# Properties

 Label 360.2.s.a.233.1 Level $360$ Weight $2$ Character 360.233 Analytic conductor $2.875$ Analytic rank $0$ Dimension $4$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$360 = 2^{3} \cdot 3^{2} \cdot 5$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 360.s (of order $$4$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$2.87461447277$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(i)$$ Coefficient field: $$\Q(\zeta_{8})$$ Defining polynomial: $$x^{4} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{19}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

## Embedding invariants

 Embedding label 233.1 Root $$-0.707107 + 0.707107i$$ of defining polynomial Character $$\chi$$ $$=$$ 360.233 Dual form 360.2.s.a.17.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+(-2.12132 + 0.707107i) q^{5} +O(q^{10})$$ $$q+(-2.12132 + 0.707107i) q^{5} +5.65685i q^{11} +(3.00000 + 3.00000i) q^{13} +4.00000i q^{19} +(-2.82843 + 2.82843i) q^{23} +(4.00000 - 3.00000i) q^{25} -1.41421 q^{29} -8.00000 q^{31} +(7.00000 - 7.00000i) q^{37} -1.41421i q^{41} +(4.00000 + 4.00000i) q^{43} +(-2.82843 - 2.82843i) q^{47} +7.00000i q^{49} +(8.48528 - 8.48528i) q^{53} +(-4.00000 - 12.0000i) q^{55} +11.3137 q^{59} -12.0000 q^{61} +(-8.48528 - 4.24264i) q^{65} +(-8.00000 + 8.00000i) q^{67} +5.65685i q^{71} +(-3.00000 - 3.00000i) q^{73} -8.00000i q^{79} +(11.3137 - 11.3137i) q^{83} +7.07107 q^{89} +(-2.82843 - 8.48528i) q^{95} +(5.00000 - 5.00000i) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + O(q^{10})$$ $$4q + 12q^{13} + 16q^{25} - 32q^{31} + 28q^{37} + 16q^{43} - 16q^{55} - 48q^{61} - 32q^{67} - 12q^{73} + 20q^{97} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/360\mathbb{Z}\right)^\times$$.

 $$n$$ $$181$$ $$217$$ $$271$$ $$281$$ $$\chi(n)$$ $$1$$ $$e\left(\frac{3}{4}\right)$$ $$1$$ $$-1$$

## 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
$$4$$ 0 0
$$5$$ −2.12132 + 0.707107i −0.948683 + 0.316228i
$$6$$ 0 0
$$7$$ 0 0 −0.707107 0.707107i $$-0.750000\pi$$
0.707107 + 0.707107i $$0.250000\pi$$
$$8$$ 0 0
$$9$$ 0 0
$$10$$ 0 0
$$11$$ 5.65685i 1.70561i 0.522233 + 0.852803i $$0.325099\pi$$
−0.522233 + 0.852803i $$0.674901\pi$$
$$12$$ 0 0
$$13$$ 3.00000 + 3.00000i 0.832050 + 0.832050i 0.987797 0.155747i $$-0.0497784\pi$$
−0.155747 + 0.987797i $$0.549778\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 0 0
$$17$$ 0 0 0.707107 0.707107i $$-0.250000\pi$$
−0.707107 + 0.707107i $$0.750000\pi$$
$$18$$ 0 0
$$19$$ 4.00000i 0.917663i 0.888523 + 0.458831i $$0.151732\pi$$
−0.888523 + 0.458831i $$0.848268\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 0 0
$$23$$ −2.82843 + 2.82843i −0.589768 + 0.589768i −0.937568 0.347801i $$-0.886929\pi$$
0.347801 + 0.937568i $$0.386929\pi$$
$$24$$ 0 0
$$25$$ 4.00000 3.00000i 0.800000 0.600000i
$$26$$ 0 0
$$27$$ 0 0
$$28$$ 0 0
$$29$$ −1.41421 −0.262613 −0.131306 0.991342i $$-0.541917\pi$$
−0.131306 + 0.991342i $$0.541917\pi$$
$$30$$ 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
$$35$$ 0 0
$$36$$ 0 0
$$37$$ 7.00000 7.00000i 1.15079 1.15079i 0.164399 0.986394i $$-0.447432\pi$$
0.986394 0.164399i $$-0.0525685\pi$$
$$38$$ 0 0
$$39$$ 0 0
$$40$$ 0 0
$$41$$ 1.41421i 0.220863i −0.993884 0.110432i $$-0.964777\pi$$
0.993884 0.110432i $$-0.0352233\pi$$
$$42$$ 0 0
$$43$$ 4.00000 + 4.00000i 0.609994 + 0.609994i 0.942944 0.332950i $$-0.108044\pi$$
−0.332950 + 0.942944i $$0.608044\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 0 0
$$47$$ −2.82843 2.82843i −0.412568 0.412568i 0.470064 0.882632i $$-0.344231\pi$$
−0.882632 + 0.470064i $$0.844231\pi$$
$$48$$ 0 0
$$49$$ 7.00000i 1.00000i
$$50$$ 0 0
$$51$$ 0 0
$$52$$ 0 0
$$53$$ 8.48528 8.48528i 1.16554 1.16554i 0.182300 0.983243i $$-0.441646\pi$$
0.983243 0.182300i $$-0.0583542\pi$$
$$54$$ 0 0
$$55$$ −4.00000 12.0000i −0.539360 1.61808i
$$56$$ 0 0
$$57$$ 0 0
$$58$$ 0 0
$$59$$ 11.3137 1.47292 0.736460 0.676481i $$-0.236496\pi$$
0.736460 + 0.676481i $$0.236496\pi$$
$$60$$ 0 0
$$61$$ −12.0000 −1.53644 −0.768221 0.640184i $$-0.778858\pi$$
−0.768221 + 0.640184i $$0.778858\pi$$
$$62$$ 0 0
$$63$$ 0 0
$$64$$ 0 0
$$65$$ −8.48528 4.24264i −1.05247 0.526235i
$$66$$ 0 0
$$67$$ −8.00000 + 8.00000i −0.977356 + 0.977356i −0.999749 0.0223937i $$-0.992871\pi$$
0.0223937 + 0.999749i $$0.492871\pi$$
$$68$$ 0 0
$$69$$ 0 0
$$70$$ 0 0
$$71$$ 5.65685i 0.671345i 0.941979 + 0.335673i $$0.108964\pi$$
−0.941979 + 0.335673i $$0.891036\pi$$
$$72$$ 0 0
$$73$$ −3.00000 3.00000i −0.351123 0.351123i 0.509404 0.860527i $$-0.329866\pi$$
−0.860527 + 0.509404i $$0.829866\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 0 0
$$77$$ 0 0
$$78$$ 0 0
$$79$$ 8.00000i 0.900070i −0.893011 0.450035i $$-0.851411\pi$$
0.893011 0.450035i $$-0.148589\pi$$
$$80$$ 0 0
$$81$$ 0 0
$$82$$ 0 0
$$83$$ 11.3137 11.3137i 1.24184 1.24184i 0.282604 0.959237i $$-0.408802\pi$$
0.959237 0.282604i $$-0.0911983\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ 0 0
$$87$$ 0 0
$$88$$ 0 0
$$89$$ 7.07107 0.749532 0.374766 0.927119i $$-0.377723\pi$$
0.374766 + 0.927119i $$0.377723\pi$$
$$90$$ 0 0
$$91$$ 0 0
$$92$$ 0 0
$$93$$ 0 0
$$94$$ 0 0
$$95$$ −2.82843 8.48528i −0.290191 0.870572i
$$96$$ 0 0
$$97$$ 5.00000 5.00000i 0.507673 0.507673i −0.406138 0.913812i $$-0.633125\pi$$
0.913812 + 0.406138i $$0.133125\pi$$
$$98$$ 0 0
$$99$$ 0 0
$$100$$ 0 0
$$101$$ 1.41421i 0.140720i 0.997522 + 0.0703598i $$0.0224147\pi$$
−0.997522 + 0.0703598i $$0.977585\pi$$
$$102$$ 0 0
$$103$$ 4.00000 + 4.00000i 0.394132 + 0.394132i 0.876157 0.482025i $$-0.160099\pi$$
−0.482025 + 0.876157i $$0.660099\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ −11.3137 11.3137i −1.09374 1.09374i −0.995126 0.0986115i $$-0.968560\pi$$
−0.0986115 0.995126i $$-0.531440\pi$$
$$108$$ 0 0
$$109$$ 4.00000i 0.383131i 0.981480 + 0.191565i $$0.0613564\pi$$
−0.981480 + 0.191565i $$0.938644\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ 0 0
$$113$$ 4.24264 4.24264i 0.399114 0.399114i −0.478806 0.877920i $$-0.658930\pi$$
0.877920 + 0.478806i $$0.158930\pi$$
$$114$$ 0 0
$$115$$ 4.00000 8.00000i 0.373002 0.746004i
$$116$$ 0 0
$$117$$ 0 0
$$118$$ 0 0
$$119$$ 0 0
$$120$$ 0 0
$$121$$ −21.0000 −1.90909
$$122$$ 0 0
$$123$$ 0 0
$$124$$ 0 0
$$125$$ −6.36396 + 9.19239i −0.569210 + 0.822192i
$$126$$ 0 0
$$127$$ 12.0000 12.0000i 1.06483 1.06483i 0.0670802 0.997748i $$-0.478632\pi$$
0.997748 0.0670802i $$-0.0213683\pi$$
$$128$$ 0 0
$$129$$ 0 0
$$130$$ 0 0
$$131$$ 5.65685i 0.494242i 0.968985 + 0.247121i $$0.0794845\pi$$
−0.968985 + 0.247121i $$0.920516\pi$$
$$132$$ 0 0
$$133$$ 0 0
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ 12.7279 + 12.7279i 1.08742 + 1.08742i 0.995793 + 0.0916263i $$0.0292065\pi$$
0.0916263 + 0.995793i $$0.470793\pi$$
$$138$$ 0 0
$$139$$ 20.0000i 1.69638i 0.529694 + 0.848189i $$0.322307\pi$$
−0.529694 + 0.848189i $$0.677693\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 0 0
$$143$$ −16.9706 + 16.9706i −1.41915 + 1.41915i
$$144$$ 0 0
$$145$$ 3.00000 1.00000i 0.249136 0.0830455i
$$146$$ 0 0
$$147$$ 0 0
$$148$$ 0 0
$$149$$ −4.24264 −0.347571 −0.173785 0.984784i $$-0.555600\pi$$
−0.173785 + 0.984784i $$0.555600\pi$$
$$150$$ 0 0
$$151$$ 8.00000 0.651031 0.325515 0.945537i $$-0.394462\pi$$
0.325515 + 0.945537i $$0.394462\pi$$
$$152$$ 0 0
$$153$$ 0 0
$$154$$ 0 0
$$155$$ 16.9706 5.65685i 1.36311 0.454369i
$$156$$ 0 0
$$157$$ −7.00000 + 7.00000i −0.558661 + 0.558661i −0.928926 0.370265i $$-0.879267\pi$$
0.370265 + 0.928926i $$0.379267\pi$$
$$158$$ 0 0
$$159$$ 0 0
$$160$$ 0 0
$$161$$ 0 0
$$162$$ 0 0
$$163$$ 12.0000 + 12.0000i 0.939913 + 0.939913i 0.998294 0.0583818i $$-0.0185941\pi$$
−0.0583818 + 0.998294i $$0.518594\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ −8.48528 8.48528i −0.656611 0.656611i 0.297966 0.954577i $$-0.403692\pi$$
−0.954577 + 0.297966i $$0.903692\pi$$
$$168$$ 0 0
$$169$$ 5.00000i 0.384615i
$$170$$ 0 0
$$171$$ 0 0
$$172$$ 0 0
$$173$$ −7.07107 + 7.07107i −0.537603 + 0.537603i −0.922824 0.385221i $$-0.874125\pi$$
0.385221 + 0.922824i $$0.374125\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 0 0
$$177$$ 0 0
$$178$$ 0 0
$$179$$ −5.65685 −0.422813 −0.211407 0.977398i $$-0.567804\pi$$
−0.211407 + 0.977398i $$0.567804\pi$$
$$180$$ 0 0
$$181$$ 4.00000 0.297318 0.148659 0.988889i $$-0.452504\pi$$
0.148659 + 0.988889i $$0.452504\pi$$
$$182$$ 0 0
$$183$$ 0 0
$$184$$ 0 0
$$185$$ −9.89949 + 19.7990i −0.727825 + 1.45565i
$$186$$ 0 0
$$187$$ 0 0
$$188$$ 0 0
$$189$$ 0 0
$$190$$ 0 0
$$191$$ 11.3137i 0.818631i 0.912393 + 0.409316i $$0.134232\pi$$
−0.912393 + 0.409316i $$0.865768\pi$$
$$192$$ 0 0
$$193$$ 5.00000 + 5.00000i 0.359908 + 0.359908i 0.863779 0.503871i $$-0.168091\pi$$
−0.503871 + 0.863779i $$0.668091\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ 12.7279 + 12.7279i 0.906827 + 0.906827i 0.996015 0.0891879i $$-0.0284272\pi$$
−0.0891879 + 0.996015i $$0.528427\pi$$
$$198$$ 0 0
$$199$$ 24.0000i 1.70131i −0.525720 0.850657i $$-0.676204\pi$$
0.525720 0.850657i $$-0.323796\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ 0 0
$$203$$ 0 0
$$204$$ 0 0
$$205$$ 1.00000 + 3.00000i 0.0698430 + 0.209529i
$$206$$ 0 0
$$207$$ 0 0
$$208$$ 0 0
$$209$$ −22.6274 −1.56517
$$210$$ 0 0
$$211$$ 20.0000 1.37686 0.688428 0.725304i $$-0.258301\pi$$
0.688428 + 0.725304i $$0.258301\pi$$
$$212$$ 0 0
$$213$$ 0 0
$$214$$ 0 0
$$215$$ −11.3137 5.65685i −0.771589 0.385794i
$$216$$ 0 0
$$217$$ 0 0
$$218$$ 0 0
$$219$$ 0 0
$$220$$ 0 0
$$221$$ 0 0
$$222$$ 0 0
$$223$$ 0 0 0.707107 0.707107i $$-0.250000\pi$$
−0.707107 + 0.707107i $$0.750000\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 0 0
$$227$$ 0 0 0.707107 0.707107i $$-0.250000\pi$$
−0.707107 + 0.707107i $$0.750000\pi$$
$$228$$ 0 0
$$229$$ 6.00000i 0.396491i 0.980152 + 0.198246i $$0.0635244\pi$$
−0.980152 + 0.198246i $$0.936476\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ 5.65685 5.65685i 0.370593 0.370593i −0.497100 0.867693i $$-0.665602\pi$$
0.867693 + 0.497100i $$0.165602\pi$$
$$234$$ 0 0
$$235$$ 8.00000 + 4.00000i 0.521862 + 0.260931i
$$236$$ 0 0
$$237$$ 0 0
$$238$$ 0 0
$$239$$ 16.9706 1.09773 0.548867 0.835910i $$-0.315059\pi$$
0.548867 + 0.835910i $$0.315059\pi$$
$$240$$ 0 0
$$241$$ 14.0000 0.901819 0.450910 0.892570i $$-0.351100\pi$$
0.450910 + 0.892570i $$0.351100\pi$$
$$242$$ 0 0
$$243$$ 0 0
$$244$$ 0 0
$$245$$ −4.94975 14.8492i −0.316228 0.948683i
$$246$$ 0 0
$$247$$ −12.0000 + 12.0000i −0.763542 + 0.763542i
$$248$$ 0 0
$$249$$ 0 0
$$250$$ 0 0
$$251$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$252$$ 0 0
$$253$$ −16.0000 16.0000i −1.00591 1.00591i
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 0 0
$$257$$ −7.07107 7.07107i −0.441081 0.441081i 0.451294 0.892375i $$-0.350963\pi$$
−0.892375 + 0.451294i $$0.850963\pi$$
$$258$$ 0 0
$$259$$ 0 0
$$260$$ 0 0
$$261$$ 0 0
$$262$$ 0 0
$$263$$ −2.82843 + 2.82843i −0.174408 + 0.174408i −0.788913 0.614505i $$-0.789356\pi$$
0.614505 + 0.788913i $$0.289356\pi$$
$$264$$ 0 0
$$265$$ −12.0000 + 24.0000i −0.737154 + 1.47431i
$$266$$ 0 0
$$267$$ 0 0
$$268$$ 0 0
$$269$$ 15.5563 0.948487 0.474244 0.880394i $$-0.342722\pi$$
0.474244 + 0.880394i $$0.342722\pi$$
$$270$$ 0 0
$$271$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ 0 0
$$275$$ 16.9706 + 22.6274i 1.02336 + 1.36448i
$$276$$ 0 0
$$277$$ 11.0000 11.0000i 0.660926 0.660926i −0.294672 0.955598i $$-0.595211\pi$$
0.955598 + 0.294672i $$0.0952105\pi$$
$$278$$ 0 0
$$279$$ 0 0
$$280$$ 0 0
$$281$$ 4.24264i 0.253095i −0.991961 0.126547i $$-0.959610\pi$$
0.991961 0.126547i $$-0.0403896\pi$$
$$282$$ 0 0
$$283$$ −16.0000 16.0000i −0.951101 0.951101i 0.0477577 0.998859i $$-0.484792\pi$$
−0.998859 + 0.0477577i $$0.984792\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ 0 0
$$288$$ 0 0
$$289$$ 17.0000i 1.00000i
$$290$$ 0 0
$$291$$ 0 0
$$292$$ 0 0
$$293$$ −1.41421 + 1.41421i −0.0826192 + 0.0826192i −0.747209 0.664589i $$-0.768606\pi$$
0.664589 + 0.747209i $$0.268606\pi$$
$$294$$ 0 0
$$295$$ −24.0000 + 8.00000i −1.39733 + 0.465778i
$$296$$ 0 0
$$297$$ 0 0
$$298$$ 0 0
$$299$$ −16.9706 −0.981433
$$300$$ 0 0
$$301$$ 0 0
$$302$$ 0 0
$$303$$ 0 0
$$304$$ 0 0
$$305$$ 25.4558 8.48528i 1.45760 0.485866i
$$306$$ 0 0
$$307$$ −20.0000 + 20.0000i −1.14146 + 1.14146i −0.153277 + 0.988183i $$0.548983\pi$$
−0.988183 + 0.153277i $$0.951017\pi$$
$$308$$ 0 0
$$309$$ 0 0
$$310$$ 0 0
$$311$$ 28.2843i 1.60385i −0.597422 0.801927i $$-0.703808\pi$$
0.597422 0.801927i $$-0.296192\pi$$
$$312$$ 0 0
$$313$$ −9.00000 9.00000i −0.508710 0.508710i 0.405420 0.914130i $$-0.367125\pi$$
−0.914130 + 0.405420i $$0.867125\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ 2.82843 + 2.82843i 0.158860 + 0.158860i 0.782062 0.623201i $$-0.214168\pi$$
−0.623201 + 0.782062i $$0.714168\pi$$
$$318$$ 0 0
$$319$$ 8.00000i 0.447914i
$$320$$ 0 0
$$321$$ 0 0
$$322$$ 0 0
$$323$$ 0 0
$$324$$ 0 0
$$325$$ 21.0000 + 3.00000i 1.16487 + 0.166410i
$$326$$ 0 0
$$327$$ 0 0
$$328$$ 0 0
$$329$$ 0 0
$$330$$ 0 0
$$331$$ −4.00000 −0.219860 −0.109930 0.993939i $$-0.535063\pi$$
−0.109930 + 0.993939i $$0.535063\pi$$
$$332$$ 0 0
$$333$$ 0 0
$$334$$ 0 0
$$335$$ 11.3137 22.6274i 0.618134 1.23627i
$$336$$ 0 0
$$337$$ 15.0000 15.0000i 0.817102 0.817102i −0.168585 0.985687i $$-0.553920\pi$$
0.985687 + 0.168585i $$0.0539198\pi$$
$$338$$ 0 0
$$339$$ 0 0
$$340$$ 0 0
$$341$$ 45.2548i 2.45069i
$$342$$ 0 0
$$343$$ 0 0
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ 0 0 0.707107 0.707107i $$-0.250000\pi$$
−0.707107 + 0.707107i $$0.750000\pi$$
$$348$$ 0 0
$$349$$ 4.00000i 0.214115i −0.994253 0.107058i $$-0.965857\pi$$
0.994253 0.107058i $$-0.0341429\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 0 0
$$353$$ 5.65685 5.65685i 0.301084 0.301084i −0.540354 0.841438i $$-0.681710\pi$$
0.841438 + 0.540354i $$0.181710\pi$$
$$354$$ 0 0
$$355$$ −4.00000 12.0000i −0.212298 0.636894i
$$356$$ 0 0
$$357$$ 0 0
$$358$$ 0 0
$$359$$ 5.65685 0.298557 0.149279 0.988795i $$-0.452305\pi$$
0.149279 + 0.988795i $$0.452305\pi$$
$$360$$ 0 0
$$361$$ 3.00000 0.157895
$$362$$ 0 0
$$363$$ 0 0
$$364$$ 0 0
$$365$$ 8.48528 + 4.24264i 0.444140 + 0.222070i
$$366$$ 0 0
$$367$$ −4.00000 + 4.00000i −0.208798 + 0.208798i −0.803757 0.594958i $$-0.797169\pi$$
0.594958 + 0.803757i $$0.297169\pi$$
$$368$$ 0 0
$$369$$ 0 0
$$370$$ 0 0
$$371$$ 0 0
$$372$$ 0 0
$$373$$ −3.00000 3.00000i −0.155334 0.155334i 0.625161 0.780496i $$-0.285033\pi$$
−0.780496 + 0.625161i $$0.785033\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ −4.24264 4.24264i −0.218507 0.218507i
$$378$$ 0 0
$$379$$ 12.0000i 0.616399i −0.951322 0.308199i $$-0.900274\pi$$
0.951322 0.308199i $$-0.0997264\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ 0 0
$$383$$ −8.48528 + 8.48528i −0.433578 + 0.433578i −0.889843 0.456266i $$-0.849187\pi$$
0.456266 + 0.889843i $$0.349187\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ 0 0
$$388$$ 0 0
$$389$$ −29.6985 −1.50577 −0.752886 0.658150i $$-0.771339\pi$$
−0.752886 + 0.658150i $$0.771339\pi$$
$$390$$ 0 0
$$391$$ 0 0
$$392$$ 0 0
$$393$$ 0 0
$$394$$ 0 0
$$395$$ 5.65685 + 16.9706i 0.284627 + 0.853882i
$$396$$ 0 0
$$397$$ −19.0000 + 19.0000i −0.953583 + 0.953583i −0.998969 0.0453868i $$-0.985548\pi$$
0.0453868 + 0.998969i $$0.485548\pi$$
$$398$$ 0 0
$$399$$ 0 0
$$400$$ 0 0
$$401$$ 26.8701i 1.34183i −0.741536 0.670913i $$-0.765902\pi$$
0.741536 0.670913i $$-0.234098\pi$$
$$402$$ 0 0
$$403$$ −24.0000 24.0000i −1.19553 1.19553i
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 0 0
$$407$$ 39.5980 + 39.5980i 1.96280 + 1.96280i
$$408$$ 0 0
$$409$$ 24.0000i 1.18672i −0.804936 0.593362i $$-0.797800\pi$$
0.804936 0.593362i $$-0.202200\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ 0 0
$$413$$ 0 0
$$414$$ 0 0
$$415$$ −16.0000 + 32.0000i −0.785409 + 1.57082i
$$416$$ 0 0
$$417$$ 0 0
$$418$$ 0 0
$$419$$ −11.3137 −0.552711 −0.276355 0.961056i $$-0.589127\pi$$
−0.276355 + 0.961056i $$0.589127\pi$$
$$420$$ 0 0
$$421$$ 38.0000 1.85201 0.926003 0.377515i $$-0.123221\pi$$
0.926003 + 0.377515i $$0.123221\pi$$
$$422$$ 0 0
$$423$$ 0 0
$$424$$ 0 0
$$425$$ 0 0
$$426$$ 0 0
$$427$$ 0 0
$$428$$ 0 0
$$429$$ 0 0
$$430$$ 0 0
$$431$$ 5.65685i 0.272481i −0.990676 0.136241i $$-0.956498\pi$$
0.990676 0.136241i $$-0.0435020\pi$$
$$432$$ 0 0
$$433$$ 11.0000 + 11.0000i 0.528626 + 0.528626i 0.920163 0.391536i $$-0.128056\pi$$
−0.391536 + 0.920163i $$0.628056\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 0 0
$$437$$ −11.3137 11.3137i −0.541208 0.541208i
$$438$$ 0 0
$$439$$ 8.00000i 0.381819i 0.981608 + 0.190910i $$0.0611437\pi$$
−0.981608 + 0.190910i $$0.938856\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 0 0
$$443$$ 11.3137 11.3137i 0.537531 0.537531i −0.385272 0.922803i $$-0.625893\pi$$
0.922803 + 0.385272i $$0.125893\pi$$
$$444$$ 0 0
$$445$$ −15.0000 + 5.00000i −0.711068 + 0.237023i
$$446$$ 0 0
$$447$$ 0 0
$$448$$ 0 0
$$449$$ 18.3848 0.867631 0.433816 0.901002i $$-0.357167\pi$$
0.433816 + 0.901002i $$0.357167\pi$$
$$450$$ 0 0
$$451$$ 8.00000 0.376705
$$452$$ 0 0
$$453$$ 0 0
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ −23.0000 + 23.0000i −1.07589 + 1.07589i −0.0790217 + 0.996873i $$0.525180\pi$$
−0.996873 + 0.0790217i $$0.974820\pi$$
$$458$$ 0 0
$$459$$ 0 0
$$460$$ 0 0
$$461$$ 32.5269i 1.51493i 0.652876 + 0.757465i $$0.273562\pi$$
−0.652876 + 0.757465i $$0.726438\pi$$
$$462$$ 0 0
$$463$$ 4.00000 + 4.00000i 0.185896 + 0.185896i 0.793919 0.608023i $$-0.208037\pi$$
−0.608023 + 0.793919i $$0.708037\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ −16.9706 16.9706i −0.785304 0.785304i 0.195416 0.980720i $$-0.437394\pi$$
−0.980720 + 0.195416i $$0.937394\pi$$
$$468$$ 0 0
$$469$$ 0 0
$$470$$ 0 0
$$471$$ 0 0
$$472$$ 0 0
$$473$$ −22.6274 + 22.6274i −1.04041 + 1.04041i
$$474$$ 0 0
$$475$$ 12.0000 + 16.0000i 0.550598 + 0.734130i
$$476$$ 0 0
$$477$$ 0 0
$$478$$ 0 0
$$479$$ 5.65685 0.258468 0.129234 0.991614i $$-0.458748\pi$$
0.129234 + 0.991614i $$0.458748\pi$$
$$480$$ 0 0
$$481$$ 42.0000 1.91504
$$482$$ 0 0
$$483$$ 0 0
$$484$$ 0 0
$$485$$ −7.07107 + 14.1421i −0.321081 + 0.642161i
$$486$$ 0 0
$$487$$ 12.0000 12.0000i 0.543772 0.543772i −0.380861 0.924632i $$-0.624372\pi$$
0.924632 + 0.380861i $$0.124372\pi$$
$$488$$ 0 0
$$489$$ 0 0
$$490$$ 0 0
$$491$$ 22.6274i 1.02116i 0.859830 + 0.510581i $$0.170569\pi$$
−0.859830 + 0.510581i $$0.829431\pi$$
$$492$$ 0 0
$$493$$ 0 0
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ 0 0
$$498$$ 0 0
$$499$$ 4.00000i 0.179065i −0.995984 0.0895323i $$-0.971463\pi$$
0.995984 0.0895323i $$-0.0285372\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ 0 0
$$503$$ −25.4558 + 25.4558i −1.13502 + 1.13502i −0.145690 + 0.989330i $$0.546540\pi$$
−0.989330 + 0.145690i $$0.953460\pi$$
$$504$$ 0 0
$$505$$ −1.00000 3.00000i −0.0444994 0.133498i
$$506$$ 0 0
$$507$$ 0 0
$$508$$ 0 0
$$509$$ 12.7279 0.564155 0.282078 0.959392i $$-0.408976\pi$$
0.282078 + 0.959392i $$0.408976\pi$$
$$510$$ 0 0
$$511$$ 0 0
$$512$$ 0 0
$$513$$ 0 0
$$514$$ 0 0
$$515$$ −11.3137 5.65685i −0.498542 0.249271i
$$516$$ 0 0
$$517$$ 16.0000 16.0000i 0.703679 0.703679i
$$518$$ 0 0
$$519$$ 0 0
$$520$$ 0 0
$$521$$ 21.2132i 0.929367i 0.885477 + 0.464684i $$0.153832\pi$$
−0.885477 + 0.464684i $$0.846168\pi$$
$$522$$ 0 0
$$523$$ −24.0000 24.0000i −1.04945 1.04945i −0.998712 0.0507346i $$-0.983844\pi$$
−0.0507346 0.998712i $$-0.516156\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ 0 0
$$528$$ 0 0
$$529$$ 7.00000i 0.304348i
$$530$$ 0 0
$$531$$ 0 0
$$532$$ 0 0
$$533$$ 4.24264 4.24264i 0.183769 0.183769i
$$534$$ 0 0
$$535$$ 32.0000 + 16.0000i 1.38348 + 0.691740i
$$536$$ 0 0
$$537$$ 0 0
$$538$$ 0 0
$$539$$ −39.5980 −1.70561
$$540$$ 0 0
$$541$$ −4.00000 −0.171973 −0.0859867 0.996296i $$-0.527404\pi$$
−0.0859867 + 0.996296i $$0.527404\pi$$
$$542$$ 0 0
$$543$$ 0 0
$$544$$ 0 0
$$545$$ −2.82843 8.48528i −0.121157 0.363470i
$$546$$ 0 0
$$547$$ 12.0000 12.0000i 0.513083 0.513083i −0.402387 0.915470i $$-0.631819\pi$$
0.915470 + 0.402387i $$0.131819\pi$$
$$548$$ 0 0
$$549$$ 0 0
$$550$$ 0 0
$$551$$ 5.65685i 0.240990i
$$552$$ 0 0
$$553$$ 0 0
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 0 0
$$557$$ 8.48528 + 8.48528i 0.359533 + 0.359533i 0.863641 0.504108i $$-0.168179\pi$$
−0.504108 + 0.863641i $$0.668179\pi$$
$$558$$ 0 0
$$559$$ 24.0000i 1.01509i
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 0 0
$$563$$ 5.65685 5.65685i 0.238408 0.238408i −0.577783 0.816191i $$-0.696082\pi$$
0.816191 + 0.577783i $$0.196082\pi$$
$$564$$ 0 0
$$565$$ −6.00000 + 12.0000i −0.252422 + 0.504844i
$$566$$ 0 0
$$567$$ 0 0
$$568$$ 0 0
$$569$$ −35.3553 −1.48217 −0.741086 0.671410i $$-0.765689\pi$$
−0.741086 + 0.671410i $$0.765689\pi$$
$$570$$ 0 0
$$571$$ 20.0000 0.836974 0.418487 0.908223i $$-0.362561\pi$$
0.418487 + 0.908223i $$0.362561\pi$$
$$572$$ 0 0
$$573$$ 0 0
$$574$$ 0 0
$$575$$ −2.82843 + 19.7990i −0.117954 + 0.825675i
$$576$$ 0 0
$$577$$ −9.00000 + 9.00000i −0.374675 + 0.374675i −0.869177 0.494502i $$-0.835351\pi$$
0.494502 + 0.869177i $$0.335351\pi$$
$$578$$ 0 0
$$579$$ 0 0
$$580$$ 0 0
$$581$$ 0 0
$$582$$ 0 0
$$583$$ 48.0000 + 48.0000i 1.98796 + 1.98796i
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ −5.65685 5.65685i −0.233483 0.233483i 0.580662 0.814145i $$-0.302794\pi$$
−0.814145 + 0.580662i $$0.802794\pi$$
$$588$$ 0 0
$$589$$ 32.0000i 1.31854i
$$590$$ 0 0
$$591$$ 0 0
$$592$$ 0 0
$$593$$ −18.3848 + 18.3848i −0.754972 + 0.754972i −0.975403 0.220430i $$-0.929254\pi$$
0.220430 + 0.975403i $$0.429254\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 0 0
$$597$$ 0 0
$$598$$ 0 0
$$599$$ 28.2843 1.15566 0.577832 0.816156i $$-0.303899\pi$$
0.577832 + 0.816156i $$0.303899\pi$$
$$600$$ 0 0
$$601$$ −32.0000 −1.30531 −0.652654 0.757656i $$-0.726344\pi$$
−0.652654 + 0.757656i $$0.726344\pi$$
$$602$$ 0 0
$$603$$ 0 0
$$604$$ 0 0
$$605$$ 44.5477 14.8492i 1.81112 0.603708i
$$606$$ 0 0
$$607$$ 8.00000 8.00000i 0.324710 0.324710i −0.525861 0.850571i $$-0.676257\pi$$
0.850571 + 0.525861i $$0.176257\pi$$
$$608$$ 0 0
$$609$$ 0 0
$$610$$ 0 0
$$611$$ 16.9706i 0.686555i
$$612$$ 0 0
$$613$$ 15.0000 + 15.0000i 0.605844 + 0.605844i 0.941857 0.336013i $$-0.109079\pi$$
−0.336013 + 0.941857i $$0.609079\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ 16.9706 + 16.9706i 0.683209 + 0.683209i 0.960722 0.277513i $$-0.0895101\pi$$
−0.277513 + 0.960722i $$0.589510\pi$$
$$618$$ 0 0
$$619$$ 12.0000i 0.482321i −0.970485 0.241160i $$-0.922472\pi$$
0.970485 0.241160i $$-0.0775280\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ 0 0
$$623$$ 0 0
$$624$$ 0 0
$$625$$ 7.00000 24.0000i 0.280000 0.960000i
$$626$$ 0 0
$$627$$ 0 0
$$628$$ 0 0
$$629$$ 0 0
$$630$$ 0 0
$$631$$ −8.00000 −0.318475 −0.159237 0.987240i $$-0.550904\pi$$
−0.159237 + 0.987240i $$0.550904\pi$$
$$632$$ 0 0
$$633$$ 0 0
$$634$$ 0 0
$$635$$ −16.9706 + 33.9411i −0.673456 + 1.34691i
$$636$$ 0 0
$$637$$ −21.0000 + 21.0000i −0.832050 + 0.832050i
$$638$$ 0 0
$$639$$ 0 0
$$640$$ 0 0
$$641$$ 26.8701i 1.06130i 0.847590 + 0.530652i $$0.178053\pi$$
−0.847590 + 0.530652i $$0.821947\pi$$
$$642$$ 0 0
$$643$$ −28.0000 28.0000i −1.10421 1.10421i −0.993897 0.110316i $$-0.964814\pi$$
−0.110316 0.993897i $$-0.535186\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ 19.7990 + 19.7990i 0.778379 + 0.778379i 0.979555 0.201176i $$-0.0644765\pi$$
−0.201176 + 0.979555i $$0.564476\pi$$
$$648$$ 0 0
$$649$$ 64.0000i 2.51222i
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 0 0
$$653$$ −25.4558 + 25.4558i −0.996164 + 0.996164i −0.999993 0.00382851i $$-0.998781\pi$$
0.00382851 + 0.999993i $$0.498781\pi$$
$$654$$ 0 0
$$655$$ −4.00000 12.0000i −0.156293 0.468879i
$$656$$ 0 0
$$657$$ 0 0
$$658$$ 0 0
$$659$$ −33.9411 −1.32216 −0.661079 0.750316i $$-0.729901\pi$$
−0.661079 + 0.750316i $$0.729901\pi$$
$$660$$ 0 0
$$661$$ −20.0000 −0.777910 −0.388955 0.921257i $$-0.627164\pi$$
−0.388955 + 0.921257i $$0.627164\pi$$
$$662$$ 0 0
$$663$$ 0 0
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 0 0
$$667$$ 4.00000 4.00000i 0.154881 0.154881i
$$668$$ 0 0
$$669$$ 0 0
$$670$$ 0 0
$$671$$ 67.8823i 2.62057i
$$672$$ 0 0
$$673$$ −19.0000 19.0000i −0.732396 0.732396i 0.238698 0.971094i $$-0.423279\pi$$
−0.971094 + 0.238698i $$0.923279\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ 0 0
$$677$$ −19.7990 19.7990i −0.760937 0.760937i 0.215555 0.976492i $$-0.430844\pi$$
−0.976492 + 0.215555i $$0.930844\pi$$
$$678$$ 0 0
$$679$$ 0 0
$$680$$ 0 0
$$681$$ 0 0
$$682$$ 0 0
$$683$$ 33.9411 33.9411i 1.29872 1.29872i 0.369484 0.929237i $$-0.379534\pi$$
0.929237 0.369484i $$-0.120466\pi$$
$$684$$ 0 0
$$685$$ −36.0000 18.0000i −1.37549 0.687745i
$$686$$ 0 0
$$687$$ 0 0
$$688$$ 0 0
$$689$$ 50.9117 1.93958
$$690$$ 0 0
$$691$$ −12.0000 −0.456502 −0.228251 0.973602i $$-0.573301\pi$$
−0.228251 + 0.973602i $$0.573301\pi$$
$$692$$ 0 0
$$693$$ 0 0
$$694$$ 0 0
$$695$$ −14.1421 42.4264i −0.536442 1.60933i
$$696$$ 0 0
$$697$$ 0 0
$$698$$ 0 0
$$699$$ 0 0
$$700$$ 0 0
$$701$$ 32.5269i 1.22852i 0.789102 + 0.614262i $$0.210546\pi$$
−0.789102 + 0.614262i $$0.789454\pi$$
$$702$$ 0 0
$$703$$ 28.0000 + 28.0000i 1.05604 + 1.05604i
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ 0 0
$$708$$ 0 0
$$709$$ 6.00000i 0.225335i −0.993633 0.112667i $$-0.964061\pi$$
0.993633 0.112667i $$-0.0359394\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ 0 0
$$713$$ 22.6274 22.6274i 0.847403 0.847403i
$$714$$ 0 0
$$715$$ 24.0000 48.0000i 0.897549 1.79510i
$$716$$ 0 0
$$717$$ 0 0
$$718$$ 0 0
$$719$$ 22.6274 0.843860 0.421930 0.906628i $$-0.361353\pi$$
0.421930 + 0.906628i $$0.361353\pi$$
$$720$$ 0 0
$$721$$ 0 0
$$722$$ 0 0
$$723$$ 0 0
$$724$$ 0 0
$$725$$ −5.65685 + 4.24264i −0.210090 + 0.157568i
$$726$$ 0 0
$$727$$ 12.0000 12.0000i 0.445055 0.445055i −0.448651 0.893707i $$-0.648096\pi$$
0.893707 + 0.448651i $$0.148096\pi$$
$$728$$ 0 0
$$729$$ 0 0
$$730$$ 0 0
$$731$$ 0 0
$$732$$ 0 0
$$733$$ 23.0000 + 23.0000i 0.849524 + 0.849524i 0.990074 0.140549i $$-0.0448868\pi$$
−0.140549 + 0.990074i $$0.544887\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ −45.2548 45.2548i −1.66698 1.66698i
$$738$$ 0 0
$$739$$ 4.00000i 0.147142i −0.997290 0.0735712i $$-0.976560\pi$$
0.997290 0.0735712i $$-0.0234396\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 0 0
$$743$$ 25.4558 25.4558i 0.933884 0.933884i −0.0640616 0.997946i $$-0.520405\pi$$
0.997946 + 0.0640616i $$0.0204054\pi$$
$$744$$ 0 0
$$745$$ 9.00000 3.00000i 0.329734 0.109911i
$$746$$ 0 0
$$747$$ 0 0
$$748$$ 0 0
$$749$$ 0 0
$$750$$ 0 0
$$751$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$752$$ 0 0
$$753$$ 0 0
$$754$$ 0 0
$$755$$ −16.9706 + 5.65685i −0.617622 + 0.205874i
$$756$$ 0 0
$$757$$ −3.00000 + 3.00000i −0.109037 + 0.109037i −0.759520 0.650484i $$-0.774566\pi$$
0.650484 + 0.759520i $$0.274566\pi$$
$$758$$ 0 0
$$759$$ 0 0
$$760$$ 0 0
$$761$$ 1.41421i 0.0512652i −0.999671 0.0256326i $$-0.991840\pi$$
0.999671 0.0256326i $$-0.00816000\pi$$
$$762$$ 0 0
$$763$$ 0 0
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ 33.9411 + 33.9411i 1.22554 + 1.22554i
$$768$$ 0 0
$$769$$ 40.0000i 1.44244i −0.692708 0.721218i $$-0.743582\pi$$
0.692708 0.721218i $$-0.256418\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ 0 0
$$773$$ 14.1421 14.1421i 0.508657 0.508657i −0.405457 0.914114i $$-0.632888\pi$$
0.914114 + 0.405457i $$0.132888\pi$$
$$774$$ 0 0
$$775$$ −32.0000 + 24.0000i −1.14947 + 0.862105i
$$776$$ 0 0
$$777$$ 0 0
$$778$$ 0 0
$$779$$ 5.65685 0.202678
$$780$$ 0 0
$$781$$ −32.0000 −1.14505
$$782$$ 0 0
$$783$$ 0 0
$$784$$ 0 0
$$785$$ 9.89949 19.7990i 0.353328 0.706656i
$$786$$ 0 0
$$787$$ 24.0000 24.0000i 0.855508 0.855508i −0.135297 0.990805i $$-0.543199\pi$$
0.990805 + 0.135297i $$0.0431990\pi$$
$$788$$ 0 0
$$789$$ 0 0
$$790$$ 0 0
$$791$$ 0 0
$$792$$ 0 0
$$793$$ −36.0000 36.0000i −1.27840 1.27840i
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 0 0
$$797$$ −29.6985 29.6985i −1.05197 1.05197i −0.998573 0.0534012i $$-0.982994\pi$$
−0.0534012 0.998573i $$-0.517006\pi$$
$$798$$ 0 0
$$799$$ 0 0
$$800$$ 0 0
$$801$$ 0 0
$$802$$ 0 0
$$803$$ 16.9706 16.9706i 0.598878 0.598878i
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 0 0