# Properties

 Label 4.4.6125.1-59.1-e Base field 4.4.6125.1 Weight $[2, 2, 2, 2]$ Level norm $59$ Level $[59, 59, -3w^{3} - 4w^{2} + 19w + 14]$ Dimension $8$ CM no Base change no

# Related objects

• L-function not available

## Base field 4.4.6125.1

Generator $$w$$, with minimal polynomial $$x^{4} - x^{3} - 9x^{2} + 9x + 11$$; narrow class number $$2$$ and class number $$1$$.

## Form

 Weight: $[2, 2, 2, 2]$ Level: $[59, 59, -3w^{3} - 4w^{2} + 19w + 14]$ Dimension: $8$ CM: no Base change: no Newspace dimension: $20$

## Hecke eigenvalues ($q$-expansion)

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

 $$x^{8} - 10x^{7} + 29x^{6} - 100x^{4} + 64x^{3} + 78x^{2} - 50x - 1$$
Norm Prime Eigenvalue
5 $[5, 5, -2w^{3} - 2w^{2} + 13w + 8]$ $\phantom{-}e$
11 $[11, 11, 2w^{3} + 3w^{2} - 12w - 11]$ $\phantom{-}\frac{105}{809}e^{7} - \frac{1150}{809}e^{6} + \frac{3755}{809}e^{5} - \frac{841}{809}e^{4} - \frac{12858}{809}e^{3} + \frac{8911}{809}e^{2} + \frac{11299}{809}e - \frac{4762}{809}$
11 $[11, 11, -w^{3} - 2w^{2} + 6w + 9]$ $-\frac{134}{809}e^{7} + \frac{1275}{809}e^{6} - \frac{3020}{809}e^{5} - \frac{3095}{809}e^{4} + \frac{13728}{809}e^{3} + \frac{1464}{809}e^{2} - \frac{11384}{809}e - \frac{102}{809}$
11 $[11, 11, w^{3} + w^{2} - 7w - 4]$ $-\frac{53}{809}e^{7} + \frac{619}{809}e^{6} - \frac{2088}{809}e^{5} + \frac{255}{809}e^{4} + \frac{8062}{809}e^{3} - \frac{5253}{809}e^{2} - \frac{6181}{809}e + \frac{1795}{809}$
11 $[11, 11, w - 1]$ $\phantom{-}\frac{11}{809}e^{7} - \frac{159}{809}e^{6} + \frac{586}{809}e^{5} - \frac{404}{809}e^{4} - \frac{330}{809}e^{3} - \frac{253}{809}e^{2} - \frac{2060}{809}e + \frac{2375}{809}$
16 $[16, 2, 2]$ $\phantom{-}\frac{91}{809}e^{7} - \frac{727}{809}e^{6} + \frac{1097}{809}e^{5} + \frac{3424}{809}e^{4} - \frac{6775}{809}e^{3} - \frac{6947}{809}e^{2} + \frac{8552}{809}e + \frac{4718}{809}$
19 $[19, 19, -w^{2} + 4]$ $\phantom{-}\frac{21}{809}e^{7} - \frac{230}{809}e^{6} + \frac{751}{809}e^{5} - \frac{330}{809}e^{4} - \frac{1439}{809}e^{3} + \frac{326}{809}e^{2} + \frac{1289}{809}e + \frac{342}{809}$
19 $[19, 19, 2w^{3} + 2w^{2} - 13w - 6]$ $-\frac{187}{809}e^{7} + \frac{1894}{809}e^{6} - \frac{5108}{809}e^{5} - \frac{2840}{809}e^{4} + \frac{21790}{809}e^{3} - \frac{2980}{809}e^{2} - \frac{21610}{809}e + \frac{4120}{809}$
19 $[19, 19, 3w^{3} + 4w^{2} - 18w - 16]$ $-\frac{48}{809}e^{7} + \frac{179}{809}e^{6} + \frac{826}{809}e^{5} - \frac{3753}{809}e^{4} + \frac{1440}{809}e^{3} + \frac{4340}{809}e^{2} - \frac{1675}{809}e + \frac{3610}{809}$
19 $[19, 19, -w^{3} - w^{2} + 5w + 3]$ $-\frac{72}{809}e^{7} + \frac{673}{809}e^{6} - \frac{1997}{809}e^{5} + \frac{1247}{809}e^{4} + \frac{3778}{809}e^{3} - \frac{5625}{809}e^{2} - \frac{1299}{809}e + \frac{4606}{809}$
49 $[49, 7, 2w^{3} + 3w^{2} - 13w - 15]$ $\phantom{-}\frac{11}{809}e^{7} - \frac{159}{809}e^{6} + \frac{1395}{809}e^{5} - \frac{5258}{809}e^{4} + \frac{3715}{809}e^{3} + \frac{11073}{809}e^{2} - \frac{6105}{809}e - \frac{861}{809}$
59 $[59, 59, -3w^{3} - 4w^{2} + 19w + 14]$ $-1$
59 $[59, 59, 3w^{3} + 4w^{2} - 18w - 17]$ $\phantom{-}\frac{433}{809}e^{7} - \frac{4126}{809}e^{6} + \frac{10785}{809}e^{5} + \frac{4175}{809}e^{4} - \frac{42114}{809}e^{3} + \frac{15929}{809}e^{2} + \frac{37172}{809}e - \frac{10284}{809}$
59 $[59, 59, 2w^{3} + 3w^{2} - 11w - 13]$ $\phantom{-}\frac{71}{809}e^{7} - \frac{585}{809}e^{6} + \frac{1576}{809}e^{5} - \frac{1578}{809}e^{4} - \frac{512}{809}e^{3} + \frac{4030}{809}e^{2} - \frac{573}{809}e - \frac{924}{809}$
59 $[59, 59, -w^{3} - 2w^{2} + 7w + 9]$ $-\frac{43}{809}e^{7} + \frac{548}{809}e^{6} - \frac{1923}{809}e^{5} + \frac{329}{809}e^{4} + \frac{7762}{809}e^{3} - \frac{8719}{809}e^{2} - \frac{2832}{809}e + \frac{11088}{809}$
71 $[71, 71, 3w^{3} + 3w^{2} - 17w - 10]$ $\phantom{-}\frac{7}{809}e^{7} + \frac{193}{809}e^{6} - \frac{1907}{809}e^{5} + \frac{3935}{809}e^{4} + \frac{3835}{809}e^{3} - \frac{11487}{809}e^{2} - \frac{5503}{809}e + \frac{3350}{809}$
71 $[71, 71, -2w^{3} - w^{2} + 14w + 2]$ $\phantom{-}\frac{62}{809}e^{7} - \frac{602}{809}e^{6} + \frac{1832}{809}e^{5} - \frac{512}{809}e^{4} - \frac{5096}{809}e^{3} + \frac{1810}{809}e^{2} + \frac{2804}{809}e + \frac{4708}{809}$
71 $[71, 71, 5w^{3} + 7w^{2} - 30w - 25]$ $-\frac{234}{809}e^{7} + \frac{1985}{809}e^{6} - \frac{4670}{809}e^{5} - \frac{599}{809}e^{4} + \frac{11874}{809}e^{3} - \frac{7562}{809}e^{2} - \frac{4424}{809}e + \frac{1621}{809}$
71 $[71, 71, -3w^{3} - 4w^{2} + 16w + 16]$ $-\frac{189}{809}e^{7} + \frac{1261}{809}e^{6} - \frac{287}{809}e^{5} - \frac{10783}{809}e^{4} + \frac{11333}{809}e^{3} + \frac{20527}{809}e^{2} - \frac{14837}{809}e - \frac{7932}{809}$
81 $[81, 3, -3]$ $-\frac{214}{809}e^{7} + \frac{1843}{809}e^{6} - \frac{4340}{809}e^{5} - \frac{2069}{809}e^{4} + \frac{16128}{809}e^{3} - \frac{3168}{809}e^{2} - \frac{17142}{809}e - \frac{1636}{809}$
 Display number of eigenvalues

## Atkin-Lehner eigenvalues

Norm Prime Eigenvalue
$59$ $[59, 59, -3w^{3} - 4w^{2} + 19w + 14]$ $1$