Newspace parameters
| Level: | \( N \) | \(=\) | \( 200 = 2^{3} \cdot 5^{2} \) |
| Weight: | \( k \) | \(=\) | \( 8 \) |
| Character orbit: | \([\chi]\) | \(=\) | 200.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(62.4770050968\) |
| Analytic rank: | \(0\) |
| Dimension: | \(3\) |
| Coefficient field: | 3.3.40101.1 |
|
|
|
| Defining polynomial: |
\( x^{3} - x^{2} - 34x + 4 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 2^{8}\cdot 5 \) |
| 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
\( x^{3} - x^{2} - 34x + 4 \)
:
| \(\beta_{1}\) | \(=\) |
\( 2\nu^{2} + 4\nu - 47 \)
|
| \(\beta_{2}\) | \(=\) |
\( -48\nu^{2} + 224\nu + 1029 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{2} + 24\beta _1 + 99 ) / 320 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( -\beta_{2} + 56\beta _1 + 3661 ) / 160 \)
|
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 | −89.5167 | 0 | 0 | 0 | 1193.94 | 0 | 5826.24 | 0 | |||||||||||||||||||||||||||
| 1.2 | 0 | −21.9866 | 0 | 0 | 0 | −1068.54 | 0 | −1703.59 | 0 | ||||||||||||||||||||||||||||
| 1.3 | 0 | 14.5033 | 0 | 0 | 0 | 1504.61 | 0 | −1976.65 | 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 | 200.8.a.o | ✓ | 3 |
| 4.b | odd | 2 | 1 | 400.8.a.bi | 3 | ||
| 5.b | even | 2 | 1 | 200.8.a.p | yes | 3 | |
| 5.c | odd | 4 | 2 | 200.8.c.j | 6 | ||
| 20.d | odd | 2 | 1 | 400.8.a.bh | 3 | ||
| 20.e | even | 4 | 2 | 400.8.c.u | 6 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 200.8.a.o | ✓ | 3 | 1.a | even | 1 | 1 | trivial |
| 200.8.a.p | yes | 3 | 5.b | even | 2 | 1 | |
| 200.8.c.j | 6 | 5.c | odd | 4 | 2 | ||
| 400.8.a.bh | 3 | 20.d | odd | 2 | 1 | ||
| 400.8.a.bi | 3 | 4.b | odd | 2 | 1 | ||
| 400.8.c.u | 6 | 20.e | even | 4 | 2 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{3} + 97T_{3}^{2} + 351T_{3} - 28545 \)
acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(200))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{3} \)
|
| $3$ |
\( T^{3} + 97 T^{2} + \cdots - 28545 \)
|
| $5$ |
\( T^{3} \)
|
| $7$ |
\( T^{3} + \cdots + 1919541112 \)
|
| $11$ |
\( T^{3} + \cdots - 33910231899 \)
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| $13$ |
\( T^{3} + \cdots - 62226542784 \)
|
| $17$ |
\( T^{3} + \cdots + 13545067874125 \)
|
| $19$ |
\( T^{3} + \cdots + 3636837484829 \)
|
| $23$ |
\( T^{3} + \cdots - 55015565119928 \)
|
| $29$ |
\( T^{3} + \cdots + 53\!\cdots\!52 \)
|
| $31$ |
\( T^{3} + \cdots - 46\!\cdots\!00 \)
|
| $37$ |
\( T^{3} + \cdots - 162472248229752 \)
|
| $41$ |
\( T^{3} + \cdots - 63\!\cdots\!41 \)
|
| $43$ |
\( T^{3} + \cdots - 12\!\cdots\!76 \)
|
| $47$ |
\( T^{3} + \cdots + 49\!\cdots\!32 \)
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| $53$ |
\( T^{3} + \cdots + 29\!\cdots\!72 \)
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| $59$ |
\( T^{3} + \cdots + 47\!\cdots\!24 \)
|
| $61$ |
\( T^{3} + \cdots - 56\!\cdots\!00 \)
|
| $67$ |
\( T^{3} + \cdots + 37\!\cdots\!89 \)
|
| $71$ |
\( T^{3} + \cdots - 49\!\cdots\!80 \)
|
| $73$ |
\( T^{3} + \cdots - 35\!\cdots\!13 \)
|
| $79$ |
\( T^{3} + \cdots + 30\!\cdots\!80 \)
|
| $83$ |
\( T^{3} + \cdots + 31\!\cdots\!77 \)
|
| $89$ |
\( T^{3} + \cdots - 39\!\cdots\!91 \)
|
| $97$ |
\( T^{3} + \cdots + 85\!\cdots\!48 \)
|
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