Newspace parameters
| Level: | \( N \) | \(=\) | \( 1728 = 2^{6} \cdot 3^{3} \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1728.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(101.955300490\) |
| Analytic rank: | \(1\) |
| Dimension: | \(3\) |
| Coefficient field: | 3.3.2708.1 |
|
|
|
| Defining polynomial: |
\( x^{3} - x^{2} - 11x - 7 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{3}\cdot 3 \) |
| Twist minimal: | no (minimal twist has level 864) |
| 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
\( x^{3} - x^{2} - 11x - 7 \)
:
| \(\beta_{1}\) | \(=\) |
\( 2\nu^{2} - 15 \)
|
| \(\beta_{2}\) | \(=\) |
\( -4\nu^{2} + 12\nu + 27 \)
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| \(\nu\) | \(=\) |
\( ( \beta_{2} + 2\beta _1 + 3 ) / 12 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta _1 + 15 ) / 2 \)
|
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 |
|
0 | 0 | 0 | −21.6142 | 0 | −14.9673 | 0 | 0 | 0 | |||||||||||||||||||||||||||
| 1.2 | 0 | 0 | 0 | 0.640862 | 0 | 18.3165 | 0 | 0 | 0 | ||||||||||||||||||||||||||||
| 1.3 | 0 | 0 | 0 | 10.9734 | 0 | −22.3492 | 0 | 0 | 0 | ||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \( +1 \) |
| \(3\) | \( -1 \) |
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 | 1728.4.a.bu | 3 | |
| 3.b | odd | 2 | 1 | 1728.4.a.ce | 3 | ||
| 4.b | odd | 2 | 1 | 1728.4.a.bv | 3 | ||
| 8.b | even | 2 | 1 | 864.4.a.s | yes | 3 | |
| 8.d | odd | 2 | 1 | 864.4.a.t | yes | 3 | |
| 12.b | even | 2 | 1 | 1728.4.a.cf | 3 | ||
| 24.f | even | 2 | 1 | 864.4.a.j | yes | 3 | |
| 24.h | odd | 2 | 1 | 864.4.a.i | ✓ | 3 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 864.4.a.i | ✓ | 3 | 24.h | odd | 2 | 1 | |
| 864.4.a.j | yes | 3 | 24.f | even | 2 | 1 | |
| 864.4.a.s | yes | 3 | 8.b | even | 2 | 1 | |
| 864.4.a.t | yes | 3 | 8.d | odd | 2 | 1 | |
| 1728.4.a.bu | 3 | 1.a | even | 1 | 1 | trivial | |
| 1728.4.a.bv | 3 | 4.b | odd | 2 | 1 | ||
| 1728.4.a.ce | 3 | 3.b | odd | 2 | 1 | ||
| 1728.4.a.cf | 3 | 12.b | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1728))\):
|
\( T_{5}^{3} + 10T_{5}^{2} - 244T_{5} + 152 \)
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\( T_{7}^{3} + 19T_{7}^{2} - 349T_{7} - 6127 \)
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\( T_{11}^{3} - 42T_{11}^{2} - 1620T_{11} + 61736 \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{3} \)
|
| $3$ |
\( T^{3} \)
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| $5$ |
\( T^{3} + 10 T^{2} + \cdots + 152 \)
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| $7$ |
\( T^{3} + 19 T^{2} + \cdots - 6127 \)
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| $11$ |
\( T^{3} - 42 T^{2} + \cdots + 61736 \)
|
| $13$ |
\( T^{3} + 15 T^{2} + \cdots - 65219 \)
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| $17$ |
\( T^{3} - 6 T^{2} + \cdots - 29160 \)
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| $19$ |
\( T^{3} - 13 T^{2} + \cdots - 47375 \)
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| $23$ |
\( T^{3} + 42 T^{2} + \cdots + 803736 \)
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| $29$ |
\( T^{3} + 328 T^{2} + \cdots - 2232832 \)
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| $31$ |
\( T^{3} - 88 T^{2} + \cdots - 2593280 \)
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| $37$ |
\( T^{3} - 99 T^{2} + \cdots - 1220321 \)
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| $41$ |
\( T^{3} - 424 T^{2} + \cdots - 1158656 \)
|
| $43$ |
\( T^{3} + 316 T^{2} + \cdots - 1941184 \)
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| $47$ |
\( T^{3} - 534 T^{2} + \cdots + 15342488 \)
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| $53$ |
\( T^{3} + 692 T^{2} + \cdots - 38325056 \)
|
| $59$ |
\( T^{3} - 186 T^{2} + \cdots + 11865448 \)
|
| $61$ |
\( T^{3} - 147 T^{2} + \cdots + 25588143 \)
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| $67$ |
\( T^{3} + 619 T^{2} + \cdots - 350147287 \)
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| $71$ |
\( T^{3} - 540 T^{2} + \cdots - 18963520 \)
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| $73$ |
\( T^{3} + 381 T^{2} + \cdots - 100255617 \)
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| $79$ |
\( T^{3} + 1241 T^{2} + \cdots + 5282347 \)
|
| $83$ |
\( T^{3} + 132 T^{2} + \cdots - 266462784 \)
|
| $89$ |
\( T^{3} - 2222 T^{2} + \cdots + 240275576 \)
|
| $97$ |
\( T^{3} + \cdots - 1364061265 \)
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