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: | \(0\) |
| Dimension: | \(3\) |
| Coefficient field: | 3.3.940.1 |
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|
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| Defining polynomial: |
\( x^{3} - 7x - 4 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 680) |
| 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
\( x^{3} - 7x - 4 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - \nu - 5 \)
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| \(\nu\) | \(=\) |
\( \beta_1 \)
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| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + \beta _1 + 5 \)
|
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 | −1.89511 | 0 | 0 | 0 | 0.513465 | 0 | 0.591429 | 0 | |||||||||||||||||||||||||||
| 1.2 | 0 | 1.60270 | 0 | 0 | 0 | 5.03404 | 0 | −0.431337 | 0 | ||||||||||||||||||||||||||||
| 1.3 | 0 | 3.29240 | 0 | 0 | 0 | −1.54751 | 0 | 7.83991 | 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.bs | 3 | |
| 4.b | odd | 2 | 1 | 3400.2.a.k | 3 | ||
| 5.b | even | 2 | 1 | 1360.2.a.p | 3 | ||
| 20.d | odd | 2 | 1 | 680.2.a.h | ✓ | 3 | |
| 20.e | even | 4 | 2 | 3400.2.e.i | 6 | ||
| 40.e | odd | 2 | 1 | 5440.2.a.bn | 3 | ||
| 40.f | even | 2 | 1 | 5440.2.a.bu | 3 | ||
| 60.h | even | 2 | 1 | 6120.2.a.bq | 3 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 680.2.a.h | ✓ | 3 | 20.d | odd | 2 | 1 | |
| 1360.2.a.p | 3 | 5.b | even | 2 | 1 | ||
| 3400.2.a.k | 3 | 4.b | odd | 2 | 1 | ||
| 3400.2.e.i | 6 | 20.e | even | 4 | 2 | ||
| 5440.2.a.bn | 3 | 40.e | odd | 2 | 1 | ||
| 5440.2.a.bu | 3 | 40.f | even | 2 | 1 | ||
| 6120.2.a.bq | 3 | 60.h | even | 2 | 1 | ||
| 6800.2.a.bs | 3 | 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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\( T_{3}^{3} - 3T_{3}^{2} - 4T_{3} + 10 \)
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\( T_{7}^{3} - 4T_{7}^{2} - 6T_{7} + 4 \)
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\( T_{11}^{3} - 2T_{11}^{2} - 10T_{11} + 16 \)
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\( T_{13}^{3} + 5T_{13}^{2} - 8T_{13} - 32 \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{3} \)
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| $3$ |
\( T^{3} - 3 T^{2} + \cdots + 10 \)
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| $5$ |
\( T^{3} \)
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| $7$ |
\( T^{3} - 4 T^{2} + \cdots + 4 \)
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| $11$ |
\( T^{3} - 2 T^{2} + \cdots + 16 \)
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| $13$ |
\( T^{3} + 5 T^{2} + \cdots - 32 \)
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| $17$ |
\( (T - 1)^{3} \)
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| $19$ |
\( T^{3} - T^{2} + \cdots + 40 \)
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| $23$ |
\( T^{3} + 4 T^{2} + \cdots - 4 \)
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| $29$ |
\( T^{3} - 7 T^{2} + \cdots + 68 \)
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| $31$ |
\( T^{3} - 3 T^{2} + \cdots + 170 \)
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| $37$ |
\( T^{3} + 12 T^{2} + \cdots - 16 \)
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| $41$ |
\( T^{3} - 10 T^{2} + \cdots + 184 \)
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| $43$ |
\( (T - 2)^{3} \)
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| $47$ |
\( T^{3} + 5 T^{2} + \cdots + 4 \)
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| $53$ |
\( T^{3} - 13 T^{2} + \cdots + 1564 \)
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| $59$ |
\( T^{3} - 21 T^{2} + \cdots + 776 \)
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| $61$ |
\( T^{3} - 13 T^{2} + \cdots - 20 \)
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| $67$ |
\( T^{3} - 14 T^{2} + \cdots + 1448 \)
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| $71$ |
\( T^{3} - 11 T^{2} + \cdots + 410 \)
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| $73$ |
\( T^{3} - 5 T^{2} + \cdots - 4 \)
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| $79$ |
\( T^{3} - 18 T^{2} + \cdots + 160 \)
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| $83$ |
\( T^{3} - 4 T^{2} + \cdots + 160 \)
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| $89$ |
\( T^{3} - 11 T^{2} + \cdots + 536 \)
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| $97$ |
\( T^{3} + 37 T^{2} + \cdots + 1684 \)
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