# Properties

 Label 5445.2.a.y.1.2 Level $5445$ Weight $2$ Character 5445.1 Self dual yes Analytic conductor $43.479$ Analytic rank $0$ Dimension $2$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$5445 = 3^{2} \cdot 5 \cdot 11^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 5445.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$43.4785439006$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\zeta_{8})^+$$ Defining polynomial: $$x^{2} - 2$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 55) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.2 Root $$1.41421$$ of defining polynomial Character $$\chi$$ $$=$$ 5445.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+2.41421 q^{2} +3.82843 q^{4} +1.00000 q^{5} +2.00000 q^{7} +4.41421 q^{8} +O(q^{10})$$ $$q+2.41421 q^{2} +3.82843 q^{4} +1.00000 q^{5} +2.00000 q^{7} +4.41421 q^{8} +2.41421 q^{10} +1.17157 q^{13} +4.82843 q^{14} +3.00000 q^{16} +6.82843 q^{17} +3.82843 q^{20} +2.82843 q^{23} +1.00000 q^{25} +2.82843 q^{26} +7.65685 q^{28} -3.65685 q^{29} -1.58579 q^{32} +16.4853 q^{34} +2.00000 q^{35} -7.65685 q^{37} +4.41421 q^{40} +6.00000 q^{41} +6.00000 q^{43} +6.82843 q^{46} -2.82843 q^{47} -3.00000 q^{49} +2.41421 q^{50} +4.48528 q^{52} -11.6569 q^{53} +8.82843 q^{56} -8.82843 q^{58} -1.65685 q^{59} +9.31371 q^{61} -9.82843 q^{64} +1.17157 q^{65} +12.4853 q^{67} +26.1421 q^{68} +4.82843 q^{70} -11.3137 q^{71} +1.17157 q^{73} -18.4853 q^{74} -4.00000 q^{79} +3.00000 q^{80} +14.4853 q^{82} -6.00000 q^{83} +6.82843 q^{85} +14.4853 q^{86} +13.3137 q^{89} +2.34315 q^{91} +10.8284 q^{92} -6.82843 q^{94} +3.65685 q^{97} -7.24264 q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{2} + 2 q^{4} + 2 q^{5} + 4 q^{7} + 6 q^{8} + O(q^{10})$$ $$2 q + 2 q^{2} + 2 q^{4} + 2 q^{5} + 4 q^{7} + 6 q^{8} + 2 q^{10} + 8 q^{13} + 4 q^{14} + 6 q^{16} + 8 q^{17} + 2 q^{20} + 2 q^{25} + 4 q^{28} + 4 q^{29} - 6 q^{32} + 16 q^{34} + 4 q^{35} - 4 q^{37} + 6 q^{40} + 12 q^{41} + 12 q^{43} + 8 q^{46} - 6 q^{49} + 2 q^{50} - 8 q^{52} - 12 q^{53} + 12 q^{56} - 12 q^{58} + 8 q^{59} - 4 q^{61} - 14 q^{64} + 8 q^{65} + 8 q^{67} + 24 q^{68} + 4 q^{70} + 8 q^{73} - 20 q^{74} - 8 q^{79} + 6 q^{80} + 12 q^{82} - 12 q^{83} + 8 q^{85} + 12 q^{86} + 4 q^{89} + 16 q^{91} + 16 q^{92} - 8 q^{94} - 4 q^{97} - 6 q^{98} + O(q^{100})$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 2.41421 1.70711 0.853553 0.521005i $$-0.174443\pi$$
0.853553 + 0.521005i $$0.174443\pi$$
$$3$$ 0 0
$$4$$ 3.82843 1.91421
$$5$$ 1.00000 0.447214
$$6$$ 0 0
$$7$$ 2.00000 0.755929 0.377964 0.925820i $$-0.376624\pi$$
0.377964 + 0.925820i $$0.376624\pi$$
$$8$$ 4.41421 1.56066
$$9$$ 0 0
$$10$$ 2.41421 0.763441
$$11$$ 0 0
$$12$$ 0 0
$$13$$ 1.17157 0.324936 0.162468 0.986714i $$-0.448055\pi$$
0.162468 + 0.986714i $$0.448055\pi$$
$$14$$ 4.82843 1.29045
$$15$$ 0 0
$$16$$ 3.00000 0.750000
$$17$$ 6.82843 1.65614 0.828068 0.560627i $$-0.189440\pi$$
0.828068 + 0.560627i $$0.189440\pi$$
$$18$$ 0 0
$$19$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$20$$ 3.82843 0.856062
$$21$$ 0 0
$$22$$ 0 0
$$23$$ 2.82843 0.589768 0.294884 0.955533i $$-0.404719\pi$$
0.294884 + 0.955533i $$0.404719\pi$$
$$24$$ 0 0
$$25$$ 1.00000 0.200000
$$26$$ 2.82843 0.554700
$$27$$ 0 0
$$28$$ 7.65685 1.44701
$$29$$ −3.65685 −0.679061 −0.339530 0.940595i $$-0.610268\pi$$
−0.339530 + 0.940595i $$0.610268\pi$$
$$30$$ 0 0
$$31$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$32$$ −1.58579 −0.280330
$$33$$ 0 0
$$34$$ 16.4853 2.82720
$$35$$ 2.00000 0.338062
$$36$$ 0 0
$$37$$ −7.65685 −1.25878 −0.629390 0.777090i $$-0.716695\pi$$
−0.629390 + 0.777090i $$0.716695\pi$$
$$38$$ 0 0
$$39$$ 0 0
$$40$$ 4.41421 0.697948
$$41$$ 6.00000 0.937043 0.468521 0.883452i $$-0.344787\pi$$
0.468521 + 0.883452i $$0.344787\pi$$
$$42$$ 0 0
$$43$$ 6.00000 0.914991 0.457496 0.889212i $$-0.348747\pi$$
0.457496 + 0.889212i $$0.348747\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 6.82843 1.00680
$$47$$ −2.82843 −0.412568 −0.206284 0.978492i $$-0.566137\pi$$
−0.206284 + 0.978492i $$0.566137\pi$$
$$48$$ 0 0
$$49$$ −3.00000 −0.428571
$$50$$ 2.41421 0.341421
$$51$$ 0 0
$$52$$ 4.48528 0.621997
$$53$$ −11.6569 −1.60119 −0.800596 0.599204i $$-0.795484\pi$$
−0.800596 + 0.599204i $$0.795484\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 8.82843 1.17975
$$57$$ 0 0
$$58$$ −8.82843 −1.15923
$$59$$ −1.65685 −0.215704 −0.107852 0.994167i $$-0.534397\pi$$
−0.107852 + 0.994167i $$0.534397\pi$$
$$60$$ 0 0
$$61$$ 9.31371 1.19250 0.596249 0.802799i $$-0.296657\pi$$
0.596249 + 0.802799i $$0.296657\pi$$
$$62$$ 0 0
$$63$$ 0 0
$$64$$ −9.82843 −1.22855
$$65$$ 1.17157 0.145316
$$66$$ 0 0
$$67$$ 12.4853 1.52532 0.762660 0.646800i $$-0.223893\pi$$
0.762660 + 0.646800i $$0.223893\pi$$
$$68$$ 26.1421 3.17020
$$69$$ 0 0
$$70$$ 4.82843 0.577107
$$71$$ −11.3137 −1.34269 −0.671345 0.741145i $$-0.734283\pi$$
−0.671345 + 0.741145i $$0.734283\pi$$
$$72$$ 0 0
$$73$$ 1.17157 0.137122 0.0685611 0.997647i $$-0.478159\pi$$
0.0685611 + 0.997647i $$0.478159\pi$$
$$74$$ −18.4853 −2.14887
$$75$$ 0 0
$$76$$ 0 0
$$77$$ 0 0
$$78$$ 0 0
$$79$$ −4.00000 −0.450035 −0.225018 0.974355i $$-0.572244\pi$$
−0.225018 + 0.974355i $$0.572244\pi$$
$$80$$ 3.00000 0.335410
$$81$$ 0 0
$$82$$ 14.4853 1.59963
$$83$$ −6.00000 −0.658586 −0.329293 0.944228i $$-0.606810\pi$$
−0.329293 + 0.944228i $$0.606810\pi$$
$$84$$ 0 0
$$85$$ 6.82843 0.740647
$$86$$ 14.4853 1.56199
$$87$$ 0 0
$$88$$ 0 0
$$89$$ 13.3137 1.41125 0.705625 0.708585i $$-0.250666\pi$$
0.705625 + 0.708585i $$0.250666\pi$$
$$90$$ 0 0
$$91$$ 2.34315 0.245628
$$92$$ 10.8284 1.12894
$$93$$ 0 0
$$94$$ −6.82843 −0.704298
$$95$$ 0 0
$$96$$ 0 0
$$97$$ 3.65685 0.371297 0.185649 0.982616i $$-0.440561\pi$$
0.185649 + 0.982616i $$0.440561\pi$$
$$98$$ −7.24264 −0.731617
$$99$$ 0 0
$$100$$ 3.82843 0.382843
$$101$$ 9.31371 0.926749 0.463374 0.886163i $$-0.346639\pi$$
0.463374 + 0.886163i $$0.346639\pi$$
$$102$$ 0 0
$$103$$ 6.82843 0.672825 0.336412 0.941715i $$-0.390786\pi$$
0.336412 + 0.941715i $$0.390786\pi$$
$$104$$ 5.17157 0.507114
$$105$$ 0 0
$$106$$ −28.1421 −2.73341
$$107$$ 7.65685 0.740216 0.370108 0.928989i $$-0.379321\pi$$
0.370108 + 0.928989i $$0.379321\pi$$
$$108$$ 0 0
$$109$$ 7.65685 0.733394 0.366697 0.930341i $$-0.380489\pi$$
0.366697 + 0.930341i $$0.380489\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ 6.00000 0.566947
$$113$$ −19.6569 −1.84916 −0.924581 0.380986i $$-0.875584\pi$$
−0.924581 + 0.380986i $$0.875584\pi$$
$$114$$ 0 0
$$115$$ 2.82843 0.263752
$$116$$ −14.0000 −1.29987
$$117$$ 0 0
$$118$$ −4.00000 −0.368230
$$119$$ 13.6569 1.25192
$$120$$ 0 0
$$121$$ 0 0
$$122$$ 22.4853 2.03572
$$123$$ 0 0
$$124$$ 0 0
$$125$$ 1.00000 0.0894427
$$126$$ 0 0
$$127$$ −4.34315 −0.385392 −0.192696 0.981259i $$-0.561723\pi$$
−0.192696 + 0.981259i $$0.561723\pi$$
$$128$$ −20.5563 −1.81694
$$129$$ 0 0
$$130$$ 2.82843 0.248069
$$131$$ −11.3137 −0.988483 −0.494242 0.869325i $$-0.664554\pi$$
−0.494242 + 0.869325i $$0.664554\pi$$
$$132$$ 0 0
$$133$$ 0 0
$$134$$ 30.1421 2.60388
$$135$$ 0 0
$$136$$ 30.1421 2.58467
$$137$$ 10.9706 0.937278 0.468639 0.883390i $$-0.344744\pi$$
0.468639 + 0.883390i $$0.344744\pi$$
$$138$$ 0 0
$$139$$ 4.00000 0.339276 0.169638 0.985506i $$-0.445740\pi$$
0.169638 + 0.985506i $$0.445740\pi$$
$$140$$ 7.65685 0.647122
$$141$$ 0 0
$$142$$ −27.3137 −2.29212
$$143$$ 0 0
$$144$$ 0 0
$$145$$ −3.65685 −0.303685
$$146$$ 2.82843 0.234082
$$147$$ 0 0
$$148$$ −29.3137 −2.40957
$$149$$ 0.343146 0.0281116 0.0140558 0.999901i $$-0.495526\pi$$
0.0140558 + 0.999901i $$0.495526\pi$$
$$150$$ 0 0
$$151$$ 12.0000 0.976546 0.488273 0.872691i $$-0.337627\pi$$
0.488273 + 0.872691i $$0.337627\pi$$
$$152$$ 0 0
$$153$$ 0 0
$$154$$ 0 0
$$155$$ 0 0
$$156$$ 0 0
$$157$$ −14.0000 −1.11732 −0.558661 0.829396i $$-0.688685\pi$$
−0.558661 + 0.829396i $$0.688685\pi$$
$$158$$ −9.65685 −0.768258
$$159$$ 0 0
$$160$$ −1.58579 −0.125367
$$161$$ 5.65685 0.445823
$$162$$ 0 0
$$163$$ 16.4853 1.29123 0.645613 0.763664i $$-0.276602\pi$$
0.645613 + 0.763664i $$0.276602\pi$$
$$164$$ 22.9706 1.79370
$$165$$ 0 0
$$166$$ −14.4853 −1.12428
$$167$$ −22.9706 −1.77752 −0.888758 0.458377i $$-0.848431\pi$$
−0.888758 + 0.458377i $$0.848431\pi$$
$$168$$ 0 0
$$169$$ −11.6274 −0.894417
$$170$$ 16.4853 1.26436
$$171$$ 0 0
$$172$$ 22.9706 1.75149
$$173$$ −22.1421 −1.68344 −0.841718 0.539918i $$-0.818455\pi$$
−0.841718 + 0.539918i $$0.818455\pi$$
$$174$$ 0 0
$$175$$ 2.00000 0.151186
$$176$$ 0 0
$$177$$ 0 0
$$178$$ 32.1421 2.40915
$$179$$ −9.65685 −0.721787 −0.360894 0.932607i $$-0.617528\pi$$
−0.360894 + 0.932607i $$0.617528\pi$$
$$180$$ 0 0
$$181$$ 21.3137 1.58424 0.792118 0.610368i $$-0.208979\pi$$
0.792118 + 0.610368i $$0.208979\pi$$
$$182$$ 5.65685 0.419314
$$183$$ 0 0
$$184$$ 12.4853 0.920427
$$185$$ −7.65685 −0.562943
$$186$$ 0 0
$$187$$ 0 0
$$188$$ −10.8284 −0.789744
$$189$$ 0 0
$$190$$ 0 0
$$191$$ −3.31371 −0.239772 −0.119886 0.992788i $$-0.538253\pi$$
−0.119886 + 0.992788i $$0.538253\pi$$
$$192$$ 0 0
$$193$$ 1.17157 0.0843317 0.0421658 0.999111i $$-0.486574\pi$$
0.0421658 + 0.999111i $$0.486574\pi$$
$$194$$ 8.82843 0.633844
$$195$$ 0 0
$$196$$ −11.4853 −0.820377
$$197$$ −10.8284 −0.771493 −0.385747 0.922605i $$-0.626056\pi$$
−0.385747 + 0.922605i $$0.626056\pi$$
$$198$$ 0 0
$$199$$ 10.3431 0.733206 0.366603 0.930377i $$-0.380521\pi$$
0.366603 + 0.930377i $$0.380521\pi$$
$$200$$ 4.41421 0.312132
$$201$$ 0 0
$$202$$ 22.4853 1.58206
$$203$$ −7.31371 −0.513322
$$204$$ 0 0
$$205$$ 6.00000 0.419058
$$206$$ 16.4853 1.14858
$$207$$ 0 0
$$208$$ 3.51472 0.243702
$$209$$ 0 0
$$210$$ 0 0
$$211$$ 16.0000 1.10149 0.550743 0.834675i $$-0.314345\pi$$
0.550743 + 0.834675i $$0.314345\pi$$
$$212$$ −44.6274 −3.06502
$$213$$ 0 0
$$214$$ 18.4853 1.26363
$$215$$ 6.00000 0.409197
$$216$$ 0 0
$$217$$ 0 0
$$218$$ 18.4853 1.25198
$$219$$ 0 0
$$220$$ 0 0
$$221$$ 8.00000 0.538138
$$222$$ 0 0
$$223$$ −10.8284 −0.725125 −0.362563 0.931959i $$-0.618098\pi$$
−0.362563 + 0.931959i $$0.618098\pi$$
$$224$$ −3.17157 −0.211910
$$225$$ 0 0
$$226$$ −47.4558 −3.15672
$$227$$ 25.3137 1.68013 0.840065 0.542486i $$-0.182517\pi$$
0.840065 + 0.542486i $$0.182517\pi$$
$$228$$ 0 0
$$229$$ 1.31371 0.0868123 0.0434062 0.999058i $$-0.486179\pi$$
0.0434062 + 0.999058i $$0.486179\pi$$
$$230$$ 6.82843 0.450253
$$231$$ 0 0
$$232$$ −16.1421 −1.05978
$$233$$ −6.14214 −0.402385 −0.201192 0.979552i $$-0.564482\pi$$
−0.201192 + 0.979552i $$0.564482\pi$$
$$234$$ 0 0
$$235$$ −2.82843 −0.184506
$$236$$ −6.34315 −0.412904
$$237$$ 0 0
$$238$$ 32.9706 2.13716
$$239$$ −23.3137 −1.50804 −0.754019 0.656852i $$-0.771887\pi$$
−0.754019 + 0.656852i $$0.771887\pi$$
$$240$$ 0 0
$$241$$ −6.00000 −0.386494 −0.193247 0.981150i $$-0.561902\pi$$
−0.193247 + 0.981150i $$0.561902\pi$$
$$242$$ 0 0
$$243$$ 0 0
$$244$$ 35.6569 2.28270
$$245$$ −3.00000 −0.191663
$$246$$ 0 0
$$247$$ 0 0
$$248$$ 0 0
$$249$$ 0 0
$$250$$ 2.41421 0.152688
$$251$$ −12.0000 −0.757433 −0.378717 0.925513i $$-0.623635\pi$$
−0.378717 + 0.925513i $$0.623635\pi$$
$$252$$ 0 0
$$253$$ 0 0
$$254$$ −10.4853 −0.657905
$$255$$ 0 0
$$256$$ −29.9706 −1.87316
$$257$$ 9.31371 0.580973 0.290487 0.956879i $$-0.406183\pi$$
0.290487 + 0.956879i $$0.406183\pi$$
$$258$$ 0 0
$$259$$ −15.3137 −0.951548
$$260$$ 4.48528 0.278165
$$261$$ 0 0
$$262$$ −27.3137 −1.68745
$$263$$ −10.9706 −0.676474 −0.338237 0.941061i $$-0.609831\pi$$
−0.338237 + 0.941061i $$0.609831\pi$$
$$264$$ 0 0
$$265$$ −11.6569 −0.716075
$$266$$ 0 0
$$267$$ 0 0
$$268$$ 47.7990 2.91979
$$269$$ −17.3137 −1.05564 −0.527818 0.849358i $$-0.676990\pi$$
−0.527818 + 0.849358i $$0.676990\pi$$
$$270$$ 0 0
$$271$$ −7.31371 −0.444276 −0.222138 0.975015i $$-0.571304\pi$$
−0.222138 + 0.975015i $$0.571304\pi$$
$$272$$ 20.4853 1.24210
$$273$$ 0 0
$$274$$ 26.4853 1.60003
$$275$$ 0 0
$$276$$ 0 0
$$277$$ −6.82843 −0.410280 −0.205140 0.978733i $$-0.565765\pi$$
−0.205140 + 0.978733i $$0.565765\pi$$
$$278$$ 9.65685 0.579180
$$279$$ 0 0
$$280$$ 8.82843 0.527599
$$281$$ 17.3137 1.03285 0.516425 0.856333i $$-0.327263\pi$$
0.516425 + 0.856333i $$0.327263\pi$$
$$282$$ 0 0
$$283$$ −32.6274 −1.93950 −0.969749 0.244103i $$-0.921507\pi$$
−0.969749 + 0.244103i $$0.921507\pi$$
$$284$$ −43.3137 −2.57020
$$285$$ 0 0
$$286$$ 0 0
$$287$$ 12.0000 0.708338
$$288$$ 0 0
$$289$$ 29.6274 1.74279
$$290$$ −8.82843 −0.518423
$$291$$ 0 0
$$292$$ 4.48528 0.262481
$$293$$ −9.17157 −0.535809 −0.267905 0.963445i $$-0.586331\pi$$
−0.267905 + 0.963445i $$0.586331\pi$$
$$294$$ 0 0
$$295$$ −1.65685 −0.0964658
$$296$$ −33.7990 −1.96453
$$297$$ 0 0
$$298$$ 0.828427 0.0479895
$$299$$ 3.31371 0.191637
$$300$$ 0 0
$$301$$ 12.0000 0.691669
$$302$$ 28.9706 1.66707
$$303$$ 0 0
$$304$$ 0 0
$$305$$ 9.31371 0.533301
$$306$$ 0 0
$$307$$ 16.3431 0.932753 0.466376 0.884586i $$-0.345559\pi$$
0.466376 + 0.884586i $$0.345559\pi$$
$$308$$ 0 0
$$309$$ 0 0
$$310$$ 0 0
$$311$$ −4.68629 −0.265735 −0.132868 0.991134i $$-0.542419\pi$$
−0.132868 + 0.991134i $$0.542419\pi$$
$$312$$ 0 0
$$313$$ −1.31371 −0.0742552 −0.0371276 0.999311i $$-0.511821\pi$$
−0.0371276 + 0.999311i $$0.511821\pi$$
$$314$$ −33.7990 −1.90739
$$315$$ 0 0
$$316$$ −15.3137 −0.861463
$$317$$ 1.31371 0.0737852 0.0368926 0.999319i $$-0.488254\pi$$
0.0368926 + 0.999319i $$0.488254\pi$$
$$318$$ 0 0
$$319$$ 0 0
$$320$$ −9.82843 −0.549426
$$321$$ 0 0
$$322$$ 13.6569 0.761067
$$323$$ 0 0
$$324$$ 0 0
$$325$$ 1.17157 0.0649872
$$326$$ 39.7990 2.20426
$$327$$ 0 0
$$328$$ 26.4853 1.46241
$$329$$ −5.65685 −0.311872
$$330$$ 0 0
$$331$$ −7.31371 −0.401998 −0.200999 0.979591i $$-0.564419\pi$$
−0.200999 + 0.979591i $$0.564419\pi$$
$$332$$ −22.9706 −1.26067
$$333$$ 0 0
$$334$$ −55.4558 −3.03441
$$335$$ 12.4853 0.682144
$$336$$ 0 0
$$337$$ 20.4853 1.11590 0.557952 0.829873i $$-0.311587\pi$$
0.557952 + 0.829873i $$0.311587\pi$$
$$338$$ −28.0711 −1.52686
$$339$$ 0 0
$$340$$ 26.1421 1.41776
$$341$$ 0 0
$$342$$ 0 0
$$343$$ −20.0000 −1.07990
$$344$$ 26.4853 1.42799
$$345$$ 0 0
$$346$$ −53.4558 −2.87380
$$347$$ 10.9706 0.588931 0.294465 0.955662i $$-0.404858\pi$$
0.294465 + 0.955662i $$0.404858\pi$$
$$348$$ 0 0
$$349$$ −26.9706 −1.44370 −0.721851 0.692049i $$-0.756708\pi$$
−0.721851 + 0.692049i $$0.756708\pi$$
$$350$$ 4.82843 0.258090
$$351$$ 0 0
$$352$$ 0 0
$$353$$ −21.3137 −1.13441 −0.567207 0.823575i $$-0.691976\pi$$
−0.567207 + 0.823575i $$0.691976\pi$$
$$354$$ 0 0
$$355$$ −11.3137 −0.600469
$$356$$ 50.9706 2.70143
$$357$$ 0 0
$$358$$ −23.3137 −1.23217
$$359$$ 0.686292 0.0362211 0.0181105 0.999836i $$-0.494235\pi$$
0.0181105 + 0.999836i $$0.494235\pi$$
$$360$$ 0 0
$$361$$ −19.0000 −1.00000
$$362$$ 51.4558 2.70446
$$363$$ 0 0
$$364$$ 8.97056 0.470185
$$365$$ 1.17157 0.0613229
$$366$$ 0 0
$$367$$ −8.48528 −0.442928 −0.221464 0.975169i $$-0.571084\pi$$
−0.221464 + 0.975169i $$0.571084\pi$$
$$368$$ 8.48528 0.442326
$$369$$ 0 0
$$370$$ −18.4853 −0.961004
$$371$$ −23.3137 −1.21039
$$372$$ 0 0
$$373$$ −35.7990 −1.85360 −0.926801 0.375554i $$-0.877453\pi$$
−0.926801 + 0.375554i $$0.877453\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ −12.4853 −0.643879
$$377$$ −4.28427 −0.220651
$$378$$ 0 0
$$379$$ 33.6569 1.72884 0.864418 0.502773i $$-0.167687\pi$$
0.864418 + 0.502773i $$0.167687\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ −8.00000 −0.409316
$$383$$ 5.85786 0.299323 0.149661 0.988737i $$-0.452182\pi$$
0.149661 + 0.988737i $$0.452182\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 2.82843 0.143963
$$387$$ 0 0
$$388$$ 14.0000 0.710742
$$389$$ −20.6274 −1.04585 −0.522926 0.852378i $$-0.675160\pi$$
−0.522926 + 0.852378i $$0.675160\pi$$
$$390$$ 0 0
$$391$$ 19.3137 0.976736
$$392$$ −13.2426 −0.668854
$$393$$ 0 0
$$394$$ −26.1421 −1.31702
$$395$$ −4.00000 −0.201262
$$396$$ 0 0
$$397$$ −9.31371 −0.467442 −0.233721 0.972304i $$-0.575090\pi$$
−0.233721 + 0.972304i $$0.575090\pi$$
$$398$$ 24.9706 1.25166
$$399$$ 0 0
$$400$$ 3.00000 0.150000
$$401$$ 5.31371 0.265354 0.132677 0.991159i $$-0.457643\pi$$
0.132677 + 0.991159i $$0.457643\pi$$
$$402$$ 0 0
$$403$$ 0 0
$$404$$ 35.6569 1.77399
$$405$$ 0 0
$$406$$ −17.6569 −0.876295
$$407$$ 0 0
$$408$$ 0 0
$$409$$ −1.02944 −0.0509024 −0.0254512 0.999676i $$-0.508102\pi$$
−0.0254512 + 0.999676i $$0.508102\pi$$
$$410$$ 14.4853 0.715377
$$411$$ 0 0
$$412$$ 26.1421 1.28793
$$413$$ −3.31371 −0.163057
$$414$$ 0 0
$$415$$ −6.00000 −0.294528
$$416$$ −1.85786 −0.0910893
$$417$$ 0 0
$$418$$ 0 0
$$419$$ 25.6569 1.25342 0.626710 0.779253i $$-0.284401\pi$$
0.626710 + 0.779253i $$0.284401\pi$$
$$420$$ 0 0
$$421$$ −6.00000 −0.292422 −0.146211 0.989253i $$-0.546708\pi$$
−0.146211 + 0.989253i $$0.546708\pi$$
$$422$$ 38.6274 1.88035
$$423$$ 0 0
$$424$$ −51.4558 −2.49892
$$425$$ 6.82843 0.331227
$$426$$ 0 0
$$427$$ 18.6274 0.901444
$$428$$ 29.3137 1.41693
$$429$$ 0 0
$$430$$ 14.4853 0.698542
$$431$$ −11.3137 −0.544962 −0.272481 0.962161i $$-0.587844\pi$$
−0.272481 + 0.962161i $$0.587844\pi$$
$$432$$ 0 0
$$433$$ −7.65685 −0.367965 −0.183982 0.982930i $$-0.558899\pi$$
−0.183982 + 0.982930i $$0.558899\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 29.3137 1.40387
$$437$$ 0 0
$$438$$ 0 0
$$439$$ 16.0000 0.763638 0.381819 0.924237i $$-0.375298\pi$$
0.381819 + 0.924237i $$0.375298\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 19.3137 0.918659
$$443$$ 26.8284 1.27466 0.637329 0.770592i $$-0.280039\pi$$
0.637329 + 0.770592i $$0.280039\pi$$
$$444$$ 0 0
$$445$$ 13.3137 0.631130
$$446$$ −26.1421 −1.23787
$$447$$ 0 0
$$448$$ −19.6569 −0.928699
$$449$$ −28.6274 −1.35101 −0.675506 0.737355i $$-0.736075\pi$$
−0.675506 + 0.737355i $$0.736075\pi$$
$$450$$ 0 0
$$451$$ 0 0
$$452$$ −75.2548 −3.53969
$$453$$ 0 0
$$454$$ 61.1127 2.86816
$$455$$ 2.34315 0.109848
$$456$$ 0 0
$$457$$ −0.485281 −0.0227005 −0.0113503 0.999936i $$-0.503613\pi$$
−0.0113503 + 0.999936i $$0.503613\pi$$
$$458$$ 3.17157 0.148198
$$459$$ 0 0
$$460$$ 10.8284 0.504878
$$461$$ 12.6274 0.588117 0.294059 0.955787i $$-0.404994\pi$$
0.294059 + 0.955787i $$0.404994\pi$$
$$462$$ 0 0
$$463$$ −6.14214 −0.285449 −0.142725 0.989762i $$-0.545586\pi$$
−0.142725 + 0.989762i $$0.545586\pi$$
$$464$$ −10.9706 −0.509296
$$465$$ 0 0
$$466$$ −14.8284 −0.686914
$$467$$ 14.8284 0.686178 0.343089 0.939303i $$-0.388527\pi$$
0.343089 + 0.939303i $$0.388527\pi$$
$$468$$ 0 0
$$469$$ 24.9706 1.15303
$$470$$ −6.82843 −0.314972
$$471$$ 0 0
$$472$$ −7.31371 −0.336641
$$473$$ 0 0
$$474$$ 0 0
$$475$$ 0 0
$$476$$ 52.2843 2.39645
$$477$$ 0 0
$$478$$ −56.2843 −2.57438
$$479$$ −36.0000 −1.64488 −0.822441 0.568850i $$-0.807388\pi$$
−0.822441 + 0.568850i $$0.807388\pi$$
$$480$$ 0 0
$$481$$ −8.97056 −0.409022
$$482$$ −14.4853 −0.659786
$$483$$ 0 0
$$484$$ 0 0
$$485$$ 3.65685 0.166049
$$486$$ 0 0
$$487$$ −24.4853 −1.10953 −0.554767 0.832006i $$-0.687193\pi$$
−0.554767 + 0.832006i $$0.687193\pi$$
$$488$$ 41.1127 1.86108
$$489$$ 0 0
$$490$$ −7.24264 −0.327189
$$491$$ −0.686292 −0.0309719 −0.0154860 0.999880i $$-0.504930\pi$$
−0.0154860 + 0.999880i $$0.504930\pi$$
$$492$$ 0 0
$$493$$ −24.9706 −1.12462
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ −22.6274 −1.01498
$$498$$ 0 0
$$499$$ 9.65685 0.432300 0.216150 0.976360i $$-0.430650\pi$$
0.216150 + 0.976360i $$0.430650\pi$$
$$500$$ 3.82843 0.171212
$$501$$ 0 0
$$502$$ −28.9706 −1.29302
$$503$$ 16.6274 0.741380 0.370690 0.928757i $$-0.379121\pi$$
0.370690 + 0.928757i $$0.379121\pi$$
$$504$$ 0 0
$$505$$ 9.31371 0.414455
$$506$$ 0 0
$$507$$ 0 0
$$508$$ −16.6274 −0.737722
$$509$$ 13.3137 0.590120 0.295060 0.955479i $$-0.404660\pi$$
0.295060 + 0.955479i $$0.404660\pi$$
$$510$$ 0 0
$$511$$ 2.34315 0.103655
$$512$$ −31.2426 −1.38074
$$513$$ 0 0
$$514$$ 22.4853 0.991783
$$515$$ 6.82843 0.300896
$$516$$ 0 0
$$517$$ 0 0
$$518$$ −36.9706 −1.62439
$$519$$ 0 0
$$520$$ 5.17157 0.226788
$$521$$ −25.3137 −1.10901 −0.554507 0.832179i $$-0.687093\pi$$
−0.554507 + 0.832179i $$0.687093\pi$$
$$522$$ 0 0
$$523$$ 41.5980 1.81895 0.909476 0.415756i $$-0.136483\pi$$
0.909476 + 0.415756i $$0.136483\pi$$
$$524$$ −43.3137 −1.89217
$$525$$ 0 0
$$526$$ −26.4853 −1.15481
$$527$$ 0 0
$$528$$ 0 0
$$529$$ −15.0000 −0.652174
$$530$$ −28.1421 −1.22242
$$531$$ 0 0
$$532$$ 0 0
$$533$$ 7.02944 0.304479
$$534$$ 0 0
$$535$$ 7.65685 0.331035
$$536$$ 55.1127 2.38051
$$537$$ 0 0
$$538$$ −41.7990 −1.80208
$$539$$ 0 0
$$540$$ 0 0
$$541$$ −6.00000 −0.257960 −0.128980 0.991647i $$-0.541170\pi$$
−0.128980 + 0.991647i $$0.541170\pi$$
$$542$$ −17.6569 −0.758427
$$543$$ 0 0
$$544$$ −10.8284 −0.464265
$$545$$ 7.65685 0.327984
$$546$$ 0 0
$$547$$ 34.0000 1.45374 0.726868 0.686778i $$-0.240975\pi$$
0.726868 + 0.686778i $$0.240975\pi$$
$$548$$ 42.0000 1.79415
$$549$$ 0 0
$$550$$ 0 0
$$551$$ 0 0
$$552$$ 0 0
$$553$$ −8.00000 −0.340195
$$554$$ −16.4853 −0.700392
$$555$$ 0 0
$$556$$ 15.3137 0.649446
$$557$$ 9.85786 0.417691 0.208846 0.977949i $$-0.433029\pi$$
0.208846 + 0.977949i $$0.433029\pi$$
$$558$$ 0 0
$$559$$ 7.02944 0.297314
$$560$$ 6.00000 0.253546
$$561$$ 0 0
$$562$$ 41.7990 1.76318
$$563$$ 0.343146 0.0144619 0.00723093 0.999974i $$-0.497698\pi$$
0.00723093 + 0.999974i $$0.497698\pi$$
$$564$$ 0 0
$$565$$ −19.6569 −0.826970
$$566$$ −78.7696 −3.31093
$$567$$ 0 0
$$568$$ −49.9411 −2.09548
$$569$$ 31.6569 1.32712 0.663562 0.748121i $$-0.269044\pi$$
0.663562 + 0.748121i $$0.269044\pi$$
$$570$$ 0 0
$$571$$ 21.9411 0.918208 0.459104 0.888383i $$-0.348171\pi$$
0.459104 + 0.888383i $$0.348171\pi$$
$$572$$ 0 0
$$573$$ 0 0
$$574$$ 28.9706 1.20921
$$575$$ 2.82843 0.117954
$$576$$ 0 0
$$577$$ −26.9706 −1.12280 −0.561400 0.827545i $$-0.689737\pi$$
−0.561400 + 0.827545i $$0.689737\pi$$
$$578$$ 71.5269 2.97513
$$579$$ 0 0
$$580$$ −14.0000 −0.581318
$$581$$ −12.0000 −0.497844
$$582$$ 0 0
$$583$$ 0 0
$$584$$ 5.17157 0.214001
$$585$$ 0 0
$$586$$ −22.1421 −0.914683
$$587$$ 2.14214 0.0884154 0.0442077 0.999022i $$-0.485924\pi$$
0.0442077 + 0.999022i $$0.485924\pi$$
$$588$$ 0 0
$$589$$ 0 0
$$590$$ −4.00000 −0.164677
$$591$$ 0 0
$$592$$ −22.9706 −0.944084
$$593$$ 3.51472 0.144332 0.0721661 0.997393i $$-0.477009\pi$$
0.0721661 + 0.997393i $$0.477009\pi$$
$$594$$ 0 0
$$595$$ 13.6569 0.559876
$$596$$ 1.31371 0.0538116
$$597$$ 0 0
$$598$$ 8.00000 0.327144
$$599$$ 5.65685 0.231133 0.115566 0.993300i $$-0.463132\pi$$
0.115566 + 0.993300i $$0.463132\pi$$
$$600$$ 0 0
$$601$$ −23.9411 −0.976579 −0.488289 0.872682i $$-0.662379\pi$$
−0.488289 + 0.872682i $$0.662379\pi$$
$$602$$ 28.9706 1.18075
$$603$$ 0 0
$$604$$ 45.9411 1.86932
$$605$$ 0 0
$$606$$ 0 0
$$607$$ −38.2843 −1.55391 −0.776955 0.629556i $$-0.783237\pi$$
−0.776955 + 0.629556i $$0.783237\pi$$
$$608$$ 0 0
$$609$$ 0 0
$$610$$ 22.4853 0.910402
$$611$$ −3.31371 −0.134058
$$612$$ 0 0
$$613$$ 25.4558 1.02815 0.514076 0.857745i $$-0.328135\pi$$
0.514076 + 0.857745i $$0.328135\pi$$
$$614$$ 39.4558 1.59231
$$615$$ 0 0
$$616$$ 0 0
$$617$$ −0.343146 −0.0138145 −0.00690726 0.999976i $$-0.502199\pi$$
−0.00690726 + 0.999976i $$0.502199\pi$$
$$618$$ 0 0
$$619$$ −14.3431 −0.576500 −0.288250 0.957555i $$-0.593073\pi$$
−0.288250 + 0.957555i $$0.593073\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ −11.3137 −0.453638
$$623$$ 26.6274 1.06680
$$624$$ 0 0
$$625$$ 1.00000 0.0400000
$$626$$ −3.17157 −0.126762
$$627$$ 0 0
$$628$$ −53.5980 −2.13879
$$629$$ −52.2843 −2.08471
$$630$$ 0 0
$$631$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$632$$ −17.6569 −0.702352
$$633$$ 0 0
$$634$$ 3.17157 0.125959
$$635$$ −4.34315 −0.172352
$$636$$ 0 0
$$637$$ −3.51472 −0.139258
$$638$$ 0 0
$$639$$ 0 0
$$640$$ −20.5563 −0.812561
$$641$$ −30.0000 −1.18493 −0.592464 0.805597i $$-0.701845\pi$$
−0.592464 + 0.805597i $$0.701845\pi$$
$$642$$ 0 0
$$643$$ −1.45584 −0.0574129 −0.0287064 0.999588i $$-0.509139\pi$$
−0.0287064 + 0.999588i $$0.509139\pi$$
$$644$$ 21.6569 0.853400
$$645$$ 0 0
$$646$$ 0 0
$$647$$ −27.1127 −1.06591 −0.532955 0.846144i $$-0.678919\pi$$
−0.532955 + 0.846144i $$0.678919\pi$$
$$648$$ 0 0
$$649$$ 0 0
$$650$$ 2.82843 0.110940
$$651$$ 0 0
$$652$$ 63.1127 2.47168
$$653$$ −11.6569 −0.456168 −0.228084 0.973641i $$-0.573246\pi$$
−0.228084 + 0.973641i $$0.573246\pi$$
$$654$$ 0 0
$$655$$ −11.3137 −0.442063
$$656$$ 18.0000 0.702782
$$657$$ 0 0
$$658$$ −13.6569 −0.532400
$$659$$ 45.9411 1.78961 0.894806 0.446455i $$-0.147314\pi$$
0.894806 + 0.446455i $$0.147314\pi$$
$$660$$ 0 0
$$661$$ 44.6274 1.73581 0.867903 0.496734i $$-0.165468\pi$$
0.867903 + 0.496734i $$0.165468\pi$$
$$662$$ −17.6569 −0.686253
$$663$$ 0 0
$$664$$ −26.4853 −1.02783
$$665$$ 0 0
$$666$$ 0 0
$$667$$ −10.3431 −0.400488
$$668$$ −87.9411 −3.40254
$$669$$ 0 0
$$670$$ 30.1421 1.16449
$$671$$ 0 0
$$672$$ 0 0
$$673$$ 12.4853 0.481272 0.240636 0.970615i $$-0.422644\pi$$
0.240636 + 0.970615i $$0.422644\pi$$
$$674$$ 49.4558 1.90497
$$675$$ 0 0
$$676$$ −44.5147 −1.71210
$$677$$ 22.8284 0.877368 0.438684 0.898641i $$-0.355445\pi$$
0.438684 + 0.898641i $$0.355445\pi$$
$$678$$ 0 0
$$679$$ 7.31371 0.280674
$$680$$ 30.1421 1.15590
$$681$$ 0 0
$$682$$ 0 0
$$683$$ 7.79899 0.298420 0.149210 0.988806i $$-0.452327\pi$$
0.149210 + 0.988806i $$0.452327\pi$$
$$684$$ 0 0
$$685$$ 10.9706 0.419164
$$686$$ −48.2843 −1.84350
$$687$$ 0 0
$$688$$ 18.0000 0.686244
$$689$$ −13.6569 −0.520285
$$690$$ 0 0
$$691$$ −39.3137 −1.49556 −0.747782 0.663944i $$-0.768881\pi$$
−0.747782 + 0.663944i $$0.768881\pi$$
$$692$$ −84.7696 −3.22245
$$693$$ 0 0
$$694$$ 26.4853 1.00537
$$695$$ 4.00000 0.151729
$$696$$ 0 0
$$697$$ 40.9706 1.55187
$$698$$ −65.1127 −2.46455
$$699$$ 0 0
$$700$$ 7.65685 0.289402
$$701$$ −12.6274 −0.476931 −0.238465 0.971151i $$-0.576644\pi$$
−0.238465 + 0.971151i $$0.576644\pi$$
$$702$$ 0 0
$$703$$ 0 0
$$704$$ 0 0
$$705$$ 0 0
$$706$$ −51.4558 −1.93657
$$707$$ 18.6274 0.700556
$$708$$ 0 0
$$709$$ 24.6274 0.924902 0.462451 0.886645i $$-0.346970\pi$$
0.462451 + 0.886645i $$0.346970\pi$$
$$710$$ −27.3137 −1.02507
$$711$$ 0 0
$$712$$ 58.7696 2.20248
$$713$$ 0 0
$$714$$ 0 0
$$715$$ 0 0
$$716$$ −36.9706 −1.38165
$$717$$ 0 0
$$718$$ 1.65685 0.0618333
$$719$$ −18.3431 −0.684084 −0.342042 0.939685i $$-0.611118\pi$$
−0.342042 + 0.939685i $$0.611118\pi$$
$$720$$ 0 0
$$721$$ 13.6569 0.508608
$$722$$ −45.8701 −1.70711
$$723$$ 0 0
$$724$$ 81.5980 3.03257
$$725$$ −3.65685 −0.135812
$$726$$ 0 0
$$727$$ −19.5147 −0.723761 −0.361880 0.932225i $$-0.617865\pi$$
−0.361880 + 0.932225i $$0.617865\pi$$
$$728$$ 10.3431 0.383342
$$729$$ 0 0
$$730$$ 2.82843 0.104685
$$731$$ 40.9706 1.51535
$$732$$ 0 0
$$733$$ 17.4558 0.644746 0.322373 0.946613i $$-0.395519\pi$$
0.322373 + 0.946613i $$0.395519\pi$$
$$734$$ −20.4853 −0.756126
$$735$$ 0 0
$$736$$ −4.48528 −0.165330
$$737$$ 0 0
$$738$$ 0 0
$$739$$ −29.9411 −1.10140 −0.550701 0.834703i $$-0.685640\pi$$
−0.550701 + 0.834703i $$0.685640\pi$$
$$740$$ −29.3137 −1.07759
$$741$$ 0 0
$$742$$ −56.2843 −2.06626
$$743$$ −49.5980 −1.81957 −0.909787 0.415076i $$-0.863755\pi$$
−0.909787 + 0.415076i $$0.863755\pi$$
$$744$$ 0 0
$$745$$ 0.343146 0.0125719
$$746$$ −86.4264 −3.16430
$$747$$ 0 0
$$748$$ 0 0
$$749$$ 15.3137 0.559551
$$750$$ 0 0
$$751$$ −16.0000 −0.583848 −0.291924 0.956441i $$-0.594295\pi$$
−0.291924 + 0.956441i $$0.594295\pi$$
$$752$$ −8.48528 −0.309426
$$753$$ 0 0
$$754$$ −10.3431 −0.376675
$$755$$ 12.0000 0.436725
$$756$$ 0 0
$$757$$ 13.3137 0.483895 0.241947 0.970289i $$-0.422214\pi$$
0.241947 + 0.970289i $$0.422214\pi$$
$$758$$ 81.2548 2.95131
$$759$$ 0 0
$$760$$ 0 0
$$761$$ −30.0000 −1.08750 −0.543750 0.839248i $$-0.682996\pi$$
−0.543750 + 0.839248i $$0.682996\pi$$
$$762$$ 0 0
$$763$$ 15.3137 0.554393
$$764$$ −12.6863 −0.458974
$$765$$ 0 0
$$766$$ 14.1421 0.510976
$$767$$ −1.94113 −0.0700900
$$768$$ 0 0
$$769$$ 18.9706 0.684096 0.342048 0.939682i $$-0.388879\pi$$
0.342048 + 0.939682i $$0.388879\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ 4.48528 0.161429
$$773$$ −26.2843 −0.945380 −0.472690 0.881229i $$-0.656717\pi$$
−0.472690 + 0.881229i $$0.656717\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ 16.1421 0.579469
$$777$$ 0 0
$$778$$ −49.7990 −1.78538
$$779$$ 0 0
$$780$$ 0 0
$$781$$ 0 0
$$782$$ 46.6274 1.66739
$$783$$ 0 0
$$784$$ −9.00000 −0.321429
$$785$$ −14.0000 −0.499681
$$786$$ 0 0
$$787$$ 14.9706 0.533643 0.266821 0.963746i $$-0.414027\pi$$
0.266821 + 0.963746i $$0.414027\pi$$
$$788$$ −41.4558 −1.47680
$$789$$ 0 0
$$790$$ −9.65685 −0.343575
$$791$$ −39.3137 −1.39783
$$792$$ 0 0
$$793$$ 10.9117 0.387485
$$794$$ −22.4853 −0.797973
$$795$$ 0 0
$$796$$ 39.5980 1.40351
$$797$$ −32.6274 −1.15572 −0.577861 0.816135i $$-0.696113\pi$$
−0.577861 + 0.816135i $$0.696113\pi$$
$$798$$ 0 0
$$799$$ −19.3137 −0.683270
$$800$$ −1.58579 −0.0560660
$$801$$ 0 0
$$802$$ 12.8284 0.452988
$$803$$ 0 0
$$804$$ 0 0
$$805$$ 5.65685 0.199378
$$806$$ 0 0
$$807$$ 0 0
$$808$$ 41.1127 1.44634
$$809$$ −10.9706 −0.385704 −0.192852 0.981228i $$-0.561774\pi$$
−0.192852 + 0.981228i $$0.561774\pi$$
$$810$$ 0 0
$$811$$ −53.9411 −1.89413 −0.947065 0.321043i $$-0.895967\pi$$
−0.947065 + 0.321043i $$0.895967\pi$$
$$812$$ −28.0000 −0.982607
$$813$$ 0 0
$$814$$ 0 0
$$815$$ 16.4853 0.577454
$$816$$ 0 0
$$817$$ 0 0
$$818$$ −2.48528 −0.0868958
$$819$$ 0 0
$$820$$ 22.9706 0.802167
$$821$$ −41.3137 −1.44186 −0.720929 0.693009i $$-0.756285\pi$$
−0.720929 + 0.693009i $$0.756285\pi$$
$$822$$ 0 0
$$823$$ 19.5147 0.680240 0.340120 0.940382i $$-0.389532\pi$$
0.340120 + 0.940382i $$0.389532\pi$$
$$824$$ 30.1421 1.05005
$$825$$ 0 0
$$826$$ −8.00000 −0.278356
$$827$$ 22.2843 0.774900 0.387450 0.921891i $$-0.373356\pi$$
0.387450 + 0.921891i $$0.373356\pi$$
$$828$$ 0 0
$$829$$ 18.0000 0.625166 0.312583 0.949890i $$-0.398806\pi$$
0.312583 + 0.949890i $$0.398806\pi$$
$$830$$ −14.4853 −0.502791
$$831$$ 0 0
$$832$$ −11.5147 −0.399201
$$833$$ −20.4853 −0.709773
$$834$$ 0 0
$$835$$ −22.9706 −0.794929
$$836$$ 0 0
$$837$$ 0 0
$$838$$ 61.9411 2.13972
$$839$$ −26.3431 −0.909466 −0.454733 0.890628i $$-0.650265\pi$$
−0.454733 + 0.890628i $$0.650265\pi$$
$$840$$ 0 0
$$841$$ −15.6274 −0.538876
$$842$$ −14.4853 −0.499196
$$843$$ 0 0
$$844$$ 61.2548 2.10848
$$845$$ −11.6274 −0.399995
$$846$$ 0 0
$$847$$ 0 0
$$848$$ −34.9706 −1.20089
$$849$$ 0 0
$$850$$ 16.4853 0.565440
$$851$$ −21.6569 −0.742387
$$852$$ 0 0
$$853$$ 15.5147 0.531214 0.265607 0.964081i $$-0.414428\pi$$
0.265607 + 0.964081i $$0.414428\pi$$
$$854$$ 44.9706 1.53886
$$855$$ 0 0
$$856$$ 33.7990 1.15523
$$857$$ 24.7696 0.846112 0.423056 0.906104i $$-0.360957\pi$$
0.423056 + 0.906104i $$0.360957\pi$$
$$858$$ 0 0
$$859$$ 24.2843 0.828569 0.414284 0.910148i $$-0.364032\pi$$
0.414284 + 0.910148i $$0.364032\pi$$
$$860$$ 22.9706 0.783290
$$861$$ 0 0
$$862$$ −27.3137 −0.930309
$$863$$ 9.17157 0.312204 0.156102 0.987741i $$-0.450107\pi$$
0.156102 + 0.987741i $$0.450107\pi$$
$$864$$ 0 0
$$865$$ −22.1421 −0.752855
$$866$$ −18.4853 −0.628155
$$867$$ 0 0
$$868$$ 0 0
$$869$$ 0 0
$$870$$ 0 0
$$871$$ 14.6274 0.495631
$$872$$ 33.7990 1.14458
$$873$$ 0 0
$$874$$ 0 0
$$875$$ 2.00000 0.0676123
$$876$$ 0 0
$$877$$ 49.4558 1.67001 0.835003 0.550246i $$-0.185466\pi$$
0.835003 + 0.550246i $$0.185466\pi$$
$$878$$ 38.6274 1.30361
$$879$$ 0 0
$$880$$ 0 0
$$881$$ 7.37258 0.248389 0.124194 0.992258i $$-0.460365\pi$$
0.124194 + 0.992258i $$0.460365\pi$$
$$882$$ 0 0
$$883$$ 37.1716 1.25092 0.625462 0.780255i $$-0.284911\pi$$
0.625462 + 0.780255i $$0.284911\pi$$
$$884$$ 30.6274 1.03011
$$885$$ 0 0
$$886$$ 64.7696 2.17598
$$887$$ 38.2843 1.28546 0.642730 0.766093i $$-0.277802\pi$$
0.642730 + 0.766093i $$0.277802\pi$$
$$888$$ 0 0
$$889$$ −8.68629 −0.291329
$$890$$ 32.1421 1.07741
$$891$$ 0 0
$$892$$ −41.4558 −1.38804
$$893$$ 0 0
$$894$$ 0 0
$$895$$ −9.65685 −0.322793
$$896$$ −41.1127 −1.37348
$$897$$ 0 0
$$898$$ −69.1127 −2.30632
$$899$$ 0 0
$$900$$ 0 0
$$901$$ −79.5980 −2.65179
$$902$$ 0 0
$$903$$ 0 0
$$904$$ −86.7696 −2.88591
$$905$$ 21.3137 0.708492
$$906$$ 0 0
$$907$$ −27.5147 −0.913611 −0.456806 0.889567i $$-0.651007\pi$$
−0.456806 + 0.889567i $$0.651007\pi$$
$$908$$ 96.9117 3.21613
$$909$$ 0 0
$$910$$ 5.65685 0.187523
$$911$$ 9.94113 0.329364 0.164682 0.986347i $$-0.447340\pi$$
0.164682 + 0.986347i $$0.447340\pi$$
$$912$$ 0 0
$$913$$ 0 0
$$914$$ −1.17157 −0.0387522
$$915$$ 0 0
$$916$$ 5.02944 0.166177
$$917$$ −22.6274 −0.747223
$$918$$ 0 0
$$919$$ 32.0000 1.05558 0.527791 0.849374i $$-0.323020\pi$$
0.527791 + 0.849374i $$0.323020\pi$$
$$920$$ 12.4853 0.411628
$$921$$ 0 0
$$922$$ 30.4853 1.00398
$$923$$ −13.2548 −0.436288
$$924$$ 0 0
$$925$$ −7.65685 −0.251756
$$926$$ −14.8284 −0.487292
$$927$$ 0 0
$$928$$ 5.79899 0.190361
$$929$$ −5.31371 −0.174337 −0.0871686 0.996194i $$-0.527782\pi$$
−0.0871686 + 0.996194i $$0.527782\pi$$
$$930$$ 0 0
$$931$$ 0 0
$$932$$ −23.5147 −0.770250
$$933$$ 0 0
$$934$$ 35.7990 1.17138
$$935$$ 0 0
$$936$$ 0 0
$$937$$ 1.45584 0.0475604 0.0237802 0.999717i $$-0.492430\pi$$
0.0237802 + 0.999717i $$0.492430\pi$$
$$938$$ 60.2843 1.96835
$$939$$ 0 0
$$940$$ −10.8284 −0.353184
$$941$$ −6.68629 −0.217967 −0.108983 0.994044i $$-0.534760\pi$$
−0.108983 + 0.994044i $$0.534760\pi$$
$$942$$ 0 0
$$943$$ 16.9706 0.552638
$$944$$ −4.97056 −0.161778
$$945$$ 0 0
$$946$$ 0 0
$$947$$ −41.1716 −1.33790 −0.668948 0.743309i $$-0.733255\pi$$
−0.668948 + 0.743309i $$0.733255\pi$$
$$948$$ 0 0
$$949$$ 1.37258 0.0445559
$$950$$ 0 0
$$951$$ 0 0
$$952$$ 60.2843 1.95382
$$953$$ 53.1716 1.72240 0.861198 0.508269i $$-0.169715\pi$$
0.861198 + 0.508269i $$0.169715\pi$$
$$954$$ 0 0
$$955$$ −3.31371 −0.107229
$$956$$ −89.2548 −2.88671
$$957$$ 0 0
$$958$$ −86.9117 −2.80799
$$959$$ 21.9411 0.708516
$$960$$ 0 0
$$961$$ −31.0000 −1.00000
$$962$$ −21.6569 −0.698245
$$963$$ 0 0
$$964$$ −22.9706 −0.739832
$$965$$ 1.17157 0.0377143
$$966$$ 0 0
$$967$$ 14.9706 0.481421 0.240710 0.970597i $$-0.422620\pi$$
0.240710 + 0.970597i $$0.422620\pi$$
$$968$$ 0 0
$$969$$ 0 0
$$970$$ 8.82843 0.283464
$$971$$ 8.68629 0.278756 0.139378 0.990239i $$-0.455490\pi$$
0.139378 + 0.990239i $$0.455490\pi$$
$$972$$ 0 0
$$973$$ 8.00000 0.256468
$$974$$ −59.1127 −1.89409
$$975$$ 0 0
$$976$$ 27.9411 0.894374
$$977$$ −32.3431 −1.03475 −0.517374 0.855759i $$-0.673091\pi$$
−0.517374 + 0.855759i $$0.673091\pi$$
$$978$$ 0 0
$$979$$ 0 0
$$980$$ −11.4853 −0.366884
$$981$$ 0 0
$$982$$ −1.65685 −0.0528723
$$983$$ 21.8579 0.697158 0.348579 0.937279i $$-0.386664\pi$$
0.348579 + 0.937279i $$0.386664\pi$$
$$984$$ 0 0
$$985$$ −10.8284 −0.345022
$$986$$ −60.2843 −1.91984
$$987$$ 0 0
$$988$$ 0 0
$$989$$ 16.9706 0.539633
$$990$$ 0 0
$$991$$ −57.9411 −1.84056 −0.920280 0.391260i $$-0.872039\pi$$
−0.920280 + 0.391260i $$0.872039\pi$$
$$992$$ 0 0
$$993$$ 0 0
$$994$$ −54.6274 −1.73268
$$995$$ 10.3431 0.327900
$$996$$ 0 0
$$997$$ 41.4558 1.31292 0.656460 0.754361i $$-0.272053\pi$$
0.656460 + 0.754361i $$0.272053\pi$$
$$998$$ 23.3137 0.737983
$$999$$ 0 0
Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000

## Twists

By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 5445.2.a.y.1.2 2
3.2 odd 2 605.2.a.d.1.1 2
11.10 odd 2 495.2.a.b.1.1 2
12.11 even 2 9680.2.a.bn.1.2 2
15.14 odd 2 3025.2.a.o.1.2 2
33.2 even 10 605.2.g.f.81.1 8
33.5 odd 10 605.2.g.l.366.2 8
33.8 even 10 605.2.g.f.251.2 8
33.14 odd 10 605.2.g.l.251.1 8
33.17 even 10 605.2.g.f.366.1 8
33.20 odd 10 605.2.g.l.81.2 8
33.26 odd 10 605.2.g.l.511.1 8
33.29 even 10 605.2.g.f.511.2 8
33.32 even 2 55.2.a.b.1.2 2
44.43 even 2 7920.2.a.ch.1.2 2
55.32 even 4 2475.2.c.l.199.1 4
55.43 even 4 2475.2.c.l.199.4 4
55.54 odd 2 2475.2.a.x.1.2 2
132.131 odd 2 880.2.a.m.1.2 2
165.32 odd 4 275.2.b.d.199.4 4
165.98 odd 4 275.2.b.d.199.1 4
165.164 even 2 275.2.a.c.1.1 2
231.230 odd 2 2695.2.a.f.1.2 2
264.131 odd 2 3520.2.a.bo.1.1 2
264.197 even 2 3520.2.a.bn.1.2 2
429.428 even 2 9295.2.a.g.1.1 2
660.263 even 4 4400.2.b.q.4049.3 4
660.527 even 4 4400.2.b.q.4049.2 4
660.659 odd 2 4400.2.a.bn.1.1 2

By twisted newform
Twist Min Dim Char Parity Ord Type
55.2.a.b.1.2 2 33.32 even 2
275.2.a.c.1.1 2 165.164 even 2
275.2.b.d.199.1 4 165.98 odd 4
275.2.b.d.199.4 4 165.32 odd 4
495.2.a.b.1.1 2 11.10 odd 2
605.2.a.d.1.1 2 3.2 odd 2
605.2.g.f.81.1 8 33.2 even 10
605.2.g.f.251.2 8 33.8 even 10
605.2.g.f.366.1 8 33.17 even 10
605.2.g.f.511.2 8 33.29 even 10
605.2.g.l.81.2 8 33.20 odd 10
605.2.g.l.251.1 8 33.14 odd 10
605.2.g.l.366.2 8 33.5 odd 10
605.2.g.l.511.1 8 33.26 odd 10
880.2.a.m.1.2 2 132.131 odd 2
2475.2.a.x.1.2 2 55.54 odd 2
2475.2.c.l.199.1 4 55.32 even 4
2475.2.c.l.199.4 4 55.43 even 4
2695.2.a.f.1.2 2 231.230 odd 2
3025.2.a.o.1.2 2 15.14 odd 2
3520.2.a.bn.1.2 2 264.197 even 2
3520.2.a.bo.1.1 2 264.131 odd 2
4400.2.a.bn.1.1 2 660.659 odd 2
4400.2.b.q.4049.2 4 660.527 even 4
4400.2.b.q.4049.3 4 660.263 even 4
5445.2.a.y.1.2 2 1.1 even 1 trivial
7920.2.a.ch.1.2 2 44.43 even 2
9295.2.a.g.1.1 2 429.428 even 2
9680.2.a.bn.1.2 2 12.11 even 2