# Properties

 Label 4.4.14272.1-13.1-a Base field 4.4.14272.1 Weight $[2, 2, 2, 2]$ Level norm $13$ Level $[13, 13, w^{3} - 2w^{2} - 4w + 2]$ Dimension $9$ CM no Base change no

# Related objects

• L-function not available

## Base field 4.4.14272.1

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

## Form

 Weight: $[2, 2, 2, 2]$ Level: $[13, 13, w^{3} - 2w^{2} - 4w + 2]$ Dimension: $9$ 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:

 $$x^{9} - 3x^{8} - 13x^{7} + 36x^{6} + 47x^{5} - 110x^{4} - 44x^{3} + 100x^{2} + 10x - 25$$
Norm Prime Eigenvalue
3 $[3, 3, w]$ $\phantom{-}e$
4 $[4, 2, -w^{3} + 3w^{2} + w - 2]$ $-\frac{4}{35}e^{8} + \frac{16}{105}e^{7} + \frac{206}{105}e^{6} - \frac{26}{15}e^{5} - \frac{1124}{105}e^{4} + \frac{97}{21}e^{3} + \frac{671}{35}e^{2} - \frac{29}{7}e - \frac{169}{21}$
11 $[11, 11, -w^{3} + 3w^{2} + 2w - 5]$ $\phantom{-}\frac{34}{105}e^{8} - \frac{19}{35}e^{7} - \frac{179}{35}e^{6} + \frac{77}{15}e^{5} + \frac{871}{35}e^{4} - \frac{122}{21}e^{3} - \frac{1207}{35}e^{2} - \frac{58}{21}e + \frac{263}{21}$
13 $[13, 13, w^{3} - 2w^{2} - 4w + 2]$ $-1$
13 $[13, 13, -w + 2]$ $-\frac{4}{35}e^{8} + \frac{16}{105}e^{7} + \frac{206}{105}e^{6} - \frac{26}{15}e^{5} - \frac{1124}{105}e^{4} + \frac{97}{21}e^{3} + \frac{671}{35}e^{2} - \frac{22}{7}e - \frac{211}{21}$
17 $[17, 17, w^{3} - 3w^{2} - 2w + 2]$ $\phantom{-}\frac{46}{105}e^{8} - \frac{73}{105}e^{7} - \frac{743}{105}e^{6} + \frac{103}{15}e^{5} + \frac{3737}{105}e^{4} - \frac{73}{7}e^{3} - \frac{1843}{35}e^{2} + \frac{8}{21}e + \frac{137}{7}$
19 $[19, 19, -w^{2} + w + 4]$ $\phantom{-}\frac{16}{105}e^{8} - \frac{11}{35}e^{7} - \frac{76}{35}e^{6} + \frac{53}{15}e^{5} + \frac{309}{35}e^{4} - \frac{176}{21}e^{3} - \frac{323}{35}e^{2} + \frac{11}{21}e + \frac{41}{21}$
19 $[19, 19, w^{3} - 2w^{2} - 3w + 2]$ $\phantom{-}\frac{16}{105}e^{8} - \frac{11}{35}e^{7} - \frac{76}{35}e^{6} + \frac{53}{15}e^{5} + \frac{309}{35}e^{4} - \frac{197}{21}e^{3} - \frac{288}{35}e^{2} + \frac{116}{21}e + \frac{41}{21}$
23 $[23, 23, w^{2} - 2w - 1]$ $-\frac{88}{105}e^{8} + \frac{199}{105}e^{7} + \frac{1289}{105}e^{6} - \frac{319}{15}e^{5} - \frac{5711}{105}e^{4} + \frac{374}{7}e^{3} + \frac{2494}{35}e^{2} - \frac{638}{21}e - \frac{165}{7}$
27 $[27, 3, w^{3} - 2w^{2} - 5w - 1]$ $\phantom{-}\frac{53}{105}e^{8} - \frac{43}{35}e^{7} - \frac{243}{35}e^{6} + \frac{199}{15}e^{5} + \frac{982}{35}e^{4} - \frac{625}{21}e^{3} - \frac{1164}{35}e^{2} + \frac{253}{21}e + \frac{292}{21}$
29 $[29, 29, -w^{3} + 2w^{2} + 5w - 1]$ $\phantom{-}\frac{94}{105}e^{8} - \frac{242}{105}e^{7} - \frac{1252}{105}e^{6} + \frac{129}{5}e^{5} + \frac{4663}{105}e^{4} - \frac{1363}{21}e^{3} - \frac{1447}{35}e^{2} + \frac{734}{21}e + \frac{268}{21}$
37 $[37, 37, -w^{3} + 2w^{2} + 5w - 4]$ $\phantom{-}\frac{6}{35}e^{8} - \frac{8}{35}e^{7} - \frac{103}{35}e^{6} + \frac{13}{5}e^{5} + \frac{562}{35}e^{4} - \frac{45}{7}e^{3} - \frac{1024}{35}e^{2} + \frac{12}{7}e + \frac{116}{7}$
41 $[41, 41, 2w^{3} - 5w^{2} - 6w + 4]$ $-\frac{8}{15}e^{8} + \frac{14}{15}e^{7} + \frac{124}{15}e^{6} - \frac{143}{15}e^{5} - \frac{601}{15}e^{4} + 18e^{3} + \frac{309}{5}e^{2} - \frac{22}{3}e - 26$
53 $[53, 53, w^{3} - 2w^{2} - 3w - 2]$ $\phantom{-}\frac{22}{105}e^{8} - \frac{37}{35}e^{7} - \frac{52}{35}e^{6} + \frac{191}{15}e^{5} - \frac{157}{35}e^{4} - \frac{809}{21}e^{3} + \frac{1074}{35}e^{2} + \frac{506}{21}e - \frac{361}{21}$
59 $[59, 59, w - 4]$ $-\frac{62}{105}e^{8} + \frac{47}{35}e^{7} + \frac{312}{35}e^{6} - \frac{226}{15}e^{5} - \frac{1473}{35}e^{4} + \frac{766}{21}e^{3} + \frac{2271}{35}e^{2} - \frac{292}{21}e - \frac{613}{21}$
67 $[67, 67, 2w^{2} - 3w - 8]$ $-\frac{118}{105}e^{8} + \frac{169}{105}e^{7} + \frac{1979}{105}e^{6} - \frac{244}{15}e^{5} - \frac{10376}{105}e^{4} + \frac{197}{7}e^{3} + \frac{5414}{35}e^{2} - \frac{278}{21}e - \frac{405}{7}$
73 $[73, 73, 2w^{3} - 5w^{2} - 6w + 5]$ $\phantom{-}\frac{13}{21}e^{8} - \frac{12}{7}e^{7} - \frac{60}{7}e^{6} + \frac{59}{3}e^{5} + \frac{241}{7}e^{4} - \frac{1093}{21}e^{3} - \frac{241}{7}e^{2} + \frac{739}{21}e + \frac{181}{21}$
89 $[89, 89, -2w^{3} + 6w^{2} + 5w - 7]$ $-\frac{89}{105}e^{8} + \frac{212}{105}e^{7} + \frac{1207}{105}e^{6} - \frac{317}{15}e^{5} - \frac{4603}{105}e^{4} + \frac{296}{7}e^{3} + \frac{1357}{35}e^{2} - \frac{283}{21}e - \frac{75}{7}$
97 $[97, 97, -2w^{3} + 6w^{2} + 3w - 7]$ $\phantom{-}\frac{11}{105}e^{8} - \frac{73}{105}e^{7} - \frac{8}{105}e^{6} + \frac{41}{5}e^{5} - \frac{988}{105}e^{4} - \frac{485}{21}e^{3} + \frac{992}{35}e^{2} + \frac{169}{21}e - \frac{205}{21}$
97 $[97, 97, -w^{3} + 3w^{2} + 3w - 1]$ $\phantom{-}\frac{1}{15}e^{8} - \frac{1}{5}e^{7} - \frac{1}{5}e^{6} + \frac{26}{15}e^{5} - \frac{31}{5}e^{4} - \frac{2}{3}e^{3} + \frac{147}{5}e^{2} - \frac{13}{3}e - \frac{46}{3}$
 Display number of eigenvalues

## Atkin-Lehner eigenvalues

Norm Prime Eigenvalue
$13$ $[13, 13, w^{3} - 2w^{2} - 4w + 2]$ $1$