# Properties

 Label 2304.3.h.b.2177.4 Level $2304$ Weight $3$ Character 2304.2177 Analytic conductor $62.779$ Analytic rank $0$ Dimension $4$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$2304 = 2^{8} \cdot 3^{2}$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 2304.h (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$62.7794529086$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\zeta_{8})$$ Defining polynomial: $$x^{4} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{17}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: no (minimal twist has level 288) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 2177.4 Root $$0.707107 - 0.707107i$$ of defining polynomial Character $$\chi$$ $$=$$ 2304.2177 Dual form 2304.3.h.b.2177.3

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+1.41421 q^{5} -8.00000 q^{7} +O(q^{10})$$ $$q+1.41421 q^{5} -8.00000 q^{7} +11.3137 q^{11} +8.00000i q^{13} +12.7279i q^{17} -32.0000i q^{19} +33.9411i q^{23} -23.0000 q^{25} -43.8406 q^{29} +40.0000 q^{31} -11.3137 q^{35} -26.0000i q^{37} -66.4680i q^{41} +16.0000i q^{43} +11.3137i q^{47} +15.0000 q^{49} +32.5269 q^{53} +16.0000 q^{55} +22.6274 q^{59} +54.0000i q^{61} +11.3137i q^{65} +80.0000i q^{67} -79.1960i q^{71} -96.0000 q^{73} -90.5097 q^{77} -104.000 q^{79} +101.823 q^{83} +18.0000i q^{85} -77.7817i q^{89} -64.0000i q^{91} -45.2548i q^{95} -80.0000 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 32q^{7} + O(q^{10})$$ $$4q - 32q^{7} - 92q^{25} + 160q^{31} + 60q^{49} + 64q^{55} - 384q^{73} - 416q^{79} - 320q^{97} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/2304\mathbb{Z}\right)^\times$$.

 $$n$$ $$1279$$ $$1793$$ $$2053$$ $$\chi(n)$$ $$1$$ $$-1$$ $$-1$$

## 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
$$5$$ 1.41421 0.282843 0.141421 0.989949i $$-0.454833\pi$$
0.141421 + 0.989949i $$0.454833\pi$$
$$6$$ 0 0
$$7$$ −8.00000 −1.14286 −0.571429 0.820652i $$-0.693611\pi$$
−0.571429 + 0.820652i $$0.693611\pi$$
$$8$$ 0 0
$$9$$ 0 0
$$10$$ 0 0
$$11$$ 11.3137 1.02852 0.514259 0.857635i $$-0.328067\pi$$
0.514259 + 0.857635i $$0.328067\pi$$
$$12$$ 0 0
$$13$$ 8.00000i 0.615385i 0.951486 + 0.307692i $$0.0995567\pi$$
−0.951486 + 0.307692i $$0.900443\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 0 0
$$17$$ 12.7279i 0.748701i 0.927287 + 0.374351i $$0.122134\pi$$
−0.927287 + 0.374351i $$0.877866\pi$$
$$18$$ 0 0
$$19$$ − 32.0000i − 1.68421i −0.539313 0.842105i $$-0.681316\pi$$
0.539313 0.842105i $$-0.318684\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 0 0
$$23$$ 33.9411i 1.47570i 0.674964 + 0.737851i $$0.264159\pi$$
−0.674964 + 0.737851i $$0.735841\pi$$
$$24$$ 0 0
$$25$$ −23.0000 −0.920000
$$26$$ 0 0
$$27$$ 0 0
$$28$$ 0 0
$$29$$ −43.8406 −1.51175 −0.755873 0.654719i $$-0.772787\pi$$
−0.755873 + 0.654719i $$0.772787\pi$$
$$30$$ 0 0
$$31$$ 40.0000 1.29032 0.645161 0.764046i $$-0.276790\pi$$
0.645161 + 0.764046i $$0.276790\pi$$
$$32$$ 0 0
$$33$$ 0 0
$$34$$ 0 0
$$35$$ −11.3137 −0.323249
$$36$$ 0 0
$$37$$ − 26.0000i − 0.702703i −0.936244 0.351351i $$-0.885722\pi$$
0.936244 0.351351i $$-0.114278\pi$$
$$38$$ 0 0
$$39$$ 0 0
$$40$$ 0 0
$$41$$ − 66.4680i − 1.62117i −0.585620 0.810586i $$-0.699149\pi$$
0.585620 0.810586i $$-0.300851\pi$$
$$42$$ 0 0
$$43$$ 16.0000i 0.372093i 0.982541 + 0.186047i $$0.0595675\pi$$
−0.982541 + 0.186047i $$0.940432\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 0 0
$$47$$ 11.3137i 0.240717i 0.992730 + 0.120359i $$0.0384044\pi$$
−0.992730 + 0.120359i $$0.961596\pi$$
$$48$$ 0 0
$$49$$ 15.0000 0.306122
$$50$$ 0 0
$$51$$ 0 0
$$52$$ 0 0
$$53$$ 32.5269 0.613715 0.306858 0.951755i $$-0.400722\pi$$
0.306858 + 0.951755i $$0.400722\pi$$
$$54$$ 0 0
$$55$$ 16.0000 0.290909
$$56$$ 0 0
$$57$$ 0 0
$$58$$ 0 0
$$59$$ 22.6274 0.383516 0.191758 0.981442i $$-0.438581\pi$$
0.191758 + 0.981442i $$0.438581\pi$$
$$60$$ 0 0
$$61$$ 54.0000i 0.885246i 0.896708 + 0.442623i $$0.145952\pi$$
−0.896708 + 0.442623i $$0.854048\pi$$
$$62$$ 0 0
$$63$$ 0 0
$$64$$ 0 0
$$65$$ 11.3137i 0.174057i
$$66$$ 0 0
$$67$$ 80.0000i 1.19403i 0.802230 + 0.597015i $$0.203647\pi$$
−0.802230 + 0.597015i $$0.796353\pi$$
$$68$$ 0 0
$$69$$ 0 0
$$70$$ 0 0
$$71$$ − 79.1960i − 1.11544i −0.830030 0.557718i $$-0.811677\pi$$
0.830030 0.557718i $$-0.188323\pi$$
$$72$$ 0 0
$$73$$ −96.0000 −1.31507 −0.657534 0.753425i $$-0.728401\pi$$
−0.657534 + 0.753425i $$0.728401\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 0 0
$$77$$ −90.5097 −1.17545
$$78$$ 0 0
$$79$$ −104.000 −1.31646 −0.658228 0.752819i $$-0.728694\pi$$
−0.658228 + 0.752819i $$0.728694\pi$$
$$80$$ 0 0
$$81$$ 0 0
$$82$$ 0 0
$$83$$ 101.823 1.22679 0.613394 0.789777i $$-0.289804\pi$$
0.613394 + 0.789777i $$0.289804\pi$$
$$84$$ 0 0
$$85$$ 18.0000i 0.211765i
$$86$$ 0 0
$$87$$ 0 0
$$88$$ 0 0
$$89$$ − 77.7817i − 0.873952i −0.899473 0.436976i $$-0.856049\pi$$
0.899473 0.436976i $$-0.143951\pi$$
$$90$$ 0 0
$$91$$ − 64.0000i − 0.703297i
$$92$$ 0 0
$$93$$ 0 0
$$94$$ 0 0
$$95$$ − 45.2548i − 0.476367i
$$96$$ 0 0
$$97$$ −80.0000 −0.824742 −0.412371 0.911016i $$-0.635299\pi$$
−0.412371 + 0.911016i $$0.635299\pi$$
$$98$$ 0 0
$$99$$ 0 0
$$100$$ 0 0
$$101$$ −123.037 −1.21818 −0.609092 0.793100i $$-0.708466\pi$$
−0.609092 + 0.793100i $$0.708466\pi$$
$$102$$ 0 0
$$103$$ 72.0000 0.699029 0.349515 0.936931i $$-0.386347\pi$$
0.349515 + 0.936931i $$0.386347\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ −181.019 −1.69177 −0.845885 0.533366i $$-0.820927\pi$$
−0.845885 + 0.533366i $$0.820927\pi$$
$$108$$ 0 0
$$109$$ 88.0000i 0.807339i 0.914905 + 0.403670i $$0.132266\pi$$
−0.914905 + 0.403670i $$0.867734\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ 0 0
$$113$$ − 137.179i − 1.21397i −0.794713 0.606985i $$-0.792379\pi$$
0.794713 0.606985i $$-0.207621\pi$$
$$114$$ 0 0
$$115$$ 48.0000i 0.417391i
$$116$$ 0 0
$$117$$ 0 0
$$118$$ 0 0
$$119$$ − 101.823i − 0.855659i
$$120$$ 0 0
$$121$$ 7.00000 0.0578512
$$122$$ 0 0
$$123$$ 0 0
$$124$$ 0 0
$$125$$ −67.8823 −0.543058
$$126$$ 0 0
$$127$$ −56.0000 −0.440945 −0.220472 0.975393i $$-0.570760\pi$$
−0.220472 + 0.975393i $$0.570760\pi$$
$$128$$ 0 0
$$129$$ 0 0
$$130$$ 0 0
$$131$$ −248.902 −1.90001 −0.950006 0.312231i $$-0.898924\pi$$
−0.950006 + 0.312231i $$0.898924\pi$$
$$132$$ 0 0
$$133$$ 256.000i 1.92481i
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ − 46.6690i − 0.340650i −0.985388 0.170325i $$-0.945518\pi$$
0.985388 0.170325i $$-0.0544818\pi$$
$$138$$ 0 0
$$139$$ 16.0000i 0.115108i 0.998342 + 0.0575540i $$0.0183301\pi$$
−0.998342 + 0.0575540i $$0.981670\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 0 0
$$143$$ 90.5097i 0.632935i
$$144$$ 0 0
$$145$$ −62.0000 −0.427586
$$146$$ 0 0
$$147$$ 0 0
$$148$$ 0 0
$$149$$ −182.434 −1.22439 −0.612193 0.790708i $$-0.709712\pi$$
−0.612193 + 0.790708i $$0.709712\pi$$
$$150$$ 0 0
$$151$$ −168.000 −1.11258 −0.556291 0.830987i $$-0.687776\pi$$
−0.556291 + 0.830987i $$0.687776\pi$$
$$152$$ 0 0
$$153$$ 0 0
$$154$$ 0 0
$$155$$ 56.5685 0.364958
$$156$$ 0 0
$$157$$ 10.0000i 0.0636943i 0.999493 + 0.0318471i $$0.0101390\pi$$
−0.999493 + 0.0318471i $$0.989861\pi$$
$$158$$ 0 0
$$159$$ 0 0
$$160$$ 0 0
$$161$$ − 271.529i − 1.68652i
$$162$$ 0 0
$$163$$ − 80.0000i − 0.490798i −0.969422 0.245399i $$-0.921081\pi$$
0.969422 0.245399i $$-0.0789189\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ 294.156i 1.76142i 0.473660 + 0.880708i $$0.342933\pi$$
−0.473660 + 0.880708i $$0.657067\pi$$
$$168$$ 0 0
$$169$$ 105.000 0.621302
$$170$$ 0 0
$$171$$ 0 0
$$172$$ 0 0
$$173$$ 55.1543 0.318811 0.159406 0.987213i $$-0.449042\pi$$
0.159406 + 0.987213i $$0.449042\pi$$
$$174$$ 0 0
$$175$$ 184.000 1.05143
$$176$$ 0 0
$$177$$ 0 0
$$178$$ 0 0
$$179$$ 135.765 0.758461 0.379230 0.925302i $$-0.376189\pi$$
0.379230 + 0.925302i $$0.376189\pi$$
$$180$$ 0 0
$$181$$ 8.00000i 0.0441989i 0.999756 + 0.0220994i $$0.00703505\pi$$
−0.999756 + 0.0220994i $$0.992965\pi$$
$$182$$ 0 0
$$183$$ 0 0
$$184$$ 0 0
$$185$$ − 36.7696i − 0.198754i
$$186$$ 0 0
$$187$$ 144.000i 0.770053i
$$188$$ 0 0
$$189$$ 0 0
$$190$$ 0 0
$$191$$ − 67.8823i − 0.355404i −0.984084 0.177702i $$-0.943134\pi$$
0.984084 0.177702i $$-0.0568664\pi$$
$$192$$ 0 0
$$193$$ −258.000 −1.33679 −0.668394 0.743808i $$-0.733018\pi$$
−0.668394 + 0.743808i $$0.733018\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ −371.938 −1.88801 −0.944005 0.329930i $$-0.892975\pi$$
−0.944005 + 0.329930i $$0.892975\pi$$
$$198$$ 0 0
$$199$$ −88.0000 −0.442211 −0.221106 0.975250i $$-0.570967\pi$$
−0.221106 + 0.975250i $$0.570967\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ 0 0
$$203$$ 350.725 1.72771
$$204$$ 0 0
$$205$$ − 94.0000i − 0.458537i
$$206$$ 0 0
$$207$$ 0 0
$$208$$ 0 0
$$209$$ − 362.039i − 1.73224i
$$210$$ 0 0
$$211$$ − 368.000i − 1.74408i −0.489438 0.872038i $$-0.662798\pi$$
0.489438 0.872038i $$-0.337202\pi$$
$$212$$ 0 0
$$213$$ 0 0
$$214$$ 0 0
$$215$$ 22.6274i 0.105244i
$$216$$ 0 0
$$217$$ −320.000 −1.47465
$$218$$ 0 0
$$219$$ 0 0
$$220$$ 0 0
$$221$$ −101.823 −0.460739
$$222$$ 0 0
$$223$$ 104.000 0.466368 0.233184 0.972433i $$-0.425086\pi$$
0.233184 + 0.972433i $$0.425086\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 0 0
$$227$$ 169.706 0.747602 0.373801 0.927509i $$-0.378054\pi$$
0.373801 + 0.927509i $$0.378054\pi$$
$$228$$ 0 0
$$229$$ 344.000i 1.50218i 0.660198 + 0.751092i $$0.270472\pi$$
−0.660198 + 0.751092i $$0.729528\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ − 80.6102i − 0.345966i −0.984925 0.172983i $$-0.944659\pi$$
0.984925 0.172983i $$-0.0553406\pi$$
$$234$$ 0 0
$$235$$ 16.0000i 0.0680851i
$$236$$ 0 0
$$237$$ 0 0
$$238$$ 0 0
$$239$$ − 271.529i − 1.13610i −0.822992 0.568052i $$-0.807697\pi$$
0.822992 0.568052i $$-0.192303\pi$$
$$240$$ 0 0
$$241$$ −272.000 −1.12863 −0.564315 0.825559i $$-0.690860\pi$$
−0.564315 + 0.825559i $$0.690860\pi$$
$$242$$ 0 0
$$243$$ 0 0
$$244$$ 0 0
$$245$$ 21.2132 0.0865845
$$246$$ 0 0
$$247$$ 256.000 1.03644
$$248$$ 0 0
$$249$$ 0 0
$$250$$ 0 0
$$251$$ 33.9411 0.135224 0.0676118 0.997712i $$-0.478462\pi$$
0.0676118 + 0.997712i $$0.478462\pi$$
$$252$$ 0 0
$$253$$ 384.000i 1.51779i
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 0 0
$$257$$ − 43.8406i − 0.170586i −0.996356 0.0852930i $$-0.972817\pi$$
0.996356 0.0852930i $$-0.0271826\pi$$
$$258$$ 0 0
$$259$$ 208.000i 0.803089i
$$260$$ 0 0
$$261$$ 0 0
$$262$$ 0 0
$$263$$ − 203.647i − 0.774322i −0.922012 0.387161i $$-0.873456\pi$$
0.922012 0.387161i $$-0.126544\pi$$
$$264$$ 0 0
$$265$$ 46.0000 0.173585
$$266$$ 0 0
$$267$$ 0 0
$$268$$ 0 0
$$269$$ −46.6690 −0.173491 −0.0867454 0.996231i $$-0.527647\pi$$
−0.0867454 + 0.996231i $$0.527647\pi$$
$$270$$ 0 0
$$271$$ 264.000 0.974170 0.487085 0.873355i $$-0.338060\pi$$
0.487085 + 0.873355i $$0.338060\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ 0 0
$$275$$ −260.215 −0.946237
$$276$$ 0 0
$$277$$ 40.0000i 0.144404i 0.997390 + 0.0722022i $$0.0230027\pi$$
−0.997390 + 0.0722022i $$0.976997\pi$$
$$278$$ 0 0
$$279$$ 0 0
$$280$$ 0 0
$$281$$ − 190.919i − 0.679426i −0.940529 0.339713i $$-0.889670\pi$$
0.940529 0.339713i $$-0.110330\pi$$
$$282$$ 0 0
$$283$$ − 224.000i − 0.791519i −0.918354 0.395760i $$-0.870481\pi$$
0.918354 0.395760i $$-0.129519\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ 531.744i 1.85277i
$$288$$ 0 0
$$289$$ 127.000 0.439446
$$290$$ 0 0
$$291$$ 0 0
$$292$$ 0 0
$$293$$ −352.139 −1.20184 −0.600920 0.799309i $$-0.705199\pi$$
−0.600920 + 0.799309i $$0.705199\pi$$
$$294$$ 0 0
$$295$$ 32.0000 0.108475
$$296$$ 0 0
$$297$$ 0 0
$$298$$ 0 0
$$299$$ −271.529 −0.908124
$$300$$ 0 0
$$301$$ − 128.000i − 0.425249i
$$302$$ 0 0
$$303$$ 0 0
$$304$$ 0 0
$$305$$ 76.3675i 0.250385i
$$306$$ 0 0
$$307$$ 432.000i 1.40717i 0.710613 + 0.703583i $$0.248418\pi$$
−0.710613 + 0.703583i $$0.751582\pi$$
$$308$$ 0 0
$$309$$ 0 0
$$310$$ 0 0
$$311$$ 203.647i 0.654813i 0.944884 + 0.327406i $$0.106175\pi$$
−0.944884 + 0.327406i $$0.893825\pi$$
$$312$$ 0 0
$$313$$ 14.0000 0.0447284 0.0223642 0.999750i $$-0.492881\pi$$
0.0223642 + 0.999750i $$0.492881\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ −589.727 −1.86034 −0.930169 0.367132i $$-0.880340\pi$$
−0.930169 + 0.367132i $$0.880340\pi$$
$$318$$ 0 0
$$319$$ −496.000 −1.55486
$$320$$ 0 0
$$321$$ 0 0
$$322$$ 0 0
$$323$$ 407.294 1.26097
$$324$$ 0 0
$$325$$ − 184.000i − 0.566154i
$$326$$ 0 0
$$327$$ 0 0
$$328$$ 0 0
$$329$$ − 90.5097i − 0.275105i
$$330$$ 0 0
$$331$$ 16.0000i 0.0483384i 0.999708 + 0.0241692i $$0.00769404\pi$$
−0.999708 + 0.0241692i $$0.992306\pi$$
$$332$$ 0 0
$$333$$ 0 0
$$334$$ 0 0
$$335$$ 113.137i 0.337723i
$$336$$ 0 0
$$337$$ 128.000 0.379822 0.189911 0.981801i $$-0.439180\pi$$
0.189911 + 0.981801i $$0.439180\pi$$
$$338$$ 0 0
$$339$$ 0 0
$$340$$ 0 0
$$341$$ 452.548 1.32712
$$342$$ 0 0
$$343$$ 272.000 0.793003
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ −350.725 −1.01073 −0.505367 0.862904i $$-0.668643\pi$$
−0.505367 + 0.862904i $$0.668643\pi$$
$$348$$ 0 0
$$349$$ − 10.0000i − 0.0286533i −0.999897 0.0143266i $$-0.995440\pi$$
0.999897 0.0143266i $$-0.00456047\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 0 0
$$353$$ 247.487i 0.701097i 0.936545 + 0.350549i $$0.114005\pi$$
−0.936545 + 0.350549i $$0.885995\pi$$
$$354$$ 0 0
$$355$$ − 112.000i − 0.315493i
$$356$$ 0 0
$$357$$ 0 0
$$358$$ 0 0
$$359$$ − 328.098i − 0.913921i −0.889487 0.456960i $$-0.848938\pi$$
0.889487 0.456960i $$-0.151062\pi$$
$$360$$ 0 0
$$361$$ −663.000 −1.83657
$$362$$ 0 0
$$363$$ 0 0
$$364$$ 0 0
$$365$$ −135.765 −0.371958
$$366$$ 0 0
$$367$$ −696.000 −1.89646 −0.948229 0.317588i $$-0.897127\pi$$
−0.948229 + 0.317588i $$0.897127\pi$$
$$368$$ 0 0
$$369$$ 0 0
$$370$$ 0 0
$$371$$ −260.215 −0.701389
$$372$$ 0 0
$$373$$ − 454.000i − 1.21716i −0.793493 0.608579i $$-0.791740\pi$$
0.793493 0.608579i $$-0.208260\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ − 350.725i − 0.930305i
$$378$$ 0 0
$$379$$ − 64.0000i − 0.168865i −0.996429 0.0844327i $$-0.973092\pi$$
0.996429 0.0844327i $$-0.0269078\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ 0 0
$$383$$ − 362.039i − 0.945271i −0.881258 0.472635i $$-0.843303\pi$$
0.881258 0.472635i $$-0.156697\pi$$
$$384$$ 0 0
$$385$$ −128.000 −0.332468
$$386$$ 0 0
$$387$$ 0 0
$$388$$ 0 0
$$389$$ −386.080 −0.992494 −0.496247 0.868181i $$-0.665289\pi$$
−0.496247 + 0.868181i $$0.665289\pi$$
$$390$$ 0 0
$$391$$ −432.000 −1.10486
$$392$$ 0 0
$$393$$ 0 0
$$394$$ 0 0
$$395$$ −147.078 −0.372350
$$396$$ 0 0
$$397$$ 662.000i 1.66751i 0.552137 + 0.833753i $$0.313812\pi$$
−0.552137 + 0.833753i $$0.686188\pi$$
$$398$$ 0 0
$$399$$ 0 0
$$400$$ 0 0
$$401$$ 171.120i 0.426733i 0.976972 + 0.213366i $$0.0684428\pi$$
−0.976972 + 0.213366i $$0.931557\pi$$
$$402$$ 0 0
$$403$$ 320.000i 0.794045i
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 0 0
$$407$$ − 294.156i − 0.722743i
$$408$$ 0 0
$$409$$ 176.000 0.430318 0.215159 0.976579i $$-0.430973\pi$$
0.215159 + 0.976579i $$0.430973\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ 0 0
$$413$$ −181.019 −0.438303
$$414$$ 0 0
$$415$$ 144.000 0.346988
$$416$$ 0 0
$$417$$ 0 0
$$418$$ 0 0
$$419$$ −79.1960 −0.189012 −0.0945059 0.995524i $$-0.530127\pi$$
−0.0945059 + 0.995524i $$0.530127\pi$$
$$420$$ 0 0
$$421$$ 488.000i 1.15914i 0.814921 + 0.579572i $$0.196780\pi$$
−0.814921 + 0.579572i $$0.803220\pi$$
$$422$$ 0 0
$$423$$ 0 0
$$424$$ 0 0
$$425$$ − 292.742i − 0.688805i
$$426$$ 0 0
$$427$$ − 432.000i − 1.01171i
$$428$$ 0 0
$$429$$ 0 0
$$430$$ 0 0
$$431$$ − 305.470i − 0.708747i −0.935104 0.354374i $$-0.884694\pi$$
0.935104 0.354374i $$-0.115306\pi$$
$$432$$ 0 0
$$433$$ 478.000 1.10393 0.551963 0.833869i $$-0.313879\pi$$
0.551963 + 0.833869i $$0.313879\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 0 0
$$437$$ 1086.12 2.48539
$$438$$ 0 0
$$439$$ −392.000 −0.892938 −0.446469 0.894799i $$-0.647319\pi$$
−0.446469 + 0.894799i $$0.647319\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 0 0
$$443$$ 33.9411 0.0766165 0.0383083 0.999266i $$-0.487803\pi$$
0.0383083 + 0.999266i $$0.487803\pi$$
$$444$$ 0 0
$$445$$ − 110.000i − 0.247191i
$$446$$ 0 0
$$447$$ 0 0
$$448$$ 0 0
$$449$$ − 125.865i − 0.280323i −0.990129 0.140161i $$-0.955238\pi$$
0.990129 0.140161i $$-0.0447622\pi$$
$$450$$ 0 0
$$451$$ − 752.000i − 1.66741i
$$452$$ 0 0
$$453$$ 0 0
$$454$$ 0 0
$$455$$ − 90.5097i − 0.198922i
$$456$$ 0 0
$$457$$ 16.0000 0.0350109 0.0175055 0.999847i $$-0.494428\pi$$
0.0175055 + 0.999847i $$0.494428\pi$$
$$458$$ 0 0
$$459$$ 0 0
$$460$$ 0 0
$$461$$ 284.257 0.616609 0.308305 0.951288i $$-0.400238\pi$$
0.308305 + 0.951288i $$0.400238\pi$$
$$462$$ 0 0
$$463$$ 568.000 1.22678 0.613391 0.789779i $$-0.289805\pi$$
0.613391 + 0.789779i $$0.289805\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ −418.607 −0.896375 −0.448188 0.893940i $$-0.647930\pi$$
−0.448188 + 0.893940i $$0.647930\pi$$
$$468$$ 0 0
$$469$$ − 640.000i − 1.36461i
$$470$$ 0 0
$$471$$ 0 0
$$472$$ 0 0
$$473$$ 181.019i 0.382705i
$$474$$ 0 0
$$475$$ 736.000i 1.54947i
$$476$$ 0 0
$$477$$ 0 0
$$478$$ 0 0
$$479$$ − 33.9411i − 0.0708583i −0.999372 0.0354291i $$-0.988720\pi$$
0.999372 0.0354291i $$-0.0112798\pi$$
$$480$$ 0 0
$$481$$ 208.000 0.432432
$$482$$ 0 0
$$483$$ 0 0
$$484$$ 0 0
$$485$$ −113.137 −0.233272
$$486$$ 0 0
$$487$$ 424.000 0.870637 0.435318 0.900277i $$-0.356636\pi$$
0.435318 + 0.900277i $$0.356636\pi$$
$$488$$ 0 0
$$489$$ 0 0
$$490$$ 0 0
$$491$$ 724.077 1.47470 0.737350 0.675511i $$-0.236077\pi$$
0.737350 + 0.675511i $$0.236077\pi$$
$$492$$ 0 0
$$493$$ − 558.000i − 1.13185i
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ 633.568i 1.27478i
$$498$$ 0 0
$$499$$ 192.000i 0.384770i 0.981320 + 0.192385i $$0.0616222\pi$$
−0.981320 + 0.192385i $$0.938378\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ 0 0
$$503$$ 441.235i 0.877206i 0.898681 + 0.438603i $$0.144527\pi$$
−0.898681 + 0.438603i $$0.855473\pi$$
$$504$$ 0 0
$$505$$ −174.000 −0.344554
$$506$$ 0 0
$$507$$ 0 0
$$508$$ 0 0
$$509$$ 250.316 0.491780 0.245890 0.969298i $$-0.420920\pi$$
0.245890 + 0.969298i $$0.420920\pi$$
$$510$$ 0 0
$$511$$ 768.000 1.50294
$$512$$ 0 0
$$513$$ 0 0
$$514$$ 0 0
$$515$$ 101.823 0.197715
$$516$$ 0 0
$$517$$ 128.000i 0.247582i
$$518$$ 0 0
$$519$$ 0 0
$$520$$ 0 0
$$521$$ 156.978i 0.301301i 0.988587 + 0.150650i $$0.0481368\pi$$
−0.988587 + 0.150650i $$0.951863\pi$$
$$522$$ 0 0
$$523$$ 576.000i 1.10134i 0.834724 + 0.550669i $$0.185627\pi$$
−0.834724 + 0.550669i $$0.814373\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ 509.117i 0.966066i
$$528$$ 0 0
$$529$$ −623.000 −1.17769
$$530$$ 0 0
$$531$$ 0 0
$$532$$ 0 0
$$533$$ 531.744 0.997644
$$534$$ 0 0
$$535$$ −256.000 −0.478505
$$536$$ 0 0
$$537$$ 0 0
$$538$$ 0 0
$$539$$ 169.706 0.314853
$$540$$ 0 0
$$541$$ 536.000i 0.990758i 0.868677 + 0.495379i $$0.164971\pi$$
−0.868677 + 0.495379i $$0.835029\pi$$
$$542$$ 0 0
$$543$$ 0 0
$$544$$ 0 0
$$545$$ 124.451i 0.228350i
$$546$$ 0 0
$$547$$ − 144.000i − 0.263254i −0.991299 0.131627i $$-0.957980\pi$$
0.991299 0.131627i $$-0.0420201\pi$$
$$548$$ 0 0
$$549$$ 0 0
$$550$$ 0 0
$$551$$ 1402.90i 2.54610i
$$552$$ 0 0
$$553$$ 832.000 1.50452
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 0 0
$$557$$ 533.159 0.957197 0.478598 0.878034i $$-0.341145\pi$$
0.478598 + 0.878034i $$0.341145\pi$$
$$558$$ 0 0
$$559$$ −128.000 −0.228980
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 0 0
$$563$$ 576.999 1.02487 0.512433 0.858727i $$-0.328744\pi$$
0.512433 + 0.858727i $$0.328744\pi$$
$$564$$ 0 0
$$565$$ − 194.000i − 0.343363i
$$566$$ 0 0
$$567$$ 0 0
$$568$$ 0 0
$$569$$ 994.192i 1.74726i 0.486589 + 0.873631i $$0.338241\pi$$
−0.486589 + 0.873631i $$0.661759\pi$$
$$570$$ 0 0
$$571$$ 416.000i 0.728546i 0.931292 + 0.364273i $$0.118683\pi$$
−0.931292 + 0.364273i $$0.881317\pi$$
$$572$$ 0 0
$$573$$ 0 0
$$574$$ 0 0
$$575$$ − 780.646i − 1.35765i
$$576$$ 0 0
$$577$$ 834.000 1.44541 0.722704 0.691158i $$-0.242899\pi$$
0.722704 + 0.691158i $$0.242899\pi$$
$$578$$ 0 0
$$579$$ 0 0
$$580$$ 0 0
$$581$$ −814.587 −1.40204
$$582$$ 0 0
$$583$$ 368.000 0.631218
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ 67.8823 0.115643 0.0578213 0.998327i $$-0.481585\pi$$
0.0578213 + 0.998327i $$0.481585\pi$$
$$588$$ 0 0
$$589$$ − 1280.00i − 2.17317i
$$590$$ 0 0
$$591$$ 0 0
$$592$$ 0 0
$$593$$ − 224.860i − 0.379190i −0.981862 0.189595i $$-0.939282\pi$$
0.981862 0.189595i $$-0.0607176\pi$$
$$594$$ 0 0
$$595$$ − 144.000i − 0.242017i
$$596$$ 0 0
$$597$$ 0 0
$$598$$ 0 0
$$599$$ − 690.136i − 1.15215i −0.817398 0.576074i $$-0.804584\pi$$
0.817398 0.576074i $$-0.195416\pi$$
$$600$$ 0 0
$$601$$ 626.000 1.04160 0.520799 0.853680i $$-0.325634\pi$$
0.520799 + 0.853680i $$0.325634\pi$$
$$602$$ 0 0
$$603$$ 0 0
$$604$$ 0 0
$$605$$ 9.89949 0.0163628
$$606$$ 0 0
$$607$$ 232.000 0.382208 0.191104 0.981570i $$-0.438793\pi$$
0.191104 + 0.981570i $$0.438793\pi$$
$$608$$ 0 0
$$609$$ 0 0
$$610$$ 0 0
$$611$$ −90.5097 −0.148134
$$612$$ 0 0
$$613$$ 666.000i 1.08646i 0.839584 + 0.543230i $$0.182799\pi$$
−0.839584 + 0.543230i $$0.817201\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ 284.257i 0.460708i 0.973107 + 0.230354i $$0.0739884\pi$$
−0.973107 + 0.230354i $$0.926012\pi$$
$$618$$ 0 0
$$619$$ − 768.000i − 1.24071i −0.784321 0.620355i $$-0.786988\pi$$
0.784321 0.620355i $$-0.213012\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ 0 0
$$623$$ 622.254i 0.998803i
$$624$$ 0 0
$$625$$ 479.000 0.766400
$$626$$ 0 0
$$627$$ 0 0
$$628$$ 0 0
$$629$$ 330.926 0.526114
$$630$$ 0 0
$$631$$ −472.000 −0.748019 −0.374010 0.927425i $$-0.622017\pi$$
−0.374010 + 0.927425i $$0.622017\pi$$
$$632$$ 0 0
$$633$$ 0 0
$$634$$ 0 0
$$635$$ −79.1960 −0.124718
$$636$$ 0 0
$$637$$ 120.000i 0.188383i
$$638$$ 0 0
$$639$$ 0 0
$$640$$ 0 0
$$641$$ − 371.938i − 0.580247i −0.956989 0.290123i $$-0.906304\pi$$
0.956989 0.290123i $$-0.0936963\pi$$
$$642$$ 0 0
$$643$$ 240.000i 0.373250i 0.982431 + 0.186625i $$0.0597550\pi$$
−0.982431 + 0.186625i $$0.940245\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ − 712.764i − 1.10164i −0.834623 0.550822i $$-0.814314\pi$$
0.834623 0.550822i $$-0.185686\pi$$
$$648$$ 0 0
$$649$$ 256.000 0.394453
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 0 0
$$653$$ 1163.90 1.78239 0.891193 0.453625i $$-0.149869\pi$$
0.891193 + 0.453625i $$0.149869\pi$$
$$654$$ 0 0
$$655$$ −352.000 −0.537405
$$656$$ 0 0
$$657$$ 0 0
$$658$$ 0 0
$$659$$ −203.647 −0.309024 −0.154512 0.987991i $$-0.549381\pi$$
−0.154512 + 0.987991i $$0.549381\pi$$
$$660$$ 0 0
$$661$$ − 1018.00i − 1.54009i −0.637989 0.770045i $$-0.720234\pi$$
0.637989 0.770045i $$-0.279766\pi$$
$$662$$ 0 0
$$663$$ 0 0
$$664$$ 0 0
$$665$$ 362.039i 0.544419i
$$666$$ 0 0
$$667$$ − 1488.00i − 2.23088i
$$668$$ 0 0
$$669$$ 0 0
$$670$$ 0 0
$$671$$ 610.940i 0.910492i
$$672$$ 0 0
$$673$$ −382.000 −0.567608 −0.283804 0.958882i $$-0.591596\pi$$
−0.283804 + 0.958882i $$0.591596\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ 0 0
$$677$$ −247.487 −0.365565 −0.182782 0.983153i $$-0.558510\pi$$
−0.182782 + 0.983153i $$0.558510\pi$$
$$678$$ 0 0
$$679$$ 640.000 0.942563
$$680$$ 0 0
$$681$$ 0 0
$$682$$ 0 0
$$683$$ 169.706 0.248471 0.124235 0.992253i $$-0.460352\pi$$
0.124235 + 0.992253i $$0.460352\pi$$
$$684$$ 0 0
$$685$$ − 66.0000i − 0.0963504i
$$686$$ 0 0
$$687$$ 0 0
$$688$$ 0 0
$$689$$ 260.215i 0.377671i
$$690$$ 0 0
$$691$$ − 1040.00i − 1.50507i −0.658555 0.752533i $$-0.728832\pi$$
0.658555 0.752533i $$-0.271168\pi$$
$$692$$ 0 0
$$693$$ 0 0
$$694$$ 0 0
$$695$$ 22.6274i 0.0325574i
$$696$$ 0 0
$$697$$ 846.000 1.21377
$$698$$ 0 0
$$699$$ 0 0
$$700$$ 0 0
$$701$$ 657.609 0.938102 0.469051 0.883171i $$-0.344596\pi$$
0.469051 + 0.883171i $$0.344596\pi$$
$$702$$ 0 0
$$703$$ −832.000 −1.18350
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ 984.293 1.39221
$$708$$ 0 0
$$709$$ 952.000i 1.34274i 0.741124 + 0.671368i $$0.234293\pi$$
−0.741124 + 0.671368i $$0.765707\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ 0 0
$$713$$ 1357.65i 1.90413i
$$714$$ 0 0
$$715$$ 128.000i 0.179021i
$$716$$ 0 0
$$717$$ 0 0
$$718$$ 0 0
$$719$$ − 192.333i − 0.267501i −0.991015 0.133750i $$-0.957298\pi$$
0.991015 0.133750i $$-0.0427020\pi$$
$$720$$ 0 0
$$721$$ −576.000 −0.798890
$$722$$ 0 0
$$723$$ 0 0
$$724$$ 0 0
$$725$$ 1008.33 1.39081
$$726$$ 0 0
$$727$$ 648.000 0.891334 0.445667 0.895199i $$-0.352967\pi$$
0.445667 + 0.895199i $$0.352967\pi$$
$$728$$ 0 0
$$729$$ 0 0
$$730$$ 0 0
$$731$$ −203.647 −0.278587
$$732$$ 0 0
$$733$$ − 1208.00i − 1.64802i −0.566574 0.824011i $$-0.691731\pi$$
0.566574 0.824011i $$-0.308269\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ 905.097i 1.22808i
$$738$$ 0 0
$$739$$ 1312.00i 1.77537i 0.460449 + 0.887686i $$0.347688\pi$$
−0.460449 + 0.887686i $$0.652312\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 0 0
$$743$$ − 610.940i − 0.822261i −0.911576 0.411131i $$-0.865134\pi$$
0.911576 0.411131i $$-0.134866\pi$$
$$744$$ 0 0
$$745$$ −258.000 −0.346309
$$746$$ 0 0
$$747$$ 0 0
$$748$$ 0 0
$$749$$ 1448.15 1.93345
$$750$$ 0 0
$$751$$ −632.000 −0.841545 −0.420772 0.907166i $$-0.638241\pi$$
−0.420772 + 0.907166i $$0.638241\pi$$
$$752$$ 0 0
$$753$$ 0 0
$$754$$ 0 0
$$755$$ −237.588 −0.314686
$$756$$ 0 0
$$757$$ − 840.000i − 1.10964i −0.831969 0.554822i $$-0.812786\pi$$
0.831969 0.554822i $$-0.187214\pi$$
$$758$$ 0 0
$$759$$ 0 0
$$760$$ 0 0
$$761$$ 521.845i 0.685736i 0.939384 + 0.342868i $$0.111398\pi$$
−0.939384 + 0.342868i $$0.888602\pi$$
$$762$$ 0 0
$$763$$ − 704.000i − 0.922674i
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ 181.019i 0.236010i
$$768$$ 0 0
$$769$$ 130.000 0.169051 0.0845254 0.996421i $$-0.473063\pi$$
0.0845254 + 0.996421i $$0.473063\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ 0 0
$$773$$ −284.257 −0.367732 −0.183866 0.982951i $$-0.558861\pi$$
−0.183866 + 0.982951i $$0.558861\pi$$
$$774$$ 0 0
$$775$$ −920.000 −1.18710
$$776$$ 0 0
$$777$$ 0 0
$$778$$ 0 0
$$779$$ −2126.98 −2.73039
$$780$$ 0 0
$$781$$ − 896.000i − 1.14725i
$$782$$ 0 0
$$783$$ 0 0
$$784$$ 0 0
$$785$$ 14.1421i 0.0180155i
$$786$$ 0 0
$$787$$ 864.000i 1.09784i 0.835875 + 0.548920i $$0.184961\pi$$
−0.835875 + 0.548920i $$0.815039\pi$$
$$788$$ 0 0
$$789$$ 0 0
$$790$$ 0 0
$$791$$ 1097.43i 1.38740i
$$792$$ 0 0
$$793$$ −432.000 −0.544767
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 0 0
$$797$$ −759.433 −0.952864 −0.476432 0.879211i $$-0.658070\pi$$
−0.476432 + 0.879211i $$0.658070\pi$$
$$798$$ 0 0
$$799$$ −144.000 −0.180225
$$800$$ 0 0
$$801$$ 0 0
$$802$$ 0 0
$$803$$ −1086.12 −1.35257
$$804$$ 0 0
$$805$$ − 384.000i − 0.477019i
$$806$$ 0 0
$$807$$ 0 0
$$808$$ 0 0
$$809$$ 906.511i 1.12053i 0.828313 + 0.560266i $$0.189301\pi$$
−0.828313 + 0.560266i $$0.810699\pi$$
$$810$$ 0 0
$$811$$ 1472.00i 1.81504i 0.420005 + 0.907522i $$0.362028\pi$$
−0.420005 + 0.907522i $$0.637972\pi$$
$$812$$ 0 0
$$813$$ 0 0
$$814$$ 0 0
$$815$$ − 113.137i − 0.138819i
$$816$$ 0 0
$$817$$ 512.000 0.626683
$$818$$ 0 0
$$819$$ 0 0
$$820$$ 0 0
$$821$$ 1189.35 1.44866 0.724332 0.689451i $$-0.242148\pi$$
0.724332 + 0.689451i $$0.242148\pi$$
$$822$$ 0 0
$$823$$ 664.000 0.806804 0.403402 0.915023i $$-0.367828\pi$$
0.403402 + 0.915023i $$0.367828\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ 248.902 0.300969 0.150485 0.988612i $$-0.451917\pi$$
0.150485 + 0.988612i $$0.451917\pi$$
$$828$$ 0 0
$$829$$ 280.000i 0.337756i 0.985637 + 0.168878i $$0.0540144\pi$$
−0.985637 + 0.168878i $$0.945986\pi$$
$$830$$ 0 0
$$831$$ 0 0
$$832$$ 0 0
$$833$$ 190.919i 0.229194i
$$834$$ 0 0
$$835$$ 416.000i 0.498204i
$$836$$ 0 0
$$837$$ 0 0
$$838$$ 0 0
$$839$$ − 554.372i − 0.660753i −0.943849 0.330376i $$-0.892824\pi$$
0.943849 0.330376i $$-0.107176\pi$$
$$840$$ 0 0
$$841$$ 1081.00 1.28537
$$842$$ 0 0
$$843$$ 0 0
$$844$$ 0 0
$$845$$ 148.492 0.175731
$$846$$ 0 0
$$847$$ −56.0000 −0.0661157
$$848$$ 0 0
$$849$$ 0 0
$$850$$ 0 0
$$851$$ 882.469 1.03698
$$852$$ 0 0
$$853$$ − 762.000i − 0.893318i −0.894704 0.446659i $$-0.852614\pi$$
0.894704 0.446659i $$-0.147386\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0 0
$$857$$ − 1537.25i − 1.79376i −0.442277 0.896879i $$-0.645829\pi$$
0.442277 0.896879i $$-0.354171\pi$$
$$858$$ 0 0
$$859$$ 1552.00i 1.80675i 0.428849 + 0.903376i $$0.358919\pi$$
−0.428849 + 0.903376i $$0.641081\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 0 0
$$863$$ − 22.6274i − 0.0262195i −0.999914 0.0131097i $$-0.995827\pi$$
0.999914 0.0131097i $$-0.00417308\pi$$
$$864$$ 0 0
$$865$$ 78.0000 0.0901734
$$866$$ 0 0
$$867$$ 0 0
$$868$$ 0 0
$$869$$ −1176.63 −1.35400
$$870$$ 0 0
$$871$$ −640.000 −0.734788
$$872$$ 0 0
$$873$$ 0 0
$$874$$ 0 0
$$875$$ 543.058 0.620638
$$876$$ 0 0
$$877$$ 86.0000i 0.0980616i 0.998797 + 0.0490308i $$0.0156132\pi$$
−0.998797 + 0.0490308i $$0.984387\pi$$
$$878$$ 0 0
$$879$$ 0 0
$$880$$ 0 0
$$881$$ 657.609i 0.746435i 0.927744 + 0.373218i $$0.121745\pi$$
−0.927744 + 0.373218i $$0.878255\pi$$
$$882$$ 0 0
$$883$$ − 496.000i − 0.561721i −0.959749 0.280861i $$-0.909380\pi$$
0.959749 0.280861i $$-0.0906199\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 0 0
$$887$$ − 1018.23i − 1.14795i −0.818872 0.573976i $$-0.805400\pi$$
0.818872 0.573976i $$-0.194600\pi$$
$$888$$ 0 0
$$889$$ 448.000 0.503937
$$890$$ 0 0
$$891$$ 0 0
$$892$$ 0 0
$$893$$ 362.039 0.405418
$$894$$ 0 0
$$895$$ 192.000 0.214525
$$896$$ 0 0
$$897$$ 0 0
$$898$$ 0 0
$$899$$ −1753.62 −1.95064
$$900$$ 0 0
$$901$$ 414.000i 0.459489i
$$902$$ 0 0
$$903$$ 0 0
$$904$$ 0 0
$$905$$ 11.3137i 0.0125013i
$$906$$ 0 0
$$907$$ − 880.000i − 0.970232i −0.874450 0.485116i $$-0.838777\pi$$
0.874450 0.485116i $$-0.161223\pi$$
$$908$$ 0 0
$$909$$ 0 0
$$910$$ 0 0
$$911$$ − 1561.29i − 1.71382i −0.515464 0.856911i $$-0.672381\pi$$
0.515464 0.856911i $$-0.327619\pi$$
$$912$$ 0 0
$$913$$ 1152.00 1.26177
$$914$$ 0 0
$$915$$ 0 0
$$916$$ 0 0
$$917$$ 1991.21 2.17144
$$918$$ 0 0
$$919$$ −264.000 −0.287269 −0.143634 0.989631i $$-0.545879\pi$$
−0.143634 + 0.989631i $$0.545879\pi$$
$$920$$ 0 0
$$921$$ 0 0
$$922$$ 0 0
$$923$$ 633.568 0.686422
$$924$$ 0 0
$$925$$ 598.000i 0.646486i
$$926$$ 0 0
$$927$$ 0 0
$$928$$ 0 0
$$929$$ 284.257i 0.305982i 0.988228 + 0.152991i $$0.0488905\pi$$
−0.988228 + 0.152991i $$0.951110\pi$$
$$930$$ 0 0
$$931$$ − 480.000i − 0.515575i
$$932$$ 0 0
$$933$$ 0 0
$$934$$ 0 0
$$935$$ 203.647i 0.217804i
$$936$$ 0 0
$$937$$ −1070.00 −1.14194 −0.570971 0.820970i $$-0.693433\pi$$
−0.570971 + 0.820970i $$0.693433\pi$$
$$938$$ 0 0
$$939$$ 0 0
$$940$$ 0 0
$$941$$ −575.585 −0.611674 −0.305837 0.952084i $$-0.598936\pi$$
−0.305837 + 0.952084i $$0.598936\pi$$
$$942$$ 0 0
$$943$$ 2256.00 2.39236
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ 610.940 0.645132 0.322566 0.946547i $$-0.395455\pi$$
0.322566 + 0.946547i $$0.395455\pi$$
$$948$$ 0 0
$$949$$ − 768.000i − 0.809273i
$$950$$ 0 0
$$951$$ 0 0
$$952$$ 0 0
$$953$$ 1039.45i 1.09071i 0.838205 + 0.545355i $$0.183605\pi$$
−0.838205 + 0.545355i $$0.816395\pi$$
$$954$$ 0 0
$$955$$ − 96.0000i − 0.100524i
$$956$$ 0 0
$$957$$ 0 0
$$958$$ 0 0
$$959$$ 373.352i 0.389314i
$$960$$ 0 0
$$961$$ 639.000 0.664932
$$962$$ 0 0
$$963$$ 0 0
$$964$$ 0 0
$$965$$ −364.867 −0.378101
$$966$$ 0 0
$$967$$ −696.000 −0.719752 −0.359876 0.933000i $$-0.617181\pi$$
−0.359876 + 0.933000i $$0.617181\pi$$
$$968$$ 0 0
$$969$$ 0 0
$$970$$ 0 0
$$971$$ 1572.61 1.61957 0.809787 0.586725i $$-0.199583\pi$$
0.809787 + 0.586725i $$0.199583\pi$$
$$972$$ 0 0
$$973$$ − 128.000i − 0.131552i
$$974$$ 0 0
$$975$$ 0 0
$$976$$ 0 0
$$977$$ 507.703i 0.519655i 0.965655 + 0.259827i $$0.0836657\pi$$
−0.965655 + 0.259827i $$0.916334\pi$$
$$978$$ 0 0
$$979$$ − 880.000i − 0.898876i
$$980$$ 0 0
$$981$$ 0 0
$$982$$ 0 0
$$983$$ 475.176i 0.483393i 0.970352 + 0.241697i $$0.0777039\pi$$
−0.970352 + 0.241697i $$0.922296\pi$$
$$984$$ 0 0
$$985$$ −526.000 −0.534010
$$986$$ 0 0
$$987$$ 0 0
$$988$$ 0 0
$$989$$ −543.058 −0.549098
$$990$$ 0 0
$$991$$ 520.000 0.524723 0.262361 0.964970i $$-0.415499\pi$$
0.262361 + 0.964970i $$0.415499\pi$$
$$992$$ 0 0
$$993$$ 0 0
$$994$$ 0 0
$$995$$ −124.451 −0.125076
$$996$$ 0 0
$$997$$ − 486.000i − 0.487462i −0.969843 0.243731i $$-0.921629\pi$$
0.969843 0.243731i $$-0.0783715\pi$$
$$998$$ 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 2304.3.h.b.2177.4 4
3.2 odd 2 inner 2304.3.h.b.2177.2 4
4.3 odd 2 2304.3.h.g.2177.4 4
8.3 odd 2 2304.3.h.g.2177.1 4
8.5 even 2 inner 2304.3.h.b.2177.1 4
12.11 even 2 2304.3.h.g.2177.2 4
16.3 odd 4 288.3.e.a.161.1 2
16.5 even 4 576.3.e.g.449.2 2
16.11 odd 4 576.3.e.b.449.2 2
16.13 even 4 288.3.e.d.161.1 yes 2
24.5 odd 2 inner 2304.3.h.b.2177.3 4
24.11 even 2 2304.3.h.g.2177.3 4
48.5 odd 4 576.3.e.g.449.1 2
48.11 even 4 576.3.e.b.449.1 2
48.29 odd 4 288.3.e.d.161.2 yes 2
48.35 even 4 288.3.e.a.161.2 yes 2

By twisted newform
Twist Min Dim Char Parity Ord Type
288.3.e.a.161.1 2 16.3 odd 4
288.3.e.a.161.2 yes 2 48.35 even 4
288.3.e.d.161.1 yes 2 16.13 even 4
288.3.e.d.161.2 yes 2 48.29 odd 4
576.3.e.b.449.1 2 48.11 even 4
576.3.e.b.449.2 2 16.11 odd 4
576.3.e.g.449.1 2 48.5 odd 4
576.3.e.g.449.2 2 16.5 even 4
2304.3.h.b.2177.1 4 8.5 even 2 inner
2304.3.h.b.2177.2 4 3.2 odd 2 inner
2304.3.h.b.2177.3 4 24.5 odd 2 inner
2304.3.h.b.2177.4 4 1.1 even 1 trivial
2304.3.h.g.2177.1 4 8.3 odd 2
2304.3.h.g.2177.2 4 12.11 even 2
2304.3.h.g.2177.3 4 24.11 even 2
2304.3.h.g.2177.4 4 4.3 odd 2