
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 | −8.56288 | 0 | 5.00000 | 0 | 7.00000 | 0 | 46.3230 | 0 | |||||||||||||||||||||||||||
| 1.2 | 0 | −0.402395 | 0 | 5.00000 | 0 | 7.00000 | 0 | −26.8381 | 0 | ||||||||||||||||||||||||||||
| 1.3 | 0 | 6.96528 | 0 | 5.00000 | 0 | 7.00000 | 0 | 21.5151 | 0 | ||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \( +1 \) |
| \(5\) | \( -1 \) |
| \(7\) | \( -1 \) |
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 | 2240.4.a.bu | 3 | |
| 4.b | odd | 2 | 1 | 2240.4.a.bx | 3 | ||
| 8.b | even | 2 | 1 | 280.4.a.h | ✓ | 3 | |
| 8.d | odd | 2 | 1 | 560.4.a.t | 3 | ||
| 40.f | even | 2 | 1 | 1400.4.a.k | 3 | ||
| 40.i | odd | 4 | 2 | 1400.4.g.k | 6 | ||
| 56.h | odd | 2 | 1 | 1960.4.a.p | 3 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 280.4.a.h | ✓ | 3 | 8.b | even | 2 | 1 | |
| 560.4.a.t | 3 | 8.d | odd | 2 | 1 | ||
| 1400.4.a.k | 3 | 40.f | even | 2 | 1 | ||
| 1400.4.g.k | 6 | 40.i | odd | 4 | 2 | ||
| 1960.4.a.p | 3 | 56.h | odd | 2 | 1 | ||
| 2240.4.a.bu | 3 | 1.a | even | 1 | 1 | trivial | |
| 2240.4.a.bx | 3 | 4.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 \(S_{4}^{\mathrm{new}}(\Gamma_0(2240))\):
|
\( T_{3}^{3} + 2T_{3}^{2} - 59T_{3} - 24 \)
|
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\( T_{11}^{3} + 10T_{11}^{2} - 2855T_{11} - 41444 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{3} \)
|
| $3$ |
\( T^{3} + 2 T^{2} + \cdots - 24 \)
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| $5$ |
\( (T - 5)^{3} \)
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| $7$ |
\( (T - 7)^{3} \)
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| $11$ |
\( T^{3} + 10 T^{2} + \cdots - 41444 \)
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| $13$ |
\( T^{3} - 64 T^{2} + \cdots + 222154 \)
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| $17$ |
\( T^{3} - 132 T^{2} + \cdots + 32150 \)
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| $19$ |
\( T^{3} + 160 T^{2} + \cdots - 190112 \)
|
| $23$ |
\( T^{3} - 40 T^{2} + \cdots + 368384 \)
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| $29$ |
\( T^{3} + 348 T^{2} + \cdots - 4436914 \)
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| $31$ |
\( T^{3} - 396 T^{2} + \cdots + 22286720 \)
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| $37$ |
\( T^{3} - 18 T^{2} + \cdots - 4740504 \)
|
| $41$ |
\( T^{3} - 478 T^{2} + \cdots + 86470176 \)
|
| $43$ |
\( T^{3} - 220 T^{2} + \cdots - 272176 \)
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| $47$ |
\( T^{3} + 270 T^{2} + \cdots - 9181688 \)
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| $53$ |
\( T^{3} + 718 T^{2} + \cdots - 45700128 \)
|
| $59$ |
\( T^{3} + 480 T^{2} + \cdots - 111751168 \)
|
| $61$ |
\( T^{3} + 358 T^{2} + \cdots + 751840 \)
|
| $67$ |
\( T^{3} - 2172 T^{2} + \cdots - 333271616 \)
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| $71$ |
\( T^{3} - 368 T^{2} + \cdots + 307200 \)
|
| $73$ |
\( T^{3} - 510 T^{2} + \cdots + 488314008 \)
|
| $79$ |
\( T^{3} + 150 T^{2} + \cdots - 17020488 \)
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| $83$ |
\( T^{3} + 1056 T^{2} + \cdots - 376174336 \)
|
| $89$ |
\( T^{3} - 422 T^{2} + \cdots - 18067296 \)
|
| $97$ |
\( T^{3} + 1172 T^{2} + \cdots - 204617466 \)
|
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