Properties

 Label 1520.2.d.b.609.2 Level $1520$ Weight $2$ Character 1520.609 Analytic conductor $12.137$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

Related objects

Newspace parameters

 Level: $$N$$ $$=$$ $$1520 = 2^{4} \cdot 5 \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1520.d (of order $$2$$, degree $$1$$, not minimal)

Newform invariants

 Self dual: no Analytic conductor: $$12.1372611072$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(i)$$ Defining polynomial: $$x^{2} + 1$$ x^2 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 95) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

 Embedding label 609.2 Root $$-1.00000i$$ of defining polynomial Character $$\chi$$ $$=$$ 1520.609 Dual form 1520.2.d.b.609.1

$q$-expansion

 $$f(q)$$ $$=$$ $$q+(-1.00000 + 2.00000i) q^{5} +2.00000i q^{7} +3.00000 q^{9} +O(q^{10})$$ $$q+(-1.00000 + 2.00000i) q^{5} +2.00000i q^{7} +3.00000 q^{9} +4.00000 q^{11} -2.00000i q^{13} -4.00000i q^{17} +1.00000 q^{19} +6.00000i q^{23} +(-3.00000 - 4.00000i) q^{25} +6.00000 q^{29} +4.00000 q^{31} +(-4.00000 - 2.00000i) q^{35} +10.0000i q^{37} -10.0000 q^{41} -2.00000i q^{43} +(-3.00000 + 6.00000i) q^{45} -6.00000i q^{47} +3.00000 q^{49} +10.0000i q^{53} +(-4.00000 + 8.00000i) q^{55} +2.00000 q^{61} +6.00000i q^{63} +(4.00000 + 2.00000i) q^{65} +8.00000i q^{67} -4.00000 q^{71} +4.00000i q^{73} +8.00000i q^{77} +4.00000 q^{79} +9.00000 q^{81} +18.0000i q^{83} +(8.00000 + 4.00000i) q^{85} +2.00000 q^{89} +4.00000 q^{91} +(-1.00000 + 2.00000i) q^{95} -6.00000i q^{97} +12.0000 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{5} + 6 q^{9}+O(q^{10})$$ 2 * q - 2 * q^5 + 6 * q^9 $$2 q - 2 q^{5} + 6 q^{9} + 8 q^{11} + 2 q^{19} - 6 q^{25} + 12 q^{29} + 8 q^{31} - 8 q^{35} - 20 q^{41} - 6 q^{45} + 6 q^{49} - 8 q^{55} + 4 q^{61} + 8 q^{65} - 8 q^{71} + 8 q^{79} + 18 q^{81} + 16 q^{85} + 4 q^{89} + 8 q^{91} - 2 q^{95} + 24 q^{99}+O(q^{100})$$ 2 * q - 2 * q^5 + 6 * q^9 + 8 * q^11 + 2 * q^19 - 6 * q^25 + 12 * q^29 + 8 * q^31 - 8 * q^35 - 20 * q^41 - 6 * q^45 + 6 * q^49 - 8 * q^55 + 4 * q^61 + 8 * q^65 - 8 * q^71 + 8 * q^79 + 18 * q^81 + 16 * q^85 + 4 * q^89 + 8 * q^91 - 2 * q^95 + 24 * q^99

Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1520\mathbb{Z}\right)^\times$$.

 $$n$$ $$191$$ $$401$$ $$1141$$ $$1217$$ $$\chi(n)$$ $$1$$ $$1$$ $$1$$ $$-1$$

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
$$3$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$4$$ 0 0
$$5$$ −1.00000 + 2.00000i −0.447214 + 0.894427i
$$6$$ 0 0
$$7$$ 2.00000i 0.755929i 0.925820 + 0.377964i $$0.123376\pi$$
−0.925820 + 0.377964i $$0.876624\pi$$
$$8$$ 0 0
$$9$$ 3.00000 1.00000
$$10$$ 0 0
$$11$$ 4.00000 1.20605 0.603023 0.797724i $$-0.293963\pi$$
0.603023 + 0.797724i $$0.293963\pi$$
$$12$$ 0 0
$$13$$ 2.00000i 0.554700i −0.960769 0.277350i $$-0.910544\pi$$
0.960769 0.277350i $$-0.0894562\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 0 0
$$17$$ 4.00000i 0.970143i −0.874475 0.485071i $$-0.838794\pi$$
0.874475 0.485071i $$-0.161206\pi$$
$$18$$ 0 0
$$19$$ 1.00000 0.229416
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 0 0
$$23$$ 6.00000i 1.25109i 0.780189 + 0.625543i $$0.215123\pi$$
−0.780189 + 0.625543i $$0.784877\pi$$
$$24$$ 0 0
$$25$$ −3.00000 4.00000i −0.600000 0.800000i
$$26$$ 0 0
$$27$$ 0 0
$$28$$ 0 0
$$29$$ 6.00000 1.11417 0.557086 0.830455i $$-0.311919\pi$$
0.557086 + 0.830455i $$0.311919\pi$$
$$30$$ 0 0
$$31$$ 4.00000 0.718421 0.359211 0.933257i $$-0.383046\pi$$
0.359211 + 0.933257i $$0.383046\pi$$
$$32$$ 0 0
$$33$$ 0 0
$$34$$ 0 0
$$35$$ −4.00000 2.00000i −0.676123 0.338062i
$$36$$ 0 0
$$37$$ 10.0000i 1.64399i 0.569495 + 0.821995i $$0.307139\pi$$
−0.569495 + 0.821995i $$0.692861\pi$$
$$38$$ 0 0
$$39$$ 0 0
$$40$$ 0 0
$$41$$ −10.0000 −1.56174 −0.780869 0.624695i $$-0.785223\pi$$
−0.780869 + 0.624695i $$0.785223\pi$$
$$42$$ 0 0
$$43$$ 2.00000i 0.304997i −0.988304 0.152499i $$-0.951268\pi$$
0.988304 0.152499i $$-0.0487319\pi$$
$$44$$ 0 0
$$45$$ −3.00000 + 6.00000i −0.447214 + 0.894427i
$$46$$ 0 0
$$47$$ 6.00000i 0.875190i −0.899172 0.437595i $$-0.855830\pi$$
0.899172 0.437595i $$-0.144170\pi$$
$$48$$ 0 0
$$49$$ 3.00000 0.428571
$$50$$ 0 0
$$51$$ 0 0
$$52$$ 0 0
$$53$$ 10.0000i 1.37361i 0.726844 + 0.686803i $$0.240986\pi$$
−0.726844 + 0.686803i $$0.759014\pi$$
$$54$$ 0 0
$$55$$ −4.00000 + 8.00000i −0.539360 + 1.07872i
$$56$$ 0 0
$$57$$ 0 0
$$58$$ 0 0
$$59$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$60$$ 0 0
$$61$$ 2.00000 0.256074 0.128037 0.991769i $$-0.459132\pi$$
0.128037 + 0.991769i $$0.459132\pi$$
$$62$$ 0 0
$$63$$ 6.00000i 0.755929i
$$64$$ 0 0
$$65$$ 4.00000 + 2.00000i 0.496139 + 0.248069i
$$66$$ 0 0
$$67$$ 8.00000i 0.977356i 0.872464 + 0.488678i $$0.162521\pi$$
−0.872464 + 0.488678i $$0.837479\pi$$
$$68$$ 0 0
$$69$$ 0 0
$$70$$ 0 0
$$71$$ −4.00000 −0.474713 −0.237356 0.971423i $$-0.576281\pi$$
−0.237356 + 0.971423i $$0.576281\pi$$
$$72$$ 0 0
$$73$$ 4.00000i 0.468165i 0.972217 + 0.234082i $$0.0752085\pi$$
−0.972217 + 0.234082i $$0.924791\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 0 0
$$77$$ 8.00000i 0.911685i
$$78$$ 0 0
$$79$$ 4.00000 0.450035 0.225018 0.974355i $$-0.427756\pi$$
0.225018 + 0.974355i $$0.427756\pi$$
$$80$$ 0 0
$$81$$ 9.00000 1.00000
$$82$$ 0 0
$$83$$ 18.0000i 1.97576i 0.155230 + 0.987878i $$0.450388\pi$$
−0.155230 + 0.987878i $$0.549612\pi$$
$$84$$ 0 0
$$85$$ 8.00000 + 4.00000i 0.867722 + 0.433861i
$$86$$ 0 0
$$87$$ 0 0
$$88$$ 0 0
$$89$$ 2.00000 0.212000 0.106000 0.994366i $$-0.466196\pi$$
0.106000 + 0.994366i $$0.466196\pi$$
$$90$$ 0 0
$$91$$ 4.00000 0.419314
$$92$$ 0 0
$$93$$ 0 0
$$94$$ 0 0
$$95$$ −1.00000 + 2.00000i −0.102598 + 0.205196i
$$96$$ 0 0
$$97$$ 6.00000i 0.609208i −0.952479 0.304604i $$-0.901476\pi$$
0.952479 0.304604i $$-0.0985241\pi$$
$$98$$ 0 0
$$99$$ 12.0000 1.20605
$$100$$ 0 0
$$101$$ −6.00000 −0.597022 −0.298511 0.954406i $$-0.596490\pi$$
−0.298511 + 0.954406i $$0.596490\pi$$
$$102$$ 0 0
$$103$$ 16.0000i 1.57653i −0.615338 0.788263i $$-0.710980\pi$$
0.615338 0.788263i $$-0.289020\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ 4.00000i 0.386695i −0.981130 0.193347i $$-0.938066\pi$$
0.981130 0.193347i $$-0.0619344\pi$$
$$108$$ 0 0
$$109$$ −6.00000 −0.574696 −0.287348 0.957826i $$-0.592774\pi$$
−0.287348 + 0.957826i $$0.592774\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ 0 0
$$113$$ 6.00000i 0.564433i 0.959351 + 0.282216i $$0.0910696\pi$$
−0.959351 + 0.282216i $$0.908930\pi$$
$$114$$ 0 0
$$115$$ −12.0000 6.00000i −1.11901 0.559503i
$$116$$ 0 0
$$117$$ 6.00000i 0.554700i
$$118$$ 0 0
$$119$$ 8.00000 0.733359
$$120$$ 0 0
$$121$$ 5.00000 0.454545
$$122$$ 0 0
$$123$$ 0 0
$$124$$ 0 0
$$125$$ 11.0000 2.00000i 0.983870 0.178885i
$$126$$ 0 0
$$127$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$128$$ 0 0
$$129$$ 0 0
$$130$$ 0 0
$$131$$ 12.0000 1.04844 0.524222 0.851581i $$-0.324356\pi$$
0.524222 + 0.851581i $$0.324356\pi$$
$$132$$ 0 0
$$133$$ 2.00000i 0.173422i
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ 12.0000i 1.02523i −0.858619 0.512615i $$-0.828677\pi$$
0.858619 0.512615i $$-0.171323\pi$$
$$138$$ 0 0
$$139$$ −4.00000 −0.339276 −0.169638 0.985506i $$-0.554260\pi$$
−0.169638 + 0.985506i $$0.554260\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 0 0
$$143$$ 8.00000i 0.668994i
$$144$$ 0 0
$$145$$ −6.00000 + 12.0000i −0.498273 + 0.996546i
$$146$$ 0 0
$$147$$ 0 0
$$148$$ 0 0
$$149$$ 10.0000 0.819232 0.409616 0.912258i $$-0.365663\pi$$
0.409616 + 0.912258i $$0.365663\pi$$
$$150$$ 0 0
$$151$$ −24.0000 −1.95309 −0.976546 0.215308i $$-0.930924\pi$$
−0.976546 + 0.215308i $$0.930924\pi$$
$$152$$ 0 0
$$153$$ 12.0000i 0.970143i
$$154$$ 0 0
$$155$$ −4.00000 + 8.00000i −0.321288 + 0.642575i
$$156$$ 0 0
$$157$$ 8.00000i 0.638470i 0.947676 + 0.319235i $$0.103426\pi$$
−0.947676 + 0.319235i $$0.896574\pi$$
$$158$$ 0 0
$$159$$ 0 0
$$160$$ 0 0
$$161$$ −12.0000 −0.945732
$$162$$ 0 0
$$163$$ 10.0000i 0.783260i −0.920123 0.391630i $$-0.871911\pi$$
0.920123 0.391630i $$-0.128089\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ 12.0000i 0.928588i 0.885681 + 0.464294i $$0.153692\pi$$
−0.885681 + 0.464294i $$0.846308\pi$$
$$168$$ 0 0
$$169$$ 9.00000 0.692308
$$170$$ 0 0
$$171$$ 3.00000 0.229416
$$172$$ 0 0
$$173$$ 6.00000i 0.456172i −0.973641 0.228086i $$-0.926753\pi$$
0.973641 0.228086i $$-0.0732467\pi$$
$$174$$ 0 0
$$175$$ 8.00000 6.00000i 0.604743 0.453557i
$$176$$ 0 0
$$177$$ 0 0
$$178$$ 0 0
$$179$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$180$$ 0 0
$$181$$ −18.0000 −1.33793 −0.668965 0.743294i $$-0.733262\pi$$
−0.668965 + 0.743294i $$0.733262\pi$$
$$182$$ 0 0
$$183$$ 0 0
$$184$$ 0 0
$$185$$ −20.0000 10.0000i −1.47043 0.735215i
$$186$$ 0 0
$$187$$ 16.0000i 1.17004i
$$188$$ 0 0
$$189$$ 0 0
$$190$$ 0 0
$$191$$ 24.0000 1.73658 0.868290 0.496058i $$-0.165220\pi$$
0.868290 + 0.496058i $$0.165220\pi$$
$$192$$ 0 0
$$193$$ 26.0000i 1.87152i −0.352636 0.935760i $$-0.614715\pi$$
0.352636 0.935760i $$-0.385285\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ 12.0000i 0.854965i 0.904024 + 0.427482i $$0.140599\pi$$
−0.904024 + 0.427482i $$0.859401\pi$$
$$198$$ 0 0
$$199$$ 16.0000 1.13421 0.567105 0.823646i $$-0.308063\pi$$
0.567105 + 0.823646i $$0.308063\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ 0 0
$$203$$ 12.0000i 0.842235i
$$204$$ 0 0
$$205$$ 10.0000 20.0000i 0.698430 1.39686i
$$206$$ 0 0
$$207$$ 18.0000i 1.25109i
$$208$$ 0 0
$$209$$ 4.00000 0.276686
$$210$$ 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
$$215$$ 4.00000 + 2.00000i 0.272798 + 0.136399i
$$216$$ 0 0
$$217$$ 8.00000i 0.543075i
$$218$$ 0 0
$$219$$ 0 0
$$220$$ 0 0
$$221$$ −8.00000 −0.538138
$$222$$ 0 0
$$223$$ 16.0000i 1.07144i −0.844396 0.535720i $$-0.820040\pi$$
0.844396 0.535720i $$-0.179960\pi$$
$$224$$ 0 0
$$225$$ −9.00000 12.0000i −0.600000 0.800000i
$$226$$ 0 0
$$227$$ 20.0000i 1.32745i 0.747978 + 0.663723i $$0.231025\pi$$
−0.747978 + 0.663723i $$0.768975\pi$$
$$228$$ 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
$$233$$ 16.0000i 1.04819i 0.851658 + 0.524097i $$0.175597\pi$$
−0.851658 + 0.524097i $$0.824403\pi$$
$$234$$ 0 0
$$235$$ 12.0000 + 6.00000i 0.782794 + 0.391397i
$$236$$ 0 0
$$237$$ 0 0
$$238$$ 0 0
$$239$$ −16.0000 −1.03495 −0.517477 0.855697i $$-0.673129\pi$$
−0.517477 + 0.855697i $$0.673129\pi$$
$$240$$ 0 0
$$241$$ −2.00000 −0.128831 −0.0644157 0.997923i $$-0.520518\pi$$
−0.0644157 + 0.997923i $$0.520518\pi$$
$$242$$ 0 0
$$243$$ 0 0
$$244$$ 0 0
$$245$$ −3.00000 + 6.00000i −0.191663 + 0.383326i
$$246$$ 0 0
$$247$$ 2.00000i 0.127257i
$$248$$ 0 0
$$249$$ 0 0
$$250$$ 0 0
$$251$$ −4.00000 −0.252478 −0.126239 0.992000i $$-0.540291\pi$$
−0.126239 + 0.992000i $$0.540291\pi$$
$$252$$ 0 0
$$253$$ 24.0000i 1.50887i
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 0 0
$$257$$ 6.00000i 0.374270i 0.982334 + 0.187135i $$0.0599201\pi$$
−0.982334 + 0.187135i $$0.940080\pi$$
$$258$$ 0 0
$$259$$ −20.0000 −1.24274
$$260$$ 0 0
$$261$$ 18.0000 1.11417
$$262$$ 0 0
$$263$$ 6.00000i 0.369976i 0.982741 + 0.184988i $$0.0592246\pi$$
−0.982741 + 0.184988i $$0.940775\pi$$
$$264$$ 0 0
$$265$$ −20.0000 10.0000i −1.22859 0.614295i
$$266$$ 0 0
$$267$$ 0 0
$$268$$ 0 0
$$269$$ 18.0000 1.09748 0.548740 0.835993i $$-0.315108\pi$$
0.548740 + 0.835993i $$0.315108\pi$$
$$270$$ 0 0
$$271$$ −24.0000 −1.45790 −0.728948 0.684569i $$-0.759990\pi$$
−0.728948 + 0.684569i $$0.759990\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ 0 0
$$275$$ −12.0000 16.0000i −0.723627 0.964836i
$$276$$ 0 0
$$277$$ 8.00000i 0.480673i −0.970690 0.240337i $$-0.922742\pi$$
0.970690 0.240337i $$-0.0772579\pi$$
$$278$$ 0 0
$$279$$ 12.0000 0.718421
$$280$$ 0 0
$$281$$ −26.0000 −1.55103 −0.775515 0.631329i $$-0.782510\pi$$
−0.775515 + 0.631329i $$0.782510\pi$$
$$282$$ 0 0
$$283$$ 14.0000i 0.832214i −0.909316 0.416107i $$-0.863394\pi$$
0.909316 0.416107i $$-0.136606\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ 20.0000i 1.18056i
$$288$$ 0 0
$$289$$ 1.00000 0.0588235
$$290$$ 0 0
$$291$$ 0 0
$$292$$ 0 0
$$293$$ 14.0000i 0.817889i −0.912559 0.408944i $$-0.865897\pi$$
0.912559 0.408944i $$-0.134103\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ 0 0
$$297$$ 0 0
$$298$$ 0 0
$$299$$ 12.0000 0.693978
$$300$$ 0 0
$$301$$ 4.00000 0.230556
$$302$$ 0 0
$$303$$ 0 0
$$304$$ 0 0
$$305$$ −2.00000 + 4.00000i −0.114520 + 0.229039i
$$306$$ 0 0
$$307$$ 20.0000i 1.14146i −0.821138 0.570730i $$-0.806660\pi$$
0.821138 0.570730i $$-0.193340\pi$$
$$308$$ 0 0
$$309$$ 0 0
$$310$$ 0 0
$$311$$ −24.0000 −1.36092 −0.680458 0.732787i $$-0.738219\pi$$
−0.680458 + 0.732787i $$0.738219\pi$$
$$312$$ 0 0
$$313$$ 24.0000i 1.35656i −0.734803 0.678280i $$-0.762726\pi$$
0.734803 0.678280i $$-0.237274\pi$$
$$314$$ 0 0
$$315$$ −12.0000 6.00000i −0.676123 0.338062i
$$316$$ 0 0
$$317$$ 22.0000i 1.23564i −0.786318 0.617822i $$-0.788015\pi$$
0.786318 0.617822i $$-0.211985\pi$$
$$318$$ 0 0
$$319$$ 24.0000 1.34374
$$320$$ 0 0
$$321$$ 0 0
$$322$$ 0 0
$$323$$ 4.00000i 0.222566i
$$324$$ 0 0
$$325$$ −8.00000 + 6.00000i −0.443760 + 0.332820i
$$326$$ 0 0
$$327$$ 0 0
$$328$$ 0 0
$$329$$ 12.0000 0.661581
$$330$$ 0 0
$$331$$ 4.00000 0.219860 0.109930 0.993939i $$-0.464937\pi$$
0.109930 + 0.993939i $$0.464937\pi$$
$$332$$ 0 0
$$333$$ 30.0000i 1.64399i
$$334$$ 0 0
$$335$$ −16.0000 8.00000i −0.874173 0.437087i
$$336$$ 0 0
$$337$$ 22.0000i 1.19842i −0.800593 0.599208i $$-0.795482\pi$$
0.800593 0.599208i $$-0.204518\pi$$
$$338$$ 0 0
$$339$$ 0 0
$$340$$ 0 0
$$341$$ 16.0000 0.866449
$$342$$ 0 0
$$343$$ 20.0000i 1.07990i
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ 6.00000i 0.322097i −0.986947 0.161048i $$-0.948512\pi$$
0.986947 0.161048i $$-0.0514875\pi$$
$$348$$ 0 0
$$349$$ −2.00000 −0.107058 −0.0535288 0.998566i $$-0.517047\pi$$
−0.0535288 + 0.998566i $$0.517047\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 0 0
$$353$$ 4.00000i 0.212899i 0.994318 + 0.106449i $$0.0339482\pi$$
−0.994318 + 0.106449i $$0.966052\pi$$
$$354$$ 0 0
$$355$$ 4.00000 8.00000i 0.212298 0.424596i
$$356$$ 0 0
$$357$$ 0 0
$$358$$ 0 0
$$359$$ −8.00000 −0.422224 −0.211112 0.977462i $$-0.567708\pi$$
−0.211112 + 0.977462i $$0.567708\pi$$
$$360$$ 0 0
$$361$$ 1.00000 0.0526316
$$362$$ 0 0
$$363$$ 0 0
$$364$$ 0 0
$$365$$ −8.00000 4.00000i −0.418739 0.209370i
$$366$$ 0 0
$$367$$ 22.0000i 1.14839i −0.818718 0.574195i $$-0.805315\pi$$
0.818718 0.574195i $$-0.194685\pi$$
$$368$$ 0 0
$$369$$ −30.0000 −1.56174
$$370$$ 0 0
$$371$$ −20.0000 −1.03835
$$372$$ 0 0
$$373$$ 6.00000i 0.310668i −0.987862 0.155334i $$-0.950355\pi$$
0.987862 0.155334i $$-0.0496454\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ 12.0000i 0.618031i
$$378$$ 0 0
$$379$$ 28.0000 1.43826 0.719132 0.694874i $$-0.244540\pi$$
0.719132 + 0.694874i $$0.244540\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ 0 0
$$383$$ 12.0000i 0.613171i −0.951843 0.306586i $$-0.900813\pi$$
0.951843 0.306586i $$-0.0991866\pi$$
$$384$$ 0 0
$$385$$ −16.0000 8.00000i −0.815436 0.407718i
$$386$$ 0 0
$$387$$ 6.00000i 0.304997i
$$388$$ 0 0
$$389$$ 30.0000 1.52106 0.760530 0.649303i $$-0.224939\pi$$
0.760530 + 0.649303i $$0.224939\pi$$
$$390$$ 0 0
$$391$$ 24.0000 1.21373
$$392$$ 0 0
$$393$$ 0 0
$$394$$ 0 0
$$395$$ −4.00000 + 8.00000i −0.201262 + 0.402524i
$$396$$ 0 0
$$397$$ 8.00000i 0.401508i 0.979642 + 0.200754i $$0.0643393\pi$$
−0.979642 + 0.200754i $$0.935661\pi$$
$$398$$ 0 0
$$399$$ 0 0
$$400$$ 0 0
$$401$$ 18.0000 0.898877 0.449439 0.893311i $$-0.351624\pi$$
0.449439 + 0.893311i $$0.351624\pi$$
$$402$$ 0 0
$$403$$ 8.00000i 0.398508i
$$404$$ 0 0
$$405$$ −9.00000 + 18.0000i −0.447214 + 0.894427i
$$406$$ 0 0
$$407$$ 40.0000i 1.98273i
$$408$$ 0 0
$$409$$ 14.0000 0.692255 0.346128 0.938187i $$-0.387496\pi$$
0.346128 + 0.938187i $$0.387496\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ 0 0
$$413$$ 0 0
$$414$$ 0 0
$$415$$ −36.0000 18.0000i −1.76717 0.883585i
$$416$$ 0 0
$$417$$ 0 0
$$418$$ 0 0
$$419$$ 36.0000 1.75872 0.879358 0.476162i $$-0.157972\pi$$
0.879358 + 0.476162i $$0.157972\pi$$
$$420$$ 0 0
$$421$$ −30.0000 −1.46211 −0.731055 0.682318i $$-0.760972\pi$$
−0.731055 + 0.682318i $$0.760972\pi$$
$$422$$ 0 0
$$423$$ 18.0000i 0.875190i
$$424$$ 0 0
$$425$$ −16.0000 + 12.0000i −0.776114 + 0.582086i
$$426$$ 0 0
$$427$$ 4.00000i 0.193574i
$$428$$ 0 0
$$429$$ 0 0
$$430$$ 0 0
$$431$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$432$$ 0 0
$$433$$ 34.0000i 1.63394i −0.576683 0.816968i $$-0.695653\pi$$
0.576683 0.816968i $$-0.304347\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 0 0
$$437$$ 6.00000i 0.287019i
$$438$$ 0 0
$$439$$ 20.0000 0.954548 0.477274 0.878755i $$-0.341625\pi$$
0.477274 + 0.878755i $$0.341625\pi$$
$$440$$ 0 0
$$441$$ 9.00000 0.428571
$$442$$ 0 0
$$443$$ 2.00000i 0.0950229i 0.998871 + 0.0475114i $$0.0151291\pi$$
−0.998871 + 0.0475114i $$0.984871\pi$$
$$444$$ 0 0
$$445$$ −2.00000 + 4.00000i −0.0948091 + 0.189618i
$$446$$ 0 0
$$447$$ 0 0
$$448$$ 0 0
$$449$$ −10.0000 −0.471929 −0.235965 0.971762i $$-0.575825\pi$$
−0.235965 + 0.971762i $$0.575825\pi$$
$$450$$ 0 0
$$451$$ −40.0000 −1.88353
$$452$$ 0 0
$$453$$ 0 0
$$454$$ 0 0
$$455$$ −4.00000 + 8.00000i −0.187523 + 0.375046i
$$456$$ 0 0
$$457$$ 8.00000i 0.374224i 0.982339 + 0.187112i $$0.0599128\pi$$
−0.982339 + 0.187112i $$0.940087\pi$$
$$458$$ 0 0
$$459$$ 0 0
$$460$$ 0 0
$$461$$ 14.0000 0.652045 0.326023 0.945362i $$-0.394291\pi$$
0.326023 + 0.945362i $$0.394291\pi$$
$$462$$ 0 0
$$463$$ 2.00000i 0.0929479i −0.998920 0.0464739i $$-0.985202\pi$$
0.998920 0.0464739i $$-0.0147984\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ 38.0000i 1.75843i −0.476425 0.879215i $$-0.658068\pi$$
0.476425 0.879215i $$-0.341932\pi$$
$$468$$ 0 0
$$469$$ −16.0000 −0.738811
$$470$$ 0 0
$$471$$ 0 0
$$472$$ 0 0
$$473$$ 8.00000i 0.367840i
$$474$$ 0 0
$$475$$ −3.00000 4.00000i −0.137649 0.183533i
$$476$$ 0 0
$$477$$ 30.0000i 1.37361i
$$478$$ 0 0
$$479$$ −24.0000 −1.09659 −0.548294 0.836286i $$-0.684723\pi$$
−0.548294 + 0.836286i $$0.684723\pi$$
$$480$$ 0 0
$$481$$ 20.0000 0.911922
$$482$$ 0 0
$$483$$ 0 0
$$484$$ 0 0
$$485$$ 12.0000 + 6.00000i 0.544892 + 0.272446i
$$486$$ 0 0
$$487$$ 20.0000i 0.906287i −0.891438 0.453143i $$-0.850303\pi$$
0.891438 0.453143i $$-0.149697\pi$$
$$488$$ 0 0
$$489$$ 0 0
$$490$$ 0 0
$$491$$ 20.0000 0.902587 0.451294 0.892375i $$-0.350963\pi$$
0.451294 + 0.892375i $$0.350963\pi$$
$$492$$ 0 0
$$493$$ 24.0000i 1.08091i
$$494$$ 0 0
$$495$$ −12.0000 + 24.0000i −0.539360 + 1.07872i
$$496$$ 0 0
$$497$$ 8.00000i 0.358849i
$$498$$ 0 0
$$499$$ 4.00000 0.179065 0.0895323 0.995984i $$-0.471463\pi$$
0.0895323 + 0.995984i $$0.471463\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ 0 0
$$503$$ 10.0000i 0.445878i −0.974832 0.222939i $$-0.928435\pi$$
0.974832 0.222939i $$-0.0715651\pi$$
$$504$$ 0 0
$$505$$ 6.00000 12.0000i 0.266996 0.533993i
$$506$$ 0 0
$$507$$ 0 0
$$508$$ 0 0
$$509$$ 6.00000 0.265945 0.132973 0.991120i $$-0.457548\pi$$
0.132973 + 0.991120i $$0.457548\pi$$
$$510$$ 0 0
$$511$$ −8.00000 −0.353899
$$512$$ 0 0
$$513$$ 0 0
$$514$$ 0 0
$$515$$ 32.0000 + 16.0000i 1.41009 + 0.705044i
$$516$$ 0 0
$$517$$ 24.0000i 1.05552i
$$518$$ 0 0
$$519$$ 0 0
$$520$$ 0 0
$$521$$ 42.0000 1.84005 0.920027 0.391856i $$-0.128167\pi$$
0.920027 + 0.391856i $$0.128167\pi$$
$$522$$ 0 0
$$523$$ 44.0000i 1.92399i 0.273075 + 0.961993i $$0.411959\pi$$
−0.273075 + 0.961993i $$0.588041\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ 16.0000i 0.696971i
$$528$$ 0 0
$$529$$ −13.0000 −0.565217
$$530$$ 0 0
$$531$$ 0 0
$$532$$ 0 0
$$533$$ 20.0000i 0.866296i
$$534$$ 0 0
$$535$$ 8.00000 + 4.00000i 0.345870 + 0.172935i
$$536$$ 0 0
$$537$$ 0 0
$$538$$ 0 0
$$539$$ 12.0000 0.516877
$$540$$ 0 0
$$541$$ −22.0000 −0.945854 −0.472927 0.881102i $$-0.656803\pi$$
−0.472927 + 0.881102i $$0.656803\pi$$
$$542$$ 0 0
$$543$$ 0 0
$$544$$ 0 0
$$545$$ 6.00000 12.0000i 0.257012 0.514024i
$$546$$ 0 0
$$547$$ 28.0000i 1.19719i 0.801050 + 0.598597i $$0.204275\pi$$
−0.801050 + 0.598597i $$0.795725\pi$$
$$548$$ 0 0
$$549$$ 6.00000 0.256074
$$550$$ 0 0
$$551$$ 6.00000 0.255609
$$552$$ 0 0
$$553$$ 8.00000i 0.340195i
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 0 0
$$557$$ 16.0000i 0.677942i 0.940797 + 0.338971i $$0.110079\pi$$
−0.940797 + 0.338971i $$0.889921\pi$$
$$558$$ 0 0
$$559$$ −4.00000 −0.169182
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 0 0
$$563$$ 12.0000i 0.505740i −0.967500 0.252870i $$-0.918626\pi$$
0.967500 0.252870i $$-0.0813744\pi$$
$$564$$ 0 0
$$565$$ −12.0000 6.00000i −0.504844 0.252422i
$$566$$ 0 0
$$567$$ 18.0000i 0.755929i
$$568$$ 0 0
$$569$$ −42.0000 −1.76073 −0.880366 0.474295i $$-0.842703\pi$$
−0.880366 + 0.474295i $$0.842703\pi$$
$$570$$ 0 0
$$571$$ −4.00000 −0.167395 −0.0836974 0.996491i $$-0.526673\pi$$
−0.0836974 + 0.996491i $$0.526673\pi$$
$$572$$ 0 0
$$573$$ 0 0
$$574$$ 0 0
$$575$$ 24.0000 18.0000i 1.00087 0.750652i
$$576$$ 0 0
$$577$$ 24.0000i 0.999133i −0.866276 0.499567i $$-0.833493\pi$$
0.866276 0.499567i $$-0.166507\pi$$
$$578$$ 0 0
$$579$$ 0 0
$$580$$ 0 0
$$581$$ −36.0000 −1.49353
$$582$$ 0 0
$$583$$ 40.0000i 1.65663i
$$584$$ 0 0
$$585$$ 12.0000 + 6.00000i 0.496139 + 0.248069i
$$586$$ 0 0
$$587$$ 18.0000i 0.742940i 0.928445 + 0.371470i $$0.121146\pi$$
−0.928445 + 0.371470i $$0.878854\pi$$
$$588$$ 0 0
$$589$$ 4.00000 0.164817
$$590$$ 0 0
$$591$$ 0 0
$$592$$ 0 0
$$593$$ 8.00000i 0.328521i −0.986417 0.164260i $$-0.947476\pi$$
0.986417 0.164260i $$-0.0525237\pi$$
$$594$$ 0 0
$$595$$ −8.00000 + 16.0000i −0.327968 + 0.655936i
$$596$$ 0 0
$$597$$ 0 0
$$598$$ 0 0
$$599$$ 12.0000 0.490307 0.245153 0.969484i $$-0.421162\pi$$
0.245153 + 0.969484i $$0.421162\pi$$
$$600$$ 0 0
$$601$$ 14.0000 0.571072 0.285536 0.958368i $$-0.407828\pi$$
0.285536 + 0.958368i $$0.407828\pi$$
$$602$$ 0 0
$$603$$ 24.0000i 0.977356i
$$604$$ 0 0
$$605$$ −5.00000 + 10.0000i −0.203279 + 0.406558i
$$606$$ 0 0
$$607$$ 44.0000i 1.78590i −0.450151 0.892952i $$-0.648630\pi$$
0.450151 0.892952i $$-0.351370\pi$$
$$608$$ 0 0
$$609$$ 0 0
$$610$$ 0 0
$$611$$ −12.0000 −0.485468
$$612$$ 0 0
$$613$$ 24.0000i 0.969351i −0.874694 0.484675i $$-0.838938\pi$$
0.874694 0.484675i $$-0.161062\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ 12.0000i 0.483102i −0.970388 0.241551i $$-0.922344\pi$$
0.970388 0.241551i $$-0.0776561\pi$$
$$618$$ 0 0
$$619$$ −20.0000 −0.803868 −0.401934 0.915669i $$-0.631662\pi$$
−0.401934 + 0.915669i $$0.631662\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ 0 0
$$623$$ 4.00000i 0.160257i
$$624$$ 0 0
$$625$$ −7.00000 + 24.0000i −0.280000 + 0.960000i
$$626$$ 0 0
$$627$$ 0 0
$$628$$ 0 0
$$629$$ 40.0000 1.59490
$$630$$ 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
$$637$$ 6.00000i 0.237729i
$$638$$ 0 0
$$639$$ −12.0000 −0.474713
$$640$$ 0 0
$$641$$ 6.00000 0.236986 0.118493 0.992955i $$-0.462194\pi$$
0.118493 + 0.992955i $$0.462194\pi$$
$$642$$ 0 0
$$643$$ 38.0000i 1.49857i −0.662246 0.749287i $$-0.730396\pi$$
0.662246 0.749287i $$-0.269604\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ 30.0000i 1.17942i 0.807614 + 0.589711i $$0.200758\pi$$
−0.807614 + 0.589711i $$0.799242\pi$$
$$648$$ 0 0
$$649$$ 0 0
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 0 0
$$653$$ 16.0000i 0.626128i 0.949732 + 0.313064i $$0.101356\pi$$
−0.949732 + 0.313064i $$0.898644\pi$$
$$654$$ 0 0
$$655$$ −12.0000 + 24.0000i −0.468879 + 0.937758i
$$656$$ 0 0
$$657$$ 12.0000i 0.468165i
$$658$$ 0 0
$$659$$ −16.0000 −0.623272 −0.311636 0.950202i $$-0.600877\pi$$
−0.311636 + 0.950202i $$0.600877\pi$$
$$660$$ 0 0
$$661$$ −30.0000 −1.16686 −0.583432 0.812162i $$-0.698291\pi$$
−0.583432 + 0.812162i $$0.698291\pi$$
$$662$$ 0 0
$$663$$ 0 0
$$664$$ 0 0
$$665$$ −4.00000 2.00000i −0.155113 0.0775567i
$$666$$ 0 0
$$667$$ 36.0000i 1.39393i
$$668$$ 0 0
$$669$$ 0 0
$$670$$ 0 0
$$671$$ 8.00000 0.308837
$$672$$ 0 0
$$673$$ 34.0000i 1.31060i 0.755367 + 0.655302i $$0.227459\pi$$
−0.755367 + 0.655302i $$0.772541\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ 0 0
$$677$$ 18.0000i 0.691796i −0.938272 0.345898i $$-0.887574\pi$$
0.938272 0.345898i $$-0.112426\pi$$
$$678$$ 0 0
$$679$$ 12.0000 0.460518
$$680$$ 0 0
$$681$$ 0 0
$$682$$ 0 0
$$683$$ 12.0000i 0.459167i −0.973289 0.229584i $$-0.926264\pi$$
0.973289 0.229584i $$-0.0737364\pi$$
$$684$$ 0 0
$$685$$ 24.0000 + 12.0000i 0.916993 + 0.458496i
$$686$$ 0 0
$$687$$ 0 0
$$688$$ 0 0
$$689$$ 20.0000 0.761939
$$690$$ 0 0
$$691$$ 12.0000 0.456502 0.228251 0.973602i $$-0.426699\pi$$
0.228251 + 0.973602i $$0.426699\pi$$
$$692$$ 0 0
$$693$$ 24.0000i 0.911685i
$$694$$ 0 0
$$695$$ 4.00000 8.00000i 0.151729 0.303457i
$$696$$ 0 0
$$697$$ 40.0000i 1.51511i
$$698$$ 0 0
$$699$$ 0 0
$$700$$ 0 0
$$701$$ −30.0000 −1.13308 −0.566542 0.824033i $$-0.691719\pi$$
−0.566542 + 0.824033i $$0.691719\pi$$
$$702$$ 0 0
$$703$$ 10.0000i 0.377157i
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ 12.0000i 0.451306i
$$708$$ 0 0
$$709$$ 38.0000 1.42712 0.713560 0.700594i $$-0.247082\pi$$
0.713560 + 0.700594i $$0.247082\pi$$
$$710$$ 0 0
$$711$$ 12.0000 0.450035
$$712$$ 0 0
$$713$$ 24.0000i 0.898807i
$$714$$ 0 0
$$715$$ 16.0000 + 8.00000i 0.598366 + 0.299183i
$$716$$ 0 0
$$717$$ 0 0
$$718$$ 0 0
$$719$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$720$$ 0 0
$$721$$ 32.0000 1.19174
$$722$$ 0 0
$$723$$ 0 0
$$724$$ 0 0
$$725$$ −18.0000 24.0000i −0.668503 0.891338i
$$726$$ 0 0
$$727$$ 2.00000i 0.0741759i 0.999312 + 0.0370879i $$0.0118082\pi$$
−0.999312 + 0.0370879i $$0.988192\pi$$
$$728$$ 0 0
$$729$$ 27.0000 1.00000
$$730$$ 0 0
$$731$$ −8.00000 −0.295891
$$732$$ 0 0
$$733$$ 12.0000i 0.443230i −0.975134 0.221615i $$-0.928867\pi$$
0.975134 0.221615i $$-0.0711328\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ 32.0000i 1.17874i
$$738$$ 0 0
$$739$$ 4.00000 0.147142 0.0735712 0.997290i $$-0.476560\pi$$
0.0735712 + 0.997290i $$0.476560\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 0 0
$$743$$ 4.00000i 0.146746i 0.997305 + 0.0733729i $$0.0233763\pi$$
−0.997305 + 0.0733729i $$0.976624\pi$$
$$744$$ 0 0
$$745$$ −10.0000 + 20.0000i −0.366372 + 0.732743i
$$746$$ 0 0
$$747$$ 54.0000i 1.97576i
$$748$$ 0 0
$$749$$ 8.00000 0.292314
$$750$$ 0 0
$$751$$ 40.0000 1.45962 0.729810 0.683650i $$-0.239608\pi$$
0.729810 + 0.683650i $$0.239608\pi$$
$$752$$ 0 0
$$753$$ 0 0
$$754$$ 0 0
$$755$$ 24.0000 48.0000i 0.873449 1.74690i
$$756$$ 0 0
$$757$$ 4.00000i 0.145382i 0.997354 + 0.0726912i $$0.0231588\pi$$
−0.997354 + 0.0726912i $$0.976841\pi$$
$$758$$ 0 0
$$759$$ 0 0
$$760$$ 0 0
$$761$$ 6.00000 0.217500 0.108750 0.994069i $$-0.465315\pi$$
0.108750 + 0.994069i $$0.465315\pi$$
$$762$$ 0 0
$$763$$ 12.0000i 0.434429i
$$764$$ 0 0
$$765$$ 24.0000 + 12.0000i 0.867722 + 0.433861i
$$766$$ 0 0
$$767$$ 0 0
$$768$$ 0 0
$$769$$ −26.0000 −0.937584 −0.468792 0.883309i $$-0.655311\pi$$
−0.468792 + 0.883309i $$0.655311\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ 0 0
$$773$$ 46.0000i 1.65451i 0.561830 + 0.827253i $$0.310097\pi$$
−0.561830 + 0.827253i $$0.689903\pi$$
$$774$$ 0 0
$$775$$ −12.0000 16.0000i −0.431053 0.574737i
$$776$$ 0 0
$$777$$ 0 0
$$778$$ 0 0
$$779$$ −10.0000 −0.358287
$$780$$ 0 0
$$781$$ −16.0000 −0.572525
$$782$$ 0 0
$$783$$ 0 0
$$784$$ 0 0
$$785$$ −16.0000 8.00000i −0.571064 0.285532i
$$786$$ 0 0
$$787$$ 44.0000i 1.56843i −0.620489 0.784215i $$-0.713066\pi$$
0.620489 0.784215i $$-0.286934\pi$$
$$788$$ 0 0
$$789$$ 0 0
$$790$$ 0 0
$$791$$ −12.0000 −0.426671
$$792$$ 0 0
$$793$$ 4.00000i 0.142044i
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 0 0
$$797$$ 6.00000i 0.212531i 0.994338 + 0.106265i $$0.0338893\pi$$
−0.994338 + 0.106265i $$0.966111\pi$$
$$798$$ 0 0
$$799$$ −24.0000 −0.849059
$$800$$ 0 0
$$801$$ 6.00000 0.212000
$$802$$ 0 0
$$803$$ 16.0000i 0.564628i
$$804$$ 0