# Properties

 Label 216.4.a.f.1.2 Level $216$ Weight $4$ Character 216.1 Self dual yes Analytic conductor $12.744$ Analytic rank $1$ Dimension $2$ CM no Inner twists $1$

# Learn more about

## 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 1.2 Root $$-0.618034$$ of defining polynomial Character $$\chi$$ $$=$$ 216.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+11.4164 q^{5} -29.8328 q^{7} +O(q^{10})$$ $$q+11.4164 q^{5} -29.8328 q^{7} -66.2492 q^{11} +39.8328 q^{13} -107.416 q^{17} +70.3313 q^{19} -6.91796 q^{23} +5.33437 q^{25} -36.6687 q^{29} -231.331 q^{31} -340.584 q^{35} +36.8359 q^{37} -429.325 q^{41} -74.3344 q^{43} -52.5836 q^{47} +546.997 q^{49} -288.170 q^{53} -756.328 q^{55} +783.745 q^{59} +439.158 q^{61} +454.748 q^{65} -218.337 q^{67} -790.492 q^{71} +1098.00 q^{73} +1976.40 q^{77} +439.827 q^{79} +50.8452 q^{83} -1226.31 q^{85} -719.745 q^{89} -1188.33 q^{91} +802.930 q^{95} -1199.64 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 4q^{5} - 6q^{7} + O(q^{10})$$ $$2q - 4q^{5} - 6q^{7} - 52q^{11} + 26q^{13} - 188q^{17} - 74q^{19} - 148q^{23} + 118q^{25} - 288q^{29} - 248q^{31} - 708q^{35} + 342q^{37} - 256q^{43} - 132q^{47} + 772q^{49} - 952q^{53} - 976q^{55} + 1004q^{59} - 34q^{61} + 668q^{65} - 866q^{67} - 776q^{71} + 1874q^{73} + 2316q^{77} + 182q^{79} + 1336q^{83} + 16q^{85} - 876q^{89} - 1518q^{91} + 3028q^{95} - 38q^{97} + O(q^{100})$$

## 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
$$5$$ 11.4164 1.02111 0.510557 0.859844i $$-0.329439\pi$$
0.510557 + 0.859844i $$0.329439\pi$$
$$6$$ 0 0
$$7$$ −29.8328 −1.61082 −0.805410 0.592718i $$-0.798055\pi$$
−0.805410 + 0.592718i $$0.798055\pi$$
$$8$$ 0 0
$$9$$ 0 0
$$10$$ 0 0
$$11$$ −66.2492 −1.81590 −0.907950 0.419079i $$-0.862353\pi$$
−0.907950 + 0.419079i $$0.862353\pi$$
$$12$$ 0 0
$$13$$ 39.8328 0.849818 0.424909 0.905236i $$-0.360306\pi$$
0.424909 + 0.905236i $$0.360306\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 0 0
$$17$$ −107.416 −1.53249 −0.766244 0.642549i $$-0.777877\pi$$
−0.766244 + 0.642549i $$0.777877\pi$$
$$18$$ 0 0
$$19$$ 70.3313 0.849216 0.424608 0.905377i $$-0.360412\pi$$
0.424608 + 0.905377i $$0.360412\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 0 0
$$23$$ −6.91796 −0.0627172 −0.0313586 0.999508i $$-0.509983\pi$$
−0.0313586 + 0.999508i $$0.509983\pi$$
$$24$$ 0 0
$$25$$ 5.33437 0.0426749
$$26$$ 0 0
$$27$$ 0 0
$$28$$ 0 0
$$29$$ −36.6687 −0.234800 −0.117400 0.993085i $$-0.537456\pi$$
−0.117400 + 0.993085i $$0.537456\pi$$
$$30$$ 0 0
$$31$$ −231.331 −1.34027 −0.670134 0.742240i $$-0.733763\pi$$
−0.670134 + 0.742240i $$0.733763\pi$$
$$32$$ 0 0
$$33$$ 0 0
$$34$$ 0 0
$$35$$ −340.584 −1.64483
$$36$$ 0 0
$$37$$ 36.8359 0.163670 0.0818350 0.996646i $$-0.473922\pi$$
0.0818350 + 0.996646i $$0.473922\pi$$
$$38$$ 0 0
$$39$$ 0 0
$$40$$ 0 0
$$41$$ −429.325 −1.63535 −0.817674 0.575681i $$-0.804737\pi$$
−0.817674 + 0.575681i $$0.804737\pi$$
$$42$$ 0 0
$$43$$ −74.3344 −0.263625 −0.131813 0.991275i $$-0.542080\pi$$
−0.131813 + 0.991275i $$0.542080\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 0 0
$$47$$ −52.5836 −0.163194 −0.0815969 0.996665i $$-0.526002\pi$$
−0.0815969 + 0.996665i $$0.526002\pi$$
$$48$$ 0 0
$$49$$ 546.997 1.59474
$$50$$ 0 0
$$51$$ 0 0
$$52$$ 0 0
$$53$$ −288.170 −0.746853 −0.373427 0.927660i $$-0.621817\pi$$
−0.373427 + 0.927660i $$0.621817\pi$$
$$54$$ 0 0
$$55$$ −756.328 −1.85424
$$56$$ 0 0
$$57$$ 0 0
$$58$$ 0 0
$$59$$ 783.745 1.72940 0.864702 0.502285i $$-0.167507\pi$$
0.864702 + 0.502285i $$0.167507\pi$$
$$60$$ 0 0
$$61$$ 439.158 0.921777 0.460889 0.887458i $$-0.347531\pi$$
0.460889 + 0.887458i $$0.347531\pi$$
$$62$$ 0 0
$$63$$ 0 0
$$64$$ 0 0
$$65$$ 454.748 0.867762
$$66$$ 0 0
$$67$$ −218.337 −0.398122 −0.199061 0.979987i $$-0.563789\pi$$
−0.199061 + 0.979987i $$0.563789\pi$$
$$68$$ 0 0
$$69$$ 0 0
$$70$$ 0 0
$$71$$ −790.492 −1.32133 −0.660663 0.750682i $$-0.729725\pi$$
−0.660663 + 0.750682i $$0.729725\pi$$
$$72$$ 0 0
$$73$$ 1098.00 1.76042 0.880211 0.474582i $$-0.157401\pi$$
0.880211 + 0.474582i $$0.157401\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 0 0
$$77$$ 1976.40 2.92509
$$78$$ 0 0
$$79$$ 439.827 0.626384 0.313192 0.949690i $$-0.398602\pi$$
0.313192 + 0.949690i $$0.398602\pi$$
$$80$$ 0 0
$$81$$ 0 0
$$82$$ 0 0
$$83$$ 50.8452 0.0672408 0.0336204 0.999435i $$-0.489296\pi$$
0.0336204 + 0.999435i $$0.489296\pi$$
$$84$$ 0 0
$$85$$ −1226.31 −1.56485
$$86$$ 0 0
$$87$$ 0 0
$$88$$ 0 0
$$89$$ −719.745 −0.857222 −0.428611 0.903489i $$-0.640997\pi$$
−0.428611 + 0.903489i $$0.640997\pi$$
$$90$$ 0 0
$$91$$ −1188.33 −1.36890
$$92$$ 0 0
$$93$$ 0 0
$$94$$ 0 0
$$95$$ 802.930 0.867147
$$96$$ 0 0
$$97$$ −1199.64 −1.25573 −0.627863 0.778324i $$-0.716070\pi$$
−0.627863 + 0.778324i $$0.716070\pi$$
$$98$$ 0 0
$$99$$ 0 0
$$100$$ 0 0
$$101$$ −245.981 −0.242337 −0.121169 0.992632i $$-0.538664\pi$$
−0.121169 + 0.992632i $$0.538664\pi$$
$$102$$ 0 0
$$103$$ 575.820 0.550847 0.275424 0.961323i $$-0.411182\pi$$
0.275424 + 0.961323i $$0.411182\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ 1015.89 0.917849 0.458924 0.888475i $$-0.348235\pi$$
0.458924 + 0.888475i $$0.348235\pi$$
$$108$$ 0 0
$$109$$ 1471.64 1.29319 0.646595 0.762834i $$-0.276193\pi$$
0.646595 + 0.762834i $$0.276193\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ 0 0
$$113$$ 1903.57 1.58472 0.792359 0.610055i $$-0.208853\pi$$
0.792359 + 0.610055i $$0.208853\pi$$
$$114$$ 0 0
$$115$$ −78.9783 −0.0640414
$$116$$ 0 0
$$117$$ 0 0
$$118$$ 0 0
$$119$$ 3204.53 2.46856
$$120$$ 0 0
$$121$$ 3057.96 2.29749
$$122$$ 0 0
$$123$$ 0 0
$$124$$ 0 0
$$125$$ −1366.15 −0.977539
$$126$$ 0 0
$$127$$ 230.310 0.160919 0.0804593 0.996758i $$-0.474361\pi$$
0.0804593 + 0.996758i $$0.474361\pi$$
$$128$$ 0 0
$$129$$ 0 0
$$130$$ 0 0
$$131$$ −794.310 −0.529764 −0.264882 0.964281i $$-0.585333\pi$$
−0.264882 + 0.964281i $$0.585333\pi$$
$$132$$ 0 0
$$133$$ −2098.18 −1.36793
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ 683.076 0.425979 0.212989 0.977055i $$-0.431680\pi$$
0.212989 + 0.977055i $$0.431680\pi$$
$$138$$ 0 0
$$139$$ −1512.64 −0.923025 −0.461513 0.887134i $$-0.652693\pi$$
−0.461513 + 0.887134i $$0.652693\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 0 0
$$143$$ −2638.89 −1.54318
$$144$$ 0 0
$$145$$ −418.625 −0.239758
$$146$$ 0 0
$$147$$ 0 0
$$148$$ 0 0
$$149$$ −2307.14 −1.26851 −0.634255 0.773124i $$-0.718693\pi$$
−0.634255 + 0.773124i $$0.718693\pi$$
$$150$$ 0 0
$$151$$ 2523.17 1.35982 0.679910 0.733296i $$-0.262019\pi$$
0.679910 + 0.733296i $$0.262019\pi$$
$$152$$ 0 0
$$153$$ 0 0
$$154$$ 0 0
$$155$$ −2640.97 −1.36857
$$156$$ 0 0
$$157$$ −2918.98 −1.48382 −0.741912 0.670497i $$-0.766081\pi$$
−0.741912 + 0.670497i $$0.766081\pi$$
$$158$$ 0 0
$$159$$ 0 0
$$160$$ 0 0
$$161$$ 206.382 0.101026
$$162$$ 0 0
$$163$$ 1096.99 0.527133 0.263567 0.964641i $$-0.415101\pi$$
0.263567 + 0.964641i $$0.415101\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ 732.729 0.339523 0.169761 0.985485i $$-0.445700\pi$$
0.169761 + 0.985485i $$0.445700\pi$$
$$168$$ 0 0
$$169$$ −610.347 −0.277809
$$170$$ 0 0
$$171$$ 0 0
$$172$$ 0 0
$$173$$ −722.675 −0.317595 −0.158798 0.987311i $$-0.550762\pi$$
−0.158798 + 0.987311i $$0.550762\pi$$
$$174$$ 0 0
$$175$$ −159.139 −0.0687417
$$176$$ 0 0
$$177$$ 0 0
$$178$$ 0 0
$$179$$ −1010.16 −0.421806 −0.210903 0.977507i $$-0.567640\pi$$
−0.210903 + 0.977507i $$0.567640\pi$$
$$180$$ 0 0
$$181$$ −4256.79 −1.74809 −0.874045 0.485844i $$-0.838512\pi$$
−0.874045 + 0.485844i $$0.838512\pi$$
$$182$$ 0 0
$$183$$ 0 0
$$184$$ 0 0
$$185$$ 420.534 0.167126
$$186$$ 0 0
$$187$$ 7116.25 2.78284
$$188$$ 0 0
$$189$$ 0 0
$$190$$ 0 0
$$191$$ −3429.39 −1.29917 −0.649585 0.760289i $$-0.725057\pi$$
−0.649585 + 0.760289i $$0.725057\pi$$
$$192$$ 0 0
$$193$$ 967.341 0.360781 0.180390 0.983595i $$-0.442264\pi$$
0.180390 + 0.983595i $$0.442264\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ 468.413 0.169406 0.0847032 0.996406i $$-0.473006\pi$$
0.0847032 + 0.996406i $$0.473006\pi$$
$$198$$ 0 0
$$199$$ −2673.47 −0.952347 −0.476174 0.879351i $$-0.657977\pi$$
−0.476174 + 0.879351i $$0.657977\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ 0 0
$$203$$ 1093.93 0.378221
$$204$$ 0 0
$$205$$ −4901.35 −1.66988
$$206$$ 0 0
$$207$$ 0 0
$$208$$ 0 0
$$209$$ −4659.39 −1.54209
$$210$$ 0 0
$$211$$ 2870.62 0.936594 0.468297 0.883571i $$-0.344868\pi$$
0.468297 + 0.883571i $$0.344868\pi$$
$$212$$ 0 0
$$213$$ 0 0
$$214$$ 0 0
$$215$$ −848.631 −0.269192
$$216$$ 0 0
$$217$$ 6901.26 2.15893
$$218$$ 0 0
$$219$$ 0 0
$$220$$ 0 0
$$221$$ −4278.70 −1.30234
$$222$$ 0 0
$$223$$ −1246.60 −0.374343 −0.187172 0.982327i $$-0.559932\pi$$
−0.187172 + 0.982327i $$0.559932\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 0 0
$$227$$ −560.461 −0.163873 −0.0819364 0.996638i $$-0.526110\pi$$
−0.0819364 + 0.996638i $$0.526110\pi$$
$$228$$ 0 0
$$229$$ −1145.38 −0.330519 −0.165260 0.986250i $$-0.552846\pi$$
−0.165260 + 0.986250i $$0.552846\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ 623.709 0.175367 0.0876836 0.996148i $$-0.472054\pi$$
0.0876836 + 0.996148i $$0.472054\pi$$
$$234$$ 0 0
$$235$$ −600.316 −0.166639
$$236$$ 0 0
$$237$$ 0 0
$$238$$ 0 0
$$239$$ −6265.80 −1.69582 −0.847910 0.530141i $$-0.822139\pi$$
−0.847910 + 0.530141i $$0.822139\pi$$
$$240$$ 0 0
$$241$$ −640.653 −0.171237 −0.0856185 0.996328i $$-0.527287\pi$$
−0.0856185 + 0.996328i $$0.527287\pi$$
$$242$$ 0 0
$$243$$ 0 0
$$244$$ 0 0
$$245$$ 6244.74 1.62842
$$246$$ 0 0
$$247$$ 2801.49 0.721679
$$248$$ 0 0
$$249$$ 0 0
$$250$$ 0 0
$$251$$ 7748.91 1.94863 0.974316 0.225183i $$-0.0722980\pi$$
0.974316 + 0.225183i $$0.0722980\pi$$
$$252$$ 0 0
$$253$$ 458.310 0.113888
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 0 0
$$257$$ −3191.76 −0.774693 −0.387347 0.921934i $$-0.626608\pi$$
−0.387347 + 0.921934i $$0.626608\pi$$
$$258$$ 0 0
$$259$$ −1098.92 −0.263643
$$260$$ 0 0
$$261$$ 0 0
$$262$$ 0 0
$$263$$ 3776.41 0.885413 0.442706 0.896667i $$-0.354018\pi$$
0.442706 + 0.896667i $$0.354018\pi$$
$$264$$ 0 0
$$265$$ −3289.87 −0.762623
$$266$$ 0 0
$$267$$ 0 0
$$268$$ 0 0
$$269$$ 4065.91 0.921572 0.460786 0.887511i $$-0.347568\pi$$
0.460786 + 0.887511i $$0.347568\pi$$
$$270$$ 0 0
$$271$$ −5508.17 −1.23468 −0.617339 0.786697i $$-0.711789\pi$$
−0.617339 + 0.786697i $$0.711789\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ 0 0
$$275$$ −353.398 −0.0774934
$$276$$ 0 0
$$277$$ −8306.91 −1.80186 −0.900928 0.433970i $$-0.857113\pi$$
−0.900928 + 0.433970i $$0.857113\pi$$
$$278$$ 0 0
$$279$$ 0 0
$$280$$ 0 0
$$281$$ 2962.67 0.628962 0.314481 0.949264i $$-0.398169\pi$$
0.314481 + 0.949264i $$0.398169\pi$$
$$282$$ 0 0
$$283$$ −4509.67 −0.947250 −0.473625 0.880727i $$-0.657055\pi$$
−0.473625 + 0.880727i $$0.657055\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ 12808.0 2.63425
$$288$$ 0 0
$$289$$ 6625.28 1.34852
$$290$$ 0 0
$$291$$ 0 0
$$292$$ 0 0
$$293$$ −1814.46 −0.361781 −0.180890 0.983503i $$-0.557898\pi$$
−0.180890 + 0.983503i $$0.557898\pi$$
$$294$$ 0 0
$$295$$ 8947.55 1.76592
$$296$$ 0 0
$$297$$ 0 0
$$298$$ 0 0
$$299$$ −275.562 −0.0532982
$$300$$ 0 0
$$301$$ 2217.60 0.424653
$$302$$ 0 0
$$303$$ 0 0
$$304$$ 0 0
$$305$$ 5013.61 0.941240
$$306$$ 0 0
$$307$$ 4395.62 0.817171 0.408585 0.912720i $$-0.366022\pi$$
0.408585 + 0.912720i $$0.366022\pi$$
$$308$$ 0 0
$$309$$ 0 0
$$310$$ 0 0
$$311$$ 1377.23 0.251111 0.125555 0.992087i $$-0.459929\pi$$
0.125555 + 0.992087i $$0.459929\pi$$
$$312$$ 0 0
$$313$$ 7347.95 1.32693 0.663467 0.748205i $$-0.269084\pi$$
0.663467 + 0.748205i $$0.269084\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ −2607.78 −0.462043 −0.231021 0.972949i $$-0.574207\pi$$
−0.231021 + 0.972949i $$0.574207\pi$$
$$318$$ 0 0
$$319$$ 2429.28 0.426374
$$320$$ 0 0
$$321$$ 0 0
$$322$$ 0 0
$$323$$ −7554.73 −1.30141
$$324$$ 0 0
$$325$$ 212.483 0.0362659
$$326$$ 0 0
$$327$$ 0 0
$$328$$ 0 0
$$329$$ 1568.72 0.262876
$$330$$ 0 0
$$331$$ −6541.33 −1.08624 −0.543118 0.839656i $$-0.682756\pi$$
−0.543118 + 0.839656i $$0.682756\pi$$
$$332$$ 0 0
$$333$$ 0 0
$$334$$ 0 0
$$335$$ −2492.63 −0.406528
$$336$$ 0 0
$$337$$ 4315.66 0.697594 0.348797 0.937198i $$-0.386590\pi$$
0.348797 + 0.937198i $$0.386590\pi$$
$$338$$ 0 0
$$339$$ 0 0
$$340$$ 0 0
$$341$$ 15325.5 2.43379
$$342$$ 0 0
$$343$$ −6085.80 −0.958025
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ −3895.76 −0.602695 −0.301347 0.953514i $$-0.597436\pi$$
−0.301347 + 0.953514i $$0.597436\pi$$
$$348$$ 0 0
$$349$$ −4877.42 −0.748087 −0.374044 0.927411i $$-0.622029\pi$$
−0.374044 + 0.927411i $$0.622029\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 0 0
$$353$$ −1739.79 −0.262322 −0.131161 0.991361i $$-0.541871\pi$$
−0.131161 + 0.991361i $$0.541871\pi$$
$$354$$ 0 0
$$355$$ −9024.58 −1.34923
$$356$$ 0 0
$$357$$ 0 0
$$358$$ 0 0
$$359$$ −188.828 −0.0277604 −0.0138802 0.999904i $$-0.504418\pi$$
−0.0138802 + 0.999904i $$0.504418\pi$$
$$360$$ 0 0
$$361$$ −1912.51 −0.278833
$$362$$ 0 0
$$363$$ 0 0
$$364$$ 0 0
$$365$$ 12535.2 1.79759
$$366$$ 0 0
$$367$$ −6338.15 −0.901495 −0.450747 0.892651i $$-0.648843\pi$$
−0.450747 + 0.892651i $$0.648843\pi$$
$$368$$ 0 0
$$369$$ 0 0
$$370$$ 0 0
$$371$$ 8596.93 1.20305
$$372$$ 0 0
$$373$$ 8641.41 1.19956 0.599779 0.800166i $$-0.295255\pi$$
0.599779 + 0.800166i $$0.295255\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ −1460.62 −0.199538
$$378$$ 0 0
$$379$$ −4143.88 −0.561627 −0.280814 0.959762i $$-0.590604\pi$$
−0.280814 + 0.959762i $$0.590604\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ 0 0
$$383$$ 4193.94 0.559531 0.279766 0.960068i $$-0.409743\pi$$
0.279766 + 0.960068i $$0.409743\pi$$
$$384$$ 0 0
$$385$$ 22563.4 2.98685
$$386$$ 0 0
$$387$$ 0 0
$$388$$ 0 0
$$389$$ −10870.4 −1.41684 −0.708422 0.705789i $$-0.750593\pi$$
−0.708422 + 0.705789i $$0.750593\pi$$
$$390$$ 0 0
$$391$$ 743.102 0.0961133
$$392$$ 0 0
$$393$$ 0 0
$$394$$ 0 0
$$395$$ 5021.24 0.639610
$$396$$ 0 0
$$397$$ −8890.87 −1.12398 −0.561990 0.827144i $$-0.689964\pi$$
−0.561990 + 0.827144i $$0.689964\pi$$
$$398$$ 0 0
$$399$$ 0 0
$$400$$ 0 0
$$401$$ −15095.4 −1.87987 −0.939936 0.341349i $$-0.889116\pi$$
−0.939936 + 0.341349i $$0.889116\pi$$
$$402$$ 0 0
$$403$$ −9214.58 −1.13898
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 0 0
$$407$$ −2440.35 −0.297208
$$408$$ 0 0
$$409$$ 1618.34 0.195652 0.0978260 0.995204i $$-0.468811\pi$$
0.0978260 + 0.995204i $$0.468811\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ 0 0
$$413$$ −23381.3 −2.78576
$$414$$ 0 0
$$415$$ 580.470 0.0686606
$$416$$ 0 0
$$417$$ 0 0
$$418$$ 0 0
$$419$$ −10011.9 −1.16733 −0.583666 0.811994i $$-0.698382\pi$$
−0.583666 + 0.811994i $$0.698382\pi$$
$$420$$ 0 0
$$421$$ 7234.43 0.837493 0.418747 0.908103i $$-0.362470\pi$$
0.418747 + 0.908103i $$0.362470\pi$$
$$422$$ 0 0
$$423$$ 0 0
$$424$$ 0 0
$$425$$ −572.999 −0.0653989
$$426$$ 0 0
$$427$$ −13101.3 −1.48482
$$428$$ 0 0
$$429$$ 0 0
$$430$$ 0 0
$$431$$ −5379.03 −0.601157 −0.300579 0.953757i $$-0.597180\pi$$
−0.300579 + 0.953757i $$0.597180\pi$$
$$432$$ 0 0
$$433$$ −1602.96 −0.177906 −0.0889530 0.996036i $$-0.528352\pi$$
−0.0889530 + 0.996036i $$0.528352\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 0 0
$$437$$ −486.549 −0.0532604
$$438$$ 0 0
$$439$$ 6725.87 0.731226 0.365613 0.930767i $$-0.380859\pi$$
0.365613 + 0.930767i $$0.380859\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 0 0
$$443$$ −1169.92 −0.125473 −0.0627367 0.998030i $$-0.519983\pi$$
−0.0627367 + 0.998030i $$0.519983\pi$$
$$444$$ 0 0
$$445$$ −8216.90 −0.875322
$$446$$ 0 0
$$447$$ 0 0
$$448$$ 0 0
$$449$$ 2996.56 0.314958 0.157479 0.987522i $$-0.449663\pi$$
0.157479 + 0.987522i $$0.449663\pi$$
$$450$$ 0 0
$$451$$ 28442.5 2.96963
$$452$$ 0 0
$$453$$ 0 0
$$454$$ 0 0
$$455$$ −13566.4 −1.39781
$$456$$ 0 0
$$457$$ −11152.1 −1.14151 −0.570757 0.821119i $$-0.693350\pi$$
−0.570757 + 0.821119i $$0.693350\pi$$
$$458$$ 0 0
$$459$$ 0 0
$$460$$ 0 0
$$461$$ 2947.17 0.297752 0.148876 0.988856i $$-0.452435\pi$$
0.148876 + 0.988856i $$0.452435\pi$$
$$462$$ 0 0
$$463$$ 6563.38 0.658804 0.329402 0.944190i $$-0.393153\pi$$
0.329402 + 0.944190i $$0.393153\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ 2727.33 0.270248 0.135124 0.990829i $$-0.456857\pi$$
0.135124 + 0.990829i $$0.456857\pi$$
$$468$$ 0 0
$$469$$ 6513.62 0.641303
$$470$$ 0 0
$$471$$ 0 0
$$472$$ 0 0
$$473$$ 4924.59 0.478717
$$474$$ 0 0
$$475$$ 375.173 0.0362402
$$476$$ 0 0
$$477$$ 0 0
$$478$$ 0 0
$$479$$ −14512.1 −1.38429 −0.692146 0.721758i $$-0.743335\pi$$
−0.692146 + 0.721758i $$0.743335\pi$$
$$480$$ 0 0
$$481$$ 1467.28 0.139090
$$482$$ 0 0
$$483$$ 0 0
$$484$$ 0 0
$$485$$ −13695.6 −1.28224
$$486$$ 0 0
$$487$$ −14189.9 −1.32034 −0.660170 0.751116i $$-0.729516\pi$$
−0.660170 + 0.751116i $$0.729516\pi$$
$$488$$ 0 0
$$489$$ 0 0
$$490$$ 0 0
$$491$$ 7215.91 0.663238 0.331619 0.943413i $$-0.392405\pi$$
0.331619 + 0.943413i $$0.392405\pi$$
$$492$$ 0 0
$$493$$ 3938.82 0.359829
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ 23582.6 2.12842
$$498$$ 0 0
$$499$$ 13032.6 1.16917 0.584587 0.811331i $$-0.301257\pi$$
0.584587 + 0.811331i $$0.301257\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ 0 0
$$503$$ −9502.84 −0.842367 −0.421184 0.906975i $$-0.638385\pi$$
−0.421184 + 0.906975i $$0.638385\pi$$
$$504$$ 0 0
$$505$$ −2808.22 −0.247454
$$506$$ 0 0
$$507$$ 0 0
$$508$$ 0 0
$$509$$ −3794.30 −0.330411 −0.165206 0.986259i $$-0.552829\pi$$
−0.165206 + 0.986259i $$0.552829\pi$$
$$510$$ 0 0
$$511$$ −32756.3 −2.83572
$$512$$ 0 0
$$513$$ 0 0
$$514$$ 0 0
$$515$$ 6573.80 0.562478
$$516$$ 0 0
$$517$$ 3483.62 0.296343
$$518$$ 0 0
$$519$$ 0 0
$$520$$ 0 0
$$521$$ −14811.2 −1.24547 −0.622735 0.782432i $$-0.713979\pi$$
−0.622735 + 0.782432i $$0.713979\pi$$
$$522$$ 0 0
$$523$$ −345.532 −0.0288892 −0.0144446 0.999896i $$-0.504598\pi$$
−0.0144446 + 0.999896i $$0.504598\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ 24848.8 2.05395
$$528$$ 0 0
$$529$$ −12119.1 −0.996067
$$530$$ 0 0
$$531$$ 0 0
$$532$$ 0 0
$$533$$ −17101.2 −1.38975
$$534$$ 0 0
$$535$$ 11597.8 0.937229
$$536$$ 0 0
$$537$$ 0 0
$$538$$ 0 0
$$539$$ −36238.1 −2.89589
$$540$$ 0 0
$$541$$ −5474.45 −0.435056 −0.217528 0.976054i $$-0.569799\pi$$
−0.217528 + 0.976054i $$0.569799\pi$$
$$542$$ 0 0
$$543$$ 0 0
$$544$$ 0 0
$$545$$ 16800.9 1.32049
$$546$$ 0 0
$$547$$ 977.278 0.0763901 0.0381951 0.999270i $$-0.487839\pi$$
0.0381951 + 0.999270i $$0.487839\pi$$
$$548$$ 0 0
$$549$$ 0 0
$$550$$ 0 0
$$551$$ −2578.96 −0.199396
$$552$$ 0 0
$$553$$ −13121.3 −1.00899
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 0 0
$$557$$ 18833.3 1.43266 0.716330 0.697762i $$-0.245821\pi$$
0.716330 + 0.697762i $$0.245821\pi$$
$$558$$ 0 0
$$559$$ −2960.95 −0.224033
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 0 0
$$563$$ 20967.0 1.56954 0.784772 0.619784i $$-0.212780\pi$$
0.784772 + 0.619784i $$0.212780\pi$$
$$564$$ 0 0
$$565$$ 21732.0 1.61818
$$566$$ 0 0
$$567$$ 0 0
$$568$$ 0 0
$$569$$ −13488.7 −0.993804 −0.496902 0.867807i $$-0.665529\pi$$
−0.496902 + 0.867807i $$0.665529\pi$$
$$570$$ 0 0
$$571$$ −2248.90 −0.164822 −0.0824110 0.996598i $$-0.526262\pi$$
−0.0824110 + 0.996598i $$0.526262\pi$$
$$572$$ 0 0
$$573$$ 0 0
$$574$$ 0 0
$$575$$ −36.9030 −0.00267645
$$576$$ 0 0
$$577$$ −1968.87 −0.142054 −0.0710270 0.997474i $$-0.522628\pi$$
−0.0710270 + 0.997474i $$0.522628\pi$$
$$578$$ 0 0
$$579$$ 0 0
$$580$$ 0 0
$$581$$ −1516.86 −0.108313
$$582$$ 0 0
$$583$$ 19091.1 1.35621
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ −4107.66 −0.288827 −0.144413 0.989517i $$-0.546129\pi$$
−0.144413 + 0.989517i $$0.546129\pi$$
$$588$$ 0 0
$$589$$ −16269.8 −1.13818
$$590$$ 0 0
$$591$$ 0 0
$$592$$ 0 0
$$593$$ 7497.21 0.519180 0.259590 0.965719i $$-0.416413\pi$$
0.259590 + 0.965719i $$0.416413\pi$$
$$594$$ 0 0
$$595$$ 36584.3 2.52069
$$596$$ 0 0
$$597$$ 0 0
$$598$$ 0 0
$$599$$ 27107.1 1.84903 0.924513 0.381151i $$-0.124472\pi$$
0.924513 + 0.381151i $$0.124472\pi$$
$$600$$ 0 0
$$601$$ 6802.17 0.461674 0.230837 0.972992i $$-0.425854\pi$$
0.230837 + 0.972992i $$0.425854\pi$$
$$602$$ 0 0
$$603$$ 0 0
$$604$$ 0 0
$$605$$ 34910.9 2.34600
$$606$$ 0 0
$$607$$ 16992.4 1.13625 0.568123 0.822943i $$-0.307670\pi$$
0.568123 + 0.822943i $$0.307670\pi$$
$$608$$ 0 0
$$609$$ 0 0
$$610$$ 0 0
$$611$$ −2094.55 −0.138685
$$612$$ 0 0
$$613$$ −13766.6 −0.907062 −0.453531 0.891241i $$-0.649836\pi$$
−0.453531 + 0.891241i $$0.649836\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ 17356.0 1.13246 0.566229 0.824248i $$-0.308402\pi$$
0.566229 + 0.824248i $$0.308402\pi$$
$$618$$ 0 0
$$619$$ 2924.89 0.189922 0.0949608 0.995481i $$-0.469727\pi$$
0.0949608 + 0.995481i $$0.469727\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ 0 0
$$623$$ 21472.0 1.38083
$$624$$ 0 0
$$625$$ −16263.3 −1.04085
$$626$$ 0 0
$$627$$ 0 0
$$628$$ 0 0
$$629$$ −3956.78 −0.250822
$$630$$ 0 0
$$631$$ 6158.40 0.388530 0.194265 0.980949i $$-0.437768\pi$$
0.194265 + 0.980949i $$0.437768\pi$$
$$632$$ 0 0
$$633$$ 0 0
$$634$$ 0 0
$$635$$ 2629.31 0.164316
$$636$$ 0 0
$$637$$ 21788.4 1.35524
$$638$$ 0 0
$$639$$ 0 0
$$640$$ 0 0
$$641$$ −11814.2 −0.727979 −0.363990 0.931403i $$-0.618586\pi$$
−0.363990 + 0.931403i $$0.618586\pi$$
$$642$$ 0 0
$$643$$ −16914.7 −1.03741 −0.518703 0.854954i $$-0.673585\pi$$
−0.518703 + 0.854954i $$0.673585\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ 28652.4 1.74103 0.870513 0.492145i $$-0.163787\pi$$
0.870513 + 0.492145i $$0.163787\pi$$
$$648$$ 0 0
$$649$$ −51922.5 −3.14042
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 0 0
$$653$$ 5498.48 0.329513 0.164757 0.986334i $$-0.447316\pi$$
0.164757 + 0.986334i $$0.447316\pi$$
$$654$$ 0 0
$$655$$ −9068.16 −0.540950
$$656$$ 0 0
$$657$$ 0 0
$$658$$ 0 0
$$659$$ 18785.1 1.11042 0.555209 0.831711i $$-0.312638\pi$$
0.555209 + 0.831711i $$0.312638\pi$$
$$660$$ 0 0
$$661$$ −17304.5 −1.01825 −0.509127 0.860691i $$-0.670032\pi$$
−0.509127 + 0.860691i $$0.670032\pi$$
$$662$$ 0 0
$$663$$ 0 0
$$664$$ 0 0
$$665$$ −23953.7 −1.39682
$$666$$ 0 0
$$667$$ 253.673 0.0147260
$$668$$ 0 0
$$669$$ 0 0
$$670$$ 0 0
$$671$$ −29093.9 −1.67385
$$672$$ 0 0
$$673$$ −13711.5 −0.785346 −0.392673 0.919678i $$-0.628450\pi$$
−0.392673 + 0.919678i $$0.628450\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ 0 0
$$677$$ 30566.9 1.73527 0.867637 0.497198i $$-0.165638\pi$$
0.867637 + 0.497198i $$0.165638\pi$$
$$678$$ 0 0
$$679$$ 35788.8 2.02275
$$680$$ 0 0
$$681$$ 0 0
$$682$$ 0 0
$$683$$ −4445.72 −0.249064 −0.124532 0.992216i $$-0.539743\pi$$
−0.124532 + 0.992216i $$0.539743\pi$$
$$684$$ 0 0
$$685$$ 7798.27 0.434973
$$686$$ 0 0
$$687$$ 0 0
$$688$$ 0 0
$$689$$ −11478.6 −0.634690
$$690$$ 0 0
$$691$$ −12198.7 −0.671580 −0.335790 0.941937i $$-0.609003\pi$$
−0.335790 + 0.941937i $$0.609003\pi$$
$$692$$ 0 0
$$693$$ 0 0
$$694$$ 0 0
$$695$$ −17268.9 −0.942515
$$696$$ 0 0
$$697$$ 46116.6 2.50615
$$698$$ 0 0
$$699$$ 0 0
$$700$$ 0 0
$$701$$ −21613.4 −1.16452 −0.582258 0.813004i $$-0.697831\pi$$
−0.582258 + 0.813004i $$0.697831\pi$$
$$702$$ 0 0
$$703$$ 2590.72 0.138991
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ 7338.32 0.390362
$$708$$ 0 0
$$709$$ −4924.85 −0.260870 −0.130435 0.991457i $$-0.541637\pi$$
−0.130435 + 0.991457i $$0.541637\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ 0 0
$$713$$ 1600.34 0.0840578
$$714$$ 0 0
$$715$$ −30126.7 −1.57577
$$716$$ 0 0
$$717$$ 0 0
$$718$$ 0 0
$$719$$ 205.354 0.0106515 0.00532573 0.999986i $$-0.498305\pi$$
0.00532573 + 0.999986i $$0.498305\pi$$
$$720$$ 0 0
$$721$$ −17178.3 −0.887316
$$722$$ 0 0
$$723$$ 0 0
$$724$$ 0 0
$$725$$ −195.605 −0.0100201
$$726$$ 0 0
$$727$$ 15796.0 0.805836 0.402918 0.915236i $$-0.367996\pi$$
0.402918 + 0.915236i $$0.367996\pi$$
$$728$$ 0 0
$$729$$ 0 0
$$730$$ 0 0
$$731$$ 7984.73 0.404003
$$732$$ 0 0
$$733$$ 32125.3 1.61880 0.809398 0.587261i $$-0.199794\pi$$
0.809398 + 0.587261i $$0.199794\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ 14464.7 0.722949
$$738$$ 0 0
$$739$$ 28938.9 1.44051 0.720255 0.693710i $$-0.244025\pi$$
0.720255 + 0.693710i $$0.244025\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 0 0
$$743$$ −36108.5 −1.78289 −0.891447 0.453124i $$-0.850309\pi$$
−0.891447 + 0.453124i $$0.850309\pi$$
$$744$$ 0 0
$$745$$ −26339.2 −1.29529
$$746$$ 0 0
$$747$$ 0 0
$$748$$ 0 0
$$749$$ −30306.9 −1.47849
$$750$$ 0 0
$$751$$ −23231.7 −1.12881 −0.564405 0.825498i $$-0.690894\pi$$
−0.564405 + 0.825498i $$0.690894\pi$$
$$752$$ 0 0
$$753$$ 0 0
$$754$$ 0 0
$$755$$ 28805.5 1.38853
$$756$$ 0 0
$$757$$ 22762.6 1.09289 0.546447 0.837494i $$-0.315980\pi$$
0.546447 + 0.837494i $$0.315980\pi$$
$$758$$ 0 0
$$759$$ 0 0
$$760$$ 0 0
$$761$$ 40518.6 1.93009 0.965044 0.262088i $$-0.0844112\pi$$
0.965044 + 0.262088i $$0.0844112\pi$$
$$762$$ 0 0
$$763$$ −43903.2 −2.08310
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ 31218.8 1.46968
$$768$$ 0 0
$$769$$ −27401.1 −1.28493 −0.642464 0.766316i $$-0.722088\pi$$
−0.642464 + 0.766316i $$0.722088\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ 0 0
$$773$$ 35698.4 1.66104 0.830520 0.556989i $$-0.188043\pi$$
0.830520 + 0.556989i $$0.188043\pi$$
$$774$$ 0 0
$$775$$ −1234.01 −0.0571959
$$776$$ 0 0
$$777$$ 0 0
$$778$$ 0 0
$$779$$ −30195.0 −1.38876
$$780$$ 0 0
$$781$$ 52369.5 2.39940
$$782$$ 0 0
$$783$$ 0 0
$$784$$ 0 0
$$785$$ −33324.3 −1.51515
$$786$$ 0 0
$$787$$ −9566.36 −0.433296 −0.216648 0.976250i $$-0.569512\pi$$
−0.216648 + 0.976250i $$0.569512\pi$$
$$788$$ 0 0
$$789$$ 0 0
$$790$$ 0 0
$$791$$ −56789.0 −2.55270
$$792$$ 0 0
$$793$$ 17492.9 0.783343
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 0 0
$$797$$ 5306.01 0.235820 0.117910 0.993024i $$-0.462381\pi$$
0.117910 + 0.993024i $$0.462381\pi$$
$$798$$ 0 0
$$799$$ 5648.34 0.250093
$$800$$ 0 0
$$801$$ 0 0
$$802$$ 0 0
$$803$$ −72741.4 −3.19675
$$804$$ 0 0
$$805$$ 2356.14 0.103159
$$806$$ 0 0
$$807$$ 0 0
$$808$$ 0 0
$$809$$ −9362.75 −0.406893 −0.203447 0.979086i $$-0.565214\pi$$
−0.203447 + 0.979086i $$0.565214\pi$$
$$810$$ 0 0
$$811$$ 32610.2 1.41196 0.705981 0.708231i $$-0.250507\pi$$
0.705981 + 0.708231i $$0.250507\pi$$
$$812$$ 0 0
$$813$$ 0 0
$$814$$ 0 0
$$815$$ 12523.7 0.538263
$$816$$ 0 0
$$817$$ −5228.03 −0.223875
$$818$$ 0 0
$$819$$ 0 0
$$820$$ 0 0
$$821$$ −18930.7 −0.804732 −0.402366 0.915479i $$-0.631812\pi$$
−0.402366 + 0.915479i $$0.631812\pi$$
$$822$$ 0 0
$$823$$ 40673.1 1.72269 0.861346 0.508018i $$-0.169622\pi$$
0.861346 + 0.508018i $$0.169622\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ 30770.2 1.29382 0.646908 0.762568i $$-0.276062\pi$$
0.646908 + 0.762568i $$0.276062\pi$$
$$828$$ 0 0
$$829$$ 10464.9 0.438434 0.219217 0.975676i $$-0.429650\pi$$
0.219217 + 0.975676i $$0.429650\pi$$
$$830$$ 0 0
$$831$$ 0 0
$$832$$ 0 0
$$833$$ −58756.4 −2.44393
$$834$$ 0 0
$$835$$ 8365.13 0.346691
$$836$$ 0 0
$$837$$ 0 0
$$838$$ 0 0
$$839$$ −22716.4 −0.934752 −0.467376 0.884059i $$-0.654800\pi$$
−0.467376 + 0.884059i $$0.654800\pi$$
$$840$$ 0 0
$$841$$ −23044.4 −0.944869
$$842$$ 0 0
$$843$$ 0 0
$$844$$ 0 0
$$845$$ −6967.97 −0.283675
$$846$$ 0 0
$$847$$ −91227.5 −3.70084
$$848$$ 0 0
$$849$$ 0 0
$$850$$ 0 0
$$851$$ −254.829 −0.0102649
$$852$$ 0 0
$$853$$ −10181.2 −0.408671 −0.204335 0.978901i $$-0.565503\pi$$
−0.204335 + 0.978901i $$0.565503\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0 0
$$857$$ −3885.59 −0.154877 −0.0774384 0.996997i $$-0.524674\pi$$
−0.0774384 + 0.996997i $$0.524674\pi$$
$$858$$ 0 0
$$859$$ −17734.9 −0.704433 −0.352217 0.935919i $$-0.614572\pi$$
−0.352217 + 0.935919i $$0.614572\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 0 0
$$863$$ −19658.7 −0.775421 −0.387711 0.921781i $$-0.626734\pi$$
−0.387711 + 0.921781i $$0.626734\pi$$
$$864$$ 0 0
$$865$$ −8250.35 −0.324301
$$866$$ 0 0
$$867$$ 0 0
$$868$$ 0 0
$$869$$ −29138.2 −1.13745
$$870$$ 0 0
$$871$$ −8697.00 −0.338331
$$872$$ 0 0
$$873$$ 0 0
$$874$$ 0 0
$$875$$ 40756.2 1.57464
$$876$$ 0 0
$$877$$ 5586.82 0.215112 0.107556 0.994199i $$-0.465697\pi$$
0.107556 + 0.994199i $$0.465697\pi$$
$$878$$ 0 0
$$879$$ 0 0
$$880$$ 0 0
$$881$$ 8050.03 0.307846 0.153923 0.988083i $$-0.450809\pi$$
0.153923 + 0.988083i $$0.450809\pi$$
$$882$$ 0 0
$$883$$ −2326.88 −0.0886815 −0.0443408 0.999016i $$-0.514119\pi$$
−0.0443408 + 0.999016i $$0.514119\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 0 0
$$887$$ 19343.5 0.732233 0.366116 0.930569i $$-0.380687\pi$$
0.366116 + 0.930569i $$0.380687\pi$$
$$888$$ 0 0
$$889$$ −6870.78 −0.259211
$$890$$ 0 0
$$891$$ 0 0
$$892$$ 0 0
$$893$$ −3698.27 −0.138587
$$894$$ 0 0
$$895$$ −11532.4 −0.430712
$$896$$ 0 0
$$897$$ 0 0
$$898$$ 0 0
$$899$$ 8482.63 0.314696
$$900$$ 0 0
$$901$$ 30954.2 1.14454
$$902$$ 0 0
$$903$$ 0 0
$$904$$ 0 0
$$905$$ −48597.2 −1.78500
$$906$$ 0 0
$$907$$ −299.317 −0.0109577 −0.00547886 0.999985i $$-0.501744\pi$$
−0.00547886 + 0.999985i $$0.501744\pi$$
$$908$$ 0 0
$$909$$ 0 0
$$910$$ 0 0
$$911$$ 21022.5 0.764551 0.382275 0.924048i $$-0.375141\pi$$
0.382275 + 0.924048i $$0.375141\pi$$
$$912$$ 0 0
$$913$$ −3368.46 −0.122103
$$914$$ 0 0
$$915$$ 0 0
$$916$$ 0 0
$$917$$ 23696.5 0.853356
$$918$$ 0 0
$$919$$ −18375.8 −0.659587 −0.329794 0.944053i $$-0.606979\pi$$
−0.329794 + 0.944053i $$0.606979\pi$$
$$920$$ 0 0
$$921$$ 0 0
$$922$$ 0 0
$$923$$ −31487.5 −1.12289
$$924$$ 0 0
$$925$$ 196.496 0.00698461
$$926$$ 0 0
$$927$$ 0 0
$$928$$ 0 0
$$929$$ 29548.0 1.04353 0.521764 0.853090i $$-0.325274\pi$$
0.521764 + 0.853090i $$0.325274\pi$$
$$930$$ 0 0
$$931$$ 38471.0 1.35428
$$932$$ 0 0
$$933$$ 0 0
$$934$$ 0 0
$$935$$ 81242.1 2.84160
$$936$$ 0 0
$$937$$ 36751.2 1.28133 0.640667 0.767819i $$-0.278658\pi$$
0.640667 + 0.767819i $$0.278658\pi$$
$$938$$ 0 0
$$939$$ 0 0
$$940$$ 0 0
$$941$$ −26024.7 −0.901575 −0.450788 0.892631i $$-0.648857\pi$$
−0.450788 + 0.892631i $$0.648857\pi$$
$$942$$ 0 0
$$943$$ 2970.05 0.102564
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ −40238.6 −1.38076 −0.690379 0.723448i $$-0.742556\pi$$
−0.690379 + 0.723448i $$0.742556\pi$$
$$948$$ 0 0
$$949$$ 43736.3 1.49604
$$950$$ 0 0
$$951$$ 0 0
$$952$$ 0 0
$$953$$ 52396.4 1.78099 0.890496 0.454991i $$-0.150357\pi$$
0.890496 + 0.454991i $$0.150357\pi$$
$$954$$ 0 0
$$955$$ −39151.3 −1.32660
$$956$$ 0 0
$$957$$ 0 0
$$958$$ 0 0
$$959$$ −20378.1 −0.686176
$$960$$ 0 0
$$961$$ 23723.2 0.796319
$$962$$ 0 0
$$963$$ 0 0
$$964$$ 0 0
$$965$$ 11043.6 0.368399
$$966$$ 0 0
$$967$$ −9007.03 −0.299531 −0.149765 0.988722i $$-0.547852\pi$$
−0.149765 + 0.988722i $$0.547852\pi$$
$$968$$ 0 0
$$969$$ 0 0
$$970$$ 0 0
$$971$$ 1110.63 0.0367062 0.0183531 0.999832i $$-0.494158\pi$$
0.0183531 + 0.999832i $$0.494158\pi$$
$$972$$ 0 0
$$973$$ 45126.3 1.48683
$$974$$ 0 0
$$975$$ 0 0
$$976$$ 0 0
$$977$$ 27544.4 0.901969 0.450984 0.892532i $$-0.351073\pi$$
0.450984 + 0.892532i $$0.351073\pi$$
$$978$$ 0 0
$$979$$ 47682.5 1.55663
$$980$$ 0 0
$$981$$ 0 0
$$982$$ 0 0
$$983$$ 8657.48 0.280906 0.140453 0.990087i $$-0.455144\pi$$
0.140453 + 0.990087i $$0.455144\pi$$
$$984$$ 0 0
$$985$$ 5347.60 0.172983
$$986$$ 0 0
$$987$$ 0 0
$$988$$ 0 0
$$989$$ 514.242 0.0165338
$$990$$ 0 0
$$991$$ −54045.9 −1.73242 −0.866208 0.499683i $$-0.833450\pi$$
−0.866208 + 0.499683i $$0.833450\pi$$
$$992$$ 0 0
$$993$$ 0 0
$$994$$ 0 0
$$995$$ −30521.4 −0.972456
$$996$$ 0 0
$$997$$ −32591.1 −1.03528 −0.517638 0.855600i $$-0.673189\pi$$
−0.517638 + 0.855600i $$0.673189\pi$$
$$998$$ 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.1.2 2
3.2 odd 2 216.4.a.g.1.1 yes 2
4.3 odd 2 432.4.a.p.1.2 2
8.3 odd 2 1728.4.a.br.1.1 2
8.5 even 2 1728.4.a.bq.1.1 2
9.2 odd 6 648.4.i.o.433.2 4
9.4 even 3 648.4.i.r.217.1 4
9.5 odd 6 648.4.i.o.217.2 4
9.7 even 3 648.4.i.r.433.1 4
12.11 even 2 432.4.a.r.1.1 2
24.5 odd 2 1728.4.a.bi.1.2 2
24.11 even 2 1728.4.a.bj.1.2 2

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