Newspace parameters
Level: | \( N \) | \(=\) | \( 380 = 2^{2} \cdot 5 \cdot 19 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 380.d (of order \(2\), degree \(1\), minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(3.03431527681\) |
Analytic rank: | \(0\) |
Dimension: | \(16\) |
Coefficient field: | \(\mathbb{Q}[x]/(x^{16} - \cdots)\) |
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
Defining polynomial: |
\( x^{16} - 13x^{8} + 256 \)
|
Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
Coefficient ring index: | \( 2^{11} \) |
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 \(1,\beta_1,\ldots,\beta_{15}\) 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
\( x^{16} - 13x^{8} + 256 \)
:
\(\beta_{1}\) | \(=\) |
\( \nu \)
|
\(\beta_{2}\) | \(=\) |
\( \nu^{2} \)
|
\(\beta_{3}\) | \(=\) |
\( \nu^{3} \)
|
\(\beta_{4}\) | \(=\) |
\( ( \nu^{12} + 19\nu^{4} ) / 48 \)
|
\(\beta_{5}\) | \(=\) |
\( ( \nu^{13} + 4\nu^{11} + 19\nu^{5} - 20\nu^{3} ) / 96 \)
|
\(\beta_{6}\) | \(=\) |
\( ( -\nu^{12} + 29\nu^{4} ) / 48 \)
|
\(\beta_{7}\) | \(=\) |
\( ( -\nu^{13} + 4\nu^{11} - 19\nu^{5} - 20\nu^{3} ) / 96 \)
|
\(\beta_{8}\) | \(=\) |
\( ( \nu^{8} - 8 ) / 3 \)
|
\(\beta_{9}\) | \(=\) |
\( ( -\nu^{14} - 19\nu^{6} ) / 96 \)
|
\(\beta_{10}\) | \(=\) |
\( ( -\nu^{13} + 29\nu^{5} ) / 48 \)
|
\(\beta_{11}\) | \(=\) |
\( ( -\nu^{15} - 19\nu^{7} ) / 96 \)
|
\(\beta_{12}\) | \(=\) |
\( ( -3\nu^{15} + 64\nu^{9} + 39\nu^{7} - 320\nu ) / 384 \)
|
\(\beta_{13}\) | \(=\) |
\( ( 3\nu^{14} + 16\nu^{10} - 39\nu^{6} - 80\nu^{2} ) / 192 \)
|
\(\beta_{14}\) | \(=\) |
\( ( -3\nu^{14} + 16\nu^{10} + 39\nu^{6} - 80\nu^{2} ) / 192 \)
|
\(\beta_{15}\) | \(=\) |
\( ( \nu^{15} - 13\nu^{7} ) / 64 \)
|
\(\nu\) | \(=\) |
\( \beta_1 \)
|
\(\nu^{2}\) | \(=\) |
\( \beta_{2} \)
|
\(\nu^{3}\) | \(=\) |
\( \beta_{3} \)
|
\(\nu^{4}\) | \(=\) |
\( \beta_{6} + \beta_{4} \)
|
\(\nu^{5}\) | \(=\) |
\( \beta_{10} - \beta_{7} + \beta_{5} \)
|
\(\nu^{6}\) | \(=\) |
\( \beta_{14} - \beta_{13} - 3\beta_{9} \)
|
\(\nu^{7}\) | \(=\) |
\( -2\beta_{15} - 3\beta_{11} \)
|
\(\nu^{8}\) | \(=\) |
\( 3\beta_{8} + 8 \)
|
\(\nu^{9}\) | \(=\) |
\( 3\beta_{15} + 6\beta_{12} + 5\beta_1 \)
|
\(\nu^{10}\) | \(=\) |
\( 6\beta_{14} + 6\beta_{13} + 5\beta_{2} \)
|
\(\nu^{11}\) | \(=\) |
\( 12\beta_{7} + 12\beta_{5} + 5\beta_{3} \)
|
\(\nu^{12}\) | \(=\) |
\( -19\beta_{6} + 29\beta_{4} \)
|
\(\nu^{13}\) | \(=\) |
\( -19\beta_{10} - 29\beta_{7} + 29\beta_{5} \)
|
\(\nu^{14}\) | \(=\) |
\( -19\beta_{14} + 19\beta_{13} - 39\beta_{9} \)
|
\(\nu^{15}\) | \(=\) |
\( 38\beta_{15} - 39\beta_{11} \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/380\mathbb{Z}\right)^\times\).
\(n\) | \(21\) | \(77\) | \(191\) |
\(\chi(n)\) | \(-1\) | \(-1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
379.1 |
|
−1.39956 | − | 0.203022i | 3.27727i | 1.91756 | + | 0.568286i | −2.23607 | 0.665359 | − | 4.58675i | 0 | −2.56838 | − | 1.18466i | −7.74048 | 3.12952 | + | 0.453972i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||
379.2 | −1.39956 | + | 0.203022i | − | 3.27727i | 1.91756 | − | 0.568286i | −2.23607 | 0.665359 | + | 4.58675i | 0 | −2.56838 | + | 1.18466i | −7.74048 | 3.12952 | − | 0.453972i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||
379.3 | −1.13320 | − | 0.846083i | 1.52380i | 0.568286 | + | 1.91756i | 2.23607 | 1.28926 | − | 1.72677i | 0 | 0.978437 | − | 2.65380i | 0.678024 | −2.53391 | − | 1.89190i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||
379.4 | −1.13320 | + | 0.846083i | − | 1.52380i | 0.568286 | − | 1.91756i | 2.23607 | 1.28926 | + | 1.72677i | 0 | 0.978437 | + | 2.65380i | 0.678024 | −2.53391 | + | 1.89190i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||
379.5 | −0.846083 | − | 1.13320i | − | 3.11095i | −0.568286 | + | 1.91756i | 2.23607 | −3.52533 | + | 2.63212i | 0 | 2.65380 | − | 0.978437i | −6.67802 | −1.89190 | − | 2.53391i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||
379.6 | −0.846083 | + | 1.13320i | 3.11095i | −0.568286 | − | 1.91756i | 2.23607 | −3.52533 | − | 2.63212i | 0 | 2.65380 | + | 0.978437i | −6.67802 | −1.89190 | + | 2.53391i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||
379.7 | −0.203022 | − | 1.39956i | 1.12228i | −1.91756 | + | 0.568286i | −2.23607 | 1.57071 | − | 0.227849i | 0 | 1.18466 | + | 2.56838i | 1.74048 | 0.453972 | + | 3.12952i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||
379.8 | −0.203022 | + | 1.39956i | − | 1.12228i | −1.91756 | − | 0.568286i | −2.23607 | 1.57071 | + | 0.227849i | 0 | 1.18466 | − | 2.56838i | 1.74048 | 0.453972 | − | 3.12952i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||
379.9 | 0.203022 | − | 1.39956i | 1.12228i | −1.91756 | − | 0.568286i | −2.23607 | 1.57071 | + | 0.227849i | 0 | −1.18466 | + | 2.56838i | 1.74048 | −0.453972 | + | 3.12952i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||
379.10 | 0.203022 | + | 1.39956i | − | 1.12228i | −1.91756 | + | 0.568286i | −2.23607 | 1.57071 | − | 0.227849i | 0 | −1.18466 | − | 2.56838i | 1.74048 | −0.453972 | − | 3.12952i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||
379.11 | 0.846083 | − | 1.13320i | − | 3.11095i | −0.568286 | − | 1.91756i | 2.23607 | −3.52533 | − | 2.63212i | 0 | −2.65380 | − | 0.978437i | −6.67802 | 1.89190 | − | 2.53391i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||
379.12 | 0.846083 | + | 1.13320i | 3.11095i | −0.568286 | + | 1.91756i | 2.23607 | −3.52533 | + | 2.63212i | 0 | −2.65380 | + | 0.978437i | −6.67802 | 1.89190 | + | 2.53391i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||
379.13 | 1.13320 | − | 0.846083i | 1.52380i | 0.568286 | − | 1.91756i | 2.23607 | 1.28926 | + | 1.72677i | 0 | −0.978437 | − | 2.65380i | 0.678024 | 2.53391 | − | 1.89190i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||
379.14 | 1.13320 | + | 0.846083i | − | 1.52380i | 0.568286 | + | 1.91756i | 2.23607 | 1.28926 | − | 1.72677i | 0 | −0.978437 | + | 2.65380i | 0.678024 | 2.53391 | + | 1.89190i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||
379.15 | 1.39956 | − | 0.203022i | 3.27727i | 1.91756 | − | 0.568286i | −2.23607 | 0.665359 | + | 4.58675i | 0 | 2.56838 | − | 1.18466i | −7.74048 | −3.12952 | + | 0.453972i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||
379.16 | 1.39956 | + | 0.203022i | − | 3.27727i | 1.91756 | + | 0.568286i | −2.23607 | 0.665359 | − | 4.58675i | 0 | 2.56838 | + | 1.18466i | −7.74048 | −3.12952 | − | 0.453972i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
95.d | odd | 2 | 1 | CM by \(\Q(\sqrt{-95}) \) |
4.b | odd | 2 | 1 | inner |
5.b | even | 2 | 1 | inner |
19.b | odd | 2 | 1 | inner |
20.d | odd | 2 | 1 | inner |
76.d | even | 2 | 1 | inner |
380.d | even | 2 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 380.2.d.a | ✓ | 16 |
4.b | odd | 2 | 1 | inner | 380.2.d.a | ✓ | 16 |
5.b | even | 2 | 1 | inner | 380.2.d.a | ✓ | 16 |
19.b | odd | 2 | 1 | inner | 380.2.d.a | ✓ | 16 |
20.d | odd | 2 | 1 | inner | 380.2.d.a | ✓ | 16 |
76.d | even | 2 | 1 | inner | 380.2.d.a | ✓ | 16 |
95.d | odd | 2 | 1 | CM | 380.2.d.a | ✓ | 16 |
380.d | even | 2 | 1 | inner | 380.2.d.a | ✓ | 16 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
380.2.d.a | ✓ | 16 | 1.a | even | 1 | 1 | trivial |
380.2.d.a | ✓ | 16 | 4.b | odd | 2 | 1 | inner |
380.2.d.a | ✓ | 16 | 5.b | even | 2 | 1 | inner |
380.2.d.a | ✓ | 16 | 19.b | odd | 2 | 1 | inner |
380.2.d.a | ✓ | 16 | 20.d | odd | 2 | 1 | inner |
380.2.d.a | ✓ | 16 | 76.d | even | 2 | 1 | inner |
380.2.d.a | ✓ | 16 | 95.d | odd | 2 | 1 | CM |
380.2.d.a | ✓ | 16 | 380.d | even | 2 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{8} + 24T_{3}^{6} + 180T_{3}^{4} + 432T_{3}^{2} + 304 \)
acting on \(S_{2}^{\mathrm{new}}(380, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{16} - 13T^{8} + 256 \)
$3$
\( (T^{8} + 24 T^{6} + 180 T^{4} + 432 T^{2} + \cdots + 304)^{2} \)
$5$
\( (T^{2} - 5)^{8} \)
$7$
\( T^{16} \)
$11$
\( (T^{4} + 44 T^{2} + 304)^{4} \)
$13$
\( (T^{8} - 104 T^{6} + 3380 T^{4} + \cdots + 109744)^{2} \)
$17$
\( T^{16} \)
$19$
\( (T^{2} + 19)^{8} \)
$23$
\( T^{16} \)
$29$
\( T^{16} \)
$31$
\( T^{16} \)
$37$
\( (T^{8} - 296 T^{6} + 27380 T^{4} + \cdots + 511024)^{2} \)
$41$
\( T^{16} \)
$43$
\( T^{16} \)
$47$
\( T^{16} \)
$53$
\( (T^{8} - 424 T^{6} + 56180 T^{4} + \cdots + 30935344)^{2} \)
$59$
\( T^{16} \)
$61$
\( (T^{4} - 244 T^{2} + 304)^{4} \)
$67$
\( (T^{8} + 536 T^{6} + 89780 T^{4} + \cdots + 43666864)^{2} \)
$71$
\( T^{16} \)
$73$
\( T^{16} \)
$79$
\( T^{16} \)
$83$
\( T^{16} \)
$89$
\( T^{16} \)
$97$
\( (T^{8} - 776 T^{6} + 188180 T^{4} + \cdots + 116480944)^{2} \)
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