# Properties

 Label 11.6.c.a Level 11 Weight 6 Character orbit 11.c Analytic conductor 1.764 Analytic rank 0 Dimension 16 CM No Inner twists 2

# Related objects

## Newspace parameters

 Level: $$N$$ = $$11$$ Weight: $$k$$ = $$6$$ Character orbit: $$[\chi]$$ = 11.c (of order $$5$$ and degree $$4$$)

## Newform invariants

 Self dual: No Analytic conductor: $$1.76422201794$$ Analytic rank: $$0$$ Dimension: $$16$$ Relative dimension: $$4$$ over $$\Q(\zeta_{5})$$ Coefficient field: $$\mathbb{Q}[x]/(x^{16} - \cdots)$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2^{4}$$ Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{15}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q$$ $$+ ( 1 - \beta_{1} - \beta_{2} + \beta_{3} + \beta_{4} + \beta_{7} ) q^{2}$$ $$+ ( -3 \beta_{6} + 3 \beta_{7} - \beta_{9} - \beta_{10} + \beta_{11} - \beta_{12} ) q^{3}$$ $$+ ( -3 + \beta_{2} - \beta_{3} + \beta_{4} + \beta_{5} - 9 \beta_{6} - 3 \beta_{7} + \beta_{8} - \beta_{14} ) q^{4}$$ $$+ ( 3 - 3 \beta_{1} - 3 \beta_{2} + 2 \beta_{3} + 2 \beta_{6} + 9 \beta_{7} - \beta_{8} + \beta_{10} - \beta_{11} + \beta_{12} - \beta_{14} - \beta_{15} ) q^{5}$$ $$+ ( -2 + 2 \beta_{1} + 2 \beta_{2} + 3 \beta_{3} + 5 \beta_{6} - 17 \beta_{7} + 7 \beta_{8} + 2 \beta_{11} + 2 \beta_{14} + \beta_{15} ) q^{6}$$ $$+ ( 5 + \beta_{2} - \beta_{3} + 3 \beta_{4} - 3 \beta_{5} + 34 \beta_{6} + 5 \beta_{7} + \beta_{8} + 3 \beta_{9} - 2 \beta_{10} - 3 \beta_{11} + \beta_{13} + 3 \beta_{14} + \beta_{15} ) q^{7}$$ $$+ ( 41 + 16 \beta_{1} - 19 \beta_{3} - 19 \beta_{4} - 5 \beta_{5} - 7 \beta_{6} + 7 \beta_{7} - 41 \beta_{8} + 4 \beta_{10} - 4 \beta_{11} - 3 \beta_{13} + 2 \beta_{14} + 2 \beta_{15} ) q^{8}$$ $$+ ( -42 + 3 \beta_{1} + 5 \beta_{2} - 3 \beta_{3} - 5 \beta_{4} + 4 \beta_{5} + 37 \beta_{6} - 5 \beta_{7} - 21 \beta_{8} + 10 \beta_{9} + 10 \beta_{10} + 5 \beta_{12} + \beta_{13} - 4 \beta_{15} ) q^{9}$$ $$+O(q^{10})$$ $$q$$ $$+ ( 1 - \beta_{1} - \beta_{2} + \beta_{3} + \beta_{4} + \beta_{7} ) q^{2}$$ $$+ ( -3 \beta_{6} + 3 \beta_{7} - \beta_{9} - \beta_{10} + \beta_{11} - \beta_{12} ) q^{3}$$ $$+ ( -3 + \beta_{2} - \beta_{3} + \beta_{4} + \beta_{5} - 9 \beta_{6} - 3 \beta_{7} + \beta_{8} - \beta_{14} ) q^{4}$$ $$+ ( 3 - 3 \beta_{1} - 3 \beta_{2} + 2 \beta_{3} + 2 \beta_{6} + 9 \beta_{7} - \beta_{8} + \beta_{10} - \beta_{11} + \beta_{12} - \beta_{14} - \beta_{15} ) q^{5}$$ $$+ ( -2 + 2 \beta_{1} + 2 \beta_{2} + 3 \beta_{3} + 5 \beta_{6} - 17 \beta_{7} + 7 \beta_{8} + 2 \beta_{11} + 2 \beta_{14} + \beta_{15} ) q^{6}$$ $$+ ( 5 + \beta_{2} - \beta_{3} + 3 \beta_{4} - 3 \beta_{5} + 34 \beta_{6} + 5 \beta_{7} + \beta_{8} + 3 \beta_{9} - 2 \beta_{10} - 3 \beta_{11} + \beta_{13} + 3 \beta_{14} + \beta_{15} ) q^{7}$$ $$+ ( 41 + 16 \beta_{1} - 19 \beta_{3} - 19 \beta_{4} - 5 \beta_{5} - 7 \beta_{6} + 7 \beta_{7} - 41 \beta_{8} + 4 \beta_{10} - 4 \beta_{11} - 3 \beta_{13} + 2 \beta_{14} + 2 \beta_{15} ) q^{8}$$ $$+ ( -42 + 3 \beta_{1} + 5 \beta_{2} - 3 \beta_{3} - 5 \beta_{4} + 4 \beta_{5} + 37 \beta_{6} - 5 \beta_{7} - 21 \beta_{8} + 10 \beta_{9} + 10 \beta_{10} + 5 \beta_{12} + \beta_{13} - 4 \beta_{15} ) q^{9}$$ $$+ ( -125 - 2 \beta_{1} + 31 \beta_{2} + 2 \beta_{4} + 5 \beta_{5} - 139 \beta_{7} + 139 \beta_{8} - 4 \beta_{9} - 10 \beta_{11} + 10 \beta_{12} + 4 \beta_{13} - 4 \beta_{14} ) q^{10}$$ $$+ ( -23 + 32 \beta_{1} + 16 \beta_{2} + 2 \beta_{3} - 35 \beta_{4} - 2 \beta_{5} - 35 \beta_{6} + 97 \beta_{7} - 69 \beta_{8} + 8 \beta_{9} - 7 \beta_{10} + 5 \beta_{11} + 5 \beta_{12} - 3 \beta_{13} - 4 \beta_{14} + \beta_{15} ) q^{11}$$ $$+ ( -52 - 33 \beta_{1} - 40 \beta_{2} + 33 \beta_{4} + 3 \beta_{5} + 9 \beta_{7} - 9 \beta_{8} - 20 \beta_{9} + 12 \beta_{11} - 12 \beta_{12} + 2 \beta_{13} - 2 \beta_{14} ) q^{12}$$ $$+ ( 105 + 20 \beta_{1} - 15 \beta_{2} - 20 \beta_{3} + 15 \beta_{4} + 3 \beta_{5} - 90 \beta_{6} + 15 \beta_{7} + 27 \beta_{8} - 5 \beta_{9} - 5 \beta_{10} - 30 \beta_{12} - 5 \beta_{13} - 3 \beta_{15} ) q^{13}$$ $$+ ( 164 - 104 \beta_{1} + 17 \beta_{3} + 17 \beta_{4} - 2 \beta_{5} - 112 \beta_{6} + 112 \beta_{7} - 164 \beta_{8} - 14 \beta_{9} - 24 \beta_{10} + 24 \beta_{11} - 14 \beta_{12} - 3 \beta_{13} - \beta_{14} - \beta_{15} ) q^{14}$$ $$+ ( 333 - 91 \beta_{2} + 91 \beta_{3} + 136 \beta_{4} - 5 \beta_{5} - 222 \beta_{6} + 333 \beta_{7} - 91 \beta_{8} - 10 \beta_{9} - 25 \beta_{10} + 10 \beta_{11} + 10 \beta_{13} + 5 \beta_{14} + 10 \beta_{15} ) q^{15}$$ $$+ ( 109 - 109 \beta_{1} - 109 \beta_{2} + 148 \beta_{3} + 630 \beta_{6} - 186 \beta_{7} + 521 \beta_{8} + 12 \beta_{10} - 4 \beta_{11} + 12 \beta_{12} + 10 \beta_{14} + 3 \beta_{15} ) q^{16}$$ $$+ ( -36 + 36 \beta_{1} + 36 \beta_{2} + 41 \beta_{3} - 223 \beta_{6} - 372 \beta_{7} - 187 \beta_{8} + 20 \beta_{10} + \beta_{11} + 20 \beta_{12} - 13 \beta_{14} - 10 \beta_{15} ) q^{17}$$ $$+ ( -49 + 138 \beta_{2} - 138 \beta_{3} + 11 \beta_{4} + 17 \beta_{5} + 137 \beta_{6} - 49 \beta_{7} + 138 \beta_{8} + 32 \beta_{9} + 18 \beta_{10} - 32 \beta_{11} - 14 \beta_{13} - 17 \beta_{14} - 14 \beta_{15} ) q^{18}$$ $$+ ( 151 + 65 \beta_{1} - 176 \beta_{3} - 176 \beta_{4} + 25 \beta_{5} - 664 \beta_{6} + 664 \beta_{7} - 151 \beta_{8} + 10 \beta_{9} + 16 \beta_{10} - 16 \beta_{11} + 10 \beta_{12} + 18 \beta_{13} - 7 \beta_{14} - 7 \beta_{15} ) q^{19}$$ $$+ ( -156 + 214 \beta_{1} + 74 \beta_{2} - 214 \beta_{3} - 74 \beta_{4} - 30 \beta_{5} + 82 \beta_{6} - 74 \beta_{7} - 1166 \beta_{8} + 4 \beta_{12} + 30 \beta_{15} ) q^{20}$$ $$+ ( -1153 + 137 \beta_{1} + 221 \beta_{2} - 137 \beta_{4} - 38 \beta_{5} - 931 \beta_{7} + 931 \beta_{8} + 2 \beta_{9} + 7 \beta_{11} - 7 \beta_{12} - 31 \beta_{13} + 31 \beta_{14} ) q^{21}$$ $$+ ( -1747 + 90 \beta_{1} + 67 \beta_{2} + 95 \beta_{3} - 150 \beta_{4} + 15 \beta_{5} + 1610 \beta_{6} - 458 \beta_{7} + 1117 \beta_{8} - 16 \beta_{9} - 8 \beta_{10} - 10 \beta_{11} - 10 \beta_{12} + 28 \beta_{13} + 30 \beta_{14} - 2 \beta_{15} ) q^{22}$$ $$+ ( -270 - 114 \beta_{1} - 84 \beta_{2} + 114 \beta_{4} - 18 \beta_{5} + 594 \beta_{7} - 594 \beta_{8} + 70 \beta_{9} - 22 \beta_{11} + 22 \beta_{12} - 18 \beta_{13} + 18 \beta_{14} ) q^{23}$$ $$+ ( 1257 + 33 \beta_{1} - 60 \beta_{2} - 33 \beta_{3} + 60 \beta_{4} - 24 \beta_{5} - 1197 \beta_{6} + 60 \beta_{7} - 288 \beta_{8} + 20 \beta_{9} + 20 \beta_{10} + 100 \beta_{12} + 33 \beta_{13} + 24 \beta_{15} ) q^{24}$$ $$+ ( 1129 - 215 \beta_{1} + 25 \beta_{3} + 25 \beta_{4} + 25 \beta_{5} + 411 \beta_{6} - 411 \beta_{7} - 1129 \beta_{8} + 35 \beta_{9} + 90 \beta_{10} - 90 \beta_{11} + 35 \beta_{12} + 25 \beta_{13} ) q^{25}$$ $$+ ( 1856 - 200 \beta_{2} + 200 \beta_{3} + 119 \beta_{4} - 2 \beta_{5} - 1036 \beta_{6} + 1856 \beta_{7} - 200 \beta_{8} + 22 \beta_{9} + 102 \beta_{10} - 22 \beta_{11} - 77 \beta_{13} + 2 \beta_{14} - 77 \beta_{15} ) q^{26}$$ $$+ ( 141 - 141 \beta_{1} - 141 \beta_{2} - 30 \beta_{3} + 2430 \beta_{6} - 834 \beta_{7} + 2289 \beta_{8} - 54 \beta_{10} + 26 \beta_{11} - 54 \beta_{12} - 45 \beta_{14} + 15 \beta_{15} ) q^{27}$$ $$+ ( -64 + 64 \beta_{1} + 64 \beta_{2} + 34 \beta_{3} - 376 \beta_{6} - 2926 \beta_{7} - 312 \beta_{8} - 100 \beta_{10} - 36 \beta_{11} - 100 \beta_{12} + 12 \beta_{14} + 40 \beta_{15} ) q^{28}$$ $$+ ( 1063 - 55 \beta_{2} + 55 \beta_{3} + 252 \beta_{4} - 7 \beta_{5} + 502 \beta_{6} + 1063 \beta_{7} - 55 \beta_{8} - 203 \beta_{9} - 94 \beta_{10} + 203 \beta_{11} + 70 \beta_{13} + 7 \beta_{14} + 70 \beta_{15} ) q^{29}$$ $$+ ( 1458 - 51 \beta_{1} + 87 \beta_{3} + 87 \beta_{4} + 21 \beta_{5} - 5010 \beta_{6} + 5010 \beta_{7} - 1458 \beta_{8} - 70 \beta_{9} - 150 \beta_{10} + 150 \beta_{11} - 70 \beta_{12} - 15 \beta_{13} - 36 \beta_{14} - 36 \beta_{15} ) q^{30}$$ $$+ ( -577 - 287 \beta_{1} + 3 \beta_{2} + 287 \beta_{3} - 3 \beta_{4} + 72 \beta_{5} + 574 \beta_{6} - 3 \beta_{7} - 1923 \beta_{8} - 155 \beta_{9} - 155 \beta_{10} - 92 \beta_{12} - 43 \beta_{13} - 72 \beta_{15} ) q^{31}$$ $$+ ( -5242 - 347 \beta_{1} - 183 \beta_{2} + 347 \beta_{4} + 97 \beta_{5} - 3779 \beta_{7} + 3779 \beta_{8} + 56 \beta_{9} + 156 \beta_{11} - 156 \beta_{12} + 75 \beta_{13} - 75 \beta_{14} ) q^{32}$$ $$+ ( -4212 - 87 \beta_{1} - 225 \beta_{2} - 13 \beta_{3} + 156 \beta_{4} - 42 \beta_{5} + 3379 \beta_{6} - 779 \beta_{7} + 1279 \beta_{8} - 74 \beta_{9} + 183 \beta_{10} - 16 \beta_{11} - 49 \beta_{12} - 96 \beta_{13} - 62 \beta_{14} - 23 \beta_{15} ) q^{33}$$ $$+ ( -849 + 218 \beta_{1} - 108 \beta_{2} - 218 \beta_{4} + 13 \beta_{5} + 1845 \beta_{7} - 1845 \beta_{8} - 10 \beta_{9} - 132 \beta_{11} + 132 \beta_{12} + 57 \beta_{13} - 57 \beta_{14} ) q^{34}$$ $$+ ( 2966 - 187 \beta_{1} + 22 \beta_{2} + 187 \beta_{3} - 22 \beta_{4} + 59 \beta_{5} - 2988 \beta_{6} - 22 \beta_{7} + 1912 \beta_{8} - 16 \beta_{9} - 16 \beta_{10} + 45 \beta_{12} - 40 \beta_{13} - 59 \beta_{15} ) q^{35}$$ $$+ ( 7016 + 599 \beta_{1} - 63 \beta_{3} - 63 \beta_{4} - 129 \beta_{5} - 2971 \beta_{6} + 2971 \beta_{7} - 7016 \beta_{8} + 188 \beta_{9} + 56 \beta_{10} - 56 \beta_{11} + 188 \beta_{12} - 74 \beta_{13} + 55 \beta_{14} + 55 \beta_{15} ) q^{36}$$ $$+ ( -553 + 879 \beta_{2} - 879 \beta_{3} - 650 \beta_{4} + 43 \beta_{5} + 544 \beta_{6} - 553 \beta_{7} + 879 \beta_{8} + 143 \beta_{9} - 20 \beta_{10} - 143 \beta_{11} + 176 \beta_{13} - 43 \beta_{14} + 176 \beta_{15} ) q^{37}$$ $$+ ( -849 + 849 \beta_{1} + 849 \beta_{2} - 349 \beta_{3} + 6769 \beta_{6} - 3721 \beta_{7} + 7618 \beta_{8} - 32 \beta_{10} - 16 \beta_{11} - 32 \beta_{12} + 120 \beta_{14} - 62 \beta_{15} ) q^{38}$$ $$+ ( 211 - 211 \beta_{1} - 211 \beta_{2} - 263 \beta_{3} - 596 \beta_{6} - 6633 \beta_{7} - 807 \beta_{8} + 120 \beta_{10} + 63 \beta_{11} + 120 \beta_{12} + 100 \beta_{14} - 81 \beta_{15} ) q^{39}$$ $$+ ( 2612 - 514 \beta_{2} + 514 \beta_{3} - 1032 \beta_{4} - 120 \beta_{5} + 4722 \beta_{6} + 2612 \beta_{7} - 514 \beta_{8} + 200 \beta_{9} + 320 \beta_{10} - 200 \beta_{11} - 130 \beta_{13} + 120 \beta_{14} - 130 \beta_{15} ) q^{40}$$ $$+ ( 3724 + 51 \beta_{1} + 500 \beta_{3} + 500 \beta_{4} - 311 \beta_{5} - 7531 \beta_{6} + 7531 \beta_{7} - 3724 \beta_{8} + 217 \beta_{9} + 205 \beta_{10} - 205 \beta_{11} + 217 \beta_{12} - 94 \beta_{13} + 217 \beta_{14} + 217 \beta_{15} ) q^{41}$$ $$+ ( -3539 - 281 \beta_{1} - 465 \beta_{2} + 281 \beta_{3} + 465 \beta_{4} - 30 \beta_{5} + 4004 \beta_{6} + 465 \beta_{7} - 5403 \beta_{8} + 110 \beta_{9} + 110 \beta_{10} + 24 \beta_{12} + 143 \beta_{13} + 30 \beta_{15} ) q^{42}$$ $$+ ( -3091 + 313 \beta_{1} - 1131 \beta_{2} - 313 \beta_{4} - 82 \beta_{5} - 3836 \beta_{7} + 3836 \beta_{8} - 42 \beta_{9} - 301 \beta_{11} + 301 \beta_{12} - 9 \beta_{13} + 9 \beta_{14} ) q^{43}$$ $$+ ( -8758 - 651 \beta_{1} + 439 \beta_{2} - 1030 \beta_{3} + 733 \beta_{4} + 73 \beta_{5} + 3934 \beta_{6} - 2303 \beta_{7} + 4757 \beta_{8} + 192 \beta_{9} - 256 \beta_{10} - 144 \beta_{11} + 164 \beta_{12} + 115 \beta_{13} - 63 \beta_{14} + 68 \beta_{15} ) q^{44}$$ $$+ ( -7451 + 486 \beta_{1} + 1729 \beta_{2} - 486 \beta_{4} + 111 \beta_{5} - 447 \beta_{7} + 447 \beta_{8} + 35 \beta_{9} + 320 \beta_{11} - 320 \beta_{12} - 44 \beta_{13} + 44 \beta_{14} ) q^{45}$$ $$+ ( 3736 + 118 \beta_{1} + 754 \beta_{2} - 118 \beta_{3} - 754 \beta_{4} + 14 \beta_{5} - 4490 \beta_{6} - 754 \beta_{7} - 996 \beta_{8} + 28 \beta_{9} + 28 \beta_{10} - 140 \beta_{12} - 170 \beta_{13} - 14 \beta_{15} ) q^{46}$$ $$+ ( 3786 + 1610 \beta_{1} - 277 \beta_{3} - 277 \beta_{4} + 328 \beta_{5} - 1580 \beta_{6} + 1580 \beta_{7} - 3786 \beta_{8} - 531 \beta_{9} - 212 \beta_{10} + 212 \beta_{11} - 531 \beta_{12} + 75 \beta_{13} - 253 \beta_{14} - 253 \beta_{15} ) q^{47}$$ $$+ ( 3426 + 233 \beta_{2} - 233 \beta_{3} - 812 \beta_{4} - 20 \beta_{5} - 4143 \beta_{6} + 3426 \beta_{7} + 233 \beta_{8} - 572 \beta_{9} - 264 \beta_{10} + 572 \beta_{11} + 55 \beta_{13} + 20 \beta_{14} + 55 \beta_{15} ) q^{48}$$ $$+ ( -173 + 173 \beta_{1} + 173 \beta_{2} - 1186 \beta_{3} + 3544 \beta_{6} - 2368 \beta_{7} + 3717 \beta_{8} + 199 \beta_{10} + 131 \beta_{11} + 199 \beta_{12} - 177 \beta_{14} - \beta_{15} ) q^{49}$$ $$+ ( 1025 - 1025 \beta_{1} - 1025 \beta_{2} - 11 \beta_{3} - 860 \beta_{6} - 4676 \beta_{7} - 1885 \beta_{8} + 10 \beta_{10} - 80 \beta_{11} + 10 \beta_{12} - 285 \beta_{14} + 95 \beta_{15} ) q^{50}$$ $$+ ( 7781 - 675 \beta_{2} + 675 \beta_{3} + 983 \beta_{4} + 197 \beta_{5} - 1915 \beta_{6} + 7781 \beta_{7} - 675 \beta_{8} + 282 \beta_{9} - 616 \beta_{10} - 282 \beta_{11} - 83 \beta_{13} - 197 \beta_{14} - 83 \beta_{15} ) q^{51}$$ $$+ ( -3818 - 1702 \beta_{1} + 2146 \beta_{3} + 2146 \beta_{4} + 400 \beta_{5} - 3208 \beta_{6} + 3208 \beta_{7} + 3818 \beta_{8} - 492 \beta_{9} + 12 \beta_{10} - 12 \beta_{11} - 492 \beta_{12} + 132 \beta_{13} - 268 \beta_{14} - 268 \beta_{15} ) q^{52}$$ $$+ ( 1703 - 1830 \beta_{1} - 631 \beta_{2} + 1830 \beta_{3} + 631 \beta_{4} - 75 \beta_{5} - 1072 \beta_{6} + 631 \beta_{7} - 6015 \beta_{8} + 461 \beta_{9} + 461 \beta_{10} + 432 \beta_{12} + 23 \beta_{13} + 75 \beta_{15} ) q^{53}$$ $$+ ( -284 - 1184 \beta_{1} - 941 \beta_{2} + 1184 \beta_{4} + 62 \beta_{5} - 8537 \beta_{7} + 8537 \beta_{8} - 236 \beta_{9} - 12 \beta_{11} + 12 \beta_{12} - 170 \beta_{13} + 170 \beta_{14} ) q^{54}$$ $$+ ( 1638 - 364 \beta_{1} - 2008 \beta_{2} + 1685 \beta_{3} + 1560 \beta_{4} - 156 \beta_{5} + 3980 \beta_{6} + 2748 \beta_{7} - 1730 \beta_{8} - 80 \beta_{9} - 535 \beta_{10} + 841 \beta_{11} - 325 \beta_{12} + 96 \beta_{13} + 381 \beta_{14} + 122 \beta_{15} ) q^{55}$$ $$+ ( 6932 + 506 \beta_{1} - 1122 \beta_{2} - 506 \beta_{4} - 198 \beta_{5} + 4606 \beta_{7} - 4606 \beta_{8} - 672 \beta_{9} + 88 \beta_{11} - 88 \beta_{12} - 154 \beta_{13} + 154 \beta_{14} ) q^{56}$$ $$+ ( -3312 - 225 \beta_{1} - 750 \beta_{2} + 225 \beta_{3} + 750 \beta_{4} - 285 \beta_{5} + 4062 \beta_{6} + 750 \beta_{7} + 2121 \beta_{8} - 141 \beta_{9} - 141 \beta_{10} - 923 \beta_{12} + 420 \beta_{13} + 285 \beta_{15} ) q^{57}$$ $$+ ( 5448 - 1086 \beta_{1} + 101 \beta_{3} + 101 \beta_{4} - 292 \beta_{5} - 6236 \beta_{6} + 6236 \beta_{7} - 5448 \beta_{8} - 62 \beta_{9} - 776 \beta_{10} + 776 \beta_{11} - 62 \beta_{12} + 37 \beta_{13} + 329 \beta_{14} + 329 \beta_{15} ) q^{58}$$ $$+ ( 4001 + 412 \beta_{2} - 412 \beta_{3} - 1975 \beta_{4} - 182 \beta_{5} + 2204 \beta_{6} + 4001 \beta_{7} + 412 \beta_{8} + 550 \beta_{9} - 92 \beta_{10} - 550 \beta_{11} - 671 \beta_{13} + 182 \beta_{14} - 671 \beta_{15} ) q^{59}$$ $$+ ( -1240 + 1240 \beta_{1} + 1240 \beta_{2} - 20 \beta_{3} + 2862 \beta_{6} - 5588 \beta_{7} + 4102 \beta_{8} + 364 \beta_{10} - 848 \beta_{11} + 364 \beta_{12} - 12 \beta_{14} + 142 \beta_{15} ) q^{60}$$ $$+ ( -2315 + 2315 \beta_{1} + 2315 \beta_{2} - 1762 \beta_{3} - 3800 \beta_{6} - 1343 \beta_{7} - 1485 \beta_{8} + 367 \beta_{10} + 141 \beta_{11} + 367 \beta_{12} + 81 \beta_{14} - 79 \beta_{15} ) q^{61}$$ $$+ ( -15176 + 722 \beta_{2} - 722 \beta_{3} - 699 \beta_{4} + 98 \beta_{5} + 5278 \beta_{6} - 15176 \beta_{7} + 722 \beta_{8} + 150 \beta_{9} + 506 \beta_{10} - 150 \beta_{11} + 643 \beta_{13} - 98 \beta_{14} + 643 \beta_{15} ) q^{62}$$ $$+ ( -2540 + 309 \beta_{1} - 1826 \beta_{3} - 1826 \beta_{4} + 649 \beta_{5} + 2892 \beta_{6} - 2892 \beta_{7} + 2540 \beta_{8} + 968 \beta_{9} + 561 \beta_{10} - 561 \beta_{11} + 968 \beta_{12} + 308 \beta_{13} - 341 \beta_{14} - 341 \beta_{15} ) q^{63}$$ $$+ ( -2065 + 4092 \beta_{1} + 2167 \beta_{2} - 4092 \beta_{3} - 2167 \beta_{4} + 11 \beta_{5} - 102 \beta_{6} - 2167 \beta_{7} + 3752 \beta_{8} + 484 \beta_{9} + 484 \beta_{10} + 56 \beta_{12} - 682 \beta_{13} - 11 \beta_{15} ) q^{64}$$ $$+ ( 458 + 2897 \beta_{1} + 2746 \beta_{2} - 2897 \beta_{4} - 265 \beta_{5} + 12574 \beta_{7} - 12574 \beta_{8} + 49 \beta_{9} - 215 \beta_{11} + 215 \beta_{12} - 9 \beta_{13} + 9 \beta_{14} ) q^{65}$$ $$+ ( 2583 + 537 \beta_{1} + 2320 \beta_{2} - 1547 \beta_{3} - 3502 \beta_{4} + 128 \beta_{5} - 8177 \beta_{6} - 719 \beta_{7} + 1226 \beta_{8} + 302 \beta_{9} + 1064 \beta_{10} - 1090 \beta_{11} + 1022 \beta_{12} - 325 \beta_{13} - 283 \beta_{14} - 570 \beta_{15} ) q^{66}$$ $$+ ( -6743 - 859 \beta_{1} + 122 \beta_{2} + 859 \beta_{4} - 131 \beta_{5} - 6259 \beta_{7} + 6259 \beta_{8} + 419 \beta_{9} - 275 \beta_{11} + 275 \beta_{12} + 353 \beta_{13} - 353 \beta_{14} ) q^{67}$$ $$+ ( -7704 + 481 \beta_{1} + 1119 \beta_{2} - 481 \beta_{3} - 1119 \beta_{4} + 359 \beta_{5} + 6585 \beta_{6} - 1119 \beta_{7} - 3914 \beta_{8} - 28 \beta_{9} - 28 \beta_{10} + 868 \beta_{12} + 161 \beta_{13} - 359 \beta_{15} ) q^{68}$$ $$+ ( -17746 - 226 \beta_{1} - 96 \beta_{3} - 96 \beta_{4} - 558 \beta_{5} + 18854 \beta_{6} - 18854 \beta_{7} + 17746 \beta_{8} + 474 \beta_{9} + 1046 \beta_{10} - 1046 \beta_{11} + 474 \beta_{12} - 140 \beta_{13} + 418 \beta_{14} + 418 \beta_{15} ) q^{69}$$ $$+ ( -3503 - 2663 \beta_{2} + 2663 \beta_{3} + 6474 \beta_{4} + 129 \beta_{5} - 2482 \beta_{6} - 3503 \beta_{7} - 2663 \beta_{8} + 364 \beta_{9} + 434 \beta_{10} - 364 \beta_{11} + 406 \beta_{13} - 129 \beta_{14} + 406 \beta_{15} ) q^{70}$$ $$+ ( 4156 - 4156 \beta_{1} - 4156 \beta_{2} + 1235 \beta_{3} - 17648 \beta_{6} + 11038 \beta_{7} - 21804 \beta_{8} - 901 \beta_{10} + 1105 \beta_{11} - 901 \beta_{12} + 571 \beta_{14} + 246 \beta_{15} ) q^{71}$$ $$+ ( 1931 - 1931 \beta_{1} - 1931 \beta_{2} + 5555 \beta_{3} + 9935 \beta_{6} + 29916 \beta_{7} + 8004 \beta_{8} - 1212 \beta_{10} + 608 \beta_{11} - 1212 \beta_{12} + 605 \beta_{14} - 108 \beta_{15} ) q^{72}$$ $$+ ( 4477 + 2066 \beta_{2} - 2066 \beta_{3} + 2323 \beta_{4} - 92 \beta_{5} - 14787 \beta_{6} + 4477 \beta_{7} + 2066 \beta_{8} - 1066 \beta_{9} + 261 \beta_{10} + 1066 \beta_{11} - 551 \beta_{13} + 92 \beta_{14} - 551 \beta_{15} ) q^{73}$$ $$+ ( 6876 + 4972 \beta_{1} - 4341 \beta_{3} - 4341 \beta_{4} - 1390 \beta_{5} + 24518 \beta_{6} - 24518 \beta_{7} - 6876 \beta_{8} - 1030 \beta_{9} - 1276 \beta_{10} + 1276 \beta_{11} - 1030 \beta_{12} - 571 \beta_{13} + 819 \beta_{14} + 819 \beta_{15} ) q^{74}$$ $$+ ( 15668 - 535 \beta_{1} - 385 \beta_{2} + 535 \beta_{3} + 385 \beta_{4} - 200 \beta_{5} - 15283 \beta_{6} + 385 \beta_{7} + 15675 \beta_{8} - 2245 \beta_{9} - 2245 \beta_{10} - 1529 \beta_{12} + 485 \beta_{13} + 200 \beta_{15} ) q^{75}$$ $$+ ( 19530 - 3886 \beta_{1} + 465 \beta_{2} + 3886 \beta_{4} - 170 \beta_{5} + 13000 \beta_{7} - 13000 \beta_{8} + 704 \beta_{9} + 488 \beta_{11} - 488 \beta_{12} + 393 \beta_{13} - 393 \beta_{14} ) q^{76}$$ $$+ ( 27579 + 1036 \beta_{1} + 595 \beta_{2} + 3857 \beta_{3} - 1470 \beta_{4} + 763 \beta_{5} - 9588 \beta_{6} + 2325 \beta_{7} - 10741 \beta_{8} - 401 \beta_{9} + 344 \beta_{10} - 175 \beta_{11} - 1264 \beta_{12} + 116 \beta_{13} - 487 \beta_{14} + 64 \beta_{15} ) q^{77}$$ $$+ ( 27326 - 2679 \beta_{1} - 5248 \beta_{2} + 2679 \beta_{4} + 273 \beta_{5} + 798 \beta_{7} - 798 \beta_{8} + 766 \beta_{9} - 164 \beta_{11} + 164 \beta_{12} - 25 \beta_{13} + 25 \beta_{14} ) q^{78}$$ $$+ ( 5135 - 2207 \beta_{1} - 5497 \beta_{2} + 2207 \beta_{3} + 5497 \beta_{4} + 172 \beta_{5} + 362 \beta_{6} + 5497 \beta_{7} + 8841 \beta_{8} + 201 \beta_{9} + 201 \beta_{10} + 2108 \beta_{12} - 855 \beta_{13} - 172 \beta_{15} ) q^{79}$$ $$+ ( -25860 - 8222 \beta_{1} + 3154 \beta_{3} + 3154 \beta_{4} + 1790 \beta_{5} - 4074 \beta_{6} + 4074 \beta_{7} + 25860 \beta_{8} + 888 \beta_{9} + 1328 \beta_{10} - 1328 \beta_{11} + 888 \beta_{12} + 316 \beta_{13} - 1474 \beta_{14} - 1474 \beta_{15} ) q^{80}$$ $$+ ( -25242 - 1520 \beta_{2} + 1520 \beta_{3} + 2414 \beta_{4} + 524 \beta_{5} + 17100 \beta_{6} - 25242 \beta_{7} - 1520 \beta_{8} - 1264 \beta_{9} + 634 \beta_{10} + 1264 \beta_{11} + 626 \beta_{13} - 524 \beta_{14} + 626 \beta_{15} ) q^{81}$$ $$+ ( 1536 - 1536 \beta_{1} - 1536 \beta_{2} + 9063 \beta_{3} - 28957 \beta_{6} + 32573 \beta_{7} - 30493 \beta_{8} - 516 \beta_{10} + 950 \beta_{11} - 516 \beta_{12} - 804 \beta_{14} - 555 \beta_{15} ) q^{82}$$ $$+ ( -2611 + 2611 \beta_{1} + 2611 \beta_{2} - 3517 \beta_{3} - 1005 \beta_{6} + 5528 \beta_{7} + 1606 \beta_{8} + 628 \beta_{10} - 1568 \beta_{11} + 628 \beta_{12} - 780 \beta_{14} + 841 \beta_{15} ) q^{83}$$ $$+ ( -19194 + 2412 \beta_{2} - 2412 \beta_{3} - 3258 \beta_{4} - 402 \beta_{5} - 19182 \beta_{6} - 19194 \beta_{7} + 2412 \beta_{8} - 696 \beta_{9} - 1380 \beta_{10} + 696 \beta_{11} - 720 \beta_{13} + 402 \beta_{14} - 720 \beta_{15} ) q^{84}$$ $$+ ( -13242 - 747 \beta_{1} - 3087 \beta_{3} - 3087 \beta_{4} - 147 \beta_{5} + 28246 \beta_{6} - 28246 \beta_{7} + 13242 \beta_{8} - 711 \beta_{9} - 748 \beta_{10} + 748 \beta_{11} - 711 \beta_{12} - 219 \beta_{13} - 72 \beta_{14} - 72 \beta_{15} ) q^{85}$$ $$+ ( -13889 + 1239 \beta_{1} - 1322 \beta_{2} - 1239 \beta_{3} + 1322 \beta_{4} + 882 \beta_{5} + 15211 \beta_{6} + 1322 \beta_{7} + 51810 \beta_{8} - 858 \beta_{9} - 858 \beta_{10} + 376 \beta_{12} + 1309 \beta_{13} - 882 \beta_{15} ) q^{86}$$ $$+ ( 10815 + 4700 \beta_{1} + 7797 \beta_{2} - 4700 \beta_{4} + 1735 \beta_{5} + 40463 \beta_{7} - 40463 \beta_{8} - 166 \beta_{9} + 1861 \beta_{11} - 1861 \beta_{12} + 318 \beta_{13} - 318 \beta_{14} ) q^{87}$$ $$+ ( 31295 + 8998 \beta_{1} + 2299 \beta_{2} - 6578 \beta_{3} - 3421 \beta_{4} - 1771 \beta_{5} - 45144 \beta_{6} + 12067 \beta_{7} - 27808 \beta_{8} - 1540 \beta_{9} - 1276 \beta_{10} + 1056 \beta_{11} - 1848 \beta_{12} - 110 \beta_{13} + 682 \beta_{14} + 1331 \beta_{15} ) q^{88}$$ $$+ ( -3047 - 3719 \beta_{1} + 2311 \beta_{2} + 3719 \beta_{4} + 56 \beta_{5} - 27170 \beta_{7} + 27170 \beta_{8} + 1222 \beta_{9} - 1419 \beta_{11} + 1419 \beta_{12} - 659 \beta_{13} + 659 \beta_{14} ) q^{89}$$ $$+ ( -18980 + 7093 \beta_{1} + 5452 \beta_{2} - 7093 \beta_{3} - 5452 \beta_{4} - 679 \beta_{5} + 13528 \beta_{6} - 5452 \beta_{7} - 47790 \beta_{8} + 1436 \beta_{9} + 1436 \beta_{10} - 690 \beta_{12} - 500 \beta_{13} + 679 \beta_{15} ) q^{90}$$ $$+ ( -44946 + 1339 \beta_{1} - 5297 \beta_{3} - 5297 \beta_{4} - 1499 \beta_{5} + 33942 \beta_{6} - 33942 \beta_{7} + 44946 \beta_{8} + 51 \beta_{9} - 597 \beta_{10} + 597 \beta_{11} + 51 \beta_{12} - 587 \beta_{13} + 912 \beta_{14} + 912 \beta_{15} ) q^{91}$$ $$+ ( -578 - 4064 \beta_{2} + 4064 \beta_{3} + 1326 \beta_{4} - 654 \beta_{5} - 4580 \beta_{6} - 578 \beta_{7} - 4064 \beta_{8} + 1496 \beta_{9} + 216 \beta_{10} - 1496 \beta_{11} + 286 \beta_{13} + 654 \beta_{14} + 286 \beta_{15} ) q^{92}$$ $$+ ( 4438 - 4438 \beta_{1} - 4438 \beta_{2} - 1853 \beta_{3} - 35318 \beta_{6} + 23244 \beta_{7} - 39756 \beta_{8} + 892 \beta_{10} - 2717 \beta_{11} + 892 \beta_{12} + 175 \beta_{14} - 960 \beta_{15} ) q^{93}$$ $$+ ( -947 + 947 \beta_{1} + 947 \beta_{2} + 1240 \beta_{3} + 18948 \beta_{6} + 54143 \beta_{7} + 19895 \beta_{8} + 1350 \beta_{10} - 1300 \beta_{11} + 1350 \beta_{12} + 307 \beta_{14} - 1389 \beta_{15} ) q^{94}$$ $$+ ( -2947 - 3165 \beta_{2} + 3165 \beta_{3} + 4478 \beta_{4} - 919 \beta_{5} + 18336 \beta_{6} - 2947 \beta_{7} - 3165 \beta_{8} + 1020 \beta_{9} + 2217 \beta_{10} - 1020 \beta_{11} + 908 \beta_{13} + 919 \beta_{14} + 908 \beta_{15} ) q^{95}$$ $$+ ( -17748 - 157 \beta_{1} + 1969 \beta_{3} + 1969 \beta_{4} + 355 \beta_{5} + 67967 \beta_{6} - 67967 \beta_{7} + 17748 \beta_{8} + 3596 \beta_{9} + 1512 \beta_{10} - 1512 \beta_{11} + 3596 \beta_{12} + 612 \beta_{13} + 257 \beta_{14} + 257 \beta_{15} ) q^{96}$$ $$+ ( 33313 - 1188 \beta_{1} + 5586 \beta_{2} + 1188 \beta_{3} - 5586 \beta_{4} - 198 \beta_{5} - 38899 \beta_{6} - 5586 \beta_{7} + 3126 \beta_{8} + 3046 \beta_{9} + 3046 \beta_{10} + 1596 \beta_{12} - 978 \beta_{13} + 198 \beta_{15} ) q^{97}$$ $$+ ( 49597 + 262 \beta_{1} - 1180 \beta_{2} - 262 \beta_{4} - 1403 \beta_{5} + 5758 \beta_{7} - 5758 \beta_{8} - 48 \beta_{9} - 1110 \beta_{11} + 1110 \beta_{12} - 1294 \beta_{13} + 1294 \beta_{14} ) q^{98}$$ $$+ ( 54518 - 6917 \beta_{1} - 3882 \beta_{2} + 6794 \beta_{3} + 4789 \beta_{4} - 293 \beta_{5} - 6321 \beta_{6} - 8675 \beta_{7} - 25899 \beta_{8} + 1535 \beta_{9} + 1444 \beta_{10} - 1517 \beta_{11} + 5006 \beta_{12} + 919 \beta_{13} + 107 \beta_{14} - 244 \beta_{15} ) q^{99}$$ $$+O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$16q$$ $$\mathstrut -\mathstrut q^{2}$$ $$\mathstrut -\mathstrut 24q^{3}$$ $$\mathstrut -\mathstrut 73q^{4}$$ $$\mathstrut -\mathstrut 10q^{5}$$ $$\mathstrut +\mathstrut 121q^{6}$$ $$\mathstrut +\mathstrut 196q^{7}$$ $$\mathstrut +\mathstrut 527q^{8}$$ $$\mathstrut -\mathstrut 530q^{9}$$ $$\mathstrut +\mathstrut O(q^{10})$$ $$16q$$ $$\mathstrut -\mathstrut q^{2}$$ $$\mathstrut -\mathstrut 24q^{3}$$ $$\mathstrut -\mathstrut 73q^{4}$$ $$\mathstrut -\mathstrut 10q^{5}$$ $$\mathstrut +\mathstrut 121q^{6}$$ $$\mathstrut +\mathstrut 196q^{7}$$ $$\mathstrut +\mathstrut 527q^{8}$$ $$\mathstrut -\mathstrut 530q^{9}$$ $$\mathstrut -\mathstrut 672q^{10}$$ $$\mathstrut -\mathstrut 692q^{11}$$ $$\mathstrut -\mathstrut 1562q^{12}$$ $$\mathstrut +\mathstrut 1162q^{13}$$ $$\mathstrut +\mathstrut 560q^{14}$$ $$\mathstrut +\mathstrut 1796q^{15}$$ $$\mathstrut +\mathstrut 6399q^{16}$$ $$\mathstrut -\mathstrut 22q^{17}$$ $$\mathstrut +\mathstrut 834q^{18}$$ $$\mathstrut -\mathstrut 3236q^{19}$$ $$\mathstrut -\mathstrut 5514q^{20}$$ $$\mathstrut -\mathstrut 7772q^{21}$$ $$\mathstrut -\mathstrut 13059q^{22}$$ $$\mathstrut -\mathstrut 10848q^{23}$$ $$\mathstrut +\mathstrut 13233q^{24}$$ $$\mathstrut +\mathstrut 15686q^{25}$$ $$\mathstrut +\mathstrut 16216q^{26}$$ $$\mathstrut +\mathstrut 22500q^{27}$$ $$\mathstrut +\mathstrut 8838q^{28}$$ $$\mathstrut +\mathstrut 13070q^{29}$$ $$\mathstrut -\mathstrut 22830q^{30}$$ $$\mathstrut -\mathstrut 14764q^{31}$$ $$\mathstrut -\mathstrut 58812q^{32}$$ $$\mathstrut -\mathstrut 48544q^{33}$$ $$\mathstrut -\mathstrut 26966q^{34}$$ $$\mathstrut +\mathstrut 43368q^{35}$$ $$\mathstrut +\mathstrut 63696q^{36}$$ $$\mathstrut +\mathstrut 4638q^{37}$$ $$\mathstrut +\mathstrut 68144q^{38}$$ $$\mathstrut +\mathstrut 20300q^{39}$$ $$\mathstrut +\mathstrut 52284q^{40}$$ $$\mathstrut -\mathstrut 14806q^{41}$$ $$\mathstrut -\mathstrut 69922q^{42}$$ $$\mathstrut -\mathstrut 24376q^{43}$$ $$\mathstrut -\mathstrut 104960q^{44}$$ $$\mathstrut -\mathstrut 97480q^{45}$$ $$\mathstrut +\mathstrut 50452q^{46}$$ $$\mathstrut +\mathstrut 40364q^{47}$$ $$\mathstrut +\mathstrut 30236q^{48}$$ $$\mathstrut +\mathstrut 32246q^{49}$$ $$\mathstrut +\mathstrut 10839q^{50}$$ $$\mathstrut +\mathstrut 75564q^{51}$$ $$\mathstrut -\mathstrut 80654q^{52}$$ $$\mathstrut -\mathstrut 11654q^{53}$$ $$\mathstrut +\mathstrut 43796q^{54}$$ $$\mathstrut +\mathstrut 7052q^{55}$$ $$\mathstrut +\mathstrut 70632q^{56}$$ $$\mathstrut -\mathstrut 40020q^{57}$$ $$\mathstrut +\mathstrut 10276q^{58}$$ $$\mathstrut +\mathstrut 70804q^{59}$$ $$\mathstrut +\mathstrut 46116q^{60}$$ $$\mathstrut -\mathstrut 31446q^{61}$$ $$\mathstrut -\mathstrut 153388q^{62}$$ $$\mathstrut -\mathstrut 7064q^{63}$$ $$\mathstrut +\mathstrut 17695q^{64}$$ $$\mathstrut -\mathstrut 41284q^{65}$$ $$\mathstrut +\mathstrut 48006q^{66}$$ $$\mathstrut -\mathstrut 64200q^{67}$$ $$\mathstrut -\mathstrut 94114q^{68}$$ $$\mathstrut -\mathstrut 62412q^{69}$$ $$\mathstrut -\mathstrut 103768q^{70}$$ $$\mathstrut -\mathstrut 184380q^{71}$$ $$\mathstrut -\mathstrut 14729q^{72}$$ $$\mathstrut -\mathstrut 1750q^{73}$$ $$\mathstrut +\mathstrut 306048q^{74}$$ $$\mathstrut +\mathstrut 246596q^{75}$$ $$\mathstrut +\mathstrut 174806q^{76}$$ $$\mathstrut +\mathstrut 384646q^{77}$$ $$\mathstrut +\mathstrut 360916q^{78}$$ $$\mathstrut +\mathstrut 24324q^{79}$$ $$\mathstrut -\mathstrut 386444q^{80}$$ $$\mathstrut -\mathstrut 259412q^{81}$$ $$\mathstrut -\mathstrut 316255q^{82}$$ $$\mathstrut -\mathstrut 46028q^{83}$$ $$\mathstrut -\mathstrut 271956q^{84}$$ $$\mathstrut +\mathstrut 63914q^{85}$$ $$\mathstrut +\mathstrut 22931q^{86}$$ $$\mathstrut -\mathstrut 47592q^{87}$$ $$\mathstrut +\mathstrut 211871q^{88}$$ $$\mathstrut +\mathstrut 148364q^{89}$$ $$\mathstrut -\mathstrut 347186q^{90}$$ $$\mathstrut -\mathstrut 258448q^{91}$$ $$\mathstrut -\mathstrut 58658q^{92}$$ $$\mathstrut -\mathstrut 387478q^{93}$$ $$\mathstrut -\mathstrut 55370q^{94}$$ $$\mathstrut -\mathstrut 4716q^{95}$$ $$\mathstrut +\mathstrut 328396q^{96}$$ $$\mathstrut +\mathstrut 484296q^{97}$$ $$\mathstrut +\mathstrut 743692q^{98}$$ $$\mathstrut +\mathstrut 724964q^{99}$$ $$\mathstrut +\mathstrut O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{16}\mathstrut -\mathstrut$$ $$5$$ $$x^{15}\mathstrut +\mathstrut$$ $$86$$ $$x^{14}\mathstrut -\mathstrut$$ $$146$$ $$x^{13}\mathstrut +\mathstrut$$ $$7205$$ $$x^{12}\mathstrut -\mathstrut$$ $$23732$$ $$x^{11}\mathstrut +\mathstrut$$ $$774165$$ $$x^{10}\mathstrut -\mathstrut$$ $$1228996$$ $$x^{9}\mathstrut +\mathstrut$$ $$44649817$$ $$x^{8}\mathstrut -\mathstrut$$ $$92720004$$ $$x^{7}\mathstrut +\mathstrut$$ $$943915592$$ $$x^{6}\mathstrut -\mathstrut$$ $$1892495035$$ $$x^{5}\mathstrut +\mathstrut$$ $$8611100939$$ $$x^{4}\mathstrut -\mathstrut$$ $$17760110698$$ $$x^{3}\mathstrut +\mathstrut$$ $$18348896492$$ $$x^{2}\mathstrut -\mathstrut$$ $$4224271656$$ $$x\mathstrut +\mathstrut$$ $$393784336$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$25\!\cdots\!53$$ $$\nu^{15}\mathstrut -\mathstrut$$ $$10\!\cdots\!45$$ $$\nu^{14}\mathstrut +\mathstrut$$ $$21\!\cdots\!82$$ $$\nu^{13}\mathstrut -\mathstrut$$ $$21\!\cdots\!94$$ $$\nu^{12}\mathstrut +\mathstrut$$ $$18\!\cdots\!97$$ $$\nu^{11}\mathstrut -\mathstrut$$ $$46\!\cdots\!28$$ $$\nu^{10}\mathstrut +\mathstrut$$ $$19\!\cdots\!45$$ $$\nu^{9}\mathstrut -\mathstrut$$ $$16\!\cdots\!76$$ $$\nu^{8}\mathstrut +\mathstrut$$ $$11\!\cdots\!17$$ $$\nu^{7}\mathstrut -\mathstrut$$ $$14\!\cdots\!44$$ $$\nu^{6}\mathstrut +\mathstrut$$ $$23\!\cdots\!48$$ $$\nu^{5}\mathstrut -\mathstrut$$ $$27\!\cdots\!55$$ $$\nu^{4}\mathstrut +\mathstrut$$ $$17\!\cdots\!07$$ $$\nu^{3}\mathstrut -\mathstrut$$ $$24\!\cdots\!30$$ $$\nu^{2}\mathstrut +\mathstrut$$ $$57\!\cdots\!92$$ $$\nu\mathstrut +\mathstrut$$ $$31\!\cdots\!52$$$$)/$$$$22\!\cdots\!92$$ $$\beta_{3}$$ $$=$$ $$($$$$-$$$$19\!\cdots\!81$$ $$\nu^{15}\mathstrut +\mathstrut$$ $$78\!\cdots\!11$$ $$\nu^{14}\mathstrut -\mathstrut$$ $$15\!\cdots\!68$$ $$\nu^{13}\mathstrut +\mathstrut$$ $$14\!\cdots\!06$$ $$\nu^{12}\mathstrut -\mathstrut$$ $$13\!\cdots\!13$$ $$\nu^{11}\mathstrut +\mathstrut$$ $$33\!\cdots\!06$$ $$\nu^{10}\mathstrut -\mathstrut$$ $$14\!\cdots\!09$$ $$\nu^{9}\mathstrut +\mathstrut$$ $$10\!\cdots\!54$$ $$\nu^{8}\mathstrut -\mathstrut$$ $$85\!\cdots\!69$$ $$\nu^{7}\mathstrut +\mathstrut$$ $$10\!\cdots\!06$$ $$\nu^{6}\mathstrut -\mathstrut$$ $$17\!\cdots\!68$$ $$\nu^{5}\mathstrut +\mathstrut$$ $$21\!\cdots\!31$$ $$\nu^{4}\mathstrut -\mathstrut$$ $$16\!\cdots\!73$$ $$\nu^{3}\mathstrut +\mathstrut$$ $$22\!\cdots\!08$$ $$\nu^{2}\mathstrut -\mathstrut$$ $$25\!\cdots\!84$$ $$\nu\mathstrut +\mathstrut$$ $$38\!\cdots\!44$$$$)/$$$$12\!\cdots\!56$$ $$\beta_{4}$$ $$=$$ $$($$$$21\!\cdots\!19$$ $$\nu^{15}\mathstrut -\mathstrut$$ $$81\!\cdots\!75$$ $$\nu^{14}\mathstrut +\mathstrut$$ $$17\!\cdots\!26$$ $$\nu^{13}\mathstrut -\mathstrut$$ $$89\!\cdots\!78$$ $$\nu^{12}\mathstrut +\mathstrut$$ $$15\!\cdots\!15$$ $$\nu^{11}\mathstrut -\mathstrut$$ $$32\!\cdots\!60$$ $$\nu^{10}\mathstrut +\mathstrut$$ $$16\!\cdots\!99$$ $$\nu^{9}\mathstrut -\mathstrut$$ $$58\!\cdots\!76$$ $$\nu^{8}\mathstrut +\mathstrut$$ $$94\!\cdots\!51$$ $$\nu^{7}\mathstrut -\mathstrut$$ $$83\!\cdots\!24$$ $$\nu^{6}\mathstrut +\mathstrut$$ $$18\!\cdots\!40$$ $$\nu^{5}\mathstrut -\mathstrut$$ $$18\!\cdots\!97$$ $$\nu^{4}\mathstrut +\mathstrut$$ $$15\!\cdots\!89$$ $$\nu^{3}\mathstrut -\mathstrut$$ $$20\!\cdots\!82$$ $$\nu^{2}\mathstrut +\mathstrut$$ $$17\!\cdots\!88$$ $$\nu\mathstrut -\mathstrut$$ $$37\!\cdots\!20$$$$)/$$$$12\!\cdots\!56$$ $$\beta_{5}$$ $$=$$ $$($$$$35\!\cdots\!61$$ $$\nu^{15}\mathstrut -\mathstrut$$ $$24\!\cdots\!13$$ $$\nu^{14}\mathstrut +\mathstrut$$ $$30\!\cdots\!94$$ $$\nu^{13}\mathstrut -\mathstrut$$ $$87\!\cdots\!26$$ $$\nu^{12}\mathstrut +\mathstrut$$ $$23\!\cdots\!01$$ $$\nu^{11}\mathstrut -\mathstrut$$ $$12\!\cdots\!56$$ $$\nu^{10}\mathstrut +\mathstrut$$ $$26\!\cdots\!97$$ $$\nu^{9}\mathstrut -\mathstrut$$ $$82\!\cdots\!28$$ $$\nu^{8}\mathstrut +\mathstrut$$ $$14\!\cdots\!61$$ $$\nu^{7}\mathstrut -\mathstrut$$ $$54\!\cdots\!44$$ $$\nu^{6}\mathstrut +\mathstrut$$ $$25\!\cdots\!52$$ $$\nu^{5}\mathstrut -\mathstrut$$ $$78\!\cdots\!51$$ $$\nu^{4}\mathstrut +\mathstrut$$ $$18\!\cdots\!19$$ $$\nu^{3}\mathstrut -\mathstrut$$ $$18\!\cdots\!62$$ $$\nu^{2}\mathstrut +\mathstrut$$ $$42\!\cdots\!60$$ $$\nu\mathstrut +\mathstrut$$ $$32\!\cdots\!44$$$$)/$$$$12\!\cdots\!56$$ $$\beta_{6}$$ $$=$$ $$($$$$-$$$$48\!\cdots\!69$$ $$\nu^{15}\mathstrut +\mathstrut$$ $$23\!\cdots\!05$$ $$\nu^{14}\mathstrut -\mathstrut$$ $$41\!\cdots\!70$$ $$\nu^{13}\mathstrut +\mathstrut$$ $$62\!\cdots\!58$$ $$\nu^{12}\mathstrut -\mathstrut$$ $$34\!\cdots\!05$$ $$\nu^{11}\mathstrut +\mathstrut$$ $$10\!\cdots\!80$$ $$\nu^{10}\mathstrut -\mathstrut$$ $$37\!\cdots\!37$$ $$\nu^{9}\mathstrut +\mathstrut$$ $$51\!\cdots\!52$$ $$\nu^{8}\mathstrut -\mathstrut$$ $$21\!\cdots\!25$$ $$\nu^{7}\mathstrut +\mathstrut$$ $$40\!\cdots\!84$$ $$\nu^{6}\mathstrut -\mathstrut$$ $$45\!\cdots\!80$$ $$\nu^{5}\mathstrut +\mathstrut$$ $$82\!\cdots\!39$$ $$\nu^{4}\mathstrut -\mathstrut$$ $$40\!\cdots\!95$$ $$\nu^{3}\mathstrut +\mathstrut$$ $$78\!\cdots\!74$$ $$\nu^{2}\mathstrut -\mathstrut$$ $$78\!\cdots\!04$$ $$\nu\mathstrut +\mathstrut$$ $$18\!\cdots\!36$$$$)/$$$$99\!\cdots\!92$$ $$\beta_{7}$$ $$=$$ $$($$$$-$$$$78\!\cdots\!15$$ $$\nu^{15}\mathstrut +\mathstrut$$ $$37\!\cdots\!17$$ $$\nu^{14}\mathstrut -\mathstrut$$ $$67\!\cdots\!40$$ $$\nu^{13}\mathstrut +\mathstrut$$ $$10\!\cdots\!58$$ $$\nu^{12}\mathstrut -\mathstrut$$ $$56\!\cdots\!79$$ $$\nu^{11}\mathstrut +\mathstrut$$ $$17\!\cdots\!50$$ $$\nu^{10}\mathstrut -\mathstrut$$ $$60\!\cdots\!55$$ $$\nu^{9}\mathstrut +\mathstrut$$ $$83\!\cdots\!22$$ $$\nu^{8}\mathstrut -\mathstrut$$ $$35\!\cdots\!23$$ $$\nu^{7}\mathstrut +\mathstrut$$ $$65\!\cdots\!78$$ $$\nu^{6}\mathstrut -\mathstrut$$ $$73\!\cdots\!12$$ $$\nu^{5}\mathstrut +\mathstrut$$ $$13\!\cdots\!45$$ $$\nu^{4}\mathstrut -\mathstrut$$ $$66\!\cdots\!31$$ $$\nu^{3}\mathstrut +\mathstrut$$ $$12\!\cdots\!72$$ $$\nu^{2}\mathstrut -\mathstrut$$ $$12\!\cdots\!56$$ $$\nu\mathstrut +\mathstrut$$ $$19\!\cdots\!24$$$$)/$$$$99\!\cdots\!92$$ $$\beta_{8}$$ $$=$$ $$($$$$-$$$$39\!\cdots\!39$$ $$\nu^{15}\mathstrut +\mathstrut$$ $$19\!\cdots\!74$$ $$\nu^{14}\mathstrut -\mathstrut$$ $$33\!\cdots\!03$$ $$\nu^{13}\mathstrut +\mathstrut$$ $$51\!\cdots\!06$$ $$\nu^{12}\mathstrut -\mathstrut$$ $$28\!\cdots\!49$$ $$\nu^{11}\mathstrut +\mathstrut$$ $$88\!\cdots\!15$$ $$\nu^{10}\mathstrut -\mathstrut$$ $$30\!\cdots\!89$$ $$\nu^{9}\mathstrut +\mathstrut$$ $$42\!\cdots\!75$$ $$\nu^{8}\mathstrut -\mathstrut$$ $$17\!\cdots\!49$$ $$\nu^{7}\mathstrut +\mathstrut$$ $$33\!\cdots\!27$$ $$\nu^{6}\mathstrut -\mathstrut$$ $$37\!\cdots\!42$$ $$\nu^{5}\mathstrut +\mathstrut$$ $$67\!\cdots\!77$$ $$\nu^{4}\mathstrut -\mathstrut$$ $$33\!\cdots\!50$$ $$\nu^{3}\mathstrut +\mathstrut$$ $$63\!\cdots\!29$$ $$\nu^{2}\mathstrut -\mathstrut$$ $$63\!\cdots\!60$$ $$\nu\mathstrut +\mathstrut$$ $$65\!\cdots\!40$$$$)/$$$$49\!\cdots\!96$$ $$\beta_{9}$$ $$=$$ $$($$$$-$$$$47\!\cdots\!26$$ $$\nu^{15}\mathstrut +\mathstrut$$ $$18\!\cdots\!92$$ $$\nu^{14}\mathstrut -\mathstrut$$ $$38\!\cdots\!80$$ $$\nu^{13}\mathstrut +\mathstrut$$ $$23\!\cdots\!61$$ $$\nu^{12}\mathstrut -\mathstrut$$ $$33\!\cdots\!19$$ $$\nu^{11}\mathstrut +\mathstrut$$ $$74\!\cdots\!53$$ $$\nu^{10}\mathstrut -\mathstrut$$ $$35\!\cdots\!87$$ $$\nu^{9}\mathstrut +\mathstrut$$ $$16\!\cdots\!15$$ $$\nu^{8}\mathstrut -\mathstrut$$ $$20\!\cdots\!25$$ $$\nu^{7}\mathstrut +\mathstrut$$ $$20\!\cdots\!57$$ $$\nu^{6}\mathstrut -\mathstrut$$ $$40\!\cdots\!17$$ $$\nu^{5}\mathstrut +\mathstrut$$ $$42\!\cdots\!45$$ $$\nu^{4}\mathstrut -\mathstrut$$ $$32\!\cdots\!49$$ $$\nu^{3}\mathstrut +\mathstrut$$ $$45\!\cdots\!90$$ $$\nu^{2}\mathstrut -\mathstrut$$ $$10\!\cdots\!24$$ $$\nu\mathstrut -\mathstrut$$ $$20\!\cdots\!72$$$$)/$$$$12\!\cdots\!56$$ $$\beta_{10}$$ $$=$$ $$($$$$-$$$$28\!\cdots\!59$$ $$\nu^{15}\mathstrut +\mathstrut$$ $$15\!\cdots\!16$$ $$\nu^{14}\mathstrut -\mathstrut$$ $$24\!\cdots\!81$$ $$\nu^{13}\mathstrut +\mathstrut$$ $$53\!\cdots\!12$$ $$\nu^{12}\mathstrut -\mathstrut$$ $$20\!\cdots\!98$$ $$\nu^{11}\mathstrut +\mathstrut$$ $$78\!\cdots\!54$$ $$\nu^{10}\mathstrut -\mathstrut$$ $$22\!\cdots\!48$$ $$\nu^{9}\mathstrut +\mathstrut$$ $$46\!\cdots\!06$$ $$\nu^{8}\mathstrut -\mathstrut$$ $$12\!\cdots\!58$$ $$\nu^{7}\mathstrut +\mathstrut$$ $$33\!\cdots\!88$$ $$\nu^{6}\mathstrut -\mathstrut$$ $$27\!\cdots\!13$$ $$\nu^{5}\mathstrut +\mathstrut$$ $$67\!\cdots\!78$$ $$\nu^{4}\mathstrut -\mathstrut$$ $$25\!\cdots\!39$$ $$\nu^{3}\mathstrut +\mathstrut$$ $$61\!\cdots\!16$$ $$\nu^{2}\mathstrut -\mathstrut$$ $$67\!\cdots\!28$$ $$\nu\mathstrut +\mathstrut$$ $$15\!\cdots\!04$$$$)/$$$$49\!\cdots\!96$$ $$\beta_{11}$$ $$=$$ $$($$$$-$$$$64\!\cdots\!06$$ $$\nu^{15}\mathstrut +\mathstrut$$ $$32\!\cdots\!06$$ $$\nu^{14}\mathstrut -\mathstrut$$ $$55\!\cdots\!37$$ $$\nu^{13}\mathstrut +\mathstrut$$ $$98\!\cdots\!99$$ $$\nu^{12}\mathstrut -\mathstrut$$ $$46\!\cdots\!36$$ $$\nu^{11}\mathstrut +\mathstrut$$ $$15\!\cdots\!65$$ $$\nu^{10}\mathstrut -\mathstrut$$ $$50\!\cdots\!34$$ $$\nu^{9}\mathstrut +\mathstrut$$ $$83\!\cdots\!07$$ $$\nu^{8}\mathstrut -\mathstrut$$ $$28\!\cdots\!54$$ $$\nu^{7}\mathstrut +\mathstrut$$ $$62\!\cdots\!65$$ $$\nu^{6}\mathstrut -\mathstrut$$ $$60\!\cdots\!50$$ $$\nu^{5}\mathstrut +\mathstrut$$ $$12\!\cdots\!73$$ $$\nu^{4}\mathstrut -\mathstrut$$ $$55\!\cdots\!29$$ $$\nu^{3}\mathstrut +\mathstrut$$ $$11\!\cdots\!32$$ $$\nu^{2}\mathstrut -\mathstrut$$ $$11\!\cdots\!44$$ $$\nu\mathstrut +\mathstrut$$ $$17\!\cdots\!80$$$$)/$$$$49\!\cdots\!96$$ $$\beta_{12}$$ $$=$$ $$($$$$-$$$$76\!\cdots\!69$$ $$\nu^{15}\mathstrut +\mathstrut$$ $$37\!\cdots\!46$$ $$\nu^{14}\mathstrut -\mathstrut$$ $$65\!\cdots\!41$$ $$\nu^{13}\mathstrut +\mathstrut$$ $$10\!\cdots\!29$$ $$\nu^{12}\mathstrut -\mathstrut$$ $$55\!\cdots\!68$$ $$\nu^{11}\mathstrut +\mathstrut$$ $$17\!\cdots\!87$$ $$\nu^{10}\mathstrut -\mathstrut$$ $$59\!\cdots\!38$$ $$\nu^{9}\mathstrut +\mathstrut$$ $$89\!\cdots\!93$$ $$\nu^{8}\mathstrut -\mathstrut$$ $$34\!\cdots\!52$$ $$\nu^{7}\mathstrut +\mathstrut$$ $$68\!\cdots\!17$$ $$\nu^{6}\mathstrut -\mathstrut$$ $$71\!\cdots\!25$$ $$\nu^{5}\mathstrut +\mathstrut$$ $$13\!\cdots\!33$$ $$\nu^{4}\mathstrut -\mathstrut$$ $$63\!\cdots\!43$$ $$\nu^{3}\mathstrut +\mathstrut$$ $$12\!\cdots\!24$$ $$\nu^{2}\mathstrut -\mathstrut$$ $$11\!\cdots\!32$$ $$\nu\mathstrut +\mathstrut$$ $$11\!\cdots\!76$$$$)/$$$$49\!\cdots\!96$$ $$\beta_{13}$$ $$=$$ $$($$$$10\!\cdots\!27$$ $$\nu^{15}\mathstrut -\mathstrut$$ $$50\!\cdots\!23$$ $$\nu^{14}\mathstrut +\mathstrut$$ $$90\!\cdots\!94$$ $$\nu^{13}\mathstrut -\mathstrut$$ $$13\!\cdots\!70$$ $$\nu^{12}\mathstrut +\mathstrut$$ $$76\!\cdots\!71$$ $$\nu^{11}\mathstrut -\mathstrut$$ $$23\!\cdots\!84$$ $$\nu^{10}\mathstrut +\mathstrut$$ $$81\!\cdots\!07$$ $$\nu^{9}\mathstrut -\mathstrut$$ $$11\!\cdots\!44$$ $$\nu^{8}\mathstrut +\mathstrut$$ $$47\!\cdots\!31$$ $$\nu^{7}\mathstrut -\mathstrut$$ $$88\!\cdots\!64$$ $$\nu^{6}\mathstrut +\mathstrut$$ $$98\!\cdots\!92$$ $$\nu^{5}\mathstrut -\mathstrut$$ $$17\!\cdots\!53$$ $$\nu^{4}\mathstrut +\mathstrut$$ $$88\!\cdots\!33$$ $$\nu^{3}\mathstrut -\mathstrut$$ $$17\!\cdots\!58$$ $$\nu^{2}\mathstrut +\mathstrut$$ $$17\!\cdots\!84$$ $$\nu\mathstrut -\mathstrut$$ $$17\!\cdots\!96$$$$)/$$$$49\!\cdots\!96$$ $$\beta_{14}$$ $$=$$ $$($$$$26\!\cdots\!26$$ $$\nu^{15}\mathstrut -\mathstrut$$ $$12\!\cdots\!10$$ $$\nu^{14}\mathstrut +\mathstrut$$ $$22\!\cdots\!01$$ $$\nu^{13}\mathstrut -\mathstrut$$ $$34\!\cdots\!20$$ $$\nu^{12}\mathstrut +\mathstrut$$ $$19\!\cdots\!44$$ $$\nu^{11}\mathstrut -\mathstrut$$ $$59\!\cdots\!15$$ $$\nu^{10}\mathstrut +\mathstrut$$ $$20\!\cdots\!54$$ $$\nu^{9}\mathstrut -\mathstrut$$ $$28\!\cdots\!26$$ $$\nu^{8}\mathstrut +\mathstrut$$ $$11\!\cdots\!30$$ $$\nu^{7}\mathstrut -\mathstrut$$ $$22\!\cdots\!41$$ $$\nu^{6}\mathstrut +\mathstrut$$ $$24\!\cdots\!29$$ $$\nu^{5}\mathstrut -\mathstrut$$ $$45\!\cdots\!58$$ $$\nu^{4}\mathstrut +\mathstrut$$ $$22\!\cdots\!92$$ $$\nu^{3}\mathstrut -\mathstrut$$ $$42\!\cdots\!36$$ $$\nu^{2}\mathstrut +\mathstrut$$ $$42\!\cdots\!31$$ $$\nu\mathstrut -\mathstrut$$ $$64\!\cdots\!38$$$$)/$$$$12\!\cdots\!99$$ $$\beta_{15}$$ $$=$$ $$($$$$-$$$$27\!\cdots\!65$$ $$\nu^{15}\mathstrut +\mathstrut$$ $$13\!\cdots\!43$$ $$\nu^{14}\mathstrut -\mathstrut$$ $$23\!\cdots\!00$$ $$\nu^{13}\mathstrut +\mathstrut$$ $$35\!\cdots\!02$$ $$\nu^{12}\mathstrut -\mathstrut$$ $$20\!\cdots\!93$$ $$\nu^{11}\mathstrut +\mathstrut$$ $$62\!\cdots\!50$$ $$\nu^{10}\mathstrut -\mathstrut$$ $$21\!\cdots\!61$$ $$\nu^{9}\mathstrut +\mathstrut$$ $$29\!\cdots\!86$$ $$\nu^{8}\mathstrut -\mathstrut$$ $$12\!\cdots\!57$$ $$\nu^{7}\mathstrut +\mathstrut$$ $$23\!\cdots\!74$$ $$\nu^{6}\mathstrut -\mathstrut$$ $$26\!\cdots\!24$$ $$\nu^{5}\mathstrut +\mathstrut$$ $$47\!\cdots\!51$$ $$\nu^{4}\mathstrut -\mathstrut$$ $$23\!\cdots\!61$$ $$\nu^{3}\mathstrut +\mathstrut$$ $$45\!\cdots\!76$$ $$\nu^{2}\mathstrut -\mathstrut$$ $$45\!\cdots\!28$$ $$\nu\mathstrut +\mathstrut$$ $$68\!\cdots\!96$$$$)/$$$$81\!\cdots\!36$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{15}\mathstrut -\mathstrut$$ $$43$$ $$\beta_{8}\mathstrut +\mathstrut$$ $$\beta_{7}\mathstrut -\mathstrut$$ $$2$$ $$\beta_{6}\mathstrut -\mathstrut$$ $$\beta_{5}\mathstrut +\mathstrut$$ $$\beta_{4}\mathstrut -\mathstrut$$ $$\beta_{2}\mathstrut +\mathstrut$$ $$3$$ $$\nu^{3}$$ $$=$$ $$-$$$$2$$ $$\beta_{14}\mathstrut -\mathstrut$$ $$4$$ $$\beta_{12}\mathstrut +\mathstrut$$ $$4$$ $$\beta_{11}\mathstrut -\mathstrut$$ $$4$$ $$\beta_{10}\mathstrut -\mathstrut$$ $$12$$ $$\beta_{8}\mathstrut +\mathstrut$$ $$34$$ $$\beta_{7}\mathstrut -\mathstrut$$ $$28$$ $$\beta_{6}\mathstrut -\mathstrut$$ $$77$$ $$\beta_{3}\mathstrut +\mathstrut$$ $$16$$ $$\beta_{2}\mathstrut +\mathstrut$$ $$16$$ $$\beta_{1}\mathstrut -\mathstrut$$ $$16$$ $$\nu^{4}$$ $$=$$ $$2$$ $$\beta_{15}\mathstrut -\mathstrut$$ $$87$$ $$\beta_{14}\mathstrut +\mathstrut$$ $$2$$ $$\beta_{13}\mathstrut +\mathstrut$$ $$4$$ $$\beta_{11}\mathstrut -\mathstrut$$ $$12$$ $$\beta_{10}\mathstrut -\mathstrut$$ $$4$$ $$\beta_{9}\mathstrut +\mathstrut$$ $$135$$ $$\beta_{8}\mathstrut -\mathstrut$$ $$833$$ $$\beta_{7}\mathstrut -\mathstrut$$ $$2576$$ $$\beta_{6}\mathstrut +\mathstrut$$ $$87$$ $$\beta_{5}\mathstrut -\mathstrut$$ $$107$$ $$\beta_{4}\mathstrut -\mathstrut$$ $$135$$ $$\beta_{3}\mathstrut +\mathstrut$$ $$135$$ $$\beta_{2}\mathstrut -\mathstrut$$ $$833$$ $$\nu^{5}$$ $$=$$ $$-$$$$14$$ $$\beta_{14}\mathstrut +\mathstrut$$ $$14$$ $$\beta_{13}\mathstrut +\mathstrut$$ $$336$$ $$\beta_{12}\mathstrut -\mathstrut$$ $$336$$ $$\beta_{11}\mathstrut +\mathstrut$$ $$16$$ $$\beta_{9}\mathstrut +\mathstrut$$ $$9007$$ $$\beta_{8}\mathstrut -\mathstrut$$ $$9007$$ $$\beta_{7}\mathstrut +\mathstrut$$ $$262$$ $$\beta_{5}\mathstrut +\mathstrut$$ $$1934$$ $$\beta_{4}\mathstrut +\mathstrut$$ $$4483$$ $$\beta_{2}\mathstrut -\mathstrut$$ $$1934$$ $$\beta_{1}\mathstrut -\mathstrut$$ $$3705$$ $$\nu^{6}$$ $$=$$ $$-$$$$724$$ $$\beta_{15}\mathstrut -\mathstrut$$ $$724$$ $$\beta_{14}\mathstrut +\mathstrut$$ $$7321$$ $$\beta_{13}\mathstrut +\mathstrut$$ $$1008$$ $$\beta_{12}\mathstrut -\mathstrut$$ $$1608$$ $$\beta_{11}\mathstrut +\mathstrut$$ $$1608$$ $$\beta_{10}\mathstrut +\mathstrut$$ $$1008$$ $$\beta_{9}\mathstrut +\mathstrut$$ $$272598$$ $$\beta_{8}\mathstrut -\mathstrut$$ $$191468$$ $$\beta_{7}\mathstrut +\mathstrut$$ $$191468$$ $$\beta_{6}\mathstrut +\mathstrut$$ $$8045$$ $$\beta_{5}\mathstrut +\mathstrut$$ $$14617$$ $$\beta_{4}\mathstrut +\mathstrut$$ $$14617$$ $$\beta_{3}\mathstrut -\mathstrut$$ $$5736$$ $$\beta_{1}\mathstrut -\mathstrut$$ $$272598$$ $$\nu^{7}$$ $$=$$ $$-$$$$4252$$ $$\beta_{15}\mathstrut +\mathstrut$$ $$25030$$ $$\beta_{13}\mathstrut +\mathstrut$$ $$3904$$ $$\beta_{12}\mathstrut +\mathstrut$$ $$25788$$ $$\beta_{10}\mathstrut +\mathstrut$$ $$25788$$ $$\beta_{9}\mathstrut -\mathstrut$$ $$282911$$ $$\beta_{8}\mathstrut +\mathstrut$$ $$353785$$ $$\beta_{7}\mathstrut +\mathstrut$$ $$518476$$ $$\beta_{6}\mathstrut +\mathstrut$$ $$4252$$ $$\beta_{5}\mathstrut +\mathstrut$$ $$353785$$ $$\beta_{4}\mathstrut +\mathstrut$$ $$549413$$ $$\beta_{3}\mathstrut -\mathstrut$$ $$353785$$ $$\beta_{2}\mathstrut -\mathstrut$$ $$549413$$ $$\beta_{1}\mathstrut -\mathstrut$$ $$164691$$ $$\nu^{8}$$ $$=$$ $$-$$$$622115$$ $$\beta_{15}\mathstrut +\mathstrut$$ $$105294$$ $$\beta_{14}\mathstrut -\mathstrut$$ $$57324$$ $$\beta_{12}\mathstrut +\mathstrut$$ $$161348$$ $$\beta_{11}\mathstrut -\mathstrut$$ $$57324$$ $$\beta_{10}\mathstrut +\mathstrut$$ $$6672519$$ $$\beta_{8}\mathstrut +\mathstrut$$ $$16558939$$ $$\beta_{7}\mathstrut +\mathstrut$$ $$8139610$$ $$\beta_{6}\mathstrut +\mathstrut$$ $$824020$$ $$\beta_{3}\mathstrut -\mathstrut$$ $$1467091$$ $$\beta_{2}\mathstrut -\mathstrut$$ $$1467091$$ $$\beta_{1}\mathstrut +\mathstrut$$ $$1467091$$ $$\nu^{9}$$ $$=$$ $$-$$$$2363932$$ $$\beta_{15}\mathstrut +\mathstrut$$ $$710642$$ $$\beta_{14}\mathstrut -\mathstrut$$ $$2363932$$ $$\beta_{13}\mathstrut +\mathstrut$$ $$2009960$$ $$\beta_{11}\mathstrut -\mathstrut$$ $$1484760$$ $$\beta_{10}\mathstrut -\mathstrut$$ $$2009960$$ $$\beta_{9}\mathstrut -\mathstrut$$ $$18818226$$ $$\beta_{8}\mathstrut +\mathstrut$$ $$72285022$$ $$\beta_{7}\mathstrut -\mathstrut$$ $$33616894$$ $$\beta_{6}\mathstrut -\mathstrut$$ $$710642$$ $$\beta_{5}\mathstrut -\mathstrut$$ $$28866391$$ $$\beta_{4}\mathstrut +\mathstrut$$ $$18818226$$ $$\beta_{3}\mathstrut -\mathstrut$$ $$18818226$$ $$\beta_{2}\mathstrut +\mathstrut$$ $$72285022$$ $$\nu^{10}$$ $$=$$ $$53543269$$ $$\beta_{14}\mathstrut -\mathstrut$$ $$53543269$$ $$\beta_{13}\mathstrut +\mathstrut$$ $$4603200$$ $$\beta_{12}\mathstrut -\mathstrut$$ $$4603200$$ $$\beta_{11}\mathstrut -\mathstrut$$ $$9980928$$ $$\beta_{9}\mathstrut -\mathstrut$$ $$738647557$$ $$\beta_{8}\mathstrut +\mathstrut$$ $$738647557$$ $$\beta_{7}\mathstrut -\mathstrut$$ $$65728725$$ $$\beta_{5}\mathstrut -\mathstrut$$ $$142452645$$ $$\beta_{4}\mathstrut +\mathstrut$$ $$40578149$$ $$\beta_{2}\mathstrut +\mathstrut$$ $$142452645$$ $$\beta_{1}\mathstrut +\mathstrut$$ $$1965217305$$ $$\nu^{11}$$ $$=$$ $$217581914$$ $$\beta_{15}\mathstrut +\mathstrut$$ $$217581914$$ $$\beta_{14}\mathstrut -\mathstrut$$ $$94872624$$ $$\beta_{13}\mathstrut -\mathstrut$$ $$58722752$$ $$\beta_{12}\mathstrut -\mathstrut$$ $$102105300$$ $$\beta_{11}\mathstrut +\mathstrut$$ $$102105300$$ $$\beta_{10}\mathstrut -\mathstrut$$ $$58722752$$ $$\beta_{9}\mathstrut -\mathstrut$$ $$2995735314$$ $$\beta_{8}\mathstrut -\mathstrut$$ $$2280079802$$ $$\beta_{7}\mathstrut +\mathstrut$$ $$2280079802$$ $$\beta_{6}\mathstrut -\mathstrut$$ $$312454538$$ $$\beta_{5}\mathstrut -\mathstrut$$ $$1774094784$$ $$\beta_{4}\mathstrut -\mathstrut$$ $$1774094784$$ $$\beta_{3}\mathstrut +\mathstrut$$ $$4176840669$$ $$\beta_{1}\mathstrut +\mathstrut$$ $$2995735314$$ $$\nu^{12}$$ $$=$$ $$4657355935$$ $$\beta_{15}\mathstrut -\mathstrut$$ $$1286346842$$ $$\beta_{13}\mathstrut -\mathstrut$$ $$929050408$$ $$\beta_{12}\mathstrut +\mathstrut$$ $$330009108$$ $$\beta_{10}\mathstrut +\mathstrut$$ $$330009108$$ $$\beta_{9}\mathstrut -\mathstrut$$ $$176806829869$$ $$\beta_{8}\mathstrut +\mathstrut$$ $$2019861251$$ $$\beta_{7}\mathstrut -\mathstrut$$ $$73142895322$$ $$\beta_{6}\mathstrut -\mathstrut$$ $$4657355935$$ $$\beta_{5}\mathstrut +\mathstrut$$ $$2019861251$$ $$\beta_{4}\mathstrut -\mathstrut$$ $$11596224332$$ $$\beta_{3}\mathstrut -\mathstrut$$ $$2019861251$$ $$\beta_{2}\mathstrut +\mathstrut$$ $$11596224332$$ $$\beta_{1}\mathstrut +\mathstrut$$ $$75162756573$$ $$\nu^{13}$$ $$=$$ $$11293502550$$ $$\beta_{15}\mathstrut -\mathstrut$$ $$19835222816$$ $$\beta_{14}\mathstrut -\mathstrut$$ $$13154027264$$ $$\beta_{12}\mathstrut +\mathstrut$$ $$7079589488$$ $$\beta_{11}\mathstrut -\mathstrut$$ $$13154027264$$ $$\beta_{10}\mathstrut -\mathstrut$$ $$344191010582$$ $$\beta_{8}\mathstrut -\mathstrut$$ $$30939301040$$ $$\beta_{7}\mathstrut -\mathstrut$$ $$509884005868$$ $$\beta_{6}\mathstrut -\mathstrut$$ $$368505500737$$ $$\beta_{3}\mathstrut +\mathstrut$$ $$165692995286$$ $$\beta_{2}\mathstrut +\mathstrut$$ $$165692995286$$ $$\beta_{1}\mathstrut -\mathstrut$$ $$165692995286$$ $$\nu^{14}$$ $$=$$ $$129838068780$$ $$\beta_{15}\mathstrut -\mathstrut$$ $$408574340305$$ $$\beta_{14}\mathstrut +\mathstrut$$ $$129838068780$$ $$\beta_{13}\mathstrut +\mathstrut$$ $$21012853800$$ $$\beta_{11}\mathstrut -\mathstrut$$ $$106428182840$$ $$\beta_{10}\mathstrut -\mathstrut$$ $$21012853800$$ $$\beta_{9}\mathstrut +\mathstrut$$ $$1291350775377$$ $$\beta_{8}\mathstrut -\mathstrut$$ $$8094359895355$$ $$\beta_{7}\mathstrut -\mathstrut$$ $$8541029006908$$ $$\beta_{6}\mathstrut +\mathstrut$$ $$408574340305$$ $$\beta_{5}\mathstrut -\mathstrut$$ $$37642698937$$ $$\beta_{4}\mathstrut -\mathstrut$$ $$1291350775377$$ $$\beta_{3}\mathstrut +\mathstrut$$ $$1291350775377$$ $$\beta_{2}\mathstrut -\mathstrut$$ $$8094359895355$$ $$\nu^{15}$$ $$=$$ $$-$$$$1256105108580$$ $$\beta_{14}\mathstrut +\mathstrut$$ $$1256105108580$$ $$\beta_{13}\mathstrut +\mathstrut$$ $$1093932232300$$ $$\beta_{12}\mathstrut -\mathstrut$$ $$1093932232300$$ $$\beta_{11}\mathstrut +\mathstrut$$ $$604767604160$$ $$\beta_{9}\mathstrut +\mathstrut$$ $$66067478443437$$ $$\beta_{8}\mathstrut -\mathstrut$$ $$66067478443437$$ $$\beta_{7}\mathstrut +\mathstrut$$ $$3060061374962$$ $$\beta_{5}\mathstrut +\mathstrut$$ $$15404294453876$$ $$\beta_{4}\mathstrut +\mathstrut$$ $$17305426183777$$ $$\beta_{2}\mathstrut -\mathstrut$$ $$15404294453876$$ $$\beta_{1}\mathstrut -\mathstrut$$ $$60157769831531$$

## Character Values

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

 $$n$$ $$2$$ $$\chi(n)$$ $$-\beta_{6}$$

## 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
 2.73971 + 8.43195i 1.27432 + 3.92196i −1.19154 − 3.66717i −2.13151 − 6.56011i 2.73971 − 8.43195i 1.27432 − 3.92196i −1.19154 + 3.66717i −2.13151 + 6.56011i 7.81150 + 5.67539i 1.13051 + 0.821361i 0.136182 + 0.0989419i −7.26917 − 5.28136i 7.81150 − 5.67539i 1.13051 − 0.821361i 0.136182 − 0.0989419i −7.26917 + 5.28136i
−6.36363 + 4.62345i −5.98507 18.4202i 9.23097 28.4100i −59.3249 43.1020i 123.251 + 89.5474i −51.6143 + 158.852i −5.17243 15.9191i −106.890 + 77.6601i 576.801
3.2 −2.52720 + 1.83612i 4.03595 + 12.4214i −6.87313 + 21.1533i 23.3906 + 16.9942i −33.0068 23.9808i 27.3683 84.2310i −52.3600 161.147i 58.5894 42.5677i −90.3160
3.3 3.92850 2.85422i −7.68717 23.6587i −2.60202 + 8.00819i 68.6567 + 49.8820i −97.7261 71.0022i 19.5777 60.2541i 60.6528 + 186.670i −304.049 + 220.905i 412.092
3.4 6.38938 4.64216i 3.63629 + 11.1914i 9.38602 28.8872i −51.9930 37.7751i 75.1856 + 54.6256i −33.5385 + 103.221i 3.96879 + 12.2147i 84.5674 61.4418i −507.561
4.1 −6.36363 4.62345i −5.98507 + 18.4202i 9.23097 + 28.4100i −59.3249 + 43.1020i 123.251 89.5474i −51.6143 158.852i −5.17243 + 15.9191i −106.890 77.6601i 576.801
4.2 −2.52720 1.83612i 4.03595 12.4214i −6.87313 21.1533i 23.3906 16.9942i −33.0068 + 23.9808i 27.3683 + 84.2310i −52.3600 + 161.147i 58.5894 + 42.5677i −90.3160
4.3 3.92850 + 2.85422i −7.68717 + 23.6587i −2.60202 8.00819i 68.6567 49.8820i −97.7261 + 71.0022i 19.5777 + 60.2541i 60.6528 186.670i −304.049 220.905i 412.092
4.4 6.38938 + 4.64216i 3.63629 11.1914i 9.38602 + 28.8872i −51.9930 + 37.7751i 75.1856 54.6256i −33.5385 103.221i 3.96879 12.2147i 84.5674 + 61.4418i −507.561
5.1 −3.29275 10.1340i 2.26410 + 1.64496i −65.9678 + 47.9284i 20.4388 62.9043i 9.21501 28.3609i 58.3278 42.3777i 427.066 + 310.282i −72.6709 223.658i −704.774
5.2 −0.740832 2.28005i −21.7558 15.8065i 21.2388 15.4309i −5.23817 + 16.1214i −19.9222 + 61.3143i 87.7560 63.7585i −112.982 82.0864i 148.378 + 456.662i 40.6382
5.3 −0.361034 1.11115i 12.7716 + 9.27912i 24.7842 18.0068i 1.31139 4.03604i 5.69949 17.5412i −146.551 + 106.476i −59.2025 43.0132i 1.92087 + 5.91183i −4.95810
5.4 2.46756 + 7.59437i 0.720120 + 0.523198i −25.6970 + 18.6700i −2.24156 + 6.89880i −2.19642 + 6.75988i 136.674 99.2995i 1.52919 + 1.11102i −74.8463 230.353i −57.9232
9.1 −3.29275 + 10.1340i 2.26410 1.64496i −65.9678 47.9284i 20.4388 + 62.9043i 9.21501 + 28.3609i 58.3278 + 42.3777i 427.066 310.282i −72.6709 + 223.658i −704.774
9.2 −0.740832 + 2.28005i −21.7558 + 15.8065i 21.2388 + 15.4309i −5.23817 16.1214i −19.9222 61.3143i 87.7560 + 63.7585i −112.982 + 82.0864i 148.378 456.662i 40.6382
9.3 −0.361034 + 1.11115i 12.7716 9.27912i 24.7842 + 18.0068i 1.31139 + 4.03604i 5.69949 + 17.5412i −146.551 106.476i −59.2025 + 43.0132i 1.92087 5.91183i −4.95810
9.4 2.46756 7.59437i 0.720120 0.523198i −25.6970 18.6700i −2.24156 6.89880i −2.19642 6.75988i 136.674 + 99.2995i 1.52919 1.11102i −74.8463 + 230.353i −57.9232
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 9.4 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.c Even 1 yes

## Hecke kernels

There are no other newforms in $$S_{6}^{\mathrm{new}}(11, [\chi])$$.