# Properties

 Label 105.4.i.d Level $105$ Weight $4$ Character orbit 105.i 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.i (of order $$3$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$6.19520055060$$ Analytic rank: $$0$$ Dimension: $$10$$ Relative dimension: $$5$$ over $$\Q(\zeta_{3})$$ Coefficient field: $$\mathbb{Q}[x]/(x^{10} - \cdots)$$ Defining polynomial: $$x^{10} - 2 x^{9} + 34 x^{8} + 16 x^{7} + 791 x^{6} - 132 x^{5} + 4906 x^{4} - 1674 x^{3} + 25257 x^{2} - 12852 x + 7056$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + ( \beta_{1} + \beta_{4} ) q^{2} + ( -3 - 3 \beta_{4} ) q^{3} + ( -5 - 5 \beta_{4} + \beta_{7} ) q^{4} + 5 \beta_{4} q^{5} + ( 3 - 3 \beta_{2} ) q^{6} + ( -4 - \beta_{2} - 2 \beta_{4} - \beta_{5} ) q^{7} + ( 6 - 5 \beta_{2} - \beta_{5} - \beta_{7} - \beta_{8} - \beta_{9} ) q^{8} + 9 \beta_{4} q^{9} +O(q^{10})$$ $$q + ( \beta_{1} + \beta_{4} ) q^{2} + ( -3 - 3 \beta_{4} ) q^{3} + ( -5 - 5 \beta_{4} + \beta_{7} ) q^{4} + 5 \beta_{4} q^{5} + ( 3 - 3 \beta_{2} ) q^{6} + ( -4 - \beta_{2} - 2 \beta_{4} - \beta_{5} ) q^{7} + ( 6 - 5 \beta_{2} - \beta_{5} - \beta_{7} - \beta_{8} - \beta_{9} ) q^{8} + 9 \beta_{4} q^{9} + ( -5 - 5 \beta_{1} + 5 \beta_{2} - 5 \beta_{4} ) q^{10} + ( -7 + 4 \beta_{1} - 4 \beta_{2} - 7 \beta_{4} + \beta_{5} - \beta_{6} + \beta_{7} - \beta_{8} - \beta_{9} ) q^{11} + ( -3 \beta_{3} + 15 \beta_{4} - 3 \beta_{7} ) q^{12} + ( 22 + 5 \beta_{2} + 4 \beta_{3} - 2 \beta_{5} - \beta_{6} - \beta_{7} - \beta_{8} - \beta_{9} ) q^{13} + ( 6 - 6 \beta_{1} - 3 \beta_{2} - 2 \beta_{3} + 12 \beta_{4} - \beta_{5} + \beta_{6} - 6 \beta_{7} - 2 \beta_{8} - 3 \beta_{9} ) q^{14} + 15 q^{15} + ( -\beta_{1} + 32 \beta_{4} + 3 \beta_{5} - 3 \beta_{6} + \beta_{7} + 2 \beta_{8} + \beta_{9} ) q^{16} + ( -25 - \beta_{1} + \beta_{2} - 25 \beta_{4} + \beta_{5} + \beta_{7} - \beta_{9} ) q^{17} + ( -9 - 9 \beta_{1} + 9 \beta_{2} - 9 \beta_{4} ) q^{18} + ( 12 \beta_{1} + 13 \beta_{4} - 2 \beta_{7} + 2 \beta_{8} - 2 \beta_{9} ) q^{19} + ( 25 + 5 \beta_{3} ) q^{20} + ( 6 - 3 \beta_{1} + 3 \beta_{2} + 12 \beta_{4} + 3 \beta_{5} + 3 \beta_{8} ) q^{21} + ( -35 - 24 \beta_{2} - 7 \beta_{3} + 2 \beta_{5} - 4 \beta_{6} + 6 \beta_{7} + 6 \beta_{8} + 6 \beta_{9} ) q^{22} + ( -2 \beta_{1} + 8 \beta_{3} + 15 \beta_{4} + 7 \beta_{7} + \beta_{8} - \beta_{9} ) q^{23} + ( -18 - 15 \beta_{1} + 15 \beta_{2} - 18 \beta_{4} + 3 \beta_{6} + 3 \beta_{8} ) q^{24} + ( -25 - 25 \beta_{4} ) q^{25} + ( 50 \beta_{1} - \beta_{3} + 4 \beta_{4} + 5 \beta_{5} - 5 \beta_{6} + 2 \beta_{7} + 2 \beta_{8} + 3 \beta_{9} ) q^{26} + 27 q^{27} + ( 15 - 23 \beta_{1} + 38 \beta_{2} - 7 \beta_{3} + 66 \beta_{4} + 3 \beta_{5} - 7 \beta_{6} - 7 \beta_{7} + 2 \beta_{8} + 7 \beta_{9} ) q^{28} + ( 71 + 5 \beta_{2} + 5 \beta_{6} - 5 \beta_{7} - 5 \beta_{8} - 5 \beta_{9} ) q^{29} + ( 15 \beta_{1} + 15 \beta_{4} ) q^{30} + ( -63 + 3 \beta_{1} - 3 \beta_{2} - 63 \beta_{4} - 7 \beta_{6} - 4 \beta_{7} - 7 \beta_{8} ) q^{31} + ( 10 - \beta_{1} + \beta_{2} + 10 \beta_{4} - 6 \beta_{5} - \beta_{6} + 8 \beta_{7} - \beta_{8} + 6 \beta_{9} ) q^{32} + ( -12 \beta_{1} - 6 \beta_{3} + 21 \beta_{4} - 6 \beta_{5} + 6 \beta_{6} - 9 \beta_{7} - 3 \beta_{8} - 3 \beta_{9} ) q^{33} + ( 49 - 35 \beta_{2} + \beta_{3} + \beta_{5} - 2 \beta_{6} + 3 \beta_{7} + 3 \beta_{8} + 3 \beta_{9} ) q^{34} + ( 10 + 5 \beta_{1} - 10 \beta_{4} - 5 \beta_{8} ) q^{35} + ( 45 + 9 \beta_{3} ) q^{36} + ( -43 \beta_{1} + 2 \beta_{3} + 26 \beta_{4} - 5 \beta_{5} + 5 \beta_{6} + 4 \beta_{7} - 7 \beta_{8} + 2 \beta_{9} ) q^{37} + ( -157 - 15 \beta_{1} + 15 \beta_{2} - 157 \beta_{4} + 4 \beta_{6} + 6 \beta_{7} + 4 \beta_{8} ) q^{38} + ( -66 + 15 \beta_{1} - 15 \beta_{2} - 66 \beta_{4} + 3 \beta_{5} + 6 \beta_{6} + 9 \beta_{7} + 6 \beta_{8} - 3 \beta_{9} ) q^{39} + ( 25 \beta_{1} + 30 \beta_{4} + 5 \beta_{5} - 5 \beta_{6} + 5 \beta_{7} + 5 \beta_{9} ) q^{40} + ( 65 + 16 \beta_{2} + 10 \beta_{3} + 7 \beta_{5} + 9 \beta_{6} - 2 \beta_{7} - 2 \beta_{8} - 2 \beta_{9} ) q^{41} + ( 18 - 9 \beta_{1} + 27 \beta_{2} + 9 \beta_{3} - 18 \beta_{4} - 3 \beta_{5} + 6 \beta_{6} + 6 \beta_{7} + 3 \beta_{8} + 3 \beta_{9} ) q^{42} + ( 34 - 70 \beta_{2} - 10 \beta_{3} - 3 \beta_{5} + \beta_{6} - 4 \beta_{7} - 4 \beta_{8} - 4 \beta_{9} ) q^{43} + ( -99 \beta_{1} - 2 \beta_{3} + 98 \beta_{4} - 17 \beta_{5} + 17 \beta_{6} - 15 \beta_{7} - 4 \beta_{8} - 13 \beta_{9} ) q^{44} + ( -45 - 45 \beta_{4} ) q^{45} + ( 81 + 40 \beta_{1} - 40 \beta_{2} + 81 \beta_{4} - 6 \beta_{6} - 5 \beta_{7} - 6 \beta_{8} ) q^{46} + ( 63 \beta_{1} - 14 \beta_{3} + 28 \beta_{4} - 9 \beta_{5} + 9 \beta_{6} - 13 \beta_{7} - 10 \beta_{8} + \beta_{9} ) q^{47} + ( 96 + 3 \beta_{2} - 3 \beta_{5} + 6 \beta_{6} - 9 \beta_{7} - 9 \beta_{8} - 9 \beta_{9} ) q^{48} + ( 87 + 21 \beta_{1} + 62 \beta_{2} - 14 \beta_{3} + 89 \beta_{4} - \beta_{5} + 7 \beta_{6} - 7 \beta_{7} - 7 \beta_{9} ) q^{49} + ( 25 - 25 \beta_{2} ) q^{50} + ( 3 \beta_{1} - 6 \beta_{3} + 75 \beta_{4} - 3 \beta_{5} + 3 \beta_{6} - 6 \beta_{7} - 3 \beta_{8} ) q^{51} + ( -467 - 6 \beta_{1} + 6 \beta_{2} - 467 \beta_{4} - 18 \beta_{5} - 6 \beta_{6} + 45 \beta_{7} - 6 \beta_{8} + 18 \beta_{9} ) q^{52} + ( 116 + 119 \beta_{1} - 119 \beta_{2} + 116 \beta_{4} + 8 \beta_{5} + 17 \beta_{6} + 18 \beta_{7} + 17 \beta_{8} - 8 \beta_{9} ) q^{53} + ( 27 \beta_{1} + 27 \beta_{4} ) q^{54} + ( 35 + 20 \beta_{2} + 10 \beta_{3} + 5 \beta_{5} - 5 \beta_{6} + 10 \beta_{7} + 10 \beta_{8} + 10 \beta_{9} ) q^{55} + ( 14 - 74 \beta_{1} + 117 \beta_{2} + 12 \beta_{3} - 284 \beta_{4} - 7 \beta_{5} + 8 \beta_{6} + 15 \beta_{7} + 3 \beta_{8} - 3 \beta_{9} ) q^{56} + ( 39 - 36 \beta_{2} + 6 \beta_{5} + 6 \beta_{6} ) q^{57} + ( 101 \beta_{1} + 5 \beta_{3} + 41 \beta_{4} + 25 \beta_{5} - 25 \beta_{6} + 20 \beta_{7} + 10 \beta_{8} + 15 \beta_{9} ) q^{58} + ( -163 - 75 \beta_{1} + 75 \beta_{2} - 163 \beta_{4} + 7 \beta_{5} + 4 \beta_{6} - 35 \beta_{7} + 4 \beta_{8} - 7 \beta_{9} ) q^{59} + ( -75 - 75 \beta_{4} + 15 \beta_{7} ) q^{60} + ( -2 \beta_{1} + 42 \beta_{3} - 38 \beta_{4} - 10 \beta_{5} + 10 \beta_{6} + 36 \beta_{7} - 4 \beta_{8} - 6 \beta_{9} ) q^{61} + ( -51 - 48 \beta_{2} - 24 \beta_{3} + 11 \beta_{5} - 14 \beta_{6} + 25 \beta_{7} + 25 \beta_{8} + 25 \beta_{9} ) q^{62} + ( 18 + 9 \beta_{1} - 18 \beta_{4} - 9 \beta_{8} ) q^{63} + ( -206 - 45 \beta_{2} - 2 \beta_{3} - 11 \beta_{5} - 6 \beta_{6} - 5 \beta_{7} - 5 \beta_{8} - 5 \beta_{9} ) q^{64} + ( -25 \beta_{1} - 20 \beta_{3} + 110 \beta_{4} + 5 \beta_{5} - 5 \beta_{6} - 10 \beta_{7} - 5 \beta_{8} + 10 \beta_{9} ) q^{65} + ( 105 - 72 \beta_{1} + 72 \beta_{2} + 105 \beta_{4} + 12 \beta_{5} - 6 \beta_{6} - 33 \beta_{7} - 6 \beta_{8} - 12 \beta_{9} ) q^{66} + ( 360 + 24 \beta_{1} - 24 \beta_{2} + 360 \beta_{4} - 17 \beta_{5} - 7 \beta_{6} - 39 \beta_{7} - 7 \beta_{8} + 17 \beta_{9} ) q^{67} + ( 6 \beta_{1} - 16 \beta_{3} + 260 \beta_{4} - 4 \beta_{5} + 4 \beta_{6} - 22 \beta_{7} + 2 \beta_{8} - 6 \beta_{9} ) q^{68} + ( 45 + 6 \beta_{2} - 24 \beta_{3} + 3 \beta_{5} + 3 \beta_{6} ) q^{69} + ( -60 + 45 \beta_{1} - 30 \beta_{2} - 5 \beta_{3} - 30 \beta_{4} + 10 \beta_{5} - 15 \beta_{6} + 20 \beta_{7} + 5 \beta_{8} + 10 \beta_{9} ) q^{70} + ( -25 + 71 \beta_{2} + 24 \beta_{3} + 24 \beta_{5} - \beta_{6} + 25 \beta_{7} + 25 \beta_{8} + 25 \beta_{9} ) q^{71} + ( 45 \beta_{1} + 54 \beta_{4} + 9 \beta_{5} - 9 \beta_{6} + 9 \beta_{7} + 9 \beta_{9} ) q^{72} + ( -466 - 180 \beta_{1} + 180 \beta_{2} - 466 \beta_{4} + 17 \beta_{5} + 19 \beta_{6} - 33 \beta_{7} + 19 \beta_{8} - 17 \beta_{9} ) q^{73} + ( 538 - 14 \beta_{1} + 14 \beta_{2} + 538 \beta_{4} + 10 \beta_{5} - 21 \beta_{6} - 47 \beta_{7} - 21 \beta_{8} - 10 \beta_{9} ) q^{74} + 75 \beta_{4} q^{75} + ( 311 - 86 \beta_{2} + 27 \beta_{3} - 26 \beta_{5} - 8 \beta_{6} - 18 \beta_{7} - 18 \beta_{8} - 18 \beta_{9} ) q^{76} + ( 259 - 193 \beta_{1} + 66 \beta_{2} + 18 \beta_{3} - 250 \beta_{4} + 6 \beta_{5} - 16 \beta_{6} - 9 \beta_{7} + 6 \beta_{8} + 13 \beta_{9} ) q^{77} + ( 12 - 150 \beta_{2} + 3 \beta_{3} - 9 \beta_{5} + 6 \beta_{6} - 15 \beta_{7} - 15 \beta_{8} - 15 \beta_{9} ) q^{78} + ( -31 \beta_{1} + 34 \beta_{3} + 33 \beta_{4} - 3 \beta_{5} + 3 \beta_{6} + 19 \beta_{7} + 12 \beta_{8} - 15 \beta_{9} ) q^{79} + ( -160 + 5 \beta_{1} - 5 \beta_{2} - 160 \beta_{4} - 10 \beta_{5} + 5 \beta_{6} + 10 \beta_{7} + 5 \beta_{8} + 10 \beta_{9} ) q^{80} + ( -81 - 81 \beta_{4} ) q^{81} + ( 220 \beta_{1} + 37 \beta_{3} - 25 \beta_{4} + 34 \beta_{5} - 34 \beta_{6} + 67 \beta_{7} + 4 \beta_{8} + 30 \beta_{9} ) q^{82} + ( -93 + 65 \beta_{2} - 70 \beta_{3} - 6 \beta_{5} + 3 \beta_{6} - 9 \beta_{7} - 9 \beta_{8} - 9 \beta_{9} ) q^{83} + ( 153 + 114 \beta_{1} - 45 \beta_{2} + 42 \beta_{3} - 45 \beta_{4} - 3 \beta_{5} - 9 \beta_{8} - 21 \beta_{9} ) q^{84} + ( 125 - 5 \beta_{2} + 10 \beta_{3} - 5 \beta_{6} + 5 \beta_{7} + 5 \beta_{8} + 5 \beta_{9} ) q^{85} + ( -117 \beta_{1} - 79 \beta_{3} + 808 \beta_{4} + 4 \beta_{5} - 4 \beta_{6} - 83 \beta_{7} + 8 \beta_{8} - 4 \beta_{9} ) q^{86} + ( -213 + 15 \beta_{1} - 15 \beta_{2} - 213 \beta_{4} - 15 \beta_{5} + 15 \beta_{7} + 15 \beta_{9} ) q^{87} + ( 894 - 27 \beta_{1} + 27 \beta_{2} + 894 \beta_{4} + 2 \beta_{5} - 7 \beta_{6} - 84 \beta_{7} - 7 \beta_{8} - 2 \beta_{9} ) q^{88} + ( 91 \beta_{1} + 12 \beta_{3} + 225 \beta_{4} + 13 \beta_{5} - 13 \beta_{6} - 6 \beta_{7} + 31 \beta_{8} - 18 \beta_{9} ) q^{89} + ( 45 - 45 \beta_{2} ) q^{90} + ( 52 - 77 \beta_{1} - 35 \beta_{2} - 66 \beta_{3} - 355 \beta_{4} - 18 \beta_{5} + 19 \beta_{6} + 12 \beta_{7} - 5 \beta_{8} - 22 \beta_{9} ) q^{91} + ( -762 + 53 \beta_{2} + 6 \beta_{3} + 3 \beta_{5} - 20 \beta_{6} + 23 \beta_{7} + 23 \beta_{8} + 23 \beta_{9} ) q^{92} + ( -9 \beta_{1} + 12 \beta_{3} + 189 \beta_{4} - 21 \beta_{5} + 21 \beta_{6} - 9 \beta_{7} - 21 \beta_{9} ) q^{93} + ( -856 - 25 \beta_{1} + 25 \beta_{2} - 856 \beta_{4} + 18 \beta_{5} - 15 \beta_{6} + 48 \beta_{7} - 15 \beta_{8} - 18 \beta_{9} ) q^{94} + ( -65 - 60 \beta_{1} + 60 \beta_{2} - 65 \beta_{4} - 10 \beta_{5} - 10 \beta_{6} + 10 \beta_{7} - 10 \beta_{8} + 10 \beta_{9} ) q^{95} + ( 3 \beta_{1} - 6 \beta_{3} - 30 \beta_{4} + 15 \beta_{5} - 15 \beta_{6} - 9 \beta_{7} + 18 \beta_{8} - 3 \beta_{9} ) q^{96} + ( -154 + 102 \beta_{2} - 74 \beta_{3} - 10 \beta_{5} + 20 \beta_{6} - 30 \beta_{7} - 30 \beta_{8} - 30 \beta_{9} ) q^{97} + ( -323 - 27 \beta_{1} + 88 \beta_{2} + 68 \beta_{3} - 1094 \beta_{4} + 6 \beta_{5} + \beta_{6} + 64 \beta_{7} + 5 \beta_{8} - 10 \beta_{9} ) q^{98} + ( 63 + 36 \beta_{2} + 18 \beta_{3} + 9 \beta_{5} - 9 \beta_{6} + 18 \beta_{7} + 18 \beta_{8} + 18 \beta_{9} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$10q - 3q^{2} - 15q^{3} - 25q^{4} - 25q^{5} + 18q^{6} - 32q^{7} + 42q^{8} - 45q^{9} + O(q^{10})$$ $$10q - 3q^{2} - 15q^{3} - 25q^{4} - 25q^{5} + 18q^{6} - 32q^{7} + 42q^{8} - 45q^{9} - 15q^{10} - 43q^{11} - 75q^{12} + 246q^{13} - 23q^{14} + 150q^{15} - 161q^{16} - 124q^{17} - 27q^{18} - 37q^{19} + 250q^{20} + 3q^{21} - 442q^{22} - 77q^{23} - 63q^{24} - 125q^{25} + 79q^{26} + 270q^{27} - 71q^{28} + 720q^{29} - 45q^{30} - 314q^{31} + 59q^{32} - 129q^{33} + 352q^{34} + 155q^{35} + 450q^{36} - 225q^{37} - 759q^{38} - 369q^{39} - 105q^{40} + 682q^{41} + 354q^{42} + 64q^{43} - 679q^{44} - 225q^{45} + 331q^{46} - 25q^{47} + 966q^{48} + 710q^{49} + 150q^{50} - 372q^{51} - 2299q^{52} + 317q^{53} - 81q^{54} + 430q^{55} + 1884q^{56} + 222q^{57} - 8q^{58} - 676q^{59} - 375q^{60} + 188q^{61} - 696q^{62} + 279q^{63} - 2206q^{64} - 615q^{65} + 663q^{66} + 1776q^{67} - 1280q^{68} + 462q^{69} - 475q^{70} - 12q^{71} - 189q^{72} - 2006q^{73} + 2729q^{74} - 375q^{75} + 2834q^{76} + 3731q^{77} - 474q^{78} - 200q^{79} - 805q^{80} - 405q^{81} + 539q^{82} - 664q^{83} + 1821q^{84} + 1240q^{85} - 4262q^{86} - 1080q^{87} + 4529q^{88} - 894q^{89} + 270q^{90} + 2016q^{91} - 7374q^{92} - 942q^{93} - 4233q^{94} - 185q^{95} + 177q^{96} - 1152q^{97} + 2539q^{98} + 774q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{10} - 2 x^{9} + 34 x^{8} + 16 x^{7} + 791 x^{6} - 132 x^{5} + 4906 x^{4} - 1674 x^{3} + 25257 x^{2} - 12852 x + 7056$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$12943327 \nu^{9} + 75909616 \nu^{8} - 373768160 \nu^{7} + 4214332012 \nu^{6} - 4661698144 \nu^{5} + 48270423888 \nu^{4} - 454016335892 \nu^{3} + 299152475424 \nu^{2} - 154004812416 \nu + 1484766936324$$$$)/ 2726355561837$$ $$\beta_{3}$$ $$=$$ $$($$$$75909616 \nu^{9} - 965660510 \nu^{8} + 4754775100 \nu^{7} - 23328533825 \nu^{6} + 59302339340 \nu^{5} - 614057145930 \nu^{4} + 1228852276606 \nu^{3} - 3805574935140 \nu^{2} + 1959124199760 \nu - 35777128730004$$$$)/ 2726355561837$$ $$\beta_{4}$$ $$=$$ $$($$$$-5891932287 \nu^{9} + 12146277730 \nu^{8} - 198200228510 \nu^{7} - 104736425072 \nu^{6} - 4542517142681 \nu^{5} + 647207513852 \nu^{4} - 27554247931158 \nu^{3} - 2849362756538 \nu^{2} - 140436264460887 \nu - 4926976726560$$$$)/ 76337955731436$$ $$\beta_{5}$$ $$=$$ $$($$$$-16488755029 \nu^{9} + 119035582802 \nu^{8} - 655311132625 \nu^{7} + 3399331150877 \nu^{6} - 12389162294075 \nu^{5} + 97601661662829 \nu^{4} - 38711293243072 \nu^{3} + 799993208735817 \nu^{2} - 976894778144850 \nu + 2042580337554330$$$$)/ 114506933597154$$ $$\beta_{6}$$ $$=$$ $$($$$$-30058669202 \nu^{9} + 236260514137 \nu^{8} - 1094116349765 \nu^{7} + 4524205545394 \nu^{6} - 9429985831651 \nu^{5} + 128328781787148 \nu^{4} + 59709531979981 \nu^{3} + 600194442164529 \nu^{2} + 256072942328586 \nu + 1698046934032386$$$$)/ 114506933597154$$ $$\beta_{7}$$ $$=$$ $$($$$$-17494590283 \nu^{9} + 37501567814 \nu^{8} - 599833439770 \nu^{7} - 255208627048 \nu^{6} - 13692815202059 \nu^{5} + 2617408475988 \nu^{4} - 89018972495962 \nu^{3} + 14724535319181 \nu^{2} - 461633838622203 \nu + 235019674123164$$$$)/ 19084488932859$$ $$\beta_{8}$$ $$=$$ $$($$$$-279383812109 \nu^{9} + 132041218924 \nu^{8} - 8344840988516 \nu^{7} - 19453355724956 \nu^{6} - 217927468728697 \nu^{5} - 273866567873244 \nu^{4} - 1142781356570312 \nu^{3} - 1067899558666308 \nu^{2} - 5106750093747669 \nu - 1081834742065392$$$$)/ 229013867194308$$ $$\beta_{9}$$ $$=$$ $$($$$$477073435099 \nu^{9} - 729818535632 \nu^{8} + 16408777816366 \nu^{7} + 13099894135486 \nu^{6} + 401473369686179 \nu^{5} + 104683569051882 \nu^{4} + 2665043905267252 \nu^{3} - 352867789435002 \nu^{2} + 12416920116994941 \nu - 3236223734104032$$$$)/ 229013867194308$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{7} - 12 \beta_{4} - 2 \beta_{2} + 2 \beta_{1} - 12$$ $$\nu^{3}$$ $$=$$ $$-\beta_{9} - \beta_{8} - \beta_{7} - \beta_{5} - 3 \beta_{3} - 24 \beta_{2} - 15$$ $$\nu^{4}$$ $$=$$ $$-3 \beta_{9} + 2 \beta_{8} - 33 \beta_{7} + \beta_{6} - \beta_{5} + 267 \beta_{4} - 30 \beta_{3} - 89 \beta_{1}$$ $$\nu^{5}$$ $$=$$ $$-4 \beta_{9} + 36 \beta_{8} - 132 \beta_{7} + 36 \beta_{6} + 4 \beta_{5} + 804 \beta_{4} + 701 \beta_{2} - 701 \beta_{1} + 804$$ $$\nu^{6}$$ $$=$$ $$72 \beta_{9} + 72 \beta_{8} + 72 \beta_{7} + 68 \beta_{6} + 140 \beta_{5} + 937 \beta_{3} + 3322 \beta_{2} + 7452$$ $$\nu^{7}$$ $$=$$ $$1141 \beta_{9} - 212 \beta_{8} + 5820 \beta_{7} - 929 \beta_{6} + 929 \beta_{5} - 31863 \beta_{4} + 4679 \beta_{3} + 22212 \beta_{1}$$ $$\nu^{8}$$ $$=$$ $$2070 \beta_{9} - 5315 \beta_{8} + 32384 \beta_{7} - 5315 \beta_{6} - 2070 \beta_{5} - 230007 \beta_{4} - 117061 \beta_{2} + 117061 \beta_{1} - 230007$$ $$\nu^{9}$$ $$=$$ $$-27964 \beta_{9} - 27964 \beta_{8} - 27964 \beta_{7} - 8560 \beta_{6} - 36524 \beta_{5} - 163320 \beta_{3} - 727621 \beta_{2} - 1151376$$

## Character values

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

 $$n$$ $$22$$ $$31$$ $$71$$ $$\chi(n)$$ $$1$$ $$\beta_{4}$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
16.1
 −2.05285 + 3.55565i −1.33997 + 2.32090i 0.269375 − 0.466571i 1.22021 − 2.11347i 2.90324 − 5.02855i −2.05285 − 3.55565i −1.33997 − 2.32090i 0.269375 + 0.466571i 1.22021 + 2.11347i 2.90324 + 5.02855i
−2.55285 + 4.42167i −1.50000 2.59808i −9.03412 15.6476i −2.50000 + 4.33013i 15.3171 −2.74187 18.3162i 51.4055 −4.50000 + 7.79423i −12.7643 22.1084i
16.2 −1.83997 + 3.18692i −1.50000 2.59808i −2.77099 4.79950i −2.50000 + 4.33013i 11.0398 5.08172 + 17.8094i −9.04535 −4.50000 + 7.79423i −9.19986 15.9346i
16.3 −0.230625 + 0.399454i −1.50000 2.59808i 3.89362 + 6.74396i −2.50000 + 4.33013i 1.38375 −18.0800 4.01437i −7.28187 −4.50000 + 7.79423i −1.15312 1.99727i
16.4 0.720214 1.24745i −1.50000 2.59808i 2.96258 + 5.13135i −2.50000 + 4.33013i −4.32128 18.2377 3.22290i 20.0582 −4.50000 + 7.79423i 3.60107 + 6.23723i
16.5 2.40324 4.16253i −1.50000 2.59808i −7.55109 13.0789i −2.50000 + 4.33013i −14.4194 −18.4976 0.916253i −34.1365 −4.50000 + 7.79423i 12.0162 + 20.8126i
46.1 −2.55285 4.42167i −1.50000 + 2.59808i −9.03412 + 15.6476i −2.50000 4.33013i 15.3171 −2.74187 + 18.3162i 51.4055 −4.50000 7.79423i −12.7643 + 22.1084i
46.2 −1.83997 3.18692i −1.50000 + 2.59808i −2.77099 + 4.79950i −2.50000 4.33013i 11.0398 5.08172 17.8094i −9.04535 −4.50000 7.79423i −9.19986 + 15.9346i
46.3 −0.230625 0.399454i −1.50000 + 2.59808i 3.89362 6.74396i −2.50000 4.33013i 1.38375 −18.0800 + 4.01437i −7.28187 −4.50000 7.79423i −1.15312 + 1.99727i
46.4 0.720214 + 1.24745i −1.50000 + 2.59808i 2.96258 5.13135i −2.50000 4.33013i −4.32128 18.2377 + 3.22290i 20.0582 −4.50000 7.79423i 3.60107 6.23723i
46.5 2.40324 + 4.16253i −1.50000 + 2.59808i −7.55109 + 13.0789i −2.50000 4.33013i −14.4194 −18.4976 + 0.916253i −34.1365 −4.50000 7.79423i 12.0162 20.8126i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 46.5 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.c even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 105.4.i.d 10
3.b odd 2 1 315.4.j.h 10
7.c even 3 1 inner 105.4.i.d 10
7.c even 3 1 735.4.a.ba 5
7.d odd 6 1 735.4.a.z 5
21.g even 6 1 2205.4.a.bs 5
21.h odd 6 1 315.4.j.h 10
21.h odd 6 1 2205.4.a.br 5

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
105.4.i.d 10 1.a even 1 1 trivial
105.4.i.d 10 7.c even 3 1 inner
315.4.j.h 10 3.b odd 2 1
315.4.j.h 10 21.h odd 6 1
735.4.a.z 5 7.d odd 6 1
735.4.a.ba 5 7.c even 3 1
2205.4.a.br 5 21.h odd 6 1
2205.4.a.bs 5 21.g even 6 1

## Hecke kernels

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$3600 + 6240 T + 15016 T^{2} - 3920 T^{3} + 7632 T^{4} + 1396 T^{5} + 890 T^{6} + 56 T^{7} + 37 T^{8} + 3 T^{9} + T^{10}$$
$3$ $$( 9 + 3 T + T^{2} )^{5}$$
$5$ $$( 25 + 5 T + T^{2} )^{5}$$
$7$ $$4747561509943 + 442921190432 T + 6335516299 T^{2} - 83295492 T^{3} - 42596141 T^{4} - 5295724 T^{5} - 124187 T^{6} - 708 T^{7} + 157 T^{8} + 32 T^{9} + T^{10}$$
$11$ $$1239151234496400 - 159901487474160 T + 17995863767104 T^{2} - 607098522960 T^{3} + 24336479716 T^{4} - 141979732 T^{5} + 13028912 T^{6} - 13004 T^{7} + 5637 T^{8} + 43 T^{9} + T^{10}$$
$13$ $$( -125988247 - 14427235 T + 566590 T^{2} - 1074 T^{3} - 123 T^{4} + T^{5} )^{2}$$
$17$ $$420843889670400 + 36111803259840 T + 3211431836944 T^{2} + 162811095072 T^{3} + 9974336368 T^{4} + 439147088 T^{5} + 18752420 T^{6} + 510304 T^{7} + 11172 T^{8} + 124 T^{9} + T^{10}$$
$19$ $$55363842907800625 + 329853081775525 T + 150603583277911 T^{2} + 4301273273910 T^{3} + 405803000893 T^{4} + 7094264539 T^{5} + 143455885 T^{6} + 855606 T^{7} + 12391 T^{8} + 37 T^{9} + T^{10}$$
$23$ $$23226765850890000 - 4603355697603600 T + 854956767240864 T^{2} - 15757510812624 T^{3} + 587890748244 T^{4} + 611117892 T^{5} + 205575352 T^{6} - 354116 T^{7} + 20309 T^{8} + 77 T^{9} + T^{10}$$
$29$ $$( -15483733056 - 531478100 T + 7718760 T^{2} + 8380 T^{3} - 360 T^{4} + T^{5} )^{2}$$
$31$ $$47\!\cdots\!00$$$$- 10916778172265241210 T + 855366855185536929 T^{2} + 5390402264723808 T^{3} + 118817222621958 T^{4} + 472042681650 T^{5} + 4570952043 T^{6} + 16108188 T^{7} + 123862 T^{8} + 314 T^{9} + T^{10}$$
$37$ $$45\!\cdots\!69$$$$+ 17393215869690041745 T + 6411976229923775163 T^{2} + 51259116910389714 T^{3} + 870274256556361 T^{4} + 3982725424767 T^{5} + 22662594801 T^{6} + 31828802 T^{7} + 176979 T^{8} + 225 T^{9} + T^{10}$$
$41$ $$( -914820763500 + 1167469508 T + 54248916 T^{2} - 154532 T^{3} - 341 T^{4} + T^{5} )^{2}$$
$43$ $$( 45414054020 - 2475134103 T + 37497540 T^{2} - 178918 T^{3} - 32 T^{4} + T^{5} )^{2}$$
$47$ $$19\!\cdots\!64$$$$-$$$$43\!\cdots\!48$$$$T +$$$$11\!\cdots\!92$$$$T^{2} + 31263960846124672 T^{3} + 3729414172928944 T^{4} - 9375650641232 T^{5} + 73997760856 T^{6} - 65314736 T^{7} + 291505 T^{8} + 25 T^{9} + T^{10}$$
$53$ $$16\!\cdots\!00$$$$-$$$$14\!\cdots\!60$$$$T +$$$$15\!\cdots\!64$$$$T^{2} - 307640108281540608 T^{3} + 96405132388377664 T^{4} - 38250479870656 T^{5} + 393040710896 T^{6} - 76091144 T^{7} + 780993 T^{8} - 317 T^{9} + T^{10}$$
$59$ $$17\!\cdots\!04$$$$-$$$$63\!\cdots\!08$$$$T +$$$$27\!\cdots\!48$$$$T^{2} + 15457739778441747424 T^{3} + 75133536317519344 T^{4} + 160694496677200 T^{5} + 349108653220 T^{6} + 393580192 T^{7} + 767716 T^{8} + 676 T^{9} + T^{10}$$
$61$ $$27\!\cdots\!00$$$$+$$$$14\!\cdots\!00$$$$T +$$$$82\!\cdots\!00$$$$T^{2} + 16937270761420800000 T^{3} + 63472207628800000 T^{4} + 88329703168000 T^{5} + 350071443200 T^{6} + 244987840 T^{7} + 703424 T^{8} - 188 T^{9} + T^{10}$$
$67$ $$55\!\cdots\!00$$$$-$$$$39\!\cdots\!20$$$$T +$$$$21\!\cdots\!69$$$$T^{2} -$$$$52\!\cdots\!92$$$$T^{3} + 1108511528597231446 T^{4} - 1490948015170436 T^{5} + 2229587081227 T^{6} - 2397547688 T^{7} + 2806646 T^{8} - 1776 T^{9} + T^{10}$$
$71$ $$( 12244368636072 + 209969685484 T - 75660576 T^{2} - 992348 T^{3} + 6 T^{4} + T^{5} )^{2}$$
$73$ $$42\!\cdots\!76$$$$+$$$$82\!\cdots\!98$$$$T +$$$$14\!\cdots\!73$$$$T^{2} +$$$$29\!\cdots\!00$$$$T^{3} + 5434932382090399330 T^{4} + 5319975443504014 T^{5} + 5975136659395 T^{6} + 4409295148 T^{7} + 4158530 T^{8} + 2006 T^{9} + T^{10}$$
$79$ $$13\!\cdots\!56$$$$-$$$$10\!\cdots\!44$$$$T +$$$$15\!\cdots\!13$$$$T^{2} - 616440938464152420 T^{3} + 8872093725718810 T^{4} - 36641243473684 T^{5} + 248138896947 T^{6} - 225964904 T^{7} + 559434 T^{8} + 200 T^{9} + T^{10}$$
$83$ $$( 217219935694608 + 548038491900 T - 508153176 T^{2} - 1528172 T^{3} + 332 T^{4} + T^{5} )^{2}$$
$89$ $$70\!\cdots\!56$$$$+$$$$19\!\cdots\!88$$$$T +$$$$63\!\cdots\!92$$$$T^{2} -$$$$19\!\cdots\!72$$$$T^{3} + 1075918480107158208 T^{4} + 675530864623320 T^{5} + 2058534452980 T^{6} + 686943432 T^{7} + 2023592 T^{8} + 894 T^{9} + T^{10}$$
$97$ $$( 207081920604160 + 806618639616 T - 591903872 T^{2} - 2146352 T^{3} + 576 T^{4} + T^{5} )^{2}$$