# Properties

 Base field 4.4.19225.1 Weight [2, 2, 2, 2] Level norm 9 Level $[9,3,-\frac{3}{2}w^{3} + \frac{11}{2}w^{2} + \frac{19}{2}w - 28]$ Label 4.4.19225.1-9.2-e Dimension 10 CM no Base change no

# Related objects

• L-function not available

## Base field 4.4.19225.1

Generator $$w$$, with minimal polynomial $$x^{4} - x^{3} - 15x^{2} + 2x + 44$$; narrow class number $$1$$ and class number $$1$$.

## Form

 Weight [2, 2, 2, 2] Level $[9,3,-\frac{3}{2}w^{3} + \frac{11}{2}w^{2} + \frac{19}{2}w - 28]$ Label 4.4.19225.1-9.2-e Dimension 10 Is CM no Is base change no Parent newspace dimension 17

## Hecke eigenvalues ($q$-expansion)

The Hecke eigenvalue field is $\Q(e)$ where $e$ is a root of the defining polynomial:
$$x^{10} - x^{9} - 30x^{8} + 19x^{7} + 315x^{6} - 119x^{5} - 1379x^{4} + 361x^{3} + 2504x^{2} - 356x - 1571$$
Norm Prime Eigenvalue
4 $[4, 2, w + 2]$ $-\frac{2107}{14912}e^{9} - \frac{401}{3728}e^{8} + \frac{30219}{7456}e^{7} + \frac{67901}{14912}e^{6} - \frac{8592}{233}e^{5} - \frac{750035}{14912}e^{4} + \frac{840669}{7456}e^{3} + \frac{2395807}{14912}e^{2} - \frac{1546725}{14912}e - \frac{2196981}{14912}$
4 $[4, 2, -\frac{1}{2}w^{3} + \frac{3}{2}w^{2} + \frac{9}{2}w - 10]$ $\phantom{-}e$
9 $[9, 3, \frac{1}{2}w^{3} - \frac{5}{2}w^{2} - \frac{5}{2}w + 17]$ $-\frac{3843}{14912}e^{9} - \frac{1233}{3728}e^{8} + \frac{56851}{7456}e^{7} + \frac{172149}{14912}e^{6} - \frac{16559}{233}e^{5} - \frac{1731515}{14912}e^{4} + \frac{1647717}{7456}e^{3} + \frac{5235623}{14912}e^{2} - \frac{3033661}{14912}e - \frac{4582221}{14912}$
9 $[9, 3, \frac{3}{2}w^{3} - \frac{11}{2}w^{2} - \frac{19}{2}w + 28]$ $-1$
11 $[11, 11, \frac{1}{2}w^{3} - \frac{3}{2}w^{2} - \frac{9}{2}w + 11]$ $-\frac{719}{14912}e^{9} - \frac{569}{3728}e^{8} + \frac{12207}{7456}e^{7} + \frac{62873}{14912}e^{6} - \frac{15845}{932}e^{5} - \frac{540343}{14912}e^{4} + \frac{437817}{7456}e^{3} + \frac{1484179}{14912}e^{2} - \frac{863873}{14912}e - \frac{1292033}{14912}$
11 $[11, 11, -w - 3]$ $\phantom{-}\frac{4559}{14912}e^{9} + \frac{307}{3728}e^{8} - \frac{64163}{7456}e^{7} - \frac{87457}{14912}e^{6} + \frac{36095}{466}e^{5} + \frac{1106343}{14912}e^{4} - \frac{1767837}{7456}e^{3} - \frac{3597595}{14912}e^{2} + \frac{3224865}{14912}e + \frac{3327417}{14912}$
25 $[25, 5, w^{3} - 3w^{2} - 7w + 15]$ $\phantom{-}\frac{4991}{14912}e^{9} + \frac{295}{3728}e^{8} - \frac{70043}{7456}e^{7} - \frac{90945}{14912}e^{6} + \frac{19614}{233}e^{5} + \frac{1158135}{14912}e^{4} - \frac{1902261}{7456}e^{3} - \frac{3676891}{14912}e^{2} + \frac{3399793}{14912}e + \frac{3233017}{14912}$
29 $[29, 29, w + 1]$ $-\frac{27}{233}e^{9} - \frac{221}{932}e^{8} + \frac{1703}{466}e^{7} + \frac{6697}{932}e^{6} - \frac{33685}{932}e^{5} - \frac{61645}{932}e^{4} + \frac{28220}{233}e^{3} + \frac{175701}{932}e^{2} - \frac{112933}{932}e - \frac{36739}{233}$
29 $[29, 29, \frac{1}{2}w^{3} - \frac{3}{2}w^{2} - \frac{9}{2}w + 9]$ $\phantom{-}\frac{3115}{14912}e^{9} + \frac{839}{3728}e^{8} - \frac{46735}{7456}e^{7} - \frac{117669}{14912}e^{6} + \frac{13917}{233}e^{5} + \frac{1176579}{14912}e^{4} - \frac{1444177}{7456}e^{3} - \frac{3469959}{14912}e^{2} + \frac{2746469}{14912}e + \frac{3057213}{14912}$
31 $[31, 31, -\frac{1}{2}w^{3} + \frac{5}{2}w^{2} + \frac{5}{2}w - 16]$ $\phantom{-}\frac{2649}{14912}e^{9} + \frac{373}{3728}e^{8} - \frac{38813}{7456}e^{7} - \frac{65943}{14912}e^{6} + \frac{45649}{932}e^{5} + \frac{721297}{14912}e^{4} - \frac{1187411}{7456}e^{3} - \frac{2140461}{14912}e^{2} + \frac{2295847}{14912}e + \frac{1887087}{14912}$
31 $[31, 31, \frac{1}{2}w^{3} - \frac{3}{2}w^{2} - \frac{9}{2}w + 5]$ $\phantom{-}\frac{9837}{14912}e^{9} + \frac{1183}{3728}e^{8} - \frac{140533}{7456}e^{7} - \frac{237787}{14912}e^{6} + \frac{160225}{932}e^{5} + \frac{2753029}{14912}e^{4} - \frac{3973491}{7456}e^{3} - \frac{8664457}{14912}e^{2} + \frac{7313603}{14912}e + \frac{7872307}{14912}$
31 $[31, 31, -w + 3]$ $-\frac{255}{466}e^{9} - \frac{127}{466}e^{8} + \frac{14427}{932}e^{7} + \frac{3230}{233}e^{6} - \frac{129723}{932}e^{5} - \frac{150693}{932}e^{4} + \frac{391717}{932}e^{3} + \frac{119710}{233}e^{2} - \frac{350791}{932}e - \frac{427381}{932}$
31 $[31, 31, \frac{3}{2}w^{3} - \frac{11}{2}w^{2} - \frac{19}{2}w + 29]$ $-\frac{1139}{7456}e^{9} + \frac{8}{233}e^{8} + \frac{15749}{3728}e^{7} + \frac{8273}{7456}e^{6} - \frac{8865}{233}e^{5} - \frac{176267}{7456}e^{4} + \frac{452007}{3728}e^{3} + \frac{682707}{7456}e^{2} - \frac{896213}{7456}e - \frac{687241}{7456}$
59 $[59, 59, 2w^{2} - w - 13]$ $-\frac{971}{7456}e^{9} - \frac{281}{932}e^{8} + \frac{14705}{3728}e^{7} + \frac{71121}{7456}e^{6} - \frac{17209}{466}e^{5} - \frac{687987}{7456}e^{4} + \frac{420235}{3728}e^{3} + \frac{2134371}{7456}e^{2} - \frac{801261}{7456}e - \frac{1918569}{7456}$
59 $[59, 59, \frac{9}{2}w^{3} - \frac{31}{2}w^{2} - \frac{61}{2}w + 85]$ $\phantom{-}\frac{1369}{1864}e^{9} + \frac{377}{932}e^{8} - \frac{19475}{932}e^{7} - \frac{36157}{1864}e^{6} + \frac{176183}{932}e^{5} + \frac{414111}{1864}e^{4} - \frac{537167}{932}e^{3} - \frac{1310635}{1864}e^{2} + \frac{975901}{1864}e + \frac{1188743}{1864}$
61 $[61, 61, 2w^{3} - 6w^{2} - 15w + 31]$ $-\frac{2057}{7456}e^{9} - \frac{493}{1864}e^{8} + \frac{30341}{3728}e^{7} + \frac{72071}{7456}e^{6} - \frac{17707}{233}e^{5} - \frac{732753}{7456}e^{4} + \frac{892603}{3728}e^{3} + \frac{2149021}{7456}e^{2} - \frac{1664967}{7456}e - \frac{1817071}{7456}$
61 $[61, 61, -\frac{3}{2}w^{3} + \frac{11}{2}w^{2} + \frac{21}{2}w - 34]$ $\phantom{-}\frac{85}{1864}e^{9} + \frac{293}{932}e^{8} - \frac{805}{466}e^{7} - \frac{15745}{1864}e^{6} + \frac{4319}{233}e^{5} + \frac{134509}{1864}e^{4} - \frac{14544}{233}e^{3} - \frac{381231}{1864}e^{2} + \frac{123511}{1864}e + \frac{321977}{1864}$
71 $[71, 71, \frac{3}{2}w^{3} - \frac{11}{2}w^{2} - \frac{19}{2}w + 32]$ $-\frac{811}{7456}e^{9} - \frac{51}{466}e^{8} + \frac{12113}{3728}e^{7} + \frac{30409}{7456}e^{6} - \frac{29043}{932}e^{5} - \frac{320651}{7456}e^{4} + \frac{386603}{3728}e^{3} + \frac{1034859}{7456}e^{2} - \frac{757253}{7456}e - \frac{978729}{7456}$
71 $[71, 71, -\frac{1}{2}w^{3} + \frac{5}{2}w^{2} + \frac{5}{2}w - 13]$ $-\frac{855}{932}e^{9} - \frac{975}{1864}e^{8} + \frac{48647}{1864}e^{7} + \frac{46349}{1864}e^{6} - \frac{219937}{932}e^{5} - \frac{132453}{466}e^{4} + \frac{1337499}{1864}e^{3} + \frac{1678561}{1864}e^{2} - \frac{602201}{932}e - \frac{1516507}{1864}$
79 $[79, 79, -3w^{3} + 10w^{2} + 19w - 51]$ $\phantom{-}\frac{2243}{14912}e^{9} + \frac{449}{3728}e^{8} - \frac{30931}{7456}e^{7} - \frac{81909}{14912}e^{6} + \frac{8384}{233}e^{5} + \frac{939899}{14912}e^{4} - \frac{767109}{7456}e^{3} - \frac{3124327}{14912}e^{2} + \frac{1449085}{14912}e + \frac{2893325}{14912}$
 Display number of eigenvalues

## Atkin-Lehner eigenvalues

Norm Prime Eigenvalue
9 $[9,3,-\frac{3}{2}w^{3} + \frac{11}{2}w^{2} + \frac{19}{2}w - 28]$ $1$