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 $2$

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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:

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 \(S_{3}^{\mathrm{new}}(192, [\chi])\).