# Properties

 Label 3600.2.w.d.1457.1 Level $3600$ Weight $2$ Character 3600.1457 Analytic conductor $28.746$ Analytic rank $0$ Dimension $4$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$3600 = 2^{4} \cdot 3^{2} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 3600.w (of order $$4$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$28.7461447277$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(i)$$ Coefficient field: $$\Q(\zeta_{8})$$ Defining polynomial: $$x^{4} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{29}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 360) Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

## Embedding invariants

 Embedding label 1457.1 Root $$0.707107 + 0.707107i$$ of defining polynomial Character $$\chi$$ $$=$$ 3600.1457 Dual form 3600.2.w.d.593.2

## $q$-expansion

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

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/3600\mathbb{Z}\right)^\times$$.

 $$n$$ $$577$$ $$901$$ $$2801$$ $$3151$$ $$\chi(n)$$ $$e\left(\frac{1}{4}\right)$$ $$1$$ $$-1$$ $$1$$

## 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
$$4$$ 0 0
$$5$$ 0 0
$$6$$ 0 0
$$7$$ 0 0 0.707107 0.707107i $$-0.250000\pi$$
−0.707107 + 0.707107i $$0.750000\pi$$
$$8$$ 0 0
$$9$$ 0 0
$$10$$ 0 0
$$11$$ 5.65685i 1.70561i −0.522233 0.852803i $$-0.674901\pi$$
0.522233 0.852803i $$-0.325099\pi$$
$$12$$ 0 0
$$13$$ −3.00000 + 3.00000i −0.832050 + 0.832050i −0.987797 0.155747i $$-0.950222\pi$$
0.155747 + 0.987797i $$0.450222\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 0 0
$$17$$ 0 0 −0.707107 0.707107i $$-0.750000\pi$$
0.707107 + 0.707107i $$0.250000\pi$$
$$18$$ 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 $$-0.113071\pi$$
−0.347801 + 0.937568i $$0.613071\pi$$
$$24$$ 0 0
$$25$$ 0 0
$$26$$ 0 0
$$27$$ 0 0
$$28$$ 0 0
$$29$$ 1.41421 0.262613 0.131306 0.991342i $$-0.458083\pi$$
0.131306 + 0.991342i $$0.458083\pi$$
$$30$$ 0 0
$$31$$ 8.00000 1.43684 0.718421 0.695608i $$-0.244865\pi$$
0.718421 + 0.695608i $$0.244865\pi$$
$$32$$ 0 0
$$33$$ 0 0
$$34$$ 0 0
$$35$$ 0 0
$$36$$ 0 0
$$37$$ −7.00000 7.00000i −1.15079 1.15079i −0.986394 0.164399i $$-0.947432\pi$$
−0.164399 0.986394i $$-0.552568\pi$$
$$38$$ 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 −0.332950 0.942944i $$-0.608044\pi$$
0.942944 + 0.332950i $$0.108044\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 0 0
$$47$$ 2.82843 2.82843i 0.412568 0.412568i −0.470064 0.882632i $$-0.655769\pi$$
0.882632 + 0.470064i $$0.155769\pi$$
$$48$$ 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 0.983243 + 0.182300i $$0.0583542\pi$$
0.182300 + 0.983243i $$0.441646\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 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
$$65$$ 0 0
$$66$$ 0 0
$$67$$ −8.00000 8.00000i −0.977356 0.977356i 0.0223937 0.999749i $$-0.492871\pi$$
−0.999749 + 0.0223937i $$0.992871\pi$$
$$68$$ 0 0
$$69$$ 0 0
$$70$$ 0 0
$$71$$ 5.65685i 0.671345i −0.941979 0.335673i $$-0.891036\pi$$
0.941979 0.335673i $$-0.108964\pi$$
$$72$$ 0 0
$$73$$ 3.00000 3.00000i 0.351123 0.351123i −0.509404 0.860527i $$-0.670134\pi$$
0.860527 + 0.509404i $$0.170134\pi$$
$$74$$ 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 −0.959237 0.282604i $$-0.908802\pi$$
−0.282604 0.959237i $$-0.591198\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ 0 0
$$87$$ 0 0
$$88$$ 0 0
$$89$$ −7.07107 −0.749532 −0.374766 0.927119i $$-0.622277\pi$$
−0.374766 + 0.927119i $$0.622277\pi$$
$$90$$ 0 0
$$91$$ 0 0
$$92$$ 0 0
$$93$$ 0 0
$$94$$ 0 0
$$95$$ 0 0
$$96$$ 0 0
$$97$$ −5.00000 5.00000i −0.507673 0.507673i 0.406138 0.913812i $$-0.366875\pi$$
−0.913812 + 0.406138i $$0.866875\pi$$
$$98$$ 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 −0.482025 0.876157i $$-0.660099\pi$$
0.876157 + 0.482025i $$0.160099\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ 11.3137 11.3137i 1.09374 1.09374i 0.0986115 0.995126i $$-0.468560\pi$$
0.995126 0.0986115i $$-0.0314401\pi$$
$$108$$ 0 0
$$109$$ 4.00000i 0.383131i −0.981480 0.191565i $$-0.938644\pi$$
0.981480 0.191565i $$-0.0613564\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ 0 0
$$113$$ 4.24264 + 4.24264i 0.399114 + 0.399114i 0.877920 0.478806i $$-0.158930\pi$$
−0.478806 + 0.877920i $$0.658930\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ 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
$$125$$ 0 0
$$126$$ 0 0
$$127$$ 12.0000 + 12.0000i 1.06483 + 1.06483i 0.997748 + 0.0670802i $$0.0213683\pi$$
0.0670802 + 0.997748i $$0.478632\pi$$
$$128$$ 0 0
$$129$$ 0 0
$$130$$ 0 0
$$131$$ 5.65685i 0.494242i −0.968985 0.247121i $$-0.920516\pi$$
0.968985 0.247121i $$-0.0794845\pi$$
$$132$$ 0 0
$$133$$ 0 0
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ 12.7279 12.7279i 1.08742 1.08742i 0.0916263 0.995793i $$-0.470793\pi$$
0.995793 0.0916263i $$-0.0292065\pi$$
$$138$$ 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
$$145$$ 0 0
$$146$$ 0 0
$$147$$ 0 0
$$148$$ 0 0
$$149$$ 4.24264 0.347571 0.173785 0.984784i $$-0.444400\pi$$
0.173785 + 0.984784i $$0.444400\pi$$
$$150$$ 0 0
$$151$$ −8.00000 −0.651031 −0.325515 0.945537i $$-0.605538\pi$$
−0.325515 + 0.945537i $$0.605538\pi$$
$$152$$ 0 0
$$153$$ 0 0
$$154$$ 0 0
$$155$$ 0 0
$$156$$ 0 0
$$157$$ 7.00000 + 7.00000i 0.558661 + 0.558661i 0.928926 0.370265i $$-0.120733\pi$$
−0.370265 + 0.928926i $$0.620733\pi$$
$$158$$ 0 0
$$159$$ 0 0
$$160$$ 0 0
$$161$$ 0 0
$$162$$ 0 0
$$163$$ 12.0000 12.0000i 0.939913 0.939913i −0.0583818 0.998294i $$-0.518594\pi$$
0.998294 + 0.0583818i $$0.0185941\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ 8.48528 8.48528i 0.656611 0.656611i −0.297966 0.954577i $$-0.596308\pi$$
0.954577 + 0.297966i $$0.0963081\pi$$
$$168$$ 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 0.385221 0.922824i $$-0.374125\pi$$
−0.922824 + 0.385221i $$0.874125\pi$$
$$174$$ 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
$$185$$ 0 0
$$186$$ 0 0
$$187$$ 0 0
$$188$$ 0 0
$$189$$ 0 0
$$190$$ 0 0
$$191$$ 11.3137i 0.818631i −0.912393 0.409316i $$-0.865768\pi$$
0.912393 0.409316i $$-0.134232\pi$$
$$192$$ 0 0
$$193$$ −5.00000 + 5.00000i −0.359908 + 0.359908i −0.863779 0.503871i $$-0.831909\pi$$
0.503871 + 0.863779i $$0.331909\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ 12.7279 12.7279i 0.906827 0.906827i −0.0891879 0.996015i $$-0.528427\pi$$
0.996015 + 0.0891879i $$0.0284272\pi$$
$$198$$ 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
$$205$$ 0 0
$$206$$ 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 $$-0.741699\pi$$
−0.688428 + 0.725304i $$0.741699\pi$$
$$212$$ 0 0
$$213$$ 0 0
$$214$$ 0 0
$$215$$ 0 0
$$216$$ 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 $$-0.750000\pi$$
0.707107 + 0.707107i $$0.250000\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 0 0
$$227$$ 0 0 −0.707107 0.707107i $$-0.750000\pi$$
0.707107 + 0.707107i $$0.250000\pi$$
$$228$$ 0 0
$$229$$ 6.00000i 0.396491i −0.980152 0.198246i $$-0.936476\pi$$
0.980152 0.198246i $$-0.0635244\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ 5.65685 + 5.65685i 0.370593 + 0.370593i 0.867693 0.497100i $$-0.165602\pi$$
−0.497100 + 0.867693i $$0.665602\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ 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
$$245$$ 0 0
$$246$$ 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 −0.892375 0.451294i $$-0.850963\pi$$
0.451294 + 0.892375i $$0.350963\pi$$
$$258$$ 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 $$-0.210644\pi$$
−0.614505 + 0.788913i $$0.710644\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ 0 0
$$267$$ 0 0
$$268$$ 0 0
$$269$$ −15.5563 −0.948487 −0.474244 0.880394i $$-0.657278\pi$$
−0.474244 + 0.880394i $$0.657278\pi$$
$$270$$ 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
$$277$$ −11.0000 11.0000i −0.660926 0.660926i 0.294672 0.955598i $$-0.404789\pi$$
−0.955598 + 0.294672i $$0.904789\pi$$
$$278$$ 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 −0.998859 0.0477577i $$-0.984792\pi$$
0.0477577 + 0.998859i $$0.484792\pi$$
$$284$$ 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 0.664589 0.747209i $$-0.268606\pi$$
−0.747209 + 0.664589i $$0.768606\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ 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
$$305$$ 0 0
$$306$$ 0 0
$$307$$ −20.0000 20.0000i −1.14146 1.14146i −0.988183 0.153277i $$-0.951017\pi$$
−0.153277 0.988183i $$-0.548983\pi$$
$$308$$ 0 0
$$309$$ 0 0
$$310$$ 0 0
$$311$$ 28.2843i 1.60385i 0.597422 + 0.801927i $$0.296192\pi$$
−0.597422 + 0.801927i $$0.703808\pi$$
$$312$$ 0 0
$$313$$ 9.00000 9.00000i 0.508710 0.508710i −0.405420 0.914130i $$-0.632875\pi$$
0.914130 + 0.405420i $$0.132875\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ 2.82843 2.82843i 0.158860 0.158860i −0.623201 0.782062i $$-0.714168\pi$$
0.782062 + 0.623201i $$0.214168\pi$$
$$318$$ 0 0
$$319$$ 8.00000i 0.447914i
$$320$$ 0 0
$$321$$ 0 0
$$322$$ 0 0
$$323$$ 0 0
$$324$$ 0 0
$$325$$ 0 0
$$326$$ 0 0
$$327$$ 0 0
$$328$$ 0 0
$$329$$ 0 0
$$330$$ 0 0
$$331$$ 4.00000 0.219860 0.109930 0.993939i $$-0.464937\pi$$
0.109930 + 0.993939i $$0.464937\pi$$
$$332$$ 0 0
$$333$$ 0 0
$$334$$ 0 0
$$335$$ 0 0
$$336$$ 0 0
$$337$$ −15.0000 15.0000i −0.817102 0.817102i 0.168585 0.985687i $$-0.446080\pi$$
−0.985687 + 0.168585i $$0.946080\pi$$
$$338$$ 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 $$-0.750000\pi$$
0.707107 + 0.707107i $$0.250000\pi$$
$$348$$ 0 0
$$349$$ 4.00000i 0.214115i 0.994253 + 0.107058i $$0.0341429\pi$$
−0.994253 + 0.107058i $$0.965857\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 0 0
$$353$$ 5.65685 + 5.65685i 0.301084 + 0.301084i 0.841438 0.540354i $$-0.181710\pi$$
−0.540354 + 0.841438i $$0.681710\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ 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
$$365$$ 0 0
$$366$$ 0 0
$$367$$ −4.00000 4.00000i −0.208798 0.208798i 0.594958 0.803757i $$-0.297169\pi$$
−0.803757 + 0.594958i $$0.797169\pi$$
$$368$$ 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 $$-0.714967\pi$$
0.780496 + 0.625161i $$0.214967\pi$$
$$374$$ 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 $$-0.150813\pi$$
−0.456266 + 0.889843i $$0.650813\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ 0 0
$$388$$ 0 0
$$389$$ 29.6985 1.50577 0.752886 0.658150i $$-0.228661\pi$$
0.752886 + 0.658150i $$0.228661\pi$$
$$390$$ 0 0
$$391$$ 0 0
$$392$$ 0 0
$$393$$ 0 0
$$394$$ 0 0
$$395$$ 0 0
$$396$$ 0 0
$$397$$ 19.0000 + 19.0000i 0.953583 + 0.953583i 0.998969 0.0453868i $$-0.0144520\pi$$
−0.0453868 + 0.998969i $$0.514452\pi$$
$$398$$ 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 $$0.202200\pi$$
−0.804936 + 0.593362i $$0.797800\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ 0 0
$$413$$ 0 0
$$414$$ 0 0
$$415$$ 0 0
$$416$$ 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 $$0.0435020\pi$$
−0.990676 + 0.136241i $$0.956498\pi$$
$$432$$ 0 0
$$433$$ −11.0000 + 11.0000i −0.528626 + 0.528626i −0.920163 0.391536i $$-0.871944\pi$$
0.391536 + 0.920163i $$0.371944\pi$$
$$434$$ 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 $$-0.374107\pi$$
−0.922803 + 0.385272i $$0.874107\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ 0 0
$$447$$ 0 0
$$448$$ 0 0
$$449$$ −18.3848 −0.867631 −0.433816 0.901002i $$-0.642833\pi$$
−0.433816 + 0.901002i $$0.642833\pi$$
$$450$$ 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 0.996873 + 0.0790217i $$0.0251796\pi$$
0.0790217 + 0.996873i $$0.474820\pi$$
$$458$$ 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 −0.608023 0.793919i $$-0.708037\pi$$
0.793919 + 0.608023i $$0.208037\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ 16.9706 16.9706i 0.785304 0.785304i −0.195416 0.980720i $$-0.562606\pi$$
0.980720 + 0.195416i $$0.0626058\pi$$
$$468$$ 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
$$475$$ 0 0
$$476$$ 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
$$485$$ 0 0
$$486$$ 0 0
$$487$$ 12.0000 + 12.0000i 0.543772 + 0.543772i 0.924632 0.380861i $$-0.124372\pi$$
−0.380861 + 0.924632i $$0.624372\pi$$
$$488$$ 0 0
$$489$$ 0 0
$$490$$ 0 0
$$491$$ 22.6274i 1.02116i −0.859830 0.510581i $$-0.829431\pi$$
0.859830 0.510581i $$-0.170569\pi$$
$$492$$ 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 0.989330 + 0.145690i $$0.0465401\pi$$
0.145690 + 0.989330i $$0.453460\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ 0 0
$$507$$ 0 0
$$508$$ 0 0
$$509$$ −12.7279 −0.564155 −0.282078 0.959392i $$-0.591024\pi$$
−0.282078 + 0.959392i $$0.591024\pi$$
$$510$$ 0 0
$$511$$ 0 0
$$512$$ 0 0
$$513$$ 0 0
$$514$$ 0 0
$$515$$ 0 0
$$516$$ 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 −0.0507346 + 0.998712i $$0.516156\pi$$
−0.998712 + 0.0507346i $$0.983844\pi$$
$$524$$ 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
$$535$$ 0 0
$$536$$ 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
$$545$$ 0 0
$$546$$ 0 0
$$547$$ 12.0000 + 12.0000i 0.513083 + 0.513083i 0.915470 0.402387i $$-0.131819\pi$$
−0.402387 + 0.915470i $$0.631819\pi$$
$$548$$ 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 −0.504108 0.863641i $$-0.668179\pi$$
0.863641 + 0.504108i $$0.168179\pi$$
$$558$$ 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 $$-0.303918\pi$$
−0.816191 + 0.577783i $$0.803918\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ 0 0
$$567$$ 0 0
$$568$$ 0 0
$$569$$ 35.3553 1.48217 0.741086 0.671410i $$-0.234311\pi$$
0.741086 + 0.671410i $$0.234311\pi$$
$$570$$ 0 0
$$571$$ −20.0000 −0.836974 −0.418487 0.908223i $$-0.637439\pi$$
−0.418487 + 0.908223i $$0.637439\pi$$
$$572$$ 0 0
$$573$$ 0 0
$$574$$ 0 0
$$575$$ 0 0
$$576$$ 0 0
$$577$$ 9.00000 + 9.00000i 0.374675 + 0.374675i 0.869177 0.494502i $$-0.164649\pi$$
−0.494502 + 0.869177i $$0.664649\pi$$
$$578$$ 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 $$-0.697206\pi$$
0.814145 + 0.580662i $$0.197206\pi$$
$$588$$ 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 0.220430 0.975403i $$-0.429254\pi$$
−0.975403 + 0.220430i $$0.929254\pi$$
$$594$$ 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
$$605$$ 0 0
$$606$$ 0 0
$$607$$ 8.00000 + 8.00000i 0.324710 + 0.324710i 0.850571 0.525861i $$-0.176257\pi$$
−0.525861 + 0.850571i $$0.676257\pi$$
$$608$$ 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 $$-0.890921\pi$$
0.336013 + 0.941857i $$0.390921\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ 16.9706 16.9706i 0.683209 0.683209i −0.277513 0.960722i $$-0.589510\pi$$
0.960722 + 0.277513i $$0.0895101\pi$$
$$618$$ 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
$$625$$ 0 0
$$626$$ 0 0
$$627$$ 0 0
$$628$$ 0 0
$$629$$ 0 0
$$630$$ 0 0
$$631$$ 8.00000 0.318475 0.159237 0.987240i $$-0.449096\pi$$
0.159237 + 0.987240i $$0.449096\pi$$
$$632$$ 0 0
$$633$$ 0 0
$$634$$ 0 0
$$635$$ 0 0
$$636$$ 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 −0.110316 + 0.993897i $$0.535186\pi$$
−0.993897 + 0.110316i $$0.964814\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ −19.7990 + 19.7990i −0.778379 + 0.778379i −0.979555 0.201176i $$-0.935524\pi$$
0.201176 + 0.979555i $$0.435524\pi$$
$$648$$ 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 0.00382851 0.999993i $$-0.498781\pi$$
−0.999993 + 0.00382851i $$0.998781\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ 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 $$-0.576721\pi$$
0.971094 + 0.238698i $$0.0767205\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ 0 0
$$677$$ −19.7990 + 19.7990i −0.760937 + 0.760937i −0.976492 0.215555i $$-0.930844\pi$$
0.215555 + 0.976492i $$0.430844\pi$$
$$678$$ 0 0
$$679$$ 0 0
$$680$$ 0 0
$$681$$ 0 0
$$682$$ 0 0
$$683$$ −33.9411 33.9411i −1.29872 1.29872i −0.929237 0.369484i $$-0.879534\pi$$
−0.369484 0.929237i $$-0.620466\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ 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 $$-0.426699\pi$$
0.228251 + 0.973602i $$0.426699\pi$$
$$692$$ 0 0
$$693$$ 0 0
$$694$$ 0 0
$$695$$ 0 0
$$696$$ 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 $$0.0359394\pi$$
−0.993633 + 0.112667i $$0.964061\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ 0 0
$$713$$ 22.6274 + 22.6274i 0.847403 + 0.847403i
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 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
$$725$$ 0 0
$$726$$ 0 0
$$727$$ 12.0000 + 12.0000i 0.445055 + 0.445055i 0.893707 0.448651i $$-0.148096\pi$$
−0.448651 + 0.893707i $$0.648096\pi$$
$$728$$ 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 $$-0.955113\pi$$
0.140549 + 0.990074i $$0.455113\pi$$
$$734$$ 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 $$-0.479595\pi$$
−0.997946 + 0.0640616i $$0.979595\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ 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
$$755$$ 0 0
$$756$$ 0 0
$$757$$ 3.00000 + 3.00000i 0.109037 + 0.109037i 0.759520 0.650484i $$-0.225434\pi$$
−0.650484 + 0.759520i $$0.725434\pi$$
$$758$$ 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 $$0.256418\pi$$
−0.692708 + 0.721218i $$0.743582\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ 0 0
$$773$$ 14.1421 + 14.1421i 0.508657 + 0.508657i 0.914114 0.405457i $$-0.132888\pi$$
−0.405457 + 0.914114i $$0.632888\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ 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
$$785$$ 0 0
$$786$$ 0 0
$$787$$ 24.0000 + 24.0000i 0.855508 + 0.855508i 0.990805 0.135297i $$-0.0431990\pi$$
−0.135297 + 0.990805i $$0.543199\pi$$
$$788$$ 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 −0.0534012 + 0.998573i $$0.517006\pi$$
−0.998573 + 0.0534012i $$0.982994\pi$$
$$798$$ 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
$$807$$ 0 0
$$808$$ 0 0
$$809$$ −32.5269 −1.14359 −0.571793