Properties

Label 20.5.b.a
Level 20
Weight 5
Character orbit 20.b
Analytic conductor 2.067
Analytic rank 0
Dimension 8
CM No
Inner twists 2

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Newspace parameters

Level: \( N \) = \( 20 = 2^{2} \cdot 5 \)
Weight: \( k \) = \( 5 \)
Character orbit: \([\chi]\) = 20.b (of order \(2\) and degree \(1\))

Newform invariants

Self dual: No
Analytic conductor: \(2.06739926168\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.246034965625.1
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{14}\cdot 5^{2} \)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q\) \( + ( 1 + \beta_{1} ) q^{2} \) \( + \beta_{3} q^{3} \) \( + ( -2 + \beta_{1} + \beta_{2} ) q^{4} \) \( -\beta_{4} q^{5} \) \( + ( 6 - 2 \beta_{4} - \beta_{5} + 2 \beta_{6} - \beta_{7} ) q^{6} \) \( + ( 1 + 2 \beta_{1} - \beta_{2} + \beta_{3} - \beta_{6} + 2 \beta_{7} ) q^{7} \) \( + ( 26 - 5 \beta_{1} - 2 \beta_{3} + 2 \beta_{4} - 3 \beta_{5} + \beta_{6} ) q^{8} \) \( + ( -41 + 4 \beta_{1} + 2 \beta_{4} + 4 \beta_{5} - 4 \beta_{6} ) q^{9} \) \(+O(q^{10})\) \( q\) \( + ( 1 + \beta_{1} ) q^{2} \) \( + \beta_{3} q^{3} \) \( + ( -2 + \beta_{1} + \beta_{2} ) q^{4} \) \( -\beta_{4} q^{5} \) \( + ( 6 - 2 \beta_{4} - \beta_{5} + 2 \beta_{6} - \beta_{7} ) q^{6} \) \( + ( 1 + 2 \beta_{1} - \beta_{2} + \beta_{3} - \beta_{6} + 2 \beta_{7} ) q^{7} \) \( + ( 26 - 5 \beta_{1} - 2 \beta_{3} + 2 \beta_{4} - 3 \beta_{5} + \beta_{6} ) q^{8} \) \( + ( -41 + 4 \beta_{1} + 2 \beta_{4} + 4 \beta_{5} - 4 \beta_{6} ) q^{9} \) \( + ( -7 - 2 \beta_{3} - \beta_{6} - \beta_{7} ) q^{10} \) \( + ( -2 - 8 \beta_{1} - 2 \beta_{2} + 2 \beta_{3} + 4 \beta_{5} + 2 \beta_{6} ) q^{11} \) \( + ( -26 + \beta_{1} + 2 \beta_{2} - 2 \beta_{3} + 10 \beta_{4} + 5 \beta_{5} + \beta_{6} - 4 \beta_{7} ) q^{12} \) \( + ( 36 - 20 \beta_{1} - 8 \beta_{2} + 6 \beta_{4} - 4 \beta_{5} - 4 \beta_{6} ) q^{13} \) \( + ( -18 + 2 \beta_{1} + 4 \beta_{2} - 12 \beta_{3} - 14 \beta_{4} + \beta_{5} + 3 \beta_{7} ) q^{14} \) \( + ( -1 - 10 \beta_{1} + 5 \beta_{2} - \beta_{3} - 3 \beta_{6} + 2 \beta_{7} ) q^{15} \) \( + ( -28 + 22 \beta_{1} - 8 \beta_{2} - 4 \beta_{3} + 12 \beta_{4} - 2 \beta_{5} + 2 \beta_{6} + 4 \beta_{7} ) q^{16} \) \( + ( -6 - 12 \beta_{1} - 16 \beta_{4} - 12 \beta_{5} + 12 \beta_{6} ) q^{17} \) \( + ( 29 - 25 \beta_{1} + 20 \beta_{3} - 16 \beta_{4} - 8 \beta_{5} - 6 \beta_{6} + 2 \beta_{7} ) q^{18} \) \( + ( 2 + 24 \beta_{1} - 6 \beta_{2} - 4 \beta_{3} - 4 \beta_{5} + 6 \beta_{6} - 8 \beta_{7} ) q^{19} \) \( + ( -40 - 5 \beta_{2} + 10 \beta_{3} + 6 \beta_{4} - 5 \beta_{5} - 5 \beta_{6} ) q^{20} \) \( + ( 34 + 88 \beta_{1} + 32 \beta_{2} + 6 \beta_{4} + 24 \beta_{5} + 8 \beta_{6} ) q^{21} \) \( + ( 100 - 4 \beta_{1} + 8 \beta_{2} + 8 \beta_{3} - 12 \beta_{4} + 26 \beta_{5} - 2 \beta_{7} ) q^{22} \) \( + ( 1 + 2 \beta_{1} + 23 \beta_{2} - 9 \beta_{3} - 16 \beta_{5} - 9 \beta_{6} - 6 \beta_{7} ) q^{23} \) \( + ( 192 - 24 \beta_{1} + 4 \beta_{2} + 56 \beta_{3} + 16 \beta_{4} + 8 \beta_{5} + 4 \beta_{6} + 4 \beta_{7} ) q^{24} \) \( + 125 q^{25} \) \( + ( -258 + 76 \beta_{1} - 32 \beta_{2} + 28 \beta_{3} - 16 \beta_{4} - 8 \beta_{5} - 2 \beta_{6} + 6 \beta_{7} ) q^{26} \) \( + ( 26 + 140 \beta_{1} - 34 \beta_{2} - 14 \beta_{3} - 24 \beta_{5} + 6 \beta_{6} - 4 \beta_{7} ) q^{27} \) \( + ( -10 - 39 \beta_{1} + 6 \beta_{2} - 58 \beta_{3} + 18 \beta_{4} + 9 \beta_{5} - 35 \beta_{6} + 4 \beta_{7} ) q^{28} \) \( + ( 134 - 72 \beta_{1} - 16 \beta_{2} + 12 \beta_{4} - 40 \beta_{5} + 24 \beta_{6} ) q^{29} \) \( + ( 170 - 10 \beta_{1} - 20 \beta_{2} - 20 \beta_{3} - 2 \beta_{4} - 25 \beta_{5} + 5 \beta_{7} ) q^{30} \) \( + ( -50 - 236 \beta_{1} + 34 \beta_{2} - 26 \beta_{3} + 56 \beta_{5} + 10 \beta_{6} - 4 \beta_{7} ) q^{31} \) \( + ( -272 - 24 \beta_{1} + 40 \beta_{2} - 16 \beta_{4} + 40 \beta_{5} + 24 \beta_{7} ) q^{32} \) \( + ( -156 - 36 \beta_{1} - 16 \beta_{2} - 44 \beta_{4} - 4 \beta_{5} - 12 \beta_{6} ) q^{33} \) \( + ( -286 - 54 \beta_{1} - 80 \beta_{3} + 48 \beta_{4} + 24 \beta_{5} + 8 \beta_{6} - 16 \beta_{7} ) q^{34} \) \( + ( -24 - 100 \beta_{1} + 20 \beta_{2} + 11 \beta_{3} + 20 \beta_{5} + 8 \beta_{6} - 12 \beta_{7} ) q^{35} \) \( + ( -142 + 47 \beta_{1} - 55 \beta_{2} - 52 \beta_{3} - 44 \beta_{4} - 70 \beta_{5} + 26 \beta_{6} - 32 \beta_{7} ) q^{36} \) \( + ( -780 - 136 \beta_{1} - 64 \beta_{2} + 102 \beta_{4} - 8 \beta_{5} - 56 \beta_{6} ) q^{37} \) \( + ( -416 + 20 \beta_{1} + 24 \beta_{2} + 56 \beta_{3} + 48 \beta_{4} + 24 \beta_{5} - 4 \beta_{6} - 12 \beta_{7} ) q^{38} \) \( + ( 46 + 148 \beta_{1} - 38 \beta_{2} + 42 \beta_{3} - 24 \beta_{5} - 30 \beta_{6} + 52 \beta_{7} ) q^{39} \) \( + ( -278 - 5 \beta_{1} - 20 \beta_{2} + 22 \beta_{3} - 30 \beta_{4} - 35 \beta_{5} + 21 \beta_{6} - 4 \beta_{7} ) q^{40} \) \( + ( 676 + 204 \beta_{1} + 64 \beta_{2} - 58 \beta_{4} + 76 \beta_{5} - 12 \beta_{6} ) q^{41} \) \( + ( 1532 - 94 \beta_{1} + 128 \beta_{2} - 20 \beta_{3} + 32 \beta_{4} + 16 \beta_{5} + 22 \beta_{6} + 6 \beta_{7} ) q^{42} \) \( + ( 22 + 60 \beta_{1} - 54 \beta_{2} + 25 \beta_{3} + 16 \beta_{5} - 6 \beta_{6} + 44 \beta_{7} ) q^{43} \) \( + ( 1004 + 82 \beta_{1} + 36 \beta_{2} + 28 \beta_{3} - 44 \beta_{4} + 42 \beta_{5} - 14 \beta_{6} - 24 \beta_{7} ) q^{44} \) \( + ( -10 + 140 \beta_{1} + 40 \beta_{2} + 33 \beta_{4} + 60 \beta_{5} - 20 \beta_{6} ) q^{45} \) \( + ( 66 - 14 \beta_{1} - 92 \beta_{2} - 12 \beta_{3} + 102 \beta_{4} - 165 \beta_{5} + 12 \beta_{6} - 3 \beta_{7} ) q^{46} \) \( + ( 69 + 394 \beta_{1} - 77 \beta_{2} + 85 \beta_{3} - 80 \beta_{5} + 19 \beta_{6} - 30 \beta_{7} ) q^{47} \) \( + ( 1136 + 152 \beta_{1} + 8 \beta_{2} - 128 \beta_{4} - 16 \beta_{5} + 128 \beta_{6} - 32 \beta_{7} ) q^{48} \) \( + ( -753 - 68 \beta_{1} - 32 \beta_{2} - 150 \beta_{4} - 4 \beta_{5} - 28 \beta_{6} ) q^{49} \) \( + ( 125 + 125 \beta_{1} ) q^{50} \) \( + ( -88 - 520 \beta_{1} + 152 \beta_{2} + 146 \beta_{3} + 72 \beta_{5} - 48 \beta_{6} + 32 \beta_{7} ) q^{51} \) \( + ( -1592 - 172 \beta_{1} + 98 \beta_{2} - 28 \beta_{3} - 132 \beta_{4} + 46 \beta_{5} + 14 \beta_{6} - 32 \beta_{7} ) q^{52} \) \( + ( 380 + 84 \beta_{1} + 56 \beta_{2} + 126 \beta_{4} - 28 \beta_{5} + 84 \beta_{6} ) q^{53} \) \( + ( -2164 + 116 \beta_{1} + 136 \beta_{2} + 40 \beta_{3} + 36 \beta_{4} + 66 \beta_{5} - 32 \beta_{6} + 6 \beta_{7} ) q^{54} \) \( + ( -20 - 120 \beta_{1} - 50 \beta_{3} + 40 \beta_{5} + 20 \beta_{7} ) q^{55} \) \( + ( 8 + 100 \beta_{1} - 124 \beta_{2} + 80 \beta_{3} + 24 \beta_{4} - 100 \beta_{5} - 136 \beta_{6} + 84 \beta_{7} ) q^{56} \) \( + ( 352 - 428 \beta_{1} - 128 \beta_{2} - 200 \beta_{4} - 172 \beta_{5} + 44 \beta_{6} ) q^{57} \) \( + ( -902 + 86 \beta_{1} - 64 \beta_{2} - 72 \beta_{3} + 96 \beta_{4} + 48 \beta_{5} + 60 \beta_{6} + 12 \beta_{7} ) q^{58} \) \( + ( -6 + 96 \beta_{1} + 42 \beta_{2} - 168 \beta_{3} - 60 \beta_{5} + 30 \beta_{6} - 96 \beta_{7} ) q^{59} \) \( + ( -1174 + 215 \beta_{1} - 30 \beta_{2} - 54 \beta_{3} + 30 \beta_{4} + 15 \beta_{5} - 37 \beta_{6} + 28 \beta_{7} ) q^{60} \) \( + ( 976 - 16 \beta_{1} - 16 \beta_{2} + 234 \beta_{4} + 16 \beta_{5} - 32 \beta_{6} ) q^{61} \) \( + ( 3124 - 180 \beta_{1} - 136 \beta_{2} + 56 \beta_{3} + 44 \beta_{4} + 134 \beta_{5} - 64 \beta_{6} + 18 \beta_{7} ) q^{62} \) \( + ( -25 + 110 \beta_{1} - 127 \beta_{2} + 9 \beta_{3} + 48 \beta_{5} + 129 \beta_{6} - 106 \beta_{7} ) q^{63} \) \( + ( 2272 - 464 \beta_{1} + 64 \beta_{2} - 224 \beta_{3} - 96 \beta_{4} + 48 \beta_{5} - 16 \beta_{6} + 32 \beta_{7} ) q^{64} \) \( + ( -70 + 180 \beta_{1} + 80 \beta_{2} - 100 \beta_{4} + 20 \beta_{5} + 60 \beta_{6} ) q^{65} \) \( + ( -1080 - 60 \beta_{1} - 64 \beta_{2} - 40 \beta_{3} - 48 \beta_{4} - 24 \beta_{5} - 68 \beta_{6} - 44 \beta_{7} ) q^{66} \) \( + ( 62 + 140 \beta_{1} + 50 \beta_{2} - 289 \beta_{3} - 80 \beta_{5} - 94 \beta_{6} + 76 \beta_{7} ) q^{67} \) \( + ( 284 - 270 \beta_{1} - 14 \beta_{2} + 256 \beta_{3} + 192 \beta_{4} + 160 \beta_{5} - 128 \beta_{6} + 96 \beta_{7} ) q^{68} \) \( + ( -298 + 8 \beta_{1} - 16 \beta_{2} + 386 \beta_{4} + 40 \beta_{5} - 56 \beta_{6} ) q^{69} \) \( + ( 1450 - 80 \beta_{1} - 80 \beta_{2} + 80 \beta_{3} + 42 \beta_{4} + 5 \beta_{5} + 30 \beta_{6} - 35 \beta_{7} ) q^{70} \) \( + ( -38 - 108 \beta_{1} + 126 \beta_{2} - 458 \beta_{3} - 48 \beta_{5} - 2 \beta_{6} - 84 \beta_{7} ) q^{71} \) \( + ( -4590 + 15 \beta_{1} - 24 \beta_{2} + 86 \beta_{3} + 314 \beta_{4} + 73 \beta_{5} - 107 \beta_{6} - 56 \beta_{7} ) q^{72} \) \( + ( -1694 + 340 \beta_{1} + 112 \beta_{2} - 216 \beta_{4} + 116 \beta_{5} - 4 \beta_{6} ) q^{73} \) \( + ( -2418 - 364 \beta_{1} - 256 \beta_{2} + 428 \beta_{3} - 224 \beta_{4} - 112 \beta_{5} - 10 \beta_{6} + 102 \beta_{7} ) q^{74} \) \( + 125 \beta_{3} q^{75} \) \( + ( 1416 - 436 \beta_{1} + 8 \beta_{2} + 200 \beta_{3} - 72 \beta_{4} - 100 \beta_{5} + 156 \beta_{6} - 32 \beta_{7} ) q^{76} \) \( + ( 420 + 180 \beta_{1} + 120 \beta_{2} + 92 \beta_{4} - 60 \beta_{5} + 180 \beta_{6} ) q^{77} \) \( + ( -1700 + 124 \beta_{1} + 152 \beta_{2} - 328 \beta_{3} - 380 \beta_{4} - 46 \beta_{5} + 40 \beta_{6} + 62 \beta_{7} ) q^{78} \) \( + ( 8 + 136 \beta_{1} - 104 \beta_{2} + 284 \beta_{3} + 24 \beta_{5} + 64 \beta_{6} - 32 \beta_{7} ) q^{79} \) \( + ( -1740 - 290 \beta_{1} - 100 \beta_{3} + 44 \beta_{4} + 30 \beta_{5} + 50 \beta_{6} - 60 \beta_{7} ) q^{80} \) \( + ( -149 - 116 \beta_{1} - 32 \beta_{2} - 482 \beta_{4} - 52 \beta_{5} + 20 \beta_{6} ) q^{81} \) \( + ( 3574 + 532 \beta_{1} + 256 \beta_{2} - 68 \beta_{3} - 48 \beta_{4} - 24 \beta_{5} - 82 \beta_{6} - 58 \beta_{7} ) q^{82} \) \( + ( 30 + 20 \beta_{1} + 122 \beta_{2} - 175 \beta_{3} - 88 \beta_{5} - 94 \beta_{6} + 36 \beta_{7} ) q^{83} \) \( + ( 6604 + 1042 \beta_{1} - 112 \beta_{2} - 252 \beta_{3} + 284 \beta_{4} - 194 \beta_{5} + 126 \beta_{6} + 64 \beta_{7} ) q^{84} \) \( + ( 1280 - 420 \beta_{1} - 120 \beta_{2} + 30 \beta_{4} - 180 \beta_{5} + 60 \beta_{6} ) q^{85} \) \( + ( -826 + 76 \beta_{1} + 216 \beta_{2} - 200 \beta_{3} - 378 \beta_{4} + 243 \beta_{5} - 26 \beta_{6} + 63 \beta_{7} ) q^{86} \) \( + ( -100 - 664 \beta_{1} + 116 \beta_{2} + 594 \beta_{3} + 144 \beta_{5} - 60 \beta_{6} + 104 \beta_{7} ) q^{87} \) \( + ( 2464 + 1024 \beta_{1} + 40 \beta_{2} + 144 \beta_{3} - 128 \beta_{5} - 8 \beta_{6} - 120 \beta_{7} ) q^{88} \) \( + ( 2938 - 168 \beta_{1} - 32 \beta_{2} - 152 \beta_{4} - 104 \beta_{5} + 72 \beta_{6} ) q^{89} \) \( + ( 2461 - 50 \beta_{1} + 160 \beta_{2} + 146 \beta_{3} - 80 \beta_{4} - 40 \beta_{5} - 7 \beta_{6} + 33 \beta_{7} ) q^{90} \) \( + ( 188 + 1032 \beta_{1} - 420 \beta_{2} + 866 \beta_{3} - 64 \beta_{5} + 108 \beta_{6} + 16 \beta_{7} ) q^{91} \) \( + ( -6470 + 303 \beta_{1} - 222 \beta_{2} + 58 \beta_{3} + 206 \beta_{4} - 153 \beta_{5} + 163 \beta_{6} + 108 \beta_{7} ) q^{92} \) \( + ( 1212 - 288 \beta_{1} - 208 \beta_{2} + 388 \beta_{4} + 128 \beta_{5} - 336 \beta_{6} ) q^{93} \) \( + ( -5290 + 314 \beta_{1} + 308 \beta_{2} + 196 \beta_{3} - 6 \beta_{4} - 59 \beta_{5} + 192 \beta_{6} - 145 \beta_{7} ) q^{94} \) \( + ( 70 + 340 \beta_{1} - 50 \beta_{2} - 220 \beta_{3} - 80 \beta_{5} - 10 \beta_{6} ) q^{95} \) \( + ( 528 + 696 \beta_{1} + 432 \beta_{2} - 304 \beta_{3} + 432 \beta_{4} + 504 \beta_{5} + 24 \beta_{6} - 192 \beta_{7} ) q^{96} \) \( + ( -306 + 1068 \beta_{1} + 240 \beta_{2} - 132 \beta_{4} + 588 \beta_{5} - 348 \beta_{6} ) q^{97} \) \( + ( -2979 - 545 \beta_{1} - 128 \beta_{2} - 188 \beta_{3} - 112 \beta_{4} - 56 \beta_{5} - 206 \beta_{6} - 150 \beta_{7} ) q^{98} \) \( + ( -102 - 792 \beta_{1} + 18 \beta_{2} - 94 \beta_{3} + 252 \beta_{5} - 66 \beta_{6} + 216 \beta_{7} ) q^{99} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(8q \) \(\mathstrut +\mathstrut 6q^{2} \) \(\mathstrut -\mathstrut 20q^{4} \) \(\mathstrut +\mathstrut 48q^{6} \) \(\mathstrut +\mathstrut 216q^{8} \) \(\mathstrut -\mathstrut 328q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(8q \) \(\mathstrut +\mathstrut 6q^{2} \) \(\mathstrut -\mathstrut 20q^{4} \) \(\mathstrut +\mathstrut 48q^{6} \) \(\mathstrut +\mathstrut 216q^{8} \) \(\mathstrut -\mathstrut 328q^{9} \) \(\mathstrut -\mathstrut 50q^{10} \) \(\mathstrut -\mathstrut 200q^{12} \) \(\mathstrut +\mathstrut 352q^{13} \) \(\mathstrut -\mathstrut 168q^{14} \) \(\mathstrut -\mathstrut 272q^{16} \) \(\mathstrut -\mathstrut 48q^{17} \) \(\mathstrut +\mathstrut 286q^{18} \) \(\mathstrut -\mathstrut 300q^{20} \) \(\mathstrut +\mathstrut 16q^{21} \) \(\mathstrut +\mathstrut 800q^{22} \) \(\mathstrut +\mathstrut 1552q^{24} \) \(\mathstrut +\mathstrut 1000q^{25} \) \(\mathstrut -\mathstrut 2172q^{26} \) \(\mathstrut +\mathstrut 40q^{28} \) \(\mathstrut +\mathstrut 1200q^{29} \) \(\mathstrut +\mathstrut 1400q^{30} \) \(\mathstrut -\mathstrut 2304q^{32} \) \(\mathstrut -\mathstrut 1120q^{33} \) \(\mathstrut -\mathstrut 2132q^{34} \) \(\mathstrut -\mathstrut 1044q^{36} \) \(\mathstrut -\mathstrut 5728q^{37} \) \(\mathstrut -\mathstrut 3360q^{38} \) \(\mathstrut -\mathstrut 2200q^{40} \) \(\mathstrut +\mathstrut 4896q^{41} \) \(\mathstrut +\mathstrut 12120q^{42} \) \(\mathstrut +\mathstrut 7920q^{44} \) \(\mathstrut -\mathstrut 400q^{45} \) \(\mathstrut +\mathstrut 728q^{46} \) \(\mathstrut +\mathstrut 8640q^{48} \) \(\mathstrut -\mathstrut 5768q^{49} \) \(\mathstrut +\mathstrut 750q^{50} \) \(\mathstrut -\mathstrut 12488q^{52} \) \(\mathstrut +\mathstrut 2592q^{53} \) \(\mathstrut -\mathstrut 17776q^{54} \) \(\mathstrut +\mathstrut 48q^{56} \) \(\mathstrut +\mathstrut 3840q^{57} \) \(\mathstrut -\mathstrut 7428q^{58} \) \(\mathstrut -\mathstrut 9800q^{60} \) \(\mathstrut +\mathstrut 7936q^{61} \) \(\mathstrut +\mathstrut 25680q^{62} \) \(\mathstrut +\mathstrut 18880q^{64} \) \(\mathstrut -\mathstrut 1200q^{65} \) \(\mathstrut -\mathstrut 8080q^{66} \) \(\mathstrut +\mathstrut 2712q^{68} \) \(\mathstrut -\mathstrut 2256q^{69} \) \(\mathstrut +\mathstrut 12000q^{70} \) \(\mathstrut -\mathstrut 36264q^{72} \) \(\mathstrut -\mathstrut 14448q^{73} \) \(\mathstrut -\mathstrut 18492q^{74} \) \(\mathstrut +\mathstrut 12000q^{76} \) \(\mathstrut +\mathstrut 2400q^{77} \) \(\mathstrut -\mathstrut 14480q^{78} \) \(\mathstrut -\mathstrut 13200q^{80} \) \(\mathstrut -\mathstrut 936q^{81} \) \(\mathstrut +\mathstrut 27412q^{82} \) \(\mathstrut +\mathstrut 50464q^{84} \) \(\mathstrut +\mathstrut 11200q^{85} \) \(\mathstrut -\mathstrut 7392q^{86} \) \(\mathstrut +\mathstrut 18080q^{88} \) \(\mathstrut +\mathstrut 23760q^{89} \) \(\mathstrut +\mathstrut 19350q^{90} \) \(\mathstrut -\mathstrut 52680q^{92} \) \(\mathstrut +\mathstrut 11360q^{93} \) \(\mathstrut -\mathstrut 43368q^{94} \) \(\mathstrut +\mathstrut 2688q^{96} \) \(\mathstrut -\mathstrut 4368q^{97} \) \(\mathstrut -\mathstrut 21474q^{98} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8}\mathstrut -\mathstrut \) \(3\) \(x^{7}\mathstrut +\mathstrut \) \(7\) \(x^{6}\mathstrut -\mathstrut \) \(21\) \(x^{5}\mathstrut +\mathstrut \) \(49\) \(x^{4}\mathstrut -\mathstrut \) \(84\) \(x^{3}\mathstrut +\mathstrut \) \(112\) \(x^{2}\mathstrut -\mathstrut \) \(192\) \(x\mathstrut +\mathstrut \) \(256\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( 2 \nu - 1 \)
\(\beta_{2}\)\(=\)\( 4 \nu^{2} - 2 \nu + 3 \)
\(\beta_{3}\)\(=\)\((\)\( \nu^{7} - 11 \nu^{6} + 15 \nu^{5} - 29 \nu^{4} + 41 \nu^{3} - 76 \nu^{2} + 256 \)\()/32\)
\(\beta_{4}\)\(=\)\((\)\( -\nu^{7} + 3 \nu^{6} - 23 \nu^{5} + 69 \nu^{4} - 97 \nu^{3} + 228 \nu^{2} - 384 \nu + 480 \)\()/32\)
\(\beta_{5}\)\(=\)\((\)\( -\nu^{7} + 3 \nu^{6} - 7 \nu^{5} + 21 \nu^{4} - 49 \nu^{3} + 52 \nu^{2} - 80 \nu + 136 \)\()/8\)
\(\beta_{6}\)\(=\)\((\)\( -\nu^{7} + \nu^{6} - \nu^{5} + 7 \nu^{4} - 7 \nu^{3} + 2 \nu^{2} + 16 \nu + 24 \)\()/4\)
\(\beta_{7}\)\(=\)\((\)\( 3 \nu^{7} - 9 \nu^{6} + 37 \nu^{5} - 47 \nu^{4} + 131 \nu^{3} - 204 \nu^{2} + 224 \nu - 208 \)\()/16\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{1}\mathstrut +\mathstrut \) \(1\)\()/2\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{2}\mathstrut +\mathstrut \) \(\beta_{1}\mathstrut -\mathstrut \) \(2\)\()/4\)
\(\nu^{3}\)\(=\)\((\)\(\beta_{6}\mathstrut -\mathstrut \) \(3\) \(\beta_{5}\mathstrut +\mathstrut \) \(2\) \(\beta_{4}\mathstrut -\mathstrut \) \(2\) \(\beta_{3}\mathstrut -\mathstrut \) \(5\) \(\beta_{1}\mathstrut +\mathstrut \) \(26\)\()/8\)
\(\nu^{4}\)\(=\)\((\)\(2\) \(\beta_{7}\mathstrut +\mathstrut \) \(\beta_{6}\mathstrut -\mathstrut \) \(\beta_{5}\mathstrut +\mathstrut \) \(6\) \(\beta_{4}\mathstrut -\mathstrut \) \(2\) \(\beta_{3}\mathstrut -\mathstrut \) \(4\) \(\beta_{2}\mathstrut +\mathstrut \) \(11\) \(\beta_{1}\mathstrut -\mathstrut \) \(14\)\()/8\)
\(\nu^{5}\)\(=\)\((\)\(3\) \(\beta_{7}\mathstrut +\mathstrut \) \(5\) \(\beta_{5}\mathstrut -\mathstrut \) \(2\) \(\beta_{4}\mathstrut +\mathstrut \) \(5\) \(\beta_{2}\mathstrut -\mathstrut \) \(3\) \(\beta_{1}\mathstrut -\mathstrut \) \(34\)\()/4\)
\(\nu^{6}\)\(=\)\((\)\(2\) \(\beta_{7}\mathstrut -\mathstrut \) \(\beta_{6}\mathstrut +\mathstrut \) \(3\) \(\beta_{5}\mathstrut -\mathstrut \) \(6\) \(\beta_{4}\mathstrut -\mathstrut \) \(14\) \(\beta_{3}\mathstrut +\mathstrut \) \(4\) \(\beta_{2}\mathstrut -\mathstrut \) \(29\) \(\beta_{1}\mathstrut +\mathstrut \) \(142\)\()/4\)
\(\nu^{7}\)\(=\)\((\)\(6\) \(\beta_{7}\mathstrut -\mathstrut \) \(17\) \(\beta_{6}\mathstrut +\mathstrut \) \(5\) \(\beta_{5}\mathstrut +\mathstrut \) \(10\) \(\beta_{4}\mathstrut -\mathstrut \) \(14\) \(\beta_{3}\mathstrut -\mathstrut \) \(13\) \(\beta_{2}\mathstrut +\mathstrut \) \(64\) \(\beta_{1}\mathstrut +\mathstrut \) \(160\)\()/4\)

Character Values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/20\mathbb{Z}\right)^\times\).

\(n\) \(11\) \(17\)
\(\chi(n)\) \(-1\) \(1\)

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
11.1
−1.02661 1.71641i
−1.02661 + 1.71641i
−0.641015 1.89449i
−0.641015 + 1.89449i
1.21760 1.58665i
1.21760 + 1.58665i
1.95003 0.444269i
1.95003 + 0.444269i
−2.05323 3.43282i 15.5779i −7.56853 + 14.0967i 11.1803 53.4761 31.9849i 37.6230i 63.9314 2.96235i −161.671 −22.9558 38.3801i
11.2 −2.05323 + 3.43282i 15.5779i −7.56853 14.0967i 11.1803 53.4761 + 31.9849i 37.6230i 63.9314 + 2.96235i −161.671 −22.9558 + 38.3801i
11.3 −1.28203 3.78898i 8.14153i −12.7128 + 9.71518i −11.1803 −30.8481 + 10.4377i 63.6032i 53.1089 + 35.7134i 14.7155 14.3335 + 42.3621i
11.4 −1.28203 + 3.78898i 8.14153i −12.7128 9.71518i −11.1803 −30.8481 10.4377i 63.6032i 53.1089 35.7134i 14.7155 14.3335 42.3621i
11.5 2.43519 3.17330i 3.20523i −4.13968 15.4552i 11.1803 −10.1712 7.80536i 30.6227i −59.1249 24.4999i 70.7265 27.2263 35.4786i
11.6 2.43519 + 3.17330i 3.20523i −4.13968 + 15.4552i 11.1803 −10.1712 + 7.80536i 30.6227i −59.1249 + 24.4999i 70.7265 27.2263 + 35.4786i
11.7 3.90006 0.888538i 12.9912i 14.4210 6.93071i −11.1803 11.5432 + 50.6665i 78.0345i 50.0846 39.8438i −87.7712 −43.6040 + 9.93415i
11.8 3.90006 + 0.888538i 12.9912i 14.4210 + 6.93071i −11.1803 11.5432 50.6665i 78.0345i 50.0846 + 39.8438i −87.7712 −43.6040 9.93415i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 11.8
Significant digits:
Format:

Inner twists

Char. orbit Parity Mult. Self Twist Proved
1.a Even 1 trivial yes
4.b Odd 1 yes

Hecke kernels

There are no other newforms in \(S_{5}^{\mathrm{new}}(20, [\chi])\).