# Properties

 Label 4.4.16317.1-9.1-d Base field 4.4.16317.1 Weight $[2, 2, 2, 2]$ Level norm $9$ Level $[9, 3, w^{3} - w^{2} - 4w]$ Dimension $8$ CM no Base change yes

# Related objects

• L-function not available

## Base field 4.4.16317.1

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

## Form

 Weight: $[2, 2, 2, 2]$ Level: $[9, 3, w^{3} - w^{2} - 4w]$ Dimension: $8$ CM: no Base change: yes Newspace dimension: $18$

## Hecke eigenvalues ($q$-expansion)

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

 $$x^{8} - 36x^{6} + 432x^{4} - 2020x^{2} + 3008$$
Norm Prime Eigenvalue
5 $[5, 5, w + 1]$ $\phantom{-}e$
5 $[5, 5, -w + 2]$ $\phantom{-}e$
7 $[7, 7, -w^{2} + 2w + 1]$ $\phantom{-}\frac{1}{37}e^{6} - \frac{53}{74}e^{4} + \frac{171}{37}e^{2} - \frac{192}{37}$
7 $[7, 7, -w^{2} + 2]$ $\phantom{-}\frac{1}{37}e^{6} - \frac{53}{74}e^{4} + \frac{171}{37}e^{2} - \frac{192}{37}$
9 $[9, 3, w^{3} - w^{2} - 4w]$ $-1$
16 $[16, 2, 2]$ $\phantom{-}\frac{7}{74}e^{6} - \frac{102}{37}e^{4} + \frac{839}{37}e^{2} - \frac{1819}{37}$
17 $[17, 17, -w^{3} + 2w^{2} + 3w - 2]$ $-\frac{3}{148}e^{7} + \frac{49}{74}e^{5} - \frac{230}{37}e^{3} + \frac{551}{37}e$
17 $[17, 17, -w^{3} + w^{2} + 4w - 2]$ $-\frac{3}{148}e^{7} + \frac{49}{74}e^{5} - \frac{230}{37}e^{3} + \frac{551}{37}e$
25 $[25, 5, w^{2} - w - 3]$ $\phantom{-}\frac{2}{37}e^{6} - \frac{53}{37}e^{4} + \frac{379}{37}e^{2} - \frac{606}{37}$
37 $[37, 37, 2w - 1]$ $-\frac{11}{74}e^{6} + \frac{155}{37}e^{4} - \frac{1218}{37}e^{2} + \frac{2462}{37}$
43 $[43, 43, -w^{3} + 2w^{2} + 2w - 2]$ $-\frac{3}{37}e^{6} + \frac{159}{74}e^{4} - \frac{587}{37}e^{2} + \frac{1316}{37}$
43 $[43, 43, w^{3} - w^{2} - 3w + 1]$ $-\frac{3}{37}e^{6} + \frac{159}{74}e^{4} - \frac{587}{37}e^{2} + \frac{1316}{37}$
59 $[59, 59, 2w - 5]$ $\phantom{-}\frac{1}{37}e^{7} - \frac{53}{74}e^{5} + \frac{171}{37}e^{3} - \frac{118}{37}e$
59 $[59, 59, -2w - 3]$ $\phantom{-}\frac{1}{37}e^{7} - \frac{53}{74}e^{5} + \frac{171}{37}e^{3} - \frac{118}{37}e$
79 $[79, 79, -w^{3} + 2w^{2} + 5w - 4]$ $\phantom{-}\frac{1}{37}e^{6} - \frac{45}{37}e^{4} + \frac{504}{37}e^{2} - \frac{1080}{37}$
79 $[79, 79, -w^{3} + w^{2} + 6w - 2]$ $\phantom{-}\frac{1}{37}e^{6} - \frac{45}{37}e^{4} + \frac{504}{37}e^{2} - \frac{1080}{37}$
83 $[83, 83, -w^{3} + 7w]$ $-\frac{1}{37}e^{7} + \frac{53}{74}e^{5} - \frac{171}{37}e^{3} + \frac{266}{37}e$
83 $[83, 83, -w^{3} + 2w^{2} + 4w - 2]$ $\phantom{-}\frac{3}{74}e^{7} - \frac{49}{37}e^{5} + \frac{460}{37}e^{3} - \frac{1102}{37}e$
83 $[83, 83, -w^{3} + w^{2} + 5w - 3]$ $\phantom{-}\frac{3}{74}e^{7} - \frac{49}{37}e^{5} + \frac{460}{37}e^{3} - \frac{1102}{37}e$
83 $[83, 83, w^{2} - 2w - 6]$ $-\frac{1}{37}e^{7} + \frac{53}{74}e^{5} - \frac{171}{37}e^{3} + \frac{266}{37}e$
 Display number of eigenvalues

## Atkin-Lehner eigenvalues

Norm Prime Eigenvalue
$9$ $[9, 3, w^{3} - w^{2} - 4w]$ $1$