Properties

Base field \(\Q(\sqrt{157}) \)
Weight [2, 2]
Level norm 9
Level $[9, 3, 3]$
Label 2.2.157.1-9.1-f
Dimension 7
CM no
Base change yes

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Base field \(\Q(\sqrt{157}) \)

Generator \(w\), with minimal polynomial \(x^{2} - x - 39\); narrow class number \(1\) and class number \(1\).

Form

Weight [2, 2]
Level $[9, 3, 3]$
Label 2.2.157.1-9.1-f
Dimension 7
Is CM no
Is base change yes
Parent newspace dimension 19

Hecke eigenvalues ($q$-expansion)

The Hecke eigenvalue field is $\Q(e)$ where $e$ is a root of the defining polynomial:
\(x^{7} \) \(\mathstrut -\mathstrut 4x^{6} \) \(\mathstrut -\mathstrut 16x^{5} \) \(\mathstrut +\mathstrut 67x^{4} \) \(\mathstrut +\mathstrut 47x^{3} \) \(\mathstrut -\mathstrut 248x^{2} \) \(\mathstrut +\mathstrut 54x \) \(\mathstrut +\mathstrut 123\)

  Show full eigenvalues   Hide large eigenvalues

Norm Prime Eigenvalue
3 $[3, 3, w + 6]$ $\phantom{-}1$
3 $[3, 3, -w + 7]$ $\phantom{-}1$
4 $[4, 2, 2]$ $\phantom{-}e$
11 $[11, 11, -3w - 17]$ $\phantom{-}\frac{2}{19}e^{6} + \frac{4}{19}e^{5} - \frac{27}{19}e^{4} - \frac{66}{19}e^{3} + \frac{2}{19}e^{2} + \frac{124}{19}e + \frac{111}{19}$
11 $[11, 11, 3w - 20]$ $\phantom{-}\frac{2}{19}e^{6} + \frac{4}{19}e^{5} - \frac{27}{19}e^{4} - \frac{66}{19}e^{3} + \frac{2}{19}e^{2} + \frac{124}{19}e + \frac{111}{19}$
13 $[13, 13, 2w - 13]$ $-\frac{21}{38}e^{6} + \frac{15}{38}e^{5} + \frac{369}{38}e^{4} - \frac{100}{19}e^{3} - \frac{1427}{38}e^{2} + \frac{693}{38}e + \frac{953}{38}$
13 $[13, 13, 2w + 11]$ $-\frac{21}{38}e^{6} + \frac{15}{38}e^{5} + \frac{369}{38}e^{4} - \frac{100}{19}e^{3} - \frac{1427}{38}e^{2} + \frac{693}{38}e + \frac{953}{38}$
17 $[17, 17, w + 7]$ $\phantom{-}\frac{1}{38}e^{6} - \frac{17}{38}e^{5} - \frac{23}{38}e^{4} + \frac{126}{19}e^{3} + \frac{191}{38}e^{2} - \frac{527}{38}e - \frac{87}{38}$
17 $[17, 17, -w + 8]$ $\phantom{-}\frac{1}{38}e^{6} - \frac{17}{38}e^{5} - \frac{23}{38}e^{4} + \frac{126}{19}e^{3} + \frac{191}{38}e^{2} - \frac{527}{38}e - \frac{87}{38}$
19 $[19, 19, -w - 4]$ $\phantom{-}\frac{5}{19}e^{6} - \frac{9}{19}e^{5} - \frac{96}{19}e^{4} + \frac{139}{19}e^{3} + \frac{461}{19}e^{2} - \frac{412}{19}e - \frac{378}{19}$
19 $[19, 19, -w + 5]$ $\phantom{-}\frac{5}{19}e^{6} - \frac{9}{19}e^{5} - \frac{96}{19}e^{4} + \frac{139}{19}e^{3} + \frac{461}{19}e^{2} - \frac{412}{19}e - \frac{378}{19}$
25 $[25, 5, 5]$ $\phantom{-}\frac{3}{19}e^{6} + \frac{6}{19}e^{5} - \frac{50}{19}e^{4} - \frac{99}{19}e^{3} + \frac{155}{19}e^{2} + \frac{205}{19}e - \frac{90}{19}$
31 $[31, 31, -6w - 35]$ $\phantom{-}\frac{7}{19}e^{6} - \frac{5}{19}e^{5} - \frac{123}{19}e^{4} + \frac{73}{19}e^{3} + \frac{463}{19}e^{2} - \frac{288}{19}e - \frac{267}{19}$
31 $[31, 31, -6w + 41]$ $\phantom{-}\frac{7}{19}e^{6} - \frac{5}{19}e^{5} - \frac{123}{19}e^{4} + \frac{73}{19}e^{3} + \frac{463}{19}e^{2} - \frac{288}{19}e - \frac{267}{19}$
37 $[37, 37, -w - 1]$ $\phantom{-}\frac{7}{38}e^{6} - \frac{5}{38}e^{5} - \frac{123}{38}e^{4} + \frac{27}{19}e^{3} + \frac{501}{38}e^{2} - \frac{117}{38}e - \frac{495}{38}$
37 $[37, 37, w - 2]$ $\phantom{-}\frac{7}{38}e^{6} - \frac{5}{38}e^{5} - \frac{123}{38}e^{4} + \frac{27}{19}e^{3} + \frac{501}{38}e^{2} - \frac{117}{38}e - \frac{495}{38}$
47 $[47, 47, 3w + 16]$ $\phantom{-}\frac{10}{19}e^{6} + \frac{1}{19}e^{5} - \frac{173}{19}e^{4} - \frac{26}{19}e^{3} + \frac{618}{19}e^{2} - \frac{121}{19}e - \frac{357}{19}$
47 $[47, 47, -3w + 19]$ $\phantom{-}\frac{10}{19}e^{6} + \frac{1}{19}e^{5} - \frac{173}{19}e^{4} - \frac{26}{19}e^{3} + \frac{618}{19}e^{2} - \frac{121}{19}e - \frac{357}{19}$
49 $[49, 7, -7]$ $-\frac{37}{38}e^{6} + \frac{21}{38}e^{5} + \frac{623}{38}e^{4} - \frac{140}{19}e^{3} - \frac{2127}{38}e^{2} + \frac{1335}{38}e + \frac{1243}{38}$
67 $[67, 67, 3w - 22]$ $-\frac{2}{19}e^{6} - \frac{4}{19}e^{5} + \frac{27}{19}e^{4} + \frac{66}{19}e^{3} - \frac{2}{19}e^{2} - \frac{162}{19}e - \frac{35}{19}$
Display number of eigenvalues

Atkin-Lehner eigenvalues

Norm Prime Eigenvalue
3 $[3, 3, w + 6]$ $-1$
3 $[3, 3, -w + 7]$ $-1$