# Properties

 Label 400.2.j.d Level $400$ Weight $2$ Character orbit 400.j Analytic conductor $3.194$ Analytic rank $0$ Dimension $18$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$400 = 2^{4} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 400.j (of order $$4$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$3.19401608085$$ Analytic rank: $$0$$ Dimension: $$18$$ Relative dimension: $$9$$ over $$\Q(i)$$ Coefficient field: $$\mathbb{Q}[x]/(x^{18} + \cdots)$$ Defining polynomial: $$x^{18} + 2 x^{16} - 4 x^{15} - 5 x^{14} - 14 x^{13} - 10 x^{12} + 6 x^{11} + 37 x^{10} + 70 x^{9} + 74 x^{8} + 24 x^{7} - 80 x^{6} - 224 x^{5} - 160 x^{4} - 256 x^{3} + 256 x^{2} + 512$$ Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2^{7}$$ Twist minimal: no (minimal twist has level 80) Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{17}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q -\beta_{7} q^{2} + \beta_{16} q^{3} + \beta_{14} q^{4} + ( -1 - \beta_{3} - \beta_{9} - \beta_{12} ) q^{6} + ( \beta_{6} + \beta_{7} + \beta_{14} ) q^{7} + ( 1 - \beta_{2} + \beta_{3} + \beta_{4} + \beta_{7} + \beta_{11} + \beta_{15} ) q^{8} + ( -\beta_{2} - \beta_{8} + \beta_{10} + \beta_{11} - \beta_{13} ) q^{9} +O(q^{10})$$ $$q -\beta_{7} q^{2} + \beta_{16} q^{3} + \beta_{14} q^{4} + ( -1 - \beta_{3} - \beta_{9} - \beta_{12} ) q^{6} + ( \beta_{6} + \beta_{7} + \beta_{14} ) q^{7} + ( 1 - \beta_{2} + \beta_{3} + \beta_{4} + \beta_{7} + \beta_{11} + \beta_{15} ) q^{8} + ( -\beta_{2} - \beta_{8} + \beta_{10} + \beta_{11} - \beta_{13} ) q^{9} + ( 1 + \beta_{2} + \beta_{3} + \beta_{4} - \beta_{5} + \beta_{6} + 2 \beta_{7} + \beta_{10} + \beta_{12} - \beta_{13} + \beta_{14} - \beta_{16} + \beta_{17} ) q^{11} + ( \beta_{1} - \beta_{3} + \beta_{4} - \beta_{8} - \beta_{9} - \beta_{12} - \beta_{17} ) q^{12} + ( -\beta_{5} + \beta_{7} - \beta_{9} - \beta_{12} + \beta_{14} - \beta_{15} ) q^{13} + ( 1 - \beta_{2} + \beta_{3} + \beta_{4} - \beta_{5} + \beta_{7} - \beta_{10} + \beta_{11} - \beta_{12} + \beta_{16} ) q^{14} + ( -1 - \beta_{1} + \beta_{2} - \beta_{4} + 2 \beta_{5} - \beta_{7} + \beta_{8} + \beta_{12} - 2 \beta_{14} + \beta_{15} + \beta_{16} - \beta_{17} ) q^{16} + ( 1 + \beta_{1} + \beta_{2} + \beta_{4} - \beta_{5} + \beta_{7} + \beta_{10} + \beta_{12} - \beta_{16} + \beta_{17} ) q^{17} + ( -2 - \beta_{1} + \beta_{2} - \beta_{4} + \beta_{5} + 2 \beta_{6} - \beta_{9} + \beta_{10} + \beta_{11} - \beta_{13} - \beta_{14} - \beta_{16} - \beta_{17} ) q^{18} + ( -\beta_{1} + \beta_{2} + \beta_{3} + \beta_{6} + \beta_{8} + \beta_{9} + \beta_{11} + 2 \beta_{12} + \beta_{15} ) q^{19} + ( 1 + \beta_{1} - 2 \beta_{2} + \beta_{3} + \beta_{4} - \beta_{5} - \beta_{6} - \beta_{8} + \beta_{9} + \beta_{14} + \beta_{16} ) q^{21} + ( -1 - \beta_{2} + 2 \beta_{3} + \beta_{4} + \beta_{7} + \beta_{8} - 2 \beta_{10} - \beta_{12} + 2 \beta_{13} - \beta_{15} + \beta_{16} ) q^{22} + ( 1 + \beta_{1} - \beta_{2} + \beta_{3} + \beta_{4} - \beta_{5} - \beta_{8} + \beta_{9} - \beta_{10} + \beta_{11} - \beta_{12} + \beta_{14} - \beta_{15} + \beta_{16} - \beta_{17} ) q^{23} + ( \beta_{1} - \beta_{2} - \beta_{3} - 2 \beta_{4} - \beta_{6} - \beta_{7} - \beta_{9} - \beta_{11} - 2 \beta_{12} + \beta_{13} - \beta_{14} - \beta_{17} ) q^{24} + ( -2 + \beta_{1} - \beta_{2} + \beta_{4} + \beta_{5} - \beta_{7} - \beta_{9} - \beta_{10} + \beta_{11} + \beta_{13} - \beta_{14} - \beta_{16} - \beta_{17} ) q^{26} + ( -\beta_{1} + \beta_{3} + \beta_{6} + 2 \beta_{12} - \beta_{13} - \beta_{14} - \beta_{17} ) q^{27} + ( -2 - \beta_{1} + 2 \beta_{2} - \beta_{3} - \beta_{4} + 2 \beta_{5} - 2 \beta_{7} + \beta_{8} - \beta_{9} + 3 \beta_{12} - 2 \beta_{14} - \beta_{17} ) q^{28} + ( -1 - \beta_{1} + \beta_{2} + \beta_{7} + \beta_{12} + \beta_{17} ) q^{29} + ( 1 - 2 \beta_{2} + \beta_{3} - \beta_{5} - \beta_{6} - \beta_{8} + \beta_{9} - \beta_{10} - \beta_{11} - \beta_{12} - \beta_{15} + 2 \beta_{16} ) q^{31} + ( -1 + 2 \beta_{2} - 3 \beta_{3} - \beta_{4} + \beta_{6} + \beta_{8} - \beta_{9} - \beta_{11} - \beta_{12} + \beta_{15} - \beta_{16} ) q^{32} + ( 3 + \beta_{1} - 2 \beta_{2} + 2 \beta_{3} + \beta_{4} - 2 \beta_{6} - \beta_{8} + \beta_{9} - \beta_{11} - \beta_{12} + \beta_{13} + \beta_{14} + \beta_{15} + \beta_{16} + \beta_{17} ) q^{33} + ( -2 - \beta_{2} + \beta_{3} - \beta_{6} - \beta_{7} + \beta_{9} - \beta_{11} - 2 \beta_{12} + 2 \beta_{13} - 2 \beta_{15} + 2 \beta_{16} ) q^{34} + ( -3 + 2 \beta_{2} - \beta_{3} - \beta_{4} + 3 \beta_{6} + \beta_{8} - \beta_{9} - 2 \beta_{10} + \beta_{11} - \beta_{12} + \beta_{13} - \beta_{15} - \beta_{16} - 2 \beta_{17} ) q^{36} + ( -1 - \beta_{3} + \beta_{5} - \beta_{6} - 3 \beta_{7} - \beta_{8} + \beta_{11} - 2 \beta_{13} - \beta_{14} ) q^{37} + ( -1 - 2 \beta_{1} + \beta_{2} + \beta_{3} + \beta_{4} + \beta_{6} + \beta_{7} + \beta_{8} + \beta_{9} + \beta_{11} + 3 \beta_{12} + \beta_{15} + \beta_{16} ) q^{38} + ( -\beta_{1} + 2 \beta_{2} + \beta_{3} + 2 \beta_{5} + \beta_{6} + 2 \beta_{7} + 2 \beta_{12} - \beta_{13} - \beta_{14} - \beta_{17} ) q^{39} + ( 1 + \beta_{1} + 2 \beta_{2} - 2 \beta_{3} + \beta_{7} - \beta_{8} + \beta_{10} - \beta_{11} - \beta_{13} + 2 \beta_{14} + \beta_{17} ) q^{41} + ( -1 - \beta_{1} - \beta_{2} + \beta_{3} - 2 \beta_{4} + \beta_{5} - \beta_{6} + \beta_{7} + \beta_{8} + \beta_{10} + \beta_{12} - \beta_{13} - \beta_{14} + \beta_{15} + \beta_{17} ) q^{42} + ( 2 + \beta_{2} - \beta_{3} - \beta_{4} - \beta_{6} + \beta_{7} + 2 \beta_{8} - 2 \beta_{11} + \beta_{13} - \beta_{14} ) q^{43} + ( 1 - \beta_{3} - \beta_{4} + 2 \beta_{5} - 3 \beta_{6} - 2 \beta_{7} + \beta_{8} + \beta_{9} - 2 \beta_{10} - \beta_{11} - \beta_{12} + \beta_{13} - \beta_{14} - \beta_{15} + \beta_{16} ) q^{44} + ( -1 - 2 \beta_{1} + \beta_{2} + \beta_{3} - 3 \beta_{4} + \beta_{5} - \beta_{7} + 2 \beta_{8} + \beta_{10} + \beta_{11} + 3 \beta_{12} - 2 \beta_{14} - \beta_{16} ) q^{46} + ( 3 - \beta_{1} + \beta_{2} + \beta_{3} + \beta_{5} - \beta_{8} + \beta_{9} + \beta_{10} - \beta_{11} - \beta_{12} - \beta_{14} + \beta_{15} + \beta_{17} ) q^{47} + ( -4 + 2 \beta_{1} - 2 \beta_{3} - 2 \beta_{4} - 2 \beta_{7} ) q^{48} + ( 2 \beta_{2} + \beta_{5} - \beta_{7} - \beta_{9} + 2 \beta_{10} - \beta_{14} + \beta_{15} - 2 \beta_{16} ) q^{49} + ( 1 - \beta_{1} - 2 \beta_{2} + 2 \beta_{3} + \beta_{4} - \beta_{5} - \beta_{8} + \beta_{9} + \beta_{10} + \beta_{11} - \beta_{12} + \beta_{13} + \beta_{14} - \beta_{15} + \beta_{16} + \beta_{17} ) q^{51} + ( 1 - \beta_{3} - 3 \beta_{4} + 2 \beta_{5} - \beta_{6} - 2 \beta_{7} + \beta_{8} + \beta_{9} - \beta_{11} - \beta_{12} + \beta_{13} - \beta_{14} - \beta_{15} + \beta_{16} - 2 \beta_{17} ) q^{52} + ( -\beta_{1} - \beta_{2} + 2 \beta_{3} - \beta_{5} + \beta_{9} + \beta_{12} - \beta_{14} - \beta_{15} + \beta_{17} ) q^{53} + ( -2 - \beta_{3} - 2 \beta_{4} + \beta_{6} + 2 \beta_{8} - \beta_{9} - 2 \beta_{10} - \beta_{11} + 2 \beta_{12} - 2 \beta_{16} ) q^{54} + ( \beta_{1} - \beta_{2} - 3 \beta_{3} + \beta_{6} - \beta_{7} - \beta_{9} - \beta_{11} - 2 \beta_{12} + \beta_{13} + \beta_{14} - 2 \beta_{16} - \beta_{17} ) q^{56} + ( \beta_{1} - 3 \beta_{2} + \beta_{4} - \beta_{5} - 2 \beta_{6} - \beta_{7} + \beta_{10} + 2 \beta_{14} + \beta_{16} + \beta_{17} ) q^{57} + ( 2 - \beta_{2} + 2 \beta_{4} + \beta_{7} - 2 \beta_{8} ) q^{58} + ( 2 - 2 \beta_{2} - \beta_{3} - 2 \beta_{5} - \beta_{6} + \beta_{7} - \beta_{8} + \beta_{9} + 2 \beta_{10} - \beta_{11} - 2 \beta_{13} + \beta_{15} + \beta_{17} ) q^{59} + ( 1 + \beta_{1} + 3 \beta_{2} - \beta_{3} + \beta_{4} - 2 \beta_{5} - \beta_{6} - \beta_{7} + \beta_{10} - \beta_{11} + \beta_{13} - \beta_{15} - \beta_{16} ) q^{61} + ( -\beta_{3} - 2 \beta_{4} - \beta_{6} - \beta_{9} - \beta_{11} + 2 \beta_{12} - 2 \beta_{13} - 2 \beta_{16} + 2 \beta_{17} ) q^{62} + ( -1 + \beta_{1} - 2 \beta_{3} + \beta_{4} - \beta_{5} + \beta_{6} - \beta_{7} - \beta_{8} - \beta_{9} - \beta_{10} - \beta_{11} - 3 \beta_{12} + \beta_{13} + 2 \beta_{14} - \beta_{15} - \beta_{16} - \beta_{17} ) q^{63} + ( 2 + 2 \beta_{1} + 4 \beta_{4} - 4 \beta_{5} + 2 \beta_{7} - 2 \beta_{8} + 2 \beta_{14} + 2 \beta_{16} ) q^{64} + ( 2 - 2 \beta_{2} + \beta_{3} - 2 \beta_{4} + 2 \beta_{5} - 3 \beta_{6} + \beta_{9} - \beta_{11} - 2 \beta_{12} + 2 \beta_{15} + 2 \beta_{16} + 2 \beta_{17} ) q^{66} + ( -2 \beta_{1} + 3 \beta_{2} + \beta_{3} + \beta_{4} - 2 \beta_{5} + \beta_{6} + 3 \beta_{7} + \beta_{13} + \beta_{14} ) q^{67} + ( -1 - \beta_{3} - \beta_{4} - 3 \beta_{6} - \beta_{8} + \beta_{9} - \beta_{11} - 5 \beta_{12} - \beta_{13} + \beta_{14} - \beta_{15} - \beta_{16} + 2 \beta_{17} ) q^{68} + ( 1 - \beta_{2} + \beta_{3} + \beta_{4} - \beta_{5} - \beta_{6} - 3 \beta_{7} + \beta_{8} + \beta_{9} - 2 \beta_{10} + 2 \beta_{12} - \beta_{14} - \beta_{16} + \beta_{17} ) q^{69} + ( 3 - \beta_{1} + \beta_{2} + 3 \beta_{3} + 2 \beta_{4} - \beta_{5} + \beta_{6} + \beta_{7} - \beta_{8} + \beta_{9} - \beta_{10} + \beta_{11} + \beta_{12} + 2 \beta_{14} + \beta_{15} + \beta_{17} ) q^{71} + ( \beta_{2} - 2 \beta_{5} + \beta_{6} - \beta_{7} + \beta_{8} + \beta_{9} - 2 \beta_{10} + 2 \beta_{11} + 3 \beta_{12} + 2 \beta_{13} - 2 \beta_{15} + \beta_{16} - 2 \beta_{17} ) q^{72} + ( -1 + \beta_{1} - 3 \beta_{2} - \beta_{4} + \beta_{5} + \beta_{7} - \beta_{10} - \beta_{12} + \beta_{16} + \beta_{17} ) q^{73} + ( 2 - \beta_{1} + 3 \beta_{2} - \beta_{3} - \beta_{4} + \beta_{5} + 3 \beta_{6} + \beta_{7} - 2 \beta_{9} + 3 \beta_{10} + 2 \beta_{12} - \beta_{13} + \beta_{14} - 3 \beta_{16} - \beta_{17} ) q^{74} + ( -2 \beta_{1} + 2 \beta_{4} + 2 \beta_{7} + 2 \beta_{12} - \beta_{13} - \beta_{14} + 2 \beta_{15} ) q^{76} + ( 1 + 2 \beta_{1} - 2 \beta_{2} - \beta_{3} - 3 \beta_{6} - \beta_{8} + \beta_{9} - 2 \beta_{10} - \beta_{11} - \beta_{12} + 2 \beta_{13} + 2 \beta_{14} - \beta_{15} ) q^{77} + ( 2 - \beta_{3} - 2 \beta_{4} + \beta_{6} + 2 \beta_{8} - \beta_{9} - 2 \beta_{10} - \beta_{11} - 2 \beta_{12} + 2 \beta_{13} - 2 \beta_{14} - 2 \beta_{16} ) q^{78} + ( -3 - \beta_{2} - 2 \beta_{3} + 3 \beta_{5} + 2 \beta_{6} - 5 \beta_{7} - \beta_{8} - \beta_{9} - \beta_{10} + \beta_{11} - \beta_{12} + \beta_{13} - \beta_{14} - \beta_{15} - 2 \beta_{17} ) q^{79} + ( 1 + \beta_{1} - \beta_{2} + \beta_{5} - 2 \beta_{6} - 2 \beta_{7} + \beta_{9} + \beta_{12} - 2 \beta_{13} - \beta_{14} + \beta_{15} + \beta_{17} ) q^{81} + ( 4 + \beta_{1} - 3 \beta_{2} + 2 \beta_{3} + 3 \beta_{4} - 3 \beta_{5} + 3 \beta_{7} - 2 \beta_{8} - \beta_{9} + \beta_{10} + \beta_{11} - 4 \beta_{12} + \beta_{13} + \beta_{14} + \beta_{16} + \beta_{17} ) q^{82} + ( \beta_{1} + 2 \beta_{2} - 3 \beta_{3} - 2 \beta_{5} + \beta_{6} - 2 \beta_{7} + 2 \beta_{10} + 2 \beta_{12} - \beta_{13} + 3 \beta_{14} - 3 \beta_{16} + \beta_{17} ) q^{83} + ( -3 + 2 \beta_{2} - \beta_{3} + \beta_{4} + 2 \beta_{5} + 3 \beta_{6} + 2 \beta_{7} + \beta_{8} - \beta_{9} + \beta_{11} + \beta_{12} - \beta_{13} - \beta_{14} + \beta_{15} - \beta_{16} ) q^{84} + ( -1 + 2 \beta_{1} - \beta_{3} + 2 \beta_{4} - 2 \beta_{5} - 2 \beta_{6} - 2 \beta_{7} + \beta_{9} - 2 \beta_{11} - \beta_{12} + 2 \beta_{17} ) q^{86} + ( -2 \beta_{1} + 2 \beta_{3} + \beta_{4} + 2 \beta_{5} + 2 \beta_{6} + 4 \beta_{7} + 2 \beta_{10} + 2 \beta_{11} + 2 \beta_{12} - 2 \beta_{13} + 2 \beta_{15} - \beta_{16} ) q^{87} + ( 7 + 2 \beta_{2} - \beta_{3} + \beta_{4} - \beta_{6} - 2 \beta_{7} - \beta_{8} + \beta_{9} + 4 \beta_{10} - \beta_{11} + \beta_{12} - 2 \beta_{13} + \beta_{15} - 3 \beta_{16} + 2 \beta_{17} ) q^{88} + ( 1 + \beta_{1} - 4 \beta_{2} + 2 \beta_{3} + 2 \beta_{4} + \beta_{5} + 2 \beta_{7} + \beta_{8} + \beta_{9} + \beta_{10} - \beta_{11} + \beta_{12} + \beta_{13} + \beta_{14} + \beta_{15} + \beta_{17} ) q^{89} + ( -1 - \beta_{1} - 2 \beta_{2} - 3 \beta_{4} + 3 \beta_{5} - 2 \beta_{6} + \beta_{8} + \beta_{9} - \beta_{10} - \beta_{11} - \beta_{12} + \beta_{13} - \beta_{14} - \beta_{15} + 3 \beta_{16} - \beta_{17} ) q^{91} + ( -6 - \beta_{1} + 2 \beta_{2} - \beta_{3} + \beta_{4} + 2 \beta_{5} + 4 \beta_{6} - 2 \beta_{7} + \beta_{8} + \beta_{9} + 2 \beta_{11} + 3 \beta_{12} - 2 \beta_{16} - \beta_{17} ) q^{92} + ( -1 - \beta_{1} + 2 \beta_{2} + \beta_{5} - 2 \beta_{6} - 2 \beta_{7} + \beta_{8} + \beta_{9} - \beta_{10} - \beta_{11} + \beta_{12} - \beta_{13} - \beta_{14} + \beta_{15} + \beta_{17} ) q^{93} + ( 3 + \beta_{2} - \beta_{4} - \beta_{5} - \beta_{6} - \beta_{7} - 2 \beta_{8} + \beta_{9} - \beta_{10} - 2 \beta_{11} - 3 \beta_{12} + 2 \beta_{14} + \beta_{16} + 2 \beta_{17} ) q^{94} + ( -2 - 2 \beta_{5} + 2 \beta_{6} + 4 \beta_{7} + 2 \beta_{10} + 2 \beta_{11} + 2 \beta_{14} - 2 \beta_{15} + 2 \beta_{16} ) q^{96} + ( -2 - \beta_{1} - 3 \beta_{4} + 2 \beta_{5} - 2 \beta_{6} - 2 \beta_{7} + \beta_{8} + \beta_{9} + \beta_{11} + \beta_{13} - 3 \beta_{14} + \beta_{15} + 3 \beta_{16} - \beta_{17} ) q^{97} + ( -2 + \beta_{1} - \beta_{4} + \beta_{5} - \beta_{7} + \beta_{9} - \beta_{10} - \beta_{11} - 4 \beta_{12} + 3 \beta_{13} + \beta_{14} + \beta_{16} - \beta_{17} ) q^{98} + ( 3 \beta_{1} + 3 \beta_{2} - 3 \beta_{3} + \beta_{4} - 4 \beta_{5} + \beta_{6} - 3 \beta_{8} - 3 \beta_{9} + 2 \beta_{10} - \beta_{11} - 4 \beta_{12} + 2 \beta_{14} - \beta_{15} - \beta_{16} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$18 q + 4 q^{2} - 4 q^{4} - 8 q^{6} - 2 q^{7} + 4 q^{8} - 10 q^{9} + O(q^{10})$$ $$18 q + 4 q^{2} - 4 q^{4} - 8 q^{6} - 2 q^{7} + 4 q^{8} - 10 q^{9} - 2 q^{11} - 4 q^{12} + 12 q^{14} + 6 q^{17} - 16 q^{18} + 2 q^{19} - 16 q^{21} - 4 q^{22} + 2 q^{23} + 4 q^{24} - 16 q^{26} + 4 q^{28} - 14 q^{29} + 4 q^{32} + 8 q^{33} - 28 q^{34} - 4 q^{36} - 8 q^{37} - 16 q^{38} - 28 q^{42} + 44 q^{43} + 44 q^{44} + 12 q^{46} + 38 q^{47} - 60 q^{48} + 8 q^{51} + 40 q^{52} - 4 q^{54} + 20 q^{56} - 24 q^{57} + 20 q^{58} - 10 q^{59} + 14 q^{61} - 6 q^{63} - 16 q^{64} + 4 q^{66} - 12 q^{67} - 36 q^{68} + 32 q^{69} + 24 q^{71} + 36 q^{72} - 14 q^{73} + 48 q^{74} - 16 q^{76} + 84 q^{78} + 16 q^{79} + 2 q^{81} + 28 q^{82} - 24 q^{84} - 36 q^{86} - 24 q^{87} + 96 q^{88} - 12 q^{89} - 52 q^{92} - 16 q^{93} + 28 q^{94} - 40 q^{96} - 18 q^{97} - 32 q^{98} - 22 q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{18} + 2 x^{16} - 4 x^{15} - 5 x^{14} - 14 x^{13} - 10 x^{12} + 6 x^{11} + 37 x^{10} + 70 x^{9} + 74 x^{8} + 24 x^{7} - 80 x^{6} - 224 x^{5} - 160 x^{4} - 256 x^{3} + 256 x^{2} + 512$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$71 \nu^{17} + 98 \nu^{16} + 286 \nu^{15} + 144 \nu^{14} - 123 \nu^{13} - 1148 \nu^{12} - 2354 \nu^{11} - 3026 \nu^{10} - 2001 \nu^{9} + 1732 \nu^{8} + 7730 \nu^{7} + 13124 \nu^{6} + 13472 \nu^{5} + 4592 \nu^{4} - 3584 \nu^{3} - 23488 \nu^{2} - 16768 \nu - 24064$$$$)/1280$$ $$\beta_{2}$$ $$=$$ $$($$$$-129 \nu^{17} - 124 \nu^{16} - 398 \nu^{15} + 116 \nu^{14} + 797 \nu^{13} + 2778 \nu^{12} + 4526 \nu^{11} + 4402 \nu^{10} + 123 \nu^{9} - 9450 \nu^{8} - 21502 \nu^{7} - 29424 \nu^{6} - 25048 \nu^{5} + 368 \nu^{4} + 22176 \nu^{3} + 64960 \nu^{2} + 44800 \nu + 58624$$$$)/1280$$ $$\beta_{3}$$ $$=$$ $$($$$$-16 \nu^{17} - 19 \nu^{16} - 53 \nu^{15} + 2 \nu^{14} + 88 \nu^{13} + 331 \nu^{12} + 559 \nu^{11} + 580 \nu^{10} + 98 \nu^{9} - 1033 \nu^{8} - 2471 \nu^{7} - 3408 \nu^{6} - 2972 \nu^{5} - 44 \nu^{4} + 2444 \nu^{3} + 7152 \nu^{2} + 4832 \nu + 6048$$$$)/160$$ $$\beta_{4}$$ $$=$$ $$($$$$-75 \nu^{17} - 89 \nu^{16} - 248 \nu^{15} - 6 \nu^{14} + 375 \nu^{13} + 1487 \nu^{12} + 2550 \nu^{11} + 2676 \nu^{10} + 583 \nu^{9} - 4379 \nu^{8} - 10894 \nu^{7} - 15406 \nu^{6} - 13500 \nu^{5} - 848 \nu^{4} + 9920 \nu^{3} + 31296 \nu^{2} + 21696 \nu + 28416$$$$)/640$$ $$\beta_{5}$$ $$=$$ $$($$$$-229 \nu^{17} - 258 \nu^{16} - 706 \nu^{15} + 120 \nu^{14} + 1377 \nu^{13} + 4800 \nu^{12} + 7846 \nu^{11} + 7678 \nu^{10} + 331 \nu^{9} - 15784 \nu^{8} - 35846 \nu^{7} - 48500 \nu^{6} - 40528 \nu^{5} + 1200 \nu^{4} + 37376 \nu^{3} + 102976 \nu^{2} + 71296 \nu + 89600$$$$)/1280$$ $$\beta_{6}$$ $$=$$ $$($$$$-178 \nu^{17} - 217 \nu^{16} - 614 \nu^{15} + 6 \nu^{14} + 954 \nu^{13} + 3793 \nu^{12} + 6492 \nu^{11} + 6810 \nu^{10} + 1284 \nu^{9} - 11749 \nu^{8} - 28628 \nu^{7} - 40294 \nu^{6} - 35116 \nu^{5} - 1872 \nu^{4} + 28832 \nu^{3} + 83776 \nu^{2} + 58816 \nu + 75264$$$$)/640$$ $$\beta_{7}$$ $$=$$ $$($$$$545 \nu^{17} + 684 \nu^{16} + 1918 \nu^{15} + 156 \nu^{14} - 2685 \nu^{13} - 11242 \nu^{12} - 19710 \nu^{11} - 21346 \nu^{10} - 5723 \nu^{9} + 32794 \nu^{8} + 84014 \nu^{7} + 120736 \nu^{6} + 108280 \nu^{5} + 10928 \nu^{4} - 79200 \nu^{3} - 245696 \nu^{2} - 175616 \nu - 222976$$$$)/1280$$ $$\beta_{8}$$ $$=$$ $$($$$$298 \nu^{17} + 337 \nu^{16} + 974 \nu^{15} - 166 \nu^{14} - 1874 \nu^{13} - 6713 \nu^{12} - 11092 \nu^{11} - 10970 \nu^{10} - 604 \nu^{9} + 22749 \nu^{8} + 52268 \nu^{7} + 71254 \nu^{6} + 59756 \nu^{5} - 1008 \nu^{4} - 57152 \nu^{3} - 154176 \nu^{2} - 110016 \nu - 133504$$$$)/640$$ $$\beta_{9}$$ $$=$$ $$($$$$159 \nu^{17} + 235 \nu^{16} + 665 \nu^{15} + 358 \nu^{14} - 257 \nu^{13} - 2621 \nu^{12} - 5511 \nu^{11} - 7306 \nu^{10} - 5065 \nu^{9} + 3611 \nu^{8} + 17383 \nu^{7} + 30068 \nu^{6} + 32638 \nu^{5} + 13104 \nu^{4} - 6096 \nu^{3} - 50784 \nu^{2} - 36384 \nu - 54848$$$$)/320$$ $$\beta_{10}$$ $$=$$ $$($$$$871 \nu^{17} + 1090 \nu^{16} + 3110 \nu^{15} + 352 \nu^{14} - 4043 \nu^{13} - 17564 \nu^{12} - 31194 \nu^{11} - 34194 \nu^{10} - 10465 \nu^{9} + 49524 \nu^{8} + 130042 \nu^{7} + 188932 \nu^{6} + 171792 \nu^{5} + 21456 \nu^{4} - 117504 \nu^{3} - 381376 \nu^{2} - 268416 \nu - 351232$$$$)/1280$$ $$\beta_{11}$$ $$=$$ $$($$$$503 \nu^{17} + 761 \nu^{16} + 2162 \nu^{15} + 1230 \nu^{14} - 679 \nu^{13} - 8175 \nu^{12} - 17512 \nu^{11} - 23616 \nu^{10} - 17047 \nu^{9} + 9903 \nu^{8} + 53232 \nu^{7} + 94050 \nu^{6} + 103336 \nu^{5} + 44320 \nu^{4} - 14672 \nu^{3} - 154432 \nu^{2} - 108032 \nu - 168960$$$$)/640$$ $$\beta_{12}$$ $$=$$ $$($$$$-129 \nu^{17} - 186 \nu^{16} - 524 \nu^{15} - 232 \nu^{14} + 321 \nu^{13} + 2280 \nu^{12} + 4568 \nu^{11} + 5762 \nu^{10} + 3435 \nu^{9} - 4196 \nu^{8} - 15672 \nu^{7} - 25600 \nu^{6} - 26316 \nu^{5} - 8624 \nu^{4} + 8592 \nu^{3} + 45504 \nu^{2} + 32320 \nu + 46208$$$$)/128$$ $$\beta_{13}$$ $$=$$ $$($$$$93 \nu^{17} + 133 \nu^{16} + 374 \nu^{15} + 162 \nu^{14} - 237 \nu^{13} - 1639 \nu^{12} - 3268 \nu^{11} - 4104 \nu^{10} - 2417 \nu^{9} + 3063 \nu^{8} + 11252 \nu^{7} + 18318 \nu^{6} + 18760 \nu^{5} + 6024 \nu^{4} - 6240 \nu^{3} - 32672 \nu^{2} - 23040 \nu - 33024$$$$)/64$$ $$\beta_{14}$$ $$=$$ $$($$$$-93 \nu^{17} - 133 \nu^{16} - 374 \nu^{15} - 162 \nu^{14} + 237 \nu^{13} + 1639 \nu^{12} + 3268 \nu^{11} + 4104 \nu^{10} + 2417 \nu^{9} - 3063 \nu^{8} - 11252 \nu^{7} - 18318 \nu^{6} - 18760 \nu^{5} - 6024 \nu^{4} + 6240 \nu^{3} + 32672 \nu^{2} + 23168 \nu + 33024$$$$)/64$$ $$\beta_{15}$$ $$=$$ $$($$$$-981 \nu^{17} - 1344 \nu^{16} - 3828 \nu^{15} - 1348 \nu^{14} + 3013 \nu^{13} + 17886 \nu^{12} + 34564 \nu^{11} + 41990 \nu^{10} + 21883 \nu^{9} - 38378 \nu^{8} - 125876 \nu^{7} - 198348 \nu^{6} - 196852 \nu^{5} - 53344 \nu^{4} + 82784 \nu^{3} + 368512 \nu^{2} + 259392 \nu + 363008$$$$)/640$$ $$\beta_{16}$$ $$=$$ $$($$$$-1039 \nu^{17} - 1484 \nu^{16} - 4198 \nu^{15} - 1844 \nu^{14} + 2547 \nu^{13} + 18238 \nu^{12} + 36566 \nu^{11} + 46182 \nu^{10} + 27653 \nu^{9} - 33350 \nu^{8} - 125222 \nu^{7} - 205144 \nu^{6} - 211368 \nu^{5} - 69792 \nu^{4} + 67376 \nu^{3} + 364800 \nu^{2} + 258560 \nu + 372864$$$$)/640$$ $$\beta_{17}$$ $$=$$ $$($$$$3253 \nu^{17} + 4754 \nu^{16} + 13458 \nu^{15} + 6632 \nu^{14} - 6769 \nu^{13} - 55664 \nu^{12} - 114262 \nu^{11} - 147678 \nu^{10} - 95003 \nu^{9} + 90616 \nu^{8} + 375990 \nu^{7} + 630772 \nu^{6} + 663696 \nu^{5} + 240976 \nu^{4} - 172352 \nu^{3} - 1092544 \nu^{2} - 772224 \nu - 1132032$$$$)/1280$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{14} + \beta_{13}$$$$)/2$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{17} - \beta_{16} + \beta_{15} - 2 \beta_{13} + \beta_{12} + 2 \beta_{10} - \beta_{8} + \beta_{7} - \beta_{4} + \beta_{2} - \beta_{1} + 1$$$$)/2$$ $$\nu^{3}$$ $$=$$ $$($$$$-\beta_{17} + \beta_{16} - \beta_{15} - \beta_{14} + \beta_{13} - \beta_{12} - 2 \beta_{10} + \beta_{8} - \beta_{7} + 2 \beta_{5} - 3 \beta_{4} - \beta_{2} - \beta_{1} - 1$$$$)/2$$ $$\nu^{4}$$ $$=$$ $$($$$$\beta_{16} - \beta_{15} + \beta_{14} + \beta_{13} + \beta_{12} + \beta_{11} + \beta_{9} + \beta_{8} + \beta_{6} - 2 \beta_{5} + \beta_{4} + 3 \beta_{3} + 2 \beta_{2} - 2 \beta_{1} + 3$$$$)/2$$ $$\nu^{5}$$ $$=$$ $$($$$$\beta_{17} + \beta_{16} + \beta_{15} + 2 \beta_{14} - 3 \beta_{12} - 2 \beta_{11} + 2 \beta_{10} + 2 \beta_{9} - 3 \beta_{8} + \beta_{7} - 2 \beta_{6} + \beta_{4} - 3 \beta_{2} - \beta_{1} + 7$$$$)/2$$ $$\nu^{6}$$ $$=$$ $$($$$$\beta_{17} - 2 \beta_{16} + 2 \beta_{15} + \beta_{14} + \beta_{13} + 4 \beta_{12} + \beta_{11} + 2 \beta_{10} - 3 \beta_{9} - 2 \beta_{8} + 5 \beta_{7} + \beta_{6} + 2 \beta_{5} - 2 \beta_{4} + 3 \beta_{3} - \beta_{2} + \beta_{1} + 6$$$$)/2$$ $$\nu^{7}$$ $$=$$ $$($$$$-2 \beta_{16} - 2 \beta_{15} + 5 \beta_{14} - 3 \beta_{13} - 2 \beta_{12} + 2 \beta_{11} + 2 \beta_{10} + 2 \beta_{9} + 2 \beta_{8} + 6 \beta_{6} - 8 \beta_{4} - 4 \beta_{3} + 10 \beta_{2} - 4 \beta_{1} - 2$$$$)/2$$ $$\nu^{8}$$ $$=$$ $$($$$$\beta_{17} + 3 \beta_{16} - \beta_{15} - 10 \beta_{14} + 15 \beta_{12} + 2 \beta_{10} + 4 \beta_{9} + 3 \beta_{8} - 5 \beta_{7} - 4 \beta_{6} + 8 \beta_{5} - 7 \beta_{4} + 4 \beta_{3} + 5 \beta_{2} - 15 \beta_{1} - 3$$$$)/2$$ $$\nu^{9}$$ $$=$$ $$($$$$\beta_{17} + 7 \beta_{16} - 7 \beta_{15} + 15 \beta_{14} + 3 \beta_{13} - 25 \beta_{12} - 8 \beta_{11} - 10 \beta_{10} + 2 \beta_{9} - 9 \beta_{8} + 5 \beta_{7} - 10 \beta_{6} - 10 \beta_{5} + 3 \beta_{4} + 12 \beta_{3} - 13 \beta_{2} + 3 \beta_{1} + 31$$$$)/2$$ $$\nu^{10}$$ $$=$$ $$($$$$-2 \beta_{17} + 3 \beta_{16} + 7 \beta_{15} - \beta_{14} + 9 \beta_{13} + \beta_{12} - \beta_{11} + 18 \beta_{10} + 7 \beta_{9} + \beta_{8} - 12 \beta_{7} + 5 \beta_{6} + 12 \beta_{5} - 13 \beta_{4} + 7 \beta_{3} + 2 \beta_{2} - 4 \beta_{1} + 23$$$$)/2$$ $$\nu^{11}$$ $$=$$ $$($$$$13 \beta_{17} + 17 \beta_{16} + 3 \beta_{15} + 18 \beta_{14} + 15 \beta_{12} + 18 \beta_{11} + 8 \beta_{10} + 14 \beta_{9} - 15 \beta_{8} + 53 \beta_{7} + 2 \beta_{6} - 10 \beta_{5} + 7 \beta_{4} + 16 \beta_{3} - \beta_{2} - 11 \beta_{1} + 27$$$$)/2$$ $$\nu^{12}$$ $$=$$ $$($$$$-3 \beta_{17} - 42 \beta_{16} - 20 \beta_{15} + 9 \beta_{14} - 21 \beta_{13} + 18 \beta_{12} - 9 \beta_{11} + 18 \beta_{10} - 29 \beta_{9} - 6 \beta_{8} + 3 \beta_{7} + 47 \beta_{6} + 18 \beta_{5} - 24 \beta_{4} - 19 \beta_{3} + 55 \beta_{2} - 29 \beta_{1} - 22$$$$)/2$$ $$\nu^{13}$$ $$=$$ $$($$$$-8 \beta_{17} + 2 \beta_{16} - 8 \beta_{15} - 19 \beta_{14} - \beta_{13} + 10 \beta_{12} - 14 \beta_{11} - 12 \beta_{10} + 40 \beta_{9} + 22 \beta_{8} - 72 \beta_{7} - 64 \beta_{6} + 30 \beta_{5} - 62 \beta_{4} + 4 \beta_{3} + 30 \beta_{2} - 24 \beta_{1} + 92$$$$)/2$$ $$\nu^{14}$$ $$=$$ $$($$$$37 \beta_{17} + 101 \beta_{16} - 7 \beta_{15} + 62 \beta_{14} + 76 \beta_{13} - 35 \beta_{12} - 42 \beta_{11} - 20 \beta_{10} + 70 \beta_{9} - 13 \beta_{8} + 15 \beta_{7} - 56 \beta_{6} - 18 \beta_{5} - 15 \beta_{4} + 100 \beta_{3} - 61 \beta_{2} - 29 \beta_{1} + 13$$$$)/2$$ $$\nu^{15}$$ $$=$$ $$($$$$45 \beta_{17} + 43 \beta_{16} + 13 \beta_{15} + 13 \beta_{14} - 117 \beta_{13} - 63 \beta_{12} + 32 \beta_{11} + 138 \beta_{10} - 4 \beta_{9} - 117 \beta_{8} + 117 \beta_{7} + 24 \beta_{6} - 14 \beta_{5} + 19 \beta_{4} + 56 \beta_{3} - 23 \beta_{2} - 35 \beta_{1} + 233$$$$)/2$$ $$\nu^{16}$$ $$=$$ $$($$$$-164 \beta_{17} - 9 \beta_{16} - 103 \beta_{15} + 7 \beta_{14} + 187 \beta_{13} - 53 \beta_{12} + 59 \beta_{11} - 104 \beta_{10} - 33 \beta_{9} + 51 \beta_{8} + 16 \beta_{7} + 139 \beta_{6} + 130 \beta_{5} - 69 \beta_{4} - 35 \beta_{3} + 86 \beta_{2} - 26 \beta_{1} - 43$$$$)/2$$ $$\nu^{17}$$ $$=$$ $$($$$$163 \beta_{17} - 173 \beta_{16} - 5 \beta_{15} + 158 \beta_{14} - 112 \beta_{13} + 439 \beta_{12} - 54 \beta_{11} + 194 \beta_{10} + 134 \beta_{9} - 37 \beta_{8} + 75 \beta_{7} + 46 \beta_{6} - 180 \beta_{5} - 9 \beta_{4} + 76 \beta_{3} + 419 \beta_{2} - 135 \beta_{1} + 265$$$$)/2$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/400\mathbb{Z}\right)^\times$$.

 $$n$$ $$101$$ $$177$$ $$351$$ $$\chi(n)$$ $$-\beta_{12}$$ $$\beta_{12}$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
43.1
 1.41303 − 0.0578659i 0.482716 − 1.32928i −1.37691 + 0.322680i −1.08900 − 0.902261i −0.480367 − 1.33013i 0.0376504 − 1.41371i −0.635486 + 1.26339i 1.41323 + 0.0526497i 0.235136 + 1.39453i 1.41303 + 0.0578659i 0.482716 + 1.32928i −1.37691 − 0.322680i −1.08900 + 0.902261i −0.480367 + 1.33013i 0.0376504 + 1.41371i −0.635486 − 1.26339i 1.41323 − 0.0526497i 0.235136 − 1.39453i
−1.29521 + 0.567819i 1.96251i 1.35516 1.47090i 0 1.11435 + 2.54187i 1.60205 1.60205i −0.920026 + 2.67461i −0.851447 0
43.2 −0.759419 + 1.19301i 1.39319i −0.846564 1.81200i 0 −1.66209 1.05801i 2.13436 2.13436i 2.80463 + 0.366101i 1.05903 0
43.3 −0.687667 1.23576i 0.614566i −1.05423 + 1.69959i 0 −0.759459 + 0.422617i −2.83610 + 2.83610i 2.82525 + 0.134028i 2.62231 0
43.4 0.0660953 + 1.41267i 0.496487i −1.99126 + 0.186742i 0 −0.701372 + 0.0328155i −1.55426 + 1.55426i −0.395417 2.80065i 2.75350 0
43.5 0.307817 1.38031i 2.85601i −1.81050 0.849763i 0 −3.94217 0.879127i 0.458895 0.458895i −1.73024 + 2.23747i −5.15678 0
43.6 0.558542 1.29924i 2.55161i −1.37606 1.45136i 0 3.31516 + 1.42518i −2.40368 + 2.40368i −2.65426 + 0.977191i −3.51070 0
43.7 1.14628 + 0.828280i 0.692712i 0.627905 + 1.89888i 0 0.573759 0.794040i 0.343872 0.343872i −0.853049 + 2.69672i 2.52015 0
43.8 1.31641 0.516777i 1.28110i 1.46588 1.36058i 0 −0.662041 1.68645i 1.13975 1.13975i 1.22659 2.54862i 1.35879 0
43.9 1.34716 + 0.430311i 2.96561i 1.62967 + 1.15939i 0 −1.27613 + 3.99515i 0.115101 0.115101i 1.69652 + 2.26315i −5.79486 0
307.1 −1.29521 0.567819i 1.96251i 1.35516 + 1.47090i 0 1.11435 2.54187i 1.60205 + 1.60205i −0.920026 2.67461i −0.851447 0
307.2 −0.759419 1.19301i 1.39319i −0.846564 + 1.81200i 0 −1.66209 + 1.05801i 2.13436 + 2.13436i 2.80463 0.366101i 1.05903 0
307.3 −0.687667 + 1.23576i 0.614566i −1.05423 1.69959i 0 −0.759459 0.422617i −2.83610 2.83610i 2.82525 0.134028i 2.62231 0
307.4 0.0660953 1.41267i 0.496487i −1.99126 0.186742i 0 −0.701372 0.0328155i −1.55426 1.55426i −0.395417 + 2.80065i 2.75350 0
307.5 0.307817 + 1.38031i 2.85601i −1.81050 + 0.849763i 0 −3.94217 + 0.879127i 0.458895 + 0.458895i −1.73024 2.23747i −5.15678 0
307.6 0.558542 + 1.29924i 2.55161i −1.37606 + 1.45136i 0 3.31516 1.42518i −2.40368 2.40368i −2.65426 0.977191i −3.51070 0
307.7 1.14628 0.828280i 0.692712i 0.627905 1.89888i 0 0.573759 + 0.794040i 0.343872 + 0.343872i −0.853049 2.69672i 2.52015 0
307.8 1.31641 + 0.516777i 1.28110i 1.46588 + 1.36058i 0 −0.662041 + 1.68645i 1.13975 + 1.13975i 1.22659 + 2.54862i 1.35879 0
307.9 1.34716 0.430311i 2.96561i 1.62967 1.15939i 0 −1.27613 3.99515i 0.115101 + 0.115101i 1.69652 2.26315i −5.79486 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 307.9 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
80.j even 4 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 400.2.j.d 18
4.b odd 2 1 1600.2.j.d 18
5.b even 2 1 80.2.j.b 18
5.c odd 4 1 80.2.s.b yes 18
5.c odd 4 1 400.2.s.d 18
15.d odd 2 1 720.2.bd.g 18
15.e even 4 1 720.2.z.g 18
16.e even 4 1 1600.2.s.d 18
16.f odd 4 1 400.2.s.d 18
20.d odd 2 1 320.2.j.b 18
20.e even 4 1 320.2.s.b 18
20.e even 4 1 1600.2.s.d 18
40.e odd 2 1 640.2.j.c 18
40.f even 2 1 640.2.j.d 18
40.i odd 4 1 640.2.s.d 18
40.k even 4 1 640.2.s.c 18
80.i odd 4 1 320.2.j.b 18
80.j even 4 1 inner 400.2.j.d 18
80.j even 4 1 640.2.j.d 18
80.k odd 4 1 80.2.s.b yes 18
80.k odd 4 1 640.2.s.d 18
80.q even 4 1 320.2.s.b 18
80.q even 4 1 640.2.s.c 18
80.s even 4 1 80.2.j.b 18
80.t odd 4 1 640.2.j.c 18
80.t odd 4 1 1600.2.j.d 18
240.t even 4 1 720.2.z.g 18
240.z odd 4 1 720.2.bd.g 18

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
80.2.j.b 18 5.b even 2 1
80.2.j.b 18 80.s even 4 1
80.2.s.b yes 18 5.c odd 4 1
80.2.s.b yes 18 80.k odd 4 1
320.2.j.b 18 20.d odd 2 1
320.2.j.b 18 80.i odd 4 1
320.2.s.b 18 20.e even 4 1
320.2.s.b 18 80.q even 4 1
400.2.j.d 18 1.a even 1 1 trivial
400.2.j.d 18 80.j even 4 1 inner
400.2.s.d 18 5.c odd 4 1
400.2.s.d 18 16.f odd 4 1
640.2.j.c 18 40.e odd 2 1
640.2.j.c 18 80.t odd 4 1
640.2.j.d 18 40.f even 2 1
640.2.j.d 18 80.j even 4 1
640.2.s.c 18 40.k even 4 1
640.2.s.c 18 80.q even 4 1
640.2.s.d 18 40.i odd 4 1
640.2.s.d 18 80.k odd 4 1
720.2.z.g 18 15.e even 4 1
720.2.z.g 18 240.t even 4 1
720.2.bd.g 18 15.d odd 2 1
720.2.bd.g 18 240.z odd 4 1
1600.2.j.d 18 4.b odd 2 1
1600.2.j.d 18 80.t odd 4 1
1600.2.s.d 18 16.e even 4 1
1600.2.s.d 18 20.e even 4 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{18} + \cdots$$ acting on $$S_{2}^{\mathrm{new}}(400, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$512 - 1024 T + 1280 T^{2} - 1280 T^{3} + 1088 T^{4} - 832 T^{5} + 608 T^{6} - 384 T^{7} + 232 T^{8} - 152 T^{9} + 116 T^{10} - 96 T^{11} + 76 T^{12} - 52 T^{13} + 34 T^{14} - 20 T^{15} + 10 T^{16} - 4 T^{17} + T^{18}$$
$3$ $$256 + 2704 T^{2} + 10624 T^{4} + 19744 T^{6} + 18624 T^{8} + 9464 T^{10} + 2656 T^{12} + 408 T^{14} + 32 T^{16} + T^{18}$$
$5$ $$T^{18}$$
$7$ $$288 - 4128 T + 29584 T^{2} - 108160 T^{3} + 239360 T^{4} - 315488 T^{5} + 244448 T^{6} - 85184 T^{7} + 17328 T^{8} - 11648 T^{9} + 14648 T^{10} - 4160 T^{11} + 352 T^{12} + 120 T^{13} + 200 T^{14} - 32 T^{15} + 2 T^{16} + 2 T^{17} + T^{18}$$
$11$ $$5431808 - 4060672 T + 1517824 T^{2} - 4217856 T^{3} + 18015744 T^{4} - 19945472 T^{5} + 11514112 T^{6} - 2158336 T^{7} + 289472 T^{8} - 209344 T^{9} + 182240 T^{10} - 27968 T^{11} + 1376 T^{12} - 64 T^{13} + 848 T^{14} - 80 T^{15} + 2 T^{16} + 2 T^{17} + T^{18}$$
$13$ $$( 8192 + 8976 T - 3136 T^{2} - 5024 T^{3} + 320 T^{4} + 920 T^{5} - 16 T^{6} - 56 T^{7} + T^{9} )^{2}$$
$17$ $$512 - 1536 T + 2304 T^{2} + 186368 T^{3} + 1493504 T^{4} + 5352960 T^{5} + 11139328 T^{6} + 12464640 T^{7} + 6499264 T^{8} - 984640 T^{9} + 14816 T^{10} + 36480 T^{11} + 66080 T^{12} - 19232 T^{13} + 2768 T^{14} - 32 T^{15} + 18 T^{16} - 6 T^{17} + T^{18}$$
$19$ $$4608 + 339456 T + 12503296 T^{2} + 7467008 T^{3} + 2215424 T^{4} - 335360 T^{5} + 13924096 T^{6} + 9073664 T^{7} + 2905024 T^{8} - 1873856 T^{9} + 675296 T^{10} + 48384 T^{11} + 16480 T^{12} - 10080 T^{13} + 2864 T^{14} - 64 T^{15} + 2 T^{16} - 2 T^{17} + T^{18}$$
$23$ $$17700587552 - 17587696352 T + 8737762576 T^{2} + 758796096 T^{3} + 938524160 T^{4} - 784929440 T^{5} + 332892064 T^{6} + 13833888 T^{7} + 11039216 T^{8} - 7905920 T^{9} + 2752216 T^{10} - 54000 T^{11} + 9120 T^{12} - 7896 T^{13} + 3480 T^{14} - 24 T^{15} + 2 T^{16} - 2 T^{17} + T^{18}$$
$29$ $$82330112 + 372641280 T + 843321600 T^{2} + 687951872 T^{3} + 281020928 T^{4} + 16436736 T^{5} + 70118656 T^{6} + 45759488 T^{7} + 14859968 T^{8} + 1819968 T^{9} + 693088 T^{10} + 360192 T^{11} + 111008 T^{12} + 15584 T^{13} + 1616 T^{14} + 320 T^{15} + 98 T^{16} + 14 T^{17} + T^{18}$$
$31$ $$16384 + 1150976 T^{2} + 16687104 T^{4} + 32610304 T^{6} + 17532416 T^{8} + 3648384 T^{10} + 327040 T^{12} + 12832 T^{14} + 196 T^{16} + T^{18}$$
$37$ $$( -757824 + 209104 T + 273280 T^{2} - 112032 T^{3} - 6528 T^{4} + 6584 T^{5} - 160 T^{6} - 136 T^{7} + 4 T^{8} + T^{9} )^{2}$$
$41$ $$242788765696 + 208906121472 T^{2} + 68836776960 T^{4} + 11517059840 T^{6} + 1077600512 T^{8} + 58367328 T^{10} + 1831744 T^{12} + 32176 T^{14} + 288 T^{16} + T^{18}$$
$43$ $$( -580696 + 1273092 T - 855208 T^{2} + 178008 T^{3} + 25304 T^{4} - 14972 T^{5} + 1620 T^{6} + 72 T^{7} - 22 T^{8} + T^{9} )^{2}$$
$47$ $$16870640672 - 178795286432 T + 947437476496 T^{2} - 866389389184 T^{3} + 419056561664 T^{4} - 124183363808 T^{5} + 28973153120 T^{6} - 7620275328 T^{7} + 2411169968 T^{8} - 643585920 T^{9} + 126078328 T^{10} - 18512576 T^{11} + 2503968 T^{12} - 394472 T^{13} + 64680 T^{14} - 8272 T^{15} + 722 T^{16} - 38 T^{17} + T^{18}$$
$53$ $$48766772224 + 92515164416 T^{2} + 54724847616 T^{4} + 11957438208 T^{6} + 1230240896 T^{8} + 67640928 T^{10} + 2087424 T^{12} + 35696 T^{14} + 308 T^{16} + T^{18}$$
$59$ $$144166720393728 + 63266193922560 T + 13881883705600 T^{2} - 2799567403008 T^{3} + 1079531100672 T^{4} + 262683543040 T^{5} + 38509938944 T^{6} - 5387559424 T^{7} + 1811701952 T^{8} + 356104896 T^{9} + 37394528 T^{10} - 1999488 T^{11} + 826528 T^{12} + 162272 T^{13} + 16016 T^{14} - 96 T^{15} + 50 T^{16} + 10 T^{17} + T^{18}$$
$61$ $$121236758528 - 124325191680 T + 63746150400 T^{2} + 35916963840 T^{3} + 28713865216 T^{4} - 15011409920 T^{5} + 5616384000 T^{6} + 2393885696 T^{7} + 974385920 T^{8} - 231668992 T^{9} + 33596800 T^{10} + 3516032 T^{11} + 740800 T^{12} - 135872 T^{13} + 14496 T^{14} + 720 T^{15} + 98 T^{16} - 14 T^{17} + T^{18}$$
$67$ $$( 745336 + 892868 T - 378936 T^{2} - 315496 T^{3} + 23960 T^{4} + 20036 T^{5} - 884 T^{6} - 264 T^{7} + 6 T^{8} + T^{9} )^{2}$$
$71$ $$( -27648 - 72640 T + 110336 T^{2} - 8704 T^{3} - 23136 T^{4} + 3344 T^{5} + 1408 T^{6} - 152 T^{7} - 12 T^{8} + T^{9} )^{2}$$
$73$ $$35535647232 + 3228962304 T + 146700544 T^{2} + 5453305856 T^{3} + 14687183360 T^{4} + 5125904896 T^{5} + 823567616 T^{6} - 175353344 T^{7} + 212641728 T^{8} + 49164608 T^{9} + 5990112 T^{10} - 758144 T^{11} + 628896 T^{12} + 125856 T^{13} + 13072 T^{14} - 544 T^{15} + 98 T^{16} + 14 T^{17} + T^{18}$$
$79$ $$( -45002752 + 3950848 T + 7267840 T^{2} - 485376 T^{3} - 296064 T^{4} + 27488 T^{5} + 2976 T^{6} - 320 T^{7} - 8 T^{8} + T^{9} )^{2}$$
$83$ $$70791088097536 + 50219869553296 T^{2} + 11697046692480 T^{4} + 1226317787168 T^{6} + 63494671808 T^{8} + 1678046840 T^{10} + 23589920 T^{12} + 177560 T^{14} + 672 T^{16} + T^{18}$$
$89$ $$( 251904 + 5727232 T - 4338688 T^{2} - 1356288 T^{3} + 330368 T^{4} + 55104 T^{5} - 2752 T^{6} - 448 T^{7} + 6 T^{8} + T^{9} )^{2}$$
$97$ $$380349381734912 - 440719049317888 T + 255335343974656 T^{2} - 73928552845312 T^{3} + 10687493231104 T^{4} - 76758095360 T^{5} + 98986508544 T^{6} - 49913260032 T^{7} + 9402360768 T^{8} + 513287616 T^{9} + 21224416 T^{10} - 9460992 T^{11} + 2501024 T^{12} + 238368 T^{13} + 11856 T^{14} - 512 T^{15} + 162 T^{16} + 18 T^{17} + T^{18}$$