Newspace parameters
| Level: | \( N \) | \(=\) | \( 6800 = 2^{4} \cdot 5^{2} \cdot 17 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 6800.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(54.2982733745\) |
| Analytic rank: | \(1\) |
| Dimension: | \(4\) |
| Coefficient field: | 4.4.11344.1 |
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| Defining polynomial: |
\( x^{4} - 2x^{3} - 4x^{2} + 4x + 3 \)
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| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 340) |
| Fricke sign: | \(+1\) |
| Sato-Tate group: | $\mathrm{SU}(2)$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) 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
\( x^{4} - 2x^{3} - 4x^{2} + 4x + 3 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - \nu - 3 \)
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| \(\beta_{3}\) | \(=\) |
\( \nu^{3} - 2\nu^{2} - 3\nu + 2 \)
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| \(\nu\) | \(=\) |
\( \beta_1 \)
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| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + \beta _1 + 3 \)
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| \(\nu^{3}\) | \(=\) |
\( \beta_{3} + 2\beta_{2} + 5\beta _1 + 4 \)
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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 | −1.78165 | 0 | 0 | 0 | 2.65910 | 0 | 0.174289 | 0 | ||||||||||||||||||||||||||||||
| 1.2 | 0 | −0.287336 | 0 | 0 | 0 | −4.67316 | 0 | −2.91744 | 0 | |||||||||||||||||||||||||||||||
| 1.3 | 0 | 1.55241 | 0 | 0 | 0 | 1.73591 | 0 | −0.590025 | 0 | |||||||||||||||||||||||||||||||
| 1.4 | 0 | 2.51658 | 0 | 0 | 0 | 0.278150 | 0 | 3.33317 | 0 | |||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \( -1 \) |
| \(5\) | \( -1 \) |
| \(17\) | \( +1 \) |
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 | 6800.2.a.bv | 4 | |
| 4.b | odd | 2 | 1 | 1700.2.a.f | 4 | ||
| 5.b | even | 2 | 1 | 6800.2.a.bu | 4 | ||
| 5.c | odd | 4 | 2 | 1360.2.e.e | 8 | ||
| 20.d | odd | 2 | 1 | 1700.2.a.g | 4 | ||
| 20.e | even | 4 | 2 | 340.2.e.a | ✓ | 8 | |
| 60.l | odd | 4 | 2 | 3060.2.g.f | 8 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 340.2.e.a | ✓ | 8 | 20.e | even | 4 | 2 | |
| 1360.2.e.e | 8 | 5.c | odd | 4 | 2 | ||
| 1700.2.a.f | 4 | 4.b | odd | 2 | 1 | ||
| 1700.2.a.g | 4 | 20.d | odd | 2 | 1 | ||
| 3060.2.g.f | 8 | 60.l | odd | 4 | 2 | ||
| 6800.2.a.bu | 4 | 5.b | even | 2 | 1 | ||
| 6800.2.a.bv | 4 | 1.a | even | 1 | 1 | trivial | |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6800))\):
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\( T_{3}^{4} - 2T_{3}^{3} - 4T_{3}^{2} + 6T_{3} + 2 \)
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\( T_{7}^{4} - 16T_{7}^{2} + 26T_{7} - 6 \)
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\( T_{11}^{4} - 18T_{11}^{2} + 14T_{11} + 30 \)
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\( T_{13}^{4} + 10T_{13}^{3} + 28T_{13}^{2} + 8T_{13} - 36 \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} \)
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| $3$ |
\( T^{4} - 2 T^{3} + \cdots + 2 \)
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| $5$ |
\( T^{4} \)
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| $7$ |
\( T^{4} - 16 T^{2} + \cdots - 6 \)
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| $11$ |
\( T^{4} - 18 T^{2} + \cdots + 30 \)
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| $13$ |
\( T^{4} + 10 T^{3} + \cdots - 36 \)
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| $17$ |
\( (T + 1)^{4} \)
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| $19$ |
\( T^{4} - 40 T^{2} + \cdots + 16 \)
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| $23$ |
\( T^{4} - 10 T^{3} + \cdots + 10 \)
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| $29$ |
\( T^{4} - 40 T^{2} + \cdots + 16 \)
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| $31$ |
\( T^{4} + 2 T^{3} + \cdots - 98 \)
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| $37$ |
\( T^{4} + 16 T^{3} + \cdots - 496 \)
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| $41$ |
\( T^{4} + 2 T^{3} + \cdots + 1320 \)
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| $43$ |
\( T^{4} - 18 T^{3} + \cdots + 188 \)
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| $47$ |
\( T^{4} + 6 T^{3} + \cdots + 36 \)
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| $53$ |
\( T^{4} + 16 T^{3} + \cdots + 1488 \)
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| $59$ |
\( T^{4} - 4 T^{3} + \cdots + 2608 \)
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| $61$ |
\( T^{4} - 6 T^{3} + \cdots + 600 \)
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| $67$ |
\( T^{4} - 2 T^{3} + \cdots + 292 \)
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| $71$ |
\( T^{4} + 20 T^{3} + \cdots - 3378 \)
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| $73$ |
\( T^{4} + 26 T^{3} + \cdots - 12088 \)
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| $79$ |
\( T^{4} + 6 T^{3} + \cdots + 502 \)
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| $83$ |
\( T^{4} - 14 T^{3} + \cdots + 7724 \)
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| $89$ |
\( T^{4} + 4 T^{3} + \cdots - 772 \)
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| $97$ |
\( T^{4} + 36 T^{3} + \cdots - 24048 \)
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