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: | 3.3.78693.1 |
|
|
|
| Defining polynomial: |
\( x^{3} - x^{2} - 68x + 48 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| 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
\( x^{3} - x^{2} - 68x + 48 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 46 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
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| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 46 \)
|
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 | −7.40831 | 0 | −5.00000 | 0 | 7.00000 | 0 | 27.8831 | 0 | |||||||||||||||||||||||||||
| 1.2 | 0 | 0.296275 | 0 | −5.00000 | 0 | 7.00000 | 0 | −26.9122 | 0 | ||||||||||||||||||||||||||||
| 1.3 | 0 | 9.11203 | 0 | −5.00000 | 0 | 7.00000 | 0 | 56.0292 | 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.bw | 3 | |
| 4.b | odd | 2 | 1 | 2240.4.a.bs | 3 | ||
| 8.b | even | 2 | 1 | 560.4.a.v | 3 | ||
| 8.d | odd | 2 | 1 | 280.4.a.i | ✓ | 3 | |
| 40.e | odd | 2 | 1 | 1400.4.a.l | 3 | ||
| 40.k | even | 4 | 2 | 1400.4.g.j | 6 | ||
| 56.e | even | 2 | 1 | 1960.4.a.o | 3 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 280.4.a.i | ✓ | 3 | 8.d | odd | 2 | 1 | |
| 560.4.a.v | 3 | 8.b | even | 2 | 1 | ||
| 1400.4.a.l | 3 | 40.e | odd | 2 | 1 | ||
| 1400.4.g.j | 6 | 40.k | even | 4 | 2 | ||
| 1960.4.a.o | 3 | 56.e | even | 2 | 1 | ||
| 2240.4.a.bs | 3 | 4.b | odd | 2 | 1 | ||
| 2240.4.a.bw | 3 | 1.a | even | 1 | 1 | trivial | |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(2240))\):
|
\( T_{3}^{3} - 2T_{3}^{2} - 67T_{3} + 20 \)
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|
\( T_{11}^{3} + 22T_{11}^{2} - 1679T_{11} - 19044 \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{3} \)
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| $3$ |
\( T^{3} - 2 T^{2} + \cdots + 20 \)
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| $5$ |
\( (T + 5)^{3} \)
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| $7$ |
\( (T - 7)^{3} \)
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| $11$ |
\( T^{3} + 22 T^{2} + \cdots - 19044 \)
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| $13$ |
\( T^{3} + 80 T^{2} + \cdots + 17166 \)
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| $17$ |
\( T^{3} - 88 T^{2} + \cdots - 8586 \)
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| $19$ |
\( T^{3} - 100 T^{2} + \cdots + 431664 \)
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| $23$ |
\( T^{3} + 136 T^{2} + \cdots - 4552800 \)
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| $29$ |
\( T^{3} + 132 T^{2} + \cdots - 336474 \)
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| $31$ |
\( T^{3} - 108 T^{2} + \cdots - 235008 \)
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| $37$ |
\( T^{3} + 438 T^{2} + \cdots - 14075064 \)
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| $41$ |
\( T^{3} - 374 T^{2} + \cdots + 5640224 \)
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| $43$ |
\( T^{3} - 140 T^{2} + \cdots + 11118288 \)
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| $47$ |
\( T^{3} + 230 T^{2} + \cdots - 40757952 \)
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| $53$ |
\( T^{3} + 238 T^{2} + \cdots - 62699136 \)
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| $59$ |
\( T^{3} + 356 T^{2} + \cdots + 8606400 \)
|
| $61$ |
\( T^{3} + 1210 T^{2} + \cdots + 51565344 \)
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| $67$ |
\( T^{3} - 636 T^{2} + \cdots + 262147072 \)
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| $71$ |
\( T^{3} - 872 T^{2} + \cdots + 610165248 \)
|
| $73$ |
\( T^{3} + 306 T^{2} + \cdots - 24653320 \)
|
| $79$ |
\( T^{3} - 1814 T^{2} + \cdots - 6393408 \)
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| $83$ |
\( T^{3} + 1764 T^{2} + \cdots - 69164352 \)
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| $89$ |
\( T^{3} - 78 T^{2} + \cdots + 189480352 \)
|
| $97$ |
\( T^{3} + \cdots + 1147113622 \)
|
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