# Properties

 Label 2160.4.a.bt.1.3 Level $2160$ Weight $4$ Character 2160.1 Self dual yes Analytic conductor $127.444$ Analytic rank $0$ Dimension $3$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$2160 = 2^{4} \cdot 3^{3} \cdot 5$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 2160.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$127.444125612$$ Analytic rank: $$0$$ Dimension: $$3$$ Coefficient field: 3.3.697.1 Defining polynomial: $$x^{3} - 7x - 5$$ x^3 - 7*x - 5 Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2^{3}\cdot 3^{2}$$ Twist minimal: no (minimal twist has level 1080) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.3 Root $$-2.16601$$ of defining polynomial Character $$\chi$$ $$=$$ 2160.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+5.00000 q^{5} +21.1457 q^{7} +O(q^{10})$$ $$q+5.00000 q^{5} +21.1457 q^{7} -67.4294 q^{11} +2.55493 q^{13} -126.114 q^{17} +103.539 q^{19} -200.434 q^{23} +25.0000 q^{25} -71.4215 q^{29} +158.130 q^{31} +105.729 q^{35} +7.18163 q^{37} +347.052 q^{41} -189.318 q^{43} +585.443 q^{47} +104.142 q^{49} -77.2791 q^{53} -337.147 q^{55} +200.790 q^{59} +681.137 q^{61} +12.7746 q^{65} +810.751 q^{67} +515.476 q^{71} +385.577 q^{73} -1425.84 q^{77} +209.405 q^{79} +887.189 q^{83} -630.572 q^{85} +548.890 q^{89} +54.0258 q^{91} +517.696 q^{95} -1523.96 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q + 15 q^{5} + 24 q^{7}+O(q^{10})$$ 3 * q + 15 * q^5 + 24 * q^7 $$3 q + 15 q^{5} + 24 q^{7} - 6 q^{11} + 48 q^{13} - 27 q^{17} + 195 q^{19} + 27 q^{23} + 75 q^{25} + 60 q^{29} + 279 q^{31} + 120 q^{35} - 138 q^{37} + 66 q^{41} - 222 q^{43} + 264 q^{47} - 237 q^{49} - 507 q^{53} - 30 q^{55} + 960 q^{59} + 543 q^{61} + 240 q^{65} + 1086 q^{67} + 1818 q^{71} + 1362 q^{73} - 1776 q^{77} + 129 q^{79} + 1569 q^{83} - 135 q^{85} - 1770 q^{89} - 1488 q^{91} + 975 q^{95} - 336 q^{97}+O(q^{100})$$ 3 * q + 15 * q^5 + 24 * q^7 - 6 * q^11 + 48 * q^13 - 27 * q^17 + 195 * q^19 + 27 * q^23 + 75 * q^25 + 60 * q^29 + 279 * q^31 + 120 * q^35 - 138 * q^37 + 66 * q^41 - 222 * q^43 + 264 * q^47 - 237 * q^49 - 507 * q^53 - 30 * q^55 + 960 * q^59 + 543 * q^61 + 240 * q^65 + 1086 * q^67 + 1818 * q^71 + 1362 * q^73 - 1776 * q^77 + 129 * q^79 + 1569 * q^83 - 135 * q^85 - 1770 * q^89 - 1488 * q^91 + 975 * q^95 - 336 * q^97

## 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
$$3$$ 0 0
$$4$$ 0 0
$$5$$ 5.00000 0.447214
$$6$$ 0 0
$$7$$ 21.1457 1.14176 0.570881 0.821033i $$-0.306602\pi$$
0.570881 + 0.821033i $$0.306602\pi$$
$$8$$ 0 0
$$9$$ 0 0
$$10$$ 0 0
$$11$$ −67.4294 −1.84825 −0.924124 0.382093i $$-0.875203\pi$$
−0.924124 + 0.382093i $$0.875203\pi$$
$$12$$ 0 0
$$13$$ 2.55493 0.0545084 0.0272542 0.999629i $$-0.491324\pi$$
0.0272542 + 0.999629i $$0.491324\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 0 0
$$17$$ −126.114 −1.79925 −0.899624 0.436665i $$-0.856160\pi$$
−0.899624 + 0.436665i $$0.856160\pi$$
$$18$$ 0 0
$$19$$ 103.539 1.25019 0.625093 0.780550i $$-0.285061\pi$$
0.625093 + 0.780550i $$0.285061\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 0 0
$$23$$ −200.434 −1.81710 −0.908551 0.417774i $$-0.862810\pi$$
−0.908551 + 0.417774i $$0.862810\pi$$
$$24$$ 0 0
$$25$$ 25.0000 0.200000
$$26$$ 0 0
$$27$$ 0 0
$$28$$ 0 0
$$29$$ −71.4215 −0.457333 −0.228666 0.973505i $$-0.573436\pi$$
−0.228666 + 0.973505i $$0.573436\pi$$
$$30$$ 0 0
$$31$$ 158.130 0.916161 0.458081 0.888911i $$-0.348537\pi$$
0.458081 + 0.888911i $$0.348537\pi$$
$$32$$ 0 0
$$33$$ 0 0
$$34$$ 0 0
$$35$$ 105.729 0.510612
$$36$$ 0 0
$$37$$ 7.18163 0.0319095 0.0159548 0.999873i $$-0.494921\pi$$
0.0159548 + 0.999873i $$0.494921\pi$$
$$38$$ 0 0
$$39$$ 0 0
$$40$$ 0 0
$$41$$ 347.052 1.32196 0.660980 0.750404i $$-0.270141\pi$$
0.660980 + 0.750404i $$0.270141\pi$$
$$42$$ 0 0
$$43$$ −189.318 −0.671413 −0.335707 0.941967i $$-0.608975\pi$$
−0.335707 + 0.941967i $$0.608975\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 0 0
$$47$$ 585.443 1.81693 0.908464 0.417963i $$-0.137256\pi$$
0.908464 + 0.417963i $$0.137256\pi$$
$$48$$ 0 0
$$49$$ 104.142 0.303622
$$50$$ 0 0
$$51$$ 0 0
$$52$$ 0 0
$$53$$ −77.2791 −0.200285 −0.100143 0.994973i $$-0.531930\pi$$
−0.100143 + 0.994973i $$0.531930\pi$$
$$54$$ 0 0
$$55$$ −337.147 −0.826561
$$56$$ 0 0
$$57$$ 0 0
$$58$$ 0 0
$$59$$ 200.790 0.443062 0.221531 0.975153i $$-0.428895\pi$$
0.221531 + 0.975153i $$0.428895\pi$$
$$60$$ 0 0
$$61$$ 681.137 1.42968 0.714841 0.699287i $$-0.246499\pi$$
0.714841 + 0.699287i $$0.246499\pi$$
$$62$$ 0 0
$$63$$ 0 0
$$64$$ 0 0
$$65$$ 12.7746 0.0243769
$$66$$ 0 0
$$67$$ 810.751 1.47834 0.739172 0.673517i $$-0.235217\pi$$
0.739172 + 0.673517i $$0.235217\pi$$
$$68$$ 0 0
$$69$$ 0 0
$$70$$ 0 0
$$71$$ 515.476 0.861631 0.430816 0.902440i $$-0.358226\pi$$
0.430816 + 0.902440i $$0.358226\pi$$
$$72$$ 0 0
$$73$$ 385.577 0.618197 0.309099 0.951030i $$-0.399973\pi$$
0.309099 + 0.951030i $$0.399973\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 0 0
$$77$$ −1425.84 −2.11026
$$78$$ 0 0
$$79$$ 209.405 0.298226 0.149113 0.988820i $$-0.452358\pi$$
0.149113 + 0.988820i $$0.452358\pi$$
$$80$$ 0 0
$$81$$ 0 0
$$82$$ 0 0
$$83$$ 887.189 1.17327 0.586637 0.809850i $$-0.300452\pi$$
0.586637 + 0.809850i $$0.300452\pi$$
$$84$$ 0 0
$$85$$ −630.572 −0.804648
$$86$$ 0 0
$$87$$ 0 0
$$88$$ 0 0
$$89$$ 548.890 0.653733 0.326867 0.945071i $$-0.394007\pi$$
0.326867 + 0.945071i $$0.394007\pi$$
$$90$$ 0 0
$$91$$ 54.0258 0.0622357
$$92$$ 0 0
$$93$$ 0 0
$$94$$ 0 0
$$95$$ 517.696 0.559100
$$96$$ 0 0
$$97$$ −1523.96 −1.59520 −0.797600 0.603186i $$-0.793897\pi$$
−0.797600 + 0.603186i $$0.793897\pi$$
$$98$$ 0 0
$$99$$ 0 0
$$100$$ 0 0
$$101$$ −347.825 −0.342672 −0.171336 0.985213i $$-0.554808\pi$$
−0.171336 + 0.985213i $$0.554808\pi$$
$$102$$ 0 0
$$103$$ 1955.17 1.87037 0.935187 0.354155i $$-0.115232\pi$$
0.935187 + 0.354155i $$0.115232\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ 1576.16 1.42405 0.712026 0.702153i $$-0.247778\pi$$
0.712026 + 0.702153i $$0.247778\pi$$
$$108$$ 0 0
$$109$$ −158.232 −0.139045 −0.0695224 0.997580i $$-0.522148\pi$$
−0.0695224 + 0.997580i $$0.522148\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ 0 0
$$113$$ −583.856 −0.486058 −0.243029 0.970019i $$-0.578141\pi$$
−0.243029 + 0.970019i $$0.578141\pi$$
$$114$$ 0 0
$$115$$ −1002.17 −0.812633
$$116$$ 0 0
$$117$$ 0 0
$$118$$ 0 0
$$119$$ −2666.78 −2.05431
$$120$$ 0 0
$$121$$ 3215.72 2.41602
$$122$$ 0 0
$$123$$ 0 0
$$124$$ 0 0
$$125$$ 125.000 0.0894427
$$126$$ 0 0
$$127$$ −2572.68 −1.79754 −0.898772 0.438416i $$-0.855540\pi$$
−0.898772 + 0.438416i $$0.855540\pi$$
$$128$$ 0 0
$$129$$ 0 0
$$130$$ 0 0
$$131$$ 1793.93 1.19646 0.598229 0.801325i $$-0.295871\pi$$
0.598229 + 0.801325i $$0.295871\pi$$
$$132$$ 0 0
$$133$$ 2189.41 1.42742
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ 1729.59 1.07860 0.539302 0.842112i $$-0.318688\pi$$
0.539302 + 0.842112i $$0.318688\pi$$
$$138$$ 0 0
$$139$$ −690.713 −0.421479 −0.210739 0.977542i $$-0.567587\pi$$
−0.210739 + 0.977542i $$0.567587\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 0 0
$$143$$ −172.277 −0.100745
$$144$$ 0 0
$$145$$ −357.108 −0.204525
$$146$$ 0 0
$$147$$ 0 0
$$148$$ 0 0
$$149$$ −2876.27 −1.58143 −0.790715 0.612185i $$-0.790291\pi$$
−0.790715 + 0.612185i $$0.790291\pi$$
$$150$$ 0 0
$$151$$ −1204.10 −0.648930 −0.324465 0.945898i $$-0.605184\pi$$
−0.324465 + 0.945898i $$0.605184\pi$$
$$152$$ 0 0
$$153$$ 0 0
$$154$$ 0 0
$$155$$ 790.650 0.409720
$$156$$ 0 0
$$157$$ −790.488 −0.401833 −0.200917 0.979608i $$-0.564392\pi$$
−0.200917 + 0.979608i $$0.564392\pi$$
$$158$$ 0 0
$$159$$ 0 0
$$160$$ 0 0
$$161$$ −4238.32 −2.07470
$$162$$ 0 0
$$163$$ 2202.61 1.05842 0.529209 0.848492i $$-0.322489\pi$$
0.529209 + 0.848492i $$0.322489\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ −1833.20 −0.849445 −0.424722 0.905324i $$-0.639628\pi$$
−0.424722 + 0.905324i $$0.639628\pi$$
$$168$$ 0 0
$$169$$ −2190.47 −0.997029
$$170$$ 0 0
$$171$$ 0 0
$$172$$ 0 0
$$173$$ −2686.62 −1.18069 −0.590347 0.807150i $$-0.701009\pi$$
−0.590347 + 0.807150i $$0.701009\pi$$
$$174$$ 0 0
$$175$$ 528.644 0.228353
$$176$$ 0 0
$$177$$ 0 0
$$178$$ 0 0
$$179$$ 4500.41 1.87920 0.939598 0.342280i $$-0.111199\pi$$
0.939598 + 0.342280i $$0.111199\pi$$
$$180$$ 0 0
$$181$$ −1600.82 −0.657390 −0.328695 0.944436i $$-0.606609\pi$$
−0.328695 + 0.944436i $$0.606609\pi$$
$$182$$ 0 0
$$183$$ 0 0
$$184$$ 0 0
$$185$$ 35.9081 0.0142704
$$186$$ 0 0
$$187$$ 8503.81 3.32546
$$188$$ 0 0
$$189$$ 0 0
$$190$$ 0 0
$$191$$ 862.280 0.326662 0.163331 0.986571i $$-0.447776\pi$$
0.163331 + 0.986571i $$0.447776\pi$$
$$192$$ 0 0
$$193$$ −768.319 −0.286554 −0.143277 0.989683i $$-0.545764\pi$$
−0.143277 + 0.989683i $$0.545764\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ 3090.06 1.11755 0.558776 0.829318i $$-0.311271\pi$$
0.558776 + 0.829318i $$0.311271\pi$$
$$198$$ 0 0
$$199$$ −3255.41 −1.15965 −0.579824 0.814742i $$-0.696879\pi$$
−0.579824 + 0.814742i $$0.696879\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ 0 0
$$203$$ −1510.26 −0.522165
$$204$$ 0 0
$$205$$ 1735.26 0.591198
$$206$$ 0 0
$$207$$ 0 0
$$208$$ 0 0
$$209$$ −6981.59 −2.31065
$$210$$ 0 0
$$211$$ 2891.42 0.943383 0.471691 0.881764i $$-0.343644\pi$$
0.471691 + 0.881764i $$0.343644\pi$$
$$212$$ 0 0
$$213$$ 0 0
$$214$$ 0 0
$$215$$ −946.592 −0.300265
$$216$$ 0 0
$$217$$ 3343.78 1.04604
$$218$$ 0 0
$$219$$ 0 0
$$220$$ 0 0
$$221$$ −322.213 −0.0980742
$$222$$ 0 0
$$223$$ 1345.31 0.403983 0.201992 0.979387i $$-0.435259\pi$$
0.201992 + 0.979387i $$0.435259\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 0 0
$$227$$ 3643.36 1.06528 0.532639 0.846343i $$-0.321200\pi$$
0.532639 + 0.846343i $$0.321200\pi$$
$$228$$ 0 0
$$229$$ 314.732 0.0908213 0.0454106 0.998968i $$-0.485540\pi$$
0.0454106 + 0.998968i $$0.485540\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ −1962.28 −0.551732 −0.275866 0.961196i $$-0.588965\pi$$
−0.275866 + 0.961196i $$0.588965\pi$$
$$234$$ 0 0
$$235$$ 2927.21 0.812555
$$236$$ 0 0
$$237$$ 0 0
$$238$$ 0 0
$$239$$ 3128.78 0.846795 0.423397 0.905944i $$-0.360837\pi$$
0.423397 + 0.905944i $$0.360837\pi$$
$$240$$ 0 0
$$241$$ 1538.75 0.411284 0.205642 0.978627i $$-0.434072\pi$$
0.205642 + 0.978627i $$0.434072\pi$$
$$242$$ 0 0
$$243$$ 0 0
$$244$$ 0 0
$$245$$ 520.712 0.135784
$$246$$ 0 0
$$247$$ 264.535 0.0681456
$$248$$ 0 0
$$249$$ 0 0
$$250$$ 0 0
$$251$$ −5699.82 −1.43334 −0.716672 0.697410i $$-0.754336\pi$$
−0.716672 + 0.697410i $$0.754336\pi$$
$$252$$ 0 0
$$253$$ 13515.1 3.35845
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 0 0
$$257$$ −1406.28 −0.341327 −0.170664 0.985329i $$-0.554591\pi$$
−0.170664 + 0.985329i $$0.554591\pi$$
$$258$$ 0 0
$$259$$ 151.861 0.0364331
$$260$$ 0 0
$$261$$ 0 0
$$262$$ 0 0
$$263$$ 7658.46 1.79559 0.897796 0.440412i $$-0.145167\pi$$
0.897796 + 0.440412i $$0.145167\pi$$
$$264$$ 0 0
$$265$$ −386.396 −0.0895702
$$266$$ 0 0
$$267$$ 0 0
$$268$$ 0 0
$$269$$ −808.242 −0.183195 −0.0915974 0.995796i $$-0.529197\pi$$
−0.0915974 + 0.995796i $$0.529197\pi$$
$$270$$ 0 0
$$271$$ 4146.22 0.929390 0.464695 0.885471i $$-0.346164\pi$$
0.464695 + 0.885471i $$0.346164\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ 0 0
$$275$$ −1685.73 −0.369649
$$276$$ 0 0
$$277$$ 1948.21 0.422588 0.211294 0.977423i $$-0.432232\pi$$
0.211294 + 0.977423i $$0.432232\pi$$
$$278$$ 0 0
$$279$$ 0 0
$$280$$ 0 0
$$281$$ 5888.41 1.25008 0.625041 0.780592i $$-0.285082\pi$$
0.625041 + 0.780592i $$0.285082\pi$$
$$282$$ 0 0
$$283$$ −826.285 −0.173560 −0.0867801 0.996227i $$-0.527658\pi$$
−0.0867801 + 0.996227i $$0.527658\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ 7338.66 1.50936
$$288$$ 0 0
$$289$$ 10991.8 2.23729
$$290$$ 0 0
$$291$$ 0 0
$$292$$ 0 0
$$293$$ −1817.15 −0.362318 −0.181159 0.983454i $$-0.557985\pi$$
−0.181159 + 0.983454i $$0.557985\pi$$
$$294$$ 0 0
$$295$$ 1003.95 0.198143
$$296$$ 0 0
$$297$$ 0 0
$$298$$ 0 0
$$299$$ −512.094 −0.0990474
$$300$$ 0 0
$$301$$ −4003.28 −0.766595
$$302$$ 0 0
$$303$$ 0 0
$$304$$ 0 0
$$305$$ 3405.68 0.639373
$$306$$ 0 0
$$307$$ 7055.67 1.31169 0.655845 0.754896i $$-0.272313\pi$$
0.655845 + 0.754896i $$0.272313\pi$$
$$308$$ 0 0
$$309$$ 0 0
$$310$$ 0 0
$$311$$ −1144.46 −0.208669 −0.104335 0.994542i $$-0.533271\pi$$
−0.104335 + 0.994542i $$0.533271\pi$$
$$312$$ 0 0
$$313$$ −5868.73 −1.05981 −0.529905 0.848057i $$-0.677772\pi$$
−0.529905 + 0.848057i $$0.677772\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ 9215.18 1.63273 0.816366 0.577534i $$-0.195985\pi$$
0.816366 + 0.577534i $$0.195985\pi$$
$$318$$ 0 0
$$319$$ 4815.91 0.845264
$$320$$ 0 0
$$321$$ 0 0
$$322$$ 0 0
$$323$$ −13057.8 −2.24939
$$324$$ 0 0
$$325$$ 63.8732 0.0109017
$$326$$ 0 0
$$327$$ 0 0
$$328$$ 0 0
$$329$$ 12379.6 2.07450
$$330$$ 0 0
$$331$$ −4947.10 −0.821502 −0.410751 0.911748i $$-0.634733\pi$$
−0.410751 + 0.911748i $$0.634733\pi$$
$$332$$ 0 0
$$333$$ 0 0
$$334$$ 0 0
$$335$$ 4053.76 0.661135
$$336$$ 0 0
$$337$$ 1879.56 0.303816 0.151908 0.988395i $$-0.451458\pi$$
0.151908 + 0.988395i $$0.451458\pi$$
$$338$$ 0 0
$$339$$ 0 0
$$340$$ 0 0
$$341$$ −10662.6 −1.69329
$$342$$ 0 0
$$343$$ −5050.82 −0.795098
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ −2086.38 −0.322775 −0.161388 0.986891i $$-0.551597\pi$$
−0.161388 + 0.986891i $$0.551597\pi$$
$$348$$ 0 0
$$349$$ 7986.00 1.22487 0.612437 0.790519i $$-0.290189\pi$$
0.612437 + 0.790519i $$0.290189\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 0 0
$$353$$ −1443.39 −0.217631 −0.108815 0.994062i $$-0.534706\pi$$
−0.108815 + 0.994062i $$0.534706\pi$$
$$354$$ 0 0
$$355$$ 2577.38 0.385333
$$356$$ 0 0
$$357$$ 0 0
$$358$$ 0 0
$$359$$ −9598.40 −1.41110 −0.705549 0.708661i $$-0.749300\pi$$
−0.705549 + 0.708661i $$0.749300\pi$$
$$360$$ 0 0
$$361$$ 3861.37 0.562964
$$362$$ 0 0
$$363$$ 0 0
$$364$$ 0 0
$$365$$ 1927.89 0.276466
$$366$$ 0 0
$$367$$ −1237.82 −0.176059 −0.0880296 0.996118i $$-0.528057\pi$$
−0.0880296 + 0.996118i $$0.528057\pi$$
$$368$$ 0 0
$$369$$ 0 0
$$370$$ 0 0
$$371$$ −1634.12 −0.228678
$$372$$ 0 0
$$373$$ −3832.31 −0.531982 −0.265991 0.963976i $$-0.585699\pi$$
−0.265991 + 0.963976i $$0.585699\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ −182.477 −0.0249285
$$378$$ 0 0
$$379$$ −8252.24 −1.11844 −0.559220 0.829019i $$-0.688899\pi$$
−0.559220 + 0.829019i $$0.688899\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ 0 0
$$383$$ 5080.20 0.677771 0.338885 0.940828i $$-0.389950\pi$$
0.338885 + 0.940828i $$0.389950\pi$$
$$384$$ 0 0
$$385$$ −7129.22 −0.943737
$$386$$ 0 0
$$387$$ 0 0
$$388$$ 0 0
$$389$$ 7382.79 0.962268 0.481134 0.876647i $$-0.340225\pi$$
0.481134 + 0.876647i $$0.340225\pi$$
$$390$$ 0 0
$$391$$ 25277.6 3.26942
$$392$$ 0 0
$$393$$ 0 0
$$394$$ 0 0
$$395$$ 1047.02 0.133371
$$396$$ 0 0
$$397$$ 7108.65 0.898673 0.449336 0.893363i $$-0.351660\pi$$
0.449336 + 0.893363i $$0.351660\pi$$
$$398$$ 0 0
$$399$$ 0 0
$$400$$ 0 0
$$401$$ 5347.35 0.665920 0.332960 0.942941i $$-0.391953\pi$$
0.332960 + 0.942941i $$0.391953\pi$$
$$402$$ 0 0
$$403$$ 404.011 0.0499385
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 0 0
$$407$$ −484.253 −0.0589767
$$408$$ 0 0
$$409$$ −223.839 −0.0270615 −0.0135307 0.999908i $$-0.504307\pi$$
−0.0135307 + 0.999908i $$0.504307\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ 0 0
$$413$$ 4245.86 0.505872
$$414$$ 0 0
$$415$$ 4435.95 0.524704
$$416$$ 0 0
$$417$$ 0 0
$$418$$ 0 0
$$419$$ −384.631 −0.0448459 −0.0224230 0.999749i $$-0.507138\pi$$
−0.0224230 + 0.999749i $$0.507138\pi$$
$$420$$ 0 0
$$421$$ 4125.76 0.477618 0.238809 0.971067i $$-0.423243\pi$$
0.238809 + 0.971067i $$0.423243\pi$$
$$422$$ 0 0
$$423$$ 0 0
$$424$$ 0 0
$$425$$ −3152.86 −0.359850
$$426$$ 0 0
$$427$$ 14403.1 1.63236
$$428$$ 0 0
$$429$$ 0 0
$$430$$ 0 0
$$431$$ −457.020 −0.0510762 −0.0255381 0.999674i $$-0.508130\pi$$
−0.0255381 + 0.999674i $$0.508130\pi$$
$$432$$ 0 0
$$433$$ −11085.8 −1.23037 −0.615187 0.788382i $$-0.710919\pi$$
−0.615187 + 0.788382i $$0.710919\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 0 0
$$437$$ −20752.8 −2.27172
$$438$$ 0 0
$$439$$ 4082.70 0.443865 0.221933 0.975062i $$-0.428764\pi$$
0.221933 + 0.975062i $$0.428764\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 0 0
$$443$$ −14975.7 −1.60613 −0.803067 0.595888i $$-0.796800\pi$$
−0.803067 + 0.595888i $$0.796800\pi$$
$$444$$ 0 0
$$445$$ 2744.45 0.292358
$$446$$ 0 0
$$447$$ 0 0
$$448$$ 0 0
$$449$$ −1863.53 −0.195870 −0.0979349 0.995193i $$-0.531224\pi$$
−0.0979349 + 0.995193i $$0.531224\pi$$
$$450$$ 0 0
$$451$$ −23401.5 −2.44331
$$452$$ 0 0
$$453$$ 0 0
$$454$$ 0 0
$$455$$ 270.129 0.0278326
$$456$$ 0 0
$$457$$ −4029.90 −0.412496 −0.206248 0.978500i $$-0.566125\pi$$
−0.206248 + 0.978500i $$0.566125\pi$$
$$458$$ 0 0
$$459$$ 0 0
$$460$$ 0 0
$$461$$ 9320.82 0.941679 0.470839 0.882219i $$-0.343951\pi$$
0.470839 + 0.882219i $$0.343951\pi$$
$$462$$ 0 0
$$463$$ −9948.81 −0.998619 −0.499309 0.866424i $$-0.666413\pi$$
−0.499309 + 0.866424i $$0.666413\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ −7946.35 −0.787395 −0.393697 0.919240i $$-0.628804\pi$$
−0.393697 + 0.919240i $$0.628804\pi$$
$$468$$ 0 0
$$469$$ 17143.9 1.68792
$$470$$ 0 0
$$471$$ 0 0
$$472$$ 0 0
$$473$$ 12765.6 1.24094
$$474$$ 0 0
$$475$$ 2588.48 0.250037
$$476$$ 0 0
$$477$$ 0 0
$$478$$ 0 0
$$479$$ −4166.01 −0.397390 −0.198695 0.980061i $$-0.563670\pi$$
−0.198695 + 0.980061i $$0.563670\pi$$
$$480$$ 0 0
$$481$$ 18.3485 0.00173934
$$482$$ 0 0
$$483$$ 0 0
$$484$$ 0 0
$$485$$ −7619.79 −0.713395
$$486$$ 0 0
$$487$$ −4662.27 −0.433815 −0.216907 0.976192i $$-0.569597\pi$$
−0.216907 + 0.976192i $$0.569597\pi$$
$$488$$ 0 0
$$489$$ 0 0
$$490$$ 0 0
$$491$$ 9609.89 0.883275 0.441638 0.897194i $$-0.354398\pi$$
0.441638 + 0.897194i $$0.354398\pi$$
$$492$$ 0 0
$$493$$ 9007.28 0.822855
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ 10900.1 0.983778
$$498$$ 0 0
$$499$$ 5439.00 0.487942 0.243971 0.969782i $$-0.421550\pi$$
0.243971 + 0.969782i $$0.421550\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ 0 0
$$503$$ −10407.2 −0.922537 −0.461268 0.887261i $$-0.652605\pi$$
−0.461268 + 0.887261i $$0.652605\pi$$
$$504$$ 0 0
$$505$$ −1739.13 −0.153248
$$506$$ 0 0
$$507$$ 0 0
$$508$$ 0 0
$$509$$ 8982.15 0.782174 0.391087 0.920354i $$-0.372099\pi$$
0.391087 + 0.920354i $$0.372099\pi$$
$$510$$ 0 0
$$511$$ 8153.32 0.705835
$$512$$ 0 0
$$513$$ 0 0
$$514$$ 0 0
$$515$$ 9775.84 0.836456
$$516$$ 0 0
$$517$$ −39476.0 −3.35813
$$518$$ 0 0
$$519$$ 0 0
$$520$$ 0 0
$$521$$ 871.580 0.0732910 0.0366455 0.999328i $$-0.488333\pi$$
0.0366455 + 0.999328i $$0.488333\pi$$
$$522$$ 0 0
$$523$$ −16215.9 −1.35578 −0.677888 0.735165i $$-0.737105\pi$$
−0.677888 + 0.735165i $$0.737105\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ −19942.5 −1.64840
$$528$$ 0 0
$$529$$ 28006.7 2.30186
$$530$$ 0 0
$$531$$ 0 0
$$532$$ 0 0
$$533$$ 886.691 0.0720579
$$534$$ 0 0
$$535$$ 7880.82 0.636856
$$536$$ 0 0
$$537$$ 0 0
$$538$$ 0 0
$$539$$ −7022.26 −0.561169
$$540$$ 0 0
$$541$$ 7195.89 0.571859 0.285929 0.958251i $$-0.407698\pi$$
0.285929 + 0.958251i $$0.407698\pi$$
$$542$$ 0 0
$$543$$ 0 0
$$544$$ 0 0
$$545$$ −791.161 −0.0621828
$$546$$ 0 0
$$547$$ 12847.2 1.00422 0.502109 0.864804i $$-0.332558\pi$$
0.502109 + 0.864804i $$0.332558\pi$$
$$548$$ 0 0
$$549$$ 0 0
$$550$$ 0 0
$$551$$ −7394.93 −0.571751
$$552$$ 0 0
$$553$$ 4428.02 0.340504
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 0 0
$$557$$ 1916.53 0.145792 0.0728958 0.997340i $$-0.476776\pi$$
0.0728958 + 0.997340i $$0.476776\pi$$
$$558$$ 0 0
$$559$$ −483.695 −0.0365977
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 0 0
$$563$$ −5124.14 −0.383582 −0.191791 0.981436i $$-0.561430\pi$$
−0.191791 + 0.981436i $$0.561430\pi$$
$$564$$ 0 0
$$565$$ −2919.28 −0.217372
$$566$$ 0 0
$$567$$ 0 0
$$568$$ 0 0
$$569$$ 17498.6 1.28924 0.644620 0.764503i $$-0.277015\pi$$
0.644620 + 0.764503i $$0.277015\pi$$
$$570$$ 0 0
$$571$$ −9253.34 −0.678179 −0.339090 0.940754i $$-0.610119\pi$$
−0.339090 + 0.940754i $$0.610119\pi$$
$$572$$ 0 0
$$573$$ 0 0
$$574$$ 0 0
$$575$$ −5010.85 −0.363420
$$576$$ 0 0
$$577$$ −17420.7 −1.25690 −0.628451 0.777849i $$-0.716311\pi$$
−0.628451 + 0.777849i $$0.716311\pi$$
$$578$$ 0 0
$$579$$ 0 0
$$580$$ 0 0
$$581$$ 18760.3 1.33960
$$582$$ 0 0
$$583$$ 5210.88 0.370176
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ 7839.38 0.551219 0.275610 0.961270i $$-0.411120\pi$$
0.275610 + 0.961270i $$0.411120\pi$$
$$588$$ 0 0
$$589$$ 16372.7 1.14537
$$590$$ 0 0
$$591$$ 0 0
$$592$$ 0 0
$$593$$ −1611.03 −0.111564 −0.0557818 0.998443i $$-0.517765\pi$$
−0.0557818 + 0.998443i $$0.517765\pi$$
$$594$$ 0 0
$$595$$ −13333.9 −0.918717
$$596$$ 0 0
$$597$$ 0 0
$$598$$ 0 0
$$599$$ 17274.0 1.17829 0.589146 0.808026i $$-0.299464\pi$$
0.589146 + 0.808026i $$0.299464\pi$$
$$600$$ 0 0
$$601$$ −26908.5 −1.82632 −0.913161 0.407599i $$-0.866366\pi$$
−0.913161 + 0.407599i $$0.866366\pi$$
$$602$$ 0 0
$$603$$ 0 0
$$604$$ 0 0
$$605$$ 16078.6 1.08048
$$606$$ 0 0
$$607$$ 5500.73 0.367822 0.183911 0.982943i $$-0.441124\pi$$
0.183911 + 0.982943i $$0.441124\pi$$
$$608$$ 0 0
$$609$$ 0 0
$$610$$ 0 0
$$611$$ 1495.76 0.0990379
$$612$$ 0 0
$$613$$ 17362.0 1.14396 0.571978 0.820269i $$-0.306176\pi$$
0.571978 + 0.820269i $$0.306176\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ −2946.26 −0.192240 −0.0961198 0.995370i $$-0.530643\pi$$
−0.0961198 + 0.995370i $$0.530643\pi$$
$$618$$ 0 0
$$619$$ 15311.6 0.994223 0.497112 0.867687i $$-0.334394\pi$$
0.497112 + 0.867687i $$0.334394\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ 0 0
$$623$$ 11606.7 0.746408
$$624$$ 0 0
$$625$$ 625.000 0.0400000
$$626$$ 0 0
$$627$$ 0 0
$$628$$ 0 0
$$629$$ −905.706 −0.0574132
$$630$$ 0 0
$$631$$ −1067.12 −0.0673241 −0.0336621 0.999433i $$-0.510717\pi$$
−0.0336621 + 0.999433i $$0.510717\pi$$
$$632$$ 0 0
$$633$$ 0 0
$$634$$ 0 0
$$635$$ −12863.4 −0.803886
$$636$$ 0 0
$$637$$ 266.076 0.0165500
$$638$$ 0 0
$$639$$ 0 0
$$640$$ 0 0
$$641$$ −8977.71 −0.553195 −0.276598 0.960986i $$-0.589207\pi$$
−0.276598 + 0.960986i $$0.589207\pi$$
$$642$$ 0 0
$$643$$ 15935.5 0.977346 0.488673 0.872467i $$-0.337481\pi$$
0.488673 + 0.872467i $$0.337481\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ −21284.1 −1.29330 −0.646648 0.762789i $$-0.723830\pi$$
−0.646648 + 0.762789i $$0.723830\pi$$
$$648$$ 0 0
$$649$$ −13539.2 −0.818889
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 0 0
$$653$$ −8173.21 −0.489805 −0.244902 0.969548i $$-0.578756\pi$$
−0.244902 + 0.969548i $$0.578756\pi$$
$$654$$ 0 0
$$655$$ 8969.63 0.535072
$$656$$ 0 0
$$657$$ 0 0
$$658$$ 0 0
$$659$$ 32350.6 1.91229 0.956145 0.292895i $$-0.0946186\pi$$
0.956145 + 0.292895i $$0.0946186\pi$$
$$660$$ 0 0
$$661$$ 26775.6 1.57557 0.787783 0.615953i $$-0.211229\pi$$
0.787783 + 0.615953i $$0.211229\pi$$
$$662$$ 0 0
$$663$$ 0 0
$$664$$ 0 0
$$665$$ 10947.1 0.638360
$$666$$ 0 0
$$667$$ 14315.3 0.831020
$$668$$ 0 0
$$669$$ 0 0
$$670$$ 0 0
$$671$$ −45928.6 −2.64241
$$672$$ 0 0
$$673$$ 13448.7 0.770297 0.385149 0.922855i $$-0.374150\pi$$
0.385149 + 0.922855i $$0.374150\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ 0 0
$$677$$ 20767.3 1.17896 0.589478 0.807784i $$-0.299333\pi$$
0.589478 + 0.807784i $$0.299333\pi$$
$$678$$ 0 0
$$679$$ −32225.2 −1.82134
$$680$$ 0 0
$$681$$ 0 0
$$682$$ 0 0
$$683$$ 11894.6 0.666373 0.333187 0.942861i $$-0.391876\pi$$
0.333187 + 0.942861i $$0.391876\pi$$
$$684$$ 0 0
$$685$$ 8647.94 0.482366
$$686$$ 0 0
$$687$$ 0 0
$$688$$ 0 0
$$689$$ −197.443 −0.0109172
$$690$$ 0 0
$$691$$ −14871.9 −0.818747 −0.409374 0.912367i $$-0.634253\pi$$
−0.409374 + 0.912367i $$0.634253\pi$$
$$692$$ 0 0
$$693$$ 0 0
$$694$$ 0 0
$$695$$ −3453.57 −0.188491
$$696$$ 0 0
$$697$$ −43768.2 −2.37853
$$698$$ 0 0
$$699$$ 0 0
$$700$$ 0 0
$$701$$ −2423.29 −0.130566 −0.0652828 0.997867i $$-0.520795\pi$$
−0.0652828 + 0.997867i $$0.520795\pi$$
$$702$$ 0 0
$$703$$ 743.580 0.0398928
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ −7355.02 −0.391250
$$708$$ 0 0
$$709$$ 8971.88 0.475241 0.237621 0.971358i $$-0.423632\pi$$
0.237621 + 0.971358i $$0.423632\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ 0 0
$$713$$ −31694.6 −1.66476
$$714$$ 0 0
$$715$$ −861.386 −0.0450545
$$716$$ 0 0
$$717$$ 0 0
$$718$$ 0 0
$$719$$ 18757.9 0.972951 0.486475 0.873694i $$-0.338282\pi$$
0.486475 + 0.873694i $$0.338282\pi$$
$$720$$ 0 0
$$721$$ 41343.5 2.13552
$$722$$ 0 0
$$723$$ 0 0
$$724$$ 0 0
$$725$$ −1785.54 −0.0914665
$$726$$ 0 0
$$727$$ 14205.6 0.724700 0.362350 0.932042i $$-0.381974\pi$$
0.362350 + 0.932042i $$0.381974\pi$$
$$728$$ 0 0
$$729$$ 0 0
$$730$$ 0 0
$$731$$ 23875.8 1.20804
$$732$$ 0 0
$$733$$ 23776.1 1.19808 0.599038 0.800720i $$-0.295550\pi$$
0.599038 + 0.800720i $$0.295550\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ −54668.4 −2.73234
$$738$$ 0 0
$$739$$ −14175.5 −0.705623 −0.352811 0.935694i $$-0.614774\pi$$
−0.352811 + 0.935694i $$0.614774\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 0 0
$$743$$ −7564.68 −0.373514 −0.186757 0.982406i $$-0.559798\pi$$
−0.186757 + 0.982406i $$0.559798\pi$$
$$744$$ 0 0
$$745$$ −14381.3 −0.707237
$$746$$ 0 0
$$747$$ 0 0
$$748$$ 0 0
$$749$$ 33329.2 1.62593
$$750$$ 0 0
$$751$$ −18602.1 −0.903863 −0.451932 0.892053i $$-0.649265\pi$$
−0.451932 + 0.892053i $$0.649265\pi$$
$$752$$ 0 0
$$753$$ 0 0
$$754$$ 0 0
$$755$$ −6020.51 −0.290210
$$756$$ 0 0
$$757$$ −4665.12 −0.223985 −0.111993 0.993709i $$-0.535723\pi$$
−0.111993 + 0.993709i $$0.535723\pi$$
$$758$$ 0 0
$$759$$ 0 0
$$760$$ 0 0
$$761$$ −36063.6 −1.71788 −0.858938 0.512079i $$-0.828875\pi$$
−0.858938 + 0.512079i $$0.828875\pi$$
$$762$$ 0 0
$$763$$ −3345.94 −0.158756
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ 513.005 0.0241506
$$768$$ 0 0
$$769$$ −28921.5 −1.35623 −0.678113 0.734958i $$-0.737202\pi$$
−0.678113 + 0.734958i $$0.737202\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ 0 0
$$773$$ −25489.9 −1.18604 −0.593019 0.805188i $$-0.702064\pi$$
−0.593019 + 0.805188i $$0.702064\pi$$
$$774$$ 0 0
$$775$$ 3953.25 0.183232
$$776$$ 0 0
$$777$$ 0 0
$$778$$ 0 0
$$779$$ 35933.5 1.65270
$$780$$ 0 0
$$781$$ −34758.3 −1.59251
$$782$$ 0 0
$$783$$ 0 0
$$784$$ 0 0
$$785$$ −3952.44 −0.179705
$$786$$ 0 0
$$787$$ −3627.06 −0.164283 −0.0821415 0.996621i $$-0.526176\pi$$
−0.0821415 + 0.996621i $$0.526176\pi$$
$$788$$ 0 0
$$789$$ 0 0
$$790$$ 0 0
$$791$$ −12346.1 −0.554962
$$792$$ 0 0
$$793$$ 1740.26 0.0779297
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 0 0
$$797$$ −15571.1 −0.692041 −0.346021 0.938227i $$-0.612467\pi$$
−0.346021 + 0.938227i $$0.612467\pi$$
$$798$$ 0 0
$$799$$ −73832.7 −3.26910
$$800$$ 0 0
$$801$$ 0 0
$$802$$ 0 0
$$803$$ −25999.2 −1.14258
$$804$$ 0 0
$$805$$ −21191.6 −0.927834
$$806$$ 0 0
$$807$$ 0 0
$$808$$ 0 0
$$809$$ −7352.63 −0.319536 −0.159768 0.987155i $$-0.551075\pi$$
−0.159768 + 0.987155i $$0.551075\pi$$
$$810$$ 0 0
$$811$$ −19743.8 −0.854869 −0.427434 0.904046i $$-0.640583\pi$$
−0.427434 + 0.904046i $$0.640583\pi$$
$$812$$ 0 0
$$813$$ 0 0
$$814$$ 0 0
$$815$$ 11013.1 0.473339
$$816$$ 0 0
$$817$$ −19601.9 −0.839392
$$818$$ 0 0
$$819$$ 0 0
$$820$$ 0 0
$$821$$ −43315.1 −1.84130 −0.920651 0.390387i $$-0.872341\pi$$
−0.920651 + 0.390387i $$0.872341\pi$$
$$822$$ 0 0
$$823$$ 23632.2 1.00093 0.500466 0.865756i $$-0.333162\pi$$
0.500466 + 0.865756i $$0.333162\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ −24109.3 −1.01374 −0.506869 0.862023i $$-0.669197\pi$$
−0.506869 + 0.862023i $$0.669197\pi$$
$$828$$ 0 0
$$829$$ 1330.02 0.0557218 0.0278609 0.999612i $$-0.491130\pi$$
0.0278609 + 0.999612i $$0.491130\pi$$
$$830$$ 0 0
$$831$$ 0 0
$$832$$ 0 0
$$833$$ −13133.8 −0.546292
$$834$$ 0 0
$$835$$ −9166.00 −0.379883
$$836$$ 0 0
$$837$$ 0 0
$$838$$ 0 0
$$839$$ 16823.3 0.692258 0.346129 0.938187i $$-0.387496\pi$$
0.346129 + 0.938187i $$0.387496\pi$$
$$840$$ 0 0
$$841$$ −19288.0 −0.790847
$$842$$ 0 0
$$843$$ 0 0
$$844$$ 0 0
$$845$$ −10952.4 −0.445885
$$846$$ 0 0
$$847$$ 67998.8 2.75852
$$848$$ 0 0
$$849$$ 0 0
$$850$$ 0 0
$$851$$ −1439.44 −0.0579829
$$852$$ 0 0
$$853$$ −11732.7 −0.470951 −0.235475 0.971880i $$-0.575665\pi$$
−0.235475 + 0.971880i $$0.575665\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0 0
$$857$$ 25843.7 1.03011 0.515055 0.857157i $$-0.327771\pi$$
0.515055 + 0.857157i $$0.327771\pi$$
$$858$$ 0 0
$$859$$ −21463.9 −0.852549 −0.426275 0.904594i $$-0.640174\pi$$
−0.426275 + 0.904594i $$0.640174\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 0 0
$$863$$ 11980.8 0.472575 0.236287 0.971683i $$-0.424069\pi$$
0.236287 + 0.971683i $$0.424069\pi$$
$$864$$ 0 0
$$865$$ −13433.1 −0.528022
$$866$$ 0 0
$$867$$ 0 0
$$868$$ 0 0
$$869$$ −14120.0 −0.551196
$$870$$ 0 0
$$871$$ 2071.41 0.0805822
$$872$$ 0 0
$$873$$ 0 0
$$874$$ 0 0
$$875$$ 2643.22 0.102122
$$876$$ 0 0
$$877$$ 3662.20 0.141008 0.0705038 0.997512i $$-0.477539\pi$$
0.0705038 + 0.997512i $$0.477539\pi$$
$$878$$ 0 0
$$879$$ 0 0
$$880$$ 0 0
$$881$$ 27016.4 1.03315 0.516575 0.856242i $$-0.327207\pi$$
0.516575 + 0.856242i $$0.327207\pi$$
$$882$$ 0 0
$$883$$ −24437.6 −0.931359 −0.465679 0.884954i $$-0.654190\pi$$
−0.465679 + 0.884954i $$0.654190\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 0 0
$$887$$ 35067.7 1.32746 0.663731 0.747971i $$-0.268972\pi$$
0.663731 + 0.747971i $$0.268972\pi$$
$$888$$ 0 0
$$889$$ −54401.2 −2.05237
$$890$$ 0 0
$$891$$ 0 0
$$892$$ 0 0
$$893$$ 60616.3 2.27150
$$894$$ 0 0
$$895$$ 22502.0 0.840402
$$896$$ 0 0
$$897$$ 0 0
$$898$$ 0 0
$$899$$ −11293.9 −0.418990
$$900$$ 0 0
$$901$$ 9746.01 0.360363
$$902$$ 0 0
$$903$$ 0 0
$$904$$ 0 0
$$905$$ −8004.08 −0.293994
$$906$$ 0 0
$$907$$ 29567.8 1.08245 0.541225 0.840878i $$-0.317961\pi$$
0.541225 + 0.840878i $$0.317961\pi$$
$$908$$ 0 0
$$909$$ 0 0
$$910$$ 0 0
$$911$$ 37564.2 1.36614 0.683072 0.730351i $$-0.260644\pi$$
0.683072 + 0.730351i $$0.260644\pi$$
$$912$$ 0 0
$$913$$ −59822.6 −2.16850
$$914$$ 0 0
$$915$$ 0 0
$$916$$ 0 0
$$917$$ 37933.9 1.36607
$$918$$ 0 0
$$919$$ −42968.8 −1.54234 −0.771170 0.636629i $$-0.780328\pi$$
−0.771170 + 0.636629i $$0.780328\pi$$
$$920$$ 0 0
$$921$$ 0 0
$$922$$ 0 0
$$923$$ 1317.00 0.0469661
$$924$$ 0 0
$$925$$ 179.541 0.00638191
$$926$$ 0 0
$$927$$ 0 0
$$928$$ 0 0
$$929$$ 25931.4 0.915804 0.457902 0.889003i $$-0.348601\pi$$
0.457902 + 0.889003i $$0.348601\pi$$
$$930$$ 0 0
$$931$$ 10782.8 0.379584
$$932$$ 0 0
$$933$$ 0 0
$$934$$ 0 0
$$935$$ 42519.1 1.48719
$$936$$ 0 0
$$937$$ −20940.0 −0.730075 −0.365037 0.930993i $$-0.618944\pi$$
−0.365037 + 0.930993i $$0.618944\pi$$
$$938$$ 0 0
$$939$$ 0 0
$$940$$ 0 0
$$941$$ −8528.77 −0.295462 −0.147731 0.989028i $$-0.547197\pi$$
−0.147731 + 0.989028i $$0.547197\pi$$
$$942$$ 0 0
$$943$$ −69560.9 −2.40214
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ 18566.9 0.637112 0.318556 0.947904i $$-0.396802\pi$$
0.318556 + 0.947904i $$0.396802\pi$$
$$948$$ 0 0
$$949$$ 985.122 0.0336970
$$950$$ 0 0
$$951$$ 0 0
$$952$$ 0 0
$$953$$ −8866.54 −0.301380 −0.150690 0.988581i $$-0.548150\pi$$
−0.150690 + 0.988581i $$0.548150\pi$$
$$954$$ 0 0
$$955$$ 4311.40 0.146088
$$956$$ 0 0
$$957$$ 0 0
$$958$$ 0 0
$$959$$ 36573.4 1.23151
$$960$$ 0 0
$$961$$ −4785.89 −0.160649
$$962$$ 0 0
$$963$$ 0 0
$$964$$ 0 0
$$965$$ −3841.60 −0.128151
$$966$$ 0 0
$$967$$ −38880.2 −1.29297 −0.646485 0.762927i $$-0.723762\pi$$
−0.646485 + 0.762927i $$0.723762\pi$$
$$968$$ 0 0
$$969$$ 0 0
$$970$$ 0 0
$$971$$ 33655.7 1.11232 0.556161 0.831075i $$-0.312274\pi$$
0.556161 + 0.831075i $$0.312274\pi$$
$$972$$ 0 0
$$973$$ −14605.6 −0.481228
$$974$$ 0 0
$$975$$ 0 0
$$976$$ 0 0
$$977$$ 6269.02 0.205285 0.102643 0.994718i $$-0.467270\pi$$
0.102643 + 0.994718i $$0.467270\pi$$
$$978$$ 0 0
$$979$$ −37011.3 −1.20826
$$980$$ 0 0
$$981$$ 0 0
$$982$$ 0 0
$$983$$ 39772.2 1.29048 0.645238 0.763982i $$-0.276758\pi$$
0.645238 + 0.763982i $$0.276758\pi$$
$$984$$ 0 0
$$985$$ 15450.3 0.499785
$$986$$ 0 0
$$987$$ 0 0
$$988$$ 0 0
$$989$$ 37945.8 1.22003
$$990$$ 0 0
$$991$$ 8799.28 0.282057 0.141028 0.990006i $$-0.454959\pi$$
0.141028 + 0.990006i $$0.454959\pi$$
$$992$$ 0 0
$$993$$ 0 0
$$994$$ 0 0
$$995$$ −16277.1 −0.518610
$$996$$ 0 0
$$997$$ 42122.5 1.33805 0.669024 0.743241i $$-0.266713\pi$$
0.669024 + 0.743241i $$0.266713\pi$$
$$998$$ 0 0
$$999$$ 0 0
Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000

## Twists

By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 2160.4.a.bt.1.3 3
3.2 odd 2 2160.4.a.bl.1.3 3
4.3 odd 2 1080.4.a.i.1.1 yes 3
12.11 even 2 1080.4.a.c.1.1 3

By twisted newform
Twist Min Dim Char Parity Ord Type
1080.4.a.c.1.1 3 12.11 even 2
1080.4.a.i.1.1 yes 3 4.3 odd 2
2160.4.a.bl.1.3 3 3.2 odd 2
2160.4.a.bt.1.3 3 1.1 even 1 trivial