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 \(5\), degree \(4\), minimal)

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: \( 1 \)
Twist minimal: yes
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})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q - 7q^{2} - 3q^{3} + 3q^{4} - 7q^{5} - 29q^{6} - 35q^{7} + 47q^{8} + 31q^{9} + O(q^{10}) \) \( 8q - 7q^{2} - 3q^{3} + 3q^{4} - 7q^{5} - 29q^{6} - 35q^{7} + 47q^{8} + 31q^{9} + 40q^{10} + 67q^{11} + 190q^{12} - 65q^{13} - 196q^{14} - 121q^{15} - 377q^{16} - 31q^{17} - 102q^{18} + 148q^{19} + 342q^{20} + 334q^{21} + 647q^{22} - 12q^{23} - 447q^{24} - 201q^{25} - 140q^{26} + 72q^{27} - 42q^{28} - 199q^{29} - 114q^{30} - 361q^{31} + 324q^{32} - 232q^{33} - 298q^{34} + 237q^{35} + 120q^{36} + 81q^{37} - 52q^{38} + 365q^{39} + 532q^{40} - 31q^{41} + 170q^{42} - 650q^{43} - 1208q^{44} + 452q^{45} + 1204q^{46} + 857q^{47} + 644q^{48} + 1375q^{49} - 147q^{50} - 246q^{51} - 590q^{52} - 1493q^{53} - 3100q^{54} - 1583q^{55} - 1560q^{56} + 102q^{57} + 1392q^{58} + 676q^{59} + 1068q^{60} - 525q^{61} + 2456q^{62} - 68q^{63} + 471q^{64} + 1790q^{65} + 1014q^{66} + 86q^{67} + 710q^{68} - 42q^{69} - 144q^{70} + 1143q^{71} + 919q^{72} - 2155q^{73} - 1476q^{74} - 160q^{75} - 242q^{76} - 2015q^{77} - 1340q^{78} - 861q^{79} - 1916q^{80} - 26q^{81} - 3497q^{82} + 52q^{83} - 84q^{84} + 2383q^{85} + 1061q^{86} + 2310q^{87} + 4543q^{88} + 3782q^{89} - 1682q^{90} + 135q^{91} - 2450q^{92} - 2077q^{93} + 702q^{94} - 1317q^{95} + 1252q^{96} - 1344q^{97} + 2740q^{98} + 2099q^{99} + O(q^{100}) \)

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

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

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 Parity Ord Mult Type
1.a even 1 1 trivial
11.c even 5 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 11.4.c.a 8
3.b odd 2 1 99.4.f.c 8
4.b odd 2 1 176.4.m.c 8
11.b odd 2 1 121.4.c.i 8
11.c even 5 1 inner 11.4.c.a 8
11.c even 5 1 121.4.a.g 4
11.c even 5 2 121.4.c.h 8
11.d odd 10 1 121.4.a.f 4
11.d odd 10 2 121.4.c.b 8
11.d odd 10 1 121.4.c.i 8
33.f even 10 1 1089.4.a.bh 4
33.h odd 10 1 99.4.f.c 8
33.h odd 10 1 1089.4.a.y 4
44.g even 10 1 1936.4.a.bl 4
44.h odd 10 1 176.4.m.c 8
44.h odd 10 1 1936.4.a.bk 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
11.4.c.a 8 1.a even 1 1 trivial
11.4.c.a 8 11.c even 5 1 inner
99.4.f.c 8 3.b odd 2 1
99.4.f.c 8 33.h odd 10 1
121.4.a.f 4 11.d odd 10 1
121.4.a.g 4 11.c even 5 1
121.4.c.b 8 11.d odd 10 2
121.4.c.h 8 11.c even 5 2
121.4.c.i 8 11.b odd 2 1
121.4.c.i 8 11.d odd 10 1
176.4.m.c 8 4.b odd 2 1
176.4.m.c 8 44.h odd 10 1
1089.4.a.y 4 33.h odd 10 1
1089.4.a.bh 4 33.f even 10 1
1936.4.a.bk 4 44.h odd 10 1
1936.4.a.bl 4 44.g even 10 1

Hecke kernels

This newform subspace is the entire newspace \(S_{4}^{\mathrm{new}}(11, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + 7 T + 15 T^{2} - 17 T^{3} - 81 T^{4} + 366 T^{5} + 1624 T^{6} - 1744 T^{7} - 17904 T^{8} - 13952 T^{9} + 103936 T^{10} + 187392 T^{11} - 331776 T^{12} - 557056 T^{13} + 3932160 T^{14} + 14680064 T^{15} + 16777216 T^{16} \)
$3$ \( 1 + 3 T - 38 T^{2} - 174 T^{3} + 778 T^{4} + 4539 T^{5} - 9341 T^{6} - 51492 T^{7} + 85048 T^{8} - 1390284 T^{9} - 6809589 T^{10} + 89341137 T^{11} + 413461098 T^{12} - 2496709818 T^{13} - 14721978582 T^{14} + 31381059609 T^{15} + 282429536481 T^{16} \)
$5$ \( 1 + 7 T - 518 T^{3} - 1266 T^{4} - 256767 T^{5} - 39725 T^{6} + 1239098 T^{7} + 146654796 T^{8} + 154887250 T^{9} - 620703125 T^{10} - 501498046875 T^{11} - 309082031250 T^{12} - 15808105468750 T^{13} + 3337860107421875 T^{15} + 59604644775390625 T^{16} \)
$7$ \( 1 + 35 T - 418 T^{2} - 32140 T^{3} - 221250 T^{4} + 13130885 T^{5} + 294623937 T^{6} - 1901444100 T^{7} - 135159872636 T^{8} - 652195326300 T^{9} + 34662211564113 T^{10} + 529878572852195 T^{11} - 3062384793221250 T^{12} - 152586626929568020 T^{13} - 680676883926567682 T^{14} + 19549105242914940245 T^{15} + \)\(19\!\cdots\!01\)\( T^{16} \)
$11$ \( 1 - 67 T + 1463 T^{2} - 67639 T^{3} + 4205960 T^{4} - 90027509 T^{5} + 2591793743 T^{6} - 157982495297 T^{7} + 3138428376721 T^{8} \)
$13$ \( 1 + 65 T - 3774 T^{2} - 299880 T^{3} + 2516892 T^{4} + 764350785 T^{5} + 39746687433 T^{6} - 593220674550 T^{7} - 134342539611120 T^{8} - 1303305821986350 T^{9} + 191849668621791297 T^{10} + 8105557420284557805 T^{11} + 58638764060091449052 T^{12} - \)\(15\!\cdots\!60\)\( T^{13} - \)\(42\!\cdots\!46\)\( T^{14} + \)\(16\!\cdots\!45\)\( T^{15} + \)\(54\!\cdots\!61\)\( T^{16} \)
$17$ \( 1 + 31 T - 792 T^{2} - 523972 T^{3} + 3004088 T^{4} - 470896367 T^{5} + 53763531151 T^{6} - 2458004448816 T^{7} + 615070592257928 T^{8} - 12076175857033008 T^{9} + 1297720942840911919 T^{10} - 55842600212681986399 T^{11} + \)\(17\!\cdots\!68\)\( T^{12} - \)\(14\!\cdots\!96\)\( T^{13} - \)\(11\!\cdots\!28\)\( T^{14} + \)\(21\!\cdots\!27\)\( T^{15} + \)\(33\!\cdots\!21\)\( T^{16} \)
$19$ \( 1 - 148 T + 11575 T^{2} - 1444790 T^{3} + 194162525 T^{4} - 16652433274 T^{5} + 1325481769157 T^{6} - 142885089670400 T^{7} + 13417456053996940 T^{8} - 980048830049273600 T^{9} + 62358457579429692317 T^{10} - \)\(53\!\cdots\!46\)\( T^{11} + \)\(42\!\cdots\!25\)\( T^{12} - \)\(21\!\cdots\!10\)\( T^{13} + \)\(12\!\cdots\!75\)\( T^{14} - \)\(10\!\cdots\!12\)\( T^{15} + \)\(48\!\cdots\!21\)\( T^{16} \)
$23$ \( ( 1 + 6 T + 27488 T^{2} + 757614 T^{3} + 422704798 T^{4} + 9217889538 T^{5} + 4069210516832 T^{6} + 10806915968778 T^{7} + 21914624432020321 T^{8} )^{2} \)
$29$ \( 1 + 199 T + 14972 T^{2} + 3155808 T^{3} + 811239992 T^{4} + 144591222505 T^{5} + 23423431670163 T^{6} + 4517385217828956 T^{7} + 793290858128995600 T^{8} + \)\(11\!\cdots\!84\)\( T^{9} + \)\(13\!\cdots\!23\)\( T^{10} + \)\(20\!\cdots\!45\)\( T^{11} + \)\(28\!\cdots\!72\)\( T^{12} + \)\(27\!\cdots\!92\)\( T^{13} + \)\(31\!\cdots\!92\)\( T^{14} + \)\(10\!\cdots\!71\)\( T^{15} + \)\(12\!\cdots\!81\)\( T^{16} \)
$31$ \( 1 + 361 T + 68292 T^{2} + 8747944 T^{3} + 1502105536 T^{4} - 10476370985 T^{5} - 64833894853287 T^{6} - 17021745180201160 T^{7} - 2614984674824755148 T^{8} - \)\(50\!\cdots\!60\)\( T^{9} - \)\(57\!\cdots\!47\)\( T^{10} - \)\(27\!\cdots\!35\)\( T^{11} + \)\(11\!\cdots\!96\)\( T^{12} + \)\(20\!\cdots\!44\)\( T^{13} + \)\(47\!\cdots\!72\)\( T^{14} + \)\(75\!\cdots\!91\)\( T^{15} + \)\(62\!\cdots\!21\)\( T^{16} \)
$37$ \( 1 - 81 T - 78732 T^{2} + 413682 T^{3} + 4985376798 T^{4} + 13237102557 T^{5} - 205379482511809 T^{6} + 1480099189059246 T^{7} + 6444772391025324588 T^{8} + 74971464223417987638 T^{9} - \)\(52\!\cdots\!81\)\( T^{10} + \)\(17\!\cdots\!89\)\( T^{11} + \)\(32\!\cdots\!38\)\( T^{12} + \)\(13\!\cdots\!26\)\( T^{13} - \)\(13\!\cdots\!28\)\( T^{14} - \)\(69\!\cdots\!97\)\( T^{15} + \)\(43\!\cdots\!61\)\( T^{16} \)
$41$ \( 1 + 31 T - 41412 T^{2} + 41956 T^{3} + 4511377452 T^{4} + 946844618625 T^{5} - 140505296388233 T^{6} + 5245993854286808 T^{7} + 10132197830913446520 T^{8} + \)\(36\!\cdots\!68\)\( T^{9} - \)\(66\!\cdots\!53\)\( T^{10} + \)\(30\!\cdots\!25\)\( T^{11} + \)\(10\!\cdots\!12\)\( T^{12} + \)\(65\!\cdots\!56\)\( T^{13} - \)\(44\!\cdots\!52\)\( T^{14} + \)\(22\!\cdots\!71\)\( T^{15} + \)\(50\!\cdots\!61\)\( T^{16} \)
$43$ \( ( 1 + 325 T + 241017 T^{2} + 48348645 T^{3} + 24393602404 T^{4} + 3844055718015 T^{5} + 1523555957980833 T^{6} + 163342598879473975 T^{7} + 39959630797262576401 T^{8} )^{2} \)
$47$ \( 1 - 857 T + 121092 T^{2} + 122368114 T^{3} - 61023105504 T^{4} + 11078984252823 T^{5} + 420607797631015 T^{6} - 1505063442345718948 T^{7} + \)\(72\!\cdots\!64\)\( T^{8} - \)\(15\!\cdots\!04\)\( T^{9} + \)\(45\!\cdots\!35\)\( T^{10} + \)\(12\!\cdots\!41\)\( T^{11} - \)\(70\!\cdots\!64\)\( T^{12} + \)\(14\!\cdots\!02\)\( T^{13} + \)\(15\!\cdots\!88\)\( T^{14} - \)\(11\!\cdots\!79\)\( T^{15} + \)\(13\!\cdots\!81\)\( T^{16} \)
$53$ \( 1 + 1493 T + 684042 T^{2} - 45129934 T^{3} - 136433012142 T^{4} - 36785759748711 T^{5} + 3910583895798949 T^{6} + 5538380615310974548 T^{7} + \)\(24\!\cdots\!28\)\( T^{8} + \)\(82\!\cdots\!96\)\( T^{9} + \)\(86\!\cdots\!21\)\( T^{10} - \)\(12\!\cdots\!63\)\( T^{11} - \)\(67\!\cdots\!22\)\( T^{12} - \)\(33\!\cdots\!38\)\( T^{13} + \)\(74\!\cdots\!38\)\( T^{14} + \)\(24\!\cdots\!29\)\( T^{15} + \)\(24\!\cdots\!81\)\( T^{16} \)
$59$ \( 1 - 676 T - 75509 T^{2} + 9379530 T^{3} + 128650837633 T^{4} - 34757410557650 T^{5} - 18754376858866031 T^{6} - 1699130425604624376 T^{7} + \)\(76\!\cdots\!28\)\( T^{8} - \)\(34\!\cdots\!04\)\( T^{9} - \)\(79\!\cdots\!71\)\( T^{10} - \)\(30\!\cdots\!50\)\( T^{11} + \)\(22\!\cdots\!73\)\( T^{12} + \)\(34\!\cdots\!70\)\( T^{13} - \)\(56\!\cdots\!89\)\( T^{14} - \)\(10\!\cdots\!84\)\( T^{15} + \)\(31\!\cdots\!61\)\( T^{16} \)
$61$ \( 1 + 525 T - 238754 T^{2} - 162289050 T^{3} - 31751133810 T^{4} - 278276538435 T^{5} + 17785841568589049 T^{6} + 2328023641412643300 T^{7} - \)\(43\!\cdots\!16\)\( T^{8} + \)\(52\!\cdots\!00\)\( T^{9} + \)\(91\!\cdots\!89\)\( T^{10} - \)\(32\!\cdots\!35\)\( T^{11} - \)\(84\!\cdots\!10\)\( T^{12} - \)\(97\!\cdots\!50\)\( T^{13} - \)\(32\!\cdots\!74\)\( T^{14} + \)\(16\!\cdots\!25\)\( T^{15} + \)\(70\!\cdots\!41\)\( T^{16} \)
$67$ \( ( 1 - 43 T + 808331 T^{2} - 155629659 T^{3} + 296445352544 T^{4} - 46807643129817 T^{5} + 73120314517049939 T^{6} - 1169880979040682721 T^{7} + \)\(81\!\cdots\!61\)\( T^{8} )^{2} \)
$71$ \( 1 - 1143 T + 717774 T^{2} - 426999060 T^{3} + 398304224128 T^{4} - 150060483633915 T^{5} - 16487601883081509 T^{6} + 18708832368626731338 T^{7} + \)\(60\!\cdots\!48\)\( T^{8} + \)\(66\!\cdots\!18\)\( T^{9} - \)\(21\!\cdots\!89\)\( T^{10} - \)\(68\!\cdots\!65\)\( T^{11} + \)\(65\!\cdots\!48\)\( T^{12} - \)\(25\!\cdots\!60\)\( T^{13} + \)\(15\!\cdots\!14\)\( T^{14} - \)\(85\!\cdots\!53\)\( T^{15} + \)\(26\!\cdots\!81\)\( T^{16} \)
$73$ \( 1 + 2155 T + 2185548 T^{2} + 1809165000 T^{3} + 1562803949820 T^{4} + 1050489744927825 T^{5} + 520439165741232123 T^{6} + \)\(31\!\cdots\!60\)\( T^{7} + \)\(21\!\cdots\!64\)\( T^{8} + \)\(12\!\cdots\!20\)\( T^{9} + \)\(78\!\cdots\!47\)\( T^{10} + \)\(61\!\cdots\!25\)\( T^{11} + \)\(35\!\cdots\!20\)\( T^{12} + \)\(16\!\cdots\!00\)\( T^{13} + \)\(75\!\cdots\!12\)\( T^{14} + \)\(29\!\cdots\!15\)\( T^{15} + \)\(52\!\cdots\!41\)\( T^{16} \)
$79$ \( 1 + 861 T - 428248 T^{2} - 602343768 T^{3} + 107104581192 T^{4} + 169355011342215 T^{5} - 103898581181934707 T^{6} - 30048205990965814356 T^{7} + \)\(59\!\cdots\!00\)\( T^{8} - \)\(14\!\cdots\!84\)\( T^{9} - \)\(25\!\cdots\!47\)\( T^{10} + \)\(20\!\cdots\!85\)\( T^{11} + \)\(63\!\cdots\!72\)\( T^{12} - \)\(17\!\cdots\!32\)\( T^{13} - \)\(61\!\cdots\!28\)\( T^{14} + \)\(60\!\cdots\!19\)\( T^{15} + \)\(34\!\cdots\!81\)\( T^{16} \)
$83$ \( 1 - 52 T - 837901 T^{2} - 580716282 T^{3} + 33426101033 T^{4} + 830942030041778 T^{5} + 400459572181036761 T^{6} - \)\(26\!\cdots\!76\)\( T^{7} - \)\(36\!\cdots\!72\)\( T^{8} - \)\(15\!\cdots\!12\)\( T^{9} + \)\(13\!\cdots\!09\)\( T^{10} + \)\(15\!\cdots\!34\)\( T^{11} + \)\(35\!\cdots\!13\)\( T^{12} - \)\(35\!\cdots\!74\)\( T^{13} - \)\(29\!\cdots\!09\)\( T^{14} - \)\(10\!\cdots\!16\)\( T^{15} + \)\(11\!\cdots\!21\)\( T^{16} \)
$89$ \( ( 1 - 1891 T + 3678143 T^{2} - 4008289261 T^{3} + 4189944324368 T^{4} - 2825719672037909 T^{5} + 1827968256479165423 T^{6} - \)\(66\!\cdots\!19\)\( T^{7} + \)\(24\!\cdots\!21\)\( T^{8} )^{2} \)
$97$ \( 1 + 1344 T + 412889 T^{2} + 1296130300 T^{3} + 2215229286991 T^{4} + 1422791121306492 T^{5} + 1824021419147257715 T^{6} + \)\(23\!\cdots\!68\)\( T^{7} + \)\(18\!\cdots\!72\)\( T^{8} + \)\(21\!\cdots\!64\)\( T^{9} + \)\(15\!\cdots\!35\)\( T^{10} + \)\(10\!\cdots\!64\)\( T^{11} + \)\(15\!\cdots\!31\)\( T^{12} + \)\(82\!\cdots\!00\)\( T^{13} + \)\(23\!\cdots\!21\)\( T^{14} + \)\(70\!\cdots\!68\)\( T^{15} + \)\(48\!\cdots\!81\)\( T^{16} \)
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