# Properties

 Label 300.7.b.e Level $300$ Weight $7$ Character orbit 300.b Analytic conductor $69.016$ Analytic rank $0$ Dimension $16$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$300 = 2^{2} \cdot 3 \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$7$$ Character orbit: $$[\chi]$$ $$=$$ 300.b (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$69.0162250860$$ Analytic rank: $$0$$ Dimension: $$16$$ Coefficient field: $$\mathbb{Q}[x]/(x^{16} + \cdots)$$ Defining polynomial: $$x^{16} + 406 x^{14} + 67561 x^{12} + 5921226 x^{10} + 291565644 x^{8} + 7924637994 x^{6} + 107459649193 x^{4} + 554332708870 x^{2} + 276002078881$$ Coefficient ring: $$\Z[a_1, \ldots, a_{17}]$$ Coefficient ring index: $$2^{30}\cdot 3^{18}\cdot 5^{26}$$ Twist minimal: no (minimal twist has level 60) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{15}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q -\beta_{3} q^{3} + ( \beta_{4} - \beta_{5} ) q^{7} + ( -187 + \beta_{8} ) q^{9} +O(q^{10})$$ $$q -\beta_{3} q^{3} + ( \beta_{4} - \beta_{5} ) q^{7} + ( -187 + \beta_{8} ) q^{9} + ( \beta_{8} - \beta_{11} + \beta_{14} ) q^{11} + ( -4 \beta_{3} - 17 \beta_{4} - 2 \beta_{5} + 2 \beta_{7} + \beta_{12} ) q^{13} + ( 2 \beta_{2} - 73 \beta_{3} + 4 \beta_{4} - \beta_{6} + 4 \beta_{7} - 4 \beta_{12} ) q^{17} + ( 1906 + \beta_{1} + 3 \beta_{8} + 3 \beta_{9} + 3 \beta_{11} - 6 \beta_{13} - 3 \beta_{14} ) q^{19} + ( -104 - 4 \beta_{1} + 4 \beta_{8} - 2 \beta_{9} - 3 \beta_{11} + 11 \beta_{13} + 11 \beta_{14} + 2 \beta_{15} ) q^{21} + ( -9 \beta_{2} - 42 \beta_{3} + 3 \beta_{4} + 3 \beta_{7} - 2 \beta_{10} - 3 \beta_{12} ) q^{23} + ( \beta_{2} + 193 \beta_{3} + 34 \beta_{4} - 23 \beta_{5} - \beta_{6} - 19 \beta_{7} - 9 \beta_{10} ) q^{27} + ( -8 + 7 \beta_{1} - 16 \beta_{8} - 9 \beta_{9} + 19 \beta_{11} - 11 \beta_{13} - 29 \beta_{14} - 3 \beta_{15} ) q^{29} + ( 4391 + 5 \beta_{1} - 24 \beta_{8} + 12 \beta_{11} + 12 \beta_{14} - 12 \beta_{15} ) q^{31} + ( -15 \beta_{2} + 57 \beta_{3} - 286 \beta_{4} - 21 \beta_{5} - 4 \beta_{6} - 16 \beta_{7} - 3 \beta_{10} + 13 \beta_{12} ) q^{33} + ( -28 \beta_{3} - 209 \beta_{4} + 78 \beta_{5} - 58 \beta_{7} - 23 \beta_{12} ) q^{37} + ( -4942 - 2 \beta_{1} + 5 \beta_{8} + 2 \beta_{9} - 45 \beta_{11} + 34 \beta_{13} + 16 \beta_{14} + 7 \beta_{15} ) q^{39} + ( -27 - 17 \beta_{1} + 82 \beta_{8} + 14 \beta_{9} + 46 \beta_{13} + 40 \beta_{14} - 12 \beta_{15} ) q^{41} + ( -1006 \beta_{3} - 353 \beta_{4} - 62 \beta_{5} - 199 \beta_{7} - 41 \beta_{12} ) q^{43} + ( -57 \beta_{2} + 206 \beta_{3} - 29 \beta_{4} + 2 \beta_{6} - 29 \beta_{7} + 40 \beta_{10} + 29 \beta_{12} ) q^{47} + ( -25302 + 53 \beta_{1} - 24 \beta_{8} - 12 \beta_{9} - 6 \beta_{11} + 24 \beta_{13} + 18 \beta_{14} - 6 \beta_{15} ) q^{49} + ( 53550 + 42 \beta_{1} + 93 \beta_{8} - 108 \beta_{9} - 6 \beta_{11} - 48 \beta_{13} - 99 \beta_{14} + 12 \beta_{15} ) q^{51} + ( -9 \beta_{2} + 591 \beta_{3} + 18 \beta_{4} - 15 \beta_{6} + 18 \beta_{7} - 94 \beta_{10} - 18 \beta_{12} ) q^{53} + ( -9 \beta_{2} - 1777 \beta_{3} + 11 \beta_{4} - 384 \beta_{5} + 11 \beta_{6} + 188 \beta_{7} - 48 \beta_{10} - 89 \beta_{12} ) q^{57} + ( 93 - 58 \beta_{1} + 120 \beta_{8} - 33 \beta_{9} + 84 \beta_{11} + 24 \beta_{13} + 214 \beta_{14} + 92 \beta_{15} ) q^{59} + ( -16961 + 85 \beta_{1} - 32 \beta_{8} + 52 \beta_{9} + 94 \beta_{11} - 104 \beta_{13} - 10 \beta_{14} - 42 \beta_{15} ) q^{61} + ( 25 \beta_{2} - 80 \beta_{3} + 1000 \beta_{4} + 118 \beta_{5} - 31 \beta_{6} - 424 \beta_{7} + 81 \beta_{10} + 111 \beta_{12} ) q^{63} + ( 1042 \beta_{3} - 1011 \beta_{4} + 276 \beta_{5} + 433 \beta_{7} + 137 \beta_{12} ) q^{67} + ( 31860 + 17 \beta_{1} + 2 \beta_{8} + 161 \beta_{9} + 164 \beta_{11} + 160 \beta_{13} - 26 \beta_{14} + 13 \beta_{15} ) q^{69} + ( 23 - 10 \beta_{1} - 325 \beta_{8} + 105 \beta_{9} + 73 \beta_{11} + 56 \beta_{13} - 243 \beta_{14} - 36 \beta_{15} ) q^{71} + ( -8820 \beta_{3} - 611 \beta_{4} - 186 \beta_{5} + 342 \beta_{7} + 510 \beta_{12} ) q^{73} + ( 130 \beta_{2} + 2730 \beta_{3} - 100 \beta_{4} - 94 \beta_{6} - 100 \beta_{7} - 8 \beta_{10} + 100 \beta_{12} ) q^{77} + ( -88484 + 453 \beta_{1} + 69 \beta_{8} - 27 \beta_{9} - 75 \beta_{11} + 54 \beta_{13} - 21 \beta_{14} + 48 \beta_{15} ) q^{79} + ( -149486 - 27 \beta_{1} - 28 \beta_{8} - 60 \beta_{9} + 96 \beta_{11} + 138 \beta_{13} + 312 \beta_{14} + 216 \beta_{15} ) q^{81} + ( 240 \beta_{2} + 8904 \beta_{3} - 515 \beta_{4} + 148 \beta_{6} - 515 \beta_{7} + 40 \beta_{10} + 515 \beta_{12} ) q^{83} + ( 117 \beta_{2} - 900 \beta_{3} + 4156 \beta_{4} + 717 \beta_{5} + 112 \beta_{6} + \beta_{7} - 42 \beta_{10} - 235 \beta_{12} ) q^{87} + ( -614 + 278 \beta_{1} + 248 \beta_{8} - 564 \beta_{9} + 832 \beta_{11} - 392 \beta_{13} - 816 \beta_{14} - 164 \beta_{15} ) q^{89} + ( -250599 + 266 \beta_{1} + 283 \beta_{8} - 113 \beta_{9} - 311 \beta_{11} + 226 \beta_{13} - 85 \beta_{14} + 198 \beta_{15} ) q^{91} + ( -117 \beta_{2} - 4916 \beta_{3} - 1163 \beta_{4} + 231 \beta_{5} + 79 \beta_{6} + 208 \beta_{7} + 345 \beta_{10} + 1277 \beta_{12} ) q^{93} + ( 5088 \beta_{3} + 3099 \beta_{4} - 452 \beta_{5} - 1392 \beta_{7} - 792 \beta_{12} ) q^{97} + ( -331212 + 444 \beta_{1} - 273 \beta_{8} - 96 \beta_{9} + 57 \beta_{11} + 438 \beta_{13} - 255 \beta_{14} + 78 \beta_{15} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$16q - 2984q^{9} + O(q^{10})$$ $$16q - 2984q^{9} + 30544q^{19} - 1736q^{21} + 70064q^{31} - 79216q^{39} - 405120q^{49} + 858240q^{51} - 271216q^{61} + 509880q^{69} - 1415408q^{79} - 2396224q^{81} - 4008224q^{91} - 5300160q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{16} + 406 x^{14} + 67561 x^{12} + 5921226 x^{10} + 291565644 x^{8} + 7924637994 x^{6} + 107459649193 x^{4} + 554332708870 x^{2} + 276002078881$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$-1252675619533905 \nu^{14} - 440214851499970685 \nu^{12} - 60044417409487298285 \nu^{10} - 3967655518433443573320 \nu^{8} - 128113849378167809763685 \nu^{6} - 1742982712345100421316515 \nu^{4} - 5454385878624956099514485 \nu^{2} + 20195725327289917161405865$$$$)/$$$$13\!\cdots\!31$$ $$\beta_{2}$$ $$=$$ $$($$$$31736063694062753024 \nu^{14} + 11107392155378242951258 \nu^{12} + 1527170924255254647520581 \nu^{10} + 103984881564544350154621806 \nu^{8} + 3622130802609414194683427709 \nu^{6} + 59901282440813916831739208637 \nu^{4} + 383973264428424358342327595306 \nu^{2} + 547634181856412916324023336407$$$$)/$$$$17\!\cdots\!99$$ $$\beta_{3}$$ $$=$$ $$($$$$-3012719447587828789270107 \nu^{15} - 7780178617160314676540495 \nu^{14} - 1035686605692988441365818296 \nu^{13} - 2338560529860861779924474115 \nu^{12} - 138707651633089086973906598842 \nu^{11} - 260263887583518326478912157555 \nu^{10} - 9040866925696791145677560368287 \nu^{9} - 12773852861458423760049154352680 \nu^{8} - 287610340205400478326841618523699 \nu^{7} - 232418845852522540232926083462245 \nu^{6} - 3604448905638560183559861792588282 \nu^{5} + 696374256365419137315089286319465 \nu^{4} + 3593482852885531974081521079385400 \nu^{3} + 40470710844905815877538566094329895 \nu^{2} + 244470983511236446047346109891489537 \nu + 93868688450966908693407260047569890$$$$)/$$$$71\!\cdots\!80$$ $$\beta_{4}$$ $$=$$ $$($$$$-486211813880 \nu^{15} - 167875593922015 \nu^{13} - 23236663947043680 \nu^{11} - 1662969825829102230 \nu^{9} - 66393637913186876010 \nu^{7} - 1480343766796965103980 \nu^{5} - 16591860818105470592750 \nu^{3} - 58849894218717041894095 \nu$$$$)/$$$$74\!\cdots\!82$$ $$\beta_{5}$$ $$=$$ $$($$$$-8446080604119162947707 \nu^{15} - 3773172436919745381719696 \nu^{13} - 685600712213175091229136642 \nu^{11} - 64596336226093644451150015287 \nu^{9} - 3304999948767333487061555576499 \nu^{7} - 86544324077021574068942699578482 \nu^{5} - 947083013722382916200647460384800 \nu^{3} - 2382591543257389678583579113890263 \nu$$$$)/$$$$76\!\cdots\!20$$ $$\beta_{6}$$ $$=$$ $$($$$$-$$$$21\!\cdots\!49$$$$\nu^{15} +$$$$12\!\cdots\!95$$$$\nu^{14} -$$$$72\!\cdots\!72$$$$\nu^{13} +$$$$38\!\cdots\!15$$$$\nu^{12} -$$$$97\!\cdots\!94$$$$\nu^{11} +$$$$46\!\cdots\!55$$$$\nu^{10} -$$$$63\!\cdots\!09$$$$\nu^{9} +$$$$26\!\cdots\!80$$$$\nu^{8} -$$$$20\!\cdots\!93$$$$\nu^{7} +$$$$64\!\cdots\!45$$$$\nu^{6} -$$$$25\!\cdots\!74$$$$\nu^{5} +$$$$37\!\cdots\!35$$$$\nu^{4} +$$$$25\!\cdots\!00$$$$\nu^{3} -$$$$49\!\cdots\!95$$$$\nu^{2} +$$$$17\!\cdots\!59$$$$\nu +$$$$11\!\cdots\!10$$$$)/$$$$71\!\cdots\!80$$ $$\beta_{7}$$ $$=$$ $$($$$$-$$$$10\!\cdots\!26$$$$\nu^{15} +$$$$38\!\cdots\!75$$$$\nu^{14} -$$$$39\!\cdots\!28$$$$\nu^{13} +$$$$11\!\cdots\!75$$$$\nu^{12} -$$$$58\!\cdots\!56$$$$\nu^{11} +$$$$13\!\cdots\!75$$$$\nu^{10} -$$$$44\!\cdots\!66$$$$\nu^{9} +$$$$63\!\cdots\!00$$$$\nu^{8} -$$$$17\!\cdots\!82$$$$\nu^{7} +$$$$11\!\cdots\!25$$$$\nu^{6} -$$$$34\!\cdots\!76$$$$\nu^{5} -$$$$34\!\cdots\!25$$$$\nu^{4} -$$$$28\!\cdots\!00$$$$\nu^{3} -$$$$20\!\cdots\!75$$$$\nu^{2} -$$$$93\!\cdots\!34$$$$\nu -$$$$46\!\cdots\!50$$$$)/$$$$35\!\cdots\!40$$ $$\beta_{8}$$ $$=$$ $$($$$$1160950726707374904349396 \nu^{15} + 8172021689111499901863175 \nu^{14} + 483440840876400339664349206 \nu^{13} + 2866364727271310786718752285 \nu^{12} + 79999887965139103135716644514 \nu^{11} + 401319103042277050452628932925 \nu^{10} + 6735025054474025215027980643814 \nu^{9} + 28377858439365959685403199832830 \nu^{8} + 306985150885386773979698324478840 \nu^{7} + 1043827462606262690215784516655425 \nu^{6} + 7444387602976715275940023550563366 \nu^{5} + 17972657146337429317121047472933005 \nu^{4} + 85708324581963455040345218912123266 \nu^{3} + 97118248255275987902846545613888075 \nu^{2} + 286533907198152937801623007671910926 \nu - 74209332895502126344760410345176728$$$$)/$$$$35\!\cdots\!64$$ $$\beta_{9}$$ $$=$$ $$($$$$-432315283487123800024697 \nu^{15} - 11090220911272616538907805 \nu^{14} - 101900478540801379572010862 \nu^{13} - 3799267219395341023767751395 \nu^{12} - 4590071253317354640350504808 \nu^{11} - 507619445614834563329665982935 \nu^{10} + 614081537973666039112923962087 \nu^{9} - 33346165318641344044947859811350 \nu^{8} + 71355120482066181477650657313885 \nu^{7} - 1115381693941368164853063574543235 \nu^{6} + 2611186129819162887825507986334088 \nu^{5} - 17788269463023239530334633119457795 \nu^{4} + 35376876499261365083822913936497188 \nu^{3} - 110866039584470066748309720286928985 \nu^{2} + 125562431117140170261915042989151663 \nu - 62977140915549025757055245811329200$$$$)/$$$$11\!\cdots\!88$$ $$\beta_{10}$$ $$=$$ $$($$$$-$$$$42\!\cdots\!31$$$$\nu^{15} +$$$$10\!\cdots\!95$$$$\nu^{14} -$$$$14\!\cdots\!68$$$$\nu^{13} +$$$$36\!\cdots\!15$$$$\nu^{12} -$$$$19\!\cdots\!86$$$$\nu^{11} +$$$$48\!\cdots\!55$$$$\nu^{10} -$$$$12\!\cdots\!71$$$$\nu^{9} +$$$$31\!\cdots\!80$$$$\nu^{8} -$$$$40\!\cdots\!67$$$$\nu^{7} +$$$$10\!\cdots\!45$$$$\nu^{6} -$$$$51\!\cdots\!06$$$$\nu^{5} +$$$$14\!\cdots\!35$$$$\nu^{4} +$$$$51\!\cdots\!00$$$$\nu^{3} +$$$$51\!\cdots\!05$$$$\nu^{2} +$$$$34\!\cdots\!21$$$$\nu -$$$$15\!\cdots\!90$$$$)/$$$$11\!\cdots\!60$$ $$\beta_{11}$$ $$=$$ $$($$$$-$$$$28\!\cdots\!25$$$$\nu^{15} -$$$$49\!\cdots\!65$$$$\nu^{14} -$$$$10\!\cdots\!50$$$$\nu^{13} -$$$$17\!\cdots\!55$$$$\nu^{12} -$$$$15\!\cdots\!00$$$$\nu^{11} -$$$$23\!\cdots\!55$$$$\nu^{10} -$$$$11\!\cdots\!75$$$$\nu^{9} -$$$$15\!\cdots\!10$$$$\nu^{8} -$$$$44\!\cdots\!25$$$$\nu^{7} -$$$$54\!\cdots\!55$$$$\nu^{6} -$$$$74\!\cdots\!00$$$$\nu^{5} -$$$$89\!\cdots\!95$$$$\nu^{4} -$$$$11\!\cdots\!50$$$$\nu^{3} -$$$$52\!\cdots\!05$$$$\nu^{2} +$$$$47\!\cdots\!25$$$$\nu -$$$$40\!\cdots\!80$$$$)/$$$$71\!\cdots\!28$$ $$\beta_{12}$$ $$=$$ $$($$$$38\!\cdots\!99$$$$\nu^{15} -$$$$21\!\cdots\!65$$$$\nu^{14} +$$$$12\!\cdots\!47$$$$\nu^{13} -$$$$63\!\cdots\!05$$$$\nu^{12} +$$$$15\!\cdots\!44$$$$\nu^{11} -$$$$70\!\cdots\!85$$$$\nu^{10} +$$$$86\!\cdots\!09$$$$\nu^{9} -$$$$34\!\cdots\!60$$$$\nu^{8} +$$$$18\!\cdots\!43$$$$\nu^{7} -$$$$62\!\cdots\!15$$$$\nu^{6} -$$$$12\!\cdots\!76$$$$\nu^{5} +$$$$18\!\cdots\!55$$$$\nu^{4} -$$$$91\!\cdots\!50$$$$\nu^{3} +$$$$10\!\cdots\!65$$$$\nu^{2} -$$$$82\!\cdots\!84$$$$\nu +$$$$25\!\cdots\!30$$$$)/$$$$80\!\cdots\!90$$ $$\beta_{13}$$ $$=$$ $$($$$$17\!\cdots\!71$$$$\nu^{15} +$$$$24\!\cdots\!80$$$$\nu^{14} +$$$$73\!\cdots\!86$$$$\nu^{13} +$$$$83\!\cdots\!00$$$$\nu^{12} +$$$$12\!\cdots\!04$$$$\nu^{11} +$$$$11\!\cdots\!60$$$$\nu^{10} +$$$$11\!\cdots\!49$$$$\nu^{9} +$$$$75\!\cdots\!40$$$$\nu^{8} +$$$$53\!\cdots\!55$$$$\nu^{7} +$$$$26\!\cdots\!60$$$$\nu^{6} +$$$$13\!\cdots\!16$$$$\nu^{5} +$$$$48\!\cdots\!60$$$$\nu^{4} +$$$$17\!\cdots\!66$$$$\nu^{3} +$$$$35\!\cdots\!60$$$$\nu^{2} +$$$$77\!\cdots\!21$$$$\nu +$$$$38\!\cdots\!20$$$$)/$$$$35\!\cdots\!64$$ $$\beta_{14}$$ $$=$$ $$($$$$-$$$$10\!\cdots\!99$$$$\nu^{15} -$$$$65\!\cdots\!15$$$$\nu^{14} -$$$$36\!\cdots\!54$$$$\nu^{13} -$$$$22\!\cdots\!25$$$$\nu^{12} -$$$$52\!\cdots\!36$$$$\nu^{11} -$$$$31\!\cdots\!05$$$$\nu^{10} -$$$$38\!\cdots\!21$$$$\nu^{9} -$$$$21\!\cdots\!70$$$$\nu^{8} -$$$$15\!\cdots\!55$$$$\nu^{7} -$$$$75\!\cdots\!05$$$$\nu^{6} -$$$$31\!\cdots\!04$$$$\nu^{5} -$$$$12\!\cdots\!05$$$$\nu^{4} -$$$$31\!\cdots\!54$$$$\nu^{3} -$$$$72\!\cdots\!55$$$$\nu^{2} -$$$$12\!\cdots\!29$$$$\nu +$$$$10\!\cdots\!76$$$$)/$$$$71\!\cdots\!28$$ $$\beta_{15}$$ $$=$$ $$($$$$-$$$$88\!\cdots\!54$$$$\nu^{15} +$$$$49\!\cdots\!05$$$$\nu^{14} -$$$$33\!\cdots\!14$$$$\nu^{13} +$$$$16\!\cdots\!55$$$$\nu^{12} -$$$$50\!\cdots\!46$$$$\nu^{11} +$$$$23\!\cdots\!35$$$$\nu^{10} -$$$$38\!\cdots\!26$$$$\nu^{9} +$$$$16\!\cdots\!30$$$$\nu^{8} -$$$$16\!\cdots\!20$$$$\nu^{7} +$$$$58\!\cdots\!35$$$$\nu^{6} -$$$$34\!\cdots\!34$$$$\nu^{5} +$$$$10\!\cdots\!75$$$$\nu^{4} -$$$$33\!\cdots\!84$$$$\nu^{3} +$$$$64\!\cdots\!85$$$$\nu^{2} -$$$$97\!\cdots\!54$$$$\nu +$$$$16\!\cdots\!08$$$$)/$$$$35\!\cdots\!64$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$-339 \beta_{15} - 561 \beta_{14} - 855 \beta_{13} - 950 \beta_{12} - 1167 \beta_{11} - 330 \beta_{9} - 78 \beta_{8} + 275 \beta_{7} + 1125 \beta_{5} + 590 \beta_{4} + 25550 \beta_{3} + 597 \beta_{1} - 411$$$$)/270000$$ $$\nu^{2}$$ $$=$$ $$($$$$-465 \beta_{15} + 40 \beta_{14} - 850 \beta_{13} + 1244 \beta_{12} + 890 \beta_{11} + 1044 \beta_{10} + 425 \beta_{9} - 505 \beta_{8} - 1244 \beta_{7} + 564 \beta_{6} - 1244 \beta_{4} + 11536 \beta_{3} + 1539 \beta_{2} + 1985 \beta_{1} - 6851210$$$$)/135000$$ $$\nu^{3}$$ $$=$$ $$($$$$29217 \beta_{15} + 33483 \beta_{14} + 57765 \beta_{13} + 39950 \beta_{12} + 86901 \beta_{11} + 26790 \beta_{9} - 20766 \beta_{8} - 100025 \beta_{7} - 75375 \beta_{5} - 44990 \beta_{4} - 1959050 \beta_{3} - 43491 \beta_{1} + 41733$$$$)/270000$$ $$\nu^{4}$$ $$=$$ $$($$$$21855 \beta_{15} + 2620 \beta_{14} + 48950 \beta_{13} - 60453 \beta_{12} - 46330 \beta_{11} - 82053 \beta_{10} - 24475 \beta_{9} + 19235 \beta_{8} + 60453 \beta_{7} - 21168 \beta_{6} + 60453 \beta_{4} - 322407 \beta_{3} - 67743 \beta_{2} - 101170 \beta_{1} + 250714495$$$$)/67500$$ $$\nu^{5}$$ $$=$$ $$($$$$-2699667 \beta_{15} - 3510433 \beta_{14} - 4850615 \beta_{13} - 2092750 \beta_{12} - 6500351 \beta_{11} - 2544890 \beta_{9} + 2128066 \beta_{8} + 12254875 \beta_{7} + 4528125 \beta_{5} + 16390150 \beta_{4} + 172774750 \beta_{3} + 3775141 \beta_{1} - 4169083$$$$)/270000$$ $$\nu^{6}$$ $$=$$ $$($$$$-4261155 \beta_{15} - 822320 \beta_{14} - 10166950 \beta_{13} + 9589078 \beta_{12} + 9344630 \beta_{11} + 18048078 \beta_{10} + 5083475 \beta_{9} - 3438835 \beta_{8} - 9589078 \beta_{7} + 2701668 \beta_{6} - 9589078 \beta_{4} + 10437182 \beta_{3} + 12991293 \beta_{2} + 19662245 \beta_{1} - 40463629820$$$$)/135000$$ $$\nu^{7}$$ $$=$$ $$($$$$50272101 \beta_{15} + 75351319 \beta_{14} + 89164625 \beta_{13} + 23604290 \beta_{12} + 101785433 \beta_{11} + 50868350 \beta_{9} - 37764958 \beta_{8} - 258732305 \beta_{7} - 38042775 \beta_{5} - 516246578 \beta_{4} - 3153826010 \beta_{3} - 69718363 \beta_{1} + 81694189$$$$)/54000$$ $$\nu^{8}$$ $$=$$ $$($$$$69224880 \beta_{15} + 21207220 \beta_{14} + 180864200 \beta_{13} - 116217423 \beta_{12} - 159656980 \beta_{11} - 302987223 \beta_{10} - 90432100 \beta_{9} + 48017660 \beta_{8} + 116217423 \beta_{7} - 25504488 \beta_{6} + 116217423 \beta_{4} + 581067963 \beta_{3} - 216157338 \beta_{2} - 314949895 \beta_{1} + 573075962845$$$$)/22500$$ $$\nu^{9}$$ $$=$$ $$($$$$-23543580147 \beta_{15} - 39230971953 \beta_{14} - 42709222815 \beta_{13} - 6979773050 \beta_{12} - 41154508191 \beta_{11} - 26060062890 \beta_{9} + 17267091306 \beta_{8} + 130072118975 \beta_{7} - 2371422375 \beta_{5} + 320075470610 \beta_{4} + 1468235742950 \beta_{3} + 33126401481 \beta_{1} - 40020821703$$$$)/270000$$ $$\nu^{10}$$ $$=$$ $$($$$$-7979228265 \beta_{15} - 3719985160 \beta_{14} - 23398426850 \beta_{13} + 9382226752 \beta_{12} + 19678441690 \beta_{11} + 35490550392 \beta_{10} + 11699213425 \beta_{9} - 4259243105 \beta_{8} - 9382226752 \beta_{7} + 1336339812 \beta_{6} - 9382226752 \beta_{4} - 141124797172 \beta_{3} + 26213282667 \beta_{2} + 36468341185 \beta_{1} - 60473560985410$$$$)/27000$$ $$\nu^{11}$$ $$=$$ $$($$$$2221526278353 \beta_{15} + 3994892201547 \beta_{14} + 4175505191085 \beta_{13} + 432931941650 \beta_{12} + 3404082051909 \beta_{11} + 2697544342710 \beta_{9} - 1667743849494 \beta_{8} - 12814381795925 \beta_{7} + 1919355958125 \beta_{5} - 36138687409730 \beta_{4} - 138534184558850 \beta_{3} - 3198515734719 \beta_{1} + 3942081164697$$$$)/270000$$ $$\nu^{12}$$ $$=$$ $$($$$$1891483563315 \beta_{15} + 1245313063360 \beta_{14} + 6273593253350 \beta_{13} - 1380226877514 \beta_{12} - 5028280189990 \beta_{11} - 8556407864514 \beta_{10} - 3136796626675 \beta_{9} + 646170499955 \beta_{8} + 1380226877514 \beta_{7} - 15786662184 \beta_{6} + 1380226877514 \beta_{4} + 49513639865634 \beta_{3} - 6629317359309 \beta_{2} - 8870932542385 \beta_{1} + 13632114789950860$$$$)/67500$$ $$\nu^{13}$$ $$=$$ $$($$$$-211290899286339 \beta_{15} - 401297883975361 \beta_{14} - 412273074365255 \beta_{13} - 28589265592150 \beta_{12} - 286494664407167 \beta_{11} - 279667699460330 \beta_{9} + 168346239152722 \beta_{8} + 1250861429380675 \beta_{7} - 322733290876875 \beta_{5} + 3908212140674830 \beta_{4} + 13194756668018350 \beta_{3} + 311781986825797 \beta_{1} - 390467511207211$$$$)/270000$$ $$\nu^{14}$$ $$=$$ $$($$$$-355620497542395 \beta_{15} - 310094154421880 \beta_{14} - 1331429303928550 \beta_{13} + 112210500831352 \beta_{12} + 1021335149506670 \beta_{11} + 1643915514457152 \beta_{10} + 665714651964275 \beta_{9} - 45526343120515 \beta_{8} - 112210500831352 \beta_{7} - 44343475356588 \beta_{6} - 112210500831352 \beta_{4} - 12240625285991212 \beta_{3} + 1337983663147137 \beta_{2} + 1740546102609455 \beta_{1} - 2505966584140153130$$$$)/135000$$ $$\nu^{15}$$ $$=$$ $$($$$$20248112657181609 \beta_{15} + 39982475201518691 \beta_{14} + 40892818533638605 \beta_{13} + 2060576082682750 \beta_{12} + 24464056468034077 \beta_{11} + 28917665356043830 \beta_{9} - 17456262291293582 \beta_{8} - 121631653176378625 \beta_{7} + 42437126489945625 \beta_{5} - 413404480665578350 \beta_{4} - 1265770357748172250 \beta_{3} - 30570465595410107 \beta_{1} + 38843425074996941$$$$)/270000$$

## Character values

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

 $$n$$ $$101$$ $$151$$ $$277$$ $$\chi(n)$$ $$-1$$ $$1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
149.1
 − 4.99240i 4.99240i 8.61329i − 8.61329i 9.98918i − 9.98918i 0.745166i − 0.745166i 6.94307i − 6.94307i − 7.89868i 7.89868i 3.23723i − 3.23723i 9.24517i − 9.24517i
0 −24.5453 11.2485i 0 0 0 414.697i 0 475.944 + 552.194i 0
149.2 0 −24.5453 + 11.2485i 0 0 0 414.697i 0 475.944 552.194i 0
149.3 0 −15.3993 22.1779i 0 0 0 437.053i 0 −254.720 + 683.051i 0
149.4 0 −15.3993 + 22.1779i 0 0 0 437.053i 0 −254.720 683.051i 0
149.5 0 −12.0655 24.1542i 0 0 0 436.815i 0 −437.848 + 582.864i 0
149.6 0 −12.0655 + 24.1542i 0 0 0 436.815i 0 −437.848 582.864i 0
149.7 0 −9.99060 25.0836i 0 0 0 134.460i 0 −529.376 + 501.201i 0
149.8 0 −9.99060 + 25.0836i 0 0 0 134.460i 0 −529.376 501.201i 0
149.9 0 9.99060 25.0836i 0 0 0 134.460i 0 −529.376 501.201i 0
149.10 0 9.99060 + 25.0836i 0 0 0 134.460i 0 −529.376 + 501.201i 0
149.11 0 12.0655 24.1542i 0 0 0 436.815i 0 −437.848 582.864i 0
149.12 0 12.0655 + 24.1542i 0 0 0 436.815i 0 −437.848 + 582.864i 0
149.13 0 15.3993 22.1779i 0 0 0 437.053i 0 −254.720 683.051i 0
149.14 0 15.3993 + 22.1779i 0 0 0 437.053i 0 −254.720 + 683.051i 0
149.15 0 24.5453 11.2485i 0 0 0 414.697i 0 475.944 552.194i 0
149.16 0 24.5453 + 11.2485i 0 0 0 414.697i 0 475.944 + 552.194i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 149.16 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
5.b even 2 1 inner
15.d odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 300.7.b.e 16
3.b odd 2 1 inner 300.7.b.e 16
5.b even 2 1 inner 300.7.b.e 16
5.c odd 4 1 60.7.g.a 8
5.c odd 4 1 300.7.g.h 8
15.d odd 2 1 inner 300.7.b.e 16
15.e even 4 1 60.7.g.a 8
15.e even 4 1 300.7.g.h 8
20.e even 4 1 240.7.l.c 8
60.l odd 4 1 240.7.l.c 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
60.7.g.a 8 5.c odd 4 1
60.7.g.a 8 15.e even 4 1
240.7.l.c 8 20.e even 4 1
240.7.l.c 8 60.l odd 4 1
300.7.b.e 16 1.a even 1 1 trivial
300.7.b.e 16 3.b odd 2 1 inner
300.7.b.e 16 5.b even 2 1 inner
300.7.b.e 16 15.d odd 2 1 inner
300.7.g.h 8 5.c odd 4 1
300.7.g.h 8 15.e even 4 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{7}^{8} + 571876 T_{7}^{6} + 112122973716 T_{7}^{4} +$$$$81\!\cdots\!36$$$$T_{7}^{2} +$$$$11\!\cdots\!96$$ acting on $$S_{7}^{\mathrm{new}}(300, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{16}$$
$3$ $$79\!\cdots\!61$$$$+$$$$22\!\cdots\!32$$$$T^{2} + 483544220254682328 T^{4} + 547621616191644 T^{6} + 805288413870 T^{8} + 1030446684 T^{10} + 1712088 T^{12} + 1492 T^{14} + T^{16}$$
$5$ $$T^{16}$$
$7$ $$($$$$11\!\cdots\!96$$$$+ 8114052473491936 T^{2} + 112122973716 T^{4} + 571876 T^{6} + T^{8} )^{2}$$
$11$ $$($$$$22\!\cdots\!00$$$$+ 5861305881531360000 T^{2} + 10809709880400 T^{4} + 6152760 T^{6} + T^{8} )^{2}$$
$13$ $$($$$$68\!\cdots\!36$$$$+ 11320002613898530816 T^{2} + 21081026021136 T^{4} + 9864136 T^{6} + T^{8} )^{2}$$
$17$ $$($$$$26\!\cdots\!00$$$$-$$$$90\!\cdots\!00$$$$T^{2} + 7295360344322400 T^{4} - 161822160 T^{6} + T^{8} )^{2}$$
$19$ $$( 1136545793559136 + 455354684384 T - 78241764 T^{2} - 7636 T^{3} + T^{4} )^{4}$$
$23$ $$($$$$17\!\cdots\!00$$$$-$$$$59\!\cdots\!00$$$$T^{2} + 143743273758558900 T^{4} - 715419540 T^{6} + T^{8} )^{2}$$
$29$ $$($$$$75\!\cdots\!00$$$$+$$$$42\!\cdots\!00$$$$T^{2} + 787104956899904400 T^{4} + 2283274440 T^{6} + T^{8} )^{2}$$
$31$ $$( 43347363402701056 + 54665955569344 T - 2254268604 T^{2} - 17516 T^{3} + T^{4} )^{4}$$
$37$ $$($$$$25\!\cdots\!36$$$$+$$$$50\!\cdots\!84$$$$T^{2} + 14277586081444421136 T^{4} + 7769113864 T^{6} + T^{8} )^{2}$$
$41$ $$($$$$64\!\cdots\!00$$$$+$$$$36\!\cdots\!00$$$$T^{2} +$$$$17\!\cdots\!00$$$$T^{4} + 24416952840 T^{6} + T^{8} )^{2}$$
$43$ $$($$$$67\!\cdots\!56$$$$+$$$$13\!\cdots\!56$$$$T^{2} +$$$$41\!\cdots\!96$$$$T^{4} + 36889997116 T^{6} + T^{8} )^{2}$$
$47$ $$($$$$88\!\cdots\!00$$$$-$$$$34\!\cdots\!00$$$$T^{2} +$$$$25\!\cdots\!00$$$$T^{4} - 35727961140 T^{6} + T^{8} )^{2}$$
$53$ $$($$$$23\!\cdots\!00$$$$-$$$$49\!\cdots\!00$$$$T^{2} +$$$$34\!\cdots\!00$$$$T^{4} - 97591865040 T^{6} + T^{8} )^{2}$$
$59$ $$($$$$11\!\cdots\!00$$$$+$$$$33\!\cdots\!00$$$$T^{2} +$$$$18\!\cdots\!00$$$$T^{4} + 265901508360 T^{6} + T^{8} )^{2}$$
$61$ $$($$$$48\!\cdots\!76$$$$+ 711714612123904 T - 62487698844 T^{2} + 67804 T^{3} + T^{4} )^{4}$$
$67$ $$($$$$87\!\cdots\!96$$$$+$$$$35\!\cdots\!36$$$$T^{2} +$$$$55\!\cdots\!16$$$$T^{4} + 210719977276 T^{6} + T^{8} )^{2}$$
$71$ $$($$$$89\!\cdots\!00$$$$+$$$$40\!\cdots\!00$$$$T^{2} +$$$$62\!\cdots\!00$$$$T^{4} + 415954038960 T^{6} + T^{8} )^{2}$$
$73$ $$($$$$53\!\cdots\!36$$$$+$$$$39\!\cdots\!84$$$$T^{2} +$$$$15\!\cdots\!36$$$$T^{4} + 748644248464 T^{6} + T^{8} )^{2}$$
$79$ $$($$$$63\!\cdots\!36$$$$- 130563026559349312 T - 567974089236 T^{2} + 353852 T^{3} + T^{4} )^{4}$$
$83$ $$($$$$61\!\cdots\!00$$$$-$$$$85\!\cdots\!00$$$$T^{2} +$$$$25\!\cdots\!00$$$$T^{4} - 2736050734260 T^{6} + T^{8} )^{2}$$
$89$ $$($$$$98\!\cdots\!00$$$$+$$$$40\!\cdots\!00$$$$T^{2} +$$$$62\!\cdots\!00$$$$T^{4} + 4108796641440 T^{6} + T^{8} )^{2}$$
$97$ $$($$$$37\!\cdots\!76$$$$+$$$$57\!\cdots\!96$$$$T^{2} +$$$$21\!\cdots\!56$$$$T^{4} + 2630378057296 T^{6} + T^{8} )^{2}$$