# Properties

 Label 600.8.f.a Level 600 Weight 8 Character orbit 600.f Analytic conductor 187.431 Analytic rank 0 Dimension 2 CM no Inner twists 2

# Related objects

## Newspace parameters

 Level: $$N$$ = $$600 = 2^{3} \cdot 3 \cdot 5^{2}$$ Weight: $$k$$ = $$8$$ Character orbit: $$[\chi]$$ = 600.f (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$187.431015290$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-1})$$ Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 24) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$i = \sqrt{-1}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q -27 i q^{3} + 120 i q^{7} -729 q^{9} +O(q^{10})$$ $$q -27 i q^{3} + 120 i q^{7} -729 q^{9} -7196 q^{11} + 9626 i q^{13} + 18674 i q^{17} -7004 q^{19} + 3240 q^{21} + 63704 i q^{23} + 19683 i q^{27} -29334 q^{29} + 87968 q^{31} + 194292 i q^{33} + 227982 i q^{37} + 259902 q^{39} -160806 q^{41} -136132 i q^{43} -1206960 i q^{47} + 809143 q^{49} + 504198 q^{51} + 398786 i q^{53} + 189108 i q^{57} -1152436 q^{59} -2070602 q^{61} -87480 i q^{63} -4073428 i q^{67} + 1720008 q^{69} -383752 q^{71} -3006010 i q^{73} -863520 i q^{77} + 4948112 q^{79} + 531441 q^{81} + 9163492 i q^{83} + 792018 i q^{87} -7304106 q^{89} -1155120 q^{91} -2375136 i q^{93} -690526 i q^{97} + 5245884 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 1458q^{9} + O(q^{10})$$ $$2q - 1458q^{9} - 14392q^{11} - 14008q^{19} + 6480q^{21} - 58668q^{29} + 175936q^{31} + 519804q^{39} - 321612q^{41} + 1618286q^{49} + 1008396q^{51} - 2304872q^{59} - 4141204q^{61} + 3440016q^{69} - 767504q^{71} + 9896224q^{79} + 1062882q^{81} - 14608212q^{89} - 2310240q^{91} + 10491768q^{99} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/600\mathbb{Z}\right)^\times$$.

 $$n$$ $$151$$ $$301$$ $$401$$ $$577$$ $$\chi(n)$$ $$1$$ $$1$$ $$1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
49.1
 1.00000i − 1.00000i
0 27.0000i 0 0 0 120.000i 0 −729.000 0
49.2 0 27.0000i 0 0 0 120.000i 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

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 600.8.f.a 2
5.b even 2 1 inner 600.8.f.a 2
5.c odd 4 1 24.8.a.b 1
5.c odd 4 1 600.8.a.b 1
15.e even 4 1 72.8.a.e 1
20.e even 4 1 48.8.a.a 1
40.i odd 4 1 192.8.a.h 1
40.k even 4 1 192.8.a.p 1
60.l odd 4 1 144.8.a.k 1
120.q odd 4 1 576.8.a.b 1
120.w even 4 1 576.8.a.c 1

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{7}^{2} + 14400$$ acting on $$S_{8}^{\mathrm{new}}(600, [\chi])$$.

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ 
$3$ $$1 + 729 T^{2}$$
$5$ 
$7$ $$1 - 1632686 T^{2} + 678223072849 T^{4}$$
$11$ $$( 1 + 7196 T + 19487171 T^{2} )^{2}$$
$13$ $$1 - 32837158 T^{2} + 3937376385699289 T^{4}$$
$17$ $$1 - 471959070 T^{2} + 168377826559400929 T^{4}$$
$19$ $$( 1 + 7004 T + 893871739 T^{2} )^{2}$$
$23$ $$1 - 2751451278 T^{2} + 11592836324538749809 T^{4}$$
$29$ $$( 1 + 29334 T + 17249876309 T^{2} )^{2}$$
$31$ $$( 1 - 87968 T + 27512614111 T^{2} )^{2}$$
$37$ $$1 - 137887961942 T^{2} +$$$$90\!\cdots\!89$$$$T^{4}$$
$41$ $$( 1 + 160806 T + 194754273881 T^{2} )^{2}$$
$43$ $$1 - 525105300790 T^{2} +$$$$73\!\cdots\!49$$$$T^{4}$$
$47$ $$1 + 443506200674 T^{2} +$$$$25\!\cdots\!69$$$$T^{4}$$
$53$ $$1 - 2190392005878 T^{2} +$$$$13\!\cdots\!69$$$$T^{4}$$
$59$ $$( 1 + 1152436 T + 2488651484819 T^{2} )^{2}$$
$61$ $$( 1 + 2070602 T + 3142742836021 T^{2} )^{2}$$
$67$ $$1 + 4471392460538 T^{2} +$$$$36\!\cdots\!29$$$$T^{4}$$
$71$ $$( 1 + 383752 T + 9095120158391 T^{2} )^{2}$$
$73$ $$1 - 13058700918094 T^{2} +$$$$12\!\cdots\!09$$$$T^{4}$$
$79$ $$( 1 - 4948112 T + 19203908986159 T^{2} )^{2}$$
$83$ $$1 + 29697483654810 T^{2} +$$$$73\!\cdots\!29$$$$T^{4}$$
$89$ $$( 1 + 7304106 T + 44231334895529 T^{2} )^{2}$$
$97$ $$1 - 161119742799550 T^{2} +$$$$65\!\cdots\!69$$$$T^{4}$$