# Properties

 Label 252.8.k.e Level $252$ Weight $8$ Character orbit 252.k Analytic conductor $78.721$ Analytic rank $0$ Dimension $16$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$252 = 2^{2} \cdot 3^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$8$$ Character orbit: $$[\chi]$$ $$=$$ 252.k (of order $$3$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$78.7210264220$$ Analytic rank: $$0$$ Dimension: $$16$$ Relative dimension: $$8$$ over $$\Q(\zeta_{3})$$ Coefficient field: $$\mathbb{Q}[x]/(x^{16} + \cdots)$$ Defining polynomial: $$x^{16} + 89566 x^{14} - 207320 x^{13} + 5161603375 x^{12} - 17143558340 x^{11} + 178819626045814 x^{10} - 993183560713460 x^{9} + 4531523101103784409 x^{8} - 25038421018919762700 x^{7} + 77467785793463920028824 x^{6} - 352418565495722251364640 x^{5} + 967025470523891143640719680 x^{4} - 1651641278575894088053178880 x^{3} + 7200793191683961412674394484736 x^{2} + 21939035854717669821876654243840 x + 34494438285389383231614325978300416$$ Coefficient ring: $$\Z[a_1, \ldots, a_{19}]$$ Coefficient ring index: $$2^{20}\cdot 3^{13}\cdot 7^{8}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + ( \beta_{2} + \beta_{4} ) q^{5} + ( 35 + 140 \beta_{1} - 3 \beta_{3} + 2 \beta_{9} ) q^{7} +O(q^{10})$$ $$q + ( \beta_{2} + \beta_{4} ) q^{5} + ( 35 + 140 \beta_{1} - 3 \beta_{3} + 2 \beta_{9} ) q^{7} -\beta_{10} q^{11} + ( -1765 - 2 \beta_{3} + 2 \beta_{5} + 5 \beta_{7} + 5 \beta_{8} + 7 \beta_{9} ) q^{13} -\beta_{12} q^{17} + ( 5289 - 5267 \beta_{1} - 10 \beta_{3} - 39 \beta_{5} + 22 \beta_{6} + 22 \beta_{7} + 29 \beta_{8} - 29 \beta_{9} ) q^{19} + ( -14 \beta_{2} - 14 \beta_{4} + \beta_{10} + \beta_{11} - \beta_{12} - \beta_{13} - \beta_{14} - \beta_{15} ) q^{23} + ( -10054 \beta_{1} - 65 \beta_{3} - 33 \beta_{5} + 7 \beta_{6} + 65 \beta_{8} + 33 \beta_{9} ) q^{25} + ( -21 \beta_{4} - 2 \beta_{11} + 5 \beta_{13} - 2 \beta_{15} ) q^{29} + ( 20582 \beta_{1} + 78 \beta_{3} + 5 \beta_{5} - 24 \beta_{6} - 78 \beta_{8} - 5 \beta_{9} ) q^{31} + ( 10 \beta_{2} + 8 \beta_{4} - 19 \beta_{10} - 11 \beta_{11} - 11 \beta_{12} - 6 \beta_{13} - \beta_{14} + 2 \beta_{15} ) q^{35} + ( -81085 + 80910 \beta_{1} + 113 \beta_{3} + 335 \beta_{5} - 175 \beta_{6} - 175 \beta_{7} - 222 \beta_{8} + 222 \beta_{9} ) q^{37} + ( 100 \beta_{4} + 56 \beta_{11} - 10 \beta_{13} ) q^{41} + ( 83805 + 475 \beta_{3} - 475 \beta_{5} - 70 \beta_{7} - 32 \beta_{8} - 507 \beta_{9} ) q^{43} + ( 178 \beta_{2} + 178 \beta_{4} - 77 \beta_{10} - 77 \beta_{11} + 23 \beta_{12} + 23 \beta_{13} + 7 \beta_{14} + 7 \beta_{15} ) q^{47} + ( 71638 - 115003 \beta_{1} - 854 \beta_{3} - 882 \beta_{5} + 98 \beta_{6} - 539 \beta_{7} + 931 \beta_{8} + 63 \beta_{9} ) q^{49} + ( 441 \beta_{2} + 176 \beta_{10} + 18 \beta_{12} + 4 \beta_{14} ) q^{53} + ( -19761 + 1638 \beta_{3} - 1638 \beta_{5} + 868 \beta_{7} + 933 \beta_{8} - 705 \beta_{9} ) q^{55} + ( -626 \beta_{2} + 84 \beta_{10} + 75 \beta_{12} - 7 \beta_{14} ) q^{59} + ( -539378 + 540456 \beta_{1} + 390 \beta_{3} - 786 \beta_{5} + 1078 \beta_{6} + 1078 \beta_{7} + 1176 \beta_{8} - 1176 \beta_{9} ) q^{61} + ( -1988 \beta_{2} - 1988 \beta_{4} + 252 \beta_{10} + 252 \beta_{11} - 65 \beta_{12} - 65 \beta_{13} + 12 \beta_{14} + 12 \beta_{15} ) q^{65} + ( -488815 \beta_{1} - 5895 \beta_{3} - 5298 \beta_{5} - 1190 \beta_{6} + 5895 \beta_{8} + 5298 \beta_{9} ) q^{67} + ( 882 \beta_{4} - 293 \beta_{11} + 75 \beta_{13} + 19 \beta_{15} ) q^{71} + ( 809240 \beta_{1} - 2145 \beta_{3} - 4389 \beta_{5} + 535 \beta_{6} + 2145 \beta_{8} + 4389 \beta_{9} ) q^{73} + ( 2579 \beta_{2} - 2379 \beta_{4} - 866 \beta_{10} - 494 \beta_{11} - 25 \beta_{12} + \beta_{13} + 6 \beta_{14} - 26 \beta_{15} ) q^{77} + ( -761232 + 762044 \beta_{1} - 1985 \beta_{3} - 8564 \beta_{5} + 812 \beta_{6} + 812 \beta_{7} + 6579 \beta_{8} - 6579 \beta_{9} ) q^{79} + ( 7042 \beta_{4} + 1372 \beta_{11} + 39 \beta_{13} - 7 \beta_{15} ) q^{83} + ( 27132 + 11064 \beta_{3} - 11064 \beta_{5} - 2786 \beta_{7} + 7254 \beta_{8} - 3810 \beta_{9} ) q^{85} + ( 3758 \beta_{2} + 3758 \beta_{4} - 980 \beta_{10} - 980 \beta_{11} - 140 \beta_{12} - 140 \beta_{13} - 84 \beta_{14} - 84 \beta_{15} ) q^{89} + ( 258209 + 1476825 \beta_{1} + 5816 \beta_{3} - 16905 \beta_{5} - 3038 \beta_{6} - 1470 \beta_{7} + 5439 \beta_{8} - 5205 \beta_{9} ) q^{91} + ( 11410 \beta_{2} + 1353 \beta_{10} - 161 \beta_{12} - 49 \beta_{14} ) q^{95} + ( -1697780 + 17080 \beta_{3} - 17080 \beta_{5} - 2905 \beta_{7} + 17595 \beta_{8} + 515 \beta_{9} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$16q + 1680q^{7} + O(q^{10})$$ $$16q + 1680q^{7} - 28280q^{13} + 42224q^{19} - 80460q^{25} + 164752q^{31} - 647980q^{37} + 1341440q^{43} + 230104q^{49} - 323120q^{55} - 4319336q^{61} - 3905760q^{67} + 6471780q^{73} - 6093104q^{79} + 456400q^{85} + 15969856q^{91} - 27141240q^{97} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{16} + 89566 x^{14} - 207320 x^{13} + 5161603375 x^{12} - 17143558340 x^{11} + 178819626045814 x^{10} - 993183560713460 x^{9} + 4531523101103784409 x^{8} - 25038421018919762700 x^{7} + 77467785793463920028824 x^{6} - 352418565495722251364640 x^{5} + 967025470523891143640719680 x^{4} - 1651641278575894088053178880 x^{3} + 7200793191683961412674394484736 x^{2} + 21939035854717669821876654243840 x + 34494438285389383231614325978300416$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$63\!\cdots\!15$$$$\nu^{15} -$$$$65\!\cdots\!00$$$$\nu^{14} +$$$$51\!\cdots\!70$$$$\nu^{13} -$$$$52\!\cdots\!20$$$$\nu^{12} +$$$$30\!\cdots\!25$$$$\nu^{11} -$$$$29\!\cdots\!48$$$$\nu^{10} +$$$$10\!\cdots\!50$$$$\nu^{9} -$$$$94\!\cdots\!00$$$$\nu^{8} +$$$$27\!\cdots\!55$$$$\nu^{7} -$$$$23\!\cdots\!00$$$$\nu^{6} +$$$$44\!\cdots\!60$$$$\nu^{5} -$$$$34\!\cdots\!40$$$$\nu^{4} +$$$$52\!\cdots\!00$$$$\nu^{3} -$$$$46\!\cdots\!80$$$$\nu^{2} +$$$$20\!\cdots\!20$$$$\nu -$$$$15\!\cdots\!72$$$$)/$$$$18\!\cdots\!28$$ $$\beta_{2}$$ $$=$$ $$($$$$-$$$$75\!\cdots\!99$$$$\nu^{15} +$$$$16\!\cdots\!32$$$$\nu^{14} -$$$$28\!\cdots\!82$$$$\nu^{13} +$$$$16\!\cdots\!80$$$$\nu^{12} -$$$$88\!\cdots\!37$$$$\nu^{11} +$$$$93\!\cdots\!44$$$$\nu^{10} +$$$$17\!\cdots\!74$$$$\nu^{9} +$$$$32\!\cdots\!52$$$$\nu^{8} +$$$$87\!\cdots\!89$$$$\nu^{7} +$$$$71\!\cdots\!80$$$$\nu^{6} +$$$$27\!\cdots\!68$$$$\nu^{5} +$$$$10\!\cdots\!60$$$$\nu^{4} +$$$$32\!\cdots\!88$$$$\nu^{3} +$$$$97\!\cdots\!24$$$$\nu^{2} +$$$$32\!\cdots\!16$$$$\nu +$$$$51\!\cdots\!36$$$$)/$$$$12\!\cdots\!80$$ $$\beta_{3}$$ $$=$$ $$($$$$-$$$$85\!\cdots\!63$$$$\nu^{15} -$$$$17\!\cdots\!60$$$$\nu^{14} -$$$$32\!\cdots\!26$$$$\nu^{13} -$$$$15\!\cdots\!64$$$$\nu^{12} -$$$$21\!\cdots\!93$$$$\nu^{11} -$$$$84\!\cdots\!88$$$$\nu^{10} -$$$$97\!\cdots\!74$$$$\nu^{9} -$$$$27\!\cdots\!04$$$$\nu^{8} -$$$$23\!\cdots\!63$$$$\nu^{7} -$$$$58\!\cdots\!72$$$$\nu^{6} -$$$$42\!\cdots\!56$$$$\nu^{5} -$$$$82\!\cdots\!64$$$$\nu^{4} -$$$$41\!\cdots\!28$$$$\nu^{3} -$$$$72\!\cdots\!08$$$$\nu^{2} -$$$$29\!\cdots\!80$$$$\nu -$$$$33\!\cdots\!64$$$$)/$$$$12\!\cdots\!80$$ $$\beta_{4}$$ $$=$$ $$($$$$-$$$$77\!\cdots\!57$$$$\nu^{15} -$$$$15\!\cdots\!52$$$$\nu^{14} -$$$$77\!\cdots\!50$$$$\nu^{13} -$$$$11\!\cdots\!28$$$$\nu^{12} -$$$$43\!\cdots\!59$$$$\nu^{11} -$$$$59\!\cdots\!60$$$$\nu^{10} -$$$$14\!\cdots\!42$$$$\nu^{9} -$$$$16\!\cdots\!80$$$$\nu^{8} -$$$$32\!\cdots\!05$$$$\nu^{7} -$$$$33\!\cdots\!84$$$$\nu^{6} -$$$$49\!\cdots\!00$$$$\nu^{5} -$$$$40\!\cdots\!08$$$$\nu^{4} -$$$$46\!\cdots\!84$$$$\nu^{3} -$$$$34\!\cdots\!80$$$$\nu^{2} -$$$$24\!\cdots\!96$$$$\nu -$$$$11\!\cdots\!84$$$$)/$$$$68\!\cdots\!40$$ $$\beta_{5}$$ $$=$$ $$($$$$30\!\cdots\!77$$$$\nu^{15} +$$$$21\!\cdots\!16$$$$\nu^{14} +$$$$50\!\cdots\!42$$$$\nu^{13} +$$$$17\!\cdots\!32$$$$\nu^{12} +$$$$30\!\cdots\!23$$$$\nu^{11} +$$$$92\!\cdots\!76$$$$\nu^{10} +$$$$12\!\cdots\!74$$$$\nu^{9} +$$$$27\!\cdots\!08$$$$\nu^{8} +$$$$28\!\cdots\!61$$$$\nu^{7} +$$$$57\!\cdots\!56$$$$\nu^{6} +$$$$48\!\cdots\!72$$$$\nu^{5} +$$$$77\!\cdots\!72$$$$\nu^{4} +$$$$46\!\cdots\!08$$$$\nu^{3} +$$$$67\!\cdots\!76$$$$\nu^{2} +$$$$30\!\cdots\!08$$$$\nu +$$$$29\!\cdots\!20$$$$)/$$$$12\!\cdots\!80$$ $$\beta_{6}$$ $$=$$ $$($$$$-$$$$82\!\cdots\!73$$$$\nu^{15} +$$$$26\!\cdots\!04$$$$\nu^{14} -$$$$19\!\cdots\!06$$$$\nu^{13} +$$$$25\!\cdots\!00$$$$\nu^{12} -$$$$81\!\cdots\!43$$$$\nu^{11} +$$$$14\!\cdots\!00$$$$\nu^{10} +$$$$67\!\cdots\!58$$$$\nu^{9} +$$$$47\!\cdots\!16$$$$\nu^{8} +$$$$23\!\cdots\!83$$$$\nu^{7} +$$$$10\!\cdots\!80$$$$\nu^{6} +$$$$60\!\cdots\!64$$$$\nu^{5} +$$$$16\!\cdots\!00$$$$\nu^{4} +$$$$69\!\cdots\!48$$$$\nu^{3} +$$$$15\!\cdots\!36$$$$\nu^{2} +$$$$63\!\cdots\!92$$$$\nu +$$$$77\!\cdots\!20$$$$)/$$$$30\!\cdots\!72$$ $$\beta_{7}$$ $$=$$ $$($$$$-$$$$21\!\cdots\!13$$$$\nu^{15} -$$$$28\!\cdots\!60$$$$\nu^{14} -$$$$20\!\cdots\!90$$$$\nu^{13} -$$$$21\!\cdots\!56$$$$\nu^{12} -$$$$11\!\cdots\!51$$$$\nu^{11} -$$$$11\!\cdots\!20$$$$\nu^{10} -$$$$38\!\cdots\!94$$$$\nu^{9} -$$$$31\!\cdots\!60$$$$\nu^{8} -$$$$85\!\cdots\!25$$$$\nu^{7} -$$$$63\!\cdots\!52$$$$\nu^{6} -$$$$13\!\cdots\!80$$$$\nu^{5} -$$$$77\!\cdots\!80$$$$\nu^{4} -$$$$12\!\cdots\!64$$$$\nu^{3} -$$$$66\!\cdots\!40$$$$\nu^{2} -$$$$63\!\cdots\!64$$$$\nu -$$$$16\!\cdots\!44$$$$)/$$$$24\!\cdots\!16$$ $$\beta_{8}$$ $$=$$ $$($$$$-$$$$38\!\cdots\!90$$$$\nu^{15} +$$$$18\!\cdots\!03$$$$\nu^{14} -$$$$25\!\cdots\!76$$$$\nu^{13} +$$$$26\!\cdots\!18$$$$\nu^{12} -$$$$12\!\cdots\!42$$$$\nu^{11} +$$$$15\!\cdots\!77$$$$\nu^{10} -$$$$28\!\cdots\!76$$$$\nu^{9} +$$$$60\!\cdots\!66$$$$\nu^{8} -$$$$51\!\cdots\!18$$$$\nu^{7} +$$$$13\!\cdots\!99$$$$\nu^{6} -$$$$41\!\cdots\!96$$$$\nu^{5} +$$$$22\!\cdots\!88$$$$\nu^{4} -$$$$20\!\cdots\!32$$$$\nu^{3} +$$$$20\!\cdots\!12$$$$\nu^{2} +$$$$26\!\cdots\!64$$$$\nu +$$$$12\!\cdots\!32$$$$)/$$$$38\!\cdots\!40$$ $$\beta_{9}$$ $$=$$ $$($$$$33\!\cdots\!33$$$$\nu^{15} +$$$$25\!\cdots\!44$$$$\nu^{14} +$$$$23\!\cdots\!58$$$$\nu^{13} -$$$$69\!\cdots\!92$$$$\nu^{12} +$$$$11\!\cdots\!47$$$$\nu^{11} -$$$$53\!\cdots\!76$$$$\nu^{10} +$$$$29\!\cdots\!86$$$$\nu^{9} -$$$$28\!\cdots\!08$$$$\nu^{8} +$$$$53\!\cdots\!89$$$$\nu^{7} -$$$$72\!\cdots\!36$$$$\nu^{6} +$$$$52\!\cdots\!68$$$$\nu^{5} -$$$$13\!\cdots\!32$$$$\nu^{4} +$$$$34\!\cdots\!12$$$$\nu^{3} -$$$$12\!\cdots\!76$$$$\nu^{2} -$$$$10\!\cdots\!28$$$$\nu -$$$$82\!\cdots\!00$$$$)/$$$$30\!\cdots\!20$$ $$\beta_{10}$$ $$=$$ $$($$$$-$$$$48\!\cdots\!03$$$$\nu^{15} +$$$$11\!\cdots\!36$$$$\nu^{14} -$$$$20\!\cdots\!42$$$$\nu^{13} +$$$$12\!\cdots\!00$$$$\nu^{12} -$$$$71\!\cdots\!05$$$$\nu^{11} +$$$$67\!\cdots\!28$$$$\nu^{10} +$$$$46\!\cdots\!58$$$$\nu^{9} +$$$$23\!\cdots\!88$$$$\nu^{8} +$$$$36\!\cdots\!73$$$$\nu^{7} +$$$$51\!\cdots\!40$$$$\nu^{6} +$$$$13\!\cdots\!12$$$$\nu^{5} +$$$$77\!\cdots\!00$$$$\nu^{4} +$$$$16\!\cdots\!24$$$$\nu^{3} +$$$$70\!\cdots\!80$$$$\nu^{2} +$$$$17\!\cdots\!72$$$$\nu +$$$$36\!\cdots\!52$$$$)/$$$$72\!\cdots\!20$$ $$\beta_{11}$$ $$=$$ $$($$$$-$$$$91\!\cdots\!09$$$$\nu^{15} -$$$$21\!\cdots\!44$$$$\nu^{14} -$$$$91\!\cdots\!30$$$$\nu^{13} -$$$$16\!\cdots\!48$$$$\nu^{12} -$$$$50\!\cdots\!15$$$$\nu^{11} -$$$$83\!\cdots\!60$$$$\nu^{10} -$$$$17\!\cdots\!18$$$$\nu^{9} -$$$$23\!\cdots\!80$$$$\nu^{8} -$$$$38\!\cdots\!85$$$$\nu^{7} -$$$$46\!\cdots\!72$$$$\nu^{6} -$$$$57\!\cdots\!60$$$$\nu^{5} -$$$$57\!\cdots\!56$$$$\nu^{4} -$$$$52\!\cdots\!84$$$$\nu^{3} -$$$$47\!\cdots\!40$$$$\nu^{2} -$$$$29\!\cdots\!00$$$$\nu -$$$$14\!\cdots\!40$$$$)/$$$$80\!\cdots\!20$$ $$\beta_{12}$$ $$=$$ $$($$$$-$$$$10\!\cdots\!97$$$$\nu^{15} +$$$$20\!\cdots\!28$$$$\nu^{14} -$$$$44\!\cdots\!54$$$$\nu^{13} +$$$$21\!\cdots\!80$$$$\nu^{12} -$$$$16\!\cdots\!07$$$$\nu^{11} +$$$$11\!\cdots\!72$$$$\nu^{10} +$$$$56\!\cdots\!02$$$$\nu^{9} +$$$$41\!\cdots\!80$$$$\nu^{8} +$$$$50\!\cdots\!07$$$$\nu^{7} +$$$$90\!\cdots\!20$$$$\nu^{6} +$$$$22\!\cdots\!20$$$$\nu^{5} +$$$$13\!\cdots\!60$$$$\nu^{4} +$$$$27\!\cdots\!72$$$$\nu^{3} +$$$$12\!\cdots\!84$$$$\nu^{2} +$$$$30\!\cdots\!68$$$$\nu +$$$$64\!\cdots\!88$$$$)/$$$$36\!\cdots\!60$$ $$\beta_{13}$$ $$=$$ $$($$$$-$$$$81\!\cdots\!09$$$$\nu^{15} -$$$$19\!\cdots\!24$$$$\nu^{14} -$$$$82\!\cdots\!90$$$$\nu^{13} -$$$$15\!\cdots\!12$$$$\nu^{12} -$$$$45\!\cdots\!59$$$$\nu^{11} -$$$$76\!\cdots\!40$$$$\nu^{10} -$$$$15\!\cdots\!86$$$$\nu^{9} -$$$$21\!\cdots\!20$$$$\nu^{8} -$$$$34\!\cdots\!85$$$$\nu^{7} -$$$$42\!\cdots\!00$$$$\nu^{6} -$$$$52\!\cdots\!80$$$$\nu^{5} -$$$$53\!\cdots\!36$$$$\nu^{4} -$$$$47\!\cdots\!56$$$$\nu^{3} -$$$$43\!\cdots\!00$$$$\nu^{2} -$$$$26\!\cdots\!16$$$$\nu -$$$$14\!\cdots\!24$$$$)/$$$$20\!\cdots\!80$$ $$\beta_{14}$$ $$=$$ $$($$$$46\!\cdots\!07$$$$\nu^{15} -$$$$64\!\cdots\!88$$$$\nu^{14} +$$$$21\!\cdots\!74$$$$\nu^{13} -$$$$66\!\cdots\!80$$$$\nu^{12} +$$$$85\!\cdots\!97$$$$\nu^{11} -$$$$37\!\cdots\!52$$$$\nu^{10} +$$$$49\!\cdots\!38$$$$\nu^{9} -$$$$12\!\cdots\!00$$$$\nu^{8} -$$$$98\!\cdots\!17$$$$\nu^{7} -$$$$28\!\cdots\!20$$$$\nu^{6} -$$$$76\!\cdots\!40$$$$\nu^{5} -$$$$42\!\cdots\!60$$$$\nu^{4} -$$$$90\!\cdots\!32$$$$\nu^{3} -$$$$38\!\cdots\!24$$$$\nu^{2} -$$$$10\!\cdots\!08$$$$\nu -$$$$20\!\cdots\!08$$$$)/$$$$72\!\cdots\!20$$ $$\beta_{15}$$ $$=$$ $$($$$$59\!\cdots\!53$$$$\nu^{15} +$$$$12\!\cdots\!28$$$$\nu^{14} +$$$$63\!\cdots\!30$$$$\nu^{13} +$$$$93\!\cdots\!84$$$$\nu^{12} +$$$$35\!\cdots\!03$$$$\nu^{11} +$$$$47\!\cdots\!80$$$$\nu^{10} +$$$$12\!\cdots\!82$$$$\nu^{9} +$$$$13\!\cdots\!40$$$$\nu^{8} +$$$$26\!\cdots\!45$$$$\nu^{7} +$$$$26\!\cdots\!80$$$$\nu^{6} +$$$$40\!\cdots\!60$$$$\nu^{5} +$$$$33\!\cdots\!92$$$$\nu^{4} +$$$$35\!\cdots\!72$$$$\nu^{3} +$$$$27\!\cdots\!00$$$$\nu^{2} +$$$$19\!\cdots\!72$$$$\nu +$$$$92\!\cdots\!88$$$$)/$$$$80\!\cdots\!20$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$5 \beta_{9} - 5 \beta_{8} + \beta_{5} + 63 \beta_{4} - 4 \beta_{3} + 63 \beta_{2}$$$$)/126$$ $$\nu^{2}$$ $$=$$ $$($$$$-3 \beta_{14} - 33 \beta_{12} - 57 \beta_{10} + 1052 \beta_{9} + 2089 \beta_{8} + 210 \beta_{6} - 1052 \beta_{5} - 2089 \beta_{3} - 75 \beta_{2} - 2821224 \beta_{1}$$$$)/126$$ $$\nu^{3}$$ $$=$$ $$($$$$3249 \beta_{15} - 864 \beta_{13} - 16263 \beta_{11} - 37634 \beta_{9} + 619508 \beta_{8} - 63756 \beta_{7} - 657142 \beta_{5} - 3148974 \beta_{4} + 657142 \beta_{3} + 9763992$$$$)/252$$ $$\nu^{4}$$ $$=$$ $$($$$$194508 \beta_{15} + 194508 \beta_{14} + 1807515 \beta_{13} + 1807515 \beta_{12} + 1523958 \beta_{11} + 1523958 \beta_{10} + 44481269 \beta_{9} - 44481269 \beta_{8} - 8008518 \beta_{7} - 8008518 \beta_{6} + 99507457 \beta_{5} + 1419603 \beta_{4} + 55026188 \beta_{3} + 1419603 \beta_{2} + 72481855152 \beta_{1} - 72489863670$$$$)/126$$ $$\nu^{5}$$ $$=$$ $$($$$$163253895 \beta_{14} - 34799100 \beta_{12} - 641970105 \beta_{10} - 28291292084 \beta_{9} - 1671004750 \beta_{8} - 4738521060 \beta_{6} + 28291292084 \beta_{5} + 1671004750 \beta_{3} - 88275468750 \beta_{2} - 226848463380 \beta_{1}$$$$)/252$$ $$\nu^{6}$$ $$=$$ $$($$$$-9066555951 \beta_{15} - 77238171597 \beta_{13} - 25403468157 \beta_{11} - 3847036593817 \beta_{9} - 2263513699340 \beta_{8} + 259376468106 \beta_{7} - 1583522894477 \beta_{5} + 154010063061 \beta_{4} + 1583522894477 \beta_{3} + 2079342683649042$$$$)/126$$ $$\nu^{7}$$ $$=$$ $$($$$$-6559965413589 \beta_{15} - 6559965413589 \beta_{14} + 1037354143224 \beta_{13} + 1037354143224 \beta_{12} + 21290387894643 \beta_{11} + 21290387894643 \beta_{10} + 980378473239862 \beta_{9} - 980378473239862 \beta_{8} + 234143264366172 \beta_{7} + 234143264366172 \beta_{6} - 158006190047134 \beta_{5} + 2697019280692590 \beta_{4} - 1138384663286996 \beta_{3} + 2697019280692590 \beta_{2} - 11070043119630684 \beta_{1} + 11304186383996856$$$$)/252$$ $$\nu^{8}$$ $$=$$ $$($$$$-376064851106226 \beta_{14} - 3022949379396255 \beta_{12} - 91083682319556 \beta_{10} + 86536889975543084 \beta_{9} + 140645163781973809 \beta_{8} + 8454274001180238 \beta_{6} - 86536889975543084 \beta_{5} - 140645163781973809 \beta_{3} + 15309519466391709 \beta_{2} - 64624346034281951232 \beta_{1}$$$$)/126$$ $$\nu^{9}$$ $$=$$ $$($$$$246759249839945283 \beta_{15} - 22201430928268788 \beta_{13} - 694649984643376461 \beta_{11} + 8560301279034334702 \beta_{9} + 43667941343384940212 \beta_{8} - 9906482796013716948 \beta_{7} - 35107640064350605510 \beta_{5} - 87466194110263468878 \beta_{4} + 35107640064350605510 \beta_{3} - 1311745391948870623584$$$$)/252$$ $$\nu^{10}$$ $$=$$ $$($$$$14827134331306300725 \beta_{15} + 14827134331306300725 \beta_{14} + 113598637693014104145 \beta_{13} + 113598637693014104145 \beta_{12} - 16582090983173550165 \beta_{11} - 16582090983173550165 \beta_{10} + 1830307028222558463005 \beta_{9} - 1830307028222558463005 \beta_{8} - 288890412414209948370 \beta_{7} - 288890412414209948370 \beta_{6} + 5055844634204412972745 \beta_{5} - 910780558498118504325 \beta_{4} + 3225537605981854509740 \beta_{3} - 910780558498118504325 \beta_{2} + 2120878281555523942808664 \beta_{1} - 2121167171967938152757034$$$$)/126$$ $$\nu^{11}$$ $$=$$ $$($$$$9069087707878824443985 \beta_{14} - 46980131941007142000 \beta_{12} - 23117280169427570921415 \beta_{10} - 1639185045393697840153172 \beta_{9} - 394192811069079931132222 \beta_{8} - 389477118410518635394380 \beta_{6} + 1639185045393697840153172 \beta_{5} + 394192811069079931132222 \beta_{3} - 2949887920578517956849774 \beta_{2} + 80140072346777186884603260 \beta_{1}$$$$)/252$$ $$\nu^{12}$$ $$=$$ $$($$$$-570344854904722906394712 \beta_{15} - 4179750441340046586172707 \beta_{13} + 1055508452617253616728622 \beta_{11} - 180982553285741181596108641 \beta_{9} - 119270408130012453769394828 \beta_{8} + 10367738460880843114320150 \beta_{7} - 61712145155728727826713813 \beta_{5} + 44856800724328041999794445 \beta_{4} + 61712145155728727826713813 \beta_{3} + 72092590552779166190391323046$$$$)/126$$ $$\nu^{13}$$ $$=$$ $$($$$$-330653118835792624750007871 \beta_{15} - 330653118835792624750007871 \beta_{14} - 31337276809321761224721684 \beta_{13} - 31337276809321761224721684 \beta_{12} + 789249226414454830343716257 \beta_{11} + 789249226414454830343716257 \beta_{10} + 43980269926038923215204989478 \beta_{9} - 43980269926038923215204989478 \beta_{8} + 14738758577922922267835468484 \beta_{7} + 14738758577922922267835468484 \beta_{6} - 16877310414766275399125206414 \beta_{5} + 101999207499498512587927356366 \beta_{4} - 60857580340805198614330195892 \beta_{3} + 101999207499498512587927356366 \beta_{2} - 4009355902253300474451102909228 \beta_{1} + 4024094660831223396718938377712$$$$)/252$$ $$\nu^{14}$$ $$=$$ $$($$$$-21651381551214403546374228027 \beta_{14} - 152025200032722756957028386645 \beta_{12} + 48572773882365353594341701543 \beta_{10} + 4403653259025256386341637790412 \beta_{9} + 6479566073895443218050065765113 \beta_{8} + 386119908152767400137052578842 \beta_{6} - 4403653259025256386341637790412 \beta_{5} - 6479566073895443218050065765113 \beta_{3} + 2007717067393559806442197886133 \beta_{2} - 2504273417306956140820027134950568 \beta_{1}$$$$)/126$$ $$\nu^{15}$$ $$=$$ $$($$$$12029613387009532791986091820365 \beta_{15} + 2479559844207740937408904095000 \beta_{13} - 27552874416468267091471725979035 \beta_{11} + 694680297219135291832406661211870 \beta_{9} + 2245802814289335485625002714126036 \beta_{8} - 546252323953915616654052360606780 \beta_{7} - 1551122517070200193792596052914166 \beta_{5} - 3582671638088609186466052350039150 \beta_{4} + 1551122517070200193792596052914166 \beta_{3} - 181854956667776876195086978795828440$$$$)/252$$

## Character values

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

 $$n$$ $$29$$ $$73$$ $$127$$ $$\chi(n)$$ $$1$$ $$-1 + \beta_{1}$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
37.1
 −77.2060 + 133.725i −92.9954 + 161.073i −73.4015 + 127.135i −50.1630 + 86.8849i 67.3630 − 116.676i 62.2223 − 107.772i 68.2530 − 118.218i 95.9276 − 166.152i −77.2060 − 133.725i −92.9954 − 161.073i −73.4015 − 127.135i −50.1630 − 86.8849i 67.3630 + 116.676i 62.2223 + 107.772i 68.2530 + 118.218i 95.9276 + 166.152i
0 0 0 −173.134 + 299.876i 0 334.186 843.720i 0 0 0
37.2 0 0 0 −161.248 + 279.290i 0 −882.807 + 210.227i 0 0 0
37.3 0 0 0 −135.624 + 234.907i 0 87.0016 + 903.313i 0 0 0
37.4 0 0 0 −117.526 + 203.561i 0 881.619 + 215.154i 0 0 0
37.5 0 0 0 117.526 203.561i 0 881.619 + 215.154i 0 0 0
37.6 0 0 0 135.624 234.907i 0 87.0016 + 903.313i 0 0 0
37.7 0 0 0 161.248 279.290i 0 −882.807 + 210.227i 0 0 0
37.8 0 0 0 173.134 299.876i 0 334.186 843.720i 0 0 0
109.1 0 0 0 −173.134 299.876i 0 334.186 + 843.720i 0 0 0
109.2 0 0 0 −161.248 279.290i 0 −882.807 210.227i 0 0 0
109.3 0 0 0 −135.624 234.907i 0 87.0016 903.313i 0 0 0
109.4 0 0 0 −117.526 203.561i 0 881.619 215.154i 0 0 0
109.5 0 0 0 117.526 + 203.561i 0 881.619 215.154i 0 0 0
109.6 0 0 0 135.624 + 234.907i 0 87.0016 903.313i 0 0 0
109.7 0 0 0 161.248 + 279.290i 0 −882.807 210.227i 0 0 0
109.8 0 0 0 173.134 + 299.876i 0 334.186 + 843.720i 0 0 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 109.8 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
7.c even 3 1 inner
21.h odd 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 252.8.k.e 16
3.b odd 2 1 inner 252.8.k.e 16
7.c even 3 1 inner 252.8.k.e 16
21.h odd 6 1 inner 252.8.k.e 16

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
252.8.k.e 16 1.a even 1 1 trivial
252.8.k.e 16 3.b odd 2 1 inner
252.8.k.e 16 7.c even 3 1 inner
252.8.k.e 16 21.h odd 6 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$10\!\cdots\!50$$$$T_{5}^{10} +$$$$11\!\cdots\!25$$$$T_{5}^{8} +$$$$78\!\cdots\!00$$$$T_{5}^{6} +$$$$40\!\cdots\!00$$$$T_{5}^{4} +$$$$12\!\cdots\!00$$$$T_{5}^{2} +$$$$25\!\cdots\!00$$">$$T_{5}^{16} + \cdots$$ acting on $$S_{8}^{\mathrm{new}}(252, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{16}$$
$3$ $$T^{16}$$
$5$ $$25\!\cdots\!00$$$$+$$$$12\!\cdots\!00$$$$T^{2} +$$$$40\!\cdots\!00$$$$T^{4} +$$$$78\!\cdots\!00$$$$T^{6} +$$$$11\!\cdots\!25$$$$T^{8} + 10973506102891850 T^{10} + 79038704955 T^{12} + 352730 T^{14} + T^{16}$$
$7$ $$($$$$45\!\cdots\!01$$$$-$$$$46\!\cdots\!80$$$$T + 200261639612415626 T^{2} + 450830497404000 T^{3} - 1232165859813 T^{4} + 547428000 T^{5} + 295274 T^{6} - 840 T^{7} + T^{8} )^{2}$$
$11$ $$12\!\cdots\!00$$$$+$$$$15\!\cdots\!00$$$$T^{2} +$$$$20\!\cdots\!00$$$$T^{4} +$$$$28\!\cdots\!00$$$$T^{6} +$$$$36\!\cdots\!25$$$$T^{8} +$$$$17\!\cdots\!50$$$$T^{10} + 6701588161920435 T^{12} + 93007250 T^{14} + T^{16}$$
$13$ $$( -247198114185804 - 527666752620 T - 76586567 T^{2} + 7070 T^{3} + T^{4} )^{4}$$
$17$ $$93\!\cdots\!00$$$$+$$$$41\!\cdots\!00$$$$T^{2} +$$$$13\!\cdots\!00$$$$T^{4} +$$$$20\!\cdots\!00$$$$T^{6} +$$$$23\!\cdots\!00$$$$T^{8} +$$$$74\!\cdots\!00$$$$T^{10} + 1797407154790106160 T^{12} + 1530777080 T^{14} + T^{16}$$
$19$ $$($$$$12\!\cdots\!16$$$$+$$$$12\!\cdots\!72$$$$T +$$$$13\!\cdots\!80$$$$T^{2} -$$$$54\!\cdots\!48$$$$T^{3} + 3147829124233809109 T^{4} - 39307778154232 T^{5} + 2001963805 T^{6} - 21112 T^{7} + T^{8} )^{2}$$
$23$ $$36\!\cdots\!00$$$$+$$$$41\!\cdots\!00$$$$T^{2} +$$$$45\!\cdots\!00$$$$T^{4} +$$$$25\!\cdots\!00$$$$T^{6} +$$$$98\!\cdots\!00$$$$T^{8} +$$$$18\!\cdots\!00$$$$T^{10} +$$$$26\!\cdots\!00$$$$T^{12} + 19627491800 T^{14} + T^{16}$$
$29$ $$($$$$13\!\cdots\!00$$$$-$$$$53\!\cdots\!00$$$$T^{2} +$$$$44\!\cdots\!25$$$$T^{4} - 118720919450 T^{6} + T^{8} )^{2}$$
$31$ $$($$$$20\!\cdots\!21$$$$-$$$$98\!\cdots\!16$$$$T +$$$$43\!\cdots\!70$$$$T^{2} -$$$$95\!\cdots\!64$$$$T^{3} + 23464431730137896419 T^{4} - 357403787721344 T^{5} + 7761417570 T^{6} - 82376 T^{7} + T^{8} )^{2}$$
$37$ $$($$$$38\!\cdots\!56$$$$+$$$$15\!\cdots\!40$$$$T +$$$$55\!\cdots\!28$$$$T^{2} +$$$$21\!\cdots\!00$$$$T^{3} +$$$$13\!\cdots\!73$$$$T^{4} + 25474786650872750 T^{5} + 177955551283 T^{6} + 323990 T^{7} + T^{8} )^{2}$$
$41$ $$($$$$13\!\cdots\!00$$$$-$$$$51\!\cdots\!00$$$$T^{2} +$$$$73\!\cdots\!40$$$$T^{4} - 448676625600 T^{6} + T^{8} )^{2}$$
$43$ $$($$$$50\!\cdots\!16$$$$+ 762587418422560 T - 83403429537 T^{2} - 335360 T^{3} + T^{4} )^{4}$$
$47$ $$10\!\cdots\!00$$$$+$$$$99\!\cdots\!00$$$$T^{2} +$$$$90\!\cdots\!00$$$$T^{4} +$$$$43\!\cdots\!00$$$$T^{6} +$$$$14\!\cdots\!00$$$$T^{8} +$$$$25\!\cdots\!00$$$$T^{10} +$$$$32\!\cdots\!60$$$$T^{12} + 2176158907440 T^{14} + T^{16}$$
$53$ $$16\!\cdots\!00$$$$+$$$$18\!\cdots\!00$$$$T^{2} +$$$$15\!\cdots\!00$$$$T^{4} +$$$$50\!\cdots\!00$$$$T^{6} +$$$$12\!\cdots\!25$$$$T^{8} +$$$$12\!\cdots\!50$$$$T^{10} +$$$$95\!\cdots\!35$$$$T^{12} + 3709369772010 T^{14} + T^{16}$$
$59$ $$18\!\cdots\!00$$$$+$$$$51\!\cdots\!00$$$$T^{2} +$$$$98\!\cdots\!00$$$$T^{4} +$$$$99\!\cdots\!00$$$$T^{6} +$$$$72\!\cdots\!25$$$$T^{8} +$$$$28\!\cdots\!50$$$$T^{10} +$$$$77\!\cdots\!35$$$$T^{12} + 10566106141010 T^{14} + T^{16}$$
$61$ $$($$$$80\!\cdots\!96$$$$-$$$$21\!\cdots\!88$$$$T +$$$$76\!\cdots\!80$$$$T^{2} +$$$$67\!\cdots\!52$$$$T^{3} +$$$$78\!\cdots\!44$$$$T^{4} + 847337381041027408 T^{5} + 6530740002780 T^{6} + 2159668 T^{7} + T^{8} )^{2}$$
$67$ $$($$$$52\!\cdots\!96$$$$+$$$$10\!\cdots\!80$$$$T +$$$$18\!\cdots\!68$$$$T^{2} +$$$$86\!\cdots\!00$$$$T^{3} +$$$$38\!\cdots\!33$$$$T^{4} + 62576466833934468000 T^{5} + 20168710544813 T^{6} + 1952880 T^{7} + T^{8} )^{2}$$
$71$ $$($$$$91\!\cdots\!00$$$$-$$$$38\!\cdots\!00$$$$T^{2} +$$$$16\!\cdots\!40$$$$T^{4} - 22650049642640 T^{6} + T^{8} )^{2}$$
$73$ $$($$$$19\!\cdots\!96$$$$-$$$$28\!\cdots\!60$$$$T +$$$$37\!\cdots\!32$$$$T^{2} -$$$$17\!\cdots\!00$$$$T^{3} +$$$$99\!\cdots\!33$$$$T^{4} - 27595785411171227250 T^{5} + 14791978436687 T^{6} - 3235890 T^{7} + T^{8} )^{2}$$
$79$ $$($$$$37\!\cdots\!01$$$$-$$$$69\!\cdots\!52$$$$T +$$$$13\!\cdots\!30$$$$T^{2} -$$$$13\!\cdots\!52$$$$T^{3} +$$$$39\!\cdots\!79$$$$T^{4} - 39904607716315614848 T^{5} + 29847704448330 T^{6} + 3046552 T^{7} + T^{8} )^{2}$$
$83$ $$($$$$10\!\cdots\!00$$$$-$$$$20\!\cdots\!00$$$$T^{2} +$$$$11\!\cdots\!65$$$$T^{4} - 193945394219570 T^{6} + T^{8} )^{2}$$
$89$ $$84\!\cdots\!00$$$$+$$$$10\!\cdots\!00$$$$T^{2} +$$$$11\!\cdots\!00$$$$T^{4} +$$$$63\!\cdots\!00$$$$T^{6} +$$$$26\!\cdots\!00$$$$T^{8} +$$$$40\!\cdots\!00$$$$T^{10} +$$$$44\!\cdots\!60$$$$T^{12} + 251844521642280 T^{14} + T^{16}$$
$97$ $$($$$$10\!\cdots\!00$$$$-$$$$71\!\cdots\!00$$$$T - 180110817122075 T^{2} + 6785310 T^{3} + T^{4} )^{4}$$