# Properties

 Label 252.4.b.g Level $252$ Weight $4$ Character orbit 252.b Analytic conductor $14.868$ Analytic rank $0$ Dimension $16$ CM no Inner twists $8$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$252 = 2^{2} \cdot 3^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 252.b (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$14.8684813214$$ Analytic rank: $$0$$ Dimension: $$16$$ Coefficient field: $$\mathbb{Q}[x]/(x^{16} - \cdots)$$ Defining polynomial: $$x^{16} - 4 x^{15} + 358 x^{14} - 2828 x^{13} + 52557 x^{12} - 549972 x^{11} + 4434734 x^{10} - 37785264 x^{9} + 272741368 x^{8} - 1739202044 x^{7} + 9778426658 x^{6} - 39463975388 x^{5} + 101978126949 x^{4} - 176540053420 x^{3} + 219245087130 x^{2} - 139977817400 x + 52705588025$$ Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2^{26}$$ 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.

 $$f(q)$$ $$=$$ $$q + \beta_{2} q^{2} + ( 3 + \beta_{5} ) q^{4} + \beta_{11} q^{5} + ( 1 + \beta_{5} + \beta_{9} ) q^{7} + ( 2 \beta_{2} - 2 \beta_{6} + \beta_{7} - \beta_{8} ) q^{8} +O(q^{10})$$ $$q + \beta_{2} q^{2} + ( 3 + \beta_{5} ) q^{4} + \beta_{11} q^{5} + ( 1 + \beta_{5} + \beta_{9} ) q^{7} + ( 2 \beta_{2} - 2 \beta_{6} + \beta_{7} - \beta_{8} ) q^{8} -\beta_{1} q^{10} + ( 2 \beta_{2} - 2 \beta_{6} + \beta_{7} ) q^{11} -\beta_{14} q^{13} + ( -2 \beta_{2} - 3 \beta_{6} - 3 \beta_{7} - \beta_{8} - \beta_{10} ) q^{14} + ( -18 + 2 \beta_{3} - 2 \beta_{4} + 4 \beta_{5} - 2 \beta_{9} ) q^{16} + ( \beta_{7} + \beta_{10} - \beta_{11} - \beta_{12} ) q^{17} + ( 2 + \beta_{1} + 2 \beta_{3} + 2 \beta_{9} - \beta_{13} ) q^{19} + ( \beta_{7} + \beta_{10} + \beta_{11} - \beta_{12} + \beta_{15} ) q^{20} + ( -17 + \beta_{3} + 3 \beta_{4} + 2 \beta_{5} - \beta_{9} ) q^{22} + ( 13 \beta_{2} - 13 \beta_{6} - 7 \beta_{7} ) q^{23} + ( -86 - 13 \beta_{4} + 13 \beta_{5} ) q^{25} + ( 2 \beta_{7} + \beta_{8} + \beta_{10} - \beta_{11} - 3 \beta_{12} - \beta_{15} ) q^{26} + ( -51 + 2 \beta_{3} - 15 \beta_{4} + 9 \beta_{5} + 4 \beta_{9} + \beta_{13} + \beta_{14} ) q^{28} + ( 21 \beta_{2} + 7 \beta_{6} + 2 \beta_{8} ) q^{29} + ( 3 - 3 \beta_{1} + 3 \beta_{3} + 3 \beta_{9} + 3 \beta_{13} ) q^{31} + ( -12 \beta_{2} + 4 \beta_{6} + 18 \beta_{7} - 2 \beta_{8} ) q^{32} + ( 2 + \beta_{1} + 2 \beta_{3} + 2 \beta_{9} - 2 \beta_{13} - 2 \beta_{14} ) q^{34} + ( -\beta_{2} + \beta_{6} - 14 \beta_{7} - 2 \beta_{8} + 3 \beta_{10} + \beta_{12} + 2 \beta_{15} ) q^{35} + ( -51 - 11 \beta_{4} + 11 \beta_{5} ) q^{37} + ( -3 \beta_{7} + 2 \beta_{8} - 5 \beta_{10} + 7 \beta_{11} + \beta_{12} - \beta_{15} ) q^{38} + ( 4 - 2 \beta_{1} + 4 \beta_{3} + 4 \beta_{9} + 4 \beta_{13} - 4 \beta_{14} ) q^{40} + ( -7 \beta_{7} - 7 \beta_{10} - \beta_{11} + 7 \beta_{12} ) q^{41} + ( 15 + 13 \beta_{3} + 28 \beta_{4} + 2 \beta_{5} - 13 \beta_{9} ) q^{43} + ( -22 \beta_{2} - 14 \beta_{6} + 9 \beta_{7} - \beta_{8} ) q^{44} + ( -151 - 7 \beta_{3} - 21 \beta_{4} + 40 \beta_{5} + 7 \beta_{9} ) q^{46} + ( 2 \beta_{7} - 4 \beta_{8} + 6 \beta_{10} + 2 \beta_{12} - 4 \beta_{15} ) q^{47} + ( -40 - 4 \beta_{1} - 31 \beta_{4} + 31 \beta_{5} - 4 \beta_{13} - \beta_{14} ) q^{49} + ( -73 \beta_{2} + 26 \beta_{6} + 13 \beta_{7} - 13 \beta_{8} ) q^{50} + ( 10 + 2 \beta_{1} + 10 \beta_{3} + 10 \beta_{9} - 10 \beta_{13} - 2 \beta_{14} ) q^{52} + ( 27 \beta_{2} + 9 \beta_{6} + 22 \beta_{8} ) q^{53} + ( 4 - 3 \beta_{1} + 4 \beta_{3} + 4 \beta_{9} + 3 \beta_{13} ) q^{55} + ( -36 \beta_{2} + 40 \beta_{6} - 3 \beta_{7} - 8 \beta_{8} - 7 \beta_{10} - 7 \beta_{11} + 3 \beta_{12} + \beta_{15} ) q^{56} + ( 156 - 2 \beta_{3} + 10 \beta_{4} + 10 \beta_{5} + 2 \beta_{9} ) q^{58} + ( 4 \beta_{7} - 8 \beta_{8} + 12 \beta_{10} + 4 \beta_{12} ) q^{59} + ( -8 \beta_{1} - 8 \beta_{13} - \beta_{14} ) q^{61} + ( 3 \beta_{8} - 3 \beta_{10} - 21 \beta_{11} - 3 \beta_{12} + 3 \beta_{15} ) q^{62} + ( 68 + 20 \beta_{3} + 44 \beta_{4} - 48 \beta_{5} - 20 \beta_{9} ) q^{64} + ( 72 \beta_{2} + 24 \beta_{6} + 56 \beta_{8} ) q^{65} + ( 21 - 17 \beta_{3} + 4 \beta_{4} + 38 \beta_{5} + 17 \beta_{9} ) q^{67} + ( \beta_{7} + 4 \beta_{8} - 3 \beta_{10} + 13 \beta_{11} - 5 \beta_{12} - 3 \beta_{15} ) q^{68} + ( -45 - 2 \beta_{1} - 25 \beta_{3} - 45 \beta_{4} + 28 \beta_{5} + 5 \beta_{9} + 10 \beta_{13} - 6 \beta_{14} ) q^{70} + ( -39 \beta_{2} + 39 \beta_{6} - 77 \beta_{7} ) q^{71} + ( 16 \beta_{1} + 16 \beta_{13} - 2 \beta_{14} ) q^{73} + ( -40 \beta_{2} + 22 \beta_{6} + 11 \beta_{7} - 11 \beta_{8} ) q^{74} + ( 8 - 6 \beta_{1} + 8 \beta_{3} + 8 \beta_{9} + 8 \beta_{14} ) q^{76} + ( 3 \beta_{2} + \beta_{6} - 4 \beta_{7} - 10 \beta_{8} - 4 \beta_{10} - 7 \beta_{11} + 4 \beta_{12} ) q^{77} + ( -37 + 5 \beta_{3} - 32 \beta_{4} - 42 \beta_{5} - 5 \beta_{9} ) q^{79} + ( 6 \beta_{7} + 8 \beta_{8} - 2 \beta_{10} - 34 \beta_{11} - 14 \beta_{12} - 2 \beta_{15} ) q^{80} + ( -14 + \beta_{1} - 14 \beta_{3} - 14 \beta_{9} + 14 \beta_{13} + 14 \beta_{14} ) q^{82} + ( 6 \beta_{7} - 12 \beta_{8} + 18 \beta_{10} + 6 \beta_{12} + 4 \beta_{15} ) q^{83} + ( 317 + 99 \beta_{4} - 99 \beta_{5} ) q^{85} + ( -4 \beta_{2} - 90 \beta_{6} + 93 \beta_{7} + 11 \beta_{8} ) q^{86} + ( -222 + 10 \beta_{3} + 22 \beta_{4} - 24 \beta_{5} - 10 \beta_{9} ) q^{88} + ( \beta_{7} + \beta_{10} + \beta_{11} - \beta_{12} ) q^{89} + ( -47 + 17 \beta_{1} - 19 \beta_{3} - 60 \beta_{4} - 28 \beta_{5} + 13 \beta_{9} - 17 \beta_{13} ) q^{91} + ( -170 \beta_{2} - 10 \beta_{6} - 9 \beta_{7} - 47 \beta_{8} ) q^{92} + ( -36 + 4 \beta_{1} - 36 \beta_{3} - 36 \beta_{9} - 28 \beta_{13} + 4 \beta_{14} ) q^{94} + ( 134 \beta_{2} - 134 \beta_{6} - 86 \beta_{7} ) q^{95} + ( 8 \beta_{1} + 8 \beta_{13} + 10 \beta_{14} ) q^{97} + ( -9 \beta_{2} + 62 \beta_{6} + 37 \beta_{7} - 30 \beta_{8} + 5 \beta_{10} + 35 \beta_{11} - 7 \beta_{12} + 3 \beta_{15} ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$16 q + 40 q^{4} + O(q^{10})$$ $$16 q + 40 q^{4} - 304 q^{16} - 312 q^{22} - 1376 q^{25} - 816 q^{28} - 816 q^{37} - 2568 q^{46} - 640 q^{49} + 2336 q^{58} + 1120 q^{64} - 424 q^{70} + 5072 q^{85} - 3536 q^{88} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{16} - 4 x^{15} + 358 x^{14} - 2828 x^{13} + 52557 x^{12} - 549972 x^{11} + 4434734 x^{10} - 37785264 x^{9} + 272741368 x^{8} - 1739202044 x^{7} + 9778426658 x^{6} - 39463975388 x^{5} + 101978126949 x^{4} - 176540053420 x^{3} + 219245087130 x^{2} - 139977817400 x + 52705588025$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$-$$$$43\!\cdots\!52$$$$\nu^{15} +$$$$37\!\cdots\!63$$$$\nu^{14} -$$$$45\!\cdots\!16$$$$\nu^{13} +$$$$14\!\cdots\!31$$$$\nu^{12} -$$$$17\!\cdots\!04$$$$\nu^{11} +$$$$25\!\cdots\!94$$$$\nu^{10} -$$$$28\!\cdots\!88$$$$\nu^{9} +$$$$25\!\cdots\!28$$$$\nu^{8} -$$$$19\!\cdots\!56$$$$\nu^{7} +$$$$15\!\cdots\!48$$$$\nu^{6} -$$$$98\!\cdots\!76$$$$\nu^{5} +$$$$57\!\cdots\!26$$$$\nu^{4} -$$$$25\!\cdots\!88$$$$\nu^{3} +$$$$73\!\cdots\!25$$$$\nu^{2} -$$$$11\!\cdots\!80$$$$\nu +$$$$10\!\cdots\!25$$$$)/$$$$11\!\cdots\!00$$ $$\beta_{2}$$ $$=$$ $$($$$$19\!\cdots\!11$$$$\nu^{15} -$$$$57\!\cdots\!92$$$$\nu^{14} +$$$$70\!\cdots\!88$$$$\nu^{13} -$$$$47\!\cdots\!92$$$$\nu^{12} +$$$$10\!\cdots\!40$$$$\nu^{11} -$$$$97\!\cdots\!56$$$$\nu^{10} +$$$$79\!\cdots\!64$$$$\nu^{9} -$$$$67\!\cdots\!96$$$$\nu^{8} +$$$$47\!\cdots\!92$$$$\nu^{7} -$$$$30\!\cdots\!12$$$$\nu^{6} +$$$$16\!\cdots\!58$$$$\nu^{5} -$$$$65\!\cdots\!32$$$$\nu^{4} +$$$$15\!\cdots\!87$$$$\nu^{3} -$$$$25\!\cdots\!60$$$$\nu^{2} +$$$$25\!\cdots\!70$$$$\nu -$$$$71\!\cdots\!00$$$$)/$$$$20\!\cdots\!00$$ $$\beta_{3}$$ $$=$$ $$($$$$-$$$$96\!\cdots\!34$$$$\nu^{15} +$$$$58\!\cdots\!51$$$$\nu^{14} -$$$$34\!\cdots\!02$$$$\nu^{13} +$$$$34\!\cdots\!72$$$$\nu^{12} -$$$$53\!\cdots\!98$$$$\nu^{11} +$$$$62\!\cdots\!28$$$$\nu^{10} -$$$$50\!\cdots\!36$$$$\nu^{9} +$$$$43\!\cdots\!86$$$$\nu^{8} -$$$$31\!\cdots\!42$$$$\nu^{7} +$$$$20\!\cdots\!66$$$$\nu^{6} -$$$$11\!\cdots\!22$$$$\nu^{5} +$$$$50\!\cdots\!62$$$$\nu^{4} -$$$$14\!\cdots\!26$$$$\nu^{3} +$$$$25\!\cdots\!65$$$$\nu^{2} -$$$$27\!\cdots\!60$$$$\nu +$$$$12\!\cdots\!00$$$$)/$$$$56\!\cdots\!00$$ $$\beta_{4}$$ $$=$$ $$($$$$42\!\cdots\!00$$$$\nu^{15} -$$$$24\!\cdots\!99$$$$\nu^{14} +$$$$15\!\cdots\!00$$$$\nu^{13} -$$$$14\!\cdots\!49$$$$\nu^{12} +$$$$23\!\cdots\!48$$$$\nu^{11} -$$$$26\!\cdots\!26$$$$\nu^{10} +$$$$21\!\cdots\!20$$$$\nu^{9} -$$$$17\!\cdots\!52$$$$\nu^{8} +$$$$13\!\cdots\!04$$$$\nu^{7} -$$$$84\!\cdots\!36$$$$\nu^{6} +$$$$48\!\cdots\!40$$$$\nu^{5} -$$$$20\!\cdots\!54$$$$\nu^{4} +$$$$52\!\cdots\!08$$$$\nu^{3} -$$$$84\!\cdots\!29$$$$\nu^{2} +$$$$89\!\cdots\!60$$$$\nu -$$$$30\!\cdots\!55$$$$)/$$$$22\!\cdots\!20$$ $$\beta_{5}$$ $$=$$ $$($$$$42\!\cdots\!20$$$$\nu^{15} -$$$$24\!\cdots\!11$$$$\nu^{14} +$$$$15\!\cdots\!40$$$$\nu^{13} -$$$$14\!\cdots\!61$$$$\nu^{12} +$$$$23\!\cdots\!92$$$$\nu^{11} -$$$$26\!\cdots\!54$$$$\nu^{10} +$$$$21\!\cdots\!00$$$$\nu^{9} -$$$$18\!\cdots\!68$$$$\nu^{8} +$$$$13\!\cdots\!36$$$$\nu^{7} -$$$$84\!\cdots\!64$$$$\nu^{6} +$$$$48\!\cdots\!40$$$$\nu^{5} -$$$$20\!\cdots\!26$$$$\nu^{4} +$$$$52\!\cdots\!52$$$$\nu^{3} -$$$$84\!\cdots\!01$$$$\nu^{2} +$$$$89\!\cdots\!40$$$$\nu -$$$$48\!\cdots\!75$$$$)/$$$$22\!\cdots\!20$$ $$\beta_{6}$$ $$=$$ $$($$$$-$$$$41\!\cdots\!21$$$$\nu^{15} +$$$$15\!\cdots\!80$$$$\nu^{14} -$$$$14\!\cdots\!08$$$$\nu^{13} +$$$$11\!\cdots\!56$$$$\nu^{12} -$$$$21\!\cdots\!88$$$$\nu^{11} +$$$$22\!\cdots\!60$$$$\nu^{10} -$$$$18\!\cdots\!04$$$$\nu^{9} +$$$$15\!\cdots\!28$$$$\nu^{8} -$$$$11\!\cdots\!36$$$$\nu^{7} +$$$$70\!\cdots\!40$$$$\nu^{6} -$$$$39\!\cdots\!18$$$$\nu^{5} +$$$$15\!\cdots\!76$$$$\nu^{4} -$$$$40\!\cdots\!65$$$$\nu^{3} +$$$$65\!\cdots\!20$$$$\nu^{2} -$$$$68\!\cdots\!50$$$$\nu +$$$$32\!\cdots\!00$$$$)/$$$$20\!\cdots\!00$$ $$\beta_{7}$$ $$=$$ $$($$$$-$$$$69\!\cdots\!10$$$$\nu^{15} +$$$$23\!\cdots\!12$$$$\nu^{14} -$$$$25\!\cdots\!99$$$$\nu^{13} +$$$$18\!\cdots\!72$$$$\nu^{12} -$$$$36\!\cdots\!54$$$$\nu^{11} +$$$$36\!\cdots\!96$$$$\nu^{10} -$$$$29\!\cdots\!72$$$$\nu^{9} +$$$$25\!\cdots\!96$$$$\nu^{8} -$$$$18\!\cdots\!32$$$$\nu^{7} +$$$$11\!\cdots\!72$$$$\nu^{6} -$$$$64\!\cdots\!44$$$$\nu^{5} +$$$$25\!\cdots\!52$$$$\nu^{4} -$$$$64\!\cdots\!44$$$$\nu^{3} +$$$$10\!\cdots\!40$$$$\nu^{2} -$$$$10\!\cdots\!75$$$$\nu +$$$$45\!\cdots\!00$$$$)/$$$$29\!\cdots\!40$$ $$\beta_{8}$$ $$=$$ $$($$$$23\!\cdots\!97$$$$\nu^{15} -$$$$30\!\cdots\!56$$$$\nu^{14} +$$$$83\!\cdots\!11$$$$\nu^{13} -$$$$44\!\cdots\!20$$$$\nu^{12} +$$$$11\!\cdots\!02$$$$\nu^{11} -$$$$99\!\cdots\!88$$$$\nu^{10} +$$$$76\!\cdots\!28$$$$\nu^{9} -$$$$67\!\cdots\!60$$$$\nu^{8} +$$$$45\!\cdots\!20$$$$\nu^{7} -$$$$28\!\cdots\!56$$$$\nu^{6} +$$$$15\!\cdots\!86$$$$\nu^{5} -$$$$50\!\cdots\!20$$$$\nu^{4} +$$$$94\!\cdots\!11$$$$\nu^{3} -$$$$14\!\cdots\!60$$$$\nu^{2} +$$$$96\!\cdots\!85$$$$\nu +$$$$15\!\cdots\!00$$$$)/$$$$10\!\cdots\!00$$ $$\beta_{9}$$ $$=$$ $$($$$$17\!\cdots\!06$$$$\nu^{15} -$$$$94\!\cdots\!54$$$$\nu^{14} +$$$$61\!\cdots\!18$$$$\nu^{13} -$$$$57\!\cdots\!23$$$$\nu^{12} +$$$$92\!\cdots\!82$$$$\nu^{11} -$$$$10\!\cdots\!42$$$$\nu^{10} +$$$$83\!\cdots\!24$$$$\nu^{9} -$$$$70\!\cdots\!74$$$$\nu^{8} +$$$$51\!\cdots\!78$$$$\nu^{7} -$$$$32\!\cdots\!54$$$$\nu^{6} +$$$$18\!\cdots\!98$$$$\nu^{5} -$$$$76\!\cdots\!08$$$$\nu^{4} +$$$$18\!\cdots\!34$$$$\nu^{3} -$$$$28\!\cdots\!90$$$$\nu^{2} +$$$$29\!\cdots\!40$$$$\nu -$$$$72\!\cdots\!25$$$$)/$$$$56\!\cdots\!00$$ $$\beta_{10}$$ $$=$$ $$($$$$-$$$$94\!\cdots\!79$$$$\nu^{15} +$$$$19\!\cdots\!24$$$$\nu^{14} -$$$$33\!\cdots\!47$$$$\nu^{13} +$$$$20\!\cdots\!36$$$$\nu^{12} -$$$$46\!\cdots\!46$$$$\nu^{11} +$$$$43\!\cdots\!72$$$$\nu^{10} -$$$$33\!\cdots\!76$$$$\nu^{9} +$$$$29\!\cdots\!68$$$$\nu^{8} -$$$$20\!\cdots\!56$$$$\nu^{7} +$$$$12\!\cdots\!84$$$$\nu^{6} -$$$$67\!\cdots\!82$$$$\nu^{5} +$$$$23\!\cdots\!56$$$$\nu^{4} -$$$$48\!\cdots\!49$$$$\nu^{3} +$$$$61\!\cdots\!60$$$$\nu^{2} -$$$$66\!\cdots\!65$$$$\nu +$$$$28\!\cdots\!00$$$$)/$$$$20\!\cdots\!00$$ $$\beta_{11}$$ $$=$$ $$($$$$62\!\cdots\!59$$$$\nu^{15} -$$$$14\!\cdots\!20$$$$\nu^{14} +$$$$22\!\cdots\!27$$$$\nu^{13} -$$$$14\!\cdots\!64$$$$\nu^{12} +$$$$31\!\cdots\!02$$$$\nu^{11} -$$$$29\!\cdots\!20$$$$\nu^{10} +$$$$23\!\cdots\!76$$$$\nu^{9} -$$$$20\!\cdots\!32$$$$\nu^{8} +$$$$14\!\cdots\!44$$$$\nu^{7} -$$$$89\!\cdots\!00$$$$\nu^{6} +$$$$48\!\cdots\!42$$$$\nu^{5} -$$$$18\!\cdots\!44$$$$\nu^{4} +$$$$41\!\cdots\!85$$$$\nu^{3} -$$$$64\!\cdots\!60$$$$\nu^{2} +$$$$74\!\cdots\!25$$$$\nu -$$$$32\!\cdots\!00$$$$)/$$$$10\!\cdots\!00$$ $$\beta_{12}$$ $$=$$ $$($$$$17\!\cdots\!49$$$$\nu^{15} -$$$$39\!\cdots\!76$$$$\nu^{14} +$$$$62\!\cdots\!07$$$$\nu^{13} -$$$$38\!\cdots\!12$$$$\nu^{12} +$$$$87\!\cdots\!98$$$$\nu^{11} -$$$$82\!\cdots\!68$$$$\nu^{10} +$$$$65\!\cdots\!76$$$$\nu^{9} -$$$$56\!\cdots\!56$$$$\nu^{8} +$$$$39\!\cdots\!72$$$$\nu^{7} -$$$$24\!\cdots\!36$$$$\nu^{6} +$$$$13\!\cdots\!02$$$$\nu^{5} -$$$$50\!\cdots\!52$$$$\nu^{4} +$$$$11\!\cdots\!31$$$$\nu^{3} -$$$$19\!\cdots\!80$$$$\nu^{2} +$$$$24\!\cdots\!85$$$$\nu -$$$$10\!\cdots\!00$$$$)/$$$$20\!\cdots\!00$$ $$\beta_{13}$$ $$=$$ $$($$$$-$$$$11\!\cdots\!36$$$$\nu^{15} +$$$$58\!\cdots\!29$$$$\nu^{14} -$$$$40\!\cdots\!28$$$$\nu^{13} +$$$$36\!\cdots\!73$$$$\nu^{12} -$$$$59\!\cdots\!72$$$$\nu^{11} +$$$$67\!\cdots\!02$$$$\nu^{10} -$$$$52\!\cdots\!04$$$$\nu^{9} +$$$$44\!\cdots\!24$$$$\nu^{8} -$$$$32\!\cdots\!08$$$$\nu^{7} +$$$$20\!\cdots\!84$$$$\nu^{6} -$$$$11\!\cdots\!08$$$$\nu^{5} +$$$$44\!\cdots\!58$$$$\nu^{4} -$$$$98\!\cdots\!84$$$$\nu^{3} +$$$$12\!\cdots\!75$$$$\nu^{2} -$$$$14\!\cdots\!40$$$$\nu +$$$$13\!\cdots\!75$$$$)/$$$$11\!\cdots\!00$$ $$\beta_{14}$$ $$=$$ $$($$$$24\!\cdots\!56$$$$\nu^{15} -$$$$13\!\cdots\!14$$$$\nu^{14} +$$$$86\!\cdots\!08$$$$\nu^{13} -$$$$81\!\cdots\!38$$$$\nu^{12} +$$$$12\!\cdots\!32$$$$\nu^{11} -$$$$14\!\cdots\!82$$$$\nu^{10} +$$$$11\!\cdots\!44$$$$\nu^{9} -$$$$99\!\cdots\!44$$$$\nu^{8} +$$$$72\!\cdots\!28$$$$\nu^{7} -$$$$45\!\cdots\!94$$$$\nu^{6} +$$$$25\!\cdots\!88$$$$\nu^{5} -$$$$10\!\cdots\!98$$$$\nu^{4} +$$$$26\!\cdots\!84$$$$\nu^{3} -$$$$42\!\cdots\!50$$$$\nu^{2} +$$$$55\!\cdots\!40$$$$\nu -$$$$24\!\cdots\!00$$$$)/$$$$21\!\cdots\!25$$ $$\beta_{15}$$ $$=$$ $$($$$$-$$$$69\!\cdots\!57$$$$\nu^{15} +$$$$12\!\cdots\!00$$$$\nu^{14} -$$$$24\!\cdots\!86$$$$\nu^{13} +$$$$14\!\cdots\!32$$$$\nu^{12} -$$$$33\!\cdots\!36$$$$\nu^{11} +$$$$30\!\cdots\!00$$$$\nu^{10} -$$$$23\!\cdots\!68$$$$\nu^{9} +$$$$20\!\cdots\!16$$$$\nu^{8} -$$$$14\!\cdots\!32$$$$\nu^{7} +$$$$88\!\cdots\!00$$$$\nu^{6} -$$$$47\!\cdots\!06$$$$\nu^{5} +$$$$16\!\cdots\!72$$$$\nu^{4} -$$$$30\!\cdots\!45$$$$\nu^{3} +$$$$32\!\cdots\!00$$$$\nu^{2} -$$$$18\!\cdots\!00$$$$\nu -$$$$15\!\cdots\!00$$$$)/$$$$14\!\cdots\!00$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{12} - 2 \beta_{11} - \beta_{10} + \beta_{7} - 3 \beta_{6} - 2 \beta_{5} + 2 \beta_{4} - 5 \beta_{2} + 2$$$$)/8$$ $$\nu^{2}$$ $$=$$ $$($$$$-4 \beta_{14} - 6 \beta_{13} - 3 \beta_{12} + 10 \beta_{11} + 3 \beta_{10} - 10 \beta_{9} - 4 \beta_{8} - 3 \beta_{7} + 9 \beta_{6} - 50 \beta_{5} + 42 \beta_{4} - 18 \beta_{3} - \beta_{2} - 2 \beta_{1} - 368$$$$)/8$$ $$\nu^{3}$$ $$=$$ $$($$$$12 \beta_{15} + 36 \beta_{14} + 66 \beta_{13} - 215 \beta_{12} + 462 \beta_{11} + 47 \beta_{10} + 114 \beta_{9} - 224 \beta_{8} - 841 \beta_{7} + 1564 \beta_{6} + 508 \beta_{5} - 388 \beta_{4} + 186 \beta_{3} + 1628 \beta_{2} + 30 \beta_{1} + 4462$$$$)/16$$ $$\nu^{4}$$ $$=$$ $$($$$$-36 \beta_{15} + 804 \beta_{14} + 1196 \beta_{13} + 820 \beta_{12} - 1636 \beta_{11} - 220 \beta_{10} + 12 \beta_{9} + 748 \beta_{8} + 3026 \beta_{7} - 5621 \beta_{6} + 4782 \beta_{5} + 482 \beta_{4} + 4124 \beta_{3} - 6203 \beta_{2} + 660 \beta_{1} + 39046$$$$)/8$$ $$\nu^{5}$$ $$=$$ $$($$$$-3240 \beta_{15} - 15240 \beta_{14} - 22450 \beta_{13} + 25851 \beta_{12} - 33578 \beta_{11} + 23669 \beta_{10} - 2210 \beta_{9} + 31972 \beta_{8} + 91613 \beta_{7} - 227570 \beta_{6} - 95040 \beta_{5} + 1000 \beta_{4} - 76810 \beta_{3} - 400182 \beta_{2} - 11790 \beta_{1} - 773946$$$$)/16$$ $$\nu^{6}$$ $$=$$ $$($$$$18750 \beta_{15} - 141186 \beta_{14} - 173172 \beta_{13} - 161927 \beta_{12} + 233324 \beta_{11} - 118573 \beta_{10} + 329416 \beta_{9} - 193358 \beta_{8} - 579503 \beta_{7} + 1383882 \beta_{6} - 295256 \beta_{5} - 791320 \beta_{4} - 529368 \beta_{3} + 2309638 \beta_{2} - 146164 \beta_{1} - 2110944$$$$)/8$$ $$\nu^{7}$$ $$=$$ $$($$$$266168 \beta_{15} + 2077460 \beta_{14} + 2633750 \beta_{13} - 805420 \beta_{12} - 489540 \beta_{11} - 3194548 \beta_{10} - 4009054 \beta_{9} - 2303488 \beta_{8} - 199550 \beta_{7} + 11144103 \beta_{6} + 5780738 \beta_{5} + 9715806 \beta_{4} + 8237250 \beta_{3} + 36486257 \beta_{2} + 2074506 \beta_{1} + 43322176$$$$)/8$$ $$\nu^{8}$$ $$=$$ $$($$$$-4847584 \beta_{15} + 17684688 \beta_{14} + 16913192 \beta_{13} + 21114556 \beta_{12} - 7210088 \beta_{11} + 51939460 \beta_{10} - 77871896 \beta_{9} + 44097328 \beta_{8} + 37622634 \beta_{7} - 239964527 \beta_{6} - 36621690 \beta_{5} + 174287130 \beta_{4} + 30924808 \beta_{3} - 651794481 \beta_{2} + 23156120 \beta_{1} - 366498678$$$$)/8$$ $$\nu^{9}$$ $$=$$ $$($$$$-48703440 \beta_{15} - 883011696 \beta_{14} - 946725174 \beta_{13} - 212327525 \beta_{12} + 1001801038 \beta_{11} + 946190405 \beta_{10} + 3072715818 \beta_{9} + 431180796 \beta_{8} - 1847708687 \beta_{7} - 349264568 \beta_{6} + 235058804 \beta_{5} - 6991690844 \beta_{4} - 2265848622 \beta_{3} - 8478589616 \beta_{2} - 1053796170 \beta_{1} + 4634714062$$$$)/16$$ $$\nu^{10}$$ $$=$$ $$($$$$904335150 \beta_{15} - 834604638 \beta_{14} - 11176738 \beta_{13} - 284385156 \beta_{12} - 7899378702 \beta_{11} - 13336545744 \beta_{10} + 9554100242 \beta_{9} - 8147035402 \beta_{8} + 12131540892 \beta_{7} + 27081904205 \beta_{6} + 14644364698 \beta_{5} - 21017586322 \beta_{4} + 4518974522 \beta_{3} + 138383558427 \beta_{2} - 1879735766 \beta_{1} + 127307491120$$$$)/8$$ $$\nu^{11}$$ $$=$$ $$($$$$-1709516380 \beta_{15} + 140781230964 \beta_{14} + 118134494434 \beta_{13} + 78604043615 \beta_{12} - 182664208222 \beta_{11} - 52853769879 \beta_{10} - 735967210550 \beta_{9} + 14926146072 \beta_{8} + 372005735461 \beta_{7} - 412006960442 \beta_{6} - 573841693696 \beta_{5} + 1648637691752 \beta_{4} + 113302270114 \beta_{3} + 104468702290 \beta_{2} + 200813588734 \beta_{1} - 5309532618958$$$$)/16$$ $$\nu^{12}$$ $$=$$ $$($$$$-59587475706 \beta_{15} - 133295785114 \beta_{14} - 202402059216 \beta_{13} - 366139102959 \beta_{12} + 1494678265780 \beta_{11} + 1263659388555 \beta_{10} + 16326659088 \beta_{9} + 540495443954 \beta_{8} - 2741166180499 \beta_{7} - 13462740024 \beta_{6} - 958027896676 \beta_{5} - 63233604956 \beta_{4} - 790626717184 \beta_{3} - 10929941133512 \beta_{2} - 99589169360 \beta_{1} - 7757738479936$$$$)/4$$ $$\nu^{13}$$ $$=$$ $$($$$$849995603080 \beta_{15} - 7099600357616 \beta_{14} - 2257804253224 \beta_{13} - 4421761574763 \beta_{12} + 3090593743942 \beta_{11} - 8381128713477 \beta_{10} + 64907719966128 \beta_{9} - 7687785843640 \beta_{8} - 9318926453931 \beta_{7} + 44583240471617 \beta_{6} + 90412717133934 \beta_{5} - 144437042900254 \beta_{4} + 22219850150368 \beta_{3} + 111275340793575 \beta_{2} - 13826805247320 \beta_{1} + 791181188266770$$$$)/8$$ $$\nu^{14}$$ $$=$$ $$($$$$6936535769800 \beta_{15} + 85734367697660 \beta_{14} + 85620421150406 \beta_{13} + 242965843464895 \beta_{12} - 681091059946938 \beta_{11} - 347445975883375 \beta_{10} - 344653095116662 \beta_{9} - 63263013712996 \beta_{8} + 1323361309955667 \beta_{7} - 919686719917747 \beta_{6} - 122143046100018 \beta_{5} + 775326991511386 \beta_{4} + 171467748656722 \beta_{3} + 2197410711886891 \beta_{2} + 108616297696162 \beta_{1} - 1297266920300076$$$$)/8$$ $$\nu^{15}$$ $$=$$ $$($$$$-345292836168636 \beta_{15} - 220267946319092 \beta_{14} - 2466449448558502 \beta_{13} - 1230338357585255 \beta_{12} + 6405173991855366 \beta_{11} + 6431225539341103 \beta_{10} - 16028645404405462 \beta_{9} + 3138728928356680 \beta_{8} - 11398692250537553 \beta_{7} - 4212910709054100 \beta_{6} - 37034519133421588 \beta_{5} + 35345314731978412 \beta_{4} - 17363388468424622 \beta_{3} - 59275303729666004 \beta_{2} + 1967418289897926 \beta_{1} - 314693562872341418$$$$)/16$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/252\mathbb{Z}\right)^\times$$.

 $$n$$ $$29$$ $$73$$ $$127$$ $$\chi(n)$$ $$1$$ $$-1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
55.1
 0.929907 + 6.47387i 0.929907 − 12.9280i 0.929907 + 12.9280i 0.929907 − 6.47387i 4.06402 − 0.600106i 4.06402 + 1.75957i 4.06402 − 1.75957i 4.06402 + 0.600106i 0.467111 − 0.600106i 0.467111 + 1.75957i 0.467111 − 1.75957i 0.467111 + 0.600106i −4.46104 + 6.47387i −4.46104 − 12.9280i −4.46104 + 12.9280i −4.46104 − 6.47387i
−2.69547 0.856992i 0 6.53113 + 4.62000i 10.3049i 0 −16.6272 8.15690i −13.6452 18.0502i 0 −8.83121 + 27.7765i
55.2 −2.69547 0.856992i 0 6.53113 + 4.62000i 10.3049i 0 16.6272 8.15690i −13.6452 18.0502i 0 8.83121 27.7765i
55.3 −2.69547 + 0.856992i 0 6.53113 4.62000i 10.3049i 0 16.6272 + 8.15690i −13.6452 + 18.0502i 0 8.83121 + 27.7765i
55.4 −2.69547 + 0.856992i 0 6.53113 4.62000i 10.3049i 0 −16.6272 + 8.15690i −13.6452 + 18.0502i 0 −8.83121 27.7765i
55.5 −1.79845 2.18302i 0 −1.53113 + 7.85211i 17.7710i 0 5.15121 + 17.7895i 19.8950 10.7792i 0 −38.7945 + 31.9604i
55.6 −1.79845 2.18302i 0 −1.53113 + 7.85211i 17.7710i 0 −5.15121 + 17.7895i 19.8950 10.7792i 0 38.7945 31.9604i
55.7 −1.79845 + 2.18302i 0 −1.53113 7.85211i 17.7710i 0 −5.15121 17.7895i 19.8950 + 10.7792i 0 38.7945 + 31.9604i
55.8 −1.79845 + 2.18302i 0 −1.53113 7.85211i 17.7710i 0 5.15121 17.7895i 19.8950 + 10.7792i 0 −38.7945 31.9604i
55.9 1.79845 2.18302i 0 −1.53113 7.85211i 17.7710i 0 5.15121 17.7895i −19.8950 10.7792i 0 −38.7945 31.9604i
55.10 1.79845 2.18302i 0 −1.53113 7.85211i 17.7710i 0 −5.15121 17.7895i −19.8950 10.7792i 0 38.7945 + 31.9604i
55.11 1.79845 + 2.18302i 0 −1.53113 + 7.85211i 17.7710i 0 −5.15121 + 17.7895i −19.8950 + 10.7792i 0 38.7945 31.9604i
55.12 1.79845 + 2.18302i 0 −1.53113 + 7.85211i 17.7710i 0 5.15121 + 17.7895i −19.8950 + 10.7792i 0 −38.7945 + 31.9604i
55.13 2.69547 0.856992i 0 6.53113 4.62000i 10.3049i 0 −16.6272 + 8.15690i 13.6452 18.0502i 0 −8.83121 27.7765i
55.14 2.69547 0.856992i 0 6.53113 4.62000i 10.3049i 0 16.6272 + 8.15690i 13.6452 18.0502i 0 8.83121 + 27.7765i
55.15 2.69547 + 0.856992i 0 6.53113 + 4.62000i 10.3049i 0 16.6272 8.15690i 13.6452 + 18.0502i 0 8.83121 27.7765i
55.16 2.69547 + 0.856992i 0 6.53113 + 4.62000i 10.3049i 0 −16.6272 8.15690i 13.6452 + 18.0502i 0 −8.83121 + 27.7765i
 $$n$$: 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
7.b odd 2 1 inner
12.b even 2 1 inner
21.c even 2 1 inner
28.d even 2 1 inner
84.h odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 252.4.b.g 16
3.b odd 2 1 inner 252.4.b.g 16
4.b odd 2 1 inner 252.4.b.g 16
7.b odd 2 1 inner 252.4.b.g 16
12.b even 2 1 inner 252.4.b.g 16
21.c even 2 1 inner 252.4.b.g 16
28.d even 2 1 inner 252.4.b.g 16
84.h odd 2 1 inner 252.4.b.g 16

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
252.4.b.g 16 1.a even 1 1 trivial
252.4.b.g 16 3.b odd 2 1 inner
252.4.b.g 16 4.b odd 2 1 inner
252.4.b.g 16 7.b odd 2 1 inner
252.4.b.g 16 12.b even 2 1 inner
252.4.b.g 16 21.c even 2 1 inner
252.4.b.g 16 28.d even 2 1 inner
252.4.b.g 16 84.h odd 2 1 inner

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{4}^{\mathrm{new}}(252, [\chi])$$:

 $$T_{5}^{4} + 422 T_{5}^{2} + 33536$$ $$T_{11}^{4} + 442 T_{11}^{2} + 37856$$ $$T_{19}^{4} - 12028 T_{19}^{2} + 23005696$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 4096 - 640 T^{2} + 88 T^{4} - 10 T^{6} + T^{8} )^{2}$$
$3$ $$T^{16}$$
$5$ $$( 33536 + 422 T^{2} + T^{4} )^{4}$$
$7$ $$( 13841287201 + 18823840 T^{2} - 8162 T^{4} + 160 T^{6} + T^{8} )^{2}$$
$11$ $$( 37856 + 442 T^{2} + T^{4} )^{4}$$
$13$ $$( 12609536 + 7648 T^{2} + T^{4} )^{4}$$
$17$ $$( 8585216 + 6118 T^{2} + T^{4} )^{4}$$
$19$ $$( 23005696 - 12028 T^{2} + T^{4} )^{4}$$
$23$ $$( 29353856 + 26842 T^{2} + T^{4} )^{4}$$
$29$ $$( 18866176 - 10448 T^{2} + T^{4} )^{4}$$
$31$ $$( 950745600 - 64080 T^{2} + T^{4} )^{4}$$
$37$ $$( -5264 + 102 T + T^{2} )^{8}$$
$41$ $$( 14785888256 + 255782 T^{2} + T^{4} )^{4}$$
$43$ $$( 115302656 + 222172 T^{2} + T^{4} )^{4}$$
$47$ $$( 70613401600 - 535680 T^{2} + T^{4} )^{4}$$
$53$ $$( 10418845696 - 293712 T^{2} + T^{4} )^{4}$$
$59$ $$( 11298144256 - 536192 T^{2} + T^{4} )^{4}$$
$61$ $$( 46920083456 + 436192 T^{2} + T^{4} )^{4}$$
$67$ $$( 25756457216 + 503548 T^{2} + T^{4} )^{4}$$
$71$ $$( 30079110656 + 1021658 T^{2} + T^{4} )^{4}$$
$73$ $$( 852404633600 + 1988480 T^{2} + T^{4} )^{4}$$
$79$ $$( 21224784896 + 508828 T^{2} + T^{4} )^{4}$$
$83$ $$( 477346594816 - 1392768 T^{2} + T^{4} )^{4}$$
$89$ $$( 3353600 + 5270 T^{2} + T^{4} )^{4}$$
$97$ $$( 39543504896 + 919168 T^{2} + T^{4} )^{4}$$