# Properties

 Label 600.8.a.b Level 600 Weight 8 Character orbit 600.a Self dual yes Analytic conductor 187.431 Analytic rank 1 Dimension 1 CM no Inner twists 1

# Related objects

## Newspace parameters

 Level: $$N$$ = $$600 = 2^{3} \cdot 3 \cdot 5^{2}$$ Weight: $$k$$ = $$8$$ Character orbit: $$[\chi]$$ = 600.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$187.431015290$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 24) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q - 27q^{3} - 120q^{7} + 729q^{9} + O(q^{10})$$ $$q - 27q^{3} - 120q^{7} + 729q^{9} - 7196q^{11} + 9626q^{13} - 18674q^{17} + 7004q^{19} + 3240q^{21} + 63704q^{23} - 19683q^{27} + 29334q^{29} + 87968q^{31} + 194292q^{33} - 227982q^{37} - 259902q^{39} - 160806q^{41} - 136132q^{43} + 1206960q^{47} - 809143q^{49} + 504198q^{51} + 398786q^{53} - 189108q^{57} + 1152436q^{59} - 2070602q^{61} - 87480q^{63} + 4073428q^{67} - 1720008q^{69} - 383752q^{71} - 3006010q^{73} + 863520q^{77} - 4948112q^{79} + 531441q^{81} + 9163492q^{83} - 792018q^{87} + 7304106q^{89} - 1155120q^{91} - 2375136q^{93} + 690526q^{97} - 5245884q^{99} + O(q^{100})$$

## 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
0 −27.0000 0 0 0 −120.000 0 729.000 0
 $$n$$: 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 600.8.a.b 1
5.b even 2 1 24.8.a.b 1
5.c odd 4 2 600.8.f.a 2
15.d odd 2 1 72.8.a.e 1
20.d odd 2 1 48.8.a.a 1
40.e odd 2 1 192.8.a.p 1
40.f even 2 1 192.8.a.h 1
60.h even 2 1 144.8.a.k 1
120.i odd 2 1 576.8.a.c 1
120.m even 2 1 576.8.a.b 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
24.8.a.b 1 5.b even 2 1
48.8.a.a 1 20.d odd 2 1
72.8.a.e 1 15.d odd 2 1
144.8.a.k 1 60.h even 2 1
192.8.a.h 1 40.f even 2 1
192.8.a.p 1 40.e odd 2 1
576.8.a.b 1 120.m even 2 1
576.8.a.c 1 120.i odd 2 1
600.8.a.b 1 1.a even 1 1 trivial
600.8.f.a 2 5.c odd 4 2

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$-1$$
$$3$$ $$1$$
$$5$$ $$1$$

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{7} + 120$$ acting on $$S_{8}^{\mathrm{new}}(\Gamma_0(600))$$.

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ 
$3$ $$1 + 27 T$$
$5$ 
$7$ $$1 + 120 T + 823543 T^{2}$$
$11$ $$1 + 7196 T + 19487171 T^{2}$$
$13$ $$1 - 9626 T + 62748517 T^{2}$$
$17$ $$1 + 18674 T + 410338673 T^{2}$$
$19$ $$1 - 7004 T + 893871739 T^{2}$$
$23$ $$1 - 63704 T + 3404825447 T^{2}$$
$29$ $$1 - 29334 T + 17249876309 T^{2}$$
$31$ $$1 - 87968 T + 27512614111 T^{2}$$
$37$ $$1 + 227982 T + 94931877133 T^{2}$$
$41$ $$1 + 160806 T + 194754273881 T^{2}$$
$43$ $$1 + 136132 T + 271818611107 T^{2}$$
$47$ $$1 - 1206960 T + 506623120463 T^{2}$$
$53$ $$1 - 398786 T + 1174711139837 T^{2}$$
$59$ $$1 - 1152436 T + 2488651484819 T^{2}$$
$61$ $$1 + 2070602 T + 3142742836021 T^{2}$$
$67$ $$1 - 4073428 T + 6060711605323 T^{2}$$
$71$ $$1 + 383752 T + 9095120158391 T^{2}$$
$73$ $$1 + 3006010 T + 11047398519097 T^{2}$$
$79$ $$1 + 4948112 T + 19203908986159 T^{2}$$
$83$ $$1 - 9163492 T + 27136050989627 T^{2}$$
$89$ $$1 - 7304106 T + 44231334895529 T^{2}$$
$97$ $$1 - 690526 T + 80798284478113 T^{2}$$