# Properties

 Label 126.2.m.a Level $126$ Weight $2$ Character orbit 126.m Analytic conductor $1.006$ Analytic rank $0$ Dimension $16$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$126 = 2 \cdot 3^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 126.m (of order $$6$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$1.00611506547$$ Analytic rank: $$0$$ Dimension: $$16$$ Relative dimension: $$8$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\mathbb{Q}[x]/(x^{16} - \cdots)$$ Defining polynomial: $$x^{16} - 6 x^{14} + 9 x^{12} + 54 x^{10} - 288 x^{8} + 486 x^{6} + 729 x^{4} - 4374 x^{2} + 6561$$ 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 $$1,\beta_1,\ldots,\beta_{15}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{10} q^{2} -\beta_{1} q^{3} + \beta_{5} q^{4} + ( \beta_{1} - \beta_{6} + \beta_{8} - \beta_{11} - \beta_{13} + \beta_{14} ) q^{5} + ( \beta_{12} - \beta_{13} ) q^{6} + ( -\beta_{3} + \beta_{14} ) q^{7} + \beta_{2} q^{8} + ( 1 + \beta_{2} + \beta_{3} + \beta_{4} - \beta_{5} - \beta_{10} ) q^{9} +O(q^{10})$$ $$q + \beta_{10} q^{2} -\beta_{1} q^{3} + \beta_{5} q^{4} + ( \beta_{1} - \beta_{6} + \beta_{8} - \beta_{11} - \beta_{13} + \beta_{14} ) q^{5} + ( \beta_{12} - \beta_{13} ) q^{6} + ( -\beta_{3} + \beta_{14} ) q^{7} + \beta_{2} q^{8} + ( 1 + \beta_{2} + \beta_{3} + \beta_{4} - \beta_{5} - \beta_{10} ) q^{9} + ( -\beta_{1} - \beta_{8} + \beta_{11} - \beta_{12} + \beta_{13} - \beta_{14} ) q^{10} + ( -1 - \beta_{1} - \beta_{3} + \beta_{4} - \beta_{5} - \beta_{8} + \beta_{10} - \beta_{11} + \beta_{12} - 2 \beta_{13} + 2 \beta_{14} - 2 \beta_{15} ) q^{11} + ( -\beta_{1} - \beta_{9} ) q^{12} + ( -\beta_{8} - \beta_{9} - \beta_{11} - \beta_{12} + \beta_{13} ) q^{13} + ( -\beta_{4} + \beta_{5} - \beta_{6} - \beta_{7} - \beta_{12} + \beta_{13} - \beta_{14} ) q^{14} + ( 1 + \beta_{1} + \beta_{2} - \beta_{4} - 2 \beta_{5} + \beta_{7} + \beta_{8} - 2 \beta_{10} + \beta_{11} + \beta_{13} - \beta_{14} + \beta_{15} ) q^{15} + ( -1 + \beta_{5} ) q^{16} + ( \beta_{6} + \beta_{9} - \beta_{13} - \beta_{14} ) q^{17} + ( -1 + \beta_{1} - \beta_{2} + \beta_{4} - \beta_{5} + \beta_{6} + \beta_{7} + \beta_{9} + \beta_{10} + \beta_{11} + \beta_{12} + \beta_{15} ) q^{18} + ( \beta_{1} + \beta_{8} + \beta_{13} - \beta_{14} ) q^{19} + ( \beta_{1} + \beta_{8} + \beta_{9} + \beta_{12} - \beta_{13} + \beta_{14} ) q^{20} + ( 2 - \beta_{4} + \beta_{6} - \beta_{7} - 2 \beta_{10} - 2 \beta_{11} + \beta_{14} ) q^{21} + ( \beta_{1} - \beta_{2} - \beta_{4} + \beta_{7} + \beta_{8} - \beta_{10} + \beta_{11} + \beta_{13} - \beta_{14} + \beta_{15} ) q^{22} + ( -4 + \beta_{1} - \beta_{2} + 2 \beta_{3} + \beta_{4} + \beta_{5} + \beta_{7} + \beta_{8} + \beta_{10} + \beta_{11} + \beta_{13} - \beta_{14} + \beta_{15} ) q^{23} + \beta_{12} q^{24} + ( -\beta_{1} - 4 \beta_{2} + \beta_{3} - \beta_{4} + 2 \beta_{5} - \beta_{6} - 2 \beta_{7} - \beta_{9} + 2 \beta_{10} - \beta_{11} - 2 \beta_{12} + \beta_{13} - \beta_{14} ) q^{25} + ( \beta_{1} + \beta_{6} - \beta_{8} + \beta_{9} + \beta_{13} + \beta_{14} ) q^{26} + ( -\beta_{1} + \beta_{9} + 3 \beta_{11} - \beta_{12} + 2 \beta_{13} - 3 \beta_{14} ) q^{27} + ( -\beta_{3} - \beta_{13} + \beta_{14} - \beta_{15} ) q^{28} + ( -2 \beta_{4} - \beta_{6} + \beta_{8} - \beta_{9} - 2 \beta_{10} - \beta_{12} + \beta_{13} - \beta_{14} ) q^{29} + ( -1 - \beta_{1} - \beta_{2} + \beta_{3} - \beta_{6} - \beta_{7} - \beta_{9} + \beta_{10} - \beta_{11} - \beta_{12} - \beta_{15} ) q^{30} + ( \beta_{1} + \beta_{6} - 2 \beta_{8} - \beta_{9} + \beta_{11} + 2 \beta_{12} - \beta_{13} + \beta_{14} ) q^{31} + ( \beta_{2} - \beta_{10} ) q^{32} + ( \beta_{1} + 3 \beta_{6} - 3 \beta_{8} + 2 \beta_{9} + 3 \beta_{11} + \beta_{12} + \beta_{13} ) q^{33} + ( -\beta_{1} + \beta_{6} - \beta_{11} - \beta_{13} ) q^{34} + ( -1 + \beta_{1} - \beta_{2} - \beta_{3} + \beta_{5} + \beta_{7} + \beta_{11} + \beta_{12} - 2 \beta_{13} - 2 \beta_{14} + \beta_{15} ) q^{35} + ( 1 + \beta_{3} + \beta_{5} - \beta_{7} - \beta_{10} - \beta_{12} + \beta_{13} - \beta_{14} + \beta_{15} ) q^{36} + ( -2 + \beta_{1} - 2 \beta_{2} + 4 \beta_{4} - 2 \beta_{5} + 2 \beta_{6} + 2 \beta_{7} - \beta_{8} + 2 \beta_{9} + 4 \beta_{10} + \beta_{11} + 3 \beta_{12} - 2 \beta_{13} + 2 \beta_{14} ) q^{37} + ( \beta_{1} + \beta_{6} - \beta_{12} + \beta_{13} - \beta_{14} ) q^{38} + ( -2 - \beta_{1} + 3 \beta_{2} - 2 \beta_{4} + 3 \beta_{5} - \beta_{6} - 2 \beta_{7} - \beta_{9} - \beta_{10} - \beta_{11} - 2 \beta_{12} + \beta_{13} - \beta_{14} ) q^{39} + ( -\beta_{1} - \beta_{6} - \beta_{9} - \beta_{12} - \beta_{14} ) q^{40} + ( -\beta_{1} + \beta_{6} - \beta_{8} + \beta_{11} - 3 \beta_{12} + \beta_{13} + 2 \beta_{14} ) q^{41} + ( -\beta_{1} - \beta_{2} - \beta_{3} - 2 \beta_{5} - 2 \beta_{8} - \beta_{9} + 2 \beta_{10} - 2 \beta_{11} - \beta_{13} + \beta_{14} - \beta_{15} ) q^{42} + ( 1 - \beta_{1} + 2 \beta_{2} - \beta_{3} - \beta_{4} + \beta_{5} - 2 \beta_{7} - \beta_{8} - \beta_{10} - \beta_{11} - \beta_{12} ) q^{43} + ( 1 - \beta_{1} + \beta_{2} + \beta_{3} - \beta_{5} - \beta_{6} - \beta_{7} - \beta_{9} - \beta_{11} - \beta_{12} - \beta_{15} ) q^{44} + ( 2 \beta_{1} - 3 \beta_{6} + 3 \beta_{8} + \beta_{9} - 3 \beta_{11} + 3 \beta_{14} ) q^{45} + ( 1 + \beta_{1} + 2 \beta_{2} + \beta_{3} + 2 \beta_{4} - \beta_{5} + \beta_{6} + \beta_{7} + \beta_{9} - 4 \beta_{10} + \beta_{11} + \beta_{12} + \beta_{15} ) q^{46} + ( -4 \beta_{1} - 3 \beta_{6} - \beta_{8} - \beta_{9} - \beta_{12} + \beta_{13} - \beta_{14} ) q^{47} -\beta_{9} q^{48} + ( \beta_{1} - \beta_{2} - \beta_{4} - \beta_{5} + \beta_{7} + 2 \beta_{8} + \beta_{9} - \beta_{10} + 3 \beta_{11} + 3 \beta_{12} - \beta_{13} + \beta_{15} ) q^{49} + ( 4 - 2 \beta_{3} + \beta_{4} - 3 \beta_{5} + \beta_{6} + \beta_{7} - \beta_{8} + \beta_{9} + 2 \beta_{12} - 2 \beta_{13} + 2 \beta_{14} - \beta_{15} ) q^{50} + ( -1 + \beta_{1} + \beta_{3} + \beta_{4} + \beta_{5} + 2 \beta_{7} + \beta_{8} + 3 \beta_{10} + \beta_{11} + \beta_{12} ) q^{51} + ( \beta_{1} - \beta_{6} - \beta_{11} - \beta_{12} + \beta_{14} ) q^{52} + ( 2 \beta_{1} + 3 \beta_{8} + \beta_{9} + \beta_{12} - 2 \beta_{13} ) q^{54} + ( \beta_{1} - 2 \beta_{6} + \beta_{8} + 2 \beta_{9} - 3 \beta_{11} + 3 \beta_{12} ) q^{55} + ( -\beta_{4} - \beta_{6} + \beta_{8} - \beta_{12} + \beta_{13} - \beta_{14} ) q^{56} + ( 3 - \beta_{1} - 2 \beta_{3} - \beta_{4} + \beta_{5} - \beta_{7} - \beta_{8} + 4 \beta_{10} - \beta_{11} - \beta_{13} + \beta_{14} - \beta_{15} ) q^{57} + ( -\beta_{1} - 2 \beta_{5} - \beta_{6} - \beta_{9} - \beta_{11} - \beta_{13} + \beta_{14} - 2 \beta_{15} ) q^{58} + ( -\beta_{1} + \beta_{6} + 2 \beta_{8} - 3 \beta_{9} + 2 \beta_{12} - 2 \beta_{14} ) q^{59} + ( 1 + \beta_{1} - \beta_{2} - \beta_{3} + \beta_{4} - 2 \beta_{5} + \beta_{6} + 2 \beta_{7} + \beta_{9} - \beta_{10} + \beta_{11} + 2 \beta_{12} - \beta_{13} + \beta_{14} ) q^{60} + ( -3 \beta_{1} + \beta_{6} + 2 \beta_{8} - 2 \beta_{9} - 2 \beta_{11} - 2 \beta_{13} ) q^{61} + ( -\beta_{1} - 2 \beta_{6} + \beta_{8} - 2 \beta_{9} - \beta_{11} - \beta_{12} + 2 \beta_{13} + 2 \beta_{14} ) q^{62} + ( -1 - \beta_{1} + 4 \beta_{2} - \beta_{3} + \beta_{4} + 3 \beta_{5} + \beta_{6} + \beta_{7} - 2 \beta_{8} + \beta_{9} - 3 \beta_{10} - \beta_{11} - \beta_{12} + \beta_{13} + \beta_{14} ) q^{63} - q^{64} + ( 3 + 2 \beta_{4} + 3 \beta_{5} + \beta_{6} - \beta_{8} + \beta_{9} - \beta_{10} + \beta_{12} - \beta_{13} + \beta_{14} ) q^{65} + ( \beta_{1} - 3 \beta_{6} + 3 \beta_{8} - \beta_{9} - 3 \beta_{11} - \beta_{12} - \beta_{13} + 3 \beta_{14} ) q^{66} + ( 5 \beta_{2} - \beta_{4} - 4 \beta_{5} - \beta_{6} + \beta_{7} + \beta_{8} - \beta_{9} + 5 \beta_{10} - \beta_{15} ) q^{67} + ( -\beta_{1} + \beta_{6} - \beta_{8} - \beta_{11} + \beta_{12} - \beta_{13} ) q^{68} + ( 2 \beta_{1} - 3 \beta_{6} + 3 \beta_{8} - 2 \beta_{9} + 3 \beta_{11} - \beta_{12} - \beta_{13} - 3 \beta_{14} ) q^{69} + ( 1 - 3 \beta_{1} + 2 \beta_{2} + \beta_{3} - \beta_{4} + \beta_{5} + \beta_{6} - 2 \beta_{7} - \beta_{9} - \beta_{10} - 2 \beta_{12} + 2 \beta_{13} - \beta_{14} ) q^{70} + ( -1 - 2 \beta_{2} - \beta_{3} + 3 \beta_{5} + \beta_{6} - \beta_{7} - \beta_{8} + \beta_{9} + \beta_{15} ) q^{71} + ( -\beta_{3} + \beta_{4} - \beta_{5} + \beta_{6} - \beta_{8} + \beta_{9} + \beta_{10} + \beta_{12} - \beta_{13} + \beta_{14} ) q^{72} + ( -\beta_{6} + \beta_{9} + 2 \beta_{11} - 2 \beta_{12} - \beta_{13} + \beta_{14} ) q^{73} + ( 2 + 2 \beta_{1} + 2 \beta_{3} + 2 \beta_{5} + \beta_{6} + \beta_{8} + \beta_{9} - 2 \beta_{10} + 2 \beta_{11} - \beta_{12} + 3 \beta_{13} - 3 \beta_{14} + 4 \beta_{15} ) q^{74} + ( -2 \beta_{1} - 3 \beta_{8} - 2 \beta_{9} - 4 \beta_{12} - \beta_{13} - 3 \beta_{14} ) q^{75} + ( \beta_{1} + \beta_{6} + \beta_{9} - \beta_{11} - \beta_{12} + \beta_{13} ) q^{76} + ( 6 - \beta_{1} + 2 \beta_{2} + \beta_{4} - 4 \beta_{5} + 2 \beta_{6} + \beta_{7} - 3 \beta_{8} + 2 \beta_{9} - 2 \beta_{10} + 3 \beta_{11} + 2 \beta_{12} + 2 \beta_{13} - 3 \beta_{14} ) q^{77} + ( -3 - \beta_{1} + \beta_{2} - 2 \beta_{3} + 2 \beta_{5} - \beta_{8} - 2 \beta_{10} - \beta_{11} + \beta_{12} - 2 \beta_{13} + 2 \beta_{14} - 2 \beta_{15} ) q^{78} + ( 1 - \beta_{1} + 8 \beta_{2} + 3 \beta_{3} - \beta_{4} + \beta_{5} - 2 \beta_{6} - 2 \beta_{7} + \beta_{8} - 2 \beta_{9} - 4 \beta_{10} - \beta_{11} - 3 \beta_{12} + 2 \beta_{13} - 2 \beta_{14} ) q^{79} + ( \beta_{6} + \beta_{9} + \beta_{11} + \beta_{12} ) q^{80} + ( 3 - 2 \beta_{1} - 5 \beta_{2} + \beta_{4} - 2 \beta_{5} - \beta_{6} - \beta_{7} - \beta_{8} - \beta_{9} + 7 \beta_{10} - 2 \beta_{11} - 2 \beta_{13} + 2 \beta_{14} - 3 \beta_{15} ) q^{81} + ( \beta_{1} - 3 \beta_{6} + \beta_{8} + 3 \beta_{9} - \beta_{11} + \beta_{12} - \beta_{13} + \beta_{14} ) q^{82} + ( \beta_{1} + 3 \beta_{6} - 2 \beta_{8} - 2 \beta_{9} - 3 \beta_{11} + 7 \beta_{12} - 4 \beta_{13} + 7 \beta_{14} ) q^{83} + ( 1 - 2 \beta_{2} - \beta_{4} + \beta_{5} + \beta_{6} - \beta_{8} + \beta_{13} + \beta_{14} ) q^{84} + ( -\beta_{1} - 2 \beta_{5} - \beta_{6} - \beta_{9} - \beta_{11} - \beta_{13} + \beta_{14} - 2 \beta_{15} ) q^{85} + ( -2 - \beta_{1} - \beta_{2} - 2 \beta_{3} - \beta_{4} + 2 \beta_{5} - \beta_{7} - \beta_{8} + \beta_{10} - \beta_{11} - \beta_{13} + \beta_{14} - \beta_{15} ) q^{86} + ( \beta_{1} - 3 \beta_{6} - \beta_{9} - 3 \beta_{11} - 2 \beta_{12} + \beta_{13} + 3 \beta_{14} ) q^{87} + ( -1 + \beta_{1} - 2 \beta_{2} - \beta_{3} + \beta_{4} - \beta_{5} + \beta_{6} + 2 \beta_{7} + \beta_{9} + \beta_{10} + \beta_{11} + 2 \beta_{12} - \beta_{13} + \beta_{14} ) q^{88} + ( 2 \beta_{11} + 2 \beta_{12} + 2 \beta_{13} + 2 \beta_{14} ) q^{89} + ( -3 \beta_{8} + 3 \beta_{11} - 2 \beta_{12} + \beta_{13} - 3 \beta_{14} ) q^{90} + ( 2 + \beta_{1} + 2 \beta_{3} - \beta_{6} + \beta_{8} + \beta_{9} + 4 \beta_{11} - 4 \beta_{12} + 3 \beta_{13} - 3 \beta_{14} + 2 \beta_{15} ) q^{91} + ( -2 + \beta_{1} + \beta_{3} + \beta_{4} - 2 \beta_{5} + \beta_{6} + \beta_{9} + \beta_{10} + \beta_{11} + \beta_{13} - \beta_{14} + 2 \beta_{15} ) q^{92} + ( -8 + \beta_{1} - 6 \beta_{2} - 2 \beta_{3} + 4 \beta_{5} + 2 \beta_{6} - \beta_{8} + 2 \beta_{9} + \beta_{11} + \beta_{12} + 2 \beta_{15} ) q^{93} + ( \beta_{1} + \beta_{6} + \beta_{9} + 3 \beta_{11} + 4 \beta_{12} - 3 \beta_{13} + \beta_{14} ) q^{94} + ( 2 - 2 \beta_{2} + 2 \beta_{3} - \beta_{4} - \beta_{6} - \beta_{7} + \beta_{8} - \beta_{9} + 2 \beta_{10} - 2 \beta_{12} + 2 \beta_{13} - 2 \beta_{14} + \beta_{15} ) q^{95} + \beta_{13} q^{96} + ( \beta_{1} - \beta_{8} + \beta_{9} - 5 \beta_{11} + \beta_{12} - 6 \beta_{13} - \beta_{14} ) q^{97} + ( 1 - 3 \beta_{1} + \beta_{3} - \beta_{5} - 4 \beta_{6} - \beta_{7} + 2 \beta_{8} - 4 \beta_{9} - \beta_{11} - \beta_{12} - \beta_{13} - \beta_{14} - \beta_{15} ) q^{98} + ( -9 - \beta_{1} - 7 \beta_{2} - 3 \beta_{3} + 2 \beta_{4} + 5 \beta_{5} + \beta_{6} + \beta_{7} - 2 \beta_{8} + \beta_{9} + 2 \beta_{10} - \beta_{11} + 3 \beta_{12} - 4 \beta_{13} + 4 \beta_{14} - 3 \beta_{15} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$16 q + 8 q^{4} + 2 q^{7} + 12 q^{9} + O(q^{10})$$ $$16 q + 8 q^{4} + 2 q^{7} + 12 q^{9} - 12 q^{11} - 6 q^{14} - 8 q^{16} - 12 q^{18} + 18 q^{21} - 48 q^{23} - 8 q^{25} + 4 q^{28} - 12 q^{29} - 24 q^{30} + 12 q^{36} - 8 q^{37} - 36 q^{39} - 12 q^{42} + 4 q^{43} + 24 q^{46} - 8 q^{49} + 60 q^{50} + 12 q^{51} - 6 q^{56} + 48 q^{57} - 12 q^{58} + 24 q^{60} + 24 q^{63} - 16 q^{64} + 84 q^{65} - 28 q^{67} + 36 q^{74} + 78 q^{77} - 24 q^{78} - 4 q^{79} + 36 q^{81} + 18 q^{84} - 12 q^{85} - 24 q^{86} + 24 q^{91} - 48 q^{92} - 96 q^{93} + 12 q^{95} - 72 q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{16} - 6 x^{14} + 9 x^{12} + 54 x^{10} - 288 x^{8} + 486 x^{6} + 729 x^{4} - 4374 x^{2} + 6561$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$-\nu^{14} + 3 \nu^{12} + 9 \nu^{10} - 81 \nu^{8} + 126 \nu^{6} + 135 \nu^{4} - 1458 \nu^{2} + 2187$$$$)/1458$$ $$\beta_{3}$$ $$=$$ $$($$$$-2 \nu^{15} - 3 \nu^{14} - 15 \nu^{13} - 36 \nu^{12} + 36 \nu^{11} + 216 \nu^{10} + 54 \nu^{9} - 162 \nu^{8} - 396 \nu^{7} - 594 \nu^{6} - 486 \nu^{5} + 5346 \nu^{4} + 3402 \nu^{3} - 729 \nu^{2} - 10935 \nu - 8748$$$$)/17496$$ $$\beta_{4}$$ $$=$$ $$($$$$2 \nu^{15} - 6 \nu^{14} + 15 \nu^{13} + 27 \nu^{12} - 36 \nu^{11} - 162 \nu^{10} - 54 \nu^{9} - 162 \nu^{8} + 396 \nu^{7} + 1242 \nu^{6} + 486 \nu^{5} - 3240 \nu^{4} - 3402 \nu^{3} + 7290 \nu^{2} + 10935 \nu + 6561$$$$)/17496$$ $$\beta_{5}$$ $$=$$ $$($$$$2 \nu^{14} - 21 \nu^{12} + 18 \nu^{10} + 108 \nu^{8} - 576 \nu^{6} + 648 \nu^{4} + 972 \nu^{2} - 3645$$$$)/5832$$ $$\beta_{6}$$ $$=$$ $$($$$$-5 \nu^{15} + 12 \nu^{13} + 144 \nu^{11} - 432 \nu^{9} + 468 \nu^{7} + 2754 \nu^{5} - 9477 \nu^{3} + 13122 \nu$$$$)/17496$$ $$\beta_{7}$$ $$=$$ $$($$$$-4 \nu^{15} + 6 \nu^{14} + 24 \nu^{13} + 9 \nu^{12} + 18 \nu^{11} + 108 \nu^{10} - 540 \nu^{9} - 486 \nu^{8} + 1638 \nu^{7} + 2160 \nu^{6} - 486 \nu^{5} + 162 \nu^{4} - 9720 \nu^{3} - 18954 \nu^{2} + 21870 \nu + 67797$$$$)/17496$$ $$\beta_{8}$$ $$=$$ $$($$$$\nu^{15} - 6 \nu^{13} + 9 \nu^{11} + 54 \nu^{9} - 288 \nu^{7} + 486 \nu^{5} + 729 \nu^{3} - 4374 \nu$$$$)/2187$$ $$\beta_{9}$$ $$=$$ $$($$$$2 \nu^{15} - 21 \nu^{13} + 18 \nu^{11} + 108 \nu^{9} - 576 \nu^{7} + 648 \nu^{5} + 972 \nu^{3} - 9477 \nu$$$$)/5832$$ $$\beta_{10}$$ $$=$$ $$($$$$-3 \nu^{14} + 10 \nu^{12} + 12 \nu^{10} - 180 \nu^{8} + 432 \nu^{6} + 198 \nu^{4} - 3483 \nu^{2} + 5832$$$$)/1944$$ $$\beta_{11}$$ $$=$$ $$($$$$-\nu^{15} + 3 \nu^{13} - 27 \nu^{9} + 45 \nu^{7} - 108 \nu^{5} - 324 \nu^{3}$$$$)/1458$$ $$\beta_{12}$$ $$=$$ $$($$$$\nu^{15} - 3 \nu^{13} - 9 \nu^{11} + 81 \nu^{9} - 126 \nu^{7} - 135 \nu^{5} + 1458 \nu^{3} - 2187 \nu$$$$)/1458$$ $$\beta_{13}$$ $$=$$ $$($$$$-5 \nu^{15} + 18 \nu^{13} - 216 \nu^{9} + 792 \nu^{7} + 54 \nu^{5} - 4617 \nu^{3} + 8748 \nu$$$$)/5832$$ $$\beta_{14}$$ $$=$$ $$($$$$-8 \nu^{15} + 21 \nu^{13} + 18 \nu^{11} - 324 \nu^{9} + 684 \nu^{7} - 4050 \nu^{3} + 3645 \nu$$$$)/5832$$ $$\beta_{15}$$ $$=$$ $$($$$$\nu^{15} - 51 \nu^{14} + 12 \nu^{13} + 198 \nu^{12} - 72 \nu^{11} - 54 \nu^{10} + 54 \nu^{9} - 2268 \nu^{8} + 198 \nu^{7} + 7398 \nu^{6} - 1782 \nu^{5} - 6804 \nu^{4} + 6075 \nu^{3} - 37179 \nu^{2} - 8748 \nu + 83106$$$$)/17496$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$-\beta_{10} - \beta_{5} + \beta_{4} + \beta_{3} + \beta_{2} + 1$$ $$\nu^{3}$$ $$=$$ $$3 \beta_{14} - 2 \beta_{13} + \beta_{12} - 3 \beta_{11} - \beta_{9} + \beta_{1}$$ $$\nu^{4}$$ $$=$$ $$-3 \beta_{15} + 2 \beta_{14} - 2 \beta_{13} - 2 \beta_{11} + 7 \beta_{10} - \beta_{9} - \beta_{8} - \beta_{7} - \beta_{6} - 2 \beta_{5} + \beta_{4} - 5 \beta_{2} - 2 \beta_{1} + 3$$ $$\nu^{5}$$ $$=$$ $$3 \beta_{14} + 3 \beta_{13} - 6 \beta_{11} - 3 \beta_{9} + 6 \beta_{8} - 3 \beta_{6} + 3 \beta_{1}$$ $$\nu^{6}$$ $$=$$ $$3 \beta_{12} + 3 \beta_{11} + 6 \beta_{10} + 3 \beta_{8} + 6 \beta_{7} + 6 \beta_{4} + 6 \beta_{3} - 15 \beta_{2} + 3 \beta_{1} - 18$$ $$\nu^{7}$$ $$=$$ $$9 \beta_{14} + 3 \beta_{13} + 12 \beta_{12} - 18 \beta_{11} + 3 \beta_{9} - 9 \beta_{8} + 9 \beta_{6} - 12 \beta_{1}$$ $$\nu^{8}$$ $$=$$ $$6 \beta_{15} - 18 \beta_{14} + 18 \beta_{13} - 15 \beta_{12} + 3 \beta_{11} + 27 \beta_{10} - 12 \beta_{9} + 15 \beta_{8} - 12 \beta_{6} + 3 \beta_{5} - 27 \beta_{4} + 3 \beta_{3} - 72 \beta_{2} + 3 \beta_{1} + 12$$ $$\nu^{9}$$ $$=$$ $$-36 \beta_{14} + 45 \beta_{13} + 45 \beta_{12} + 45 \beta_{11} + 18 \beta_{9} - 9 \beta_{8} + 45 \beta_{6}$$ $$\nu^{10}$$ $$=$$ $$63 \beta_{15} - 108 \beta_{14} + 108 \beta_{13} - 63 \beta_{12} + 45 \beta_{11} - 153 \beta_{10} - 45 \beta_{9} + 90 \beta_{8} + 27 \beta_{7} - 45 \beta_{6} - 54 \beta_{5} - 63 \beta_{4} + 90 \beta_{3} + 72 \beta_{2} + 45 \beta_{1} - 18$$ $$\nu^{11}$$ $$=$$ $$-90 \beta_{13} + 126 \beta_{12} + 54 \beta_{11} + 36 \beta_{9} - 135 \beta_{8} + 270 \beta_{6} - 90 \beta_{1}$$ $$\nu^{12}$$ $$=$$ $$-54 \beta_{15} - 72 \beta_{14} + 72 \beta_{13} - 162 \beta_{12} - 90 \beta_{11} - 36 \beta_{10} - 126 \beta_{9} + 36 \beta_{8} - 126 \beta_{7} - 126 \beta_{6} - 522 \beta_{5} - 252 \beta_{4} - 54 \beta_{3} + 126 \beta_{2} - 90 \beta_{1} + 405$$ $$\nu^{13}$$ $$=$$ $$-270 \beta_{14} - 54 \beta_{12} + 378 \beta_{11} - 486 \beta_{9} + 270 \beta_{8} + 108 \beta_{6} - 243 \beta_{1}$$ $$\nu^{14}$$ $$=$$ $$-486 \beta_{15} + 540 \beta_{14} - 540 \beta_{13} + 540 \beta_{12} - 513 \beta_{10} + 54 \beta_{9} - 54 \beta_{8} + 486 \beta_{7} + 54 \beta_{6} - 1107 \beta_{5} + 297 \beta_{4} - 297 \beta_{3} + 1377 \beta_{2} - 1053$$ $$\nu^{15}$$ $$=$$ $$-729 \beta_{14} - 756 \beta_{13} - 1161 \beta_{12} - 729 \beta_{11} - 1161 \beta_{9} - 162 \beta_{6} - 1917 \beta_{1}$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/126\mathbb{Z}\right)^\times$$.

 $$n$$ $$29$$ $$73$$ $$\chi(n)$$ $$\beta_{5}$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
41.1
 1.62181 − 0.608059i 1.40917 + 1.00709i −1.40917 − 1.00709i −1.62181 + 0.608059i 1.69547 + 0.354107i 0.0967785 − 1.72934i −0.0967785 + 1.72934i −1.69547 − 0.354107i 1.62181 + 0.608059i 1.40917 − 1.00709i −1.40917 + 1.00709i −1.62181 − 0.608059i 1.69547 − 0.354107i 0.0967785 + 1.72934i −0.0967785 − 1.72934i −1.69547 + 0.354107i
−0.866025 + 0.500000i −1.62181 + 0.608059i 0.500000 0.866025i 1.94556 3.36980i 1.10050 1.33750i 0.343982 + 2.62329i 1.00000i 2.26053 1.97231i 3.89111i
41.2 −0.866025 + 0.500000i −1.40917 1.00709i 0.500000 0.866025i −1.17468 + 2.03460i 1.72392 + 0.167584i −2.63145 + 0.274725i 1.00000i 0.971521 + 2.83834i 2.34936i
41.3 −0.866025 + 0.500000i 1.40917 + 1.00709i 0.500000 0.866025i 1.17468 2.03460i −1.72392 0.167584i 1.55364 2.14154i 1.00000i 0.971521 + 2.83834i 2.34936i
41.4 −0.866025 + 0.500000i 1.62181 0.608059i 0.500000 0.866025i −1.94556 + 3.36980i −1.10050 + 1.33750i 2.09985 + 1.60954i 1.00000i 2.26053 1.97231i 3.89111i
41.5 0.866025 0.500000i −1.69547 0.354107i 0.500000 0.866025i 0.895175 1.55049i −1.64537 + 0.541068i 0.0213944 2.64566i 1.00000i 2.74922 + 1.20075i 1.79035i
41.6 0.866025 0.500000i −0.0967785 + 1.72934i 0.500000 0.866025i −0.183299 + 0.317483i 0.780860 + 1.54605i 2.53871 + 0.744936i 1.00000i −2.98127 0.334727i 0.366598i
41.7 0.866025 0.500000i 0.0967785 1.72934i 0.500000 0.866025i 0.183299 0.317483i −0.780860 1.54605i −0.624224 + 2.57106i 1.00000i −2.98127 0.334727i 0.366598i
41.8 0.866025 0.500000i 1.69547 + 0.354107i 0.500000 0.866025i −0.895175 + 1.55049i 1.64537 0.541068i −2.30191 1.30430i 1.00000i 2.74922 + 1.20075i 1.79035i
83.1 −0.866025 0.500000i −1.62181 0.608059i 0.500000 + 0.866025i 1.94556 + 3.36980i 1.10050 + 1.33750i 0.343982 2.62329i 1.00000i 2.26053 + 1.97231i 3.89111i
83.2 −0.866025 0.500000i −1.40917 + 1.00709i 0.500000 + 0.866025i −1.17468 2.03460i 1.72392 0.167584i −2.63145 0.274725i 1.00000i 0.971521 2.83834i 2.34936i
83.3 −0.866025 0.500000i 1.40917 1.00709i 0.500000 + 0.866025i 1.17468 + 2.03460i −1.72392 + 0.167584i 1.55364 + 2.14154i 1.00000i 0.971521 2.83834i 2.34936i
83.4 −0.866025 0.500000i 1.62181 + 0.608059i 0.500000 + 0.866025i −1.94556 3.36980i −1.10050 1.33750i 2.09985 1.60954i 1.00000i 2.26053 + 1.97231i 3.89111i
83.5 0.866025 + 0.500000i −1.69547 + 0.354107i 0.500000 + 0.866025i 0.895175 + 1.55049i −1.64537 0.541068i 0.0213944 + 2.64566i 1.00000i 2.74922 1.20075i 1.79035i
83.6 0.866025 + 0.500000i −0.0967785 1.72934i 0.500000 + 0.866025i −0.183299 0.317483i 0.780860 1.54605i 2.53871 0.744936i 1.00000i −2.98127 + 0.334727i 0.366598i
83.7 0.866025 + 0.500000i 0.0967785 + 1.72934i 0.500000 + 0.866025i 0.183299 + 0.317483i −0.780860 + 1.54605i −0.624224 2.57106i 1.00000i −2.98127 + 0.334727i 0.366598i
83.8 0.866025 + 0.500000i 1.69547 0.354107i 0.500000 + 0.866025i −0.895175 1.55049i 1.64537 + 0.541068i −2.30191 + 1.30430i 1.00000i 2.74922 1.20075i 1.79035i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 83.8 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 inner
9.d odd 6 1 inner
63.o even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 126.2.m.a 16
3.b odd 2 1 378.2.m.a 16
4.b odd 2 1 1008.2.cc.b 16
7.b odd 2 1 inner 126.2.m.a 16
7.c even 3 1 882.2.l.a 16
7.c even 3 1 882.2.t.b 16
7.d odd 6 1 882.2.l.a 16
7.d odd 6 1 882.2.t.b 16
9.c even 3 1 378.2.m.a 16
9.c even 3 1 1134.2.d.a 16
9.d odd 6 1 inner 126.2.m.a 16
9.d odd 6 1 1134.2.d.a 16
12.b even 2 1 3024.2.cc.b 16
21.c even 2 1 378.2.m.a 16
21.g even 6 1 2646.2.l.b 16
21.g even 6 1 2646.2.t.a 16
21.h odd 6 1 2646.2.l.b 16
21.h odd 6 1 2646.2.t.a 16
28.d even 2 1 1008.2.cc.b 16
36.f odd 6 1 3024.2.cc.b 16
36.h even 6 1 1008.2.cc.b 16
63.g even 3 1 2646.2.l.b 16
63.h even 3 1 2646.2.t.a 16
63.i even 6 1 882.2.t.b 16
63.j odd 6 1 882.2.t.b 16
63.k odd 6 1 2646.2.l.b 16
63.l odd 6 1 378.2.m.a 16
63.l odd 6 1 1134.2.d.a 16
63.n odd 6 1 882.2.l.a 16
63.o even 6 1 inner 126.2.m.a 16
63.o even 6 1 1134.2.d.a 16
63.s even 6 1 882.2.l.a 16
63.t odd 6 1 2646.2.t.a 16
84.h odd 2 1 3024.2.cc.b 16
252.s odd 6 1 1008.2.cc.b 16
252.bi even 6 1 3024.2.cc.b 16

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
126.2.m.a 16 1.a even 1 1 trivial
126.2.m.a 16 7.b odd 2 1 inner
126.2.m.a 16 9.d odd 6 1 inner
126.2.m.a 16 63.o even 6 1 inner
378.2.m.a 16 3.b odd 2 1
378.2.m.a 16 9.c even 3 1
378.2.m.a 16 21.c even 2 1
378.2.m.a 16 63.l odd 6 1
882.2.l.a 16 7.c even 3 1
882.2.l.a 16 7.d odd 6 1
882.2.l.a 16 63.n odd 6 1
882.2.l.a 16 63.s even 6 1
882.2.t.b 16 7.c even 3 1
882.2.t.b 16 7.d odd 6 1
882.2.t.b 16 63.i even 6 1
882.2.t.b 16 63.j odd 6 1
1008.2.cc.b 16 4.b odd 2 1
1008.2.cc.b 16 28.d even 2 1
1008.2.cc.b 16 36.h even 6 1
1008.2.cc.b 16 252.s odd 6 1
1134.2.d.a 16 9.c even 3 1
1134.2.d.a 16 9.d odd 6 1
1134.2.d.a 16 63.l odd 6 1
1134.2.d.a 16 63.o even 6 1
2646.2.l.b 16 21.g even 6 1
2646.2.l.b 16 21.h odd 6 1
2646.2.l.b 16 63.g even 3 1
2646.2.l.b 16 63.k odd 6 1
2646.2.t.a 16 21.g even 6 1
2646.2.t.a 16 21.h odd 6 1
2646.2.t.a 16 63.h even 3 1
2646.2.t.a 16 63.t odd 6 1
3024.2.cc.b 16 12.b even 2 1
3024.2.cc.b 16 36.f odd 6 1
3024.2.cc.b 16 84.h odd 2 1
3024.2.cc.b 16 252.bi even 6 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{2}^{\mathrm{new}}(126, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 - T^{2} + T^{4} )^{4}$$
$3$ $$6561 - 4374 T^{2} + 729 T^{4} + 486 T^{6} - 288 T^{8} + 54 T^{10} + 9 T^{12} - 6 T^{14} + T^{16}$$
$5$ $$1296 + 10368 T^{2} + 77436 T^{4} + 42336 T^{6} + 16461 T^{8} + 3096 T^{10} + 423 T^{12} + 24 T^{14} + T^{16}$$
$7$ $$5764801 - 1647086 T + 705894 T^{2} + 134456 T^{3} - 139258 T^{4} + 76146 T^{5} - 5096 T^{6} - 4634 T^{7} + 3483 T^{8} - 662 T^{9} - 104 T^{10} + 222 T^{11} - 58 T^{12} + 8 T^{13} + 6 T^{14} - 2 T^{15} + T^{16}$$
$11$ $$( 1296 - 2592 T + 972 T^{2} + 1512 T^{3} + 261 T^{4} - 126 T^{5} - 9 T^{6} + 6 T^{7} + T^{8} )^{2}$$
$13$ $$331776 - 497664 T^{2} + 559872 T^{4} - 238464 T^{6} + 73296 T^{8} - 9936 T^{10} + 972 T^{12} - 36 T^{14} + T^{16}$$
$17$ $$( 576 - 1296 T^{2} + 477 T^{4} - 42 T^{6} + T^{8} )^{2}$$
$19$ $$( 1521 + 1854 T^{2} + 594 T^{4} + 54 T^{6} + T^{8} )^{2}$$
$23$ $$( 443556 + 227772 T + 17010 T^{2} - 11286 T^{3} - 981 T^{4} + 792 T^{5} + 225 T^{6} + 24 T^{7} + T^{8} )^{2}$$
$29$ $$( 20736 - 10368 T - 2592 T^{2} + 2160 T^{3} + 612 T^{4} - 180 T^{5} - 18 T^{6} + 6 T^{7} + T^{8} )^{2}$$
$31$ $$557256278016 - 92876046336 T^{2} + 10400182272 T^{4} - 631535616 T^{6} + 27632016 T^{8} - 730944 T^{10} + 13932 T^{12} - 144 T^{14} + T^{16}$$
$37$ $$( 1336 - 184 T - 102 T^{2} + 2 T^{3} + T^{4} )^{4}$$
$41$ $$73499483897856 + 6883786653696 T^{2} + 448658922240 T^{4} + 13938763392 T^{6} + 307258425 T^{8} + 4294314 T^{10} + 43695 T^{12} + 258 T^{14} + T^{16}$$
$43$ $$( 10816 - 15392 T + 17848 T^{2} - 6188 T^{3} + 1921 T^{4} - 218 T^{5} + 43 T^{6} - 2 T^{7} + T^{8} )^{2}$$
$47$ $$1485512441856 + 399459631104 T^{2} + 87539678208 T^{4} + 4759817472 T^{6} + 186073488 T^{8} + 3258432 T^{10} + 41292 T^{12} + 240 T^{14} + T^{16}$$
$53$ $$T^{16}$$
$59$ $$1296 + 2901744 T^{2} + 6496369452 T^{4} + 1422558828 T^{6} + 287789589 T^{8} + 5027598 T^{10} + 68787 T^{12} + 294 T^{14} + T^{16}$$
$61$ $$2425818710016 - 545450360832 T^{2} + 96222587904 T^{4} - 5193676800 T^{6} + 202203801 T^{8} - 3371184 T^{10} + 40635 T^{12} - 240 T^{14} + T^{16}$$
$67$ $$( 824464 + 1507280 T + 2654812 T^{2} + 209684 T^{3} + 36469 T^{4} + 1766 T^{5} + 307 T^{6} + 14 T^{7} + T^{8} )^{2}$$
$71$ $$( 82944 + 31104 T^{2} + 2745 T^{4} + 90 T^{6} + T^{8} )^{2}$$
$73$ $$( 1710864 + 246816 T^{2} + 12069 T^{4} + 222 T^{6} + T^{8} )^{2}$$
$79$ $$( 1444804 + 935156 T + 450226 T^{2} + 105170 T^{3} + 19399 T^{4} + 1298 T^{5} + 133 T^{6} + 2 T^{7} + T^{8} )^{2}$$
$83$ $$337116351515590656 + 10214329542377472 T^{2} + 207879529033728 T^{4} + 2256409253376 T^{6} + 17587710864 T^{8} + 88712784 T^{10} + 326268 T^{12} + 708 T^{14} + T^{16}$$
$89$ $$( 186624 - 155520 T^{2} + 12960 T^{4} - 216 T^{6} + T^{8} )^{2}$$
$97$ $$4512402164941056 - 390697151362560 T^{2} + 24485300891568 T^{4} - 714581204256 T^{6} + 15192293193 T^{8} - 85999734 T^{10} + 353727 T^{12} - 702 T^{14} + T^{16}$$