# Properties

 Label 4864.2.a.bk.1.1 Level $4864$ Weight $2$ Character 4864.1 Self dual yes Analytic conductor $38.839$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$4864 = 2^{8} \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 4864.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$38.8392355432$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\sqrt{3}, \sqrt{19})$$ Defining polynomial: $$x^{4} - 11 x^{2} + 16$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 1216) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Root $$-3.04547$$ of defining polynomial Character $$\chi$$ $$=$$ 4864.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-3.04547 q^{5} -0.418627 q^{7} -3.00000 q^{9} +O(q^{10})$$ $$q-3.04547 q^{5} -0.418627 q^{7} -3.00000 q^{9} +5.27492 q^{11} -2.62685 q^{13} +3.27492 q^{17} -1.00000 q^{19} -3.46410 q^{23} +4.27492 q^{25} -2.62685 q^{29} +6.09095 q^{31} +1.27492 q^{35} -9.55505 q^{37} -4.54983 q^{41} +2.72508 q^{43} +9.13642 q^{45} -0.418627 q^{47} -6.82475 q^{49} -10.3923 q^{53} -16.0646 q^{55} -6.54983 q^{59} +3.04547 q^{61} +1.25588 q^{63} +8.00000 q^{65} +6.54983 q^{67} +11.3446 q^{71} +3.27492 q^{73} -2.20822 q^{77} +13.0192 q^{79} +9.00000 q^{81} -17.0997 q^{83} -9.97368 q^{85} +10.0000 q^{89} +1.09967 q^{91} +3.04547 q^{95} +16.5498 q^{97} -15.8248 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 12q^{9} + O(q^{10})$$ $$4q - 12q^{9} + 6q^{11} - 2q^{17} - 4q^{19} + 2q^{25} - 10q^{35} + 12q^{41} + 26q^{43} + 18q^{49} + 4q^{59} + 32q^{65} - 4q^{67} - 2q^{73} + 36q^{81} - 8q^{83} + 40q^{89} - 56q^{91} + 36q^{97} - 18q^{99} + O(q^{100})$$

## 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 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$4$$ 0 0
$$5$$ −3.04547 −1.36198 −0.680989 0.732294i $$-0.738450\pi$$
−0.680989 + 0.732294i $$0.738450\pi$$
$$6$$ 0 0
$$7$$ −0.418627 −0.158226 −0.0791130 0.996866i $$-0.525209\pi$$
−0.0791130 + 0.996866i $$0.525209\pi$$
$$8$$ 0 0
$$9$$ −3.00000 −1.00000
$$10$$ 0 0
$$11$$ 5.27492 1.59045 0.795224 0.606316i $$-0.207353\pi$$
0.795224 + 0.606316i $$0.207353\pi$$
$$12$$ 0 0
$$13$$ −2.62685 −0.728557 −0.364278 0.931290i $$-0.618684\pi$$
−0.364278 + 0.931290i $$0.618684\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 0 0
$$17$$ 3.27492 0.794284 0.397142 0.917757i $$-0.370002\pi$$
0.397142 + 0.917757i $$0.370002\pi$$
$$18$$ 0 0
$$19$$ −1.00000 −0.229416
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 0 0
$$23$$ −3.46410 −0.722315 −0.361158 0.932505i $$-0.617618\pi$$
−0.361158 + 0.932505i $$0.617618\pi$$
$$24$$ 0 0
$$25$$ 4.27492 0.854983
$$26$$ 0 0
$$27$$ 0 0
$$28$$ 0 0
$$29$$ −2.62685 −0.487793 −0.243897 0.969801i $$-0.578426\pi$$
−0.243897 + 0.969801i $$0.578426\pi$$
$$30$$ 0 0
$$31$$ 6.09095 1.09397 0.546983 0.837143i $$-0.315776\pi$$
0.546983 + 0.837143i $$0.315776\pi$$
$$32$$ 0 0
$$33$$ 0 0
$$34$$ 0 0
$$35$$ 1.27492 0.215500
$$36$$ 0 0
$$37$$ −9.55505 −1.57084 −0.785420 0.618963i $$-0.787553\pi$$
−0.785420 + 0.618963i $$0.787553\pi$$
$$38$$ 0 0
$$39$$ 0 0
$$40$$ 0 0
$$41$$ −4.54983 −0.710565 −0.355282 0.934759i $$-0.615615\pi$$
−0.355282 + 0.934759i $$0.615615\pi$$
$$42$$ 0 0
$$43$$ 2.72508 0.415571 0.207786 0.978174i $$-0.433374\pi$$
0.207786 + 0.978174i $$0.433374\pi$$
$$44$$ 0 0
$$45$$ 9.13642 1.36198
$$46$$ 0 0
$$47$$ −0.418627 −0.0610630 −0.0305315 0.999534i $$-0.509720\pi$$
−0.0305315 + 0.999534i $$0.509720\pi$$
$$48$$ 0 0
$$49$$ −6.82475 −0.974965
$$50$$ 0 0
$$51$$ 0 0
$$52$$ 0 0
$$53$$ −10.3923 −1.42749 −0.713746 0.700404i $$-0.753003\pi$$
−0.713746 + 0.700404i $$0.753003\pi$$
$$54$$ 0 0
$$55$$ −16.0646 −2.16615
$$56$$ 0 0
$$57$$ 0 0
$$58$$ 0 0
$$59$$ −6.54983 −0.852716 −0.426358 0.904555i $$-0.640204\pi$$
−0.426358 + 0.904555i $$0.640204\pi$$
$$60$$ 0 0
$$61$$ 3.04547 0.389933 0.194967 0.980810i $$-0.437540\pi$$
0.194967 + 0.980810i $$0.437540\pi$$
$$62$$ 0 0
$$63$$ 1.25588 0.158226
$$64$$ 0 0
$$65$$ 8.00000 0.992278
$$66$$ 0 0
$$67$$ 6.54983 0.800190 0.400095 0.916474i $$-0.368977\pi$$
0.400095 + 0.916474i $$0.368977\pi$$
$$68$$ 0 0
$$69$$ 0 0
$$70$$ 0 0
$$71$$ 11.3446 1.34636 0.673181 0.739478i $$-0.264928\pi$$
0.673181 + 0.739478i $$0.264928\pi$$
$$72$$ 0 0
$$73$$ 3.27492 0.383300 0.191650 0.981463i $$-0.438616\pi$$
0.191650 + 0.981463i $$0.438616\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 0 0
$$77$$ −2.20822 −0.251650
$$78$$ 0 0
$$79$$ 13.0192 1.46477 0.732385 0.680891i $$-0.238407\pi$$
0.732385 + 0.680891i $$0.238407\pi$$
$$80$$ 0 0
$$81$$ 9.00000 1.00000
$$82$$ 0 0
$$83$$ −17.0997 −1.87693 −0.938466 0.345371i $$-0.887753\pi$$
−0.938466 + 0.345371i $$0.887753\pi$$
$$84$$ 0 0
$$85$$ −9.97368 −1.08180
$$86$$ 0 0
$$87$$ 0 0
$$88$$ 0 0
$$89$$ 10.0000 1.06000 0.529999 0.847998i $$-0.322192\pi$$
0.529999 + 0.847998i $$0.322192\pi$$
$$90$$ 0 0
$$91$$ 1.09967 0.115277
$$92$$ 0 0
$$93$$ 0 0
$$94$$ 0 0
$$95$$ 3.04547 0.312459
$$96$$ 0 0
$$97$$ 16.5498 1.68038 0.840191 0.542291i $$-0.182443\pi$$
0.840191 + 0.542291i $$0.182443\pi$$
$$98$$ 0 0
$$99$$ −15.8248 −1.59045
$$100$$ 0 0
$$101$$ 13.8564 1.37876 0.689382 0.724398i $$-0.257882\pi$$
0.689382 + 0.724398i $$0.257882\pi$$
$$102$$ 0 0
$$103$$ −6.09095 −0.600159 −0.300080 0.953914i $$-0.597013\pi$$
−0.300080 + 0.953914i $$0.597013\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ 1.45017 0.140193 0.0700964 0.997540i $$-0.477669\pi$$
0.0700964 + 0.997540i $$0.477669\pi$$
$$108$$ 0 0
$$109$$ −3.46410 −0.331801 −0.165900 0.986143i $$-0.553053\pi$$
−0.165900 + 0.986143i $$0.553053\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ 0 0
$$113$$ 12.5498 1.18059 0.590295 0.807188i $$-0.299012\pi$$
0.590295 + 0.807188i $$0.299012\pi$$
$$114$$ 0 0
$$115$$ 10.5498 0.983777
$$116$$ 0 0
$$117$$ 7.88054 0.728557
$$118$$ 0 0
$$119$$ −1.37097 −0.125676
$$120$$ 0 0
$$121$$ 16.8248 1.52952
$$122$$ 0 0
$$123$$ 0 0
$$124$$ 0 0
$$125$$ 2.20822 0.197509
$$126$$ 0 0
$$127$$ −17.4356 −1.54716 −0.773579 0.633699i $$-0.781536\pi$$
−0.773579 + 0.633699i $$0.781536\pi$$
$$128$$ 0 0
$$129$$ 0 0
$$130$$ 0 0
$$131$$ −5.27492 −0.460872 −0.230436 0.973088i $$-0.574015\pi$$
−0.230436 + 0.973088i $$0.574015\pi$$
$$132$$ 0 0
$$133$$ 0.418627 0.0362995
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ 12.7251 1.08718 0.543589 0.839352i $$-0.317065\pi$$
0.543589 + 0.839352i $$0.317065\pi$$
$$138$$ 0 0
$$139$$ 2.72508 0.231139 0.115569 0.993299i $$-0.463131\pi$$
0.115569 + 0.993299i $$0.463131\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 0 0
$$143$$ −13.8564 −1.15873
$$144$$ 0 0
$$145$$ 8.00000 0.664364
$$146$$ 0 0
$$147$$ 0 0
$$148$$ 0 0
$$149$$ 16.0646 1.31607 0.658033 0.752989i $$-0.271389\pi$$
0.658033 + 0.752989i $$0.271389\pi$$
$$150$$ 0 0
$$151$$ 0.837253 0.0681347 0.0340674 0.999420i $$-0.489154\pi$$
0.0340674 + 0.999420i $$0.489154\pi$$
$$152$$ 0 0
$$153$$ −9.82475 −0.794284
$$154$$ 0 0
$$155$$ −18.5498 −1.48996
$$156$$ 0 0
$$157$$ 12.1819 0.972221 0.486111 0.873897i $$-0.338415\pi$$
0.486111 + 0.873897i $$0.338415\pi$$
$$158$$ 0 0
$$159$$ 0 0
$$160$$ 0 0
$$161$$ 1.45017 0.114289
$$162$$ 0 0
$$163$$ −4.00000 −0.313304 −0.156652 0.987654i $$-0.550070\pi$$
−0.156652 + 0.987654i $$0.550070\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ 19.1101 1.47878 0.739392 0.673275i $$-0.235113\pi$$
0.739392 + 0.673275i $$0.235113\pi$$
$$168$$ 0 0
$$169$$ −6.09967 −0.469205
$$170$$ 0 0
$$171$$ 3.00000 0.229416
$$172$$ 0 0
$$173$$ −8.71780 −0.662802 −0.331401 0.943490i $$-0.607521\pi$$
−0.331401 + 0.943490i $$0.607521\pi$$
$$174$$ 0 0
$$175$$ −1.78959 −0.135281
$$176$$ 0 0
$$177$$ 0 0
$$178$$ 0 0
$$179$$ 21.0997 1.57706 0.788532 0.614994i $$-0.210842\pi$$
0.788532 + 0.614994i $$0.210842\pi$$
$$180$$ 0 0
$$181$$ −3.46410 −0.257485 −0.128742 0.991678i $$-0.541094\pi$$
−0.128742 + 0.991678i $$0.541094\pi$$
$$182$$ 0 0
$$183$$ 0 0
$$184$$ 0 0
$$185$$ 29.0997 2.13945
$$186$$ 0 0
$$187$$ 17.2749 1.26327
$$188$$ 0 0
$$189$$ 0 0
$$190$$ 0 0
$$191$$ 11.7633 0.851161 0.425580 0.904921i $$-0.360070\pi$$
0.425580 + 0.904921i $$0.360070\pi$$
$$192$$ 0 0
$$193$$ 11.0997 0.798972 0.399486 0.916739i $$-0.369189\pi$$
0.399486 + 0.916739i $$0.369189\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ −5.25370 −0.374310 −0.187155 0.982330i $$-0.559927\pi$$
−0.187155 + 0.982330i $$0.559927\pi$$
$$198$$ 0 0
$$199$$ 14.2750 1.01193 0.505965 0.862554i $$-0.331136\pi$$
0.505965 + 0.862554i $$0.331136\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ 0 0
$$203$$ 1.09967 0.0771816
$$204$$ 0 0
$$205$$ 13.8564 0.967773
$$206$$ 0 0
$$207$$ 10.3923 0.722315
$$208$$ 0 0
$$209$$ −5.27492 −0.364874
$$210$$ 0 0
$$211$$ 26.5498 1.82777 0.913883 0.405978i $$-0.133069\pi$$
0.913883 + 0.405978i $$0.133069\pi$$
$$212$$ 0 0
$$213$$ 0 0
$$214$$ 0 0
$$215$$ −8.29917 −0.565999
$$216$$ 0 0
$$217$$ −2.54983 −0.173094
$$218$$ 0 0
$$219$$ 0 0
$$220$$ 0 0
$$221$$ −8.60271 −0.578681
$$222$$ 0 0
$$223$$ 17.4356 1.16757 0.583787 0.811907i $$-0.301570\pi$$
0.583787 + 0.811907i $$0.301570\pi$$
$$224$$ 0 0
$$225$$ −12.8248 −0.854983
$$226$$ 0 0
$$227$$ −4.00000 −0.265489 −0.132745 0.991150i $$-0.542379\pi$$
−0.132745 + 0.991150i $$0.542379\pi$$
$$228$$ 0 0
$$229$$ 17.7391 1.17224 0.586118 0.810226i $$-0.300656\pi$$
0.586118 + 0.810226i $$0.300656\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ 20.3746 1.33478 0.667392 0.744707i $$-0.267411\pi$$
0.667392 + 0.744707i $$0.267411\pi$$
$$234$$ 0 0
$$235$$ 1.27492 0.0831664
$$236$$ 0 0
$$237$$ 0 0
$$238$$ 0 0
$$239$$ −25.6197 −1.65720 −0.828600 0.559842i $$-0.810862\pi$$
−0.828600 + 0.559842i $$0.810862\pi$$
$$240$$ 0 0
$$241$$ −12.5498 −0.808406 −0.404203 0.914669i $$-0.632451\pi$$
−0.404203 + 0.914669i $$0.632451\pi$$
$$242$$ 0 0
$$243$$ 0 0
$$244$$ 0 0
$$245$$ 20.7846 1.32788
$$246$$ 0 0
$$247$$ 2.62685 0.167142
$$248$$ 0 0
$$249$$ 0 0
$$250$$ 0 0
$$251$$ −10.7251 −0.676961 −0.338481 0.940973i $$-0.609913\pi$$
−0.338481 + 0.940973i $$0.609913\pi$$
$$252$$ 0 0
$$253$$ −18.2728 −1.14880
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 0 0
$$257$$ 7.45017 0.464729 0.232364 0.972629i $$-0.425354\pi$$
0.232364 + 0.972629i $$0.425354\pi$$
$$258$$ 0 0
$$259$$ 4.00000 0.248548
$$260$$ 0 0
$$261$$ 7.88054 0.487793
$$262$$ 0 0
$$263$$ −1.25588 −0.0774409 −0.0387204 0.999250i $$-0.512328\pi$$
−0.0387204 + 0.999250i $$0.512328\pi$$
$$264$$ 0 0
$$265$$ 31.6495 1.94421
$$266$$ 0 0
$$267$$ 0 0
$$268$$ 0 0
$$269$$ −5.13861 −0.313306 −0.156653 0.987654i $$-0.550071\pi$$
−0.156653 + 0.987654i $$0.550071\pi$$
$$270$$ 0 0
$$271$$ −27.8279 −1.69042 −0.845212 0.534431i $$-0.820526\pi$$
−0.845212 + 0.534431i $$0.820526\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ 0 0
$$275$$ 22.5498 1.35981
$$276$$ 0 0
$$277$$ −16.0646 −0.965230 −0.482615 0.875833i $$-0.660313\pi$$
−0.482615 + 0.875833i $$0.660313\pi$$
$$278$$ 0 0
$$279$$ −18.2728 −1.09397
$$280$$ 0 0
$$281$$ −11.0997 −0.662151 −0.331075 0.943604i $$-0.607411\pi$$
−0.331075 + 0.943604i $$0.607411\pi$$
$$282$$ 0 0
$$283$$ −26.3746 −1.56781 −0.783903 0.620883i $$-0.786774\pi$$
−0.783903 + 0.620883i $$0.786774\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ 1.90468 0.112430
$$288$$ 0 0
$$289$$ −6.27492 −0.369113
$$290$$ 0 0
$$291$$ 0 0
$$292$$ 0 0
$$293$$ 30.3397 1.77246 0.886231 0.463244i $$-0.153315\pi$$
0.886231 + 0.463244i $$0.153315\pi$$
$$294$$ 0 0
$$295$$ 19.9474 1.16138
$$296$$ 0 0
$$297$$ 0 0
$$298$$ 0 0
$$299$$ 9.09967 0.526247
$$300$$ 0 0
$$301$$ −1.14079 −0.0657542
$$302$$ 0 0
$$303$$ 0 0
$$304$$ 0 0
$$305$$ −9.27492 −0.531080
$$306$$ 0 0
$$307$$ 1.45017 0.0827653 0.0413827 0.999143i $$-0.486824\pi$$
0.0413827 + 0.999143i $$0.486824\pi$$
$$308$$ 0 0
$$309$$ 0 0
$$310$$ 0 0
$$311$$ 24.7824 1.40528 0.702641 0.711544i $$-0.252004\pi$$
0.702641 + 0.711544i $$0.252004\pi$$
$$312$$ 0 0
$$313$$ −15.0997 −0.853484 −0.426742 0.904373i $$-0.640339\pi$$
−0.426742 + 0.904373i $$0.640339\pi$$
$$314$$ 0 0
$$315$$ −3.82475 −0.215500
$$316$$ 0 0
$$317$$ 8.71780 0.489640 0.244820 0.969569i $$-0.421271\pi$$
0.244820 + 0.969569i $$0.421271\pi$$
$$318$$ 0 0
$$319$$ −13.8564 −0.775810
$$320$$ 0 0
$$321$$ 0 0
$$322$$ 0 0
$$323$$ −3.27492 −0.182221
$$324$$ 0 0
$$325$$ −11.2296 −0.622904
$$326$$ 0 0
$$327$$ 0 0
$$328$$ 0 0
$$329$$ 0.175248 0.00966175
$$330$$ 0 0
$$331$$ 9.45017 0.519428 0.259714 0.965686i $$-0.416372\pi$$
0.259714 + 0.965686i $$0.416372\pi$$
$$332$$ 0 0
$$333$$ 28.6652 1.57084
$$334$$ 0 0
$$335$$ −19.9474 −1.08984
$$336$$ 0 0
$$337$$ 33.6495 1.83301 0.916503 0.400029i $$-0.131000\pi$$
0.916503 + 0.400029i $$0.131000\pi$$
$$338$$ 0 0
$$339$$ 0 0
$$340$$ 0 0
$$341$$ 32.1293 1.73990
$$342$$ 0 0
$$343$$ 5.78741 0.312491
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ 7.82475 0.420055 0.210027 0.977696i $$-0.432645\pi$$
0.210027 + 0.977696i $$0.432645\pi$$
$$348$$ 0 0
$$349$$ −12.7156 −0.680651 −0.340326 0.940308i $$-0.610537\pi$$
−0.340326 + 0.940308i $$0.610537\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 0 0
$$353$$ −10.0000 −0.532246 −0.266123 0.963939i $$-0.585743\pi$$
−0.266123 + 0.963939i $$0.585743\pi$$
$$354$$ 0 0
$$355$$ −34.5498 −1.83371
$$356$$ 0 0
$$357$$ 0 0
$$358$$ 0 0
$$359$$ 12.6005 0.665030 0.332515 0.943098i $$-0.392103\pi$$
0.332515 + 0.943098i $$0.392103\pi$$
$$360$$ 0 0
$$361$$ 1.00000 0.0526316
$$362$$ 0 0
$$363$$ 0 0
$$364$$ 0 0
$$365$$ −9.97368 −0.522046
$$366$$ 0 0
$$367$$ 15.6460 0.816715 0.408357 0.912822i $$-0.366102\pi$$
0.408357 + 0.912822i $$0.366102\pi$$
$$368$$ 0 0
$$369$$ 13.6495 0.710565
$$370$$ 0 0
$$371$$ 4.35050 0.225867
$$372$$ 0 0
$$373$$ 0.952341 0.0493104 0.0246552 0.999696i $$-0.492151\pi$$
0.0246552 + 0.999696i $$0.492151\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ 6.90033 0.355385
$$378$$ 0 0
$$379$$ 23.6495 1.21479 0.607397 0.794399i $$-0.292214\pi$$
0.607397 + 0.794399i $$0.292214\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ 0 0
$$383$$ −32.1293 −1.64173 −0.820864 0.571124i $$-0.806508\pi$$
−0.820864 + 0.571124i $$0.806508\pi$$
$$384$$ 0 0
$$385$$ 6.72508 0.342742
$$386$$ 0 0
$$387$$ −8.17525 −0.415571
$$388$$ 0 0
$$389$$ −4.71998 −0.239313 −0.119656 0.992815i $$-0.538179\pi$$
−0.119656 + 0.992815i $$0.538179\pi$$
$$390$$ 0 0
$$391$$ −11.3446 −0.573723
$$392$$ 0 0
$$393$$ 0 0
$$394$$ 0 0
$$395$$ −39.6495 −1.99498
$$396$$ 0 0
$$397$$ −32.6630 −1.63931 −0.819654 0.572859i $$-0.805834\pi$$
−0.819654 + 0.572859i $$0.805834\pi$$
$$398$$ 0 0
$$399$$ 0 0
$$400$$ 0 0
$$401$$ 35.0997 1.75279 0.876397 0.481590i $$-0.159940\pi$$
0.876397 + 0.481590i $$0.159940\pi$$
$$402$$ 0 0
$$403$$ −16.0000 −0.797017
$$404$$ 0 0
$$405$$ −27.4093 −1.36198
$$406$$ 0 0
$$407$$ −50.4021 −2.49834
$$408$$ 0 0
$$409$$ −18.0000 −0.890043 −0.445021 0.895520i $$-0.646804\pi$$
−0.445021 + 0.895520i $$0.646804\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ 0 0
$$413$$ 2.74194 0.134922
$$414$$ 0 0
$$415$$ 52.0766 2.55634
$$416$$ 0 0
$$417$$ 0 0
$$418$$ 0 0
$$419$$ −12.0000 −0.586238 −0.293119 0.956076i $$-0.594693\pi$$
−0.293119 + 0.956076i $$0.594693\pi$$
$$420$$ 0 0
$$421$$ −24.2487 −1.18181 −0.590905 0.806741i $$-0.701229\pi$$
−0.590905 + 0.806741i $$0.701229\pi$$
$$422$$ 0 0
$$423$$ 1.25588 0.0610630
$$424$$ 0 0
$$425$$ 14.0000 0.679100
$$426$$ 0 0
$$427$$ −1.27492 −0.0616976
$$428$$ 0 0
$$429$$ 0 0
$$430$$ 0 0
$$431$$ 1.67451 0.0806582 0.0403291 0.999186i $$-0.487159\pi$$
0.0403291 + 0.999186i $$0.487159\pi$$
$$432$$ 0 0
$$433$$ −17.6495 −0.848181 −0.424091 0.905620i $$-0.639406\pi$$
−0.424091 + 0.905620i $$0.639406\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 0 0
$$437$$ 3.46410 0.165710
$$438$$ 0 0
$$439$$ 31.2920 1.49349 0.746743 0.665113i $$-0.231617\pi$$
0.746743 + 0.665113i $$0.231617\pi$$
$$440$$ 0 0
$$441$$ 20.4743 0.974965
$$442$$ 0 0
$$443$$ −10.3746 −0.492911 −0.246456 0.969154i $$-0.579266\pi$$
−0.246456 + 0.969154i $$0.579266\pi$$
$$444$$ 0 0
$$445$$ −30.4547 −1.44369
$$446$$ 0 0
$$447$$ 0 0
$$448$$ 0 0
$$449$$ −20.5498 −0.969807 −0.484903 0.874568i $$-0.661145\pi$$
−0.484903 + 0.874568i $$0.661145\pi$$
$$450$$ 0 0
$$451$$ −24.0000 −1.13012
$$452$$ 0 0
$$453$$ 0 0
$$454$$ 0 0
$$455$$ −3.34901 −0.157004
$$456$$ 0 0
$$457$$ 15.2749 0.714530 0.357265 0.934003i $$-0.383709\pi$$
0.357265 + 0.934003i $$0.383709\pi$$
$$458$$ 0 0
$$459$$ 0 0
$$460$$ 0 0
$$461$$ −12.7156 −0.592225 −0.296113 0.955153i $$-0.595690\pi$$
−0.296113 + 0.955153i $$0.595690\pi$$
$$462$$ 0 0
$$463$$ −2.93039 −0.136187 −0.0680933 0.997679i $$-0.521692\pi$$
−0.0680933 + 0.997679i $$0.521692\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ 8.17525 0.378305 0.189153 0.981948i $$-0.439426\pi$$
0.189153 + 0.981948i $$0.439426\pi$$
$$468$$ 0 0
$$469$$ −2.74194 −0.126611
$$470$$ 0 0
$$471$$ 0 0
$$472$$ 0 0
$$473$$ 14.3746 0.660944
$$474$$ 0 0
$$475$$ −4.27492 −0.196147
$$476$$ 0 0
$$477$$ 31.1769 1.42749
$$478$$ 0 0
$$479$$ −17.3205 −0.791394 −0.395697 0.918381i $$-0.629497\pi$$
−0.395697 + 0.918381i $$0.629497\pi$$
$$480$$ 0 0
$$481$$ 25.0997 1.14445
$$482$$ 0 0
$$483$$ 0 0
$$484$$ 0 0
$$485$$ −50.4021 −2.28864
$$486$$ 0 0
$$487$$ −39.0575 −1.76986 −0.884931 0.465722i $$-0.845795\pi$$
−0.884931 + 0.465722i $$0.845795\pi$$
$$488$$ 0 0
$$489$$ 0 0
$$490$$ 0 0
$$491$$ 12.0000 0.541552 0.270776 0.962642i $$-0.412720\pi$$
0.270776 + 0.962642i $$0.412720\pi$$
$$492$$ 0 0
$$493$$ −8.60271 −0.387447
$$494$$ 0 0
$$495$$ 48.1939 2.16615
$$496$$ 0 0
$$497$$ −4.74917 −0.213029
$$498$$ 0 0
$$499$$ 26.7251 1.19638 0.598190 0.801355i $$-0.295887\pi$$
0.598190 + 0.801355i $$0.295887\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ 0 0
$$503$$ 41.6843 1.85861 0.929306 0.369311i $$-0.120406\pi$$
0.929306 + 0.369311i $$0.120406\pi$$
$$504$$ 0 0
$$505$$ −42.1993 −1.87785
$$506$$ 0 0
$$507$$ 0 0
$$508$$ 0 0
$$509$$ 4.30136 0.190654 0.0953271 0.995446i $$-0.469610\pi$$
0.0953271 + 0.995446i $$0.469610\pi$$
$$510$$ 0 0
$$511$$ −1.37097 −0.0606480
$$512$$ 0 0
$$513$$ 0 0
$$514$$ 0 0
$$515$$ 18.5498 0.817403
$$516$$ 0 0
$$517$$ −2.20822 −0.0971175
$$518$$ 0 0
$$519$$ 0 0
$$520$$ 0 0
$$521$$ 27.0997 1.18726 0.593629 0.804739i $$-0.297695\pi$$
0.593629 + 0.804739i $$0.297695\pi$$
$$522$$ 0 0
$$523$$ −5.45017 −0.238319 −0.119160 0.992875i $$-0.538020\pi$$
−0.119160 + 0.992875i $$0.538020\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ 19.9474 0.868920
$$528$$ 0 0
$$529$$ −11.0000 −0.478261
$$530$$ 0 0
$$531$$ 19.6495 0.852716
$$532$$ 0 0
$$533$$ 11.9517 0.517687
$$534$$ 0 0
$$535$$ −4.41644 −0.190939
$$536$$ 0 0
$$537$$ 0 0
$$538$$ 0 0
$$539$$ −36.0000 −1.55063
$$540$$ 0 0
$$541$$ 29.0838 1.25041 0.625205 0.780461i $$-0.285015\pi$$
0.625205 + 0.780461i $$0.285015\pi$$
$$542$$ 0 0
$$543$$ 0 0
$$544$$ 0 0
$$545$$ 10.5498 0.451905
$$546$$ 0 0
$$547$$ −32.0000 −1.36822 −0.684111 0.729378i $$-0.739809\pi$$
−0.684111 + 0.729378i $$0.739809\pi$$
$$548$$ 0 0
$$549$$ −9.13642 −0.389933
$$550$$ 0 0
$$551$$ 2.62685 0.111907
$$552$$ 0 0
$$553$$ −5.45017 −0.231765
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 0 0
$$557$$ 29.9210 1.26779 0.633897 0.773417i $$-0.281454\pi$$
0.633897 + 0.773417i $$0.281454\pi$$
$$558$$ 0 0
$$559$$ −7.15838 −0.302767
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 0 0
$$563$$ 24.0000 1.01148 0.505740 0.862686i $$-0.331220\pi$$
0.505740 + 0.862686i $$0.331220\pi$$
$$564$$ 0 0
$$565$$ −38.2202 −1.60794
$$566$$ 0 0
$$567$$ −3.76764 −0.158226
$$568$$ 0 0
$$569$$ 19.4502 0.815393 0.407697 0.913117i $$-0.366332\pi$$
0.407697 + 0.913117i $$0.366332\pi$$
$$570$$ 0 0
$$571$$ 41.0997 1.71997 0.859984 0.510321i $$-0.170474\pi$$
0.859984 + 0.510321i $$0.170474\pi$$
$$572$$ 0 0
$$573$$ 0 0
$$574$$ 0 0
$$575$$ −14.8087 −0.617567
$$576$$ 0 0
$$577$$ 23.2749 0.968947 0.484474 0.874806i $$-0.339011\pi$$
0.484474 + 0.874806i $$0.339011\pi$$
$$578$$ 0 0
$$579$$ 0 0
$$580$$ 0 0
$$581$$ 7.15838 0.296980
$$582$$ 0 0
$$583$$ −54.8185 −2.27035
$$584$$ 0 0
$$585$$ −24.0000 −0.992278
$$586$$ 0 0
$$587$$ 37.2749 1.53850 0.769250 0.638948i $$-0.220630\pi$$
0.769250 + 0.638948i $$0.220630\pi$$
$$588$$ 0 0
$$589$$ −6.09095 −0.250973
$$590$$ 0 0
$$591$$ 0 0
$$592$$ 0 0
$$593$$ 19.0997 0.784329 0.392165 0.919895i $$-0.371726\pi$$
0.392165 + 0.919895i $$0.371726\pi$$
$$594$$ 0 0
$$595$$ 4.17525 0.171168
$$596$$ 0 0
$$597$$ 0 0
$$598$$ 0 0
$$599$$ 9.43996 0.385706 0.192853 0.981228i $$-0.438226\pi$$
0.192853 + 0.981228i $$0.438226\pi$$
$$600$$ 0 0
$$601$$ −11.0997 −0.452765 −0.226382 0.974038i $$-0.572690\pi$$
−0.226382 + 0.974038i $$0.572690\pi$$
$$602$$ 0 0
$$603$$ −19.6495 −0.800190
$$604$$ 0 0
$$605$$ −51.2394 −2.08318
$$606$$ 0 0
$$607$$ 29.3873 1.19279 0.596397 0.802689i $$-0.296598\pi$$
0.596397 + 0.802689i $$0.296598\pi$$
$$608$$ 0 0
$$609$$ 0 0
$$610$$ 0 0
$$611$$ 1.09967 0.0444878
$$612$$ 0 0
$$613$$ −29.9210 −1.20850 −0.604250 0.796795i $$-0.706527\pi$$
−0.604250 + 0.796795i $$0.706527\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ −12.7251 −0.512293 −0.256146 0.966638i $$-0.582453\pi$$
−0.256146 + 0.966638i $$0.582453\pi$$
$$618$$ 0 0
$$619$$ −4.00000 −0.160774 −0.0803868 0.996764i $$-0.525616\pi$$
−0.0803868 + 0.996764i $$0.525616\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ 0 0
$$623$$ −4.18627 −0.167719
$$624$$ 0 0
$$625$$ −28.0997 −1.12399
$$626$$ 0 0
$$627$$ 0 0
$$628$$ 0 0
$$629$$ −31.2920 −1.24769
$$630$$ 0 0
$$631$$ −13.4378 −0.534950 −0.267475 0.963565i $$-0.586189\pi$$
−0.267475 + 0.963565i $$0.586189\pi$$
$$632$$ 0 0
$$633$$ 0 0
$$634$$ 0 0
$$635$$ 53.0997 2.10720
$$636$$ 0 0
$$637$$ 17.9276 0.710317
$$638$$ 0 0
$$639$$ −34.0339 −1.34636
$$640$$ 0 0
$$641$$ −12.5498 −0.495689 −0.247844 0.968800i $$-0.579722\pi$$
−0.247844 + 0.968800i $$0.579722\pi$$
$$642$$ 0 0
$$643$$ −44.9244 −1.77165 −0.885823 0.464023i $$-0.846405\pi$$
−0.885823 + 0.464023i $$0.846405\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ −25.6197 −1.00721 −0.503607 0.863933i $$-0.667994\pi$$
−0.503607 + 0.863933i $$0.667994\pi$$
$$648$$ 0 0
$$649$$ −34.5498 −1.35620
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 0 0
$$653$$ 16.0646 0.628657 0.314329 0.949314i $$-0.398221\pi$$
0.314329 + 0.949314i $$0.398221\pi$$
$$654$$ 0 0
$$655$$ 16.0646 0.627697
$$656$$ 0 0
$$657$$ −9.82475 −0.383300
$$658$$ 0 0
$$659$$ −13.4502 −0.523944 −0.261972 0.965075i $$-0.584373\pi$$
−0.261972 + 0.965075i $$0.584373\pi$$
$$660$$ 0 0
$$661$$ −9.55505 −0.371648 −0.185824 0.982583i $$-0.559495\pi$$
−0.185824 + 0.982583i $$0.559495\pi$$
$$662$$ 0 0
$$663$$ 0 0
$$664$$ 0 0
$$665$$ −1.27492 −0.0494392
$$666$$ 0 0
$$667$$ 9.09967 0.352341
$$668$$ 0 0
$$669$$ 0 0
$$670$$ 0 0
$$671$$ 16.0646 0.620168
$$672$$ 0 0
$$673$$ −47.0997 −1.81556 −0.907779 0.419448i $$-0.862224\pi$$
−0.907779 + 0.419448i $$0.862224\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ 0 0
$$677$$ −0.722166 −0.0277551 −0.0138775 0.999904i $$-0.504418\pi$$
−0.0138775 + 0.999904i $$0.504418\pi$$
$$678$$ 0 0
$$679$$ −6.92820 −0.265880
$$680$$ 0 0
$$681$$ 0 0
$$682$$ 0 0
$$683$$ −22.1993 −0.849434 −0.424717 0.905326i $$-0.639626\pi$$
−0.424717 + 0.905326i $$0.639626\pi$$
$$684$$ 0 0
$$685$$ −38.7539 −1.48071
$$686$$ 0 0
$$687$$ 0 0
$$688$$ 0 0
$$689$$ 27.2990 1.04001
$$690$$ 0 0
$$691$$ 23.4743 0.893003 0.446501 0.894783i $$-0.352670\pi$$
0.446501 + 0.894783i $$0.352670\pi$$
$$692$$ 0 0
$$693$$ 6.62466 0.251650
$$694$$ 0 0
$$695$$ −8.29917 −0.314806
$$696$$ 0 0
$$697$$ −14.9003 −0.564390
$$698$$ 0 0
$$699$$ 0 0
$$700$$ 0 0
$$701$$ 21.0148 0.793717 0.396859 0.917880i $$-0.370100\pi$$
0.396859 + 0.917880i $$0.370100\pi$$
$$702$$ 0 0
$$703$$ 9.55505 0.360376
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ −5.80066 −0.218156
$$708$$ 0 0
$$709$$ 38.2202 1.43539 0.717695 0.696358i $$-0.245197\pi$$
0.717695 + 0.696358i $$0.245197\pi$$
$$710$$ 0 0
$$711$$ −39.0575 −1.46477
$$712$$ 0 0
$$713$$ −21.0997 −0.790189
$$714$$ 0 0
$$715$$ 42.1993 1.57817
$$716$$ 0 0
$$717$$ 0 0
$$718$$ 0 0
$$719$$ 51.6580 1.92652 0.963259 0.268575i $$-0.0865526\pi$$
0.963259 + 0.268575i $$0.0865526\pi$$
$$720$$ 0 0
$$721$$ 2.54983 0.0949608
$$722$$ 0 0
$$723$$ 0 0
$$724$$ 0 0
$$725$$ −11.2296 −0.417055
$$726$$ 0 0
$$727$$ 10.9260 0.405224 0.202612 0.979259i $$-0.435057\pi$$
0.202612 + 0.979259i $$0.435057\pi$$
$$728$$ 0 0
$$729$$ −27.0000 −1.00000
$$730$$ 0 0
$$731$$ 8.92442 0.330082
$$732$$ 0 0
$$733$$ 19.1101 0.705848 0.352924 0.935652i $$-0.385187\pi$$
0.352924 + 0.935652i $$0.385187\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ 34.5498 1.27266
$$738$$ 0 0
$$739$$ 7.82475 0.287838 0.143919 0.989589i $$-0.454029\pi$$
0.143919 + 0.989589i $$0.454029\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 0 0
$$743$$ 52.9139 1.94122 0.970611 0.240655i $$-0.0773622\pi$$
0.970611 + 0.240655i $$0.0773622\pi$$
$$744$$ 0 0
$$745$$ −48.9244 −1.79245
$$746$$ 0 0
$$747$$ 51.2990 1.87693
$$748$$ 0 0
$$749$$ −0.607078 −0.0221822
$$750$$ 0 0
$$751$$ −36.3155 −1.32517 −0.662586 0.748986i $$-0.730541\pi$$
−0.662586 + 0.748986i $$0.730541\pi$$
$$752$$ 0 0
$$753$$ 0 0
$$754$$ 0 0
$$755$$ −2.54983 −0.0927980
$$756$$ 0 0
$$757$$ 3.04547 0.110690 0.0553448 0.998467i $$-0.482374\pi$$
0.0553448 + 0.998467i $$0.482374\pi$$
$$758$$ 0 0
$$759$$ 0 0
$$760$$ 0 0
$$761$$ −3.27492 −0.118716 −0.0593578 0.998237i $$-0.518905\pi$$
−0.0593578 + 0.998237i $$0.518905\pi$$
$$762$$ 0 0
$$763$$ 1.45017 0.0524995
$$764$$ 0 0
$$765$$ 29.9210 1.08180
$$766$$ 0 0
$$767$$ 17.2054 0.621252
$$768$$ 0 0
$$769$$ −13.8248 −0.498533 −0.249267 0.968435i $$-0.580190\pi$$
−0.249267 + 0.968435i $$0.580190\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ 0 0
$$773$$ 34.7561 1.25009 0.625045 0.780589i $$-0.285081\pi$$
0.625045 + 0.780589i $$0.285081\pi$$
$$774$$ 0 0
$$775$$ 26.0383 0.935324
$$776$$ 0 0
$$777$$ 0 0
$$778$$ 0 0
$$779$$ 4.54983 0.163015
$$780$$ 0 0
$$781$$ 59.8421 2.14132
$$782$$ 0 0
$$783$$ 0 0
$$784$$ 0 0
$$785$$ −37.0997 −1.32414
$$786$$ 0 0
$$787$$ −14.1993 −0.506152 −0.253076 0.967446i $$-0.581442\pi$$
−0.253076 + 0.967446i $$0.581442\pi$$
$$788$$ 0 0
$$789$$ 0 0
$$790$$ 0 0
$$791$$ −5.25370 −0.186800
$$792$$ 0 0
$$793$$ −8.00000 −0.284088
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 0 0
$$797$$ −24.2487 −0.858933 −0.429467 0.903083i $$-0.641298\pi$$
−0.429467 + 0.903083i $$0.641298\pi$$
$$798$$ 0 0
$$799$$ −1.37097 −0.0485014
$$800$$ 0 0
$$801$$ −30.0000 −1.06000
$$802$$ 0 0
$$803$$ 17.2749 0.609619
$$804$$ 0 0
$$805$$ −4.41644 −0.155659
$$806$$ 0 0
$$807$$ 0 0
$$808$$ 0 0
$$809$$ 5.82475 0.204787 0.102394 0.994744i $$-0.467350\pi$$
0.102394 + 0.994744i $$0.467350\pi$$
$$810$$ 0 0
$$811$$ −47.2990 −1.66089 −0.830446 0.557099i $$-0.811915\pi$$
−0.830446 + 0.557099i $$0.811915\pi$$
$$812$$ 0 0
$$813$$ 0 0
$$814$$ 0 0
$$815$$ 12.1819 0.426713
$$816$$ 0 0
$$817$$ −2.72508 −0.0953386
$$818$$ 0 0
$$819$$ −3.29901 −0.115277
$$820$$ 0 0
$$821$$ 26.5720 0.927370 0.463685 0.886000i $$-0.346527\pi$$
0.463685 + 0.886000i $$0.346527\pi$$
$$822$$ 0 0
$$823$$ 21.4334 0.747122 0.373561 0.927606i $$-0.378137\pi$$
0.373561 + 0.927606i $$0.378137\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ −47.6495 −1.65694 −0.828468 0.560037i $$-0.810787\pi$$
−0.828468 + 0.560037i $$0.810787\pi$$
$$828$$ 0 0
$$829$$ 25.3161 0.879266 0.439633 0.898178i $$-0.355109\pi$$
0.439633 + 0.898178i $$0.355109\pi$$
$$830$$ 0 0
$$831$$ 0 0
$$832$$ 0 0
$$833$$ −22.3505 −0.774399
$$834$$ 0 0
$$835$$ −58.1993 −2.01407
$$836$$ 0 0
$$837$$ 0 0
$$838$$ 0 0
$$839$$ 47.6602 1.64541 0.822706 0.568467i $$-0.192463\pi$$
0.822706 + 0.568467i $$0.192463\pi$$
$$840$$ 0 0
$$841$$ −22.0997 −0.762058
$$842$$ 0 0
$$843$$ 0 0
$$844$$ 0 0
$$845$$ 18.5764 0.639047
$$846$$ 0 0
$$847$$ −7.04329 −0.242010
$$848$$ 0 0
$$849$$ 0 0
$$850$$ 0 0
$$851$$ 33.0997 1.13464
$$852$$ 0 0
$$853$$ 32.9665 1.12875 0.564376 0.825518i $$-0.309117\pi$$
0.564376 + 0.825518i $$0.309117\pi$$
$$854$$ 0 0
$$855$$ −9.13642 −0.312459
$$856$$ 0 0
$$857$$ 16.1993 0.553359 0.276679 0.960962i $$-0.410766\pi$$
0.276679 + 0.960962i $$0.410766\pi$$
$$858$$ 0 0
$$859$$ −55.4743 −1.89276 −0.946379 0.323060i $$-0.895289\pi$$
−0.946379 + 0.323060i $$0.895289\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 0 0
$$863$$ −41.7994 −1.42287 −0.711434 0.702753i $$-0.751954\pi$$
−0.711434 + 0.702753i $$0.751954\pi$$
$$864$$ 0 0
$$865$$ 26.5498 0.902721
$$866$$ 0 0
$$867$$ 0 0
$$868$$ 0 0
$$869$$ 68.6750 2.32964
$$870$$ 0 0
$$871$$ −17.2054 −0.582983
$$872$$ 0 0
$$873$$ −49.6495 −1.68038
$$874$$ 0 0
$$875$$ −0.924421 −0.0312511
$$876$$ 0 0
$$877$$ 31.4071 1.06054 0.530271 0.847828i $$-0.322090\pi$$
0.530271 + 0.847828i $$0.322090\pi$$
$$878$$ 0 0
$$879$$ 0 0
$$880$$ 0 0
$$881$$ 1.82475 0.0614774 0.0307387 0.999527i $$-0.490214\pi$$
0.0307387 + 0.999527i $$0.490214\pi$$
$$882$$ 0 0
$$883$$ 13.6254 0.458532 0.229266 0.973364i $$-0.426367\pi$$
0.229266 + 0.973364i $$0.426367\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 0 0
$$887$$ −0.837253 −0.0281122 −0.0140561 0.999901i $$-0.504474\pi$$
−0.0140561 + 0.999901i $$0.504474\pi$$
$$888$$ 0 0
$$889$$ 7.29901 0.244801
$$890$$ 0 0
$$891$$ 47.4743 1.59045
$$892$$ 0 0
$$893$$ 0.418627 0.0140088
$$894$$ 0 0
$$895$$ −64.2585 −2.14793
$$896$$ 0 0
$$897$$ 0 0
$$898$$ 0 0
$$899$$ −16.0000 −0.533630
$$900$$ 0 0
$$901$$ −34.0339 −1.13383
$$902$$ 0 0
$$903$$ 0 0
$$904$$ 0 0
$$905$$ 10.5498 0.350688
$$906$$ 0 0
$$907$$ −23.6495 −0.785269 −0.392634 0.919695i $$-0.628436\pi$$
−0.392634 + 0.919695i $$0.628436\pi$$
$$908$$ 0 0
$$909$$ −41.5692 −1.37876
$$910$$ 0 0
$$911$$ 34.0339 1.12759 0.563797 0.825913i $$-0.309340\pi$$
0.563797 + 0.825913i $$0.309340\pi$$
$$912$$ 0 0
$$913$$ −90.1993 −2.98516
$$914$$ 0 0
$$915$$ 0 0
$$916$$ 0 0
$$917$$ 2.20822 0.0729219
$$918$$ 0 0
$$919$$ −0.115088 −0.00379639 −0.00189820 0.999998i $$-0.500604\pi$$
−0.00189820 + 0.999998i $$0.500604\pi$$
$$920$$ 0 0
$$921$$ 0 0
$$922$$ 0 0
$$923$$ −29.8007 −0.980901
$$924$$ 0 0
$$925$$ −40.8471 −1.34304
$$926$$ 0 0
$$927$$ 18.2728 0.600159
$$928$$ 0 0
$$929$$ −7.09967 −0.232933 −0.116466 0.993195i $$-0.537157\pi$$
−0.116466 + 0.993195i $$0.537157\pi$$
$$930$$ 0 0
$$931$$ 6.82475 0.223672
$$932$$ 0 0
$$933$$ 0 0
$$934$$ 0 0
$$935$$ −52.6103 −1.72054
$$936$$ 0 0
$$937$$ 14.9244 0.487560 0.243780 0.969831i $$-0.421613\pi$$
0.243780 + 0.969831i $$0.421613\pi$$
$$938$$ 0 0
$$939$$ 0 0
$$940$$ 0 0
$$941$$ 31.1769 1.01634 0.508169 0.861257i $$-0.330322\pi$$
0.508169 + 0.861257i $$0.330322\pi$$
$$942$$ 0 0
$$943$$ 15.7611 0.513252
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ 30.1993 0.981347 0.490673 0.871344i $$-0.336751\pi$$
0.490673 + 0.871344i $$0.336751\pi$$
$$948$$ 0 0
$$949$$ −8.60271 −0.279256
$$950$$ 0 0
$$951$$ 0 0
$$952$$ 0 0
$$953$$ 45.2990 1.46738 0.733689 0.679485i $$-0.237797\pi$$
0.733689 + 0.679485i $$0.237797\pi$$
$$954$$ 0 0
$$955$$ −35.8248 −1.15926
$$956$$ 0 0
$$957$$ 0 0
$$958$$ 0 0
$$959$$ −5.32706 −0.172020
$$960$$ 0 0
$$961$$ 6.09967 0.196764
$$962$$ 0 0
$$963$$ −4.35050 −0.140193
$$964$$ 0 0
$$965$$ −33.8038 −1.08818
$$966$$ 0 0
$$967$$ −55.5407 −1.78607 −0.893034 0.449988i $$-0.851428\pi$$
−0.893034 + 0.449988i $$0.851428\pi$$
$$968$$ 0 0
$$969$$ 0 0
$$970$$ 0 0
$$971$$ 44.0000 1.41203 0.706014 0.708198i $$-0.250492\pi$$
0.706014 + 0.708198i $$0.250492\pi$$
$$972$$ 0 0
$$973$$ −1.14079 −0.0365721
$$974$$ 0 0
$$975$$ 0 0
$$976$$ 0 0
$$977$$ −28.1993 −0.902177 −0.451088 0.892479i $$-0.648964\pi$$
−0.451088 + 0.892479i $$0.648964\pi$$
$$978$$ 0 0
$$979$$ 52.7492 1.68587
$$980$$ 0 0
$$981$$ 10.3923 0.331801
$$982$$ 0 0
$$983$$ 22.6893 0.723676 0.361838 0.932241i $$-0.382149\pi$$
0.361838 + 0.932241i $$0.382149\pi$$
$$984$$ 0 0
$$985$$ 16.0000 0.509802
$$986$$ 0 0
$$987$$ 0 0
$$988$$ 0 0
$$989$$ −9.43996 −0.300173
$$990$$ 0 0
$$991$$ 47.0531 1.49469 0.747345 0.664436i $$-0.231328\pi$$
0.747345 + 0.664436i $$0.231328\pi$$
$$992$$ 0 0
$$993$$ 0 0
$$994$$ 0 0
$$995$$ −43.4743 −1.37823
$$996$$ 0 0
$$997$$ −42.1029 −1.33341 −0.666707 0.745320i $$-0.732297\pi$$
−0.666707 + 0.745320i $$0.732297\pi$$
$$998$$ 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 4864.2.a.bk.1.1 4
4.3 odd 2 4864.2.a.bj.1.1 4
8.3 odd 2 inner 4864.2.a.bk.1.4 4
8.5 even 2 4864.2.a.bj.1.4 4
16.3 odd 4 1216.2.c.i.609.7 yes 8
16.5 even 4 1216.2.c.i.609.2 yes 8
16.11 odd 4 1216.2.c.i.609.1 8
16.13 even 4 1216.2.c.i.609.8 yes 8

By twisted newform
Twist Min Dim Char Parity Ord Type
1216.2.c.i.609.1 8 16.11 odd 4
1216.2.c.i.609.2 yes 8 16.5 even 4
1216.2.c.i.609.7 yes 8 16.3 odd 4
1216.2.c.i.609.8 yes 8 16.13 even 4
4864.2.a.bj.1.1 4 4.3 odd 2
4864.2.a.bj.1.4 4 8.5 even 2
4864.2.a.bk.1.1 4 1.1 even 1 trivial
4864.2.a.bk.1.4 4 8.3 odd 2 inner