Properties

Label 22.4.c.b
Level 22
Weight 4
Character orbit 22.c
Analytic conductor 1.298
Analytic rank 0
Dimension 8
CM No
Inner twists 2

Related objects

Downloads

Learn more about

Newspace parameters

Level: \( N \) = \( 22 = 2 \cdot 11 \)
Weight: \( k \) = \( 4 \)
Character orbit: \([\chi]\) = 22.c (of order \(5\) and degree \(4\))

Newform invariants

Self dual: No
Analytic conductor: \(1.29804202013\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(2\) over \(\Q(\zeta_{5})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
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.

\(f(q)\) \(=\) \( q\) \( -2 \beta_{3} q^{2} \) \( + ( 1 + \beta_{3} - \beta_{4} - \beta_{5} ) q^{3} \) \( + 4 \beta_{2} q^{4} \) \( + ( -5 - 8 \beta_{2} - 8 \beta_{3} + 4 \beta_{4} + \beta_{5} - \beta_{6} ) q^{5} \) \( + ( 2 - 2 \beta_{4} - 2 \beta_{7} ) q^{6} \) \( + ( 1 + 2 \beta_{1} + 2 \beta_{3} - 2 \beta_{4} + 2 \beta_{5} + \beta_{6} + \beta_{7} ) q^{7} \) \( + 8 \beta_{4} q^{8} \) \( + ( 7 - 3 \beta_{1} + 7 \beta_{2} + 28 \beta_{3} - 2 \beta_{5} + 2 \beta_{7} ) q^{9} \) \(+O(q^{10})\) \( q\) \( -2 \beta_{3} q^{2} \) \( + ( 1 + \beta_{3} - \beta_{4} - \beta_{5} ) q^{3} \) \( + 4 \beta_{2} q^{4} \) \( + ( -5 - 8 \beta_{2} - 8 \beta_{3} + 4 \beta_{4} + \beta_{5} - \beta_{6} ) q^{5} \) \( + ( 2 - 2 \beta_{4} - 2 \beta_{7} ) q^{6} \) \( + ( 1 + 2 \beta_{1} + 2 \beta_{3} - 2 \beta_{4} + 2 \beta_{5} + \beta_{6} + \beta_{7} ) q^{7} \) \( + 8 \beta_{4} q^{8} \) \( + ( 7 - 3 \beta_{1} + 7 \beta_{2} + 28 \beta_{3} - 2 \beta_{5} + 2 \beta_{7} ) q^{9} \) \( + ( -8 - 2 \beta_{1} + 8 \beta_{2} - 8 \beta_{4} + 2 \beta_{7} ) q^{10} \) \( + ( -13 - 6 \beta_{2} + 8 \beta_{3} - 15 \beta_{4} - 3 \beta_{5} + 3 \beta_{6} + 2 \beta_{7} ) q^{11} \) \( + ( 8 + 4 \beta_{2} - 4 \beta_{4} + 4 \beta_{6} ) q^{12} \) \( + ( 4 - 3 \beta_{1} + 4 \beta_{2} + 10 \beta_{3} + 3 \beta_{5} - 3 \beta_{7} ) q^{13} \) \( + ( -2 + 6 \beta_{1} + 4 \beta_{3} - 4 \beta_{4} + 4 \beta_{5} - 6 \beta_{6} ) q^{14} \) \( + ( -9 + 7 \beta_{1} - 12 \beta_{2} - 50 \beta_{3} + 50 \beta_{4} + 7 \beta_{5} - 9 \beta_{6} - 9 \beta_{7} ) q^{15} \) \( + ( -16 - 16 \beta_{2} - 16 \beta_{3} + 16 \beta_{4} ) q^{16} \) \( + ( 30 + \beta_{2} + \beta_{3} - 26 \beta_{4} - 4 \beta_{5} + 4 \beta_{6} + 3 \beta_{7} ) q^{17} \) \( + ( 2 - 6 \beta_{1} - 56 \beta_{2} - 14 \beta_{3} + 14 \beta_{4} - 6 \beta_{5} + 2 \beta_{6} + 2 \beta_{7} ) q^{18} \) \( + ( -55 - 2 \beta_{1} - 57 \beta_{3} + 36 \beta_{4} - 8 \beta_{5} + 2 \beta_{6} ) q^{19} \) \( + ( 16 - 4 \beta_{1} + 16 \beta_{2} + 32 \beta_{3} - 4 \beta_{5} + 4 \beta_{7} ) q^{20} \) \( + ( 66 + 3 \beta_{1} + 144 \beta_{2} - 144 \beta_{4} + 2 \beta_{6} - 3 \beta_{7} ) q^{21} \) \( + ( 26 + 6 \beta_{1} + 14 \beta_{2} + 62 \beta_{3} - 42 \beta_{4} - 4 \beta_{6} - 6 \beta_{7} ) q^{22} \) \( + ( 42 - 6 \beta_{1} - 70 \beta_{2} + 70 \beta_{4} - 6 \beta_{6} + 6 \beta_{7} ) q^{23} \) \( + ( 8 + 8 \beta_{1} + 8 \beta_{2} ) q^{24} \) \( + ( -1 - 9 \beta_{1} - 10 \beta_{3} + 9 \beta_{4} - 2 \beta_{5} + 9 \beta_{6} ) q^{25} \) \( + ( 12 - 6 \beta_{1} - 20 \beta_{2} - 8 \beta_{3} + 8 \beta_{4} - 6 \beta_{5} + 12 \beta_{6} + 12 \beta_{7} ) q^{26} \) \( + ( -107 - 31 \beta_{2} - 31 \beta_{3} + 109 \beta_{4} - 2 \beta_{5} + 2 \beta_{6} + 4 \beta_{7} ) q^{27} \) \( + ( -4 - 8 \beta_{4} + 12 \beta_{5} - 12 \beta_{6} - 4 \beta_{7} ) q^{28} \) \( + ( -9 + 10 \beta_{1} - 120 \beta_{2} + 50 \beta_{3} - 50 \beta_{4} + 10 \beta_{5} - 9 \beta_{6} - 9 \beta_{7} ) q^{29} \) \( + ( -96 - 4 \beta_{1} - 100 \beta_{3} + 76 \beta_{4} + 14 \beta_{5} + 4 \beta_{6} ) q^{30} \) \( + ( 74 + 25 \beta_{1} + 74 \beta_{2} + 94 \beta_{3} + 21 \beta_{5} - 21 \beta_{7} ) q^{31} \) \( -32 q^{32} \) \( + ( -50 - 20 \beta_{1} + 111 \beta_{2} + 138 \beta_{3} - 124 \beta_{4} + 11 \beta_{5} - 5 \beta_{6} + 9 \beta_{7} ) q^{33} \) \( + ( 46 + 8 \beta_{1} + 50 \beta_{2} - 50 \beta_{4} - 6 \beta_{6} - 8 \beta_{7} ) q^{34} \) \( + ( -60 + 3 \beta_{1} - 60 \beta_{2} + 154 \beta_{3} - 16 \beta_{5} + 16 \beta_{7} ) q^{35} \) \( + ( -20 - 8 \beta_{1} - 28 \beta_{3} - 84 \beta_{4} - 12 \beta_{5} + 8 \beta_{6} ) q^{36} \) \( + ( 27 - 32 \beta_{1} + 80 \beta_{2} - 48 \beta_{3} + 48 \beta_{4} - 32 \beta_{5} + 27 \beta_{6} + 27 \beta_{7} ) q^{37} \) \( + ( -68 + 42 \beta_{2} + 42 \beta_{3} + 72 \beta_{4} - 4 \beta_{5} + 4 \beta_{6} - 12 \beta_{7} ) q^{38} \) \( + ( -160 - 344 \beta_{2} - 344 \beta_{3} + 164 \beta_{4} - 4 \beta_{5} + 4 \beta_{6} + 13 \beta_{7} ) q^{39} \) \( + ( -8 \beta_{1} - 64 \beta_{2} - 32 \beta_{3} + 32 \beta_{4} - 8 \beta_{5} ) q^{40} \) \( + ( 57 + 24 \beta_{1} + 81 \beta_{3} - 35 \beta_{4} + 17 \beta_{5} - 24 \beta_{6} ) q^{41} \) \( + ( 288 + 10 \beta_{1} + 288 \beta_{2} + 160 \beta_{3} + 6 \beta_{5} - 6 \beta_{7} ) q^{42} \) \( + ( 255 + 7 \beta_{1} + 167 \beta_{2} - 167 \beta_{4} - 14 \beta_{6} - 7 \beta_{7} ) q^{43} \) \( + ( 84 + 4 \beta_{1} - 40 \beta_{2} + 24 \beta_{3} - 56 \beta_{4} + 12 \beta_{5} - 12 \beta_{7} ) q^{44} \) \( + ( 313 + 28 \beta_{1} - 18 \beta_{2} + 18 \beta_{4} - 5 \beta_{6} - 28 \beta_{7} ) q^{45} \) \( + ( -140 - 24 \beta_{1} - 140 \beta_{2} - 236 \beta_{3} - 12 \beta_{5} + 12 \beta_{7} ) q^{46} \) \( + ( -169 - 9 \beta_{1} - 178 \beta_{3} - 20 \beta_{4} + 12 \beta_{5} + 9 \beta_{6} ) q^{47} \) \( + ( -16 + 16 \beta_{1} - 16 \beta_{3} + 16 \beta_{4} + 16 \beta_{5} - 16 \beta_{6} - 16 \beta_{7} ) q^{48} \) \( + ( -234 - 177 \beta_{2} - 177 \beta_{3} + 239 \beta_{4} - 5 \beta_{5} + 5 \beta_{6} - 2 \beta_{7} ) q^{49} \) \( + ( 2 \beta_{2} + 2 \beta_{3} + 18 \beta_{4} - 18 \beta_{5} + 18 \beta_{6} + 14 \beta_{7} ) q^{50} \) \( + ( 2 - 32 \beta_{1} + 169 \beta_{2} + 234 \beta_{3} - 234 \beta_{4} - 32 \beta_{5} + 2 \beta_{6} + 2 \beta_{7} ) q^{51} \) \( + ( -28 + 12 \beta_{1} - 16 \beta_{3} - 24 \beta_{4} - 12 \beta_{5} - 12 \beta_{6} ) q^{52} \) \( + ( -50 - 47 \beta_{1} - 50 \beta_{2} - 72 \beta_{3} - 39 \beta_{5} + 39 \beta_{7} ) q^{53} \) \( + ( -226 + 4 \beta_{1} - 156 \beta_{2} + 156 \beta_{4} - 8 \beta_{6} - 4 \beta_{7} ) q^{54} \) \( + ( 45 + 50 \beta_{1} + 192 \beta_{2} + 118 \beta_{3} + 84 \beta_{4} + \beta_{5} - 5 \beta_{6} - 14 \beta_{7} ) q^{55} \) \( + ( 24 - 24 \beta_{1} + 16 \beta_{2} - 16 \beta_{4} + 8 \beta_{6} + 24 \beta_{7} ) q^{56} \) \( + ( -8 + 18 \beta_{1} - 8 \beta_{2} + 251 \beta_{3} + 47 \beta_{5} - 47 \beta_{7} ) q^{57} \) \( + ( 98 + 2 \beta_{1} + 100 \beta_{3} - 340 \beta_{4} + 20 \beta_{5} - 2 \beta_{6} ) q^{58} \) \( + ( -26 + 40 \beta_{1} + 203 \beta_{2} - 69 \beta_{3} + 69 \beta_{4} + 40 \beta_{5} - 26 \beta_{6} - 26 \beta_{7} ) q^{59} \) \( + ( -144 + 48 \beta_{2} + 48 \beta_{3} + 152 \beta_{4} - 8 \beta_{5} + 8 \beta_{6} + 36 \beta_{7} ) q^{60} \) \( + ( -273 - 228 \beta_{2} - 228 \beta_{3} + 246 \beta_{4} + 27 \beta_{5} - 27 \beta_{6} - 28 \beta_{7} ) q^{61} \) \( + ( -8 + 50 \beta_{1} - 188 \beta_{2} - 148 \beta_{3} + 148 \beta_{4} + 50 \beta_{5} - 8 \beta_{6} - 8 \beta_{7} ) q^{62} \) \( + ( 311 - 71 \beta_{1} + 240 \beta_{3} - 468 \beta_{4} - 23 \beta_{5} + 71 \beta_{6} ) q^{63} \) \( + 64 \beta_{3} q^{64} \) \( + ( 69 - 27 \beta_{1} + 106 \beta_{2} - 106 \beta_{4} + 53 \beta_{6} + 27 \beta_{7} ) q^{65} \) \( + ( 270 - 50 \beta_{1} - 28 \beta_{2} + 338 \beta_{3} - 26 \beta_{4} - 40 \beta_{5} + 22 \beta_{6} + 62 \beta_{7} ) q^{66} \) \( + ( 91 - 59 \beta_{1} - 43 \beta_{2} + 43 \beta_{4} + 51 \beta_{6} + 59 \beta_{7} ) q^{67} \) \( + ( 100 + 4 \beta_{1} + 100 \beta_{2} - 4 \beta_{3} + 16 \beta_{5} - 16 \beta_{7} ) q^{68} \) \( + ( -268 + 88 \beta_{1} - 180 \beta_{3} + 526 \beta_{4} - 18 \beta_{5} - 88 \beta_{6} ) q^{69} \) \( + ( -38 + 6 \beta_{1} - 308 \beta_{2} + 120 \beta_{3} - 120 \beta_{4} + 6 \beta_{5} - 38 \beta_{6} - 38 \beta_{7} ) q^{70} \) \( + ( 211 + 334 \beta_{2} + 334 \beta_{3} - 262 \beta_{4} + 51 \beta_{5} - 51 \beta_{6} - 32 \beta_{7} ) q^{71} \) \( + ( 184 + 224 \beta_{2} + 224 \beta_{3} - 168 \beta_{4} - 16 \beta_{5} + 16 \beta_{6} - 8 \beta_{7} ) q^{72} \) \( + ( 45 + 17 \beta_{1} + 418 \beta_{2} + 205 \beta_{3} - 205 \beta_{4} + 17 \beta_{5} + 45 \beta_{6} + 45 \beta_{7} ) q^{73} \) \( + ( -86 - 10 \beta_{1} - 96 \beta_{3} + 256 \beta_{4} - 64 \beta_{5} + 10 \beta_{6} ) q^{74} \) \( + ( -461 - 4 \beta_{1} - 461 \beta_{2} - 460 \beta_{3} - \beta_{5} + \beta_{7} ) q^{75} \) \( + ( -120 + 8 \beta_{1} - 228 \beta_{2} + 228 \beta_{4} + 24 \beta_{6} - 8 \beta_{7} ) q^{76} \) \( + ( -135 - 8 \beta_{1} - 118 \beta_{2} - 642 \beta_{3} + 90 \beta_{4} - 68 \beta_{5} + 11 \beta_{6} - 53 \beta_{7} ) q^{77} \) \( + ( -354 + 8 \beta_{1} + 360 \beta_{2} - 360 \beta_{4} - 26 \beta_{6} - 8 \beta_{7} ) q^{78} \) \( + ( 102 - 19 \beta_{1} + 102 \beta_{2} + 206 \beta_{3} - 55 \beta_{5} + 55 \beta_{7} ) q^{79} \) \( + ( -48 - 16 \beta_{1} - 64 \beta_{3} - 64 \beta_{4} - 16 \beta_{5} + 16 \beta_{6} ) q^{80} \) \( + ( -6 + 30 \beta_{1} - 446 \beta_{2} - 174 \beta_{3} + 174 \beta_{4} + 30 \beta_{5} - 6 \beta_{6} - 6 \beta_{7} ) q^{81} \) \( + ( 22 - 92 \beta_{2} - 92 \beta_{3} - 70 \beta_{4} + 48 \beta_{5} - 48 \beta_{6} - 14 \beta_{7} ) q^{82} \) \( + ( 457 + 214 \beta_{2} + 214 \beta_{3} - 445 \beta_{4} - 12 \beta_{5} + 12 \beta_{6} - 92 \beta_{7} ) q^{83} \) \( + ( -8 + 20 \beta_{1} - 320 \beta_{2} - 576 \beta_{3} + 576 \beta_{4} + 20 \beta_{5} - 8 \beta_{6} - 8 \beta_{7} ) q^{84} \) \( + ( -165 + 55 \beta_{1} - 110 \beta_{3} + 226 \beta_{4} + 44 \beta_{5} - 55 \beta_{6} ) q^{85} \) \( + ( 334 - 14 \beta_{1} + 334 \beta_{2} - 204 \beta_{3} + 14 \beta_{5} - 14 \beta_{7} ) q^{86} \) \( + ( 122 - 39 \beta_{1} - 196 \beta_{2} + 196 \beta_{4} - 110 \beta_{6} + 39 \beta_{7} ) q^{87} \) \( + ( 128 + 8 \beta_{1} + 64 \beta_{2} - 56 \beta_{3} - 192 \beta_{4} + 8 \beta_{5} + 16 \beta_{6} + 16 \beta_{7} ) q^{88} \) \( + ( -521 + 61 \beta_{1} - 435 \beta_{2} + 435 \beta_{4} - 26 \beta_{6} - 61 \beta_{7} ) q^{89} \) \( + ( -36 + 46 \beta_{1} - 36 \beta_{2} - 672 \beta_{3} + 56 \beta_{5} - 56 \beta_{7} ) q^{90} \) \( + ( 688 - 28 \beta_{1} + 660 \beta_{3} + 238 \beta_{4} + 3 \beta_{5} + 28 \beta_{6} ) q^{91} \) \( + ( 24 - 48 \beta_{1} + 472 \beta_{2} + 280 \beta_{3} - 280 \beta_{4} - 48 \beta_{5} + 24 \beta_{6} + 24 \beta_{7} ) q^{92} \) \( + ( 1156 + 122 \beta_{2} + 122 \beta_{3} - 1036 \beta_{4} - 120 \beta_{5} + 120 \beta_{6} + 69 \beta_{7} ) q^{93} \) \( + ( 58 + 396 \beta_{2} + 396 \beta_{3} - 40 \beta_{4} - 18 \beta_{5} + 18 \beta_{6} + 42 \beta_{7} ) q^{94} \) \( + ( 15 - 59 \beta_{1} + 440 \beta_{2} - 40 \beta_{3} + 40 \beta_{4} - 59 \beta_{5} + 15 \beta_{6} + 15 \beta_{7} ) q^{95} \) \( + ( -32 - 32 \beta_{3} + 32 \beta_{4} + 32 \beta_{5} ) q^{96} \) \( + ( -809 + 10 \beta_{1} - 809 \beta_{2} + 132 \beta_{3} - 10 \beta_{5} + 10 \beta_{7} ) q^{97} \) \( + ( -474 + 10 \beta_{1} - 124 \beta_{2} + 124 \beta_{4} + 4 \beta_{6} - 10 \beta_{7} ) q^{98} \) \( + ( -736 + 2 \beta_{1} - 506 \beta_{2} - 770 \beta_{3} + 583 \beta_{4} + 104 \beta_{5} + 23 \beta_{6} + 57 \beta_{7} ) q^{99} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(8q \) \(\mathstrut +\mathstrut 4q^{2} \) \(\mathstrut +\mathstrut 3q^{3} \) \(\mathstrut -\mathstrut 8q^{4} \) \(\mathstrut +\mathstrut 5q^{5} \) \(\mathstrut +\mathstrut 14q^{6} \) \(\mathstrut -\mathstrut q^{7} \) \(\mathstrut +\mathstrut 16q^{8} \) \(\mathstrut -\mathstrut 21q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(8q \) \(\mathstrut +\mathstrut 4q^{2} \) \(\mathstrut +\mathstrut 3q^{3} \) \(\mathstrut -\mathstrut 8q^{4} \) \(\mathstrut +\mathstrut 5q^{5} \) \(\mathstrut +\mathstrut 14q^{6} \) \(\mathstrut -\mathstrut q^{7} \) \(\mathstrut +\mathstrut 16q^{8} \) \(\mathstrut -\mathstrut 21q^{9} \) \(\mathstrut -\mathstrut 100q^{10} \) \(\mathstrut -\mathstrut 155q^{11} \) \(\mathstrut +\mathstrut 32q^{12} \) \(\mathstrut +\mathstrut 7q^{13} \) \(\mathstrut +\mathstrut 2q^{14} \) \(\mathstrut +\mathstrut 211q^{15} \) \(\mathstrut -\mathstrut 32q^{16} \) \(\mathstrut +\mathstrut 161q^{17} \) \(\mathstrut +\mathstrut 162q^{18} \) \(\mathstrut -\mathstrut 272q^{19} \) \(\mathstrut +\mathstrut 20q^{20} \) \(\mathstrut -\mathstrut 50q^{21} \) \(\mathstrut +\mathstrut 628q^{23} \) \(\mathstrut +\mathstrut 56q^{24} \) \(\mathstrut -\mathstrut 17q^{25} \) \(\mathstrut +\mathstrut 96q^{26} \) \(\mathstrut -\mathstrut 528q^{27} \) \(\mathstrut +\mathstrut 16q^{28} \) \(\mathstrut +\mathstrut 33q^{29} \) \(\mathstrut -\mathstrut 422q^{30} \) \(\mathstrut +\mathstrut 323q^{31} \) \(\mathstrut -\mathstrut 256q^{32} \) \(\mathstrut -\mathstrut 1144q^{33} \) \(\mathstrut +\mathstrut 208q^{34} \) \(\mathstrut -\mathstrut 697q^{35} \) \(\mathstrut -\mathstrut 324q^{36} \) \(\mathstrut +\mathstrut 49q^{37} \) \(\mathstrut -\mathstrut 576q^{38} \) \(\mathstrut +\mathstrut 391q^{39} \) \(\mathstrut +\mathstrut 240q^{40} \) \(\mathstrut +\mathstrut 361q^{41} \) \(\mathstrut +\mathstrut 1430q^{42} \) \(\mathstrut +\mathstrut 1442q^{43} \) \(\mathstrut +\mathstrut 620q^{44} \) \(\mathstrut +\mathstrut 2652q^{45} \) \(\mathstrut -\mathstrut 416q^{46} \) \(\mathstrut -\mathstrut 1069q^{47} \) \(\mathstrut +\mathstrut 48q^{48} \) \(\mathstrut -\mathstrut 709q^{49} \) \(\mathstrut -\mathstrut 76q^{50} \) \(\mathstrut -\mathstrut 1332q^{51} \) \(\mathstrut -\mathstrut 192q^{52} \) \(\mathstrut -\mathstrut 281q^{53} \) \(\mathstrut -\mathstrut 1144q^{54} \) \(\mathstrut -\mathstrut 7q^{55} \) \(\mathstrut +\mathstrut 48q^{56} \) \(\mathstrut -\mathstrut 438q^{57} \) \(\mathstrut -\mathstrut 66q^{58} \) \(\mathstrut -\mathstrut 128q^{59} \) \(\mathstrut -\mathstrut 1116q^{60} \) \(\mathstrut -\mathstrut 617q^{61} \) \(\mathstrut +\mathstrut 1044q^{62} \) \(\mathstrut +\mathstrut 694q^{63} \) \(\mathstrut -\mathstrut 128q^{64} \) \(\mathstrut -\mathstrut 138q^{65} \) \(\mathstrut +\mathstrut 1248q^{66} \) \(\mathstrut +\mathstrut 578q^{67} \) \(\mathstrut +\mathstrut 644q^{68} \) \(\mathstrut -\mathstrut 310q^{69} \) \(\mathstrut +\mathstrut 34q^{70} \) \(\mathstrut +\mathstrut 115q^{71} \) \(\mathstrut +\mathstrut 168q^{72} \) \(\mathstrut -\mathstrut 1487q^{73} \) \(\mathstrut -\mathstrut 98q^{74} \) \(\mathstrut -\mathstrut 1852q^{75} \) \(\mathstrut -\mathstrut 128q^{76} \) \(\mathstrut +\mathstrut 553q^{77} \) \(\mathstrut -\mathstrut 4152q^{78} \) \(\mathstrut +\mathstrut 71q^{79} \) \(\mathstrut -\mathstrut 480q^{80} \) \(\mathstrut +\mathstrut 1630q^{81} \) \(\mathstrut +\mathstrut 658q^{82} \) \(\mathstrut +\mathstrut 1942q^{83} \) \(\mathstrut +\mathstrut 2960q^{84} \) \(\mathstrut -\mathstrut 329q^{85} \) \(\mathstrut +\mathstrut 2426q^{86} \) \(\mathstrut +\mathstrut 2122q^{87} \) \(\mathstrut +\mathstrut 560q^{88} \) \(\mathstrut -\mathstrut 2202q^{89} \) \(\mathstrut +\mathstrut 1286q^{90} \) \(\mathstrut +\mathstrut 4523q^{91} \) \(\mathstrut -\mathstrut 2088q^{92} \) \(\mathstrut +\mathstrut 6019q^{93} \) \(\mathstrut -\mathstrut 1332q^{94} \) \(\mathstrut -\mathstrut 793q^{95} \) \(\mathstrut -\mathstrut 96q^{96} \) \(\mathstrut -\mathstrut 5128q^{97} \) \(\mathstrut -\mathstrut 3292q^{98} \) \(\mathstrut -\mathstrut 2213q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8}\mathstrut -\mathstrut \) \(x^{7}\mathstrut +\mathstrut \) \(71\) \(x^{6}\mathstrut -\mathstrut \) \(141\) \(x^{5}\mathstrut +\mathstrut \) \(2911\) \(x^{4}\mathstrut +\mathstrut \) \(2710\) \(x^{3}\mathstrut +\mathstrut \) \(75340\) \(x^{2}\mathstrut +\mathstrut \) \(169400\) \(x\mathstrut +\mathstrut \) \(5856400\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\(22554025143\) \(\nu^{7}\mathstrut +\mathstrut \) \(151013926876\) \(\nu^{6}\mathstrut -\mathstrut \) \(1356924294556\) \(\nu^{5}\mathstrut +\mathstrut \) \(28230146638036\) \(\nu^{4}\mathstrut -\mathstrut \) \(111071208987556\) \(\nu^{3}\mathstrut +\mathstrut \) \(8011043329394989\) \(\nu^{2}\mathstrut -\mathstrut \) \(4059658693406480\) \(\nu\mathstrut +\mathstrut \) \(48229141584676800\)\()/\)\(342693986325717620\)
\(\beta_{3}\)\(=\)\((\)\(14112299722\) \(\nu^{7}\mathstrut +\mathstrut \) \(955320989289\) \(\nu^{6}\mathstrut +\mathstrut \) \(1256420838616\) \(\nu^{5}\mathstrut -\mathstrut \) \(5250112809736\) \(\nu^{4}\mathstrut +\mathstrut \) \(12754848436776\) \(\nu^{3}\mathstrut -\mathstrut \) \(1313471371492594\) \(\nu^{2}\mathstrut +\mathstrut \) \(8498101996878245\) \(\nu\mathstrut -\mathstrut \) \(66484077419717320\)\()/\)\(171346993162858810\)
\(\beta_{4}\)\(=\)\((\)\(-\)\(123788106187\) \(\nu^{7}\mathstrut +\mathstrut \) \(5732748190805\) \(\nu^{6}\mathstrut -\mathstrut \) \(11662655347575\) \(\nu^{5}\mathstrut +\mathstrut \) \(374269556401525\) \(\nu^{4}\mathstrut -\mathstrut \) \(772584075715295\) \(\nu^{3}\mathstrut +\mathstrut \) \(13504262211521188\) \(\nu^{2}\mathstrut +\mathstrut \) \(13624366122837560\) \(\nu\mathstrut +\mathstrut \) \(426480737760227920\)\()/\)\(685387972651435240\)
\(\beta_{5}\)\(=\)\((\)\(15778904729\) \(\nu^{7}\mathstrut -\mathstrut \) \(268932734519\) \(\nu^{6}\mathstrut +\mathstrut \) \(2855478562109\) \(\nu^{5}\mathstrut -\mathstrut \) \(16065997834439\) \(\nu^{4}\mathstrut +\mathstrut \) \(722720174659769\) \(\nu^{3}\mathstrut -\mathstrut \) \(523534449789100\) \(\nu^{2}\mathstrut +\mathstrut \) \(4037135429586600\) \(\nu\mathstrut -\mathstrut \) \(12007762986133200\)\()/\)\(31153998756883420\)
\(\beta_{6}\)\(=\)\((\)\(-\)\(15728307097\) \(\nu^{7}\mathstrut -\mathstrut \) \(23011705883\) \(\nu^{6}\mathstrut -\mathstrut \) \(3489045673392\) \(\nu^{5}\mathstrut -\mathstrut \) \(619537249643\) \(\nu^{4}\mathstrut -\mathstrut \) \(38031215360567\) \(\nu^{3}\mathstrut -\mathstrut \) \(53736302216310\) \(\nu^{2}\mathstrut +\mathstrut \) \(582451195267480\) \(\nu\mathstrut -\mathstrut \) \(22785233088495405\)\()/\)\(7788499689220855\)
\(\beta_{7}\)\(=\)\((\)\(254952731119\) \(\nu^{7}\mathstrut -\mathstrut \) \(130622718559\) \(\nu^{6}\mathstrut +\mathstrut \) \(16218883337689\) \(\nu^{5}\mathstrut -\mathstrut \) \(18738040845679\) \(\nu^{4}\mathstrut +\mathstrut \) \(629078544513089\) \(\nu^{3}\mathstrut +\mathstrut \) \(1043207365589370\) \(\nu^{2}\mathstrut +\mathstrut \) \(20338656497650260\) \(\nu\mathstrut +\mathstrut \) \(32952393866979400\)\()/\)\(31153998756883420\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(-\)\(\beta_{7}\mathstrut -\mathstrut \) \(\beta_{6}\mathstrut +\mathstrut \) \(\beta_{5}\mathstrut -\mathstrut \) \(8\) \(\beta_{4}\mathstrut +\mathstrut \) \(8\) \(\beta_{3}\mathstrut +\mathstrut \) \(54\) \(\beta_{2}\mathstrut +\mathstrut \) \(\beta_{1}\mathstrut -\mathstrut \) \(1\)
\(\nu^{3}\)\(=\)\(8\) \(\beta_{6}\mathstrut +\mathstrut \) \(47\) \(\beta_{5}\mathstrut +\mathstrut \) \(46\) \(\beta_{4}\mathstrut +\mathstrut \) \(8\) \(\beta_{3}\mathstrut -\mathstrut \) \(8\) \(\beta_{1}\mathstrut +\mathstrut \) \(16\)
\(\nu^{4}\)\(=\)\(109\) \(\beta_{7}\mathstrut +\mathstrut \) \(16\) \(\beta_{6}\mathstrut -\mathstrut \) \(16\) \(\beta_{5}\mathstrut +\mathstrut \) \(2226\) \(\beta_{4}\mathstrut -\mathstrut \) \(3034\) \(\beta_{3}\mathstrut -\mathstrut \) \(3034\) \(\beta_{2}\mathstrut -\mathstrut \) \(2210\)
\(\nu^{5}\)\(=\)\(-\)\(824\) \(\beta_{7}\mathstrut -\mathstrut \) \(3143\) \(\beta_{6}\mathstrut -\mathstrut \) \(1608\) \(\beta_{4}\mathstrut +\mathstrut \) \(1608\) \(\beta_{2}\mathstrut +\mathstrut \) \(824\) \(\beta_{1}\mathstrut -\mathstrut \) \(7549\)
\(\nu^{6}\)\(=\)\(-\)\(2432\) \(\beta_{7}\mathstrut +\mathstrut \) \(2432\) \(\beta_{5}\mathstrut +\mathstrut \) \(176314\) \(\beta_{3}\mathstrut +\mathstrut \) \(63048\) \(\beta_{2}\mathstrut -\mathstrut \) \(6725\) \(\beta_{1}\mathstrut +\mathstrut \) \(63048\)
\(\nu^{7}\)\(=\)\(185471\) \(\beta_{7}\mathstrut +\mathstrut \) \(185471\) \(\beta_{6}\mathstrut -\mathstrut \) \(119991\) \(\beta_{5}\mathstrut +\mathstrut \) \(185128\) \(\beta_{4}\mathstrut -\mathstrut \) \(185128\) \(\beta_{3}\mathstrut -\mathstrut \) \(513934\) \(\beta_{2}\mathstrut -\mathstrut \) \(119991\) \(\beta_{1}\mathstrut +\mathstrut \) \(185471\)

Character Values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/22\mathbb{Z}\right)^\times\).

\(n\) \(13\)
\(\chi(n)\) \(\beta_{2}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
3.1
−4.79501 + 3.48378i
5.60402 4.07156i
−2.53202 7.79275i
2.22300 + 6.84169i
−2.53202 + 7.79275i
2.22300 6.84169i
−4.79501 3.48378i
5.60402 + 4.07156i
1.61803 1.17557i −1.33153 4.09803i 1.23607 3.80423i 6.52241 + 4.73881i −6.97198 5.06544i −8.05890 + 24.8027i −2.47214 7.60845i 6.82261 4.95692i 16.1243
3.2 1.61803 1.17557i 2.64055 + 8.12677i 1.23607 3.80423i −10.3036 7.48598i 13.8261 + 10.0452i 7.24988 22.3128i −2.47214 7.60845i −37.2284 + 27.0480i −25.4718
5.1 −0.618034 1.90211i −6.12891 4.45291i −3.23607 + 2.35114i 1.67119 5.14341i −4.68207 + 14.4099i 17.9196 13.0193i 6.47214 + 4.70228i 9.39163 + 28.9045i −10.8162
5.2 −0.618034 1.90211i 6.31989 + 4.59167i −3.23607 + 2.35114i 4.60996 14.1880i 4.82797 14.8590i −17.6106 + 12.7948i 6.47214 + 4.70228i 10.5141 + 32.3592i −29.8363
9.1 −0.618034 + 1.90211i −6.12891 + 4.45291i −3.23607 2.35114i 1.67119 + 5.14341i −4.68207 14.4099i 17.9196 + 13.0193i 6.47214 4.70228i 9.39163 28.9045i −10.8162
9.2 −0.618034 + 1.90211i 6.31989 4.59167i −3.23607 2.35114i 4.60996 + 14.1880i 4.82797 + 14.8590i −17.6106 12.7948i 6.47214 4.70228i 10.5141 32.3592i −29.8363
15.1 1.61803 + 1.17557i −1.33153 + 4.09803i 1.23607 + 3.80423i 6.52241 4.73881i −6.97198 + 5.06544i −8.05890 24.8027i −2.47214 + 7.60845i 6.82261 + 4.95692i 16.1243
15.2 1.61803 + 1.17557i 2.64055 8.12677i 1.23607 + 3.80423i −10.3036 + 7.48598i 13.8261 10.0452i 7.24988 + 22.3128i −2.47214 + 7.60845i −37.2284 27.0480i −25.4718
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 15.2
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

This newform can be constructed as the kernel of the linear operator \(T_{3}^{8} - \cdots\) acting on \(S_{4}^{\mathrm{new}}(22, [\chi])\).