Base field 4.4.7053.1
Generator \(w\), with minimal polynomial \(x^{4} - 2x^{3} - 4x^{2} + 3x + 3\); narrow class number \(1\) and class number \(1\).
Form
| Weight: | $[2, 2, 2, 2]$ |
| Level: | $[31, 31, w^{3} - 2w^{2} - 4w + 2]$ |
| Dimension: | $11$ |
| CM: | no |
| Base change: | no |
| Newspace dimension: | $17$ |
Hecke eigenvalues ($q$-expansion)
The Hecke eigenvalue field is $\Q(e)$ where $e$ is a root of the defining polynomial:
| \(x^{11} - 3x^{10} - 18x^{9} + 51x^{8} + 117x^{7} - 291x^{6} - 358x^{5} + 634x^{4} + 556x^{3} - 352x^{2} - 390x - 82\) |
Show full eigenvalues Hide large eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| 3 | $[3, 3, w]$ | $\phantom{-}e$ |
| 7 | $[7, 7, w^{2} - w - 2]$ | $\phantom{-}\frac{1}{29}e^{10} - \frac{23}{29}e^{9} + \frac{7}{29}e^{8} + \frac{433}{29}e^{7} - \frac{220}{29}e^{6} - \frac{2764}{29}e^{5} + \frac{924}{29}e^{4} + \frac{6833}{29}e^{3} - \frac{384}{29}e^{2} - \frac{4707}{29}e - \frac{1428}{29}$ |
| 9 | $[9, 3, w^{2} - 2w - 1]$ | $\phantom{-}\frac{26}{1421}e^{10} + \frac{127}{1421}e^{9} - \frac{17}{29}e^{8} - \frac{1908}{1421}e^{7} + \frac{7127}{1421}e^{6} + \frac{1255}{203}e^{5} - \frac{18142}{1421}e^{4} - \frac{15830}{1421}e^{3} + \frac{2776}{1421}e^{2} + \frac{1835}{203}e + \frac{8228}{1421}$ |
| 13 | $[13, 13, w^{3} - 3w^{2} - w + 5]$ | $\phantom{-}\frac{445}{1421}e^{10} - \frac{2144}{1421}e^{9} - \frac{127}{29}e^{8} + \frac{36317}{1421}e^{7} + \frac{25408}{1421}e^{6} - \frac{29278}{203}e^{5} - \frac{31882}{1421}e^{4} + \frac{442952}{1421}e^{3} + \frac{26088}{1421}e^{2} - \frac{37746}{203}e - \frac{77790}{1421}$ |
| 13 | $[13, 13, w^{3} - 3w^{2} - w + 2]$ | $\phantom{-}\frac{88}{1421}e^{10} - \frac{226}{1421}e^{9} - \frac{33}{29}e^{8} + \frac{4145}{1421}e^{7} + \frac{9147}{1421}e^{6} - \frac{3560}{203}e^{5} - \frac{13199}{1421}e^{4} + \frac{49608}{1421}e^{3} - \frac{14652}{1421}e^{2} - \frac{1066}{203}e + \frac{7736}{1421}$ |
| 16 | $[16, 2, 2]$ | $-\frac{1195}{1421}e^{10} + \frac{3618}{1421}e^{9} + \frac{401}{29}e^{8} - \frac{58887}{1421}e^{7} - \frac{106821}{1421}e^{6} + \frac{45372}{203}e^{5} + \frac{230893}{1421}e^{4} - \frac{651564}{1421}e^{3} - \frac{201044}{1421}e^{2} + \frac{52912}{203}e + \frac{142461}{1421}$ |
| 17 | $[17, 17, -w^{3} + 2w^{2} + 2w - 2]$ | $\phantom{-}\frac{450}{1421}e^{10} - \frac{316}{1421}e^{9} - \frac{176}{29}e^{8} + \frac{2174}{1421}e^{7} + \frac{56237}{1421}e^{6} + \frac{1468}{203}e^{5} - \frac{144132}{1421}e^{4} - \frac{84004}{1421}e^{3} + \frac{102700}{1421}e^{2} + \frac{14286}{203}e + \frac{21732}{1421}$ |
| 19 | $[19, 19, -w^{3} + 2w^{2} + 2w - 1]$ | $\phantom{-}\frac{877}{1421}e^{10} - \frac{2220}{1421}e^{9} - \frac{318}{29}e^{8} + \frac{36642}{1421}e^{7} + \frac{93435}{1421}e^{6} - \frac{28632}{203}e^{5} - \frac{220950}{1421}e^{4} + \frac{409258}{1421}e^{3} + \frac{181520}{1421}e^{2} - \frac{30836}{203}e - \frac{91486}{1421}$ |
| 29 | $[29, 29, -2w^{3} + 5w^{2} + 4w - 7]$ | $\phantom{-}\frac{97}{1421}e^{10} - \frac{1767}{1421}e^{9} + \frac{18}{29}e^{8} + \frac{31358}{1421}e^{7} - \frac{25168}{1421}e^{6} - \frac{26900}{203}e^{5} + \frac{113002}{1421}e^{4} + \frac{443648}{1421}e^{3} - \frac{103542}{1421}e^{2} - \frac{42850}{203}e - \frac{71178}{1421}$ |
| 31 | $[31, 31, w^{3} - 2w^{2} - 4w + 2]$ | $-1$ |
| 47 | $[47, 47, -w^{3} + 4w^{2} - w - 5]$ | $\phantom{-}\frac{547}{1421}e^{10} - \frac{2083}{1421}e^{9} - \frac{158}{29}e^{8} + \frac{33532}{1421}e^{7} + \frac{31069}{1421}e^{6} - \frac{25432}{203}e^{5} - \frac{31130}{1421}e^{4} + \frac{359644}{1421}e^{3} - \frac{842}{1421}e^{2} - \frac{28564}{203}e - \frac{49446}{1421}$ |
| 53 | $[53, 53, -w^{3} + 4w^{2} - w - 7]$ | $\phantom{-}\frac{94}{203}e^{10} - \frac{103}{203}e^{9} - \frac{254}{29}e^{8} + \frac{1175}{203}e^{7} + \frac{11510}{203}e^{6} - \frac{386}{29}e^{5} - \frac{29550}{203}e^{4} - \frac{4108}{203}e^{3} + \frac{21904}{203}e^{2} + \frac{1238}{29}e + \frac{1546}{203}$ |
| 67 | $[67, 67, 2w^{3} - 3w^{2} - 8w + 1]$ | $\phantom{-}\frac{332}{1421}e^{10} - \frac{1111}{1421}e^{9} - \frac{110}{29}e^{8} + \frac{18157}{1421}e^{7} + \frac{28373}{1421}e^{6} - \frac{13800}{203}e^{5} - \frac{52832}{1421}e^{4} + \frac{186124}{1421}e^{3} + \frac{18614}{1421}e^{2} - \frac{11422}{203}e - \frac{23262}{1421}$ |
| 67 | $[67, 67, 2w^{3} - 5w^{2} - 4w + 4]$ | $-\frac{75}{1421}e^{10} - \frac{421}{1421}e^{9} + \frac{39}{29}e^{8} + \frac{9111}{1421}e^{7} - \frac{14820}{1421}e^{6} - \frac{9718}{203}e^{5} + \frac{35390}{1421}e^{4} + \frac{202520}{1421}e^{3} + \frac{16040}{1421}e^{2} - \frac{26132}{203}e - \frac{68988}{1421}$ |
| 71 | $[71, 71, w^{3} - 3w^{2} - 2w + 2]$ | $-\frac{985}{1421}e^{10} + \frac{2239}{1421}e^{9} + \frac{373}{29}e^{8} - \frac{37789}{1421}e^{7} - \frac{115060}{1421}e^{6} + \frac{30399}{203}e^{5} + \frac{283848}{1421}e^{4} - \frac{445596}{1421}e^{3} - \frac{234588}{1421}e^{2} + \frac{33879}{203}e + \frac{114804}{1421}$ |
| 79 | $[79, 79, w^{2} - 3w - 4]$ | $\phantom{-}\frac{330}{1421}e^{10} - \frac{137}{1421}e^{9} - \frac{160}{29}e^{8} + \frac{3110}{1421}e^{7} + \frac{62366}{1421}e^{6} - \frac{3200}{203}e^{5} - \frac{189820}{1421}e^{4} + \frac{49614}{1421}e^{3} + \frac{182362}{1421}e^{2} - \frac{2678}{203}e - \frac{36356}{1421}$ |
| 83 | $[83, 83, 2w^{2} - 3w - 4]$ | $-\frac{811}{1421}e^{10} + \frac{4182}{1421}e^{9} + \frac{228}{29}e^{8} - \frac{71545}{1421}e^{7} - \frac{47142}{1421}e^{6} + \frac{58848}{203}e^{5} + \frac{90621}{1421}e^{4} - \frac{929084}{1421}e^{3} - \frac{182562}{1421}e^{2} + \frac{88602}{203}e + \frac{233704}{1421}$ |
| 89 | $[89, 89, -3w^{3} + 7w^{2} + 8w - 11]$ | $-\frac{185}{1421}e^{10} - \frac{849}{1421}e^{9} + \frac{102}{29}e^{8} + \frac{17074}{1421}e^{7} - \frac{42240}{1421}e^{6} - \frac{16636}{203}e^{5} + \frac{128978}{1421}e^{4} + \frac{311030}{1421}e^{3} - \frac{93535}{1421}e^{2} - \frac{34036}{203}e - \frac{55922}{1421}$ |
| 101 | $[101, 101, w^{3} - 2w^{2} - 3w - 2]$ | $-\frac{48}{29}e^{10} + \frac{118}{29}e^{9} + \frac{824}{29}e^{8} - \frac{1847}{29}e^{7} - \frac{4723}{29}e^{6} + \frac{9364}{29}e^{5} + \frac{10574}{29}e^{4} - \frac{17220}{29}e^{3} - \frac{7726}{29}e^{2} + \frac{7624}{29}e + \frac{3004}{29}$ |
| 103 | $[103, 103, -2w^{3} + 4w^{2} + 5w - 1]$ | $\phantom{-}\frac{13}{1421}e^{10} - \frac{647}{1421}e^{9} + \frac{6}{29}e^{8} + \frac{13256}{1421}e^{7} - \frac{7094}{1421}e^{6} - \frac{12872}{203}e^{5} + \frac{40664}{1421}e^{4} + \frac{223708}{1421}e^{3} - \frac{59715}{1421}e^{2} - \frac{19078}{203}e - \frac{29990}{1421}$ |
Atkin-Lehner eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| $31$ | $[31, 31, w^{3} - 2w^{2} - 4w + 2]$ | $1$ |