Properties

Label 384.3.i.c
Level $384$
Weight $3$
Character orbit 384.i
Analytic conductor $10.463$
Analytic rank $0$
Dimension $20$
CM no
Inner twists $4$

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Newspace parameters

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

Newform invariants

Self dual: no
Analytic conductor: \(10.4632421514\)
Analytic rank: \(0\)
Dimension: \(20\)
Relative dimension: \(10\) over \(\Q(i)\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} - \cdots)\)
Defining polynomial: \(x^{20} - 2 x^{18} + 6 x^{16} - 24 x^{14} - 24 x^{12} + 1216 x^{10} - 384 x^{8} - 6144 x^{6} + 24576 x^{4} - 131072 x^{2} + 1048576\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{23} \)
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 \(1,\beta_1,\ldots,\beta_{19}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -\beta_{5} q^{3} + ( \beta_{8} - \beta_{9} - \beta_{10} ) q^{5} -\beta_{6} q^{7} -\beta_{16} q^{9} +O(q^{10})\) \( q -\beta_{5} q^{3} + ( \beta_{8} - \beta_{9} - \beta_{10} ) q^{5} -\beta_{6} q^{7} -\beta_{16} q^{9} + ( \beta_{10} - \beta_{11} + \beta_{13} + \beta_{15} - \beta_{18} ) q^{11} + ( -5 + 5 \beta_{1} - \beta_{3} + \beta_{4} + \beta_{5} ) q^{13} + ( 6 + \beta_{12} - \beta_{14} + \beta_{15} ) q^{15} + ( \beta_{2} - \beta_{4} + \beta_{5} + \beta_{8} - \beta_{9} + \beta_{12} + \beta_{13} - \beta_{14} ) q^{17} + ( -4 + 4 \beta_{1} + 3 \beta_{4} + 3 \beta_{5} - \beta_{6} - \beta_{7} - \beta_{12} - \beta_{14} + \beta_{16} - \beta_{19} ) q^{19} + ( -4 - 4 \beta_{1} + \beta_{6} - \beta_{7} - 2 \beta_{8} - 2 \beta_{9} - \beta_{10} - 3 \beta_{11} + 4 \beta_{13} - \beta_{14} + \beta_{15} - \beta_{16} - \beta_{18} ) q^{21} + ( 3 \beta_{2} - \beta_{4} + \beta_{5} - \beta_{8} + \beta_{9} + 3 \beta_{10} - 3 \beta_{13} - 3 \beta_{17} ) q^{23} + ( 3 \beta_{1} - \beta_{3} - \beta_{4} - \beta_{5} - \beta_{8} - \beta_{9} - \beta_{11} - \beta_{16} + \beta_{19} ) q^{25} + ( -\beta_{9} - 3 \beta_{10} - 3 \beta_{11} - 3 \beta_{13} ) q^{27} + ( 2 \beta_{2} - 2 \beta_{3} + \beta_{4} - \beta_{5} - \beta_{12} + \beta_{14} - 2 \beta_{15} - \beta_{16} + \beta_{17} - 2 \beta_{18} - \beta_{19} ) q^{29} + ( 4 + 3 \beta_{3} - \beta_{7} - 3 \beta_{11} - 2 \beta_{12} - 2 \beta_{14} ) q^{31} + ( 1 + \beta_{2} - 3 \beta_{3} + 2 \beta_{4} + 2 \beta_{5} - 2 \beta_{7} - 2 \beta_{8} - 2 \beta_{9} - 4 \beta_{10} + 3 \beta_{11} - \beta_{12} + \beta_{13} - 2 \beta_{15} - 4 \beta_{17} ) q^{33} + ( 7 \beta_{2} + \beta_{3} - 3 \beta_{4} + 3 \beta_{5} + 2 \beta_{12} - 2 \beta_{14} + \beta_{15} + 2 \beta_{16} - 5 \beta_{17} + \beta_{18} + 2 \beta_{19} ) q^{35} + ( 7 + 7 \beta_{1} + 2 \beta_{6} - 2 \beta_{7} + \beta_{8} + \beta_{9} + 3 \beta_{11} - \beta_{12} - \beta_{14} - \beta_{16} + \beta_{19} ) q^{37} + ( 10 \beta_{1} - \beta_{2} + \beta_{4} + 5 \beta_{5} - \beta_{6} + \beta_{8} + 5 \beta_{9} - \beta_{10} + \beta_{13} + \beta_{16} + \beta_{17} - \beta_{18} + \beta_{19} ) q^{39} + ( \beta_{2} - 2 \beta_{3} - \beta_{4} + \beta_{5} - \beta_{8} + \beta_{9} - 2 \beta_{10} - 2 \beta_{11} - \beta_{13} - 2 \beta_{16} + 2 \beta_{17} - 4 \beta_{18} - 2 \beta_{19} ) q^{41} + ( 12 + 12 \beta_{1} - \beta_{6} + \beta_{7} + 3 \beta_{8} + 3 \beta_{9} + 2 \beta_{11} - \beta_{12} - \beta_{14} - \beta_{16} + \beta_{19} ) q^{43} + ( -1 + \beta_{1} - 2 \beta_{2} - 7 \beta_{4} - 3 \beta_{5} - \beta_{6} - \beta_{7} + \beta_{12} - \beta_{15} - 7 \beta_{17} - \beta_{18} + \beta_{19} ) q^{45} + ( 5 \beta_{2} + \beta_{3} - \beta_{4} + \beta_{5} + \beta_{8} - \beta_{9} - \beta_{10} - \beta_{11} + 2 \beta_{12} + 5 \beta_{13} - 2 \beta_{14} + 2 \beta_{15} - \beta_{17} ) q^{47} + ( -19 + 3 \beta_{3} - \beta_{4} - \beta_{5} - 4 \beta_{7} + \beta_{8} + \beta_{9} - 3 \beta_{11} - \beta_{12} - \beta_{14} ) q^{49} + ( 8 - 8 \beta_{1} - 5 \beta_{2} + 3 \beta_{3} - 7 \beta_{4} + \beta_{5} - \beta_{6} - \beta_{7} - \beta_{12} - \beta_{14} + \beta_{16} - \beta_{17} - \beta_{19} ) q^{51} + ( -3 \beta_{8} + 3 \beta_{9} + \beta_{10} + 2 \beta_{12} - 6 \beta_{13} - 2 \beta_{14} - 2 \beta_{16} - 2 \beta_{19} ) q^{53} + ( -4 \beta_{1} - 3 \beta_{3} - 2 \beta_{4} - 2 \beta_{5} - 2 \beta_{8} - 2 \beta_{9} - 3 \beta_{11} + 2 \beta_{16} - 2 \beta_{19} ) q^{55} + ( 19 \beta_{1} - 7 \beta_{2} + 7 \beta_{4} + 3 \beta_{5} + 2 \beta_{6} + 7 \beta_{8} + 3 \beta_{9} + 2 \beta_{10} + 7 \beta_{13} + \beta_{16} - 2 \beta_{17} - 2 \beta_{18} ) q^{57} + ( -3 \beta_{8} + 3 \beta_{9} + 12 \beta_{10} + 2 \beta_{12} - 2 \beta_{14} - 2 \beta_{16} - 2 \beta_{19} ) q^{59} + ( 7 - 7 \beta_{1} - 3 \beta_{3} + 7 \beta_{4} + 7 \beta_{5} - 2 \beta_{6} - 2 \beta_{7} + \beta_{12} + \beta_{14} - \beta_{16} + \beta_{19} ) q^{61} + ( -16 - 7 \beta_{2} - 3 \beta_{3} + \beta_{4} + 5 \beta_{5} - \beta_{7} - \beta_{8} - 5 \beta_{9} - 5 \beta_{10} + 3 \beta_{11} - 4 \beta_{12} - 7 \beta_{13} - 5 \beta_{17} ) q^{63} + ( -\beta_{2} + 3 \beta_{4} - 3 \beta_{5} - 3 \beta_{8} + 3 \beta_{9} + 6 \beta_{10} - 2 \beta_{12} - \beta_{13} + 2 \beta_{14} + 6 \beta_{17} ) q^{65} + ( 16 - 16 \beta_{1} - 2 \beta_{3} + \beta_{4} + \beta_{5} + 2 \beta_{12} + 2 \beta_{14} - 2 \beta_{16} + 2 \beta_{19} ) q^{67} + ( 3 + 3 \beta_{1} - 3 \beta_{6} + 3 \beta_{7} + 6 \beta_{8} - 6 \beta_{10} + 6 \beta_{13} + \beta_{14} - 3 \beta_{15} + \beta_{16} + 3 \beta_{18} ) q^{69} + ( -\beta_{3} + 4 \beta_{4} - 4 \beta_{5} + 4 \beta_{8} - 4 \beta_{9} + 4 \beta_{10} - \beta_{11} + 2 \beta_{16} - 4 \beta_{17} - 2 \beta_{18} + 2 \beta_{19} ) q^{71} + ( -20 \beta_{1} - 3 \beta_{3} - 7 \beta_{4} - 7 \beta_{5} + 4 \beta_{6} - 7 \beta_{8} - 7 \beta_{9} - 3 \beta_{11} ) q^{73} + ( -8 - 8 \beta_{1} - \beta_{6} + \beta_{7} - 7 \beta_{8} + 2 \beta_{9} - 5 \beta_{10} - 3 \beta_{11} - \beta_{12} - \beta_{13} - \beta_{14} + \beta_{15} - \beta_{16} - \beta_{18} + \beta_{19} ) q^{75} + ( 6 \beta_{2} + 4 \beta_{3} - 8 \beta_{4} + 8 \beta_{5} + 2 \beta_{12} - 2 \beta_{14} + 4 \beta_{15} + 2 \beta_{16} + 14 \beta_{17} + 4 \beta_{18} + 2 \beta_{19} ) q^{77} + ( -20 + \beta_{3} - 2 \beta_{4} - 2 \beta_{5} + \beta_{7} + 2 \beta_{8} + 2 \beta_{9} - \beta_{11} + 4 \beta_{12} + 4 \beta_{14} ) q^{79} + ( -3 + 6 \beta_{2} - 6 \beta_{4} + 6 \beta_{7} + 6 \beta_{8} - 6 \beta_{10} + 3 \beta_{12} + 6 \beta_{13} - \beta_{14} + 6 \beta_{15} - 6 \beta_{17} ) q^{81} + ( 9 \beta_{2} - \beta_{3} + 8 \beta_{4} - 8 \beta_{5} - 4 \beta_{12} + 4 \beta_{14} - \beta_{15} - 4 \beta_{16} + \beta_{17} - \beta_{18} - 4 \beta_{19} ) q^{83} + ( -4 - 4 \beta_{1} - 4 \beta_{6} + 4 \beta_{7} - 6 \beta_{8} - 6 \beta_{9} + 2 \beta_{11} ) q^{85} + ( 6 \beta_{1} - 3 \beta_{2} + 3 \beta_{3} + 9 \beta_{4} - 5 \beta_{5} + 9 \beta_{8} - 5 \beta_{9} - 15 \beta_{10} + 3 \beta_{11} + 3 \beta_{13} - \beta_{16} + 15 \beta_{17} + \beta_{18} - 5 \beta_{19} ) q^{87} + ( 14 \beta_{2} + 4 \beta_{3} + 8 \beta_{4} - 8 \beta_{5} + 8 \beta_{8} - 8 \beta_{9} - 6 \beta_{10} + 4 \beta_{11} - 14 \beta_{13} + 3 \beta_{16} + 6 \beta_{17} + 8 \beta_{18} + 3 \beta_{19} ) q^{89} + ( 4 + 4 \beta_{1} + \beta_{6} - \beta_{7} - 2 \beta_{8} - 2 \beta_{9} + 6 \beta_{11} + 3 \beta_{12} + 3 \beta_{14} + 3 \beta_{16} - 3 \beta_{19} ) q^{91} + ( 7 - 7 \beta_{1} - 4 \beta_{2} + 3 \beta_{3} + 10 \beta_{4} - 4 \beta_{5} + 4 \beta_{6} + 4 \beta_{7} - 3 \beta_{12} + \beta_{14} + 2 \beta_{15} - \beta_{16} - 11 \beta_{17} + 2 \beta_{18} - 3 \beta_{19} ) q^{93} + ( 11 \beta_{2} + 5 \beta_{4} - 5 \beta_{5} - 5 \beta_{8} + 5 \beta_{9} + 13 \beta_{10} - 4 \beta_{12} + 11 \beta_{13} + 4 \beta_{14} + 13 \beta_{17} ) q^{95} + ( 18 + 2 \beta_{3} + 6 \beta_{4} + 6 \beta_{5} + 8 \beta_{7} - 6 \beta_{8} - 6 \beta_{9} - 2 \beta_{11} + \beta_{12} + \beta_{14} ) q^{97} + ( -24 + 24 \beta_{1} - 18 \beta_{2} + 9 \beta_{4} - 9 \beta_{5} + 4 \beta_{12} + \beta_{15} + 18 \beta_{17} + \beta_{18} + 4 \beta_{19} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20q - 6q^{3} + O(q^{10}) \) \( 20q - 6q^{3} - 92q^{13} + 116q^{15} - 52q^{19} - 48q^{21} + 18q^{27} + 80q^{31} + 60q^{33} + 116q^{37} + 172q^{43} - 60q^{45} - 364q^{49} + 128q^{51} + 244q^{61} - 296q^{63} + 356q^{67} + 20q^{69} - 146q^{75} - 384q^{79} - 188q^{81} - 48q^{85} + 136q^{91} + 132q^{93} + 472q^{97} - 452q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{20} - 2 x^{18} + 6 x^{16} - 24 x^{14} - 24 x^{12} + 1216 x^{10} - 384 x^{8} - 6144 x^{6} + 24576 x^{4} - 131072 x^{2} + 1048576\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( -33 \nu^{18} - 42 \nu^{16} + 1090 \nu^{14} - 528 \nu^{12} - 11816 \nu^{10} + 6496 \nu^{8} - 64512 \nu^{6} + 1111040 \nu^{4} + 1482752 \nu^{2} - 16842752 \)\()/10158080\)
\(\beta_{2}\)\(=\)\((\)\( -3 \nu^{19} - 90 \nu^{17} + 430 \nu^{15} + 1288 \nu^{13} - 5816 \nu^{11} - 3904 \nu^{9} - 19328 \nu^{7} + 278528 \nu^{5} + 2621440 \nu^{3} - 1638400 \nu \)\()/4063232\)
\(\beta_{3}\)\(=\)\((\)\( 4 \nu^{18} + 51 \nu^{16} - 74 \nu^{14} - 790 \nu^{12} + 2736 \nu^{10} + 5240 \nu^{8} + 45664 \nu^{6} + 11520 \nu^{4} - 942080 \nu^{2} + 2064384 \)\()/507904\)
\(\beta_{4}\)\(=\)\((\)\(-4 \nu^{19} - 112 \nu^{18} - 101 \nu^{17} + 542 \nu^{16} - 410 \nu^{15} - 980 \nu^{14} + 1946 \nu^{13} - 5372 \nu^{12} - 368 \nu^{11} + 13216 \nu^{10} + 24248 \nu^{9} - 64976 \nu^{8} - 17696 \nu^{7} + 29632 \nu^{6} - 398080 \nu^{5} + 750080 \nu^{4} + 1181696 \nu^{3} - 7708672 \nu^{2} + 753664 \nu + 18481152\)\()/10158080\)
\(\beta_{5}\)\(=\)\((\)\(4 \nu^{19} - 112 \nu^{18} + 101 \nu^{17} + 542 \nu^{16} + 410 \nu^{15} - 980 \nu^{14} - 1946 \nu^{13} - 5372 \nu^{12} + 368 \nu^{11} + 13216 \nu^{10} - 24248 \nu^{9} - 64976 \nu^{8} + 17696 \nu^{7} + 29632 \nu^{6} + 398080 \nu^{5} + 750080 \nu^{4} - 1181696 \nu^{3} - 7708672 \nu^{2} - 753664 \nu + 18481152\)\()/10158080\)
\(\beta_{6}\)\(=\)\((\)\( -187 \nu^{18} - 858 \nu^{16} + 6590 \nu^{14} - 21592 \nu^{12} - 90104 \nu^{10} - 159936 \nu^{8} - 782208 \nu^{6} + 3809280 \nu^{4} + 4169728 \nu^{2} - 139460608 \)\()/10158080\)
\(\beta_{7}\)\(=\)\((\)\( -43 \nu^{18} - 106 \nu^{16} - 258 \nu^{14} - 1400 \nu^{12} + 7432 \nu^{10} - 17984 \nu^{8} - 125824 \nu^{6} + 706560 \nu^{4} - 1171456 \nu^{2} + 1966080 \)\()/2031616\)
\(\beta_{8}\)\(=\)\((\)\(79 \nu^{19} + 592 \nu^{18} - 34 \nu^{17} - 3112 \nu^{16} - 2030 \nu^{15} + 2320 \nu^{14} - 4896 \nu^{13} + 18832 \nu^{12} - 12392 \nu^{11} - 77056 \nu^{10} + 68512 \nu^{9} + 518336 \nu^{8} + 78336 \nu^{7} - 1806592 \nu^{6} + 10240 \nu^{5} - 8714240 \nu^{4} - 6332416 \nu^{3} + 46989312 \nu^{2} + 10878976 \nu - 96468992\)\()/40632320\)
\(\beta_{9}\)\(=\)\((\)\(-79 \nu^{19} + 592 \nu^{18} + 34 \nu^{17} - 3112 \nu^{16} + 2030 \nu^{15} + 2320 \nu^{14} + 4896 \nu^{13} + 18832 \nu^{12} + 12392 \nu^{11} - 77056 \nu^{10} - 68512 \nu^{9} + 518336 \nu^{8} - 78336 \nu^{7} - 1806592 \nu^{6} - 10240 \nu^{5} - 8714240 \nu^{4} + 6332416 \nu^{3} + 46989312 \nu^{2} - 10878976 \nu - 96468992\)\()/40632320\)
\(\beta_{10}\)\(=\)\((\)\( -117 \nu^{19} + 282 \nu^{17} + 2210 \nu^{15} - 8552 \nu^{13} - 18184 \nu^{11} + 45504 \nu^{9} - 161408 \nu^{7} + 3051520 \nu^{5} - 7176192 \nu^{3} - 18546688 \nu \)\()/20316160\)
\(\beta_{11}\)\(=\)\((\)\( 29 \nu^{18} - 100 \nu^{16} - 22 \nu^{14} + 1180 \nu^{12} - 280 \nu^{10} + 11696 \nu^{8} + 6976 \nu^{6} - 372224 \nu^{4} + 3276800 \nu^{2} - 327680 \)\()/1015808\)
\(\beta_{12}\)\(=\)\((\)\(-53 \nu^{19} - 48 \nu^{18} + 394 \nu^{17} + 320 \nu^{16} - 670 \nu^{15} - 2272 \nu^{14} - 392 \nu^{13} - 2624 \nu^{12} + 32824 \nu^{11} + 2944 \nu^{10} - 31936 \nu^{9} + 55040 \nu^{8} + 184960 \nu^{7} + 655360 \nu^{6} + 1164288 \nu^{5} - 2379776 \nu^{4} - 7356416 \nu^{3} - 7733248 \nu^{2} + 31326208 \nu + 15990784\)\()/8126464\)
\(\beta_{13}\)\(=\)\((\)\( 7 \nu^{19} - 38 \nu^{17} + 154 \nu^{15} + 136 \nu^{13} - 1640 \nu^{11} + 8448 \nu^{9} - 7808 \nu^{7} + 22016 \nu^{5} + 845824 \nu^{3} - 3457024 \nu \)\()/1015808\)
\(\beta_{14}\)\(=\)\((\)\(53 \nu^{19} - 48 \nu^{18} - 394 \nu^{17} + 320 \nu^{16} + 670 \nu^{15} - 2272 \nu^{14} + 392 \nu^{13} - 2624 \nu^{12} - 32824 \nu^{11} + 2944 \nu^{10} + 31936 \nu^{9} + 55040 \nu^{8} - 184960 \nu^{7} + 655360 \nu^{6} - 1164288 \nu^{5} - 2379776 \nu^{4} + 7356416 \nu^{3} - 7733248 \nu^{2} - 31326208 \nu + 15990784\)\()/8126464\)
\(\beta_{15}\)\(=\)\((\)\(-25 \nu^{19} + 84 \nu^{18} + 1234 \nu^{17} - 808 \nu^{16} - 4022 \nu^{15} + 504 \nu^{14} - 5800 \nu^{13} + 11040 \nu^{12} + 22296 \nu^{11} - 23008 \nu^{10} - 69568 \nu^{9} + 4864 \nu^{8} + 772736 \nu^{7} - 337408 \nu^{6} - 1160192 \nu^{5} - 1581056 \nu^{4} - 14385152 \nu^{3} + 20643840 \nu^{2} + 47972352 \nu - 17825792\)\()/8126464\)
\(\beta_{16}\)\(=\)\((\)\(-539 \nu^{19} - 1700 \nu^{18} - 386 \nu^{17} - 2840 \nu^{16} + 2190 \nu^{15} + 35560 \nu^{14} - 27464 \nu^{13} - 22240 \nu^{12} + 28232 \nu^{11} - 101280 \nu^{10} - 311552 \nu^{9} - 995840 \nu^{8} - 523136 \nu^{7} - 3965440 \nu^{6} + 5811200 \nu^{5} + 23490560 \nu^{4} - 4603904 \nu^{3} + 18513920 \nu^{2} - 48889856 \nu - 399769600\)\()/40632320\)
\(\beta_{17}\)\(=\)\((\)\( 17 \nu^{19} - 37 \nu^{17} - 80 \nu^{15} + 302 \nu^{13} + 904 \nu^{11} + 8936 \nu^{9} + 6368 \nu^{7} - 250240 \nu^{5} + 686592 \nu^{3} + 2428928 \nu \)\()/1269760\)
\(\beta_{18}\)\(=\)\((\)\(597 \nu^{19} - 740 \nu^{18} - 962 \nu^{17} - 40 \nu^{16} - 530 \nu^{15} + 3400 \nu^{14} + 33272 \nu^{13} + 8000 \nu^{12} - 39736 \nu^{11} - 103840 \nu^{10} + 562176 \nu^{9} - 443520 \nu^{8} - 2165632 \nu^{7} - 1966080 \nu^{6} - 4244480 \nu^{5} + 6983680 \nu^{4} + 41500672 \nu^{3} - 27852800 \nu^{2} - 27000832 \nu - 76021760\)\()/40632320\)
\(\beta_{19}\)\(=\)\((\)\(-539 \nu^{19} + 1700 \nu^{18} - 386 \nu^{17} + 2840 \nu^{16} + 2190 \nu^{15} - 35560 \nu^{14} - 27464 \nu^{13} + 22240 \nu^{12} + 28232 \nu^{11} + 101280 \nu^{10} - 311552 \nu^{9} + 995840 \nu^{8} - 523136 \nu^{7} + 3965440 \nu^{6} + 5811200 \nu^{5} - 23490560 \nu^{4} - 4603904 \nu^{3} - 18513920 \nu^{2} - 48889856 \nu + 399769600\)\()/40632320\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{17} - \beta_{13} + \beta_{10} + \beta_{2}\)\()/4\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{11} + \beta_{5} + \beta_{4} + 2 \beta_{1}\)\()/2\)
\(\nu^{3}\)\(=\)\((\)\(\beta_{19} + \beta_{18} + \beta_{17} + \beta_{16} + \beta_{15} + \beta_{13} + \beta_{9} - \beta_{8} + \beta_{3} + 2 \beta_{2}\)\()/2\)
\(\nu^{4}\)\(=\)\((\)\(\beta_{19} - \beta_{16} - \beta_{14} - \beta_{12} - 2 \beta_{9} - 2 \beta_{8} + 2 \beta_{7} - 2 \beta_{5} - 2 \beta_{4} + 12 \beta_{1}\)\()/2\)
\(\nu^{5}\)\(=\)\((\)\(\beta_{19} + 2 \beta_{18} - 5 \beta_{17} + \beta_{16} - 3 \beta_{14} + 5 \beta_{13} + 3 \beta_{12} + \beta_{11} - 5 \beta_{10} - 11 \beta_{9} + 11 \beta_{8} + 3 \beta_{5} - 3 \beta_{4} + \beta_{3} + 7 \beta_{2}\)\()/2\)
\(\nu^{6}\)\(=\)\(3 \beta_{14} + 3 \beta_{12} + 3 \beta_{11} - 7 \beta_{9} - 7 \beta_{8} + \beta_{7} + \beta_{6} - 9 \beta_{5} - 9 \beta_{4} + \beta_{3} + 4 \beta_{1} + 4\)
\(\nu^{7}\)\(=\)\(-3 \beta_{19} - 13 \beta_{18} + 2 \beta_{17} - 3 \beta_{16} + 3 \beta_{15} - 2 \beta_{14} + 16 \beta_{13} + 2 \beta_{12} - 8 \beta_{11} - 3 \beta_{10} - \beta_{9} + \beta_{8} - 8 \beta_{5} + 8 \beta_{4} - 5 \beta_{3} + 7 \beta_{2}\)
\(\nu^{8}\)\(=\)\(-\beta_{19} + \beta_{16} + 9 \beta_{14} + 9 \beta_{12} - 4 \beta_{11} + 28 \beta_{9} + 28 \beta_{8} + 6 \beta_{7} - 16 \beta_{6} + 14 \beta_{5} + 14 \beta_{4} + 14 \beta_{3} + 148 \beta_{1} + 28\)
\(\nu^{9}\)\(=\)\(-\beta_{19} + 4 \beta_{18} + 37 \beta_{17} - \beta_{16} + 10 \beta_{15} - 7 \beta_{14} + 31 \beta_{13} + 7 \beta_{12} - 3 \beta_{11} + 95 \beta_{10} + 19 \beta_{9} - 19 \beta_{8} - 81 \beta_{5} + 81 \beta_{4} + 7 \beta_{3} - 17 \beta_{2}\)
\(\nu^{10}\)\(=\)\(-38 \beta_{19} + 38 \beta_{16} - 12 \beta_{14} - 12 \beta_{12} + 42 \beta_{11} + 18 \beta_{9} + 18 \beta_{8} + 30 \beta_{7} - 54 \beta_{6} + 14 \beta_{5} + 14 \beta_{4} + 78 \beta_{3} - 200 \beta_{1} - 576\)
\(\nu^{11}\)\(=\)\(56 \beta_{19} + 26 \beta_{18} + 30 \beta_{17} + 56 \beta_{16} - 18 \beta_{15} - 86 \beta_{14} + 182 \beta_{13} + 86 \beta_{12} + 22 \beta_{11} - 156 \beta_{10} + 148 \beta_{9} - 148 \beta_{8} + 18 \beta_{5} - 18 \beta_{4} + 4 \beta_{3} - 240 \beta_{2}\)
\(\nu^{12}\)\(=\)\(-26 \beta_{19} + 26 \beta_{16} - 18 \beta_{14} - 18 \beta_{12} - 236 \beta_{11} - 100 \beta_{9} - 100 \beta_{8} - 88 \beta_{7} - 276 \beta_{6} - 328 \beta_{5} - 328 \beta_{4} - 256 \beta_{3} + 168 \beta_{1} - 1160\)
\(\nu^{13}\)\(=\)\(-486 \beta_{19} - 100 \beta_{18} - 942 \beta_{17} - 486 \beta_{16} - 248 \beta_{15} - 150 \beta_{14} - 82 \beta_{13} + 150 \beta_{12} + 74 \beta_{11} - 662 \beta_{10} + 122 \beta_{9} - 122 \beta_{8} - 466 \beta_{5} + 466 \beta_{4} - 174 \beta_{3} + 34 \beta_{2}\)
\(\nu^{14}\)\(=\)\(-712 \beta_{19} + 712 \beta_{16} - 92 \beta_{14} - 92 \beta_{12} - 300 \beta_{11} + 388 \beta_{9} + 388 \beta_{8} - 1148 \beta_{7} - 332 \beta_{6} - 492 \beta_{5} - 492 \beta_{4} + 916 \beta_{3} - 2240 \beta_{1} + 7024\)
\(\nu^{15}\)\(=\)\(-1308 \beta_{19} - 1596 \beta_{18} + 2616 \beta_{17} - 1308 \beta_{16} - 908 \beta_{15} + 288 \beta_{14} - 1440 \beta_{13} - 288 \beta_{12} - 344 \beta_{11} + 4348 \beta_{10} + 5132 \beta_{9} - 5132 \beta_{8} + 1656 \beta_{5} - 1656 \beta_{4} - 1252 \beta_{3} - 1068 \beta_{2}\)
\(\nu^{16}\)\(=\)\(36 \beta_{19} - 36 \beta_{16} - 1940 \beta_{14} - 1940 \beta_{12} + 1152 \beta_{11} + 2416 \beta_{9} + 2416 \beta_{8} - 5832 \beta_{7} - 1904 \beta_{6} + 11192 \beta_{5} + 11192 \beta_{4} + 264 \beta_{3} + 13232 \beta_{1} - 21584\)
\(\nu^{17}\)\(=\)\(340 \beta_{19} + 9200 \beta_{18} - 2164 \beta_{17} + 340 \beta_{16} + 7096 \beta_{15} + 1100 \beta_{14} + 1636 \beta_{13} - 1100 \beta_{12} + 1052 \beta_{11} + 7748 \beta_{10} + 15716 \beta_{9} - 15716 \beta_{8} + 6708 \beta_{5} - 6708 \beta_{4} + 8148 \beta_{3} + 708 \beta_{2}\)
\(\nu^{18}\)\(=\)\(7096 \beta_{19} - 7096 \beta_{16} - 16912 \beta_{14} - 16912 \beta_{12} - 6824 \beta_{11} - 9576 \beta_{9} - 9576 \beta_{8} - 6936 \beta_{7} + 10104 \beta_{6} - 21112 \beta_{5} - 21112 \beta_{4} + 6888 \beta_{3} - 61472 \beta_{1} - 28416\)
\(\nu^{19}\)\(=\)\(-10944 \beta_{19} + 10200 \beta_{18} - 15480 \beta_{17} - 10944 \beta_{16} - 10040 \beta_{15} - 6664 \beta_{14} + 18600 \beta_{13} + 6664 \beta_{12} + 10120 \beta_{11} - 63952 \beta_{10} - 62448 \beta_{9} + 62448 \beta_{8} + 97368 \beta_{5} - 97368 \beta_{4} + 80 \beta_{3} - 9600 \beta_{2}\)

Character values

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

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

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
161.1
−1.96139 + 0.391068i
1.85381 0.750590i
1.28499 + 1.53258i
−1.28499 1.53258i
−0.312316 1.97546i
1.96139 0.391068i
−1.21144 + 1.59136i
−1.85381 + 0.750590i
0.312316 + 1.97546i
1.21144 1.59136i
−1.96139 0.391068i
1.85381 + 0.750590i
1.28499 1.53258i
−1.28499 + 1.53258i
−0.312316 + 1.97546i
1.96139 + 0.391068i
−1.21144 1.59136i
−1.85381 0.750590i
0.312316 1.97546i
1.21144 + 1.59136i
0 −2.99548 0.164573i 0 −3.61305 3.61305i 0 12.2792i 0 8.94583 + 0.985948i 0
161.2 0 −2.59524 + 1.50491i 0 −2.59897 2.59897i 0 7.30027i 0 4.47050 7.81118i 0
161.3 0 −2.17774 2.06336i 0 −3.17955 3.17955i 0 6.03979i 0 0.485128 + 8.98692i 0
161.4 0 −2.06336 2.17774i 0 3.17955 + 3.17955i 0 6.03979i 0 −0.485128 + 8.98692i 0
161.5 0 −1.18505 + 2.75602i 0 0.00985921 + 0.00985921i 0 6.42277i 0 −6.19134 6.53203i 0
161.6 0 −0.164573 2.99548i 0 3.61305 + 3.61305i 0 12.2792i 0 −8.94583 + 0.985948i 0
161.7 0 1.14944 + 2.77106i 0 −4.80434 4.80434i 0 7.36187i 0 −6.35757 + 6.37035i 0
161.8 0 1.50491 2.59524i 0 2.59897 + 2.59897i 0 7.30027i 0 −4.47050 7.81118i 0
161.9 0 2.75602 1.18505i 0 −0.00985921 0.00985921i 0 6.42277i 0 6.19134 6.53203i 0
161.10 0 2.77106 + 1.14944i 0 4.80434 + 4.80434i 0 7.36187i 0 6.35757 + 6.37035i 0
353.1 0 −2.99548 + 0.164573i 0 −3.61305 + 3.61305i 0 12.2792i 0 8.94583 0.985948i 0
353.2 0 −2.59524 1.50491i 0 −2.59897 + 2.59897i 0 7.30027i 0 4.47050 + 7.81118i 0
353.3 0 −2.17774 + 2.06336i 0 −3.17955 + 3.17955i 0 6.03979i 0 0.485128 8.98692i 0
353.4 0 −2.06336 + 2.17774i 0 3.17955 3.17955i 0 6.03979i 0 −0.485128 8.98692i 0
353.5 0 −1.18505 2.75602i 0 0.00985921 0.00985921i 0 6.42277i 0 −6.19134 + 6.53203i 0
353.6 0 −0.164573 + 2.99548i 0 3.61305 3.61305i 0 12.2792i 0 −8.94583 0.985948i 0
353.7 0 1.14944 2.77106i 0 −4.80434 + 4.80434i 0 7.36187i 0 −6.35757 6.37035i 0
353.8 0 1.50491 + 2.59524i 0 2.59897 2.59897i 0 7.30027i 0 −4.47050 + 7.81118i 0
353.9 0 2.75602 + 1.18505i 0 −0.00985921 + 0.00985921i 0 6.42277i 0 6.19134 + 6.53203i 0
353.10 0 2.77106 1.14944i 0 4.80434 4.80434i 0 7.36187i 0 6.35757 6.37035i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 353.10
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
16.e even 4 1 inner
48.i 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.i.c 20
3.b odd 2 1 inner 384.3.i.c 20
4.b odd 2 1 384.3.i.d 20
8.b even 2 1 192.3.i.b 20
8.d odd 2 1 48.3.i.b 20
12.b even 2 1 384.3.i.d 20
16.e even 4 1 192.3.i.b 20
16.e even 4 1 inner 384.3.i.c 20
16.f odd 4 1 48.3.i.b 20
16.f odd 4 1 384.3.i.d 20
24.f even 2 1 48.3.i.b 20
24.h odd 2 1 192.3.i.b 20
48.i odd 4 1 192.3.i.b 20
48.i odd 4 1 inner 384.3.i.c 20
48.k even 4 1 48.3.i.b 20
48.k even 4 1 384.3.i.d 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
48.3.i.b 20 8.d odd 2 1
48.3.i.b 20 16.f odd 4 1
48.3.i.b 20 24.f even 2 1
48.3.i.b 20 48.k even 4 1
192.3.i.b 20 8.b even 2 1
192.3.i.b 20 16.e even 4 1
192.3.i.b 20 24.h odd 2 1
192.3.i.b 20 48.i odd 4 1
384.3.i.c 20 1.a even 1 1 trivial
384.3.i.c 20 3.b odd 2 1 inner
384.3.i.c 20 16.e even 4 1 inner
384.3.i.c 20 48.i odd 4 1 inner
384.3.i.d 20 4.b odd 2 1
384.3.i.d 20 12.b even 2 1
384.3.i.d 20 16.f odd 4 1
384.3.i.d 20 48.k even 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(384, [\chi])\):

\( T_{5}^{20} + 3404 T_{5}^{16} + 3190384 T_{5}^{12} + 1068787520 T_{5}^{8} + 108375444480 T_{5}^{4} + 4096 \)
\(T_{19}^{10} + \cdots\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{20} \)
$3$ \( 3486784401 + 2324522934 T + 774840978 T^{2} + 143489070 T^{3} + 34543665 T^{4} + 9920232 T^{5} + 1889568 T^{6} + 1137240 T^{7} + 913518 T^{8} + 578988 T^{9} + 225180 T^{10} + 64332 T^{11} + 11278 T^{12} + 1560 T^{13} + 288 T^{14} + 168 T^{15} + 65 T^{16} + 30 T^{17} + 18 T^{18} + 6 T^{19} + T^{20} \)
$5$ \( 4096 + 108375444480 T^{4} + 1068787520 T^{8} + 3190384 T^{12} + 3404 T^{16} + T^{20} \)
$7$ \( ( 655360000 + 62587904 T^{2} + 2308480 T^{4} + 40676 T^{6} + 336 T^{8} + T^{10} )^{2} \)
$11$ \( 2240545423360000 + 8039082785243136 T^{4} + 204108821294144 T^{8} + 12405334576 T^{12} + 207308 T^{16} + T^{20} \)
$13$ \( ( 33620000 - 46477600 T + 32126224 T^{2} - 960768 T^{3} - 352112 T^{4} + 191600 T^{5} + 85320 T^{6} + 12736 T^{7} + 1058 T^{8} + 46 T^{9} + T^{10} )^{2} \)
$17$ \( ( 15510536192 + 1356865536 T^{2} + 34106368 T^{4} + 297760 T^{6} + 952 T^{8} + T^{10} )^{2} \)
$19$ \( ( 23975244288 + 21827527680 T + 9936102400 T^{2} + 2439206208 T^{3} + 353223624 T^{4} + 25536424 T^{5} + 942404 T^{6} + 3952 T^{7} + 338 T^{8} + 26 T^{9} + T^{10} )^{2} \)
$23$ \( ( -2157878476800 + 60056123392 T^{2} - 495575104 T^{4} + 1632624 T^{6} - 2236 T^{8} + T^{10} )^{2} \)
$29$ \( \)\(14\!\cdots\!00\)\( + \)\(63\!\cdots\!56\)\( T^{4} + 979627163704944448 T^{8} + 14592942824560 T^{12} + 8250700 T^{16} + T^{20} \)
$31$ \( ( -6473680 + 937016 T + 7516 T^{2} - 2750 T^{3} - 20 T^{4} + T^{5} )^{4} \)
$37$ \( ( 93878430976800 + 17120760301920 T + 1561170283024 T^{2} + 60745982336 T^{3} + 998599952 T^{4} + 6429040 T^{5} + 4219528 T^{6} + 130144 T^{7} + 1682 T^{8} - 58 T^{9} + T^{10} )^{2} \)
$41$ \( ( -89172136396800 + 4759931137024 T^{2} - 24298249280 T^{4} + 24250736 T^{6} - 8644 T^{8} + T^{10} )^{2} \)
$43$ \( ( 398518394892800 - 94334096030720 T + 11165007422464 T^{2} - 692164733120 T^{3} + 26089764040 T^{4} - 583167128 T^{5} + 8195716 T^{6} - 111760 T^{7} + 3698 T^{8} - 86 T^{9} + T^{10} )^{2} \)
$47$ \( ( 2199023255552 + 146867748864 T^{2} + 2561671168 T^{4} + 6660224 T^{6} + 4944 T^{8} + T^{10} )^{2} \)
$53$ \( \)\(51\!\cdots\!00\)\( + \)\(21\!\cdots\!76\)\( T^{4} + \)\(31\!\cdots\!08\)\( T^{8} + 185846408858736 T^{12} + 33387084 T^{16} + T^{20} \)
$59$ \( \)\(55\!\cdots\!00\)\( + \)\(78\!\cdots\!96\)\( T^{4} + \)\(17\!\cdots\!92\)\( T^{8} + 2609014014580272 T^{12} + 96029644 T^{16} + T^{20} \)
$61$ \( ( 7907544324449568 - 1330247085982368 T + 111890445197584 T^{2} - 5334667962752 T^{3} + 157007740304 T^{4} - 2668194448 T^{5} + 26687560 T^{6} - 245344 T^{7} + 7442 T^{8} - 122 T^{9} + T^{10} )^{2} \)
$67$ \( ( 1442777382684800 + 245738443228160 T + 20927477518336 T^{2} - 934037268512 T^{3} + 18053629992 T^{4} - 143429896 T^{5} + 16045188 T^{6} - 702440 T^{7} + 15842 T^{8} - 178 T^{9} + T^{10} )^{2} \)
$71$ \( ( -6303938155520000 + 49273291498496 T^{2} - 91544227136 T^{4} + 55549616 T^{6} - 12876 T^{8} + T^{10} )^{2} \)
$73$ \( ( 900192010240000 + 76360353726464 T^{2} + 137600509952 T^{4} + 77003072 T^{6} + 16160 T^{8} + T^{10} )^{2} \)
$79$ \( ( 147403248 + 3773624 T - 263044 T^{2} - 3534 T^{3} + 96 T^{4} + T^{5} )^{4} \)
$83$ \( \)\(53\!\cdots\!00\)\( + \)\(82\!\cdots\!96\)\( T^{4} + \)\(34\!\cdots\!04\)\( T^{8} + 36252904681183792 T^{12} + 433330892 T^{16} + T^{20} \)
$89$ \( ( -15016301280992460800 + 15202762675533824 T^{2} - 5223733537856 T^{4} + 780108464 T^{6} - 49740 T^{8} + T^{10} )^{2} \)
$97$ \( ( -2657552000 + 30283520 T + 1148408 T^{2} - 11780 T^{3} - 118 T^{4} + T^{5} )^{4} \)
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