# Properties

 Base field 4.4.16997.1 Weight [2, 2, 2, 2] Level norm 13 Level $[13, 13, -w^{2} + 3]$ Label 4.4.16997.1-13.1-b Dimension 9 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 $[13, 13, -w^{2} + 3]$ Label 4.4.16997.1-13.1-b Dimension 9 Is CM no Is base change no Parent newspace dimension 26

## Hecke eigenvalues ($q$-expansion)

The Hecke eigenvalue field is $\Q(e)$ where $e$ is a root of the defining polynomial:
$$x^{9} + 2x^{8} - 24x^{7} - 34x^{6} + 173x^{5} + 154x^{4} - 444x^{3} - 306x^{2} + 368x + 245$$
Norm Prime Eigenvalue
5 $[5, 5, w]$ $\phantom{-}\frac{49787}{798007}e^{8} + \frac{188199}{798007}e^{7} - \frac{130483}{114001}e^{6} - \frac{3378358}{798007}e^{5} + \frac{3830616}{798007}e^{4} + \frac{15002032}{798007}e^{3} - \frac{2900951}{798007}e^{2} - \frac{18763825}{798007}e - \frac{789127}{114001}$
5 $[5, 5, -w^{2} + w + 2]$ $\phantom{-}e$
7 $[7, 7, -w^{2} + 2]$ $\phantom{-}\frac{56307}{798007}e^{8} + \frac{167084}{798007}e^{7} - \frac{156881}{114001}e^{6} - \frac{2770342}{798007}e^{5} + \frac{5122178}{798007}e^{4} + \frac{10396889}{798007}e^{3} - \frac{4181843}{798007}e^{2} - \frac{11055246}{798007}e - \frac{875003}{114001}$
13 $[13, 13, -w^{2} + 3]$ $\phantom{-}1$
13 $[13, 13, w^{2} - w - 4]$ $\phantom{-}\frac{18976}{798007}e^{8} + \frac{41357}{798007}e^{7} - \frac{61303}{114001}e^{6} - \frac{556872}{798007}e^{5} + \frac{3052054}{798007}e^{4} + \frac{645896}{798007}e^{3} - \frac{9302245}{798007}e^{2} + \frac{2397956}{798007}e + \frac{1243547}{114001}$
16 $[16, 2, 2]$ $-\frac{146439}{798007}e^{8} - \frac{416172}{798007}e^{7} + \frac{412420}{114001}e^{6} + \frac{6743584}{798007}e^{5} - \frac{13891106}{798007}e^{4} - \frac{24052818}{798007}e^{3} + \frac{13459300}{798007}e^{2} + \frac{24933215}{798007}e + \frac{1338415}{114001}$
19 $[19, 19, -w^{2} + w + 1]$ $\phantom{-}\frac{8027}{798007}e^{8} - \frac{53534}{798007}e^{7} - \frac{41137}{114001}e^{6} + \frac{1417309}{798007}e^{5} + \frac{2725656}{798007}e^{4} - \frac{10153691}{798007}e^{3} - \frac{7827353}{798007}e^{2} + \frac{16080101}{798007}e + \frac{1421259}{114001}$
23 $[23, 23, -w^{3} + 4w + 2]$ $-\frac{79661}{798007}e^{8} - \frac{347255}{798007}e^{7} + \frac{177790}{114001}e^{6} + \frac{6084539}{798007}e^{5} - \frac{2502229}{798007}e^{4} - \frac{25183838}{798007}e^{3} - \frac{8108710}{798007}e^{2} + \frac{28227487}{798007}e + \frac{2376046}{114001}$
25 $[25, 5, -w^{3} + w^{2} + 3w - 1]$ $-\frac{26433}{798007}e^{8} - \frac{8028}{798007}e^{7} + \frac{109574}{114001}e^{6} + \frac{64161}{798007}e^{5} - \frac{6450565}{798007}e^{4} - \frac{215083}{798007}e^{3} + \frac{15191504}{798007}e^{2} + \frac{793577}{798007}e - \frac{1281921}{114001}$
29 $[29, 29, w^{3} - w^{2} - 4w + 1]$ $\phantom{-}\frac{1237}{114001}e^{8} + \frac{1152}{114001}e^{7} - \frac{30827}{114001}e^{6} - \frac{30678}{114001}e^{5} + \frac{180801}{114001}e^{4} + \frac{335037}{114001}e^{3} + \frac{11633}{114001}e^{2} - \frac{1129150}{114001}e - \frac{701469}{114001}$
29 $[29, 29, -w + 3]$ $-\frac{101777}{798007}e^{8} - \frac{361308}{798007}e^{7} + \frac{265794}{114001}e^{6} + \frac{6280053}{798007}e^{5} - \frac{7517728}{798007}e^{4} - \frac{26351091}{798007}e^{3} + \frac{3567980}{798007}e^{2} + \frac{32689486}{798007}e + \frac{1616572}{114001}$
31 $[31, 31, -w^{3} + w^{2} + 5w - 2]$ $-\frac{45506}{798007}e^{8} - \frac{318949}{798007}e^{7} + \frac{59262}{114001}e^{6} + \frac{5920937}{798007}e^{5} + \frac{3106994}{798007}e^{4} - \frac{26797826}{798007}e^{3} - \frac{18610416}{798007}e^{2} + \frac{32025986}{798007}e + \frac{3174251}{114001}$
37 $[37, 37, w^{3} - 4w - 1]$ $-\frac{5818}{114001}e^{8} + \frac{2231}{114001}e^{7} + \frac{179088}{114001}e^{6} - \frac{88506}{114001}e^{5} - \frac{1658232}{114001}e^{4} + \frac{707922}{114001}e^{3} + \frac{4909629}{114001}e^{2} - \frac{1269792}{114001}e - \frac{3935088}{114001}$
37 $[37, 37, w^{3} - 3w + 1]$ $-\frac{83942}{798007}e^{8} - \frac{216505}{798007}e^{7} + \frac{249011}{114001}e^{6} + \frac{3541960}{798007}e^{5} - \frac{9439839}{798007}e^{4} - \frac{13388044}{798007}e^{3} + \frac{14200664}{798007}e^{2} + \frac{16561340}{798007}e - \frac{465082}{114001}$
53 $[53, 53, -w^{3} + 2w^{2} + 4w - 6]$ $\phantom{-}\frac{156874}{798007}e^{8} + \frac{193280}{798007}e^{7} - \frac{539820}{114001}e^{6} - \frac{2221061}{798007}e^{5} + \frac{25660349}{798007}e^{4} + \frac{883278}{798007}e^{3} - \frac{50438034}{798007}e^{2} + \frac{6192105}{798007}e + \frac{3690907}{114001}$
59 $[59, 59, w^{2} + w - 4]$ $\phantom{-}\frac{24205}{114001}e^{8} + \frac{117005}{114001}e^{7} - \frac{363409}{114001}e^{6} - \frac{2115390}{114001}e^{5} + \frac{560527}{114001}e^{4} + \frac{9308542}{114001}e^{3} + \frac{3165758}{114001}e^{2} - \frac{11025028}{114001}e - \frac{6920035}{114001}$
61 $[61, 61, -2w^{2} + w + 8]$ $-\frac{210519}{798007}e^{8} - \frac{580726}{798007}e^{7} + \frac{612417}{114001}e^{6} + \frac{9438239}{798007}e^{5} - \frac{22878781}{798007}e^{4} - \frac{33884313}{798007}e^{3} + \frac{35536150}{798007}e^{2} + \frac{34812926}{798007}e - \frac{525625}{114001}$
73 $[73, 73, -w^{3} + 5w - 1]$ $-\frac{345073}{798007}e^{8} - \frac{851185}{798007}e^{7} + \frac{1041586}{114001}e^{6} + \frac{13760312}{798007}e^{5} - \frac{40656415}{798007}e^{4} - \frac{50091819}{798007}e^{3} + \frac{59768939}{798007}e^{2} + \frac{54383043}{798007}e - \frac{712111}{114001}$
79 $[79, 79, 2w^{2} + w - 6]$ $-\frac{28776}{798007}e^{8} + \frac{54025}{798007}e^{7} + \frac{136650}{114001}e^{6} - \frac{1409603}{798007}e^{5} - \frac{9291826}{798007}e^{4} + \frac{9325996}{798007}e^{3} + \frac{29155743}{798007}e^{2} - \frac{16212036}{798007}e - \frac{4323283}{114001}$
79 $[79, 79, 2w^{3} - w^{2} - 9w + 3]$ $\phantom{-}\frac{52180}{798007}e^{8} + \frac{166374}{798007}e^{7} - \frac{132933}{114001}e^{6} - \frac{2859498}{798007}e^{5} + \frac{2664820}{798007}e^{4} + \frac{12364137}{798007}e^{3} + \frac{10125040}{798007}e^{2} - \frac{19471888}{798007}e - \frac{3862514}{114001}$
 Display number of eigenvalues

## Atkin-Lehner eigenvalues

Norm Prime Eigenvalue
13 $[13, 13, -w^{2} + 3]$ $-1$