# Properties

 Label 4.4.13676.1-16.3-b Base field 4.4.13676.1 Weight $[2, 2, 2, 2]$ Level norm $16$ Level $[16, 4, w^{3} + w^{2} - 5w - 2]$ Dimension $12$ CM no Base change no

# Related objects

• L-function not available

## Base field 4.4.13676.1

Generator $$w$$, with minimal polynomial $$x^{4} - x^{3} - 6x^{2} + 7x + 1$$; narrow class number $$1$$ and class number $$1$$.

## Form

 Weight: $[2, 2, 2, 2]$ Level: $[16, 4, w^{3} + w^{2} - 5w - 2]$ Dimension: $12$ CM: no Base change: no Newspace dimension: $18$

## Hecke eigenvalues ($q$-expansion)

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

 $$x^{12} - 4x^{11} - 10x^{10} + 51x^{9} + 25x^{8} - 222x^{7} - 7x^{6} + 430x^{5} - 19x^{4} - 374x^{3} - 10x^{2} + 112x + 16$$
Norm Prime Eigenvalue
2 $[2, 2, -w + 1]$ $\phantom{-}e$
4 $[4, 2, -w^{2} - w + 3]$ $\phantom{-}0$
5 $[5, 5, w^{3} - 5w + 1]$ $\phantom{-}\frac{3}{10}e^{10} - \frac{6}{5}e^{9} - \frac{13}{5}e^{8} + \frac{137}{10}e^{7} + \frac{37}{10}e^{6} - 48e^{5} + \frac{13}{2}e^{4} + 62e^{3} - \frac{87}{10}e^{2} - \frac{116}{5}e - \frac{8}{5}$
11 $[11, 11, -w^{3} + 6w - 2]$ $-\frac{2}{5}e^{11} + \frac{12}{5}e^{10} + \frac{3}{5}e^{9} - \frac{131}{5}e^{8} + \frac{138}{5}e^{7} + \frac{431}{5}e^{6} - 122e^{5} - 112e^{4} + \frac{798}{5}e^{3} + \frac{317}{5}e^{2} - \frac{282}{5}e - \frac{68}{5}$
13 $[13, 13, -w^{2} + 4]$ $\phantom{-}\frac{1}{2}e^{11} - \frac{5}{2}e^{10} - 2e^{9} + \frac{53}{2}e^{8} - 20e^{7} - \frac{161}{2}e^{6} + \frac{201}{2}e^{5} + \frac{167}{2}e^{4} - \frac{265}{2}e^{3} - \frac{53}{2}e^{2} + 50e + 6$
17 $[17, 17, 2w^{3} + w^{2} - 10w - 2]$ $-\frac{3}{10}e^{11} + \frac{17}{10}e^{10} + \frac{3}{5}e^{9} - \frac{177}{10}e^{8} + \frac{89}{5}e^{7} + \frac{103}{2}e^{6} - \frac{147}{2}e^{5} - \frac{95}{2}e^{4} + \frac{797}{10}e^{3} + \frac{77}{10}e^{2} - \frac{77}{5}e + 2$
37 $[37, 37, w^{3} + w^{2} - 6w - 3]$ $\phantom{-}\frac{1}{10}e^{11} - \frac{7}{10}e^{10} + \frac{15}{2}e^{8} - \frac{39}{5}e^{7} - \frac{227}{10}e^{6} + \frac{57}{2}e^{5} + \frac{39}{2}e^{4} - \frac{279}{10}e^{3} + \frac{53}{10}e^{2} + 8e - \frac{22}{5}$
37 $[37, 37, -w^{3} + 6w - 4]$ $-\frac{3}{5}e^{11} + \frac{29}{10}e^{10} + \frac{16}{5}e^{9} - \frac{162}{5}e^{8} + \frac{161}{10}e^{7} + \frac{221}{2}e^{6} - 100e^{5} - \frac{289}{2}e^{4} + \frac{712}{5}e^{3} + \frac{669}{10}e^{2} - \frac{274}{5}e - 8$
41 $[41, 41, -w^{3} + 5w - 5]$ $-\frac{7}{10}e^{11} + \frac{41}{10}e^{10} + \frac{6}{5}e^{9} - \frac{439}{10}e^{8} + \frac{232}{5}e^{7} + \frac{1367}{10}e^{6} - \frac{413}{2}e^{5} - \frac{305}{2}e^{4} + \frac{2733}{10}e^{3} + \frac{671}{10}e^{2} - \frac{504}{5}e - \frac{98}{5}$
43 $[43, 43, w^{3} - 4w + 4]$ $-\frac{6}{5}e^{11} + \frac{27}{5}e^{10} + 8e^{9} - 61e^{8} + \frac{68}{5}e^{7} + \frac{1062}{5}e^{6} - 133e^{5} - 289e^{4} + \frac{974}{5}e^{3} + \frac{762}{5}e^{2} - 72e - \frac{116}{5}$
43 $[43, 43, -2w^{3} + 11w - 2]$ $-\frac{1}{10}e^{11} + \frac{6}{5}e^{10} - 2e^{9} - \frac{23}{2}e^{8} + \frac{303}{10}e^{7} + \frac{126}{5}e^{6} - \frac{211}{2}e^{5} + 2e^{4} + \frac{1249}{10}e^{3} - \frac{129}{5}e^{2} - 43e + \frac{2}{5}$
47 $[47, 47, -2w^{3} - w^{2} + 12w]$ $\phantom{-}\frac{1}{10}e^{11} - \frac{14}{5}e^{9} + \frac{21}{10}e^{8} + \frac{45}{2}e^{7} - \frac{107}{5}e^{6} - \frac{131}{2}e^{5} + 59e^{4} + \frac{631}{10}e^{3} - 48e^{2} - \frac{29}{5}e + \frac{46}{5}$
47 $[47, 47, 2w - 1]$ $-\frac{1}{2}e^{11} + \frac{13}{5}e^{10} + \frac{13}{5}e^{9} - \frac{297}{10}e^{8} + \frac{129}{10}e^{7} + \frac{532}{5}e^{6} - \frac{145}{2}e^{5} - 155e^{4} + \frac{165}{2}e^{3} + \frac{468}{5}e^{2} - \frac{87}{5}e - \frac{86}{5}$
61 $[61, 61, -w^{3} + w^{2} + 6w - 5]$ $\phantom{-}\frac{1}{2}e^{11} - 2e^{10} - 5e^{9} + \frac{49}{2}e^{8} + \frac{27}{2}e^{7} - 98e^{6} - \frac{23}{2}e^{5} + 160e^{4} - \frac{1}{2}e^{3} - 98e^{2} + 7e + 14$
71 $[71, 71, w^{3} + w^{2} - 4w - 3]$ $-\frac{3}{5}e^{10} + \frac{7}{5}e^{9} + \frac{41}{5}e^{8} - \frac{87}{5}e^{7} - \frac{197}{5}e^{6} + 70e^{5} + 84e^{4} - 108e^{3} - \frac{373}{5}e^{2} + \frac{262}{5}e + \frac{96}{5}$
71 $[71, 71, 2w^{3} + 2w^{2} - 9w - 2]$ $-\frac{4}{5}e^{11} + \frac{17}{5}e^{10} + \frac{29}{5}e^{9} - \frac{193}{5}e^{8} + \frac{23}{5}e^{7} + \frac{674}{5}e^{6} - 77e^{5} - 179e^{4} + \frac{576}{5}e^{3} + \frac{417}{5}e^{2} - \frac{176}{5}e - \frac{32}{5}$
79 $[79, 79, w^{3} + w^{2} - 6w - 5]$ $\phantom{-}\frac{1}{5}e^{11} - \frac{6}{5}e^{10} - \frac{4}{5}e^{9} + \frac{73}{5}e^{8} - \frac{49}{5}e^{7} - \frac{283}{5}e^{6} + 57e^{5} + 82e^{4} - \frac{454}{5}e^{3} - \frac{166}{5}e^{2} + \frac{216}{5}e + \frac{4}{5}$
81 $[81, 3, -3]$ $-\frac{1}{2}e^{11} + \frac{17}{10}e^{10} + \frac{26}{5}e^{9} - \frac{189}{10}e^{8} - \frac{91}{5}e^{7} + \frac{633}{10}e^{6} + \frac{87}{2}e^{5} - \frac{153}{2}e^{4} - \frac{143}{2}e^{3} + \frac{197}{10}e^{2} + \frac{196}{5}e + \frac{58}{5}$
83 $[83, 83, -3w^{3} - w^{2} + 16w + 3]$ $-\frac{2}{5}e^{11} + 2e^{10} + \frac{11}{5}e^{9} - \frac{112}{5}e^{8} + 10e^{7} + \frac{373}{5}e^{6} - 64e^{5} - 83e^{4} + \frac{443}{5}e^{3} + 15e^{2} - \frac{154}{5}e + \frac{36}{5}$
97 $[97, 97, w^{3} + w^{2} - 6w + 1]$ $\phantom{-}\frac{1}{5}e^{11} - \frac{3}{10}e^{10} - \frac{17}{5}e^{9} + \frac{24}{5}e^{8} + \frac{193}{10}e^{7} - \frac{51}{2}e^{6} - 40e^{5} + \frac{101}{2}e^{4} + \frac{96}{5}e^{3} - \frac{353}{10}e^{2} + \frac{58}{5}e + 8$
 Display number of eigenvalues

## Atkin-Lehner eigenvalues

Norm Prime Eigenvalue
$4$ $[4, 2, -w^{2} - w + 3]$ $1$