# Properties

 Label 4.4.13768.1-13.1-c Base field 4.4.13768.1 Weight $[2, 2, 2, 2]$ Level norm $13$ Level $[13, 13, -w^{2} + 3]$ Dimension $15$ CM no Base change no

# Related objects

• L-function not available

## Base field 4.4.13768.1

Generator $$w$$, with minimal polynomial $$x^{4} - x^{3} - 5x^{2} + 2x + 2$$; narrow class number $$1$$ and class number $$1$$.

## Form

 Weight: $[2, 2, 2, 2]$ Level: $[13, 13, -w^{2} + 3]$ Dimension: $15$ CM: no Base change: no Newspace dimension: $27$

## Hecke eigenvalues ($q$-expansion)

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

 $$x^{15} - 6x^{14} - 7x^{13} + 99x^{12} - 66x^{11} - 567x^{10} + 730x^{9} + 1359x^{8} - 2232x^{7} - 1376x^{6} + 2784x^{5} + 349x^{4} - 1291x^{3} + 189x^{2} + 81x - 15$$
Norm Prime Eigenvalue
2 $[2, 2, -w]$ $\phantom{-}e$
3 $[3, 3, w + 1]$ $-\frac{1}{8}e^{14} + \frac{3}{4}e^{13} + \frac{7}{8}e^{12} - \frac{95}{8}e^{11} + \frac{55}{8}e^{10} + \frac{127}{2}e^{9} - \frac{141}{2}e^{8} - \frac{1091}{8}e^{7} + \frac{709}{4}e^{6} + 129e^{5} - 167e^{4} - 55e^{3} + 52e^{2} + 11e - \frac{19}{8}$
4 $[4, 2, -w^{2} - w + 1]$ $-\frac{1}{4}e^{14} + \frac{5}{4}e^{13} + \frac{21}{8}e^{12} - \frac{41}{2}e^{11} + \frac{5}{8}e^{10} + \frac{465}{4}e^{9} - \frac{311}{4}e^{8} - 275e^{7} + \frac{993}{4}e^{6} + \frac{2263}{8}e^{5} - \frac{565}{2}e^{4} - \frac{427}{4}e^{3} + \frac{917}{8}e^{2} + \frac{39}{8}e - \frac{61}{8}$
13 $[13, 13, -w^{2} + 3]$ $-1$
17 $[17, 17, -w + 3]$ $\phantom{-}\frac{1}{8}e^{14} - \frac{3}{8}e^{13} - \frac{9}{4}e^{12} + 7e^{11} + \frac{57}{4}e^{10} - 48e^{9} - \frac{71}{2}e^{8} + \frac{1193}{8}e^{7} + \frac{113}{8}e^{6} - \frac{1683}{8}e^{5} + \frac{279}{4}e^{4} + \frac{929}{8}e^{3} - \frac{157}{2}e^{2} - \frac{21}{2}e + 9$
27 $[27, 3, w^{3} - 2w^{2} - 3w + 5]$ $...$
31 $[31, 31, -w^{2} - 2w + 1]$ $\phantom{-}\frac{1}{8}e^{14} - \frac{5}{4}e^{13} + \frac{9}{8}e^{12} + 19e^{11} - \frac{161}{4}e^{10} - \frac{377}{4}e^{9} + \frac{531}{2}e^{8} + \frac{1385}{8}e^{7} - \frac{2657}{4}e^{6} - 124e^{5} + \frac{5705}{8}e^{4} + \frac{193}{8}e^{3} - \frac{2299}{8}e^{2} + \frac{53}{4}e + 16$
41 $[41, 41, -w^{2} + 5]$ $-\frac{5}{8}e^{14} + \frac{11}{4}e^{13} + 8e^{12} - \frac{187}{4}e^{11} - \frac{163}{8}e^{10} + 282e^{9} - \frac{345}{4}e^{8} - \frac{5959}{8}e^{7} + \frac{1751}{4}e^{6} + \frac{7269}{8}e^{5} - \frac{5183}{8}e^{4} - \frac{3555}{8}e^{3} + \frac{1371}{4}e^{2} + \frac{261}{8}e - \frac{171}{8}$
43 $[43, 43, w^{3} - w^{2} - 5w - 1]$ $...$
43 $[43, 43, w^{3} - w^{2} - 3w + 1]$ $-\frac{1}{8}e^{14} + e^{13} - \frac{1}{2}e^{12} - \frac{113}{8}e^{11} + \frac{57}{2}e^{10} + \frac{235}{4}e^{9} - \frac{737}{4}e^{8} - \frac{429}{8}e^{7} + 410e^{6} - \frac{479}{8}e^{5} - 357e^{4} + \frac{285}{4}e^{3} + \frac{855}{8}e^{2} + \frac{45}{8}e - \frac{25}{2}$
59 $[59, 59, -w - 3]$ $-\frac{1}{4}e^{14} + \frac{11}{8}e^{13} + \frac{21}{8}e^{12} - \frac{191}{8}e^{11} + \frac{15}{8}e^{10} + \frac{297}{2}e^{9} - \frac{193}{2}e^{8} - 410e^{7} + \frac{2699}{8}e^{6} + \frac{4263}{8}e^{5} - 424e^{4} - \frac{2509}{8}e^{3} + \frac{365}{2}e^{2} + \frac{275}{4}e - \frac{93}{8}$
59 $[59, 59, -2w^{3} + 2w^{2} + 9w - 5]$ $...$
59 $[59, 59, -w^{3} - 3w^{2} - w + 3]$ $\phantom{-}\frac{1}{2}e^{14} - \frac{5}{2}e^{13} - \frac{25}{4}e^{12} + \frac{183}{4}e^{11} + \frac{19}{2}e^{10} - 307e^{9} + \frac{297}{2}e^{8} + \frac{1877}{2}e^{7} - \frac{1405}{2}e^{6} - \frac{5339}{4}e^{5} + \frac{4407}{4}e^{4} + \frac{3009}{4}e^{3} - \frac{1193}{2}e^{2} - \frac{225}{4}e + 33$
59 $[59, 59, w^{3} - 7w + 1]$ $\phantom{-}\frac{1}{4}e^{14} - \frac{7}{8}e^{13} - 4e^{12} + \frac{31}{2}e^{11} + \frac{167}{8}e^{10} - 99e^{9} - \frac{137}{4}e^{8} + 281e^{7} - \frac{195}{8}e^{6} - \frac{1453}{4}e^{5} + \frac{925}{8}e^{4} + \frac{781}{4}e^{3} - \frac{339}{4}e^{2} - \frac{269}{8}e + \frac{45}{8}$
61 $[61, 61, -w^{3} + w^{2} + 2w + 5]$ $...$
61 $[61, 61, -w^{3} - w^{2} + 4w + 3]$ $-\frac{1}{4}e^{13} + \frac{1}{2}e^{12} + \frac{47}{8}e^{11} - \frac{87}{8}e^{10} - \frac{207}{4}e^{9} + 86e^{8} + \frac{837}{4}e^{7} - \frac{1181}{4}e^{6} - 383e^{5} + \frac{3273}{8}e^{4} + \frac{2243}{8}e^{3} - \frac{1509}{8}e^{2} - \frac{95}{2}e + \frac{65}{8}$
67 $[67, 67, 4w^{3} - 4w^{2} - 18w + 7]$ $...$
73 $[73, 73, -2w^{3} + 6w - 3]$ $...$
73 $[73, 73, -2w + 3]$ $-\frac{1}{2}e^{14} + \frac{7}{4}e^{13} + \frac{17}{2}e^{12} - \frac{261}{8}e^{11} - \frac{395}{8}e^{10} + \frac{897}{4}e^{9} + 106e^{8} - \frac{2821}{4}e^{7} - \frac{61}{4}e^{6} + \frac{2059}{2}e^{5} - \frac{1631}{8}e^{4} - \frac{4825}{8}e^{3} + \frac{1479}{8}e^{2} + \frac{129}{2}e - \frac{107}{8}$
97 $[97, 97, w^{3} - 2w^{2} - 5w + 3]$ $-\frac{3}{4}e^{13} + \frac{9}{4}e^{12} + 13e^{11} - \frac{79}{2}e^{10} - \frac{163}{2}e^{9} + 249e^{8} + 234e^{7} - \frac{2767}{4}e^{6} - \frac{1363}{4}e^{5} + \frac{3423}{4}e^{4} + 239e^{3} - \frac{1589}{4}e^{2} - 51e + \frac{55}{2}$
 Display number of eigenvalues

## Atkin-Lehner eigenvalues

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