Properties

Base field \(\Q(\sqrt{393}) \)
Weight [2, 2]
Level norm 4
Level $[4, 2, 2]$
Label 2.2.393.1-4.1-f
Dimension 11
CM no
Base change yes

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Base field \(\Q(\sqrt{393}) \)

Generator \(w\), with minimal polynomial \(x^{2} - x - 98\); narrow class number \(2\) and class number \(1\).

Form

Weight [2, 2]
Level $[4, 2, 2]$
Label 2.2.393.1-4.1-f
Dimension 11
Is CM no
Is base change yes
Parent newspace dimension 36

Hecke eigenvalues ($q$-expansion)

The Hecke eigenvalue field is $\Q(e)$ where $e$ is a root of the defining polynomial:
\(x^{11} \) \(\mathstrut -\mathstrut x^{10} \) \(\mathstrut -\mathstrut 33x^{9} \) \(\mathstrut +\mathstrut 34x^{8} \) \(\mathstrut +\mathstrut 386x^{7} \) \(\mathstrut -\mathstrut 400x^{6} \) \(\mathstrut -\mathstrut 1866x^{5} \) \(\mathstrut +\mathstrut 1848x^{4} \) \(\mathstrut +\mathstrut 3272x^{3} \) \(\mathstrut -\mathstrut 2624x^{2} \) \(\mathstrut -\mathstrut 1920x \) \(\mathstrut +\mathstrut 1024\)

  Show full eigenvalues   Hide large eigenvalues

Norm Prime Eigenvalue
2 $[2, 2, -17w - 160]$ $-1$
2 $[2, 2, -17w + 177]$ $-1$
3 $[3, 3, -842w + 8767]$ $\phantom{-}e$
7 $[7, 7, -2w + 21]$ $\phantom{-}\frac{109}{2432}e^{10} - \frac{253}{2432}e^{9} - \frac{3405}{2432}e^{8} + \frac{4045}{1216}e^{7} + \frac{17605}{1216}e^{6} - \frac{5505}{152}e^{5} - \frac{60721}{1216}e^{4} + \frac{44383}{304}e^{3} + \frac{153}{304}e^{2} - \frac{4853}{38}e + \frac{868}{19}$
7 $[7, 7, 2w + 19]$ $\phantom{-}\frac{109}{2432}e^{10} - \frac{253}{2432}e^{9} - \frac{3405}{2432}e^{8} + \frac{4045}{1216}e^{7} + \frac{17605}{1216}e^{6} - \frac{5505}{152}e^{5} - \frac{60721}{1216}e^{4} + \frac{44383}{304}e^{3} + \frac{153}{304}e^{2} - \frac{4853}{38}e + \frac{868}{19}$
13 $[13, 13, -12w - 113]$ $\phantom{-}\frac{135}{2432}e^{10} - \frac{199}{2432}e^{9} - \frac{4167}{2432}e^{8} + \frac{3303}{1216}e^{7} + \frac{21591}{1216}e^{6} - \frac{4619}{152}e^{5} - \frac{78627}{1216}e^{4} + \frac{37761}{304}e^{3} + \frac{8295}{304}e^{2} - \frac{4021}{38}e + \frac{637}{19}$
13 $[13, 13, 12w - 125]$ $\phantom{-}\frac{135}{2432}e^{10} - \frac{199}{2432}e^{9} - \frac{4167}{2432}e^{8} + \frac{3303}{1216}e^{7} + \frac{21591}{1216}e^{6} - \frac{4619}{152}e^{5} - \frac{78627}{1216}e^{4} + \frac{37761}{304}e^{3} + \frac{8295}{304}e^{2} - \frac{4021}{38}e + \frac{637}{19}$
17 $[17, 17, 182w - 1895]$ $-\frac{25}{1216}e^{10} + \frac{65}{1216}e^{9} + \frac{721}{1216}e^{8} - \frac{1093}{608}e^{7} - \frac{3289}{608}e^{6} + \frac{1545}{76}e^{5} + \frac{7805}{608}e^{4} - \frac{12773}{152}e^{3} + \frac{4375}{152}e^{2} + \frac{2827}{38}e - \frac{725}{19}$
17 $[17, 17, 182w + 1713]$ $-\frac{25}{1216}e^{10} + \frac{65}{1216}e^{9} + \frac{721}{1216}e^{8} - \frac{1093}{608}e^{7} - \frac{3289}{608}e^{6} + \frac{1545}{76}e^{5} + \frac{7805}{608}e^{4} - \frac{12773}{152}e^{3} + \frac{4375}{152}e^{2} + \frac{2827}{38}e - \frac{725}{19}$
23 $[23, 23, -512w - 4819]$ $-\frac{203}{2432}e^{10} + \frac{467}{2432}e^{9} + \frac{6195}{2432}e^{8} - \frac{7495}{1216}e^{7} - \frac{30987}{1216}e^{6} + \frac{5127}{76}e^{5} + \frac{99431}{1216}e^{4} - \frac{83367}{304}e^{3} + \frac{7633}{304}e^{2} + \frac{4656}{19}e - \frac{1832}{19}$
23 $[23, 23, 512w - 5331]$ $-\frac{203}{2432}e^{10} + \frac{467}{2432}e^{9} + \frac{6195}{2432}e^{8} - \frac{7495}{1216}e^{7} - \frac{30987}{1216}e^{6} + \frac{5127}{76}e^{5} + \frac{99431}{1216}e^{4} - \frac{83367}{304}e^{3} + \frac{7633}{304}e^{2} + \frac{4656}{19}e - \frac{1832}{19}$
25 $[25, 5, -5]$ $\phantom{-}\frac{69}{2432}e^{10} - \frac{149}{2432}e^{9} - \frac{2069}{2432}e^{8} + \frac{2509}{1216}e^{7} + \frac{10245}{1216}e^{6} - \frac{3565}{152}e^{5} - \frac{33337}{1216}e^{4} + \frac{29935}{304}e^{3} - \frac{979}{304}e^{2} - \frac{1784}{19}e + \frac{554}{19}$
29 $[29, 29, 22w - 229]$ $\phantom{-}\frac{277}{2432}e^{10} - \frac{629}{2432}e^{9} - \frac{8469}{2432}e^{8} + \frac{10101}{1216}e^{7} + \frac{42741}{1216}e^{6} - \frac{13805}{152}e^{5} - \frac{141929}{1216}e^{4} + \frac{112011}{304}e^{3} - \frac{3939}{304}e^{2} - \frac{12549}{38}e + \frac{2183}{19}$
29 $[29, 29, 22w + 207]$ $\phantom{-}\frac{277}{2432}e^{10} - \frac{629}{2432}e^{9} - \frac{8469}{2432}e^{8} + \frac{10101}{1216}e^{7} + \frac{42741}{1216}e^{6} - \frac{13805}{152}e^{5} - \frac{141929}{1216}e^{4} + \frac{112011}{304}e^{3} - \frac{3939}{304}e^{2} - \frac{12549}{38}e + \frac{2183}{19}$
43 $[43, 43, 114w + 1073]$ $-\frac{117}{1216}e^{10} + \frac{213}{1216}e^{9} + \frac{3581}{1216}e^{8} - \frac{3425}{608}e^{7} - \frac{18241}{608}e^{6} + \frac{4677}{76}e^{5} + \frac{63249}{608}e^{4} - \frac{37727}{152}e^{3} - \frac{2591}{152}e^{2} + \frac{8149}{38}e - \frac{1512}{19}$
43 $[43, 43, 114w - 1187]$ $-\frac{117}{1216}e^{10} + \frac{213}{1216}e^{9} + \frac{3581}{1216}e^{8} - \frac{3425}{608}e^{7} - \frac{18241}{608}e^{6} + \frac{4677}{76}e^{5} + \frac{63249}{608}e^{4} - \frac{37727}{152}e^{3} - \frac{2591}{152}e^{2} + \frac{8149}{38}e - \frac{1512}{19}$
47 $[47, 47, 8w - 83]$ $-\frac{213}{2432}e^{10} + \frac{493}{2432}e^{9} + \frac{6605}{2432}e^{8} - \frac{7689}{1216}e^{7} - \frac{34005}{1216}e^{6} + \frac{5113}{76}e^{5} + \frac{118969}{1216}e^{4} - \frac{80785}{304}e^{3} - \frac{5361}{304}e^{2} + \frac{4344}{19}e - \frac{1312}{19}$
47 $[47, 47, -8w - 75]$ $-\frac{213}{2432}e^{10} + \frac{493}{2432}e^{9} + \frac{6605}{2432}e^{8} - \frac{7689}{1216}e^{7} - \frac{34005}{1216}e^{6} + \frac{5113}{76}e^{5} + \frac{118969}{1216}e^{4} - \frac{80785}{304}e^{3} - \frac{5361}{304}e^{2} + \frac{4344}{19}e - \frac{1312}{19}$
61 $[61, 61, -1172w - 11031]$ $\phantom{-}\frac{35}{2432}e^{10} - \frac{91}{2432}e^{9} - \frac{1131}{2432}e^{8} + \frac{1439}{1216}e^{7} + \frac{5851}{1216}e^{6} - \frac{977}{76}e^{5} - \frac{18223}{1216}e^{4} + \frac{15739}{304}e^{3} - \frac{3541}{304}e^{2} - \frac{808}{19}e + \frac{517}{19}$
61 $[61, 61, 1172w - 12203]$ $\phantom{-}\frac{35}{2432}e^{10} - \frac{91}{2432}e^{9} - \frac{1131}{2432}e^{8} + \frac{1439}{1216}e^{7} + \frac{5851}{1216}e^{6} - \frac{977}{76}e^{5} - \frac{18223}{1216}e^{4} + \frac{15739}{304}e^{3} - \frac{3541}{304}e^{2} - \frac{808}{19}e + \frac{517}{19}$
Display number of eigenvalues

Atkin-Lehner eigenvalues

Norm Prime Eigenvalue
2 $[2, 2, -17w - 160]$ $1$
2 $[2, 2, -17w + 177]$ $1$