Properties

Base field \(\Q(\sqrt{197}) \)
Weight [2, 2]
Level norm 1
Level $[1, 1, 1]$
Label 2.2.197.1-1.1-a
Dimension 8
CM no
Base change yes

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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 $[1, 1, 1]$
Label 2.2.197.1-1.1-a
Dimension 8
Is CM no
Is base change yes
Parent newspace dimension 8

Hecke eigenvalues ($q$-expansion)

The Hecke eigenvalue field is $\Q(e)$ where $e$ is a root of the defining polynomial:
\(x^{8} \) \(\mathstrut -\mathstrut 8x^{7} \) \(\mathstrut +\mathstrut 4x^{6} \) \(\mathstrut +\mathstrut 103x^{5} \) \(\mathstrut -\mathstrut 186x^{4} \) \(\mathstrut -\mathstrut 346x^{3} \) \(\mathstrut +\mathstrut 867x^{2} \) \(\mathstrut +\mathstrut 101x \) \(\mathstrut -\mathstrut 724\)

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Norm Prime Eigenvalue
4 $[4, 2, 2]$ $\phantom{-}e$
7 $[7, 7, w - 7]$ $-\frac{4}{23}e^{7} + \frac{26}{23}e^{6} + e^{5} - \frac{366}{23}e^{4} + \frac{149}{23}e^{3} + \frac{1527}{23}e^{2} - \frac{798}{23}e - \frac{1555}{23}$
7 $[7, 7, w + 6]$ $-\frac{4}{23}e^{7} + \frac{26}{23}e^{6} + e^{5} - \frac{366}{23}e^{4} + \frac{149}{23}e^{3} + \frac{1527}{23}e^{2} - \frac{798}{23}e - \frac{1555}{23}$
9 $[9, 3, 3]$ $-\frac{1}{4}e^{7} + \frac{5}{4}e^{6} + \frac{11}{4}e^{5} - \frac{37}{2}e^{4} - 4e^{3} + \frac{157}{2}e^{2} - \frac{81}{4}e - 73$
19 $[19, 19, w + 5]$ $\phantom{-}\frac{1}{23}e^{7} - \frac{18}{23}e^{6} + e^{5} + \frac{287}{23}e^{4} - \frac{388}{23}e^{3} - \frac{1434}{23}e^{2} + \frac{1154}{23}e + \frac{1809}{23}$
19 $[19, 19, w - 6]$ $\phantom{-}\frac{1}{23}e^{7} - \frac{18}{23}e^{6} + e^{5} + \frac{287}{23}e^{4} - \frac{388}{23}e^{3} - \frac{1434}{23}e^{2} + \frac{1154}{23}e + \frac{1809}{23}$
23 $[23, 23, w + 8]$ $\phantom{-}\frac{9}{23}e^{7} - \frac{47}{23}e^{6} - 4e^{5} + \frac{674}{23}e^{4} + \frac{96}{23}e^{3} - \frac{2763}{23}e^{2} + \frac{772}{23}e + \frac{2619}{23}$
23 $[23, 23, -w + 9]$ $\phantom{-}\frac{9}{23}e^{7} - \frac{47}{23}e^{6} - 4e^{5} + \frac{674}{23}e^{4} + \frac{96}{23}e^{3} - \frac{2763}{23}e^{2} + \frac{772}{23}e + \frac{2619}{23}$
25 $[25, 5, 5]$ $\phantom{-}\frac{16}{23}e^{7} - \frac{81}{23}e^{6} - 7e^{5} + \frac{1142}{23}e^{4} + \frac{163}{23}e^{3} - \frac{4682}{23}e^{2} + \frac{1329}{23}e + \frac{4702}{23}$
29 $[29, 29, -w - 4]$ $\phantom{-}\frac{9}{92}e^{7} - \frac{1}{92}e^{6} - \frac{9}{4}e^{5} - \frac{77}{46}e^{4} + \frac{392}{23}e^{3} + \frac{999}{46}e^{2} - \frac{3391}{92}e - \frac{1076}{23}$
29 $[29, 29, w - 5]$ $\phantom{-}\frac{9}{92}e^{7} - \frac{1}{92}e^{6} - \frac{9}{4}e^{5} - \frac{77}{46}e^{4} + \frac{392}{23}e^{3} + \frac{999}{46}e^{2} - \frac{3391}{92}e - \frac{1076}{23}$
37 $[37, 37, -w - 3]$ $\phantom{-}\frac{31}{92}e^{7} - \frac{167}{92}e^{6} - \frac{11}{4}e^{5} + \frac{1125}{46}e^{4} - \frac{63}{23}e^{3} - \frac{4471}{46}e^{2} + \frac{3827}{92}e + \frac{2238}{23}$
37 $[37, 37, w - 4]$ $\phantom{-}\frac{31}{92}e^{7} - \frac{167}{92}e^{6} - \frac{11}{4}e^{5} + \frac{1125}{46}e^{4} - \frac{63}{23}e^{3} - \frac{4471}{46}e^{2} + \frac{3827}{92}e + \frac{2238}{23}$
41 $[41, 41, -w - 9]$ $\phantom{-}\frac{49}{92}e^{7} - \frac{353}{92}e^{6} - \frac{9}{4}e^{5} + \frac{2627}{46}e^{4} - \frac{751}{23}e^{3} - \frac{11857}{46}e^{2} + \frac{13329}{92}e + \frac{6664}{23}$
41 $[41, 41, w - 10]$ $\phantom{-}\frac{49}{92}e^{7} - \frac{353}{92}e^{6} - \frac{9}{4}e^{5} + \frac{2627}{46}e^{4} - \frac{751}{23}e^{3} - \frac{11857}{46}e^{2} + \frac{13329}{92}e + \frac{6664}{23}$
43 $[43, 43, -w - 2]$ $\phantom{-}\frac{8}{23}e^{7} - \frac{52}{23}e^{6} - 2e^{5} + \frac{755}{23}e^{4} - \frac{321}{23}e^{3} - \frac{3307}{23}e^{2} + \frac{1642}{23}e + \frac{3593}{23}$
43 $[43, 43, w - 3]$ $\phantom{-}\frac{8}{23}e^{7} - \frac{52}{23}e^{6} - 2e^{5} + \frac{755}{23}e^{4} - \frac{321}{23}e^{3} - \frac{3307}{23}e^{2} + \frac{1642}{23}e + \frac{3593}{23}$
47 $[47, 47, -w - 1]$ $-\frac{24}{23}e^{7} + \frac{133}{23}e^{6} + 9e^{5} - \frac{1874}{23}e^{4} + \frac{112}{23}e^{3} + \frac{7782}{23}e^{2} - \frac{2741}{23}e - \frac{7927}{23}$
47 $[47, 47, w - 2]$ $-\frac{24}{23}e^{7} + \frac{133}{23}e^{6} + 9e^{5} - \frac{1874}{23}e^{4} + \frac{112}{23}e^{3} + \frac{7782}{23}e^{2} - \frac{2741}{23}e - \frac{7927}{23}$
53 $[53, 53, 2w - 13]$ $\phantom{-}\frac{113}{92}e^{7} - \frac{585}{92}e^{6} - \frac{49}{4}e^{5} + \frac{4175}{46}e^{4} + \frac{217}{23}e^{3} - \frac{17219}{46}e^{2} + \frac{10825}{92}e + \frac{8261}{23}$
Display number of eigenvalues

Atkin-Lehner eigenvalues

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