# Properties

 Label 216.4.a.f.1.1 Level $216$ Weight $4$ Character 216.1 Self dual yes Analytic conductor $12.744$ Analytic rank $1$ Dimension $2$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$216 = 2^{3} \cdot 3^{3}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 216.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$12.7444125612$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\zeta_{10})^+$$ Defining polynomial: $$x^{2} - x - 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2^{2}\cdot 3$$ Twist minimal: yes Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Root $$1.61803$$ of defining polynomial Character $$\chi$$ $$=$$ 216.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-15.4164 q^{5} +23.8328 q^{7} +O(q^{10})$$ $$q-15.4164 q^{5} +23.8328 q^{7} +14.2492 q^{11} -13.8328 q^{13} -80.5836 q^{17} -144.331 q^{19} -141.082 q^{23} +112.666 q^{25} -251.331 q^{29} -16.6687 q^{31} -367.416 q^{35} +305.164 q^{37} +429.325 q^{41} -181.666 q^{43} -79.4164 q^{47} +225.003 q^{49} -663.830 q^{53} -219.672 q^{55} +220.255 q^{59} -473.158 q^{61} +213.252 q^{65} -647.663 q^{67} +14.4922 q^{71} +776.003 q^{73} +339.599 q^{77} -257.827 q^{79} +1285.15 q^{83} +1242.31 q^{85} -156.255 q^{89} -329.675 q^{91} +2225.07 q^{95} +1161.64 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 4q^{5} - 6q^{7} + O(q^{10})$$ $$2q - 4q^{5} - 6q^{7} - 52q^{11} + 26q^{13} - 188q^{17} - 74q^{19} - 148q^{23} + 118q^{25} - 288q^{29} - 248q^{31} - 708q^{35} + 342q^{37} - 256q^{43} - 132q^{47} + 772q^{49} - 952q^{53} - 976q^{55} + 1004q^{59} - 34q^{61} + 668q^{65} - 866q^{67} - 776q^{71} + 1874q^{73} + 2316q^{77} + 182q^{79} + 1336q^{83} + 16q^{85} - 876q^{89} - 1518q^{91} + 3028q^{95} - 38q^{97} + O(q^{100})$$

## 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
$$5$$ −15.4164 −1.37889 −0.689443 0.724340i $$-0.742145\pi$$
−0.689443 + 0.724340i $$0.742145\pi$$
$$6$$ 0 0
$$7$$ 23.8328 1.28685 0.643426 0.765509i $$-0.277513\pi$$
0.643426 + 0.765509i $$0.277513\pi$$
$$8$$ 0 0
$$9$$ 0 0
$$10$$ 0 0
$$11$$ 14.2492 0.390573 0.195286 0.980746i $$-0.437436\pi$$
0.195286 + 0.980746i $$0.437436\pi$$
$$12$$ 0 0
$$13$$ −13.8328 −0.295118 −0.147559 0.989053i $$-0.547142\pi$$
−0.147559 + 0.989053i $$0.547142\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 0 0
$$17$$ −80.5836 −1.14967 −0.574835 0.818269i $$-0.694934\pi$$
−0.574835 + 0.818269i $$0.694934\pi$$
$$18$$ 0 0
$$19$$ −144.331 −1.74273 −0.871365 0.490636i $$-0.836765\pi$$
−0.871365 + 0.490636i $$0.836765\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 0 0
$$23$$ −141.082 −1.27903 −0.639514 0.768780i $$-0.720864\pi$$
−0.639514 + 0.768780i $$0.720864\pi$$
$$24$$ 0 0
$$25$$ 112.666 0.901325
$$26$$ 0 0
$$27$$ 0 0
$$28$$ 0 0
$$29$$ −251.331 −1.60935 −0.804673 0.593718i $$-0.797659\pi$$
−0.804673 + 0.593718i $$0.797659\pi$$
$$30$$ 0 0
$$31$$ −16.6687 −0.0965740 −0.0482870 0.998834i $$-0.515376\pi$$
−0.0482870 + 0.998834i $$0.515376\pi$$
$$32$$ 0 0
$$33$$ 0 0
$$34$$ 0 0
$$35$$ −367.416 −1.77442
$$36$$ 0 0
$$37$$ 305.164 1.35591 0.677955 0.735103i $$-0.262866\pi$$
0.677955 + 0.735103i $$0.262866\pi$$
$$38$$ 0 0
$$39$$ 0 0
$$40$$ 0 0
$$41$$ 429.325 1.63535 0.817674 0.575681i $$-0.195263\pi$$
0.817674 + 0.575681i $$0.195263\pi$$
$$42$$ 0 0
$$43$$ −181.666 −0.644273 −0.322137 0.946693i $$-0.604401\pi$$
−0.322137 + 0.946693i $$0.604401\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 0 0
$$47$$ −79.4164 −0.246470 −0.123235 0.992378i $$-0.539327\pi$$
−0.123235 + 0.992378i $$0.539327\pi$$
$$48$$ 0 0
$$49$$ 225.003 0.655986
$$50$$ 0 0
$$51$$ 0 0
$$52$$ 0 0
$$53$$ −663.830 −1.72045 −0.860227 0.509912i $$-0.829678\pi$$
−0.860227 + 0.509912i $$0.829678\pi$$
$$54$$ 0 0
$$55$$ −219.672 −0.538555
$$56$$ 0 0
$$57$$ 0 0
$$58$$ 0 0
$$59$$ 220.255 0.486014 0.243007 0.970025i $$-0.421866\pi$$
0.243007 + 0.970025i $$0.421866\pi$$
$$60$$ 0 0
$$61$$ −473.158 −0.993142 −0.496571 0.867996i $$-0.665408\pi$$
−0.496571 + 0.867996i $$0.665408\pi$$
$$62$$ 0 0
$$63$$ 0 0
$$64$$ 0 0
$$65$$ 213.252 0.406934
$$66$$ 0 0
$$67$$ −647.663 −1.18096 −0.590482 0.807051i $$-0.701062\pi$$
−0.590482 + 0.807051i $$0.701062\pi$$
$$68$$ 0 0
$$69$$ 0 0
$$70$$ 0 0
$$71$$ 14.4922 0.0242241 0.0121121 0.999927i $$-0.496145\pi$$
0.0121121 + 0.999927i $$0.496145\pi$$
$$72$$ 0 0
$$73$$ 776.003 1.24417 0.622084 0.782950i $$-0.286286\pi$$
0.622084 + 0.782950i $$0.286286\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 0 0
$$77$$ 339.599 0.502609
$$78$$ 0 0
$$79$$ −257.827 −0.367187 −0.183593 0.983002i $$-0.558773\pi$$
−0.183593 + 0.983002i $$0.558773\pi$$
$$80$$ 0 0
$$81$$ 0 0
$$82$$ 0 0
$$83$$ 1285.15 1.69957 0.849784 0.527132i $$-0.176733\pi$$
0.849784 + 0.527132i $$0.176733\pi$$
$$84$$ 0 0
$$85$$ 1242.31 1.58526
$$86$$ 0 0
$$87$$ 0 0
$$88$$ 0 0
$$89$$ −156.255 −0.186102 −0.0930508 0.995661i $$-0.529662\pi$$
−0.0930508 + 0.995661i $$0.529662\pi$$
$$90$$ 0 0
$$91$$ −329.675 −0.379773
$$92$$ 0 0
$$93$$ 0 0
$$94$$ 0 0
$$95$$ 2225.07 2.40302
$$96$$ 0 0
$$97$$ 1161.64 1.21595 0.607975 0.793956i $$-0.291982\pi$$
0.607975 + 0.793956i $$0.291982\pi$$
$$98$$ 0 0
$$99$$ 0 0
$$100$$ 0 0
$$101$$ 1685.98 1.66100 0.830502 0.557016i $$-0.188054\pi$$
0.830502 + 0.557016i $$0.188054\pi$$
$$102$$ 0 0
$$103$$ −765.820 −0.732607 −0.366304 0.930495i $$-0.619377\pi$$
−0.366304 + 0.930495i $$0.619377\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ −1747.89 −1.57920 −0.789602 0.613619i $$-0.789713\pi$$
−0.789602 + 0.613619i $$0.789713\pi$$
$$108$$ 0 0
$$109$$ −1211.64 −1.06472 −0.532358 0.846519i $$-0.678694\pi$$
−0.532358 + 0.846519i $$0.678694\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ 0 0
$$113$$ 964.426 0.802881 0.401440 0.915885i $$-0.368510\pi$$
0.401440 + 0.915885i $$0.368510\pi$$
$$114$$ 0 0
$$115$$ 2174.98 1.76363
$$116$$ 0 0
$$117$$ 0 0
$$118$$ 0 0
$$119$$ −1920.53 −1.47945
$$120$$ 0 0
$$121$$ −1127.96 −0.847453
$$122$$ 0 0
$$123$$ 0 0
$$124$$ 0 0
$$125$$ 190.152 0.136061
$$126$$ 0 0
$$127$$ −2238.31 −1.56392 −0.781960 0.623329i $$-0.785780\pi$$
−0.781960 + 0.623329i $$0.785780\pi$$
$$128$$ 0 0
$$129$$ 0 0
$$130$$ 0 0
$$131$$ 1674.31 1.11668 0.558340 0.829612i $$-0.311438\pi$$
0.558340 + 0.829612i $$0.311438\pi$$
$$132$$ 0 0
$$133$$ −3439.82 −2.24263
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ −95.0758 −0.0592911 −0.0296455 0.999560i $$-0.509438\pi$$
−0.0296455 + 0.999560i $$0.509438\pi$$
$$138$$ 0 0
$$139$$ 1170.64 0.714334 0.357167 0.934041i $$-0.383743\pi$$
0.357167 + 0.934041i $$0.383743\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 0 0
$$143$$ −197.107 −0.115265
$$144$$ 0 0
$$145$$ 3874.63 2.21910
$$146$$ 0 0
$$147$$ 0 0
$$148$$ 0 0
$$149$$ 859.136 0.472370 0.236185 0.971708i $$-0.424103\pi$$
0.236185 + 0.971708i $$0.424103\pi$$
$$150$$ 0 0
$$151$$ 2898.83 1.56227 0.781137 0.624359i $$-0.214640\pi$$
0.781137 + 0.624359i $$0.214640\pi$$
$$152$$ 0 0
$$153$$ 0 0
$$154$$ 0 0
$$155$$ 256.972 0.133164
$$156$$ 0 0
$$157$$ −1309.02 −0.665419 −0.332710 0.943029i $$-0.607963\pi$$
−0.332710 + 0.943029i $$0.607963\pi$$
$$158$$ 0 0
$$159$$ 0 0
$$160$$ 0 0
$$161$$ −3362.38 −1.64592
$$162$$ 0 0
$$163$$ −190.988 −0.0917749 −0.0458874 0.998947i $$-0.514612\pi$$
−0.0458874 + 0.998947i $$0.514612\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ −1440.73 −0.667587 −0.333793 0.942646i $$-0.608329\pi$$
−0.333793 + 0.942646i $$0.608329\pi$$
$$168$$ 0 0
$$169$$ −2005.65 −0.912905
$$170$$ 0 0
$$171$$ 0 0
$$172$$ 0 0
$$173$$ −1581.33 −0.694947 −0.347474 0.937690i $$-0.612960\pi$$
−0.347474 + 0.937690i $$0.612960\pi$$
$$174$$ 0 0
$$175$$ 2685.14 1.15987
$$176$$ 0 0
$$177$$ 0 0
$$178$$ 0 0
$$179$$ −741.836 −0.309762 −0.154881 0.987933i $$-0.549499\pi$$
−0.154881 + 0.987933i $$0.549499\pi$$
$$180$$ 0 0
$$181$$ 626.786 0.257396 0.128698 0.991684i $$-0.458920\pi$$
0.128698 + 0.991684i $$0.458920\pi$$
$$182$$ 0 0
$$183$$ 0 0
$$184$$ 0 0
$$185$$ −4704.53 −1.86964
$$186$$ 0 0
$$187$$ −1148.25 −0.449030
$$188$$ 0 0
$$189$$ 0 0
$$190$$ 0 0
$$191$$ −182.615 −0.0691808 −0.0345904 0.999402i $$-0.511013\pi$$
−0.0345904 + 0.999402i $$0.511013\pi$$
$$192$$ 0 0
$$193$$ 1718.66 0.640994 0.320497 0.947250i $$-0.396150\pi$$
0.320497 + 0.947250i $$0.396150\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ 119.587 0.0432497 0.0216249 0.999766i $$-0.493116\pi$$
0.0216249 + 0.999766i $$0.493116\pi$$
$$198$$ 0 0
$$199$$ 707.467 0.252015 0.126008 0.992029i $$-0.459784\pi$$
0.126008 + 0.992029i $$0.459784\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ 0 0
$$203$$ −5989.93 −2.07099
$$204$$ 0 0
$$205$$ −6618.65 −2.25496
$$206$$ 0 0
$$207$$ 0 0
$$208$$ 0 0
$$209$$ −2056.61 −0.680663
$$210$$ 0 0
$$211$$ −2388.62 −0.779332 −0.389666 0.920956i $$-0.627410\pi$$
−0.389666 + 0.920956i $$0.627410\pi$$
$$212$$ 0 0
$$213$$ 0 0
$$214$$ 0 0
$$215$$ 2800.63 0.888379
$$216$$ 0 0
$$217$$ −397.263 −0.124276
$$218$$ 0 0
$$219$$ 0 0
$$220$$ 0 0
$$221$$ 1114.70 0.339288
$$222$$ 0 0
$$223$$ 5622.60 1.68842 0.844209 0.536014i $$-0.180071\pi$$
0.844209 + 0.536014i $$0.180071\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 0 0
$$227$$ 3464.46 1.01297 0.506485 0.862249i $$-0.330944\pi$$
0.506485 + 0.862249i $$0.330944\pi$$
$$228$$ 0 0
$$229$$ −6082.62 −1.75524 −0.877622 0.479354i $$-0.840871\pi$$
−0.877622 + 0.479354i $$0.840871\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ 5024.29 1.41267 0.706335 0.707877i $$-0.250347\pi$$
0.706335 + 0.707877i $$0.250347\pi$$
$$234$$ 0 0
$$235$$ 1224.32 0.339853
$$236$$ 0 0
$$237$$ 0 0
$$238$$ 0 0
$$239$$ −2670.20 −0.722682 −0.361341 0.932434i $$-0.617681\pi$$
−0.361341 + 0.932434i $$0.617681\pi$$
$$240$$ 0 0
$$241$$ 754.653 0.201707 0.100854 0.994901i $$-0.467843\pi$$
0.100854 + 0.994901i $$0.467843\pi$$
$$242$$ 0 0
$$243$$ 0 0
$$244$$ 0 0
$$245$$ −3468.74 −0.904529
$$246$$ 0 0
$$247$$ 1996.51 0.514311
$$248$$ 0 0
$$249$$ 0 0
$$250$$ 0 0
$$251$$ −1588.91 −0.399566 −0.199783 0.979840i $$-0.564024\pi$$
−0.199783 + 0.979840i $$0.564024\pi$$
$$252$$ 0 0
$$253$$ −2010.31 −0.499554
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 0 0
$$257$$ 4911.76 1.19217 0.596083 0.802923i $$-0.296723\pi$$
0.596083 + 0.802923i $$0.296723\pi$$
$$258$$ 0 0
$$259$$ 7272.92 1.74485
$$260$$ 0 0
$$261$$ 0 0
$$262$$ 0 0
$$263$$ −5400.41 −1.26617 −0.633087 0.774081i $$-0.718212\pi$$
−0.633087 + 0.774081i $$0.718212\pi$$
$$264$$ 0 0
$$265$$ 10233.9 2.37231
$$266$$ 0 0
$$267$$ 0 0
$$268$$ 0 0
$$269$$ 3234.09 0.733034 0.366517 0.930411i $$-0.380550\pi$$
0.366517 + 0.930411i $$0.380550\pi$$
$$270$$ 0 0
$$271$$ −6205.83 −1.39106 −0.695530 0.718497i $$-0.744830\pi$$
−0.695530 + 0.718497i $$0.744830\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ 0 0
$$275$$ 1605.40 0.352033
$$276$$ 0 0
$$277$$ 1030.91 0.223615 0.111808 0.993730i $$-0.464336\pi$$
0.111808 + 0.993730i $$0.464336\pi$$
$$278$$ 0 0
$$279$$ 0 0
$$280$$ 0 0
$$281$$ 3821.33 0.811250 0.405625 0.914040i $$-0.367054\pi$$
0.405625 + 0.914040i $$0.367054\pi$$
$$282$$ 0 0
$$283$$ −4402.33 −0.924705 −0.462353 0.886696i $$-0.652995\pi$$
−0.462353 + 0.886696i $$0.652995\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ 10232.0 2.10445
$$288$$ 0 0
$$289$$ 1580.72 0.321741
$$290$$ 0 0
$$291$$ 0 0
$$292$$ 0 0
$$293$$ −5973.54 −1.19105 −0.595526 0.803336i $$-0.703056\pi$$
−0.595526 + 0.803336i $$0.703056\pi$$
$$294$$ 0 0
$$295$$ −3395.55 −0.670157
$$296$$ 0 0
$$297$$ 0 0
$$298$$ 0 0
$$299$$ 1951.56 0.377464
$$300$$ 0 0
$$301$$ −4329.60 −0.829084
$$302$$ 0 0
$$303$$ 0 0
$$304$$ 0 0
$$305$$ 7294.39 1.36943
$$306$$ 0 0
$$307$$ −219.622 −0.0408290 −0.0204145 0.999792i $$-0.506499\pi$$
−0.0204145 + 0.999792i $$0.506499\pi$$
$$308$$ 0 0
$$309$$ 0 0
$$310$$ 0 0
$$311$$ −957.227 −0.174532 −0.0872659 0.996185i $$-0.527813\pi$$
−0.0872659 + 0.996185i $$0.527813\pi$$
$$312$$ 0 0
$$313$$ 1874.05 0.338427 0.169214 0.985579i $$-0.445877\pi$$
0.169214 + 0.985579i $$0.445877\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ 2919.78 0.517322 0.258661 0.965968i $$-0.416719\pi$$
0.258661 + 0.965968i $$0.416719\pi$$
$$318$$ 0 0
$$319$$ −3581.28 −0.628567
$$320$$ 0 0
$$321$$ 0 0
$$322$$ 0 0
$$323$$ 11630.7 2.00356
$$324$$ 0 0
$$325$$ −1558.48 −0.265997
$$326$$ 0 0
$$327$$ 0 0
$$328$$ 0 0
$$329$$ −1892.72 −0.317170
$$330$$ 0 0
$$331$$ −6648.67 −1.10406 −0.552030 0.833824i $$-0.686147\pi$$
−0.552030 + 0.833824i $$0.686147\pi$$
$$332$$ 0 0
$$333$$ 0 0
$$334$$ 0 0
$$335$$ 9984.63 1.62841
$$336$$ 0 0
$$337$$ 3886.34 0.628197 0.314098 0.949390i $$-0.398298\pi$$
0.314098 + 0.949390i $$0.398298\pi$$
$$338$$ 0 0
$$339$$ 0 0
$$340$$ 0 0
$$341$$ −237.517 −0.0377192
$$342$$ 0 0
$$343$$ −2812.20 −0.442695
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ 4207.76 0.650963 0.325481 0.945548i $$-0.394474\pi$$
0.325481 + 0.945548i $$0.394474\pi$$
$$348$$ 0 0
$$349$$ 3011.42 0.461885 0.230942 0.972967i $$-0.425819\pi$$
0.230942 + 0.972967i $$0.425819\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 0 0
$$353$$ 2499.79 0.376914 0.188457 0.982081i $$-0.439651\pi$$
0.188457 + 0.982081i $$0.439651\pi$$
$$354$$ 0 0
$$355$$ −223.418 −0.0334023
$$356$$ 0 0
$$357$$ 0 0
$$358$$ 0 0
$$359$$ −8319.17 −1.22303 −0.611517 0.791231i $$-0.709440\pi$$
−0.611517 + 0.791231i $$0.709440\pi$$
$$360$$ 0 0
$$361$$ 13972.5 2.03711
$$362$$ 0 0
$$363$$ 0 0
$$364$$ 0 0
$$365$$ −11963.2 −1.71557
$$366$$ 0 0
$$367$$ −4459.85 −0.634339 −0.317169 0.948369i $$-0.602732\pi$$
−0.317169 + 0.948369i $$0.602732\pi$$
$$368$$ 0 0
$$369$$ 0 0
$$370$$ 0 0
$$371$$ −15820.9 −2.21397
$$372$$ 0 0
$$373$$ −1179.41 −0.163719 −0.0818596 0.996644i $$-0.526086\pi$$
−0.0818596 + 0.996644i $$0.526086\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ 3476.62 0.474947
$$378$$ 0 0
$$379$$ 8413.88 1.14035 0.570174 0.821524i $$-0.306876\pi$$
0.570174 + 0.821524i $$0.306876\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ 0 0
$$383$$ −1601.94 −0.213722 −0.106861 0.994274i $$-0.534080\pi$$
−0.106861 + 0.994274i $$0.534080\pi$$
$$384$$ 0 0
$$385$$ −5235.40 −0.693041
$$386$$ 0 0
$$387$$ 0 0
$$388$$ 0 0
$$389$$ 5202.43 0.678081 0.339041 0.940772i $$-0.389898\pi$$
0.339041 + 0.940772i $$0.389898\pi$$
$$390$$ 0 0
$$391$$ 11368.9 1.47046
$$392$$ 0 0
$$393$$ 0 0
$$394$$ 0 0
$$395$$ 3974.76 0.506309
$$396$$ 0 0
$$397$$ 4310.87 0.544979 0.272489 0.962159i $$-0.412153\pi$$
0.272489 + 0.962159i $$0.412153\pi$$
$$398$$ 0 0
$$399$$ 0 0
$$400$$ 0 0
$$401$$ −6240.59 −0.777157 −0.388579 0.921416i $$-0.627034\pi$$
−0.388579 + 0.921416i $$0.627034\pi$$
$$402$$ 0 0
$$403$$ 230.576 0.0285007
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 0 0
$$407$$ 4348.35 0.529582
$$408$$ 0 0
$$409$$ 2047.66 0.247556 0.123778 0.992310i $$-0.460499\pi$$
0.123778 + 0.992310i $$0.460499\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ 0 0
$$413$$ 5249.31 0.625427
$$414$$ 0 0
$$415$$ −19812.5 −2.34351
$$416$$ 0 0
$$417$$ 0 0
$$418$$ 0 0
$$419$$ −5960.12 −0.694919 −0.347459 0.937695i $$-0.612956\pi$$
−0.347459 + 0.937695i $$0.612956\pi$$
$$420$$ 0 0
$$421$$ 311.567 0.0360685 0.0180342 0.999837i $$-0.494259\pi$$
0.0180342 + 0.999837i $$0.494259\pi$$
$$422$$ 0 0
$$423$$ 0 0
$$424$$ 0 0
$$425$$ −9079.00 −1.03623
$$426$$ 0 0
$$427$$ −11276.7 −1.27803
$$428$$ 0 0
$$429$$ 0 0
$$430$$ 0 0
$$431$$ −17105.0 −1.91164 −0.955821 0.293950i $$-0.905030\pi$$
−0.955821 + 0.293950i $$0.905030\pi$$
$$432$$ 0 0
$$433$$ 2582.96 0.286672 0.143336 0.989674i $$-0.454217\pi$$
0.143336 + 0.989674i $$0.454217\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 0 0
$$437$$ 20362.5 2.22900
$$438$$ 0 0
$$439$$ −6797.87 −0.739054 −0.369527 0.929220i $$-0.620480\pi$$
−0.369527 + 0.929220i $$0.620480\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 0 0
$$443$$ −10454.1 −1.12119 −0.560596 0.828089i $$-0.689428\pi$$
−0.560596 + 0.828089i $$0.689428\pi$$
$$444$$ 0 0
$$445$$ 2408.90 0.256613
$$446$$ 0 0
$$447$$ 0 0
$$448$$ 0 0
$$449$$ −17208.6 −1.80873 −0.904367 0.426755i $$-0.859657\pi$$
−0.904367 + 0.426755i $$0.859657\pi$$
$$450$$ 0 0
$$451$$ 6117.55 0.638723
$$452$$ 0 0
$$453$$ 0 0
$$454$$ 0 0
$$455$$ 5082.40 0.523663
$$456$$ 0 0
$$457$$ 15788.1 1.61605 0.808025 0.589148i $$-0.200537\pi$$
0.808025 + 0.589148i $$0.200537\pi$$
$$458$$ 0 0
$$459$$ 0 0
$$460$$ 0 0
$$461$$ −5183.17 −0.523654 −0.261827 0.965115i $$-0.584325\pi$$
−0.261827 + 0.965115i $$0.584325\pi$$
$$462$$ 0 0
$$463$$ −5833.38 −0.585530 −0.292765 0.956184i $$-0.594575\pi$$
−0.292765 + 0.956184i $$0.594575\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ −6315.33 −0.625779 −0.312889 0.949790i $$-0.601297\pi$$
−0.312889 + 0.949790i $$0.601297\pi$$
$$468$$ 0 0
$$469$$ −15435.6 −1.51972
$$470$$ 0 0
$$471$$ 0 0
$$472$$ 0 0
$$473$$ −2588.59 −0.251636
$$474$$ 0 0
$$475$$ −16261.2 −1.57077
$$476$$ 0 0
$$477$$ 0 0
$$478$$ 0 0
$$479$$ −9735.88 −0.928692 −0.464346 0.885654i $$-0.653711\pi$$
−0.464346 + 0.885654i $$0.653711\pi$$
$$480$$ 0 0
$$481$$ −4221.28 −0.400153
$$482$$ 0 0
$$483$$ 0 0
$$484$$ 0 0
$$485$$ −17908.4 −1.67665
$$486$$ 0 0
$$487$$ 13447.9 1.25130 0.625649 0.780104i $$-0.284834\pi$$
0.625649 + 0.780104i $$0.284834\pi$$
$$488$$ 0 0
$$489$$ 0 0
$$490$$ 0 0
$$491$$ 7028.09 0.645974 0.322987 0.946403i $$-0.395313\pi$$
0.322987 + 0.946403i $$0.395313\pi$$
$$492$$ 0 0
$$493$$ 20253.2 1.85022
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ 345.391 0.0311728
$$498$$ 0 0
$$499$$ 1655.44 0.148513 0.0742563 0.997239i $$-0.476342\pi$$
0.0742563 + 0.997239i $$0.476342\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ 0 0
$$503$$ −1909.16 −0.169235 −0.0846174 0.996414i $$-0.526967\pi$$
−0.0846174 + 0.996414i $$0.526967\pi$$
$$504$$ 0 0
$$505$$ −25991.8 −2.29033
$$506$$ 0 0
$$507$$ 0 0
$$508$$ 0 0
$$509$$ −8865.70 −0.772034 −0.386017 0.922492i $$-0.626149\pi$$
−0.386017 + 0.922492i $$0.626149\pi$$
$$510$$ 0 0
$$511$$ 18494.3 1.60106
$$512$$ 0 0
$$513$$ 0 0
$$514$$ 0 0
$$515$$ 11806.2 1.01018
$$516$$ 0 0
$$517$$ −1131.62 −0.0962644
$$518$$ 0 0
$$519$$ 0 0
$$520$$ 0 0
$$521$$ −9256.80 −0.778403 −0.389201 0.921153i $$-0.627249\pi$$
−0.389201 + 0.921153i $$0.627249\pi$$
$$522$$ 0 0
$$523$$ 13607.5 1.13770 0.568849 0.822442i $$-0.307389\pi$$
0.568849 + 0.822442i $$0.307389\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ 1343.23 0.111028
$$528$$ 0 0
$$529$$ 7737.14 0.635912
$$530$$ 0 0
$$531$$ 0 0
$$532$$ 0 0
$$533$$ −5938.77 −0.482621
$$534$$ 0 0
$$535$$ 26946.2 2.17754
$$536$$ 0 0
$$537$$ 0 0
$$538$$ 0 0
$$539$$ 3206.12 0.256210
$$540$$ 0 0
$$541$$ −483.548 −0.0384276 −0.0192138 0.999815i $$-0.506116\pi$$
−0.0192138 + 0.999815i $$0.506116\pi$$
$$542$$ 0 0
$$543$$ 0 0
$$544$$ 0 0
$$545$$ 18679.1 1.46812
$$546$$ 0 0
$$547$$ −4711.28 −0.368263 −0.184131 0.982902i $$-0.558947\pi$$
−0.184131 + 0.982902i $$0.558947\pi$$
$$548$$ 0 0
$$549$$ 0 0
$$550$$ 0 0
$$551$$ 36275.0 2.80466
$$552$$ 0 0
$$553$$ −6144.73 −0.472515
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 0 0
$$557$$ 21650.7 1.64698 0.823492 0.567327i $$-0.192023\pi$$
0.823492 + 0.567327i $$0.192023\pi$$
$$558$$ 0 0
$$559$$ 2512.95 0.190137
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 0 0
$$563$$ −14023.0 −1.04973 −0.524866 0.851185i $$-0.675884\pi$$
−0.524866 + 0.851185i $$0.675884\pi$$
$$564$$ 0 0
$$565$$ −14868.0 −1.10708
$$566$$ 0 0
$$567$$ 0 0
$$568$$ 0 0
$$569$$ −4875.33 −0.359200 −0.179600 0.983740i $$-0.557480\pi$$
−0.179600 + 0.983740i $$0.557480\pi$$
$$570$$ 0 0
$$571$$ −25969.1 −1.90328 −0.951640 0.307215i $$-0.900603\pi$$
−0.951640 + 0.307215i $$0.900603\pi$$
$$572$$ 0 0
$$573$$ 0 0
$$574$$ 0 0
$$575$$ −15895.1 −1.15282
$$576$$ 0 0
$$577$$ −23113.1 −1.66761 −0.833806 0.552058i $$-0.813843\pi$$
−0.833806 + 0.552058i $$0.813843\pi$$
$$578$$ 0 0
$$579$$ 0 0
$$580$$ 0 0
$$581$$ 30628.9 2.18709
$$582$$ 0 0
$$583$$ −9459.06 −0.671963
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ 5471.66 0.384735 0.192368 0.981323i $$-0.438383\pi$$
0.192368 + 0.981323i $$0.438383\pi$$
$$588$$ 0 0
$$589$$ 2405.82 0.168302
$$590$$ 0 0
$$591$$ 0 0
$$592$$ 0 0
$$593$$ −22609.2 −1.56568 −0.782841 0.622222i $$-0.786230\pi$$
−0.782841 + 0.622222i $$0.786230\pi$$
$$594$$ 0 0
$$595$$ 29607.7 2.04000
$$596$$ 0 0
$$597$$ 0 0
$$598$$ 0 0
$$599$$ 3708.89 0.252990 0.126495 0.991967i $$-0.459627\pi$$
0.126495 + 0.991967i $$0.459627\pi$$
$$600$$ 0 0
$$601$$ −10478.2 −0.711170 −0.355585 0.934644i $$-0.615718\pi$$
−0.355585 + 0.934644i $$0.615718\pi$$
$$602$$ 0 0
$$603$$ 0 0
$$604$$ 0 0
$$605$$ 17389.1 1.16854
$$606$$ 0 0
$$607$$ 10713.6 0.716392 0.358196 0.933646i $$-0.383392\pi$$
0.358196 + 0.933646i $$0.383392\pi$$
$$608$$ 0 0
$$609$$ 0 0
$$610$$ 0 0
$$611$$ 1098.55 0.0727376
$$612$$ 0 0
$$613$$ 7860.62 0.517924 0.258962 0.965887i $$-0.416619\pi$$
0.258962 + 0.965887i $$0.416619\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ 8528.00 0.556442 0.278221 0.960517i $$-0.410255\pi$$
0.278221 + 0.960517i $$0.410255\pi$$
$$618$$ 0 0
$$619$$ −8022.89 −0.520949 −0.260474 0.965481i $$-0.583879\pi$$
−0.260474 + 0.965481i $$0.583879\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ 0 0
$$623$$ −3724.01 −0.239485
$$624$$ 0 0
$$625$$ −17014.7 −1.08894
$$626$$ 0 0
$$627$$ 0 0
$$628$$ 0 0
$$629$$ −24591.2 −1.55885
$$630$$ 0 0
$$631$$ −3984.40 −0.251373 −0.125687 0.992070i $$-0.540113\pi$$
−0.125687 + 0.992070i $$0.540113\pi$$
$$632$$ 0 0
$$633$$ 0 0
$$634$$ 0 0
$$635$$ 34506.7 2.15647
$$636$$ 0 0
$$637$$ −3112.43 −0.193593
$$638$$ 0 0
$$639$$ 0 0
$$640$$ 0 0
$$641$$ 14750.2 0.908892 0.454446 0.890774i $$-0.349837\pi$$
0.454446 + 0.890774i $$0.349837\pi$$
$$642$$ 0 0
$$643$$ 9810.74 0.601707 0.300854 0.953670i $$-0.402728\pi$$
0.300854 + 0.953670i $$0.402728\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ 5683.55 0.345353 0.172677 0.984979i $$-0.444758\pi$$
0.172677 + 0.984979i $$0.444758\pi$$
$$648$$ 0 0
$$649$$ 3138.47 0.189824
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 0 0
$$653$$ 3405.52 0.204086 0.102043 0.994780i $$-0.467462\pi$$
0.102043 + 0.994780i $$0.467462\pi$$
$$654$$ 0 0
$$655$$ −25811.8 −1.53977
$$656$$ 0 0
$$657$$ 0 0
$$658$$ 0 0
$$659$$ −19049.1 −1.12602 −0.563011 0.826449i $$-0.690357\pi$$
−0.563011 + 0.826449i $$0.690357\pi$$
$$660$$ 0 0
$$661$$ 20422.5 1.20173 0.600864 0.799351i $$-0.294823\pi$$
0.600864 + 0.799351i $$0.294823\pi$$
$$662$$ 0 0
$$663$$ 0 0
$$664$$ 0 0
$$665$$ 53029.7 3.09233
$$666$$ 0 0
$$667$$ 35458.3 2.05840
$$668$$ 0 0
$$669$$ 0 0
$$670$$ 0 0
$$671$$ −6742.13 −0.387894
$$672$$ 0 0
$$673$$ 8613.45 0.493349 0.246675 0.969098i $$-0.420662\pi$$
0.246675 + 0.969098i $$0.420662\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ 0 0
$$677$$ 10469.1 0.594329 0.297164 0.954826i $$-0.403959\pi$$
0.297164 + 0.954826i $$0.403959\pi$$
$$678$$ 0 0
$$679$$ 27685.2 1.56475
$$680$$ 0 0
$$681$$ 0 0
$$682$$ 0 0
$$683$$ −9490.28 −0.531677 −0.265839 0.964018i $$-0.585649\pi$$
−0.265839 + 0.964018i $$0.585649\pi$$
$$684$$ 0 0
$$685$$ 1465.73 0.0817556
$$686$$ 0 0
$$687$$ 0 0
$$688$$ 0 0
$$689$$ 9182.63 0.507737
$$690$$ 0 0
$$691$$ 15814.7 0.870652 0.435326 0.900273i $$-0.356633\pi$$
0.435326 + 0.900273i $$0.356633\pi$$
$$692$$ 0 0
$$693$$ 0 0
$$694$$ 0 0
$$695$$ −18047.1 −0.984985
$$696$$ 0 0
$$697$$ −34596.6 −1.88011
$$698$$ 0 0
$$699$$ 0 0
$$700$$ 0 0
$$701$$ −15790.6 −0.850791 −0.425395 0.905008i $$-0.639865\pi$$
−0.425395 + 0.905008i $$0.639865\pi$$
$$702$$ 0 0
$$703$$ −44044.7 −2.36298
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ 40181.7 2.13746
$$708$$ 0 0
$$709$$ 27542.9 1.45895 0.729474 0.684009i $$-0.239765\pi$$
0.729474 + 0.684009i $$0.239765\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ 0 0
$$713$$ 2351.66 0.123521
$$714$$ 0 0
$$715$$ 3038.68 0.158937
$$716$$ 0 0
$$717$$ 0 0
$$718$$ 0 0
$$719$$ 19578.6 1.01552 0.507761 0.861498i $$-0.330473\pi$$
0.507761 + 0.861498i $$0.330473\pi$$
$$720$$ 0 0
$$721$$ −18251.7 −0.942756
$$722$$ 0 0
$$723$$ 0 0
$$724$$ 0 0
$$725$$ −28316.4 −1.45054
$$726$$ 0 0
$$727$$ 20948.0 1.06866 0.534330 0.845276i $$-0.320564\pi$$
0.534330 + 0.845276i $$0.320564\pi$$
$$728$$ 0 0
$$729$$ 0 0
$$730$$ 0 0
$$731$$ 14639.3 0.740702
$$732$$ 0 0
$$733$$ −825.349 −0.0415893 −0.0207946 0.999784i $$-0.506620\pi$$
−0.0207946 + 0.999784i $$0.506620\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ −9228.69 −0.461253
$$738$$ 0 0
$$739$$ 22821.1 1.13598 0.567988 0.823037i $$-0.307722\pi$$
0.567988 + 0.823037i $$0.307722\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 0 0
$$743$$ 12056.5 0.595301 0.297650 0.954675i $$-0.403797\pi$$
0.297650 + 0.954675i $$0.403797\pi$$
$$744$$ 0 0
$$745$$ −13244.8 −0.651345
$$746$$ 0 0
$$747$$ 0 0
$$748$$ 0 0
$$749$$ −41657.1 −2.03220
$$750$$ 0 0
$$751$$ −7722.32 −0.375221 −0.187611 0.982243i $$-0.560074\pi$$
−0.187611 + 0.982243i $$0.560074\pi$$
$$752$$ 0 0
$$753$$ 0 0
$$754$$ 0 0
$$755$$ −44689.5 −2.15420
$$756$$ 0 0
$$757$$ −1440.60 −0.0691671 −0.0345835 0.999402i $$-0.511010\pi$$
−0.0345835 + 0.999402i $$0.511010\pi$$
$$758$$ 0 0
$$759$$ 0 0
$$760$$ 0 0
$$761$$ −30722.6 −1.46346 −0.731729 0.681595i $$-0.761286\pi$$
−0.731729 + 0.681595i $$0.761286\pi$$
$$762$$ 0 0
$$763$$ −28876.8 −1.37013
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ −3046.75 −0.143431
$$768$$ 0 0
$$769$$ −40280.9 −1.88890 −0.944451 0.328652i $$-0.893406\pi$$
−0.944451 + 0.328652i $$0.893406\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ 0 0
$$773$$ −4282.45 −0.199261 −0.0996306 0.995024i $$-0.531766\pi$$
−0.0996306 + 0.995024i $$0.531766\pi$$
$$774$$ 0 0
$$775$$ −1877.99 −0.0870446
$$776$$ 0 0
$$777$$ 0 0
$$778$$ 0 0
$$779$$ −61965.0 −2.84997
$$780$$ 0 0
$$781$$ 206.503 0.00946128
$$782$$ 0 0
$$783$$ 0 0
$$784$$ 0 0
$$785$$ 20180.3 0.917537
$$786$$ 0 0
$$787$$ −12571.6 −0.569416 −0.284708 0.958614i $$-0.591897\pi$$
−0.284708 + 0.958614i $$0.591897\pi$$
$$788$$ 0 0
$$789$$ 0 0
$$790$$ 0 0
$$791$$ 22985.0 1.03319
$$792$$ 0 0
$$793$$ 6545.11 0.293094
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 0 0
$$797$$ −28074.0 −1.24772 −0.623860 0.781536i $$-0.714436\pi$$
−0.623860 + 0.781536i $$0.714436\pi$$
$$798$$ 0 0
$$799$$ 6399.66 0.283359
$$800$$ 0 0
$$801$$ 0 0
$$802$$ 0 0
$$803$$ 11057.4 0.485939
$$804$$ 0 0
$$805$$ 51835.9 2.26953
$$806$$ 0 0
$$807$$ 0 0
$$808$$ 0 0
$$809$$ −293.254 −0.0127445 −0.00637223 0.999980i $$-0.502028\pi$$
−0.00637223 + 0.999980i $$0.502028\pi$$
$$810$$ 0 0
$$811$$ −45634.2 −1.97588 −0.987938 0.154851i $$-0.950510\pi$$
−0.987938 + 0.154851i $$0.950510\pi$$
$$812$$ 0 0
$$813$$ 0 0
$$814$$ 0 0
$$815$$ 2944.34 0.126547
$$816$$ 0 0
$$817$$ 26220.0 1.12279
$$818$$ 0 0
$$819$$ 0 0
$$820$$ 0 0
$$821$$ 5406.68 0.229835 0.114917 0.993375i $$-0.463340\pi$$
0.114917 + 0.993375i $$0.463340\pi$$
$$822$$ 0 0
$$823$$ 1228.88 0.0520487 0.0260243 0.999661i $$-0.491715\pi$$
0.0260243 + 0.999661i $$0.491715\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ 45769.8 1.92451 0.962256 0.272147i $$-0.0877337\pi$$
0.962256 + 0.272147i $$0.0877337\pi$$
$$828$$ 0 0
$$829$$ −13630.9 −0.571076 −0.285538 0.958367i $$-0.592172\pi$$
−0.285538 + 0.958367i $$0.592172\pi$$
$$830$$ 0 0
$$831$$ 0 0
$$832$$ 0 0
$$833$$ −18131.6 −0.754167
$$834$$ 0 0
$$835$$ 22210.9 0.920525
$$836$$ 0 0
$$837$$ 0 0
$$838$$ 0 0
$$839$$ −44987.6 −1.85119 −0.925593 0.378520i $$-0.876433\pi$$
−0.925593 + 0.378520i $$0.876433\pi$$
$$840$$ 0 0
$$841$$ 38778.4 1.59000
$$842$$ 0 0
$$843$$ 0 0
$$844$$ 0 0
$$845$$ 30920.0 1.25879
$$846$$ 0 0
$$847$$ −26882.5 −1.09055
$$848$$ 0 0
$$849$$ 0 0
$$850$$ 0 0
$$851$$ −43053.2 −1.73425
$$852$$ 0 0
$$853$$ 26043.2 1.04537 0.522685 0.852526i $$-0.324930\pi$$
0.522685 + 0.852526i $$0.324930\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0 0
$$857$$ −13706.4 −0.546327 −0.273163 0.961968i $$-0.588070\pi$$
−0.273163 + 0.961968i $$0.588070\pi$$
$$858$$ 0 0
$$859$$ −45319.1 −1.80008 −0.900039 0.435810i $$-0.856462\pi$$
−0.900039 + 0.435810i $$0.856462\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 0 0
$$863$$ −28057.3 −1.10670 −0.553350 0.832949i $$-0.686651\pi$$
−0.553350 + 0.832949i $$0.686651\pi$$
$$864$$ 0 0
$$865$$ 24378.4 0.958253
$$866$$ 0 0
$$867$$ 0 0
$$868$$ 0 0
$$869$$ −3673.83 −0.143413
$$870$$ 0 0
$$871$$ 8959.00 0.348524
$$872$$ 0 0
$$873$$ 0 0
$$874$$ 0 0
$$875$$ 4531.85 0.175091
$$876$$ 0 0
$$877$$ 3923.18 0.151056 0.0755282 0.997144i $$-0.475936\pi$$
0.0755282 + 0.997144i $$0.475936\pi$$
$$878$$ 0 0
$$879$$ 0 0
$$880$$ 0 0
$$881$$ 2441.97 0.0933849 0.0466924 0.998909i $$-0.485132\pi$$
0.0466924 + 0.998909i $$0.485132\pi$$
$$882$$ 0 0
$$883$$ 44576.9 1.69890 0.849452 0.527667i $$-0.176933\pi$$
0.849452 + 0.527667i $$0.176933\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 0 0
$$887$$ −17095.5 −0.647137 −0.323568 0.946205i $$-0.604883\pi$$
−0.323568 + 0.946205i $$0.604883\pi$$
$$888$$ 0 0
$$889$$ −53345.2 −2.01253
$$890$$ 0 0
$$891$$ 0 0
$$892$$ 0 0
$$893$$ 11462.3 0.429530
$$894$$ 0 0
$$895$$ 11436.4 0.427126
$$896$$ 0 0
$$897$$ 0 0
$$898$$ 0 0
$$899$$ 4189.37 0.155421
$$900$$ 0 0
$$901$$ 53493.8 1.97795
$$902$$ 0 0
$$903$$ 0 0
$$904$$ 0 0
$$905$$ −9662.79 −0.354919
$$906$$ 0 0
$$907$$ −33142.7 −1.21332 −0.606662 0.794960i $$-0.707492\pi$$
−0.606662 + 0.794960i $$0.707492\pi$$
$$908$$ 0 0
$$909$$ 0 0
$$910$$ 0 0
$$911$$ −32750.5 −1.19108 −0.595539 0.803326i $$-0.703061\pi$$
−0.595539 + 0.803326i $$0.703061\pi$$
$$912$$ 0 0
$$913$$ 18312.5 0.663805
$$914$$ 0 0
$$915$$ 0 0
$$916$$ 0 0
$$917$$ 39903.5 1.43700
$$918$$ 0 0
$$919$$ 6095.76 0.218804 0.109402 0.993998i $$-0.465106\pi$$
0.109402 + 0.993998i $$0.465106\pi$$
$$920$$ 0 0
$$921$$ 0 0
$$922$$ 0 0
$$923$$ −200.468 −0.00714897
$$924$$ 0 0
$$925$$ 34381.5 1.22212
$$926$$ 0 0
$$927$$ 0 0
$$928$$ 0 0
$$929$$ −24708.0 −0.872597 −0.436298 0.899802i $$-0.643711\pi$$
−0.436298 + 0.899802i $$0.643711\pi$$
$$930$$ 0 0
$$931$$ −32475.0 −1.14321
$$932$$ 0 0
$$933$$ 0 0
$$934$$ 0 0
$$935$$ 17701.9 0.619161
$$936$$ 0 0
$$937$$ −44713.2 −1.55893 −0.779465 0.626446i $$-0.784509\pi$$
−0.779465 + 0.626446i $$0.784509\pi$$
$$938$$ 0 0
$$939$$ 0 0
$$940$$ 0 0
$$941$$ 10172.7 0.352414 0.176207 0.984353i $$-0.443617\pi$$
0.176207 + 0.984353i $$0.443617\pi$$
$$942$$ 0 0
$$943$$ −60570.1 −2.09166
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ 13346.6 0.457978 0.228989 0.973429i $$-0.426458\pi$$
0.228989 + 0.973429i $$0.426458\pi$$
$$948$$ 0 0
$$949$$ −10734.3 −0.367176
$$950$$ 0 0
$$951$$ 0 0
$$952$$ 0 0
$$953$$ −36232.4 −1.23157 −0.615783 0.787916i $$-0.711160\pi$$
−0.615783 + 0.787916i $$0.711160\pi$$
$$954$$ 0 0
$$955$$ 2815.26 0.0953924
$$956$$ 0 0
$$957$$ 0 0
$$958$$ 0 0
$$959$$ −2265.92 −0.0762988
$$960$$ 0 0
$$961$$ −29513.2 −0.990673
$$962$$ 0 0
$$963$$ 0 0
$$964$$ 0 0
$$965$$ −26495.6 −0.883857
$$966$$ 0 0
$$967$$ 5429.03 0.180544 0.0902718 0.995917i $$-0.471226\pi$$
0.0902718 + 0.995917i $$0.471226\pi$$
$$968$$ 0 0
$$969$$ 0 0
$$970$$ 0 0
$$971$$ 5645.37 0.186579 0.0932897 0.995639i $$-0.470262\pi$$
0.0932897 + 0.995639i $$0.470262\pi$$
$$972$$ 0 0
$$973$$ 27899.7 0.919242
$$974$$ 0 0
$$975$$ 0 0
$$976$$ 0 0
$$977$$ 51103.6 1.67344 0.836719 0.547632i $$-0.184471\pi$$
0.836719 + 0.547632i $$0.184471\pi$$
$$978$$ 0 0
$$979$$ −2226.52 −0.0726863
$$980$$ 0 0
$$981$$ 0 0
$$982$$ 0 0
$$983$$ 32082.5 1.04097 0.520485 0.853871i $$-0.325751\pi$$
0.520485 + 0.853871i $$0.325751\pi$$
$$984$$ 0 0
$$985$$ −1843.60 −0.0596364
$$986$$ 0 0
$$987$$ 0 0
$$988$$ 0 0
$$989$$ 25629.8 0.824043
$$990$$ 0 0
$$991$$ −27696.1 −0.887785 −0.443893 0.896080i $$-0.646403\pi$$
−0.443893 + 0.896080i $$0.646403\pi$$
$$992$$ 0 0
$$993$$ 0 0
$$994$$ 0 0
$$995$$ −10906.6 −0.347500
$$996$$ 0 0
$$997$$ −41284.9 −1.31144 −0.655720 0.755004i $$-0.727635\pi$$
−0.655720 + 0.755004i $$0.727635\pi$$
$$998$$ 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 216.4.a.f.1.1 2
3.2 odd 2 216.4.a.g.1.2 yes 2
4.3 odd 2 432.4.a.p.1.1 2
8.3 odd 2 1728.4.a.br.1.2 2
8.5 even 2 1728.4.a.bq.1.2 2
9.2 odd 6 648.4.i.o.433.1 4
9.4 even 3 648.4.i.r.217.2 4
9.5 odd 6 648.4.i.o.217.1 4
9.7 even 3 648.4.i.r.433.2 4
12.11 even 2 432.4.a.r.1.2 2
24.5 odd 2 1728.4.a.bi.1.1 2
24.11 even 2 1728.4.a.bj.1.1 2

By twisted newform
Twist Min Dim Char Parity Ord Type
216.4.a.f.1.1 2 1.1 even 1 trivial
216.4.a.g.1.2 yes 2 3.2 odd 2
432.4.a.p.1.1 2 4.3 odd 2
432.4.a.r.1.2 2 12.11 even 2
648.4.i.o.217.1 4 9.5 odd 6
648.4.i.o.433.1 4 9.2 odd 6
648.4.i.r.217.2 4 9.4 even 3
648.4.i.r.433.2 4 9.7 even 3
1728.4.a.bi.1.1 2 24.5 odd 2
1728.4.a.bj.1.1 2 24.11 even 2
1728.4.a.bq.1.2 2 8.5 even 2
1728.4.a.br.1.2 2 8.3 odd 2