# Properties

 Label 684.6.k.f Level $684$ Weight $6$ Character orbit 684.k Analytic conductor $109.703$ Analytic rank $0$ Dimension $18$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$684 = 2^{2} \cdot 3^{2} \cdot 19$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 684.k (of order $$3$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$109.702532752$$ Analytic rank: $$0$$ Dimension: $$18$$ Relative dimension: $$9$$ over $$\Q(\zeta_{3})$$ Coefficient field: $$\mathbb{Q}[x]/(x^{18} - \cdots)$$ Defining polynomial: $$x^{18} - 7 x^{17} + 17616 x^{16} - 456301 x^{15} + 216301789 x^{14} - 6076762674 x^{13} + 1365504992251 x^{12} - 54625271848675 x^{11} + 6304911261795867 x^{10} - 242577082836320134 x^{9} + 18704695568091019759 x^{8} - 704026968823840581411 x^{7} + 38243816152807979345695 x^{6} - 1065755622605542033556002 x^{5} + 37300806349205873742869889 x^{4} - 655224799021744228503297133 x^{3} + 19168823714556940699533734824 x^{2} - 236484931518202417328772433599 x + 5519876026771720332419776049541$$ Coefficient ring: $$\Z[a_1, \ldots, a_{19}]$$ Coefficient ring index: $$2^{16}\cdot 3^{7}\cdot 5^{2}$$ Twist minimal: no (minimal twist has level 76) Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{17}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( -\beta_{1} - \beta_{2} - \beta_{4} ) q^{5} + ( 19 + \beta_{3} - \beta_{9} - \beta_{13} ) q^{7} +O(q^{10})$$ $$q + ( -\beta_{1} - \beta_{2} - \beta_{4} ) q^{5} + ( 19 + \beta_{3} - \beta_{9} - \beta_{13} ) q^{7} + ( 18 - \beta_{10} ) q^{11} + ( 25 - 3 \beta_{1} - 25 \beta_{2} - 2 \beta_{5} - \beta_{11} ) q^{13} + ( -\beta_{1} - 21 \beta_{2} - \beta_{4} - 5 \beta_{5} + \beta_{7} + 5 \beta_{9} - \beta_{10} + \beta_{15} - \beta_{17} ) q^{17} + ( -170 - 2 \beta_{1} + 243 \beta_{2} + \beta_{3} - 3 \beta_{4} - 5 \beta_{5} + \beta_{6} + \beta_{7} + 3 \beta_{9} - 2 \beta_{10} + 2 \beta_{13} - \beta_{14} + \beta_{15} - \beta_{16} + \beta_{17} ) q^{19} + ( 384 - 383 \beta_{2} - 6 \beta_{3} - 4 \beta_{5} + \beta_{6} - \beta_{7} - \beta_{8} - \beta_{9} + \beta_{11} + \beta_{14} - \beta_{15} + \beta_{16} - 2 \beta_{17} ) q^{23} + ( -787 + 10 \beta_{1} + 787 \beta_{2} - 10 \beta_{3} + 11 \beta_{5} + 2 \beta_{6} + 4 \beta_{7} + \beta_{8} - \beta_{11} ) q^{25} + ( 810 + 12 \beta_{1} - 811 \beta_{2} - 6 \beta_{3} + 25 \beta_{5} - 2 \beta_{6} + 3 \beta_{7} - 4 \beta_{8} + \beta_{9} + 5 \beta_{11} - \beta_{14} + \beta_{15} - \beta_{16} + 2 \beta_{17} ) q^{29} + ( -558 + \beta_{2} + 11 \beta_{3} + 13 \beta_{4} + \beta_{5} + 2 \beta_{6} - \beta_{8} - 4 \beta_{9} + 9 \beta_{10} + 7 \beta_{11} - 2 \beta_{12} - 11 \beta_{13} - 6 \beta_{14} + 2 \beta_{15} + \beta_{17} ) q^{31} + ( -97 \beta_{1} - 1744 \beta_{2} - 97 \beta_{4} - 9 \beta_{5} + 5 \beta_{7} - 7 \beta_{8} + 9 \beta_{9} - 5 \beta_{10} + \beta_{12} - 15 \beta_{13} - 6 \beta_{14} + 7 \beta_{16} ) q^{35} + ( 1488 + 20 \beta_{3} - 42 \beta_{4} - 4 \beta_{6} - 32 \beta_{9} + 8 \beta_{10} - 3 \beta_{11} + 4 \beta_{12} - 20 \beta_{13} + 3 \beta_{14} - 2 \beta_{16} ) q^{37} + ( -55 \beta_{1} + 814 \beta_{2} - 55 \beta_{4} - 21 \beta_{5} - 16 \beta_{7} + 5 \beta_{8} + 21 \beta_{9} + 16 \beta_{10} - 5 \beta_{16} ) q^{41} + ( -16 \beta_{1} - 935 \beta_{2} - 16 \beta_{4} - 54 \beta_{5} + 11 \beta_{7} + 54 \beta_{9} - 11 \beta_{10} - 5 \beta_{12} + 29 \beta_{13} + 5 \beta_{14} + \beta_{15} - \beta_{17} ) q^{43} + ( -4183 + 19 \beta_{1} + 4183 \beta_{2} - 30 \beta_{3} - \beta_{5} - \beta_{6} + 32 \beta_{7} + 11 \beta_{8} - 12 \beta_{11} ) q^{47} + ( 6995 + \beta_{2} + 14 \beta_{3} - 44 \beta_{4} + \beta_{5} + 2 \beta_{6} - \beta_{8} - 162 \beta_{9} - 9 \beta_{10} + 9 \beta_{11} - 2 \beta_{12} - 14 \beta_{13} - 8 \beta_{14} + 2 \beta_{15} - 25 \beta_{16} + \beta_{17} ) q^{49} + ( 2352 + 228 \beta_{1} - 2353 \beta_{2} + 66 \beta_{3} - 135 \beta_{5} + 10 \beta_{6} - 13 \beta_{7} - 6 \beta_{8} + \beta_{9} - 4 \beta_{11} - \beta_{14} + \beta_{15} - \beta_{16} + 2 \beta_{17} ) q^{53} + ( -202 \beta_{1} + 159 \beta_{2} - 202 \beta_{4} + 91 \beta_{5} + 30 \beta_{7} - 2 \beta_{8} - 91 \beta_{9} - 30 \beta_{10} - 4 \beta_{12} - 71 \beta_{13} + 10 \beta_{14} + 10 \beta_{15} + 2 \beta_{16} - 10 \beta_{17} ) q^{55} + ( -102 \beta_{1} + 8331 \beta_{2} - 102 \beta_{4} + 98 \beta_{5} - 10 \beta_{7} + 14 \beta_{8} - 98 \beta_{9} + 10 \beta_{10} + 17 \beta_{12} - 129 \beta_{13} - 3 \beta_{14} - 11 \beta_{15} - 14 \beta_{16} + 11 \beta_{17} ) q^{59} + ( -817 - 19 \beta_{1} + 827 \beta_{2} + 120 \beta_{3} - 34 \beta_{5} + 4 \beta_{6} + 2 \beta_{7} + 12 \beta_{8} - 10 \beta_{9} - 6 \beta_{11} + 10 \beta_{14} - 10 \beta_{15} + 10 \beta_{16} - 20 \beta_{17} ) q^{61} + ( -10501 - 10 \beta_{2} - 186 \beta_{3} + 29 \beta_{4} - 10 \beta_{5} + 10 \beta_{6} + 10 \beta_{8} + 4 \beta_{9} + 10 \beta_{10} - 35 \beta_{11} - 10 \beta_{12} + 186 \beta_{13} + 25 \beta_{14} - 20 \beta_{15} + 30 \beta_{16} - 10 \beta_{17} ) q^{65} + ( -2870 - 103 \beta_{1} + 2860 \beta_{2} + 29 \beta_{3} - 178 \beta_{5} - 19 \beta_{6} - 27 \beta_{7} + 9 \beta_{8} + 10 \beta_{9} + 44 \beta_{11} - 10 \beta_{14} + 10 \beta_{15} - 10 \beta_{16} + 20 \beta_{17} ) q^{67} + ( 172 \beta_{1} + 6033 \beta_{2} + 172 \beta_{4} + 382 \beta_{5} - 13 \beta_{7} + 18 \beta_{8} - 382 \beta_{9} + 13 \beta_{10} + \beta_{12} + 201 \beta_{13} - 33 \beta_{14} - 11 \beta_{15} - 18 \beta_{16} + 11 \beta_{17} ) q^{71} + ( 700 \beta_{1} - 1855 \beta_{2} + 700 \beta_{4} - 433 \beta_{5} + 17 \beta_{7} + 8 \beta_{8} + 433 \beta_{9} - 17 \beta_{10} + 6 \beta_{12} + 154 \beta_{13} - 27 \beta_{14} + \beta_{15} - 8 \beta_{16} - \beta_{17} ) q^{73} + ( 2005 + 11 \beta_{2} - 6 \beta_{3} - 325 \beta_{4} + 11 \beta_{5} - 26 \beta_{6} - 11 \beta_{8} - 487 \beta_{9} - 39 \beta_{10} - 2 \beta_{11} + 26 \beta_{12} + 6 \beta_{13} + 13 \beta_{14} + 22 \beta_{15} + 11 \beta_{17} ) q^{77} + ( 540 \beta_{1} + 3557 \beta_{2} + 540 \beta_{4} + 468 \beta_{5} - 57 \beta_{7} - 6 \beta_{8} - 468 \beta_{9} + 57 \beta_{10} - 3 \beta_{12} + 21 \beta_{13} + 7 \beta_{14} + \beta_{15} + 6 \beta_{16} - \beta_{17} ) q^{79} + ( -4603 - 11 \beta_{2} + 117 \beta_{3} + 68 \beta_{4} - 11 \beta_{5} - 15 \beta_{6} + 11 \beta_{8} + 78 \beta_{9} + 41 \beta_{10} + 41 \beta_{11} + 15 \beta_{12} - 117 \beta_{13} - 52 \beta_{14} - 22 \beta_{15} + 31 \beta_{16} - 11 \beta_{17} ) q^{83} + ( -5905 - 303 \beta_{1} + 5905 \beta_{2} + 446 \beta_{5} - 12 \beta_{6} + 144 \beta_{7} + 32 \beta_{8} - 81 \beta_{11} ) q^{85} + ( 17475 + 347 \beta_{1} - 17464 \beta_{2} + 24 \beta_{3} - 348 \beta_{5} - 16 \beta_{6} - 81 \beta_{7} - 39 \beta_{8} - 11 \beta_{9} - 34 \beta_{11} + 11 \beta_{14} - 11 \beta_{15} + 11 \beta_{16} - 22 \beta_{17} ) q^{89} + ( -596 - 1170 \beta_{1} + 586 \beta_{2} - 50 \beta_{3} + 20 \beta_{5} + 40 \beta_{6} - 86 \beta_{7} - 40 \beta_{8} + 10 \beta_{9} - 18 \beta_{11} - 10 \beta_{14} + 10 \beta_{15} - 10 \beta_{16} + 20 \beta_{17} ) q^{91} + ( -11266 - 334 \beta_{1} + 16200 \beta_{2} - 123 \beta_{3} + 541 \beta_{4} + 189 \beta_{5} - 20 \beta_{6} + 97 \beta_{7} + 51 \beta_{8} + 836 \beta_{9} - 98 \beta_{10} - 69 \beta_{11} + 21 \beta_{12} + 36 \beta_{13} + 27 \beta_{14} - 35 \beta_{15} + 42 \beta_{16} - 46 \beta_{17} ) q^{95} + ( 721 \beta_{1} + 12320 \beta_{2} + 721 \beta_{4} + 316 \beta_{5} - 5 \beta_{7} + 21 \beta_{8} - 316 \beta_{9} + 5 \beta_{10} + 50 \beta_{12} + 134 \beta_{13} - 19 \beta_{14} - 45 \beta_{15} - 21 \beta_{16} + 45 \beta_{17} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$18q - 11q^{5} + 336q^{7} + O(q^{10})$$ $$18q - 11q^{5} + 336q^{7} + 320q^{11} + 227q^{13} - 179q^{17} - 868q^{19} + 3425q^{23} - 7054q^{25} + 7349q^{29} - 9960q^{31} - 15888q^{35} + 26444q^{37} + 7311q^{41} - 8283q^{43} - 37603q^{47} + 124738q^{49} + 20337q^{53} + 716q^{55} + 74455q^{59} - 7569q^{61} - 188998q^{65} - 26177q^{67} + 53463q^{71} - 14103q^{73} + 31960q^{77} + 31825q^{79} - 82600q^{83} - 50787q^{85} + 155197q^{89} - 2800q^{91} - 49315q^{95} + 111241q^{97} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{18} - 7 x^{17} + 17616 x^{16} - 456301 x^{15} + 216301789 x^{14} - 6076762674 x^{13} + 1365504992251 x^{12} - 54625271848675 x^{11} + 6304911261795867 x^{10} - 242577082836320134 x^{9} + 18704695568091019759 x^{8} - 704026968823840581411 x^{7} + 38243816152807979345695 x^{6} - 1065755622605542033556002 x^{5} + 37300806349205873742869889 x^{4} - 655224799021744228503297133 x^{3} + 19168823714556940699533734824 x^{2} - 236484931518202417328772433599 x + 5519876026771720332419776049541$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$14\!\cdots\!75$$$$\nu^{17} +$$$$24\!\cdots\!90$$$$\nu^{16} +$$$$24\!\cdots\!10$$$$\nu^{15} -$$$$40\!\cdots\!25$$$$\nu^{14} +$$$$28\!\cdots\!50$$$$\nu^{13} -$$$$54\!\cdots\!60$$$$\nu^{12} +$$$$17\!\cdots\!25$$$$\nu^{11} -$$$$54\!\cdots\!24$$$$\nu^{10} +$$$$74\!\cdots\!65$$$$\nu^{9} -$$$$22\!\cdots\!05$$$$\nu^{8} +$$$$19\!\cdots\!00$$$$\nu^{7} -$$$$58\!\cdots\!85$$$$\nu^{6} +$$$$34\!\cdots\!68$$$$\nu^{5} -$$$$57\!\cdots\!70$$$$\nu^{4} +$$$$19\!\cdots\!25$$$$\nu^{3} +$$$$19\!\cdots\!70$$$$\nu^{2} +$$$$32\!\cdots\!74$$$$\nu -$$$$55\!\cdots\!83$$$$)/$$$$31\!\cdots\!64$$ $$\beta_{2}$$ $$=$$ $$($$$$-$$$$14\!\cdots\!75$$$$\nu^{17} -$$$$24\!\cdots\!90$$$$\nu^{16} -$$$$24\!\cdots\!10$$$$\nu^{15} +$$$$40\!\cdots\!25$$$$\nu^{14} -$$$$28\!\cdots\!50$$$$\nu^{13} +$$$$54\!\cdots\!60$$$$\nu^{12} -$$$$17\!\cdots\!25$$$$\nu^{11} +$$$$54\!\cdots\!24$$$$\nu^{10} -$$$$74\!\cdots\!65$$$$\nu^{9} +$$$$22\!\cdots\!05$$$$\nu^{8} -$$$$19\!\cdots\!00$$$$\nu^{7} +$$$$58\!\cdots\!85$$$$\nu^{6} -$$$$34\!\cdots\!68$$$$\nu^{5} +$$$$57\!\cdots\!70$$$$\nu^{4} -$$$$19\!\cdots\!25$$$$\nu^{3} -$$$$19\!\cdots\!70$$$$\nu^{2} -$$$$75\!\cdots\!10$$$$\nu +$$$$55\!\cdots\!83$$$$)/$$$$31\!\cdots\!64$$ $$\beta_{3}$$ $$=$$ $$($$$$42\!\cdots\!41$$$$\nu^{17} +$$$$29\!\cdots\!74$$$$\nu^{16} +$$$$90\!\cdots\!94$$$$\nu^{15} +$$$$50\!\cdots\!69$$$$\nu^{14} +$$$$48\!\cdots\!94$$$$\nu^{13} +$$$$57\!\cdots\!36$$$$\nu^{12} +$$$$36\!\cdots\!43$$$$\nu^{11} +$$$$33\!\cdots\!32$$$$\nu^{10} -$$$$36\!\cdots\!89$$$$\nu^{9} +$$$$12\!\cdots\!85$$$$\nu^{8} -$$$$18\!\cdots\!12$$$$\nu^{7} +$$$$30\!\cdots\!97$$$$\nu^{6} -$$$$63\!\cdots\!56$$$$\nu^{5} +$$$$46\!\cdots\!50$$$$\nu^{4} -$$$$50\!\cdots\!25$$$$\nu^{3} +$$$$28\!\cdots\!62$$$$\nu^{2} -$$$$16\!\cdots\!14$$$$\nu +$$$$12\!\cdots\!35$$$$)/$$$$20\!\cdots\!00$$ $$\beta_{4}$$ $$=$$ $$($$$$36\!\cdots\!35$$$$\nu^{17} -$$$$23\!\cdots\!70$$$$\nu^{16} +$$$$70\!\cdots\!10$$$$\nu^{15} -$$$$51\!\cdots\!25$$$$\nu^{14} +$$$$95\!\cdots\!10$$$$\nu^{13} -$$$$65\!\cdots\!60$$$$\nu^{12} +$$$$69\!\cdots\!09$$$$\nu^{11} -$$$$46\!\cdots\!24$$$$\nu^{10} +$$$$37\!\cdots\!45$$$$\nu^{9} -$$$$21\!\cdots\!05$$$$\nu^{8} +$$$$12\!\cdots\!60$$$$\nu^{7} -$$$$60\!\cdots\!73$$$$\nu^{6} +$$$$29\!\cdots\!08$$$$\nu^{5} -$$$$10\!\cdots\!70$$$$\nu^{4} +$$$$37\!\cdots\!05$$$$\nu^{3} -$$$$66\!\cdots\!90$$$$\nu^{2} +$$$$90\!\cdots\!38$$$$\nu -$$$$27\!\cdots\!79$$$$)/$$$$10\!\cdots\!76$$ $$\beta_{5}$$ $$=$$ $$($$$$14\!\cdots\!03$$$$\nu^{17} -$$$$12\!\cdots\!58$$$$\nu^{16} -$$$$15\!\cdots\!98$$$$\nu^{15} -$$$$22\!\cdots\!73$$$$\nu^{14} +$$$$63\!\cdots\!02$$$$\nu^{13} -$$$$25\!\cdots\!12$$$$\nu^{12} -$$$$74\!\cdots\!31$$$$\nu^{11} -$$$$14\!\cdots\!44$$$$\nu^{10} +$$$$22\!\cdots\!13$$$$\nu^{9} -$$$$55\!\cdots\!45$$$$\nu^{8} +$$$$94\!\cdots\!04$$$$\nu^{7} -$$$$13\!\cdots\!49$$$$\nu^{6} +$$$$30\!\cdots\!52$$$$\nu^{5} -$$$$20\!\cdots\!50$$$$\nu^{4} +$$$$23\!\cdots\!25$$$$\nu^{3} -$$$$12\!\cdots\!54$$$$\nu^{2} +$$$$87\!\cdots\!38$$$$\nu -$$$$50\!\cdots\!95$$$$)/$$$$40\!\cdots\!00$$ $$\beta_{6}$$ $$=$$ $$($$$$50\!\cdots\!61$$$$\nu^{17} -$$$$20\!\cdots\!46$$$$\nu^{16} +$$$$26\!\cdots\!74$$$$\nu^{15} -$$$$37\!\cdots\!51$$$$\nu^{14} +$$$$74\!\cdots\!74$$$$\nu^{13} -$$$$42\!\cdots\!44$$$$\nu^{12} +$$$$40\!\cdots\!03$$$$\nu^{11} -$$$$25\!\cdots\!28$$$$\nu^{10} +$$$$55\!\cdots\!31$$$$\nu^{9} -$$$$97\!\cdots\!15$$$$\nu^{8} +$$$$20\!\cdots\!48$$$$\nu^{7} -$$$$24\!\cdots\!63$$$$\nu^{6} +$$$$58\!\cdots\!24$$$$\nu^{5} -$$$$35\!\cdots\!50$$$$\nu^{4} +$$$$43\!\cdots\!75$$$$\nu^{3} -$$$$19\!\cdots\!98$$$$\nu^{2} +$$$$16\!\cdots\!06$$$$\nu -$$$$77\!\cdots\!65$$$$)/$$$$20\!\cdots\!00$$ $$\beta_{7}$$ $$=$$ $$($$$$34\!\cdots\!11$$$$\nu^{17} +$$$$23\!\cdots\!54$$$$\nu^{16} +$$$$61\!\cdots\!74$$$$\nu^{15} +$$$$28\!\cdots\!99$$$$\nu^{14} +$$$$67\!\cdots\!74$$$$\nu^{13} +$$$$31\!\cdots\!56$$$$\nu^{12} +$$$$37\!\cdots\!53$$$$\nu^{11} +$$$$12\!\cdots\!72$$$$\nu^{10} +$$$$11\!\cdots\!81$$$$\nu^{9} +$$$$54\!\cdots\!35$$$$\nu^{8} +$$$$24\!\cdots\!48$$$$\nu^{7} +$$$$13\!\cdots\!87$$$$\nu^{6} +$$$$16\!\cdots\!24$$$$\nu^{5} +$$$$31\!\cdots\!50$$$$\nu^{4} -$$$$54\!\cdots\!75$$$$\nu^{3} +$$$$25\!\cdots\!02$$$$\nu^{2} +$$$$80\!\cdots\!06$$$$\nu +$$$$14\!\cdots\!85$$$$)/$$$$44\!\cdots\!00$$ $$\beta_{8}$$ $$=$$ $$($$$$-$$$$33\!\cdots\!69$$$$\nu^{17} +$$$$47\!\cdots\!34$$$$\nu^{16} +$$$$94\!\cdots\!54$$$$\nu^{15} +$$$$83\!\cdots\!79$$$$\nu^{14} +$$$$18\!\cdots\!54$$$$\nu^{13} +$$$$95\!\cdots\!76$$$$\nu^{12} +$$$$28\!\cdots\!13$$$$\nu^{11} +$$$$55\!\cdots\!12$$$$\nu^{10} -$$$$73\!\cdots\!99$$$$\nu^{9} +$$$$20\!\cdots\!35$$$$\nu^{8} -$$$$32\!\cdots\!92$$$$\nu^{7} +$$$$51\!\cdots\!27$$$$\nu^{6} -$$$$10\!\cdots\!96$$$$\nu^{5} +$$$$75\!\cdots\!50$$$$\nu^{4} -$$$$85\!\cdots\!75$$$$\nu^{3} +$$$$45\!\cdots\!42$$$$\nu^{2} -$$$$30\!\cdots\!74$$$$\nu +$$$$19\!\cdots\!85$$$$)/$$$$40\!\cdots\!00$$ $$\beta_{9}$$ $$=$$ $$($$$$17\!\cdots\!59$$$$\nu^{17} -$$$$80\!\cdots\!18$$$$\nu^{16} +$$$$28\!\cdots\!30$$$$\nu^{15} -$$$$79\!\cdots\!05$$$$\nu^{14} +$$$$33\!\cdots\!70$$$$\nu^{13} -$$$$10\!\cdots\!72$$$$\nu^{12} +$$$$19\!\cdots\!93$$$$\nu^{11} -$$$$93\!\cdots\!04$$$$\nu^{10} +$$$$81\!\cdots\!21$$$$\nu^{9} -$$$$37\!\cdots\!29$$$$\nu^{8} +$$$$21\!\cdots\!12$$$$\nu^{7} -$$$$10\!\cdots\!89$$$$\nu^{6} +$$$$40\!\cdots\!28$$$$\nu^{5} -$$$$12\!\cdots\!66$$$$\nu^{4} +$$$$26\!\cdots\!85$$$$\nu^{3} -$$$$68\!\cdots\!62$$$$\nu^{2} +$$$$10\!\cdots\!06$$$$\nu -$$$$24\!\cdots\!79$$$$)/$$$$20\!\cdots\!00$$ $$\beta_{10}$$ $$=$$ $$($$$$-$$$$80\!\cdots\!81$$$$\nu^{17} +$$$$16\!\cdots\!62$$$$\nu^{16} -$$$$13\!\cdots\!70$$$$\nu^{15} +$$$$55\!\cdots\!95$$$$\nu^{14} -$$$$16\!\cdots\!30$$$$\nu^{13} +$$$$69\!\cdots\!48$$$$\nu^{12} -$$$$10\!\cdots\!87$$$$\nu^{11} +$$$$54\!\cdots\!36$$$$\nu^{10} -$$$$47\!\cdots\!39$$$$\nu^{9} +$$$$22\!\cdots\!11$$$$\nu^{8} -$$$$13\!\cdots\!08$$$$\nu^{7} +$$$$63\!\cdots\!51$$$$\nu^{6} -$$$$26\!\cdots\!52$$$$\nu^{5} +$$$$89\!\cdots\!94$$$$\nu^{4} -$$$$19\!\cdots\!15$$$$\nu^{3} +$$$$49\!\cdots\!58$$$$\nu^{2} -$$$$74\!\cdots\!54$$$$\nu +$$$$23\!\cdots\!61$$$$)/$$$$54\!\cdots\!00$$ $$\beta_{11}$$ $$=$$ $$($$$$31\!\cdots\!17$$$$\nu^{17} -$$$$12\!\cdots\!62$$$$\nu^{16} +$$$$10\!\cdots\!78$$$$\nu^{15} -$$$$22\!\cdots\!47$$$$\nu^{14} +$$$$37\!\cdots\!78$$$$\nu^{13} -$$$$25\!\cdots\!68$$$$\nu^{12} +$$$$15\!\cdots\!91$$$$\nu^{11} -$$$$15\!\cdots\!16$$$$\nu^{10} +$$$$28\!\cdots\!07$$$$\nu^{9} -$$$$57\!\cdots\!55$$$$\nu^{8} +$$$$11\!\cdots\!56$$$$\nu^{7} -$$$$14\!\cdots\!11$$$$\nu^{6} +$$$$32\!\cdots\!28$$$$\nu^{5} -$$$$20\!\cdots\!50$$$$\nu^{4} +$$$$24\!\cdots\!75$$$$\nu^{3} -$$$$11\!\cdots\!06$$$$\nu^{2} +$$$$92\!\cdots\!82$$$$\nu -$$$$48\!\cdots\!05$$$$)/$$$$20\!\cdots\!00$$ $$\beta_{12}$$ $$=$$ $$($$$$-$$$$51\!\cdots\!95$$$$\nu^{17} -$$$$17\!\cdots\!34$$$$\nu^{16} -$$$$90\!\cdots\!46$$$$\nu^{15} -$$$$10\!\cdots\!31$$$$\nu^{14} -$$$$10\!\cdots\!06$$$$\nu^{13} -$$$$85\!\cdots\!96$$$$\nu^{12} -$$$$58\!\cdots\!09$$$$\nu^{11} +$$$$55\!\cdots\!08$$$$\nu^{10} -$$$$21\!\cdots\!33$$$$\nu^{9} +$$$$27\!\cdots\!21$$$$\nu^{8} -$$$$52\!\cdots\!60$$$$\nu^{7} +$$$$10\!\cdots\!13$$$$\nu^{6} -$$$$75\!\cdots\!28$$$$\nu^{5} +$$$$71\!\cdots\!94$$$$\nu^{4} -$$$$43\!\cdots\!65$$$$\nu^{3} +$$$$36\!\cdots\!10$$$$\nu^{2} -$$$$18\!\cdots\!98$$$$\nu -$$$$41\!\cdots\!29$$$$)/$$$$20\!\cdots\!00$$ $$\beta_{13}$$ $$=$$ $$($$$$81\!\cdots\!55$$$$\nu^{17} +$$$$24\!\cdots\!46$$$$\nu^{16} +$$$$14\!\cdots\!74$$$$\nu^{15} +$$$$11\!\cdots\!39$$$$\nu^{14} +$$$$16\!\cdots\!14$$$$\nu^{13} +$$$$78\!\cdots\!24$$$$\nu^{12} +$$$$94\!\cdots\!21$$$$\nu^{11} -$$$$11\!\cdots\!52$$$$\nu^{10} +$$$$35\!\cdots\!77$$$$\nu^{9} -$$$$56\!\cdots\!49$$$$\nu^{8} +$$$$88\!\cdots\!40$$$$\nu^{7} -$$$$19\!\cdots\!97$$$$\nu^{6} +$$$$13\!\cdots\!32$$$$\nu^{5} -$$$$16\!\cdots\!86$$$$\nu^{4} +$$$$83\!\cdots\!85$$$$\nu^{3} -$$$$60\!\cdots\!90$$$$\nu^{2} +$$$$36\!\cdots\!62$$$$\nu -$$$$54\!\cdots\!99$$$$)/$$$$20\!\cdots\!00$$ $$\beta_{14}$$ $$=$$ $$($$$$-$$$$31\!\cdots\!15$$$$\nu^{17} -$$$$11\!\cdots\!98$$$$\nu^{16} -$$$$55\!\cdots\!62$$$$\nu^{15} -$$$$75\!\cdots\!07$$$$\nu^{14} -$$$$62\!\cdots\!82$$$$\nu^{13} -$$$$66\!\cdots\!12$$$$\nu^{12} -$$$$35\!\cdots\!73$$$$\nu^{11} +$$$$26\!\cdots\!76$$$$\nu^{10} -$$$$12\!\cdots\!01$$$$\nu^{9} +$$$$14\!\cdots\!37$$$$\nu^{8} -$$$$31\!\cdots\!20$$$$\nu^{7} +$$$$57\!\cdots\!61$$$$\nu^{6} -$$$$44\!\cdots\!16$$$$\nu^{5} +$$$$38\!\cdots\!18$$$$\nu^{4} -$$$$25\!\cdots\!05$$$$\nu^{3} +$$$$12\!\cdots\!70$$$$\nu^{2} -$$$$10\!\cdots\!06$$$$\nu -$$$$33\!\cdots\!13$$$$)/$$$$20\!\cdots\!00$$ $$\beta_{15}$$ $$=$$ $$($$$$91\!\cdots\!65$$$$\nu^{17} +$$$$52\!\cdots\!66$$$$\nu^{16} +$$$$16\!\cdots\!34$$$$\nu^{15} +$$$$56\!\cdots\!49$$$$\nu^{14} +$$$$18\!\cdots\!74$$$$\nu^{13} +$$$$58\!\cdots\!04$$$$\nu^{12} +$$$$11\!\cdots\!31$$$$\nu^{11} +$$$$15\!\cdots\!08$$$$\nu^{10} +$$$$37\!\cdots\!47$$$$\nu^{9} +$$$$39\!\cdots\!81$$$$\nu^{8} +$$$$85\!\cdots\!20$$$$\nu^{7} +$$$$35\!\cdots\!13$$$$\nu^{6} +$$$$10\!\cdots\!32$$$$\nu^{5} +$$$$17\!\cdots\!34$$$$\nu^{4} +$$$$56\!\cdots\!35$$$$\nu^{3} +$$$$14\!\cdots\!30$$$$\nu^{2} +$$$$26\!\cdots\!82$$$$\nu +$$$$86\!\cdots\!31$$$$)/$$$$44\!\cdots\!00$$ $$\beta_{16}$$ $$=$$ $$($$$$-$$$$59\!\cdots\!13$$$$\nu^{17} +$$$$27\!\cdots\!26$$$$\nu^{16} -$$$$97\!\cdots\!10$$$$\nu^{15} +$$$$27\!\cdots\!35$$$$\nu^{14} -$$$$11\!\cdots\!90$$$$\nu^{13} +$$$$34\!\cdots\!04$$$$\nu^{12} -$$$$65\!\cdots\!51$$$$\nu^{11} +$$$$32\!\cdots\!28$$$$\nu^{10} -$$$$27\!\cdots\!47$$$$\nu^{9} +$$$$12\!\cdots\!03$$$$\nu^{8} -$$$$75\!\cdots\!84$$$$\nu^{7} +$$$$35\!\cdots\!23$$$$\nu^{6} -$$$$13\!\cdots\!96$$$$\nu^{5} +$$$$43\!\cdots\!62$$$$\nu^{4} -$$$$91\!\cdots\!95$$$$\nu^{3} +$$$$23\!\cdots\!34$$$$\nu^{2} -$$$$36\!\cdots\!42$$$$\nu +$$$$87\!\cdots\!53$$$$)/$$$$18\!\cdots\!00$$ $$\beta_{17}$$ $$=$$ $$($$$$-$$$$16\!\cdots\!45$$$$\nu^{17} -$$$$27\!\cdots\!46$$$$\nu^{16} -$$$$27\!\cdots\!94$$$$\nu^{15} +$$$$13\!\cdots\!91$$$$\nu^{14} -$$$$31\!\cdots\!34$$$$\nu^{13} +$$$$25\!\cdots\!76$$$$\nu^{12} -$$$$18\!\cdots\!31$$$$\nu^{11} +$$$$46\!\cdots\!52$$$$\nu^{10} -$$$$71\!\cdots\!47$$$$\nu^{9} +$$$$19\!\cdots\!59$$$$\nu^{8} -$$$$18\!\cdots\!60$$$$\nu^{7} +$$$$59\!\cdots\!47$$$$\nu^{6} -$$$$29\!\cdots\!72$$$$\nu^{5} +$$$$63\!\cdots\!26$$$$\nu^{4} -$$$$19\!\cdots\!35$$$$\nu^{3} +$$$$31\!\cdots\!10$$$$\nu^{2} -$$$$80\!\cdots\!82$$$$\nu +$$$$92\!\cdots\!09$$$$)/$$$$44\!\cdots\!00$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{2} + \beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{16} + \beta_{14} + 10 \beta_{13} - 2 \beta_{12} + 4 \beta_{10} + 11 \beta_{9} - \beta_{8} - 4 \beta_{7} - 11 \beta_{5} - 10 \beta_{4} - 3910 \beta_{2} - 8 \beta_{1} - 1$$ $$\nu^{3}$$ $$=$$ $$-45 \beta_{17} - 23 \beta_{16} - 90 \beta_{15} + 162 \beta_{14} + 180 \beta_{13} + 56 \beta_{12} - 204 \beta_{11} + 173 \beta_{10} + 2352 \beta_{9} + 42 \beta_{8} - 12 \beta_{7} - 62 \beta_{6} - 78 \beta_{5} + 6836 \beta_{4} - 150 \beta_{3} - 11778 \beta_{2} - 27 \beta_{1} + 59110$$ $$\nu^{4}$$ $$=$$ $$-5330 \beta_{17} + 2477 \beta_{16} - 2935 \beta_{15} + 3217 \beta_{14} + 660 \beta_{13} + 236 \beta_{12} - 13488 \beta_{11} + 668 \beta_{10} + 6767 \beta_{9} + 9346 \beta_{8} + 36839 \beta_{7} + 22396 \beta_{6} + 12571 \beta_{5} + 27404 \beta_{4} - 117220 \beta_{3} + 26581845 \beta_{2} + 87614 \beta_{1} - 26366473$$ $$\nu^{5}$$ $$=$$ $$-401865 \beta_{17} + 618262 \beta_{16} + 361890 \beta_{15} - 1450833 \beta_{14} - 1737650 \beta_{13} - 990114 \beta_{12} - 65400 \beta_{11} - 1992667 \beta_{10} - 29841333 \beta_{9} - 559337 \beta_{8} + 2178592 \beta_{7} + 112600 \beta_{6} + 29915283 \beta_{5} - 58169896 \beta_{4} - 584600 \beta_{3} - 426784388 \beta_{2} - 57800201 \beta_{1} - 132423474$$ $$\nu^{6}$$ $$=$$ $$31315085 \beta_{17} - 76163096 \beta_{16} + 69508015 \beta_{15} - 93640341 \beta_{14} - 1153549930 \beta_{13} + 229106402 \beta_{12} + 118343378 \beta_{11} - 335818799 \beta_{10} + 390197179 \beta_{9} - 37142582 \beta_{8} + 12518787 \beta_{7} - 234711056 \beta_{6} + 212948963 \beta_{5} + 509175164 \beta_{4} + 1141365280 \beta_{3} - 2925963682 \beta_{2} - 348115830 \beta_{1} + 224305429259$$ $$\nu^{7}$$ $$=$$ $$6309274755 \beta_{17} - 3586945138 \beta_{16} + 3519764325 \beta_{15} - 3665815433 \beta_{14} - 8038341850 \beta_{13} + 1624543466 \beta_{12} + 21988137508 \beta_{11} - 2308868177 \beta_{10} + 6399110487 \beta_{9} + 1692351110 \beta_{8} - 19645263215 \beta_{7} + 10553055044 \beta_{6} - 318705636015 \beta_{5} + 4786513634 \beta_{4} + 21984548660 \beta_{3} + 5455991947371 \beta_{2} + 531719887067 \beta_{1} - 3897627431452$$ $$\nu^{8}$$ $$=$$ $$393961777125 \beta_{17} + 341214452599 \beta_{16} - 318154178895 \beta_{15} + 320234613514 \beta_{14} + 10816039649080 \beta_{13} - 2388623579288 \beta_{12} + 172588755816 \beta_{11} + 2882899101841 \beta_{10} - 9648455892436 \beta_{9} - 353196172585 \beta_{8} - 3049471957315 \beta_{7} + 91001084328 \beta_{6} + 7133118511120 \beta_{5} - 9755043427570 \beta_{4} + 143893620840 \beta_{3} - 2016868010504734 \beta_{2} - 5515556188430 \beta_{1} - 37467138222115$$ $$\nu^{9}$$ $$=$$ $$-22127547800730 \beta_{17} - 14750449644371 \beta_{16} - 53881807088895 \beta_{15} + 175288367873139 \beta_{14} + 186062739185550 \beta_{13} + 116319190463462 \beta_{12} - 196959046671990 \beta_{11} + 207098998615736 \beta_{10} + 3210906350642289 \beta_{9} + 22206011818386 \beta_{8} - 26737236422001 \beta_{7} - 137436543836570 \beta_{6} + 50236839135951 \beta_{5} + 4955338649420474 \beta_{4} - 87926148797250 \beta_{3} - 18373865210941314 \beta_{2} - 68803869493341 \beta_{1} + 58104465221952634$$ $$\nu^{10}$$ $$=$$ $$-6823966142941505 \beta_{17} + 3129582980107109 \beta_{16} - 3816949676806120 \beta_{15} + 5030908932281944 \beta_{14} + 1373182814104950 \beta_{13} + 1270826549022602 \beta_{12} - 10800958165256670 \beta_{11} + 1941052482069281 \beta_{10} + 29251066041472994 \beta_{9} + 6784935250716436 \beta_{8} + 26070548075692274 \beta_{7} + 22827731876941840 \beta_{6} - 129381369584227874 \beta_{5} + 49992816239549168 \beta_{4} - 103653423603263200 \beta_{3} + 19489060090149136635 \beta_{2} + 116219744004274868 \beta_{1} - 18995875778841124495$$ $$\nu^{11}$$ $$=$$ $$-291324051049630230 \beta_{17} + 435236271386758648 \beta_{16} + 178463802818182755 \beta_{15} - 1363452816555497442 \beta_{14} - 320966313736419560 \beta_{13} - 1495747413470505876 \beta_{12} - 107956516491093864 \beta_{11} - 1606780987199050588 \beta_{10} - 33409953464240419302 \beta_{9} - 326588030905587059 \beta_{8} + 1904968307317826269 \beta_{7} + 258675484746993028 \beta_{6} + 32128845558943178556 \beta_{5} - 48280637717525080018 \beta_{4} - 1135334706363478940 \beta_{3} - 430905369741671869010 \beta_{2} - 47275288442409848807 \beta_{1} - 212155171333636760778$$ $$\nu^{12}$$ $$=$$ $$27741035763656749010 \beta_{17} - 56689475399047316678 \beta_{16} + 63964991954400764800 \beta_{15} - 59696515246395079068 \beta_{14} - 982555761185393393020 \beta_{13} + 225699409277359025036 \beta_{12} + 73207030204285636382 \beta_{11} - 259003676713831737362 \beta_{10} + 1111984111595987677552 \beta_{9} - 35153401283213833094 \beta_{8} + 21134575242515637564 \beta_{7} - 242138276123631385844 \beta_{6} + 424884186437601377666 \beta_{5} + 798843098743258804496 \beta_{4} + 971846820774068762380 \beta_{3} - 6429390379318354322314 \beta_{2} - 574985949666187138680 \beta_{1} + 188228988341031716909573$$ $$\nu^{13}$$ $$=$$ $$4130477416906616220210 \beta_{17} - 2644533563315802032062 \beta_{16} + 2691099187716892458720 \beta_{15} - 2542919457897069012092 \beta_{14} - 12747915895452788958700 \beta_{13} + 3051037830476363756444 \beta_{12} + 18201750304866532376098 \beta_{11} - 3241308101372963283818 \beta_{10} + 18942355406400660366528 \beta_{9} + 1532298375644694091694 \beta_{8} - 14083643767621838171084 \beta_{7} + 12932482726745488646564 \beta_{6} - 331042437899991773111946 \beta_{5} + 14161592341933574320016 \beta_{4} + 10395892292789461356260 \beta_{3} + 7170028968721410193814955 \beta_{2} + 468481893078233255247605 \beta_{1} - 4758240003910549640505712$$ $$\nu^{14}$$ $$=$$ $$341188885281995187122430 \beta_{17} + 274440679854538109523187 \beta_{16} - 254062643278178266848900 \beta_{15} + 2263651066983092781817 \beta_{14} + 9287052660975992910073750 \beta_{13} - 2429516986052749020144734 \beta_{12} + 248133785351474438335110 \beta_{11} + 2182608657327805357580398 \beta_{10} - 16497762506497425303149113 \beta_{9} - 281624632221207070086367 \beta_{8} - 2403431033177387075255548 \beta_{7} + 203160373234212491990340 \beta_{6} + 11988146778910988200861963 \beta_{5} - 15890566635588144001157530 \beta_{4} + 56794750264282724241780 \beta_{3} - 1753284143901542532201691576 \beta_{2} - 9404774397146732018364296 \beta_{1} - 83802291180889699837217581$$ $$\nu^{15}$$ $$=$$ $$-11452782306149694570807735 \beta_{17} - 17206515081204492088934261 \beta_{16} - 36364485602484253207354710 \beta_{15} + 145938403001377912891664274 \beta_{14} + 63350524523338461668227800 \beta_{13} + 133912001556206641065164372 \beta_{12} - 159968395058009277109437018 \beta_{11} + 156595788292382867321547191 \beta_{10} + 3087828745250362085041309044 \beta_{9} + 11749125350132427309813384 \beta_{8} - 34565588547885796354647204 \beta_{7} - 168991339029425612002004354 \beta_{6} + 198580820451991007929828164 \beta_{5} + 4384018613220557854604324708 \beta_{4} + 77050956838251253745145750 \beta_{3} - 27070543697342880031691843496 \beta_{2} - 190458953255450876301284703 \beta_{1} + 79347541599974342604552975286$$ $$\nu^{16}$$ $$=$$ $$-5529309966906002734616942360 \beta_{17} + 2344079061208546100795176895 \beta_{16} - 3203644915866289218667311535 \beta_{15} + 4987113500970486800693662405 \beta_{14} - 106189919347833706986907440 \beta_{13} + 2435425329245786936130787760 \beta_{12} - 7400861267318891304279675162 \beta_{11} + 2242268796322425086497632290 \beta_{10} + 48739737128504272422756884135 \beta_{9} + 5976629786443277052816886372 \beta_{8} + 20106277878392699522757969323 \beta_{7} + 21941660534647357807198002784 \beta_{6} - 183476911922537769481642621043 \beta_{5} + 72057431171717922412936025660 \beta_{4} - 89107674978664594980856639840 \beta_{3} + 18033151004386345092810152494371 \beta_{2} + 179837637377674244658196553870 \beta_{1} - 16972598059467072594968318901397$$ $$\nu^{17}$$ $$=$$ $$-$$$$23\!\cdots\!35$$$$\beta_{17} +$$$$41\!\cdots\!12$$$$\beta_{16} +$$$$88\!\cdots\!30$$$$\beta_{15} -$$$$11\!\cdots\!33$$$$\beta_{14} +$$$$12\!\cdots\!50$$$$\beta_{13} -$$$$17\!\cdots\!94$$$$\beta_{12} -$$$$10\!\cdots\!42$$$$\beta_{11} -$$$$10\!\cdots\!57$$$$\beta_{10} -$$$$33\!\cdots\!53$$$$\beta_{9} -$$$$27\!\cdots\!75$$$$\beta_{8} +$$$$14\!\cdots\!60$$$$\beta_{7} +$$$$39\!\cdots\!24$$$$\beta_{6} +$$$$30\!\cdots\!05$$$$\beta_{5} -$$$$44\!\cdots\!44$$$$\beta_{4} -$$$$15\!\cdots\!00$$$$\beta_{3} -$$$$56\!\cdots\!46$$$$\beta_{2} -$$$$42\!\cdots\!45$$$$\beta_{1} -$$$$29\!\cdots\!46$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/684\mathbb{Z}\right)^\times$$.

 $$n$$ $$325$$ $$343$$ $$533$$ $$\chi(n)$$ $$-1 + \beta_{2}$$ $$1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
505.1
 −49.5928 − 87.6293i −34.6401 − 61.7304i −30.6056 − 54.7425i −11.6095 − 21.8402i 16.4645 + 26.7852i 19.0963 + 31.3437i 19.6530 + 32.3080i 26.2470 + 43.7291i 48.4871 + 82.2500i −49.5928 + 87.6293i −34.6401 + 61.7304i −30.6056 + 54.7425i −11.6095 + 21.8402i 16.4645 − 26.7852i 19.0963 − 31.3437i 19.6530 − 32.3080i 26.2470 − 43.7291i 48.4871 − 82.2500i
0 0 0 −50.5928 + 87.6293i 0 95.5451 0 0 0
505.2 0 0 0 −35.6401 + 61.7304i 0 252.315 0 0 0
505.3 0 0 0 −31.6056 + 54.7425i 0 −80.2775 0 0 0
505.4 0 0 0 −12.6095 + 21.8402i 0 −40.7176 0 0 0
505.5 0 0 0 15.4645 26.7852i 0 132.225 0 0 0
505.6 0 0 0 18.0963 31.3437i 0 −208.885 0 0 0
505.7 0 0 0 18.6530 32.3080i 0 15.8772 0 0 0
505.8 0 0 0 25.2470 43.7291i 0 −187.942 0 0 0
505.9 0 0 0 47.4871 82.2500i 0 189.860 0 0 0
577.1 0 0 0 −50.5928 87.6293i 0 95.5451 0 0 0
577.2 0 0 0 −35.6401 61.7304i 0 252.315 0 0 0
577.3 0 0 0 −31.6056 54.7425i 0 −80.2775 0 0 0
577.4 0 0 0 −12.6095 21.8402i 0 −40.7176 0 0 0
577.5 0 0 0 15.4645 + 26.7852i 0 132.225 0 0 0
577.6 0 0 0 18.0963 + 31.3437i 0 −208.885 0 0 0
577.7 0 0 0 18.6530 + 32.3080i 0 15.8772 0 0 0
577.8 0 0 0 25.2470 + 43.7291i 0 −187.942 0 0 0
577.9 0 0 0 47.4871 + 82.2500i 0 189.860 0 0 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 577.9 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
19.c even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 684.6.k.f 18
3.b odd 2 1 76.6.e.a 18
12.b even 2 1 304.6.i.d 18
19.c even 3 1 inner 684.6.k.f 18
57.h odd 6 1 76.6.e.a 18
228.m even 6 1 304.6.i.d 18

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
76.6.e.a 18 3.b odd 2 1
76.6.e.a 18 57.h odd 6 1
304.6.i.d 18 12.b even 2 1
304.6.i.d 18 228.m even 6 1
684.6.k.f 18 1.a even 1 1 trivial
684.6.k.f 18 19.c even 3 1 inner

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{6}^{\mathrm{new}}(684, [\chi])$$:

 $$13\!\cdots\!01$$$$T_{5}^{12} -$$$$38\!\cdots\!24$$$$T_{5}^{11} +$$$$57\!\cdots\!80$$$$T_{5}^{10} -$$$$18\!\cdots\!16$$$$T_{5}^{9} +$$$$16\!\cdots\!40$$$$T_{5}^{8} -$$$$56\!\cdots\!24$$$$T_{5}^{7} +$$$$33\!\cdots\!96$$$$T_{5}^{6} -$$$$85\!\cdots\!40$$$$T_{5}^{5} +$$$$32\!\cdots\!64$$$$T_{5}^{4} -$$$$51\!\cdots\!56$$$$T_{5}^{3} +$$$$17\!\cdots\!60$$$$T_{5}^{2} -$$$$19\!\cdots\!64$$$$T_{5} +$$$$53\!\cdots\!96$$">$$T_{5}^{18} + \cdots$$ $$19\!\cdots\!68$$$$T_{7}^{3} +$$$$19\!\cdots\!52$$$$T_{7}^{2} +$$$$53\!\cdots\!40$$$$T_{7} -$$$$12\!\cdots\!00$$">$$T_{7}^{9} - \cdots$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{18}$$
$3$ $$T^{18}$$
$5$ $$53\!\cdots\!96$$$$-$$$$19\!\cdots\!64$$$$T +$$$$17\!\cdots\!60$$$$T^{2} -$$$$51\!\cdots\!56$$$$T^{3} +$$$$32\!\cdots\!64$$$$T^{4} -$$$$85\!\cdots\!40$$$$T^{5} +$$$$33\!\cdots\!96$$$$T^{6} -$$$$56\!\cdots\!24$$$$T^{7} + 16796878410512090940 T^{8} - 182235863003254716 T^{9} + 5792633935619680 T^{10} - 38635011538024 T^{11} + 1306014959701 T^{12} - 3086592365 T^{13} + 211569494 T^{14} - 174581 T^{15} + 17650 T^{16} + 11 T^{17} + T^{18}$$
$7$ $$( -1233055145631744000 + 53167521425264640 T + 1928158954033152 T^{2} - 19652044114368 T^{3} - 334655586976 T^{4} + 2542736868 T^{5} + 13902400 T^{6} - 92704 T^{7} - 168 T^{8} + T^{9} )^{2}$$
$11$ $$( -$$$$15\!\cdots\!96$$$$+ 30973025188546606224 T + 2007063245270803884 T^{2} - 8001493002002373 T^{3} - 31276588367368 T^{4} + 134453704519 T^{5} + 146426924 T^{6} - 687587 T^{7} - 160 T^{8} + T^{9} )^{2}$$
$13$ $$60\!\cdots\!64$$$$+$$$$71\!\cdots\!76$$$$T +$$$$81\!\cdots\!20$$$$T^{2} +$$$$18\!\cdots\!84$$$$T^{3} +$$$$98\!\cdots\!00$$$$T^{4} +$$$$65\!\cdots\!04$$$$T^{5} +$$$$74\!\cdots\!24$$$$T^{6} +$$$$17\!\cdots\!76$$$$T^{7} +$$$$30\!\cdots\!40$$$$T^{8} +$$$$19\!\cdots\!40$$$$T^{9} +$$$$91\!\cdots\!68$$$$T^{10} + 15606742310908401088 T^{11} + 1766913125495305309 T^{12} - 49775665263507 T^{13} + 2463225262058 T^{14} - 100940479 T^{15} + 1961430 T^{16} - 227 T^{17} + T^{18}$$
$17$ $$16\!\cdots\!84$$$$+$$$$28\!\cdots\!92$$$$T +$$$$83\!\cdots\!04$$$$T^{2} +$$$$63\!\cdots\!00$$$$T^{3} +$$$$15\!\cdots\!04$$$$T^{4} +$$$$10\!\cdots\!76$$$$T^{5} +$$$$19\!\cdots\!16$$$$T^{6} +$$$$84\!\cdots\!84$$$$T^{7} +$$$$13\!\cdots\!40$$$$T^{8} +$$$$46\!\cdots\!68$$$$T^{9} +$$$$71\!\cdots\!40$$$$T^{10} +$$$$16\!\cdots\!60$$$$T^{11} +$$$$24\!\cdots\!29$$$$T^{12} + 37790308727158707 T^{13} + 63161431950210 T^{14} + 5114803983 T^{15} + 9862822 T^{16} + 179 T^{17} + T^{18}$$
$19$ $$34\!\cdots\!99$$$$+$$$$12\!\cdots\!68$$$$T +$$$$41\!\cdots\!72$$$$T^{2} +$$$$24\!\cdots\!24$$$$T^{3} +$$$$30\!\cdots\!48$$$$T^{4} +$$$$20\!\cdots\!76$$$$T^{5} +$$$$14\!\cdots\!03$$$$T^{6} +$$$$12\!\cdots\!80$$$$T^{7} +$$$$60\!\cdots\!84$$$$T^{8} +$$$$54\!\cdots\!88$$$$T^{9} +$$$$24\!\cdots\!16$$$$T^{10} +$$$$20\!\cdots\!80$$$$T^{11} + 96465687567916357097 T^{12} + 54428798269344676 T^{13} + 32274638969252 T^{14} + 10562267824 T^{15} + 7248128 T^{16} + 868 T^{17} + T^{18}$$
$23$ $$75\!\cdots\!00$$$$-$$$$45\!\cdots\!00$$$$T +$$$$32\!\cdots\!00$$$$T^{2} +$$$$24\!\cdots\!00$$$$T^{3} +$$$$50\!\cdots\!96$$$$T^{4} +$$$$59\!\cdots\!96$$$$T^{5} +$$$$22\!\cdots\!32$$$$T^{6} -$$$$75\!\cdots\!08$$$$T^{7} +$$$$75\!\cdots\!28$$$$T^{8} -$$$$10\!\cdots\!04$$$$T^{9} +$$$$13\!\cdots\!72$$$$T^{10} -$$$$16\!\cdots\!72$$$$T^{11} +$$$$15\!\cdots\!81$$$$T^{12} - 1959012574736536137 T^{13} + 1113768359604122 T^{14} - 89398709041 T^{15} + 44751186 T^{16} - 3425 T^{17} + T^{18}$$
$29$ $$96\!\cdots\!24$$$$-$$$$14\!\cdots\!80$$$$T +$$$$18\!\cdots\!96$$$$T^{2} -$$$$10\!\cdots\!80$$$$T^{3} +$$$$69\!\cdots\!64$$$$T^{4} -$$$$30\!\cdots\!76$$$$T^{5} +$$$$14\!\cdots\!88$$$$T^{6} -$$$$52\!\cdots\!68$$$$T^{7} +$$$$18\!\cdots\!24$$$$T^{8} -$$$$43\!\cdots\!64$$$$T^{9} +$$$$11\!\cdots\!88$$$$T^{10} -$$$$20\!\cdots\!64$$$$T^{11} +$$$$44\!\cdots\!49$$$$T^{12} - 61471755887734205509 T^{13} + 10053122334757166 T^{14} - 861197973469 T^{15} + 125209066 T^{16} - 7349 T^{17} + T^{18}$$
$31$ $$( -$$$$91\!\cdots\!60$$$$-$$$$78\!\cdots\!92$$$$T -$$$$49\!\cdots\!88$$$$T^{2} -$$$$44\!\cdots\!00$$$$T^{3} + 49426332686599495056 T^{4} + 6591076477990692 T^{5} - 989114300688 T^{6} - 167296276 T^{7} + 4980 T^{8} + T^{9} )^{2}$$
$37$ $$( -$$$$70\!\cdots\!00$$$$-$$$$14\!\cdots\!80$$$$T +$$$$38\!\cdots\!28$$$$T^{2} +$$$$31\!\cdots\!88$$$$T^{3} - 91986213539390854248 T^{4} - 4630799241237276 T^{5} + 3092826215800 T^{6} - 174702136 T^{7} - 13222 T^{8} + T^{9} )^{2}$$
$41$ $$33\!\cdots\!25$$$$+$$$$74\!\cdots\!25$$$$T +$$$$16\!\cdots\!25$$$$T^{2} +$$$$26\!\cdots\!00$$$$T^{3} +$$$$31\!\cdots\!14$$$$T^{4} +$$$$72\!\cdots\!30$$$$T^{5} +$$$$45\!\cdots\!06$$$$T^{6} +$$$$69\!\cdots\!44$$$$T^{7} +$$$$22\!\cdots\!23$$$$T^{8} +$$$$12\!\cdots\!27$$$$T^{9} +$$$$55\!\cdots\!99$$$$T^{10} +$$$$64\!\cdots\!52$$$$T^{11} +$$$$11\!\cdots\!54$$$$T^{12} -$$$$45\!\cdots\!54$$$$T^{13} + 102037756916691790 T^{14} - 1992308609992 T^{15} + 386322313 T^{16} - 7311 T^{17} + T^{18}$$
$43$ $$51\!\cdots\!00$$$$+$$$$14\!\cdots\!00$$$$T +$$$$38\!\cdots\!00$$$$T^{2} +$$$$62\!\cdots\!00$$$$T^{3} +$$$$90\!\cdots\!00$$$$T^{4} +$$$$41\!\cdots\!40$$$$T^{5} +$$$$18\!\cdots\!76$$$$T^{6} +$$$$47\!\cdots\!96$$$$T^{7} +$$$$10\!\cdots\!80$$$$T^{8} +$$$$14\!\cdots\!84$$$$T^{9} +$$$$18\!\cdots\!84$$$$T^{10} +$$$$17\!\cdots\!12$$$$T^{11} +$$$$17\!\cdots\!89$$$$T^{12} +$$$$12\!\cdots\!91$$$$T^{13} + 112961728986806062 T^{14} + 4760975065791 T^{15} + 379698838 T^{16} + 8283 T^{17} + T^{18}$$
$47$ $$16\!\cdots\!16$$$$-$$$$23\!\cdots\!44$$$$T +$$$$96\!\cdots\!24$$$$T^{2} -$$$$78\!\cdots\!88$$$$T^{3} +$$$$21\!\cdots\!36$$$$T^{4} +$$$$67\!\cdots\!76$$$$T^{5} +$$$$40\!\cdots\!32$$$$T^{6} +$$$$26\!\cdots\!32$$$$T^{7} +$$$$39\!\cdots\!12$$$$T^{8} +$$$$20\!\cdots\!40$$$$T^{9} +$$$$20\!\cdots\!08$$$$T^{10} +$$$$95\!\cdots\!48$$$$T^{11} +$$$$70\!\cdots\!29$$$$T^{12} +$$$$25\!\cdots\!99$$$$T^{13} + 1438596564851851442 T^{14} + 41903660395535 T^{15} + 1912969374 T^{16} + 37603 T^{17} + T^{18}$$
$53$ $$50\!\cdots\!64$$$$-$$$$39\!\cdots\!44$$$$T +$$$$17\!\cdots\!60$$$$T^{2} -$$$$65\!\cdots\!96$$$$T^{3} +$$$$33\!\cdots\!16$$$$T^{4} -$$$$11\!\cdots\!92$$$$T^{5} +$$$$37\!\cdots\!08$$$$T^{6} -$$$$73\!\cdots\!92$$$$T^{7} +$$$$27\!\cdots\!96$$$$T^{8} -$$$$69\!\cdots\!00$$$$T^{9} +$$$$10\!\cdots\!88$$$$T^{10} -$$$$24\!\cdots\!56$$$$T^{11} +$$$$25\!\cdots\!69$$$$T^{12} -$$$$59\!\cdots\!49$$$$T^{13} + 3460270021546904626 T^{14} - 43357881060629 T^{15} + 2267852878 T^{16} - 20337 T^{17} + T^{18}$$
$59$ $$21\!\cdots\!89$$$$-$$$$40\!\cdots\!55$$$$T +$$$$71\!\cdots\!53$$$$T^{2} -$$$$47\!\cdots\!00$$$$T^{3} +$$$$44\!\cdots\!82$$$$T^{4} -$$$$25\!\cdots\!38$$$$T^{5} +$$$$17\!\cdots\!26$$$$T^{6} -$$$$79\!\cdots\!68$$$$T^{7} +$$$$39\!\cdots\!87$$$$T^{8} -$$$$13\!\cdots\!77$$$$T^{9} +$$$$53\!\cdots\!07$$$$T^{10} -$$$$15\!\cdots\!52$$$$T^{11} +$$$$47\!\cdots\!62$$$$T^{12} -$$$$10\!\cdots\!18$$$$T^{13} + 23327287366044483930 T^{14} - 357651739695276 T^{15} + 7129477477 T^{16} - 74455 T^{17} + T^{18}$$
$61$ $$21\!\cdots\!56$$$$-$$$$20\!\cdots\!04$$$$T +$$$$11\!\cdots\!76$$$$T^{2} -$$$$16\!\cdots\!40$$$$T^{3} +$$$$57\!\cdots\!80$$$$T^{4} -$$$$63\!\cdots\!04$$$$T^{5} +$$$$81\!\cdots\!92$$$$T^{6} -$$$$33\!\cdots\!76$$$$T^{7} +$$$$26\!\cdots\!64$$$$T^{8} -$$$$57\!\cdots\!12$$$$T^{9} +$$$$57\!\cdots\!12$$$$T^{10} -$$$$69\!\cdots\!56$$$$T^{11} +$$$$74\!\cdots\!81$$$$T^{12} -$$$$16\!\cdots\!47$$$$T^{13} + 6930515824571977902 T^{14} - 1910399499327 T^{15} + 3182325786 T^{16} + 7569 T^{17} + T^{18}$$
$67$ $$92\!\cdots\!81$$$$+$$$$49\!\cdots\!29$$$$T +$$$$58\!\cdots\!85$$$$T^{2} +$$$$13\!\cdots\!60$$$$T^{3} +$$$$15\!\cdots\!62$$$$T^{4} +$$$$27\!\cdots\!90$$$$T^{5} +$$$$25\!\cdots\!86$$$$T^{6} +$$$$31\!\cdots\!92$$$$T^{7} +$$$$27\!\cdots\!35$$$$T^{8} +$$$$30\!\cdots\!31$$$$T^{9} +$$$$21\!\cdots\!79$$$$T^{10} +$$$$17\!\cdots\!60$$$$T^{11} +$$$$10\!\cdots\!10$$$$T^{12} +$$$$82\!\cdots\!10$$$$T^{13} + 38408940894285316426 T^{14} + 181133911740404 T^{15} + 7768368661 T^{16} + 26177 T^{17} + T^{18}$$
$71$ $$17\!\cdots\!00$$$$+$$$$39\!\cdots\!80$$$$T +$$$$13\!\cdots\!96$$$$T^{2} +$$$$43\!\cdots\!60$$$$T^{3} +$$$$28\!\cdots\!72$$$$T^{4} +$$$$21\!\cdots\!92$$$$T^{5} +$$$$27\!\cdots\!00$$$$T^{6} +$$$$71\!\cdots\!28$$$$T^{7} +$$$$42\!\cdots\!72$$$$T^{8} +$$$$65\!\cdots\!32$$$$T^{9} +$$$$41\!\cdots\!64$$$$T^{10} +$$$$44\!\cdots\!68$$$$T^{11} +$$$$19\!\cdots\!33$$$$T^{12} +$$$$39\!\cdots\!97$$$$T^{13} + 53973233604256978870 T^{14} - 53391804438599 T^{15} + 10847901010 T^{16} - 53463 T^{17} + T^{18}$$
$73$ $$43\!\cdots\!25$$$$+$$$$74\!\cdots\!75$$$$T +$$$$29\!\cdots\!25$$$$T^{2} +$$$$13\!\cdots\!20$$$$T^{3} +$$$$17\!\cdots\!06$$$$T^{4} +$$$$62\!\cdots\!86$$$$T^{5} +$$$$33\!\cdots\!42$$$$T^{6} +$$$$67\!\cdots\!72$$$$T^{7} +$$$$29\!\cdots\!47$$$$T^{8} +$$$$45\!\cdots\!81$$$$T^{9} +$$$$17\!\cdots\!35$$$$T^{10} +$$$$15\!\cdots\!60$$$$T^{11} +$$$$50\!\cdots\!46$$$$T^{12} +$$$$28\!\cdots\!26$$$$T^{13} +$$$$11\!\cdots\!74$$$$T^{14} + 307313940776492 T^{15} + 12536846917 T^{16} + 14103 T^{17} + T^{18}$$
$79$ $$12\!\cdots\!44$$$$+$$$$90\!\cdots\!32$$$$T +$$$$33\!\cdots\!84$$$$T^{2} -$$$$34\!\cdots\!16$$$$T^{3} +$$$$49\!\cdots\!56$$$$T^{4} -$$$$22\!\cdots\!72$$$$T^{5} +$$$$11\!\cdots\!08$$$$T^{6} -$$$$26\!\cdots\!68$$$$T^{7} +$$$$12\!\cdots\!16$$$$T^{8} -$$$$20\!\cdots\!36$$$$T^{9} +$$$$86\!\cdots\!68$$$$T^{10} -$$$$76\!\cdots\!32$$$$T^{11} +$$$$31\!\cdots\!41$$$$T^{12} -$$$$18\!\cdots\!25$$$$T^{13} + 83272908481639449190 T^{14} - 246647107958493 T^{15} + 11209186574 T^{16} - 31825 T^{17} + T^{18}$$
$83$ $$( -$$$$24\!\cdots\!00$$$$+$$$$20\!\cdots\!20$$$$T -$$$$98\!\cdots\!28$$$$T^{2} -$$$$39\!\cdots\!37$$$$T^{3} +$$$$15\!\cdots\!20$$$$T^{4} + 38020731823258649847 T^{5} - 511917348889560 T^{6} - 11706552339 T^{7} + 41300 T^{8} + T^{9} )^{2}$$
$89$ $$86\!\cdots\!44$$$$-$$$$57\!\cdots\!40$$$$T +$$$$64\!\cdots\!92$$$$T^{2} -$$$$27\!\cdots\!72$$$$T^{3} +$$$$26\!\cdots\!88$$$$T^{4} -$$$$11\!\cdots\!40$$$$T^{5} +$$$$49\!\cdots\!48$$$$T^{6} -$$$$13\!\cdots\!36$$$$T^{7} +$$$$35\!\cdots\!68$$$$T^{8} -$$$$69\!\cdots\!20$$$$T^{9} +$$$$14\!\cdots\!24$$$$T^{10} -$$$$22\!\cdots\!88$$$$T^{11} +$$$$35\!\cdots\!61$$$$T^{12} -$$$$38\!\cdots\!45$$$$T^{13} +$$$$41\!\cdots\!70$$$$T^{14} - 3121899045395049 T^{15} + 29541644542 T^{16} - 155197 T^{17} + T^{18}$$
$97$ $$93\!\cdots\!25$$$$+$$$$95\!\cdots\!75$$$$T +$$$$10\!\cdots\!25$$$$T^{2} +$$$$26\!\cdots\!00$$$$T^{3} +$$$$17\!\cdots\!14$$$$T^{4} +$$$$38\!\cdots\!90$$$$T^{5} +$$$$22\!\cdots\!50$$$$T^{6} +$$$$28\!\cdots\!48$$$$T^{7} +$$$$71\!\cdots\!71$$$$T^{8} +$$$$31\!\cdots\!17$$$$T^{9} +$$$$10\!\cdots\!83$$$$T^{10} +$$$$19\!\cdots\!24$$$$T^{11} +$$$$12\!\cdots\!86$$$$T^{12} -$$$$12\!\cdots\!34$$$$T^{13} +$$$$85\!\cdots\!22$$$$T^{14} - 1646469577269280 T^{15} + 42695596097 T^{16} - 111241 T^{17} + T^{18}$$