Properties

Label 4.4.15188.1-13.1-a
Base field 4.4.15188.1
Weight $[2, 2, 2, 2]$
Level norm $13$
Level $[13, 13, -w^{3} + w^{2} + 6w - 3]$
Dimension $18$
CM no
Base change no

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Base field 4.4.15188.1

Generator \(w\), with minimal polynomial \(x^{4} - x^{3} - 7x^{2} + x + 2\); narrow class number \(2\) and class number \(1\).

Form

Weight: $[2, 2, 2, 2]$
Level: $[13, 13, -w^{3} + w^{2} + 6w - 3]$
Dimension: $18$
CM: no
Base change: no
Newspace dimension: $36$

Hecke eigenvalues ($q$-expansion)

The Hecke eigenvalue field is $\Q(e)$ where $e$ is a root of the defining polynomial:

\(x^{18} - 4x^{17} - 19x^{16} + 90x^{15} + 124x^{14} - 814x^{13} - 222x^{12} + 3802x^{11} - 1023x^{10} - 9774x^{9} + 5785x^{8} + 13540x^{7} - 11139x^{6} - 8912x^{5} + 9483x^{4} + 1598x^{3} - 2988x^{2} + 436x + 40\)

  Show full eigenvalues   Hide large eigenvalues

Norm Prime Eigenvalue
2 $[2, 2, w]$ $-\frac{17892337}{19430084}e^{17} + \frac{49752117}{19430084}e^{16} + \frac{100283421}{4857521}e^{15} - \frac{560781335}{9715042}e^{14} - \frac{899579169}{4857521}e^{13} + \frac{5090393611}{9715042}e^{12} + \frac{4126726013}{4857521}e^{11} - \frac{23950068733}{9715042}e^{10} - \frac{40771065907}{19430084}e^{9} + \frac{125059468565}{19430084}e^{8} + \frac{12790775120}{4857521}e^{7} - \frac{2362512209}{255659}e^{6} - \frac{23737116371}{19430084}e^{5} + \frac{130467874855}{19430084}e^{4} - \frac{1644009338}{4857521}e^{3} - \frac{18657141899}{9715042}e^{2} + \frac{1617822337}{4857521}e + \frac{150118723}{4857521}$
2 $[2, 2, -\frac{1}{2}w^{3} + w^{2} + \frac{3}{2}w]$ $\phantom{-}e$
11 $[11, 11, \frac{1}{2}w^{3} - w^{2} - \frac{5}{2}w + 4]$ $-\frac{9041561}{9715042}e^{17} + \frac{24394923}{9715042}e^{16} + \frac{101429639}{4857521}e^{15} - \frac{273823492}{4857521}e^{14} - \frac{910155190}{4857521}e^{13} + \frac{2473967285}{4857521}e^{12} + \frac{4173813415}{4857521}e^{11} - \frac{11581612074}{4857521}e^{10} - \frac{20601373283}{9715042}e^{9} + \frac{60166868135}{9715042}e^{8} + \frac{12941673877}{4857521}e^{7} - \frac{2261783112}{255659}e^{6} - \frac{12294724513}{9715042}e^{5} + \frac{62113632311}{9715042}e^{4} - \frac{1396029250}{4857521}e^{3} - \frac{8800353852}{4857521}e^{2} + \frac{1508705483}{4857521}e + \frac{117212480}{4857521}$
11 $[11, 11, -w^{3} + w^{2} + 6w + 1]$ $\phantom{-}\frac{19791728}{4857521}e^{17} - \frac{53120522}{4857521}e^{16} - \frac{445985714}{4857521}e^{15} + \frac{1194768496}{4857521}e^{14} + \frac{4026273003}{4857521}e^{13} - \frac{10819937141}{4857521}e^{12} - \frac{18621738169}{4857521}e^{11} + \frac{50799358593}{4857521}e^{10} + \frac{46515077964}{4857521}e^{9} - \frac{132432128813}{4857521}e^{8} - \frac{59469809434}{4857521}e^{7} + \frac{10004516786}{255659}e^{6} + \frac{29138356766}{4857521}e^{5} - \frac{138358767373}{4857521}e^{4} + \frac{6114970300}{4857521}e^{3} + \frac{39734529269}{4857521}e^{2} - \frac{7071866945}{4857521}e - \frac{605089910}{4857521}$
13 $[13, 13, -w^{3} + w^{2} + 6w - 3]$ $-1$
19 $[19, 19, -\frac{1}{2}w^{3} + w^{2} + \frac{5}{2}w]$ $-\frac{18899797}{4857521}e^{17} + \frac{53723936}{4857521}e^{16} + \frac{422993153}{4857521}e^{15} - \frac{1213756943}{4857521}e^{14} - \frac{3787801760}{4857521}e^{13} + \frac{11046476590}{4857521}e^{12} + \frac{17344305869}{4857521}e^{11} - \frac{52133850000}{4857521}e^{10} - \frac{42724887602}{4857521}e^{9} + \frac{136618570450}{4857521}e^{8} + \frac{53220580585}{4857521}e^{7} - \frac{10371934163}{255659}e^{6} - \frac{23662097739}{4857521}e^{5} + \frac{144087574192}{4857521}e^{4} - \frac{8378261980}{4857521}e^{3} - \frac{41519335691}{4857521}e^{2} + \frac{7397362898}{4857521}e + \frac{648867954}{4857521}$
23 $[23, 23, -\frac{1}{2}w^{3} + 2w^{2} - \frac{1}{2}w - 2]$ $-\frac{50717567}{9715042}e^{17} + \frac{136530567}{9715042}e^{16} + \frac{570927638}{4857521}e^{15} - \frac{1536017185}{4857521}e^{14} - \frac{5147637353}{4857521}e^{13} + \frac{13919576082}{4857521}e^{12} + \frac{23762350328}{4857521}e^{11} - \frac{65420120412}{4857521}e^{10} - \frac{118347187809}{9715042}e^{9} + \frac{341616538775}{9715042}e^{8} + \frac{75236831548}{4857521}e^{7} - \frac{12930100655}{255659}e^{6} - \frac{72619738611}{9715042}e^{5} + \frac{358507641269}{9715042}e^{4} - \frac{8191525246}{4857521}e^{3} - \frac{51618860860}{4857521}e^{2} + \frac{9012068158}{4857521}e + \frac{809699324}{4857521}$
31 $[31, 31, -w^{3} + w^{2} + 6w - 1]$ $-\frac{12983168}{4857521}e^{17} + \frac{35213504}{4857521}e^{16} + \frac{292842727}{4857521}e^{15} - \frac{793168005}{4857521}e^{14} - \frac{2649113389}{4857521}e^{13} + \frac{7194494000}{4857521}e^{12} + \frac{12299080998}{4857521}e^{11} - \frac{33832584848}{4857521}e^{10} - \frac{30929927620}{4857521}e^{9} + \frac{88327485974}{4857521}e^{8} + \frac{40034754341}{4857521}e^{7} - \frac{6679320639}{255659}e^{6} - \frac{20259898376}{4857521}e^{5} + \frac{92382009564}{4857521}e^{4} - \frac{3559251140}{4857521}e^{3} - \frac{26456668523}{4857521}e^{2} + \frac{4625656656}{4857521}e + \frac{381107924}{4857521}$
31 $[31, 31, \frac{1}{2}w^{3} - \frac{7}{2}w]$ $-\frac{11783539}{4857521}e^{17} + \frac{30460793}{4857521}e^{16} + \frac{266994464}{4857521}e^{15} - \frac{683745399}{4857521}e^{14} - \frac{2425989843}{4857521}e^{13} + \frac{6178166158}{4857521}e^{12} + \frac{11306526328}{4857521}e^{11} - \frac{28938122802}{4857521}e^{10} - \frac{28515340405}{4857521}e^{9} + \frac{75271985655}{4857521}e^{8} + \frac{36987148408}{4857521}e^{7} - \frac{5675794711}{255659}e^{6} - \frac{18834504313}{4857521}e^{5} + \frac{78386808089}{4857521}e^{4} - \frac{3040158512}{4857521}e^{3} - \frac{22472015903}{4857521}e^{2} + \frac{4098695180}{4857521}e + \frac{332966518}{4857521}$
43 $[43, 43, \frac{1}{2}w^{3} - w^{2} - \frac{9}{2}w + 2]$ $-\frac{27238387}{9715042}e^{17} + \frac{70249387}{9715042}e^{16} + \frac{307825204}{4857521}e^{15} - \frac{786799165}{4857521}e^{14} - \frac{2788118166}{4857521}e^{13} + \frac{7093260210}{4857521}e^{12} + \frac{12942332118}{4857521}e^{11} - \frac{33144053076}{4857521}e^{10} - \frac{64977700215}{9715042}e^{9} + \frac{171962856001}{9715042}e^{8} + \frac{41985760714}{4857521}e^{7} - \frac{6461290837}{255659}e^{6} - \frac{43139144233}{9715042}e^{5} + \frac{177526007327}{9715042}e^{4} - \frac{2898445399}{4857521}e^{3} - \frac{25205636384}{4857521}e^{2} + \frac{4368161548}{4857521}e + \frac{348260252}{4857521}$
67 $[67, 67, w^{2} - w - 5]$ $-\frac{11662167}{9715042}e^{17} + \frac{33832343}{9715042}e^{16} + \frac{130194272}{4857521}e^{15} - \frac{382714748}{4857521}e^{14} - \frac{1163147006}{4857521}e^{13} + \frac{3489530803}{4857521}e^{12} + \frac{5314906737}{4857521}e^{11} - \frac{16505296588}{4857521}e^{10} - \frac{26141071905}{9715042}e^{9} + \frac{86695544819}{9715042}e^{8} + \frac{16267902747}{4857521}e^{7} - \frac{3294467854}{255659}e^{6} - \frac{14510215273}{9715042}e^{5} + \frac{91304343029}{9715042}e^{4} - \frac{2524738418}{4857521}e^{3} - \frac{13033569974}{4857521}e^{2} + \frac{2250878899}{4857521}e + \frac{219292600}{4857521}$
67 $[67, 67, -\frac{1}{2}w^{3} + \frac{11}{2}w + 2]$ $-\frac{2873343}{255659}e^{17} + \frac{7662380}{255659}e^{16} + \frac{64730712}{255659}e^{15} - \frac{172255614}{255659}e^{14} - \frac{583966624}{255659}e^{13} + \frac{1559521018}{255659}e^{12} + \frac{2696983254}{255659}e^{11} - \frac{7322753354}{255659}e^{10} - \frac{6718949180}{255659}e^{9} + \frac{19104112153}{255659}e^{8} + \frac{8549913891}{255659}e^{7} - \frac{27461726382}{255659}e^{6} - \frac{4144689776}{255659}e^{5} + \frac{20031204843}{255659}e^{4} - \frac{904841356}{255659}e^{3} - \frac{5767972549}{255659}e^{2} + \frac{1013667361}{255659}e + \frac{90613786}{255659}$
73 $[73, 73, w^{2} + w + 1]$ $\phantom{-}\frac{30976413}{9715042}e^{17} - \frac{88165477}{9715042}e^{16} - \frac{346232444}{4857521}e^{15} + \frac{994246401}{4857521}e^{14} + \frac{3097041115}{4857521}e^{13} - \frac{9029007630}{4857521}e^{12} - \frac{14169363393}{4857521}e^{11} + \frac{42492908052}{4857521}e^{10} + \frac{69784631583}{9715042}e^{9} - \frac{221910517443}{9715042}e^{8} - \frac{43501960254}{4857521}e^{7} + \frac{8387641954}{255659}e^{6} + \frac{38982709077}{9715042}e^{5} - \frac{232144898027}{9715042}e^{4} + \frac{6572327349}{4857521}e^{3} + \frac{33475070449}{4857521}e^{2} - \frac{5861395993}{4857521}e - \frac{584213322}{4857521}$
79 $[79, 79, \frac{1}{2}w^{3} + w^{2} - \frac{5}{2}w - 2]$ $-\frac{14894510}{4857521}e^{17} + \frac{37752616}{4857521}e^{16} + \frac{338114049}{4857521}e^{15} - \frac{848330538}{4857521}e^{14} - \frac{3077132530}{4857521}e^{13} + \frac{7683985517}{4857521}e^{12} + \frac{14359060025}{4857521}e^{11} - \frac{36159328831}{4857521}e^{10} - \frac{36263587006}{4857521}e^{9} + \frac{94806116175}{4857521}e^{8} + \frac{47242010435}{4857521}e^{7} - \frac{7236204882}{255659}e^{6} - \frac{24643069770}{4857521}e^{5} + \frac{101600718868}{4857521}e^{4} - \frac{3128261126}{4857521}e^{3} - \frac{29768700008}{4857521}e^{2} + \frac{5045381670}{4857521}e + \frac{504686160}{4857521}$
81 $[81, 3, -3]$ $\phantom{-}\frac{8118361}{4857521}e^{17} - \frac{22702427}{4857521}e^{16} - \frac{181857085}{4857521}e^{15} + \frac{512747839}{4857521}e^{14} + \frac{1628731896}{4857521}e^{13} - \frac{4666906839}{4857521}e^{12} - \frac{7447902884}{4857521}e^{11} + \frac{22040136479}{4857521}e^{10} + \frac{18268912829}{4857521}e^{9} - \frac{57832595586}{4857521}e^{8} - \frac{22511802579}{4857521}e^{7} + \frac{4396818764}{255659}e^{6} + \frac{9586568203}{4857521}e^{5} - \frac{61056336969}{4857521}e^{4} + \frac{4018303005}{4857521}e^{3} + \frac{17471363745}{4857521}e^{2} - \frac{3291906168}{4857521}e - \frac{226943542}{4857521}$
83 $[83, 83, 2w^{3} - 2w^{2} - 12w + 5]$ $\phantom{-}\frac{5404095}{9715042}e^{17} - \frac{18014803}{9715042}e^{16} - \frac{58214089}{4857521}e^{15} + \frac{204450657}{4857521}e^{14} + \frac{494052666}{4857521}e^{13} - \frac{1870093122}{4857521}e^{12} - \frac{2084774869}{4857521}e^{11} + \frac{8870036089}{4857521}e^{10} + \frac{8881264353}{9715042}e^{9} - \frac{46696796833}{9715042}e^{8} - \frac{3806341762}{4857521}e^{7} + \frac{1779249138}{255659}e^{6} - \frac{2391235695}{9715042}e^{5} - \frac{49622441293}{9715042}e^{4} + \frac{3816371435}{4857521}e^{3} + \frac{7200962840}{4857521}e^{2} - \frac{1587481386}{4857521}e - \frac{121776152}{4857521}$
83 $[83, 83, -\frac{1}{2}w^{3} - w^{2} + \frac{1}{2}w + 2]$ $-\frac{6631332}{4857521}e^{17} + \frac{18820312}{4857521}e^{16} + \frac{148945552}{4857521}e^{15} - \frac{424128376}{4857521}e^{14} - \frac{1342285212}{4857521}e^{13} + \frac{3843211476}{4857521}e^{12} + \frac{6214994314}{4857521}e^{11} - \frac{18009334591}{4857521}e^{10} - \frac{15610795476}{4857521}e^{9} + \frac{46682806579}{4857521}e^{8} + \frac{20176306588}{4857521}e^{7} - \frac{3490496531}{255659}e^{6} - \frac{10005108856}{4857521}e^{5} + \frac{47608261587}{4857521}e^{4} - \frac{2232124342}{4857521}e^{3} - \frac{13488221406}{4857521}e^{2} + \frac{2573432392}{4857521}e + \frac{207219868}{4857521}$
89 $[89, 89, -\frac{3}{2}w^{3} + 2w^{2} + \frac{17}{2}w - 2]$ $\phantom{-}\frac{49917367}{4857521}e^{17} - \frac{138978206}{4857521}e^{16} - \frac{1118643249}{4857521}e^{15} + \frac{3131578549}{4857521}e^{14} + \frac{10031131560}{4857521}e^{13} - \frac{28415303572}{4857521}e^{12} - \frac{45997003278}{4857521}e^{11} + \frac{133660185474}{4857521}e^{10} + \frac{113470011077}{4857521}e^{9} - \frac{348976377624}{4857521}e^{8} - \frac{141676105837}{4857521}e^{7} + \frac{26384396507}{255659}e^{6} + \frac{63778654430}{4857521}e^{5} - \frac{364768529456}{4857521}e^{4} + \frac{20931345709}{4857521}e^{3} + \frac{104513867107}{4857521}e^{2} - \frac{18937211130}{4857521}e - \frac{1551889810}{4857521}$
89 $[89, 89, \frac{3}{2}w^{3} - 2w^{2} - \frac{21}{2}w + 2]$ $\phantom{-}\frac{42491978}{4857521}e^{17} - \frac{112832468}{4857521}e^{16} - \frac{958531823}{4857521}e^{15} + \frac{2535331850}{4857521}e^{14} + \frac{8665287847}{4857521}e^{13} - \frac{22939113380}{4857521}e^{12} - \frac{40149949191}{4857521}e^{11} + \frac{107619920066}{4857521}e^{10} + \frac{100551049103}{4857521}e^{9} - \frac{280452888266}{4857521}e^{8} - \frac{129148253580}{4857521}e^{7} + \frac{21188067198}{255659}e^{6} + \frac{64205061659}{4857521}e^{5} - \frac{293149276226}{4857521}e^{4} + \frac{12215900453}{4857521}e^{3} + \frac{84191876052}{4857521}e^{2} - \frac{14934935074}{4857521}e - \frac{1250541380}{4857521}$
97 $[97, 97, \frac{1}{2}w^{3} - w^{2} - \frac{1}{2}w - 2]$ $-\frac{35002899}{9715042}e^{17} + \frac{97462769}{9715042}e^{16} + \frac{392056502}{4857521}e^{15} - \frac{1098465417}{4857521}e^{14} - \frac{3513361199}{4857521}e^{13} + \frac{9973697505}{4857521}e^{12} + \frac{16093717871}{4857521}e^{11} - \frac{46965559943}{4857521}e^{10} - \frac{79291682059}{9715042}e^{9} + \frac{245681494747}{9715042}e^{8} + \frac{49442067105}{4857521}e^{7} - \frac{9312674348}{255659}e^{6} - \frac{44753484655}{9715042}e^{5} + \frac{258462447159}{9715042}e^{4} - \frac{6927183772}{4857521}e^{3} - \frac{37175628881}{4857521}e^{2} + \frac{6357780633}{4857521}e + \frac{572087654}{4857521}$
Display number of eigenvalues

Atkin-Lehner eigenvalues

Norm Prime Eigenvalue
$13$ $[13, 13, -w^{3} + w^{2} + 6w - 3]$ $1$