# Properties

 Label 3840.2.d.o.2689.1 Level $3840$ Weight $2$ Character 3840.2689 Analytic conductor $30.663$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$3840 = 2^{8} \cdot 3 \cdot 5$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 3840.d (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$30.6625543762$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(i)$$ Defining polynomial: $$x^{2} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 60) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 2689.1 Root $$1.00000i$$ of defining polynomial Character $$\chi$$ $$=$$ 3840.2689 Dual form 3840.2.d.o.2689.2

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-1.00000 q^{3} +(2.00000 - 1.00000i) q^{5} +4.00000i q^{7} +1.00000 q^{9} +O(q^{10})$$ $$q-1.00000 q^{3} +(2.00000 - 1.00000i) q^{5} +4.00000i q^{7} +1.00000 q^{9} +4.00000i q^{11} +(-2.00000 + 1.00000i) q^{15} -4.00000i q^{17} -4.00000i q^{21} -4.00000i q^{23} +(3.00000 - 4.00000i) q^{25} -1.00000 q^{27} +6.00000i q^{29} +4.00000 q^{31} -4.00000i q^{33} +(4.00000 + 8.00000i) q^{35} +8.00000 q^{37} +10.0000 q^{41} +4.00000 q^{43} +(2.00000 - 1.00000i) q^{45} +4.00000i q^{47} -9.00000 q^{49} +4.00000i q^{51} -12.0000 q^{53} +(4.00000 + 8.00000i) q^{55} +4.00000i q^{59} +2.00000i q^{61} +4.00000i q^{63} -4.00000 q^{67} +4.00000i q^{69} +8.00000i q^{73} +(-3.00000 + 4.00000i) q^{75} -16.0000 q^{77} +12.0000 q^{79} +1.00000 q^{81} -4.00000 q^{83} +(-4.00000 - 8.00000i) q^{85} -6.00000i q^{87} -10.0000 q^{89} -4.00000 q^{93} -8.00000i q^{97} +4.00000i q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 2q^{3} + 4q^{5} + 2q^{9} + O(q^{10})$$ $$2q - 2q^{3} + 4q^{5} + 2q^{9} - 4q^{15} + 6q^{25} - 2q^{27} + 8q^{31} + 8q^{35} + 16q^{37} + 20q^{41} + 8q^{43} + 4q^{45} - 18q^{49} - 24q^{53} + 8q^{55} - 8q^{67} - 6q^{75} - 32q^{77} + 24q^{79} + 2q^{81} - 8q^{83} - 8q^{85} - 20q^{89} - 8q^{93} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/3840\mathbb{Z}\right)^\times$$.

 $$n$$ $$511$$ $$1537$$ $$2561$$ $$2821$$ $$\chi(n)$$ $$1$$ $$-1$$ $$1$$ $$-1$$

## 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 −0.577350
$$4$$ 0 0
$$5$$ 2.00000 1.00000i 0.894427 0.447214i
$$6$$ 0 0
$$7$$ 4.00000i 1.51186i 0.654654 + 0.755929i $$0.272814\pi$$
−0.654654 + 0.755929i $$0.727186\pi$$
$$8$$ 0 0
$$9$$ 1.00000 0.333333
$$10$$ 0 0
$$11$$ 4.00000i 1.20605i 0.797724 + 0.603023i $$0.206037\pi$$
−0.797724 + 0.603023i $$0.793963\pi$$
$$12$$ 0 0
$$13$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$14$$ 0 0
$$15$$ −2.00000 + 1.00000i −0.516398 + 0.258199i
$$16$$ 0 0
$$17$$ 4.00000i 0.970143i −0.874475 0.485071i $$-0.838794\pi$$
0.874475 0.485071i $$-0.161206\pi$$
$$18$$ 0 0
$$19$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$20$$ 0 0
$$21$$ 4.00000i 0.872872i
$$22$$ 0 0
$$23$$ 4.00000i 0.834058i −0.908893 0.417029i $$-0.863071\pi$$
0.908893 0.417029i $$-0.136929\pi$$
$$24$$ 0 0
$$25$$ 3.00000 4.00000i 0.600000 0.800000i
$$26$$ 0 0
$$27$$ −1.00000 −0.192450
$$28$$ 0 0
$$29$$ 6.00000i 1.11417i 0.830455 + 0.557086i $$0.188081\pi$$
−0.830455 + 0.557086i $$0.811919\pi$$
$$30$$ 0 0
$$31$$ 4.00000 0.718421 0.359211 0.933257i $$-0.383046\pi$$
0.359211 + 0.933257i $$0.383046\pi$$
$$32$$ 0 0
$$33$$ 4.00000i 0.696311i
$$34$$ 0 0
$$35$$ 4.00000 + 8.00000i 0.676123 + 1.35225i
$$36$$ 0 0
$$37$$ 8.00000 1.31519 0.657596 0.753371i $$-0.271573\pi$$
0.657596 + 0.753371i $$0.271573\pi$$
$$38$$ 0 0
$$39$$ 0 0
$$40$$ 0 0
$$41$$ 10.0000 1.56174 0.780869 0.624695i $$-0.214777\pi$$
0.780869 + 0.624695i $$0.214777\pi$$
$$42$$ 0 0
$$43$$ 4.00000 0.609994 0.304997 0.952353i $$-0.401344\pi$$
0.304997 + 0.952353i $$0.401344\pi$$
$$44$$ 0 0
$$45$$ 2.00000 1.00000i 0.298142 0.149071i
$$46$$ 0 0
$$47$$ 4.00000i 0.583460i 0.956501 + 0.291730i $$0.0942309\pi$$
−0.956501 + 0.291730i $$0.905769\pi$$
$$48$$ 0 0
$$49$$ −9.00000 −1.28571
$$50$$ 0 0
$$51$$ 4.00000i 0.560112i
$$52$$ 0 0
$$53$$ −12.0000 −1.64833 −0.824163 0.566352i $$-0.808354\pi$$
−0.824163 + 0.566352i $$0.808354\pi$$
$$54$$ 0 0
$$55$$ 4.00000 + 8.00000i 0.539360 + 1.07872i
$$56$$ 0 0
$$57$$ 0 0
$$58$$ 0 0
$$59$$ 4.00000i 0.520756i 0.965507 + 0.260378i $$0.0838471\pi$$
−0.965507 + 0.260378i $$0.916153\pi$$
$$60$$ 0 0
$$61$$ 2.00000i 0.256074i 0.991769 + 0.128037i $$0.0408676\pi$$
−0.991769 + 0.128037i $$0.959132\pi$$
$$62$$ 0 0
$$63$$ 4.00000i 0.503953i
$$64$$ 0 0
$$65$$ 0 0
$$66$$ 0 0
$$67$$ −4.00000 −0.488678 −0.244339 0.969690i $$-0.578571\pi$$
−0.244339 + 0.969690i $$0.578571\pi$$
$$68$$ 0 0
$$69$$ 4.00000i 0.481543i
$$70$$ 0 0
$$71$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$72$$ 0 0
$$73$$ 8.00000i 0.936329i 0.883641 + 0.468165i $$0.155085\pi$$
−0.883641 + 0.468165i $$0.844915\pi$$
$$74$$ 0 0
$$75$$ −3.00000 + 4.00000i −0.346410 + 0.461880i
$$76$$ 0 0
$$77$$ −16.0000 −1.82337
$$78$$ 0 0
$$79$$ 12.0000 1.35011 0.675053 0.737769i $$-0.264121\pi$$
0.675053 + 0.737769i $$0.264121\pi$$
$$80$$ 0 0
$$81$$ 1.00000 0.111111
$$82$$ 0 0
$$83$$ −4.00000 −0.439057 −0.219529 0.975606i $$-0.570452\pi$$
−0.219529 + 0.975606i $$0.570452\pi$$
$$84$$ 0 0
$$85$$ −4.00000 8.00000i −0.433861 0.867722i
$$86$$ 0 0
$$87$$ 6.00000i 0.643268i
$$88$$ 0 0
$$89$$ −10.0000 −1.06000 −0.529999 0.847998i $$-0.677808\pi$$
−0.529999 + 0.847998i $$0.677808\pi$$
$$90$$ 0 0
$$91$$ 0 0
$$92$$ 0 0
$$93$$ −4.00000 −0.414781
$$94$$ 0 0
$$95$$ 0 0
$$96$$ 0 0
$$97$$ 8.00000i 0.812277i −0.913812 0.406138i $$-0.866875\pi$$
0.913812 0.406138i $$-0.133125\pi$$
$$98$$ 0 0
$$99$$ 4.00000i 0.402015i
$$100$$ 0 0
$$101$$ 2.00000i 0.199007i 0.995037 + 0.0995037i $$0.0317255\pi$$
−0.995037 + 0.0995037i $$0.968274\pi$$
$$102$$ 0 0
$$103$$ 4.00000i 0.394132i 0.980390 + 0.197066i $$0.0631413\pi$$
−0.980390 + 0.197066i $$0.936859\pi$$
$$104$$ 0 0
$$105$$ −4.00000 8.00000i −0.390360 0.780720i
$$106$$ 0 0
$$107$$ −12.0000 −1.16008 −0.580042 0.814587i $$-0.696964\pi$$
−0.580042 + 0.814587i $$0.696964\pi$$
$$108$$ 0 0
$$109$$ 2.00000i 0.191565i 0.995402 + 0.0957826i $$0.0305354\pi$$
−0.995402 + 0.0957826i $$0.969465\pi$$
$$110$$ 0 0
$$111$$ −8.00000 −0.759326
$$112$$ 0 0
$$113$$ 12.0000i 1.12887i 0.825479 + 0.564433i $$0.190905\pi$$
−0.825479 + 0.564433i $$0.809095\pi$$
$$114$$ 0 0
$$115$$ −4.00000 8.00000i −0.373002 0.746004i
$$116$$ 0 0
$$117$$ 0 0
$$118$$ 0 0
$$119$$ 16.0000 1.46672
$$120$$ 0 0
$$121$$ −5.00000 −0.454545
$$122$$ 0 0
$$123$$ −10.0000 −0.901670
$$124$$ 0 0
$$125$$ 2.00000 11.0000i 0.178885 0.983870i
$$126$$ 0 0
$$127$$ 4.00000i 0.354943i −0.984126 0.177471i $$-0.943208\pi$$
0.984126 0.177471i $$-0.0567917\pi$$
$$128$$ 0 0
$$129$$ −4.00000 −0.352180
$$130$$ 0 0
$$131$$ 12.0000i 1.04844i 0.851581 + 0.524222i $$0.175644\pi$$
−0.851581 + 0.524222i $$0.824356\pi$$
$$132$$ 0 0
$$133$$ 0 0
$$134$$ 0 0
$$135$$ −2.00000 + 1.00000i −0.172133 + 0.0860663i
$$136$$ 0 0
$$137$$ 12.0000i 1.02523i 0.858619 + 0.512615i $$0.171323\pi$$
−0.858619 + 0.512615i $$0.828677\pi$$
$$138$$ 0 0
$$139$$ 16.0000i 1.35710i 0.734553 + 0.678551i $$0.237392\pi$$
−0.734553 + 0.678551i $$0.762608\pi$$
$$140$$ 0 0
$$141$$ 4.00000i 0.336861i
$$142$$ 0 0
$$143$$ 0 0
$$144$$ 0 0
$$145$$ 6.00000 + 12.0000i 0.498273 + 0.996546i
$$146$$ 0 0
$$147$$ 9.00000 0.742307
$$148$$ 0 0
$$149$$ 2.00000i 0.163846i −0.996639 0.0819232i $$-0.973894\pi$$
0.996639 0.0819232i $$-0.0261062\pi$$
$$150$$ 0 0
$$151$$ 20.0000 1.62758 0.813788 0.581161i $$-0.197401\pi$$
0.813788 + 0.581161i $$0.197401\pi$$
$$152$$ 0 0
$$153$$ 4.00000i 0.323381i
$$154$$ 0 0
$$155$$ 8.00000 4.00000i 0.642575 0.321288i
$$156$$ 0 0
$$157$$ 8.00000 0.638470 0.319235 0.947676i $$-0.396574\pi$$
0.319235 + 0.947676i $$0.396574\pi$$
$$158$$ 0 0
$$159$$ 12.0000 0.951662
$$160$$ 0 0
$$161$$ 16.0000 1.26098
$$162$$ 0 0
$$163$$ −20.0000 −1.56652 −0.783260 0.621694i $$-0.786445\pi$$
−0.783260 + 0.621694i $$0.786445\pi$$
$$164$$ 0 0
$$165$$ −4.00000 8.00000i −0.311400 0.622799i
$$166$$ 0 0
$$167$$ 12.0000i 0.928588i 0.885681 + 0.464294i $$0.153692\pi$$
−0.885681 + 0.464294i $$0.846308\pi$$
$$168$$ 0 0
$$169$$ −13.0000 −1.00000
$$170$$ 0 0
$$171$$ 0 0
$$172$$ 0 0
$$173$$ 4.00000 0.304114 0.152057 0.988372i $$-0.451410\pi$$
0.152057 + 0.988372i $$0.451410\pi$$
$$174$$ 0 0
$$175$$ 16.0000 + 12.0000i 1.20949 + 0.907115i
$$176$$ 0 0
$$177$$ 4.00000i 0.300658i
$$178$$ 0 0
$$179$$ 4.00000i 0.298974i −0.988764 0.149487i $$-0.952238\pi$$
0.988764 0.149487i $$-0.0477622\pi$$
$$180$$ 0 0
$$181$$ 10.0000i 0.743294i 0.928374 + 0.371647i $$0.121207\pi$$
−0.928374 + 0.371647i $$0.878793\pi$$
$$182$$ 0 0
$$183$$ 2.00000i 0.147844i
$$184$$ 0 0
$$185$$ 16.0000 8.00000i 1.17634 0.588172i
$$186$$ 0 0
$$187$$ 16.0000 1.17004
$$188$$ 0 0
$$189$$ 4.00000i 0.290957i
$$190$$ 0 0
$$191$$ 24.0000 1.73658 0.868290 0.496058i $$-0.165220\pi$$
0.868290 + 0.496058i $$0.165220\pi$$
$$192$$ 0 0
$$193$$ 16.0000i 1.15171i 0.817554 + 0.575853i $$0.195330\pi$$
−0.817554 + 0.575853i $$0.804670\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ 4.00000 0.284988 0.142494 0.989796i $$-0.454488\pi$$
0.142494 + 0.989796i $$0.454488\pi$$
$$198$$ 0 0
$$199$$ −12.0000 −0.850657 −0.425329 0.905039i $$-0.639842\pi$$
−0.425329 + 0.905039i $$0.639842\pi$$
$$200$$ 0 0
$$201$$ 4.00000 0.282138
$$202$$ 0 0
$$203$$ −24.0000 −1.68447
$$204$$ 0 0
$$205$$ 20.0000 10.0000i 1.39686 0.698430i
$$206$$ 0 0
$$207$$ 4.00000i 0.278019i
$$208$$ 0 0
$$209$$ 0 0
$$210$$ 0 0
$$211$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$212$$ 0 0
$$213$$ 0 0
$$214$$ 0 0
$$215$$ 8.00000 4.00000i 0.545595 0.272798i
$$216$$ 0 0
$$217$$ 16.0000i 1.08615i
$$218$$ 0 0
$$219$$ 8.00000i 0.540590i
$$220$$ 0 0
$$221$$ 0 0
$$222$$ 0 0
$$223$$ 20.0000i 1.33930i −0.742677 0.669650i $$-0.766444\pi$$
0.742677 0.669650i $$-0.233556\pi$$
$$224$$ 0 0
$$225$$ 3.00000 4.00000i 0.200000 0.266667i
$$226$$ 0 0
$$227$$ −20.0000 −1.32745 −0.663723 0.747978i $$-0.731025\pi$$
−0.663723 + 0.747978i $$0.731025\pi$$
$$228$$ 0 0
$$229$$ 26.0000i 1.71813i 0.511868 + 0.859064i $$0.328954\pi$$
−0.511868 + 0.859064i $$0.671046\pi$$
$$230$$ 0 0
$$231$$ 16.0000 1.05272
$$232$$ 0 0
$$233$$ 20.0000i 1.31024i −0.755523 0.655122i $$-0.772617\pi$$
0.755523 0.655122i $$-0.227383\pi$$
$$234$$ 0 0
$$235$$ 4.00000 + 8.00000i 0.260931 + 0.521862i
$$236$$ 0 0
$$237$$ −12.0000 −0.779484
$$238$$ 0 0
$$239$$ −8.00000 −0.517477 −0.258738 0.965947i $$-0.583307\pi$$
−0.258738 + 0.965947i $$0.583307\pi$$
$$240$$ 0 0
$$241$$ −2.00000 −0.128831 −0.0644157 0.997923i $$-0.520518\pi$$
−0.0644157 + 0.997923i $$0.520518\pi$$
$$242$$ 0 0
$$243$$ −1.00000 −0.0641500
$$244$$ 0 0
$$245$$ −18.0000 + 9.00000i −1.14998 + 0.574989i
$$246$$ 0 0
$$247$$ 0 0
$$248$$ 0 0
$$249$$ 4.00000 0.253490
$$250$$ 0 0
$$251$$ 12.0000i 0.757433i 0.925513 + 0.378717i $$0.123635\pi$$
−0.925513 + 0.378717i $$0.876365\pi$$
$$252$$ 0 0
$$253$$ 16.0000 1.00591
$$254$$ 0 0
$$255$$ 4.00000 + 8.00000i 0.250490 + 0.500979i
$$256$$ 0 0
$$257$$ 12.0000i 0.748539i −0.927320 0.374270i $$-0.877893\pi$$
0.927320 0.374270i $$-0.122107\pi$$
$$258$$ 0 0
$$259$$ 32.0000i 1.98838i
$$260$$ 0 0
$$261$$ 6.00000i 0.371391i
$$262$$ 0 0
$$263$$ 4.00000i 0.246651i −0.992366 0.123325i $$-0.960644\pi$$
0.992366 0.123325i $$-0.0393559\pi$$
$$264$$ 0 0
$$265$$ −24.0000 + 12.0000i −1.47431 + 0.737154i
$$266$$ 0 0
$$267$$ 10.0000 0.611990
$$268$$ 0 0
$$269$$ 6.00000i 0.365826i −0.983129 0.182913i $$-0.941447\pi$$
0.983129 0.182913i $$-0.0585527\pi$$
$$270$$ 0 0
$$271$$ 4.00000 0.242983 0.121491 0.992592i $$-0.461232\pi$$
0.121491 + 0.992592i $$0.461232\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ 0 0
$$275$$ 16.0000 + 12.0000i 0.964836 + 0.723627i
$$276$$ 0 0
$$277$$ 32.0000 1.92269 0.961347 0.275340i $$-0.0887905\pi$$
0.961347 + 0.275340i $$0.0887905\pi$$
$$278$$ 0 0
$$279$$ 4.00000 0.239474
$$280$$ 0 0
$$281$$ −6.00000 −0.357930 −0.178965 0.983855i $$-0.557275\pi$$
−0.178965 + 0.983855i $$0.557275\pi$$
$$282$$ 0 0
$$283$$ −28.0000 −1.66443 −0.832214 0.554455i $$-0.812927\pi$$
−0.832214 + 0.554455i $$0.812927\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ 40.0000i 2.36113i
$$288$$ 0 0
$$289$$ 1.00000 0.0588235
$$290$$ 0 0
$$291$$ 8.00000i 0.468968i
$$292$$ 0 0
$$293$$ 12.0000 0.701047 0.350524 0.936554i $$-0.386004\pi$$
0.350524 + 0.936554i $$0.386004\pi$$
$$294$$ 0 0
$$295$$ 4.00000 + 8.00000i 0.232889 + 0.465778i
$$296$$ 0 0
$$297$$ 4.00000i 0.232104i
$$298$$ 0 0
$$299$$ 0 0
$$300$$ 0 0
$$301$$ 16.0000i 0.922225i
$$302$$ 0 0
$$303$$ 2.00000i 0.114897i
$$304$$ 0 0
$$305$$ 2.00000 + 4.00000i 0.114520 + 0.229039i
$$306$$ 0 0
$$307$$ 12.0000 0.684876 0.342438 0.939540i $$-0.388747\pi$$
0.342438 + 0.939540i $$0.388747\pi$$
$$308$$ 0 0
$$309$$ 4.00000i 0.227552i
$$310$$ 0 0
$$311$$ −24.0000 −1.36092 −0.680458 0.732787i $$-0.738219\pi$$
−0.680458 + 0.732787i $$0.738219\pi$$
$$312$$ 0 0
$$313$$ 16.0000i 0.904373i −0.891923 0.452187i $$-0.850644\pi$$
0.891923 0.452187i $$-0.149356\pi$$
$$314$$ 0 0
$$315$$ 4.00000 + 8.00000i 0.225374 + 0.450749i
$$316$$ 0 0
$$317$$ 4.00000 0.224662 0.112331 0.993671i $$-0.464168\pi$$
0.112331 + 0.993671i $$0.464168\pi$$
$$318$$ 0 0
$$319$$ −24.0000 −1.34374
$$320$$ 0 0
$$321$$ 12.0000 0.669775
$$322$$ 0 0
$$323$$ 0 0
$$324$$ 0 0
$$325$$ 0 0
$$326$$ 0 0
$$327$$ 2.00000i 0.110600i
$$328$$ 0 0
$$329$$ −16.0000 −0.882109
$$330$$ 0 0
$$331$$ 8.00000i 0.439720i 0.975531 + 0.219860i $$0.0705600\pi$$
−0.975531 + 0.219860i $$0.929440\pi$$
$$332$$ 0 0
$$333$$ 8.00000 0.438397
$$334$$ 0 0
$$335$$ −8.00000 + 4.00000i −0.437087 + 0.218543i
$$336$$ 0 0
$$337$$ 8.00000i 0.435788i 0.975972 + 0.217894i $$0.0699187\pi$$
−0.975972 + 0.217894i $$0.930081\pi$$
$$338$$ 0 0
$$339$$ 12.0000i 0.651751i
$$340$$ 0 0
$$341$$ 16.0000i 0.866449i
$$342$$ 0 0
$$343$$ 8.00000i 0.431959i
$$344$$ 0 0
$$345$$ 4.00000 + 8.00000i 0.215353 + 0.430706i
$$346$$ 0 0
$$347$$ −12.0000 −0.644194 −0.322097 0.946707i $$-0.604388\pi$$
−0.322097 + 0.946707i $$0.604388\pi$$
$$348$$ 0 0
$$349$$ 14.0000i 0.749403i 0.927146 + 0.374701i $$0.122255\pi$$
−0.927146 + 0.374701i $$0.877745\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 0 0
$$353$$ 28.0000i 1.49029i −0.666903 0.745145i $$-0.732380\pi$$
0.666903 0.745145i $$-0.267620\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ 0 0
$$357$$ −16.0000 −0.846810
$$358$$ 0 0
$$359$$ 24.0000 1.26667 0.633336 0.773877i $$-0.281685\pi$$
0.633336 + 0.773877i $$0.281685\pi$$
$$360$$ 0 0
$$361$$ 19.0000 1.00000
$$362$$ 0 0
$$363$$ 5.00000 0.262432
$$364$$ 0 0
$$365$$ 8.00000 + 16.0000i 0.418739 + 0.837478i
$$366$$ 0 0
$$367$$ 4.00000i 0.208798i −0.994535 0.104399i $$-0.966708\pi$$
0.994535 0.104399i $$-0.0332919\pi$$
$$368$$ 0 0
$$369$$ 10.0000 0.520579
$$370$$ 0 0
$$371$$ 48.0000i 2.49204i
$$372$$ 0 0
$$373$$ 24.0000 1.24267 0.621336 0.783544i $$-0.286590\pi$$
0.621336 + 0.783544i $$0.286590\pi$$
$$374$$ 0 0
$$375$$ −2.00000 + 11.0000i −0.103280 + 0.568038i
$$376$$ 0 0
$$377$$ 0 0
$$378$$ 0 0
$$379$$ 8.00000i 0.410932i 0.978664 + 0.205466i $$0.0658711\pi$$
−0.978664 + 0.205466i $$0.934129\pi$$
$$380$$ 0 0
$$381$$ 4.00000i 0.204926i
$$382$$ 0 0
$$383$$ 28.0000i 1.43073i −0.698749 0.715367i $$-0.746260\pi$$
0.698749 0.715367i $$-0.253740\pi$$
$$384$$ 0 0
$$385$$ −32.0000 + 16.0000i −1.63087 + 0.815436i
$$386$$ 0 0
$$387$$ 4.00000 0.203331
$$388$$ 0 0
$$389$$ 34.0000i 1.72387i −0.507020 0.861934i $$-0.669253\pi$$
0.507020 0.861934i $$-0.330747\pi$$
$$390$$ 0 0
$$391$$ −16.0000 −0.809155
$$392$$ 0 0
$$393$$ 12.0000i 0.605320i
$$394$$ 0 0
$$395$$ 24.0000 12.0000i 1.20757 0.603786i
$$396$$ 0 0
$$397$$ −8.00000 −0.401508 −0.200754 0.979642i $$-0.564339\pi$$
−0.200754 + 0.979642i $$0.564339\pi$$
$$398$$ 0 0
$$399$$ 0 0
$$400$$ 0 0
$$401$$ −14.0000 −0.699127 −0.349563 0.936913i $$-0.613670\pi$$
−0.349563 + 0.936913i $$0.613670\pi$$
$$402$$ 0 0
$$403$$ 0 0
$$404$$ 0 0
$$405$$ 2.00000 1.00000i 0.0993808 0.0496904i
$$406$$ 0 0
$$407$$ 32.0000i 1.58618i
$$408$$ 0 0
$$409$$ 26.0000 1.28562 0.642809 0.766027i $$-0.277769\pi$$
0.642809 + 0.766027i $$0.277769\pi$$
$$410$$ 0 0
$$411$$ 12.0000i 0.591916i
$$412$$ 0 0
$$413$$ −16.0000 −0.787309
$$414$$ 0 0
$$415$$ −8.00000 + 4.00000i −0.392705 + 0.196352i
$$416$$ 0 0
$$417$$ 16.0000i 0.783523i
$$418$$ 0 0
$$419$$ 28.0000i 1.36789i 0.729534 + 0.683945i $$0.239737\pi$$
−0.729534 + 0.683945i $$0.760263\pi$$
$$420$$ 0 0
$$421$$ 10.0000i 0.487370i −0.969854 0.243685i $$-0.921644\pi$$
0.969854 0.243685i $$-0.0783563\pi$$
$$422$$ 0 0
$$423$$ 4.00000i 0.194487i
$$424$$ 0 0
$$425$$ −16.0000 12.0000i −0.776114 0.582086i
$$426$$ 0 0
$$427$$ −8.00000 −0.387147
$$428$$ 0 0
$$429$$ 0 0
$$430$$ 0 0
$$431$$ 8.00000 0.385346 0.192673 0.981263i $$-0.438284\pi$$
0.192673 + 0.981263i $$0.438284\pi$$
$$432$$ 0 0
$$433$$ 16.0000i 0.768911i 0.923144 + 0.384455i $$0.125611\pi$$
−0.923144 + 0.384455i $$0.874389\pi$$
$$434$$ 0 0
$$435$$ −6.00000 12.0000i −0.287678 0.575356i
$$436$$ 0 0
$$437$$ 0 0
$$438$$ 0 0
$$439$$ −12.0000 −0.572729 −0.286364 0.958121i $$-0.592447\pi$$
−0.286364 + 0.958121i $$0.592447\pi$$
$$440$$ 0 0
$$441$$ −9.00000 −0.428571
$$442$$ 0 0
$$443$$ 36.0000 1.71041 0.855206 0.518289i $$-0.173431\pi$$
0.855206 + 0.518289i $$0.173431\pi$$
$$444$$ 0 0
$$445$$ −20.0000 + 10.0000i −0.948091 + 0.474045i
$$446$$ 0 0
$$447$$ 2.00000i 0.0945968i
$$448$$ 0 0
$$449$$ −18.0000 −0.849473 −0.424736 0.905317i $$-0.639633\pi$$
−0.424736 + 0.905317i $$0.639633\pi$$
$$450$$ 0 0
$$451$$ 40.0000i 1.88353i
$$452$$ 0 0
$$453$$ −20.0000 −0.939682
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ 40.0000i 1.87112i 0.353166 + 0.935561i $$0.385105\pi$$
−0.353166 + 0.935561i $$0.614895\pi$$
$$458$$ 0 0
$$459$$ 4.00000i 0.186704i
$$460$$ 0 0
$$461$$ 6.00000i 0.279448i 0.990190 + 0.139724i $$0.0446215\pi$$
−0.990190 + 0.139724i $$0.955378\pi$$
$$462$$ 0 0
$$463$$ 12.0000i 0.557687i 0.960337 + 0.278844i $$0.0899511\pi$$
−0.960337 + 0.278844i $$0.910049\pi$$
$$464$$ 0 0
$$465$$ −8.00000 + 4.00000i −0.370991 + 0.185496i
$$466$$ 0 0
$$467$$ 12.0000 0.555294 0.277647 0.960683i $$-0.410445\pi$$
0.277647 + 0.960683i $$0.410445\pi$$
$$468$$ 0 0
$$469$$ 16.0000i 0.738811i
$$470$$ 0 0
$$471$$ −8.00000 −0.368621
$$472$$ 0 0
$$473$$ 16.0000i 0.735681i
$$474$$ 0 0
$$475$$ 0 0
$$476$$ 0 0
$$477$$ −12.0000 −0.549442
$$478$$ 0 0
$$479$$ 16.0000 0.731059 0.365529 0.930800i $$-0.380888\pi$$
0.365529 + 0.930800i $$0.380888\pi$$
$$480$$ 0 0
$$481$$ 0 0
$$482$$ 0 0
$$483$$ −16.0000 −0.728025
$$484$$ 0 0
$$485$$ −8.00000 16.0000i −0.363261 0.726523i
$$486$$ 0 0
$$487$$ 12.0000i 0.543772i −0.962329 0.271886i $$-0.912353\pi$$
0.962329 0.271886i $$-0.0876473\pi$$
$$488$$ 0 0
$$489$$ 20.0000 0.904431
$$490$$ 0 0
$$491$$ 36.0000i 1.62466i 0.583200 + 0.812329i $$0.301800\pi$$
−0.583200 + 0.812329i $$0.698200\pi$$
$$492$$ 0 0
$$493$$ 24.0000 1.08091
$$494$$ 0 0
$$495$$ 4.00000 + 8.00000i 0.179787 + 0.359573i
$$496$$ 0 0
$$497$$ 0 0
$$498$$ 0 0
$$499$$ 24.0000i 1.07439i −0.843459 0.537194i $$-0.819484\pi$$
0.843459 0.537194i $$-0.180516\pi$$
$$500$$ 0 0
$$501$$ 12.0000i 0.536120i
$$502$$ 0 0
$$503$$ 36.0000i 1.60516i −0.596544 0.802580i $$-0.703460\pi$$
0.596544 0.802580i $$-0.296540\pi$$
$$504$$ 0 0
$$505$$ 2.00000 + 4.00000i 0.0889988 + 0.177998i
$$506$$ 0 0
$$507$$ 13.0000 0.577350
$$508$$ 0 0
$$509$$ 42.0000i 1.86162i −0.365507 0.930809i $$-0.619104\pi$$
0.365507 0.930809i $$-0.380896\pi$$
$$510$$ 0 0
$$511$$ −32.0000 −1.41560
$$512$$ 0 0
$$513$$ 0 0
$$514$$ 0 0
$$515$$ 4.00000 + 8.00000i 0.176261 + 0.352522i
$$516$$ 0 0
$$517$$ −16.0000 −0.703679
$$518$$ 0 0
$$519$$ −4.00000 −0.175581
$$520$$ 0 0
$$521$$ 38.0000 1.66481 0.832405 0.554168i $$-0.186963\pi$$
0.832405 + 0.554168i $$0.186963\pi$$
$$522$$ 0 0
$$523$$ −28.0000 −1.22435 −0.612177 0.790721i $$-0.709706\pi$$
−0.612177 + 0.790721i $$0.709706\pi$$
$$524$$ 0 0
$$525$$ −16.0000 12.0000i −0.698297 0.523723i
$$526$$ 0 0
$$527$$ 16.0000i 0.696971i
$$528$$ 0 0
$$529$$ 7.00000 0.304348
$$530$$ 0 0
$$531$$ 4.00000i 0.173585i
$$532$$ 0 0
$$533$$ 0 0
$$534$$ 0 0
$$535$$ −24.0000 + 12.0000i −1.03761 + 0.518805i
$$536$$ 0 0
$$537$$ 4.00000i 0.172613i
$$538$$ 0 0
$$539$$ 36.0000i 1.55063i
$$540$$ 0 0
$$541$$ 2.00000i 0.0859867i −0.999075 0.0429934i $$-0.986311\pi$$
0.999075 0.0429934i $$-0.0136894\pi$$
$$542$$ 0 0
$$543$$ 10.0000i 0.429141i
$$544$$ 0 0
$$545$$ 2.00000 + 4.00000i 0.0856706 + 0.171341i
$$546$$ 0 0
$$547$$ 28.0000 1.19719 0.598597 0.801050i $$-0.295725\pi$$
0.598597 + 0.801050i $$0.295725\pi$$
$$548$$ 0 0
$$549$$ 2.00000i 0.0853579i
$$550$$ 0 0
$$551$$ 0 0
$$552$$ 0 0
$$553$$ 48.0000i 2.04117i
$$554$$ 0 0
$$555$$ −16.0000 + 8.00000i −0.679162 + 0.339581i
$$556$$ 0 0
$$557$$ −12.0000 −0.508456 −0.254228 0.967144i $$-0.581821\pi$$
−0.254228 + 0.967144i $$0.581821\pi$$
$$558$$ 0 0
$$559$$ 0 0
$$560$$ 0 0
$$561$$ −16.0000 −0.675521
$$562$$ 0 0
$$563$$ −20.0000 −0.842900 −0.421450 0.906852i $$-0.638479\pi$$
−0.421450 + 0.906852i $$0.638479\pi$$
$$564$$ 0 0
$$565$$ 12.0000 + 24.0000i 0.504844 + 1.00969i
$$566$$ 0 0
$$567$$ 4.00000i 0.167984i
$$568$$ 0 0
$$569$$ 26.0000 1.08998 0.544988 0.838444i $$-0.316534\pi$$
0.544988 + 0.838444i $$0.316534\pi$$
$$570$$ 0 0
$$571$$ 40.0000i 1.67395i −0.547243 0.836974i $$-0.684323\pi$$
0.547243 0.836974i $$-0.315677\pi$$
$$572$$ 0 0
$$573$$ −24.0000 −1.00261
$$574$$ 0 0
$$575$$ −16.0000 12.0000i −0.667246 0.500435i
$$576$$ 0 0
$$577$$ 32.0000i 1.33218i 0.745873 + 0.666089i $$0.232033\pi$$
−0.745873 + 0.666089i $$0.767967\pi$$
$$578$$ 0 0
$$579$$ 16.0000i 0.664937i
$$580$$ 0 0
$$581$$ 16.0000i 0.663792i
$$582$$ 0 0
$$583$$ 48.0000i 1.98796i
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ 36.0000 1.48588 0.742940 0.669359i $$-0.233431\pi$$
0.742940 + 0.669359i $$0.233431\pi$$
$$588$$ 0 0
$$589$$ 0 0
$$590$$ 0 0
$$591$$ −4.00000 −0.164538
$$592$$ 0 0
$$593$$ 36.0000i 1.47834i −0.673517 0.739171i $$-0.735217\pi$$
0.673517 0.739171i $$-0.264783\pi$$
$$594$$ 0 0
$$595$$ 32.0000 16.0000i 1.31187 0.655936i
$$596$$ 0 0
$$597$$ 12.0000 0.491127
$$598$$ 0 0
$$599$$ −24.0000 −0.980613 −0.490307 0.871550i $$-0.663115\pi$$
−0.490307 + 0.871550i $$0.663115\pi$$
$$600$$ 0 0
$$601$$ −38.0000 −1.55005 −0.775026 0.631929i $$-0.782263\pi$$
−0.775026 + 0.631929i $$0.782263\pi$$
$$602$$ 0 0
$$603$$ −4.00000 −0.162893
$$604$$ 0 0
$$605$$ −10.0000 + 5.00000i −0.406558 + 0.203279i
$$606$$ 0 0
$$607$$ 28.0000i 1.13648i 0.822861 + 0.568242i $$0.192376\pi$$
−0.822861 + 0.568242i $$0.807624\pi$$
$$608$$ 0 0
$$609$$ 24.0000 0.972529
$$610$$ 0 0
$$611$$ 0 0
$$612$$ 0 0
$$613$$ −8.00000 −0.323117 −0.161558 0.986863i $$-0.551652\pi$$
−0.161558 + 0.986863i $$0.551652\pi$$
$$614$$ 0 0
$$615$$ −20.0000 + 10.0000i −0.806478 + 0.403239i
$$616$$ 0 0
$$617$$ 44.0000i 1.77137i −0.464283 0.885687i $$-0.653688\pi$$
0.464283 0.885687i $$-0.346312\pi$$
$$618$$ 0 0
$$619$$ 16.0000i 0.643094i −0.946894 0.321547i $$-0.895797\pi$$
0.946894 0.321547i $$-0.104203\pi$$
$$620$$ 0 0
$$621$$ 4.00000i 0.160514i
$$622$$ 0 0
$$623$$ 40.0000i 1.60257i
$$624$$ 0 0
$$625$$ −7.00000 24.0000i −0.280000 0.960000i
$$626$$ 0 0
$$627$$ 0 0
$$628$$ 0 0
$$629$$ 32.0000i 1.27592i
$$630$$ 0 0
$$631$$ 28.0000 1.11466 0.557331 0.830290i $$-0.311825\pi$$
0.557331 + 0.830290i $$0.311825\pi$$
$$632$$ 0 0
$$633$$ 0 0
$$634$$ 0 0
$$635$$ −4.00000 8.00000i −0.158735 0.317470i
$$636$$ 0 0
$$637$$ 0 0
$$638$$ 0 0
$$639$$ 0 0
$$640$$ 0 0
$$641$$ 14.0000 0.552967 0.276483 0.961019i $$-0.410831\pi$$
0.276483 + 0.961019i $$0.410831\pi$$
$$642$$ 0 0
$$643$$ −36.0000 −1.41970 −0.709851 0.704352i $$-0.751238\pi$$
−0.709851 + 0.704352i $$0.751238\pi$$
$$644$$ 0 0
$$645$$ −8.00000 + 4.00000i −0.315000 + 0.157500i
$$646$$ 0 0
$$647$$ 12.0000i 0.471769i 0.971781 + 0.235884i $$0.0757987\pi$$
−0.971781 + 0.235884i $$0.924201\pi$$
$$648$$ 0 0
$$649$$ −16.0000 −0.628055
$$650$$ 0 0
$$651$$ 16.0000i 0.627089i
$$652$$ 0 0
$$653$$ −4.00000 −0.156532 −0.0782660 0.996933i $$-0.524938\pi$$
−0.0782660 + 0.996933i $$0.524938\pi$$
$$654$$ 0 0
$$655$$ 12.0000 + 24.0000i 0.468879 + 0.937758i
$$656$$ 0 0
$$657$$ 8.00000i 0.312110i
$$658$$ 0 0
$$659$$ 28.0000i 1.09073i −0.838200 0.545363i $$-0.816392\pi$$
0.838200 0.545363i $$-0.183608\pi$$
$$660$$ 0 0
$$661$$ 22.0000i 0.855701i 0.903850 + 0.427850i $$0.140729\pi$$
−0.903850 + 0.427850i $$0.859271\pi$$
$$662$$ 0 0
$$663$$ 0 0
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 0 0
$$667$$ 24.0000 0.929284
$$668$$ 0 0
$$669$$ 20.0000i 0.773245i
$$670$$ 0 0
$$671$$ −8.00000 −0.308837
$$672$$ 0 0
$$673$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$674$$ 0 0
$$675$$ −3.00000 + 4.00000i −0.115470 + 0.153960i
$$676$$ 0 0
$$677$$ −4.00000 −0.153732 −0.0768662 0.997041i $$-0.524491\pi$$
−0.0768662 + 0.997041i $$0.524491\pi$$
$$678$$ 0 0
$$679$$ 32.0000 1.22805
$$680$$ 0 0
$$681$$ 20.0000 0.766402
$$682$$ 0 0
$$683$$ −44.0000 −1.68361 −0.841807 0.539779i $$-0.818508\pi$$
−0.841807 + 0.539779i $$0.818508\pi$$
$$684$$ 0 0
$$685$$ 12.0000 + 24.0000i 0.458496 + 0.916993i
$$686$$ 0 0
$$687$$ 26.0000i 0.991962i
$$688$$ 0 0
$$689$$ 0 0
$$690$$ 0 0
$$691$$ 32.0000i 1.21734i −0.793424 0.608669i $$-0.791704\pi$$
0.793424 0.608669i $$-0.208296\pi$$
$$692$$ 0 0
$$693$$ −16.0000 −0.607790
$$694$$ 0 0
$$695$$ 16.0000 + 32.0000i 0.606915 + 1.21383i
$$696$$ 0 0
$$697$$ 40.0000i 1.51511i
$$698$$ 0 0
$$699$$ 20.0000i 0.756469i
$$700$$ 0 0
$$701$$ 26.0000i 0.982006i 0.871158 + 0.491003i $$0.163370\pi$$
−0.871158 + 0.491003i $$0.836630\pi$$
$$702$$ 0 0
$$703$$ 0 0
$$704$$ 0 0
$$705$$ −4.00000 8.00000i −0.150649 0.301297i
$$706$$ 0 0
$$707$$ −8.00000 −0.300871
$$708$$ 0 0
$$709$$ 6.00000i 0.225335i −0.993633 0.112667i $$-0.964061\pi$$
0.993633 0.112667i $$-0.0359394\pi$$
$$710$$ 0 0
$$711$$ 12.0000 0.450035
$$712$$ 0 0
$$713$$ 16.0000i 0.599205i
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 0 0
$$717$$ 8.00000 0.298765
$$718$$ 0 0
$$719$$ −8.00000 −0.298350 −0.149175 0.988811i $$-0.547662\pi$$
−0.149175 + 0.988811i $$0.547662\pi$$
$$720$$ 0 0
$$721$$ −16.0000 −0.595871
$$722$$ 0 0
$$723$$ 2.00000 0.0743808
$$724$$ 0 0
$$725$$ 24.0000 + 18.0000i 0.891338 + 0.668503i
$$726$$ 0 0
$$727$$ 12.0000i 0.445055i −0.974926 0.222528i $$-0.928569\pi$$
0.974926 0.222528i $$-0.0714308\pi$$
$$728$$ 0 0
$$729$$ 1.00000 0.0370370
$$730$$ 0 0
$$731$$ 16.0000i 0.591781i
$$732$$ 0 0
$$733$$ 16.0000 0.590973 0.295487 0.955347i $$-0.404518\pi$$
0.295487 + 0.955347i $$0.404518\pi$$
$$734$$ 0 0
$$735$$ 18.0000 9.00000i 0.663940 0.331970i
$$736$$ 0 0
$$737$$ 16.0000i 0.589368i
$$738$$ 0 0
$$739$$ 40.0000i 1.47142i −0.677295 0.735712i $$-0.736848\pi$$
0.677295 0.735712i $$-0.263152\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 0 0
$$743$$ 36.0000i 1.32071i −0.750953 0.660356i $$-0.770405\pi$$
0.750953 0.660356i $$-0.229595\pi$$
$$744$$ 0 0
$$745$$ −2.00000 4.00000i −0.0732743 0.146549i
$$746$$ 0 0
$$747$$ −4.00000 −0.146352
$$748$$ 0 0
$$749$$ 48.0000i 1.75388i
$$750$$ 0 0
$$751$$ −28.0000 −1.02173 −0.510867 0.859660i $$-0.670676\pi$$
−0.510867 + 0.859660i $$0.670676\pi$$
$$752$$ 0 0
$$753$$ 12.0000i 0.437304i
$$754$$ 0 0
$$755$$ 40.0000 20.0000i 1.45575 0.727875i
$$756$$ 0 0
$$757$$ −16.0000 −0.581530 −0.290765 0.956795i $$-0.593910\pi$$
−0.290765 + 0.956795i $$0.593910\pi$$
$$758$$ 0 0
$$759$$ −16.0000 −0.580763
$$760$$ 0 0
$$761$$ −42.0000 −1.52250 −0.761249 0.648459i $$-0.775414\pi$$
−0.761249 + 0.648459i $$0.775414\pi$$
$$762$$ 0 0
$$763$$ −8.00000 −0.289619
$$764$$ 0 0
$$765$$ −4.00000 8.00000i −0.144620 0.289241i
$$766$$ 0 0
$$767$$ 0 0
$$768$$ 0 0
$$769$$ 18.0000 0.649097 0.324548 0.945869i $$-0.394788\pi$$
0.324548 + 0.945869i $$0.394788\pi$$
$$770$$ 0 0
$$771$$ 12.0000i 0.432169i
$$772$$ 0 0
$$773$$ −12.0000 −0.431610 −0.215805 0.976436i $$-0.569238\pi$$
−0.215805 + 0.976436i $$0.569238\pi$$
$$774$$ 0 0
$$775$$ 12.0000 16.0000i 0.431053 0.574737i
$$776$$ 0 0
$$777$$ 32.0000i 1.14799i
$$778$$ 0 0
$$779$$ 0 0
$$780$$ 0 0
$$781$$ 0 0
$$782$$ 0 0
$$783$$ 6.00000i 0.214423i
$$784$$ 0 0
$$785$$ 16.0000 8.00000i 0.571064 0.285532i
$$786$$ 0 0
$$787$$ 28.0000 0.998092 0.499046 0.866575i $$-0.333684\pi$$
0.499046 + 0.866575i $$0.333684\pi$$
$$788$$ 0 0
$$789$$ 4.00000i 0.142404i
$$790$$ 0 0
$$791$$ −48.0000 −1.70668
$$792$$ 0 0
$$793$$ 0 0
$$794$$ 0 0
$$795$$ 24.0000 12.0000i 0.851192 0.425596i
$$796$$ 0 0
$$797$$ 28.0000 0.991811 0.495905 0.868377i $$-0.334836\pi$$
0.495905 + 0.868377i $$0.334836\pi$$
$$798$$ 0 0
$$799$$ 16.0000 0.566039
$$800$$ 0 0
$$801$$ −10.0000 −0.353333
$$802$$ 0 0
$$803$$ −32.0000 −1.12926