Newspace parameters
| Level: | \( N \) | \(=\) | \( 2 \) |
| Weight: | \( k \) | \(=\) | \( 84 \) |
| Character orbit: | \([\chi]\) | \(=\) | 2.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(87.2544256533\) |
| Analytic rank: | \(1\) |
| 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: |
\( x^{3} - x^{2} - 76392209863211857938006422774x + 4214151671129618412000783695211690286445664 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2^{24}\cdot 3^{11}\cdot 5^{2}\cdot 7 \) |
| 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
\( x^{3} - x^{2} - 76392209863211857938006422774x + 4214151671129618412000783695211690286445664 \)
:
| \(\beta_{1}\) | \(=\) |
\( 311040\nu - 103680 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 163840\nu^{2} + 13557284114987572480\nu - 8344066442659091722136686534106880 ) / 95350401 \)
|
| \(\nu\) | \(=\) |
\( ( \beta _1 + 103680 ) / 311040 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 23170147443\beta_{2} - 10591628214834041\beta _1 + 2027608145566158190339201513794600960 ) / 39813120 \)
|
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 |
|
2.19902e12 | −8.12053e19 | 4.83570e24 | −9.96318e28 | −1.78572e32 | −5.66053e34 | 1.06338e37 | 2.60347e39 | −2.19093e41 | |||||||||||||||||||||||||||
| 1.2 | 2.19902e12 | −2.35615e19 | 4.83570e24 | 9.47163e28 | −5.18122e31 | 1.10671e35 | 1.06338e37 | −3.43570e39 | 2.08283e41 | ||||||||||||||||||||||||||||
| 1.3 | 2.19902e12 | 8.79012e19 | 4.83570e24 | −8.42086e28 | 1.93297e32 | 1.88466e34 | 1.06338e37 | 3.73579e39 | −1.85177e41 | ||||||||||||||||||||||||||||
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 | 2.84.a.b | ✓ | 3 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 2.84.a.b | ✓ | 3 | 1.a | even | 1 | 1 | trivial |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{3} + \cdots - 16\!\cdots\!88 \)
acting on \(S_{84}^{\mathrm{new}}(\Gamma_0(2))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T - 2199023255552)^{3} \)
|
| $3$ |
\( T^{3} + \cdots - 16\!\cdots\!88 \)
|
| $5$ |
\( T^{3} + \cdots - 79\!\cdots\!00 \)
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| $7$ |
\( T^{3} + \cdots + 11\!\cdots\!64 \)
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| $11$ |
\( T^{3} + \cdots + 54\!\cdots\!72 \)
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| $13$ |
\( T^{3} + \cdots - 80\!\cdots\!88 \)
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| $17$ |
\( T^{3} + \cdots + 58\!\cdots\!04 \)
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| $19$ |
\( T^{3} + \cdots + 29\!\cdots\!00 \)
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| $23$ |
\( T^{3} + \cdots - 75\!\cdots\!68 \)
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| $29$ |
\( T^{3} + \cdots + 49\!\cdots\!00 \)
|
| $31$ |
\( T^{3} + \cdots - 11\!\cdots\!08 \)
|
| $37$ |
\( T^{3} + \cdots - 27\!\cdots\!96 \)
|
| $41$ |
\( T^{3} + \cdots + 97\!\cdots\!72 \)
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| $43$ |
\( T^{3} + \cdots + 10\!\cdots\!32 \)
|
| $47$ |
\( T^{3} + \cdots - 33\!\cdots\!36 \)
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| $53$ |
\( T^{3} + \cdots + 47\!\cdots\!52 \)
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| $59$ |
\( T^{3} + \cdots + 15\!\cdots\!00 \)
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| $61$ |
\( T^{3} + \cdots - 11\!\cdots\!08 \)
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| $67$ |
\( T^{3} + \cdots + 15\!\cdots\!24 \)
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| $71$ |
\( T^{3} + \cdots - 16\!\cdots\!68 \)
|
| $73$ |
\( T^{3} + \cdots - 15\!\cdots\!28 \)
|
| $79$ |
\( T^{3} + \cdots - 58\!\cdots\!00 \)
|
| $83$ |
\( T^{3} + \cdots - 79\!\cdots\!48 \)
|
| $89$ |
\( T^{3} + \cdots + 72\!\cdots\!00 \)
|
| $97$ |
\( T^{3} + \cdots - 76\!\cdots\!16 \)
|
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