# Properties

 Base field 5.5.160801.1 Weight [2, 2, 2, 2, 2] Level norm 19 Level $[19, 19, -w^{3} + w^{2} + 4w - 2]$ Label 5.5.160801.1-19.1-a Dimension 11 CM no Base change no

# Related objects

• L-function not available

## Base field 5.5.160801.1

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

## Form

 Weight [2, 2, 2, 2, 2] Level $[19, 19, -w^{3} + w^{2} + 4w - 2]$ Label 5.5.160801.1-19.1-a Dimension 11 Is CM no Is base change no Parent newspace dimension 30

## Hecke eigenvalues ($q$-expansion)

The Hecke eigenvalue field is $\Q(e)$ where $e$ is a root of the defining polynomial:
$$x^{11}$$ $$\mathstrut +\mathstrut 5x^{10}$$ $$\mathstrut -\mathstrut 8x^{9}$$ $$\mathstrut -\mathstrut 70x^{8}$$ $$\mathstrut -\mathstrut 33x^{7}$$ $$\mathstrut +\mathstrut 265x^{6}$$ $$\mathstrut +\mathstrut 276x^{5}$$ $$\mathstrut -\mathstrut 323x^{4}$$ $$\mathstrut -\mathstrut 451x^{3}$$ $$\mathstrut +\mathstrut 77x^{2}$$ $$\mathstrut +\mathstrut 204x$$ $$\mathstrut +\mathstrut 44$$
Norm Prime Eigenvalue
3 $[3, 3, w^{4} - w^{3} - 5w^{2} + 3w + 3]$ $\phantom{-}e$
9 $[9, 3, -w^{4} + 5w^{2} - 3]$ $\phantom{-}\frac{6023}{11964}e^{10} + \frac{23261}{11964}e^{9} - \frac{12159}{1994}e^{8} - \frac{165535}{5982}e^{7} + \frac{163121}{11964}e^{6} + \frac{437815}{3988}e^{5} + \frac{23851}{1994}e^{4} - \frac{1813045}{11964}e^{3} - \frac{431287}{11964}e^{2} + \frac{756449}{11964}e + \frac{96461}{5982}$
9 $[9, 3, -w^{4} + w^{3} + 5w^{2} - 3w - 2]$ $-\frac{6941}{11964}e^{10} - \frac{25835}{11964}e^{9} + \frac{14677}{1994}e^{8} + \frac{185533}{5982}e^{7} - \frac{239663}{11964}e^{6} - \frac{499861}{3988}e^{5} - \frac{1117}{1994}e^{4} + \frac{2115439}{11964}e^{3} + \frac{389773}{11964}e^{2} - \frac{895487}{11964}e - \frac{125633}{5982}$
13 $[13, 13, -w^{4} + w^{3} + 4w^{2} - 3w - 1]$ $...$
17 $[17, 17, w^{4} - w^{3} - 5w^{2} + 3w + 1]$ $-\frac{2955}{3988}e^{10} - \frac{10045}{3988}e^{9} + \frac{19937}{1994}e^{8} + \frac{71293}{1994}e^{7} - \frac{136637}{3988}e^{6} - \frac{561401}{3988}e^{5} + \frac{70341}{1994}e^{4} + \frac{747093}{3988}e^{3} - \frac{35261}{3988}e^{2} - \frac{265321}{3988}e - \frac{21787}{1994}$
19 $[19, 19, -w^{3} + w^{2} + 4w - 2]$ $-1$
23 $[23, 23, -w^{2} + 3]$ $-\frac{2018}{997}e^{10} - \frac{6929}{997}e^{9} + \frac{26923}{997}e^{8} + \frac{98693}{997}e^{7} - \frac{87704}{997}e^{6} - \frac{392758}{997}e^{5} + \frac{63738}{997}e^{4} + \frac{541809}{997}e^{3} + \frac{39127}{997}e^{2} - \frac{220590}{997}e - \frac{52040}{997}$
31 $[31, 31, w^{3} - 4w + 2]$ $\phantom{-}\frac{3533}{5982}e^{10} + \frac{12995}{5982}e^{9} - \frac{7271}{997}e^{8} - \frac{91528}{2991}e^{7} + \frac{104627}{5982}e^{6} + \frac{237499}{1994}e^{5} + \frac{12089}{997}e^{4} - \frac{964519}{5982}e^{3} - \frac{299293}{5982}e^{2} + \frac{414899}{5982}e + \frac{83801}{2991}$
32 $[32, 2, 2]$ $-\frac{2801}{1994}e^{10} - \frac{9535}{1994}e^{9} + \frac{18761}{997}e^{8} + \frac{68251}{997}e^{7} - \frac{121555}{1994}e^{6} - \frac{550819}{1994}e^{5} + \frac{34092}{997}e^{4} + \frac{786655}{1994}e^{3} + \frac{120367}{1994}e^{2} - \frac{343873}{1994}e - \frac{50902}{997}$
37 $[37, 37, w^{3} - 3w - 1]$ $-\frac{24209}{11964}e^{10} - \frac{84575}{11964}e^{9} + \frac{53375}{1994}e^{8} + \frac{604165}{5982}e^{7} - \frac{1014239}{11964}e^{6} - \frac{1612081}{3988}e^{5} + \frac{102319}{1994}e^{4} + \frac{6716683}{11964}e^{3} + \frac{722857}{11964}e^{2} - \frac{2705123}{11964}e - \frac{406511}{5982}$
53 $[53, 53, -2w^{4} + w^{3} + 9w^{2} - 3w - 2]$ $-\frac{7531}{3988}e^{10} - \frac{25769}{3988}e^{9} + \frac{50893}{1994}e^{8} + \frac{185327}{1994}e^{7} - \frac{343229}{3988}e^{6} - \frac{1507081}{3988}e^{5} + \frac{138775}{1994}e^{4} + \frac{2154621}{3988}e^{3} + \frac{136219}{3988}e^{2} - \frac{909101}{3988}e - \frac{102951}{1994}$
59 $[59, 59, -w^{4} + 5w^{2} + w - 4]$ $\phantom{-}\frac{390}{997}e^{10} + \frac{1680}{997}e^{9} - \frac{4053}{997}e^{8} - \frac{23267}{997}e^{7} + \frac{1474}{997}e^{6} + \frac{85764}{997}e^{5} + \frac{41516}{997}e^{4} - \frac{98343}{997}e^{3} - \frac{62788}{997}e^{2} + \frac{25796}{997}e + \frac{20134}{997}$
61 $[61, 61, -w^{4} + w^{3} + 5w^{2} - 4w]$ $-\frac{10513}{11964}e^{10} - \frac{36007}{11964}e^{9} + \frac{22863}{1994}e^{8} + \frac{250757}{5982}e^{7} - \frac{417019}{11964}e^{6} - \frac{629153}{3988}e^{5} + \frac{36085}{1994}e^{4} + \frac{2287307}{11964}e^{3} + \frac{243305}{11964}e^{2} - \frac{657139}{11964}e - \frac{98677}{5982}$
67 $[67, 67, -w^{4} + 6w^{2} + 2w - 4]$ $\phantom{-}\frac{11699}{1994}e^{10} + \frac{40349}{1994}e^{9} - \frac{77827}{997}e^{8} - \frac{287482}{997}e^{7} + \frac{502509}{1994}e^{6} + \frac{2290273}{1994}e^{5} - \frac{173906}{997}e^{4} - \frac{3165873}{1994}e^{3} - \frac{245725}{1994}e^{2} + \frac{1288363}{1994}e + \frac{148551}{997}$
71 $[71, 71, 2w^{4} - w^{3} - 9w^{2} + 4w + 5]$ $-\frac{4556}{997}e^{10} - \frac{15101}{997}e^{9} + \frac{62241}{997}e^{8} + \frac{214783}{997}e^{7} - \frac{217918}{997}e^{6} - \frac{853009}{997}e^{5} + \frac{216354}{997}e^{4} + \frac{1174862}{997}e^{3} + \frac{4113}{997}e^{2} - \frac{474910}{997}e - \frac{95698}{997}$
79 $[79, 79, 2w^{4} - w^{3} - 10w^{2} + 2w + 7]$ $\phantom{-}\frac{18145}{5982}e^{10} + \frac{61981}{5982}e^{9} - \frac{41021}{997}e^{8} - \frac{446573}{2991}e^{7} + \frac{837547}{5982}e^{6} + \frac{1215011}{1994}e^{5} - \frac{114129}{997}e^{4} - \frac{5244113}{5982}e^{3} - \frac{410627}{5982}e^{2} + \frac{2245057}{5982}e + \frac{309301}{2991}$
83 $[83, 83, -w^{4} + 2w^{3} + 5w^{2} - 7w - 2]$ $\phantom{-}\frac{1202}{997}e^{10} + \frac{3644}{997}e^{9} - \frac{17814}{997}e^{8} - \frac{53171}{997}e^{7} + \frac{75831}{997}e^{6} + \frac{223314}{997}e^{5} - \frac{115968}{997}e^{4} - \frac{334496}{997}e^{3} + \frac{42947}{997}e^{2} + \frac{142198}{997}e + \frac{27430}{997}$
83 $[83, 83, -w^{4} + w^{3} + 4w^{2} - 3w + 3]$ $-\frac{5481}{1994}e^{10} - \frac{19239}{1994}e^{9} + \frac{36598}{997}e^{8} + \frac{138812}{997}e^{7} - \frac{238951}{1994}e^{6} - \frac{1136337}{1994}e^{5} + \frac{84354}{997}e^{4} + \frac{1645129}{1994}e^{3} + \frac{146949}{1994}e^{2} - \frac{705455}{1994}e - \frac{95411}{997}$
83 $[83, 83, w^{4} - w^{3} - 5w^{2} + 4w - 1]$ $-\frac{4883}{997}e^{10} - \frac{16663}{997}e^{9} + \frac{65785}{997}e^{8} + \frac{238824}{997}e^{7} - \frac{219476}{997}e^{6} - \frac{964354}{997}e^{5} + \frac{167295}{997}e^{4} + \frac{1368515}{997}e^{3} + \frac{109630}{997}e^{2} - \frac{581422}{997}e - \frac{143732}{997}$
83 $[83, 83, -w^{4} + w^{3} + 4w^{2} - 4w - 2]$ $\phantom{-}\frac{1706}{997}e^{10} + \frac{5585}{997}e^{9} - \frac{23880}{997}e^{8} - \frac{80877}{997}e^{7} + \frac{88120}{997}e^{6} + \frac{333718}{997}e^{5} - \frac{92564}{997}e^{4} - \frac{488259}{997}e^{3} - \frac{15018}{997}e^{2} + \frac{216304}{997}e + \frac{45504}{997}$
 Display number of eigenvalues

## Atkin-Lehner eigenvalues

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