# Properties

 Base field $\Q(\sqrt{393})$ Weight [2, 2] Level norm 1 Level $[1, 1, 1]$ Label 2.2.393.1-1.1-e Dimension 16 CM no Base change yes

# Related objects

• L-function not available

## Base field $\Q(\sqrt{393})$

Generator $w$, with minimal polynomial $x^{2} - x - 98$; narrow class number $2$ and class number $1$.

## Form

 Weight [2, 2] Level $[1, 1, 1]$ Label 2.2.393.1-1.1-e Dimension 16 Is CM no Is base change yes Parent newspace dimension 31

## Hecke eigenvalues ($q$-expansion)

The Hecke eigenvalue field is $\Q(e)$ where $e$ is a root of the defining polynomial:
$x^{16}$ $\mathstrut -\mathstrut 30x^{14}$ $\mathstrut +\mathstrut 378x^{12}$ $\mathstrut -\mathstrut 2602x^{10}$ $\mathstrut +\mathstrut 10654x^{8}$ $\mathstrut -\mathstrut 26443x^{6}$ $\mathstrut +\mathstrut 38661x^{4}$ $\mathstrut -\mathstrut 30305x^{2}$ $\mathstrut +\mathstrut 9710$
Norm Prime Eigenvalue
2 $[2, 2, -17w - 160]$ $\phantom{-}e$
2 $[2, 2, -17w + 177]$ $\phantom{-}e$
3 $[3, 3, -842w + 8767]$ $-\frac{1}{116}e^{14} + \frac{21}{116}e^{12} - \frac{189}{116}e^{10} + \frac{1017}{116}e^{8} - \frac{3589}{116}e^{6} + \frac{129}{2}e^{4} - \frac{7283}{116}e^{2} + \frac{1113}{58}$
7 $[7, 7, -2w + 21]$ $\phantom{-}\frac{7}{58}e^{14} - \frac{88}{29}e^{12} + \frac{1787}{58}e^{10} - \frac{4705}{29}e^{8} + \frac{27443}{58}e^{6} - \frac{1511}{2}e^{4} + \frac{17646}{29}e^{2} - \frac{5587}{29}$
7 $[7, 7, 2w + 19]$ $\phantom{-}\frac{7}{58}e^{14} - \frac{88}{29}e^{12} + \frac{1787}{58}e^{10} - \frac{4705}{29}e^{8} + \frac{27443}{58}e^{6} - \frac{1511}{2}e^{4} + \frac{17646}{29}e^{2} - \frac{5587}{29}$
13 $[13, 13, -12w - 113]$ $-\frac{7}{58}e^{14} + \frac{88}{29}e^{12} - \frac{1787}{58}e^{10} + \frac{4705}{29}e^{8} - \frac{27443}{58}e^{6} + \frac{1509}{2}e^{4} - \frac{17414}{29}e^{2} + \frac{5239}{29}$
13 $[13, 13, 12w - 125]$ $-\frac{7}{58}e^{14} + \frac{88}{29}e^{12} - \frac{1787}{58}e^{10} + \frac{4705}{29}e^{8} - \frac{27443}{58}e^{6} + \frac{1509}{2}e^{4} - \frac{17414}{29}e^{2} + \frac{5239}{29}$
17 $[17, 17, 182w - 1895]$ $\phantom{-}\frac{11}{116}e^{15} - \frac{289}{116}e^{13} + \frac{3123}{116}e^{11} - \frac{17973}{116}e^{9} + \frac{59431}{116}e^{7} - 971e^{5} + \frac{112419}{116}e^{3} - \frac{22393}{58}e$
17 $[17, 17, 182w + 1713]$ $\phantom{-}\frac{11}{116}e^{15} - \frac{289}{116}e^{13} + \frac{3123}{116}e^{11} - \frac{17973}{116}e^{9} + \frac{59431}{116}e^{7} - 971e^{5} + \frac{112419}{116}e^{3} - \frac{22393}{58}e$
23 $[23, 23, -512w - 4819]$ $-\frac{5}{29}e^{15} + \frac{134}{29}e^{13} - \frac{1467}{29}e^{11} + \frac{8449}{29}e^{9} - \frac{27428}{29}e^{7} + 1714e^{5} - \frac{46014}{29}e^{3} + \frac{16553}{29}e$
23 $[23, 23, 512w - 5331]$ $-\frac{5}{29}e^{15} + \frac{134}{29}e^{13} - \frac{1467}{29}e^{11} + \frac{8449}{29}e^{9} - \frac{27428}{29}e^{7} + 1714e^{5} - \frac{46014}{29}e^{3} + \frac{16553}{29}e$
25 $[25, 5, -5]$ $\phantom{-}\frac{15}{116}e^{14} - \frac{431}{116}e^{12} + \frac{5039}{116}e^{10} - \frac{30799}{116}e^{8} + \frac{105223}{116}e^{6} - \frac{3427}{2}e^{4} + \frac{189981}{116}e^{2} - \frac{34733}{58}$
29 $[29, 29, 22w - 229]$ $\phantom{-}\frac{7}{116}e^{15} - \frac{205}{116}e^{13} + \frac{2483}{116}e^{11} - \frac{15993}{116}e^{9} + \frac{58531}{116}e^{7} - 1036e^{5} + \frac{126555}{116}e^{3} - \frac{26119}{58}e$
29 $[29, 29, 22w + 207]$ $\phantom{-}\frac{7}{116}e^{15} - \frac{205}{116}e^{13} + \frac{2483}{116}e^{11} - \frac{15993}{116}e^{9} + \frac{58531}{116}e^{7} - 1036e^{5} + \frac{126555}{116}e^{3} - \frac{26119}{58}e$
43 $[43, 43, 114w + 1073]$ $-\frac{13}{29}e^{14} + \frac{691}{58}e^{12} - \frac{3762}{29}e^{10} + \frac{43291}{58}e^{8} - \frac{70640}{29}e^{6} + \frac{8951}{2}e^{4} - \frac{246401}{58}e^{2} + \frac{46164}{29}$
43 $[43, 43, 114w - 1187]$ $-\frac{13}{29}e^{14} + \frac{691}{58}e^{12} - \frac{3762}{29}e^{10} + \frac{43291}{58}e^{8} - \frac{70640}{29}e^{6} + \frac{8951}{2}e^{4} - \frac{246401}{58}e^{2} + \frac{46164}{29}$
47 $[47, 47, 8w - 83]$ $-\frac{1}{58}e^{15} + \frac{21}{58}e^{13} - \frac{131}{58}e^{11} - \frac{85}{58}e^{9} + \frac{4067}{58}e^{7} - 281e^{5} + \frac{25023}{58}e^{3} - \frac{6514}{29}e$
47 $[47, 47, -8w - 75]$ $-\frac{1}{58}e^{15} + \frac{21}{58}e^{13} - \frac{131}{58}e^{11} - \frac{85}{58}e^{9} + \frac{4067}{58}e^{7} - 281e^{5} + \frac{25023}{58}e^{3} - \frac{6514}{29}e$
61 $[61, 61, -1172w - 11031]$ $-\frac{25}{116}e^{14} + \frac{641}{116}e^{12} - \frac{6697}{116}e^{10} + \frac{36793}{116}e^{8} - \frac{114433}{116}e^{6} + \frac{3477}{2}e^{4} - \frac{187875}{116}e^{2} + \frac{35829}{58}$
61 $[61, 61, 1172w - 12203]$ $-\frac{25}{116}e^{14} + \frac{641}{116}e^{12} - \frac{6697}{116}e^{10} + \frac{36793}{116}e^{8} - \frac{114433}{116}e^{6} + \frac{3477}{2}e^{4} - \frac{187875}{116}e^{2} + \frac{35829}{58}$
 Display number of eigenvalues

## Atkin-Lehner eigenvalues

This form has no Atkin-Lehner eigenvalues since the level is $(1)$.