Properties

 Label 11.5.d.a Level 11 Weight 5 Character orbit 11.d Analytic conductor 1.137 Analytic rank 0 Dimension 12 CM No Inner twists 2

Related objects

Newspace parameters

 Level: $$N$$ = $$11$$ Weight: $$k$$ = $$5$$ Character orbit: $$[\chi]$$ = 11.d (of order $$10$$ and degree $$4$$)

Newform invariants

 Self dual: No Analytic conductor: $$1.13706959392$$ Analytic rank: $$0$$ Dimension: $$12$$ Relative dimension: $$3$$ over $$\Q(\zeta_{10})$$ Coefficient field: $$\mathbb{Q}[x]/(x^{12} + \cdots)$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Sato-Tate group: $\mathrm{SU}(2)[C_{10}]$

$q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{11}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q$$ $$+ ( -1 - \beta_{2} - 2 \beta_{3} - \beta_{4} + \beta_{7} ) q^{2}$$ $$+ ( 1 + \beta_{2} + 3 \beta_{3} + 3 \beta_{4} + \beta_{9} ) q^{3}$$ $$+ ( -2 + \beta_{1} - 2 \beta_{3} - 7 \beta_{4} + \beta_{6} - \beta_{11} ) q^{4}$$ $$+ ( 4 \beta_{2} + 4 \beta_{4} - \beta_{6} - 2 \beta_{7} + 2 \beta_{8} - \beta_{9} + \beta_{10} + \beta_{11} ) q^{5}$$ $$+ ( 12 - 6 \beta_{1} - \beta_{2} + 6 \beta_{3} + 13 \beta_{4} - \beta_{5} - 6 \beta_{8} - 3 \beta_{9} - 2 \beta_{10} + 2 \beta_{11} ) q^{6}$$ $$+ ( -17 + 5 \beta_{1} - 24 \beta_{2} - 7 \beta_{3} - 12 \beta_{4} + 4 \beta_{5} - 5 \beta_{6} - 5 \beta_{7} + 5 \beta_{8} - 2 \beta_{9} - 2 \beta_{10} - \beta_{11} ) q^{7}$$ $$+ ( -19 + 5 \beta_{1} + 19 \beta_{2} + 7 \beta_{3} - 7 \beta_{4} - 5 \beta_{5} - 5 \beta_{7} - 2 \beta_{8} + 2 \beta_{9} + 6 \beta_{10} + \beta_{11} ) q^{8}$$ $$+ ( 7 - 12 \beta_{1} + 38 \beta_{2} + 7 \beta_{4} + \beta_{5} + 15 \beta_{6} + 6 \beta_{7} + 3 \beta_{8} + 4 \beta_{9} - \beta_{10} - 4 \beta_{11} ) q^{9}$$ $$+O(q^{10})$$ $$q$$ $$+ ( -1 - \beta_{2} - 2 \beta_{3} - \beta_{4} + \beta_{7} ) q^{2}$$ $$+ ( 1 + \beta_{2} + 3 \beta_{3} + 3 \beta_{4} + \beta_{9} ) q^{3}$$ $$+ ( -2 + \beta_{1} - 2 \beta_{3} - 7 \beta_{4} + \beta_{6} - \beta_{11} ) q^{4}$$ $$+ ( 4 \beta_{2} + 4 \beta_{4} - \beta_{6} - 2 \beta_{7} + 2 \beta_{8} - \beta_{9} + \beta_{10} + \beta_{11} ) q^{5}$$ $$+ ( 12 - 6 \beta_{1} - \beta_{2} + 6 \beta_{3} + 13 \beta_{4} - \beta_{5} - 6 \beta_{8} - 3 \beta_{9} - 2 \beta_{10} + 2 \beta_{11} ) q^{6}$$ $$+ ( -17 + 5 \beta_{1} - 24 \beta_{2} - 7 \beta_{3} - 12 \beta_{4} + 4 \beta_{5} - 5 \beta_{6} - 5 \beta_{7} + 5 \beta_{8} - 2 \beta_{9} - 2 \beta_{10} - \beta_{11} ) q^{7}$$ $$+ ( -19 + 5 \beta_{1} + 19 \beta_{2} + 7 \beta_{3} - 7 \beta_{4} - 5 \beta_{5} - 5 \beta_{7} - 2 \beta_{8} + 2 \beta_{9} + 6 \beta_{10} + \beta_{11} ) q^{8}$$ $$+ ( 7 - 12 \beta_{1} + 38 \beta_{2} + 7 \beta_{4} + \beta_{5} + 15 \beta_{6} + 6 \beta_{7} + 3 \beta_{8} + 4 \beta_{9} - \beta_{10} - 4 \beta_{11} ) q^{9}$$ $$+ ( 17 + 9 \beta_{1} - 7 \beta_{2} + 41 \beta_{3} + 34 \beta_{4} + 2 \beta_{5} - 5 \beta_{6} - 5 \beta_{7} + 5 \beta_{9} - 5 \beta_{10} - 4 \beta_{11} ) q^{10}$$ $$+ ( -10 + 7 \beta_{1} + 7 \beta_{2} - 9 \beta_{3} - 51 \beta_{4} - 3 \beta_{5} - \beta_{6} + 18 \beta_{7} + 2 \beta_{8} - \beta_{9} + 5 \beta_{10} + 7 \beta_{11} ) q^{11}$$ $$+ ( -2 - 9 \beta_{1} - 127 \beta_{2} - 127 \beta_{3} + 6 \beta_{5} - 6 \beta_{6} + 24 \beta_{7} - 18 \beta_{8} - \beta_{9} - \beta_{10} ) q^{12}$$ $$+ ( 16 - 6 \beta_{2} - 12 \beta_{3} - 28 \beta_{4} - 5 \beta_{5} - 28 \beta_{6} - 15 \beta_{7} + 28 \beta_{8} + \beta_{9} - 3 \beta_{10} + \beta_{11} ) q^{13}$$ $$+ ( 101 - 14 \beta_{1} + 101 \beta_{2} + 59 \beta_{3} + 59 \beta_{4} + 38 \beta_{6} + 24 \beta_{7} - 19 \beta_{8} - 5 \beta_{9} ) q^{14}$$ $$+ ( 94 + 18 \beta_{1} + 94 \beta_{3} - 51 \beta_{4} + 18 \beta_{6} - 9 \beta_{7} - 9 \beta_{8} + \beta_{10} - 3 \beta_{11} ) q^{15}$$ $$+ ( 9 \beta_{1} + 130 \beta_{2} - 15 \beta_{3} + 130 \beta_{4} + 3 \beta_{5} - 21 \beta_{6} - 42 \beta_{7} + 33 \beta_{8} - 3 \beta_{9} ) q^{16}$$ $$+ ( -176 - 29 \beta_{1} - 71 \beta_{2} - 88 \beta_{3} - 105 \beta_{4} + \beta_{5} + 5 \beta_{6} - 29 \beta_{8} + 9 \beta_{9} + 8 \beta_{10} - 8 \beta_{11} ) q^{17}$$ $$+ ( -313 + 30 \beta_{1} - 200 \beta_{2} + 113 \beta_{3} - 100 \beta_{4} - 20 \beta_{5} - 30 \beta_{6} - 36 \beta_{7} + 36 \beta_{8} + 6 \beta_{9} + 10 \beta_{10} + 7 \beta_{11} ) q^{18}$$ $$+ ( -68 + 15 \beta_{1} + 68 \beta_{2} - 48 \beta_{3} + 48 \beta_{4} + 27 \beta_{5} - 15 \beta_{7} - 28 \beta_{8} - 11 \beta_{9} - 32 \beta_{10} - 5 \beta_{11} ) q^{19}$$ $$+ ( 120 - 44 \beta_{1} + 96 \beta_{2} + 120 \beta_{4} + 2 \beta_{5} + 20 \beta_{6} + 22 \beta_{7} - 24 \beta_{8} - 14 \beta_{9} - 2 \beta_{10} + 14 \beta_{11} ) q^{20}$$ $$+ ( 172 - 21 \beta_{1} + 106 \beta_{2} + 238 \beta_{3} + 344 \beta_{4} - 14 \beta_{5} + 12 \beta_{6} + 12 \beta_{7} - 27 \beta_{9} + 27 \beta_{10} + 28 \beta_{11} ) q^{21}$$ $$+ ( -35 + 19 \beta_{1} + 162 \beta_{2} + 51 \beta_{3} - 349 \beta_{4} + 6 \beta_{5} - 20 \beta_{6} - 25 \beta_{7} + 40 \beta_{8} + 13 \beta_{9} - 21 \beta_{10} - 36 \beta_{11} ) q^{22}$$ $$+ ( -40 + 8 \beta_{1} - 362 \beta_{2} - 362 \beta_{3} - 28 \beta_{5} - 8 \beta_{6} - 8 \beta_{7} + 16 \beta_{8} + 8 \beta_{9} + 8 \beta_{10} ) q^{23}$$ $$+ ( 113 - 236 \beta_{2} - 472 \beta_{3} - 585 \beta_{4} + 25 \beta_{5} + 54 \beta_{6} + 51 \beta_{7} - 54 \beta_{8} - 3 \beta_{9} + 19 \beta_{10} - 3 \beta_{11} ) q^{24}$$ $$+ ( 297 + 43 \beta_{1} + 297 \beta_{2} + 383 \beta_{3} + 383 \beta_{4} + 10 \beta_{5} - 66 \beta_{6} - 23 \beta_{7} + 33 \beta_{8} + 9 \beta_{9} - 10 \beta_{11} ) q^{25}$$ $$+ ( 525 - 13 \beta_{1} + 525 \beta_{3} + 352 \beta_{4} - 13 \beta_{6} + 22 \beta_{7} + 22 \beta_{8} - 13 \beta_{10} + 35 \beta_{11} ) q^{26}$$ $$+ ( -30 \beta_{1} + 182 \beta_{2} - 220 \beta_{3} + 182 \beta_{4} - 23 \beta_{5} + 51 \beta_{6} + 102 \beta_{7} - 72 \beta_{8} + 46 \beta_{9} - 23 \beta_{10} - 23 \beta_{11} ) q^{27}$$ $$+ ( -472 + 62 \beta_{1} - 76 \beta_{2} - 236 \beta_{3} - 396 \beta_{4} + 16 \beta_{5} - 26 \beta_{6} + 62 \beta_{8} + 18 \beta_{9} + 2 \beta_{10} - 2 \beta_{11} ) q^{28}$$ $$+ ( -502 - 96 \beta_{1} - 640 \beta_{2} - 138 \beta_{3} - 320 \beta_{4} + 22 \beta_{5} + 96 \beta_{6} + 95 \beta_{7} - 95 \beta_{8} + 14 \beta_{9} - 11 \beta_{10} - 18 \beta_{11} ) q^{29}$$ $$+ ( -430 - 81 \beta_{1} + 430 \beta_{2} - 231 \beta_{3} + 231 \beta_{4} - 37 \beta_{5} + 81 \beta_{7} + 75 \beta_{8} + 14 \beta_{9} + 46 \beta_{10} + 9 \beta_{11} ) q^{30}$$ $$+ ( -35 + 208 \beta_{1} + 258 \beta_{2} - 35 \beta_{4} - 38 \beta_{5} - 111 \beta_{6} - 104 \beta_{7} + 97 \beta_{8} - 25 \beta_{9} + 38 \beta_{10} + 25 \beta_{11} ) q^{31}$$ $$+ ( 89 + 45 \beta_{1} - 166 \beta_{2} + 344 \beta_{3} + 178 \beta_{4} + 33 \beta_{5} - 41 \beta_{6} - 41 \beta_{7} + 31 \beta_{9} - 31 \beta_{10} - 66 \beta_{11} ) q^{32}$$ $$+ ( 549 - 117 \beta_{1} + 158 \beta_{2} + 372 \beta_{3} - 169 \beta_{4} + 47 \beta_{5} + 78 \beta_{6} - 18 \beta_{7} - 189 \beta_{8} - 54 \beta_{9} - 5 \beta_{10} + 37 \beta_{11} ) q^{33}$$ $$+ ( -36 + 37 \beta_{1} - 290 \beta_{2} - 290 \beta_{3} + 13 \beta_{5} + 84 \beta_{6} - 158 \beta_{7} + 74 \beta_{8} - 29 \beta_{9} - 29 \beta_{10} ) q^{34}$$ $$+ ( 174 - 49 \beta_{2} - 98 \beta_{3} - 272 \beta_{4} - 21 \beta_{5} - 3 \beta_{6} - 99 \beta_{7} + 3 \beta_{8} - 11 \beta_{9} - 43 \beta_{10} - 11 \beta_{11} ) q^{35}$$ $$+ ( 558 - 78 \beta_{1} + 558 \beta_{2} + 965 \beta_{3} + 965 \beta_{4} - 67 \beta_{5} - 66 \beta_{6} - 144 \beta_{7} + 33 \beta_{8} - 23 \beta_{9} + 67 \beta_{11} ) q^{36}$$ $$+ ( 34 - 166 \beta_{1} + 34 \beta_{3} + 6 \beta_{4} - 166 \beta_{6} - 11 \beta_{7} - 11 \beta_{8} + 61 \beta_{10} - 32 \beta_{11} ) q^{37}$$ $$+ ( 47 \beta_{1} + 33 \beta_{2} - 757 \beta_{3} + 33 \beta_{4} + 68 \beta_{5} + 29 \beta_{6} + 58 \beta_{7} - 105 \beta_{8} - 108 \beta_{9} + 40 \beta_{10} + 40 \beta_{11} ) q^{38}$$ $$+ ( -934 + 183 \beta_{1} - 318 \beta_{2} - 467 \beta_{3} - 616 \beta_{4} - 28 \beta_{5} + 84 \beta_{6} + 183 \beta_{8} - 59 \beta_{9} - 31 \beta_{10} + 31 \beta_{11} ) q^{39}$$ $$+ ( -334 - 42 \beta_{1} - 440 \beta_{2} - 106 \beta_{3} - 220 \beta_{4} + 12 \beta_{5} + 42 \beta_{6} + 144 \beta_{7} - 144 \beta_{8} - 48 \beta_{9} - 6 \beta_{10} + 18 \beta_{11} ) q^{40}$$ $$+ ( -330 + 79 \beta_{1} + 330 \beta_{2} + 381 \beta_{3} - 381 \beta_{4} - 19 \beta_{5} - 79 \beta_{7} + 128 \beta_{8} + 18 \beta_{9} + 2 \beta_{10} - 17 \beta_{11} ) q^{41}$$ $$+ ( -15 - 725 \beta_{2} - 15 \beta_{4} + 79 \beta_{5} - 189 \beta_{6} - 189 \beta_{8} + 147 \beta_{9} - 79 \beta_{10} - 147 \beta_{11} ) q^{42}$$ $$+ ( 516 - 189 \beta_{1} + 615 \beta_{2} + 417 \beta_{3} + 1032 \beta_{4} - 14 \beta_{5} + 188 \beta_{6} + 188 \beta_{7} + 61 \beta_{9} - 61 \beta_{10} + 28 \beta_{11} ) q^{43}$$ $$+ ( -251 + 3 \beta_{1} + 740 \beta_{2} + 224 \beta_{3} + 382 \beta_{4} - 149 \beta_{5} - 13 \beta_{6} - 107 \beta_{7} + 224 \beta_{8} + 75 \beta_{9} + 109 \beta_{10} + 58 \beta_{11} ) q^{44}$$ $$+ ( 350 + 9 \beta_{1} - 146 \beta_{2} - 146 \beta_{3} + 87 \beta_{5} - 57 \beta_{6} + 39 \beta_{7} + 18 \beta_{8} + 61 \beta_{9} + 61 \beta_{10} ) q^{45}$$ $$+ ( 224 - 18 \beta_{2} - 36 \beta_{3} - 260 \beta_{4} - 60 \beta_{5} + 258 \beta_{6} + 206 \beta_{7} - 258 \beta_{8} + 56 \beta_{9} + 52 \beta_{10} + 56 \beta_{11} ) q^{46}$$ $$+ ( -503 + 294 \beta_{1} - 503 \beta_{2} - 942 \beta_{3} - 942 \beta_{4} + 129 \beta_{5} - 198 \beta_{6} + 96 \beta_{7} + 99 \beta_{8} + 80 \beta_{9} - 129 \beta_{11} ) q^{47}$$ $$+ ( -103 + 189 \beta_{1} - 103 \beta_{3} - 443 \beta_{4} + 189 \beta_{6} + 99 \beta_{7} + 99 \beta_{8} - 121 \beta_{10} - 145 \beta_{11} ) q^{48}$$ $$+ ( -154 \beta_{1} - 370 \beta_{2} + 481 \beta_{3} - 370 \beta_{4} - 88 \beta_{5} + 21 \beta_{6} + 42 \beta_{7} + 112 \beta_{8} + 21 \beta_{9} + 67 \beta_{10} + 67 \beta_{11} ) q^{49}$$ $$+ ( 1398 - 92 \beta_{1} + 527 \beta_{2} + 699 \beta_{3} + 871 \beta_{4} - 39 \beta_{5} - 248 \beta_{6} - 92 \beta_{8} - 70 \beta_{9} - 31 \beta_{10} + 31 \beta_{11} ) q^{50}$$ $$+ ( 1030 - 45 \beta_{1} + 416 \beta_{2} - 614 \beta_{3} + 208 \beta_{4} + 50 \beta_{5} + 45 \beta_{6} - 147 \beta_{7} + 147 \beta_{8} - 54 \beta_{9} - 25 \beta_{10} + 2 \beta_{11} ) q^{51}$$ $$+ ( 1018 - 234 \beta_{1} - 1018 \beta_{2} - 664 \beta_{3} + 664 \beta_{4} + 42 \beta_{5} + 234 \beta_{7} - 396 \beta_{8} - 36 \beta_{9} - 12 \beta_{10} + 30 \beta_{11} ) q^{52}$$ $$+ ( 236 - 210 \beta_{1} + 756 \beta_{2} + 236 \beta_{4} + 29 \beta_{5} + 262 \beta_{6} + 105 \beta_{7} + 52 \beta_{8} - 99 \beta_{9} - 29 \beta_{10} + 99 \beta_{11} ) q^{53}$$ $$+ ( -985 + 195 \beta_{1} - 809 \beta_{2} - 1161 \beta_{3} - 1970 \beta_{4} - 64 \beta_{5} - 228 \beta_{6} - 228 \beta_{7} - 136 \beta_{9} + 136 \beta_{10} + 128 \beta_{11} ) q^{54}$$ $$+ ( -186 + 293 \beta_{1} - 1093 \beta_{2} + 497 \beta_{3} + 521 \beta_{4} - 25 \beta_{5} - 177 \beta_{6} - 48 \beta_{7} - 31 \beta_{8} + 32 \beta_{9} - 39 \beta_{10} - 15 \beta_{11} ) q^{55}$$ $$+ ( -1148 + 22 \beta_{1} + 1134 \beta_{2} + 1134 \beta_{3} - 28 \beta_{5} - 66 \beta_{6} + 22 \beta_{7} + 44 \beta_{8} - 44 \beta_{9} - 44 \beta_{10} ) q^{56}$$ $$+ ( -890 + 793 \beta_{2} + 1586 \beta_{3} + 2476 \beta_{4} + 68 \beta_{5} - 519 \beta_{6} - 261 \beta_{7} + 519 \beta_{8} - 73 \beta_{9} - 78 \beta_{10} - 73 \beta_{11} ) q^{57}$$ $$+ ( -1127 - 548 \beta_{1} - 1127 \beta_{2} - 1765 \beta_{3} - 1765 \beta_{4} + 60 \beta_{5} + 742 \beta_{6} + 194 \beta_{7} - 371 \beta_{8} - 89 \beta_{9} - 60 \beta_{11} ) q^{58}$$ $$+ ( -1636 + 115 \beta_{1} - 1636 \beta_{3} - 766 \beta_{4} + 115 \beta_{6} - 308 \beta_{7} - 308 \beta_{8} + 43 \beta_{10} + 171 \beta_{11} ) q^{59}$$ $$+ ( 330 \beta_{1} + 268 \beta_{2} + 2162 \beta_{3} + 268 \beta_{4} - 276 \beta_{6} - 552 \beta_{7} + 222 \beta_{8} + 142 \beta_{9} - 142 \beta_{10} - 142 \beta_{11} ) q^{60}$$ $$+ ( 836 - 426 \beta_{1} - 116 \beta_{2} + 418 \beta_{3} + 952 \beta_{4} + 20 \beta_{5} + 469 \beta_{6} - 426 \beta_{8} + 169 \beta_{9} + 149 \beta_{10} - 149 \beta_{11} ) q^{61}$$ $$+ ( 1595 + 435 \beta_{1} + 4112 \beta_{2} + 2517 \beta_{3} + 2056 \beta_{4} - 170 \beta_{5} - 435 \beta_{6} - 484 \beta_{7} + 484 \beta_{8} + 188 \beta_{9} + 85 \beta_{10} - 9 \beta_{11} ) q^{62}$$ $$+ ( 456 + 429 \beta_{1} - 456 \beta_{2} + 1265 \beta_{3} - 1265 \beta_{4} - 22 \beta_{5} - 429 \beta_{7} + 72 \beta_{8} - 11 \beta_{9} + 66 \beta_{10} + 44 \beta_{11} ) q^{63}$$ $$+ ( -862 - 176 \beta_{1} - 1157 \beta_{2} - 862 \beta_{4} - 121 \beta_{5} + 473 \beta_{6} + 88 \beta_{7} + 297 \beta_{8} - 196 \beta_{9} + 121 \beta_{10} + 196 \beta_{11} ) q^{64}$$ $$+ ( -378 - 22 \beta_{1} + 368 \beta_{2} - 1124 \beta_{3} - 756 \beta_{4} + 93 \beta_{5} - 149 \beta_{6} - 149 \beta_{7} + 32 \beta_{9} - 32 \beta_{10} - 186 \beta_{11} ) q^{65}$$ $$+ ( -1967 - 129 \beta_{1} - 2846 \beta_{2} - 5012 \beta_{3} - 157 \beta_{4} + 445 \beta_{5} - 24 \beta_{6} + 663 \beta_{7} + 81 \beta_{8} - 90 \beta_{9} - 210 \beta_{10} - 206 \beta_{11} ) q^{66}$$ $$+ ( 2285 - 416 \beta_{1} + 1521 \beta_{2} + 1521 \beta_{3} - 139 \beta_{5} - 79 \beta_{6} + 911 \beta_{7} - 832 \beta_{8} - 152 \beta_{9} - 152 \beta_{10} ) q^{67}$$ $$+ ( -419 + 167 \beta_{2} + 334 \beta_{3} + 753 \beta_{4} - 12 \beta_{5} - 151 \beta_{6} + 25 \beta_{7} + 151 \beta_{8} + 33 \beta_{9} + 54 \beta_{10} + 33 \beta_{11} ) q^{68}$$ $$+ ( -1508 + 252 \beta_{1} - 1508 \beta_{2} - 1090 \beta_{3} - 1090 \beta_{4} - 430 \beta_{5} + 252 \beta_{7} - 100 \beta_{9} + 430 \beta_{11} ) q^{69}$$ $$+ ( -343 - 46 \beta_{1} - 343 \beta_{3} + 1821 \beta_{4} - 46 \beta_{6} + 299 \beta_{7} + 299 \beta_{8} + 212 \beta_{10} + 259 \beta_{11} ) q^{70}$$ $$+ ( -195 \beta_{1} - 2769 \beta_{2} - 659 \beta_{3} - 2769 \beta_{4} + 193 \beta_{5} + 139 \beta_{6} + 278 \beta_{7} - 83 \beta_{8} - 64 \beta_{9} - 129 \beta_{10} - 129 \beta_{11} ) q^{71}$$ $$+ ( 5842 - 420 \beta_{1} + 2667 \beta_{2} + 2921 \beta_{3} + 3175 \beta_{4} + 220 \beta_{5} - 369 \beta_{6} - 420 \beta_{8} + 215 \beta_{9} - 5 \beta_{10} + 5 \beta_{11} ) q^{72}$$ $$+ ( 2059 + 505 \beta_{1} + 174 \beta_{2} - 1885 \beta_{3} + 87 \beta_{4} - 52 \beta_{5} - 505 \beta_{6} - 112 \beta_{7} + 112 \beta_{8} + 134 \beta_{9} + 26 \beta_{10} - 41 \beta_{11} ) q^{73}$$ $$+ ( 2837 + 306 \beta_{1} - 2837 \beta_{2} - 37 \beta_{3} + 37 \beta_{4} + 186 \beta_{5} - 306 \beta_{7} + 401 \beta_{8} - 23 \beta_{9} - 326 \beta_{10} - 140 \beta_{11} ) q^{74}$$ $$+ ( 261 - 408 \beta_{1} + 1869 \beta_{2} + 261 \beta_{4} - 168 \beta_{5} + 393 \beta_{6} + 204 \beta_{7} - 15 \beta_{8} + 192 \beta_{9} + 168 \beta_{10} - 192 \beta_{11} ) q^{75}$$ $$+ ( -3205 + 574 \beta_{1} - 2261 \beta_{2} - 4149 \beta_{3} - 6410 \beta_{4} + 29 \beta_{5} + 371 \beta_{6} + 371 \beta_{7} + 72 \beta_{9} - 72 \beta_{10} - 58 \beta_{11} ) q^{76}$$ $$+ ( 1758 - 234 \beta_{1} + 1262 \beta_{2} + 1800 \beta_{3} + 2500 \beta_{4} - 214 \beta_{5} + 640 \beta_{6} + 547 \beta_{7} - 389 \beta_{8} - 108 \beta_{9} + 23 \beta_{10} - 102 \beta_{11} ) q^{77}$$ $$+ ( -1710 + 168 \beta_{1} + 4567 \beta_{2} + 4567 \beta_{3} - 295 \beta_{5} + 147 \beta_{6} - 483 \beta_{7} + 336 \beta_{8} + 407 \beta_{9} + 407 \beta_{10} ) q^{78}$$ $$+ ( -1901 + 948 \beta_{2} + 1896 \beta_{3} + 3797 \beta_{4} + 234 \beta_{5} - 69 \beta_{6} - 40 \beta_{7} + 69 \beta_{8} + 17 \beta_{9} + 268 \beta_{10} + 17 \beta_{11} ) q^{79}$$ $$+ ( -2204 + 100 \beta_{1} - 2204 \beta_{2} - 2860 \beta_{3} - 2860 \beta_{4} + 232 \beta_{5} - 264 \beta_{6} - 164 \beta_{7} + 132 \beta_{8} + 212 \beta_{9} - 232 \beta_{11} ) q^{80}$$ $$+ ( -3378 + 228 \beta_{1} - 3378 \beta_{3} - 2734 \beta_{4} + 228 \beta_{6} - 240 \beta_{7} - 240 \beta_{8} - 354 \beta_{10} - 158 \beta_{11} ) q^{81}$$ $$+ ( -64 \beta_{1} + 2941 \beta_{2} + 4310 \beta_{3} + 2941 \beta_{4} - 297 \beta_{5} - 388 \beta_{6} - 776 \beta_{7} + 840 \beta_{8} + 133 \beta_{9} + 164 \beta_{10} + 164 \beta_{11} ) q^{82}$$ $$+ ( -2156 + 361 \beta_{1} - 4070 \beta_{2} - 1078 \beta_{3} + 1914 \beta_{4} - 17 \beta_{5} - 280 \beta_{6} + 361 \beta_{8} - 219 \beta_{9} - 202 \beta_{10} + 202 \beta_{11} ) q^{83}$$ $$+ ( 2936 - 852 \beta_{1} + 2308 \beta_{2} - 628 \beta_{3} + 1154 \beta_{4} + 364 \beta_{5} + 852 \beta_{6} + 678 \beta_{7} - 678 \beta_{8} - 540 \beta_{9} - 182 \beta_{10} + 88 \beta_{11} ) q^{84}$$ $$+ ( -194 - 255 \beta_{1} + 194 \beta_{2} + 1602 \beta_{3} - 1602 \beta_{4} - 78 \beta_{5} + 255 \beta_{7} - 693 \beta_{8} + 87 \beta_{9} - 18 \beta_{10} - 96 \beta_{11} ) q^{85}$$ $$+ ( -2297 - 374 \beta_{1} - 4946 \beta_{2} - 2297 \beta_{4} + 71 \beta_{5} - 610 \beta_{6} + 187 \beta_{7} - 984 \beta_{8} - 117 \beta_{9} - 71 \beta_{10} + 117 \beta_{11} ) q^{86}$$ $$+ ( 1664 - 447 \beta_{1} + 2291 \beta_{2} + 1037 \beta_{3} + 3328 \beta_{4} - 169 \beta_{5} - 369 \beta_{6} - 369 \beta_{7} - 234 \beta_{9} + 234 \beta_{10} + 338 \beta_{11} ) q^{87}$$ $$+ ( -1496 + 814 \beta_{1} - 847 \beta_{2} - 440 \beta_{3} + 1496 \beta_{4} - 341 \beta_{5} - 957 \beta_{6} - 1716 \beta_{7} + 671 \beta_{8} - 132 \beta_{9} + 77 \beta_{10} + 506 \beta_{11} ) q^{88}$$ $$+ ( -80 + 497 \beta_{1} - 437 \beta_{2} - 437 \beta_{3} + 466 \beta_{5} - 90 \beta_{6} - 904 \beta_{7} + 994 \beta_{8} - 185 \beta_{9} - 185 \beta_{10} ) q^{89}$$ $$+ ( 929 - 468 \beta_{2} - 936 \beta_{3} - 1865 \beta_{4} - 5 \beta_{5} + 555 \beta_{6} + 513 \beta_{7} - 555 \beta_{8} - 274 \beta_{9} - 553 \beta_{10} - 274 \beta_{11} ) q^{90}$$ $$+ ( 1279 - 483 \beta_{1} + 1279 \beta_{2} + 1121 \beta_{3} + 1121 \beta_{4} + 301 \beta_{5} - 490 \beta_{6} - 973 \beta_{7} + 245 \beta_{8} - 37 \beta_{9} - 301 \beta_{11} ) q^{91}$$ $$+ ( 870 - 38 \beta_{1} + 870 \beta_{3} + 1422 \beta_{4} - 38 \beta_{6} + 22 \beta_{7} + 22 \beta_{8} + 318 \beta_{10} - 314 \beta_{11} ) q^{92}$$ $$+ ( -21 \beta_{1} - 5090 \beta_{2} - 2802 \beta_{3} - 5090 \beta_{4} + 61 \beta_{5} + 471 \beta_{6} + 942 \beta_{7} - 921 \beta_{8} - 429 \beta_{9} + 368 \beta_{10} + 368 \beta_{11} ) q^{93}$$ $$+ ( 2102 + 506 \beta_{1} + 6506 \beta_{2} + 1051 \beta_{3} - 4404 \beta_{4} - 533 \beta_{5} + 1099 \beta_{6} + 506 \beta_{8} - 468 \beta_{9} + 65 \beta_{10} - 65 \beta_{11} ) q^{94}$$ $$+ ( 2046 - 798 \beta_{1} + 2638 \beta_{2} + 592 \beta_{3} + 1319 \beta_{4} - 362 \beta_{5} + 798 \beta_{6} + 261 \beta_{7} - 261 \beta_{8} + 208 \beta_{9} + 181 \beta_{10} + 77 \beta_{11} ) q^{95}$$ $$+ ( -1222 - 810 \beta_{1} + 1222 \beta_{2} - 3315 \beta_{3} + 3315 \beta_{4} - 391 \beta_{5} + 810 \beta_{7} + 279 \beta_{8} + 143 \beta_{9} + 496 \beta_{10} + 105 \beta_{11} ) q^{96}$$ $$+ ( 2929 + 2496 \beta_{1} + 1500 \beta_{2} + 2929 \beta_{4} + 728 \beta_{5} - 1460 \beta_{6} - 1248 \beta_{7} + 1036 \beta_{8} + 348 \beta_{9} - 728 \beta_{10} - 348 \beta_{11} ) q^{97}$$ $$+ ( 630 - 868 \beta_{1} - 1858 \beta_{2} + 3118 \beta_{3} + 1260 \beta_{4} + 46 \beta_{5} + 39 \beta_{6} + 39 \beta_{7} + 313 \beta_{9} - 313 \beta_{10} - 92 \beta_{11} ) q^{98}$$ $$+ ( 493 - 630 \beta_{1} - 36 \beta_{2} + 5485 \beta_{3} - 52 \beta_{4} + 105 \beta_{5} + 387 \beta_{6} - 267 \beta_{7} - 114 \beta_{8} + 849 \beta_{9} + 364 \beta_{10} - 113 \beta_{11} ) q^{99}$$ $$+O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$12q$$ $$\mathstrut -\mathstrut 5q^{2}$$ $$\mathstrut -\mathstrut 6q^{3}$$ $$\mathstrut +\mathstrut 7q^{4}$$ $$\mathstrut -\mathstrut 18q^{5}$$ $$\mathstrut +\mathstrut 75q^{6}$$ $$\mathstrut -\mathstrut 80q^{7}$$ $$\mathstrut -\mathstrut 245q^{8}$$ $$\mathstrut +\mathstrut q^{9}$$ $$\mathstrut +\mathstrut O(q^{10})$$ $$12q$$ $$\mathstrut -\mathstrut 5q^{2}$$ $$\mathstrut -\mathstrut 6q^{3}$$ $$\mathstrut +\mathstrut 7q^{4}$$ $$\mathstrut -\mathstrut 18q^{5}$$ $$\mathstrut +\mathstrut 75q^{6}$$ $$\mathstrut -\mathstrut 80q^{7}$$ $$\mathstrut -\mathstrut 245q^{8}$$ $$\mathstrut +\mathstrut q^{9}$$ $$\mathstrut -\mathstrut 43q^{11}$$ $$\mathstrut +\mathstrut 594q^{12}$$ $$\mathstrut +\mathstrut 250q^{13}$$ $$\mathstrut +\mathstrut 610q^{14}$$ $$\mathstrut +\mathstrut 1134q^{15}$$ $$\mathstrut -\mathstrut 633q^{16}$$ $$\mathstrut -\mathstrut 1250q^{17}$$ $$\mathstrut -\mathstrut 3150q^{18}$$ $$\mathstrut -\mathstrut 1025q^{19}$$ $$\mathstrut +\mathstrut 752q^{20}$$ $$\mathstrut -\mathstrut 35q^{22}$$ $$\mathstrut +\mathstrut 1684q^{23}$$ $$\mathstrut +\mathstrut 5345q^{24}$$ $$\mathstrut +\mathstrut 197q^{25}$$ $$\mathstrut +\mathstrut 3490q^{26}$$ $$\mathstrut -\mathstrut 687q^{27}$$ $$\mathstrut -\mathstrut 3580q^{28}$$ $$\mathstrut -\mathstrut 2690q^{29}$$ $$\mathstrut -\mathstrut 6740q^{30}$$ $$\mathstrut -\mathstrut 1136q^{31}$$ $$\mathstrut +\mathstrut 5939q^{33}$$ $$\mathstrut +\mathstrut 2370q^{34}$$ $$\mathstrut +\mathstrut 3610q^{35}$$ $$\mathstrut -\mathstrut 514q^{36}$$ $$\mathstrut -\mathstrut 336q^{37}$$ $$\mathstrut +\mathstrut 1900q^{38}$$ $$\mathstrut -\mathstrut 6880q^{39}$$ $$\mathstrut -\mathstrut 2340q^{40}$$ $$\mathstrut -\mathstrut 4550q^{41}$$ $$\mathstrut +\mathstrut 1310q^{42}$$ $$\mathstrut -\mathstrut 6268q^{44}$$ $$\mathstrut +\mathstrut 5136q^{45}$$ $$\mathstrut +\mathstrut 4150q^{46}$$ $$\mathstrut +\mathstrut 24q^{47}$$ $$\mathstrut +\mathstrut 344q^{48}$$ $$\mathstrut +\mathstrut 827q^{49}$$ $$\mathstrut +\mathstrut 8895q^{50}$$ $$\mathstrut +\mathstrut 13155q^{51}$$ $$\mathstrut +\mathstrut 14070q^{52}$$ $$\mathstrut +\mathstrut 414q^{53}$$ $$\mathstrut -\mathstrut 2738q^{55}$$ $$\mathstrut -\mathstrut 21340q^{56}$$ $$\mathstrut -\mathstrut 26925q^{57}$$ $$\mathstrut +\mathstrut 2980q^{58}$$ $$\mathstrut -\mathstrut 10011q^{59}$$ $$\mathstrut -\mathstrut 6856q^{60}$$ $$\mathstrut +\mathstrut 9460q^{61}$$ $$\mathstrut -\mathstrut 6200q^{62}$$ $$\mathstrut +\mathstrut 9150q^{63}$$ $$\mathstrut -\mathstrut 2633q^{64}$$ $$\mathstrut -\mathstrut 3210q^{66}$$ $$\mathstrut +\mathstrut 12154q^{67}$$ $$\mathstrut -\mathstrut 9400q^{68}$$ $$\mathstrut -\mathstrut 9022q^{69}$$ $$\mathstrut -\mathstrut 9380q^{70}$$ $$\mathstrut +\mathstrut 17574q^{71}$$ $$\mathstrut +\mathstrut 43045q^{72}$$ $$\mathstrut +\mathstrut 27950q^{73}$$ $$\mathstrut +\mathstrut 43270q^{74}$$ $$\mathstrut -\mathstrut 1761q^{75}$$ $$\mathstrut +\mathstrut 4090q^{77}$$ $$\mathstrut -\mathstrut 42920q^{78}$$ $$\mathstrut -\mathstrut 41540q^{79}$$ $$\mathstrut -\mathstrut 2308q^{80}$$ $$\mathstrut -\mathstrut 21080q^{81}$$ $$\mathstrut -\mathstrut 28175q^{82}$$ $$\mathstrut -\mathstrut 18665q^{83}$$ $$\mathstrut +\mathstrut 26250q^{84}$$ $$\mathstrut -\mathstrut 4230q^{85}$$ $$\mathstrut -\mathstrut 10125q^{86}$$ $$\mathstrut -\mathstrut 15125q^{88}$$ $$\mathstrut +\mathstrut 5554q^{89}$$ $$\mathstrut +\mathstrut 18400q^{90}$$ $$\mathstrut +\mathstrut 7390q^{91}$$ $$\mathstrut +\mathstrut 3904q^{92}$$ $$\mathstrut +\mathstrut 36898q^{93}$$ $$\mathstrut +\mathstrut 18920q^{94}$$ $$\mathstrut +\mathstrut 14110q^{95}$$ $$\mathstrut -\mathstrut 21140q^{96}$$ $$\mathstrut +\mathstrut 20769q^{97}$$ $$\mathstrut -\mathstrut 3269q^{99}$$ $$\mathstrut +\mathstrut O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{12}\mathstrut +\mathstrut$$ $$115$$ $$x^{10}\mathstrut +\mathstrut$$ $$5030$$ $$x^{8}\mathstrut +\mathstrut$$ $$102975$$ $$x^{6}\mathstrut +\mathstrut$$ $$953170$$ $$x^{4}\mathstrut +\mathstrut$$ $$2910655$$ $$x^{2}\mathstrut +\mathstrut$$ $$73205$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$-$$$$13052$$ $$\nu^{11}\mathstrut -\mathstrut$$ $$117304$$ $$\nu^{10}\mathstrut -\mathstrut$$ $$1219776$$ $$\nu^{9}\mathstrut -\mathstrut$$ $$10477907$$ $$\nu^{8}\mathstrut -\mathstrut$$ $$40135806$$ $$\nu^{7}\mathstrut -\mathstrut$$ $$321264966$$ $$\nu^{6}\mathstrut -\mathstrut$$ $$549385674$$ $$\nu^{5}\mathstrut -\mathstrut$$ $$3857093625$$ $$\nu^{4}\mathstrut -\mathstrut$$ $$2823880454$$ $$\nu^{3}\mathstrut -\mathstrut$$ $$13654448170$$ $$\nu^{2}\mathstrut -\mathstrut$$ $$5016493020$$ $$\nu\mathstrut -\mathstrut$$ $$1427754383$$$$)/$$$$1339827192$$ $$\beta_{3}$$ $$=$$ $$($$$$13052$$ $$\nu^{11}\mathstrut -\mathstrut$$ $$117304$$ $$\nu^{10}\mathstrut +\mathstrut$$ $$1219776$$ $$\nu^{9}\mathstrut -\mathstrut$$ $$10477907$$ $$\nu^{8}\mathstrut +\mathstrut$$ $$40135806$$ $$\nu^{7}\mathstrut -\mathstrut$$ $$321264966$$ $$\nu^{6}\mathstrut +\mathstrut$$ $$549385674$$ $$\nu^{5}\mathstrut -\mathstrut$$ $$3857093625$$ $$\nu^{4}\mathstrut +\mathstrut$$ $$2823880454$$ $$\nu^{3}\mathstrut -\mathstrut$$ $$13654448170$$ $$\nu^{2}\mathstrut +\mathstrut$$ $$5016493020$$ $$\nu\mathstrut -\mathstrut$$ $$1427754383$$$$)/$$$$1339827192$$ $$\beta_{4}$$ $$=$$ $$($$$$-$$$$31452$$ $$\nu^{11}\mathstrut +\mathstrut$$ $$117304$$ $$\nu^{10}\mathstrut -\mathstrut$$ $$2737673$$ $$\nu^{9}\mathstrut +\mathstrut$$ $$10477907$$ $$\nu^{8}\mathstrut -\mathstrut$$ $$79586472$$ $$\nu^{7}\mathstrut +\mathstrut$$ $$321264966$$ $$\nu^{6}\mathstrut -\mathstrut$$ $$827426403$$ $$\nu^{5}\mathstrut +\mathstrut$$ $$3857093625$$ $$\nu^{4}\mathstrut -\mathstrut$$ $$1134660378$$ $$\nu^{3}\mathstrut +\mathstrut$$ $$13654448170$$ $$\nu^{2}\mathstrut +\mathstrut$$ $$7973305285$$ $$\nu\mathstrut +\mathstrut$$ $$757840787$$$$)/$$$$1339827192$$ $$\beta_{5}$$ $$=$$ $$($$$$45643 \nu^{10} + 4016197 \nu^{8} + 120327945 \nu^{6} + 1386775659 \nu^{4} + 4387794733 \nu^{2} - 676101899$$$$)/60901236$$ $$\beta_{6}$$ $$=$$ $$($$$$10664$$ $$\nu^{11}\mathstrut +\mathstrut$$ $$25564$$ $$\nu^{10}\mathstrut +\mathstrut$$ $$952537$$ $$\nu^{9}\mathstrut +\mathstrut$$ $$2319614$$ $$\nu^{8}\mathstrut +\mathstrut$$ $$29205906$$ $$\nu^{7}\mathstrut +\mathstrut$$ $$72240366$$ $$\nu^{6}\mathstrut +\mathstrut$$ $$350644875$$ $$\nu^{5}\mathstrut +\mathstrut$$ $$874263126$$ $$\nu^{4}\mathstrut +\mathstrut$$ $$1241313470$$ $$\nu^{3}\mathstrut +\mathstrut$$ $$2997579640$$ $$\nu^{2}\mathstrut +\mathstrut$$ $$129795853$$ $$\nu\mathstrut +\mathstrut$$ $$86861060$$$$)/$$$$121802472$$ $$\beta_{7}$$ $$=$$ $$($$$$10664$$ $$\nu^{11}\mathstrut -\mathstrut$$ $$25564$$ $$\nu^{10}\mathstrut +\mathstrut$$ $$952537$$ $$\nu^{9}\mathstrut -\mathstrut$$ $$2319614$$ $$\nu^{8}\mathstrut +\mathstrut$$ $$29205906$$ $$\nu^{7}\mathstrut -\mathstrut$$ $$72240366$$ $$\nu^{6}\mathstrut +\mathstrut$$ $$350644875$$ $$\nu^{5}\mathstrut -\mathstrut$$ $$874263126$$ $$\nu^{4}\mathstrut +\mathstrut$$ $$1241313470$$ $$\nu^{3}\mathstrut -\mathstrut$$ $$2997579640$$ $$\nu^{2}\mathstrut +\mathstrut$$ $$129795853$$ $$\nu\mathstrut -\mathstrut$$ $$86861060$$$$)/$$$$121802472$$ $$\beta_{8}$$ $$=$$ $$($$$$10664$$ $$\nu^{11}\mathstrut -\mathstrut$$ $$79937$$ $$\nu^{10}\mathstrut +\mathstrut$$ $$952537$$ $$\nu^{9}\mathstrut -\mathstrut$$ $$7147008$$ $$\nu^{8}\mathstrut +\mathstrut$$ $$29205906$$ $$\nu^{7}\mathstrut -\mathstrut$$ $$219213027$$ $$\nu^{6}\mathstrut +\mathstrut$$ $$350644875$$ $$\nu^{5}\mathstrut -\mathstrut$$ $$2622222042$$ $$\nu^{4}\mathstrut +\mathstrut$$ $$1241313470$$ $$\nu^{3}\mathstrut -\mathstrut$$ $$9047202395$$ $$\nu^{2}\mathstrut +\mathstrut$$ $$68894617$$ $$\nu\mathstrut -\mathstrut$$ $$209313060$$$$)/$$$$121802472$$ $$\beta_{9}$$ $$=$$ $$($$$$-$$$$169940$$ $$\nu^{11}\mathstrut -\mathstrut$$ $$259930$$ $$\nu^{10}\mathstrut -\mathstrut$$ $$15878131$$ $$\nu^{9}\mathstrut -\mathstrut$$ $$23961014$$ $$\nu^{8}\mathstrut -\mathstrut$$ $$530879748$$ $$\nu^{7}\mathstrut -\mathstrut$$ $$766470936$$ $$\nu^{6}\mathstrut -\mathstrut$$ $$7708194267$$ $$\nu^{5}\mathstrut -\mathstrut$$ $$9732816588$$ $$\nu^{4}\mathstrut -\mathstrut$$ $$46854588548$$ $$\nu^{3}\mathstrut -\mathstrut$$ $$36929122516$$ $$\nu^{2}\mathstrut -\mathstrut$$ $$104516796175$$ $$\nu\mathstrut +\mathstrut$$ $$4416548158$$$$)/$$$$1339827192$$ $$\beta_{10}$$ $$=$$ $$($$$$169940$$ $$\nu^{11}\mathstrut -\mathstrut$$ $$259930$$ $$\nu^{10}\mathstrut +\mathstrut$$ $$15878131$$ $$\nu^{9}\mathstrut -\mathstrut$$ $$23961014$$ $$\nu^{8}\mathstrut +\mathstrut$$ $$530879748$$ $$\nu^{7}\mathstrut -\mathstrut$$ $$766470936$$ $$\nu^{6}\mathstrut +\mathstrut$$ $$7708194267$$ $$\nu^{5}\mathstrut -\mathstrut$$ $$9732816588$$ $$\nu^{4}\mathstrut +\mathstrut$$ $$46854588548$$ $$\nu^{3}\mathstrut -\mathstrut$$ $$36929122516$$ $$\nu^{2}\mathstrut +\mathstrut$$ $$104516796175$$ $$\nu\mathstrut +\mathstrut$$ $$4416548158$$$$)/$$$$1339827192$$ $$\beta_{11}$$ $$=$$ $$($$$$-$$$$380623$$ $$\nu^{11}\mathstrut +\mathstrut$$ $$502073$$ $$\nu^{10}\mathstrut -\mathstrut$$ $$35561311$$ $$\nu^{9}\mathstrut +\mathstrut$$ $$44178167$$ $$\nu^{8}\mathstrut -\mathstrut$$ $$1181014629$$ $$\nu^{7}\mathstrut +\mathstrut$$ $$1323607395$$ $$\nu^{6}\mathstrut -\mathstrut$$ $$16702393701$$ $$\nu^{5}\mathstrut +\mathstrut$$ $$15254532249$$ $$\nu^{4}\mathstrut -\mathstrut$$ $$93304347409$$ $$\nu^{3}\mathstrut +\mathstrut$$ $$48265742063$$ $$\nu^{2}\mathstrut -\mathstrut$$ $$169552623955$$ $$\nu\mathstrut -\mathstrut$$ $$7437120889$$$$)/$$$$1339827192$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{10}\mathstrut +\mathstrut$$ $$\beta_{9}\mathstrut -\mathstrut$$ $$2$$ $$\beta_{8}\mathstrut +\mathstrut$$ $$\beta_{7}\mathstrut +\mathstrut$$ $$\beta_{6}\mathstrut -\mathstrut$$ $$\beta_{5}\mathstrut +\mathstrut$$ $$\beta_{3}\mathstrut +\mathstrut$$ $$\beta_{2}\mathstrut -\mathstrut$$ $$\beta_{1}\mathstrut -\mathstrut$$ $$19$$ $$\nu^{3}$$ $$=$$ $$\beta_{10}\mathstrut -\mathstrut$$ $$\beta_{9}\mathstrut -\mathstrut$$ $$5$$ $$\beta_{7}\mathstrut -\mathstrut$$ $$5$$ $$\beta_{6}\mathstrut -\mathstrut$$ $$24$$ $$\beta_{4}\mathstrut -\mathstrut$$ $$9$$ $$\beta_{3}\mathstrut -\mathstrut$$ $$15$$ $$\beta_{2}\mathstrut -\mathstrut$$ $$25$$ $$\beta_{1}\mathstrut -\mathstrut$$ $$12$$ $$\nu^{4}$$ $$=$$ $$-$$$$31$$ $$\beta_{10}\mathstrut -\mathstrut$$ $$31$$ $$\beta_{9}\mathstrut +\mathstrut$$ $$90$$ $$\beta_{8}\mathstrut -\mathstrut$$ $$51$$ $$\beta_{7}\mathstrut -\mathstrut$$ $$39$$ $$\beta_{6}\mathstrut +\mathstrut$$ $$33$$ $$\beta_{5}\mathstrut -\mathstrut$$ $$113$$ $$\beta_{3}\mathstrut -\mathstrut$$ $$113$$ $$\beta_{2}\mathstrut +\mathstrut$$ $$45$$ $$\beta_{1}\mathstrut +\mathstrut$$ $$476$$ $$\nu^{5}$$ $$=$$ $$32$$ $$\beta_{11}\mathstrut -\mathstrut$$ $$33$$ $$\beta_{10}\mathstrut +\mathstrut$$ $$33$$ $$\beta_{9}\mathstrut +\mathstrut$$ $$271$$ $$\beta_{7}\mathstrut +\mathstrut$$ $$271$$ $$\beta_{6}\mathstrut -\mathstrut$$ $$16$$ $$\beta_{5}\mathstrut +\mathstrut$$ $$1300$$ $$\beta_{4}\mathstrut +\mathstrut$$ $$677$$ $$\beta_{3}\mathstrut +\mathstrut$$ $$623$$ $$\beta_{2}\mathstrut +\mathstrut$$ $$682$$ $$\beta_{1}\mathstrut +\mathstrut$$ $$650$$ $$\nu^{6}$$ $$=$$ $$954$$ $$\beta_{10}\mathstrut +\mathstrut$$ $$954$$ $$\beta_{9}\mathstrut -\mathstrut$$ $$3628$$ $$\beta_{8}\mathstrut +\mathstrut$$ $$2292$$ $$\beta_{7}\mathstrut +\mathstrut$$ $$1336$$ $$\beta_{6}\mathstrut -\mathstrut$$ $$1142$$ $$\beta_{5}\mathstrut +\mathstrut$$ $$5450$$ $$\beta_{3}\mathstrut +\mathstrut$$ $$5450$$ $$\beta_{2}\mathstrut -\mathstrut$$ $$1814$$ $$\beta_{1}\mathstrut -\mathstrut$$ $$12905$$ $$\nu^{7}$$ $$=$$ $$-$$$$2096$$ $$\beta_{11}\mathstrut +\mathstrut$$ $$1056$$ $$\beta_{10}\mathstrut -\mathstrut$$ $$1056$$ $$\beta_{9}\mathstrut -\mathstrut$$ $$11834$$ $$\beta_{7}\mathstrut -\mathstrut$$ $$11834$$ $$\beta_{6}\mathstrut +\mathstrut$$ $$1048$$ $$\beta_{5}\mathstrut -\mathstrut$$ $$56748$$ $$\beta_{4}\mathstrut -\mathstrut$$ $$34702$$ $$\beta_{3}\mathstrut -\mathstrut$$ $$22046$$ $$\beta_{2}\mathstrut -\mathstrut$$ $$19683$$ $$\beta_{1}\mathstrut -\mathstrut$$ $$28374$$ $$\nu^{8}$$ $$=$$ $$-$$$$30477$$ $$\beta_{10}\mathstrut -\mathstrut$$ $$30477$$ $$\beta_{9}\mathstrut +\mathstrut$$ $$140790$$ $$\beta_{8}\mathstrut -\mathstrut$$ $$95925$$ $$\beta_{7}\mathstrut -\mathstrut$$ $$44865$$ $$\beta_{6}\mathstrut +\mathstrut$$ $$40191$$ $$\beta_{5}\mathstrut -\mathstrut$$ $$226923$$ $$\beta_{3}\mathstrut -\mathstrut$$ $$226923$$ $$\beta_{2}\mathstrut +\mathstrut$$ $$70395$$ $$\beta_{1}\mathstrut +\mathstrut$$ $$369011$$ $$\nu^{9}$$ $$=$$ $$99264$$ $$\beta_{11}\mathstrut -\mathstrut$$ $$36579$$ $$\beta_{10}\mathstrut +\mathstrut$$ $$36579$$ $$\beta_{9}\mathstrut +\mathstrut$$ $$478665$$ $$\beta_{7}\mathstrut +\mathstrut$$ $$478665$$ $$\beta_{6}\mathstrut -\mathstrut$$ $$49632$$ $$\beta_{5}\mathstrut +\mathstrut$$ $$2301516$$ $$\beta_{4}\mathstrut +\mathstrut$$ $$1545459$$ $$\beta_{3}\mathstrut +\mathstrut$$ $$756057$$ $$\beta_{2}\mathstrut +\mathstrut$$ $$596465$$ $$\beta_{1}\mathstrut +\mathstrut$$ $$1150758$$ $$\nu^{10}$$ $$=$$ $$1012445$$ $$\beta_{10}\mathstrut +\mathstrut$$ $$1012445$$ $$\beta_{9}\mathstrut -\mathstrut$$ $$5366098$$ $$\beta_{8}\mathstrut +\mathstrut$$ $$3851627$$ $$\beta_{7}\mathstrut +\mathstrut$$ $$1514471$$ $$\beta_{6}\mathstrut -\mathstrut$$ $$1431005$$ $$\beta_{5}\mathstrut +\mathstrut$$ $$8936705$$ $$\beta_{3}\mathstrut +\mathstrut$$ $$8936705$$ $$\beta_{2}\mathstrut -\mathstrut$$ $$2683049$$ $$\beta_{1}\mathstrut -\mathstrut$$ $$11069600$$ $$\nu^{11}$$ $$=$$ $$-$$$$4178328$$ $$\beta_{11}\mathstrut +\mathstrut$$ $$1343903$$ $$\beta_{10}\mathstrut -\mathstrut$$ $$1343903$$ $$\beta_{9}\mathstrut -\mathstrut$$ $$18668485$$ $$\beta_{7}\mathstrut -\mathstrut$$ $$18668485$$ $$\beta_{6}\mathstrut +\mathstrut$$ $$2089164$$ $$\beta_{5}\mathstrut -\mathstrut$$ $$90111516$$ $$\beta_{4}\mathstrut -\mathstrut$$ $$64217769$$ $$\beta_{3}\mathstrut -\mathstrut$$ $$25893747$$ $$\beta_{2}\mathstrut -\mathstrut$$ $$18898340$$ $$\beta_{1}\mathstrut -\mathstrut$$ $$45055758$$

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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
2.1
 4.56289i 2.38108i − 5.04186i − 4.56289i − 2.38108i 5.04186i 6.10049i 0.159251i − 5.08417i − 6.10049i − 0.159251i 5.08417i
−5.45760 1.77328i −10.4040 + 7.55891i 13.6966 + 9.95117i −9.33639 28.7345i 70.1847 22.8044i −33.6132 + 46.2646i −3.13669 4.31729i 26.0747 80.2495i 173.377i
2.2 −3.38257 1.09906i 12.7857 9.28936i −2.71041 1.96923i 5.48545 + 16.8825i −53.4582 + 17.3696i −19.6356 + 27.0262i 40.4526 + 55.6782i 52.1517 160.506i 63.1351i
2.3 3.67706 + 1.19475i −1.64569 + 1.19566i −0.850950 0.618251i −1.76709 5.43854i −7.47979 + 2.43033i −2.52826 + 3.47985i −38.7511 53.3363i −23.7517 + 73.1002i 22.1090i
6.1 −5.45760 + 1.77328i −10.4040 7.55891i 13.6966 9.95117i −9.33639 + 28.7345i 70.1847 + 22.8044i −33.6132 46.2646i −3.13669 + 4.31729i 26.0747 + 80.2495i 173.377i
6.2 −3.38257 + 1.09906i 12.7857 + 9.28936i −2.71041 + 1.96923i 5.48545 16.8825i −53.4582 17.3696i −19.6356 27.0262i 40.4526 55.6782i 52.1517 + 160.506i 63.1351i
6.3 3.67706 1.19475i −1.64569 1.19566i −0.850950 + 0.618251i −1.76709 + 5.43854i −7.47979 2.43033i −2.52826 3.47985i −38.7511 + 53.3363i −23.7517 73.1002i 22.1090i
7.1 −2.46775 + 3.39656i −2.26281 + 6.96422i −0.502581 1.54679i 14.7329 10.7041i −18.0704 24.8717i 47.9990 15.5958i −57.3923 18.6479i 22.1503 + 16.0931i 76.4560i
7.2 1.02443 1.41001i 2.68168 8.25338i 4.00561 + 12.3280i −8.06057 + 5.85635i −8.89011 12.2362i −56.0338 + 18.2065i 48.0070 + 15.5984i 4.60358 + 3.34469i 17.3649i
7.3 4.10644 5.65202i −4.15494 + 12.7876i −10.1383 31.2024i −10.0543 + 7.30486i 55.2138 + 75.9952i 23.8119 7.73696i −111.679 36.2869i −80.7285 58.6527i 86.8239i
8.1 −2.46775 3.39656i −2.26281 6.96422i −0.502581 + 1.54679i 14.7329 + 10.7041i −18.0704 + 24.8717i 47.9990 + 15.5958i −57.3923 + 18.6479i 22.1503 16.0931i 76.4560i
8.2 1.02443 + 1.41001i 2.68168 + 8.25338i 4.00561 12.3280i −8.06057 5.85635i −8.89011 + 12.2362i −56.0338 18.2065i 48.0070 15.5984i 4.60358 3.34469i 17.3649i
8.3 4.10644 + 5.65202i −4.15494 12.7876i −10.1383 + 31.2024i −10.0543 7.30486i 55.2138 75.9952i 23.8119 + 7.73696i −111.679 + 36.2869i −80.7285 + 58.6527i 86.8239i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 8.3 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

Inner twists

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

Hecke kernels

There are no other newforms in $$S_{5}^{\mathrm{new}}(11, [\chi])$$.