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: | $[41,41,w - 10]$ |
| Dimension: | $81$ |
| CM: | no |
| Base change: | no |
| Newspace dimension: | $157$ |
Hecke eigenvalues ($q$-expansion)
The Hecke eigenvalue field is $\Q(e)$ where $e$ is a root of the defining polynomial:
| \(x^{81} - 197x^{79} - 18x^{78} + 18514x^{77} + 3339x^{76} - 1104932x^{75} - 294605x^{74} + 47030284x^{73} + 16456370x^{72} - 1520071996x^{71} - 653517637x^{70} + 38787050086x^{69} + 19642511268x^{68} - 802036085833x^{67} - 464503221408x^{66} + 13689995155669x^{65} + 8869897261723x^{64} - 195507384540730x^{63} - 139294491116702x^{62} + 2359502527088380x^{61} + 1823069013979925x^{60} - 24245527928768139x^{59} - 20080816073412168x^{58} + 213318614989355362x^{57} + 187507745455804675x^{56} - 1613608228138541051x^{55} - 1492193475977244038x^{54} + 10524563385655878000x^{53} + 10158752120222852446x^{52} - 59302975068715031858x^{51} - 59314058402740742260x^{50} + 288983066714808174544x^{49} + 297441992737232762078x^{48} - 1218204048932443941267x^{47} - 1281700324006645857853x^{46} + 4440671131447710426281x^{45} + 4743989354859875635469x^{44} - 13983219623409062143582x^{43} - 15064233678617317445750x^{42} + 37972939862529676534555x^{41} + 40955708498482002309654x^{40} - 88727601570015799139637x^{39} - 95060866246555459075784x^{38} + 177866057440171878867445x^{37} + 187665512715945622161408x^{36} - 304801281675155112712381x^{35} - 313637215060299987830414x^{34} + 444580114160969363057410x^{33} + 441207728285372866174962x^{32} - 549102841587111692440661x^{31} - 518839184372376637792904x^{30} + 570784266907063005863427x^{29} + 505839783758969115672821x^{28} - 495737597283058751431093x^{27} - 404870813169379718723849x^{26} + 356641459022530731528847x^{25} + 262942659172679106552602x^{24} - 210326212560841755786466x^{23} - 136637051602128428975068x^{22} + 100408346942565881950433x^{21} + 55859755304185240918362x^{20} - 38213165206384581918177x^{19} - 17596490149420119312328x^{18} + 11379571592991260737021x^{17} + 4159699281147625547503x^{16} - 2592156264989222753777x^{15} - 711868218421045829888x^{14} + 439365144798769106729x^{13} + 83442759894791271488x^{12} - 53542156378009854888x^{11} - 6009297976481752480x^{10} + 4480238069061348871x^{9} + 184175716807282922x^{8} - 239649961372801650x^{7} + 5978398498793906x^{6} + 7097834755407401x^{5} - 642072368003840x^{4} - 71333971013797x^{3} + 14019920756512x^{2} - 777784123388x + 14706718153\) |
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 | $[37, 37, w - 4]$ | $...$ |
| 41 | $[41, 41, -w - 9]$ | $...$ |
| 41 | $[41, 41, w - 10]$ | $-1$ |
| 43 | $[43, 43, -w - 2]$ | $...$ |
| 43 | $[43, 43, w - 3]$ | $...$ |
| 47 | $[47, 47, -w - 1]$ | $...$ |
| 47 | $[47, 47, w - 2]$ | $...$ |
Atkin-Lehner eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| $41$ | $[41,41,w - 10]$ | $1$ |