# Properties

 Label 4400.2.a.bn.1.1 Level $4400$ Weight $2$ Character 4400.1 Self dual yes Analytic conductor $35.134$ Analytic rank $0$ Dimension $2$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$4400 = 2^{4} \cdot 5^{2} \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 4400.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$35.1341768894$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\zeta_{8})^+$$ Defining polynomial: $$x^{2} - 2$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 55) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Root $$-1.41421$$ of defining polynomial Character $$\chi$$ $$=$$ 4400.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-2.82843 q^{3} -2.00000 q^{7} +5.00000 q^{9} +O(q^{10})$$ $$q-2.82843 q^{3} -2.00000 q^{7} +5.00000 q^{9} -1.00000 q^{11} +1.17157 q^{13} -6.82843 q^{17} +5.65685 q^{21} -2.82843 q^{23} -5.65685 q^{27} -3.65685 q^{29} +2.82843 q^{33} +7.65685 q^{37} -3.31371 q^{39} +6.00000 q^{41} -6.00000 q^{43} +2.82843 q^{47} -3.00000 q^{49} +19.3137 q^{51} -11.6569 q^{53} -1.65685 q^{59} -9.31371 q^{61} -10.0000 q^{63} +12.4853 q^{67} +8.00000 q^{69} -11.3137 q^{71} +1.17157 q^{73} +2.00000 q^{77} -4.00000 q^{79} +1.00000 q^{81} -6.00000 q^{83} +10.3431 q^{87} -13.3137 q^{89} -2.34315 q^{91} -3.65685 q^{97} -5.00000 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 4 q^{7} + 10 q^{9} + O(q^{10})$$ $$2 q - 4 q^{7} + 10 q^{9} - 2 q^{11} + 8 q^{13} - 8 q^{17} + 4 q^{29} + 4 q^{37} + 16 q^{39} + 12 q^{41} - 12 q^{43} - 6 q^{49} + 16 q^{51} - 12 q^{53} + 8 q^{59} + 4 q^{61} - 20 q^{63} + 8 q^{67} + 16 q^{69} + 8 q^{73} + 4 q^{77} - 8 q^{79} + 2 q^{81} - 12 q^{83} + 32 q^{87} - 4 q^{89} - 16 q^{91} + 4 q^{97} - 10 q^{99} + O(q^{100})$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 0 0
$$3$$ −2.82843 −1.63299 −0.816497 0.577350i $$-0.804087\pi$$
−0.816497 + 0.577350i $$0.804087\pi$$
$$4$$ 0 0
$$5$$ 0 0
$$6$$ 0 0
$$7$$ −2.00000 −0.755929 −0.377964 0.925820i $$-0.623376\pi$$
−0.377964 + 0.925820i $$0.623376\pi$$
$$8$$ 0 0
$$9$$ 5.00000 1.66667
$$10$$ 0 0
$$11$$ −1.00000 −0.301511
$$12$$ 0 0
$$13$$ 1.17157 0.324936 0.162468 0.986714i $$-0.448055\pi$$
0.162468 + 0.986714i $$0.448055\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 0 0
$$17$$ −6.82843 −1.65614 −0.828068 0.560627i $$-0.810560\pi$$
−0.828068 + 0.560627i $$0.810560\pi$$
$$18$$ 0 0
$$19$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$20$$ 0 0
$$21$$ 5.65685 1.23443
$$22$$ 0 0
$$23$$ −2.82843 −0.589768 −0.294884 0.955533i $$-0.595281\pi$$
−0.294884 + 0.955533i $$0.595281\pi$$
$$24$$ 0 0
$$25$$ 0 0
$$26$$ 0 0
$$27$$ −5.65685 −1.08866
$$28$$ 0 0
$$29$$ −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 $$0.5\pi$$
$$32$$ 0 0
$$33$$ 2.82843 0.492366
$$34$$ 0 0
$$35$$ 0 0
$$36$$ 0 0
$$37$$ 7.65685 1.25878 0.629390 0.777090i $$-0.283305\pi$$
0.629390 + 0.777090i $$0.283305\pi$$
$$38$$ 0 0
$$39$$ −3.31371 −0.530618
$$40$$ 0 0
$$41$$ 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 $$-0.651253\pi$$
−0.457496 + 0.889212i $$0.651253\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 0 0
$$47$$ 2.82843 0.412568 0.206284 0.978492i $$-0.433863\pi$$
0.206284 + 0.978492i $$0.433863\pi$$
$$48$$ 0 0
$$49$$ −3.00000 −0.428571
$$50$$ 0 0
$$51$$ 19.3137 2.70446
$$52$$ 0 0
$$53$$ −11.6569 −1.60119 −0.800596 0.599204i $$-0.795484\pi$$
−0.800596 + 0.599204i $$0.795484\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 0 0
$$57$$ 0 0
$$58$$ 0 0
$$59$$ −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 $$-0.703343\pi$$
−0.596249 + 0.802799i $$0.703343\pi$$
$$62$$ 0 0
$$63$$ −10.0000 −1.25988
$$64$$ 0 0
$$65$$ 0 0
$$66$$ 0 0
$$67$$ 12.4853 1.52532 0.762660 0.646800i $$-0.223893\pi$$
0.762660 + 0.646800i $$0.223893\pi$$
$$68$$ 0 0
$$69$$ 8.00000 0.963087
$$70$$ 0 0
$$71$$ −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 $$0.478159\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 0 0
$$77$$ 2.00000 0.227921
$$78$$ 0 0
$$79$$ −4.00000 −0.450035 −0.225018 0.974355i $$-0.572244\pi$$
−0.225018 + 0.974355i $$0.572244\pi$$
$$80$$ 0 0
$$81$$ 1.00000 0.111111
$$82$$ 0 0
$$83$$ −6.00000 −0.658586 −0.329293 0.944228i $$-0.606810\pi$$
−0.329293 + 0.944228i $$0.606810\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ 0 0
$$87$$ 10.3431 1.10890
$$88$$ 0 0
$$89$$ −13.3137 −1.41125 −0.705625 0.708585i $$-0.749334\pi$$
−0.705625 + 0.708585i $$0.749334\pi$$
$$90$$ 0 0
$$91$$ −2.34315 −0.245628
$$92$$ 0 0
$$93$$ 0 0
$$94$$ 0 0
$$95$$ 0 0
$$96$$ 0 0
$$97$$ −3.65685 −0.371297 −0.185649 0.982616i $$-0.559439\pi$$
−0.185649 + 0.982616i $$0.559439\pi$$
$$98$$ 0 0
$$99$$ −5.00000 −0.502519
$$100$$ 0 0
$$101$$ 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 $$0.390786\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ 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 $$-0.619511\pi$$
−0.366697 + 0.930341i $$0.619511\pi$$
$$110$$ 0 0
$$111$$ −21.6569 −2.05558
$$112$$ 0 0
$$113$$ −19.6569 −1.84916 −0.924581 0.380986i $$-0.875584\pi$$
−0.924581 + 0.380986i $$0.875584\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ 0 0
$$117$$ 5.85786 0.541560
$$118$$ 0 0
$$119$$ 13.6569 1.25192
$$120$$ 0 0
$$121$$ 1.00000 0.0909091
$$122$$ 0 0
$$123$$ −16.9706 −1.53018
$$124$$ 0 0
$$125$$ 0 0
$$126$$ 0 0
$$127$$ 4.34315 0.385392 0.192696 0.981259i $$-0.438277\pi$$
0.192696 + 0.981259i $$0.438277\pi$$
$$128$$ 0 0
$$129$$ 16.9706 1.49417
$$130$$ 0 0
$$131$$ 11.3137 0.988483 0.494242 0.869325i $$-0.335446\pi$$
0.494242 + 0.869325i $$0.335446\pi$$
$$132$$ 0 0
$$133$$ 0 0
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ 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 $$0.445740\pi$$
$$140$$ 0 0
$$141$$ −8.00000 −0.673722
$$142$$ 0 0
$$143$$ −1.17157 −0.0979718
$$144$$ 0 0
$$145$$ 0 0
$$146$$ 0 0
$$147$$ 8.48528 0.699854
$$148$$ 0 0
$$149$$ 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
$$153$$ −34.1421 −2.76023
$$154$$ 0 0
$$155$$ 0 0
$$156$$ 0 0
$$157$$ 14.0000 1.11732 0.558661 0.829396i $$-0.311315\pi$$
0.558661 + 0.829396i $$0.311315\pi$$
$$158$$ 0 0
$$159$$ 32.9706 2.61474
$$160$$ 0 0
$$161$$ 5.65685 0.445823
$$162$$ 0 0
$$163$$ 16.4853 1.29123 0.645613 0.763664i $$-0.276602\pi$$
0.645613 + 0.763664i $$0.276602\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ −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
$$170$$ 0 0
$$171$$ 0 0
$$172$$ 0 0
$$173$$ 22.1421 1.68344 0.841718 0.539918i $$-0.181545\pi$$
0.841718 + 0.539918i $$0.181545\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 0 0
$$177$$ 4.68629 0.352243
$$178$$ 0 0
$$179$$ −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 $$0.208979\pi$$
$$182$$ 0 0
$$183$$ 26.3431 1.94734
$$184$$ 0 0
$$185$$ 0 0
$$186$$ 0 0
$$187$$ 6.82843 0.499344
$$188$$ 0 0
$$189$$ 11.3137 0.822951
$$190$$ 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 $$0.486574\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ 10.8284 0.771493 0.385747 0.922605i $$-0.373944\pi$$
0.385747 + 0.922605i $$0.373944\pi$$
$$198$$ 0 0
$$199$$ −10.3431 −0.733206 −0.366603 0.930377i $$-0.619479\pi$$
−0.366603 + 0.930377i $$0.619479\pi$$
$$200$$ 0 0
$$201$$ −35.3137 −2.49084
$$202$$ 0 0
$$203$$ 7.31371 0.513322
$$204$$ 0 0
$$205$$ 0 0
$$206$$ 0 0
$$207$$ −14.1421 −0.982946
$$208$$ 0 0
$$209$$ 0 0
$$210$$ 0 0
$$211$$ 16.0000 1.10149 0.550743 0.834675i $$-0.314345\pi$$
0.550743 + 0.834675i $$0.314345\pi$$
$$212$$ 0 0
$$213$$ 32.0000 2.19260
$$214$$ 0 0
$$215$$ 0 0
$$216$$ 0 0
$$217$$ 0 0
$$218$$ 0 0
$$219$$ −3.31371 −0.223920
$$220$$ 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 $$0.618098\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 0 0
$$227$$ 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 $$0.486179\pi$$
$$230$$ 0 0
$$231$$ −5.65685 −0.372194
$$232$$ 0 0
$$233$$ 6.14214 0.402385 0.201192 0.979552i $$-0.435518\pi$$
0.201192 + 0.979552i $$0.435518\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ 0 0
$$237$$ 11.3137 0.734904
$$238$$ 0 0
$$239$$ 23.3137 1.50804 0.754019 0.656852i $$-0.228113\pi$$
0.754019 + 0.656852i $$0.228113\pi$$
$$240$$ 0 0
$$241$$ 6.00000 0.386494 0.193247 0.981150i $$-0.438098\pi$$
0.193247 + 0.981150i $$0.438098\pi$$
$$242$$ 0 0
$$243$$ 14.1421 0.907218
$$244$$ 0 0
$$245$$ 0 0
$$246$$ 0 0
$$247$$ 0 0
$$248$$ 0 0
$$249$$ 16.9706 1.07547
$$250$$ 0 0
$$251$$ −12.0000 −0.757433 −0.378717 0.925513i $$-0.623635\pi$$
−0.378717 + 0.925513i $$0.623635\pi$$
$$252$$ 0 0
$$253$$ 2.82843 0.177822
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 0 0
$$257$$ 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
$$260$$ 0 0
$$261$$ −18.2843 −1.13177
$$262$$ 0 0
$$263$$ −10.9706 −0.676474 −0.338237 0.941061i $$-0.609831\pi$$
−0.338237 + 0.941061i $$0.609831\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ 0 0
$$267$$ 37.6569 2.30456
$$268$$ 0 0
$$269$$ 17.3137 1.05564 0.527818 0.849358i $$-0.323010\pi$$
0.527818 + 0.849358i $$0.323010\pi$$
$$270$$ 0 0
$$271$$ −7.31371 −0.444276 −0.222138 0.975015i $$-0.571304\pi$$
−0.222138 + 0.975015i $$0.571304\pi$$
$$272$$ 0 0
$$273$$ 6.62742 0.401110
$$274$$ 0 0
$$275$$ 0 0
$$276$$ 0 0
$$277$$ −6.82843 −0.410280 −0.205140 0.978733i $$-0.565765\pi$$
−0.205140 + 0.978733i $$0.565765\pi$$
$$278$$ 0 0
$$279$$ 0 0
$$280$$ 0 0
$$281$$ 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 $$-0.0784935\pi$$
0.969749 + 0.244103i $$0.0784935\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ −12.0000 −0.708338
$$288$$ 0 0
$$289$$ 29.6274 1.74279
$$290$$ 0 0
$$291$$ 10.3431 0.606326
$$292$$ 0 0
$$293$$ 9.17157 0.535809 0.267905 0.963445i $$-0.413669\pi$$
0.267905 + 0.963445i $$0.413669\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ 0 0
$$297$$ 5.65685 0.328244
$$298$$ 0 0
$$299$$ −3.31371 −0.191637
$$300$$ 0 0
$$301$$ 12.0000 0.691669
$$302$$ 0 0
$$303$$ −26.3431 −1.51337
$$304$$ 0 0
$$305$$ 0 0
$$306$$ 0 0
$$307$$ −16.3431 −0.932753 −0.466376 0.884586i $$-0.654441\pi$$
−0.466376 + 0.884586i $$0.654441\pi$$
$$308$$ 0 0
$$309$$ −19.3137 −1.09872
$$310$$ 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 $$-0.488179\pi$$
0.0371276 + 0.999311i $$0.488179\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ 1.31371 0.0737852 0.0368926 0.999319i $$-0.488254\pi$$
0.0368926 + 0.999319i $$0.488254\pi$$
$$318$$ 0 0
$$319$$ 3.65685 0.204745
$$320$$ 0 0
$$321$$ −21.6569 −1.20877
$$322$$ 0 0
$$323$$ 0 0
$$324$$ 0 0
$$325$$ 0 0
$$326$$ 0 0
$$327$$ 21.6569 1.19763
$$328$$ 0 0
$$329$$ −5.65685 −0.311872
$$330$$ 0 0
$$331$$ 7.31371 0.401998 0.200999 0.979591i $$-0.435581\pi$$
0.200999 + 0.979591i $$0.435581\pi$$
$$332$$ 0 0
$$333$$ 38.2843 2.09797
$$334$$ 0 0
$$335$$ 0 0
$$336$$ 0 0
$$337$$ 20.4853 1.11590 0.557952 0.829873i $$-0.311587\pi$$
0.557952 + 0.829873i $$0.311587\pi$$
$$338$$ 0 0
$$339$$ 55.5980 3.01967
$$340$$ 0 0
$$341$$ 0 0
$$342$$ 0 0
$$343$$ 20.0000 1.07990
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ 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 $$-0.243292\pi$$
0.721851 + 0.692049i $$0.243292\pi$$
$$350$$ 0 0
$$351$$ −6.62742 −0.353745
$$352$$ 0 0
$$353$$ −21.3137 −1.13441 −0.567207 0.823575i $$-0.691976\pi$$
−0.567207 + 0.823575i $$0.691976\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ 0 0
$$357$$ −38.6274 −2.04438
$$358$$ 0 0
$$359$$ −0.686292 −0.0362211 −0.0181105 0.999836i $$-0.505765\pi$$
−0.0181105 + 0.999836i $$0.505765\pi$$
$$360$$ 0 0
$$361$$ −19.0000 −1.00000
$$362$$ 0 0
$$363$$ −2.82843 −0.148454
$$364$$ 0 0
$$365$$ 0 0
$$366$$ 0 0
$$367$$ −8.48528 −0.442928 −0.221464 0.975169i $$-0.571084\pi$$
−0.221464 + 0.975169i $$0.571084\pi$$
$$368$$ 0 0
$$369$$ 30.0000 1.56174
$$370$$ 0 0
$$371$$ 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
$$376$$ 0 0
$$377$$ −4.28427 −0.220651
$$378$$ 0 0
$$379$$ −33.6569 −1.72884 −0.864418 0.502773i $$-0.832313\pi$$
−0.864418 + 0.502773i $$0.832313\pi$$
$$380$$ 0 0
$$381$$ −12.2843 −0.629342
$$382$$ 0 0
$$383$$ −5.85786 −0.299323 −0.149661 0.988737i $$-0.547818\pi$$
−0.149661 + 0.988737i $$0.547818\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ −30.0000 −1.52499
$$388$$ 0 0
$$389$$ 20.6274 1.04585 0.522926 0.852378i $$-0.324840\pi$$
0.522926 + 0.852378i $$0.324840\pi$$
$$390$$ 0 0
$$391$$ 19.3137 0.976736
$$392$$ 0 0
$$393$$ −32.0000 −1.61419
$$394$$ 0 0
$$395$$ 0 0
$$396$$ 0 0
$$397$$ 9.31371 0.467442 0.233721 0.972304i $$-0.424910\pi$$
0.233721 + 0.972304i $$0.424910\pi$$
$$398$$ 0 0
$$399$$ 0 0
$$400$$ 0 0
$$401$$ −5.31371 −0.265354 −0.132677 0.991159i $$-0.542357\pi$$
−0.132677 + 0.991159i $$0.542357\pi$$
$$402$$ 0 0
$$403$$ 0 0
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 0 0
$$407$$ −7.65685 −0.379536
$$408$$ 0 0
$$409$$ 1.02944 0.0509024 0.0254512 0.999676i $$-0.491898\pi$$
0.0254512 + 0.999676i $$0.491898\pi$$
$$410$$ 0 0
$$411$$ −31.0294 −1.53057
$$412$$ 0 0
$$413$$ 3.31371 0.163057
$$414$$ 0 0
$$415$$ 0 0
$$416$$ 0 0
$$417$$ −11.3137 −0.554035
$$418$$ 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 $$0.546708\pi$$
$$422$$ 0 0
$$423$$ 14.1421 0.687614
$$424$$ 0 0
$$425$$ 0 0
$$426$$ 0 0
$$427$$ 18.6274 0.901444
$$428$$ 0 0
$$429$$ 3.31371 0.159987
$$430$$ 0 0
$$431$$ 11.3137 0.544962 0.272481 0.962161i $$-0.412156\pi$$
0.272481 + 0.962161i $$0.412156\pi$$
$$432$$ 0 0
$$433$$ 7.65685 0.367965 0.183982 0.982930i $$-0.441101\pi$$
0.183982 + 0.982930i $$0.441101\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 0 0
$$437$$ 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
$$441$$ −15.0000 −0.714286
$$442$$ 0 0
$$443$$ −26.8284 −1.27466 −0.637329 0.770592i $$-0.719961\pi$$
−0.637329 + 0.770592i $$0.719961\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ 0 0
$$447$$ −0.970563 −0.0459060
$$448$$ 0 0
$$449$$ 28.6274 1.35101 0.675506 0.737355i $$-0.263925\pi$$
0.675506 + 0.737355i $$0.263925\pi$$
$$450$$ 0 0
$$451$$ −6.00000 −0.282529
$$452$$ 0 0
$$453$$ −33.9411 −1.59469
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ −0.485281 −0.0227005 −0.0113503 0.999936i $$-0.503613\pi$$
−0.0113503 + 0.999936i $$0.503613\pi$$
$$458$$ 0 0
$$459$$ 38.6274 1.80297
$$460$$ 0 0
$$461$$ 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 $$0.545586\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ −14.8284 −0.686178 −0.343089 0.939303i $$-0.611473\pi$$
−0.343089 + 0.939303i $$0.611473\pi$$
$$468$$ 0 0
$$469$$ −24.9706 −1.15303
$$470$$ 0 0
$$471$$ −39.5980 −1.82458
$$472$$ 0 0
$$473$$ 6.00000 0.275880
$$474$$ 0 0
$$475$$ 0 0
$$476$$ 0 0
$$477$$ −58.2843 −2.66865
$$478$$ 0 0
$$479$$ 36.0000 1.64488 0.822441 0.568850i $$-0.192612\pi$$
0.822441 + 0.568850i $$0.192612\pi$$
$$480$$ 0 0
$$481$$ 8.97056 0.409022
$$482$$ 0 0
$$483$$ −16.0000 −0.728025
$$484$$ 0 0
$$485$$ 0 0
$$486$$ 0 0
$$487$$ −24.4853 −1.10953 −0.554767 0.832006i $$-0.687193\pi$$
−0.554767 + 0.832006i $$0.687193\pi$$
$$488$$ 0 0
$$489$$ −46.6274 −2.10856
$$490$$ 0 0
$$491$$ 0.686292 0.0309719 0.0154860 0.999880i $$-0.495070\pi$$
0.0154860 + 0.999880i $$0.495070\pi$$
$$492$$ 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 $$-0.569350\pi$$
−0.216150 + 0.976360i $$0.569350\pi$$
$$500$$ 0 0
$$501$$ 64.9706 2.90267
$$502$$ 0 0
$$503$$ 16.6274 0.741380 0.370690 0.928757i $$-0.379121\pi$$
0.370690 + 0.928757i $$0.379121\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ 0 0
$$507$$ 32.8873 1.46058
$$508$$ 0 0
$$509$$ −13.3137 −0.590120 −0.295060 0.955479i $$-0.595340\pi$$
−0.295060 + 0.955479i $$0.595340\pi$$
$$510$$ 0 0
$$511$$ −2.34315 −0.103655
$$512$$ 0 0
$$513$$ 0 0
$$514$$ 0 0
$$515$$ 0 0
$$516$$ 0 0
$$517$$ −2.82843 −0.124394
$$518$$ 0 0
$$519$$ −62.6274 −2.74904
$$520$$ 0 0
$$521$$ 25.3137 1.10901 0.554507 0.832179i $$-0.312907\pi$$
0.554507 + 0.832179i $$0.312907\pi$$
$$522$$ 0 0
$$523$$ −41.5980 −1.81895 −0.909476 0.415756i $$-0.863517\pi$$
−0.909476 + 0.415756i $$0.863517\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ 0 0
$$528$$ 0 0
$$529$$ −15.0000 −0.652174
$$530$$ 0 0
$$531$$ −8.28427 −0.359507
$$532$$ 0 0
$$533$$ 7.02944 0.304479
$$534$$ 0 0
$$535$$ 0 0
$$536$$ 0 0
$$537$$ 27.3137 1.17867
$$538$$ 0 0
$$539$$ 3.00000 0.129219
$$540$$ 0 0
$$541$$ 6.00000 0.257960 0.128980 0.991647i $$-0.458830\pi$$
0.128980 + 0.991647i $$0.458830\pi$$
$$542$$ 0 0
$$543$$ −60.2843 −2.58705
$$544$$ 0 0
$$545$$ 0 0
$$546$$ 0 0
$$547$$ −34.0000 −1.45374 −0.726868 0.686778i $$-0.759025\pi$$
−0.726868 + 0.686778i $$0.759025\pi$$
$$548$$ 0 0
$$549$$ −46.5685 −1.98750
$$550$$ 0 0
$$551$$ 0 0
$$552$$ 0 0
$$553$$ 8.00000 0.340195
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 0 0
$$557$$ −9.85786 −0.417691 −0.208846 0.977949i $$-0.566971\pi$$
−0.208846 + 0.977949i $$0.566971\pi$$
$$558$$ 0 0
$$559$$ −7.02944 −0.297314
$$560$$ 0 0
$$561$$ −19.3137 −0.815425
$$562$$ 0 0
$$563$$ 0.343146 0.0144619 0.00723093 0.999974i $$-0.497698\pi$$
0.00723093 + 0.999974i $$0.497698\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ 0 0
$$567$$ −2.00000 −0.0839921
$$568$$ 0 0
$$569$$ 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
$$573$$ 9.37258 0.391545
$$574$$ 0 0
$$575$$ 0 0
$$576$$ 0 0
$$577$$ 26.9706 1.12280 0.561400 0.827545i $$-0.310263\pi$$
0.561400 + 0.827545i $$0.310263\pi$$
$$578$$ 0 0
$$579$$ −3.31371 −0.137713
$$580$$ 0 0
$$581$$ 12.0000 0.497844
$$582$$ 0 0
$$583$$ 11.6569 0.482778
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ −2.14214 −0.0884154 −0.0442077 0.999022i $$-0.514076\pi$$
−0.0442077 + 0.999022i $$0.514076\pi$$
$$588$$ 0 0
$$589$$ 0 0
$$590$$ 0 0
$$591$$ −30.6274 −1.25984
$$592$$ 0 0
$$593$$ −3.51472 −0.144332 −0.0721661 0.997393i $$-0.522991\pi$$
−0.0721661 + 0.997393i $$0.522991\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 0 0
$$597$$ 29.2548 1.19732
$$598$$ 0 0
$$599$$ 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 $$-0.337621\pi$$
0.488289 + 0.872682i $$0.337621\pi$$
$$602$$ 0 0
$$603$$ 62.4264 2.54220
$$604$$ 0 0
$$605$$ 0 0
$$606$$ 0 0
$$607$$ 38.2843 1.55391 0.776955 0.629556i $$-0.216763\pi$$
0.776955 + 0.629556i $$0.216763\pi$$
$$608$$ 0 0
$$609$$ −20.6863 −0.838251
$$610$$ 0 0
$$611$$ 3.31371 0.134058
$$612$$ 0 0
$$613$$ 25.4558 1.02815 0.514076 0.857745i $$-0.328135\pi$$
0.514076 + 0.857745i $$0.328135\pi$$
$$614$$ 0 0
$$615$$ 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 $$-0.406927\pi$$
0.288250 + 0.957555i $$0.406927\pi$$
$$620$$ 0 0
$$621$$ 16.0000 0.642058
$$622$$ 0 0
$$623$$ 26.6274 1.06680
$$624$$ 0 0
$$625$$ 0 0
$$626$$ 0 0
$$627$$ 0 0
$$628$$ 0 0
$$629$$ −52.2843 −2.08471
$$630$$ 0 0
$$631$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$632$$ 0 0
$$633$$ −45.2548 −1.79872
$$634$$ 0 0
$$635$$ 0 0
$$636$$ 0 0
$$637$$ −3.51472 −0.139258
$$638$$ 0 0
$$639$$ −56.5685 −2.23782
$$640$$ 0 0
$$641$$ 30.0000 1.18493 0.592464 0.805597i $$-0.298155\pi$$
0.592464 + 0.805597i $$0.298155\pi$$
$$642$$ 0 0
$$643$$ −1.45584 −0.0574129 −0.0287064 0.999588i $$-0.509139\pi$$
−0.0287064 + 0.999588i $$0.509139\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ 27.1127 1.06591 0.532955 0.846144i $$-0.321081\pi$$
0.532955 + 0.846144i $$0.321081\pi$$
$$648$$ 0 0
$$649$$ 1.65685 0.0650372
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 0 0
$$653$$ −11.6569 −0.456168 −0.228084 0.973641i $$-0.573246\pi$$
−0.228084 + 0.973641i $$0.573246\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ 0 0
$$657$$ 5.85786 0.228537
$$658$$ 0 0
$$659$$ −45.9411 −1.78961 −0.894806 0.446455i $$-0.852686\pi$$
−0.894806 + 0.446455i $$0.852686\pi$$
$$660$$ 0 0
$$661$$ 44.6274 1.73581 0.867903 0.496734i $$-0.165468\pi$$
0.867903 + 0.496734i $$0.165468\pi$$
$$662$$ 0 0
$$663$$ 22.6274 0.878776
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 0 0
$$667$$ 10.3431 0.400488
$$668$$ 0 0
$$669$$ 30.6274 1.18412
$$670$$ 0 0
$$671$$ 9.31371 0.359552
$$672$$ 0 0
$$673$$ 12.4853 0.481272 0.240636 0.970615i $$-0.422644\pi$$
0.240636 + 0.970615i $$0.422644\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ 0 0
$$677$$ −22.8284 −0.877368 −0.438684 0.898641i $$-0.644555\pi$$
−0.438684 + 0.898641i $$0.644555\pi$$
$$678$$ 0 0
$$679$$ 7.31371 0.280674
$$680$$ 0 0
$$681$$ −71.5980 −2.74364
$$682$$ 0 0
$$683$$ −7.79899 −0.298420 −0.149210 0.988806i $$-0.547673\pi$$
−0.149210 + 0.988806i $$0.547673\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ 0 0
$$687$$ −3.71573 −0.141764
$$688$$ 0 0
$$689$$ −13.6569 −0.520285
$$690$$ 0 0
$$691$$ 39.3137 1.49556 0.747782 0.663944i $$-0.231119\pi$$
0.747782 + 0.663944i $$0.231119\pi$$
$$692$$ 0 0
$$693$$ 10.0000 0.379869
$$694$$ 0 0
$$695$$ 0 0
$$696$$ 0 0
$$697$$ −40.9706 −1.55187
$$698$$ 0 0
$$699$$ −17.3726 −0.657091
$$700$$ 0 0
$$701$$ −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
$$706$$ 0 0
$$707$$ −18.6274 −0.700556
$$708$$ 0 0
$$709$$ 24.6274 0.924902 0.462451 0.886645i $$-0.346970\pi$$
0.462451 + 0.886645i $$0.346970\pi$$
$$710$$ 0 0
$$711$$ −20.0000 −0.750059
$$712$$ 0 0
$$713$$ 0 0
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 0 0
$$717$$ −65.9411 −2.46262
$$718$$ 0 0
$$719$$ −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
$$722$$ 0 0
$$723$$ −16.9706 −0.631142
$$724$$ 0 0
$$725$$ 0 0
$$726$$ 0 0
$$727$$ −19.5147 −0.723761 −0.361880 0.932225i $$-0.617865\pi$$
−0.361880 + 0.932225i $$0.617865\pi$$
$$728$$ 0 0
$$729$$ −43.0000 −1.59259
$$730$$ 0 0
$$731$$ 40.9706 1.51535
$$732$$ 0 0
$$733$$ 17.4558 0.644746 0.322373 0.946613i $$-0.395519\pi$$
0.322373 + 0.946613i $$0.395519\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ −12.4853 −0.459901
$$738$$ 0 0
$$739$$ −29.9411 −1.10140 −0.550701 0.834703i $$-0.685640\pi$$
−0.550701 + 0.834703i $$0.685640\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 0 0
$$743$$ −49.5980 −1.81957 −0.909787 0.415076i $$-0.863755\pi$$
−0.909787 + 0.415076i $$0.863755\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ 0 0
$$747$$ −30.0000 −1.09764
$$748$$ 0 0
$$749$$ −15.3137 −0.559551
$$750$$ 0 0
$$751$$ 16.0000 0.583848 0.291924 0.956441i $$-0.405705\pi$$
0.291924 + 0.956441i $$0.405705\pi$$
$$752$$ 0 0
$$753$$ 33.9411 1.23688
$$754$$ 0 0
$$755$$ 0 0
$$756$$ 0 0
$$757$$ −13.3137 −0.483895 −0.241947 0.970289i $$-0.577786\pi$$
−0.241947 + 0.970289i $$0.577786\pi$$
$$758$$ 0 0
$$759$$ −8.00000 −0.290382
$$760$$ 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
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ −1.94113 −0.0700900
$$768$$ 0 0
$$769$$ −18.9706 −0.684096 −0.342048 0.939682i $$-0.611121\pi$$
−0.342048 + 0.939682i $$0.611121\pi$$
$$770$$ 0 0
$$771$$ −26.3431 −0.948725
$$772$$ 0 0
$$773$$ −26.2843 −0.945380 −0.472690 0.881229i $$-0.656717\pi$$
−0.472690 + 0.881229i $$0.656717\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ 0 0
$$777$$ 43.3137 1.55387
$$778$$ 0 0
$$779$$ 0 0
$$780$$ 0 0
$$781$$ 11.3137 0.404836
$$782$$ 0 0
$$783$$ 20.6863 0.739268
$$784$$ 0 0
$$785$$ 0 0
$$786$$ 0 0
$$787$$ −14.9706 −0.533643 −0.266821 0.963746i $$-0.585973\pi$$
−0.266821 + 0.963746i $$0.585973\pi$$
$$788$$ 0 0
$$789$$ 31.0294 1.10468
$$790$$ 0 0
$$791$$ 39.3137 1.39783
$$792$$ 0 0
$$793$$ −10.9117 −0.387485
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 0 0
$$797$$ −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
$$800$$ 0 0
$$801$$ −66.5685 −2.35208
$$802$$ 0 0
$$803$$ −1.17157 −0.0413439
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 0 0
$$807$$ −48.9706 −1.72385
$$808$$ 0 0
$$809$$ −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 $$0.895967\pi$$
$$812$$ 0 0
$$813$$ 20.6863 0.725500
$$814$$ 0 0
$$815$$ 0 0
$$816$$ 0 0
$$817$$ 0 0
$$818$$ 0 0
$$819$$ −11.7157 −0.409381
$$820$$ 0 0
$$821$$ −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 $$0.389532\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ 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 $$0.398806\pi$$
$$830$$ 0 0
$$831$$ 19.3137 0.669985
$$832$$ 0 0
$$833$$ 20.4853 0.709773
$$834$$ 0 0
$$835$$ 0 0
$$836$$ 0 0
$$837$$ 0 0
$$838$$ 0 0
$$839$$ −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
$$842$$ 0 0
$$843$$ −48.9706 −1.68664
$$844$$ 0 0
$$845$$ 0 0
$$846$$ 0 0
$$847$$ −2.00000 −0.0687208
$$848$$ 0 0
$$849$$ −92.2843 −3.16719
$$850$$ 0 0
$$851$$ −21.6569 −0.742387
$$852$$ 0 0
$$853$$ 15.5147 0.531214 0.265607 0.964081i $$-0.414428\pi$$
0.265607 + 0.964081i $$0.414428\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0 0
$$857$$ −24.7696 −0.846112 −0.423056 0.906104i $$-0.639043\pi$$
−0.423056 + 0.906104i $$0.639043\pi$$
$$858$$ 0 0
$$859$$ −24.2843 −0.828569 −0.414284 0.910148i $$-0.635968\pi$$
−0.414284 + 0.910148i $$0.635968\pi$$
$$860$$ 0 0
$$861$$ 33.9411 1.15671
$$862$$ 0 0
$$863$$ −9.17157 −0.312204 −0.156102 0.987741i $$-0.549893\pi$$
−0.156102 + 0.987741i $$0.549893\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ 0 0
$$867$$ −83.7990 −2.84596
$$868$$ 0 0
$$869$$ 4.00000 0.135691
$$870$$ 0 0
$$871$$ 14.6274 0.495631
$$872$$ 0 0
$$873$$ −18.2843 −0.618829
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 0 0
$$877$$ 49.4558 1.67001 0.835003 0.550246i $$-0.185466\pi$$
0.835003 + 0.550246i $$0.185466\pi$$
$$878$$ 0 0
$$879$$ −25.9411 −0.874972
$$880$$ 0 0
$$881$$ −7.37258 −0.248389 −0.124194 0.992258i $$-0.539635\pi$$
−0.124194 + 0.992258i $$0.539635\pi$$
$$882$$ 0 0
$$883$$ 37.1716 1.25092 0.625462 0.780255i $$-0.284911\pi$$
0.625462 + 0.780255i $$0.284911\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 0 0
$$887$$ 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
$$890$$ 0 0
$$891$$ −1.00000 −0.0335013
$$892$$ 0 0
$$893$$ 0 0
$$894$$ 0 0
$$895$$ 0 0
$$896$$ 0 0
$$897$$ 9.37258 0.312941
$$898$$ 0 0
$$899$$ 0 0
$$900$$ 0 0
$$901$$ 79.5980 2.65179
$$902$$ 0 0
$$903$$ −33.9411 −1.12949
$$904$$ 0 0
$$905$$ 0 0
$$906$$ 0 0
$$907$$ −27.5147 −0.913611 −0.456806 0.889567i $$-0.651007\pi$$
−0.456806 + 0.889567i $$0.651007\pi$$
$$908$$ 0 0
$$909$$ 46.5685 1.54458
$$910$$ 0 0
$$911$$ 9.94113 0.329364 0.164682 0.986347i $$-0.447340\pi$$
0.164682 + 0.986347i $$0.447340\pi$$
$$912$$ 0 0
$$913$$ 6.00000 0.198571
$$914$$ 0 0
$$915$$ 0 0
$$916$$ 0 0
$$917$$ −22.6274 −0.747223
$$918$$ 0 0
$$919$$ 32.0000 1.05558 0.527791 0.849374i $$-0.323020\pi$$
0.527791 + 0.849374i $$0.323020\pi$$
$$920$$ 0 0
$$921$$ 46.2254 1.52318
$$922$$ 0 0
$$923$$ −13.2548 −0.436288
$$924$$ 0 0
$$925$$ 0 0
$$926$$ 0 0
$$927$$ 34.1421 1.12137
$$928$$ 0 0
$$929$$ 5.31371 0.174337 0.0871686 0.996194i $$-0.472218\pi$$
0.0871686 + 0.996194i $$0.472218\pi$$
$$930$$ 0 0
$$931$$ 0 0
$$932$$ 0 0
$$933$$ 13.2548 0.433944
$$934$$ 0 0
$$935$$ 0 0
$$936$$ 0 0
$$937$$ 1.45584 0.0475604 0.0237802 0.999717i $$-0.492430\pi$$
0.0237802 + 0.999717i $$0.492430\pi$$
$$938$$ 0 0
$$939$$ −3.71573 −0.121258
$$940$$ 0 0
$$941$$ −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
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ 41.1716 1.33790 0.668948 0.743309i $$-0.266745\pi$$
0.668948 + 0.743309i $$0.266745\pi$$
$$948$$ 0 0
$$949$$ 1.37258 0.0445559
$$950$$ 0 0
$$951$$ −3.71573 −0.120491
$$952$$ 0 0
$$953$$ −53.1716 −1.72240 −0.861198 0.508269i $$-0.830285\pi$$
−0.861198 + 0.508269i $$0.830285\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ 0 0
$$957$$ −10.3431 −0.334346
$$958$$ 0 0
$$959$$ −21.9411 −0.708516
$$960$$ 0 0
$$961$$ −31.0000 −1.00000
$$962$$ 0 0
$$963$$ 38.2843 1.23369
$$964$$ 0 0
$$965$$ 0 0
$$966$$ 0 0
$$967$$ −14.9706 −0.481421 −0.240710 0.970597i $$-0.577380\pi$$
−0.240710 + 0.970597i $$0.577380\pi$$
$$968$$ 0 0
$$969$$ 0 0
$$970$$ 0 0
$$971$$ 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
$$974$$ 0 0
$$975$$ 0 0
$$976$$ 0 0
$$977$$ −32.3431 −1.03475 −0.517374 0.855759i $$-0.673091\pi$$
−0.517374 + 0.855759i $$0.673091\pi$$
$$978$$ 0 0
$$979$$ 13.3137 0.425508
$$980$$ 0 0
$$981$$ −38.2843 −1.22232
$$982$$ 0 0
$$983$$ −21.8579 −0.697158 −0.348579 0.937279i $$-0.613336\pi$$
−0.348579 + 0.937279i $$0.613336\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ 0 0
$$987$$ 16.0000 0.509286
$$988$$ 0 0
$$989$$ 16.9706 0.539633
$$990$$ 0 0
$$991$$ 57.9411 1.84056 0.920280 0.391260i $$-0.127961\pi$$
0.920280 + 0.391260i $$0.127961\pi$$
$$992$$ 0 0
$$993$$ −20.6863 −0.656460
$$994$$ 0 0
$$995$$ 0 0
$$996$$ 0 0
$$997$$ 41.4558 1.31292 0.656460 0.754361i $$-0.272053\pi$$
0.656460 + 0.754361i $$0.272053\pi$$
$$998$$ 0 0
$$999$$ −43.3137 −1.37039
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 4400.2.a.bn.1.1 2
4.3 odd 2 275.2.a.c.1.1 2
5.2 odd 4 4400.2.b.q.4049.3 4
5.3 odd 4 4400.2.b.q.4049.2 4
5.4 even 2 880.2.a.m.1.2 2
12.11 even 2 2475.2.a.x.1.2 2
15.14 odd 2 7920.2.a.ch.1.2 2
20.3 even 4 275.2.b.d.199.4 4
20.7 even 4 275.2.b.d.199.1 4
20.19 odd 2 55.2.a.b.1.2 2
40.19 odd 2 3520.2.a.bn.1.2 2
40.29 even 2 3520.2.a.bo.1.1 2
44.43 even 2 3025.2.a.o.1.2 2
55.54 odd 2 9680.2.a.bn.1.2 2
60.23 odd 4 2475.2.c.l.199.1 4
60.47 odd 4 2475.2.c.l.199.4 4
60.59 even 2 495.2.a.b.1.1 2
140.139 even 2 2695.2.a.f.1.2 2
220.19 even 10 605.2.g.l.251.1 8
220.39 even 10 605.2.g.l.366.2 8
220.59 odd 10 605.2.g.f.511.2 8
220.79 even 10 605.2.g.l.81.2 8
220.119 odd 10 605.2.g.f.81.1 8
220.139 even 10 605.2.g.l.511.1 8
220.159 odd 10 605.2.g.f.366.1 8
220.179 odd 10 605.2.g.f.251.2 8
220.219 even 2 605.2.a.d.1.1 2
260.259 odd 2 9295.2.a.g.1.1 2
660.659 odd 2 5445.2.a.y.1.2 2

By twisted newform
Twist Min Dim Char Parity Ord Type
55.2.a.b.1.2 2 20.19 odd 2
275.2.a.c.1.1 2 4.3 odd 2
275.2.b.d.199.1 4 20.7 even 4
275.2.b.d.199.4 4 20.3 even 4
495.2.a.b.1.1 2 60.59 even 2
605.2.a.d.1.1 2 220.219 even 2
605.2.g.f.81.1 8 220.119 odd 10
605.2.g.f.251.2 8 220.179 odd 10
605.2.g.f.366.1 8 220.159 odd 10
605.2.g.f.511.2 8 220.59 odd 10
605.2.g.l.81.2 8 220.79 even 10
605.2.g.l.251.1 8 220.19 even 10
605.2.g.l.366.2 8 220.39 even 10
605.2.g.l.511.1 8 220.139 even 10
880.2.a.m.1.2 2 5.4 even 2
2475.2.a.x.1.2 2 12.11 even 2
2475.2.c.l.199.1 4 60.23 odd 4
2475.2.c.l.199.4 4 60.47 odd 4
2695.2.a.f.1.2 2 140.139 even 2
3025.2.a.o.1.2 2 44.43 even 2
3520.2.a.bn.1.2 2 40.19 odd 2
3520.2.a.bo.1.1 2 40.29 even 2
4400.2.a.bn.1.1 2 1.1 even 1 trivial
4400.2.b.q.4049.2 4 5.3 odd 4
4400.2.b.q.4049.3 4 5.2 odd 4
5445.2.a.y.1.2 2 660.659 odd 2
7920.2.a.ch.1.2 2 15.14 odd 2
9295.2.a.g.1.1 2 260.259 odd 2
9680.2.a.bn.1.2 2 55.54 odd 2