# Properties

 Label 384.3.l.a Level $384$ Weight $3$ Character orbit 384.l Analytic conductor $10.463$ Analytic rank $0$ Dimension $16$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$384 = 2^{7} \cdot 3$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 384.l (of order $$4$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$10.4632421514$$ Analytic rank: $$0$$ Dimension: $$16$$ Relative dimension: $$8$$ over $$\Q(i)$$ Coefficient field: $$\mathbb{Q}[x]/(x^{16} - \cdots)$$ Defining polynomial: $$x^{16} - 6 x^{14} - 4 x^{13} + 10 x^{12} + 56 x^{11} + 88 x^{10} - 128 x^{9} - 496 x^{8} - 512 x^{7} + 1408 x^{6} + 3584 x^{5} + 2560 x^{4} - 4096 x^{3} - 24576 x^{2} + 65536$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{24}$$ Twist minimal: no (minimal twist has level 48) Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{15}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q -\beta_{2} q^{3} + \beta_{9} q^{5} -\beta_{5} q^{7} -3 \beta_{4} q^{9} +O(q^{10})$$ $$q -\beta_{2} q^{3} + \beta_{9} q^{5} -\beta_{5} q^{7} -3 \beta_{4} q^{9} + ( -2 - 2 \beta_{4} + \beta_{7} - \beta_{10} ) q^{11} + ( \beta_{1} - \beta_{5} + \beta_{6} - \beta_{7} - \beta_{8} - \beta_{13} ) q^{13} + \beta_{14} q^{15} + ( \beta_{1} - 2 \beta_{2} - 2 \beta_{3} + \beta_{5} - \beta_{6} + \beta_{7} - \beta_{9} ) q^{17} + ( 2 + \beta_{1} - 2 \beta_{4} - \beta_{9} + \beta_{11} + 2 \beta_{12} + \beta_{15} ) q^{19} -\beta_{12} q^{21} + ( -8 + 2 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} - 2 \beta_{5} + 2 \beta_{6} - 2 \beta_{7} + 2 \beta_{9} + 2 \beta_{12} - 2 \beta_{13} - 2 \beta_{14} + 2 \beta_{15} ) q^{23} + ( -4 \beta_{2} + 4 \beta_{3} - 5 \beta_{4} - 2 \beta_{8} - \beta_{10} + \beta_{11} - \beta_{12} - \beta_{13} - \beta_{14} - \beta_{15} ) q^{25} + 3 \beta_{3} q^{27} + ( -2 + \beta_{1} - 2 \beta_{4} - \beta_{5} - \beta_{6} - \beta_{7} - \beta_{8} + 2 \beta_{10} - 2 \beta_{13} - 3 \beta_{14} + \beta_{15} ) q^{29} + ( 8 \beta_{4} + \beta_{7} - \beta_{8} + \beta_{9} - \beta_{10} + \beta_{11} + 2 \beta_{15} ) q^{31} + ( \beta_{1} + 2 \beta_{2} + 2 \beta_{3} + \beta_{5} + \beta_{6} - \beta_{7} + \beta_{9} + \beta_{10} + \beta_{11} + \beta_{12} - \beta_{13} - \beta_{14} + \beta_{15} ) q^{33} + ( -6 + 2 \beta_{1} + 6 \beta_{4} - 2 \beta_{5} + 2 \beta_{8} - \beta_{9} - \beta_{11} + 2 \beta_{15} ) q^{35} + ( 6 - \beta_{1} - 6 \beta_{4} + \beta_{5} - 3 \beta_{6} - \beta_{8} - \beta_{9} + 2 \beta_{11} - \beta_{12} + 3 \beta_{14} - \beta_{15} ) q^{37} + ( 2 \beta_{1} - \beta_{5} + \beta_{6} + \beta_{7} - \beta_{9} - \beta_{10} - \beta_{11} ) q^{39} + ( 6 \beta_{2} - 6 \beta_{3} + \beta_{7} - 3 \beta_{8} + \beta_{9} - 2 \beta_{12} - 2 \beta_{13} + \beta_{14} + \beta_{15} ) q^{41} + ( -10 - \beta_{1} - 10 \beta_{4} - 2 \beta_{5} + \beta_{7} - 2 \beta_{8} + \beta_{10} - 2 \beta_{13} - 2 \beta_{14} + 3 \beta_{15} ) q^{43} + 3 \beta_{7} q^{45} + ( -6 \beta_{2} + 6 \beta_{3} - 24 \beta_{4} + 2 \beta_{7} + 2 \beta_{8} + 2 \beta_{9} + 2 \beta_{12} + 2 \beta_{13} ) q^{47} + ( 7 - 8 \beta_{2} - 8 \beta_{3} - 4 \beta_{5} - 2 \beta_{6} - 4 \beta_{7} + 4 \beta_{9} + \beta_{10} + \beta_{11} + \beta_{12} - \beta_{13} - \beta_{14} + \beta_{15} ) q^{49} + ( 6 + \beta_{1} - 6 \beta_{4} - 2 \beta_{5} + 2 \beta_{6} + 2 \beta_{8} + 5 \beta_{9} - \beta_{11} + 2 \beta_{12} - 2 \beta_{14} + \beta_{15} ) q^{51} + ( 10 + \beta_{1} - 8 \beta_{2} - 10 \beta_{4} + 3 \beta_{5} + 3 \beta_{6} - 3 \beta_{8} + 3 \beta_{9} + 2 \beta_{11} + 2 \beta_{12} - 3 \beta_{14} + \beta_{15} ) q^{53} + ( -16 - 2 \beta_{1} - 4 \beta_{2} - 4 \beta_{3} + 2 \beta_{5} - 4 \beta_{6} + 7 \beta_{7} - 7 \beta_{9} + \beta_{10} + \beta_{11} ) q^{55} + ( -2 \beta_{2} + 2 \beta_{3} + \beta_{7} - \beta_{8} + \beta_{9} + 2 \beta_{10} - 2 \beta_{11} - 3 \beta_{14} + \beta_{15} ) q^{57} + ( 8 + 2 \beta_{1} + 8 \beta_{4} - 2 \beta_{5} + 4 \beta_{6} + 2 \beta_{7} - 2 \beta_{8} - 2 \beta_{10} + 4 \beta_{14} - 2 \beta_{15} ) q^{59} + ( 2 + \beta_{1} - 16 \beta_{3} + 2 \beta_{4} - \beta_{5} - \beta_{6} + \beta_{7} - \beta_{8} - 6 \beta_{10} + \beta_{13} - 2 \beta_{15} ) q^{61} + 3 \beta_{8} q^{63} + ( -2 + \beta_{1} + 10 \beta_{2} + 10 \beta_{3} - 3 \beta_{5} + 3 \beta_{6} + \beta_{7} - \beta_{9} - 2 \beta_{12} + 2 \beta_{13} + 2 \beta_{14} - 2 \beta_{15} ) q^{65} + ( -20 - 2 \beta_{1} + 4 \beta_{2} + 20 \beta_{4} + 2 \beta_{5} - 2 \beta_{8} + 4 \beta_{9} + 4 \beta_{11} - 2 \beta_{15} ) q^{67} + ( -6 + 2 \beta_{1} + 8 \beta_{2} + 6 \beta_{4} + 2 \beta_{5} - 2 \beta_{8} - 2 \beta_{9} - 2 \beta_{11} + 2 \beta_{15} ) q^{69} + ( 32 + 2 \beta_{5} ) q^{71} + ( 8 \beta_{2} - 8 \beta_{3} + 6 \beta_{4} - 4 \beta_{7} - 4 \beta_{9} - \beta_{10} + \beta_{11} + \beta_{12} + \beta_{13} + 7 \beta_{14} - 5 \beta_{15} ) q^{73} + ( -12 + 3 \beta_{1} + 5 \beta_{3} - 12 \beta_{4} + 2 \beta_{6} - 6 \beta_{7} - 2 \beta_{13} - \beta_{15} ) q^{75} + ( -14 - \beta_{1} + 24 \beta_{3} - 14 \beta_{4} + 5 \beta_{5} + \beta_{6} + 5 \beta_{8} + 2 \beta_{10} + \beta_{14} + \beta_{15} ) q^{77} + ( 12 \beta_{2} - 12 \beta_{3} - 8 \beta_{4} - 6 \beta_{7} - 5 \beta_{8} - 6 \beta_{9} - 2 \beta_{10} + 2 \beta_{11} ) q^{79} -9 q^{81} + ( 10 - 2 \beta_{1} - 10 \beta_{4} - 4 \beta_{6} + \beta_{9} - 3 \beta_{11} - 4 \beta_{12} + 4 \beta_{14} - 2 \beta_{15} ) q^{83} + ( -10 - 4 \beta_{1} - 32 \beta_{2} + 10 \beta_{4} - 4 \beta_{5} - 2 \beta_{6} + 4 \beta_{8} - 6 \beta_{9} - 2 \beta_{11} - 2 \beta_{12} + 2 \beta_{14} - 4 \beta_{15} ) q^{85} + ( 2 \beta_{2} + 2 \beta_{3} + \beta_{6} - 3 \beta_{7} + 3 \beta_{9} - 3 \beta_{10} - 3 \beta_{11} - 2 \beta_{12} + 2 \beta_{13} + 2 \beta_{14} - 2 \beta_{15} ) q^{87} + ( -12 \beta_{2} + 12 \beta_{3} + 10 \beta_{4} + 8 \beta_{8} + 2 \beta_{10} - 2 \beta_{11} - 4 \beta_{14} + 4 \beta_{15} ) q^{89} + ( 30 - 5 \beta_{1} - 12 \beta_{3} + 30 \beta_{4} - 8 \beta_{6} - 5 \beta_{7} + 3 \beta_{10} + 2 \beta_{13} - 6 \beta_{14} + 3 \beta_{15} ) q^{91} + ( -\beta_{1} - 8 \beta_{3} + 5 \beta_{5} - \beta_{6} + 2 \beta_{7} + 5 \beta_{8} + 4 \beta_{10} + \beta_{13} ) q^{93} + ( -10 \beta_{2} + 10 \beta_{3} + 40 \beta_{4} - 4 \beta_{7} - 4 \beta_{9} + 2 \beta_{10} - 2 \beta_{11} - 2 \beta_{12} - 2 \beta_{13} - 4 \beta_{15} ) q^{95} + ( -2 \beta_{1} + 12 \beta_{2} + 12 \beta_{3} - 2 \beta_{5} + 2 \beta_{6} - 4 \beta_{10} - 4 \beta_{11} + 2 \beta_{12} - 2 \beta_{13} - 2 \beta_{14} + 2 \beta_{15} ) q^{97} + ( -6 + 6 \beta_{4} - 3 \beta_{9} - 3 \beta_{11} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$16q + O(q^{10})$$ $$16q - 32q^{11} + 32q^{19} - 128q^{23} - 32q^{29} - 96q^{35} + 96q^{37} - 160q^{43} + 112q^{49} + 96q^{51} + 160q^{53} - 256q^{55} + 128q^{59} + 32q^{61} - 32q^{65} - 320q^{67} - 96q^{69} + 512q^{71} - 192q^{75} - 224q^{77} - 144q^{81} + 160q^{83} - 160q^{85} + 480q^{91} - 96q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{16} - 6 x^{14} - 4 x^{13} + 10 x^{12} + 56 x^{11} + 88 x^{10} - 128 x^{9} - 496 x^{8} - 512 x^{7} + 1408 x^{6} + 3584 x^{5} + 2560 x^{4} - 4096 x^{3} - 24576 x^{2} + 65536$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$14 \nu^{15} - 15 \nu^{14} - 80 \nu^{13} - 126 \nu^{12} + 80 \nu^{11} + 1258 \nu^{10} + 1392 \nu^{9} - 2184 \nu^{8} - 10752 \nu^{7} - 16752 \nu^{6} + 16960 \nu^{5} + 82304 \nu^{4} + 84480 \nu^{3} - 166400 \nu^{2} - 620544 \nu - 163840$$$$)/61440$$ $$\beta_{2}$$ $$=$$ $$($$$$-81 \nu^{15} - 268 \nu^{14} - 218 \nu^{13} + 588 \nu^{12} + 2310 \nu^{11} + 1616 \nu^{10} - 9208 \nu^{9} - 30752 \nu^{8} - 38416 \nu^{7} + 22336 \nu^{6} + 142976 \nu^{5} + 146432 \nu^{4} - 195072 \nu^{3} - 976896 \nu^{2} - 966656 \nu + 180224$$$$)/245760$$ $$\beta_{3}$$ $$=$$ $$($$$$131 \nu^{15} + 88 \nu^{14} - 1122 \nu^{13} - 2268 \nu^{12} - 610 \nu^{11} + 9944 \nu^{10} + 27688 \nu^{9} + 1472 \nu^{8} - 117584 \nu^{7} - 278656 \nu^{6} - 125056 \nu^{5} + 588288 \nu^{4} + 1316352 \nu^{3} + 741376 \nu^{2} - 4317184 \nu - 7716864$$$$)/245760$$ $$\beta_{4}$$ $$=$$ $$($$$$-347 \nu^{15} - 626 \nu^{14} + 1234 \nu^{13} + 4536 \nu^{12} + 5530 \nu^{11} - 11868 \nu^{10} - 59096 \nu^{9} - 66544 \nu^{8} + 88528 \nu^{7} + 450272 \nu^{6} + 454272 \nu^{5} - 499456 \nu^{4} - 2271744 \nu^{3} - 3177472 \nu^{2} + 3719168 \nu + 10231808$$$$)/368640$$ $$\beta_{5}$$ $$=$$ $$($$$$-47 \nu^{15} - 110 \nu^{14} + 90 \nu^{13} + 528 \nu^{12} + 610 \nu^{11} - 1684 \nu^{10} - 8376 \nu^{9} - 11728 \nu^{8} + 5136 \nu^{7} + 52256 \nu^{6} + 60800 \nu^{5} - 73472 \nu^{4} - 350720 \nu^{3} - 537600 \nu^{2} + 172032 \nu + 1228800$$$$)/40960$$ $$\beta_{6}$$ $$=$$ $$($$$$91 \nu^{15} + 1260 \nu^{14} + 3590 \nu^{13} + 2316 \nu^{12} - 9170 \nu^{11} - 28288 \nu^{10} - 4872 \nu^{9} + 162144 \nu^{8} + 452592 \nu^{7} + 476352 \nu^{6} - 428800 \nu^{5} - 1854464 \nu^{4} - 1497600 \nu^{3} + 4352000 \nu^{2} + 14905344 \nu + 14909440$$$$)/122880$$ $$\beta_{7}$$ $$=$$ $$($$$$-751 \nu^{15} - 448 \nu^{14} + 6626 \nu^{13} + 14076 \nu^{12} + 5306 \nu^{11} - 54888 \nu^{10} - 156376 \nu^{9} - 19520 \nu^{8} + 641360 \nu^{7} + 1598464 \nu^{6} + 829440 \nu^{5} - 3026432 \nu^{4} - 7073280 \nu^{3} - 4517888 \nu^{2} + 22466560 \nu + 41107456$$$$)/368640$$ $$\beta_{8}$$ $$=$$ $$($$$$1063 \nu^{15} + 2242 \nu^{14} - 2666 \nu^{13} - 14184 \nu^{12} - 21122 \nu^{11} + 26652 \nu^{10} + 189400 \nu^{9} + 278960 \nu^{8} - 106640 \nu^{7} - 1269472 \nu^{6} - 1748352 \nu^{5} + 775424 \nu^{4} + 6971904 \nu^{3} + 12336128 \nu^{2} - 4268032 \nu - 26214400$$$$)/368640$$ $$\beta_{9}$$ $$=$$ $$($$$$-545 \nu^{15} - 1574 \nu^{14} - 302 \nu^{13} + 5256 \nu^{12} + 12838 \nu^{11} - 1188 \nu^{10} - 82544 \nu^{9} - 190768 \nu^{8} - 127664 \nu^{7} + 372128 \nu^{6} + 897600 \nu^{5} + 303872 \nu^{4} - 2511360 \nu^{3} - 7066624 \nu^{2} - 3770368 \nu + 6053888$$$$)/184320$$ $$\beta_{10}$$ $$=$$ $$($$$$-134 \nu^{15} - 20 \nu^{14} + 1153 \nu^{13} + 2232 \nu^{12} + 622 \nu^{11} - 9756 \nu^{10} - 25118 \nu^{9} + 1448 \nu^{8} + 113704 \nu^{7} + 257408 \nu^{6} + 102288 \nu^{5} - 502912 \nu^{4} - 1184256 \nu^{3} - 538624 \nu^{2} + 3969536 \nu + 6232064$$$$)/46080$$ $$\beta_{11}$$ $$=$$ $$($$$$1417 \nu^{15} + 4300 \nu^{14} + 1186 \nu^{13} - 13356 \nu^{12} - 35366 \nu^{11} + 528 \nu^{10} + 219304 \nu^{9} + 508256 \nu^{8} + 401488 \nu^{7} - 933184 \nu^{6} - 2421504 \nu^{5} - 922624 \nu^{4} + 6455808 \nu^{3} + 18839552 \nu^{2} + 11743232 \nu - 13975552$$$$)/368640$$ $$\beta_{12}$$ $$=$$ $$($$$$411 \nu^{15} - 178 \nu^{14} - 4586 \nu^{13} - 7776 \nu^{12} + 198 \nu^{11} + 38228 \nu^{10} + 84104 \nu^{9} - 37808 \nu^{8} - 478864 \nu^{7} - 985376 \nu^{6} - 282112 \nu^{5} + 2124032 \nu^{4} + 4176384 \nu^{3} + 1102848 \nu^{2} - 16609280 \nu - 25722880$$$$)/122880$$ $$\beta_{13}$$ $$=$$ $$($$$$-1229 \nu^{15} + 844 \nu^{14} + 13750 \nu^{13} + 23436 \nu^{12} - 2786 \nu^{11} - 118848 \nu^{10} - 248264 \nu^{9} + 154016 \nu^{8} + 1503088 \nu^{7} + 2985920 \nu^{6} + 638208 \nu^{5} - 6613504 \nu^{4} - 12980736 \nu^{3} - 1914880 \nu^{2} + 52957184 \nu + 79364096$$$$)/368640$$ $$\beta_{14}$$ $$=$$ $$($$$$-151 \nu^{15} - 271 \nu^{14} + 512 \nu^{13} + 1878 \nu^{12} + 2402 \nu^{11} - 4854 \nu^{10} - 25204 \nu^{9} - 30296 \nu^{8} + 32192 \nu^{7} + 186640 \nu^{6} + 194400 \nu^{5} - 199808 \nu^{4} - 975360 \nu^{3} - 1435136 \nu^{2} + 1352704 \nu + 4022272$$$$)/30720$$ $$\beta_{15}$$ $$=$$ $$($$$$-4331 \nu^{15} - 7634 \nu^{14} + 15322 \nu^{13} + 56088 \nu^{12} + 66634 \nu^{11} - 143484 \nu^{10} - 729800 \nu^{9} - 826480 \nu^{8} + 1085200 \nu^{7} + 5587424 \nu^{6} + 5662464 \nu^{5} - 6193408 \nu^{4} - 28205568 \nu^{3} - 39095296 \nu^{2} + 45215744 \nu + 128368640$$$$)/368640$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{11} - \beta_{9} - \beta_{8} - \beta_{6} + 2 \beta_{4} - 2 \beta_{3} - \beta_{1}$$$$)/8$$ $$\nu^{2}$$ $$=$$ $$($$$$2 \beta_{15} - 2 \beta_{14} - 2 \beta_{13} - \beta_{11} + \beta_{10} - \beta_{9} - 3 \beta_{7} + 2 \beta_{6} - 2 \beta_{4} - 4 \beta_{2} + 6$$$$)/8$$ $$\nu^{3}$$ $$=$$ $$($$$$-\beta_{15} - 3 \beta_{14} - 4 \beta_{12} - \beta_{11} - \beta_{10} + \beta_{9} + \beta_{8} + 3 \beta_{7} - 3 \beta_{6} - \beta_{5} + 10 \beta_{4} + 6 \beta_{3} - 6 \beta_{2} - 3 \beta_{1} + 6$$$$)/8$$ $$\nu^{4}$$ $$=$$ $$($$$$2 \beta_{15} - 3 \beta_{14} - \beta_{13} + \beta_{12} + \beta_{11} + 2 \beta_{10} - 4 \beta_{9} - \beta_{8} - 5 \beta_{7} + \beta_{5} + 22 \beta_{4} + 4 \beta_{3} - 12 \beta_{2} + 3 \beta_{1} + 8$$$$)/4$$ $$\nu^{5}$$ $$=$$ $$($$$$-4 \beta_{14} - 4 \beta_{13} - 8 \beta_{12} - \beta_{11} - 7 \beta_{9} - 7 \beta_{8} - 4 \beta_{7} - \beta_{6} + 6 \beta_{5} - 14 \beta_{4} - 6 \beta_{3} + 40 \beta_{2} - \beta_{1} - 40$$$$)/4$$ $$\nu^{6}$$ $$=$$ $$($$$$8 \beta_{15} + 2 \beta_{14} + 4 \beta_{13} + 2 \beta_{12} - 9 \beta_{11} - 13 \beta_{10} - 7 \beta_{9} + 8 \beta_{8} - 7 \beta_{7} + 8 \beta_{6} - 4 \beta_{5} - 2 \beta_{4} + 36 \beta_{3} - 48 \beta_{2} + 8 \beta_{1} - 102$$$$)/4$$ $$\nu^{7}$$ $$=$$ $$($$$$-11 \beta_{15} - 13 \beta_{14} - 4 \beta_{13} - 8 \beta_{12} + 3 \beta_{11} + 5 \beta_{10} + 57 \beta_{9} + 13 \beta_{8} - 19 \beta_{7} + 17 \beta_{6} - 11 \beta_{5} + 90 \beta_{4} - 42 \beta_{3} + 22 \beta_{2} + 21 \beta_{1} - 182$$$$)/4$$ $$\nu^{8}$$ $$=$$ $$($$$$-8 \beta_{15} + 3 \beta_{14} + 21 \beta_{13} + 9 \beta_{12} - 26 \beta_{11} + 3 \beta_{10} - 35 \beta_{9} + 9 \beta_{8} + 24 \beta_{7} - 12 \beta_{6} + 9 \beta_{5} - 16 \beta_{4} + 24 \beta_{3} + 72 \beta_{2} + 17 \beta_{1} - 146$$$$)/2$$ $$\nu^{9}$$ $$=$$ $$($$$$-19 \beta_{15} + 71 \beta_{14} + 12 \beta_{13} + 16 \beta_{12} + 42 \beta_{11} - 45 \beta_{10} + 38 \beta_{9} + 12 \beta_{8} + 23 \beta_{7} + 38 \beta_{6} + 35 \beta_{5} + 20 \beta_{4} + 108 \beta_{3} + 270 \beta_{2} + 30 \beta_{1} - 138$$$$)/2$$ $$\nu^{10}$$ $$=$$ $$($$$$50 \beta_{15} + 42 \beta_{14} + 38 \beta_{13} + 80 \beta_{12} - 5 \beta_{11} + 17 \beta_{10} + 107 \beta_{9} + 56 \beta_{8} + 53 \beta_{7} + 6 \beta_{6} - 128 \beta_{5} - 1018 \beta_{4} - 320 \beta_{3} - 116 \beta_{2} + 32 \beta_{1} - 786$$$$)/2$$ $$\nu^{11}$$ $$=$$ $$($$$$-215 \beta_{15} + 219 \beta_{14} + 48 \beta_{13} + 36 \beta_{12} - 143 \beta_{11} - 75 \beta_{10} + 175 \beta_{9} + 59 \beta_{8} - 47 \beta_{7} + 183 \beta_{6} - 199 \beta_{5} + 1006 \beta_{4} - 198 \beta_{3} + 446 \beta_{2} + 39 \beta_{1} + 1210$$$$)/2$$ $$\nu^{12}$$ $$=$$ $$-38 \beta_{15} - 13 \beta_{14} + 165 \beta_{13} + 227 \beta_{12} + 175 \beta_{11} + 42 \beta_{10} + 152 \beta_{9} + 17 \beta_{8} + 541 \beta_{7} - 132 \beta_{6} + 71 \beta_{5} - 238 \beta_{4} + 516 \beta_{3} + 180 \beta_{2} - 123 \beta_{1} + 8$$ $$\nu^{13}$$ $$=$$ $$192 \beta_{15} + 300 \beta_{14} - 452 \beta_{13} + 16 \beta_{12} + 161 \beta_{11} + 304 \beta_{10} - 353 \beta_{9} - 9 \beta_{8} - 52 \beta_{7} + 33 \beta_{6} + 10 \beta_{5} - 1394 \beta_{4} - 250 \beta_{3} - 312 \beta_{2} + 73 \beta_{1} + 3224$$ $$\nu^{14}$$ $$=$$ $$-280 \beta_{15} + 390 \beta_{14} + 484 \beta_{13} - 426 \beta_{12} + 25 \beta_{11} - 371 \beta_{10} + 495 \beta_{9} - 888 \beta_{8} + 63 \beta_{7} - 416 \beta_{6} - 692 \beta_{5} - 4926 \beta_{4} - 500 \beta_{3} - 112 \beta_{2} - 1096 \beta_{1} + 7174$$ $$\nu^{15}$$ $$=$$ $$211 \beta_{15} - 779 \beta_{14} - 1244 \beta_{13} + 920 \beta_{12} + 565 \beta_{11} + 11 \beta_{10} - 977 \beta_{9} - 1245 \beta_{8} - 781 \beta_{7} + 591 \beta_{6} - 605 \beta_{5} + 14390 \beta_{4} - 2838 \beta_{3} - 10182 \beta_{2} - 149 \beta_{1} + 4502$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/384\mathbb{Z}\right)^\times$$.

 $$n$$ $$127$$ $$133$$ $$257$$ $$\chi(n)$$ $$-1$$ $$-\beta_{4}$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
31.1
 0.125358 + 1.99607i 1.78012 − 0.911682i −1.25564 − 1.55672i −1.87459 + 0.697079i 1.80398 + 0.863518i −1.96679 − 0.362960i −0.455024 − 1.94755i 1.84258 − 0.777752i 0.125358 − 1.99607i 1.78012 + 0.911682i −1.25564 + 1.55672i −1.87459 − 0.697079i 1.80398 − 0.863518i −1.96679 + 0.362960i −0.455024 + 1.94755i 1.84258 + 0.777752i
0 −1.22474 1.22474i 0 −3.32679 3.32679i 0 −4.04088 0 3.00000i 0
31.2 0 −1.22474 1.22474i 0 −1.00772 1.00772i 0 10.0236 0 3.00000i 0
31.3 0 −1.22474 1.22474i 0 −0.909023 0.909023i 0 −0.654713 0 3.00000i 0
31.4 0 −1.22474 1.22474i 0 5.24354 + 5.24354i 0 −5.32796 0 3.00000i 0
31.5 0 1.22474 + 1.22474i 0 −6.49473 6.49473i 0 3.94273 0 3.00000i 0
31.6 0 1.22474 + 1.22474i 0 −1.69930 1.69930i 0 −5.74280 0 3.00000i 0
31.7 0 1.22474 + 1.22474i 0 3.40572 + 3.40572i 0 12.1303 0 3.00000i 0
31.8 0 1.22474 + 1.22474i 0 4.78830 + 4.78830i 0 −10.3302 0 3.00000i 0
223.1 0 −1.22474 + 1.22474i 0 −3.32679 + 3.32679i 0 −4.04088 0 3.00000i 0
223.2 0 −1.22474 + 1.22474i 0 −1.00772 + 1.00772i 0 10.0236 0 3.00000i 0
223.3 0 −1.22474 + 1.22474i 0 −0.909023 + 0.909023i 0 −0.654713 0 3.00000i 0
223.4 0 −1.22474 + 1.22474i 0 5.24354 5.24354i 0 −5.32796 0 3.00000i 0
223.5 0 1.22474 1.22474i 0 −6.49473 + 6.49473i 0 3.94273 0 3.00000i 0
223.6 0 1.22474 1.22474i 0 −1.69930 + 1.69930i 0 −5.74280 0 3.00000i 0
223.7 0 1.22474 1.22474i 0 3.40572 3.40572i 0 12.1303 0 3.00000i 0
223.8 0 1.22474 1.22474i 0 4.78830 4.78830i 0 −10.3302 0 3.00000i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 223.8 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
16.f odd 4 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 384.3.l.a 16
3.b odd 2 1 1152.3.m.f 16
4.b odd 2 1 384.3.l.b 16
8.b even 2 1 48.3.l.a 16
8.d odd 2 1 192.3.l.a 16
12.b even 2 1 1152.3.m.c 16
16.e even 4 1 192.3.l.a 16
16.e even 4 1 384.3.l.b 16
16.f odd 4 1 48.3.l.a 16
16.f odd 4 1 inner 384.3.l.a 16
24.f even 2 1 576.3.m.c 16
24.h odd 2 1 144.3.m.c 16
48.i odd 4 1 576.3.m.c 16
48.i odd 4 1 1152.3.m.c 16
48.k even 4 1 144.3.m.c 16
48.k even 4 1 1152.3.m.f 16

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
48.3.l.a 16 8.b even 2 1
48.3.l.a 16 16.f odd 4 1
144.3.m.c 16 24.h odd 2 1
144.3.m.c 16 48.k even 4 1
192.3.l.a 16 8.d odd 2 1
192.3.l.a 16 16.e even 4 1
384.3.l.a 16 1.a even 1 1 trivial
384.3.l.a 16 16.f odd 4 1 inner
384.3.l.b 16 4.b odd 2 1
384.3.l.b 16 16.e even 4 1
576.3.m.c 16 24.f even 2 1
576.3.m.c 16 48.i odd 4 1
1152.3.m.c 16 12.b even 2 1
1152.3.m.c 16 48.i odd 4 1
1152.3.m.f 16 3.b odd 2 1
1152.3.m.f 16 48.k even 4 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{7}^{8} - 224 T_{7}^{6} - 448 T_{7}^{5} + 13704 T_{7}^{4} + 53248 T_{7}^{3} - 136576 T_{7}^{2} - 720640 T_{7} - 400880$$ acting on $$S_{3}^{\mathrm{new}}(384, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{16}$$
$3$ $$( 9 + T^{4} )^{4}$$
$5$ $$2117472256 + 5171462144 T + 6315081728 T^{2} + 4258177024 T^{3} + 1663893504 T^{4} + 283883520 T^{5} + 12517376 T^{6} + 4026368 T^{7} + 6221952 T^{8} + 518400 T^{9} + 512 T^{10} - 8064 T^{11} + 6656 T^{12} - 32 T^{13} + T^{16}$$
$7$ $$( -400880 - 720640 T - 136576 T^{2} + 53248 T^{3} + 13704 T^{4} - 448 T^{5} - 224 T^{6} + T^{8} )^{2}$$
$11$ $$25620118503424 - 29481536323584 T + 16962470019072 T^{2} + 22391525736448 T^{3} + 10536909537280 T^{4} + 2206640635904 T^{5} + 269438418944 T^{6} + 21404090368 T^{7} + 3292530688 T^{8} + 584835072 T^{9} + 68780032 T^{10} + 4242432 T^{11} + 151552 T^{12} + 4608 T^{13} + 512 T^{14} + 32 T^{15} + T^{16}$$
$13$ $$27957043852960000 + 3777806164172800 T + 255245502513152 T^{2} - 75155778920448 T^{3} + 18426489989376 T^{4} + 1347500722176 T^{5} + 114873139200 T^{6} - 37570686976 T^{7} + 5634806368 T^{8} - 8847872 T^{9} + 5120000 T^{10} - 1717760 T^{11} + 271248 T^{12} - 3200 T^{13} + T^{16}$$
$17$ $$( 816881920 + 9390080 T - 53986304 T^{2} + 1044480 T^{3} + 508448 T^{4} - 2944 T^{5} - 1344 T^{6} + T^{8} )^{2}$$
$19$ $$10598900522979229696 - 1221045122401566720 T + 70335181828915200 T^{2} + 89104253114777600 T^{3} + 23939487244337152 T^{4} + 1508224010715136 T^{5} + 41927575470080 T^{6} - 1219751600128 T^{7} + 149480318464 T^{8} + 9271351296 T^{9} + 294903808 T^{10} - 15014400 T^{11} + 559552 T^{12} + 14208 T^{13} + 512 T^{14} - 32 T^{15} + T^{16}$$
$23$ $$( -35037900800 + 425492480 T + 777799680 T^{2} + 42348544 T^{3} - 1089152 T^{4} - 109056 T^{5} - 736 T^{6} + 64 T^{7} + T^{8} )^{2}$$
$29$ $$56446323002698240000 + 10320873959271219200 T + 943555165480583168 T^{2} - 1809064899469205504 T^{3} + 1336868519252525056 T^{4} - 34131461984319488 T^{5} + 401836469485568 T^{6} + 22033099003904 T^{7} + 4665816875136 T^{8} - 92505709824 T^{9} + 1050931712 T^{10} + 64121984 T^{11} + 3957248 T^{12} - 45280 T^{13} + 512 T^{14} + 32 T^{15} + T^{16}$$
$31$ $$41\!\cdots\!00$$$$+$$$$88\!\cdots\!60$$$$T^{2} + 7691970571770562816 T^{4} + 26353243415873536 T^{6} + 46734989650528 T^{8} + 46823575040 T^{10} + 26696080 T^{12} + 8064 T^{14} + T^{16}$$
$37$ $$38\!\cdots\!04$$$$+$$$$73\!\cdots\!12$$$$T + 70869372871016480768 T^{2} - 2884347249537718272 T^{3} + 703119961976168704 T^{4} + 97908324877949952 T^{5} + 6893443090702336 T^{6} - 6367638655488 T^{7} - 968313092000 T^{8} + 28675020544 T^{9} + 14234605568 T^{10} - 224961664 T^{11} + 1620496 T^{12} - 14528 T^{13} + 4608 T^{14} - 96 T^{15} + T^{16}$$
$41$ $$94\!\cdots\!00$$$$+$$$$70\!\cdots\!60$$$$T^{2} +$$$$21\!\cdots\!56$$$$T^{4} + 365640212557922304 T^{6} + 356212041131520 T^{8} + 207527260160 T^{10} + 70874688 T^{12} + 13056 T^{14} + T^{16}$$
$43$ $$92\!\cdots\!00$$$$+$$$$28\!\cdots\!80$$$$T +$$$$44\!\cdots\!68$$$$T^{2} +$$$$32\!\cdots\!48$$$$T^{3} +$$$$14\!\cdots\!24$$$$T^{4} + 5239225685693136896 T^{5} + 245950489383796736 T^{6} + 12945845326176256 T^{7} + 537795837003264 T^{8} + 15508493912064 T^{9} + 306484011008 T^{10} + 4149187072 T^{11} + 46122944 T^{12} + 682624 T^{13} + 12800 T^{14} + 160 T^{15} + T^{16}$$
$47$ $$11\!\cdots\!00$$$$+$$$$37\!\cdots\!36$$$$T^{2} + 4369756732867477504 T^{4} + 22735051785764864 T^{6} + 57150256209920 T^{8} + 69921902592 T^{10} + 41678592 T^{12} + 11200 T^{14} + T^{16}$$
$53$ $$96\!\cdots\!00$$$$-$$$$20\!\cdots\!60$$$$T +$$$$21\!\cdots\!52$$$$T^{2} -$$$$19\!\cdots\!20$$$$T^{3} + 91449263140998676480 T^{4} - 5591746172921030656 T^{5} + 1685959386162102272 T^{6} - 18190042770372608 T^{7} + 100041121243264 T^{8} - 1029094838016 T^{9} + 299678376448 T^{10} - 3156153984 T^{11} + 16960000 T^{12} - 153504 T^{13} + 12800 T^{14} - 160 T^{15} + T^{16}$$
$59$ $$23\!\cdots\!00$$$$+$$$$18\!\cdots\!00$$$$T +$$$$69\!\cdots\!32$$$$T^{2} -$$$$16\!\cdots\!76$$$$T^{3} + 97922953659373584384 T^{4} - 714303909442617344 T^{5} + 215142153171501056 T^{6} - 32123905497890816 T^{7} + 2068957458595840 T^{8} - 72789256044544 T^{9} + 1586277384192 T^{10} - 20765343744 T^{11} + 158699520 T^{12} - 675840 T^{13} + 8192 T^{14} - 128 T^{15} + T^{16}$$
$61$ $$12\!\cdots\!00$$$$+$$$$90\!\cdots\!60$$$$T +$$$$33\!\cdots\!72$$$$T^{2} -$$$$22\!\cdots\!24$$$$T^{3} +$$$$35\!\cdots\!44$$$$T^{4} +$$$$20\!\cdots\!88$$$$T^{5} + 735866880468475904 T^{6} - 49175921370646016 T^{7} + 3370685859892320 T^{8} + 12153910400256 T^{9} + 42705459200 T^{10} - 2795215232 T^{11} + 115399952 T^{12} + 157120 T^{13} + 512 T^{14} - 32 T^{15} + T^{16}$$
$67$ $$21\!\cdots\!96$$$$-$$$$11\!\cdots\!04$$$$T +$$$$30\!\cdots\!48$$$$T^{2} +$$$$28\!\cdots\!28$$$$T^{3} +$$$$13\!\cdots\!96$$$$T^{4} - 621625262253015040 T^{5} + 579595725034225664 T^{6} + 50262831949414400 T^{7} + 2061397582544896 T^{8} + 36863866568704 T^{9} + 475717435392 T^{10} + 9110568960 T^{11} + 255286272 T^{12} + 4611072 T^{13} + 51200 T^{14} + 320 T^{15} + T^{16}$$
$71$ $$( 290924400640 - 109021003776 T + 16299499520 T^{2} - 1284915200 T^{3} + 59283584 T^{4} - 1659392 T^{5} + 27776 T^{6} - 256 T^{7} + T^{8} )^{2}$$
$73$ $$98\!\cdots\!16$$$$+$$$$13\!\cdots\!52$$$$T^{2} +$$$$45\!\cdots\!44$$$$T^{4} + 53883999480140791808 T^{6} + 25520342188187648 T^{8} + 5899646435328 T^{10} + 709382400 T^{12} + 42496 T^{14} + T^{16}$$
$79$ $$18\!\cdots\!00$$$$+$$$$25\!\cdots\!40$$$$T^{2} +$$$$11\!\cdots\!36$$$$T^{4} + 1719443636116761600 T^{6} + 4201068991559776 T^{8} + 2270484756224 T^{10} + 458363920 T^{12} + 36928 T^{14} + T^{16}$$
$83$ $$14\!\cdots\!76$$$$-$$$$42\!\cdots\!84$$$$T +$$$$63\!\cdots\!28$$$$T^{2} +$$$$13\!\cdots\!24$$$$T^{3} +$$$$12\!\cdots\!56$$$$T^{4} +$$$$39\!\cdots\!64$$$$T^{5} + 6381120662228434944 T^{6} - 5903701162754048 T^{7} + 105517061441536 T^{8} + 17298459828224 T^{9} + 474865041408 T^{10} - 3104705536 T^{11} + 3373056 T^{12} + 206336 T^{13} + 12800 T^{14} - 160 T^{15} + T^{16}$$
$89$ $$52\!\cdots\!00$$$$+$$$$68\!\cdots\!60$$$$T^{2} +$$$$23\!\cdots\!56$$$$T^{4} + 30661655762820161536 T^{6} + 18282328190707200 T^{8} + 5304960677376 T^{10} + 743027648 T^{12} + 45728 T^{14} + T^{16}$$
$97$ $$( 409778579046400 + 2337541980160 T - 1019025981440 T^{2} - 1982349312 T^{3} + 383621120 T^{4} + 116224 T^{5} - 37056 T^{6} + T^{8} )^{2}$$