# Properties

 Label 1200.2.h.n.1151.3 Level $1200$ Weight $2$ Character 1200.1151 Analytic conductor $9.582$ Analytic rank $0$ Dimension $4$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1200 = 2^{4} \cdot 3 \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1200.h (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$9.58204824255$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\sqrt{-2}, \sqrt{-3})$$ Defining polynomial: $$x^{4} - 2 x^{2} + 4$$ Coefficient ring: $$\Z[a_1, \ldots, a_{17}]$$ Coefficient ring index: $$2^{4}$$ Twist minimal: no (minimal twist has level 240) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 1151.3 Root $$1.22474 - 0.707107i$$ of defining polynomial Character $$\chi$$ $$=$$ 1200.1151 Dual form 1200.2.h.n.1151.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+(1.00000 + 1.41421i) q^{3} +(-1.00000 + 2.82843i) q^{9} +O(q^{10})$$ $$q+(1.00000 + 1.41421i) q^{3} +(-1.00000 + 2.82843i) q^{9} -4.89898 q^{11} -4.89898 q^{13} -3.46410i q^{17} +3.46410i q^{19} -6.00000 q^{23} +(-5.00000 + 1.41421i) q^{27} -2.82843i q^{29} +3.46410i q^{31} +(-4.89898 - 6.92820i) q^{33} -4.89898 q^{37} +(-4.89898 - 6.92820i) q^{39} +5.65685i q^{41} +8.48528i q^{43} +6.00000 q^{47} +7.00000 q^{49} +(4.89898 - 3.46410i) q^{51} -10.3923i q^{53} +(-4.89898 + 3.46410i) q^{57} +4.89898 q^{59} -2.00000 q^{61} +8.48528i q^{67} +(-6.00000 - 8.48528i) q^{69} +9.79796 q^{71} -9.79796 q^{73} +10.3923i q^{79} +(-7.00000 - 5.65685i) q^{81} -6.00000 q^{83} +(4.00000 - 2.82843i) q^{87} +5.65685i q^{89} +(-4.89898 + 3.46410i) q^{93} +(4.89898 - 13.8564i) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + 4q^{3} - 4q^{9} + O(q^{10})$$ $$4q + 4q^{3} - 4q^{9} - 24q^{23} - 20q^{27} + 24q^{47} + 28q^{49} - 8q^{61} - 24q^{69} - 28q^{81} - 24q^{83} + 16q^{87} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1200\mathbb{Z}\right)^\times$$.

 $$n$$ $$401$$ $$577$$ $$751$$ $$901$$ $$\chi(n)$$ $$-1$$ $$1$$ $$-1$$ $$1$$

## 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
$$3$$ 1.00000 + 1.41421i 0.577350 + 0.816497i
$$4$$ 0 0
$$5$$ 0 0
$$6$$ 0 0
$$7$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$8$$ 0 0
$$9$$ −1.00000 + 2.82843i −0.333333 + 0.942809i
$$10$$ 0 0
$$11$$ −4.89898 −1.47710 −0.738549 0.674200i $$-0.764489\pi$$
−0.738549 + 0.674200i $$0.764489\pi$$
$$12$$ 0 0
$$13$$ −4.89898 −1.35873 −0.679366 0.733799i $$-0.737745\pi$$
−0.679366 + 0.733799i $$0.737745\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 0 0
$$17$$ 3.46410i 0.840168i −0.907485 0.420084i $$-0.862001\pi$$
0.907485 0.420084i $$-0.137999\pi$$
$$18$$ 0 0
$$19$$ 3.46410i 0.794719i 0.917663 + 0.397360i $$0.130073\pi$$
−0.917663 + 0.397360i $$0.869927\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 0 0
$$23$$ −6.00000 −1.25109 −0.625543 0.780189i $$-0.715123\pi$$
−0.625543 + 0.780189i $$0.715123\pi$$
$$24$$ 0 0
$$25$$ 0 0
$$26$$ 0 0
$$27$$ −5.00000 + 1.41421i −0.962250 + 0.272166i
$$28$$ 0 0
$$29$$ 2.82843i 0.525226i −0.964901 0.262613i $$-0.915416\pi$$
0.964901 0.262613i $$-0.0845842\pi$$
$$30$$ 0 0
$$31$$ 3.46410i 0.622171i 0.950382 + 0.311086i $$0.100693\pi$$
−0.950382 + 0.311086i $$0.899307\pi$$
$$32$$ 0 0
$$33$$ −4.89898 6.92820i −0.852803 1.20605i
$$34$$ 0 0
$$35$$ 0 0
$$36$$ 0 0
$$37$$ −4.89898 −0.805387 −0.402694 0.915335i $$-0.631926\pi$$
−0.402694 + 0.915335i $$0.631926\pi$$
$$38$$ 0 0
$$39$$ −4.89898 6.92820i −0.784465 1.10940i
$$40$$ 0 0
$$41$$ 5.65685i 0.883452i 0.897150 + 0.441726i $$0.145634\pi$$
−0.897150 + 0.441726i $$0.854366\pi$$
$$42$$ 0 0
$$43$$ 8.48528i 1.29399i 0.762493 + 0.646997i $$0.223975\pi$$
−0.762493 + 0.646997i $$0.776025\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 0 0
$$47$$ 6.00000 0.875190 0.437595 0.899172i $$-0.355830\pi$$
0.437595 + 0.899172i $$0.355830\pi$$
$$48$$ 0 0
$$49$$ 7.00000 1.00000
$$50$$ 0 0
$$51$$ 4.89898 3.46410i 0.685994 0.485071i
$$52$$ 0 0
$$53$$ 10.3923i 1.42749i −0.700404 0.713746i $$-0.746997\pi$$
0.700404 0.713746i $$-0.253003\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 0 0
$$57$$ −4.89898 + 3.46410i −0.648886 + 0.458831i
$$58$$ 0 0
$$59$$ 4.89898 0.637793 0.318896 0.947790i $$-0.396688\pi$$
0.318896 + 0.947790i $$0.396688\pi$$
$$60$$ 0 0
$$61$$ −2.00000 −0.256074 −0.128037 0.991769i $$-0.540868\pi$$
−0.128037 + 0.991769i $$0.540868\pi$$
$$62$$ 0 0
$$63$$ 0 0
$$64$$ 0 0
$$65$$ 0 0
$$66$$ 0 0
$$67$$ 8.48528i 1.03664i 0.855186 + 0.518321i $$0.173443\pi$$
−0.855186 + 0.518321i $$0.826557\pi$$
$$68$$ 0 0
$$69$$ −6.00000 8.48528i −0.722315 1.02151i
$$70$$ 0 0
$$71$$ 9.79796 1.16280 0.581402 0.813617i $$-0.302504\pi$$
0.581402 + 0.813617i $$0.302504\pi$$
$$72$$ 0 0
$$73$$ −9.79796 −1.14676 −0.573382 0.819288i $$-0.694369\pi$$
−0.573382 + 0.819288i $$0.694369\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 0 0
$$77$$ 0 0
$$78$$ 0 0
$$79$$ 10.3923i 1.16923i 0.811312 + 0.584613i $$0.198754\pi$$
−0.811312 + 0.584613i $$0.801246\pi$$
$$80$$ 0 0
$$81$$ −7.00000 5.65685i −0.777778 0.628539i
$$82$$ 0 0
$$83$$ −6.00000 −0.658586 −0.329293 0.944228i $$-0.606810\pi$$
−0.329293 + 0.944228i $$0.606810\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ 0 0
$$87$$ 4.00000 2.82843i 0.428845 0.303239i
$$88$$ 0 0
$$89$$ 5.65685i 0.599625i 0.953998 + 0.299813i $$0.0969242\pi$$
−0.953998 + 0.299813i $$0.903076\pi$$
$$90$$ 0 0
$$91$$ 0 0
$$92$$ 0 0
$$93$$ −4.89898 + 3.46410i −0.508001 + 0.359211i
$$94$$ 0 0
$$95$$ 0 0
$$96$$ 0 0
$$97$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$98$$ 0 0
$$99$$ 4.89898 13.8564i 0.492366 1.39262i
$$100$$ 0 0
$$101$$ 14.1421i 1.40720i −0.710599 0.703598i $$-0.751576\pi$$
0.710599 0.703598i $$-0.248424\pi$$
$$102$$ 0 0
$$103$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ −18.0000 −1.74013 −0.870063 0.492941i $$-0.835922\pi$$
−0.870063 + 0.492941i $$0.835922\pi$$
$$108$$ 0 0
$$109$$ 2.00000 0.191565 0.0957826 0.995402i $$-0.469465\pi$$
0.0957826 + 0.995402i $$0.469465\pi$$
$$110$$ 0 0
$$111$$ −4.89898 6.92820i −0.464991 0.657596i
$$112$$ 0 0
$$113$$ 3.46410i 0.325875i 0.986636 + 0.162938i $$0.0520969\pi$$
−0.986636 + 0.162938i $$0.947903\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ 0 0
$$117$$ 4.89898 13.8564i 0.452911 1.28103i
$$118$$ 0 0
$$119$$ 0 0
$$120$$ 0 0
$$121$$ 13.0000 1.18182
$$122$$ 0 0
$$123$$ −8.00000 + 5.65685i −0.721336 + 0.510061i
$$124$$ 0 0
$$125$$ 0 0
$$126$$ 0 0
$$127$$ 16.9706i 1.50589i −0.658081 0.752947i $$-0.728632\pi$$
0.658081 0.752947i $$-0.271368\pi$$
$$128$$ 0 0
$$129$$ −12.0000 + 8.48528i −1.05654 + 0.747087i
$$130$$ 0 0
$$131$$ 4.89898 0.428026 0.214013 0.976831i $$-0.431347\pi$$
0.214013 + 0.976831i $$0.431347\pi$$
$$132$$ 0 0
$$133$$ 0 0
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ 3.46410i 0.295958i −0.988990 0.147979i $$-0.952723\pi$$
0.988990 0.147979i $$-0.0472768\pi$$
$$138$$ 0 0
$$139$$ 17.3205i 1.46911i 0.678551 + 0.734553i $$0.262608\pi$$
−0.678551 + 0.734553i $$0.737392\pi$$
$$140$$ 0 0
$$141$$ 6.00000 + 8.48528i 0.505291 + 0.714590i
$$142$$ 0 0
$$143$$ 24.0000 2.00698
$$144$$ 0 0
$$145$$ 0 0
$$146$$ 0 0
$$147$$ 7.00000 + 9.89949i 0.577350 + 0.816497i
$$148$$ 0 0
$$149$$ 2.82843i 0.231714i 0.993266 + 0.115857i $$0.0369614\pi$$
−0.993266 + 0.115857i $$0.963039\pi$$
$$150$$ 0 0
$$151$$ 3.46410i 0.281905i 0.990016 + 0.140952i $$0.0450164\pi$$
−0.990016 + 0.140952i $$0.954984\pi$$
$$152$$ 0 0
$$153$$ 9.79796 + 3.46410i 0.792118 + 0.280056i
$$154$$ 0 0
$$155$$ 0 0
$$156$$ 0 0
$$157$$ 4.89898 0.390981 0.195491 0.980706i $$-0.437370\pi$$
0.195491 + 0.980706i $$0.437370\pi$$
$$158$$ 0 0
$$159$$ 14.6969 10.3923i 1.16554 0.824163i
$$160$$ 0 0
$$161$$ 0 0
$$162$$ 0 0
$$163$$ 8.48528i 0.664619i −0.943170 0.332309i $$-0.892172\pi$$
0.943170 0.332309i $$-0.107828\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ 6.00000 0.464294 0.232147 0.972681i $$-0.425425\pi$$
0.232147 + 0.972681i $$0.425425\pi$$
$$168$$ 0 0
$$169$$ 11.0000 0.846154
$$170$$ 0 0
$$171$$ −9.79796 3.46410i −0.749269 0.264906i
$$172$$ 0 0
$$173$$ 3.46410i 0.263371i 0.991292 + 0.131685i $$0.0420389\pi$$
−0.991292 + 0.131685i $$0.957961\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 0 0
$$177$$ 4.89898 + 6.92820i 0.368230 + 0.520756i
$$178$$ 0 0
$$179$$ −24.4949 −1.83083 −0.915417 0.402506i $$-0.868139\pi$$
−0.915417 + 0.402506i $$0.868139\pi$$
$$180$$ 0 0
$$181$$ −10.0000 −0.743294 −0.371647 0.928374i $$-0.621207\pi$$
−0.371647 + 0.928374i $$0.621207\pi$$
$$182$$ 0 0
$$183$$ −2.00000 2.82843i −0.147844 0.209083i
$$184$$ 0 0
$$185$$ 0 0
$$186$$ 0 0
$$187$$ 16.9706i 1.24101i
$$188$$ 0 0
$$189$$ 0 0
$$190$$ 0 0
$$191$$ −19.5959 −1.41791 −0.708955 0.705253i $$-0.750833\pi$$
−0.708955 + 0.705253i $$0.750833\pi$$
$$192$$ 0 0
$$193$$ 19.5959 1.41055 0.705273 0.708936i $$-0.250825\pi$$
0.705273 + 0.708936i $$0.250825\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ 24.2487i 1.72765i 0.503793 + 0.863825i $$0.331938\pi$$
−0.503793 + 0.863825i $$0.668062\pi$$
$$198$$ 0 0
$$199$$ 10.3923i 0.736691i 0.929689 + 0.368345i $$0.120076\pi$$
−0.929689 + 0.368345i $$0.879924\pi$$
$$200$$ 0 0
$$201$$ −12.0000 + 8.48528i −0.846415 + 0.598506i
$$202$$ 0 0
$$203$$ 0 0
$$204$$ 0 0
$$205$$ 0 0
$$206$$ 0 0
$$207$$ 6.00000 16.9706i 0.417029 1.17954i
$$208$$ 0 0
$$209$$ 16.9706i 1.17388i
$$210$$ 0 0
$$211$$ 24.2487i 1.66935i 0.550743 + 0.834675i $$0.314345\pi$$
−0.550743 + 0.834675i $$0.685655\pi$$
$$212$$ 0 0
$$213$$ 9.79796 + 13.8564i 0.671345 + 0.949425i
$$214$$ 0 0
$$215$$ 0 0
$$216$$ 0 0
$$217$$ 0 0
$$218$$ 0 0
$$219$$ −9.79796 13.8564i −0.662085 0.936329i
$$220$$ 0 0
$$221$$ 16.9706i 1.14156i
$$222$$ 0 0
$$223$$ 16.9706i 1.13643i −0.822879 0.568216i $$-0.807634\pi$$
0.822879 0.568216i $$-0.192366\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 0 0
$$227$$ 6.00000 0.398234 0.199117 0.979976i $$-0.436193\pi$$
0.199117 + 0.979976i $$0.436193\pi$$
$$228$$ 0 0
$$229$$ 10.0000 0.660819 0.330409 0.943838i $$-0.392813\pi$$
0.330409 + 0.943838i $$0.392813\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ 3.46410i 0.226941i 0.993541 + 0.113470i $$0.0361967\pi$$
−0.993541 + 0.113470i $$0.963803\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ 0 0
$$237$$ −14.6969 + 10.3923i −0.954669 + 0.675053i
$$238$$ 0 0
$$239$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$240$$ 0 0
$$241$$ 10.0000 0.644157 0.322078 0.946713i $$-0.395619\pi$$
0.322078 + 0.946713i $$0.395619\pi$$
$$242$$ 0 0
$$243$$ 1.00000 15.5563i 0.0641500 0.997940i
$$244$$ 0 0
$$245$$ 0 0
$$246$$ 0 0
$$247$$ 16.9706i 1.07981i
$$248$$ 0 0
$$249$$ −6.00000 8.48528i −0.380235 0.537733i
$$250$$ 0 0
$$251$$ 14.6969 0.927663 0.463831 0.885924i $$-0.346474\pi$$
0.463831 + 0.885924i $$0.346474\pi$$
$$252$$ 0 0
$$253$$ 29.3939 1.84798
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 0 0
$$257$$ 3.46410i 0.216085i −0.994146 0.108042i $$-0.965542\pi$$
0.994146 0.108042i $$-0.0344582\pi$$
$$258$$ 0 0
$$259$$ 0 0
$$260$$ 0 0
$$261$$ 8.00000 + 2.82843i 0.495188 + 0.175075i
$$262$$ 0 0
$$263$$ −6.00000 −0.369976 −0.184988 0.982741i $$-0.559225\pi$$
−0.184988 + 0.982741i $$0.559225\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ 0 0
$$267$$ −8.00000 + 5.65685i −0.489592 + 0.346194i
$$268$$ 0 0
$$269$$ 19.7990i 1.20717i 0.797300 + 0.603583i $$0.206261\pi$$
−0.797300 + 0.603583i $$0.793739\pi$$
$$270$$ 0 0
$$271$$ 10.3923i 0.631288i −0.948878 0.315644i $$-0.897780\pi$$
0.948878 0.315644i $$-0.102220\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ 0 0
$$275$$ 0 0
$$276$$ 0 0
$$277$$ 14.6969 0.883053 0.441527 0.897248i $$-0.354437\pi$$
0.441527 + 0.897248i $$0.354437\pi$$
$$278$$ 0 0
$$279$$ −9.79796 3.46410i −0.586588 0.207390i
$$280$$ 0 0
$$281$$ 5.65685i 0.337460i 0.985662 + 0.168730i $$0.0539665\pi$$
−0.985662 + 0.168730i $$0.946033\pi$$
$$282$$ 0 0
$$283$$ 8.48528i 0.504398i 0.967675 + 0.252199i $$0.0811537\pi$$
−0.967675 + 0.252199i $$0.918846\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ 0 0
$$288$$ 0 0
$$289$$ 5.00000 0.294118
$$290$$ 0 0
$$291$$ 0 0
$$292$$ 0 0
$$293$$ 17.3205i 1.01187i 0.862570 + 0.505937i $$0.168853\pi$$
−0.862570 + 0.505937i $$0.831147\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ 0 0
$$297$$ 24.4949 6.92820i 1.42134 0.402015i
$$298$$ 0 0
$$299$$ 29.3939 1.69989
$$300$$ 0 0
$$301$$ 0 0
$$302$$ 0 0
$$303$$ 20.0000 14.1421i 1.14897 0.812444i
$$304$$ 0 0
$$305$$ 0 0
$$306$$ 0 0
$$307$$ 8.48528i 0.484281i 0.970241 + 0.242140i $$0.0778494\pi$$
−0.970241 + 0.242140i $$0.922151\pi$$
$$308$$ 0 0
$$309$$ 0 0
$$310$$ 0 0
$$311$$ −9.79796 −0.555591 −0.277796 0.960640i $$-0.589604\pi$$
−0.277796 + 0.960640i $$0.589604\pi$$
$$312$$ 0 0
$$313$$ 9.79796 0.553813 0.276907 0.960897i $$-0.410691\pi$$
0.276907 + 0.960897i $$0.410691\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ 17.3205i 0.972817i −0.873732 0.486408i $$-0.838307\pi$$
0.873732 0.486408i $$-0.161693\pi$$
$$318$$ 0 0
$$319$$ 13.8564i 0.775810i
$$320$$ 0 0
$$321$$ −18.0000 25.4558i −1.00466 1.42081i
$$322$$ 0 0
$$323$$ 12.0000 0.667698
$$324$$ 0 0
$$325$$ 0 0
$$326$$ 0 0
$$327$$ 2.00000 + 2.82843i 0.110600 + 0.156412i
$$328$$ 0 0
$$329$$ 0 0
$$330$$ 0 0
$$331$$ 17.3205i 0.952021i −0.879440 0.476011i $$-0.842082\pi$$
0.879440 0.476011i $$-0.157918\pi$$
$$332$$ 0 0
$$333$$ 4.89898 13.8564i 0.268462 0.759326i
$$334$$ 0 0
$$335$$ 0 0
$$336$$ 0 0
$$337$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$338$$ 0 0
$$339$$ −4.89898 + 3.46410i −0.266076 + 0.188144i
$$340$$ 0 0
$$341$$ 16.9706i 0.919007i
$$342$$ 0 0
$$343$$ 0 0
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ 6.00000 0.322097 0.161048 0.986947i $$-0.448512\pi$$
0.161048 + 0.986947i $$0.448512\pi$$
$$348$$ 0 0
$$349$$ −14.0000 −0.749403 −0.374701 0.927146i $$-0.622255\pi$$
−0.374701 + 0.927146i $$0.622255\pi$$
$$350$$ 0 0
$$351$$ 24.4949 6.92820i 1.30744 0.369800i
$$352$$ 0 0
$$353$$ 31.1769i 1.65938i 0.558225 + 0.829690i $$0.311483\pi$$
−0.558225 + 0.829690i $$0.688517\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ 0 0
$$357$$ 0 0
$$358$$ 0 0
$$359$$ 9.79796 0.517116 0.258558 0.965996i $$-0.416753\pi$$
0.258558 + 0.965996i $$0.416753\pi$$
$$360$$ 0 0
$$361$$ 7.00000 0.368421
$$362$$ 0 0
$$363$$ 13.0000 + 18.3848i 0.682323 + 0.964951i
$$364$$ 0 0
$$365$$ 0 0
$$366$$ 0 0
$$367$$ 16.9706i 0.885856i −0.896557 0.442928i $$-0.853940\pi$$
0.896557 0.442928i $$-0.146060\pi$$
$$368$$ 0 0
$$369$$ −16.0000 5.65685i −0.832927 0.294484i
$$370$$ 0 0
$$371$$ 0 0
$$372$$ 0 0
$$373$$ −14.6969 −0.760979 −0.380489 0.924785i $$-0.624244\pi$$
−0.380489 + 0.924785i $$0.624244\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ 13.8564i 0.713641i
$$378$$ 0 0
$$379$$ 10.3923i 0.533817i −0.963722 0.266908i $$-0.913998\pi$$
0.963722 0.266908i $$-0.0860021\pi$$
$$380$$ 0 0
$$381$$ 24.0000 16.9706i 1.22956 0.869428i
$$382$$ 0 0
$$383$$ −6.00000 −0.306586 −0.153293 0.988181i $$-0.548988\pi$$
−0.153293 + 0.988181i $$0.548988\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ −24.0000 8.48528i −1.21999 0.431331i
$$388$$ 0 0
$$389$$ 2.82843i 0.143407i 0.997426 + 0.0717035i $$0.0228435\pi$$
−0.997426 + 0.0717035i $$0.977156\pi$$
$$390$$ 0 0
$$391$$ 20.7846i 1.05112i
$$392$$ 0 0
$$393$$ 4.89898 + 6.92820i 0.247121 + 0.349482i
$$394$$ 0 0
$$395$$ 0 0
$$396$$ 0 0
$$397$$ 4.89898 0.245873 0.122936 0.992415i $$-0.460769\pi$$
0.122936 + 0.992415i $$0.460769\pi$$
$$398$$ 0 0
$$399$$ 0 0
$$400$$ 0 0
$$401$$ 22.6274i 1.12996i −0.825105 0.564980i $$-0.808884\pi$$
0.825105 0.564980i $$-0.191116\pi$$
$$402$$ 0 0
$$403$$ 16.9706i 0.845364i
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 0 0
$$407$$ 24.0000 1.18964
$$408$$ 0 0
$$409$$ −10.0000 −0.494468 −0.247234 0.968956i $$-0.579522\pi$$
−0.247234 + 0.968956i $$0.579522\pi$$
$$410$$ 0 0
$$411$$ 4.89898 3.46410i 0.241649 0.170872i
$$412$$ 0 0
$$413$$ 0 0
$$414$$ 0 0
$$415$$ 0 0
$$416$$ 0 0
$$417$$ −24.4949 + 17.3205i −1.19952 + 0.848189i
$$418$$ 0 0
$$419$$ 14.6969 0.717992 0.358996 0.933339i $$-0.383119\pi$$
0.358996 + 0.933339i $$0.383119\pi$$
$$420$$ 0 0
$$421$$ −26.0000 −1.26716 −0.633581 0.773676i $$-0.718416\pi$$
−0.633581 + 0.773676i $$0.718416\pi$$
$$422$$ 0 0
$$423$$ −6.00000 + 16.9706i −0.291730 + 0.825137i
$$424$$ 0 0
$$425$$ 0 0
$$426$$ 0 0
$$427$$ 0 0
$$428$$ 0 0
$$429$$ 24.0000 + 33.9411i 1.15873 + 1.63869i
$$430$$ 0 0
$$431$$ 19.5959 0.943902 0.471951 0.881625i $$-0.343550\pi$$
0.471951 + 0.881625i $$0.343550\pi$$
$$432$$ 0 0
$$433$$ −19.5959 −0.941720 −0.470860 0.882208i $$-0.656056\pi$$
−0.470860 + 0.882208i $$0.656056\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 0 0
$$437$$ 20.7846i 0.994263i
$$438$$ 0 0
$$439$$ 17.3205i 0.826663i −0.910581 0.413331i $$-0.864365\pi$$
0.910581 0.413331i $$-0.135635\pi$$
$$440$$ 0 0
$$441$$ −7.00000 + 19.7990i −0.333333 + 0.942809i
$$442$$ 0 0
$$443$$ −30.0000 −1.42534 −0.712672 0.701498i $$-0.752515\pi$$
−0.712672 + 0.701498i $$0.752515\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ 0 0
$$447$$ −4.00000 + 2.82843i −0.189194 + 0.133780i
$$448$$ 0 0
$$449$$ 22.6274i 1.06785i −0.845531 0.533927i $$-0.820716\pi$$
0.845531 0.533927i $$-0.179284\pi$$
$$450$$ 0 0
$$451$$ 27.7128i 1.30495i
$$452$$ 0 0
$$453$$ −4.89898 + 3.46410i −0.230174 + 0.162758i
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ −9.79796 −0.458329 −0.229165 0.973388i $$-0.573599\pi$$
−0.229165 + 0.973388i $$0.573599\pi$$
$$458$$ 0 0
$$459$$ 4.89898 + 17.3205i 0.228665 + 0.808452i
$$460$$ 0 0
$$461$$ 36.7696i 1.71253i 0.516538 + 0.856264i $$0.327221\pi$$
−0.516538 + 0.856264i $$0.672779\pi$$
$$462$$ 0 0
$$463$$ 16.9706i 0.788689i −0.918963 0.394344i $$-0.870972\pi$$
0.918963 0.394344i $$-0.129028\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ −18.0000 −0.832941 −0.416470 0.909149i $$-0.636733\pi$$
−0.416470 + 0.909149i $$0.636733\pi$$
$$468$$ 0 0
$$469$$ 0 0
$$470$$ 0 0
$$471$$ 4.89898 + 6.92820i 0.225733 + 0.319235i
$$472$$ 0 0
$$473$$ 41.5692i 1.91135i
$$474$$ 0 0
$$475$$ 0 0
$$476$$ 0 0
$$477$$ 29.3939 + 10.3923i 1.34585 + 0.475831i
$$478$$ 0 0
$$479$$ −19.5959 −0.895360 −0.447680 0.894194i $$-0.647750\pi$$
−0.447680 + 0.894194i $$0.647750\pi$$
$$480$$ 0 0
$$481$$ 24.0000 1.09431
$$482$$ 0 0
$$483$$ 0 0
$$484$$ 0 0
$$485$$ 0 0
$$486$$ 0 0
$$487$$ 33.9411i 1.53802i 0.639237 + 0.769010i $$0.279250\pi$$
−0.639237 + 0.769010i $$0.720750\pi$$
$$488$$ 0 0
$$489$$ 12.0000 8.48528i 0.542659 0.383718i
$$490$$ 0 0
$$491$$ −24.4949 −1.10544 −0.552720 0.833367i $$-0.686410\pi$$
−0.552720 + 0.833367i $$0.686410\pi$$
$$492$$ 0 0
$$493$$ −9.79796 −0.441278
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ 0 0
$$498$$ 0 0
$$499$$ 3.46410i 0.155074i 0.996989 + 0.0775372i $$0.0247057\pi$$
−0.996989 + 0.0775372i $$0.975294\pi$$
$$500$$ 0 0
$$501$$ 6.00000 + 8.48528i 0.268060 + 0.379094i
$$502$$ 0 0
$$503$$ −6.00000 −0.267527 −0.133763 0.991013i $$-0.542706\pi$$
−0.133763 + 0.991013i $$0.542706\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ 0 0
$$507$$ 11.0000 + 15.5563i 0.488527 + 0.690882i
$$508$$ 0 0
$$509$$ 36.7696i 1.62978i −0.579614 0.814891i $$-0.696797\pi$$
0.579614 0.814891i $$-0.303203\pi$$
$$510$$ 0 0
$$511$$ 0 0
$$512$$ 0 0
$$513$$ −4.89898 17.3205i −0.216295 0.764719i
$$514$$ 0 0
$$515$$ 0 0
$$516$$ 0 0
$$517$$ −29.3939 −1.29274
$$518$$ 0 0
$$519$$ −4.89898 + 3.46410i −0.215041 + 0.152057i
$$520$$ 0 0
$$521$$ 5.65685i 0.247831i −0.992293 0.123916i $$-0.960455\pi$$
0.992293 0.123916i $$-0.0395452\pi$$
$$522$$ 0 0
$$523$$ 42.4264i 1.85518i 0.373603 + 0.927589i $$0.378122\pi$$
−0.373603 + 0.927589i $$0.621878\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ 12.0000 0.522728
$$528$$ 0 0
$$529$$ 13.0000 0.565217
$$530$$ 0 0
$$531$$ −4.89898 + 13.8564i −0.212598 + 0.601317i
$$532$$ 0 0
$$533$$ 27.7128i 1.20038i
$$534$$ 0 0
$$535$$ 0 0
$$536$$ 0 0
$$537$$ −24.4949 34.6410i −1.05703 1.49487i
$$538$$ 0 0
$$539$$ −34.2929 −1.47710
$$540$$ 0 0
$$541$$ −34.0000 −1.46177 −0.730887 0.682498i $$-0.760893\pi$$
−0.730887 + 0.682498i $$0.760893\pi$$
$$542$$ 0 0
$$543$$ −10.0000 14.1421i −0.429141 0.606897i
$$544$$ 0 0
$$545$$ 0 0
$$546$$ 0 0
$$547$$ 25.4558i 1.08841i −0.838951 0.544207i $$-0.816831\pi$$
0.838951 0.544207i $$-0.183169\pi$$
$$548$$ 0 0
$$549$$ 2.00000 5.65685i 0.0853579 0.241429i
$$550$$ 0 0
$$551$$ 9.79796 0.417407
$$552$$ 0 0
$$553$$ 0 0
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 0 0
$$557$$ 3.46410i 0.146779i −0.997303 0.0733893i $$-0.976618\pi$$
0.997303 0.0733893i $$-0.0233816\pi$$
$$558$$ 0 0
$$559$$ 41.5692i 1.75819i
$$560$$ 0 0
$$561$$ −24.0000 + 16.9706i −1.01328 + 0.716498i
$$562$$ 0 0
$$563$$ 18.0000 0.758610 0.379305 0.925272i $$-0.376163\pi$$
0.379305 + 0.925272i $$0.376163\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ 0 0
$$567$$ 0 0
$$568$$ 0 0
$$569$$ 28.2843i 1.18574i 0.805299 + 0.592869i $$0.202005\pi$$
−0.805299 + 0.592869i $$0.797995\pi$$
$$570$$ 0 0
$$571$$ 38.1051i 1.59465i 0.603550 + 0.797325i $$0.293752\pi$$
−0.603550 + 0.797325i $$0.706248\pi$$
$$572$$ 0 0
$$573$$ −19.5959 27.7128i −0.818631 1.15772i
$$574$$ 0 0
$$575$$ 0 0
$$576$$ 0 0
$$577$$ 19.5959 0.815789 0.407894 0.913029i $$-0.366263\pi$$
0.407894 + 0.913029i $$0.366263\pi$$
$$578$$ 0 0
$$579$$ 19.5959 + 27.7128i 0.814379 + 1.15171i
$$580$$ 0 0
$$581$$ 0 0
$$582$$ 0 0
$$583$$ 50.9117i 2.10855i
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ −18.0000 −0.742940 −0.371470 0.928445i $$-0.621146\pi$$
−0.371470 + 0.928445i $$0.621146\pi$$
$$588$$ 0 0
$$589$$ −12.0000 −0.494451
$$590$$ 0 0
$$591$$ −34.2929 + 24.2487i −1.41062 + 0.997459i
$$592$$ 0 0
$$593$$ 24.2487i 0.995775i −0.867242 0.497888i $$-0.834109\pi$$
0.867242 0.497888i $$-0.165891\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 0 0
$$597$$ −14.6969 + 10.3923i −0.601506 + 0.425329i
$$598$$ 0 0
$$599$$ −9.79796 −0.400334 −0.200167 0.979762i $$-0.564148\pi$$
−0.200167 + 0.979762i $$0.564148\pi$$
$$600$$ 0 0
$$601$$ 10.0000 0.407909 0.203954 0.978980i $$-0.434621\pi$$
0.203954 + 0.978980i $$0.434621\pi$$
$$602$$ 0 0
$$603$$ −24.0000 8.48528i −0.977356 0.345547i
$$604$$ 0 0
$$605$$ 0 0
$$606$$ 0 0
$$607$$ 16.9706i 0.688814i 0.938820 + 0.344407i $$0.111920\pi$$
−0.938820 + 0.344407i $$0.888080\pi$$
$$608$$ 0 0
$$609$$ 0 0
$$610$$ 0 0
$$611$$ −29.3939 −1.18915
$$612$$ 0 0
$$613$$ −14.6969 −0.593604 −0.296802 0.954939i $$-0.595920\pi$$
−0.296802 + 0.954939i $$0.595920\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ 24.2487i 0.976216i 0.872783 + 0.488108i $$0.162313\pi$$
−0.872783 + 0.488108i $$0.837687\pi$$
$$618$$ 0 0
$$619$$ 10.3923i 0.417702i −0.977947 0.208851i $$-0.933028\pi$$
0.977947 0.208851i $$-0.0669724\pi$$
$$620$$ 0 0
$$621$$ 30.0000 8.48528i 1.20386 0.340503i
$$622$$ 0 0
$$623$$ 0 0
$$624$$ 0 0
$$625$$ 0 0
$$626$$ 0 0
$$627$$ 24.0000 16.9706i 0.958468 0.677739i
$$628$$ 0 0
$$629$$ 16.9706i 0.676661i
$$630$$ 0 0
$$631$$ 38.1051i 1.51694i −0.651707 0.758470i $$-0.725947\pi$$
0.651707 0.758470i $$-0.274053\pi$$
$$632$$ 0 0
$$633$$ −34.2929 + 24.2487i −1.36302 + 0.963800i
$$634$$ 0 0
$$635$$ 0 0
$$636$$ 0 0
$$637$$ −34.2929 −1.35873
$$638$$ 0 0
$$639$$ −9.79796 + 27.7128i −0.387601 + 1.09630i
$$640$$ 0 0
$$641$$ 22.6274i 0.893729i 0.894602 + 0.446865i $$0.147459\pi$$
−0.894602 + 0.446865i $$0.852541\pi$$
$$642$$ 0 0
$$643$$ 8.48528i 0.334627i −0.985904 0.167313i $$-0.946491\pi$$
0.985904 0.167313i $$-0.0535092\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ −42.0000 −1.65119 −0.825595 0.564263i $$-0.809160\pi$$
−0.825595 + 0.564263i $$0.809160\pi$$
$$648$$ 0 0
$$649$$ −24.0000 −0.942082
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 0 0
$$653$$ 31.1769i 1.22005i 0.792383 + 0.610023i $$0.208840\pi$$
−0.792383 + 0.610023i $$0.791160\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ 0 0
$$657$$ 9.79796 27.7128i 0.382255 1.08118i
$$658$$ 0 0
$$659$$ 14.6969 0.572511 0.286256 0.958153i $$-0.407589\pi$$
0.286256 + 0.958153i $$0.407589\pi$$
$$660$$ 0 0
$$661$$ 22.0000 0.855701 0.427850 0.903850i $$-0.359271\pi$$
0.427850 + 0.903850i $$0.359271\pi$$
$$662$$ 0 0
$$663$$ −24.0000 + 16.9706i −0.932083 + 0.659082i
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 0 0
$$667$$ 16.9706i 0.657103i
$$668$$ 0 0
$$669$$ 24.0000 16.9706i 0.927894 0.656120i
$$670$$ 0 0
$$671$$ 9.79796 0.378246
$$672$$ 0 0
$$673$$ −39.1918 −1.51073 −0.755367 0.655302i $$-0.772541\pi$$
−0.755367 + 0.655302i $$0.772541\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ 0 0
$$677$$ 10.3923i 0.399409i 0.979856 + 0.199704i $$0.0639982\pi$$
−0.979856 + 0.199704i $$0.936002\pi$$
$$678$$ 0 0
$$679$$ 0 0
$$680$$ 0 0
$$681$$ 6.00000 + 8.48528i 0.229920 + 0.325157i
$$682$$ 0 0
$$683$$ −6.00000 −0.229584 −0.114792 0.993390i $$-0.536620\pi$$
−0.114792 + 0.993390i $$0.536620\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ 0 0
$$687$$ 10.0000 + 14.1421i 0.381524 + 0.539556i
$$688$$ 0 0
$$689$$ 50.9117i 1.93958i
$$690$$ 0 0
$$691$$ 31.1769i 1.18603i −0.805193 0.593013i $$-0.797938\pi$$
0.805193 0.593013i $$-0.202062\pi$$
$$692$$ 0 0
$$693$$ 0 0
$$694$$ 0 0
$$695$$ 0 0
$$696$$ 0 0
$$697$$ 19.5959 0.742248
$$698$$ 0 0
$$699$$ −4.89898 + 3.46410i −0.185296 + 0.131024i
$$700$$ 0 0
$$701$$ 19.7990i 0.747798i −0.927470 0.373899i $$-0.878021\pi$$
0.927470 0.373899i $$-0.121979\pi$$
$$702$$ 0 0
$$703$$ 16.9706i 0.640057i
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ 0 0
$$708$$ 0 0
$$709$$ −38.0000 −1.42712 −0.713560 0.700594i $$-0.752918\pi$$
−0.713560 + 0.700594i $$0.752918\pi$$
$$710$$ 0 0
$$711$$ −29.3939 10.3923i −1.10236 0.389742i
$$712$$ 0 0
$$713$$ 20.7846i 0.778390i
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 0 0
$$717$$ 0 0
$$718$$ 0 0
$$719$$ 19.5959 0.730804 0.365402 0.930850i $$-0.380931\pi$$
0.365402 + 0.930850i $$0.380931\pi$$
$$720$$ 0 0
$$721$$ 0 0
$$722$$ 0 0
$$723$$ 10.0000 + 14.1421i 0.371904 + 0.525952i
$$724$$ 0 0
$$725$$ 0 0
$$726$$ 0 0
$$727$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$728$$ 0 0
$$729$$ 23.0000 14.1421i 0.851852 0.523783i
$$730$$ 0 0
$$731$$ 29.3939 1.08717
$$732$$ 0 0
$$733$$ −44.0908 −1.62853 −0.814266 0.580492i $$-0.802860\pi$$
−0.814266 + 0.580492i $$0.802860\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ 41.5692i 1.53122i
$$738$$ 0 0
$$739$$ 24.2487i 0.892003i −0.895032 0.446002i $$-0.852848\pi$$
0.895032 0.446002i $$-0.147152\pi$$
$$740$$ 0 0
$$741$$ 24.0000 16.9706i 0.881662 0.623429i
$$742$$ 0 0
$$743$$ −6.00000 −0.220119 −0.110059 0.993925i $$-0.535104\pi$$
−0.110059 + 0.993925i $$0.535104\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ 0 0
$$747$$ 6.00000 16.9706i 0.219529 0.620920i
$$748$$ 0 0
$$749$$ 0 0
$$750$$ 0 0
$$751$$ 31.1769i 1.13766i 0.822455 + 0.568831i $$0.192604\pi$$
−0.822455 + 0.568831i $$0.807396\pi$$
$$752$$ 0 0
$$753$$ 14.6969 + 20.7846i 0.535586 + 0.757433i
$$754$$ 0 0
$$755$$ 0 0
$$756$$ 0 0
$$757$$ 14.6969 0.534169 0.267085 0.963673i $$-0.413940\pi$$
0.267085 + 0.963673i $$0.413940\pi$$
$$758$$ 0 0
$$759$$ 29.3939 + 41.5692i 1.06693 + 1.50887i
$$760$$ 0 0
$$761$$ 39.5980i 1.43543i −0.696339 0.717713i $$-0.745189\pi$$
0.696339 0.717713i $$-0.254811\pi$$
$$762$$ 0 0
$$763$$ 0 0
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ −24.0000 −0.866590
$$768$$ 0 0
$$769$$ 22.0000 0.793340 0.396670 0.917961i $$-0.370166\pi$$
0.396670 + 0.917961i $$0.370166\pi$$
$$770$$ 0 0
$$771$$ 4.89898 3.46410i 0.176432 0.124757i
$$772$$ 0 0
$$773$$ 24.2487i 0.872166i −0.899907 0.436083i $$-0.856365\pi$$
0.899907 0.436083i $$-0.143635\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ 0 0
$$777$$ 0 0
$$778$$ 0 0
$$779$$ −19.5959 −0.702097
$$780$$ 0 0
$$781$$ −48.0000 −1.71758
$$782$$ 0 0
$$783$$ 4.00000 + 14.1421i 0.142948 + 0.505399i
$$784$$ 0 0
$$785$$ 0 0
$$786$$ 0 0
$$787$$ 25.4558i 0.907403i −0.891154 0.453701i $$-0.850103\pi$$
0.891154 0.453701i $$-0.149897\pi$$
$$788$$ 0 0
$$789$$ −6.00000 8.48528i −0.213606 0.302084i
$$790$$ 0 0
$$791$$ 0 0
$$792$$ 0 0
$$793$$ 9.79796 0.347936
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 0 0
$$797$$ 17.3205i 0.613524i −0.951786 0.306762i $$-0.900754\pi$$
0.951786 0.306762i $$-0.0992455\pi$$
$$798$$ 0 0
$$799$$ 20.7846i 0.735307i
$$800$$ 0 0
$$801$$ −16.0000 5.65685i −0.565332 0.199875i
$$802$$ 0 0
$$803$$ 48.0000 1.69388
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 0 0
$$807$$ −28.0000