Newspace parameters
| Level: | \( N \) | \(=\) | \( 840 = 2^{3} \cdot 3 \cdot 5 \cdot 7 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 840.bg (of order \(3\), degree \(2\), minimal) |
Newform invariants
| Self dual: | no |
| Analytic conductor: | \(6.70743376979\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Relative dimension: | \(3\) over \(\Q(\zeta_{3})\) |
| Coefficient field: | 6.0.29428272.1 |
| Defining polynomial: |
\( x^{6} - 6x^{4} - 4x^{3} - 42x^{2} + 343 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{3}]$ |
$q$-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
\( x^{6} - 6x^{4} - 4x^{3} - 42x^{2} + 343 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( 3\nu^{5} + 35\nu^{4} + 31\nu^{3} + 121\nu^{2} - 217\nu - 1519 ) / 490 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 5\nu^{5} + 7\nu^{4} + 19\nu^{3} - 13\nu^{2} + 203\nu - 147 ) / 490 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -5\nu^{5} - 7\nu^{4} - 19\nu^{3} + 13\nu^{2} + 287\nu + 147 ) / 490 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -\nu^{5} - 7\nu^{4} - \nu^{3} + 11\nu^{2} + 91\nu + 245 ) / 70 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -13\nu^{5} - 35\nu^{4} + 29\nu^{3} - 81\nu^{2} + 637\nu + 1519 ) / 490 \)
|
| \(\nu\) | \(=\) |
\( \beta_{3} + \beta_{2} \)
|
| \(\nu^{2}\) | \(=\) |
\( -\beta_{5} + 2\beta_{4} - \beta_{2} + 2\beta _1 + 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( 5\beta_{5} + \beta_{4} - 11\beta_{3} + \beta_{2} + 4\beta _1 - 3 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{5} - 9\beta_{4} + 18\beta_{3} + 5\beta_{2} + 5\beta _1 + 40 \)
|
| \(\nu^{5}\) | \(=\) |
\( -23\beta_{5} + 14\beta_{4} - 24\beta_{3} + 44\beta_{2} - 17\beta _1 - 10 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/840\mathbb{Z}\right)^\times\).
| \(n\) | \(241\) | \(281\) | \(337\) | \(421\) | \(631\) |
| \(\chi(n)\) | \(-1 - \beta_{3}\) | \(1\) | \(1\) | \(1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 121.1 |
|
0 | −0.500000 | − | 0.866025i | 0 | 0.500000 | − | 0.866025i | 0 | −2.56022 | − | 0.667305i | 0 | −0.500000 | + | 0.866025i | 0 | ||||||||||||||||||||||||||||
| 121.2 | 0 | −0.500000 | − | 0.866025i | 0 | 0.500000 | − | 0.866025i | 0 | −0.0741344 | + | 2.64471i | 0 | −0.500000 | + | 0.866025i | 0 | |||||||||||||||||||||||||||||
| 121.3 | 0 | −0.500000 | − | 0.866025i | 0 | 0.500000 | − | 0.866025i | 0 | 2.63435 | − | 0.245357i | 0 | −0.500000 | + | 0.866025i | 0 | |||||||||||||||||||||||||||||
| 361.1 | 0 | −0.500000 | + | 0.866025i | 0 | 0.500000 | + | 0.866025i | 0 | −2.56022 | + | 0.667305i | 0 | −0.500000 | − | 0.866025i | 0 | |||||||||||||||||||||||||||||
| 361.2 | 0 | −0.500000 | + | 0.866025i | 0 | 0.500000 | + | 0.866025i | 0 | −0.0741344 | − | 2.64471i | 0 | −0.500000 | − | 0.866025i | 0 | |||||||||||||||||||||||||||||
| 361.3 | 0 | −0.500000 | + | 0.866025i | 0 | 0.500000 | + | 0.866025i | 0 | 2.63435 | + | 0.245357i | 0 | −0.500000 | − | 0.866025i | 0 | |||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 7.c | even | 3 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 840.2.bg.j | ✓ | 6 |
| 3.b | odd | 2 | 1 | 2520.2.bi.n | 6 | ||
| 4.b | odd | 2 | 1 | 1680.2.bg.v | 6 | ||
| 7.c | even | 3 | 1 | inner | 840.2.bg.j | ✓ | 6 |
| 7.c | even | 3 | 1 | 5880.2.a.bv | 3 | ||
| 7.d | odd | 6 | 1 | 5880.2.a.bu | 3 | ||
| 21.h | odd | 6 | 1 | 2520.2.bi.n | 6 | ||
| 28.g | odd | 6 | 1 | 1680.2.bg.v | 6 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 840.2.bg.j | ✓ | 6 | 1.a | even | 1 | 1 | trivial |
| 840.2.bg.j | ✓ | 6 | 7.c | even | 3 | 1 | inner |
| 1680.2.bg.v | 6 | 4.b | odd | 2 | 1 | ||
| 1680.2.bg.v | 6 | 28.g | odd | 6 | 1 | ||
| 2520.2.bi.n | 6 | 3.b | odd | 2 | 1 | ||
| 2520.2.bi.n | 6 | 21.h | odd | 6 | 1 | ||
| 5880.2.a.bu | 3 | 7.d | odd | 6 | 1 | ||
| 5880.2.a.bv | 3 | 7.c | even | 3 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{11}^{6} - 3T_{11}^{5} + 45T_{11}^{4} - 144T_{11}^{3} + 1674T_{11}^{2} - 4536T_{11} + 15876 \)
acting on \(S_{2}^{\mathrm{new}}(840, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{6} \)
$3$
\( (T^{2} + T + 1)^{3} \)
$5$
\( (T^{2} - T + 1)^{3} \)
$7$
\( T^{6} - 6 T^{4} - 4 T^{3} - 42 T^{2} + \cdots + 343 \)
$11$
\( T^{6} - 3 T^{5} + 45 T^{4} + \cdots + 15876 \)
$13$
\( (T^{3} + 3 T^{2} - 15 T + 3)^{2} \)
$17$
\( T^{6} + 6 T^{5} + 90 T^{4} + \cdots + 102400 \)
$19$
\( (T^{2} + T + 1)^{3} \)
$23$
\( T^{6} - 3 T^{5} + 33 T^{4} + 28 T^{3} + \cdots + 484 \)
$29$
\( (T^{3} - 6 T^{2} - 6 T + 48)^{2} \)
$31$
\( T^{6} + 12 T^{5} + 171 T^{4} + \cdots + 202500 \)
$37$
\( T^{6} + 3 T^{5} + 96 T^{4} + \cdots + 145161 \)
$41$
\( (T^{3} + 9 T^{2} - 58)^{2} \)
$43$
\( (T^{3} - 39 T + 88)^{2} \)
$47$
\( T^{6} + 3 T^{5} + 81 T^{4} + \cdots + 19600 \)
$53$
\( T^{6} - 15 T^{5} + 165 T^{4} + \cdots + 3136 \)
$59$
\( T^{6} + 30 T^{5} + 618 T^{4} + \cdots + 705600 \)
$61$
\( T^{6} \)
$67$
\( T^{6} + 39 T^{4} + 176 T^{3} + \cdots + 7744 \)
$71$
\( (T^{3} + 12 T^{2} - 186 T - 2100)^{2} \)
$73$
\( T^{6} + 18 T^{5} + 243 T^{4} + \cdots + 3364 \)
$79$
\( T^{6} + 6 T^{5} + 123 T^{4} + \cdots + 1024 \)
$83$
\( (T^{3} - 6 T^{2} - 6 T + 48)^{2} \)
$89$
\( T^{6} + 90 T^{4} + 584 T^{3} + \cdots + 85264 \)
$97$
\( (T - 8)^{6} \)
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