Newspace parameters
| Level: | \( N \) | \(=\) | \( 768 = 2^{8} \cdot 3 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 768.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(45.3134668844\) |
| Analytic rank: | \(1\) |
| Dimension: | \(3\) |
| Coefficient field: | 3.3.1436.1 |
|
|
|
| Defining polynomial: |
\( x^{3} - 11x - 12 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{5} \) |
| Twist minimal: | no (minimal twist has level 24) |
| 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
\( x^{3} - 11x - 12 \)
:
| \(\beta_{1}\) | \(=\) |
\( 4\nu^{2} - 29 \)
|
| \(\beta_{2}\) | \(=\) |
\( 4\nu^{2} - 8\nu - 29 \)
|
| \(\nu\) | \(=\) |
\( ( -\beta_{2} + \beta_1 ) / 8 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta _1 + 29 ) / 4 \)
|
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 | −3.00000 | 0 | −18.5422 | 0 | 9.32669 | 0 | 9.00000 | 0 | |||||||||||||||||||||||||||
| 1.2 | 0 | −3.00000 | 0 | −0.612661 | 0 | −22.7441 | 0 | 9.00000 | 0 | ||||||||||||||||||||||||||||
| 1.3 | 0 | −3.00000 | 0 | 9.15486 | 0 | 27.4175 | 0 | 9.00000 | 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 | 768.4.a.q | 3 | |
| 3.b | odd | 2 | 1 | 2304.4.a.bw | 3 | ||
| 4.b | odd | 2 | 1 | 768.4.a.s | 3 | ||
| 8.b | even | 2 | 1 | 768.4.a.t | 3 | ||
| 8.d | odd | 2 | 1 | 768.4.a.r | 3 | ||
| 12.b | even | 2 | 1 | 2304.4.a.bv | 3 | ||
| 16.e | even | 4 | 2 | 96.4.d.a | 6 | ||
| 16.f | odd | 4 | 2 | 24.4.d.a | ✓ | 6 | |
| 24.f | even | 2 | 1 | 2304.4.a.bt | 3 | ||
| 24.h | odd | 2 | 1 | 2304.4.a.bu | 3 | ||
| 48.i | odd | 4 | 2 | 288.4.d.d | 6 | ||
| 48.k | even | 4 | 2 | 72.4.d.d | 6 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 24.4.d.a | ✓ | 6 | 16.f | odd | 4 | 2 | |
| 72.4.d.d | 6 | 48.k | even | 4 | 2 | ||
| 96.4.d.a | 6 | 16.e | even | 4 | 2 | ||
| 288.4.d.d | 6 | 48.i | odd | 4 | 2 | ||
| 768.4.a.q | 3 | 1.a | even | 1 | 1 | trivial | |
| 768.4.a.r | 3 | 8.d | odd | 2 | 1 | ||
| 768.4.a.s | 3 | 4.b | odd | 2 | 1 | ||
| 768.4.a.t | 3 | 8.b | even | 2 | 1 | ||
| 2304.4.a.bt | 3 | 24.f | even | 2 | 1 | ||
| 2304.4.a.bu | 3 | 24.h | odd | 2 | 1 | ||
| 2304.4.a.bv | 3 | 12.b | even | 2 | 1 | ||
| 2304.4.a.bw | 3 | 3.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(768))\):
|
\( T_{5}^{3} + 10T_{5}^{2} - 164T_{5} - 104 \)
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\( T_{7}^{3} - 14T_{7}^{2} - 580T_{7} + 5816 \)
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\( T_{11}^{3} - 2816T_{11} - 49152 \)
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\( T_{19}^{3} + 28T_{19}^{2} - 11088T_{19} + 274752 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{3} \)
|
| $3$ |
\( (T + 3)^{3} \)
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| $5$ |
\( T^{3} + 10 T^{2} + \cdots - 104 \)
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| $7$ |
\( T^{3} - 14 T^{2} + \cdots + 5816 \)
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| $11$ |
\( T^{3} - 2816T - 49152 \)
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| $13$ |
\( T^{3} + 52 T^{2} + \cdots - 55872 \)
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| $17$ |
\( T^{3} - 26 T^{2} + \cdots + 477576 \)
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| $19$ |
\( T^{3} + 28 T^{2} + \cdots + 274752 \)
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| $23$ |
\( T^{3} - 164 T^{2} + \cdots - 45504 \)
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| $29$ |
\( T^{3} + 174 T^{2} + \cdots - 61368 \)
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| $31$ |
\( T^{3} - 318 T^{2} + \cdots + 3749624 \)
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| $37$ |
\( T^{3} + 296 T^{2} + \cdots - 82944 \)
|
| $41$ |
\( T^{3} + 118 T^{2} + \cdots - 19985976 \)
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| $43$ |
\( T^{3} - 260 T^{2} + \cdots + 8601408 \)
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| $47$ |
\( T^{3} - 204 T^{2} + \cdots + 1964736 \)
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| $53$ |
\( T^{3} + 1086 T^{2} + \cdots + 20665224 \)
|
| $59$ |
\( T^{3} + 196 T^{2} + \cdots - 8523584 \)
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| $61$ |
\( T^{3} + 1536 T^{2} + \cdots + 104841216 \)
|
| $67$ |
\( T^{3} - 660 T^{2} + \cdots + 32228928 \)
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| $71$ |
\( T^{3} + 852 T^{2} + \cdots - 85084992 \)
|
| $73$ |
\( T^{3} + 478 T^{2} + \cdots - 120833304 \)
|
| $79$ |
\( T^{3} - 22 T^{2} + \cdots + 7902616 \)
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| $83$ |
\( T^{3} + 1136 T^{2} + \cdots + 37953536 \)
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| $89$ |
\( T^{3} - 110 T^{2} + \cdots - 1423656 \)
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| $97$ |
\( T^{3} + 1222 T^{2} + \cdots - 74802424 \)
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