# Properties

 Label 2240.2.k.f Level $2240$ Weight $2$ Character orbit 2240.k Analytic conductor $17.886$ Analytic rank $0$ Dimension $16$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$2240 = 2^{6} \cdot 5 \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2240.k (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$17.8864900528$$ Analytic rank: $$0$$ Dimension: $$16$$ Coefficient field: $$\mathbb{Q}[x]/(x^{16} - \cdots)$$ Defining polynomial: $$x^{16} - 4 x^{15} - 6 x^{14} + 68 x^{13} - 126 x^{12} - 148 x^{11} + 1006 x^{10} - 1516 x^{9} - 179 x^{8} + 3992 x^{7} - 5596 x^{6} - 488 x^{5} + 16080 x^{4} - 33776 x^{3} + 37344 x^{2} - 22336 x + 5956$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{11}$$ Twist minimal: no (minimal twist has level 1120) 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.

 $$f(q)$$ $$=$$ $$q + \beta_{9} q^{3} + \beta_{4} q^{5} -\beta_{11} q^{7} + ( 1 - \beta_{1} - \beta_{8} + \beta_{9} ) q^{9} +O(q^{10})$$ $$q + \beta_{9} q^{3} + \beta_{4} q^{5} -\beta_{11} q^{7} + ( 1 - \beta_{1} - \beta_{8} + \beta_{9} ) q^{9} + ( -\beta_{2} + \beta_{3} - \beta_{7} + \beta_{11} - \beta_{14} ) q^{11} + ( \beta_{3} - \beta_{7} ) q^{13} + \beta_{7} q^{15} + ( -\beta_{3} + \beta_{6} + \beta_{7} - \beta_{11} - \beta_{13} + \beta_{14} ) q^{17} + ( \beta_{6} + \beta_{8} - \beta_{10} ) q^{19} + ( \beta_{3} - \beta_{4} - \beta_{6} - \beta_{9} + \beta_{11} - \beta_{12} ) q^{21} + ( \beta_{2} + \beta_{3} - \beta_{6} + \beta_{13} ) q^{23} - q^{25} + ( 2 + \beta_{4} + \beta_{6} + \beta_{9} + \beta_{12} - \beta_{15} ) q^{27} + ( -2 + \beta_{1} + \beta_{4} + \beta_{6} + \beta_{8} - \beta_{9} + \beta_{11} + \beta_{12} + \beta_{14} - \beta_{15} ) q^{29} + ( -2 \beta_{1} - \beta_{8} + 2 \beta_{9} + \beta_{10} - \beta_{11} + \beta_{13} - \beta_{14} ) q^{31} + ( 2 \beta_{2} - \beta_{3} - 2 \beta_{4} + \beta_{7} - 2 \beta_{11} + 2 \beta_{14} ) q^{33} -\beta_{13} q^{35} + ( -2 + \beta_{1} + \beta_{8} + 2 \beta_{9} + \beta_{10} + \beta_{11} + \beta_{13} + \beta_{14} ) q^{37} + ( \beta_{2} + \beta_{3} - 3 \beta_{4} - \beta_{6} - \beta_{7} - \beta_{12} - \beta_{15} ) q^{39} + ( -\beta_{2} - \beta_{3} - 2 \beta_{4} + \beta_{5} - 2 \beta_{7} ) q^{41} + ( -\beta_{2} - \beta_{3} + \beta_{4} - 2 \beta_{5} + \beta_{6} + 2 \beta_{7} - \beta_{12} - 2 \beta_{13} - \beta_{15} ) q^{43} + ( -\beta_{2} + \beta_{4} - \beta_{5} + \beta_{7} ) q^{45} + ( -2 + 2 \beta_{1} + \beta_{4} - \beta_{6} - \beta_{9} + 2 \beta_{10} - \beta_{11} + \beta_{12} - \beta_{14} - \beta_{15} ) q^{47} + ( -1 + \beta_{1} - \beta_{2} + \beta_{4} - \beta_{5} - \beta_{8} - 2 \beta_{9} + \beta_{12} - \beta_{13} + \beta_{14} ) q^{49} + ( -\beta_{2} + \beta_{3} + \beta_{4} - \beta_{6} - \beta_{7} + 2 \beta_{11} - \beta_{12} - 2 \beta_{14} - \beta_{15} ) q^{51} + ( -\beta_{4} + 2 \beta_{6} + 2 \beta_{8} - 2 \beta_{10} + 2 \beta_{11} - \beta_{12} + \beta_{13} + 2 \beta_{14} + \beta_{15} ) q^{53} + ( \beta_{8} + \beta_{9} + \beta_{10} + \beta_{13} ) q^{55} + ( 2 - \beta_{1} - \beta_{4} - \beta_{8} - 2 \beta_{9} - \beta_{10} + \beta_{11} - \beta_{12} + \beta_{14} + \beta_{15} ) q^{57} + ( \beta_{6} - 2 \beta_{8} - 2 \beta_{10} + \beta_{11} - \beta_{13} + \beta_{14} ) q^{59} + ( \beta_{2} - \beta_{3} + \beta_{4} + \beta_{5} - \beta_{6} + 2 \beta_{7} - 3 \beta_{11} + \beta_{12} + 2 \beta_{13} + 3 \beta_{14} + \beta_{15} ) q^{61} + ( -4 + 2 \beta_{1} + \beta_{2} + \beta_{3} + \beta_{4} + \beta_{6} + 2 \beta_{8} - \beta_{9} + \beta_{11} - \beta_{12} - \beta_{15} ) q^{63} + ( -\beta_{6} + \beta_{9} + \beta_{10} ) q^{65} + ( -\beta_{4} + 2 \beta_{7} - \beta_{11} + \beta_{12} + \beta_{13} + \beta_{14} + \beta_{15} ) q^{67} + ( -\beta_{2} + \beta_{3} + 2 \beta_{4} - \beta_{5} ) q^{69} + ( 2 \beta_{2} - 2 \beta_{3} - 5 \beta_{4} + 4 \beta_{5} - \beta_{6} - \beta_{11} + \beta_{12} + 2 \beta_{13} + \beta_{14} + \beta_{15} ) q^{71} + ( -2 \beta_{4} + 2 \beta_{5} - \beta_{6} - 2 \beta_{7} - \beta_{11} + \beta_{13} + \beta_{14} ) q^{73} -\beta_{9} q^{75} + ( -\beta_{1} + \beta_{3} + \beta_{4} + 2 \beta_{5} + \beta_{6} - \beta_{7} + \beta_{8} + \beta_{10} + \beta_{12} + 3 \beta_{13} + \beta_{15} ) q^{77} + ( -\beta_{2} - \beta_{3} - \beta_{7} + \beta_{11} - \beta_{14} ) q^{79} + ( 1 + \beta_{4} - \beta_{6} + 2 \beta_{9} + \beta_{11} + \beta_{12} - 2 \beta_{13} + \beta_{14} - \beta_{15} ) q^{81} + ( 6 - \beta_{4} - \beta_{6} - 2 \beta_{8} - 2 \beta_{9} - 2 \beta_{10} - \beta_{12} - 2 \beta_{13} + \beta_{15} ) q^{83} + ( -\beta_{9} - \beta_{10} + \beta_{11} - \beta_{13} + \beta_{14} ) q^{85} + ( -2 - 2 \beta_{1} + \beta_{4} - \beta_{6} - 2 \beta_{8} - \beta_{9} + \beta_{12} - 2 \beta_{13} - \beta_{15} ) q^{87} + ( -\beta_{2} - \beta_{3} + \beta_{4} - 3 \beta_{5} + \beta_{6} - \beta_{11} - \beta_{12} - 2 \beta_{13} + \beta_{14} - \beta_{15} ) q^{89} + ( -4 - \beta_{2} + \beta_{3} + 3 \beta_{4} + \beta_{6} - \beta_{7} + \beta_{8} - 2 \beta_{9} - \beta_{10} + \beta_{11} + \beta_{12} + \beta_{13} + \beta_{14} + \beta_{15} ) q^{91} + ( 2 + \beta_{1} + \beta_{4} - \beta_{8} + 2 \beta_{9} - \beta_{10} + \beta_{11} + \beta_{12} - 2 \beta_{13} + \beta_{14} - \beta_{15} ) q^{93} + ( \beta_{2} + \beta_{3} ) q^{95} + ( -2 \beta_{2} + \beta_{3} + 4 \beta_{4} - 2 \beta_{5} + \beta_{6} + \beta_{7} - \beta_{11} - \beta_{13} + \beta_{14} ) q^{97} + ( \beta_{4} - \beta_{6} - 2 \beta_{7} + \beta_{11} - \beta_{12} - \beta_{14} - \beta_{15} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$16q - 4q^{7} + 16q^{9} + O(q^{10})$$ $$16q - 4q^{7} + 16q^{9} - 8q^{19} - 4q^{21} - 16q^{25} + 48q^{27} - 8q^{29} - 16q^{37} - 8q^{47} - 4q^{49} - 16q^{53} + 8q^{55} + 16q^{57} - 8q^{59} - 60q^{63} + 8q^{65} + 8q^{77} + 40q^{81} + 64q^{83} - 16q^{87} - 64q^{91} + 48q^{93} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{16} - 4 x^{15} - 6 x^{14} + 68 x^{13} - 126 x^{12} - 148 x^{11} + 1006 x^{10} - 1516 x^{9} - 179 x^{8} + 3992 x^{7} - 5596 x^{6} - 488 x^{5} + 16080 x^{4} - 33776 x^{3} + 37344 x^{2} - 22336 x + 5956$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$-12758377455 \nu^{15} - 17029973070 \nu^{14} + 255451555872 \nu^{13} - 5782883602 \nu^{12} - 2428092530782 \nu^{11} + 2461866146935 \nu^{10} + 10812644798656 \nu^{9} - 20882203036101 \nu^{8} - 17931607412769 \nu^{7} + 69829732643649 \nu^{6} - 13633178133740 \nu^{5} - 109539021238893 \nu^{4} + 106762977907130 \nu^{3} + 7255442894980 \nu^{2} - 55212568945784 \nu + 634026321716742$$$$)/ 216025654874206$$ $$\beta_{2}$$ $$=$$ $$($$$$8765614024466674 \nu^{15} - 62502845947521146 \nu^{14} + 27930788945903613 \nu^{13} + 919541822694668216 \nu^{12} - 2758329551827048716 \nu^{11} - 505933606394946956 \nu^{10} + 16874220098891849632 \nu^{9} - 31541580432531125900 \nu^{8} + 2468356936003309764 \nu^{7} + 71827760037341267022 \nu^{6} - 102679857907601843147 \nu^{5} + 8812409656846222588 \nu^{4} + 240534886957138604770 \nu^{3} - 612154820371672153200 \nu^{2} + 735676972740082807394 \nu - 348289662532115458336$$$$)/ 64532047726642313144$$ $$\beta_{3}$$ $$=$$ $$($$$$13090040637728505 \nu^{15} + 211916728286895534 \nu^{14} - 725493091669104857 \nu^{13} - 1961008466169258786 \nu^{12} + 12493927886363852188 \nu^{11} - 10299818010203611792 \nu^{10} - 52349917243107468542 \nu^{9} + 149934458133588569104 \nu^{8} - 74189346620384657089 \nu^{7} - 263650896553140225122 \nu^{6} + 546458846557001737209 \nu^{5} - 205329180934372231738 \nu^{4} - 836795779768439795796 \nu^{3} + 2402362524933101244484 \nu^{2} - 2957390654429469965602 \nu + 1427468805432647561708$$$$)/ 64532047726642313144$$ $$\beta_{4}$$ $$=$$ $$($$$$-1210752049 \nu^{15} - 1110096336 \nu^{14} + 20403779147 \nu^{13} - 19050599618 \nu^{12} - 146738265037 \nu^{11} + 351660968602 \nu^{10} + 157695428815 \nu^{9} - 1793479248628 \nu^{8} + 1916781072920 \nu^{7} + 1497184388698 \nu^{6} - 6855726785840 \nu^{5} + 5953850442270 \nu^{4} + 1659909458620 \nu^{3} - 20164201672372 \nu^{2} + 32148639488212 \nu - 17489786211320$$$$)/ 4256018862188$$ $$\beta_{5}$$ $$=$$ $$($$$$-3343558 \nu^{15} - 1238164 \nu^{14} + 52258753 \nu^{13} - 74255495 \nu^{12} - 312877523 \nu^{11} + 939062890 \nu^{10} - 23564043 \nu^{9} - 3862442118 \nu^{8} + 4919305197 \nu^{7} + 1699089560 \nu^{6} - 14234786412 \nu^{5} + 16165231231 \nu^{4} - 5856993922 \nu^{3} - 37472206390 \nu^{2} + 73775229484 \nu - 42814645170$$$$)/ 3918178202$$ $$\beta_{6}$$ $$=$$ $$($$$$-120223170196860373 \nu^{15} + 508409725558160580 \nu^{14} + 1001904479712438592 \nu^{13} - 9006400866078907244 \nu^{12} + 12390972778503174636 \nu^{11} + 31494971175260942572 \nu^{10} - 125744891278723942282 \nu^{9} + 118832891793713150884 \nu^{8} + 160017238475171890871 \nu^{7} - 495008163163051562664 \nu^{6} + 391831367090758893822 \nu^{5} + 538294739675650401264 \nu^{4} - 2039725726045424389930 \nu^{3} + 3212935115563632993032 \nu^{2} - 2117068501808309500776 \nu + 331964725632513472440$$$$)/$$$$12\!\cdots\!88$$ $$\beta_{7}$$ $$=$$ $$($$$$126425665182626895 \nu^{15} - 402469074925196481 \nu^{14} - 1121956011275831728 \nu^{13} + 7733154061744690654 \nu^{12} - 9125291680193799000 \nu^{11} - 27685781977964858108 \nu^{10} + 103545419009722591998 \nu^{9} - 96472901443840712450 \nu^{8} - 114632157105709674301 \nu^{7} + 394060154218144271303 \nu^{6} - 329971656220426141850 \nu^{5} - 366229213299615695132 \nu^{4} + 1676643046921451745682 \nu^{3} - 2720002409034269825638 \nu^{2} + 2374127347238370132368 \nu - 908825391783385858004$$$$)/$$$$12\!\cdots\!88$$ $$\beta_{8}$$ $$=$$ $$($$$$82713379168634325 \nu^{15} - 281075459079638160 \nu^{14} - 713483502120422931 \nu^{13} + 5336541030249633080 \nu^{12} - 6695141944048570864 \nu^{11} - 19147893177766628976 \nu^{10} + 73841771752378564118 \nu^{9} - 68228822294103924196 \nu^{8} - 91976769270153237685 \nu^{7} + 294773493670649913012 \nu^{6} - 220270109894291825885 \nu^{5} - 310073024820485450084 \nu^{4} + 1201588731759913835968 \nu^{3} - 1840543284179116451012 \nu^{2} + 1393734925792015031754 \nu - 394988529315488072880$$$$)/ 64532047726642313144$$ $$\beta_{9}$$ $$=$$ $$($$$$237140810440379735 \nu^{15} - 694839951210633181 \nu^{14} - 2223510643919990830 \nu^{13} + 13842456444243259968 \nu^{12} - 14492654803433050080 \nu^{11} - 53007947937536874348 \nu^{10} + 183385760798826199310 \nu^{9} - 153337931243226373826 \nu^{8} - 237649344704454034305 \nu^{7} + 719125568618894204991 \nu^{6} - 517634899815047395024 \nu^{5} - 821736504755162748938 \nu^{4} + 3021381137616653188334 \nu^{3} - 4556629515839990044858 \nu^{2} + 3426054626530645624092 \nu - 1020322781709869753760$$$$)/$$$$12\!\cdots\!88$$ $$\beta_{10}$$ $$=$$ $$($$$$259293226844765315 \nu^{15} - 770370709610531096 \nu^{14} - 2305263887028898338 \nu^{13} + 15278820432296986152 \nu^{12} - 17498219145135478052 \nu^{11} - 55549252514026466004 \nu^{10} + 205855473162409076806 \nu^{9} - 189152648431021147452 \nu^{8} - 234485299461395158061 \nu^{7} + 807831922352873380748 \nu^{6} - 649630715069165603172 \nu^{5} - 728970739234563805676 \nu^{4} + 3399968126198016637530 \nu^{3} - 5424839780231186441632 \nu^{2} + 4533424520138778259396 \nu - 1310279850884661310672$$$$)/$$$$12\!\cdots\!88$$ $$\beta_{11}$$ $$=$$ $$($$$$-401514885070314967 \nu^{15} + 1118370145453479424 \nu^{14} + 3999070438841951874 \nu^{13} - 22929190127933666378 \nu^{12} + 20193867782511255404 \nu^{11} + 95284425048363017136 \nu^{10} - 293551487608489295942 \nu^{9} + 197696087447360859392 \nu^{8} + 449419601267458200241 \nu^{7} - 1107409032339395042580 \nu^{6} + 632064008668323899948 \nu^{5} + 1485218623630154093798 \nu^{4} - 4785700310140283855202 \nu^{3} + 6761685223629213815172 \nu^{2} - 4307250443939268057452 \nu + 958797510564913058372$$$$)/$$$$12\!\cdots\!88$$ $$\beta_{12}$$ $$=$$ $$($$$$-203138371139447396 \nu^{15} + 707031214890825801 \nu^{14} + 1605940222342174914 \nu^{13} - 12971112886976321418 \nu^{12} + 18772666473353955162 \nu^{11} + 40117807089961648770 \nu^{10} - 184898054661417274554 \nu^{9} + 212344977649881363120 \nu^{8} + 158314411051918264382 \nu^{7} - 748686664357395711185 \nu^{6} + 753019705595276023700 \nu^{5} + 546154222509972678626 \nu^{4} - 3060392513637979096840 \nu^{3} + 5296744972024004608390 \nu^{2} - 4644440188818386285000 \nu + 1678744867755471500432$$$$)/ 64532047726642313144$$ $$\beta_{13}$$ $$=$$ $$($$$$-622859702467771945 \nu^{15} + 1867679735746360758 \nu^{14} + 5650730542224720896 \nu^{13} - 36854430938284936774 \nu^{12} + 41116165628762327132 \nu^{11} + 136242982429528198460 \nu^{10} - 492786652970957868114 \nu^{9} + 438963481118088233728 \nu^{8} + 586766801271558866667 \nu^{7} - 1921017645189911857042 \nu^{6} + 1497237271104639140046 \nu^{5} + 1916936457289342249746 \nu^{4} - 8097992310736121571546 \nu^{3} + 12701471705105446312128 \nu^{2} - 10075478303001224763280 \nu + 3239428024377916922436$$$$)/$$$$12\!\cdots\!88$$ $$\beta_{14}$$ $$=$$ $$($$$$-625676353274793981 \nu^{15} + 2046731257544845448 \nu^{14} + 5284503975155719586 \nu^{13} - 38894072028060292708 \nu^{12} + 50274671850866341900 \nu^{11} + 132799764376717123880 \nu^{10} - 538509129785875372338 \nu^{9} + 543808656389170520872 \nu^{8} + 561969906032701808571 \nu^{7} - 2136373410007872469824 \nu^{6} + 1862585432143242059608 \nu^{5} + 1900518047840479370596 \nu^{4} - 8817764898518887072334 \nu^{3} + 14421858692828249207256 \nu^{2} - 11871046424796416720100 \nu + 4139600485325221870320$$$$)/$$$$12\!\cdots\!88$$ $$\beta_{15}$$ $$=$$ $$($$$$773830196453774649 \nu^{15} - 2257663355263442372 \nu^{14} - 7170009218278348232 \nu^{13} + 45269222028181278742 \nu^{12} - 47961230184040499388 \nu^{11} - 173776796765668563960 \nu^{10} + 601569813669377153970 \nu^{9} - 494725854442072439064 \nu^{8} - 777850522470759953827 \nu^{7} + 2319785730056291618072 \nu^{6} - 1659946859124546953910 \nu^{5} - 2510620628878174669634 \nu^{4} + 9819322295097294277378 \nu^{3} - 15062240195278282876764 \nu^{2} + 11281234388048318993104 \nu - 3185578392552200053596$$$$)/$$$$12\!\cdots\!88$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{13} + \beta_{10} + \beta_{9} + \beta_{7}$$$$)/2$$ $$\nu^{2}$$ $$=$$ $$($$$$-\beta_{14} + \beta_{11} - \beta_{10} - \beta_{9} + 2 \beta_{8} + \beta_{7} + \beta_{6} + 2 \beta_{4} + \beta_{3} - \beta_{2} + 4$$$$)/2$$ $$\nu^{3}$$ $$=$$ $$($$$$-3 \beta_{15} - 8 \beta_{14} + 9 \beta_{13} - \beta_{12} + 2 \beta_{11} + 11 \beta_{10} - 5 \beta_{8} + 4 \beta_{7} - 5 \beta_{6} - 5 \beta_{4} + 3 \beta_{3} - 3 \beta_{2} + 3 \beta_{1} - 22$$$$)/4$$ $$\nu^{4}$$ $$=$$ $$($$$$\beta_{15} - 5 \beta_{14} - 7 \beta_{13} - 3 \beta_{12} + 13 \beta_{11} - 12 \beta_{10} - 11 \beta_{9} + 13 \beta_{8} - 5 \beta_{7} + 4 \beta_{5} - 5 \beta_{4} + 8 \beta_{3} - 2 \beta_{2} - 3 \beta_{1} + 22$$$$)/2$$ $$\nu^{5}$$ $$=$$ $$($$$$-21 \beta_{15} - 46 \beta_{14} + 37 \beta_{13} + 13 \beta_{12} + 75 \beta_{10} + 24 \beta_{9} - 79 \beta_{8} - 8 \beta_{7} - 39 \beta_{6} + 22 \beta_{5} - 75 \beta_{4} + 7 \beta_{3} - 13 \beta_{2} + 31 \beta_{1} - 182$$$$)/4$$ $$\nu^{6}$$ $$=$$ $$($$$$15 \beta_{15} - 82 \beta_{13} - 19 \beta_{12} + 98 \beta_{11} - 130 \beta_{10} - 43 \beta_{9} + 100 \beta_{8} - 31 \beta_{7} + 3 \beta_{6} + 14 \beta_{5} - 21 \beta_{4} + 40 \beta_{3} + 7 \beta_{2} - 25 \beta_{1} + 214$$$$)/2$$ $$\nu^{7}$$ $$=$$ $$($$$$-81 \beta_{15} - 144 \beta_{14} + 263 \beta_{13} + 151 \beta_{12} - 174 \beta_{11} + 581 \beta_{10} + 320 \beta_{9} - 749 \beta_{8} + 48 \beta_{7} - 215 \beta_{6} + 81 \beta_{5} - 283 \beta_{4} - 109 \beta_{3} - 177 \beta_{2} + 286 \beta_{1} - 1544$$$$)/4$$ $$\nu^{8}$$ $$=$$ $$($$$$179 \beta_{15} + 231 \beta_{14} - 757 \beta_{13} - 225 \beta_{12} + 653 \beta_{11} - 1232 \beta_{10} - 463 \beta_{9} + 907 \beta_{8} - 81 \beta_{7} + 124 \beta_{6} - 74 \beta_{5} + 281 \beta_{4} + 252 \beta_{3} + 208 \beta_{2} - 205 \beta_{1} + 2002$$$$)/2$$ $$\nu^{9}$$ $$=$$ $$($$$$-553 \beta_{15} + 46 \beta_{14} + 2417 \beta_{13} + 1181 \beta_{12} - 3100 \beta_{11} + 5499 \beta_{10} + 2580 \beta_{9} - 6271 \beta_{8} + 916 \beta_{7} - 1191 \beta_{6} + 316 \beta_{5} - 1839 \beta_{4} - 1881 \beta_{3} - 1529 \beta_{2} + 2449 \beta_{1} - 14066$$$$)/4$$ $$\nu^{10}$$ $$=$$ $$($$$$1161 \beta_{15} + 2128 \beta_{14} - 6430 \beta_{13} - 2141 \beta_{12} + 4718 \beta_{11} - 10312 \beta_{10} - 4485 \beta_{9} + 8492 \beta_{8} - 1041 \beta_{7} + 1531 \beta_{6} - 1270 \beta_{5} + 4613 \beta_{4} + 2350 \beta_{3} + 2981 \beta_{2} - 2169 \beta_{1} + 18554$$$$)/2$$ $$\nu^{11}$$ $$=$$ $$($$$$-5851 \beta_{15} + 3292 \beta_{14} + 25033 \beta_{13} + 11801 \beta_{12} - 35758 \beta_{11} + 54695 \beta_{10} + 24584 \beta_{9} - 53811 \beta_{8} + 8872 \beta_{7} - 7513 \beta_{6} + 2785 \beta_{5} - 21121 \beta_{4} - 22107 \beta_{3} - 15347 \beta_{2} + 18288 \beta_{1} - 121468$$$$)/4$$ $$\nu^{12}$$ $$=$$ $$($$$$8415 \beta_{15} + 11375 \beta_{14} - 49763 \beta_{13} - 17893 \beta_{12} + 41993 \beta_{11} - 85620 \beta_{10} - 38113 \beta_{9} + 77845 \beta_{8} - 12523 \beta_{7} + 14390 \beta_{6} - 12908 \beta_{5} + 53341 \beta_{4} + 25804 \beta_{3} + 31616 \beta_{2} - 23965 \beta_{1} + 174470$$$$)/2$$ $$\nu^{13}$$ $$=$$ $$($$$$-47307 \beta_{15} + 26662 \beta_{14} + 261031 \beta_{13} + 116311 \beta_{12} - 346640 \beta_{11} + 514361 \beta_{10} + 228868 \beta_{9} - 466245 \beta_{8} + 95516 \beta_{7} - 50433 \beta_{6} + 39340 \beta_{5} - 263453 \beta_{4} - 226323 \beta_{3} - 173323 \beta_{2} + 141931 \beta_{1} - 1043830$$$$)/4$$ $$\nu^{14}$$ $$=$$ $$($$$$69001 \beta_{15} + 17414 \beta_{14} - 384702 \beta_{13} - 164929 \beta_{12} + 427592 \beta_{11} - 738564 \beta_{10} - 333975 \beta_{9} + 706460 \beta_{8} - 137783 \beta_{7} + 115051 \beta_{6} - 114152 \beta_{5} + 544801 \beta_{4} + 298730 \beta_{3} + 313623 \beta_{2} - 231165 \beta_{1} + 1588582$$$$)/2$$ $$\nu^{15}$$ $$=$$ $$($$$$-373861 \beta_{15} + 223888 \beta_{14} + 2459443 \beta_{13} + 1059647 \beta_{12} - 3146942 \beta_{11} + 4620993 \beta_{10} + 2034216 \beta_{9} - 4088325 \beta_{8} + 956432 \beta_{7} - 388587 \beta_{6} + 541235 \beta_{5} - 3161903 \beta_{4} - 2184813 \beta_{3} - 1894373 \beta_{2} + 1209820 \beta_{1} - 9147324$$$$)/4$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/2240\mathbb{Z}\right)^\times$$.

 $$n$$ $$897$$ $$1471$$ $$1541$$ $$1921$$ $$\chi(n)$$ $$1$$ $$-1$$ $$1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1791.1
 −0.00830917 + 1.38001i −0.00830917 − 1.38001i 2.23037 + 0.887196i 2.23037 − 0.887196i −1.65334 + 0.722492i −1.65334 − 0.722492i 1.23885 + 0.371381i 1.23885 − 0.371381i −3.01485 + 0.0876273i −3.01485 − 0.0876273i 0.715495 − 0.510550i 0.715495 + 0.510550i 1.40610 − 1.29043i 1.40610 + 1.29043i 1.08568 − 1.64772i 1.08568 + 1.64772i
0 −2.76002 0 1.00000i 0 −2.62068 + 0.363381i 0 4.61769 0
1791.2 0 −2.76002 0 1.00000i 0 −2.62068 0.363381i 0 4.61769 0
1791.3 0 −1.77439 0 1.00000i 0 0.829909 2.51222i 0 0.148464 0
1791.4 0 −1.77439 0 1.00000i 0 0.829909 + 2.51222i 0 0.148464 0
1791.5 0 −1.44498 0 1.00000i 0 2.48080 0.919594i 0 −0.912023 0
1791.6 0 −1.44498 0 1.00000i 0 2.48080 + 0.919594i 0 −0.912023 0
1791.7 0 −0.742762 0 1.00000i 0 −2.53467 0.758567i 0 −2.44831 0
1791.8 0 −0.742762 0 1.00000i 0 −2.53467 + 0.758567i 0 −2.44831 0
1791.9 0 −0.175255 0 1.00000i 0 0.684520 + 2.55567i 0 −2.96929 0
1791.10 0 −0.175255 0 1.00000i 0 0.684520 2.55567i 0 −2.96929 0
1791.11 0 1.02110 0 1.00000i 0 1.49445 + 2.18326i 0 −1.95735 0
1791.12 0 1.02110 0 1.00000i 0 1.49445 2.18326i 0 −1.95735 0
1791.13 0 2.58086 0 1.00000i 0 −0.319247 2.62642i 0 3.66083 0
1791.14 0 2.58086 0 1.00000i 0 −0.319247 + 2.62642i 0 3.66083 0
1791.15 0 3.29545 0 1.00000i 0 −2.01507 + 1.71449i 0 7.85998 0
1791.16 0 3.29545 0 1.00000i 0 −2.01507 1.71449i 0 7.85998 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1791.16 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
28.d even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2240.2.k.f 16
4.b odd 2 1 2240.2.k.g 16
7.b odd 2 1 2240.2.k.g 16
8.b even 2 1 1120.2.k.a 16
8.d odd 2 1 1120.2.k.b yes 16
28.d even 2 1 inner 2240.2.k.f 16
56.e even 2 1 1120.2.k.a 16
56.h odd 2 1 1120.2.k.b yes 16

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1120.2.k.a 16 8.b even 2 1
1120.2.k.a 16 56.e even 2 1
1120.2.k.b yes 16 8.d odd 2 1
1120.2.k.b yes 16 56.h odd 2 1
2240.2.k.f 16 1.a even 1 1 trivial
2240.2.k.f 16 28.d even 2 1 inner
2240.2.k.g 16 4.b odd 2 1
2240.2.k.g 16 7.b odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(2240, [\chi])$$:

 $$T_{3}^{8} - 16 T_{3}^{6} - 8 T_{3}^{5} + 69 T_{3}^{4} + 64 T_{3}^{3} - 50 T_{3}^{2} - 56 T_{3} - 8$$ $$T_{19}^{8} + \cdots$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{16}$$
$3$ $$( -8 - 56 T - 50 T^{2} + 64 T^{3} + 69 T^{4} - 8 T^{5} - 16 T^{6} + T^{8} )^{2}$$
$5$ $$( 1 + T^{2} )^{8}$$
$7$ $$5764801 + 3294172 T + 1176490 T^{2} + 873964 T^{3} + 307328 T^{4} + 56252 T^{5} + 29302 T^{6} + 7532 T^{7} + 190 T^{8} + 1076 T^{9} + 598 T^{10} + 164 T^{11} + 128 T^{12} + 52 T^{13} + 10 T^{14} + 4 T^{15} + T^{16}$$
$11$ $$4096 + 458240 T^{2} + 1692944 T^{4} + 1334888 T^{6} + 400289 T^{8} + 54956 T^{10} + 3542 T^{12} + 100 T^{14} + T^{16}$$
$13$ $$20736 + 158976 T^{2} + 438880 T^{4} + 539664 T^{6} + 289329 T^{8} + 53268 T^{10} + 3574 T^{12} + 100 T^{14} + T^{16}$$
$17$ $$131974144 + 155093760 T^{2} + 71941840 T^{4} + 16930648 T^{6} + 2177601 T^{8} + 155700 T^{10} + 6134 T^{12} + 124 T^{14} + T^{16}$$
$19$ $$( 2048 - 3072 T - 1152 T^{2} + 1792 T^{3} + 384 T^{4} - 256 T^{5} - 60 T^{6} + 4 T^{7} + T^{8} )^{2}$$
$23$ $$4734976 + 13099008 T^{2} + 12598272 T^{4} + 5132288 T^{6} + 995264 T^{8} + 98400 T^{10} + 4948 T^{12} + 116 T^{14} + T^{16}$$
$29$ $$( 272 - 4192 T - 30504 T^{2} + 9528 T^{3} + 4617 T^{4} - 436 T^{5} - 138 T^{6} + 4 T^{7} + T^{8} )^{2}$$
$31$ $$( 32768 - 8192 T - 26368 T^{2} + 3584 T^{3} + 4320 T^{4} - 32 T^{5} - 128 T^{6} + T^{8} )^{2}$$
$37$ $$( 122624 - 46080 T - 50944 T^{2} + 12672 T^{3} + 3840 T^{4} - 640 T^{5} - 112 T^{6} + 8 T^{7} + T^{8} )^{2}$$
$41$ $$53557067776 + 38860750848 T^{2} + 10589323264 T^{4} + 1357856768 T^{6} + 85502976 T^{8} + 2714368 T^{10} + 44048 T^{12} + 344 T^{14} + T^{16}$$
$43$ $$122466402304 + 183466065920 T^{2} + 39227764736 T^{4} + 3304139776 T^{6} + 142480448 T^{8} + 3445600 T^{10} + 47156 T^{12} + 340 T^{14} + T^{16}$$
$47$ $$( -231568 + 766912 T - 287030 T^{2} - 14468 T^{3} + 13581 T^{4} - 224 T^{5} - 204 T^{6} + 4 T^{7} + T^{8} )^{2}$$
$53$ $$( -665856 - 1459200 T - 464000 T^{2} + 129664 T^{3} + 20480 T^{4} - 2176 T^{5} - 280 T^{6} + 8 T^{7} + T^{8} )^{2}$$
$59$ $$( -672768 + 1970688 T - 637760 T^{2} - 21184 T^{3} + 22832 T^{4} - 448 T^{5} - 268 T^{6} + 4 T^{7} + T^{8} )^{2}$$
$61$ $$647532301582336 + 168587098128384 T^{2} + 10287941124096 T^{4} + 294979448832 T^{6} + 4750933504 T^{8} + 45476608 T^{10} + 256384 T^{12} + 784 T^{14} + T^{16}$$
$67$ $$16384 + 647168 T^{2} + 7105536 T^{4} + 21862400 T^{6} + 21440192 T^{8} + 2491232 T^{10} + 57492 T^{12} + 436 T^{14} + T^{16}$$
$71$ $$22419997720576 + 70103100030976 T^{2} + 6607642955776 T^{4} + 252540254208 T^{6} + 4963401472 T^{8} + 53264640 T^{10} + 307696 T^{12} + 888 T^{14} + T^{16}$$
$73$ $$113154195456 + 245725986816 T^{2} + 79220592640 T^{4} + 9381095424 T^{6} + 492927744 T^{8} + 11843584 T^{10} + 124848 T^{12} + 584 T^{14} + T^{16}$$
$79$ $$1196606464 + 1999695104 T^{2} + 1075124688 T^{4} + 249816632 T^{6} + 26891185 T^{8} + 1353180 T^{10} + 30278 T^{12} + 292 T^{14} + T^{16}$$
$83$ $$( -12271744 - 4208512 T + 2571168 T^{2} - 141504 T^{3} - 50056 T^{4} + 5576 T^{5} + 102 T^{6} - 32 T^{7} + T^{8} )^{2}$$
$89$ $$718886928384 + 2794048192512 T^{2} + 510216503296 T^{4} + 36256497664 T^{6} + 1238941952 T^{8} + 21196800 T^{10} + 180256 T^{12} + 704 T^{14} + T^{16}$$
$97$ $$335984794092544 + 56407587773184 T^{2} + 3798419954896 T^{4} + 133405156696 T^{6} + 2676287009 T^{8} + 31426644 T^{10} + 210582 T^{12} + 732 T^{14} + T^{16}$$