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: | $[4, 4, 165w + 1553]$ |
| Dimension: | $20$ |
| CM: | no |
| Base change: | no |
| Newspace dimension: | $20$ |
Hecke eigenvalues ($q$-expansion)
The Hecke eigenvalue field is $\Q(e)$ where $e$ is a root of the defining polynomial:
| \(x^{20} - 31x^{18} + 403x^{16} - 2855x^{14} + 12007x^{12} - 30720x^{10} + 47425x^{8} - 42732x^{6} + 20826x^{4} - 4644x^{2} + 324\) |
Show full eigenvalues Hide large eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| 2 | $[2, 2, -17w - 160]$ | $\phantom{-}e$ |
| 2 | $[2, 2, -17w + 177]$ | $\phantom{-}0$ |
| 3 | $[3, 3, -842w + 8767]$ | $-\frac{17}{108}e^{18} + \frac{253}{54}e^{16} - \frac{6227}{108}e^{14} + \frac{10216}{27}e^{12} - \frac{153785}{108}e^{10} + \frac{110857}{36}e^{8} - \frac{98648}{27}e^{6} + \frac{39055}{18}e^{4} - \frac{1588}{3}e^{2} + 36$ |
| 7 | $[7, 7, -2w + 21]$ | $\phantom{-}\frac{35}{108}e^{18} - \frac{523}{54}e^{16} + \frac{12941}{108}e^{14} - \frac{21385}{27}e^{12} + \frac{325235}{108}e^{10} - \frac{238027}{36}e^{8} + \frac{216692}{27}e^{6} - \frac{88621}{18}e^{4} + \frac{3773}{3}e^{2} - 90$ |
| 7 | $[7, 7, 2w + 19]$ | $-\frac{31}{108}e^{18} + \frac{232}{27}e^{16} - \frac{11497}{108}e^{14} + \frac{38035}{54}e^{12} - \frac{289387}{108}e^{10} + \frac{211889}{36}e^{8} - \frac{386657}{54}e^{6} + \frac{79913}{18}e^{4} - \frac{3523}{3}e^{2} + 90$ |
| 13 | $[13, 13, -12w - 113]$ | $\phantom{-}\frac{31}{108}e^{18} - \frac{232}{27}e^{16} + \frac{11497}{108}e^{14} - \frac{38035}{54}e^{12} + \frac{289387}{108}e^{10} - \frac{211889}{36}e^{8} + \frac{386657}{54}e^{6} - \frac{79931}{18}e^{4} + \frac{3541}{3}e^{2} - 94$ |
| 13 | $[13, 13, 12w - 125]$ | $-\frac{35}{108}e^{18} + \frac{523}{54}e^{16} - \frac{12941}{108}e^{14} + \frac{21385}{27}e^{12} - \frac{325235}{108}e^{10} + \frac{238027}{36}e^{8} - \frac{216692}{27}e^{6} + \frac{88639}{18}e^{4} - \frac{3797}{3}e^{2} + 98$ |
| 17 | $[17, 17, 182w - 1895]$ | $\phantom{-}\frac{10}{27}e^{19} - \frac{298}{27}e^{17} + \frac{7343}{54}e^{15} - \frac{24119}{27}e^{13} + \frac{181673}{54}e^{11} - \frac{21824}{3}e^{9} + \frac{464669}{54}e^{7} - \frac{90757}{18}e^{5} + \frac{3505}{3}e^{3} - 64e$ |
| 17 | $[17, 17, 182w + 1713]$ | $-\frac{1}{6}e^{19} + \frac{89}{18}e^{17} - \frac{1091}{18}e^{15} + \frac{7127}{18}e^{13} - \frac{26683}{18}e^{11} + \frac{28699}{9}e^{9} - \frac{22621}{6}e^{7} + \frac{20239}{9}e^{5} - \frac{1675}{3}e^{3} + 33e$ |
| 23 | $[23, 23, -512w - 4819]$ | $\phantom{-}\frac{1}{54}e^{19} - \frac{31}{54}e^{17} + \frac{403}{54}e^{15} - \frac{2855}{54}e^{13} + \frac{12007}{54}e^{11} - \frac{5117}{9}e^{9} + \frac{47119}{54}e^{7} - \frac{2275}{3}e^{5} + \frac{931}{3}e^{3} - 31e$ |
| 23 | $[23, 23, 512w - 5331]$ | $\phantom{-}\frac{5}{54}e^{19} - \frac{73}{27}e^{17} + \frac{877}{27}e^{15} - \frac{5576}{27}e^{13} + \frac{20077}{27}e^{11} - \frac{27077}{18}e^{9} + \frac{86393}{54}e^{7} - \frac{4727}{6}e^{5} + \frac{415}{3}e^{3} - 10e$ |
| 25 | $[25, 5, -5]$ | $-\frac{35}{108}e^{18} + \frac{523}{54}e^{16} - \frac{12941}{108}e^{14} + \frac{21385}{27}e^{12} - \frac{325235}{108}e^{10} + \frac{238027}{36}e^{8} - \frac{216692}{27}e^{6} + \frac{88639}{18}e^{4} - \frac{3794}{3}e^{2} + 95$ |
| 29 | $[29, 29, 22w - 229]$ | $\phantom{-}\frac{31}{54}e^{19} - \frac{464}{27}e^{17} + \frac{5753}{27}e^{15} - \frac{38143}{27}e^{13} + \frac{145706}{27}e^{11} - \frac{215003}{18}e^{9} + \frac{795301}{54}e^{7} - \frac{168217}{18}e^{5} + \frac{7775}{3}e^{3} - 227e$ |
| 29 | $[29, 29, 22w + 207]$ | $-\frac{16}{27}e^{19} + \frac{478}{27}e^{17} - \frac{11825}{54}e^{15} + \frac{39083}{27}e^{13} - \frac{297329}{54}e^{11} + \frac{108916}{9}e^{9} - \frac{794603}{54}e^{7} + \frac{54227}{6}e^{5} - \frac{6874}{3}e^{3} + 151e$ |
| 43 | $[43, 43, 114w + 1073]$ | $-\frac{31}{108}e^{18} + \frac{232}{27}e^{16} - \frac{11497}{108}e^{14} + \frac{38035}{54}e^{12} - \frac{289351}{108}e^{10} + \frac{211685}{36}e^{8} - \frac{384875}{54}e^{6} + \frac{78575}{18}e^{4} - \frac{3373}{3}e^{2} + 84$ |
| 43 | $[43, 43, 114w - 1187]$ | $\phantom{-}\frac{35}{108}e^{18} - \frac{523}{54}e^{16} + \frac{12941}{108}e^{14} - \frac{21385}{27}e^{12} + \frac{325199}{108}e^{10} - \frac{237823}{36}e^{8} + \frac{215801}{27}e^{6} - \frac{87283}{18}e^{4} + \frac{3629}{3}e^{2} - 90$ |
| 47 | $[47, 47, 8w - 83]$ | $-\frac{10}{27}e^{19} + \frac{593}{54}e^{17} - \frac{7259}{54}e^{15} + \frac{47281}{54}e^{13} - \frac{175997}{54}e^{11} + \frac{124757}{18}e^{9} - \frac{216112}{27}e^{7} + \frac{27295}{6}e^{5} - \frac{3118}{3}e^{3} + 78e$ |
| 47 | $[47, 47, -8w - 75]$ | $\phantom{-}\frac{17}{54}e^{19} - \frac{253}{27}e^{17} + \frac{6227}{54}e^{15} - \frac{20432}{27}e^{13} + \frac{153767}{54}e^{11} - \frac{110743}{18}e^{9} + \frac{196126}{27}e^{7} - \frac{37909}{9}e^{5} + 911e^{3} - 25e$ |
| 61 | $[61, 61, -1172w - 11031]$ | $-\frac{7}{108}e^{18} + \frac{101}{54}e^{16} - \frac{2401}{108}e^{14} + \frac{3791}{27}e^{12} - \frac{54823}{108}e^{10} + \frac{38159}{36}e^{8} - \frac{33721}{27}e^{6} + \frac{14513}{18}e^{4} - \frac{763}{3}e^{2} + 18$ |
| 61 | $[61, 61, 1172w - 12203]$ | $-\frac{25}{108}e^{18} + \frac{190}{27}e^{16} - \frac{9583}{108}e^{14} + \frac{32359}{54}e^{12} - \frac{252229}{108}e^{10} + \frac{190019}{36}e^{8} - \frac{357503}{54}e^{6} + \frac{75191}{18}e^{4} - \frac{3199}{3}e^{2} + 72$ |
Atkin-Lehner eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| $2$ | $[2, 2, -17w + 177]$ | $-1$ |