# Properties

 Label 21.7.b.a Level $21$ Weight $7$ Character orbit 21.b Analytic conductor $4.831$ Analytic rank $0$ Dimension $12$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$21 = 3 \cdot 7$$ Weight: $$k$$ $$=$$ $$7$$ Character orbit: $$[\chi]$$ $$=$$ 21.b (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$4.83113575602$$ Analytic rank: $$0$$ Dimension: $$12$$ Coefficient field: $$\mathbb{Q}[x]/(x^{12} + \cdots)$$ Defining polynomial: $$x^{12} + 642 x^{10} + 155265 x^{8} + 17813036 x^{6} + 1003321428 x^{4} + 26369892864 x^{2} + 256461520896$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{2}\cdot 3^{8}\cdot 7^{6}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + \beta_{1} q^{2} + ( 4 - \beta_{3} ) q^{3} + ( -43 + \beta_{2} ) q^{4} + ( \beta_{1} - \beta_{10} ) q^{5} + ( 29 + 5 \beta_{1} + \beta_{5} - \beta_{6} + \beta_{9} ) q^{6} + ( \beta_{2} + \beta_{3} + \beta_{4} ) q^{7} + ( 1 - 32 \beta_{1} + 3 \beta_{3} - \beta_{4} + \beta_{6} + \beta_{8} - 2 \beta_{9} ) q^{8} + ( -56 + \beta_{1} - 3 \beta_{2} - 5 \beta_{3} + \beta_{5} + 2 \beta_{6} + \beta_{11} ) q^{9} +O(q^{10})$$ $$q + \beta_{1} q^{2} + ( 4 - \beta_{3} ) q^{3} + ( -43 + \beta_{2} ) q^{4} + ( \beta_{1} - \beta_{10} ) q^{5} + ( 29 + 5 \beta_{1} + \beta_{5} - \beta_{6} + \beta_{9} ) q^{6} + ( \beta_{2} + \beta_{3} + \beta_{4} ) q^{7} + ( 1 - 32 \beta_{1} + 3 \beta_{3} - \beta_{4} + \beta_{6} + \beta_{8} - 2 \beta_{9} ) q^{8} + ( -56 + \beta_{1} - 3 \beta_{2} - 5 \beta_{3} + \beta_{5} + 2 \beta_{6} + \beta_{11} ) q^{9} + ( -87 - \beta_{1} + 2 \beta_{2} + 10 \beta_{3} - \beta_{4} - \beta_{5} - 3 \beta_{6} + 3 \beta_{7} ) q^{10} + ( -4 - 7 \beta_{1} - 7 \beta_{3} + 2 \beta_{4} - 2 \beta_{6} + \beta_{7} + \beta_{8} + 2 \beta_{9} - 2 \beta_{10} - 2 \beta_{11} ) q^{11} + ( -129 - 3 \beta_{1} + 18 \beta_{2} + 42 \beta_{3} + 6 \beta_{4} - 5 \beta_{5} - 3 \beta_{6} + \beta_{7} + \beta_{9} + 3 \beta_{10} + 2 \beta_{11} ) q^{12} + ( 28 + 4 \beta_{1} - 4 \beta_{2} - 18 \beta_{3} - 10 \beta_{4} + 7 \beta_{5} - 3 \beta_{7} ) q^{13} + ( -1 - 32 \beta_{1} - \beta_{3} - \beta_{4} + \beta_{6} + \beta_{7} + \beta_{8} - 4 \beta_{9} - 3 \beta_{10} - 2 \beta_{11} ) q^{14} + ( -52 + 78 \beta_{1} + 15 \beta_{2} + 2 \beta_{3} + 9 \beta_{4} + 3 \beta_{5} + \beta_{6} - 8 \beta_{7} - 3 \beta_{8} + 2 \beta_{9} + 3 \beta_{11} ) q^{15} + ( 456 + 20 \beta_{1} - 59 \beta_{2} - 104 \beta_{3} - 19 \beta_{4} + 8 \beta_{5} + 15 \beta_{6} - 3 \beta_{7} ) q^{16} + ( -13 + 145 \beta_{1} - 44 \beta_{3} + 3 \beta_{4} - 3 \beta_{5} - 3 \beta_{6} + 2 \beta_{7} - 6 \beta_{8} - 4 \beta_{9} - 13 \beta_{10} - 4 \beta_{11} ) q^{17} + ( -31 + 143 \beta_{1} + 3 \beta_{2} - 37 \beta_{3} + 48 \beta_{4} - 16 \beta_{5} - 2 \beta_{6} + 3 \beta_{7} - 3 \beta_{8} + 24 \beta_{9} + 6 \beta_{10} - 4 \beta_{11} ) q^{18} + ( 974 - 32 \beta_{1} + 8 \beta_{2} + 46 \beta_{3} - 18 \beta_{4} - 23 \beta_{5} - 9 \beta_{7} ) q^{19} + ( 40 - 263 \beta_{1} + 94 \beta_{3} - 26 \beta_{4} + 3 \beta_{5} + 26 \beta_{6} - 4 \beta_{7} + 2 \beta_{8} - 38 \beta_{9} + 20 \beta_{10} + 8 \beta_{11} ) q^{20} + ( -459 + 33 \beta_{1} + 18 \beta_{2} + 7 \beta_{5} - 3 \beta_{6} + 4 \beta_{7} - 3 \beta_{8} - 2 \beta_{9} + 9 \beta_{10} - \beta_{11} ) q^{21} + ( 1355 - 34 \beta_{1} - 52 \beta_{2} + 156 \beta_{3} - 47 \beta_{4} + 2 \beta_{5} - 45 \beta_{6} + 9 \beta_{7} ) q^{22} + ( -13 + 343 \beta_{1} - 7 \beta_{3} + 17 \beta_{4} + 6 \beta_{5} - 17 \beta_{6} + 4 \beta_{7} - 5 \beta_{8} + 38 \beta_{9} + 62 \beta_{10} - 8 \beta_{11} ) q^{23} + ( 1018 - 784 \beta_{1} + 24 \beta_{2} + 74 \beta_{3} + 33 \beta_{4} + 16 \beta_{5} + 12 \beta_{6} - 2 \beta_{7} + 18 \beta_{8} - 57 \beta_{9} + 39 \beta_{10} - 6 \beta_{11} ) q^{24} + ( -6487 - 20 \beta_{1} - 44 \beta_{2} - 156 \beta_{3} + 2 \beta_{4} - 98 \beta_{5} + 66 \beta_{6} + 12 \beta_{7} ) q^{25} + ( 58 + 101 \beta_{1} + 196 \beta_{3} + 4 \beta_{4} + 21 \beta_{5} - 4 \beta_{6} - 10 \beta_{7} - 4 \beta_{8} + 70 \beta_{9} - 42 \beta_{10} + 20 \beta_{11} ) q^{26} + ( 9598 + 184 \beta_{1} - 102 \beta_{2} + 145 \beta_{3} + 24 \beta_{4} + 37 \beta_{5} + 2 \beta_{6} + 9 \beta_{7} - 6 \beta_{8} - 36 \beta_{9} - 60 \beta_{10} - 2 \beta_{11} ) q^{27} + ( 3381 + 49 \beta_{1} - 93 \beta_{2} - 142 \beta_{3} - 44 \beta_{4} + 49 \beta_{5} ) q^{28} + ( -87 - 854 \beta_{1} - 362 \beta_{3} + 5 \beta_{4} - 48 \beta_{5} - 5 \beta_{6} - 7 \beta_{7} + 4 \beta_{8} - 72 \beta_{9} + 38 \beta_{10} + 14 \beta_{11} ) q^{29} + ( -8498 - 540 \beta_{1} + 315 \beta_{2} + 137 \beta_{3} + 105 \beta_{4} - 10 \beta_{5} + 15 \beta_{6} + 14 \beta_{7} + 15 \beta_{8} - 10 \beta_{9} - 186 \beta_{10} - 20 \beta_{11} ) q^{30} + ( -761 + 38 \beta_{1} - 80 \beta_{2} - 38 \beta_{3} - 133 \beta_{4} + 122 \beta_{5} - 57 \beta_{6} - 27 \beta_{7} ) q^{31} + ( -119 + 1572 \beta_{1} - 257 \beta_{3} + 115 \beta_{4} - 6 \beta_{5} - 115 \beta_{6} - 4 \beta_{7} + 5 \beta_{8} + 226 \beta_{9} - 60 \beta_{10} + 8 \beta_{11} ) q^{32} + ( -5558 + 1099 \beta_{1} + 162 \beta_{2} - 60 \beta_{3} + 42 \beta_{4} + 35 \beta_{5} + 88 \beta_{6} - 21 \beta_{7} + 12 \beta_{8} + 32 \beta_{9} + 111 \beta_{10} - 6 \beta_{11} ) q^{33} + ( -14129 + 120 \beta_{1} + 474 \beta_{2} - 230 \beta_{3} - 215 \beta_{4} + 174 \beta_{5} - 75 \beta_{6} + 21 \beta_{7} ) q^{34} + ( 120 + 165 \beta_{1} + 323 \beta_{3} - 62 \beta_{4} + 21 \beta_{5} + 62 \beta_{6} - 8 \beta_{7} - \beta_{8} - 66 \beta_{9} - 46 \beta_{10} + 16 \beta_{11} ) q^{35} + ( -16270 + 506 \beta_{1} + 444 \beta_{2} + 89 \beta_{3} + 204 \beta_{4} - 112 \beta_{5} - 182 \beta_{6} + 69 \beta_{7} + 3 \beta_{8} - 48 \beta_{9} - 114 \beta_{10} - 28 \beta_{11} ) q^{36} + ( 17809 - 126 \beta_{1} - 62 \beta_{2} + 340 \beta_{3} + 43 \beta_{4} - 156 \beta_{5} - 15 \beta_{6} + 45 \beta_{7} ) q^{37} + ( -94 + 165 \beta_{1} - 456 \beta_{3} - 8 \beta_{4} - 69 \beta_{5} + 8 \beta_{6} - 18 \beta_{7} + 8 \beta_{8} - 118 \beta_{9} - 162 \beta_{10} + 36 \beta_{11} ) q^{38} + ( 13314 - 1497 \beta_{1} - 351 \beta_{2} - 9 \beta_{3} + 75 \beta_{4} - 121 \beta_{5} - 9 \beta_{6} - 31 \beta_{7} + 104 \beta_{9} + 150 \beta_{10} + 31 \beta_{11} ) q^{39} + ( 15664 + 435 \beta_{1} - 762 \beta_{2} - 1868 \beta_{3} - 134 \beta_{4} + 81 \beta_{5} + 288 \beta_{6} + 66 \beta_{7} ) q^{40} + ( 181 - 1101 \beta_{1} + 736 \beta_{3} - 83 \beta_{4} + 93 \beta_{5} + 83 \beta_{6} + 44 \beta_{7} + 2 \beta_{8} - 68 \beta_{9} + 121 \beta_{10} - 88 \beta_{11} ) q^{41} + ( -3133 - 1437 \beta_{1} + 177 \beta_{2} + 68 \beta_{3} - 93 \beta_{4} + 84 \beta_{5} + 88 \beta_{6} - 38 \beta_{7} + 18 \beta_{8} - 37 \beta_{9} + 93 \beta_{10} + 6 \beta_{11} ) q^{42} + ( 2775 - 330 \beta_{1} - 678 \beta_{2} + 1360 \beta_{3} + 385 \beta_{4} - 120 \beta_{5} - 105 \beta_{6} - 105 \beta_{7} ) q^{43} + ( 428 + 1498 \beta_{1} + 1252 \beta_{3} - 180 \beta_{4} + 96 \beta_{5} + 180 \beta_{6} - 28 \beta_{7} + 12 \beta_{8} - 112 \beta_{9} + 364 \beta_{10} + 56 \beta_{11} ) q^{44} + ( -275 + 661 \beta_{1} - 516 \beta_{2} - 194 \beta_{3} - 75 \beta_{4} - 128 \beta_{5} - 283 \beta_{6} + 27 \beta_{7} - 114 \beta_{8} + 156 \beta_{9} + 291 \beta_{10} + 64 \beta_{11} ) q^{45} + ( -34279 - 456 \beta_{1} + 1062 \beta_{2} + 1556 \beta_{3} + 221 \beta_{4} - 192 \beta_{5} - 141 \beta_{6} - 123 \beta_{7} ) q^{46} + ( 394 - 6662 \beta_{1} + 1570 \beta_{3} - 170 \beta_{4} + 192 \beta_{5} + 170 \beta_{6} + 80 \beta_{7} + 14 \beta_{8} - 116 \beta_{9} - 216 \beta_{10} - 160 \beta_{11} ) q^{47} + ( 72078 + 1371 \beta_{1} - 1620 \beta_{2} - 192 \beta_{3} - 459 \beta_{4} + 277 \beta_{5} + 342 \beta_{6} - 110 \beta_{7} + 24 \beta_{8} - 53 \beta_{9} - 63 \beta_{10} + 38 \beta_{11} ) q^{48} + 16807 q^{49} + ( -1560 - 3163 \beta_{1} - 4868 \beta_{3} + 572 \beta_{4} - 426 \beta_{5} - 572 \beta_{6} + 68 \beta_{7} - 44 \beta_{8} + 156 \beta_{9} + 84 \beta_{10} - 136 \beta_{11} ) q^{50} + ( -30899 - 1237 \beta_{1} + 186 \beta_{2} - 325 \beta_{3} - 675 \beta_{4} + 184 \beta_{5} + 135 \beta_{6} - 134 \beta_{7} - 111 \beta_{8} + 258 \beta_{9} - 162 \beta_{10} + 126 \beta_{11} ) q^{51} + ( -10352 - 921 \beta_{1} + 1146 \beta_{2} + 3440 \beta_{3} + 842 \beta_{4} - 759 \beta_{5} - 252 \beta_{6} + 90 \beta_{7} ) q^{52} + ( -105 + 5336 \beta_{1} - 786 \beta_{3} - 141 \beta_{4} - 132 \beta_{5} + 141 \beta_{6} - 9 \beta_{7} - 48 \beta_{8} - 528 \beta_{9} - 148 \beta_{10} + 18 \beta_{11} ) q^{53} + ( -21505 + 15980 \beta_{1} + 102 \beta_{2} - 802 \beta_{3} - 510 \beta_{4} + 350 \beta_{5} + 205 \beta_{6} + 126 \beta_{7} - 102 \beta_{8} + 99 \beta_{9} + 114 \beta_{10} - 52 \beta_{11} ) q^{54} + ( -62499 + 698 \beta_{1} + 1444 \beta_{2} - 3070 \beta_{3} + 693 \beta_{4} - 82 \beta_{5} + 789 \beta_{6} - 9 \beta_{7} ) q^{55} + ( 225 + 5583 \beta_{1} + 1009 \beta_{3} + 29 \beta_{4} + 147 \beta_{5} - 29 \beta_{6} + 20 \beta_{7} - 29 \beta_{8} + 312 \beta_{9} - 60 \beta_{10} - 40 \beta_{11} ) q^{56} + ( -36858 + 3273 \beta_{1} - 297 \beta_{2} - 1761 \beta_{3} - 255 \beta_{4} - 583 \beta_{5} - 351 \beta_{6} - 169 \beta_{7} + 96 \beta_{8} + 32 \beta_{9} + 318 \beta_{10} + 97 \beta_{11} ) q^{57} + ( 95338 + 1322 \beta_{1} - 1954 \beta_{2} - 4324 \beta_{3} - 600 \beta_{4} + 1202 \beta_{5} + 360 \beta_{6} - 240 \beta_{7} ) q^{58} + ( -1125 + 1704 \beta_{1} - 3332 \beta_{3} + 371 \beta_{4} - 315 \beta_{5} - 371 \beta_{6} + 62 \beta_{7} + 178 \beta_{8} - 12 \beta_{9} - 72 \beta_{10} - 124 \beta_{11} ) q^{59} + ( 57155 - 17980 \beta_{1} - 603 \beta_{2} + 1173 \beta_{3} - 510 \beta_{4} + 16 \beta_{5} - 334 \beta_{6} + 141 \beta_{7} + 123 \beta_{8} - 722 \beta_{9} - 264 \beta_{10} - 48 \beta_{11} ) q^{60} + ( 76974 + 512 \beta_{1} + 2000 \beta_{2} + 146 \beta_{3} + 396 \beta_{4} + 845 \beta_{5} - 258 \beta_{6} - 75 \beta_{7} ) q^{61} + ( 1716 + 252 \beta_{1} + 5272 \beta_{3} - 376 \beta_{4} + 480 \beta_{5} + 376 \beta_{6} - 190 \beta_{7} - 80 \beta_{8} + 588 \beta_{9} - 78 \beta_{10} + 380 \beta_{11} ) q^{62} + ( 416 - 3985 \beta_{1} + 444 \beta_{2} + 566 \beta_{3} - 120 \beta_{4} - 49 \beta_{5} + 25 \beta_{6} + 123 \beta_{7} - 24 \beta_{8} + 96 \beta_{9} - 222 \beta_{10} - \beta_{11} ) q^{63} + ( -112046 - 1958 \beta_{1} + 639 \beta_{2} + 8136 \beta_{3} + 1205 \beta_{4} - 1406 \beta_{5} - 1005 \beta_{6} + 453 \beta_{7} ) q^{64} + ( 855 + 13146 \beta_{1} + 3142 \beta_{3} - 253 \beta_{4} + 324 \beta_{5} + 253 \beta_{6} + 23 \beta_{7} + 136 \beta_{8} + 96 \beta_{9} - 710 \beta_{10} - 46 \beta_{11} ) q^{65} + ( -118547 - 11538 \beta_{1} + 504 \beta_{2} - 1528 \beta_{3} + 939 \beta_{4} - 778 \beta_{5} - 183 \beta_{6} - 133 \beta_{7} + 162 \beta_{8} + 398 \beta_{9} - 966 \beta_{10} - 260 \beta_{11} ) q^{66} + ( -18639 + 1058 \beta_{1} - 202 \beta_{2} + 956 \beta_{3} + 39 \beta_{4} + 1598 \beta_{5} - 1011 \beta_{6} + 471 \beta_{7} ) q^{67} + ( 2166 - 35668 \beta_{1} + 6342 \beta_{3} - 882 \beta_{4} + 480 \beta_{5} + 882 \beta_{6} - 162 \beta_{7} + 90 \beta_{8} - 480 \beta_{9} + 542 \beta_{10} + 324 \beta_{11} ) q^{68} + ( 12263 + 11611 \beta_{1} - 456 \beta_{2} - 1028 \beta_{3} - 471 \beta_{4} - 271 \beta_{5} - 21 \beta_{6} + 512 \beta_{7} + 78 \beta_{8} + 732 \beta_{9} + 129 \beta_{10} - 252 \beta_{11} ) q^{69} + ( -33383 + 707 \beta_{1} - 578 \beta_{2} - 3756 \beta_{3} - 11 \beta_{4} - 91 \beta_{5} + 777 \beta_{6} + 21 \beta_{7} ) q^{70} + ( -2162 - 12701 \beta_{1} - 7645 \beta_{3} + 1008 \beta_{4} - 822 \beta_{5} - 1008 \beta_{6} - 245 \beta_{7} - 33 \beta_{8} + 862 \beta_{9} - 606 \beta_{10} + 490 \beta_{11} ) q^{71} + ( -45270 - 30210 \beta_{1} + 81 \beta_{2} + 444 \beta_{3} - 423 \beta_{4} + 828 \beta_{5} + 417 \beta_{6} + 435 \beta_{7} + 252 \beta_{8} - 1206 \beta_{9} + 1638 \beta_{10} - 300 \beta_{11} ) q^{72} + ( 49818 - 128 \beta_{2} + 180 \beta_{3} + 972 \beta_{4} - 114 \beta_{5} + 264 \beta_{6} - 150 \beta_{7} ) q^{73} + ( -874 + 20318 \beta_{1} - 3562 \beta_{3} - 58 \beta_{4} - 438 \beta_{5} + 58 \beta_{6} + 28 \beta_{7} - 62 \beta_{8} - 1048 \beta_{9} + 996 \beta_{10} - 56 \beta_{11} ) q^{74} + ( 52146 + 30618 \beta_{1} - 234 \beta_{2} + 5895 \beta_{3} - 2460 \beta_{4} + 898 \beta_{5} + 588 \beta_{6} - 260 \beta_{7} + 324 \beta_{8} - 1376 \beta_{9} - 1428 \beta_{10} + 266 \beta_{11} ) q^{75} + ( 50438 - 211 \beta_{1} - 948 \beta_{2} - 1896 \beta_{3} - 1850 \beta_{4} - 85 \beta_{5} + 156 \beta_{6} - 282 \beta_{7} ) q^{76} + ( 547 + 7851 \beta_{1} + 1772 \beta_{3} - 237 \beta_{4} + 147 \beta_{5} + 237 \beta_{6} - 8 \beta_{7} + 90 \beta_{8} - 164 \beta_{9} + 661 \beta_{10} + 16 \beta_{11} ) q^{77} + ( 155353 + 31776 \beta_{1} + 21 \beta_{2} - 29 \beta_{3} + 2784 \beta_{4} - 1362 \beta_{5} - 784 \beta_{6} - 145 \beta_{7} - 351 \beta_{8} + 346 \beta_{9} - 570 \beta_{10} - 132 \beta_{11} ) q^{78} + ( -22564 - 808 \beta_{1} + 5220 \beta_{2} + 2748 \beta_{3} + 2032 \beta_{4} - 1552 \beta_{5} + 300 \beta_{6} + 444 \beta_{7} ) q^{79} + ( -1480 + 42929 \beta_{1} - 4158 \beta_{3} + 1402 \beta_{4} - 141 \beta_{5} - 1402 \beta_{6} - 102 \beta_{7} - 634 \beta_{8} + 2726 \beta_{9} + 2402 \beta_{10} + 204 \beta_{11} ) q^{80} + ( 48664 - 18818 \beta_{1} - 174 \beta_{2} - 7904 \beta_{3} - 939 \beta_{4} + 1672 \beta_{5} + 275 \beta_{6} - 369 \beta_{7} - 420 \beta_{8} + 72 \beta_{9} - 636 \beta_{10} - 248 \beta_{11} ) q^{81} + ( 93401 + 582 \beta_{1} - 2894 \beta_{2} - 6482 \beta_{3} - 2717 \beta_{4} + 300 \beta_{5} + 867 \beta_{6} - 585 \beta_{7} ) q^{82} + ( -1179 - 12634 \beta_{1} - 1594 \beta_{3} + 793 \beta_{4} + 165 \beta_{5} - 793 \beta_{6} + 358 \beta_{7} + 104 \beta_{8} + 1200 \beta_{9} - 1132 \beta_{10} - 716 \beta_{11} ) q^{83} + ( 115354 - 9537 \beta_{1} - 2115 \beta_{2} - 2401 \beta_{3} + 294 \beta_{4} + 231 \beta_{5} + 573 \beta_{6} - 78 \beta_{7} - 15 \beta_{8} + 333 \beta_{9} - 249 \beta_{10} - 54 \beta_{11} ) q^{84} + ( -247607 - 2218 \beta_{1} + 1494 \beta_{2} + 5692 \beta_{3} + 113 \beta_{4} - 1132 \beta_{5} - 381 \beta_{6} - 705 \beta_{7} ) q^{85} + ( -698 + 44934 \beta_{1} - 2674 \beta_{3} - 162 \beta_{4} - 150 \beta_{5} + 162 \beta_{6} + 280 \beta_{7} - 678 \beta_{8} - 1184 \beta_{9} - 3360 \beta_{10} - 560 \beta_{11} ) q^{86} + ( -280881 - 34816 \beta_{1} - 2526 \beta_{2} - 218 \beta_{3} - 1011 \beta_{4} + 292 \beta_{5} + 625 \beta_{6} + 263 \beta_{7} + 240 \beta_{8} - 1600 \beta_{9} - 456 \beta_{10} + 390 \beta_{11} ) q^{87} + ( -132802 - 1124 \beta_{1} - 4386 \beta_{2} - 904 \beta_{3} - 1136 \beta_{4} - 1124 \beta_{5} + 672 \beta_{6} - 672 \beta_{7} ) q^{88} + ( 1793 - 55735 \beta_{1} + 4164 \beta_{3} - 575 \beta_{4} + 303 \beta_{5} + 575 \beta_{6} - 306 \beta_{7} - 634 \beta_{8} + 68 \beta_{9} + 2299 \beta_{10} + 612 \beta_{11} ) q^{89} + ( -62947 + 17687 \beta_{1} + 10398 \beta_{2} + 9134 \beta_{3} + 2469 \beta_{4} + 467 \beta_{5} + 505 \beta_{6} - 969 \beta_{7} - 516 \beta_{8} - 546 \beta_{9} + 2490 \beta_{10} + 716 \beta_{11} ) q^{90} + ( -103005 - 14 \beta_{1} + 1188 \beta_{2} + 1216 \beta_{3} - 513 \beta_{4} + 763 \beta_{5} - 567 \beta_{6} - 210 \beta_{7} ) q^{91} + ( 610 - 66840 \beta_{1} + 2322 \beta_{3} - 1102 \beta_{4} + 90 \beta_{5} + 1102 \beta_{6} + 336 \beta_{7} + 742 \beta_{8} - 2696 \beta_{9} + 776 \beta_{10} - 672 \beta_{11} ) q^{92} + ( 34177 - 36942 \beta_{1} - 3456 \beta_{2} + 1370 \beta_{3} + 2403 \beta_{4} - 1886 \beta_{5} - 279 \beta_{6} - 275 \beta_{7} - 48 \beta_{8} + 1888 \beta_{9} + 1314 \beta_{10} - 22 \beta_{11} ) q^{93} + ( 670910 + 272 \beta_{1} - 10092 \beta_{2} - 6928 \beta_{3} - 5106 \beta_{4} - 424 \beta_{5} + 426 \beta_{6} + 270 \beta_{7} ) q^{94} + ( 1685 + 23828 \beta_{1} + 7190 \beta_{3} - 655 \beta_{4} + 720 \beta_{5} + 655 \beta_{6} + 205 \beta_{7} + 940 \beta_{8} - 280 \beta_{9} + 292 \beta_{10} - 410 \beta_{11} ) q^{95} + ( -98730 + 107114 \beta_{1} + 426 \beta_{2} - 4520 \beta_{3} + 6789 \beta_{4} - 986 \beta_{5} - 710 \beta_{6} - 100 \beta_{7} - 468 \beta_{8} + 3161 \beta_{9} + 663 \beta_{10} - 150 \beta_{11} ) q^{96} + ( 135098 + 224 \beta_{1} - 10160 \beta_{2} + 2168 \beta_{3} + 2004 \beta_{4} - 328 \beta_{5} - 204 \beta_{6} + 756 \beta_{7} ) q^{97} + 16807 \beta_{1} q^{98} + ( 436198 + 29323 \beta_{1} - 570 \beta_{2} + 6319 \beta_{3} - 606 \beta_{4} + 1042 \beta_{5} - 1474 \beta_{6} + 1113 \beta_{7} + 237 \beta_{8} + 714 \beta_{9} + 3090 \beta_{10} - 32 \beta_{11} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$12 q + 52 q^{3} - 516 q^{4} + 350 q^{6} - 644 q^{9} + O(q^{10})$$ $$12 q + 52 q^{3} - 516 q^{4} + 350 q^{6} - 644 q^{9} - 1092 q^{10} - 1726 q^{12} + 384 q^{13} - 632 q^{15} + 5892 q^{16} - 40 q^{18} + 11304 q^{19} - 5488 q^{21} + 15312 q^{22} + 12138 q^{24} - 77292 q^{25} + 114796 q^{27} + 41160 q^{28} - 101908 q^{30} - 9360 q^{31} - 65744 q^{33} - 169008 q^{34} - 195652 q^{36} + 212016 q^{37} + 159544 q^{39} + 196644 q^{40} - 37730 q^{42} + 28080 q^{43} - 4760 q^{45} - 418512 q^{46} + 865742 q^{48} + 201684 q^{49} - 371880 q^{51} - 138300 q^{52} - 254170 q^{54} - 732144 q^{55} - 440624 q^{57} + 1164240 q^{58} + 677660 q^{60} + 926736 q^{61} + 2744 q^{63} - 1380108 q^{64} - 1414588 q^{66} - 223104 q^{67} + 153048 q^{69} - 382788 q^{70} - 540192 q^{72} + 600984 q^{73} + 594716 q^{75} + 604596 q^{76} + 1866140 q^{78} - 276864 q^{79} + 617596 q^{81} + 1138200 q^{82} + 1398754 q^{84} - 3002472 q^{85} - 3372824 q^{87} - 1599048 q^{88} - 788032 q^{90} - 1243032 q^{91} + 408168 q^{93} + 8059296 q^{94} - 1141658 q^{96} + 1621416 q^{97} + 5211904 q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{12} + 642 x^{10} + 155265 x^{8} + 17813036 x^{6} + 1003321428 x^{4} + 26369892864 x^{2} + 256461520896$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2} + 107$$ $$\beta_{3}$$ $$=$$ $$($$$$-8717 \nu^{11} + 39992 \nu^{10} - 4877786 \nu^{9} + 23410928 \nu^{8} - 988016909 \nu^{7} + 4963685432 \nu^{6} - 91199451516 \nu^{5} + 465298015776 \nu^{4} - 3869925659460 \nu^{3} + 18284751065952 \nu^{2} - 58019012432640 \nu + 231704485963776$$$$)/ 838396403712$$ $$\beta_{4}$$ $$=$$ $$($$$$8717 \nu^{11} + 90904 \nu^{10} + 4877786 \nu^{9} + 14227504 \nu^{8} + 988016909 \nu^{7} - 7060434152 \nu^{6} + 91199451516 \nu^{5} - 1690333280736 \nu^{4} + 3869925659460 \nu^{3} - 106787303818272 \nu^{2} + 58019012432640 \nu - 1920833907406848$$$$)/ 838396403712$$ $$\beta_{5}$$ $$=$$ $$($$$$-8717 \nu^{11} - 289192 \nu^{10} - 4877786 \nu^{9} - 163888720 \nu^{8} - 988016909 \nu^{7} - 33639776680 \nu^{6} - 91199451516 \nu^{5} - 3119382867936 \nu^{4} - 3869925659460 \nu^{3} - 129447717958944 \nu^{2} - 58438210634496 \nu - 1857542063118336$$$$)/ 419198201856$$ $$\beta_{6}$$ $$=$$ $$($$$$-43585 \nu^{11} + 269368 \nu^{10} - 24388930 \nu^{9} + 107895664 \nu^{8} - 4940084545 \nu^{7} + 10582264888 \nu^{6} - 455997257580 \nu^{5} - 104547702624 \nu^{4} - 19349628297300 \nu^{3} - 30739378559904 \nu^{2} - 290933458566912 \nu - 525309830019072$$$$)/ 838396403712$$ $$\beta_{7}$$ $$=$$ $$($$$$-8717 \nu^{11} - 1078816 \nu^{10} - 4877786 \nu^{9} - 618140608 \nu^{8} - 988016909 \nu^{7} - 126929581600 \nu^{6} - 91199451516 \nu^{5} - 11431899856128 \nu^{4} - 3869925659460 \nu^{3} - 435891166897536 \nu^{2} - 58438210634496 \nu - 5591050360971264$$$$)/ 419198201856$$ $$\beta_{8}$$ $$=$$ $$($$$$-53833 \nu^{11} - 39992 \nu^{10} - 32188498 \nu^{9} - 23410928 \nu^{8} - 6963320137 \nu^{7} - 4963685432 \nu^{6} - 656997677004 \nu^{5} - 465298015776 \nu^{4} - 24121328749812 \nu^{3} - 18284751065952 \nu^{2} - 156046906439424 \nu - 231704485963776$$$$)/ 838396403712$$ $$\beta_{9}$$ $$=$$ $$($$$$-66143 \nu^{11} + 129224 \nu^{10} - 38044286 \nu^{9} + 70245008 \nu^{8} - 7927736159 \nu^{7} + 13785034952 \nu^{6} - 738896370324 \nu^{5} + 1258190804832 \nu^{4} - 29894528044332 \nu^{3} + 56308713695136 \nu^{2} - 406599919665408 \nu + 929885722859520$$$$)/ 838396403712$$ $$\beta_{10}$$ $$=$$ $$($$$$-15539 \nu^{11} - 8910438 \nu^{9} - 1831850931 \nu^{7} - 165238859332 \nu^{5} - 6307998860604 \nu^{3} - 80801728729344 \nu$$$$)/ 139732733952$$ $$\beta_{11}$$ $$=$$ $$($$$$149263 \nu^{11} - 1288024 \nu^{10} + 104296798 \nu^{9} - 735207472 \nu^{8} + 26450785807 \nu^{7} - 150641987416 \nu^{6} + 2929763188308 \nu^{5} - 13620686692512 \nu^{4} + 134787719825388 \nu^{3} - 528769382724576 \nu^{2} + 2044783165420800 \nu - 6985183452162048$$$$)/ 838396403712$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{2} - 107$$ $$\nu^{3}$$ $$=$$ $$-2 \beta_{9} + \beta_{8} + \beta_{6} - \beta_{4} + 3 \beta_{3} - 160 \beta_{1} + 1$$ $$\nu^{4}$$ $$=$$ $$-3 \beta_{7} + 15 \beta_{6} + 8 \beta_{5} - 19 \beta_{4} - 104 \beta_{3} - 251 \beta_{2} + 20 \beta_{1} + 16904$$ $$\nu^{5}$$ $$=$$ $$8 \beta_{11} - 60 \beta_{10} + 738 \beta_{9} - 251 \beta_{8} - 4 \beta_{7} - 371 \beta_{6} - 6 \beta_{5} + 371 \beta_{4} - 1025 \beta_{3} + 30244 \beta_{1} - 375$$ $$\nu^{6}$$ $$=$$ $$1413 \beta_{7} - 5805 \beta_{6} - 3966 \beta_{5} + 7285 \beta_{4} + 41416 \beta_{3} + 56383 \beta_{2} - 8358 \beta_{1} - 3153838$$ $$\nu^{7}$$ $$=$$ $$-2960 \beta_{11} + 29472 \beta_{10} - 209182 \beta_{9} + 56383 \beta_{8} + 1480 \beta_{7} + 102823 \beta_{6} - 288 \beta_{5} - 102823 \beta_{4} + 253805 \beta_{3} - 6165258 \beta_{1} + 99287$$ $$\nu^{8}$$ $$=$$ $$-453357 \beta_{7} + 1670049 \beta_{6} + 1298200 \beta_{5} - 2105949 \beta_{4} - 12145880 \beta_{3} - 12641423 \beta_{2} + 2514892 \beta_{1} + 636751766$$ $$\nu^{9}$$ $$=$$ $$871800 \beta_{11} - 9572868 \beta_{10} + 53984434 \beta_{9} - 12641423 \beta_{8} - 435900 \beta_{7} - 26001815 \beta_{6} + 554502 \beta_{5} + 26001815 \beta_{4} - 58465437 \beta_{3} + 1314534716 \beta_{1} - 24021011$$ $$\nu^{10}$$ $$=$$ $$124917849 \beta_{7} - 432817833 \beta_{6} - 361948690 \beta_{5} + 550835465 \beta_{4} + 3188986312 \beta_{3} + 2865198511 \beta_{2} - 669848674 \beta_{1} - 134853558850$$ $$\nu^{11}$$ $$=$$ $$-236035264 \beta_{11} + 2643975480 \beta_{10} - 13331938422 \beta_{9} + 2865198511 \beta_{8} + 118017632 \beta_{7} + 6327741175 \beta_{6} - 220210404 \beta_{5} - 6327741175 \beta_{4} + 13286927101 \beta_{3} - 288837920798 \beta_{1} + 5651285103$$

## Character values

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

 $$n$$ $$8$$ $$10$$ $$\chi(n)$$ $$-1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
8.1
 − 15.2632i − 12.1723i − 11.9090i − 7.64160i − 5.92187i − 5.05795i 5.05795i 5.92187i 7.64160i 11.9090i 12.1723i 15.2632i
15.2632i 26.9946 + 0.538759i −168.965 118.492i 8.22319 412.025i −129.642 1602.11i 728.419 + 29.0872i −1808.56
8.2 12.1723i −18.4482 19.7146i −84.1649 4.50929i −239.971 + 224.557i 129.642 245.453i −48.3270 + 727.396i −54.8885
8.3 11.9090i −13.4494 + 23.4118i −77.8243 201.466i 278.811 + 160.169i −129.642 164.633i −367.226 629.751i 2399.25
8.4 7.64160i −4.69404 + 26.5888i 5.60598 237.150i 203.181 + 35.8700i 129.642 531.901i −684.932 249.618i −1812.21
8.5 5.92187i 25.5592 + 8.70200i 28.9315 144.537i 51.5321 151.359i 129.642 550.328i 577.550 + 444.833i 855.929
8.6 5.05795i 10.0378 25.0648i 38.4172 24.8176i −126.776 50.7707i −129.642 518.021i −527.485 503.190i −125.526
8.7 5.05795i 10.0378 + 25.0648i 38.4172 24.8176i −126.776 + 50.7707i −129.642 518.021i −527.485 + 503.190i −125.526
8.8 5.92187i 25.5592 8.70200i 28.9315 144.537i 51.5321 + 151.359i 129.642 550.328i 577.550 444.833i 855.929
8.9 7.64160i −4.69404 26.5888i 5.60598 237.150i 203.181 35.8700i 129.642 531.901i −684.932 + 249.618i −1812.21
8.10 11.9090i −13.4494 23.4118i −77.8243 201.466i 278.811 160.169i −129.642 164.633i −367.226 + 629.751i 2399.25
8.11 12.1723i −18.4482 + 19.7146i −84.1649 4.50929i −239.971 224.557i 129.642 245.453i −48.3270 727.396i −54.8885
8.12 15.2632i 26.9946 0.538759i −168.965 118.492i 8.22319 + 412.025i −129.642 1602.11i 728.419 29.0872i −1808.56
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 8.12 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 21.7.b.a 12
3.b odd 2 1 inner 21.7.b.a 12
4.b odd 2 1 336.7.d.a 12
7.b odd 2 1 147.7.b.b 12
12.b even 2 1 336.7.d.a 12
21.c even 2 1 147.7.b.b 12

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
21.7.b.a 12 1.a even 1 1 trivial
21.7.b.a 12 3.b odd 2 1 inner
147.7.b.b 12 7.b odd 2 1
147.7.b.b 12 21.c even 2 1
336.7.d.a 12 4.b odd 2 1
336.7.d.a 12 12.b even 2 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{7}^{\mathrm{new}}(21, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$256461520896 + 26369892864 T^{2} + 1003321428 T^{4} + 17813036 T^{6} + 155265 T^{8} + 642 T^{10} + T^{12}$$
$3$ $$150094635296999121 - 10706338868921748 T + 472787044069194 T^{2} - 30390812839116 T^{3} + 1396220395635 T^{4} - 55534403520 T^{5} + 2309848164 T^{6} - 76178880 T^{7} + 2627235 T^{8} - 78444 T^{9} + 1674 T^{10} - 52 T^{11} + T^{12}$$
$5$ $$83\!\cdots\!00$$$$+$$$$42\!\cdots\!00$$$$T^{2} + 738425820008670000 T^{4} + 111931245096800 T^{6} + 6042194604 T^{8} + 132396 T^{10} + T^{12}$$
$7$ $$( -16807 + T^{2} )^{6}$$
$11$ $$19\!\cdots\!64$$$$+$$$$83\!\cdots\!16$$$$T^{2} +$$$$12\!\cdots\!72$$$$T^{4} + 8636921772315251264 T^{6} + 22710805391748 T^{8} + 9565092 T^{10} + T^{12}$$
$13$ $$( -2574877218622502128 - 7392538099344432 T + 16228433562612 T^{2} + 6302573448 T^{3} - 9739692 T^{4} - 192 T^{5} + T^{6} )^{2}$$
$17$ $$11\!\cdots\!04$$$$+$$$$10\!\cdots\!28$$$$T^{2} +$$$$27\!\cdots\!92$$$$T^{4} +$$$$24\!\cdots\!44$$$$T^{6} + 9542391068326212 T^{8} + 165385596 T^{10} + T^{12}$$
$19$ $$( -$$$$16\!\cdots\!72$$$$- 1614784352083337088 T + 507083837895348 T^{2} + 432085652424 T^{3} - 102302352 T^{4} - 5652 T^{5} + T^{6} )^{2}$$
$23$ $$47\!\cdots\!16$$$$+$$$$51\!\cdots\!24$$$$T^{2} +$$$$23\!\cdots\!80$$$$T^{4} +$$$$37\!\cdots\!84$$$$T^{6} + 266145170853608100 T^{8} + 848162964 T^{10} + T^{12}$$
$29$ $$57\!\cdots\!64$$$$+$$$$56\!\cdots\!32$$$$T^{2} +$$$$58\!\cdots\!28$$$$T^{4} +$$$$23\!\cdots\!00$$$$T^{6} + 4339677865109828640 T^{8} + 3577363776 T^{10} + T^{12}$$
$31$ $$( -$$$$72\!\cdots\!68$$$$-$$$$50\!\cdots\!64$$$$T + 887083000203543360 T^{2} + 10794530462176 T^{3} - 2088767868 T^{4} + 4680 T^{5} + T^{6} )^{2}$$
$37$ $$($$$$14\!\cdots\!68$$$$-$$$$38\!\cdots\!32$$$$T - 2477938164281512560 T^{2} + 94429381867264 T^{3} + 1857447768 T^{4} - 106008 T^{5} + T^{6} )^{2}$$
$41$ $$19\!\cdots\!96$$$$+$$$$13\!\cdots\!24$$$$T^{2} +$$$$11\!\cdots\!48$$$$T^{4} +$$$$18\!\cdots\!04$$$$T^{6} + 95012024604887472516 T^{8} + 17953755708 T^{10} + T^{12}$$
$43$ $$( -$$$$98\!\cdots\!52$$$$-$$$$73\!\cdots\!96$$$$T +$$$$19\!\cdots\!52$$$$T^{2} + 747262144358528 T^{3} - 28179176280 T^{4} - 14040 T^{5} + T^{6} )^{2}$$
$47$ $$26\!\cdots\!84$$$$+$$$$85\!\cdots\!16$$$$T^{2} +$$$$81\!\cdots\!00$$$$T^{4} +$$$$21\!\cdots\!28$$$$T^{6} +$$$$20\!\cdots\!56$$$$T^{8} + 79708865808 T^{10} + T^{12}$$
$53$ $$12\!\cdots\!24$$$$+$$$$23\!\cdots\!36$$$$T^{2} +$$$$14\!\cdots\!44$$$$T^{4} +$$$$34\!\cdots\!84$$$$T^{6} +$$$$30\!\cdots\!84$$$$T^{8} + 104867217264 T^{10} + T^{12}$$
$59$ $$54\!\cdots\!04$$$$+$$$$78\!\cdots\!16$$$$T^{2} +$$$$31\!\cdots\!20$$$$T^{4} +$$$$28\!\cdots\!04$$$$T^{6} +$$$$10\!\cdots\!84$$$$T^{8} + 168344383296 T^{10} + T^{12}$$
$61$ $$( -$$$$17\!\cdots\!56$$$$-$$$$82\!\cdots\!64$$$$T +$$$$18\!\cdots\!20$$$$T^{2} + 19069462420915416 T^{3} - 34152329580 T^{4} - 463368 T^{5} + T^{6} )^{2}$$
$67$ $$($$$$77\!\cdots\!64$$$$+$$$$40\!\cdots\!04$$$$T +$$$$30\!\cdots\!32$$$$T^{2} - 53849187613945056 T^{3} - 399730016328 T^{4} + 111552 T^{5} + T^{6} )^{2}$$
$71$ $$12\!\cdots\!64$$$$+$$$$65\!\cdots\!80$$$$T^{2} +$$$$12\!\cdots\!28$$$$T^{4} +$$$$10\!\cdots\!72$$$$T^{6} +$$$$50\!\cdots\!52$$$$T^{8} + 1135842182244 T^{10} + T^{12}$$
$73$ $$($$$$59\!\cdots\!48$$$$-$$$$13\!\cdots\!80$$$$T -$$$$41\!\cdots\!96$$$$T^{2} + 15600374008896928 T^{3} - 35574255540 T^{4} - 300492 T^{5} + T^{6} )^{2}$$
$79$ $$( -$$$$26\!\cdots\!24$$$$-$$$$75\!\cdots\!60$$$$T +$$$$88\!\cdots\!76$$$$T^{2} + 71018904251930624 T^{3} - 713296879056 T^{4} + 138432 T^{5} + T^{6} )^{2}$$
$83$ $$16\!\cdots\!96$$$$+$$$$17\!\cdots\!84$$$$T^{2} +$$$$20\!\cdots\!92$$$$T^{4} +$$$$49\!\cdots\!04$$$$T^{6} +$$$$44\!\cdots\!76$$$$T^{8} + 1501060755600 T^{10} + T^{12}$$
$89$ $$37\!\cdots\!36$$$$+$$$$45\!\cdots\!88$$$$T^{2} +$$$$21\!\cdots\!04$$$$T^{4} +$$$$50\!\cdots\!80$$$$T^{6} +$$$$62\!\cdots\!04$$$$T^{8} + 3963680885148 T^{10} + T^{12}$$
$97$ $$( -$$$$98\!\cdots\!08$$$$-$$$$58\!\cdots\!48$$$$T +$$$$10\!\cdots\!16$$$$T^{2} + 1504994309876277664 T^{3} - 2030571670404 T^{4} - 810708 T^{5} + T^{6} )^{2}$$