# Properties

 Label 3200.2.a.bs.1.3 Level $3200$ Weight $2$ Character 3200.1 Self dual yes Analytic conductor $25.552$ Analytic rank $0$ Dimension $3$ 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: $$3$$ Coefficient field: 3.3.148.1 Defining polynomial: $$x^{3} - x^{2} - 3x + 1$$ x^3 - x^2 - 3*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 1.3 Root $$0.311108$$ of defining polynomial Character $$\chi$$ $$=$$ 3200.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+2.90321 q^{3} -3.52543 q^{7} +5.42864 q^{9} +O(q^{10})$$ $$q+2.90321 q^{3} -3.52543 q^{7} +5.42864 q^{9} +3.80642 q^{11} +2.62222 q^{13} +5.80642 q^{17} -5.05086 q^{19} -10.2351 q^{21} -0.474572 q^{23} +7.05086 q^{27} -2.00000 q^{29} +2.75557 q^{31} +11.0509 q^{33} +7.18421 q^{37} +7.61285 q^{39} +5.18421 q^{41} -1.95407 q^{43} +5.33185 q^{47} +5.42864 q^{49} +16.8573 q^{51} +5.37778 q^{53} -14.6637 q^{57} +5.05086 q^{59} -12.2351 q^{61} -19.1383 q^{63} -7.76049 q^{67} -1.37778 q^{69} +4.85728 q^{71} -6.66370 q^{73} -13.4193 q^{77} +5.24443 q^{79} +4.18421 q^{81} +12.1476 q^{83} -5.80642 q^{87} -12.1017 q^{89} -9.24443 q^{91} +8.00000 q^{93} +13.8064 q^{97} +20.6637 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q + 2 q^{3} - 4 q^{7} + 3 q^{9}+O(q^{10})$$ 3 * q + 2 * q^3 - 4 * q^7 + 3 * q^9 $$3 q + 2 q^{3} - 4 q^{7} + 3 q^{9} - 2 q^{11} + 8 q^{13} + 4 q^{17} - 2 q^{19} - 4 q^{21} - 8 q^{23} + 8 q^{27} - 6 q^{29} + 8 q^{31} + 20 q^{33} + 8 q^{37} - 4 q^{39} + 2 q^{41} + 14 q^{43} - 4 q^{47} + 3 q^{49} + 24 q^{51} + 16 q^{53} - 4 q^{57} + 2 q^{59} - 10 q^{61} - 24 q^{63} + 10 q^{67} - 4 q^{69} - 12 q^{71} + 20 q^{73} + 16 q^{79} - q^{81} + 30 q^{83} - 4 q^{87} - 10 q^{89} - 28 q^{91} + 24 q^{93} + 28 q^{97} + 22 q^{99}+O(q^{100})$$ 3 * q + 2 * q^3 - 4 * q^7 + 3 * q^9 - 2 * q^11 + 8 * q^13 + 4 * q^17 - 2 * q^19 - 4 * q^21 - 8 * q^23 + 8 * q^27 - 6 * q^29 + 8 * q^31 + 20 * q^33 + 8 * q^37 - 4 * q^39 + 2 * q^41 + 14 * q^43 - 4 * q^47 + 3 * q^49 + 24 * q^51 + 16 * q^53 - 4 * q^57 + 2 * q^59 - 10 * q^61 - 24 * q^63 + 10 * q^67 - 4 * q^69 - 12 * q^71 + 20 * q^73 + 16 * q^79 - q^81 + 30 * q^83 - 4 * q^87 - 10 * q^89 - 28 * q^91 + 24 * q^93 + 28 * q^97 + 22 * q^99

## 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
$$3$$ 2.90321 1.67617 0.838085 0.545540i $$-0.183675\pi$$
0.838085 + 0.545540i $$0.183675\pi$$
$$4$$ 0 0
$$5$$ 0 0
$$6$$ 0 0
$$7$$ −3.52543 −1.33249 −0.666243 0.745735i $$-0.732099\pi$$
−0.666243 + 0.745735i $$0.732099\pi$$
$$8$$ 0 0
$$9$$ 5.42864 1.80955
$$10$$ 0 0
$$11$$ 3.80642 1.14768 0.573840 0.818967i $$-0.305453\pi$$
0.573840 + 0.818967i $$0.305453\pi$$
$$12$$ 0 0
$$13$$ 2.62222 0.727272 0.363636 0.931541i $$-0.381535\pi$$
0.363636 + 0.931541i $$0.381535\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 0 0
$$17$$ 5.80642 1.40826 0.704132 0.710069i $$-0.251336\pi$$
0.704132 + 0.710069i $$0.251336\pi$$
$$18$$ 0 0
$$19$$ −5.05086 −1.15875 −0.579373 0.815063i $$-0.696702\pi$$
−0.579373 + 0.815063i $$0.696702\pi$$
$$20$$ 0 0
$$21$$ −10.2351 −2.23347
$$22$$ 0 0
$$23$$ −0.474572 −0.0989552 −0.0494776 0.998775i $$-0.515756\pi$$
−0.0494776 + 0.998775i $$0.515756\pi$$
$$24$$ 0 0
$$25$$ 0 0
$$26$$ 0 0
$$27$$ 7.05086 1.35694
$$28$$ 0 0
$$29$$ −2.00000 −0.371391 −0.185695 0.982607i $$-0.559454\pi$$
−0.185695 + 0.982607i $$0.559454\pi$$
$$30$$ 0 0
$$31$$ 2.75557 0.494915 0.247457 0.968899i $$-0.420405\pi$$
0.247457 + 0.968899i $$0.420405\pi$$
$$32$$ 0 0
$$33$$ 11.0509 1.92371
$$34$$ 0 0
$$35$$ 0 0
$$36$$ 0 0
$$37$$ 7.18421 1.18108 0.590538 0.807010i $$-0.298915\pi$$
0.590538 + 0.807010i $$0.298915\pi$$
$$38$$ 0 0
$$39$$ 7.61285 1.21903
$$40$$ 0 0
$$41$$ 5.18421 0.809637 0.404819 0.914397i $$-0.367335\pi$$
0.404819 + 0.914397i $$0.367335\pi$$
$$42$$ 0 0
$$43$$ −1.95407 −0.297992 −0.148996 0.988838i $$-0.547604\pi$$
−0.148996 + 0.988838i $$0.547604\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 0 0
$$47$$ 5.33185 0.777730 0.388865 0.921295i $$-0.372867\pi$$
0.388865 + 0.921295i $$0.372867\pi$$
$$48$$ 0 0
$$49$$ 5.42864 0.775520
$$50$$ 0 0
$$51$$ 16.8573 2.36049
$$52$$ 0 0
$$53$$ 5.37778 0.738695 0.369348 0.929291i $$-0.379581\pi$$
0.369348 + 0.929291i $$0.379581\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 0 0
$$57$$ −14.6637 −1.94225
$$58$$ 0 0
$$59$$ 5.05086 0.657565 0.328783 0.944406i $$-0.393362\pi$$
0.328783 + 0.944406i $$0.393362\pi$$
$$60$$ 0 0
$$61$$ −12.2351 −1.56654 −0.783270 0.621682i $$-0.786450\pi$$
−0.783270 + 0.621682i $$0.786450\pi$$
$$62$$ 0 0
$$63$$ −19.1383 −2.41120
$$64$$ 0 0
$$65$$ 0 0
$$66$$ 0 0
$$67$$ −7.76049 −0.948095 −0.474047 0.880499i $$-0.657207\pi$$
−0.474047 + 0.880499i $$0.657207\pi$$
$$68$$ 0 0
$$69$$ −1.37778 −0.165866
$$70$$ 0 0
$$71$$ 4.85728 0.576453 0.288226 0.957562i $$-0.406934\pi$$
0.288226 + 0.957562i $$0.406934\pi$$
$$72$$ 0 0
$$73$$ −6.66370 −0.779927 −0.389964 0.920830i $$-0.627512\pi$$
−0.389964 + 0.920830i $$0.627512\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 0 0
$$77$$ −13.4193 −1.52927
$$78$$ 0 0
$$79$$ 5.24443 0.590045 0.295022 0.955490i $$-0.404673\pi$$
0.295022 + 0.955490i $$0.404673\pi$$
$$80$$ 0 0
$$81$$ 4.18421 0.464912
$$82$$ 0 0
$$83$$ 12.1476 1.33338 0.666689 0.745336i $$-0.267711\pi$$
0.666689 + 0.745336i $$0.267711\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ 0 0
$$87$$ −5.80642 −0.622514
$$88$$ 0 0
$$89$$ −12.1017 −1.28278 −0.641389 0.767216i $$-0.721642\pi$$
−0.641389 + 0.767216i $$0.721642\pi$$
$$90$$ 0 0
$$91$$ −9.24443 −0.969080
$$92$$ 0 0
$$93$$ 8.00000 0.829561
$$94$$ 0 0
$$95$$ 0 0
$$96$$ 0 0
$$97$$ 13.8064 1.40183 0.700915 0.713245i $$-0.252775\pi$$
0.700915 + 0.713245i $$0.252775\pi$$
$$98$$ 0 0
$$99$$ 20.6637 2.07678
$$100$$ 0 0
$$101$$ 19.7146 1.96167 0.980836 0.194836i $$-0.0624173\pi$$
0.980836 + 0.194836i $$0.0624173\pi$$
$$102$$ 0 0
$$103$$ 3.13828 0.309223 0.154612 0.987975i $$-0.450587\pi$$
0.154612 + 0.987975i $$0.450587\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ −5.39207 −0.521272 −0.260636 0.965437i $$-0.583932\pi$$
−0.260636 + 0.965437i $$0.583932\pi$$
$$108$$ 0 0
$$109$$ 8.62222 0.825858 0.412929 0.910763i $$-0.364506\pi$$
0.412929 + 0.910763i $$0.364506\pi$$
$$110$$ 0 0
$$111$$ 20.8573 1.97969
$$112$$ 0 0
$$113$$ 5.51114 0.518444 0.259222 0.965818i $$-0.416534\pi$$
0.259222 + 0.965818i $$0.416534\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ 0 0
$$117$$ 14.2351 1.31603
$$118$$ 0 0
$$119$$ −20.4701 −1.87649
$$120$$ 0 0
$$121$$ 3.48886 0.317169
$$122$$ 0 0
$$123$$ 15.0509 1.35709
$$124$$ 0 0
$$125$$ 0 0
$$126$$ 0 0
$$127$$ −10.2810 −0.912291 −0.456145 0.889905i $$-0.650770\pi$$
−0.456145 + 0.889905i $$0.650770\pi$$
$$128$$ 0 0
$$129$$ −5.67307 −0.499486
$$130$$ 0 0
$$131$$ −4.66370 −0.407470 −0.203735 0.979026i $$-0.565308\pi$$
−0.203735 + 0.979026i $$0.565308\pi$$
$$132$$ 0 0
$$133$$ 17.8064 1.54401
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ 11.3461 0.969366 0.484683 0.874690i $$-0.338935\pi$$
0.484683 + 0.874690i $$0.338935\pi$$
$$138$$ 0 0
$$139$$ −11.8064 −1.00141 −0.500704 0.865619i $$-0.666925\pi$$
−0.500704 + 0.865619i $$0.666925\pi$$
$$140$$ 0 0
$$141$$ 15.4795 1.30361
$$142$$ 0 0
$$143$$ 9.98126 0.834675
$$144$$ 0 0
$$145$$ 0 0
$$146$$ 0 0
$$147$$ 15.7605 1.29990
$$148$$ 0 0
$$149$$ −5.47949 −0.448898 −0.224449 0.974486i $$-0.572058\pi$$
−0.224449 + 0.974486i $$0.572058\pi$$
$$150$$ 0 0
$$151$$ −23.6128 −1.92159 −0.960793 0.277266i $$-0.910572\pi$$
−0.960793 + 0.277266i $$0.910572\pi$$
$$152$$ 0 0
$$153$$ 31.5210 2.54832
$$154$$ 0 0
$$155$$ 0 0
$$156$$ 0 0
$$157$$ 0.815792 0.0651073 0.0325536 0.999470i $$-0.489636\pi$$
0.0325536 + 0.999470i $$0.489636\pi$$
$$158$$ 0 0
$$159$$ 15.6128 1.23818
$$160$$ 0 0
$$161$$ 1.67307 0.131856
$$162$$ 0 0
$$163$$ 10.9032 0.854005 0.427003 0.904250i $$-0.359569\pi$$
0.427003 + 0.904250i $$0.359569\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ 6.57628 0.508888 0.254444 0.967088i $$-0.418108\pi$$
0.254444 + 0.967088i $$0.418108\pi$$
$$168$$ 0 0
$$169$$ −6.12399 −0.471076
$$170$$ 0 0
$$171$$ −27.4193 −2.09680
$$172$$ 0 0
$$173$$ −10.5303 −0.800608 −0.400304 0.916382i $$-0.631095\pi$$
−0.400304 + 0.916382i $$0.631095\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 0 0
$$177$$ 14.6637 1.10219
$$178$$ 0 0
$$179$$ 6.29529 0.470532 0.235266 0.971931i $$-0.424404\pi$$
0.235266 + 0.971931i $$0.424404\pi$$
$$180$$ 0 0
$$181$$ −0.488863 −0.0363369 −0.0181684 0.999835i $$-0.505784\pi$$
−0.0181684 + 0.999835i $$0.505784\pi$$
$$182$$ 0 0
$$183$$ −35.5210 −2.62579
$$184$$ 0 0
$$185$$ 0 0
$$186$$ 0 0
$$187$$ 22.1017 1.61624
$$188$$ 0 0
$$189$$ −24.8573 −1.80810
$$190$$ 0 0
$$191$$ 10.4889 0.758947 0.379474 0.925203i $$-0.376105\pi$$
0.379474 + 0.925203i $$0.376105\pi$$
$$192$$ 0 0
$$193$$ 13.8064 0.993808 0.496904 0.867805i $$-0.334470\pi$$
0.496904 + 0.867805i $$0.334470\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ 16.7239 1.19153 0.595765 0.803159i $$-0.296849\pi$$
0.595765 + 0.803159i $$0.296849\pi$$
$$198$$ 0 0
$$199$$ −20.8573 −1.47853 −0.739267 0.673413i $$-0.764828\pi$$
−0.739267 + 0.673413i $$0.764828\pi$$
$$200$$ 0 0
$$201$$ −22.5303 −1.58917
$$202$$ 0 0
$$203$$ 7.05086 0.494873
$$204$$ 0 0
$$205$$ 0 0
$$206$$ 0 0
$$207$$ −2.57628 −0.179064
$$208$$ 0 0
$$209$$ −19.2257 −1.32987
$$210$$ 0 0
$$211$$ −4.66370 −0.321063 −0.160531 0.987031i $$-0.551321\pi$$
−0.160531 + 0.987031i $$0.551321\pi$$
$$212$$ 0 0
$$213$$ 14.1017 0.966233
$$214$$ 0 0
$$215$$ 0 0
$$216$$ 0 0
$$217$$ −9.71456 −0.659467
$$218$$ 0 0
$$219$$ −19.3461 −1.30729
$$220$$ 0 0
$$221$$ 15.2257 1.02419
$$222$$ 0 0
$$223$$ −26.4558 −1.77161 −0.885807 0.464054i $$-0.846394\pi$$
−0.885807 + 0.464054i $$0.846394\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 0 0
$$227$$ 12.3225 0.817872 0.408936 0.912563i $$-0.365900\pi$$
0.408936 + 0.912563i $$0.365900\pi$$
$$228$$ 0 0
$$229$$ 13.2257 0.873979 0.436989 0.899467i $$-0.356045\pi$$
0.436989 + 0.899467i $$0.356045\pi$$
$$230$$ 0 0
$$231$$ −38.9590 −2.56331
$$232$$ 0 0
$$233$$ −6.66370 −0.436554 −0.218277 0.975887i $$-0.570044\pi$$
−0.218277 + 0.975887i $$0.570044\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ 0 0
$$237$$ 15.2257 0.989015
$$238$$ 0 0
$$239$$ −22.9590 −1.48509 −0.742547 0.669794i $$-0.766382\pi$$
−0.742547 + 0.669794i $$0.766382\pi$$
$$240$$ 0 0
$$241$$ −14.0415 −0.904492 −0.452246 0.891893i $$-0.649377\pi$$
−0.452246 + 0.891893i $$0.649377\pi$$
$$242$$ 0 0
$$243$$ −9.00492 −0.577666
$$244$$ 0 0
$$245$$ 0 0
$$246$$ 0 0
$$247$$ −13.2444 −0.842723
$$248$$ 0 0
$$249$$ 35.2672 2.23497
$$250$$ 0 0
$$251$$ −24.9304 −1.57359 −0.786797 0.617212i $$-0.788262\pi$$
−0.786797 + 0.617212i $$0.788262\pi$$
$$252$$ 0 0
$$253$$ −1.80642 −0.113569
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 0 0
$$257$$ −25.7146 −1.60403 −0.802015 0.597304i $$-0.796239\pi$$
−0.802015 + 0.597304i $$0.796239\pi$$
$$258$$ 0 0
$$259$$ −25.3274 −1.57377
$$260$$ 0 0
$$261$$ −10.8573 −0.672049
$$262$$ 0 0
$$263$$ −2.57628 −0.158860 −0.0794302 0.996840i $$-0.525310\pi$$
−0.0794302 + 0.996840i $$0.525310\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ 0 0
$$267$$ −35.1338 −2.15016
$$268$$ 0 0
$$269$$ 25.7462 1.56977 0.784887 0.619639i $$-0.212721\pi$$
0.784887 + 0.619639i $$0.212721\pi$$
$$270$$ 0 0
$$271$$ −30.1847 −1.83359 −0.916795 0.399359i $$-0.869233\pi$$
−0.916795 + 0.399359i $$0.869233\pi$$
$$272$$ 0 0
$$273$$ −26.8385 −1.62434
$$274$$ 0 0
$$275$$ 0 0
$$276$$ 0 0
$$277$$ −10.5303 −0.632707 −0.316354 0.948641i $$-0.602459\pi$$
−0.316354 + 0.948641i $$0.602459\pi$$
$$278$$ 0 0
$$279$$ 14.9590 0.895571
$$280$$ 0 0
$$281$$ −7.93978 −0.473647 −0.236824 0.971553i $$-0.576106\pi$$
−0.236824 + 0.971553i $$0.576106\pi$$
$$282$$ 0 0
$$283$$ 10.9032 0.648129 0.324064 0.946035i $$-0.394951\pi$$
0.324064 + 0.946035i $$0.394951\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ −18.2766 −1.07883
$$288$$ 0 0
$$289$$ 16.7146 0.983209
$$290$$ 0 0
$$291$$ 40.0830 2.34971
$$292$$ 0 0
$$293$$ 4.42864 0.258724 0.129362 0.991597i $$-0.458707\pi$$
0.129362 + 0.991597i $$0.458707\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ 0 0
$$297$$ 26.8385 1.55733
$$298$$ 0 0
$$299$$ −1.24443 −0.0719673
$$300$$ 0 0
$$301$$ 6.88892 0.397071
$$302$$ 0 0
$$303$$ 57.2355 3.28810
$$304$$ 0 0
$$305$$ 0 0
$$306$$ 0 0
$$307$$ −5.27163 −0.300868 −0.150434 0.988620i $$-0.548067\pi$$
−0.150434 + 0.988620i $$0.548067\pi$$
$$308$$ 0 0
$$309$$ 9.11108 0.518311
$$310$$ 0 0
$$311$$ −0.387152 −0.0219534 −0.0109767 0.999940i $$-0.503494\pi$$
−0.0109767 + 0.999940i $$0.503494\pi$$
$$312$$ 0 0
$$313$$ 11.3461 0.641322 0.320661 0.947194i $$-0.396095\pi$$
0.320661 + 0.947194i $$0.396095\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ −16.7239 −0.939309 −0.469655 0.882850i $$-0.655622\pi$$
−0.469655 + 0.882850i $$0.655622\pi$$
$$318$$ 0 0
$$319$$ −7.61285 −0.426238
$$320$$ 0 0
$$321$$ −15.6543 −0.873740
$$322$$ 0 0
$$323$$ −29.3274 −1.63182
$$324$$ 0 0
$$325$$ 0 0
$$326$$ 0 0
$$327$$ 25.0321 1.38428
$$328$$ 0 0
$$329$$ −18.7971 −1.03632
$$330$$ 0 0
$$331$$ −3.33630 −0.183379 −0.0916897 0.995788i $$-0.529227\pi$$
−0.0916897 + 0.995788i $$0.529227\pi$$
$$332$$ 0 0
$$333$$ 39.0005 2.13721
$$334$$ 0 0
$$335$$ 0 0
$$336$$ 0 0
$$337$$ −3.61285 −0.196804 −0.0984022 0.995147i $$-0.531373\pi$$
−0.0984022 + 0.995147i $$0.531373\pi$$
$$338$$ 0 0
$$339$$ 16.0000 0.869001
$$340$$ 0 0
$$341$$ 10.4889 0.568004
$$342$$ 0 0
$$343$$ 5.53972 0.299117
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ −15.7605 −0.846067 −0.423034 0.906114i $$-0.639035\pi$$
−0.423034 + 0.906114i $$0.639035\pi$$
$$348$$ 0 0
$$349$$ −0.285442 −0.0152794 −0.00763968 0.999971i $$-0.502432\pi$$
−0.00763968 + 0.999971i $$0.502432\pi$$
$$350$$ 0 0
$$351$$ 18.4889 0.986862
$$352$$ 0 0
$$353$$ −4.38715 −0.233505 −0.116752 0.993161i $$-0.537248\pi$$
−0.116752 + 0.993161i $$0.537248\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ 0 0
$$357$$ −59.4291 −3.14532
$$358$$ 0 0
$$359$$ 20.5906 1.08673 0.543364 0.839497i $$-0.317150\pi$$
0.543364 + 0.839497i $$0.317150\pi$$
$$360$$ 0 0
$$361$$ 6.51114 0.342691
$$362$$ 0 0
$$363$$ 10.1289 0.531630
$$364$$ 0 0
$$365$$ 0 0
$$366$$ 0 0
$$367$$ 16.5575 0.864297 0.432148 0.901802i $$-0.357756\pi$$
0.432148 + 0.901802i $$0.357756\pi$$
$$368$$ 0 0
$$369$$ 28.1432 1.46508
$$370$$ 0 0
$$371$$ −18.9590 −0.984302
$$372$$ 0 0
$$373$$ −24.8988 −1.28921 −0.644605 0.764516i $$-0.722978\pi$$
−0.644605 + 0.764516i $$0.722978\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ −5.24443 −0.270102
$$378$$ 0 0
$$379$$ −9.31756 −0.478611 −0.239305 0.970944i $$-0.576920\pi$$
−0.239305 + 0.970944i $$0.576920\pi$$
$$380$$ 0 0
$$381$$ −29.8479 −1.52915
$$382$$ 0 0
$$383$$ −32.5575 −1.66361 −0.831806 0.555066i $$-0.812693\pi$$
−0.831806 + 0.555066i $$0.812693\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ −10.6079 −0.539231
$$388$$ 0 0
$$389$$ −18.9906 −0.962863 −0.481432 0.876484i $$-0.659883\pi$$
−0.481432 + 0.876484i $$0.659883\pi$$
$$390$$ 0 0
$$391$$ −2.75557 −0.139355
$$392$$ 0 0
$$393$$ −13.5397 −0.682988
$$394$$ 0 0
$$395$$ 0 0
$$396$$ 0 0
$$397$$ 32.9906 1.65575 0.827876 0.560911i $$-0.189549\pi$$
0.827876 + 0.560911i $$0.189549\pi$$
$$398$$ 0 0
$$399$$ 51.6958 2.58803
$$400$$ 0 0
$$401$$ −16.1017 −0.804081 −0.402041 0.915622i $$-0.631699\pi$$
−0.402041 + 0.915622i $$0.631699\pi$$
$$402$$ 0 0
$$403$$ 7.22570 0.359938
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 0 0
$$407$$ 27.3461 1.35550
$$408$$ 0 0
$$409$$ −33.3876 −1.65091 −0.825456 0.564466i $$-0.809082\pi$$
−0.825456 + 0.564466i $$0.809082\pi$$
$$410$$ 0 0
$$411$$ 32.9403 1.62482
$$412$$ 0 0
$$413$$ −17.8064 −0.876197
$$414$$ 0 0
$$415$$ 0 0
$$416$$ 0 0
$$417$$ −34.2766 −1.67853
$$418$$ 0 0
$$419$$ −27.4193 −1.33952 −0.669760 0.742578i $$-0.733603\pi$$
−0.669760 + 0.742578i $$0.733603\pi$$
$$420$$ 0 0
$$421$$ −12.2351 −0.596301 −0.298150 0.954519i $$-0.596370\pi$$
−0.298150 + 0.954519i $$0.596370\pi$$
$$422$$ 0 0
$$423$$ 28.9447 1.40734
$$424$$ 0 0
$$425$$ 0 0
$$426$$ 0 0
$$427$$ 43.1338 2.08739
$$428$$ 0 0
$$429$$ 28.9777 1.39906
$$430$$ 0 0
$$431$$ 28.4701 1.37136 0.685679 0.727904i $$-0.259505\pi$$
0.685679 + 0.727904i $$0.259505\pi$$
$$432$$ 0 0
$$433$$ 34.0098 1.63441 0.817204 0.576348i $$-0.195523\pi$$
0.817204 + 0.576348i $$0.195523\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 0 0
$$437$$ 2.39700 0.114664
$$438$$ 0 0
$$439$$ 14.8385 0.708205 0.354103 0.935207i $$-0.384786\pi$$
0.354103 + 0.935207i $$0.384786\pi$$
$$440$$ 0 0
$$441$$ 29.4701 1.40334
$$442$$ 0 0
$$443$$ −13.0968 −0.622247 −0.311124 0.950369i $$-0.600705\pi$$
−0.311124 + 0.950369i $$0.600705\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ 0 0
$$447$$ −15.9081 −0.752429
$$448$$ 0 0
$$449$$ −21.3876 −1.00934 −0.504672 0.863311i $$-0.668387\pi$$
−0.504672 + 0.863311i $$0.668387\pi$$
$$450$$ 0 0
$$451$$ 19.7333 0.929205
$$452$$ 0 0
$$453$$ −68.5531 −3.22091
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ 17.9813 0.841128 0.420564 0.907263i $$-0.361832\pi$$
0.420564 + 0.907263i $$0.361832\pi$$
$$458$$ 0 0
$$459$$ 40.9403 1.91093
$$460$$ 0 0
$$461$$ −10.7368 −0.500064 −0.250032 0.968238i $$-0.580441\pi$$
−0.250032 + 0.968238i $$0.580441\pi$$
$$462$$ 0 0
$$463$$ 9.30327 0.432360 0.216180 0.976354i $$-0.430640\pi$$
0.216180 + 0.976354i $$0.430640\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ 8.70964 0.403034 0.201517 0.979485i $$-0.435413\pi$$
0.201517 + 0.979485i $$0.435413\pi$$
$$468$$ 0 0
$$469$$ 27.3590 1.26332
$$470$$ 0 0
$$471$$ 2.36842 0.109131
$$472$$ 0 0
$$473$$ −7.43801 −0.342000
$$474$$ 0 0
$$475$$ 0 0
$$476$$ 0 0
$$477$$ 29.1941 1.33670
$$478$$ 0 0
$$479$$ −23.2257 −1.06121 −0.530605 0.847619i $$-0.678035\pi$$
−0.530605 + 0.847619i $$0.678035\pi$$
$$480$$ 0 0
$$481$$ 18.8385 0.858964
$$482$$ 0 0
$$483$$ 4.85728 0.221014
$$484$$ 0 0
$$485$$ 0 0
$$486$$ 0 0
$$487$$ −32.8528 −1.48870 −0.744352 0.667787i $$-0.767241\pi$$
−0.744352 + 0.667787i $$0.767241\pi$$
$$488$$ 0 0
$$489$$ 31.6543 1.43146
$$490$$ 0 0
$$491$$ 2.94914 0.133093 0.0665465 0.997783i $$-0.478802\pi$$
0.0665465 + 0.997783i $$0.478802\pi$$
$$492$$ 0 0
$$493$$ −11.6128 −0.523016
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ −17.1240 −0.768116
$$498$$ 0 0
$$499$$ 35.0321 1.56825 0.784127 0.620601i $$-0.213111\pi$$
0.784127 + 0.620601i $$0.213111\pi$$
$$500$$ 0 0
$$501$$ 19.0923 0.852983
$$502$$ 0 0
$$503$$ −16.2908 −0.726373 −0.363186 0.931717i $$-0.618311\pi$$
−0.363186 + 0.931717i $$0.618311\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ 0 0
$$507$$ −17.7792 −0.789603
$$508$$ 0 0
$$509$$ −26.0000 −1.15243 −0.576215 0.817298i $$-0.695471\pi$$
−0.576215 + 0.817298i $$0.695471\pi$$
$$510$$ 0 0
$$511$$ 23.4924 1.03924
$$512$$ 0 0
$$513$$ −35.6128 −1.57235
$$514$$ 0 0
$$515$$ 0 0
$$516$$ 0 0
$$517$$ 20.2953 0.892586
$$518$$ 0 0
$$519$$ −30.5718 −1.34195
$$520$$ 0 0
$$521$$ 11.7146 0.513224 0.256612 0.966514i $$-0.417394\pi$$
0.256612 + 0.966514i $$0.417394\pi$$
$$522$$ 0 0
$$523$$ −38.0370 −1.66324 −0.831622 0.555342i $$-0.812587\pi$$
−0.831622 + 0.555342i $$0.812587\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ 16.0000 0.696971
$$528$$ 0 0
$$529$$ −22.7748 −0.990208
$$530$$ 0 0
$$531$$ 27.4193 1.18990
$$532$$ 0 0
$$533$$ 13.5941 0.588826
$$534$$ 0 0
$$535$$ 0 0
$$536$$ 0 0
$$537$$ 18.2766 0.788691
$$538$$ 0 0
$$539$$ 20.6637 0.890049
$$540$$ 0 0
$$541$$ −11.5111 −0.494902 −0.247451 0.968900i $$-0.579593\pi$$
−0.247451 + 0.968900i $$0.579593\pi$$
$$542$$ 0 0
$$543$$ −1.41927 −0.0609068
$$544$$ 0 0
$$545$$ 0 0
$$546$$ 0 0
$$547$$ 30.2208 1.29215 0.646073 0.763275i $$-0.276410\pi$$
0.646073 + 0.763275i $$0.276410\pi$$
$$548$$ 0 0
$$549$$ −66.4197 −2.83473
$$550$$ 0 0
$$551$$ 10.1017 0.430347
$$552$$ 0 0
$$553$$ −18.4889 −0.786226
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 0 0
$$557$$ 9.75605 0.413377 0.206688 0.978407i $$-0.433731\pi$$
0.206688 + 0.978407i $$0.433731\pi$$
$$558$$ 0 0
$$559$$ −5.12399 −0.216721
$$560$$ 0 0
$$561$$ 64.1659 2.70909
$$562$$ 0 0
$$563$$ −4.50622 −0.189914 −0.0949572 0.995481i $$-0.530271\pi$$
−0.0949572 + 0.995481i $$0.530271\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ 0 0
$$567$$ −14.7511 −0.619489
$$568$$ 0 0
$$569$$ 5.30465 0.222383 0.111191 0.993799i $$-0.464533\pi$$
0.111191 + 0.993799i $$0.464533\pi$$
$$570$$ 0 0
$$571$$ −16.3970 −0.686193 −0.343096 0.939300i $$-0.611476\pi$$
−0.343096 + 0.939300i $$0.611476\pi$$
$$572$$ 0 0
$$573$$ 30.4514 1.27213
$$574$$ 0 0
$$575$$ 0 0
$$576$$ 0 0
$$577$$ 39.8163 1.65757 0.828786 0.559565i $$-0.189032\pi$$
0.828786 + 0.559565i $$0.189032\pi$$
$$578$$ 0 0
$$579$$ 40.0830 1.66579
$$580$$ 0 0
$$581$$ −42.8256 −1.77671
$$582$$ 0 0
$$583$$ 20.4701 0.847786
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ −15.1699 −0.626130 −0.313065 0.949732i $$-0.601356\pi$$
−0.313065 + 0.949732i $$0.601356\pi$$
$$588$$ 0 0
$$589$$ −13.9180 −0.573480
$$590$$ 0 0
$$591$$ 48.5531 1.99721
$$592$$ 0 0
$$593$$ −8.00000 −0.328521 −0.164260 0.986417i $$-0.552524\pi$$
−0.164260 + 0.986417i $$0.552524\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 0 0
$$597$$ −60.5531 −2.47827
$$598$$ 0 0
$$599$$ −1.83500 −0.0749762 −0.0374881 0.999297i $$-0.511936\pi$$
−0.0374881 + 0.999297i $$0.511936\pi$$
$$600$$ 0 0
$$601$$ −36.1432 −1.47431 −0.737156 0.675723i $$-0.763832\pi$$
−0.737156 + 0.675723i $$0.763832\pi$$
$$602$$ 0 0
$$603$$ −42.1289 −1.71562
$$604$$ 0 0
$$605$$ 0 0
$$606$$ 0 0
$$607$$ 17.5353 0.711735 0.355867 0.934536i $$-0.384185\pi$$
0.355867 + 0.934536i $$0.384185\pi$$
$$608$$ 0 0
$$609$$ 20.4701 0.829491
$$610$$ 0 0
$$611$$ 13.9813 0.565621
$$612$$ 0 0
$$613$$ 33.9309 1.37046 0.685228 0.728329i $$-0.259703\pi$$
0.685228 + 0.728329i $$0.259703\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ −9.68598 −0.389943 −0.194971 0.980809i $$-0.562461\pi$$
−0.194971 + 0.980809i $$0.562461\pi$$
$$618$$ 0 0
$$619$$ −41.4005 −1.66403 −0.832014 0.554755i $$-0.812812\pi$$
−0.832014 + 0.554755i $$0.812812\pi$$
$$620$$ 0 0
$$621$$ −3.34614 −0.134276
$$622$$ 0 0
$$623$$ 42.6637 1.70929
$$624$$ 0 0
$$625$$ 0 0
$$626$$ 0 0
$$627$$ −55.8163 −2.22909
$$628$$ 0 0
$$629$$ 41.7146 1.66327
$$630$$ 0 0
$$631$$ 27.8163 1.10735 0.553674 0.832733i $$-0.313225\pi$$
0.553674 + 0.832733i $$0.313225\pi$$
$$632$$ 0 0
$$633$$ −13.5397 −0.538155
$$634$$ 0 0
$$635$$ 0 0
$$636$$ 0 0
$$637$$ 14.2351 0.564014
$$638$$ 0 0
$$639$$ 26.3684 1.04312
$$640$$ 0 0
$$641$$ 42.8988 1.69440 0.847200 0.531275i $$-0.178287\pi$$
0.847200 + 0.531275i $$0.178287\pi$$
$$642$$ 0 0
$$643$$ 27.9639 1.10279 0.551395 0.834245i $$-0.314096\pi$$
0.551395 + 0.834245i $$0.314096\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ −7.13828 −0.280635 −0.140317 0.990107i $$-0.544812\pi$$
−0.140317 + 0.990107i $$0.544812\pi$$
$$648$$ 0 0
$$649$$ 19.2257 0.754675
$$650$$ 0 0
$$651$$ −28.2034 −1.10538
$$652$$ 0 0
$$653$$ −17.7649 −0.695196 −0.347598 0.937644i $$-0.613003\pi$$
−0.347598 + 0.937644i $$0.613003\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ 0 0
$$657$$ −36.1748 −1.41131
$$658$$ 0 0
$$659$$ −20.1936 −0.786630 −0.393315 0.919404i $$-0.628672\pi$$
−0.393315 + 0.919404i $$0.628672\pi$$
$$660$$ 0 0
$$661$$ 22.0701 0.858426 0.429213 0.903203i $$-0.358791\pi$$
0.429213 + 0.903203i $$0.358791\pi$$
$$662$$ 0 0
$$663$$ 44.2034 1.71672
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 0 0
$$667$$ 0.949145 0.0367510
$$668$$ 0 0
$$669$$ −76.8069 −2.96953
$$670$$ 0 0
$$671$$ −46.5718 −1.79789
$$672$$ 0 0
$$673$$ −26.0098 −1.00261 −0.501303 0.865272i $$-0.667146\pi$$
−0.501303 + 0.865272i $$0.667146\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ 0 0
$$677$$ −7.86665 −0.302340 −0.151170 0.988508i $$-0.548304\pi$$
−0.151170 + 0.988508i $$0.548304\pi$$
$$678$$ 0 0
$$679$$ −48.6735 −1.86792
$$680$$ 0 0
$$681$$ 35.7748 1.37089
$$682$$ 0 0
$$683$$ 18.2494 0.698292 0.349146 0.937068i $$-0.386472\pi$$
0.349146 + 0.937068i $$0.386472\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ 0 0
$$687$$ 38.3970 1.46494
$$688$$ 0 0
$$689$$ 14.1017 0.537232
$$690$$ 0 0
$$691$$ 25.2543 0.960718 0.480359 0.877072i $$-0.340506\pi$$
0.480359 + 0.877072i $$0.340506\pi$$
$$692$$ 0 0
$$693$$ −72.8484 −2.76728
$$694$$ 0 0
$$695$$ 0 0
$$696$$ 0 0
$$697$$ 30.1017 1.14018
$$698$$ 0 0
$$699$$ −19.3461 −0.731738
$$700$$ 0 0
$$701$$ 34.8069 1.31464 0.657319 0.753612i $$-0.271690\pi$$
0.657319 + 0.753612i $$0.271690\pi$$
$$702$$ 0 0
$$703$$ −36.2864 −1.36857
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ −69.5022 −2.61390
$$708$$ 0 0
$$709$$ −7.51114 −0.282087 −0.141043 0.990003i $$-0.545046\pi$$
−0.141043 + 0.990003i $$0.545046\pi$$
$$710$$ 0 0
$$711$$ 28.4701 1.06771
$$712$$ 0 0
$$713$$ −1.30772 −0.0489744
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 0 0
$$717$$ −66.6548 −2.48927
$$718$$ 0 0
$$719$$ −26.7556 −0.997814 −0.498907 0.866655i $$-0.666265\pi$$
−0.498907 + 0.866655i $$0.666265\pi$$
$$720$$ 0 0
$$721$$ −11.0638 −0.412036
$$722$$ 0 0
$$723$$ −40.7654 −1.51608
$$724$$ 0 0
$$725$$ 0 0
$$726$$ 0 0
$$727$$ −25.6271 −0.950458 −0.475229 0.879862i $$-0.657635\pi$$
−0.475229 + 0.879862i $$0.657635\pi$$
$$728$$ 0 0
$$729$$ −38.6958 −1.43318
$$730$$ 0 0
$$731$$ −11.3461 −0.419652
$$732$$ 0 0
$$733$$ −19.6543 −0.725949 −0.362975 0.931799i $$-0.618239\pi$$
−0.362975 + 0.931799i $$0.618239\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ −29.5397 −1.08811
$$738$$ 0 0
$$739$$ 32.5433 1.19712 0.598562 0.801077i $$-0.295739\pi$$
0.598562 + 0.801077i $$0.295739\pi$$
$$740$$ 0 0
$$741$$ −38.4514 −1.41255
$$742$$ 0 0
$$743$$ 23.1383 0.848861 0.424430 0.905461i $$-0.360474\pi$$
0.424430 + 0.905461i $$0.360474\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ 0 0
$$747$$ 65.9452 2.41281
$$748$$ 0 0
$$749$$ 19.0094 0.694587
$$750$$ 0 0
$$751$$ 2.48886 0.0908199 0.0454099 0.998968i $$-0.485541\pi$$
0.0454099 + 0.998968i $$0.485541\pi$$
$$752$$ 0 0
$$753$$ −72.3783 −2.63761
$$754$$ 0 0
$$755$$ 0 0
$$756$$ 0 0
$$757$$ 21.2859 0.773650 0.386825 0.922153i $$-0.373572\pi$$
0.386825 + 0.922153i $$0.373572\pi$$
$$758$$ 0 0
$$759$$ −5.24443 −0.190361
$$760$$ 0 0
$$761$$ 19.1240 0.693244 0.346622 0.938005i $$-0.387329\pi$$
0.346622 + 0.938005i $$0.387329\pi$$
$$762$$ 0 0
$$763$$ −30.3970 −1.10045
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ 13.2444 0.478229
$$768$$ 0 0
$$769$$ −33.9625 −1.22472 −0.612360 0.790579i $$-0.709780\pi$$
−0.612360 + 0.790579i $$0.709780\pi$$
$$770$$ 0 0
$$771$$ −74.6548 −2.68863
$$772$$ 0 0
$$773$$ −0.133353 −0.00479638 −0.00239819 0.999997i $$-0.500763\pi$$
−0.00239819 + 0.999997i $$0.500763\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ 0 0
$$777$$ −73.5308 −2.63790
$$778$$ 0 0
$$779$$ −26.1847 −0.938164
$$780$$ 0 0
$$781$$ 18.4889 0.661584
$$782$$ 0 0
$$783$$ −14.1017 −0.503954
$$784$$ 0 0
$$785$$ 0 0
$$786$$ 0 0
$$787$$ −1.12537 −0.0401150 −0.0200575 0.999799i $$-0.506385\pi$$
−0.0200575 + 0.999799i $$0.506385\pi$$
$$788$$ 0 0
$$789$$ −7.47949 −0.266277
$$790$$ 0 0
$$791$$ −19.4291 −0.690820
$$792$$ 0 0
$$793$$ −32.0830 −1.13930
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 0 0
$$797$$ −5.11108 −0.181044 −0.0905218 0.995894i $$-0.528853\pi$$
−0.0905218 + 0.995894i $$0.528853\pi$$
$$798$$ 0 0
$$799$$ 30.9590 1.09525
$$800$$ 0 0
$$801$$ −65.6958 −2.32125
$$802$$ 0 0
$$803$$ −25.3649 −0.895107
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 0 0
$$807$$ 74.7467 2.63121
$$808$$ 0 0
$$809$$ 18.7368 0.658752 0.329376 0.944199i $$-0.393162\pi$$
0.329376 + 0.944199i $$0.393162\pi$$
$$810$$ 0 0
$$811$$ 37.5210 1.31754 0.658770 0.752344i $$-0.271077\pi$$
0.658770 + 0.752344i $$0.271077\pi$$
$$812$$ 0 0
$$813$$ −87.6325 −3.07341
$$814$$ 0 0
$$815$$ 0 0
$$816$$ 0 0
$$817$$ 9.86971 0.345297
$$818$$ 0 0
$$819$$ −50.1847 −1.75359
$$820$$ 0 0
$$821$$ 18.4001 0.642166 0.321083 0.947051i $$-0.395953\pi$$
0.321083 + 0.947051i $$0.395953\pi$$
$$822$$ 0 0
$$823$$ 0.649413 0.0226371 0.0113186 0.999936i $$-0.496397\pi$$
0.0113186 + 0.999936i $$0.496397\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ −29.8622 −1.03841 −0.519205 0.854650i $$-0.673772\pi$$
−0.519205 + 0.854650i $$0.673772\pi$$
$$828$$ 0 0
$$829$$ 21.1753 0.735449 0.367725 0.929935i $$-0.380137\pi$$
0.367725 + 0.929935i $$0.380137\pi$$
$$830$$ 0 0
$$831$$ −30.5718 −1.06053
$$832$$ 0 0
$$833$$ 31.5210 1.09214
$$834$$ 0 0
$$835$$ 0 0
$$836$$ 0 0
$$837$$ 19.4291 0.671568
$$838$$ 0 0
$$839$$ 43.8163 1.51271 0.756353 0.654164i $$-0.226979\pi$$
0.756353 + 0.654164i $$0.226979\pi$$
$$840$$ 0 0
$$841$$ −25.0000 −0.862069
$$842$$ 0 0
$$843$$ −23.0509 −0.793914
$$844$$ 0 0
$$845$$ 0 0
$$846$$ 0 0
$$847$$ −12.2997 −0.422624
$$848$$ 0 0
$$849$$ 31.6543 1.08637
$$850$$ 0 0
$$851$$ −3.40943 −0.116874
$$852$$ 0 0
$$853$$ 37.7846 1.29372 0.646860 0.762608i $$-0.276082\pi$$
0.646860 + 0.762608i $$0.276082\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0 0
$$857$$ −19.5299 −0.667128 −0.333564 0.942727i $$-0.608251\pi$$
−0.333564 + 0.942727i $$0.608251\pi$$
$$858$$ 0 0
$$859$$ 33.2543 1.13462 0.567311 0.823504i $$-0.307984\pi$$
0.567311 + 0.823504i $$0.307984\pi$$
$$860$$ 0 0
$$861$$ −53.0607 −1.80830
$$862$$ 0 0
$$863$$ 44.9733 1.53091 0.765454 0.643490i $$-0.222514\pi$$
0.765454 + 0.643490i $$0.222514\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ 0 0
$$867$$ 48.5259 1.64803
$$868$$ 0 0
$$869$$ 19.9625 0.677182
$$870$$ 0 0
$$871$$ −20.3497 −0.689523
$$872$$ 0 0
$$873$$ 74.9501 2.53668
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 0 0
$$877$$ −8.30819 −0.280548 −0.140274 0.990113i $$-0.544798\pi$$
−0.140274 + 0.990113i $$0.544798\pi$$
$$878$$ 0 0
$$879$$ 12.8573 0.433665
$$880$$ 0 0
$$881$$ −39.3689 −1.32637 −0.663186 0.748455i $$-0.730796\pi$$
−0.663186 + 0.748455i $$0.730796\pi$$
$$882$$ 0 0
$$883$$ 7.29036 0.245340 0.122670 0.992447i $$-0.460854\pi$$
0.122670 + 0.992447i $$0.460854\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 0 0
$$887$$ 11.1097 0.373027 0.186514 0.982452i $$-0.440281\pi$$
0.186514 + 0.982452i $$0.440281\pi$$
$$888$$ 0 0
$$889$$ 36.2449 1.21562
$$890$$ 0 0
$$891$$ 15.9269 0.533570
$$892$$ 0 0
$$893$$ −26.9304 −0.901192
$$894$$ 0 0
$$895$$ 0 0
$$896$$ 0 0
$$897$$ −3.61285 −0.120629
$$898$$ 0 0
$$899$$ −5.51114 −0.183807
$$900$$ 0 0
$$901$$ 31.2257 1.04028
$$902$$ 0 0
$$903$$ 20.0000 0.665558
$$904$$ 0 0
$$905$$ 0 0
$$906$$ 0 0
$$907$$ −17.5383 −0.582351 −0.291175 0.956670i $$-0.594046\pi$$
−0.291175 + 0.956670i $$0.594046\pi$$
$$908$$ 0 0
$$909$$ 107.023 3.54974
$$910$$ 0 0
$$911$$ −44.4701 −1.47336 −0.736681 0.676241i $$-0.763608\pi$$
−0.736681 + 0.676241i $$0.763608\pi$$
$$912$$ 0 0
$$913$$ 46.2391 1.53029
$$914$$ 0 0
$$915$$ 0 0
$$916$$ 0 0
$$917$$ 16.4415 0.542948
$$918$$ 0 0
$$919$$ −7.87955 −0.259922 −0.129961 0.991519i $$-0.541485\pi$$
−0.129961 + 0.991519i $$0.541485\pi$$
$$920$$ 0 0
$$921$$ −15.3047 −0.504306
$$922$$ 0 0
$$923$$ 12.7368 0.419238
$$924$$ 0 0
$$925$$ 0 0
$$926$$ 0 0
$$927$$ 17.0366 0.559554
$$928$$ 0 0
$$929$$ 15.7560 0.516939 0.258470 0.966019i $$-0.416782\pi$$
0.258470 + 0.966019i $$0.416782\pi$$
$$930$$ 0 0
$$931$$ −27.4193 −0.898630
$$932$$ 0 0
$$933$$ −1.12399 −0.0367976
$$934$$ 0 0
$$935$$ 0 0
$$936$$ 0 0
$$937$$ −10.2766 −0.335720 −0.167860 0.985811i $$-0.553686\pi$$
−0.167860 + 0.985811i $$0.553686\pi$$
$$938$$ 0 0
$$939$$ 32.9403 1.07496
$$940$$ 0 0
$$941$$ 2.53341 0.0825869 0.0412934 0.999147i $$-0.486852\pi$$
0.0412934 + 0.999147i $$0.486852\pi$$
$$942$$ 0 0
$$943$$ −2.46028 −0.0801178
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ −1.06821 −0.0347121 −0.0173560 0.999849i $$-0.505525\pi$$
−0.0173560 + 0.999849i $$0.505525\pi$$
$$948$$ 0 0
$$949$$ −17.4737 −0.567219
$$950$$ 0 0
$$951$$ −48.5531 −1.57444
$$952$$ 0 0
$$953$$ −51.6958 −1.67459 −0.837296 0.546750i $$-0.815865\pi$$
−0.837296 + 0.546750i $$0.815865\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ 0 0
$$957$$ −22.1017 −0.714447
$$958$$ 0 0
$$959$$ −40.0000 −1.29167
$$960$$ 0 0
$$961$$ −23.4068 −0.755059
$$962$$ 0 0
$$963$$ −29.2716 −0.943265
$$964$$ 0 0
$$965$$ 0 0
$$966$$ 0 0
$$967$$ −52.2623 −1.68064 −0.840320 0.542090i $$-0.817633\pi$$
−0.840320 + 0.542090i $$0.817633\pi$$
$$968$$ 0 0
$$969$$ −85.1437 −2.73521
$$970$$ 0 0
$$971$$ 59.3560 1.90482 0.952412 0.304813i $$-0.0985941\pi$$
0.952412 + 0.304813i $$0.0985941\pi$$
$$972$$ 0 0
$$973$$ 41.6227 1.33436
$$974$$ 0 0
$$975$$ 0 0
$$976$$ 0 0
$$977$$ 47.8707 1.53152 0.765759 0.643128i $$-0.222363\pi$$
0.765759 + 0.643128i $$0.222363\pi$$
$$978$$ 0 0
$$979$$ −46.0642 −1.47222
$$980$$ 0 0
$$981$$ 46.8069 1.49443
$$982$$ 0 0
$$983$$ 6.01429 0.191826 0.0959130 0.995390i $$-0.469423\pi$$
0.0959130 + 0.995390i $$0.469423\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ 0 0
$$987$$ −54.5718 −1.73704
$$988$$ 0 0
$$989$$ 0.927346 0.0294879
$$990$$ 0 0
$$991$$ 19.4291 0.617186 0.308593 0.951194i $$-0.400142\pi$$
0.308593 + 0.951194i $$0.400142\pi$$
$$992$$ 0 0
$$993$$ −9.68598 −0.307375
$$994$$ 0 0
$$995$$ 0 0
$$996$$ 0 0
$$997$$ 32.1334 1.01767 0.508837 0.860863i $$-0.330076\pi$$
0.508837 + 0.860863i $$0.330076\pi$$
$$998$$ 0 0
$$999$$ 50.6548 1.60265
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.bs.1.3 3
4.3 odd 2 3200.2.a.br.1.1 3
5.2 odd 4 640.2.c.a.129.1 6
5.3 odd 4 640.2.c.a.129.6 yes 6
5.4 even 2 3200.2.a.bq.1.1 3
8.3 odd 2 3200.2.a.bu.1.3 3
8.5 even 2 3200.2.a.bp.1.1 3
20.3 even 4 640.2.c.b.129.1 yes 6
20.7 even 4 640.2.c.b.129.6 yes 6
20.19 odd 2 3200.2.a.bt.1.3 3
40.3 even 4 640.2.c.c.129.6 yes 6
40.13 odd 4 640.2.c.d.129.1 yes 6
40.19 odd 2 3200.2.a.bo.1.1 3
40.27 even 4 640.2.c.c.129.1 yes 6
40.29 even 2 3200.2.a.bv.1.3 3
40.37 odd 4 640.2.c.d.129.6 yes 6
80.3 even 4 1280.2.f.k.129.6 6
80.13 odd 4 1280.2.f.i.129.2 6
80.27 even 4 1280.2.f.k.129.5 6
80.37 odd 4 1280.2.f.i.129.1 6
80.43 even 4 1280.2.f.j.129.1 6
80.53 odd 4 1280.2.f.l.129.5 6
80.67 even 4 1280.2.f.j.129.2 6
80.77 odd 4 1280.2.f.l.129.6 6

By twisted newform
Twist Min Dim Char Parity Ord Type
640.2.c.a.129.1 6 5.2 odd 4
640.2.c.a.129.6 yes 6 5.3 odd 4
640.2.c.b.129.1 yes 6 20.3 even 4
640.2.c.b.129.6 yes 6 20.7 even 4
640.2.c.c.129.1 yes 6 40.27 even 4
640.2.c.c.129.6 yes 6 40.3 even 4
640.2.c.d.129.1 yes 6 40.13 odd 4
640.2.c.d.129.6 yes 6 40.37 odd 4
1280.2.f.i.129.1 6 80.37 odd 4
1280.2.f.i.129.2 6 80.13 odd 4
1280.2.f.j.129.1 6 80.43 even 4
1280.2.f.j.129.2 6 80.67 even 4
1280.2.f.k.129.5 6 80.27 even 4
1280.2.f.k.129.6 6 80.3 even 4
1280.2.f.l.129.5 6 80.53 odd 4
1280.2.f.l.129.6 6 80.77 odd 4
3200.2.a.bo.1.1 3 40.19 odd 2
3200.2.a.bp.1.1 3 8.5 even 2
3200.2.a.bq.1.1 3 5.4 even 2
3200.2.a.br.1.1 3 4.3 odd 2
3200.2.a.bs.1.3 3 1.1 even 1 trivial
3200.2.a.bt.1.3 3 20.19 odd 2
3200.2.a.bu.1.3 3 8.3 odd 2
3200.2.a.bv.1.3 3 40.29 even 2