Properties

Base field \(\Q(\sqrt{197}) \)
Weight [2, 2]
Level norm 16
Level $[16, 4, 4]$
Label 2.2.197.1-16.1-e
Dimension 8
CM no
Base change yes

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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 $[16, 4, 4]$
Label 2.2.197.1-16.1-e
Dimension 8
Is CM no
Is base change yes
Parent newspace dimension 40

Hecke eigenvalues ($q$-expansion)

The Hecke eigenvalue field is $\Q(e)$ where $e$ is a root of the defining polynomial:
\(x^{8} \) \(\mathstrut +\mathstrut x^{7} \) \(\mathstrut -\mathstrut 31x^{6} \) \(\mathstrut -\mathstrut 20x^{5} \) \(\mathstrut +\mathstrut 282x^{4} \) \(\mathstrut +\mathstrut 148x^{3} \) \(\mathstrut -\mathstrut 733x^{2} \) \(\mathstrut -\mathstrut 468x \) \(\mathstrut -\mathstrut 36\)

  Show full eigenvalues   Hide large eigenvalues

Norm Prime Eigenvalue
4 $[4, 2, 2]$ $\phantom{-}0$
7 $[7, 7, w - 7]$ $\phantom{-}e$
7 $[7, 7, w + 6]$ $\phantom{-}e$
9 $[9, 3, 3]$ $\phantom{-}\frac{843}{153928}e^{7} + \frac{3345}{153928}e^{6} - \frac{21683}{153928}e^{5} - \frac{31021}{76964}e^{4} + \frac{89789}{76964}e^{3} + \frac{21414}{19241}e^{2} - \frac{363095}{153928}e + \frac{308539}{76964}$
19 $[19, 19, w + 5]$ $-\frac{1129}{115446}e^{7} - \frac{4069}{115446}e^{6} + \frac{34015}{115446}e^{5} + \frac{49648}{57723}e^{4} - \frac{51534}{19241}e^{3} - \frac{295691}{57723}e^{2} + \frac{747757}{115446}e + \frac{95999}{19241}$
19 $[19, 19, w - 6]$ $-\frac{1129}{115446}e^{7} - \frac{4069}{115446}e^{6} + \frac{34015}{115446}e^{5} + \frac{49648}{57723}e^{4} - \frac{51534}{19241}e^{3} - \frac{295691}{57723}e^{2} + \frac{747757}{115446}e + \frac{95999}{19241}$
23 $[23, 23, w + 8]$ $-\frac{1199}{57723}e^{7} + \frac{1405}{57723}e^{6} + \frac{41339}{57723}e^{5} - \frac{40670}{57723}e^{4} - \frac{135755}{19241}e^{3} + \frac{214840}{57723}e^{2} + \frac{1180439}{57723}e + \frac{70732}{19241}$
23 $[23, 23, -w + 9]$ $-\frac{1199}{57723}e^{7} + \frac{1405}{57723}e^{6} + \frac{41339}{57723}e^{5} - \frac{40670}{57723}e^{4} - \frac{135755}{19241}e^{3} + \frac{214840}{57723}e^{2} + \frac{1180439}{57723}e + \frac{70732}{19241}$
25 $[25, 5, 5]$ $-\frac{2333}{115446}e^{7} - \frac{2273}{115446}e^{6} + \frac{67631}{115446}e^{5} + \frac{11945}{57723}e^{4} - \frac{97902}{19241}e^{3} - \frac{23416}{57723}e^{2} + \frac{1563287}{115446}e + \frac{161393}{19241}$
29 $[29, 29, -w - 4]$ $-\frac{20129}{461784}e^{7} - \frac{19067}{461784}e^{6} + \frac{587129}{461784}e^{5} + \frac{144083}{230892}e^{4} - \frac{801601}{76964}e^{3} - \frac{165187}{57723}e^{2} + \frac{10066589}{461784}e + \frac{565053}{76964}$
29 $[29, 29, w - 5]$ $-\frac{20129}{461784}e^{7} - \frac{19067}{461784}e^{6} + \frac{587129}{461784}e^{5} + \frac{144083}{230892}e^{4} - \frac{801601}{76964}e^{3} - \frac{165187}{57723}e^{2} + \frac{10066589}{461784}e + \frac{565053}{76964}$
37 $[37, 37, -w - 3]$ $\phantom{-}\frac{2509}{461784}e^{7} + \frac{11599}{461784}e^{6} - \frac{95941}{461784}e^{5} - \frac{162535}{230892}e^{4} + \frac{214309}{76964}e^{3} + \frac{286820}{57723}e^{2} - \frac{5319625}{461784}e - \frac{494917}{76964}$
37 $[37, 37, w - 4]$ $\phantom{-}\frac{2509}{461784}e^{7} + \frac{11599}{461784}e^{6} - \frac{95941}{461784}e^{5} - \frac{162535}{230892}e^{4} + \frac{214309}{76964}e^{3} + \frac{286820}{57723}e^{2} - \frac{5319625}{461784}e - \frac{494917}{76964}$
41 $[41, 41, -w - 9]$ $-\frac{15593}{461784}e^{7} - \frac{4355}{461784}e^{6} + \frac{481961}{461784}e^{5} + \frac{14963}{230892}e^{4} - \frac{719985}{76964}e^{3} - \frac{92074}{57723}e^{2} + \frac{11305901}{461784}e + \frac{830585}{76964}$
41 $[41, 41, w - 10]$ $-\frac{15593}{461784}e^{7} - \frac{4355}{461784}e^{6} + \frac{481961}{461784}e^{5} + \frac{14963}{230892}e^{4} - \frac{719985}{76964}e^{3} - \frac{92074}{57723}e^{2} + \frac{11305901}{461784}e + \frac{830585}{76964}$
43 $[43, 43, -w - 2]$ $\phantom{-}\frac{6733}{115446}e^{7} + \frac{4531}{115446}e^{6} - \frac{198151}{115446}e^{5} - \frac{24700}{57723}e^{4} + \frac{275855}{19241}e^{3} + \frac{66638}{57723}e^{2} - \frac{3923059}{115446}e - \frac{110417}{19241}$
43 $[43, 43, w - 3]$ $\phantom{-}\frac{6733}{115446}e^{7} + \frac{4531}{115446}e^{6} - \frac{198151}{115446}e^{5} - \frac{24700}{57723}e^{4} + \frac{275855}{19241}e^{3} + \frac{66638}{57723}e^{2} - \frac{3923059}{115446}e - \frac{110417}{19241}$
47 $[47, 47, -w - 1]$ $-\frac{1199}{115446}e^{7} + \frac{1405}{115446}e^{6} + \frac{41339}{115446}e^{5} - \frac{20335}{57723}e^{4} - \frac{77498}{19241}e^{3} + \frac{49697}{57723}e^{2} + \frac{1873115}{115446}e + \frac{131571}{19241}$
47 $[47, 47, w - 2]$ $-\frac{1199}{115446}e^{7} + \frac{1405}{115446}e^{6} + \frac{41339}{115446}e^{5} - \frac{20335}{57723}e^{4} - \frac{77498}{19241}e^{3} + \frac{49697}{57723}e^{2} + \frac{1873115}{115446}e + \frac{131571}{19241}$
53 $[53, 53, 2w - 13]$ $-\frac{16115}{461784}e^{7} - \frac{9713}{461784}e^{6} + \frac{506891}{461784}e^{5} + \frac{71969}{230892}e^{4} - \frac{740983}{76964}e^{3} - \frac{89722}{57723}e^{2} + \frac{10105943}{461784}e + \frac{402075}{76964}$
Display number of eigenvalues

Atkin-Lehner eigenvalues

Norm Prime Eigenvalue
4 $[4, 2, 2]$ $1$