# Properties

 Label 5520.2.be.b.1471.7 Level $5520$ Weight $2$ Character 5520.1471 Analytic conductor $44.077$ Analytic rank $0$ Dimension $16$ CM no Inner twists $2$

# Learn more

## Newspace parameters

 Level: $$N$$ $$=$$ $$5520 = 2^{4} \cdot 3 \cdot 5 \cdot 23$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 5520.be (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$44.0774219157$$ Analytic rank: $$0$$ Dimension: $$16$$ Coefficient field: $$\mathbb{Q}[x]/(x^{16} - \cdots)$$ Defining polynomial: $$x^{16} - 4 x^{15} + 8 x^{14} + 28 x^{13} + 373 x^{12} - 920 x^{11} + 1088 x^{10} - 168 x^{9} + 16460 x^{8} - 45408 x^{7} + 62624 x^{6} - 18048 x^{5} + 2160 x^{4} - 1664 x^{3} + 6272 x^{2} - 896 x + 64$$ Coefficient ring: $$\Z[a_1, \ldots, a_{19}]$$ Coefficient ring index: $$2^{8}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 1471.7 Root $$0.476829 + 0.476829i$$ of defining polynomial Character $$\chi$$ $$=$$ 5520.1471 Dual form 5520.2.be.b.1471.15

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-1.00000i q^{3} -1.00000i q^{5} +3.79952 q^{7} -1.00000 q^{9} +O(q^{10})$$ $$q-1.00000i q^{3} -1.00000i q^{5} +3.79952 q^{7} -1.00000 q^{9} +3.69013 q^{11} +2.60963 q^{13} -1.00000 q^{15} +1.60151i q^{17} -8.20535 q^{19} -3.79952i q^{21} +(1.19801 - 4.64379i) q^{23} -1.00000 q^{25} +1.00000i q^{27} +0.706420 q^{29} -6.46598i q^{31} -3.69013i q^{33} -3.79952i q^{35} -5.01787i q^{37} -2.60963i q^{39} -10.6396 q^{41} -5.94134 q^{43} +1.00000i q^{45} -4.46270i q^{47} +7.43635 q^{49} +1.60151 q^{51} -7.25579i q^{53} -3.69013i q^{55} +8.20535i q^{57} +6.04730i q^{59} +3.36597i q^{61} -3.79952 q^{63} -2.60963i q^{65} +13.5383 q^{67} +(-4.64379 - 1.19801i) q^{69} -8.69292i q^{71} -0.516947 q^{73} +1.00000i q^{75} +14.0207 q^{77} -3.23175 q^{79} +1.00000 q^{81} +4.10586 q^{83} +1.60151 q^{85} -0.706420i q^{87} -5.11034i q^{89} +9.91534 q^{91} -6.46598 q^{93} +8.20535i q^{95} -19.4018i q^{97} -3.69013 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$16q + 8q^{7} - 16q^{9} + O(q^{10})$$ $$16q + 8q^{7} - 16q^{9} - 8q^{11} + 8q^{13} - 16q^{15} - 12q^{23} - 16q^{25} - 4q^{29} + 4q^{41} + 20q^{49} + 4q^{51} - 8q^{63} + 16q^{67} + 40q^{73} + 24q^{77} - 32q^{79} + 16q^{81} + 4q^{85} + 48q^{91} + 8q^{99} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/5520\mathbb{Z}\right)^\times$$.

 $$n$$ $$1201$$ $$1381$$ $$1841$$ $$4417$$ $$4831$$ $$\chi(n)$$ $$-1$$ $$1$$ $$1$$ $$1$$ $$-1$$

## 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
$$3$$ 1.00000i 0.577350i
$$4$$ 0 0
$$5$$ 1.00000i 0.447214i
$$6$$ 0 0
$$7$$ 3.79952 1.43608 0.718042 0.696000i $$-0.245039\pi$$
0.718042 + 0.696000i $$0.245039\pi$$
$$8$$ 0 0
$$9$$ −1.00000 −0.333333
$$10$$ 0 0
$$11$$ 3.69013 1.11262 0.556308 0.830976i $$-0.312217\pi$$
0.556308 + 0.830976i $$0.312217\pi$$
$$12$$ 0 0
$$13$$ 2.60963 0.723781 0.361891 0.932221i $$-0.382131\pi$$
0.361891 + 0.932221i $$0.382131\pi$$
$$14$$ 0 0
$$15$$ −1.00000 −0.258199
$$16$$ 0 0
$$17$$ 1.60151i 0.388424i 0.980960 + 0.194212i $$0.0622150\pi$$
−0.980960 + 0.194212i $$0.937785\pi$$
$$18$$ 0 0
$$19$$ −8.20535 −1.88244 −0.941218 0.337801i $$-0.890317\pi$$
−0.941218 + 0.337801i $$0.890317\pi$$
$$20$$ 0 0
$$21$$ 3.79952i 0.829123i
$$22$$ 0 0
$$23$$ 1.19801 4.64379i 0.249802 0.968297i
$$24$$ 0 0
$$25$$ −1.00000 −0.200000
$$26$$ 0 0
$$27$$ 1.00000i 0.192450i
$$28$$ 0 0
$$29$$ 0.706420 0.131179 0.0655894 0.997847i $$-0.479107\pi$$
0.0655894 + 0.997847i $$0.479107\pi$$
$$30$$ 0 0
$$31$$ 6.46598i 1.16132i −0.814145 0.580662i $$-0.802794\pi$$
0.814145 0.580662i $$-0.197206\pi$$
$$32$$ 0 0
$$33$$ 3.69013i 0.642369i
$$34$$ 0 0
$$35$$ 3.79952i 0.642236i
$$36$$ 0 0
$$37$$ 5.01787i 0.824933i −0.910973 0.412466i $$-0.864667\pi$$
0.910973 0.412466i $$-0.135333\pi$$
$$38$$ 0 0
$$39$$ 2.60963i 0.417875i
$$40$$ 0 0
$$41$$ −10.6396 −1.66162 −0.830812 0.556553i $$-0.812124\pi$$
−0.830812 + 0.556553i $$0.812124\pi$$
$$42$$ 0 0
$$43$$ −5.94134 −0.906045 −0.453023 0.891499i $$-0.649654\pi$$
−0.453023 + 0.891499i $$0.649654\pi$$
$$44$$ 0 0
$$45$$ 1.00000i 0.149071i
$$46$$ 0 0
$$47$$ 4.46270i 0.650952i −0.945550 0.325476i $$-0.894476\pi$$
0.945550 0.325476i $$-0.105524\pi$$
$$48$$ 0 0
$$49$$ 7.43635 1.06234
$$50$$ 0 0
$$51$$ 1.60151 0.224257
$$52$$ 0 0
$$53$$ 7.25579i 0.996660i −0.866987 0.498330i $$-0.833947\pi$$
0.866987 0.498330i $$-0.166053\pi$$
$$54$$ 0 0
$$55$$ 3.69013i 0.497577i
$$56$$ 0 0
$$57$$ 8.20535i 1.08682i
$$58$$ 0 0
$$59$$ 6.04730i 0.787291i 0.919262 + 0.393646i $$0.128786\pi$$
−0.919262 + 0.393646i $$0.871214\pi$$
$$60$$ 0 0
$$61$$ 3.36597i 0.430968i 0.976507 + 0.215484i $$0.0691329\pi$$
−0.976507 + 0.215484i $$0.930867\pi$$
$$62$$ 0 0
$$63$$ −3.79952 −0.478695
$$64$$ 0 0
$$65$$ 2.60963i 0.323685i
$$66$$ 0 0
$$67$$ 13.5383 1.65397 0.826984 0.562226i $$-0.190055\pi$$
0.826984 + 0.562226i $$0.190055\pi$$
$$68$$ 0 0
$$69$$ −4.64379 1.19801i −0.559047 0.144223i
$$70$$ 0 0
$$71$$ 8.69292i 1.03166i −0.856691 0.515830i $$-0.827484\pi$$
0.856691 0.515830i $$-0.172516\pi$$
$$72$$ 0 0
$$73$$ −0.516947 −0.0605040 −0.0302520 0.999542i $$-0.509631\pi$$
−0.0302520 + 0.999542i $$0.509631\pi$$
$$74$$ 0 0
$$75$$ 1.00000i 0.115470i
$$76$$ 0 0
$$77$$ 14.0207 1.59781
$$78$$ 0 0
$$79$$ −3.23175 −0.363600 −0.181800 0.983336i $$-0.558192\pi$$
−0.181800 + 0.983336i $$0.558192\pi$$
$$80$$ 0 0
$$81$$ 1.00000 0.111111
$$82$$ 0 0
$$83$$ 4.10586 0.450677 0.225338 0.974281i $$-0.427651\pi$$
0.225338 + 0.974281i $$0.427651\pi$$
$$84$$ 0 0
$$85$$ 1.60151 0.173709
$$86$$ 0 0
$$87$$ 0.706420i 0.0757361i
$$88$$ 0 0
$$89$$ 5.11034i 0.541695i −0.962622 0.270848i $$-0.912696\pi$$
0.962622 0.270848i $$-0.0873040\pi$$
$$90$$ 0 0
$$91$$ 9.91534 1.03941
$$92$$ 0 0
$$93$$ −6.46598 −0.670491
$$94$$ 0 0
$$95$$ 8.20535i 0.841851i
$$96$$ 0 0
$$97$$ 19.4018i 1.96996i −0.172675 0.984979i $$-0.555241\pi$$
0.172675 0.984979i $$-0.444759\pi$$
$$98$$ 0 0
$$99$$ −3.69013 −0.370872
$$100$$ 0 0
$$101$$ 13.1514 1.30861 0.654305 0.756231i $$-0.272961\pi$$
0.654305 + 0.756231i $$0.272961\pi$$
$$102$$ 0 0
$$103$$ 5.37041 0.529162 0.264581 0.964363i $$-0.414766\pi$$
0.264581 + 0.964363i $$0.414766\pi$$
$$104$$ 0 0
$$105$$ −3.79952 −0.370795
$$106$$ 0 0
$$107$$ 17.9219 1.73258 0.866290 0.499542i $$-0.166498\pi$$
0.866290 + 0.499542i $$0.166498\pi$$
$$108$$ 0 0
$$109$$ 1.27328i 0.121958i −0.998139 0.0609792i $$-0.980578\pi$$
0.998139 0.0609792i $$-0.0194223\pi$$
$$110$$ 0 0
$$111$$ −5.01787 −0.476275
$$112$$ 0 0
$$113$$ 3.32431i 0.312725i −0.987700 0.156362i $$-0.950023\pi$$
0.987700 0.156362i $$-0.0499768\pi$$
$$114$$ 0 0
$$115$$ −4.64379 1.19801i −0.433036 0.111715i
$$116$$ 0 0
$$117$$ −2.60963 −0.241260
$$118$$ 0 0
$$119$$ 6.08498i 0.557810i
$$120$$ 0 0
$$121$$ 2.61707 0.237915
$$122$$ 0 0
$$123$$ 10.6396i 0.959339i
$$124$$ 0 0
$$125$$ 1.00000i 0.0894427i
$$126$$ 0 0
$$127$$ 15.7160i 1.39457i 0.716793 + 0.697286i $$0.245609\pi$$
−0.716793 + 0.697286i $$0.754391\pi$$
$$128$$ 0 0
$$129$$ 5.94134i 0.523106i
$$130$$ 0 0
$$131$$ 10.5445i 0.921280i −0.887587 0.460640i $$-0.847620\pi$$
0.887587 0.460640i $$-0.152380\pi$$
$$132$$ 0 0
$$133$$ −31.1764 −2.70333
$$134$$ 0 0
$$135$$ 1.00000 0.0860663
$$136$$ 0 0
$$137$$ 1.67864i 0.143416i 0.997426 + 0.0717081i $$0.0228450\pi$$
−0.997426 + 0.0717081i $$0.977155\pi$$
$$138$$ 0 0
$$139$$ 13.7161i 1.16338i 0.813410 + 0.581691i $$0.197609\pi$$
−0.813410 + 0.581691i $$0.802391\pi$$
$$140$$ 0 0
$$141$$ −4.46270 −0.375827
$$142$$ 0 0
$$143$$ 9.62988 0.805291
$$144$$ 0 0
$$145$$ 0.706420i 0.0586650i
$$146$$ 0 0
$$147$$ 7.43635i 0.613340i
$$148$$ 0 0
$$149$$ 22.3923i 1.83445i −0.398368 0.917226i $$-0.630423\pi$$
0.398368 0.917226i $$-0.369577\pi$$
$$150$$ 0 0
$$151$$ 22.9206i 1.86526i 0.360840 + 0.932628i $$0.382490\pi$$
−0.360840 + 0.932628i $$0.617510\pi$$
$$152$$ 0 0
$$153$$ 1.60151i 0.129475i
$$154$$ 0 0
$$155$$ −6.46598 −0.519360
$$156$$ 0 0
$$157$$ 14.5054i 1.15765i −0.815450 0.578827i $$-0.803511\pi$$
0.815450 0.578827i $$-0.196489\pi$$
$$158$$ 0 0
$$159$$ −7.25579 −0.575422
$$160$$ 0 0
$$161$$ 4.55185 17.6442i 0.358736 1.39056i
$$162$$ 0 0
$$163$$ 6.70939i 0.525520i 0.964861 + 0.262760i $$0.0846327\pi$$
−0.964861 + 0.262760i $$0.915367\pi$$
$$164$$ 0 0
$$165$$ −3.69013 −0.287276
$$166$$ 0 0
$$167$$ 5.17703i 0.400611i 0.979734 + 0.200305i $$0.0641934\pi$$
−0.979734 + 0.200305i $$0.935807\pi$$
$$168$$ 0 0
$$169$$ −6.18983 −0.476141
$$170$$ 0 0
$$171$$ 8.20535 0.627478
$$172$$ 0 0
$$173$$ 14.0638 1.06925 0.534624 0.845090i $$-0.320453\pi$$
0.534624 + 0.845090i $$0.320453\pi$$
$$174$$ 0 0
$$175$$ −3.79952 −0.287217
$$176$$ 0 0
$$177$$ 6.04730 0.454543
$$178$$ 0 0
$$179$$ 11.9606i 0.893975i 0.894540 + 0.446988i $$0.147503\pi$$
−0.894540 + 0.446988i $$0.852497\pi$$
$$180$$ 0 0
$$181$$ 1.90385i 0.141512i 0.997494 + 0.0707561i $$0.0225412\pi$$
−0.997494 + 0.0707561i $$0.977459\pi$$
$$182$$ 0 0
$$183$$ 3.36597 0.248819
$$184$$ 0 0
$$185$$ −5.01787 −0.368921
$$186$$ 0 0
$$187$$ 5.90980i 0.432167i
$$188$$ 0 0
$$189$$ 3.79952i 0.276374i
$$190$$ 0 0
$$191$$ 16.6906 1.20769 0.603845 0.797102i $$-0.293635\pi$$
0.603845 + 0.797102i $$0.293635\pi$$
$$192$$ 0 0
$$193$$ −11.9024 −0.856751 −0.428376 0.903601i $$-0.640914\pi$$
−0.428376 + 0.903601i $$0.640914\pi$$
$$194$$ 0 0
$$195$$ −2.60963 −0.186879
$$196$$ 0 0
$$197$$ 15.3515 1.09375 0.546876 0.837214i $$-0.315817\pi$$
0.546876 + 0.837214i $$0.315817\pi$$
$$198$$ 0 0
$$199$$ −17.4361 −1.23601 −0.618006 0.786173i $$-0.712059\pi$$
−0.618006 + 0.786173i $$0.712059\pi$$
$$200$$ 0 0
$$201$$ 13.5383i 0.954919i
$$202$$ 0 0
$$203$$ 2.68406 0.188384
$$204$$ 0 0
$$205$$ 10.6396i 0.743101i
$$206$$ 0 0
$$207$$ −1.19801 + 4.64379i −0.0832672 + 0.322766i
$$208$$ 0 0
$$209$$ −30.2788 −2.09443
$$210$$ 0 0
$$211$$ 7.11509i 0.489823i −0.969545 0.244911i $$-0.921241\pi$$
0.969545 0.244911i $$-0.0787589\pi$$
$$212$$ 0 0
$$213$$ −8.69292 −0.595629
$$214$$ 0 0
$$215$$ 5.94134i 0.405196i
$$216$$ 0 0
$$217$$ 24.5676i 1.66776i
$$218$$ 0 0
$$219$$ 0.516947i 0.0349320i
$$220$$ 0 0
$$221$$ 4.17936i 0.281134i
$$222$$ 0 0
$$223$$ 2.92403i 0.195808i 0.995196 + 0.0979038i $$0.0312137\pi$$
−0.995196 + 0.0979038i $$0.968786\pi$$
$$224$$ 0 0
$$225$$ 1.00000 0.0666667
$$226$$ 0 0
$$227$$ 8.86711 0.588531 0.294265 0.955724i $$-0.404925\pi$$
0.294265 + 0.955724i $$0.404925\pi$$
$$228$$ 0 0
$$229$$ 7.81117i 0.516177i 0.966121 + 0.258088i $$0.0830926\pi$$
−0.966121 + 0.258088i $$0.916907\pi$$
$$230$$ 0 0
$$231$$ 14.0207i 0.922496i
$$232$$ 0 0
$$233$$ −5.09372 −0.333701 −0.166850 0.985982i $$-0.553360\pi$$
−0.166850 + 0.985982i $$0.553360\pi$$
$$234$$ 0 0
$$235$$ −4.46270 −0.291114
$$236$$ 0 0
$$237$$ 3.23175i 0.209925i
$$238$$ 0 0
$$239$$ 19.5551i 1.26491i −0.774595 0.632457i $$-0.782046\pi$$
0.774595 0.632457i $$-0.217954\pi$$
$$240$$ 0 0
$$241$$ 8.39986i 0.541083i −0.962708 0.270541i $$-0.912797\pi$$
0.962708 0.270541i $$-0.0872027\pi$$
$$242$$ 0 0
$$243$$ 1.00000i 0.0641500i
$$244$$ 0 0
$$245$$ 7.43635i 0.475091i
$$246$$ 0 0
$$247$$ −21.4129 −1.36247
$$248$$ 0 0
$$249$$ 4.10586i 0.260198i
$$250$$ 0 0
$$251$$ 7.84091 0.494914 0.247457 0.968899i $$-0.420405\pi$$
0.247457 + 0.968899i $$0.420405\pi$$
$$252$$ 0 0
$$253$$ 4.42080 17.1362i 0.277933 1.07734i
$$254$$ 0 0
$$255$$ 1.60151i 0.100291i
$$256$$ 0 0
$$257$$ −5.18308 −0.323312 −0.161656 0.986847i $$-0.551683\pi$$
−0.161656 + 0.986847i $$0.551683\pi$$
$$258$$ 0 0
$$259$$ 19.0655i 1.18467i
$$260$$ 0 0
$$261$$ −0.706420 −0.0437263
$$262$$ 0 0
$$263$$ 6.41083 0.395309 0.197654 0.980272i $$-0.436668\pi$$
0.197654 + 0.980272i $$0.436668\pi$$
$$264$$ 0 0
$$265$$ −7.25579 −0.445720
$$266$$ 0 0
$$267$$ −5.11034 −0.312748
$$268$$ 0 0
$$269$$ 28.1418 1.71584 0.857919 0.513785i $$-0.171757\pi$$
0.857919 + 0.513785i $$0.171757\pi$$
$$270$$ 0 0
$$271$$ 23.8196i 1.44694i −0.690356 0.723470i $$-0.742546\pi$$
0.690356 0.723470i $$-0.257454\pi$$
$$272$$ 0 0
$$273$$ 9.91534i 0.600104i
$$274$$ 0 0
$$275$$ −3.69013 −0.222523
$$276$$ 0 0
$$277$$ −22.1168 −1.32887 −0.664436 0.747345i $$-0.731328\pi$$
−0.664436 + 0.747345i $$0.731328\pi$$
$$278$$ 0 0
$$279$$ 6.46598i 0.387108i
$$280$$ 0 0
$$281$$ 10.5452i 0.629075i −0.949245 0.314538i $$-0.898151\pi$$
0.949245 0.314538i $$-0.101849\pi$$
$$282$$ 0 0
$$283$$ 13.3512 0.793648 0.396824 0.917895i $$-0.370112\pi$$
0.396824 + 0.917895i $$0.370112\pi$$
$$284$$ 0 0
$$285$$ 8.20535 0.486043
$$286$$ 0 0
$$287$$ −40.4253 −2.38623
$$288$$ 0 0
$$289$$ 14.4352 0.849127
$$290$$ 0 0
$$291$$ −19.4018 −1.13736
$$292$$ 0 0
$$293$$ 13.0456i 0.762129i 0.924548 + 0.381065i $$0.124443\pi$$
−0.924548 + 0.381065i $$0.875557\pi$$
$$294$$ 0 0
$$295$$ 6.04730 0.352087
$$296$$ 0 0
$$297$$ 3.69013i 0.214123i
$$298$$ 0 0
$$299$$ 3.12635 12.1186i 0.180802 0.700835i
$$300$$ 0 0
$$301$$ −22.5742 −1.30116
$$302$$ 0 0
$$303$$ 13.1514i 0.755526i
$$304$$ 0 0
$$305$$ 3.36597 0.192735
$$306$$ 0 0
$$307$$ 24.4269i 1.39412i −0.717015 0.697058i $$-0.754492\pi$$
0.717015 0.697058i $$-0.245508\pi$$
$$308$$ 0 0
$$309$$ 5.37041i 0.305512i
$$310$$ 0 0
$$311$$ 2.74305i 0.155544i −0.996971 0.0777721i $$-0.975219\pi$$
0.996971 0.0777721i $$-0.0247807\pi$$
$$312$$ 0 0
$$313$$ 3.88307i 0.219484i 0.993960 + 0.109742i $$0.0350025\pi$$
−0.993960 + 0.109742i $$0.964998\pi$$
$$314$$ 0 0
$$315$$ 3.79952i 0.214079i
$$316$$ 0 0
$$317$$ −26.5346 −1.49033 −0.745166 0.666879i $$-0.767630\pi$$
−0.745166 + 0.666879i $$0.767630\pi$$
$$318$$ 0 0
$$319$$ 2.60678 0.145952
$$320$$ 0 0
$$321$$ 17.9219i 1.00031i
$$322$$ 0 0
$$323$$ 13.1410i 0.731183i
$$324$$ 0 0
$$325$$ −2.60963 −0.144756
$$326$$ 0 0
$$327$$ −1.27328 −0.0704127
$$328$$ 0 0
$$329$$ 16.9561i 0.934821i
$$330$$ 0 0
$$331$$ 28.5918i 1.57155i 0.618515 + 0.785773i $$0.287735\pi$$
−0.618515 + 0.785773i $$0.712265\pi$$
$$332$$ 0 0
$$333$$ 5.01787i 0.274978i
$$334$$ 0 0
$$335$$ 13.5383i 0.739677i
$$336$$ 0 0
$$337$$ 6.76479i 0.368501i 0.982879 + 0.184251i $$0.0589858\pi$$
−0.982879 + 0.184251i $$0.941014\pi$$
$$338$$ 0 0
$$339$$ −3.32431 −0.180552
$$340$$ 0 0
$$341$$ 23.8603i 1.29211i
$$342$$ 0 0
$$343$$ 1.65793 0.0895200
$$344$$ 0 0
$$345$$ −1.19801 + 4.64379i −0.0644985 + 0.250013i
$$346$$ 0 0
$$347$$ 29.0461i 1.55927i 0.626231 + 0.779637i $$0.284597\pi$$
−0.626231 + 0.779637i $$0.715403\pi$$
$$348$$ 0 0
$$349$$ −20.4208 −1.09310 −0.546550 0.837427i $$-0.684059\pi$$
−0.546550 + 0.837427i $$0.684059\pi$$
$$350$$ 0 0
$$351$$ 2.60963i 0.139292i
$$352$$ 0 0
$$353$$ −18.6709 −0.993751 −0.496876 0.867822i $$-0.665519\pi$$
−0.496876 + 0.867822i $$0.665519\pi$$
$$354$$ 0 0
$$355$$ −8.69292 −0.461372
$$356$$ 0 0
$$357$$ 6.08498 0.322052
$$358$$ 0 0
$$359$$ 5.96281 0.314705 0.157353 0.987542i $$-0.449704\pi$$
0.157353 + 0.987542i $$0.449704\pi$$
$$360$$ 0 0
$$361$$ 48.3277 2.54356
$$362$$ 0 0
$$363$$ 2.61707i 0.137360i
$$364$$ 0 0
$$365$$ 0.516947i 0.0270582i
$$366$$ 0 0
$$367$$ −23.4914 −1.22624 −0.613121 0.789989i $$-0.710086\pi$$
−0.613121 + 0.789989i $$0.710086\pi$$
$$368$$ 0 0
$$369$$ 10.6396 0.553875
$$370$$ 0 0
$$371$$ 27.5685i 1.43129i
$$372$$ 0 0
$$373$$ 20.8948i 1.08189i 0.841057 + 0.540946i $$0.181934\pi$$
−0.841057 + 0.540946i $$0.818066\pi$$
$$374$$ 0 0
$$375$$ 1.00000 0.0516398
$$376$$ 0 0
$$377$$ 1.84349 0.0949447
$$378$$ 0 0
$$379$$ 8.21337 0.421892 0.210946 0.977498i $$-0.432346\pi$$
0.210946 + 0.977498i $$0.432346\pi$$
$$380$$ 0 0
$$381$$ 15.7160 0.805156
$$382$$ 0 0
$$383$$ 7.84776 0.401002 0.200501 0.979694i $$-0.435743\pi$$
0.200501 + 0.979694i $$0.435743\pi$$
$$384$$ 0 0
$$385$$ 14.0207i 0.714562i
$$386$$ 0 0
$$387$$ 5.94134 0.302015
$$388$$ 0 0
$$389$$ 25.0255i 1.26884i −0.772988 0.634421i $$-0.781239\pi$$
0.772988 0.634421i $$-0.218761\pi$$
$$390$$ 0 0
$$391$$ 7.43709 + 1.91862i 0.376110 + 0.0970290i
$$392$$ 0 0
$$393$$ −10.5445 −0.531901
$$394$$ 0 0
$$395$$ 3.23175i 0.162607i
$$396$$ 0 0
$$397$$ 19.5249 0.979928 0.489964 0.871743i $$-0.337010\pi$$
0.489964 + 0.871743i $$0.337010\pi$$
$$398$$ 0 0
$$399$$ 31.1764i 1.56077i
$$400$$ 0 0
$$401$$ 27.8201i 1.38927i 0.719363 + 0.694634i $$0.244434\pi$$
−0.719363 + 0.694634i $$0.755566\pi$$
$$402$$ 0 0
$$403$$ 16.8738i 0.840545i
$$404$$ 0 0
$$405$$ 1.00000i 0.0496904i
$$406$$ 0 0
$$407$$ 18.5166i 0.917834i
$$408$$ 0 0
$$409$$ −18.0574 −0.892882 −0.446441 0.894813i $$-0.647309\pi$$
−0.446441 + 0.894813i $$0.647309\pi$$
$$410$$ 0 0
$$411$$ 1.67864 0.0828014
$$412$$ 0 0
$$413$$ 22.9768i 1.13062i
$$414$$ 0 0
$$415$$ 4.10586i 0.201549i
$$416$$ 0 0
$$417$$ 13.7161 0.671679
$$418$$ 0 0
$$419$$ −11.4005 −0.556952 −0.278476 0.960443i $$-0.589829\pi$$
−0.278476 + 0.960443i $$0.589829\pi$$
$$420$$ 0 0
$$421$$ 20.5611i 1.00209i 0.865422 + 0.501044i $$0.167051\pi$$
−0.865422 + 0.501044i $$0.832949\pi$$
$$422$$ 0 0
$$423$$ 4.46270i 0.216984i
$$424$$ 0 0
$$425$$ 1.60151i 0.0776848i
$$426$$ 0 0
$$427$$ 12.7891i 0.618906i
$$428$$ 0 0
$$429$$ 9.62988i 0.464935i
$$430$$ 0 0
$$431$$ −3.10368 −0.149499 −0.0747496 0.997202i $$-0.523816\pi$$
−0.0747496 + 0.997202i $$0.523816\pi$$
$$432$$ 0 0
$$433$$ 9.48758i 0.455944i −0.973668 0.227972i $$-0.926791\pi$$
0.973668 0.227972i $$-0.0732095\pi$$
$$434$$ 0 0
$$435$$ −0.706420 −0.0338702
$$436$$ 0 0
$$437$$ −9.83005 + 38.1039i −0.470235 + 1.82276i
$$438$$ 0 0
$$439$$ 4.16012i 0.198551i 0.995060 + 0.0992757i $$0.0316526\pi$$
−0.995060 + 0.0992757i $$0.968347\pi$$
$$440$$ 0 0
$$441$$ −7.43635 −0.354112
$$442$$ 0 0
$$443$$ 31.5136i 1.49726i 0.662990 + 0.748629i $$0.269287\pi$$
−0.662990 + 0.748629i $$0.730713\pi$$
$$444$$ 0 0
$$445$$ −5.11034 −0.242254
$$446$$ 0 0
$$447$$ −22.3923 −1.05912
$$448$$ 0 0
$$449$$ −15.5729 −0.734930 −0.367465 0.930037i $$-0.619774\pi$$
−0.367465 + 0.930037i $$0.619774\pi$$
$$450$$ 0 0
$$451$$ −39.2615 −1.84875
$$452$$ 0 0
$$453$$ 22.9206 1.07691
$$454$$ 0 0
$$455$$ 9.91534i 0.464838i
$$456$$ 0 0
$$457$$ 30.9828i 1.44931i −0.689110 0.724657i $$-0.741998\pi$$
0.689110 0.724657i $$-0.258002\pi$$
$$458$$ 0 0
$$459$$ −1.60151 −0.0747523
$$460$$ 0 0
$$461$$ 15.7446 0.733301 0.366651 0.930359i $$-0.380504\pi$$
0.366651 + 0.930359i $$0.380504\pi$$
$$462$$ 0 0
$$463$$ 1.71750i 0.0798191i −0.999203 0.0399096i $$-0.987293\pi$$
0.999203 0.0399096i $$-0.0127070\pi$$
$$464$$ 0 0
$$465$$ 6.46598i 0.299853i
$$466$$ 0 0
$$467$$ 35.3495 1.63578 0.817891 0.575373i $$-0.195143\pi$$
0.817891 + 0.575373i $$0.195143\pi$$
$$468$$ 0 0
$$469$$ 51.4391 2.37524
$$470$$ 0 0
$$471$$ −14.5054 −0.668372
$$472$$ 0 0
$$473$$ −21.9243 −1.00808
$$474$$ 0 0
$$475$$ 8.20535 0.376487
$$476$$ 0 0
$$477$$ 7.25579i 0.332220i
$$478$$ 0 0
$$479$$ 27.0253 1.23482 0.617408 0.786643i $$-0.288183\pi$$
0.617408 + 0.786643i $$0.288183\pi$$
$$480$$ 0 0
$$481$$ 13.0948i 0.597071i
$$482$$ 0 0
$$483$$ −17.6442 4.55185i −0.802838 0.207116i
$$484$$ 0 0
$$485$$ −19.4018 −0.880992
$$486$$ 0 0
$$487$$ 42.0999i 1.90773i 0.300237 + 0.953865i $$0.402934\pi$$
−0.300237 + 0.953865i $$0.597066\pi$$
$$488$$ 0 0
$$489$$ 6.70939 0.303409
$$490$$ 0 0
$$491$$ 14.3694i 0.648480i 0.945975 + 0.324240i $$0.105109\pi$$
−0.945975 + 0.324240i $$0.894891\pi$$
$$492$$ 0 0
$$493$$ 1.13134i 0.0509530i
$$494$$ 0 0
$$495$$ 3.69013i 0.165859i
$$496$$ 0 0
$$497$$ 33.0289i 1.48155i
$$498$$ 0 0
$$499$$ 11.1409i 0.498734i 0.968409 + 0.249367i $$0.0802226\pi$$
−0.968409 + 0.249367i $$0.919777\pi$$
$$500$$ 0 0
$$501$$ 5.17703 0.231293
$$502$$ 0 0
$$503$$ 15.8161 0.705207 0.352603 0.935773i $$-0.385297\pi$$
0.352603 + 0.935773i $$0.385297\pi$$
$$504$$ 0 0
$$505$$ 13.1514i 0.585228i
$$506$$ 0 0
$$507$$ 6.18983i 0.274900i
$$508$$ 0 0
$$509$$ −31.3061 −1.38762 −0.693809 0.720159i $$-0.744069\pi$$
−0.693809 + 0.720159i $$0.744069\pi$$
$$510$$ 0 0
$$511$$ −1.96415 −0.0868888
$$512$$ 0 0
$$513$$ 8.20535i 0.362275i
$$514$$ 0 0
$$515$$ 5.37041i 0.236648i
$$516$$ 0 0
$$517$$ 16.4679i 0.724259i
$$518$$ 0 0
$$519$$ 14.0638i 0.617331i
$$520$$ 0 0
$$521$$ 23.5026i 1.02967i −0.857290 0.514833i $$-0.827854\pi$$
0.857290 0.514833i $$-0.172146\pi$$
$$522$$ 0 0
$$523$$ −35.4909 −1.55191 −0.775954 0.630789i $$-0.782731\pi$$
−0.775954 + 0.630789i $$0.782731\pi$$
$$524$$ 0 0
$$525$$ 3.79952i 0.165825i
$$526$$ 0 0
$$527$$ 10.3554 0.451087
$$528$$ 0 0
$$529$$ −20.1296 11.1266i −0.875198 0.483764i
$$530$$ 0 0
$$531$$ 6.04730i 0.262430i
$$532$$ 0 0
$$533$$ −27.7654 −1.20265
$$534$$ 0 0
$$535$$ 17.9219i 0.774833i
$$536$$ 0 0
$$537$$ 11.9606 0.516137
$$538$$ 0 0
$$539$$ 27.4411 1.18197
$$540$$ 0 0
$$541$$ 31.7968 1.36705 0.683525 0.729927i $$-0.260446\pi$$
0.683525 + 0.729927i $$0.260446\pi$$
$$542$$ 0 0
$$543$$ 1.90385 0.0817022
$$544$$ 0 0
$$545$$ −1.27328 −0.0545415
$$546$$ 0 0
$$547$$ 21.8699i 0.935090i −0.883969 0.467545i $$-0.845139\pi$$
0.883969 0.467545i $$-0.154861\pi$$
$$548$$ 0 0
$$549$$ 3.36597i 0.143656i
$$550$$ 0 0
$$551$$ −5.79642 −0.246936
$$552$$ 0 0
$$553$$ −12.2791 −0.522160
$$554$$ 0 0
$$555$$ 5.01787i 0.212997i
$$556$$ 0 0
$$557$$ 1.91446i 0.0811182i −0.999177 0.0405591i $$-0.987086\pi$$
0.999177 0.0405591i $$-0.0129139\pi$$
$$558$$ 0 0
$$559$$ −15.5047 −0.655779
$$560$$ 0 0
$$561$$ 5.90980 0.249512
$$562$$ 0 0
$$563$$ 33.0420 1.39255 0.696277 0.717773i $$-0.254838\pi$$
0.696277 + 0.717773i $$0.254838\pi$$
$$564$$ 0 0
$$565$$ −3.32431 −0.139855
$$566$$ 0 0
$$567$$ 3.79952 0.159565
$$568$$ 0 0
$$569$$ 10.1677i 0.426250i −0.977025 0.213125i $$-0.931636\pi$$
0.977025 0.213125i $$-0.0683642\pi$$
$$570$$ 0 0
$$571$$ −22.6137 −0.946355 −0.473177 0.880967i $$-0.656893\pi$$
−0.473177 + 0.880967i $$0.656893\pi$$
$$572$$ 0 0
$$573$$ 16.6906i 0.697260i
$$574$$ 0 0
$$575$$ −1.19801 + 4.64379i −0.0499603 + 0.193659i
$$576$$ 0 0
$$577$$ −10.0069 −0.416591 −0.208295 0.978066i $$-0.566792\pi$$
−0.208295 + 0.978066i $$0.566792\pi$$
$$578$$ 0 0
$$579$$ 11.9024i 0.494646i
$$580$$ 0 0
$$581$$ 15.6003 0.647210
$$582$$ 0 0
$$583$$ 26.7748i 1.10890i
$$584$$ 0 0
$$585$$ 2.60963i 0.107895i
$$586$$ 0 0
$$587$$ 20.6747i 0.853336i −0.904408 0.426668i $$-0.859687\pi$$
0.904408 0.426668i $$-0.140313\pi$$
$$588$$ 0 0
$$589$$ 53.0556i 2.18612i
$$590$$ 0 0
$$591$$ 15.3515i 0.631478i
$$592$$ 0 0
$$593$$ −34.2092 −1.40480 −0.702402 0.711780i $$-0.747889\pi$$
−0.702402 + 0.711780i $$0.747889\pi$$
$$594$$ 0 0
$$595$$ 6.08498 0.249460
$$596$$ 0 0
$$597$$ 17.4361i 0.713612i
$$598$$ 0 0
$$599$$ 29.2726i 1.19605i 0.801479 + 0.598023i $$0.204047\pi$$
−0.801479 + 0.598023i $$0.795953\pi$$
$$600$$ 0 0
$$601$$ 32.7471 1.33578 0.667891 0.744259i $$-0.267197\pi$$
0.667891 + 0.744259i $$0.267197\pi$$
$$602$$ 0 0
$$603$$ −13.5383 −0.551323
$$604$$ 0 0
$$605$$ 2.61707i 0.106399i
$$606$$ 0 0
$$607$$ 29.9681i 1.21637i 0.793796 + 0.608185i $$0.208102\pi$$
−0.793796 + 0.608185i $$0.791898\pi$$
$$608$$ 0 0
$$609$$ 2.68406i 0.108763i
$$610$$ 0 0
$$611$$ 11.6460i 0.471146i
$$612$$ 0 0
$$613$$ 12.6769i 0.512014i −0.966675 0.256007i $$-0.917593\pi$$
0.966675 0.256007i $$-0.0824070\pi$$
$$614$$ 0 0
$$615$$ 10.6396 0.429029
$$616$$ 0 0
$$617$$ 6.68220i 0.269015i −0.990913 0.134508i $$-0.957055\pi$$
0.990913 0.134508i $$-0.0429453\pi$$
$$618$$ 0 0
$$619$$ −12.1774 −0.489451 −0.244725 0.969592i $$-0.578698\pi$$
−0.244725 + 0.969592i $$0.578698\pi$$
$$620$$ 0 0
$$621$$ 4.64379 + 1.19801i 0.186349 + 0.0480743i
$$622$$ 0 0
$$623$$ 19.4169i 0.777920i
$$624$$ 0 0
$$625$$ 1.00000 0.0400000
$$626$$ 0 0
$$627$$ 30.2788i 1.20922i
$$628$$ 0 0
$$629$$ 8.03619 0.320424
$$630$$ 0 0
$$631$$ −20.6325 −0.821365 −0.410683 0.911778i $$-0.634710\pi$$
−0.410683 + 0.911778i $$0.634710\pi$$
$$632$$ 0 0
$$633$$ −7.11509 −0.282799
$$634$$ 0 0
$$635$$ 15.7160 0.623671
$$636$$ 0 0
$$637$$ 19.4061 0.768899
$$638$$ 0 0
$$639$$ 8.69292i 0.343887i
$$640$$ 0 0
$$641$$ 18.8025i 0.742655i 0.928502 + 0.371328i $$0.121097\pi$$
−0.928502 + 0.371328i $$0.878903\pi$$
$$642$$ 0 0
$$643$$ 6.17101 0.243361 0.121680 0.992569i $$-0.461172\pi$$
0.121680 + 0.992569i $$0.461172\pi$$
$$644$$ 0 0
$$645$$ 5.94134 0.233940
$$646$$ 0 0
$$647$$ 11.0800i 0.435601i 0.975993 + 0.217801i $$0.0698882\pi$$
−0.975993 + 0.217801i $$0.930112\pi$$
$$648$$ 0 0
$$649$$ 22.3153i 0.875953i
$$650$$ 0 0
$$651$$ −24.5676 −0.962881
$$652$$ 0 0
$$653$$ −14.6281 −0.572441 −0.286221 0.958164i $$-0.592399\pi$$
−0.286221 + 0.958164i $$0.592399\pi$$
$$654$$ 0 0
$$655$$ −10.5445 −0.412009
$$656$$ 0 0
$$657$$ 0.516947 0.0201680
$$658$$ 0 0
$$659$$ −19.1684 −0.746694 −0.373347 0.927692i $$-0.621790\pi$$
−0.373347 + 0.927692i $$0.621790\pi$$
$$660$$ 0 0
$$661$$ 32.1112i 1.24898i −0.781033 0.624490i $$-0.785307\pi$$
0.781033 0.624490i $$-0.214693\pi$$
$$662$$ 0 0
$$663$$ 4.17936 0.162313
$$664$$ 0 0
$$665$$ 31.1764i 1.20897i
$$666$$ 0 0
$$667$$ 0.846295 3.28046i 0.0327687 0.127020i
$$668$$ 0 0
$$669$$ 2.92403 0.113050
$$670$$ 0 0
$$671$$ 12.4209i 0.479502i
$$672$$ 0 0
$$673$$ 0.150913 0.00581729 0.00290864 0.999996i $$-0.499074\pi$$
0.00290864 + 0.999996i $$0.499074\pi$$
$$674$$ 0 0
$$675$$ 1.00000i 0.0384900i
$$676$$ 0 0
$$677$$ 41.2608i 1.58578i 0.609365 + 0.792890i $$0.291425\pi$$
−0.609365 + 0.792890i $$0.708575\pi$$
$$678$$ 0 0
$$679$$ 73.7177i 2.82902i
$$680$$ 0 0
$$681$$ 8.86711i 0.339788i
$$682$$ 0 0
$$683$$ 20.7486i 0.793925i −0.917835 0.396963i $$-0.870064\pi$$
0.917835 0.396963i $$-0.129936\pi$$
$$684$$ 0 0
$$685$$ 1.67864 0.0641377
$$686$$ 0 0
$$687$$ 7.81117 0.298015
$$688$$ 0 0
$$689$$ 18.9349i 0.721364i
$$690$$ 0 0
$$691$$ 4.48560i 0.170640i −0.996354 0.0853202i $$-0.972809\pi$$
0.996354 0.0853202i $$-0.0271913\pi$$
$$692$$ 0 0
$$693$$ −14.0207 −0.532603
$$694$$ 0 0
$$695$$ 13.7161 0.520280
$$696$$ 0 0
$$697$$ 17.0394i 0.645415i
$$698$$ 0 0
$$699$$ 5.09372i 0.192662i
$$700$$ 0 0
$$701$$ 1.49486i 0.0564601i 0.999601 + 0.0282301i $$0.00898710\pi$$
−0.999601 + 0.0282301i $$0.991013\pi$$
$$702$$ 0 0
$$703$$ 41.1734i 1.55288i
$$704$$ 0 0
$$705$$ 4.46270i 0.168075i
$$706$$ 0 0
$$707$$ 49.9689 1.87927
$$708$$ 0 0
$$709$$ 48.1753i 1.80926i 0.426194 + 0.904632i $$0.359854\pi$$
−0.426194 + 0.904632i $$0.640146\pi$$
$$710$$ 0 0
$$711$$ 3.23175 0.121200
$$712$$ 0 0
$$713$$ −30.0267 7.74629i −1.12451 0.290101i
$$714$$ 0 0
$$715$$ 9.62988i 0.360137i
$$716$$ 0 0
$$717$$ −19.5551 −0.730299
$$718$$ 0 0
$$719$$ 13.1017i 0.488610i 0.969698 + 0.244305i $$0.0785599\pi$$
−0.969698 + 0.244305i $$0.921440\pi$$
$$720$$ 0 0
$$721$$ 20.4050 0.759921
$$722$$ 0 0
$$723$$ −8.39986 −0.312394
$$724$$ 0 0
$$725$$ −0.706420 −0.0262358
$$726$$ 0 0
$$727$$ 7.30395 0.270888 0.135444 0.990785i $$-0.456754\pi$$
0.135444 + 0.990785i $$0.456754\pi$$
$$728$$ 0 0
$$729$$ −1.00000 −0.0370370
$$730$$ 0 0
$$731$$ 9.51513i 0.351930i
$$732$$ 0 0
$$733$$ 17.8028i 0.657563i 0.944406 + 0.328781i $$0.106638\pi$$
−0.944406 + 0.328781i $$0.893362\pi$$
$$734$$ 0 0
$$735$$ −7.43635 −0.274294
$$736$$ 0 0
$$737$$ 49.9581 1.84023
$$738$$ 0 0
$$739$$ 30.6415i 1.12716i 0.826060 + 0.563582i $$0.190577\pi$$
−0.826060 + 0.563582i $$0.809423\pi$$
$$740$$ 0 0
$$741$$ 21.4129i 0.786623i
$$742$$ 0 0
$$743$$ −23.7573 −0.871572 −0.435786 0.900050i $$-0.643530\pi$$
−0.435786 + 0.900050i $$0.643530\pi$$
$$744$$ 0 0
$$745$$ −22.3923 −0.820392
$$746$$ 0 0
$$747$$ −4.10586 −0.150226
$$748$$ 0 0
$$749$$ 68.0948 2.48813
$$750$$ 0 0
$$751$$ 11.6741 0.425996 0.212998 0.977053i $$-0.431677\pi$$
0.212998 + 0.977053i $$0.431677\pi$$
$$752$$ 0 0
$$753$$ 7.84091i 0.285739i
$$754$$ 0 0
$$755$$ 22.9206 0.834167
$$756$$ 0 0
$$757$$ 16.7159i 0.607551i −0.952744 0.303776i $$-0.901753\pi$$
0.952744 0.303776i $$-0.0982473\pi$$
$$758$$ 0 0
$$759$$ −17.1362 4.42080i −0.622004 0.160465i
$$760$$ 0 0
$$761$$ 25.0602 0.908430 0.454215 0.890892i $$-0.349920\pi$$
0.454215 + 0.890892i $$0.349920\pi$$
$$762$$ 0 0
$$763$$ 4.83786i 0.175142i
$$764$$ 0 0
$$765$$ −1.60151 −0.0579029
$$766$$ 0 0
$$767$$ 15.7812i 0.569826i
$$768$$ 0 0
$$769$$ 21.2483i 0.766234i −0.923700 0.383117i $$-0.874851\pi$$
0.923700 0.383117i $$-0.125149\pi$$
$$770$$ 0 0
$$771$$ 5.18308i 0.186664i
$$772$$ 0 0
$$773$$ 19.8965i 0.715628i −0.933793 0.357814i $$-0.883522\pi$$
0.933793 0.357814i $$-0.116478\pi$$
$$774$$ 0 0
$$775$$ 6.46598i 0.232265i
$$776$$ 0 0
$$777$$ −19.0655 −0.683971
$$778$$ 0 0
$$779$$ 87.3015 3.12790
$$780$$ 0 0
$$781$$ 32.0780i 1.14784i
$$782$$ 0 0
$$783$$ 0.706420i 0.0252454i
$$784$$ 0 0
$$785$$ −14.5054 −0.517719
$$786$$ 0 0
$$787$$ −9.36272 −0.333745 −0.166873 0.985978i $$-0.553367\pi$$
−0.166873 + 0.985978i $$0.553367\pi$$
$$788$$ 0 0
$$789$$ 6.41083i 0.228232i
$$790$$ 0 0
$$791$$ 12.6308i 0.449099i
$$792$$ 0 0
$$793$$ 8.78393i 0.311926i
$$794$$ 0 0
$$795$$ 7.25579i 0.257336i
$$796$$ 0 0
$$797$$ 43.4609i 1.53946i 0.638367 + 0.769732i $$0.279610\pi$$
−0.638367 + 0.769732i $$0.720390\pi$$
$$798$$ 0 0
$$799$$ 7.14708 0.252845
$$800$$ 0 0
$$801$$ 5.11034i 0.180565i
$$802$$ 0 0
$$803$$ −1.90760 −0.0673178
$$804$$ 0 0
$$805$$