Properties

Label 11.4.c.a
Level 11
Weight 4
Character orbit 11.c
Analytic conductor 0.649
Analytic rank 0
Dimension 8
CM No
Inner twists 2

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Newspace parameters

Level: \( N \) = \( 11 \)
Weight: \( k \) = \( 4 \)
Character orbit: \([\chi]\) = 11.c (of order \(5\) and degree \(4\))

Newform invariants

Self dual: No
Analytic conductor: \(0.649021010063\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(2\) over \(\Q(\zeta_{5})\)
Coefficient field: 8.0.29283765625.1
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

$q$-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.

\(f(q)\) \(=\) \( q\) \( + ( -1 + \beta_{1} + \beta_{2} - \beta_{4} ) q^{2} \) \( + ( -2 - 2 \beta_{2} - 2 \beta_{4} + \beta_{5} - 2 \beta_{6} ) q^{3} \) \( + ( 3 - 2 \beta_{1} + 5 \beta_{4} - 2 \beta_{5} + 3 \beta_{6} + \beta_{7} ) q^{4} \) \( + ( -2 \beta_{1} + \beta_{2} + \beta_{3} + 3 \beta_{4} + \beta_{5} - 2 \beta_{6} - 2 \beta_{7} ) q^{5} \) \( + ( -2 \beta_{1} + 2 \beta_{2} - 3 \beta_{3} + 4 \beta_{4} - 3 \beta_{5} + 13 \beta_{6} - 2 \beta_{7} ) q^{6} \) \( + ( -6 + 3 \beta_{1} + 3 \beta_{5} - 6 \beta_{6} + 5 \beta_{7} ) q^{7} \) \( + ( -7 + 6 \beta_{1} - 7 \beta_{2} + 6 \beta_{3} - 26 \beta_{4} + 3 \beta_{5} - 26 \beta_{6} ) q^{8} \) \( + ( 3 \beta_{1} - 14 \beta_{2} ) q^{9} \) \(+O(q^{10})\) \( q\) \( + ( -1 + \beta_{1} + \beta_{2} - \beta_{4} ) q^{2} \) \( + ( -2 - 2 \beta_{2} - 2 \beta_{4} + \beta_{5} - 2 \beta_{6} ) q^{3} \) \( + ( 3 - 2 \beta_{1} + 5 \beta_{4} - 2 \beta_{5} + 3 \beta_{6} + \beta_{7} ) q^{4} \) \( + ( -2 \beta_{1} + \beta_{2} + \beta_{3} + 3 \beta_{4} + \beta_{5} - 2 \beta_{6} - 2 \beta_{7} ) q^{5} \) \( + ( -2 \beta_{1} + 2 \beta_{2} - 3 \beta_{3} + 4 \beta_{4} - 3 \beta_{5} + 13 \beta_{6} - 2 \beta_{7} ) q^{6} \) \( + ( -6 + 3 \beta_{1} + 3 \beta_{5} - 6 \beta_{6} + 5 \beta_{7} ) q^{7} \) \( + ( -7 + 6 \beta_{1} - 7 \beta_{2} + 6 \beta_{3} - 26 \beta_{4} + 3 \beta_{5} - 26 \beta_{6} ) q^{8} \) \( + ( 3 \beta_{1} - 14 \beta_{2} ) q^{9} \) \( + ( 16 + 28 \beta_{2} - 6 \beta_{3} + 28 \beta_{6} ) q^{10} \) \( + ( 25 - 4 \beta_{1} + 24 \beta_{2} - 7 \beta_{3} + 36 \beta_{4} - 9 \beta_{5} + 19 \beta_{6} - 10 \beta_{7} ) q^{11} \) \( + ( 15 - 24 \beta_{2} + 9 \beta_{3} - 5 \beta_{4} + 5 \beta_{5} - 24 \beta_{6} + 5 \beta_{7} ) q^{12} \) \( + ( -7 - 4 \beta_{1} - \beta_{2} + 7 \beta_{3} - 14 \beta_{4} + 7 \beta_{7} ) q^{13} \) \( + ( -50 - 4 \beta_{1} - 50 \beta_{2} - 4 \beta_{3} - 22 \beta_{4} + 4 \beta_{5} - 22 \beta_{6} ) q^{14} \) \( + ( -25 + 2 \beta_{1} - 12 \beta_{4} + 2 \beta_{5} - 25 \beta_{6} - \beta_{7} ) q^{15} \) \( + ( -5 \beta_{1} + 75 \beta_{2} - 11 \beta_{3} + 80 \beta_{4} - 11 \beta_{5} + 50 \beta_{6} - 5 \beta_{7} ) q^{16} \) \( + ( 7 \beta_{1} - 18 \beta_{2} + 9 \beta_{3} - 25 \beta_{4} + 9 \beta_{5} + 45 \beta_{6} + 7 \beta_{7} ) q^{17} \) \( + ( -14 - 3 \beta_{1} + 13 \beta_{4} - 3 \beta_{5} - 14 \beta_{6} - 14 \beta_{7} ) q^{18} \) \( + ( -19 - 12 \beta_{1} - 19 \beta_{2} - 12 \beta_{3} - 58 \beta_{4} - 6 \beta_{5} - 58 \beta_{6} ) q^{19} \) \( + ( 48 - 42 \beta_{2} + 10 \beta_{3} + 38 \beta_{4} + 10 \beta_{7} ) q^{20} \) \( + ( 67 + 39 \beta_{2} + 7 \beta_{3} + 9 \beta_{4} - 9 \beta_{5} + 39 \beta_{6} - 9 \beta_{7} ) q^{21} \) \( + ( 82 + 19 \beta_{1} + 7 \beta_{2} + 25 \beta_{3} - 6 \beta_{4} + 29 \beta_{5} - 38 \beta_{6} + 31 \beta_{7} ) q^{22} \) \( + ( -4 + 26 \beta_{2} - 36 \beta_{3} + 10 \beta_{4} - 10 \beta_{5} + 26 \beta_{6} - 10 \beta_{7} ) q^{23} \) \( + ( -92 + 4 \beta_{1} - 13 \beta_{2} - 25 \beta_{3} - 67 \beta_{4} - 25 \beta_{7} ) q^{24} \) \( + ( -2 + 19 \beta_{1} - 2 \beta_{2} + 19 \beta_{3} + 30 \beta_{4} - 12 \beta_{5} + 30 \beta_{6} ) q^{25} \) \( + ( -50 + 4 \beta_{1} - 72 \beta_{4} + 4 \beta_{5} - 50 \beta_{6} - 8 \beta_{7} ) q^{26} \) \( + ( 38 \beta_{1} + 38 \beta_{3} - 38 \beta_{4} + 38 \beta_{5} - 55 \beta_{6} + 38 \beta_{7} ) q^{27} \) \( + ( -6 \beta_{1} + 6 \beta_{2} - 6 \beta_{3} + 12 \beta_{4} - 6 \beta_{5} + 12 \beta_{6} - 6 \beta_{7} ) q^{28} \) \( + ( 12 - 31 \beta_{1} + 104 \beta_{4} - 31 \beta_{5} + 12 \beta_{6} + \beta_{7} ) q^{29} \) \( + ( 22 - 28 \beta_{1} + 22 \beta_{2} - 28 \beta_{3} + 78 \beta_{4} - 18 \beta_{5} + 78 \beta_{6} ) q^{30} \) \( + ( -89 - 40 \beta_{1} - 45 \beta_{2} - 41 \beta_{3} - 48 \beta_{4} - 41 \beta_{7} ) q^{31} \) \( + ( 39 - 27 \beta_{2} + 43 \beta_{3} - 38 \beta_{4} + 38 \beta_{5} - 27 \beta_{6} + 38 \beta_{7} ) q^{32} \) \( + ( -25 - 29 \beta_{1} - 68 \beta_{2} - 4 \beta_{3} + 52 \beta_{4} - 2 \beta_{5} + 25 \beta_{6} - \beta_{7} ) q^{33} \) \( + ( -18 - 9 \beta_{2} + 34 \beta_{3} + 27 \beta_{4} - 27 \beta_{5} - 9 \beta_{6} - 27 \beta_{7} ) q^{34} \) \( + ( 85 + 3 \beta_{1} + 132 \beta_{2} + 4 \beta_{3} + 81 \beta_{4} + 4 \beta_{7} ) q^{35} \) \( + ( 15 - 25 \beta_{1} + 15 \beta_{2} - 25 \beta_{3} + 3 \beta_{4} - 33 \beta_{5} + 3 \beta_{6} ) q^{36} \) \( + ( 8 + 35 \beta_{1} + 10 \beta_{4} + 35 \beta_{5} + 8 \beta_{6} + 17 \beta_{7} ) q^{37} \) \( + ( -7 \beta_{1} - 11 \beta_{2} - 40 \beta_{3} - 4 \beta_{4} - 40 \beta_{5} + 101 \beta_{6} - 7 \beta_{7} ) q^{38} \) \( + ( 5 \beta_{1} - 49 \beta_{2} + 5 \beta_{3} - 54 \beta_{4} + 5 \beta_{5} - 87 \beta_{6} + 5 \beta_{7} ) q^{39} \) \( + ( -4 + 58 \beta_{1} - 218 \beta_{4} + 58 \beta_{5} - 4 \beta_{6} - 4 \beta_{7} ) q^{40} \) \( + ( -26 + 62 \beta_{1} - 26 \beta_{2} + 62 \beta_{3} - 66 \beta_{4} + 47 \beta_{5} - 66 \beta_{6} ) q^{41} \) \( + ( 92 + 58 \beta_{1} + 172 \beta_{2} + 32 \beta_{3} + 60 \beta_{4} + 32 \beta_{7} ) q^{42} \) \( + ( -177 - 168 \beta_{2} - 39 \beta_{3} + 31 \beta_{4} - 31 \beta_{5} - 168 \beta_{6} - 31 \beta_{7} ) q^{43} \) \( + ( -225 + 14 \beta_{1} + 81 \beta_{2} - 69 \beta_{3} - 192 \beta_{4} - 40 \beta_{5} - 39 \beta_{6} - 42 \beta_{7} ) q^{44} \) \( + ( 96 + 95 \beta_{2} + 11 \beta_{3} - 54 \beta_{4} + 54 \beta_{5} + 95 \beta_{6} + 54 \beta_{7} ) q^{45} \) \( + ( 146 - 14 \beta_{1} - 264 \beta_{2} + 62 \beta_{3} + 84 \beta_{4} + 62 \beta_{7} ) q^{46} \) \( + ( 241 - 9 \beta_{1} + 241 \beta_{2} - 9 \beta_{3} + 161 \beta_{4} + 28 \beta_{5} + 161 \beta_{6} ) q^{47} \) \( + ( 204 - 81 \beta_{1} + 219 \beta_{4} - 81 \beta_{5} + 204 \beta_{6} - 20 \beta_{7} ) q^{48} \) \( + ( -112 \beta_{1} - 285 \beta_{2} - 43 \beta_{3} - 173 \beta_{4} - 43 \beta_{5} - 165 \beta_{6} - 112 \beta_{7} ) q^{49} \) \( + ( -21 \beta_{1} + 109 \beta_{2} + 23 \beta_{3} + 130 \beta_{4} + 23 \beta_{5} - 200 \beta_{6} - 21 \beta_{7} ) q^{50} \) \( + ( -54 + 20 \beta_{1} - 47 \beta_{4} + 20 \beta_{5} - 54 \beta_{6} + 56 \beta_{7} ) q^{51} \) \( + ( 144 - 6 \beta_{1} + 144 \beta_{2} - 6 \beta_{3} + 366 \beta_{4} - 8 \beta_{5} + 366 \beta_{6} ) q^{52} \) \( + ( -197 + 2 \beta_{1} + 95 \beta_{2} + 41 \beta_{3} - 238 \beta_{4} + 41 \beta_{7} ) q^{53} \) \( + ( -397 + 55 \beta_{2} - 93 \beta_{3} + 38 \beta_{4} - 38 \beta_{5} + 55 \beta_{6} - 38 \beta_{7} ) q^{54} \) \( + ( -140 - 4 \beta_{1} - 119 \beta_{2} + 59 \beta_{3} + 47 \beta_{4} - 53 \beta_{5} + 162 \beta_{6} - 10 \beta_{7} ) q^{55} \) \( + ( -334 - 242 \beta_{2} - 14 \beta_{3} - 44 \beta_{4} + 44 \beta_{5} - 242 \beta_{6} + 44 \beta_{7} ) q^{56} \) \( + ( 30 + 25 \beta_{1} + 92 \beta_{2} - 27 \beta_{3} + 57 \beta_{4} - 27 \beta_{7} ) q^{57} \) \( + ( -114 + 44 \beta_{1} - 114 \beta_{2} + 44 \beta_{3} - 510 \beta_{4} + 168 \beta_{5} - 510 \beta_{6} ) q^{58} \) \( + ( 169 - 32 \beta_{1} + 136 \beta_{4} - 32 \beta_{5} + 169 \beta_{6} + 2 \beta_{7} ) q^{59} \) \( + ( 42 \beta_{1} - 186 \beta_{2} + 100 \beta_{3} - 228 \beta_{4} + 100 \beta_{5} - 270 \beta_{6} + 42 \beta_{7} ) q^{60} \) \( + ( 42 \beta_{1} + 189 \beta_{2} - 105 \beta_{3} + 147 \beta_{4} - 105 \beta_{5} + 84 \beta_{6} + 42 \beta_{7} ) q^{61} \) \( + ( 502 - 90 \beta_{1} + 190 \beta_{4} - 90 \beta_{5} + 502 \beta_{6} - 4 \beta_{7} ) q^{62} \) \( + ( 19 + 33 \beta_{1} + 19 \beta_{2} + 33 \beta_{3} + 16 \beta_{4} - 19 \beta_{5} + 16 \beta_{6} ) q^{63} \) \( + ( 165 + 37 \beta_{1} + 311 \beta_{2} - 22 \beta_{3} + 187 \beta_{4} - 22 \beta_{7} ) q^{64} \) \( + ( 127 - 156 \beta_{2} - 44 \beta_{3} + 13 \beta_{4} - 13 \beta_{5} - 156 \beta_{6} - 13 \beta_{7} ) q^{65} \) \( + ( -60 + 3 \beta_{1} - 128 \beta_{2} + 30 \beta_{3} - 401 \beta_{4} + 81 \beta_{5} - 204 \beta_{6} - 64 \beta_{7} ) q^{66} \) \( + ( 204 + 309 \beta_{2} + 46 \beta_{3} + 63 \beta_{4} - 63 \beta_{5} + 309 \beta_{6} - 63 \beta_{7} ) q^{67} \) \( + ( 99 + 11 \beta_{1} + 36 \beta_{2} - 59 \beta_{3} + 158 \beta_{4} - 59 \beta_{7} ) q^{68} \) \( + ( 242 - 36 \beta_{1} + 242 \beta_{2} - 36 \beta_{3} + 384 \beta_{4} - 66 \beta_{5} + 384 \beta_{6} ) q^{69} \) \( + ( -74 + 86 \beta_{1} - 128 \beta_{4} + 86 \beta_{5} - 74 \beta_{6} + 128 \beta_{7} ) q^{70} \) \( + ( 120 \beta_{1} - 253 \beta_{2} + 43 \beta_{3} - 373 \beta_{4} + 43 \beta_{5} - 10 \beta_{6} + 120 \beta_{7} ) q^{71} \) \( + ( -72 \beta_{1} - 104 \beta_{2} - 27 \beta_{3} - 32 \beta_{4} - 27 \beta_{5} - 283 \beta_{6} - 72 \beta_{7} ) q^{72} \) \( + ( -480 + 44 \beta_{1} - 293 \beta_{4} + 44 \beta_{5} - 480 \beta_{6} - 51 \beta_{7} ) q^{73} \) \( + ( -180 - 10 \beta_{1} - 180 \beta_{2} - 10 \beta_{3} + 100 \beta_{4} - 26 \beta_{5} + 100 \beta_{6} ) q^{74} \) \( + ( -107 - 29 \beta_{1} - 275 \beta_{2} + 13 \beta_{3} - 120 \beta_{4} + 13 \beta_{7} ) q^{75} \) \( + ( 12 - 75 \beta_{2} + 222 \beta_{3} - 125 \beta_{4} + 125 \beta_{5} - 75 \beta_{6} + 125 \beta_{7} ) q^{76} \) \( + ( -336 + 19 \beta_{1} - 4 \beta_{2} - 52 \beta_{3} + 82 \beta_{4} + 95 \beta_{5} - 302 \beta_{6} + 97 \beta_{7} ) q^{77} \) \( + ( -132 + 136 \beta_{2} - 92 \beta_{3} + 54 \beta_{4} - 54 \beta_{5} + 136 \beta_{6} - 54 \beta_{7} ) q^{78} \) \( + ( -81 - 128 \beta_{1} + 143 \beta_{2} - 13 \beta_{3} - 68 \beta_{4} - 13 \beta_{7} ) q^{79} \) \( + ( -78 + 14 \beta_{1} - 78 \beta_{2} + 14 \beta_{3} + 286 \beta_{4} - 262 \beta_{5} + 286 \beta_{6} ) q^{80} \) \( + ( 100 \beta_{4} - 174 \beta_{7} ) q^{81} \) \( + ( -88 \beta_{1} + 664 \beta_{2} - 175 \beta_{3} + 752 \beta_{4} - 175 \beta_{5} + 595 \beta_{6} - 88 \beta_{7} ) q^{82} \) \( + ( -14 \beta_{1} - 283 \beta_{2} + 194 \beta_{3} - 269 \beta_{4} + 194 \beta_{5} + 235 \beta_{6} - 14 \beta_{7} ) q^{83} \) \( + ( 12 - 6 \beta_{1} + 66 \beta_{4} - 6 \beta_{5} + 12 \beta_{6} + 12 \beta_{7} ) q^{84} \) \( + ( 279 - 197 \beta_{1} + 279 \beta_{2} - 197 \beta_{3} - 27 \beta_{4} + 10 \beta_{5} - 27 \beta_{6} ) q^{85} \) \( + ( 120 - 208 \beta_{1} - 81 \beta_{2} - 129 \beta_{3} + 249 \beta_{4} - 129 \beta_{7} ) q^{86} \) \( + ( 195 - 303 \beta_{2} + 137 \beta_{3} - 43 \beta_{4} + 43 \beta_{5} - 303 \beta_{6} + 43 \beta_{7} ) q^{87} \) \( + ( 935 - 33 \beta_{1} + 165 \beta_{2} + 44 \beta_{3} + 561 \beta_{4} - 154 \beta_{5} + 462 \beta_{6} + 198 \beta_{7} ) q^{88} \) \( + ( 281 - 428 \beta_{2} + 57 \beta_{3} - 25 \beta_{4} + 25 \beta_{5} - 428 \beta_{6} + 25 \beta_{7} ) q^{89} \) \( + ( -392 + 150 \beta_{1} - 386 \beta_{2} + 84 \beta_{3} - 476 \beta_{4} + 84 \beta_{7} ) q^{90} \) \( + ( 44 + 8 \beta_{1} + 44 \beta_{2} + 8 \beta_{3} + 21 \beta_{4} - 21 \beta_{5} + 21 \beta_{6} ) q^{91} \) \( + ( -906 + 142 \beta_{1} - 1292 \beta_{4} + 142 \beta_{5} - 906 \beta_{6} - 118 \beta_{7} ) q^{92} \) \( + ( 85 \beta_{1} + 547 \beta_{2} - 45 \beta_{3} + 462 \beta_{4} - 45 \beta_{5} + 97 \beta_{6} + 85 \beta_{7} ) q^{93} \) \( + ( 250 \beta_{1} - 162 \beta_{2} + 142 \beta_{3} - 412 \beta_{4} + 142 \beta_{5} + 10 \beta_{6} + 250 \beta_{7} ) q^{94} \) \( + ( -113 + 60 \beta_{1} + 458 \beta_{4} + 60 \beta_{5} - 113 \beta_{6} - 157 \beta_{7} ) q^{95} \) \( + ( -123 + 65 \beta_{1} - 123 \beta_{2} + 65 \beta_{3} - 519 \beta_{4} + 109 \beta_{5} - 519 \beta_{6} ) q^{96} \) \( + ( -471 + 144 \beta_{1} - 594 \beta_{2} - 50 \beta_{3} - 421 \beta_{4} - 50 \beta_{7} ) q^{97} \) \( + ( 843 + 1071 \beta_{2} - 191 \beta_{3} + 242 \beta_{4} - 242 \beta_{5} + 1071 \beta_{6} - 242 \beta_{7} ) q^{98} \) \( + ( 368 + 41 \beta_{1} + 95 \beta_{2} - 96 \beta_{3} + 203 \beta_{4} + 95 \beta_{5} + 391 \beta_{6} - 13 \beta_{7} ) q^{99} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(8q \) \(\mathstrut -\mathstrut 7q^{2} \) \(\mathstrut -\mathstrut 3q^{3} \) \(\mathstrut +\mathstrut 3q^{4} \) \(\mathstrut -\mathstrut 7q^{5} \) \(\mathstrut -\mathstrut 29q^{6} \) \(\mathstrut -\mathstrut 35q^{7} \) \(\mathstrut +\mathstrut 47q^{8} \) \(\mathstrut +\mathstrut 31q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(8q \) \(\mathstrut -\mathstrut 7q^{2} \) \(\mathstrut -\mathstrut 3q^{3} \) \(\mathstrut +\mathstrut 3q^{4} \) \(\mathstrut -\mathstrut 7q^{5} \) \(\mathstrut -\mathstrut 29q^{6} \) \(\mathstrut -\mathstrut 35q^{7} \) \(\mathstrut +\mathstrut 47q^{8} \) \(\mathstrut +\mathstrut 31q^{9} \) \(\mathstrut +\mathstrut 40q^{10} \) \(\mathstrut +\mathstrut 67q^{11} \) \(\mathstrut +\mathstrut 190q^{12} \) \(\mathstrut -\mathstrut 65q^{13} \) \(\mathstrut -\mathstrut 196q^{14} \) \(\mathstrut -\mathstrut 121q^{15} \) \(\mathstrut -\mathstrut 377q^{16} \) \(\mathstrut -\mathstrut 31q^{17} \) \(\mathstrut -\mathstrut 102q^{18} \) \(\mathstrut +\mathstrut 148q^{19} \) \(\mathstrut +\mathstrut 342q^{20} \) \(\mathstrut +\mathstrut 334q^{21} \) \(\mathstrut +\mathstrut 647q^{22} \) \(\mathstrut -\mathstrut 12q^{23} \) \(\mathstrut -\mathstrut 447q^{24} \) \(\mathstrut -\mathstrut 201q^{25} \) \(\mathstrut -\mathstrut 140q^{26} \) \(\mathstrut +\mathstrut 72q^{27} \) \(\mathstrut -\mathstrut 42q^{28} \) \(\mathstrut -\mathstrut 199q^{29} \) \(\mathstrut -\mathstrut 114q^{30} \) \(\mathstrut -\mathstrut 361q^{31} \) \(\mathstrut +\mathstrut 324q^{32} \) \(\mathstrut -\mathstrut 232q^{33} \) \(\mathstrut -\mathstrut 298q^{34} \) \(\mathstrut +\mathstrut 237q^{35} \) \(\mathstrut +\mathstrut 120q^{36} \) \(\mathstrut +\mathstrut 81q^{37} \) \(\mathstrut -\mathstrut 52q^{38} \) \(\mathstrut +\mathstrut 365q^{39} \) \(\mathstrut +\mathstrut 532q^{40} \) \(\mathstrut -\mathstrut 31q^{41} \) \(\mathstrut +\mathstrut 170q^{42} \) \(\mathstrut -\mathstrut 650q^{43} \) \(\mathstrut -\mathstrut 1208q^{44} \) \(\mathstrut +\mathstrut 452q^{45} \) \(\mathstrut +\mathstrut 1204q^{46} \) \(\mathstrut +\mathstrut 857q^{47} \) \(\mathstrut +\mathstrut 644q^{48} \) \(\mathstrut +\mathstrut 1375q^{49} \) \(\mathstrut -\mathstrut 147q^{50} \) \(\mathstrut -\mathstrut 246q^{51} \) \(\mathstrut -\mathstrut 590q^{52} \) \(\mathstrut -\mathstrut 1493q^{53} \) \(\mathstrut -\mathstrut 3100q^{54} \) \(\mathstrut -\mathstrut 1583q^{55} \) \(\mathstrut -\mathstrut 1560q^{56} \) \(\mathstrut +\mathstrut 102q^{57} \) \(\mathstrut +\mathstrut 1392q^{58} \) \(\mathstrut +\mathstrut 676q^{59} \) \(\mathstrut +\mathstrut 1068q^{60} \) \(\mathstrut -\mathstrut 525q^{61} \) \(\mathstrut +\mathstrut 2456q^{62} \) \(\mathstrut -\mathstrut 68q^{63} \) \(\mathstrut +\mathstrut 471q^{64} \) \(\mathstrut +\mathstrut 1790q^{65} \) \(\mathstrut +\mathstrut 1014q^{66} \) \(\mathstrut +\mathstrut 86q^{67} \) \(\mathstrut +\mathstrut 710q^{68} \) \(\mathstrut -\mathstrut 42q^{69} \) \(\mathstrut -\mathstrut 144q^{70} \) \(\mathstrut +\mathstrut 1143q^{71} \) \(\mathstrut +\mathstrut 919q^{72} \) \(\mathstrut -\mathstrut 2155q^{73} \) \(\mathstrut -\mathstrut 1476q^{74} \) \(\mathstrut -\mathstrut 160q^{75} \) \(\mathstrut -\mathstrut 242q^{76} \) \(\mathstrut -\mathstrut 2015q^{77} \) \(\mathstrut -\mathstrut 1340q^{78} \) \(\mathstrut -\mathstrut 861q^{79} \) \(\mathstrut -\mathstrut 1916q^{80} \) \(\mathstrut -\mathstrut 26q^{81} \) \(\mathstrut -\mathstrut 3497q^{82} \) \(\mathstrut +\mathstrut 52q^{83} \) \(\mathstrut -\mathstrut 84q^{84} \) \(\mathstrut +\mathstrut 2383q^{85} \) \(\mathstrut +\mathstrut 1061q^{86} \) \(\mathstrut +\mathstrut 2310q^{87} \) \(\mathstrut +\mathstrut 4543q^{88} \) \(\mathstrut +\mathstrut 3782q^{89} \) \(\mathstrut -\mathstrut 1682q^{90} \) \(\mathstrut +\mathstrut 135q^{91} \) \(\mathstrut -\mathstrut 2450q^{92} \) \(\mathstrut -\mathstrut 2077q^{93} \) \(\mathstrut +\mathstrut 702q^{94} \) \(\mathstrut -\mathstrut 1317q^{95} \) \(\mathstrut +\mathstrut 1252q^{96} \) \(\mathstrut -\mathstrut 1344q^{97} \) \(\mathstrut +\mathstrut 2740q^{98} \) \(\mathstrut +\mathstrut 2099q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8}\mathstrut -\mathstrut \) \(x^{7}\mathstrut +\mathstrut \) \(10\) \(x^{6}\mathstrut -\mathstrut \) \(19\) \(x^{5}\mathstrut +\mathstrut \) \(109\) \(x^{4}\mathstrut +\mathstrut \) \(171\) \(x^{3}\mathstrut +\mathstrut \) \(810\) \(x^{2}\mathstrut +\mathstrut \) \(729\) \(x\mathstrut +\mathstrut \) \(6561\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( \nu^{6} + 280 \nu \)\()/981\)
\(\beta_{3}\)\(=\)\((\)\( \nu^{5} + 171 \)\()/109\)
\(\beta_{4}\)\(=\)\((\)\( \nu^{7} + 1261 \nu^{2} \)\()/8829\)
\(\beta_{5}\)\(=\)\((\)\( \nu^{7} - 10 \nu^{6} + 19 \nu^{5} - 109 \nu^{4} + 1090 \nu^{3} - 810 \nu^{2} - 729 \nu - 6561 \)\()/8829\)
\(\beta_{6}\)\(=\)\((\)\( -19 \nu^{7} + 19 \nu^{6} - 190 \nu^{5} + 1090 \nu^{4} - 2071 \nu^{3} - 3249 \nu^{2} - 15390 \nu - 13851 \)\()/79461\)
\(\beta_{7}\)\(=\)\((\)\( 10 \nu^{7} + 3781 \nu^{2} \)\()/8829\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(-\)\(\beta_{7}\mathstrut +\mathstrut \) \(10\) \(\beta_{4}\)
\(\nu^{3}\)\(=\)\(9\) \(\beta_{6}\mathstrut +\mathstrut \) \(10\) \(\beta_{5}\mathstrut +\mathstrut \) \(9\) \(\beta_{4}\mathstrut +\mathstrut \) \(9\) \(\beta_{2}\mathstrut +\mathstrut \) \(9\)
\(\nu^{4}\)\(=\)\(19\) \(\beta_{7}\mathstrut +\mathstrut \) \(90\) \(\beta_{6}\mathstrut +\mathstrut \) \(19\) \(\beta_{5}\mathstrut -\mathstrut \) \(19\) \(\beta_{4}\mathstrut +\mathstrut \) \(19\) \(\beta_{3}\mathstrut +\mathstrut \) \(19\) \(\beta_{1}\)
\(\nu^{5}\)\(=\)\(109\) \(\beta_{3}\mathstrut -\mathstrut \) \(171\)
\(\nu^{6}\)\(=\)\(981\) \(\beta_{2}\mathstrut -\mathstrut \) \(280\) \(\beta_{1}\)
\(\nu^{7}\)\(=\)\(1261\) \(\beta_{7}\mathstrut -\mathstrut \) \(3781\) \(\beta_{4}\)

Character Values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/11\mathbb{Z}\right)^\times\).

\(n\) \(2\)
\(\chi(n)\) \(\beta_{4}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
3.1
−2.05602 + 1.49379i
2.86504 2.08157i
−2.05602 1.49379i
2.86504 + 2.08157i
−1.09435 3.36805i
0.785330 + 2.41700i
−1.09435 + 3.36805i
0.785330 2.41700i
−4.17405 + 3.03263i 1.40336 + 4.31911i 5.75376 17.7083i 6.98613 + 5.07572i −18.9560 13.7723i 0.513765 1.58121i 16.9313 + 52.1091i 5.15818 3.74763i −44.5532
3.2 0.747004 0.542730i −0.476313 1.46594i −2.20868 + 6.79761i −7.05908 5.12872i −1.15142 0.836554i 0.239524 0.737179i 4.32202 + 13.3018i 19.9214 14.4737i −8.05666
4.1 −4.17405 3.03263i 1.40336 4.31911i 5.75376 + 17.7083i 6.98613 5.07572i −18.9560 + 13.7723i 0.513765 + 1.58121i 16.9313 52.1091i 5.15818 + 3.74763i −44.5532
4.2 0.747004 + 0.542730i −0.476313 + 1.46594i −2.20868 6.79761i −7.05908 + 5.12872i −1.15142 + 0.836554i 0.239524 + 0.737179i 4.32202 13.3018i 19.9214 + 14.4737i −8.05666
5.1 −0.976313 3.00478i 1.24700 + 0.906001i −1.60339 + 1.16493i −5.33576 + 16.4218i 1.50487 4.63152i 7.73807 5.62204i −15.3824 11.1760i −7.60928 23.4190i 54.5532
5.2 0.903364 + 2.78027i −3.67405 2.66936i −0.441690 + 0.320907i 1.90871 5.87440i 4.10252 12.6263i −25.9914 + 18.8838i 17.6291 + 12.8083i −1.97025 6.06380i 18.0567
9.1 −0.976313 + 3.00478i 1.24700 0.906001i −1.60339 1.16493i −5.33576 16.4218i 1.50487 + 4.63152i 7.73807 + 5.62204i −15.3824 + 11.1760i −7.60928 + 23.4190i 54.5532
9.2 0.903364 2.78027i −3.67405 + 2.66936i −0.441690 0.320907i 1.90871 + 5.87440i 4.10252 + 12.6263i −25.9914 18.8838i 17.6291 12.8083i −1.97025 + 6.06380i 18.0567
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 9.2
Significant digits:
Format:

Inner twists

Char. orbit Parity Mult. Self Twist Proved
1.a Even 1 trivial yes
11.c Even 1 yes

Hecke kernels

There are no other newforms in \(S_{4}^{\mathrm{new}}(11, [\chi])\).