Newspace parameters
| Level: | \( N \) | \(=\) | \( 200 = 2^{3} \cdot 5^{2} \) |
| Weight: | \( k \) | \(=\) | \( 6 \) |
| Character orbit: | \([\chi]\) | \(=\) | 200.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(32.0767639626\) |
| Analytic rank: | \(0\) |
| Dimension: | \(3\) |
| Coefficient field: | 3.3.47217.1 |
| Defining polynomial: |
\( x^{3} - x^{2} - 38x - 24 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 2^{4}\cdot 5 \) |
| Twist minimal: | yes |
| 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
\( x^{3} - x^{2} - 38x - 24 \)
:
| \(\beta_{1}\) | \(=\) |
\( -\nu^{2} + 3\nu + 25 \)
|
| \(\beta_{2}\) | \(=\) |
\( 3\nu^{2} + 31\nu - 87 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{2} + 3\beta _1 + 12 ) / 40 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 3\beta_{2} - 31\beta _1 + 1036 ) / 40 \)
|
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 | −22.6278 | 0 | 0 | 0 | 107.252 | 0 | 269.020 | 0 | |||||||||||||||||||||||||||
| 1.2 | 0 | 2.53448 | 0 | 0 | 0 | −247.370 | 0 | −236.576 | 0 | ||||||||||||||||||||||||||||
| 1.3 | 0 | 19.0934 | 0 | 0 | 0 | 210.117 | 0 | 121.557 | 0 | ||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(-1\) |
| \(5\) | \(1\) |
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 | 200.6.a.h | ✓ | 3 |
| 4.b | odd | 2 | 1 | 400.6.a.y | 3 | ||
| 5.b | even | 2 | 1 | 200.6.a.i | yes | 3 | |
| 5.c | odd | 4 | 2 | 200.6.c.g | 6 | ||
| 20.d | odd | 2 | 1 | 400.6.a.x | 3 | ||
| 20.e | even | 4 | 2 | 400.6.c.p | 6 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 200.6.a.h | ✓ | 3 | 1.a | even | 1 | 1 | trivial |
| 200.6.a.i | yes | 3 | 5.b | even | 2 | 1 | |
| 200.6.c.g | 6 | 5.c | odd | 4 | 2 | ||
| 400.6.a.x | 3 | 20.d | odd | 2 | 1 | ||
| 400.6.a.y | 3 | 4.b | odd | 2 | 1 | ||
| 400.6.c.p | 6 | 20.e | even | 4 | 2 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{3} + T_{3}^{2} - 441T_{3} + 1095 \)
acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(200))\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{3} \)
$3$
\( T^{3} + T^{2} - 441 T + 1095 \)
$5$
\( T^{3} \)
$7$
\( T^{3} - 70 T^{2} - 55972 T + 5574616 \)
$11$
\( T^{3} + 19 T^{2} - 84401 T - 1542819 \)
$13$
\( T^{3} - 196 T^{2} + \cdots - 51768768 \)
$17$
\( T^{3} + 1223 T^{2} + \cdots - 654046475 \)
$19$
\( T^{3} - 1221 T^{2} + \cdots + 7033288181 \)
$23$
\( T^{3} - 2490 T^{2} + \cdots + 50437840744 \)
$29$
\( T^{3} - 11912 T^{2} + \cdots - 55993024512 \)
$31$
\( T^{3} - 7442 T^{2} + \cdots + 14434811400 \)
$37$
\( T^{3} + 14766 T^{2} + \cdots - 435216240216 \)
$41$
\( T^{3} - 3223 T^{2} + \cdots - 86972031621 \)
$43$
\( T^{3} - 41060 T^{2} + \cdots - 660952050112 \)
$47$
\( T^{3} + 29188 T^{2} + \cdots - 324350860864 \)
$53$
\( T^{3} - 12878 T^{2} + \cdots + 22325036001624 \)
$59$
\( T^{3} - 64912 T^{2} + \cdots - 8618337690624 \)
$61$
\( T^{3} - 22478 T^{2} + \cdots + 28184724893400 \)
$67$
\( T^{3} + 26499 T^{2} + \cdots + 1261173627437 \)
$71$
\( T^{3} - 86676 T^{2} + \cdots + 31597978616640 \)
$73$
\( T^{3} + \cdots - 160799137840701 \)
$79$
\( T^{3} - 21982 T^{2} + \cdots + 26059657326840 \)
$83$
\( T^{3} + \cdots - 289154781250911 \)
$89$
\( T^{3} + \cdots - 190810675088559 \)
$97$
\( T^{3} + \cdots - 166573103686456 \)
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