Newspace parameters
| Level: | \( N \) | \(=\) | \( 70 = 2 \cdot 5 \cdot 7 \) |
| Weight: | \( k \) | \(=\) | \( 18 \) |
| Character orbit: | \([\chi]\) | \(=\) | 70.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(128.255461141\) |
| Analytic rank: | \(0\) |
| 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: |
\( x^{4} - x^{3} - 24827115x^{2} + 13898107909x + 88638955281254 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{13}]\) |
| Coefficient ring index: | \( 2^{5}\cdot 3^{4}\cdot 5^{2}\cdot 7^{2} \) |
| 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
\( x^{4} - x^{3} - 24827115x^{2} + 13898107909x + 88638955281254 \)
:
| \(\beta_{1}\) | \(=\) |
\( 3\nu - 1 \)
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| \(\beta_{2}\) | \(=\) |
\( ( -\nu^{3} + 1958\nu^{2} + 19322209\nu - 34715537222 ) / 5850 \)
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| \(\beta_{3}\) | \(=\) |
\( ( \nu^{3} + 3892\nu^{2} - 14419909\nu - 37905001678 ) / 1950 \)
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| \(\nu\) | \(=\) |
\( ( \beta _1 + 1 ) / 3 \)
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| \(\nu^{2}\) | \(=\) |
\( ( \beta_{3} + 3\beta_{2} - 838\beta _1 + 37240464 ) / 3 \)
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| \(\nu^{3}\) | \(=\) |
\( ( 1958\beta_{3} - 11676\beta_{2} + 17681405\beta _1 - 31210460945 ) / 3 \)
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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 |
|
−256.000 | −15591.4 | 65536.0 | −390625. | 3.99140e6 | 5.76480e6 | −1.67772e7 | 1.13951e8 | 1.00000e8 | ||||||||||||||||||||||||||||||
| 1.2 | −256.000 | −6067.63 | 65536.0 | −390625. | 1.55331e6 | 5.76480e6 | −1.67772e7 | −9.23240e7 | 1.00000e8 | |||||||||||||||||||||||||||||||
| 1.3 | −256.000 | 7073.06 | 65536.0 | −390625. | −1.81070e6 | 5.76480e6 | −1.67772e7 | −7.91121e7 | 1.00000e8 | |||||||||||||||||||||||||||||||
| 1.4 | −256.000 | 10965.0 | 65536.0 | −390625. | −2.80703e6 | 5.76480e6 | −1.67772e7 | −8.90977e6 | 1.00000e8 | |||||||||||||||||||||||||||||||
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 | 70.18.a.a | ✓ | 4 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 70.18.a.a | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{4} + 3621T_{3}^{3} - 218527173T_{3}^{2} - 26664355737T_{3} + 7336991526244920 \)
acting on \(S_{18}^{\mathrm{new}}(\Gamma_0(70))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T + 256)^{4} \)
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| $3$ |
\( T^{4} + \cdots + 73\!\cdots\!20 \)
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| $5$ |
\( (T + 390625)^{4} \)
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| $7$ |
\( (T - 5764801)^{4} \)
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| $11$ |
\( T^{4} + \cdots + 64\!\cdots\!00 \)
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| $13$ |
\( T^{4} + \cdots + 16\!\cdots\!26 \)
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| $17$ |
\( T^{4} + \cdots + 53\!\cdots\!58 \)
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| $19$ |
\( T^{4} + \cdots + 11\!\cdots\!40 \)
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| $23$ |
\( T^{4} + \cdots + 28\!\cdots\!00 \)
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| $29$ |
\( T^{4} + \cdots + 41\!\cdots\!94 \)
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| $31$ |
\( T^{4} + \cdots - 16\!\cdots\!12 \)
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| $37$ |
\( T^{4} + \cdots + 13\!\cdots\!00 \)
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| $41$ |
\( T^{4} + \cdots - 12\!\cdots\!88 \)
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| $43$ |
\( T^{4} + \cdots - 30\!\cdots\!68 \)
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| $47$ |
\( T^{4} + \cdots + 51\!\cdots\!44 \)
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| $53$ |
\( T^{4} + \cdots + 10\!\cdots\!72 \)
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| $59$ |
\( T^{4} + \cdots - 14\!\cdots\!92 \)
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| $61$ |
\( T^{4} + \cdots + 15\!\cdots\!12 \)
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| $67$ |
\( T^{4} + \cdots - 97\!\cdots\!68 \)
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| $71$ |
\( T^{4} + \cdots - 21\!\cdots\!00 \)
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| $73$ |
\( T^{4} + \cdots - 42\!\cdots\!76 \)
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| $79$ |
\( T^{4} + \cdots + 52\!\cdots\!44 \)
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| $83$ |
\( T^{4} + \cdots - 53\!\cdots\!92 \)
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| $89$ |
\( T^{4} + \cdots - 27\!\cdots\!40 \)
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| $97$ |
\( T^{4} + \cdots - 54\!\cdots\!50 \)
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