# Properties

 Label 1840.3.k.c 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: $$x^{16} - 64 x^{14} - 16 x^{13} + 2252 x^{12} + 648 x^{11} - 30106 x^{10} + 12360 x^{9} + 374528 x^{8} + 196544 x^{7} + 1261236 x^{6} - 4237944 x^{5} + 38013345 x^{4} - 8913096 x^{3} + 327235800 x^{2} + 72566336 x + 1535848276$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{6}$$ Twist minimal: no (minimal twist has level 460) 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.

 $$f(q)$$ $$=$$ $$q -\beta_{3} q^{3} -\beta_{4} q^{5} -\beta_{7} q^{7} + ( 4 + \beta_{2} ) q^{9} +O(q^{10})$$ $$q -\beta_{3} q^{3} -\beta_{4} q^{5} -\beta_{7} q^{7} + ( 4 + \beta_{2} ) q^{9} + ( 2 \beta_{4} - \beta_{9} ) q^{11} + ( -1 - \beta_{11} ) q^{13} + \beta_{1} q^{15} + ( \beta_{1} - \beta_{10} ) q^{17} + ( -\beta_{7} - \beta_{8} - \beta_{9} ) q^{19} + ( -\beta_{6} + \beta_{7} + \beta_{9} ) q^{21} + ( 1 - \beta_{1} + \beta_{3} - \beta_{5} - \beta_{7} - \beta_{15} ) q^{23} -5 q^{25} + ( -3 - \beta_{2} - 6 \beta_{3} + \beta_{5} - \beta_{11} + \beta_{13} + \beta_{14} ) q^{27} + ( 5 + 4 \beta_{3} + 2 \beta_{5} - \beta_{11} + \beta_{13} ) q^{29} + ( -1 + 2 \beta_{3} + \beta_{5} - \beta_{11} + \beta_{12} + \beta_{13} ) q^{31} + ( -4 \beta_{1} - 6 \beta_{4} + \beta_{6} + 3 \beta_{7} + \beta_{8} + \beta_{9} + \beta_{10} - \beta_{15} ) q^{33} + ( -2 + \beta_{13} ) q^{35} + ( 2 \beta_{4} - 2 \beta_{7} - 2 \beta_{9} - \beta_{10} + \beta_{15} ) q^{37} + ( -2 + 2 \beta_{3} - \beta_{5} - \beta_{11} + \beta_{12} + 2 \beta_{13} - \beta_{14} ) q^{39} + ( 12 + \beta_{2} + 2 \beta_{3} + \beta_{5} - \beta_{12} - \beta_{13} + \beta_{14} ) q^{41} + ( -2 \beta_{1} - 4 \beta_{4} - \beta_{6} - \beta_{8} - \beta_{9} + 2 \beta_{10} - 2 \beta_{15} ) q^{43} + ( -5 \beta_{4} + \beta_{6} + \beta_{7} + 2 \beta_{9} - \beta_{15} ) q^{45} + ( 20 + \beta_{2} + \beta_{3} - \beta_{5} + 2 \beta_{12} - \beta_{13} - \beta_{14} ) q^{47} + ( 2 \beta_{11} + \beta_{13} - \beta_{14} ) q^{49} + ( -12 \beta_{4} + \beta_{6} + \beta_{7} + 4 \beta_{9} + 2 \beta_{10} - 2 \beta_{15} ) q^{51} + ( 2 \beta_{1} - 8 \beta_{4} + \beta_{6} - \beta_{8} + 3 \beta_{9} - \beta_{10} + \beta_{15} ) q^{53} + ( 8 + \beta_{2} + \beta_{3} - \beta_{5} + \beta_{11} - \beta_{13} ) q^{55} + ( \beta_{1} - 8 \beta_{4} + 5 \beta_{7} + 6 \beta_{9} + \beta_{10} - 2 \beta_{15} ) q^{57} + ( 6 + \beta_{2} + 6 \beta_{3} - \beta_{5} + \beta_{11} + \beta_{12} + \beta_{13} ) q^{59} + ( 2 \beta_{1} + 6 \beta_{4} - 2 \beta_{6} - 3 \beta_{9} + 2 \beta_{10} + 2 \beta_{15} ) q^{61} + ( -3 \beta_{1} + 8 \beta_{4} - 2 \beta_{6} - 3 \beta_{7} - 2 \beta_{8} - 2 \beta_{9} - \beta_{15} ) q^{63} + ( 2 \beta_{4} + \beta_{6} - \beta_{7} + \beta_{8} - \beta_{9} - \beta_{10} ) q^{65} + ( 4 \beta_{1} - 6 \beta_{4} + 2 \beta_{6} + 2 \beta_{7} + 4 \beta_{9} - \beta_{10} - 3 \beta_{15} ) q^{67} + ( -15 + 2 \beta_{1} - \beta_{2} - 2 \beta_{3} + 10 \beta_{4} + \beta_{5} - 3 \beta_{6} - 2 \beta_{7} - \beta_{8} - 2 \beta_{9} - 2 \beta_{11} + \beta_{12} ) q^{69} + ( 16 + \beta_{2} - 12 \beta_{3} - \beta_{5} + \beta_{12} - 3 \beta_{13} + \beta_{14} ) q^{71} + ( -16 - \beta_{2} + \beta_{3} - 3 \beta_{5} - 5 \beta_{13} + \beta_{14} ) q^{73} + 5 \beta_{3} q^{75} + ( 21 - 9 \beta_{3} - 2 \beta_{5} + \beta_{11} - 2 \beta_{13} ) q^{77} + ( -2 \beta_{4} - \beta_{6} + 6 \beta_{7} + \beta_{8} - 3 \beta_{9} + 4 \beta_{10} ) q^{79} + ( 44 + 5 \beta_{2} + 14 \beta_{3} - 4 \beta_{5} + 3 \beta_{11} + 2 \beta_{12} - 3 \beta_{14} ) q^{81} + ( 2 \beta_{1} + 8 \beta_{4} - \beta_{6} - 6 \beta_{7} - 3 \beta_{8} - \beta_{9} - \beta_{10} - \beta_{15} ) q^{83} + ( -2 + 4 \beta_{3} + \beta_{11} + \beta_{12} - \beta_{14} ) q^{85} + ( -46 - 3 \beta_{2} - 5 \beta_{3} - \beta_{5} + 3 \beta_{13} - \beta_{14} ) q^{87} + ( -12 \beta_{1} + 10 \beta_{4} + \beta_{6} + \beta_{8} - 5 \beta_{9} - 4 \beta_{15} ) q^{89} + ( -8 \beta_{1} - 8 \beta_{7} - 5 \beta_{9} + 2 \beta_{15} ) q^{91} + ( -25 + \beta_{2} + 2 \beta_{3} - 3 \beta_{5} - 3 \beta_{11} + 2 \beta_{12} + 5 \beta_{13} - \beta_{14} ) q^{93} + ( -5 + \beta_{3} - 3 \beta_{5} - \beta_{13} - \beta_{14} ) q^{95} + ( -3 \beta_{1} + 8 \beta_{4} - 2 \beta_{6} + 2 \beta_{9} + 4 \beta_{10} - \beta_{15} ) q^{97} + ( 12 \beta_{1} + 38 \beta_{4} - \beta_{6} - 5 \beta_{7} - 10 \beta_{9} - 4 \beta_{10} + 8 \beta_{15} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$16q + 64q^{9} + O(q^{10})$$ $$16q + 64q^{9} - 12q^{13} + 14q^{23} - 80q^{25} - 48q^{27} + 90q^{29} - 10q^{31} - 30q^{35} - 20q^{39} + 186q^{41} + 320q^{47} + 2q^{49} + 120q^{55} + 90q^{59} - 232q^{69} + 238q^{71} - 280q^{73} + 324q^{77} + 704q^{81} - 30q^{85} - 724q^{87} - 380q^{93} - 80q^{95} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{16} - 64 x^{14} - 16 x^{13} + 2252 x^{12} + 648 x^{11} - 30106 x^{10} + 12360 x^{9} + 374528 x^{8} + 196544 x^{7} + 1261236 x^{6} - 4237944 x^{5} + 38013345 x^{4} - 8913096 x^{3} + 327235800 x^{2} + 72566336 x + 1535848276$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$23\!\cdots\!29$$$$\nu^{15} -$$$$42\!\cdots\!34$$$$\nu^{14} -$$$$12\!\cdots\!60$$$$\nu^{13} +$$$$23\!\cdots\!76$$$$\nu^{12} +$$$$11\!\cdots\!72$$$$\nu^{11} -$$$$79\!\cdots\!84$$$$\nu^{10} -$$$$26\!\cdots\!08$$$$\nu^{9} +$$$$74\!\cdots\!49$$$$\nu^{8} -$$$$57\!\cdots\!86$$$$\nu^{7} -$$$$12\!\cdots\!80$$$$\nu^{6} -$$$$12\!\cdots\!44$$$$\nu^{5} -$$$$16\!\cdots\!66$$$$\nu^{4} +$$$$83\!\cdots\!91$$$$\nu^{3} -$$$$56\!\cdots\!87$$$$\nu^{2} +$$$$61\!\cdots\!34$$$$\nu -$$$$21\!\cdots\!52$$$$)/$$$$97\!\cdots\!90$$ $$\beta_{2}$$ $$=$$ $$($$$$23\!\cdots\!29$$$$\nu^{15} -$$$$42\!\cdots\!34$$$$\nu^{14} -$$$$12\!\cdots\!60$$$$\nu^{13} +$$$$23\!\cdots\!76$$$$\nu^{12} +$$$$11\!\cdots\!72$$$$\nu^{11} -$$$$79\!\cdots\!84$$$$\nu^{10} -$$$$26\!\cdots\!08$$$$\nu^{9} +$$$$74\!\cdots\!49$$$$\nu^{8} -$$$$57\!\cdots\!86$$$$\nu^{7} -$$$$12\!\cdots\!80$$$$\nu^{6} -$$$$12\!\cdots\!44$$$$\nu^{5} -$$$$16\!\cdots\!66$$$$\nu^{4} +$$$$83\!\cdots\!91$$$$\nu^{3} -$$$$76\!\cdots\!42$$$$\nu^{2} +$$$$61\!\cdots\!34$$$$\nu -$$$$60\!\cdots\!12$$$$)/$$$$48\!\cdots\!45$$ $$\beta_{3}$$ $$=$$ $$($$$$-$$$$34\!\cdots\!21$$$$\nu^{15} +$$$$46\!\cdots\!58$$$$\nu^{14} +$$$$21\!\cdots\!76$$$$\nu^{13} +$$$$53\!\cdots\!16$$$$\nu^{12} -$$$$73\!\cdots\!40$$$$\nu^{11} -$$$$20\!\cdots\!64$$$$\nu^{10} +$$$$89\!\cdots\!58$$$$\nu^{9} -$$$$48\!\cdots\!76$$$$\nu^{8} -$$$$11\!\cdots\!90$$$$\nu^{7} -$$$$79\!\cdots\!96$$$$\nu^{6} -$$$$69\!\cdots\!16$$$$\nu^{5} +$$$$12\!\cdots\!36$$$$\nu^{4} -$$$$16\!\cdots\!77$$$$\nu^{3} +$$$$47\!\cdots\!98$$$$\nu^{2} -$$$$32\!\cdots\!94$$$$\nu -$$$$12\!\cdots\!88$$$$)/$$$$19\!\cdots\!80$$ $$\beta_{4}$$ $$=$$ $$($$$$-$$$$34\!\cdots\!21$$$$\nu^{15} +$$$$46\!\cdots\!58$$$$\nu^{14} +$$$$21\!\cdots\!76$$$$\nu^{13} +$$$$53\!\cdots\!16$$$$\nu^{12} -$$$$73\!\cdots\!40$$$$\nu^{11} -$$$$20\!\cdots\!64$$$$\nu^{10} +$$$$89\!\cdots\!58$$$$\nu^{9} -$$$$48\!\cdots\!76$$$$\nu^{8} -$$$$11\!\cdots\!90$$$$\nu^{7} -$$$$79\!\cdots\!96$$$$\nu^{6} -$$$$69\!\cdots\!16$$$$\nu^{5} +$$$$12\!\cdots\!36$$$$\nu^{4} -$$$$16\!\cdots\!77$$$$\nu^{3} +$$$$47\!\cdots\!98$$$$\nu^{2} -$$$$22\!\cdots\!74$$$$\nu -$$$$12\!\cdots\!88$$$$)/$$$$19\!\cdots\!80$$ $$\beta_{5}$$ $$=$$ $$($$$$-$$$$14\!\cdots\!29$$$$\nu^{15} -$$$$59\!\cdots\!16$$$$\nu^{14} +$$$$15\!\cdots\!93$$$$\nu^{13} +$$$$31\!\cdots\!88$$$$\nu^{12} -$$$$79\!\cdots\!61$$$$\nu^{11} -$$$$10\!\cdots\!24$$$$\nu^{10} +$$$$22\!\cdots\!75$$$$\nu^{9} +$$$$76\!\cdots\!88$$$$\nu^{8} -$$$$40\!\cdots\!79$$$$\nu^{7} -$$$$15\!\cdots\!52$$$$\nu^{6} +$$$$33\!\cdots\!83$$$$\nu^{5} -$$$$16\!\cdots\!00$$$$\nu^{4} -$$$$12\!\cdots\!76$$$$\nu^{3} -$$$$53\!\cdots\!48$$$$\nu^{2} +$$$$27\!\cdots\!40$$$$\nu -$$$$33\!\cdots\!12$$$$)/$$$$34\!\cdots\!80$$ $$\beta_{6}$$ $$=$$ $$($$$$-$$$$18\!\cdots\!44$$$$\nu^{15} -$$$$21\!\cdots\!33$$$$\nu^{14} +$$$$81\!\cdots\!10$$$$\nu^{13} +$$$$13\!\cdots\!45$$$$\nu^{12} -$$$$20\!\cdots\!14$$$$\nu^{11} -$$$$10\!\cdots\!25$$$$\nu^{10} -$$$$11\!\cdots\!54$$$$\nu^{9} -$$$$14\!\cdots\!25$$$$\nu^{8} -$$$$28\!\cdots\!18$$$$\nu^{7} +$$$$10\!\cdots\!97$$$$\nu^{6} -$$$$67\!\cdots\!66$$$$\nu^{5} -$$$$65\!\cdots\!09$$$$\nu^{4} -$$$$17\!\cdots\!26$$$$\nu^{3} +$$$$23\!\cdots\!64$$$$\nu^{2} -$$$$16\!\cdots\!08$$$$\nu +$$$$93\!\cdots\!56$$$$)/$$$$34\!\cdots\!80$$ $$\beta_{7}$$ $$=$$ $$($$$$-$$$$20\!\cdots\!08$$$$\nu^{15} +$$$$98\!\cdots\!41$$$$\nu^{14} +$$$$15\!\cdots\!94$$$$\nu^{13} -$$$$59\!\cdots\!17$$$$\nu^{12} -$$$$60\!\cdots\!50$$$$\nu^{11} +$$$$20\!\cdots\!57$$$$\nu^{10} +$$$$10\!\cdots\!34$$$$\nu^{9} -$$$$32\!\cdots\!63$$$$\nu^{8} -$$$$93\!\cdots\!14$$$$\nu^{7} +$$$$42\!\cdots\!31$$$$\nu^{6} -$$$$10\!\cdots\!22$$$$\nu^{5} +$$$$68\!\cdots\!13$$$$\nu^{4} -$$$$51\!\cdots\!74$$$$\nu^{3} +$$$$36\!\cdots\!04$$$$\nu^{2} -$$$$46\!\cdots\!80$$$$\nu +$$$$72\!\cdots\!04$$$$)/$$$$34\!\cdots\!80$$ $$\beta_{8}$$ $$=$$ $$($$$$31\!\cdots\!80$$$$\nu^{15} -$$$$32\!\cdots\!25$$$$\nu^{14} -$$$$18\!\cdots\!14$$$$\nu^{13} +$$$$21\!\cdots\!01$$$$\nu^{12} +$$$$63\!\cdots\!66$$$$\nu^{11} -$$$$77\!\cdots\!21$$$$\nu^{10} -$$$$78\!\cdots\!30$$$$\nu^{9} +$$$$12\!\cdots\!75$$$$\nu^{8} +$$$$45\!\cdots\!46$$$$\nu^{7} -$$$$15\!\cdots\!19$$$$\nu^{6} -$$$$20\!\cdots\!30$$$$\nu^{5} +$$$$28\!\cdots\!47$$$$\nu^{4} +$$$$26\!\cdots\!30$$$$\nu^{3} -$$$$17\!\cdots\!96$$$$\nu^{2} +$$$$14\!\cdots\!40$$$$\nu -$$$$54\!\cdots\!00$$$$)/$$$$34\!\cdots\!80$$ $$\beta_{9}$$ $$=$$ $$($$$$16\!\cdots\!59$$$$\nu^{15} -$$$$29\!\cdots\!08$$$$\nu^{14} -$$$$11\!\cdots\!96$$$$\nu^{13} +$$$$20\!\cdots\!48$$$$\nu^{12} +$$$$42\!\cdots\!46$$$$\nu^{11} -$$$$88\!\cdots\!36$$$$\nu^{10} -$$$$70\!\cdots\!06$$$$\nu^{9} +$$$$22\!\cdots\!48$$$$\nu^{8} +$$$$72\!\cdots\!94$$$$\nu^{7} -$$$$28\!\cdots\!36$$$$\nu^{6} +$$$$21\!\cdots\!26$$$$\nu^{5} +$$$$47\!\cdots\!04$$$$\nu^{4} +$$$$20\!\cdots\!31$$$$\nu^{3} -$$$$38\!\cdots\!40$$$$\nu^{2} -$$$$20\!\cdots\!78$$$$\nu +$$$$32\!\cdots\!44$$$$)/$$$$17\!\cdots\!40$$ $$\beta_{10}$$ $$=$$ $$($$$$60\!\cdots\!66$$$$\nu^{15} +$$$$51\!\cdots\!07$$$$\nu^{14} -$$$$42\!\cdots\!58$$$$\nu^{13} -$$$$58\!\cdots\!35$$$$\nu^{12} +$$$$15\!\cdots\!46$$$$\nu^{11} +$$$$26\!\cdots\!39$$$$\nu^{10} -$$$$23\!\cdots\!22$$$$\nu^{9} -$$$$48\!\cdots\!61$$$$\nu^{8} +$$$$26\!\cdots\!30$$$$\nu^{7} +$$$$79\!\cdots\!49$$$$\nu^{6} +$$$$59\!\cdots\!86$$$$\nu^{5} -$$$$82\!\cdots\!41$$$$\nu^{4} -$$$$13\!\cdots\!16$$$$\nu^{3} -$$$$62\!\cdots\!00$$$$\nu^{2} +$$$$16\!\cdots\!44$$$$\nu +$$$$41\!\cdots\!16$$$$)/$$$$34\!\cdots\!80$$ $$\beta_{11}$$ $$=$$ $$($$$$-$$$$32\!\cdots\!82$$$$\nu^{15} +$$$$27\!\cdots\!58$$$$\nu^{14} +$$$$23\!\cdots\!93$$$$\nu^{13} -$$$$13\!\cdots\!24$$$$\nu^{12} -$$$$88\!\cdots\!67$$$$\nu^{11} +$$$$39\!\cdots\!00$$$$\nu^{10} +$$$$14\!\cdots\!29$$$$\nu^{9} -$$$$10\!\cdots\!72$$$$\nu^{8} -$$$$18\!\cdots\!13$$$$\nu^{7} +$$$$48\!\cdots\!04$$$$\nu^{6} +$$$$71\!\cdots\!09$$$$\nu^{5} +$$$$30\!\cdots\!92$$$$\nu^{4} -$$$$15\!\cdots\!49$$$$\nu^{3} +$$$$17\!\cdots\!02$$$$\nu^{2} -$$$$18\!\cdots\!34$$$$\nu +$$$$87\!\cdots\!64$$$$)/$$$$17\!\cdots\!40$$ $$\beta_{12}$$ $$=$$ $$($$$$-$$$$66\!\cdots\!45$$$$\nu^{15} +$$$$54\!\cdots\!24$$$$\nu^{14} +$$$$39\!\cdots\!05$$$$\nu^{13} -$$$$34\!\cdots\!80$$$$\nu^{12} -$$$$14\!\cdots\!53$$$$\nu^{11} +$$$$11\!\cdots\!64$$$$\nu^{10} +$$$$18\!\cdots\!07$$$$\nu^{9} -$$$$16\!\cdots\!08$$$$\nu^{8} -$$$$13\!\cdots\!79$$$$\nu^{7} +$$$$11\!\cdots\!84$$$$\nu^{6} -$$$$92\!\cdots\!13$$$$\nu^{5} +$$$$18\!\cdots\!80$$$$\nu^{4} -$$$$60\!\cdots\!04$$$$\nu^{3} +$$$$69\!\cdots\!84$$$$\nu^{2} -$$$$21\!\cdots\!04$$$$\nu +$$$$10\!\cdots\!68$$$$)/$$$$34\!\cdots\!80$$ $$\beta_{13}$$ $$=$$ $$($$$$-$$$$67\!\cdots\!87$$$$\nu^{15} -$$$$14\!\cdots\!24$$$$\nu^{14} +$$$$47\!\cdots\!37$$$$\nu^{13} +$$$$11\!\cdots\!68$$$$\nu^{12} -$$$$17\!\cdots\!85$$$$\nu^{11} -$$$$41\!\cdots\!76$$$$\nu^{10} +$$$$28\!\cdots\!59$$$$\nu^{9} +$$$$44\!\cdots\!48$$$$\nu^{8} -$$$$38\!\cdots\!91$$$$\nu^{7} -$$$$41\!\cdots\!04$$$$\nu^{6} +$$$$11\!\cdots\!35$$$$\nu^{5} -$$$$30\!\cdots\!56$$$$\nu^{4} -$$$$18\!\cdots\!22$$$$\nu^{3} -$$$$15\!\cdots\!60$$$$\nu^{2} +$$$$29\!\cdots\!80$$$$\nu -$$$$20\!\cdots\!12$$$$)/$$$$34\!\cdots\!80$$ $$\beta_{14}$$ $$=$$ $$($$$$-$$$$59\!\cdots\!01$$$$\nu^{15} +$$$$11\!\cdots\!38$$$$\nu^{14} +$$$$34\!\cdots\!10$$$$\nu^{13} -$$$$65\!\cdots\!40$$$$\nu^{12} -$$$$10\!\cdots\!32$$$$\nu^{11} +$$$$22\!\cdots\!16$$$$\nu^{10} +$$$$86\!\cdots\!40$$$$\nu^{9} -$$$$43\!\cdots\!00$$$$\nu^{8} -$$$$27\!\cdots\!52$$$$\nu^{7} +$$$$14\!\cdots\!64$$$$\nu^{6} -$$$$24\!\cdots\!00$$$$\nu^{5} +$$$$61\!\cdots\!72$$$$\nu^{4} -$$$$30\!\cdots\!83$$$$\nu^{3} +$$$$43\!\cdots\!66$$$$\nu^{2} -$$$$98\!\cdots\!46$$$$\nu +$$$$15\!\cdots\!28$$$$)/$$$$17\!\cdots\!40$$ $$\beta_{15}$$ $$=$$ $$($$$$64\!\cdots\!75$$$$\nu^{15} -$$$$21\!\cdots\!42$$$$\nu^{14} -$$$$43\!\cdots\!42$$$$\nu^{13} +$$$$10\!\cdots\!92$$$$\nu^{12} +$$$$15\!\cdots\!04$$$$\nu^{11} -$$$$47\!\cdots\!92$$$$\nu^{10} -$$$$23\!\cdots\!34$$$$\nu^{9} +$$$$29\!\cdots\!16$$$$\nu^{8} +$$$$25\!\cdots\!36$$$$\nu^{7} -$$$$19\!\cdots\!44$$$$\nu^{6} +$$$$91\!\cdots\!40$$$$\nu^{5} -$$$$99\!\cdots\!56$$$$\nu^{4} +$$$$11\!\cdots\!63$$$$\nu^{3} +$$$$14\!\cdots\!62$$$$\nu^{2} +$$$$14\!\cdots\!02$$$$\nu +$$$$16\!\cdots\!20$$$$)/$$$$17\!\cdots\!40$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$-\beta_{4} + \beta_{3}$$ $$\nu^{2}$$ $$=$$ $$\beta_{2} - 2 \beta_{1} + 8$$ $$\nu^{3}$$ $$=$$ $$-3 \beta_{15} - \beta_{14} - \beta_{13} + \beta_{11} + 6 \beta_{9} + 3 \beta_{7} + 3 \beta_{6} - \beta_{5} - 37 \beta_{4} + 9 \beta_{3} + \beta_{2} + 3$$ $$\nu^{4}$$ $$=$$ $$-12 \beta_{15} - 3 \beta_{14} + 2 \beta_{12} + 3 \beta_{11} + 12 \beta_{10} + 16 \beta_{9} + 8 \beta_{8} + 4 \beta_{7} - 4 \beta_{5} - 24 \beta_{4} + 14 \beta_{3} + 2 \beta_{2} - 80 \beta_{1} - 51$$ $$\nu^{5}$$ $$=$$ $$-105 \beta_{15} + 12 \beta_{14} + 14 \beta_{13} + \beta_{11} + 50 \beta_{10} + 260 \beta_{9} + 25 \beta_{8} + 180 \beta_{7} + 105 \beta_{6} + 8 \beta_{5} - 1125 \beta_{4} - 403 \beta_{3} + 10 \beta_{2} - 70 \beta_{1} + 121$$ $$\nu^{6}$$ $$=$$ $$-492 \beta_{15} + 76 \beta_{14} + 2 \beta_{13} - 67 \beta_{12} - 88 \beta_{11} + 438 \beta_{10} + 902 \beta_{9} + 202 \beta_{8} + 470 \beta_{7} + 54 \beta_{6} + 105 \beta_{5} - 1992 \beta_{4} - 96 \beta_{3} - 1029 \beta_{2} - 1886 \beta_{1} - 9948$$ $$\nu^{7}$$ $$=$$ $$-2093 \beta_{15} + 1833 \beta_{14} + 1717 \beta_{13} + 6 \beta_{12} - 2546 \beta_{11} + 1554 \beta_{10} + 5040 \beta_{9} + 1036 \beta_{8} + 4452 \beta_{7} + 1694 \beta_{6} + 2005 \beta_{5} - 17921 \beta_{4} - 30306 \beta_{3} - 2749 \beta_{2} - 4368 \beta_{1} - 10828$$ $$\nu^{8}$$ $$=$$ $$-7032 \beta_{15} + 9718 \beta_{14} - 476 \beta_{13} - 5252 \beta_{12} - 8558 \beta_{11} + 3808 \beta_{10} + 17352 \beta_{9} + 1272 \beta_{8} + 11040 \beta_{7} + 3464 \beta_{6} + 12708 \beta_{5} - 60064 \beta_{4} - 63452 \beta_{3} - 55267 \beta_{2} - 5576 \beta_{1} - 467398$$ $$\nu^{9}$$ $$=$$ $$28053 \beta_{15} + 78623 \beta_{14} + 72687 \beta_{13} - 80 \beta_{12} - 130637 \beta_{11} - 1824 \beta_{10} - 75510 \beta_{9} + 1314 \beta_{8} - 47247 \beta_{7} - 39969 \beta_{6} + 100283 \beta_{5} + 425261 \beta_{4} - 1124603 \beta_{3} - 196087 \beta_{2} - 57408 \beta_{1} - 1205919$$ $$\nu^{10}$$ $$=$$ $$401930 \beta_{15} + 404837 \beta_{14} + 32552 \beta_{13} - 167844 \beta_{12} - 363817 \beta_{11} - 385430 \beta_{10} - 679140 \beta_{9} - 191720 \beta_{8} - 294010 \beta_{7} + 15500 \beta_{6} + 512402 \beta_{5} + 1117380 \beta_{4} - 3723658 \beta_{3} - 1680140 \beta_{2} + 2017660 \beta_{1} - 13138755$$ $$\nu^{11}$$ $$=$$ $$4719737 \beta_{15} + 1871644 \beta_{14} + 1472966 \beta_{13} - 152960 \beta_{12} - 3126813 \beta_{11} - 2668270 \beta_{10} - 11292204 \beta_{9} - 1695265 \beta_{8} - 8956222 \beta_{7} - 4319887 \beta_{6} + 2526084 \beta_{5} + 46905189 \beta_{4} - 22889857 \beta_{3} - 6671334 \beta_{2} + 6479946 \beta_{1} - 49274749$$ $$\nu^{12}$$ $$=$$ $$35357024 \beta_{15} + 7306256 \beta_{14} + 3282590 \beta_{13} - 1523959 \beta_{12} - 8191892 \beta_{11} - 27317252 \beta_{10} - 69695660 \beta_{9} - 12986388 \beta_{8} - 37465716 \beta_{7} - 8593372 \beta_{6} + 8891241 \beta_{5} + 192058512 \beta_{4} - 93874956 \beta_{3} - 19001651 \beta_{2} + 119396812 \beta_{1} - 121656184$$ $$\nu^{13}$$ $$=$$ $$238683133 \beta_{15} - 5803841 \beta_{14} - 19627253 \beta_{13} - 7811938 \beta_{12} + 17346326 \beta_{11} - 155666914 \beta_{10} - 544563500 \beta_{9} - 97260826 \beta_{8} - 422615622 \beta_{7} - 182130754 \beta_{6} - 4851545 \beta_{5} + 2069826045 \beta_{4} + 226480374 \beta_{3} - 47695775 \beta_{2} + 504243844 \beta_{1} - 546243096$$ $$\nu^{14}$$ $$=$$ $$1439013442 \beta_{15} - 228382132 \beta_{14} + 68953924 \beta_{13} + 144852348 \beta_{12} + 139890736 \beta_{11} - 1015933716 \beta_{10} - 2988873050 \beta_{9} - 496930214 \beta_{8} - 1770634388 \beta_{7} - 573467386 \beta_{6} - 294366192 \beta_{5} + 9491328720 \beta_{4} + 1346596748 \beta_{3} + 1299106089 \beta_{2} + 4014142154 \beta_{1} + 11047082428$$ $$\nu^{15}$$ $$=$$ $$6968471215 \beta_{15} - 3180242887 \beta_{14} - 2976448991 \beta_{13} + 10488064 \beta_{12} + 5374132165 \beta_{11} - 4918684580 \beta_{10} - 15147940310 \beta_{9} - 2891101520 \beta_{8} - 10933071245 \beta_{7} - 4220391785 \beta_{6} - 4092112867 \beta_{5} + 52749131295 \beta_{4} + 45082668763 \beta_{3} + 8735228059 \beta_{2} + 18359505520 \beta_{1} + 57576702111$$

## 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
 5.89296 − 2.23607i 5.89296 + 2.23607i 3.41677 − 2.23607i 3.41677 + 2.23607i 2.47022 − 2.23607i 2.47022 + 2.23607i 0.613622 − 2.23607i 0.613622 + 2.23607i −0.886481 − 2.23607i −0.886481 + 2.23607i −1.83366 − 2.23607i −1.83366 + 2.23607i −4.53300 − 2.23607i −4.53300 + 2.23607i −5.14043 − 2.23607i −5.14043 + 2.23607i
0 −5.89296 0 2.23607i 0 1.21480i 0 25.7269 0
321.2 0 −5.89296 0 2.23607i 0 1.21480i 0 25.7269 0
321.3 0 −3.41677 0 2.23607i 0 11.7428i 0 2.67432 0
321.4 0 −3.41677 0 2.23607i 0 11.7428i 0 2.67432 0
321.5 0 −2.47022 0 2.23607i 0 8.25674i 0 −2.89803 0
321.6 0 −2.47022 0 2.23607i 0 8.25674i 0 −2.89803 0
321.7 0 −0.613622 0 2.23607i 0 2.35348i 0 −8.62347 0
321.8 0 −0.613622 0 2.23607i 0 2.35348i 0 −8.62347 0
321.9 0 0.886481 0 2.23607i 0 9.10808i 0 −8.21415 0
321.10 0 0.886481 0 2.23607i 0 9.10808i 0 −8.21415 0
321.11 0 1.83366 0 2.23607i 0 0.167846i 0 −5.63770 0
321.12 0 1.83366 0 2.23607i 0 0.167846i 0 −5.63770 0
321.13 0 4.53300 0 2.23607i 0 5.73038i 0 11.5481 0
321.14 0 4.53300 0 2.23607i 0 5.73038i 0 11.5481 0
321.15 0 5.14043 0 2.23607i 0 7.88014i 0 17.4240 0
321.16 0 5.14043 0 2.23607i 0 7.88014i 0 17.4240 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.c 16
4.b odd 2 1 460.3.f.a 16
12.b even 2 1 4140.3.d.a 16
20.d odd 2 1 2300.3.f.e 16
20.e even 4 2 2300.3.d.b 32
23.b odd 2 1 inner 1840.3.k.c 16
92.b even 2 1 460.3.f.a 16
276.h odd 2 1 4140.3.d.a 16
460.g even 2 1 2300.3.f.e 16
460.k odd 4 2 2300.3.d.b 32

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
460.3.f.a 16 4.b odd 2 1
460.3.f.a 16 92.b even 2 1
1840.3.k.c 16 1.a even 1 1 trivial
1840.3.k.c 16 23.b odd 2 1 inner
2300.3.d.b 32 20.e even 4 2
2300.3.d.b 32 460.k odd 4 2
2300.3.f.e 16 20.d odd 2 1
2300.3.f.e 16 460.g even 2 1
4140.3.d.a 16 12.b even 2 1
4140.3.d.a 16 276.h odd 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{8} - 52 T_{3}^{6} + 8 T_{3}^{5} + 724 T_{3}^{4} + 12 T_{3}^{3} - 2557 T_{3}^{2} + 472 T_{3} + 1156$$ acting on $$S_{3}^{\mathrm{new}}(1840, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{16}$$
$3$ $$( 1156 + 472 T - 2557 T^{2} + 12 T^{3} + 724 T^{4} + 8 T^{5} - 52 T^{6} + T^{8} )^{2}$$
$5$ $$( 5 + T^{2} )^{8}$$
$7$ $$366186496 + 13341794000 T^{2} + 12272252524 T^{4} + 2524614569 T^{6} + 155886747 T^{8} + 4281913 T^{10} + 58685 T^{12} + 391 T^{14} + T^{16}$$
$11$ $$538982656 + 20926554176 T^{2} + 62650963968 T^{4} + 23451968484 T^{6} + 1682709097 T^{8} + 32012450 T^{10} + 249475 T^{12} + 842 T^{14} + T^{16}$$
$13$ $$( 12532480 + 55269430 T - 8146861 T^{2} - 1210348 T^{3} + 209672 T^{4} + 408 T^{5} - 832 T^{6} + 6 T^{7} + T^{8} )^{2}$$
$17$ $$645469713961216 + 1095995238952528 T^{2} + 183863288669160 T^{4} + 7139592083589 T^{6} + 103619670159 T^{8} + 612883733 T^{10} + 1669657 T^{12} + 2107 T^{14} + T^{16}$$
$19$ $$483658046479878400 + 209327202365092160 T^{2} + 19764941764557056 T^{4} + 263607893757396 T^{6} + 1388771053737 T^{8} + 3650498706 T^{10} + 5083907 T^{12} + 3578 T^{14} + T^{16}$$
$23$ $$61\!\cdots\!61$$$$-$$$$16\!\cdots\!26$$$$T + 18495943020625150924 T^{2} - 1274362337954270538 T^{3} + 79976503315956308 T^{4} - 3955052048886094 T^{5} + 202509260091572 T^{6} - 10158790780410 T^{7} + 461135845846 T^{8} - 19203763290 T^{9} + 723658292 T^{10} - 26716846 T^{11} + 1021268 T^{12} - 30762 T^{13} + 844 T^{14} - 14 T^{15} + T^{16}$$
$29$ $$( -519358880 + 539930480 T + 343011096 T^{2} - 46866036 T^{3} + 432474 T^{4} + 110777 T^{5} - 2411 T^{6} - 45 T^{7} + T^{8} )^{2}$$
$31$ $$( -162152849534 - 39818067751 T - 2322982667 T^{2} + 53506228 T^{3} + 5788622 T^{4} - 20974 T^{5} - 4394 T^{6} + 5 T^{7} + T^{8} )^{2}$$
$37$ $$216591737307136 + 166004797403668480 T^{2} + 22822976416410624 T^{4} + 848256676333312 T^{6} + 5630424572096 T^{8} + 14212671024 T^{10} + 15172360 T^{12} + 6637 T^{14} + T^{16}$$
$41$ $$( 580968030770 + 39640831075 T - 3870477691 T^{2} - 240145676 T^{3} + 4624142 T^{4} + 285874 T^{5} - 2586 T^{6} - 93 T^{7} + T^{8} )^{2}$$
$43$ $$17\!\cdots\!56$$$$+$$$$18\!\cdots\!20$$$$T^{2} +$$$$66\!\cdots\!24$$$$T^{4} + 1049280834589435904 T^{6} + 875691541868032 T^{8} + 416545461888 T^{10} + 113608080 T^{12} + 16556 T^{14} + T^{16}$$
$47$ $$( -66361863793216 + 1164169651008 T + 103994227520 T^{2} - 2253983984 T^{3} - 41346840 T^{4} + 1249104 T^{5} - 997 T^{6} - 160 T^{7} + T^{8} )^{2}$$
$53$ $$55\!\cdots\!76$$$$+$$$$11\!\cdots\!76$$$$T^{2} + 82687734869125993472 T^{4} + 264266260655688960 T^{6} + 408063006271872 T^{8} + 306592228448 T^{10} + 110793796 T^{12} + 18049 T^{14} + T^{16}$$
$59$ $$( 199085136256 + 130516990272 T - 1515485024 T^{2} - 516164720 T^{3} + 9479032 T^{4} + 399932 T^{5} - 7890 T^{6} - 45 T^{7} + T^{8} )^{2}$$
$61$ $$21\!\cdots\!76$$$$+$$$$90\!\cdots\!72$$$$T^{2} +$$$$15\!\cdots\!64$$$$T^{4} + 14634885952073347364 T^{6} + 7721448036303089 T^{8} + 2321502596466 T^{10} + 380794547 T^{12} + 31266 T^{14} + T^{16}$$
$67$ $$11\!\cdots\!96$$$$+$$$$10\!\cdots\!32$$$$T^{2} +$$$$88\!\cdots\!68$$$$T^{4} + 22176717437650435072 T^{6} + 15511162691402304 T^{8} + 4555411323968 T^{10} + 629483568 T^{12} + 40765 T^{14} + T^{16}$$
$71$ $$( -318523111508570 + 24525587575845 T + 44577347181 T^{2} - 14923409512 T^{3} + 80275002 T^{4} + 2470446 T^{5} - 18498 T^{6} - 119 T^{7} + T^{8} )^{2}$$
$73$ $$( -71365498673344 + 873793190848 T + 664172759456 T^{2} + 16845279680 T^{3} - 25382248 T^{4} - 3676292 T^{5} - 19845 T^{6} + 140 T^{7} + T^{8} )^{2}$$
$79$ $$17\!\cdots\!16$$$$+$$$$28\!\cdots\!96$$$$T^{2} +$$$$12\!\cdots\!24$$$$T^{4} +$$$$14\!\cdots\!80$$$$T^{6} + 63613047675516672 T^{8} + 12545839071936 T^{10} + 1215192880 T^{12} + 56440 T^{14} + T^{16}$$
$83$ $$14\!\cdots\!16$$$$+$$$$10\!\cdots\!52$$$$T^{2} +$$$$78\!\cdots\!00$$$$T^{4} + 87029274096726457344 T^{6} + 38243523522303296 T^{8} + 8309783161072 T^{10} + 936578764 T^{12} + 51025 T^{14} + T^{16}$$
$89$ $$43\!\cdots\!16$$$$+$$$$67\!\cdots\!16$$$$T^{2} +$$$$32\!\cdots\!36$$$$T^{4} +$$$$74\!\cdots\!64$$$$T^{6} + 986394349960649472 T^{8} + 78193127349184 T^{10} + 3666695728 T^{12} + 93608 T^{14} + T^{16}$$
$97$ $$31\!\cdots\!56$$$$+$$$$13\!\cdots\!40$$$$T^{2} +$$$$16\!\cdots\!44$$$$T^{4} + 61013760166167497248 T^{6} + 36744767797548921 T^{8} + 8800917147286 T^{10} + 986183835 T^{12} + 51438 T^{14} + T^{16}$$