# Properties

 Label 140.2.e.b.29.1 Level $140$ Weight $2$ Character 140.29 Analytic conductor $1.118$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$140 = 2^{2} \cdot 5 \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 140.e (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$1.11790562830$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(i)$$ Defining polynomial: $$x^{2} + 1$$ x^2 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 29.1 Root $$-1.00000i$$ of defining polynomial Character $$\chi$$ $$=$$ 140.29 Dual form 140.2.e.b.29.2

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+(1.00000 - 2.00000i) q^{5} -1.00000i q^{7} +3.00000 q^{9} +O(q^{10})$$ $$q+(1.00000 - 2.00000i) q^{5} -1.00000i q^{7} +3.00000 q^{9} +4.00000i q^{13} -4.00000i q^{17} -4.00000 q^{19} +8.00000i q^{23} +(-3.00000 - 4.00000i) q^{25} -2.00000 q^{29} -8.00000 q^{31} +(-2.00000 - 1.00000i) q^{35} +8.00000i q^{37} +6.00000 q^{41} +8.00000i q^{43} +(3.00000 - 6.00000i) q^{45} -8.00000i q^{47} -1.00000 q^{49} +4.00000 q^{59} -6.00000 q^{61} -3.00000i q^{63} +(8.00000 + 4.00000i) q^{65} -8.00000i q^{67} +12.0000 q^{71} -4.00000i q^{73} +4.00000 q^{79} +9.00000 q^{81} +(-8.00000 - 4.00000i) q^{85} +10.0000 q^{89} +4.00000 q^{91} +(-4.00000 + 8.00000i) q^{95} +12.0000i q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{5} + 6 q^{9}+O(q^{10})$$ 2 * q + 2 * q^5 + 6 * q^9 $$2 q + 2 q^{5} + 6 q^{9} - 8 q^{19} - 6 q^{25} - 4 q^{29} - 16 q^{31} - 4 q^{35} + 12 q^{41} + 6 q^{45} - 2 q^{49} + 8 q^{59} - 12 q^{61} + 16 q^{65} + 24 q^{71} + 8 q^{79} + 18 q^{81} - 16 q^{85} + 20 q^{89} + 8 q^{91} - 8 q^{95}+O(q^{100})$$ 2 * q + 2 * q^5 + 6 * q^9 - 8 * q^19 - 6 * q^25 - 4 * q^29 - 16 * q^31 - 4 * q^35 + 12 * q^41 + 6 * q^45 - 2 * q^49 + 8 * q^59 - 12 * q^61 + 16 * q^65 + 24 * q^71 + 8 * q^79 + 18 * q^81 - 16 * q^85 + 20 * q^89 + 8 * q^91 - 8 * q^95

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/140\mathbb{Z}\right)^\times$$.

 $$n$$ $$57$$ $$71$$ $$101$$ $$\chi(n)$$ $$-1$$ $$1$$ $$1$$

## 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 1.00000 $$0$$
−1.00000 $$\pi$$
$$4$$ 0 0
$$5$$ 1.00000 2.00000i 0.447214 0.894427i
$$6$$ 0 0
$$7$$ 1.00000i 0.377964i
$$8$$ 0 0
$$9$$ 3.00000 1.00000
$$10$$ 0 0
$$11$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$12$$ 0 0
$$13$$ 4.00000i 1.10940i 0.832050 + 0.554700i $$0.187167\pi$$
−0.832050 + 0.554700i $$0.812833\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 0 0
$$17$$ 4.00000i 0.970143i −0.874475 0.485071i $$-0.838794\pi$$
0.874475 0.485071i $$-0.161206\pi$$
$$18$$ 0 0
$$19$$ −4.00000 −0.917663 −0.458831 0.888523i $$-0.651732\pi$$
−0.458831 + 0.888523i $$0.651732\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 0 0
$$23$$ 8.00000i 1.66812i 0.551677 + 0.834058i $$0.313988\pi$$
−0.551677 + 0.834058i $$0.686012\pi$$
$$24$$ 0 0
$$25$$ −3.00000 4.00000i −0.600000 0.800000i
$$26$$ 0 0
$$27$$ 0 0
$$28$$ 0 0
$$29$$ −2.00000 −0.371391 −0.185695 0.982607i $$-0.559454\pi$$
−0.185695 + 0.982607i $$0.559454\pi$$
$$30$$ 0 0
$$31$$ −8.00000 −1.43684 −0.718421 0.695608i $$-0.755135\pi$$
−0.718421 + 0.695608i $$0.755135\pi$$
$$32$$ 0 0
$$33$$ 0 0
$$34$$ 0 0
$$35$$ −2.00000 1.00000i −0.338062 0.169031i
$$36$$ 0 0
$$37$$ 8.00000i 1.31519i 0.753371 + 0.657596i $$0.228427\pi$$
−0.753371 + 0.657596i $$0.771573\pi$$
$$38$$ 0 0
$$39$$ 0 0
$$40$$ 0 0
$$41$$ 6.00000 0.937043 0.468521 0.883452i $$-0.344787\pi$$
0.468521 + 0.883452i $$0.344787\pi$$
$$42$$ 0 0
$$43$$ 8.00000i 1.21999i 0.792406 + 0.609994i $$0.208828\pi$$
−0.792406 + 0.609994i $$0.791172\pi$$
$$44$$ 0 0
$$45$$ 3.00000 6.00000i 0.447214 0.894427i
$$46$$ 0 0
$$47$$ 8.00000i 1.16692i −0.812142 0.583460i $$-0.801699\pi$$
0.812142 0.583460i $$-0.198301\pi$$
$$48$$ 0 0
$$49$$ −1.00000 −0.142857
$$50$$ 0 0
$$51$$ 0 0
$$52$$ 0 0
$$53$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 0 0
$$57$$ 0 0
$$58$$ 0 0
$$59$$ 4.00000 0.520756 0.260378 0.965507i $$-0.416153\pi$$
0.260378 + 0.965507i $$0.416153\pi$$
$$60$$ 0 0
$$61$$ −6.00000 −0.768221 −0.384111 0.923287i $$-0.625492\pi$$
−0.384111 + 0.923287i $$0.625492\pi$$
$$62$$ 0 0
$$63$$ 3.00000i 0.377964i
$$64$$ 0 0
$$65$$ 8.00000 + 4.00000i 0.992278 + 0.496139i
$$66$$ 0 0
$$67$$ 8.00000i 0.977356i −0.872464 0.488678i $$-0.837479\pi$$
0.872464 0.488678i $$-0.162521\pi$$
$$68$$ 0 0
$$69$$ 0 0
$$70$$ 0 0
$$71$$ 12.0000 1.42414 0.712069 0.702109i $$-0.247758\pi$$
0.712069 + 0.702109i $$0.247758\pi$$
$$72$$ 0 0
$$73$$ 4.00000i 0.468165i −0.972217 0.234082i $$-0.924791\pi$$
0.972217 0.234082i $$-0.0752085\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 0 0
$$77$$ 0 0
$$78$$ 0 0
$$79$$ 4.00000 0.450035 0.225018 0.974355i $$-0.427756\pi$$
0.225018 + 0.974355i $$0.427756\pi$$
$$80$$ 0 0
$$81$$ 9.00000 1.00000
$$82$$ 0 0
$$83$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$84$$ 0 0
$$85$$ −8.00000 4.00000i −0.867722 0.433861i
$$86$$ 0 0
$$87$$ 0 0
$$88$$ 0 0
$$89$$ 10.0000 1.06000 0.529999 0.847998i $$-0.322192\pi$$
0.529999 + 0.847998i $$0.322192\pi$$
$$90$$ 0 0
$$91$$ 4.00000 0.419314
$$92$$ 0 0
$$93$$ 0 0
$$94$$ 0 0
$$95$$ −4.00000 + 8.00000i −0.410391 + 0.820783i
$$96$$ 0 0
$$97$$ 12.0000i 1.21842i 0.793011 + 0.609208i $$0.208512\pi$$
−0.793011 + 0.609208i $$0.791488\pi$$
$$98$$ 0 0
$$99$$ 0 0
$$100$$ 0 0
$$101$$ −18.0000 −1.79107 −0.895533 0.444994i $$-0.853206\pi$$
−0.895533 + 0.444994i $$0.853206\pi$$
$$102$$ 0 0
$$103$$ 8.00000i 0.788263i −0.919054 0.394132i $$-0.871045\pi$$
0.919054 0.394132i $$-0.128955\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ 8.00000i 0.773389i −0.922208 0.386695i $$-0.873617\pi$$
0.922208 0.386695i $$-0.126383\pi$$
$$108$$ 0 0
$$109$$ −14.0000 −1.34096 −0.670478 0.741929i $$-0.733911\pi$$
−0.670478 + 0.741929i $$0.733911\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ 0 0
$$113$$ 16.0000i 1.50515i −0.658505 0.752577i $$-0.728811\pi$$
0.658505 0.752577i $$-0.271189\pi$$
$$114$$ 0 0
$$115$$ 16.0000 + 8.00000i 1.49201 + 0.746004i
$$116$$ 0 0
$$117$$ 12.0000i 1.10940i
$$118$$ 0 0
$$119$$ −4.00000 −0.366679
$$120$$ 0 0
$$121$$ −11.0000 −1.00000
$$122$$ 0 0
$$123$$ 0 0
$$124$$ 0 0
$$125$$ −11.0000 + 2.00000i −0.983870 + 0.178885i
$$126$$ 0 0
$$127$$ 8.00000i 0.709885i 0.934888 + 0.354943i $$0.115500\pi$$
−0.934888 + 0.354943i $$0.884500\pi$$
$$128$$ 0 0
$$129$$ 0 0
$$130$$ 0 0
$$131$$ −4.00000 −0.349482 −0.174741 0.984614i $$-0.555909\pi$$
−0.174741 + 0.984614i $$0.555909\pi$$
$$132$$ 0 0
$$133$$ 4.00000i 0.346844i
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ 8.00000i 0.683486i 0.939793 + 0.341743i $$0.111017\pi$$
−0.939793 + 0.341743i $$0.888983\pi$$
$$138$$ 0 0
$$139$$ −4.00000 −0.339276 −0.169638 0.985506i $$-0.554260\pi$$
−0.169638 + 0.985506i $$0.554260\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 0 0
$$143$$ 0 0
$$144$$ 0 0
$$145$$ −2.00000 + 4.00000i −0.166091 + 0.332182i
$$146$$ 0 0
$$147$$ 0 0
$$148$$ 0 0
$$149$$ −6.00000 −0.491539 −0.245770 0.969328i $$-0.579041\pi$$
−0.245770 + 0.969328i $$0.579041\pi$$
$$150$$ 0 0
$$151$$ 12.0000 0.976546 0.488273 0.872691i $$-0.337627\pi$$
0.488273 + 0.872691i $$0.337627\pi$$
$$152$$ 0 0
$$153$$ 12.0000i 0.970143i
$$154$$ 0 0
$$155$$ −8.00000 + 16.0000i −0.642575 + 1.28515i
$$156$$ 0 0
$$157$$ 12.0000i 0.957704i 0.877896 + 0.478852i $$0.158947\pi$$
−0.877896 + 0.478852i $$0.841053\pi$$
$$158$$ 0 0
$$159$$ 0 0
$$160$$ 0 0
$$161$$ 8.00000 0.630488
$$162$$ 0 0
$$163$$ 24.0000i 1.87983i −0.341415 0.939913i $$-0.610906\pi$$
0.341415 0.939913i $$-0.389094\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ 24.0000i 1.85718i −0.371113 0.928588i $$-0.621024\pi$$
0.371113 0.928588i $$-0.378976\pi$$
$$168$$ 0 0
$$169$$ −3.00000 −0.230769
$$170$$ 0 0
$$171$$ −12.0000 −0.917663
$$172$$ 0 0
$$173$$ 12.0000i 0.912343i 0.889892 + 0.456172i $$0.150780\pi$$
−0.889892 + 0.456172i $$0.849220\pi$$
$$174$$ 0 0
$$175$$ −4.00000 + 3.00000i −0.302372 + 0.226779i
$$176$$ 0 0
$$177$$ 0 0
$$178$$ 0 0
$$179$$ 8.00000 0.597948 0.298974 0.954261i $$-0.403356\pi$$
0.298974 + 0.954261i $$0.403356\pi$$
$$180$$ 0 0
$$181$$ 14.0000 1.04061 0.520306 0.853980i $$-0.325818\pi$$
0.520306 + 0.853980i $$0.325818\pi$$
$$182$$ 0 0
$$183$$ 0 0
$$184$$ 0 0
$$185$$ 16.0000 + 8.00000i 1.17634 + 0.588172i
$$186$$ 0 0
$$187$$ 0 0
$$188$$ 0 0
$$189$$ 0 0
$$190$$ 0 0
$$191$$ −20.0000 −1.44715 −0.723575 0.690246i $$-0.757502\pi$$
−0.723575 + 0.690246i $$0.757502\pi$$
$$192$$ 0 0
$$193$$ 16.0000i 1.15171i −0.817554 0.575853i $$-0.804670\pi$$
0.817554 0.575853i $$-0.195330\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ 16.0000i 1.13995i −0.821661 0.569976i $$-0.806952\pi$$
0.821661 0.569976i $$-0.193048\pi$$
$$198$$ 0 0
$$199$$ 24.0000 1.70131 0.850657 0.525720i $$-0.176204\pi$$
0.850657 + 0.525720i $$0.176204\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ 0 0
$$203$$ 2.00000i 0.140372i
$$204$$ 0 0
$$205$$ 6.00000 12.0000i 0.419058 0.838116i
$$206$$ 0 0
$$207$$ 24.0000i 1.66812i
$$208$$ 0 0
$$209$$ 0 0
$$210$$ 0 0
$$211$$ 24.0000 1.65223 0.826114 0.563503i $$-0.190547\pi$$
0.826114 + 0.563503i $$0.190547\pi$$
$$212$$ 0 0
$$213$$ 0 0
$$214$$ 0 0
$$215$$ 16.0000 + 8.00000i 1.09119 + 0.545595i
$$216$$ 0 0
$$217$$ 8.00000i 0.543075i
$$218$$ 0 0
$$219$$ 0 0
$$220$$ 0 0
$$221$$ 16.0000 1.07628
$$222$$ 0 0
$$223$$ 8.00000i 0.535720i −0.963458 0.267860i $$-0.913684\pi$$
0.963458 0.267860i $$-0.0863164\pi$$
$$224$$ 0 0
$$225$$ −9.00000 12.0000i −0.600000 0.800000i
$$226$$ 0 0
$$227$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$228$$ 0 0
$$229$$ 14.0000 0.925146 0.462573 0.886581i $$-0.346926\pi$$
0.462573 + 0.886581i $$0.346926\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ 8.00000i 0.524097i −0.965055 0.262049i $$-0.915602\pi$$
0.965055 0.262049i $$-0.0843981\pi$$
$$234$$ 0 0
$$235$$ −16.0000 8.00000i −1.04372 0.521862i
$$236$$ 0 0
$$237$$ 0 0
$$238$$ 0 0
$$239$$ −20.0000 −1.29369 −0.646846 0.762620i $$-0.723912\pi$$
−0.646846 + 0.762620i $$0.723912\pi$$
$$240$$ 0 0
$$241$$ −18.0000 −1.15948 −0.579741 0.814801i $$-0.696846\pi$$
−0.579741 + 0.814801i $$0.696846\pi$$
$$242$$ 0 0
$$243$$ 0 0
$$244$$ 0 0
$$245$$ −1.00000 + 2.00000i −0.0638877 + 0.127775i
$$246$$ 0 0
$$247$$ 16.0000i 1.01806i
$$248$$ 0 0
$$249$$ 0 0
$$250$$ 0 0
$$251$$ −4.00000 −0.252478 −0.126239 0.992000i $$-0.540291\pi$$
−0.126239 + 0.992000i $$0.540291\pi$$
$$252$$ 0 0
$$253$$ 0 0
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 0 0
$$257$$ 12.0000i 0.748539i 0.927320 + 0.374270i $$0.122107\pi$$
−0.927320 + 0.374270i $$0.877893\pi$$
$$258$$ 0 0
$$259$$ 8.00000 0.497096
$$260$$ 0 0
$$261$$ −6.00000 −0.371391
$$262$$ 0 0
$$263$$ 24.0000i 1.47990i 0.672660 + 0.739952i $$0.265152\pi$$
−0.672660 + 0.739952i $$0.734848\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ 0 0
$$267$$ 0 0
$$268$$ 0 0
$$269$$ 10.0000 0.609711 0.304855 0.952399i $$-0.401392\pi$$
0.304855 + 0.952399i $$0.401392\pi$$
$$270$$ 0 0
$$271$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ 0 0
$$275$$ 0 0
$$276$$ 0 0
$$277$$ 16.0000i 0.961347i −0.876900 0.480673i $$-0.840392\pi$$
0.876900 0.480673i $$-0.159608\pi$$
$$278$$ 0 0
$$279$$ −24.0000 −1.43684
$$280$$ 0 0
$$281$$ 6.00000 0.357930 0.178965 0.983855i $$-0.442725\pi$$
0.178965 + 0.983855i $$0.442725\pi$$
$$282$$ 0 0
$$283$$ 32.0000i 1.90220i 0.308879 + 0.951101i $$0.400046\pi$$
−0.308879 + 0.951101i $$0.599954\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ 6.00000i 0.354169i
$$288$$ 0 0
$$289$$ 1.00000 0.0588235
$$290$$ 0 0
$$291$$ 0 0
$$292$$ 0 0
$$293$$ 12.0000i 0.701047i 0.936554 + 0.350524i $$0.113996\pi$$
−0.936554 + 0.350524i $$0.886004\pi$$
$$294$$ 0 0
$$295$$ 4.00000 8.00000i 0.232889 0.465778i
$$296$$ 0 0
$$297$$ 0 0
$$298$$ 0 0
$$299$$ −32.0000 −1.85061
$$300$$ 0 0
$$301$$ 8.00000 0.461112
$$302$$ 0 0
$$303$$ 0 0
$$304$$ 0 0
$$305$$ −6.00000 + 12.0000i −0.343559 + 0.687118i
$$306$$ 0 0
$$307$$ 16.0000i 0.913168i 0.889680 + 0.456584i $$0.150927\pi$$
−0.889680 + 0.456584i $$0.849073\pi$$
$$308$$ 0 0
$$309$$ 0 0
$$310$$ 0 0
$$311$$ −24.0000 −1.36092 −0.680458 0.732787i $$-0.738219\pi$$
−0.680458 + 0.732787i $$0.738219\pi$$
$$312$$ 0 0
$$313$$ 4.00000i 0.226093i −0.993590 0.113047i $$-0.963939\pi$$
0.993590 0.113047i $$-0.0360610\pi$$
$$314$$ 0 0
$$315$$ −6.00000 3.00000i −0.338062 0.169031i
$$316$$ 0 0
$$317$$ 16.0000i 0.898650i −0.893368 0.449325i $$-0.851665\pi$$
0.893368 0.449325i $$-0.148335\pi$$
$$318$$ 0 0
$$319$$ 0 0
$$320$$ 0 0
$$321$$ 0 0
$$322$$ 0 0
$$323$$ 16.0000i 0.890264i
$$324$$ 0 0
$$325$$ 16.0000 12.0000i 0.887520 0.665640i
$$326$$ 0 0
$$327$$ 0 0
$$328$$ 0 0
$$329$$ −8.00000 −0.441054
$$330$$ 0 0
$$331$$ −16.0000 −0.879440 −0.439720 0.898135i $$-0.644922\pi$$
−0.439720 + 0.898135i $$0.644922\pi$$
$$332$$ 0 0
$$333$$ 24.0000i 1.31519i
$$334$$ 0 0
$$335$$ −16.0000 8.00000i −0.874173 0.437087i
$$336$$ 0 0
$$337$$ 8.00000i 0.435788i 0.975972 + 0.217894i $$0.0699187\pi$$
−0.975972 + 0.217894i $$0.930081\pi$$
$$338$$ 0 0
$$339$$ 0 0
$$340$$ 0 0
$$341$$ 0 0
$$342$$ 0 0
$$343$$ 1.00000i 0.0539949i
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ 24.0000i 1.28839i −0.764862 0.644194i $$-0.777193\pi$$
0.764862 0.644194i $$-0.222807\pi$$
$$348$$ 0 0
$$349$$ 6.00000 0.321173 0.160586 0.987022i $$-0.448662\pi$$
0.160586 + 0.987022i $$0.448662\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 0 0
$$353$$ 12.0000i 0.638696i −0.947638 0.319348i $$-0.896536\pi$$
0.947638 0.319348i $$-0.103464\pi$$
$$354$$ 0 0
$$355$$ 12.0000 24.0000i 0.636894 1.27379i
$$356$$ 0 0
$$357$$ 0 0
$$358$$ 0 0
$$359$$ 20.0000 1.05556 0.527780 0.849381i $$-0.323025\pi$$
0.527780 + 0.849381i $$0.323025\pi$$
$$360$$ 0 0
$$361$$ −3.00000 −0.157895
$$362$$ 0 0
$$363$$ 0 0
$$364$$ 0 0
$$365$$ −8.00000 4.00000i −0.418739 0.209370i
$$366$$ 0 0
$$367$$ 8.00000i 0.417597i 0.977959 + 0.208798i $$0.0669552\pi$$
−0.977959 + 0.208798i $$0.933045\pi$$
$$368$$ 0 0
$$369$$ 18.0000 0.937043
$$370$$ 0 0
$$371$$ 0 0
$$372$$ 0 0
$$373$$ 32.0000i 1.65690i 0.560065 + 0.828449i $$0.310776\pi$$
−0.560065 + 0.828449i $$0.689224\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ 8.00000i 0.412021i
$$378$$ 0 0
$$379$$ 8.00000 0.410932 0.205466 0.978664i $$-0.434129\pi$$
0.205466 + 0.978664i $$0.434129\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ 0 0
$$383$$ 24.0000i 1.22634i 0.789950 + 0.613171i $$0.210106\pi$$
−0.789950 + 0.613171i $$0.789894\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ 24.0000i 1.21999i
$$388$$ 0 0
$$389$$ −6.00000 −0.304212 −0.152106 0.988364i $$-0.548606\pi$$
−0.152106 + 0.988364i $$0.548606\pi$$
$$390$$ 0 0
$$391$$ 32.0000 1.61831
$$392$$ 0 0
$$393$$ 0 0
$$394$$ 0 0
$$395$$ 4.00000 8.00000i 0.201262 0.402524i
$$396$$ 0 0
$$397$$ 28.0000i 1.40528i −0.711546 0.702640i $$-0.752005\pi$$
0.711546 0.702640i $$-0.247995\pi$$
$$398$$ 0 0
$$399$$ 0 0
$$400$$ 0 0
$$401$$ 18.0000 0.898877 0.449439 0.893311i $$-0.351624\pi$$
0.449439 + 0.893311i $$0.351624\pi$$
$$402$$ 0 0
$$403$$ 32.0000i 1.59403i
$$404$$ 0 0
$$405$$ 9.00000 18.0000i 0.447214 0.894427i
$$406$$ 0 0
$$407$$ 0 0
$$408$$ 0 0
$$409$$ −10.0000 −0.494468 −0.247234 0.968956i $$-0.579522\pi$$
−0.247234 + 0.968956i $$0.579522\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ 0 0
$$413$$ 4.00000i 0.196827i
$$414$$ 0 0
$$415$$ 0 0
$$416$$ 0 0
$$417$$ 0 0
$$418$$ 0 0
$$419$$ 28.0000 1.36789 0.683945 0.729534i $$-0.260263\pi$$
0.683945 + 0.729534i $$0.260263\pi$$
$$420$$ 0 0
$$421$$ 10.0000 0.487370 0.243685 0.969854i $$-0.421644\pi$$
0.243685 + 0.969854i $$0.421644\pi$$
$$422$$ 0 0
$$423$$ 24.0000i 1.16692i
$$424$$ 0 0
$$425$$ −16.0000 + 12.0000i −0.776114 + 0.582086i
$$426$$ 0 0
$$427$$ 6.00000i 0.290360i
$$428$$ 0 0
$$429$$ 0 0
$$430$$ 0 0
$$431$$ 20.0000 0.963366 0.481683 0.876346i $$-0.340026\pi$$
0.481683 + 0.876346i $$0.340026\pi$$
$$432$$ 0 0
$$433$$ 28.0000i 1.34559i 0.739827 + 0.672797i $$0.234907\pi$$
−0.739827 + 0.672797i $$0.765093\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 0 0
$$437$$ 32.0000i 1.53077i
$$438$$ 0 0
$$439$$ −8.00000 −0.381819 −0.190910 0.981608i $$-0.561144\pi$$
−0.190910 + 0.981608i $$0.561144\pi$$
$$440$$ 0 0
$$441$$ −3.00000 −0.142857
$$442$$ 0 0
$$443$$ 24.0000i 1.14027i 0.821549 + 0.570137i $$0.193110\pi$$
−0.821549 + 0.570137i $$0.806890\pi$$
$$444$$ 0 0
$$445$$ 10.0000 20.0000i 0.474045 0.948091i
$$446$$ 0 0
$$447$$ 0 0
$$448$$ 0 0
$$449$$ −2.00000 −0.0943858 −0.0471929 0.998886i $$-0.515028\pi$$
−0.0471929 + 0.998886i $$0.515028\pi$$
$$450$$ 0 0
$$451$$ 0 0
$$452$$ 0 0
$$453$$ 0 0
$$454$$ 0 0
$$455$$ 4.00000 8.00000i 0.187523 0.375046i
$$456$$ 0 0
$$457$$ 16.0000i 0.748448i 0.927338 + 0.374224i $$0.122091\pi$$
−0.927338 + 0.374224i $$0.877909\pi$$
$$458$$ 0 0
$$459$$ 0 0
$$460$$ 0 0
$$461$$ −26.0000 −1.21094 −0.605470 0.795868i $$-0.707015\pi$$
−0.605470 + 0.795868i $$0.707015\pi$$
$$462$$ 0 0
$$463$$ 8.00000i 0.371792i −0.982569 0.185896i $$-0.940481\pi$$
0.982569 0.185896i $$-0.0595187\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ 16.0000i 0.740392i −0.928954 0.370196i $$-0.879291\pi$$
0.928954 0.370196i $$-0.120709\pi$$
$$468$$ 0 0
$$469$$ −8.00000 −0.369406
$$470$$ 0 0
$$471$$ 0 0
$$472$$ 0 0
$$473$$ 0 0
$$474$$ 0 0
$$475$$ 12.0000 + 16.0000i 0.550598 + 0.734130i
$$476$$ 0 0
$$477$$ 0 0
$$478$$ 0 0
$$479$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$480$$ 0 0
$$481$$ −32.0000 −1.45907
$$482$$ 0 0
$$483$$ 0 0
$$484$$ 0 0
$$485$$ 24.0000 + 12.0000i 1.08978 + 0.544892i
$$486$$ 0 0
$$487$$ 8.00000i 0.362515i 0.983436 + 0.181257i $$0.0580167\pi$$
−0.983436 + 0.181257i $$0.941983\pi$$
$$488$$ 0 0
$$489$$ 0 0
$$490$$ 0 0
$$491$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$492$$ 0 0
$$493$$ 8.00000i 0.360302i
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ 12.0000i 0.538274i
$$498$$ 0 0
$$499$$ 8.00000 0.358129 0.179065 0.983837i $$-0.442693\pi$$
0.179065 + 0.983837i $$0.442693\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ 0 0
$$503$$ 8.00000i 0.356702i −0.983967 0.178351i $$-0.942924\pi$$
0.983967 0.178351i $$-0.0570763\pi$$
$$504$$ 0 0
$$505$$ −18.0000 + 36.0000i −0.800989 + 1.60198i
$$506$$ 0 0
$$507$$ 0 0
$$508$$ 0 0
$$509$$ 22.0000 0.975133 0.487566 0.873086i $$-0.337885\pi$$
0.487566 + 0.873086i $$0.337885\pi$$
$$510$$ 0 0
$$511$$ −4.00000 −0.176950
$$512$$ 0 0
$$513$$ 0 0
$$514$$ 0 0
$$515$$ −16.0000 8.00000i −0.705044 0.352522i
$$516$$ 0 0
$$517$$ 0 0
$$518$$ 0 0
$$519$$ 0 0
$$520$$ 0 0
$$521$$ −6.00000 −0.262865 −0.131432 0.991325i $$-0.541958\pi$$
−0.131432 + 0.991325i $$0.541958\pi$$
$$522$$ 0 0
$$523$$ 32.0000i 1.39926i −0.714504 0.699631i $$-0.753348\pi$$
0.714504 0.699631i $$-0.246652\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ 32.0000i 1.39394i
$$528$$ 0 0
$$529$$ −41.0000 −1.78261
$$530$$ 0 0
$$531$$ 12.0000 0.520756
$$532$$ 0 0
$$533$$ 24.0000i 1.03956i
$$534$$ 0 0
$$535$$ −16.0000 8.00000i −0.691740 0.345870i
$$536$$ 0 0
$$537$$ 0 0
$$538$$ 0 0
$$539$$ 0 0
$$540$$ 0 0
$$541$$ 14.0000 0.601907 0.300954 0.953639i $$-0.402695\pi$$
0.300954 + 0.953639i $$0.402695\pi$$
$$542$$ 0 0
$$543$$ 0 0
$$544$$ 0 0
$$545$$ −14.0000 + 28.0000i −0.599694 + 1.19939i
$$546$$ 0 0
$$547$$ 8.00000i 0.342055i 0.985266 + 0.171028i $$0.0547087\pi$$
−0.985266 + 0.171028i $$0.945291\pi$$
$$548$$ 0 0
$$549$$ −18.0000 −0.768221
$$550$$ 0 0
$$551$$ 8.00000 0.340811
$$552$$ 0 0
$$553$$ 4.00000i 0.170097i
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 0 0
$$557$$ 16.0000i 0.677942i 0.940797 + 0.338971i $$0.110079\pi$$
−0.940797 + 0.338971i $$0.889921\pi$$
$$558$$ 0 0
$$559$$ −32.0000 −1.35346
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 0 0
$$563$$ 16.0000i 0.674320i −0.941447 0.337160i $$-0.890534\pi$$
0.941447 0.337160i $$-0.109466\pi$$
$$564$$ 0 0
$$565$$ −32.0000 16.0000i −1.34625 0.673125i
$$566$$ 0 0
$$567$$ 9.00000i 0.377964i
$$568$$ 0 0
$$569$$ −42.0000 −1.76073 −0.880366 0.474295i $$-0.842703\pi$$
−0.880366 + 0.474295i $$0.842703\pi$$
$$570$$ 0 0
$$571$$ −40.0000 −1.67395 −0.836974 0.547243i $$-0.815677\pi$$
−0.836974 + 0.547243i $$0.815677\pi$$
$$572$$ 0 0
$$573$$ 0 0
$$574$$ 0 0
$$575$$ 32.0000 24.0000i 1.33449 1.00087i
$$576$$ 0 0
$$577$$ 4.00000i 0.166522i 0.996528 + 0.0832611i $$0.0265335\pi$$
−0.996528 + 0.0832611i $$0.973466\pi$$
$$578$$ 0 0
$$579$$ 0 0
$$580$$ 0 0
$$581$$ 0 0
$$582$$ 0 0
$$583$$ 0 0
$$584$$ 0 0
$$585$$ 24.0000 + 12.0000i 0.992278 + 0.496139i
$$586$$ 0 0
$$587$$ 32.0000i 1.32078i 0.750922 + 0.660391i $$0.229609\pi$$
−0.750922 + 0.660391i $$0.770391\pi$$
$$588$$ 0 0
$$589$$ 32.0000 1.31854
$$590$$ 0 0
$$591$$ 0 0
$$592$$ 0 0
$$593$$ 36.0000i 1.47834i −0.673517 0.739171i $$-0.735217\pi$$
0.673517 0.739171i $$-0.264783\pi$$
$$594$$ 0 0
$$595$$ −4.00000 + 8.00000i −0.163984 + 0.327968i
$$596$$ 0 0
$$597$$ 0 0
$$598$$ 0 0
$$599$$ 36.0000 1.47092 0.735460 0.677568i $$-0.236966\pi$$
0.735460 + 0.677568i $$0.236966\pi$$
$$600$$ 0 0
$$601$$ −10.0000 −0.407909 −0.203954 0.978980i $$-0.565379\pi$$
−0.203954 + 0.978980i $$0.565379\pi$$
$$602$$ 0 0
$$603$$ 24.0000i 0.977356i
$$604$$ 0 0
$$605$$ −11.0000 + 22.0000i −0.447214 + 0.894427i
$$606$$ 0 0
$$607$$ 40.0000i 1.62355i 0.583970 + 0.811775i $$0.301498\pi$$
−0.583970 + 0.811775i $$0.698502\pi$$
$$608$$ 0 0
$$609$$ 0 0
$$610$$ 0 0
$$611$$ 32.0000 1.29458
$$612$$ 0 0
$$613$$ 24.0000i 0.969351i −0.874694 0.484675i $$-0.838938\pi$$
0.874694 0.484675i $$-0.161062\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ 8.00000i 0.322068i −0.986949 0.161034i $$-0.948517\pi$$
0.986949 0.161034i $$-0.0514829\pi$$
$$618$$ 0 0
$$619$$ −4.00000 −0.160774 −0.0803868 0.996764i $$-0.525616\pi$$
−0.0803868 + 0.996764i $$0.525616\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ 0 0
$$623$$ 10.0000i 0.400642i
$$624$$ 0 0
$$625$$ −7.00000 + 24.0000i −0.280000 + 0.960000i
$$626$$ 0 0
$$627$$ 0 0
$$628$$ 0 0
$$629$$ 32.0000 1.27592
$$630$$ 0 0
$$631$$ −36.0000 −1.43314 −0.716569 0.697517i $$-0.754288\pi$$
−0.716569 + 0.697517i $$0.754288\pi$$
$$632$$ 0 0
$$633$$ 0 0
$$634$$ 0 0
$$635$$ 16.0000 + 8.00000i 0.634941 + 0.317470i
$$636$$ 0 0
$$637$$ 4.00000i 0.158486i
$$638$$ 0 0
$$639$$ 36.0000 1.42414
$$640$$ 0 0
$$641$$ −2.00000 −0.0789953 −0.0394976 0.999220i $$-0.512576\pi$$
−0.0394976 + 0.999220i $$0.512576\pi$$
$$642$$ 0 0
$$643$$ 16.0000i 0.630978i 0.948929 + 0.315489i $$0.102169\pi$$
−0.948929 + 0.315489i $$0.897831\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ 24.0000i 0.943537i −0.881722 0.471769i $$-0.843616\pi$$
0.881722 0.471769i $$-0.156384\pi$$
$$648$$ 0 0
$$649$$ 0 0
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 0 0
$$653$$ 24.0000i 0.939193i −0.882881 0.469596i $$-0.844399\pi$$
0.882881 0.469596i $$-0.155601\pi$$
$$654$$ 0 0
$$655$$ −4.00000 + 8.00000i −0.156293 + 0.312586i
$$656$$ 0 0
$$657$$ 12.0000i 0.468165i
$$658$$ 0 0
$$659$$ −24.0000 −0.934907 −0.467454 0.884018i $$-0.654829\pi$$
−0.467454 + 0.884018i $$0.654829\pi$$
$$660$$ 0 0
$$661$$ 34.0000 1.32245 0.661223 0.750189i $$-0.270038\pi$$
0.661223 + 0.750189i $$0.270038\pi$$
$$662$$ 0 0
$$663$$ 0 0
$$664$$ 0 0
$$665$$ 8.00000 + 4.00000i 0.310227 + 0.155113i
$$666$$ 0 0
$$667$$ 16.0000i 0.619522i
$$668$$ 0 0
$$669$$ 0 0
$$670$$ 0 0
$$671$$ 0 0
$$672$$ 0 0
$$673$$ 8.00000i 0.308377i 0.988041 + 0.154189i $$0.0492764\pi$$
−0.988041 + 0.154189i $$0.950724\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ 0 0
$$677$$ 4.00000i 0.153732i −0.997041 0.0768662i $$-0.975509\pi$$
0.997041 0.0768662i $$-0.0244914\pi$$
$$678$$ 0 0
$$679$$ 12.0000 0.460518
$$680$$ 0 0
$$681$$ 0 0
$$682$$ 0 0
$$683$$ 8.00000i 0.306111i −0.988218 0.153056i $$-0.951089\pi$$
0.988218 0.153056i $$-0.0489114\pi$$
$$684$$ 0 0
$$685$$ 16.0000 + 8.00000i 0.611329 + 0.305664i
$$686$$ 0 0
$$687$$ 0 0
$$688$$ 0 0
$$689$$ 0 0
$$690$$ 0 0
$$691$$ 28.0000 1.06517 0.532585 0.846376i $$-0.321221\pi$$
0.532585 + 0.846376i $$0.321221\pi$$
$$692$$ 0 0
$$693$$ 0 0
$$694$$ 0 0
$$695$$ −4.00000 + 8.00000i −0.151729 + 0.303457i
$$696$$ 0 0
$$697$$ 24.0000i 0.909065i
$$698$$ 0 0
$$699$$ 0 0
$$700$$ 0 0
$$701$$ 2.00000 0.0755390 0.0377695 0.999286i $$-0.487975\pi$$
0.0377695 + 0.999286i $$0.487975\pi$$
$$702$$ 0 0
$$703$$ 32.0000i 1.20690i
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ 18.0000i 0.676960i
$$708$$ 0 0
$$709$$ 6.00000 0.225335 0.112667 0.993633i $$-0.464061\pi$$
0.112667 + 0.993633i $$0.464061\pi$$
$$710$$ 0 0
$$711$$ 12.0000 0.450035
$$712$$ 0 0
$$713$$ 64.0000i 2.39682i
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 0 0
$$717$$ 0 0
$$718$$ 0 0
$$719$$ −32.0000 −1.19340 −0.596699 0.802465i $$-0.703521\pi$$
−0.596699 + 0.802465i $$0.703521\pi$$
$$720$$ 0 0
$$721$$ −8.00000 −0.297936
$$722$$ 0 0
$$723$$ 0 0
$$724$$ 0 0
$$725$$ 6.00000 + 8.00000i 0.222834 + 0.297113i
$$726$$ 0 0
$$727$$ 8.00000i 0.296704i 0.988935 + 0.148352i $$0.0473968\pi$$
−0.988935 + 0.148352i $$0.952603\pi$$
$$728$$ 0 0
$$729$$ 27.0000 1.00000
$$730$$ 0 0
$$731$$ 32.0000 1.18356
$$732$$ 0 0
$$733$$ 20.0000i 0.738717i 0.929287 + 0.369358i $$0.120423\pi$$
−0.929287 + 0.369358i $$0.879577\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ 0 0
$$738$$ 0 0
$$739$$ 8.00000 0.294285 0.147142 0.989115i $$-0.452992\pi$$
0.147142 + 0.989115i $$0.452992\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 0 0
$$743$$ 24.0000i 0.880475i −0.897881 0.440237i $$-0.854894\pi$$
0.897881 0.440237i $$-0.145106\pi$$
$$744$$ 0 0
$$745$$ −6.00000 + 12.0000i −0.219823 + 0.439646i
$$746$$ 0 0
$$747$$ 0 0
$$748$$ 0 0
$$749$$ −8.00000 −0.292314
$$750$$ 0 0
$$751$$ −20.0000 −0.729810 −0.364905 0.931045i $$-0.618899\pi$$
−0.364905 + 0.931045i $$0.618899\pi$$
$$752$$ 0 0
$$753$$ 0 0
$$754$$ 0 0
$$755$$ 12.0000 24.0000i 0.436725 0.873449i
$$756$$ 0 0
$$757$$ 24.0000i 0.872295i 0.899875 + 0.436147i $$0.143657\pi$$
−0.899875 + 0.436147i $$0.856343\pi$$
$$758$$ 0 0
$$759$$ 0 0
$$760$$ 0 0
$$761$$ 10.0000 0.362500 0.181250 0.983437i $$-0.441986\pi$$
0.181250 + 0.983437i $$0.441986\pi$$
$$762$$ 0 0
$$763$$ 14.0000i 0.506834i
$$764$$ 0 0
$$765$$ −24.0000 12.0000i −0.867722 0.433861i
$$766$$ 0 0
$$767$$ 16.0000i 0.577727i
$$768$$ 0 0
$$769$$ 2.00000 0.0721218 0.0360609 0.999350i $$-0.488519\pi$$
0.0360609 + 0.999350i $$0.488519\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ 0 0
$$773$$ 36.0000i 1.29483i 0.762138 + 0.647415i $$0.224150\pi$$
−0.762138 + 0.647415i $$0.775850\pi$$
$$774$$ 0 0
$$775$$ 24.0000 + 32.0000i 0.862105 + 1.14947i
$$776$$ 0 0
$$777$$ 0 0
$$778$$ 0 0
$$779$$ −24.0000 −0.859889
$$780$$ 0 0
$$781$$ 0 0
$$782$$ 0 0
$$783$$ 0 0
$$784$$ 0 0
$$785$$ 24.0000 + 12.0000i 0.856597 + 0.428298i
$$786$$ 0 0
$$787$$ 48.0000i 1.71102i 0.517790 + 0.855508i $$0.326755\pi$$
−0.517790 + 0.855508i $$0.673245\pi$$
$$788$$ 0 0
$$789$$ 0 0
$$790$$ 0 0
$$791$$ −16.0000 −0.568895
$$792$$ 0 0
$$793$$ 24.0000i 0.852265i
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 0 0
$$797$$ 28.0000i 0.991811i −0.868377 0.495905i $$-0.834836\pi$$
0.868377 0.495905i $$-0.165164\pi$$
$$798$$ 0 0
$$799$$ −32.0000 −1.13208
$$800$$ 0 0
$$801$$ 30.0000 1.06000
$$802$$ 0 0
$$803$$ 0 0
$$804$$ 0 0
$$805$$ 8.00000 16.0000i 0.281963 0.563926i
$$806$$ 0