# Properties

 Label 4704.2.a.bs.1.3 Level $4704$ Weight $2$ Character 4704.1 Self dual yes Analytic conductor $37.562$ Analytic rank $0$ Dimension $3$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$4704 = 2^{5} \cdot 3 \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 4704.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$37.5616291108$$ Analytic rank: $$0$$ Dimension: $$3$$ Coefficient field: 3.3.621.1 Defining polynomial: $$x^{3} - 6x - 3$$ x^3 - 6*x - 3 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 672) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.3 Root $$-2.14510$$ of defining polynomial Character $$\chi$$ $$=$$ 4704.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-1.00000 q^{3} +2.74657 q^{5} +1.00000 q^{9} +O(q^{10})$$ $$q-1.00000 q^{3} +2.74657 q^{5} +1.00000 q^{9} -1.54364 q^{11} +6.03677 q^{13} -2.74657 q^{15} +7.49314 q^{17} +6.03677 q^{19} +7.49314 q^{23} +2.54364 q^{25} -1.00000 q^{27} +1.25343 q^{29} -5.29021 q^{31} +1.54364 q^{33} +4.94950 q^{37} -6.03677 q^{39} -5.08727 q^{41} -3.45636 q^{43} +2.74657 q^{45} -9.49314 q^{47} -7.49314 q^{51} -3.83384 q^{53} -4.23970 q^{55} -6.03677 q^{57} -5.54364 q^{59} -14.5804 q^{61} +16.5804 q^{65} +4.03677 q^{67} -7.49314 q^{69} +5.49314 q^{71} +12.5436 q^{73} -2.54364 q^{75} +7.79707 q^{79} +1.00000 q^{81} +6.52991 q^{83} +20.5804 q^{85} -1.25343 q^{87} -9.49314 q^{89} +5.29021 q^{93} +16.5804 q^{95} +1.54364 q^{97} -1.54364 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q - 3 q^{3} + 3 q^{9}+O(q^{10})$$ 3 * q - 3 * q^3 + 3 * q^9 $$3 q - 3 q^{3} + 3 q^{9} - 3 q^{13} + 6 q^{17} - 3 q^{19} + 6 q^{23} + 3 q^{25} - 3 q^{27} + 12 q^{29} - 3 q^{31} + 3 q^{37} + 3 q^{39} - 6 q^{41} - 15 q^{43} - 12 q^{47} - 6 q^{51} + 6 q^{53} + 12 q^{55} + 3 q^{57} - 12 q^{59} - 18 q^{61} + 24 q^{65} - 9 q^{67} - 6 q^{69} + 33 q^{73} - 3 q^{75} + 27 q^{79} + 3 q^{81} - 18 q^{83} + 36 q^{85} - 12 q^{87} - 12 q^{89} + 3 q^{93} + 24 q^{95}+O(q^{100})$$ 3 * q - 3 * q^3 + 3 * q^9 - 3 * q^13 + 6 * q^17 - 3 * q^19 + 6 * q^23 + 3 * q^25 - 3 * q^27 + 12 * q^29 - 3 * q^31 + 3 * q^37 + 3 * q^39 - 6 * q^41 - 15 * q^43 - 12 * q^47 - 6 * q^51 + 6 * q^53 + 12 * q^55 + 3 * q^57 - 12 * q^59 - 18 * q^61 + 24 * q^65 - 9 * q^67 - 6 * q^69 + 33 * q^73 - 3 * q^75 + 27 * q^79 + 3 * q^81 - 18 * q^83 + 36 * q^85 - 12 * q^87 - 12 * q^89 + 3 * q^93 + 24 * q^95

## Coefficient data

For each $$n$$ we display the coefficients of the $$q$$-expansion $$a_n$$, the Satake parameters $$\alpha_p$$, and the Satake angles $$\theta_p = \textrm{Arg}(\alpha_p)$$.

Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 0 0
$$3$$ −1.00000 −0.577350
$$4$$ 0 0
$$5$$ 2.74657 1.22830 0.614151 0.789188i $$-0.289498\pi$$
0.614151 + 0.789188i $$0.289498\pi$$
$$6$$ 0 0
$$7$$ 0 0
$$8$$ 0 0
$$9$$ 1.00000 0.333333
$$10$$ 0 0
$$11$$ −1.54364 −0.465424 −0.232712 0.972546i $$-0.574760\pi$$
−0.232712 + 0.972546i $$0.574760\pi$$
$$12$$ 0 0
$$13$$ 6.03677 1.67430 0.837150 0.546974i $$-0.184220\pi$$
0.837150 + 0.546974i $$0.184220\pi$$
$$14$$ 0 0
$$15$$ −2.74657 −0.709161
$$16$$ 0 0
$$17$$ 7.49314 1.81735 0.908676 0.417501i $$-0.137094\pi$$
0.908676 + 0.417501i $$0.137094\pi$$
$$18$$ 0 0
$$19$$ 6.03677 1.38493 0.692465 0.721451i $$-0.256525\pi$$
0.692465 + 0.721451i $$0.256525\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 0 0
$$23$$ 7.49314 1.56243 0.781213 0.624264i $$-0.214601\pi$$
0.781213 + 0.624264i $$0.214601\pi$$
$$24$$ 0 0
$$25$$ 2.54364 0.508727
$$26$$ 0 0
$$27$$ −1.00000 −0.192450
$$28$$ 0 0
$$29$$ 1.25343 0.232756 0.116378 0.993205i $$-0.462872\pi$$
0.116378 + 0.993205i $$0.462872\pi$$
$$30$$ 0 0
$$31$$ −5.29021 −0.950149 −0.475074 0.879946i $$-0.657579\pi$$
−0.475074 + 0.879946i $$0.657579\pi$$
$$32$$ 0 0
$$33$$ 1.54364 0.268713
$$34$$ 0 0
$$35$$ 0 0
$$36$$ 0 0
$$37$$ 4.94950 0.813693 0.406846 0.913497i $$-0.366628\pi$$
0.406846 + 0.913497i $$0.366628\pi$$
$$38$$ 0 0
$$39$$ −6.03677 −0.966657
$$40$$ 0 0
$$41$$ −5.08727 −0.794499 −0.397249 0.917711i $$-0.630035\pi$$
−0.397249 + 0.917711i $$0.630035\pi$$
$$42$$ 0 0
$$43$$ −3.45636 −0.527090 −0.263545 0.964647i $$-0.584892\pi$$
−0.263545 + 0.964647i $$0.584892\pi$$
$$44$$ 0 0
$$45$$ 2.74657 0.409434
$$46$$ 0 0
$$47$$ −9.49314 −1.38472 −0.692358 0.721554i $$-0.743428\pi$$
−0.692358 + 0.721554i $$0.743428\pi$$
$$48$$ 0 0
$$49$$ 0 0
$$50$$ 0 0
$$51$$ −7.49314 −1.04925
$$52$$ 0 0
$$53$$ −3.83384 −0.526619 −0.263309 0.964711i $$-0.584814\pi$$
−0.263309 + 0.964711i $$0.584814\pi$$
$$54$$ 0 0
$$55$$ −4.23970 −0.571682
$$56$$ 0 0
$$57$$ −6.03677 −0.799590
$$58$$ 0 0
$$59$$ −5.54364 −0.721720 −0.360860 0.932620i $$-0.617517\pi$$
−0.360860 + 0.932620i $$0.617517\pi$$
$$60$$ 0 0
$$61$$ −14.5804 −1.86683 −0.933415 0.358798i $$-0.883187\pi$$
−0.933415 + 0.358798i $$0.883187\pi$$
$$62$$ 0 0
$$63$$ 0 0
$$64$$ 0 0
$$65$$ 16.5804 2.05655
$$66$$ 0 0
$$67$$ 4.03677 0.493170 0.246585 0.969121i $$-0.420691\pi$$
0.246585 + 0.969121i $$0.420691\pi$$
$$68$$ 0 0
$$69$$ −7.49314 −0.902068
$$70$$ 0 0
$$71$$ 5.49314 0.651915 0.325958 0.945384i $$-0.394313\pi$$
0.325958 + 0.945384i $$0.394313\pi$$
$$72$$ 0 0
$$73$$ 12.5436 1.46812 0.734061 0.679084i $$-0.237623\pi$$
0.734061 + 0.679084i $$0.237623\pi$$
$$74$$ 0 0
$$75$$ −2.54364 −0.293714
$$76$$ 0 0
$$77$$ 0 0
$$78$$ 0 0
$$79$$ 7.79707 0.877239 0.438619 0.898673i $$-0.355468\pi$$
0.438619 + 0.898673i $$0.355468\pi$$
$$80$$ 0 0
$$81$$ 1.00000 0.111111
$$82$$ 0 0
$$83$$ 6.52991 0.716751 0.358375 0.933578i $$-0.383331\pi$$
0.358375 + 0.933578i $$0.383331\pi$$
$$84$$ 0 0
$$85$$ 20.5804 2.23226
$$86$$ 0 0
$$87$$ −1.25343 −0.134382
$$88$$ 0 0
$$89$$ −9.49314 −1.00627 −0.503135 0.864208i $$-0.667820\pi$$
−0.503135 + 0.864208i $$0.667820\pi$$
$$90$$ 0 0
$$91$$ 0 0
$$92$$ 0 0
$$93$$ 5.29021 0.548569
$$94$$ 0 0
$$95$$ 16.5804 1.70111
$$96$$ 0 0
$$97$$ 1.54364 0.156733 0.0783663 0.996925i $$-0.475030\pi$$
0.0783663 + 0.996925i $$0.475030\pi$$
$$98$$ 0 0
$$99$$ −1.54364 −0.155141
$$100$$ 0 0
$$101$$ −2.58041 −0.256760 −0.128380 0.991725i $$-0.540978\pi$$
−0.128380 + 0.991725i $$0.540978\pi$$
$$102$$ 0 0
$$103$$ −6.54364 −0.644764 −0.322382 0.946610i $$-0.604484\pi$$
−0.322382 + 0.946610i $$0.604484\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ 4.52991 0.437923 0.218961 0.975734i $$-0.429733\pi$$
0.218961 + 0.975734i $$0.429733\pi$$
$$108$$ 0 0
$$109$$ −6.44264 −0.617093 −0.308546 0.951209i $$-0.599842\pi$$
−0.308546 + 0.951209i $$0.599842\pi$$
$$110$$ 0 0
$$111$$ −4.94950 −0.469786
$$112$$ 0 0
$$113$$ 8.00000 0.752577 0.376288 0.926503i $$-0.377200\pi$$
0.376288 + 0.926503i $$0.377200\pi$$
$$114$$ 0 0
$$115$$ 20.5804 1.91913
$$116$$ 0 0
$$117$$ 6.03677 0.558100
$$118$$ 0 0
$$119$$ 0 0
$$120$$ 0 0
$$121$$ −8.61718 −0.783380
$$122$$ 0 0
$$123$$ 5.08727 0.458704
$$124$$ 0 0
$$125$$ −6.74657 −0.603431
$$126$$ 0 0
$$127$$ −3.79707 −0.336935 −0.168468 0.985707i $$-0.553882\pi$$
−0.168468 + 0.985707i $$0.553882\pi$$
$$128$$ 0 0
$$129$$ 3.45636 0.304316
$$130$$ 0 0
$$131$$ −1.54364 −0.134868 −0.0674341 0.997724i $$-0.521481\pi$$
−0.0674341 + 0.997724i $$0.521481\pi$$
$$132$$ 0 0
$$133$$ 0 0
$$134$$ 0 0
$$135$$ −2.74657 −0.236387
$$136$$ 0 0
$$137$$ 1.49314 0.127567 0.0637836 0.997964i $$-0.479683\pi$$
0.0637836 + 0.997964i $$0.479683\pi$$
$$138$$ 0 0
$$139$$ 2.94950 0.250173 0.125087 0.992146i $$-0.460079\pi$$
0.125087 + 0.992146i $$0.460079\pi$$
$$140$$ 0 0
$$141$$ 9.49314 0.799466
$$142$$ 0 0
$$143$$ −9.31859 −0.779259
$$144$$ 0 0
$$145$$ 3.44264 0.285895
$$146$$ 0 0
$$147$$ 0 0
$$148$$ 0 0
$$149$$ −1.49314 −0.122323 −0.0611613 0.998128i $$-0.519480\pi$$
−0.0611613 + 0.998128i $$0.519480\pi$$
$$150$$ 0 0
$$151$$ −9.73284 −0.792047 −0.396024 0.918240i $$-0.629610\pi$$
−0.396024 + 0.918240i $$0.629610\pi$$
$$152$$ 0 0
$$153$$ 7.49314 0.605784
$$154$$ 0 0
$$155$$ −14.5299 −1.16707
$$156$$ 0 0
$$157$$ 9.08727 0.725243 0.362622 0.931936i $$-0.381882\pi$$
0.362622 + 0.931936i $$0.381882\pi$$
$$158$$ 0 0
$$159$$ 3.83384 0.304043
$$160$$ 0 0
$$161$$ 0 0
$$162$$ 0 0
$$163$$ −19.5667 −1.53258 −0.766290 0.642494i $$-0.777900\pi$$
−0.766290 + 0.642494i $$0.777900\pi$$
$$164$$ 0 0
$$165$$ 4.23970 0.330061
$$166$$ 0 0
$$167$$ 11.8990 0.920772 0.460386 0.887719i $$-0.347711\pi$$
0.460386 + 0.887719i $$0.347711\pi$$
$$168$$ 0 0
$$169$$ 23.4426 1.80328
$$170$$ 0 0
$$171$$ 6.03677 0.461644
$$172$$ 0 0
$$173$$ 2.58041 0.196185 0.0980925 0.995177i $$-0.468726\pi$$
0.0980925 + 0.995177i $$0.468726\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 0 0
$$177$$ 5.54364 0.416685
$$178$$ 0 0
$$179$$ 2.98627 0.223205 0.111602 0.993753i $$-0.464402\pi$$
0.111602 + 0.993753i $$0.464402\pi$$
$$180$$ 0 0
$$181$$ −10.5436 −0.783702 −0.391851 0.920029i $$-0.628165\pi$$
−0.391851 + 0.920029i $$0.628165\pi$$
$$182$$ 0 0
$$183$$ 14.5804 1.07781
$$184$$ 0 0
$$185$$ 13.5941 0.999461
$$186$$ 0 0
$$187$$ −11.5667 −0.845840
$$188$$ 0 0
$$189$$ 0 0
$$190$$ 0 0
$$191$$ −17.1608 −1.24171 −0.620857 0.783924i $$-0.713215\pi$$
−0.620857 + 0.783924i $$0.713215\pi$$
$$192$$ 0 0
$$193$$ 10.8990 0.784527 0.392264 0.919853i $$-0.371692\pi$$
0.392264 + 0.919853i $$0.371692\pi$$
$$194$$ 0 0
$$195$$ −16.5804 −1.18735
$$196$$ 0 0
$$197$$ 5.41959 0.386130 0.193065 0.981186i $$-0.438157\pi$$
0.193065 + 0.981186i $$0.438157\pi$$
$$198$$ 0 0
$$199$$ 22.9863 1.62945 0.814727 0.579845i $$-0.196887\pi$$
0.814727 + 0.579845i $$0.196887\pi$$
$$200$$ 0 0
$$201$$ −4.03677 −0.284732
$$202$$ 0 0
$$203$$ 0 0
$$204$$ 0 0
$$205$$ −13.9725 −0.975885
$$206$$ 0 0
$$207$$ 7.49314 0.520809
$$208$$ 0 0
$$209$$ −9.31859 −0.644580
$$210$$ 0 0
$$211$$ −3.08727 −0.212537 −0.106268 0.994337i $$-0.533890\pi$$
−0.106268 + 0.994337i $$0.533890\pi$$
$$212$$ 0 0
$$213$$ −5.49314 −0.376384
$$214$$ 0 0
$$215$$ −9.49314 −0.647427
$$216$$ 0 0
$$217$$ 0 0
$$218$$ 0 0
$$219$$ −12.5436 −0.847620
$$220$$ 0 0
$$221$$ 45.2344 3.04279
$$222$$ 0 0
$$223$$ 22.7466 1.52322 0.761611 0.648034i $$-0.224409\pi$$
0.761611 + 0.648034i $$0.224409\pi$$
$$224$$ 0 0
$$225$$ 2.54364 0.169576
$$226$$ 0 0
$$227$$ −22.6035 −1.50024 −0.750122 0.661299i $$-0.770005\pi$$
−0.750122 + 0.661299i $$0.770005\pi$$
$$228$$ 0 0
$$229$$ −11.3554 −0.750383 −0.375192 0.926947i $$-0.622423\pi$$
−0.375192 + 0.926947i $$0.622423\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ 3.41959 0.224025 0.112012 0.993707i $$-0.464270\pi$$
0.112012 + 0.993707i $$0.464270\pi$$
$$234$$ 0 0
$$235$$ −26.0735 −1.70085
$$236$$ 0 0
$$237$$ −7.79707 −0.506474
$$238$$ 0 0
$$239$$ 7.49314 0.484691 0.242345 0.970190i $$-0.422083\pi$$
0.242345 + 0.970190i $$0.422083\pi$$
$$240$$ 0 0
$$241$$ −18.6035 −1.19835 −0.599177 0.800617i $$-0.704505\pi$$
−0.599177 + 0.800617i $$0.704505\pi$$
$$242$$ 0 0
$$243$$ −1.00000 −0.0641500
$$244$$ 0 0
$$245$$ 0 0
$$246$$ 0 0
$$247$$ 36.4426 2.31879
$$248$$ 0 0
$$249$$ −6.52991 −0.413816
$$250$$ 0 0
$$251$$ −24.5299 −1.54831 −0.774157 0.632994i $$-0.781826\pi$$
−0.774157 + 0.632994i $$0.781826\pi$$
$$252$$ 0 0
$$253$$ −11.5667 −0.727191
$$254$$ 0 0
$$255$$ −20.5804 −1.28880
$$256$$ 0 0
$$257$$ −17.4931 −1.09119 −0.545596 0.838048i $$-0.683697\pi$$
−0.545596 + 0.838048i $$0.683697\pi$$
$$258$$ 0 0
$$259$$ 0 0
$$260$$ 0 0
$$261$$ 1.25343 0.0775855
$$262$$ 0 0
$$263$$ 13.4931 0.832022 0.416011 0.909359i $$-0.363428\pi$$
0.416011 + 0.909359i $$0.363428\pi$$
$$264$$ 0 0
$$265$$ −10.5299 −0.646847
$$266$$ 0 0
$$267$$ 9.49314 0.580971
$$268$$ 0 0
$$269$$ 6.34071 0.386600 0.193300 0.981140i $$-0.438081\pi$$
0.193300 + 0.981140i $$0.438081\pi$$
$$270$$ 0 0
$$271$$ 20.2397 1.22947 0.614737 0.788732i $$-0.289262\pi$$
0.614737 + 0.788732i $$0.289262\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ 0 0
$$275$$ −3.92645 −0.236774
$$276$$ 0 0
$$277$$ −2.54364 −0.152832 −0.0764162 0.997076i $$-0.524348\pi$$
−0.0764162 + 0.997076i $$0.524348\pi$$
$$278$$ 0 0
$$279$$ −5.29021 −0.316716
$$280$$ 0 0
$$281$$ 26.5804 1.58565 0.792827 0.609447i $$-0.208608\pi$$
0.792827 + 0.609447i $$0.208608\pi$$
$$282$$ 0 0
$$283$$ 20.5436 1.22119 0.610596 0.791942i $$-0.290930\pi$$
0.610596 + 0.791942i $$0.290930\pi$$
$$284$$ 0 0
$$285$$ −16.5804 −0.982139
$$286$$ 0 0
$$287$$ 0 0
$$288$$ 0 0
$$289$$ 39.1471 2.30277
$$290$$ 0 0
$$291$$ −1.54364 −0.0904896
$$292$$ 0 0
$$293$$ 5.25343 0.306909 0.153454 0.988156i $$-0.450960\pi$$
0.153454 + 0.988156i $$0.450960\pi$$
$$294$$ 0 0
$$295$$ −15.2260 −0.886491
$$296$$ 0 0
$$297$$ 1.54364 0.0895709
$$298$$ 0 0
$$299$$ 45.2344 2.61597
$$300$$ 0 0
$$301$$ 0 0
$$302$$ 0 0
$$303$$ 2.58041 0.148241
$$304$$ 0 0
$$305$$ −40.0461 −2.29303
$$306$$ 0 0
$$307$$ −14.8485 −0.847449 −0.423724 0.905791i $$-0.639277\pi$$
−0.423724 + 0.905791i $$0.639277\pi$$
$$308$$ 0 0
$$309$$ 6.54364 0.372255
$$310$$ 0 0
$$311$$ 1.89900 0.107682 0.0538412 0.998550i $$-0.482854\pi$$
0.0538412 + 0.998550i $$0.482854\pi$$
$$312$$ 0 0
$$313$$ −0.0872743 −0.00493303 −0.00246652 0.999997i $$-0.500785\pi$$
−0.00246652 + 0.999997i $$0.500785\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ 8.23970 0.462788 0.231394 0.972860i $$-0.425671\pi$$
0.231394 + 0.972860i $$0.425671\pi$$
$$318$$ 0 0
$$319$$ −1.93484 −0.108330
$$320$$ 0 0
$$321$$ −4.52991 −0.252835
$$322$$ 0 0
$$323$$ 45.2344 2.51691
$$324$$ 0 0
$$325$$ 15.3554 0.851762
$$326$$ 0 0
$$327$$ 6.44264 0.356279
$$328$$ 0 0
$$329$$ 0 0
$$330$$ 0 0
$$331$$ 5.52991 0.303951 0.151976 0.988384i $$-0.451436\pi$$
0.151976 + 0.988384i $$0.451436\pi$$
$$332$$ 0 0
$$333$$ 4.94950 0.271231
$$334$$ 0 0
$$335$$ 11.0873 0.605763
$$336$$ 0 0
$$337$$ 7.17455 0.390823 0.195411 0.980721i $$-0.437396\pi$$
0.195411 + 0.980721i $$0.437396\pi$$
$$338$$ 0 0
$$339$$ −8.00000 −0.434500
$$340$$ 0 0
$$341$$ 8.16616 0.442222
$$342$$ 0 0
$$343$$ 0 0
$$344$$ 0 0
$$345$$ −20.5804 −1.10801
$$346$$ 0 0
$$347$$ −24.0735 −1.29234 −0.646168 0.763195i $$-0.723629\pi$$
−0.646168 + 0.763195i $$0.723629\pi$$
$$348$$ 0 0
$$349$$ −31.1608 −1.66800 −0.834000 0.551764i $$-0.813955\pi$$
−0.834000 + 0.551764i $$0.813955\pi$$
$$350$$ 0 0
$$351$$ −6.03677 −0.322219
$$352$$ 0 0
$$353$$ −10.1745 −0.541537 −0.270768 0.962645i $$-0.587278\pi$$
−0.270768 + 0.962645i $$0.587278\pi$$
$$354$$ 0 0
$$355$$ 15.0873 0.800749
$$356$$ 0 0
$$357$$ 0 0
$$358$$ 0 0
$$359$$ 32.2481 1.70199 0.850995 0.525174i $$-0.176000\pi$$
0.850995 + 0.525174i $$0.176000\pi$$
$$360$$ 0 0
$$361$$ 17.4426 0.918033
$$362$$ 0 0
$$363$$ 8.61718 0.452285
$$364$$ 0 0
$$365$$ 34.4520 1.80330
$$366$$ 0 0
$$367$$ −3.87062 −0.202045 −0.101022 0.994884i $$-0.532211\pi$$
−0.101022 + 0.994884i $$0.532211\pi$$
$$368$$ 0 0
$$369$$ −5.08727 −0.264833
$$370$$ 0 0
$$371$$ 0 0
$$372$$ 0 0
$$373$$ 34.6907 1.79622 0.898109 0.439773i $$-0.144941\pi$$
0.898109 + 0.439773i $$0.144941\pi$$
$$374$$ 0 0
$$375$$ 6.74657 0.348391
$$376$$ 0 0
$$377$$ 7.56668 0.389704
$$378$$ 0 0
$$379$$ −13.4564 −0.691207 −0.345603 0.938381i $$-0.612326\pi$$
−0.345603 + 0.938381i $$0.612326\pi$$
$$380$$ 0 0
$$381$$ 3.79707 0.194530
$$382$$ 0 0
$$383$$ −4.17455 −0.213309 −0.106655 0.994296i $$-0.534014\pi$$
−0.106655 + 0.994296i $$0.534014\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ −3.45636 −0.175697
$$388$$ 0 0
$$389$$ −18.6540 −0.945793 −0.472897 0.881118i $$-0.656792\pi$$
−0.472897 + 0.881118i $$0.656792\pi$$
$$390$$ 0 0
$$391$$ 56.1471 2.83948
$$392$$ 0 0
$$393$$ 1.54364 0.0778662
$$394$$ 0 0
$$395$$ 21.4152 1.07751
$$396$$ 0 0
$$397$$ −1.45636 −0.0730928 −0.0365464 0.999332i $$-0.511636\pi$$
−0.0365464 + 0.999332i $$0.511636\pi$$
$$398$$ 0 0
$$399$$ 0 0
$$400$$ 0 0
$$401$$ −29.2344 −1.45989 −0.729947 0.683503i $$-0.760455\pi$$
−0.729947 + 0.683503i $$0.760455\pi$$
$$402$$ 0 0
$$403$$ −31.9358 −1.59083
$$404$$ 0 0
$$405$$ 2.74657 0.136478
$$406$$ 0 0
$$407$$ −7.64023 −0.378712
$$408$$ 0 0
$$409$$ 26.0598 1.28858 0.644288 0.764783i $$-0.277154\pi$$
0.644288 + 0.764783i $$0.277154\pi$$
$$410$$ 0 0
$$411$$ −1.49314 −0.0736510
$$412$$ 0 0
$$413$$ 0 0
$$414$$ 0 0
$$415$$ 17.9348 0.880387
$$416$$ 0 0
$$417$$ −2.94950 −0.144438
$$418$$ 0 0
$$419$$ −32.8853 −1.60655 −0.803275 0.595608i $$-0.796911\pi$$
−0.803275 + 0.595608i $$0.796911\pi$$
$$420$$ 0 0
$$421$$ −17.1976 −0.838160 −0.419080 0.907949i $$-0.637647\pi$$
−0.419080 + 0.907949i $$0.637647\pi$$
$$422$$ 0 0
$$423$$ −9.49314 −0.461572
$$424$$ 0 0
$$425$$ 19.0598 0.924537
$$426$$ 0 0
$$427$$ 0 0
$$428$$ 0 0
$$429$$ 9.31859 0.449906
$$430$$ 0 0
$$431$$ −2.98627 −0.143844 −0.0719219 0.997410i $$-0.522913\pi$$
−0.0719219 + 0.997410i $$0.522913\pi$$
$$432$$ 0 0
$$433$$ 18.4426 0.886297 0.443148 0.896448i $$-0.353862\pi$$
0.443148 + 0.896448i $$0.353862\pi$$
$$434$$ 0 0
$$435$$ −3.44264 −0.165062
$$436$$ 0 0
$$437$$ 45.2344 2.16385
$$438$$ 0 0
$$439$$ −0.572020 −0.0273010 −0.0136505 0.999907i $$-0.504345\pi$$
−0.0136505 + 0.999907i $$0.504345\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 0 0
$$443$$ −36.5299 −1.73559 −0.867794 0.496924i $$-0.834463\pi$$
−0.867794 + 0.496924i $$0.834463\pi$$
$$444$$ 0 0
$$445$$ −26.0735 −1.23600
$$446$$ 0 0
$$447$$ 1.49314 0.0706229
$$448$$ 0 0
$$449$$ −22.5530 −1.06434 −0.532170 0.846638i $$-0.678623\pi$$
−0.532170 + 0.846638i $$0.678623\pi$$
$$450$$ 0 0
$$451$$ 7.85291 0.369779
$$452$$ 0 0
$$453$$ 9.73284 0.457289
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ 20.2344 0.946524 0.473262 0.880922i $$-0.343076\pi$$
0.473262 + 0.880922i $$0.343076\pi$$
$$458$$ 0 0
$$459$$ −7.49314 −0.349750
$$460$$ 0 0
$$461$$ 25.4931 1.18733 0.593667 0.804711i $$-0.297680\pi$$
0.593667 + 0.804711i $$0.297680\pi$$
$$462$$ 0 0
$$463$$ 7.70446 0.358057 0.179028 0.983844i $$-0.442705\pi$$
0.179028 + 0.983844i $$0.442705\pi$$
$$464$$ 0 0
$$465$$ 14.5299 0.673808
$$466$$ 0 0
$$467$$ −36.1471 −1.67269 −0.836344 0.548205i $$-0.815311\pi$$
−0.836344 + 0.548205i $$0.815311\pi$$
$$468$$ 0 0
$$469$$ 0 0
$$470$$ 0 0
$$471$$ −9.08727 −0.418719
$$472$$ 0 0
$$473$$ 5.33537 0.245321
$$474$$ 0 0
$$475$$ 15.3554 0.704552
$$476$$ 0 0
$$477$$ −3.83384 −0.175540
$$478$$ 0 0
$$479$$ −26.5804 −1.21449 −0.607245 0.794515i $$-0.707725\pi$$
−0.607245 + 0.794515i $$0.707725\pi$$
$$480$$ 0 0
$$481$$ 29.8790 1.36237
$$482$$ 0 0
$$483$$ 0 0
$$484$$ 0 0
$$485$$ 4.23970 0.192515
$$486$$ 0 0
$$487$$ 40.4236 1.83177 0.915883 0.401444i $$-0.131492\pi$$
0.915883 + 0.401444i $$0.131492\pi$$
$$488$$ 0 0
$$489$$ 19.5667 0.884836
$$490$$ 0 0
$$491$$ −14.6309 −0.660284 −0.330142 0.943931i $$-0.607097\pi$$
−0.330142 + 0.943931i $$0.607097\pi$$
$$492$$ 0 0
$$493$$ 9.39214 0.423000
$$494$$ 0 0
$$495$$ −4.23970 −0.190561
$$496$$ 0 0
$$497$$ 0 0
$$498$$ 0 0
$$499$$ 36.1103 1.61652 0.808260 0.588826i $$-0.200410\pi$$
0.808260 + 0.588826i $$0.200410\pi$$
$$500$$ 0 0
$$501$$ −11.8990 −0.531608
$$502$$ 0 0
$$503$$ −19.4931 −0.869156 −0.434578 0.900634i $$-0.643102\pi$$
−0.434578 + 0.900634i $$0.643102\pi$$
$$504$$ 0 0
$$505$$ −7.08727 −0.315380
$$506$$ 0 0
$$507$$ −23.4426 −1.04112
$$508$$ 0 0
$$509$$ −23.9074 −1.05968 −0.529838 0.848099i $$-0.677747\pi$$
−0.529838 + 0.848099i $$0.677747\pi$$
$$510$$ 0 0
$$511$$ 0 0
$$512$$ 0 0
$$513$$ −6.03677 −0.266530
$$514$$ 0 0
$$515$$ −17.9725 −0.791965
$$516$$ 0 0
$$517$$ 14.6540 0.644480
$$518$$ 0 0
$$519$$ −2.58041 −0.113267
$$520$$ 0 0
$$521$$ −7.08727 −0.310499 −0.155250 0.987875i $$-0.549618\pi$$
−0.155250 + 0.987875i $$0.549618\pi$$
$$522$$ 0 0
$$523$$ 28.7475 1.25704 0.628520 0.777793i $$-0.283661\pi$$
0.628520 + 0.777793i $$0.283661\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ −39.6402 −1.72676
$$528$$ 0 0
$$529$$ 33.1471 1.44118
$$530$$ 0 0
$$531$$ −5.54364 −0.240573
$$532$$ 0 0
$$533$$ −30.7107 −1.33023
$$534$$ 0 0
$$535$$ 12.4417 0.537902
$$536$$ 0 0
$$537$$ −2.98627 −0.128867
$$538$$ 0 0
$$539$$ 0 0
$$540$$ 0 0
$$541$$ −7.02305 −0.301944 −0.150972 0.988538i $$-0.548240\pi$$
−0.150972 + 0.988538i $$0.548240\pi$$
$$542$$ 0 0
$$543$$ 10.5436 0.452471
$$544$$ 0 0
$$545$$ −17.6951 −0.757976
$$546$$ 0 0
$$547$$ −5.59414 −0.239188 −0.119594 0.992823i $$-0.538159\pi$$
−0.119594 + 0.992823i $$0.538159\pi$$
$$548$$ 0 0
$$549$$ −14.5804 −0.622277
$$550$$ 0 0
$$551$$ 7.56668 0.322352
$$552$$ 0 0
$$553$$ 0 0
$$554$$ 0 0
$$555$$ −13.5941 −0.577039
$$556$$ 0 0
$$557$$ 28.6456 1.21375 0.606876 0.794797i $$-0.292423\pi$$
0.606876 + 0.794797i $$0.292423\pi$$
$$558$$ 0 0
$$559$$ −20.8653 −0.882507
$$560$$ 0 0
$$561$$ 11.5667 0.488346
$$562$$ 0 0
$$563$$ 19.8927 0.838379 0.419189 0.907899i $$-0.362314\pi$$
0.419189 + 0.907899i $$0.362314\pi$$
$$564$$ 0 0
$$565$$ 21.9725 0.924392
$$566$$ 0 0
$$567$$ 0 0
$$568$$ 0 0
$$569$$ 35.6402 1.49412 0.747058 0.664759i $$-0.231466\pi$$
0.747058 + 0.664759i $$0.231466\pi$$
$$570$$ 0 0
$$571$$ 27.6770 1.15825 0.579123 0.815240i $$-0.303395\pi$$
0.579123 + 0.815240i $$0.303395\pi$$
$$572$$ 0 0
$$573$$ 17.1608 0.716904
$$574$$ 0 0
$$575$$ 19.0598 0.794849
$$576$$ 0 0
$$577$$ −6.01373 −0.250355 −0.125177 0.992134i $$-0.539950\pi$$
−0.125177 + 0.992134i $$0.539950\pi$$
$$578$$ 0 0
$$579$$ −10.8990 −0.452947
$$580$$ 0 0
$$581$$ 0 0
$$582$$ 0 0
$$583$$ 5.91806 0.245101
$$584$$ 0 0
$$585$$ 16.5804 0.685516
$$586$$ 0 0
$$587$$ −15.5436 −0.641555 −0.320777 0.947155i $$-0.603944\pi$$
−0.320777 + 0.947155i $$0.603944\pi$$
$$588$$ 0 0
$$589$$ −31.9358 −1.31589
$$590$$ 0 0
$$591$$ −5.41959 −0.222932
$$592$$ 0 0
$$593$$ −48.6540 −1.99798 −0.998989 0.0449488i $$-0.985688\pi$$
−0.998989 + 0.0449488i $$0.985688\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 0 0
$$597$$ −22.9863 −0.940766
$$598$$ 0 0
$$599$$ 20.9127 0.854471 0.427235 0.904140i $$-0.359488\pi$$
0.427235 + 0.904140i $$0.359488\pi$$
$$600$$ 0 0
$$601$$ −7.98627 −0.325767 −0.162883 0.986645i $$-0.552079\pi$$
−0.162883 + 0.986645i $$0.552079\pi$$
$$602$$ 0 0
$$603$$ 4.03677 0.164390
$$604$$ 0 0
$$605$$ −23.6677 −0.962228
$$606$$ 0 0
$$607$$ −2.70979 −0.109987 −0.0549936 0.998487i $$-0.517514\pi$$
−0.0549936 + 0.998487i $$0.517514\pi$$
$$608$$ 0 0
$$609$$ 0 0
$$610$$ 0 0
$$611$$ −57.3079 −2.31843
$$612$$ 0 0
$$613$$ −16.5069 −0.666706 −0.333353 0.942802i $$-0.608180\pi$$
−0.333353 + 0.942802i $$0.608180\pi$$
$$614$$ 0 0
$$615$$ 13.9725 0.563427
$$616$$ 0 0
$$617$$ −23.1608 −0.932420 −0.466210 0.884674i $$-0.654381\pi$$
−0.466210 + 0.884674i $$0.654381\pi$$
$$618$$ 0 0
$$619$$ 29.4289 1.18285 0.591424 0.806361i $$-0.298566\pi$$
0.591424 + 0.806361i $$0.298566\pi$$
$$620$$ 0 0
$$621$$ −7.49314 −0.300689
$$622$$ 0 0
$$623$$ 0 0
$$624$$ 0 0
$$625$$ −31.2481 −1.24992
$$626$$ 0 0
$$627$$ 9.31859 0.372149
$$628$$ 0 0
$$629$$ 37.0873 1.47877
$$630$$ 0 0
$$631$$ 35.7054 1.42141 0.710705 0.703491i $$-0.248376\pi$$
0.710705 + 0.703491i $$0.248376\pi$$
$$632$$ 0 0
$$633$$ 3.08727 0.122708
$$634$$ 0 0
$$635$$ −10.4289 −0.413859
$$636$$ 0 0
$$637$$ 0 0
$$638$$ 0 0
$$639$$ 5.49314 0.217305
$$640$$ 0 0
$$641$$ −3.36282 −0.132824 −0.0664118 0.997792i $$-0.521155\pi$$
−0.0664118 + 0.997792i $$0.521155\pi$$
$$642$$ 0 0
$$643$$ −10.2113 −0.402695 −0.201348 0.979520i $$-0.564532\pi$$
−0.201348 + 0.979520i $$0.564532\pi$$
$$644$$ 0 0
$$645$$ 9.49314 0.373792
$$646$$ 0 0
$$647$$ 20.5530 0.808020 0.404010 0.914755i $$-0.367616\pi$$
0.404010 + 0.914755i $$0.367616\pi$$
$$648$$ 0 0
$$649$$ 8.55736 0.335906
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 0 0
$$653$$ −11.3544 −0.444333 −0.222167 0.975009i $$-0.571313\pi$$
−0.222167 + 0.975009i $$0.571313\pi$$
$$654$$ 0 0
$$655$$ −4.23970 −0.165659
$$656$$ 0 0
$$657$$ 12.5436 0.489374
$$658$$ 0 0
$$659$$ −21.2344 −0.827174 −0.413587 0.910465i $$-0.635724\pi$$
−0.413587 + 0.910465i $$0.635724\pi$$
$$660$$ 0 0
$$661$$ −28.9220 −1.12494 −0.562469 0.826819i $$-0.690148\pi$$
−0.562469 + 0.826819i $$0.690148\pi$$
$$662$$ 0 0
$$663$$ −45.2344 −1.75676
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 0 0
$$667$$ 9.39214 0.363665
$$668$$ 0 0
$$669$$ −22.7466 −0.879433
$$670$$ 0 0
$$671$$ 22.5069 0.868868
$$672$$ 0 0
$$673$$ −34.2618 −1.32070 −0.660348 0.750960i $$-0.729591\pi$$
−0.660348 + 0.750960i $$0.729591\pi$$
$$674$$ 0 0
$$675$$ −2.54364 −0.0979046
$$676$$ 0 0
$$677$$ 34.7466 1.33542 0.667710 0.744422i $$-0.267275\pi$$
0.667710 + 0.744422i $$0.267275\pi$$
$$678$$ 0 0
$$679$$ 0 0
$$680$$ 0 0
$$681$$ 22.6035 0.866166
$$682$$ 0 0
$$683$$ −0.281814 −0.0107833 −0.00539166 0.999985i $$-0.501716\pi$$
−0.00539166 + 0.999985i $$0.501716\pi$$
$$684$$ 0 0
$$685$$ 4.10100 0.156691
$$686$$ 0 0
$$687$$ 11.3554 0.433234
$$688$$ 0 0
$$689$$ −23.1440 −0.881718
$$690$$ 0 0
$$691$$ −34.4152 −1.30922 −0.654608 0.755969i $$-0.727166\pi$$
−0.654608 + 0.755969i $$0.727166\pi$$
$$692$$ 0 0
$$693$$ 0 0
$$694$$ 0 0
$$695$$ 8.10100 0.307288
$$696$$ 0 0
$$697$$ −38.1196 −1.44388
$$698$$ 0 0
$$699$$ −3.41959 −0.129341
$$700$$ 0 0
$$701$$ −0.498472 −0.0188270 −0.00941352 0.999956i $$-0.502996\pi$$
−0.00941352 + 0.999956i $$0.502996\pi$$
$$702$$ 0 0
$$703$$ 29.8790 1.12691
$$704$$ 0 0
$$705$$ 26.0735 0.981987
$$706$$ 0 0
$$707$$ 0 0
$$708$$ 0 0
$$709$$ −4.17455 −0.156778 −0.0783892 0.996923i $$-0.524978\pi$$
−0.0783892 + 0.996923i $$0.524978\pi$$
$$710$$ 0 0
$$711$$ 7.79707 0.292413
$$712$$ 0 0
$$713$$ −39.6402 −1.48454
$$714$$ 0 0
$$715$$ −25.5941 −0.957166
$$716$$ 0 0
$$717$$ −7.49314 −0.279836
$$718$$ 0 0
$$719$$ 0.986273 0.0367818 0.0183909 0.999831i $$-0.494146\pi$$
0.0183909 + 0.999831i $$0.494146\pi$$
$$720$$ 0 0
$$721$$ 0 0
$$722$$ 0 0
$$723$$ 18.6035 0.691870
$$724$$ 0 0
$$725$$ 3.18828 0.118410
$$726$$ 0 0
$$727$$ 34.9304 1.29550 0.647749 0.761854i $$-0.275711\pi$$
0.647749 + 0.761854i $$0.275711\pi$$
$$728$$ 0 0
$$729$$ 1.00000 0.0370370
$$730$$ 0 0
$$731$$ −25.8990 −0.957909
$$732$$ 0 0
$$733$$ 5.12405 0.189261 0.0946305 0.995512i $$-0.469833\pi$$
0.0946305 + 0.995512i $$0.469833\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ −6.23131 −0.229533
$$738$$ 0 0
$$739$$ 16.9220 0.622488 0.311244 0.950330i $$-0.399254\pi$$
0.311244 + 0.950330i $$0.399254\pi$$
$$740$$ 0 0
$$741$$ −36.4426 −1.33875
$$742$$ 0 0
$$743$$ −6.81172 −0.249898 −0.124949 0.992163i $$-0.539877\pi$$
−0.124949 + 0.992163i $$0.539877\pi$$
$$744$$ 0 0
$$745$$ −4.10100 −0.150249
$$746$$ 0 0
$$747$$ 6.52991 0.238917
$$748$$ 0 0
$$749$$ 0 0
$$750$$ 0 0
$$751$$ 26.1755 0.955157 0.477578 0.878589i $$-0.341515\pi$$
0.477578 + 0.878589i $$0.341515\pi$$
$$752$$ 0 0
$$753$$ 24.5299 0.893920
$$754$$ 0 0
$$755$$ −26.7319 −0.972874
$$756$$ 0 0
$$757$$ 49.7138 1.80688 0.903439 0.428717i $$-0.141034\pi$$
0.903439 + 0.428717i $$0.141034\pi$$
$$758$$ 0 0
$$759$$ 11.5667 0.419844
$$760$$ 0 0
$$761$$ −12.6540 −0.458706 −0.229353 0.973343i $$-0.573661\pi$$
−0.229353 + 0.973343i $$0.573661\pi$$
$$762$$ 0 0
$$763$$ 0 0
$$764$$ 0 0
$$765$$ 20.5804 0.744086
$$766$$ 0 0
$$767$$ −33.4657 −1.20838
$$768$$ 0 0
$$769$$ −20.2344 −0.729670 −0.364835 0.931072i $$-0.618875\pi$$
−0.364835 + 0.931072i $$0.618875\pi$$
$$770$$ 0 0
$$771$$ 17.4931 0.630000
$$772$$ 0 0
$$773$$ −19.7412 −0.710043 −0.355021 0.934858i $$-0.615526\pi$$
−0.355021 + 0.934858i $$0.615526\pi$$
$$774$$ 0 0
$$775$$ −13.4564 −0.483367
$$776$$ 0 0
$$777$$ 0 0
$$778$$ 0 0
$$779$$ −30.7107 −1.10033
$$780$$ 0 0
$$781$$ −8.47941 −0.303417
$$782$$ 0 0
$$783$$ −1.25343 −0.0447940
$$784$$ 0 0
$$785$$ 24.9588 0.890818
$$786$$ 0 0
$$787$$ 16.9127 0.602874 0.301437 0.953486i $$-0.402534\pi$$
0.301437 + 0.953486i $$0.402534\pi$$
$$788$$ 0 0
$$789$$ −13.4931 −0.480368
$$790$$ 0 0
$$791$$ 0 0
$$792$$ 0 0
$$793$$ −88.0186 −3.12563
$$794$$ 0 0
$$795$$ 10.5299 0.373457
$$796$$ 0 0
$$797$$ 3.48475 0.123436 0.0617180 0.998094i $$-0.480342\pi$$
0.0617180 + 0.998094i $$0.480342\pi$$
$$798$$ 0 0
$$799$$ −71.1334 −2.51652
$$800$$ 0 0
$$801$$ −9.49314 −0.335423
$$802$$ 0 0
$$803$$ −19.3628 −0.683299
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 0 0
$$807$$ −6.34071 −0.223203
$$808$$ 0 0
$$809$$ −19.1883 −0.674624 −0.337312 0.941393i $$-0.609518\pi$$
−0.337312 + 0.941393i $$0.609518\pi$$
$$810$$ 0 0
$$811$$ 48.8285 1.71460 0.857300 0.514817i $$-0.172140\pi$$
0.857300 + 0.514817i $$0.172140\pi$$
$$812$$ 0 0
$$813$$ −20.2397 −0.709837
$$814$$ 0 0
$$815$$ −53.7412 −1.88247
$$816$$ 0 0
$$817$$ −20.8653 −0.729984
$$818$$ 0 0
$$819$$ 0 0
$$820$$ 0 0
$$821$$ −26.5446 −0.926412 −0.463206 0.886251i $$-0.653301\pi$$
−0.463206 + 0.886251i $$0.653301\pi$$
$$822$$ 0 0
$$823$$ −16.9588 −0.591147 −0.295574 0.955320i $$-0.595511\pi$$
−0.295574 + 0.955320i $$0.595511\pi$$
$$824$$ 0 0
$$825$$ 3.92645 0.136702
$$826$$ 0 0
$$827$$ −42.4289 −1.47540 −0.737699 0.675130i $$-0.764088\pi$$
−0.737699 + 0.675130i $$0.764088\pi$$
$$828$$ 0 0
$$829$$ −31.9358 −1.10918 −0.554588 0.832125i $$-0.687124\pi$$
−0.554588 + 0.832125i $$0.687124\pi$$
$$830$$ 0 0
$$831$$ 2.54364 0.0882378
$$832$$ 0 0
$$833$$ 0 0
$$834$$ 0 0
$$835$$ 32.6814 1.13099
$$836$$ 0 0
$$837$$ 5.29021 0.182856
$$838$$ 0 0
$$839$$ −49.5392 −1.71028 −0.855142 0.518394i $$-0.826530\pi$$
−0.855142 + 0.518394i $$0.826530\pi$$
$$840$$ 0 0
$$841$$ −27.4289 −0.945824
$$842$$ 0 0
$$843$$ −26.5804 −0.915478
$$844$$ 0 0
$$845$$ 64.3868 2.21497
$$846$$ 0 0
$$847$$ 0 0
$$848$$ 0 0
$$849$$ −20.5436 −0.705056
$$850$$ 0 0
$$851$$ 37.0873 1.27134
$$852$$ 0 0
$$853$$ 11.4564 0.392258 0.196129 0.980578i $$-0.437163\pi$$
0.196129 + 0.980578i $$0.437163\pi$$
$$854$$ 0 0
$$855$$ 16.5804 0.567038
$$856$$ 0 0
$$857$$ −32.2481 −1.10157 −0.550787 0.834646i $$-0.685672\pi$$
−0.550787 + 0.834646i $$0.685672\pi$$
$$858$$ 0 0
$$859$$ 3.08727 0.105336 0.0526682 0.998612i $$-0.483227\pi$$
0.0526682 + 0.998612i $$0.483227\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 0 0
$$863$$ 18.1471 0.617734 0.308867 0.951105i $$-0.400050\pi$$
0.308867 + 0.951105i $$0.400050\pi$$
$$864$$ 0 0
$$865$$ 7.08727 0.240975
$$866$$ 0 0
$$867$$ −39.1471 −1.32951
$$868$$ 0 0
$$869$$ −12.0358 −0.408288
$$870$$ 0 0
$$871$$ 24.3691 0.825715
$$872$$ 0 0
$$873$$ 1.54364 0.0522442
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 0 0
$$877$$ 8.88527 0.300034 0.150017 0.988683i $$-0.452067\pi$$
0.150017 + 0.988683i $$0.452067\pi$$
$$878$$ 0 0
$$879$$ −5.25343 −0.177194
$$880$$ 0 0
$$881$$ −41.5667 −1.40042 −0.700209 0.713938i $$-0.746910\pi$$
−0.700209 + 0.713938i $$0.746910\pi$$
$$882$$ 0 0
$$883$$ 28.7182 0.966444 0.483222 0.875498i $$-0.339466\pi$$
0.483222 + 0.875498i $$0.339466\pi$$
$$884$$ 0 0
$$885$$ 15.2260 0.511816
$$886$$ 0 0
$$887$$ 41.8148 1.40400 0.702001 0.712176i $$-0.252290\pi$$
0.702001 + 0.712176i $$0.252290\pi$$
$$888$$ 0 0
$$889$$ 0 0
$$890$$ 0 0
$$891$$ −1.54364 −0.0517138
$$892$$ 0 0
$$893$$ −57.3079 −1.91774
$$894$$ 0 0
$$895$$ 8.20200 0.274163
$$896$$ 0 0
$$897$$ −45.2344 −1.51033
$$898$$ 0 0
$$899$$ −6.63091 −0.221153
$$900$$ 0 0
$$901$$ −28.7275 −0.957052
$$902$$ 0 0
$$903$$ 0 0
$$904$$ 0 0
$$905$$ −28.9588 −0.962624
$$906$$ 0 0
$$907$$ −57.1701 −1.89830 −0.949152 0.314819i $$-0.898056\pi$$
−0.949152 + 0.314819i $$0.898056\pi$$
$$908$$ 0 0
$$909$$ −2.58041 −0.0855868
$$910$$ 0 0
$$911$$ −24.4059 −0.808602 −0.404301 0.914626i $$-0.632485\pi$$
−0.404301 + 0.914626i $$0.632485\pi$$
$$912$$ 0 0
$$913$$ −10.0798 −0.333593
$$914$$ 0 0
$$915$$ 40.0461 1.32388
$$916$$ 0 0
$$917$$ 0 0
$$918$$ 0 0
$$919$$ 47.7045 1.57362 0.786812 0.617192i $$-0.211730\pi$$
0.786812 + 0.617192i $$0.211730\pi$$
$$920$$ 0 0
$$921$$ 14.8485 0.489275
$$922$$ 0 0
$$923$$ 33.1608 1.09150
$$924$$ 0 0
$$925$$ 12.5897 0.413948
$$926$$ 0 0
$$927$$ −6.54364 −0.214921
$$928$$ 0 0
$$929$$ 39.2344 1.28724 0.643619 0.765346i $$-0.277432\pi$$
0.643619 + 0.765346i $$0.277432\pi$$
$$930$$ 0 0
$$931$$ 0 0
$$932$$ 0 0
$$933$$ −1.89900 −0.0621704
$$934$$ 0 0
$$935$$ −31.7687 −1.03895
$$936$$ 0 0
$$937$$ 15.3491 0.501433 0.250717 0.968061i $$-0.419334\pi$$
0.250717 + 0.968061i $$0.419334\pi$$
$$938$$ 0 0
$$939$$ 0.0872743 0.00284809
$$940$$ 0 0
$$941$$ −12.1662 −0.396605 −0.198303 0.980141i $$-0.563543\pi$$
−0.198303 + 0.980141i $$0.563543\pi$$
$$942$$ 0 0
$$943$$ −38.1196 −1.24135
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ −19.0598 −0.619361 −0.309680 0.950841i $$-0.600222\pi$$
−0.309680 + 0.950841i $$0.600222\pi$$
$$948$$ 0 0
$$949$$ 75.7231 2.45808
$$950$$ 0 0
$$951$$ −8.23970 −0.267191
$$952$$ 0 0
$$953$$ 17.1334 0.555004 0.277502 0.960725i $$-0.410493\pi$$
0.277502 + 0.960725i $$0.410493\pi$$
$$954$$ 0 0
$$955$$ −47.1334 −1.52520
$$956$$ 0 0
$$957$$ 1.93484 0.0625446
$$958$$ 0 0
$$959$$ 0 0
$$960$$ 0 0
$$961$$ −3.01373 −0.0972170
$$962$$ 0 0
$$963$$ 4.52991 0.145974
$$964$$ 0 0
$$965$$ 29.9348 0.963637
$$966$$ 0 0
$$967$$ 26.7833 0.861294 0.430647 0.902520i $$-0.358285\pi$$
0.430647 + 0.902520i $$0.358285\pi$$
$$968$$ 0 0
$$969$$ −45.2344 −1.45314
$$970$$ 0 0
$$971$$ 11.5162 0.369572 0.184786 0.982779i $$-0.440841\pi$$
0.184786 + 0.982779i $$0.440841\pi$$
$$972$$ 0 0
$$973$$ 0 0
$$974$$ 0 0
$$975$$ −15.3554 −0.491765
$$976$$ 0 0
$$977$$ 21.3354 0.682579 0.341289 0.939958i $$-0.389136\pi$$
0.341289 + 0.939958i $$0.389136\pi$$
$$978$$ 0 0
$$979$$ 14.6540 0.468343
$$980$$ 0 0
$$981$$ −6.44264 −0.205698
$$982$$ 0 0
$$983$$ −50.3952 −1.60736 −0.803678 0.595064i $$-0.797127\pi$$
−0.803678 + 0.595064i $$0.797127\pi$$
$$984$$ 0 0
$$985$$ 14.8853 0.474284
$$986$$ 0 0
$$987$$ 0 0
$$988$$ 0 0
$$989$$ −25.8990 −0.823540
$$990$$ 0 0
$$991$$ 12.6823 0.402868 0.201434 0.979502i $$-0.435440\pi$$
0.201434 + 0.979502i $$0.435440\pi$$
$$992$$ 0 0
$$993$$ −5.52991 −0.175486
$$994$$ 0 0
$$995$$ 63.1334 2.00146
$$996$$ 0 0
$$997$$ −40.3416 −1.27763 −0.638816 0.769359i $$-0.720576\pi$$
−0.638816 + 0.769359i $$0.720576\pi$$
$$998$$ 0 0
$$999$$ −4.94950 −0.156595
Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000

## Twists

By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 4704.2.a.bs.1.3 3
4.3 odd 2 4704.2.a.bu.1.3 3
7.2 even 3 672.2.q.l.193.1 yes 6
7.4 even 3 672.2.q.l.289.1 yes 6
7.6 odd 2 4704.2.a.bv.1.1 3
8.3 odd 2 9408.2.a.eh.1.1 3
8.5 even 2 9408.2.a.ej.1.1 3
21.2 odd 6 2016.2.s.v.865.3 6
21.11 odd 6 2016.2.s.v.289.3 6
28.11 odd 6 672.2.q.k.289.1 yes 6
28.23 odd 6 672.2.q.k.193.1 6
28.27 even 2 4704.2.a.bt.1.1 3
56.11 odd 6 1344.2.q.z.961.3 6
56.13 odd 2 9408.2.a.eg.1.3 3
56.27 even 2 9408.2.a.ei.1.3 3
56.37 even 6 1344.2.q.y.193.3 6
56.51 odd 6 1344.2.q.z.193.3 6
56.53 even 6 1344.2.q.y.961.3 6
84.11 even 6 2016.2.s.u.289.3 6
84.23 even 6 2016.2.s.u.865.3 6

By twisted newform
Twist Min Dim Char Parity Ord Type
672.2.q.k.193.1 6 28.23 odd 6
672.2.q.k.289.1 yes 6 28.11 odd 6
672.2.q.l.193.1 yes 6 7.2 even 3
672.2.q.l.289.1 yes 6 7.4 even 3
1344.2.q.y.193.3 6 56.37 even 6
1344.2.q.y.961.3 6 56.53 even 6
1344.2.q.z.193.3 6 56.51 odd 6
1344.2.q.z.961.3 6 56.11 odd 6
2016.2.s.u.289.3 6 84.11 even 6
2016.2.s.u.865.3 6 84.23 even 6
2016.2.s.v.289.3 6 21.11 odd 6
2016.2.s.v.865.3 6 21.2 odd 6
4704.2.a.bs.1.3 3 1.1 even 1 trivial
4704.2.a.bt.1.1 3 28.27 even 2
4704.2.a.bu.1.3 3 4.3 odd 2
4704.2.a.bv.1.1 3 7.6 odd 2
9408.2.a.eg.1.3 3 56.13 odd 2
9408.2.a.eh.1.1 3 8.3 odd 2
9408.2.a.ei.1.3 3 56.27 even 2
9408.2.a.ej.1.1 3 8.5 even 2