# Properties

 Label 4032.2.v.e.1583.7 Level 4032 Weight 2 Character 4032.1583 Analytic conductor 32.196 Analytic rank 0 Dimension 40 CM no Inner twists 4

# Related objects

## Newspace parameters

 Level: $$N$$ = $$4032 = 2^{6} \cdot 3^{2} \cdot 7$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 4032.v (of order $$4$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$32.1956820950$$ Analytic rank: $$0$$ Dimension: $$40$$ Relative dimension: $$20$$ over $$\Q(i)$$ Coefficient ring index: multiple of None Twist minimal: no (minimal twist has level 1008) Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

## Embedding invariants

 Embedding label 1583.7 Character $$\chi$$ = 4032.1583 Dual form 4032.2.v.e.3599.7

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+(-0.925496 - 0.925496i) q^{5} -1.00000 q^{7} +O(q^{10})$$ $$q+(-0.925496 - 0.925496i) q^{5} -1.00000 q^{7} +(-1.72403 + 1.72403i) q^{11} +(0.328273 + 0.328273i) q^{13} -2.34181i q^{17} +(1.77976 - 1.77976i) q^{19} +6.17143i q^{23} -3.28691i q^{25} +(0.122671 - 0.122671i) q^{29} -1.74700i q^{31} +(0.925496 + 0.925496i) q^{35} +(-1.68105 + 1.68105i) q^{37} -2.88812 q^{41} +(-2.77330 - 2.77330i) q^{43} +5.92184 q^{47} +1.00000 q^{49} +(0.973689 + 0.973689i) q^{53} +3.19117 q^{55} +(-8.33124 + 8.33124i) q^{59} +(4.28808 + 4.28808i) q^{61} -0.607630i q^{65} +(-1.78259 + 1.78259i) q^{67} +8.57053i q^{71} +6.41750i q^{73} +(1.72403 - 1.72403i) q^{77} -5.38299i q^{79} +(-3.46360 - 3.46360i) q^{83} +(-2.16734 + 2.16734i) q^{85} +1.51391 q^{89} +(-0.328273 - 0.328273i) q^{91} -3.29433 q^{95} +15.7177 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$40q - 40q^{7} + O(q^{10})$$ $$40q - 40q^{7} - 24q^{13} + 32q^{19} - 8q^{37} - 32q^{43} + 40q^{49} - 48q^{55} - 24q^{61} + 64q^{85} + 24q^{91} + 64q^{97} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/4032\mathbb{Z}\right)^\times$$.

 $$n$$ $$127$$ $$577$$ $$1793$$ $$3781$$ $$\chi(n)$$ $$-1$$ $$1$$ $$-1$$ $$e\left(\frac{3}{4}\right)$$

## 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
$$3$$ 0 0
$$4$$ 0 0
$$5$$ −0.925496 0.925496i −0.413894 0.413894i 0.469198 0.883093i $$-0.344543\pi$$
−0.883093 + 0.469198i $$0.844543\pi$$
$$6$$ 0 0
$$7$$ −1.00000 −0.377964
$$8$$ 0 0
$$9$$ 0 0
$$10$$ 0 0
$$11$$ −1.72403 + 1.72403i −0.519815 + 0.519815i −0.917515 0.397701i $$-0.869808\pi$$
0.397701 + 0.917515i $$0.369808\pi$$
$$12$$ 0 0
$$13$$ 0.328273 + 0.328273i 0.0910464 + 0.0910464i 0.751163 0.660117i $$-0.229493\pi$$
−0.660117 + 0.751163i $$0.729493\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 0 0
$$17$$ 2.34181i 0.567973i −0.958828 0.283986i $$-0.908343\pi$$
0.958828 0.283986i $$-0.0916570\pi$$
$$18$$ 0 0
$$19$$ 1.77976 1.77976i 0.408306 0.408306i −0.472842 0.881147i $$-0.656772\pi$$
0.881147 + 0.472842i $$0.156772\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 0 0
$$23$$ 6.17143i 1.28683i 0.765517 + 0.643416i $$0.222483\pi$$
−0.765517 + 0.643416i $$0.777517\pi$$
$$24$$ 0 0
$$25$$ 3.28691i 0.657383i
$$26$$ 0 0
$$27$$ 0 0
$$28$$ 0 0
$$29$$ 0.122671 0.122671i 0.0227794 0.0227794i −0.695625 0.718405i $$-0.744873\pi$$
0.718405 + 0.695625i $$0.244873\pi$$
$$30$$ 0 0
$$31$$ 1.74700i 0.313770i −0.987617 0.156885i $$-0.949855\pi$$
0.987617 0.156885i $$-0.0501452\pi$$
$$32$$ 0 0
$$33$$ 0 0
$$34$$ 0 0
$$35$$ 0.925496 + 0.925496i 0.156437 + 0.156437i
$$36$$ 0 0
$$37$$ −1.68105 + 1.68105i −0.276363 + 0.276363i −0.831655 0.555292i $$-0.812606\pi$$
0.555292 + 0.831655i $$0.312606\pi$$
$$38$$ 0 0
$$39$$ 0 0
$$40$$ 0 0
$$41$$ −2.88812 −0.451048 −0.225524 0.974238i $$-0.572409\pi$$
−0.225524 + 0.974238i $$0.572409\pi$$
$$42$$ 0 0
$$43$$ −2.77330 2.77330i −0.422924 0.422924i 0.463285 0.886209i $$-0.346670\pi$$
−0.886209 + 0.463285i $$0.846670\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 0 0
$$47$$ 5.92184 0.863789 0.431894 0.901924i $$-0.357845\pi$$
0.431894 + 0.901924i $$0.357845\pi$$
$$48$$ 0 0
$$49$$ 1.00000 0.142857
$$50$$ 0 0
$$51$$ 0 0
$$52$$ 0 0
$$53$$ 0.973689 + 0.973689i 0.133746 + 0.133746i 0.770811 0.637064i $$-0.219851\pi$$
−0.637064 + 0.770811i $$0.719851\pi$$
$$54$$ 0 0
$$55$$ 3.19117 0.430297
$$56$$ 0 0
$$57$$ 0 0
$$58$$ 0 0
$$59$$ −8.33124 + 8.33124i −1.08464 + 1.08464i −0.0885647 + 0.996070i $$0.528228\pi$$
−0.996070 + 0.0885647i $$0.971772\pi$$
$$60$$ 0 0
$$61$$ 4.28808 + 4.28808i 0.549033 + 0.549033i 0.926161 0.377128i $$-0.123088\pi$$
−0.377128 + 0.926161i $$0.623088\pi$$
$$62$$ 0 0
$$63$$ 0 0
$$64$$ 0 0
$$65$$ 0.607630i 0.0753672i
$$66$$ 0 0
$$67$$ −1.78259 + 1.78259i −0.217778 + 0.217778i −0.807561 0.589783i $$-0.799213\pi$$
0.589783 + 0.807561i $$0.299213\pi$$
$$68$$ 0 0
$$69$$ 0 0
$$70$$ 0 0
$$71$$ 8.57053i 1.01713i 0.861022 + 0.508567i $$0.169825\pi$$
−0.861022 + 0.508567i $$0.830175\pi$$
$$72$$ 0 0
$$73$$ 6.41750i 0.751112i 0.926800 + 0.375556i $$0.122548\pi$$
−0.926800 + 0.375556i $$0.877452\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 0 0
$$77$$ 1.72403 1.72403i 0.196471 0.196471i
$$78$$ 0 0
$$79$$ 5.38299i 0.605633i −0.953049 0.302817i $$-0.902073\pi$$
0.953049 0.302817i $$-0.0979270\pi$$
$$80$$ 0 0
$$81$$ 0 0
$$82$$ 0 0
$$83$$ −3.46360 3.46360i −0.380179 0.380179i 0.490987 0.871167i $$-0.336636\pi$$
−0.871167 + 0.490987i $$0.836636\pi$$
$$84$$ 0 0
$$85$$ −2.16734 + 2.16734i −0.235081 + 0.235081i
$$86$$ 0 0
$$87$$ 0 0
$$88$$ 0 0
$$89$$ 1.51391 0.160474 0.0802368 0.996776i $$-0.474432\pi$$
0.0802368 + 0.996776i $$0.474432\pi$$
$$90$$ 0 0
$$91$$ −0.328273 0.328273i −0.0344123 0.0344123i
$$92$$ 0 0
$$93$$ 0 0
$$94$$ 0 0
$$95$$ −3.29433 −0.337991
$$96$$ 0 0
$$97$$ 15.7177 1.59589 0.797943 0.602732i $$-0.205921\pi$$
0.797943 + 0.602732i $$0.205921\pi$$
$$98$$ 0 0
$$99$$ 0 0
$$100$$ 0 0
$$101$$ 13.6438 + 13.6438i 1.35760 + 1.35760i 0.876861 + 0.480744i $$0.159633\pi$$
0.480744 + 0.876861i $$0.340367\pi$$
$$102$$ 0 0
$$103$$ 11.9238 1.17489 0.587445 0.809264i $$-0.300134\pi$$
0.587445 + 0.809264i $$0.300134\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ −13.6108 + 13.6108i −1.31580 + 1.31580i −0.398740 + 0.917064i $$0.630553\pi$$
−0.917064 + 0.398740i $$0.869447\pi$$
$$108$$ 0 0
$$109$$ 4.97865 + 4.97865i 0.476868 + 0.476868i 0.904129 0.427260i $$-0.140521\pi$$
−0.427260 + 0.904129i $$0.640521\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ 0 0
$$113$$ 0.0768901i 0.00723322i 0.999993 + 0.00361661i $$0.00115120\pi$$
−0.999993 + 0.00361661i $$0.998849\pi$$
$$114$$ 0 0
$$115$$ 5.71163 5.71163i 0.532612 0.532612i
$$116$$ 0 0
$$117$$ 0 0
$$118$$ 0 0
$$119$$ 2.34181i 0.214673i
$$120$$ 0 0
$$121$$ 5.05544i 0.459585i
$$122$$ 0 0
$$123$$ 0 0
$$124$$ 0 0
$$125$$ −7.66951 + 7.66951i −0.685981 + 0.685981i
$$126$$ 0 0
$$127$$ 15.0611i 1.33646i 0.743956 + 0.668229i $$0.232947\pi$$
−0.743956 + 0.668229i $$0.767053\pi$$
$$128$$ 0 0
$$129$$ 0 0
$$130$$ 0 0
$$131$$ 3.87027 + 3.87027i 0.338147 + 0.338147i 0.855670 0.517522i $$-0.173146\pi$$
−0.517522 + 0.855670i $$0.673146\pi$$
$$132$$ 0 0
$$133$$ −1.77976 + 1.77976i −0.154325 + 0.154325i
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ 5.47491 0.467753 0.233877 0.972266i $$-0.424859\pi$$
0.233877 + 0.972266i $$0.424859\pi$$
$$138$$ 0 0
$$139$$ −11.1117 11.1117i −0.942478 0.942478i 0.0559554 0.998433i $$-0.482180\pi$$
−0.998433 + 0.0559554i $$0.982180\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 0 0
$$143$$ −1.13190 −0.0946545
$$144$$ 0 0
$$145$$ −0.227063 −0.0188565
$$146$$ 0 0
$$147$$ 0 0
$$148$$ 0 0
$$149$$ 10.2234 + 10.2234i 0.837535 + 0.837535i 0.988534 0.150999i $$-0.0482490\pi$$
−0.150999 + 0.988534i $$0.548249\pi$$
$$150$$ 0 0
$$151$$ −5.98993 −0.487454 −0.243727 0.969844i $$-0.578370\pi$$
−0.243727 + 0.969844i $$0.578370\pi$$
$$152$$ 0 0
$$153$$ 0 0
$$154$$ 0 0
$$155$$ −1.61684 + 1.61684i −0.129868 + 0.129868i
$$156$$ 0 0
$$157$$ 9.57922 + 9.57922i 0.764505 + 0.764505i 0.977133 0.212628i $$-0.0682023\pi$$
−0.212628 + 0.977133i $$0.568202\pi$$
$$158$$ 0 0
$$159$$ 0 0
$$160$$ 0 0
$$161$$ 6.17143i 0.486377i
$$162$$ 0 0
$$163$$ 7.81085 7.81085i 0.611793 0.611793i −0.331620 0.943413i $$-0.607595\pi$$
0.943413 + 0.331620i $$0.107595\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ 5.28108i 0.408662i −0.978902 0.204331i $$-0.934498\pi$$
0.978902 0.204331i $$-0.0655019\pi$$
$$168$$ 0 0
$$169$$ 12.7845i 0.983421i
$$170$$ 0 0
$$171$$ 0 0
$$172$$ 0 0
$$173$$ −11.8893 + 11.8893i −0.903929 + 0.903929i −0.995773 0.0918444i $$-0.970724\pi$$
0.0918444 + 0.995773i $$0.470724\pi$$
$$174$$ 0 0
$$175$$ 3.28691i 0.248467i
$$176$$ 0 0
$$177$$ 0 0
$$178$$ 0 0
$$179$$ 6.38837 + 6.38837i 0.477489 + 0.477489i 0.904328 0.426839i $$-0.140373\pi$$
−0.426839 + 0.904328i $$0.640373\pi$$
$$180$$ 0 0
$$181$$ 5.56367 5.56367i 0.413545 0.413545i −0.469427 0.882971i $$-0.655539\pi$$
0.882971 + 0.469427i $$0.155539\pi$$
$$182$$ 0 0
$$183$$ 0 0
$$184$$ 0 0
$$185$$ 3.11161 0.228770
$$186$$ 0 0
$$187$$ 4.03735 + 4.03735i 0.295240 + 0.295240i
$$188$$ 0 0
$$189$$ 0 0
$$190$$ 0 0
$$191$$ 2.43241 0.176003 0.0880014 0.996120i $$-0.471952\pi$$
0.0880014 + 0.996120i $$0.471952\pi$$
$$192$$ 0 0
$$193$$ 24.8507 1.78879 0.894397 0.447273i $$-0.147605\pi$$
0.894397 + 0.447273i $$0.147605\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ −1.77151 1.77151i −0.126215 0.126215i 0.641178 0.767393i $$-0.278446\pi$$
−0.767393 + 0.641178i $$0.778446\pi$$
$$198$$ 0 0
$$199$$ 13.8970 0.985130 0.492565 0.870276i $$-0.336059\pi$$
0.492565 + 0.870276i $$0.336059\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ 0 0
$$203$$ −0.122671 + 0.122671i −0.00860980 + 0.00860980i
$$204$$ 0 0
$$205$$ 2.67294 + 2.67294i 0.186686 + 0.186686i
$$206$$ 0 0
$$207$$ 0 0
$$208$$ 0 0
$$209$$ 6.13673i 0.424487i
$$210$$ 0 0
$$211$$ 0.243974 0.243974i 0.0167959 0.0167959i −0.698659 0.715455i $$-0.746220\pi$$
0.715455 + 0.698659i $$0.246220\pi$$
$$212$$ 0 0
$$213$$ 0 0
$$214$$ 0 0
$$215$$ 5.13335i 0.350091i
$$216$$ 0 0
$$217$$ 1.74700i 0.118594i
$$218$$ 0 0
$$219$$ 0 0
$$220$$ 0 0
$$221$$ 0.768752 0.768752i 0.0517119 0.0517119i
$$222$$ 0 0
$$223$$ 7.07187i 0.473568i 0.971562 + 0.236784i $$0.0760933\pi$$
−0.971562 + 0.236784i $$0.923907\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 0 0
$$227$$ −2.15348 2.15348i −0.142932 0.142932i 0.632020 0.774952i $$-0.282226\pi$$
−0.774952 + 0.632020i $$0.782226\pi$$
$$228$$ 0 0
$$229$$ −14.4509 + 14.4509i −0.954943 + 0.954943i −0.999028 0.0440845i $$-0.985963\pi$$
0.0440845 + 0.999028i $$0.485963\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ −17.7969 −1.16591 −0.582955 0.812504i $$-0.698104\pi$$
−0.582955 + 0.812504i $$0.698104\pi$$
$$234$$ 0 0
$$235$$ −5.48063 5.48063i −0.357517 0.357517i
$$236$$ 0 0
$$237$$ 0 0
$$238$$ 0 0
$$239$$ 9.62147 0.622361 0.311180 0.950351i $$-0.399276\pi$$
0.311180 + 0.950351i $$0.399276\pi$$
$$240$$ 0 0
$$241$$ 27.3786 1.76361 0.881805 0.471614i $$-0.156329\pi$$
0.881805 + 0.471614i $$0.156329\pi$$
$$242$$ 0 0
$$243$$ 0 0
$$244$$ 0 0
$$245$$ −0.925496 0.925496i −0.0591278 0.0591278i
$$246$$ 0 0
$$247$$ 1.16849 0.0743495
$$248$$ 0 0
$$249$$ 0 0
$$250$$ 0 0
$$251$$ 6.56293 6.56293i 0.414249 0.414249i −0.468967 0.883216i $$-0.655374\pi$$
0.883216 + 0.468967i $$0.155374\pi$$
$$252$$ 0 0
$$253$$ −10.6397 10.6397i −0.668914 0.668914i
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 0 0
$$257$$ 13.6808i 0.853383i 0.904397 + 0.426691i $$0.140321\pi$$
−0.904397 + 0.426691i $$0.859679\pi$$
$$258$$ 0 0
$$259$$ 1.68105 1.68105i 0.104455 0.104455i
$$260$$ 0 0
$$261$$ 0 0
$$262$$ 0 0
$$263$$ 26.5064i 1.63445i −0.576317 0.817226i $$-0.695511\pi$$
0.576317 0.817226i $$-0.304489\pi$$
$$264$$ 0 0
$$265$$ 1.80229i 0.110714i
$$266$$ 0 0
$$267$$ 0 0
$$268$$ 0 0
$$269$$ −11.7053 + 11.7053i −0.713684 + 0.713684i −0.967304 0.253620i $$-0.918379\pi$$
0.253620 + 0.967304i $$0.418379\pi$$
$$270$$ 0 0
$$271$$ 26.7904i 1.62740i 0.581286 + 0.813700i $$0.302550\pi$$
−0.581286 + 0.813700i $$0.697450\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ 0 0
$$275$$ 5.66674 + 5.66674i 0.341717 + 0.341717i
$$276$$ 0 0
$$277$$ −0.376534 + 0.376534i −0.0226237 + 0.0226237i −0.718328 0.695704i $$-0.755092\pi$$
0.695704 + 0.718328i $$0.255092\pi$$
$$278$$ 0 0
$$279$$ 0 0
$$280$$ 0 0
$$281$$ −10.8174 −0.645314 −0.322657 0.946516i $$-0.604576\pi$$
−0.322657 + 0.946516i $$0.604576\pi$$
$$282$$ 0 0
$$283$$ −0.954650 0.954650i −0.0567481 0.0567481i 0.678163 0.734911i $$-0.262776\pi$$
−0.734911 + 0.678163i $$0.762776\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ 2.88812 0.170480
$$288$$ 0 0
$$289$$ 11.5159 0.677407
$$290$$ 0 0
$$291$$ 0 0
$$292$$ 0 0
$$293$$ 14.4106 + 14.4106i 0.841877 + 0.841877i 0.989103 0.147226i $$-0.0470346\pi$$
−0.147226 + 0.989103i $$0.547035\pi$$
$$294$$ 0 0
$$295$$ 15.4211 0.897849
$$296$$ 0 0
$$297$$ 0 0
$$298$$ 0 0
$$299$$ −2.02591 + 2.02591i −0.117161 + 0.117161i
$$300$$ 0 0
$$301$$ 2.77330 + 2.77330i 0.159850 + 0.159850i
$$302$$ 0 0
$$303$$ 0 0
$$304$$ 0 0
$$305$$ 7.93721i 0.454483i
$$306$$ 0 0
$$307$$ −15.9801 + 15.9801i −0.912034 + 0.912034i −0.996432 0.0843977i $$-0.973103\pi$$
0.0843977 + 0.996432i $$0.473103\pi$$
$$308$$ 0 0
$$309$$ 0 0
$$310$$ 0 0
$$311$$ 7.61653i 0.431894i 0.976405 + 0.215947i $$0.0692838\pi$$
−0.976405 + 0.215947i $$0.930716\pi$$
$$312$$ 0 0
$$313$$ 30.7549i 1.73837i −0.494490 0.869184i $$-0.664645\pi$$
0.494490 0.869184i $$-0.335355\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ 4.52751 4.52751i 0.254291 0.254291i −0.568437 0.822727i $$-0.692452\pi$$
0.822727 + 0.568437i $$0.192452\pi$$
$$318$$ 0 0
$$319$$ 0.422976i 0.0236821i
$$320$$ 0 0
$$321$$ 0 0
$$322$$ 0 0
$$323$$ −4.16787 4.16787i −0.231906 0.231906i
$$324$$ 0 0
$$325$$ 1.07900 1.07900i 0.0598524 0.0598524i
$$326$$ 0 0
$$327$$ 0 0
$$328$$ 0 0
$$329$$ −5.92184 −0.326481
$$330$$ 0 0
$$331$$ −11.4312 11.4312i −0.628318 0.628318i 0.319326 0.947645i $$-0.396543\pi$$
−0.947645 + 0.319326i $$0.896543\pi$$
$$332$$ 0 0
$$333$$ 0 0
$$334$$ 0 0
$$335$$ 3.29956 0.180274
$$336$$ 0 0
$$337$$ −10.7569 −0.585964 −0.292982 0.956118i $$-0.594648\pi$$
−0.292982 + 0.956118i $$0.594648\pi$$
$$338$$ 0 0
$$339$$ 0 0
$$340$$ 0 0
$$341$$ 3.01187 + 3.01187i 0.163102 + 0.163102i
$$342$$ 0 0
$$343$$ −1.00000 −0.0539949
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ −16.3589 + 16.3589i −0.878193 + 0.878193i −0.993348 0.115154i $$-0.963264\pi$$
0.115154 + 0.993348i $$0.463264\pi$$
$$348$$ 0 0
$$349$$ −25.8389 25.8389i −1.38312 1.38312i −0.839013 0.544112i $$-0.816867\pi$$
−0.544112 0.839013i $$-0.683133\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 0 0
$$353$$ 24.5770i 1.30810i 0.756451 + 0.654051i $$0.226932\pi$$
−0.756451 + 0.654051i $$0.773068\pi$$
$$354$$ 0 0
$$355$$ 7.93199 7.93199i 0.420986 0.420986i
$$356$$ 0 0
$$357$$ 0 0
$$358$$ 0 0
$$359$$ 17.5485i 0.926176i 0.886312 + 0.463088i $$0.153259\pi$$
−0.886312 + 0.463088i $$0.846741\pi$$
$$360$$ 0 0
$$361$$ 12.6649i 0.666573i
$$362$$ 0 0
$$363$$ 0 0
$$364$$ 0 0
$$365$$ 5.93937 5.93937i 0.310881 0.310881i
$$366$$ 0 0
$$367$$ 15.7849i 0.823964i 0.911192 + 0.411982i $$0.135163\pi$$
−0.911192 + 0.411982i $$0.864837\pi$$
$$368$$ 0 0
$$369$$ 0 0
$$370$$ 0 0
$$371$$ −0.973689 0.973689i −0.0505514 0.0505514i
$$372$$ 0 0
$$373$$ −11.5837 + 11.5837i −0.599781 + 0.599781i −0.940254 0.340473i $$-0.889413\pi$$
0.340473 + 0.940254i $$0.389413\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ 0.0805389 0.00414796
$$378$$ 0 0
$$379$$ 1.84371 + 1.84371i 0.0947049 + 0.0947049i 0.752872 0.658167i $$-0.228668\pi$$
−0.658167 + 0.752872i $$0.728668\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ 0 0
$$383$$ 4.04073 0.206472 0.103236 0.994657i $$-0.467080\pi$$
0.103236 + 0.994657i $$0.467080\pi$$
$$384$$ 0 0
$$385$$ −3.19117 −0.162637
$$386$$ 0 0
$$387$$ 0 0
$$388$$ 0 0
$$389$$ −2.92444 2.92444i −0.148275 0.148275i 0.629072 0.777347i $$-0.283435\pi$$
−0.777347 + 0.629072i $$0.783435\pi$$
$$390$$ 0 0
$$391$$ 14.4523 0.730885
$$392$$ 0 0
$$393$$ 0 0
$$394$$ 0 0
$$395$$ −4.98193 + 4.98193i −0.250668 + 0.250668i
$$396$$ 0 0
$$397$$ −21.9555 21.9555i −1.10191 1.10191i −0.994180 0.107735i $$-0.965640\pi$$
−0.107735 0.994180i $$-0.534360\pi$$
$$398$$ 0 0
$$399$$ 0 0
$$400$$ 0 0
$$401$$ 3.18398i 0.159000i 0.996835 + 0.0795001i $$0.0253324\pi$$
−0.996835 + 0.0795001i $$0.974668\pi$$
$$402$$ 0 0
$$403$$ 0.573491 0.573491i 0.0285676 0.0285676i
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 0 0
$$407$$ 5.79636i 0.287315i
$$408$$ 0 0
$$409$$ 15.1001i 0.746650i 0.927701 + 0.373325i $$0.121782\pi$$
−0.927701 + 0.373325i $$0.878218\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ 0 0
$$413$$ 8.33124 8.33124i 0.409954 0.409954i
$$414$$ 0 0
$$415$$ 6.41109i 0.314708i
$$416$$ 0 0
$$417$$ 0 0
$$418$$ 0 0
$$419$$ −2.19389 2.19389i −0.107178 0.107178i 0.651484 0.758662i $$-0.274147\pi$$
−0.758662 + 0.651484i $$0.774147\pi$$
$$420$$ 0 0
$$421$$ −16.8539 + 16.8539i −0.821410 + 0.821410i −0.986310 0.164900i $$-0.947270\pi$$
0.164900 + 0.986310i $$0.447270\pi$$
$$422$$ 0 0
$$423$$ 0 0
$$424$$ 0 0
$$425$$ −7.69733 −0.373376
$$426$$ 0 0
$$427$$ −4.28808 4.28808i −0.207515 0.207515i
$$428$$ 0 0
$$429$$ 0 0
$$430$$ 0 0
$$431$$ 19.9661 0.961735 0.480868 0.876793i $$-0.340322\pi$$
0.480868 + 0.876793i $$0.340322\pi$$
$$432$$ 0 0
$$433$$ −3.83212 −0.184160 −0.0920800 0.995752i $$-0.529352\pi$$
−0.0920800 + 0.995752i $$0.529352\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 0 0
$$437$$ 10.9837 + 10.9837i 0.525421 + 0.525421i
$$438$$ 0 0
$$439$$ 23.0120 1.09830 0.549151 0.835723i $$-0.314951\pi$$
0.549151 + 0.835723i $$0.314951\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 0 0
$$443$$ −13.6982 + 13.6982i −0.650821 + 0.650821i −0.953191 0.302370i $$-0.902222\pi$$
0.302370 + 0.953191i $$0.402222\pi$$
$$444$$ 0 0
$$445$$ −1.40111 1.40111i −0.0664191 0.0664191i
$$446$$ 0 0
$$447$$ 0 0
$$448$$ 0 0
$$449$$ 27.3194i 1.28928i −0.764486 0.644640i $$-0.777007\pi$$
0.764486 0.644640i $$-0.222993\pi$$
$$450$$ 0 0
$$451$$ 4.97920 4.97920i 0.234462 0.234462i
$$452$$ 0 0
$$453$$ 0 0
$$454$$ 0 0
$$455$$ 0.607630i 0.0284861i
$$456$$ 0 0
$$457$$ 30.4516i 1.42447i −0.701942 0.712234i $$-0.747684\pi$$
0.701942 0.712234i $$-0.252316\pi$$
$$458$$ 0 0
$$459$$ 0 0
$$460$$ 0 0
$$461$$ −17.8016 + 17.8016i −0.829101 + 0.829101i −0.987392 0.158291i $$-0.949402\pi$$
0.158291 + 0.987392i $$0.449402\pi$$
$$462$$ 0 0
$$463$$ 8.43314i 0.391921i 0.980612 + 0.195961i $$0.0627825\pi$$
−0.980612 + 0.195961i $$0.937218\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ −3.17711 3.17711i −0.147019 0.147019i 0.629766 0.776785i $$-0.283151\pi$$
−0.776785 + 0.629766i $$0.783151\pi$$
$$468$$ 0 0
$$469$$ 1.78259 1.78259i 0.0823124 0.0823124i
$$470$$ 0 0
$$471$$ 0 0
$$472$$ 0 0
$$473$$ 9.56249 0.439684
$$474$$ 0 0
$$475$$ −5.84993 5.84993i −0.268413 0.268413i
$$476$$ 0 0
$$477$$ 0 0
$$478$$ 0 0
$$479$$ −0.550076 −0.0251336 −0.0125668 0.999921i $$-0.504000\pi$$
−0.0125668 + 0.999921i $$0.504000\pi$$
$$480$$ 0 0
$$481$$ −1.10368 −0.0503237
$$482$$ 0 0
$$483$$ 0 0
$$484$$ 0 0
$$485$$ −14.5466 14.5466i −0.660528 0.660528i
$$486$$ 0 0
$$487$$ −18.1678 −0.823262 −0.411631 0.911351i $$-0.635041\pi$$
−0.411631 + 0.911351i $$0.635041\pi$$
$$488$$ 0 0
$$489$$ 0 0
$$490$$ 0 0
$$491$$ 12.4565 12.4565i 0.562156 0.562156i −0.367764 0.929919i $$-0.619876\pi$$
0.929919 + 0.367764i $$0.119876\pi$$
$$492$$ 0 0
$$493$$ −0.287272 0.287272i −0.0129381 0.0129381i
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ 8.57053i 0.384441i
$$498$$ 0 0
$$499$$ 23.1835 23.1835i 1.03784 1.03784i 0.0385798 0.999256i $$-0.487717\pi$$
0.999256 0.0385798i $$-0.0122834\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ 0 0
$$503$$ 39.4375i 1.75843i −0.476426 0.879215i $$-0.658068\pi$$
0.476426 0.879215i $$-0.341932\pi$$
$$504$$ 0 0
$$505$$ 25.2545i 1.12381i
$$506$$ 0 0
$$507$$ 0 0
$$508$$ 0 0
$$509$$ 1.26193 1.26193i 0.0559340 0.0559340i −0.678587 0.734520i $$-0.737407\pi$$
0.734520 + 0.678587i $$0.237407\pi$$
$$510$$ 0 0
$$511$$ 6.41750i 0.283894i
$$512$$ 0 0
$$513$$ 0 0
$$514$$ 0 0
$$515$$ −11.0355 11.0355i −0.486280 0.486280i
$$516$$ 0 0
$$517$$ −10.2094 + 10.2094i −0.449010 + 0.449010i
$$518$$ 0 0
$$519$$ 0 0
$$520$$ 0 0
$$521$$ −27.2672 −1.19460 −0.597300 0.802018i $$-0.703760\pi$$
−0.597300 + 0.802018i $$0.703760\pi$$
$$522$$ 0 0
$$523$$ 31.1608 + 31.1608i 1.36257 + 1.36257i 0.870633 + 0.491933i $$0.163709\pi$$
0.491933 + 0.870633i $$0.336291\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ −4.09113 −0.178213
$$528$$ 0 0
$$529$$ −15.0865 −0.655936
$$530$$ 0 0
$$531$$ 0 0
$$532$$ 0 0
$$533$$ −0.948090 0.948090i −0.0410663 0.0410663i
$$534$$ 0 0
$$535$$ 25.1935 1.08921
$$536$$ 0 0
$$537$$ 0 0
$$538$$ 0 0
$$539$$ −1.72403 + 1.72403i −0.0742592 + 0.0742592i
$$540$$ 0 0
$$541$$ 12.9117 + 12.9117i 0.555119 + 0.555119i 0.927914 0.372795i $$-0.121600\pi$$
−0.372795 + 0.927914i $$0.621600\pi$$
$$542$$ 0 0
$$543$$ 0 0
$$544$$ 0 0
$$545$$ 9.21544i 0.394746i
$$546$$ 0 0
$$547$$ 17.6998 17.6998i 0.756790 0.756790i −0.218947 0.975737i $$-0.570262\pi$$
0.975737 + 0.218947i $$0.0702622\pi$$
$$548$$ 0 0
$$549$$ 0 0
$$550$$ 0 0
$$551$$ 0.436650i 0.0186019i
$$552$$ 0 0
$$553$$ 5.38299i 0.228908i
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 0 0
$$557$$ −3.90087 + 3.90087i −0.165285 + 0.165285i −0.784903 0.619618i $$-0.787287\pi$$
0.619618 + 0.784903i $$0.287287\pi$$
$$558$$ 0 0
$$559$$ 1.82079i 0.0770114i
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 0 0
$$563$$ 27.4905 + 27.4905i 1.15859 + 1.15859i 0.984780 + 0.173807i $$0.0556068\pi$$
0.173807 + 0.984780i $$0.444393\pi$$
$$564$$ 0 0
$$565$$ 0.0711615 0.0711615i 0.00299379 0.00299379i
$$566$$ 0 0
$$567$$ 0 0
$$568$$ 0 0
$$569$$ 19.2030 0.805031 0.402516 0.915413i $$-0.368136\pi$$
0.402516 + 0.915413i $$0.368136\pi$$
$$570$$ 0 0
$$571$$ −23.2363 23.2363i −0.972408 0.972408i 0.0272217 0.999629i $$-0.491334\pi$$
−0.999629 + 0.0272217i $$0.991334\pi$$
$$572$$ 0 0
$$573$$ 0 0
$$574$$ 0 0
$$575$$ 20.2850 0.845941
$$576$$ 0 0
$$577$$ 25.0676 1.04358 0.521789 0.853074i $$-0.325265\pi$$
0.521789 + 0.853074i $$0.325265\pi$$
$$578$$ 0 0
$$579$$ 0 0
$$580$$ 0 0
$$581$$ 3.46360 + 3.46360i 0.143694 + 0.143694i
$$582$$ 0 0
$$583$$ −3.35734 −0.139047
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ 6.53626 6.53626i 0.269780 0.269780i −0.559231 0.829012i $$-0.688904\pi$$
0.829012 + 0.559231i $$0.188904\pi$$
$$588$$ 0 0
$$589$$ −3.10924 3.10924i −0.128114 0.128114i
$$590$$ 0 0
$$591$$ 0 0
$$592$$ 0 0
$$593$$ 15.1162i 0.620749i −0.950614 0.310375i $$-0.899545\pi$$
0.950614 0.310375i $$-0.100455\pi$$
$$594$$ 0 0
$$595$$ 2.16734 2.16734i 0.0888521 0.0888521i
$$596$$ 0 0
$$597$$ 0 0
$$598$$ 0 0
$$599$$ 29.1264i 1.19007i −0.803700 0.595035i $$-0.797138\pi$$
0.803700 0.595035i $$-0.202862\pi$$
$$600$$ 0 0
$$601$$ 14.2202i 0.580053i 0.957019 + 0.290026i $$0.0936641\pi$$
−0.957019 + 0.290026i $$0.906336\pi$$
$$602$$ 0 0
$$603$$ 0 0
$$604$$ 0 0
$$605$$ 4.67879 4.67879i 0.190220 0.190220i
$$606$$ 0 0
$$607$$ 26.8584i 1.09015i 0.838387 + 0.545075i $$0.183499\pi$$
−0.838387 + 0.545075i $$0.816501\pi$$
$$608$$ 0 0
$$609$$ 0 0
$$610$$ 0 0
$$611$$ 1.94398 + 1.94398i 0.0786448 + 0.0786448i
$$612$$ 0 0
$$613$$ −28.2483 + 28.2483i −1.14094 + 1.14094i −0.152659 + 0.988279i $$0.548784\pi$$
−0.988279 + 0.152659i $$0.951216\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ −31.4617 −1.26660 −0.633300 0.773906i $$-0.718300\pi$$
−0.633300 + 0.773906i $$0.718300\pi$$
$$618$$ 0 0
$$619$$ −13.3846 13.3846i −0.537973 0.537973i 0.384961 0.922933i $$-0.374215\pi$$
−0.922933 + 0.384961i $$0.874215\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ 0 0
$$623$$ −1.51391 −0.0606533
$$624$$ 0 0
$$625$$ −2.23838 −0.0895353
$$626$$ 0 0
$$627$$ 0 0
$$628$$ 0 0
$$629$$ 3.93670 + 3.93670i 0.156966 + 0.156966i
$$630$$ 0 0
$$631$$ −15.5279 −0.618154 −0.309077 0.951037i $$-0.600020\pi$$
−0.309077 + 0.951037i $$0.600020\pi$$
$$632$$ 0 0
$$633$$ 0 0
$$634$$ 0 0
$$635$$ 13.9390 13.9390i 0.553152 0.553152i
$$636$$ 0 0
$$637$$ 0.328273 + 0.328273i 0.0130066 + 0.0130066i
$$638$$ 0 0
$$639$$ 0 0
$$640$$ 0 0
$$641$$ 33.1261i 1.30840i −0.756320 0.654201i $$-0.773005\pi$$
0.756320 0.654201i $$-0.226995\pi$$
$$642$$ 0 0
$$643$$ −28.5683 + 28.5683i −1.12662 + 1.12662i −0.135902 + 0.990722i $$0.543393\pi$$
−0.990722 + 0.135902i $$0.956607\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ 48.7052i 1.91480i 0.288765 + 0.957400i $$0.406755\pi$$
−0.288765 + 0.957400i $$0.593245\pi$$
$$648$$ 0 0
$$649$$ 28.7266i 1.12762i
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 0 0
$$653$$ −2.67453 + 2.67453i −0.104662 + 0.104662i −0.757499 0.652836i $$-0.773579\pi$$
0.652836 + 0.757499i $$0.273579\pi$$
$$654$$ 0 0
$$655$$ 7.16384i 0.279915i
$$656$$ 0 0
$$657$$ 0 0
$$658$$ 0 0
$$659$$ −20.0589 20.0589i −0.781384 0.781384i 0.198680 0.980064i $$-0.436334\pi$$
−0.980064 + 0.198680i $$0.936334\pi$$
$$660$$ 0 0
$$661$$ 3.44403 3.44403i 0.133957 0.133957i −0.636949 0.770906i $$-0.719804\pi$$
0.770906 + 0.636949i $$0.219804\pi$$
$$662$$ 0 0
$$663$$ 0 0
$$664$$ 0 0
$$665$$ 3.29433 0.127749
$$666$$ 0 0
$$667$$ 0.757054 + 0.757054i 0.0293132 + 0.0293132i
$$668$$ 0 0
$$669$$ 0 0
$$670$$ 0 0
$$671$$ −14.7856 −0.570791
$$672$$ 0 0
$$673$$ −32.1701 −1.24007 −0.620033 0.784576i $$-0.712881\pi$$
−0.620033 + 0.784576i $$0.712881\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ 0 0
$$677$$ −3.59090 3.59090i −0.138009 0.138009i 0.634727 0.772736i $$-0.281113\pi$$
−0.772736 + 0.634727i $$0.781113\pi$$
$$678$$ 0 0
$$679$$ −15.7177 −0.603188
$$680$$ 0 0
$$681$$ 0 0
$$682$$ 0 0
$$683$$ 4.35732 4.35732i 0.166728 0.166728i −0.618811 0.785540i $$-0.712385\pi$$
0.785540 + 0.618811i $$0.212385\pi$$
$$684$$ 0 0
$$685$$ −5.06701 5.06701i −0.193600 0.193600i
$$686$$ 0 0
$$687$$ 0 0
$$688$$ 0 0
$$689$$ 0.639271i 0.0243543i
$$690$$ 0 0
$$691$$ 9.12084 9.12084i 0.346973 0.346973i −0.512008 0.858981i $$-0.671098\pi$$
0.858981 + 0.512008i $$0.171098\pi$$
$$692$$ 0 0
$$693$$ 0 0
$$694$$ 0 0
$$695$$ 20.5676i 0.780173i
$$696$$ 0 0
$$697$$ 6.76343i 0.256183i
$$698$$ 0 0
$$699$$ 0 0
$$700$$ 0 0
$$701$$ 13.1832 13.1832i 0.497923 0.497923i −0.412868 0.910791i $$-0.635473\pi$$
0.910791 + 0.412868i $$0.135473\pi$$
$$702$$ 0 0
$$703$$ 5.98374i 0.225681i
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ −13.6438 13.6438i −0.513126 0.513126i
$$708$$ 0 0
$$709$$ 5.95604 5.95604i 0.223684 0.223684i −0.586364 0.810048i $$-0.699441\pi$$
0.810048 + 0.586364i $$0.199441\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ 0 0
$$713$$ 10.7815 0.403769
$$714$$ 0 0
$$715$$ 1.04757 + 1.04757i 0.0391770 + 0.0391770i
$$716$$ 0 0
$$717$$ 0 0
$$718$$ 0 0
$$719$$ 38.3917 1.43177 0.715884 0.698220i $$-0.246024\pi$$
0.715884 + 0.698220i $$0.246024\pi$$
$$720$$ 0 0
$$721$$ −11.9238 −0.444067
$$722$$ 0 0
$$723$$ 0 0
$$724$$ 0 0
$$725$$ −0.403208 0.403208i −0.0149748 0.0149748i
$$726$$ 0 0
$$727$$ 5.05200 0.187368 0.0936842 0.995602i $$-0.470136\pi$$
0.0936842 + 0.995602i $$0.470136\pi$$
$$728$$ 0 0
$$729$$ 0 0
$$730$$ 0 0
$$731$$ −6.49454 + 6.49454i −0.240209 + 0.240209i
$$732$$ 0 0
$$733$$ 2.73691 + 2.73691i 0.101090 + 0.101090i 0.755843 0.654753i $$-0.227227\pi$$
−0.654753 + 0.755843i $$0.727227\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ 6.14648i 0.226408i
$$738$$ 0 0
$$739$$ −18.5031 + 18.5031i −0.680648 + 0.680648i −0.960146 0.279498i $$-0.909832\pi$$
0.279498 + 0.960146i $$0.409832\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 0 0
$$743$$ 41.1110i 1.50822i −0.656751 0.754108i $$-0.728070\pi$$
0.656751 0.754108i $$-0.271930\pi$$
$$744$$ 0 0
$$745$$ 18.9235i 0.693302i
$$746$$ 0 0
$$747$$ 0 0
$$748$$ 0 0
$$749$$ 13.6108 13.6108i 0.497327 0.497327i
$$750$$ 0 0
$$751$$ 27.5241i 1.00437i 0.864760 + 0.502185i $$0.167470\pi$$
−0.864760 + 0.502185i $$0.832530\pi$$
$$752$$ 0 0
$$753$$ 0 0
$$754$$ 0 0
$$755$$ 5.54366 + 5.54366i 0.201754 + 0.201754i
$$756$$ 0 0
$$757$$ 6.27349 6.27349i 0.228014 0.228014i −0.583849 0.811863i $$-0.698454\pi$$
0.811863 + 0.583849i $$0.198454\pi$$
$$758$$ 0 0
$$759$$ 0 0
$$760$$ 0 0
$$761$$ −23.6884 −0.858704 −0.429352 0.903137i $$-0.641258\pi$$
−0.429352 + 0.903137i $$0.641258\pi$$
$$762$$ 0 0
$$763$$ −4.97865 4.97865i −0.180239 0.180239i
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ −5.46983 −0.197504
$$768$$ 0 0
$$769$$ −4.74756 −0.171201 −0.0856006 0.996330i $$-0.527281\pi$$
−0.0856006 + 0.996330i $$0.527281\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ 0 0
$$773$$ −18.2666 18.2666i −0.657005 0.657005i 0.297666 0.954670i $$-0.403792\pi$$
−0.954670 + 0.297666i $$0.903792\pi$$
$$774$$ 0 0
$$775$$ −5.74223 −0.206267
$$776$$ 0 0
$$777$$ 0 0
$$778$$ 0 0
$$779$$ −5.14017 + 5.14017i −0.184166 + 0.184166i
$$780$$ 0 0
$$781$$ −14.7759 14.7759i −0.528722 0.528722i
$$782$$ 0 0
$$783$$ 0 0
$$784$$ 0 0
$$785$$ 17.7311i 0.632849i
$$786$$ 0 0
$$787$$ −28.1972 + 28.1972i −1.00512 + 1.00512i −0.00513510 + 0.999987i $$0.501635\pi$$
−0.999987 + 0.00513510i $$0.998365\pi$$
$$788$$ 0 0
$$789$$ 0 0
$$790$$ 0 0
$$791$$ 0.0768901i 0.00273390i
$$792$$ 0 0
$$793$$ 2.81532i 0.0999749i
$$794$$ 0 0