Base field \(\Q(\sqrt{357}) \)
Generator \(w\), with minimal polynomial \(x^{2} - x - 89\); narrow class number \(4\) and class number \(2\).
Form
| Weight: | $[2, 2]$ |
| Level: | $[1, 1, 1]$ |
| Dimension: | $12$ |
| CM: | no |
| Base change: | yes |
| Newspace dimension: | $38$ |
Hecke eigenvalues ($q$-expansion)
The Hecke eigenvalue field is $\Q(e)$ where $e$ is a root of the defining polynomial:
| \(x^{12} - 140x^{10} + 7246x^{8} - 168516x^{6} + 1618137x^{4} - 3479624x^{2} + 1763584\) |
Show full eigenvalues Hide large eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| 3 | $[3, 3, w + 1]$ | $\phantom{-}\frac{15311139}{1393457988512}e^{11} - \frac{935243319}{696728994256}e^{9} + \frac{2502959962}{43545562141}e^{7} - \frac{715165736465}{696728994256}e^{5} + \frac{9282077472085}{1393457988512}e^{3} - \frac{1239955667803}{174182248564}e$ |
| 4 | $[4, 2, 2]$ | $-\frac{121809}{8394325232}e^{10} + \frac{13386039}{8394325232}e^{8} - \frac{491046079}{8394325232}e^{6} + \frac{7008535345}{8394325232}e^{4} - \frac{4000351657}{1049290654}e^{2} + \frac{3262213561}{524645327}$ |
| 7 | $[7, 7, w + 3]$ | $-\frac{1398525}{348364497128}e^{11} + \frac{683924177}{1393457988512}e^{9} - \frac{29655804789}{1393457988512}e^{7} + \frac{552511262739}{1393457988512}e^{5} - \frac{3952164556715}{1393457988512}e^{3} + \frac{454877787181}{174182248564}e$ |
| 11 | $[11, 11, w + 3]$ | $-\frac{15311139}{1393457988512}e^{11} + \frac{935243319}{696728994256}e^{9} - \frac{2502959962}{43545562141}e^{7} + \frac{715165736465}{696728994256}e^{5} - \frac{9282077472085}{1393457988512}e^{3} + \frac{1414137916367}{174182248564}e$ |
| 11 | $[11, 11, w + 7]$ | $-\frac{15311139}{1393457988512}e^{11} + \frac{935243319}{696728994256}e^{9} - \frac{2502959962}{43545562141}e^{7} + \frac{715165736465}{696728994256}e^{5} - \frac{9282077472085}{1393457988512}e^{3} + \frac{1414137916367}{174182248564}e$ |
| 17 | $[17, 17, w + 8]$ | $-\frac{565797}{8394325232}e^{10} + \frac{64348299}{8394325232}e^{8} - \frac{2407411435}{8394325232}e^{6} + \frac{31990528597}{8394325232}e^{4} - \frac{4860115326}{524645327}e^{2} + \frac{6474916558}{524645327}$ |
| 23 | $[23, 23, w + 4]$ | $\phantom{-}\frac{48066725}{1393457988512}e^{11} - \frac{2692974409}{696728994256}e^{9} + \frac{50022642981}{348364497128}e^{7} - \frac{1383643827579}{696728994256}e^{5} + \frac{9979603023479}{1393457988512}e^{3} - \frac{2101982506069}{174182248564}e$ |
| 23 | $[23, 23, w + 18]$ | $\phantom{-}\frac{48066725}{1393457988512}e^{11} - \frac{2692974409}{696728994256}e^{9} + \frac{50022642981}{348364497128}e^{7} - \frac{1383643827579}{696728994256}e^{5} + \frac{9979603023479}{1393457988512}e^{3} - \frac{2101982506069}{174182248564}e$ |
| 25 | $[25, 5, 5]$ | $\phantom{-}\frac{121809}{8394325232}e^{10} - \frac{13386039}{8394325232}e^{8} + \frac{491046079}{8394325232}e^{6} - \frac{7008535345}{8394325232}e^{4} + \frac{4000351657}{1049290654}e^{2} + \frac{1459594382}{524645327}$ |
| 29 | $[29, 29, w + 1]$ | $-\frac{16377793}{696728994256}e^{11} + \frac{878865545}{348364497128}e^{9} - \frac{29998963285}{348364497128}e^{7} + \frac{334239045557}{348364497128}e^{5} - \frac{348762775697}{696728994256}e^{3} + \frac{343922294851}{87091124282}e$ |
| 29 | $[29, 29, w + 27]$ | $-\frac{16377793}{696728994256}e^{11} + \frac{878865545}{348364497128}e^{9} - \frac{29998963285}{348364497128}e^{7} + \frac{334239045557}{348364497128}e^{5} - \frac{348762775697}{696728994256}e^{3} + \frac{343922294851}{87091124282}e$ |
| 31 | $[31, 31, w + 13]$ | $\phantom{-}\frac{411639}{43545562141}e^{11} - \frac{1142172793}{1393457988512}e^{9} + \frac{18138195659}{1393457988512}e^{7} + \frac{553464848105}{1393457988512}e^{5} - \frac{13565024592835}{1393457988512}e^{3} + \frac{2861083941765}{174182248564}e$ |
| 31 | $[31, 31, w + 17]$ | $\phantom{-}\frac{411639}{43545562141}e^{11} - \frac{1142172793}{1393457988512}e^{9} + \frac{18138195659}{1393457988512}e^{7} + \frac{553464848105}{1393457988512}e^{5} - \frac{13565024592835}{1393457988512}e^{3} + \frac{2861083941765}{174182248564}e$ |
| 43 | $[43, 43, -w - 11]$ | $-\frac{36999}{1049290654}e^{10} + \frac{4246855}{1049290654}e^{8} - \frac{159697113}{1049290654}e^{6} + \frac{2081832771}{1049290654}e^{4} - \frac{1731744556}{524645327}e^{2} - \frac{1006069964}{524645327}$ |
| 43 | $[43, 43, w - 12]$ | $-\frac{36999}{1049290654}e^{10} + \frac{4246855}{1049290654}e^{8} - \frac{159697113}{1049290654}e^{6} + \frac{2081832771}{1049290654}e^{4} - \frac{1731744556}{524645327}e^{2} - \frac{1006069964}{524645327}$ |
| 47 | $[47, 47, -w - 6]$ | $-\frac{1382715}{8394325232}e^{10} + \frac{38178499}{2098581308}e^{8} - \frac{5468646107}{8394325232}e^{6} + \frac{33571330019}{4197162616}e^{4} - \frac{6029545128}{524645327}e^{2} + \frac{715286628}{524645327}$ |
| 47 | $[47, 47, w - 7]$ | $-\frac{1382715}{8394325232}e^{10} + \frac{38178499}{2098581308}e^{8} - \frac{5468646107}{8394325232}e^{6} + \frac{33571330019}{4197162616}e^{4} - \frac{6029545128}{524645327}e^{2} + \frac{715286628}{524645327}$ |
| 59 | $[59, 59, -w - 5]$ | $-\frac{671177}{16788650464}e^{10} + \frac{64816539}{16788650464}e^{8} - \frac{1784806263}{16788650464}e^{6} + \frac{9061375245}{16788650464}e^{4} + \frac{17344197603}{2098581308}e^{2} - \frac{7835408576}{524645327}$ |
| 59 | $[59, 59, w - 6]$ | $-\frac{671177}{16788650464}e^{10} + \frac{64816539}{16788650464}e^{8} - \frac{1784806263}{16788650464}e^{6} + \frac{9061375245}{16788650464}e^{4} + \frac{17344197603}{2098581308}e^{2} - \frac{7835408576}{524645327}$ |
| 61 | $[61, 61, w + 16]$ | $-\frac{50859987}{1393457988512}e^{11} + \frac{5748356139}{1393457988512}e^{9} - \frac{216856133599}{1393457988512}e^{7} + \frac{3086911675781}{1393457988512}e^{5} - \frac{5762855553055}{696728994256}e^{3} + \frac{99191437381}{87091124282}e$ |
Atkin-Lehner eigenvalues
This form has no Atkin-Lehner eigenvalues since the level is \((1)\).