Base field \(\Q(\sqrt{197}) \)
Generator \(w\), with minimal polynomial \(x^{2} - x - 49\); narrow class number \(1\) and class number \(1\).
Form
| Weight: | $[2, 2]$ |
| Level: | $[47, 47, -w - 1]$ |
| Dimension: | $93$ |
| CM: | no |
| Base change: | no |
| Newspace dimension: | $179$ |
Hecke eigenvalues ($q$-expansion)
The Hecke eigenvalue field is $\Q(e)$ where $e$ is a root of the defining polynomial:
| \(x^{93} + 3x^{92} - 224x^{91} - 668x^{90} + 24126x^{89} + 71538x^{88} - 1664242x^{87} - 4908713x^{86} + 82629072x^{85} + 242566815x^{84} - 3146098637x^{83} - 9199159119x^{82} + 95573985599x^{81} + 278623397032x^{80} - 2379625862100x^{79} - 6925042469005x^{78} + 49503062532294x^{77} + 144025863904264x^{76} - 872771296407822x^{75} - 2543348011615881x^{74} + 13182766291834037x^{73} + 38563484692798023x^{72} - 172008892408099705x^{71} - 506458608279227461x^{70} + 1951194411116513949x^{69} + 5800962214578422417x^{68} - 19335606232628619923x^{67} - 58265676077251442575x^{66} + 167984675760923382317x^{65} + 515407193985686406166x^{64} - 1282609454545673462669x^{63} - 4028727486232019393083x^{62} + 8618523410218846831343x^{61} + 27897674485714730837941x^{60} - 50982914197417575235660x^{59} - 171453282439011567947311x^{58} + 265278984305373962936153x^{57} + 936308569470745245990218x^{56} - 1211396172570035797101780x^{55} - 4546198973764966044245268x^{54} + 4834641800578392985014771x^{53} + 19626830827043520139172565x^{52} - 16744424961743462432160151x^{51} - 75300892717299799081180320x^{50} + 49724744140731937965649518x^{49} + 256467103092415811008243926x^{48} - 123848529738490275904600041x^{47} - 774151532889320737110372119x^{46} + 246861277688215034351090599x^{45} + 2066303106051145575017091214x^{44} - 344036008619738182505161824x^{43} - 4862374600866082178062209388x^{42} + 116805362882776592635152300x^{41} + 10049902182738347076889202734x^{40} + 1108932442592013840452734451x^{39} - 18159626143341085141732562018x^{38} - 4332579545367409896286673102x^{37} + 28521232297405408026451595517x^{36} + 10355337110044389658707225459x^{35} - 38655491503040725929458753521x^{34} - 18821597081272696263981241628x^{33} + 44799808665711030691485129042x^{32} + 27528125764031036856497935511x^{31} - 43877158951960184437355784045x^{30} - 33023640533044035691543121004x^{29} + 35743239637704819856323799292x^{28} + 32646973636742341818082664525x^{27} - 23670797982231830933409541742x^{26} - 26519101838870492071575198472x^{25} + 12284694867130826081437227408x^{24} + 17555098568888275503682625768x^{23} - 4651247104375390495465167381x^{22} - 9349728328959893856591673796x^{21} + 1041992938451338398758249611x^{20} + 3936229073686323140698563544x^{19} + 34666065937337016650171627x^{18} - 1279819309090000535112069507x^{17} - 131799306131184551242171053x^{16} + 311801435155848869901137141x^{15} + 54624708250522861795495178x^{14} - 54737286367952706807256817x^{13} - 12409613450601611828624974x^{12} + 6585752907736855111836597x^{11} + 1680321438753040613156806x^{10} - 510433876792738396556668x^{9} - 128922798811904507977917x^{8} + 23719418672391754011890x^{7} + 4918149307707234795211x^{6} - 604471238956727225158x^{5} - 77879998218708780721x^{4} + 7679038033082466292x^{3} + 294265861726766428x^{2} - 36419868090133175x + 689412834908779\) |
Show full eigenvalues Hide large eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| 4 | $[4, 2, 2]$ | $\phantom{-}e$ |
| 7 | $[7, 7, w - 7]$ | $...$ |
| 7 | $[7, 7, w + 6]$ | $...$ |
| 9 | $[9, 3, 3]$ | $...$ |
| 19 | $[19, 19, w + 5]$ | $...$ |
| 19 | $[19, 19, w - 6]$ | $...$ |
| 23 | $[23, 23, w + 8]$ | $...$ |
| 23 | $[23, 23, -w + 9]$ | $...$ |
| 25 | $[25, 5, 5]$ | $...$ |
| 29 | $[29, 29, -w - 4]$ | $...$ |
| 29 | $[29, 29, w - 5]$ | $...$ |
| 37 | $[37, 37, -w - 3]$ | $...$ |
| 37 | $[37, 37, w - 4]$ | $...$ |
Atkin-Lehner eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| $47$ | $[47, 47, -w - 1]$ | $1$ |