# Properties

 Label 300.8.i.c Level $300$ Weight $8$ Character orbit 300.i Analytic conductor $93.716$ Analytic rank $0$ Dimension $16$ CM no Inner twists $8$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$300 = 2^{2} \cdot 3 \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$8$$ Character orbit: $$[\chi]$$ $$=$$ 300.i (of order $$4$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$93.7155076452$$ Analytic rank: $$0$$ Dimension: $$16$$ Relative dimension: $$8$$ over $$\Q(i)$$ Coefficient field: $$\mathbb{Q}[x]/(x^{16} - \cdots)$$ Defining polynomial: $$x^{16} - 8 x^{15} + 132 x^{14} - 784 x^{13} + 5524236 x^{12} - 33135588 x^{11} - 49457570 x^{10} + 551013008 x^{9} + 10299424335549 x^{8} - 41201367881396 x^{7} - 506171935260830 x^{6} + 1662723272081664 x^{5} + 7501118196131608896 x^{4} - 15005055667263471212 x^{3} - 559914287081811190962 x^{2} + 567417098903075834864 x + 1831773137685647789586721$$ Coefficient ring: $$\Z[a_1, \ldots, a_{19}]$$ Coefficient ring index: $$2^{44}\cdot 3^{12}\cdot 5^{8}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q -\beta_{6} q^{3} + ( 6 \beta_{4} + 3 \beta_{5} - \beta_{11} ) q^{7} + ( -573 \beta_{1} - \beta_{7} ) q^{9} +O(q^{10})$$ $$q -\beta_{6} q^{3} + ( 6 \beta_{4} + 3 \beta_{5} - \beta_{11} ) q^{7} + ( -573 \beta_{1} - \beta_{7} ) q^{9} + \beta_{8} q^{11} + ( -16 \beta_{6} + 8 \beta_{10} - 26 \beta_{12} ) q^{13} + ( -\beta_{3} - 60 \beta_{4} + 15 \beta_{5} ) q^{17} + ( -7172 \beta_{1} + 8 \beta_{7} - 4 \beta_{9} ) q^{19} + ( 13494 - 3 \beta_{8} + 9 \beta_{13} + \beta_{15} ) q^{21} + ( \beta_{2} + 24 \beta_{6} + 6 \beta_{10} ) q^{23} + ( 3 \beta_{3} + 489 \beta_{4} + 138 \beta_{5} + 117 \beta_{11} ) q^{27} + ( -36 \beta_{7} - 9 \beta_{9} + 26 \beta_{14} ) q^{29} + ( 54544 - 44 \beta_{13} - 22 \beta_{15} ) q^{31} + ( -3 \beta_{2} - 312 \beta_{6} - 105 \beta_{10} + 612 \beta_{12} ) q^{33} + ( -1464 \beta_{4} - 732 \beta_{5} - 322 \beta_{11} ) q^{37} + ( 65568 \beta_{1} - 24 \beta_{7} - 26 \beta_{9} - 78 \beta_{14} ) q^{39} + ( 44 \beta_{8} + 252 \beta_{13} - 63 \beta_{15} ) q^{41} + ( 1994 \beta_{6} - 997 \beta_{10} - 1326 \beta_{12} ) q^{43} + ( 7 \beta_{3} - 1632 \beta_{4} + 408 \beta_{5} ) q^{47} + ( 460795 \beta_{1} + 70 \beta_{7} - 35 \beta_{9} ) q^{49} + ( 123660 - 117 \beta_{8} + 60 \beta_{13} - 204 \beta_{15} ) q^{51} + ( -8 \beta_{2} - 7248 \beta_{6} - 1812 \beta_{10} ) q^{53} + ( -36 \beta_{3} + 6308 \beta_{4} + 7092 \beta_{5} - 1404 \beta_{11} ) q^{57} + ( -432 \beta_{7} - 108 \beta_{9} + 91 \beta_{14} ) q^{59} + ( 797054 - 302 \beta_{13} - 151 \beta_{15} ) q^{61} + ( 39 \beta_{2} - 12930 \beta_{6} + 1122 \beta_{10} - 666 \beta_{12} ) q^{63} + ( -5038 \beta_{4} - 2519 \beta_{5} + 6266 \beta_{11} ) q^{67} + ( 49140 \beta_{1} - 87 \beta_{7} - 204 \beta_{9} + 117 \beta_{14} ) q^{69} + ( -46 \beta_{8} + 720 \beta_{13} - 180 \beta_{15} ) q^{71} + ( 38632 \beta_{6} - 19316 \beta_{10} + 6292 \beta_{12} ) q^{73} + ( 14 \beta_{3} - 34728 \beta_{4} + 8682 \beta_{5} ) q^{77} + ( -1300936 \beta_{1} + 444 \beta_{7} - 222 \beta_{9} ) q^{79} + ( 668511 + 702 \beta_{8} + 312 \beta_{13} + 495 \beta_{15} ) q^{81} + ( -6 \beta_{2} - 87588 \beta_{6} - 21897 \beta_{10} ) q^{83} + ( 159 \beta_{3} + 1632 \beta_{4} + 20679 \beta_{5} - 12753 \beta_{11} ) q^{87} + ( 504 \beta_{7} + 126 \beta_{9} - 872 \beta_{14} ) q^{89} + ( -2468544 + 416 \beta_{13} + 208 \beta_{15} ) q^{91} + ( -198 \beta_{2} - 59296 \beta_{6} - 39006 \beta_{10} - 7722 \beta_{12} ) q^{93} + ( 43688 \beta_{4} + 21844 \beta_{5} - 4412 \beta_{11} ) q^{97} + ( -1437480 \beta_{1} + 108 \beta_{7} + 1224 \beta_{9} + 1485 \beta_{14} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$16q + O(q^{10})$$ $$16q + 215904q^{21} + 872704q^{31} + 1978560q^{51} + 12752864q^{61} + 10696176q^{81} - 39496704q^{91} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{16} - 8 x^{15} + 132 x^{14} - 784 x^{13} + 5524236 x^{12} - 33135588 x^{11} - 49457570 x^{10} + 551013008 x^{9} + 10299424335549 x^{8} - 41201367881396 x^{7} - 506171935260830 x^{6} + 1662723272081664 x^{5} + 7501118196131608896 x^{4} - 15005055667263471212 x^{3} - 559914287081811190962 x^{2} + 567417098903075834864 x + 1831773137685647789586721$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$-127077663550772 \nu^{14} + 889543644855404 \nu^{13} - 18575791285636901 \nu^{12} + 99890680330701154 \nu^{11} - 729761235766306475869 \nu^{10} + 3647911715052036672562 \nu^{9} - 9117910351391012676864 \nu^{8} + 14585233568523692863986 \nu^{7} - 1492651709261143673520544469 \nu^{6} + 4477919399634508344917218354 \nu^{5} + 35031584283286251950704707794 \nu^{4} - 77526357845150826006522794058 \nu^{3} - 1611312411360702025050329347065403 \nu^{2} + 1611351920858187960485940331725082 \nu + 51696753313793325928362747543089343$$$$)/$$$$73\!\cdots\!00$$ $$\beta_{2}$$ $$=$$ $$($$$$22\!\cdots\!41$$$$\nu^{14} -$$$$15\!\cdots\!87$$$$\nu^{13} +$$$$46\!\cdots\!92$$$$\nu^{12} -$$$$27\!\cdots\!21$$$$\nu^{11} +$$$$10\!\cdots\!59$$$$\nu^{10} -$$$$50\!\cdots\!76$$$$\nu^{9} +$$$$18\!\cdots\!87$$$$\nu^{8} -$$$$72\!\cdots\!97$$$$\nu^{7} +$$$$10\!\cdots\!81$$$$\nu^{6} -$$$$30\!\cdots\!94$$$$\nu^{5} +$$$$24\!\cdots\!23$$$$\nu^{4} -$$$$47\!\cdots\!31$$$$\nu^{3} +$$$$24\!\cdots\!06$$$$\nu^{2} -$$$$24\!\cdots\!83$$$$\nu +$$$$95\!\cdots\!97$$$$)/$$$$11\!\cdots\!00$$ $$\beta_{3}$$ $$=$$ $$($$$$-$$$$19\!\cdots\!95$$$$\nu^{14} +$$$$13\!\cdots\!65$$$$\nu^{13} +$$$$77\!\cdots\!55$$$$\nu^{12} -$$$$48\!\cdots\!75$$$$\nu^{11} -$$$$87\!\cdots\!31$$$$\nu^{10} +$$$$44\!\cdots\!75$$$$\nu^{9} +$$$$47\!\cdots\!60$$$$\nu^{8} -$$$$19\!\cdots\!85$$$$\nu^{7} -$$$$12\!\cdots\!35$$$$\nu^{6} +$$$$38\!\cdots\!61$$$$\nu^{5} +$$$$55\!\cdots\!60$$$$\nu^{4} -$$$$11\!\cdots\!25$$$$\nu^{3} -$$$$54\!\cdots\!10$$$$\nu^{2} +$$$$54\!\cdots\!80$$$$\nu +$$$$18\!\cdots\!67$$$$)/$$$$48\!\cdots\!00$$ $$\beta_{4}$$ $$=$$ $$($$$$-$$$$36\!\cdots\!52$$$$\nu^{15} -$$$$94\!\cdots\!10$$$$\nu^{14} +$$$$21\!\cdots\!40$$$$\nu^{13} -$$$$11\!\cdots\!75$$$$\nu^{12} -$$$$14\!\cdots\!58$$$$\nu^{11} -$$$$49\!\cdots\!13$$$$\nu^{10} +$$$$89\!\cdots\!90$$$$\nu^{9} +$$$$25\!\cdots\!70$$$$\nu^{8} -$$$$32\!\cdots\!90$$$$\nu^{7} -$$$$76\!\cdots\!57$$$$\nu^{6} +$$$$10\!\cdots\!94$$$$\nu^{5} +$$$$80\!\cdots\!40$$$$\nu^{4} -$$$$19\!\cdots\!70$$$$\nu^{3} -$$$$32\!\cdots\!25$$$$\nu^{2} +$$$$32\!\cdots\!66$$$$\nu +$$$$43\!\cdots\!15$$$$)/$$$$14\!\cdots\!00$$ $$\beta_{5}$$ $$=$$ $$($$$$-$$$$36\!\cdots\!52$$$$\nu^{15} +$$$$51\!\cdots\!40$$$$\nu^{14} +$$$$11\!\cdots\!90$$$$\nu^{13} -$$$$85\!\cdots\!25$$$$\nu^{12} -$$$$14\!\cdots\!08$$$$\nu^{11} +$$$$25\!\cdots\!17$$$$\nu^{10} +$$$$51\!\cdots\!40$$$$\nu^{9} -$$$$56\!\cdots\!30$$$$\nu^{8} -$$$$29\!\cdots\!40$$$$\nu^{7} +$$$$39\!\cdots\!93$$$$\nu^{6} +$$$$66\!\cdots\!64$$$$\nu^{5} -$$$$69\!\cdots\!60$$$$\nu^{4} -$$$$16\!\cdots\!20$$$$\nu^{3} +$$$$16\!\cdots\!75$$$$\nu^{2} +$$$$27\!\cdots\!66$$$$\nu -$$$$23\!\cdots\!95$$$$)/$$$$36\!\cdots\!00$$ $$\beta_{6}$$ $$=$$ $$($$$$23\!\cdots\!32$$$$\nu^{15} +$$$$36\!\cdots\!63$$$$\nu^{14} +$$$$68\!\cdots\!19$$$$\nu^{13} +$$$$10\!\cdots\!16$$$$\nu^{12} +$$$$97\!\cdots\!25$$$$\nu^{11} +$$$$56\!\cdots\!85$$$$\nu^{10} +$$$$98\!\cdots\!12$$$$\nu^{9} +$$$$56\!\cdots\!01$$$$\nu^{8} +$$$$11\!\cdots\!09$$$$\nu^{7} +$$$$92\!\cdots\!55$$$$\nu^{6} +$$$$21\!\cdots\!74$$$$\nu^{5} +$$$$91\!\cdots\!89$$$$\nu^{4} +$$$$33\!\cdots\!07$$$$\nu^{3} +$$$$17\!\cdots\!98$$$$\nu^{2} +$$$$12\!\cdots\!75$$$$\nu +$$$$38\!\cdots\!71$$$$)/$$$$23\!\cdots\!00$$ $$\beta_{7}$$ $$=$$ $$($$$$-$$$$21\!\cdots\!94$$$$\nu^{15} +$$$$88\!\cdots\!83$$$$\nu^{14} -$$$$61\!\cdots\!87$$$$\nu^{13} -$$$$39\!\cdots\!02$$$$\nu^{12} -$$$$34\!\cdots\!95$$$$\nu^{11} +$$$$30\!\cdots\!35$$$$\nu^{10} -$$$$23\!\cdots\!58$$$$\nu^{9} -$$$$29\!\cdots\!03$$$$\nu^{8} -$$$$77\!\cdots\!83$$$$\nu^{7} +$$$$13\!\cdots\!25$$$$\nu^{6} -$$$$20\!\cdots\!68$$$$\nu^{5} -$$$$39\!\cdots\!51$$$$\nu^{4} -$$$$59\!\cdots\!11$$$$\nu^{3} -$$$$18\!\cdots\!56$$$$\nu^{2} -$$$$16\!\cdots\!65$$$$\nu -$$$$11\!\cdots\!67$$$$)/$$$$11\!\cdots\!00$$ $$\beta_{8}$$ $$=$$ $$($$$$40\!\cdots\!14$$$$\nu^{15} -$$$$30\!\cdots\!55$$$$\nu^{14} +$$$$68\!\cdots\!91$$$$\nu^{13} -$$$$43\!\cdots\!74$$$$\nu^{12} -$$$$85\!\cdots\!21$$$$\nu^{11} +$$$$47\!\cdots\!01$$$$\nu^{10} -$$$$21\!\cdots\!30$$$$\nu^{9} +$$$$94\!\cdots\!39$$$$\nu^{8} -$$$$43\!\cdots\!61$$$$\nu^{7} +$$$$15\!\cdots\!31$$$$\nu^{6} -$$$$12\!\cdots\!48$$$$\nu^{5} +$$$$30\!\cdots\!95$$$$\nu^{4} -$$$$51\!\cdots\!37$$$$\nu^{3} +$$$$77\!\cdots\!28$$$$\nu^{2} -$$$$25\!\cdots\!03$$$$\nu +$$$$12\!\cdots\!15$$$$)/$$$$12\!\cdots\!00$$ $$\beta_{9}$$ $$=$$ $$($$$$-$$$$21\!\cdots\!94$$$$\nu^{15} -$$$$17\!\cdots\!01$$$$\nu^{14} -$$$$42\!\cdots\!99$$$$\nu^{13} +$$$$18\!\cdots\!26$$$$\nu^{12} -$$$$48\!\cdots\!07$$$$\nu^{11} -$$$$54\!\cdots\!33$$$$\nu^{10} -$$$$19\!\cdots\!94$$$$\nu^{9} +$$$$90\!\cdots\!89$$$$\nu^{8} -$$$$12\!\cdots\!91$$$$\nu^{7} -$$$$16\!\cdots\!43$$$$\nu^{6} -$$$$19\!\cdots\!80$$$$\nu^{5} +$$$$93\!\cdots\!17$$$$\nu^{4} -$$$$85\!\cdots\!87$$$$\nu^{3} +$$$$39\!\cdots\!28$$$$\nu^{2} -$$$$21\!\cdots\!61$$$$\nu +$$$$22\!\cdots\!29$$$$)/$$$$59\!\cdots\!00$$ $$\beta_{10}$$ $$=$$ $$($$$$-$$$$47\!\cdots\!64$$$$\nu^{15} +$$$$56\!\cdots\!83$$$$\nu^{14} -$$$$18\!\cdots\!01$$$$\nu^{13} +$$$$11\!\cdots\!76$$$$\nu^{12} -$$$$21\!\cdots\!79$$$$\nu^{11} +$$$$22\!\cdots\!21$$$$\nu^{10} -$$$$36\!\cdots\!48$$$$\nu^{9} +$$$$58\!\cdots\!61$$$$\nu^{8} -$$$$29\!\cdots\!71$$$$\nu^{7} +$$$$23\!\cdots\!59$$$$\nu^{6} -$$$$55\!\cdots\!54$$$$\nu^{5} +$$$$93\!\cdots\!49$$$$\nu^{4} -$$$$12\!\cdots\!33$$$$\nu^{3} +$$$$40\!\cdots\!98$$$$\nu^{2} -$$$$25\!\cdots\!17$$$$\nu +$$$$38\!\cdots\!11$$$$)/$$$$11\!\cdots\!00$$ $$\beta_{11}$$ $$=$$ $$($$$$-$$$$14\!\cdots\!52$$$$\nu^{15} +$$$$11\!\cdots\!90$$$$\nu^{14} +$$$$15\!\cdots\!40$$$$\nu^{13} -$$$$10\!\cdots\!25$$$$\nu^{12} -$$$$72\!\cdots\!58$$$$\nu^{11} +$$$$40\!\cdots\!57$$$$\nu^{10} +$$$$87\!\cdots\!90$$$$\nu^{9} -$$$$39\!\cdots\!30$$$$\nu^{8} -$$$$11\!\cdots\!90$$$$\nu^{7} +$$$$40\!\cdots\!93$$$$\nu^{6} +$$$$13\!\cdots\!74$$$$\nu^{5} -$$$$33\!\cdots\!60$$$$\nu^{4} -$$$$51\!\cdots\!70$$$$\nu^{3} +$$$$77\!\cdots\!25$$$$\nu^{2} +$$$$56\!\cdots\!66$$$$\nu -$$$$28\!\cdots\!75$$$$)/$$$$92\!\cdots\!00$$ $$\beta_{12}$$ $$=$$ $$($$$$22\!\cdots\!52$$$$\nu^{15} -$$$$17\!\cdots\!90$$$$\nu^{14} +$$$$13\!\cdots\!40$$$$\nu^{13} -$$$$87\!\cdots\!95$$$$\nu^{12} +$$$$11\!\cdots\!98$$$$\nu^{11} -$$$$62\!\cdots\!57$$$$\nu^{10} +$$$$54\!\cdots\!70$$$$\nu^{9} -$$$$24\!\cdots\!70$$$$\nu^{8} +$$$$16\!\cdots\!10$$$$\nu^{7} -$$$$58\!\cdots\!73$$$$\nu^{6} +$$$$82\!\cdots\!86$$$$\nu^{5} -$$$$20\!\cdots\!20$$$$\nu^{4} +$$$$67\!\cdots\!30$$$$\nu^{3} -$$$$10\!\cdots\!25$$$$\nu^{2} +$$$$32\!\cdots\!54$$$$\nu -$$$$16\!\cdots\!05$$$$)/$$$$92\!\cdots\!00$$ $$\beta_{13}$$ $$=$$ $$($$$$73\!\cdots\!10$$$$\nu^{15} +$$$$10\!\cdots\!05$$$$\nu^{14} +$$$$20\!\cdots\!19$$$$\nu^{13} +$$$$29\!\cdots\!74$$$$\nu^{12} +$$$$24\!\cdots\!51$$$$\nu^{11} +$$$$56\!\cdots\!73$$$$\nu^{10} +$$$$12\!\cdots\!90$$$$\nu^{9} +$$$$11\!\cdots\!91$$$$\nu^{8} +$$$$99\!\cdots\!51$$$$\nu^{7} +$$$$65\!\cdots\!99$$$$\nu^{6} -$$$$46\!\cdots\!08$$$$\nu^{5} +$$$$10\!\cdots\!15$$$$\nu^{4} -$$$$10\!\cdots\!13$$$$\nu^{3} +$$$$13\!\cdots\!32$$$$\nu^{2} -$$$$73\!\cdots\!87$$$$\nu +$$$$38\!\cdots\!99$$$$)/$$$$24\!\cdots\!00$$ $$\beta_{14}$$ $$=$$ $$($$$$-$$$$54\!\cdots\!50$$$$\nu^{15} +$$$$41\!\cdots\!25$$$$\nu^{14} -$$$$88\!\cdots\!49$$$$\nu^{13} +$$$$56\!\cdots\!06$$$$\nu^{12} -$$$$35\!\cdots\!21$$$$\nu^{11} +$$$$19\!\cdots\!37$$$$\nu^{10} -$$$$31\!\cdots\!70$$$$\nu^{9} +$$$$13\!\cdots\!99$$$$\nu^{8} -$$$$71\!\cdots\!01$$$$\nu^{7} +$$$$24\!\cdots\!31$$$$\nu^{6} -$$$$18\!\cdots\!12$$$$\nu^{5} +$$$$45\!\cdots\!35$$$$\nu^{4} -$$$$58\!\cdots\!17$$$$\nu^{3} +$$$$88\!\cdots\!28$$$$\nu^{2} +$$$$23\!\cdots\!57$$$$\nu -$$$$11\!\cdots\!49$$$$)/$$$$13\!\cdots\!00$$ $$\beta_{15}$$ $$=$$ $$($$$$-$$$$73\!\cdots\!10$$$$\nu^{15} +$$$$36\!\cdots\!35$$$$\nu^{14} -$$$$53\!\cdots\!99$$$$\nu^{13} +$$$$65\!\cdots\!26$$$$\nu^{12} -$$$$30\!\cdots\!11$$$$\nu^{11} +$$$$15\!\cdots\!67$$$$\nu^{10} -$$$$11\!\cdots\!30$$$$\nu^{9} +$$$$24\!\cdots\!69$$$$\nu^{8} -$$$$24\!\cdots\!71$$$$\nu^{7} +$$$$14\!\cdots\!01$$$$\nu^{6} -$$$$16\!\cdots\!32$$$$\nu^{5} +$$$$21\!\cdots\!85$$$$\nu^{4} +$$$$39\!\cdots\!73$$$$\nu^{3} +$$$$23\!\cdots\!28$$$$\nu^{2} +$$$$69\!\cdots\!67$$$$\nu -$$$$30\!\cdots\!99$$$$)/$$$$12\!\cdots\!00$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$-4 \beta_{15} + 12 \beta_{14} + 16 \beta_{13} - 9 \beta_{12} - 9 \beta_{11} + 36 \beta_{10} - 4 \beta_{9} - 12 \beta_{8} - 16 \beta_{7} - 72 \beta_{6} - 36 \beta_{5} - 72 \beta_{4} + 1440$$$$)/2880$$ $$\nu^{2}$$ $$=$$ $$($$$$-4 \beta_{15} + 12 \beta_{14} + 16 \beta_{13} - 9 \beta_{12} - 9 \beta_{11} + 36 \beta_{10} - 124 \beta_{9} - 12 \beta_{8} + 224 \beta_{7} - 72 \beta_{6} - 348 \beta_{5} + 1176 \beta_{4} + 24 \beta_{3} - 3247200 \beta_{1} - 36000$$$$)/2880$$ $$\nu^{3}$$ $$=$$ $$($$$$-3275 \beta_{15} - 6675 \beta_{14} + 13100 \beta_{13} - 6561 \beta_{12} + 6669 \beta_{11} + 72144 \beta_{10} + 3335 \beta_{9} - 6225 \beta_{8} + 13880 \beta_{7} - 144288 \beta_{6} + 72342 \beta_{5} + 146088 \beta_{4} + 18 \beta_{3} - 2435400 \beta_{1} - 27360$$$$)/1440$$ $$\nu^{4}$$ $$=$$ $$($$$$-283696 \beta_{15} - 26712 \beta_{14} - 488816 \beta_{13} - 26235 \beta_{12} + 26685 \beta_{11} - 886740 \beta_{10} + 22704 \beta_{9} - 24888 \beta_{8} + 36816 \beta_{7} - 5278200 \beta_{6} + 297828 \beta_{5} + 550728 \beta_{4} - 576 \beta_{3} + 60240 \beta_{2} + 243540000 \beta_{1} - 3972006720$$$$)/2880$$ $$\nu^{5}$$ $$=$$ $$($$$$10613830 \beta_{15} - 13509990 \beta_{14} - 46514320 \beta_{13} - 2093229 \beta_{12} - 3151629 \beta_{11} - 343252284 \beta_{10} + 9638130 \beta_{9} + 16734990 \beta_{8} + 38415720 \beta_{7} + 668875368 \beta_{6} + 328755984 \beta_{5} + 657394968 \beta_{4} - 1500 \beta_{3} + 150600 \beta_{2} + 616968000 \beta_{1} - 9929925360$$$$)/2880$$ $$\nu^{6}$$ $$=$$ $$($$$$21160307 \beta_{15} - 10115796 \beta_{14} - 8534978 \beta_{13} - 1553526 \beta_{12} - 2380401 \beta_{11} - 237786696 \beta_{10} + 124793107 \beta_{9} + 12566796 \beta_{8} - 206368622 \beta_{7} + 581348592 \beta_{6} - 1019694708 \beta_{5} + 5557004208 \beta_{4} - 30000492 \beta_{3} - 903600 \beta_{2} + 1293418615500 \beta_{1} + 186206125500$$$$)/720$$ $$\nu^{7}$$ $$=$$ $$($$$$13185881800 \beta_{15} + 23022350100 \beta_{14} - 51663833200 \beta_{13} - 43975205937 \beta_{12} + 44415598383 \beta_{11} - 634831558152 \beta_{10} - 16848910260 \beta_{9} + 15078770400 \beta_{8} - 77272735680 \beta_{7} + 1271205671304 \beta_{6} - 709014588120 \beta_{5} - 1311679252728 \beta_{4} - 420001596 \beta_{3} - 13177500 \beta_{2} + 18105695546400 \beta_{1} + 2641640431680$$$$)/2880$$ $$\nu^{8}$$ $$=$$ $$($$$$265210410416 \beta_{15} + 30759388752 \beta_{14} + 426689854336 \beta_{13} - 58623961935 \beta_{12} + 59235629985 \beta_{11} + 3318089103500 \beta_{10} - 79608829184 \beta_{9} + 20026814448 \beta_{8} + 10988071264 \beta_{7} + 18343528745960 \beta_{6} - 734190478508 \beta_{5} - 2602751384728 \beta_{4} + 4480385696 \beta_{3} - 73527153280 \beta_{2} - 604366103880000 \beta_{1} + 2340591862918560$$$$)/960$$ $$\nu^{9}$$ $$=$$ $$($$$$-13499174754773 \beta_{15} + 8315363193369 \beta_{14} + 64034289011192 \beta_{13} + 64482795692073 \beta_{12} + 58120206802353 \beta_{11} + 686674299316968 \beta_{10} - 8445602785883 \beta_{9} - 16915672608369 \beta_{8} - 31529172278972 \beta_{7} - 1204740241387776 \beta_{6} - 561727386202338 \beta_{5} - 1131430822810776 \beta_{4} + 31502605050 \beta_{3} - 496268435880 \beta_{2} - 4133786987325600 \beta_{1} + 15791049300616920$$$$)/1440$$ $$\nu^{10}$$ $$=$$ $$($$$$-571923207396520 \beta_{15} + 82461262578060 \beta_{14} - 231187507188920 \beta_{13} + 646146952696701 \beta_{12} + 579869199564201 \beta_{11} + 4356221165607996 \beta_{10} - 1201859761862920 \beta_{9} - 169606977414060 \beta_{8} + 1922852120609720 \beta_{7} - 22203548183067192 \beta_{6} + 19756521055186044 \beta_{5} - 112684807466123352 \beta_{4} + 388753003436520 \beta_{3} + 39722295621600 \beta_{2} - 9741819141537714000 \beta_{1} - 3970963705702230000$$$$)/2880$$ $$\nu^{11}$$ $$=$$ $$($$$$-19967617936705750 \beta_{15} - 51896876595944250 \beta_{14} + 65832619332655000 \beta_{13} + 229813019230996863 \beta_{12} - 283761716334839037 \beta_{11} + 1869923179167693048 \beta_{10} + 43669120605733450 \beta_{9} - 16848646450565250 \beta_{8} + 211651665821557600 \beta_{7} - 3817138576574721096 \beta_{6} + 2560243232946401076 \beta_{5} + 4283545690663316808 \beta_{4} + 2137559351140752 \beta_{3} + 227570733967500 \beta_{2} - 53504020027852218000 \beta_{1} - 22129773784599949920$$$$)/2880$$ $$\nu^{12}$$ $$=$$ $$($$$$-436052253977733442 \beta_{15} - 78071702496132924 \beta_{14} - 715963819693841732 \beta_{13} + 342941899289119785 \beta_{12} - 427236481707660135 \beta_{11} - 8883606091718936340 \beta_{10} + 309086020611578808 \beta_{9} - 24806302932702576 \beta_{8} - 168366630606639768 \beta_{7} - 52370630394383293920 \beta_{6} + 1970738236706241636 \beta_{5} + 13996317891782363616 \beta_{4} - 25696339546374252 \beta_{3} + 166456745697291300 \beta_{2} + 2039684247271435050000 \beta_{1} - 3401029687728582627480$$$$)/720$$ $$\nu^{13}$$ $$=$$ $$($$$$71032215297707471968 \beta_{15} - 12920582727114629904 \beta_{14} - 347910384421000919872 \beta_{13} - 557273473142073257709 \beta_{12} - 396826793995780425429 \beta_{11} - 4649163381743910575484 \beta_{10} + 20678891870658875208 \beta_{9} + 81615804539392079904 \beta_{8} + 45712519367006413392 \beta_{7} + 7475755692236163558648 \beta_{6} + 2954133924287738551284 \beta_{5} + 6180573570994182178968 \beta_{4} - 695891580986386200 \beta_{3} + 4324893313670741760 \beta_{2} + 53727144905852578274400 \beta_{1} - 88138332223415892660960$$$$)/2880$$ $$\nu^{14}$$ $$=$$ $$($$$$2495849132528586359252 \beta_{15} - 85704984282846216756 \beta_{14} + 1552432499469863912992 \beta_{13} - 3921697881540511298469 \beta_{12} - 2751849209151570572469 \beta_{11} - 13515418227810508978764 \beta_{10} + 2430635780910372658132 \beta_{9} + 572809883976797399556 \beta_{8} - 4279083648103017705152 \beta_{7} + 129465264766380728789688 \beta_{6} - 60645105594449251670220 \beta_{5} + 367231722933355847769624 \beta_{4} - 1114845342901192687512 \beta_{3} - 243284772870815828880 \beta_{2} + 18873501650667020810296800 \beta_{1} + 16013957804460283708368000$$$$)/2880$$ $$\nu^{15}$$ $$=$$ $$($$$$7755245763692026128875 \beta_{15} + 64915893156064773679875 \beta_{14} + 12796143577264372922500 \beta_{13} - 314446689066447580961061 \beta_{12} + 515663113930844925111489 \beta_{11} - 2215684596413578582184256 \beta_{10} - 57851971746334811002275 \beta_{9} - 369606243832889424375 \beta_{8} - 283589369792517344539200 \beta_{7} + 4831444017375664678228512 \beta_{6} - 4138252727877283095479214 \beta_{5} - 6446949917308027131678408 \beta_{4} - 4174532358869708680410 \beta_{3} - 950155478388755730000 \beta_{2} + 70304301950170865717151000 \beta_{1} + 60823046839845090456930240$$$$)/1440$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/300\mathbb{Z}\right)^\times$$.

 $$n$$ $$101$$ $$151$$ $$277$$ $$\chi(n)$$ $$-1$$ $$1$$ $$-\beta_{1}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
257.1
 27.4835 + 30.5542i −19.4848 − 16.4140i 20.4848 + 16.4140i −26.4835 − 23.4128i −26.4835 − 30.5542i 20.4848 + 23.5555i −19.4848 − 23.5555i 27.4835 + 23.4128i 27.4835 − 30.5542i −19.4848 + 16.4140i 20.4848 − 16.4140i −26.4835 + 23.4128i −26.4835 + 30.5542i 20.4848 − 23.5555i −19.4848 + 23.5555i 27.4835 − 23.4128i
0 −46.0669 + 8.05216i 0 0 0 −565.187 565.187i 0 2057.33 741.876i 0
257.2 0 −40.5650 23.2698i 0 0 0 208.115 + 208.115i 0 1104.03 + 1887.88i 0
257.3 0 −23.2698 40.5650i 0 0 0 −208.115 208.115i 0 −1104.03 + 1887.88i 0
257.4 0 −8.05216 + 46.0669i 0 0 0 −565.187 565.187i 0 −2057.33 741.876i 0
257.5 0 8.05216 46.0669i 0 0 0 565.187 + 565.187i 0 −2057.33 741.876i 0
257.6 0 23.2698 + 40.5650i 0 0 0 208.115 + 208.115i 0 −1104.03 + 1887.88i 0
257.7 0 40.5650 + 23.2698i 0 0 0 −208.115 208.115i 0 1104.03 + 1887.88i 0
257.8 0 46.0669 8.05216i 0 0 0 565.187 + 565.187i 0 2057.33 741.876i 0
293.1 0 −46.0669 8.05216i 0 0 0 −565.187 + 565.187i 0 2057.33 + 741.876i 0
293.2 0 −40.5650 + 23.2698i 0 0 0 208.115 208.115i 0 1104.03 1887.88i 0
293.3 0 −23.2698 + 40.5650i 0 0 0 −208.115 + 208.115i 0 −1104.03 1887.88i 0
293.4 0 −8.05216 46.0669i 0 0 0 −565.187 + 565.187i 0 −2057.33 + 741.876i 0
293.5 0 8.05216 + 46.0669i 0 0 0 565.187 565.187i 0 −2057.33 + 741.876i 0
293.6 0 23.2698 40.5650i 0 0 0 208.115 208.115i 0 −1104.03 1887.88i 0
293.7 0 40.5650 23.2698i 0 0 0 −208.115 + 208.115i 0 1104.03 1887.88i 0
293.8 0 46.0669 + 8.05216i 0 0 0 565.187 565.187i 0 2057.33 + 741.876i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 293.8 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
5.b even 2 1 inner
5.c odd 4 2 inner
15.d odd 2 1 inner
15.e even 4 2 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 300.8.i.c 16
3.b odd 2 1 inner 300.8.i.c 16
5.b even 2 1 inner 300.8.i.c 16
5.c odd 4 2 inner 300.8.i.c 16
15.d odd 2 1 inner 300.8.i.c 16
15.e even 4 2 inner 300.8.i.c 16

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
300.8.i.c 16 1.a even 1 1 trivial
300.8.i.c 16 3.b odd 2 1 inner
300.8.i.c 16 5.b even 2 1 inner
300.8.i.c 16 5.c odd 4 2 inner
300.8.i.c 16 15.d odd 2 1 inner
300.8.i.c 16 15.e even 4 2 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{7}^{8} + 415661203008 T_{7}^{4} +$$$$30\!\cdots\!16$$ acting on $$S_{8}^{\mathrm{new}}(300, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{16}$$
$3$ $$52\!\cdots\!21$$$$- 61173549603433732284 T^{4} + 11211740220006 T^{8} - 2674044 T^{12} + T^{16}$$
$5$ $$T^{16}$$
$7$ $$($$$$30\!\cdots\!16$$$$+ 415661203008 T^{4} + T^{8} )^{2}$$
$11$ $$( 273576376627200 + 54637920 T^{2} + T^{4} )^{4}$$
$13$ $$($$$$43\!\cdots\!76$$$$+ 38009942833102848 T^{4} + T^{8} )^{2}$$
$17$ $$($$$$41\!\cdots\!00$$$$+ 1475216350887744000 T^{4} + T^{8} )^{2}$$
$19$ $$( 891908011738521856 + 2094567968 T^{2} + T^{4} )^{4}$$
$23$ $$($$$$32\!\cdots\!00$$$$+ 1558411553255220000 T^{4} + T^{8} )^{2}$$
$29$ $$($$$$44\!\cdots\!00$$$$- 44758861920 T^{2} + T^{4} )^{4}$$
$31$ $$( -27149305664 - 109088 T + T^{2} )^{8}$$
$37$ $$($$$$71\!\cdots\!56$$$$+$$$$26\!\cdots\!68$$$$T^{4} + T^{8} )^{2}$$
$41$ $$($$$$23\!\cdots\!00$$$$+ 980340950880 T^{2} + T^{4} )^{4}$$
$43$ $$($$$$27\!\cdots\!16$$$$+$$$$15\!\cdots\!08$$$$T^{4} + T^{8} )^{2}$$
$47$ $$($$$$12\!\cdots\!00$$$$+$$$$91\!\cdots\!00$$$$T^{4} + T^{8} )^{2}$$
$53$ $$($$$$68\!\cdots\!00$$$$+$$$$18\!\cdots\!00$$$$T^{4} + T^{8} )^{2}$$
$59$ $$($$$$14\!\cdots\!00$$$$- 2402493590880 T^{2} + T^{4} )^{4}$$
$61$ $$( -783848281484 - 1594108 T + T^{2} )^{8}$$
$67$ $$($$$$22\!\cdots\!56$$$$+$$$$87\!\cdots\!68$$$$T^{4} + T^{8} )^{2}$$
$71$ $$($$$$41\!\cdots\!00$$$$+ 6188564085120 T^{2} + T^{4} )^{4}$$
$73$ $$($$$$91\!\cdots\!96$$$$+$$$$66\!\cdots\!28$$$$T^{4} + T^{8} )^{2}$$
$79$ $$($$$$18\!\cdots\!16$$$$+ 9519780699392 T^{2} + T^{4} )^{4}$$
$83$ $$($$$$16\!\cdots\!00$$$$+$$$$14\!\cdots\!00$$$$T^{4} + T^{8} )^{2}$$
$89$ $$($$$$36\!\cdots\!00$$$$- 40870985973120 T^{2} + T^{4} )^{4}$$
$97$ $$($$$$18\!\cdots\!56$$$$+$$$$91\!\cdots\!68$$$$T^{4} + T^{8} )^{2}$$