# Properties

 Label 4.4.16225.1-36.1-a Base field 4.4.16225.1 Weight $[2, 2, 2, 2]$ Level norm $36$ Level $[36, 6, -w]$ Dimension $11$ CM no Base change no

# Related objects

• L-function not available

## Base field 4.4.16225.1

Generator $$w$$, with minimal polynomial $$x^{4} - x^{3} - 13x^{2} + 6x + 36$$; narrow class number $$2$$ and class number $$1$$.

## Form

 Weight: $[2, 2, 2, 2]$ Level: $[36, 6, -w]$ Dimension: $11$ CM: no Base change: no Newspace dimension: $52$

## Hecke eigenvalues ($q$-expansion)

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

 $$x^{11} - 4x^{10} - 16x^{9} + 80x^{8} + 7x^{7} - 334x^{6} + 215x^{5} + 362x^{4} - 283x^{3} - 81x^{2} + 81x - 10$$
Norm Prime Eigenvalue
4 $[4, 2, \frac{1}{3}w^{3} - \frac{4}{3}w^{2} - \frac{4}{3}w + 6]$ $-1$
4 $[4, 2, -\frac{1}{3}w^{3} - \frac{2}{3}w^{2} + \frac{10}{3}w + 7]$ $\phantom{-}e$
9 $[9, 3, -\frac{1}{2}w^{3} + \frac{3}{2}w^{2} + \frac{7}{2}w - 9]$ $-1$
9 $[9, 3, \frac{1}{3}w^{3} + \frac{2}{3}w^{2} - \frac{7}{3}w - 5]$ $\phantom{-}\frac{13862}{78891}e^{10} - \frac{13794}{26297}e^{9} - \frac{261848}{78891}e^{8} + \frac{857771}{78891}e^{7} + \frac{885817}{78891}e^{6} - \frac{3993691}{78891}e^{5} - \frac{24437}{26297}e^{4} + \frac{5707762}{78891}e^{3} - \frac{1292608}{78891}e^{2} - \frac{2204131}{78891}e + \frac{608015}{78891}$
11 $[11, 11, -\frac{1}{6}w^{3} + \frac{1}{6}w^{2} + \frac{1}{6}w]$ $\phantom{-}\frac{10372}{78891}e^{10} - \frac{12590}{26297}e^{9} - \frac{168082}{78891}e^{8} + \frac{747907}{78891}e^{7} + \frac{128567}{78891}e^{6} - \frac{2995244}{78891}e^{5} + \frac{612225}{26297}e^{4} + \frac{2883101}{78891}e^{3} - \frac{1865534}{78891}e^{2} - \frac{574004}{78891}e + \frac{230272}{78891}$
19 $[19, 19, w + 1]$ $\phantom{-}\frac{7454}{26297}e^{10} - \frac{30440}{26297}e^{9} - \frac{123320}{26297}e^{8} + \frac{616310}{26297}e^{7} + \frac{133262}{26297}e^{6} - \frac{2694502}{26297}e^{5} + \frac{1237753}{26297}e^{4} + \frac{3348313}{26297}e^{3} - \frac{1624790}{26297}e^{2} - \frac{1175850}{26297}e + \frac{432860}{26297}$
19 $[19, 19, \frac{1}{6}w^{3} - \frac{1}{6}w^{2} - \frac{13}{6}w + 2]$ $\phantom{-}\frac{23777}{78891}e^{10} - \frac{31757}{26297}e^{9} - \frac{394640}{78891}e^{8} + \frac{1913366}{78891}e^{7} + \frac{481438}{78891}e^{6} - \frac{8119522}{78891}e^{5} + \frac{1090386}{26297}e^{4} + \frac{9189085}{78891}e^{3} - \frac{3163948}{78891}e^{2} - \frac{2144656}{78891}e + \frac{655025}{78891}$
25 $[25, 5, -\frac{1}{3}w^{3} + \frac{1}{3}w^{2} + \frac{7}{3}w - 1]$ $\phantom{-}\frac{15600}{26297}e^{10} - \frac{56382}{26297}e^{9} - \frac{263077}{26297}e^{8} + \frac{1128540}{26297}e^{7} + \frac{388190}{26297}e^{6} - \frac{4710797}{26297}e^{5} + \frac{2079884}{26297}e^{4} + \frac{5186126}{26297}e^{3} - \frac{2633368}{26297}e^{2} - \frac{1486193}{26297}e + \frac{529527}{26297}$
29 $[29, 29, -\frac{1}{6}w^{3} + \frac{1}{6}w^{2} + \frac{13}{6}w]$ $\phantom{-}\frac{9097}{78891}e^{10} - \frac{2204}{26297}e^{9} - \frac{214978}{78891}e^{8} + \frac{153499}{78891}e^{7} + \frac{1451894}{78891}e^{6} - \frac{863120}{78891}e^{5} - \frac{1220666}{26297}e^{4} + \frac{1684232}{78891}e^{3} + \frac{2987068}{78891}e^{2} - \frac{1034231}{78891}e - \frac{248045}{78891}$
29 $[29, 29, w - 1]$ $-\frac{44555}{78891}e^{10} + \frac{39332}{26297}e^{9} + \frac{847364}{78891}e^{8} - \frac{2372099}{78891}e^{7} - \frac{3037405}{78891}e^{6} + \frac{9877300}{78891}e^{5} + \frac{653780}{26297}e^{4} - \frac{10524130}{78891}e^{3} - \frac{358727}{78891}e^{2} + \frac{2222200}{78891}e + \frac{132880}{78891}$
31 $[31, 31, \frac{1}{3}w^{3} - \frac{1}{3}w^{2} - \frac{10}{3}w + 3]$ $\phantom{-}\frac{60263}{78891}e^{10} - \frac{57892}{26297}e^{9} - \frac{1131095}{78891}e^{8} + \frac{3529085}{78891}e^{7} + \frac{3777628}{78891}e^{6} - \frac{15358867}{78891}e^{5} - \frac{318773}{26297}e^{4} + \frac{18755710}{78891}e^{3} - \frac{2383981}{78891}e^{2} - \frac{5943388}{78891}e + \frac{1354712}{78891}$
31 $[31, 31, \frac{1}{6}w^{3} - \frac{1}{6}w^{2} - \frac{1}{6}w + 2]$ $\phantom{-}\frac{13862}{78891}e^{10} - \frac{13794}{26297}e^{9} - \frac{261848}{78891}e^{8} + \frac{857771}{78891}e^{7} + \frac{885817}{78891}e^{6} - \frac{3993691}{78891}e^{5} - \frac{24437}{26297}e^{4} + \frac{5707762}{78891}e^{3} - \frac{1213717}{78891}e^{2} - \frac{2204131}{78891}e + \frac{371342}{78891}$
41 $[41, 41, -\frac{1}{2}w^{3} - \frac{1}{2}w^{2} + \frac{9}{2}w + 5]$ $\phantom{-}\frac{36568}{78891}e^{10} - \frac{34723}{26297}e^{9} - \frac{692725}{78891}e^{8} + \frac{2107741}{78891}e^{7} + \frac{2440982}{78891}e^{6} - \frac{9041039}{78891}e^{5} - \frac{512599}{26297}e^{4} + \frac{10620554}{78891}e^{3} + \frac{607345}{78891}e^{2} - \frac{3119690}{78891}e - \frac{158018}{78891}$
41 $[41, 41, -\frac{1}{2}w^{3} + \frac{3}{2}w^{2} + \frac{5}{2}w - 8]$ $-\frac{17164}{78891}e^{10} + \frac{24171}{26297}e^{9} + \frac{286546}{78891}e^{8} - \frac{1467433}{78891}e^{7} - \frac{394271}{78891}e^{6} + \frac{6418430}{78891}e^{5} - \frac{635920}{26297}e^{4} - \frac{7942136}{78891}e^{3} + \frac{1039706}{78891}e^{2} + \frac{2393228}{78891}e + \frac{97607}{78891}$
59 $[59, 59, \frac{1}{3}w^{3} - \frac{1}{3}w^{2} - \frac{10}{3}w - 1]$ $-\frac{7037}{26297}e^{10} + \frac{20002}{26297}e^{9} + \frac{138534}{26297}e^{8} - \frac{423203}{26297}e^{7} - \frac{561489}{26297}e^{6} + \frac{2095430}{26297}e^{5} + \frac{503340}{26297}e^{4} - \frac{3389525}{26297}e^{3} + \frac{153658}{26297}e^{2} + \frac{1552278}{26297}e - \frac{207060}{26297}$
59 $[59, 59, -\frac{1}{3}w^{3} + \frac{4}{3}w^{2} + \frac{7}{3}w - 7]$ $\phantom{-}\frac{20862}{26297}e^{10} - \frac{74773}{26297}e^{9} - \frac{369444}{26297}e^{8} + \frac{1523972}{26297}e^{7} + \frac{869183}{26297}e^{6} - \frac{6769318}{26297}e^{5} + \frac{1306345}{26297}e^{4} + \frac{8606661}{26297}e^{3} - \frac{1959096}{26297}e^{2} - \frac{2616471}{26297}e + \frac{527025}{26297}$
59 $[59, 59, \frac{1}{2}w^{3} - \frac{1}{2}w^{2} - \frac{5}{2}w + 2]$ $\phantom{-}\frac{14564}{78891}e^{10} - \frac{12273}{26297}e^{9} - \frac{264614}{78891}e^{8} + \frac{727106}{78891}e^{7} + \frac{741559}{78891}e^{6} - \frac{2823901}{78891}e^{5} + \frac{175588}{26297}e^{4} + \frac{2598526}{78891}e^{3} - \frac{1161814}{78891}e^{2} - \frac{1045438}{78891}e - \frac{67525}{78891}$
79 $[79, 79, w^{2} - 11]$ $\phantom{-}\frac{37519}{78891}e^{10} - \frac{45811}{26297}e^{9} - \frac{662758}{78891}e^{8} + \frac{2774593}{78891}e^{7} + \frac{1560485}{78891}e^{6} - \frac{11937602}{78891}e^{5} + \frac{606959}{26297}e^{4} + \frac{13727801}{78891}e^{3} - \frac{1156052}{78891}e^{2} - \frac{2896886}{78891}e - \frac{502730}{78891}$
79 $[79, 79, \frac{1}{6}w^{3} - \frac{7}{6}w^{2} - \frac{7}{6}w + 3]$ $\phantom{-}\frac{20155}{26297}e^{10} - \frac{66220}{26297}e^{9} - \frac{360365}{26297}e^{8} + \frac{1339217}{26297}e^{7} + \frac{909730}{26297}e^{6} - \frac{5778724}{26297}e^{5} + \frac{1036777}{26297}e^{4} + \frac{6992895}{26297}e^{3} - \frac{1952369}{26297}e^{2} - \frac{2291343}{26297}e + \frac{701665}{26297}$
89 $[89, 89, -\frac{1}{6}w^{3} + \frac{1}{6}w^{2} + \frac{19}{6}w - 5]$ $-\frac{6491}{78891}e^{10} + \frac{16616}{26297}e^{9} + \frac{75023}{78891}e^{8} - \frac{1006943}{78891}e^{7} + \frac{518441}{78891}e^{6} + \frac{4442836}{78891}e^{5} - \frac{1096683}{26297}e^{4} - \frac{5378302}{78891}e^{3} + \frac{2508163}{78891}e^{2} + \frac{998383}{78891}e - \frac{600995}{78891}$
 Display number of eigenvalues

## Atkin-Lehner eigenvalues

Norm Prime Eigenvalue
$4$ $[4, 2, \frac{1}{3}w^{3} - \frac{4}{3}w^{2} - \frac{4}{3}w + 6]$ $1$
$9$ $[9, 3, -\frac{1}{2}w^{3} + \frac{3}{2}w^{2} + \frac{7}{2}w - 9]$ $1$