# Properties

 Label 1840.3.k.d Level $1840$ Weight $3$ Character orbit 1840.k Analytic conductor $50.136$ Analytic rank $0$ Dimension $16$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1840 = 2^{4} \cdot 5 \cdot 23$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 1840.k (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$50.1363686423$$ Analytic rank: $$0$$ Dimension: $$16$$ Coefficient field: $$\mathbb{Q}[x]/(x^{16} + \cdots)$$ Defining polynomial: $$x^{16} + 78 x^{14} + 2165 x^{12} + 28310 x^{10} + 184804 x^{8} + 569634 x^{6} + 696037 x^{4} + 285578 x^{2} + 529$$ Coefficient ring: $$\Z[a_1, \ldots, a_{23}]$$ Coefficient ring index: $$2^{9}$$ Twist minimal: no (minimal twist has level 230) 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.

 $$f(q)$$ $$=$$ $$q + \beta_{4} q^{3} + \beta_{2} q^{5} + ( \beta_{1} + \beta_{5} ) q^{7} + ( 5 + \beta_{9} + \beta_{10} - \beta_{12} ) q^{9} +O(q^{10})$$ $$q + \beta_{4} q^{3} + \beta_{2} q^{5} + ( \beta_{1} + \beta_{5} ) q^{7} + ( 5 + \beta_{9} + \beta_{10} - \beta_{12} ) q^{9} + ( \beta_{1} - \beta_{2} - \beta_{3} + \beta_{5} + \beta_{7} ) q^{11} + ( 1 - \beta_{4} + 2 \beta_{6} - \beta_{9} - \beta_{10} - \beta_{12} - \beta_{14} + \beta_{15} ) q^{13} + ( -\beta_{2} - \beta_{3} + \beta_{5} - \beta_{8} - \beta_{13} ) q^{15} + ( -2 \beta_{2} + 3 \beta_{5} + \beta_{7} + \beta_{11} - \beta_{13} ) q^{17} + ( \beta_{1} - 3 \beta_{2} + \beta_{3} - \beta_{5} - \beta_{7} - 2 \beta_{8} + \beta_{11} ) q^{19} + ( -3 \beta_{1} + \beta_{2} - 4 \beta_{3} + \beta_{5} + 2 \beta_{7} - 5 \beta_{8} - 2 \beta_{11} - 3 \beta_{13} ) q^{21} + ( -1 - \beta_{1} + 2 \beta_{2} - \beta_{4} + \beta_{6} + \beta_{7} - 2 \beta_{8} - \beta_{9} + \beta_{12} - \beta_{14} - \beta_{15} ) q^{23} -5 q^{25} + ( 8 + 2 \beta_{1} + \beta_{2} + \beta_{3} + \beta_{4} - \beta_{5} - 3 \beta_{6} + 2 \beta_{8} + 4 \beta_{9} + \beta_{11} + \beta_{13} + 2 \beta_{15} ) q^{27} + ( -6 + 6 \beta_{1} + 3 \beta_{2} + 3 \beta_{3} - \beta_{4} - 3 \beta_{5} + 2 \beta_{6} + 6 \beta_{8} + \beta_{9} + 2 \beta_{10} + 3 \beta_{11} + \beta_{12} + 3 \beta_{13} - 2 \beta_{14} + \beta_{15} ) q^{29} + ( 8 - 2 \beta_{1} - \beta_{2} - \beta_{3} + 5 \beta_{4} + \beta_{5} - 3 \beta_{6} - 2 \beta_{8} + 2 \beta_{9} - \beta_{11} + \beta_{12} - \beta_{13} - 3 \beta_{15} ) q^{31} + ( -5 \beta_{1} - \beta_{2} - 8 \beta_{3} + \beta_{5} + 4 \beta_{7} - 9 \beta_{8} - 5 \beta_{11} - 5 \beta_{13} ) q^{33} + ( -4 + 2 \beta_{1} + \beta_{2} + \beta_{3} - 2 \beta_{4} - \beta_{5} + 2 \beta_{6} + 2 \beta_{8} + \beta_{11} + \beta_{12} + \beta_{13} + \beta_{15} ) q^{35} + ( -5 \beta_{1} + 3 \beta_{2} - 4 \beta_{3} - 2 \beta_{5} + 5 \beta_{7} - \beta_{8} - 6 \beta_{11} ) q^{37} + ( -18 - 8 \beta_{1} - 4 \beta_{2} - 4 \beta_{3} + 2 \beta_{4} + 4 \beta_{5} + 7 \beta_{6} - 8 \beta_{8} - 2 \beta_{9} - 5 \beta_{10} - 4 \beta_{11} + \beta_{12} - 4 \beta_{13} + 4 \beta_{14} ) q^{39} + ( -10 - 2 \beta_{1} - \beta_{2} - \beta_{3} - 3 \beta_{4} + \beta_{5} - 2 \beta_{8} - \beta_{9} + 2 \beta_{10} - \beta_{11} + \beta_{12} - \beta_{13} + 4 \beta_{14} - \beta_{15} ) q^{41} + ( -\beta_{1} - 4 \beta_{2} + 2 \beta_{3} + 2 \beta_{5} - \beta_{7} - \beta_{11} + 5 \beta_{13} ) q^{43} + ( \beta_{1} + 3 \beta_{2} - 2 \beta_{3} + 2 \beta_{5} + 2 \beta_{7} + \beta_{8} ) q^{45} + ( 10 + 4 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} - \beta_{4} - 2 \beta_{5} + 5 \beta_{6} + 4 \beta_{8} + 2 \beta_{9} + 4 \beta_{10} + 2 \beta_{11} + 2 \beta_{13} + 2 \beta_{14} + 2 \beta_{15} ) q^{47} + ( -1 - 2 \beta_{1} - \beta_{2} - \beta_{3} - \beta_{4} + \beta_{5} + 4 \beta_{6} - 2 \beta_{8} + 3 \beta_{10} - \beta_{11} - \beta_{13} + 2 \beta_{14} - 3 \beta_{15} ) q^{49} + ( 2 \beta_{1} + 12 \beta_{2} - 2 \beta_{3} - 2 \beta_{5} - 4 \beta_{7} + 4 \beta_{8} - \beta_{11} ) q^{51} + ( -12 \beta_{2} - 6 \beta_{3} - \beta_{5} + \beta_{7} - 6 \beta_{8} - 3 \beta_{11} + \beta_{13} ) q^{53} + ( -2 + 2 \beta_{1} + \beta_{2} + \beta_{3} - \beta_{4} - \beta_{5} + 2 \beta_{8} - 2 \beta_{9} - \beta_{10} + \beta_{11} + 3 \beta_{12} + \beta_{13} ) q^{55} + ( 13 \beta_{1} + 3 \beta_{2} + 10 \beta_{3} - 11 \beta_{5} - 2 \beta_{7} + 11 \beta_{8} + 7 \beta_{11} + 5 \beta_{13} ) q^{57} + ( -10 - 6 \beta_{1} - 3 \beta_{2} - 3 \beta_{3} - 3 \beta_{4} + 3 \beta_{5} + 4 \beta_{6} - 6 \beta_{8} + 3 \beta_{10} - 3 \beta_{11} - 8 \beta_{12} - 3 \beta_{13} - \beta_{15} ) q^{59} + ( -5 \beta_{1} - 3 \beta_{2} - 2 \beta_{3} + 9 \beta_{5} + 8 \beta_{7} - 9 \beta_{8} - 2 \beta_{11} - 5 \beta_{13} ) q^{61} + ( -2 \beta_{1} + 19 \beta_{2} - 11 \beta_{3} - 2 \beta_{5} + 5 \beta_{7} - 4 \beta_{11} ) q^{63} + ( 4 \beta_{1} + 3 \beta_{2} - 5 \beta_{5} - 2 \beta_{7} + 2 \beta_{8} + 2 \beta_{11} + 3 \beta_{13} ) q^{65} + ( -13 \beta_{1} - 10 \beta_{2} + 2 \beta_{3} + 3 \beta_{5} + 6 \beta_{7} - 8 \beta_{8} - 10 \beta_{11} - 4 \beta_{13} ) q^{67} + ( -21 + 11 \beta_{1} + 2 \beta_{2} - \beta_{3} - 5 \beta_{4} - 4 \beta_{5} - 3 \beta_{7} + 7 \beta_{8} - 6 \beta_{9} + \beta_{10} - \beta_{11} + 6 \beta_{12} + 2 \beta_{13} + 2 \beta_{14} - 3 \beta_{15} ) q^{69} + ( -16 + 2 \beta_{1} + \beta_{2} + \beta_{3} - 13 \beta_{4} - \beta_{5} + 7 \beta_{6} + 2 \beta_{8} - 2 \beta_{9} - 2 \beta_{10} + \beta_{11} - \beta_{12} + \beta_{13} - 4 \beta_{14} + \beta_{15} ) q^{71} + ( -3 + 7 \beta_{4} + 8 \beta_{6} + 7 \beta_{9} + \beta_{10} - \beta_{12} + 3 \beta_{14} + 3 \beta_{15} ) q^{73} -5 \beta_{4} q^{75} + ( -62 - 8 \beta_{1} - 4 \beta_{2} - 4 \beta_{3} + 2 \beta_{4} + 4 \beta_{5} - 4 \beta_{6} - 8 \beta_{8} - 4 \beta_{9} - 2 \beta_{10} - 4 \beta_{11} + 4 \beta_{12} - 4 \beta_{13} + 2 \beta_{14} - 8 \beta_{15} ) q^{77} + ( 9 \beta_{1} + 9 \beta_{2} + \beta_{3} - 17 \beta_{5} + \beta_{7} + 14 \beta_{8} + 8 \beta_{11} + 6 \beta_{13} ) q^{79} + ( -5 + 4 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} + 6 \beta_{4} - 2 \beta_{5} - 20 \beta_{6} + 4 \beta_{8} + 8 \beta_{9} - 2 \beta_{10} + 2 \beta_{11} + 2 \beta_{13} - 4 \beta_{14} + 2 \beta_{15} ) q^{81} + ( -6 \beta_{1} - 3 \beta_{2} - 7 \beta_{3} - 8 \beta_{5} + 3 \beta_{7} - 4 \beta_{8} + 6 \beta_{11} - 8 \beta_{13} ) q^{83} + ( 2 + 2 \beta_{1} + \beta_{2} + \beta_{3} - 3 \beta_{4} - \beta_{5} - 2 \beta_{6} + 2 \beta_{8} - 2 \beta_{9} - 3 \beta_{10} + \beta_{11} + \beta_{13} - 2 \beta_{14} + 3 \beta_{15} ) q^{85} + ( 14 + 6 \beta_{1} + 3 \beta_{2} + 3 \beta_{3} - 11 \beta_{4} - 3 \beta_{5} - 8 \beta_{6} + 6 \beta_{8} + 6 \beta_{9} + 10 \beta_{10} + 3 \beta_{11} + 4 \beta_{12} + 3 \beta_{13} + 2 \beta_{14} - 6 \beta_{15} ) q^{87} + ( 13 \beta_{1} - 13 \beta_{2} + 10 \beta_{3} + 7 \beta_{5} - 10 \beta_{7} + 9 \beta_{8} + 8 \beta_{11} - 3 \beta_{13} ) q^{89} + ( 19 \beta_{1} - 3 \beta_{2} + 9 \beta_{3} + \beta_{5} + 3 \beta_{7} + 16 \beta_{8} + 7 \beta_{11} - 4 \beta_{13} ) q^{91} + ( 57 + 12 \beta_{1} + 6 \beta_{2} + 6 \beta_{3} + 7 \beta_{4} - 6 \beta_{5} - 20 \beta_{6} + 12 \beta_{8} + 3 \beta_{9} + 7 \beta_{10} + 6 \beta_{11} - \beta_{12} + 6 \beta_{13} - 3 \beta_{14} + 3 \beta_{15} ) q^{93} + ( 14 + 6 \beta_{1} + 3 \beta_{2} + 3 \beta_{3} - 3 \beta_{4} - 3 \beta_{5} - 5 \beta_{6} + 6 \beta_{8} + 4 \beta_{9} + 7 \beta_{10} + 3 \beta_{11} - \beta_{12} + 3 \beta_{13} ) q^{95} + ( -13 \beta_{1} - 9 \beta_{2} - 6 \beta_{3} + 7 \beta_{5} + 10 \beta_{7} - 3 \beta_{8} - \beta_{11} + 3 \beta_{13} ) q^{97} + ( -4 \beta_{1} + 38 \beta_{2} - 14 \beta_{3} + 4 \beta_{5} + 6 \beta_{7} - 8 \beta_{8} - 12 \beta_{11} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$16q + 64q^{9} + O(q^{10})$$ $$16q + 64q^{9} + 24q^{13} - 4q^{23} - 80q^{25} + 96q^{27} - 108q^{29} + 116q^{31} - 60q^{35} - 248q^{39} - 156q^{41} + 128q^{47} - 28q^{49} - 204q^{59} - 268q^{69} - 236q^{71} - 112q^{73} - 936q^{77} - 136q^{81} + 60q^{85} + 152q^{87} + 856q^{93} + 160q^{95} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{16} + 78 x^{14} + 2165 x^{12} + 28310 x^{10} + 184804 x^{8} + 569634 x^{6} + 696037 x^{4} + 285578 x^{2} + 529$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$2 \nu$$ $$\beta_{2}$$ $$=$$ $$($$$$-249266 \nu^{15} - 19450521 \nu^{13} - 540300528 \nu^{11} - 7075107694 \nu^{9} - 46294879486 \nu^{7} - 143284317912 \nu^{5} - 176966386308 \nu^{3} - 76734676377 \nu$$$$)/ 1473983980$$ $$\beta_{3}$$ $$=$$ $$($$$$2028139777 \nu^{15} + 159386627629 \nu^{13} + 4483504013264 \nu^{11} + 59956611391678 \nu^{9} + 407018725514790 \nu^{7} + 1350423473295780 \nu^{5} + 1906976451293273 \nu^{3} + 890813001228721 \nu$$$$)/ 3401955025840$$ $$\beta_{4}$$ $$=$$ $$($$$$3266665611 \nu^{14} + 251231034675 \nu^{12} + 6803101750504 \nu^{10} + 85389448480554 \nu^{8} + 517393068635482 \nu^{6} + 1353800445494324 \nu^{4} + 988318890938067 \nu^{2} + 38708131230591$$$$)/ 6803910051680$$ $$\beta_{5}$$ $$=$$ $$($$$$651909559 \nu^{15} + 49847669703 \nu^{13} + 1334253632328 \nu^{11} + 16362993734546 \nu^{9} + 94189913828650 \nu^{7} + 212449351122740 \nu^{5} + 42951096904351 \nu^{3} - 93749086185413 \nu$$$$)/ 485993575120$$ $$\beta_{6}$$ $$=$$ $$($$$$-328071 \nu^{14} - 25254583 \nu^{12} - 684448696 \nu^{10} - 8586649314 \nu^{8} - 51832164194 \nu^{6} - 133890163508 \nu^{4} - 92129883455 \nu^{2} - 885659459$$$$)/ 252679840$$ $$\beta_{7}$$ $$=$$ $$($$$$229620557 \nu^{15} + 17699955559 \nu^{13} + 480902133464 \nu^{11} + 6059544792898 \nu^{9} + 36877647447520 \nu^{7} + 97036544153580 \nu^{5} + 71790826408533 \nu^{3} + 4166619657821 \nu$$$$)/ 121498393780$$ $$\beta_{8}$$ $$=$$ $$($$$$-3312811321 \nu^{15} - 255261199013 \nu^{13} - 6931819756236 \nu^{11} - 87315212918534 \nu^{9} - 531693729398674 \nu^{7} - 1405115322439008 \nu^{5} - 1071494275482145 \nu^{3} - 105891864922749 \nu$$$$)/ 1700977512920$$ $$\beta_{9}$$ $$=$$ $$($$$$-14365389855 \nu^{14} - 1105876709199 \nu^{12} - 29972322851736 \nu^{10} - 375976473404450 \nu^{8} - 2268003524367106 \nu^{6} - 5840712005282452 \nu^{4} - 3940822109052359 \nu^{2} + 44834433467221$$$$)/ 6803910051680$$ $$\beta_{10}$$ $$=$$ $$($$$$-11655955868 \nu^{15} + 10501228463 \nu^{14} - 902137906104 \nu^{13} + 808854205323 \nu^{12} - 24691247910968 \nu^{11} + 21944081936944 \nu^{10} - 315121017964672 \nu^{9} + 275725471799962 \nu^{8} - 1965954530430392 \nu^{7} + 1667894705250978 \nu^{6} - 5493551145777624 \nu^{5} + 4320316051003516 \nu^{4} - 5127616258362740 \nu^{3} + 2993798596228719 \nu^{2} - 1269801964558832 \nu + 41218966537631$$$$)/ 6803910051680$$ $$\beta_{11}$$ $$=$$ $$($$$$191987899 \nu^{15} + 14808207659 \nu^{13} + 402812007992 \nu^{11} + 5087217548666 \nu^{9} + 31116072823834 \nu^{7} + 83034850253508 \nu^{5} + 65752777188627 \nu^{3} + 8504894308903 \nu$$$$)/ 57175714720$$ $$\beta_{12}$$ $$=$$ $$($$$$-2913988967 \nu^{15} + 3973258621 \nu^{14} - 225534476526 \nu^{13} + 305594639244 \nu^{12} - 6172811977742 \nu^{11} + 8272935439550 \nu^{10} - 78780254491168 \nu^{9} + 103683543905204 \nu^{8} - 491488632607598 \nu^{7} + 625310254500318 \nu^{6} - 1373387786444406 \nu^{5} + 1611860000746766 \nu^{4} - 1281904064590685 \nu^{3} + 1091222027116891 \nu^{2} - 317450491139708 \nu - 2167987268690$$$$)/ 1700977512920$$ $$\beta_{13}$$ $$=$$ $$($$$$33188908451 \nu^{15} + 2553066418411 \nu^{13} + 69108994984088 \nu^{11} + 865212237602154 \nu^{9} + 5200964736292826 \nu^{7} + 13281722071508452 \nu^{5} + 8606845528408123 \nu^{3} - 392465157363513 \nu$$$$)/ 6803910051680$$ $$\beta_{14}$$ $$=$$ $$($$$$5827977934 \nu^{15} + 9278078059 \nu^{14} + 451068953052 \nu^{13} + 713586557253 \nu^{12} + 12345623955484 \nu^{11} + 19311664420068 \nu^{10} + 157560508982336 \nu^{9} + 241748603007606 \nu^{8} + 982977265215196 \nu^{7} + 1453931394806290 \nu^{6} + 2746775572888812 \nu^{5} + 3725622196294520 \nu^{4} + 2563808129181370 \nu^{3} + 2489579385232911 \nu^{2} + 634900982279416 \nu - 1041206773123$$$$)/ 3401955025840$$ $$\beta_{15}$$ $$=$$ $$($$$$-5827977934 \nu^{15} + 19413839169 \nu^{14} - 451068953052 \nu^{13} + 1494915649571 \nu^{12} - 12345623955484 \nu^{11} + 40536908099620 \nu^{10} - 157560508982336 \nu^{9} + 508965980646546 \nu^{8} - 982977265215196 \nu^{7} + 3075443167460942 \nu^{6} - 2746775572888812 \nu^{5} + 7948451110010704 \nu^{4} - 2563808129181370 \nu^{3} + 5438422689717229 \nu^{2} - 634900982279416 \nu + 25507426770635$$$$)/ 3401955025840$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$$$/2$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{13} + \beta_{12} + \beta_{11} + \beta_{10} + 3 \beta_{9} + 2 \beta_{8} - 3 \beta_{6} - \beta_{5} - 3 \beta_{4} + \beta_{3} + \beta_{2} + 2 \beta_{1} - 18$$$$)/2$$ $$\nu^{3}$$ $$=$$ $$($$$$7 \beta_{13} - 7 \beta_{11} + 5 \beta_{8} - 5 \beta_{7} + 2 \beta_{5} + 7 \beta_{3} - 10 \beta_{2} - 20 \beta_{1}$$$$)/2$$ $$\nu^{4}$$ $$=$$ $$($$$$14 \beta_{15} + 14 \beta_{14} - 45 \beta_{13} - 47 \beta_{12} - 45 \beta_{11} - 43 \beta_{10} - 79 \beta_{9} - 90 \beta_{8} + 127 \beta_{6} + 45 \beta_{5} + 117 \beta_{4} - 45 \beta_{3} - 45 \beta_{2} - 90 \beta_{1} + 400$$$$)/2$$ $$\nu^{5}$$ $$=$$ $$($$$$-349 \beta_{13} + 353 \beta_{11} - 223 \beta_{8} + 193 \beta_{7} - 28 \beta_{5} - 313 \beta_{3} + 342 \beta_{2} + 611 \beta_{1}$$$$)/2$$ $$\nu^{6}$$ $$=$$ $$-359 \beta_{15} - 325 \beta_{14} + 868 \beta_{13} + 898 \beta_{12} + 868 \beta_{11} + 872 \beta_{10} + 1235 \beta_{9} + 1736 \beta_{8} - 2404 \beta_{6} - 868 \beta_{5} - 2129 \beta_{4} + 868 \beta_{3} + 868 \beta_{2} + 1736 \beta_{1} - 5998$$ $$\nu^{7}$$ $$=$$ $$($$$$13910 \beta_{13} - 14070 \beta_{11} + 8514 \beta_{8} - 6966 \beta_{7} + 52 \beta_{5} + 11798 \beta_{3} - 12044 \beta_{2} - 21063 \beta_{1}$$$$)/2$$ $$\nu^{8}$$ $$=$$ $$($$$$28822 \beta_{15} + 25026 \beta_{14} - 64379 \beta_{13} - 66115 \beta_{12} - 64379 \beta_{11} - 66439 \beta_{10} - 84671 \beta_{9} - 128758 \beta_{8} + 176117 \beta_{6} + 64379 \beta_{5} + 154427 \beta_{4} - 64379 \beta_{3} - 64379 \beta_{2} - 128758 \beta_{1} + 406476$$$$)/2$$ $$\nu^{9}$$ $$=$$ $$($$$$-520505 \beta_{13} + 526825 \beta_{11} - 313151 \beta_{8} + 250339 \beta_{7} + 12974 \beta_{5} - 430885 \beta_{3} + 434946 \beta_{2} + 752686 \beta_{1}$$$$)/2$$ $$\nu^{10}$$ $$=$$ $$($$$$-1081278 \beta_{15} - 923762 \beta_{14} + 2355559 \beta_{13} + 2410481 \beta_{12} + 2355559 \beta_{11} + 2458153 \beta_{10} + 3010165 \beta_{9} + 4711118 \beta_{8} - 6400113 \beta_{6} - 2355559 \beta_{5} - 5600419 \beta_{4} + 2355559 \beta_{3} + 2355559 \beta_{2} + 4711118 \beta_{1} - 14401944$$$$)/2$$ $$\nu^{11}$$ $$=$$ $$($$$$19083917 \beta_{13} - 19330033 \beta_{11} + 11400387 \beta_{8} - 9029821 \beta_{7} - 679532 \beta_{5} + 15643385 \beta_{3} - 15800198 \beta_{2} - 27187083 \beta_{1}$$$$)/2$$ $$\nu^{12}$$ $$=$$ $$19844966 \beta_{15} + 16848718 \beta_{14} - 42873518 \beta_{13} - 43809734 \beta_{12} - 42873518 \beta_{11} - 44933550 \beta_{10} - 54218408 \beta_{9} - 85747036 \beta_{8} + 116111238 \beta_{6} + 42873518 \beta_{5} + 101571480 \beta_{4} - 42873518 \beta_{3} - 42873518 \beta_{2} - 85747036 \beta_{1} + 259136963$$ $$\nu^{13}$$ $$=$$ $$($$$$-694927048 \beta_{13} + 704244200 \beta_{11} - 413915040 \beta_{8} + 326698624 \beta_{7} + 27487912 \beta_{5} - 567460000 \beta_{3} + 574139472 \beta_{2} + 985154605 \beta_{1}$$$$)/2$$ $$\nu^{14}$$ $$=$$ $$($$$$-1446079476 \beta_{15} - 1224795972 \beta_{14} + 3115174881 \beta_{13} + 3181503425 \beta_{12} + 3115174881 \beta_{11} + 3270129841 \beta_{10} + 3924573439 \beta_{9} + 6230349762 \beta_{8} - 8424721587 \beta_{6} - 3115174881 \beta_{5} - 7369713143 \beta_{4} + 3115174881 \beta_{3} + 3115174881 \beta_{2} + 6230349762 \beta_{1} - 18751391662$$$$)/2$$ $$\nu^{15}$$ $$=$$ $$($$$$25246432839 \beta_{13} - 25591957271 \beta_{11} + 15018968005 \beta_{8} - 11839142477 \beta_{7} - 1035217254 \beta_{5} + 20585412727 \beta_{3} - 20850847362 \beta_{2} - 35732448956 \beta_{1}$$$$)/2$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1840\mathbb{Z}\right)^\times$$.

 $$n$$ $$737$$ $$1151$$ $$1201$$ $$1381$$ $$\chi(n)$$ $$1$$ $$1$$ $$-1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
321.1
 − 3.47734i 3.47734i 1.00527i − 1.00527i 1.01877i − 1.01877i 2.26343i − 2.26343i − 2.98291i 2.98291i 3.68124i − 3.68124i − 6.02373i 6.02373i 0.0431371i − 0.0431371i
0 −4.76369 0 2.23607i 0 7.05858i 0 13.6927 0
321.2 0 −4.76369 0 2.23607i 0 7.05858i 0 13.6927 0
321.3 0 −4.30716 0 2.23607i 0 1.47532i 0 9.55167 0
321.4 0 −4.30716 0 2.23607i 0 1.47532i 0 9.55167 0
321.5 0 −2.34854 0 2.23607i 0 7.61815i 0 −3.48436 0
321.6 0 −2.34854 0 2.23607i 0 7.61815i 0 −3.48436 0
321.7 0 −1.43837 0 2.23607i 0 10.1866i 0 −6.93108 0
321.8 0 −1.43837 0 2.23607i 0 10.1866i 0 −6.93108 0
321.9 0 0.278523 0 2.23607i 0 8.51262i 0 −8.92243 0
321.10 0 0.278523 0 2.23607i 0 8.51262i 0 −8.92243 0
321.11 0 3.36596 0 2.23607i 0 1.16919i 0 2.32968 0
321.12 0 3.36596 0 2.23607i 0 1.16919i 0 2.32968 0
321.13 0 3.79379 0 2.23607i 0 7.10180i 0 5.39287 0
321.14 0 3.79379 0 2.23607i 0 7.10180i 0 5.39287 0
321.15 0 5.41949 0 2.23607i 0 8.24199i 0 20.3709 0
321.16 0 5.41949 0 2.23607i 0 8.24199i 0 20.3709 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 321.16 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
23.b odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1840.3.k.d 16
4.b odd 2 1 230.3.d.a 16
12.b even 2 1 2070.3.c.a 16
20.d odd 2 1 1150.3.d.b 16
20.e even 4 2 1150.3.c.c 32
23.b odd 2 1 inner 1840.3.k.d 16
92.b even 2 1 230.3.d.a 16
276.h odd 2 1 2070.3.c.a 16
460.g even 2 1 1150.3.d.b 16
460.k odd 4 2 1150.3.c.c 32

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
230.3.d.a 16 4.b odd 2 1
230.3.d.a 16 92.b even 2 1
1150.3.c.c 32 20.e even 4 2
1150.3.c.c 32 460.k odd 4 2
1150.3.d.b 16 20.d odd 2 1
1150.3.d.b 16 460.g even 2 1
1840.3.k.d 16 1.a even 1 1 trivial
1840.3.k.d 16 23.b odd 2 1 inner
2070.3.c.a 16 12.b even 2 1
2070.3.c.a 16 276.h odd 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{8} - 52 T_{3}^{6} - 16 T_{3}^{5} + 829 T_{3}^{4} + 456 T_{3}^{3} - 4114 T_{3}^{2} - 3704 T_{3} + 1336$$ acting on $$S_{3}^{\mathrm{new}}(1840, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{16}$$
$3$ $$( 1336 - 3704 T - 4114 T^{2} + 456 T^{3} + 829 T^{4} - 16 T^{5} - 52 T^{6} + T^{8} )^{2}$$
$5$ $$( 5 + T^{2} )^{8}$$
$7$ $$221645107264 + 285091956800 T^{2} + 100475819056 T^{4} + 8103516608 T^{6} + 299518572 T^{8} + 6012916 T^{10} + 67901 T^{12} + 406 T^{14} + T^{16}$$
$11$ $$21342021632671744 + 1858395528359936 T^{2} + 66311017788672 T^{4} + 1277708760960 T^{6} + 14656413328 T^{8} + 103091648 T^{10} + 435976 T^{12} + 1016 T^{14} + T^{16}$$
$13$ $$( -343464224 - 219072992 T - 35098468 T^{2} + 212612 T^{3} + 299633 T^{4} + 5112 T^{5} - 898 T^{6} - 12 T^{7} + T^{8} )^{2}$$
$17$ $$17115756259840000 + 7339457014528000 T^{2} + 359932722681600 T^{4} + 7295803595520 T^{6} + 75620799984 T^{8} + 424642616 T^{10} + 1272985 T^{12} + 1858 T^{14} + T^{16}$$
$19$ $$11\!\cdots\!24$$$$+ 7426573318442000384 T^{2} + 117628148722456832 T^{4} + 821281616987520 T^{6} + 2997759304848 T^{8} + 6125065152 T^{10} + 7019336 T^{12} + 4184 T^{14} + T^{16}$$
$23$ $$61\!\cdots\!61$$$$+ 46371345298154999236 T - 13981530387628964798 T^{2} + 1118018684633959212 T^{3} + 24625359187522136 T^{4} - 2483084721289948 T^{5} + 124487227993214 T^{6} + 4023615463020 T^{7} - 284833923122 T^{8} + 7606078380 T^{9} + 444849854 T^{10} - 16773532 T^{11} + 314456 T^{12} + 26988 T^{13} - 638 T^{14} + 4 T^{15} + T^{16}$$
$29$ $$( -61767459836 - 2966407060 T + 1296604821 T^{2} + 104891262 T^{3} + 156267 T^{4} - 153376 T^{5} - 2309 T^{6} + 54 T^{7} + T^{8} )^{2}$$
$31$ $$( 229759835104 + 20345840768 T - 1198716119 T^{2} - 97845314 T^{3} + 2467787 T^{4} + 141380 T^{5} - 2573 T^{6} - 58 T^{7} + T^{8} )^{2}$$
$37$ $$27\!\cdots\!04$$$$+$$$$21\!\cdots\!52$$$$T^{2} +$$$$15\!\cdots\!80$$$$T^{4} + 370281351257423200 T^{6} + 418684864678616 T^{8} + 253723445496 T^{10} + 84388753 T^{12} + 14482 T^{14} + T^{16}$$
$41$ $$( -212194449184 - 18286974656 T + 3161705921 T^{2} + 211053646 T^{3} - 4997857 T^{4} - 430172 T^{5} - 4161 T^{6} + 78 T^{7} + T^{8} )^{2}$$
$43$ $$18\!\cdots\!24$$$$+$$$$15\!\cdots\!60$$$$T^{2} + 3954975273611339584 T^{4} + 35443215524837696 T^{6} + 105348919606192 T^{8} + 122794231488 T^{10} + 62675724 T^{12} + 13412 T^{14} + T^{16}$$
$47$ $$( 13232824136 - 19369312968 T + 3951598214 T^{2} - 166721408 T^{3} - 7463547 T^{4} + 510024 T^{5} - 5764 T^{6} - 64 T^{7} + T^{8} )^{2}$$
$53$ $$16\!\cdots\!64$$$$+$$$$25\!\cdots\!92$$$$T^{2} +$$$$14\!\cdots\!04$$$$T^{4} + 405693095099781120 T^{6} + 569359307713296 T^{8} + 413528876744 T^{10} + 146364745 T^{12} + 21250 T^{14} + T^{16}$$
$59$ $$( -42922529206784 - 6511297990656 T + 33913547968 T^{2} + 6885625216 T^{3} + 38371048 T^{4} - 1682848 T^{5} - 14463 T^{6} + 102 T^{7} + T^{8} )^{2}$$
$61$ $$54\!\cdots\!84$$$$+$$$$31\!\cdots\!96$$$$T^{2} +$$$$71\!\cdots\!48$$$$T^{4} + 8134549162935653888 T^{6} + 4974850707546128 T^{8} + 1677363078528 T^{10} + 306610520 T^{12} + 28128 T^{14} + T^{16}$$
$67$ $$20\!\cdots\!04$$$$+$$$$10\!\cdots\!72$$$$T^{2} +$$$$22\!\cdots\!76$$$$T^{4} +$$$$16\!\cdots\!64$$$$T^{6} + 60762039405722412 T^{8} + 11205152610740 T^{10} + 1088116077 T^{12} + 52678 T^{14} + T^{16}$$
$71$ $$( -24390990617024 - 2657914180464 T + 49408197993 T^{2} + 4679164574 T^{3} + 21207483 T^{4} - 1424892 T^{5} - 11133 T^{6} + 118 T^{7} + T^{8} )^{2}$$
$73$ $$( 1317400530416 + 141690861952 T - 13436169136 T^{2} - 890866888 T^{3} + 59158553 T^{4} - 283856 T^{5} - 14814 T^{6} + 56 T^{7} + T^{8} )^{2}$$
$79$ $$18\!\cdots\!24$$$$+$$$$18\!\cdots\!48$$$$T^{2} +$$$$30\!\cdots\!52$$$$T^{4} +$$$$15\!\cdots\!36$$$$T^{6} + 368416958397233808 T^{8} + 43217476931904 T^{10} + 2664485320 T^{12} + 82216 T^{14} + T^{16}$$
$83$ $$39\!\cdots\!64$$$$+$$$$62\!\cdots\!20$$$$T^{2} +$$$$39\!\cdots\!44$$$$T^{4} +$$$$13\!\cdots\!56$$$$T^{6} + 261063543358679972 T^{8} + 29965030291348 T^{10} + 1987717669 T^{12} + 69862 T^{14} + T^{16}$$
$89$ $$54\!\cdots\!00$$$$+$$$$10\!\cdots\!00$$$$T^{2} +$$$$51\!\cdots\!00$$$$T^{4} +$$$$45\!\cdots\!80$$$$T^{6} + 150481148161409424 T^{8} + 23169014835136 T^{10} + 1794255880 T^{12} + 67928 T^{14} + T^{16}$$
$97$ $$37\!\cdots\!04$$$$+$$$$70\!\cdots\!28$$$$T^{2} +$$$$22\!\cdots\!32$$$$T^{4} +$$$$14\!\cdots\!56$$$$T^{6} + 360397864589537808 T^{8} + 44029769741824 T^{10} + 2741887320 T^{12} + 83856 T^{14} + T^{16}$$