Newspace parameters
Level: | \( N \) | \(=\) | \( 141 = 3 \cdot 47 \) |
Weight: | \( k \) | \(=\) | \( 4 \) |
Character orbit: | \([\chi]\) | \(=\) | 141.c (of order \(2\), degree \(1\), minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(8.31926931081\) |
Analytic rank: | \(0\) |
Dimension: | \(10\) |
Coefficient field: | 10.0.2239697333984375.1 |
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
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Defining polynomial: | \( x^{10} - 9x^{5} + 32 \) |
Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
Coefficient ring index: | \( 2\cdot 3^{4} \) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{U}(1)[D_{2}]$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{9}\) 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 \( x^{10} - 9x^{5} + 32 \) :
\(\beta_{1}\) | \(=\) | \( ( -\nu^{9} + 9\nu^{4} - 16\nu ) / 16 \) |
\(\beta_{2}\) | \(=\) | \( ( \nu^{9} + 8\nu^{6} + 7\nu^{4} - 24\nu ) / 16 \) |
\(\beta_{3}\) | \(=\) | \( ( -\nu^{7} - 4\nu^{3} + 9\nu^{2} ) / 4 \) |
\(\beta_{4}\) | \(=\) | \( ( \nu^{8} - 9\nu^{3} - 8\nu^{2} ) / 8 \) |
\(\beta_{5}\) | \(=\) | \( ( -\nu^{9} - 8\nu^{6} + 9\nu^{4} + 40\nu ) / 16 \) |
\(\beta_{6}\) | \(=\) | \( ( -\nu^{8} + 2\nu^{7} - 8\nu^{5} + \nu^{3} - 2\nu^{2} + 40 ) / 8 \) |
\(\beta_{7}\) | \(=\) | \( ( \nu^{8} + 2\nu^{7} - 5\nu^{3} - 6\nu^{2} ) / 4 \) |
\(\beta_{8}\) | \(=\) | \( ( \nu^{8} - 2\nu^{7} - 8\nu^{5} - \nu^{3} + 2\nu^{2} + 40 ) / 8 \) |
\(\beta_{9}\) | \(=\) | \( ( -3\nu^{9} + \nu^{8} - 2\nu^{7} + 11\nu^{4} - \nu^{3} + 2\nu^{2} + 8\nu ) / 8 \) |
\(\nu\) | \(=\) | \( ( 2\beta_{9} - \beta_{8} + \beta_{6} + 4\beta_{5} + 4\beta_{2} - 12\beta_1 ) / 18 \) |
\(\nu^{2}\) | \(=\) | \( ( \beta_{8} + 4\beta_{7} - \beta_{6} - 10\beta_{4} + 6\beta_{3} ) / 18 \) |
\(\nu^{3}\) | \(=\) | \( ( 5\beta_{8} + 2\beta_{7} - 5\beta_{6} - 14\beta_{4} - 6\beta_{3} ) / 18 \) |
\(\nu^{4}\) | \(=\) | \( ( -2\beta_{9} + \beta_{8} - \beta_{6} + 14\beta_{5} + 14\beta_{2} + 12\beta_1 ) / 18 \) |
\(\nu^{5}\) | \(=\) | \( ( -\beta_{8} - \beta_{6} + 10 ) / 2 \) |
\(\nu^{6}\) | \(=\) | \( ( 14\beta_{9} - 7\beta_{8} + 7\beta_{6} - 8\beta_{5} + 28\beta_{2} - 48\beta_1 ) / 18 \) |
\(\nu^{7}\) | \(=\) | \( ( -11\beta_{8} + 28\beta_{7} + 11\beta_{6} - 34\beta_{4} + 6\beta_{3} ) / 18 \) |
\(\nu^{8}\) | \(=\) | \( ( 53\beta_{8} + 50\beta_{7} - 53\beta_{6} - 62\beta_{4} - 6\beta_{3} ) / 18 \) |
\(\nu^{9}\) | \(=\) | \( ( -50\beta_{9} + 25\beta_{8} - 25\beta_{6} + 62\beta_{5} + 62\beta_{2} + 12\beta_1 ) / 18 \) |
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/141\mathbb{Z}\right)^\times\).
\(n\) | \(52\) | \(95\) |
\(\chi(n)\) | \(-1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
140.1 |
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− | 5.64034i | 2.87961 | + | 4.32526i | −23.8135 | 0 | 24.3959 | − | 16.2420i | −14.7562 | 89.1934i | −10.4157 | + | 24.9101i | 0 | |||||||||||||||||||||||||||||||||||||||||
140.2 | − | 4.81700i | −3.22372 | − | 4.07525i | −15.2035 | 0 | −19.6305 | + | 15.5286i | −8.03159 | 34.6990i | −6.21529 | + | 26.2749i | 0 | ||||||||||||||||||||||||||||||||||||||||||
140.3 | − | 4.30927i | 5.00341 | + | 1.40209i | −10.5698 | 0 | 6.04199 | − | 21.5611i | 31.9076 | 11.0740i | 23.0683 | + | 14.0305i | 0 | ||||||||||||||||||||||||||||||||||||||||||
140.4 | − | 2.15372i | −4.87197 | − | 1.80662i | 3.36148 | 0 | −3.89095 | + | 10.4929i | 27.7516 | − | 24.4695i | 20.4723 | + | 17.6036i | 0 | |||||||||||||||||||||||||||||||||||||||||
140.5 | − | 1.33220i | 0.212671 | − | 5.19180i | 6.22524 | 0 | −6.91652 | − | 0.283321i | −36.8714 | − | 18.9509i | −26.9095 | − | 2.20829i | 0 | |||||||||||||||||||||||||||||||||||||||||
140.6 | 1.33220i | 0.212671 | + | 5.19180i | 6.22524 | 0 | −6.91652 | + | 0.283321i | −36.8714 | 18.9509i | −26.9095 | + | 2.20829i | 0 | |||||||||||||||||||||||||||||||||||||||||||
140.7 | 2.15372i | −4.87197 | + | 1.80662i | 3.36148 | 0 | −3.89095 | − | 10.4929i | 27.7516 | 24.4695i | 20.4723 | − | 17.6036i | 0 | |||||||||||||||||||||||||||||||||||||||||||
140.8 | 4.30927i | 5.00341 | − | 1.40209i | −10.5698 | 0 | 6.04199 | + | 21.5611i | 31.9076 | − | 11.0740i | 23.0683 | − | 14.0305i | 0 | ||||||||||||||||||||||||||||||||||||||||||
140.9 | 4.81700i | −3.22372 | + | 4.07525i | −15.2035 | 0 | −19.6305 | − | 15.5286i | −8.03159 | − | 34.6990i | −6.21529 | − | 26.2749i | 0 | ||||||||||||||||||||||||||||||||||||||||||
140.10 | 5.64034i | 2.87961 | − | 4.32526i | −23.8135 | 0 | 24.3959 | + | 16.2420i | −14.7562 | − | 89.1934i | −10.4157 | − | 24.9101i | 0 | ||||||||||||||||||||||||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
47.b | odd | 2 | 1 | CM by \(\Q(\sqrt{-47}) \) |
3.b | odd | 2 | 1 | inner |
141.c | even | 2 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 141.4.c.a | ✓ | 10 |
3.b | odd | 2 | 1 | inner | 141.4.c.a | ✓ | 10 |
47.b | odd | 2 | 1 | CM | 141.4.c.a | ✓ | 10 |
141.c | even | 2 | 1 | inner | 141.4.c.a | ✓ | 10 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
141.4.c.a | ✓ | 10 | 1.a | even | 1 | 1 | trivial |
141.4.c.a | ✓ | 10 | 3.b | odd | 2 | 1 | inner |
141.4.c.a | ✓ | 10 | 47.b | odd | 2 | 1 | CM |
141.4.c.a | ✓ | 10 | 141.c | even | 2 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{10} + 80T_{2}^{8} + 2240T_{2}^{6} + 25600T_{2}^{4} + 102400T_{2}^{2} + 112847 \)
acting on \(S_{4}^{\mathrm{new}}(141, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{10} + 80 T^{8} + 2240 T^{6} + \cdots + 112847 \)
$3$
\( T^{10} - 1540 T^{5} + \cdots + 14348907 \)
$5$
\( T^{10} \)
$7$
\( (T^{5} - 1715 T^{3} + 588245 T + 3869440)^{2} \)
$11$
\( T^{10} \)
$13$
\( T^{10} \)
$17$
\( T^{10} + 49130 T^{8} + \cdots + 71\!\cdots\!72 \)
$19$
\( T^{10} \)
$23$
\( T^{10} \)
$29$
\( T^{10} \)
$31$
\( T^{10} \)
$37$
\( (T^{5} - 253265 T^{3} + \cdots + 1139571232270)^{2} \)
$41$
\( T^{10} \)
$43$
\( T^{10} \)
$47$
\( (T^{2} + 103823)^{5} \)
$53$
\( T^{10} + 1488770 T^{8} + \cdots + 33\!\cdots\!28 \)
$59$
\( T^{10} + 2053790 T^{8} + \cdots + 10\!\cdots\!00 \)
$61$
\( (T^{5} - 1134905 T^{3} + \cdots + 45922923778102)^{2} \)
$67$
\( T^{10} \)
$71$
\( T^{10} + 3579110 T^{8} + \cdots + 12\!\cdots\!00 \)
$73$
\( T^{10} \)
$79$
\( (T^{5} - 2465195 T^{3} + \cdots - 283163708513936)^{2} \)
$83$
\( (T^{2} + 699548)^{5} \)
$89$
\( T^{10} + 7049690 T^{8} + \cdots + 19\!\cdots\!00 \)
$97$
\( (T^{5} - 4563365 T^{3} + \cdots + 13\!\cdots\!10)^{2} \)
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