# Properties

 Label 4.4.16997.1-13.2-a Base field 4.4.16997.1 Weight $[2, 2, 2, 2]$ Level norm $13$ Level $[13, 13, w^{2} - w - 4]$ Dimension $12$ CM no Base change no

# Related objects

• L-function not available

## Base field 4.4.16997.1

Generator $$w$$, with minimal polynomial $$x^{4} - 6x^{2} - x + 5$$; narrow class number $$1$$ and class number $$1$$.

## Form

 Weight: $[2, 2, 2, 2]$ Level: $[13, 13, w^{2} - w - 4]$ Dimension: $12$ CM: no Base change: no Newspace dimension: $26$

## Hecke eigenvalues ($q$-expansion)

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

 $$x^{12} + 6x^{11} - 17x^{10} - 125x^{9} + 122x^{8} + 924x^{7} - 611x^{6} - 2701x^{5} + 1871x^{4} + 2290x^{3} - 1645x^{2} - 369x + 245$$
Norm Prime Eigenvalue
5 $[5, 5, w]$ $-\frac{4656319424}{128610195783}e^{11} - \frac{31969863416}{128610195783}e^{10} + \frac{50834089283}{128610195783}e^{9} + \frac{207666087164}{42870065261}e^{8} - \frac{14923334167}{128610195783}e^{7} - \frac{4253657976541}{128610195783}e^{6} - \frac{322807029001}{42870065261}e^{5} + \frac{11303681560166}{128610195783}e^{4} + \frac{1730929025221}{128610195783}e^{3} - \frac{8069723405779}{128610195783}e^{2} - \frac{769785260564}{128610195783}e + \frac{827236896574}{128610195783}$
5 $[5, 5, -w^{2} + w + 2]$ $\phantom{-}e$
7 $[7, 7, -w^{2} + 2]$ $\phantom{-}\frac{9158256107}{128610195783}e^{11} + \frac{63471137930}{128610195783}e^{10} - \frac{96434026751}{128610195783}e^{9} - \frac{411173965623}{42870065261}e^{8} - \frac{38424912215}{128610195783}e^{7} + \frac{8388038053462}{128610195783}e^{6} + \frac{769062539146}{42870065261}e^{5} - \frac{22184768486306}{128610195783}e^{4} - \frac{4092049908418}{128610195783}e^{3} + \frac{15808557627901}{128610195783}e^{2} + \frac{1232475992540}{128610195783}e - \frac{1720648254100}{128610195783}$
13 $[13, 13, -w^{2} + 3]$ $-\frac{30907939444}{128610195783}e^{11} - \frac{213842339341}{128610195783}e^{10} + \frac{328642399336}{128610195783}e^{9} + \frac{1387689370644}{42870065261}e^{8} + \frac{57317553625}{128610195783}e^{7} - \frac{28461930067832}{128610195783}e^{6} - \frac{2419732198820}{42870065261}e^{5} + \frac{76544103172612}{128610195783}e^{4} + \frac{12427441211192}{128610195783}e^{3} - \frac{58798675644686}{128610195783}e^{2} - \frac{3356573551768}{128610195783}e + \frac{8187657998006}{128610195783}$
13 $[13, 13, w^{2} - w - 4]$ $\phantom{-}1$
16 $[16, 2, 2]$ $-\frac{22212954440}{128610195783}e^{11} - \frac{153563806457}{128610195783}e^{10} + \frac{237057142412}{128610195783}e^{9} + \frac{996616469460}{42870065261}e^{8} + \frac{19793006915}{128610195783}e^{7} - \frac{20447242081666}{128610195783}e^{6} - \frac{1686287923587}{42870065261}e^{5} + \frac{54970441682822}{128610195783}e^{4} + \frac{8582046563626}{128610195783}e^{3} - \frac{41913270443212}{128610195783}e^{2} - \frac{2237751057155}{128610195783}e + \frac{5385051375709}{128610195783}$
19 $[19, 19, -w^{2} + w + 1]$ $\phantom{-}\frac{1895838832}{128610195783}e^{11} + \frac{14283923173}{128610195783}e^{10} - \frac{10503247345}{128610195783}e^{9} - \frac{85102041942}{42870065261}e^{8} - \frac{164871826039}{128610195783}e^{7} + \frac{1539798132014}{128610195783}e^{6} + \frac{450772381227}{42870065261}e^{5} - \frac{3318299911147}{128610195783}e^{4} - \frac{2683439745980}{128610195783}e^{3} + \frac{1127182386209}{128610195783}e^{2} + \frac{1292679020314}{128610195783}e - \frac{53568035567}{128610195783}$
23 $[23, 23, -w^{3} + 4w + 2]$ $-\frac{362462582}{42870065261}e^{11} - \frac{1981259515}{42870065261}e^{10} + \frac{7735024845}{42870065261}e^{9} + \frac{45683940043}{42870065261}e^{8} - \frac{66664391897}{42870065261}e^{7} - \frac{366377386976}{42870065261}e^{6} + \frac{321877495492}{42870065261}e^{5} + \frac{1135930536388}{42870065261}e^{4} - \frac{865747201543}{42870065261}e^{3} - \frac{1012661073991}{42870065261}e^{2} + \frac{784464179660}{42870065261}e + \frac{147622697696}{42870065261}$
25 $[25, 5, -w^{3} + w^{2} + 3w - 1]$ $\phantom{-}\frac{6898678237}{128610195783}e^{11} + \frac{45893093152}{128610195783}e^{10} - \frac{84922266094}{128610195783}e^{9} - \frac{301698760002}{42870065261}e^{8} + \frac{205534599602}{128610195783}e^{7} + \frac{6256074678404}{128610195783}e^{6} + \frac{72906748640}{42870065261}e^{5} - \frac{16803905452273}{128610195783}e^{4} + \frac{509196000112}{128610195783}e^{3} + \frac{12252962188001}{128610195783}e^{2} - \frac{889202936636}{128610195783}e - \frac{1858167566771}{128610195783}$
29 $[29, 29, w^{3} - w^{2} - 4w + 1]$ $-\frac{2450127661}{128610195783}e^{11} - \frac{19851890389}{128610195783}e^{10} + \frac{6621194431}{128610195783}e^{9} + \frac{121906967750}{42870065261}e^{8} + \frac{394044722509}{128610195783}e^{7} - \frac{2319008174375}{128610195783}e^{6} - \frac{1090195511536}{42870065261}e^{5} + \frac{5746564876084}{128610195783}e^{4} + \frac{7978807504076}{128610195783}e^{3} - \frac{4400223760010}{128610195783}e^{2} - \frac{4246847928679}{128610195783}e + \frac{481008341423}{128610195783}$
29 $[29, 29, -w + 3]$ $\phantom{-}\frac{6019179246}{42870065261}e^{11} + \frac{40567279502}{42870065261}e^{10} - \frac{71680711470}{42870065261}e^{9} - \frac{802039221897}{42870065261}e^{8} + \frac{129679443647}{42870065261}e^{7} + \frac{5591826858198}{42870065261}e^{6} + \frac{539359751854}{42870065261}e^{5} - \frac{15423759579214}{42870065261}e^{4} - \frac{432453636372}{42870065261}e^{3} + \frac{12410164205701}{42870065261}e^{2} - \frac{392202297445}{42870065261}e - \frac{1865870949532}{42870065261}$
31 $[31, 31, -w^{3} + w^{2} + 5w - 2]$ $-\frac{59732643}{42870065261}e^{11} - \frac{117255901}{42870065261}e^{10} + \frac{3071134727}{42870065261}e^{9} + \frac{8353553986}{42870065261}e^{8} - \frac{37668512380}{42870065261}e^{7} - \frac{104095545568}{42870065261}e^{6} + \frac{177755306845}{42870065261}e^{5} + \frac{414575057684}{42870065261}e^{4} - \frac{349011767772}{42870065261}e^{3} - \frac{452560131869}{42870065261}e^{2} + \frac{213409764616}{42870065261}e + \frac{113362469374}{42870065261}$
37 $[37, 37, w^{3} - 4w - 1]$ $\phantom{-}\frac{24342841877}{128610195783}e^{11} + \frac{168769602794}{128610195783}e^{10} - \frac{258167689238}{128610195783}e^{9} - \frac{1098502373333}{42870065261}e^{8} - \frac{81575035538}{128610195783}e^{7} + \frac{22637419925752}{128610195783}e^{6} + \frac{2045621662191}{42870065261}e^{5} - \frac{61406517527417}{128610195783}e^{4} - \frac{11259410050012}{128610195783}e^{3} + \frac{48105672840166}{128610195783}e^{2} + \frac{3526811135795}{128610195783}e - \frac{6732013574188}{128610195783}$
37 $[37, 37, w^{3} - 3w + 1]$ $\phantom{-}\frac{4025726269}{128610195783}e^{11} + \frac{29489719510}{128610195783}e^{10} - \frac{31613794171}{128610195783}e^{9} - \frac{186987945815}{42870065261}e^{8} - \frac{226653854662}{128610195783}e^{7} + \frac{3729975976100}{128610195783}e^{6} + \frac{810106119831}{42870065261}e^{5} - \frac{9754375755742}{128610195783}e^{4} - \frac{5360803232366}{128610195783}e^{3} + \frac{7190974587380}{128610195783}e^{2} + \frac{2324518707388}{128610195783}e - \frac{500258863565}{128610195783}$
53 $[53, 53, -w^{3} + 2w^{2} + 4w - 6]$ $\phantom{-}\frac{362462582}{42870065261}e^{11} + \frac{1981259515}{42870065261}e^{10} - \frac{7735024845}{42870065261}e^{9} - \frac{45683940043}{42870065261}e^{8} + \frac{66664391897}{42870065261}e^{7} + \frac{366377386976}{42870065261}e^{6} - \frac{321877495492}{42870065261}e^{5} - \frac{1135930536388}{42870065261}e^{4} + \frac{865747201543}{42870065261}e^{3} + \frac{969791008730}{42870065261}e^{2} - \frac{913074375443}{42870065261}e - \frac{19012501913}{42870065261}$
59 $[59, 59, w^{2} + w - 4]$ $-\frac{7510103020}{128610195783}e^{11} - \frac{52230421762}{128610195783}e^{10} + \frac{76247014726}{128610195783}e^{9} + \frac{333589834027}{42870065261}e^{8} + \frac{49928839069}{128610195783}e^{7} - \frac{6711795943364}{128610195783}e^{6} - \frac{581320235754}{42870065261}e^{5} + \frac{17596090720429}{128610195783}e^{4} + \frac{2051290900832}{128610195783}e^{3} - \frac{12862225205057}{128610195783}e^{2} + \frac{1566714924758}{128610195783}e + \frac{1405137304517}{128610195783}$
61 $[61, 61, -2w^{2} + w + 8]$ $\phantom{-}\frac{6761985082}{42870065261}e^{11} + \frac{46648272113}{42870065261}e^{10} - \frac{73423533456}{42870065261}e^{9} - \frac{914823854563}{42870065261}e^{8} - \frac{363973988}{42870065261}e^{7} + \frac{6295167365728}{42870065261}e^{6} + \frac{1615106087631}{42870065261}e^{5} - \frac{16991567528638}{42870065261}e^{4} - \frac{2977859967588}{42870065261}e^{3} + \frac{12906005584123}{42870065261}e^{2} + \frac{772818781717}{42870065261}e - \frac{1429941179714}{42870065261}$
73 $[73, 73, -w^{3} + 5w - 1]$ $\phantom{-}\frac{19929681331}{42870065261}e^{11} + \frac{137672475939}{42870065261}e^{10} - \frac{214385333059}{42870065261}e^{9} - \frac{2690116003381}{42870065261}e^{8} - \frac{8971143907}{42870065261}e^{7} + \frac{18474493098321}{42870065261}e^{6} + \frac{4637535977776}{42870065261}e^{5} - \frac{50018342049570}{42870065261}e^{4} - \frac{8286235126029}{42870065261}e^{3} + \frac{39211588750787}{42870065261}e^{2} + \frac{2543665510077}{42870065261}e - \frac{5953272256474}{42870065261}$
79 $[79, 79, 2w^{2} + w - 6]$ $-\frac{11429904323}{128610195783}e^{11} - \frac{82115341373}{128610195783}e^{10} + \frac{99733960196}{128610195783}e^{9} + \frac{522313095531}{42870065261}e^{8} + \frac{446187923444}{128610195783}e^{7} - \frac{10388641575703}{128610195783}e^{6} - \frac{1841920544967}{42870065261}e^{5} + \frac{26520811009823}{128610195783}e^{4} + \frac{11792701400845}{128610195783}e^{3} - \frac{17641440054991}{128610195783}e^{2} - \frac{6038827278971}{128610195783}e + \frac{2011339022542}{128610195783}$
79 $[79, 79, 2w^{3} - w^{2} - 9w + 3]$ $\phantom{-}\frac{29628158549}{128610195783}e^{11} + \frac{207832494143}{128610195783}e^{10} - \frac{295567096067}{128610195783}e^{9} - \frac{1339532342037}{42870065261}e^{8} - \frac{407333204447}{128610195783}e^{7} + \frac{27305013608833}{128610195783}e^{6} + \frac{3062876821036}{42870065261}e^{5} - \frac{73312887936521}{128610195783}e^{4} - \frac{17207867717854}{128610195783}e^{3} + \frac{57638673508951}{128610195783}e^{2} + \frac{4950908965082}{128610195783}e - \frac{8925471220276}{128610195783}$
 Display number of eigenvalues

## Atkin-Lehner eigenvalues

Norm Prime Eigenvalue
$13$ $[13, 13, w^{2} - w - 4]$ $-1$