# Properties

 Label 5472.2.a.bs.1.4 Level $5472$ Weight $2$ Character 5472.1 Self dual yes Analytic conductor $43.694$ Analytic rank $0$ Dimension $4$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$5472 = 2^{5} \cdot 3^{2} \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 5472.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$43.6941399860$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: 4.4.15317.1 Defining polynomial: $$x^{4} - 2x^{3} - 4x^{2} + 5x + 2$$ x^4 - 2*x^3 - 4*x^2 + 5*x + 2 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 608) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.4 Root $$-1.69353$$ of defining polynomial Character $$\chi$$ $$=$$ 5472.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+4.20608 q^{5} +1.74252 q^{7} +O(q^{10})$$ $$q+4.20608 q^{5} +1.74252 q^{7} -6.20608 q^{11} +4.82550 q^{13} +5.12957 q^{17} +1.00000 q^{19} -0.463560 q^{23} +12.6911 q^{25} -4.82550 q^{29} -1.63806 q^{31} +7.32919 q^{35} +3.63806 q^{37} +5.28906 q^{41} +1.08298 q^{43} +0.555087 q^{47} -3.96362 q^{49} -2.65955 q^{53} -26.1033 q^{55} +1.85061 q^{59} -9.32919 q^{61} +20.2964 q^{65} +4.72751 q^{67} +11.4850 q^{71} -0.00646614 q^{73} -10.8142 q^{77} +6.76117 q^{79} +11.8972 q^{83} +21.5754 q^{85} +7.31909 q^{89} +8.40853 q^{91} +4.20608 q^{95} -10.4122 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - q^{5} - q^{7}+O(q^{10})$$ 4 * q - q^5 - q^7 $$4 q - q^{5} - q^{7} - 7 q^{11} + 10 q^{13} - 5 q^{17} + 4 q^{19} + 8 q^{23} + 17 q^{25} - 10 q^{29} - 6 q^{31} - 5 q^{35} + 14 q^{37} + 2 q^{41} + 3 q^{43} + 3 q^{47} + 7 q^{49} - 4 q^{53} - 35 q^{55} - 20 q^{59} - 3 q^{61} + 12 q^{65} + 8 q^{67} + 30 q^{71} + 9 q^{73} + 7 q^{77} + 10 q^{79} - 4 q^{83} + 19 q^{85} + 16 q^{89} + 10 q^{91} - q^{95} - 6 q^{97}+O(q^{100})$$ 4 * q - q^5 - q^7 - 7 * q^11 + 10 * q^13 - 5 * q^17 + 4 * q^19 + 8 * q^23 + 17 * q^25 - 10 * q^29 - 6 * q^31 - 5 * q^35 + 14 * q^37 + 2 * q^41 + 3 * q^43 + 3 * q^47 + 7 * q^49 - 4 * q^53 - 35 * q^55 - 20 * q^59 - 3 * q^61 + 12 * q^65 + 8 * q^67 + 30 * q^71 + 9 * q^73 + 7 * q^77 + 10 * q^79 - 4 * q^83 + 19 * q^85 + 16 * q^89 + 10 * q^91 - q^95 - 6 * 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
$$5$$ 4.20608 1.88102 0.940508 0.339770i $$-0.110349\pi$$
0.940508 + 0.339770i $$0.110349\pi$$
$$6$$ 0 0
$$7$$ 1.74252 0.658611 0.329306 0.944223i $$-0.393185\pi$$
0.329306 + 0.944223i $$0.393185\pi$$
$$8$$ 0 0
$$9$$ 0 0
$$10$$ 0 0
$$11$$ −6.20608 −1.87120 −0.935602 0.353056i $$-0.885142\pi$$
−0.935602 + 0.353056i $$0.885142\pi$$
$$12$$ 0 0
$$13$$ 4.82550 1.33835 0.669176 0.743104i $$-0.266647\pi$$
0.669176 + 0.743104i $$0.266647\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 0 0
$$17$$ 5.12957 1.24410 0.622052 0.782976i $$-0.286299\pi$$
0.622052 + 0.782976i $$0.286299\pi$$
$$18$$ 0 0
$$19$$ 1.00000 0.229416
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 0 0
$$23$$ −0.463560 −0.0966590 −0.0483295 0.998831i $$-0.515390\pi$$
−0.0483295 + 0.998831i $$0.515390\pi$$
$$24$$ 0 0
$$25$$ 12.6911 2.53822
$$26$$ 0 0
$$27$$ 0 0
$$28$$ 0 0
$$29$$ −4.82550 −0.896072 −0.448036 0.894015i $$-0.647876\pi$$
−0.448036 + 0.894015i $$0.647876\pi$$
$$30$$ 0 0
$$31$$ −1.63806 −0.294205 −0.147102 0.989121i $$-0.546995\pi$$
−0.147102 + 0.989121i $$0.546995\pi$$
$$32$$ 0 0
$$33$$ 0 0
$$34$$ 0 0
$$35$$ 7.32919 1.23886
$$36$$ 0 0
$$37$$ 3.63806 0.598094 0.299047 0.954238i $$-0.403331\pi$$
0.299047 + 0.954238i $$0.403331\pi$$
$$38$$ 0 0
$$39$$ 0 0
$$40$$ 0 0
$$41$$ 5.28906 0.826012 0.413006 0.910728i $$-0.364479\pi$$
0.413006 + 0.910728i $$0.364479\pi$$
$$42$$ 0 0
$$43$$ 1.08298 0.165152 0.0825762 0.996585i $$-0.473685\pi$$
0.0825762 + 0.996585i $$0.473685\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 0 0
$$47$$ 0.555087 0.0809677 0.0404839 0.999180i $$-0.487110\pi$$
0.0404839 + 0.999180i $$0.487110\pi$$
$$48$$ 0 0
$$49$$ −3.96362 −0.566231
$$50$$ 0 0
$$51$$ 0 0
$$52$$ 0 0
$$53$$ −2.65955 −0.365317 −0.182658 0.983176i $$-0.558470\pi$$
−0.182658 + 0.983176i $$0.558470\pi$$
$$54$$ 0 0
$$55$$ −26.1033 −3.51977
$$56$$ 0 0
$$57$$ 0 0
$$58$$ 0 0
$$59$$ 1.85061 0.240929 0.120465 0.992718i $$-0.461562\pi$$
0.120465 + 0.992718i $$0.461562\pi$$
$$60$$ 0 0
$$61$$ −9.32919 −1.19448 −0.597240 0.802063i $$-0.703736\pi$$
−0.597240 + 0.802063i $$0.703736\pi$$
$$62$$ 0 0
$$63$$ 0 0
$$64$$ 0 0
$$65$$ 20.2964 2.51746
$$66$$ 0 0
$$67$$ 4.72751 0.577557 0.288778 0.957396i $$-0.406751\pi$$
0.288778 + 0.957396i $$0.406751\pi$$
$$68$$ 0 0
$$69$$ 0 0
$$70$$ 0 0
$$71$$ 11.4850 1.36302 0.681512 0.731807i $$-0.261323\pi$$
0.681512 + 0.731807i $$0.261323\pi$$
$$72$$ 0 0
$$73$$ −0.00646614 −0.000756804 0 −0.000378402 1.00000i $$-0.500120\pi$$
−0.000378402 1.00000i $$0.500120\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 0 0
$$77$$ −10.8142 −1.23240
$$78$$ 0 0
$$79$$ 6.76117 0.760691 0.380345 0.924844i $$-0.375805\pi$$
0.380345 + 0.924844i $$0.375805\pi$$
$$80$$ 0 0
$$81$$ 0 0
$$82$$ 0 0
$$83$$ 11.8972 1.30589 0.652944 0.757406i $$-0.273534\pi$$
0.652944 + 0.757406i $$0.273534\pi$$
$$84$$ 0 0
$$85$$ 21.5754 2.34018
$$86$$ 0 0
$$87$$ 0 0
$$88$$ 0 0
$$89$$ 7.31909 0.775822 0.387911 0.921697i $$-0.373197\pi$$
0.387911 + 0.921697i $$0.373197\pi$$
$$90$$ 0 0
$$91$$ 8.40853 0.881454
$$92$$ 0 0
$$93$$ 0 0
$$94$$ 0 0
$$95$$ 4.20608 0.431535
$$96$$ 0 0
$$97$$ −10.4122 −1.05720 −0.528598 0.848873i $$-0.677282\pi$$
−0.528598 + 0.848873i $$0.677282\pi$$
$$98$$ 0 0
$$99$$ 0 0
$$100$$ 0 0
$$101$$ −0.246211 −0.0244989 −0.0122495 0.999925i $$-0.503899\pi$$
−0.0122495 + 0.999925i $$0.503899\pi$$
$$102$$ 0 0
$$103$$ 0.710942 0.0700512 0.0350256 0.999386i $$-0.488849\pi$$
0.0350256 + 0.999386i $$0.488849\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ 11.6976 1.13085 0.565424 0.824800i $$-0.308712\pi$$
0.565424 + 0.824800i $$0.308712\pi$$
$$108$$ 0 0
$$109$$ −2.99145 −0.286529 −0.143264 0.989684i $$-0.545760\pi$$
−0.143264 + 0.989684i $$0.545760\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ 0 0
$$113$$ −6.41216 −0.603206 −0.301603 0.953434i $$-0.597522\pi$$
−0.301603 + 0.953434i $$0.597522\pi$$
$$114$$ 0 0
$$115$$ −1.94977 −0.181817
$$116$$ 0 0
$$117$$ 0 0
$$118$$ 0 0
$$119$$ 8.93839 0.819381
$$120$$ 0 0
$$121$$ 27.5155 2.50140
$$122$$ 0 0
$$123$$ 0 0
$$124$$ 0 0
$$125$$ 32.3495 2.89343
$$126$$ 0 0
$$127$$ −18.1434 −1.60997 −0.804984 0.593297i $$-0.797826\pi$$
−0.804984 + 0.593297i $$0.797826\pi$$
$$128$$ 0 0
$$129$$ 0 0
$$130$$ 0 0
$$131$$ −12.4652 −1.08909 −0.544546 0.838731i $$-0.683298\pi$$
−0.544546 + 0.838731i $$0.683298\pi$$
$$132$$ 0 0
$$133$$ 1.74252 0.151096
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ −8.41863 −0.719252 −0.359626 0.933097i $$-0.617096\pi$$
−0.359626 + 0.933097i $$0.617096\pi$$
$$138$$ 0 0
$$139$$ 11.3292 0.960929 0.480465 0.877014i $$-0.340468\pi$$
0.480465 + 0.877014i $$0.340468\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 0 0
$$143$$ −29.9474 −2.50433
$$144$$ 0 0
$$145$$ −20.2964 −1.68553
$$146$$ 0 0
$$147$$ 0 0
$$148$$ 0 0
$$149$$ 0.321808 0.0263635 0.0131818 0.999913i $$-0.495804\pi$$
0.0131818 + 0.999913i $$0.495804\pi$$
$$150$$ 0 0
$$151$$ −2.95715 −0.240650 −0.120325 0.992735i $$-0.538394\pi$$
−0.120325 + 0.992735i $$0.538394\pi$$
$$152$$ 0 0
$$153$$ 0 0
$$154$$ 0 0
$$155$$ −6.88983 −0.553404
$$156$$ 0 0
$$157$$ 6.52789 0.520982 0.260491 0.965476i $$-0.416116\pi$$
0.260491 + 0.965476i $$0.416116\pi$$
$$158$$ 0 0
$$159$$ 0 0
$$160$$ 0 0
$$161$$ −0.807764 −0.0636607
$$162$$ 0 0
$$163$$ 2.05023 0.160586 0.0802931 0.996771i $$-0.474414\pi$$
0.0802931 + 0.996771i $$0.474414\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ 10.2462 0.792876 0.396438 0.918062i $$-0.370246\pi$$
0.396438 + 0.918062i $$0.370246\pi$$
$$168$$ 0 0
$$169$$ 10.2854 0.791187
$$170$$ 0 0
$$171$$ 0 0
$$172$$ 0 0
$$173$$ −8.05023 −0.612047 −0.306024 0.952024i $$-0.598999\pi$$
−0.306024 + 0.952024i $$0.598999\pi$$
$$174$$ 0 0
$$175$$ 22.1146 1.67170
$$176$$ 0 0
$$177$$ 0 0
$$178$$ 0 0
$$179$$ −1.75379 −0.131084 −0.0655422 0.997850i $$-0.520878\pi$$
−0.0655422 + 0.997850i $$0.520878\pi$$
$$180$$ 0 0
$$181$$ −11.0203 −0.819133 −0.409567 0.912280i $$-0.634320\pi$$
−0.409567 + 0.912280i $$0.634320\pi$$
$$182$$ 0 0
$$183$$ 0 0
$$184$$ 0 0
$$185$$ 15.3020 1.12502
$$186$$ 0 0
$$187$$ −31.8345 −2.32797
$$188$$ 0 0
$$189$$ 0 0
$$190$$ 0 0
$$191$$ 13.2276 0.957113 0.478556 0.878057i $$-0.341160\pi$$
0.478556 + 0.878057i $$0.341160\pi$$
$$192$$ 0 0
$$193$$ 1.47211 0.105965 0.0529824 0.998595i $$-0.483127\pi$$
0.0529824 + 0.998595i $$0.483127\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ −7.84698 −0.559074 −0.279537 0.960135i $$-0.590181\pi$$
−0.279537 + 0.960135i $$0.590181\pi$$
$$198$$ 0 0
$$199$$ −11.8057 −0.836882 −0.418441 0.908244i $$-0.637423\pi$$
−0.418441 + 0.908244i $$0.637423\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ 0 0
$$203$$ −8.40853 −0.590163
$$204$$ 0 0
$$205$$ 22.2462 1.55374
$$206$$ 0 0
$$207$$ 0 0
$$208$$ 0 0
$$209$$ −6.20608 −0.429284
$$210$$ 0 0
$$211$$ −14.5745 −1.00335 −0.501674 0.865056i $$-0.667282\pi$$
−0.501674 + 0.865056i $$0.667282\pi$$
$$212$$ 0 0
$$213$$ 0 0
$$214$$ 0 0
$$215$$ 4.55509 0.310654
$$216$$ 0 0
$$217$$ −2.85436 −0.193767
$$218$$ 0 0
$$219$$ 0 0
$$220$$ 0 0
$$221$$ 24.7527 1.66505
$$222$$ 0 0
$$223$$ 26.6713 1.78604 0.893021 0.450014i $$-0.148581\pi$$
0.893021 + 0.450014i $$0.148581\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 0 0
$$227$$ 4.01656 0.266589 0.133294 0.991076i $$-0.457444\pi$$
0.133294 + 0.991076i $$0.457444\pi$$
$$228$$ 0 0
$$229$$ 23.1834 1.53200 0.766002 0.642838i $$-0.222243\pi$$
0.766002 + 0.642838i $$0.222243\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ −22.2692 −1.45891 −0.729453 0.684031i $$-0.760225\pi$$
−0.729453 + 0.684031i $$0.760225\pi$$
$$234$$ 0 0
$$235$$ 2.33474 0.152302
$$236$$ 0 0
$$237$$ 0 0
$$238$$ 0 0
$$239$$ 23.0818 1.49304 0.746519 0.665364i $$-0.231724\pi$$
0.746519 + 0.665364i $$0.231724\pi$$
$$240$$ 0 0
$$241$$ −0.565184 −0.0364067 −0.0182033 0.999834i $$-0.505795\pi$$
−0.0182033 + 0.999834i $$0.505795\pi$$
$$242$$ 0 0
$$243$$ 0 0
$$244$$ 0 0
$$245$$ −16.6713 −1.06509
$$246$$ 0 0
$$247$$ 4.82550 0.307039
$$248$$ 0 0
$$249$$ 0 0
$$250$$ 0 0
$$251$$ −15.8441 −1.00007 −0.500037 0.866004i $$-0.666680\pi$$
−0.500037 + 0.866004i $$0.666680\pi$$
$$252$$ 0 0
$$253$$ 2.87689 0.180869
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 0 0
$$257$$ −12.0632 −0.752479 −0.376240 0.926522i $$-0.622783\pi$$
−0.376240 + 0.926522i $$0.622783\pi$$
$$258$$ 0 0
$$259$$ 6.33940 0.393911
$$260$$ 0 0
$$261$$ 0 0
$$262$$ 0 0
$$263$$ −23.2135 −1.43140 −0.715702 0.698406i $$-0.753893\pi$$
−0.715702 + 0.698406i $$0.753893\pi$$
$$264$$ 0 0
$$265$$ −11.1863 −0.687167
$$266$$ 0 0
$$267$$ 0 0
$$268$$ 0 0
$$269$$ −26.2964 −1.60332 −0.801661 0.597779i $$-0.796050\pi$$
−0.801661 + 0.597779i $$0.796050\pi$$
$$270$$ 0 0
$$271$$ 12.9560 0.787020 0.393510 0.919320i $$-0.371261\pi$$
0.393510 + 0.919320i $$0.371261\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ 0 0
$$275$$ −78.7622 −4.74954
$$276$$ 0 0
$$277$$ −10.8645 −0.652782 −0.326391 0.945235i $$-0.605833\pi$$
−0.326391 + 0.945235i $$0.605833\pi$$
$$278$$ 0 0
$$279$$ 0 0
$$280$$ 0 0
$$281$$ 15.8470 0.945352 0.472676 0.881236i $$-0.343288\pi$$
0.472676 + 0.881236i $$0.343288\pi$$
$$282$$ 0 0
$$283$$ −27.6740 −1.64505 −0.822525 0.568729i $$-0.807435\pi$$
−0.822525 + 0.568729i $$0.807435\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ 9.21630 0.544021
$$288$$ 0 0
$$289$$ 9.31251 0.547795
$$290$$ 0 0
$$291$$ 0 0
$$292$$ 0 0
$$293$$ 21.3179 1.24541 0.622703 0.782458i $$-0.286034\pi$$
0.622703 + 0.782458i $$0.286034\pi$$
$$294$$ 0 0
$$295$$ 7.78382 0.453192
$$296$$ 0 0
$$297$$ 0 0
$$298$$ 0 0
$$299$$ −2.23691 −0.129364
$$300$$ 0 0
$$301$$ 1.88711 0.108771
$$302$$ 0 0
$$303$$ 0 0
$$304$$ 0 0
$$305$$ −39.2393 −2.24684
$$306$$ 0 0
$$307$$ −26.6584 −1.52147 −0.760737 0.649060i $$-0.775162\pi$$
−0.760737 + 0.649060i $$0.775162\pi$$
$$308$$ 0 0
$$309$$ 0 0
$$310$$ 0 0
$$311$$ −20.0690 −1.13801 −0.569004 0.822335i $$-0.692671\pi$$
−0.569004 + 0.822335i $$0.692671\pi$$
$$312$$ 0 0
$$313$$ −11.1397 −0.629651 −0.314826 0.949150i $$-0.601946\pi$$
−0.314826 + 0.949150i $$0.601946\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ −33.1349 −1.86104 −0.930520 0.366242i $$-0.880644\pi$$
−0.930520 + 0.366242i $$0.880644\pi$$
$$318$$ 0 0
$$319$$ 29.9474 1.67673
$$320$$ 0 0
$$321$$ 0 0
$$322$$ 0 0
$$323$$ 5.12957 0.285417
$$324$$ 0 0
$$325$$ 61.2410 3.39704
$$326$$ 0 0
$$327$$ 0 0
$$328$$ 0 0
$$329$$ 0.967250 0.0533262
$$330$$ 0 0
$$331$$ 4.30241 0.236482 0.118241 0.992985i $$-0.462274\pi$$
0.118241 + 0.992985i $$0.462274\pi$$
$$332$$ 0 0
$$333$$ 0 0
$$334$$ 0 0
$$335$$ 19.8843 1.08639
$$336$$ 0 0
$$337$$ 31.3393 1.70716 0.853580 0.520962i $$-0.174427\pi$$
0.853580 + 0.520962i $$0.174427\pi$$
$$338$$ 0 0
$$339$$ 0 0
$$340$$ 0 0
$$341$$ 10.1660 0.550517
$$342$$ 0 0
$$343$$ −19.1043 −1.03154
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ 21.6782 1.16375 0.581873 0.813280i $$-0.302320\pi$$
0.581873 + 0.813280i $$0.302320\pi$$
$$348$$ 0 0
$$349$$ 4.50486 0.241140 0.120570 0.992705i $$-0.461528\pi$$
0.120570 + 0.992705i $$0.461528\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 0 0
$$353$$ 22.4158 1.19307 0.596536 0.802586i $$-0.296543\pi$$
0.596536 + 0.802586i $$0.296543\pi$$
$$354$$ 0 0
$$355$$ 48.3070 2.56387
$$356$$ 0 0
$$357$$ 0 0
$$358$$ 0 0
$$359$$ −11.4940 −0.606629 −0.303314 0.952891i $$-0.598093\pi$$
−0.303314 + 0.952891i $$0.598093\pi$$
$$360$$ 0 0
$$361$$ 1.00000 0.0526316
$$362$$ 0 0
$$363$$ 0 0
$$364$$ 0 0
$$365$$ −0.0271971 −0.00142356
$$366$$ 0 0
$$367$$ −19.7944 −1.03326 −0.516630 0.856209i $$-0.672814\pi$$
−0.516630 + 0.856209i $$0.672814\pi$$
$$368$$ 0 0
$$369$$ 0 0
$$370$$ 0 0
$$371$$ −4.63431 −0.240602
$$372$$ 0 0
$$373$$ −16.6199 −0.860546 −0.430273 0.902699i $$-0.641583\pi$$
−0.430273 + 0.902699i $$0.641583\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ −23.2854 −1.19926
$$378$$ 0 0
$$379$$ 16.3956 0.842185 0.421093 0.907018i $$-0.361647\pi$$
0.421093 + 0.907018i $$0.361647\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ 0 0
$$383$$ 30.6915 1.56826 0.784131 0.620595i $$-0.213109\pi$$
0.784131 + 0.620595i $$0.213109\pi$$
$$384$$ 0 0
$$385$$ −45.4855 −2.31816
$$386$$ 0 0
$$387$$ 0 0
$$388$$ 0 0
$$389$$ 2.24905 0.114031 0.0570156 0.998373i $$-0.481842\pi$$
0.0570156 + 0.998373i $$0.481842\pi$$
$$390$$ 0 0
$$391$$ −2.37787 −0.120254
$$392$$ 0 0
$$393$$ 0 0
$$394$$ 0 0
$$395$$ 28.4380 1.43087
$$396$$ 0 0
$$397$$ −4.12582 −0.207069 −0.103535 0.994626i $$-0.533015\pi$$
−0.103535 + 0.994626i $$0.533015\pi$$
$$398$$ 0 0
$$399$$ 0 0
$$400$$ 0 0
$$401$$ 22.3992 1.11856 0.559282 0.828977i $$-0.311077\pi$$
0.559282 + 0.828977i $$0.311077\pi$$
$$402$$ 0 0
$$403$$ −7.90447 −0.393750
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 0 0
$$407$$ −22.5781 −1.11916
$$408$$ 0 0
$$409$$ 13.1992 0.652658 0.326329 0.945256i $$-0.394188\pi$$
0.326329 + 0.945256i $$0.394188\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ 0 0
$$413$$ 3.22473 0.158679
$$414$$ 0 0
$$415$$ 50.0406 2.45640
$$416$$ 0 0
$$417$$ 0 0
$$418$$ 0 0
$$419$$ 31.5984 1.54368 0.771842 0.635814i $$-0.219336\pi$$
0.771842 + 0.635814i $$0.219336\pi$$
$$420$$ 0 0
$$421$$ −25.8587 −1.26028 −0.630139 0.776482i $$-0.717002\pi$$
−0.630139 + 0.776482i $$0.717002\pi$$
$$422$$ 0 0
$$423$$ 0 0
$$424$$ 0 0
$$425$$ 65.1000 3.15782
$$426$$ 0 0
$$427$$ −16.2563 −0.786698
$$428$$ 0 0
$$429$$ 0 0
$$430$$ 0 0
$$431$$ 31.8972 1.53643 0.768217 0.640189i $$-0.221144\pi$$
0.768217 + 0.640189i $$0.221144\pi$$
$$432$$ 0 0
$$433$$ 23.1006 1.11014 0.555071 0.831803i $$-0.312691\pi$$
0.555071 + 0.831803i $$0.312691\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 0 0
$$437$$ −0.463560 −0.0221751
$$438$$ 0 0
$$439$$ 15.7944 0.753826 0.376913 0.926249i $$-0.376986\pi$$
0.376913 + 0.926249i $$0.376986\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 0 0
$$443$$ −40.5584 −1.92699 −0.963494 0.267729i $$-0.913727\pi$$
−0.963494 + 0.267729i $$0.913727\pi$$
$$444$$ 0 0
$$445$$ 30.7847 1.45933
$$446$$ 0 0
$$447$$ 0 0
$$448$$ 0 0
$$449$$ 30.4026 1.43479 0.717393 0.696669i $$-0.245335\pi$$
0.717393 + 0.696669i $$0.245335\pi$$
$$450$$ 0 0
$$451$$ −32.8243 −1.54564
$$452$$ 0 0
$$453$$ 0 0
$$454$$ 0 0
$$455$$ 35.3670 1.65803
$$456$$ 0 0
$$457$$ 1.33516 0.0624561 0.0312280 0.999512i $$-0.490058\pi$$
0.0312280 + 0.999512i $$0.490058\pi$$
$$458$$ 0 0
$$459$$ 0 0
$$460$$ 0 0
$$461$$ 23.0604 1.07403 0.537016 0.843572i $$-0.319552\pi$$
0.537016 + 0.843572i $$0.319552\pi$$
$$462$$ 0 0
$$463$$ 9.29916 0.432168 0.216084 0.976375i $$-0.430671\pi$$
0.216084 + 0.976375i $$0.430671\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ 18.8344 0.871553 0.435777 0.900055i $$-0.356474\pi$$
0.435777 + 0.900055i $$0.356474\pi$$
$$468$$ 0 0
$$469$$ 8.23778 0.380385
$$470$$ 0 0
$$471$$ 0 0
$$472$$ 0 0
$$473$$ −6.72104 −0.309034
$$474$$ 0 0
$$475$$ 12.6911 0.582309
$$476$$ 0 0
$$477$$ 0 0
$$478$$ 0 0
$$479$$ −10.9701 −0.501236 −0.250618 0.968086i $$-0.580634\pi$$
−0.250618 + 0.968086i $$0.580634\pi$$
$$480$$ 0 0
$$481$$ 17.5555 0.800460
$$482$$ 0 0
$$483$$ 0 0
$$484$$ 0 0
$$485$$ −43.7944 −1.98860
$$486$$ 0 0
$$487$$ 19.3895 0.878623 0.439311 0.898335i $$-0.355223\pi$$
0.439311 + 0.898335i $$0.355223\pi$$
$$488$$ 0 0
$$489$$ 0 0
$$490$$ 0 0
$$491$$ −33.1562 −1.49632 −0.748160 0.663518i $$-0.769062\pi$$
−0.748160 + 0.663518i $$0.769062\pi$$
$$492$$ 0 0
$$493$$ −24.7527 −1.11481
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ 20.0129 0.897703
$$498$$ 0 0
$$499$$ −25.8814 −1.15861 −0.579306 0.815110i $$-0.696676\pi$$
−0.579306 + 0.815110i $$0.696676\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ 0 0
$$503$$ −10.3178 −0.460048 −0.230024 0.973185i $$-0.573880\pi$$
−0.230024 + 0.973185i $$0.573880\pi$$
$$504$$ 0 0
$$505$$ −1.03558 −0.0460829
$$506$$ 0 0
$$507$$ 0 0
$$508$$ 0 0
$$509$$ 6.21618 0.275527 0.137764 0.990465i $$-0.456009\pi$$
0.137764 + 0.990465i $$0.456009\pi$$
$$510$$ 0 0
$$511$$ −0.0112674 −0.000498440 0
$$512$$ 0 0
$$513$$ 0 0
$$514$$ 0 0
$$515$$ 2.99028 0.131767
$$516$$ 0 0
$$517$$ −3.44491 −0.151507
$$518$$ 0 0
$$519$$ 0 0
$$520$$ 0 0
$$521$$ −22.8917 −1.00290 −0.501451 0.865186i $$-0.667200\pi$$
−0.501451 + 0.865186i $$0.667200\pi$$
$$522$$ 0 0
$$523$$ 9.02628 0.394692 0.197346 0.980334i $$-0.436768\pi$$
0.197346 + 0.980334i $$0.436768\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ −8.40256 −0.366021
$$528$$ 0 0
$$529$$ −22.7851 −0.990657
$$530$$ 0 0
$$531$$ 0 0
$$532$$ 0 0
$$533$$ 25.5223 1.10550
$$534$$ 0 0
$$535$$ 49.2010 2.12715
$$536$$ 0 0
$$537$$ 0 0
$$538$$ 0 0
$$539$$ 24.5985 1.05953
$$540$$ 0 0
$$541$$ −21.2937 −0.915489 −0.457744 0.889084i $$-0.651342\pi$$
−0.457744 + 0.889084i $$0.651342\pi$$
$$542$$ 0 0
$$543$$ 0 0
$$544$$ 0 0
$$545$$ −12.5823 −0.538966
$$546$$ 0 0
$$547$$ 3.60803 0.154268 0.0771341 0.997021i $$-0.475423\pi$$
0.0771341 + 0.997021i $$0.475423\pi$$
$$548$$ 0 0
$$549$$ 0 0
$$550$$ 0 0
$$551$$ −4.82550 −0.205573
$$552$$ 0 0
$$553$$ 11.7815 0.501000
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 0 0
$$557$$ 34.8645 1.47725 0.738627 0.674114i $$-0.235474\pi$$
0.738627 + 0.674114i $$0.235474\pi$$
$$558$$ 0 0
$$559$$ 5.22590 0.221032
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 0 0
$$563$$ −41.3167 −1.74129 −0.870647 0.491909i $$-0.836299\pi$$
−0.870647 + 0.491909i $$0.836299\pi$$
$$564$$ 0 0
$$565$$ −26.9701 −1.13464
$$566$$ 0 0
$$567$$ 0 0
$$568$$ 0 0
$$569$$ −41.0835 −1.72231 −0.861154 0.508344i $$-0.830258\pi$$
−0.861154 + 0.508344i $$0.830258\pi$$
$$570$$ 0 0
$$571$$ −36.8243 −1.54105 −0.770525 0.637410i $$-0.780006\pi$$
−0.770525 + 0.637410i $$0.780006\pi$$
$$572$$ 0 0
$$573$$ 0 0
$$574$$ 0 0
$$575$$ −5.88310 −0.245342
$$576$$ 0 0
$$577$$ 15.7004 0.653617 0.326809 0.945091i $$-0.394027\pi$$
0.326809 + 0.945091i $$0.394027\pi$$
$$578$$ 0 0
$$579$$ 0 0
$$580$$ 0 0
$$581$$ 20.7311 0.860072
$$582$$ 0 0
$$583$$ 16.5054 0.683582
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ −34.3099 −1.41612 −0.708060 0.706152i $$-0.750429\pi$$
−0.708060 + 0.706152i $$0.750429\pi$$
$$588$$ 0 0
$$589$$ −1.63806 −0.0674952
$$590$$ 0 0
$$591$$ 0 0
$$592$$ 0 0
$$593$$ −7.11584 −0.292213 −0.146106 0.989269i $$-0.546674\pi$$
−0.146106 + 0.989269i $$0.546674\pi$$
$$594$$ 0 0
$$595$$ 37.5956 1.54127
$$596$$ 0 0
$$597$$ 0 0
$$598$$ 0 0
$$599$$ 13.8340 0.565244 0.282622 0.959231i $$-0.408796\pi$$
0.282622 + 0.959231i $$0.408796\pi$$
$$600$$ 0 0
$$601$$ 39.2140 1.59957 0.799785 0.600286i $$-0.204947\pi$$
0.799785 + 0.600286i $$0.204947\pi$$
$$602$$ 0 0
$$603$$ 0 0
$$604$$ 0 0
$$605$$ 115.732 4.70518
$$606$$ 0 0
$$607$$ 35.1733 1.42764 0.713821 0.700328i $$-0.246963\pi$$
0.713821 + 0.700328i $$0.246963\pi$$
$$608$$ 0 0
$$609$$ 0 0
$$610$$ 0 0
$$611$$ 2.67857 0.108363
$$612$$ 0 0
$$613$$ 6.24905 0.252397 0.126198 0.992005i $$-0.459722\pi$$
0.126198 + 0.992005i $$0.459722\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ 42.0378 1.69238 0.846189 0.532883i $$-0.178891\pi$$
0.846189 + 0.532883i $$0.178891\pi$$
$$618$$ 0 0
$$619$$ 31.0633 1.24854 0.624269 0.781209i $$-0.285397\pi$$
0.624269 + 0.781209i $$0.285397\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ 0 0
$$623$$ 12.7537 0.510965
$$624$$ 0 0
$$625$$ 72.6090 2.90436
$$626$$ 0 0
$$627$$ 0 0
$$628$$ 0 0
$$629$$ 18.6617 0.744091
$$630$$ 0 0
$$631$$ −40.0378 −1.59388 −0.796940 0.604059i $$-0.793549\pi$$
−0.796940 + 0.604059i $$0.793549\pi$$
$$632$$ 0 0
$$633$$ 0 0
$$634$$ 0 0
$$635$$ −76.3127 −3.02838
$$636$$ 0 0
$$637$$ −19.1264 −0.757817
$$638$$ 0 0
$$639$$ 0 0
$$640$$ 0 0
$$641$$ 20.3619 0.804248 0.402124 0.915585i $$-0.368272\pi$$
0.402124 + 0.915585i $$0.368272\pi$$
$$642$$ 0 0
$$643$$ 13.7041 0.540435 0.270218 0.962799i $$-0.412904\pi$$
0.270218 + 0.962799i $$0.412904\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ −26.9588 −1.05986 −0.529930 0.848041i $$-0.677782\pi$$
−0.529930 + 0.848041i $$0.677782\pi$$
$$648$$ 0 0
$$649$$ −11.4850 −0.450827
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 0 0
$$653$$ −30.7842 −1.20468 −0.602339 0.798240i $$-0.705765\pi$$
−0.602339 + 0.798240i $$0.705765\pi$$
$$654$$ 0 0
$$655$$ −52.4298 −2.04860
$$656$$ 0 0
$$657$$ 0 0
$$658$$ 0 0
$$659$$ −7.57854 −0.295218 −0.147609 0.989046i $$-0.547158\pi$$
−0.147609 + 0.989046i $$0.547158\pi$$
$$660$$ 0 0
$$661$$ 47.8191 1.85995 0.929974 0.367626i $$-0.119829\pi$$
0.929974 + 0.367626i $$0.119829\pi$$
$$662$$ 0 0
$$663$$ 0 0
$$664$$ 0 0
$$665$$ 7.32919 0.284214
$$666$$ 0 0
$$667$$ 2.23691 0.0866135
$$668$$ 0 0
$$669$$ 0 0
$$670$$ 0 0
$$671$$ 57.8977 2.23512
$$672$$ 0 0
$$673$$ 12.2389 0.471777 0.235888 0.971780i $$-0.424200\pi$$
0.235888 + 0.971780i $$0.424200\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ 0 0
$$677$$ −23.3609 −0.897832 −0.448916 0.893574i $$-0.648190\pi$$
−0.448916 + 0.893574i $$0.648190\pi$$
$$678$$ 0 0
$$679$$ −18.1434 −0.696280
$$680$$ 0 0
$$681$$ 0 0
$$682$$ 0 0
$$683$$ −24.7239 −0.946033 −0.473016 0.881054i $$-0.656835\pi$$
−0.473016 + 0.881054i $$0.656835\pi$$
$$684$$ 0 0
$$685$$ −35.4094 −1.35293
$$686$$ 0 0
$$687$$ 0 0
$$688$$ 0 0
$$689$$ −12.8336 −0.488922
$$690$$ 0 0
$$691$$ 6.78837 0.258242 0.129121 0.991629i $$-0.458785\pi$$
0.129121 + 0.991629i $$0.458785\pi$$
$$692$$ 0 0
$$693$$ 0 0
$$694$$ 0 0
$$695$$ 47.6515 1.80752
$$696$$ 0 0
$$697$$ 27.1306 1.02764
$$698$$ 0 0
$$699$$ 0 0
$$700$$ 0 0
$$701$$ −26.9175 −1.01666 −0.508330 0.861162i $$-0.669737\pi$$
−0.508330 + 0.861162i $$0.669737\pi$$
$$702$$ 0 0
$$703$$ 3.63806 0.137212
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ −0.429028 −0.0161353
$$708$$ 0 0
$$709$$ −21.5984 −0.811146 −0.405573 0.914063i $$-0.632928\pi$$
−0.405573 + 0.914063i $$0.632928\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ 0 0
$$713$$ 0.759341 0.0284376
$$714$$ 0 0
$$715$$ −125.961 −4.71069
$$716$$ 0 0
$$717$$ 0 0
$$718$$ 0 0
$$719$$ 22.2575 0.830064 0.415032 0.909807i $$-0.363770\pi$$
0.415032 + 0.909807i $$0.363770\pi$$
$$720$$ 0 0
$$721$$ 1.23883 0.0461365
$$722$$ 0 0
$$723$$ 0 0
$$724$$ 0 0
$$725$$ −61.2410 −2.27443
$$726$$ 0 0
$$727$$ −51.9118 −1.92530 −0.962651 0.270745i $$-0.912730\pi$$
−0.962651 + 0.270745i $$0.912730\pi$$
$$728$$ 0 0
$$729$$ 0 0
$$730$$ 0 0
$$731$$ 5.55520 0.205467
$$732$$ 0 0
$$733$$ 20.0575 0.740840 0.370420 0.928864i $$-0.379214\pi$$
0.370420 + 0.928864i $$0.379214\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ −29.3393 −1.08073
$$738$$ 0 0
$$739$$ 7.36465 0.270913 0.135457 0.990783i $$-0.456750\pi$$
0.135457 + 0.990783i $$0.456750\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 0 0
$$743$$ 2.85436 0.104716 0.0523581 0.998628i $$-0.483326\pi$$
0.0523581 + 0.998628i $$0.483326\pi$$
$$744$$ 0 0
$$745$$ 1.35355 0.0495902
$$746$$ 0 0
$$747$$ 0 0
$$748$$ 0 0
$$749$$ 20.3833 0.744790
$$750$$ 0 0
$$751$$ −22.3725 −0.816385 −0.408193 0.912896i $$-0.633841\pi$$
−0.408193 + 0.912896i $$0.633841\pi$$
$$752$$ 0 0
$$753$$ 0 0
$$754$$ 0 0
$$755$$ −12.4380 −0.452666
$$756$$ 0 0
$$757$$ 6.07326 0.220736 0.110368 0.993891i $$-0.464797\pi$$
0.110368 + 0.993891i $$0.464797\pi$$
$$758$$ 0 0
$$759$$ 0 0
$$760$$ 0 0
$$761$$ −13.1166 −0.475478 −0.237739 0.971329i $$-0.576406\pi$$
−0.237739 + 0.971329i $$0.576406\pi$$
$$762$$ 0 0
$$763$$ −5.21267 −0.188711
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ 8.93012 0.322448
$$768$$ 0 0
$$769$$ −44.5791 −1.60757 −0.803783 0.594923i $$-0.797182\pi$$
−0.803783 + 0.594923i $$0.797182\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ 0 0
$$773$$ 4.43920 0.159667 0.0798334 0.996808i $$-0.474561\pi$$
0.0798334 + 0.996808i $$0.474561\pi$$
$$774$$ 0 0
$$775$$ −20.7889 −0.746758
$$776$$ 0 0
$$777$$ 0 0
$$778$$ 0 0
$$779$$ 5.28906 0.189500
$$780$$ 0 0
$$781$$ −71.2771 −2.55050
$$782$$ 0 0
$$783$$ 0 0
$$784$$ 0 0
$$785$$ 27.4568 0.979977
$$786$$ 0 0
$$787$$ 0.315342 0.0112407 0.00562036 0.999984i $$-0.498211\pi$$
0.00562036 + 0.999984i $$0.498211\pi$$
$$788$$ 0 0
$$789$$ 0 0
$$790$$ 0 0
$$791$$ −11.1733 −0.397278
$$792$$ 0 0
$$793$$ −45.0180 −1.59864
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 0 0
$$797$$ −6.16712 −0.218451 −0.109225 0.994017i $$-0.534837\pi$$
−0.109225 + 0.994017i $$0.534837\pi$$
$$798$$ 0 0
$$799$$ 2.84736 0.100732
$$800$$ 0 0
$$801$$ 0 0
$$802$$ 0 0
$$803$$ 0.0401294 0.00141614
$$804$$ 0 0
$$805$$ −3.39752 −0.119747
$$806$$ 0 0
$$807$$ 0 0
$$808$$ 0 0
$$809$$ −20.4057 −0.717426 −0.358713 0.933448i $$-0.616784\pi$$
−0.358713 + 0.933448i $$0.616784\pi$$
$$810$$ 0 0
$$811$$ −36.5763 −1.28437 −0.642184 0.766551i $$-0.721971\pi$$
−0.642184 + 0.766551i $$0.721971\pi$$
$$812$$ 0 0
$$813$$ 0 0
$$814$$ 0 0
$$815$$ 8.62342 0.302065
$$816$$ 0 0
$$817$$ 1.08298 0.0378885
$$818$$ 0 0
$$819$$ 0 0
$$820$$ 0 0
$$821$$ −19.6686 −0.686439 −0.343219 0.939255i $$-0.611517\pi$$
−0.343219 + 0.939255i $$0.611517\pi$$
$$822$$ 0 0
$$823$$ −36.7328 −1.28042 −0.640212 0.768198i $$-0.721154\pi$$
−0.640212 + 0.768198i $$0.721154\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ −27.4920 −0.955991 −0.477995 0.878362i $$-0.658636\pi$$
−0.477995 + 0.878362i $$0.658636\pi$$
$$828$$ 0 0
$$829$$ −12.1016 −0.420307 −0.210153 0.977668i $$-0.567396\pi$$
−0.210153 + 0.977668i $$0.567396\pi$$
$$830$$ 0 0
$$831$$ 0 0
$$832$$ 0 0
$$833$$ −20.3317 −0.704451
$$834$$ 0 0
$$835$$ 43.0964 1.49141
$$836$$ 0 0
$$837$$ 0 0
$$838$$ 0 0
$$839$$ 51.4731 1.77705 0.888524 0.458829i $$-0.151731\pi$$
0.888524 + 0.458829i $$0.151731\pi$$
$$840$$ 0 0
$$841$$ −5.71457 −0.197054
$$842$$ 0 0
$$843$$ 0 0
$$844$$ 0 0
$$845$$ 43.2613 1.48824
$$846$$ 0 0
$$847$$ 47.9463 1.64745
$$848$$ 0 0
$$849$$ 0 0
$$850$$ 0 0
$$851$$ −1.68646 −0.0578112
$$852$$ 0 0
$$853$$ −42.6187 −1.45924 −0.729619 0.683854i $$-0.760303\pi$$
−0.729619 + 0.683854i $$0.760303\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0 0
$$857$$ −3.45501 −0.118021 −0.0590105 0.998257i $$-0.518795\pi$$
−0.0590105 + 0.998257i $$0.518795\pi$$
$$858$$ 0 0
$$859$$ 17.1161 0.583994 0.291997 0.956419i $$-0.405680\pi$$
0.291997 + 0.956419i $$0.405680\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 0 0
$$863$$ 13.0130 0.442969 0.221485 0.975164i $$-0.428910\pi$$
0.221485 + 0.975164i $$0.428910\pi$$
$$864$$ 0 0
$$865$$ −33.8599 −1.15127
$$866$$ 0 0
$$867$$ 0 0
$$868$$ 0 0
$$869$$ −41.9604 −1.42341
$$870$$ 0 0
$$871$$ 22.8126 0.772974
$$872$$ 0 0
$$873$$ 0 0
$$874$$ 0 0
$$875$$ 56.3697 1.90564
$$876$$ 0 0
$$877$$ 0.908097 0.0306642 0.0153321 0.999882i $$-0.495119\pi$$
0.0153321 + 0.999882i $$0.495119\pi$$
$$878$$ 0 0
$$879$$ 0 0
$$880$$ 0 0
$$881$$ −29.1332 −0.981523 −0.490761 0.871294i $$-0.663281\pi$$
−0.490761 + 0.871294i $$0.663281\pi$$
$$882$$ 0 0
$$883$$ −31.4780 −1.05932 −0.529660 0.848210i $$-0.677681\pi$$
−0.529660 + 0.848210i $$0.677681\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 0 0
$$887$$ −37.0933 −1.24547 −0.622736 0.782432i $$-0.713979\pi$$
−0.622736 + 0.782432i $$0.713979\pi$$
$$888$$ 0 0
$$889$$ −31.6153 −1.06034
$$890$$ 0 0
$$891$$ 0 0
$$892$$ 0 0
$$893$$ 0.555087 0.0185753
$$894$$ 0 0
$$895$$ −7.37658 −0.246572
$$896$$ 0 0
$$897$$ 0 0
$$898$$ 0 0
$$899$$ 7.90447 0.263629
$$900$$ 0 0
$$901$$ −13.6423 −0.454492
$$902$$ 0 0
$$903$$ 0 0
$$904$$ 0 0
$$905$$ −46.3523 −1.54080
$$906$$ 0 0
$$907$$ −24.0295 −0.797886 −0.398943 0.916976i $$-0.630623\pi$$
−0.398943 + 0.916976i $$0.630623\pi$$
$$908$$ 0 0
$$909$$ 0 0
$$910$$ 0 0
$$911$$ 45.7644 1.51624 0.758121 0.652114i $$-0.226118\pi$$
0.758121 + 0.652114i $$0.226118\pi$$
$$912$$ 0 0
$$913$$ −73.8350 −2.44358
$$914$$ 0 0
$$915$$ 0 0
$$916$$ 0 0
$$917$$ −21.7209 −0.717288
$$918$$ 0 0
$$919$$ −2.91536 −0.0961688 −0.0480844 0.998843i $$-0.515312\pi$$
−0.0480844 + 0.998843i $$0.515312\pi$$
$$920$$ 0 0
$$921$$ 0 0
$$922$$ 0 0
$$923$$ 55.4210 1.82421
$$924$$ 0 0
$$925$$ 46.1711 1.51810
$$926$$ 0 0
$$927$$ 0 0
$$928$$ 0 0
$$929$$ 14.4417 0.473815 0.236908 0.971532i $$-0.423866\pi$$
0.236908 + 0.971532i $$0.423866\pi$$
$$930$$ 0 0
$$931$$ −3.96362 −0.129902
$$932$$ 0 0
$$933$$ 0 0
$$934$$ 0 0
$$935$$ −133.899 −4.37896
$$936$$ 0 0
$$937$$ 39.3960 1.28701 0.643505 0.765442i $$-0.277479\pi$$
0.643505 + 0.765442i $$0.277479\pi$$
$$938$$ 0 0
$$939$$ 0 0
$$940$$ 0 0
$$941$$ −37.5954 −1.22558 −0.612788 0.790247i $$-0.709952\pi$$
−0.612788 + 0.790247i $$0.709952\pi$$
$$942$$ 0 0
$$943$$ −2.45180 −0.0798415
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ 21.3522 0.693854 0.346927 0.937892i $$-0.387225\pi$$
0.346927 + 0.937892i $$0.387225\pi$$
$$948$$ 0 0
$$949$$ −0.0312023 −0.00101287
$$950$$ 0 0
$$951$$ 0 0
$$952$$ 0 0
$$953$$ −37.5726 −1.21709 −0.608547 0.793518i $$-0.708247\pi$$
−0.608547 + 0.793518i $$0.708247\pi$$
$$954$$ 0 0
$$955$$ 55.6362 1.80035
$$956$$ 0 0
$$957$$ 0 0
$$958$$ 0 0
$$959$$ −14.6696 −0.473707
$$960$$ 0 0
$$961$$ −28.3167 −0.913444
$$962$$ 0 0
$$963$$ 0 0
$$964$$ 0 0
$$965$$ 6.19182 0.199322
$$966$$ 0 0
$$967$$ 11.5676 0.371990 0.185995 0.982551i $$-0.440449\pi$$
0.185995 + 0.982551i $$0.440449\pi$$
$$968$$ 0 0
$$969$$ 0 0
$$970$$ 0 0
$$971$$ 60.2009 1.93194 0.965970 0.258656i $$-0.0832796\pi$$
0.965970 + 0.258656i $$0.0832796\pi$$
$$972$$ 0 0
$$973$$ 19.7414 0.632879
$$974$$ 0 0
$$975$$ 0 0
$$976$$ 0 0
$$977$$ −41.2738 −1.32047 −0.660233 0.751061i $$-0.729542\pi$$
−0.660233 + 0.751061i $$0.729542\pi$$
$$978$$ 0 0
$$979$$ −45.4229 −1.45172
$$980$$ 0 0
$$981$$ 0 0
$$982$$ 0 0
$$983$$ −16.5610 −0.528214 −0.264107 0.964493i $$-0.585077\pi$$
−0.264107 + 0.964493i $$0.585077\pi$$
$$984$$ 0 0
$$985$$ −33.0050 −1.05163
$$986$$ 0 0
$$987$$ 0 0
$$988$$ 0 0
$$989$$ −0.502025 −0.0159635
$$990$$ 0 0
$$991$$ 35.6243 1.13164 0.565821 0.824528i $$-0.308559\pi$$
0.565821 + 0.824528i $$0.308559\pi$$
$$992$$ 0 0
$$993$$ 0 0
$$994$$ 0 0
$$995$$ −49.6557 −1.57419
$$996$$ 0 0
$$997$$ 41.0563 1.30027 0.650133 0.759821i $$-0.274713\pi$$
0.650133 + 0.759821i $$0.274713\pi$$
$$998$$ 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 5472.2.a.bs.1.4 4
3.2 odd 2 608.2.a.j.1.3 yes 4
4.3 odd 2 5472.2.a.bt.1.4 4
12.11 even 2 608.2.a.i.1.1 4
24.5 odd 2 1216.2.a.w.1.2 4
24.11 even 2 1216.2.a.x.1.4 4

By twisted newform
Twist Min Dim Char Parity Ord Type
608.2.a.i.1.1 4 12.11 even 2
608.2.a.j.1.3 yes 4 3.2 odd 2
1216.2.a.w.1.2 4 24.5 odd 2
1216.2.a.x.1.4 4 24.11 even 2
5472.2.a.bs.1.4 4 1.1 even 1 trivial
5472.2.a.bt.1.4 4 4.3 odd 2