# Properties

 Label 14.9.d.a Level $14$ Weight $9$ Character orbit 14.d Analytic conductor $5.703$ Analytic rank $0$ Dimension $12$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$14 = 2 \cdot 7$$ Weight: $$k$$ $$=$$ $$9$$ Character orbit: $$[\chi]$$ $$=$$ 14.d (of order $$6$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$5.70330054086$$ Analytic rank: $$0$$ Dimension: $$12$$ Relative dimension: $$6$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\mathbb{Q}[x]/(x^{12} - \cdots)$$ Defining polynomial: $$x^{12} - 2 x^{11} + 1771 x^{10} + 26038 x^{9} + 2442597 x^{8} + 26522276 x^{7} + 1175865280 x^{6} + 6901058684 x^{5} + 370996492174 x^{4} + 1285719886320 x^{3} + 55526550982200 x^{2} - 240706289358000 x + 3600954063202500$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{24}\cdot 3^{4}\cdot 7^{3}$$ 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.

 $$f(q)$$ $$=$$ $$q + ( -\beta_{2} + \beta_{3} ) q^{2} + ( 18 - 9 \beta_{1} + \beta_{2} + \beta_{4} ) q^{3} -128 \beta_{1} q^{4} + ( 93 + 93 \beta_{1} - 7 \beta_{2} - 7 \beta_{3} - 2 \beta_{5} + \beta_{7} + \beta_{9} ) q^{5} + ( -42 + 85 \beta_{1} - 9 \beta_{2} + 18 \beta_{3} + \beta_{8} ) q^{6} + ( 38 - 295 \beta_{1} + 51 \beta_{2} - 66 \beta_{3} - 8 \beta_{4} - \beta_{5} + 2 \beta_{9} + \beta_{11} ) q^{7} + 128 \beta_{2} q^{8} + ( 3933 - 3933 \beta_{1} + 49 \beta_{2} + 6 \beta_{3} + 57 \beta_{4} - 30 \beta_{5} + 4 \beta_{6} - 2 \beta_{7} + 2 \beta_{8} - \beta_{9} - 2 \beta_{10} + \beta_{11} ) q^{9} +O(q^{10})$$ $$q + ( -\beta_{2} + \beta_{3} ) q^{2} + ( 18 - 9 \beta_{1} + \beta_{2} + \beta_{4} ) q^{3} -128 \beta_{1} q^{4} + ( 93 + 93 \beta_{1} - 7 \beta_{2} - 7 \beta_{3} - 2 \beta_{5} + \beta_{7} + \beta_{9} ) q^{5} + ( -42 + 85 \beta_{1} - 9 \beta_{2} + 18 \beta_{3} + \beta_{8} ) q^{6} + ( 38 - 295 \beta_{1} + 51 \beta_{2} - 66 \beta_{3} - 8 \beta_{4} - \beta_{5} + 2 \beta_{9} + \beta_{11} ) q^{7} + 128 \beta_{2} q^{8} + ( 3933 - 3933 \beta_{1} + 49 \beta_{2} + 6 \beta_{3} + 57 \beta_{4} - 30 \beta_{5} + 4 \beta_{6} - 2 \beta_{7} + 2 \beta_{8} - \beta_{9} - 2 \beta_{10} + \beta_{11} ) q^{9} + ( 1960 - 982 \beta_{1} - 184 \beta_{2} + 99 \beta_{3} + 14 \beta_{4} - 2 \beta_{5} + 2 \beta_{6} - 2 \beta_{9} + 4 \beta_{11} ) q^{10} + ( 4 + 1714 \beta_{1} - 24 \beta_{2} + 585 \beta_{3} - 26 \beta_{4} + 49 \beta_{5} + 5 \beta_{6} + 5 \beta_{7} + 12 \beta_{8} + 13 \beta_{9} + 6 \beta_{10} - 4 \beta_{11} ) q^{11} + ( -1152 - 1152 \beta_{1} - 128 \beta_{5} ) q^{12} + ( 1471 - 2932 \beta_{1} + 826 \beta_{2} - 1719 \beta_{3} - 36 \beta_{4} + 41 \beta_{5} - 15 \beta_{6} + 5 \beta_{7} + 20 \beta_{9} + 5 \beta_{11} ) q^{13} + ( 1273 + 6910 \beta_{1} + 211 \beta_{2} + 7 \beta_{3} - 52 \beta_{4} + 66 \beta_{5} - 10 \beta_{6} + 14 \beta_{7} - 15 \beta_{8} - 18 \beta_{9} - 3 \beta_{10} + 14 \beta_{11} ) q^{14} + ( -21283 + 32 \beta_{1} - 1182 \beta_{2} - 18 \beta_{3} + 33 \beta_{4} + 51 \beta_{5} + 6 \beta_{6} + 8 \beta_{7} - 20 \beta_{8} + 13 \beta_{9} - 40 \beta_{10} - 30 \beta_{11} ) q^{15} + ( -16384 + 16384 \beta_{1} ) q^{16} + ( 19224 - 9648 \beta_{1} - 2387 \beta_{2} + 1163 \beta_{3} - 61 \beta_{4} + 9 \beta_{5} - 9 \beta_{6} - \beta_{7} + 9 \beta_{9} + 90 \beta_{10} - 18 \beta_{11} ) q^{17} + ( 24 + 2790 \beta_{1} + 114 \beta_{2} + 4056 \beta_{3} + 118 \beta_{4} - 230 \beta_{5} - 10 \beta_{6} - 60 \beta_{7} + 40 \beta_{8} - 126 \beta_{9} + 20 \beta_{10} + 8 \beta_{11} ) q^{18} + ( 22534 + 22602 \beta_{1} - 801 \beta_{2} - 822 \beta_{3} - 7 \beta_{4} + 759 \beta_{5} + 14 \beta_{6} - 70 \beta_{7} - 54 \beta_{8} - 63 \beta_{9} - 54 \beta_{10} - 21 \beta_{11} ) q^{19} + ( 11904 - 23808 \beta_{1} - 1152 \beta_{2} + 1792 \beta_{3} - 256 \beta_{4} + 256 \beta_{5} - 128 \beta_{9} ) q^{20} + ( -89813 + 70024 \beta_{1} + 5627 \beta_{2} - 6735 \beta_{3} + 499 \beta_{4} - 346 \beta_{5} + 16 \beta_{6} - 97 \beta_{7} - 164 \beta_{8} + 49 \beta_{9} - 16 \beta_{10} - 47 \beta_{11} ) q^{21} + ( -78478 + 17 \beta_{1} - 1745 \beta_{2} + 42 \beta_{3} - 528 \beta_{4} - 570 \beta_{5} - 14 \beta_{6} - 114 \beta_{7} - 45 \beta_{8} - 78 \beta_{9} - 90 \beta_{10} + 70 \beta_{11} ) q^{22} + ( 26600 - 26506 \beta_{1} + 2060 \beta_{2} - 4065 \beta_{3} - 2039 \beta_{4} + 1045 \beta_{5} - 68 \beta_{6} + 89 \beta_{7} + 60 \beta_{8} - 38 \beta_{9} - 60 \beta_{10} - 17 \beta_{11} ) q^{23} + ( 10880 - 5504 \beta_{1} + 2304 \beta_{2} - 1152 \beta_{3} + 128 \beta_{10} ) q^{24} + ( 192 + 139580 \beta_{1} + 26 \beta_{2} - 1884 \beta_{3} + 38 \beta_{4} - 58 \beta_{5} - 30 \beta_{6} + 196 \beta_{7} + 360 \beta_{8} + 374 \beta_{9} + 180 \beta_{10} + 24 \beta_{11} ) q^{25} + ( 108798 + 108774 \beta_{1} + 1586 \beta_{2} + 1676 \beta_{3} + 30 \beta_{4} - 460 \beta_{5} - 60 \beta_{6} + 350 \beta_{7} - 36 \beta_{8} + 320 \beta_{9} - 36 \beta_{10} + 90 \beta_{11} ) q^{26} + ( 272352 - 544728 \beta_{1} - 6357 \beta_{2} + 19137 \beta_{3} + 3255 \beta_{4} - 3342 \beta_{5} + 261 \beta_{6} - 87 \beta_{7} + 150 \beta_{8} + 186 \beta_{9} - 87 \beta_{11} ) q^{27} + ( -37760 + 32896 \beta_{1} - 8576 \beta_{2} + 768 \beta_{3} - 128 \beta_{4} + 1152 \beta_{5} + 128 \beta_{6} + 256 \beta_{7} - 128 \beta_{11} ) q^{28} + ( -362999 + 10 \beta_{1} - 12070 \beta_{2} + 183 \beta_{3} + 2310 \beta_{4} + 2127 \beta_{5} - 61 \beta_{6} + 399 \beta_{7} - 132 \beta_{8} + 108 \beta_{9} - 264 \beta_{10} + 305 \beta_{11} ) q^{29} + ( 152885 - 152818 \beta_{1} + 24687 \beta_{2} - 17877 \beta_{3} + 6838 \beta_{4} - 3440 \beta_{5} + 56 \beta_{6} - 398 \beta_{7} + 95 \beta_{8} + 356 \beta_{9} - 95 \beta_{10} + 14 \beta_{11} ) q^{30} + ( 502162 - 251147 \beta_{1} + 1797 \beta_{2} - 1311 \beta_{3} - 825 \beta_{4} - 66 \beta_{5} + 66 \beta_{6} + 81 \beta_{7} - 66 \beta_{9} + 132 \beta_{11} ) q^{31} -16384 \beta_{3} q^{32} + ( 275337 + 275049 \beta_{1} + 49608 \beta_{2} + 49680 \beta_{3} + 24 \beta_{4} - 3228 \beta_{5} - 48 \beta_{6} - 540 \beta_{7} + 240 \beta_{8} - 564 \beta_{9} + 240 \beta_{10} + 72 \beta_{11} ) q^{33} + ( 152238 - 304472 \beta_{1} - 16541 \beta_{2} + 12324 \beta_{3} - 10380 \beta_{4} + 10382 \beta_{5} - 6 \beta_{6} + 2 \beta_{7} + 578 \beta_{9} + 2 \beta_{11} ) q^{34} + ( -955373 + 1032616 \beta_{1} - 49561 \beta_{2} - 34259 \beta_{3} - 4213 \beta_{4} - 5792 \beta_{5} - 175 \beta_{6} - 255 \beta_{7} + 480 \beta_{8} - 239 \beta_{9} - 30 \beta_{10} + 350 \beta_{11} ) q^{35} + ( -503424 - 2560 \beta_{2} - 384 \beta_{3} - 3840 \beta_{4} - 3456 \beta_{5} + 128 \beta_{6} + 128 \beta_{7} + 256 \beta_{8} + 256 \beta_{9} + 512 \beta_{10} - 640 \beta_{11} ) q^{36} + ( 21790 - 22296 \beta_{1} + 73208 \beta_{2} - 79941 \beta_{3} - 6587 \beta_{4} + 3184 \beta_{5} + 292 \beta_{6} + 515 \beta_{7} - 360 \beta_{8} - 734 \beta_{9} + 360 \beta_{10} + 73 \beta_{11} ) q^{37} + ( 142473 - 70773 \beta_{1} - 42632 \beta_{2} + 24779 \beta_{3} + 6926 \beta_{4} + 126 \beta_{5} - 126 \beta_{6} - 448 \beta_{7} + 126 \beta_{9} - 675 \beta_{10} - 252 \beta_{11} ) q^{38} + ( -1162 + 223745 \beta_{1} - 4559 \beta_{2} + 11943 \beta_{3} - 4597 \beta_{4} + 9137 \beta_{5} + 95 \beta_{6} - 929 \beta_{7} - 2248 \beta_{8} - 1801 \beta_{9} - 1124 \beta_{10} - 76 \beta_{11} ) q^{39} + ( -125696 - 125184 \beta_{1} + 12672 \beta_{2} + 11904 \beta_{3} - 256 \beta_{4} - 1536 \beta_{5} + 512 \beta_{6} - 256 \beta_{7} - 768 \beta_{11} ) q^{40} + ( 263301 - 526518 \beta_{1} - 30768 \beta_{2} + 91291 \beta_{3} + 14564 \beta_{4} - 13937 \beta_{5} - 1881 \beta_{6} + 627 \beta_{7} - 1170 \beta_{8} - 2090 \beta_{9} + 627 \beta_{11} ) q^{41} + ( 215018 + 680338 \beta_{1} + 14241 \beta_{2} - 94818 \beta_{3} + 3390 \beta_{4} + 19812 \beta_{5} - 684 \beta_{6} - 466 \beta_{7} + 300 \beta_{8} + 592 \beta_{9} + 200 \beta_{10} + 426 \beta_{11} ) q^{42} + ( -1079876 - 876 \beta_{1} + 20284 \beta_{2} - 846 \beta_{3} + 5254 \beta_{4} + 6100 \beta_{5} + 282 \beta_{6} - 2578 \beta_{7} + 1440 \beta_{8} - 866 \beta_{9} + 2880 \beta_{10} - 1410 \beta_{11} ) q^{43} + ( 219392 - 219904 \beta_{1} + 81792 \beta_{2} - 75264 \beta_{3} + 6272 \beta_{4} - 2944 \beta_{5} - 512 \beta_{6} + 1024 \beta_{7} - 768 \beta_{8} - 640 \beta_{9} + 768 \beta_{10} - 128 \beta_{11} ) q^{44} + ( -934008 + 468471 \beta_{1} - 500013 \beta_{2} + 241905 \beta_{3} - 16203 \beta_{4} + 87 \beta_{5} - 87 \beta_{6} + 597 \beta_{7} + 87 \beta_{9} - 2760 \beta_{10} - 174 \beta_{11} ) q^{45} + ( -1053 + 392109 \beta_{1} + 10864 \beta_{2} + 36777 \beta_{3} + 10576 \beta_{4} - 21584 \beta_{5} + 720 \beta_{6} + 800 \beta_{7} - 1530 \beta_{8} + 2032 \beta_{9} - 765 \beta_{10} - 576 \beta_{11} ) q^{46} + ( 1021449 + 1021089 \beta_{1} + 184493 \beta_{2} + 185033 \beta_{3} + 180 \beta_{4} + 23377 \beta_{5} - 360 \beta_{6} + 1951 \beta_{7} + 1771 \beta_{9} + 540 \beta_{11} ) q^{47} + ( -147456 + 294912 \beta_{1} - 16384 \beta_{2} - 16384 \beta_{4} + 16384 \beta_{5} ) q^{48} + ( 733846 - 2075776 \beta_{1} + 51382 \beta_{2} - 313371 \beta_{3} + 12374 \beta_{4} - 27735 \beta_{5} + 669 \beta_{6} + 2441 \beta_{7} + 2790 \beta_{8} - 696 \beta_{9} + 936 \beta_{10} - 593 \beta_{11} ) q^{49} + ( 247184 - 436 \beta_{1} - 140514 \beta_{2} + 1104 \beta_{3} - 20864 \beta_{4} - 21968 \beta_{5} - 368 \beta_{6} + 1136 \beta_{7} - 300 \beta_{8} + 16 \beta_{9} - 600 \beta_{10} + 1840 \beta_{11} ) q^{50} + ( -445840 + 445772 \beta_{1} + 779073 \beta_{2} - 810294 \beta_{3} - 30975 \beta_{4} + 15303 \beta_{5} + 492 \beta_{6} - 4162 \beta_{7} + 178 \beta_{8} + 3793 \beta_{9} - 178 \beta_{10} + 123 \beta_{11} ) q^{51} + ( -375296 + 187008 \beta_{1} - 216704 \beta_{2} + 110976 \beta_{3} + 5248 \beta_{4} - 640 \beta_{5} + 640 \beta_{6} + 1920 \beta_{7} - 640 \beta_{9} + 1280 \beta_{11} ) q^{52} + ( 678 - 2757588 \beta_{1} - 781 \beta_{2} + 311616 \beta_{3} - 463 \beta_{4} + 1403 \beta_{5} - 795 \beta_{6} + 1078 \beta_{7} + 720 \beta_{8} + 1679 \beta_{9} + 360 \beta_{10} + 636 \beta_{11} ) q^{53} + ( -1090545 - 1093824 \beta_{1} + 273447 \beta_{2} + 275085 \beta_{3} + 546 \beta_{4} - 4788 \beta_{5} - 1092 \beta_{6} - 5022 \beta_{7} + 2187 \beta_{8} - 5568 \beta_{9} + 2187 \beta_{10} + 1638 \beta_{11} ) q^{54} + ( -1226802 + 2452158 \beta_{1} - 97518 \beta_{2} + 247191 \beta_{3} + 27249 \beta_{4} - 29592 \beta_{5} + 7029 \beta_{6} - 2343 \beta_{7} + 3240 \beta_{8} + 8055 \beta_{9} - 2343 \beta_{11} ) q^{55} + ( 884480 - 1047424 \beta_{1} + 8064 \beta_{2} - 35072 \beta_{3} + 8448 \beta_{4} - 1792 \beta_{5} + 1792 \beta_{6} - 4096 \beta_{7} + 384 \beta_{8} - 1792 \beta_{9} - 1536 \beta_{10} - 512 \beta_{11} ) q^{56} + ( 8354419 + 148 \beta_{1} - 554652 \beta_{2} + 2688 \beta_{3} + 40256 \beta_{4} + 37568 \beta_{5} - 896 \beta_{6} + 8264 \beta_{7} - 1940 \beta_{8} + 2788 \beta_{9} - 3880 \beta_{10} + 4480 \beta_{11} ) q^{57} + ( 1530490 - 1529006 \beta_{1} + 373716 \beta_{2} - 351822 \beta_{3} + 22570 \beta_{4} - 11792 \beta_{5} + 1352 \beta_{6} + 3566 \beta_{7} + 2160 \beta_{8} - 4580 \beta_{9} - 2160 \beta_{10} + 338 \beta_{11} ) q^{58} + ( 3461670 - 1732701 \beta_{1} - 708091 \beta_{2} + 370810 \beta_{3} + 33529 \beta_{4} + 384 \beta_{5} - 384 \beta_{6} - 6566 \beta_{7} + 384 \beta_{9} + 4500 \beta_{10} - 768 \beta_{11} ) q^{59} + ( 4096 + 2720128 \beta_{1} + 4992 \beta_{2} + 160896 \beta_{3} + 6528 \beta_{4} - 10752 \beta_{5} - 3840 \beta_{6} + 640 \beta_{7} + 5120 \beta_{8} - 1024 \beta_{9} + 2560 \beta_{10} + 3072 \beta_{11} ) q^{60} + ( -4744216 - 4742414 \beta_{1} + 596898 \beta_{2} + 594357 \beta_{3} - 847 \beta_{4} - 58506 \beta_{5} + 1694 \beta_{6} + 9665 \beta_{7} - 108 \beta_{8} + 10512 \beta_{9} - 108 \beta_{10} - 2541 \beta_{11} ) q^{61} + ( -139260 + 277071 \beta_{1} - 256109 \beta_{2} + 497104 \beta_{3} - 7476 \beta_{4} + 7314 \beta_{5} + 486 \beta_{6} - 162 \beta_{7} - 1125 \beta_{8} - 4386 \beta_{9} - 162 \beta_{11} ) q^{62} + ( 2580873 - 4439835 \beta_{1} - 245281 \beta_{2} - 1314978 \beta_{3} - 63837 \beta_{4} + 41766 \beta_{5} - 424 \beta_{6} + 1610 \beta_{7} - 11360 \beta_{8} + 7669 \beta_{9} - 2860 \beta_{10} - 2989 \beta_{11} ) q^{63} + 2097152 q^{64} + ( -7744345 + 7749221 \beta_{1} + 1449181 \beta_{2} - 1376526 \beta_{3} + 69009 \beta_{4} - 31770 \beta_{5} - 7292 \beta_{6} + 3486 \beta_{7} + 1230 \beta_{8} + 1983 \beta_{9} - 1230 \beta_{10} - 1823 \beta_{11} ) q^{65} + ( -12430332 + 6214044 \beta_{1} - 562890 \beta_{2} + 260601 \beta_{3} - 41688 \beta_{4} + 1128 \beta_{5} - 1128 \beta_{6} + 1536 \beta_{7} + 1128 \beta_{9} + 4500 \beta_{10} - 2256 \beta_{11} ) q^{66} + ( 4792 - 6295585 \beta_{1} - 52333 \beta_{2} - 66342 \beta_{3} - 54201 \beta_{4} + 105600 \beta_{5} + 4670 \beta_{6} - 2370 \beta_{7} + 13320 \beta_{8} - 1938 \beta_{9} + 6660 \beta_{10} - 3736 \beta_{11} ) q^{67} + ( -1234944 - 1225728 \beta_{1} + 148864 \beta_{2} + 152320 \beta_{3} + 1152 \beta_{4} + 6656 \beta_{5} - 2304 \beta_{6} + 1280 \beta_{7} - 11520 \beta_{8} + 128 \beta_{9} - 11520 \beta_{10} + 3456 \beta_{11} ) q^{68} + ( -10285503 + 20575806 \beta_{1} - 891627 \beta_{2} + 1665750 \beta_{3} - 60906 \beta_{4} + 65214 \beta_{5} - 12924 \beta_{6} + 4308 \beta_{7} - 3816 \beta_{8} - 20811 \beta_{9} + 4308 \beta_{11} ) q^{69} + ( 10859297 - 6389687 \beta_{1} - 62154 \beta_{2} - 941193 \beta_{3} - 16848 \beta_{4} - 59678 \beta_{5} - 1274 \beta_{6} + 6682 \beta_{7} - 4770 \beta_{8} - 890 \beta_{9} + 6795 \beta_{10} - 574 \beta_{11} ) q^{70} + ( 3794318 + 11012 \beta_{1} - 461056 \beta_{2} - 6348 \beta_{3} - 80078 \beta_{4} - 73730 \beta_{5} + 2116 \beta_{6} - 30136 \beta_{7} - 6780 \beta_{8} - 11894 \beta_{9} - 13560 \beta_{10} - 10580 \beta_{11} ) q^{71} + ( 357120 - 360192 \beta_{1} + 488448 \beta_{2} - 518400 \beta_{3} - 29440 \beta_{4} + 14336 \beta_{5} + 1024 \beta_{6} - 8448 \beta_{7} - 2560 \beta_{8} + 7680 \beta_{9} + 2560 \beta_{10} + 256 \beta_{11} ) q^{72} + ( 1039454 - 529157 \beta_{1} - 1391300 \beta_{2} + 688938 \beta_{3} - 13424 \beta_{4} - 520 \beta_{5} + 520 \beta_{6} + 16938 \beta_{7} - 520 \beta_{9} + 17820 \beta_{10} + 1040 \beta_{11} ) q^{73} + ( -4944 + 9812328 \beta_{1} - 47192 \beta_{2} - 29527 \beta_{3} - 49544 \beta_{4} + 95560 \beta_{5} + 5880 \beta_{6} - 7024 \beta_{7} - 5184 \beta_{8} - 10520 \beta_{9} - 2592 \beta_{10} - 4704 \beta_{11} ) q^{74} + ( 544488 + 551820 \beta_{1} + 1716816 \beta_{2} + 1713198 \beta_{3} - 1206 \beta_{4} + 66578 \beta_{5} + 2412 \beta_{6} - 36258 \beta_{7} - 4920 \beta_{8} - 35052 \beta_{9} - 4920 \beta_{10} - 3618 \beta_{11} ) q^{75} + ( 2893056 - 5777408 \beta_{1} - 6272 \beta_{2} + 207744 \beta_{3} + 97152 \beta_{4} - 96256 \beta_{5} - 2688 \beta_{6} + 896 \beta_{7} + 6912 \beta_{8} + 8960 \beta_{9} + 896 \beta_{11} ) q^{76} + ( -7196425 - 2387839 \beta_{1} - 932094 \beta_{2} - 864558 \beta_{3} + 116426 \beta_{4} + 92933 \beta_{5} - 4571 \beta_{6} - 20348 \beta_{7} - 2820 \beta_{8} - 25300 \beta_{9} - 1320 \beta_{10} + 14749 \beta_{11} ) q^{77} + ( -2120202 + 6474 \beta_{1} - 219000 \beta_{2} - 5346 \beta_{3} + 132048 \beta_{4} + 137394 \beta_{5} + 1782 \beta_{6} - 4214 \beta_{7} - 2910 \beta_{8} + 566 \beta_{9} - 5820 \beta_{10} - 8910 \beta_{11} ) q^{78} + ( 9941786 - 9950148 \beta_{1} + 1463548 \beta_{2} - 1534479 \beta_{3} - 61129 \beta_{4} + 23213 \beta_{5} + 19604 \beta_{6} + 12247 \beta_{7} + 1440 \beta_{8} - 26950 \beta_{9} - 1440 \beta_{10} + 4901 \beta_{11} ) q^{79} + ( -3047424 + 1523712 \beta_{1} + 262144 \beta_{2} - 114688 \beta_{3} + 32768 \beta_{4} - 16384 \beta_{7} ) q^{80} + ( 3732 - 14952396 \beta_{1} + 270453 \beta_{2} + 617535 \beta_{3} + 275307 \beta_{4} - 543333 \beta_{5} - 12135 \beta_{6} + 9495 \beta_{7} - 2244 \beta_{8} + 11709 \beta_{9} - 1122 \beta_{10} + 9708 \beta_{11} ) q^{81} + ( -5174990 - 5187882 \beta_{1} + 256170 \beta_{2} + 239868 \beta_{3} - 5434 \beta_{4} + 36900 \beta_{5} + 10868 \beta_{6} + 34694 \beta_{7} + 23760 \beta_{8} + 40128 \beta_{9} + 23760 \beta_{10} - 16302 \beta_{11} ) q^{82} + ( -10582284 + 21164100 \beta_{1} + 1806 \beta_{2} - 408776 \beta_{3} - 202060 \beta_{4} + 201016 \beta_{5} + 3132 \beta_{6} - 1044 \beta_{7} + 1620 \beta_{8} + 45670 \beta_{9} - 1044 \beta_{11} ) q^{83} + ( 8963072 + 2532992 \beta_{1} - 904320 \beta_{2} + 209664 \beta_{3} - 44288 \beta_{4} - 19584 \beta_{5} - 6016 \beta_{6} + 18688 \beta_{7} + 2048 \beta_{8} + 12416 \beta_{9} - 18944 \beta_{10} + 3968 \beta_{11} ) q^{84} + ( -5389614 - 13578 \beta_{1} - 444505 \beta_{2} + 5787 \beta_{3} - 197464 \beta_{4} - 203251 \beta_{5} - 1929 \beta_{6} + 87805 \beta_{7} + 9720 \beta_{8} + 41009 \beta_{9} + 19440 \beta_{10} + 9645 \beta_{11} ) q^{85} + ( -2512058 + 2521864 \beta_{1} + 920454 \beta_{2} - 1242158 \beta_{3} - 326296 \beta_{4} + 166592 \beta_{5} - 9184 \beta_{6} - 15752 \beta_{7} + 5214 \beta_{8} + 22640 \beta_{9} - 5214 \beta_{10} - 2296 \beta_{11} ) q^{86} + ( 37144020 - 18557007 \beta_{1} - 2105007 \beta_{2} + 1065009 \beta_{3} + 25011 \beta_{4} - 3927 \beta_{5} + 3927 \beta_{6} - 12723 \beta_{7} - 3927 \beta_{9} - 37860 \beta_{10} + 7854 \beta_{11} ) q^{87} + ( 2176 + 10043008 \beta_{1} - 69376 \beta_{2} + 143232 \beta_{3} - 72960 \beta_{4} + 140544 \beta_{5} + 8960 \beta_{6} + 4608 \beta_{7} + 11520 \beta_{8} + 14592 \beta_{9} + 5760 \beta_{10} - 7168 \beta_{11} ) q^{88} + ( 17951043 + 17921463 \beta_{1} + 401988 \beta_{2} + 438582 \beta_{3} + 12198 \beta_{4} + 31116 \beta_{5} - 24396 \beta_{6} + 13254 \beta_{7} + 5184 \beta_{8} + 1056 \beta_{9} + 5184 \beta_{10} + 36594 \beta_{11} ) q^{89} + ( 31651002 - 63318792 \beta_{1} + 715242 \beta_{2} - 688518 \beta_{3} + 371580 \beta_{4} - 372774 \beta_{5} + 3582 \beta_{6} - 1194 \beta_{7} - 14400 \beta_{8} + 4374 \beta_{9} - 1194 \beta_{11} ) q^{90} + ( -33926493 + 23407448 \beta_{1} + 2598668 \beta_{2} - 1296393 \beta_{3} + 235888 \beta_{4} - 139147 \beta_{5} + 14843 \beta_{6} + 24819 \beta_{7} + 47682 \beta_{8} + 46844 \beta_{9} + 19440 \beta_{10} - 17967 \beta_{11} ) q^{91} + ( -3392768 - 12032 \beta_{1} - 395264 \beta_{2} + 6528 \beta_{3} + 133760 \beta_{4} + 127232 \beta_{5} - 2176 \beta_{6} - 16256 \beta_{7} + 7680 \beta_{8} - 11392 \beta_{9} + 15360 \beta_{10} + 10880 \beta_{11} ) q^{92} + ( -1852470 + 1852128 \beta_{1} + 1248544 \beta_{2} - 635247 \beta_{3} + 610999 \beta_{4} - 303776 \beta_{5} - 4596 \beta_{6} - 9831 \beta_{7} - 2640 \beta_{8} + 13278 \beta_{9} + 2640 \beta_{10} - 1149 \beta_{11} ) q^{93} + ( -49267235 + 24642995 \beta_{1} - 2046916 \beta_{2} + 1024335 \beta_{3} + 1754 \beta_{4} - 3542 \beta_{5} + 3542 \beta_{6} + 11520 \beta_{7} - 3542 \beta_{9} - 25839 \beta_{10} + 7084 \beta_{11} ) q^{94} + ( -34430 - 29282540 \beta_{1} - 246560 \beta_{2} + 1110960 \beta_{3} - 252070 \beta_{4} + 495875 \beta_{5} + 13775 \beta_{6} - 6380 \beta_{7} - 57840 \beta_{8} - 4495 \beta_{9} - 28920 \beta_{10} - 11020 \beta_{11} ) q^{95} + ( -704512 - 688128 \beta_{1} - 147456 \beta_{2} - 147456 \beta_{3} - 16384 \beta_{8} - 16384 \beta_{10} ) q^{96} + ( 16623857 - 33253410 \beta_{1} - 903596 \beta_{2} + 1483143 \beta_{3} - 156888 \beta_{4} + 146615 \beta_{5} + 30819 \beta_{6} - 10273 \beta_{7} + 14850 \beta_{8} - 83302 \beta_{9} - 10273 \beta_{11} ) q^{97} + ( 36919238 + 4837514 \beta_{1} + 1408545 \beta_{2} + 789777 \beta_{3} - 68066 \beta_{4} - 267280 \beta_{5} + 18616 \beta_{6} - 24246 \beta_{7} + 18420 \beta_{8} - 12940 \beta_{9} + 27120 \beta_{10} - 8906 \beta_{11} ) q^{98} + ( -28914951 - 41448 \beta_{1} + 867366 \beta_{2} + 18279 \beta_{3} + 56070 \beta_{4} + 37791 \beta_{5} - 6093 \beta_{6} - 124593 \beta_{7} + 29262 \beta_{8} - 71436 \beta_{9} + 58524 \beta_{10} + 30465 \beta_{11} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$12q + 162q^{3} - 768q^{4} + 1674q^{5} - 1308q^{7} + 23604q^{9} + O(q^{10})$$ $$12q + 162q^{3} - 768q^{4} + 1674q^{5} - 1308q^{7} + 23604q^{9} + 17664q^{10} + 10302q^{11} - 20736q^{12} + 56832q^{14} - 255468q^{15} - 98304q^{16} + 173178q^{17} + 16896q^{18} + 405978q^{19} - 656910q^{21} - 941568q^{22} + 158934q^{23} + 98304q^{24} + 838668q^{25} + 1958400q^{26} - 255744q^{28} - 4355256q^{29} + 916992q^{30} + 4520250q^{31} + 4954482q^{33} - 5270790q^{35} - 6042624q^{36} + 134214q^{37} + 1278720q^{38} + 1335384q^{39} - 2260992q^{40} + 6660096q^{42} - 12961896q^{43} + 1318656q^{44} - 8415396q^{45} + 2345472q^{46} + 18385002q^{47} - 3659172q^{49} + 2970624q^{50} - 2673894q^{51} - 3369984q^{52} - 16540506q^{53} - 19646208q^{54} + 4325376q^{56} + 100263780q^{57} + 9176064q^{58} + 31163922q^{59} + 16349952q^{60} - 85390158q^{61} + 4361988q^{63} + 25165824q^{64} - 46506264q^{65} - 111873024q^{66} - 37750362q^{67} - 22166784q^{68} + 92031744q^{70} + 45506424q^{71} + 2162688q^{72} + 9414786q^{73} + 58837248q^{74} + 9837540q^{75} - 100614066q^{77} - 25463808q^{78} + 59730294q^{79} - 27426816q^{80} - 89677422q^{81} - 93259776q^{82} + 122616576q^{84} - 64652220q^{85} - 15144960q^{86} + 334229724q^{87} + 60260352q^{88} + 323014482q^{89} - 266861424q^{91} - 40687104q^{92} - 11119662q^{93} - 443440128q^{94} - 175918350q^{95} - 12582912q^{96} + 472166400q^{98} - 346906296q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{12} - 2 x^{11} + 1771 x^{10} + 26038 x^{9} + 2442597 x^{8} + 26522276 x^{7} + 1175865280 x^{6} + 6901058684 x^{5} + 370996492174 x^{4} + 1285719886320 x^{3} + 55526550982200 x^{2} - 240706289358000 x + 3600954063202500$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$24\!\cdots\!14$$$$\nu^{11} +$$$$17\!\cdots\!42$$$$\nu^{10} +$$$$40\!\cdots\!04$$$$\nu^{9} +$$$$10\!\cdots\!77$$$$\nu^{8} +$$$$62\!\cdots\!18$$$$\nu^{7} +$$$$11\!\cdots\!79$$$$\nu^{6} +$$$$28\!\cdots\!40$$$$\nu^{5} +$$$$34\!\cdots\!51$$$$\nu^{4} +$$$$83\!\cdots\!16$$$$\nu^{3} +$$$$11\!\cdots\!10$$$$\nu^{2} +$$$$97\!\cdots\!00$$$$\nu +$$$$10\!\cdots\!00$$$$)/$$$$12\!\cdots\!50$$ $$\beta_{2}$$ $$=$$ $$($$$$-202090052997757213384 \nu^{11} - 1989997430335076733412 \nu^{10} - 255656756837032256753104 \nu^{9} - 8227695102102134864192572 \nu^{8} - 492429624969135883257007488 \nu^{7} - 4647099311970982545404004644 \nu^{6} - 177379614919651411062138004000 \nu^{5} - 210613223065982349309540476456 \nu^{4} - 78770721438988807079781895174936 \nu^{3} + 216994393067807114987978096647200 \nu^{2} - 2348955365028536232136922784228000 \nu + 150034933393435109178188851023054000$$$$)/$$$$14\!\cdots\!75$$ $$\beta_{3}$$ $$=$$ $$($$$$41\!\cdots\!32$$$$\nu^{11} -$$$$21\!\cdots\!04$$$$\nu^{10} +$$$$64\!\cdots\!52$$$$\nu^{9} +$$$$13\!\cdots\!76$$$$\nu^{8} +$$$$73\!\cdots\!84$$$$\nu^{7} +$$$$12\!\cdots\!52$$$$\nu^{6} +$$$$34\!\cdots\!20$$$$\nu^{5} +$$$$53\!\cdots\!88$$$$\nu^{4} +$$$$95\!\cdots\!08$$$$\nu^{3} +$$$$12\!\cdots\!80$$$$\nu^{2} +$$$$26\!\cdots\!00$$$$\nu +$$$$11\!\cdots\!00$$$$)/$$$$22\!\cdots\!75$$ $$\beta_{4}$$ $$=$$ $$($$$$41\!\cdots\!63$$$$\nu^{11} +$$$$83\!\cdots\!49$$$$\nu^{10} -$$$$72\!\cdots\!77$$$$\nu^{9} +$$$$15\!\cdots\!69$$$$\nu^{8} +$$$$16\!\cdots\!11$$$$\nu^{7} +$$$$15\!\cdots\!13$$$$\nu^{6} +$$$$46\!\cdots\!90$$$$\nu^{5} +$$$$47\!\cdots\!42$$$$\nu^{4} +$$$$10\!\cdots\!12$$$$\nu^{3} +$$$$11\!\cdots\!10$$$$\nu^{2} -$$$$24\!\cdots\!00$$$$\nu +$$$$17\!\cdots\!00$$$$)/$$$$44\!\cdots\!50$$ $$\beta_{5}$$ $$=$$ $$($$$$-$$$$11\!\cdots\!87$$$$\nu^{11} +$$$$19\!\cdots\!04$$$$\nu^{10} -$$$$19\!\cdots\!37$$$$\nu^{9} -$$$$11\!\cdots\!26$$$$\nu^{8} -$$$$19\!\cdots\!99$$$$\nu^{7} -$$$$55\!\cdots\!02$$$$\nu^{6} -$$$$76\!\cdots\!80$$$$\nu^{5} -$$$$49\!\cdots\!08$$$$\nu^{4} -$$$$21\!\cdots\!18$$$$\nu^{3} -$$$$94\!\cdots\!70$$$$\nu^{2} -$$$$10\!\cdots\!00$$$$\nu -$$$$24\!\cdots\!00$$$$)/$$$$14\!\cdots\!50$$ $$\beta_{6}$$ $$=$$ $$($$$$54\!\cdots\!58$$$$\nu^{11} +$$$$66\!\cdots\!09$$$$\nu^{10} -$$$$71\!\cdots\!82$$$$\nu^{9} +$$$$14\!\cdots\!54$$$$\nu^{8} +$$$$80\!\cdots\!26$$$$\nu^{7} +$$$$13\!\cdots\!08$$$$\nu^{6} -$$$$16\!\cdots\!10$$$$\nu^{5} +$$$$51\!\cdots\!97$$$$\nu^{4} -$$$$15\!\cdots\!58$$$$\nu^{3} +$$$$12\!\cdots\!10$$$$\nu^{2} -$$$$97\!\cdots\!00$$$$\nu +$$$$77\!\cdots\!00$$$$)/$$$$44\!\cdots\!50$$ $$\beta_{7}$$ $$=$$ $$($$$$-$$$$15\!\cdots\!52$$$$\nu^{11} +$$$$11\!\cdots\!26$$$$\nu^{10} -$$$$44\!\cdots\!26$$$$\nu^{9} +$$$$13\!\cdots\!11$$$$\nu^{8} -$$$$28\!\cdots\!28$$$$\nu^{7} +$$$$15\!\cdots\!17$$$$\nu^{6} -$$$$16\!\cdots\!68$$$$\nu^{5} +$$$$55\!\cdots\!67$$$$\nu^{4} -$$$$45\!\cdots\!60$$$$\nu^{3} +$$$$16\!\cdots\!98$$$$\nu^{2} -$$$$93\!\cdots\!80$$$$\nu +$$$$57\!\cdots\!00$$$$)/$$$$89\!\cdots\!75$$ $$\beta_{8}$$ $$=$$ $$($$$$21\!\cdots\!74$$$$\nu^{11} +$$$$33\!\cdots\!02$$$$\nu^{10} +$$$$18\!\cdots\!04$$$$\nu^{9} +$$$$10\!\cdots\!87$$$$\nu^{8} +$$$$21\!\cdots\!78$$$$\nu^{7} +$$$$61\!\cdots\!49$$$$\nu^{6} -$$$$55\!\cdots\!80$$$$\nu^{5} +$$$$23\!\cdots\!41$$$$\nu^{4} -$$$$22\!\cdots\!24$$$$\nu^{3} +$$$$57\!\cdots\!30$$$$\nu^{2} -$$$$13\!\cdots\!00$$$$\nu +$$$$62\!\cdots\!00$$$$)/$$$$44\!\cdots\!50$$ $$\beta_{9}$$ $$=$$ $$($$$$-$$$$21\!\cdots\!04$$$$\nu^{11} -$$$$13\!\cdots\!92$$$$\nu^{10} -$$$$27\!\cdots\!84$$$$\nu^{9} -$$$$10\!\cdots\!02$$$$\nu^{8} -$$$$39\!\cdots\!88$$$$\nu^{7} -$$$$88\!\cdots\!04$$$$\nu^{6} -$$$$10\!\cdots\!20$$$$\nu^{5} -$$$$31\!\cdots\!86$$$$\nu^{4} -$$$$25\!\cdots\!96$$$$\nu^{3} -$$$$84\!\cdots\!80$$$$\nu^{2} +$$$$33\!\cdots\!00$$$$\nu -$$$$53\!\cdots\!50$$$$)/$$$$44\!\cdots\!75$$ $$\beta_{10}$$ $$=$$ $$($$$$17\!\cdots\!06$$$$\nu^{11} -$$$$59\!\cdots\!66$$$$\nu^{10} +$$$$41\!\cdots\!44$$$$\nu^{9} -$$$$51\!\cdots\!51$$$$\nu^{8} +$$$$38\!\cdots\!90$$$$\nu^{7} -$$$$49\!\cdots\!77$$$$\nu^{6} +$$$$19\!\cdots\!96$$$$\nu^{5} -$$$$22\!\cdots\!81$$$$\nu^{4} +$$$$51\!\cdots\!08$$$$\nu^{3} -$$$$70\!\cdots\!86$$$$\nu^{2} +$$$$82\!\cdots\!60$$$$\nu -$$$$72\!\cdots\!50$$$$)/$$$$29\!\cdots\!50$$ $$\beta_{11}$$ $$=$$ $$($$$$-$$$$33\!\cdots\!77$$$$\nu^{11} +$$$$91\!\cdots\!69$$$$\nu^{10} -$$$$64\!\cdots\!47$$$$\nu^{9} +$$$$37\!\cdots\!64$$$$\nu^{8} -$$$$59\!\cdots\!49$$$$\nu^{7} +$$$$56\!\cdots\!78$$$$\nu^{6} -$$$$22\!\cdots\!70$$$$\nu^{5} +$$$$18\!\cdots\!57$$$$\nu^{4} -$$$$70\!\cdots\!88$$$$\nu^{3} +$$$$57\!\cdots\!20$$$$\nu^{2} -$$$$54\!\cdots\!00$$$$\nu +$$$$23\!\cdots\!50$$$$)/$$$$44\!\cdots\!50$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$4 \beta_{11} + \beta_{10} + \beta_{9} + 2 \beta_{8} + 2 \beta_{7} - 5 \beta_{6} - 35 \beta_{5} + 19 \beta_{4} + 21 \beta_{3} + 17 \beta_{2} + 108 \beta_{1} + 3$$$$)/336$$ $$\nu^{2}$$ $$=$$ $$($$$$-6 \beta_{11} - 49 \beta_{10} + 108 \beta_{9} + 49 \beta_{8} - 90 \beta_{7} - 24 \beta_{6} - 256 \beta_{5} + 530 \beta_{4} - 7068 \beta_{3} + 7610 \beta_{2} + 198122 \beta_{1} - 198061$$$$)/336$$ $$\nu^{3}$$ $$=$$ $$($$$$-4450 \beta_{11} - 3438 \beta_{10} - 1302 \beta_{9} - 1719 \beta_{8} - 5274 \beta_{7} + 890 \beta_{6} + 25454 \beta_{5} + 22784 \beta_{4} - 2670 \beta_{3} + 313294 \beta_{2} + 3499 \beta_{1} - 2483606$$$$)/336$$ $$\nu^{4}$$ $$=$$ $$($$$$-37984 \beta_{11} - 38339 \beta_{10} - 166368 \beta_{9} - 76678 \beta_{8} - 97428 \beta_{7} + 47480 \beta_{6} + 790952 \beta_{5} - 409720 \beta_{4} + 6263988 \beta_{3} - 390728 \beta_{2} - 104488813 \beta_{1} - 57331$$$$)/168$$ $$\nu^{5}$$ $$=$$ $$($$$$580054 \beta_{11} + 1425607 \beta_{10} - 3624328 \beta_{9} - 1425607 \beta_{8} + 1884166 \beta_{7} + 2320216 \beta_{6} + 18640520 \beta_{5} - 39021202 \beta_{4} + 277743927 \beta_{3} - 317925237 \beta_{2} - 2919875442 \beta_{1} + 2917289727$$$$)/168$$ $$\nu^{6}$$ $$=$$ $$($$$$13549640 \beta_{11} + 17760170 \beta_{10} + 15048932 \beta_{9} + 8880085 \beta_{8} + 38227648 \beta_{7} - 2709928 \beta_{6} - 111399896 \beta_{5} - 103270112 \beta_{4} + 8129784 \beta_{3} - 1685863026 \beta_{2} - 14299941 \beta_{1} + 21328938000$$$$)/24$$ $$\nu^{7}$$ $$=$$ $$($$$$3581059988 \beta_{11} + 2441634696 \beta_{10} + 10040555165 \beta_{9} + 4883269392 \beta_{8} + 6363175078 \beta_{7} - 4476324985 \beta_{6} - 64992079103 \beta_{5} + 33838937047 \beta_{4} - 471889366788 \beta_{3} + 32048407053 \beta_{2} + 5420021051233 \beta_{1} + 4232164690$$$$)/168$$ $$\nu^{8}$$ $$=$$ $$($$$$-8497690459 \beta_{11} - 25897143496 \beta_{10} + 67862605303 \beta_{9} + 25897143496 \beta_{8} - 42369533926 \beta_{7} - 33990761836 \beta_{6} - 315210326596 \beta_{5} + 655913724569 \beta_{4} - 4754139016128 \beta_{3} + 5427048121615 \beta_{2} + 59944346353347 \beta_{1} - 59901453828933$$$$)/42$$ $$\nu^{9}$$ $$=$$ $$($$$$-7353839084605 \beta_{11} - 8363749520578 \beta_{10} - 6546479969827 \beta_{9} - 4181874760289 \beta_{8} - 17505263390417 \beta_{7} + 1470767816921 \beta_{6} + 57019626639011 \beta_{5} + 52607323188248 \beta_{4} - 4412303450763 \beta_{3} + 859734419949511 \beta_{2} + 7123410394131 \beta_{1} - 9480413951692911$$$$)/168$$ $$\nu^{10}$$ $$=$$ $$($$$$-117706616186580 \beta_{11} - 87304514607604 \beta_{10} - 369567931992129 \beta_{9} - 174609029215208 \beta_{8} - 228923947066032 \beta_{7} + 147133270233225 \beta_{6} + 2244966554105279 \beta_{5} - 1166623258122607 \beta_{4} + 16200521022635094 \beta_{3} - 1107769950029317 \beta_{2} - 199915136669237473 \beta_{1} - 146157822700894$$$$)/84$$ $$\nu^{11}$$ $$=$$ $$($$$$2469376638310483 \beta_{11} + 7135763060222049 \beta_{10} - 18719084672796138 \beta_{9} - 7135763060222049 \beta_{8} + 11310954757864689 \beta_{7} + 9877506553241932 \beta_{6} + 89202919479220984 \beta_{5} - 185813968873373417 \beta_{4} + 1355576672891212071 \beta_{3} - 1546329395041206454 \beta_{2} - 16261684963502488357 \beta_{1} + 16249610447165645342$$$$)/168$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/14\mathbb{Z}\right)^\times$$.

 $$n$$ $$3$$ $$\chi(n)$$ $$1 - \beta_{1}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
3.1
 3.92504 − 6.79836i −12.5688 + 21.7698i 9.14374 − 15.8374i 20.6104 − 35.6982i −9.54988 + 16.5409i −10.5605 + 18.2913i 3.92504 + 6.79836i −12.5688 − 21.7698i 9.14374 + 15.8374i 20.6104 + 35.6982i −9.54988 − 16.5409i −10.5605 − 18.2913i
−5.65685 9.79796i −90.8670 52.4621i −64.0000 + 110.851i 32.9005 18.9951i 1187.08i 1399.84 + 1950.71i 1448.15 2224.04 + 3852.15i −372.227 214.905i
3.2 −5.65685 9.79796i 24.5958 + 14.2004i −64.0000 + 110.851i 753.641 435.115i 321.318i −1836.02 1547.20i 1448.15 −2877.20 4983.45i −8526.48 4922.77i
3.3 −5.65685 9.79796i 123.742 + 71.4423i −64.0000 + 110.851i −758.365 + 437.842i 1616.56i 855.887 + 2243.27i 1448.15 6927.51 + 11998.8i 8579.92 + 4953.62i
3.4 5.65685 + 9.79796i −81.1886 46.8743i −64.0000 + 110.851i 928.109 535.844i 1060.64i 2212.86 931.703i −1448.15 1113.90 + 1929.33i 10500.4 + 6062.38i
3.5 5.65685 + 9.79796i −21.7328 12.5474i −64.0000 + 110.851i −492.158 + 284.147i 283.915i −1740.97 + 1653.43i −1448.15 −2965.62 5136.61i −5568.13 3214.76i
3.6 5.65685 + 9.79796i 126.451 + 73.0064i −64.0000 + 110.851i 372.872 215.278i 1651.95i −1545.60 1837.37i −1448.15 7379.38 + 12781.5i 4218.56 + 2435.59i
5.1 −5.65685 + 9.79796i −90.8670 + 52.4621i −64.0000 110.851i 32.9005 + 18.9951i 1187.08i 1399.84 1950.71i 1448.15 2224.04 3852.15i −372.227 + 214.905i
5.2 −5.65685 + 9.79796i 24.5958 14.2004i −64.0000 110.851i 753.641 + 435.115i 321.318i −1836.02 + 1547.20i 1448.15 −2877.20 + 4983.45i −8526.48 + 4922.77i
5.3 −5.65685 + 9.79796i 123.742 71.4423i −64.0000 110.851i −758.365 437.842i 1616.56i 855.887 2243.27i 1448.15 6927.51 11998.8i 8579.92 4953.62i
5.4 5.65685 9.79796i −81.1886 + 46.8743i −64.0000 110.851i 928.109 + 535.844i 1060.64i 2212.86 + 931.703i −1448.15 1113.90 1929.33i 10500.4 6062.38i
5.5 5.65685 9.79796i −21.7328 + 12.5474i −64.0000 110.851i −492.158 284.147i 283.915i −1740.97 1653.43i −1448.15 −2965.62 + 5136.61i −5568.13 + 3214.76i
5.6 5.65685 9.79796i 126.451 73.0064i −64.0000 110.851i 372.872 + 215.278i 1651.95i −1545.60 + 1837.37i −1448.15 7379.38 12781.5i 4218.56 2435.59i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 5.6 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 14.9.d.a 12
3.b odd 2 1 126.9.n.b 12
4.b odd 2 1 112.9.s.c 12
7.b odd 2 1 98.9.d.b 12
7.c even 3 1 98.9.b.c 12
7.c even 3 1 98.9.d.b 12
7.d odd 6 1 inner 14.9.d.a 12
7.d odd 6 1 98.9.b.c 12
21.g even 6 1 126.9.n.b 12
28.f even 6 1 112.9.s.c 12

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
14.9.d.a 12 1.a even 1 1 trivial
14.9.d.a 12 7.d odd 6 1 inner
98.9.b.c 12 7.c even 3 1
98.9.b.c 12 7.d odd 6 1
98.9.d.b 12 7.b odd 2 1
98.9.d.b 12 7.c even 3 1
112.9.s.c 12 4.b odd 2 1
112.9.s.c 12 28.f even 6 1
126.9.n.b 12 3.b odd 2 1
126.9.n.b 12 21.g even 6 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{9}^{\mathrm{new}}(14, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 16384 + 128 T^{2} + T^{4} )^{3}$$
$3$ $$21\!\cdots\!81$$$$+$$$$40\!\cdots\!18$$$$T - 29585263980998660859 T^{2} - 612264379343032206 T^{3} + 42067423004909094 T^{4} + 530856468284346 T^{5} - 3701363994855 T^{6} - 65573971146 T^{7} + 378913158 T^{8} + 4391982 T^{9} - 18363 T^{10} - 162 T^{11} + T^{12}$$
$5$ $$57\!\cdots\!25$$$$-$$$$27\!\cdots\!50$$$$T +$$$$46\!\cdots\!25$$$$T^{2} -$$$$96\!\cdots\!50$$$$T^{3} -$$$$13\!\cdots\!50$$$$T^{4} +$$$$47\!\cdots\!50$$$$T^{5} + 516425178513746025 T^{6} - 1612081221903450 T^{7} + 18486646794 T^{8} + 1881848862 T^{9} - 190071 T^{10} - 1674 T^{11} + T^{12}$$
$7$ $$36\!\cdots\!01$$$$+$$$$83\!\cdots\!08$$$$T +$$$$29\!\cdots\!18$$$$T^{2} -$$$$42\!\cdots\!12$$$$T^{3} +$$$$18\!\cdots\!15$$$$T^{4} -$$$$10\!\cdots\!28$$$$T^{5} + 16422695158604759052 T^{6} - 17664438572780328 T^{7} + 56377019557215 T^{8} - 2234556212 T^{9} + 2685018 T^{10} + 1308 T^{11} + T^{12}$$
$11$ $$48\!\cdots\!81$$$$-$$$$66\!\cdots\!38$$$$T +$$$$10\!\cdots\!61$$$$T^{2} +$$$$23\!\cdots\!18$$$$T^{3} +$$$$57\!\cdots\!98$$$$T^{4} +$$$$38\!\cdots\!02$$$$T^{5} +$$$$51\!\cdots\!01$$$$T^{6} +$$$$18\!\cdots\!58$$$$T^{7} + 326160370076618934 T^{8} + 6525798653934 T^{9} + 740755377 T^{10} - 10302 T^{11} + T^{12}$$
$13$ $$35\!\cdots\!24$$$$+$$$$39\!\cdots\!40$$$$T^{2} +$$$$25\!\cdots\!12$$$$T^{4} +$$$$64\!\cdots\!16$$$$T^{6} + 7906434877372039776 T^{8} + 4612551504 T^{10} + T^{12}$$
$17$ $$30\!\cdots\!01$$$$-$$$$20\!\cdots\!18$$$$T +$$$$51\!\cdots\!97$$$$T^{2} -$$$$42\!\cdots\!02$$$$T^{3} -$$$$36\!\cdots\!14$$$$T^{4} +$$$$69\!\cdots\!22$$$$T^{5} +$$$$28\!\cdots\!53$$$$T^{6} -$$$$98\!\cdots\!18$$$$T^{7} +$$$$25\!\cdots\!46$$$$T^{8} + 4491697539933198 T^{9} - 15940009863 T^{10} - 173178 T^{11} + T^{12}$$
$19$ $$94\!\cdots\!25$$$$-$$$$32\!\cdots\!50$$$$T +$$$$46\!\cdots\!25$$$$T^{2} -$$$$34\!\cdots\!50$$$$T^{3} +$$$$14\!\cdots\!46$$$$T^{4} -$$$$31\!\cdots\!14$$$$T^{5} +$$$$36\!\cdots\!41$$$$T^{6} -$$$$15\!\cdots\!66$$$$T^{7} -$$$$74\!\cdots\!82$$$$T^{8} + 5967426272557302 T^{9} + 40240488069 T^{10} - 405978 T^{11} + T^{12}$$
$23$ $$25\!\cdots\!41$$$$+$$$$93\!\cdots\!54$$$$T +$$$$60\!\cdots\!61$$$$T^{2} +$$$$67\!\cdots\!62$$$$T^{3} +$$$$53\!\cdots\!86$$$$T^{4} +$$$$58\!\cdots\!42$$$$T^{5} +$$$$30\!\cdots\!49$$$$T^{6} -$$$$11\!\cdots\!94$$$$T^{7} +$$$$60\!\cdots\!10$$$$T^{8} - 66695318786854170 T^{9} + 257114823633 T^{10} - 158934 T^{11} + T^{12}$$
$29$ $$( -$$$$79\!\cdots\!12$$$$-$$$$14\!\cdots\!08$$$$T -$$$$74\!\cdots\!56$$$$T^{2} - 940925167313519808 T^{3} + 819765973332 T^{4} + 2177628 T^{5} + T^{6} )^{2}$$
$31$ $$95\!\cdots\!29$$$$-$$$$52\!\cdots\!14$$$$T +$$$$12\!\cdots\!85$$$$T^{2} -$$$$16\!\cdots\!82$$$$T^{3} +$$$$12\!\cdots\!22$$$$T^{4} -$$$$64\!\cdots\!18$$$$T^{5} +$$$$22\!\cdots\!77$$$$T^{6} -$$$$56\!\cdots\!82$$$$T^{7} +$$$$97\!\cdots\!78$$$$T^{8} - 11797369442909678250 T^{9} + 9420779822373 T^{10} - 4520250 T^{11} + T^{12}$$
$37$ $$20\!\cdots\!21$$$$-$$$$67\!\cdots\!46$$$$T +$$$$17\!\cdots\!41$$$$T^{2} -$$$$22\!\cdots\!98$$$$T^{3} +$$$$24\!\cdots\!54$$$$T^{4} -$$$$15\!\cdots\!66$$$$T^{5} +$$$$11\!\cdots\!93$$$$T^{6} -$$$$45\!\cdots\!34$$$$T^{7} +$$$$38\!\cdots\!14$$$$T^{8} - 8596405628770429042 T^{9} + 7075094642901 T^{10} - 134214 T^{11} + T^{12}$$
$41$ $$50\!\cdots\!96$$$$+$$$$29\!\cdots\!88$$$$T^{2} +$$$$27\!\cdots\!12$$$$T^{4} +$$$$96\!\cdots\!64$$$$T^{6} +$$$$12\!\cdots\!48$$$$T^{8} + 63126434226384 T^{10} + T^{12}$$
$43$ $$( -$$$$32\!\cdots\!64$$$$+$$$$76\!\cdots\!12$$$$T +$$$$98\!\cdots\!12$$$$T^{2} -$$$$14\!\cdots\!44$$$$T^{3} - 19549380996084 T^{4} + 6480948 T^{5} + T^{6} )^{2}$$
$47$ $$85\!\cdots\!49$$$$-$$$$10\!\cdots\!06$$$$T +$$$$28\!\cdots\!69$$$$T^{2} +$$$$12\!\cdots\!86$$$$T^{3} -$$$$56\!\cdots\!22$$$$T^{4} -$$$$19\!\cdots\!94$$$$T^{5} +$$$$12\!\cdots\!49$$$$T^{6} -$$$$81\!\cdots\!38$$$$T^{7} -$$$$33\!\cdots\!58$$$$T^{8} +$$$$32\!\cdots\!58$$$$T^{9} + 95082902343189 T^{10} - 18385002 T^{11} + T^{12}$$
$53$ $$10\!\cdots\!41$$$$+$$$$10\!\cdots\!06$$$$T +$$$$56\!\cdots\!73$$$$T^{2} -$$$$35\!\cdots\!46$$$$T^{3} +$$$$19\!\cdots\!26$$$$T^{4} +$$$$89\!\cdots\!78$$$$T^{5} +$$$$31\!\cdots\!45$$$$T^{6} +$$$$11\!\cdots\!82$$$$T^{7} +$$$$59\!\cdots\!86$$$$T^{8} +$$$$12\!\cdots\!18$$$$T^{9} + 204949252399797 T^{10} + 16540506 T^{11} + T^{12}$$
$59$ $$28\!\cdots\!01$$$$-$$$$15\!\cdots\!82$$$$T +$$$$26\!\cdots\!69$$$$T^{2} +$$$$46\!\cdots\!98$$$$T^{3} -$$$$55\!\cdots\!90$$$$T^{4} +$$$$51\!\cdots\!74$$$$T^{5} +$$$$12\!\cdots\!21$$$$T^{6} -$$$$11\!\cdots\!22$$$$T^{7} -$$$$29\!\cdots\!70$$$$T^{8} +$$$$63\!\cdots\!34$$$$T^{9} + 120060743343381 T^{10} - 31163922 T^{11} + T^{12}$$
$61$ $$13\!\cdots\!69$$$$+$$$$11\!\cdots\!10$$$$T -$$$$20\!\cdots\!11$$$$T^{2} -$$$$18\!\cdots\!90$$$$T^{3} +$$$$26\!\cdots\!50$$$$T^{4} +$$$$40\!\cdots\!74$$$$T^{5} +$$$$10\!\cdots\!65$$$$T^{6} -$$$$96\!\cdots\!22$$$$T^{7} -$$$$42\!\cdots\!38$$$$T^{8} +$$$$28\!\cdots\!34$$$$T^{9} + 2767349780555961 T^{10} + 85390158 T^{11} + T^{12}$$
$67$ $$23\!\cdots\!89$$$$-$$$$14\!\cdots\!54$$$$T +$$$$79\!\cdots\!13$$$$T^{2} -$$$$11\!\cdots\!50$$$$T^{3} +$$$$26\!\cdots\!86$$$$T^{4} +$$$$16\!\cdots\!14$$$$T^{5} +$$$$62\!\cdots\!73$$$$T^{6} +$$$$24\!\cdots\!94$$$$T^{7} +$$$$36\!\cdots\!30$$$$T^{8} +$$$$11\!\cdots\!02$$$$T^{9} + 1574633884496769 T^{10} + 37750362 T^{11} + T^{12}$$
$71$ $$( -$$$$21\!\cdots\!64$$$$+$$$$79\!\cdots\!68$$$$T +$$$$15\!\cdots\!48$$$$T^{2} +$$$$20\!\cdots\!72$$$$T^{3} - 2426256761646132 T^{4} - 22753212 T^{5} + T^{6} )^{2}$$
$73$ $$26\!\cdots\!49$$$$-$$$$10\!\cdots\!18$$$$T +$$$$17\!\cdots\!69$$$$T^{2} -$$$$14\!\cdots\!14$$$$T^{3} +$$$$60\!\cdots\!06$$$$T^{4} -$$$$71\!\cdots\!66$$$$T^{5} -$$$$11\!\cdots\!95$$$$T^{6} +$$$$20\!\cdots\!02$$$$T^{7} +$$$$40\!\cdots\!74$$$$T^{8} +$$$$18\!\cdots\!86$$$$T^{9} - 1980813382799319 T^{10} - 9414786 T^{11} + T^{12}$$
$79$ $$92\!\cdots\!25$$$$+$$$$14\!\cdots\!50$$$$T +$$$$25\!\cdots\!25$$$$T^{2} -$$$$41\!\cdots\!50$$$$T^{3} +$$$$48\!\cdots\!50$$$$T^{4} -$$$$24\!\cdots\!50$$$$T^{5} +$$$$96\!\cdots\!25$$$$T^{6} -$$$$22\!\cdots\!90$$$$T^{7} +$$$$43\!\cdots\!14$$$$T^{8} -$$$$48\!\cdots\!38$$$$T^{9} + 7328718281480049 T^{10} - 59730294 T^{11} + T^{12}$$
$83$ $$17\!\cdots\!84$$$$+$$$$39\!\cdots\!56$$$$T^{2} +$$$$19\!\cdots\!00$$$$T^{4} +$$$$37\!\cdots\!52$$$$T^{6} +$$$$31\!\cdots\!64$$$$T^{8} + 9881353210358592 T^{10} + T^{12}$$
$89$ $$13\!\cdots\!21$$$$-$$$$58\!\cdots\!90$$$$T -$$$$39\!\cdots\!03$$$$T^{2} +$$$$21\!\cdots\!70$$$$T^{3} +$$$$11\!\cdots\!10$$$$T^{4} -$$$$58\!\cdots\!62$$$$T^{5} -$$$$19\!\cdots\!35$$$$T^{6} +$$$$38\!\cdots\!90$$$$T^{7} +$$$$31\!\cdots\!98$$$$T^{8} -$$$$22\!\cdots\!22$$$$T^{9} + 41683197526157529 T^{10} - 323014482 T^{11} + T^{12}$$
$97$ $$52\!\cdots\!16$$$$+$$$$19\!\cdots\!52$$$$T^{2} +$$$$25\!\cdots\!56$$$$T^{4} +$$$$13\!\cdots\!68$$$$T^{6} +$$$$32\!\cdots\!76$$$$T^{8} + 32297903455130832 T^{10} + T^{12}$$