Newspace parameters
Level: | \( N \) | \(=\) | \( 2023 = 7 \cdot 17^{2} \) |
Weight: | \( k \) | \(=\) | \( 1 \) |
Character orbit: | \([\chi]\) | \(=\) | 2023.l (of order \(8\), degree \(4\), not minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(1.00960852056\) |
Analytic rank: | \(0\) |
Dimension: | \(16\) |
Relative dimension: | \(4\) over \(\Q(\zeta_{8})\) |
Coefficient field: | 16.0.109951162777600000000.1 |
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
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Defining polynomial: | \( x^{16} + 47x^{8} + 1 \) |
Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
Coefficient ring index: | \( 1 \) |
Twist minimal: | no (minimal twist has level 119) |
Projective image: | \(D_{5}\) |
Projective field: | Galois closure of 5.1.14161.1 |
Artin image: | $D_5\times C_{16}$ |
Artin field: | Galois closure of \(\mathbb{Q}[x]/(x^{80} - \cdots)\) |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{15}\) 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 \( x^{16} + 47x^{8} + 1 \) :
\(\beta_{1}\) | \(=\) | \( \nu \) |
\(\beta_{2}\) | \(=\) | \( ( \nu^{8} + 13 ) / 21 \) |
\(\beta_{3}\) | \(=\) | \( ( \nu^{9} + 34\nu ) / 21 \) |
\(\beta_{4}\) | \(=\) | \( ( -\nu^{10} - 34\nu^{2} ) / 21 \) |
\(\beta_{5}\) | \(=\) | \( ( \nu^{10} + 55\nu^{2} ) / 21 \) |
\(\beta_{6}\) | \(=\) | \( ( \nu^{11} + 55\nu^{3} ) / 21 \) |
\(\beta_{7}\) | \(=\) | \( ( -2\nu^{11} - 89\nu^{3} ) / 21 \) |
\(\beta_{8}\) | \(=\) | \( ( -2\nu^{12} - 89\nu^{4} ) / 21 \) |
\(\beta_{9}\) | \(=\) | \( ( -\nu^{12} - 48\nu^{4} ) / 7 \) |
\(\beta_{10}\) | \(=\) | \( ( -\nu^{13} - 48\nu^{5} ) / 7 \) |
\(\beta_{11}\) | \(=\) | \( ( -5\nu^{13} - 233\nu^{5} ) / 21 \) |
\(\beta_{12}\) | \(=\) | \( ( 5\nu^{14} + 233\nu^{6} ) / 21 \) |
\(\beta_{13}\) | \(=\) | \( ( -8\nu^{14} - 377\nu^{6} ) / 21 \) |
\(\beta_{14}\) | \(=\) | \( ( -8\nu^{15} - 377\nu^{7} ) / 21 \) |
\(\beta_{15}\) | \(=\) | \( ( 13\nu^{15} + 610\nu^{7} ) / 21 \) |
\(\nu\) | \(=\) | \( \beta_1 \) |
\(\nu^{2}\) | \(=\) | \( \beta_{5} + \beta_{4} \) |
\(\nu^{3}\) | \(=\) | \( \beta_{7} + 2\beta_{6} \) |
\(\nu^{4}\) | \(=\) | \( -2\beta_{9} + 3\beta_{8} \) |
\(\nu^{5}\) | \(=\) | \( 3\beta_{11} - 5\beta_{10} \) |
\(\nu^{6}\) | \(=\) | \( -5\beta_{13} - 8\beta_{12} \) |
\(\nu^{7}\) | \(=\) | \( -8\beta_{15} - 13\beta_{14} \) |
\(\nu^{8}\) | \(=\) | \( 21\beta_{2} - 13 \) |
\(\nu^{9}\) | \(=\) | \( 21\beta_{3} - 34\beta_1 \) |
\(\nu^{10}\) | \(=\) | \( -34\beta_{5} - 55\beta_{4} \) |
\(\nu^{11}\) | \(=\) | \( -55\beta_{7} - 89\beta_{6} \) |
\(\nu^{12}\) | \(=\) | \( 89\beta_{9} - 144\beta_{8} \) |
\(\nu^{13}\) | \(=\) | \( -144\beta_{11} + 233\beta_{10} \) |
\(\nu^{14}\) | \(=\) | \( 233\beta_{13} + 377\beta_{12} \) |
\(\nu^{15}\) | \(=\) | \( 377\beta_{15} + 610\beta_{14} \) |
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2023\mathbb{Z}\right)^\times\).
\(n\) | \(290\) | \(1737\) |
\(\chi(n)\) | \(-1\) | \(-\beta_{5}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
468.1 |
|
−0.437016 | + | 0.437016i | −0.236511 | + | 0.570989i | 0.618034i | 1.49487 | + | 0.619195i | −0.146172 | − | 0.352891i | −0.923880 | + | 0.382683i | −0.707107 | − | 0.707107i | 0.437016 | + | 0.437016i | −0.923880 | + | 0.382683i | ||||||||||||||||||||||||||||||||||||||||||||||||||
468.2 | −0.437016 | + | 0.437016i | 0.236511 | − | 0.570989i | 0.618034i | −1.49487 | − | 0.619195i | 0.146172 | + | 0.352891i | 0.923880 | − | 0.382683i | −0.707107 | − | 0.707107i | 0.437016 | + | 0.437016i | 0.923880 | − | 0.382683i | |||||||||||||||||||||||||||||||||||||||||||||||||||
468.3 | 1.14412 | − | 1.14412i | −0.619195 | + | 1.49487i | − | 1.61803i | 0.570989 | + | 0.236511i | 1.00188 | + | 2.41875i | 0.923880 | − | 0.382683i | −0.707107 | − | 0.707107i | −1.14412 | − | 1.14412i | 0.923880 | − | 0.382683i | ||||||||||||||||||||||||||||||||||||||||||||||||||
468.4 | 1.14412 | − | 1.14412i | 0.619195 | − | 1.49487i | − | 1.61803i | −0.570989 | − | 0.236511i | −1.00188 | − | 2.41875i | −0.923880 | + | 0.382683i | −0.707107 | − | 0.707107i | −1.14412 | − | 1.14412i | −0.923880 | + | 0.382683i | ||||||||||||||||||||||||||||||||||||||||||||||||||
1266.1 | −1.14412 | + | 1.14412i | −1.49487 | − | 0.619195i | − | 1.61803i | −0.236511 | + | 0.570989i | 2.41875 | − | 1.00188i | −0.382683 | − | 0.923880i | 0.707107 | + | 0.707107i | 1.14412 | + | 1.14412i | −0.382683 | − | 0.923880i | ||||||||||||||||||||||||||||||||||||||||||||||||||
1266.2 | −1.14412 | + | 1.14412i | 1.49487 | + | 0.619195i | − | 1.61803i | 0.236511 | − | 0.570989i | −2.41875 | + | 1.00188i | 0.382683 | + | 0.923880i | 0.707107 | + | 0.707107i | 1.14412 | + | 1.14412i | 0.382683 | + | 0.923880i | ||||||||||||||||||||||||||||||||||||||||||||||||||
1266.3 | 0.437016 | − | 0.437016i | −0.570989 | − | 0.236511i | 0.618034i | −0.619195 | + | 1.49487i | −0.352891 | + | 0.146172i | 0.382683 | + | 0.923880i | 0.707107 | + | 0.707107i | −0.437016 | − | 0.437016i | 0.382683 | + | 0.923880i | |||||||||||||||||||||||||||||||||||||||||||||||||||
1266.4 | 0.437016 | − | 0.437016i | 0.570989 | + | 0.236511i | 0.618034i | 0.619195 | − | 1.49487i | 0.352891 | − | 0.146172i | −0.382683 | − | 0.923880i | 0.707107 | + | 0.707107i | −0.437016 | − | 0.437016i | −0.382683 | − | 0.923880i | |||||||||||||||||||||||||||||||||||||||||||||||||||
1868.1 | −1.14412 | − | 1.14412i | −1.49487 | + | 0.619195i | 1.61803i | −0.236511 | − | 0.570989i | 2.41875 | + | 1.00188i | −0.382683 | + | 0.923880i | 0.707107 | − | 0.707107i | 1.14412 | − | 1.14412i | −0.382683 | + | 0.923880i | |||||||||||||||||||||||||||||||||||||||||||||||||||
1868.2 | −1.14412 | − | 1.14412i | 1.49487 | − | 0.619195i | 1.61803i | 0.236511 | + | 0.570989i | −2.41875 | − | 1.00188i | 0.382683 | − | 0.923880i | 0.707107 | − | 0.707107i | 1.14412 | − | 1.14412i | 0.382683 | − | 0.923880i | |||||||||||||||||||||||||||||||||||||||||||||||||||
1868.3 | 0.437016 | + | 0.437016i | −0.570989 | + | 0.236511i | − | 0.618034i | −0.619195 | − | 1.49487i | −0.352891 | − | 0.146172i | 0.382683 | − | 0.923880i | 0.707107 | − | 0.707107i | −0.437016 | + | 0.437016i | 0.382683 | − | 0.923880i | ||||||||||||||||||||||||||||||||||||||||||||||||||
1868.4 | 0.437016 | + | 0.437016i | 0.570989 | − | 0.236511i | − | 0.618034i | 0.619195 | + | 1.49487i | 0.352891 | + | 0.146172i | −0.382683 | + | 0.923880i | 0.707107 | − | 0.707107i | −0.437016 | + | 0.437016i | −0.382683 | + | 0.923880i | ||||||||||||||||||||||||||||||||||||||||||||||||||
1889.1 | −0.437016 | − | 0.437016i | −0.236511 | − | 0.570989i | − | 0.618034i | 1.49487 | − | 0.619195i | −0.146172 | + | 0.352891i | −0.923880 | − | 0.382683i | −0.707107 | + | 0.707107i | 0.437016 | − | 0.437016i | −0.923880 | − | 0.382683i | ||||||||||||||||||||||||||||||||||||||||||||||||||
1889.2 | −0.437016 | − | 0.437016i | 0.236511 | + | 0.570989i | − | 0.618034i | −1.49487 | + | 0.619195i | 0.146172 | − | 0.352891i | 0.923880 | + | 0.382683i | −0.707107 | + | 0.707107i | 0.437016 | − | 0.437016i | 0.923880 | + | 0.382683i | ||||||||||||||||||||||||||||||||||||||||||||||||||
1889.3 | 1.14412 | + | 1.14412i | −0.619195 | − | 1.49487i | 1.61803i | 0.570989 | − | 0.236511i | 1.00188 | − | 2.41875i | 0.923880 | + | 0.382683i | −0.707107 | + | 0.707107i | −1.14412 | + | 1.14412i | 0.923880 | + | 0.382683i | |||||||||||||||||||||||||||||||||||||||||||||||||||
1889.4 | 1.14412 | + | 1.14412i | 0.619195 | + | 1.49487i | 1.61803i | −0.570989 | + | 0.236511i | −1.00188 | + | 2.41875i | −0.923880 | − | 0.382683i | −0.707107 | + | 0.707107i | −1.14412 | + | 1.14412i | −0.923880 | − | 0.382683i | |||||||||||||||||||||||||||||||||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
119.d | odd | 2 | 1 | CM by \(\Q(\sqrt{-119}) \) |
7.b | odd | 2 | 1 | inner |
17.b | even | 2 | 1 | inner |
17.c | even | 4 | 2 | inner |
17.d | even | 8 | 4 | inner |
119.f | odd | 4 | 2 | inner |
119.l | odd | 8 | 4 | inner |
Twists
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{8} + 7T_{2}^{4} + 1 \)
acting on \(S_{1}^{\mathrm{new}}(2023, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( (T^{8} + 7 T^{4} + 1)^{2} \)
$3$
\( T^{16} + 47T^{8} + 1 \)
$5$
\( T^{16} + 47T^{8} + 1 \)
$7$
\( (T^{8} + 1)^{2} \)
$11$
\( T^{16} \)
$13$
\( T^{16} \)
$17$
\( T^{16} \)
$19$
\( T^{16} \)
$23$
\( T^{16} \)
$29$
\( T^{16} \)
$31$
\( T^{16} + 47T^{8} + 1 \)
$37$
\( T^{16} \)
$41$
\( T^{16} + 47T^{8} + 1 \)
$43$
\( (T^{8} + 7 T^{4} + 1)^{2} \)
$47$
\( T^{16} \)
$53$
\( (T^{8} + 7 T^{4} + 1)^{2} \)
$59$
\( T^{16} \)
$61$
\( T^{16} + 47T^{8} + 1 \)
$67$
\( (T^{2} - T - 1)^{8} \)
$71$
\( T^{16} \)
$73$
\( T^{16} + 47T^{8} + 1 \)
$79$
\( T^{16} \)
$83$
\( T^{16} \)
$89$
\( T^{16} \)
$97$
\( T^{16} + 47T^{8} + 1 \)
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