# Properties

 Label 21.9.f.b Level $21$ Weight $9$ Character orbit 21.f Analytic conductor $8.555$ Analytic rank $0$ Dimension $12$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$21 = 3 \cdot 7$$ Weight: $$k$$ $$=$$ $$9$$ Character orbit: $$[\chi]$$ $$=$$ 21.f (of order $$6$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$8.55495081128$$ Analytic rank: $$0$$ Dimension: $$12$$ Relative dimension: $$6$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\mathbb{Q}[x]/(x^{12} - \cdots)$$ Defining polynomial: $$x^{12} - 3 x^{11} + 1331 x^{10} + 1482 x^{9} + 1359381 x^{8} + 1222227 x^{7} + 474169529 x^{6} + 1911324162 x^{5} + 121870629463 x^{4} + 259296590901 x^{3} + 9508380016437 x^{2} - 12075311354004 x + 556282980265104$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{8}\cdot 3^{5}\cdot 7^{5}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + ( 1 + \beta_{2} - \beta_{3} ) q^{2} + ( 54 + 27 \beta_{2} ) q^{3} + ( -\beta_{1} + 186 \beta_{2} + \beta_{5} ) q^{4} + ( 76 + 6 \beta_{1} - 76 \beta_{2} + 3 \beta_{3} + \beta_{4} + \beta_{6} + \beta_{8} ) q^{5} + ( 27 + 27 \beta_{1} + 54 \beta_{2} - 27 \beta_{3} ) q^{6} + ( 101 + 48 \beta_{1} + 443 \beta_{2} + 29 \beta_{3} + 2 \beta_{4} + 5 \beta_{5} + 4 \beta_{6} + 2 \beta_{8} + \beta_{9} + \beta_{10} ) q^{7} + ( 186 + 222 \beta_{1} + 221 \beta_{3} - \beta_{4} + \beta_{5} + 2 \beta_{6} + 3 \beta_{8} + \beta_{9} + \beta_{10} - \beta_{11} ) q^{8} + ( 2187 + 2187 \beta_{2} ) q^{9} +O(q^{10})$$ $$q + ( 1 + \beta_{2} - \beta_{3} ) q^{2} + ( 54 + 27 \beta_{2} ) q^{3} + ( -\beta_{1} + 186 \beta_{2} + \beta_{5} ) q^{4} + ( 76 + 6 \beta_{1} - 76 \beta_{2} + 3 \beta_{3} + \beta_{4} + \beta_{6} + \beta_{8} ) q^{5} + ( 27 + 27 \beta_{1} + 54 \beta_{2} - 27 \beta_{3} ) q^{6} + ( 101 + 48 \beta_{1} + 443 \beta_{2} + 29 \beta_{3} + 2 \beta_{4} + 5 \beta_{5} + 4 \beta_{6} + 2 \beta_{8} + \beta_{9} + \beta_{10} ) q^{7} + ( 186 + 222 \beta_{1} + 221 \beta_{3} - \beta_{4} + \beta_{5} + 2 \beta_{6} + 3 \beta_{8} + \beta_{9} + \beta_{10} - \beta_{11} ) q^{8} + ( 2187 + 2187 \beta_{2} ) q^{9} + ( -1908 - 292 \beta_{1} - 954 \beta_{2} - 574 \beta_{3} - 2 \beta_{4} - \beta_{5} - 4 \beta_{6} - \beta_{7} + 6 \beta_{8} - 5 \beta_{9} - 3 \beta_{10} + 2 \beta_{11} ) q^{10} + ( -68 \beta_{1} - 3054 \beta_{2} - \beta_{4} - 10 \beta_{5} + 7 \beta_{6} + 2 \beta_{7} + 3 \beta_{8} - \beta_{9} + 2 \beta_{10} ) q^{11} + ( -5022 - 54 \beta_{1} + 5022 \beta_{2} - 27 \beta_{3} - 27 \beta_{4} + 27 \beta_{5} ) q^{12} + ( 1795 - 253 \beta_{1} + 3590 \beta_{2} + 239 \beta_{3} + 11 \beta_{4} + 20 \beta_{5} - 9 \beta_{6} - 14 \beta_{9} + 7 \beta_{10} ) q^{13} + ( -21040 + 1249 \beta_{1} - 11915 \beta_{2} + 595 \beta_{3} - 67 \beta_{4} - 13 \beta_{5} + 24 \beta_{6} - 3 \beta_{7} + 21 \beta_{8} - 18 \beta_{9} - 5 \beta_{10} + \beta_{11} ) q^{14} + ( 6156 + 243 \beta_{1} + 243 \beta_{3} + 81 \beta_{4} + 27 \beta_{5} + 27 \beta_{6} + 54 \beta_{8} ) q^{15} + ( -49910 - 37 \beta_{1} - 49910 \beta_{2} - 43 \beta_{3} - 336 \beta_{4} - 299 \beta_{5} + 5 \beta_{6} + 3 \beta_{7} - 45 \beta_{8} - 37 \beta_{9} - 8 \beta_{10} - 3 \beta_{11} ) q^{16} + ( 7928 - 874 \beta_{1} + 3964 \beta_{2} - 1748 \beta_{3} + 60 \beta_{4} + 30 \beta_{5} + 66 \beta_{8} ) q^{17} + ( 2187 \beta_{1} + 2187 \beta_{2} ) q^{18} + ( 26051 + 1651 \beta_{1} - 26051 \beta_{2} + 817 \beta_{3} - 14 \beta_{4} - 24 \beta_{5} - 38 \beta_{6} - 2 \beta_{7} - 53 \beta_{8} + 17 \beta_{9} + 49 \beta_{10} - 2 \beta_{11} ) q^{19} + ( 105026 - 2000 \beta_{1} + 210052 \beta_{2} + 2049 \beta_{3} + 417 \beta_{4} + 785 \beta_{5} - 28 \beta_{6} + 2 \beta_{7} - \beta_{8} + 49 \beta_{9} - 25 \beta_{10} - \beta_{11} ) q^{20} + ( -6507 + 1809 \beta_{1} + 14688 \beta_{2} + 2052 \beta_{3} + 54 \beta_{4} + 216 \beta_{5} + 162 \beta_{6} + 135 \beta_{8} + 54 \beta_{9} + 27 \beta_{10} ) q^{21} + ( 38542 - 6780 \beta_{1} - 6749 \beta_{3} + 719 \beta_{4} + 54 \beta_{5} + 24 \beta_{6} + 83 \beta_{8} - 31 \beta_{9} - 32 \beta_{10} + 27 \beta_{11} ) q^{22} + ( -7312 + 166 \beta_{1} - 7312 \beta_{2} - 7600 \beta_{3} - 306 \beta_{4} - 184 \beta_{5} + 258 \beta_{6} - 4 \beta_{7} - 88 \beta_{8} + 166 \beta_{9} + 34 \beta_{10} + 4 \beta_{11} ) q^{23} + ( 10044 + 6021 \beta_{1} + 5022 \beta_{2} + 11934 \beta_{3} - 81 \beta_{4} - 54 \beta_{5} + 27 \beta_{6} + 27 \beta_{7} + 108 \beta_{8} + 54 \beta_{9} + 27 \beta_{10} - 54 \beta_{11} ) q^{24} + ( 544 \beta_{1} - 131939 \beta_{2} + 59 \beta_{4} - 470 \beta_{5} - 101 \beta_{6} - 54 \beta_{7} - 21 \beta_{8} - 5 \beta_{9} + 74 \beta_{10} ) q^{25} + ( 102753 + 8866 \beta_{1} - 102753 \beta_{2} + 4472 \beta_{3} - 85 \beta_{4} - 543 \beta_{5} - 628 \beta_{6} + 3 \beta_{7} - 553 \beta_{8} - 78 \beta_{9} - 231 \beta_{10} + 3 \beta_{11} ) q^{26} + ( 59049 + 118098 \beta_{2} ) q^{27} + ( -462518 - 17252 \beta_{1} - 221746 \beta_{2} + 11817 \beta_{3} - 1600 \beta_{4} - 284 \beta_{5} - 307 \beta_{6} + 85 \beta_{7} - 404 \beta_{8} - 30 \beta_{9} - 99 \beta_{10} - 26 \beta_{11} ) q^{28} + ( 56410 - 215 \beta_{1} - 287 \beta_{3} + 951 \beta_{4} - 529 \beta_{5} - 459 \beta_{6} - 998 \beta_{8} + 72 \beta_{9} + 74 \beta_{10} - 64 \beta_{11} ) q^{29} + ( -77274 - 216 \beta_{1} - 77274 \beta_{2} - 23247 \beta_{3} - 27 \beta_{4} - 243 \beta_{5} - 324 \beta_{6} - 81 \beta_{7} + 189 \beta_{8} - 216 \beta_{9} - 27 \beta_{10} + 81 \beta_{11} ) q^{30} + ( 85370 - 5541 \beta_{1} + 42685 \beta_{2} - 11228 \beta_{3} + 179 \beta_{4} + 155 \beta_{5} + 137 \beta_{6} - 64 \beta_{7} - 200 \beta_{8} + 73 \beta_{9} + 70 \beta_{10} + 128 \beta_{11} ) q^{31} + ( -80065 \beta_{1} - 189246 \beta_{2} - 227 \beta_{4} - 1018 \beta_{5} - 361 \beta_{6} + 123 \beta_{7} - 294 \beta_{8} + 104 \beta_{9} - 539 \beta_{10} ) q^{32} + ( 82458 - 3591 \beta_{1} - 82458 \beta_{2} - 1809 \beta_{3} + 486 \beta_{4} - 108 \beta_{5} + 378 \beta_{6} + 54 \beta_{7} + 297 \beta_{8} + 27 \beta_{9} + 135 \beta_{10} + 54 \beta_{11} ) q^{33} + ( 397036 + 5180 \beta_{1} + 794072 \beta_{2} - 5342 \beta_{3} + 994 \beta_{4} + 1292 \beta_{5} - 672 \beta_{6} - 60 \beta_{7} + 30 \beta_{8} - 162 \beta_{9} + 96 \beta_{10} + 30 \beta_{11} ) q^{34} + ( 33296 + 37171 \beta_{1} - 475442 \beta_{2} + 32907 \beta_{3} + 3119 \beta_{4} + 1546 \beta_{5} - 685 \beta_{6} - 188 \beta_{7} - 476 \beta_{8} + 148 \beta_{9} + 59 \beta_{10} + 58 \beta_{11} ) q^{35} + ( -406782 - 2187 \beta_{1} - 2187 \beta_{3} - 2187 \beta_{4} ) q^{36} + ( 394651 + 315 \beta_{1} + 394651 \beta_{2} - 26515 \beta_{3} - 1156 \beta_{4} - 1796 \beta_{5} - 628 \beta_{6} + 300 \beta_{7} + 643 \beta_{8} + 315 \beta_{9} + 3 \beta_{10} - 300 \beta_{11} ) q^{37} + ( -671954 - 27333 \beta_{1} - 335977 \beta_{2} - 53876 \beta_{3} - 4290 \beta_{4} - 2215 \beta_{5} - 400 \beta_{6} + 5 \beta_{7} - 2158 \beta_{8} - 395 \beta_{9} - 265 \beta_{10} - 10 \beta_{11} ) q^{38} + ( -19548 \beta_{1} + 145395 \beta_{2} + 189 \beta_{4} + 972 \beta_{5} + 81 \beta_{6} + 135 \beta_{8} - 189 \beta_{9} + 756 \beta_{10} ) q^{39} + ( 410526 + 311199 \beta_{1} - 410526 \beta_{2} + 155455 \beta_{3} + 2370 \beta_{4} + 2755 \beta_{5} + 5125 \beta_{6} - 209 \beta_{7} + 5045 \beta_{8} + 289 \beta_{9} + 658 \beta_{10} - 209 \beta_{11} ) q^{40} + ( 84182 - 39036 \beta_{1} + 168364 \beta_{2} + 39556 \beta_{3} - 2756 \beta_{4} - 2736 \beta_{5} + 2812 \beta_{6} + 128 \beta_{7} - 64 \beta_{8} + 520 \beta_{9} - 292 \beta_{10} - 64 \beta_{11} ) q^{41} + ( -814455 + 51732 \beta_{1} - 889785 \beta_{2} + 50274 \beta_{3} - 2187 \beta_{4} - 1674 \beta_{5} + 1080 \beta_{6} - 108 \beta_{7} + 1701 \beta_{8} - 621 \beta_{9} + 216 \beta_{10} - 27 \beta_{11} ) q^{42} + ( -819007 - 48555 \beta_{1} - 48857 \beta_{3} - 619 \beta_{4} + 3408 \beta_{5} + 3633 \beta_{6} + 6656 \beta_{8} + 302 \beta_{9} + 379 \beta_{10} + 6 \beta_{11} ) q^{43} + ( 2456246 - 817 \beta_{1} + 2456246 \beta_{2} - 196077 \beta_{3} + 13746 \beta_{4} + 13483 \beta_{5} - 711 \beta_{6} - 257 \beta_{7} + 151 \beta_{8} - 817 \beta_{9} - 112 \beta_{10} + 257 \beta_{11} ) q^{44} + ( 332424 + 6561 \beta_{1} + 166212 \beta_{2} + 13122 \beta_{3} + 4374 \beta_{4} + 2187 \beta_{5} + 2187 \beta_{8} ) q^{45} + ( -80424 \beta_{1} + 3282280 \beta_{2} + 92 \beta_{4} + 11468 \beta_{5} + 6388 \beta_{6} + 132 \beta_{7} + 3240 \beta_{8} - 224 \beta_{9} + 764 \beta_{10} ) q^{46} + ( -245618 + 43874 \beta_{1} + 245618 \beta_{2} + 22066 \beta_{3} - 3892 \beta_{4} + 348 \beta_{5} - 3544 \beta_{6} + 192 \beta_{7} - 3478 \beta_{8} - 258 \beta_{9} - 582 \beta_{10} + 192 \beta_{11} ) q^{47} + ( -1347570 - 54 \beta_{1} - 2695140 \beta_{2} - 1161 \beta_{3} - 8937 \beta_{4} - 14715 \beta_{5} + 2268 \beta_{6} + 162 \beta_{7} - 81 \beta_{8} - 1215 \beta_{9} + 567 \beta_{10} - 81 \beta_{11} ) q^{48} + ( -61208 + 118493 \beta_{1} - 1366827 \beta_{2} + 245925 \beta_{3} - 145 \beta_{4} - 1528 \beta_{5} - 4905 \beta_{6} + 148 \beta_{7} - 2658 \beta_{8} - 22 \beta_{9} + 443 \beta_{10} + 226 \beta_{11} ) q^{49} + ( 32315 - 274729 \beta_{1} - 273026 \beta_{3} - 20329 \beta_{4} - 4946 \beta_{5} - 6296 \beta_{6} - 9477 \beta_{8} - 1703 \beta_{9} - 2056 \beta_{10} + 291 \beta_{11} ) q^{50} + ( 321084 + 321084 \beta_{2} - 70794 \beta_{3} + 2430 \beta_{4} + 648 \beta_{5} - 1782 \beta_{6} + 1782 \beta_{8} ) q^{51} + ( -2852120 - 79796 \beta_{1} - 1426060 \beta_{2} - 157604 \beta_{3} + 3025 \beta_{4} + 1221 \beta_{5} - 1145 \beta_{6} + 151 \beta_{7} + 676 \beta_{8} - 994 \beta_{9} - 713 \beta_{10} - 302 \beta_{11} ) q^{52} + ( -51811 \beta_{1} + 5862 \beta_{2} + 520 \beta_{4} - 18177 \beta_{5} - 7942 \beta_{6} - 892 \beta_{7} - 3711 \beta_{8} + 372 \beta_{9} - 596 \beta_{10} ) q^{53} + ( -59049 + 118098 \beta_{1} + 59049 \beta_{2} + 59049 \beta_{3} ) q^{54} + ( -2449314 - 156167 \beta_{1} - 4898628 \beta_{2} + 156035 \beta_{3} - 2707 \beta_{4} - 17431 \beta_{5} - 11425 \beta_{6} - 376 \beta_{7} + 188 \beta_{8} - 132 \beta_{9} + 160 \beta_{10} + 188 \beta_{11} ) q^{55} + ( 3812840 + 4083 \beta_{1} - 3819058 \beta_{2} + 510180 \beta_{3} + 22009 \beta_{4} - 9930 \beta_{5} + 9223 \beta_{6} + 1111 \beta_{7} + 2196 \beta_{8} + 854 \beta_{9} - 1937 \beta_{10} - 662 \beta_{11} ) q^{56} + ( 2110131 + 67959 \beta_{1} + 66177 \beta_{3} - 1701 \beta_{4} - 1134 \beta_{5} + 243 \beta_{6} - 2916 \beta_{8} + 1782 \beta_{9} + 2187 \beta_{10} - 162 \beta_{11} ) q^{57} + ( 555080 - 516 \beta_{1} + 555080 \beta_{2} - 478099 \beta_{3} - 20029 \beta_{4} - 9945 \beta_{5} + 10336 \beta_{6} - 831 \beta_{7} - 10021 \beta_{8} - 516 \beta_{9} + 63 \beta_{10} + 831 \beta_{11} ) q^{58} + ( -10860112 + 83068 \beta_{1} - 5430056 \beta_{2} + 156638 \beta_{3} + 307 \beta_{4} + 680 \beta_{5} + 4431 \beta_{6} + 318 \beta_{7} + 4211 \beta_{8} + 4749 \beta_{9} + 3060 \beta_{10} - 636 \beta_{11} ) q^{59} + ( -165321 \beta_{1} + 8507106 \beta_{2} - 729 \beta_{4} + 30456 \beta_{5} - 3483 \beta_{6} + 81 \beta_{7} - 2106 \beta_{8} + 648 \beta_{9} - 2673 \beta_{10} ) q^{60} + ( -1341796 - 19540 \beta_{1} + 1341796 \beta_{2} - 9384 \beta_{3} - 6620 \beta_{4} - 12992 \beta_{5} - 19612 \beta_{6} + 572 \beta_{7} - 19412 \beta_{8} - 772 \beta_{9} - 1744 \beta_{10} + 572 \beta_{11} ) q^{61} + ( 2547335 + 50983 \beta_{1} + 5094670 \beta_{2} - 49649 \beta_{3} + 27002 \beta_{4} + 50952 \beta_{5} - 1384 \beta_{6} - 572 \beta_{7} + 286 \beta_{8} + 1334 \beta_{9} - 524 \beta_{10} + 286 \beta_{11} ) q^{62} + ( -747954 + 41553 \beta_{1} + 220887 \beta_{2} + 102789 \beta_{3} + 6561 \beta_{5} + 4374 \beta_{6} + 6561 \beta_{8} + 2187 \beta_{9} ) q^{63} + ( 22847302 - 183820 \beta_{1} - 184405 \beta_{3} + 52349 \beta_{4} + 2385 \beta_{5} + 2946 \beta_{6} + 5211 \beta_{8} + 585 \beta_{9} + 609 \beta_{10} - 489 \beta_{11} ) q^{64} + ( 5143254 + 5780 \beta_{1} + 5143254 \beta_{2} - 140418 \beta_{3} - 8006 \beta_{4} - 9744 \beta_{5} + 2022 \beta_{6} + 1080 \beta_{7} + 2678 \beta_{8} + 5780 \beta_{9} + 940 \beta_{10} - 1080 \beta_{11} ) q^{65} + ( 2081268 - 183924 \beta_{1} + 1040634 \beta_{2} - 364446 \beta_{3} + 39582 \beta_{4} + 20115 \beta_{5} - 972 \beta_{6} - 729 \beta_{7} + 3726 \beta_{8} - 1701 \beta_{9} - 891 \beta_{10} + 1458 \beta_{11} ) q^{66} + ( -703972 \beta_{1} + 9235739 \beta_{2} - 1613 \beta_{4} - 20254 \beta_{5} - 7573 \beta_{6} + 1266 \beta_{7} - 4593 \beta_{8} + 347 \beta_{9} - 2654 \beta_{10} ) q^{67} + ( -1991092 + 730330 \beta_{1} + 1991092 \beta_{2} + 365094 \beta_{3} - 12240 \beta_{4} + 16262 \beta_{5} + 4022 \beta_{6} - 862 \beta_{7} + 4742 \beta_{8} + 142 \beta_{9} - 436 \beta_{10} - 862 \beta_{11} ) q^{68} + ( -197424 + 210600 \beta_{1} - 394848 \beta_{2} - 205200 \beta_{3} - 9072 \beta_{4} - 8478 \beta_{5} + 12744 \beta_{6} - 216 \beta_{7} + 108 \beta_{8} + 5400 \beta_{9} - 2646 \beta_{10} + 108 \beta_{11} ) q^{69} + ( -15568842 + 51962 \beta_{1} - 13488516 \beta_{2} - 916315 \beta_{3} - 76775 \beta_{4} - 10343 \beta_{5} - 1124 \beta_{6} - 3253 \beta_{7} - 15311 \beta_{8} - 512 \beta_{9} + 2085 \beta_{10} + 221 \beta_{11} ) q^{70} + ( 18514154 + 413522 \beta_{1} + 410626 \beta_{3} + 13850 \beta_{4} + 12120 \beta_{5} + 14214 \beta_{6} + 22324 \beta_{8} + 2896 \beta_{9} + 3698 \beta_{10} + 312 \beta_{11} ) q^{71} + ( 406782 + 2187 \beta_{1} + 406782 \beta_{2} + 483327 \beta_{3} - 4374 \beta_{4} - 6561 \beta_{5} - 2187 \beta_{6} + 2187 \beta_{7} + 2187 \beta_{8} + 2187 \beta_{9} - 2187 \beta_{11} ) q^{72} + ( -21976894 + 502324 \beta_{1} - 10988447 \beta_{2} + 1005818 \beta_{3} + 26621 \beta_{4} + 13168 \beta_{5} - 639 \beta_{6} + 54 \beta_{7} - 15519 \beta_{8} - 585 \beta_{9} - 408 \beta_{10} - 108 \beta_{11} ) q^{73} + ( -6039 \beta_{1} + 11801571 \beta_{2} + 1332 \beta_{4} - 67543 \beta_{5} + 15348 \beta_{6} + 1441 \beta_{7} + 8340 \beta_{8} - 2773 \beta_{9} + 9651 \beta_{10} ) q^{74} + ( 3562353 + 31509 \beta_{1} - 3562353 \beta_{2} + 14823 \beta_{3} + 11826 \beta_{4} - 14580 \beta_{5} - 2754 \beta_{6} - 1458 \beta_{7} - 3159 \beta_{8} + 1863 \beta_{9} + 4131 \beta_{10} - 1458 \beta_{11} ) q^{75} + ( 4214696 - 623645 \beta_{1} + 8429392 \beta_{2} + 624624 \beta_{3} + 29568 \beta_{4} + 51893 \beta_{5} - 13204 \beta_{6} + 3686 \beta_{7} - 1843 \beta_{8} + 979 \beta_{9} - 1411 \beta_{10} - 1843 \beta_{11} ) q^{76} + ( 5228924 - 14196 \beta_{1} - 5170302 \beta_{2} - 536341 \beta_{3} + 68163 \beta_{4} + 79692 \beta_{5} - 10057 \beta_{6} - 1336 \beta_{7} + 23609 \beta_{8} - 6834 \beta_{9} + 4182 \beta_{10} + 2228 \beta_{11} ) q^{77} + ( 8322993 + 353889 \beta_{1} + 362232 \beta_{3} - 4617 \beta_{4} - 16794 \beta_{5} - 23112 \beta_{6} - 29781 \beta_{8} - 8343 \beta_{9} - 10368 \beta_{10} + 243 \beta_{11} ) q^{78} + ( -4993457 - 9601 \beta_{1} - 4993457 \beta_{2} + 167358 \beta_{3} - 59573 \beta_{4} - 58858 \beta_{5} - 4633 \beta_{6} - 2916 \beta_{7} - 2052 \beta_{8} - 9601 \beta_{9} - 1337 \beta_{10} + 2916 \beta_{11} ) q^{79} + ( -81323444 - 484091 \beta_{1} - 40661722 \beta_{2} - 939882 \beta_{3} - 244927 \beta_{4} - 123516 \beta_{5} - 12583 \beta_{6} - 1567 \beta_{7} + 15348 \beta_{8} - 14150 \beta_{9} - 8911 \beta_{10} + 3134 \beta_{11} ) q^{80} + 4782969 \beta_{2} q^{81} + ( 18031858 - 668596 \beta_{1} - 18031858 \beta_{2} - 333750 \beta_{3} + 73924 \beta_{4} - 29732 \beta_{5} + 44192 \beta_{6} + 2300 \beta_{7} + 42988 \beta_{8} - 1096 \beta_{9} - 988 \beta_{10} + 2300 \beta_{11} ) q^{82} + ( 14590518 + 916717 \beta_{1} + 29181036 \beta_{2} - 937633 \beta_{3} + 4091 \beta_{4} + 47222 \beta_{5} + 28603 \beta_{6} - 12 \beta_{7} + 6 \beta_{8} - 20916 \beta_{9} + 10461 \beta_{10} + 6 \beta_{11} ) q^{83} + ( -18988830 - 1252530 \beta_{1} - 18475128 \beta_{2} - 145935 \beta_{3} - 84618 \beta_{4} - 45144 \beta_{5} - 7533 \beta_{6} + 2997 \beta_{7} - 18387 \beta_{8} - 3483 \beta_{9} - 4536 \beta_{10} + 891 \beta_{11} ) q^{84} + ( 21232008 - 1041514 \beta_{1} - 1035582 \beta_{3} - 13322 \beta_{4} + 2640 \beta_{5} - 1722 \beta_{6} + 8768 \beta_{8} - 5932 \beta_{9} - 7502 \beta_{10} - 348 \beta_{11} ) q^{85} + ( 20250081 - 19996 \beta_{1} + 20250081 \beta_{2} + 1934602 \beta_{3} + 37709 \beta_{4} + 6161 \beta_{5} - 47784 \beta_{6} + 299 \beta_{7} + 27489 \beta_{8} - 19996 \beta_{9} - 4059 \beta_{10} - 299 \beta_{11} ) q^{86} + ( 3046140 - 3807 \beta_{1} + 1523070 \beta_{2} - 15498 \beta_{3} + 49572 \beta_{4} + 24003 \beta_{5} + 2214 \beta_{6} + 1728 \beta_{7} - 41283 \beta_{8} + 3942 \beta_{9} + 2052 \beta_{10} - 3456 \beta_{11} ) q^{87} + ( 4110325 \beta_{1} + 83283158 \beta_{2} + 3507 \beta_{4} + 154414 \beta_{5} + 12953 \beta_{6} - 6531 \beta_{7} + 8230 \beta_{8} + 3024 \beta_{9} - 5565 \beta_{10} ) q^{88} + ( 7410672 - 3026680 \beta_{1} - 7410672 \beta_{2} - 1516858 \beta_{3} - 27234 \beta_{4} + 103832 \beta_{5} + 76598 \beta_{6} - 256 \beta_{7} + 69818 \beta_{8} + 7036 \beta_{9} + 20852 \beta_{10} - 256 \beta_{11} ) q^{89} + ( -2086398 + 621108 \beta_{1} - 4172796 \beta_{2} - 627669 \beta_{3} + 2187 \beta_{4} - 17496 \beta_{5} - 17496 \beta_{6} - 4374 \beta_{7} + 2187 \beta_{8} - 6561 \beta_{9} + 4374 \beta_{10} + 2187 \beta_{11} ) q^{90} + ( -9724995 + 47619 \beta_{1} - 36719025 \beta_{2} - 1238543 \beta_{3} + 24290 \beta_{4} + 73518 \beta_{5} - 14034 \beta_{6} + 7898 \beta_{7} - 465 \beta_{8} + 6169 \beta_{9} - 10841 \beta_{10} - 6478 \beta_{11} ) q^{91} + ( 28493760 + 3727040 \beta_{1} + 3708528 \beta_{3} + 15760 \beta_{4} + 7520 \beta_{5} + 22016 \beta_{6} + 9456 \beta_{8} + 18512 \beta_{9} + 22528 \beta_{10} - 2448 \beta_{11} ) q^{92} + ( 3457485 + 3861 \beta_{1} + 3457485 \beta_{2} - 454734 \beta_{3} + 8937 \beta_{4} + 14418 \beta_{5} + 12717 \beta_{6} - 5184 \beta_{7} - 3672 \beta_{8} + 3861 \beta_{9} + 1809 \beta_{10} + 5184 \beta_{11} ) q^{93} + ( -22237196 + 1075842 \beta_{1} - 11118598 \beta_{2} + 2091440 \beta_{3} + 87432 \beta_{4} + 45918 \beta_{5} + 26740 \beta_{6} + 3382 \beta_{7} + 21540 \beta_{8} + 30122 \beta_{9} + 18954 \beta_{10} - 6764 \beta_{11} ) q^{94} + ( 2719092 \beta_{1} + 17236350 \beta_{2} - 1260 \beta_{4} - 102964 \beta_{5} + 58532 \beta_{6} + 1716 \beta_{7} + 28636 \beta_{8} - 456 \beta_{9} + 108 \beta_{10} ) q^{95} + ( 5109642 - 4340871 \beta_{1} - 5109642 \beta_{2} - 2164563 \beta_{3} + 5994 \beta_{4} - 34911 \beta_{5} - 28917 \beta_{6} + 3321 \beta_{7} - 20493 \beta_{8} - 11745 \beta_{9} - 31914 \beta_{10} + 3321 \beta_{11} ) q^{96} + ( 14411544 - 107457 \beta_{1} + 28823088 \beta_{2} + 123493 \beta_{3} - 102383 \beta_{4} - 147532 \beta_{5} + 74737 \beta_{6} - 5420 \beta_{7} + 2710 \beta_{8} + 16036 \beta_{9} - 6663 \beta_{10} + 2710 \beta_{11} ) q^{97} + ( -51754297 - 755451 \beta_{1} - 108546144 \beta_{2} - 1110768 \beta_{3} - 3523 \beta_{4} - 196873 \beta_{5} - 12756 \beta_{6} - 659 \beta_{7} - 34911 \beta_{8} + 14974 \beta_{9} + 14999 \beta_{10} - 1787 \beta_{11} ) q^{98} + ( 6679098 - 142155 \beta_{1} - 146529 \beta_{3} + 41553 \beta_{4} + 13122 \beta_{5} + 15309 \beta_{6} + 17496 \beta_{8} + 4374 \beta_{9} + 6561 \beta_{10} + 4374 \beta_{11} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$12 q + 3 q^{2} + 486 q^{3} - 1117 q^{4} + 1389 q^{5} - 1226 q^{7} + 3558 q^{8} + 13122 q^{9} + O(q^{10})$$ $$12 q + 3 q^{2} + 486 q^{3} - 1117 q^{4} + 1389 q^{5} - 1226 q^{7} + 3558 q^{8} + 13122 q^{9} - 19731 q^{10} + 18081 q^{11} - 90477 q^{12} - 175188 q^{14} + 75006 q^{15} - 298849 q^{16} + 63306 q^{17} - 6561 q^{18} + 476265 q^{19} - 154953 q^{21} + 419258 q^{22} - 66384 q^{23} + 144099 q^{24} + 792375 q^{25} + 1890762 q^{26} - 4229651 q^{28} + 671106 q^{29} - 532737 q^{30} + 717240 q^{31} + 895149 q^{32} + 1464561 q^{33} + 3454614 q^{35} - 4885758 q^{36} + 2289443 q^{37} - 6275988 q^{38} - 930933 q^{39} + 8773863 q^{40} - 4123629 q^{42} - 10119958 q^{43} + 14123901 q^{44} + 3037743 q^{45} - 19926544 q^{46} - 4198782 q^{47} + 8565642 q^{49} - 1162248 q^{50} + 1709262 q^{51} - 26384844 q^{52} - 209511 q^{53} - 531441 q^{54} + 70081743 q^{56} + 25718310 q^{57} + 1938935 q^{58} - 97052259 q^{59} - 51464727 q^{60} - 24195864 q^{61} - 9869931 q^{63} + 272849458 q^{64} + 30440562 q^{65} + 16979949 q^{66} - 57546057 q^{67} - 32476050 q^{68} - 108189027 q^{70} + 224566620 q^{71} + 3890673 q^{72} - 193344135 q^{73} - 71015400 q^{74} + 64182375 q^{75} + 92041617 q^{77} + 102101148 q^{78} - 29314488 q^{79} - 735358533 q^{80} - 28697814 q^{81} + 321118926 q^{82} - 120939804 q^{84} + 248647620 q^{85} + 127268268 q^{86} + 27179793 q^{87} - 487070897 q^{88} + 119863098 q^{89} + 100090965 q^{91} + 364039200 q^{92} + 19365480 q^{93} - 191086092 q^{94} - 95584488 q^{95} + 72507069 q^{96} + 24200217 q^{98} + 79086294 q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{12} - 3 x^{11} + 1331 x^{10} + 1482 x^{9} + 1359381 x^{8} + 1222227 x^{7} + 474169529 x^{6} + 1911324162 x^{5} + 121870629463 x^{4} + 259296590901 x^{3} + 9508380016437 x^{2} - 12075311354004 x + 556282980265104$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$-$$$$26\!\cdots\!73$$$$\nu^{11} -$$$$96\!\cdots\!45$$$$\nu^{10} -$$$$30\!\cdots\!99$$$$\nu^{9} -$$$$13\!\cdots\!94$$$$\nu^{8} -$$$$35\!\cdots\!65$$$$\nu^{7} -$$$$13\!\cdots\!99$$$$\nu^{6} -$$$$11\!\cdots\!73$$$$\nu^{5} -$$$$45\!\cdots\!66$$$$\nu^{4} -$$$$44\!\cdots\!75$$$$\nu^{3} -$$$$12\!\cdots\!17$$$$\nu^{2} -$$$$23\!\cdots\!65$$$$\nu -$$$$90\!\cdots\!64$$$$)/$$$$74\!\cdots\!16$$ $$\beta_{3}$$ $$=$$ $$($$$$-1330423089573140230247524396 \nu^{11} + 5711838431028074153179425546 \nu^{10} - 1672324312091638034918640508487 \nu^{9} + 83336889443940209931451286472 \nu^{8} - 1699505513779915732058712320631492 \nu^{7} + 977721146013857882476939945878366 \nu^{6} - 519954109922107136680929721906873760 \nu^{5} - 1622787012481741851118092624420065664 \nu^{4} - 149959771553751480026963800865474292616 \nu^{3} + 19639955411004356498644947234120041154 \nu^{2} - 11915524893493396659536813206028898821259 \nu + 18699924734382295688486372151716709334188$$$$)/$$$$94\!\cdots\!99$$ $$\beta_{4}$$ $$=$$ $$($$$$1903946143676024717726475182 \nu^{11} + 27111101612375796360425395317 \nu^{10} + 2357332281488749378638067988935 \nu^{9} + 36265448492925795548534428822656 \nu^{8} + 2567438902951105180949272635907578 \nu^{7} + 35986272131157334566353218174273542 \nu^{6} + 826648371356137582403677720019937256 \nu^{5} + 5682696287360525788075009148584927908 \nu^{4} + 271497813884849939193311209702724013266 \nu^{3} + 225241186192403232253971311095259850777 \nu^{2} + 3322973008396847750858420244832665900828 \nu - 1164205744482793429197063939085717778389413$$$$)/$$$$31\!\cdots\!33$$ $$\beta_{5}$$ $$=$$ $$($$$$11\!\cdots\!69$$$$\nu^{11} +$$$$38\!\cdots\!41$$$$\nu^{10} +$$$$13\!\cdots\!07$$$$\nu^{9} +$$$$56\!\cdots\!22$$$$\nu^{8} +$$$$15\!\cdots\!77$$$$\nu^{7} +$$$$57\!\cdots\!27$$$$\nu^{6} +$$$$49\!\cdots\!89$$$$\nu^{5} +$$$$21\!\cdots\!02$$$$\nu^{4} +$$$$18\!\cdots\!75$$$$\nu^{3} +$$$$52\!\cdots\!41$$$$\nu^{2} +$$$$13\!\cdots\!53$$$$\nu +$$$$37\!\cdots\!52$$$$)/$$$$82\!\cdots\!24$$ $$\beta_{6}$$ $$=$$ $$($$$$92\!\cdots\!16$$$$\nu^{11} +$$$$70\!\cdots\!27$$$$\nu^{10} -$$$$75\!\cdots\!66$$$$\nu^{9} +$$$$10\!\cdots\!80$$$$\nu^{8} +$$$$53\!\cdots\!20$$$$\nu^{7} +$$$$10\!\cdots\!95$$$$\nu^{6} -$$$$74\!\cdots\!92$$$$\nu^{5} +$$$$36\!\cdots\!92$$$$\nu^{4} -$$$$66\!\cdots\!06$$$$\nu^{3} +$$$$54\!\cdots\!67$$$$\nu^{2} -$$$$32\!\cdots\!16$$$$\nu +$$$$24\!\cdots\!76$$$$)/$$$$35\!\cdots\!68$$ $$\beta_{7}$$ $$=$$ $$($$$$44\!\cdots\!50$$$$\nu^{11} +$$$$30\!\cdots\!17$$$$\nu^{10} +$$$$64\!\cdots\!42$$$$\nu^{9} +$$$$50\!\cdots\!88$$$$\nu^{8} +$$$$73\!\cdots\!02$$$$\nu^{7} +$$$$53\!\cdots\!41$$$$\nu^{6} +$$$$46\!\cdots\!88$$$$\nu^{5} +$$$$24\!\cdots\!56$$$$\nu^{4} +$$$$18\!\cdots\!28$$$$\nu^{3} +$$$$55\!\cdots\!13$$$$\nu^{2} +$$$$90\!\cdots\!44$$$$\nu +$$$$43\!\cdots\!60$$$$)/$$$$11\!\cdots\!56$$ $$\beta_{8}$$ $$=$$ $$($$$$19\!\cdots\!27$$$$\nu^{11} -$$$$40\!\cdots\!18$$$$\nu^{10} +$$$$27\!\cdots\!56$$$$\nu^{9} -$$$$38\!\cdots\!16$$$$\nu^{8} +$$$$26\!\cdots\!75$$$$\nu^{7} -$$$$37\!\cdots\!80$$$$\nu^{6} +$$$$90\!\cdots\!32$$$$\nu^{5} -$$$$64\!\cdots\!62$$$$\nu^{4} +$$$$13\!\cdots\!39$$$$\nu^{3} -$$$$14\!\cdots\!68$$$$\nu^{2} +$$$$85\!\cdots\!24$$$$\nu -$$$$29\!\cdots\!92$$$$)/$$$$11\!\cdots\!56$$ $$\beta_{9}$$ $$=$$ $$($$$$-$$$$95\!\cdots\!82$$$$\nu^{11} +$$$$10\!\cdots\!13$$$$\nu^{10} -$$$$13\!\cdots\!82$$$$\nu^{9} -$$$$43\!\cdots\!68$$$$\nu^{8} -$$$$13\!\cdots\!10$$$$\nu^{7} -$$$$44\!\cdots\!19$$$$\nu^{6} -$$$$50\!\cdots\!84$$$$\nu^{5} -$$$$33\!\cdots\!44$$$$\nu^{4} -$$$$12\!\cdots\!80$$$$\nu^{3} -$$$$56\!\cdots\!59$$$$\nu^{2} -$$$$93\!\cdots\!88$$$$\nu -$$$$49\!\cdots\!36$$$$)/$$$$35\!\cdots\!68$$ $$\beta_{10}$$ $$=$$ $$($$$$-$$$$18\!\cdots\!86$$$$\nu^{11} +$$$$74\!\cdots\!21$$$$\nu^{10} -$$$$21\!\cdots\!66$$$$\nu^{9} -$$$$28\!\cdots\!36$$$$\nu^{8} -$$$$20\!\cdots\!10$$$$\nu^{7} -$$$$78\!\cdots\!47$$$$\nu^{6} -$$$$38\!\cdots\!36$$$$\nu^{5} -$$$$38\!\cdots\!60$$$$\nu^{4} -$$$$63\!\cdots\!08$$$$\nu^{3} +$$$$47\!\cdots\!05$$$$\nu^{2} +$$$$11\!\cdots\!92$$$$\nu +$$$$17\!\cdots\!68$$$$)/$$$$59\!\cdots\!28$$ $$\beta_{11}$$ $$=$$ $$($$$$-$$$$37\!\cdots\!79$$$$\nu^{11} +$$$$16\!\cdots\!15$$$$\nu^{10} -$$$$46\!\cdots\!38$$$$\nu^{9} -$$$$16\!\cdots\!56$$$$\nu^{8} -$$$$48\!\cdots\!31$$$$\nu^{7} -$$$$12\!\cdots\!35$$$$\nu^{6} -$$$$14\!\cdots\!76$$$$\nu^{5} -$$$$76\!\cdots\!50$$$$\nu^{4} -$$$$41\!\cdots\!17$$$$\nu^{3} -$$$$45\!\cdots\!19$$$$\nu^{2} -$$$$76\!\cdots\!92$$$$\nu +$$$$12\!\cdots\!52$$$$)/$$$$35\!\cdots\!68$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$-\beta_{5} - \beta_{4} - 3 \beta_{3} - 441 \beta_{2} - 441$$ $$\nu^{3}$$ $$=$$ $$\beta_{11} - \beta_{10} - \beta_{9} - 3 \beta_{8} - 2 \beta_{6} - \beta_{5} - 2 \beta_{4} - 739 \beta_{3} - 740 \beta_{1} - 998$$ $$\nu^{4}$$ $$=$$ $$-33 \beta_{10} + 8 \beta_{9} - 50 \beta_{8} + \beta_{7} - 91 \beta_{6} + 1004 \beta_{5} - 9 \beta_{4} + 325175 \beta_{2} - 3745 \beta_{1}$$ $$\nu^{5}$$ $$=$$ $$-1142 \beta_{11} + 144 \beta_{10} + 1862 \beta_{9} + 1430 \beta_{8} + 1142 \beta_{7} - 710 \beta_{6} + 1500 \beta_{5} + 2786 \beta_{4} + 645553 \beta_{3} + 1293844 \beta_{2} + 1862 \beta_{1} + 1293844$$ $$\nu^{6}$$ $$=$$ $$-3506 \beta_{11} + 58546 \beta_{10} + 47538 \beta_{9} + 139158 \beta_{8} + 115364 \beta_{6} + 78834 \beta_{5} + 1086445 \beta_{4} + 4292223 \beta_{3} + 4339761 \beta_{1} + 283224229$$ $$\nu^{7}$$ $$=$$ $$1980795 \beta_{10} - 213456 \beta_{9} + 2768022 \beta_{8} - 1126971 \beta_{7} + 4195617 \beta_{6} - 1344093 \beta_{5} + 1340427 \beta_{4} - 1539240306 \beta_{2} + 593763343 \beta_{1}$$ $$\nu^{8}$$ $$=$$ $$5693475 \beta_{11} - 11877864 \beta_{10} - 65082795 \beta_{9} - 73091667 \beta_{8} - 5693475 \beta_{7} + 13702347 \beta_{6} - 957355069 \beta_{5} - 1018568872 \beta_{4} - 4875593493 \beta_{3} - 259845486819 \beta_{2} - 65082795 \beta_{1} - 259845486819$$ $$\nu^{9}$$ $$=$$ $$1077958192 \beta_{11} - 2295970672 \beta_{10} - 2052368176 \beta_{9} - 4562761872 \beta_{8} - 3794770016 \beta_{6} - 1986004336 \beta_{5} - 7528952996 \beta_{4} - 556966017901 \beta_{3} - 559018386077 \beta_{1} - 1812479642816$$ $$\nu^{10}$$ $$=$$ $$-55296351348 \beta_{10} + 11967736640 \beta_{9} - 81075419624 \beta_{8} + 7425404788 \beta_{7} - 142757697820 \beta_{6} + 816461444429 \beta_{5} - 19393141428 \beta_{4} + 243016370791985 \beta_{2} - 5442078503659 \beta_{1}$$ $$\nu^{11}$$ $$=$$ $$-1023600157301 \beta_{11} + 257272951392 \beta_{10} + 2309964914261 \beta_{9} + 1789287974453 \beta_{8} + 1023600157301 \beta_{7} - 502923217493 \beta_{6} + 6003682500624 \beta_{5} + 7535697523685 \beta_{4} + 526787865683227 \beta_{3} + 2084504130864190 \beta_{2} + 2309964914261 \beta_{1} + 2084504130864190$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/21\mathbb{Z}\right)^\times$$.

 $$n$$ $$8$$ $$10$$ $$\chi(n)$$ $$1$$ $$1 + \beta_{2}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
10.1
 15.7115 − 27.2131i 9.75345 − 16.8935i 3.88954 − 6.73687i −5.65790 + 9.79977i −7.37103 + 12.7670i −14.8255 + 25.6786i 15.7115 + 27.2131i 9.75345 + 16.8935i 3.88954 + 6.73687i −5.65790 − 9.79977i −7.37103 − 12.7670i −14.8255 − 25.6786i
−15.2115 26.3471i 40.5000 + 23.3827i −334.779 + 579.854i 890.914 514.369i 1422.74i 2247.80 + 843.914i 12581.7 1093.50 + 1894.00i −27104.3 15648.6i
10.2 −9.25345 16.0274i 40.5000 + 23.3827i −43.2526 + 74.9157i −255.810 + 147.692i 865.482i −2399.31 + 89.9896i −3136.82 1093.50 + 1894.00i 4734.26 + 2733.32i
10.3 −3.38954 5.87085i 40.5000 + 23.3827i 105.022 181.904i −231.653 + 133.745i 317.026i 2354.84 468.558i −3159.35 1093.50 + 1894.00i 1570.39 + 906.668i
10.4 6.15790 + 10.6658i 40.5000 + 23.3827i 52.1605 90.3446i 599.787 346.287i 575.953i −419.022 2364.15i 4437.64 1093.50 + 1894.00i 7386.86 + 4264.81i
10.5 7.87103 + 13.6330i 40.5000 + 23.3827i 4.09363 7.09038i −872.651 + 503.826i 736.184i −1325.41 + 2002.02i 4158.85 1093.50 + 1894.00i −13737.3 7931.26i
10.6 15.3255 + 26.5446i 40.5000 + 23.3827i −341.745 + 591.919i 563.914 325.576i 1433.41i −1071.89 + 2148.45i −13103.0 1093.50 + 1894.00i 17284.6 + 9979.26i
19.1 −15.2115 + 26.3471i 40.5000 23.3827i −334.779 579.854i 890.914 + 514.369i 1422.74i 2247.80 843.914i 12581.7 1093.50 1894.00i −27104.3 + 15648.6i
19.2 −9.25345 + 16.0274i 40.5000 23.3827i −43.2526 74.9157i −255.810 147.692i 865.482i −2399.31 89.9896i −3136.82 1093.50 1894.00i 4734.26 2733.32i
19.3 −3.38954 + 5.87085i 40.5000 23.3827i 105.022 + 181.904i −231.653 133.745i 317.026i 2354.84 + 468.558i −3159.35 1093.50 1894.00i 1570.39 906.668i
19.4 6.15790 10.6658i 40.5000 23.3827i 52.1605 + 90.3446i 599.787 + 346.287i 575.953i −419.022 + 2364.15i 4437.64 1093.50 1894.00i 7386.86 4264.81i
19.5 7.87103 13.6330i 40.5000 23.3827i 4.09363 + 7.09038i −872.651 503.826i 736.184i −1325.41 2002.02i 4158.85 1093.50 1894.00i −13737.3 + 7931.26i
19.6 15.3255 26.5446i 40.5000 23.3827i −341.745 591.919i 563.914 + 325.576i 1433.41i −1071.89 2148.45i −13103.0 1093.50 1894.00i 17284.6 9979.26i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 19.6 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.d odd 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 21.9.f.b 12
3.b odd 2 1 63.9.m.d 12
7.c even 3 1 147.9.d.b 12
7.d odd 6 1 inner 21.9.f.b 12
7.d odd 6 1 147.9.d.b 12
21.g even 6 1 63.9.m.d 12

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
21.9.f.b 12 1.a even 1 1 trivial
21.9.f.b 12 7.d odd 6 1 inner
63.9.m.d 12 3.b odd 2 1
63.9.m.d 12 21.g even 6 1
147.9.d.b 12 7.c even 3 1
147.9.d.b 12 7.d odd 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$10\!\cdots\!20$$$$T_{2}^{2} +$$$$29\!\cdots\!92$$$$T_{2} +$$$$51\!\cdots\!84$$">$$T_{2}^{12} - \cdots$$ acting on $$S_{9}^{\mathrm{new}}(21, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$514460785065984 + 29360680353792 T + 10471792619520 T^{2} - 318009317376 T^{3} + 125660146432 T^{4} - 1781568384 T^{5} + 479886464 T^{6} - 4329648 T^{7} + 1372044 T^{8} - 4146 T^{9} + 1331 T^{10} - 3 T^{11} + T^{12}$$
$3$ $$( 2187 - 81 T + T^{2} )^{6}$$
$5$ $$13\!\cdots\!00$$$$+$$$$98\!\cdots\!00$$$$T +$$$$13\!\cdots\!00$$$$T^{2} -$$$$71\!\cdots\!00$$$$T^{3} -$$$$12\!\cdots\!00$$$$T^{4} +$$$$39\!\cdots\!70$$$$T^{5} + 542698230245598333 T^{6} - 1810282372382583 T^{7} + 515242824390 T^{8} + 1731401001 T^{9} - 603402 T^{10} - 1389 T^{11} + T^{12}$$
$7$ $$36\!\cdots\!01$$$$+$$$$78\!\cdots\!26$$$$T -$$$$39\!\cdots\!83$$$$T^{2} -$$$$58\!\cdots\!26$$$$T^{3} -$$$$17\!\cdots\!42$$$$T^{4} +$$$$82\!\cdots\!82$$$$T^{5} +$$$$49\!\cdots\!93$$$$T^{6} + 143196420021710482 T^{7} - 52350496282742 T^{8} - 30597397026 T^{9} - 3531283 T^{10} + 1226 T^{11} + T^{12}$$
$11$ $$90\!\cdots\!00$$$$-$$$$53\!\cdots\!00$$$$T +$$$$25\!\cdots\!60$$$$T^{2} -$$$$37\!\cdots\!80$$$$T^{3} +$$$$44\!\cdots\!24$$$$T^{4} +$$$$35\!\cdots\!42$$$$T^{5} +$$$$74\!\cdots\!27$$$$T^{6} -$$$$11\!\cdots\!87$$$$T^{7} + 328498760957963340 T^{8} - 3489707953551 T^{9} + 851197268 T^{10} - 18081 T^{11} + T^{12}$$
$13$ $$95\!\cdots\!44$$$$+$$$$27\!\cdots\!24$$$$T^{2} +$$$$67\!\cdots\!44$$$$T^{4} +$$$$39\!\cdots\!43$$$$T^{6} + 8349458011051423131 T^{8} + 5387688009 T^{10} + T^{12}$$
$17$ $$19\!\cdots\!16$$$$+$$$$63\!\cdots\!88$$$$T +$$$$86\!\cdots\!08$$$$T^{2} +$$$$57\!\cdots\!40$$$$T^{3} +$$$$18\!\cdots\!76$$$$T^{4} +$$$$10\!\cdots\!96$$$$T^{5} -$$$$76\!\cdots\!24$$$$T^{6} -$$$$95\!\cdots\!56$$$$T^{7} + 30201085137680178336 T^{8} + 422294836108392 T^{9} - 5334808920 T^{10} - 63306 T^{11} + T^{12}$$
$19$ $$60\!\cdots\!24$$$$+$$$$81\!\cdots\!88$$$$T +$$$$42\!\cdots\!64$$$$T^{2} +$$$$79\!\cdots\!44$$$$T^{3} -$$$$64\!\cdots\!24$$$$T^{4} -$$$$31\!\cdots\!74$$$$T^{5} +$$$$23\!\cdots\!17$$$$T^{6} +$$$$47\!\cdots\!25$$$$T^{7} -$$$$41\!\cdots\!50$$$$T^{8} - 4697863690624335 T^{9} + 85473420114 T^{10} - 476265 T^{11} + T^{12}$$
$23$ $$28\!\cdots\!00$$$$-$$$$87\!\cdots\!00$$$$T +$$$$16\!\cdots\!40$$$$T^{2} +$$$$21\!\cdots\!20$$$$T^{3} +$$$$55\!\cdots\!56$$$$T^{4} +$$$$44\!\cdots\!16$$$$T^{5} +$$$$71\!\cdots\!64$$$$T^{6} +$$$$41\!\cdots\!80$$$$T^{7} +$$$$66\!\cdots\!64$$$$T^{8} + 19488097166529024 T^{9} + 307326586208 T^{10} + 66384 T^{11} + T^{12}$$
$29$ $$( -$$$$92\!\cdots\!80$$$$-$$$$16\!\cdots\!52$$$$T +$$$$10\!\cdots\!04$$$$T^{2} + 475973011611930309 T^{3} - 1982708619821 T^{4} - 335553 T^{5} + T^{6} )^{2}$$
$31$ $$61\!\cdots\!69$$$$-$$$$56\!\cdots\!80$$$$T -$$$$65\!\cdots\!85$$$$T^{2} +$$$$61\!\cdots\!00$$$$T^{3} +$$$$60\!\cdots\!94$$$$T^{4} +$$$$34\!\cdots\!40$$$$T^{5} -$$$$11\!\cdots\!41$$$$T^{6} -$$$$85\!\cdots\!76$$$$T^{7} +$$$$18\!\cdots\!14$$$$T^{8} + 1203989022581890680 T^{9} - 1507164030057 T^{10} - 717240 T^{11} + T^{12}$$
$37$ $$66\!\cdots\!00$$$$+$$$$34\!\cdots\!00$$$$T +$$$$11\!\cdots\!60$$$$T^{2} +$$$$36\!\cdots\!40$$$$T^{3} +$$$$14\!\cdots\!96$$$$T^{4} +$$$$42\!\cdots\!10$$$$T^{5} +$$$$68\!\cdots\!75$$$$T^{6} -$$$$38\!\cdots\!89$$$$T^{7} +$$$$14\!\cdots\!04$$$$T^{8} - 7236021253430129385 T^{9} + 17182287134024 T^{10} - 2289443 T^{11} + T^{12}$$
$41$ $$52\!\cdots\!36$$$$+$$$$22\!\cdots\!64$$$$T^{2} +$$$$16\!\cdots\!72$$$$T^{4} +$$$$49\!\cdots\!96$$$$T^{6} +$$$$67\!\cdots\!72$$$$T^{8} + 43079833241832 T^{10} + T^{12}$$
$43$ $$($$$$39\!\cdots\!96$$$$+$$$$35\!\cdots\!48$$$$T -$$$$17\!\cdots\!70$$$$T^{2} -$$$$18\!\cdots\!63$$$$T^{3} - 26377786210915 T^{4} + 5059979 T^{5} + T^{6} )^{2}$$
$47$ $$59\!\cdots\!00$$$$+$$$$27\!\cdots\!00$$$$T +$$$$46\!\cdots\!20$$$$T^{2} +$$$$22\!\cdots\!80$$$$T^{3} -$$$$74\!\cdots\!84$$$$T^{4} -$$$$32\!\cdots\!80$$$$T^{5} -$$$$80\!\cdots\!00$$$$T^{6} +$$$$38\!\cdots\!80$$$$T^{7} +$$$$61\!\cdots\!76$$$$T^{8} -$$$$13\!\cdots\!80$$$$T^{9} - 26247667694832 T^{10} + 4198782 T^{11} + T^{12}$$
$53$ $$11\!\cdots\!44$$$$+$$$$30\!\cdots\!32$$$$T +$$$$28\!\cdots\!32$$$$T^{2} -$$$$13\!\cdots\!36$$$$T^{3} +$$$$34\!\cdots\!92$$$$T^{4} -$$$$28\!\cdots\!48$$$$T^{5} +$$$$16\!\cdots\!81$$$$T^{6} -$$$$14\!\cdots\!77$$$$T^{7} +$$$$42\!\cdots\!14$$$$T^{8} -$$$$12\!\cdots\!81$$$$T^{9} + 220766769405446 T^{10} + 209511 T^{11} + T^{12}$$
$59$ $$57\!\cdots\!00$$$$-$$$$17\!\cdots\!00$$$$T -$$$$24\!\cdots\!12$$$$T^{2} +$$$$80\!\cdots\!36$$$$T^{3} +$$$$14\!\cdots\!12$$$$T^{4} +$$$$60\!\cdots\!56$$$$T^{5} +$$$$13\!\cdots\!51$$$$T^{6} +$$$$19\!\cdots\!35$$$$T^{7} +$$$$18\!\cdots\!88$$$$T^{8} +$$$$11\!\cdots\!59$$$$T^{9} + 4274276003847828 T^{10} + 97052259 T^{11} + T^{12}$$
$61$ $$74\!\cdots\!00$$$$-$$$$14\!\cdots\!00$$$$T +$$$$12\!\cdots\!00$$$$T^{2} -$$$$51\!\cdots\!00$$$$T^{3} +$$$$11\!\cdots\!00$$$$T^{4} -$$$$95\!\cdots\!40$$$$T^{5} -$$$$66\!\cdots\!72$$$$T^{6} +$$$$12\!\cdots\!76$$$$T^{7} +$$$$69\!\cdots\!00$$$$T^{8} -$$$$85\!\cdots\!08$$$$T^{9} - 158147996769840 T^{10} + 24195864 T^{11} + T^{12}$$
$67$ $$35\!\cdots\!84$$$$+$$$$12\!\cdots\!12$$$$T +$$$$26\!\cdots\!40$$$$T^{2} +$$$$36\!\cdots\!08$$$$T^{3} +$$$$37\!\cdots\!80$$$$T^{4} +$$$$27\!\cdots\!32$$$$T^{5} +$$$$16\!\cdots\!37$$$$T^{6} +$$$$69\!\cdots\!21$$$$T^{7} +$$$$27\!\cdots\!42$$$$T^{8} +$$$$84\!\cdots\!41$$$$T^{9} + 2939449607113026 T^{10} + 57546057 T^{11} + T^{12}$$
$71$ $$( -$$$$15\!\cdots\!12$$$$+$$$$52\!\cdots\!56$$$$T +$$$$59\!\cdots\!12$$$$T^{2} -$$$$85\!\cdots\!84$$$$T^{3} + 4662393949524436 T^{4} - 112283310 T^{5} + T^{6} )^{2}$$
$73$ $$39\!\cdots\!96$$$$-$$$$61\!\cdots\!12$$$$T +$$$$29\!\cdots\!20$$$$T^{2} +$$$$49\!\cdots\!96$$$$T^{3} +$$$$23\!\cdots\!28$$$$T^{4} -$$$$12\!\cdots\!08$$$$T^{5} -$$$$37\!\cdots\!57$$$$T^{6} +$$$$51\!\cdots\!95$$$$T^{7} +$$$$15\!\cdots\!56$$$$T^{8} +$$$$70\!\cdots\!55$$$$T^{9} + 16085602226180208 T^{10} + 193344135 T^{11} + T^{12}$$
$79$ $$78\!\cdots\!81$$$$-$$$$25\!\cdots\!64$$$$T +$$$$92\!\cdots\!51$$$$T^{2} +$$$$28\!\cdots\!44$$$$T^{3} +$$$$35\!\cdots\!34$$$$T^{4} +$$$$57\!\cdots\!92$$$$T^{5} +$$$$68\!\cdots\!23$$$$T^{6} +$$$$11\!\cdots\!80$$$$T^{7} +$$$$75\!\cdots\!10$$$$T^{8} +$$$$50\!\cdots\!48$$$$T^{9} + 3517482916252011 T^{10} + 29314488 T^{11} + T^{12}$$
$83$ $$12\!\cdots\!16$$$$+$$$$28\!\cdots\!52$$$$T^{2} +$$$$23\!\cdots\!56$$$$T^{4} +$$$$99\!\cdots\!99$$$$T^{6} +$$$$21\!\cdots\!67$$$$T^{8} + 23341793249943165 T^{10} + T^{12}$$
$89$ $$55\!\cdots\!64$$$$-$$$$57\!\cdots\!04$$$$T +$$$$23\!\cdots\!44$$$$T^{2} -$$$$34\!\cdots\!56$$$$T^{3} -$$$$17\!\cdots\!04$$$$T^{4} +$$$$96\!\cdots\!04$$$$T^{5} +$$$$92\!\cdots\!32$$$$T^{6} -$$$$22\!\cdots\!56$$$$T^{7} +$$$$11\!\cdots\!20$$$$T^{8} +$$$$18\!\cdots\!64$$$$T^{9} - 10701808714001400 T^{10} - 119863098 T^{11} + T^{12}$$
$97$ $$21\!\cdots\!84$$$$+$$$$51\!\cdots\!44$$$$T^{2} +$$$$47\!\cdots\!40$$$$T^{4} +$$$$21\!\cdots\!91$$$$T^{6} +$$$$45\!\cdots\!11$$$$T^{8} + 41657428191481425 T^{10} + T^{12}$$