Base field 4.4.16225.1
Generator \(w\), with minimal polynomial \(x^{4} - x^{3} - 13x^{2} + 6x + 36\); narrow class number \(2\) and class number \(1\).
Form
| Weight: | $[2, 2, 2, 2]$ |
| Level: | $[25, 5, -\frac{1}{3}w^{3} + \frac{1}{3}w^{2} + \frac{7}{3}w - 1]$ |
| Dimension: | $24$ |
| CM: | no |
| Base change: | yes |
| Newspace dimension: | $50$ |
Hecke eigenvalues ($q$-expansion)
The Hecke eigenvalue field is $\Q(e)$ where $e$ is a root of the defining polynomial:
| \(x^{24} - 66x^{22} + 1847x^{20} - 28728x^{18} + 273519x^{16} - 1653134x^{14} + 6356249x^{12} - 15138196x^{10} + 21014688x^{8} - 15146516x^{6} + 4394960x^{4} - 260624x^{2} + 144\) |
Show full eigenvalues Hide large eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| 4 | $[4, 2, \frac{1}{3}w^{3} - \frac{4}{3}w^{2} - \frac{4}{3}w + 6]$ | $\phantom{-}e$ |
| 4 | $[4, 2, -\frac{1}{3}w^{3} - \frac{2}{3}w^{2} + \frac{10}{3}w + 7]$ | $\phantom{-}e$ |
| 9 | $[9, 3, -\frac{1}{2}w^{3} + \frac{3}{2}w^{2} + \frac{7}{2}w - 9]$ | $\phantom{-}\frac{315071241079559}{8849428462974687632}e^{22} - \frac{41218247098381733}{17698856925949375264}e^{20} + \frac{3240268133395004}{50280843539628907}e^{18} - \frac{8745743885340675687}{8849428462974687632}e^{16} + \frac{81804757365846105017}{8849428462974687632}e^{14} - \frac{966957844922047557501}{17698856925949375264}e^{12} + \frac{903185576514836180451}{4424714231487343816}e^{10} - \frac{23534206856896201950}{50280843539628907}e^{8} + \frac{2727316820874263435353}{4424714231487343816}e^{6} - \frac{227129337237445841538}{553089278935917977}e^{4} + \frac{10500773552889891755}{100561687079257814}e^{2} - \frac{4963976647318282793}{1106178557871835954}$ |
| 9 | $[9, 3, \frac{1}{3}w^{3} + \frac{2}{3}w^{2} - \frac{7}{3}w - 5]$ | $\phantom{-}\frac{315071241079559}{8849428462974687632}e^{22} - \frac{41218247098381733}{17698856925949375264}e^{20} + \frac{3240268133395004}{50280843539628907}e^{18} - \frac{8745743885340675687}{8849428462974687632}e^{16} + \frac{81804757365846105017}{8849428462974687632}e^{14} - \frac{966957844922047557501}{17698856925949375264}e^{12} + \frac{903185576514836180451}{4424714231487343816}e^{10} - \frac{23534206856896201950}{50280843539628907}e^{8} + \frac{2727316820874263435353}{4424714231487343816}e^{6} - \frac{227129337237445841538}{553089278935917977}e^{4} + \frac{10500773552889891755}{100561687079257814}e^{2} - \frac{4963976647318282793}{1106178557871835954}$ |
| 11 | $[11, 11, -\frac{1}{6}w^{3} + \frac{1}{6}w^{2} + \frac{1}{6}w]$ | $-\frac{25220777727873693}{35397713851898750528}e^{23} + \frac{1664915525740384205}{35397713851898750528}e^{21} - \frac{2118596426705010585}{1608986993268125024}e^{19} + \frac{362690301756049705057}{17698856925949375264}e^{17} - \frac{6912802639561386818641}{35397713851898750528}e^{15} + \frac{41840651078088438210741}{35397713851898750528}e^{13} - \frac{20152998078796318998043}{4424714231487343816}e^{11} + \frac{8754751739166890127301}{804493496634062512}e^{9} - \frac{134315576099311641557797}{8849428462974687632}e^{7} + \frac{24362167621498153245945}{2212357115743671908}e^{5} - \frac{161935760162901109658}{50280843539628907}e^{3} + \frac{416871632978001764061}{2212357115743671908}e$ |
| 19 | $[19, 19, w + 1]$ | $\phantom{-}\frac{3693469858115171}{8849428462974687632}e^{23} - \frac{243554844769518533}{8849428462974687632}e^{21} + \frac{309488993927147025}{402246748317031256}e^{19} - \frac{52886260996607704849}{4424714231487343816}e^{17} + \frac{1005589638190643631999}{8849428462974687632}e^{15} - \frac{6066914146981707976565}{8849428462974687632}e^{13} + \frac{1454626653723194789986}{553089278935917977}e^{11} - \frac{1255876514538311052577}{201123374158515628}e^{9} + \frac{4773172893207767183875}{553089278935917977}e^{7} - \frac{3419602236034985941124}{553089278935917977}e^{5} + \frac{89928656872285924048}{50280843539628907}e^{3} - \frac{64605311507434799130}{553089278935917977}e$ |
| 19 | $[19, 19, \frac{1}{6}w^{3} - \frac{1}{6}w^{2} - \frac{13}{6}w + 2]$ | $\phantom{-}\frac{3693469858115171}{8849428462974687632}e^{23} - \frac{243554844769518533}{8849428462974687632}e^{21} + \frac{309488993927147025}{402246748317031256}e^{19} - \frac{52886260996607704849}{4424714231487343816}e^{17} + \frac{1005589638190643631999}{8849428462974687632}e^{15} - \frac{6066914146981707976565}{8849428462974687632}e^{13} + \frac{1454626653723194789986}{553089278935917977}e^{11} - \frac{1255876514538311052577}{201123374158515628}e^{9} + \frac{4773172893207767183875}{553089278935917977}e^{7} - \frac{3419602236034985941124}{553089278935917977}e^{5} + \frac{89928656872285924048}{50280843539628907}e^{3} - \frac{64605311507434799130}{553089278935917977}e$ |
| 25 | $[25, 5, -\frac{1}{3}w^{3} + \frac{1}{3}w^{2} + \frac{7}{3}w - 1]$ | $-1$ |
| 29 | $[29, 29, -\frac{1}{6}w^{3} + \frac{1}{6}w^{2} + \frac{13}{6}w]$ | $\phantom{-}\frac{29312333470014479}{35397713851898750528}e^{23} - \frac{1933340590544733849}{35397713851898750528}e^{21} + \frac{2457360659762033259}{1608986993268125024}e^{19} - \frac{420047871369950711273}{17698856925949375264}e^{17} + \frac{7989809057992366984595}{35397713851898750528}e^{15} - \frac{48226326065685180832209}{35397713851898750528}e^{13} + \frac{23139672671214415221579}{4424714231487343816}e^{11} - \frac{9996215245783517040699}{804493496634062512}e^{9} + \frac{152039559056428009971397}{8849428462974687632}e^{7} - \frac{27174284618020899858211}{2212357115743671908}e^{5} + \frac{351000033034267212571}{100561687079257814}e^{3} - \frac{433477445745700035725}{2212357115743671908}e$ |
| 29 | $[29, 29, w - 1]$ | $\phantom{-}\frac{29312333470014479}{35397713851898750528}e^{23} - \frac{1933340590544733849}{35397713851898750528}e^{21} + \frac{2457360659762033259}{1608986993268125024}e^{19} - \frac{420047871369950711273}{17698856925949375264}e^{17} + \frac{7989809057992366984595}{35397713851898750528}e^{15} - \frac{48226326065685180832209}{35397713851898750528}e^{13} + \frac{23139672671214415221579}{4424714231487343816}e^{11} - \frac{9996215245783517040699}{804493496634062512}e^{9} + \frac{152039559056428009971397}{8849428462974687632}e^{7} - \frac{27174284618020899858211}{2212357115743671908}e^{5} + \frac{351000033034267212571}{100561687079257814}e^{3} - \frac{433477445745700035725}{2212357115743671908}e$ |
| 31 | $[31, 31, \frac{1}{3}w^{3} - \frac{1}{3}w^{2} - \frac{10}{3}w + 3]$ | $\phantom{-}\frac{34366794856667971}{35397713851898750528}e^{23} - \frac{2270840277898020791}{35397713851898750528}e^{21} + \frac{2892773230192395971}{1608986993268125024}e^{19} - \frac{495826447991839304055}{17698856925949375264}e^{17} + \frac{9462965905248671243127}{35397713851898750528}e^{15} - \frac{57359069913891291487127}{35397713851898750528}e^{13} + \frac{27671357310463334400265}{4424714231487343816}e^{11} - \frac{12041641528488361528419}{804493496634062512}e^{9} + \frac{185084920272210899585103}{8849428462974687632}e^{7} - \frac{33634257952608778275479}{2212357115743671908}e^{5} + \frac{448967675102752794939}{100561687079257814}e^{3} - \frac{605844392461410243791}{2212357115743671908}e$ |
| 31 | $[31, 31, \frac{1}{6}w^{3} - \frac{1}{6}w^{2} - \frac{1}{6}w + 2]$ | $\phantom{-}\frac{34366794856667971}{35397713851898750528}e^{23} - \frac{2270840277898020791}{35397713851898750528}e^{21} + \frac{2892773230192395971}{1608986993268125024}e^{19} - \frac{495826447991839304055}{17698856925949375264}e^{17} + \frac{9462965905248671243127}{35397713851898750528}e^{15} - \frac{57359069913891291487127}{35397713851898750528}e^{13} + \frac{27671357310463334400265}{4424714231487343816}e^{11} - \frac{12041641528488361528419}{804493496634062512}e^{9} + \frac{185084920272210899585103}{8849428462974687632}e^{7} - \frac{33634257952608778275479}{2212357115743671908}e^{5} + \frac{448967675102752794939}{100561687079257814}e^{3} - \frac{605844392461410243791}{2212357115743671908}e$ |
| 41 | $[41, 41, -\frac{1}{2}w^{3} - \frac{1}{2}w^{2} + \frac{9}{2}w + 5]$ | $\phantom{-}\frac{15864038081123035}{17698856925949375264}e^{23} - \frac{523033240869849319}{8849428462974687632}e^{21} + \frac{1329253756727093953}{804493496634062512}e^{19} - \frac{113581524702944721587}{4424714231487343816}e^{17} + \frac{4320195243756273273499}{17698856925949375264}e^{15} - \frac{13037639388840684030359}{8849428462974687632}e^{13} + \frac{25025932704487294076375}{4424714231487343816}e^{11} - \frac{2703959074889367796885}{201123374158515628}e^{9} + \frac{41167931922161626625211}{2212357115743671908}e^{7} - \frac{7374253607203616903508}{553089278935917977}e^{5} + \frac{191240383685274905991}{50280843539628907}e^{3} - \frac{111181373206845104525}{553089278935917977}e$ |
| 41 | $[41, 41, -\frac{1}{2}w^{3} + \frac{3}{2}w^{2} + \frac{5}{2}w - 8]$ | $\phantom{-}\frac{15864038081123035}{17698856925949375264}e^{23} - \frac{523033240869849319}{8849428462974687632}e^{21} + \frac{1329253756727093953}{804493496634062512}e^{19} - \frac{113581524702944721587}{4424714231487343816}e^{17} + \frac{4320195243756273273499}{17698856925949375264}e^{15} - \frac{13037639388840684030359}{8849428462974687632}e^{13} + \frac{25025932704487294076375}{4424714231487343816}e^{11} - \frac{2703959074889367796885}{201123374158515628}e^{9} + \frac{41167931922161626625211}{2212357115743671908}e^{7} - \frac{7374253607203616903508}{553089278935917977}e^{5} + \frac{191240383685274905991}{50280843539628907}e^{3} - \frac{111181373206845104525}{553089278935917977}e$ |
| 59 | $[59, 59, \frac{1}{3}w^{3} - \frac{1}{3}w^{2} - \frac{10}{3}w - 1]$ | $-\frac{100259862439835}{17698856925949375264}e^{22} + \frac{4349785875061375}{17698856925949375264}e^{20} - \frac{2011654629403797}{804493496634062512}e^{18} - \frac{391279700449985169}{8849428462974687632}e^{16} + \frac{24599142849888969689}{17698856925949375264}e^{14} - \frac{276937718945008884017}{17698856925949375264}e^{12} + \frac{421240989377896541259}{4424714231487343816}e^{10} - \frac{134008496060025548021}{402246748317031256}e^{8} + \frac{2874252899693594438641}{4424714231487343816}e^{6} - \frac{346005442815212115047}{553089278935917977}e^{4} + \frac{10966702054003044132}{50280843539628907}e^{2} - \frac{7917677555318062497}{1106178557871835954}$ |
| 59 | $[59, 59, -\frac{1}{3}w^{3} + \frac{4}{3}w^{2} + \frac{7}{3}w - 7]$ | $-\frac{100259862439835}{17698856925949375264}e^{22} + \frac{4349785875061375}{17698856925949375264}e^{20} - \frac{2011654629403797}{804493496634062512}e^{18} - \frac{391279700449985169}{8849428462974687632}e^{16} + \frac{24599142849888969689}{17698856925949375264}e^{14} - \frac{276937718945008884017}{17698856925949375264}e^{12} + \frac{421240989377896541259}{4424714231487343816}e^{10} - \frac{134008496060025548021}{402246748317031256}e^{8} + \frac{2874252899693594438641}{4424714231487343816}e^{6} - \frac{346005442815212115047}{553089278935917977}e^{4} + \frac{10966702054003044132}{50280843539628907}e^{2} - \frac{7917677555318062497}{1106178557871835954}$ |
| 59 | $[59, 59, \frac{1}{2}w^{3} - \frac{1}{2}w^{2} - \frac{5}{2}w + 2]$ | $\phantom{-}\frac{2751673075969853}{35397713851898750528}e^{22} - \frac{181710824335935233}{35397713851898750528}e^{20} + \frac{231370619534721261}{1608986993268125024}e^{18} - \frac{39636489713824269185}{17698856925949375264}e^{16} + \frac{755464451982864362425}{35397713851898750528}e^{14} - \frac{4560712894895584136241}{35397713851898750528}e^{12} + \frac{2176133553271183329273}{4424714231487343816}e^{10} - \frac{921838695830948920393}{804493496634062512}e^{8} + \frac{13322681190401515002401}{8849428462974687632}e^{6} - \frac{2112449823325101581593}{2212357115743671908}e^{4} + \frac{10393894658953560727}{50280843539628907}e^{2} - \frac{21255752325454804857}{2212357115743671908}$ |
| 79 | $[79, 79, w^{2} - 11]$ | $\phantom{-}\frac{4714917557044067}{35397713851898750528}e^{22} - \frac{302661602282379759}{35397713851898750528}e^{20} + \frac{372073409109745687}{1608986993268125024}e^{18} - \frac{61040818862256750299}{17698856925949375264}e^{16} + \frac{1103569157396012717567}{35397713851898750528}e^{14} - \frac{6250165790596840290407}{35397713851898750528}e^{12} + \frac{2764044666030025023553}{4424714231487343816}e^{10} - \frac{1073667272532951853719}{804493496634062512}e^{8} + \frac{14269619352840684374327}{8849428462974687632}e^{6} - \frac{2225763356671299248833}{2212357115743671908}e^{4} + \frac{14175647667917352761}{50280843539628907}e^{2} - \frac{30174990899917534783}{2212357115743671908}$ |
| 79 | $[79, 79, \frac{1}{6}w^{3} - \frac{7}{6}w^{2} - \frac{7}{6}w + 3]$ | $\phantom{-}\frac{4714917557044067}{35397713851898750528}e^{22} - \frac{302661602282379759}{35397713851898750528}e^{20} + \frac{372073409109745687}{1608986993268125024}e^{18} - \frac{61040818862256750299}{17698856925949375264}e^{16} + \frac{1103569157396012717567}{35397713851898750528}e^{14} - \frac{6250165790596840290407}{35397713851898750528}e^{12} + \frac{2764044666030025023553}{4424714231487343816}e^{10} - \frac{1073667272532951853719}{804493496634062512}e^{8} + \frac{14269619352840684374327}{8849428462974687632}e^{6} - \frac{2225763356671299248833}{2212357115743671908}e^{4} + \frac{14175647667917352761}{50280843539628907}e^{2} - \frac{30174990899917534783}{2212357115743671908}$ |
| 89 | $[89, 89, -\frac{1}{6}w^{3} + \frac{1}{6}w^{2} + \frac{19}{6}w - 5]$ | $\phantom{-}\frac{5010191021036401}{35397713851898750528}e^{23} - \frac{329994393889025787}{35397713851898750528}e^{21} + \frac{418717081012541325}{1608986993268125024}e^{19} - \frac{71418496261622513747}{17698856925949375264}e^{17} + \frac{1354588006525840987213}{35397713851898750528}e^{15} - \frac{8142544365580038125683}{35397713851898750528}e^{13} + \frac{3880129187600925253675}{4424714231487343816}e^{11} - \frac{1653407222254260677585}{804493496634062512}e^{9} + \frac{24326138390432553881867}{8849428462974687632}e^{7} - \frac{3929268208658731053189}{2212357115743671908}e^{5} + \frac{31207094341679634349}{100561687079257814}e^{3} + \frac{103682632881778553585}{2212357115743671908}e$ |
Atkin-Lehner eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| $25$ | $[25, 5, -\frac{1}{3}w^{3} + \frac{1}{3}w^{2} + \frac{7}{3}w - 1]$ | $1$ |