# Properties

 Base field $$\Q(\sqrt{197})$$ Weight [2, 2] Level norm 9 Level $[9, 3, 3]$ Label 2.2.197.1-9.1-e Dimension 6 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 $[9, 3, 3]$ Label 2.2.197.1-9.1-e Dimension 6 Is CM no Is base change no Parent newspace dimension 30

## Hecke eigenvalues ($q$-expansion)

The Hecke eigenvalue field is $\Q(e)$ where $e$ is a root of the defining polynomial:
$$x^{6}$$ $$\mathstrut +\mathstrut 3x^{5}$$ $$\mathstrut -\mathstrut 28x^{4}$$ $$\mathstrut -\mathstrut 72x^{3}$$ $$\mathstrut +\mathstrut 226x^{2}$$ $$\mathstrut +\mathstrut 383x$$ $$\mathstrut -\mathstrut 577$$
Norm Prime Eigenvalue
4 $[4, 2, 2]$ $-\frac{3}{49}e^{5} + \frac{4}{49}e^{4} + \frac{83}{49}e^{3} - \frac{111}{49}e^{2} - \frac{540}{49}e + \frac{701}{49}$
7 $[7, 7, w - 7]$ $-\frac{1}{49}e^{5} - \frac{1}{49}e^{4} + \frac{30}{49}e^{3} + \frac{12}{49}e^{2} - \frac{250}{49}e + \frac{19}{49}$
7 $[7, 7, w + 6]$ $\phantom{-}e$
9 $[9, 3, 3]$ $\phantom{-}1$
19 $[19, 19, w + 5]$ $-\frac{1}{112}e^{5} + \frac{1}{28}e^{4} + \frac{2}{7}e^{3} - \frac{9}{14}e^{2} - \frac{85}{56}e + \frac{535}{112}$
19 $[19, 19, w - 6]$ $-\frac{57}{784}e^{5} + \frac{5}{196}e^{4} + \frac{99}{49}e^{3} - \frac{135}{98}e^{2} - \frac{5333}{392}e + \frac{12479}{784}$
23 $[23, 23, w + 8]$ $-\frac{73}{1568}e^{5} - \frac{27}{392}e^{4} + \frac{87}{98}e^{3} + \frac{183}{196}e^{2} - \frac{2349}{784}e - \frac{3009}{1568}$
23 $[23, 23, -w + 9]$ $\phantom{-}\frac{361}{1568}e^{5} - \frac{13}{392}e^{4} - \frac{599}{98}e^{3} + \frac{561}{196}e^{2} + \frac{29277}{784}e - \frac{56223}{1568}$
25 $[25, 5, 5]$ $\phantom{-}\frac{1}{49}e^{5} - \frac{6}{49}e^{4} - \frac{23}{49}e^{3} + \frac{135}{49}e^{2} + \frac{138}{49}e - \frac{663}{49}$
29 $[29, 29, -w - 4]$ $-\frac{47}{1568}e^{5} + \frac{11}{392}e^{4} + \frac{61}{98}e^{3} - \frac{199}{196}e^{2} - \frac{1227}{784}e + \frac{8873}{1568}$
29 $[29, 29, w - 5]$ $\phantom{-}\frac{111}{1568}e^{5} + \frac{5}{392}e^{4} - \frac{181}{98}e^{3} + \frac{103}{196}e^{2} + \frac{7659}{784}e - \frac{19497}{1568}$
37 $[37, 37, -w - 3]$ $\phantom{-}\frac{33}{784}e^{5} - \frac{25}{196}e^{4} - \frac{75}{49}e^{3} + \frac{367}{98}e^{2} + \frac{5021}{392}e - \frac{19527}{784}$
37 $[37, 37, w - 4]$ $\phantom{-}\frac{143}{784}e^{5} - \frac{15}{196}e^{4} - \frac{234}{49}e^{3} + \frac{251}{98}e^{2} + \frac{11155}{392}e - \frac{17865}{784}$
41 $[41, 41, -w - 9]$ $-\frac{361}{1568}e^{5} + \frac{13}{392}e^{4} + \frac{599}{98}e^{3} - \frac{561}{196}e^{2} - \frac{29277}{784}e + \frac{49951}{1568}$
41 $[41, 41, w - 10]$ $\phantom{-}\frac{73}{1568}e^{5} + \frac{27}{392}e^{4} - \frac{87}{98}e^{3} - \frac{183}{196}e^{2} + \frac{2349}{784}e - \frac{3263}{1568}$
43 $[43, 43, -w - 2]$ $-\frac{15}{98}e^{5} + \frac{3}{49}e^{4} + \frac{190}{49}e^{3} - \frac{155}{49}e^{2} - \frac{1070}{49}e + \frac{3001}{98}$
43 $[43, 43, w - 3]$ $\phantom{-}\frac{13}{98}e^{5} + \frac{3}{49}e^{4} - \frac{167}{49}e^{3} + \frac{20}{49}e^{2} + \frac{932}{49}e - \frac{1675}{98}$
47 $[47, 47, -w - 1]$ $\phantom{-}\frac{3}{1568}e^{5} - \frac{15}{392}e^{4} - \frac{17}{98}e^{3} + \frac{363}{196}e^{2} + \frac{2223}{784}e - \frac{28981}{1568}$
47 $[47, 47, w - 2]$ $-\frac{67}{1568}e^{5} - \frac{1}{392}e^{4} + \frac{137}{98}e^{3} - \frac{267}{196}e^{2} - \frac{8655}{784}e + \frac{20789}{1568}$
53 $[53, 53, 2w - 13]$ $\phantom{-}\frac{57}{784}e^{5} + \frac{23}{196}e^{4} - \frac{106}{49}e^{3} - \frac{159}{98}e^{2} + \frac{5837}{392}e - \frac{4527}{784}$
 Display number of eigenvalues

## Atkin-Lehner eigenvalues

Norm Prime Eigenvalue
9 $[9, 3, 3]$ $-1$