Newspace parameters
| Level: | \( N \) | \(=\) | \( 600 = 2^{3} \cdot 3 \cdot 5^{2} \) |
| Weight: | \( k \) | \(=\) | \( 6 \) |
| Character orbit: | \([\chi]\) | \(=\) | 600.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(96.2302918878\) |
| Analytic rank: | \(1\) |
| Dimension: | \(3\) |
| Coefficient field: | 3.3.246388.1 |
|
|
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| Defining polynomial: |
\( x^{3} - x^{2} - 89x + 183 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| Coefficient ring index: | \( 2^{5}\cdot 5 \) |
| 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
\( x^{3} - x^{2} - 89x + 183 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( -8\nu^{2} + 16\nu + 471 ) / 3 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 8\nu^{2} + 44\nu - 492 ) / 3 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{2} + \beta _1 + 7 ) / 20 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 4\beta_{2} - 11\beta _1 + 2383 ) / 40 \)
|
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 | 9.00000 | 0 | 0 | 0 | −94.1318 | 0 | 81.0000 | 0 | |||||||||||||||||||||||||||
| 1.2 | 0 | 9.00000 | 0 | 0 | 0 | −21.5548 | 0 | 81.0000 | 0 | ||||||||||||||||||||||||||||
| 1.3 | 0 | 9.00000 | 0 | 0 | 0 | 196.687 | 0 | 81.0000 | 0 | ||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \( +1 \) |
| \(3\) | \( -1 \) |
| \(5\) | \( -1 \) |
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 | 600.6.a.u | yes | 3 |
| 5.b | even | 2 | 1 | 600.6.a.p | ✓ | 3 | |
| 5.c | odd | 4 | 2 | 600.6.f.o | 6 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 600.6.a.p | ✓ | 3 | 5.b | even | 2 | 1 | |
| 600.6.a.u | yes | 3 | 1.a | even | 1 | 1 | trivial |
| 600.6.f.o | 6 | 5.c | odd | 4 | 2 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{7}^{3} - 81T_{7}^{2} - 20725T_{7} - 399075 \)
acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(600))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{3} \)
|
| $3$ |
\( (T - 9)^{3} \)
|
| $5$ |
\( T^{3} \)
|
| $7$ |
\( T^{3} - 81 T^{2} + \cdots - 399075 \)
|
| $11$ |
\( T^{3} - 38 T^{2} + \cdots + 687480 \)
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| $13$ |
\( T^{3} + 713 T^{2} + \cdots - 809372709 \)
|
| $17$ |
\( T^{3} + \cdots - 3731232408 \)
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| $19$ |
\( T^{3} + \cdots - 10898987009 \)
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| $23$ |
\( T^{3} + \cdots + 3059368488 \)
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| $29$ |
\( T^{3} + \cdots - 65540735544 \)
|
| $31$ |
\( T^{3} + \cdots - 30382097115 \)
|
| $37$ |
\( T^{3} + \cdots - 556723838376 \)
|
| $41$ |
\( T^{3} + \cdots - 503596157760 \)
|
| $43$ |
\( T^{3} + \cdots + 598620517581 \)
|
| $47$ |
\( T^{3} + \cdots - 607569920600 \)
|
| $53$ |
\( T^{3} + \cdots + 1898584398720 \)
|
| $59$ |
\( T^{3} + \cdots - 19463994415080 \)
|
| $61$ |
\( T^{3} + \cdots - 318660736299 \)
|
| $67$ |
\( T^{3} + \cdots + 5442746686001 \)
|
| $71$ |
\( T^{3} + \cdots - 171119857803840 \)
|
| $73$ |
\( T^{3} + \cdots - 231250667106600 \)
|
| $79$ |
\( T^{3} + \cdots + 238255782882816 \)
|
| $83$ |
\( T^{3} + \cdots + 59289716448792 \)
|
| $89$ |
\( T^{3} + \cdots - 162093891612672 \)
|
| $97$ |
\( T^{3} + \cdots - 834679641090457 \)
|
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