# Properties

 Label 108.6.b.b Level 108 Weight 6 Character orbit 108.b Analytic conductor 17.321 Analytic rank 0 Dimension 16 CM no Inner twists 4

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$108 = 2^{2} \cdot 3^{3}$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 108.b (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$17.3214525398$$ Analytic rank: $$0$$ Dimension: $$16$$ Coefficient field: $$\mathbb{Q}[x]/(x^{16} + \cdots)$$ Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$2^{30}\cdot 3^{32}\cdot 5^{2}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + \beta_{1} q^{2} + ( 6 - \beta_{4} ) q^{4} + ( -3 \beta_{1} + \beta_{3} ) q^{5} + ( \beta_{4} - \beta_{10} ) q^{7} + ( 6 \beta_{1} - \beta_{2} + \beta_{3} - \beta_{7} ) q^{8} +O(q^{10})$$ $$q + \beta_{1} q^{2} + ( 6 - \beta_{4} ) q^{4} + ( -3 \beta_{1} + \beta_{3} ) q^{5} + ( \beta_{4} - \beta_{10} ) q^{7} + ( 6 \beta_{1} - \beta_{2} + \beta_{3} - \beta_{7} ) q^{8} + ( 90 + 3 \beta_{4} + \beta_{10} - \beta_{14} ) q^{10} + ( -\beta_{2} + 2 \beta_{7} + \beta_{13} ) q^{11} + ( 56 - \beta_{4} + \beta_{5} ) q^{13} + ( \beta_{1} + \beta_{2} - 6 \beta_{3} + \beta_{6} - 2 \beta_{7} - \beta_{8} + 3 \beta_{9} ) q^{14} + ( 11 - 6 \beta_{4} + \beta_{5} - \beta_{10} + 2 \beta_{11} + \beta_{12} - 2 \beta_{14} ) q^{16} + ( 8 \beta_{1} + 4 \beta_{2} + 8 \beta_{3} + 2 \beta_{6} + \beta_{8} - 6 \beta_{9} ) q^{17} + ( -4 + 2 \beta_{4} + 6 \beta_{10} + 3 \beta_{11} + 4 \beta_{12} - 5 \beta_{14} ) q^{19} + ( 86 \beta_{1} + 5 \beta_{2} + 12 \beta_{3} - \beta_{6} - 6 \beta_{7} - 10 \beta_{9} + \beta_{13} ) q^{20} + ( 3 - 3 \beta_{4} + 2 \beta_{5} - 4 \beta_{10} - 3 \beta_{11} - 5 \beta_{12} - 2 \beta_{14} - 2 \beta_{15} ) q^{22} + ( 68 \beta_{1} + 5 \beta_{2} - \beta_{3} - 2 \beta_{6} - 9 \beta_{7} - 2 \beta_{8} + \beta_{9} + 5 \beta_{13} ) q^{23} + ( 614 + 17 \beta_{4} - \beta_{5} + 2 \beta_{10} - 3 \beta_{11} + 6 \beta_{12} + 2 \beta_{14} + \beta_{15} ) q^{25} + ( 52 \beta_{1} - 2 \beta_{2} + 8 \beta_{6} + 2 \beta_{7} + 2 \beta_{8} - 4 \beta_{9} + 6 \beta_{13} ) q^{26} + ( 724 + 13 \beta_{4} - 2 \beta_{5} - 30 \beta_{10} - 10 \beta_{12} + 4 \beta_{14} + 4 \beta_{15} ) q^{28} + ( 15 \beta_{1} - 2 \beta_{2} + \beta_{3} + 12 \beta_{6} - 7 \beta_{8} - 3 \beta_{9} ) q^{29} + ( 12 - 23 \beta_{4} + \beta_{10} + 2 \beta_{11} - 6 \beta_{12} + 17 \beta_{14} + 3 \beta_{15} ) q^{31} + ( 2 \beta_{1} - 7 \beta_{2} + 33 \beta_{3} + 6 \beta_{6} + 3 \beta_{7} - 4 \beta_{8} + 14 \beta_{13} ) q^{32} + ( -388 - 28 \beta_{4} - 6 \beta_{5} + 18 \beta_{10} - 8 \beta_{12} - 16 \beta_{14} - 10 \beta_{15} ) q^{34} + ( 457 \beta_{1} - 34 \beta_{2} + 26 \beta_{3} + 4 \beta_{6} - 71 \beta_{7} + 4 \beta_{8} + 22 \beta_{9} + 14 \beta_{13} ) q^{35} + ( -4454 + 3 \beta_{4} - 3 \beta_{5} + 16 \beta_{10} - 16 \beta_{11} + 48 \beta_{12} + 16 \beta_{14} ) q^{37} + ( -30 \beta_{1} + 6 \beta_{2} + 120 \beta_{3} - 8 \beta_{6} + 42 \beta_{7} - 6 \beta_{8} - 28 \beta_{9} + 14 \beta_{13} ) q^{38} + ( -1025 - 120 \beta_{4} - 3 \beta_{5} + 39 \beta_{10} - 2 \beta_{11} - 23 \beta_{12} - 14 \beta_{14} ) q^{40} + ( 406 \beta_{1} + 4 \beta_{2} + 86 \beta_{3} + 6 \beta_{6} - \beta_{8} - 70 \beta_{9} ) q^{41} + ( 8 + 146 \beta_{4} - 6 \beta_{10} + 40 \beta_{11} - 4 \beta_{12} + 50 \beta_{14} + 2 \beta_{15} ) q^{43} + ( 2 \beta_{1} + 9 \beta_{2} + 10 \beta_{3} + 25 \beta_{6} - 148 \beta_{7} + 8 \beta_{8} + 14 \beta_{9} - \beta_{13} ) q^{44} + ( 2244 - 74 \beta_{4} + 2 \beta_{5} - 12 \beta_{10} - 12 \beta_{11} - 68 \beta_{12} + 6 \beta_{14} + 6 \beta_{15} ) q^{46} + ( 122 \beta_{1} + 42 \beta_{2} - 14 \beta_{3} - 12 \beta_{6} + 56 \beta_{7} - 12 \beta_{8} - 2 \beta_{9} + 18 \beta_{13} ) q^{47} + ( -3162 + 377 \beta_{4} - 9 \beta_{5} + 30 \beta_{10} - 53 \beta_{11} + 90 \beta_{12} + 30 \beta_{14} + 23 \beta_{15} ) q^{49} + ( 606 \beta_{1} + 18 \beta_{2} - 8 \beta_{3} + 102 \beta_{7} - 2 \beta_{8} - 28 \beta_{9} - 14 \beta_{13} ) q^{50} + ( 948 - 82 \beta_{4} - 56 \beta_{10} - 56 \beta_{12} - 40 \beta_{14} - 40 \beta_{15} ) q^{52} + ( -954 \beta_{1} + 62 \beta_{2} - 84 \beta_{3} + 32 \beta_{6} + 15 \beta_{8} + 77 \beta_{9} ) q^{53} + ( -8 - 53 \beta_{4} + 121 \beta_{10} + 40 \beta_{11} + 20 \beta_{12} + 6 \beta_{14} + 6 \beta_{15} ) q^{55} + ( 790 \beta_{1} - \beta_{2} - 411 \beta_{3} - 4 \beta_{6} - 125 \beta_{7} - 24 \beta_{8} + 16 \beta_{9} - 4 \beta_{13} ) q^{56} + ( -636 + 16 \beta_{4} - 10 \beta_{5} - 206 \beta_{10} - 48 \beta_{12} - 68 \beta_{14} + 18 \beta_{15} ) q^{58} + ( -1824 \beta_{1} + 6 \beta_{2} - 88 \beta_{3} + 140 \beta_{7} - 88 \beta_{9} - 6 \beta_{13} ) q^{59} + ( 4860 - 364 \beta_{4} + 12 \beta_{5} + 52 \beta_{10} - 30 \beta_{11} + 156 \beta_{12} + 52 \beta_{14} - 22 \beta_{15} ) q^{61} + ( 73 \beta_{1} - 67 \beta_{2} - 282 \beta_{3} - 17 \beta_{6} + 156 \beta_{7} + 19 \beta_{8} + 49 \beta_{9} - 2 \beta_{13} ) q^{62} + ( 5605 - 22 \beta_{4} + 15 \beta_{5} - 119 \beta_{10} - 26 \beta_{11} - 137 \beta_{12} - 86 \beta_{14} - 16 \beta_{15} ) q^{64} + ( 30 \beta_{1} - 64 \beta_{2} + 274 \beta_{3} - 94 \beta_{6} + 15 \beta_{8} - 28 \beta_{9} ) q^{65} + ( 48 - 430 \beta_{4} + 30 \beta_{10} + 112 \beta_{11} + 80 \beta_{12} + 16 \beta_{14} + 64 \beta_{15} ) q^{67} + ( -404 \beta_{1} + 30 \beta_{2} + 512 \beta_{3} - 54 \beta_{6} - 388 \beta_{7} + 8 \beta_{8} + 100 \beta_{9} - 50 \beta_{13} ) q^{68} + ( 14423 - 445 \beta_{4} + 44 \beta_{5} - 72 \beta_{10} + 89 \beta_{11} - 121 \beta_{12} - 60 \beta_{14} - 60 \beta_{15} ) q^{70} + ( 1132 \beta_{1} - 73 \beta_{2} + 119 \beta_{3} + 30 \beta_{6} - 45 \beta_{7} + 30 \beta_{8} + 89 \beta_{9} - 77 \beta_{13} ) q^{71} + ( -2533 + 717 \beta_{4} + 35 \beta_{5} + 62 \beta_{10} - 109 \beta_{11} + 186 \beta_{12} + 62 \beta_{14} + 47 \beta_{15} ) q^{73} + ( -4522 \beta_{1} + 6 \beta_{2} + 320 \beta_{3} + 40 \beta_{6} + 698 \beta_{7} - 6 \beta_{8} + 12 \beta_{9} - 82 \beta_{13} ) q^{74} + ( 3032 - 80 \beta_{4} + 16 \beta_{5} + 120 \beta_{10} - 96 \beta_{11} - 104 \beta_{12} - 136 \beta_{14} + 24 \beta_{15} ) q^{76} + ( 1687 \beta_{1} - 108 \beta_{2} - 817 \beta_{3} - 44 \beta_{6} - 32 \beta_{8} + 150 \beta_{9} ) q^{77} + ( -244 + 1648 \beta_{4} + 134 \beta_{10} + 160 \beta_{11} + 46 \beta_{12} - 31 \beta_{14} - 99 \beta_{15} ) q^{79} + ( -1050 \beta_{1} - 85 \beta_{2} + 247 \beta_{3} - 82 \beta_{6} - 499 \beta_{7} + 44 \beta_{8} - 240 \beta_{9} - 10 \beta_{13} ) q^{80} + ( -14024 - 606 \beta_{4} - 10 \beta_{5} + 96 \beta_{10} - 24 \beta_{12} - 178 \beta_{14} - 6 \beta_{15} ) q^{82} + ( 748 \beta_{1} + 95 \beta_{2} + 78 \beta_{3} + 12 \beta_{6} + 16 \beta_{7} + 12 \beta_{8} + 66 \beta_{9} - 155 \beta_{13} ) q^{83} + ( -23420 + 1093 \beta_{4} + 27 \beta_{5} + 76 \beta_{10} - 146 \beta_{11} + 228 \beta_{12} + 76 \beta_{14} + 70 \beta_{15} ) q^{85} + ( 210 \beta_{1} - 38 \beta_{2} - 692 \beta_{3} - 82 \beta_{6} + 1096 \beta_{7} + 6 \beta_{8} + 594 \beta_{9} + 76 \beta_{13} ) q^{86} + ( 15631 + 36 \beta_{4} - 35 \beta_{5} - 97 \beta_{10} + 182 \beta_{11} - 175 \beta_{12} - 70 \beta_{14} - 96 \beta_{15} ) q^{88} + ( -1744 \beta_{1} - 180 \beta_{2} + 20 \beta_{3} - 148 \beta_{6} - 16 \beta_{8} + 306 \beta_{9} ) q^{89} + ( 316 - 772 \beta_{4} - 408 \beta_{10} + 81 \beta_{11} - 56 \beta_{12} + 323 \beta_{14} + 130 \beta_{15} ) q^{91} + ( 2384 \beta_{1} - 76 \beta_{2} - 864 \beta_{3} - 28 \beta_{6} - 976 \beta_{7} + 72 \beta_{8} - 232 \beta_{9} - 12 \beta_{13} ) q^{92} + ( 4410 - 146 \beta_{4} - 12 \beta_{5} - 24 \beta_{10} - 170 \beta_{11} - 214 \beta_{12} + 60 \beta_{14} + 60 \beta_{15} ) q^{94} + ( -9328 \beta_{1} + 79 \beta_{2} - 475 \beta_{3} + 10 \beta_{6} - 295 \beta_{7} + 10 \beta_{8} - 485 \beta_{9} - 129 \beta_{13} ) q^{95} + ( 163 - 2308 \beta_{4} - 60 \beta_{5} + 72 \beta_{10} + 76 \beta_{11} + 216 \beta_{12} + 72 \beta_{14} - 148 \beta_{15} ) q^{97} + ( -3322 \beta_{1} + 386 \beta_{2} - 504 \beta_{3} + 48 \beta_{6} + 1670 \beta_{7} - 18 \beta_{8} - 700 \beta_{9} - 174 \beta_{13} ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$16q + 94q^{4} + O(q^{10})$$ $$16q + 94q^{4} + 1454q^{10} + 896q^{13} + 178q^{16} + 30q^{22} + 9888q^{25} + 11454q^{28} - 6172q^{34} - 71008q^{37} - 16618q^{40} + 35304q^{46} - 49376q^{49} + 14876q^{52} - 10492q^{58} + 77888q^{61} + 89206q^{64} + 229398q^{70} - 38032q^{73} + 48960q^{76} - 224488q^{82} - 371264q^{85} + 249102q^{88} + 68772q^{94} - 976q^{97} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{16} + 30 x^{14} + 619 x^{12} + 5604 x^{10} + 40971 x^{8} - 4866 x^{6} + 568069 x^{4} - 7909632 x^{2} + 20340100$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$8580011824 \nu^{15} + 1068480910 \nu^{14} + 249341355770 \nu^{13} + 130620941625 \nu^{12} + 5081598023376 \nu^{11} + 4765378422410 \nu^{10} + 43568188779276 \nu^{9} + 87307080887660 \nu^{8} + 313660285662984 \nu^{7} + 893607005984080 \nu^{6} - 246986959442814 \nu^{5} + 3950333051441935 \nu^{4} + 2311687085732536 \nu^{3} + 3726439934534280 \nu^{2} - 84413038481484808 \nu - 46105280481940900$$$$)/ 27320067658741440$$ $$\beta_{2}$$ $$=$$ $$($$$$10356042052 \nu^{15} - 512818688367 \nu^{14} - 559634572300 \nu^{13} - 16112948528313 \nu^{12} - 26629723336920 \nu^{11} - 350893430964918 \nu^{10} - 476483650046136 \nu^{9} - 3594820177768626 \nu^{8} - 3823073798507004 \nu^{7} - 30327806025643095 \nu^{6} - 2361397091643468 \nu^{5} - 66333038546990385 \nu^{4} + 56068965106536208 \nu^{3} - 759555828747254340 \nu^{2} + 429507259594744208 \nu + 2890609923869254620$$$$)/ 21856054126993152$$ $$\beta_{3}$$ $$=$$ $$($$$$102960141888 \nu^{15} + 993088055285 \nu^{14} + 2992096269240 \nu^{13} + 30244077699495 \nu^{12} + 60979176280512 \nu^{11} + 643457956723450 \nu^{10} + 522818265351312 \nu^{9} + 6250733902328230 \nu^{8} + 3763923427955808 \nu^{7} + 50839289998983845 \nu^{6} - 2963843513313768 \nu^{5} + 117563191362615095 \nu^{4} + 27740245028790432 \nu^{3} + 1116890596304218860 \nu^{2} - 1012956461777817696 \nu - 4428604990499706500$$$$)/ 109280270634965760$$ $$\beta_{4}$$ $$=$$ $$($$$$9163365761 \nu^{15} + 19341597480 \nu^{14} + 350667454149 \nu^{13} + 550630309632 \nu^{12} + 7905413917662 \nu^{11} + 10834073957808 \nu^{10} + 96990250254366 \nu^{9} + 90891309067488 \nu^{8} + 825057021454845 \nu^{7} + 492567892577352 \nu^{6} + 3546416484979497 \nu^{5} - 2046376334945664 \nu^{4} + 13149961404225860 \nu^{3} - 9460196030544096 \nu^{2} - 53564135734156332 \nu - 37401213497760288$$$$)/ 5464013531748288$$ $$\beta_{5}$$ $$=$$ $$($$$$9163365761 \nu^{15} + 58012811010 \nu^{14} + 350667454149 \nu^{13} + 7239682944882 \nu^{12} + 7905413917662 \nu^{11} + 161786304141444 \nu^{10} + 96990250254366 \nu^{9} + 2053625267133108 \nu^{8} + 825057021454845 \nu^{7} + 7222446589773474 \nu^{6} + 3546416484979497 \nu^{5} - 17248010042058894 \nu^{4} + 13149961404225860 \nu^{3} - 268232566609178280 \nu^{2} - 53564135734156332 \nu - 320653757486630040$$$$)/ 5464013531748288$$ $$\beta_{6}$$ $$=$$ $$($$$$55695042027 \nu^{15} + 975870014470 \nu^{14} + 2694274061535 \nu^{13} + 36674968957590 \nu^{12} + 73939938358158 \nu^{11} + 710473409590100 \nu^{10} + 944150072791878 \nu^{9} + 8176357852336100 \nu^{8} + 7288124533437627 \nu^{7} + 55897067394860110 \nu^{6} + 975850689011823 \nu^{5} + 275007599550638350 \nu^{4} - 51592709697309972 \nu^{3} - 220447805379729000 \nu^{2} - 1212188382345308724 \nu - 1683748425978368200$$$$)/ 27320067658741440$$ $$\beta_{7}$$ $$=$$ $$($$$$71549127163 \nu^{15} + 1068480910 \nu^{14} + 2035013416385 \nu^{13} + 130620941625 \nu^{12} + 41452865084262 \nu^{11} + 4765378422410 \nu^{10} + 343512000853362 \nu^{9} + 87307080887660 \nu^{8} + 2546755795631283 \nu^{7} + 893607005984080 \nu^{6} - 3713598838595943 \nu^{5} + 3950333051441935 \nu^{4} + 55944700569709132 \nu^{3} + 3726439934534280 \nu^{2} - 940341586904762476 \nu - 46105280481940900$$$$)/ 27320067658741440$$ $$\beta_{8}$$ $$=$$ $$($$$$77070036758 \nu^{15} - 4413231616935 \nu^{14} + 4391182699990 \nu^{13} - 142982602560045 \nu^{12} + 127553484622812 \nu^{11} - 2992941165965550 \nu^{10} + 1714027390466652 \nu^{9} - 30897061225657170 \nu^{8} + 13321607924223318 \nu^{7} - 244751572465114215 \nu^{6} + 2939649215794902 \nu^{5} - 659724892538353005 \nu^{4} - 112432167737550088 \nu^{3} - 4750102174348651140 \nu^{2} - 2086724610764678216 \nu + 20165632490557022700$$$$)/ 27320067658741440$$ $$\beta_{9}$$ $$=$$ $$($$$$68640094592 \nu^{15} - 214858520889 \nu^{14} + 1994730846160 \nu^{13} - 8034253852599 \nu^{12} + 40652784187008 \nu^{11} - 201125343365322 \nu^{10} + 348545510234208 \nu^{9} - 2577214409958078 \nu^{8} + 2509282285303872 \nu^{7} - 23750684490754785 \nu^{6} - 1975895675542512 \nu^{5} - 83557700654440431 \nu^{4} + 18493496685860288 \nu^{3} - 280020006265764828 \nu^{2} - 675304307851878464 \nu + 1586521261425442980$$$$)/ 21856054126993152$$ $$\beta_{10}$$ $$=$$ $$($$$$197752231459 \nu^{15} + 77366389920 \nu^{14} + 5720648825457 \nu^{13} + 2202521238528 \nu^{12} + 126675512685102 \nu^{11} + 43336295831232 \nu^{10} + 1324695912905490 \nu^{9} + 363565236269952 \nu^{8} + 12030939274314747 \nu^{7} + 1970271570309408 \nu^{6} + 33843251673208953 \nu^{5} - 8185505339782656 \nu^{4} + 227884103467684660 \nu^{3} - 37840784122176384 \nu^{2} - 779967511320439932 \nu - 149604853991041152$$$$)/ 21856054126993152$$ $$\beta_{11}$$ $$=$$ $$($$$$-255661874717 \nu^{15} - 72011660196 \nu^{14} - 7569237235779 \nu^{13} - 2319750209892 \nu^{12} - 139913144743434 \nu^{11} - 44324503433928 \nu^{10} - 952195798773822 \nu^{9} - 449680780871208 \nu^{8} - 2568347734803405 \nu^{7} - 1412379809487972 \nu^{6} + 46601071704677997 \nu^{5} + 2081241797592348 \nu^{4} - 54489074005977548 \nu^{3} + 66917652510931152 \nu^{2} - 147081596436745260 \nu - 1550222927392082064$$$$)/ 21856054126993152$$ $$\beta_{12}$$ $$=$$ $$($$$$-376776329737 \nu^{15} - 397541409048 \nu^{14} - 11873706511995 \nu^{13} - 10778148249912 \nu^{12} - 252500518393338 \nu^{11} - 214705063950768 \nu^{10} - 2556162483663318 \nu^{9} - 1645595092147248 \nu^{8} - 20174920678535457 \nu^{7} - 10967141373189912 \nu^{6} - 43334129619730755 \nu^{5} + 53136053783293896 \nu^{4} - 368298916257938908 \nu^{3} + 131050183833372384 \nu^{2} + 1182977833928478612 \nu + 4169535886848445344$$$$)/ 21856054126993152$$ $$\beta_{13}$$ $$=$$ $$($$$$-2514369237472 \nu^{15} - 2572641289115 \nu^{14} - 85268253719840 \nu^{13} - 81609710174565 \nu^{12} - 1760738758337568 \nu^{11} - 1792590182203870 \nu^{10} - 18325740937675968 \nu^{9} - 18672557535944410 \nu^{8} - 119753016368466912 \nu^{7} - 158787886176088115 \nu^{6} - 62477968355329248 \nu^{5} - 363267857146487405 \nu^{4} + 1703776013392893152 \nu^{3} - 3827590663212545940 \nu^{2} + 16289150957518926784 \nu + 14821891863201800300$$$$)/ 109280270634965760$$ $$\beta_{14}$$ $$=$$ $$($$$$225638294345 \nu^{15} - 79151299828 \nu^{14} + 7443744491583 \nu^{13} - 2163444914740 \nu^{12} + 163637632583826 \nu^{11} - 43006893297000 \nu^{10} + 1797198579770214 \nu^{9} - 334860054736200 \nu^{8} + 15308491675496073 \nu^{7} - 2156235490583220 \nu^{6} + 47586736296887103 \nu^{5} + 10220259853846092 \nu^{4} + 274174523766718172 \nu^{3} + 28148494659258128 \nu^{2} - 972015862300580388 \nu + 716214114452082224$$$$)/ 7285351375664384$$ $$\beta_{15}$$ $$=$$ $$($$$$-842117283421 \nu^{15} + 1165850578524 \nu^{14} - 30011954301315 \nu^{13} + 32920589606556 \nu^{12} - 645859635473802 \nu^{11} + 649056229865784 \nu^{10} - 7159571815053246 \nu^{9} + 5367362999448024 \nu^{8} - 55371997107913485 \nu^{7} + 30111965315462556 \nu^{6} - 180369583334009811 \nu^{5} - 128886843638930148 \nu^{4} - 896086603876432588 \nu^{3} - 538534893443890992 \nu^{2} + 3281023090549259988 \nu - 3987612699502726800$$$$)/ 21856054126993152$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$19 \beta_{15} + 52 \beta_{14} - 24 \beta_{12} - 15 \beta_{11} - 64 \beta_{10} - 240 \beta_{7} - 200 \beta_{4} + 240 \beta_{1} + 62$$$$)/6480$$ $$\nu^{2}$$ $$=$$ $$($$$$5 \beta_{15} + 20 \beta_{14} + 60 \beta_{12} - 25 \beta_{11} + 20 \beta_{10} + 66 \beta_{9} - 16 \beta_{8} - 24 \beta_{6} + 80 \beta_{4} - 198 \beta_{3} - 56 \beta_{2} + 318 \beta_{1} - 8130$$$$)/2160$$ $$\nu^{3}$$ $$=$$ $$($$$$-52 \beta_{15} - 76 \beta_{14} - 48 \beta_{12} - 120 \beta_{11} + 232 \beta_{10} - 1215 \beta_{9} + 3600 \beta_{7} - 40 \beta_{4} - 1215 \beta_{3} - 26685 \beta_{1} - 56$$$$)/6480$$ $$\nu^{4}$$ $$=$$ $$($$$$-15 \beta_{15} + 120 \beta_{14} + 360 \beta_{12} - 105 \beta_{11} + 120 \beta_{10} - 648 \beta_{9} + 208 \beta_{8} + 192 \beta_{6} - 240 \beta_{4} + 4824 \beta_{3} + 608 \beta_{2} - 12024 \beta_{1} - 91350$$$$)/2160$$ $$\nu^{5}$$ $$=$$ $$($$$$-3334 \beta_{15} - 7732 \beta_{14} + 2400 \beta_{13} + 2304 \beta_{12} + 210 \beta_{11} + 10984 \beta_{10} + 13275 \beta_{9} - 300 \beta_{8} - 30960 \beta_{7} - 300 \beta_{6} + 27560 \beta_{4} + 12975 \beta_{3} - 900 \beta_{2} + 292485 \beta_{1} - 8972$$$$)/6480$$ $$\nu^{6}$$ $$=$$ $$($$$$-2535 \beta_{15} - 5260 \beta_{14} - 15780 \beta_{12} + 7795 \beta_{11} - 5260 \beta_{10} + 2484 \beta_{9} - 1224 \beta_{8} + 864 \beta_{6} - 160 \beta_{5} - 40400 \beta_{4} - 29052 \beta_{3} - 1584 \beta_{2} + 70092 \beta_{1} + 3222790$$$$)/2160$$ $$\nu^{7}$$ $$=$$ $$($$$$62722 \beta_{15} + 159316 \beta_{14} - 10080 \beta_{13} - 37152 \beta_{12} + 22290 \beta_{11} - 277192 \beta_{10} + 25515 \beta_{9} - 2520 \beta_{8} - 127440 \beta_{7} - 2520 \beta_{6} - 349160 \beta_{4} + 22995 \beta_{3} + 22680 \beta_{2} + 629865 \beta_{1} + 162596$$$$)/6480$$ $$\nu^{8}$$ $$=$$ $$($$$$44805 \beta_{15} + 68760 \beta_{14} + 206280 \beta_{12} - 113565 \beta_{11} + 68760 \beta_{10} + 59088 \beta_{9} - 25568 \beta_{8} - 29952 \beta_{6} - 9120 \beta_{5} + 726000 \beta_{4} - 597744 \beta_{3} - 81088 \beta_{2} + 1748784 \beta_{1} - 38898510$$$$)/2160$$ $$\nu^{9}$$ $$=$$ $$($$$$-379946 \beta_{15} - 1114628 \beta_{14} - 501120 \beta_{13} + 289536 \beta_{12} - 155610 \beta_{11} + 2771336 \beta_{10} - 3465585 \beta_{9} + 57780 \beta_{8} + 8659440 \beta_{7} + 57780 \beta_{6} + 876040 \beta_{4} - 3407805 \beta_{3} + 212220 \beta_{2} - 76374495 \beta_{1} - 1049428$$$$)/6480$$ $$\nu^{10}$$ $$=$$ $$($$$$-165935 \beta_{15} - 69900 \beta_{14} - 209700 \beta_{12} + 235835 \beta_{11} - 69900 \beta_{10} - 2443884 \beta_{9} + 796264 \beta_{8} + 127296 \beta_{6} + 200000 \beta_{5} - 2854960 \beta_{4} + 17073252 \beta_{3} + 1719824 \beta_{2} - 37354452 \beta_{1} + 23948470$$$$)/2160$$ $$\nu^{11}$$ $$=$$ $$($$$$-6305902 \beta_{15} - 14271196 \beta_{14} + 11689920 \beta_{13} + 2874432 \beta_{12} - 2216430 \beta_{11} + 16454392 \beta_{10} + 56666115 \beta_{9} + 510840 \beta_{8} - 133415280 \beta_{7} + 510840 \beta_{6} + 47476280 \beta_{4} + 57176955 \beta_{3} - 14244120 \beta_{2} + 1225788705 \beta_{1} - 15486236$$$$)/6480$$ $$\nu^{12}$$ $$=$$ $$($$$$-2593905 \beta_{15} - 4597560 \beta_{14} - 13792680 \beta_{12} + 7191465 \beta_{11} - 4597560 \beta_{10} + 12234024 \beta_{9} - 3096304 \beta_{8} + 1688064 \beta_{6} + 185280 \beta_{5} - 41687760 \beta_{4} - 57849912 \beta_{3} - 4504544 \beta_{2} + 74860632 \beta_{1} + 2693267670$$$$)/720$$ $$\nu^{13}$$ $$=$$ $$($$$$183929966 \beta_{15} + 496673228 \beta_{14} - 93506400 \beta_{13} - 129359136 \beta_{12} + 54024990 \beta_{11} - 1028336696 \beta_{10} - 255757905 \beta_{9} - 21375900 \beta_{8} + 526693200 \beta_{7} - 21375900 \beta_{6} - 833363800 \beta_{4} - 277133805 \beta_{3} + 200385900 \beta_{2} - 5252467695 \beta_{1} + 497219068$$$$)/6480$$ $$\nu^{14}$$ $$=$$ $$($$$$188417785 \beta_{15} + 265125940 \beta_{14} + 795377820 \beta_{12} - 453543725 \beta_{11} + 265125940 \beta_{10} - 53142276 \beta_{9} - 16075784 \beta_{8} - 88150656 \beta_{6} - 80920800 \beta_{5} + 3095605360 \beta_{4} - 660375252 \beta_{3} - 120302224 \beta_{2} + 3410663172 \beta_{1} - 152142237690$$$$)/2160$$ $$\nu^{15}$$ $$=$$ $$($$$$-1869056678 \beta_{15} - 5836123724 \beta_{14} - 1223903520 \beta_{13} + 1787937888 \beta_{12} - 391191270 \beta_{11} + 15823806968 \beta_{10} - 8109244125 \beta_{9} + 108737640 \beta_{8} + 20175410160 \beta_{7} + 108737640 \beta_{6} + 3252170200 \beta_{4} - 8000506485 \beta_{3} + 680215320 \beta_{2} - 178329920175 \beta_{1} - 5526051244$$$$)/6480$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/108\mathbb{Z}\right)^\times$$.

 $$n$$ $$29$$ $$55$$ $$\chi(n)$$ $$-1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
107.1
 −1.73205 + 3.53958i −1.73205 − 3.53958i 1.73205 − 0.353400i 1.73205 + 0.353400i −1.73205 + 3.22829i −1.73205 − 3.22829i 1.73205 + 1.98106i 1.73205 − 1.98106i −1.73205 − 1.98106i −1.73205 + 1.98106i 1.73205 − 3.22829i 1.73205 + 3.22829i −1.73205 + 0.353400i −1.73205 − 0.353400i 1.73205 − 3.53958i 1.73205 + 3.53958i
−5.65395 0.181119i 0 31.9344 + 2.04808i 69.1814i 0 238.886i −180.185 17.3637i 0 12.5301 391.148i
107.2 −5.65395 + 0.181119i 0 31.9344 2.04808i 69.1814i 0 238.886i −180.185 + 17.3637i 0 12.5301 + 391.148i
107.3 −5.08799 2.47231i 0 19.7753 + 25.1582i 33.1536i 0 59.4073i −38.4178 176.896i 0 −81.9660 + 168.685i
107.4 −5.08799 + 2.47231i 0 19.7753 25.1582i 33.1536i 0 59.4073i −38.4178 + 176.896i 0 −81.9660 168.685i
107.5 −3.82945 4.16357i 0 −2.67058 + 31.8884i 46.0468i 0 134.772i 142.996 110.996i 0 191.719 176.334i
107.6 −3.82945 + 4.16357i 0 −2.67058 31.8884i 46.0468i 0 134.772i 142.996 + 110.996i 0 191.719 + 176.334i
107.7 −1.79734 5.36373i 0 −25.5392 + 19.2809i 44.9719i 0 28.5094i 149.320 + 102.331i 0 241.217 80.8297i
107.8 −1.79734 + 5.36373i 0 −25.5392 19.2809i 44.9719i 0 28.5094i 149.320 102.331i 0 241.217 + 80.8297i
107.9 1.79734 5.36373i 0 −25.5392 19.2809i 44.9719i 0 28.5094i −149.320 + 102.331i 0 241.217 + 80.8297i
107.10 1.79734 + 5.36373i 0 −25.5392 + 19.2809i 44.9719i 0 28.5094i −149.320 102.331i 0 241.217 80.8297i
107.11 3.82945 4.16357i 0 −2.67058 31.8884i 46.0468i 0 134.772i −142.996 110.996i 0 191.719 + 176.334i
107.12 3.82945 + 4.16357i 0 −2.67058 + 31.8884i 46.0468i 0 134.772i −142.996 + 110.996i 0 191.719 176.334i
107.13 5.08799 2.47231i 0 19.7753 25.1582i 33.1536i 0 59.4073i 38.4178 176.896i 0 −81.9660 168.685i
107.14 5.08799 + 2.47231i 0 19.7753 + 25.1582i 33.1536i 0 59.4073i 38.4178 + 176.896i 0 −81.9660 + 168.685i
107.15 5.65395 0.181119i 0 31.9344 2.04808i 69.1814i 0 238.886i 180.185 17.3637i 0 12.5301 + 391.148i
107.16 5.65395 + 0.181119i 0 31.9344 + 2.04808i 69.1814i 0 238.886i 180.185 + 17.3637i 0 12.5301 391.148i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 107.16 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
4.b odd 2 1 inner
12.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 108.6.b.b 16
3.b odd 2 1 inner 108.6.b.b 16
4.b odd 2 1 inner 108.6.b.b 16
12.b even 2 1 inner 108.6.b.b 16

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
108.6.b.b 16 1.a even 1 1 trivial
108.6.b.b 16 3.b odd 2 1 inner
108.6.b.b 16 4.b odd 2 1 inner
108.6.b.b 16 12.b even 2 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{8} + 10028 T_{5}^{6} + 33930078 T_{5}^{4} + 47031065276 T_{5}^{2} +$$$$22\!\cdots\!25$$ acting on $$S_{6}^{\mathrm{new}}(108, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 - 47 T^{2} + 1060 T^{4} - 30080 T^{6} + 762880 T^{8} - 30801920 T^{10} + 1111490560 T^{12} - 50465865728 T^{14} + 1099511627776 T^{16}$$
$3$ 1
$5$ $$( 1 - 14972 T^{2} + 119342578 T^{4} - 617133972224 T^{6} + 2271820128564475 T^{8} - 6026698947500000000 T^{10} +$$$$11\!\cdots\!50$$$$T^{12} -$$$$13\!\cdots\!00$$$$T^{14} +$$$$90\!\cdots\!25$$$$T^{16} )^{2}$$
$7$ $$( 1 - 54884 T^{2} + 1251149938 T^{4} - 15826547698400 T^{6} + 189674874534565723 T^{8} -$$$$44\!\cdots\!00$$$$T^{10} +$$$$99\!\cdots\!38$$$$T^{12} -$$$$12\!\cdots\!16$$$$T^{14} +$$$$63\!\cdots\!01$$$$T^{16} )^{2}$$
$11$ $$( 1 + 741148 T^{2} + 299964078250 T^{4} + 79656648391789648 T^{6} +$$$$15\!\cdots\!99$$$$T^{8} +$$$$20\!\cdots\!48$$$$T^{10} +$$$$20\!\cdots\!50$$$$T^{12} +$$$$12\!\cdots\!48$$$$T^{14} +$$$$45\!\cdots\!01$$$$T^{16} )^{2}$$
$13$ $$( 1 - 224 T + 554236 T^{2} + 77263648 T^{3} + 200668790230 T^{4} + 28687451656864 T^{5} + 76406139088422364 T^{6} - 11465640035156329568 T^{7} +$$$$19\!\cdots\!01$$$$T^{8} )^{4}$$
$17$ $$( 1 - 4946936 T^{2} + 15515554849948 T^{4} - 34385249721915417800 T^{6} +$$$$55\!\cdots\!90$$$$T^{8} -$$$$69\!\cdots\!00$$$$T^{10} +$$$$63\!\cdots\!48$$$$T^{12} -$$$$40\!\cdots\!64$$$$T^{14} +$$$$16\!\cdots\!01$$$$T^{16} )^{2}$$
$19$ $$( 1 - 9713816 T^{2} + 55349997016060 T^{4} -$$$$21\!\cdots\!72$$$$T^{6} +$$$$61\!\cdots\!02$$$$T^{8} -$$$$13\!\cdots\!72$$$$T^{10} +$$$$20\!\cdots\!60$$$$T^{12} -$$$$22\!\cdots\!16$$$$T^{14} +$$$$14\!\cdots\!01$$$$T^{16} )^{2}$$
$23$ $$( 1 + 29994040 T^{2} + 487786705761244 T^{4} +$$$$51\!\cdots\!44$$$$T^{6} +$$$$39\!\cdots\!14$$$$T^{8} +$$$$21\!\cdots\!56$$$$T^{10} +$$$$83\!\cdots\!44$$$$T^{12} +$$$$21\!\cdots\!60$$$$T^{14} +$$$$29\!\cdots\!01$$$$T^{16} )^{2}$$
$29$ $$( 1 - 26575640 T^{2} + 859790443267708 T^{4} -$$$$25\!\cdots\!88$$$$T^{6} +$$$$44\!\cdots\!26$$$$T^{8} -$$$$10\!\cdots\!88$$$$T^{10} +$$$$15\!\cdots\!08$$$$T^{12} -$$$$19\!\cdots\!40$$$$T^{14} +$$$$31\!\cdots\!01$$$$T^{16} )^{2}$$
$31$ $$( 1 - 160216820 T^{2} + 12412163878108546 T^{4} -$$$$60\!\cdots\!08$$$$T^{6} +$$$$20\!\cdots\!87$$$$T^{8} -$$$$49\!\cdots\!08$$$$T^{10} +$$$$83\!\cdots\!46$$$$T^{12} -$$$$88\!\cdots\!20$$$$T^{14} +$$$$45\!\cdots\!01$$$$T^{16} )^{2}$$
$37$ $$( 1 + 17752 T + 254119924 T^{2} + 2137611581224 T^{3} + 19827261455867542 T^{4} +$$$$14\!\cdots\!68$$$$T^{5} +$$$$12\!\cdots\!76$$$$T^{6} +$$$$59\!\cdots\!36$$$$T^{7} +$$$$23\!\cdots\!01$$$$T^{8} )^{4}$$
$41$ $$( 1 - 667999304 T^{2} + 215857510496501980 T^{4} -$$$$43\!\cdots\!84$$$$T^{6} +$$$$60\!\cdots\!14$$$$T^{8} -$$$$58\!\cdots\!84$$$$T^{10} +$$$$38\!\cdots\!80$$$$T^{12} -$$$$16\!\cdots\!04$$$$T^{14} +$$$$32\!\cdots\!01$$$$T^{16} )^{2}$$
$43$ $$( 1 - 271892936 T^{2} + 81822516500494204 T^{4} -$$$$14\!\cdots\!72$$$$T^{6} +$$$$26\!\cdots\!02$$$$T^{8} -$$$$30\!\cdots\!28$$$$T^{10} +$$$$38\!\cdots\!04$$$$T^{12} -$$$$27\!\cdots\!64$$$$T^{14} +$$$$21\!\cdots\!01$$$$T^{16} )^{2}$$
$47$ $$( 1 + 1268760328 T^{2} + 775645167851419804 T^{4} +$$$$30\!\cdots\!92$$$$T^{6} +$$$$81\!\cdots\!58$$$$T^{8} +$$$$15\!\cdots\!08$$$$T^{10} +$$$$21\!\cdots\!04$$$$T^{12} +$$$$18\!\cdots\!72$$$$T^{14} +$$$$76\!\cdots\!01$$$$T^{16} )^{2}$$
$53$ $$( 1 - 1665548972 T^{2} + 1494189343182800962 T^{4} -$$$$95\!\cdots\!20$$$$T^{6} +$$$$46\!\cdots\!75$$$$T^{8} -$$$$16\!\cdots\!80$$$$T^{10} +$$$$45\!\cdots\!62$$$$T^{12} -$$$$89\!\cdots\!28$$$$T^{14} +$$$$93\!\cdots\!01$$$$T^{16} )^{2}$$
$59$ $$( 1 + 4822767208 T^{2} + 10641078380871180220 T^{4} +$$$$14\!\cdots\!92$$$$T^{6} +$$$$12\!\cdots\!22$$$$T^{8} +$$$$71\!\cdots\!92$$$$T^{10} +$$$$27\!\cdots\!20$$$$T^{12} +$$$$64\!\cdots\!08$$$$T^{14} +$$$$68\!\cdots\!01$$$$T^{16} )^{2}$$
$61$ $$( 1 - 19472 T + 1261489684 T^{2} - 43949943879536 T^{3} + 833752337244254902 T^{4} -$$$$37\!\cdots\!36$$$$T^{5} +$$$$89\!\cdots\!84$$$$T^{6} -$$$$11\!\cdots\!72$$$$T^{7} +$$$$50\!\cdots\!01$$$$T^{8} )^{4}$$
$67$ $$( 1 - 1447449416 T^{2} + 2915553445416739516 T^{4} -$$$$52\!\cdots\!68$$$$T^{6} +$$$$74\!\cdots\!58$$$$T^{8} -$$$$95\!\cdots\!32$$$$T^{10} +$$$$96\!\cdots\!16$$$$T^{12} -$$$$87\!\cdots\!84$$$$T^{14} +$$$$11\!\cdots\!01$$$$T^{16} )^{2}$$
$71$ $$( 1 + 9830208232 T^{2} + 46886702724166517020 T^{4} +$$$$14\!\cdots\!88$$$$T^{6} +$$$$30\!\cdots\!22$$$$T^{8} +$$$$46\!\cdots\!88$$$$T^{10} +$$$$49\!\cdots\!20$$$$T^{12} +$$$$33\!\cdots\!32$$$$T^{14} +$$$$11\!\cdots\!01$$$$T^{16} )^{2}$$
$73$ $$( 1 + 9508 T + 3097043794 T^{2} - 21796113866240 T^{3} + 7953703459188333211 T^{4} -$$$$45\!\cdots\!20$$$$T^{5} +$$$$13\!\cdots\!06$$$$T^{6} +$$$$84\!\cdots\!56$$$$T^{7} +$$$$18\!\cdots\!01$$$$T^{8} )^{4}$$
$79$ $$( 1 + 99394360 T^{2} + 19778301042529431580 T^{4} -$$$$22\!\cdots\!96$$$$T^{6} +$$$$20\!\cdots\!66$$$$T^{8} -$$$$21\!\cdots\!96$$$$T^{10} +$$$$17\!\cdots\!80$$$$T^{12} +$$$$84\!\cdots\!60$$$$T^{14} +$$$$80\!\cdots\!01$$$$T^{16} )^{2}$$
$83$ $$( 1 + 20559730876 T^{2} +$$$$21\!\cdots\!14$$$$T^{4} +$$$$14\!\cdots\!68$$$$T^{6} +$$$$70\!\cdots\!39$$$$T^{8} +$$$$23\!\cdots\!32$$$$T^{10} +$$$$52\!\cdots\!14$$$$T^{12} +$$$$76\!\cdots\!24$$$$T^{14} +$$$$57\!\cdots\!01$$$$T^{16} )^{2}$$
$89$ $$( 1 - 30119375768 T^{2} +$$$$43\!\cdots\!12$$$$T^{4} -$$$$41\!\cdots\!60$$$$T^{6} +$$$$27\!\cdots\!18$$$$T^{8} -$$$$12\!\cdots\!60$$$$T^{10} +$$$$42\!\cdots\!12$$$$T^{12} -$$$$91\!\cdots\!68$$$$T^{14} +$$$$94\!\cdots\!01$$$$T^{16} )^{2}$$
$97$ $$( 1 + 244 T + 6580710202 T^{2} + 1109461517844208 T^{3} - 18256445891874654653 T^{4} +$$$$95\!\cdots\!56$$$$T^{5} +$$$$48\!\cdots\!98$$$$T^{6} +$$$$15\!\cdots\!92$$$$T^{7} +$$$$54\!\cdots\!01$$$$T^{8} )^{4}$$