Properties

 Label 21.4.g.a Level $21$ Weight $4$ Character orbit 21.g Analytic conductor $1.239$ Analytic rank $0$ Dimension $12$ CM no Inner twists $4$

Related objects

Newspace parameters

 Level: $$N$$ $$=$$ $$21 = 3 \cdot 7$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 21.g (of order $$6$$, degree $$2$$, minimal)

Newform invariants

 Self dual: no Analytic conductor: $$1.23904011012$$ Analytic rank: $$0$$ Dimension: $$12$$ Relative dimension: $$6$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\mathbb{Q}[x]/(x^{12} - \cdots)$$ Defining polynomial: $$x^{12} - x^{11} - 29 x^{9} + 6 x^{8} - 49 x^{7} + 1564 x^{6} - 441 x^{5} + 486 x^{4} - 21141 x^{3} - 59049 x + 531441$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$3^{3}$$ 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 $$1,\beta_1,\ldots,\beta_{11}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( \beta_{2} - \beta_{6} ) q^{2} + ( \beta_{7} + \beta_{8} ) q^{3} + ( -2 \beta_{4} + \beta_{7} + \beta_{10} - \beta_{11} ) q^{4} + \beta_{5} q^{5} + ( 2 - \beta_{2} - \beta_{3} + 4 \beta_{4} - \beta_{5} + 2 \beta_{6} - \beta_{7} + \beta_{9} - \beta_{10} ) q^{6} + ( -5 + 3 \beta_{3} - \beta_{4} - \beta_{7} - \beta_{8} - \beta_{10} + 3 \beta_{11} ) q^{7} + ( \beta_{1} - 2 \beta_{2} - \beta_{5} - \beta_{7} - 2 \beta_{8} - \beta_{9} + \beta_{11} ) q^{8} + ( \beta_{1} - 3 \beta_{3} + \beta_{5} - \beta_{7} + \beta_{8} - 2 \beta_{9} - 2 \beta_{11} ) q^{9} +O(q^{10})$$ $$q + ( \beta_{2} - \beta_{6} ) q^{2} + ( \beta_{7} + \beta_{8} ) q^{3} + ( -2 \beta_{4} + \beta_{7} + \beta_{10} - \beta_{11} ) q^{4} + \beta_{5} q^{5} + ( 2 - \beta_{2} - \beta_{3} + 4 \beta_{4} - \beta_{5} + 2 \beta_{6} - \beta_{7} + \beta_{9} - \beta_{10} ) q^{6} + ( -5 + 3 \beta_{3} - \beta_{4} - \beta_{7} - \beta_{8} - \beta_{10} + 3 \beta_{11} ) q^{7} + ( \beta_{1} - 2 \beta_{2} - \beta_{5} - \beta_{7} - 2 \beta_{8} - \beta_{9} + \beta_{11} ) q^{8} + ( \beta_{1} - 3 \beta_{3} + \beta_{5} - \beta_{7} + \beta_{8} - 2 \beta_{9} - 2 \beta_{11} ) q^{9} + ( -6 \beta_{1} - 2 \beta_{3} - 5 \beta_{7} - 4 \beta_{8} + \beta_{10} - \beta_{11} ) q^{10} + ( 2 \beta_{1} - 2 \beta_{5} + 4 \beta_{6} - 2 \beta_{7} + 2 \beta_{8} + \beta_{9} - 4 \beta_{11} ) q^{11} + ( -10 + 2 \beta_{1} - 4 \beta_{2} + 2 \beta_{3} + 10 \beta_{4} + \beta_{5} - 4 \beta_{6} - 2 \beta_{7} - \beta_{8} - 4 \beta_{10} + 5 \beta_{11} ) q^{12} + ( -9 + 2 \beta_{1} + 3 \beta_{3} - 18 \beta_{4} + 5 \beta_{7} + 3 \beta_{10} - 2 \beta_{11} ) q^{13} + ( -4 \beta_{1} - 10 \beta_{2} + \beta_{5} + 9 \beta_{6} + 4 \beta_{7} - \beta_{8} + 2 \beta_{9} + 5 \beta_{11} ) q^{14} + ( 3 + 4 \beta_{1} + 12 \beta_{2} + \beta_{5} + \beta_{7} + 2 \beta_{8} + \beta_{9} + 4 \beta_{11} ) q^{15} + ( 24 + 3 \beta_{1} - \beta_{3} + 24 \beta_{4} + 3 \beta_{7} + 2 \beta_{8} - 2 \beta_{11} ) q^{16} + ( -9 \beta_{1} + 8 \beta_{2} - 4 \beta_{6} + 9 \beta_{7} + 9 \beta_{8} ) q^{17} + ( 6 \beta_{1} - 6 \beta_{4} - 3 \beta_{6} + 6 \beta_{7} + 6 \beta_{10} - 18 \beta_{11} ) q^{18} + ( 17 + \beta_{1} + \beta_{3} - 17 \beta_{4} - \beta_{7} - 2 \beta_{10} + 2 \beta_{11} ) q^{19} + ( 3 \beta_{1} + 12 \beta_{2} + 3 \beta_{5} - 24 \beta_{6} - 3 \beta_{7} - 3 \beta_{9} - 3 \beta_{11} ) q^{20} + ( 13 - 16 \beta_{1} + 16 \beta_{2} - 5 \beta_{3} - 31 \beta_{4} - 3 \beta_{5} - 20 \beta_{6} - \beta_{7} + \beta_{9} + 4 \beta_{10} + 8 \beta_{11} ) q^{21} + ( -16 + 3 \beta_{1} - 5 \beta_{3} + 8 \beta_{7} + 16 \beta_{8} + 5 \beta_{10} + 3 \beta_{11} ) q^{22} + ( 10 \beta_{1} + 4 \beta_{2} - 4 \beta_{6} - 10 \beta_{7} - 5 \beta_{8} - 5 \beta_{11} ) q^{23} + ( 40 - 10 \beta_{1} - 20 \beta_{2} + 10 \beta_{3} + 20 \beta_{4} + 10 \beta_{6} - 3 \beta_{7} - 8 \beta_{8} - 2 \beta_{9} - 5 \beta_{10} + 5 \beta_{11} ) q^{24} + ( 11 \beta_{1} + 13 \beta_{4} - 4 \beta_{7} - 11 \beta_{8} - 15 \beta_{10} - 7 \beta_{11} ) q^{25} + ( 19 \beta_{2} - 7 \beta_{5} + 19 \beta_{6} - 9 \beta_{8} + 9 \beta_{11} ) q^{26} + ( 21 - 24 \beta_{2} + 3 \beta_{3} + 42 \beta_{4} + 3 \beta_{5} + 48 \beta_{6} - 3 \beta_{7} - 3 \beta_{9} + 3 \beta_{10} ) q^{27} + ( -6 - 7 \beta_{1} - 9 \beta_{3} + 80 \beta_{4} - 11 \beta_{7} + 10 \beta_{8} - 4 \beta_{10} + 12 \beta_{11} ) q^{28} + ( 7 \beta_{1} - 52 \beta_{2} + 5 \beta_{5} - 7 \beta_{7} - 14 \beta_{8} + 5 \beta_{9} + 7 \beta_{11} ) q^{29} + ( -138 + 4 \beta_{1} - 12 \beta_{2} + 18 \beta_{3} - 138 \beta_{4} - 5 \beta_{5} + 12 \beta_{6} + 4 \beta_{7} - 7 \beta_{8} + 10 \beta_{9} + 7 \beta_{11} ) q^{30} + ( -110 - 12 \beta_{1} - 4 \beta_{3} - 55 \beta_{4} - 10 \beta_{7} - 8 \beta_{8} + 2 \beta_{10} - 2 \beta_{11} ) q^{31} + ( 9 \beta_{1} + 10 \beta_{5} - 9 \beta_{7} + 9 \beta_{8} - 5 \beta_{9} - 18 \beta_{11} ) q^{32} + ( -49 - 10 \beta_{1} - 16 \beta_{2} - 10 \beta_{3} + 49 \beta_{4} - 5 \beta_{5} - 16 \beta_{6} + 10 \beta_{7} + 7 \beta_{8} + 20 \beta_{10} - 16 \beta_{11} ) q^{33} + ( -4 + 4 \beta_{1} - 14 \beta_{3} - 8 \beta_{4} - 10 \beta_{7} - 14 \beta_{10} - 4 \beta_{11} ) q^{34} + ( -9 \beta_{1} - 12 \beta_{2} - 3 \beta_{5} + 36 \beta_{6} + 9 \beta_{7} + 3 \beta_{8} - 13 \beta_{9} + 6 \beta_{11} ) q^{35} + ( 66 - \beta_{1} + 48 \beta_{2} + 3 \beta_{3} - 4 \beta_{5} - 8 \beta_{7} - 16 \beta_{8} - 4 \beta_{9} - 3 \beta_{10} - \beta_{11} ) q^{36} + ( 135 + 19 \beta_{1} + 11 \beta_{3} + 135 \beta_{4} + 19 \beta_{7} + 4 \beta_{8} - 4 \beta_{11} ) q^{37} + ( -3 \beta_{1} + 26 \beta_{2} - 13 \beta_{6} + 3 \beta_{7} + 3 \beta_{8} + 3 \beta_{9} ) q^{38} + ( -5 \beta_{1} + 63 \beta_{4} + 4 \beta_{5} - 36 \beta_{6} - 16 \beta_{7} + \beta_{8} - 2 \beta_{9} - 15 \beta_{10} + 25 \beta_{11} ) q^{39} + ( 132 - 13 \beta_{1} - 13 \beta_{3} - 132 \beta_{4} + 13 \beta_{7} + 26 \beta_{8} + 26 \beta_{10} ) q^{40} + ( -6 \beta_{1} + 32 \beta_{2} - 16 \beta_{5} - 64 \beta_{6} + 6 \beta_{7} + 16 \beta_{9} + 6 \beta_{11} ) q^{41} + ( 14 + 30 \beta_{1} + 47 \beta_{2} + 14 \beta_{3} - 140 \beta_{4} + 17 \beta_{5} - 43 \beta_{6} + 5 \beta_{7} - 3 \beta_{8} - 8 \beta_{9} - 7 \beta_{10} - 6 \beta_{11} ) q^{42} + ( -87 + 18 \beta_{1} + 31 \beta_{3} - 13 \beta_{7} - 26 \beta_{8} - 31 \beta_{10} + 18 \beta_{11} ) q^{43} + ( -22 \beta_{1} - 12 \beta_{2} - 3 \beta_{5} + 12 \beta_{6} + 22 \beta_{7} + 11 \beta_{8} + 6 \beta_{9} + 11 \beta_{11} ) q^{44} + ( 270 - 30 \beta_{3} + 135 \beta_{4} + 6 \beta_{7} + 21 \beta_{8} + 12 \beta_{9} + 15 \beta_{10} - 15 \beta_{11} ) q^{45} + ( -10 \beta_{1} - 100 \beta_{4} + 24 \beta_{7} + 10 \beta_{8} + 34 \beta_{10} - 14 \beta_{11} ) q^{46} + ( 20 \beta_{2} + 14 \beta_{5} + 20 \beta_{6} + 27 \beta_{8} - 27 \beta_{11} ) q^{47} + ( 44 + 9 \beta_{1} - 4 \beta_{2} - 4 \beta_{3} + 88 \beta_{4} + 5 \beta_{5} + 8 \beta_{6} + 19 \beta_{7} - 5 \beta_{9} - 4 \beta_{10} - 9 \beta_{11} ) q^{48} + ( 42 - 21 \beta_{1} - 7 \beta_{3} + 175 \beta_{4} + 14 \beta_{7} - 21 \beta_{8} + 35 \beta_{10} - 42 \beta_{11} ) q^{49} + ( -15 \beta_{1} - 7 \beta_{2} - 7 \beta_{5} + 15 \beta_{7} + 30 \beta_{8} - 7 \beta_{9} - 15 \beta_{11} ) q^{50} + ( -219 + 5 \beta_{1} - 12 \beta_{2} - 39 \beta_{3} - 219 \beta_{4} + 5 \beta_{5} + 12 \beta_{6} - 13 \beta_{7} + 13 \beta_{8} - 10 \beta_{9} - 22 \beta_{11} ) q^{51} + ( -308 + 58 \beta_{1} + 40 \beta_{3} - 154 \beta_{4} + 38 \beta_{7} + 18 \beta_{8} - 20 \beta_{10} + 20 \beta_{11} ) q^{52} + ( -28 \beta_{1} - 6 \beta_{5} + 40 \beta_{6} + 28 \beta_{7} - 28 \beta_{8} + 3 \beta_{9} + 56 \beta_{11} ) q^{53} + ( -228 + 15 \beta_{1} - 21 \beta_{2} + 15 \beta_{3} + 228 \beta_{4} + 3 \beta_{5} - 21 \beta_{6} - 15 \beta_{7} - 15 \beta_{8} - 30 \beta_{10} + 33 \beta_{11} ) q^{54} + ( -150 - 28 \beta_{1} + 19 \beta_{3} - 300 \beta_{4} - 9 \beta_{7} + 19 \beta_{10} + 28 \beta_{11} ) q^{55} + ( 45 \beta_{1} - 66 \beta_{2} - 6 \beta_{5} + 2 \beta_{6} - 45 \beta_{7} - 15 \beta_{8} + 23 \beta_{9} - 30 \beta_{11} ) q^{56} + ( 30 - 7 \beta_{1} + 12 \beta_{2} - 6 \beta_{3} - \beta_{5} + 20 \beta_{7} + 40 \beta_{8} - \beta_{9} + 6 \beta_{10} - 7 \beta_{11} ) q^{57} + ( 436 - 93 \beta_{1} - 25 \beta_{3} + 436 \beta_{4} - 93 \beta_{7} - 34 \beta_{8} + 34 \beta_{11} ) q^{58} + ( 54 \beta_{1} - 32 \beta_{2} + 16 \beta_{6} - 54 \beta_{7} - 54 \beta_{8} - 19 \beta_{9} ) q^{59} + ( -6 \beta_{1} + 144 \beta_{4} - 24 \beta_{5} + 72 \beta_{6} - 21 \beta_{7} - 6 \beta_{8} + 12 \beta_{9} - 27 \beta_{10} + 39 \beta_{11} ) q^{60} + ( 114 + 38 \beta_{1} + 38 \beta_{3} - 114 \beta_{4} - 38 \beta_{7} - 89 \beta_{8} - 76 \beta_{10} - 13 \beta_{11} ) q^{61} + ( 6 \beta_{1} - 31 \beta_{2} + 22 \beta_{5} + 62 \beta_{6} - 6 \beta_{7} - 22 \beta_{9} - 6 \beta_{11} ) q^{62} + ( -45 + 23 \beta_{1} - 72 \beta_{2} + 6 \beta_{3} - 282 \beta_{4} - 25 \beta_{5} + 48 \beta_{6} - 11 \beta_{7} + 23 \beta_{8} + 20 \beta_{9} - 30 \beta_{10} - 7 \beta_{11} ) q^{63} + ( -84 - 79 \beta_{1} - 61 \beta_{3} - 18 \beta_{7} - 36 \beta_{8} + 61 \beta_{10} - 79 \beta_{11} ) q^{64} + ( 12 \beta_{1} + 84 \beta_{2} + 14 \beta_{5} - 84 \beta_{6} - 12 \beta_{7} - 6 \beta_{8} - 28 \beta_{9} - 6 \beta_{11} ) q^{65} + ( 332 + 25 \beta_{1} - 40 \beta_{2} - 28 \beta_{3} + 166 \beta_{4} + 20 \beta_{6} - 21 \beta_{7} - 7 \beta_{8} - 19 \beta_{9} + 14 \beta_{10} - 14 \beta_{11} ) q^{66} + ( -62 \beta_{1} - 181 \beta_{4} - 23 \beta_{7} + 62 \beta_{8} + 39 \beta_{10} + 85 \beta_{11} ) q^{67} + ( -48 \beta_{2} + 6 \beta_{5} - 48 \beta_{6} - 30 \beta_{8} + 30 \beta_{11} ) q^{68} + ( 143 - 15 \beta_{1} - 4 \beta_{2} + 11 \beta_{3} + 286 \beta_{4} - 19 \beta_{5} + 8 \beta_{6} + 11 \beta_{7} + 19 \beta_{9} + 11 \beta_{10} + 15 \beta_{11} ) q^{69} + ( -252 + 119 \beta_{1} + 49 \beta_{3} + 168 \beta_{4} + 63 \beta_{7} - 56 \beta_{10} - 14 \beta_{11} ) q^{70} + ( -18 \beta_{1} + 68 \beta_{2} + 4 \beta_{5} + 18 \beta_{7} + 36 \beta_{8} + 4 \beta_{9} - 18 \beta_{11} ) q^{71} + ( -456 - 54 \beta_{1} + 66 \beta_{2} + 24 \beta_{3} - 456 \beta_{4} + 9 \beta_{5} - 66 \beta_{6} + 54 \beta_{7} + 15 \beta_{8} - 18 \beta_{9} + 39 \beta_{11} ) q^{72} + ( -290 + 38 \beta_{1} - 22 \beta_{3} - 145 \beta_{4} + 49 \beta_{7} + 60 \beta_{8} + 11 \beta_{10} - 11 \beta_{11} ) q^{73} + ( -11 \beta_{1} - 38 \beta_{5} - 67 \beta_{6} + 11 \beta_{7} - 11 \beta_{8} + 19 \beta_{9} + 22 \beta_{11} ) q^{74} + ( -147 + 3 \beta_{1} + 60 \beta_{2} + 3 \beta_{3} + 147 \beta_{4} + 18 \beta_{5} + 60 \beta_{6} - 3 \beta_{7} - \beta_{8} - 6 \beta_{10} - 75 \beta_{11} ) q^{75} + ( 18 + 6 \beta_{1} + 12 \beta_{3} + 36 \beta_{4} + 18 \beta_{7} + 12 \beta_{10} - 6 \beta_{11} ) q^{76} + ( -15 \beta_{1} + 148 \beta_{2} + 23 \beta_{5} - 52 \beta_{6} + 15 \beta_{7} + 54 \beta_{8} + 18 \beta_{9} - 39 \beta_{11} ) q^{77} + ( 336 - 57 \beta_{1} - 141 \beta_{2} - 30 \beta_{3} + 21 \beta_{5} + 3 \beta_{7} + 6 \beta_{8} + 21 \beta_{9} + 30 \beta_{10} - 57 \beta_{11} ) q^{78} + ( 325 + 88 \beta_{1} - 12 \beta_{3} + 325 \beta_{4} + 88 \beta_{7} + 50 \beta_{8} - 50 \beta_{11} ) q^{79} + ( 15 \beta_{1} + 72 \beta_{2} - 36 \beta_{6} - 15 \beta_{7} - 15 \beta_{8} + 37 \beta_{9} ) q^{80} + ( -57 \beta_{1} - 144 \beta_{4} + 42 \beta_{5} - 72 \beta_{6} + 75 \beta_{7} + 6 \beta_{8} - 21 \beta_{9} + 81 \beta_{10} + 33 \beta_{11} ) q^{81} + ( 296 - 4 \beta_{1} - 4 \beta_{3} - 296 \beta_{4} + 4 \beta_{7} + 52 \beta_{8} + 8 \beta_{10} + 44 \beta_{11} ) q^{82} + ( -27 \beta_{1} - 56 \beta_{2} + 13 \beta_{5} + 112 \beta_{6} + 27 \beta_{7} - 13 \beta_{9} + 27 \beta_{11} ) q^{83} + ( 220 + 13 \beta_{1} - 20 \beta_{2} - 20 \beta_{3} - 166 \beta_{4} - 19 \beta_{5} + 88 \beta_{6} + 17 \beta_{7} - 56 \beta_{8} - 3 \beta_{9} + 58 \beta_{10} - 59 \beta_{11} ) q^{84} + ( 54 + 17 \beta_{1} - 12 \beta_{3} + 29 \beta_{7} + 58 \beta_{8} + 12 \beta_{10} + 17 \beta_{11} ) q^{85} + ( -62 \beta_{1} - 133 \beta_{2} - 5 \beta_{5} + 133 \beta_{6} + 62 \beta_{7} + 31 \beta_{8} + 10 \beta_{9} + 31 \beta_{11} ) q^{86} + ( 200 + 112 \beta_{1} + 224 \beta_{2} + 146 \beta_{3} + 100 \beta_{4} - 112 \beta_{6} + 15 \beta_{7} - 58 \beta_{8} - 16 \beta_{9} - 73 \beta_{10} + 73 \beta_{11} ) q^{87} + ( 74 \beta_{1} + 124 \beta_{4} - 53 \beta_{7} - 74 \beta_{8} - 127 \beta_{10} - 21 \beta_{11} ) q^{88} + ( -104 \beta_{2} - 48 \beta_{5} - 104 \beta_{6} + 51 \beta_{8} - 51 \beta_{11} ) q^{89} + ( -18 - 15 \beta_{1} + 144 \beta_{2} - 3 \beta_{3} - 36 \beta_{4} - 6 \beta_{5} - 288 \beta_{6} - 108 \beta_{7} + 6 \beta_{9} - 3 \beta_{10} + 15 \beta_{11} ) q^{90} + ( 143 - 77 \beta_{1} - 55 \beta_{3} + 499 \beta_{4} - 75 \beta_{7} + 9 \beta_{8} + 2 \beta_{10} + 29 \beta_{11} ) q^{91} + ( -6 \beta_{1} + 144 \beta_{2} - 14 \beta_{5} + 6 \beta_{7} + 12 \beta_{8} - 14 \beta_{9} - 6 \beta_{11} ) q^{92} + ( -276 + 8 \beta_{1} - 24 \beta_{2} + 36 \beta_{3} - 276 \beta_{4} - 10 \beta_{5} + 24 \beta_{6} - 102 \beta_{7} - 69 \beta_{8} + 20 \beta_{9} + 14 \beta_{11} ) q^{93} + ( -184 - 98 \beta_{1} - 96 \beta_{3} - 92 \beta_{4} - 50 \beta_{7} - 2 \beta_{8} + 48 \beta_{10} - 48 \beta_{11} ) q^{94} + ( -3 \beta_{1} + 32 \beta_{5} + 36 \beta_{6} + 3 \beta_{7} - 3 \beta_{8} - 16 \beta_{9} + 6 \beta_{11} ) q^{95} + ( -228 - 27 \beta_{1} + 60 \beta_{2} - 27 \beta_{3} + 228 \beta_{4} - 12 \beta_{5} + 60 \beta_{6} + 27 \beta_{7} + 42 \beta_{8} + 54 \beta_{10} + 60 \beta_{11} ) q^{96} + ( -156 + 37 \beta_{1} - 81 \beta_{3} - 312 \beta_{4} - 44 \beta_{7} - 81 \beta_{10} - 37 \beta_{11} ) q^{97} + ( 42 \beta_{1} + 63 \beta_{2} + 7 \beta_{5} - 140 \beta_{6} - 42 \beta_{7} - 63 \beta_{8} - 84 \beta_{9} + 21 \beta_{11} ) q^{98} + ( -303 + 75 \beta_{1} - 132 \beta_{2} + 87 \beta_{3} - 18 \beta_{5} - 93 \beta_{7} - 186 \beta_{8} - 18 \beta_{9} - 87 \beta_{10} + 75 \beta_{11} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$12 q - 3 q^{3} + 14 q^{4} - 56 q^{7} - 3 q^{9} + O(q^{10})$$ $$12 q - 3 q^{3} + 14 q^{4} - 56 q^{7} - 3 q^{9} + 30 q^{10} - 192 q^{12} + 6 q^{15} + 134 q^{16} + 66 q^{18} + 300 q^{19} + 357 q^{21} - 268 q^{22} + 414 q^{24} - 42 q^{25} - 602 q^{28} - 822 q^{30} - 930 q^{31} - 855 q^{33} + 852 q^{36} + 764 q^{37} - 426 q^{39} + 2298 q^{40} + 966 q^{42} - 1012 q^{43} + 2367 q^{45} + 608 q^{46} - 336 q^{49} - 1341 q^{51} - 3000 q^{52} - 4158 q^{54} + 270 q^{57} + 2870 q^{58} - 918 q^{60} + 2358 q^{61} + 1071 q^{63} - 548 q^{64} + 2934 q^{66} + 792 q^{67} - 4242 q^{70} - 2712 q^{72} - 2904 q^{73} - 2418 q^{75} + 4296 q^{78} + 1674 q^{79} + 837 q^{81} + 5040 q^{82} + 3864 q^{84} + 348 q^{85} + 1638 q^{87} - 554 q^{88} - 1218 q^{91} - 1479 q^{93} - 1356 q^{94} - 4410 q^{96} - 3354 q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{12} - x^{11} - 29 x^{9} + 6 x^{8} - 49 x^{7} + 1564 x^{6} - 441 x^{5} + 486 x^{4} - 21141 x^{3} - 59049 x + 531441$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$-538 \nu^{11} - 22601 \nu^{10} + 146502 \nu^{9} - 1327 \nu^{8} + 161148 \nu^{7} - 632573 \nu^{6} + 6980468 \nu^{5} - 17528769 \nu^{4} + 61874280 \nu^{3} - 81309015 \nu^{2} - 145536102 \nu + 142839531$$$$)/ 489398112$$ $$\beta_{2}$$ $$=$$ $$($$$$-661 \nu^{11} - 17114 \nu^{10} - 13815 \nu^{9} + 226448 \nu^{8} + 617943 \nu^{7} + 441628 \nu^{6} - 2790145 \nu^{5} - 21396420 \nu^{4} - 82640331 \nu^{3} + 112284954 \nu^{2} + 410108427 \nu + 575491554$$$$)/ 489398112$$ $$\beta_{3}$$ $$=$$ $$($$$$109 \nu^{11} - 3322 \nu^{10} - 26109 \nu^{9} + 69172 \nu^{8} + 83625 \nu^{7} - 33772 \nu^{6} + 26593 \nu^{5} - 1141056 \nu^{4} - 12229137 \nu^{3} + 61499898 \nu^{2} - 15136227 \nu - 271271106$$$$)/69914016$$ $$\beta_{4}$$ $$=$$ $$($$$$-857 \nu^{11} + 5426 \nu^{10} - 627 \nu^{9} + 6088 \nu^{8} - 24405 \nu^{7} + 289700 \nu^{6} - 658001 \nu^{5} + 2301324 \nu^{4} - 3432699 \nu^{3} - 5390226 \nu^{2} + 4113747 \nu + 173722158$$$$)/ 163132704$$ $$\beta_{5}$$ $$=$$ $$($$$$-4301 \nu^{11} + 145853 \nu^{10} - 231543 \nu^{9} - 1385435 \nu^{8} - 5417499 \nu^{7} + 15945431 \nu^{6} + 12125077 \nu^{5} + 198180063 \nu^{4} - 245384073 \nu^{3} - 938815677 \nu^{2} - 3646715337 \nu + 8693961417$$$$)/ 489398112$$ $$\beta_{6}$$ $$=$$ $$($$$$-4345 \nu^{11} + 19933 \nu^{10} + 136449 \nu^{9} + 345029 \nu^{8} - 1557771 \nu^{7} - 3453101 \nu^{6} - 8432491 \nu^{5} + 16879851 \nu^{4} + 90039843 \nu^{3} + 303484887 \nu^{2} - 575773677 \nu - 1180448559$$$$)/ 489398112$$ $$\beta_{7}$$ $$=$$ $$($$$$-1523 \nu^{11} + 209 \nu^{10} + 6255 \nu^{9} + 6421 \nu^{8} - 82569 \nu^{7} - 227449 \nu^{6} - 641129 \nu^{5} + 1005399 \nu^{4} + 7836021 \nu^{3} - 1371249 \nu^{2} + 13338513 \nu - 151814979$$$$)/54377568$$ $$\beta_{8}$$ $$=$$ $$($$$$-1523 \nu^{11} + 209 \nu^{10} + 6255 \nu^{9} + 6421 \nu^{8} - 82569 \nu^{7} - 227449 \nu^{6} - 641129 \nu^{5} + 1005399 \nu^{4} + 7836021 \nu^{3} - 1371249 \nu^{2} - 149794191 \nu - 151814979$$$$)/54377568$$ $$\beta_{9}$$ $$=$$ $$($$$$22376 \nu^{11} + 116845 \nu^{10} - 61380 \nu^{9} - 2138089 \nu^{8} - 3679350 \nu^{7} + 8057845 \nu^{6} + 34199906 \nu^{5} + 126399249 \nu^{4} - 75013614 \nu^{3} - 784745901 \nu^{2} - 2180220300 \nu + 5512873689$$$$)/ 489398112$$ $$\beta_{10}$$ $$=$$ $$($$$$635 \nu^{11} - 221 \nu^{10} + 4041 \nu^{9} + 4103 \nu^{8} + 30441 \nu^{7} - 164387 \nu^{6} + 335465 \nu^{5} + 2637 \nu^{4} + 2622699 \nu^{3} + 8961597 \nu^{2} + 22720743 \nu - 66193929$$$$)/13226976$$ $$\beta_{11}$$ $$=$$ $$($$$$-12970 \nu^{11} - 10169 \nu^{10} + 146502 \nu^{9} + 359201 \nu^{8} + 86556 \nu^{7} - 23405 \nu^{6} - 12463180 \nu^{5} - 12046257 \nu^{4} + 55832328 \nu^{3} + 181515897 \nu^{2} - 145536102 \nu + 876936699$$$$)/ 244699056$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$-\beta_{8} + \beta_{7}$$$$)/3$$ $$\nu^{2}$$ $$=$$ $$($$$$2 \beta_{11} + 2 \beta_{9} - \beta_{8} + \beta_{7} - \beta_{5} + 3 \beta_{3} - \beta_{1}$$$$)/3$$ $$\nu^{3}$$ $$=$$ $$($$$$2 \beta_{11} - 3 \beta_{10} - \beta_{9} - 6 \beta_{8} - 3 \beta_{7} - \beta_{5} + 3 \beta_{3} - 24 \beta_{2} + 2 \beta_{1} + 21$$$$)/3$$ $$\nu^{4}$$ $$=$$ $$($$$$11 \beta_{11} + 27 \beta_{10} - 7 \beta_{9} + 2 \beta_{8} + 25 \beta_{7} - 24 \beta_{6} + 14 \beta_{5} - 48 \beta_{4} - 19 \beta_{1}$$$$)/3$$ $$\nu^{5}$$ $$=$$ $$($$$$-33 \beta_{11} + 12 \beta_{9} - 10 \beta_{8} + 52 \beta_{7} - 6 \beta_{5} + 216 \beta_{4} + 72 \beta_{3} + 138 \beta_{1} + 216$$$$)/3$$ $$\nu^{6}$$ $$=$$ $$($$$$134 \beta_{11} - 114 \beta_{10} + 8 \beta_{9} - 280 \beta_{8} - 140 \beta_{7} + 8 \beta_{5} + 114 \beta_{3} - 360 \beta_{2} + 134 \beta_{1} - 1629$$$$)/3$$ $$\nu^{7}$$ $$=$$ $$($$$$260 \beta_{11} + 552 \beta_{10} - 10 \beta_{9} + 681 \beta_{8} - 129 \beta_{7} - 912 \beta_{6} + 20 \beta_{5} + 984 \beta_{4} - 406 \beta_{1}$$$$)/3$$ $$\nu^{8}$$ $$=$$ $$($$$$-1282 \beta_{11} - 1582 \beta_{9} - 493 \beta_{8} - 1013 \beta_{7} + 1176 \beta_{6} + 791 \beta_{5} + 3648 \beta_{4} - 27 \beta_{3} - 1176 \beta_{2} + 2537 \beta_{1} + 3648$$$$)/3$$ $$\nu^{9}$$ $$=$$ $$($$$$1824 \beta_{11} + 2709 \beta_{10} + 681 \beta_{9} + 974 \beta_{8} + 487 \beta_{7} + 681 \beta_{5} - 2709 \beta_{3} + 8424 \beta_{2} + 1824 \beta_{1} - 29619$$$$)/3$$ $$\nu^{10}$$ $$=$$ $$($$$$-10351 \beta_{11} - 4467 \beta_{10} + 3905 \beta_{9} + 8606 \beta_{8} - 13073 \beta_{7} + 11088 \beta_{6} - 7810 \beta_{5} + 114192 \beta_{4} + 7409 \beta_{1}$$$$)/3$$ $$\nu^{11}$$ $$=$$ $$($$$$-3799 \beta_{11} - 20608 \beta_{9} - 25200 \beta_{8} - 96240 \beta_{7} + 40080 \beta_{6} + 10304 \beta_{5} - 144912 \beta_{4} - 45840 \beta_{3} - 40080 \beta_{2} - 38242 \beta_{1} - 144912$$$$)/3$$

Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/21\mathbb{Z}\right)^\times$$.

 $$n$$ $$8$$ $$10$$ $$\chi(n)$$ $$-1$$ $$1 + \beta_{4}$$

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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
5.1
 2.70662 + 1.29391i 0.00299931 − 3.00000i −2.23014 + 2.00661i 2.85284 − 0.928053i −2.59957 − 1.49740i −0.232749 + 2.99096i 2.70662 − 1.29391i 0.00299931 + 3.00000i −2.23014 − 2.00661i 2.85284 + 0.928053i −2.59957 + 1.49740i −0.232749 − 2.99096i
−3.93653 2.27276i −2.24112 + 4.68800i 6.33084 + 10.9653i −5.80193 + 10.0492i 19.4769 13.3609i −18.4018 2.09174i 21.1897i −16.9548 21.0128i 45.6790 26.3728i
5.2 −2.24076 1.29370i 5.19615 + 0.00519496i −0.652660 1.13044i 8.05907 13.9587i −11.6366 6.73392i −5.67909 + 17.6280i 24.0767i 26.9999 + 0.0539876i −36.1169 + 20.8521i
5.3 −1.65310 0.954416i −3.47555 3.86271i −2.17818 3.77272i 0.623706 1.08029i 2.05878 + 9.70256i 10.0808 15.5363i 23.5862i −2.84113 + 26.8501i −2.06209 + 1.19055i
5.4 1.65310 + 0.954416i 1.60743 + 4.94127i −2.17818 3.77272i −0.623706 + 1.08029i −2.05878 + 9.70256i 10.0808 15.5363i 23.5862i −21.8323 + 15.8855i −2.06209 + 1.19055i
5.5 2.24076 + 1.29370i 2.59358 4.50260i −0.652660 1.13044i −8.05907 + 13.9587i 11.6366 6.73392i −5.67909 + 17.6280i 24.0767i −13.5467 23.3556i −36.1169 + 20.8521i
5.6 3.93653 + 2.27276i −5.18049 0.403134i 6.33084 + 10.9653i 5.80193 10.0492i −19.4769 13.3609i −18.4018 2.09174i 21.1897i 26.6750 + 4.17686i 45.6790 26.3728i
17.1 −3.93653 + 2.27276i −2.24112 4.68800i 6.33084 10.9653i −5.80193 10.0492i 19.4769 + 13.3609i −18.4018 + 2.09174i 21.1897i −16.9548 + 21.0128i 45.6790 + 26.3728i
17.2 −2.24076 + 1.29370i 5.19615 0.00519496i −0.652660 + 1.13044i 8.05907 + 13.9587i −11.6366 + 6.73392i −5.67909 17.6280i 24.0767i 26.9999 0.0539876i −36.1169 20.8521i
17.3 −1.65310 + 0.954416i −3.47555 + 3.86271i −2.17818 + 3.77272i 0.623706 + 1.08029i 2.05878 9.70256i 10.0808 + 15.5363i 23.5862i −2.84113 26.8501i −2.06209 1.19055i
17.4 1.65310 0.954416i 1.60743 4.94127i −2.17818 + 3.77272i −0.623706 1.08029i −2.05878 9.70256i 10.0808 + 15.5363i 23.5862i −21.8323 15.8855i −2.06209 1.19055i
17.5 2.24076 1.29370i 2.59358 + 4.50260i −0.652660 + 1.13044i −8.05907 13.9587i 11.6366 + 6.73392i −5.67909 17.6280i 24.0767i −13.5467 + 23.3556i −36.1169 20.8521i
17.6 3.93653 2.27276i −5.18049 + 0.403134i 6.33084 10.9653i 5.80193 + 10.0492i −19.4769 + 13.3609i −18.4018 + 2.09174i 21.1897i 26.6750 4.17686i 45.6790 + 26.3728i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 17.6 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
7.d odd 6 1 inner
21.g even 6 1 inner

Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 21.4.g.a 12
3.b odd 2 1 inner 21.4.g.a 12
4.b odd 2 1 336.4.bc.d 12
7.b odd 2 1 147.4.g.d 12
7.c even 3 1 147.4.c.a 12
7.c even 3 1 147.4.g.d 12
7.d odd 6 1 inner 21.4.g.a 12
7.d odd 6 1 147.4.c.a 12
12.b even 2 1 336.4.bc.d 12
21.c even 2 1 147.4.g.d 12
21.g even 6 1 inner 21.4.g.a 12
21.g even 6 1 147.4.c.a 12
21.h odd 6 1 147.4.c.a 12
21.h odd 6 1 147.4.g.d 12
28.f even 6 1 336.4.bc.d 12
84.j odd 6 1 336.4.bc.d 12

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
21.4.g.a 12 1.a even 1 1 trivial
21.4.g.a 12 3.b odd 2 1 inner
21.4.g.a 12 7.d odd 6 1 inner
21.4.g.a 12 21.g even 6 1 inner
147.4.c.a 12 7.c even 3 1
147.4.c.a 12 7.d odd 6 1
147.4.c.a 12 21.g even 6 1
147.4.c.a 12 21.h odd 6 1
147.4.g.d 12 7.b odd 2 1
147.4.g.d 12 7.c even 3 1
147.4.g.d 12 21.c even 2 1
147.4.g.d 12 21.h odd 6 1
336.4.bc.d 12 4.b odd 2 1
336.4.bc.d 12 12.b even 2 1
336.4.bc.d 12 28.f even 6 1
336.4.bc.d 12 84.j odd 6 1

Hecke kernels

This newform subspace is the entire newspace $$S_{4}^{\mathrm{new}}(21, [\chi])$$.

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$254016 - 119952 T^{2} + 41020 T^{4} - 6370 T^{6} + 723 T^{8} - 31 T^{10} + T^{12}$$
$3$ $$387420489 + 43046721 T + 3188646 T^{2} + 177147 T^{3} - 144342 T^{4} - 69255 T^{5} - 36018 T^{6} - 2565 T^{7} - 198 T^{8} + 9 T^{9} + 6 T^{10} + 3 T^{11} + T^{12}$$
$5$ $$2962842624 + 1937507040 T^{2} + 1245448953 T^{4} + 13986756 T^{6} + 121221 T^{8} + 396 T^{10} + T^{12}$$
$7$ $$( 40353607 + 3294172 T + 163268 T^{2} + 10780 T^{3} + 476 T^{4} + 28 T^{5} + T^{6} )^{2}$$
$11$ $$453620224546062336 - 2045138759862912 T^{2} + 7035594641593 T^{4} - 8503453924 T^{6} + 7487013 T^{8} - 3244 T^{10} + T^{12}$$
$13$ $$( 82121472 + 1731204 T^{2} + 4335 T^{4} + T^{6} )^{2}$$
$17$ $$17696515773733945344 + 61355067231370752 T^{2} + 161143714802448 T^{4} + 170413288668 T^{6} + 135747117 T^{8} + 12261 T^{10} + T^{12}$$
$19$ $$( 243972972 - 60952662 T + 6428709 T^{2} - 337950 T^{3} + 9753 T^{4} - 150 T^{5} + T^{6} )^{2}$$
$23$ $$89\!\cdots\!76$$$$- 6200136676834232832 T^{2} + 2919015626694160 T^{4} - 746172164500 T^{6} + 139414053 T^{8} - 14311 T^{10} + T^{12}$$
$29$ $$( 14683734245376 + 3697274560 T^{2} + 120001 T^{4} + T^{6} )^{2}$$
$31$ $$( 33414175107 - 4755813831 T + 176555736 T^{2} + 6984765 T^{3} + 87096 T^{4} + 465 T^{5} + T^{6} )^{2}$$
$37$ $$( 3418867564324 + 49455684446 T + 1421726885 T^{2} - 13915390 T^{3} + 119177 T^{4} - 382 T^{5} + T^{6} )^{2}$$
$41$ $$( -4591113633792 + 4941510336 T^{2} - 172788 T^{4} + T^{6} )^{2}$$
$43$ $$( -6662944 - 23284 T + 253 T^{2} + T^{3} )^{4}$$
$47$ $$11\!\cdots\!04$$$$+$$$$85\!\cdots\!56$$$$T^{2} + 47280242457173456640 T^{4} + 1306125205474320 T^{6} + 26259713937 T^{8} + 185553 T^{10} + T^{12}$$
$53$ $$13\!\cdots\!76$$$$-$$$$30\!\cdots\!92$$$$T^{2} +$$$$50\!\cdots\!89$$$$T^{4} - 37040421831506548 T^{6} + 198439484133 T^{8} - 531100 T^{10} + T^{12}$$
$59$ $$38\!\cdots\!24$$$$+$$$$44\!\cdots\!36$$$$T^{2} +$$$$47\!\cdots\!89$$$$T^{4} + 39639172481816796 T^{6} + 253702703277 T^{8} + 570420 T^{10} + T^{12}$$
$61$ $$( 6477700166618112 - 28202272530048 T - 13856716944 T^{2} + 238521132 T^{3} + 261039 T^{4} - 1179 T^{5} + T^{6} )^{2}$$
$67$ $$( 9665143560577444 + 52759337639610 T + 249067250073 T^{2} + 409138304 T^{3} + 693471 T^{4} - 396 T^{5} + T^{6} )^{2}$$
$71$ $$( 6720226523136 + 3717765184 T^{2} + 225148 T^{4} + T^{6} )^{2}$$
$73$ $$( 4047431204396592 - 4635670445244 T - 51563164023 T^{2} + 61084188 T^{3} + 744837 T^{4} + 1452 T^{5} + T^{6} )^{2}$$
$79$ $$( 363201760969609 - 4951859118549 T + 83464610850 T^{2} + 179364515 T^{3} + 960402 T^{4} - 837 T^{5} + T^{6} )^{2}$$
$83$ $$( -388952511994368 + 75760581456 T^{2} - 567987 T^{4} + T^{6} )^{2}$$
$89$ $$88\!\cdots\!24$$$$+$$$$70\!\cdots\!32$$$$T^{2} +$$$$53\!\cdots\!72$$$$T^{4} + 1923321552966019836 T^{6} + 5981516897085 T^{8} + 2594253 T^{10} + T^{12}$$
$97$ $$( 9887068459035648 + 312291915984 T^{2} + 2159691 T^{4} + T^{6} )^{2}$$