# Properties

 Label 4704.2.a.bv.1.1 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 1.1 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 $$-0.710502\pi$$
−0.614151 + 0.789188i $$0.710502\pi$$
$$6$$ 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 $$-0.815780\pi$$
−0.837150 + 0.546974i $$0.815780\pi$$
$$14$$ 0 0
$$15$$ −2.74657 −0.709161
$$16$$ 0 0
$$17$$ −7.49314 −1.81735 −0.908676 0.417501i $$-0.862906\pi$$
−0.908676 + 0.417501i $$0.862906\pi$$
$$18$$ 0 0
$$19$$ −6.03677 −1.38493 −0.692465 0.721451i $$-0.743475\pi$$
−0.692465 + 0.721451i $$0.743475\pi$$
$$20$$ 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 $$-0.342421\pi$$
0.475074 + 0.879946i $$0.342421\pi$$
$$32$$ 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 $$-0.369965\pi$$
0.397249 + 0.917711i $$0.369965\pi$$
$$42$$ 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 $$-0.256572\pi$$
0.692358 + 0.721554i $$0.256572\pi$$
$$48$$ 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 $$-0.382483\pi$$
0.360860 + 0.932620i $$0.382483\pi$$
$$60$$ 0 0
$$61$$ 14.5804 1.86683 0.933415 0.358798i $$-0.116813\pi$$
0.933415 + 0.358798i $$0.116813\pi$$
$$62$$ 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 $$-0.762377\pi$$
−0.734061 + 0.679084i $$0.762377\pi$$
$$74$$ 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 $$-0.616669\pi$$
−0.358375 + 0.933578i $$0.616669\pi$$
$$84$$ 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 $$-0.332180\pi$$
0.503135 + 0.864208i $$0.332180\pi$$
$$90$$ 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 $$-0.524970\pi$$
−0.0783663 + 0.996925i $$0.524970\pi$$
$$98$$ 0 0
$$99$$ −1.54364 −0.155141
$$100$$ 0 0
$$101$$ 2.58041 0.256760 0.128380 0.991725i $$-0.459022\pi$$
0.128380 + 0.991725i $$0.459022\pi$$
$$102$$ 0 0
$$103$$ 6.54364 0.644764 0.322382 0.946610i $$-0.395516\pi$$
0.322382 + 0.946610i $$0.395516\pi$$
$$104$$ 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 $$-0.478519\pi$$
0.0674341 + 0.997724i $$0.478519\pi$$
$$132$$ 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 $$-0.539921\pi$$
−0.125087 + 0.992146i $$0.539921\pi$$
$$140$$ 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 $$-0.618118\pi$$
−0.362622 + 0.931936i $$0.618118\pi$$
$$158$$ 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 $$-0.652289\pi$$
−0.460386 + 0.887719i $$0.652289\pi$$
$$168$$ 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 $$-0.531274\pi$$
−0.0980925 + 0.995177i $$0.531274\pi$$
$$174$$ 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 $$-0.371835\pi$$
0.391851 + 0.920029i $$0.371835\pi$$
$$182$$ 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 $$-0.803113\pi$$
−0.814727 + 0.579845i $$0.803113\pi$$
$$200$$ 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 $$-0.775591\pi$$
−0.761611 + 0.648034i $$0.775591\pi$$
$$224$$ 0 0
$$225$$ 2.54364 0.169576
$$226$$ 0 0
$$227$$ 22.6035 1.50024 0.750122 0.661299i $$-0.229995\pi$$
0.750122 + 0.661299i $$0.229995\pi$$
$$228$$ 0 0
$$229$$ 11.3554 0.750383 0.375192 0.926947i $$-0.377577\pi$$
0.375192 + 0.926947i $$0.377577\pi$$
$$230$$ 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 $$-0.295495\pi$$
0.599177 + 0.800617i $$0.295495\pi$$
$$242$$ 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 $$-0.218174\pi$$
0.774157 + 0.632994i $$0.218174\pi$$
$$252$$ 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 $$-0.316303\pi$$
0.545596 + 0.838048i $$0.316303\pi$$
$$258$$ 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 $$-0.561919\pi$$
−0.193300 + 0.981140i $$0.561919\pi$$
$$270$$ 0 0
$$271$$ −20.2397 −1.22947 −0.614737 0.788732i $$-0.710738\pi$$
−0.614737 + 0.788732i $$0.710738\pi$$
$$272$$ 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 $$-0.709070\pi$$
−0.610596 + 0.791942i $$0.709070\pi$$
$$284$$ 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 $$-0.549040\pi$$
−0.153454 + 0.988156i $$0.549040\pi$$
$$294$$ 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 $$-0.360723\pi$$
0.423724 + 0.905791i $$0.360723\pi$$
$$308$$ 0 0
$$309$$ 6.54364 0.372255
$$310$$ 0 0
$$311$$ −1.89900 −0.107682 −0.0538412 0.998550i $$-0.517146\pi$$
−0.0538412 + 0.998550i $$0.517146\pi$$
$$312$$ 0 0
$$313$$ 0.0872743 0.00493303 0.00246652 0.999997i $$-0.499215\pi$$
0.00246652 + 0.999997i $$0.499215\pi$$
$$314$$ 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 $$-0.186045\pi$$
0.834000 + 0.551764i $$0.186045\pi$$
$$350$$ 0 0
$$351$$ −6.03677 −0.322219
$$352$$ 0 0
$$353$$ 10.1745 0.541537 0.270768 0.962645i $$-0.412722\pi$$
0.270768 + 0.962645i $$0.412722\pi$$
$$354$$ 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 $$-0.467789\pi$$
0.101022 + 0.994884i $$0.467789\pi$$
$$368$$ 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 $$-0.465986\pi$$
0.106655 + 0.994296i $$0.465986\pi$$
$$384$$ 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 $$-0.488364\pi$$
0.0365464 + 0.999332i $$0.488364\pi$$
$$398$$ 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 $$-0.722846\pi$$
−0.644288 + 0.764783i $$0.722846\pi$$
$$410$$ 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 $$-0.203089\pi$$
0.803275 + 0.595608i $$0.203089\pi$$
$$420$$ 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 $$-0.646138\pi$$
−0.443148 + 0.896448i $$0.646138\pi$$
$$434$$ 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 $$-0.495655\pi$$
0.0136505 + 0.999907i $$0.495655\pi$$
$$440$$ 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 $$-0.702320\pi$$
−0.593667 + 0.804711i $$0.702320\pi$$
$$462$$ 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 $$-0.184689\pi$$
0.836344 + 0.548205i $$0.184689\pi$$
$$468$$ 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 $$-0.292275\pi$$
0.607245 + 0.794515i $$0.292275\pi$$
$$480$$ 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 $$-0.356898\pi$$
0.434578 + 0.900634i $$0.356898\pi$$
$$504$$ 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 $$-0.322253\pi$$
0.529838 + 0.848099i $$0.322253\pi$$
$$510$$ 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 $$-0.450382\pi$$
0.155250 + 0.987875i $$0.450382\pi$$
$$522$$ 0 0
$$523$$ −28.7475 −1.25704 −0.628520 0.777793i $$-0.716339\pi$$
−0.628520 + 0.777793i $$0.716339\pi$$
$$524$$ 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 $$-0.637686\pi$$
−0.419189 + 0.907899i $$0.637686\pi$$
$$564$$ 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 $$-0.460050\pi$$
0.125177 + 0.992134i $$0.460050\pi$$
$$578$$ 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 $$-0.396056\pi$$
0.320777 + 0.947155i $$0.396056\pi$$
$$588$$ 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 $$-0.0143125\pi$$
0.998989 + 0.0449488i $$0.0143125\pi$$
$$594$$ 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 $$-0.447921\pi$$
0.162883 + 0.986645i $$0.447921\pi$$
$$602$$ 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 $$-0.482486\pi$$
0.0549936 + 0.998487i $$0.482486\pi$$
$$608$$ 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 $$-0.701434\pi$$
−0.591424 + 0.806361i $$0.701434\pi$$
$$620$$ 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 $$-0.435468\pi$$
0.201348 + 0.979520i $$0.435468\pi$$
$$644$$ 0 0
$$645$$ 9.49314 0.373792
$$646$$ 0 0
$$647$$ −20.5530 −0.808020 −0.404010 0.914755i $$-0.632384\pi$$
−0.404010 + 0.914755i $$0.632384\pi$$
$$648$$ 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 $$-0.309852\pi$$
0.562469 + 0.826819i $$0.309852\pi$$
$$662$$ 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 $$-0.732725\pi$$
−0.667710 + 0.744422i $$0.732725\pi$$
$$678$$ 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 $$-0.272834\pi$$
0.654608 + 0.755969i $$0.272834\pi$$
$$692$$ 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 $$-0.505854\pi$$
−0.0183909 + 0.999831i $$0.505854\pi$$
$$720$$ 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 $$-0.724289\pi$$
−0.647749 + 0.761854i $$0.724289\pi$$
$$728$$ 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 $$-0.530167\pi$$
−0.0946305 + 0.995512i $$0.530167\pi$$
$$734$$ 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 $$-0.426339\pi$$
0.229353 + 0.973343i $$0.426339\pi$$
$$762$$ 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 $$-0.381125\pi$$
0.364835 + 0.931072i $$0.381125\pi$$
$$770$$ 0 0
$$771$$ 17.4931 0.630000
$$772$$ 0 0
$$773$$ 19.7412 0.710043 0.355021 0.934858i $$-0.384474\pi$$
0.355021 + 0.934858i $$0.384474\pi$$
$$774$$ 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 $$-0.597466\pi$$
−0.301437 + 0.953486i $$0.597466\pi$$
$$788$$ 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 $$-0.519658\pi$$
−0.0617180 + 0.998094i $$0.519658\pi$$
$$798$$ 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 $$-0.827860\pi$$
−0.857300 + 0.514817i $$0.827860\pi$$
$$812$$ 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 $$-0.312876\pi$$
0.554588 + 0.832125i $$0.312876\pi$$
$$830$$ 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 $$-0.173470\pi$$
0.855142 + 0.518394i $$0.173470\pi$$
$$840$$ 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 $$-0.562837\pi$$
−0.196129 + 0.980578i $$0.562837\pi$$
$$854$$ 0 0
$$855$$ 16.5804 0.567038
$$856$$ 0 0
$$857$$ 32.2481 1.10157 0.550787 0.834646i $$-0.314328\pi$$
0.550787 + 0.834646i $$0.314328\pi$$
$$858$$ 0 0
$$859$$ −3.08727 −0.105336 −0.0526682 0.998612i $$-0.516773\pi$$
−0.0526682 + 0.998612i $$0.516773\pi$$
$$860$$ 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 $$-0.253090\pi$$
0.700209 + 0.713938i $$0.253090\pi$$
$$882$$ 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 $$-0.747710\pi$$
−0.702001 + 0.712176i $$0.747710\pi$$
$$888$$ 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 $$-0.722568\pi$$
−0.643619 + 0.765346i $$0.722568\pi$$
$$930$$ 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 $$-0.580666\pi$$
−0.250717 + 0.968061i $$0.580666\pi$$
$$938$$ 0 0
$$939$$ 0.0872743 0.00284809
$$940$$ 0 0
$$941$$ 12.1662 0.396605 0.198303 0.980141i $$-0.436457\pi$$
0.198303 + 0.980141i $$0.436457\pi$$
$$942$$ 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 $$-0.559159\pi$$
−0.184786 + 0.982779i $$0.559159\pi$$
$$972$$ 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 $$-0.202873\pi$$
0.803678 + 0.595064i $$0.202873\pi$$
$$984$$ 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 $$-0.279424\pi$$
0.638816 + 0.769359i $$0.279424\pi$$
$$998$$ 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.bv.1.1 3
4.3 odd 2 4704.2.a.bt.1.1 3
7.3 odd 6 672.2.q.l.289.1 yes 6
7.5 odd 6 672.2.q.l.193.1 yes 6
7.6 odd 2 4704.2.a.bs.1.3 3
8.3 odd 2 9408.2.a.ei.1.3 3
8.5 even 2 9408.2.a.eg.1.3 3
21.5 even 6 2016.2.s.v.865.3 6
21.17 even 6 2016.2.s.v.289.3 6
28.3 even 6 672.2.q.k.289.1 yes 6
28.19 even 6 672.2.q.k.193.1 6
28.27 even 2 4704.2.a.bu.1.3 3
56.3 even 6 1344.2.q.z.961.3 6
56.5 odd 6 1344.2.q.y.193.3 6
56.13 odd 2 9408.2.a.ej.1.1 3
56.19 even 6 1344.2.q.z.193.3 6
56.27 even 2 9408.2.a.eh.1.1 3
56.45 odd 6 1344.2.q.y.961.3 6
84.47 odd 6 2016.2.s.u.865.3 6
84.59 odd 6 2016.2.s.u.289.3 6

By twisted newform
Twist Min Dim Char Parity Ord Type
672.2.q.k.193.1 6 28.19 even 6
672.2.q.k.289.1 yes 6 28.3 even 6
672.2.q.l.193.1 yes 6 7.5 odd 6
672.2.q.l.289.1 yes 6 7.3 odd 6
1344.2.q.y.193.3 6 56.5 odd 6
1344.2.q.y.961.3 6 56.45 odd 6
1344.2.q.z.193.3 6 56.19 even 6
1344.2.q.z.961.3 6 56.3 even 6
2016.2.s.u.289.3 6 84.59 odd 6
2016.2.s.u.865.3 6 84.47 odd 6
2016.2.s.v.289.3 6 21.17 even 6
2016.2.s.v.865.3 6 21.5 even 6
4704.2.a.bs.1.3 3 7.6 odd 2
4704.2.a.bt.1.1 3 4.3 odd 2
4704.2.a.bu.1.3 3 28.27 even 2
4704.2.a.bv.1.1 3 1.1 even 1 trivial
9408.2.a.eg.1.3 3 8.5 even 2
9408.2.a.eh.1.1 3 56.27 even 2
9408.2.a.ei.1.3 3 8.3 odd 2
9408.2.a.ej.1.1 3 56.13 odd 2