# Properties

 Base field $$\Q(\sqrt{197})$$ Weight [2, 2] Level norm 7 Level $[7,7,-w - 6]$ Label 2.2.197.1-7.2-a Dimension 10 CM no Base change no

# Related objects

• L-function not available

## Base field $$\Q(\sqrt{197})$$

Generator $$w$$, with minimal polynomial $$x^{2} - x - 49$$; narrow class number $$1$$ and class number $$1$$.

## Form

 Weight [2, 2] Level $[7,7,-w - 6]$ Label 2.2.197.1-7.2-a Dimension 10 Is CM no Is base change no Parent newspace dimension 23

## Hecke eigenvalues ($q$-expansion)

The Hecke eigenvalue field is $\Q(e)$ where $e$ is a root of the defining polynomial:
$$x^{10}$$ $$\mathstrut +\mathstrut 7x^{9}$$ $$\mathstrut +\mathstrut 6x^{8}$$ $$\mathstrut -\mathstrut 46x^{7}$$ $$\mathstrut -\mathstrut 83x^{6}$$ $$\mathstrut +\mathstrut 34x^{5}$$ $$\mathstrut +\mathstrut 93x^{4}$$ $$\mathstrut -\mathstrut 7x^{3}$$ $$\mathstrut -\mathstrut 20x^{2}$$ $$\mathstrut +\mathstrut x$$ $$\mathstrut +\mathstrut 1$$
Norm Prime Eigenvalue
4 $[4, 2, 2]$ $\phantom{-}e$
7 $[7, 7, w - 7]$ $\phantom{-}\frac{255}{47}e^{9} + \frac{1835}{47}e^{8} + \frac{1911}{47}e^{7} - \frac{11206}{47}e^{6} - \frac{23236}{47}e^{5} + \frac{3126}{47}e^{4} + \frac{22600}{47}e^{3} + \frac{3391}{47}e^{2} - \frac{2662}{47}e - \frac{384}{47}$
7 $[7, 7, w + 6]$ $\phantom{-}1$
9 $[9, 3, 3]$ $-\frac{294}{47}e^{9} - \frac{2182}{47}e^{8} - \frac{2660}{47}e^{7} + \frac{12561}{47}e^{6} + \frac{29790}{47}e^{5} + \frac{1499}{47}e^{4} - \frac{28271}{47}e^{3} - \frac{8953}{47}e^{2} + \frac{3496}{47}e + \frac{926}{47}$
19 $[19, 19, w + 5]$ $\phantom{-}\frac{102}{47}e^{9} + \frac{828}{47}e^{8} + \frac{1413}{47}e^{7} - \frac{3956}{47}e^{6} - \frac{13515}{47}e^{5} - \frac{6138}{47}e^{4} + \frac{11860}{47}e^{3} + \frac{8820}{47}e^{2} - \frac{1281}{47}e - \frac{1009}{47}$
19 $[19, 19, w - 6]$ $-\frac{32}{47}e^{9} - \frac{210}{47}e^{8} - \frac{106}{47}e^{7} + \frac{1489}{47}e^{6} + \frac{2031}{47}e^{5} - \frac{1721}{47}e^{4} - \frac{2132}{47}e^{3} + \frac{675}{47}e^{2} + \frac{1}{47}e - \frac{53}{47}$
23 $[23, 23, w + 8]$ $-\frac{273}{47}e^{9} - \frac{2006}{47}e^{8} - \frac{2329}{47}e^{7} + \frac{11788}{47}e^{6} + \frac{26749}{47}e^{5} - \frac{290}{47}e^{4} - \frac{25644}{47}e^{3} - \frac{6504}{47}e^{2} + \frac{3253}{47}e + \frac{645}{47}$
23 $[23, 23, -w + 9]$ $\phantom{-}\frac{439}{47}e^{9} + \frac{3254}{47}e^{8} + \frac{3954}{47}e^{7} - \frac{18722}{47}e^{6} - \frac{44326}{47}e^{5} - \frac{2359}{47}e^{4} + \frac{41909}{47}e^{3} + \frac{13692}{47}e^{2} - \frac{5288}{47}e - \frac{1595}{47}$
25 $[25, 5, 5]$ $\phantom{-}\frac{118}{47}e^{9} + \frac{839}{47}e^{8} + \frac{808}{47}e^{7} - \frac{5288}{47}e^{6} - \frac{10324}{47}e^{5} + \frac{2595}{47}e^{4} + \frac{10576}{47}e^{3} + \frac{234}{47}e^{2} - \frac{1493}{47}e - \frac{66}{47}$
29 $[29, 29, -w - 4]$ $\phantom{-}\frac{160}{47}e^{9} + \frac{1191}{47}e^{8} + \frac{1470}{47}e^{7} - \frac{6834}{47}e^{6} - \frac{16406}{47}e^{5} - \frac{983}{47}e^{4} + \frac{15877}{47}e^{3} + \frac{5085}{47}e^{2} - \frac{2684}{47}e - \frac{534}{47}$
29 $[29, 29, w - 5]$ $-\frac{229}{47}e^{9} - \frac{1635}{47}e^{8} - \frac{1631}{47}e^{7} + \frac{10099}{47}e^{6} + \frac{20214}{47}e^{5} - \frac{3640}{47}e^{4} - \frac{19305}{47}e^{3} - \frac{1845}{47}e^{2} + \frac{2153}{47}e + \frac{148}{47}$
37 $[37, 37, -w - 3]$ $-\frac{713}{47}e^{9} - \frac{5293}{47}e^{8} - \frac{6489}{47}e^{7} + \frac{30276}{47}e^{6} + \frac{72265}{47}e^{5} + \frac{5010}{47}e^{4} - \frac{67273}{47}e^{3} - \frac{23202}{47}e^{2} + \frac{7673}{47}e + \frac{2560}{47}$
37 $[37, 37, w - 4]$ $-\frac{385}{47}e^{9} - \frac{2788}{47}e^{8} - \frac{2982}{47}e^{7} + \frac{16976}{47}e^{6} + \frac{35949}{47}e^{5} - \frac{4363}{47}e^{4} - \frac{36067}{47}e^{3} - \frac{5810}{47}e^{2} + \frac{5113}{47}e + \frac{671}{47}$
41 $[41, 41, -w - 9]$ $\phantom{-}\frac{419}{47}e^{9} + \frac{3111}{47}e^{8} + \frac{3829}{47}e^{7} - \frac{17715}{47}e^{6} - \frac{42428}{47}e^{5} - \frac{3323}{47}e^{4} + \frac{38767}{47}e^{3} + \frac{13121}{47}e^{2} - \frac{4271}{47}e - \frac{1258}{47}$
41 $[41, 41, w - 10]$ $\phantom{-}\frac{257}{47}e^{9} + \frac{1948}{47}e^{8} + \frac{2605}{47}e^{7} - \frac{10738}{47}e^{6} - \frac{27778}{47}e^{5} - \frac{4542}{47}e^{4} + \frac{25330}{47}e^{3} + \frac{10813}{47}e^{2} - \frac{2618}{47}e - \frac{1353}{47}$
43 $[43, 43, -w - 2]$ $\phantom{-}\frac{339}{47}e^{9} + \frac{2539}{47}e^{8} + \frac{3235}{47}e^{7} - \frac{14298}{47}e^{6} - \frac{35353}{47}e^{5} - \frac{3889}{47}e^{4} + \frac{32779}{47}e^{3} + \frac{12388}{47}e^{2} - \frac{3775}{47}e - \frac{1743}{47}$
43 $[43, 43, w - 3]$ $-\frac{388}{47}e^{9} - \frac{2840}{47}e^{8} - \frac{3224}{47}e^{7} + \frac{16885}{47}e^{6} + \frac{37592}{47}e^{5} - \frac{1697}{47}e^{4} - \frac{36966}{47}e^{3} - \frac{8248}{47}e^{2} + \frac{5047}{47}e + \frac{926}{47}$
47 $[47, 47, -w - 1]$ $-\frac{641}{47}e^{9} - \frac{4703}{47}e^{8} - \frac{5428}{47}e^{7} + \frac{27666}{47}e^{6} + \frac{62490}{47}e^{5} - \frac{835}{47}e^{4} - \frac{59750}{47}e^{3} - \frac{15356}{47}e^{2} + \frac{7330}{47}e + \frac{1469}{47}$
47 $[47, 47, w - 2]$ $-\frac{234}{47}e^{9} - \frac{1659}{47}e^{8} - \frac{1580}{47}e^{7} + \frac{10433}{47}e^{6} + \frac{20195}{47}e^{5} - \frac{4915}{47}e^{4} - \frac{19973}{47}e^{3} - \frac{989}{47}e^{2} + \frac{2466}{47}e + \frac{244}{47}$
53 $[53, 53, 2w - 13]$ $-\frac{290}{47}e^{9} - \frac{2238}{47}e^{8} - \frac{3246}{47}e^{7} + \frac{11711}{47}e^{6} + \frac{33114}{47}e^{5} + \frac{9428}{47}e^{4} - \frac{28639}{47}e^{3} - \frac{15917}{47}e^{2} + \frac{2738}{47}e + \frac{1526}{47}$
 Display number of eigenvalues

## Atkin-Lehner eigenvalues

Norm Prime Eigenvalue
7 $[7,7,-w - 6]$ $-1$