Properties

Label 21.11.f.a
Level $21$
Weight $11$
Character orbit 21.f
Analytic conductor $13.343$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 21 = 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 11 \)
Character orbit: \([\chi]\) \(=\) 21.f (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(13.3425023061\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(6\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
Defining polynomial: \(x^{12} - x^{11} + 3773 x^{10} + 44516 x^{9} + 11068388 x^{8} + 100480832 x^{7} + 11177140432 x^{6} + 67553728512 x^{5} + 8140577253696 x^{4} + 32158758551040 x^{3} + 2022248850888960 x^{2} - 9516227338368000 x + 300517801880601600\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{10}\cdot 3^{7}\cdot 7^{7} \)
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.

\(f(q)\) \(=\) \( q + ( -\beta_{1} + 2 \beta_{2} ) q^{2} + ( -81 - 81 \beta_{2} ) q^{3} + ( -236 + 236 \beta_{2} + 5 \beta_{3} + \beta_{5} + \beta_{7} ) q^{4} + ( -138 - 16 \beta_{1} + 69 \beta_{2} + 32 \beta_{3} + \beta_{4} + \beta_{9} ) q^{5} + ( 162 + 81 \beta_{1} - 324 \beta_{2} + 81 \beta_{3} ) q^{6} + ( 3508 + 169 \beta_{1} - 3707 \beta_{2} - 72 \beta_{3} - 7 \beta_{5} - 2 \beta_{6} - 8 \beta_{7} + \beta_{8} - 2 \beta_{9} - 2 \beta_{10} - 3 \beta_{11} ) q^{7} + ( 8272 + 24 \beta_{1} - \beta_{2} - 22 \beta_{3} - 4 \beta_{4} - 13 \beta_{5} - 2 \beta_{6} - 2 \beta_{7} + \beta_{8} - 6 \beta_{9} - 7 \beta_{10} - 2 \beta_{11} ) q^{8} + 19683 \beta_{2} q^{9} +O(q^{10})\) \( q + ( -\beta_{1} + 2 \beta_{2} ) q^{2} + ( -81 - 81 \beta_{2} ) q^{3} + ( -236 + 236 \beta_{2} + 5 \beta_{3} + \beta_{5} + \beta_{7} ) q^{4} + ( -138 - 16 \beta_{1} + 69 \beta_{2} + 32 \beta_{3} + \beta_{4} + \beta_{9} ) q^{5} + ( 162 + 81 \beta_{1} - 324 \beta_{2} + 81 \beta_{3} ) q^{6} + ( 3508 + 169 \beta_{1} - 3707 \beta_{2} - 72 \beta_{3} - 7 \beta_{5} - 2 \beta_{6} - 8 \beta_{7} + \beta_{8} - 2 \beta_{9} - 2 \beta_{10} - 3 \beta_{11} ) q^{7} + ( 8272 + 24 \beta_{1} - \beta_{2} - 22 \beta_{3} - 4 \beta_{4} - 13 \beta_{5} - 2 \beta_{6} - 2 \beta_{7} + \beta_{8} - 6 \beta_{9} - 7 \beta_{10} - 2 \beta_{11} ) q^{8} + 19683 \beta_{2} q^{9} + ( 19972 + 91 \beta_{1} + 19963 \beta_{2} - 41 \beta_{3} + 29 \beta_{4} - 47 \beta_{5} - 2 \beta_{6} + 48 \beta_{7} + \beta_{9} + 5 \beta_{10} + 10 \beta_{11} ) q^{10} + ( 39677 - 39677 \beta_{2} + 1900 \beta_{3} + 19 \beta_{4} - 36 \beta_{5} - 27 \beta_{6} - 27 \beta_{7} + 18 \beta_{8} + 11 \beta_{9} - 9 \beta_{11} ) q^{11} + ( 38232 + 405 \beta_{1} - 19116 \beta_{2} - 810 \beta_{3} - 162 \beta_{5} - 81 \beta_{7} ) q^{12} + ( -1195 + 177 \beta_{1} + 2352 \beta_{2} + 200 \beta_{3} + 56 \beta_{4} + 67 \beta_{5} - 79 \beta_{6} + 157 \beta_{7} + 61 \beta_{8} - 53 \beta_{9} - 28 \beta_{10} ) q^{13} + ( 125351 - 8582 \beta_{1} - 210626 \beta_{2} + 8180 \beta_{3} - 29 \beta_{4} - 187 \beta_{5} + 47 \beta_{6} - 299 \beta_{7} + 46 \beta_{8} + 91 \beta_{9} + 52 \beta_{10} - 26 \beta_{11} ) q^{14} + ( 16767 + 3888 \beta_{1} - 3888 \beta_{3} - 162 \beta_{4} - 81 \beta_{9} ) q^{15} + ( -30 - 13043 \beta_{1} + 227352 \beta_{2} + 11 \beta_{3} - 59 \beta_{4} + 30 \beta_{5} + 49 \beta_{6} + 143 \beta_{7} + 41 \beta_{8} + 100 \beta_{9} + 60 \beta_{10} + 41 \beta_{11} ) q^{16} + ( -113018 - 3268 \beta_{1} - 112934 \beta_{2} + 1592 \beta_{3} - 34 \beta_{4} + 240 \beta_{5} + 232 \beta_{6} - 36 \beta_{7} + 204 \beta_{9} + 380 \beta_{10} + 120 \beta_{11} ) q^{17} + ( -39366 + 39366 \beta_{2} - 19683 \beta_{3} ) q^{18} + ( 107979 + 13973 \beta_{1} - 53940 \beta_{2} - 27946 \beta_{3} - 447 \beta_{4} - 1031 \beta_{5} - 66 \beta_{6} - 466 \beta_{7} + 99 \beta_{8} - 414 \beta_{9} + 66 \beta_{10} - 99 \beta_{11} ) q^{19} + ( -31840 - 43190 \beta_{1} + 63185 \beta_{2} - 43364 \beta_{3} + 98 \beta_{4} + 693 \beta_{5} + 76 \beta_{6} + 1212 \beta_{7} + 321 \beta_{8} - 186 \beta_{9} - 49 \beta_{10} ) q^{20} + ( -584577 - 19359 \beta_{1} + 316305 \beta_{2} - 1944 \beta_{3} + 243 \beta_{4} + 1215 \beta_{5} + 81 \beta_{6} + 729 \beta_{7} + 81 \beta_{8} + 162 \beta_{9} + 324 \beta_{11} ) q^{21} + ( 2438149 - 48209 \beta_{1} + 315 \beta_{2} + 47992 \beta_{3} - 92 \beta_{4} - 4082 \beta_{5} + 217 \beta_{6} + 217 \beta_{7} + 98 \beta_{8} + 388 \beta_{9} + 553 \beta_{10} - 196 \beta_{11} ) q^{22} + ( 2 - 10506 \beta_{1} - 722012 \beta_{2} + 496 \beta_{3} - 280 \beta_{4} - 2 \beta_{5} - 500 \beta_{6} - 1024 \beta_{7} + 494 \beta_{8} + 774 \beta_{9} - 4 \beta_{10} + 494 \beta_{11} ) q^{23} + ( -670194 - 3645 \beta_{1} - 669951 \beta_{2} + 1701 \beta_{3} + 405 \beta_{4} + 1215 \beta_{5} + 567 \beta_{6} - 729 \beta_{7} + 486 \beta_{9} + 891 \beta_{10} + 243 \beta_{11} ) q^{24} + ( 2098446 + 152 \beta_{1} - 2098598 \beta_{2} + 225036 \beta_{3} + 663 \beta_{4} - 5460 \beta_{5} - 1475 \beta_{6} - 5019 \beta_{7} + 1186 \beta_{8} - 301 \beta_{9} - 152 \beta_{10} - 593 \beta_{11} ) q^{25} + ( 465843 + 70813 \beta_{1} - 232722 \beta_{2} - 141131 \beta_{3} - 2675 \beta_{4} - 1689 \beta_{5} - 431 \beta_{6} - 645 \beta_{7} + 894 \beta_{8} - 2707 \beta_{9} - 64 \beta_{10} - 894 \beta_{11} ) q^{26} + ( 1594323 - 3188646 \beta_{2} ) q^{27} + ( -403330 - 132540 \beta_{1} + 7064187 \beta_{2} + 321146 \beta_{3} + 3179 \beta_{4} + 5948 \beta_{5} - 1387 \beta_{6} + 8687 \beta_{7} + 652 \beta_{8} + 1670 \beta_{9} - 279 \beta_{10} - 647 \beta_{11} ) q^{28} + ( 7929665 + 139734 \beta_{1} - 716 \beta_{2} - 139580 \beta_{3} + 4802 \beta_{4} - 16562 \beta_{5} - 154 \beta_{6} - 154 \beta_{7} - 562 \beta_{8} + 2093 \beta_{9} + 100 \beta_{10} + 1124 \beta_{11} ) q^{29} + ( 81 - 11502 \beta_{1} - 4851738 \beta_{2} - 729 \beta_{3} - 2835 \beta_{4} - 81 \beta_{5} + 567 \beta_{6} - 11583 \beta_{7} - 810 \beta_{8} + 2025 \beta_{9} - 162 \beta_{10} - 810 \beta_{11} ) q^{30} + ( 523559 + 156436 \beta_{1} + 519947 \beta_{2} - 76412 \beta_{3} + 7898 \beta_{4} - 6310 \beta_{5} - 5645 \beta_{6} + 1869 \beta_{7} - 4441 \beta_{9} - 7678 \beta_{10} - 829 \beta_{11} ) q^{31} + ( -8302602 + 1311 \beta_{1} + 8301291 \beta_{2} - 206601 \beta_{3} - 3927 \beta_{4} + 3695 \beta_{5} + 3939 \beta_{6} + 1945 \beta_{7} - 878 \beta_{8} - 5226 \beta_{9} - 1311 \beta_{10} + 439 \beta_{11} ) q^{32} + ( -6426945 + 153171 \beta_{1} + 3212379 \beta_{2} - 306342 \beta_{3} - 1701 \beta_{4} + 5103 \beta_{5} + 1458 \beta_{6} + 1458 \beta_{7} - 2187 \beta_{8} - 2430 \beta_{9} - 1458 \beta_{10} + 2187 \beta_{11} ) q^{33} + ( -1792666 + 273030 \beta_{1} + 3592814 \beta_{2} + 274932 \beta_{3} - 2452 \beta_{4} + 22000 \beta_{5} + 550 \beta_{6} + 45902 \beta_{7} - 5580 \beta_{8} - 3732 \beta_{9} + 1226 \beta_{10} ) q^{34} + ( -11451002 + 553453 \beta_{1} + 9828664 \beta_{2} - 312254 \beta_{3} - 2558 \beta_{4} + 31961 \beta_{5} + 3659 \beta_{6} - 7769 \beta_{7} - 3455 \beta_{8} + 11603 \beta_{9} + 958 \beta_{10} - 1304 \beta_{11} ) q^{35} + ( -4645188 - 98415 \beta_{1} + 98415 \beta_{3} + 19683 \beta_{5} ) q^{36} + ( -43 - 58537 \beta_{1} - 14576808 \beta_{2} - 5080 \beta_{3} - 3459 \beta_{4} + 43 \beta_{5} + 5166 \beta_{6} - 51080 \beta_{7} - 5037 \beta_{8} - 1578 \beta_{9} + 86 \beta_{10} - 5037 \beta_{11} ) q^{37} + ( -17677178 - 941233 \beta_{1} - 17667797 \beta_{2} + 465926 \beta_{3} - 14857 \beta_{4} + 26979 \beta_{5} + 7390 \beta_{6} - 22716 \beta_{7} + 4263 \beta_{9} + 5399 \beta_{10} - 5118 \beta_{11} ) q^{38} + ( 292248 - 3078 \beta_{1} - 289170 \beta_{2} - 41796 \beta_{3} - 2511 \beta_{4} - 18144 \beta_{5} + 8667 \beta_{6} - 20007 \beta_{7} - 9882 \beta_{8} + 6723 \beta_{9} + 3078 \beta_{10} + 4941 \beta_{11} ) q^{39} + ( -67597798 + 210533 \beta_{1} + 33799486 \beta_{2} - 434469 \beta_{3} + 5197 \beta_{4} + 30856 \beta_{5} + 3685 \beta_{6} + 16015 \beta_{7} - 12229 \beta_{8} + 10056 \beta_{9} + 9718 \beta_{10} + 12229 \beta_{11} ) q^{40} + ( 23986506 + 614612 \beta_{1} - 47986188 \beta_{2} + 609728 \beta_{3} + 2272 \beta_{4} + 33948 \beta_{5} + 2612 \beta_{6} + 63012 \beta_{7} + 8292 \beta_{8} + 1604 \beta_{9} - 1136 \beta_{10} ) q^{41} + ( -27212517 + 1356102 \beta_{1} + 23964255 \beta_{2} - 626292 \beta_{3} + 1296 \beta_{4} + 39366 \beta_{5} - 10125 \beta_{6} + 33291 \beta_{7} - 5346 \beta_{8} - 13284 \beta_{9} - 2997 \beta_{10} + 5832 \beta_{11} ) q^{42} + ( 58380497 + 133837 \beta_{1} - 6246 \beta_{2} - 125198 \beta_{3} - 13754 \beta_{4} + 349 \beta_{5} - 8639 \beta_{6} - 8639 \beta_{7} + 2393 \beta_{8} - 24155 \beta_{9} - 28310 \beta_{10} - 4786 \beta_{11} ) q^{43} + ( -1142 - 3292465 \beta_{1} + 22820416 \beta_{2} + 7727 \beta_{3} + 13489 \beta_{4} + 1142 \beta_{5} - 5443 \beta_{6} + 43705 \beta_{7} + 8869 \beta_{8} - 4620 \beta_{9} + 2284 \beta_{10} + 8869 \beta_{11} ) q^{44} + ( -1358127 - 629856 \beta_{1} - 1358127 \beta_{2} + 314928 \beta_{3} + 19683 \beta_{4} ) q^{45} + ( -12147488 - 21448 \beta_{1} + 12168936 \beta_{2} + 1999088 \beta_{3} + 8424 \beta_{4} - 50744 \beta_{5} - 7784 \beta_{6} - 41000 \beta_{7} - 23408 \beta_{8} + 30512 \beta_{9} + 21448 \beta_{10} + 11704 \beta_{11} ) q^{46} + ( -21019846 + 2383118 \beta_{1} + 10508708 \beta_{2} - 4731964 \beta_{3} + 27914 \beta_{4} - 144954 \beta_{5} - 9804 \beta_{6} - 73692 \beta_{7} + 31842 \beta_{8} + 15680 \beta_{9} - 24468 \beta_{10} - 31842 \beta_{11} ) q^{47} + ( 18424584 + 1050732 \beta_{1} - 36831915 \beta_{2} + 1058022 \beta_{3} - 1782 \beta_{4} - 14013 \beta_{5} - 5508 \beta_{6} - 20736 \beta_{7} - 9963 \beta_{8} - 17010 \beta_{9} + 891 \beta_{10} ) q^{48} + ( 15316875 + 4985057 \beta_{1} + 23578765 \beta_{2} - 2745078 \beta_{3} - 17108 \beta_{4} + 3549 \beta_{5} + 31325 \beta_{6} - 34391 \beta_{7} + 6657 \beta_{8} - 51317 \beta_{9} - 9954 \beta_{10} - 6398 \beta_{11} ) q^{49} + ( 283930523 - 4323330 \beta_{1} + 24127 \beta_{2} + 4303597 \beta_{3} - 18860 \beta_{4} - 258650 \beta_{5} + 19733 \beta_{6} + 19733 \beta_{7} + 4394 \beta_{8} + 30036 \beta_{9} + 54805 \beta_{10} - 8788 \beta_{11} ) q^{50} + ( 16524 + 383940 \beta_{1} + 27449766 \beta_{2} + 6804 \beta_{3} + 7290 \beta_{4} - 16524 \beta_{5} - 39852 \beta_{6} + 25272 \beta_{7} - 9720 \beta_{8} - 17010 \beta_{9} - 33048 \beta_{10} - 9720 \beta_{11} ) q^{51} + ( -87266006 - 1570219 \beta_{1} - 87304607 \beta_{2} + 804410 \beta_{3} - 26415 \beta_{4} + 57532 \beta_{5} - 20745 \beta_{6} - 65410 \beta_{7} - 7878 \beta_{9} - 2889 \beta_{10} + 30723 \beta_{11} ) q^{52} + ( -115199953 + 64524 \beta_{1} + 115135429 \beta_{2} + 4275040 \beta_{3} + 41437 \beta_{4} + 191752 \beta_{5} - 62520 \beta_{6} + 191084 \beta_{7} + 127712 \beta_{8} - 44170 \beta_{9} - 64524 \beta_{10} - 63856 \beta_{11} ) q^{53} + ( 6377292 - 1594323 \beta_{1} - 3188646 \beta_{2} + 3188646 \beta_{3} ) q^{54} + ( 108709251 + 1696304 \beta_{1} - 217400350 \beta_{2} + 1695816 \beta_{3} - 12752 \beta_{4} - 105192 \beta_{5} + 13240 \beta_{6} - 210872 \beta_{7} - 18640 \beta_{8} + 163859 \beta_{9} + 6376 \beta_{10} ) q^{55} + ( 138402474 + 6721197 \beta_{1} + 50543017 \beta_{2} - 6363301 \beta_{3} - 34307 \beta_{4} - 289443 \beta_{5} - 36869 \beta_{6} - 13545 \beta_{7} + 61740 \beta_{8} - 41034 \beta_{9} + 30947 \beta_{10} + 3675 \beta_{11} ) q^{56} + ( -13115439 - 3395439 \beta_{1} - 16038 \beta_{2} + 3403458 \beta_{3} + 83106 \beta_{4} + 121257 \beta_{5} - 8019 \beta_{6} - 8019 \beta_{7} - 8019 \beta_{8} + 25515 \beta_{9} - 16038 \beta_{10} + 16038 \beta_{11} ) q^{57} + ( -27041 - 22609584 \beta_{1} - 166903258 \beta_{2} + 13393 \beta_{3} + 120891 \beta_{4} + 27041 \beta_{5} + 40689 \beta_{6} - 48057 \beta_{7} + 40434 \beta_{8} - 80457 \beta_{9} + 54082 \beta_{10} + 40434 \beta_{11} ) q^{58} + ( 76421315 - 1194594 \beta_{1} + 76494725 \beta_{2} + 560592 \beta_{3} - 161133 \beta_{4} - 382722 \beta_{5} + 100613 \beta_{6} + 458865 \beta_{7} + 76143 \beta_{9} + 127816 \beta_{10} + 2733 \beta_{11} ) q^{59} + ( 7723026 - 40095 \beta_{1} - 7682931 \beta_{2} + 10549359 \beta_{3} + 3159 \beta_{4} - 154305 \beta_{5} - 2187 \beta_{6} - 140211 \beta_{7} - 52002 \beta_{8} + 44226 \beta_{9} + 40095 \beta_{10} + 26001 \beta_{11} ) q^{60} + ( -285257936 + 14255744 \beta_{1} + 142650596 \beta_{2} - 28585196 \beta_{3} + 47244 \beta_{4} + 560088 \beta_{5} - 4268 \beta_{6} + 301672 \beta_{7} - 30452 \beta_{8} + 86232 \beta_{9} + 77976 \beta_{10} + 30452 \beta_{11} ) q^{61} + ( 97188760 - 5897697 \beta_{1} - 194586698 \beta_{2} - 5927211 \beta_{3} + 100100 \beta_{4} - 316188 \beta_{5} - 70586 \beta_{6} - 661890 \beta_{7} + 179664 \beta_{8} - 138716 \beta_{9} - 50050 \beta_{10} ) q^{62} + ( 73004247 + 1377810 \beta_{1} - 3897234 \beta_{2} + 1889568 \beta_{3} - 59049 \beta_{4} - 157464 \beta_{5} + 19683 \beta_{6} - 19683 \beta_{7} - 39366 \beta_{8} + 39366 \beta_{10} - 19683 \beta_{11} ) q^{63} + ( -39336820 - 4531958 \beta_{1} - 44829 \beta_{2} + 4520012 \beta_{3} - 255444 \beta_{4} + 723409 \beta_{5} + 11946 \beta_{6} + 11946 \beta_{7} - 56775 \beta_{8} - 103830 \beta_{9} + 92613 \beta_{10} + 113550 \beta_{11} ) q^{64} + ( -46304 - 27501952 \beta_{1} + 453436600 \beta_{2} - 56020 \beta_{3} - 214206 \beta_{4} + 46304 \beta_{5} + 148628 \beta_{6} + 665992 \beta_{7} - 9716 \beta_{8} + 204490 \beta_{9} + 92608 \beta_{10} - 9716 \beta_{11} ) q^{65} + ( -197472492 + 7800219 \beta_{1} - 197549037 \beta_{2} - 3861837 \beta_{3} + 36693 \beta_{4} + 313065 \beta_{5} - 78246 \beta_{6} - 365796 \beta_{7} - 52731 \beta_{9} - 79947 \beta_{10} + 23814 \beta_{11} ) q^{66} + ( -371935440 + 15316 \beta_{1} + 371920124 \beta_{2} + 23349324 \beta_{3} - 220557 \beta_{4} + 117684 \beta_{5} + 220673 \beta_{6} + 39021 \beta_{7} - 126694 \beta_{8} - 235757 \beta_{9} - 15316 \beta_{10} + 63347 \beta_{11} ) q^{67} + ( 472261340 + 21098218 \beta_{1} - 236133148 \beta_{2} - 42133814 \beta_{3} - 72274 \beta_{4} - 801096 \beta_{5} - 17570 \beta_{6} - 403026 \beta_{7} + 57666 \beta_{8} - 94800 \beta_{9} - 45052 \beta_{10} - 57666 \beta_{11} ) q^{68} + ( -58403106 + 811134 \beta_{1} + 116925768 \beta_{2} + 810648 \beta_{3} - 80352 \beta_{4} + 83106 \beta_{5} + 80838 \beta_{6} + 165726 \beta_{7} - 120042 \beta_{8} - 188568 \beta_{9} + 40176 \beta_{10} ) q^{69} + ( 284307401 + 32175810 \beta_{1} - 682944218 \beta_{2} - 2745611 \beta_{3} + 318025 \beta_{4} - 915553 \beta_{5} - 110329 \beta_{6} - 635191 \beta_{7} - 140982 \beta_{8} + 404113 \beta_{9} - 81606 \beta_{10} + 82186 \beta_{11} ) q^{70} + ( -320076922 - 19840530 \beta_{1} + 119884 \beta_{2} + 19847164 \beta_{3} + 142388 \beta_{4} + 1172590 \beta_{5} - 6634 \beta_{6} - 6634 \beta_{7} + 126518 \beta_{8} + 57926 \beta_{9} - 146420 \beta_{10} - 253036 \beta_{11} ) q^{71} + ( 39366 + 413343 \beta_{1} + 162817776 \beta_{2} + 19683 \beta_{3} - 19683 \beta_{4} - 39366 \beta_{5} - 98415 \beta_{6} + 216513 \beta_{7} - 19683 \beta_{8} - 78732 \beta_{10} - 19683 \beta_{11} ) q^{72} + ( -103887158 - 13762326 \beta_{1} - 103735808 \beta_{2} + 6805488 \beta_{3} + 317725 \beta_{4} + 37946 \beta_{5} - 98041 \beta_{6} - 186437 \beta_{7} - 148491 \beta_{9} - 347432 \beta_{10} - 299841 \beta_{11} ) q^{73} + ( -65521352 - 35529 \beta_{1} + 65556881 \beta_{2} + 53005208 \beta_{3} + 204899 \beta_{4} + 461179 \beta_{5} - 54276 \beta_{6} + 491114 \beta_{7} - 11188 \beta_{8} + 391051 \beta_{9} + 35529 \beta_{10} + 5594 \beta_{11} ) q^{74} + ( -339912531 + 18167571 \beta_{1} + 169902684 \beta_{2} - 36372078 \beta_{3} + 6399 \beta_{4} + 848799 \beta_{5} + 83754 \beta_{6} + 370818 \beta_{7} - 144099 \beta_{8} - 17010 \beta_{9} - 46818 \beta_{10} + 144099 \beta_{11} ) q^{75} + ( -510188380 + 27792379 \beta_{1} + 1020705923 \beta_{2} + 27970129 \beta_{3} + 17558 \beta_{4} + 35424 \beta_{5} - 195308 \beta_{6} + 248598 \beta_{7} - 151413 \beta_{8} + 13218 \beta_{9} - 8779 \beta_{10} ) q^{76} + ( -74918014 + 45817858 \beta_{1} - 1153298381 \beta_{2} - 12552904 \beta_{3} - 77827 \beta_{4} + 1763750 \beta_{5} + 139672 \beta_{6} + 791560 \beta_{7} + 65470 \beta_{8} - 118073 \beta_{9} - 79696 \beta_{10} - 152198 \beta_{11} ) q^{77} + ( -56583765 - 17167464 \beta_{1} - 104733 \beta_{2} + 17199783 \beta_{3} + 503172 \beta_{4} + 189054 \beta_{5} - 32319 \beta_{6} - 32319 \beta_{7} - 72414 \beta_{8} + 186948 \beta_{9} - 24543 \beta_{10} + 144828 \beta_{11} ) q^{78} + ( 184749 - 332145 \beta_{1} + 1376780981 \beta_{2} - 25192 \beta_{3} + 65662 \beta_{4} - 184749 \beta_{5} - 344306 \beta_{6} + 670872 \beta_{7} - 209941 \beta_{8} - 275603 \beta_{9} - 369498 \beta_{10} - 209941 \beta_{11} ) q^{79} + ( -304237094 - 872367 \beta_{1} - 304582913 \beta_{2} + 609093 \beta_{3} + 748163 \beta_{4} - 1024737 \beta_{5} - 188467 \beta_{6} + 951543 \beta_{7} - 73194 \beta_{9} - 31115 \beta_{10} + 272625 \beta_{11} ) q^{80} + ( -387420489 + 387420489 \beta_{2} ) q^{81} + ( 1656034396 + 3068886 \beta_{1} - 828060196 \beta_{2} - 6128056 \beta_{3} - 444388 \beta_{4} - 700620 \beta_{5} + 54092 \beta_{6} - 393308 \beta_{7} - 76280 \beta_{8} - 476292 \beta_{9} - 63808 \beta_{10} + 76280 \beta_{11} ) q^{82} + ( -621805070 + 189903 \beta_{1} + 1243833016 \beta_{2} - 24630 \beta_{3} - 434628 \beta_{4} - 609669 \beta_{5} + 649161 \beta_{6} - 1433871 \beta_{7} - 437409 \beta_{8} + 187015 \beta_{9} + 217314 \beta_{10} ) q^{83} + ( 604869282 + 36748161 \beta_{1} - 1111781376 \beta_{2} - 41237100 \beta_{3} - 362961 \beta_{4} - 1185435 \beta_{5} + 82539 \beta_{6} - 925506 \beta_{7} - 53217 \beta_{8} - 125388 \beta_{9} - 66744 \beta_{10} + 105219 \beta_{11} ) q^{84} + ( 1299383848 - 15597514 \beta_{1} + 72420 \beta_{2} + 15676548 \beta_{3} + 45556 \beta_{4} - 1928674 \beta_{5} - 79034 \beta_{6} - 79034 \beta_{7} + 151454 \beta_{8} - 135290 \beta_{9} - 388556 \beta_{10} - 302908 \beta_{11} ) q^{85} + ( 8629 - 57940785 \beta_{1} - 33247588 \beta_{2} + 135287 \beta_{3} - 188035 \beta_{4} - 8629 \beta_{5} - 152545 \beta_{6} - 3304199 \beta_{7} + 126658 \beta_{8} + 314693 \beta_{9} - 17258 \beta_{10} + 126658 \beta_{11} ) q^{86} + ( -642315339 - 22669956 \beta_{1} - 642141351 \beta_{2} + 11247984 \beta_{3} - 641439 \beta_{4} + 1353996 \beta_{5} + 95418 \beta_{6} - 1316574 \beta_{7} + 37422 \beta_{9} + 16848 \beta_{10} - 136566 \beta_{11} ) q^{87} + ( -1666209998 - 127579 \beta_{1} + 1666337577 \beta_{2} + 16287385 \beta_{3} - 132061 \beta_{4} - 363495 \beta_{5} - 625175 \beta_{6} - 112577 \beta_{7} + 246678 \beta_{8} - 761718 \beta_{9} + 127579 \beta_{10} - 123339 \beta_{11} ) q^{88} + ( 1031318176 + 57113564 \beta_{1} - 515626238 \beta_{2} - 114107896 \beta_{3} - 536814 \beta_{4} + 2191212 \beta_{5} - 83544 \beta_{6} + 1128456 \beta_{7} + 184932 \beta_{8} - 554658 \beta_{9} - 35688 \beta_{10} - 184932 \beta_{11} ) q^{89} + ( -393128559 + 1003833 \beta_{1} + 786040605 \beta_{2} + 984150 \beta_{3} + 118098 \beta_{4} + 944784 \beta_{5} - 98415 \beta_{6} + 1869885 \beta_{7} + 196830 \beta_{8} - 511758 \beta_{9} - 59049 \beta_{10} ) q^{90} + ( 1309061945 + 57271623 \beta_{1} - 3156931554 \beta_{2} + 11729066 \beta_{3} + 1417805 \beta_{4} + 829471 \beta_{5} + 30844 \beta_{6} + 2747104 \beta_{7} + 236097 \beta_{8} + 773858 \beta_{9} + 272730 \beta_{10} - 48415 \beta_{11} ) q^{91} + ( 1714322336 - 17272328 \beta_{1} - 410392 \beta_{2} + 17306872 \beta_{3} - 128608 \beta_{4} - 4695664 \beta_{5} - 34544 \beta_{6} - 34544 \beta_{7} - 375848 \beta_{8} - 133392 \beta_{9} + 272216 \beta_{10} + 751696 \beta_{11} ) q^{92} + ( -359721 - 18793539 \beta_{1} - 126639693 \beta_{2} - 292572 \beta_{3} - 834786 \beta_{4} + 359721 \beta_{5} + 1012014 \beta_{6} - 813888 \beta_{7} + 67149 \beta_{8} + 901935 \beta_{9} + 719442 \beta_{10} + 67149 \beta_{11} ) q^{93} + ( -3015667516 - 134429210 \beta_{1} - 3015417874 \beta_{2} + 67089784 \beta_{3} + 36430 \beta_{4} + 3201078 \beta_{5} + 974684 \beta_{6} - 2309608 \beta_{7} + 891470 \beta_{9} + 1699726 \beta_{10} + 641828 \beta_{11} ) q^{94} + ( -5571724086 - 228456 \beta_{1} + 5571952542 \beta_{2} - 55262832 \beta_{3} - 181004 \beta_{4} + 2444880 \beta_{5} + 749844 \beta_{6} + 2271084 \beta_{7} - 804504 \beta_{8} + 616292 \beta_{9} + 228456 \beta_{10} + 402252 \beta_{11} ) q^{95} + ( 1344879774 - 16805313 \beta_{1} - 672227262 \beta_{2} + 33292053 \beta_{3} + 599643 \beta_{4} - 456840 \beta_{5} - 177309 \beta_{6} - 15795 \beta_{7} + 106677 \beta_{8} + 847584 \beta_{9} + 495882 \beta_{10} - 106677 \beta_{11} ) q^{96} + ( -2077147394 + 165303 \beta_{1} + 4152919356 \beta_{2} - 176306 \beta_{3} + 461476 \beta_{4} + 3442271 \beta_{5} - 119867 \beta_{6} + 6542933 \beta_{7} + 1033823 \beta_{8} + 2053241 \beta_{9} - 230738 \beta_{10} ) q^{97} + ( 2758816823 - 39928021 \beta_{1} - 6153524216 \beta_{2} - 18148620 \beta_{3} - 1406965 \beta_{4} + 4993625 \beta_{5} - 10185 \beta_{6} - 2816051 \beta_{7} - 68726 \beta_{8} - 1019109 \beta_{9} + 162456 \beta_{10} + 235774 \beta_{11} ) q^{98} + ( 780785244 - 37220553 \beta_{1} + 354294 \beta_{2} + 37043406 \beta_{3} + 39366 \beta_{4} - 531441 \beta_{5} + 177147 \beta_{6} + 177147 \beta_{7} + 177147 \beta_{8} + 373977 \beta_{9} + 354294 \beta_{10} - 354294 \beta_{11} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 11 q^{2} - 1458 q^{3} - 1421 q^{4} - 1287 q^{5} + 20090 q^{7} + 99310 q^{8} + 118098 q^{9} + O(q^{10}) \) \( 12 q + 11 q^{2} - 1458 q^{3} - 1421 q^{4} - 1287 q^{5} + 20090 q^{7} + 99310 q^{8} + 118098 q^{9} + 359661 q^{10} + 236165 q^{11} + 345303 q^{12} + 223832 q^{14} + 208494 q^{15} + 1350895 q^{16} - 2038782 q^{17} - 216513 q^{18} + 1012389 q^{19} - 5131917 q^{21} + 29162074 q^{22} - 4341928 q^{23} - 12066165 q^{24} + 12365439 q^{25} + 4398486 q^{26} + 37097165 q^{28} + 95444234 q^{29} - 29132541 q^{30} + 9658932 q^{31} - 49606359 q^{32} - 57388095 q^{33} - 77592186 q^{35} - 55939086 q^{36} - 87545045 q^{37} - 319584648 q^{38} + 1789695 q^{39} - 607773201 q^{40} - 180831609 q^{42} + 700816306 q^{43} + 133692537 q^{44} - 25332021 q^{45} - 74981552 q^{46} - 181842702 q^{47} + 333171090 q^{49} + 3398506976 q^{50} + 165141342 q^{51} - 1573473588 q^{52} - 695152867 q^{53} + 52612659 q^{54} + 1977142699 q^{56} - 164007018 q^{57} - 1023653321 q^{58} + 1373785545 q^{59} + 35593425 q^{60} - 2524633584 q^{61} + 851624361 q^{63} - 482439838 q^{64} + 2692260666 q^{65} - 3543191991 q^{66} - 2255105709 q^{67} + 4313617758 q^{68} - 650925555 q^{70} - 3878911780 q^{71} + 977359365 q^{72} - 1888675383 q^{73} - 445804820 q^{74} - 3004801677 q^{75} - 7759715803 q^{77} - 712554732 q^{78} + 8260659900 q^{79} - 5477579577 q^{80} - 2324522934 q^{81} + 14912212206 q^{82} + 665315532 q^{84} + 15563062356 q^{85} - 257575928 q^{86} - 11596474431 q^{87} - 10015831801 q^{88} + 9451951530 q^{89} - 3182715375 q^{91} + 20533179072 q^{92} - 782373492 q^{93} - 54481927140 q^{94} - 33375267288 q^{95} + 12054345237 q^{96} - 3841749499 q^{98} + 9296871390 q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{12} - x^{11} + 3773 x^{10} + 44516 x^{9} + 11068388 x^{8} + 100480832 x^{7} + 11177140432 x^{6} + 67553728512 x^{5} + 8140577253696 x^{4} + 32158758551040 x^{3} + 2022248850888960 x^{2} - 9516227338368000 x + 300517801880601600\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\(1384939568965668757870948270385 \nu^{11} + 30939208497149042210289341565671 \nu^{10} + 4886399126893414182835445154657729 \nu^{9} + 176884871480071443148950077847466488 \nu^{8} + 15722540066229425871610817726608387896 \nu^{7} + 463924213053373362229050601898654785368 \nu^{6} + 15354581899287105170527414678643745962752 \nu^{5} + 371822140683736914632099524036220976390592 \nu^{4} + 10616729307056126194515245452044993007933632 \nu^{3} + 292455912871080090271143948118656334853154816 \nu^{2} + 1935949925752984955269837587921625276026693120 \nu + 55118906115663215373528054482094718715374748160\)\()/ \)\(54\!\cdots\!60\)\( \)
\(\beta_{3}\)\(=\)\((\)\(191045580664523457813188786 \nu^{11} - 2003462651682392258751050081 \nu^{10} + 681061618643441527716771936943 \nu^{9} + 2325655218590542211500912208671 \nu^{8} + 1919456328128215714329955619881138 \nu^{7} - 739273699589372816090093574652508 \nu^{6} + 1644626994942574450426518484164145232 \nu^{5} - 3885897100459464607566764006649161568 \nu^{4} + 1465270902769165573038188935457216589696 \nu^{3} - 5110892255954574925195258157822219234880 \nu^{2} + 403663833092046541005765155155043739000960 \nu - 2459863087797764840446919958226100284896000\)\()/ \)\(32\!\cdots\!60\)\( \)
\(\beta_{4}\)\(=\)\((\)\(-\)\(40\!\cdots\!73\)\( \nu^{11} - \)\(11\!\cdots\!24\)\( \nu^{10} - \)\(77\!\cdots\!62\)\( \nu^{9} + \)\(75\!\cdots\!41\)\( \nu^{8} - \)\(57\!\cdots\!06\)\( \nu^{7} + \)\(35\!\cdots\!08\)\( \nu^{6} + \)\(64\!\cdots\!20\)\( \nu^{5} + \)\(10\!\cdots\!40\)\( \nu^{4} - \)\(39\!\cdots\!32\)\( \nu^{3} + \)\(66\!\cdots\!88\)\( \nu^{2} + \)\(74\!\cdots\!40\)\( \nu + \)\(25\!\cdots\!60\)\(\)\()/ \)\(81\!\cdots\!40\)\( \)
\(\beta_{5}\)\(=\)\((\)\(-1177275765666193307085520123 \nu^{11} - 7240731112887982761209967302 \nu^{10} - 4102828139354119261987315740464 \nu^{9} - 72013727106447543917185863564623 \nu^{8} - 12403599849279248056548601314644234 \nu^{7} - 161354275253827920854969785504525916 \nu^{6} - 10531127114857082960124356942191111696 \nu^{5} - 18325777234487751465056089942401736416 \nu^{4} - 8147373027236903639018504049201501726528 \nu^{3} + 21106719143130345861414389398811769637440 \nu^{2} - 461769069754504325168491885565577729782400 \nu + 122657185560079924958464340719897077587493120\)\()/ \)\(10\!\cdots\!20\)\( \)
\(\beta_{6}\)\(=\)\((\)\(\)\(38\!\cdots\!59\)\( \nu^{11} - \)\(65\!\cdots\!97\)\( \nu^{10} + \)\(28\!\cdots\!65\)\( \nu^{9} - \)\(22\!\cdots\!10\)\( \nu^{8} + \)\(58\!\cdots\!64\)\( \nu^{7} - \)\(59\!\cdots\!56\)\( \nu^{6} + \)\(10\!\cdots\!92\)\( \nu^{5} - \)\(51\!\cdots\!68\)\( \nu^{4} + \)\(90\!\cdots\!88\)\( \nu^{3} - \)\(29\!\cdots\!28\)\( \nu^{2} + \)\(64\!\cdots\!60\)\( \nu - \)\(41\!\cdots\!20\)\(\)\()/ \)\(16\!\cdots\!80\)\( \)
\(\beta_{7}\)\(=\)\((\)\(-\)\(51\!\cdots\!55\)\( \nu^{11} - \)\(11\!\cdots\!47\)\( \nu^{10} - \)\(18\!\cdots\!73\)\( \nu^{9} - \)\(67\!\cdots\!66\)\( \nu^{8} - \)\(58\!\cdots\!72\)\( \nu^{7} - \)\(17\!\cdots\!96\)\( \nu^{6} - \)\(58\!\cdots\!04\)\( \nu^{5} - \)\(16\!\cdots\!64\)\( \nu^{4} - \)\(40\!\cdots\!24\)\( \nu^{3} - \)\(11\!\cdots\!12\)\( \nu^{2} - \)\(10\!\cdots\!20\)\( \nu - \)\(21\!\cdots\!20\)\(\)\()/ \)\(19\!\cdots\!20\)\( \)
\(\beta_{8}\)\(=\)\((\)\(-\)\(32\!\cdots\!91\)\( \nu^{11} - \)\(30\!\cdots\!39\)\( \nu^{10} - \)\(15\!\cdots\!53\)\( \nu^{9} - \)\(27\!\cdots\!06\)\( \nu^{8} - \)\(46\!\cdots\!08\)\( \nu^{7} - \)\(88\!\cdots\!92\)\( \nu^{6} - \)\(65\!\cdots\!52\)\( \nu^{5} - \)\(96\!\cdots\!12\)\( \nu^{4} - \)\(43\!\cdots\!56\)\( \nu^{3} - \)\(53\!\cdots\!20\)\( \nu^{2} - \)\(12\!\cdots\!80\)\( \nu - \)\(74\!\cdots\!20\)\(\)\()/ \)\(33\!\cdots\!20\)\( \)
\(\beta_{9}\)\(=\)\((\)\(-\)\(19\!\cdots\!31\)\( \nu^{11} - \)\(26\!\cdots\!01\)\( \nu^{10} - \)\(67\!\cdots\!31\)\( \nu^{9} - \)\(21\!\cdots\!52\)\( \nu^{8} - \)\(20\!\cdots\!60\)\( \nu^{7} - \)\(52\!\cdots\!68\)\( \nu^{6} - \)\(19\!\cdots\!16\)\( \nu^{5} - \)\(48\!\cdots\!56\)\( \nu^{4} - \)\(14\!\cdots\!40\)\( \nu^{3} - \)\(23\!\cdots\!12\)\( \nu^{2} - \)\(37\!\cdots\!60\)\( \nu - \)\(19\!\cdots\!60\)\(\)\()/ \)\(16\!\cdots\!80\)\( \)
\(\beta_{10}\)\(=\)\((\)\(-\)\(41\!\cdots\!91\)\( \nu^{11} + \)\(18\!\cdots\!30\)\( \nu^{10} - \)\(16\!\cdots\!12\)\( \nu^{9} + \)\(45\!\cdots\!41\)\( \nu^{8} - \)\(40\!\cdots\!74\)\( \nu^{7} + \)\(13\!\cdots\!80\)\( \nu^{6} - \)\(35\!\cdots\!44\)\( \nu^{5} + \)\(11\!\cdots\!96\)\( \nu^{4} - \)\(25\!\cdots\!28\)\( \nu^{3} + \)\(67\!\cdots\!44\)\( \nu^{2} - \)\(17\!\cdots\!80\)\( \nu + \)\(76\!\cdots\!00\)\(\)\()/ \)\(27\!\cdots\!80\)\( \)
\(\beta_{11}\)\(=\)\((\)\(\)\(27\!\cdots\!63\)\( \nu^{11} - \)\(57\!\cdots\!82\)\( \nu^{10} + \)\(99\!\cdots\!88\)\( \nu^{9} - \)\(69\!\cdots\!49\)\( \nu^{8} + \)\(25\!\cdots\!30\)\( \nu^{7} - \)\(29\!\cdots\!96\)\( \nu^{6} + \)\(19\!\cdots\!28\)\( \nu^{5} - \)\(33\!\cdots\!32\)\( \nu^{4} + \)\(11\!\cdots\!60\)\( \nu^{3} - \)\(23\!\cdots\!24\)\( \nu^{2} - \)\(13\!\cdots\!20\)\( \nu - \)\(54\!\cdots\!00\)\(\)\()/ \)\(81\!\cdots\!40\)\( \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{7} + \beta_{5} + 9 \beta_{3} + 1256 \beta_{2} - 1256\)
\(\nu^{3}\)\(=\)\(2 \beta_{11} + 7 \beta_{10} + 6 \beta_{9} - \beta_{8} + 2 \beta_{7} + 2 \beta_{6} + 19 \beta_{5} + 4 \beta_{4} + 2112 \beta_{3} + \beta_{2} - 2114 \beta_{1} - 11720\)
\(\nu^{4}\)\(=\)\(49 \beta_{11} + 92 \beta_{10} + 100 \beta_{9} + 49 \beta_{8} - 3041 \beta_{7} + 89 \beta_{6} + 46 \beta_{5} - 51 \beta_{4} + 3 \beta_{3} - 2658392 \beta_{2} - 45107 \beta_{1} - 46\)
\(\nu^{5}\)\(=\)\(-4985 \beta_{11} - 11807 \beta_{10} - 18490 \beta_{9} + 9970 \beta_{8} - 85383 \beta_{7} + 8659 \beta_{6} - 78561 \beta_{5} - 7671 \beta_{4} - 5498137 \beta_{3} - 57917301 \beta_{2} + 11807 \beta_{1} + 57905494\)
\(\nu^{6}\)\(=\)\(-420370 \beta_{11} - 873419 \beta_{10} - 285046 \beta_{9} + 210185 \beta_{8} - 221078 \beta_{7} - 221078 \beta_{6} - 8977391 \beta_{5} + 314220 \beta_{4} - 173238180 \beta_{3} - 10893 \beta_{2} + 173459258 \beta_{1} + 6929260332\)
\(\nu^{7}\)\(=\)\(-16872609 \beta_{11} - 41224652 \beta_{10} - 6142124 \beta_{9} - 16872609 \beta_{8} + 296313649 \beta_{7} - 44964369 \beta_{6} - 20612326 \beta_{5} - 10730485 \beta_{4} + 3739717 \beta_{3} + 220948491176 \beta_{2} + 15570248355 \beta_{1} + 20612326\)
\(\nu^{8}\)\(=\)\(740063673 \beta_{11} + 1554437295 \beta_{10} + 165376074 \beta_{9} - 1480127346 \beta_{8} + 26446653055 \beta_{7} - 888683571 \beta_{6} + 25632279433 \beta_{5} - 250188825 \beta_{4} + 607602795489 \beta_{3} + 19678593513893 \beta_{2} - 1554437295 \beta_{1} - 19677039076598\)
\(\nu^{9}\)\(=\)\(105427992482 \beta_{11} + 238563843427 \beta_{10} + 158675910438 \beta_{9} - 52713996241 \beta_{8} + 61949949062 \beta_{7} + 61949949062 \beta_{6} + 1093570332775 \beta_{5} + 69552024628 \beta_{4} + 46051155982788 \beta_{3} + 9235952821 \beta_{2} - 46113105931850 \beta_{1} - 772687374238988\)
\(\nu^{10}\)\(=\)\(2468471109865 \beta_{11} + 5532921404972 \beta_{10} + 3298579732876 \beta_{9} + 2468471109865 \beta_{8} - 76568357146745 \beta_{7} + 5830910997593 \beta_{6} + 2766460702486 \beta_{5} - 830108623011 \beta_{4} - 297989592621 \beta_{3} - 58200093179211752 \beta_{2} - 2038230432502331 \beta_{1} - 2766460702486\)
\(\nu^{11}\)\(=\)\(-162499451784257 \beta_{11} - 350726880646247 \beta_{10} - 357292586394394 \beta_{9} + 324998903568514 \beta_{8} - 3639117913422711 \beta_{7} + 213955405939723 \beta_{6} - 3450890484560721 \beta_{5} - 110260555843935 \beta_{4} - 139222261518191209 \beta_{3} - 2591709677523949965 \beta_{2} + 350726880646247 \beta_{1} + 2591358950643303718\)

Character values

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

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

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
28.0722 + 48.6225i
14.5243 + 25.1569i
5.98749 + 10.3706i
−10.2102 17.6846i
−15.0682 26.0989i
−22.8056 39.5005i
28.0722 48.6225i
14.5243 25.1569i
5.98749 10.3706i
−10.2102 + 17.6846i
−15.0682 + 26.0989i
−22.8056 + 39.5005i
−27.0722 46.8904i −121.500 70.1481i −953.808 + 1652.04i −1899.28 + 1096.55i 7596.25i 16751.6 1363.71i 47842.8 9841.50 + 17046.0i 102836. + 59372.1i
10.2 −13.5243 23.4248i −121.500 70.1481i 146.185 253.199i −1211.43 + 699.418i 3794.82i −12945.1 10719.1i −35606.0 9841.50 + 17046.0i 32767.5 + 18918.3i
10.3 −4.98749 8.63859i −121.500 70.1481i 462.250 800.640i 2618.43 1511.75i 1399.45i 13450.1 + 10078.2i −19436.3 9841.50 + 17046.0i −26118.8 15079.7i
10.4 11.2102 + 19.4166i −121.500 70.1481i 260.663 451.481i −354.602 + 204.730i 3145.50i −12291.7 + 11462.5i 34646.8 9841.50 + 17046.0i −7950.32 4590.12i
10.5 16.0682 + 27.8309i −121.500 70.1481i −4.37327 + 7.57472i −4433.83 + 2559.87i 4508.61i 10835.6 12847.8i 32626.6 9841.50 + 17046.0i −142487. 82265.1i
10.6 23.8056 + 41.2326i −121.500 70.1481i −621.416 + 1076.32i 4637.22 2677.30i 6679.68i −5755.42 15790.8i −10418.9 9841.50 + 17046.0i 220784. + 127470.i
19.1 −27.0722 + 46.8904i −121.500 + 70.1481i −953.808 1652.04i −1899.28 1096.55i 7596.25i 16751.6 + 1363.71i 47842.8 9841.50 17046.0i 102836. 59372.1i
19.2 −13.5243 + 23.4248i −121.500 + 70.1481i 146.185 + 253.199i −1211.43 699.418i 3794.82i −12945.1 + 10719.1i −35606.0 9841.50 17046.0i 32767.5 18918.3i
19.3 −4.98749 + 8.63859i −121.500 + 70.1481i 462.250 + 800.640i 2618.43 + 1511.75i 1399.45i 13450.1 10078.2i −19436.3 9841.50 17046.0i −26118.8 + 15079.7i
19.4 11.2102 19.4166i −121.500 + 70.1481i 260.663 + 451.481i −354.602 204.730i 3145.50i −12291.7 11462.5i 34646.8 9841.50 17046.0i −7950.32 + 4590.12i
19.5 16.0682 27.8309i −121.500 + 70.1481i −4.37327 7.57472i −4433.83 2559.87i 4508.61i 10835.6 + 12847.8i 32626.6 9841.50 17046.0i −142487. + 82265.1i
19.6 23.8056 41.2326i −121.500 + 70.1481i −621.416 1076.32i 4637.22 + 2677.30i 6679.68i −5755.42 + 15790.8i −10418.9 9841.50 17046.0i 220784. 127470.i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 19.6
Significant digits:
Format:

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 21.11.f.a 12
3.b odd 2 1 63.11.m.b 12
7.c even 3 1 147.11.d.a 12
7.d odd 6 1 inner 21.11.f.a 12
7.d odd 6 1 147.11.d.a 12
21.g even 6 1 63.11.m.b 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
21.11.f.a 12 1.a even 1 1 trivial
21.11.f.a 12 7.d odd 6 1 inner
63.11.m.b 12 3.b odd 2 1
63.11.m.b 12 21.g even 6 1
147.11.d.a 12 7.c even 3 1
147.11.d.a 12 7.d odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(81\!\cdots\!16\)\( T_{2}^{4} - \)\(35\!\cdots\!20\)\( T_{2}^{3} + \)\(23\!\cdots\!60\)\( T_{2}^{2} + \)\(14\!\cdots\!00\)\( T_{2} + \)\(25\!\cdots\!00\)\( \)">\(T_{2}^{12} - \cdots\) acting on \(S_{11}^{\mathrm{new}}(21, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 251144346540441600 + 14809298020761600 T + 2339921213378560 T^{2} - 35941500446720 T^{3} + 8190073487616 T^{4} - 66881573248 T^{5} + 12108530192 T^{6} - 152858328 T^{7} + 11481368 T^{8} - 59914 T^{9} + 3843 T^{10} - 11 T^{11} + T^{12} \)
$3$ \( ( 19683 + 243 T + T^{2} )^{6} \)
$5$ \( \)\(10\!\cdots\!00\)\( + \)\(61\!\cdots\!00\)\( T + \)\(13\!\cdots\!00\)\( T^{2} + \)\(12\!\cdots\!00\)\( T^{3} + \)\(33\!\cdots\!00\)\( T^{4} - \)\(13\!\cdots\!50\)\( T^{5} - \)\(63\!\cdots\!35\)\( T^{6} + 1693007896207023765 T^{7} + 1005052632410574 T^{8} - 45306946971 T^{9} - 34651410 T^{10} + 1287 T^{11} + T^{12} \)
$7$ \( \)\(50\!\cdots\!01\)\( - \)\(36\!\cdots\!10\)\( T + \)\(22\!\cdots\!05\)\( T^{2} - \)\(78\!\cdots\!98\)\( T^{3} + \)\(16\!\cdots\!10\)\( T^{4} - \)\(71\!\cdots\!30\)\( T^{5} + \)\(97\!\cdots\!49\)\( T^{6} - \)\(25\!\cdots\!70\)\( T^{7} + 200650326945180210 T^{8} - 3500703863902 T^{9} + 35218505 T^{10} - 20090 T^{11} + T^{12} \)
$11$ \( \)\(26\!\cdots\!00\)\( + \)\(32\!\cdots\!00\)\( T + \)\(41\!\cdots\!60\)\( T^{2} + \)\(44\!\cdots\!20\)\( T^{3} + \)\(65\!\cdots\!36\)\( T^{4} - \)\(40\!\cdots\!90\)\( T^{5} + \)\(94\!\cdots\!67\)\( T^{6} - \)\(43\!\cdots\!35\)\( T^{7} + \)\(39\!\cdots\!08\)\( T^{8} - 11198487115113955 T^{9} + 96012196932 T^{10} - 236165 T^{11} + T^{12} \)
$13$ \( \)\(22\!\cdots\!00\)\( + \)\(45\!\cdots\!60\)\( T^{2} + \)\(19\!\cdots\!16\)\( T^{4} + \)\(31\!\cdots\!07\)\( T^{6} + \)\(23\!\cdots\!79\)\( T^{8} + 803804815113 T^{10} + T^{12} \)
$17$ \( \)\(73\!\cdots\!00\)\( - \)\(10\!\cdots\!00\)\( T + \)\(52\!\cdots\!80\)\( T^{2} + \)\(12\!\cdots\!00\)\( T^{3} - \)\(54\!\cdots\!64\)\( T^{4} - \)\(37\!\cdots\!08\)\( T^{5} + \)\(25\!\cdots\!32\)\( T^{6} + \)\(60\!\cdots\!88\)\( T^{7} + \)\(18\!\cdots\!68\)\( T^{8} - 11164389903383850072 T^{9} - 4090465634088 T^{10} + 2038782 T^{11} + T^{12} \)
$19$ \( \)\(62\!\cdots\!24\)\( - \)\(70\!\cdots\!12\)\( T - \)\(80\!\cdots\!36\)\( T^{2} + \)\(11\!\cdots\!44\)\( T^{3} + \)\(15\!\cdots\!84\)\( T^{4} + \)\(62\!\cdots\!22\)\( T^{5} - \)\(45\!\cdots\!83\)\( T^{6} - \)\(37\!\cdots\!31\)\( T^{7} + \)\(10\!\cdots\!06\)\( T^{8} + 12218418340863979797 T^{9} - 11727252949566 T^{10} - 1012389 T^{11} + T^{12} \)
$23$ \( \)\(61\!\cdots\!00\)\( - \)\(36\!\cdots\!00\)\( T + \)\(77\!\cdots\!00\)\( T^{2} - \)\(57\!\cdots\!00\)\( T^{3} + \)\(70\!\cdots\!00\)\( T^{4} - \)\(44\!\cdots\!00\)\( T^{5} + \)\(28\!\cdots\!80\)\( T^{6} - \)\(44\!\cdots\!40\)\( T^{7} + \)\(75\!\cdots\!76\)\( T^{8} - 13553040975168174592 T^{9} + 117223473942528 T^{10} + 4341928 T^{11} + T^{12} \)
$29$ \( ( -\)\(47\!\cdots\!00\)\( - \)\(25\!\cdots\!00\)\( T + \)\(69\!\cdots\!80\)\( T^{2} + \)\(35\!\cdots\!65\)\( T^{3} - 616346059060829 T^{4} - 47722117 T^{5} + T^{6} )^{2} \)
$31$ \( \)\(28\!\cdots\!49\)\( - \)\(12\!\cdots\!44\)\( T + \)\(21\!\cdots\!71\)\( T^{2} - \)\(17\!\cdots\!48\)\( T^{3} + \)\(50\!\cdots\!54\)\( T^{4} + \)\(14\!\cdots\!44\)\( T^{5} - \)\(70\!\cdots\!77\)\( T^{6} - \)\(12\!\cdots\!88\)\( T^{7} + \)\(74\!\cdots\!86\)\( T^{8} + \)\(30\!\cdots\!24\)\( T^{9} - 3136356770364549 T^{10} - 9658932 T^{11} + T^{12} \)
$37$ \( \)\(16\!\cdots\!00\)\( - \)\(41\!\cdots\!00\)\( T + \)\(14\!\cdots\!00\)\( T^{2} + \)\(12\!\cdots\!00\)\( T^{3} + \)\(96\!\cdots\!00\)\( T^{4} + \)\(34\!\cdots\!50\)\( T^{5} + \)\(95\!\cdots\!75\)\( T^{6} + \)\(14\!\cdots\!75\)\( T^{7} + \)\(19\!\cdots\!00\)\( T^{8} + \)\(14\!\cdots\!75\)\( T^{9} + 16022979214987040 T^{10} + 87545045 T^{11} + T^{12} \)
$41$ \( \)\(14\!\cdots\!84\)\( + \)\(13\!\cdots\!28\)\( T^{2} + \)\(14\!\cdots\!00\)\( T^{4} + \)\(53\!\cdots\!20\)\( T^{6} + \)\(75\!\cdots\!00\)\( T^{8} + 46081130786465448 T^{10} + T^{12} \)
$43$ \( ( -\)\(26\!\cdots\!24\)\( - \)\(25\!\cdots\!28\)\( T - \)\(18\!\cdots\!66\)\( T^{2} + \)\(51\!\cdots\!01\)\( T^{3} + 11344569640419509 T^{4} - 350408153 T^{5} + T^{6} )^{2} \)
$47$ \( \)\(26\!\cdots\!00\)\( + \)\(43\!\cdots\!00\)\( T + \)\(24\!\cdots\!00\)\( T^{2} + \)\(16\!\cdots\!00\)\( T^{3} - \)\(11\!\cdots\!00\)\( T^{4} - \)\(11\!\cdots\!00\)\( T^{5} + \)\(76\!\cdots\!80\)\( T^{6} + \)\(97\!\cdots\!40\)\( T^{7} + \)\(26\!\cdots\!24\)\( T^{8} - \)\(32\!\cdots\!56\)\( T^{9} - 168051993351717360 T^{10} + 181842702 T^{11} + T^{12} \)
$53$ \( \)\(35\!\cdots\!00\)\( + \)\(20\!\cdots\!00\)\( T + \)\(10\!\cdots\!00\)\( T^{2} + \)\(27\!\cdots\!40\)\( T^{3} + \)\(76\!\cdots\!16\)\( T^{4} + \)\(12\!\cdots\!68\)\( T^{5} + \)\(30\!\cdots\!85\)\( T^{6} + \)\(40\!\cdots\!23\)\( T^{7} + \)\(79\!\cdots\!14\)\( T^{8} + \)\(61\!\cdots\!27\)\( T^{9} + 1176704486916078870 T^{10} + 695152867 T^{11} + T^{12} \)
$59$ \( \)\(39\!\cdots\!64\)\( - \)\(34\!\cdots\!44\)\( T + \)\(11\!\cdots\!00\)\( T^{2} - \)\(12\!\cdots\!32\)\( T^{3} - \)\(86\!\cdots\!48\)\( T^{4} + \)\(24\!\cdots\!72\)\( T^{5} + \)\(62\!\cdots\!67\)\( T^{6} - \)\(41\!\cdots\!57\)\( T^{7} + \)\(15\!\cdots\!48\)\( T^{8} + \)\(26\!\cdots\!15\)\( T^{9} - 1278614517030125652 T^{10} - 1373785545 T^{11} + T^{12} \)
$61$ \( \)\(95\!\cdots\!00\)\( - \)\(81\!\cdots\!00\)\( T + \)\(18\!\cdots\!20\)\( T^{2} + \)\(41\!\cdots\!40\)\( T^{3} - \)\(34\!\cdots\!24\)\( T^{4} - \)\(84\!\cdots\!24\)\( T^{5} + \)\(58\!\cdots\!28\)\( T^{6} + \)\(12\!\cdots\!56\)\( T^{7} - \)\(14\!\cdots\!92\)\( T^{8} - \)\(63\!\cdots\!76\)\( T^{9} - 396538115374076112 T^{10} + 2524633584 T^{11} + T^{12} \)
$67$ \( \)\(86\!\cdots\!00\)\( + \)\(16\!\cdots\!00\)\( T + \)\(48\!\cdots\!40\)\( T^{2} + \)\(49\!\cdots\!40\)\( T^{3} + \)\(11\!\cdots\!76\)\( T^{4} + \)\(12\!\cdots\!92\)\( T^{5} + \)\(15\!\cdots\!57\)\( T^{6} + \)\(89\!\cdots\!17\)\( T^{7} + \)\(55\!\cdots\!98\)\( T^{8} + \)\(16\!\cdots\!21\)\( T^{9} + 9566271847740082578 T^{10} + 2255105709 T^{11} + T^{12} \)
$71$ \( ( \)\(36\!\cdots\!44\)\( + \)\(33\!\cdots\!72\)\( T - \)\(25\!\cdots\!56\)\( T^{2} - \)\(12\!\cdots\!64\)\( T^{3} - 6659082852468435116 T^{4} + 1939455890 T^{5} + T^{6} )^{2} \)
$73$ \( \)\(10\!\cdots\!64\)\( + \)\(10\!\cdots\!16\)\( T - \)\(21\!\cdots\!76\)\( T^{2} - \)\(58\!\cdots\!36\)\( T^{3} + \)\(31\!\cdots\!64\)\( T^{4} - \)\(47\!\cdots\!60\)\( T^{5} - \)\(52\!\cdots\!53\)\( T^{6} + \)\(92\!\cdots\!95\)\( T^{7} + \)\(83\!\cdots\!36\)\( T^{8} - \)\(17\!\cdots\!17\)\( T^{9} - 8277694708284713136 T^{10} + 1888675383 T^{11} + T^{12} \)
$79$ \( \)\(42\!\cdots\!25\)\( - \)\(95\!\cdots\!00\)\( T + \)\(14\!\cdots\!35\)\( T^{2} - \)\(12\!\cdots\!00\)\( T^{3} + \)\(86\!\cdots\!86\)\( T^{4} - \)\(42\!\cdots\!80\)\( T^{5} + \)\(16\!\cdots\!83\)\( T^{6} - \)\(47\!\cdots\!60\)\( T^{7} + \)\(12\!\cdots\!78\)\( T^{8} - \)\(26\!\cdots\!40\)\( T^{9} + 60296050378786294743 T^{10} - 8260659900 T^{11} + T^{12} \)
$83$ \( \)\(78\!\cdots\!04\)\( + \)\(41\!\cdots\!96\)\( T^{2} + \)\(68\!\cdots\!64\)\( T^{4} + \)\(41\!\cdots\!47\)\( T^{6} + \)\(80\!\cdots\!91\)\( T^{8} + 51604835358977047461 T^{10} + T^{12} \)
$89$ \( \)\(17\!\cdots\!44\)\( + \)\(93\!\cdots\!36\)\( T - \)\(13\!\cdots\!40\)\( T^{2} - \)\(69\!\cdots\!92\)\( T^{3} + \)\(10\!\cdots\!32\)\( T^{4} - \)\(74\!\cdots\!68\)\( T^{5} + \)\(24\!\cdots\!72\)\( T^{6} - \)\(38\!\cdots\!92\)\( T^{7} + \)\(10\!\cdots\!68\)\( T^{8} + \)\(47\!\cdots\!60\)\( T^{9} - 19956926066127955032 T^{10} - 9451951530 T^{11} + T^{12} \)
$97$ \( \)\(28\!\cdots\!00\)\( + \)\(71\!\cdots\!40\)\( T^{2} + \)\(46\!\cdots\!96\)\( T^{4} + \)\(10\!\cdots\!95\)\( T^{6} + \)\(11\!\cdots\!07\)\( T^{8} + \)\(56\!\cdots\!05\)\( T^{10} + T^{12} \)
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