# Properties

 Label 192.3.l.a Level $192$ Weight $3$ Character orbit 192.l Analytic conductor $5.232$ Analytic rank $0$ Dimension $16$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$192 = 2^{6} \cdot 3$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 192.l (of order $$4$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$5.23162107572$$ 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 $$\left(\mathbb{Z}/192\mathbb{Z}\right)^\times$$.

 $$n$$ $$65$$ $$127$$ $$133$$ $$\chi(n)$$ $$1$$ $$-1$$ $$-\beta_{4}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
79.1
 −1.87459 + 0.697079i −1.25564 − 1.55672i 1.78012 − 0.911682i 0.125358 + 1.99607i 1.84258 − 0.777752i −0.455024 − 1.94755i −1.96679 − 0.362960i 1.80398 + 0.863518i −1.87459 − 0.697079i −1.25564 + 1.55672i 1.78012 + 0.911682i 0.125358 − 1.99607i 1.84258 + 0.777752i −0.455024 + 1.94755i −1.96679 + 0.362960i 1.80398 − 0.863518i
0 −1.22474 1.22474i 0 −5.24354 5.24354i 0 5.32796 0 3.00000i 0
79.2 0 −1.22474 1.22474i 0 0.909023 + 0.909023i 0 0.654713 0 3.00000i 0
79.3 0 −1.22474 1.22474i 0 1.00772 + 1.00772i 0 −10.0236 0 3.00000i 0
79.4 0 −1.22474 1.22474i 0 3.32679 + 3.32679i 0 4.04088 0 3.00000i 0
79.5 0 1.22474 + 1.22474i 0 −4.78830 4.78830i 0 10.3302 0 3.00000i 0
79.6 0 1.22474 + 1.22474i 0 −3.40572 3.40572i 0 −12.1303 0 3.00000i 0
79.7 0 1.22474 + 1.22474i 0 1.69930 + 1.69930i 0 5.74280 0 3.00000i 0
79.8 0 1.22474 + 1.22474i 0 6.49473 + 6.49473i 0 −3.94273 0 3.00000i 0
175.1 0 −1.22474 + 1.22474i 0 −5.24354 + 5.24354i 0 5.32796 0 3.00000i 0
175.2 0 −1.22474 + 1.22474i 0 0.909023 0.909023i 0 0.654713 0 3.00000i 0
175.3 0 −1.22474 + 1.22474i 0 1.00772 1.00772i 0 −10.0236 0 3.00000i 0
175.4 0 −1.22474 + 1.22474i 0 3.32679 3.32679i 0 4.04088 0 3.00000i 0
175.5 0 1.22474 1.22474i 0 −4.78830 + 4.78830i 0 10.3302 0 3.00000i 0
175.6 0 1.22474 1.22474i 0 −3.40572 + 3.40572i 0 −12.1303 0 3.00000i 0
175.7 0 1.22474 1.22474i 0 1.69930 1.69930i 0 5.74280 0 3.00000i 0
175.8 0 1.22474 1.22474i 0 6.49473 6.49473i 0 −3.94273 0 3.00000i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 175.8 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 192.3.l.a 16
3.b odd 2 1 576.3.m.c 16
4.b odd 2 1 48.3.l.a 16
8.b even 2 1 384.3.l.b 16
8.d odd 2 1 384.3.l.a 16
12.b even 2 1 144.3.m.c 16
16.e even 4 1 48.3.l.a 16
16.e even 4 1 384.3.l.a 16
16.f odd 4 1 inner 192.3.l.a 16
16.f odd 4 1 384.3.l.b 16
24.f even 2 1 1152.3.m.f 16
24.h odd 2 1 1152.3.m.c 16
48.i odd 4 1 144.3.m.c 16
48.i odd 4 1 1152.3.m.f 16
48.k even 4 1 576.3.m.c 16
48.k even 4 1 1152.3.m.c 16

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

## Hecke kernels

This newform subspace is the entire newspace $$S_{3}^{\mathrm{new}}(192, [\chi])$$.