# Properties

 Label 6012.2.a.d.1.1 Level $6012$ Weight $2$ Character 6012.1 Self dual yes Analytic conductor $48.006$ Analytic rank $0$ Dimension $5$ CM no Inner twists $1$

# Learn more about

## Newspace parameters

 Level: $$N$$ $$=$$ $$6012 = 2^{2} \cdot 3^{2} \cdot 167$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 6012.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$48.0060616952$$ Analytic rank: $$0$$ Dimension: $$5$$ Coefficient field: 5.5.826865.1 Defining polynomial: $$x^{5} - 2 x^{4} - 5 x^{3} + 6 x^{2} + 6 x - 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 668) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Root $$-0.873948$$ of defining polynomial Character $$\chi$$ $$=$$ 6012.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-2.00000 q^{5} -1.42676 q^{7} +O(q^{10})$$ $$q-2.00000 q^{5} -1.42676 q^{7} -2.19055 q^{11} +1.74790 q^{13} -3.26510 q^{17} +5.89213 q^{19} -6.12900 q^{23} -1.00000 q^{25} +0.321133 q^{29} +1.75308 q^{31} +2.85353 q^{35} -2.47243 q^{37} +5.98963 q^{41} -3.49579 q^{43} -2.39634 q^{47} -4.96434 q^{49} +1.51720 q^{53} +4.38110 q^{55} -5.65526 q^{59} -4.51463 q^{61} -3.49579 q^{65} -9.61442 q^{67} +6.66956 q^{71} +2.86390 q^{73} +3.12540 q^{77} +10.4997 q^{79} +7.24369 q^{83} +6.53020 q^{85} +0.256634 q^{89} -2.49384 q^{91} -11.7843 q^{95} +5.83544 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$5q - 10q^{5} + 9q^{7} + O(q^{10})$$ $$5q - 10q^{5} + 9q^{7} - 5q^{11} - 4q^{13} + 2q^{17} + 5q^{19} - 6q^{23} - 5q^{25} + 5q^{29} + 9q^{31} - 18q^{35} + 8q^{37} + 4q^{41} + 8q^{43} - 13q^{47} + 14q^{49} + 2q^{53} + 10q^{55} - 4q^{59} + 11q^{61} + 8q^{65} + 28q^{67} - 2q^{71} + 8q^{73} + 12q^{77} - 10q^{79} - 2q^{83} - 4q^{85} + 17q^{89} - 12q^{91} - 10q^{95} - 27q^{97} + O(q^{100})$$

## 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
$$3$$ 0 0
$$4$$ 0 0
$$5$$ −2.00000 −0.894427 −0.447214 0.894427i $$-0.647584\pi$$
−0.447214 + 0.894427i $$0.647584\pi$$
$$6$$ 0 0
$$7$$ −1.42676 −0.539266 −0.269633 0.962963i $$-0.586902\pi$$
−0.269633 + 0.962963i $$0.586902\pi$$
$$8$$ 0 0
$$9$$ 0 0
$$10$$ 0 0
$$11$$ −2.19055 −0.660476 −0.330238 0.943898i $$-0.607129\pi$$
−0.330238 + 0.943898i $$0.607129\pi$$
$$12$$ 0 0
$$13$$ 1.74790 0.484779 0.242390 0.970179i $$-0.422069\pi$$
0.242390 + 0.970179i $$0.422069\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 0 0
$$17$$ −3.26510 −0.791903 −0.395951 0.918271i $$-0.629585\pi$$
−0.395951 + 0.918271i $$0.629585\pi$$
$$18$$ 0 0
$$19$$ 5.89213 1.35175 0.675874 0.737018i $$-0.263767\pi$$
0.675874 + 0.737018i $$0.263767\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 0 0
$$23$$ −6.12900 −1.27798 −0.638992 0.769213i $$-0.720648\pi$$
−0.638992 + 0.769213i $$0.720648\pi$$
$$24$$ 0 0
$$25$$ −1.00000 −0.200000
$$26$$ 0 0
$$27$$ 0 0
$$28$$ 0 0
$$29$$ 0.321133 0.0596329 0.0298164 0.999555i $$-0.490508\pi$$
0.0298164 + 0.999555i $$0.490508\pi$$
$$30$$ 0 0
$$31$$ 1.75308 0.314863 0.157431 0.987530i $$-0.449679\pi$$
0.157431 + 0.987530i $$0.449679\pi$$
$$32$$ 0 0
$$33$$ 0 0
$$34$$ 0 0
$$35$$ 2.85353 0.482334
$$36$$ 0 0
$$37$$ −2.47243 −0.406465 −0.203232 0.979131i $$-0.565145\pi$$
−0.203232 + 0.979131i $$0.565145\pi$$
$$38$$ 0 0
$$39$$ 0 0
$$40$$ 0 0
$$41$$ 5.98963 0.935423 0.467712 0.883881i $$-0.345079\pi$$
0.467712 + 0.883881i $$0.345079\pi$$
$$42$$ 0 0
$$43$$ −3.49579 −0.533104 −0.266552 0.963821i $$-0.585884\pi$$
−0.266552 + 0.963821i $$0.585884\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 0 0
$$47$$ −2.39634 −0.349541 −0.174771 0.984609i $$-0.555918\pi$$
−0.174771 + 0.984609i $$0.555918\pi$$
$$48$$ 0 0
$$49$$ −4.96434 −0.709192
$$50$$ 0 0
$$51$$ 0 0
$$52$$ 0 0
$$53$$ 1.51720 0.208404 0.104202 0.994556i $$-0.466771\pi$$
0.104202 + 0.994556i $$0.466771\pi$$
$$54$$ 0 0
$$55$$ 4.38110 0.590747
$$56$$ 0 0
$$57$$ 0 0
$$58$$ 0 0
$$59$$ −5.65526 −0.736252 −0.368126 0.929776i $$-0.620001\pi$$
−0.368126 + 0.929776i $$0.620001\pi$$
$$60$$ 0 0
$$61$$ −4.51463 −0.578039 −0.289019 0.957323i $$-0.593329\pi$$
−0.289019 + 0.957323i $$0.593329\pi$$
$$62$$ 0 0
$$63$$ 0 0
$$64$$ 0 0
$$65$$ −3.49579 −0.433600
$$66$$ 0 0
$$67$$ −9.61442 −1.17459 −0.587294 0.809374i $$-0.699807\pi$$
−0.587294 + 0.809374i $$0.699807\pi$$
$$68$$ 0 0
$$69$$ 0 0
$$70$$ 0 0
$$71$$ 6.66956 0.791532 0.395766 0.918351i $$-0.370479\pi$$
0.395766 + 0.918351i $$0.370479\pi$$
$$72$$ 0 0
$$73$$ 2.86390 0.335194 0.167597 0.985856i $$-0.446399\pi$$
0.167597 + 0.985856i $$0.446399\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 0 0
$$77$$ 3.12540 0.356172
$$78$$ 0 0
$$79$$ 10.4997 1.18131 0.590656 0.806924i $$-0.298869\pi$$
0.590656 + 0.806924i $$0.298869\pi$$
$$80$$ 0 0
$$81$$ 0 0
$$82$$ 0 0
$$83$$ 7.24369 0.795098 0.397549 0.917581i $$-0.369861\pi$$
0.397549 + 0.917581i $$0.369861\pi$$
$$84$$ 0 0
$$85$$ 6.53020 0.708299
$$86$$ 0 0
$$87$$ 0 0
$$88$$ 0 0
$$89$$ 0.256634 0.0272032 0.0136016 0.999907i $$-0.495670\pi$$
0.0136016 + 0.999907i $$0.495670\pi$$
$$90$$ 0 0
$$91$$ −2.49384 −0.261425
$$92$$ 0 0
$$93$$ 0 0
$$94$$ 0 0
$$95$$ −11.7843 −1.20904
$$96$$ 0 0
$$97$$ 5.83544 0.592499 0.296250 0.955111i $$-0.404264\pi$$
0.296250 + 0.955111i $$0.404264\pi$$
$$98$$ 0 0
$$99$$ 0 0
$$100$$ 0 0
$$101$$ 5.58843 0.556069 0.278035 0.960571i $$-0.410317\pi$$
0.278035 + 0.960571i $$0.410317\pi$$
$$102$$ 0 0
$$103$$ 6.17359 0.608302 0.304151 0.952624i $$-0.401627\pi$$
0.304151 + 0.952624i $$0.401627\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ 11.9570 1.15592 0.577962 0.816064i $$-0.303848\pi$$
0.577962 + 0.816064i $$0.303848\pi$$
$$108$$ 0 0
$$109$$ −15.1784 −1.45382 −0.726911 0.686731i $$-0.759045\pi$$
−0.726911 + 0.686731i $$0.759045\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ 0 0
$$113$$ 0.710405 0.0668293 0.0334146 0.999442i $$-0.489362\pi$$
0.0334146 + 0.999442i $$0.489362\pi$$
$$114$$ 0 0
$$115$$ 12.2580 1.14306
$$116$$ 0 0
$$117$$ 0 0
$$118$$ 0 0
$$119$$ 4.65853 0.427046
$$120$$ 0 0
$$121$$ −6.20149 −0.563772
$$122$$ 0 0
$$123$$ 0 0
$$124$$ 0 0
$$125$$ 12.0000 1.07331
$$126$$ 0 0
$$127$$ 11.5165 1.02192 0.510962 0.859603i $$-0.329289\pi$$
0.510962 + 0.859603i $$0.329289\pi$$
$$128$$ 0 0
$$129$$ 0 0
$$130$$ 0 0
$$131$$ −8.02599 −0.701234 −0.350617 0.936519i $$-0.614028\pi$$
−0.350617 + 0.936519i $$0.614028\pi$$
$$132$$ 0 0
$$133$$ −8.40668 −0.728951
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ 15.6656 1.33840 0.669201 0.743081i $$-0.266636\pi$$
0.669201 + 0.743081i $$0.266636\pi$$
$$138$$ 0 0
$$139$$ −2.22163 −0.188436 −0.0942182 0.995552i $$-0.530035\pi$$
−0.0942182 + 0.995552i $$0.530035\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 0 0
$$143$$ −3.82886 −0.320185
$$144$$ 0 0
$$145$$ −0.642266 −0.0533373
$$146$$ 0 0
$$147$$ 0 0
$$148$$ 0 0
$$149$$ −6.02599 −0.493668 −0.246834 0.969058i $$-0.579390\pi$$
−0.246834 + 0.969058i $$0.579390\pi$$
$$150$$ 0 0
$$151$$ −10.9684 −0.892596 −0.446298 0.894884i $$-0.647258\pi$$
−0.446298 + 0.894884i $$0.647258\pi$$
$$152$$ 0 0
$$153$$ 0 0
$$154$$ 0 0
$$155$$ −3.50616 −0.281622
$$156$$ 0 0
$$157$$ 20.0474 1.59996 0.799980 0.600027i $$-0.204843\pi$$
0.799980 + 0.600027i $$0.204843\pi$$
$$158$$ 0 0
$$159$$ 0 0
$$160$$ 0 0
$$161$$ 8.74463 0.689174
$$162$$ 0 0
$$163$$ 5.21770 0.408682 0.204341 0.978900i $$-0.434495\pi$$
0.204341 + 0.978900i $$0.434495\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ 1.00000 0.0773823
$$168$$ 0 0
$$169$$ −9.94486 −0.764989
$$170$$ 0 0
$$171$$ 0 0
$$172$$ 0 0
$$173$$ 21.5165 1.63587 0.817935 0.575311i $$-0.195119\pi$$
0.817935 + 0.575311i $$0.195119\pi$$
$$174$$ 0 0
$$175$$ 1.42676 0.107853
$$176$$ 0 0
$$177$$ 0 0
$$178$$ 0 0
$$179$$ −10.1501 −0.758656 −0.379328 0.925262i $$-0.623845\pi$$
−0.379328 + 0.925262i $$0.623845\pi$$
$$180$$ 0 0
$$181$$ −3.64003 −0.270561 −0.135281 0.990807i $$-0.543194\pi$$
−0.135281 + 0.990807i $$0.543194\pi$$
$$182$$ 0 0
$$183$$ 0 0
$$184$$ 0 0
$$185$$ 4.94486 0.363553
$$186$$ 0 0
$$187$$ 7.15236 0.523033
$$188$$ 0 0
$$189$$ 0 0
$$190$$ 0 0
$$191$$ 17.4861 1.26525 0.632626 0.774457i $$-0.281977\pi$$
0.632626 + 0.774457i $$0.281977\pi$$
$$192$$ 0 0
$$193$$ 4.75052 0.341950 0.170975 0.985275i $$-0.445308\pi$$
0.170975 + 0.985275i $$0.445308\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ 2.67780 0.190785 0.0953927 0.995440i $$-0.469589\pi$$
0.0953927 + 0.995440i $$0.469589\pi$$
$$198$$ 0 0
$$199$$ 25.6656 1.81939 0.909693 0.415281i $$-0.136317\pi$$
0.909693 + 0.415281i $$0.136317\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ 0 0
$$203$$ −0.458181 −0.0321580
$$204$$ 0 0
$$205$$ −11.9793 −0.836668
$$206$$ 0 0
$$207$$ 0 0
$$208$$ 0 0
$$209$$ −12.9070 −0.892796
$$210$$ 0 0
$$211$$ 9.73234 0.670002 0.335001 0.942218i $$-0.391263\pi$$
0.335001 + 0.942218i $$0.391263\pi$$
$$212$$ 0 0
$$213$$ 0 0
$$214$$ 0 0
$$215$$ 6.99159 0.476822
$$216$$ 0 0
$$217$$ −2.50123 −0.169795
$$218$$ 0 0
$$219$$ 0 0
$$220$$ 0 0
$$221$$ −5.70706 −0.383898
$$222$$ 0 0
$$223$$ 21.5995 1.44641 0.723205 0.690633i $$-0.242668\pi$$
0.723205 + 0.690633i $$0.242668\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 0 0
$$227$$ 14.0401 0.931875 0.465938 0.884818i $$-0.345717\pi$$
0.465938 + 0.884818i $$0.345717\pi$$
$$228$$ 0 0
$$229$$ 5.77020 0.381305 0.190653 0.981658i $$-0.438940\pi$$
0.190653 + 0.981658i $$0.438940\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ 19.4705 1.27556 0.637778 0.770221i $$-0.279854\pi$$
0.637778 + 0.770221i $$0.279854\pi$$
$$234$$ 0 0
$$235$$ 4.79267 0.312639
$$236$$ 0 0
$$237$$ 0 0
$$238$$ 0 0
$$239$$ 14.7014 0.950954 0.475477 0.879728i $$-0.342275\pi$$
0.475477 + 0.879728i $$0.342275\pi$$
$$240$$ 0 0
$$241$$ −21.6813 −1.39661 −0.698306 0.715799i $$-0.746063\pi$$
−0.698306 + 0.715799i $$0.746063\pi$$
$$242$$ 0 0
$$243$$ 0 0
$$244$$ 0 0
$$245$$ 9.92869 0.634321
$$246$$ 0 0
$$247$$ 10.2988 0.655299
$$248$$ 0 0
$$249$$ 0 0
$$250$$ 0 0
$$251$$ 6.71130 0.423613 0.211807 0.977312i $$-0.432065\pi$$
0.211807 + 0.977312i $$0.432065\pi$$
$$252$$ 0 0
$$253$$ 13.4259 0.844077
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 0 0
$$257$$ 26.1000 1.62807 0.814037 0.580813i $$-0.197265\pi$$
0.814037 + 0.580813i $$0.197265\pi$$
$$258$$ 0 0
$$259$$ 3.52757 0.219193
$$260$$ 0 0
$$261$$ 0 0
$$262$$ 0 0
$$263$$ −16.1595 −0.996439 −0.498219 0.867051i $$-0.666013\pi$$
−0.498219 + 0.867051i $$0.666013\pi$$
$$264$$ 0 0
$$265$$ −3.03440 −0.186402
$$266$$ 0 0
$$267$$ 0 0
$$268$$ 0 0
$$269$$ −10.9418 −0.667131 −0.333566 0.942727i $$-0.608252\pi$$
−0.333566 + 0.942727i $$0.608252\pi$$
$$270$$ 0 0
$$271$$ 9.49579 0.576828 0.288414 0.957506i $$-0.406872\pi$$
0.288414 + 0.957506i $$0.406872\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ 0 0
$$275$$ 2.19055 0.132095
$$276$$ 0 0
$$277$$ 1.86063 0.111795 0.0558973 0.998437i $$-0.482198\pi$$
0.0558973 + 0.998437i $$0.482198\pi$$
$$278$$ 0 0
$$279$$ 0 0
$$280$$ 0 0
$$281$$ −4.01042 −0.239242 −0.119621 0.992820i $$-0.538168\pi$$
−0.119621 + 0.992820i $$0.538168\pi$$
$$282$$ 0 0
$$283$$ −7.71234 −0.458451 −0.229225 0.973373i $$-0.573619\pi$$
−0.229225 + 0.973373i $$0.573619\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ −8.54579 −0.504442
$$288$$ 0 0
$$289$$ −6.33913 −0.372890
$$290$$ 0 0
$$291$$ 0 0
$$292$$ 0 0
$$293$$ −9.05585 −0.529048 −0.264524 0.964379i $$-0.585215\pi$$
−0.264524 + 0.964379i $$0.585215\pi$$
$$294$$ 0 0
$$295$$ 11.3105 0.658524
$$296$$ 0 0
$$297$$ 0 0
$$298$$ 0 0
$$299$$ −10.7129 −0.619540
$$300$$ 0 0
$$301$$ 4.98767 0.287485
$$302$$ 0 0
$$303$$ 0 0
$$304$$ 0 0
$$305$$ 9.02926 0.517014
$$306$$ 0 0
$$307$$ 12.1632 0.694192 0.347096 0.937830i $$-0.387168\pi$$
0.347096 + 0.937830i $$0.387168\pi$$
$$308$$ 0 0
$$309$$ 0 0
$$310$$ 0 0
$$311$$ −24.4952 −1.38900 −0.694499 0.719494i $$-0.744374\pi$$
−0.694499 + 0.719494i $$0.744374\pi$$
$$312$$ 0 0
$$313$$ 22.5593 1.27513 0.637563 0.770398i $$-0.279943\pi$$
0.637563 + 0.770398i $$0.279943\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ −13.9546 −0.783770 −0.391885 0.920014i $$-0.628177\pi$$
−0.391885 + 0.920014i $$0.628177\pi$$
$$318$$ 0 0
$$319$$ −0.703457 −0.0393861
$$320$$ 0 0
$$321$$ 0 0
$$322$$ 0 0
$$323$$ −19.2384 −1.07045
$$324$$ 0 0
$$325$$ −1.74790 −0.0969559
$$326$$ 0 0
$$327$$ 0 0
$$328$$ 0 0
$$329$$ 3.41901 0.188496
$$330$$ 0 0
$$331$$ −21.5826 −1.18629 −0.593145 0.805096i $$-0.702114\pi$$
−0.593145 + 0.805096i $$0.702114\pi$$
$$332$$ 0 0
$$333$$ 0 0
$$334$$ 0 0
$$335$$ 19.2288 1.05058
$$336$$ 0 0
$$337$$ 16.7494 0.912399 0.456199 0.889878i $$-0.349210\pi$$
0.456199 + 0.889878i $$0.349210\pi$$
$$338$$ 0 0
$$339$$ 0 0
$$340$$ 0 0
$$341$$ −3.84021 −0.207959
$$342$$ 0 0
$$343$$ 17.0703 0.921709
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ 12.7273 0.683239 0.341620 0.939838i $$-0.389025\pi$$
0.341620 + 0.939838i $$0.389025\pi$$
$$348$$ 0 0
$$349$$ 21.8665 1.17049 0.585244 0.810857i $$-0.300999\pi$$
0.585244 + 0.810857i $$0.300999\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 0 0
$$353$$ −27.7352 −1.47620 −0.738099 0.674692i $$-0.764276\pi$$
−0.738099 + 0.674692i $$0.764276\pi$$
$$354$$ 0 0
$$355$$ −13.3391 −0.707967
$$356$$ 0 0
$$357$$ 0 0
$$358$$ 0 0
$$359$$ 8.30402 0.438269 0.219135 0.975695i $$-0.429677\pi$$
0.219135 + 0.975695i $$0.429677\pi$$
$$360$$ 0 0
$$361$$ 15.7172 0.827220
$$362$$ 0 0
$$363$$ 0 0
$$364$$ 0 0
$$365$$ −5.72780 −0.299807
$$366$$ 0 0
$$367$$ 12.4528 0.650031 0.325015 0.945709i $$-0.394631\pi$$
0.325015 + 0.945709i $$0.394631\pi$$
$$368$$ 0 0
$$369$$ 0 0
$$370$$ 0 0
$$371$$ −2.16469 −0.112385
$$372$$ 0 0
$$373$$ 17.6189 0.912272 0.456136 0.889910i $$-0.349233\pi$$
0.456136 + 0.889910i $$0.349233\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ 0.561307 0.0289088
$$378$$ 0 0
$$379$$ 12.1749 0.625383 0.312691 0.949855i $$-0.398769\pi$$
0.312691 + 0.949855i $$0.398769\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ 0 0
$$383$$ −16.9672 −0.866981 −0.433491 0.901158i $$-0.642718\pi$$
−0.433491 + 0.901158i $$0.642718\pi$$
$$384$$ 0 0
$$385$$ −6.25080 −0.318570
$$386$$ 0 0
$$387$$ 0 0
$$388$$ 0 0
$$389$$ −14.9358 −0.757275 −0.378637 0.925545i $$-0.623607\pi$$
−0.378637 + 0.925545i $$0.623607\pi$$
$$390$$ 0 0
$$391$$ 20.0118 1.01204
$$392$$ 0 0
$$393$$ 0 0
$$394$$ 0 0
$$395$$ −20.9995 −1.05660
$$396$$ 0 0
$$397$$ 27.3638 1.37335 0.686674 0.726966i $$-0.259070\pi$$
0.686674 + 0.726966i $$0.259070\pi$$
$$398$$ 0 0
$$399$$ 0 0
$$400$$ 0 0
$$401$$ −16.3411 −0.816035 −0.408017 0.912974i $$-0.633780\pi$$
−0.408017 + 0.912974i $$0.633780\pi$$
$$402$$ 0 0
$$403$$ 3.06421 0.152639
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 0 0
$$407$$ 5.41598 0.268460
$$408$$ 0 0
$$409$$ −19.2236 −0.950544 −0.475272 0.879839i $$-0.657650\pi$$
−0.475272 + 0.879839i $$0.657650\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ 0 0
$$413$$ 8.06872 0.397036
$$414$$ 0 0
$$415$$ −14.4874 −0.711158
$$416$$ 0 0
$$417$$ 0 0
$$418$$ 0 0
$$419$$ −1.15541 −0.0564456 −0.0282228 0.999602i $$-0.508985\pi$$
−0.0282228 + 0.999602i $$0.508985\pi$$
$$420$$ 0 0
$$421$$ 4.80899 0.234376 0.117188 0.993110i $$-0.462612\pi$$
0.117188 + 0.993110i $$0.462612\pi$$
$$422$$ 0 0
$$423$$ 0 0
$$424$$ 0 0
$$425$$ 3.26510 0.158381
$$426$$ 0 0
$$427$$ 6.44131 0.311717
$$428$$ 0 0
$$429$$ 0 0
$$430$$ 0 0
$$431$$ 34.3708 1.65558 0.827791 0.561036i $$-0.189597\pi$$
0.827791 + 0.561036i $$0.189597\pi$$
$$432$$ 0 0
$$433$$ −20.0626 −0.964145 −0.482072 0.876131i $$-0.660116\pi$$
−0.482072 + 0.876131i $$0.660116\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 0 0
$$437$$ −36.1128 −1.72751
$$438$$ 0 0
$$439$$ 1.90270 0.0908108 0.0454054 0.998969i $$-0.485542\pi$$
0.0454054 + 0.998969i $$0.485542\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 0 0
$$443$$ 13.8782 0.659373 0.329687 0.944090i $$-0.393057\pi$$
0.329687 + 0.944090i $$0.393057\pi$$
$$444$$ 0 0
$$445$$ −0.513269 −0.0243313
$$446$$ 0 0
$$447$$ 0 0
$$448$$ 0 0
$$449$$ −14.6316 −0.690509 −0.345254 0.938509i $$-0.612207\pi$$
−0.345254 + 0.938509i $$0.612207\pi$$
$$450$$ 0 0
$$451$$ −13.1206 −0.617824
$$452$$ 0 0
$$453$$ 0 0
$$454$$ 0 0
$$455$$ 4.98767 0.233826
$$456$$ 0 0
$$457$$ −15.1270 −0.707613 −0.353807 0.935319i $$-0.615113\pi$$
−0.353807 + 0.935319i $$0.615113\pi$$
$$458$$ 0 0
$$459$$ 0 0
$$460$$ 0 0
$$461$$ 0.617968 0.0287816 0.0143908 0.999896i $$-0.495419\pi$$
0.0143908 + 0.999896i $$0.495419\pi$$
$$462$$ 0 0
$$463$$ −6.12321 −0.284570 −0.142285 0.989826i $$-0.545445\pi$$
−0.142285 + 0.989826i $$0.545445\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ 22.0319 1.01952 0.509758 0.860318i $$-0.329735\pi$$
0.509758 + 0.860318i $$0.329735\pi$$
$$468$$ 0 0
$$469$$ 13.7175 0.633416
$$470$$ 0 0
$$471$$ 0 0
$$472$$ 0 0
$$473$$ 7.65771 0.352102
$$474$$ 0 0
$$475$$ −5.89213 −0.270349
$$476$$ 0 0
$$477$$ 0 0
$$478$$ 0 0
$$479$$ 6.88989 0.314807 0.157404 0.987534i $$-0.449688\pi$$
0.157404 + 0.987534i $$0.449688\pi$$
$$480$$ 0 0
$$481$$ −4.32155 −0.197046
$$482$$ 0 0
$$483$$ 0 0
$$484$$ 0 0
$$485$$ −11.6709 −0.529948
$$486$$ 0 0
$$487$$ −20.9486 −0.949272 −0.474636 0.880182i $$-0.657420\pi$$
−0.474636 + 0.880182i $$0.657420\pi$$
$$488$$ 0 0
$$489$$ 0 0
$$490$$ 0 0
$$491$$ −13.3184 −0.601051 −0.300525 0.953774i $$-0.597162\pi$$
−0.300525 + 0.953774i $$0.597162\pi$$
$$492$$ 0 0
$$493$$ −1.04853 −0.0472234
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ −9.51589 −0.426846
$$498$$ 0 0
$$499$$ 15.2496 0.682665 0.341333 0.939943i $$-0.389122\pi$$
0.341333 + 0.939943i $$0.389122\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ 0 0
$$503$$ 26.7387 1.19222 0.596110 0.802903i $$-0.296712\pi$$
0.596110 + 0.802903i $$0.296712\pi$$
$$504$$ 0 0
$$505$$ −11.1769 −0.497364
$$506$$ 0 0
$$507$$ 0 0
$$508$$ 0 0
$$509$$ −0.156638 −0.00694286 −0.00347143 0.999994i $$-0.501105\pi$$
−0.00347143 + 0.999994i $$0.501105\pi$$
$$510$$ 0 0
$$511$$ −4.08611 −0.180759
$$512$$ 0 0
$$513$$ 0 0
$$514$$ 0 0
$$515$$ −12.3472 −0.544082
$$516$$ 0 0
$$517$$ 5.24929 0.230864
$$518$$ 0 0
$$519$$ 0 0
$$520$$ 0 0
$$521$$ 32.1200 1.40720 0.703602 0.710594i $$-0.251574\pi$$
0.703602 + 0.710594i $$0.251574\pi$$
$$522$$ 0 0
$$523$$ 4.73691 0.207131 0.103565 0.994623i $$-0.466975\pi$$
0.103565 + 0.994623i $$0.466975\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ −5.72399 −0.249341
$$528$$ 0 0
$$529$$ 14.5646 0.633244
$$530$$ 0 0
$$531$$ 0 0
$$532$$ 0 0
$$533$$ 10.4693 0.453474
$$534$$ 0 0
$$535$$ −23.9139 −1.03389
$$536$$ 0 0
$$537$$ 0 0
$$538$$ 0 0
$$539$$ 10.8746 0.468404
$$540$$ 0 0
$$541$$ 15.3824 0.661342 0.330671 0.943746i $$-0.392725\pi$$
0.330671 + 0.943746i $$0.392725\pi$$
$$542$$ 0 0
$$543$$ 0 0
$$544$$ 0 0
$$545$$ 30.3567 1.30034
$$546$$ 0 0
$$547$$ 19.2682 0.823848 0.411924 0.911218i $$-0.364857\pi$$
0.411924 + 0.911218i $$0.364857\pi$$
$$548$$ 0 0
$$549$$ 0 0
$$550$$ 0 0
$$551$$ 1.89216 0.0806086
$$552$$ 0 0
$$553$$ −14.9806 −0.637041
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 0 0
$$557$$ 10.0313 0.425039 0.212519 0.977157i $$-0.431833\pi$$
0.212519 + 0.977157i $$0.431833\pi$$
$$558$$ 0 0
$$559$$ −6.11029 −0.258438
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 0 0
$$563$$ 8.61733 0.363177 0.181589 0.983375i $$-0.441876\pi$$
0.181589 + 0.983375i $$0.441876\pi$$
$$564$$ 0 0
$$565$$ −1.42081 −0.0597739
$$566$$ 0 0
$$567$$ 0 0
$$568$$ 0 0
$$569$$ 22.8782 0.959105 0.479552 0.877513i $$-0.340799\pi$$
0.479552 + 0.877513i $$0.340799\pi$$
$$570$$ 0 0
$$571$$ −36.9981 −1.54832 −0.774162 0.632987i $$-0.781829\pi$$
−0.774162 + 0.632987i $$0.781829\pi$$
$$572$$ 0 0
$$573$$ 0 0
$$574$$ 0 0
$$575$$ 6.12900 0.255597
$$576$$ 0 0
$$577$$ 9.42378 0.392317 0.196158 0.980572i $$-0.437153\pi$$
0.196158 + 0.980572i $$0.437153\pi$$
$$578$$ 0 0
$$579$$ 0 0
$$580$$ 0 0
$$581$$ −10.3350 −0.428770
$$582$$ 0 0
$$583$$ −3.32351 −0.137646
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ 6.06497 0.250328 0.125164 0.992136i $$-0.460054\pi$$
0.125164 + 0.992136i $$0.460054\pi$$
$$588$$ 0 0
$$589$$ 10.3294 0.425615
$$590$$ 0 0
$$591$$ 0 0
$$592$$ 0 0
$$593$$ −30.0669 −1.23470 −0.617351 0.786688i $$-0.711794\pi$$
−0.617351 + 0.786688i $$0.711794\pi$$
$$594$$ 0 0
$$595$$ −9.31705 −0.381962
$$596$$ 0 0
$$597$$ 0 0
$$598$$ 0 0
$$599$$ 8.74168 0.357175 0.178588 0.983924i $$-0.442847\pi$$
0.178588 + 0.983924i $$0.442847\pi$$
$$600$$ 0 0
$$601$$ −5.47200 −0.223208 −0.111604 0.993753i $$-0.535599\pi$$
−0.111604 + 0.993753i $$0.535599\pi$$
$$602$$ 0 0
$$603$$ 0 0
$$604$$ 0 0
$$605$$ 12.4030 0.504253
$$606$$ 0 0
$$607$$ 8.28118 0.336123 0.168061 0.985777i $$-0.446249\pi$$
0.168061 + 0.985777i $$0.446249\pi$$
$$608$$ 0 0
$$609$$ 0 0
$$610$$ 0 0
$$611$$ −4.18855 −0.169450
$$612$$ 0 0
$$613$$ 46.5063 1.87837 0.939186 0.343408i $$-0.111581\pi$$
0.939186 + 0.343408i $$0.111581\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ −34.1883 −1.37637 −0.688185 0.725535i $$-0.741592\pi$$
−0.688185 + 0.725535i $$0.741592\pi$$
$$618$$ 0 0
$$619$$ −23.3216 −0.937373 −0.468686 0.883365i $$-0.655273\pi$$
−0.468686 + 0.883365i $$0.655273\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ 0 0
$$623$$ −0.366157 −0.0146698
$$624$$ 0 0
$$625$$ −19.0000 −0.760000
$$626$$ 0 0
$$627$$ 0 0
$$628$$ 0 0
$$629$$ 8.07272 0.321881
$$630$$ 0 0
$$631$$ −29.0863 −1.15791 −0.578953 0.815361i $$-0.696538\pi$$
−0.578953 + 0.815361i $$0.696538\pi$$
$$632$$ 0 0
$$633$$ 0 0
$$634$$ 0 0
$$635$$ −23.0330 −0.914037
$$636$$ 0 0
$$637$$ −8.67716 −0.343802
$$638$$ 0 0
$$639$$ 0 0
$$640$$ 0 0
$$641$$ −31.0603 −1.22681 −0.613404 0.789769i $$-0.710200\pi$$
−0.613404 + 0.789769i $$0.710200\pi$$
$$642$$ 0 0
$$643$$ 14.9145 0.588169 0.294085 0.955779i $$-0.404985\pi$$
0.294085 + 0.955779i $$0.404985\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ −28.9951 −1.13991 −0.569957 0.821675i $$-0.693040\pi$$
−0.569957 + 0.821675i $$0.693040\pi$$
$$648$$ 0 0
$$649$$ 12.3881 0.486277
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 0 0
$$653$$ −7.42152 −0.290426 −0.145213 0.989400i $$-0.546387\pi$$
−0.145213 + 0.989400i $$0.546387\pi$$
$$654$$ 0 0
$$655$$ 16.0520 0.627203
$$656$$ 0 0
$$657$$ 0 0
$$658$$ 0 0
$$659$$ 1.40447 0.0547102 0.0273551 0.999626i $$-0.491292\pi$$
0.0273551 + 0.999626i $$0.491292\pi$$
$$660$$ 0 0
$$661$$ 13.1593 0.511837 0.255919 0.966698i $$-0.417622\pi$$
0.255919 + 0.966698i $$0.417622\pi$$
$$662$$ 0 0
$$663$$ 0 0
$$664$$ 0 0
$$665$$ 16.8134 0.651994
$$666$$ 0 0
$$667$$ −1.96822 −0.0762099
$$668$$ 0 0
$$669$$ 0 0
$$670$$ 0 0
$$671$$ 9.88952 0.381781
$$672$$ 0 0
$$673$$ −24.1775 −0.931974 −0.465987 0.884791i $$-0.654301\pi$$
−0.465987 + 0.884791i $$0.654301\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ 0 0
$$677$$ −13.6811 −0.525806 −0.262903 0.964822i $$-0.584680\pi$$
−0.262903 + 0.964822i $$0.584680\pi$$
$$678$$ 0 0
$$679$$ −8.32580 −0.319515
$$680$$ 0 0
$$681$$ 0 0
$$682$$ 0 0
$$683$$ 14.3047 0.547355 0.273678 0.961822i $$-0.411760\pi$$
0.273678 + 0.961822i $$0.411760\pi$$
$$684$$ 0 0
$$685$$ −31.3312 −1.19710
$$686$$ 0 0
$$687$$ 0 0
$$688$$ 0 0
$$689$$ 2.65191 0.101030
$$690$$ 0 0
$$691$$ 27.8109 1.05798 0.528989 0.848629i $$-0.322571\pi$$
0.528989 + 0.848629i $$0.322571\pi$$
$$692$$ 0 0
$$693$$ 0 0
$$694$$ 0 0
$$695$$ 4.44326 0.168543
$$696$$ 0 0
$$697$$ −19.5567 −0.740764
$$698$$ 0 0
$$699$$ 0 0
$$700$$ 0 0
$$701$$ −10.7440 −0.405796 −0.202898 0.979200i $$-0.565036\pi$$
−0.202898 + 0.979200i $$0.565036\pi$$
$$702$$ 0 0
$$703$$ −14.5679 −0.549437
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ −7.97337 −0.299869
$$708$$ 0 0
$$709$$ −39.8043 −1.49488 −0.747440 0.664329i $$-0.768717\pi$$
−0.747440 + 0.664329i $$0.768717\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ 0 0
$$713$$ −10.7446 −0.402390
$$714$$ 0 0
$$715$$ 7.65771 0.286382
$$716$$ 0 0
$$717$$ 0 0
$$718$$ 0 0
$$719$$ −23.3327 −0.870162 −0.435081 0.900391i $$-0.643280\pi$$
−0.435081 + 0.900391i $$0.643280\pi$$
$$720$$ 0 0
$$721$$ −8.80826 −0.328037
$$722$$ 0 0
$$723$$ 0 0
$$724$$ 0 0
$$725$$ −0.321133 −0.0119266
$$726$$ 0 0
$$727$$ −21.2593 −0.788464 −0.394232 0.919011i $$-0.628989\pi$$
−0.394232 + 0.919011i $$0.628989\pi$$
$$728$$ 0 0
$$729$$ 0 0
$$730$$ 0 0
$$731$$ 11.4141 0.422166
$$732$$ 0 0
$$733$$ −40.8733 −1.50969 −0.754846 0.655902i $$-0.772288\pi$$
−0.754846 + 0.655902i $$0.772288\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ 21.0609 0.775787
$$738$$ 0 0
$$739$$ 38.5912 1.41960 0.709799 0.704404i $$-0.248786\pi$$
0.709799 + 0.704404i $$0.248786\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 0 0
$$743$$ −18.3330 −0.672572 −0.336286 0.941760i $$-0.609171\pi$$
−0.336286 + 0.941760i $$0.609171\pi$$
$$744$$ 0 0
$$745$$ 12.0520 0.441551
$$746$$ 0 0
$$747$$ 0 0
$$748$$ 0 0
$$749$$ −17.0598 −0.623350
$$750$$ 0 0
$$751$$ −41.3572 −1.50915 −0.754573 0.656216i $$-0.772156\pi$$
−0.754573 + 0.656216i $$0.772156\pi$$
$$752$$ 0 0
$$753$$ 0 0
$$754$$ 0 0
$$755$$ 21.9368 0.798362
$$756$$ 0 0
$$757$$ 14.2655 0.518490 0.259245 0.965812i $$-0.416526\pi$$
0.259245 + 0.965812i $$0.416526\pi$$
$$758$$ 0 0
$$759$$ 0 0
$$760$$ 0 0
$$761$$ 41.8123 1.51569 0.757847 0.652432i $$-0.226251\pi$$
0.757847 + 0.652432i $$0.226251\pi$$
$$762$$ 0 0
$$763$$ 21.6559 0.783997
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ −9.88481 −0.356920
$$768$$ 0 0
$$769$$ 3.27230 0.118002 0.0590010 0.998258i $$-0.481208\pi$$
0.0590010 + 0.998258i $$0.481208\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ 0 0
$$773$$ −23.3460 −0.839696 −0.419848 0.907594i $$-0.637917\pi$$
−0.419848 + 0.907594i $$0.637917\pi$$
$$774$$ 0 0
$$775$$ −1.75308 −0.0629726
$$776$$ 0 0
$$777$$ 0 0
$$778$$ 0 0
$$779$$ 35.2917 1.26446
$$780$$ 0 0
$$781$$ −14.6100 −0.522787
$$782$$ 0 0
$$783$$ 0 0
$$784$$ 0 0
$$785$$ −40.0949 −1.43105
$$786$$ 0 0
$$787$$ 44.5309 1.58736 0.793678 0.608338i $$-0.208163\pi$$
0.793678 + 0.608338i $$0.208163\pi$$
$$788$$ 0 0
$$789$$ 0 0
$$790$$ 0 0
$$791$$ −1.01358 −0.0360388
$$792$$ 0 0
$$793$$ −7.89110 −0.280221
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 0 0
$$797$$ −7.60786 −0.269484 −0.134742 0.990881i $$-0.543021\pi$$
−0.134742 + 0.990881i $$0.543021\pi$$
$$798$$ 0 0
$$799$$ 7.82427 0.276803
$$800$$ 0 0
$$801$$ 0 0
$$802$$ 0 0
$$803$$ −6.27351 −0.221387
$$804$$ 0 0
$$805$$ −17.4893 −0.616416
$$806$$ 0 0
$$807$$ 0 0
$$808$$ 0 0
$$809$$ −7.40132 −0.260217 −0.130108 0.991500i $$-0.541532\pi$$
−0.130108 + 0.991500i $$0.541532\pi$$
$$810$$ 0 0
$$811$$ −5.69610 −0.200017 −0.100009 0.994987i $$-0.531887\pi$$
−0.100009 + 0.994987i $$0.531887\pi$$
$$812$$ 0 0
$$813$$ 0 0
$$814$$ 0 0
$$815$$ −10.4354 −0.365536
$$816$$ 0 0
$$817$$ −20.5977 −0.720621
$$818$$ 0 0
$$819$$ 0 0
$$820$$ 0 0
$$821$$ 45.1962 1.57736 0.788680 0.614804i $$-0.210765\pi$$
0.788680 + 0.614804i $$0.210765\pi$$
$$822$$ 0 0
$$823$$ 13.9450 0.486093 0.243047 0.970015i $$-0.421853\pi$$
0.243047 + 0.970015i $$0.421853\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ −2.90541 −0.101031 −0.0505155 0.998723i $$-0.516086\pi$$
−0.0505155 + 0.998723i $$0.516086\pi$$
$$828$$ 0 0
$$829$$ 31.2634 1.08582 0.542912 0.839790i $$-0.317322\pi$$
0.542912 + 0.839790i $$0.317322\pi$$
$$830$$ 0 0
$$831$$ 0 0
$$832$$ 0 0
$$833$$ 16.2091 0.561611
$$834$$ 0 0
$$835$$ −2.00000 −0.0692129
$$836$$ 0 0
$$837$$ 0 0
$$838$$ 0 0
$$839$$ 44.4886 1.53592 0.767958 0.640500i $$-0.221273\pi$$
0.767958 + 0.640500i $$0.221273\pi$$
$$840$$ 0 0
$$841$$ −28.8969 −0.996444
$$842$$ 0 0
$$843$$ 0 0
$$844$$ 0 0
$$845$$ 19.8897 0.684227
$$846$$ 0 0
$$847$$ 8.84806 0.304023
$$848$$ 0 0
$$849$$ 0 0
$$850$$ 0 0
$$851$$ 15.1535 0.519455
$$852$$ 0 0
$$853$$ −17.2717 −0.591371 −0.295686 0.955285i $$-0.595548\pi$$
−0.295686 + 0.955285i $$0.595548\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0 0
$$857$$ 35.0334 1.19672 0.598359 0.801228i $$-0.295820\pi$$
0.598359 + 0.801228i $$0.295820\pi$$
$$858$$ 0 0
$$859$$ 38.9649 1.32947 0.664733 0.747081i $$-0.268546\pi$$
0.664733 + 0.747081i $$0.268546\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 0 0
$$863$$ −32.0638 −1.09146 −0.545732 0.837960i $$-0.683748\pi$$
−0.545732 + 0.837960i $$0.683748\pi$$
$$864$$ 0 0
$$865$$ −43.0330 −1.46317
$$866$$ 0 0
$$867$$ 0 0
$$868$$ 0 0
$$869$$ −23.0002 −0.780228
$$870$$ 0 0
$$871$$ −16.8050 −0.569416
$$872$$ 0 0
$$873$$ 0 0
$$874$$ 0 0
$$875$$ −17.1212 −0.578801
$$876$$ 0 0
$$877$$ 53.9033 1.82018 0.910092 0.414406i $$-0.136011\pi$$
0.910092 + 0.414406i $$0.136011\pi$$
$$878$$ 0 0
$$879$$ 0 0
$$880$$ 0 0
$$881$$ −49.8614 −1.67987 −0.839937 0.542684i $$-0.817408\pi$$
−0.839937 + 0.542684i $$0.817408\pi$$
$$882$$ 0 0
$$883$$ 12.9835 0.436930 0.218465 0.975845i $$-0.429895\pi$$
0.218465 + 0.975845i $$0.429895\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 0 0
$$887$$ −11.8245 −0.397027 −0.198513 0.980098i $$-0.563611\pi$$
−0.198513 + 0.980098i $$0.563611\pi$$
$$888$$ 0 0
$$889$$ −16.4313 −0.551089
$$890$$ 0 0
$$891$$ 0 0
$$892$$ 0 0
$$893$$ −14.1195 −0.472492
$$894$$ 0 0
$$895$$ 20.3002 0.678562
$$896$$ 0 0
$$897$$ 0 0
$$898$$ 0 0
$$899$$ 0.562972 0.0187762
$$900$$ 0 0
$$901$$ −4.95381 −0.165036
$$902$$ 0 0
$$903$$ 0 0
$$904$$ 0 0
$$905$$ 7.28005 0.241997
$$906$$ 0 0
$$907$$ 1.08780 0.0361198 0.0180599 0.999837i $$-0.494251\pi$$
0.0180599 + 0.999837i $$0.494251\pi$$
$$908$$ 0 0
$$909$$ 0 0
$$910$$ 0 0
$$911$$ 9.16245 0.303566 0.151783 0.988414i $$-0.451499\pi$$
0.151783 + 0.988414i $$0.451499\pi$$
$$912$$ 0 0
$$913$$ −15.8677 −0.525143
$$914$$ 0 0
$$915$$ 0 0
$$916$$ 0 0
$$917$$ 11.4512 0.378152
$$918$$ 0 0
$$919$$ −23.2997 −0.768587 −0.384293 0.923211i $$-0.625555\pi$$
−0.384293 + 0.923211i $$0.625555\pi$$
$$920$$ 0 0
$$921$$ 0 0
$$922$$ 0 0
$$923$$ 11.6577 0.383718
$$924$$ 0 0
$$925$$ 2.47243 0.0812929
$$926$$ 0 0
$$927$$ 0 0
$$928$$ 0 0
$$929$$ −48.9106 −1.60471 −0.802353 0.596850i $$-0.796419\pi$$
−0.802353 + 0.596850i $$0.796419\pi$$
$$930$$ 0 0
$$931$$ −29.2506 −0.958648
$$932$$ 0 0
$$933$$ 0 0
$$934$$ 0 0
$$935$$ −14.3047 −0.467815
$$936$$ 0 0
$$937$$ 29.3682 0.959418 0.479709 0.877428i $$-0.340742\pi$$
0.479709 + 0.877428i $$0.340742\pi$$
$$938$$ 0 0
$$939$$ 0 0
$$940$$ 0 0
$$941$$ 33.9716 1.10744 0.553721 0.832702i $$-0.313207\pi$$
0.553721 + 0.832702i $$0.313207\pi$$
$$942$$ 0 0
$$943$$ −36.7104 −1.19546
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ −36.7945 −1.19566 −0.597830 0.801623i $$-0.703970\pi$$
−0.597830 + 0.801623i $$0.703970\pi$$
$$948$$ 0 0
$$949$$ 5.00580 0.162495
$$950$$ 0 0
$$951$$ 0 0
$$952$$ 0 0
$$953$$ 5.19613 0.168319 0.0841596 0.996452i $$-0.473179\pi$$
0.0841596 + 0.996452i $$0.473179\pi$$
$$954$$ 0 0
$$955$$ −34.9723 −1.13168
$$956$$ 0 0
$$957$$ 0 0
$$958$$ 0 0
$$959$$ −22.3511 −0.721755
$$960$$ 0 0
$$961$$ −27.9267 −0.900861
$$962$$ 0 0
$$963$$ 0 0
$$964$$ 0 0
$$965$$ −9.50105 −0.305849
$$966$$ 0 0
$$967$$ 49.3421 1.58673 0.793367 0.608743i $$-0.208326\pi$$
0.793367 + 0.608743i $$0.208326\pi$$
$$968$$ 0 0
$$969$$ 0 0
$$970$$ 0 0
$$971$$ 43.3773 1.39204 0.696021 0.718021i $$-0.254952\pi$$
0.696021 + 0.718021i $$0.254952\pi$$
$$972$$ 0 0
$$973$$ 3.16975 0.101617
$$974$$ 0 0
$$975$$ 0 0
$$976$$ 0 0
$$977$$ −55.2452 −1.76745 −0.883725 0.468006i $$-0.844972\pi$$
−0.883725 + 0.468006i $$0.844972\pi$$
$$978$$ 0 0
$$979$$ −0.562170 −0.0179670
$$980$$ 0 0
$$981$$ 0 0
$$982$$ 0 0
$$983$$ 12.0299 0.383694 0.191847 0.981425i $$-0.438552\pi$$
0.191847 + 0.981425i $$0.438552\pi$$
$$984$$ 0 0
$$985$$ −5.35560 −0.170644
$$986$$ 0 0
$$987$$ 0 0
$$988$$ 0 0
$$989$$ 21.4257 0.681298
$$990$$ 0 0
$$991$$ 44.9737 1.42864 0.714318 0.699822i $$-0.246737\pi$$
0.714318 + 0.699822i $$0.246737\pi$$
$$992$$ 0 0
$$993$$ 0 0
$$994$$ 0 0
$$995$$ −51.3312 −1.62731
$$996$$ 0 0
$$997$$ 44.0823 1.39610 0.698050 0.716049i $$-0.254051\pi$$
0.698050 + 0.716049i $$0.254051\pi$$
$$998$$ 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 6012.2.a.d.1.1 5
3.2 odd 2 668.2.a.b.1.3 5
12.11 even 2 2672.2.a.j.1.3 5

By twisted newform
Twist Min Dim Char Parity Ord Type
668.2.a.b.1.3 5 3.2 odd 2
2672.2.a.j.1.3 5 12.11 even 2
6012.2.a.d.1.1 5 1.1 even 1 trivial