Properties

Label 572.2.b.b
Level $572$
Weight $2$
Character orbit 572.b
Analytic conductor $4.567$
Analytic rank $0$
Dimension $20$
CM discriminant -143
Inner twists $8$

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Newspace parameters

Level: \( N \) \(=\) \( 572 = 2^{2} \cdot 11 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 572.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(4.56744299562\)
Analytic rank: \(0\)
Dimension: \(20\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} - \cdots)\)
Defining polynomial: \(x^{20} - 53 x^{10} + 1024\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{14} \)
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 \(1,\beta_1,\ldots,\beta_{19}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -\beta_{14} q^{2} -\beta_{9} q^{3} + \beta_{12} q^{4} + ( -\beta_{3} + \beta_{5} ) q^{6} + ( -\beta_{11} + \beta_{15} ) q^{7} -\beta_{15} q^{8} + ( -3 + \beta_{4} - \beta_{18} ) q^{9} +O(q^{10})\) \( q -\beta_{14} q^{2} -\beta_{9} q^{3} + \beta_{12} q^{4} + ( -\beta_{3} + \beta_{5} ) q^{6} + ( -\beta_{11} + \beta_{15} ) q^{7} -\beta_{15} q^{8} + ( -3 + \beta_{4} - \beta_{18} ) q^{9} + ( \beta_{3} - \beta_{8} ) q^{11} + ( \beta_{2} + \beta_{7} ) q^{12} + ( \beta_{3} + \beta_{8} ) q^{13} + ( \beta_{4} - \beta_{16} ) q^{14} + ( \beta_{6} - \beta_{9} + \beta_{16} ) q^{16} + ( -\beta_{10} + \beta_{11} - \beta_{13} + 3 \beta_{14} - \beta_{15} ) q^{18} + ( \beta_{10} + \beta_{14} - \beta_{19} ) q^{19} + ( 2 \beta_{5} + \beta_{10} + \beta_{11} + \beta_{14} + \beta_{15} + \beta_{19} ) q^{21} + ( -\beta_{2} + \beta_{6} + \beta_{9} ) q^{22} + ( -\beta_{4} + 2 \beta_{7} + 2 \beta_{12} + \beta_{18} ) q^{23} + ( \beta_{1} - \beta_{5} - \beta_{10} - \beta_{11} - \beta_{13} ) q^{24} + 5 q^{25} + ( -\beta_{2} - \beta_{6} - \beta_{9} ) q^{26} + ( \beta_{4} + 3 \beta_{9} - 2 \beta_{12} + \beta_{18} ) q^{27} + ( -\beta_{1} - \beta_{3} + 2 \beta_{8} - \beta_{10} ) q^{28} + ( -\beta_{3} - 2 \beta_{8} ) q^{32} + ( -2 \beta_{1} + \beta_{10} - 3 \beta_{14} + \beta_{19} ) q^{33} + ( 1 - \beta_{6} + \beta_{9} - 3 \beta_{12} + \beta_{16} + \beta_{17} ) q^{36} + ( 1 + \beta_{4} - \beta_{7} - \beta_{12} - \beta_{17} - 2 \beta_{18} ) q^{38} + ( 2 \beta_{1} + \beta_{10} - 3 \beta_{14} - \beta_{19} ) q^{39} + ( -2 \beta_{5} - \beta_{11} + 2 \beta_{13} + \beta_{15} ) q^{41} + ( -3 - 2 \beta_{4} - 2 \beta_{6} + 3 \beta_{7} + 2 \beta_{9} - \beta_{12} - \beta_{16} - \beta_{17} + 2 \beta_{18} ) q^{42} + ( -\beta_{5} + \beta_{11} + \beta_{13} ) q^{44} + ( 2 \beta_{1} - \beta_{10} - \beta_{11} + \beta_{13} - \beta_{15} ) q^{46} + ( -3 + 2 \beta_{4} - \beta_{7} + \beta_{17} ) q^{48} + ( -7 - 2 \beta_{2} + \beta_{9} - 2 \beta_{16} ) q^{49} -5 \beta_{14} q^{50} + ( 3 \beta_{5} + \beta_{11} + \beta_{13} ) q^{52} + ( 2 \beta_{2} + 2 \beta_{6} - \beta_{9} + 2 \beta_{16} ) q^{53} + ( -2 \beta_{1} + 3 \beta_{3} - 3 \beta_{5} - \beta_{10} - \beta_{11} + \beta_{13} + 3 \beta_{15} ) q^{54} + ( 5 + \beta_{2} - 2 \beta_{6} - 2 \beta_{9} + \beta_{17} ) q^{56} + ( -2 \beta_{3} - 2 \beta_{5} - 2 \beta_{8} - \beta_{11} - 2 \beta_{13} - 3 \beta_{15} ) q^{57} + ( -2 \beta_{1} - 2 \beta_{3} + 2 \beta_{8} + \beta_{10} + 3 \beta_{11} + 5 \beta_{14} - 3 \beta_{15} - \beta_{19} ) q^{63} + ( \beta_{2} + 2 \beta_{6} + 2 \beta_{9} ) q^{64} + ( 5 - \beta_{4} + \beta_{7} + 3 \beta_{12} - \beta_{17} + 2 \beta_{18} ) q^{66} + ( 2 \beta_{2} - \beta_{4} - 2 \beta_{6} + 2 \beta_{7} - \beta_{9} - 4 \beta_{12} + 2 \beta_{16} - \beta_{18} ) q^{69} + ( -\beta_{1} + 3 \beta_{3} - 2 \beta_{8} - \beta_{14} + 3 \beta_{15} + 2 \beta_{19} ) q^{72} + ( 2 \beta_{1} + \beta_{10} + 5 \beta_{14} + \beta_{19} ) q^{73} -5 \beta_{9} q^{75} + ( \beta_{1} + 2 \beta_{11} - 2 \beta_{13} - \beta_{14} - \beta_{15} - 2 \beta_{19} ) q^{76} + ( 3 \beta_{4} - 2 \beta_{7} + 4 \beta_{12} - \beta_{18} ) q^{77} + ( -7 + \beta_{4} - \beta_{7} + 3 \beta_{12} - \beta_{17} - 2 \beta_{18} ) q^{78} + ( 9 - 2 \beta_{2} - 3 \beta_{4} - 4 \beta_{6} + \beta_{9} - 2 \beta_{16} + 3 \beta_{18} ) q^{81} + ( -\beta_{4} + 2 \beta_{6} - 2 \beta_{7} - 2 \beta_{9} - \beta_{16} ) q^{82} + ( -4 \beta_{5} + \beta_{11} - 2 \beta_{13} + \beta_{15} ) q^{83} + ( 4 \beta_{1} + 3 \beta_{3} + 2 \beta_{8} - \beta_{10} - 2 \beta_{11} + 2 \beta_{13} + 3 \beta_{14} + 3 \beta_{15} - 2 \beta_{19} ) q^{84} + ( -2 \beta_{4} - \beta_{7} ) q^{88} + ( \beta_{4} - 4 \beta_{7} - 2 \beta_{12} - 3 \beta_{18} ) q^{91} + ( -7 + 3 \beta_{6} - 3 \beta_{9} + \beta_{16} + \beta_{17} ) q^{92} + ( -4 \beta_{1} - \beta_{10} + 3 \beta_{14} + 2 \beta_{19} ) q^{96} + ( -\beta_{3} + \beta_{5} + 4 \beta_{8} + 2 \beta_{11} + 2 \beta_{13} + 7 \beta_{14} ) q^{98} + ( -3 \beta_{3} + 4 \beta_{5} + 3 \beta_{8} + \beta_{11} + 2 \beta_{13} - 3 \beta_{15} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20q - 60q^{9} + O(q^{10}) \) \( 20q - 60q^{9} + 100q^{25} + 10q^{36} + 30q^{38} - 50q^{42} - 70q^{48} - 140q^{49} + 90q^{56} + 110q^{66} - 130q^{78} + 180q^{81} - 150q^{92} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{20} - 53 x^{10} + 1024\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( 2 \nu \)
\(\beta_{2}\)\(=\)\((\)\( \nu^{14} + 11 \nu^{4} \)\()/48\)
\(\beta_{3}\)\(=\)\((\)\( \nu^{15} + 11 \nu^{5} \)\()/96\)
\(\beta_{4}\)\(=\)\((\)\( \nu^{12} - 13 \nu^{2} \)\()/12\)
\(\beta_{5}\)\(=\)\((\)\( -\nu^{13} + 37 \nu^{3} \)\()/24\)
\(\beta_{6}\)\(=\)\((\)\( \nu^{16} - 4 \nu^{14} + 11 \nu^{6} + 148 \nu^{4} \)\()/192\)
\(\beta_{7}\)\(=\)\((\)\( \nu^{12} - 37 \nu^{2} \)\()/12\)
\(\beta_{8}\)\(=\)\((\)\( -\nu^{15} + 37 \nu^{5} \)\()/48\)
\(\beta_{9}\)\(=\)\((\)\( -\nu^{16} - 4 \nu^{14} - 11 \nu^{6} + 148 \nu^{4} \)\()/192\)
\(\beta_{10}\)\(=\)\((\)\( \nu^{11} - 25 \nu \)\()/6\)
\(\beta_{11}\)\(=\)\((\)\( \nu^{17} + 8 \nu^{13} + 11 \nu^{7} - 104 \nu^{3} \)\()/192\)
\(\beta_{12}\)\(=\)\((\)\( -\nu^{18} + 53 \nu^{8} \)\()/256\)
\(\beta_{13}\)\(=\)\((\)\( -\nu^{17} + 8 \nu^{13} - 11 \nu^{7} - 104 \nu^{3} \)\()/192\)
\(\beta_{14}\)\(=\)\((\)\( -\nu^{19} + 53 \nu^{9} \)\()/512\)
\(\beta_{15}\)\(=\)\((\)\( -\nu^{17} + 53 \nu^{7} \)\()/128\)
\(\beta_{16}\)\(=\)\((\)\( -5 \nu^{16} + 137 \nu^{6} \)\()/192\)
\(\beta_{17}\)\(=\)\((\)\( \nu^{10} - 28 \)\()/3\)
\(\beta_{18}\)\(=\)\((\)\( 7 \nu^{18} - 115 \nu^{8} + 768 \nu^{2} \)\()/768\)
\(\beta_{19}\)\(=\)\((\)\( -7 \nu^{19} + 115 \nu^{9} + 768 \nu \)\()/768\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)\(/2\)
\(\nu^{2}\)\(=\)\((\)\(-\beta_{7} + \beta_{4}\)\()/2\)
\(\nu^{3}\)\(=\)\((\)\(\beta_{13} + \beta_{11} + 2 \beta_{5}\)\()/2\)
\(\nu^{4}\)\(=\)\((\)\(\beta_{9} + \beta_{6} + 2 \beta_{2}\)\()/2\)
\(\nu^{5}\)\(=\)\(\beta_{8} + 2 \beta_{3}\)
\(\nu^{6}\)\(=\)\((\)\(2 \beta_{16} - 5 \beta_{9} + 5 \beta_{6}\)\()/2\)
\(\nu^{7}\)\(=\)\((\)\(4 \beta_{15} - 3 \beta_{13} + 3 \beta_{11}\)\()/2\)
\(\nu^{8}\)\(=\)\((\)\(6 \beta_{18} + 14 \beta_{12} + 3 \beta_{7} - 3 \beta_{4}\)\()/2\)
\(\nu^{9}\)\(=\)\((\)\(-6 \beta_{19} + 28 \beta_{14} + 3 \beta_{1}\)\()/2\)
\(\nu^{10}\)\(=\)\(3 \beta_{17} + 28\)
\(\nu^{11}\)\(=\)\((\)\(12 \beta_{10} + 25 \beta_{1}\)\()/2\)
\(\nu^{12}\)\(=\)\((\)\(-13 \beta_{7} + 37 \beta_{4}\)\()/2\)
\(\nu^{13}\)\(=\)\((\)\(37 \beta_{13} + 37 \beta_{11} + 26 \beta_{5}\)\()/2\)
\(\nu^{14}\)\(=\)\((\)\(-11 \beta_{9} - 11 \beta_{6} + 74 \beta_{2}\)\()/2\)
\(\nu^{15}\)\(=\)\(-11 \beta_{8} + 74 \beta_{3}\)
\(\nu^{16}\)\(=\)\((\)\(-22 \beta_{16} - 137 \beta_{9} + 137 \beta_{6}\)\()/2\)
\(\nu^{17}\)\(=\)\((\)\(-44 \beta_{15} - 159 \beta_{13} + 159 \beta_{11}\)\()/2\)
\(\nu^{18}\)\(=\)\((\)\(318 \beta_{18} + 230 \beta_{12} + 159 \beta_{7} - 159 \beta_{4}\)\()/2\)
\(\nu^{19}\)\(=\)\((\)\(-318 \beta_{19} + 460 \beta_{14} + 159 \beta_{1}\)\()/2\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/572\mathbb{Z}\right)^\times\).

\(n\) \(287\) \(353\) \(365\)
\(\chi(n)\) \(-1\) \(-1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
571.1
1.41171 0.0841020i
1.41171 + 0.0841020i
1.19153 0.761743i
1.19153 + 0.761743i
1.09266 0.897823i
1.09266 + 0.897823i
0.516228 1.31663i
0.516228 + 1.31663i
0.356257 1.36861i
0.356257 + 1.36861i
−0.356257 1.36861i
−0.356257 + 1.36861i
−0.516228 1.31663i
−0.516228 + 1.31663i
−1.09266 0.897823i
−1.09266 + 0.897823i
−1.19153 0.761743i
−1.19153 + 0.761743i
−1.41171 0.0841020i
−1.41171 + 0.0841020i
−1.41171 0.0841020i 2.49267i 1.98585 + 0.237455i 0 −0.209639 + 3.51893i 4.72571i −2.78348 0.502232i −3.21342 0
571.2 −1.41171 + 0.0841020i 2.49267i 1.98585 0.237455i 0 −0.209639 3.51893i 4.72571i −2.78348 + 0.502232i −3.21342 0
571.3 −1.19153 0.761743i 0.602681i 0.839496 + 1.81528i 0 −0.459088 + 0.718113i 3.72449i 0.382491 2.80245i 2.63678 0
571.4 −1.19153 + 0.761743i 0.602681i 0.839496 1.81528i 0 −0.459088 0.718113i 3.72449i 0.382491 + 2.80245i 2.63678 0
571.5 −1.09266 0.897823i 3.43055i 0.387829 + 1.96204i 0 3.08003 3.74844i 0.803843i 1.33779 2.49205i −8.76868 0
571.6 −1.09266 + 0.897823i 3.43055i 0.387829 1.96204i 0 3.08003 + 3.74844i 0.803843i 1.33779 + 2.49205i −8.76868 0
571.7 −0.516228 1.31663i 1.51752i −1.46702 + 1.35936i 0 −1.99800 + 0.783385i 2.42385i 2.54709 + 1.22977i 0.697144 0
571.8 −0.516228 + 1.31663i 1.51752i −1.46702 1.35936i 0 −1.99800 0.783385i 2.42385i 2.54709 1.22977i 0.697144 0
571.9 −0.356257 1.36861i 3.05807i −1.74616 + 0.975150i 0 −4.18530 + 1.08946i 5.22251i 1.95668 + 2.04240i −6.35182 0
571.10 −0.356257 + 1.36861i 3.05807i −1.74616 0.975150i 0 −4.18530 1.08946i 5.22251i 1.95668 2.04240i −6.35182 0
571.11 0.356257 1.36861i 3.05807i −1.74616 0.975150i 0 4.18530 + 1.08946i 5.22251i −1.95668 + 2.04240i −6.35182 0
571.12 0.356257 + 1.36861i 3.05807i −1.74616 + 0.975150i 0 4.18530 1.08946i 5.22251i −1.95668 2.04240i −6.35182 0
571.13 0.516228 1.31663i 1.51752i −1.46702 1.35936i 0 1.99800 + 0.783385i 2.42385i −2.54709 + 1.22977i 0.697144 0
571.14 0.516228 + 1.31663i 1.51752i −1.46702 + 1.35936i 0 1.99800 0.783385i 2.42385i −2.54709 1.22977i 0.697144 0
571.15 1.09266 0.897823i 3.43055i 0.387829 1.96204i 0 −3.08003 3.74844i 0.803843i −1.33779 2.49205i −8.76868 0
571.16 1.09266 + 0.897823i 3.43055i 0.387829 + 1.96204i 0 −3.08003 + 3.74844i 0.803843i −1.33779 + 2.49205i −8.76868 0
571.17 1.19153 0.761743i 0.602681i 0.839496 1.81528i 0 0.459088 + 0.718113i 3.72449i −0.382491 2.80245i 2.63678 0
571.18 1.19153 + 0.761743i 0.602681i 0.839496 + 1.81528i 0 0.459088 0.718113i 3.72449i −0.382491 + 2.80245i 2.63678 0
571.19 1.41171 0.0841020i 2.49267i 1.98585 0.237455i 0 0.209639 + 3.51893i 4.72571i 2.78348 0.502232i −3.21342 0
571.20 1.41171 + 0.0841020i 2.49267i 1.98585 + 0.237455i 0 0.209639 3.51893i 4.72571i 2.78348 + 0.502232i −3.21342 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 571.20
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
143.d odd 2 1 CM by \(\Q(\sqrt{-143}) \)
4.b odd 2 1 inner
11.b odd 2 1 inner
13.b even 2 1 inner
44.c even 2 1 inner
52.b odd 2 1 inner
572.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 572.2.b.b 20
4.b odd 2 1 inner 572.2.b.b 20
11.b odd 2 1 inner 572.2.b.b 20
13.b even 2 1 inner 572.2.b.b 20
44.c even 2 1 inner 572.2.b.b 20
52.b odd 2 1 inner 572.2.b.b 20
143.d odd 2 1 CM 572.2.b.b 20
572.b even 2 1 inner 572.2.b.b 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
572.2.b.b 20 1.a even 1 1 trivial
572.2.b.b 20 4.b odd 2 1 inner
572.2.b.b 20 11.b odd 2 1 inner
572.2.b.b 20 13.b even 2 1 inner
572.2.b.b 20 44.c even 2 1 inner
572.2.b.b 20 52.b odd 2 1 inner
572.2.b.b 20 143.d odd 2 1 CM
572.2.b.b 20 572.b even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{10} + 30 T_{3}^{8} + 315 T_{3}^{6} + 1350 T_{3}^{4} + 2025 T_{3}^{2} + 572 \) acting on \(S_{2}^{\mathrm{new}}(572, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1024 - 53 T^{10} + T^{20} \)
$3$ \( ( 572 + 2025 T^{2} + 1350 T^{4} + 315 T^{6} + 30 T^{8} + T^{10} )^{2} \)
$5$ \( T^{20} \)
$7$ \( ( 32076 + 60025 T^{2} + 17150 T^{4} + 1715 T^{6} + 70 T^{8} + T^{10} )^{2} \)
$11$ \( ( 11 + T^{2} )^{10} \)
$13$ \( ( -13 + T^{2} )^{10} \)
$17$ \( T^{20} \)
$19$ \( ( 19404 + 3258025 T^{2} + 342950 T^{4} + 12635 T^{6} + 190 T^{8} + T^{10} )^{2} \)
$23$ \( ( 3391388 + 6996025 T^{2} + 608350 T^{4} + 18515 T^{6} + 230 T^{8} + T^{10} )^{2} \)
$29$ \( T^{20} \)
$31$ \( T^{20} \)
$37$ \( T^{20} \)
$41$ \( ( -463331700 + 70644025 T^{2} - 3446050 T^{4} + 58835 T^{6} - 410 T^{8} + T^{10} )^{2} \)
$43$ \( T^{20} \)
$47$ \( T^{20} \)
$53$ \( ( 24858 + 14045 T - 265 T^{3} + T^{5} )^{4} \)
$59$ \( T^{20} \)
$61$ \( T^{20} \)
$67$ \( T^{20} \)
$71$ \( T^{20} \)
$73$ \( ( -7761264628 + 709956025 T^{2} - 19450850 T^{4} + 186515 T^{6} - 730 T^{8} + T^{10} )^{2} \)
$79$ \( T^{20} \)
$83$ \( ( 529494284 + 1186458025 T^{2} + 28589350 T^{4} + 241115 T^{6} + 830 T^{8} + T^{10} )^{2} \)
$89$ \( T^{20} \)
$97$ \( T^{20} \)
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