Newspace parameters
| Level: | \( N \) | \(=\) | \( 2 \) |
| Weight: | \( k \) | \(=\) | \( 70 \) |
| Character orbit: | \([\chi]\) | \(=\) | 2.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(60.3029906584\) |
| Analytic rank: | \(0\) |
| Dimension: | \(3\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{3} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{3} - x^{2} - 90823593763357992496952650x - 190864638684342453433100893828514036648 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2^{24}\cdot 3^{5}\cdot 5^{4}\cdot 7\cdot 23 \) |
| 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 \(1,\beta_1,\beta_2\) 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
\( x^{3} - x^{2} - 90823593763357992496952650x - 190864638684342453433100893828514036648 \)
:
| \(\beta_{1}\) | \(=\) |
\( 3840\nu - 1280 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 819200\nu^{2} - 2582307687984721920\nu - 49601792007294384199773079285760 ) / 21186639 \)
|
| \(\nu\) | \(=\) |
\( ( \beta _1 + 1280 ) / 3840 \)
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| \(\nu^{2}\) | \(=\) |
\( ( 21186639\beta_{2} + 672475960412688\beta _1 + 49601792007295244969002407526400 ) / 819200 \)
|
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 |
|
1.71799e10 | −3.94412e16 | 2.95148e20 | 9.11410e23 | −6.77595e26 | 2.34869e29 | 5.07060e30 | 7.21225e32 | 1.56579e34 | |||||||||||||||||||||||||||
| 1.2 | 1.71799e10 | −1.64001e16 | 2.95148e20 | −2.12060e24 | −2.81752e26 | −1.60552e29 | 5.07060e30 | −5.65421e32 | −3.64316e34 | ||||||||||||||||||||||||||||
| 1.3 | 1.71799e10 | 3.22422e16 | 2.95148e20 | 6.21700e23 | 5.53917e26 | 1.85055e28 | 5.07060e30 | 2.05176e32 | 1.06807e34 | ||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(-1\) |
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 | 2.70.a.b | ✓ | 3 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 2.70.a.b | ✓ | 3 | 1.a | even | 1 | 1 | trivial |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{3} + \cdots - 20\!\cdots\!96 \)
acting on \(S_{70}^{\mathrm{new}}(\Gamma_0(2))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T - 17179869184)^{3} \)
|
| $3$ |
\( T^{3} + \cdots - 20\!\cdots\!96 \)
|
| $5$ |
\( T^{3} + \cdots + 12\!\cdots\!00 \)
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| $7$ |
\( T^{3} + \cdots + 69\!\cdots\!52 \)
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| $11$ |
\( T^{3} + \cdots - 98\!\cdots\!48 \)
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| $13$ |
\( T^{3} + \cdots + 18\!\cdots\!24 \)
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| $17$ |
\( T^{3} + \cdots - 11\!\cdots\!88 \)
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| $19$ |
\( T^{3} + \cdots - 21\!\cdots\!00 \)
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| $23$ |
\( T^{3} + \cdots - 31\!\cdots\!16 \)
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| $29$ |
\( T^{3} + \cdots - 79\!\cdots\!00 \)
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| $31$ |
\( T^{3} + \cdots - 14\!\cdots\!68 \)
|
| $37$ |
\( T^{3} + \cdots - 34\!\cdots\!08 \)
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| $41$ |
\( T^{3} + \cdots - 19\!\cdots\!08 \)
|
| $43$ |
\( T^{3} + \cdots - 15\!\cdots\!76 \)
|
| $47$ |
\( T^{3} + \cdots + 37\!\cdots\!12 \)
|
| $53$ |
\( T^{3} + \cdots - 11\!\cdots\!16 \)
|
| $59$ |
\( T^{3} + \cdots - 14\!\cdots\!00 \)
|
| $61$ |
\( T^{3} + \cdots + 12\!\cdots\!72 \)
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| $67$ |
\( T^{3} + \cdots - 19\!\cdots\!28 \)
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| $71$ |
\( T^{3} + \cdots - 62\!\cdots\!08 \)
|
| $73$ |
\( T^{3} + \cdots - 27\!\cdots\!36 \)
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| $79$ |
\( T^{3} + \cdots - 26\!\cdots\!00 \)
|
| $83$ |
\( T^{3} + \cdots - 20\!\cdots\!56 \)
|
| $89$ |
\( T^{3} + \cdots + 72\!\cdots\!00 \)
|
| $97$ |
\( T^{3} + \cdots - 76\!\cdots\!28 \)
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