Newspace parameters
Level: | \( N \) | \(=\) | \( 143 = 11 \cdot 13 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 143.e (of order \(3\), degree \(2\), minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(1.14186074890\) |
Analytic rank: | \(0\) |
Dimension: | \(6\) |
Relative dimension: | \(3\) over \(\Q(\zeta_{3})\) |
Coefficient field: | 6.0.3518667.1 |
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
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Defining polynomial: | \( x^{6} - x^{5} + 7x^{4} - 8x^{3} + 43x^{2} - 42x + 49 \) |
Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
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 \( x^{6} - x^{5} + 7x^{4} - 8x^{3} + 43x^{2} - 42x + 49 \) :
\(\beta_{1}\) | \(=\) | \( \nu \) |
\(\beta_{2}\) | \(=\) | \( ( -\nu^{5} + 7\nu^{4} - 49\nu^{3} + 43\nu^{2} - 42\nu + 294 ) / 259 \) |
\(\beta_{3}\) | \(=\) | \( ( -6\nu^{5} + 5\nu^{4} - 35\nu^{3} - \nu^{2} - 215\nu - 49 ) / 259 \) |
\(\beta_{4}\) | \(=\) | \( ( -5\nu^{5} + 35\nu^{4} + 14\nu^{3} + 215\nu^{2} - 210\nu + 952 ) / 259 \) |
\(\beta_{5}\) | \(=\) | \( ( 18\nu^{5} + 22\nu^{4} + 105\nu^{3} + 3\nu^{2} + 608\nu + 147 ) / 259 \) |
\(\nu\) | \(=\) | \( \beta_1 \) |
\(\nu^{2}\) | \(=\) | \( -\beta_{5} + \beta_{4} - 4\beta_{3} + \beta_{2} - 5 \) |
\(\nu^{3}\) | \(=\) | \( \beta_{4} - 5\beta_{2} + 2 \) |
\(\nu^{4}\) | \(=\) | \( 7\beta_{5} + 21\beta_{3} + \beta_1 \) |
\(\nu^{5}\) | \(=\) | \( 6\beta_{5} - 6\beta_{4} - 25\beta_{3} + 29\beta_{2} - 35\beta _1 - 19 \) |
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/143\mathbb{Z}\right)^\times\).
\(n\) | \(67\) | \(79\) |
\(\chi(n)\) | \(-1 - \beta_{3}\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
100.1 |
|
−1.25351 | + | 2.17114i | −1.14257 | + | 1.97899i | −2.14257 | − | 3.71104i | −1.22188 | −2.86445 | − | 4.96137i | −0.389062 | − | 0.673875i | 5.72889 | −1.11094 | − | 1.92420i | 1.53163 | − | 2.65287i | ||||||||||||||||||||||
100.2 | 0.610938 | − | 1.05818i | 1.25351 | − | 2.17114i | 0.253509 | + | 0.439091i | −2.28514 | −1.53163 | − | 2.65287i | 0.142571 | + | 0.246941i | 3.06327 | −1.64257 | − | 2.84502i | −1.39608 | + | 2.41808i | |||||||||||||||||||||||
100.3 | 1.14257 | − | 1.97899i | −0.610938 | + | 1.05818i | −1.61094 | − | 2.79023i | 2.50702 | 1.39608 | + | 2.41808i | −2.25351 | − | 3.90319i | −2.79216 | 0.753509 | + | 1.30512i | 2.86445 | − | 4.96137i | |||||||||||||||||||||||
133.1 | −1.25351 | − | 2.17114i | −1.14257 | − | 1.97899i | −2.14257 | + | 3.71104i | −1.22188 | −2.86445 | + | 4.96137i | −0.389062 | + | 0.673875i | 5.72889 | −1.11094 | + | 1.92420i | 1.53163 | + | 2.65287i | |||||||||||||||||||||||
133.2 | 0.610938 | + | 1.05818i | 1.25351 | + | 2.17114i | 0.253509 | − | 0.439091i | −2.28514 | −1.53163 | + | 2.65287i | 0.142571 | − | 0.246941i | 3.06327 | −1.64257 | + | 2.84502i | −1.39608 | − | 2.41808i | |||||||||||||||||||||||
133.3 | 1.14257 | + | 1.97899i | −0.610938 | − | 1.05818i | −1.61094 | + | 2.79023i | 2.50702 | 1.39608 | − | 2.41808i | −2.25351 | + | 3.90319i | −2.79216 | 0.753509 | − | 1.30512i | 2.86445 | + | 4.96137i | |||||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
13.c | even | 3 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 143.2.e.b | ✓ | 6 |
13.c | even | 3 | 1 | inner | 143.2.e.b | ✓ | 6 |
13.c | even | 3 | 1 | 1859.2.a.f | 3 | ||
13.e | even | 6 | 1 | 1859.2.a.g | 3 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
143.2.e.b | ✓ | 6 | 1.a | even | 1 | 1 | trivial |
143.2.e.b | ✓ | 6 | 13.c | even | 3 | 1 | inner |
1859.2.a.f | 3 | 13.c | even | 3 | 1 | ||
1859.2.a.g | 3 | 13.e | even | 6 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{6} - T_{2}^{5} + 7T_{2}^{4} - 8T_{2}^{3} + 43T_{2}^{2} - 42T_{2} + 49 \)
acting on \(S_{2}^{\mathrm{new}}(143, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{6} - T^{5} + 7 T^{4} - 8 T^{3} + \cdots + 49 \)
$3$
\( T^{6} + T^{5} + 7 T^{4} + 8 T^{3} + \cdots + 49 \)
$5$
\( (T^{3} + T^{2} - 6 T - 7)^{2} \)
$7$
\( T^{6} + 5 T^{5} + 23 T^{4} + 12 T^{3} + \cdots + 1 \)
$11$
\( (T^{2} + T + 1)^{3} \)
$13$
\( (T^{2} + 5 T + 13)^{3} \)
$17$
\( T^{6} - 7 T^{5} + 39 T^{4} - 84 T^{3} + \cdots + 49 \)
$19$
\( T^{6} + 2 T^{5} + 47 T^{4} + \cdots + 5929 \)
$23$
\( T^{6} + 3 T^{5} + 25 T^{4} - 50 T^{3} + \cdots + 1 \)
$29$
\( T^{6} - 4 T^{5} + 93 T^{4} + \cdots + 2401 \)
$31$
\( (T^{3} - T^{2} - 25 T - 31)^{2} \)
$37$
\( T^{6} + 12 T^{5} + 115 T^{4} + \cdots + 961 \)
$41$
\( T^{6} + T^{5} + 83 T^{4} - 144 T^{3} + \cdots + 961 \)
$43$
\( T^{6} - 4 T^{5} + 131 T^{4} + \cdots + 337561 \)
$47$
\( (T^{3} + 8 T^{2} - 61 T - 259)^{2} \)
$53$
\( (T^{3} + 9 T^{2} + 8 T - 49)^{2} \)
$59$
\( T^{6} + 30 T^{5} + 619 T^{4} + \cdots + 625681 \)
$61$
\( T^{6} - 18 T^{5} + 273 T^{4} + \cdots + 49 \)
$67$
\( T^{6} + 15 T^{5} + 169 T^{4} + \cdots + 121 \)
$71$
\( T^{6} - 11 T^{5} + 87 T^{4} + \cdots + 961 \)
$73$
\( (T + 2)^{6} \)
$79$
\( (T^{3} - 12 T^{2} - 28 T + 88)^{2} \)
$83$
\( (T^{3} + 14 T^{2} - 17 T - 341)^{2} \)
$89$
\( T^{6} - 17 T^{5} + 199 T^{4} + \cdots + 22801 \)
$97$
\( T^{6} + 11 T^{5} + 182 T^{4} + \cdots + 49 \)
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