# Properties

 Label 4.4.8789.1-29.1-b Base field 4.4.8789.1 Weight $[2, 2, 2, 2]$ Level norm $29$ Level $[29, 29, w^{3} - 2w^{2} - 5w]$ Dimension $13$ CM no Base change no

# Related objects

• L-function not available

## Base field 4.4.8789.1

Generator $$w$$, with minimal polynomial $$x^{4} - x^{3} - 6x^{2} - 2x + 1$$; narrow class number $$1$$ and class number $$1$$.

## Form

 Weight: $[2, 2, 2, 2]$ Level: $[29, 29, w^{3} - 2w^{2} - 5w]$ Dimension: $13$ CM: no Base change: no Newspace dimension: $20$

## Hecke eigenvalues ($q$-expansion)

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

 $$x^{13} - 6x^{12} - 24x^{11} + 194x^{10} + 33x^{9} - 1825x^{8} + 1785x^{7} + 5153x^{6} - 8740x^{5} - 4x^{4} + 5872x^{3} - 2464x^{2} - 64x + 64$$
Norm Prime Eigenvalue
5 $[5, 5, -w^{3} + 2w^{2} + 3w]$ $\phantom{-}e$
7 $[7, 7, w - 1]$ $\phantom{-}\frac{375600423}{24395599600}e^{12} - \frac{2105432117}{24395599600}e^{11} - \frac{4962367373}{12197799800}e^{10} + \frac{1388020855}{487911992}e^{9} + \frac{42363607809}{24395599600}e^{8} - \frac{341661443341}{12197799800}e^{7} + \frac{187574767243}{12197799800}e^{6} + \frac{277576735397}{3049449950}e^{5} - \frac{2311653247803}{24395599600}e^{4} - \frac{325607552457}{6098899900}e^{3} + \frac{82071637279}{1219779980}e^{2} - \frac{4898486867}{1524724975}e - \frac{3598387411}{1524724975}$
11 $[11, 11, -w^{3} + 2w^{2} + 4w]$ $-\frac{535916973}{24395599600}e^{12} + \frac{2656026287}{24395599600}e^{11} + \frac{7705431293}{12197799800}e^{10} - \frac{1738610079}{487911992}e^{9} - \frac{101730179459}{24395599600}e^{8} + \frac{420486970471}{12197799800}e^{7} - \frac{64879481803}{12197799800}e^{6} - \frac{325232517957}{3049449950}e^{5} + \frac{1993184997513}{24395599600}e^{4} + \frac{295430581067}{6098899900}e^{3} - \frac{16639307337}{304944995}e^{2} + \frac{21777623437}{1524724975}e - \frac{1383008524}{1524724975}$
13 $[13, 13, -2w^{3} + 3w^{2} + 10w - 2]$ $\phantom{-}\frac{103960867}{9758239840}e^{12} - \frac{73047851}{1219779980}e^{11} - \frac{343732133}{1219779980}e^{10} + \frac{1916233855}{975823984}e^{9} + \frac{11828438231}{9758239840}e^{8} - \frac{186219162833}{9758239840}e^{7} + \frac{103029262269}{9758239840}e^{6} + \frac{580259406329}{9758239840}e^{5} - \frac{322438638471}{4879119920}e^{4} - \frac{7522117981}{304944995}e^{3} + \frac{5867973691}{121977998}e^{2} - \frac{3955342214}{304944995}e - \frac{272156472}{304944995}$
16 $[16, 2, 2]$ $\phantom{-}\frac{568471241}{12197799800}e^{12} - \frac{381247503}{1524724975}e^{11} - \frac{15492595127}{12197799800}e^{10} + \frac{1997809019}{243955996}e^{9} + \frac{79809698453}{12197799800}e^{8} - \frac{968617830649}{12197799800}e^{7} + \frac{103107091243}{3049449950}e^{6} + \frac{754325252633}{3049449950}e^{5} - \frac{3003535544231}{12197799800}e^{4} - \frac{1391306413921}{12197799800}e^{3} + \frac{41062029525}{243955996}e^{2} - \frac{39705033023}{1524724975}e - \frac{102047614}{1524724975}$
17 $[17, 17, -w^{3} + 2w^{2} + 5w - 3]$ $-\frac{8727613}{304944995}e^{12} + \frac{397941063}{2439559960}e^{11} + \frac{898035961}{1219779980}e^{10} - \frac{325467373}{60988999}e^{9} - \frac{3108895741}{1219779980}e^{8} + \frac{126010532007}{2439559960}e^{7} - \frac{17448564479}{487911992}e^{6} - \frac{390812712141}{2439559960}e^{5} + \frac{99585065127}{487911992}e^{4} + \frac{20856534946}{304944995}e^{3} - \frac{92978822647}{609889990}e^{2} + \frac{6376390761}{304944995}e + \frac{2617651584}{304944995}$
17 $[17, 17, -w^{3} + w^{2} + 5w]$ $\phantom{-}\frac{777796361}{24395599600}e^{12} - \frac{1946057557}{12197799800}e^{11} - \frac{11308538731}{12197799800}e^{10} + \frac{2578573249}{487911992}e^{9} + \frac{155913557113}{24395599600}e^{8} - \frac{1282572844409}{24395599600}e^{7} + \frac{104729643847}{24395599600}e^{6} + \frac{4291808154487}{24395599600}e^{5} - \frac{1296514155653}{12197799800}e^{4} - \frac{1586742195943}{12197799800}e^{3} + \frac{88747944177}{1219779980}e^{2} + \frac{42442355697}{3049449950}e + \frac{2242696883}{1524724975}$
17 $[17, 17, -w^{2} + 2w + 1]$ $\phantom{-}\frac{1330576321}{48791199200}e^{12} - \frac{466552909}{3049449950}e^{11} - \frac{1096712791}{1524724975}e^{10} + \frac{4897014761}{975823984}e^{9} + \frac{148796134093}{48791199200}e^{8} - \frac{2383227628819}{48791199200}e^{7} + \frac{1327935635407}{48791199200}e^{6} + \frac{7497306795267}{48791199200}e^{5} - \frac{4059982288093}{24395599600}e^{4} - \frac{445619215147}{6098899900}e^{3} + \frac{13524912997}{121977998}e^{2} - \frac{28002577672}{1524724975}e + \frac{648622104}{1524724975}$
19 $[19, 19, w^{2} - w - 2]$ $-\frac{583065129}{48791199200}e^{12} + \frac{459147937}{6098899900}e^{11} + \frac{417062549}{1524724975}e^{10} - \frac{2366295473}{975823984}e^{9} + \frac{2022543243}{48791199200}e^{8} + \frac{1102399303651}{48791199200}e^{7} - \frac{1249157631983}{48791199200}e^{6} - \frac{3007386994643}{48791199200}e^{5} + \frac{2891798550817}{24395599600}e^{4} - \frac{50865013737}{6098899900}e^{3} - \frac{23900126691}{304944995}e^{2} + \frac{86575283321}{3049449950}e + \frac{4179998919}{1524724975}$
29 $[29, 29, w^{3} - 2w^{2} - 5w]$ $-1$
29 $[29, 29, w^{2} - w - 3]$ $-\frac{1664931697}{48791199200}e^{12} + \frac{259035934}{1524724975}e^{11} + \frac{3031841259}{3049449950}e^{10} - \frac{5470651241}{975823984}e^{9} - \frac{337443220301}{48791199200}e^{8} + \frac{2697096211923}{48791199200}e^{7} - \frac{181323502079}{48791199200}e^{6} - \frac{8806748062739}{48791199200}e^{5} + \frac{2714122271221}{24395599600}e^{4} + \frac{718396828899}{6098899900}e^{3} - \frac{24134348087}{304944995}e^{2} - \frac{2948695437}{3049449950}e + \frac{2262502527}{1524724975}$
31 $[31, 31, -w^{3} + 2w^{2} + 3w - 2]$ $\phantom{-}\frac{816127807}{24395599600}e^{12} - \frac{2241151029}{12197799800}e^{11} - \frac{5469315381}{6098899900}e^{10} + \frac{2942836411}{487911992}e^{9} + \frac{102552317631}{24395599600}e^{8} - \frac{1435329718103}{24395599600}e^{7} + \frac{709523941979}{24395599600}e^{6} + \frac{4556590052379}{24395599600}e^{5} - \frac{585027352149}{3049449950}e^{4} - \frac{304036744489}{3049449950}e^{3} + \frac{40159861193}{304944995}e^{2} - \frac{10739262808}{1524724975}e - \frac{3538381834}{1524724975}$
43 $[43, 43, 2w^{3} - 3w^{2} - 11w]$ $-\frac{66421847}{2439559960}e^{12} + \frac{51159039}{304944995}e^{11} + \frac{767521871}{1219779980}e^{10} - \frac{1320510035}{243955996}e^{9} - \frac{244224771}{2439559960}e^{8} + \frac{24724194349}{487911992}e^{7} - \frac{137590391443}{2439559960}e^{6} - \frac{68707867115}{487911992}e^{5} + \frac{79933227098}{304944995}e^{4} - \frac{2796313027}{243955996}e^{3} - \frac{105246508579}{609889990}e^{2} + \frac{22788122064}{304944995}e + \frac{859676754}{304944995}$
47 $[47, 47, w^{3} - 7w - 4]$ $-\frac{420754963}{6098899900}e^{12} + \frac{2199013077}{6098899900}e^{11} + \frac{11808691551}{6098899900}e^{10} - \frac{2903783573}{243955996}e^{9} - \frac{70278093079}{6098899900}e^{8} + \frac{179278416473}{1524724975}e^{7} - \frac{195998996391}{6098899900}e^{6} - \frac{1179041486503}{3049449950}e^{5} + \frac{1875394916093}{6098899900}e^{4} + \frac{395804152217}{1524724975}e^{3} - \frac{277187364101}{1219779980}e^{2} - \frac{25974575367}{1524724975}e + \frac{21397819164}{1524724975}$
53 $[53, 53, -2w^{3} + 3w^{2} + 9w - 1]$ $-\frac{7698307}{4879119920}e^{12} + \frac{23719223}{2439559960}e^{11} + \frac{10195557}{243955996}e^{10} - \frac{161259825}{487911992}e^{9} - \frac{874920851}{4879119920}e^{8} + \frac{16986996407}{4879119920}e^{7} - \frac{7818032907}{4879119920}e^{6} - \frac{65303870951}{4879119920}e^{5} + \frac{6305339797}{609889990}e^{4} + \frac{20329442357}{1219779980}e^{3} - \frac{12503660183}{1219779980}e^{2} - \frac{4264081279}{304944995}e + \frac{123524490}{60988999}$
61 $[61, 61, -w - 3]$ $-\frac{162112127}{3049449950}e^{12} + \frac{7378638629}{24395599600}e^{11} + \frac{16753599581}{12197799800}e^{10} - \frac{2413751839}{243955996}e^{9} - \frac{60400685139}{12197799800}e^{8} + \frac{2334358926989}{24395599600}e^{7} - \frac{1558554212077}{24395599600}e^{6} - \frac{7196694615927}{24395599600}e^{5} + \frac{8983215914321}{24395599600}e^{4} + \frac{172270580366}{1524724975}e^{3} - \frac{164314579493}{609889990}e^{2} + \frac{91685926504}{1524724975}e + \frac{17047937642}{1524724975}$
73 $[73, 73, -w^{3} + 2w^{2} + 3w - 3]$ $\phantom{-}\frac{3029961757}{24395599600}e^{12} - \frac{4013761387}{6098899900}e^{11} - \frac{41749501527}{12197799800}e^{10} + \frac{10542618717}{487911992}e^{9} + \frac{454915637081}{24395599600}e^{8} - \frac{5140473679423}{24395599600}e^{7} + \frac{1927020555569}{24395599600}e^{6} + \frac{16272891355389}{24395599600}e^{5} - \frac{954913108782}{1524724975}e^{4} - \frac{4222398772521}{12197799800}e^{3} + \frac{110768521005}{243955996}e^{2} - \frac{172720651721}{3049449950}e - \frac{15479481939}{1524724975}$
73 $[73, 73, w^{3} - w^{2} - 7w - 1]$ $-\frac{1778300237}{24395599600}e^{12} + \frac{4690741719}{12197799800}e^{11} + \frac{3066000344}{1524724975}e^{10} - \frac{6154089715}{487911992}e^{9} - \frac{269125039621}{24395599600}e^{8} + \frac{2993275898653}{24395599600}e^{7} - \frac{1098899306549}{24395599600}e^{6} - \frac{9402631122929}{24395599600}e^{5} + \frac{1095487222469}{3049449950}e^{4} + \frac{283604081932}{1524724975}e^{3} - \frac{306552555299}{1219779980}e^{2} + \frac{69983148238}{1524724975}e + \frac{10644961514}{1524724975}$
81 $[81, 3, -3]$ $-\frac{52969605}{975823984}e^{12} + \frac{374895807}{1219779980}e^{11} + \frac{1716810589}{1219779980}e^{10} - \frac{4899661517}{487911992}e^{9} - \frac{5125932937}{975823984}e^{8} + \frac{472443525199}{4879119920}e^{7} - \frac{305346058183}{4879119920}e^{6} - \frac{1442640051187}{4879119920}e^{5} + \frac{880792107437}{2439559960}e^{4} + \frac{123682718019}{1219779980}e^{3} - \frac{150320848229}{609889990}e^{2} + \frac{20259090393}{304944995}e + \frac{3340813816}{304944995}$
83 $[83, 83, -2w^{3} + 3w^{2} + 9w + 1]$ $-\frac{224776811}{6098899900}e^{12} + \frac{311659011}{1524724975}e^{11} + \frac{5974903547}{6098899900}e^{10} - \frac{818361153}{121977998}e^{9} - \frac{26444270763}{6098899900}e^{8} + \frac{398873554599}{6098899900}e^{7} - \frac{108162695701}{3049449950}e^{6} - \frac{630452317341}{3049449950}e^{5} + \frac{1386488254571}{6098899900}e^{4} + \frac{634844688121}{6098899900}e^{3} - \frac{104147063171}{609889990}e^{2} + \frac{21050478301}{1524724975}e + \frac{12793758108}{1524724975}$
 Display number of eigenvalues

## Atkin-Lehner eigenvalues

Norm Prime Eigenvalue
$29$ $[29, 29, w^{3} - 2w^{2} - 5w]$ $1$