Newspace parameters
| Level: | \( N \) | \(=\) | \( 600 = 2^{3} \cdot 3 \cdot 5^{2} \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 600.k (of order \(2\), degree \(1\), minimal) |
Newform invariants
| Self dual: | no |
| Analytic conductor: | \(4.79102412128\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Coefficient field: | 6.0.399424.1 |
| Defining polynomial: |
\( x^{6} - 2x^{5} + 3x^{4} - 6x^{3} + 6x^{2} - 8x + 8 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 2 \) |
| Twist minimal: | no (minimal twist has level 120) |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{2}]$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{5}\) 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
\( x^{6} - 2x^{5} + 3x^{4} - 6x^{3} + 6x^{2} - 8x + 8 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{5} + 3\nu^{3} + 2\nu - 8 ) / 4 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{5} + 3\nu^{3} - 4\nu^{2} + 2\nu - 8 ) / 4 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -\nu^{5} - \nu^{3} + 2\nu^{2} + 4 ) / 2 \)
|
| \(\beta_{5}\) | \(=\) |
\( -\nu^{5} + \nu^{4} - 2\nu^{3} + 3\nu^{2} - 2\nu + 5 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( -\beta_{3} + \beta_{2} \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{4} + \beta_{3} + \beta_{2} - \beta _1 + 2 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{5} - \beta_{4} + 2\beta_{3} + \beta _1 + 1 \)
|
| \(\nu^{5}\) | \(=\) |
\( -3\beta_{4} - 3\beta_{3} + \beta_{2} + \beta _1 + 2 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/600\mathbb{Z}\right)^\times\).
| \(n\) | \(151\) | \(301\) | \(401\) | \(577\) |
| \(\chi(n)\) | \(1\) | \(-1\) | \(1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 301.1 |
|
−1.40680 | − | 0.144584i | 1.00000i | 1.95819 | + | 0.406803i | 0 | 0.144584 | − | 1.40680i | 3.62721 | −2.69597 | − | 0.855416i | −1.00000 | 0 | ||||||||||||||||||||||||||||
| 301.2 | −1.40680 | + | 0.144584i | − | 1.00000i | 1.95819 | − | 0.406803i | 0 | 0.144584 | + | 1.40680i | 3.62721 | −2.69597 | + | 0.855416i | −1.00000 | 0 | ||||||||||||||||||||||||||||
| 301.3 | −0.264658 | − | 1.38923i | − | 1.00000i | −1.85991 | + | 0.735342i | 0 | −1.38923 | + | 0.264658i | −0.941367 | 1.51380 | + | 2.38923i | −1.00000 | 0 | ||||||||||||||||||||||||||||
| 301.4 | −0.264658 | + | 1.38923i | 1.00000i | −1.85991 | − | 0.735342i | 0 | −1.38923 | − | 0.264658i | −0.941367 | 1.51380 | − | 2.38923i | −1.00000 | 0 | |||||||||||||||||||||||||||||
| 301.5 | 0.671462 | − | 1.24464i | 1.00000i | −1.09828 | − | 1.67146i | 0 | 1.24464 | + | 0.671462i | −4.68585 | −2.81783 | + | 0.244644i | −1.00000 | 0 | |||||||||||||||||||||||||||||
| 301.6 | 0.671462 | + | 1.24464i | − | 1.00000i | −1.09828 | + | 1.67146i | 0 | 1.24464 | − | 0.671462i | −4.68585 | −2.81783 | − | 0.244644i | −1.00000 | 0 | ||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 8.b | even | 2 | 1 | inner |
Twists
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{7}^{3} + 2T_{7}^{2} - 16T_{7} - 16 \)
acting on \(S_{2}^{\mathrm{new}}(600, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{6} + 2 T^{5} + 3 T^{4} + 6 T^{3} + \cdots + 8 \)
$3$
\( (T^{2} + 1)^{3} \)
$5$
\( T^{6} \)
$7$
\( (T^{3} + 2 T^{2} - 16 T - 16)^{2} \)
$11$
\( T^{6} + 64 T^{4} + 1088 T^{2} + \cdots + 4096 \)
$13$
\( T^{6} + 56 T^{4} + 784 T^{2} + \cdots + 256 \)
$17$
\( (T^{3} + 6 T^{2} - 16 T - 32)^{2} \)
$19$
\( T^{6} + 40 T^{4} + 272 T^{2} + \cdots + 256 \)
$23$
\( (T^{3} - 4 T^{2} - 12 T + 16)^{2} \)
$29$
\( (T^{2} + 4)^{3} \)
$31$
\( (T^{3} + 6 T^{2} - 16 T - 64)^{2} \)
$37$
\( T^{6} + 136 T^{4} + 5648 T^{2} + \cdots + 65536 \)
$41$
\( (T^{3} + 10 T^{2} - 36 T - 232)^{2} \)
$43$
\( T^{6} + 144 T^{4} + 5120 T^{2} + \cdots + 16384 \)
$47$
\( (T^{3} + 4 T^{2} - 92 T - 496)^{2} \)
$53$
\( (T^{2} + 4)^{3} \)
$59$
\( T^{6} + 80 T^{4} + 576 T^{2} + \cdots + 1024 \)
$61$
\( T^{6} + 256 T^{4} + 17408 T^{2} + \cdots + 262144 \)
$67$
\( (T^{2} + 16)^{3} \)
$71$
\( (T^{3} + 4 T^{2} - 112 T + 64)^{2} \)
$73$
\( (T - 6)^{6} \)
$79$
\( (T^{3} - 18 T^{2} + 80 T - 64)^{2} \)
$83$
\( T^{6} + 224 T^{4} + 8448 T^{2} + \cdots + 65536 \)
$89$
\( (T^{3} + 14 T^{2} - 4 T - 184)^{2} \)
$97$
\( (T^{3} + 18 T^{2} - 4 T - 328)^{2} \)
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