# Properties

 Label 400.2.q.e Level $400$ Weight $2$ Character orbit 400.q Analytic conductor $3.194$ Analytic rank $0$ Dimension $12$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$400 = 2^{4} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 400.q (of order $$4$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$3.19401608085$$ Analytic rank: $$0$$ Dimension: $$12$$ Relative dimension: $$6$$ over $$\Q(i)$$ Coefficient field: 12.0.4767670494822400.1 Defining polynomial: $$x^{12} - 4 x^{11} + 7 x^{10} - 4 x^{9} - 8 x^{8} + 24 x^{7} - 38 x^{6} + 48 x^{5} - 32 x^{4} - 32 x^{3} + 112 x^{2} - 128 x + 64$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{11}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q -\beta_{6} q^{2} -\beta_{7} q^{3} + \beta_{10} q^{4} + ( -\beta_{3} - \beta_{4} - 2 \beta_{5} - \beta_{6} - \beta_{9} + \beta_{10} - \beta_{11} ) q^{6} + ( -1 + \beta_{2} - \beta_{4} + \beta_{6} ) q^{7} + ( \beta_{1} + \beta_{3} + 2 \beta_{4} + 3 \beta_{5} + \beta_{6} - \beta_{7} - \beta_{8} - \beta_{10} ) q^{8} + ( -1 - \beta_{1} - \beta_{3} - 2 \beta_{4} - 2 \beta_{5} + \beta_{8} - \beta_{9} + \beta_{10} ) q^{9} +O(q^{10})$$ $$q -\beta_{6} q^{2} -\beta_{7} q^{3} + \beta_{10} q^{4} + ( -\beta_{3} - \beta_{4} - 2 \beta_{5} - \beta_{6} - \beta_{9} + \beta_{10} - \beta_{11} ) q^{6} + ( -1 + \beta_{2} - \beta_{4} + \beta_{6} ) q^{7} + ( \beta_{1} + \beta_{3} + 2 \beta_{4} + 3 \beta_{5} + \beta_{6} - \beta_{7} - \beta_{8} - \beta_{10} ) q^{8} + ( -1 - \beta_{1} - \beta_{3} - 2 \beta_{4} - 2 \beta_{5} + \beta_{8} - \beta_{9} + \beta_{10} ) q^{9} + ( -1 + \beta_{1} + \beta_{3} + 2 \beta_{5} + 2 \beta_{6} - \beta_{7} + \beta_{8} + \beta_{9} - \beta_{10} ) q^{11} + ( 2 \beta_{1} - 2 \beta_{2} + \beta_{4} + 2 \beta_{5} - \beta_{6} - \beta_{8} ) q^{12} + ( \beta_{2} + \beta_{4} + \beta_{5} + \beta_{6} - 2 \beta_{7} ) q^{13} + ( -2 + 2 \beta_{5} + 2 \beta_{6} + \beta_{8} - \beta_{10} ) q^{14} + ( -\beta_{1} - \beta_{3} - \beta_{4} - \beta_{5} + \beta_{7} - 2 \beta_{11} ) q^{16} + ( 1 - \beta_{1} + \beta_{3} - \beta_{5} - \beta_{8} + \beta_{9} + \beta_{10} ) q^{17} + ( 1 + \beta_{1} - \beta_{2} - \beta_{3} + \beta_{4} - \beta_{5} - \beta_{6} - \beta_{7} - \beta_{9} + \beta_{11} ) q^{18} + ( 1 + 2 \beta_{1} - \beta_{2} - \beta_{3} + 2 \beta_{5} - 2 \beta_{8} - \beta_{11} ) q^{19} + ( -1 + \beta_{2} + 2 \beta_{3} + \beta_{4} + 4 \beta_{5} + 3 \beta_{6} + 2 \beta_{9} - 2 \beta_{10} ) q^{21} + ( 1 - \beta_{1} - \beta_{2} - \beta_{5} - 2 \beta_{6} + \beta_{7} - 2 \beta_{9} + \beta_{10} ) q^{22} + ( -1 - 2 \beta_{1} + 3 \beta_{2} + \beta_{3} - 2 \beta_{4} + 4 \beta_{6} - \beta_{7} + 2 \beta_{8} + 2 \beta_{9} - 2 \beta_{10} - \beta_{11} ) q^{23} + ( 1 - \beta_{2} + 2 \beta_{3} + \beta_{4} + \beta_{9} - \beta_{11} ) q^{24} + ( -2 - 2 \beta_{3} - \beta_{4} - 2 \beta_{5} - \beta_{6} - \beta_{8} - 2 \beta_{9} + \beta_{10} - 2 \beta_{11} ) q^{26} + ( -2 - \beta_{1} + \beta_{2} - 2 \beta_{3} - 2 \beta_{4} - 2 \beta_{5} + 2 \beta_{6} + \beta_{8} - \beta_{9} - \beta_{10} - \beta_{11} ) q^{27} + ( 1 - \beta_{1} - \beta_{2} - \beta_{3} + \beta_{4} - 3 \beta_{5} - \beta_{6} + \beta_{7} - \beta_{9} + \beta_{11} ) q^{28} + ( -2 \beta_{1} + 3 \beta_{2} - 3 \beta_{4} - 3 \beta_{5} - \beta_{6} + 2 \beta_{8} - 2 \beta_{11} ) q^{29} + ( 2 \beta_{6} + \beta_{7} + 2 \beta_{8} + 2 \beta_{9} + \beta_{11} ) q^{31} + ( 1 + 2 \beta_{1} - \beta_{2} + 2 \beta_{4} - \beta_{6} + 2 \beta_{7} - \beta_{8} + \beta_{9} - \beta_{10} + \beta_{11} ) q^{32} + ( -1 - 2 \beta_{1} + \beta_{2} - \beta_{3} - 2 \beta_{4} - 5 \beta_{5} + 2 \beta_{8} - 2 \beta_{9} + 2 \beta_{10} ) q^{33} + ( 1 + \beta_{1} + 3 \beta_{2} + \beta_{3} + 3 \beta_{5} + 3 \beta_{6} - \beta_{7} + \beta_{9} - 2 \beta_{10} - \beta_{11} ) q^{34} + ( -1 - \beta_{1} - \beta_{2} - \beta_{3} - 3 \beta_{4} - 5 \beta_{5} - 3 \beta_{6} - \beta_{7} - \beta_{9} + 2 \beta_{10} - \beta_{11} ) q^{36} + ( 1 - 2 \beta_{1} + \beta_{2} - 3 \beta_{4} - 2 \beta_{5} + \beta_{6} + 2 \beta_{8} + 2 \beta_{9} + 2 \beta_{10} ) q^{37} + ( -3 + \beta_{1} - \beta_{2} + 2 \beta_{3} + \beta_{4} + \beta_{5} + \beta_{6} + \beta_{7} + \beta_{8} + 2 \beta_{9} - 2 \beta_{10} - 2 \beta_{11} ) q^{38} + ( -1 - \beta_{2} - \beta_{4} - 4 \beta_{5} + \beta_{6} + \beta_{7} - \beta_{11} ) q^{39} + ( 1 + \beta_{2} + 3 \beta_{3} + 4 \beta_{4} + 5 \beta_{5} + 2 \beta_{6} - 2 \beta_{7} + 2 \beta_{11} ) q^{41} + ( -2 \beta_{1} + 2 \beta_{2} + 2 \beta_{6} + 2 \beta_{7} + \beta_{8} - \beta_{10} ) q^{42} + ( -\beta_{1} - \beta_{2} + 2 \beta_{6} + \beta_{8} + \beta_{9} + \beta_{10} + 2 \beta_{11} ) q^{43} + ( 5 + \beta_{1} - \beta_{2} - \beta_{3} + 2 \beta_{4} + \beta_{5} - 2 \beta_{6} - \beta_{7} - \beta_{8} + \beta_{9} + \beta_{11} ) q^{44} + ( -1 - \beta_{1} + \beta_{2} - 3 \beta_{3} - 2 \beta_{4} - 3 \beta_{5} - 2 \beta_{6} + 3 \beta_{7} + \beta_{8} - 3 \beta_{9} + \beta_{10} + \beta_{11} ) q^{46} + ( 2 \beta_{2} - \beta_{3} + \beta_{4} - \beta_{6} - \beta_{7} + 2 \beta_{8} - 2 \beta_{9} + \beta_{11} ) q^{47} + ( 6 + \beta_{1} + \beta_{3} + \beta_{4} + 3 \beta_{5} - 2 \beta_{6} + \beta_{7} - 2 \beta_{8} ) q^{48} + ( -3 \beta_{2} - \beta_{3} + 4 \beta_{4} - 4 \beta_{6} - 2 \beta_{8} - 2 \beta_{9} ) q^{49} + ( 2 + 3 \beta_{1} - 3 \beta_{2} + 2 \beta_{3} + 4 \beta_{4} + 4 \beta_{5} - 2 \beta_{6} - 3 \beta_{8} + \beta_{9} + \beta_{10} + \beta_{11} ) q^{51} + ( -1 + 3 \beta_{1} - 3 \beta_{2} - \beta_{3} + \beta_{4} + \beta_{5} + \beta_{6} + \beta_{7} - \beta_{8} + \beta_{9} - \beta_{10} - \beta_{11} ) q^{52} + ( 1 + 2 \beta_{1} - 2 \beta_{3} - 2 \beta_{4} + \beta_{5} - 2 \beta_{8} - 2 \beta_{11} ) q^{53} + ( -2 - 2 \beta_{2} - 3 \beta_{3} - 2 \beta_{4} - 6 \beta_{5} - 2 \beta_{6} + 2 \beta_{7} + \beta_{8} - \beta_{9} + \beta_{11} ) q^{54} + ( 1 - \beta_{1} + \beta_{2} - \beta_{3} - 3 \beta_{4} - 5 \beta_{5} - \beta_{6} - \beta_{7} + \beta_{9} + \beta_{11} ) q^{56} + ( -\beta_{1} + 3 \beta_{3} - 2 \beta_{4} - 2 \beta_{7} + 3 \beta_{8} + 3 \beta_{9} - \beta_{10} - 2 \beta_{11} ) q^{57} + ( 2 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} + \beta_{4} + 4 \beta_{5} - 3 \beta_{6} + 2 \beta_{7} - \beta_{8} - 2 \beta_{9} + 3 \beta_{10} + 2 \beta_{11} ) q^{58} + ( \beta_{1} + \beta_{2} - 2 \beta_{3} + 4 \beta_{4} + 2 \beta_{5} - 2 \beta_{7} + \beta_{8} + \beta_{9} - \beta_{10} ) q^{59} + ( -2 \beta_{1} + \beta_{3} + \beta_{4} + \beta_{6} + 2 \beta_{8} ) q^{61} + ( -1 - \beta_{1} + \beta_{2} + 3 \beta_{3} + 2 \beta_{4} + \beta_{5} - \beta_{7} - \beta_{8} - \beta_{9} - \beta_{10} + 3 \beta_{11} ) q^{62} + ( 3 + \beta_{2} + \beta_{3} + 2 \beta_{4} - 2 \beta_{6} + 2 \beta_{7} - 2 \beta_{8} + 2 \beta_{9} - 2 \beta_{11} ) q^{63} + ( -3 - 2 \beta_{1} + \beta_{2} + 4 \beta_{3} - 2 \beta_{4} + 2 \beta_{5} + 3 \beta_{6} + \beta_{8} + 3 \beta_{9} - \beta_{10} + \beta_{11} ) q^{64} + ( 2 + 2 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} + \beta_{4} + 2 \beta_{5} - 2 \beta_{6} - 2 \beta_{7} - 2 \beta_{8} - 2 \beta_{9} + 2 \beta_{11} ) q^{66} + ( -5 + \beta_{1} - \beta_{3} + 4 \beta_{5} - \beta_{7} + \beta_{8} - \beta_{9} + \beta_{10} ) q^{67} + ( -5 - \beta_{1} - \beta_{2} - \beta_{3} - \beta_{4} + 3 \beta_{5} - \beta_{6} + 3 \beta_{7} + \beta_{8} - \beta_{9} - \beta_{11} ) q^{68} + ( 3 - 2 \beta_{1} + 4 \beta_{2} + 2 \beta_{3} + 2 \beta_{4} + \beta_{5} + 4 \beta_{6} + 2 \beta_{7} - 2 \beta_{8} ) q^{69} + ( 2 \beta_{1} - \beta_{3} + \beta_{4} - \beta_{6} - \beta_{7} - 2 \beta_{10} + \beta_{11} ) q^{71} + ( 3 + 3 \beta_{1} - \beta_{2} - \beta_{3} - \beta_{4} - \beta_{5} - \beta_{6} - \beta_{7} - \beta_{9} + 2 \beta_{10} - \beta_{11} ) q^{72} + ( -3 - \beta_{1} + \beta_{2} + 2 \beta_{3} - 2 \beta_{4} + 2 \beta_{7} + \beta_{8} + \beta_{9} - \beta_{10} + 2 \beta_{11} ) q^{73} + ( 2 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} + 4 \beta_{4} + 4 \beta_{5} - 2 \beta_{7} - \beta_{8} - 2 \beta_{9} - \beta_{10} + 2 \beta_{11} ) q^{74} + ( 6 - 2 \beta_{2} + 2 \beta_{3} + \beta_{4} + 2 \beta_{5} - \beta_{6} + 4 \beta_{7} - 2 \beta_{8} + \beta_{10} + 2 \beta_{11} ) q^{76} + ( 4 - 2 \beta_{1} - \beta_{3} + 3 \beta_{4} - 2 \beta_{5} - 3 \beta_{6} - 2 \beta_{8} + 2 \beta_{9} - 2 \beta_{10} ) q^{77} + ( 3 + \beta_{1} - \beta_{2} + \beta_{3} - 2 \beta_{4} + \beta_{5} + \beta_{7} + \beta_{9} - 2 \beta_{10} + \beta_{11} ) q^{78} + ( -2 + 2 \beta_{1} - 4 \beta_{2} + 3 \beta_{3} - \beta_{4} - 7 \beta_{6} + 2 \beta_{7} + 2 \beta_{10} + 2 \beta_{11} ) q^{79} + ( -\beta_{1} + 2 \beta_{2} + \beta_{3} - 2 \beta_{4} + 2 \beta_{7} - \beta_{8} - \beta_{9} - \beta_{10} + 2 \beta_{11} ) q^{81} + ( 2 - 2 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} + \beta_{4} + 2 \beta_{5} - 2 \beta_{7} - 2 \beta_{8} - 2 \beta_{9} - 2 \beta_{11} ) q^{82} + ( -2 - \beta_{1} - 3 \beta_{2} - 2 \beta_{3} - 4 \beta_{4} - 4 \beta_{5} - 6 \beta_{6} + \beta_{7} - \beta_{8} - 3 \beta_{9} + 3 \beta_{10} ) q^{83} + ( -1 - \beta_{1} + \beta_{2} - \beta_{3} + \beta_{4} + \beta_{5} + \beta_{6} + \beta_{7} + \beta_{9} - 2 \beta_{10} + 3 \beta_{11} ) q^{84} + ( 1 - \beta_{1} + 3 \beta_{2} + \beta_{3} + \beta_{4} - 3 \beta_{5} + \beta_{6} - 3 \beta_{7} - \beta_{9} - 2 \beta_{10} + \beta_{11} ) q^{86} + ( 5 + 2 \beta_{1} + \beta_{2} - 2 \beta_{3} - \beta_{4} - \beta_{6} - \beta_{7} - 4 \beta_{8} - 4 \beta_{9} + 2 \beta_{10} - \beta_{11} ) q^{87} + ( -4 - \beta_{1} + 2 \beta_{2} + \beta_{3} - 2 \beta_{4} - 5 \beta_{5} - 3 \beta_{6} - \beta_{7} + 2 \beta_{10} - 2 \beta_{11} ) q^{88} + ( -1 - \beta_{1} - 2 \beta_{2} + \beta_{3} - 2 \beta_{4} + 3 \beta_{5} + 2 \beta_{6} - 2 \beta_{7} - \beta_{8} + \beta_{9} + \beta_{10} + 2 \beta_{11} ) q^{89} + ( 2 \beta_{3} + 2 \beta_{5} + 2 \beta_{6} + 2 \beta_{9} - 2 \beta_{10} ) q^{91} + ( -4 - 2 \beta_{2} + 2 \beta_{4} + 2 \beta_{5} + 2 \beta_{6} - 2 \beta_{7} + \beta_{8} + 2 \beta_{9} - \beta_{10} + 4 \beta_{11} ) q^{92} + ( -2 + \beta_{2} - \beta_{3} + 2 \beta_{4} + 3 \beta_{5} + 2 \beta_{7} ) q^{93} + ( -3 - \beta_{1} - 3 \beta_{2} - 3 \beta_{3} - \beta_{5} - 4 \beta_{6} - \beta_{7} - 2 \beta_{8} - 3 \beta_{9} + 4 \beta_{10} + \beta_{11} ) q^{94} + ( -1 + \beta_{2} + 2 \beta_{3} + 2 \beta_{4} + 6 \beta_{5} - \beta_{6} + \beta_{8} + 3 \beta_{9} - \beta_{10} - \beta_{11} ) q^{96} + ( -1 + 2 \beta_{1} - 3 \beta_{2} + 5 \beta_{3} + 4 \beta_{4} + 8 \beta_{5} + 6 \beta_{6} - 2 \beta_{7} - 2 \beta_{8} + 2 \beta_{9} - 2 \beta_{10} + 2 \beta_{11} ) q^{97} + ( 4 - 2 \beta_{3} - 2 \beta_{4} - 8 \beta_{5} - \beta_{6} - 2 \beta_{8} + 2 \beta_{9} + 2 \beta_{10} - 2 \beta_{11} ) q^{98} + ( -1 - \beta_{1} + 2 \beta_{2} - 3 \beta_{3} - 6 \beta_{4} - 4 \beta_{5} + 2 \beta_{6} + \beta_{8} + \beta_{9} + \beta_{10} + 2 \beta_{11} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$12q - 2q^{2} + 2q^{3} - 2q^{4} + 6q^{6} - 12q^{7} - 2q^{8} + O(q^{10})$$ $$12q - 2q^{2} + 2q^{3} - 2q^{4} + 6q^{6} - 12q^{7} - 2q^{8} - 2q^{11} + 6q^{12} - 4q^{13} - 14q^{14} + 2q^{16} + 18q^{18} + 14q^{19} - 20q^{21} + 20q^{22} - 12q^{23} + 14q^{24} - 16q^{26} - 10q^{27} + 10q^{28} - 4q^{31} - 2q^{32} + 6q^{34} + 2q^{36} + 8q^{37} - 28q^{38} - 10q^{42} + 44q^{44} - 10q^{46} + 58q^{48} - 4q^{49} + 10q^{51} + 16q^{53} - 10q^{54} + 6q^{56} + 16q^{57} - 4q^{58} - 20q^{59} + 4q^{61} - 22q^{62} - 38q^{64} + 32q^{66} - 50q^{67} - 50q^{68} + 54q^{72} - 40q^{73} - 10q^{74} + 60q^{76} + 8q^{77} + 48q^{78} - 12q^{79} - 8q^{81} + 12q^{82} - 2q^{83} - 34q^{84} + 6q^{86} + 64q^{87} - 56q^{88} - 50q^{92} - 44q^{93} - 32q^{94} - 34q^{96} + 30q^{98} - 12q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{12} - 4 x^{11} + 7 x^{10} - 4 x^{9} - 8 x^{8} + 24 x^{7} - 38 x^{6} + 48 x^{5} - 32 x^{4} - 32 x^{3} + 112 x^{2} - 128 x + 64$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$-\nu^{2} + \nu$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{11} - 2 \nu^{10} - \nu^{9} + 2 \nu^{8} - 6 \nu^{5} + 4 \nu^{4} + 16 \nu^{3} - 16 \nu^{2} - 16 \nu + 16$$$$)/16$$ $$\beta_{3}$$ $$=$$ $$($$$$-\nu^{11} + 2 \nu^{10} + \nu^{9} - 2 \nu^{8} + 6 \nu^{5} - 4 \nu^{4} - 16 \nu^{3} + 16 \nu^{2} + 48 \nu - 32$$$$)/16$$ $$\beta_{4}$$ $$=$$ $$($$$$\nu^{11} - 4 \nu^{10} + 7 \nu^{9} - 4 \nu^{8} - 8 \nu^{7} + 24 \nu^{6} - 38 \nu^{5} + 48 \nu^{4} - 32 \nu^{3} - 32 \nu^{2} + 112 \nu - 128$$$$)/32$$ $$\beta_{5}$$ $$=$$ $$($$$$-\nu^{11} + 6 \nu^{10} - 11 \nu^{9} + 2 \nu^{8} + 12 \nu^{7} - 24 \nu^{6} + 38 \nu^{5} - 60 \nu^{4} + 40 \nu^{3} + 64 \nu^{2} - 144 \nu + 64$$$$)/32$$ $$\beta_{6}$$ $$=$$ $$($$$$\nu^{11} - 3 \nu^{10} + \nu^{9} + 7 \nu^{8} - 10 \nu^{7} + 12 \nu^{6} - 14 \nu^{5} + 10 \nu^{4} + 28 \nu^{3} - 72 \nu^{2} + 48 \nu + 16$$$$)/16$$ $$\beta_{7}$$ $$=$$ $$($$$$\nu^{11} - 2 \nu^{10} + 4 \nu^{8} - 5 \nu^{7} + 6 \nu^{6} - 10 \nu^{5} + 4 \nu^{4} + 18 \nu^{3} - 36 \nu^{2} + 16 \nu + 8$$$$)/8$$ $$\beta_{8}$$ $$=$$ $$($$$$\nu^{11} - 8 \nu^{10} + 19 \nu^{9} - 8 \nu^{8} - 36 \nu^{7} + 64 \nu^{6} - 86 \nu^{5} + 104 \nu^{4} - 72 \nu^{3} - 144 \nu^{2} + 400 \nu - 320$$$$)/32$$ $$\beta_{9}$$ $$=$$ $$($$$$-\nu^{11} + 6 \nu^{10} - 7 \nu^{9} - 6 \nu^{8} + 24 \nu^{7} - 32 \nu^{6} + 38 \nu^{5} - 44 \nu^{4} + 128 \nu^{2} - 192 \nu + 80$$$$)/16$$ $$\beta_{10}$$ $$=$$ $$($$$$2 \nu^{11} - 7 \nu^{10} + 10 \nu^{9} - \nu^{8} - 20 \nu^{7} + 40 \nu^{6} - 52 \nu^{5} + 58 \nu^{4} - 16 \nu^{3} - 96 \nu^{2} + 192 \nu - 144$$$$)/16$$ $$\beta_{11}$$ $$=$$ $$($$$$9 \nu^{11} - 22 \nu^{10} + 15 \nu^{9} + 22 \nu^{8} - 56 \nu^{7} + 80 \nu^{6} - 118 \nu^{5} + 124 \nu^{4} + 80 \nu^{3} - 336 \nu^{2} + 304 \nu - 64$$$$)/32$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{3} + \beta_{2} + 1$$$$)/2$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{3} + \beta_{2} - 2 \beta_{1} + 1$$$$)/2$$ $$\nu^{3}$$ $$=$$ $$($$$$2 \beta_{9} + 2 \beta_{8} - 2 \beta_{7} + 4 \beta_{6} - 2 \beta_{4} + \beta_{3} + 3 \beta_{2} - 2 \beta_{1} - 1$$$$)/2$$ $$\nu^{4}$$ $$=$$ $$($$$$2 \beta_{11} - 2 \beta_{10} - 4 \beta_{7} + 2 \beta_{6} + 2 \beta_{5} + 4 \beta_{4} + \beta_{3} + \beta_{2} + 1$$$$)/2$$ $$\nu^{5}$$ $$=$$ $$($$$$2 \beta_{11} + 2 \beta_{10} - 2 \beta_{9} - 4 \beta_{8} - 4 \beta_{7} - 2 \beta_{6} + 2 \beta_{5} + \beta_{3} - \beta_{2} + 2 \beta_{1} - 3$$$$)/2$$ $$\nu^{6}$$ $$=$$ $$($$$$4 \beta_{10} - 2 \beta_{9} - 6 \beta_{8} - 2 \beta_{7} + 4 \beta_{5} + 6 \beta_{4} - \beta_{3} - 5 \beta_{2} + 2 \beta_{1} + 7$$$$)/2$$ $$\nu^{7}$$ $$=$$ $$($$$$-2 \beta_{11} + 2 \beta_{10} + 2 \beta_{9} - 4 \beta_{8} + 6 \beta_{6} - 2 \beta_{5} + 8 \beta_{4} + 3 \beta_{3} + \beta_{2} + 6 \beta_{1} - 1$$$$)/2$$ $$\nu^{8}$$ $$=$$ $$($$$$-4 \beta_{10} - 2 \beta_{9} - 6 \beta_{8} + 6 \beta_{7} - 4 \beta_{5} + 14 \beta_{4} + 5 \beta_{3} - 7 \beta_{2} - 2 \beta_{1} - 11$$$$)/2$$ $$\nu^{9}$$ $$=$$ $$($$$$-2 \beta_{11} + 2 \beta_{10} + 14 \beta_{9} + 8 \beta_{8} + 4 \beta_{7} + 14 \beta_{6} - 18 \beta_{5} - 20 \beta_{4} + \beta_{3} - 5 \beta_{2} - 6 \beta_{1} - 31$$$$)/2$$ $$\nu^{10}$$ $$=$$ $$($$$$12 \beta_{11} - 16 \beta_{10} + 14 \beta_{9} + 10 \beta_{8} - 2 \beta_{7} + 12 \beta_{6} + 8 \beta_{5} + 22 \beta_{4} + 3 \beta_{3} - 25 \beta_{2} + 14 \beta_{1} + 3$$$$)/2$$ $$\nu^{11}$$ $$=$$ $$($$$$26 \beta_{11} - 2 \beta_{10} + 2 \beta_{9} - 16 \beta_{8} + 12 \beta_{7} - 46 \beta_{6} + 10 \beta_{5} + 12 \beta_{4} + 15 \beta_{3} - 35 \beta_{2} + 38 \beta_{1} - 9$$$$)/2$$

## Character values

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

 $$n$$ $$101$$ $$177$$ $$351$$ $$\chi(n)$$ $$-\beta_{5}$$ $$-1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
149.1
 −0.507829 + 1.31989i 0.719139 + 1.21772i 1.35979 + 0.388551i −1.41313 + 0.0554252i 1.22306 − 0.710021i 0.618969 − 1.27156i −0.507829 − 1.31989i 0.719139 − 1.21772i 1.35979 − 0.388551i −1.41313 − 0.0554252i 1.22306 + 0.710021i 0.618969 + 1.27156i
−1.31989 + 0.507829i 0.0623209 + 0.0623209i 1.48422 1.34056i 0 −0.113905 0.0506084i −0.375877 −1.27824 + 2.52312i 2.99223i 0
149.2 −1.21772 0.719139i −1.66783 1.66783i 0.965679 + 1.75142i 0 0.831547 + 3.23035i 1.87372 0.0835873 2.82719i 2.56332i 0
149.3 −0.388551 1.35979i 1.03997 + 1.03997i −1.69806 + 1.05670i 0 1.01006 1.81822i 1.49668 2.09667 + 1.89842i 0.836925i 0
149.4 −0.0554252 + 1.41313i 0.488516 + 0.488516i −1.99386 0.156646i 0 −0.717411 + 0.663259i −4.71540 0.331870 2.80889i 2.52270i 0
149.5 0.710021 1.22306i −1.09156 1.09156i −0.991741 1.73679i 0 −2.11008 + 0.560012i −0.973926 −2.82835 0.0202025i 0.616985i 0
149.6 1.27156 0.618969i 2.16859 + 2.16859i 1.23375 1.57412i 0 4.09979 + 1.41521i −3.30519 0.594467 2.76525i 6.40553i 0
349.1 −1.31989 0.507829i 0.0623209 0.0623209i 1.48422 + 1.34056i 0 −0.113905 + 0.0506084i −0.375877 −1.27824 2.52312i 2.99223i 0
349.2 −1.21772 + 0.719139i −1.66783 + 1.66783i 0.965679 1.75142i 0 0.831547 3.23035i 1.87372 0.0835873 + 2.82719i 2.56332i 0
349.3 −0.388551 + 1.35979i 1.03997 1.03997i −1.69806 1.05670i 0 1.01006 + 1.81822i 1.49668 2.09667 1.89842i 0.836925i 0
349.4 −0.0554252 1.41313i 0.488516 0.488516i −1.99386 + 0.156646i 0 −0.717411 0.663259i −4.71540 0.331870 + 2.80889i 2.52270i 0
349.5 0.710021 + 1.22306i −1.09156 + 1.09156i −0.991741 + 1.73679i 0 −2.11008 0.560012i −0.973926 −2.82835 + 0.0202025i 0.616985i 0
349.6 1.27156 + 0.618969i 2.16859 2.16859i 1.23375 + 1.57412i 0 4.09979 1.41521i −3.30519 0.594467 + 2.76525i 6.40553i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 349.6 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.q 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.q.e 12
4.b odd 2 1 1600.2.q.e 12
5.b even 2 1 400.2.q.f 12
5.c odd 4 1 400.2.l.f 12
5.c odd 4 1 400.2.l.g yes 12
16.e even 4 1 400.2.q.f 12
16.f odd 4 1 1600.2.q.f 12
20.d odd 2 1 1600.2.q.f 12
20.e even 4 1 1600.2.l.f 12
20.e even 4 1 1600.2.l.g 12
80.i odd 4 1 400.2.l.f 12
80.j even 4 1 1600.2.l.f 12
80.k odd 4 1 1600.2.q.e 12
80.q even 4 1 inner 400.2.q.e 12
80.s even 4 1 1600.2.l.g 12
80.t odd 4 1 400.2.l.g yes 12

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
400.2.l.f 12 5.c odd 4 1
400.2.l.f 12 80.i odd 4 1
400.2.l.g yes 12 5.c odd 4 1
400.2.l.g yes 12 80.t odd 4 1
400.2.q.e 12 1.a even 1 1 trivial
400.2.q.e 12 80.q even 4 1 inner
400.2.q.f 12 5.b even 2 1
400.2.q.f 12 16.e even 4 1
1600.2.l.f 12 20.e even 4 1
1600.2.l.f 12 80.j even 4 1
1600.2.l.g 12 20.e even 4 1
1600.2.l.g 12 80.s even 4 1
1600.2.q.e 12 4.b odd 2 1
1600.2.q.e 12 80.k odd 4 1
1600.2.q.f 12 16.f odd 4 1
1600.2.q.f 12 20.d odd 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{12} - \cdots$$ acting on $$S_{2}^{\mathrm{new}}(400, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$64 + 64 T + 48 T^{2} + 32 T^{3} + 16 T^{4} + 8 T^{5} + 10 T^{6} + 4 T^{7} + 4 T^{8} + 4 T^{9} + 3 T^{10} + 2 T^{11} + T^{12}$$
$3$ $$1 - 18 T + 162 T^{2} - 282 T^{3} + 243 T^{4} + 20 T^{5} + 36 T^{6} - 60 T^{7} + 51 T^{8} + 6 T^{9} + 2 T^{10} - 2 T^{11} + T^{12}$$
$5$ $$T^{12}$$
$7$ $$( 16 + 48 T - 40 T^{3} - 2 T^{4} + 6 T^{5} + T^{6} )^{2}$$
$11$ $$85849 - 294758 T + 506018 T^{2} - 383110 T^{3} + 162307 T^{4} - 30308 T^{5} + 2212 T^{6} - 484 T^{7} + 619 T^{8} - 94 T^{9} + 2 T^{10} + 2 T^{11} + T^{12}$$
$13$ $$256 - 3328 T + 21632 T^{2} - 30784 T^{3} + 22800 T^{4} - 5888 T^{5} + 832 T^{6} - 384 T^{7} + 488 T^{8} - 112 T^{9} + 8 T^{10} + 4 T^{11} + T^{12}$$
$17$ $$677329 + 3032370 T^{2} + 908063 T^{4} + 97884 T^{6} + 4879 T^{8} + 114 T^{10} + T^{12}$$
$19$ $$29997529 - 20494934 T + 7001282 T^{2} - 1036582 T^{3} + 792387 T^{4} - 486580 T^{5} + 165412 T^{6} - 25044 T^{7} + 2219 T^{8} - 254 T^{9} + 98 T^{10} - 14 T^{11} + T^{12}$$
$23$ $$( -2872 + 760 T + 1228 T^{2} - 328 T^{3} - 74 T^{4} + 6 T^{5} + T^{6} )^{2}$$
$29$ $$428655616 - 33788928 T + 1331712 T^{2} + 4592512 T^{3} + 5399440 T^{4} + 92800 T^{5} + 512 T^{6} + 2208 T^{7} + 7960 T^{8} + 32 T^{9} + T^{12}$$
$31$ $$( 2152 + 3688 T + 1100 T^{2} - 248 T^{3} - 82 T^{4} + 2 T^{5} + T^{6} )^{2}$$
$37$ $$6801664 - 9847808 T + 7129088 T^{2} + 8940288 T^{3} + 4476736 T^{4} + 781568 T^{5} + 51840 T^{6} - 4704 T^{7} + 3652 T^{8} + 512 T^{9} + 32 T^{10} - 8 T^{11} + T^{12}$$
$41$ $$86397025 + 111297670 T^{2} + 17569359 T^{4} + 1016212 T^{6} + 24687 T^{8} + 262 T^{10} + T^{12}$$
$43$ $$26214400 - 52428800 T + 52428800 T^{2} - 26214400 T^{3} + 7356416 T^{4} - 806912 T^{5} + 8192 T^{6} - 3840 T^{7} + 7108 T^{8} - 128 T^{9} + T^{12}$$
$47$ $$484704256 + 275005440 T^{2} + 31780624 T^{4} + 1451648 T^{6} + 30408 T^{8} + 288 T^{10} + T^{12}$$
$53$ $$7225000000 - 4216000000 T + 1230080000 T^{2} - 168787200 T^{3} + 15410224 T^{4} - 2509056 T^{5} + 812032 T^{6} - 106368 T^{7} + 7436 T^{8} - 352 T^{9} + 128 T^{10} - 16 T^{11} + T^{12}$$
$59$ $$56712564736 + 33926946816 T + 10147995648 T^{2} + 963770368 T^{3} + 71756800 T^{4} + 14528000 T^{5} + 4040192 T^{6} + 403136 T^{7} + 21380 T^{8} + 712 T^{9} + 200 T^{10} + 20 T^{11} + T^{12}$$
$61$ $$473344 + 374272 T + 147968 T^{2} - 236544 T^{3} + 628544 T^{4} + 249344 T^{5} + 59776 T^{6} - 33248 T^{7} + 9476 T^{8} + 72 T^{9} + 8 T^{10} - 4 T^{11} + T^{12}$$
$67$ $$38626225 + 6152850 T + 490050 T^{2} + 17786890 T^{3} + 37824659 T^{4} + 22084652 T^{5} + 7133348 T^{6} + 1460572 T^{7} + 203043 T^{8} + 19386 T^{9} + 1250 T^{10} + 50 T^{11} + T^{12}$$
$71$ $$95257600 + 132730880 T^{2} + 18877456 T^{4} + 1009152 T^{6} + 24008 T^{8} + 256 T^{10} + T^{12}$$
$73$ $$( -13879 - 24748 T - 11705 T^{2} - 1624 T^{3} + 31 T^{4} + 20 T^{5} + T^{6} )^{2}$$
$79$ $$( 1250320 + 571120 T + 45632 T^{2} - 3976 T^{3} - 450 T^{4} + 6 T^{5} + T^{6} )^{2}$$
$83$ $$4583881 - 12601926 T + 17322498 T^{2} - 13230166 T^{3} + 5853795 T^{4} - 1106836 T^{5} + 14084 T^{6} - 13044 T^{7} + 40923 T^{8} - 494 T^{9} + 2 T^{10} + 2 T^{11} + T^{12}$$
$89$ $$2165692369 + 2006433682 T^{2} + 240821727 T^{4} + 7595612 T^{6} + 96847 T^{8} + 530 T^{10} + T^{12}$$
$97$ $$1406550016 + 2955411456 T^{2} + 319368192 T^{4} + 10398464 T^{6} + 133648 T^{8} + 648 T^{10} + T^{12}$$