# Properties

 Label 3200.2.a.bf.1.2 Level $3200$ Weight $2$ Character 3200.1 Self dual yes Analytic conductor $25.552$ Analytic rank $0$ Dimension $2$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$3200 = 2^{7} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 3200.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$25.5521286468$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\zeta_{10})^+$$ Defining polynomial: $$x^{2} - x - 1$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 640) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.2 Root $$-0.618034$$ of defining polynomial Character $$\chi$$ $$=$$ 3200.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+1.23607 q^{3} +3.23607 q^{7} -1.47214 q^{9} +O(q^{10})$$ $$q+1.23607 q^{3} +3.23607 q^{7} -1.47214 q^{9} +2.00000 q^{11} -4.47214 q^{13} -4.47214 q^{17} +4.47214 q^{19} +4.00000 q^{21} +4.76393 q^{23} -5.52786 q^{27} +2.00000 q^{29} +6.47214 q^{31} +2.47214 q^{33} +6.94427 q^{37} -5.52786 q^{39} +12.4721 q^{41} -7.70820 q^{43} +7.23607 q^{47} +3.47214 q^{49} -5.52786 q^{51} +0.472136 q^{53} +5.52786 q^{57} +8.47214 q^{59} +6.00000 q^{61} -4.76393 q^{63} +7.70820 q^{67} +5.88854 q^{69} -2.47214 q^{71} -4.47214 q^{73} +6.47214 q^{77} +12.9443 q^{79} -2.41641 q^{81} -3.70820 q^{83} +2.47214 q^{87} -14.9443 q^{89} -14.4721 q^{91} +8.00000 q^{93} +16.4721 q^{97} -2.94427 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{3} + 2 q^{7} + 6 q^{9} + O(q^{10})$$ $$2 q - 2 q^{3} + 2 q^{7} + 6 q^{9} + 4 q^{11} + 8 q^{21} + 14 q^{23} - 20 q^{27} + 4 q^{29} + 4 q^{31} - 4 q^{33} - 4 q^{37} - 20 q^{39} + 16 q^{41} - 2 q^{43} + 10 q^{47} - 2 q^{49} - 20 q^{51} - 8 q^{53} + 20 q^{57} + 8 q^{59} + 12 q^{61} - 14 q^{63} + 2 q^{67} - 24 q^{69} + 4 q^{71} + 4 q^{77} + 8 q^{79} + 22 q^{81} + 6 q^{83} - 4 q^{87} - 12 q^{89} - 20 q^{91} + 16 q^{93} + 24 q^{97} + 12 q^{99} + O(q^{100})$$

## 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
$$3$$ 1.23607 0.713644 0.356822 0.934172i $$-0.383860\pi$$
0.356822 + 0.934172i $$0.383860\pi$$
$$4$$ 0 0
$$5$$ 0 0
$$6$$ 0 0
$$7$$ 3.23607 1.22312 0.611559 0.791199i $$-0.290543\pi$$
0.611559 + 0.791199i $$0.290543\pi$$
$$8$$ 0 0
$$9$$ −1.47214 −0.490712
$$10$$ 0 0
$$11$$ 2.00000 0.603023 0.301511 0.953463i $$-0.402509\pi$$
0.301511 + 0.953463i $$0.402509\pi$$
$$12$$ 0 0
$$13$$ −4.47214 −1.24035 −0.620174 0.784465i $$-0.712938\pi$$
−0.620174 + 0.784465i $$0.712938\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 0 0
$$17$$ −4.47214 −1.08465 −0.542326 0.840168i $$-0.682456\pi$$
−0.542326 + 0.840168i $$0.682456\pi$$
$$18$$ 0 0
$$19$$ 4.47214 1.02598 0.512989 0.858395i $$-0.328538\pi$$
0.512989 + 0.858395i $$0.328538\pi$$
$$20$$ 0 0
$$21$$ 4.00000 0.872872
$$22$$ 0 0
$$23$$ 4.76393 0.993348 0.496674 0.867937i $$-0.334554\pi$$
0.496674 + 0.867937i $$0.334554\pi$$
$$24$$ 0 0
$$25$$ 0 0
$$26$$ 0 0
$$27$$ −5.52786 −1.06384
$$28$$ 0 0
$$29$$ 2.00000 0.371391 0.185695 0.982607i $$-0.440546\pi$$
0.185695 + 0.982607i $$0.440546\pi$$
$$30$$ 0 0
$$31$$ 6.47214 1.16243 0.581215 0.813750i $$-0.302578\pi$$
0.581215 + 0.813750i $$0.302578\pi$$
$$32$$ 0 0
$$33$$ 2.47214 0.430344
$$34$$ 0 0
$$35$$ 0 0
$$36$$ 0 0
$$37$$ 6.94427 1.14163 0.570816 0.821078i $$-0.306627\pi$$
0.570816 + 0.821078i $$0.306627\pi$$
$$38$$ 0 0
$$39$$ −5.52786 −0.885167
$$40$$ 0 0
$$41$$ 12.4721 1.94782 0.973910 0.226934i $$-0.0728701\pi$$
0.973910 + 0.226934i $$0.0728701\pi$$
$$42$$ 0 0
$$43$$ −7.70820 −1.17549 −0.587745 0.809046i $$-0.699984\pi$$
−0.587745 + 0.809046i $$0.699984\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 0 0
$$47$$ 7.23607 1.05549 0.527744 0.849403i $$-0.323038\pi$$
0.527744 + 0.849403i $$0.323038\pi$$
$$48$$ 0 0
$$49$$ 3.47214 0.496019
$$50$$ 0 0
$$51$$ −5.52786 −0.774056
$$52$$ 0 0
$$53$$ 0.472136 0.0648529 0.0324264 0.999474i $$-0.489677\pi$$
0.0324264 + 0.999474i $$0.489677\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 0 0
$$57$$ 5.52786 0.732183
$$58$$ 0 0
$$59$$ 8.47214 1.10298 0.551489 0.834182i $$-0.314060\pi$$
0.551489 + 0.834182i $$0.314060\pi$$
$$60$$ 0 0
$$61$$ 6.00000 0.768221 0.384111 0.923287i $$-0.374508\pi$$
0.384111 + 0.923287i $$0.374508\pi$$
$$62$$ 0 0
$$63$$ −4.76393 −0.600199
$$64$$ 0 0
$$65$$ 0 0
$$66$$ 0 0
$$67$$ 7.70820 0.941707 0.470853 0.882211i $$-0.343946\pi$$
0.470853 + 0.882211i $$0.343946\pi$$
$$68$$ 0 0
$$69$$ 5.88854 0.708897
$$70$$ 0 0
$$71$$ −2.47214 −0.293389 −0.146694 0.989182i $$-0.546863\pi$$
−0.146694 + 0.989182i $$0.546863\pi$$
$$72$$ 0 0
$$73$$ −4.47214 −0.523424 −0.261712 0.965146i $$-0.584287\pi$$
−0.261712 + 0.965146i $$0.584287\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 0 0
$$77$$ 6.47214 0.737568
$$78$$ 0 0
$$79$$ 12.9443 1.45634 0.728172 0.685394i $$-0.240370\pi$$
0.728172 + 0.685394i $$0.240370\pi$$
$$80$$ 0 0
$$81$$ −2.41641 −0.268490
$$82$$ 0 0
$$83$$ −3.70820 −0.407028 −0.203514 0.979072i $$-0.565236\pi$$
−0.203514 + 0.979072i $$0.565236\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ 0 0
$$87$$ 2.47214 0.265041
$$88$$ 0 0
$$89$$ −14.9443 −1.58409 −0.792045 0.610463i $$-0.790983\pi$$
−0.792045 + 0.610463i $$0.790983\pi$$
$$90$$ 0 0
$$91$$ −14.4721 −1.51709
$$92$$ 0 0
$$93$$ 8.00000 0.829561
$$94$$ 0 0
$$95$$ 0 0
$$96$$ 0 0
$$97$$ 16.4721 1.67249 0.836246 0.548354i $$-0.184746\pi$$
0.836246 + 0.548354i $$0.184746\pi$$
$$98$$ 0 0
$$99$$ −2.94427 −0.295910
$$100$$ 0 0
$$101$$ 10.0000 0.995037 0.497519 0.867453i $$-0.334245\pi$$
0.497519 + 0.867453i $$0.334245\pi$$
$$102$$ 0 0
$$103$$ −3.23607 −0.318859 −0.159430 0.987209i $$-0.550966\pi$$
−0.159430 + 0.987209i $$0.550966\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ −17.2361 −1.66627 −0.833137 0.553067i $$-0.813457\pi$$
−0.833137 + 0.553067i $$0.813457\pi$$
$$108$$ 0 0
$$109$$ −14.9443 −1.43140 −0.715701 0.698407i $$-0.753893\pi$$
−0.715701 + 0.698407i $$0.753893\pi$$
$$110$$ 0 0
$$111$$ 8.58359 0.814719
$$112$$ 0 0
$$113$$ 2.94427 0.276974 0.138487 0.990364i $$-0.455776\pi$$
0.138487 + 0.990364i $$0.455776\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ 0 0
$$117$$ 6.58359 0.608653
$$118$$ 0 0
$$119$$ −14.4721 −1.32666
$$120$$ 0 0
$$121$$ −7.00000 −0.636364
$$122$$ 0 0
$$123$$ 15.4164 1.39005
$$124$$ 0 0
$$125$$ 0 0
$$126$$ 0 0
$$127$$ 20.1803 1.79072 0.895358 0.445348i $$-0.146920\pi$$
0.895358 + 0.445348i $$0.146920\pi$$
$$128$$ 0 0
$$129$$ −9.52786 −0.838882
$$130$$ 0 0
$$131$$ −14.9443 −1.30569 −0.652844 0.757493i $$-0.726424\pi$$
−0.652844 + 0.757493i $$0.726424\pi$$
$$132$$ 0 0
$$133$$ 14.4721 1.25489
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ −2.00000 −0.170872 −0.0854358 0.996344i $$-0.527228\pi$$
−0.0854358 + 0.996344i $$0.527228\pi$$
$$138$$ 0 0
$$139$$ −20.4721 −1.73642 −0.868212 0.496194i $$-0.834731\pi$$
−0.868212 + 0.496194i $$0.834731\pi$$
$$140$$ 0 0
$$141$$ 8.94427 0.753244
$$142$$ 0 0
$$143$$ −8.94427 −0.747958
$$144$$ 0 0
$$145$$ 0 0
$$146$$ 0 0
$$147$$ 4.29180 0.353981
$$148$$ 0 0
$$149$$ −6.94427 −0.568897 −0.284448 0.958691i $$-0.591810\pi$$
−0.284448 + 0.958691i $$0.591810\pi$$
$$150$$ 0 0
$$151$$ 23.4164 1.90560 0.952800 0.303598i $$-0.0981881\pi$$
0.952800 + 0.303598i $$0.0981881\pi$$
$$152$$ 0 0
$$153$$ 6.58359 0.532252
$$154$$ 0 0
$$155$$ 0 0
$$156$$ 0 0
$$157$$ 5.05573 0.403491 0.201746 0.979438i $$-0.435339\pi$$
0.201746 + 0.979438i $$0.435339\pi$$
$$158$$ 0 0
$$159$$ 0.583592 0.0462819
$$160$$ 0 0
$$161$$ 15.4164 1.21498
$$162$$ 0 0
$$163$$ −11.7082 −0.917057 −0.458529 0.888680i $$-0.651623\pi$$
−0.458529 + 0.888680i $$0.651623\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ −14.6525 −1.13384 −0.566921 0.823772i $$-0.691866\pi$$
−0.566921 + 0.823772i $$0.691866\pi$$
$$168$$ 0 0
$$169$$ 7.00000 0.538462
$$170$$ 0 0
$$171$$ −6.58359 −0.503460
$$172$$ 0 0
$$173$$ −10.9443 −0.832078 −0.416039 0.909347i $$-0.636582\pi$$
−0.416039 + 0.909347i $$0.636582\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 0 0
$$177$$ 10.4721 0.787134
$$178$$ 0 0
$$179$$ 20.4721 1.53016 0.765080 0.643936i $$-0.222700\pi$$
0.765080 + 0.643936i $$0.222700\pi$$
$$180$$ 0 0
$$181$$ −10.9443 −0.813481 −0.406741 0.913544i $$-0.633335\pi$$
−0.406741 + 0.913544i $$0.633335\pi$$
$$182$$ 0 0
$$183$$ 7.41641 0.548237
$$184$$ 0 0
$$185$$ 0 0
$$186$$ 0 0
$$187$$ −8.94427 −0.654070
$$188$$ 0 0
$$189$$ −17.8885 −1.30120
$$190$$ 0 0
$$191$$ 11.4164 0.826062 0.413031 0.910717i $$-0.364470\pi$$
0.413031 + 0.910717i $$0.364470\pi$$
$$192$$ 0 0
$$193$$ 11.5279 0.829794 0.414897 0.909868i $$-0.363818\pi$$
0.414897 + 0.909868i $$0.363818\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ 0.472136 0.0336383 0.0168191 0.999859i $$-0.494646\pi$$
0.0168191 + 0.999859i $$0.494646\pi$$
$$198$$ 0 0
$$199$$ −0.944272 −0.0669377 −0.0334688 0.999440i $$-0.510655\pi$$
−0.0334688 + 0.999440i $$0.510655\pi$$
$$200$$ 0 0
$$201$$ 9.52786 0.672044
$$202$$ 0 0
$$203$$ 6.47214 0.454255
$$204$$ 0 0
$$205$$ 0 0
$$206$$ 0 0
$$207$$ −7.01316 −0.487448
$$208$$ 0 0
$$209$$ 8.94427 0.618688
$$210$$ 0 0
$$211$$ 1.05573 0.0726793 0.0363397 0.999339i $$-0.488430\pi$$
0.0363397 + 0.999339i $$0.488430\pi$$
$$212$$ 0 0
$$213$$ −3.05573 −0.209375
$$214$$ 0 0
$$215$$ 0 0
$$216$$ 0 0
$$217$$ 20.9443 1.42179
$$218$$ 0 0
$$219$$ −5.52786 −0.373538
$$220$$ 0 0
$$221$$ 20.0000 1.34535
$$222$$ 0 0
$$223$$ 8.76393 0.586876 0.293438 0.955978i $$-0.405201\pi$$
0.293438 + 0.955978i $$0.405201\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 0 0
$$227$$ −10.1803 −0.675693 −0.337846 0.941201i $$-0.609698\pi$$
−0.337846 + 0.941201i $$0.609698\pi$$
$$228$$ 0 0
$$229$$ −2.94427 −0.194563 −0.0972815 0.995257i $$-0.531015\pi$$
−0.0972815 + 0.995257i $$0.531015\pi$$
$$230$$ 0 0
$$231$$ 8.00000 0.526361
$$232$$ 0 0
$$233$$ −15.5279 −1.01726 −0.508632 0.860984i $$-0.669849\pi$$
−0.508632 + 0.860984i $$0.669849\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ 0 0
$$237$$ 16.0000 1.03931
$$238$$ 0 0
$$239$$ −12.9443 −0.837295 −0.418648 0.908149i $$-0.637496\pi$$
−0.418648 + 0.908149i $$0.637496\pi$$
$$240$$ 0 0
$$241$$ 26.3607 1.69804 0.849020 0.528360i $$-0.177193\pi$$
0.849020 + 0.528360i $$0.177193\pi$$
$$242$$ 0 0
$$243$$ 13.5967 0.872232
$$244$$ 0 0
$$245$$ 0 0
$$246$$ 0 0
$$247$$ −20.0000 −1.27257
$$248$$ 0 0
$$249$$ −4.58359 −0.290473
$$250$$ 0 0
$$251$$ 2.00000 0.126239 0.0631194 0.998006i $$-0.479895\pi$$
0.0631194 + 0.998006i $$0.479895\pi$$
$$252$$ 0 0
$$253$$ 9.52786 0.599012
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 0 0
$$257$$ 2.94427 0.183659 0.0918293 0.995775i $$-0.470729\pi$$
0.0918293 + 0.995775i $$0.470729\pi$$
$$258$$ 0 0
$$259$$ 22.4721 1.39635
$$260$$ 0 0
$$261$$ −2.94427 −0.182246
$$262$$ 0 0
$$263$$ 17.7082 1.09193 0.545967 0.837806i $$-0.316162\pi$$
0.545967 + 0.837806i $$0.316162\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ 0 0
$$267$$ −18.4721 −1.13048
$$268$$ 0 0
$$269$$ 23.8885 1.45651 0.728255 0.685306i $$-0.240332\pi$$
0.728255 + 0.685306i $$0.240332\pi$$
$$270$$ 0 0
$$271$$ −24.3607 −1.47981 −0.739903 0.672714i $$-0.765129\pi$$
−0.739903 + 0.672714i $$0.765129\pi$$
$$272$$ 0 0
$$273$$ −17.8885 −1.08266
$$274$$ 0 0
$$275$$ 0 0
$$276$$ 0 0
$$277$$ −10.9443 −0.657578 −0.328789 0.944403i $$-0.606640\pi$$
−0.328789 + 0.944403i $$0.606640\pi$$
$$278$$ 0 0
$$279$$ −9.52786 −0.570418
$$280$$ 0 0
$$281$$ 3.52786 0.210455 0.105227 0.994448i $$-0.466443\pi$$
0.105227 + 0.994448i $$0.466443\pi$$
$$282$$ 0 0
$$283$$ 8.29180 0.492896 0.246448 0.969156i $$-0.420737\pi$$
0.246448 + 0.969156i $$0.420737\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ 40.3607 2.38242
$$288$$ 0 0
$$289$$ 3.00000 0.176471
$$290$$ 0 0
$$291$$ 20.3607 1.19356
$$292$$ 0 0
$$293$$ −23.8885 −1.39558 −0.697792 0.716301i $$-0.745834\pi$$
−0.697792 + 0.716301i $$0.745834\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ 0 0
$$297$$ −11.0557 −0.641518
$$298$$ 0 0
$$299$$ −21.3050 −1.23210
$$300$$ 0 0
$$301$$ −24.9443 −1.43776
$$302$$ 0 0
$$303$$ 12.3607 0.710102
$$304$$ 0 0
$$305$$ 0 0
$$306$$ 0 0
$$307$$ −0.291796 −0.0166537 −0.00832684 0.999965i $$-0.502651\pi$$
−0.00832684 + 0.999965i $$0.502651\pi$$
$$308$$ 0 0
$$309$$ −4.00000 −0.227552
$$310$$ 0 0
$$311$$ −15.4164 −0.874184 −0.437092 0.899417i $$-0.643992\pi$$
−0.437092 + 0.899417i $$0.643992\pi$$
$$312$$ 0 0
$$313$$ 28.8328 1.62973 0.814864 0.579653i $$-0.196812\pi$$
0.814864 + 0.579653i $$0.196812\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ −20.4721 −1.14983 −0.574915 0.818213i $$-0.694965\pi$$
−0.574915 + 0.818213i $$0.694965\pi$$
$$318$$ 0 0
$$319$$ 4.00000 0.223957
$$320$$ 0 0
$$321$$ −21.3050 −1.18913
$$322$$ 0 0
$$323$$ −20.0000 −1.11283
$$324$$ 0 0
$$325$$ 0 0
$$326$$ 0 0
$$327$$ −18.4721 −1.02151
$$328$$ 0 0
$$329$$ 23.4164 1.29099
$$330$$ 0 0
$$331$$ −15.8885 −0.873313 −0.436657 0.899628i $$-0.643838\pi$$
−0.436657 + 0.899628i $$0.643838\pi$$
$$332$$ 0 0
$$333$$ −10.2229 −0.560212
$$334$$ 0 0
$$335$$ 0 0
$$336$$ 0 0
$$337$$ −5.05573 −0.275403 −0.137702 0.990474i $$-0.543971\pi$$
−0.137702 + 0.990474i $$0.543971\pi$$
$$338$$ 0 0
$$339$$ 3.63932 0.197661
$$340$$ 0 0
$$341$$ 12.9443 0.700972
$$342$$ 0 0
$$343$$ −11.4164 −0.616428
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ −17.2361 −0.925281 −0.462640 0.886546i $$-0.653098\pi$$
−0.462640 + 0.886546i $$0.653098\pi$$
$$348$$ 0 0
$$349$$ 14.9443 0.799949 0.399974 0.916526i $$-0.369019\pi$$
0.399974 + 0.916526i $$0.369019\pi$$
$$350$$ 0 0
$$351$$ 24.7214 1.31953
$$352$$ 0 0
$$353$$ −5.05573 −0.269089 −0.134545 0.990908i $$-0.542957\pi$$
−0.134545 + 0.990908i $$0.542957\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ 0 0
$$357$$ −17.8885 −0.946762
$$358$$ 0 0
$$359$$ −15.0557 −0.794611 −0.397305 0.917686i $$-0.630055\pi$$
−0.397305 + 0.917686i $$0.630055\pi$$
$$360$$ 0 0
$$361$$ 1.00000 0.0526316
$$362$$ 0 0
$$363$$ −8.65248 −0.454137
$$364$$ 0 0
$$365$$ 0 0
$$366$$ 0 0
$$367$$ 18.2918 0.954824 0.477412 0.878680i $$-0.341575\pi$$
0.477412 + 0.878680i $$0.341575\pi$$
$$368$$ 0 0
$$369$$ −18.3607 −0.955819
$$370$$ 0 0
$$371$$ 1.52786 0.0793227
$$372$$ 0 0
$$373$$ 5.05573 0.261776 0.130888 0.991397i $$-0.458217\pi$$
0.130888 + 0.991397i $$0.458217\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ −8.94427 −0.460653
$$378$$ 0 0
$$379$$ −15.5279 −0.797613 −0.398806 0.917035i $$-0.630575\pi$$
−0.398806 + 0.917035i $$0.630575\pi$$
$$380$$ 0 0
$$381$$ 24.9443 1.27793
$$382$$ 0 0
$$383$$ −10.2918 −0.525886 −0.262943 0.964811i $$-0.584693\pi$$
−0.262943 + 0.964811i $$0.584693\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ 11.3475 0.576827
$$388$$ 0 0
$$389$$ −11.8885 −0.602773 −0.301387 0.953502i $$-0.597449\pi$$
−0.301387 + 0.953502i $$0.597449\pi$$
$$390$$ 0 0
$$391$$ −21.3050 −1.07744
$$392$$ 0 0
$$393$$ −18.4721 −0.931796
$$394$$ 0 0
$$395$$ 0 0
$$396$$ 0 0
$$397$$ −12.4721 −0.625959 −0.312979 0.949760i $$-0.601327\pi$$
−0.312979 + 0.949760i $$0.601327\pi$$
$$398$$ 0 0
$$399$$ 17.8885 0.895547
$$400$$ 0 0
$$401$$ −2.00000 −0.0998752 −0.0499376 0.998752i $$-0.515902\pi$$
−0.0499376 + 0.998752i $$0.515902\pi$$
$$402$$ 0 0
$$403$$ −28.9443 −1.44182
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 0 0
$$407$$ 13.8885 0.688430
$$408$$ 0 0
$$409$$ 17.4164 0.861186 0.430593 0.902546i $$-0.358304\pi$$
0.430593 + 0.902546i $$0.358304\pi$$
$$410$$ 0 0
$$411$$ −2.47214 −0.121941
$$412$$ 0 0
$$413$$ 27.4164 1.34907
$$414$$ 0 0
$$415$$ 0 0
$$416$$ 0 0
$$417$$ −25.3050 −1.23919
$$418$$ 0 0
$$419$$ 4.47214 0.218478 0.109239 0.994016i $$-0.465159\pi$$
0.109239 + 0.994016i $$0.465159\pi$$
$$420$$ 0 0
$$421$$ 28.8328 1.40523 0.702613 0.711572i $$-0.252017\pi$$
0.702613 + 0.711572i $$0.252017\pi$$
$$422$$ 0 0
$$423$$ −10.6525 −0.517941
$$424$$ 0 0
$$425$$ 0 0
$$426$$ 0 0
$$427$$ 19.4164 0.939626
$$428$$ 0 0
$$429$$ −11.0557 −0.533776
$$430$$ 0 0
$$431$$ −24.3607 −1.17341 −0.586706 0.809800i $$-0.699576\pi$$
−0.586706 + 0.809800i $$0.699576\pi$$
$$432$$ 0 0
$$433$$ 0.472136 0.0226894 0.0113447 0.999936i $$-0.496389\pi$$
0.0113447 + 0.999936i $$0.496389\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 0 0
$$437$$ 21.3050 1.01915
$$438$$ 0 0
$$439$$ 15.0557 0.718571 0.359285 0.933228i $$-0.383020\pi$$
0.359285 + 0.933228i $$0.383020\pi$$
$$440$$ 0 0
$$441$$ −5.11146 −0.243403
$$442$$ 0 0
$$443$$ −4.65248 −0.221046 −0.110523 0.993874i $$-0.535253\pi$$
−0.110523 + 0.993874i $$0.535253\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ 0 0
$$447$$ −8.58359 −0.405990
$$448$$ 0 0
$$449$$ −1.41641 −0.0668444 −0.0334222 0.999441i $$-0.510641\pi$$
−0.0334222 + 0.999441i $$0.510641\pi$$
$$450$$ 0 0
$$451$$ 24.9443 1.17458
$$452$$ 0 0
$$453$$ 28.9443 1.35992
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ −3.88854 −0.181898 −0.0909492 0.995856i $$-0.528990\pi$$
−0.0909492 + 0.995856i $$0.528990\pi$$
$$458$$ 0 0
$$459$$ 24.7214 1.15389
$$460$$ 0 0
$$461$$ −41.7771 −1.94575 −0.972876 0.231325i $$-0.925694\pi$$
−0.972876 + 0.231325i $$0.925694\pi$$
$$462$$ 0 0
$$463$$ −1.12461 −0.0522651 −0.0261326 0.999658i $$-0.508319\pi$$
−0.0261326 + 0.999658i $$0.508319\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ 41.5967 1.92487 0.962434 0.271516i $$-0.0875249\pi$$
0.962434 + 0.271516i $$0.0875249\pi$$
$$468$$ 0 0
$$469$$ 24.9443 1.15182
$$470$$ 0 0
$$471$$ 6.24922 0.287949
$$472$$ 0 0
$$473$$ −15.4164 −0.708847
$$474$$ 0 0
$$475$$ 0 0
$$476$$ 0 0
$$477$$ −0.695048 −0.0318241
$$478$$ 0 0
$$479$$ 12.9443 0.591439 0.295719 0.955275i $$-0.404441\pi$$
0.295719 + 0.955275i $$0.404441\pi$$
$$480$$ 0 0
$$481$$ −31.0557 −1.41602
$$482$$ 0 0
$$483$$ 19.0557 0.867066
$$484$$ 0 0
$$485$$ 0 0
$$486$$ 0 0
$$487$$ −12.7639 −0.578389 −0.289194 0.957270i $$-0.593387\pi$$
−0.289194 + 0.957270i $$0.593387\pi$$
$$488$$ 0 0
$$489$$ −14.4721 −0.654453
$$490$$ 0 0
$$491$$ 21.0557 0.950232 0.475116 0.879923i $$-0.342406\pi$$
0.475116 + 0.879923i $$0.342406\pi$$
$$492$$ 0 0
$$493$$ −8.94427 −0.402830
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ −8.00000 −0.358849
$$498$$ 0 0
$$499$$ −14.5836 −0.652851 −0.326426 0.945223i $$-0.605844\pi$$
−0.326426 + 0.945223i $$0.605844\pi$$
$$500$$ 0 0
$$501$$ −18.1115 −0.809160
$$502$$ 0 0
$$503$$ −5.12461 −0.228495 −0.114248 0.993452i $$-0.536446\pi$$
−0.114248 + 0.993452i $$0.536446\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ 0 0
$$507$$ 8.65248 0.384270
$$508$$ 0 0
$$509$$ 16.8328 0.746101 0.373051 0.927811i $$-0.378312\pi$$
0.373051 + 0.927811i $$0.378312\pi$$
$$510$$ 0 0
$$511$$ −14.4721 −0.640210
$$512$$ 0 0
$$513$$ −24.7214 −1.09147
$$514$$ 0 0
$$515$$ 0 0
$$516$$ 0 0
$$517$$ 14.4721 0.636484
$$518$$ 0 0
$$519$$ −13.5279 −0.593807
$$520$$ 0 0
$$521$$ −15.8885 −0.696090 −0.348045 0.937478i $$-0.613154\pi$$
−0.348045 + 0.937478i $$0.613154\pi$$
$$522$$ 0 0
$$523$$ 20.0689 0.877551 0.438776 0.898597i $$-0.355412\pi$$
0.438776 + 0.898597i $$0.355412\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ −28.9443 −1.26083
$$528$$ 0 0
$$529$$ −0.304952 −0.0132588
$$530$$ 0 0
$$531$$ −12.4721 −0.541245
$$532$$ 0 0
$$533$$ −55.7771 −2.41597
$$534$$ 0 0
$$535$$ 0 0
$$536$$ 0 0
$$537$$ 25.3050 1.09199
$$538$$ 0 0
$$539$$ 6.94427 0.299111
$$540$$ 0 0
$$541$$ −23.8885 −1.02705 −0.513524 0.858075i $$-0.671660\pi$$
−0.513524 + 0.858075i $$0.671660\pi$$
$$542$$ 0 0
$$543$$ −13.5279 −0.580536
$$544$$ 0 0
$$545$$ 0 0
$$546$$ 0 0
$$547$$ 20.6525 0.883036 0.441518 0.897252i $$-0.354440\pi$$
0.441518 + 0.897252i $$0.354440\pi$$
$$548$$ 0 0
$$549$$ −8.83282 −0.376975
$$550$$ 0 0
$$551$$ 8.94427 0.381039
$$552$$ 0 0
$$553$$ 41.8885 1.78128
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 0 0
$$557$$ 13.0557 0.553189 0.276594 0.960987i $$-0.410794\pi$$
0.276594 + 0.960987i $$0.410794\pi$$
$$558$$ 0 0
$$559$$ 34.4721 1.45802
$$560$$ 0 0
$$561$$ −11.0557 −0.466773
$$562$$ 0 0
$$563$$ 4.29180 0.180878 0.0904388 0.995902i $$-0.471173\pi$$
0.0904388 + 0.995902i $$0.471173\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ 0 0
$$567$$ −7.81966 −0.328395
$$568$$ 0 0
$$569$$ −34.3607 −1.44047 −0.720237 0.693728i $$-0.755967\pi$$
−0.720237 + 0.693728i $$0.755967\pi$$
$$570$$ 0 0
$$571$$ 16.8328 0.704431 0.352216 0.935919i $$-0.385428\pi$$
0.352216 + 0.935919i $$0.385428\pi$$
$$572$$ 0 0
$$573$$ 14.1115 0.589515
$$574$$ 0 0
$$575$$ 0 0
$$576$$ 0 0
$$577$$ −19.8885 −0.827971 −0.413985 0.910283i $$-0.635864\pi$$
−0.413985 + 0.910283i $$0.635864\pi$$
$$578$$ 0 0
$$579$$ 14.2492 0.592178
$$580$$ 0 0
$$581$$ −12.0000 −0.497844
$$582$$ 0 0
$$583$$ 0.944272 0.0391077
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ 8.65248 0.357126 0.178563 0.983928i $$-0.442855\pi$$
0.178563 + 0.983928i $$0.442855\pi$$
$$588$$ 0 0
$$589$$ 28.9443 1.19263
$$590$$ 0 0
$$591$$ 0.583592 0.0240058
$$592$$ 0 0
$$593$$ 14.0000 0.574911 0.287456 0.957794i $$-0.407191\pi$$
0.287456 + 0.957794i $$0.407191\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 0 0
$$597$$ −1.16718 −0.0477697
$$598$$ 0 0
$$599$$ −21.8885 −0.894342 −0.447171 0.894449i $$-0.647568\pi$$
−0.447171 + 0.894449i $$0.647568\pi$$
$$600$$ 0 0
$$601$$ −22.3607 −0.912111 −0.456056 0.889951i $$-0.650738\pi$$
−0.456056 + 0.889951i $$0.650738\pi$$
$$602$$ 0 0
$$603$$ −11.3475 −0.462107
$$604$$ 0 0
$$605$$ 0 0
$$606$$ 0 0
$$607$$ −6.87539 −0.279063 −0.139532 0.990218i $$-0.544560\pi$$
−0.139532 + 0.990218i $$0.544560\pi$$
$$608$$ 0 0
$$609$$ 8.00000 0.324176
$$610$$ 0 0
$$611$$ −32.3607 −1.30917
$$612$$ 0 0
$$613$$ 19.5279 0.788723 0.394362 0.918955i $$-0.370966\pi$$
0.394362 + 0.918955i $$0.370966\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ −25.4164 −1.02323 −0.511613 0.859216i $$-0.670952\pi$$
−0.511613 + 0.859216i $$0.670952\pi$$
$$618$$ 0 0
$$619$$ −20.4721 −0.822845 −0.411422 0.911445i $$-0.634968\pi$$
−0.411422 + 0.911445i $$0.634968\pi$$
$$620$$ 0 0
$$621$$ −26.3344 −1.05676
$$622$$ 0 0
$$623$$ −48.3607 −1.93753
$$624$$ 0 0
$$625$$ 0 0
$$626$$ 0 0
$$627$$ 11.0557 0.441523
$$628$$ 0 0
$$629$$ −31.0557 −1.23827
$$630$$ 0 0
$$631$$ −34.4721 −1.37231 −0.686157 0.727453i $$-0.740704\pi$$
−0.686157 + 0.727453i $$0.740704\pi$$
$$632$$ 0 0
$$633$$ 1.30495 0.0518672
$$634$$ 0 0
$$635$$ 0 0
$$636$$ 0 0
$$637$$ −15.5279 −0.615236
$$638$$ 0 0
$$639$$ 3.63932 0.143969
$$640$$ 0 0
$$641$$ 0.472136 0.0186482 0.00932412 0.999957i $$-0.497032\pi$$
0.00932412 + 0.999957i $$0.497032\pi$$
$$642$$ 0 0
$$643$$ 22.1803 0.874707 0.437354 0.899290i $$-0.355916\pi$$
0.437354 + 0.899290i $$0.355916\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ −12.7639 −0.501802 −0.250901 0.968013i $$-0.580727\pi$$
−0.250901 + 0.968013i $$0.580727\pi$$
$$648$$ 0 0
$$649$$ 16.9443 0.665121
$$650$$ 0 0
$$651$$ 25.8885 1.01465
$$652$$ 0 0
$$653$$ −49.4164 −1.93381 −0.966907 0.255130i $$-0.917882\pi$$
−0.966907 + 0.255130i $$0.917882\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ 0 0
$$657$$ 6.58359 0.256850
$$658$$ 0 0
$$659$$ −21.4164 −0.834265 −0.417132 0.908846i $$-0.636965\pi$$
−0.417132 + 0.908846i $$0.636965\pi$$
$$660$$ 0 0
$$661$$ −35.8885 −1.39590 −0.697951 0.716145i $$-0.745905\pi$$
−0.697951 + 0.716145i $$0.745905\pi$$
$$662$$ 0 0
$$663$$ 24.7214 0.960098
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 0 0
$$667$$ 9.52786 0.368920
$$668$$ 0 0
$$669$$ 10.8328 0.418821
$$670$$ 0 0
$$671$$ 12.0000 0.463255
$$672$$ 0 0
$$673$$ 23.3050 0.898340 0.449170 0.893446i $$-0.351720\pi$$
0.449170 + 0.893446i $$0.351720\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ 0 0
$$677$$ −30.3607 −1.16686 −0.583428 0.812165i $$-0.698289\pi$$
−0.583428 + 0.812165i $$0.698289\pi$$
$$678$$ 0 0
$$679$$ 53.3050 2.04566
$$680$$ 0 0
$$681$$ −12.5836 −0.482204
$$682$$ 0 0
$$683$$ 24.2918 0.929500 0.464750 0.885442i $$-0.346144\pi$$
0.464750 + 0.885442i $$0.346144\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ 0 0
$$687$$ −3.63932 −0.138849
$$688$$ 0 0
$$689$$ −2.11146 −0.0804401
$$690$$ 0 0
$$691$$ 30.0000 1.14125 0.570627 0.821209i $$-0.306700\pi$$
0.570627 + 0.821209i $$0.306700\pi$$
$$692$$ 0 0
$$693$$ −9.52786 −0.361934
$$694$$ 0 0
$$695$$ 0 0
$$696$$ 0 0
$$697$$ −55.7771 −2.11271
$$698$$ 0 0
$$699$$ −19.1935 −0.725965
$$700$$ 0 0
$$701$$ 9.05573 0.342030 0.171015 0.985268i $$-0.445295\pi$$
0.171015 + 0.985268i $$0.445295\pi$$
$$702$$ 0 0
$$703$$ 31.0557 1.17129
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ 32.3607 1.21705
$$708$$ 0 0
$$709$$ 18.0000 0.676004 0.338002 0.941145i $$-0.390249\pi$$
0.338002 + 0.941145i $$0.390249\pi$$
$$710$$ 0 0
$$711$$ −19.0557 −0.714646
$$712$$ 0 0
$$713$$ 30.8328 1.15470
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 0 0
$$717$$ −16.0000 −0.597531
$$718$$ 0 0
$$719$$ 22.8328 0.851520 0.425760 0.904836i $$-0.360007\pi$$
0.425760 + 0.904836i $$0.360007\pi$$
$$720$$ 0 0
$$721$$ −10.4721 −0.390003
$$722$$ 0 0
$$723$$ 32.5836 1.21180
$$724$$ 0 0
$$725$$ 0 0
$$726$$ 0 0
$$727$$ −1.70820 −0.0633538 −0.0316769 0.999498i $$-0.510085\pi$$
−0.0316769 + 0.999498i $$0.510085\pi$$
$$728$$ 0 0
$$729$$ 24.0557 0.890953
$$730$$ 0 0
$$731$$ 34.4721 1.27500
$$732$$ 0 0
$$733$$ 51.8885 1.91655 0.958274 0.285853i $$-0.0922768\pi$$
0.958274 + 0.285853i $$0.0922768\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ 15.4164 0.567871
$$738$$ 0 0
$$739$$ −49.1935 −1.80961 −0.904806 0.425824i $$-0.859984\pi$$
−0.904806 + 0.425824i $$0.859984\pi$$
$$740$$ 0 0
$$741$$ −24.7214 −0.908162
$$742$$ 0 0
$$743$$ 30.6525 1.12453 0.562265 0.826957i $$-0.309930\pi$$
0.562265 + 0.826957i $$0.309930\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ 0 0
$$747$$ 5.45898 0.199734
$$748$$ 0 0
$$749$$ −55.7771 −2.03805
$$750$$ 0 0
$$751$$ 16.3607 0.597010 0.298505 0.954408i $$-0.403512\pi$$
0.298505 + 0.954408i $$0.403512\pi$$
$$752$$ 0 0
$$753$$ 2.47214 0.0900896
$$754$$ 0 0
$$755$$ 0 0
$$756$$ 0 0
$$757$$ 19.8885 0.722861 0.361431 0.932399i $$-0.382288\pi$$
0.361431 + 0.932399i $$0.382288\pi$$
$$758$$ 0 0
$$759$$ 11.7771 0.427481
$$760$$ 0 0
$$761$$ 3.88854 0.140960 0.0704798 0.997513i $$-0.477547\pi$$
0.0704798 + 0.997513i $$0.477547\pi$$
$$762$$ 0 0
$$763$$ −48.3607 −1.75077
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ −37.8885 −1.36808
$$768$$ 0 0
$$769$$ −14.9443 −0.538904 −0.269452 0.963014i $$-0.586843\pi$$
−0.269452 + 0.963014i $$0.586843\pi$$
$$770$$ 0 0
$$771$$ 3.63932 0.131067
$$772$$ 0 0
$$773$$ 18.3607 0.660388 0.330194 0.943913i $$-0.392886\pi$$
0.330194 + 0.943913i $$0.392886\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ 0 0
$$777$$ 27.7771 0.996497
$$778$$ 0 0
$$779$$ 55.7771 1.99842
$$780$$ 0 0
$$781$$ −4.94427 −0.176920
$$782$$ 0 0
$$783$$ −11.0557 −0.395099
$$784$$ 0 0
$$785$$ 0 0
$$786$$ 0 0
$$787$$ −44.0689 −1.57089 −0.785443 0.618934i $$-0.787565\pi$$
−0.785443 + 0.618934i $$0.787565\pi$$
$$788$$ 0 0
$$789$$ 21.8885 0.779253
$$790$$ 0 0
$$791$$ 9.52786 0.338772
$$792$$ 0 0
$$793$$ −26.8328 −0.952861
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 0 0
$$797$$ −25.4164 −0.900295 −0.450148 0.892954i $$-0.648629\pi$$
−0.450148 + 0.892954i $$0.648629\pi$$
$$798$$ 0 0
$$799$$ −32.3607 −1.14484
$$800$$ 0 0
$$801$$ 22.0000 0.777332
$$802$$ 0 0
$$803$$ −8.94427 −0.315637
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 0 0
$$807$$ 29.5279 1.03943
$$808$$ 0 0
$$809$$ −12.1115 −0.425816 −0.212908 0.977072i $$-0.568293\pi$$
−0.212908 + 0.977072i $$0.568293\pi$$
$$810$$ 0 0
$$811$$ 2.00000 0.0702295 0.0351147 0.999383i $$-0.488820\pi$$
0.0351147 + 0.999383i $$0.488820\pi$$
$$812$$ 0 0
$$813$$ −30.1115 −1.05605
$$814$$ 0 0
$$815$$ 0 0
$$816$$ 0 0
$$817$$ −34.4721 −1.20603
$$818$$ 0 0
$$819$$ 21.3050 0.744455
$$820$$ 0 0
$$821$$ −21.0557 −0.734850 −0.367425 0.930053i $$-0.619761\pi$$
−0.367425 + 0.930053i $$0.619761\pi$$
$$822$$ 0 0
$$823$$ 1.70820 0.0595442 0.0297721 0.999557i $$-0.490522\pi$$
0.0297721 + 0.999557i $$0.490522\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ −6.18034 −0.214911 −0.107456 0.994210i $$-0.534270\pi$$
−0.107456 + 0.994210i $$0.534270\pi$$
$$828$$ 0 0
$$829$$ 38.0000 1.31979 0.659897 0.751356i $$-0.270600\pi$$
0.659897 + 0.751356i $$0.270600\pi$$
$$830$$ 0 0
$$831$$ −13.5279 −0.469276
$$832$$ 0 0
$$833$$ −15.5279 −0.538009
$$834$$ 0 0
$$835$$ 0 0
$$836$$ 0 0
$$837$$ −35.7771 −1.23664
$$838$$ 0 0
$$839$$ 4.00000 0.138095 0.0690477 0.997613i $$-0.478004\pi$$
0.0690477 + 0.997613i $$0.478004\pi$$
$$840$$ 0 0
$$841$$ −25.0000 −0.862069
$$842$$ 0 0
$$843$$ 4.36068 0.150190
$$844$$ 0 0
$$845$$ 0 0
$$846$$ 0 0
$$847$$ −22.6525 −0.778348
$$848$$ 0 0
$$849$$ 10.2492 0.351752
$$850$$ 0 0
$$851$$ 33.0820 1.13404
$$852$$ 0 0
$$853$$ −7.52786 −0.257749 −0.128875 0.991661i $$-0.541136\pi$$
−0.128875 + 0.991661i $$0.541136\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0 0
$$857$$ −24.8328 −0.848273 −0.424136 0.905598i $$-0.639422\pi$$
−0.424136 + 0.905598i $$0.639422\pi$$
$$858$$ 0 0
$$859$$ −49.4164 −1.68607 −0.843033 0.537862i $$-0.819232\pi$$
−0.843033 + 0.537862i $$0.819232\pi$$
$$860$$ 0 0
$$861$$ 49.8885 1.70020
$$862$$ 0 0
$$863$$ −18.2918 −0.622660 −0.311330 0.950302i $$-0.600774\pi$$
−0.311330 + 0.950302i $$0.600774\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ 0 0
$$867$$ 3.70820 0.125937
$$868$$ 0 0
$$869$$ 25.8885 0.878209
$$870$$ 0 0
$$871$$ −34.4721 −1.16804
$$872$$ 0 0
$$873$$ −24.2492 −0.820712
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 0 0
$$877$$ 51.8885 1.75215 0.876076 0.482173i $$-0.160152\pi$$
0.876076 + 0.482173i $$0.160152\pi$$
$$878$$ 0 0
$$879$$ −29.5279 −0.995950
$$880$$ 0 0
$$881$$ −24.4721 −0.824487 −0.412244 0.911074i $$-0.635255\pi$$
−0.412244 + 0.911074i $$0.635255\pi$$
$$882$$ 0 0
$$883$$ −27.7082 −0.932455 −0.466228 0.884665i $$-0.654387\pi$$
−0.466228 + 0.884665i $$0.654387\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 0 0
$$887$$ −11.5967 −0.389381 −0.194690 0.980865i $$-0.562370\pi$$
−0.194690 + 0.980865i $$0.562370\pi$$
$$888$$ 0 0
$$889$$ 65.3050 2.19026
$$890$$ 0 0
$$891$$ −4.83282 −0.161905
$$892$$ 0 0
$$893$$ 32.3607 1.08291
$$894$$ 0 0
$$895$$ 0 0
$$896$$ 0 0
$$897$$ −26.3344 −0.879279
$$898$$ 0 0
$$899$$ 12.9443 0.431716
$$900$$ 0 0
$$901$$ −2.11146 −0.0703428
$$902$$ 0 0
$$903$$ −30.8328 −1.02605
$$904$$ 0 0
$$905$$ 0 0
$$906$$ 0 0
$$907$$ −9.23607 −0.306679 −0.153339 0.988174i $$-0.549003\pi$$
−0.153339 + 0.988174i $$0.549003\pi$$
$$908$$ 0 0
$$909$$ −14.7214 −0.488277
$$910$$ 0 0
$$911$$ −53.3050 −1.76607 −0.883036 0.469305i $$-0.844504\pi$$
−0.883036 + 0.469305i $$0.844504\pi$$
$$912$$ 0 0
$$913$$ −7.41641 −0.245447
$$914$$ 0 0
$$915$$ 0 0
$$916$$ 0 0
$$917$$ −48.3607 −1.59701
$$918$$ 0 0
$$919$$ −5.88854 −0.194245 −0.0971226 0.995272i $$-0.530964\pi$$
−0.0971226 + 0.995272i $$0.530964\pi$$
$$920$$ 0 0
$$921$$ −0.360680 −0.0118848
$$922$$ 0 0
$$923$$ 11.0557 0.363904
$$924$$ 0 0
$$925$$ 0 0
$$926$$ 0 0
$$927$$ 4.76393 0.156468
$$928$$ 0 0
$$929$$ 24.4721 0.802905 0.401452 0.915880i $$-0.368506\pi$$
0.401452 + 0.915880i $$0.368506\pi$$
$$930$$ 0 0
$$931$$ 15.5279 0.508905
$$932$$ 0 0
$$933$$ −19.0557 −0.623857
$$934$$ 0 0
$$935$$ 0 0
$$936$$ 0 0
$$937$$ 3.52786 0.115250 0.0576251 0.998338i $$-0.481647\pi$$
0.0576251 + 0.998338i $$0.481647\pi$$
$$938$$ 0 0
$$939$$ 35.6393 1.16305
$$940$$ 0 0
$$941$$ −44.8328 −1.46151 −0.730754 0.682641i $$-0.760831\pi$$
−0.730754 + 0.682641i $$0.760831\pi$$
$$942$$ 0 0
$$943$$ 59.4164 1.93486
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ 21.8197 0.709044 0.354522 0.935048i $$-0.384644\pi$$
0.354522 + 0.935048i $$0.384644\pi$$
$$948$$ 0 0
$$949$$ 20.0000 0.649227
$$950$$ 0 0
$$951$$ −25.3050 −0.820569
$$952$$ 0 0
$$953$$ −45.7771 −1.48287 −0.741433 0.671027i $$-0.765853\pi$$
−0.741433 + 0.671027i $$0.765853\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ 0 0
$$957$$ 4.94427 0.159826
$$958$$ 0 0
$$959$$ −6.47214 −0.208996
$$960$$ 0 0
$$961$$ 10.8885 0.351243
$$962$$ 0 0
$$963$$ 25.3738 0.817660
$$964$$ 0 0
$$965$$ 0 0
$$966$$ 0 0
$$967$$ 1.34752 0.0433335 0.0216667 0.999765i $$-0.493103\pi$$
0.0216667 + 0.999765i $$0.493103\pi$$
$$968$$ 0 0
$$969$$ −24.7214 −0.794164
$$970$$ 0 0
$$971$$ −23.8885 −0.766620 −0.383310 0.923620i $$-0.625216\pi$$
−0.383310 + 0.923620i $$0.625216\pi$$
$$972$$ 0 0
$$973$$ −66.2492 −2.12385
$$974$$ 0 0
$$975$$ 0 0
$$976$$ 0 0
$$977$$ 39.3050 1.25748 0.628738 0.777617i $$-0.283572\pi$$
0.628738 + 0.777617i $$0.283572\pi$$
$$978$$ 0 0
$$979$$ −29.8885 −0.955242
$$980$$ 0 0
$$981$$ 22.0000 0.702406
$$982$$ 0 0
$$983$$ −39.0132 −1.24433 −0.622163 0.782888i $$-0.713746\pi$$
−0.622163 + 0.782888i $$0.713746\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ 0 0
$$987$$ 28.9443 0.921306
$$988$$ 0 0
$$989$$ −36.7214 −1.16767
$$990$$ 0 0
$$991$$ −17.5279 −0.556791 −0.278395 0.960467i $$-0.589803\pi$$
−0.278395 + 0.960467i $$0.589803\pi$$
$$992$$ 0 0
$$993$$ −19.6393 −0.623235
$$994$$ 0 0
$$995$$ 0 0
$$996$$ 0 0
$$997$$ −34.5836 −1.09527 −0.547637 0.836716i $$-0.684472\pi$$
−0.547637 + 0.836716i $$0.684472\pi$$
$$998$$ 0 0
$$999$$ −38.3870 −1.21451
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 3200.2.a.bf.1.2 2
4.3 odd 2 3200.2.a.bk.1.1 2
5.2 odd 4 3200.2.c.x.2049.2 4
5.3 odd 4 3200.2.c.x.2049.3 4
5.4 even 2 640.2.a.l.1.1 yes 2
8.3 odd 2 3200.2.a.be.1.2 2
8.5 even 2 3200.2.a.bl.1.1 2
15.14 odd 2 5760.2.a.bw.1.1 2
20.3 even 4 3200.2.c.v.2049.2 4
20.7 even 4 3200.2.c.v.2049.3 4
20.19 odd 2 640.2.a.j.1.2 yes 2
40.3 even 4 3200.2.c.w.2049.3 4
40.13 odd 4 3200.2.c.u.2049.2 4
40.19 odd 2 640.2.a.k.1.1 yes 2
40.27 even 4 3200.2.c.w.2049.2 4
40.29 even 2 640.2.a.i.1.2 2
40.37 odd 4 3200.2.c.u.2049.3 4
60.59 even 2 5760.2.a.cd.1.2 2
80.19 odd 4 1280.2.d.k.641.3 4
80.29 even 4 1280.2.d.m.641.2 4
80.59 odd 4 1280.2.d.k.641.2 4
80.69 even 4 1280.2.d.m.641.3 4
120.29 odd 2 5760.2.a.ch.1.1 2
120.59 even 2 5760.2.a.ci.1.2 2

By twisted newform
Twist Min Dim Char Parity Ord Type
640.2.a.i.1.2 2 40.29 even 2
640.2.a.j.1.2 yes 2 20.19 odd 2
640.2.a.k.1.1 yes 2 40.19 odd 2
640.2.a.l.1.1 yes 2 5.4 even 2
1280.2.d.k.641.2 4 80.59 odd 4
1280.2.d.k.641.3 4 80.19 odd 4
1280.2.d.m.641.2 4 80.29 even 4
1280.2.d.m.641.3 4 80.69 even 4
3200.2.a.be.1.2 2 8.3 odd 2
3200.2.a.bf.1.2 2 1.1 even 1 trivial
3200.2.a.bk.1.1 2 4.3 odd 2
3200.2.a.bl.1.1 2 8.5 even 2
3200.2.c.u.2049.2 4 40.13 odd 4
3200.2.c.u.2049.3 4 40.37 odd 4
3200.2.c.v.2049.2 4 20.3 even 4
3200.2.c.v.2049.3 4 20.7 even 4
3200.2.c.w.2049.2 4 40.27 even 4
3200.2.c.w.2049.3 4 40.3 even 4
3200.2.c.x.2049.2 4 5.2 odd 4
3200.2.c.x.2049.3 4 5.3 odd 4
5760.2.a.bw.1.1 2 15.14 odd 2
5760.2.a.cd.1.2 2 60.59 even 2
5760.2.a.ch.1.1 2 120.29 odd 2
5760.2.a.ci.1.2 2 120.59 even 2