# Properties

 Label 525.2.t.h Level 525 Weight 2 Character orbit 525.t Analytic conductor 4.192 Analytic rank 0 Dimension 20 CM no Inner twists 4

# Related objects

## Newspace parameters

 Level: $$N$$ = $$525 = 3 \cdot 5^{2} \cdot 7$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 525.t (of order $$6$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$4.19214610612$$ Analytic rank: $$0$$ Dimension: $$20$$ Relative dimension: $$10$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\mathbb{Q}[x]/(x^{20} - \cdots)$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$3^{2}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + \beta_{3} q^{2} -\beta_{1} q^{3} + ( 1 - \beta_{1} - \beta_{4} + \beta_{11} - \beta_{17} - \beta_{18} ) q^{4} + ( \beta_{1} + \beta_{4} - \beta_{9} + \beta_{13} + \beta_{18} ) q^{6} + ( 1 + \beta_{1} - \beta_{5} - \beta_{6} - \beta_{7} - \beta_{9} - \beta_{10} - \beta_{11} + \beta_{13} - \beta_{17} + 2 \beta_{18} ) q^{7} + ( \beta_{1} - \beta_{2} + \beta_{4} - \beta_{6} - \beta_{7} + \beta_{8} - 2 \beta_{10} + \beta_{12} + 2 \beta_{13} + \beta_{14} + 3 \beta_{18} + \beta_{19} ) q^{8} + ( -1 + 2 \beta_{1} - \beta_{5} - \beta_{9} - \beta_{10} + \beta_{11} + \beta_{13} + \beta_{15} + 2 \beta_{18} + \beta_{19} ) q^{9} +O(q^{10})$$ $$q + \beta_{3} q^{2} -\beta_{1} q^{3} + ( 1 - \beta_{1} - \beta_{4} + \beta_{11} - \beta_{17} - \beta_{18} ) q^{4} + ( \beta_{1} + \beta_{4} - \beta_{9} + \beta_{13} + \beta_{18} ) q^{6} + ( 1 + \beta_{1} - \beta_{5} - \beta_{6} - \beta_{7} - \beta_{9} - \beta_{10} - \beta_{11} + \beta_{13} - \beta_{17} + 2 \beta_{18} ) q^{7} + ( \beta_{1} - \beta_{2} + \beta_{4} - \beta_{6} - \beta_{7} + \beta_{8} - 2 \beta_{10} + \beta_{12} + 2 \beta_{13} + \beta_{14} + 3 \beta_{18} + \beta_{19} ) q^{8} + ( -1 + 2 \beta_{1} - \beta_{5} - \beta_{9} - \beta_{10} + \beta_{11} + \beta_{13} + \beta_{15} + 2 \beta_{18} + \beta_{19} ) q^{9} + ( -\beta_{9} + \beta_{12} + \beta_{15} ) q^{11} + ( 1 - 2 \beta_{1} + \beta_{2} - \beta_{4} + \beta_{7} + \beta_{8} + \beta_{10} + \beta_{11} - 2 \beta_{12} - \beta_{13} - 2 \beta_{18} ) q^{12} + ( 1 - \beta_{5} - \beta_{6} - \beta_{7} - 2 \beta_{11} + \beta_{18} ) q^{13} + ( \beta_{1} - \beta_{2} + \beta_{4} - \beta_{5} - \beta_{6} - \beta_{12} + \beta_{13} + \beta_{14} + \beta_{18} + \beta_{19} ) q^{14} + ( -3 - \beta_{1} - 2 \beta_{4} + \beta_{6} + 3 \beta_{7} + 2 \beta_{9} + 2 \beta_{10} + 3 \beta_{11} - 2 \beta_{13} - 5 \beta_{18} ) q^{16} + ( 3 \beta_{1} - 2 \beta_{2} + \beta_{4} - \beta_{5} - \beta_{6} - \beta_{7} - \beta_{9} - 2 \beta_{10} + \beta_{12} + 2 \beta_{13} + \beta_{15} + 4 \beta_{18} + 2 \beta_{19} ) q^{17} + ( 1 + \beta_{1} - \beta_{3} + 2 \beta_{4} - \beta_{5} - \beta_{6} - \beta_{7} - 2 \beta_{8} + \beta_{9} - 3 \beta_{11} - \beta_{12} + 2 \beta_{13} + \beta_{14} - \beta_{17} + 2 \beta_{18} ) q^{18} + ( \beta_{1} + \beta_{4} - \beta_{5} - 2 \beta_{8} + \beta_{9} + \beta_{10} - \beta_{11} + \beta_{13} + \beta_{16} - \beta_{17} ) q^{19} + ( -1 - \beta_{1} + \beta_{2} - \beta_{3} + 2 \beta_{5} + \beta_{6} + \beta_{9} + \beta_{11} - \beta_{12} - \beta_{13} - \beta_{15} + \beta_{17} - 2 \beta_{18} - \beta_{19} ) q^{21} + ( -2 + \beta_{4} + \beta_{6} + \beta_{7} - 2 \beta_{8} + 2 \beta_{9} + 2 \beta_{10} + \beta_{16} - 2 \beta_{18} ) q^{22} + ( -2 \beta_{2} - \beta_{5} - 2 \beta_{8} + \beta_{9} - \beta_{10} + \beta_{13} + \beta_{18} ) q^{23} + ( 2 + 2 \beta_{1} - 2 \beta_{2} + \beta_{3} + \beta_{4} - \beta_{5} - 2 \beta_{6} - 3 \beta_{7} + \beta_{8} - 3 \beta_{9} - 4 \beta_{10} - \beta_{11} + \beta_{12} + 3 \beta_{13} + \beta_{14} + 6 \beta_{18} ) q^{24} + ( 2 \beta_{1} + \beta_{3} - \beta_{4} + \beta_{6} - \beta_{9} + \beta_{10} - \beta_{13} - 2 \beta_{14} + \beta_{18} ) q^{26} + ( -1 - \beta_{1} - 2 \beta_{3} - \beta_{4} + \beta_{9} + 2 \beta_{11} - \beta_{13} + \beta_{14} - 2 \beta_{15} - 3 \beta_{18} - \beta_{19} ) q^{27} + ( -1 - \beta_{1} - \beta_{4} - 2 \beta_{7} + 3 \beta_{8} - 2 \beta_{9} - 2 \beta_{10} + 3 \beta_{11} - \beta_{13} + \beta_{18} ) q^{28} + ( -2 \beta_{1} + 2 \beta_{2} + \beta_{7} + \beta_{8} + \beta_{9} + \beta_{10} - 2 \beta_{12} - \beta_{18} ) q^{29} + ( 4 - 2 \beta_{1} + \beta_{5} - \beta_{7} - 2 \beta_{11} ) q^{31} + ( \beta_{1} - \beta_{3} - \beta_{4} + \beta_{5} + 2 \beta_{6} - \beta_{7} - \beta_{8} - 2 \beta_{10} + 2 \beta_{12} + \beta_{13} + \beta_{14} + 2 \beta_{15} + 3 \beta_{18} ) q^{32} + ( \beta_{1} - \beta_{7} + \beta_{9} - \beta_{10} - \beta_{11} + \beta_{13} - \beta_{15} + \beta_{16} - \beta_{17} + \beta_{18} + \beta_{19} ) q^{33} + ( 4 - 4 \beta_{1} - 2 \beta_{4} - 2 \beta_{6} - \beta_{7} + \beta_{8} + \beta_{9} + \beta_{10} - 4 \beta_{11} - 2 \beta_{13} - \beta_{16} - 2 \beta_{17} - \beta_{18} ) q^{34} + ( -2 - \beta_{1} + 2 \beta_{2} - \beta_{4} - \beta_{5} + \beta_{6} + \beta_{7} - \beta_{9} + 2 \beta_{10} - 2 \beta_{12} - 3 \beta_{13} - \beta_{14} - \beta_{16} - 3 \beta_{18} - \beta_{19} ) q^{36} + ( 1 - 3 \beta_{1} - \beta_{4} + \beta_{5} - \beta_{7} + \beta_{11} - 2 \beta_{16} - 2 \beta_{17} - \beta_{18} ) q^{37} + ( \beta_{1} + 2 \beta_{2} + \beta_{3} - \beta_{4} - \beta_{5} + \beta_{7} + \beta_{8} - \beta_{9} - \beta_{12} - \beta_{13} + \beta_{14} + \beta_{15} + \beta_{18} + 2 \beta_{19} ) q^{38} + ( -2 + \beta_{1} + 2 \beta_{3} + 2 \beta_{5} + \beta_{6} + \beta_{8} - 2 \beta_{10} + 2 \beta_{11} + 2 \beta_{12} - \beta_{13} - 2 \beta_{14} - \beta_{15} + 2 \beta_{17} ) q^{39} + ( -\beta_{1} + \beta_{2} - \beta_{4} + 2 \beta_{5} + \beta_{6} - \beta_{7} - \beta_{8} + 2 \beta_{9} - 2 \beta_{10} + \beta_{12} - 2 \beta_{13} - 2 \beta_{15} - 3 \beta_{18} - \beta_{19} ) q^{41} + ( -3 + 2 \beta_{1} + 2 \beta_{4} - 2 \beta_{5} - \beta_{6} + \beta_{7} + \beta_{8} - \beta_{9} + \beta_{10} - 2 \beta_{11} - \beta_{12} - \beta_{14} + 2 \beta_{17} + 2 \beta_{18} - \beta_{19} ) q^{42} + ( -2 \beta_{1} - 2 \beta_{4} + \beta_{5} - \beta_{6} + 2 \beta_{7} - \beta_{8} + \beta_{9} + \beta_{10} - 2 \beta_{18} ) q^{43} + ( -2 \beta_{1} + \beta_{2} - 2 \beta_{4} + \beta_{6} + \beta_{7} - \beta_{8} + \beta_{9} - 2 \beta_{13} - \beta_{15} - 2 \beta_{18} - \beta_{19} ) q^{44} + ( \beta_{1} + \beta_{4} - 2 \beta_{6} + \beta_{8} - 2 \beta_{11} - \beta_{13} + 3 \beta_{18} ) q^{46} + ( -\beta_{2} + \beta_{4} - 2 \beta_{6} - \beta_{7} + \beta_{8} - 3 \beta_{10} + 2 \beta_{12} + \beta_{13} - \beta_{15} + \beta_{18} + \beta_{19} ) q^{47} + ( -4 \beta_{1} - \beta_{2} + 2 \beta_{3} - 2 \beta_{4} - \beta_{5} - 2 \beta_{6} + \beta_{7} + 2 \beta_{8} + 3 \beta_{10} + 4 \beta_{11} - \beta_{12} - \beta_{13} - \beta_{14} - \beta_{16} - 2 \beta_{17} - 3 \beta_{18} ) q^{48} + ( 1 - \beta_{1} + 2 \beta_{4} - 3 \beta_{7} - 3 \beta_{11} + 2 \beta_{18} ) q^{49} + ( -1 + \beta_{1} - \beta_{2} - \beta_{3} - 2 \beta_{5} - \beta_{6} + 2 \beta_{7} + \beta_{9} + \beta_{10} - \beta_{11} - \beta_{15} + 2 \beta_{16} + 2 \beta_{17} - \beta_{19} ) q^{51} + ( 5 - 3 \beta_{1} + \beta_{4} + 2 \beta_{5} + \beta_{6} - \beta_{8} + 2 \beta_{9} + 2 \beta_{10} - 3 \beta_{11} - \beta_{13} + 2 \beta_{16} + \beta_{17} - 2 \beta_{18} ) q^{52} + ( -\beta_{1} + 2 \beta_{3} + \beta_{4} - \beta_{8} + \beta_{9} - \beta_{12} + \beta_{13} - 2 \beta_{14} + \beta_{15} - \beta_{18} ) q^{53} + ( -3 - \beta_{3} + 2 \beta_{5} + \beta_{6} - \beta_{7} + 2 \beta_{8} - 2 \beta_{10} - \beta_{11} + 2 \beta_{14} - 2 \beta_{16} + 2 \beta_{17} + 2 \beta_{18} ) q^{54} + ( 2 \beta_{1} - 2 \beta_{2} + \beta_{3} + 3 \beta_{4} + \beta_{5} - 2 \beta_{7} + 4 \beta_{13} + \beta_{14} + 2 \beta_{15} + 3 \beta_{18} ) q^{56} + ( 2 - 2 \beta_{1} + \beta_{2} - \beta_{4} + \beta_{5} - \beta_{6} + 2 \beta_{8} - \beta_{10} - \beta_{12} + \beta_{13} + \beta_{14} - 2 \beta_{16} + \beta_{18} + \beta_{19} ) q^{57} + ( -2 + 7 \beta_{1} + 2 \beta_{4} - 2 \beta_{5} + \beta_{6} - \beta_{7} - 4 \beta_{9} - 4 \beta_{10} + 2 \beta_{11} + 4 \beta_{13} + 5 \beta_{18} ) q^{58} + ( -\beta_{1} - 2 \beta_{2} - 2 \beta_{3} + \beta_{4} + \beta_{5} - \beta_{7} - \beta_{8} + \beta_{9} + \beta_{12} + \beta_{13} - 2 \beta_{14} - \beta_{15} - \beta_{18} - 2 \beta_{19} ) q^{59} + ( -3 - \beta_{1} + \beta_{4} + 2 \beta_{6} - \beta_{7} - 2 \beta_{8} + \beta_{9} + \beta_{10} - 3 \beta_{11} + \beta_{13} - 2 \beta_{18} ) q^{61} + ( 4 \beta_{3} - \beta_{5} - \beta_{6} + \beta_{7} + 2 \beta_{8} - 2 \beta_{9} + 2 \beta_{10} - 2 \beta_{14} + \beta_{18} ) q^{62} + ( 1 - \beta_{1} + \beta_{2} - \beta_{3} - \beta_{4} + \beta_{5} + \beta_{6} + \beta_{10} - 3 \beta_{11} + \beta_{12} - 4 \beta_{13} + 2 \beta_{14} - 4 \beta_{18} - \beta_{19} ) q^{63} + ( -7 + \beta_{4} + \beta_{6} + 5 \beta_{7} - 6 \beta_{8} + 6 \beta_{9} + 6 \beta_{10} + \beta_{16} - 6 \beta_{18} ) q^{64} + ( 3 + \beta_{3} - \beta_{4} - \beta_{5} - 2 \beta_{6} + 2 \beta_{8} - \beta_{9} - \beta_{10} - \beta_{11} + 2 \beta_{13} + \beta_{14} + \beta_{15} - 2 \beta_{16} - \beta_{17} + 3 \beta_{18} + 2 \beta_{19} ) q^{66} + ( 1 - 2 \beta_{1} - 2 \beta_{4} - \beta_{6} + 2 \beta_{8} - 3 \beta_{11} - 2 \beta_{13} - \beta_{17} - \beta_{18} ) q^{67} + ( 2 \beta_{1} - \beta_{2} - \beta_{4} + \beta_{5} + 2 \beta_{6} - \beta_{7} - \beta_{8} - \beta_{9} + 2 \beta_{12} + \beta_{15} + 2 \beta_{18} - \beta_{19} ) q^{68} + ( -7 + 3 \beta_{1} - 2 \beta_{3} + \beta_{4} - \beta_{5} + \beta_{6} + 3 \beta_{7} + 2 \beta_{8} - \beta_{9} + 2 \beta_{10} + 6 \beta_{11} - \beta_{13} + \beta_{14} + 2 \beta_{15} + 2 \beta_{16} + 4 \beta_{17} + \beta_{18} + \beta_{19} ) q^{69} + ( -2 \beta_{2} + \beta_{4} - 2 \beta_{5} - 3 \beta_{6} + \beta_{8} - \beta_{9} - 3 \beta_{10} + 2 \beta_{12} + 2 \beta_{13} + 2 \beta_{14} + 3 \beta_{18} + 2 \beta_{19} ) q^{71} + ( -3 + 7 \beta_{1} + \beta_{2} - 2 \beta_{3} + 6 \beta_{4} + \beta_{5} + \beta_{6} - 2 \beta_{7} - 2 \beta_{9} - 2 \beta_{10} - \beta_{11} + \beta_{13} - \beta_{15} + 4 \beta_{16} + 4 \beta_{17} + 5 \beta_{18} - \beta_{19} ) q^{72} + ( 2 + 3 \beta_{1} + \beta_{4} - \beta_{5} - \beta_{7} + \beta_{8} - 2 \beta_{9} - 2 \beta_{10} - \beta_{11} + \beta_{13} + 3 \beta_{18} ) q^{73} + ( -5 \beta_{3} + \beta_{5} - \beta_{6} - \beta_{7} + \beta_{8} - 2 \beta_{10} + 2 \beta_{12} - \beta_{13} + 5 \beta_{14} - 2 \beta_{15} - \beta_{18} ) q^{74} + ( -1 + 4 \beta_{1} + 4 \beta_{4} - 2 \beta_{7} - 2 \beta_{8} - 2 \beta_{9} - 2 \beta_{10} + 2 \beta_{11} + 4 \beta_{13} + 6 \beta_{18} ) q^{76} + ( -4 \beta_{1} - \beta_{2} - \beta_{4} + \beta_{6} + \beta_{7} - 3 \beta_{8} + 2 \beta_{9} + \beta_{10} - \beta_{13} + 2 \beta_{14} - \beta_{15} - 4 \beta_{18} - 3 \beta_{19} ) q^{77} + ( 5 - 4 \beta_{1} + \beta_{2} + \beta_{5} + 2 \beta_{6} + 4 \beta_{7} - 5 \beta_{8} + 5 \beta_{9} + 6 \beta_{10} - \beta_{12} - \beta_{13} - 2 \beta_{14} + \beta_{16} - 8 \beta_{18} - 2 \beta_{19} ) q^{78} + ( 2 + 2 \beta_{4} + 2 \beta_{5} + 2 \beta_{6} + \beta_{7} + \beta_{9} + \beta_{10} - 3 \beta_{11} - \beta_{13} + \beta_{16} + \beta_{17} - \beta_{18} ) q^{79} + ( -1 - 2 \beta_{1} + \beta_{3} + \beta_{5} - 2 \beta_{6} + 3 \beta_{8} - \beta_{10} + 3 \beta_{11} + \beta_{12} - 3 \beta_{13} - \beta_{14} + \beta_{17} - 2 \beta_{18} ) q^{81} + ( -3 + 6 \beta_{1} + 5 \beta_{4} - 2 \beta_{5} - \beta_{6} + \beta_{7} - 4 \beta_{8} + 2 \beta_{9} + 2 \beta_{10} - 2 \beta_{11} + 2 \beta_{13} - \beta_{16} + \beta_{17} + 4 \beta_{18} ) q^{82} + ( -\beta_{1} + \beta_{2} - 4 \beta_{3} + \beta_{4} + \beta_{5} - 2 \beta_{6} + 3 \beta_{8} - 2 \beta_{9} + 2 \beta_{10} + \beta_{12} - 2 \beta_{13} + 2 \beta_{14} - 2 \beta_{15} - 4 \beta_{18} - \beta_{19} ) q^{83} + ( 5 - \beta_{3} - \beta_{4} - 2 \beta_{7} + \beta_{8} - 3 \beta_{9} - 2 \beta_{10} - \beta_{11} - \beta_{13} - 4 \beta_{14} - \beta_{18} ) q^{84} + ( -3 \beta_{1} + 2 \beta_{2} - 3 \beta_{4} - 2 \beta_{5} + 3 \beta_{7} + 2 \beta_{8} - \beta_{9} + \beta_{10} - \beta_{13} + 2 \beta_{18} ) q^{86} + ( -\beta_{1} + \beta_{3} - 2 \beta_{4} - 3 \beta_{7} - 3 \beta_{9} - 3 \beta_{10} + \beta_{11} + \beta_{14} - 4 \beta_{16} - 2 \beta_{17} + \beta_{18} ) q^{87} + ( -3 + 4 \beta_{1} + 4 \beta_{4} + 2 \beta_{5} + 2 \beta_{7} - \beta_{8} + 2 \beta_{11} + \beta_{13} + 3 \beta_{17} + 4 \beta_{18} ) q^{88} + ( \beta_{2} + 2 \beta_{3} - \beta_{4} + 2 \beta_{6} + \beta_{7} - \beta_{8} + 3 \beta_{10} - 2 \beta_{12} - \beta_{13} - 4 \beta_{14} + \beta_{15} - \beta_{18} - \beta_{19} ) q^{89} + ( 3 - 4 \beta_{1} + \beta_{4} + 2 \beta_{5} + \beta_{6} - \beta_{7} - 4 \beta_{8} + 2 \beta_{9} + 2 \beta_{10} - 3 \beta_{11} + 2 \beta_{13} - 3 \beta_{17} - 2 \beta_{18} ) q^{91} + ( 2 \beta_{1} + 2 \beta_{2} - \beta_{4} + \beta_{5} + 2 \beta_{6} + 2 \beta_{10} - 2 \beta_{12} - 2 \beta_{13} - 2 \beta_{14} - \beta_{18} ) q^{92} + ( 1 + 2 \beta_{1} + \beta_{2} + \beta_{3} - \beta_{5} + \beta_{6} + \beta_{7} - \beta_{9} - \beta_{10} - 2 \beta_{11} + \beta_{15} + \beta_{16} + \beta_{17} + 3 \beta_{18} + \beta_{19} ) q^{93} + ( 1 + \beta_{6} - 2 \beta_{16} - \beta_{17} - \beta_{18} ) q^{94} + ( 6 - 2 \beta_{1} - \beta_{2} - 2 \beta_{3} - 3 \beta_{4} + 3 \beta_{5} + 3 \beta_{6} - 4 \beta_{7} + 2 \beta_{8} - 4 \beta_{10} + 5 \beta_{11} + 2 \beta_{12} + \beta_{13} + 4 \beta_{14} - \beta_{15} + \beta_{16} - \beta_{17} + \beta_{19} ) q^{96} + ( 1 - 2 \beta_{1} - 3 \beta_{4} - \beta_{5} + \beta_{7} + \beta_{8} + \beta_{9} + \beta_{10} - 2 \beta_{11} - 2 \beta_{13} - 4 \beta_{18} ) q^{97} + ( 3 \beta_{1} + \beta_{3} + \beta_{8} - 3 \beta_{9} + 3 \beta_{10} - 2 \beta_{13} - 3 \beta_{14} - \beta_{18} ) q^{98} + ( 7 - 2 \beta_{1} + \beta_{2} - 3 \beta_{4} - 2 \beta_{6} + \beta_{7} + 2 \beta_{8} - 2 \beta_{9} - \beta_{10} - \beta_{12} - \beta_{13} + \beta_{14} - 2 \beta_{16} - \beta_{19} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$20q - 3q^{3} + 14q^{4} - 7q^{9} + O(q^{10})$$ $$20q - 3q^{3} + 14q^{4} - 7q^{9} + 21q^{12} - 18q^{16} - 14q^{18} - 9q^{21} - 20q^{22} + 18q^{24} + 10q^{28} + 42q^{31} - 12q^{33} - 36q^{36} - 24q^{37} - 33q^{42} - 36q^{43} - 8q^{46} - 4q^{49} + 21q^{51} + 84q^{52} - 75q^{54} - 6q^{57} + 4q^{58} - 90q^{61} + 5q^{63} - 120q^{64} + 6q^{66} - 20q^{67} + 35q^{72} + 48q^{73} + 108q^{78} + 46q^{79} + 29q^{81} - 36q^{82} + 75q^{84} - 69q^{87} - 4q^{88} - 30q^{91} + 30q^{93} + 6q^{94} + 135q^{96} + 94q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{20} - 3 x^{19} + 8 x^{18} - 15 x^{17} + 18 x^{16} - 45 x^{15} + 59 x^{14} - 147 x^{13} + 271 x^{12} - 330 x^{11} + 879 x^{10} - 990 x^{9} + 2439 x^{8} - 3969 x^{7} + 4779 x^{6} - 10935 x^{5} + 13122 x^{4} - 32805 x^{3} + 52488 x^{2} - 59049 x + 59049$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$1471510 \nu^{19} - 69953127 \nu^{18} + 93678038 \nu^{17} - 218139213 \nu^{16} + 320405724 \nu^{15} - 97508871 \nu^{14} + 1767246764 \nu^{13} - 809376264 \nu^{12} + 4114757071 \nu^{11} - 7746805296 \nu^{10} + 1616990253 \nu^{9} - 31989998178 \nu^{8} + 12952089549 \nu^{7} - 56264911038 \nu^{6} + 111034459314 \nu^{5} - 27083215323 \nu^{4} + 330348211866 \nu^{3} - 307571047911 \nu^{2} + 762498870948 \nu - 1156143537891$$$$)/ 74996560260$$ $$\beta_{3}$$ $$=$$ $$($$$$2890481 \nu^{19} - 61404921 \nu^{18} + 102401257 \nu^{17} - 162636525 \nu^{16} + 390130047 \nu^{15} - 39501603 \nu^{14} + 1446509806 \nu^{13} - 1426349307 \nu^{12} + 2462906579 \nu^{11} - 8943284355 \nu^{10} + 79947147 \nu^{9} - 25788817935 \nu^{8} + 23153332023 \nu^{7} - 36086204970 \nu^{6} + 118545320313 \nu^{5} - 18265918041 \nu^{4} + 282366698139 \nu^{3} - 384817926195 \nu^{2} + 667140004431 \nu - 1076681692251$$$$)/ 74996560260$$ $$\beta_{4}$$ $$=$$ $$($$$$-797782 \nu^{19} - 4425285 \nu^{18} - 10337801 \nu^{17} - 9682413 \nu^{16} - 12388275 \nu^{15} + 55743165 \nu^{14} + 185399917 \nu^{13} + 389001240 \nu^{12} + 436225433 \nu^{11} + 389540124 \nu^{10} - 1008193641 \nu^{9} - 2616620643 \nu^{8} - 6001155927 \nu^{7} - 6852248433 \nu^{6} - 6078679992 \nu^{5} + 7755820794 \nu^{4} + 18733941873 \nu^{3} + 34933185009 \nu^{2} + 23573220291 \nu + 69903131301$$$$)/ 18749140065$$ $$\beta_{5}$$ $$=$$ $$($$$$1369114 \nu^{19} + 43032333 \nu^{18} - 52765180 \nu^{17} + 119927013 \nu^{16} - 252295776 \nu^{15} - 47743686 \nu^{14} - 1185877945 \nu^{13} + 427593096 \nu^{12} - 2012692430 \nu^{11} + 5734852011 \nu^{10} + 1710251115 \nu^{9} + 21042452988 \nu^{8} - 8041960332 \nu^{7} + 29349876528 \nu^{6} - 76966783362 \nu^{5} - 5494819761 \nu^{4} - 202226178285 \nu^{3} + 188700818211 \nu^{2} - 439084978524 \nu + 728009491662$$$$)/ 18749140065$$ $$\beta_{6}$$ $$=$$ $$($$$$2231098 \nu^{19} - 38618346 \nu^{18} + 35313338 \nu^{17} - 116786112 \nu^{16} + 157974687 \nu^{15} - 39060783 \nu^{14} + 1014330014 \nu^{13} - 121443942 \nu^{12} + 2420390266 \nu^{11} - 3276627279 \nu^{10} + 396724398 \nu^{9} - 17107428627 \nu^{8} + 2961624825 \nu^{7} - 34642649997 \nu^{6} + 43424726940 \nu^{5} - 18493149285 \nu^{4} + 167613002406 \nu^{3} - 143258717664 \nu^{2} + 400992606405 \nu - 415240815177$$$$)/ 18749140065$$ $$\beta_{7}$$ $$=$$ $$($$$$3771427 \nu^{19} - 21206079 \nu^{18} + 5337977 \nu^{17} - 61097823 \nu^{16} + 24603903 \nu^{15} - 53013492 \nu^{14} + 488779931 \nu^{13} + 215780727 \nu^{12} + 1617435454 \nu^{11} - 264913221 \nu^{10} + 1000006017 \nu^{9} - 7726695273 \nu^{8} - 3087399690 \nu^{7} - 21741102648 \nu^{6} + 2304569475 \nu^{5} - 18988372350 \nu^{4} + 76767604524 \nu^{3} - 39004101801 \nu^{2} + 189333893475 \nu - 3633009408$$$$)/ 18749140065$$ $$\beta_{8}$$ $$=$$ $$($$$$-16385029 \nu^{19} - 140514804 \nu^{18} + 102757399 \nu^{17} - 407033892 \nu^{16} + 718150113 \nu^{15} + 413985888 \nu^{14} + 4119760882 \nu^{13} + 367498977 \nu^{12} + 7913646398 \nu^{11} - 15698361039 \nu^{10} - 8983963626 \nu^{9} - 70805775267 \nu^{8} + 1916096814 \nu^{7} - 112549620312 \nu^{6} + 209265926679 \nu^{5} + 44930137212 \nu^{4} + 642661320213 \nu^{3} - 443919473244 \nu^{2} + 1490921987733 \nu - 2201360498100$$$$)/ 74996560260$$ $$\beta_{9}$$ $$=$$ $$($$$$-23044883 \nu^{19} - 34101699 \nu^{18} - 31187623 \nu^{17} - 73217703 \nu^{16} + 205204923 \nu^{15} + 588597903 \nu^{14} + 1480271006 \nu^{13} + 1904777937 \nu^{12} + 2126965909 \nu^{11} - 3595870971 \nu^{10} - 10347169503 \nu^{9} - 25927654563 \nu^{8} - 27812453535 \nu^{7} - 36111305898 \nu^{6} + 41562628065 \nu^{5} + 82826670285 \nu^{4} + 176850057159 \nu^{3} + 151394106159 \nu^{2} + 421851928095 \nu - 543176221113$$$$)/ 74996560260$$ $$\beta_{10}$$ $$=$$ $$($$$$30237802 \nu^{19} - 54259353 \nu^{18} + 214884218 \nu^{17} - 236951109 \nu^{16} + 402401106 \nu^{15} - 958273299 \nu^{14} + 658007144 \nu^{13} - 4508313216 \nu^{12} + 2230782901 \nu^{11} - 10632740088 \nu^{10} + 14372467593 \nu^{9} - 14529961764 \nu^{8} + 72824563953 \nu^{7} - 21864020634 \nu^{6} + 172090521378 \nu^{5} - 130816437021 \nu^{4} + 280445931126 \nu^{3} - 844707861963 \nu^{2} + 648246653586 \nu - 1626318995895$$$$)/ 74996560260$$ $$\beta_{11}$$ $$=$$ $$($$$$8368807 \nu^{19} - 7140033 \nu^{18} + 20556833 \nu^{17} - 38614374 \nu^{16} - 13831884 \nu^{15} - 216742059 \nu^{14} + 21733274 \nu^{13} - 387779451 \nu^{12} + 888879001 \nu^{11} + 419155827 \nu^{10} + 3830738928 \nu^{9} - 525114144 \nu^{8} + 5256258948 \nu^{7} - 11676129804 \nu^{6} - 2290610907 \nu^{5} - 36378032391 \nu^{4} + 26276498091 \nu^{3} - 72813177558 \nu^{2} + 102097203786 \nu + 40868304975$$$$)/ 18749140065$$ $$\beta_{12}$$ $$=$$ $$($$$$37886113 \nu^{19} - 154722717 \nu^{18} + 251556047 \nu^{17} - 527337261 \nu^{16} + 628251039 \nu^{15} - 997996401 \nu^{14} + 3314094746 \nu^{13} - 3240833919 \nu^{12} + 9729553519 \nu^{11} - 14749563837 \nu^{10} + 15444124467 \nu^{9} - 60132122631 \nu^{8} + 53032589037 \nu^{7} - 133364843766 \nu^{6} + 216274312197 \nu^{5} - 172355910669 \nu^{4} + 706501544409 \nu^{3} - 955238518287 \nu^{2} + 1703644474959 \nu - 2018497456485$$$$)/ 74996560260$$ $$\beta_{13}$$ $$=$$ $$($$$$-38985829 \nu^{19} + 89896524 \nu^{18} - 218228033 \nu^{17} + 285292200 \nu^{16} - 475229043 \nu^{15} + 977759172 \nu^{14} - 1629620774 \nu^{13} + 4185457533 \nu^{12} - 4351958746 \nu^{11} + 11408523885 \nu^{10} - 14427172338 \nu^{9} + 29056424445 \nu^{8} - 65352604662 \nu^{7} + 55706534460 \nu^{6} - 168003632277 \nu^{5} + 141521530104 \nu^{4} - 414992157651 \nu^{3} + 844895880540 \nu^{2} - 906591231459 \nu + 1324570004424$$$$)/ 74996560260$$ $$\beta_{14}$$ $$=$$ $$($$$$-13359008 \nu^{19} + 19113527 \nu^{18} - 18111052 \nu^{17} + 89227951 \nu^{16} + 66762696 \nu^{15} + 345761901 \nu^{14} - 261593596 \nu^{13} + 4499734 \nu^{12} - 2559235259 \nu^{11} - 2233525978 \nu^{10} - 6526917327 \nu^{9} + 4355149386 \nu^{8} + 1448555013 \nu^{7} + 35617343346 \nu^{6} + 28688453748 \nu^{5} + 67695711039 \nu^{4} - 66332338884 \nu^{3} + 54227986677 \nu^{2} - 265428080604 \nu - 229996215855$$$$)/ 24998853420$$ $$\beta_{15}$$ $$=$$ $$($$$$-12455246 \nu^{19} + 44968920 \nu^{18} - 80649223 \nu^{17} + 145859691 \nu^{16} - 182259135 \nu^{15} + 332441235 \nu^{14} - 860709214 \nu^{13} + 1242762765 \nu^{12} - 2646333596 \nu^{11} + 4113200487 \nu^{10} - 5498123943 \nu^{9} + 14947347996 \nu^{8} - 20424162816 \nu^{7} + 34981237206 \nu^{6} - 59740191711 \nu^{5} + 64752740712 \nu^{4} - 181698751341 \nu^{3} + 320598980667 \nu^{2} - 470439065862 \nu + 553787129583$$$$)/ 18749140065$$ $$\beta_{16}$$ $$=$$ $$($$$$-512306 \nu^{19} + 1702863 \nu^{18} - 2952415 \nu^{17} + 5590548 \nu^{16} - 6383691 \nu^{15} + 13684149 \nu^{14} - 32722690 \nu^{13} + 44283756 \nu^{12} - 104042195 \nu^{11} + 140771301 \nu^{10} - 217352190 \nu^{9} + 572521473 \nu^{8} - 696840777 \nu^{7} + 1432221048 \nu^{6} - 2133087372 \nu^{5} + 2424280509 \nu^{4} - 7173837495 \nu^{3} + 11255719176 \nu^{2} - 18850993029 \nu + 19282234437$$$$)/ 635564070$$ $$\beta_{17}$$ $$=$$ $$($$$$-31370234 \nu^{19} + 71949081 \nu^{18} - 107067631 \nu^{17} + 265648533 \nu^{16} - 139016052 \nu^{15} + 755506368 \nu^{14} - 1399344568 \nu^{13} + 1282566402 \nu^{12} - 6120978737 \nu^{11} + 2647174671 \nu^{10} - 12309010131 \nu^{9} + 24036532128 \nu^{8} - 17859371256 \nu^{7} + 83249628768 \nu^{6} - 42320584836 \nu^{5} + 134960919147 \nu^{4} - 316042358877 \nu^{3} + 400022908101 \nu^{2} - 845971744572 \nu + 376958974500$$$$)/ 37498280130$$ $$\beta_{18}$$ $$=$$ $$($$$$5988796 \nu^{19} - 15464541 \nu^{18} + 28972577 \nu^{17} - 54823470 \nu^{16} + 53284752 \nu^{15} - 157342113 \nu^{14} + 280811726 \nu^{13} - 459689232 \nu^{12} + 1060287379 \nu^{11} - 1175147475 \nu^{10} + 2586668262 \nu^{9} - 5051753775 \nu^{8} + 7179888393 \nu^{7} - 14095046520 \nu^{6} + 18378290718 \nu^{5} - 27846329121 \nu^{4} + 67241845359 \nu^{3} - 112388246010 \nu^{2} + 172096283151 \nu - 164723228181$$$$)/ 6249713355$$ $$\beta_{19}$$ $$=$$ $$($$$$-18932536 \nu^{19} + 53426298 \nu^{18} - 66027095 \nu^{17} + 192757263 \nu^{16} - 105204501 \nu^{15} + 443751174 \nu^{14} - 1089797630 \nu^{13} + 633496971 \nu^{12} - 4439535220 \nu^{11} + 1844096811 \nu^{10} - 7529255835 \nu^{9} + 19552231593 \nu^{8} - 8362208052 \nu^{7} + 62194582998 \nu^{6} - 29791529622 \nu^{5} + 85830622434 \nu^{4} - 232182960705 \nu^{3} + 249441222141 \nu^{2} - 646646642199 \nu + 266926260897$$$$)/ 18749140065$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{19} + 2 \beta_{18} + \beta_{15} + \beta_{13} + \beta_{11} - \beta_{10} - \beta_{9} - \beta_{5} + 2 \beta_{1} - 1$$ $$\nu^{3}$$ $$=$$ $$\beta_{19} + 3 \beta_{18} + 2 \beta_{15} - \beta_{14} + \beta_{13} - 2 \beta_{11} - \beta_{9} + \beta_{4} + 2 \beta_{3} + \beta_{1} + 1$$ $$\nu^{4}$$ $$=$$ $$-2 \beta_{18} + \beta_{17} - \beta_{14} - 3 \beta_{13} + \beta_{12} + 3 \beta_{11} - \beta_{10} + 3 \beta_{8} - 2 \beta_{6} + \beta_{5} + \beta_{3} - 2 \beta_{1} - 1$$ $$\nu^{5}$$ $$=$$ $$3 \beta_{18} + \beta_{17} - \beta_{16} + 4 \beta_{14} + 2 \beta_{13} - 2 \beta_{12} + 5 \beta_{11} + 3 \beta_{9} - \beta_{8} + 3 \beta_{4} - 2 \beta_{3} + \beta_{2} + 2 \beta_{1} + 4$$ $$\nu^{6}$$ $$=$$ $$\beta_{19} - \beta_{18} + 5 \beta_{16} - 4 \beta_{14} - 4 \beta_{13} - 5 \beta_{12} + 8 \beta_{10} + 4 \beta_{9} - 3 \beta_{8} + 9 \beta_{7} - \beta_{6} - 3 \beta_{5} - 2 \beta_{4} + 5 \beta_{2} + 13 \beta_{1} + 2$$ $$\nu^{7}$$ $$=$$ $$18 \beta_{19} + 53 \beta_{18} + 3 \beta_{17} + 6 \beta_{16} + 9 \beta_{15} - 2 \beta_{14} + 22 \beta_{13} + 7 \beta_{12} - \beta_{11} - 26 \beta_{10} - 19 \beta_{9} + 8 \beta_{8} - 12 \beta_{7} - 10 \beta_{6} - 14 \beta_{5} + 15 \beta_{4} - 2 \beta_{3} - 14 \beta_{2} + 45 \beta_{1} - 1$$ $$\nu^{8}$$ $$=$$ $$19 \beta_{19} + 7 \beta_{18} - 9 \beta_{17} - 9 \beta_{16} + 19 \beta_{15} + 6 \beta_{13} + 22 \beta_{11} - 2 \beta_{10} - 9 \beta_{9} + 7 \beta_{8} + 2 \beta_{7} + 12 \beta_{6} + 8 \beta_{5} + 5 \beta_{4} + 24 \beta_{3} + 3 \beta_{2} + 11 \beta_{1} - 13$$ $$\nu^{9}$$ $$=$$ $$2 \beta_{19} - 15 \beta_{18} + 38 \beta_{17} + 19 \beta_{16} + 4 \beta_{15} - 14 \beta_{14} - 52 \beta_{13} + 25 \beta_{12} + 54 \beta_{11} - 11 \beta_{10} + 13 \beta_{9} + 14 \beta_{8} + 4 \beta_{7} + 15 \beta_{6} + 33 \beta_{5} + 4 \beta_{4} + 28 \beta_{3} + 25 \beta_{2} + 11 \beta_{1} - 65$$ $$\nu^{10}$$ $$=$$ $$84 \beta_{18} + 49 \beta_{17} + 18 \beta_{15} - 7 \beta_{14} + 11 \beta_{13} - 83 \beta_{12} + 39 \beta_{11} + 37 \beta_{10} + 46 \beta_{9} - 11 \beta_{8} + 11 \beta_{7} - 14 \beta_{6} - 26 \beta_{5} + 74 \beta_{4} + 7 \beta_{3} + 40 \beta_{1} - 49$$ $$\nu^{11}$$ $$=$$ $$37 \beta_{19} + 78 \beta_{18} - 72 \beta_{17} + 72 \beta_{16} - 37 \beta_{15} + 30 \beta_{14} + 6 \beta_{13} - 76 \beta_{12} + 61 \beta_{11} - 17 \beta_{10} + 44 \beta_{9} + 49 \beta_{8} + 49 \beta_{7} - 99 \beta_{6} - 33 \beta_{5} - 33 \beta_{4} - 15 \beta_{3} + 38 \beta_{2} + 60 \beta_{1} + 133$$ $$\nu^{12}$$ $$=$$ $$235 \beta_{19} + 466 \beta_{18} + 104 \beta_{16} - 28 \beta_{14} + 232 \beta_{13} + 226 \beta_{12} - 194 \beta_{10} + 38 \beta_{9} - 32 \beta_{8} - 70 \beta_{7} - 124 \beta_{6} - 114 \beta_{5} + 222 \beta_{4} - 226 \beta_{2} + 538 \beta_{1} - 7$$ $$\nu^{13}$$ $$=$$ $$100 \beta_{19} - 166 \beta_{18} - 4 \beta_{17} - 8 \beta_{16} + 50 \beta_{15} - 48 \beta_{14} - 262 \beta_{13} - 118 \beta_{12} + 208 \beta_{11} + 68 \beta_{10} - 50 \beta_{9} - 26 \beta_{8} - 80 \beta_{7} + 407 \beta_{6} + 148 \beta_{5} + 141 \beta_{4} - 48 \beta_{3} + 236 \beta_{2} + 448 \beta_{1} - 412$$ $$\nu^{14}$$ $$=$$ $$166 \beta_{19} + 816 \beta_{18} + 564 \beta_{17} + 564 \beta_{16} + 166 \beta_{15} + 74 \beta_{13} + 1033 \beta_{11} - 270 \beta_{10} - 296 \beta_{9} + 26 \beta_{8} - 44 \beta_{7} + 210 \beta_{6} - 16 \beta_{5} + 622 \beta_{4} + 792 \beta_{3} + 222 \beta_{2} + 1118 \beta_{1} - 1597$$ $$\nu^{15}$$ $$=$$ $$270 \beta_{19} + 255 \beta_{18} + 472 \beta_{17} + 236 \beta_{16} + 540 \beta_{15} - 480 \beta_{14} - 588 \beta_{13} - 550 \beta_{12} + 1328 \beta_{11} + 602 \beta_{10} + 1086 \beta_{9} + 52 \beta_{8} + 734 \beta_{7} + 132 \beta_{6} - 90 \beta_{5} - 338 \beta_{4} + 960 \beta_{3} - 550 \beta_{2} - 666 \beta_{1} - 1136$$ $$\nu^{16}$$ $$=$$ $$1274 \beta_{18} - 992 \beta_{17} - 1575 \beta_{15} + 1148 \beta_{14} - 119 \beta_{13} - 50 \beta_{12} - 1761 \beta_{11} - 1752 \beta_{10} + 1802 \beta_{9} + 119 \beta_{8} - 1094 \beta_{7} - 1727 \beta_{6} + 658 \beta_{5} + 616 \beta_{4} - 1148 \beta_{3} - 32 \beta_{1} + 992$$ $$\nu^{17}$$ $$=$$ $$1787 \beta_{19} + 1176 \beta_{18} - 1144 \beta_{17} + 1144 \beta_{16} - 1787 \beta_{15} - 1078 \beta_{14} + 1834 \beta_{13} - 204 \beta_{12} + 753 \beta_{11} + 557 \beta_{10} + 1936 \beta_{9} - 2289 \beta_{8} - 325 \beta_{7} - 1518 \beta_{6} - 2400 \beta_{5} + 3090 \beta_{4} + 539 \beta_{3} + 102 \beta_{2} + 3418 \beta_{1} + 1897$$ $$\nu^{18}$$ $$=$$ $$-350 \beta_{19} + 1482 \beta_{18} + 4865 \beta_{16} - 943 \beta_{14} - 2085 \beta_{13} - 1127 \beta_{12} + 2334 \beta_{10} + 249 \beta_{9} - 1207 \beta_{8} - 3377 \beta_{7} + 5711 \beta_{6} + 1152 \beta_{5} + 2474 \beta_{4} + 1127 \beta_{2} + 8590 \beta_{1} + 497$$ $$\nu^{19}$$ $$=$$ $$2684 \beta_{19} + 8072 \beta_{18} + 3601 \beta_{17} + 7202 \beta_{16} + 1342 \beta_{15} + 4310 \beta_{14} + 2202 \beta_{13} + 2391 \beta_{12} + 18921 \beta_{11} - 3318 \beta_{10} - 927 \beta_{9} - 2580 \beta_{8} + 2540 \beta_{7} + 782 \beta_{6} - 5858 \beta_{5} + 6585 \beta_{4} + 4310 \beta_{3} - 4782 \beta_{2} + 21473 \beta_{1} - 41443$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/525\mathbb{Z}\right)^\times$$.

 $$n$$ $$127$$ $$176$$ $$451$$ $$\chi(n)$$ $$1$$ $$-1$$ $$1 - \beta_{11}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
26.1
 −1.63084 − 0.583411i 0.742502 + 1.56483i 0.803015 − 1.53466i 1.71408 + 0.248842i −1.39625 − 1.02493i 0.189492 + 1.72165i 0.641538 − 1.60886i 1.73056 + 0.0718963i −0.983931 − 1.42544i −0.310170 + 1.70405i −1.63084 + 0.583411i 0.742502 − 1.56483i 0.803015 + 1.53466i 1.71408 − 0.248842i −1.39625 + 1.02493i 0.189492 − 1.72165i 0.641538 + 1.60886i 1.73056 − 0.0718963i −0.983931 + 1.42544i −0.310170 − 1.70405i
−2.36764 1.36696i 1.63084 + 0.583411i 2.73715 + 4.74088i 0 −3.06374 3.61059i 2.24882 1.39384i 9.49844i 2.31926 + 1.90290i 0
26.2 −1.94891 1.12521i −0.742502 1.56483i 1.53217 + 2.65380i 0 −0.313682 + 3.88518i −1.42897 + 2.22667i 2.39522i −1.89738 + 2.32378i 0
26.3 −1.46613 0.846473i −0.803015 + 1.53466i 0.433034 + 0.750036i 0 2.47637 1.57028i 1.71236 + 2.01688i 1.91969i −1.71033 2.46470i 0
26.4 −0.780577 0.450666i −1.71408 0.248842i −0.593800 1.02849i 0 1.22583 + 0.966719i 0.105498 2.64365i 2.87309i 2.87616 + 0.853070i 0
26.5 −0.766266 0.442404i 1.39625 + 1.02493i −0.608557 1.05405i 0 −0.616465 1.40308i −2.63771 0.206062i 2.84653i 0.899028 + 2.86212i 0
26.6 0.766266 + 0.442404i −0.189492 1.72165i −0.608557 1.05405i 0 0.616465 1.40308i −2.63771 0.206062i 2.84653i −2.92819 + 0.652481i 0
26.7 0.780577 + 0.450666i −0.641538 + 1.60886i −0.593800 1.02849i 0 −1.22583 + 0.966719i 0.105498 2.64365i 2.87309i −2.17686 2.06429i 0
26.8 1.46613 + 0.846473i −1.73056 0.0718963i 0.433034 + 0.750036i 0 −2.47637 1.57028i 1.71236 + 2.01688i 1.91969i 2.98966 + 0.248842i 0
26.9 1.94891 + 1.12521i 0.983931 + 1.42544i 1.53217 + 2.65380i 0 0.313682 + 3.88518i −1.42897 + 2.22667i 2.39522i −1.06376 + 2.80507i 0
26.10 2.36764 + 1.36696i 0.310170 1.70405i 2.73715 + 4.74088i 0 3.06374 3.61059i 2.24882 1.39384i 9.49844i −2.80759 1.05709i 0
101.1 −2.36764 + 1.36696i 1.63084 0.583411i 2.73715 4.74088i 0 −3.06374 + 3.61059i 2.24882 + 1.39384i 9.49844i 2.31926 1.90290i 0
101.2 −1.94891 + 1.12521i −0.742502 + 1.56483i 1.53217 2.65380i 0 −0.313682 3.88518i −1.42897 2.22667i 2.39522i −1.89738 2.32378i 0
101.3 −1.46613 + 0.846473i −0.803015 1.53466i 0.433034 0.750036i 0 2.47637 + 1.57028i 1.71236 2.01688i 1.91969i −1.71033 + 2.46470i 0
101.4 −0.780577 + 0.450666i −1.71408 + 0.248842i −0.593800 + 1.02849i 0 1.22583 0.966719i 0.105498 + 2.64365i 2.87309i 2.87616 0.853070i 0
101.5 −0.766266 + 0.442404i 1.39625 1.02493i −0.608557 + 1.05405i 0 −0.616465 + 1.40308i −2.63771 + 0.206062i 2.84653i 0.899028 2.86212i 0
101.6 0.766266 0.442404i −0.189492 + 1.72165i −0.608557 + 1.05405i 0 0.616465 + 1.40308i −2.63771 + 0.206062i 2.84653i −2.92819 0.652481i 0
101.7 0.780577 0.450666i −0.641538 1.60886i −0.593800 + 1.02849i 0 −1.22583 0.966719i 0.105498 + 2.64365i 2.87309i −2.17686 + 2.06429i 0
101.8 1.46613 0.846473i −1.73056 + 0.0718963i 0.433034 0.750036i 0 −2.47637 + 1.57028i 1.71236 2.01688i 1.91969i 2.98966 0.248842i 0
101.9 1.94891 1.12521i 0.983931 1.42544i 1.53217 2.65380i 0 0.313682 3.88518i −1.42897 2.22667i 2.39522i −1.06376 2.80507i 0
101.10 2.36764 1.36696i 0.310170 + 1.70405i 2.73715 4.74088i 0 3.06374 + 3.61059i 2.24882 + 1.39384i 9.49844i −2.80759 + 1.05709i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 101.10 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
7.d odd 6 1 inner
21.g even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 525.2.t.h 20
3.b odd 2 1 inner 525.2.t.h 20
5.b even 2 1 525.2.t.i yes 20
5.c odd 4 2 525.2.q.g 40
7.d odd 6 1 inner 525.2.t.h 20
15.d odd 2 1 525.2.t.i yes 20
15.e even 4 2 525.2.q.g 40
21.g even 6 1 inner 525.2.t.h 20
35.i odd 6 1 525.2.t.i yes 20
35.k even 12 2 525.2.q.g 40
105.p even 6 1 525.2.t.i yes 20
105.w odd 12 2 525.2.q.g 40

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
525.2.q.g 40 5.c odd 4 2
525.2.q.g 40 15.e even 4 2
525.2.q.g 40 35.k even 12 2
525.2.q.g 40 105.w odd 12 2
525.2.t.h 20 1.a even 1 1 trivial
525.2.t.h 20 3.b odd 2 1 inner
525.2.t.h 20 7.d odd 6 1 inner
525.2.t.h 20 21.g even 6 1 inner
525.2.t.i yes 20 5.b even 2 1
525.2.t.i yes 20 15.d odd 2 1
525.2.t.i yes 20 35.i odd 6 1
525.2.t.i yes 20 105.p even 6 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(525, [\chi])$$:

 $$T_{2}^{20} - \cdots$$ $$T_{13}^{10} + 72 T_{13}^{8} + 1950 T_{13}^{6} + 24651 T_{13}^{4} + 142515 T_{13}^{2} + 286443$$ $$T_{37}^{10} + \cdots$$

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ $$1 + 3 T^{2} + 2 T^{4} + 5 T^{6} + 9 T^{8} - 29 T^{10} + 47 T^{12} + 294 T^{14} + 40 T^{16} - 144 T^{18} + 1017 T^{20} - 576 T^{22} + 640 T^{24} + 18816 T^{26} + 12032 T^{28} - 29696 T^{30} + 36864 T^{32} + 81920 T^{34} + 131072 T^{36} + 786432 T^{38} + 1048576 T^{40}$$
$3$ $$1 + 3 T + 8 T^{2} + 15 T^{3} + 18 T^{4} + 45 T^{5} + 59 T^{6} + 147 T^{7} + 271 T^{8} + 330 T^{9} + 879 T^{10} + 990 T^{11} + 2439 T^{12} + 3969 T^{13} + 4779 T^{14} + 10935 T^{15} + 13122 T^{16} + 32805 T^{17} + 52488 T^{18} + 59049 T^{19} + 59049 T^{20}$$
$5$ 
$7$ $$( 1 + T^{2} + 13 T^{3} + 7 T^{4} + 133 T^{5} + 49 T^{6} + 637 T^{7} + 343 T^{8} + 16807 T^{10} )^{2}$$
$11$ $$1 + 60 T^{2} + 2114 T^{4} + 53114 T^{6} + 1038747 T^{8} + 16310488 T^{10} + 209475293 T^{12} + 2203457790 T^{14} + 19212402091 T^{16} + 148735009794 T^{18} + 1341299566641 T^{20} + 17996936185074 T^{22} + 281288779014331 T^{24} + 3903559885910190 T^{26} + 44902889404627133 T^{28} + 423052052705515288 T^{30} + 3260033061033808587 T^{32} + 20170032660940262474 T^{34} + 97137728931591548354 T^{36} +$$$$33\!\cdots\!60$$$$T^{38} +$$$$67\!\cdots\!01$$$$T^{40}$$
$13$ $$( 1 - 58 T^{2} + 2067 T^{4} - 50385 T^{6} + 943419 T^{8} - 13724229 T^{10} + 159437811 T^{12} - 1439045985 T^{14} + 9977014203 T^{16} - 47312381818 T^{18} + 137858491849 T^{20} )^{2}$$
$17$ $$1 - 68 T^{2} + 1606 T^{4} - 19422 T^{6} + 550335 T^{8} - 18001236 T^{10} + 282479529 T^{12} - 4249504470 T^{14} + 113206321539 T^{16} - 2028696191474 T^{18} + 28292432369185 T^{20} - 586293199335986 T^{22} + 9455105181258819 T^{24} - 102572707360433430 T^{26} + 1970508676351925289 T^{28} - 36290381976542954964 T^{30} +$$$$32\!\cdots\!35$$$$T^{32} -$$$$32\!\cdots\!38$$$$T^{34} +$$$$78\!\cdots\!86$$$$T^{36} -$$$$95\!\cdots\!12$$$$T^{38} +$$$$40\!\cdots\!01$$$$T^{40}$$
$19$ $$( 1 + 59 T^{2} + 1650 T^{4} + 38478 T^{6} + 13230 T^{7} + 896922 T^{8} + 780570 T^{9} + 18506640 T^{10} + 14830830 T^{11} + 323788842 T^{12} + 90744570 T^{13} + 5014491438 T^{14} + 77625703650 T^{16} + 1002030219419 T^{18} + 6131066257801 T^{20} )^{2}$$
$23$ $$1 + 99 T^{2} + 4250 T^{4} + 102299 T^{6} + 1576890 T^{8} + 17142031 T^{10} + 181616261 T^{12} + 10191303513 T^{14} + 591206119363 T^{16} + 19834960319622 T^{18} + 492118045969095 T^{20} + 10492694009080038 T^{22} + 165443711648661283 T^{24} + 1508678675615778057 T^{26} + 14222548341961254341 T^{28} +$$$$71\!\cdots\!19$$$$T^{30} +$$$$34\!\cdots\!90$$$$T^{32} +$$$$11\!\cdots\!91$$$$T^{34} +$$$$26\!\cdots\!50$$$$T^{36} +$$$$32\!\cdots\!31$$$$T^{38} +$$$$17\!\cdots\!01$$$$T^{40}$$
$29$ $$( 1 - 132 T^{2} + 8782 T^{4} - 413831 T^{6} + 16000213 T^{8} - 513486626 T^{10} + 13456179133 T^{12} - 292694803511 T^{14} + 5223738405022 T^{16} - 66032526510852 T^{18} + 420707233300201 T^{20} )^{2}$$
$31$ $$( 1 - 21 T + 326 T^{2} - 3759 T^{3} + 37146 T^{4} - 316851 T^{5} + 2454105 T^{6} - 17262699 T^{7} + 113223339 T^{8} - 689960868 T^{9} + 3971445087 T^{10} - 21388786908 T^{11} + 108807628779 T^{12} - 514273065909 T^{13} + 2266417503705 T^{14} - 9071175123501 T^{15} + 32967211734426 T^{16} - 103419916443249 T^{17} + 278042478205766 T^{18} - 555232065374091 T^{19} + 819628286980801 T^{20} )^{2}$$
$37$ $$( 1 + 12 T - 49 T^{2} - 490 T^{3} + 7278 T^{4} + 25228 T^{5} - 430924 T^{6} - 718902 T^{7} + 17278462 T^{8} - 5308020 T^{9} - 787415472 T^{10} - 196396740 T^{11} + 23654214478 T^{12} - 36414543006 T^{13} - 807620954764 T^{14} + 1749409347196 T^{15} + 18673356804702 T^{16} - 46516619795170 T^{17} - 172111493242129 T^{18} + 1559540877540924 T^{19} + 4808584372417849 T^{20} )^{2}$$
$41$ $$( 1 + 149 T^{2} + 12123 T^{4} + 648375 T^{6} + 26231484 T^{8} + 1031235516 T^{10} + 44095124604 T^{12} + 1832152788375 T^{14} + 57585513713643 T^{16} + 1189753859139029 T^{18} + 13422659310152401 T^{20} )^{2}$$
$43$ $$( 1 + 9 T + 103 T^{2} + 589 T^{3} + 6250 T^{4} + 34300 T^{5} + 268750 T^{6} + 1089061 T^{7} + 8189221 T^{8} + 30769209 T^{9} + 147008443 T^{10} )^{4}$$
$47$ $$1 - 320 T^{2} + 55258 T^{4} - 6481134 T^{6} + 564305583 T^{8} - 37299364464 T^{10} + 1825870594845 T^{12} - 57025827049134 T^{14} + 130434887196171 T^{16} + 112418780517368770 T^{18} - 7605195418763792351 T^{20} +$$$$24\!\cdots\!30$$$$T^{22} +$$$$63\!\cdots\!51$$$$T^{24} -$$$$61\!\cdots\!86$$$$T^{26} +$$$$43\!\cdots\!45$$$$T^{28} -$$$$19\!\cdots\!36$$$$T^{30} +$$$$65\!\cdots\!03$$$$T^{32} -$$$$16\!\cdots\!46$$$$T^{34} +$$$$31\!\cdots\!18$$$$T^{36} -$$$$40\!\cdots\!80$$$$T^{38} +$$$$27\!\cdots\!01$$$$T^{40}$$
$53$ $$1 + 348 T^{2} + 65006 T^{4} + 8168402 T^{6} + 754331535 T^{8} + 52520483476 T^{10} + 2690516083409 T^{12} + 86297805646650 T^{14} - 153773100809741 T^{16} - 235981740156615306 T^{18} - 17360136557008258575 T^{20} -$$$$66\!\cdots\!54$$$$T^{22} -$$$$12\!\cdots\!21$$$$T^{24} +$$$$19\!\cdots\!50$$$$T^{26} +$$$$16\!\cdots\!49$$$$T^{28} +$$$$91\!\cdots\!24$$$$T^{30} +$$$$37\!\cdots\!35$$$$T^{32} +$$$$11\!\cdots\!38$$$$T^{34} +$$$$25\!\cdots\!26$$$$T^{36} +$$$$37\!\cdots\!72$$$$T^{38} +$$$$30\!\cdots\!01$$$$T^{40}$$
$59$ $$1 - 281 T^{2} + 43822 T^{4} - 4385565 T^{6} + 294312414 T^{8} - 11205257169 T^{10} - 28140925407 T^{12} + 33950080736241 T^{14} - 2072476947516237 T^{16} + 52044476827656370 T^{18} - 492615579799524713 T^{20} +$$$$18\!\cdots\!70$$$$T^{22} -$$$$25\!\cdots\!57$$$$T^{24} +$$$$14\!\cdots\!81$$$$T^{26} -$$$$41\!\cdots\!47$$$$T^{28} -$$$$57\!\cdots\!69$$$$T^{30} +$$$$52\!\cdots\!34$$$$T^{32} -$$$$27\!\cdots\!65$$$$T^{34} +$$$$94\!\cdots\!02$$$$T^{36} -$$$$21\!\cdots\!01$$$$T^{38} +$$$$26\!\cdots\!01$$$$T^{40}$$
$61$ $$( 1 + 45 T + 1163 T^{2} + 21960 T^{3} + 334221 T^{4} + 4323609 T^{5} + 49194321 T^{6} + 504078669 T^{7} + 4728952065 T^{8} + 41065652565 T^{9} + 332029517802 T^{10} + 2505004806465 T^{11} + 17596430633865 T^{12} + 114416280368289 T^{13} + 681136746668961 T^{14} + 3651704168370309 T^{15} + 17219191039307781 T^{16} + 69014632679021160 T^{17} + 222955605015837803 T^{18} + 526236574177536345 T^{19} + 713342911662882601 T^{20} )^{2}$$
$67$ $$( 1 + 10 T - 194 T^{2} - 1284 T^{3} + 29439 T^{4} + 97824 T^{5} - 3224136 T^{6} - 5037786 T^{7} + 270996189 T^{8} + 90868948 T^{9} - 19829421434 T^{10} + 6088219516 T^{11} + 1216501892421 T^{12} - 1515179630718 T^{13} - 64969954656456 T^{14} + 132074638467168 T^{15} + 2663004312673191 T^{16} - 7781953701234732 T^{17} - 78777129445988354 T^{18} + 272065343962949470 T^{19} + 1822837804551761449 T^{20} )^{2}$$
$71$ $$( 1 - 417 T^{2} + 83080 T^{4} - 10852019 T^{6} + 1067389843 T^{8} - 84042150656 T^{10} + 5380712198563 T^{12} - 275768045033939 T^{14} + 10642571588156680 T^{16} - 269279222529482337 T^{18} + 3255243551009881201 T^{20} )^{2}$$
$73$ $$( 1 - 24 T + 572 T^{2} - 9120 T^{3} + 139305 T^{4} - 1702236 T^{5} + 20269650 T^{6} - 207027012 T^{7} + 2089522809 T^{8} - 18747463404 T^{9} + 169031552934 T^{10} - 1368564828492 T^{11} + 11135067049161 T^{12} - 80537027127204 T^{13} + 575622405685650 T^{14} - 3528857096181948 T^{15} + 21081614393189145 T^{16} - 100752274494164640 T^{17} + 461295172563414332 T^{18} - 1412918080998429912 T^{19} + 4297625829703557649 T^{20} )^{2}$$
$79$ $$( 1 - 23 T + 19 T^{2} + 1650 T^{3} + 22269 T^{4} - 308997 T^{5} - 2262141 T^{6} + 18480543 T^{7} + 256079211 T^{8} - 288694607 T^{9} - 26891047898 T^{10} - 22806873953 T^{11} + 1598190355851 T^{12} + 9111628440177 T^{13} - 88110575183421 T^{14} - 950801196121803 T^{15} + 5413314546997149 T^{16} + 31686449827162350 T^{17} + 28825067388224659 T^{18} - 2756586707600221337 T^{19} + 9468276082626847201 T^{20} )^{2}$$
$83$ $$( 1 + 200 T^{2} + 27258 T^{4} + 1670937 T^{6} + 63680961 T^{8} - 932587014 T^{10} + 438698140329 T^{12} + 79299864516777 T^{14} + 8911740697292202 T^{16} + 450458446427808200 T^{18} + 15516041187205853449 T^{20} )^{2}$$
$89$ $$1 - 548 T^{2} + 149638 T^{4} - 27617742 T^{6} + 3982290303 T^{8} - 496827970068 T^{10} + 57439702492761 T^{12} - 6298039134882534 T^{14} + 650541332844392067 T^{16} - 62975586452546418242 T^{18} +$$$$57\!\cdots\!53$$$$T^{20} -$$$$49\!\cdots\!82$$$$T^{22} +$$$$40\!\cdots\!47$$$$T^{24} -$$$$31\!\cdots\!74$$$$T^{26} +$$$$22\!\cdots\!41$$$$T^{28} -$$$$15\!\cdots\!68$$$$T^{30} +$$$$98\!\cdots\!63$$$$T^{32} -$$$$54\!\cdots\!22$$$$T^{34} +$$$$23\!\cdots\!18$$$$T^{36} -$$$$67\!\cdots\!88$$$$T^{38} +$$$$97\!\cdots\!01$$$$T^{40}$$
$97$ $$( 1 - 673 T^{2} + 225150 T^{4} - 48497952 T^{6} + 7395278745 T^{8} - 830846466435 T^{10} + 69582177711705 T^{12} - 4293488820532512 T^{14} + 187543646909764350 T^{16} - 5274592809015694753 T^{18} + 73742412689492826049 T^{20} )^{2}$$