# Properties

 Label 105.3.k.c Level 105 Weight 3 Character orbit 105.k Analytic conductor 2.861 Analytic rank 0 Dimension 16 CM no Inner twists 8

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$105 = 3 \cdot 5 \cdot 7$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 105.k (of order $$4$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$2.86104277578$$ Analytic rank: $$0$$ Dimension: $$16$$ Relative dimension: $$8$$ over $$\Q(i)$$ Coefficient field: $$\mathbb{Q}[x]/(x^{16} + \cdots)$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ 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 $$1,\beta_1,\ldots,\beta_{15}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q -\beta_{1} q^{2} + ( \beta_{7} + \beta_{10} - \beta_{13} + \beta_{15} ) q^{3} + 3 \beta_{3} q^{4} + ( -\beta_{7} + 2 \beta_{8} + 2 \beta_{10} - \beta_{11} + \beta_{14} - 2 \beta_{15} ) q^{5} + ( -\beta_{6} - \beta_{10} - \beta_{13} + \beta_{14} + \beta_{15} ) q^{6} + ( 2 - \beta_{1} - \beta_{2} + 2 \beta_{3} + \beta_{4} + \beta_{5} - 2 \beta_{6} - \beta_{9} - 2 \beta_{11} - 2 \beta_{12} + 2 \beta_{14} ) q^{7} + 7 \beta_{9} q^{8} + ( -2 - 3 \beta_{1} - 3 \beta_{2} - 2 \beta_{3} - \beta_{4} - \beta_{5} - \beta_{12} ) q^{9} +O(q^{10})$$ $$q -\beta_{1} q^{2} + ( \beta_{7} + \beta_{10} - \beta_{13} + \beta_{15} ) q^{3} + 3 \beta_{3} q^{4} + ( -\beta_{7} + 2 \beta_{8} + 2 \beta_{10} - \beta_{11} + \beta_{14} - 2 \beta_{15} ) q^{5} + ( -\beta_{6} - \beta_{10} - \beta_{13} + \beta_{14} + \beta_{15} ) q^{6} + ( 2 - \beta_{1} - \beta_{2} + 2 \beta_{3} + \beta_{4} + \beta_{5} - 2 \beta_{6} - \beta_{9} - 2 \beta_{11} - 2 \beta_{12} + 2 \beta_{14} ) q^{7} + 7 \beta_{9} q^{8} + ( -2 - 3 \beta_{1} - 3 \beta_{2} - 2 \beta_{3} - \beta_{4} - \beta_{5} - \beta_{12} ) q^{9} + ( -\beta_{6} - \beta_{8} + \beta_{10} - \beta_{11} + 2 \beta_{13} ) q^{10} + ( -6 \beta_{1} - 2 \beta_{2} + 2 \beta_{3} + 2 \beta_{4} - 2 \beta_{5} - 4 \beta_{9} ) q^{11} + ( -3 \beta_{6} - 3 \beta_{7} + 3 \beta_{10} + 3 \beta_{14} ) q^{12} + ( 4 \beta_{6} - \beta_{8} - 8 \beta_{10} + 4 \beta_{11} - 7 \beta_{13} + 4 \beta_{14} + 7 \beta_{15} ) q^{13} + ( -1 - 3 \beta_{1} - \beta_{2} - \beta_{4} - \beta_{5} + 2 \beta_{6} + 4 \beta_{7} - 2 \beta_{8} + 2 \beta_{9} - 2 \beta_{10} + 2 \beta_{11} + 2 \beta_{15} ) q^{14} + ( 6 + 3 \beta_{1} - \beta_{2} + 2 \beta_{3} + 3 \beta_{4} + 4 \beta_{5} - 8 \beta_{9} ) q^{15} -5 q^{16} + ( 2 \beta_{6} - 2 \beta_{11} - 2 \beta_{14} ) q^{17} + ( 2 \beta_{2} - \beta_{3} - 3 \beta_{4} - 2 \beta_{5} - 2 \beta_{9} ) q^{18} + ( -3 \beta_{6} + 11 \beta_{8} + 7 \beta_{10} - 3 \beta_{11} - 4 \beta_{13} - 4 \beta_{14} - 3 \beta_{15} ) q^{19} + ( -9 \beta_{7} + 3 \beta_{8} + 3 \beta_{10} - 9 \beta_{11} - 6 \beta_{14} - 3 \beta_{15} ) q^{20} + ( -4 - 6 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} - 2 \beta_{4} + 2 \beta_{5} - 3 \beta_{6} - 3 \beta_{8} - 8 \beta_{9} + 3 \beta_{10} + 3 \beta_{11} - 2 \beta_{12} + 3 \beta_{13} - 3 \beta_{14} + 3 \beta_{15} ) q^{21} + ( -4 + 2 \beta_{1} + 2 \beta_{2} - 4 \beta_{3} - 2 \beta_{4} - 2 \beta_{5} + 2 \beta_{9} + 4 \beta_{12} ) q^{22} + ( 3 + 3 \beta_{3} + 6 \beta_{4} + 20 \beta_{9} ) q^{23} + ( -7 \beta_{6} + 7 \beta_{7} + 7 \beta_{13} + 7 \beta_{14} - 7 \beta_{15} ) q^{24} + ( 14 + 4 \beta_{1} + 7 \beta_{2} - 2 \beta_{3} - 7 \beta_{4} - 7 \beta_{5} + 4 \beta_{9} + 8 \beta_{12} ) q^{25} + ( 4 \beta_{6} + 4 \beta_{8} - 3 \beta_{10} - 4 \beta_{11} + 7 \beta_{13} - 7 \beta_{14} - 4 \beta_{15} ) q^{26} + ( -\beta_{6} + 2 \beta_{7} + 3 \beta_{8} - 2 \beta_{10} + 12 \beta_{11} - 3 \beta_{13} + \beta_{14} - 3 \beta_{15} ) q^{27} + ( -6 - 3 \beta_{2} + 6 \beta_{3} + 3 \beta_{4} + 3 \beta_{5} - 6 \beta_{8} + 6 \beta_{13} - 6 \beta_{15} ) q^{28} + ( 2 + 16 \beta_{1} + 2 \beta_{2} + 2 \beta_{4} + 2 \beta_{5} - 14 \beta_{9} ) q^{29} + ( -8 - 3 \beta_{1} + \beta_{2} - \beta_{3} - \beta_{4} + 4 \beta_{5} + 2 \beta_{9} - \beta_{12} ) q^{30} + ( 6 \beta_{6} - 6 \beta_{8} + 4 \beta_{10} + 6 \beta_{11} + 10 \beta_{13} - 10 \beta_{14} - 6 \beta_{15} ) q^{31} + 33 \beta_{1} q^{32} + ( 6 \beta_{6} - 18 \beta_{8} - 12 \beta_{10} + 6 \beta_{11} - 12 \beta_{13} + 6 \beta_{14} + 12 \beta_{15} ) q^{33} + ( 2 \beta_{6} - 2 \beta_{8} - 2 \beta_{10} + 2 \beta_{11} + 2 \beta_{15} ) q^{34} + ( 4 - 8 \beta_{1} + 6 \beta_{2} - 2 \beta_{3} + 2 \beta_{4} + 6 \beta_{5} + 10 \beta_{6} + 3 \beta_{7} + 4 \beta_{8} + 8 \beta_{9} - \beta_{10} - 7 \beta_{11} + 5 \beta_{13} - 8 \beta_{14} - 4 \beta_{15} ) q^{35} + ( 6 + 9 \beta_{1} + 6 \beta_{2} - 6 \beta_{3} - 6 \beta_{4} + 6 \beta_{5} + 3 \beta_{9} + 3 \beta_{12} ) q^{36} + ( -24 - 6 \beta_{1} - 6 \beta_{2} - 24 \beta_{3} + 6 \beta_{4} + 6 \beta_{5} - 6 \beta_{9} - 12 \beta_{12} ) q^{37} + ( -4 \beta_{6} - 8 \beta_{7} - 3 \beta_{8} - 4 \beta_{11} - 3 \beta_{13} - 4 \beta_{14} + 3 \beta_{15} ) q^{38} + ( 1 + 33 \beta_{1} + 12 \beta_{2} - 14 \beta_{3} - 10 \beta_{4} - 10 \beta_{5} - 21 \beta_{9} + 11 \beta_{12} ) q^{39} + ( 14 \beta_{6} - 21 \beta_{8} - 14 \beta_{10} + 14 \beta_{11} + 7 \beta_{13} ) q^{40} + ( 6 \beta_{6} + 6 \beta_{8} + 6 \beta_{10} - 6 \beta_{11} - 6 \beta_{15} ) q^{41} + ( -8 - 4 \beta_{2} - 6 \beta_{3} + 2 \beta_{4} - 6 \beta_{6} + 3 \beta_{8} - 2 \beta_{9} + 3 \beta_{11} - 4 \beta_{12} - 3 \beta_{13} + 6 \beta_{14} - 3 \beta_{15} ) q^{42} + ( 4 + 10 \beta_{2} - 4 \beta_{3} - 10 \beta_{4} - 10 \beta_{5} ) q^{43} + ( -6 - 18 \beta_{1} - 6 \beta_{2} - 6 \beta_{4} - 6 \beta_{5} + 12 \beta_{9} ) q^{44} + ( 20 \beta_{6} + 13 \beta_{7} - 6 \beta_{8} - 6 \beta_{10} + 18 \beta_{11} - 5 \beta_{13} - 8 \beta_{14} + 26 \beta_{15} ) q^{45} + ( 20 + 3 \beta_{1} + 3 \beta_{9} + 6 \beta_{12} ) q^{46} + ( 4 \beta_{6} + 28 \beta_{7} - 14 \beta_{8} - 28 \beta_{10} + 24 \beta_{11} + 14 \beta_{13} - 4 \beta_{14} + 14 \beta_{15} ) q^{47} + ( -5 \beta_{7} - 5 \beta_{10} + 5 \beta_{13} - 5 \beta_{15} ) q^{48} + ( -8 \beta_{1} - 16 \beta_{2} + \beta_{3} + 16 \beta_{4} + 16 \beta_{5} + 8 \beta_{8} - 8 \beta_{9} + 4 \beta_{10} - 16 \beta_{12} - 4 \beta_{13} - 4 \beta_{14} ) q^{49} + ( 4 - 13 \beta_{1} + \beta_{2} + 3 \beta_{3} + 7 \beta_{4} + \beta_{5} - 2 \beta_{9} ) q^{50} + ( -8 - 6 \beta_{1} - 2 \beta_{2} + 2 \beta_{3} + 2 \beta_{4} - 2 \beta_{5} - 4 \beta_{9} + 2 \beta_{12} ) q^{51} + ( -21 \beta_{6} + 12 \beta_{8} - 21 \beta_{11} - 12 \beta_{13} + 21 \beta_{14} - 12 \beta_{15} ) q^{52} + ( 10 + 10 \beta_{3} + 20 \beta_{4} - 10 \beta_{9} ) q^{53} + ( -10 \beta_{6} - 17 \beta_{7} + 12 \beta_{8} + 15 \beta_{10} - 18 \beta_{11} - 5 \beta_{13} - 5 \beta_{14} - 10 \beta_{15} ) q^{54} + ( -8 \beta_{6} + 12 \beta_{8} + 28 \beta_{10} - 8 \beta_{11} + 16 \beta_{13} - 20 \beta_{14} - 10 \beta_{15} ) q^{55} + ( 21 \beta_{1} + 7 \beta_{2} - 7 \beta_{3} - 7 \beta_{4} + 7 \beta_{5} + 14 \beta_{9} + 14 \beta_{10} - 14 \beta_{13} + 14 \beta_{14} ) q^{56} + ( 18 - 18 \beta_{1} - 18 \beta_{2} + 33 \beta_{3} + 3 \beta_{4} - 12 \beta_{5} - 3 \beta_{9} - 6 \beta_{12} ) q^{57} + ( -14 - 2 \beta_{2} + 14 \beta_{3} + 2 \beta_{4} + 2 \beta_{5} ) q^{58} + ( -18 \beta_{6} - 26 \beta_{7} + 18 \beta_{8} + 13 \beta_{10} - 8 \beta_{11} + 5 \beta_{13} + 5 \beta_{14} - 18 \beta_{15} ) q^{59} + ( -6 - 18 \beta_{1} + 6 \beta_{2} + 18 \beta_{3} - 3 \beta_{4} - 9 \beta_{5} + 3 \beta_{9} + 15 \beta_{12} ) q^{60} + ( -17 \beta_{6} + 17 \beta_{8} + 9 \beta_{10} - 17 \beta_{11} - 8 \beta_{13} + 8 \beta_{14} + 17 \beta_{15} ) q^{61} + ( -10 \beta_{6} - 12 \beta_{7} + 6 \beta_{8} + 12 \beta_{10} - 2 \beta_{11} - 6 \beta_{13} + 10 \beta_{14} - 6 \beta_{15} ) q^{62} + ( -9 - 9 \beta_{4} - 12 \beta_{6} - 22 \beta_{7} + 18 \beta_{8} + 36 \beta_{9} + 2 \beta_{10} - 12 \beta_{11} - 2 \beta_{13} - 12 \beta_{14} + 2 \beta_{15} ) q^{63} + 13 \beta_{3} q^{64} + ( -15 - 45 \beta_{1} - 5 \beta_{2} - 10 \beta_{3} - 25 \beta_{4} - 5 \beta_{5} + 15 \beta_{9} ) q^{65} + ( 6 \beta_{6} + 6 \beta_{8} - 6 \beta_{10} - 6 \beta_{11} - 6 \beta_{13} + 6 \beta_{14} - 6 \beta_{15} ) q^{66} + ( 20 + 18 \beta_{1} + 18 \beta_{2} + 20 \beta_{3} - 18 \beta_{4} - 18 \beta_{5} + 18 \beta_{9} + 36 \beta_{12} ) q^{67} + ( -6 \beta_{8} - 6 \beta_{13} + 6 \beta_{15} ) q^{68} + ( -5 \beta_{6} + 32 \beta_{7} - 36 \beta_{8} - 18 \beta_{10} + 23 \beta_{13} + 23 \beta_{14} - 5 \beta_{15} ) q^{69} + ( 8 - 2 \beta_{1} - 6 \beta_{2} - 14 \beta_{3} + 6 \beta_{4} + 6 \beta_{5} + 8 \beta_{6} - 7 \beta_{8} - 2 \beta_{9} - 3 \beta_{10} + 8 \beta_{11} - 4 \beta_{12} + 4 \beta_{13} - 5 \beta_{14} + 10 \beta_{15} ) q^{70} + ( -19 \beta_{1} + 4 \beta_{2} - 4 \beta_{3} - 4 \beta_{4} + 4 \beta_{5} - 23 \beta_{9} ) q^{71} + ( 7 - 21 \beta_{1} - 7 \beta_{2} + 14 \beta_{4} + 21 \beta_{5} - 14 \beta_{9} - 28 \beta_{12} ) q^{72} + ( 16 \beta_{6} - 8 \beta_{8} - 32 \beta_{10} + 16 \beta_{11} - 24 \beta_{13} + 16 \beta_{14} + 24 \beta_{15} ) q^{73} + ( -6 + 18 \beta_{1} - 6 \beta_{2} - 6 \beta_{4} - 6 \beta_{5} - 24 \beta_{9} ) q^{74} + ( -9 \beta_{6} - 5 \beta_{7} + 21 \beta_{8} + 34 \beta_{10} - 39 \beta_{11} - 37 \beta_{13} + 15 \beta_{14} - 5 \beta_{15} ) q^{75} + ( -12 \beta_{6} + 12 \beta_{8} + 21 \beta_{10} - 12 \beta_{11} + 9 \beta_{13} - 9 \beta_{14} + 12 \beta_{15} ) q^{76} + ( -8 - 66 \beta_{1} - 16 \beta_{2} + 8 \beta_{3} - 16 \beta_{5} - 8 \beta_{7} + 4 \beta_{8} + 8 \beta_{10} - 8 \beta_{11} - 4 \beta_{13} - 4 \beta_{15} ) q^{77} + ( -21 - \beta_{2} + 32 \beta_{3} + 12 \beta_{4} + \beta_{5} - 14 \beta_{9} ) q^{78} + ( -13 \beta_{1} - 26 \beta_{2} + 26 \beta_{4} + 26 \beta_{5} - 13 \beta_{9} - 26 \beta_{12} ) q^{79} + ( 5 \beta_{7} - 10 \beta_{8} - 10 \beta_{10} + 5 \beta_{11} - 5 \beta_{14} + 10 \beta_{15} ) q^{80} + ( 56 + 21 \beta_{1} - 7 \beta_{2} + 7 \beta_{3} + 7 \beta_{4} - 7 \beta_{5} + 28 \beta_{9} - 8 \beta_{12} ) q^{81} + ( -6 \beta_{8} + 6 \beta_{13} + 6 \beta_{15} ) q^{82} + ( 14 \beta_{6} + 28 \beta_{7} - 33 \beta_{8} + 14 \beta_{11} - 33 \beta_{13} + 14 \beta_{14} + 33 \beta_{15} ) q^{83} + ( 6 - 18 \beta_{1} - 12 \beta_{3} + 12 \beta_{4} + 12 \beta_{5} + 9 \beta_{6} + 9 \beta_{7} - 9 \beta_{8} + 18 \beta_{9} - 18 \beta_{10} + 27 \beta_{11} - 6 \beta_{12} + 9 \beta_{13} + 9 \beta_{14} + 9 \beta_{15} ) q^{84} + ( 16 - 4 \beta_{1} - 2 \beta_{2} + 2 \beta_{3} + 2 \beta_{4} + 2 \beta_{5} - 4 \beta_{9} - 8 \beta_{12} ) q^{85} + ( -14 \beta_{1} - 10 \beta_{2} + 10 \beta_{3} + 10 \beta_{4} - 10 \beta_{5} - 4 \beta_{9} ) q^{86} + ( 32 \beta_{6} - 10 \beta_{7} - 6 \beta_{8} + 10 \beta_{10} - 6 \beta_{11} + 6 \beta_{13} - 32 \beta_{14} + 6 \beta_{15} ) q^{87} + ( 28 + 14 \beta_{2} - 28 \beta_{3} - 14 \beta_{4} - 14 \beta_{5} ) q^{88} + ( -34 \beta_{6} + 4 \beta_{7} + 34 \beta_{8} - 2 \beta_{10} + 38 \beta_{11} + 36 \beta_{13} + 36 \beta_{14} - 34 \beta_{15} ) q^{89} + ( -12 \beta_{6} - 5 \beta_{7} + 18 \beta_{8} - 8 \beta_{10} + 3 \beta_{11} + 14 \beta_{13} - 5 \beta_{15} ) q^{90} + ( -52 - 6 \beta_{1} + 7 \beta_{6} - 7 \beta_{8} - 6 \beta_{9} - 43 \beta_{10} + 7 \beta_{11} - 12 \beta_{12} - 36 \beta_{13} + 36 \beta_{14} - 7 \beta_{15} ) q^{91} + ( -9 + 42 \beta_{1} - 18 \beta_{2} + 9 \beta_{3} - 18 \beta_{5} ) q^{92} + ( 6 - 10 \beta_{2} + 2 \beta_{3} + 18 \beta_{4} + 10 \beta_{5} + 58 \beta_{9} ) q^{93} + ( 4 \beta_{6} + 24 \beta_{8} + 10 \beta_{10} + 4 \beta_{11} - 14 \beta_{13} - 14 \beta_{14} + 4 \beta_{15} ) q^{94} + ( 2 + 81 \beta_{1} + 13 \beta_{2} - 11 \beta_{3} - 9 \beta_{4} + 13 \beta_{5} - 51 \beta_{9} ) q^{95} + ( 33 \beta_{6} + 33 \beta_{10} + 33 \beta_{13} - 33 \beta_{14} - 33 \beta_{15} ) q^{96} + ( -14 \beta_{6} + 4 \beta_{8} - 14 \beta_{11} - 4 \beta_{13} + 14 \beta_{14} - 4 \beta_{15} ) q^{97} + ( -8 - 8 \beta_{3} - 16 \beta_{4} - 4 \beta_{6} - 8 \beta_{7} + \beta_{9} - 4 \beta_{11} - 4 \beta_{14} ) q^{98} + ( 18 \beta_{2} - 72 \beta_{3} - 18 \beta_{4} - 18 \beta_{5} + 18 \beta_{9} + 18 \beta_{12} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$16q + 32q^{7} + O(q^{10})$$ $$16q + 32q^{7} + 80q^{15} - 80q^{16} + 8q^{18} - 64q^{21} - 64q^{22} + 224q^{25} - 96q^{28} - 128q^{30} + 96q^{36} - 384q^{37} - 112q^{42} + 64q^{43} + 320q^{46} - 128q^{51} + 408q^{57} - 224q^{58} - 120q^{60} - 72q^{63} + 320q^{67} + 128q^{70} + 56q^{72} - 424q^{78} + 896q^{81} + 256q^{85} + 448q^{88} - 832q^{91} + 32q^{93} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{16} + 433 x^{8} + 16$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$\nu^{10} + 479 \nu^{2}$$$$)/210$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{10} + \nu^{8} + 374 \nu^{2} + 164$$$$)/105$$ $$\beta_{3}$$ $$=$$ $$($$$$-\nu^{12} - 437 \nu^{4}$$$$)/84$$ $$\beta_{4}$$ $$=$$ $$($$$$-2 \nu^{12} + \nu^{8} - 853 \nu^{4} + 164$$$$)/105$$ $$\beta_{5}$$ $$=$$ $$($$$$4 \nu^{12} - 3 \nu^{10} + 1706 \nu^{4} - 1227 \nu^{2}$$$$)/210$$ $$\beta_{6}$$ $$=$$ $$($$$$-5 \nu^{13} + 2 \nu^{11} - 2185 \nu^{5} + 958 \nu^{3}$$$$)/420$$ $$\beta_{7}$$ $$=$$ $$($$$$-13 \nu^{13} + 4 \nu^{9} - 5597 \nu^{5} + 1916 \nu$$$$)/840$$ $$\beta_{8}$$ $$=$$ $$($$$$13 \nu^{13} + 4 \nu^{9} + 5597 \nu^{5} + 1916 \nu$$$$)/840$$ $$\beta_{9}$$ $$=$$ $$($$$$23 \nu^{14} + 9967 \nu^{6}$$$$)/840$$ $$\beta_{10}$$ $$=$$ $$($$$$-23 \nu^{13} - 4 \nu^{9} - 9967 \nu^{5} - 1076 \nu$$$$)/840$$ $$\beta_{11}$$ $$=$$ $$($$$$23 \nu^{13} - 4 \nu^{9} + 9967 \nu^{5} - 1076 \nu$$$$)/840$$ $$\beta_{12}$$ $$=$$ $$($$$$3 \nu^{14} - \nu^{10} + 1297 \nu^{6} - 409 \nu^{2}$$$$)/70$$ $$\beta_{13}$$ $$=$$ $$($$$$23 \nu^{15} - 10 \nu^{13} + 9967 \nu^{7} - 4370 \nu^{5}$$$$)/840$$ $$\beta_{14}$$ $$=$$ $$($$$$21 \nu^{15} - 4 \nu^{13} + 4 \nu^{11} + 9093 \nu^{7} - 1748 \nu^{5} + 1748 \nu^{3}$$$$)/336$$ $$\beta_{15}$$ $$=$$ $$($$$$21 \nu^{15} - 4 \nu^{13} - 4 \nu^{11} + 9093 \nu^{7} - 1748 \nu^{5} - 1748 \nu^{3}$$$$)/336$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{11} + \beta_{10} + \beta_{8} + \beta_{7}$$$$)/2$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{5} + \beta_{4} - \beta_{2} + 5 \beta_{1}$$$$)/2$$ $$\nu^{3}$$ $$=$$ $$($$$$2 \beta_{15} - 2 \beta_{14} + 5 \beta_{11} - 5 \beta_{10} - 5 \beta_{8} + 5 \beta_{7} + 10 \beta_{6}$$$$)/2$$ $$\nu^{4}$$ $$=$$ $$($$$$-5 \beta_{5} + 5 \beta_{4} - 16 \beta_{3} - 5 \beta_{2} - 5 \beta_{1}$$$$)/2$$ $$\nu^{5}$$ $$=$$ $$($$$$13 \beta_{11} - 13 \beta_{10} - 23 \beta_{8} + 23 \beta_{7}$$$$)/2$$ $$\nu^{6}$$ $$=$$ $$($$$$-46 \beta_{12} + 72 \beta_{9} + 23 \beta_{5} + 23 \beta_{4} - 23 \beta_{2} - 23 \beta_{1}$$$$)/2$$ $$\nu^{7}$$ $$=$$ $$($$$$-46 \beta_{15} - 46 \beta_{14} + 210 \beta_{13} + 59 \beta_{11} - 59 \beta_{10} - 59 \beta_{8} + 59 \beta_{7}$$$$)/2$$ $$\nu^{8}$$ $$=$$ $$($$$$105 \beta_{5} + 105 \beta_{4} + 105 \beta_{2} + 105 \beta_{1} - 328$$$$)/2$$ $$\nu^{9}$$ $$=$$ $$($$$$-479 \beta_{11} - 479 \beta_{10} - 269 \beta_{8} - 269 \beta_{7}$$$$)/2$$ $$\nu^{10}$$ $$=$$ $$($$$$-479 \beta_{5} - 479 \beta_{4} + 479 \beta_{2} - 1975 \beta_{1}$$$$)/2$$ $$\nu^{11}$$ $$=$$ $$($$$$-958 \beta_{15} + 958 \beta_{14} - 2185 \beta_{11} + 2185 \beta_{10} + 2185 \beta_{8} - 2185 \beta_{7} - 4370 \beta_{6}$$$$)/2$$ $$\nu^{12}$$ $$=$$ $$($$$$2185 \beta_{5} - 2185 \beta_{4} + 6824 \beta_{3} + 2185 \beta_{2} + 2185 \beta_{1}$$$$)/2$$ $$\nu^{13}$$ $$=$$ $$($$$$-5597 \beta_{11} + 5597 \beta_{10} + 9967 \beta_{8} - 9967 \beta_{7}$$$$)/2$$ $$\nu^{14}$$ $$=$$ $$($$$$19934 \beta_{12} - 31128 \beta_{9} - 9967 \beta_{5} - 9967 \beta_{4} + 9967 \beta_{2} + 9967 \beta_{1}$$$$)/2$$ $$\nu^{15}$$ $$=$$ $$($$$$19934 \beta_{15} + 19934 \beta_{14} - 90930 \beta_{13} - 25531 \beta_{11} + 25531 \beta_{10} + 25531 \beta_{8} - 25531 \beta_{7}$$$$)/2$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/105\mathbb{Z}\right)^\times$$.

 $$n$$ $$22$$ $$31$$ $$71$$ $$\chi(n)$$ $$\beta_{3}$$ $$-1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
62.1
 −1.97320 + 0.817327i −0.611750 + 0.253395i 0.611750 − 0.253395i 1.97320 − 0.817327i −0.817327 − 1.97320i 0.253395 + 0.611750i −0.253395 − 0.611750i 0.817327 + 1.97320i −1.97320 − 0.817327i −0.611750 − 0.253395i 0.611750 + 0.253395i 1.97320 + 0.817327i −0.817327 + 1.97320i 0.253395 − 0.611750i −0.253395 + 0.611750i 0.817327 − 1.97320i
−0.707107 + 0.707107i −2.87568 + 0.854662i 3.00000i −4.57796 2.01054i 1.42908 2.63775i 3.94887 5.77983i −4.94975 4.94975i 7.53910 4.91548i 4.65877 1.81544i
62.2 −0.707107 + 0.707107i −0.152778 2.99611i 3.00000i −4.24762 + 2.63775i 2.22660 + 2.01054i 4.33402 + 5.49694i −4.94975 4.94975i −8.95332 + 0.915476i 1.13835 4.86869i
62.3 −0.707107 + 0.707107i 0.152778 + 2.99611i 3.00000i 4.24762 2.63775i −2.22660 2.01054i 5.49694 + 4.33402i −4.94975 4.94975i −8.95332 + 0.915476i −1.13835 + 4.86869i
62.4 −0.707107 + 0.707107i 2.87568 0.854662i 3.00000i 4.57796 + 2.01054i −1.42908 + 2.63775i −5.77983 + 3.94887i −4.94975 4.94975i 7.53910 4.91548i −4.65877 + 1.81544i
62.5 0.707107 0.707107i −2.99611 0.152778i 3.00000i 4.24762 2.63775i −2.22660 + 2.01054i 4.33402 + 5.49694i 4.94975 + 4.94975i 8.95332 + 0.915476i 1.13835 4.86869i
62.6 0.707107 0.707107i −0.854662 + 2.87568i 3.00000i −4.57796 2.01054i 1.42908 + 2.63775i −5.77983 + 3.94887i 4.94975 + 4.94975i −7.53910 4.91548i −4.65877 + 1.81544i
62.7 0.707107 0.707107i 0.854662 2.87568i 3.00000i 4.57796 + 2.01054i −1.42908 2.63775i 3.94887 5.77983i 4.94975 + 4.94975i −7.53910 4.91548i 4.65877 1.81544i
62.8 0.707107 0.707107i 2.99611 + 0.152778i 3.00000i −4.24762 + 2.63775i 2.22660 2.01054i 5.49694 + 4.33402i 4.94975 + 4.94975i 8.95332 + 0.915476i −1.13835 + 4.86869i
83.1 −0.707107 0.707107i −2.87568 0.854662i 3.00000i −4.57796 + 2.01054i 1.42908 + 2.63775i 3.94887 + 5.77983i −4.94975 + 4.94975i 7.53910 + 4.91548i 4.65877 + 1.81544i
83.2 −0.707107 0.707107i −0.152778 + 2.99611i 3.00000i −4.24762 2.63775i 2.22660 2.01054i 4.33402 5.49694i −4.94975 + 4.94975i −8.95332 0.915476i 1.13835 + 4.86869i
83.3 −0.707107 0.707107i 0.152778 2.99611i 3.00000i 4.24762 + 2.63775i −2.22660 + 2.01054i 5.49694 4.33402i −4.94975 + 4.94975i −8.95332 0.915476i −1.13835 4.86869i
83.4 −0.707107 0.707107i 2.87568 + 0.854662i 3.00000i 4.57796 2.01054i −1.42908 2.63775i −5.77983 3.94887i −4.94975 + 4.94975i 7.53910 + 4.91548i −4.65877 1.81544i
83.5 0.707107 + 0.707107i −2.99611 + 0.152778i 3.00000i 4.24762 + 2.63775i −2.22660 2.01054i 4.33402 5.49694i 4.94975 4.94975i 8.95332 0.915476i 1.13835 + 4.86869i
83.6 0.707107 + 0.707107i −0.854662 2.87568i 3.00000i −4.57796 + 2.01054i 1.42908 2.63775i −5.77983 3.94887i 4.94975 4.94975i −7.53910 + 4.91548i −4.65877 1.81544i
83.7 0.707107 + 0.707107i 0.854662 + 2.87568i 3.00000i 4.57796 2.01054i −1.42908 + 2.63775i 3.94887 + 5.77983i 4.94975 4.94975i −7.53910 + 4.91548i 4.65877 + 1.81544i
83.8 0.707107 + 0.707107i 2.99611 0.152778i 3.00000i −4.24762 2.63775i 2.22660 + 2.01054i 5.49694 4.33402i 4.94975 4.94975i 8.95332 0.915476i −1.13835 4.86869i
 $$n$$: 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
3.b odd 2 1 inner
5.c odd 4 1 inner
7.b odd 2 1 inner
15.e even 4 1 inner
21.c even 2 1 inner
35.f even 4 1 inner
105.k odd 4 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 105.3.k.c 16
3.b odd 2 1 inner 105.3.k.c 16
5.c odd 4 1 inner 105.3.k.c 16
7.b odd 2 1 inner 105.3.k.c 16
15.e even 4 1 inner 105.3.k.c 16
21.c even 2 1 inner 105.3.k.c 16
35.f even 4 1 inner 105.3.k.c 16
105.k odd 4 1 inner 105.3.k.c 16

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
105.3.k.c 16 1.a even 1 1 trivial
105.3.k.c 16 3.b odd 2 1 inner
105.3.k.c 16 5.c odd 4 1 inner
105.3.k.c 16 7.b odd 2 1 inner
105.3.k.c 16 15.e even 4 1 inner
105.3.k.c 16 21.c even 2 1 inner
105.3.k.c 16 35.f even 4 1 inner
105.3.k.c 16 105.k odd 4 1 inner

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{3}^{\mathrm{new}}(105, [\chi])$$:

 $$T_{2}^{4} + 1$$ $$T_{11}^{4} + 200 T_{11}^{2} + 1296$$ $$T_{13}^{8} + 208844 T_{13}^{4} + 4685402500$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 + 17 T^{4} + 256 T^{8} )^{4}$$
$3$ $$1 - 224 T^{4} + 23490 T^{8} - 1469664 T^{12} + 43046721 T^{16}$$
$5$ $$( 1 - 56 T^{2} + 2000 T^{4} - 35000 T^{6} + 390625 T^{8} )^{2}$$
$7$ $$( 1 - 16 T + 128 T^{2} - 208 T^{3} - 958 T^{4} - 10192 T^{5} + 307328 T^{6} - 1882384 T^{7} + 5764801 T^{8} )^{2}$$
$11$ $$( 1 - 284 T^{2} + 40742 T^{4} - 4158044 T^{6} + 214358881 T^{8} )^{4}$$
$13$ $$( 1 - 74400 T^{4} + 2876017858 T^{8} - 60690365642400 T^{12} + 665416609183179841 T^{16} )^{2}$$
$17$ $$( 1 + 280836 T^{4} + 33000880390 T^{8} + 1959043816700676 T^{12} + 48661191875666868481 T^{16} )^{2}$$
$19$ $$( 1 + 696 T^{2} + 371920 T^{4} + 90703416 T^{6} + 16983563041 T^{8} )^{4}$$
$23$ $$( 1 + 107644 T^{4} - 83134815354 T^{8} + 8429707699587964 T^{12} +$$$$61\!\cdots\!61$$$$T^{16} )^{2}$$
$29$ $$( 1 + 2444 T^{2} + 2801222 T^{4} + 1728594764 T^{6} + 500246412961 T^{8} )^{4}$$
$31$ $$( 1 - 2596 T^{2} + 3425222 T^{4} - 2397460516 T^{6} + 852891037441 T^{8} )^{4}$$
$37$ $$( 1 + 96 T + 4608 T^{2} + 183264 T^{3} + 6996962 T^{4} + 250888416 T^{5} + 8636133888 T^{6} + 246309735264 T^{7} + 3512479453921 T^{8} )^{4}$$
$41$ $$( 1 + 5716 T^{2} + 13775622 T^{4} + 16152049876 T^{6} + 7984925229121 T^{8} )^{4}$$
$43$ $$( 1 - 16 T + 128 T^{2} - 2896 T^{3} - 2716702 T^{4} - 5354704 T^{5} + 437606528 T^{6} - 101141808784 T^{7} + 11688200277601 T^{8} )^{4}$$
$47$ $$( 1 - 5511420 T^{4} + 7619082080518 T^{8} -$$$$13\!\cdots\!20$$$$T^{12} +$$$$56\!\cdots\!21$$$$T^{16} )^{2}$$
$53$ $$( 1 - 19870076 T^{4} + 198820934324166 T^{8} -$$$$12\!\cdots\!36$$$$T^{12} +$$$$38\!\cdots\!21$$$$T^{16} )^{2}$$
$59$ $$( 1 - 10776 T^{2} + 52783792 T^{4} - 130576682136 T^{6} + 146830437604321 T^{8} )^{4}$$
$61$ $$( 1 - 8200 T^{2} + 36103376 T^{4} - 113535896200 T^{6} + 191707312997281 T^{8} )^{4}$$
$67$ $$( 1 - 80 T + 3200 T^{2} + 17520 T^{3} - 22069342 T^{4} + 78647280 T^{5} + 64483587200 T^{6} - 7236670573520 T^{7} + 406067677556641 T^{8} )^{4}$$
$71$ $$( 1 - 17504 T^{2} + 126269762 T^{4} - 444806064224 T^{6} + 645753531245761 T^{8} )^{4}$$
$73$ $$( 1 - 10432892 T^{4} + 418480444620678 T^{8} -$$$$84\!\cdots\!52$$$$T^{12} +$$$$65\!\cdots\!61$$$$T^{16} )^{2}$$
$79$ $$( 1 - 6736 T^{2} + 38950081 T^{4} )^{8}$$
$83$ $$( 1 - 55419104 T^{4} + 5263624308845250 T^{8} -$$$$12\!\cdots\!64$$$$T^{12} +$$$$50\!\cdots\!81$$$$T^{16} )^{2}$$
$89$ $$( 1 - 1108 T^{2} - 101931898 T^{4} - 69518403028 T^{6} + 3936588805702081 T^{8} )^{4}$$
$97$ $$( 1 + 284585220 T^{4} + 35721324461420038 T^{8} +$$$$22\!\cdots\!20$$$$T^{12} +$$$$61\!\cdots\!21$$$$T^{16} )^{2}$$