# Properties

 Label 5.8.a.a Level 5 Weight 8 Character orbit 5.a Self dual yes Analytic conductor 1.562 Analytic rank 1 Dimension 1 CM no Inner twists 1

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$5$$ Weight: $$k$$ $$=$$ $$8$$ Character orbit: $$[\chi]$$ $$=$$ 5.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$1.56192512742$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q - 14q^{2} - 48q^{3} + 68q^{4} + 125q^{5} + 672q^{6} - 1644q^{7} + 840q^{8} + 117q^{9} + O(q^{10})$$ $$q - 14q^{2} - 48q^{3} + 68q^{4} + 125q^{5} + 672q^{6} - 1644q^{7} + 840q^{8} + 117q^{9} - 1750q^{10} + 172q^{11} - 3264q^{12} + 3862q^{13} + 23016q^{14} - 6000q^{15} - 20464q^{16} - 12254q^{17} - 1638q^{18} - 25940q^{19} + 8500q^{20} + 78912q^{21} - 2408q^{22} + 12972q^{23} - 40320q^{24} + 15625q^{25} - 54068q^{26} + 99360q^{27} - 111792q^{28} - 81610q^{29} + 84000q^{30} - 156888q^{31} + 178976q^{32} - 8256q^{33} + 171556q^{34} - 205500q^{35} + 7956q^{36} + 110126q^{37} + 363160q^{38} - 185376q^{39} + 105000q^{40} + 467882q^{41} - 1104768q^{42} - 499208q^{43} + 11696q^{44} + 14625q^{45} - 181608q^{46} - 396884q^{47} + 982272q^{48} + 1879193q^{49} - 218750q^{50} + 588192q^{51} + 262616q^{52} - 1280498q^{53} - 1391040q^{54} + 21500q^{55} - 1380960q^{56} + 1245120q^{57} + 1142540q^{58} - 1337420q^{59} - 408000q^{60} - 923978q^{61} + 2196432q^{62} - 192348q^{63} + 113728q^{64} + 482750q^{65} + 115584q^{66} - 797304q^{67} - 833272q^{68} - 622656q^{69} + 2877000q^{70} + 5103392q^{71} + 98280q^{72} - 4267478q^{73} - 1541764q^{74} - 750000q^{75} - 1763920q^{76} - 282768q^{77} + 2595264q^{78} - 960q^{79} - 2558000q^{80} - 5025159q^{81} - 6550348q^{82} + 6140832q^{83} + 5366016q^{84} - 1531750q^{85} + 6988912q^{86} + 3917280q^{87} + 144480q^{88} + 2010570q^{89} - 204750q^{90} - 6349128q^{91} + 882096q^{92} + 7530624q^{93} + 5556376q^{94} - 3242500q^{95} - 8590848q^{96} - 4881934q^{97} - 26308702q^{98} + 20124q^{99} + O(q^{100})$$

## Embeddings

For each embedding $$\iota_m$$ of the coefficient field, the values $$\iota_m(a_n)$$ are shown below.

For more information on an embedded modular form you can click on its label.

Label $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 0
−14.0000 −48.0000 68.0000 125.000 672.000 −1644.00 840.000 117.000 −1750.00
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

This newform does not admit any (nontrivial) inner twists.

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 5.8.a.a 1
3.b odd 2 1 45.8.a.f 1
4.b odd 2 1 80.8.a.d 1
5.b even 2 1 25.8.a.a 1
5.c odd 4 2 25.8.b.a 2
7.b odd 2 1 245.8.a.a 1
8.b even 2 1 320.8.a.h 1
8.d odd 2 1 320.8.a.a 1
11.b odd 2 1 605.8.a.c 1
15.d odd 2 1 225.8.a.b 1
15.e even 4 2 225.8.b.b 2
20.d odd 2 1 400.8.a.e 1
20.e even 4 2 400.8.c.e 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
5.8.a.a 1 1.a even 1 1 trivial
25.8.a.a 1 5.b even 2 1
25.8.b.a 2 5.c odd 4 2
45.8.a.f 1 3.b odd 2 1
80.8.a.d 1 4.b odd 2 1
225.8.a.b 1 15.d odd 2 1
225.8.b.b 2 15.e even 4 2
245.8.a.a 1 7.b odd 2 1
320.8.a.a 1 8.d odd 2 1
320.8.a.h 1 8.b even 2 1
400.8.a.e 1 20.d odd 2 1
400.8.c.e 2 20.e even 4 2
605.8.a.c 1 11.b odd 2 1

## Atkin-Lehner signs

$$p$$ Sign
$$5$$ $$-1$$

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2} + 14$$ acting on $$S_{8}^{\mathrm{new}}(\Gamma_0(5))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + 14 T + 128 T^{2}$$
$3$ $$1 + 48 T + 2187 T^{2}$$
$5$ $$1 - 125 T$$
$7$ $$1 + 1644 T + 823543 T^{2}$$
$11$ $$1 - 172 T + 19487171 T^{2}$$
$13$ $$1 - 3862 T + 62748517 T^{2}$$
$17$ $$1 + 12254 T + 410338673 T^{2}$$
$19$ $$1 + 25940 T + 893871739 T^{2}$$
$23$ $$1 - 12972 T + 3404825447 T^{2}$$
$29$ $$1 + 81610 T + 17249876309 T^{2}$$
$31$ $$1 + 156888 T + 27512614111 T^{2}$$
$37$ $$1 - 110126 T + 94931877133 T^{2}$$
$41$ $$1 - 467882 T + 194754273881 T^{2}$$
$43$ $$1 + 499208 T + 271818611107 T^{2}$$
$47$ $$1 + 396884 T + 506623120463 T^{2}$$
$53$ $$1 + 1280498 T + 1174711139837 T^{2}$$
$59$ $$1 + 1337420 T + 2488651484819 T^{2}$$
$61$ $$1 + 923978 T + 3142742836021 T^{2}$$
$67$ $$1 + 797304 T + 6060711605323 T^{2}$$
$71$ $$1 - 5103392 T + 9095120158391 T^{2}$$
$73$ $$1 + 4267478 T + 11047398519097 T^{2}$$
$79$ $$1 + 960 T + 19203908986159 T^{2}$$
$83$ $$1 - 6140832 T + 27136050989627 T^{2}$$
$89$ $$1 - 2010570 T + 44231334895529 T^{2}$$
$97$ $$1 + 4881934 T + 80798284478113 T^{2}$$