# Properties

 Label 3800.2.d.d.3649.2 Level $3800$ Weight $2$ Character 3800.3649 Analytic conductor $30.343$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$3800 = 2^{3} \cdot 5^{2} \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 3800.d (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$30.3431527681$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(i)$$ Defining polynomial: $$x^{2} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 152) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 3649.2 Root $$1.00000i$$ of defining polynomial Character $$\chi$$ $$=$$ 3800.3649 Dual form 3800.2.d.d.3649.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+2.00000i q^{3} -3.00000i q^{7} -1.00000 q^{9} +O(q^{10})$$ $$q+2.00000i q^{3} -3.00000i q^{7} -1.00000 q^{9} -3.00000 q^{11} +4.00000i q^{13} +5.00000i q^{17} +1.00000 q^{19} +6.00000 q^{21} +4.00000i q^{27} -2.00000 q^{29} +8.00000 q^{31} -6.00000i q^{33} -10.0000i q^{37} -8.00000 q^{39} +6.00000 q^{41} +7.00000i q^{43} -9.00000i q^{47} -2.00000 q^{49} -10.0000 q^{51} +8.00000i q^{53} +2.00000i q^{57} -14.0000 q^{59} -5.00000 q^{61} +3.00000i q^{63} -6.00000 q^{71} +15.0000i q^{73} +9.00000i q^{77} +4.00000 q^{79} -11.0000 q^{81} -4.00000i q^{83} -4.00000i q^{87} +12.0000 q^{91} +16.0000i q^{93} +16.0000i q^{97} +3.00000 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{9} + O(q^{10})$$ $$2 q - 2 q^{9} - 6 q^{11} + 2 q^{19} + 12 q^{21} - 4 q^{29} + 16 q^{31} - 16 q^{39} + 12 q^{41} - 4 q^{49} - 20 q^{51} - 28 q^{59} - 10 q^{61} - 12 q^{71} + 8 q^{79} - 22 q^{81} + 24 q^{91} + 6 q^{99} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/3800\mathbb{Z}\right)^\times$$.

 $$n$$ $$401$$ $$951$$ $$1901$$ $$1977$$ $$\chi(n)$$ $$1$$ $$1$$ $$1$$ $$-1$$

## Coefficient data

For each $$n$$ we display the coefficients of the $$q$$-expansion $$a_n$$, the Satake parameters $$\alpha_p$$, and the Satake angles $$\theta_p = \textrm{Arg}(\alpha_p)$$.

Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 0 0
$$3$$ 2.00000i 1.15470i 0.816497 + 0.577350i $$0.195913\pi$$
−0.816497 + 0.577350i $$0.804087\pi$$
$$4$$ 0 0
$$5$$ 0 0
$$6$$ 0 0
$$7$$ − 3.00000i − 1.13389i −0.823754 0.566947i $$-0.808125\pi$$
0.823754 0.566947i $$-0.191875\pi$$
$$8$$ 0 0
$$9$$ −1.00000 −0.333333
$$10$$ 0 0
$$11$$ −3.00000 −0.904534 −0.452267 0.891883i $$-0.649385\pi$$
−0.452267 + 0.891883i $$0.649385\pi$$
$$12$$ 0 0
$$13$$ 4.00000i 1.10940i 0.832050 + 0.554700i $$0.187167\pi$$
−0.832050 + 0.554700i $$0.812833\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 0 0
$$17$$ 5.00000i 1.21268i 0.795206 + 0.606339i $$0.207363\pi$$
−0.795206 + 0.606339i $$0.792637\pi$$
$$18$$ 0 0
$$19$$ 1.00000 0.229416
$$20$$ 0 0
$$21$$ 6.00000 1.30931
$$22$$ 0 0
$$23$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$24$$ 0 0
$$25$$ 0 0
$$26$$ 0 0
$$27$$ 4.00000i 0.769800i
$$28$$ 0 0
$$29$$ −2.00000 −0.371391 −0.185695 0.982607i $$-0.559454\pi$$
−0.185695 + 0.982607i $$0.559454\pi$$
$$30$$ 0 0
$$31$$ 8.00000 1.43684 0.718421 0.695608i $$-0.244865\pi$$
0.718421 + 0.695608i $$0.244865\pi$$
$$32$$ 0 0
$$33$$ − 6.00000i − 1.04447i
$$34$$ 0 0
$$35$$ 0 0
$$36$$ 0 0
$$37$$ − 10.0000i − 1.64399i −0.569495 0.821995i $$-0.692861\pi$$
0.569495 0.821995i $$-0.307139\pi$$
$$38$$ 0 0
$$39$$ −8.00000 −1.28103
$$40$$ 0 0
$$41$$ 6.00000 0.937043 0.468521 0.883452i $$-0.344787\pi$$
0.468521 + 0.883452i $$0.344787\pi$$
$$42$$ 0 0
$$43$$ 7.00000i 1.06749i 0.845645 + 0.533745i $$0.179216\pi$$
−0.845645 + 0.533745i $$0.820784\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 0 0
$$47$$ − 9.00000i − 1.31278i −0.754420 0.656392i $$-0.772082\pi$$
0.754420 0.656392i $$-0.227918\pi$$
$$48$$ 0 0
$$49$$ −2.00000 −0.285714
$$50$$ 0 0
$$51$$ −10.0000 −1.40028
$$52$$ 0 0
$$53$$ 8.00000i 1.09888i 0.835532 + 0.549442i $$0.185160\pi$$
−0.835532 + 0.549442i $$0.814840\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 0 0
$$57$$ 2.00000i 0.264906i
$$58$$ 0 0
$$59$$ −14.0000 −1.82264 −0.911322 0.411693i $$-0.864937\pi$$
−0.911322 + 0.411693i $$0.864937\pi$$
$$60$$ 0 0
$$61$$ −5.00000 −0.640184 −0.320092 0.947386i $$-0.603714\pi$$
−0.320092 + 0.947386i $$0.603714\pi$$
$$62$$ 0 0
$$63$$ 3.00000i 0.377964i
$$64$$ 0 0
$$65$$ 0 0
$$66$$ 0 0
$$67$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$68$$ 0 0
$$69$$ 0 0
$$70$$ 0 0
$$71$$ −6.00000 −0.712069 −0.356034 0.934473i $$-0.615871\pi$$
−0.356034 + 0.934473i $$0.615871\pi$$
$$72$$ 0 0
$$73$$ 15.0000i 1.75562i 0.479012 + 0.877809i $$0.340995\pi$$
−0.479012 + 0.877809i $$0.659005\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 0 0
$$77$$ 9.00000i 1.02565i
$$78$$ 0 0
$$79$$ 4.00000 0.450035 0.225018 0.974355i $$-0.427756\pi$$
0.225018 + 0.974355i $$0.427756\pi$$
$$80$$ 0 0
$$81$$ −11.0000 −1.22222
$$82$$ 0 0
$$83$$ − 4.00000i − 0.439057i −0.975606 0.219529i $$-0.929548\pi$$
0.975606 0.219529i $$-0.0704519\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ 0 0
$$87$$ − 4.00000i − 0.428845i
$$88$$ 0 0
$$89$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$90$$ 0 0
$$91$$ 12.0000 1.25794
$$92$$ 0 0
$$93$$ 16.0000i 1.65912i
$$94$$ 0 0
$$95$$ 0 0
$$96$$ 0 0
$$97$$ 16.0000i 1.62455i 0.583272 + 0.812277i $$0.301772\pi$$
−0.583272 + 0.812277i $$0.698228\pi$$
$$98$$ 0 0
$$99$$ 3.00000 0.301511
$$100$$ 0 0
$$101$$ −18.0000 −1.79107 −0.895533 0.444994i $$-0.853206\pi$$
−0.895533 + 0.444994i $$0.853206\pi$$
$$102$$ 0 0
$$103$$ 14.0000i 1.37946i 0.724066 + 0.689730i $$0.242271\pi$$
−0.724066 + 0.689730i $$0.757729\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ 10.0000i 0.966736i 0.875417 + 0.483368i $$0.160587\pi$$
−0.875417 + 0.483368i $$0.839413\pi$$
$$108$$ 0 0
$$109$$ −12.0000 −1.14939 −0.574696 0.818367i $$-0.694880\pi$$
−0.574696 + 0.818367i $$0.694880\pi$$
$$110$$ 0 0
$$111$$ 20.0000 1.89832
$$112$$ 0 0
$$113$$ − 2.00000i − 0.188144i −0.995565 0.0940721i $$-0.970012\pi$$
0.995565 0.0940721i $$-0.0299884\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ 0 0
$$117$$ − 4.00000i − 0.369800i
$$118$$ 0 0
$$119$$ 15.0000 1.37505
$$120$$ 0 0
$$121$$ −2.00000 −0.181818
$$122$$ 0 0
$$123$$ 12.0000i 1.08200i
$$124$$ 0 0
$$125$$ 0 0
$$126$$ 0 0
$$127$$ − 6.00000i − 0.532414i −0.963916 0.266207i $$-0.914230\pi$$
0.963916 0.266207i $$-0.0857705\pi$$
$$128$$ 0 0
$$129$$ −14.0000 −1.23263
$$130$$ 0 0
$$131$$ −9.00000 −0.786334 −0.393167 0.919467i $$-0.628621\pi$$
−0.393167 + 0.919467i $$0.628621\pi$$
$$132$$ 0 0
$$133$$ − 3.00000i − 0.260133i
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ 21.0000i 1.79415i 0.441877 + 0.897076i $$0.354313\pi$$
−0.441877 + 0.897076i $$0.645687\pi$$
$$138$$ 0 0
$$139$$ −5.00000 −0.424094 −0.212047 0.977259i $$-0.568013\pi$$
−0.212047 + 0.977259i $$0.568013\pi$$
$$140$$ 0 0
$$141$$ 18.0000 1.51587
$$142$$ 0 0
$$143$$ − 12.0000i − 1.00349i
$$144$$ 0 0
$$145$$ 0 0
$$146$$ 0 0
$$147$$ − 4.00000i − 0.329914i
$$148$$ 0 0
$$149$$ −17.0000 −1.39269 −0.696347 0.717705i $$-0.745193\pi$$
−0.696347 + 0.717705i $$0.745193\pi$$
$$150$$ 0 0
$$151$$ 2.00000 0.162758 0.0813788 0.996683i $$-0.474068\pi$$
0.0813788 + 0.996683i $$0.474068\pi$$
$$152$$ 0 0
$$153$$ − 5.00000i − 0.404226i
$$154$$ 0 0
$$155$$ 0 0
$$156$$ 0 0
$$157$$ 14.0000i 1.11732i 0.829396 + 0.558661i $$0.188685\pi$$
−0.829396 + 0.558661i $$0.811315\pi$$
$$158$$ 0 0
$$159$$ −16.0000 −1.26888
$$160$$ 0 0
$$161$$ 0 0
$$162$$ 0 0
$$163$$ − 4.00000i − 0.313304i −0.987654 0.156652i $$-0.949930\pi$$
0.987654 0.156652i $$-0.0500701\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ − 2.00000i − 0.154765i −0.997001 0.0773823i $$-0.975344\pi$$
0.997001 0.0773823i $$-0.0246562\pi$$
$$168$$ 0 0
$$169$$ −3.00000 −0.230769
$$170$$ 0 0
$$171$$ −1.00000 −0.0764719
$$172$$ 0 0
$$173$$ 2.00000i 0.152057i 0.997106 + 0.0760286i $$0.0242240\pi$$
−0.997106 + 0.0760286i $$0.975776\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 0 0
$$177$$ − 28.0000i − 2.10461i
$$178$$ 0 0
$$179$$ 18.0000 1.34538 0.672692 0.739923i $$-0.265138\pi$$
0.672692 + 0.739923i $$0.265138\pi$$
$$180$$ 0 0
$$181$$ 2.00000 0.148659 0.0743294 0.997234i $$-0.476318\pi$$
0.0743294 + 0.997234i $$0.476318\pi$$
$$182$$ 0 0
$$183$$ − 10.0000i − 0.739221i
$$184$$ 0 0
$$185$$ 0 0
$$186$$ 0 0
$$187$$ − 15.0000i − 1.09691i
$$188$$ 0 0
$$189$$ 12.0000 0.872872
$$190$$ 0 0
$$191$$ −15.0000 −1.08536 −0.542681 0.839939i $$-0.682591\pi$$
−0.542681 + 0.839939i $$0.682591\pi$$
$$192$$ 0 0
$$193$$ 24.0000i 1.72756i 0.503871 + 0.863779i $$0.331909\pi$$
−0.503871 + 0.863779i $$0.668091\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ 18.0000i 1.28245i 0.767354 + 0.641223i $$0.221573\pi$$
−0.767354 + 0.641223i $$0.778427\pi$$
$$198$$ 0 0
$$199$$ 15.0000 1.06332 0.531661 0.846957i $$-0.321568\pi$$
0.531661 + 0.846957i $$0.321568\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ 0 0
$$203$$ 6.00000i 0.421117i
$$204$$ 0 0
$$205$$ 0 0
$$206$$ 0 0
$$207$$ 0 0
$$208$$ 0 0
$$209$$ −3.00000 −0.207514
$$210$$ 0 0
$$211$$ −6.00000 −0.413057 −0.206529 0.978441i $$-0.566217\pi$$
−0.206529 + 0.978441i $$0.566217\pi$$
$$212$$ 0 0
$$213$$ − 12.0000i − 0.822226i
$$214$$ 0 0
$$215$$ 0 0
$$216$$ 0 0
$$217$$ − 24.0000i − 1.62923i
$$218$$ 0 0
$$219$$ −30.0000 −2.02721
$$220$$ 0 0
$$221$$ −20.0000 −1.34535
$$222$$ 0 0
$$223$$ − 22.0000i − 1.47323i −0.676313 0.736614i $$-0.736423\pi$$
0.676313 0.736614i $$-0.263577\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 0 0
$$227$$ − 20.0000i − 1.32745i −0.747978 0.663723i $$-0.768975\pi$$
0.747978 0.663723i $$-0.231025\pi$$
$$228$$ 0 0
$$229$$ −1.00000 −0.0660819 −0.0330409 0.999454i $$-0.510519\pi$$
−0.0330409 + 0.999454i $$0.510519\pi$$
$$230$$ 0 0
$$231$$ −18.0000 −1.18431
$$232$$ 0 0
$$233$$ 13.0000i 0.851658i 0.904804 + 0.425829i $$0.140018\pi$$
−0.904804 + 0.425829i $$0.859982\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ 0 0
$$237$$ 8.00000i 0.519656i
$$238$$ 0 0
$$239$$ 3.00000 0.194054 0.0970269 0.995282i $$-0.469067\pi$$
0.0970269 + 0.995282i $$0.469067\pi$$
$$240$$ 0 0
$$241$$ 18.0000 1.15948 0.579741 0.814801i $$-0.303154\pi$$
0.579741 + 0.814801i $$0.303154\pi$$
$$242$$ 0 0
$$243$$ − 10.0000i − 0.641500i
$$244$$ 0 0
$$245$$ 0 0
$$246$$ 0 0
$$247$$ 4.00000i 0.254514i
$$248$$ 0 0
$$249$$ 8.00000 0.506979
$$250$$ 0 0
$$251$$ −13.0000 −0.820553 −0.410276 0.911961i $$-0.634568\pi$$
−0.410276 + 0.911961i $$0.634568\pi$$
$$252$$ 0 0
$$253$$ 0 0
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 0 0
$$257$$ − 24.0000i − 1.49708i −0.663090 0.748539i $$-0.730755\pi$$
0.663090 0.748539i $$-0.269245\pi$$
$$258$$ 0 0
$$259$$ −30.0000 −1.86411
$$260$$ 0 0
$$261$$ 2.00000 0.123797
$$262$$ 0 0
$$263$$ 5.00000i 0.308313i 0.988046 + 0.154157i $$0.0492660\pi$$
−0.988046 + 0.154157i $$0.950734\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ 0 0
$$267$$ 0 0
$$268$$ 0 0
$$269$$ −4.00000 −0.243884 −0.121942 0.992537i $$-0.538912\pi$$
−0.121942 + 0.992537i $$0.538912\pi$$
$$270$$ 0 0
$$271$$ −8.00000 −0.485965 −0.242983 0.970031i $$-0.578126\pi$$
−0.242983 + 0.970031i $$0.578126\pi$$
$$272$$ 0 0
$$273$$ 24.0000i 1.45255i
$$274$$ 0 0
$$275$$ 0 0
$$276$$ 0 0
$$277$$ 9.00000i 0.540758i 0.962754 + 0.270379i $$0.0871489\pi$$
−0.962754 + 0.270379i $$0.912851\pi$$
$$278$$ 0 0
$$279$$ −8.00000 −0.478947
$$280$$ 0 0
$$281$$ −10.0000 −0.596550 −0.298275 0.954480i $$-0.596411\pi$$
−0.298275 + 0.954480i $$0.596411\pi$$
$$282$$ 0 0
$$283$$ − 13.0000i − 0.772770i −0.922338 0.386385i $$-0.873724\pi$$
0.922338 0.386385i $$-0.126276\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ − 18.0000i − 1.06251i
$$288$$ 0 0
$$289$$ −8.00000 −0.470588
$$290$$ 0 0
$$291$$ −32.0000 −1.87587
$$292$$ 0 0
$$293$$ 4.00000i 0.233682i 0.993151 + 0.116841i $$0.0372769\pi$$
−0.993151 + 0.116841i $$0.962723\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ 0 0
$$297$$ − 12.0000i − 0.696311i
$$298$$ 0 0
$$299$$ 0 0
$$300$$ 0 0
$$301$$ 21.0000 1.21042
$$302$$ 0 0
$$303$$ − 36.0000i − 2.06815i
$$304$$ 0 0
$$305$$ 0 0
$$306$$ 0 0
$$307$$ 16.0000i 0.913168i 0.889680 + 0.456584i $$0.150927\pi$$
−0.889680 + 0.456584i $$0.849073\pi$$
$$308$$ 0 0
$$309$$ −28.0000 −1.59286
$$310$$ 0 0
$$311$$ 31.0000 1.75785 0.878924 0.476961i $$-0.158262\pi$$
0.878924 + 0.476961i $$0.158262\pi$$
$$312$$ 0 0
$$313$$ − 14.0000i − 0.791327i −0.918396 0.395663i $$-0.870515\pi$$
0.918396 0.395663i $$-0.129485\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ − 18.0000i − 1.01098i −0.862832 0.505490i $$-0.831312\pi$$
0.862832 0.505490i $$-0.168688\pi$$
$$318$$ 0 0
$$319$$ 6.00000 0.335936
$$320$$ 0 0
$$321$$ −20.0000 −1.11629
$$322$$ 0 0
$$323$$ 5.00000i 0.278207i
$$324$$ 0 0
$$325$$ 0 0
$$326$$ 0 0
$$327$$ − 24.0000i − 1.32720i
$$328$$ 0 0
$$329$$ −27.0000 −1.48856
$$330$$ 0 0
$$331$$ −8.00000 −0.439720 −0.219860 0.975531i $$-0.570560\pi$$
−0.219860 + 0.975531i $$0.570560\pi$$
$$332$$ 0 0
$$333$$ 10.0000i 0.547997i
$$334$$ 0 0
$$335$$ 0 0
$$336$$ 0 0
$$337$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$338$$ 0 0
$$339$$ 4.00000 0.217250
$$340$$ 0 0
$$341$$ −24.0000 −1.29967
$$342$$ 0 0
$$343$$ − 15.0000i − 0.809924i
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ 27.0000i 1.44944i 0.689046 + 0.724718i $$0.258030\pi$$
−0.689046 + 0.724718i $$0.741970\pi$$
$$348$$ 0 0
$$349$$ 19.0000 1.01705 0.508523 0.861048i $$-0.330192\pi$$
0.508523 + 0.861048i $$0.330192\pi$$
$$350$$ 0 0
$$351$$ −16.0000 −0.854017
$$352$$ 0 0
$$353$$ − 2.00000i − 0.106449i −0.998583 0.0532246i $$-0.983050\pi$$
0.998583 0.0532246i $$-0.0169499\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ 0 0
$$357$$ 30.0000i 1.58777i
$$358$$ 0 0
$$359$$ 11.0000 0.580558 0.290279 0.956942i $$-0.406252\pi$$
0.290279 + 0.956942i $$0.406252\pi$$
$$360$$ 0 0
$$361$$ 1.00000 0.0526316
$$362$$ 0 0
$$363$$ − 4.00000i − 0.209946i
$$364$$ 0 0
$$365$$ 0 0
$$366$$ 0 0
$$367$$ 16.0000i 0.835193i 0.908633 + 0.417597i $$0.137127\pi$$
−0.908633 + 0.417597i $$0.862873\pi$$
$$368$$ 0 0
$$369$$ −6.00000 −0.312348
$$370$$ 0 0
$$371$$ 24.0000 1.24602
$$372$$ 0 0
$$373$$ − 4.00000i − 0.207112i −0.994624 0.103556i $$-0.966978\pi$$
0.994624 0.103556i $$-0.0330221\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ − 8.00000i − 0.412021i
$$378$$ 0 0
$$379$$ −18.0000 −0.924598 −0.462299 0.886724i $$-0.652975\pi$$
−0.462299 + 0.886724i $$0.652975\pi$$
$$380$$ 0 0
$$381$$ 12.0000 0.614779
$$382$$ 0 0
$$383$$ − 16.0000i − 0.817562i −0.912633 0.408781i $$-0.865954\pi$$
0.912633 0.408781i $$-0.134046\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ − 7.00000i − 0.355830i
$$388$$ 0 0
$$389$$ 29.0000 1.47036 0.735179 0.677873i $$-0.237098\pi$$
0.735179 + 0.677873i $$0.237098\pi$$
$$390$$ 0 0
$$391$$ 0 0
$$392$$ 0 0
$$393$$ − 18.0000i − 0.907980i
$$394$$ 0 0
$$395$$ 0 0
$$396$$ 0 0
$$397$$ 13.0000i 0.652451i 0.945292 + 0.326226i $$0.105777\pi$$
−0.945292 + 0.326226i $$0.894223\pi$$
$$398$$ 0 0
$$399$$ 6.00000 0.300376
$$400$$ 0 0
$$401$$ 24.0000 1.19850 0.599251 0.800561i $$-0.295465\pi$$
0.599251 + 0.800561i $$0.295465\pi$$
$$402$$ 0 0
$$403$$ 32.0000i 1.59403i
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 0 0
$$407$$ 30.0000i 1.48704i
$$408$$ 0 0
$$409$$ 8.00000 0.395575 0.197787 0.980245i $$-0.436624\pi$$
0.197787 + 0.980245i $$0.436624\pi$$
$$410$$ 0 0
$$411$$ −42.0000 −2.07171
$$412$$ 0 0
$$413$$ 42.0000i 2.06668i
$$414$$ 0 0
$$415$$ 0 0
$$416$$ 0 0
$$417$$ − 10.0000i − 0.489702i
$$418$$ 0 0
$$419$$ 12.0000 0.586238 0.293119 0.956076i $$-0.405307\pi$$
0.293119 + 0.956076i $$0.405307\pi$$
$$420$$ 0 0
$$421$$ 20.0000 0.974740 0.487370 0.873195i $$-0.337956\pi$$
0.487370 + 0.873195i $$0.337956\pi$$
$$422$$ 0 0
$$423$$ 9.00000i 0.437595i
$$424$$ 0 0
$$425$$ 0 0
$$426$$ 0 0
$$427$$ 15.0000i 0.725901i
$$428$$ 0 0
$$429$$ 24.0000 1.15873
$$430$$ 0 0
$$431$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$432$$ 0 0
$$433$$ 26.0000i 1.24948i 0.780833 + 0.624740i $$0.214795\pi$$
−0.780833 + 0.624740i $$0.785205\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 0 0
$$437$$ 0 0
$$438$$ 0 0
$$439$$ 26.0000 1.24091 0.620456 0.784241i $$-0.286947\pi$$
0.620456 + 0.784241i $$0.286947\pi$$
$$440$$ 0 0
$$441$$ 2.00000 0.0952381
$$442$$ 0 0
$$443$$ − 19.0000i − 0.902717i −0.892343 0.451359i $$-0.850940\pi$$
0.892343 0.451359i $$-0.149060\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ 0 0
$$447$$ − 34.0000i − 1.60814i
$$448$$ 0 0
$$449$$ −24.0000 −1.13263 −0.566315 0.824189i $$-0.691631\pi$$
−0.566315 + 0.824189i $$0.691631\pi$$
$$450$$ 0 0
$$451$$ −18.0000 −0.847587
$$452$$ 0 0
$$453$$ 4.00000i 0.187936i
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ − 29.0000i − 1.35656i −0.734802 0.678281i $$-0.762725\pi$$
0.734802 0.678281i $$-0.237275\pi$$
$$458$$ 0 0
$$459$$ −20.0000 −0.933520
$$460$$ 0 0
$$461$$ 21.0000 0.978068 0.489034 0.872265i $$-0.337349\pi$$
0.489034 + 0.872265i $$0.337349\pi$$
$$462$$ 0 0
$$463$$ 37.0000i 1.71954i 0.510685 + 0.859768i $$0.329392\pi$$
−0.510685 + 0.859768i $$0.670608\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ − 13.0000i − 0.601568i −0.953692 0.300784i $$-0.902752\pi$$
0.953692 0.300784i $$-0.0972484\pi$$
$$468$$ 0 0
$$469$$ 0 0
$$470$$ 0 0
$$471$$ −28.0000 −1.29017
$$472$$ 0 0
$$473$$ − 21.0000i − 0.965581i
$$474$$ 0 0
$$475$$ 0 0
$$476$$ 0 0
$$477$$ − 8.00000i − 0.366295i
$$478$$ 0 0
$$479$$ 28.0000 1.27935 0.639676 0.768644i $$-0.279068\pi$$
0.639676 + 0.768644i $$0.279068\pi$$
$$480$$ 0 0
$$481$$ 40.0000 1.82384
$$482$$ 0 0
$$483$$ 0 0
$$484$$ 0 0
$$485$$ 0 0
$$486$$ 0 0
$$487$$ − 2.00000i − 0.0906287i −0.998973 0.0453143i $$-0.985571\pi$$
0.998973 0.0453143i $$-0.0144289\pi$$
$$488$$ 0 0
$$489$$ 8.00000 0.361773
$$490$$ 0 0
$$491$$ −20.0000 −0.902587 −0.451294 0.892375i $$-0.649037\pi$$
−0.451294 + 0.892375i $$0.649037\pi$$
$$492$$ 0 0
$$493$$ − 10.0000i − 0.450377i
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ 18.0000i 0.807410i
$$498$$ 0 0
$$499$$ 29.0000 1.29822 0.649109 0.760695i $$-0.275142\pi$$
0.649109 + 0.760695i $$0.275142\pi$$
$$500$$ 0 0
$$501$$ 4.00000 0.178707
$$502$$ 0 0
$$503$$ − 12.0000i − 0.535054i −0.963550 0.267527i $$-0.913794\pi$$
0.963550 0.267527i $$-0.0862064\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ 0 0
$$507$$ − 6.00000i − 0.266469i
$$508$$ 0 0
$$509$$ 8.00000 0.354594 0.177297 0.984157i $$-0.443265\pi$$
0.177297 + 0.984157i $$0.443265\pi$$
$$510$$ 0 0
$$511$$ 45.0000 1.99068
$$512$$ 0 0
$$513$$ 4.00000i 0.176604i
$$514$$ 0 0
$$515$$ 0 0
$$516$$ 0 0
$$517$$ 27.0000i 1.18746i
$$518$$ 0 0
$$519$$ −4.00000 −0.175581
$$520$$ 0 0
$$521$$ 4.00000 0.175243 0.0876216 0.996154i $$-0.472073\pi$$
0.0876216 + 0.996154i $$0.472073\pi$$
$$522$$ 0 0
$$523$$ − 26.0000i − 1.13690i −0.822718 0.568450i $$-0.807543\pi$$
0.822718 0.568450i $$-0.192457\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ 40.0000i 1.74243i
$$528$$ 0 0
$$529$$ 23.0000 1.00000
$$530$$ 0 0
$$531$$ 14.0000 0.607548
$$532$$ 0 0
$$533$$ 24.0000i 1.03956i
$$534$$ 0 0
$$535$$ 0 0
$$536$$ 0 0
$$537$$ 36.0000i 1.55351i
$$538$$ 0 0
$$539$$ 6.00000 0.258438
$$540$$ 0 0
$$541$$ 19.0000 0.816874 0.408437 0.912787i $$-0.366074\pi$$
0.408437 + 0.912787i $$0.366074\pi$$
$$542$$ 0 0
$$543$$ 4.00000i 0.171656i
$$544$$ 0 0
$$545$$ 0 0
$$546$$ 0 0
$$547$$ − 20.0000i − 0.855138i −0.903983 0.427569i $$-0.859370\pi$$
0.903983 0.427569i $$-0.140630\pi$$
$$548$$ 0 0
$$549$$ 5.00000 0.213395
$$550$$ 0 0
$$551$$ −2.00000 −0.0852029
$$552$$ 0 0
$$553$$ − 12.0000i − 0.510292i
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 0 0
$$557$$ − 7.00000i − 0.296600i −0.988942 0.148300i $$-0.952620\pi$$
0.988942 0.148300i $$-0.0473800\pi$$
$$558$$ 0 0
$$559$$ −28.0000 −1.18427
$$560$$ 0 0
$$561$$ 30.0000 1.26660
$$562$$ 0 0
$$563$$ − 14.0000i − 0.590030i −0.955493 0.295015i $$-0.904675\pi$$
0.955493 0.295015i $$-0.0953246\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ 0 0
$$567$$ 33.0000i 1.38587i
$$568$$ 0 0
$$569$$ −8.00000 −0.335377 −0.167689 0.985840i $$-0.553630\pi$$
−0.167689 + 0.985840i $$0.553630\pi$$
$$570$$ 0 0
$$571$$ −28.0000 −1.17176 −0.585882 0.810397i $$-0.699252\pi$$
−0.585882 + 0.810397i $$0.699252\pi$$
$$572$$ 0 0
$$573$$ − 30.0000i − 1.25327i
$$574$$ 0 0
$$575$$ 0 0
$$576$$ 0 0
$$577$$ − 37.0000i − 1.54033i −0.637845 0.770165i $$-0.720174\pi$$
0.637845 0.770165i $$-0.279826\pi$$
$$578$$ 0 0
$$579$$ −48.0000 −1.99481
$$580$$ 0 0
$$581$$ −12.0000 −0.497844
$$582$$ 0 0
$$583$$ − 24.0000i − 0.993978i
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ − 29.0000i − 1.19696i −0.801138 0.598479i $$-0.795772\pi$$
0.801138 0.598479i $$-0.204228\pi$$
$$588$$ 0 0
$$589$$ 8.00000 0.329634
$$590$$ 0 0
$$591$$ −36.0000 −1.48084
$$592$$ 0 0
$$593$$ 18.0000i 0.739171i 0.929197 + 0.369586i $$0.120500\pi$$
−0.929197 + 0.369586i $$0.879500\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 0 0
$$597$$ 30.0000i 1.22782i
$$598$$ 0 0
$$599$$ −8.00000 −0.326871 −0.163436 0.986554i $$-0.552258\pi$$
−0.163436 + 0.986554i $$0.552258\pi$$
$$600$$ 0 0
$$601$$ −22.0000 −0.897399 −0.448699 0.893683i $$-0.648113\pi$$
−0.448699 + 0.893683i $$0.648113\pi$$
$$602$$ 0 0
$$603$$ 0 0
$$604$$ 0 0
$$605$$ 0 0
$$606$$ 0 0
$$607$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$608$$ 0 0
$$609$$ −12.0000 −0.486265
$$610$$ 0 0
$$611$$ 36.0000 1.45640
$$612$$ 0 0
$$613$$ 31.0000i 1.25208i 0.779792 + 0.626039i $$0.215325\pi$$
−0.779792 + 0.626039i $$0.784675\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ 33.0000i 1.32853i 0.747497 + 0.664265i $$0.231255\pi$$
−0.747497 + 0.664265i $$0.768745\pi$$
$$618$$ 0 0
$$619$$ 20.0000 0.803868 0.401934 0.915669i $$-0.368338\pi$$
0.401934 + 0.915669i $$0.368338\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ 0 0
$$623$$ 0 0
$$624$$ 0 0
$$625$$ 0 0
$$626$$ 0 0
$$627$$ − 6.00000i − 0.239617i
$$628$$ 0 0
$$629$$ 50.0000 1.99363
$$630$$ 0 0
$$631$$ 25.0000 0.995234 0.497617 0.867397i $$-0.334208\pi$$
0.497617 + 0.867397i $$0.334208\pi$$
$$632$$ 0 0
$$633$$ − 12.0000i − 0.476957i
$$634$$ 0 0
$$635$$ 0 0
$$636$$ 0 0
$$637$$ − 8.00000i − 0.316972i
$$638$$ 0 0
$$639$$ 6.00000 0.237356
$$640$$ 0 0
$$641$$ −24.0000 −0.947943 −0.473972 0.880540i $$-0.657180\pi$$
−0.473972 + 0.880540i $$0.657180\pi$$
$$642$$ 0 0
$$643$$ 43.0000i 1.69575i 0.530193 + 0.847877i $$0.322120\pi$$
−0.530193 + 0.847877i $$0.677880\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ 49.0000i 1.92639i 0.268806 + 0.963194i $$0.413371\pi$$
−0.268806 + 0.963194i $$0.586629\pi$$
$$648$$ 0 0
$$649$$ 42.0000 1.64864
$$650$$ 0 0
$$651$$ 48.0000 1.88127
$$652$$ 0 0
$$653$$ − 21.0000i − 0.821794i −0.911682 0.410897i $$-0.865216\pi$$
0.911682 0.410897i $$-0.134784\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ 0 0
$$657$$ − 15.0000i − 0.585206i
$$658$$ 0 0
$$659$$ −6.00000 −0.233727 −0.116863 0.993148i $$-0.537284\pi$$
−0.116863 + 0.993148i $$0.537284\pi$$
$$660$$ 0 0
$$661$$ −16.0000 −0.622328 −0.311164 0.950356i $$-0.600719\pi$$
−0.311164 + 0.950356i $$0.600719\pi$$
$$662$$ 0 0
$$663$$ − 40.0000i − 1.55347i
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 0 0
$$667$$ 0 0
$$668$$ 0 0
$$669$$ 44.0000 1.70114
$$670$$ 0 0
$$671$$ 15.0000 0.579069
$$672$$ 0 0
$$673$$ − 6.00000i − 0.231283i −0.993291 0.115642i $$-0.963108\pi$$
0.993291 0.115642i $$-0.0368924\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ 0 0
$$677$$ − 38.0000i − 1.46046i −0.683202 0.730229i $$-0.739413\pi$$
0.683202 0.730229i $$-0.260587\pi$$
$$678$$ 0 0
$$679$$ 48.0000 1.84207
$$680$$ 0 0
$$681$$ 40.0000 1.53280
$$682$$ 0 0
$$683$$ 28.0000i 1.07139i 0.844411 + 0.535695i $$0.179950\pi$$
−0.844411 + 0.535695i $$0.820050\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ 0 0
$$687$$ − 2.00000i − 0.0763048i
$$688$$ 0 0
$$689$$ −32.0000 −1.21910
$$690$$ 0 0
$$691$$ 15.0000 0.570627 0.285313 0.958434i $$-0.407902\pi$$
0.285313 + 0.958434i $$0.407902\pi$$
$$692$$ 0 0
$$693$$ − 9.00000i − 0.341882i
$$694$$ 0 0
$$695$$ 0 0
$$696$$ 0 0
$$697$$ 30.0000i 1.13633i
$$698$$ 0 0
$$699$$ −26.0000 −0.983410
$$700$$ 0 0
$$701$$ 30.0000 1.13308 0.566542 0.824033i $$-0.308281\pi$$
0.566542 + 0.824033i $$0.308281\pi$$
$$702$$ 0 0
$$703$$ − 10.0000i − 0.377157i
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ 54.0000i 2.03088i
$$708$$ 0 0
$$709$$ 6.00000 0.225335 0.112667 0.993633i $$-0.464061\pi$$
0.112667 + 0.993633i $$0.464061\pi$$
$$710$$ 0 0
$$711$$ −4.00000 −0.150012
$$712$$ 0 0
$$713$$ 0 0
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 0 0
$$717$$ 6.00000i 0.224074i
$$718$$ 0 0
$$719$$ −29.0000 −1.08152 −0.540759 0.841178i $$-0.681863\pi$$
−0.540759 + 0.841178i $$0.681863\pi$$
$$720$$ 0 0
$$721$$ 42.0000 1.56416
$$722$$ 0 0
$$723$$ 36.0000i 1.33885i
$$724$$ 0 0
$$725$$ 0 0
$$726$$ 0 0
$$727$$ 7.00000i 0.259616i 0.991539 + 0.129808i $$0.0414360\pi$$
−0.991539 + 0.129808i $$0.958564\pi$$
$$728$$ 0 0
$$729$$ −13.0000 −0.481481
$$730$$ 0 0
$$731$$ −35.0000 −1.29452
$$732$$ 0 0
$$733$$ 14.0000i 0.517102i 0.965998 + 0.258551i $$0.0832450\pi$$
−0.965998 + 0.258551i $$0.916755\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ 0 0
$$738$$ 0 0
$$739$$ 35.0000 1.28750 0.643748 0.765238i $$-0.277379\pi$$
0.643748 + 0.765238i $$0.277379\pi$$
$$740$$ 0 0
$$741$$ −8.00000 −0.293887
$$742$$ 0 0
$$743$$ 36.0000i 1.32071i 0.750953 + 0.660356i $$0.229595\pi$$
−0.750953 + 0.660356i $$0.770405\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ 0 0
$$747$$ 4.00000i 0.146352i
$$748$$ 0 0
$$749$$ 30.0000 1.09618
$$750$$ 0 0
$$751$$ −40.0000 −1.45962 −0.729810 0.683650i $$-0.760392\pi$$
−0.729810 + 0.683650i $$0.760392\pi$$
$$752$$ 0 0
$$753$$ − 26.0000i − 0.947493i
$$754$$ 0 0
$$755$$ 0 0
$$756$$ 0 0
$$757$$ − 37.0000i − 1.34479i −0.740193 0.672394i $$-0.765266\pi$$
0.740193 0.672394i $$-0.234734\pi$$
$$758$$ 0 0
$$759$$ 0 0
$$760$$ 0 0
$$761$$ −7.00000 −0.253750 −0.126875 0.991919i $$-0.540495\pi$$
−0.126875 + 0.991919i $$0.540495\pi$$
$$762$$ 0 0
$$763$$ 36.0000i 1.30329i
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ − 56.0000i − 2.02204i
$$768$$ 0 0
$$769$$ 1.00000 0.0360609 0.0180305 0.999837i $$-0.494260\pi$$
0.0180305 + 0.999837i $$0.494260\pi$$
$$770$$ 0 0
$$771$$ 48.0000 1.72868
$$772$$ 0 0
$$773$$ − 34.0000i − 1.22290i −0.791285 0.611448i $$-0.790588\pi$$
0.791285 0.611448i $$-0.209412\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ 0 0
$$777$$ − 60.0000i − 2.15249i
$$778$$ 0 0
$$779$$ 6.00000 0.214972
$$780$$ 0 0
$$781$$ 18.0000 0.644091
$$782$$ 0 0
$$783$$ − 8.00000i − 0.285897i
$$784$$ 0 0
$$785$$ 0 0
$$786$$ 0 0
$$787$$ − 28.0000i − 0.998092i −0.866575 0.499046i $$-0.833684\pi$$
0.866575 0.499046i $$-0.166316\pi$$
$$788$$ 0 0
$$789$$ −10.0000 −0.356009
$$790$$ 0 0
$$791$$ −6.00000 −0.213335
$$792$$ 0 0
$$793$$ − 20.0000i − 0.710221i
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 0 0
$$797$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$798$$ 0 0
$$799$$ 45.0000 1.59199
$$800$$ 0 0
$$801$$ 0 0
$$802$$ 0 0
$$803$$ − 45.0000i − 1.58802i
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 0 0
$$807$$ − 8.00000i − 0.281613i
$$808$$ 0 0
$$809$$ −15.0000 −0.527372 −0.263686 0.964609i $$-0.584938\pi$$
−0.263686 + 0.964609i $$0.584938\pi$$
$$810$$ 0 0
$$811$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$812$$ 0 0
$$813$$ − 16.0000i − 0.561144i
$$814$$ 0 0
$$815$$ 0 0
$$816$$ 0 0
$$817$$ 7.00000i 0.244899i
$$818$$ 0 0
$$819$$ −12.0000 −0.419314
$$820$$ 0 0
$$821$$ −11.0000 −0.383903 −0.191951 0.981404i $$-0.561482\pi$$
−0.191951 + 0.981404i $$0.561482\pi$$
$$822$$ 0 0
$$823$$ − 37.0000i − 1.28974i −0.764293 0.644869i $$-0.776912\pi$$
0.764293 0.644869i $$-0.223088\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ 8.00000i 0.278187i 0.990279 + 0.139094i $$0.0444189\pi$$
−0.990279 + 0.139094i $$0.955581\pi$$
$$828$$ 0 0
$$829$$ −24.0000 −0.833554 −0.416777 0.909009i $$-0.636840\pi$$
−0.416777 + 0.909009i $$0.636840\pi$$
$$830$$ 0 0
$$831$$ −18.0000 −0.624413
$$832$$ 0 0
$$833$$ − 10.0000i − 0.346479i
$$834$$ 0 0
$$835$$ 0 0
$$836$$ 0 0
$$837$$ 32.0000i 1.10608i
$$838$$ 0 0
$$839$$ −30.0000 −1.03572 −0.517858 0.855467i $$-0.673270\pi$$
−0.517858 + 0.855467i $$0.673270\pi$$
$$840$$ 0 0
$$841$$ −25.0000 −0.862069
$$842$$ 0 0
$$843$$ − 20.0000i − 0.688837i
$$844$$ 0 0
$$845$$ 0 0
$$846$$ 0 0
$$847$$ 6.00000i 0.206162i
$$848$$ 0 0
$$849$$ 26.0000 0.892318
$$850$$ 0 0
$$851$$ 0 0
$$852$$ 0 0
$$853$$ 30.0000i 1.02718i 0.858036 + 0.513590i $$0.171685\pi$$
−0.858036 + 0.513590i $$0.828315\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0 0
$$857$$ − 38.0000i − 1.29806i −0.760765 0.649028i $$-0.775176\pi$$
0.760765 0.649028i $$-0.224824\pi$$
$$858$$ 0 0
$$859$$ −17.0000 −0.580033 −0.290016 0.957022i $$-0.593661\pi$$
−0.290016 + 0.957022i $$0.593661\pi$$
$$860$$ 0 0
$$861$$ 36.0000 1.22688
$$862$$ 0 0
$$863$$ 30.0000i 1.02121i 0.859815 + 0.510606i $$0.170579\pi$$
−0.859815 + 0.510606i $$0.829421\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ 0 0
$$867$$ − 16.0000i − 0.543388i
$$868$$ 0 0
$$869$$ −12.0000 −0.407072
$$870$$ 0 0
$$871$$ 0 0
$$872$$ 0 0
$$873$$ − 16.0000i − 0.541518i
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 0 0
$$877$$ 34.0000i 1.14810i 0.818821 + 0.574049i $$0.194628\pi$$
−0.818821 + 0.574049i $$0.805372\pi$$
$$878$$ 0 0
$$879$$ −8.00000 −0.269833
$$880$$ 0 0
$$881$$ 37.0000 1.24656 0.623281 0.781998i $$-0.285799\pi$$
0.623281 + 0.781998i $$0.285799\pi$$
$$882$$ 0 0
$$883$$ − 41.0000i − 1.37976i −0.723924 0.689880i $$-0.757663\pi$$
0.723924 0.689880i $$-0.242337\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 0 0
$$887$$ 38.0000i 1.27592i 0.770072 + 0.637958i $$0.220220\pi$$
−0.770072 + 0.637958i $$0.779780\pi$$
$$888$$ 0 0
$$889$$ −18.0000 −0.603701
$$890$$ 0 0
$$891$$ 33.0000 1.10554
$$892$$ 0 0
$$893$$ − 9.00000i − 0.301174i
$$894$$ 0 0
$$895$$ 0 0
$$896$$ 0 0
$$897$$ 0 0
$$898$$ 0 0
$$899$$ −16.0000 −0.533630
$$900$$ 0 0
$$901$$ −40.0000 −1.33259
$$902$$ 0 0
$$903$$ 42.0000i 1.39767i
$$904$$ 0 0
$$905$$ 0 0
$$906$$ 0 0
$$907$$ 4.00000i 0.132818i 0.997792 + 0.0664089i $$0.0211542\pi$$
−0.997792 + 0.0664089i $$0.978846\pi$$
$$908$$ 0 0
$$909$$ 18.0000 0.597022
$$910$$ 0 0
$$911$$ −2.00000 −0.0662630 −0.0331315 0.999451i $$-0.510548\pi$$
−0.0331315 + 0.999451i $$0.510548\pi$$
$$912$$ 0 0
$$913$$ 12.0000i 0.397142i
$$914$$ 0 0
$$915$$ 0 0
$$916$$ 0 0
$$917$$ 27.0000i 0.891619i
$$918$$ 0 0
$$919$$ −4.00000 −0.131948 −0.0659739 0.997821i $$-0.521015\pi$$
−0.0659739 + 0.997821i $$0.521015\pi$$
$$920$$ 0 0
$$921$$ −32.0000 −1.05444
$$922$$ 0 0
$$923$$ − 24.0000i − 0.789970i
$$924$$ 0 0
$$925$$ 0 0
$$926$$ 0 0
$$927$$ − 14.0000i − 0.459820i
$$928$$ 0 0
$$929$$ −54.0000 −1.77168 −0.885841 0.463988i $$-0.846418\pi$$
−0.885841 + 0.463988i $$0.846418\pi$$
$$930$$ 0 0
$$931$$ −2.00000 −0.0655474
$$932$$ 0 0
$$933$$ 62.0000i 2.02979i
$$934$$ 0 0
$$935$$ 0 0
$$936$$ 0 0
$$937$$ − 7.00000i − 0.228680i −0.993442 0.114340i $$-0.963525\pi$$
0.993442 0.114340i $$-0.0364753\pi$$
$$938$$ 0 0
$$939$$ 28.0000 0.913745
$$940$$ 0 0
$$941$$ 14.0000 0.456387 0.228193 0.973616i $$-0.426718\pi$$
0.228193 + 0.973616i $$0.426718\pi$$
$$942$$ 0 0
$$943$$ 0 0
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ 20.0000i 0.649913i 0.945729 + 0.324956i $$0.105350\pi$$
−0.945729 + 0.324956i $$0.894650\pi$$
$$948$$ 0 0
$$949$$ −60.0000 −1.94768
$$950$$ 0 0
$$951$$ 36.0000 1.16738
$$952$$ 0 0
$$953$$ − 4.00000i − 0.129573i −0.997899 0.0647864i $$-0.979363\pi$$
0.997899 0.0647864i $$-0.0206366\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ 0 0
$$957$$ 12.0000i 0.387905i
$$958$$ 0 0
$$959$$ 63.0000 2.03438
$$960$$ 0 0
$$961$$ 33.0000 1.06452
$$962$$ 0 0
$$963$$ − 10.0000i − 0.322245i
$$964$$ 0 0
$$965$$ 0 0
$$966$$ 0 0
$$967$$ − 48.0000i − 1.54358i −0.635880 0.771788i $$-0.719363\pi$$
0.635880 0.771788i $$-0.280637\pi$$
$$968$$ 0 0
$$969$$ −10.0000 −0.321246
$$970$$ 0 0
$$971$$ 20.0000 0.641831 0.320915 0.947108i $$-0.396010\pi$$
0.320915 + 0.947108i $$0.396010\pi$$
$$972$$ 0 0
$$973$$ 15.0000i 0.480878i
$$974$$ 0 0
$$975$$ 0 0
$$976$$ 0 0
$$977$$ 52.0000i 1.66363i 0.555055 + 0.831814i $$0.312697\pi$$
−0.555055 + 0.831814i $$0.687303\pi$$
$$978$$ 0 0
$$979$$ 0 0
$$980$$ 0 0
$$981$$ 12.0000 0.383131
$$982$$ 0 0
$$983$$ − 20.0000i − 0.637901i −0.947771 0.318950i $$-0.896670\pi$$
0.947771 0.318950i $$-0.103330\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ 0 0
$$987$$ − 54.0000i − 1.71884i
$$988$$ 0 0
$$989$$ 0 0
$$990$$ 0 0
$$991$$ −10.0000 −0.317660 −0.158830 0.987306i $$-0.550772\pi$$
−0.158830 + 0.987306i $$0.550772\pi$$
$$992$$ 0 0
$$993$$ − 16.0000i − 0.507745i
$$994$$ 0 0
$$995$$ 0 0
$$996$$ 0 0
$$997$$ 13.0000i 0.411714i 0.978582 + 0.205857i $$0.0659982\pi$$
−0.978582 + 0.205857i $$0.934002\pi$$
$$998$$ 0 0
$$999$$ 40.0000 1.26554
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 3800.2.d.d.3649.2 2
5.2 odd 4 3800.2.a.i.1.1 1
5.3 odd 4 152.2.a.a.1.1 1
5.4 even 2 inner 3800.2.d.d.3649.1 2
15.8 even 4 1368.2.a.h.1.1 1
20.3 even 4 304.2.a.e.1.1 1
20.7 even 4 7600.2.a.b.1.1 1
35.13 even 4 7448.2.a.s.1.1 1
40.3 even 4 1216.2.a.d.1.1 1
40.13 odd 4 1216.2.a.p.1.1 1
60.23 odd 4 2736.2.a.p.1.1 1
95.18 even 4 2888.2.a.f.1.1 1
380.303 odd 4 5776.2.a.b.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
152.2.a.a.1.1 1 5.3 odd 4
304.2.a.e.1.1 1 20.3 even 4
1216.2.a.d.1.1 1 40.3 even 4
1216.2.a.p.1.1 1 40.13 odd 4
1368.2.a.h.1.1 1 15.8 even 4
2736.2.a.p.1.1 1 60.23 odd 4
2888.2.a.f.1.1 1 95.18 even 4
3800.2.a.i.1.1 1 5.2 odd 4
3800.2.d.d.3649.1 2 5.4 even 2 inner
3800.2.d.d.3649.2 2 1.1 even 1 trivial
5776.2.a.b.1.1 1 380.303 odd 4
7448.2.a.s.1.1 1 35.13 even 4
7600.2.a.b.1.1 1 20.7 even 4