# Properties

 Base field $\Q(\sqrt{393})$ Weight [2, 2] Level norm 6 Level $[6, 6, -5w - 47]$ Label 2.2.393.1-6.1-h Dimension 10 CM no Base change no

# 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 $[6, 6, -5w - 47]$ Label 2.2.393.1-6.1-h Dimension 10 Is CM no Is base change no Parent newspace dimension 44

## Hecke eigenvalues ($q$-expansion)

The Hecke eigenvalue field is $\Q(e)$ where $e$ is a root of the defining polynomial:
$x^{10}$ $\mathstrut +\mathstrut 2x^{9}$ $\mathstrut -\mathstrut 14x^{8}$ $\mathstrut -\mathstrut 24x^{7}$ $\mathstrut +\mathstrut 71x^{6}$ $\mathstrut +\mathstrut 89x^{5}$ $\mathstrut -\mathstrut 162x^{4}$ $\mathstrut -\mathstrut 96x^{3}$ $\mathstrut +\mathstrut 154x^{2}$ $\mathstrut -\mathstrut 12x$ $\mathstrut -\mathstrut 8$
Norm Prime Eigenvalue
2 $[2, 2, -17w - 160]$ $\phantom{-}e$
2 $[2, 2, -17w + 177]$ $-1$
3 $[3, 3, -842w + 8767]$ $\phantom{-}1$
7 $[7, 7, -2w + 21]$ $-\frac{3}{4}e^{9} - \frac{1}{2}e^{8} + \frac{19}{2}e^{7} + 4e^{6} - \frac{149}{4}e^{5} - \frac{19}{4}e^{4} + \frac{85}{2}e^{3} - 11e^{2} + \frac{7}{2}e - 2$
7 $[7, 7, 2w + 19]$ $\phantom{-}\frac{3}{2}e^{9} + \frac{5}{2}e^{8} - 18e^{7} - 27e^{6} + \frac{135}{2}e^{5} + 85e^{4} - \frac{167}{2}e^{3} - 70e^{2} + 28e + 5$
13 $[13, 13, -12w - 113]$ $-\frac{3}{4}e^{9} + \frac{1}{2}e^{8} + \frac{21}{2}e^{7} - 8e^{6} - \frac{185}{4}e^{5} + \frac{157}{4}e^{4} + \frac{123}{2}e^{3} - 58e^{2} + \frac{11}{2}e - 1$
13 $[13, 13, 12w - 125]$ $\phantom{-}\frac{3}{4}e^{9} - \frac{5}{2}e^{8} - \frac{23}{2}e^{7} + 34e^{6} + \frac{213}{4}e^{5} - \frac{585}{4}e^{4} - \frac{125}{2}e^{3} + 197e^{2} - \frac{91}{2}e - 9$
17 $[17, 17, 182w - 1895]$ $\phantom{-}\frac{1}{4}e^{9} + e^{8} - \frac{5}{2}e^{7} - 12e^{6} + \frac{27}{4}e^{5} + \frac{179}{4}e^{4} - 5e^{3} - 52e^{2} + \frac{15}{2}e + 4$
17 $[17, 17, 182w + 1713]$ $\phantom{-}\frac{11}{4}e^{9} + \frac{5}{2}e^{8} - \frac{69}{2}e^{7} - 23e^{6} + \frac{541}{4}e^{5} + \frac{199}{4}e^{4} - \frac{329}{2}e^{3} + 4e^{2} + \frac{31}{2}e - 1$
23 $[23, 23, -512w - 4819]$ $-3e^{9} - \frac{5}{2}e^{8} + 37e^{7} + 21e^{6} - 140e^{5} - \frac{65}{2}e^{4} + \frac{301}{2}e^{3} - 39e^{2} + 19e + 1$
23 $[23, 23, 512w - 5331]$ $-\frac{7}{4}e^{9} - \frac{9}{2}e^{8} + \frac{39}{2}e^{7} + 51e^{6} - \frac{261}{4}e^{5} - \frac{695}{4}e^{4} + \frac{141}{2}e^{3} + 167e^{2} - \frac{83}{2}e - 5$
25 $[25, 5, -5]$ $\phantom{-}\frac{3}{2}e^{8} + e^{7} - 19e^{6} - 8e^{5} + \frac{151}{2}e^{4} + \frac{21}{2}e^{3} - 93e^{2} + 19e + 7$
29 $[29, 29, 22w - 229]$ $\phantom{-}\frac{7}{4}e^{9} + 3e^{8} - \frac{41}{2}e^{7} - 32e^{6} + \frac{289}{4}e^{5} + \frac{389}{4}e^{4} - 72e^{3} - 69e^{2} + \frac{11}{2}e - 2$
29 $[29, 29, 22w + 207]$ $-e^{9} - \frac{5}{2}e^{8} + 11e^{7} + 28e^{6} - 35e^{5} - \frac{187}{2}e^{4} + \frac{57}{2}e^{3} + 89e^{2} - 3e - 14$
43 $[43, 43, 114w + 1073]$ $\phantom{-}\frac{1}{2}e^{9} + 5e^{8} - 2e^{7} - 61e^{6} - \frac{29}{2}e^{5} + \frac{463}{2}e^{4} + 52e^{3} - 271e^{2} + 18e + 23$
43 $[43, 43, 114w - 1187]$ $-2e^{9} - \frac{9}{2}e^{8} + 22e^{7} + 50e^{6} - 70e^{5} - \frac{331}{2}e^{4} + \frac{117}{2}e^{3} + 153e^{2} - 15e - 11$
47 $[47, 47, 8w - 83]$ $\phantom{-}\frac{3}{4}e^{9} + \frac{1}{2}e^{8} - \frac{19}{2}e^{7} - 4e^{6} + \frac{153}{4}e^{5} + \frac{23}{4}e^{4} - \frac{99}{2}e^{3} + 6e^{2} + \frac{7}{2}e + 1$
47 $[47, 47, -8w - 75]$ $-\frac{11}{4}e^{9} - \frac{3}{2}e^{8} + \frac{71}{2}e^{7} + 10e^{6} - \frac{581}{4}e^{5} + \frac{13}{4}e^{4} + \frac{377}{2}e^{3} - 72e^{2} - \frac{29}{2}e + 8$
61 $[61, 61, -1172w - 11031]$ $-\frac{7}{4}e^{9} - 6e^{8} + \frac{35}{2}e^{7} + 69e^{6} - \frac{177}{4}e^{5} - \frac{957}{4}e^{4} + 9e^{3} + 239e^{2} - \frac{3}{2}e - 20$
61 $[61, 61, 1172w - 12203]$ $-2e^{9} + \frac{3}{2}e^{8} + 28e^{7} - 24e^{6} - 125e^{5} + \frac{235}{2}e^{4} + \frac{353}{2}e^{3} - 177e^{2} - 5e + 15$
 Display number of eigenvalues

## Atkin-Lehner eigenvalues

Norm Prime Eigenvalue
2 $[2, 2, -17w + 177]$ $1$
3 $[3, 3, -842w + 8767]$ $-1$