Newspace parameters
| Level: | \( N \) | \(=\) | \( 140 = 2^{2} \cdot 5 \cdot 7 \) |
| Weight: | \( k \) | \(=\) | \( 10 \) |
| Character orbit: | \([\chi]\) | \(=\) | 140.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(72.1050170629\) |
| Analytic rank: | \(1\) |
| Dimension: | \(4\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{4} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} - x^{3} - 9593x^{2} - 193749x - 162558 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{13}]\) |
| Coefficient ring index: | \( 2^{5}\cdot 3\cdot 5 \) |
| Twist minimal: | yes |
| 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
\( x^{4} - x^{3} - 9593x^{2} - 193749x - 162558 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( -\nu^{3} + 17\nu^{2} + 8271\nu + 68763 ) / 525 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -2\nu^{3} + 104\nu^{2} + 18922\nu - 198579 ) / 105 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -26\nu^{3} + 92\nu^{2} + 266146\nu + 3457338 ) / 1575 \)
|
| \(\nu\) | \(=\) |
\( ( 3\beta_{3} + \beta_{2} - 36\beta _1 + 21 ) / 120 \)
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| \(\nu^{2}\) | \(=\) |
\( ( -51\beta_{3} + 73\beta_{2} - 288\beta _1 + 287733 ) / 60 \)
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| \(\nu^{3}\) | \(=\) |
\( ( 23079\beta_{3} + 10753\beta_{2} - 370548\beta _1 + 18208173 ) / 120 \)
|
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 | −149.145 | 0 | −625.000 | 0 | 2401.00 | 0 | 2561.29 | 0 | ||||||||||||||||||||||||||||||
| 1.2 | 0 | −148.222 | 0 | −625.000 | 0 | 2401.00 | 0 | 2286.89 | 0 | |||||||||||||||||||||||||||||||
| 1.3 | 0 | 128.185 | 0 | −625.000 | 0 | 2401.00 | 0 | −3251.53 | 0 | |||||||||||||||||||||||||||||||
| 1.4 | 0 | 212.182 | 0 | −625.000 | 0 | 2401.00 | 0 | 25338.3 | 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 | 140.10.a.c | ✓ | 4 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 140.10.a.c | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{4} - 43T_{3}^{3} - 51909T_{3}^{2} + 563607T_{3} + 601271640 \)
acting on \(S_{10}^{\mathrm{new}}(\Gamma_0(140))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} \)
|
| $3$ |
\( T^{4} - 43 T^{3} + \cdots + 601271640 \)
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| $5$ |
\( (T + 625)^{4} \)
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| $7$ |
\( (T - 2401)^{4} \)
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| $11$ |
\( T^{4} + \cdots + 16\!\cdots\!00 \)
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| $13$ |
\( T^{4} + \cdots - 28\!\cdots\!18 \)
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| $17$ |
\( T^{4} + \cdots + 72\!\cdots\!50 \)
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| $19$ |
\( T^{4} + \cdots + 15\!\cdots\!96 \)
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| $23$ |
\( T^{4} + \cdots + 45\!\cdots\!88 \)
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| $29$ |
\( T^{4} + \cdots - 56\!\cdots\!58 \)
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| $31$ |
\( T^{4} + \cdots + 82\!\cdots\!08 \)
|
| $37$ |
\( T^{4} + \cdots - 62\!\cdots\!00 \)
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| $41$ |
\( T^{4} + \cdots + 31\!\cdots\!56 \)
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| $43$ |
\( T^{4} + \cdots + 44\!\cdots\!00 \)
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| $47$ |
\( T^{4} + \cdots - 20\!\cdots\!12 \)
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| $53$ |
\( T^{4} + \cdots + 83\!\cdots\!00 \)
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| $59$ |
\( T^{4} + \cdots + 23\!\cdots\!00 \)
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| $61$ |
\( T^{4} + \cdots - 73\!\cdots\!00 \)
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| $67$ |
\( T^{4} + \cdots + 51\!\cdots\!00 \)
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| $71$ |
\( T^{4} + \cdots + 11\!\cdots\!00 \)
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| $73$ |
\( T^{4} + \cdots - 46\!\cdots\!00 \)
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| $79$ |
\( T^{4} + \cdots - 21\!\cdots\!48 \)
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| $83$ |
\( T^{4} + \cdots - 21\!\cdots\!40 \)
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| $89$ |
\( T^{4} + \cdots - 12\!\cdots\!60 \)
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| $97$ |
\( T^{4} + \cdots + 61\!\cdots\!46 \)
|
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