# Properties

 Label 4.4.14400.1-16.1-g Base field $$\Q(\sqrt{5}, \sqrt{6})$$ Weight $[2, 2, 2, 2]$ Level norm $16$ Level $[16, 2, 2]$ Dimension $8$ CM no Base change yes

# Related objects

• L-function not available

## Base field $$\Q(\sqrt{5}, \sqrt{6})$$

Generator $$w$$, with minimal polynomial $$x^{4} - 2x^{3} - 13x^{2} + 14x + 19$$; narrow class number $$2$$ and class number $$1$$.

## Form

 Weight: $[2, 2, 2, 2]$ Level: $[16, 2, 2]$ Dimension: $8$ CM: no Base change: yes Newspace dimension: $23$

## Hecke eigenvalues ($q$-expansion)

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

 $$x^{8} - 8x^{7} - 112x^{6} + 720x^{5} + 2888x^{4} - 17184x^{3} - 16192x^{2} + 116800x - 85616$$
Norm Prime Eigenvalue
4 $[4, 2, -\frac{2}{19}w^{3} + \frac{3}{19}w^{2} + \frac{37}{19}w - 3]$ $\phantom{-}0$
5 $[5, 5, \frac{3}{19}w^{3} + \frac{5}{19}w^{2} - \frac{27}{19}w - 1]$ $-\frac{10739}{587681136}e^{7} + \frac{24817}{195893712}e^{6} + \frac{1170301}{293840568}e^{5} - \frac{1467049}{73460142}e^{4} - \frac{37459127}{146920284}e^{3} + \frac{15067991}{20988612}e^{2} + \frac{92356465}{24486714}e - \frac{216059303}{36730071}$
5 $[5, 5, -\frac{3}{19}w^{3} + \frac{14}{19}w^{2} + \frac{8}{19}w - 2]$ $-\frac{10739}{587681136}e^{7} + \frac{24817}{195893712}e^{6} + \frac{1170301}{293840568}e^{5} - \frac{1467049}{73460142}e^{4} - \frac{37459127}{146920284}e^{3} + \frac{15067991}{20988612}e^{2} + \frac{92356465}{24486714}e - \frac{216059303}{36730071}$
9 $[9, 3, -\frac{2}{19}w^{3} + \frac{3}{19}w^{2} + \frac{37}{19}w - 4]$ $\phantom{-}\frac{175}{677052}e^{7} - \frac{709}{451368}e^{6} - \frac{20221}{677052}e^{5} + \frac{19348}{169263}e^{4} + \frac{127444}{169263}e^{3} - \frac{679367}{338526}e^{2} - \frac{279997}{56421}e + \frac{1149262}{169263}$
19 $[19, 19, -w]$ $\phantom{-}\frac{10739}{587681136}e^{7} - \frac{24817}{195893712}e^{6} - \frac{1170301}{293840568}e^{5} + \frac{1467049}{73460142}e^{4} + \frac{37459127}{146920284}e^{3} - \frac{15067991}{20988612}e^{2} - \frac{67869751}{24486714}e + \frac{216059303}{36730071}$
19 $[19, 19, \frac{4}{19}w^{3} - \frac{6}{19}w^{2} - \frac{55}{19}w + 1]$ $-\frac{314539}{587681136}e^{7} + \frac{640229}{195893712}e^{6} + \frac{18722129}{293840568}e^{5} - \frac{18261113}{73460142}e^{4} - \frac{258701911}{146920284}e^{3} + \frac{99309499}{20988612}e^{2} + \frac{310907147}{24486714}e - \frac{788299153}{36730071}$
19 $[19, 19, \frac{4}{19}w^{3} - \frac{6}{19}w^{2} - \frac{55}{19}w + 2]$ $\phantom{-}\frac{10739}{587681136}e^{7} - \frac{24817}{195893712}e^{6} - \frac{1170301}{293840568}e^{5} + \frac{1467049}{73460142}e^{4} + \frac{37459127}{146920284}e^{3} - \frac{15067991}{20988612}e^{2} - \frac{67869751}{24486714}e + \frac{216059303}{36730071}$
19 $[19, 19, w - 1]$ $-\frac{314539}{587681136}e^{7} + \frac{640229}{195893712}e^{6} + \frac{18722129}{293840568}e^{5} - \frac{18261113}{73460142}e^{4} - \frac{258701911}{146920284}e^{3} + \frac{99309499}{20988612}e^{2} + \frac{310907147}{24486714}e - \frac{788299153}{36730071}$
29 $[29, 29, -\frac{5}{19}w^{3} - \frac{2}{19}w^{2} + \frac{64}{19}w + 2]$ $\phantom{-}\frac{150499}{587681136}e^{7} - \frac{92977}{97946856}e^{6} - \frac{9348005}{293840568}e^{5} + \frac{11145509}{293840568}e^{4} + \frac{127636165}{146920284}e^{3} - \frac{7733308}{36730071}e^{2} - \frac{24609053}{3498102}e + \frac{168303287}{73460142}$
29 $[29, 29, \frac{10}{19}w^{3} - \frac{34}{19}w^{2} - \frac{71}{19}w + 9]$ $-\frac{128501}{587681136}e^{7} + \frac{53161}{48973428}e^{6} + \frac{9269665}{293840568}e^{5} - \frac{29558359}{293840568}e^{4} - \frac{182487959}{146920284}e^{3} + \frac{224576335}{73460142}e^{2} + \frac{359600095}{24486714}e - \frac{253894879}{10494306}$
29 $[29, 29, \frac{4}{19}w^{3} - \frac{6}{19}w^{2} - \frac{55}{19}w + 4]$ $\phantom{-}\frac{150499}{587681136}e^{7} - \frac{92977}{97946856}e^{6} - \frac{9348005}{293840568}e^{5} + \frac{11145509}{293840568}e^{4} + \frac{127636165}{146920284}e^{3} - \frac{7733308}{36730071}e^{2} - \frac{24609053}{3498102}e + \frac{168303287}{73460142}$
29 $[29, 29, -w + 3]$ $-\frac{128501}{587681136}e^{7} + \frac{53161}{48973428}e^{6} + \frac{9269665}{293840568}e^{5} - \frac{29558359}{293840568}e^{4} - \frac{182487959}{146920284}e^{3} + \frac{224576335}{73460142}e^{2} + \frac{359600095}{24486714}e - \frac{253894879}{10494306}$
49 $[49, 7, \frac{5}{19}w^{3} - \frac{17}{19}w^{2} - \frac{64}{19}w + 11]$ $-\frac{175}{677052}e^{7} + \frac{709}{451368}e^{6} + \frac{20221}{677052}e^{5} - \frac{19348}{169263}e^{4} - \frac{127444}{169263}e^{3} + \frac{679367}{338526}e^{2} + \frac{279997}{56421}e - \frac{1487788}{169263}$
49 $[49, 7, \frac{6}{19}w^{3} - \frac{9}{19}w^{2} - \frac{92}{19}w - 1]$ $-\frac{175}{677052}e^{7} + \frac{709}{451368}e^{6} + \frac{20221}{677052}e^{5} - \frac{19348}{169263}e^{4} - \frac{127444}{169263}e^{3} + \frac{679367}{338526}e^{2} + \frac{279997}{56421}e - \frac{1487788}{169263}$
71 $[71, 71, \frac{3}{19}w^{3} + \frac{5}{19}w^{2} - \frac{46}{19}w - 1]$ $\phantom{-}\frac{68707}{587681136}e^{7} - \frac{62437}{48973428}e^{6} - \frac{3229523}{293840568}e^{5} + \frac{17722687}{146920284}e^{4} + \frac{33897523}{146920284}e^{3} - \frac{102888292}{36730071}e^{2} - \frac{9140383}{3498102}e + \frac{539298769}{36730071}$
71 $[71, 71, \frac{6}{19}w^{3} - \frac{9}{19}w^{2} - \frac{73}{19}w]$ $-\frac{133661}{587681136}e^{7} + \frac{17907}{10882984}e^{6} + \frac{7989067}{293840568}e^{5} - \frac{10126327}{73460142}e^{4} - \frac{128882237}{146920284}e^{3} + \frac{207618739}{73460142}e^{2} + \frac{27341313}{2720746}e - \frac{85579364}{5247153}$
71 $[71, 71, \frac{6}{19}w^{3} - \frac{9}{19}w^{2} - \frac{73}{19}w + 4]$ $\phantom{-}\frac{68707}{587681136}e^{7} - \frac{62437}{48973428}e^{6} - \frac{3229523}{293840568}e^{5} + \frac{17722687}{146920284}e^{4} + \frac{33897523}{146920284}e^{3} - \frac{102888292}{36730071}e^{2} - \frac{9140383}{3498102}e + \frac{539298769}{36730071}$
71 $[71, 71, -\frac{3}{19}w^{3} + \frac{14}{19}w^{2} + \frac{27}{19}w - 3]$ $-\frac{133661}{587681136}e^{7} + \frac{17907}{10882984}e^{6} + \frac{7989067}{293840568}e^{5} - \frac{10126327}{73460142}e^{4} - \frac{128882237}{146920284}e^{3} + \frac{207618739}{73460142}e^{2} + \frac{27341313}{2720746}e - \frac{85579364}{5247153}$
101 $[101, 101, \frac{7}{19}w^{3} - \frac{1}{19}w^{2} - \frac{63}{19}w - 3]$ $\phantom{-}\frac{5833}{20988612}e^{7} + \frac{44713}{97946856}e^{6} - \frac{13368095}{293840568}e^{5} - \frac{33525109}{293840568}e^{4} + \frac{63090577}{36730071}e^{3} + \frac{111663554}{36730071}e^{2} - \frac{402586207}{24486714}e - \frac{583989463}{73460142}$
101 $[101, 101, -\frac{9}{19}w^{3} + \frac{23}{19}w^{2} + \frac{81}{19}w - 5]$ $\phantom{-}\frac{5833}{20988612}e^{7} + \frac{44713}{97946856}e^{6} - \frac{13368095}{293840568}e^{5} - \frac{33525109}{293840568}e^{4} + \frac{63090577}{36730071}e^{3} + \frac{111663554}{36730071}e^{2} - \frac{402586207}{24486714}e - \frac{583989463}{73460142}$
 Display number of eigenvalues

## Atkin-Lehner eigenvalues

Norm Prime Eigenvalue
$4$ $[4, 2, -\frac{2}{19}w^{3} + \frac{3}{19}w^{2} + \frac{37}{19}w - 3]$ $1$