# Properties

 Label 38.10.c.a Level $38$ Weight $10$ Character orbit 38.c Analytic conductor $19.571$ Analytic rank $0$ Dimension $14$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$38 = 2 \cdot 19$$ Weight: $$k$$ $$=$$ $$10$$ Character orbit: $$[\chi]$$ $$=$$ 38.c (of order $$3$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$19.5713617742$$ Analytic rank: $$0$$ Dimension: $$14$$ Relative dimension: $$7$$ over $$\Q(\zeta_{3})$$ Coefficient field: $$\mathbb{Q}[x]/(x^{14} - \cdots)$$ Defining polynomial: $$x^{14} - 3 x^{13} + 93094 x^{12} + 3763497 x^{11} + 6525212431 x^{10} + 244392759522 x^{9} + 198778066371639 x^{8} + 9216645483921129 x^{7} + 4530154246073222607 x^{6} + 142122715974381300066 x^{5} + 8185019559882294055671 x^{4} - 135327799885026117969495 x^{3} + 2044710563339570147369550 x^{2} - 9649457865335314261798875 x + 37683939484136051622500625$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2^{10}\cdot 3^{6}\cdot 5^{2}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{13}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( 16 - 16 \beta_{2} ) q^{2} + ( 24 - \beta_{1} - 24 \beta_{2} ) q^{3} -256 \beta_{2} q^{4} + ( 130 - \beta_{1} - 130 \beta_{2} + \beta_{5} + \beta_{7} ) q^{5} + ( -384 \beta_{2} - 16 \beta_{3} ) q^{6} + ( 261 + 7 \beta_{1} - 7 \beta_{3} + \beta_{8} ) q^{7} -4096 q^{8} + ( -7476 \beta_{2} - 16 \beta_{3} - \beta_{4} - \beta_{6} + 2 \beta_{7} + \beta_{9} - \beta_{12} ) q^{9} +O(q^{10})$$ $$q + ( 16 - 16 \beta_{2} ) q^{2} + ( 24 - \beta_{1} - 24 \beta_{2} ) q^{3} -256 \beta_{2} q^{4} + ( 130 - \beta_{1} - 130 \beta_{2} + \beta_{5} + \beta_{7} ) q^{5} + ( -384 \beta_{2} - 16 \beta_{3} ) q^{6} + ( 261 + 7 \beta_{1} - 7 \beta_{3} + \beta_{8} ) q^{7} -4096 q^{8} + ( -7476 \beta_{2} - 16 \beta_{3} - \beta_{4} - \beta_{6} + 2 \beta_{7} + \beta_{9} - \beta_{12} ) q^{9} + ( -2080 \beta_{2} - 16 \beta_{3} + 16 \beta_{7} ) q^{10} + ( 1082 + 76 \beta_{1} - 76 \beta_{3} + \beta_{4} + 11 \beta_{5} - \beta_{6} - 2 \beta_{8} - \beta_{10} ) q^{11} + ( -6144 + 256 \beta_{1} - 256 \beta_{3} ) q^{12} + ( 913 \beta_{2} + 10 \beta_{3} + 3 \beta_{4} - 5 \beta_{6} - 29 \beta_{7} + 5 \beta_{9} + 5 \beta_{11} + 3 \beta_{12} ) q^{13} + ( 4176 + 112 \beta_{1} - 4176 \beta_{2} + 16 \beta_{8} + 16 \beta_{11} ) q^{14} + ( -32624 \beta_{2} - 273 \beta_{3} - \beta_{4} - 2 \beta_{6} + 53 \beta_{7} + 2 \beta_{9} - 22 \beta_{11} - \beta_{12} - 9 \beta_{13} ) q^{15} + ( -65536 + 65536 \beta_{2} ) q^{16} + ( 45184 - 831 \beta_{1} - 45184 \beta_{2} - 55 \beta_{5} - 55 \beta_{7} + 12 \beta_{8} - 4 \beta_{9} - 10 \beta_{10} + 12 \beta_{11} - 7 \beta_{12} - 10 \beta_{13} ) q^{17} + ( -119616 + 256 \beta_{1} - 256 \beta_{3} - 16 \beta_{4} - 32 \beta_{5} - 16 \beta_{6} ) q^{18} + ( 40603 - 760 \beta_{1} + 81624 \beta_{2} + 1178 \beta_{3} + 19 \beta_{5} + 19 \beta_{6} - 57 \beta_{7} + 19 \beta_{8} - 19 \beta_{10} + 38 \beta_{11} - 19 \beta_{13} ) q^{19} + ( -33280 + 256 \beta_{1} - 256 \beta_{3} - 256 \beta_{5} ) q^{20} + ( -175320 + 362 \beta_{1} + 175320 \beta_{2} - 412 \beta_{5} - 412 \beta_{7} + 54 \beta_{8} + 7 \beta_{9} + 54 \beta_{11} + 20 \beta_{12} ) q^{21} + ( 17312 + 1216 \beta_{1} - 17312 \beta_{2} + 176 \beta_{5} + 176 \beta_{7} - 32 \beta_{8} - 16 \beta_{9} - 16 \beta_{10} - 32 \beta_{11} - 16 \beta_{12} - 16 \beta_{13} ) q^{22} + ( 493275 \beta_{2} + 1262 \beta_{3} + 76 \beta_{4} - 24 \beta_{6} - 183 \beta_{7} + 24 \beta_{9} + 3 \beta_{11} + 76 \beta_{12} - 7 \beta_{13} ) q^{23} + ( -98304 + 4096 \beta_{1} + 98304 \beta_{2} ) q^{24} + ( -343700 \beta_{2} - 2660 \beta_{3} - 52 \beta_{4} - 2 \beta_{6} + 353 \beta_{7} + 2 \beta_{9} - \beta_{11} - 52 \beta_{12} - 62 \beta_{13} ) q^{25} + ( 14608 - 160 \beta_{1} + 160 \beta_{3} + 48 \beta_{4} + 464 \beta_{5} - 80 \beta_{6} - 80 \beta_{8} ) q^{26} + ( -137078 + 7423 \beta_{1} - 7423 \beta_{3} - \beta_{4} + 262 \beta_{5} - 107 \beta_{6} + 52 \beta_{8} + 36 \beta_{10} ) q^{27} + ( -66816 \beta_{2} + 1792 \beta_{3} + 256 \beta_{11} ) q^{28} + ( -1003803 \beta_{2} - 7164 \beta_{3} - 165 \beta_{4} - 110 \beta_{6} - 707 \beta_{7} + 110 \beta_{9} + 429 \beta_{11} - 165 \beta_{12} - 44 \beta_{13} ) q^{29} + ( -521984 + 4368 \beta_{1} - 4368 \beta_{3} - 16 \beta_{4} - 848 \beta_{5} - 32 \beta_{6} + 352 \beta_{8} + 144 \beta_{10} ) q^{30} + ( -1336580 + 8556 \beta_{1} - 8556 \beta_{3} + 65 \beta_{4} - 1100 \beta_{5} - 78 \beta_{6} + 132 \beta_{8} + 154 \beta_{10} ) q^{31} + 1048576 \beta_{2} q^{32} + ( -1976749 + 491 \beta_{1} + 1976749 \beta_{2} + 3339 \beta_{5} + 3339 \beta_{7} - 622 \beta_{8} - 299 \beta_{9} - 138 \beta_{10} - 622 \beta_{11} - 48 \beta_{12} - 138 \beta_{13} ) q^{33} + ( -722944 \beta_{2} - 13296 \beta_{3} - 112 \beta_{4} + 64 \beta_{6} - 880 \beta_{7} - 64 \beta_{9} + 192 \beta_{11} - 112 \beta_{12} - 160 \beta_{13} ) q^{34} + ( 354861 + 34561 \beta_{1} - 354861 \beta_{2} - 568 \beta_{5} - 568 \beta_{7} + 1099 \beta_{8} + 15 \beta_{9} - 28 \beta_{10} + 1099 \beta_{11} + 250 \beta_{12} - 28 \beta_{13} ) q^{35} + ( -1913856 + 4096 \beta_{1} + 1913856 \beta_{2} - 512 \beta_{5} - 512 \beta_{7} - 256 \beta_{9} + 256 \beta_{12} ) q^{36} + ( -480675 - 17933 \beta_{1} + 17933 \beta_{3} + 183 \beta_{4} - 5610 \beta_{5} - 149 \beta_{6} + 1381 \beta_{8} + 84 \beta_{10} ) q^{37} + ( 1955632 - 18848 \beta_{1} - 649648 \beta_{2} + 6688 \beta_{3} + 1216 \beta_{5} + 304 \beta_{7} - 304 \beta_{8} + 304 \beta_{9} + 304 \beta_{11} - 304 \beta_{13} ) q^{38} + ( 321530 + 7449 \beta_{1} - 7449 \beta_{3} + 158 \beta_{4} + 3421 \beta_{5} - 516 \beta_{6} - 1708 \beta_{8} - 603 \beta_{10} ) q^{39} + ( -532480 + 4096 \beta_{1} + 532480 \beta_{2} - 4096 \beta_{5} - 4096 \beta_{7} ) q^{40} + ( 513421 - 7946 \beta_{1} - 513421 \beta_{2} - 3128 \beta_{5} - 3128 \beta_{7} + 160 \beta_{8} + 657 \beta_{9} - 272 \beta_{10} + 160 \beta_{11} + 15 \beta_{12} - 272 \beta_{13} ) q^{41} + ( 2805120 \beta_{2} + 5792 \beta_{3} + 320 \beta_{4} - 112 \beta_{6} - 6592 \beta_{7} + 112 \beta_{9} + 864 \beta_{11} + 320 \beta_{12} ) q^{42} + ( 2838866 - 13531 \beta_{1} - 2838866 \beta_{2} + 1487 \beta_{5} + 1487 \beta_{7} - 1552 \beta_{8} + 942 \beta_{9} - 583 \beta_{10} - 1552 \beta_{11} - 596 \beta_{12} - 583 \beta_{13} ) q^{43} + ( -276992 \beta_{2} + 19456 \beta_{3} - 256 \beta_{4} + 256 \beta_{6} + 2816 \beta_{7} - 256 \beta_{9} - 512 \beta_{11} - 256 \beta_{12} - 256 \beta_{13} ) q^{44} + ( -5555829 + 8007 \beta_{1} - 8007 \beta_{3} - 1257 \beta_{4} - 12036 \beta_{5} + 762 \beta_{6} + 2895 \beta_{8} ) q^{45} + ( 7892400 - 20192 \beta_{1} + 20192 \beta_{3} + 1216 \beta_{4} + 2928 \beta_{5} - 384 \beta_{6} - 48 \beta_{8} + 112 \beta_{10} ) q^{46} + ( 8595314 \beta_{2} - 61623 \beta_{3} + 1131 \beta_{4} - 310 \beta_{6} - 1027 \beta_{7} + 310 \beta_{9} - 64 \beta_{11} + 1131 \beta_{12} + 1047 \beta_{13} ) q^{47} + ( 1572864 \beta_{2} + 65536 \beta_{3} ) q^{48} + ( -5673684 - 35761 \beta_{1} + 35761 \beta_{3} - 932 \beta_{4} + 16876 \beta_{5} - 175 \beta_{6} + 1661 \beta_{8} - 476 \beta_{10} ) q^{49} + ( -5499200 + 42560 \beta_{1} - 42560 \beta_{3} - 832 \beta_{4} - 5648 \beta_{5} - 32 \beta_{6} + 16 \beta_{8} + 992 \beta_{10} ) q^{50} + ( -23044307 \beta_{2} + 4558 \beta_{3} - 1556 \beta_{4} + 537 \beta_{6} + 15349 \beta_{7} - 537 \beta_{9} + 35 \beta_{11} - 1556 \beta_{12} + 483 \beta_{13} ) q^{51} + ( 233728 - 2560 \beta_{1} - 233728 \beta_{2} + 7424 \beta_{5} + 7424 \beta_{7} - 1280 \beta_{8} - 1280 \beta_{9} - 1280 \beta_{11} - 768 \beta_{12} ) q^{52} + ( 7753074 \beta_{2} - 154369 \beta_{3} + 2610 \beta_{4} + 2413 \beta_{6} - 8183 \beta_{7} - 2413 \beta_{9} - 1856 \beta_{11} + 2610 \beta_{12} + 1088 \beta_{13} ) q^{53} + ( -2193248 + 118768 \beta_{1} + 2193248 \beta_{2} + 4192 \beta_{5} + 4192 \beta_{7} + 832 \beta_{8} - 1712 \beta_{9} + 576 \beta_{10} + 832 \beta_{11} + 16 \beta_{12} + 576 \beta_{13} ) q^{54} + ( 20552305 - 233817 \beta_{1} - 20552305 \beta_{2} + 17892 \beta_{5} + 17892 \beta_{7} - 1019 \beta_{8} + 1560 \beta_{9} + 140 \beta_{10} - 1019 \beta_{11} - 1118 \beta_{12} + 140 \beta_{13} ) q^{55} + ( -1069056 - 28672 \beta_{1} + 28672 \beta_{3} - 4096 \beta_{8} ) q^{56} + ( 34444302 - 313937 \beta_{1} - 21413057 \beta_{2} + 227639 \beta_{3} - 627 \beta_{4} + 11134 \beta_{5} + 1976 \beta_{6} + 35017 \beta_{7} - 855 \beta_{8} + 798 \beta_{9} + 1121 \beta_{11} - 1748 \beta_{12} + 1482 \beta_{13} ) q^{57} + ( -16060848 + 114624 \beta_{1} - 114624 \beta_{3} - 2640 \beta_{4} + 11312 \beta_{5} - 1760 \beta_{6} - 6864 \beta_{8} + 704 \beta_{10} ) q^{58} + ( 23467378 + 53675 \beta_{1} - 23467378 \beta_{2} + 9972 \beta_{5} + 9972 \beta_{7} - 5128 \beta_{8} + 5207 \beta_{9} + 2286 \beta_{10} - 5128 \beta_{11} - 857 \beta_{12} + 2286 \beta_{13} ) q^{59} + ( -8351744 + 69888 \beta_{1} + 8351744 \beta_{2} - 13568 \beta_{5} - 13568 \beta_{7} + 5632 \beta_{8} - 512 \beta_{9} + 2304 \beta_{10} + 5632 \beta_{11} + 256 \beta_{12} + 2304 \beta_{13} ) q^{60} + ( 7071348 \beta_{2} + 68047 \beta_{3} + 2006 \beta_{4} + 2276 \beta_{6} - 32613 \beta_{7} - 2276 \beta_{9} - 14782 \beta_{11} + 2006 \beta_{12} + 968 \beta_{13} ) q^{61} + ( -21385280 + 136896 \beta_{1} + 21385280 \beta_{2} - 17600 \beta_{5} - 17600 \beta_{7} + 2112 \beta_{8} - 1248 \beta_{9} + 2464 \beta_{10} + 2112 \beta_{11} - 1040 \beta_{12} + 2464 \beta_{13} ) q^{62} + ( 19821532 \beta_{2} + 271460 \beta_{3} + 1196 \beta_{4} - 740 \beta_{6} - 39088 \beta_{7} + 740 \beta_{9} - 4924 \beta_{11} + 1196 \beta_{12} + 5040 \beta_{13} ) q^{63} + 16777216 q^{64} + ( 56357365 - 192398 \beta_{1} + 192398 \beta_{3} + 3097 \beta_{4} + 32011 \beta_{5} + 8397 \beta_{6} - 10771 \beta_{8} - 2022 \beta_{10} ) q^{65} + ( 31627984 \beta_{2} + 7856 \beta_{3} - 768 \beta_{4} + 4784 \beta_{6} + 53424 \beta_{7} - 4784 \beta_{9} - 9952 \beta_{11} - 768 \beta_{12} - 2208 \beta_{13} ) q^{66} + ( -24580445 \beta_{2} - 172348 \beta_{3} - 3422 \beta_{4} + 273 \beta_{6} - 17670 \beta_{7} - 273 \beta_{9} - 9633 \beta_{11} - 3422 \beta_{12} + 422 \beta_{13} ) q^{67} + ( -11567104 + 212736 \beta_{1} - 212736 \beta_{3} - 1792 \beta_{4} + 14080 \beta_{5} + 1024 \beta_{6} - 3072 \beta_{8} + 2560 \beta_{10} ) q^{68} + ( 45495766 - 1103555 \beta_{1} + 1103555 \beta_{3} + 1194 \beta_{4} - 18795 \beta_{5} - 484 \beta_{6} - 14006 \beta_{8} - 6972 \beta_{10} ) q^{69} + ( -5677776 \beta_{2} + 552976 \beta_{3} + 4000 \beta_{4} - 240 \beta_{6} - 9088 \beta_{7} + 240 \beta_{9} + 17584 \beta_{11} + 4000 \beta_{12} - 448 \beta_{13} ) q^{70} + ( -10578540 - 221949 \beta_{1} + 10578540 \beta_{2} + 46089 \beta_{5} + 46089 \beta_{7} - 11522 \beta_{8} - 344 \beta_{9} - 943 \beta_{10} - 11522 \beta_{11} + 1232 \beta_{12} - 943 \beta_{13} ) q^{71} + ( 30621696 \beta_{2} + 65536 \beta_{3} + 4096 \beta_{4} + 4096 \beta_{6} - 8192 \beta_{7} - 4096 \beta_{9} + 4096 \beta_{12} ) q^{72} + ( 8454772 - 135553 \beta_{1} - 8454772 \beta_{2} - 46468 \beta_{5} - 46468 \beta_{7} + 8289 \beta_{8} - 3371 \beta_{9} - 4172 \beta_{10} + 8289 \beta_{11} + 9856 \beta_{12} - 4172 \beta_{13} ) q^{73} + ( -7690800 - 286928 \beta_{1} + 7690800 \beta_{2} - 89760 \beta_{5} - 89760 \beta_{7} + 22096 \beta_{8} - 2384 \beta_{9} + 1344 \beta_{10} + 22096 \beta_{11} - 2928 \beta_{12} + 1344 \beta_{13} ) q^{74} + ( -79885426 + 547864 \beta_{1} - 547864 \beta_{3} - 7715 \beta_{4} - 149629 \beta_{5} + 3707 \beta_{6} + 17426 \beta_{8} + 2751 \beta_{10} ) q^{75} + ( 20895744 - 107008 \beta_{1} - 31290112 \beta_{2} - 194560 \beta_{3} + 14592 \beta_{5} - 4864 \beta_{6} + 19456 \beta_{7} - 9728 \beta_{8} + 4864 \beta_{9} + 4864 \beta_{10} - 4864 \beta_{11} ) q^{76} + ( -69234740 + 699230 \beta_{1} - 699230 \beta_{3} - 5294 \beta_{4} - 24416 \beta_{5} - 5421 \beta_{6} + 874 \beta_{8} + 3920 \beta_{10} ) q^{77} + ( 5144480 + 119184 \beta_{1} - 5144480 \beta_{2} + 54736 \beta_{5} + 54736 \beta_{7} - 27328 \beta_{8} - 8256 \beta_{9} - 9648 \beta_{10} - 27328 \beta_{11} - 2528 \beta_{12} - 9648 \beta_{13} ) q^{78} + ( 17475288 - 187385 \beta_{1} - 17475288 \beta_{2} + 110931 \beta_{5} + 110931 \beta_{7} + 58606 \beta_{8} + 4180 \beta_{9} - 237 \beta_{10} + 58606 \beta_{11} - 7684 \beta_{12} - 237 \beta_{13} ) q^{79} + ( 8519680 \beta_{2} + 65536 \beta_{3} - 65536 \beta_{7} ) q^{80} + ( -52863585 + 546286 \beta_{1} + 52863585 \beta_{2} - 48578 \beta_{5} - 48578 \beta_{7} - 10710 \beta_{8} + 7049 \beta_{9} - 7992 \beta_{10} - 10710 \beta_{11} - 10379 \beta_{12} - 7992 \beta_{13} ) q^{81} + ( -8214736 \beta_{2} - 127136 \beta_{3} + 240 \beta_{4} - 10512 \beta_{6} - 50048 \beta_{7} + 10512 \beta_{9} + 2560 \beta_{11} + 240 \beta_{12} - 4352 \beta_{13} ) q^{82} + ( -52777144 + 1036356 \beta_{1} - 1036356 \beta_{3} + 9144 \beta_{4} + 71445 \beta_{5} - 18570 \beta_{6} + 47434 \beta_{8} + 1211 \beta_{10} ) q^{83} + ( 44881920 - 92672 \beta_{1} + 92672 \beta_{3} + 5120 \beta_{4} + 105472 \beta_{5} - 1792 \beta_{6} - 13824 \beta_{8} ) q^{84} + ( 90138963 \beta_{2} - 1378662 \beta_{3} + 3435 \beta_{4} - 9825 \beta_{6} + 183963 \beta_{7} + 9825 \beta_{9} - 12147 \beta_{11} + 3435 \beta_{12} - 6036 \beta_{13} ) q^{85} + ( -45421856 \beta_{2} - 216496 \beta_{3} - 9536 \beta_{4} - 15072 \beta_{6} + 23792 \beta_{7} + 15072 \beta_{9} - 24832 \beta_{11} - 9536 \beta_{12} - 9328 \beta_{13} ) q^{86} + ( -214119311 + 2980818 \beta_{1} - 2980818 \beta_{3} - 3556 \beta_{4} + 151111 \beta_{5} - 17270 \beta_{6} - 28985 \beta_{8} - 3657 \beta_{10} ) q^{87} + ( -4431872 - 311296 \beta_{1} + 311296 \beta_{3} - 4096 \beta_{4} - 45056 \beta_{5} + 4096 \beta_{6} + 8192 \beta_{8} + 4096 \beta_{10} ) q^{88} + ( -234909380 \beta_{2} + 2682647 \beta_{3} - 3531 \beta_{4} - 21130 \beta_{6} + 89927 \beta_{7} + 21130 \beta_{9} + 69848 \beta_{11} - 3531 \beta_{12} - 8242 \beta_{13} ) q^{89} + ( -88893264 + 128112 \beta_{1} + 88893264 \beta_{2} - 192576 \beta_{5} - 192576 \beta_{7} + 46320 \beta_{8} + 12192 \beta_{9} + 46320 \beta_{11} + 20112 \beta_{12} ) q^{90} + ( -163911552 \beta_{2} - 2147360 \beta_{3} - 3398 \beta_{4} - 8504 \beta_{6} + 172948 \beta_{7} + 8504 \beta_{9} + 11120 \beta_{11} - 3398 \beta_{12} - 7224 \beta_{13} ) q^{91} + ( 126278400 - 323072 \beta_{1} - 126278400 \beta_{2} + 46848 \beta_{5} + 46848 \beta_{7} - 768 \beta_{8} - 6144 \beta_{9} + 1792 \beta_{10} - 768 \beta_{11} - 19456 \beta_{12} + 1792 \beta_{13} ) q^{92} + ( -262946619 + 657849 \beta_{1} + 262946619 \beta_{2} - 477968 \beta_{5} - 477968 \beta_{7} + 27717 \beta_{8} + 1013 \beta_{9} - 4272 \beta_{10} + 27717 \beta_{11} + 21433 \beta_{12} - 4272 \beta_{13} ) q^{93} + ( 137525024 + 985968 \beta_{1} - 985968 \beta_{3} + 18096 \beta_{4} + 16432 \beta_{5} - 4960 \beta_{6} + 1024 \beta_{8} - 16752 \beta_{10} ) q^{94} + ( 224752216 - 1261353 \beta_{1} - 107291974 \beta_{2} - 1715244 \beta_{3} + 17043 \beta_{4} + 157073 \beta_{5} + 2812 \beta_{6} + 203053 \beta_{7} - 45410 \beta_{8} - 26904 \beta_{9} - 9101 \beta_{10} + 21736 \beta_{11} + 190 \beta_{12} - 5643 \beta_{13} ) q^{95} + ( 25165824 - 1048576 \beta_{1} + 1048576 \beta_{3} ) q^{96} + ( 35089256 + 1240489 \beta_{1} - 35089256 \beta_{2} - 225124 \beta_{5} - 225124 \beta_{7} + 27797 \beta_{8} - 9256 \beta_{9} + 4852 \beta_{10} + 27797 \beta_{11} - 12553 \beta_{12} + 4852 \beta_{13} ) q^{97} + ( -90778944 - 572176 \beta_{1} + 90778944 \beta_{2} + 270016 \beta_{5} + 270016 \beta_{7} + 26576 \beta_{8} - 2800 \beta_{9} - 7616 \beta_{10} + 26576 \beta_{11} + 14912 \beta_{12} - 7616 \beta_{13} ) q^{98} + ( 75721781 \beta_{2} + 2679539 \beta_{3} - 3922 \beta_{4} + 29289 \beta_{6} + 540422 \beta_{7} - 29289 \beta_{9} - 111383 \beta_{11} - 3922 \beta_{12} - 16686 \beta_{13} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$14q + 112q^{2} + 165q^{3} - 1792q^{4} + 909q^{5} - 2640q^{6} + 3692q^{7} - 57344q^{8} - 52286q^{9} + O(q^{10})$$ $$14q + 112q^{2} + 165q^{3} - 1792q^{4} + 909q^{5} - 2640q^{6} + 3692q^{7} - 57344q^{8} - 52286q^{9} - 14544q^{10} + 15656q^{11} - 84480q^{12} + 6423q^{13} + 29536q^{14} - 227715q^{15} - 458752q^{16} + 313667q^{17} - 1673152q^{18} + 1134224q^{19} - 465408q^{20} - 1227046q^{21} + 125248q^{22} + 3449345q^{23} - 675840q^{24} - 2398648q^{25} + 205536q^{26} - 1873854q^{27} - 472576q^{28} - 7002615q^{29} - 7286880q^{30} - 18666588q^{31} + 7340032q^{32} - 13827668q^{33} - 5018672q^{34} + 2584932q^{35} - 13385216q^{36} - 6866080q^{37} + 22760176q^{38} + 4568410q^{39} - 3723264q^{40} + 3564107q^{41} + 19632736q^{42} + 19837521q^{43} - 2003968q^{44} - 77788260q^{45} + 110379040q^{46} + 60353825q^{47} + 10813440q^{48} - 79579650q^{49} - 76756736q^{50} - 161350373q^{51} + 1644288q^{52} + 54744235q^{53} - 14990832q^{54} + 143199990q^{55} - 15122432q^{56} + 330686241q^{57} - 224083680q^{58} + 164456585q^{59} - 58295040q^{60} + 49328881q^{61} - 149332704q^{62} + 138012360q^{63} + 234881024q^{64} + 788015550q^{65} + 221242688q^{66} - 171522309q^{67} - 160597504q^{68} + 630323350q^{69} - 41358912q^{70} - 74596055q^{71} + 214163456q^{72} + 58695287q^{73} - 54928640q^{74} - 1115757144q^{75} + 73801472q^{76} - 965186644q^{77} + 36547280q^{78} + 121854617q^{79} + 59572224q^{80} - 368486747q^{81} - 57025712q^{82} - 732607256q^{83} + 628247552q^{84} + 634697565q^{85} - 317400336q^{86} - 2979036210q^{87} - 64126976q^{88} - 1652463181q^{89} - 622306080q^{90} - 1141270092q^{91} + 883032320q^{92} - 1839612746q^{93} + 1931322400q^{94} + 2397253503q^{95} + 346030080q^{96} + 248805607q^{97} - 636637200q^{98} + 520684712q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{14} - 3 x^{13} + 93094 x^{12} + 3763497 x^{11} + 6525212431 x^{10} + 244392759522 x^{9} + 198778066371639 x^{8} + 9216645483921129 x^{7} + 4530154246073222607 x^{6} + 142122715974381300066 x^{5} + 8185019559882294055671 x^{4} - 135327799885026117969495 x^{3} + 2044710563339570147369550 x^{2} - 9649457865335314261798875 x + 37683939484136051622500625$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$-$$$$28\!\cdots\!82$$$$\nu^{13} +$$$$19\!\cdots\!36$$$$\nu^{12} -$$$$26\!\cdots\!78$$$$\nu^{11} -$$$$11\!\cdots\!94$$$$\nu^{10} -$$$$18\!\cdots\!12$$$$\nu^{9} -$$$$74\!\cdots\!89$$$$\nu^{8} -$$$$57\!\cdots\!18$$$$\nu^{7} -$$$$27\!\cdots\!18$$$$\nu^{6} -$$$$13\!\cdots\!64$$$$\nu^{5} -$$$$43\!\cdots\!82$$$$\nu^{4} -$$$$25\!\cdots\!82$$$$\nu^{3} +$$$$29\!\cdots\!30$$$$\nu^{2} -$$$$62\!\cdots\!50$$$$\nu +$$$$29\!\cdots\!50$$$$)/$$$$24\!\cdots\!75$$ $$\beta_{3}$$ $$=$$ $$($$$$13\!\cdots\!34$$$$\nu^{13} +$$$$26\!\cdots\!78$$$$\nu^{12} +$$$$12\!\cdots\!56$$$$\nu^{11} +$$$$77\!\cdots\!18$$$$\nu^{10} +$$$$87\!\cdots\!39$$$$\nu^{9} +$$$$52\!\cdots\!28$$$$\nu^{8} +$$$$27\!\cdots\!36$$$$\nu^{7} +$$$$18\!\cdots\!06$$$$\nu^{6} +$$$$63\!\cdots\!58$$$$\nu^{5} +$$$$32\!\cdots\!04$$$$\nu^{4} +$$$$17\!\cdots\!44$$$$\nu^{3} +$$$$83\!\cdots\!50$$$$\nu^{2} +$$$$45\!\cdots\!25$$$$\nu -$$$$21\!\cdots\!50$$$$)/$$$$49\!\cdots\!25$$ $$\beta_{4}$$ $$=$$ $$($$$$-$$$$11\!\cdots\!68$$$$\nu^{13} -$$$$17\!\cdots\!81$$$$\nu^{12} -$$$$10\!\cdots\!12$$$$\nu^{11} -$$$$62\!\cdots\!61$$$$\nu^{10} -$$$$71\!\cdots\!28$$$$\nu^{9} -$$$$41\!\cdots\!06$$$$\nu^{8} -$$$$21\!\cdots\!72$$$$\nu^{7} -$$$$14\!\cdots\!37$$$$\nu^{6} -$$$$48\!\cdots\!16$$$$\nu^{5} -$$$$25\!\cdots\!58$$$$\nu^{4} -$$$$31\!\cdots\!88$$$$\nu^{3} -$$$$63\!\cdots\!25$$$$\nu^{2} +$$$$30\!\cdots\!00$$$$\nu -$$$$78\!\cdots\!25$$$$)/$$$$31\!\cdots\!00$$ $$\beta_{5}$$ $$=$$ $$($$$$17\!\cdots\!24$$$$\nu^{13} +$$$$34\!\cdots\!83$$$$\nu^{12} +$$$$16\!\cdots\!16$$$$\nu^{11} +$$$$10\!\cdots\!23$$$$\nu^{10} +$$$$11\!\cdots\!04$$$$\nu^{9} +$$$$68\!\cdots\!58$$$$\nu^{8} +$$$$35\!\cdots\!96$$$$\nu^{7} +$$$$23\!\cdots\!91$$$$\nu^{6} +$$$$82\!\cdots\!88$$$$\nu^{5} +$$$$42\!\cdots\!94$$$$\nu^{4} +$$$$20\!\cdots\!84$$$$\nu^{3} +$$$$10\!\cdots\!75$$$$\nu^{2} -$$$$51\!\cdots\!00$$$$\nu +$$$$39\!\cdots\!75$$$$)/$$$$21\!\cdots\!00$$ $$\beta_{6}$$ $$=$$ $$($$$$-$$$$50\!\cdots\!63$$$$\nu^{13} -$$$$10\!\cdots\!71$$$$\nu^{12} -$$$$47\!\cdots\!42$$$$\nu^{11} -$$$$29\!\cdots\!26$$$$\nu^{10} -$$$$33\!\cdots\!73$$$$\nu^{9} -$$$$20\!\cdots\!96$$$$\nu^{8} -$$$$10\!\cdots\!02$$$$\nu^{7} -$$$$70\!\cdots\!92$$$$\nu^{6} -$$$$24\!\cdots\!31$$$$\nu^{5} -$$$$12\!\cdots\!28$$$$\nu^{4} -$$$$69\!\cdots\!08$$$$\nu^{3} -$$$$32\!\cdots\!50$$$$\nu^{2} +$$$$15\!\cdots\!00$$$$\nu -$$$$23\!\cdots\!75$$$$)/$$$$39\!\cdots\!00$$ $$\beta_{7}$$ $$=$$ $$($$$$62\!\cdots\!63$$$$\nu^{13} -$$$$27\!\cdots\!69$$$$\nu^{12} +$$$$58\!\cdots\!87$$$$\nu^{11} +$$$$22\!\cdots\!91$$$$\nu^{10} +$$$$40\!\cdots\!18$$$$\nu^{9} +$$$$14\!\cdots\!06$$$$\nu^{8} +$$$$12\!\cdots\!47$$$$\nu^{7} +$$$$55\!\cdots\!07$$$$\nu^{6} +$$$$28\!\cdots\!46$$$$\nu^{5} +$$$$84\!\cdots\!98$$$$\nu^{4} +$$$$48\!\cdots\!43$$$$\nu^{3} -$$$$96\!\cdots\!65$$$$\nu^{2} +$$$$12\!\cdots\!75$$$$\nu -$$$$57\!\cdots\!25$$$$)/$$$$23\!\cdots\!00$$ $$\beta_{8}$$ $$=$$ $$($$$$28\!\cdots\!24$$$$\nu^{13} +$$$$56\!\cdots\!33$$$$\nu^{12} +$$$$26\!\cdots\!16$$$$\nu^{11} +$$$$16\!\cdots\!73$$$$\nu^{10} +$$$$19\!\cdots\!04$$$$\nu^{9} +$$$$11\!\cdots\!58$$$$\nu^{8} +$$$$59\!\cdots\!96$$$$\nu^{7} +$$$$39\!\cdots\!41$$$$\nu^{6} +$$$$13\!\cdots\!88$$$$\nu^{5} +$$$$71\!\cdots\!94$$$$\nu^{4} +$$$$34\!\cdots\!84$$$$\nu^{3} +$$$$18\!\cdots\!25$$$$\nu^{2} -$$$$85\!\cdots\!00$$$$\nu +$$$$43\!\cdots\!25$$$$)/$$$$31\!\cdots\!00$$ $$\beta_{9}$$ $$=$$ $$($$$$-$$$$41\!\cdots\!97$$$$\nu^{13} +$$$$29\!\cdots\!56$$$$\nu^{12} -$$$$38\!\cdots\!38$$$$\nu^{11} -$$$$16\!\cdots\!24$$$$\nu^{10} -$$$$26\!\cdots\!52$$$$\nu^{9} -$$$$10\!\cdots\!69$$$$\nu^{8} -$$$$82\!\cdots\!28$$$$\nu^{7} -$$$$39\!\cdots\!78$$$$\nu^{6} -$$$$18\!\cdots\!44$$$$\nu^{5} -$$$$63\!\cdots\!47$$$$\nu^{4} -$$$$36\!\cdots\!22$$$$\nu^{3} +$$$$42\!\cdots\!30$$$$\nu^{2} -$$$$68\!\cdots\!25$$$$\nu +$$$$68\!\cdots\!75$$$$)/$$$$44\!\cdots\!00$$ $$\beta_{10}$$ $$=$$ $$($$$$-$$$$61\!\cdots\!28$$$$\nu^{13} -$$$$12\!\cdots\!51$$$$\nu^{12} -$$$$57\!\cdots\!52$$$$\nu^{11} -$$$$36\!\cdots\!31$$$$\nu^{10} -$$$$40\!\cdots\!88$$$$\nu^{9} -$$$$24\!\cdots\!26$$$$\nu^{8} -$$$$12\!\cdots\!12$$$$\nu^{7} -$$$$84\!\cdots\!27$$$$\nu^{6} -$$$$29\!\cdots\!36$$$$\nu^{5} -$$$$15\!\cdots\!18$$$$\nu^{4} -$$$$73\!\cdots\!48$$$$\nu^{3} -$$$$38\!\cdots\!75$$$$\nu^{2} +$$$$18\!\cdots\!00$$$$\nu -$$$$86\!\cdots\!75$$$$)/$$$$31\!\cdots\!00$$ $$\beta_{11}$$ $$=$$ $$($$$$76\!\cdots\!93$$$$\nu^{13} -$$$$47\!\cdots\!59$$$$\nu^{12} +$$$$71\!\cdots\!57$$$$\nu^{11} +$$$$26\!\cdots\!01$$$$\nu^{10} +$$$$49\!\cdots\!98$$$$\nu^{9} +$$$$17\!\cdots\!66$$$$\nu^{8} +$$$$15\!\cdots\!17$$$$\nu^{7} +$$$$65\!\cdots\!77$$$$\nu^{6} +$$$$34\!\cdots\!06$$$$\nu^{5} +$$$$97\!\cdots\!78$$$$\nu^{4} +$$$$56\!\cdots\!73$$$$\nu^{3} -$$$$13\!\cdots\!15$$$$\nu^{2} +$$$$14\!\cdots\!25$$$$\nu -$$$$66\!\cdots\!75$$$$)/$$$$35\!\cdots\!00$$ $$\beta_{12}$$ $$=$$ $$($$$$10\!\cdots\!77$$$$\nu^{13} -$$$$16\!\cdots\!96$$$$\nu^{12} +$$$$94\!\cdots\!33$$$$\nu^{11} +$$$$40\!\cdots\!84$$$$\nu^{10} +$$$$66\!\cdots\!82$$$$\nu^{9} +$$$$26\!\cdots\!04$$$$\nu^{8} +$$$$20\!\cdots\!73$$$$\nu^{7} +$$$$99\!\cdots\!48$$$$\nu^{6} +$$$$46\!\cdots\!54$$$$\nu^{5} +$$$$15\!\cdots\!52$$$$\nu^{4} +$$$$89\!\cdots\!77$$$$\nu^{3} -$$$$10\!\cdots\!80$$$$\nu^{2} +$$$$22\!\cdots\!25$$$$\nu -$$$$16\!\cdots\!00$$$$)/$$$$35\!\cdots\!00$$ $$\beta_{13}$$ $$=$$ $$($$$$-$$$$15\!\cdots\!31$$$$\nu^{13} +$$$$10\!\cdots\!53$$$$\nu^{12} -$$$$14\!\cdots\!19$$$$\nu^{11} -$$$$53\!\cdots\!67$$$$\nu^{10} -$$$$10\!\cdots\!66$$$$\nu^{9} -$$$$34\!\cdots\!22$$$$\nu^{8} -$$$$30\!\cdots\!39$$$$\nu^{7} -$$$$13\!\cdots\!59$$$$\nu^{6} -$$$$69\!\cdots\!02$$$$\nu^{5} -$$$$19\!\cdots\!26$$$$\nu^{4} -$$$$11\!\cdots\!91$$$$\nu^{3} +$$$$27\!\cdots\!05$$$$\nu^{2} -$$$$28\!\cdots\!75$$$$\nu +$$$$13\!\cdots\!25$$$$)/$$$$35\!\cdots\!00$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$-\beta_{12} + \beta_{9} + 2 \beta_{7} - \beta_{6} - \beta_{4} + 32 \beta_{3} - 26583 \beta_{2}$$ $$\nu^{3}$$ $$=$$ $$-36 \beta_{10} - 52 \beta_{8} + 35 \beta_{6} - 406 \beta_{5} - 71 \beta_{4} + 47365 \beta_{3} - 47365 \beta_{1} - 845938$$ $$\nu^{4}$$ $$=$$ $$-11448 \beta_{13} + 52030 \beta_{12} - 15702 \beta_{11} - 11448 \beta_{10} - 45184 \beta_{9} - 15702 \beta_{8} - 198740 \beta_{7} - 198740 \beta_{5} + 1258162311 \beta_{2} - 3000674 \beta_{1} - 1258162311$$ $$\nu^{5}$$ $$=$$ $$-3830256 \beta_{13} + 6051722 \beta_{12} - 4180162 \beta_{11} + 1545560 \beta_{9} - 52288204 \beta_{7} - 1545560 \beta_{6} + 6051722 \beta_{4} - 2391399703 \beta_{3} + 79216825858 \beta_{2}$$ $$\nu^{6}$$ $$=$$ $$1140296184 \beta_{10} + 1542167838 \beta_{8} + 2009199649 \beta_{6} + 14985660806 \beta_{5} + 2870453191 \beta_{4} - 231036642746 \beta_{3} + 231036642746 \beta_{1} + 63436713684549$$ $$\nu^{7}$$ $$=$$ $$303066446004 \beta_{13} - 441616433321 \beta_{12} + 310103988814 \beta_{11} + 303066446004 \beta_{10} - 53842768955 \beta_{9} + 310103988814 \beta_{8} + 4253033625010 \beta_{7} + 4253033625010 \beta_{5} - 6098966233032172 \beta_{2} + 126779676568195 \beta_{1} + 6098966233032172$$ $$\nu^{8}$$ $$=$$ $$86792913157104 \beta_{13} - 164655104212204 \beta_{12} + 114641442952860 \beta_{11} + 91734153370816 \beta_{9} + 1049564155594568 \beta_{7} - 91734153370816 \beta_{6} - 164655104212204 \beta_{4} + 16403947718430164 \beta_{3} - 3358973998434191805 \beta_{2}$$ $$\nu^{9}$$ $$=$$ $$-21544491993243552 \beta_{10} - 21966237278809684 \beta_{8} + 1135553561210672 \beta_{6} - 300375500284767736 \beta_{5} - 30179326293356516 \beta_{4} + 7014786199629432541 \beta_{3} - 7014786199629432541 \beta_{1} - 433058402539560590740$$ $$\nu^{10}$$ $$=$$ $$-5989779950101012080 \beta_{13} + 9721406783474515261 \beta_{12} - 7720137404613420396 \beta_{11} - 5989779950101012080 \beta_{10} - 4332008114915798785 \beta_{9} - 7720137404613420396 \beta_{8} - 70704033345864156458 \beta_{7} - 70704033345864156458 \beta_{5} + 185669641663948224365475 \beta_{2} - 1114164007994019899684 \beta_{1} - 185669641663948224365475$$ $$\nu^{11}$$ $$=$$ $$-1454573470541008381956 \beta_{13} + 1992602456070742974875 \beta_{12} - 1505142940081529604856 \beta_{11} - 39838062196026226477 \beta_{9} - 19964099963070813809326 \beta_{7} + 39838062196026226477 \beta_{6} + 1992602456070742974875 \beta_{4} - 401997508921039388957425 \beta_{3} + 29415434886674421320913334 \beta_{2}$$ $$\nu^{12}$$ $$=$$ $$394936272346976129420712 \beta_{10} + 497594733901654897316178 \beta_{8} + 212496394375571142831808 \beta_{6} + 4652485075170438320471516 \beta_{5} + 585852083740684460804746 \beta_{4} - 73673617936125002170717142 \beta_{3} + 73673617936125002170717142 \beta_{1} + 10632255945340624213017967395$$ $$\nu^{13}$$ $$=$$ $$95478728427033554332308240 \beta_{13} - 129006727772996463202723214 \beta_{12} + 100683609110753134601251126 \beta_{11} + 95478728427033554332308240 \beta_{10} + 7851138257618391745600120 \beta_{9} + 100683609110753134601251126 \beta_{8} + 1288107084366398291979612292 \beta_{7} + 1288107084366398291979612292 \beta_{5} - 1945194676776195593638651177270 \beta_{2} + 23668844752122450048904633555 \beta_{1} + 1945194676776195593638651177270$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/38\mathbb{Z}\right)^\times$$.

 $$n$$ $$21$$ $$\chi(n)$$ $$-\beta_{2}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
7.1
 125.259 + 216.954i 97.2191 + 168.388i 5.46583 + 9.46709i 2.70706 + 4.68876i −25.7605 − 44.6185i −98.6185 − 170.812i −104.772 − 181.470i 125.259 − 216.954i 97.2191 − 168.388i 5.46583 − 9.46709i 2.70706 − 4.68876i −25.7605 + 44.6185i −98.6185 + 170.812i −104.772 + 181.470i
8.00000 + 13.8564i −113.259 196.170i −128.000 + 221.703i 563.006 + 975.155i 1812.14 3138.71i 3883.37 −4096.00 −15813.5 + 27389.8i −9008.10 + 15602.5i
7.2 8.00000 + 13.8564i −85.2191 147.604i −128.000 + 221.703i −867.647 1502.81i 1363.51 2361.66i −2874.94 −4096.00 −4683.10 + 8111.37i 13882.3 24044.9i
7.3 8.00000 + 13.8564i 6.53417 + 11.3175i −128.000 + 221.703i 80.6193 + 139.637i −104.547 + 181.080i −6270.17 −4096.00 9756.11 16898.1i −1289.91 + 2234.19i
7.4 8.00000 + 13.8564i 9.29294 + 16.0959i −128.000 + 221.703i 830.940 + 1439.23i −148.687 + 257.534i 10512.6 −4096.00 9668.78 16746.8i −13295.0 + 23027.7i
7.5 8.00000 + 13.8564i 37.7605 + 65.4031i −128.000 + 221.703i −802.961 1390.77i −604.168 + 1046.45i 1586.11 −4096.00 6989.79 12106.7i 12847.4 22252.3i
7.6 8.00000 + 13.8564i 110.618 + 191.597i −128.000 + 221.703i 1160.30 + 2009.69i −1769.90 + 3065.55i −7717.55 −4096.00 −14631.4 + 25342.3i −18564.7 + 32155.1i
7.7 8.00000 + 13.8564i 116.772 + 202.254i −128.000 + 221.703i −509.754 882.920i −1868.34 + 3236.07i 2726.62 −4096.00 −17429.7 + 30189.1i 8156.07 14126.7i
11.1 8.00000 13.8564i −113.259 + 196.170i −128.000 221.703i 563.006 975.155i 1812.14 + 3138.71i 3883.37 −4096.00 −15813.5 27389.8i −9008.10 15602.5i
11.2 8.00000 13.8564i −85.2191 + 147.604i −128.000 221.703i −867.647 + 1502.81i 1363.51 + 2361.66i −2874.94 −4096.00 −4683.10 8111.37i 13882.3 + 24044.9i
11.3 8.00000 13.8564i 6.53417 11.3175i −128.000 221.703i 80.6193 139.637i −104.547 181.080i −6270.17 −4096.00 9756.11 + 16898.1i −1289.91 2234.19i
11.4 8.00000 13.8564i 9.29294 16.0959i −128.000 221.703i 830.940 1439.23i −148.687 257.534i 10512.6 −4096.00 9668.78 + 16746.8i −13295.0 23027.7i
11.5 8.00000 13.8564i 37.7605 65.4031i −128.000 221.703i −802.961 + 1390.77i −604.168 1046.45i 1586.11 −4096.00 6989.79 + 12106.7i 12847.4 + 22252.3i
11.6 8.00000 13.8564i 110.618 191.597i −128.000 221.703i 1160.30 2009.69i −1769.90 3065.55i −7717.55 −4096.00 −14631.4 25342.3i −18564.7 32155.1i
11.7 8.00000 13.8564i 116.772 202.254i −128.000 221.703i −509.754 + 882.920i −1868.34 3236.07i 2726.62 −4096.00 −17429.7 30189.1i 8156.07 + 14126.7i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 11.7 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
19.c even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 38.10.c.a 14
19.c even 3 1 inner 38.10.c.a 14

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
38.10.c.a 14 1.a even 1 1 trivial
38.10.c.a 14 19.c even 3 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$24\!\cdots\!31$$$$T_{3}^{8} -$$$$16\!\cdots\!37$$$$T_{3}^{7} +$$$$50\!\cdots\!71$$$$T_{3}^{6} -$$$$42\!\cdots\!70$$$$T_{3}^{5} +$$$$34\!\cdots\!31$$$$T_{3}^{4} -$$$$95\!\cdots\!05$$$$T_{3}^{3} +$$$$19\!\cdots\!50$$$$T_{3}^{2} -$$$$19\!\cdots\!25$$$$T_{3} +$$$$13\!\cdots\!25$$">$$T_{3}^{14} - \cdots$$ acting on $$S_{10}^{\mathrm{new}}(38, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 256 - 16 T + T^{2} )^{7}$$
$3$ $$13\!\cdots\!25$$$$-$$$$19\!\cdots\!25$$$$T +$$$$19\!\cdots\!50$$$$T^{2} -$$$$95\!\cdots\!05$$$$T^{3} +$$$$34\!\cdots\!31$$$$T^{4} -$$$$42\!\cdots\!70$$$$T^{5} + 5059196462369139471 T^{6} - 16216679100196737 T^{7} + 249179222409831 T^{8} - 742632812562 T^{9} + 7242607471 T^{10} - 11474337 T^{11} + 108646 T^{12} - 165 T^{13} + T^{14}$$
$5$ $$39\!\cdots\!00$$$$-$$$$23\!\cdots\!00$$$$T +$$$$15\!\cdots\!00$$$$T^{2} +$$$$21\!\cdots\!00$$$$T^{3} +$$$$23\!\cdots\!00$$$$T^{4} -$$$$17\!\cdots\!00$$$$T^{5} +$$$$24\!\cdots\!30$$$$T^{6} -$$$$14\!\cdots\!60$$$$T^{7} +$$$$12\!\cdots\!31$$$$T^{8} - 8141006578628469 T^{9} + 44485997698932 T^{10} - 2034733169 T^{11} + 8448402 T^{12} - 909 T^{13} + T^{14}$$
$7$ $$($$$$24\!\cdots\!96$$$$-$$$$17\!\cdots\!72$$$$T - 3220763217736372224 T^{2} + 3056513233706912 T^{3} + 51393160912 T^{4} - 119638854 T^{5} - 1846 T^{6} + T^{7} )^{2}$$
$11$ $$($$$$10\!\cdots\!52$$$$+$$$$68\!\cdots\!76$$$$T -$$$$89\!\cdots\!92$$$$T^{2} + 1118488751184792439 T^{3} + 184254991124168 T^{4} - 5761234416 T^{5} - 7828 T^{6} + T^{7} )^{2}$$
$13$ $$62\!\cdots\!96$$$$+$$$$30\!\cdots\!00$$$$T +$$$$15\!\cdots\!40$$$$T^{2} +$$$$18\!\cdots\!12$$$$T^{3} +$$$$58\!\cdots\!08$$$$T^{4} +$$$$36\!\cdots\!48$$$$T^{5} +$$$$16\!\cdots\!52$$$$T^{6} +$$$$46\!\cdots\!92$$$$T^{7} +$$$$17\!\cdots\!17$$$$T^{8} +$$$$62\!\cdots\!97$$$$T^{9} +$$$$13\!\cdots\!82$$$$T^{10} - 188297367382787 T^{11} + 42491222982 T^{12} - 6423 T^{13} + T^{14}$$
$17$ $$59\!\cdots\!84$$$$-$$$$40\!\cdots\!88$$$$T +$$$$22\!\cdots\!48$$$$T^{2} -$$$$29\!\cdots\!64$$$$T^{3} +$$$$30\!\cdots\!24$$$$T^{4} -$$$$16\!\cdots\!08$$$$T^{5} +$$$$82\!\cdots\!84$$$$T^{6} -$$$$22\!\cdots\!36$$$$T^{7} +$$$$99\!\cdots\!81$$$$T^{8} -$$$$20\!\cdots\!43$$$$T^{9} +$$$$83\!\cdots\!82$$$$T^{10} - 89713176851330559 T^{11} + 361296358822 T^{12} - 313667 T^{13} + T^{14}$$
$19$ $$36\!\cdots\!59$$$$-$$$$12\!\cdots\!04$$$$T +$$$$25\!\cdots\!39$$$$T^{2} -$$$$24\!\cdots\!32$$$$T^{3} +$$$$10\!\cdots\!52$$$$T^{4} +$$$$56\!\cdots\!92$$$$T^{5} -$$$$25\!\cdots\!64$$$$T^{6} +$$$$60\!\cdots\!28$$$$T^{7} -$$$$77\!\cdots\!16$$$$T^{8} +$$$$54\!\cdots\!12$$$$T^{9} +$$$$30\!\cdots\!68$$$$T^{10} - 223621835454267072 T^{11} + 727494745261 T^{12} - 1134224 T^{13} + T^{14}$$
$23$ $$74\!\cdots\!84$$$$-$$$$13\!\cdots\!20$$$$T +$$$$70\!\cdots\!68$$$$T^{2} +$$$$69\!\cdots\!24$$$$T^{3} +$$$$24\!\cdots\!32$$$$T^{4} +$$$$86\!\cdots\!08$$$$T^{5} +$$$$19\!\cdots\!34$$$$T^{6} -$$$$15\!\cdots\!08$$$$T^{7} +$$$$11\!\cdots\!71$$$$T^{8} -$$$$27\!\cdots\!97$$$$T^{9} +$$$$48\!\cdots\!08$$$$T^{10} - 19256056538509130981 T^{11} + 14095422273922 T^{12} - 3449345 T^{13} + T^{14}$$
$29$ $$16\!\cdots\!00$$$$+$$$$79\!\cdots\!00$$$$T +$$$$23\!\cdots\!00$$$$T^{2} +$$$$43\!\cdots\!80$$$$T^{3} +$$$$56\!\cdots\!84$$$$T^{4} +$$$$53\!\cdots\!44$$$$T^{5} +$$$$37\!\cdots\!50$$$$T^{6} +$$$$19\!\cdots\!52$$$$T^{7} +$$$$76\!\cdots\!43$$$$T^{8} +$$$$20\!\cdots\!03$$$$T^{9} +$$$$39\!\cdots\!32$$$$T^{10} +$$$$40\!\cdots\!51$$$$T^{11} + 79657539025182 T^{12} + 7002615 T^{13} + T^{14}$$
$31$ $$( -$$$$39\!\cdots\!32$$$$+$$$$93\!\cdots\!44$$$$T +$$$$44\!\cdots\!20$$$$T^{2} +$$$$24\!\cdots\!44$$$$T^{3} -$$$$41\!\cdots\!04$$$$T^{4} - 36942369733674 T^{5} + 9333294 T^{6} + T^{7} )^{2}$$
$37$ $$($$$$58\!\cdots\!00$$$$+$$$$16\!\cdots\!40$$$$T +$$$$66\!\cdots\!92$$$$T^{2} +$$$$66\!\cdots\!40$$$$T^{3} -$$$$36\!\cdots\!28$$$$T^{4} - 549361381442394 T^{5} + 3433040 T^{6} + T^{7} )^{2}$$
$41$ $$25\!\cdots\!25$$$$+$$$$13\!\cdots\!25$$$$T +$$$$55\!\cdots\!04$$$$T^{2} +$$$$97\!\cdots\!83$$$$T^{3} +$$$$12\!\cdots\!89$$$$T^{4} +$$$$25\!\cdots\!62$$$$T^{5} +$$$$47\!\cdots\!81$$$$T^{6} +$$$$33\!\cdots\!75$$$$T^{7} +$$$$35\!\cdots\!33$$$$T^{8} +$$$$11\!\cdots\!54$$$$T^{9} +$$$$18\!\cdots\!61$$$$T^{10} +$$$$31\!\cdots\!19$$$$T^{11} + 517939724407888 T^{12} - 3564107 T^{13} + T^{14}$$
$43$ $$17\!\cdots\!00$$$$+$$$$68\!\cdots\!00$$$$T +$$$$30\!\cdots\!96$$$$T^{2} +$$$$65\!\cdots\!40$$$$T^{3} +$$$$53\!\cdots\!88$$$$T^{4} +$$$$39\!\cdots\!80$$$$T^{5} +$$$$57\!\cdots\!24$$$$T^{6} +$$$$20\!\cdots\!44$$$$T^{7} +$$$$14\!\cdots\!57$$$$T^{8} +$$$$16\!\cdots\!79$$$$T^{9} +$$$$20\!\cdots\!58$$$$T^{10} +$$$$74\!\cdots\!35$$$$T^{11} + 2008776744034626 T^{12} - 19837521 T^{13} + T^{14}$$
$47$ $$27\!\cdots\!00$$$$-$$$$71\!\cdots\!00$$$$T +$$$$14\!\cdots\!00$$$$T^{2} -$$$$11\!\cdots\!00$$$$T^{3} +$$$$83\!\cdots\!04$$$$T^{4} -$$$$32\!\cdots\!68$$$$T^{5} +$$$$14\!\cdots\!30$$$$T^{6} -$$$$44\!\cdots\!92$$$$T^{7} +$$$$17\!\cdots\!19$$$$T^{8} -$$$$39\!\cdots\!77$$$$T^{9} +$$$$11\!\cdots\!16$$$$T^{10} -$$$$19\!\cdots\!49$$$$T^{11} + 5213280844969450 T^{12} - 60353825 T^{13} + T^{14}$$
$53$ $$31\!\cdots\!16$$$$-$$$$17\!\cdots\!40$$$$T +$$$$29\!\cdots\!12$$$$T^{2} +$$$$74\!\cdots\!68$$$$T^{3} +$$$$15\!\cdots\!68$$$$T^{4} -$$$$24\!\cdots\!24$$$$T^{5} +$$$$13\!\cdots\!88$$$$T^{6} -$$$$93\!\cdots\!76$$$$T^{7} +$$$$58\!\cdots\!13$$$$T^{8} -$$$$46\!\cdots\!19$$$$T^{9} +$$$$13\!\cdots\!58$$$$T^{10} -$$$$51\!\cdots\!27$$$$T^{11} + 14558935501981222 T^{12} - 54744235 T^{13} + T^{14}$$
$59$ $$77\!\cdots\!25$$$$-$$$$60\!\cdots\!05$$$$T +$$$$33\!\cdots\!34$$$$T^{2} -$$$$99\!\cdots\!45$$$$T^{3} +$$$$23\!\cdots\!11$$$$T^{4} -$$$$31\!\cdots\!90$$$$T^{5} +$$$$40\!\cdots\!39$$$$T^{6} -$$$$31\!\cdots\!05$$$$T^{7} +$$$$40\!\cdots\!39$$$$T^{8} -$$$$23\!\cdots\!50$$$$T^{9} +$$$$21\!\cdots\!63$$$$T^{10} -$$$$72\!\cdots\!05$$$$T^{11} + 61286865561161710 T^{12} - 164456585 T^{13} + T^{14}$$
$61$ $$77\!\cdots\!96$$$$-$$$$20\!\cdots\!32$$$$T +$$$$91\!\cdots\!08$$$$T^{2} -$$$$78\!\cdots\!52$$$$T^{3} +$$$$34\!\cdots\!04$$$$T^{4} -$$$$25\!\cdots\!64$$$$T^{5} +$$$$87\!\cdots\!26$$$$T^{6} -$$$$36\!\cdots\!24$$$$T^{7} +$$$$12\!\cdots\!47$$$$T^{8} -$$$$43\!\cdots\!65$$$$T^{9} +$$$$11\!\cdots\!52$$$$T^{10} -$$$$14\!\cdots\!73$$$$T^{11} + 39502891430524714 T^{12} - 49328881 T^{13} + T^{14}$$
$67$ $$20\!\cdots\!25$$$$+$$$$38\!\cdots\!25$$$$T +$$$$49\!\cdots\!50$$$$T^{2} +$$$$34\!\cdots\!25$$$$T^{3} +$$$$17\!\cdots\!75$$$$T^{4} +$$$$57\!\cdots\!70$$$$T^{5} +$$$$14\!\cdots\!99$$$$T^{6} +$$$$24\!\cdots\!45$$$$T^{7} +$$$$32\!\cdots\!95$$$$T^{8} +$$$$28\!\cdots\!34$$$$T^{9} +$$$$20\!\cdots\!03$$$$T^{10} +$$$$92\!\cdots\!85$$$$T^{11} + 49754267260894686 T^{12} + 171522309 T^{13} + T^{14}$$
$71$ $$37\!\cdots\!36$$$$+$$$$59\!\cdots\!60$$$$T +$$$$78\!\cdots\!52$$$$T^{2} +$$$$24\!\cdots\!32$$$$T^{3} +$$$$67\!\cdots\!28$$$$T^{4} +$$$$10\!\cdots\!84$$$$T^{5} +$$$$16\!\cdots\!08$$$$T^{6} +$$$$17\!\cdots\!16$$$$T^{7} +$$$$23\!\cdots\!53$$$$T^{8} +$$$$19\!\cdots\!99$$$$T^{9} +$$$$16\!\cdots\!78$$$$T^{10} +$$$$66\!\cdots\!47$$$$T^{11} + 42280143186527662 T^{12} + 74596055 T^{13} + T^{14}$$
$73$ $$13\!\cdots\!25$$$$-$$$$96\!\cdots\!75$$$$T +$$$$32\!\cdots\!00$$$$T^{2} -$$$$24\!\cdots\!65$$$$T^{3} +$$$$58\!\cdots\!69$$$$T^{4} -$$$$38\!\cdots\!46$$$$T^{5} +$$$$43\!\cdots\!29$$$$T^{6} -$$$$15\!\cdots\!73$$$$T^{7} +$$$$14\!\cdots\!45$$$$T^{8} -$$$$38\!\cdots\!10$$$$T^{9} +$$$$32\!\cdots\!37$$$$T^{10} -$$$$28\!\cdots\!49$$$$T^{11} + 196902381877295512 T^{12} - 58695287 T^{13} + T^{14}$$
$79$ $$39\!\cdots\!16$$$$-$$$$86\!\cdots\!56$$$$T +$$$$19\!\cdots\!36$$$$T^{2} +$$$$10\!\cdots\!28$$$$T^{3} +$$$$17\!\cdots\!92$$$$T^{4} -$$$$42\!\cdots\!60$$$$T^{5} +$$$$48\!\cdots\!24$$$$T^{6} -$$$$43\!\cdots\!68$$$$T^{7} +$$$$37\!\cdots\!17$$$$T^{8} -$$$$25\!\cdots\!53$$$$T^{9} +$$$$21\!\cdots\!66$$$$T^{10} -$$$$63\!\cdots\!77$$$$T^{11} + 545938609054756702 T^{12} - 121854617 T^{13} + T^{14}$$
$83$ $$( -$$$$53\!\cdots\!64$$$$-$$$$15\!\cdots\!32$$$$T +$$$$69\!\cdots\!40$$$$T^{2} +$$$$19\!\cdots\!67$$$$T^{3} -$$$$28\!\cdots\!68$$$$T^{4} - 774257301415475700 T^{5} + 366303628 T^{6} + T^{7} )^{2}$$
$89$ $$16\!\cdots\!56$$$$+$$$$59\!\cdots\!32$$$$T +$$$$66\!\cdots\!20$$$$T^{2} -$$$$13\!\cdots\!56$$$$T^{3} +$$$$11\!\cdots\!68$$$$T^{4} -$$$$83\!\cdots\!88$$$$T^{5} +$$$$43\!\cdots\!80$$$$T^{6} +$$$$71\!\cdots\!16$$$$T^{7} +$$$$12\!\cdots\!01$$$$T^{8} +$$$$59\!\cdots\!85$$$$T^{9} +$$$$23\!\cdots\!54$$$$T^{10} +$$$$20\!\cdots\!17$$$$T^{11} + 3175422927574697086 T^{12} + 1652463181 T^{13} + T^{14}$$
$97$ $$12\!\cdots\!25$$$$-$$$$11\!\cdots\!95$$$$T +$$$$12\!\cdots\!44$$$$T^{2} -$$$$26\!\cdots\!37$$$$T^{3} +$$$$20\!\cdots\!09$$$$T^{4} -$$$$16\!\cdots\!98$$$$T^{5} +$$$$30\!\cdots\!41$$$$T^{6} -$$$$16\!\cdots\!45$$$$T^{7} +$$$$17\!\cdots\!93$$$$T^{8} -$$$$39\!\cdots\!86$$$$T^{9} +$$$$69\!\cdots\!01$$$$T^{10} -$$$$27\!\cdots\!21$$$$T^{11} + 990743885770228468 T^{12} - 248805607 T^{13} + T^{14}$$