Properties

Base field \(\Q(\sqrt{157}) \)
Weight [2, 2]
Level norm 11
Level $[11,11,3w - 20]$
Label 2.2.157.1-11.2-a
Dimension 13
CM no
Base change no

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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 $[11,11,3w - 20]$
Label 2.2.157.1-11.2-a
Dimension 13
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:
\(x^{13} \) \(\mathstrut -\mathstrut 16x^{11} \) \(\mathstrut -\mathstrut 6x^{10} \) \(\mathstrut +\mathstrut 90x^{9} \) \(\mathstrut +\mathstrut 57x^{8} \) \(\mathstrut -\mathstrut 209x^{7} \) \(\mathstrut -\mathstrut 159x^{6} \) \(\mathstrut +\mathstrut 201x^{5} \) \(\mathstrut +\mathstrut 166x^{4} \) \(\mathstrut -\mathstrut 58x^{3} \) \(\mathstrut -\mathstrut 54x^{2} \) \(\mathstrut -\mathstrut 5x \) \(\mathstrut +\mathstrut 1\)

  Show full eigenvalues   Hide large eigenvalues

Norm Prime Eigenvalue
3 $[3, 3, w + 6]$ $\phantom{-}e$
3 $[3, 3, -w + 7]$ $-\frac{47}{53}e^{12} + \frac{26}{53}e^{11} + \frac{741}{53}e^{10} - \frac{111}{53}e^{9} - \frac{4234}{53}e^{8} - \frac{586}{53}e^{7} + \frac{10499}{53}e^{6} + \frac{2866}{53}e^{5} - \frac{11585}{53}e^{4} - \frac{3396}{53}e^{3} + \frac{4811}{53}e^{2} + \frac{949}{53}e - \frac{246}{53}$
4 $[4, 2, 2]$ $\phantom{-}\frac{28}{53}e^{12} - \frac{20}{53}e^{11} - \frac{411}{53}e^{10} + \frac{118}{53}e^{9} + \frac{2095}{53}e^{8} + \frac{92}{53}e^{7} - \frac{4199}{53}e^{6} - \frac{1006}{53}e^{5} + \frac{3106}{53}e^{4} + \frac{1430}{53}e^{3} - \frac{480}{53}e^{2} - \frac{677}{53}e - \frac{35}{53}$
11 $[11, 11, -3w - 17]$ $\phantom{-}\frac{93}{53}e^{12} - \frac{21}{53}e^{11} - \frac{1473}{53}e^{10} - \frac{210}{53}e^{9} + \frac{8255}{53}e^{8} + \frac{3160}{53}e^{7} - \frac{19347}{53}e^{6} - \frac{8895}{53}e^{5} + \frac{19421}{53}e^{4} + \frac{8312}{53}e^{3} - \frac{6758}{53}e^{2} - \frac{2065}{53}e + \frac{56}{53}$
11 $[11, 11, 3w - 20]$ $\phantom{-}1$
13 $[13, 13, 2w - 13]$ $-\frac{34}{53}e^{12} - \frac{6}{53}e^{11} + \frac{571}{53}e^{10} + \frac{258}{53}e^{9} - \frac{3373}{53}e^{8} - \frac{2103}{53}e^{7} + \frac{8381}{53}e^{6} + \frac{5613}{53}e^{5} - \frac{8958}{53}e^{4} - \frac{5560}{53}e^{3} + \frac{3460}{53}e^{2} + \frac{1530}{53}e - \frac{249}{53}$
13 $[13, 13, 2w + 11]$ $\phantom{-}\frac{34}{53}e^{12} + \frac{6}{53}e^{11} - \frac{571}{53}e^{10} - \frac{258}{53}e^{9} + \frac{3373}{53}e^{8} + \frac{2103}{53}e^{7} - \frac{8381}{53}e^{6} - \frac{5613}{53}e^{5} + \frac{8958}{53}e^{4} + \frac{5507}{53}e^{3} - \frac{3354}{53}e^{2} - \frac{1265}{53}e + \frac{37}{53}$
17 $[17, 17, w + 7]$ $-\frac{91}{53}e^{12} + \frac{12}{53}e^{11} + \frac{1455}{53}e^{10} + \frac{332}{53}e^{9} - \frac{8253}{53}e^{8} - \frac{3744}{53}e^{7} + \frac{19755}{53}e^{6} + \frac{10080}{53}e^{5} - \frac{20827}{53}e^{4} - \frac{9285}{53}e^{3} + \frac{8344}{53}e^{2} + \frac{2134}{53}e - \frac{562}{53}$
17 $[17, 17, -w + 8]$ $\phantom{-}\frac{3}{53}e^{12} + \frac{66}{53}e^{11} - \frac{133}{53}e^{10} - \frac{930}{53}e^{9} + \frac{1063}{53}e^{8} + \frac{4583}{53}e^{7} - \frac{2992}{53}e^{6} - \frac{8849}{53}e^{5} + \frac{3880}{53}e^{4} + \frac{6093}{53}e^{3} - \frac{2285}{53}e^{2} - \frac{665}{53}e + \frac{301}{53}$
19 $[19, 19, -w - 4]$ $-\frac{168}{53}e^{12} + \frac{67}{53}e^{11} + \frac{2625}{53}e^{10} - \frac{19}{53}e^{9} - \frac{14584}{53}e^{8} - \frac{3838}{53}e^{7} + \frac{34045}{53}e^{6} + \frac{12767}{53}e^{5} - \frac{34059}{53}e^{4} - \frac{13138}{53}e^{3} + \frac{11678}{53}e^{2} + \frac{3532}{53}e - \frac{55}{53}$
19 $[19, 19, -w + 5]$ $...$
25 $[25, 5, 5]$ $...$
31 $[31, 31, -6w - 35]$ $-\frac{84}{53}e^{12} + \frac{60}{53}e^{11} + \frac{1286}{53}e^{10} - \frac{354}{53}e^{9} - \frac{7080}{53}e^{8} - \frac{541}{53}e^{7} + \frac{16519}{53}e^{6} + \frac{5085}{53}e^{5} - \frac{16102}{53}e^{4} - \frac{7682}{53}e^{3} + \frac{4885}{53}e^{2} + \frac{2985}{53}e - \frac{1}{53}$
31 $[31, 31, -6w + 41]$ $\phantom{-}\frac{5}{53}e^{12} + \frac{57}{53}e^{11} - \frac{151}{53}e^{10} - \frac{861}{53}e^{9} + \frac{1171}{53}e^{8} + \frac{4635}{53}e^{7} - \frac{3538}{53}e^{6} - \frac{10473}{53}e^{5} + \frac{5071}{53}e^{4} + \frac{10155}{53}e^{3} - \frac{2819}{53}e^{2} - \frac{3246}{53}e - \frac{99}{53}$
37 $[37, 37, -w - 1]$ $-\frac{119}{53}e^{12} + \frac{85}{53}e^{11} + \frac{1760}{53}e^{10} - \frac{528}{53}e^{9} - \frac{9129}{53}e^{8} + \frac{33}{53}e^{7} + \frac{19078}{53}e^{6} + \frac{2235}{53}e^{5} - \frac{15824}{53}e^{4} - \frac{2606}{53}e^{3} + \frac{3842}{53}e^{2} + \frac{1009}{53}e + \frac{3}{53}$
37 $[37, 37, w - 2]$ $...$
47 $[47, 47, 3w + 16]$ $-\frac{142}{53}e^{12} + \frac{56}{53}e^{11} + \frac{2232}{53}e^{10} + \frac{30}{53}e^{9} - \frac{12597}{53}e^{8} - \frac{3745}{53}e^{7} + \frac{30339}{53}e^{6} + \frac{12802}{53}e^{5} - \frac{32038}{53}e^{4} - \frac{13756}{53}e^{3} + \frac{12792}{53}e^{2} + \frac{3846}{53}e - \frac{697}{53}$
47 $[47, 47, -3w + 19]$ $-\frac{71}{53}e^{12} + \frac{28}{53}e^{11} + \frac{1063}{53}e^{10} + \frac{15}{53}e^{9} - \frac{5477}{53}e^{8} - \frac{1687}{53}e^{7} + \frac{11009}{53}e^{6} + \frac{4970}{53}e^{5} - \frac{8334}{53}e^{4} - \frac{4970}{53}e^{3} + \frac{1202}{53}e^{2} + \frac{1923}{53}e + \frac{367}{53}$
49 $[49, 7, -7]$ $-\frac{325}{53}e^{12} + \frac{111}{53}e^{11} + \frac{5151}{53}e^{10} + \frac{156}{53}e^{9} - \frac{29051}{53}e^{8} - \frac{8185}{53}e^{7} + \frac{69115}{53}e^{6} + \frac{26354}{53}e^{5} - \frac{70657}{53}e^{4} - \frac{27520}{53}e^{3} + \frac{25136}{53}e^{2} + \frac{8106}{53}e - \frac{561}{53}$
67 $[67, 67, 3w - 22]$ $-\frac{16}{53}e^{12} - \frac{34}{53}e^{11} + \frac{250}{53}e^{10} + \frac{614}{53}e^{9} - \frac{1129}{53}e^{8} - \frac{3596}{53}e^{7} + \frac{923}{53}e^{6} + \frac{7904}{53}e^{5} + \frac{2397}{53}e^{4} - \frac{7003}{53}e^{3} - \frac{3254}{53}e^{2} + \frac{2416}{53}e + \frac{550}{53}$
Display number of eigenvalues

Atkin-Lehner eigenvalues

Norm Prime Eigenvalue
11 $[11,11,3w - 20]$ $-1$