# Properties

 Label 108.5.d.b Level 108 Weight 5 Character orbit 108.d Analytic conductor 11.164 Analytic rank 0 Dimension 16 CM no Inner twists 4

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$108 = 2^{2} \cdot 3^{3}$$ Weight: $$k$$ $$=$$ $$5$$ Character orbit: $$[\chi]$$ $$=$$ 108.d (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$11.1639560131$$ Analytic rank: $$0$$ Dimension: $$16$$ Coefficient field: $$\mathbb{Q}[x]/(x^{16} + \cdots)$$ Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$2^{32}\cdot 3^{24}$$ 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.

 $$f(q)$$ $$=$$ $$q -\beta_{7} q^{2} + ( 2 + \beta_{2} ) q^{4} + ( -\beta_{7} + \beta_{9} ) q^{5} + \beta_{5} q^{7} + ( -2 \beta_{7} - \beta_{10} ) q^{8} +O(q^{10})$$ $$q -\beta_{7} q^{2} + ( 2 + \beta_{2} ) q^{4} + ( -\beta_{7} + \beta_{9} ) q^{5} + \beta_{5} q^{7} + ( -2 \beta_{7} - \beta_{10} ) q^{8} + ( 11 - \beta_{5} + \beta_{6} ) q^{10} + ( \beta_{7} - \beta_{11} ) q^{11} + ( 11 - \beta_{1} - \beta_{2} ) q^{13} + ( -\beta_{7} + 2 \beta_{9} + \beta_{10} - \beta_{11} + \beta_{12} ) q^{14} + ( 6 - \beta_{1} + \beta_{2} + \beta_{4} - \beta_{5} + \beta_{8} ) q^{16} + ( -6 \beta_{7} - \beta_{9} - 2 \beta_{10} + \beta_{15} ) q^{17} + ( \beta_{3} + \beta_{4} - 2 \beta_{5} + 2 \beta_{6} + 2 \beta_{8} ) q^{19} + ( -13 \beta_{7} + 6 \beta_{9} - \beta_{10} + \beta_{11} - \beta_{12} + \beta_{13} + \beta_{14} - 2 \beta_{15} ) q^{20} + ( 25 - 2 \beta_{1} + 2 \beta_{2} + \beta_{3} - 2 \beta_{4} + 5 \beta_{5} + \beta_{6} ) q^{22} + ( -6 \beta_{7} + \beta_{11} - 2 \beta_{12} - \beta_{13} - 2 \beta_{14} + \beta_{15} ) q^{23} + ( 175 - 2 \beta_{1} + 14 \beta_{2} - 3 \beta_{3} - 2 \beta_{5} + 6 \beta_{6} - 2 \beta_{8} ) q^{25} + ( -8 \beta_{7} - 2 \beta_{9} - \beta_{10} + 5 \beta_{11} + 3 \beta_{12} - 2 \beta_{13} - \beta_{14} ) q^{26} + ( 114 - \beta_{1} + 2 \beta_{2} - \beta_{3} + 5 \beta_{4} + 11 \beta_{5} + 4 \beta_{6} - \beta_{8} ) q^{28} + ( 48 \beta_{7} - 4 \beta_{9} + 12 \beta_{10} - 4 \beta_{11} - 4 \beta_{12} + 4 \beta_{14} ) q^{29} + ( 8 + 32 \beta_{2} + 5 \beta_{3} + 2 \beta_{5} + 2 \beta_{6} + 2 \beta_{8} ) q^{31} + ( 3 \beta_{7} - 26 \beta_{9} - 3 \beta_{10} + 5 \beta_{11} + 3 \beta_{12} + 3 \beta_{13} + \beta_{14} - 2 \beta_{15} ) q^{32} + ( 97 + 8 \beta_{2} - 2 \beta_{3} - 8 \beta_{4} + \beta_{5} - \beta_{6} + 4 \beta_{8} ) q^{34} + ( 107 \beta_{7} + 12 \beta_{10} - \beta_{11} + 2 \beta_{13} - 8 \beta_{14} + 6 \beta_{15} ) q^{35} + ( 9 - 3 \beta_{1} + 13 \beta_{2} - \beta_{3} + 2 \beta_{5} - 6 \beta_{6} + 2 \beta_{8} ) q^{37} + ( 8 \beta_{7} - 32 \beta_{9} + 4 \beta_{13} + 5 \beta_{14} - 8 \beta_{15} ) q^{38} + ( -44 - 4 \beta_{2} + 8 \beta_{3} + 16 \beta_{4} - 20 \beta_{5} + 4 \beta_{6} ) q^{40} + ( 154 \beta_{7} - 8 \beta_{9} - 4 \beta_{11} - 4 \beta_{12} - 8 \beta_{13} + 4 \beta_{14} + 6 \beta_{15} ) q^{41} + ( -16 - 64 \beta_{2} - 9 \beta_{3} + 2 \beta_{4} + 8 \beta_{5} - 2 \beta_{6} - 2 \beta_{8} ) q^{43} + ( -25 \beta_{7} + 14 \beta_{9} - \beta_{10} + 5 \beta_{11} + 11 \beta_{12} - 11 \beta_{13} - 3 \beta_{14} - 2 \beta_{15} ) q^{44} + ( -111 - 2 \beta_{1} + 10 \beta_{2} - 5 \beta_{3} - 26 \beta_{4} - 23 \beta_{5} - 3 \beta_{6} - 4 \beta_{8} ) q^{46} + ( -22 \beta_{7} - 12 \beta_{10} - 7 \beta_{11} - 14 \beta_{12} - \beta_{13} - 6 \beta_{14} + \beta_{15} ) q^{47} + ( -526 + 6 \beta_{1} - 122 \beta_{2} + 16 \beta_{3} ) q^{49} + ( -201 \beta_{7} + 84 \beta_{9} - 22 \beta_{10} + 14 \beta_{11} + 2 \beta_{12} + 16 \beta_{13} + 2 \beta_{14} - 8 \beta_{15} ) q^{50} + ( -330 + 6 \beta_{1} - 5 \beta_{2} - 14 \beta_{3} + 34 \beta_{4} - 6 \beta_{5} - 4 \beta_{6} - 2 \beta_{8} ) q^{52} + ( -328 \beta_{7} + 14 \beta_{9} - 8 \beta_{10} - 4 \beta_{11} - 4 \beta_{12} + 16 \beta_{13} + 4 \beta_{14} + 10 \beta_{15} ) q^{53} + ( -8 - 32 \beta_{2} - 9 \beta_{3} - 10 \beta_{4} + 4 \beta_{5} - 10 \beta_{6} - 10 \beta_{8} ) q^{55} + ( -141 \beta_{7} + 74 \beta_{9} + 6 \beta_{10} - 5 \beta_{11} + 13 \beta_{12} + 5 \beta_{13} + 7 \beta_{14} - 6 \beta_{15} ) q^{56} + ( -732 + 16 \beta_{1} - 16 \beta_{2} + 24 \beta_{3} - 48 \beta_{4} + 20 \beta_{5} - 4 \beta_{6} ) q^{58} + ( -201 \beta_{7} + 24 \beta_{10} - 25 \beta_{11} + 4 \beta_{12} - 26 \beta_{13} - 12 \beta_{14} + 10 \beta_{15} ) q^{59} + ( -71 + 15 \beta_{1} + 143 \beta_{2} - 12 \beta_{3} + 8 \beta_{5} - 24 \beta_{6} + 8 \beta_{8} ) q^{61} + ( 4 \beta_{7} - 24 \beta_{9} - 28 \beta_{10} - 4 \beta_{11} + 4 \beta_{12} - 28 \beta_{13} + 4 \beta_{14} - 8 \beta_{15} ) q^{62} + ( 312 + 6 \beta_{1} - 10 \beta_{2} - 4 \beta_{3} + 58 \beta_{4} + 18 \beta_{5} - 28 \beta_{6} + 2 \beta_{8} ) q^{64} + ( 94 \beta_{7} + 45 \beta_{9} - 2 \beta_{10} - 4 \beta_{11} - 4 \beta_{12} - 8 \beta_{13} + 4 \beta_{14} + 7 \beta_{15} ) q^{65} + ( 48 + 192 \beta_{2} + 18 \beta_{3} + 5 \beta_{4} - 28 \beta_{5} - 12 \beta_{6} - 12 \beta_{8} ) q^{67} + ( -79 \beta_{7} - 94 \beta_{9} - 3 \beta_{10} - 5 \beta_{11} + 5 \beta_{12} + 35 \beta_{13} - 5 \beta_{14} - 6 \beta_{15} ) q^{68} + ( 1649 - 2 \beta_{1} - 78 \beta_{2} - 35 \beta_{3} - 82 \beta_{4} + 5 \beta_{5} + \beta_{6} - 24 \beta_{8} ) q^{70} + ( 650 \beta_{7} - 24 \beta_{10} - 6 \beta_{11} - 16 \beta_{12} + 52 \beta_{13} - 4 \beta_{15} ) q^{71} + ( 5 + 22 \beta_{1} + 22 \beta_{2} - 4 \beta_{3} - 8 \beta_{5} + 24 \beta_{6} - 8 \beta_{8} ) q^{73} + ( 24 \beta_{7} - 94 \beta_{9} - 15 \beta_{10} + 11 \beta_{11} + 13 \beta_{12} + 6 \beta_{13} - 7 \beta_{14} + 8 \beta_{15} ) q^{74} + ( -526 - \beta_{1} - 8 \beta_{2} + 31 \beta_{3} + 85 \beta_{4} + 31 \beta_{5} - 32 \beta_{6} - \beta_{8} ) q^{76} + ( 915 \beta_{7} + 21 \beta_{9} + 20 \beta_{10} - 4 \beta_{11} - 4 \beta_{12} - 48 \beta_{13} + 4 \beta_{14} - 4 \beta_{15} ) q^{77} + ( -56 - 224 \beta_{2} - 25 \beta_{3} - 20 \beta_{4} - 79 \beta_{5} + 6 \beta_{6} + 6 \beta_{8} ) q^{79} + ( 68 \beta_{7} - 8 \beta_{9} - 16 \beta_{10} + 20 \beta_{11} - 20 \beta_{12} - 60 \beta_{13} + 20 \beta_{14} - 8 \beta_{15} ) q^{80} + ( -2432 + 16 \beta_{1} - 96 \beta_{2} + 12 \beta_{3} - 96 \beta_{4} + 24 \beta_{5} - 8 \beta_{6} + 24 \beta_{8} ) q^{82} + ( -94 \beta_{7} + 60 \beta_{10} + 22 \beta_{11} + 28 \beta_{12} - 24 \beta_{13} - 12 \beta_{14} + 16 \beta_{15} ) q^{83} + ( -1112 - 6 \beta_{1} - 246 \beta_{2} + 31 \beta_{3} + 2 \beta_{5} - 6 \beta_{6} + 2 \beta_{8} ) q^{85} + ( -14 \beta_{7} + 44 \beta_{9} + 70 \beta_{10} - 6 \beta_{11} + 6 \beta_{12} + 60 \beta_{13} - 2 \beta_{14} + 8 \beta_{15} ) q^{86} + ( -1844 - 4 \beta_{1} - 32 \beta_{3} + 116 \beta_{4} + 24 \beta_{5} + 20 \beta_{6} - 12 \beta_{8} ) q^{88} + ( -1390 \beta_{7} - 89 \beta_{9} + 70 \beta_{10} - 8 \beta_{11} - 8 \beta_{12} + 96 \beta_{13} + 8 \beta_{14} - 23 \beta_{15} ) q^{89} + ( -16 - 64 \beta_{2} - 5 \beta_{3} + 43 \beta_{4} + 22 \beta_{5} + 6 \beta_{6} + 6 \beta_{8} ) q^{91} + ( 119 \beta_{7} + 14 \beta_{9} - 41 \beta_{10} + 37 \beta_{11} - 21 \beta_{12} + 13 \beta_{13} - 35 \beta_{14} + 14 \beta_{15} ) q^{92} + ( -259 - 42 \beta_{1} + 50 \beta_{2} + 15 \beta_{3} - 130 \beta_{4} - 91 \beta_{5} - 7 \beta_{6} - 4 \beta_{8} ) q^{94} + ( -1448 \beta_{7} - 72 \beta_{10} + 75 \beta_{11} - 26 \beta_{12} - 73 \beta_{13} + 22 \beta_{14} - 23 \beta_{15} ) q^{95} + ( 981 - 36 \beta_{1} + 172 \beta_{2} - 23 \beta_{3} + 6 \beta_{5} - 18 \beta_{6} + 6 \beta_{8} ) q^{97} + ( 540 \beta_{7} + 12 \beta_{9} + 134 \beta_{10} - 30 \beta_{11} - 18 \beta_{12} - 116 \beta_{13} + 6 \beta_{14} ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$16q + 28q^{4} + O(q^{10})$$ $$16q + 28q^{4} + 176q^{10} + 176q^{13} + 88q^{16} + 384q^{22} + 2736q^{25} + 1812q^{28} + 1520q^{34} + 80q^{37} - 688q^{40} - 1824q^{46} - 7904q^{49} - 5236q^{52} - 11584q^{58} - 1648q^{61} + 5056q^{64} + 26688q^{70} + 80q^{73} - 8388q^{76} - 38464q^{82} - 16832q^{85} - 29520q^{88} - 4512q^{94} + 14864q^{97} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{16} + 38 x^{14} + 1016 x^{12} + 13512 x^{10} + 130640 x^{8} + 569472 x^{6} + 1783808 x^{4} + 352256 x^{2} + 65536$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$-31526031 \nu^{14} - 1478004994 \nu^{12} - 34969469560 \nu^{10} - 518036651256 \nu^{8} - 4125035344112 \nu^{6} - 36728960741376 \nu^{4} - 169615122399232 \nu^{2} - 560383182118912$$$$)/ 9072826085376$$ $$\beta_{2}$$ $$=$$ $$($$$$22125963 \nu^{14} + 810251098 \nu^{12} + 21483034168 \nu^{10} + 275776021848 \nu^{8} + 2670612406832 \nu^{6} + 11426488988928 \nu^{4} + 42847095737344 \nu^{2} + 35527188072448$$$$)/ 2268206521344$$ $$\beta_{3}$$ $$=$$ $$($$$$55772295 \nu^{14} + 2016497426 \nu^{12} + 53270230904 \nu^{10} + 669847068600 \nu^{8} + 6417247127152 \nu^{6} + 24668040322560 \nu^{4} + 86052727752704 \nu^{2} - 54341410430464$$$$)/ 567051630336$$ $$\beta_{4}$$ $$=$$ $$($$$$336655503 \nu^{14} + 12700746162 \nu^{12} + 338874029208 \nu^{10} + 4466455855992 \nu^{8} + 43034314712688 \nu^{6} + 183107704096512 \nu^{4} + 586008280737792 \nu^{2} + 61283674128384$$$$)/ 2016183574528$$ $$\beta_{5}$$ $$=$$ $$($$$$5389537731 \nu^{14} + 205346757098 \nu^{12} + 5490588520184 \nu^{10} + 73163083212312 \nu^{8} + 705848697514672 \nu^{6} + 3070851986567424 \nu^{4} + 9296357327919104 \nu^{2} + 975274043457536$$$$)/ 18145652170752$$ $$\beta_{6}$$ $$=$$ $$($$$$5752396299 \nu^{14} + 214867475674 \nu^{12} + 5697961181752 \nu^{10} + 73822365326424 \nu^{8} + 698772611343152 \nu^{6} + 2793782286461184 \nu^{4} + 8311410549225472 \nu^{2} - 975085761093632$$$$)/ 18145652170752$$ $$\beta_{7}$$ $$=$$ $$($$$$736076377 \nu^{15} + 27811759278 \nu^{13} + 742057419752 \nu^{11} + 9793364237576 \nu^{9} + 94235408393872 \nu^{7} + 400964425500928 \nu^{5} + 1240302616969728 \nu^{3} - 12225716432896 \nu$$$$)/ 27218478256128$$ $$\beta_{8}$$ $$=$$ $$($$$$3544178403 \nu^{14} + 135483347818 \nu^{12} + 3637189030264 \nu^{10} + 48837038113560 \nu^{8} + 477434654409392 \nu^{6} + 2151704842462464 \nu^{4} + 7070030810220544 \nu^{2} + 2168664507645952$$$$)/ 9072826085376$$ $$\beta_{9}$$ $$=$$ $$($$$$21174584519 \nu^{15} + 801681331074 \nu^{13} + 21390001763416 \nu^{11} + 282726138030136 \nu^{9} + 2716360618555376 \nu^{7} + 11557905817313024 \nu^{5} + 35110154205044736 \nu^{3} + 323296419487744 \nu$$$$)/ 108873913024512$$ $$\beta_{10}$$ $$=$$ $$($$$$4319670871 \nu^{15} + 163390571778 \nu^{13} + 4358708917496 \nu^{11} + 57545083803896 \nu^{9} + 552823897601776 \nu^{7} + 2343933237913600 \nu^{5} + 7146123589074432 \nu^{3} - 530311192121344 \nu$$$$)/ 13609239128064$$ $$\beta_{11}$$ $$=$$ $$($$$$-39104187809 \nu^{15} - 1500703996398 \nu^{13} - 40295155484776 \nu^{11} - 543226320658696 \nu^{9} - 5302351460587664 \nu^{7} - 24032631381185792 \nu^{5} - 76650707290251264 \nu^{3} - 31324629737660416 \nu$$$$)/ 108873913024512$$ $$\beta_{12}$$ $$=$$ $$($$$$40828295945 \nu^{15} + 1588698201534 \nu^{13} + 42879935422312 \nu^{11} + 588927554877256 \nu^{9} + 5820851472606224 \nu^{7} + 27890207958042368 \nu^{5} + 91471110079451136 \nu^{3} + 68043692147138560 \nu$$$$)/ 108873913024512$$ $$\beta_{13}$$ $$=$$ $$($$$$-5825845645 \nu^{15} - 220083807270 \nu^{13} - 5872149996680 \nu^{11} - 77486461375016 \nu^{9} - 745716488182480 \nu^{7} - 3172966386699520 \nu^{5} - 9854134880263680 \nu^{3} - 363609072300032 \nu$$$$)/ 13609239128064$$ $$\beta_{14}$$ $$=$$ $$($$$$826772793 \nu^{15} + 31470960102 \nu^{13} + 842058646200 \nu^{11} + 11229447493704 \nu^{9} + 108851393709648 \nu^{7} + 480287270553216 \nu^{5} + 1530890719268352 \nu^{3} + 505026864599040 \nu$$$$)/ 1008091787264$$ $$\beta_{15}$$ $$=$$ $$($$$$-30611053565 \nu^{15} - 1158020226150 \nu^{13} - 30900770575624 \nu^{11} - 408252459978280 \nu^{9} - 3926943140555984 \nu^{7} - 16742001436051712 \nu^{5} - 51568612005548544 \nu^{3} - 3288361928396800 \nu$$$$)/ 27218478256128$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$-2 \beta_{15} - 2 \beta_{14} - 13 \beta_{13} + 2 \beta_{12} + 2 \beta_{11} - 2 \beta_{10} - 206 \beta_{7}$$$$)/432$$ $$\nu^{2}$$ $$=$$ $$($$$$-6 \beta_{8} - 6 \beta_{6} - 6 \beta_{5} + 40 \beta_{4} - 9 \beta_{3} + 24 \beta_{2} - 1020$$$$)/216$$ $$\nu^{3}$$ $$=$$ $$($$$$38 \beta_{15} + 26 \beta_{14} - 5 \beta_{13} - 26 \beta_{12} - 26 \beta_{11} + 2 \beta_{10} + 96 \beta_{9} + 14 \beta_{7}$$$$)/216$$ $$\nu^{4}$$ $$=$$ $$($$$$66 \beta_{8} + 78 \beta_{6} - 6 \beta_{5} - 308 \beta_{4} - 9 \beta_{3} + 372 \beta_{2} + 12 \beta_{1} - 7848$$$$)/108$$ $$\nu^{5}$$ $$=$$ $$($$$$-290 \beta_{15} - 2 \beta_{14} + 1061 \beta_{13} - 46 \beta_{12} - 70 \beta_{11} - 266 \beta_{10} - 648 \beta_{9} + 12286 \beta_{7}$$$$)/108$$ $$\nu^{6}$$ $$=$$ $$($$$$22 \beta_{8} - 66 \beta_{6} + 22 \beta_{5} + 285 \beta_{3} - 2328 \beta_{2} - 136 \beta_{1} + 45388$$$$)/18$$ $$\nu^{7}$$ $$=$$ $$($$$$-778 \beta_{15} - 1594 \beta_{14} - 8285 \beta_{13} + 2410 \beta_{12} + 2962 \beta_{11} + 5222 \beta_{10} - 7080 \beta_{9} - 119614 \beta_{7}$$$$)/54$$ $$\nu^{8}$$ $$=$$ $$($$$$-7578 \beta_{8} - 6354 \beta_{6} + 3870 \beta_{5} + 23576 \beta_{4} - 5877 \beta_{3} + 48744 \beta_{2} + 2592 \beta_{1} - 621288$$$$)/27$$ $$\nu^{9}$$ $$=$$ $$($$$$36574 \beta_{15} + 12274 \beta_{14} - 23467 \beta_{13} - 12274 \beta_{12} - 12274 \beta_{11} - 36326 \beta_{10} + 146880 \beta_{9} + 154522 \beta_{7}$$$$)/27$$ $$\nu^{10}$$ $$=$$ $$($$$$140844 \beta_{8} + 163236 \beta_{6} - 103236 \beta_{5} - 439880 \beta_{4} - 53370 \beta_{3} + 287016 \beta_{2} + 58968 \beta_{1} - 11942760$$$$)/27$$ $$\nu^{11}$$ $$=$$ $$($$$$-569636 \beta_{15} + 38524 \beta_{14} + 1958798 \beta_{13} - 274396 \beta_{12} - 465484 \beta_{11} - 378548 \beta_{10} - 1496976 \beta_{9} + 18980692 \beta_{7}$$$$)/27$$ $$\nu^{12}$$ $$=$$ $$($$$$69512 \beta_{8} - 208536 \beta_{6} + 69512 \beta_{5} + 1029588 \beta_{3} - 8805984 \beta_{2} - 847328 \beta_{1} + 152703488$$$$)/9$$ $$\nu^{13}$$ $$=$$ $$($$$$-3273608 \beta_{15} - 4697768 \beta_{14} - 27463084 \beta_{13} + 9781736 \beta_{12} + 14031560 \beta_{11} + 24750424 \beta_{10} - 30282336 \beta_{9} - 450395432 \beta_{7}$$$$)/27$$ $$\nu^{14}$$ $$=$$ $$($$$$-60988272 \beta_{8} - 53055984 \beta_{6} + 42608208 \beta_{5} + 163331456 \beta_{4} - 36299592 \beta_{3} + 451622784 \beta_{2} + 26599872 \beta_{1} - 4437300384$$$$)/27$$ $$\nu^{15}$$ $$=$$ $$($$$$292631920 \beta_{15} + 72188176 \beta_{14} - 237941896 \beta_{13} - 72188176 \beta_{12} - 72188176 \beta_{11} - 368699312 \beta_{10} + 1220340480 \beta_{9} + 1808260144 \beta_{7}$$$$)/27$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
55.1
 −0.222504 − 0.385387i −0.222504 + 0.385387i −1.12787 − 1.95353i −1.12787 + 1.95353i 1.78073 − 3.08431i 1.78073 + 3.08431i −2.23772 − 3.87585i −2.23772 + 3.87585i 2.23772 − 3.87585i 2.23772 + 3.87585i −1.78073 − 3.08431i −1.78073 + 3.08431i 1.12787 − 1.95353i 1.12787 + 1.95353i 0.222504 − 0.385387i 0.222504 + 0.385387i
−3.98139 0.385387i 0 15.7030 + 3.06876i 22.8093 0 82.1204i −61.3369 18.2696i 0 −90.8127 8.79041i
55.2 −3.98139 + 0.385387i 0 15.7030 3.06876i 22.8093 0 82.1204i −61.3369 + 18.2696i 0 −90.8127 + 8.79041i
55.3 −3.49052 1.95353i 0 8.36746 + 13.6377i −47.1174 0 67.4003i −2.56528 63.9486i 0 164.464 + 92.0451i
55.4 −3.49052 + 1.95353i 0 8.36746 13.6377i −47.1174 0 67.4003i −2.56528 + 63.9486i 0 164.464 92.0451i
55.5 −2.54696 3.08431i 0 −3.02598 + 15.7113i 3.58334 0 16.2605i 56.1655 30.6829i 0 −9.12663 11.0521i
55.6 −2.54696 + 3.08431i 0 −3.02598 15.7113i 3.58334 0 16.2605i 56.1655 + 30.6829i 0 −9.12663 + 11.0521i
55.7 −0.988828 3.87585i 0 −14.0444 + 7.66510i 20.7568 0 5.38785i 43.5963 + 46.8547i 0 −20.5250 80.4504i
55.8 −0.988828 + 3.87585i 0 −14.0444 7.66510i 20.7568 0 5.38785i 43.5963 46.8547i 0 −20.5250 + 80.4504i
55.9 0.988828 3.87585i 0 −14.0444 7.66510i −20.7568 0 5.38785i −43.5963 + 46.8547i 0 −20.5250 + 80.4504i
55.10 0.988828 + 3.87585i 0 −14.0444 + 7.66510i −20.7568 0 5.38785i −43.5963 46.8547i 0 −20.5250 80.4504i
55.11 2.54696 3.08431i 0 −3.02598 15.7113i −3.58334 0 16.2605i −56.1655 30.6829i 0 −9.12663 + 11.0521i
55.12 2.54696 + 3.08431i 0 −3.02598 + 15.7113i −3.58334 0 16.2605i −56.1655 + 30.6829i 0 −9.12663 11.0521i
55.13 3.49052 1.95353i 0 8.36746 13.6377i 47.1174 0 67.4003i 2.56528 63.9486i 0 164.464 92.0451i
55.14 3.49052 + 1.95353i 0 8.36746 + 13.6377i 47.1174 0 67.4003i 2.56528 + 63.9486i 0 164.464 + 92.0451i
55.15 3.98139 0.385387i 0 15.7030 3.06876i −22.8093 0 82.1204i 61.3369 18.2696i 0 −90.8127 + 8.79041i
55.16 3.98139 + 0.385387i 0 15.7030 + 3.06876i −22.8093 0 82.1204i 61.3369 + 18.2696i 0 −90.8127 8.79041i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 55.16 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 108.5.d.b 16
3.b odd 2 1 inner 108.5.d.b 16
4.b odd 2 1 inner 108.5.d.b 16
12.b even 2 1 inner 108.5.d.b 16

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{8} - 3184 T_{5}^{6} + 2376384 T_{5}^{4} - 527623168 T_{5}^{2} + 6389764096$$ acting on $$S_{5}^{\mathrm{new}}(108, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 - 14 T^{2} + 76 T^{4} - 992 T^{6} - 2816 T^{8} - 253952 T^{10} + 4980736 T^{12} - 234881024 T^{14} + 4294967296 T^{16}$$
$3$ 1
$5$ $$( 1 + 1816 T^{2} + 1373884 T^{4} + 428961832 T^{6} + 50788147846 T^{8} + 167563215625000 T^{10} + 209638061523437500 T^{12} +$$$$10\!\cdots\!00$$$$T^{14} +$$$$23\!\cdots\!25$$$$T^{16} )^{2}$$
$7$ $$( 1 - 7628 T^{2} + 28548058 T^{4} - 90794922320 T^{6} + 251781425444899 T^{8} - 523414658985258320 T^{10} +$$$$94\!\cdots\!58$$$$T^{12} -$$$$14\!\cdots\!28$$$$T^{14} +$$$$11\!\cdots\!01$$$$T^{16} )^{2}$$
$11$ $$( 1 - 66488 T^{2} + 2449218172 T^{4} - 58905964133768 T^{6} + 1015821535494509830 T^{8} -$$$$12\!\cdots\!08$$$$T^{10} +$$$$11\!\cdots\!92$$$$T^{12} -$$$$65\!\cdots\!08$$$$T^{14} +$$$$21\!\cdots\!21$$$$T^{16} )^{2}$$
$13$ $$( 1 - 44 T + 67090 T^{2} - 966368 T^{3} + 2192145307 T^{4} - 27600436448 T^{5} + 54727374071890 T^{6} - 1025115745389164 T^{7} + 665416609183179841 T^{8} )^{4}$$
$17$ $$( 1 + 422296 T^{2} + 87803746876 T^{4} + 11972376766816936 T^{6} +$$$$11\!\cdots\!30$$$$T^{8} +$$$$83\!\cdots\!76$$$$T^{10} +$$$$42\!\cdots\!56$$$$T^{12} +$$$$14\!\cdots\!16$$$$T^{14} +$$$$23\!\cdots\!61$$$$T^{16} )^{2}$$
$19$ $$( 1 - 298556 T^{2} + 58331731402 T^{4} - 10017587005489136 T^{6} +$$$$14\!\cdots\!39$$$$T^{8} -$$$$17\!\cdots\!76$$$$T^{10} +$$$$16\!\cdots\!62$$$$T^{12} -$$$$14\!\cdots\!76$$$$T^{14} +$$$$83\!\cdots\!61$$$$T^{16} )^{2}$$
$23$ $$( 1 - 1184696 T^{2} + 509745640444 T^{4} - 84604026939000968 T^{6} +$$$$71\!\cdots\!22$$$$T^{8} -$$$$66\!\cdots\!08$$$$T^{10} +$$$$31\!\cdots\!84$$$$T^{12} -$$$$56\!\cdots\!36$$$$T^{14} +$$$$37\!\cdots\!21$$$$T^{16} )^{2}$$
$29$ $$( 1 + 1223560 T^{2} + 603752953756 T^{4} - 126384872678073416 T^{6} -$$$$34\!\cdots\!30$$$$T^{8} -$$$$63\!\cdots\!76$$$$T^{10} +$$$$15\!\cdots\!76$$$$T^{12} +$$$$15\!\cdots\!60$$$$T^{14} +$$$$62\!\cdots\!41$$$$T^{16} )^{2}$$
$31$ $$( 1 - 4495112 T^{2} + 10589743228444 T^{4} - 16346503421024387384 T^{6} +$$$$17\!\cdots\!66$$$$T^{8} -$$$$13\!\cdots\!44$$$$T^{10} +$$$$77\!\cdots\!64$$$$T^{12} -$$$$27\!\cdots\!52$$$$T^{14} +$$$$52\!\cdots\!61$$$$T^{16} )^{2}$$
$37$ $$( 1 - 20 T + 5679442 T^{2} - 534101024 T^{3} + 14604995159131 T^{4} - 1000991309240864 T^{5} + 19948923334735992082 T^{6} -$$$$13\!\cdots\!20$$$$T^{7} +$$$$12\!\cdots\!41$$$$T^{8} )^{4}$$
$41$ $$( 1 + 8498440 T^{2} + 39457953574684 T^{4} +$$$$14\!\cdots\!68$$$$T^{6} +$$$$47\!\cdots\!42$$$$T^{8} +$$$$11\!\cdots\!28$$$$T^{10} +$$$$25\!\cdots\!44$$$$T^{12} +$$$$43\!\cdots\!40$$$$T^{14} +$$$$40\!\cdots\!81$$$$T^{16} )^{2}$$
$43$ $$( 1 - 16979720 T^{2} + 140301823474588 T^{4} -$$$$77\!\cdots\!28$$$$T^{6} +$$$$30\!\cdots\!14$$$$T^{8} -$$$$90\!\cdots\!28$$$$T^{10} +$$$$19\!\cdots\!88$$$$T^{12} -$$$$27\!\cdots\!20$$$$T^{14} +$$$$18\!\cdots\!01$$$$T^{16} )^{2}$$
$47$ $$( 1 - 11004728 T^{2} + 98757031754236 T^{4} -$$$$67\!\cdots\!20$$$$T^{6} +$$$$36\!\cdots\!06$$$$T^{8} -$$$$16\!\cdots\!20$$$$T^{10} +$$$$55\!\cdots\!56$$$$T^{12} -$$$$14\!\cdots\!68$$$$T^{14} +$$$$32\!\cdots\!41$$$$T^{16} )^{2}$$
$53$ $$( 1 + 25978312 T^{2} + 440128249758748 T^{4} +$$$$51\!\cdots\!52$$$$T^{6} +$$$$46\!\cdots\!86$$$$T^{8} +$$$$32\!\cdots\!72$$$$T^{10} +$$$$17\!\cdots\!08$$$$T^{12} +$$$$62\!\cdots\!72$$$$T^{14} +$$$$15\!\cdots\!41$$$$T^{16} )^{2}$$
$59$ $$( 1 - 31826744 T^{2} + 538590143907196 T^{4} -$$$$92\!\cdots\!92$$$$T^{6} +$$$$13\!\cdots\!74$$$$T^{8} -$$$$13\!\cdots\!32$$$$T^{10} +$$$$11\!\cdots\!36$$$$T^{12} -$$$$10\!\cdots\!84$$$$T^{14} +$$$$46\!\cdots\!81$$$$T^{16} )^{2}$$
$61$ $$( 1 + 412 T + 9607138 T^{2} + 3267599488 T^{3} + 216368249231659 T^{4} + 45242662962529408 T^{5} +$$$$18\!\cdots\!78$$$$T^{6} +$$$$10\!\cdots\!52$$$$T^{7} +$$$$36\!\cdots\!61$$$$T^{8} )^{4}$$
$67$ $$( 1 - 38213180 T^{2} + 1702311028453834 T^{4} -$$$$38\!\cdots\!28$$$$T^{6} +$$$$99\!\cdots\!27$$$$T^{8} -$$$$15\!\cdots\!48$$$$T^{10} +$$$$28\!\cdots\!54$$$$T^{12} -$$$$25\!\cdots\!80$$$$T^{14} +$$$$27\!\cdots\!61$$$$T^{16} )^{2}$$
$71$ $$( 1 - 102214088 T^{2} + 3903098982427036 T^{4} -$$$$62\!\cdots\!96$$$$T^{6} +$$$$63\!\cdots\!58$$$$T^{8} -$$$$40\!\cdots\!56$$$$T^{10} +$$$$16\!\cdots\!56$$$$T^{12} -$$$$27\!\cdots\!28$$$$T^{14} +$$$$17\!\cdots\!41$$$$T^{16} )^{2}$$
$73$ $$( 1 - 20 T + 67167226 T^{2} + 662878096 T^{3} + 2724353301452611 T^{4} + 18824571923829136 T^{5} +$$$$54\!\cdots\!06$$$$T^{6} -$$$$45\!\cdots\!20$$$$T^{7} +$$$$65\!\cdots\!61$$$$T^{8} )^{4}$$
$79$ $$( 1 - 164533964 T^{2} + 15212503116962266 T^{4} -$$$$95\!\cdots\!24$$$$T^{6} +$$$$43\!\cdots\!71$$$$T^{8} -$$$$14\!\cdots\!64$$$$T^{10} +$$$$35\!\cdots\!86$$$$T^{12} -$$$$57\!\cdots\!84$$$$T^{14} +$$$$52\!\cdots\!41$$$$T^{16} )^{2}$$
$83$ $$( 1 - 185955656 T^{2} + 18958921421058076 T^{4} -$$$$13\!\cdots\!16$$$$T^{6} +$$$$73\!\cdots\!26$$$$T^{8} -$$$$30\!\cdots\!56$$$$T^{10} +$$$$96\!\cdots\!56$$$$T^{12} -$$$$21\!\cdots\!76$$$$T^{14} +$$$$25\!\cdots\!61$$$$T^{16} )^{2}$$
$89$ $$( 1 + 103821208 T^{2} + 12140769019811644 T^{4} +$$$$81\!\cdots\!28$$$$T^{6} +$$$$66\!\cdots\!06$$$$T^{8} +$$$$32\!\cdots\!68$$$$T^{10} +$$$$18\!\cdots\!84$$$$T^{12} +$$$$63\!\cdots\!28$$$$T^{14} +$$$$24\!\cdots\!21$$$$T^{16} )^{2}$$
$97$ $$( 1 - 3716 T + 259176202 T^{2} - 799969063952 T^{3} + 30273937496959507 T^{4} - 70820686053913578512 T^{5} +$$$$20\!\cdots\!22$$$$T^{6} -$$$$25\!\cdots\!56$$$$T^{7} +$$$$61\!\cdots\!21$$$$T^{8} )^{4}$$