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: | 5.5.1981136.1 |
|
|
|
| Defining polynomial: |
\( x^{5} - 2x^{4} - 8x^{3} + 10x^{2} + 18x - 2 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 3400) |
| 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
\( x^{5} - 2x^{4} - 8x^{3} + 10x^{2} + 18x - 2 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - \nu - 4 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{3} - \nu^{2} - 5\nu + 1 \)
|
| \(\beta_{4}\) | \(=\) |
\( \nu^{4} - 6\nu^{2} - 3\nu + 1 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + \beta _1 + 4 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{3} + \beta_{2} + 6\beta _1 + 3 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{4} + 6\beta_{2} + 9\beta _1 + 23 \)
|
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 | −1.77780 | 0 | 0 | 0 | 3.30243 | 0 | 0.160589 | 0 | |||||||||||||||||||||||||||||||||
| 1.2 | 0 | −1.50269 | 0 | 0 | 0 | −4.25913 | 0 | −0.741913 | 0 | ||||||||||||||||||||||||||||||||||
| 1.3 | 0 | 0.894534 | 0 | 0 | 0 | 1.04880 | 0 | −2.19981 | 0 | ||||||||||||||||||||||||||||||||||
| 1.4 | 0 | 2.32094 | 0 | 0 | 0 | −3.69585 | 0 | 2.38676 | 0 | ||||||||||||||||||||||||||||||||||
| 1.5 | 0 | 3.06502 | 0 | 0 | 0 | 4.60374 | 0 | 6.39437 | 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.cg | 5 | |
| 4.b | odd | 2 | 1 | 3400.2.a.r | ✓ | 5 | |
| 5.b | even | 2 | 1 | 6800.2.a.bx | 5 | ||
| 20.d | odd | 2 | 1 | 3400.2.a.w | yes | 5 | |
| 20.e | even | 4 | 2 | 3400.2.e.n | 10 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 3400.2.a.r | ✓ | 5 | 4.b | odd | 2 | 1 | |
| 3400.2.a.w | yes | 5 | 20.d | odd | 2 | 1 | |
| 3400.2.e.n | 10 | 20.e | even | 4 | 2 | ||
| 6800.2.a.bx | 5 | 5.b | even | 2 | 1 | ||
| 6800.2.a.cg | 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(6800))\):
|
\( T_{3}^{5} - 3T_{3}^{4} - 6T_{3}^{3} + 16T_{3}^{2} + 11T_{3} - 17 \)
|
|
\( T_{7}^{5} - T_{7}^{4} - 32T_{7}^{3} + 30T_{7}^{2} + 243T_{7} - 251 \)
|
|
\( T_{11}^{5} - 6T_{11}^{4} - 10T_{11}^{3} + 84T_{11}^{2} - 92T_{11} + 28 \)
|
|
\( T_{13}^{5} + 3T_{13}^{4} - 46T_{13}^{3} - 186T_{13}^{2} + 141T_{13} + 875 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{5} \)
|
| $3$ |
\( T^{5} - 3 T^{4} + \cdots - 17 \)
|
| $5$ |
\( T^{5} \)
|
| $7$ |
\( T^{5} - T^{4} + \cdots - 251 \)
|
| $11$ |
\( T^{5} - 6 T^{4} + \cdots + 28 \)
|
| $13$ |
\( T^{5} + 3 T^{4} + \cdots + 875 \)
|
| $17$ |
\( (T - 1)^{5} \)
|
| $19$ |
\( T^{5} - 6 T^{4} + \cdots + 80 \)
|
| $23$ |
\( T^{5} - 10 T^{4} + \cdots - 1412 \)
|
| $29$ |
\( T^{5} - 6 T^{4} + \cdots - 400 \)
|
| $31$ |
\( T^{5} - 11 T^{4} + \cdots - 1609 \)
|
| $37$ |
\( T^{5} + 2 T^{4} + \cdots + 11504 \)
|
| $41$ |
\( T^{5} - 4 T^{4} + \cdots - 656 \)
|
| $43$ |
\( T^{5} - 8 T^{4} + \cdots - 19600 \)
|
| $47$ |
\( T^{5} - 10 T^{4} + \cdots + 4960 \)
|
| $53$ |
\( T^{5} + 19 T^{4} + \cdots - 625 \)
|
| $59$ |
\( T^{5} - 228 T^{3} + \cdots - 19696 \)
|
| $61$ |
\( T^{5} + 14 T^{4} + \cdots + 167008 \)
|
| $67$ |
\( T^{5} - 80 T^{3} + \cdots + 640 \)
|
| $71$ |
\( T^{5} - 5 T^{4} + \cdots - 79 \)
|
| $73$ |
\( T^{5} + 26 T^{4} + \cdots - 5648 \)
|
| $79$ |
\( T^{5} + 27 T^{4} + \cdots - 37325 \)
|
| $83$ |
\( T^{5} - 20 T^{4} + \cdots - 8384 \)
|
| $89$ |
\( T^{5} - 10 T^{4} + \cdots - 8720 \)
|
| $97$ |
\( T^{5} - 24 T^{4} + \cdots + 99776 \)
|
| show more | |
| show less |