# 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 $8$

# Related objects

## 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: Complex embeddings Normalized embeddings Satake parameters Satake angles

## 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

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