# Properties

 Label 84.9.m.b Level $84$ Weight $9$ Character orbit 84.m Analytic conductor $34.220$ Analytic rank $0$ Dimension $12$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$84 = 2^{2} \cdot 3 \cdot 7$$ Weight: $$k$$ $$=$$ $$9$$ Character orbit: $$[\chi]$$ $$=$$ 84.m (of order $$6$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$34.2198032451$$ Analytic rank: $$0$$ Dimension: $$12$$ Relative dimension: $$6$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\mathbb{Q}[x]/(x^{12} - \cdots)$$ Defining polynomial: $$x^{12} - 3 x^{11} + 148097 x^{10} + 46071824 x^{9} + 21578502553 x^{8} + 3561445462121 x^{7} + 576413321817541 x^{6} + 47217566733462528 x^{5} + 5214056955297543333 x^{4} + 358752845334081085965 x^{3} + 30962072851910211245661 x^{2} + 1221542968331193193318500 x + 45396580558961892385326096$$ Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$2^{20}\cdot 3^{10}\cdot 7^{4}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{11}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( 54 - 27 \beta_{1} ) q^{3} + ( 16 + 16 \beta_{1} - \beta_{3} ) q^{5} + ( 114 - 196 \beta_{1} + \beta_{3} + \beta_{5} ) q^{7} + ( 2187 - 2187 \beta_{1} ) q^{9} +O(q^{10})$$ $$q + ( 54 - 27 \beta_{1} ) q^{3} + ( 16 + 16 \beta_{1} - \beta_{3} ) q^{5} + ( 114 - 196 \beta_{1} + \beta_{3} + \beta_{5} ) q^{7} + ( 2187 - 2187 \beta_{1} ) q^{9} + ( -2987 \beta_{1} - \beta_{2} + 2 \beta_{3} + \beta_{5} - \beta_{6} + \beta_{7} ) q^{11} + ( 1915 - 3832 \beta_{1} - 2 \beta_{2} + \beta_{3} - 3 \beta_{4} + \beta_{5} + \beta_{9} - \beta_{10} - \beta_{11} ) q^{13} + ( 1296 - 27 \beta_{2} - 27 \beta_{3} ) q^{15} + ( -22860 + 11427 \beta_{1} - 3 \beta_{2} - \beta_{3} - 15 \beta_{4} - 15 \beta_{5} + \beta_{8} + \beta_{9} + 3 \beta_{10} ) q^{17} + ( 4118 + 4120 \beta_{1} + 3 \beta_{2} + 71 \beta_{3} + 23 \beta_{4} - 4 \beta_{5} + \beta_{6} - 2 \beta_{7} + 5 \beta_{9} + 3 \beta_{11} ) q^{19} + ( 864 - 8370 \beta_{1} + 27 \beta_{2} + 27 \beta_{3} + 27 \beta_{4} + 54 \beta_{5} ) q^{21} + ( -10427 + 10411 \beta_{1} - 207 \beta_{2} + 96 \beta_{3} - 43 \beta_{4} - 10 \beta_{5} + 2 \beta_{6} + \beta_{8} + 9 \beta_{9} - 8 \beta_{10} - 4 \beta_{11} ) q^{23} + ( -36 + 146524 \beta_{1} + 99 \beta_{2} - 206 \beta_{3} + 31 \beta_{4} + 79 \beta_{5} + 27 \beta_{6} - 27 \beta_{7} + 15 \beta_{8} - 8 \beta_{9} + 7 \beta_{10} + 14 \beta_{11} ) q^{25} + ( 59049 - 118098 \beta_{1} ) q^{27} + ( -47985 + 6 \beta_{1} + 77 \beta_{2} + 92 \beta_{3} + 71 \beta_{4} + 45 \beta_{5} + 32 \beta_{7} - \beta_{8} + 8 \beta_{10} - 8 \beta_{11} ) q^{29} + ( 160501 - 80227 \beta_{1} - 454 \beta_{2} + 9 \beta_{3} + 25 \beta_{4} - 61 \beta_{5} - 32 \beta_{6} - 32 \beta_{7} + 16 \beta_{8} - 9 \beta_{9} + 5 \beta_{10} ) q^{31} + ( -80649 - 80649 \beta_{1} + 81 \beta_{3} + 27 \beta_{4} + 54 \beta_{5} - 27 \beta_{6} + 54 \beta_{7} ) q^{33} + ( -74398 - 516226 \beta_{1} + 382 \beta_{2} - 160 \beta_{3} - 110 \beta_{4} - 50 \beta_{5} - 98 \beta_{6} + 63 \beta_{7} + 19 \beta_{8} + 17 \beta_{9} - 28 \beta_{10} - 35 \beta_{11} ) q^{35} + ( -342911 + 342810 \beta_{1} - 919 \beta_{2} + 416 \beta_{3} - 281 \beta_{4} - 88 \beta_{5} + 42 \beta_{6} + 23 \beta_{8} + 38 \beta_{9} - 24 \beta_{10} - 12 \beta_{11} ) q^{37} + ( -27 - 155169 \beta_{1} - 81 \beta_{2} + 81 \beta_{3} - 27 \beta_{4} + 135 \beta_{5} + 27 \beta_{9} - 27 \beta_{10} - 54 \beta_{11} ) q^{39} + ( 118027 - 236351 \beta_{1} - 2240 \beta_{2} + 2210 \beta_{3} - 40 \beta_{4} + 490 \beta_{5} + 64 \beta_{6} - 32 \beta_{7} + 42 \beta_{8} - 12 \beta_{9} + 63 \beta_{10} + 63 \beta_{11} ) q^{41} + ( 642727 + 158 \beta_{1} + 1505 \beta_{2} + 1528 \beta_{3} + 818 \beta_{4} + 88 \beta_{5} + 3 \beta_{7} + 111 \beta_{8} + 40 \beta_{9} - 24 \beta_{10} + 24 \beta_{11} ) q^{43} + ( 69984 - 34992 \beta_{1} - 2187 \beta_{2} ) q^{45} + ( 671980 + 672922 \beta_{1} - 147 \beta_{2} - 4858 \beta_{3} + 1296 \beta_{4} + 1554 \beta_{5} + 96 \beta_{6} - 192 \beta_{7} + 216 \beta_{8} - 69 \beta_{9} + 69 \beta_{11} ) q^{47} + ( -1805158 + 782782 \beta_{1} + 1599 \beta_{2} - 2177 \beta_{3} - 70 \beta_{4} + 267 \beta_{5} + 182 \beta_{6} - 133 \beta_{7} + 165 \beta_{8} + 106 \beta_{9} + 112 \beta_{10} + 91 \beta_{11} ) q^{49} + ( -925992 + 925830 \beta_{1} - 189 \beta_{2} + 54 \beta_{3} - 810 \beta_{4} - 324 \beta_{5} + 162 \beta_{8} - 81 \beta_{9} + 162 \beta_{10} + 81 \beta_{11} ) q^{51} + ( -586 - 917970 \beta_{1} - 1915 \beta_{2} + 3500 \beta_{3} + 1953 \beta_{4} + 2666 \beta_{5} + 126 \beta_{6} - 126 \beta_{7} - 41 \beta_{8} + 366 \beta_{9} - 23 \beta_{10} - 46 \beta_{11} ) q^{53} + ( 553847 - 1109548 \beta_{1} - 15285 \beta_{2} + 14770 \beta_{3} - 2097 \beta_{4} + 1807 \beta_{5} - 196 \beta_{6} + 98 \beta_{7} + 301 \beta_{8} + 214 \beta_{9} - 140 \beta_{10} - 140 \beta_{11} ) q^{55} + ( 333612 - 27 \beta_{1} + 2079 \beta_{2} + 1836 \beta_{3} + 432 \beta_{4} - 918 \beta_{5} - 81 \beta_{7} + 135 \beta_{8} + 216 \beta_{9} - 81 \beta_{10} + 81 \beta_{11} ) q^{57} + ( 836666 - 419074 \beta_{1} - 4454 \beta_{2} - 523 \beta_{3} - 2411 \beta_{4} - 3508 \beta_{5} + 161 \beta_{6} + 161 \beta_{7} + 379 \beta_{8} + 523 \beta_{9} - 231 \beta_{10} ) q^{59} + ( -2065735 - 2065505 \beta_{1} - 270 \beta_{2} - 8614 \beta_{3} + 3595 \beta_{4} + 1926 \beta_{5} - 286 \beta_{6} + 572 \beta_{7} - 177 \beta_{8} + 668 \beta_{9} - 447 \beta_{11} ) q^{61} + ( -179334 - 249318 \beta_{1} + 2187 \beta_{2} + 2187 \beta_{4} + 2187 \beta_{5} ) q^{63} + ( 844385 - 845297 \beta_{1} - 14559 \beta_{2} + 6954 \beta_{3} - 3505 \beta_{4} - 234 \beta_{5} - 604 \beta_{6} + 201 \beta_{8} + 213 \beta_{9} + 48 \beta_{10} + 24 \beta_{11} ) q^{65} + ( -1832 - 6133611 \beta_{1} + 5374 \beta_{2} - 11685 \beta_{3} - 533 \beta_{4} + 5592 \beta_{5} - 825 \beta_{6} + 825 \beta_{7} + 967 \beta_{8} - 627 \beta_{9} + 185 \beta_{10} + 370 \beta_{11} ) q^{67} + ( -281745 + 562518 \beta_{1} - 8586 \beta_{2} + 8181 \beta_{3} - 1323 \beta_{4} + 513 \beta_{5} + 108 \beta_{6} - 54 \beta_{7} + 81 \beta_{8} + 324 \beta_{9} - 324 \beta_{10} - 324 \beta_{11} ) q^{69} + ( -2506548 + 291 \beta_{1} + 11532 \beta_{2} + 12477 \beta_{3} + 1177 \beta_{4} + 1179 \beta_{5} - 164 \beta_{7} - 185 \beta_{8} - 192 \beta_{9} + 469 \beta_{10} - 469 \beta_{11} ) q^{71} + ( 10573482 - 5286290 \beta_{1} - 26527 \beta_{2} + 276 \beta_{3} + 1482 \beta_{4} + 553 \beta_{5} - 27 \beta_{6} - 27 \beta_{7} + 66 \beta_{8} - 276 \beta_{9} + 268 \beta_{10} ) q^{73} + ( 3954015 + 3956931 \beta_{1} - 216 \beta_{2} - 8235 \beta_{3} + 2592 \beta_{4} + 3240 \beta_{5} + 729 \beta_{6} - 1458 \beta_{7} + 783 \beta_{8} - 432 \beta_{9} + 567 \beta_{11} ) q^{75} + ( -5708008 + 5018006 \beta_{1} + 50349 \beta_{2} - 34779 \beta_{3} - 2233 \beta_{4} - 2134 \beta_{5} + 476 \beta_{6} + 126 \beta_{7} + 135 \beta_{8} - 73 \beta_{9} - 455 \beta_{10} + 497 \beta_{11} ) q^{77} + ( 1421832 - 1421127 \beta_{1} - 9759 \beta_{2} + 5049 \beta_{3} + 4220 \beta_{4} + 266 \beta_{5} + 1458 \beta_{6} - 378 \beta_{8} + 148 \beta_{9} - 514 \beta_{10} - 257 \beta_{11} ) q^{79} -4782969 \beta_{1} q^{81} + ( -11714524 + 23433674 \beta_{1} - 46625 \beta_{2} + 48254 \beta_{3} + 7848 \beta_{4} - 1410 \beta_{5} + 6 \beta_{6} - 3 \beta_{7} - 1161 \beta_{8} - 468 \beta_{9} + 786 \beta_{10} + 786 \beta_{11} ) q^{83} + ( 1657571 + 289 \beta_{1} + 37010 \beta_{2} + 36120 \beta_{3} - 1124 \beta_{4} - 1184 \beta_{5} - 1326 \beta_{7} + 484 \beta_{8} - 16 \beta_{9} - 695 \beta_{10} + 695 \beta_{11} ) q^{85} + ( -2591244 + 1296189 \beta_{1} + 6642 \beta_{2} + 405 \beta_{3} + 3348 \beta_{4} + 945 \beta_{5} + 864 \beta_{6} + 864 \beta_{7} + 162 \beta_{8} - 405 \beta_{9} + 648 \beta_{10} ) q^{87} + ( 4623970 + 4614982 \beta_{1} + 1382 \beta_{2} - 45802 \beta_{3} - 18250 \beta_{4} - 16952 \beta_{5} - 128 \beta_{6} + 256 \beta_{7} - 1766 \beta_{8} - 542 \beta_{9} - 384 \beta_{11} ) q^{89} + ( -22957097 + 12810079 \beta_{1} + 43725 \beta_{2} - 10051 \beta_{3} - 372 \beta_{4} - 2304 \beta_{5} + 1043 \beta_{6} - 2170 \beta_{7} - 165 \beta_{8} - 155 \beta_{9} + 476 \beta_{10} - 1414 \beta_{11} ) q^{91} + ( 6500790 - 6499521 \beta_{1} - 24273 \beta_{2} + 12501 \beta_{3} - 972 \beta_{4} - 3834 \beta_{5} - 2592 \beta_{6} + 756 \beta_{8} - 810 \beta_{9} + 270 \beta_{10} + 135 \beta_{11} ) q^{93} + ( 9314 - 36923483 \beta_{1} + 36923 \beta_{2} - 68893 \beta_{3} - 7461 \beta_{4} - 30201 \beta_{5} - 154 \beta_{6} + 154 \beta_{7} - 3227 \beta_{8} + 447 \beta_{9} - 527 \beta_{10} - 1054 \beta_{11} ) q^{95} + ( -23088145 + 46181460 \beta_{1} - 18981 \beta_{2} + 20345 \beta_{3} + 1131 \beta_{4} - 8463 \beta_{5} - 746 \beta_{6} + 373 \beta_{7} + 288 \beta_{8} - 1652 \beta_{9} + 646 \beta_{10} + 646 \beta_{11} ) q^{97} + ( -6532569 + 2187 \beta_{2} + 2187 \beta_{3} + 2187 \beta_{4} + 2187 \beta_{5} + 2187 \beta_{7} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$12 q + 486 q^{3} + 285 q^{5} + 198 q^{7} + 13122 q^{9} + O(q^{10})$$ $$12 q + 486 q^{3} + 285 q^{5} + 198 q^{7} + 13122 q^{9} - 17919 q^{11} + 15390 q^{15} - 205782 q^{17} + 74313 q^{19} - 39609 q^{21} - 62832 q^{23} + 878679 q^{25} - 575454 q^{29} + 1442952 q^{31} - 1451439 q^{33} - 3989514 q^{35} - 2058621 q^{37} - 930933 q^{39} + 7721322 q^{43} + 623295 q^{45} + 12088194 q^{47} - 16964694 q^{49} - 5556114 q^{51} - 5506743 q^{53} + 4012902 q^{57} + 7511901 q^{59} - 37215576 q^{61} - 3641355 q^{63} + 5047122 q^{65} - 36824553 q^{67} - 30011556 q^{71} + 95080185 q^{73} + 71172999 q^{75} - 38333727 q^{77} + 8514456 q^{79} - 28697814 q^{81} + 20121540 q^{85} - 23305887 q^{87} + 83038554 q^{89} - 198538635 q^{91} + 38959704 q^{93} - 221605224 q^{95} - 78377706 q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{12} - 3 x^{11} + 148097 x^{10} + 46071824 x^{9} + 21578502553 x^{8} + 3561445462121 x^{7} + 576413321817541 x^{6} + 47217566733462528 x^{5} + 5214056955297543333 x^{4} + 358752845334081085965 x^{3} + 30962072851910211245661 x^{2} + 1221542968331193193318500 x + 45396580558961892385326096$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$-$$$$15\!\cdots\!43$$$$\nu^{11} +$$$$92\!\cdots\!05$$$$\nu^{10} -$$$$23\!\cdots\!79$$$$\nu^{9} -$$$$71\!\cdots\!84$$$$\nu^{8} -$$$$33\!\cdots\!87$$$$\nu^{7} -$$$$56\!\cdots\!07$$$$\nu^{6} -$$$$91\!\cdots\!59$$$$\nu^{5} -$$$$69\!\cdots\!16$$$$\nu^{4} -$$$$71\!\cdots\!87$$$$\nu^{3} -$$$$34\!\cdots\!19$$$$\nu^{2} -$$$$40\!\cdots\!11$$$$\nu -$$$$14\!\cdots\!20$$$$)/$$$$14\!\cdots\!72$$ $$\beta_{2}$$ $$=$$ $$($$$$34\!\cdots\!83$$$$\nu^{11} +$$$$29\!\cdots\!80$$$$\nu^{10} +$$$$46\!\cdots\!52$$$$\nu^{9} +$$$$20\!\cdots\!52$$$$\nu^{8} +$$$$81\!\cdots\!19$$$$\nu^{7} +$$$$16\!\cdots\!64$$$$\nu^{6} +$$$$21\!\cdots\!88$$$$\nu^{5} +$$$$20\!\cdots\!24$$$$\nu^{4} +$$$$12\!\cdots\!07$$$$\nu^{3} +$$$$15\!\cdots\!80$$$$\nu^{2} +$$$$76\!\cdots\!20$$$$\nu +$$$$41\!\cdots\!04$$$$)/$$$$20\!\cdots\!24$$ $$\beta_{3}$$ $$=$$ $$($$$$-$$$$42\!\cdots\!13$$$$\nu^{11} +$$$$53\!\cdots\!03$$$$\nu^{10} -$$$$65\!\cdots\!28$$$$\nu^{9} -$$$$11\!\cdots\!48$$$$\nu^{8} -$$$$70\!\cdots\!05$$$$\nu^{7} -$$$$50\!\cdots\!73$$$$\nu^{6} -$$$$10\!\cdots\!56$$$$\nu^{5} -$$$$52\!\cdots\!20$$$$\nu^{4} -$$$$10\!\cdots\!09$$$$\nu^{3} -$$$$23\!\cdots\!13$$$$\nu^{2} -$$$$15\!\cdots\!76$$$$\nu +$$$$17\!\cdots\!08$$$$)/$$$$20\!\cdots\!24$$ $$\beta_{4}$$ $$=$$ $$($$$$49\!\cdots\!23$$$$\nu^{11} -$$$$91\!\cdots\!62$$$$\nu^{10} +$$$$76\!\cdots\!76$$$$\nu^{9} +$$$$96\!\cdots\!68$$$$\nu^{8} +$$$$68\!\cdots\!87$$$$\nu^{7} -$$$$16\!\cdots\!22$$$$\nu^{6} +$$$$26\!\cdots\!28$$$$\nu^{5} -$$$$16\!\cdots\!76$$$$\nu^{4} -$$$$16\!\cdots\!53$$$$\nu^{3} -$$$$12\!\cdots\!74$$$$\nu^{2} -$$$$53\!\cdots\!68$$$$\nu -$$$$86\!\cdots\!88$$$$)/$$$$20\!\cdots\!24$$ $$\beta_{5}$$ $$=$$ $$($$$$92\!\cdots\!02$$$$\nu^{11} +$$$$28\!\cdots\!05$$$$\nu^{10} +$$$$13\!\cdots\!72$$$$\nu^{9} +$$$$47\!\cdots\!16$$$$\nu^{8} +$$$$20\!\cdots\!54$$$$\nu^{7} +$$$$37\!\cdots\!49$$$$\nu^{6} +$$$$52\!\cdots\!36$$$$\nu^{5} +$$$$46\!\cdots\!04$$$$\nu^{4} +$$$$40\!\cdots\!42$$$$\nu^{3} +$$$$36\!\cdots\!69$$$$\nu^{2} +$$$$25\!\cdots\!92$$$$\nu +$$$$10\!\cdots\!68$$$$)/$$$$20\!\cdots\!24$$ $$\beta_{6}$$ $$=$$ $$($$$$15\!\cdots\!68$$$$\nu^{11} +$$$$16\!\cdots\!81$$$$\nu^{10} +$$$$16\!\cdots\!88$$$$\nu^{9} +$$$$10\!\cdots\!68$$$$\nu^{8} +$$$$31\!\cdots\!12$$$$\nu^{7} +$$$$69\!\cdots\!69$$$$\nu^{6} +$$$$30\!\cdots\!64$$$$\nu^{5} +$$$$71\!\cdots\!92$$$$\nu^{4} +$$$$14\!\cdots\!32$$$$\nu^{3} +$$$$12\!\cdots\!93$$$$\nu^{2} -$$$$23\!\cdots\!32$$$$\nu +$$$$78\!\cdots\!92$$$$)/$$$$28\!\cdots\!32$$ $$\beta_{7}$$ $$=$$ $$($$$$-$$$$18\!\cdots\!39$$$$\nu^{11} +$$$$54\!\cdots\!96$$$$\nu^{10} -$$$$31\!\cdots\!12$$$$\nu^{9} -$$$$67\!\cdots\!76$$$$\nu^{8} -$$$$20\!\cdots\!19$$$$\nu^{7} +$$$$29\!\cdots\!64$$$$\nu^{6} +$$$$30\!\cdots\!12$$$$\nu^{5} +$$$$80\!\cdots\!20$$$$\nu^{4} -$$$$16\!\cdots\!43$$$$\nu^{3} +$$$$62\!\cdots\!40$$$$\nu^{2} +$$$$14\!\cdots\!40$$$$\nu -$$$$44\!\cdots\!40$$$$)/$$$$20\!\cdots\!24$$ $$\beta_{8}$$ $$=$$ $$($$$$-$$$$13\!\cdots\!45$$$$\nu^{11} -$$$$88\!\cdots\!70$$$$\nu^{10} -$$$$18\!\cdots\!26$$$$\nu^{9} -$$$$76\!\cdots\!12$$$$\nu^{8} -$$$$31\!\cdots\!13$$$$\nu^{7} -$$$$62\!\cdots\!98$$$$\nu^{6} -$$$$86\!\cdots\!22$$$$\nu^{5} -$$$$84\!\cdots\!84$$$$\nu^{4} -$$$$65\!\cdots\!53$$$$\nu^{3} -$$$$65\!\cdots\!10$$$$\nu^{2} -$$$$33\!\cdots\!42$$$$\nu -$$$$20\!\cdots\!48$$$$)/$$$$50\!\cdots\!56$$ $$\beta_{9}$$ $$=$$ $$($$$$14\!\cdots\!29$$$$\nu^{11} -$$$$18\!\cdots\!95$$$$\nu^{10} +$$$$21\!\cdots\!76$$$$\nu^{9} +$$$$38\!\cdots\!84$$$$\nu^{8} +$$$$23\!\cdots\!05$$$$\nu^{7} +$$$$14\!\cdots\!13$$$$\nu^{6} +$$$$33\!\cdots\!60$$$$\nu^{5} -$$$$57\!\cdots\!76$$$$\nu^{4} +$$$$31\!\cdots\!85$$$$\nu^{3} -$$$$40\!\cdots\!19$$$$\nu^{2} +$$$$11\!\cdots\!00$$$$\nu -$$$$79\!\cdots\!00$$$$)/$$$$50\!\cdots\!56$$ $$\beta_{10}$$ $$=$$ $$($$$$17\!\cdots\!71$$$$\nu^{11} +$$$$84\!\cdots\!72$$$$\nu^{10} +$$$$24\!\cdots\!56$$$$\nu^{9} +$$$$92\!\cdots\!56$$$$\nu^{8} +$$$$39\!\cdots\!67$$$$\nu^{7} +$$$$76\!\cdots\!04$$$$\nu^{6} +$$$$11\!\cdots\!92$$$$\nu^{5} +$$$$10\!\cdots\!88$$$$\nu^{4} +$$$$85\!\cdots\!79$$$$\nu^{3} +$$$$82\!\cdots\!64$$$$\nu^{2} +$$$$49\!\cdots\!36$$$$\nu +$$$$25\!\cdots\!72$$$$)/$$$$20\!\cdots\!24$$ $$\beta_{11}$$ $$=$$ $$($$$$25\!\cdots\!43$$$$\nu^{11} -$$$$24\!\cdots\!56$$$$\nu^{10} +$$$$37\!\cdots\!88$$$$\nu^{9} +$$$$81\!\cdots\!48$$$$\nu^{8} +$$$$43\!\cdots\!47$$$$\nu^{7} +$$$$40\!\cdots\!52$$$$\nu^{6} +$$$$71\!\cdots\!04$$$$\nu^{5} +$$$$64\!\cdots\!84$$$$\nu^{4} +$$$$56\!\cdots\!23$$$$\nu^{3} +$$$$62\!\cdots\!24$$$$\nu^{2} +$$$$20\!\cdots\!76$$$$\nu -$$$$99\!\cdots\!36$$$$)/$$$$20\!\cdots\!24$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$-\beta_{9} + 2 \beta_{8} + 11 \beta_{5} - 2 \beta_{4} - 13 \beta_{3} + 5 \beta_{2} + 89 \beta_{1} - 4$$$$)/168$$ $$\nu^{2}$$ $$=$$ $$($$$$-49 \beta_{11} - 98 \beta_{10} + 379 \beta_{9} + \beta_{8} - 357 \beta_{6} + 460 \beta_{5} - 3002 \beta_{4} - 3851 \beta_{3} + 6992 \beta_{2} + 8294560 \beta_{1} - 8295551$$$$)/168$$ $$\nu^{3}$$ $$=$$ $$($$$$35084 \beta_{11} - 35084 \beta_{10} + 789666 \beta_{9} - 376181 \beta_{8} - 275352 \beta_{7} - 5422393 \beta_{5} - 4661151 \beta_{4} + 5080849 \beta_{3} + 6316864 \beta_{2} - 1577112 \beta_{1} - 5863038373$$$$)/504$$ $$\nu^{4}$$ $$=$$ $$($$$$53732714 \beta_{11} + 26866357 \beta_{10} + 42409788 \beta_{9} - 238180922 \beta_{8} - 173747049 \beta_{7} + 173747049 \beta_{6} - 2066402749 \beta_{5} - 33838617 \beta_{4} + 4692219244 \beta_{3} - 2102267237 \beta_{2} - 3614068160130 \beta_{1} + 602856833$$$$)/504$$ $$\nu^{5}$$ $$=$$ $$($$$$8924207908 \beta_{11} + 17848415816 \beta_{10} - 91153716975 \beta_{9} - 55734876919 \beta_{8} + 65807686308 \beta_{6} + 94177321174 \beta_{5} + 1002318143439 \beta_{4} + 1210261948916 \beta_{3} - 2191405794871 \beta_{2} - 1370554196910783 \beta_{1} + 1370856620314903$$$$)/504$$ $$\nu^{6}$$ $$=$$ $$($$$$-4764138914563 \beta_{11} + 4764138914563 \beta_{10} - 56310552509697 \beta_{9} + 33032234588515 \beta_{8} + 31905883209375 \beta_{7} + 422083627352531 \beta_{5} + 414091380503085 \beta_{4} - 424790094382343 \beta_{3} - 523661159309681 \beta_{2} + 127139160601290 \beta_{1} + 662497632731591534$$$$)/504$$ $$\nu^{7}$$ $$=$$ $$($$$$-1274799945134296 \beta_{11} - 637399972567148 \beta_{10} - 2616524333882439 \beta_{9} + 8096770470173170 \beta_{8} + 4510915272315424 \beta_{7} - 4510915272315424 \beta_{6} + 59380301078836581 \beta_{5} - 3967404870918330 \beta_{4} - 142373657159854427 \beta_{3} + 63760920304128115 \beta_{2} + 93756818996606666983 \beta_{1} - 18419862770187484$$$$)/168$$ $$\nu^{8}$$ $$=$$ $$($$$$-299961991155128967 \beta_{11} - 599923982310257934 \beta_{10} + 2761901930293169945 \beta_{9} + 1318099722270022371 \beta_{8} - 2053490526212980771 \beta_{6} - 1548366975424523396 \beta_{5} - 28339508650667420450 \beta_{4} - 34636588569638199789 \beta_{3} + 62731235547575166284 \beta_{2} + 42654693205291567086156 \beta_{1} - 42663397124831251231461$$$$)/168$$ $$\nu^{9}$$ $$=$$ $$($$$$386927738491689098956 \beta_{11} - 386927738491689098956 \beta_{10} + 5106521524739302481418 \beta_{9} - 2870639556898517562943 \beta_{8} - 2698426008365629430364 \beta_{7} - 37616807779741072724195 \beta_{5} - 35962269976141724572005 \beta_{4} + 37577453901333862646939 \beta_{3} + 46328470459955060889212 \beta_{2} - 11234728377028026706260 \beta_{1} - 56060198450193432047925767$$$$)/504$$ $$\nu^{10}$$ $$=$$ $$($$$$116599898223201223155010 \beta_{11} + 58299949111600611577505 \beta_{10} + 207707048764039563694464 \beta_{9} - 691597561928202070173914 \beta_{8} - 402328591279221355789845 \beta_{7} + 402328591279221355789845 \beta_{6} - 5215740975238483537968789 \beta_{5} + 271146812509037522861491 \beta_{4} + 12389252477810540374275648 \beta_{3} - 5548582252247487287233637 \beta_{2} - 8359417814642448384326770846 \beta_{1} + 1601078639144926471555309$$$$)/168$$ $$\nu^{11}$$ $$=$$ $$($$$$76888622884619494428173372 \beta_{11} + 153777245769238988856346744 \beta_{10} - 720087653429857536714139773 \beta_{9} - 359918693937678787472782481 \beta_{8} + 533709909997056195659524968 \beta_{6} + 460831206198717043031890610 \beta_{5} + 7478895624047354471311168581 \beta_{4} + 9122968496657117312610538480 \beta_{3} - 16522731615401460258748188305 \beta_{2} - 11087334140948660911575610655853 \beta_{1} + 11089623656734234304489941337537$$$$)/504$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/84\mathbb{Z}\right)^\times$$.

 $$n$$ $$29$$ $$43$$ $$73$$ $$\chi(n)$$ $$1$$ $$1$$ $$1 - \beta_{1}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
61.1
 44.6586 + 77.3509i −72.3408 − 125.298i −28.8366 − 49.9465i −122.377 − 211.963i 221.993 + 384.503i −41.5970 − 72.0480i 44.6586 − 77.3509i −72.3408 + 125.298i −28.8366 + 49.9465i −122.377 + 211.963i 221.993 − 384.503i −41.5970 + 72.0480i
0 40.5000 23.3827i 0 −939.615 542.487i 0 1451.57 + 1912.52i 0 1093.50 1894.00i 0
61.2 0 40.5000 23.3827i 0 −225.043 129.928i 0 597.275 2325.52i 0 1093.50 1894.00i 0
61.3 0 40.5000 23.3827i 0 −203.366 117.413i 0 1130.34 2118.29i 0 1093.50 1894.00i 0
61.4 0 40.5000 23.3827i 0 −133.903 77.3091i 0 −2234.88 + 877.558i 0 1093.50 1894.00i 0
61.5 0 40.5000 23.3827i 0 632.551 + 365.204i 0 984.437 + 2189.91i 0 1093.50 1894.00i 0
61.6 0 40.5000 23.3827i 0 1011.88 + 584.207i 0 −1829.74 1554.62i 0 1093.50 1894.00i 0
73.1 0 40.5000 + 23.3827i 0 −939.615 + 542.487i 0 1451.57 1912.52i 0 1093.50 + 1894.00i 0
73.2 0 40.5000 + 23.3827i 0 −225.043 + 129.928i 0 597.275 + 2325.52i 0 1093.50 + 1894.00i 0
73.3 0 40.5000 + 23.3827i 0 −203.366 + 117.413i 0 1130.34 + 2118.29i 0 1093.50 + 1894.00i 0
73.4 0 40.5000 + 23.3827i 0 −133.903 + 77.3091i 0 −2234.88 877.558i 0 1093.50 + 1894.00i 0
73.5 0 40.5000 + 23.3827i 0 632.551 365.204i 0 984.437 2189.91i 0 1093.50 + 1894.00i 0
73.6 0 40.5000 + 23.3827i 0 1011.88 584.207i 0 −1829.74 + 1554.62i 0 1093.50 + 1894.00i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 73.6 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.d odd 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 84.9.m.b 12
3.b odd 2 1 252.9.z.d 12
7.d odd 6 1 inner 84.9.m.b 12
21.g even 6 1 252.9.z.d 12

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
84.9.m.b 12 1.a even 1 1 trivial
84.9.m.b 12 7.d odd 6 1 inner
252.9.z.d 12 3.b odd 2 1
252.9.z.d 12 21.g even 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$21\!\cdots\!34$$$$T_{5}^{8} -$$$$31\!\cdots\!83$$$$T_{5}^{7} -$$$$67\!\cdots\!83$$$$T_{5}^{6} +$$$$32\!\cdots\!10$$$$T_{5}^{5} +$$$$20\!\cdots\!00$$$$T_{5}^{4} +$$$$85\!\cdots\!00$$$$T_{5}^{3} +$$$$17\!\cdots\!00$$$$T_{5}^{2} +$$$$17\!\cdots\!00$$$$T_{5} +$$$$76\!\cdots\!00$$">$$T_{5}^{12} - \cdots$$ acting on $$S_{9}^{\mathrm{new}}(84, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{12}$$
$3$ $$( 2187 - 81 T + T^{2} )^{6}$$
$5$ $$76\!\cdots\!00$$$$+$$$$17\!\cdots\!00$$$$T +$$$$17\!\cdots\!00$$$$T^{2} +$$$$85\!\cdots\!00$$$$T^{3} +$$$$20\!\cdots\!00$$$$T^{4} + 32049897802747435710 T^{5} - 679960365860892483 T^{6} - 311260236586983 T^{7} + 2118409775334 T^{8} + 455337945 T^{9} - 1570602 T^{10} - 285 T^{11} + T^{12}$$
$7$ $$36\!\cdots\!01$$$$-$$$$12\!\cdots\!98$$$$T +$$$$93\!\cdots\!49$$$$T^{2} +$$$$58\!\cdots\!98$$$$T^{3} +$$$$19\!\cdots\!38$$$$T^{4} +$$$$84\!\cdots\!86$$$$T^{5} +$$$$73\!\cdots\!77$$$$T^{6} + 146629745246641986 T^{7} + 57901709320938 T^{8} + 30443535598 T^{9} + 8501949 T^{10} - 198 T^{11} + T^{12}$$
$11$ $$16\!\cdots\!00$$$$-$$$$15\!\cdots\!00$$$$T +$$$$35\!\cdots\!40$$$$T^{2} -$$$$53\!\cdots\!40$$$$T^{3} +$$$$29\!\cdots\!76$$$$T^{4} -$$$$97\!\cdots\!10$$$$T^{5} +$$$$17\!\cdots\!75$$$$T^{6} +$$$$24\!\cdots\!97$$$$T^{7} + 551021544776781324 T^{8} + 6277714424145 T^{9} + 1057716756 T^{10} + 17919 T^{11} + T^{12}$$
$13$ $$21\!\cdots\!64$$$$+$$$$15\!\cdots\!24$$$$T^{2} +$$$$19\!\cdots\!60$$$$T^{4} +$$$$66\!\cdots\!15$$$$T^{6} + 9052853898338449947 T^{8} + 5107913481 T^{10} + T^{12}$$
$17$ $$10\!\cdots\!00$$$$+$$$$10\!\cdots\!80$$$$T +$$$$38\!\cdots\!52$$$$T^{2} +$$$$32\!\cdots\!08$$$$T^{3} -$$$$39\!\cdots\!60$$$$T^{4} -$$$$61\!\cdots\!56$$$$T^{5} +$$$$42\!\cdots\!08$$$$T^{6} +$$$$94\!\cdots\!28$$$$T^{7} + 94176297436197260448 T^{8} - 4882203625831704 T^{9} - 9609714264 T^{10} + 205782 T^{11} + T^{12}$$
$19$ $$83\!\cdots\!96$$$$-$$$$98\!\cdots\!20$$$$T +$$$$46\!\cdots\!36$$$$T^{2} -$$$$91\!\cdots\!20$$$$T^{3} +$$$$54\!\cdots\!00$$$$T^{4} +$$$$54\!\cdots\!22$$$$T^{5} -$$$$42\!\cdots\!39$$$$T^{6} -$$$$27\!\cdots\!31$$$$T^{7} +$$$$27\!\cdots\!26$$$$T^{8} + 4528382876267697 T^{9} - 59095810446 T^{10} - 74313 T^{11} + T^{12}$$
$23$ $$63\!\cdots\!64$$$$+$$$$26\!\cdots\!92$$$$T +$$$$91\!\cdots\!48$$$$T^{2} +$$$$10\!\cdots\!48$$$$T^{3} +$$$$98\!\cdots\!40$$$$T^{4} +$$$$29\!\cdots\!20$$$$T^{5} +$$$$19\!\cdots\!40$$$$T^{6} +$$$$29\!\cdots\!44$$$$T^{7} +$$$$31\!\cdots\!08$$$$T^{8} + 17273214136659456 T^{9} + 202735409760 T^{10} + 62832 T^{11} + T^{12}$$
$29$ $$($$$$23\!\cdots\!00$$$$-$$$$48\!\cdots\!00$$$$T +$$$$25\!\cdots\!00$$$$T^{2} + 55337485683990069 T^{3} - 1129194137997 T^{4} + 287727 T^{5} + T^{6} )^{2}$$
$31$ $$15\!\cdots\!81$$$$+$$$$45\!\cdots\!68$$$$T +$$$$44\!\cdots\!99$$$$T^{2} +$$$$76\!\cdots\!68$$$$T^{3} -$$$$56\!\cdots\!42$$$$T^{4} -$$$$10\!\cdots\!44$$$$T^{5} +$$$$87\!\cdots\!19$$$$T^{6} -$$$$14\!\cdots\!24$$$$T^{7} +$$$$71\!\cdots\!94$$$$T^{8} + 4411658625554488392 T^{9} - 2363347360953 T^{10} - 1442952 T^{11} + T^{12}$$
$37$ $$40\!\cdots\!84$$$$-$$$$57\!\cdots\!36$$$$T +$$$$19\!\cdots\!84$$$$T^{2} -$$$$10\!\cdots\!40$$$$T^{3} +$$$$40\!\cdots\!20$$$$T^{4} -$$$$16\!\cdots\!98$$$$T^{5} +$$$$53\!\cdots\!67$$$$T^{6} +$$$$58\!\cdots\!27$$$$T^{7} +$$$$23\!\cdots\!20$$$$T^{8} + 4623250969555980215 T^{9} + 8223725397576 T^{10} + 2058621 T^{11} + T^{12}$$
$41$ $$15\!\cdots\!64$$$$+$$$$47\!\cdots\!12$$$$T^{2} +$$$$13\!\cdots\!56$$$$T^{4} +$$$$13\!\cdots\!60$$$$T^{6} +$$$$46\!\cdots\!08$$$$T^{8} + 44050323101928 T^{10} + T^{12}$$
$43$ $$($$$$51\!\cdots\!00$$$$+$$$$67\!\cdots\!20$$$$T +$$$$21\!\cdots\!90$$$$T^{2} + 66060179371490808637 T^{3} - 28688639889699 T^{4} - 3860661 T^{5} + T^{6} )^{2}$$
$47$ $$18\!\cdots\!00$$$$+$$$$13\!\cdots\!00$$$$T -$$$$32\!\cdots\!20$$$$T^{2} -$$$$27\!\cdots\!60$$$$T^{3} +$$$$34\!\cdots\!04$$$$T^{4} +$$$$20\!\cdots\!68$$$$T^{5} -$$$$51\!\cdots\!44$$$$T^{6} -$$$$65\!\cdots\!72$$$$T^{7} +$$$$27\!\cdots\!28$$$$T^{8} +$$$$11\!\cdots\!88$$$$T^{9} - 44822205254640 T^{10} - 12088194 T^{11} + T^{12}$$
$53$ $$11\!\cdots\!56$$$$-$$$$57\!\cdots\!88$$$$T +$$$$54\!\cdots\!04$$$$T^{2} +$$$$95\!\cdots\!20$$$$T^{3} +$$$$63\!\cdots\!84$$$$T^{4} +$$$$19\!\cdots\!28$$$$T^{5} +$$$$15\!\cdots\!49$$$$T^{6} +$$$$28\!\cdots\!23$$$$T^{7} +$$$$29\!\cdots\!86$$$$T^{8} + 30645411150217665987 T^{9} + 216103066924230 T^{10} + 5506743 T^{11} + T^{12}$$
$59$ $$26\!\cdots\!24$$$$+$$$$21\!\cdots\!72$$$$T -$$$$49\!\cdots\!40$$$$T^{2} -$$$$45\!\cdots\!36$$$$T^{3} +$$$$72\!\cdots\!24$$$$T^{4} +$$$$47\!\cdots\!00$$$$T^{5} -$$$$55\!\cdots\!09$$$$T^{6} -$$$$28\!\cdots\!77$$$$T^{7} +$$$$34\!\cdots\!76$$$$T^{8} +$$$$51\!\cdots\!99$$$$T^{9} - 661561622457132 T^{10} - 7511901 T^{11} + T^{12}$$
$61$ $$93\!\cdots\!00$$$$+$$$$47\!\cdots\!20$$$$T +$$$$91\!\cdots\!32$$$$T^{2} +$$$$49\!\cdots\!00$$$$T^{3} -$$$$33\!\cdots\!24$$$$T^{4} -$$$$35\!\cdots\!80$$$$T^{5} +$$$$16\!\cdots\!40$$$$T^{6} +$$$$20\!\cdots\!64$$$$T^{7} +$$$$16\!\cdots\!20$$$$T^{8} -$$$$30\!\cdots\!16$$$$T^{9} - 348892366584624 T^{10} + 37215576 T^{11} + T^{12}$$
$67$ $$65\!\cdots\!44$$$$+$$$$62\!\cdots\!56$$$$T +$$$$95\!\cdots\!12$$$$T^{2} +$$$$70\!\cdots\!20$$$$T^{3} +$$$$79\!\cdots\!44$$$$T^{4} +$$$$51\!\cdots\!60$$$$T^{5} +$$$$34\!\cdots\!61$$$$T^{6} +$$$$12\!\cdots\!93$$$$T^{7} +$$$$45\!\cdots\!62$$$$T^{8} +$$$$80\!\cdots\!65$$$$T^{9} + 2671447791162786 T^{10} + 36824553 T^{11} + T^{12}$$
$71$ $$( -$$$$52\!\cdots\!20$$$$+$$$$24\!\cdots\!24$$$$T +$$$$77\!\cdots\!36$$$$T^{2} -$$$$30\!\cdots\!64$$$$T^{3} - 2133160405822764 T^{4} + 15005778 T^{5} + T^{6} )^{2}$$
$73$ $$94\!\cdots\!00$$$$-$$$$22\!\cdots\!00$$$$T +$$$$11\!\cdots\!20$$$$T^{2} +$$$$15\!\cdots\!20$$$$T^{3} -$$$$10\!\cdots\!84$$$$T^{4} -$$$$99\!\cdots\!68$$$$T^{5} +$$$$97\!\cdots\!99$$$$T^{6} +$$$$33\!\cdots\!67$$$$T^{7} -$$$$17\!\cdots\!20$$$$T^{8} +$$$$60\!\cdots\!15$$$$T^{9} + 2949383153409456 T^{10} - 95080185 T^{11} + T^{12}$$
$79$ $$47\!\cdots\!25$$$$+$$$$21\!\cdots\!40$$$$T +$$$$86\!\cdots\!91$$$$T^{2} +$$$$74\!\cdots\!60$$$$T^{3} +$$$$57\!\cdots\!34$$$$T^{4} +$$$$18\!\cdots\!36$$$$T^{5} +$$$$74\!\cdots\!95$$$$T^{6} +$$$$86\!\cdots\!36$$$$T^{7} +$$$$69\!\cdots\!18$$$$T^{8} +$$$$35\!\cdots\!24$$$$T^{9} + 3118285003434843 T^{10} - 8514456 T^{11} + T^{12}$$
$83$ $$78\!\cdots\!04$$$$+$$$$88\!\cdots\!56$$$$T^{2} +$$$$31\!\cdots\!56$$$$T^{4} +$$$$30\!\cdots\!71$$$$T^{6} +$$$$10\!\cdots\!83$$$$T^{8} + 17102803239551805 T^{10} + T^{12}$$
$89$ $$11\!\cdots\!96$$$$+$$$$13\!\cdots\!08$$$$T +$$$$33\!\cdots\!76$$$$T^{2} -$$$$22\!\cdots\!96$$$$T^{3} -$$$$16\!\cdots\!72$$$$T^{4} +$$$$14\!\cdots\!40$$$$T^{5} +$$$$70\!\cdots\!08$$$$T^{6} -$$$$75\!\cdots\!00$$$$T^{7} +$$$$56\!\cdots\!88$$$$T^{8} +$$$$82\!\cdots\!76$$$$T^{9} - 7665223344778872 T^{10} - 83038554 T^{11} + T^{12}$$
$97$ $$53\!\cdots\!00$$$$+$$$$16\!\cdots\!00$$$$T^{2} +$$$$64\!\cdots\!00$$$$T^{4} +$$$$99\!\cdots\!75$$$$T^{6} +$$$$66\!\cdots\!19$$$$T^{8} + 17993651605183569 T^{10} + T^{12}$$