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

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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:
\(x^{6} \) \(\mathstrut +\mathstrut 3x^{5} \) \(\mathstrut -\mathstrut 28x^{4} \) \(\mathstrut -\mathstrut 72x^{3} \) \(\mathstrut +\mathstrut 226x^{2} \) \(\mathstrut +\mathstrut 383x \) \(\mathstrut -\mathstrut 577\)

  Show full eigenvalues   Hide large eigenvalues

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 $[9, 3, 3]$ $-1$