# Properties

 Label 105.4.d.b Level $105$ Weight $4$ Character orbit 105.d Analytic conductor $6.195$ Analytic rank $0$ Dimension $10$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$105 = 3 \cdot 5 \cdot 7$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 105.d (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$6.19520055060$$ Analytic rank: $$0$$ Dimension: $$10$$ Coefficient field: $$\mathbb{Q}[x]/(x^{10} + \cdots)$$ Defining polynomial: $$x^{10} + 37 x^{8} + 398 x^{6} + 1149 x^{4} + 1040 x^{2} + 100$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2^{9}\cdot 5$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{9}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{6} q^{2} -3 \beta_{2} q^{3} + ( -5 - \beta_{1} + \beta_{5} ) q^{4} + ( -2 + \beta_{1} - \beta_{6} - \beta_{8} ) q^{5} -3 \beta_{1} q^{6} -7 \beta_{2} q^{7} + ( -1 + \beta_{1} + 7 \beta_{2} + \beta_{3} - \beta_{4} - \beta_{5} - 6 \beta_{6} - \beta_{7} - 3 \beta_{8} + \beta_{9} ) q^{8} -9 q^{9} +O(q^{10})$$ $$q + \beta_{6} q^{2} -3 \beta_{2} q^{3} + ( -5 - \beta_{1} + \beta_{5} ) q^{4} + ( -2 + \beta_{1} - \beta_{6} - \beta_{8} ) q^{5} -3 \beta_{1} q^{6} -7 \beta_{2} q^{7} + ( -1 + \beta_{1} + 7 \beta_{2} + \beta_{3} - \beta_{4} - \beta_{5} - 6 \beta_{6} - \beta_{7} - 3 \beta_{8} + \beta_{9} ) q^{8} -9 q^{9} + ( 8 + 4 \beta_{1} - 12 \beta_{2} + 2 \beta_{3} - \beta_{4} - 3 \beta_{5} - 4 \beta_{6} - \beta_{7} - 2 \beta_{8} + \beta_{9} ) q^{10} + ( 14 - 8 \beta_{1} - 2 \beta_{3} ) q^{11} + ( 15 \beta_{2} - 3 \beta_{6} + 3 \beta_{7} ) q^{12} + ( -1 + \beta_{1} - \beta_{2} + \beta_{3} - \beta_{4} - \beta_{5} - 7 \beta_{6} - \beta_{7} - 3 \beta_{8} - \beta_{9} ) q^{13} -7 \beta_{1} q^{14} + ( 3 \beta_{1} + 6 \beta_{2} - 3 \beta_{4} + 3 \beta_{6} ) q^{15} + ( 32 + 6 \beta_{1} + \beta_{2} + 5 \beta_{3} - 3 \beta_{4} - 4 \beta_{5} + \beta_{6} - \beta_{7} + \beta_{8} - \beta_{9} ) q^{16} + ( -1 + \beta_{1} - 19 \beta_{2} + \beta_{3} - \beta_{4} - \beta_{5} + 5 \beta_{6} - 9 \beta_{7} - 3 \beta_{8} - 3 \beta_{9} ) q^{17} -9 \beta_{6} q^{18} + ( -31 - \beta_{1} + \beta_{2} + 3 \beta_{3} - 3 \beta_{4} + 7 \beta_{5} + \beta_{6} - \beta_{7} + \beta_{8} - \beta_{9} ) q^{19} + ( 38 - 5 \beta_{1} - 27 \beta_{2} + \beta_{4} - 5 \beta_{5} + 23 \beta_{6} + 4 \beta_{8} - 5 \beta_{9} ) q^{20} -21 q^{21} + ( -2 + 2 \beta_{1} + 104 \beta_{2} + 2 \beta_{3} - 2 \beta_{4} - 2 \beta_{5} + 8 \beta_{6} + 8 \beta_{7} - 6 \beta_{8} + 2 \beta_{9} ) q^{22} + ( 3 - 3 \beta_{1} - 31 \beta_{2} - 3 \beta_{3} + 3 \beta_{4} + 3 \beta_{5} - 9 \beta_{6} + 5 \beta_{7} + 9 \beta_{8} - 5 \beta_{9} ) q^{23} + ( 21 + 18 \beta_{1} + 3 \beta_{2} + 3 \beta_{3} - 9 \beta_{4} + 3 \beta_{5} + 3 \beta_{6} - 3 \beta_{7} + 3 \beta_{8} - 3 \beta_{9} ) q^{24} + ( -42 - 3 \beta_{1} - 13 \beta_{2} - 5 \beta_{3} + 9 \beta_{4} - 5 \beta_{5} + 15 \beta_{6} + 5 \beta_{7} - \beta_{8} + 5 \beta_{9} ) q^{25} + ( 85 + 9 \beta_{1} - \beta_{2} + 3 \beta_{3} + 3 \beta_{4} - 13 \beta_{5} - \beta_{6} + \beta_{7} - \beta_{8} + \beta_{9} ) q^{26} + 27 \beta_{2} q^{27} + ( 35 \beta_{2} - 7 \beta_{6} + 7 \beta_{7} ) q^{28} + ( -74 - 10 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} + 6 \beta_{4} + 4 \beta_{5} - 2 \beta_{6} + 2 \beta_{7} - 2 \beta_{8} + 2 \beta_{9} ) q^{29} + ( -36 + 12 \beta_{1} - 24 \beta_{2} + 3 \beta_{3} - 6 \beta_{4} + 3 \beta_{5} + 12 \beta_{6} - 9 \beta_{7} + 3 \beta_{8} - 6 \beta_{9} ) q^{30} + ( 67 + 27 \beta_{1} + \beta_{2} + 3 \beta_{3} - 3 \beta_{4} + 9 \beta_{5} + \beta_{6} - \beta_{7} + \beta_{8} - \beta_{9} ) q^{31} + ( 2 - 2 \beta_{1} + 14 \beta_{2} - 2 \beta_{3} + 2 \beta_{4} + 2 \beta_{5} + 27 \beta_{6} - 10 \beta_{7} + 6 \beta_{8} - 6 \beta_{9} ) q^{32} + ( -42 \beta_{2} - 24 \beta_{6} + 6 \beta_{9} ) q^{33} + ( -23 - 83 \beta_{1} - 11 \beta_{2} - 7 \beta_{3} + 33 \beta_{4} - 17 \beta_{5} - 11 \beta_{6} + 11 \beta_{7} - 11 \beta_{8} + 11 \beta_{9} ) q^{34} + ( 7 \beta_{1} + 14 \beta_{2} - 7 \beta_{4} + 7 \beta_{6} ) q^{35} + ( 45 + 9 \beta_{1} - 9 \beta_{5} ) q^{36} + ( -2 + 2 \beta_{1} + 84 \beta_{2} + 2 \beta_{3} - 2 \beta_{4} - 2 \beta_{5} - 26 \beta_{6} + 8 \beta_{7} - 6 \beta_{8} + 18 \beta_{9} ) q^{37} + ( -3 + 3 \beta_{1} - 17 \beta_{2} + 3 \beta_{3} - 3 \beta_{4} - 3 \beta_{5} - 81 \beta_{6} - 17 \beta_{7} - 9 \beta_{8} - \beta_{9} ) q^{38} + ( -3 + 21 \beta_{1} + 3 \beta_{2} - 3 \beta_{3} - 9 \beta_{4} + 3 \beta_{5} + 3 \beta_{6} - 3 \beta_{7} + 3 \beta_{8} - 3 \beta_{9} ) q^{39} + ( -211 - 33 \beta_{1} - 15 \beta_{2} - 3 \beta_{3} + 18 \beta_{4} + 12 \beta_{5} + 45 \beta_{6} + 14 \beta_{7} + \beta_{8} + 6 \beta_{9} ) q^{40} + ( 161 - 35 \beta_{1} - 3 \beta_{2} - 17 \beta_{3} + 9 \beta_{4} + 17 \beta_{5} - 3 \beta_{6} + 3 \beta_{7} - 3 \beta_{8} + 3 \beta_{9} ) q^{41} -21 \beta_{6} q^{42} + ( 2 - 2 \beta_{1} - 116 \beta_{2} - 2 \beta_{3} + 2 \beta_{4} + 2 \beta_{5} - 14 \beta_{6} + 6 \beta_{8} + 14 \beta_{9} ) q^{43} + ( -64 + 114 \beta_{1} + 12 \beta_{2} + 4 \beta_{3} - 36 \beta_{4} + 16 \beta_{5} + 12 \beta_{6} - 12 \beta_{7} + 12 \beta_{8} - 12 \beta_{9} ) q^{44} + ( 18 - 9 \beta_{1} + 9 \beta_{6} + 9 \beta_{8} ) q^{45} + ( 123 - 7 \beta_{1} - 3 \beta_{2} - 15 \beta_{3} + 9 \beta_{4} + 13 \beta_{5} - 3 \beta_{6} + 3 \beta_{7} - 3 \beta_{8} + 3 \beta_{9} ) q^{46} + ( 6 - 6 \beta_{1} - 136 \beta_{2} - 6 \beta_{3} + 6 \beta_{4} + 6 \beta_{5} + 42 \beta_{6} - 4 \beta_{7} + 18 \beta_{8} + 10 \beta_{9} ) q^{47} + ( 3 - 3 \beta_{1} - 96 \beta_{2} - 3 \beta_{3} + 3 \beta_{4} + 3 \beta_{5} + 18 \beta_{6} - 12 \beta_{7} + 9 \beta_{8} - 15 \beta_{9} ) q^{48} -49 q^{49} + ( -239 - 17 \beta_{1} + 35 \beta_{2} + 3 \beta_{3} - 13 \beta_{4} + 23 \beta_{5} - 20 \beta_{6} + 21 \beta_{7} - \beta_{8} + 9 \beta_{9} ) q^{50} + ( -57 - 15 \beta_{1} + 3 \beta_{2} - 9 \beta_{3} - 9 \beta_{4} + 27 \beta_{5} + 3 \beta_{6} - 3 \beta_{7} + 3 \beta_{8} - 3 \beta_{9} ) q^{51} + ( 7 - 7 \beta_{1} - 59 \beta_{2} - 7 \beta_{3} + 7 \beta_{4} + 7 \beta_{5} + 129 \beta_{6} + 13 \beta_{7} + 21 \beta_{8} - 19 \beta_{9} ) q^{52} + ( 11 - 11 \beta_{1} + 5 \beta_{2} - 11 \beta_{3} + 11 \beta_{4} + 11 \beta_{5} - 3 \beta_{6} - 3 \beta_{7} + 33 \beta_{8} - 9 \beta_{9} ) q^{53} + 27 \beta_{1} q^{54} + ( -60 + 64 \beta_{1} + 54 \beta_{2} + 6 \beta_{3} - 8 \beta_{4} + 6 \beta_{5} + 46 \beta_{6} - 38 \beta_{7} - 18 \beta_{8} - 2 \beta_{9} ) q^{55} + ( 49 + 42 \beta_{1} + 7 \beta_{2} + 7 \beta_{3} - 21 \beta_{4} + 7 \beta_{5} + 7 \beta_{6} - 7 \beta_{7} + 7 \beta_{8} - 7 \beta_{9} ) q^{56} + ( 3 - 3 \beta_{1} + 93 \beta_{2} - 3 \beta_{3} + 3 \beta_{4} + 3 \beta_{5} - 3 \beta_{6} + 21 \beta_{7} + 9 \beta_{8} - 9 \beta_{9} ) q^{57} + ( -8 + 8 \beta_{1} + 82 \beta_{2} + 8 \beta_{3} - 8 \beta_{4} - 8 \beta_{5} - 134 \beta_{6} + 10 \beta_{7} - 24 \beta_{8} + 16 \beta_{9} ) q^{58} + ( -164 - 40 \beta_{1} - 12 \beta_{2} - 28 \beta_{3} + 36 \beta_{4} - 40 \beta_{5} - 12 \beta_{6} + 12 \beta_{7} - 12 \beta_{8} + 12 \beta_{9} ) q^{59} + ( -81 - 69 \beta_{1} - 114 \beta_{2} - 15 \beta_{3} + 12 \beta_{4} - 15 \beta_{6} - 15 \beta_{7} - 3 \beta_{8} ) q^{60} + ( 112 + 20 \beta_{1} - 16 \beta_{2} + 4 \beta_{3} + 48 \beta_{4} + 2 \beta_{5} - 16 \beta_{6} + 16 \beta_{7} - 16 \beta_{8} + 16 \beta_{9} ) q^{61} + ( -5 + 5 \beta_{1} - 393 \beta_{2} + 5 \beta_{3} - 5 \beta_{4} - 5 \beta_{5} + 29 \beta_{6} - 49 \beta_{7} - 15 \beta_{8} + \beta_{9} ) q^{62} + 63 \beta_{2} q^{63} + ( -11 - 59 \beta_{1} - 10 \beta_{2} + 14 \beta_{3} + 30 \beta_{4} - 17 \beta_{5} - 10 \beta_{6} + 10 \beta_{7} - 10 \beta_{8} + 10 \beta_{9} ) q^{64} + ( -337 - \beta_{1} + 77 \beta_{2} - 19 \beta_{3} + 23 \beta_{4} + \beta_{5} + 39 \beta_{6} + 17 \beta_{7} + 11 \beta_{8} + 3 \beta_{9} ) q^{65} + ( 312 - 24 \beta_{1} + 6 \beta_{2} + 6 \beta_{3} - 18 \beta_{4} - 24 \beta_{5} + 6 \beta_{6} - 6 \beta_{7} + 6 \beta_{8} - 6 \beta_{9} ) q^{66} + ( 12 - 12 \beta_{1} - 70 \beta_{2} - 12 \beta_{3} + 12 \beta_{4} + 12 \beta_{5} - 52 \beta_{6} + 10 \beta_{7} + 36 \beta_{8} - 24 \beta_{9} ) q^{67} + ( -9 + 9 \beta_{1} + 897 \beta_{2} + 9 \beta_{3} - 9 \beta_{4} - 9 \beta_{5} - 33 \beta_{6} + 89 \beta_{7} - 27 \beta_{8} + 21 \beta_{9} ) q^{68} + ( -93 + 27 \beta_{1} - 9 \beta_{2} - 15 \beta_{3} + 27 \beta_{4} - 15 \beta_{5} - 9 \beta_{6} + 9 \beta_{7} - 9 \beta_{8} + 9 \beta_{9} ) q^{69} + ( -84 + 28 \beta_{1} - 56 \beta_{2} + 7 \beta_{3} - 14 \beta_{4} + 7 \beta_{5} + 28 \beta_{6} - 21 \beta_{7} + 7 \beta_{8} - 14 \beta_{9} ) q^{70} + ( 316 + 24 \beta_{1} + 8 \beta_{2} + 42 \beta_{3} - 24 \beta_{4} + 10 \beta_{5} + 8 \beta_{6} - 8 \beta_{7} + 8 \beta_{8} - 8 \beta_{9} ) q^{71} + ( 9 - 9 \beta_{1} - 63 \beta_{2} - 9 \beta_{3} + 9 \beta_{4} + 9 \beta_{5} + 54 \beta_{6} + 9 \beta_{7} + 27 \beta_{8} - 9 \beta_{9} ) q^{72} + ( 13 - 13 \beta_{1} - 99 \beta_{2} - 13 \beta_{3} + 13 \beta_{4} + 13 \beta_{5} + 115 \beta_{6} - 11 \beta_{7} + 39 \beta_{8} - 11 \beta_{9} ) q^{73} + ( 266 + 176 \beta_{1} + 28 \beta_{2} + 36 \beta_{3} - 84 \beta_{4} - 18 \beta_{5} + 28 \beta_{6} - 28 \beta_{7} + 28 \beta_{8} - 28 \beta_{9} ) q^{74} + ( -39 - 45 \beta_{1} + 126 \beta_{2} + 15 \beta_{3} - 3 \beta_{4} - 15 \beta_{5} - 9 \beta_{6} - 15 \beta_{7} - 27 \beta_{8} + 15 \beta_{9} ) q^{75} + ( 871 - 49 \beta_{1} - 7 \beta_{2} + 21 \beta_{3} + 21 \beta_{4} - 71 \beta_{5} - 7 \beta_{6} + 7 \beta_{7} - 7 \beta_{8} + 7 \beta_{9} ) q^{76} + ( -98 \beta_{2} - 56 \beta_{6} + 14 \beta_{9} ) q^{77} + ( -3 + 3 \beta_{1} - 255 \beta_{2} + 3 \beta_{3} - 3 \beta_{4} - 3 \beta_{5} + 27 \beta_{6} - 39 \beta_{7} - 9 \beta_{8} - 9 \beta_{9} ) q^{78} + ( 194 - 82 \beta_{1} + 14 \beta_{2} + 10 \beta_{3} - 42 \beta_{4} - 22 \beta_{5} + 14 \beta_{6} - 14 \beta_{7} + 14 \beta_{8} - 14 \beta_{9} ) q^{79} + ( -393 - 36 \beta_{1} + 70 \beta_{2} + 15 \beta_{3} - 30 \beta_{4} + 20 \beta_{5} - 184 \beta_{6} + 25 \beta_{7} - 9 \beta_{8} - 10 \beta_{9} ) q^{80} + 81 q^{81} + ( -37 + 37 \beta_{1} + 317 \beta_{2} + 37 \beta_{3} - 37 \beta_{4} - 37 \beta_{5} - 23 \beta_{6} + 13 \beta_{7} - 111 \beta_{8} + 49 \beta_{9} ) q^{82} + ( -4 + 4 \beta_{1} + 312 \beta_{2} + 4 \beta_{3} - 4 \beta_{4} - 4 \beta_{5} + 44 \beta_{6} + 16 \beta_{7} - 12 \beta_{8} - 4 \beta_{9} ) q^{83} + ( 105 + 21 \beta_{1} - 21 \beta_{5} ) q^{84} + ( -207 + 205 \beta_{1} + 353 \beta_{2} + 15 \beta_{3} + 21 \beta_{4} - 25 \beta_{5} + 23 \beta_{6} + 55 \beta_{7} + 19 \beta_{8} + 5 \beta_{9} ) q^{85} + ( 206 - 136 \beta_{1} + 12 \beta_{2} + 4 \beta_{3} - 36 \beta_{4} - 6 \beta_{5} + 12 \beta_{6} - 12 \beta_{7} + 12 \beta_{8} - 12 \beta_{9} ) q^{86} + ( -6 + 6 \beta_{1} + 222 \beta_{2} + 6 \beta_{3} - 6 \beta_{4} - 6 \beta_{5} - 30 \beta_{6} + 12 \beta_{7} - 18 \beta_{8} + 6 \beta_{9} ) q^{87} + ( -16 + 16 \beta_{1} - 602 \beta_{2} + 16 \beta_{3} - 16 \beta_{4} - 16 \beta_{5} + 102 \beta_{6} - 130 \beta_{7} - 48 \beta_{8} - 32 \beta_{9} ) q^{88} + ( -71 + 57 \beta_{1} - 15 \beta_{2} + 35 \beta_{3} + 45 \beta_{4} - 7 \beta_{5} - 15 \beta_{6} + 15 \beta_{7} - 15 \beta_{8} + 15 \beta_{9} ) q^{89} + ( -72 - 36 \beta_{1} + 108 \beta_{2} - 18 \beta_{3} + 9 \beta_{4} + 27 \beta_{5} + 36 \beta_{6} + 9 \beta_{7} + 18 \beta_{8} - 9 \beta_{9} ) q^{90} + ( -7 + 49 \beta_{1} + 7 \beta_{2} - 7 \beta_{3} - 21 \beta_{4} + 7 \beta_{5} + 7 \beta_{6} - 7 \beta_{7} + 7 \beta_{8} - 7 \beta_{9} ) q^{91} + ( -7 + 7 \beta_{1} - 271 \beta_{2} + 7 \beta_{3} - 7 \beta_{4} - 7 \beta_{5} - 75 \beta_{6} + 33 \beta_{7} - 21 \beta_{8} + 3 \beta_{9} ) q^{92} + ( 3 - 3 \beta_{1} - 201 \beta_{2} - 3 \beta_{3} + 3 \beta_{4} + 3 \beta_{5} + 81 \beta_{6} + 27 \beta_{7} + 9 \beta_{8} - 9 \beta_{9} ) q^{93} + ( -450 - 280 \beta_{1} - 24 \beta_{3} + 58 \beta_{5} ) q^{94} + ( 153 - 97 \beta_{1} - 77 \beta_{2} + 9 \beta_{3} - 3 \beta_{4} - 41 \beta_{5} + 89 \beta_{6} + 23 \beta_{7} + 27 \beta_{8} - 43 \beta_{9} ) q^{95} + ( 42 - 81 \beta_{1} - 6 \beta_{2} - 18 \beta_{3} + 18 \beta_{4} + 30 \beta_{5} - 6 \beta_{6} + 6 \beta_{7} - 6 \beta_{8} + 6 \beta_{9} ) q^{96} + ( 1 - \beta_{1} - 91 \beta_{2} - \beta_{3} + \beta_{4} + \beta_{5} - 161 \beta_{6} - 19 \beta_{7} + 3 \beta_{8} - 15 \beta_{9} ) q^{97} -49 \beta_{6} q^{98} + ( -126 + 72 \beta_{1} + 18 \beta_{3} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$10q - 54q^{4} - 14q^{5} - 6q^{6} - 90q^{9} + O(q^{10})$$ $$10q - 54q^{4} - 14q^{5} - 6q^{6} - 90q^{9} + 92q^{10} + 132q^{11} - 14q^{14} + 310q^{16} - 348q^{19} + 366q^{20} - 210q^{21} + 198q^{24} - 374q^{25} + 892q^{26} - 740q^{29} - 378q^{30} + 684q^{31} - 224q^{34} + 486q^{36} - 12q^{39} - 2156q^{40} + 1604q^{41} - 580q^{44} + 126q^{45} + 1280q^{46} - 490q^{49} - 2504q^{50} - 648q^{51} + 54q^{54} - 452q^{55} + 462q^{56} - 1408q^{59} - 852q^{60} + 1300q^{61} - 150q^{64} - 3296q^{65} + 3036q^{66} - 696q^{69} - 882q^{70} + 2940q^{71} + 2624q^{74} - 408q^{75} + 8740q^{76} + 1640q^{79} - 4126q^{80} + 810q^{81} + 1134q^{84} - 1704q^{85} + 1664q^{86} - 572q^{89} - 828q^{90} - 28q^{91} - 5080q^{94} + 1268q^{95} + 330q^{96} - 1188q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{10} + 37 x^{8} + 398 x^{6} + 1149 x^{4} + 1040 x^{2} + 100$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$-16 \nu^{8} - 561 \nu^{6} - 5285 \nu^{4} - 8248 \nu^{2} - 880$$$$)/105$$ $$\beta_{2}$$ $$=$$ $$($$$$57 \nu^{9} + 1999 \nu^{7} + 18816 \nu^{5} + 28813 \nu^{3} + 790 \nu$$$$)/700$$ $$\beta_{3}$$ $$=$$ $$($$$$73 \nu^{8} + 2574 \nu^{6} + 24563 \nu^{4} + 41065 \nu^{2} + 6535$$$$)/105$$ $$\beta_{4}$$ $$=$$ $$($$$$88 \nu^{8} + 3096 \nu^{6} + 29414 \nu^{4} + 48052 \nu^{2} - 210 \nu + 5995$$$$)/105$$ $$\beta_{5}$$ $$=$$ $$($$$$-130 \nu^{8} - 4566 \nu^{6} - 43148 \nu^{4} - 67876 \nu^{2} - 5575$$$$)/105$$ $$\beta_{6}$$ $$=$$ $$($$$$157 \nu^{9} + 5514 \nu^{7} + 52136 \nu^{5} + 82603 \nu^{3} + 7690 \nu$$$$)/525$$ $$\beta_{7}$$ $$=$$ $$($$$$949 \nu^{9} + 33483 \nu^{7} + 320432 \nu^{5} + 548881 \nu^{3} + 134830 \nu$$$$)/2100$$ $$\beta_{8}$$ $$=$$ $$($$$$-1223 \nu^{9} + 660 \nu^{8} - 43161 \nu^{7} + 23220 \nu^{6} - 413224 \nu^{5} + 220080 \nu^{4} - 706907 \nu^{3} + 350940 \nu^{2} - 139610 \nu + 34200$$$$)/2100$$ $$\beta_{9}$$ $$=$$ $$($$$$-1373 \nu^{9} - 48591 \nu^{7} - 468664 \nu^{5} - 838937 \nu^{3} - 228710 \nu$$$$)/2100$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$3 \beta_{9} - 3 \beta_{8} + 3 \beta_{7} - 3 \beta_{6} - \beta_{5} - \beta_{4} + \beta_{3} - 3 \beta_{2} + \beta_{1} - 1$$$$)/20$$ $$\nu^{2}$$ $$=$$ $$($$$$-\beta_{9} + \beta_{8} - \beta_{7} + \beta_{6} + 2 \beta_{5} - 3 \beta_{4} + 3 \beta_{3} + \beta_{2} - 17 \beta_{1} - 68$$$$)/10$$ $$\nu^{3}$$ $$=$$ $$($$$$-26 \beta_{9} + 36 \beta_{8} - 11 \beta_{7} + 16 \beta_{6} + 12 \beta_{5} + 12 \beta_{4} - 12 \beta_{3} + 51 \beta_{2} - 12 \beta_{1} + 12$$$$)/10$$ $$\nu^{4}$$ $$=$$ $$($$$$53 \beta_{9} - 53 \beta_{8} + 53 \beta_{7} - 53 \beta_{6} - 51 \beta_{5} + 159 \beta_{4} - 129 \beta_{3} - 53 \beta_{2} + 591 \beta_{1} + 2059$$$$)/20$$ $$\nu^{5}$$ $$=$$ $$($$$$933 \beta_{9} - 1323 \beta_{8} + 283 \beta_{7} - 133 \beta_{6} - 441 \beta_{5} - 441 \beta_{4} + 441 \beta_{3} - 3053 \beta_{2} + 441 \beta_{1} - 441$$$$)/20$$ $$\nu^{6}$$ $$=$$ $$($$$$-1177 \beta_{9} + 1177 \beta_{8} - 1177 \beta_{7} + 1177 \beta_{6} + 639 \beta_{5} - 3531 \beta_{4} + 2741 \beta_{3} + 1177 \beta_{2} - 9679 \beta_{1} - 35351$$$$)/20$$ $$\nu^{7}$$ $$=$$ $$($$$$-4283 \beta_{9} + 6033 \beta_{8} - 1028 \beta_{7} - 1112 \beta_{6} + 2011 \beta_{5} + 2011 \beta_{4} - 2011 \beta_{3} + 18548 \beta_{2} - 2011 \beta_{1} + 2011$$$$)/5$$ $$\nu^{8}$$ $$=$$ $$($$$$24793 \beta_{9} - 24793 \beta_{8} + 24793 \beta_{7} - 24793 \beta_{6} - 7621 \beta_{5} + 74379 \beta_{4} - 56589 \beta_{3} - 24793 \beta_{2} + 161551 \beta_{1} + 628389$$$$)/20$$ $$\nu^{9}$$ $$=$$ $$($$$$159539 \beta_{9} - 222969 \beta_{8} + 30934 \beta_{7} + 91881 \beta_{6} - 74323 \beta_{5} - 74323 \beta_{4} + 74323 \beta_{3} - 822694 \beta_{2} + 74323 \beta_{1} - 74323$$$$)/10$$

## Character values

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

 $$n$$ $$22$$ $$31$$ $$71$$ $$\chi(n)$$ $$-1$$ $$1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
64.1
 3.71490i − 1.37042i − 4.40248i 1.35311i 0.329739i − 0.329739i − 1.35311i 4.40248i 1.37042i − 3.71490i
5.18660i 3.00000i −18.9008 −4.24321 + 10.3438i −15.5598 7.00000i 56.5383i −9.00000 53.6494 + 22.0078i
64.2 4.88936i 3.00000i −15.9059 −9.63020 + 5.67972i 14.6681 7.00000i 38.6546i −9.00000 27.7702 + 47.0855i
64.3 3.33774i 3.00000i −3.14050 10.1427 4.70380i −10.0132 7.00000i 16.2197i −9.00000 −15.7000 33.8537i
64.4 2.20666i 3.00000i 3.13065 −1.50045 11.0792i 6.61998 7.00000i 24.5616i −9.00000 −24.4480 + 3.31098i
64.5 0.428319i 3.00000i 7.81654 −1.76884 + 11.0395i 1.28496 7.00000i 6.77452i −9.00000 4.72844 + 0.757628i
64.6 0.428319i 3.00000i 7.81654 −1.76884 11.0395i 1.28496 7.00000i 6.77452i −9.00000 4.72844 0.757628i
64.7 2.20666i 3.00000i 3.13065 −1.50045 + 11.0792i 6.61998 7.00000i 24.5616i −9.00000 −24.4480 3.31098i
64.8 3.33774i 3.00000i −3.14050 10.1427 + 4.70380i −10.0132 7.00000i 16.2197i −9.00000 −15.7000 + 33.8537i
64.9 4.88936i 3.00000i −15.9059 −9.63020 5.67972i 14.6681 7.00000i 38.6546i −9.00000 27.7702 47.0855i
64.10 5.18660i 3.00000i −18.9008 −4.24321 10.3438i −15.5598 7.00000i 56.5383i −9.00000 53.6494 22.0078i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 64.10 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 105.4.d.b 10
3.b odd 2 1 315.4.d.b 10
5.b even 2 1 inner 105.4.d.b 10
5.c odd 4 1 525.4.a.w 5
5.c odd 4 1 525.4.a.x 5
15.d odd 2 1 315.4.d.b 10
15.e even 4 1 1575.4.a.bo 5
15.e even 4 1 1575.4.a.bp 5

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
105.4.d.b 10 1.a even 1 1 trivial
105.4.d.b 10 5.b even 2 1 inner
315.4.d.b 10 3.b odd 2 1
315.4.d.b 10 15.d odd 2 1
525.4.a.w 5 5.c odd 4 1
525.4.a.x 5 5.c odd 4 1
1575.4.a.bo 5 15.e even 4 1
1575.4.a.bp 5 15.e even 4 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{10} + 67 T_{2}^{8} + 1523 T_{2}^{6} + 13329 T_{2}^{4} + 37280 T_{2}^{2} + 6400$$ acting on $$S_{4}^{\mathrm{new}}(105, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$6400 + 37280 T^{2} + 13329 T^{4} + 1523 T^{6} + 67 T^{8} + T^{10}$$
$3$ $$( 9 + T^{2} )^{5}$$
$5$ $$30517578125 + 3417968750 T + 556640625 T^{2} + 18000000 T^{3} + 353750 T^{4} - 184700 T^{5} + 2830 T^{6} + 1152 T^{7} + 285 T^{8} + 14 T^{9} + T^{10}$$
$7$ $$( 49 + T^{2} )^{5}$$
$11$ $$( -55852416 + 1472448 T + 140456 T^{2} - 2100 T^{3} - 66 T^{4} + T^{5} )^{2}$$
$13$ $$1712383850521600 + 20084730574080 T^{2} + 36081658496 T^{4} + 25849248 T^{6} + 8308 T^{8} + T^{10}$$
$17$ $$53832671581241344 + 2085927144738816 T^{2} + 3681475123712 T^{4} + 694030608 T^{6} + 45544 T^{8} + T^{10}$$
$19$ $$( 784374624 - 40426032 T - 1513744 T^{2} - 3640 T^{3} + 174 T^{4} + T^{5} )^{2}$$
$23$ $$238300185600000000 + 4736362037760000 T^{2} + 5601652211200 T^{4} + 921968016 T^{6} + 52808 T^{8} + T^{10}$$
$29$ $$( -1150048 - 14986160 T + 1029840 T^{2} + 38440 T^{3} + 370 T^{4} + T^{5} )^{2}$$
$31$ $$( 52737095200 - 1618226480 T + 13826640 T^{2} - 6904 T^{3} - 342 T^{4} + T^{5} )^{2}$$
$37$ $$99\!\cdots\!64$$$$+ 28828609951449153536 T^{2} + 1752974284050432 T^{4} + 39259401728 T^{6} + 345664 T^{8} + T^{10}$$
$41$ $$( 531402107648 - 12602553024 T + 74825096 T^{2} + 47852 T^{3} - 802 T^{4} + T^{5} )^{2}$$
$43$ $$17\!\cdots\!00$$$$+ 20629646614603694080 T^{2} + 1345199401488384 T^{4} + 31766978048 T^{6} + 309312 T^{8} + T^{10}$$
$47$ $$43\!\cdots\!00$$$$+$$$$17\!\cdots\!00$$$$T^{2} + 7318958884495360 T^{4} + 98765465856 T^{6} + 536048 T^{8} + T^{10}$$
$53$ $$21\!\cdots\!24$$$$+ 87341928855767905536 T^{2} + 4262799772404352 T^{4} + 73000876448 T^{6} + 491764 T^{8} + T^{10}$$
$59$ $$( 33724261457920 + 47252515840 T - 380481408 T^{2} - 599408 T^{3} + 704 T^{4} + T^{5} )^{2}$$
$61$ $$( 1105143174112 + 47217562320 T + 214899120 T^{2} - 445560 T^{3} - 650 T^{4} + T^{5} )^{2}$$
$67$ $$24\!\cdots\!64$$$$+$$$$46\!\cdots\!40$$$$T^{2} + 20650444285009920 T^{4} + 244281647360 T^{6} + 990640 T^{8} + T^{10}$$
$71$ $$( 237519904000 - 40906747200 T + 376607000 T^{2} + 92060 T^{3} - 1470 T^{4} + T^{5} )^{2}$$
$73$ $$37\!\cdots\!24$$$$+$$$$45\!\cdots\!76$$$$T^{2} + 17951407658069632 T^{4} + 273800955808 T^{6} + 1491284 T^{8} + T^{10}$$
$79$ $$( -43229481181184 + 35855795200 T + 590031680 T^{2} - 728400 T^{3} - 820 T^{4} + T^{5} )^{2}$$
$83$ $$16\!\cdots\!96$$$$+$$$$11\!\cdots\!80$$$$T^{2} + 16188157928407040 T^{4} + 246934425600 T^{6} + 974720 T^{8} + T^{10}$$
$89$ $$( -1125486676224 + 41779572288 T - 206349496 T^{2} - 1347380 T^{3} + 286 T^{4} + T^{5} )^{2}$$
$97$ $$39\!\cdots\!56$$$$+$$$$21\!\cdots\!16$$$$T^{2} + 340713117665820288 T^{4} + 1579332498848 T^{6} + 2344596 T^{8} + T^{10}$$