Newspace parameters
Level: | \( N \) | \(=\) | \( 546 = 2 \cdot 3 \cdot 7 \cdot 13 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 546.j (of order \(3\), degree \(2\), minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(4.35983195036\) |
Analytic rank: | \(0\) |
Dimension: | \(8\) |
Relative dimension: | \(4\) over \(\Q(\zeta_{3})\) |
Coefficient field: | 8.0.447703281.1 |
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
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Defining polynomial: | \( x^{8} - x^{7} - 2x^{6} + 2x^{5} + 3x^{4} + 4x^{3} - 8x^{2} - 8x + 16 \) |
Coefficient ring: | \(\Z[a_1, \ldots, a_{13}]\) |
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 \(1,\beta_1,\ldots,\beta_{7}\) 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 \( x^{8} - x^{7} - 2x^{6} + 2x^{5} + 3x^{4} + 4x^{3} - 8x^{2} - 8x + 16 \) :
\(\beta_{1}\) | \(=\) | \( \nu \) |
\(\beta_{2}\) | \(=\) | \( ( \nu^{7} - \nu^{6} - 2\nu^{5} + 2\nu^{4} + 3\nu^{3} + 4\nu^{2} - 8\nu - 8 ) / 8 \) |
\(\beta_{3}\) | \(=\) | \( ( \nu^{6} + \nu^{5} - 2\nu^{3} - \nu^{2} + 6\nu + 4 ) / 4 \) |
\(\beta_{4}\) | \(=\) | \( ( -\nu^{7} - \nu^{6} + 2\nu^{4} + \nu^{3} - 6\nu^{2} - 4\nu ) / 4 \) |
\(\beta_{5}\) | \(=\) | \( ( -\nu^{7} + \nu^{6} + 6\nu^{5} + 2\nu^{4} - 3\nu^{3} - 12\nu^{2} + 4\nu + 32 ) / 8 \) |
\(\beta_{6}\) | \(=\) | \( ( -\nu^{7} + 2\nu^{5} + \nu^{4} - \nu^{3} - 5\nu^{2} - \nu + 8 ) / 2 \) |
\(\beta_{7}\) | \(=\) | \( ( 2\nu^{7} + \nu^{6} - 5\nu^{5} - 4\nu^{4} + 4\nu^{3} + 17\nu^{2} + 2\nu - 24 ) / 4 \) |
\(\nu\) | \(=\) | \( \beta_1 \) |
\(\nu^{2}\) | \(=\) | \( \beta_{6} - \beta_{5} - \beta_{4} + \beta_{2} + \beta _1 + 1 \) |
\(\nu^{3}\) | \(=\) | \( \beta_{7} + 2\beta_{5} + \beta_{4} - \beta_{3} + \beta _1 - 1 \) |
\(\nu^{4}\) | \(=\) | \( \beta_{6} - \beta_{5} + 2\beta_{3} + 3\beta_{2} + \beta _1 + 1 \) |
\(\nu^{5}\) | \(=\) | \( \beta_{6} + \beta_{5} - 2\beta_{4} - 2\beta_{3} + \beta_{2} + 2\beta _1 - 5 \) |
\(\nu^{6}\) | \(=\) | \( 2\beta_{7} + 2\beta_{5} + 3\beta_{4} + 4\beta_{3} - 5\beta_1 \) |
\(\nu^{7}\) | \(=\) | \( -\beta_{7} - 4\beta_{6} + 4\beta_{5} - \beta_{3} - 2\beta _1 - 5 \) |
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/546\mathbb{Z}\right)^\times\).
\(n\) | \(157\) | \(365\) | \(379\) |
\(\chi(n)\) | \(\beta_{3}\) | \(1\) | \(\beta_{3}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
289.1 |
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1.00000 | 0.500000 | − | 0.866025i | 1.00000 | −1.97513 | + | 3.42102i | 0.500000 | − | 0.866025i | −1.15207 | + | 2.38175i | 1.00000 | −0.500000 | − | 0.866025i | −1.97513 | + | 3.42102i | ||||||||||||||||||||||||||||||
289.2 | 1.00000 | 0.500000 | − | 0.866025i | 1.00000 | −0.228205 | + | 0.395262i | 0.500000 | − | 0.866025i | −0.369922 | − | 2.61976i | 1.00000 | −0.500000 | − | 0.866025i | −0.228205 | + | 0.395262i | |||||||||||||||||||||||||||||||
289.3 | 1.00000 | 0.500000 | − | 0.866025i | 1.00000 | 1.14553 | − | 1.98411i | 0.500000 | − | 0.866025i | 2.63641 | − | 0.222079i | 1.00000 | −0.500000 | − | 0.866025i | 1.14553 | − | 1.98411i | |||||||||||||||||||||||||||||||
289.4 | 1.00000 | 0.500000 | − | 0.866025i | 1.00000 | 2.05781 | − | 3.56422i | 0.500000 | − | 0.866025i | −2.61442 | − | 0.405935i | 1.00000 | −0.500000 | − | 0.866025i | 2.05781 | − | 3.56422i | |||||||||||||||||||||||||||||||
529.1 | 1.00000 | 0.500000 | + | 0.866025i | 1.00000 | −1.97513 | − | 3.42102i | 0.500000 | + | 0.866025i | −1.15207 | − | 2.38175i | 1.00000 | −0.500000 | + | 0.866025i | −1.97513 | − | 3.42102i | |||||||||||||||||||||||||||||||
529.2 | 1.00000 | 0.500000 | + | 0.866025i | 1.00000 | −0.228205 | − | 0.395262i | 0.500000 | + | 0.866025i | −0.369922 | + | 2.61976i | 1.00000 | −0.500000 | + | 0.866025i | −0.228205 | − | 0.395262i | |||||||||||||||||||||||||||||||
529.3 | 1.00000 | 0.500000 | + | 0.866025i | 1.00000 | 1.14553 | + | 1.98411i | 0.500000 | + | 0.866025i | 2.63641 | + | 0.222079i | 1.00000 | −0.500000 | + | 0.866025i | 1.14553 | + | 1.98411i | |||||||||||||||||||||||||||||||
529.4 | 1.00000 | 0.500000 | + | 0.866025i | 1.00000 | 2.05781 | + | 3.56422i | 0.500000 | + | 0.866025i | −2.61442 | + | 0.405935i | 1.00000 | −0.500000 | + | 0.866025i | 2.05781 | + | 3.56422i | |||||||||||||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
91.h | even | 3 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 546.2.j.d | ✓ | 8 |
3.b | odd | 2 | 1 | 1638.2.m.g | 8 | ||
7.c | even | 3 | 1 | 546.2.k.b | yes | 8 | |
13.c | even | 3 | 1 | 546.2.k.b | yes | 8 | |
21.h | odd | 6 | 1 | 1638.2.p.i | 8 | ||
39.i | odd | 6 | 1 | 1638.2.p.i | 8 | ||
91.h | even | 3 | 1 | inner | 546.2.j.d | ✓ | 8 |
273.s | odd | 6 | 1 | 1638.2.m.g | 8 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
546.2.j.d | ✓ | 8 | 1.a | even | 1 | 1 | trivial |
546.2.j.d | ✓ | 8 | 91.h | even | 3 | 1 | inner |
546.2.k.b | yes | 8 | 7.c | even | 3 | 1 | |
546.2.k.b | yes | 8 | 13.c | even | 3 | 1 | |
1638.2.m.g | 8 | 3.b | odd | 2 | 1 | ||
1638.2.m.g | 8 | 273.s | odd | 6 | 1 | ||
1638.2.p.i | 8 | 21.h | odd | 6 | 1 | ||
1638.2.p.i | 8 | 39.i | odd | 6 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{8} - 2T_{5}^{7} + 21T_{5}^{6} - 26T_{5}^{5} + 332T_{5}^{4} - 442T_{5}^{3} + 1189T_{5}^{2} + 510T_{5} + 289 \)
acting on \(S_{2}^{\mathrm{new}}(546, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( (T - 1)^{8} \)
$3$
\( (T^{2} - T + 1)^{4} \)
$5$
\( T^{8} - 2 T^{7} + 21 T^{6} - 26 T^{5} + \cdots + 289 \)
$7$
\( T^{8} + 3 T^{7} + 2 T^{6} - 21 T^{5} + \cdots + 2401 \)
$11$
\( T^{8} + 6 T^{7} + 46 T^{6} + \cdots + 4489 \)
$13$
\( T^{8} + 11 T^{7} + 62 T^{6} + \cdots + 28561 \)
$17$
\( (T^{4} + 4 T^{3} - 14 T^{2} + 9 T - 1)^{2} \)
$19$
\( T^{8} - 6 T^{7} + 31 T^{6} - 62 T^{5} + \cdots + 169 \)
$23$
\( (T^{4} - 10 T^{3} - 10 T^{2} + 155 T - 167)^{2} \)
$29$
\( T^{8} - 2 T^{7} + 54 T^{6} + \cdots + 4489 \)
$31$
\( T^{8} - 6 T^{7} + 84 T^{6} + \cdots + 210681 \)
$37$
\( (T^{4} - 28 T^{3} + 260 T^{2} - 823 T + 211)^{2} \)
$41$
\( T^{8} + 92 T^{6} + 198 T^{5} + \cdots + 1912689 \)
$43$
\( T^{8} + 6 T^{7} + 74 T^{6} + \cdots + 4489 \)
$47$
\( T^{8} - T^{7} + 57 T^{6} - 172 T^{5} + \cdots + 2601 \)
$53$
\( T^{8} - 7 T^{7} + 93 T^{6} + \cdots + 6561 \)
$59$
\( (T^{4} + 2 T^{3} - 143 T^{2} - 660 T + 311)^{2} \)
$61$
\( T^{8} - 24 T^{7} + 440 T^{6} + \cdots + 17438976 \)
$67$
\( T^{8} + 15 T^{7} + 196 T^{6} + \cdots + 27889 \)
$71$
\( T^{8} - 6 T^{7} + 131 T^{6} + \cdots + 674041 \)
$73$
\( T^{8} - T^{7} + 8 T^{6} - 3 T^{5} + 53 T^{4} + \cdots + 1 \)
$79$
\( T^{8} + 12 T^{7} + 221 T^{6} + \cdots + 6985449 \)
$83$
\( (T^{4} + 16 T^{3} - 86 T^{2} - 1329 T + 1919)^{2} \)
$89$
\( (T^{4} + 25 T^{3} + 170 T^{2} + 180 T + 27)^{2} \)
$97$
\( T^{8} + T^{7} + 189 T^{6} + \cdots + 45873529 \)
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