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: | \(5\) |
| Coefficient field: | 5.5.563792.1 |
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| Defining polynomial: |
\( x^{5} - x^{4} - 8x^{3} + 2x^{2} + 12x - 2 \)
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| 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 \(1,\beta_1,\beta_2,\beta_3,\beta_4\) 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
\( x^{5} - x^{4} - 8x^{3} + 2x^{2} + 12x - 2 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
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| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - \nu - 3 \)
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| \(\beta_{3}\) | \(=\) |
\( \nu^{4} - 2\nu^{3} - 5\nu^{2} + 5\nu + 3 \)
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| \(\beta_{4}\) | \(=\) |
\( \nu^{4} - 3\nu^{3} - 3\nu^{2} + 9\nu - 1 \)
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| \(\nu\) | \(=\) |
\( \beta_1 \)
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| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + \beta _1 + 3 \)
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| \(\nu^{3}\) | \(=\) |
\( -\beta_{4} + \beta_{3} + 2\beta_{2} + 6\beta _1 + 2 \)
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| \(\nu^{4}\) | \(=\) |
\( -2\beta_{4} + 3\beta_{3} + 9\beta_{2} + 12\beta _1 + 16 \)
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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 | −1.83529 | 0 | 0 | 0 | 3.41244 | 0 | 0.368274 | 0 | |||||||||||||||||||||||||||||||||
| 1.2 | 0 | −1.66285 | 0 | 0 | 0 | 1.13950 | 0 | −0.234926 | 0 | ||||||||||||||||||||||||||||||||||
| 1.3 | 0 | 0.165176 | 0 | 0 | 0 | −3.45441 | 0 | −2.97272 | 0 | ||||||||||||||||||||||||||||||||||
| 1.4 | 0 | 1.31440 | 0 | 0 | 0 | 1.12767 | 0 | −1.27236 | 0 | ||||||||||||||||||||||||||||||||||
| 1.5 | 0 | 3.01856 | 0 | 0 | 0 | −4.22519 | 0 | 6.11173 | 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.cc | 5 | |
| 4.b | odd | 2 | 1 | 3400.2.a.t | 5 | ||
| 5.b | even | 2 | 1 | 6800.2.a.cb | 5 | ||
| 5.c | odd | 4 | 2 | 1360.2.e.f | 10 | ||
| 20.d | odd | 2 | 1 | 3400.2.a.u | 5 | ||
| 20.e | even | 4 | 2 | 680.2.e.b | ✓ | 10 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 680.2.e.b | ✓ | 10 | 20.e | even | 4 | 2 | |
| 1360.2.e.f | 10 | 5.c | odd | 4 | 2 | ||
| 3400.2.a.t | 5 | 4.b | odd | 2 | 1 | ||
| 3400.2.a.u | 5 | 20.d | odd | 2 | 1 | ||
| 6800.2.a.cb | 5 | 5.b | even | 2 | 1 | ||
| 6800.2.a.cc | 5 | 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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\( T_{3}^{5} - T_{3}^{4} - 8T_{3}^{3} + 2T_{3}^{2} + 12T_{3} - 2 \)
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\( T_{7}^{5} + 2T_{7}^{4} - 20T_{7}^{3} - 18T_{7}^{2} + 98T_{7} - 64 \)
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\( T_{11}^{5} - 4T_{11}^{4} - 2T_{11}^{3} + 10T_{11}^{2} + 2T_{11} - 4 \)
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\( T_{13}^{5} - 3T_{13}^{4} - 30T_{13}^{3} + 108T_{13}^{2} - 84T_{13} + 4 \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{5} \)
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| $3$ |
\( T^{5} - T^{4} - 8 T^{3} + \cdots - 2 \)
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| $5$ |
\( T^{5} \)
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| $7$ |
\( T^{5} + 2 T^{4} + \cdots - 64 \)
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| $11$ |
\( T^{5} - 4 T^{4} + \cdots - 4 \)
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| $13$ |
\( T^{5} - 3 T^{4} + \cdots + 4 \)
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| $17$ |
\( (T + 1)^{5} \)
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| $19$ |
\( T^{5} - 3 T^{4} + \cdots - 2224 \)
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| $23$ |
\( T^{5} + 4 T^{4} + \cdots - 32 \)
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| $29$ |
\( T^{5} + 17 T^{4} + \cdots + 3568 \)
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| $31$ |
\( T^{5} - 11 T^{4} + \cdots + 54 \)
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| $37$ |
\( T^{5} - 20 T^{4} + \cdots + 3200 \)
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| $41$ |
\( T^{5} + 22 T^{4} + \cdots - 2752 \)
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| $43$ |
\( T^{5} - 22 T^{4} + \cdots - 1776 \)
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| $47$ |
\( T^{5} - 5 T^{4} + \cdots - 876 \)
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| $53$ |
\( T^{5} - 7 T^{4} + \cdots + 7312 \)
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| $59$ |
\( T^{5} - 3 T^{4} + \cdots + 3504 \)
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| $61$ |
\( T^{5} + 13 T^{4} + \cdots - 568 \)
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| $67$ |
\( T^{5} - 28 T^{4} + \cdots - 14264 \)
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| $71$ |
\( T^{5} - 33 T^{4} + \cdots + 4406 \)
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| $73$ |
\( T^{5} - 21 T^{4} + \cdots + 1864 \)
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| $79$ |
\( T^{5} - 20 T^{4} + \cdots - 64 \)
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| $83$ |
\( T^{5} - 18 T^{4} + \cdots - 4112 \)
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| $89$ |
\( T^{5} + 25 T^{4} + \cdots - 23748 \)
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| $97$ |
\( T^{5} - 21 T^{4} + \cdots - 15376 \)
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