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

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Newspace parameters

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

Newform invariants

Self dual: No
Analytic conductor: \(1.23904011012\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(6\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 3^{3} \)
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.

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

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

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

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:

Inner twists

Char. orbit Parity Mult. Self Twist Proved
1.a Even 1 trivial yes
3.b Odd 1 yes
7.d Odd 1 yes
21.g Even 1 yes

Hecke kernels

There are no other newforms in \(S_{4}^{\mathrm{new}}(21, [\chi])\).