# Properties

 Label 1216.2.c.i.609.2 Level $1216$ Weight $2$ Character 1216.609 Analytic conductor $9.710$ Analytic rank $0$ Dimension $8$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1216 = 2^{6} \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1216.c (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$9.70980888579$$ Analytic rank: $$0$$ Dimension: $$8$$ Coefficient field: 8.0.2702336256.1 Defining polynomial: $$x^{8} + 9 x^{6} + 56 x^{4} + 225 x^{2} + 625$$ Coefficient ring: $$\Z[a_1, \ldots, a_{19}]$$ Coefficient ring index: $$2^{6}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 609.2 Root $$0.656712 + 2.13746i$$ of defining polynomial Character $$\chi$$ $$=$$ 1216.609 Dual form 1216.2.c.i.609.8

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-3.04547i q^{5} +0.418627 q^{7} +3.00000 q^{9} +O(q^{10})$$ $$q-3.04547i q^{5} +0.418627 q^{7} +3.00000 q^{9} +5.27492i q^{11} +2.62685i q^{13} +3.27492 q^{17} +1.00000i q^{19} +3.46410 q^{23} -4.27492 q^{25} +2.62685i q^{29} +6.09095 q^{31} -1.27492i q^{35} -9.55505i q^{37} +4.54983 q^{41} +2.72508i q^{43} -9.13642i q^{45} -0.418627 q^{47} -6.82475 q^{49} -10.3923i q^{53} +16.0646 q^{55} -6.54983i q^{59} -3.04547i q^{61} +1.25588 q^{63} +8.00000 q^{65} -6.54983i q^{67} -11.3446 q^{71} -3.27492 q^{73} +2.20822i q^{77} +13.0192 q^{79} +9.00000 q^{81} +17.0997i q^{83} -9.97368i q^{85} -10.0000 q^{89} +1.09967i q^{91} +3.04547 q^{95} +16.5498 q^{97} +15.8248i q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q + 24q^{9} + O(q^{10})$$ $$8q + 24q^{9} - 4q^{17} - 4q^{25} - 24q^{41} + 36q^{49} + 64q^{65} + 4q^{73} + 72q^{81} - 80q^{89} + 72q^{97} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1216\mathbb{Z}\right)^\times$$.

 $$n$$ $$191$$ $$705$$ $$837$$ $$\chi(n)$$ $$1$$ $$1$$ $$-1$$

## 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 1.00000 $$0$$
−1.00000 $$\pi$$
$$4$$ 0 0
$$5$$ − 3.04547i − 1.36198i −0.732294 0.680989i $$-0.761550\pi$$
0.732294 0.680989i $$-0.238450\pi$$
$$6$$ 0 0
$$7$$ 0.418627 0.158226 0.0791130 0.996866i $$-0.474791\pi$$
0.0791130 + 0.996866i $$0.474791\pi$$
$$8$$ 0 0
$$9$$ 3.00000 1.00000
$$10$$ 0 0
$$11$$ 5.27492i 1.59045i 0.606316 + 0.795224i $$0.292647\pi$$
−0.606316 + 0.795224i $$0.707353\pi$$
$$12$$ 0 0
$$13$$ 2.62685i 0.728557i 0.931290 + 0.364278i $$0.118684\pi$$
−0.931290 + 0.364278i $$0.881316\pi$$
$$14$$ 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
$$19$$ 1.00000i 0.229416i
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 0 0
$$23$$ 3.46410 0.722315 0.361158 0.932505i $$-0.382382\pi$$
0.361158 + 0.932505i $$0.382382\pi$$
$$24$$ 0 0
$$25$$ −4.27492 −0.854983
$$26$$ 0 0
$$27$$ 0 0
$$28$$ 0 0
$$29$$ 2.62685i 0.487793i 0.969801 + 0.243897i $$0.0784258\pi$$
−0.969801 + 0.243897i $$0.921574\pi$$
$$30$$ 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
$$35$$ − 1.27492i − 0.215500i
$$36$$ 0 0
$$37$$ − 9.55505i − 1.57084i −0.618963 0.785420i $$-0.712447\pi$$
0.618963 0.785420i $$-0.287553\pi$$
$$38$$ 0 0
$$39$$ 0 0
$$40$$ 0 0
$$41$$ 4.54983 0.710565 0.355282 0.934759i $$-0.384385\pi$$
0.355282 + 0.934759i $$0.384385\pi$$
$$42$$ 0 0
$$43$$ 2.72508i 0.415571i 0.978174 + 0.207786i $$0.0666256\pi$$
−0.978174 + 0.207786i $$0.933374\pi$$
$$44$$ 0 0
$$45$$ − 9.13642i − 1.36198i
$$46$$ 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
$$53$$ − 10.3923i − 1.42749i −0.700404 0.713746i $$-0.746997\pi$$
0.700404 0.713746i $$-0.253003\pi$$
$$54$$ 0 0
$$55$$ 16.0646 2.16615
$$56$$ 0 0
$$57$$ 0 0
$$58$$ 0 0
$$59$$ − 6.54983i − 0.852716i −0.904555 0.426358i $$-0.859796\pi$$
0.904555 0.426358i $$-0.140204\pi$$
$$60$$ 0 0
$$61$$ − 3.04547i − 0.389933i −0.980810 0.194967i $$-0.937540\pi$$
0.980810 0.194967i $$-0.0624598\pi$$
$$62$$ 0 0
$$63$$ 1.25588 0.158226
$$64$$ 0 0
$$65$$ 8.00000 0.992278
$$66$$ 0 0
$$67$$ − 6.54983i − 0.800190i −0.916474 0.400095i $$-0.868977\pi$$
0.916474 0.400095i $$-0.131023\pi$$
$$68$$ 0 0
$$69$$ 0 0
$$70$$ 0 0
$$71$$ −11.3446 −1.34636 −0.673181 0.739478i $$-0.735072\pi$$
−0.673181 + 0.739478i $$0.735072\pi$$
$$72$$ 0 0
$$73$$ −3.27492 −0.383300 −0.191650 0.981463i $$-0.561384\pi$$
−0.191650 + 0.981463i $$0.561384\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 0 0
$$77$$ 2.20822i 0.251650i
$$78$$ 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
$$83$$ 17.0997i 1.87693i 0.345371 + 0.938466i $$0.387753\pi$$
−0.345371 + 0.938466i $$0.612247\pi$$
$$84$$ 0 0
$$85$$ − 9.97368i − 1.08180i
$$86$$ 0 0
$$87$$ 0 0
$$88$$ 0 0
$$89$$ −10.0000 −1.06000 −0.529999 0.847998i $$-0.677808\pi$$
−0.529999 + 0.847998i $$0.677808\pi$$
$$90$$ 0 0
$$91$$ 1.09967i 0.115277i
$$92$$ 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
$$99$$ 15.8248i 1.59045i
$$100$$ 0 0
$$101$$ 13.8564i 1.37876i 0.724398 + 0.689382i $$0.242118\pi$$
−0.724398 + 0.689382i $$0.757882\pi$$
$$102$$ 0 0
$$103$$ 6.09095 0.600159 0.300080 0.953914i $$-0.402987\pi$$
0.300080 + 0.953914i $$0.402987\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ 1.45017i 0.140193i 0.997540 + 0.0700964i $$0.0223307\pi$$
−0.997540 + 0.0700964i $$0.977669\pi$$
$$108$$ 0 0
$$109$$ 3.46410i 0.331801i 0.986143 + 0.165900i $$0.0530530\pi$$
−0.986143 + 0.165900i $$0.946947\pi$$
$$110$$ 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
$$115$$ − 10.5498i − 0.983777i
$$116$$ 0 0
$$117$$ 7.88054i 0.728557i
$$118$$ 0 0
$$119$$ 1.37097 0.125676
$$120$$ 0 0
$$121$$ −16.8248 −1.52952
$$122$$ 0 0
$$123$$ 0 0
$$124$$ 0 0
$$125$$ − 2.20822i − 0.197509i
$$126$$ 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
$$131$$ 5.27492i 0.460872i 0.973088 + 0.230436i $$0.0740152\pi$$
−0.973088 + 0.230436i $$0.925985\pi$$
$$132$$ 0 0
$$133$$ 0.418627i 0.0362995i
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ −12.7251 −1.08718 −0.543589 0.839352i $$-0.682935\pi$$
−0.543589 + 0.839352i $$0.682935\pi$$
$$138$$ 0 0
$$139$$ 2.72508i 0.231139i 0.993299 + 0.115569i $$0.0368692\pi$$
−0.993299 + 0.115569i $$0.963131\pi$$
$$140$$ 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
$$149$$ 16.0646i 1.31607i 0.752989 + 0.658033i $$0.228611\pi$$
−0.752989 + 0.658033i $$0.771389\pi$$
$$150$$ 0 0
$$151$$ −0.837253 −0.0681347 −0.0340674 0.999420i $$-0.510846\pi$$
−0.0340674 + 0.999420i $$0.510846\pi$$
$$152$$ 0 0
$$153$$ 9.82475 0.794284
$$154$$ 0 0
$$155$$ − 18.5498i − 1.48996i
$$156$$ 0 0
$$157$$ − 12.1819i − 0.972221i −0.873897 0.486111i $$-0.838415\pi$$
0.873897 0.486111i $$-0.161585\pi$$
$$158$$ 0 0
$$159$$ 0 0
$$160$$ 0 0
$$161$$ 1.45017 0.114289
$$162$$ 0 0
$$163$$ 4.00000i 0.313304i 0.987654 + 0.156652i $$0.0500701\pi$$
−0.987654 + 0.156652i $$0.949930\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ −19.1101 −1.47878 −0.739392 0.673275i $$-0.764887\pi$$
−0.739392 + 0.673275i $$0.764887\pi$$
$$168$$ 0 0
$$169$$ 6.09967 0.469205
$$170$$ 0 0
$$171$$ 3.00000i 0.229416i
$$172$$ 0 0
$$173$$ 8.71780i 0.662802i 0.943490 + 0.331401i $$0.107521\pi$$
−0.943490 + 0.331401i $$0.892479\pi$$
$$174$$ 0 0
$$175$$ −1.78959 −0.135281
$$176$$ 0 0
$$177$$ 0 0
$$178$$ 0 0
$$179$$ − 21.0997i − 1.57706i −0.614994 0.788532i $$-0.710842\pi$$
0.614994 0.788532i $$-0.289158\pi$$
$$180$$ 0 0
$$181$$ − 3.46410i − 0.257485i −0.991678 0.128742i $$-0.958906\pi$$
0.991678 0.128742i $$-0.0410940\pi$$
$$182$$ 0 0
$$183$$ 0 0
$$184$$ 0 0
$$185$$ −29.0997 −2.13945
$$186$$ 0 0
$$187$$ 17.2749i 1.26327i
$$188$$ 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
$$197$$ − 5.25370i − 0.374310i −0.982330 0.187155i $$-0.940073\pi$$
0.982330 0.187155i $$-0.0599267\pi$$
$$198$$ 0 0
$$199$$ −14.2750 −1.01193 −0.505965 0.862554i $$-0.668864\pi$$
−0.505965 + 0.862554i $$0.668864\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ 0 0
$$203$$ 1.09967i 0.0771816i
$$204$$ 0 0
$$205$$ − 13.8564i − 0.967773i
$$206$$ 0 0
$$207$$ 10.3923 0.722315
$$208$$ 0 0
$$209$$ −5.27492 −0.364874
$$210$$ 0 0
$$211$$ − 26.5498i − 1.82777i −0.405978 0.913883i $$-0.633069\pi$$
0.405978 0.913883i $$-0.366931\pi$$
$$212$$ 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
$$221$$ 8.60271i 0.578681i
$$222$$ 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
$$227$$ 4.00000i 0.265489i 0.991150 + 0.132745i $$0.0423790\pi$$
−0.991150 + 0.132745i $$0.957621\pi$$
$$228$$ 0 0
$$229$$ 17.7391i 1.17224i 0.810226 + 0.586118i $$0.199344\pi$$
−0.810226 + 0.586118i $$0.800656\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ −20.3746 −1.33478 −0.667392 0.744707i $$-0.732589\pi$$
−0.667392 + 0.744707i $$0.732589\pi$$
$$234$$ 0 0
$$235$$ 1.27492i 0.0831664i
$$236$$ 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
$$245$$ 20.7846i 1.32788i
$$246$$ 0 0
$$247$$ −2.62685 −0.167142
$$248$$ 0 0
$$249$$ 0 0
$$250$$ 0 0
$$251$$ − 10.7251i − 0.676961i −0.940973 0.338481i $$-0.890087\pi$$
0.940973 0.338481i $$-0.109913\pi$$
$$252$$ 0 0
$$253$$ 18.2728i 1.14880i
$$254$$ 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
$$259$$ − 4.00000i − 0.248548i
$$260$$ 0 0
$$261$$ 7.88054i 0.487793i
$$262$$ 0 0
$$263$$ 1.25588 0.0774409 0.0387204 0.999250i $$-0.487672\pi$$
0.0387204 + 0.999250i $$0.487672\pi$$
$$264$$ 0 0
$$265$$ −31.6495 −1.94421
$$266$$ 0 0
$$267$$ 0 0
$$268$$ 0 0
$$269$$ 5.13861i 0.313306i 0.987654 + 0.156653i $$0.0500705\pi$$
−0.987654 + 0.156653i $$0.949929\pi$$
$$270$$ 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
$$275$$ − 22.5498i − 1.35981i
$$276$$ 0 0
$$277$$ − 16.0646i − 0.965230i −0.875833 0.482615i $$-0.839687\pi$$
0.875833 0.482615i $$-0.160313\pi$$
$$278$$ 0 0
$$279$$ 18.2728 1.09397
$$280$$ 0 0
$$281$$ 11.0997 0.662151 0.331075 0.943604i $$-0.392589\pi$$
0.331075 + 0.943604i $$0.392589\pi$$
$$282$$ 0 0
$$283$$ − 26.3746i − 1.56781i −0.620883 0.783903i $$-0.713226\pi$$
0.620883 0.783903i $$-0.286774\pi$$
$$284$$ 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
$$293$$ 30.3397i 1.77246i 0.463244 + 0.886231i $$0.346685\pi$$
−0.463244 + 0.886231i $$0.653315\pi$$
$$294$$ 0 0
$$295$$ −19.9474 −1.16138
$$296$$ 0 0
$$297$$ 0 0
$$298$$ 0 0
$$299$$ 9.09967i 0.526247i
$$300$$ 0 0
$$301$$ 1.14079i 0.0657542i
$$302$$ 0 0
$$303$$ 0 0
$$304$$ 0 0
$$305$$ −9.27492 −0.531080
$$306$$ 0 0
$$307$$ − 1.45017i − 0.0827653i −0.999143 0.0413827i $$-0.986824\pi$$
0.999143 0.0413827i $$-0.0131763\pi$$
$$308$$ 0 0
$$309$$ 0 0
$$310$$ 0 0
$$311$$ −24.7824 −1.40528 −0.702641 0.711544i $$-0.747996\pi$$
−0.702641 + 0.711544i $$0.747996\pi$$
$$312$$ 0 0
$$313$$ 15.0997 0.853484 0.426742 0.904373i $$-0.359661\pi$$
0.426742 + 0.904373i $$0.359661\pi$$
$$314$$ 0 0
$$315$$ − 3.82475i − 0.215500i
$$316$$ 0 0
$$317$$ − 8.71780i − 0.489640i −0.969569 0.244820i $$-0.921271\pi$$
0.969569 0.244820i $$-0.0787289\pi$$
$$318$$ 0 0
$$319$$ −13.8564 −0.775810
$$320$$ 0 0
$$321$$ 0 0
$$322$$ 0 0
$$323$$ 3.27492i 0.182221i
$$324$$ 0 0
$$325$$ − 11.2296i − 0.622904i
$$326$$ 0 0
$$327$$ 0 0
$$328$$ 0 0
$$329$$ −0.175248 −0.00966175
$$330$$ 0 0
$$331$$ 9.45017i 0.519428i 0.965686 + 0.259714i $$0.0836283\pi$$
−0.965686 + 0.259714i $$0.916372\pi$$
$$332$$ 0 0
$$333$$ − 28.6652i − 1.57084i
$$334$$ 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
$$341$$ 32.1293i 1.73990i
$$342$$ 0 0
$$343$$ −5.78741 −0.312491
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ 7.82475i 0.420055i 0.977696 + 0.210027i $$0.0673553\pi$$
−0.977696 + 0.210027i $$0.932645\pi$$
$$348$$ 0 0
$$349$$ 12.7156i 0.680651i 0.940308 + 0.340326i $$0.110537\pi$$
−0.940308 + 0.340326i $$0.889463\pi$$
$$350$$ 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
$$355$$ 34.5498i 1.83371i
$$356$$ 0 0
$$357$$ 0 0
$$358$$ 0 0
$$359$$ −12.6005 −0.665030 −0.332515 0.943098i $$-0.607897\pi$$
−0.332515 + 0.943098i $$0.607897\pi$$
$$360$$ 0 0
$$361$$ −1.00000 −0.0526316
$$362$$ 0 0
$$363$$ 0 0
$$364$$ 0 0
$$365$$ 9.97368i 0.522046i
$$366$$ 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
$$371$$ − 4.35050i − 0.225867i
$$372$$ 0 0
$$373$$ 0.952341i 0.0493104i 0.999696 + 0.0246552i $$0.00784878\pi$$
−0.999696 + 0.0246552i $$0.992151\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ −6.90033 −0.355385
$$378$$ 0 0
$$379$$ 23.6495i 1.21479i 0.794399 + 0.607397i $$0.207786\pi$$
−0.794399 + 0.607397i $$0.792214\pi$$
$$380$$ 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
$$387$$ 8.17525i 0.415571i
$$388$$ 0 0
$$389$$ − 4.71998i − 0.239313i −0.992815 0.119656i $$-0.961821\pi$$
0.992815 0.119656i $$-0.0381793\pi$$
$$390$$ 0 0
$$391$$ 11.3446 0.573723
$$392$$ 0 0
$$393$$ 0 0
$$394$$ 0 0
$$395$$ − 39.6495i − 1.99498i
$$396$$ 0 0
$$397$$ 32.6630i 1.63931i 0.572859 + 0.819654i $$0.305834\pi$$
−0.572859 + 0.819654i $$0.694166\pi$$
$$398$$ 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
$$403$$ 16.0000i 0.797017i
$$404$$ 0 0
$$405$$ − 27.4093i − 1.36198i
$$406$$ 0 0
$$407$$ 50.4021 2.49834
$$408$$ 0 0
$$409$$ 18.0000 0.890043 0.445021 0.895520i $$-0.353196\pi$$
0.445021 + 0.895520i $$0.353196\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ 0 0
$$413$$ − 2.74194i − 0.134922i
$$414$$ 0 0
$$415$$ 52.0766 2.55634
$$416$$ 0 0
$$417$$ 0 0
$$418$$ 0 0
$$419$$ 12.0000i 0.586238i 0.956076 + 0.293119i $$0.0946933\pi$$
−0.956076 + 0.293119i $$0.905307\pi$$
$$420$$ 0 0
$$421$$ − 24.2487i − 1.18181i −0.806741 0.590905i $$-0.798771\pi$$
0.806741 0.590905i $$-0.201229\pi$$
$$422$$ 0 0
$$423$$ −1.25588 −0.0610630
$$424$$ 0 0
$$425$$ −14.0000 −0.679100
$$426$$ 0 0
$$427$$ − 1.27492i − 0.0616976i
$$428$$ 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
$$437$$ 3.46410i 0.165710i
$$438$$ 0 0
$$439$$ −31.2920 −1.49349 −0.746743 0.665113i $$-0.768383\pi$$
−0.746743 + 0.665113i $$0.768383\pi$$
$$440$$ 0 0
$$441$$ −20.4743 −0.974965
$$442$$ 0 0
$$443$$ − 10.3746i − 0.492911i −0.969154 0.246456i $$-0.920734\pi$$
0.969154 0.246456i $$-0.0792660\pi$$
$$444$$ 0 0
$$445$$ 30.4547i 1.44369i
$$446$$ 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
$$451$$ 24.0000i 1.13012i
$$452$$ 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 $$-0.616291\pi$$
−0.357265 + 0.934003i $$0.616291\pi$$
$$458$$ 0 0
$$459$$ 0 0
$$460$$ 0 0
$$461$$ 12.7156i 0.592225i 0.955153 + 0.296113i $$0.0956904\pi$$
−0.955153 + 0.296113i $$0.904310\pi$$
$$462$$ 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
$$467$$ − 8.17525i − 0.378305i −0.981948 0.189153i $$-0.939426\pi$$
0.981948 0.189153i $$-0.0605741\pi$$
$$468$$ 0 0
$$469$$ − 2.74194i − 0.126611i
$$470$$ 0 0
$$471$$ 0 0
$$472$$ 0 0
$$473$$ −14.3746 −0.660944
$$474$$ 0 0
$$475$$ − 4.27492i − 0.196147i
$$476$$ 0 0
$$477$$ − 31.1769i − 1.42749i
$$478$$ 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
$$485$$ − 50.4021i − 2.28864i
$$486$$ 0 0
$$487$$ 39.0575 1.76986 0.884931 0.465722i $$-0.154205\pi$$
0.884931 + 0.465722i $$0.154205\pi$$
$$488$$ 0 0
$$489$$ 0 0
$$490$$ 0 0
$$491$$ 12.0000i 0.541552i 0.962642 + 0.270776i $$0.0872803\pi$$
−0.962642 + 0.270776i $$0.912720\pi$$
$$492$$ 0 0
$$493$$ 8.60271i 0.387447i
$$494$$ 0 0
$$495$$ 48.1939 2.16615
$$496$$ 0 0
$$497$$ −4.74917 −0.213029
$$498$$ 0 0
$$499$$ − 26.7251i − 1.19638i −0.801355 0.598190i $$-0.795887\pi$$
0.801355 0.598190i $$-0.204113\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ 0 0
$$503$$ −41.6843 −1.85861 −0.929306 0.369311i $$-0.879594\pi$$
−0.929306 + 0.369311i $$0.879594\pi$$
$$504$$ 0 0
$$505$$ 42.1993 1.87785
$$506$$ 0 0
$$507$$ 0 0
$$508$$ 0 0
$$509$$ − 4.30136i − 0.190654i −0.995446 0.0953271i $$-0.969610\pi$$
0.995446 0.0953271i $$-0.0303897\pi$$
$$510$$ 0 0
$$511$$ −1.37097 −0.0606480
$$512$$ 0 0
$$513$$ 0 0
$$514$$ 0 0
$$515$$ − 18.5498i − 0.817403i
$$516$$ 0 0
$$517$$ − 2.20822i − 0.0971175i
$$518$$ 0 0
$$519$$ 0 0
$$520$$ 0 0
$$521$$ −27.0997 −1.18726 −0.593629 0.804739i $$-0.702305\pi$$
−0.593629 + 0.804739i $$0.702305\pi$$
$$522$$ 0 0
$$523$$ − 5.45017i − 0.238319i −0.992875 0.119160i $$-0.961980\pi$$
0.992875 0.119160i $$-0.0380200\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ 19.9474 0.868920
$$528$$ 0 0
$$529$$ −11.0000 −0.478261
$$530$$ 0 0
$$531$$ − 19.6495i − 0.852716i
$$532$$ 0 0
$$533$$ 11.9517i 0.517687i
$$534$$ 0 0
$$535$$ 4.41644 0.190939
$$536$$ 0 0
$$537$$ 0 0
$$538$$ 0 0
$$539$$ − 36.0000i − 1.55063i
$$540$$ 0 0
$$541$$ − 29.0838i − 1.25041i −0.780461 0.625205i $$-0.785015\pi$$
0.780461 0.625205i $$-0.214985\pi$$
$$542$$ 0 0
$$543$$ 0 0
$$544$$ 0 0
$$545$$ 10.5498 0.451905
$$546$$ 0 0
$$547$$ 32.0000i 1.36822i 0.729378 + 0.684111i $$0.239809\pi$$
−0.729378 + 0.684111i $$0.760191\pi$$
$$548$$ 0 0
$$549$$ − 9.13642i − 0.389933i
$$550$$ 0 0
$$551$$ −2.62685 −0.111907
$$552$$ 0 0
$$553$$ 5.45017 0.231765
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 0 0
$$557$$ − 29.9210i − 1.26779i −0.773417 0.633897i $$-0.781454\pi$$
0.773417 0.633897i $$-0.218546\pi$$
$$558$$ 0 0
$$559$$ −7.15838 −0.302767
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 0 0
$$563$$ − 24.0000i − 1.01148i −0.862686 0.505740i $$-0.831220\pi$$
0.862686 0.505740i $$-0.168780\pi$$
$$564$$ 0 0
$$565$$ − 38.2202i − 1.60794i
$$566$$ 0 0
$$567$$ 3.76764 0.158226
$$568$$ 0 0
$$569$$ −19.4502 −0.815393 −0.407697 0.913117i $$-0.633668\pi$$
−0.407697 + 0.913117i $$0.633668\pi$$
$$570$$ 0 0
$$571$$ 41.0997i 1.71997i 0.510321 + 0.859984i $$0.329526\pi$$
−0.510321 + 0.859984i $$0.670474\pi$$
$$572$$ 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
$$581$$ 7.15838i 0.296980i
$$582$$ 0 0
$$583$$ 54.8185 2.27035
$$584$$ 0 0
$$585$$ 24.0000 0.992278
$$586$$ 0 0
$$587$$ 37.2749i 1.53850i 0.638948 + 0.769250i $$0.279370\pi$$
−0.638948 + 0.769250i $$0.720630\pi$$
$$588$$ 0 0
$$589$$ 6.09095i 0.250973i
$$590$$ 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
$$595$$ − 4.17525i − 0.171168i
$$596$$ 0 0
$$597$$ 0 0
$$598$$ 0 0
$$599$$ −9.43996 −0.385706 −0.192853 0.981228i $$-0.561774\pi$$
−0.192853 + 0.981228i $$0.561774\pi$$
$$600$$ 0 0
$$601$$ 11.0997 0.452765 0.226382 0.974038i $$-0.427310\pi$$
0.226382 + 0.974038i $$0.427310\pi$$
$$602$$ 0 0
$$603$$ − 19.6495i − 0.800190i
$$604$$ 0 0
$$605$$ 51.2394i 2.08318i
$$606$$ 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
$$611$$ − 1.09967i − 0.0444878i
$$612$$ 0 0
$$613$$ − 29.9210i − 1.20850i −0.796795 0.604250i $$-0.793473\pi$$
0.796795 0.604250i $$-0.206527\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ 12.7251 0.512293 0.256146 0.966638i $$-0.417547\pi$$
0.256146 + 0.966638i $$0.417547\pi$$
$$618$$ 0 0
$$619$$ − 4.00000i − 0.160774i −0.996764 0.0803868i $$-0.974384\pi$$
0.996764 0.0803868i $$-0.0256155\pi$$
$$620$$ 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
$$629$$ − 31.2920i − 1.24769i
$$630$$ 0 0
$$631$$ 13.4378 0.534950 0.267475 0.963565i $$-0.413811\pi$$
0.267475 + 0.963565i $$0.413811\pi$$
$$632$$ 0 0
$$633$$ 0 0
$$634$$ 0 0
$$635$$ 53.0997i 2.10720i
$$636$$ 0 0
$$637$$ − 17.9276i − 0.710317i
$$638$$ 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
$$643$$ 44.9244i 1.77165i 0.464023 + 0.885823i $$0.346405\pi$$
−0.464023 + 0.885823i $$0.653595\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ 25.6197 1.00721 0.503607 0.863933i $$-0.332006\pi$$
0.503607 + 0.863933i $$0.332006\pi$$
$$648$$ 0 0
$$649$$ 34.5498 1.35620
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 0 0
$$653$$ − 16.0646i − 0.628657i −0.949314 0.314329i $$-0.898221\pi$$
0.949314 0.314329i $$-0.101779\pi$$
$$654$$ 0 0
$$655$$ 16.0646 0.627697
$$656$$ 0 0
$$657$$ −9.82475 −0.383300
$$658$$ 0 0
$$659$$ 13.4502i 0.523944i 0.965075 + 0.261972i $$0.0843728\pi$$
−0.965075 + 0.261972i $$0.915627\pi$$
$$660$$ 0 0
$$661$$ − 9.55505i − 0.371648i −0.982583 0.185824i $$-0.940505\pi$$
0.982583 0.185824i $$-0.0594955\pi$$
$$662$$ 0 0
$$663$$ 0 0
$$664$$ 0 0
$$665$$ 1.27492 0.0494392
$$666$$ 0 0
$$667$$ 9.09967i 0.352341i
$$668$$ 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
$$677$$ − 0.722166i − 0.0277551i −0.999904 0.0138775i $$-0.995582\pi$$
0.999904 0.0138775i $$-0.00441750\pi$$
$$678$$ 0 0
$$679$$ 6.92820 0.265880
$$680$$ 0 0
$$681$$ 0 0
$$682$$ 0 0
$$683$$ − 22.1993i − 0.849434i −0.905326 0.424717i $$-0.860374\pi$$
0.905326 0.424717i $$-0.139626\pi$$
$$684$$ 0 0
$$685$$ 38.7539i 1.48071i
$$686$$ 0 0
$$687$$ 0 0
$$688$$ 0 0
$$689$$ 27.2990 1.04001
$$690$$ 0 0
$$691$$ − 23.4743i − 0.893003i −0.894783 0.446501i $$-0.852670\pi$$
0.894783 0.446501i $$-0.147330\pi$$
$$692$$ 0 0
$$693$$ 6.62466i 0.251650i
$$694$$ 0 0
$$695$$ 8.29917 0.314806
$$696$$ 0 0
$$697$$ 14.9003 0.564390
$$698$$ 0 0
$$699$$ 0 0
$$700$$ 0 0
$$701$$ − 21.0148i − 0.793717i −0.917880 0.396859i $$-0.870100\pi$$
0.917880 0.396859i $$-0.129900\pi$$
$$702$$ 0 0
$$703$$ 9.55505 0.360376
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ 5.80066i 0.218156i
$$708$$ 0 0
$$709$$ 38.2202i 1.43539i 0.696358 + 0.717695i $$0.254803\pi$$
−0.696358 + 0.717695i $$0.745197\pi$$
$$710$$ 0 0
$$711$$ 39.0575 1.46477
$$712$$ 0 0
$$713$$ 21.0997 0.790189
$$714$$ 0 0
$$715$$ 42.1993i 1.57817i
$$716$$ 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
$$725$$ − 11.2296i − 0.417055i
$$726$$ 0 0
$$727$$ −10.9260 −0.405224 −0.202612 0.979259i $$-0.564943\pi$$
−0.202612 + 0.979259i $$0.564943\pi$$
$$728$$ 0 0
$$729$$ 27.0000 1.00000
$$730$$ 0 0
$$731$$ 8.92442i 0.330082i
$$732$$ 0 0
$$733$$ − 19.1101i − 0.705848i −0.935652 0.352924i $$-0.885187\pi$$
0.935652 0.352924i $$-0.114813\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ 34.5498 1.27266
$$738$$ 0 0
$$739$$ − 7.82475i − 0.287838i −0.989589 0.143919i $$-0.954029\pi$$
0.989589 0.143919i $$-0.0459705\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 0 0
$$743$$ −52.9139 −1.94122 −0.970611 0.240655i $$-0.922638\pi$$
−0.970611 + 0.240655i $$0.922638\pi$$
$$744$$ 0 0
$$745$$ 48.9244 1.79245
$$746$$ 0 0
$$747$$ 51.2990i 1.87693i
$$748$$ 0 0
$$749$$ 0.607078i 0.0221822i
$$750$$ 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
$$755$$ 2.54983i 0.0927980i
$$756$$ 0 0
$$757$$ 3.04547i 0.110690i 0.998467 + 0.0553448i $$0.0176258\pi$$
−0.998467 + 0.0553448i $$0.982374\pi$$
$$758$$ 0 0
$$759$$ 0 0
$$760$$ 0 0
$$761$$ 3.27492 0.118716 0.0593578 0.998237i $$-0.481095\pi$$
0.0593578 + 0.998237i $$0.481095\pi$$
$$762$$ 0 0
$$763$$ 1.45017i 0.0524995i
$$764$$ 0 0
$$765$$ − 29.9210i − 1.08180i
$$766$$ 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
$$773$$ 34.7561i 1.25009i 0.780589 + 0.625045i $$0.214919\pi$$
−0.780589 + 0.625045i $$0.785081\pi$$
$$774$$ 0 0
$$775$$ −26.0383 −0.935324
$$776$$ 0 0
$$777$$ 0 0
$$778$$ 0 0
$$779$$ 4.54983i 0.163015i
$$780$$ 0 0
$$781$$ − 59.8421i − 2.14132i
$$782$$ 0 0
$$783$$ 0 0
$$784$$ 0 0
$$785$$ −37.0997 −1.32414
$$786$$ 0 0
$$787$$ 14.1993i 0.506152i 0.967446 + 0.253076i $$0.0814422\pi$$
−0.967446 + 0.253076i $$0.918558\pi$$
$$788$$ 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
$$797$$ 24.2487i 0.858933i 0.903083 + 0.429467i $$0.141298\pi$$
−0.903083 + 0.429467i $$0.858702\pi$$
$$798$$ 0 0
$$799$$ −1.37097 −0.0485014
$$800$$ 0 0
$$801$$ −30.0000 −1.06000
$$802$$ 0 0
$$803$$ − 17.2749i − 0.609619i
$$804$$ 0 0
$$805$$ − 4.41644i − 0.155659i
$$806$$ 0 0
$$807$$ 0 0
$$808$$ 0 0
$$809$$ −5.82475 −0.204787 −0.102394 0.994744i $$-0.532650\pi$$
−0.102394 + 0.994744i $$0.532650\pi$$
$$810$$ 0 0
$$811$$ − 47.2990i − 1.66089i −0.557099 0.830446i $$-0.688085\pi$$
0.557099 0.830446i $$-0.311915\pi$$
$$812$$ 0 0
$$813$$ 0 0
$$814$$ 0 0
$$815$$ 12.1819 0.426713
$$816$$ 0 0
$$817$$ −2.72508 −0.0953386
$$818$$ 0 0
$$819$$ 3.29901i 0.115277i
$$820$$ 0 0
$$821$$ 26.5720i 0.927370i 0.886000 + 0.463685i $$0.153473\pi$$
−0.886000 + 0.463685i $$0.846527\pi$$
$$822$$ 0 0
$$823$$ −21.4334 −0.747122 −0.373561 0.927606i $$-0.621863\pi$$
−0.373561 + 0.927606i $$0.621863\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ − 47.6495i − 1.65694i −0.560037 0.828468i $$-0.689213\pi$$
0.560037 0.828468i $$-0.310787\pi$$
$$828$$ 0 0
$$829$$ − 25.3161i − 0.879266i −0.898178 0.439633i $$-0.855109\pi$$
0.898178 0.439633i $$-0.144891\pi$$
$$830$$ 0 0
$$831$$ 0 0
$$832$$ 0 0
$$833$$ −22.3505 −0.774399
$$834$$ 0 0
$$835$$ 58.1993i 2.01407i
$$836$$ 0 0
$$837$$ 0 0
$$838$$ 0 0
$$839$$ −47.6602 −1.64541 −0.822706 0.568467i $$-0.807537\pi$$
−0.822706 + 0.568467i $$0.807537\pi$$
$$840$$ 0 0
$$841$$ 22.0997 0.762058
$$842$$ 0 0
$$843$$ 0 0
$$844$$ 0 0
$$845$$ − 18.5764i − 0.639047i
$$846$$ 0 0
$$847$$ −7.04329 −0.242010
$$848$$ 0 0
$$849$$ 0 0
$$850$$ 0 0
$$851$$ − 33.0997i − 1.13464i
$$852$$ 0 0
$$853$$ 32.9665i 1.12875i 0.825518 + 0.564376i $$0.190883\pi$$
−0.825518 + 0.564376i $$0.809117\pi$$
$$854$$ 0 0
$$855$$ 9.13642 0.312459
$$856$$ 0 0
$$857$$ −16.1993 −0.553359 −0.276679 0.960962i $$-0.589234\pi$$
−0.276679 + 0.960962i $$0.589234\pi$$
$$858$$ 0 0
$$859$$ − 55.4743i − 1.89276i −0.323060 0.946379i $$-0.604711\pi$$
0.323060 0.946379i $$-0.395289\pi$$
$$860$$ 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
$$869$$ 68.6750i 2.32964i
$$870$$ 0 0
$$871$$ 17.2054 0.582983
$$872$$ 0 0
$$873$$ 49.6495 1.68038
$$874$$ 0 0
$$875$$ − 0.924421i − 0.0312511i
$$876$$ 0 0
$$877$$ − 31.4071i − 1.06054i −0.847828 0.530271i $$-0.822090\pi$$
0.847828 0.530271i $$-0.177910\pi$$
$$878$$ 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
$$883$$ − 13.6254i − 0.458532i −0.973364 0.229266i $$-0.926367\pi$$
0.973364 0.229266i $$-0.0736325\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 0 0
$$887$$ 0.837253 0.0281122 0.0140561 0.999901i $$-0.495526\pi$$
0.0140561 + 0.999901i $$0.495526\pi$$
$$888$$ 0 0
$$889$$ −7.29901 −0.244801
$$890$$ 0 0
$$891$$ 47.4743i 1.59045i
$$892$$ 0 0
$$893$$ − 0.418627i − 0.0140088i
$$894$$ 0 0
$$895$$ −64.2585 −2.14793
$$896$$ 0 0
$$897$$ 0 0
$$898$$ 0 0
$$899$$ 16.0000i 0.533630i
$$900$$ 0 0
$$901$$ − 34.0339i − 1.13383i
$$902$$ 0 0
$$903$$ 0 0
$$904$$ 0 0
$$905$$ −10.5498 −0.350688
$$906$$ 0 0
$$907$$ − 23.6495i − 0.785269i −0.919695 0.392634i $$-0.871564\pi$$
0.919695 0.392634i $$-0.128436\pi$$
$$908$$ 0 0
$$909$$ 41.5692i 1.37876i
$$910$$ 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
$$917$$ 2.20822i 0.0729219i
$$918$$ 0 0
$$919$$ 0.115088 0.00379639 0.00189820 0.999998i $$-0.499396\pi$$
0.00189820 + 0.999998i $$0.499396\pi$$
$$920$$ 0 0
$$921$$ 0 0
$$922$$ 0 0
$$923$$ − 29.8007i − 0.980901i
$$924$$ 0 0
$$925$$ 40.8471i 1.34304i
$$926$$ 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
$$931$$ − 6.82475i − 0.223672i
$$932$$ 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 $$-0.578387\pi$$
−0.243780 + 0.969831i $$0.578387\pi$$
$$938$$ 0 0
$$939$$ 0 0
$$940$$ 0 0
$$941$$ − 31.1769i − 1.01634i −0.861257 0.508169i $$-0.830322\pi$$
0.861257 0.508169i $$-0.169678\pi$$
$$942$$ 0 0
$$943$$ 15.7611 0.513252
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ − 30.1993i − 0.981347i −0.871344 0.490673i $$-0.836751\pi$$
0.871344 0.490673i $$-0.163249\pi$$
$$948$$ 0 0
$$949$$ − 8.60271i − 0.279256i
$$950$$ 0 0
$$951$$ 0 0
$$952$$ 0 0
$$953$$ −45.2990 −1.46738 −0.733689 0.679485i $$-0.762203\pi$$
−0.733689 + 0.679485i $$0.762203\pi$$
$$954$$ 0 0
$$955$$ − 35.8248i − 1.15926i
$$956$$ 0 0
$$957$$ 0 0
$$958$$ 0 0
$$959$$ −5.32706 −0.172020
$$960$$ 0 0
$$961$$ 6.09967 0.196764
$$962$$ 0 0
$$963$$ 4.35050i 0.140193i
$$964$$ 0 0
$$965$$ − 33.8038i − 1.08818i
$$966$$ 0 0
$$967$$ 55.5407 1.78607 0.893034 0.449988i $$-0.148572\pi$$
0.893034 + 0.449988i $$0.148572\pi$$
$$968$$ 0 0
$$969$$ 0 0
$$970$$ 0 0
$$971$$ 44.0000i 1.41203i 0.708198 + 0.706014i $$0.249508\pi$$
−0.708198 + 0.706014i $$0.750492\pi$$
$$972$$ 0 0
$$973$$ 1.14079i 0.0365721i
$$974$$ 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
$$979$$ − 52.7492i − 1.68587i
$$980$$ 0 0
$$981$$ 10.3923i 0.331801i
$$982$$ 0 0
$$983$$ −22.6893 −0.723676 −0.361838 0.932241i $$-0.617851\pi$$
−0.361838 + 0.932241i $$0.617851\pi$$
$$984$$ 0 0
$$985$$ −16.0000 −0.509802
$$986$$ 0 0
$$987$$ 0 0
$$988$$ 0 0
$$989$$ 9.43996i 0.300173i
$$990$$ 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
$$995$$ 43.4743i 1.37823i
$$996$$ 0 0
$$997$$ − 42.1029i − 1.33341i −0.745320 0.666707i $$-0.767703\pi$$
0.745320 0.666707i $$-0.232297\pi$$
$$998$$ 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 1216.2.c.i.609.2 yes 8
4.3 odd 2 inner 1216.2.c.i.609.1 8
8.3 odd 2 inner 1216.2.c.i.609.7 yes 8
8.5 even 2 inner 1216.2.c.i.609.8 yes 8
16.3 odd 4 4864.2.a.bj.1.1 4
16.5 even 4 4864.2.a.bj.1.4 4
16.11 odd 4 4864.2.a.bk.1.4 4
16.13 even 4 4864.2.a.bk.1.1 4

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