Newspace parameters
| Level: | \( N \) | \(=\) | \( 3 \) |
| Weight: | \( k \) | \(=\) | \( 44 \) |
| Character orbit: | \([\chi]\) | \(=\) | 3.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(35.1331186037\) |
| 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: |
\( x^{3} - x^{2} - 908401710x + 974756489742 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2^{13}\cdot 3^{5}\cdot 5\cdot 11 \) |
| 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
\( x^{3} - x^{2} - 908401710x + 974756489742 \)
:
| \(\beta_{1}\) | \(=\) |
\( 96\nu - 32 \)
|
| \(\beta_{2}\) | \(=\) |
\( 2304\nu^{2} + 3705792\nu - 1395306262592 \)
|
| \(\nu\) | \(=\) |
\( ( \beta _1 + 32 ) / 96 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{2} - 38602\beta _1 + 1395305027328 ) / 2304 \)
|
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 |
|
−1.22147e6 | 1.04604e10 | −7.30411e12 | −8.84097e14 | −1.27770e16 | 1.48034e18 | 1.96659e19 | 1.09419e20 | 1.07990e21 | |||||||||||||||||||||||||||
| 1.2 | 1.51590e6 | 1.04604e10 | −6.49815e12 | 1.66864e15 | 1.58568e16 | −2.14679e18 | −2.31845e19 | 1.09419e20 | 2.52949e21 | ||||||||||||||||||||||||||||
| 1.3 | 4.56260e6 | 1.04604e10 | 1.20212e13 | −1.29230e15 | 4.77264e16 | −9.66721e17 | 1.47149e19 | 1.09419e20 | −5.89624e21 | ||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \(-1\) |
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 | 3.44.a.a | ✓ | 3 |
| 3.b | odd | 2 | 1 | 9.44.a.a | 3 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 3.44.a.a | ✓ | 3 | 1.a | even | 1 | 1 | trivial |
| 9.44.a.a | 3 | 3.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{3} - 4857024T_{2}^{2} - 508269450240T_{2} + 8448203181300645888 \)
acting on \(S_{44}^{\mathrm{new}}(\Gamma_0(3))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{3} + \cdots + 84\!\cdots\!88 \)
|
| $3$ |
\( (T - 10460353203)^{3} \)
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| $5$ |
\( T^{3} + \cdots - 19\!\cdots\!00 \)
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| $7$ |
\( T^{3} + \cdots - 30\!\cdots\!20 \)
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| $11$ |
\( T^{3} + \cdots + 74\!\cdots\!76 \)
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| $13$ |
\( T^{3} + \cdots - 37\!\cdots\!44 \)
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| $17$ |
\( T^{3} + \cdots - 21\!\cdots\!16 \)
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| $19$ |
\( T^{3} + \cdots - 10\!\cdots\!00 \)
|
| $23$ |
\( T^{3} + \cdots - 34\!\cdots\!20 \)
|
| $29$ |
\( T^{3} + \cdots - 49\!\cdots\!00 \)
|
| $31$ |
\( T^{3} + \cdots - 12\!\cdots\!00 \)
|
| $37$ |
\( T^{3} + \cdots - 14\!\cdots\!20 \)
|
| $41$ |
\( T^{3} + \cdots + 12\!\cdots\!40 \)
|
| $43$ |
\( T^{3} + \cdots + 44\!\cdots\!48 \)
|
| $47$ |
\( T^{3} + \cdots - 64\!\cdots\!64 \)
|
| $53$ |
\( T^{3} + \cdots - 10\!\cdots\!40 \)
|
| $59$ |
\( T^{3} + \cdots + 15\!\cdots\!00 \)
|
| $61$ |
\( T^{3} + \cdots + 81\!\cdots\!88 \)
|
| $67$ |
\( T^{3} + \cdots + 18\!\cdots\!04 \)
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| $71$ |
\( T^{3} + \cdots - 39\!\cdots\!48 \)
|
| $73$ |
\( T^{3} + \cdots - 65\!\cdots\!40 \)
|
| $79$ |
\( T^{3} + \cdots - 44\!\cdots\!00 \)
|
| $83$ |
\( T^{3} + \cdots + 34\!\cdots\!44 \)
|
| $89$ |
\( T^{3} + \cdots + 12\!\cdots\!00 \)
|
| $97$ |
\( T^{3} + \cdots + 10\!\cdots\!24 \)
|
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