Properties

Label 1110.2.x.c
Level $1110$
Weight $2$
Character orbit 1110.x
Analytic conductor $8.863$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 1110 = 2 \cdot 3 \cdot 5 \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1110.x (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(8.86339462436\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
Defining polynomial: \(x^{16} - 8 x^{13} + 398 x^{12} - 136 x^{11} + 32 x^{10} - 824 x^{9} + 17825 x^{8} - 11480 x^{7} + 3104 x^{6} + 11296 x^{5} + 68320 x^{4} - 21120 x^{3} + 4608 x^{2} + 13824 x + 20736\)
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 1 \)
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 \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( -\beta_{6} + \beta_{10} ) q^{2} + \beta_{9} q^{3} -\beta_{9} q^{4} -\beta_{10} q^{5} + \beta_{6} q^{6} + \beta_{2} q^{7} -\beta_{6} q^{8} + ( -1 - \beta_{9} ) q^{9} +O(q^{10})\) \( q + ( -\beta_{6} + \beta_{10} ) q^{2} + \beta_{9} q^{3} -\beta_{9} q^{4} -\beta_{10} q^{5} + \beta_{6} q^{6} + \beta_{2} q^{7} -\beta_{6} q^{8} + ( -1 - \beta_{9} ) q^{9} - q^{10} + ( -1 + \beta_{6} - \beta_{10} + \beta_{13} ) q^{11} + ( 1 + \beta_{9} ) q^{12} + ( -2 - \beta_{1} - \beta_{3} - \beta_{4} - \beta_{9} ) q^{13} + ( -1 - \beta_{9} + \beta_{14} ) q^{14} + ( -\beta_{6} + \beta_{10} ) q^{15} + ( -1 - \beta_{9} ) q^{16} + ( -\beta_{3} + \beta_{6} - \beta_{8} - \beta_{10} - \beta_{11} + \beta_{12} + \beta_{15} ) q^{17} -\beta_{10} q^{18} + ( -\beta_{3} - \beta_{4} - \beta_{7} ) q^{19} + ( \beta_{6} - \beta_{10} ) q^{20} + ( -\beta_{2} - \beta_{6} + \beta_{10} - \beta_{13} ) q^{21} + ( \beta_{6} - \beta_{10} + \beta_{12} ) q^{22} + ( -\beta_{1} + \beta_{3} - \beta_{4} - \beta_{5} - \beta_{7} - \beta_{8} + \beta_{11} + \beta_{15} ) q^{23} + \beta_{10} q^{24} + ( 1 + \beta_{9} ) q^{25} + ( \beta_{1} - \beta_{4} + \beta_{6} - \beta_{7} - 2 \beta_{10} ) q^{26} + q^{27} + ( \beta_{2} + \beta_{6} - \beta_{10} + \beta_{13} ) q^{28} + ( -1 + \beta_{3} - \beta_{4} - \beta_{6} - \beta_{9} + \beta_{14} ) q^{29} -\beta_{9} q^{30} + ( \beta_{4} - \beta_{5} - \beta_{6} - \beta_{8} + \beta_{9} + \beta_{11} + \beta_{14} + \beta_{15} ) q^{31} -\beta_{10} q^{32} + ( \beta_{2} - \beta_{9} ) q^{33} + ( -\beta_{2} + \beta_{3} + \beta_{9} + \beta_{11} - \beta_{15} ) q^{34} + \beta_{12} q^{35} - q^{36} + ( 1 - \beta_{1} + 2 \beta_{3} - \beta_{5} - \beta_{6} + \beta_{10} + \beta_{11} - \beta_{12} ) q^{37} + ( -\beta_{3} - \beta_{4} ) q^{38} + ( 1 + \beta_{3} - \beta_{7} - \beta_{9} ) q^{39} + \beta_{9} q^{40} + ( 1 + \beta_{1} + \beta_{2} + \beta_{3} - \beta_{4} + 2 \beta_{6} - \beta_{10} + \beta_{12} - \beta_{14} ) q^{41} + ( 1 + \beta_{9} - \beta_{12} - \beta_{14} ) q^{42} + ( -2 - \beta_{1} - \beta_{3} - \beta_{5} - \beta_{6} - \beta_{7} - 3 \beta_{9} + \beta_{14} + \beta_{15} ) q^{43} + ( -\beta_{2} + \beta_{9} ) q^{44} + \beta_{6} q^{45} + ( -\beta_{7} - \beta_{8} + \beta_{11} + \beta_{15} ) q^{46} + ( -5 - \beta_{1} - 2 \beta_{3} - \beta_{4} - \beta_{6} + \beta_{7} + \beta_{10} - \beta_{13} ) q^{47} + q^{48} + ( -2 - \beta_{3} + \beta_{5} - \beta_{6} - 2 \beta_{9} - \beta_{10} - \beta_{11} + \beta_{12} + 2 \beta_{14} - \beta_{15} ) q^{49} + \beta_{10} q^{50} + ( -1 + \beta_{5} - \beta_{6} - \beta_{9} + \beta_{14} - \beta_{15} ) q^{51} + ( -1 - \beta_{3} + \beta_{7} + \beta_{9} ) q^{52} + ( -1 - 2 \beta_{1} - \beta_{2} - \beta_{3} - 2 \beta_{4} - 2 \beta_{6} - \beta_{7} - \beta_{9} + \beta_{12} - \beta_{13} + 2 \beta_{14} ) q^{53} + ( -\beta_{6} + \beta_{10} ) q^{54} + ( 1 + \beta_{9} + \beta_{10} - \beta_{12} - \beta_{14} ) q^{55} + ( -1 - \beta_{9} + \beta_{12} + \beta_{14} ) q^{56} + ( -\beta_{1} + \beta_{4} ) q^{57} + ( -1 + \beta_{1} + \beta_{2} + \beta_{4} + \beta_{6} - \beta_{9} - \beta_{10} + \beta_{13} ) q^{58} + ( 1 + 2 \beta_{1} + \beta_{2} - 2 \beta_{4} + \beta_{5} - \beta_{6} - \beta_{7} - \beta_{9} + \beta_{10} - \beta_{11} - \beta_{13} + \beta_{15} ) q^{59} -\beta_{6} q^{60} + ( -1 - \beta_{1} + \beta_{2} - \beta_{3} - \beta_{4} + 2 \beta_{6} - \beta_{10} - \beta_{12} + 2 \beta_{13} - \beta_{14} ) q^{61} + ( -1 - \beta_{1} + \beta_{2} + \beta_{3} - \beta_{4} + 2 \beta_{6} - \beta_{8} - \beta_{9} + \beta_{11} + \beta_{13} + \beta_{15} ) q^{62} + ( \beta_{6} - \beta_{10} + \beta_{13} ) q^{63} - q^{64} + ( \beta_{3} + \beta_{6} + \beta_{7} + \beta_{10} ) q^{65} + ( -1 - \beta_{6} - \beta_{9} + \beta_{14} ) q^{66} + ( 1 - \beta_{1} - \beta_{2} + 2 \beta_{6} - \beta_{7} - 3 \beta_{9} - \beta_{10} + \beta_{12} - \beta_{14} ) q^{67} + ( 1 - \beta_{5} + \beta_{6} + \beta_{9} - \beta_{14} + \beta_{15} ) q^{68} + ( \beta_{1} + \beta_{4} - \beta_{11} - \beta_{15} ) q^{69} -\beta_{2} q^{70} + ( -1 + 2 \beta_{1} - \beta_{2} - \beta_{4} + \beta_{5} - 2 \beta_{6} + \beta_{7} + \beta_{8} + \beta_{10} - \beta_{11} - \beta_{12} + \beta_{14} ) q^{71} + ( \beta_{6} - \beta_{10} ) q^{72} + ( 2 + \beta_{1} + \beta_{3} - \beta_{5} - \beta_{6} - \beta_{7} - \beta_{9} + 2 \beta_{12} - 2 \beta_{13} + \beta_{14} - \beta_{15} ) q^{73} + ( \beta_{2} + \beta_{4} - \beta_{6} - \beta_{7} - \beta_{8} - \beta_{9} + \beta_{10} + \beta_{15} ) q^{74} - q^{75} + ( \beta_{1} - \beta_{4} ) q^{76} + ( 1 - \beta_{2} - \beta_{5} + 2 \beta_{6} - \beta_{8} + 8 \beta_{9} - \beta_{10} + \beta_{12} - \beta_{14} + \beta_{15} ) q^{77} + ( -\beta_{1} - \beta_{3} + \beta_{4} - 2 \beta_{6} + \beta_{10} ) q^{78} + ( -1 + 2 \beta_{1} + \beta_{4} + \beta_{5} - \beta_{7} - \beta_{8} - 5 \beta_{10} - \beta_{11} - \beta_{12} - \beta_{14} ) q^{79} + \beta_{6} q^{80} + \beta_{9} q^{81} + ( \beta_{1} - 2 \beta_{2} + \beta_{4} - 2 \beta_{6} + \beta_{7} + \beta_{9} + \beta_{10} - \beta_{13} + \beta_{14} ) q^{82} + ( -3 - \beta_{1} + \beta_{2} - \beta_{3} - \beta_{4} + \beta_{5} - \beta_{7} - \beta_{8} - 3 \beta_{9} - 2 \beta_{10} + \beta_{12} + \beta_{13} + 2 \beta_{14} ) q^{83} + ( -\beta_{6} + \beta_{10} - \beta_{13} ) q^{84} + ( 1 - \beta_{3} + \beta_{5} - \beta_{6} - \beta_{8} + \beta_{10} - \beta_{11} - \beta_{13} + \beta_{15} ) q^{85} + ( -1 - \beta_{1} + \beta_{2} - \beta_{3} - \beta_{4} + \beta_{5} - \beta_{7} - \beta_{8} - \beta_{9} - 2 \beta_{10} + \beta_{13} ) q^{86} + ( 1 + \beta_{1} - \beta_{7} + \beta_{9} + \beta_{10} - \beta_{12} - \beta_{14} ) q^{87} + ( 1 + \beta_{6} + \beta_{9} - \beta_{14} ) q^{88} + ( \beta_{5} - 4 \beta_{6} - 2 \beta_{7} - \beta_{8} + 4 \beta_{10} - 2 \beta_{11} + 2 \beta_{12} + 2 \beta_{15} ) q^{89} + ( 1 + \beta_{9} ) q^{90} + ( -\beta_{2} + 2 \beta_{3} - \beta_{5} + \beta_{6} - \beta_{7} - \beta_{10} + \beta_{11} + \beta_{13} - \beta_{15} ) q^{91} + ( -\beta_{1} - \beta_{4} + \beta_{11} + \beta_{15} ) q^{92} + ( -1 - \beta_{1} - \beta_{3} + \beta_{7} + \beta_{10} - \beta_{11} - \beta_{12} - \beta_{14} - \beta_{15} ) q^{93} + ( 2 \beta_{1} + \beta_{3} - 2 \beta_{4} + 5 \beta_{6} - \beta_{7} - 5 \beta_{10} - \beta_{12} ) q^{94} + ( \beta_{1} + \beta_{3} + \beta_{4} + \beta_{7} ) q^{95} + ( -\beta_{6} + \beta_{10} ) q^{96} + ( -2 + \beta_{1} - \beta_{3} + \beta_{5} - \beta_{6} + \beta_{7} + 2 \beta_{8} - 3 \beta_{9} - 2 \beta_{11} + \beta_{14} - \beta_{15} ) q^{97} + ( -2 + \beta_{2} - \beta_{5} + 2 \beta_{6} + \beta_{8} - \beta_{9} - 2 \beta_{10} + 2 \beta_{13} - \beta_{15} ) q^{98} + ( 1 - \beta_{2} - \beta_{6} + \beta_{9} + \beta_{10} - \beta_{13} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16q - 8q^{3} + 8q^{4} - 2q^{7} - 8q^{9} + O(q^{10}) \) \( 16q - 8q^{3} + 8q^{4} - 2q^{7} - 8q^{9} - 16q^{10} - 12q^{11} + 8q^{12} - 12q^{13} - 8q^{16} - 6q^{17} + 6q^{19} - 2q^{21} - 6q^{22} + 8q^{25} + 16q^{27} + 2q^{28} + 8q^{30} + 6q^{33} - 4q^{34} - 6q^{35} - 16q^{36} + 18q^{37} + 12q^{38} + 12q^{39} - 8q^{40} + 6q^{42} - 6q^{44} - 4q^{46} - 60q^{47} + 16q^{48} - 4q^{49} - 12q^{52} + 12q^{53} + 6q^{55} - 6q^{56} - 6q^{57} - 12q^{58} + 12q^{59} - 4q^{62} + 4q^{63} - 16q^{64} + 22q^{67} - 6q^{69} + 2q^{70} + 2q^{71} + 24q^{73} - 8q^{74} - 16q^{75} + 6q^{76} - 58q^{77} - 36q^{79} - 8q^{81} - 8q^{83} - 4q^{84} + 8q^{85} - 2q^{86} - 42q^{89} + 8q^{90} + 6q^{92} - 6q^{93} + 6q^{94} - 6q^{95} - 12q^{98} + 6q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{16} - 8 x^{13} + 398 x^{12} - 136 x^{11} + 32 x^{10} - 824 x^{9} + 17825 x^{8} - 11480 x^{7} + 3104 x^{6} + 11296 x^{5} + 68320 x^{4} - 21120 x^{3} + 4608 x^{2} + 13824 x + 20736\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\(-1530959369985179 \nu^{15} + 144221513243906511 \nu^{14} - 5416617017878650 \nu^{13} + 13564455996723616 \nu^{12} - 1760611670334921058 \nu^{11} + 57525527169244138346 \nu^{10} - 21975684768648034276 \nu^{9} + 7040915420426720392 \nu^{8} - 144492360739287515491 \nu^{7} + 2543329135127441269831 \nu^{6} - 1796736578446055251522 \nu^{5} + 500306947410025218232 \nu^{4} + 1631362233957992249680 \nu^{3} + 7698677138785417176912 \nu^{2} - 3856588704212001518304 \nu + 661805318585190819456\)\()/ \)\(10\!\cdots\!96\)\( \)
\(\beta_{3}\)\(=\)\((\)\(127293331 \nu^{15} - 1300364410 \nu^{14} - 327477924 \nu^{13} + 632205952 \nu^{12} + 62442603850 \nu^{11} - 527849729220 \nu^{10} + 38715136056 \nu^{9} + 528335974664 \nu^{8} + 3598350196483 \nu^{7} - 22874477793826 \nu^{6} + 8198332782700 \nu^{5} + 24300910264088 \nu^{4} - 9660318458736 \nu^{3} - 55363330646368 \nu^{2} - 18798157493184 \nu + 35657093478528\)\()/ 61574442553344 \)
\(\beta_{4}\)\(=\)\((\)\(534978343510032389 \nu^{15} + 3031037355826035918 \nu^{14} + 875195508130966404 \nu^{13} - 442479116991981136 \nu^{12} + 190886417351030365030 \nu^{11} + 1118342632735371230428 \nu^{10} - 88567561334350853912 \nu^{9} + 1008325742277487625048 \nu^{8} + 7494258933450441031093 \nu^{7} + 44015401690483414912310 \nu^{6} - 23418703189733189631980 \nu^{5} + 60108843042165598194776 \nu^{4} + 83093589089348057212784 \nu^{3} + 112425053477879411711520 \nu^{2} + 7499987310702785503680 \nu + 85824450678934957944960\)\()/ \)\(13\!\cdots\!24\)\( \)
\(\beta_{5}\)\(=\)\((\)\(10700867622035989 \nu^{15} + 20469769793225850 \nu^{14} - 118821587524745172 \nu^{13} - 104347183077777296 \nu^{12} + 4093164236879373926 \nu^{11} + 7547227817160187460 \nu^{10} - 49488664514622898120 \nu^{9} + 729122466455084440 \nu^{8} + 172851352386229182245 \nu^{7} + 302400607706169550498 \nu^{6} - 2247431010834146979172 \nu^{5} + 1278935854363324482904 \nu^{4} + 940120878572826309232 \nu^{3} - 1509214972701578971296 \nu^{2} - 4785398391578079031488 \nu + 2436178448118868790400\)\()/ \)\(12\!\cdots\!84\)\( \)
\(\beta_{6}\)\(=\)\((\)\(64095230699611555 \nu^{15} - 142132866778083225 \nu^{14} - 21127762762007220 \nu^{13} - 517758107561451788 \nu^{12} + 26660677797335797250 \nu^{11} - 65232032042968064614 \nu^{10} + 12965976438420258608 \nu^{9} - 56599213538485841984 \nu^{8} + 1265996169141964180091 \nu^{7} - 3300980319073898653121 \nu^{6} + 1455318563806066674980 \nu^{5} + 422850728547815474860 \nu^{4} + 3113914324013679787696 \nu^{3} - 14383331984622559681008 \nu^{2} + 2265275126018130626880 \nu + 438440410228363557696\)\()/ \)\(73\!\cdots\!12\)\( \)
\(\beta_{7}\)\(=\)\((\)\(706056167671395937 \nu^{15} - 878884150311744826 \nu^{14} - 260643983819730636 \nu^{13} - 7012890829654040528 \nu^{12} + 288020585038319880046 \nu^{11} - 440695991704882696308 \nu^{10} + 53072035195042679880 \nu^{9} - 1104544823825578084360 \nu^{8} + 13462953239513413889233 \nu^{7} - 22489195029273411066610 \nu^{6} + 9680174825062153479940 \nu^{5} - 12164722914778335798856 \nu^{4} + 53375456957530510240944 \nu^{3} - 49732324599897798632800 \nu^{2} + 285145929980374654656 \nu - 20338327180524971376000\)\()/ \)\(45\!\cdots\!08\)\( \)
\(\beta_{8}\)\(=\)\((\)\(-866848462326922883 \nu^{15} + 1724075804842637550 \nu^{14} + 75534189332384868 \nu^{13} + 6064740370710470896 \nu^{12} - 359289657798277154698 \nu^{11} + 808550804354067676124 \nu^{10} - 220586903641146542296 \nu^{9} + 434654635875441676312 \nu^{8} - 16981837959674826320851 \nu^{7} + 42602210405302515390742 \nu^{6} - 19481953993644469684492 \nu^{5} - 13667554329408512484008 \nu^{4} - 42077180238251852712464 \nu^{3} + 201501503649934004743200 \nu^{2} - 68109142339365536494656 \nu + 94605122336474680889472\)\()/ \)\(45\!\cdots\!08\)\( \)
\(\beta_{9}\)\(=\)\((\)\(30952338089 \nu^{15} - 2291279958 \nu^{14} + 23406559380 \nu^{13} - 241724102080 \nu^{12} + 12307650852286 \nu^{11} - 5333484849404 \nu^{10} + 10491769944808 \nu^{9} - 26201599034344 \nu^{8} + 542215378892473 \nu^{7} - 420103144798414 \nu^{6} + 507816657717124 \nu^{5} + 202067620964744 \nu^{4} + 1677247353486896 \nu^{3} - 479827648182432 \nu^{2} + 1139168325548736 \nu - 342088009340544\)\()/ 1108339965960192 \)
\(\beta_{10}\)\(=\)\((\)\(8277821246039251345 \nu^{15} - 1069956687020064778 \nu^{14} - 6062074711652071836 \nu^{13} - 67972960984575943568 \nu^{12} + 3295457814157605997582 \nu^{11} - 1507556524163398912980 \nu^{10} - 1971794985597486417816 \nu^{9} - 6643789584067641400456 \nu^{8} + 145535512226094679974529 \nu^{7} - 110017905771431487502786 \nu^{6} - 62336446233260993649740 \nu^{5} + 140343675174725762457080 \nu^{4} + 445323061445070455500848 \nu^{3} - 341014762895045102831968 \nu^{2} - 186705906654009953225280 \nu + 99432626283841039585920\)\()/ \)\(27\!\cdots\!48\)\( \)
\(\beta_{11}\)\(=\)\((\)\(20491015691222859 \nu^{15} - 8797854961976146 \nu^{14} + 840940721725308 \nu^{13} - 193868250183388704 \nu^{12} + 8208569521292694938 \nu^{11} - 6409260572237152724 \nu^{10} + 2388725757748455352 \nu^{9} - 28685196991019999096 \nu^{8} + 371146069277417890043 \nu^{7} - 434778322676715395642 \nu^{6} + 201804768251729001932 \nu^{5} - 185687701098860808392 \nu^{4} + 1569502775855893887248 \nu^{3} - 2307352441308614553568 \nu^{2} + 631470103918161921600 \nu + 1174471704893458175616\)\()/ \)\(61\!\cdots\!92\)\( \)
\(\beta_{12}\)\(=\)\((\)\(37713572008252950065 \nu^{15} - 9150523980650602254 \nu^{14} + 26819172307727240364 \nu^{13} - 382301877527278028752 \nu^{12} + 15064102153198468100206 \nu^{11} - 8982575253625339740812 \nu^{10} + 13855305349959381364984 \nu^{9} - 66776611889224819477960 \nu^{8} + 684033017776146639969793 \nu^{7} - 617703877788011946849670 \nu^{6} + 796881873111354196127068 \nu^{5} - 1274328714541687923579016 \nu^{4} + 3133669130121760726283312 \nu^{3} - 1171782117161650033886496 \nu^{2} + 2082877132061060238328128 \nu - 3251799495038671367965056\)\()/ \)\(81\!\cdots\!44\)\( \)
\(\beta_{13}\)\(=\)\((\)\(-32233852064214283643 \nu^{15} - 1447130651899742514 \nu^{14} + 7606724890778034516 \nu^{13} + 256204828363328157232 \nu^{12} - 12837606475687777016602 \nu^{11} + 3742382225893527103580 \nu^{10} + 2279210047787807741768 \nu^{9} + 25053060284897634960088 \nu^{8} - 580597424736170833608907 \nu^{7} + 338374659215727264337078 \nu^{6} + 80963261576940802489316 \nu^{5} - 452329605434605393872296 \nu^{4} - 2434707268224801493035536 \nu^{3} + 782463446069713588296480 \nu^{2} + 1217389021138426435966656 \nu - 646721166022934260671360\)\()/ \)\(40\!\cdots\!72\)\( \)
\(\beta_{14}\)\(=\)\((\)\(-11377471515672630747 \nu^{15} + 1895858636927254498 \nu^{14} - 2347508706449505852 \nu^{13} + 91113197408932289856 \nu^{12} - 4549791263172465226874 \nu^{11} + 2314690387392466052660 \nu^{10} - 1583189593158895230136 \nu^{9} + 9834685102883627637944 \nu^{8} - 206866234769671836236555 \nu^{7} + 165124681727517889866314 \nu^{6} - 108384275821541845383692 \nu^{5} - 89341080974607469297048 \nu^{4} - 842146197892531822896272 \nu^{3} + 345501854689074816410080 \nu^{2} - 527887710392917929187392 \nu - 25451228263470410318976\)\()/ \)\(13\!\cdots\!24\)\( \)
\(\beta_{15}\)\(=\)\((\)\(2674284548076744781 \nu^{15} + 266877543439171740 \nu^{14} - 55221235005243384 \nu^{13} - 21167093206834735904 \nu^{12} + 1059815246809641183974 \nu^{11} - 255985829948085710560 \nu^{10} + 26655360822159754352 \nu^{9} - 2074873382610806500904 \nu^{8} + 46440011829275538421229 \nu^{7} - 25176035595572165921564 \nu^{6} + 4271197278554646569288 \nu^{5} + 38981107023236786949544 \nu^{4} + 138090820236910874950960 \nu^{3} - 1389874959082773374016 \nu^{2} - 3763717694555072181120 \nu + 61143983368062831930240\)\()/ \)\(22\!\cdots\!04\)\( \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{11} - \beta_{8} - 6 \beta_{6} - \beta_{4} + 2 \beta_{3}\)
\(\nu^{3}\)\(=\)\(4 \beta_{15} + 2 \beta_{14} + 2 \beta_{13} - 8 \beta_{10} - 8 \beta_{9} - 2 \beta_{8} + 12 \beta_{7} + 6 \beta_{6} + 12 \beta_{4} - 2 \beta_{3} - \beta_{1} - 4\)
\(\nu^{4}\)\(=\)\(4 \beta_{14} + 8 \beta_{12} + 17 \beta_{11} + 4 \beta_{10} - 4 \beta_{9} + 17 \beta_{8} - 24 \beta_{7} - 2 \beta_{6} + 3 \beta_{4} + 44 \beta_{3} + 24 \beta_{1} - 76\)
\(\nu^{5}\)\(=\)\(-34 \beta_{14} + 34 \beta_{13} - 48 \beta_{11} - 196 \beta_{10} + 196 \beta_{9} - 4 \beta_{8} + 30 \beta_{7} + 86 \beta_{6} + 96 \beta_{5} - 30 \beta_{4} - 48 \beta_{3} - 191 \beta_{1} + 144\)
\(\nu^{6}\)\(=\)\(16 \beta_{15} + 8 \beta_{14} - 96 \beta_{13} - 287 \beta_{11} + 96 \beta_{10} - 176 \beta_{9} + 287 \beta_{8} - 76 \beta_{7} + 1042 \beta_{6} - 16 \beta_{5} + 499 \beta_{4} - 862 \beta_{3} - 192 \beta_{2} - 76 \beta_{1} - 92\)
\(\nu^{7}\)\(=\)\(-1916 \beta_{15} - 574 \beta_{14} - 574 \beta_{13} + 32 \beta_{12} + 160 \beta_{11} + 3992 \beta_{10} + 3992 \beta_{9} + 958 \beta_{8} - 3400 \beta_{7} - 2962 \beta_{6} - 3400 \beta_{4} + 1582 \beta_{3} + 32 \beta_{2} + 711 \beta_{1} + 1604\)
\(\nu^{8}\)\(=\)\(656 \beta_{15} - 1916 \beta_{14} + 320 \beta_{13} - 3832 \beta_{12} - 5675 \beta_{11} - 5020 \beta_{10} + 1916 \beta_{9} - 5675 \beta_{8} + 10052 \beta_{7} + 2670 \beta_{6} + 656 \beta_{5} - 1341 \beta_{4} - 17068 \beta_{3} - 10052 \beta_{1} + 21996\)
\(\nu^{9}\)\(=\)\(11350 \beta_{14} - 11350 \beta_{13} + 1312 \beta_{12} + 18328 \beta_{11} + 78396 \beta_{10} - 78396 \beta_{9} + 4668 \beta_{8} - 15630 \beta_{7} - 29434 \beta_{6} - 36656 \beta_{5} + 15630 \beta_{4} + 18328 \beta_{3} - 1312 \beta_{2} + 63233 \beta_{1} - 60312\)
\(\nu^{10}\)\(=\)\(-18912 \beta_{15} - 9336 \beta_{14} + 36656 \beta_{13} + 89629 \beta_{11} - 36656 \beta_{10} + 123584 \beta_{9} - 89629 \beta_{8} + 19736 \beta_{7} - 333406 \beta_{6} + 18912 \beta_{5} - 200797 \beta_{4} + 310162 \beta_{3} + 73312 \beta_{2} + 19736 \beta_{1} + 66460\)
\(\nu^{11}\)\(=\)\(693636 \beta_{15} + 179258 \beta_{14} + 179258 \beta_{13} - 37824 \beta_{12} - 119680 \beta_{11} - 1464632 \beta_{10} - 1464632 \beta_{9} - 346818 \beta_{8} + 1197068 \beta_{7} + 1166958 \beta_{6} + 1197068 \beta_{4} - 784482 \beta_{3} - 37824 \beta_{2} - 332401 \beta_{1} - 476932\)
\(\nu^{12}\)\(=\)\(-475488 \beta_{15} + 693636 \beta_{14} - 239360 \beta_{13} + 1387272 \beta_{12} + 2093369 \beta_{11} + 2840900 \beta_{10} - 693636 \beta_{9} + 2093369 \beta_{8} - 3999648 \beta_{7} - 1540130 \beta_{6} - 475488 \beta_{5} + 240291 \beta_{4} + 6333308 \beta_{3} + 3999648 \beta_{1} - 7636684\)
\(\nu^{13}\)\(=\)\(-4186738 \beta_{14} + 4186738 \beta_{13} - 950976 \beta_{12} - 6551456 \beta_{11} - 28675396 \beta_{10} + 28675396 \beta_{9} - 2856324 \beta_{8} + 6945030 \beta_{7} + 8864246 \beta_{6} + 13102912 \beta_{5} - 6945030 \beta_{4} - 6551456 \beta_{3} + 950976 \beta_{2} - 22844495 \beta_{1} + 23997888\)
\(\nu^{14}\)\(=\)\(11145904 \beta_{15} + 5712648 \beta_{14} - 13102912 \beta_{13} - 29381815 \beta_{11} + 13102912 \beta_{10} - 62847184 \beta_{9} + 29381815 \beta_{8} - 1886980 \beta_{7} + 118251410 \beta_{6} - 11145904 \beta_{5} + 79548003 \beta_{4} - 110816798 \beta_{3} - 26205824 \beta_{2} - 1886980 \beta_{1} - 34279916\)
\(\nu^{15}\)\(=\)\(-247839420 \beta_{15} - 58763630 \beta_{14} - 58763630 \beta_{13} + 22291808 \beta_{12} + 65137760 \beta_{11} + 525081720 \beta_{10} + 525081720 \beta_{9} + 123919710 \beta_{8} - 437869024 \beta_{7} - 454996210 \beta_{6} - 437869024 \beta_{4} + 352157742 \beta_{3} + 22291808 \beta_{2} + 143509991 \beta_{1} + 128849140\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1110\mathbb{Z}\right)^\times\).

\(n\) \(371\) \(631\) \(667\)
\(\chi(n)\) \(1\) \(-\beta_{9}\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
751.1
0.607149 + 0.607149i
−3.13757 3.13757i
1.71606 + 1.71606i
−0.917697 0.917697i
1.14718 1.14718i
−1.91988 + 1.91988i
2.96426 2.96426i
−0.459512 + 0.459512i
0.607149 0.607149i
−3.13757 + 3.13757i
1.71606 1.71606i
−0.917697 + 0.917697i
1.14718 + 1.14718i
−1.91988 1.91988i
2.96426 + 2.96426i
−0.459512 0.459512i
−0.866025 + 0.500000i −0.500000 + 0.866025i 0.500000 0.866025i 0.866025 + 0.500000i 1.00000i −1.98397 + 3.43634i 1.00000i −0.500000 0.866025i −1.00000
751.2 −0.866025 + 0.500000i −0.500000 + 0.866025i 0.500000 0.866025i 0.866025 + 0.500000i 1.00000i −0.907693 + 1.57217i 1.00000i −0.500000 0.866025i −1.00000
751.3 −0.866025 + 0.500000i −0.500000 + 0.866025i 0.500000 0.866025i 0.866025 + 0.500000i 1.00000i 0.166951 0.289168i 1.00000i −0.500000 0.866025i −1.00000
751.4 −0.866025 + 0.500000i −0.500000 + 0.866025i 0.500000 0.866025i 0.866025 + 0.500000i 1.00000i 1.35869 2.35332i 1.00000i −0.500000 0.866025i −1.00000
751.5 0.866025 0.500000i −0.500000 + 0.866025i 0.500000 0.866025i −0.866025 0.500000i 1.00000i −1.41697 + 2.45427i 1.00000i −0.500000 0.866025i −1.00000
751.6 0.866025 0.500000i −0.500000 + 0.866025i 0.500000 0.866025i −0.866025 0.500000i 1.00000i −0.861658 + 1.49244i 1.00000i −0.500000 0.866025i −1.00000
751.7 0.866025 0.500000i −0.500000 + 0.866025i 0.500000 0.866025i −0.866025 0.500000i 1.00000i 0.293092 0.507650i 1.00000i −0.500000 0.866025i −1.00000
751.8 0.866025 0.500000i −0.500000 + 0.866025i 0.500000 0.866025i −0.866025 0.500000i 1.00000i 2.35156 4.07303i 1.00000i −0.500000 0.866025i −1.00000
841.1 −0.866025 0.500000i −0.500000 0.866025i 0.500000 + 0.866025i 0.866025 0.500000i 1.00000i −1.98397 3.43634i 1.00000i −0.500000 + 0.866025i −1.00000
841.2 −0.866025 0.500000i −0.500000 0.866025i 0.500000 + 0.866025i 0.866025 0.500000i 1.00000i −0.907693 1.57217i 1.00000i −0.500000 + 0.866025i −1.00000
841.3 −0.866025 0.500000i −0.500000 0.866025i 0.500000 + 0.866025i 0.866025 0.500000i 1.00000i 0.166951 + 0.289168i 1.00000i −0.500000 + 0.866025i −1.00000
841.4 −0.866025 0.500000i −0.500000 0.866025i 0.500000 + 0.866025i 0.866025 0.500000i 1.00000i 1.35869 + 2.35332i 1.00000i −0.500000 + 0.866025i −1.00000
841.5 0.866025 + 0.500000i −0.500000 0.866025i 0.500000 + 0.866025i −0.866025 + 0.500000i 1.00000i −1.41697 2.45427i 1.00000i −0.500000 + 0.866025i −1.00000
841.6 0.866025 + 0.500000i −0.500000 0.866025i 0.500000 + 0.866025i −0.866025 + 0.500000i 1.00000i −0.861658 1.49244i 1.00000i −0.500000 + 0.866025i −1.00000
841.7 0.866025 + 0.500000i −0.500000 0.866025i 0.500000 + 0.866025i −0.866025 + 0.500000i 1.00000i 0.293092 + 0.507650i 1.00000i −0.500000 + 0.866025i −1.00000
841.8 0.866025 + 0.500000i −0.500000 0.866025i 0.500000 + 0.866025i −0.866025 + 0.500000i 1.00000i 2.35156 + 4.07303i 1.00000i −0.500000 + 0.866025i −1.00000
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 841.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
37.e even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1110.2.x.c 16
37.e even 6 1 inner 1110.2.x.c 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1110.2.x.c 16 1.a even 1 1 trivial
1110.2.x.c 16 37.e even 6 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(T_{7}^{16} + \cdots\) acting on \(S_{2}^{\mathrm{new}}(1110, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 - T^{2} + T^{4} )^{4} \)
$3$ \( ( 1 + T + T^{2} )^{8} \)
$5$ \( ( 1 - T^{2} + T^{4} )^{4} \)
$7$ \( 7744 - 27456 T + 98576 T^{2} - 72016 T^{3} + 122140 T^{4} + 83412 T^{5} + 164354 T^{6} + 73410 T^{7} + 51193 T^{8} + 13560 T^{9} + 8012 T^{10} + 1726 T^{11} + 767 T^{12} + 80 T^{13} + 32 T^{14} + 2 T^{15} + T^{16} \)
$11$ \( ( 144 - 240 T - 152 T^{2} + 284 T^{3} + 73 T^{4} - 86 T^{5} - 14 T^{6} + 6 T^{7} + T^{8} )^{2} \)
$13$ \( 62001 - 1162332 T + 7880928 T^{2} - 11576640 T^{3} - 1254422 T^{4} + 11871732 T^{5} + 7069568 T^{6} + 406956 T^{7} - 609061 T^{8} - 80316 T^{9} + 45312 T^{10} + 10236 T^{11} - 966 T^{12} - 384 T^{13} + 16 T^{14} + 12 T^{15} + T^{16} \)
$17$ \( 224280576 - 1699356672 T + 5057058816 T^{2} - 5797057536 T^{3} + 2948551936 T^{4} - 372929280 T^{5} - 199654400 T^{6} + 46318656 T^{7} + 12825840 T^{8} - 2097120 T^{9} - 377280 T^{10} + 53832 T^{11} + 9089 T^{12} - 726 T^{13} - 109 T^{14} + 6 T^{15} + T^{16} \)
$19$ \( 20736 - 20736 T - 81792 T^{2} + 88704 T^{3} + 322288 T^{4} + 9216 T^{5} - 232792 T^{6} - 21060 T^{7} + 134681 T^{8} + 33258 T^{9} - 16590 T^{10} - 4464 T^{11} + 1911 T^{12} + 300 T^{13} - 38 T^{14} - 6 T^{15} + T^{16} \)
$23$ \( 732893184 + 66831482880 T^{2} + 14841653056 T^{4} + 1362786720 T^{6} + 66931636 T^{8} + 1898600 T^{10} + 31141 T^{12} + 274 T^{14} + T^{16} \)
$29$ \( 1021953024 + 1549366272 T^{2} + 617678800 T^{4} + 106306872 T^{6} + 9482913 T^{8} + 468620 T^{10} + 12822 T^{12} + 180 T^{14} + T^{16} \)
$31$ \( 210923421696 + 142552301568 T^{2} + 33037210624 T^{4} + 3439038976 T^{6} + 172573264 T^{8} + 4441448 T^{10} + 59281 T^{12} + 390 T^{14} + T^{16} \)
$37$ \( 3512479453921 - 1708773788394 T + 397687593395 T^{2} - 89176328702 T^{3} + 22484309517 T^{4} - 4882341364 T^{5} + 933507410 T^{6} - 167003200 T^{7} + 28068154 T^{8} - 4513600 T^{9} + 681890 T^{10} - 96388 T^{11} + 11997 T^{12} - 1286 T^{13} + 155 T^{14} - 18 T^{15} + T^{16} \)
$41$ \( 1018387759104 + 15113060352 T + 162382897152 T^{2} - 1147041792 T^{3} + 17655847168 T^{4} - 134268672 T^{5} + 986986944 T^{6} - 4819648 T^{7} + 39538512 T^{8} - 114912 T^{9} + 942288 T^{10} - 1872 T^{11} + 16236 T^{12} - 16 T^{13} + 156 T^{14} + T^{16} \)
$43$ \( 20350734336 + 59770798080 T^{2} + 33331876672 T^{4} + 3845808480 T^{6} + 190613236 T^{8} + 4795640 T^{10} + 63085 T^{12} + 406 T^{14} + T^{16} \)
$47$ \( ( -243648 + 118224 T + 178990 T^{2} + 4630 T^{3} - 13811 T^{4} - 1180 T^{5} + 208 T^{6} + 30 T^{7} + T^{8} )^{2} \)
$53$ \( 21403873050624 + 8150810886144 T + 4995640590336 T^{2} + 328070823936 T^{3} + 304852430848 T^{4} + 9160996864 T^{5} + 12647042304 T^{6} - 152830080 T^{7} + 296424592 T^{8} - 9613728 T^{9} + 5201888 T^{10} - 225408 T^{11} + 52620 T^{12} - 1936 T^{13} + 344 T^{14} - 12 T^{15} + T^{16} \)
$59$ \( 32303399897664 + 2010905979264 T - 4100998334832 T^{2} - 257887113120 T^{3} + 379059035692 T^{4} + 24724935228 T^{5} - 15030364862 T^{6} - 1091443038 T^{7} + 425935269 T^{8} + 40221378 T^{9} - 4731792 T^{10} - 487074 T^{11} + 44975 T^{12} + 3360 T^{13} - 232 T^{14} - 12 T^{15} + T^{16} \)
$61$ \( 1018387759104 + 22088318976 T - 162031214592 T^{2} - 3517839360 T^{3} + 17652584704 T^{4} + 92574720 T^{5} - 986865984 T^{6} - 1251072 T^{7} + 39528528 T^{8} - 21888 T^{9} - 942160 T^{10} + 576 T^{11} + 16236 T^{12} - 156 T^{14} + T^{16} \)
$67$ \( 4188342343936 - 4381044926976 T + 3161477089664 T^{2} - 1315332670720 T^{3} + 427598032816 T^{4} - 96641251872 T^{5} + 19382254040 T^{6} - 3074877600 T^{7} + 485542585 T^{8} - 61883766 T^{9} + 7857902 T^{10} - 755336 T^{11} + 76091 T^{12} - 5656 T^{13} + 470 T^{14} - 22 T^{15} + T^{16} \)
$71$ \( 56070144 - 764135424 T + 7742255616 T^{2} - 33537917184 T^{3} + 107561900992 T^{4} - 63782475648 T^{5} + 28872960944 T^{6} - 5399477920 T^{7} + 915977196 T^{8} - 82356564 T^{9} + 9662270 T^{10} - 581838 T^{11} + 73699 T^{12} - 2538 T^{13} + 309 T^{14} - 2 T^{15} + T^{16} \)
$73$ \( ( 22479232 + 4284800 T - 1378432 T^{2} - 213344 T^{3} + 30470 T^{4} + 3058 T^{5} - 301 T^{6} - 12 T^{7} + T^{8} )^{2} \)
$79$ \( 19917869147136 + 31652055733248 T + 21815306483712 T^{2} + 8023368440832 T^{3} + 1502587946176 T^{4} + 60587310528 T^{5} - 22957750464 T^{6} + 506374080 T^{7} + 1972100452 T^{8} + 450827412 T^{9} + 45858242 T^{10} + 1715502 T^{11} - 64733 T^{12} - 5580 T^{13} + 277 T^{14} + 36 T^{15} + T^{16} \)
$83$ \( 215266305024 - 620009717760 T + 1941570155520 T^{2} + 380025939456 T^{3} + 204281250496 T^{4} + 17901587008 T^{5} + 8968927424 T^{6} + 641132816 T^{7} + 259606132 T^{8} + 9765524 T^{9} + 4310490 T^{10} + 125714 T^{11} + 50443 T^{12} + 732 T^{13} + 301 T^{14} + 8 T^{15} + T^{16} \)
$89$ \( 214352087629824 + 1523191770390528 T + 3625837009311744 T^{2} + 127124076893184 T^{3} - 92113984579328 T^{4} - 3606222137856 T^{5} + 1723029300480 T^{6} + 86516814528 T^{7} - 12915306960 T^{8} - 757207776 T^{9} + 70500512 T^{10} + 5129796 T^{11} - 154383 T^{12} - 17010 T^{13} + 183 T^{14} + 42 T^{15} + T^{16} \)
$97$ \( 80703056646144 + 24110581899264 T^{2} + 2609354037568 T^{4} + 129472139744 T^{6} + 3164931188 T^{8} + 39890040 T^{10} + 262077 T^{12} + 838 T^{14} + T^{16} \)
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