Newspace parameters
| Level: | \( N \) | \(=\) | \( 570 = 2 \cdot 3 \cdot 5 \cdot 19 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 570.q (of order \(6\), degree \(2\), minimal) |
Newform invariants
| Self dual: | no |
| Analytic conductor: | \(4.55147291521\) |
| Analytic rank: | \(0\) |
| Dimension: | \(8\) |
| Relative dimension: | \(4\) over \(\Q(\zeta_{6})\) |
| Coefficient field: | \(\Q(\zeta_{24})\) |
| Defining polynomial: |
\( x^{8} - x^{4} + 1 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 3^{2} \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{6}]$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.
Basis of coefficient ring
| \(\beta_{1}\) | \(=\) |
\( \zeta_{24}^{2} \)
|
| \(\beta_{2}\) | \(=\) |
\( \zeta_{24}^{4} \)
|
| \(\beta_{3}\) | \(=\) |
\( \zeta_{24}^{5} + \zeta_{24} \)
|
| \(\beta_{4}\) | \(=\) |
\( \zeta_{24}^{6} \)
|
| \(\beta_{5}\) | \(=\) |
\( \zeta_{24}^{7} + \zeta_{24}^{3} \)
|
| \(\beta_{6}\) | \(=\) |
\( -\zeta_{24}^{5} + 2\zeta_{24} \)
|
| \(\beta_{7}\) | \(=\) |
\( -\zeta_{24}^{7} + 2\zeta_{24}^{3} \)
|
| \(\zeta_{24}\) | \(=\) |
\( ( \beta_{6} + \beta_{3} ) / 3 \)
|
| \(\zeta_{24}^{2}\) | \(=\) |
\( \beta_1 \)
|
| \(\zeta_{24}^{3}\) | \(=\) |
\( ( \beta_{7} + \beta_{5} ) / 3 \)
|
| \(\zeta_{24}^{4}\) | \(=\) |
\( \beta_{2} \)
|
| \(\zeta_{24}^{5}\) | \(=\) |
\( ( -\beta_{6} + 2\beta_{3} ) / 3 \)
|
| \(\zeta_{24}^{6}\) | \(=\) |
\( \beta_{4} \)
|
| \(\zeta_{24}^{7}\) | \(=\) |
\( ( -\beta_{7} + 2\beta_{5} ) / 3 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/570\mathbb{Z}\right)^\times\).
| \(n\) | \(191\) | \(211\) | \(457\) |
| \(\chi(n)\) | \(1\) | \(-1 + \beta_{2}\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 49.1 |
|
−0.866025 | − | 0.500000i | −0.866025 | − | 0.500000i | 0.500000 | + | 0.866025i | −2.03906 | − | 0.917738i | 0.500000 | + | 0.866025i | 3.00000i | − | 1.00000i | 0.500000 | + | 0.866025i | 1.30701 | + | 1.81431i | |||||||||||||||||||||||||||
| 49.2 | −0.866025 | − | 0.500000i | −0.866025 | − | 0.500000i | 0.500000 | + | 0.866025i | 1.30701 | − | 1.81431i | 0.500000 | + | 0.866025i | 3.00000i | − | 1.00000i | 0.500000 | + | 0.866025i | −2.03906 | + | 0.917738i | ||||||||||||||||||||||||||||
| 49.3 | 0.866025 | + | 0.500000i | 0.866025 | + | 0.500000i | 0.500000 | + | 0.866025i | 0.917738 | − | 2.03906i | 0.500000 | + | 0.866025i | − | 3.00000i | 1.00000i | 0.500000 | + | 0.866025i | 1.81431 | − | 1.30701i | ||||||||||||||||||||||||||||
| 49.4 | 0.866025 | + | 0.500000i | 0.866025 | + | 0.500000i | 0.500000 | + | 0.866025i | 1.81431 | + | 1.30701i | 0.500000 | + | 0.866025i | − | 3.00000i | 1.00000i | 0.500000 | + | 0.866025i | 0.917738 | + | 2.03906i | ||||||||||||||||||||||||||||
| 349.1 | −0.866025 | + | 0.500000i | −0.866025 | + | 0.500000i | 0.500000 | − | 0.866025i | −2.03906 | + | 0.917738i | 0.500000 | − | 0.866025i | − | 3.00000i | 1.00000i | 0.500000 | − | 0.866025i | 1.30701 | − | 1.81431i | ||||||||||||||||||||||||||||
| 349.2 | −0.866025 | + | 0.500000i | −0.866025 | + | 0.500000i | 0.500000 | − | 0.866025i | 1.30701 | + | 1.81431i | 0.500000 | − | 0.866025i | − | 3.00000i | 1.00000i | 0.500000 | − | 0.866025i | −2.03906 | − | 0.917738i | ||||||||||||||||||||||||||||
| 349.3 | 0.866025 | − | 0.500000i | 0.866025 | − | 0.500000i | 0.500000 | − | 0.866025i | 0.917738 | + | 2.03906i | 0.500000 | − | 0.866025i | 3.00000i | − | 1.00000i | 0.500000 | − | 0.866025i | 1.81431 | + | 1.30701i | ||||||||||||||||||||||||||||
| 349.4 | 0.866025 | − | 0.500000i | 0.866025 | − | 0.500000i | 0.500000 | − | 0.866025i | 1.81431 | − | 1.30701i | 0.500000 | − | 0.866025i | 3.00000i | − | 1.00000i | 0.500000 | − | 0.866025i | 0.917738 | − | 2.03906i | ||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 5.b | even | 2 | 1 | inner |
| 19.c | even | 3 | 1 | inner |
| 95.i | even | 6 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 570.2.q.a | ✓ | 8 |
| 3.b | odd | 2 | 1 | 1710.2.t.a | 8 | ||
| 5.b | even | 2 | 1 | inner | 570.2.q.a | ✓ | 8 |
| 15.d | odd | 2 | 1 | 1710.2.t.a | 8 | ||
| 19.c | even | 3 | 1 | inner | 570.2.q.a | ✓ | 8 |
| 57.h | odd | 6 | 1 | 1710.2.t.a | 8 | ||
| 95.i | even | 6 | 1 | inner | 570.2.q.a | ✓ | 8 |
| 285.n | odd | 6 | 1 | 1710.2.t.a | 8 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 570.2.q.a | ✓ | 8 | 1.a | even | 1 | 1 | trivial |
| 570.2.q.a | ✓ | 8 | 5.b | even | 2 | 1 | inner |
| 570.2.q.a | ✓ | 8 | 19.c | even | 3 | 1 | inner |
| 570.2.q.a | ✓ | 8 | 95.i | even | 6 | 1 | inner |
| 1710.2.t.a | 8 | 3.b | odd | 2 | 1 | ||
| 1710.2.t.a | 8 | 15.d | odd | 2 | 1 | ||
| 1710.2.t.a | 8 | 57.h | odd | 6 | 1 | ||
| 1710.2.t.a | 8 | 285.n | odd | 6 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{7}^{2} + 9 \)
acting on \(S_{2}^{\mathrm{new}}(570, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( (T^{4} - T^{2} + 1)^{2} \)
$3$
\( (T^{4} - T^{2} + 1)^{2} \)
$5$
\( T^{8} - 4 T^{7} + 8 T^{6} + 8 T^{5} + \cdots + 625 \)
$7$
\( (T^{2} + 9)^{4} \)
$11$
\( (T - 2)^{8} \)
$13$
\( T^{8} - 14 T^{6} + 171 T^{4} + \cdots + 625 \)
$17$
\( (T^{4} - 24 T^{2} + 576)^{2} \)
$19$
\( (T^{2} - 2 T + 19)^{4} \)
$23$
\( (T^{4} - 6 T^{2} + 36)^{2} \)
$29$
\( (T^{4} + 4 T^{3} + 66 T^{2} - 200 T + 2500)^{2} \)
$31$
\( (T^{2} + 6 T - 15)^{4} \)
$37$
\( (T^{4} + 110 T^{2} + 1849)^{2} \)
$41$
\( (T^{4} + 6 T^{2} + 36)^{2} \)
$43$
\( T^{8} - 30 T^{6} + 891 T^{4} + \cdots + 81 \)
$47$
\( T^{8} - 84 T^{6} + 6156 T^{4} + \cdots + 810000 \)
$53$
\( T^{8} - 176 T^{6} + 29376 T^{4} + \cdots + 2560000 \)
$59$
\( (T^{4} - 4 T^{3} + 108 T^{2} + 368 T + 8464)^{2} \)
$61$
\( (T^{4} - 6 T^{3} + 81 T^{2} + 270 T + 2025)^{2} \)
$67$
\( T^{8} - 14 T^{6} + 171 T^{4} + \cdots + 625 \)
$71$
\( (T^{4} - 12 T^{3} + 114 T^{2} - 360 T + 900)^{2} \)
$73$
\( T^{8} - 146 T^{6} + 20691 T^{4} + \cdots + 390625 \)
$79$
\( (T^{4} - 14 T^{3} + 243 T^{2} + 658 T + 2209)^{2} \)
$83$
\( (T^{4} + 180 T^{2} + 324)^{2} \)
$89$
\( (T^{4} + 28 T^{3} + 594 T^{2} + \cdots + 36100)^{2} \)
$97$
\( T^{8} - 464 T^{6} + \cdots + 1600000000 \)
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