Newspace parameters
| Level: | \( N \) | \(=\) | \( 600 = 2^{3} \cdot 3 \cdot 5^{2} \) |
| Weight: | \( k \) | \(=\) | \( 8 \) |
| Character orbit: | \([\chi]\) | \(=\) | 600.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(187.431015290\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{4} - \cdots)\) |
|
|
|
| Defining polynomial: |
\( x^{4} - 2x^{3} - 1845x^{2} + 1542x + 205929 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| Coefficient ring index: | \( 2^{10}\cdot 3\cdot 5^{2} \) |
| 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 \(1,\beta_1,\beta_2,\beta_3\) 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
\( x^{4} - 2x^{3} - 1845x^{2} + 1542x + 205929 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( \nu^{3} + 4\nu^{2} - 1836\nu - 4389 ) / 15 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{3} + 4\nu^{2} - 1116\nu - 4749 ) / 15 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -\nu^{3} + 56\nu^{2} + 1656\nu - 50931 ) / 15 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{2} - \beta _1 + 24 ) / 48 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 4\beta_{3} + \beta_{2} + 3\beta _1 + 14776 ) / 16 \)
|
| \(\nu^{3}\) | \(=\) |
\( -\beta_{3} + 38\beta_{2} - 24\beta _1 + 1613 \)
|
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 | −27.0000 | 0 | 0 | 0 | −810.235 | 0 | 729.000 | 0 | ||||||||||||||||||||||||||||||
| 1.2 | 0 | −27.0000 | 0 | 0 | 0 | −449.363 | 0 | 729.000 | 0 | |||||||||||||||||||||||||||||||
| 1.3 | 0 | −27.0000 | 0 | 0 | 0 | 122.068 | 0 | 729.000 | 0 | |||||||||||||||||||||||||||||||
| 1.4 | 0 | −27.0000 | 0 | 0 | 0 | 1685.53 | 0 | 729.000 | 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 | 600.8.a.q | ✓ | 4 |
| 5.b | even | 2 | 1 | 600.8.a.r | yes | 4 | |
| 5.c | odd | 4 | 2 | 600.8.f.n | 8 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 600.8.a.q | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
| 600.8.a.r | yes | 4 | 5.b | even | 2 | 1 | |
| 600.8.f.n | 8 | 5.c | odd | 4 | 2 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{7}^{4} - 548T_{7}^{3} - 1707010T_{7}^{2} - 398967700T_{7} + 74911031625 \)
acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(600))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} \)
|
| $3$ |
\( (T + 27)^{4} \)
|
| $5$ |
\( T^{4} \)
|
| $7$ |
\( T^{4} + \cdots + 74911031625 \)
|
| $11$ |
\( T^{4} + \cdots - 56231212418160 \)
|
| $13$ |
\( T^{4} + \cdots - 890690873316543 \)
|
| $17$ |
\( T^{4} + \cdots + 81\!\cdots\!76 \)
|
| $19$ |
\( T^{4} + \cdots - 11\!\cdots\!11 \)
|
| $23$ |
\( T^{4} + \cdots + 28\!\cdots\!44 \)
|
| $29$ |
\( T^{4} + \cdots + 13\!\cdots\!24 \)
|
| $31$ |
\( T^{4} + \cdots + 28\!\cdots\!85 \)
|
| $37$ |
\( T^{4} + \cdots + 99\!\cdots\!16 \)
|
| $41$ |
\( T^{4} + \cdots - 29\!\cdots\!60 \)
|
| $43$ |
\( T^{4} + \cdots - 28\!\cdots\!39 \)
|
| $47$ |
\( T^{4} + \cdots - 21\!\cdots\!00 \)
|
| $53$ |
\( T^{4} + \cdots + 26\!\cdots\!60 \)
|
| $59$ |
\( T^{4} + \cdots - 57\!\cdots\!40 \)
|
| $61$ |
\( T^{4} + \cdots + 23\!\cdots\!37 \)
|
| $67$ |
\( T^{4} + \cdots + 60\!\cdots\!89 \)
|
| $71$ |
\( T^{4} + \cdots + 16\!\cdots\!60 \)
|
| $73$ |
\( T^{4} + \cdots - 10\!\cdots\!00 \)
|
| $79$ |
\( T^{4} + \cdots + 11\!\cdots\!48 \)
|
| $83$ |
\( T^{4} + \cdots + 26\!\cdots\!28 \)
|
| $89$ |
\( T^{4} + \cdots - 31\!\cdots\!12 \)
|
| $97$ |
\( T^{4} + \cdots - 19\!\cdots\!51 \)
|
| show more | |
| show less |