Properties

Label 21.5.h.b
Level $21$
Weight $5$
Character orbit 21.h
Analytic conductor $2.171$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 21 = 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 21.h (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(2.17076922476\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
Defining polynomial: \(x^{16} - 103 x^{14} + 7227 x^{12} - 270898 x^{10} + 7374256 x^{8} - 115494792 x^{6} + 1245573504 x^{4} - 2908017504 x^{2} + 5639409216\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 3^{7} \)
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 \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{1} q^{2} + ( -\beta_{2} + \beta_{7} ) q^{3} + ( -10 \beta_{2} - \beta_{7} - \beta_{9} + \beta_{10} ) q^{4} + ( \beta_{4} - \beta_{13} ) q^{5} + ( 2 + \beta_{3} - \beta_{5} - \beta_{6} - \beta_{8} - \beta_{9} - \beta_{12} ) q^{6} + ( -12 - 4 \beta_{2} + 3 \beta_{4} - \beta_{5} + 2 \beta_{7} + 2 \beta_{8} ) q^{7} + ( 1 - 4 \beta_{3} - 2 \beta_{4} - \beta_{5} - \beta_{6} - 2 \beta_{7} + 2 \beta_{12} + \beta_{13} + \beta_{14} + \beta_{15} ) q^{8} + ( 5 - 4 \beta_{1} + 5 \beta_{2} + 3 \beta_{5} + 3 \beta_{8} + 4 \beta_{10} + \beta_{11} - 3 \beta_{15} ) q^{9} +O(q^{10})\) \( q + \beta_{1} q^{2} + ( -\beta_{2} + \beta_{7} ) q^{3} + ( -10 \beta_{2} - \beta_{7} - \beta_{9} + \beta_{10} ) q^{4} + ( \beta_{4} - \beta_{13} ) q^{5} + ( 2 + \beta_{3} - \beta_{5} - \beta_{6} - \beta_{8} - \beta_{9} - \beta_{12} ) q^{6} + ( -12 - 4 \beta_{2} + 3 \beta_{4} - \beta_{5} + 2 \beta_{7} + 2 \beta_{8} ) q^{7} + ( 1 - 4 \beta_{3} - 2 \beta_{4} - \beta_{5} - \beta_{6} - 2 \beta_{7} + 2 \beta_{12} + \beta_{13} + \beta_{14} + \beta_{15} ) q^{8} + ( 5 - 4 \beta_{1} + 5 \beta_{2} + 3 \beta_{5} + 3 \beta_{8} + 4 \beta_{10} + \beta_{11} - 3 \beta_{15} ) q^{9} + ( 17 \beta_{2} + \beta_{5} + 6 \beta_{6} + \beta_{7} - 6 \beta_{8} + 6 \beta_{9} - 6 \beta_{10} + \beta_{15} ) q^{10} + ( 6 \beta_{1} + \beta_{2} + 6 \beta_{3} + \beta_{5} + \beta_{7} + \beta_{9} - \beta_{10} - 2 \beta_{11} - 2 \beta_{12} - 3 \beta_{14} + \beta_{15} ) q^{11} + ( 57 + 2 \beta_{1} + 57 \beta_{2} - 14 \beta_{4} + \beta_{5} - 4 \beta_{6} - 3 \beta_{8} - 6 \beta_{10} - 2 \beta_{11} + 3 \beta_{13} + 3 \beta_{15} ) q^{12} + ( -32 - 4 \beta_{4} + 5 \beta_{5} + 5 \beta_{6} - 4 \beta_{7} + 2 \beta_{8} - 3 \beta_{9} - 3 \beta_{15} ) q^{13} + ( -2 - 13 \beta_{1} - 3 \beta_{2} + 5 \beta_{3} - 4 \beta_{4} + 4 \beta_{5} - 3 \beta_{6} - 12 \beta_{7} + 6 \beta_{8} + 3 \beta_{9} - 2 \beta_{10} + 6 \beta_{11} + 2 \beta_{12} + 3 \beta_{13} + 2 \beta_{14} + \beta_{15} ) q^{14} + ( -50 + 2 \beta_{3} - 10 \beta_{5} - 10 \beta_{6} - 6 \beta_{8} + 4 \beta_{9} + 4 \beta_{12} - 3 \beta_{13} - 3 \beta_{14} + 4 \beta_{15} ) q^{15} + ( 16 + 16 \beta_{2} + 23 \beta_{4} - 4 \beta_{5} + 12 \beta_{6} + 8 \beta_{8} + 7 \beta_{10} - 8 \beta_{15} ) q^{16} + ( -22 \beta_{1} + 3 \beta_{2} - 22 \beta_{3} - 4 \beta_{5} + 7 \beta_{6} + 13 \beta_{7} - 7 \beta_{8} - 4 \beta_{9} + 4 \beta_{10} - 6 \beta_{11} - 6 \beta_{12} + 2 \beta_{14} - 4 \beta_{15} ) q^{17} + ( -13 \beta_{1} + 110 \beta_{2} - 13 \beta_{3} + 6 \beta_{5} - 9 \beta_{6} + 3 \beta_{7} + 9 \beta_{8} + 11 \beta_{9} - 11 \beta_{10} + \beta_{11} + \beta_{12} + 9 \beta_{14} + 6 \beta_{15} ) q^{18} + ( -166 - 166 \beta_{2} + 13 \beta_{4} - 4 \beta_{5} - \beta_{6} - 5 \beta_{8} + 10 \beta_{10} + 5 \beta_{15} ) q^{19} + ( -1 + 30 \beta_{3} + 28 \beta_{4} + \beta_{5} + \beta_{6} + 28 \beta_{7} - 8 \beta_{8} - 8 \beta_{9} - 2 \beta_{12} - 3 \beta_{13} - 3 \beta_{14} - 9 \beta_{15} ) q^{20} + ( 61 + 18 \beta_{1} - 157 \beta_{2} - 22 \beta_{3} + 6 \beta_{5} - 12 \beta_{6} - 14 \beta_{7} + 16 \beta_{8} - 9 \beta_{9} + 6 \beta_{10} + 3 \beta_{11} + \beta_{12} - 9 \beta_{13} - 6 \beta_{14} + 4 \beta_{15} ) q^{21} + ( 215 - 37 \beta_{4} - 13 \beta_{5} - 13 \beta_{6} - 37 \beta_{7} - 9 \beta_{8} - 32 \beta_{9} + 4 \beta_{15} ) q^{22} + ( -4 + 42 \beta_{1} - 4 \beta_{2} - 33 \beta_{4} - 9 \beta_{5} + 13 \beta_{6} + 4 \beta_{8} + 13 \beta_{10} + 8 \beta_{11} - 2 \beta_{13} - 4 \beta_{15} ) q^{23} + ( 90 \beta_{1} - 78 \beta_{2} + 90 \beta_{3} - 2 \beta_{5} + 10 \beta_{6} - 3 \beta_{7} - 10 \beta_{8} - 21 \beta_{9} + 21 \beta_{10} - 6 \beta_{11} - 6 \beta_{12} - 6 \beta_{14} - 2 \beta_{15} ) q^{24} + ( -256 \beta_{2} + 16 \beta_{5} - 3 \beta_{6} + 32 \beta_{7} + 3 \beta_{8} + 13 \beta_{9} - 13 \beta_{10} + 16 \beta_{15} ) q^{25} + ( 4 + 29 \beta_{1} + 4 \beta_{2} + 22 \beta_{4} + 3 \beta_{5} - 7 \beta_{6} - 4 \beta_{8} - 7 \beta_{10} - 8 \beta_{11} - 5 \beta_{13} + 4 \beta_{15} ) q^{26} + ( 55 - 88 \beta_{3} + 3 \beta_{4} + 12 \beta_{5} + 12 \beta_{6} + 3 \beta_{7} - 15 \beta_{8} + 2 \beta_{9} + 4 \beta_{12} + 9 \beta_{13} + 9 \beta_{14} - 27 \beta_{15} ) q^{27} + ( 92 + 320 \beta_{2} + 33 \beta_{4} + 24 \beta_{5} + 78 \beta_{7} + 22 \beta_{8} + 56 \beta_{9} - 21 \beta_{10} ) q^{28} + ( 1 - 58 \beta_{3} - \beta_{5} - \beta_{6} - 2 \beta_{8} - 2 \beta_{9} + 2 \beta_{12} + 5 \beta_{13} + 5 \beta_{14} - \beta_{15} ) q^{29} + ( 55 - 134 \beta_{1} + 55 \beta_{2} + 15 \beta_{4} - 52 \beta_{5} + 34 \beta_{6} - 18 \beta_{8} + 21 \beta_{10} - \beta_{11} + 6 \beta_{13} + 18 \beta_{15} ) q^{30} + ( 294 \beta_{2} - 13 \beta_{5} - 20 \beta_{6} - 16 \beta_{7} + 20 \beta_{8} - 23 \beta_{9} + 23 \beta_{10} - 13 \beta_{15} ) q^{31} + ( 42 \beta_{1} - 7 \beta_{2} + 42 \beta_{3} + 20 \beta_{5} - 27 \beta_{6} - 62 \beta_{7} + 27 \beta_{8} + 20 \beta_{9} - 20 \beta_{10} + 14 \beta_{11} + 14 \beta_{12} - 5 \beta_{14} + 20 \beta_{15} ) q^{32} + ( -76 - 76 \beta_{1} - 76 \beta_{2} - 3 \beta_{4} + 9 \beta_{5} + 8 \beta_{6} + 17 \beta_{8} - 43 \beta_{10} + 13 \beta_{11} - 17 \beta_{15} ) q^{33} + ( -477 - 102 \beta_{4} - 25 \beta_{5} - 25 \beta_{6} - 102 \beta_{7} - 51 \beta_{8} - 25 \beta_{9} - 26 \beta_{15} ) q^{34} + ( 13 - 38 \beta_{1} + 16 \beta_{2} + 104 \beta_{3} - 51 \beta_{4} + 2 \beta_{5} - 12 \beta_{6} - 76 \beta_{7} + 10 \beta_{8} + 26 \beta_{9} - 15 \beta_{10} - 32 \beta_{11} - 6 \beta_{12} + 5 \beta_{13} + \beta_{14} + 39 \beta_{15} ) q^{35} + ( -634 + 160 \beta_{3} + 75 \beta_{4} + 54 \beta_{5} + 54 \beta_{6} + 75 \beta_{7} + 24 \beta_{8} + 19 \beta_{9} - 22 \beta_{12} - 30 \beta_{15} ) q^{36} + ( 189 + 189 \beta_{2} + 23 \beta_{4} - 36 \beta_{5} - 4 \beta_{6} - 40 \beta_{8} - 9 \beta_{10} + 40 \beta_{15} ) q^{37} + ( -299 \beta_{1} - 13 \beta_{2} - 299 \beta_{3} - 6 \beta_{5} - 7 \beta_{6} + 4 \beta_{7} + 7 \beta_{8} - 6 \beta_{9} + 6 \beta_{10} + 26 \beta_{11} + 26 \beta_{12} + \beta_{14} - 6 \beta_{15} ) q^{38} + ( -94 \beta_{1} + 348 \beta_{2} - 94 \beta_{3} + 10 \beta_{5} + 17 \beta_{6} - 4 \beta_{7} - 17 \beta_{8} + 21 \beta_{9} - 21 \beta_{10} - 14 \beta_{11} - 14 \beta_{12} - 30 \beta_{14} + 10 \beta_{15} ) q^{39} + ( 702 + 702 \beta_{2} - 99 \beta_{4} + 56 \beta_{5} - 38 \beta_{6} + 18 \beta_{8} - 5 \beta_{10} - 18 \beta_{15} ) q^{40} + ( 2 + 220 \beta_{3} + 78 \beta_{4} - 2 \beta_{5} - 2 \beta_{6} + 78 \beta_{7} - 26 \beta_{8} - 26 \beta_{9} + 4 \beta_{12} - 2 \beta_{13} - 2 \beta_{14} - 24 \beta_{15} ) q^{41} + ( -552 + 115 \beta_{1} - 884 \beta_{2} - 188 \beta_{3} + 9 \beta_{4} + 12 \beta_{5} + 40 \beta_{6} + 6 \beta_{7} - 10 \beta_{8} + 44 \beta_{9} - 20 \beta_{10} - 10 \beta_{11} - \beta_{12} + 30 \beta_{13} + 27 \beta_{14} + 45 \beta_{15} ) q^{42} + ( 50 + 72 \beta_{4} + 57 \beta_{5} + 57 \beta_{6} + 72 \beta_{7} + 26 \beta_{8} + 77 \beta_{9} - 31 \beta_{15} ) q^{43} + ( 21 + 366 \beta_{1} + 21 \beta_{2} - 136 \beta_{4} - 59 \beta_{5} + 38 \beta_{6} - 21 \beta_{8} + 38 \beta_{10} - 42 \beta_{11} + \beta_{13} + 21 \beta_{15} ) q^{44} + ( 364 \beta_{1} + 142 \beta_{2} + 364 \beta_{3} - 48 \beta_{5} + 3 \beta_{6} - 93 \beta_{7} - 3 \beta_{8} - 20 \beta_{9} + 20 \beta_{10} + 35 \beta_{11} + 35 \beta_{12} + 27 \beta_{14} - 48 \beta_{15} ) q^{45} + ( -1161 \beta_{2} - 19 \beta_{5} - 80 \beta_{6} + 18 \beta_{7} + 80 \beta_{8} - 43 \beta_{9} + 43 \beta_{10} - 19 \beta_{15} ) q^{46} + ( -26 + 46 \beta_{1} - 26 \beta_{2} + 25 \beta_{4} + 33 \beta_{5} - 7 \beta_{6} + 26 \beta_{8} - 7 \beta_{10} + 52 \beta_{11} + 22 \beta_{13} - 26 \beta_{15} ) q^{47} + ( 1665 - 262 \beta_{3} - 4 \beta_{4} - 45 \beta_{5} - 45 \beta_{6} - 4 \beta_{7} + 4 \beta_{8} - 90 \beta_{9} + 10 \beta_{12} - 39 \beta_{13} - 39 \beta_{14} + 49 \beta_{15} ) q^{48} + ( 791 + 770 \beta_{2} - 42 \beta_{4} - 49 \beta_{5} - 98 \beta_{6} + 21 \beta_{7} - 49 \beta_{8} - 77 \beta_{9} - 14 \beta_{10} + 147 \beta_{15} ) q^{49} + ( -32 - 185 \beta_{3} + 142 \beta_{4} + 32 \beta_{5} + 32 \beta_{6} + 142 \beta_{7} - 29 \beta_{8} - 29 \beta_{9} - 64 \beta_{12} - 23 \beta_{13} - 23 \beta_{14} - 61 \beta_{15} ) q^{50} + ( 948 - 294 \beta_{1} + 948 \beta_{2} - 9 \beta_{4} + 46 \beta_{5} - 2 \beta_{6} + 44 \beta_{8} + 39 \beta_{10} - 24 \beta_{11} - 33 \beta_{13} - 44 \beta_{15} ) q^{51} + ( -1168 \beta_{2} - 36 \beta_{5} + 50 \beta_{6} - 54 \beta_{7} - 50 \beta_{8} + 32 \beta_{9} - 32 \beta_{10} - 36 \beta_{15} ) q^{52} + ( 236 \beta_{1} + 18 \beta_{2} + 236 \beta_{3} - 6 \beta_{5} + 24 \beta_{6} - 35 \beta_{7} - 24 \beta_{8} - 6 \beta_{9} + 6 \beta_{10} - 36 \beta_{11} - 36 \beta_{12} + 71 \beta_{14} - 6 \beta_{15} ) q^{53} + ( -2333 + 49 \beta_{1} - 2333 \beta_{2} + 165 \beta_{4} - 9 \beta_{5} - 24 \beta_{6} - 33 \beta_{8} + 113 \beta_{10} - 28 \beta_{11} - 45 \beta_{13} + 33 \beta_{15} ) q^{54} + ( -2186 + 40 \beta_{4} + 18 \beta_{5} + 18 \beta_{6} + 40 \beta_{7} + 45 \beta_{8} - 32 \beta_{9} + 27 \beta_{15} ) q^{55} + ( -30 - 244 \beta_{1} - 17 \beta_{2} + 306 \beta_{3} + 80 \beta_{4} + 4 \beta_{5} + 39 \beta_{6} + 114 \beta_{7} - \beta_{8} - 18 \beta_{9} + 26 \beta_{10} + 34 \beta_{11} - 26 \beta_{12} - 74 \beta_{13} - 47 \beta_{14} - 48 \beta_{15} ) q^{56} + ( 774 + 70 \beta_{3} - 224 \beta_{4} - 54 \beta_{5} - 54 \beta_{6} - 224 \beta_{7} - 34 \beta_{8} - 69 \beta_{9} + 29 \beta_{12} + 48 \beta_{13} + 48 \beta_{14} + 20 \beta_{15} ) q^{57} + ( -1586 - 1586 \beta_{2} + 85 \beta_{4} + 12 \beta_{5} + 6 \beta_{6} + 18 \beta_{8} + 91 \beta_{10} - 18 \beta_{15} ) q^{58} + ( -584 \beta_{1} - 5 \beta_{2} - 584 \beta_{3} - 13 \beta_{5} + 8 \beta_{6} + 83 \beta_{7} - 8 \beta_{8} - 13 \beta_{9} + 13 \beta_{10} + 10 \beta_{11} + 10 \beta_{12} - 49 \beta_{14} - 13 \beta_{15} ) q^{59} + ( -208 \beta_{1} + 2294 \beta_{2} - 208 \beta_{3} + 16 \beta_{5} - 90 \beta_{6} + 3 \beta_{7} + 90 \beta_{8} - 41 \beta_{9} + 41 \beta_{10} + 64 \beta_{11} + 64 \beta_{12} + 16 \beta_{15} ) q^{60} + ( 1895 + 1895 \beta_{2} + 175 \beta_{4} - 83 \beta_{5} + 124 \beta_{6} + 41 \beta_{8} - 32 \beta_{10} - 41 \beta_{15} ) q^{61} + ( 16 + 269 \beta_{3} - 230 \beta_{4} - 16 \beta_{5} - 16 \beta_{6} - 230 \beta_{7} + 46 \beta_{8} + 46 \beta_{9} + 32 \beta_{12} + 76 \beta_{13} + 76 \beta_{14} + 62 \beta_{15} ) q^{62} + ( -1360 + 276 \beta_{1} - 1116 \beta_{2} - 20 \beta_{3} - 153 \beta_{4} - 39 \beta_{5} - 114 \beta_{6} + 87 \beta_{7} - 27 \beta_{8} + 58 \beta_{9} - 111 \beta_{10} - 24 \beta_{11} - 22 \beta_{12} + 9 \beta_{13} - 36 \beta_{14} - 18 \beta_{15} ) q^{63} + ( 460 - 57 \beta_{4} + 24 \beta_{5} + 24 \beta_{6} - 57 \beta_{7} - 20 \beta_{8} + 7 \beta_{9} - 44 \beta_{15} ) q^{64} + ( -16 + 306 \beta_{1} - 16 \beta_{2} + 232 \beta_{4} + 104 \beta_{5} - 88 \beta_{6} + 16 \beta_{8} - 88 \beta_{10} + 32 \beta_{11} + 48 \beta_{13} - 16 \beta_{15} ) q^{65} + ( 362 \beta_{1} + 2219 \beta_{2} + 362 \beta_{3} + 118 \beta_{5} - 46 \beta_{6} + 309 \beta_{7} + 46 \beta_{8} + 207 \beta_{9} - 207 \beta_{10} - 47 \beta_{11} - 47 \beta_{12} - 30 \beta_{14} + 118 \beta_{15} ) q^{66} + ( -920 \beta_{2} + 67 \beta_{5} + 190 \beta_{6} + 79 \beta_{7} - 190 \beta_{8} + 202 \beta_{9} - 202 \beta_{10} + 67 \beta_{15} ) q^{67} + ( 54 + 52 \beta_{1} + 54 \beta_{2} - 146 \beta_{4} - 66 \beta_{5} + 12 \beta_{6} - 54 \beta_{8} + 12 \beta_{10} - 108 \beta_{11} + 56 \beta_{13} + 54 \beta_{15} ) q^{68} + ( 3258 - 150 \beta_{3} + 12 \beta_{4} + 47 \beta_{5} + 47 \beta_{6} + 12 \beta_{7} - 38 \beta_{8} + 174 \beta_{9} - 81 \beta_{12} + 30 \beta_{13} + 30 \beta_{14} - 85 \beta_{15} ) q^{69} + ( 2366 + 3143 \beta_{2} - 49 \beta_{4} + 35 \beta_{5} + 294 \beta_{6} - 420 \beta_{7} + 42 \beta_{8} - 119 \beta_{9} + 112 \beta_{10} - 343 \beta_{15} ) q^{70} + ( 66 - 90 \beta_{3} - 300 \beta_{4} - 66 \beta_{5} - 66 \beta_{6} - 300 \beta_{7} + 98 \beta_{8} + 98 \beta_{9} + 132 \beta_{12} - 60 \beta_{13} - 60 \beta_{14} + 164 \beta_{15} ) q^{71} + ( 2709 - 432 \beta_{1} + 2709 \beta_{2} - 180 \beta_{4} + 57 \beta_{5} - 132 \beta_{6} - 75 \beta_{8} - 234 \beta_{10} + 72 \beta_{11} + 63 \beta_{13} + 75 \beta_{15} ) q^{72} + ( -1545 \beta_{2} + 88 \beta_{5} + 28 \beta_{6} - 103 \beta_{7} - 28 \beta_{8} - 163 \beta_{9} + 163 \beta_{10} + 88 \beta_{15} ) q^{73} + ( 7 \beta_{1} - 23 \beta_{2} + 7 \beta_{3} - 44 \beta_{5} + 21 \beta_{6} + 194 \beta_{7} - 21 \beta_{8} - 44 \beta_{9} + 44 \beta_{10} + 46 \beta_{11} + 46 \beta_{12} - 85 \beta_{14} - 44 \beta_{15} ) q^{74} + ( -2502 - 394 \beta_{1} - 2502 \beta_{2} - 224 \beta_{4} + 43 \beta_{5} + 14 \beta_{6} + 57 \beta_{8} + 135 \beta_{10} + 16 \beta_{11} + 48 \beta_{13} - 57 \beta_{15} ) q^{75} + ( -5344 + 498 \beta_{4} + 46 \beta_{5} + 46 \beta_{6} + 498 \beta_{7} + 130 \beta_{8} + 284 \beta_{9} + 84 \beta_{15} ) q^{76} + ( 31 - 166 \beta_{1} - 34 \beta_{2} - 368 \beta_{3} + 132 \beta_{4} - 13 \beta_{5} - 83 \beta_{6} + 11 \beta_{7} - 30 \beta_{8} - 64 \beta_{9} - 18 \beta_{10} + 68 \beta_{11} + 130 \beta_{12} + 83 \beta_{13} + 116 \beta_{14} - 33 \beta_{15} ) q^{77} + ( -1918 + 307 \beta_{3} - 12 \beta_{4} - 134 \beta_{5} - 134 \beta_{6} - 12 \beta_{7} - 75 \beta_{8} - 145 \beta_{9} + 20 \beta_{12} - 51 \beta_{13} - 51 \beta_{14} + 59 \beta_{15} ) q^{78} + ( -5084 - 5084 \beta_{2} - 246 \beta_{4} + 166 \beta_{5} + 27 \beta_{6} + 193 \beta_{8} - 107 \beta_{10} - 193 \beta_{15} ) q^{79} + ( 474 \beta_{1} + 83 \beta_{2} + 474 \beta_{3} + 108 \beta_{5} - 25 \beta_{6} - 358 \beta_{7} + 25 \beta_{8} + 108 \beta_{9} - 108 \beta_{10} - 166 \beta_{11} - 166 \beta_{12} + 117 \beta_{14} + 108 \beta_{15} ) q^{80} + ( -56 \beta_{1} + 2998 \beta_{2} - 56 \beta_{3} - 138 \beta_{5} + 144 \beta_{6} + 69 \beta_{7} - 144 \beta_{8} - 254 \beta_{9} + 254 \beta_{10} - 103 \beta_{11} - 103 \beta_{12} + 18 \beta_{14} - 138 \beta_{15} ) q^{81} + ( 6098 + 6098 \beta_{2} - 310 \beta_{4} - 136 \beta_{5} - 110 \beta_{6} - 246 \beta_{8} - 336 \beta_{10} + 246 \beta_{15} ) q^{82} + ( -49 - 112 \beta_{3} + 22 \beta_{4} + 49 \beta_{5} + 49 \beta_{6} + 22 \beta_{7} + 62 \beta_{8} + 62 \beta_{9} - 98 \beta_{12} - 159 \beta_{13} - 159 \beta_{14} + 13 \beta_{15} ) q^{83} + ( -2997 - 404 \beta_{1} - 5164 \beta_{2} - 382 \beta_{3} + 448 \beta_{4} - 25 \beta_{5} + 127 \beta_{6} + 64 \beta_{8} - 204 \beta_{9} + 402 \beta_{10} + 110 \beta_{11} + 46 \beta_{12} - 57 \beta_{13} + 39 \beta_{14} - 299 \beta_{15} ) q^{84} + ( 1629 + 411 \beta_{4} - 419 \beta_{5} - 419 \beta_{6} + 411 \beta_{7} + 12 \beta_{8} - 32 \beta_{9} + 431 \beta_{15} ) q^{85} + ( -72 - 373 \beta_{1} - 72 \beta_{2} + 486 \beta_{4} + 155 \beta_{5} - 83 \beta_{6} + 72 \beta_{8} - 83 \beta_{10} + 144 \beta_{11} - 165 \beta_{13} - 72 \beta_{15} ) q^{86} + ( 112 \beta_{1} + 130 \beta_{2} + 112 \beta_{3} - 32 \beta_{5} + 8 \beta_{6} - 3 \beta_{7} - 8 \beta_{8} - 115 \beta_{9} + 115 \beta_{10} - 70 \beta_{11} - 70 \beta_{12} + 6 \beta_{14} - 32 \beta_{15} ) q^{87} + ( -7146 \beta_{2} - 278 \beta_{5} + 116 \beta_{6} - 717 \beta_{7} - 116 \beta_{8} - 323 \beta_{9} + 323 \beta_{10} - 278 \beta_{15} ) q^{88} + ( -23 + 416 \beta_{1} - 23 \beta_{2} - 75 \beta_{4} - 85 \beta_{5} + 108 \beta_{6} + 23 \beta_{8} + 108 \beta_{10} + 46 \beta_{11} - 226 \beta_{13} - 23 \beta_{15} ) q^{89} + ( 8771 + 136 \beta_{3} - 84 \beta_{4} + 78 \beta_{5} + 78 \beta_{6} - 84 \beta_{7} + 255 \beta_{8} + 61 \beta_{9} + 137 \beta_{12} + 27 \beta_{13} + 27 \beta_{14} + 177 \beta_{15} ) q^{90} + ( 528 + 3410 \beta_{2} - 6 \beta_{4} + 65 \beta_{5} - 98 \beta_{6} - 102 \beta_{7} + 17 \beta_{8} + 28 \beta_{9} + 259 \beta_{10} - 147 \beta_{15} ) q^{91} + ( 46 - 264 \beta_{3} - 30 \beta_{4} - 46 \beta_{5} - 46 \beta_{6} - 30 \beta_{7} - 90 \beta_{8} - 90 \beta_{9} + 92 \beta_{12} + 254 \beta_{13} + 254 \beta_{14} - 44 \beta_{15} ) q^{92} + ( 1045 + 704 \beta_{1} + 1045 \beta_{2} + 308 \beta_{4} + 157 \beta_{5} - 106 \beta_{6} + 51 \beta_{8} - 117 \beta_{10} + 7 \beta_{11} - 69 \beta_{13} - 51 \beta_{15} ) q^{93} + ( -867 \beta_{2} + 299 \beta_{5} - 364 \beta_{6} + 858 \beta_{7} + 364 \beta_{8} + 195 \beta_{9} - 195 \beta_{10} + 299 \beta_{15} ) q^{94} + ( -138 \beta_{1} + 34 \beta_{2} - 138 \beta_{3} - 137 \beta_{5} + 171 \beta_{6} + 541 \beta_{7} - 171 \beta_{8} - 137 \beta_{9} + 137 \beta_{10} - 68 \beta_{11} - 68 \beta_{12} - 96 \beta_{14} - 137 \beta_{15} ) q^{95} + ( -4966 + 824 \beta_{1} - 4966 \beta_{2} + 21 \beta_{4} - 202 \beta_{5} + 16 \beta_{6} - 186 \beta_{8} - 211 \beta_{10} + 16 \beta_{11} + 78 \beta_{13} + 186 \beta_{15} ) q^{96} + ( -1782 - 305 \beta_{4} + 124 \beta_{5} + 124 \beta_{6} - 305 \beta_{7} + 2 \beta_{8} - 185 \beta_{9} - 122 \beta_{15} ) q^{97} + ( -21 + 966 \beta_{1} + 42 \beta_{2} + 469 \beta_{3} - 154 \beta_{4} - 126 \beta_{5} + 210 \beta_{6} + 616 \beta_{7} - 189 \beta_{8} - 147 \beta_{9} + 147 \beta_{10} - 84 \beta_{11} - 126 \beta_{12} + 175 \beta_{13} - 112 \beta_{14} - 168 \beta_{15} ) q^{98} + ( -3053 - 922 \beta_{3} + 120 \beta_{4} + 123 \beta_{5} + 123 \beta_{6} + 120 \beta_{7} + 285 \beta_{8} + 290 \beta_{9} - 8 \beta_{12} - 126 \beta_{13} - 126 \beta_{14} + 162 \beta_{15} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16q + 8q^{3} + 78q^{4} + 28q^{6} - 168q^{7} + 62q^{9} + O(q^{10}) \) \( 16q + 8q^{3} + 78q^{4} + 28q^{6} - 168q^{7} + 62q^{9} - 98q^{10} + 436q^{12} - 492q^{13} - 856q^{15} + 150q^{16} - 884q^{18} - 1326q^{19} + 2170q^{21} + 3244q^{22} + 630q^{24} + 2094q^{25} + 1028q^{27} - 854q^{28} + 340q^{30} - 2504q^{31} - 616q^{33} - 7728q^{34} - 9644q^{36} + 1342q^{37} - 2626q^{39} + 5754q^{40} - 1582q^{42} + 1460q^{43} - 1330q^{45} + 8844q^{46} + 25864q^{48} + 5572q^{49} + 7794q^{51} + 9536q^{52} - 18578q^{54} - 35140q^{55} + 11696q^{57} - 12446q^{58} - 18890q^{60} + 15012q^{61} - 13076q^{63} + 7660q^{64} - 17192q^{66} + 8658q^{67} + 53676q^{69} + 13790q^{70} + 21312q^{72} + 12322q^{73} - 19514q^{75} - 84520q^{76} - 32120q^{78} - 40168q^{79} - 23986q^{81} + 47348q^{82} - 5992q^{84} + 22536q^{85} - 1162q^{87} + 56430q^{88} + 139636q^{90} - 17878q^{91} + 8556q^{93} + 6468q^{94} - 40894q^{96} - 28268q^{97} - 47812q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{16} - 103 x^{14} + 7227 x^{12} - 270898 x^{10} + 7374256 x^{8} - 115494792 x^{6} + 1245573504 x^{4} - 2908017504 x^{2} + 5639409216\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\(-805319043514 \nu^{14} + 80361724322821 \nu^{12} - 5587950577515285 \nu^{10} + 202461849561777583 \nu^{8} - 5446135918837629556 \nu^{6} + 80467830722179252872 \nu^{4} - 907204248570216420432 \nu^{2} + 391646080902383389152\)\()/ \)\(17\!\cdots\!16\)\( \)
\(\beta_{3}\)\(=\)\((\)\(-805319043514 \nu^{15} + 80361724322821 \nu^{13} - 5587950577515285 \nu^{11} + 202461849561777583 \nu^{9} - 5446135918837629556 \nu^{7} + 80467830722179252872 \nu^{5} - 907204248570216420432 \nu^{3} + 391646080902383389152 \nu\)\()/ \)\(17\!\cdots\!16\)\( \)
\(\beta_{4}\)\(=\)\((\)\(-6321739747609 \nu^{15} - 94449672964612 \nu^{14} + 429246974607973 \nu^{13} + 6712279566892468 \nu^{12} - 38372021913949581 \nu^{11} - 397955562800194884 \nu^{10} + 1203885006206065612 \nu^{9} + 7279332122620291264 \nu^{8} - 49612101043375649332 \nu^{7} - 122134963715006887648 \nu^{6} + 234375516743104232496 \nu^{5} - 935747929193683773600 \nu^{4} - 549341601510783633552 \nu^{3} - 5825871483117175357056 \nu^{2} - 315767011778113132395648 \nu + 12576097242870408140544\)\()/ \)\(55\!\cdots\!12\)\( \)
\(\beta_{5}\)\(=\)\((\)\(-6321739747609 \nu^{15} - 98639052169252 \nu^{14} + 429246974607973 \nu^{13} + 12437033129559676 \nu^{12} - 38372021913949581 \nu^{11} - 1119363343681169964 \nu^{10} + 1203885006206065612 \nu^{9} + 55315054452012603400 \nu^{8} - 49612101043375649332 \nu^{7} - 2025190544087176326976 \nu^{6} + 234375516743104232496 \nu^{5} + 39010908881134238393664 \nu^{4} - 549341601510783633552 \nu^{3} - 495726374824244211844224 \nu^{2} - 315767011778113132395648 \nu + 452155990731001604869248\)\()/ \)\(55\!\cdots\!12\)\( \)
\(\beta_{6}\)\(=\)\((\)\(6321739747609 \nu^{15} + 110831311084076 \nu^{14} - 429246974607973 \nu^{13} - 20147918440724948 \nu^{12} + 38372021913949581 \nu^{11} + 1435670287592271972 \nu^{10} - 1203885006206065612 \nu^{9} - 68256104062193307704 \nu^{8} + 49612101043375649332 \nu^{7} + 1616100295058211874112 \nu^{6} - 234375516743104232496 \nu^{5} - 28613163938975394046272 \nu^{4} + 549341601510783633552 \nu^{3} + 173356528647599427570048 \nu^{2} + 315767011778113132395648 \nu - 1113682782702408683635584\)\()/ \)\(55\!\cdots\!12\)\( \)
\(\beta_{7}\)\(=\)\((\)\(-77376912306302 \nu^{15} - 18998427328522 \nu^{14} + 7637689986396467 \nu^{13} + 2919215791853626 \nu^{12} - 526824418509818007 \nu^{11} - 202987600221861210 \nu^{10} + 18945258963662567345 \nu^{9} + 8971608294619825072 \nu^{8} - 507347369722288259660 \nu^{7} - 197836048352900378536 \nu^{6} + 7745493595266117275364 \nu^{5} + 2923070207363415871632 \nu^{4} - 84064995383405584009392 \nu^{3} - 2841681697868694070656 \nu^{2} + 196152069983548451700816 \nu + 14226915037251732645312\)\()/ \)\(27\!\cdots\!56\)\( \)
\(\beta_{8}\)\(=\)\((\)\(161075564360213 \nu^{15} + 127149780827396 \nu^{14} - 15704626947400907 \nu^{13} - 7115746942871348 \nu^{12} + 1092020858933585595 \nu^{11} + 529478554383915972 \nu^{10} - 39094402933531200302 \nu^{9} - 13594650857918322656 \nu^{8} + 1064306840487952168652 \nu^{7} + 732906487396756344512 \nu^{6} - 15725362707275338783224 \nu^{5} - 15563785284292982571072 \nu^{4} + 168679332368321951652336 \nu^{3} + 334478281880395743846336 \nu^{2} - 76537128188983771005984 \nu - 1050169973142986622912384\)\()/ \)\(55\!\cdots\!12\)\( \)
\(\beta_{9}\)\(=\)\((\)\(161075564360213 \nu^{15} + 215202916713528 \nu^{14} - 15704626947400907 \nu^{13} - 19977595949332296 \nu^{12} + 1092020858933585595 \nu^{11} + 1306249761262412952 \nu^{10} - 39094402933531200302 \nu^{9} - 40982320542571814304 \nu^{8} + 1064306840487952168652 \nu^{7} + 919161450888299683632 \nu^{6} - 15725362707275338783224 \nu^{5} - 7978546543051389013632 \nu^{4} + 168679332368321951652336 \nu^{3} + 18700534921880363174784 \nu^{2} - 76537128188983771005984 \nu + 1249263695776735067861376\)\()/ \)\(55\!\cdots\!12\)\( \)
\(\beta_{10}\)\(=\)\((\)\(6321739747609 \nu^{15} - 492819382147164 \nu^{14} - 429246974607973 \nu^{13} + 52721790270962028 \nu^{12} + 38372021913949581 \nu^{11} - 3748900319674026588 \nu^{10} - 1203885006206065612 \nu^{9} + 145409154882066784896 \nu^{8} + 49612101043375649332 \nu^{7} - 4007695730290408864032 \nu^{6} - 234375516743104232496 \nu^{5} + 64816829032528581119136 \nu^{4} + 549341601510783633552 \nu^{3} - 686560517921195165485440 \nu^{2} + 315767011778113132395648 \nu + 1603567065162021512926464\)\()/ \)\(55\!\cdots\!12\)\( \)
\(\beta_{11}\)\(=\)\((\)\(-59901524063219 \nu^{15} - 174880886372968 \nu^{14} + 5846108600691887 \nu^{13} + 13820871442188070 \nu^{12} - 363593359708009071 \nu^{11} - 859623667940280270 \nu^{10} + 11407389349723905092 \nu^{9} + 21930915251333067958 \nu^{8} - 209291429167028173028 \nu^{7} - 491336612136214059496 \nu^{6} + 2220820726023469752336 \nu^{5} + 4584276049851494884152 \nu^{4} - 5205275837916498115632 \nu^{3} - 62415773241530269903776 \nu^{2} + 391498220799881990391744 \nu + 144295716109861493745120\)\()/ \)\(27\!\cdots\!56\)\( \)
\(\beta_{12}\)\(=\)\((\)\(-55949115474713 \nu^{15} + 51619834308255 \nu^{14} + 5118995444625851 \nu^{13} - 4992137542047363 \nu^{12} - 355949225730737835 \nu^{11} + 313324732170438795 \nu^{10} + 12200524608296085962 \nu^{9} - 9830259869532560340 \nu^{8} - 346915713845951978636 \nu^{7} + 189882650841475572060 \nu^{6} + 5125754361007219667832 \nu^{5} - 1913780988020985264720 \nu^{4} - 64956062208233394345168 \nu^{3} + 4485620031945037478640 \nu^{2} + 24947629246869435604512 \nu - 49975764111290691830016\)\()/ \)\(92\!\cdots\!52\)\( \)
\(\beta_{13}\)\(=\)\((\)\(-341327704587575 \nu^{15} - 94449672964612 \nu^{14} + 32099060125324043 \nu^{13} + 6712279566892468 \nu^{12} - 2071808502592380675 \nu^{11} - 397955562800194884 \nu^{10} + 65000984248225690100 \nu^{9} + 7279332122620291264 \nu^{8} - 1387356410265731679020 \nu^{7} - 122134963715006887648 \nu^{6} + 12654563511841432750800 \nu^{5} - 935747929193683773600 \nu^{4} - 29660428199224143039600 \nu^{3} - 5825871483117175357056 \nu^{2} - 1223764228859739589717632 \nu + 12576097242870408140544\)\()/ \)\(55\!\cdots\!12\)\( \)
\(\beta_{14}\)\(=\)\((\)\(-199118570462318 \nu^{15} - 18998427328522 \nu^{14} + 23081790900629129 \nu^{13} + 2919215791853626 \nu^{12} - 1685091215368565205 \nu^{11} - 202987600221861210 \nu^{10} + 69555989973991327439 \nu^{9} + 8971608294619825072 \nu^{8} - 1958262061773825165908 \nu^{7} - 197836048352900378536 \nu^{6} + 32855706634924223446716 \nu^{5} + 2923070207363415871632 \nu^{4} - 343393935698374187416272 \nu^{3} - 2841681697868694070656 \nu^{2} + 802590173831901669892272 \nu + 14226915037251732645312\)\()/ \)\(27\!\cdots\!56\)\( \)
\(\beta_{15}\)\(=\)\((\)\(161075564360213 \nu^{15} - 739692280405892 \nu^{14} - 15704626947400907 \nu^{13} + 64745476344002804 \nu^{12} + 1092020858933585595 \nu^{11} - 4247520605378080836 \nu^{10} - 39094402933531200302 \nu^{9} + 130244617536069961184 \nu^{8} + 1064306840487952168652 \nu^{7} - 3205489119547478962304 \nu^{6} - 15725362707275338783224 \nu^{5} + 38273509283943815493696 \nu^{4} + 168679332368321951652336 \nu^{3} - 387706521438839797516224 \nu^{2} - 76537128188983771005984 \nu - 76003940681626824655488\)\()/ \)\(55\!\cdots\!12\)\( \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{10} - \beta_{9} - \beta_{7} - 26 \beta_{2}\)
\(\nu^{3}\)\(=\)\(\beta_{15} + \beta_{14} + \beta_{13} + 2 \beta_{12} - 2 \beta_{7} - \beta_{6} - \beta_{5} - 2 \beta_{4} - 36 \beta_{3} + 1\)
\(\nu^{4}\)\(=\)\(-8 \beta_{15} + 55 \beta_{10} + 8 \beta_{8} + 12 \beta_{6} - 4 \beta_{5} + 71 \beta_{4} - 976 \beta_{2} - 976\)
\(\nu^{5}\)\(=\)\(20 \beta_{15} + 59 \beta_{14} + 142 \beta_{12} + 142 \beta_{11} - 20 \beta_{10} + 20 \beta_{9} + 91 \beta_{8} - 190 \beta_{7} - 91 \beta_{6} + 20 \beta_{5} - 1494 \beta_{3} - 71 \beta_{2} - 1494 \beta_{1}\)
\(\nu^{6}\)\(=\)\(276 \beta_{15} + 2871 \beta_{9} + 940 \beta_{8} + 4087 \beta_{7} + 664 \beta_{6} + 664 \beta_{5} + 4087 \beta_{4} - 41780\)
\(\nu^{7}\)\(=\)\(-4087 \beta_{15} - 3259 \beta_{13} + 8174 \beta_{11} - 1604 \beta_{10} + 4087 \beta_{8} - 1604 \beta_{6} + 5691 \beta_{5} + 12158 \beta_{4} - 4087 \beta_{2} - 67722 \beta_{1} - 4087\)
\(\nu^{8}\)\(=\)\(56236 \beta_{15} - 148427 \beta_{10} + 148427 \beta_{9} + 14900 \beta_{8} + 219563 \beta_{7} - 14900 \beta_{6} + 56236 \beta_{5} + 1934236 \beta_{2}\)
\(\nu^{9}\)\(=\)\(-317135 \beta_{15} - 174863 \beta_{14} - 174863 \beta_{13} - 439126 \beta_{12} - 97572 \beta_{9} - 97572 \beta_{8} + 687142 \beta_{7} + 219563 \beta_{6} + 219563 \beta_{5} + 687142 \beta_{4} + 3241626 \beta_{3} - 219563\)
\(\nu^{10}\)\(=\)\(2321752 \beta_{15} - 7632983 \beta_{10} - 2321752 \beta_{8} - 3077916 \beta_{6} + 756164 \beta_{5} - 11467063 \beta_{4} + 93768476 \beta_{2} + 93768476\)
\(\nu^{11}\)\(=\)\(-5399668 \beta_{15} - 9198571 \beta_{14} - 22934126 \beta_{12} - 22934126 \beta_{11} + 5399668 \beta_{10} - 5399668 \beta_{9} - 16866731 \beta_{8} + 36864638 \beta_{7} + 16866731 \beta_{6} - 5399668 \beta_{5} + 160001874 \beta_{3} + 11467063 \beta_{2} + 160001874 \beta_{1}\)
\(\nu^{12}\)\(=\)\(-37880532 \beta_{15} - 391263459 \beta_{9} - 162334604 \beta_{8} - 591478595 \beta_{7} - 124454072 \beta_{6} - 124454072 \beta_{5} - 591478595 \beta_{4} + 4661161900\)
\(\nu^{13}\)\(=\)\(591478595 \beta_{15} + 477836999 \beta_{13} - 1182957190 \beta_{11} + 286788676 \beta_{10} - 591478595 \beta_{8} + 286788676 \beta_{6} - 878267271 \beta_{5} - 1929681622 \beta_{4} + 591478595 \beta_{2} + 8027287194 \beta_{1} + 591478595\)
\(\nu^{14}\)\(=\)\(-8418044924 \beta_{15} + 20017474303 \beta_{10} - 20017474303 \beta_{9} - 1900609252 \beta_{8} - 30336128479 \beta_{7} + 1900609252 \beta_{6} - 8418044924 \beta_{5} - 234738533852 \beta_{2}\)
\(\nu^{15}\)\(=\)\(45271609075 \beta_{15} + 24634300723 \beta_{14} + 24634300723 \beta_{13} + 60672256958 \beta_{12} + 14935480596 \beta_{9} + 14935480596 \beta_{8} - 99776870990 \beta_{7} - 30336128479 \beta_{6} - 30336128479 \beta_{5} - 99776870990 \beta_{4} - 406127317026 \beta_{3} + 30336128479\)

Character values

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

\(n\) \(8\) \(10\)
\(\chi(n)\) \(-1\) \(-1 - \beta_{2}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2.1
−6.18666 3.57187i
−4.46939 2.58040i
−4.14633 2.39389i
−1.34450 0.776250i
1.34450 + 0.776250i
4.14633 + 2.39389i
4.46939 + 2.58040i
6.18666 + 3.57187i
−6.18666 + 3.57187i
−4.46939 + 2.58040i
−4.14633 + 2.39389i
−1.34450 + 0.776250i
1.34450 0.776250i
4.14633 2.39389i
4.46939 2.58040i
6.18666 3.57187i
−6.18666 3.57187i −7.57871 4.85420i 17.5165 + 30.3395i 13.6392 + 7.87461i 29.5483 + 57.1014i −44.8988 + 19.6239i 135.967i 33.8736 + 73.5771i −56.2542 97.4351i
2.2 −4.46939 2.58040i 8.54954 2.81164i 5.31698 + 9.20927i −31.7353 18.3224i −45.4664 9.49493i −18.9163 45.2015i 27.6932i 65.1893 48.0765i 94.5584 + 163.780i
2.3 −4.14633 2.39389i 2.75140 + 8.56912i 3.46138 + 5.99529i 30.3128 + 17.5011i 9.10526 42.1170i 48.8998 3.13209i 43.4597i −65.8596 + 47.1542i −83.7912 145.131i
2.4 −1.34450 0.776250i −7.02686 + 5.62345i −6.79487 11.7691i −23.4142 13.5182i 13.8128 2.10615i −27.0847 + 40.8340i 45.9381i 17.7536 79.0304i 20.9870 + 36.3506i
2.5 1.34450 + 0.776250i 8.38348 3.27372i −6.79487 11.7691i 23.4142 + 13.5182i 13.8128 + 2.10615i −27.0847 + 40.8340i 45.9381i 59.5656 54.8903i 20.9870 + 36.3506i
2.6 4.14633 + 2.39389i 6.04537 + 6.66734i 3.46138 + 5.99529i −30.3128 17.5011i 9.10526 + 42.1170i 48.8998 3.13209i 43.4597i −7.90694 + 80.6132i −83.7912 145.131i
2.7 4.46939 + 2.58040i −6.70973 + 5.99830i 5.31698 + 9.20927i 31.7353 + 18.3224i −45.4664 + 9.49493i −18.9163 45.2015i 27.6932i 9.04086 80.4939i 94.5584 + 163.780i
2.8 6.18666 + 3.57187i −0.414505 8.99045i 17.5165 + 30.3395i −13.6392 7.87461i 29.5483 57.1014i −44.8988 + 19.6239i 135.967i −80.6564 + 7.45317i −56.2542 97.4351i
11.1 −6.18666 + 3.57187i −7.57871 + 4.85420i 17.5165 30.3395i 13.6392 7.87461i 29.5483 57.1014i −44.8988 19.6239i 135.967i 33.8736 73.5771i −56.2542 + 97.4351i
11.2 −4.46939 + 2.58040i 8.54954 + 2.81164i 5.31698 9.20927i −31.7353 + 18.3224i −45.4664 + 9.49493i −18.9163 + 45.2015i 27.6932i 65.1893 + 48.0765i 94.5584 163.780i
11.3 −4.14633 + 2.39389i 2.75140 8.56912i 3.46138 5.99529i 30.3128 17.5011i 9.10526 + 42.1170i 48.8998 + 3.13209i 43.4597i −65.8596 47.1542i −83.7912 + 145.131i
11.4 −1.34450 + 0.776250i −7.02686 5.62345i −6.79487 + 11.7691i −23.4142 + 13.5182i 13.8128 + 2.10615i −27.0847 40.8340i 45.9381i 17.7536 + 79.0304i 20.9870 36.3506i
11.5 1.34450 0.776250i 8.38348 + 3.27372i −6.79487 + 11.7691i 23.4142 13.5182i 13.8128 2.10615i −27.0847 40.8340i 45.9381i 59.5656 + 54.8903i 20.9870 36.3506i
11.6 4.14633 2.39389i 6.04537 6.66734i 3.46138 5.99529i −30.3128 + 17.5011i 9.10526 42.1170i 48.8998 + 3.13209i 43.4597i −7.90694 80.6132i −83.7912 + 145.131i
11.7 4.46939 2.58040i −6.70973 5.99830i 5.31698 9.20927i 31.7353 18.3224i −45.4664 9.49493i −18.9163 + 45.2015i 27.6932i 9.04086 + 80.4939i 94.5584 163.780i
11.8 6.18666 3.57187i −0.414505 + 8.99045i 17.5165 30.3395i −13.6392 + 7.87461i 29.5483 + 57.1014i −44.8988 19.6239i 135.967i −80.6564 7.45317i −56.2542 + 97.4351i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 11.8
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 21.5.h.b 16
3.b odd 2 1 inner 21.5.h.b 16
7.b odd 2 1 147.5.h.b 16
7.c even 3 1 inner 21.5.h.b 16
7.c even 3 1 147.5.b.c 8
7.d odd 6 1 147.5.b.f 8
7.d odd 6 1 147.5.h.b 16
21.c even 2 1 147.5.h.b 16
21.g even 6 1 147.5.b.f 8
21.g even 6 1 147.5.h.b 16
21.h odd 6 1 inner 21.5.h.b 16
21.h odd 6 1 147.5.b.c 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
21.5.h.b 16 1.a even 1 1 trivial
21.5.h.b 16 3.b odd 2 1 inner
21.5.h.b 16 7.c even 3 1 inner
21.5.h.b 16 21.h odd 6 1 inner
147.5.b.c 8 7.c even 3 1
147.5.b.c 8 21.h odd 6 1
147.5.b.f 8 7.d odd 6 1
147.5.b.f 8 21.g even 6 1
147.5.h.b 16 7.b odd 2 1
147.5.h.b 16 7.d odd 6 1
147.5.h.b 16 21.c even 2 1
147.5.h.b 16 21.g even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(T_{2}^{16} - \cdots\) acting on \(S_{5}^{\mathrm{new}}(21, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 5639409216 - 2908017504 T^{2} + 1245573504 T^{4} - 115494792 T^{6} + 7374256 T^{8} - 270898 T^{10} + 7227 T^{12} - 103 T^{14} + T^{16} \)
$3$ \( 1853020188851841 - 183014339639688 T + 282429536481 T^{2} - 627621192180 T^{3} + 361463316237 T^{4} - 61068948192 T^{5} + 7273301526 T^{6} - 403186572 T^{7} + 29575530 T^{8} - 4977612 T^{9} + 1108566 T^{10} - 114912 T^{11} + 8397 T^{12} - 180 T^{13} + T^{14} - 8 T^{15} + T^{16} \)
$5$ \( \)\(88\!\cdots\!16\)\( - \)\(61\!\cdots\!24\)\( T^{2} + 3016060690773138561 T^{4} - 6896058986680899 T^{6} + 11177868938314 T^{8} - 11243531563 T^{10} + 8240634 T^{12} - 3547 T^{14} + T^{16} \)
$7$ \( ( 33232930569601 + 1162668124884 T + 12307850135 T^{2} - 526596924 T^{3} - 19275228 T^{4} - 219324 T^{5} + 2135 T^{6} + 84 T^{7} + T^{8} )^{2} \)
$11$ \( \)\(23\!\cdots\!36\)\( - \)\(57\!\cdots\!72\)\( T^{2} + \)\(10\!\cdots\!89\)\( T^{4} - \)\(72\!\cdots\!95\)\( T^{6} + 368603345865319966 T^{8} - 29834307230299 T^{10} + 1772602830 T^{12} - 49255 T^{14} + T^{16} \)
$13$ \( ( 114515728 - 1759704 T - 25046 T^{2} + 123 T^{3} + T^{4} )^{4} \)
$17$ \( \)\(15\!\cdots\!16\)\( - \)\(13\!\cdots\!52\)\( T^{2} + \)\(81\!\cdots\!12\)\( T^{4} - \)\(25\!\cdots\!24\)\( T^{6} + \)\(59\!\cdots\!13\)\( T^{8} - 7744269460819806 T^{10} + 71483950683 T^{12} - 320190 T^{14} + T^{16} \)
$19$ \( ( 277999631912943616 + 9729554956268160 T + 371041661855081 T^{2} - 369092879181 T^{3} + 16112841420 T^{4} + 75286737 T^{5} + 381680 T^{6} + 663 T^{7} + T^{8} )^{2} \)
$23$ \( \)\(53\!\cdots\!96\)\( - \)\(16\!\cdots\!68\)\( T^{2} + \)\(39\!\cdots\!72\)\( T^{4} - \)\(37\!\cdots\!80\)\( T^{6} + \)\(25\!\cdots\!81\)\( T^{8} - 732964885764275862 T^{10} + 1519083112851 T^{12} - 1456782 T^{14} + T^{16} \)
$29$ \( ( 43734935370689127936 + 2935135874614608 T^{2} + 62942341336 T^{4} + 472213 T^{6} + T^{8} )^{2} \)
$31$ \( ( 42915473425920809169 - 583735053955138992 T + 8960460932725596 T^{2} - 2522687119032 T^{3} + 129277610905 T^{4} - 16823792 T^{5} + 1723284 T^{6} + 1252 T^{7} + T^{8} )^{2} \)
$37$ \( ( \)\(66\!\cdots\!44\)\( - 14154002595162093924 T + 275363602547633901 T^{2} - 628410655962867 T^{3} + 1558866218290 T^{4} - 377232455 T^{5} + 1529010 T^{6} - 671 T^{7} + T^{8} )^{2} \)
$41$ \( ( \)\(27\!\cdots\!76\)\( + 35768790900903493440 T^{2} + 32416802249776 T^{4} + 9934876 T^{6} + T^{8} )^{2} \)
$43$ \( ( 885324127312 + 1737835060 T - 3850284 T^{2} - 365 T^{3} + T^{4} )^{4} \)
$47$ \( \)\(77\!\cdots\!76\)\( - \)\(82\!\cdots\!00\)\( T^{2} + \)\(77\!\cdots\!60\)\( T^{4} - \)\(12\!\cdots\!92\)\( T^{6} + \)\(18\!\cdots\!01\)\( T^{8} - \)\(66\!\cdots\!10\)\( T^{10} + 197109277981731 T^{12} - 15501246 T^{14} + T^{16} \)
$53$ \( \)\(68\!\cdots\!96\)\( - \)\(40\!\cdots\!28\)\( T^{2} + \)\(15\!\cdots\!49\)\( T^{4} - \)\(35\!\cdots\!87\)\( T^{6} + \)\(60\!\cdots\!34\)\( T^{8} - \)\(69\!\cdots\!91\)\( T^{10} + 588431159454906 T^{12} - 30342651 T^{14} + T^{16} \)
$59$ \( \)\(21\!\cdots\!96\)\( - \)\(12\!\cdots\!08\)\( T^{2} + \)\(61\!\cdots\!17\)\( T^{4} - \)\(36\!\cdots\!11\)\( T^{6} + \)\(17\!\cdots\!78\)\( T^{8} - \)\(18\!\cdots\!99\)\( T^{10} + 1497278120716278 T^{12} - 44267415 T^{14} + T^{16} \)
$61$ \( ( \)\(29\!\cdots\!64\)\( + \)\(76\!\cdots\!12\)\( T + \)\(19\!\cdots\!76\)\( T^{2} + 312272635573494864 T^{3} + 318589265015949 T^{4} - 98296807482 T^{5} + 55108751 T^{6} - 7506 T^{7} + T^{8} )^{2} \)
$67$ \( ( \)\(30\!\cdots\!84\)\( + \)\(54\!\cdots\!32\)\( T + \)\(70\!\cdots\!97\)\( T^{2} + 1385265415736239623 T^{3} + 1465187785851396 T^{4} + 168357464865 T^{5} + 59066504 T^{6} - 4329 T^{7} + T^{8} )^{2} \)
$71$ \( ( \)\(10\!\cdots\!24\)\( + \)\(16\!\cdots\!72\)\( T^{2} + 8128946327751648 T^{4} + 155086092 T^{6} + T^{8} )^{2} \)
$73$ \( ( \)\(30\!\cdots\!84\)\( - \)\(29\!\cdots\!26\)\( T + \)\(23\!\cdots\!79\)\( T^{2} - 5596374382545810531 T^{3} + 1626437390284240 T^{4} - 203558217239 T^{5} + 58704516 T^{6} - 6161 T^{7} + T^{8} )^{2} \)
$79$ \( ( \)\(45\!\cdots\!49\)\( + \)\(15\!\cdots\!32\)\( T + \)\(68\!\cdots\!60\)\( T^{2} + 2479501502776808264 T^{3} + 10369822932436285 T^{4} + 2038558036160 T^{5} + 302093344 T^{6} + 20084 T^{7} + T^{8} )^{2} \)
$83$ \( ( \)\(51\!\cdots\!44\)\( + \)\(24\!\cdots\!12\)\( T^{2} + 3194595703525464 T^{4} + 116812773 T^{6} + T^{8} )^{2} \)
$89$ \( \)\(32\!\cdots\!56\)\( - \)\(14\!\cdots\!36\)\( T^{2} + \)\(64\!\cdots\!96\)\( T^{4} - \)\(13\!\cdots\!16\)\( T^{6} + \)\(24\!\cdots\!29\)\( T^{8} - \)\(10\!\cdots\!82\)\( T^{10} + 35537146704474579 T^{12} - 202431178 T^{14} + T^{16} \)
$97$ \( ( 368225255164464 - 161925558552 T - 30719906 T^{2} + 7067 T^{3} + T^{4} )^{4} \)
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