Newspace parameters
| Level: | \( N \) | \(=\) | \( 2 \) |
| Weight: | \( k \) | \(=\) | \( 76 \) |
| Character orbit: | \([\chi]\) | \(=\) | 2.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(71.2456785644\) |
| 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: |
\( x^{3} - x^{2} - 69170275937846809816619130x + 194820240227429941309097792198860377672 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2^{24}\cdot 3^{8}\cdot 5^{3}\cdot 7\cdot 11 \) |
| 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
\( x^{3} - x^{2} - 69170275937846809816619130x + 194820240227429941309097792198860377672 \)
:
| \(\beta_{1}\) | \(=\) |
\( 103680\nu - 34560 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 163840\nu^{2} + 692190646650118400\nu - 7555238673104778277118802266880 ) / 2049237 \)
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| \(\nu\) | \(=\) |
\( ( \beta _1 + 34560 ) / 103680 \)
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| \(\nu^{2}\) | \(=\) |
\( ( 165988197\beta_{2} - 540773942695405\beta _1 + 611974332521468351299163430420480 ) / 13271040 \)
|
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 |
|
1.37439e11 | −8.29405e17 | 1.88895e22 | −2.22623e26 | −1.13993e29 | −1.55415e31 | 2.59615e33 | 7.96464e34 | −3.05970e37 | |||||||||||||||||||||||||||
| 1.2 | 1.37439e11 | 5.02117e17 | 1.88895e22 | −7.52245e25 | 6.90104e28 | 6.86531e31 | 2.59615e33 | −3.56145e35 | −1.03388e37 | ||||||||||||||||||||||||||||
| 1.3 | 1.37439e11 | 7.85531e17 | 1.88895e22 | 1.02529e26 | 1.07962e29 | −9.01778e31 | 2.59615e33 | 8.79144e33 | 1.40915e37 | ||||||||||||||||||||||||||||
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 | 2.76.a.b | ✓ | 3 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 2.76.a.b | ✓ | 3 | 1.a | even | 1 | 1 | trivial |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{3} + \cdots + 32\!\cdots\!52 \)
acting on \(S_{76}^{\mathrm{new}}(\Gamma_0(2))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T - 137438953472)^{3} \)
|
| $3$ |
\( T^{3} + \cdots + 32\!\cdots\!52 \)
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| $5$ |
\( T^{3} + \cdots - 17\!\cdots\!00 \)
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| $7$ |
\( T^{3} + \cdots - 96\!\cdots\!96 \)
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| $11$ |
\( T^{3} + \cdots + 50\!\cdots\!52 \)
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| $13$ |
\( T^{3} + \cdots - 29\!\cdots\!08 \)
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| $17$ |
\( T^{3} + \cdots + 42\!\cdots\!24 \)
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| $19$ |
\( T^{3} + \cdots - 18\!\cdots\!00 \)
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| $23$ |
\( T^{3} + \cdots + 27\!\cdots\!12 \)
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| $29$ |
\( T^{3} + \cdots - 22\!\cdots\!00 \)
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| $31$ |
\( T^{3} + \cdots - 24\!\cdots\!88 \)
|
| $37$ |
\( T^{3} + \cdots - 90\!\cdots\!56 \)
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| $41$ |
\( T^{3} + \cdots - 83\!\cdots\!28 \)
|
| $43$ |
\( T^{3} + \cdots + 78\!\cdots\!92 \)
|
| $47$ |
\( T^{3} + \cdots - 10\!\cdots\!56 \)
|
| $53$ |
\( T^{3} + \cdots - 24\!\cdots\!88 \)
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| $59$ |
\( T^{3} + \cdots - 12\!\cdots\!00 \)
|
| $61$ |
\( T^{3} + \cdots + 25\!\cdots\!32 \)
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| $67$ |
\( T^{3} + \cdots - 78\!\cdots\!96 \)
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| $71$ |
\( T^{3} + \cdots - 31\!\cdots\!68 \)
|
| $73$ |
\( T^{3} + \cdots - 11\!\cdots\!28 \)
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| $79$ |
\( T^{3} + \cdots + 85\!\cdots\!00 \)
|
| $83$ |
\( T^{3} + \cdots - 57\!\cdots\!68 \)
|
| $89$ |
\( T^{3} + \cdots - 56\!\cdots\!00 \)
|
| $97$ |
\( T^{3} + \cdots - 73\!\cdots\!76 \)
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