Properties

Label 380.2.d.a
Level $380$
Weight $2$
Character orbit 380.d
Analytic conductor $3.034$
Analytic rank $0$
Dimension $16$
CM discriminant -95
Inner twists $8$

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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: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
Defining polynomial: \(x^{16} - 13 x^{8} + 256\)
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.

\(f(q)\) \(=\) \( q + \beta_{1} q^{2} -\beta_{5} q^{3} + \beta_{2} q^{4} -\beta_{6} q^{5} + ( -\beta_{4} + \beta_{6} + \beta_{9} ) q^{6} + \beta_{3} q^{8} + ( -3 - \beta_{2} + \beta_{9} + \beta_{13} ) q^{9} +O(q^{10})\) \( q + \beta_{1} q^{2} -\beta_{5} q^{3} + \beta_{2} q^{4} -\beta_{6} q^{5} + ( -\beta_{4} + \beta_{6} + \beta_{9} ) q^{6} + \beta_{3} q^{8} + ( -3 - \beta_{2} + \beta_{9} + \beta_{13} ) q^{9} -\beta_{10} q^{10} + ( -\beta_{2} + \beta_{9} - \beta_{13} ) q^{11} + ( -\beta_{5} + \beta_{7} + \beta_{10} + \beta_{11} ) q^{12} + ( \beta_{1} - \beta_{11} - \beta_{12} - \beta_{15} ) q^{13} + ( \beta_{1} - \beta_{11} + \beta_{12} + \beta_{15} ) q^{15} + ( \beta_{4} + \beta_{6} ) q^{16} + ( -3 \beta_{1} - \beta_{3} + \beta_{5} + \beta_{7} + \beta_{11} + \beta_{15} ) q^{18} + ( 2 \beta_{4} - \beta_{6} ) q^{19} + ( \beta_{9} + \beta_{13} - \beta_{14} ) q^{20} + ( -\beta_{3} - \beta_{5} - \beta_{7} + \beta_{11} - \beta_{15} ) q^{22} + ( -\beta_{8} + \beta_{9} - \beta_{13} + \beta_{14} ) q^{24} + 5 q^{25} + ( 2 + \beta_{2} + \beta_{8} - \beta_{13} - \beta_{14} ) q^{26} + ( -3 \beta_{1} + 3 \beta_{5} - \beta_{11} + \beta_{12} - \beta_{15} ) q^{27} + ( -2 + \beta_{2} + \beta_{8} + \beta_{13} + \beta_{14} ) q^{30} + ( \beta_{5} - \beta_{7} + \beta_{10} ) q^{32} + ( -3 \beta_{1} - 2 \beta_{3} + \beta_{5} - 2 \beta_{10} - \beta_{11} - \beta_{12} + \beta_{15} ) q^{33} + ( -4 - 3 \beta_{2} + \beta_{4} - 3 \beta_{6} - \beta_{8} ) q^{36} + ( 2 \beta_{3} - \beta_{5} + 2 \beta_{7} + 2 \beta_{10} ) q^{37} + ( 2 \beta_{5} - 2 \beta_{7} - \beta_{10} ) q^{38} + ( 3 \beta_{2} - 4 \beta_{4} + 2 \beta_{6} + \beta_{9} + \beta_{13} - 2 \beta_{14} ) q^{39} + ( \beta_{11} + 2 \beta_{15} ) q^{40} + ( 4 - 3 \beta_{4} + \beta_{6} - \beta_{8} ) q^{44} + ( 3 \beta_{2} + 3 \beta_{6} + \beta_{9} - \beta_{13} + 2 \beta_{14} ) q^{45} + ( \beta_{1} + \beta_{11} - 2 \beta_{12} - 3 \beta_{15} ) q^{48} -7 q^{49} + 5 \beta_{1} q^{50} + ( \beta_{1} + \beta_{3} - 2 \beta_{5} - 2 \beta_{7} + 2 \beta_{12} + \beta_{15} ) q^{52} + ( 2 \beta_{3} - \beta_{5} - 2 \beta_{7} + 2 \beta_{10} ) q^{53} + ( 6 - 3 \beta_{2} + 3 \beta_{4} - 3 \beta_{6} + \beta_{8} - 3 \beta_{9} + \beta_{13} + \beta_{14} ) q^{54} + ( -\beta_{2} - 3 \beta_{9} + \beta_{13} + 2 \beta_{14} ) q^{55} + ( 5 \beta_{1} - \beta_{11} - \beta_{12} - 3 \beta_{15} ) q^{57} + ( -3 \beta_{1} + \beta_{3} + 2 \beta_{5} + 2 \beta_{7} + 2 \beta_{12} + \beta_{15} ) q^{60} + ( -\beta_{2} - 3 \beta_{9} - \beta_{13} - 2 \beta_{14} ) q^{61} + ( -3 \beta_{9} - \beta_{13} + \beta_{14} ) q^{64} + ( -2 \beta_{3} + \beta_{5} - 4 \beta_{7} - 2 \beta_{10} ) q^{65} + ( -6 - 3 \beta_{2} - \beta_{4} - 3 \beta_{6} + \beta_{8} + \beta_{9} + \beta_{13} - 3 \beta_{14} ) q^{66} + ( 5 \beta_{1} - \beta_{11} + \beta_{12} + 3 \beta_{15} ) q^{67} + ( -3 \beta_{1} - 3 \beta_{3} + \beta_{5} - \beta_{7} - 3 \beta_{10} - 2 \beta_{12} - \beta_{15} ) q^{72} + ( 3 \beta_{4} + \beta_{6} + \beta_{9} - 2 \beta_{13} + 2 \beta_{14} ) q^{74} -5 \beta_{5} q^{75} + ( -3 \beta_{9} + \beta_{13} - \beta_{14} ) q^{76} + ( 3 \beta_{3} - 5 \beta_{5} + 3 \beta_{7} + 2 \beta_{10} + \beta_{11} + 3 \beta_{15} ) q^{78} + ( -8 - \beta_{8} ) q^{80} + ( 9 + 3 \beta_{2} + 2 \beta_{6} - 3 \beta_{9} - 3 \beta_{13} ) q^{81} + ( 5 \beta_{1} - 3 \beta_{5} + 3 \beta_{7} + \beta_{10} - 2 \beta_{12} - \beta_{15} ) q^{88} + ( 3 \beta_{3} + \beta_{5} + \beta_{7} + 3 \beta_{10} + \beta_{11} - 3 \beta_{15} ) q^{90} + ( -1 - 2 \beta_{8} ) q^{95} + ( 8 + \beta_{2} - \beta_{8} - 2 \beta_{13} - 2 \beta_{14} ) q^{96} + ( -2 \beta_{3} + \beta_{5} + 4 \beta_{7} - 2 \beta_{10} ) q^{97} -7 \beta_{1} q^{98} + ( 2 + 3 \beta_{2} + 4 \beta_{4} - 2 \beta_{6} + 4 \beta_{8} - 3 \beta_{9} + 3 \beta_{13} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16q - 48q^{9} + O(q^{10}) \) \( 16q - 48q^{9} + 8q^{24} + 80q^{25} + 24q^{26} - 40q^{30} - 56q^{36} + 72q^{44} - 112q^{49} + 88q^{54} - 104q^{66} - 120q^{80} + 144q^{81} + 136q^{96} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{16} - 13 x^{8} + 256\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} \)
\(\beta_{3}\)\(=\)\( \nu^{3} \)
\(\beta_{4}\)\(=\)\((\)\( \nu^{12} + 19 \nu^{4} \)\()/48\)
\(\beta_{5}\)\(=\)\((\)\( \nu^{13} + 4 \nu^{11} + 19 \nu^{5} - 20 \nu^{3} \)\()/96\)
\(\beta_{6}\)\(=\)\((\)\( -\nu^{12} + 29 \nu^{4} \)\()/48\)
\(\beta_{7}\)\(=\)\((\)\( -\nu^{13} + 4 \nu^{11} - 19 \nu^{5} - 20 \nu^{3} \)\()/96\)
\(\beta_{8}\)\(=\)\((\)\( \nu^{8} - 8 \)\()/3\)
\(\beta_{9}\)\(=\)\((\)\( -\nu^{14} - 19 \nu^{6} \)\()/96\)
\(\beta_{10}\)\(=\)\((\)\( -\nu^{13} + 29 \nu^{5} \)\()/48\)
\(\beta_{11}\)\(=\)\((\)\( -\nu^{15} - 19 \nu^{7} \)\()/96\)
\(\beta_{12}\)\(=\)\((\)\( -3 \nu^{15} + 64 \nu^{9} + 39 \nu^{7} - 320 \nu \)\()/384\)
\(\beta_{13}\)\(=\)\((\)\( 3 \nu^{14} + 16 \nu^{10} - 39 \nu^{6} - 80 \nu^{2} \)\()/192\)
\(\beta_{14}\)\(=\)\((\)\( -3 \nu^{14} + 16 \nu^{10} + 39 \nu^{6} - 80 \nu^{2} \)\()/192\)
\(\beta_{15}\)\(=\)\((\)\( \nu^{15} - 13 \nu^{7} \)\()/64\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2}\)
\(\nu^{3}\)\(=\)\(\beta_{3}\)
\(\nu^{4}\)\(=\)\(\beta_{6} + \beta_{4}\)
\(\nu^{5}\)\(=\)\(\beta_{10} - \beta_{7} + \beta_{5}\)
\(\nu^{6}\)\(=\)\(\beta_{14} - \beta_{13} - 3 \beta_{9}\)
\(\nu^{7}\)\(=\)\(-2 \beta_{15} - 3 \beta_{11}\)
\(\nu^{8}\)\(=\)\(3 \beta_{8} + 8\)
\(\nu^{9}\)\(=\)\(3 \beta_{15} + 6 \beta_{12} + 5 \beta_{1}\)
\(\nu^{10}\)\(=\)\(6 \beta_{14} + 6 \beta_{13} + 5 \beta_{2}\)
\(\nu^{11}\)\(=\)\(12 \beta_{7} + 12 \beta_{5} + 5 \beta_{3}\)
\(\nu^{12}\)\(=\)\(-19 \beta_{6} + 29 \beta_{4}\)
\(\nu^{13}\)\(=\)\(-19 \beta_{10} - 29 \beta_{7} + 29 \beta_{5}\)
\(\nu^{14}\)\(=\)\(-19 \beta_{14} + 19 \beta_{13} - 39 \beta_{9}\)
\(\nu^{15}\)\(=\)\(38 \beta_{15} - 39 \beta_{11}\)

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
−1.39956 + 0.203022i
−1.13320 0.846083i
−1.13320 + 0.846083i
−0.846083 1.13320i
−0.846083 + 1.13320i
−0.203022 1.39956i
−0.203022 + 1.39956i
0.203022 1.39956i
0.203022 + 1.39956i
0.846083 1.13320i
0.846083 + 1.13320i
1.13320 0.846083i
1.13320 + 0.846083i
1.39956 0.203022i
1.39956 + 0.203022i
−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
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 379.16
Significant digits:
Format:

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 \( T_{3}^{8} + 24 T_{3}^{6} + 180 T_{3}^{4} + 432 T_{3}^{2} + 304 \) acting on \(S_{2}^{\mathrm{new}}(380, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 256 - 13 T^{8} + T^{16} \)
$3$ \( ( 304 + 432 T^{2} + 180 T^{4} + 24 T^{6} + T^{8} )^{2} \)
$5$ \( ( -5 + T^{2} )^{8} \)
$7$ \( T^{16} \)
$11$ \( ( 304 + 44 T^{2} + T^{4} )^{4} \)
$13$ \( ( 109744 - 35152 T^{2} + 3380 T^{4} - 104 T^{6} + T^{8} )^{2} \)
$17$ \( T^{16} \)
$19$ \( ( 19 + T^{2} )^{8} \)
$23$ \( T^{16} \)
$29$ \( T^{16} \)
$31$ \( T^{16} \)
$37$ \( ( 511024 - 810448 T^{2} + 27380 T^{4} - 296 T^{6} + T^{8} )^{2} \)
$41$ \( T^{16} \)
$43$ \( T^{16} \)
$47$ \( T^{16} \)
$53$ \( ( 30935344 - 2382032 T^{2} + 56180 T^{4} - 424 T^{6} + T^{8} )^{2} \)
$59$ \( T^{16} \)
$61$ \( ( 304 - 244 T^{2} + T^{4} )^{4} \)
$67$ \( ( 43666864 + 4812208 T^{2} + 89780 T^{4} + 536 T^{6} + T^{8} )^{2} \)
$71$ \( T^{16} \)
$73$ \( T^{16} \)
$79$ \( T^{16} \)
$83$ \( T^{16} \)
$89$ \( T^{16} \)
$97$ \( ( 116480944 - 14602768 T^{2} + 188180 T^{4} - 776 T^{6} + T^{8} )^{2} \)
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