Base field \(\Q(\sqrt{42}) \)
Generator \(w\), with minimal polynomial \(x^{2} - 42\); narrow class number \(4\) and class number \(2\).
Form
| Weight: | $[2, 2]$ |
| Level: | $[14, 14, w]$ |
| Dimension: | $16$ |
| CM: | no |
| Base change: | no |
| Newspace dimension: | $52$ |
Hecke eigenvalues ($q$-expansion)
The Hecke eigenvalue field is $\Q(e)$ where $e$ is a root of the defining polynomial:
| \(x^{16} + 150x^{14} + 8253x^{12} + 211248x^{10} + 2736828x^{8} + 17799264x^{6} + 52056432x^{4} + 51508224x^{2} + 419904\) |
Show full eigenvalues Hide large eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| 2 | $[2, 2, w]$ | $\phantom{-}\frac{89}{262020096}e^{15} + \frac{731}{14556672}e^{13} + \frac{78601}{29113344}e^{11} + \frac{80801}{1213056}e^{9} + \frac{25471}{31104}e^{7} + \frac{41749}{8424}e^{5} + \frac{97979}{7488}e^{3} + \frac{1303}{117}e$ |
| 3 | $[3, 3, w]$ | $\phantom{-}\frac{1591}{5036608512}e^{15} + \frac{61037}{1259152128}e^{13} + \frac{4630025}{1678869504}e^{11} + \frac{2299225}{31090176}e^{9} + \frac{1810841}{1793664}e^{7} + \frac{8921741}{1295424}e^{5} + \frac{26711717}{1295424}e^{3} + \frac{4161061}{215904}e$ |
| 7 | $[7, 7, w - 7]$ | $-1$ |
| 11 | $[11, 11, w + 3]$ | $\phantom{-}\frac{1591}{5036608512}e^{15} + \frac{61037}{1259152128}e^{13} + \frac{4630025}{1678869504}e^{11} + \frac{2299225}{31090176}e^{9} + \frac{1810841}{1793664}e^{7} + \frac{8921741}{1295424}e^{5} + \frac{26711717}{1295424}e^{3} + \frac{4376965}{215904}e$ |
| 11 | $[11, 11, w + 8]$ | $\phantom{-}\frac{2557}{2518304256}e^{15} + \frac{95603}{629576064}e^{13} + \frac{6990443}{839434752}e^{11} + \frac{9927329}{46635264}e^{9} + \frac{311767}{112104}e^{7} + \frac{6046231}{323856}e^{5} + \frac{38119097}{647712}e^{3} + \frac{7031665}{107952}e$ |
| 13 | $[13, 13, w + 4]$ | $-\frac{943}{15109825536}e^{15} - \frac{27743}{2518304256}e^{13} - \frac{139127}{186541056}e^{11} - \frac{1706905}{69952896}e^{9} - \frac{240113}{597888}e^{7} - \frac{6215333}{1943136}e^{5} - \frac{14195371}{1295424}e^{3} - \frac{255565}{17992}e$ |
| 13 | $[13, 13, w + 9]$ | $\phantom{-}\frac{1811}{1888728192}e^{15} + \frac{367483}{2518304256}e^{13} + \frac{3442609}{419717376}e^{11} + \frac{60528019}{279811584}e^{9} + \frac{2599667}{896832}e^{7} + \frac{75679057}{3886272}e^{5} + \frac{9524605}{161928}e^{3} + \frac{13716779}{215904}e$ |
| 17 | $[17, 17, -w - 5]$ | $\phantom{-}\frac{7}{387431424}e^{14} + \frac{259}{32285952}e^{12} + \frac{37363}{43047936}e^{10} + \frac{265187}{7174656}e^{8} + \frac{397775}{597888}e^{6} + \frac{54165}{11072}e^{4} + \frac{364711}{33216}e^{2} + \frac{15471}{5536}$ |
| 17 | $[17, 17, -w + 5]$ | $\phantom{-}\frac{7}{387431424}e^{14} + \frac{259}{32285952}e^{12} + \frac{37363}{43047936}e^{10} + \frac{265187}{7174656}e^{8} + \frac{397775}{597888}e^{6} + \frac{54165}{11072}e^{4} + \frac{364711}{33216}e^{2} + \frac{15471}{5536}$ |
| 19 | $[19, 19, w + 2]$ | $-\frac{14101}{10073217024}e^{15} - \frac{1051481}{5036608512}e^{13} - \frac{38219095}{3357739008}e^{11} - \frac{8931359}{31090176}e^{9} - \frac{13102549}{3587328}e^{7} - \frac{30136739}{1295424}e^{5} - \frac{174784327}{2590848}e^{3} - \frac{15207181}{215904}e$ |
| 19 | $[19, 19, w + 17]$ | $\phantom{-}\frac{11951}{10073217024}e^{15} + \frac{913261}{5036608512}e^{13} + \frac{34446173}{3357739008}e^{11} + \frac{1415677}{5181696}e^{9} + \frac{13294979}{3587328}e^{7} + \frac{5429975}{215904}e^{5} + \frac{193895621}{2590848}e^{3} + \frac{1839061}{26988}e$ |
| 25 | $[25, 5, 5]$ | $-\frac{101}{96857856}e^{14} - \frac{4741}{32285952}e^{12} - \frac{78371}{10761984}e^{10} - \frac{562579}{3587328}e^{8} - \frac{236195}{149472}e^{6} - \frac{372769}{49824}e^{4} - \frac{133303}{8304}e^{2} - \frac{24403}{2768}$ |
| 29 | $[29, 29, w + 10]$ | $\phantom{-}\frac{4135}{3777456384}e^{15} + \frac{273157}{1678869504}e^{13} + \frac{7421749}{839434752}e^{11} + \frac{124892711}{559623168}e^{9} + \frac{5162921}{1793664}e^{7} + \frac{148974101}{7772544}e^{5} + \frac{818797}{13494}e^{3} + \frac{29436959}{431808}e$ |
| 29 | $[29, 29, w + 19]$ | $-\frac{31937}{15109825536}e^{15} - \frac{175361}{559623168}e^{13} - \frac{28441471}{1678869504}e^{11} - \frac{236721295}{559623168}e^{9} - \frac{2392351}{448416}e^{7} - \frac{264535333}{7772544}e^{5} - \frac{14383957}{143936}e^{3} - \frac{43863775}{431808}e$ |
| 41 | $[41, 41, -w - 1]$ | $\phantom{-}\frac{8399}{3357739008}e^{14} + \frac{148949}{419717376}e^{12} + \frac{19926233}{1119246336}e^{10} + \frac{72665011}{186541056}e^{8} + \frac{1571911}{398592}e^{6} + \frac{44634401}{2590848}e^{4} + \frac{20290045}{863616}e^{2} - \frac{625249}{143936}$ |
| 41 | $[41, 41, w - 1]$ | $-\frac{2219}{1119246336}e^{14} - \frac{121987}{419717376}e^{12} - \frac{17205463}{1119246336}e^{10} - \frac{68371709}{186541056}e^{8} - \frac{1670393}{398592}e^{6} - \frac{56057839}{2590848}e^{4} - \frac{33165011}{863616}e^{2} - \frac{918865}{143936}$ |
| 47 | $[47, 47, -2w + 11]$ | $\phantom{-}\frac{19609}{10073217024}e^{14} + \frac{57991}{209858688}e^{12} + \frac{15534245}{1119246336}e^{10} + \frac{18931945}{62180352}e^{8} + \frac{3703705}{1195776}e^{6} + \frac{35065937}{2590848}e^{4} + \frac{14305033}{863616}e^{2} - \frac{1316217}{143936}$ |
| 47 | $[47, 47, 4w - 25]$ | $-\frac{8611}{10073217024}e^{14} - \frac{5773}{46635264}e^{12} - \frac{797599}{124360704}e^{10} - \frac{27524645}{186541056}e^{8} - \frac{1907947}{1195776}e^{6} - \frac{2209887}{287872}e^{4} - \frac{3869169}{287872}e^{2} - \frac{565273}{143936}$ |
| 53 | $[53, 53, w + 25]$ | $-\frac{47509}{15109825536}e^{15} - \frac{1179647}{2518304256}e^{13} - \frac{4756805}{186541056}e^{11} - \frac{89836211}{139905792}e^{9} - \frac{1621643}{199296}e^{7} - \frac{99860729}{1943136}e^{5} - \frac{186569989}{1295424}e^{3} - \frac{4811335}{35984}e$ |
| 53 | $[53, 53, w + 28]$ | $-\frac{425}{1888728192}e^{15} - \frac{83617}{2518304256}e^{13} - \frac{748903}{419717376}e^{11} - \frac{12413161}{279811584}e^{9} - \frac{254323}{448416}e^{7} - \frac{15159769}{3886272}e^{5} - \frac{4520531}{323856}e^{3} - \frac{3999101}{215904}e$ |
Atkin-Lehner eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| $2$ | $[2, 2, w]$ | $-\frac{89}{262020096}e^{15} - \frac{731}{14556672}e^{13} - \frac{78601}{29113344}e^{11} - \frac{80801}{1213056}e^{9} - \frac{25471}{31104}e^{7} - \frac{41749}{8424}e^{5} - \frac{97979}{7488}e^{3} - \frac{1303}{117}e$ |
| $7$ | $[7, 7, w - 7]$ | $1$ |