# Properties

 Label 2304.3.e.g.1025.4 Level $2304$ Weight $3$ Character 2304.1025 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.e (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$62.7794529086$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\sqrt{-2}, \sqrt{3})$$ Defining polynomial: $$x^{4} + 4 x^{2} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{17}]$$ Coefficient ring index: $$2^{2}\cdot 3^{3}$$ Twist minimal: no (minimal twist has level 576) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 1025.4 Root $$1.93185i$$ of defining polynomial Character $$\chi$$ $$=$$ 2304.1025 Dual form 2304.3.e.g.1025.2

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+7.34847i q^{5} +10.3923 q^{7} +O(q^{10})$$ $$q+7.34847i q^{5} +10.3923 q^{7} -8.48528i q^{11} -10.3923 q^{13} -21.2132i q^{17} -20.0000 q^{19} +14.6969i q^{23} -29.0000 q^{25} +36.7423i q^{29} +51.9615 q^{31} +76.3675i q^{35} -41.5692 q^{37} +72.1249i q^{41} +40.0000 q^{43} +73.4847i q^{47} +59.0000 q^{49} +36.7423i q^{53} +62.3538 q^{55} -33.9411i q^{59} -76.3675i q^{65} -100.000 q^{67} +73.4847i q^{71} -20.0000 q^{73} -88.1816i q^{77} -51.9615 q^{79} +127.279i q^{83} +155.885 q^{85} +12.7279i q^{89} -108.000 q^{91} -146.969i q^{95} +40.0000 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + O(q^{10})$$ $$4q - 80q^{19} - 116q^{25} + 160q^{43} + 236q^{49} - 400q^{67} - 80q^{73} - 432q^{91} + 160q^{97} + O(q^{100})$$

## 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
$$5$$ 7.34847i 1.46969i 0.678233 + 0.734847i $$0.262746\pi$$
−0.678233 + 0.734847i $$0.737254\pi$$
$$6$$ 0 0
$$7$$ 10.3923 1.48461 0.742307 0.670059i $$-0.233731\pi$$
0.742307 + 0.670059i $$0.233731\pi$$
$$8$$ 0 0
$$9$$ 0 0
$$10$$ 0 0
$$11$$ − 8.48528i − 0.771389i −0.922627 0.385695i $$-0.873962\pi$$
0.922627 0.385695i $$-0.126038\pi$$
$$12$$ 0 0
$$13$$ −10.3923 −0.799408 −0.399704 0.916644i $$-0.630887\pi$$
−0.399704 + 0.916644i $$0.630887\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 0 0
$$17$$ − 21.2132i − 1.24784i −0.781490 0.623918i $$-0.785540\pi$$
0.781490 0.623918i $$-0.214460\pi$$
$$18$$ 0 0
$$19$$ −20.0000 −1.05263 −0.526316 0.850289i $$-0.676427\pi$$
−0.526316 + 0.850289i $$0.676427\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 0 0
$$23$$ 14.6969i 0.638997i 0.947587 + 0.319499i $$0.103514\pi$$
−0.947587 + 0.319499i $$0.896486\pi$$
$$24$$ 0 0
$$25$$ −29.0000 −1.16000
$$26$$ 0 0
$$27$$ 0 0
$$28$$ 0 0
$$29$$ 36.7423i 1.26698i 0.773752 + 0.633489i $$0.218378\pi$$
−0.773752 + 0.633489i $$0.781622\pi$$
$$30$$ 0 0
$$31$$ 51.9615 1.67618 0.838089 0.545533i $$-0.183673\pi$$
0.838089 + 0.545533i $$0.183673\pi$$
$$32$$ 0 0
$$33$$ 0 0
$$34$$ 0 0
$$35$$ 76.3675i 2.18193i
$$36$$ 0 0
$$37$$ −41.5692 −1.12349 −0.561746 0.827310i $$-0.689870\pi$$
−0.561746 + 0.827310i $$0.689870\pi$$
$$38$$ 0 0
$$39$$ 0 0
$$40$$ 0 0
$$41$$ 72.1249i 1.75914i 0.475766 + 0.879572i $$0.342171\pi$$
−0.475766 + 0.879572i $$0.657829\pi$$
$$42$$ 0 0
$$43$$ 40.0000 0.930233 0.465116 0.885250i $$-0.346013\pi$$
0.465116 + 0.885250i $$0.346013\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 0 0
$$47$$ 73.4847i 1.56350i 0.623589 + 0.781752i $$0.285674\pi$$
−0.623589 + 0.781752i $$0.714326\pi$$
$$48$$ 0 0
$$49$$ 59.0000 1.20408
$$50$$ 0 0
$$51$$ 0 0
$$52$$ 0 0
$$53$$ 36.7423i 0.693252i 0.938003 + 0.346626i $$0.112673\pi$$
−0.938003 + 0.346626i $$0.887327\pi$$
$$54$$ 0 0
$$55$$ 62.3538 1.13371
$$56$$ 0 0
$$57$$ 0 0
$$58$$ 0 0
$$59$$ − 33.9411i − 0.575273i −0.957740 0.287637i $$-0.907130\pi$$
0.957740 0.287637i $$-0.0928695\pi$$
$$60$$ 0 0
$$61$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$62$$ 0 0
$$63$$ 0 0
$$64$$ 0 0
$$65$$ − 76.3675i − 1.17489i
$$66$$ 0 0
$$67$$ −100.000 −1.49254 −0.746269 0.665645i $$-0.768157\pi$$
−0.746269 + 0.665645i $$0.768157\pi$$
$$68$$ 0 0
$$69$$ 0 0
$$70$$ 0 0
$$71$$ 73.4847i 1.03500i 0.855685 + 0.517498i $$0.173136\pi$$
−0.855685 + 0.517498i $$0.826864\pi$$
$$72$$ 0 0
$$73$$ −20.0000 −0.273973 −0.136986 0.990573i $$-0.543742\pi$$
−0.136986 + 0.990573i $$0.543742\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 0 0
$$77$$ − 88.1816i − 1.14522i
$$78$$ 0 0
$$79$$ −51.9615 −0.657741 −0.328870 0.944375i $$-0.606668\pi$$
−0.328870 + 0.944375i $$0.606668\pi$$
$$80$$ 0 0
$$81$$ 0 0
$$82$$ 0 0
$$83$$ 127.279i 1.53348i 0.641955 + 0.766742i $$0.278124\pi$$
−0.641955 + 0.766742i $$0.721876\pi$$
$$84$$ 0 0
$$85$$ 155.885 1.83394
$$86$$ 0 0
$$87$$ 0 0
$$88$$ 0 0
$$89$$ 12.7279i 0.143010i 0.997440 + 0.0715052i $$0.0227802\pi$$
−0.997440 + 0.0715052i $$0.977220\pi$$
$$90$$ 0 0
$$91$$ −108.000 −1.18681
$$92$$ 0 0
$$93$$ 0 0
$$94$$ 0 0
$$95$$ − 146.969i − 1.54705i
$$96$$ 0 0
$$97$$ 40.0000 0.412371 0.206186 0.978513i $$-0.433895\pi$$
0.206186 + 0.978513i $$0.433895\pi$$
$$98$$ 0 0
$$99$$ 0 0
$$100$$ 0 0
$$101$$ − 183.712i − 1.81893i −0.415783 0.909464i $$-0.636492\pi$$
0.415783 0.909464i $$-0.363508\pi$$
$$102$$ 0 0
$$103$$ −93.5307 −0.908065 −0.454033 0.890985i $$-0.650015\pi$$
−0.454033 + 0.890985i $$0.650015\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ 169.706i 1.58603i 0.609200 + 0.793017i $$0.291491\pi$$
−0.609200 + 0.793017i $$0.708509\pi$$
$$108$$ 0 0
$$109$$ 51.9615 0.476711 0.238356 0.971178i $$-0.423392\pi$$
0.238356 + 0.971178i $$0.423392\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ 0 0
$$113$$ 63.6396i 0.563182i 0.959534 + 0.281591i $$0.0908622\pi$$
−0.959534 + 0.281591i $$0.909138\pi$$
$$114$$ 0 0
$$115$$ −108.000 −0.939130
$$116$$ 0 0
$$117$$ 0 0
$$118$$ 0 0
$$119$$ − 220.454i − 1.85256i
$$120$$ 0 0
$$121$$ 49.0000 0.404959
$$122$$ 0 0
$$123$$ 0 0
$$124$$ 0 0
$$125$$ − 29.3939i − 0.235151i
$$126$$ 0 0
$$127$$ 10.3923 0.0818292 0.0409146 0.999163i $$-0.486973\pi$$
0.0409146 + 0.999163i $$0.486973\pi$$
$$128$$ 0 0
$$129$$ 0 0
$$130$$ 0 0
$$131$$ 50.9117i 0.388639i 0.980938 + 0.194319i $$0.0622498\pi$$
−0.980938 + 0.194319i $$0.937750\pi$$
$$132$$ 0 0
$$133$$ −207.846 −1.56275
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ − 106.066i − 0.774205i −0.922037 0.387102i $$-0.873476\pi$$
0.922037 0.387102i $$-0.126524\pi$$
$$138$$ 0 0
$$139$$ −172.000 −1.23741 −0.618705 0.785623i $$-0.712342\pi$$
−0.618705 + 0.785623i $$0.712342\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 0 0
$$143$$ 88.1816i 0.616655i
$$144$$ 0 0
$$145$$ −270.000 −1.86207
$$146$$ 0 0
$$147$$ 0 0
$$148$$ 0 0
$$149$$ 110.227i 0.739779i 0.929076 + 0.369889i $$0.120604\pi$$
−0.929076 + 0.369889i $$0.879396\pi$$
$$150$$ 0 0
$$151$$ −155.885 −1.03235 −0.516174 0.856484i $$-0.672644\pi$$
−0.516174 + 0.856484i $$0.672644\pi$$
$$152$$ 0 0
$$153$$ 0 0
$$154$$ 0 0
$$155$$ 381.838i 2.46347i
$$156$$ 0 0
$$157$$ 166.277 1.05909 0.529544 0.848282i $$-0.322363\pi$$
0.529544 + 0.848282i $$0.322363\pi$$
$$158$$ 0 0
$$159$$ 0 0
$$160$$ 0 0
$$161$$ 152.735i 0.948665i
$$162$$ 0 0
$$163$$ 160.000 0.981595 0.490798 0.871274i $$-0.336705\pi$$
0.490798 + 0.871274i $$0.336705\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ − 58.7878i − 0.352022i −0.984388 0.176011i $$-0.943680\pi$$
0.984388 0.176011i $$-0.0563195\pi$$
$$168$$ 0 0
$$169$$ −61.0000 −0.360947
$$170$$ 0 0
$$171$$ 0 0
$$172$$ 0 0
$$173$$ 36.7423i 0.212384i 0.994346 + 0.106192i $$0.0338657\pi$$
−0.994346 + 0.106192i $$0.966134\pi$$
$$174$$ 0 0
$$175$$ −301.377 −1.72215
$$176$$ 0 0
$$177$$ 0 0
$$178$$ 0 0
$$179$$ 50.9117i 0.284423i 0.989836 + 0.142211i $$0.0454213\pi$$
−0.989836 + 0.142211i $$0.954579\pi$$
$$180$$ 0 0
$$181$$ 259.808 1.43540 0.717701 0.696352i $$-0.245195\pi$$
0.717701 + 0.696352i $$0.245195\pi$$
$$182$$ 0 0
$$183$$ 0 0
$$184$$ 0 0
$$185$$ − 305.470i − 1.65119i
$$186$$ 0 0
$$187$$ −180.000 −0.962567
$$188$$ 0 0
$$189$$ 0 0
$$190$$ 0 0
$$191$$ − 293.939i − 1.53895i −0.638679 0.769473i $$-0.720519\pi$$
0.638679 0.769473i $$-0.279481\pi$$
$$192$$ 0 0
$$193$$ −10.0000 −0.0518135 −0.0259067 0.999664i $$-0.508247\pi$$
−0.0259067 + 0.999664i $$0.508247\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ 22.0454i 0.111906i 0.998433 + 0.0559528i $$0.0178196\pi$$
−0.998433 + 0.0559528i $$0.982180\pi$$
$$198$$ 0 0
$$199$$ −51.9615 −0.261113 −0.130557 0.991441i $$-0.541676\pi$$
−0.130557 + 0.991441i $$0.541676\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ 0 0
$$203$$ 381.838i 1.88097i
$$204$$ 0 0
$$205$$ −530.008 −2.58540
$$206$$ 0 0
$$207$$ 0 0
$$208$$ 0 0
$$209$$ 169.706i 0.811989i
$$210$$ 0 0
$$211$$ −172.000 −0.815166 −0.407583 0.913168i $$-0.633628\pi$$
−0.407583 + 0.913168i $$0.633628\pi$$
$$212$$ 0 0
$$213$$ 0 0
$$214$$ 0 0
$$215$$ 293.939i 1.36716i
$$216$$ 0 0
$$217$$ 540.000 2.48848
$$218$$ 0 0
$$219$$ 0 0
$$220$$ 0 0
$$221$$ 220.454i 0.997530i
$$222$$ 0 0
$$223$$ 10.3923 0.0466023 0.0233011 0.999728i $$-0.492582\pi$$
0.0233011 + 0.999728i $$0.492582\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 0 0
$$227$$ 42.4264i 0.186900i 0.995624 + 0.0934502i $$0.0297896\pi$$
−0.995624 + 0.0934502i $$0.970210\pi$$
$$228$$ 0 0
$$229$$ −259.808 −1.13453 −0.567266 0.823535i $$-0.691999\pi$$
−0.567266 + 0.823535i $$0.691999\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ 106.066i 0.455219i 0.973752 + 0.227609i $$0.0730910\pi$$
−0.973752 + 0.227609i $$0.926909\pi$$
$$234$$ 0 0
$$235$$ −540.000 −2.29787
$$236$$ 0 0
$$237$$ 0 0
$$238$$ 0 0
$$239$$ 146.969i 0.614935i 0.951559 + 0.307467i $$0.0994815\pi$$
−0.951559 + 0.307467i $$0.900519\pi$$
$$240$$ 0 0
$$241$$ 140.000 0.580913 0.290456 0.956888i $$-0.406193\pi$$
0.290456 + 0.956888i $$0.406193\pi$$
$$242$$ 0 0
$$243$$ 0 0
$$244$$ 0 0
$$245$$ 433.560i 1.76963i
$$246$$ 0 0
$$247$$ 207.846 0.841482
$$248$$ 0 0
$$249$$ 0 0
$$250$$ 0 0
$$251$$ 246.073i 0.980371i 0.871618 + 0.490186i $$0.163071\pi$$
−0.871618 + 0.490186i $$0.836929\pi$$
$$252$$ 0 0
$$253$$ 124.708 0.492916
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 0 0
$$257$$ 21.2132i 0.0825416i 0.999148 + 0.0412708i $$0.0131406\pi$$
−0.999148 + 0.0412708i $$0.986859\pi$$
$$258$$ 0 0
$$259$$ −432.000 −1.66795
$$260$$ 0 0
$$261$$ 0 0
$$262$$ 0 0
$$263$$ − 88.1816i − 0.335291i −0.985847 0.167646i $$-0.946384\pi$$
0.985847 0.167646i $$-0.0536165\pi$$
$$264$$ 0 0
$$265$$ −270.000 −1.01887
$$266$$ 0 0
$$267$$ 0 0
$$268$$ 0 0
$$269$$ 110.227i 0.409766i 0.978786 + 0.204883i $$0.0656814\pi$$
−0.978786 + 0.204883i $$0.934319\pi$$
$$270$$ 0 0
$$271$$ 467.654 1.72566 0.862830 0.505495i $$-0.168690\pi$$
0.862830 + 0.505495i $$0.168690\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ 0 0
$$275$$ 246.073i 0.894811i
$$276$$ 0 0
$$277$$ −322.161 −1.16304 −0.581519 0.813533i $$-0.697541\pi$$
−0.581519 + 0.813533i $$0.697541\pi$$
$$278$$ 0 0
$$279$$ 0 0
$$280$$ 0 0
$$281$$ − 140.007i − 0.498246i −0.968472 0.249123i $$-0.919858\pi$$
0.968472 0.249123i $$-0.0801424\pi$$
$$282$$ 0 0
$$283$$ 80.0000 0.282686 0.141343 0.989961i $$-0.454858\pi$$
0.141343 + 0.989961i $$0.454858\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ 749.544i 2.61165i
$$288$$ 0 0
$$289$$ −161.000 −0.557093
$$290$$ 0 0
$$291$$ 0 0
$$292$$ 0 0
$$293$$ − 315.984i − 1.07844i −0.842164 0.539222i $$-0.818718\pi$$
0.842164 0.539222i $$-0.181282\pi$$
$$294$$ 0 0
$$295$$ 249.415 0.845476
$$296$$ 0 0
$$297$$ 0 0
$$298$$ 0 0
$$299$$ − 152.735i − 0.510820i
$$300$$ 0 0
$$301$$ 415.692 1.38104
$$302$$ 0 0
$$303$$ 0 0
$$304$$ 0 0
$$305$$ 0 0
$$306$$ 0 0
$$307$$ 380.000 1.23779 0.618893 0.785476i $$-0.287582\pi$$
0.618893 + 0.785476i $$0.287582\pi$$
$$308$$ 0 0
$$309$$ 0 0
$$310$$ 0 0
$$311$$ − 146.969i − 0.472570i −0.971684 0.236285i $$-0.924070\pi$$
0.971684 0.236285i $$-0.0759300\pi$$
$$312$$ 0 0
$$313$$ −310.000 −0.990415 −0.495208 0.868775i $$-0.664908\pi$$
−0.495208 + 0.868775i $$0.664908\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ − 242.499i − 0.764983i −0.923959 0.382491i $$-0.875066\pi$$
0.923959 0.382491i $$-0.124934\pi$$
$$318$$ 0 0
$$319$$ 311.769 0.977333
$$320$$ 0 0
$$321$$ 0 0
$$322$$ 0 0
$$323$$ 424.264i 1.31351i
$$324$$ 0 0
$$325$$ 301.377 0.927313
$$326$$ 0 0
$$327$$ 0 0
$$328$$ 0 0
$$329$$ 763.675i 2.32120i
$$330$$ 0 0
$$331$$ 500.000 1.51057 0.755287 0.655394i $$-0.227497\pi$$
0.755287 + 0.655394i $$0.227497\pi$$
$$332$$ 0 0
$$333$$ 0 0
$$334$$ 0 0
$$335$$ − 734.847i − 2.19357i
$$336$$ 0 0
$$337$$ 100.000 0.296736 0.148368 0.988932i $$-0.452598\pi$$
0.148368 + 0.988932i $$0.452598\pi$$
$$338$$ 0 0
$$339$$ 0 0
$$340$$ 0 0
$$341$$ − 440.908i − 1.29299i
$$342$$ 0 0
$$343$$ 103.923 0.302983
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ 42.4264i 0.122266i 0.998130 + 0.0611332i $$0.0194714\pi$$
−0.998130 + 0.0611332i $$0.980529\pi$$
$$348$$ 0 0
$$349$$ −207.846 −0.595548 −0.297774 0.954636i $$-0.596244\pi$$
−0.297774 + 0.954636i $$0.596244\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 0 0
$$353$$ − 615.183i − 1.74273i −0.490637 0.871364i $$-0.663236\pi$$
0.490637 0.871364i $$-0.336764\pi$$
$$354$$ 0 0
$$355$$ −540.000 −1.52113
$$356$$ 0 0
$$357$$ 0 0
$$358$$ 0 0
$$359$$ 514.393i 1.43285i 0.697665 + 0.716425i $$0.254223\pi$$
−0.697665 + 0.716425i $$0.745777\pi$$
$$360$$ 0 0
$$361$$ 39.0000 0.108033
$$362$$ 0 0
$$363$$ 0 0
$$364$$ 0 0
$$365$$ − 146.969i − 0.402656i
$$366$$ 0 0
$$367$$ 218.238 0.594655 0.297328 0.954776i $$-0.403905\pi$$
0.297328 + 0.954776i $$0.403905\pi$$
$$368$$ 0 0
$$369$$ 0 0
$$370$$ 0 0
$$371$$ 381.838i 1.02921i
$$372$$ 0 0
$$373$$ 665.108 1.78313 0.891565 0.452893i $$-0.149608\pi$$
0.891565 + 0.452893i $$0.149608\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ − 381.838i − 1.01283i
$$378$$ 0 0
$$379$$ −92.0000 −0.242744 −0.121372 0.992607i $$-0.538729\pi$$
−0.121372 + 0.992607i $$0.538729\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ 0 0
$$383$$ 146.969i 0.383732i 0.981421 + 0.191866i $$0.0614539\pi$$
−0.981421 + 0.191866i $$0.938546\pi$$
$$384$$ 0 0
$$385$$ 648.000 1.68312
$$386$$ 0 0
$$387$$ 0 0
$$388$$ 0 0
$$389$$ 257.196i 0.661173i 0.943776 + 0.330587i $$0.107247\pi$$
−0.943776 + 0.330587i $$0.892753\pi$$
$$390$$ 0 0
$$391$$ 311.769 0.797364
$$392$$ 0 0
$$393$$ 0 0
$$394$$ 0 0
$$395$$ − 381.838i − 0.966678i
$$396$$ 0 0
$$397$$ 41.5692 0.104708 0.0523542 0.998629i $$-0.483328\pi$$
0.0523542 + 0.998629i $$0.483328\pi$$
$$398$$ 0 0
$$399$$ 0 0
$$400$$ 0 0
$$401$$ − 224.860i − 0.560748i −0.959891 0.280374i $$-0.909542\pi$$
0.959891 0.280374i $$-0.0904585\pi$$
$$402$$ 0 0
$$403$$ −540.000 −1.33995
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 0 0
$$407$$ 352.727i 0.866650i
$$408$$ 0 0
$$409$$ −368.000 −0.899756 −0.449878 0.893090i $$-0.648532\pi$$
−0.449878 + 0.893090i $$0.648532\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ 0 0
$$413$$ − 352.727i − 0.854059i
$$414$$ 0 0
$$415$$ −935.307 −2.25375
$$416$$ 0 0
$$417$$ 0 0
$$418$$ 0 0
$$419$$ 432.749i 1.03281i 0.856343 + 0.516407i $$0.172731\pi$$
−0.856343 + 0.516407i $$0.827269\pi$$
$$420$$ 0 0
$$421$$ 51.9615 0.123424 0.0617120 0.998094i $$-0.480344\pi$$
0.0617120 + 0.998094i $$0.480344\pi$$
$$422$$ 0 0
$$423$$ 0 0
$$424$$ 0 0
$$425$$ 615.183i 1.44749i
$$426$$ 0 0
$$427$$ 0 0
$$428$$ 0 0
$$429$$ 0 0
$$430$$ 0 0
$$431$$ 367.423i 0.852491i 0.904608 + 0.426245i $$0.140164\pi$$
−0.904608 + 0.426245i $$0.859836\pi$$
$$432$$ 0 0
$$433$$ −470.000 −1.08545 −0.542725 0.839910i $$-0.682607\pi$$
−0.542725 + 0.839910i $$0.682607\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 0 0
$$437$$ − 293.939i − 0.672629i
$$438$$ 0 0
$$439$$ −155.885 −0.355090 −0.177545 0.984113i $$-0.556816\pi$$
−0.177545 + 0.984113i $$0.556816\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 0 0
$$443$$ − 551.543i − 1.24502i −0.782612 0.622509i $$-0.786113\pi$$
0.782612 0.622509i $$-0.213887\pi$$
$$444$$ 0 0
$$445$$ −93.5307 −0.210181
$$446$$ 0 0
$$447$$ 0 0
$$448$$ 0 0
$$449$$ − 691.550i − 1.54020i −0.637922 0.770101i $$-0.720206\pi$$
0.637922 0.770101i $$-0.279794\pi$$
$$450$$ 0 0
$$451$$ 612.000 1.35698
$$452$$ 0 0
$$453$$ 0 0
$$454$$ 0 0
$$455$$ − 793.635i − 1.74425i
$$456$$ 0 0
$$457$$ −680.000 −1.48796 −0.743982 0.668199i $$-0.767065\pi$$
−0.743982 + 0.668199i $$0.767065\pi$$
$$458$$ 0 0
$$459$$ 0 0
$$460$$ 0 0
$$461$$ − 330.681i − 0.717313i −0.933470 0.358656i $$-0.883235\pi$$
0.933470 0.358656i $$-0.116765\pi$$
$$462$$ 0 0
$$463$$ 322.161 0.695813 0.347907 0.937529i $$-0.386893\pi$$
0.347907 + 0.937529i $$0.386893\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ − 466.690i − 0.999337i −0.866217 0.499669i $$-0.833455\pi$$
0.866217 0.499669i $$-0.166545\pi$$
$$468$$ 0 0
$$469$$ −1039.23 −2.21584
$$470$$ 0 0
$$471$$ 0 0
$$472$$ 0 0
$$473$$ − 339.411i − 0.717571i
$$474$$ 0 0
$$475$$ 580.000 1.22105
$$476$$ 0 0
$$477$$ 0 0
$$478$$ 0 0
$$479$$ 220.454i 0.460238i 0.973162 + 0.230119i $$0.0739116\pi$$
−0.973162 + 0.230119i $$0.926088\pi$$
$$480$$ 0 0
$$481$$ 432.000 0.898129
$$482$$ 0 0
$$483$$ 0 0
$$484$$ 0 0
$$485$$ 293.939i 0.606059i
$$486$$ 0 0
$$487$$ −633.931 −1.30171 −0.650853 0.759204i $$-0.725588\pi$$
−0.650853 + 0.759204i $$0.725588\pi$$
$$488$$ 0 0
$$489$$ 0 0
$$490$$ 0 0
$$491$$ − 220.617i − 0.449322i −0.974437 0.224661i $$-0.927872\pi$$
0.974437 0.224661i $$-0.0721275\pi$$
$$492$$ 0 0
$$493$$ 779.423 1.58098
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ 763.675i 1.53657i
$$498$$ 0 0
$$499$$ −640.000 −1.28257 −0.641283 0.767305i $$-0.721597\pi$$
−0.641283 + 0.767305i $$0.721597\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ 0 0
$$503$$ − 73.4847i − 0.146093i −0.997329 0.0730464i $$-0.976728\pi$$
0.997329 0.0730464i $$-0.0232721\pi$$
$$504$$ 0 0
$$505$$ 1350.00 2.67327
$$506$$ 0 0
$$507$$ 0 0
$$508$$ 0 0
$$509$$ 624.620i 1.22715i 0.789636 + 0.613576i $$0.210269\pi$$
−0.789636 + 0.613576i $$0.789731\pi$$
$$510$$ 0 0
$$511$$ −207.846 −0.406744
$$512$$ 0 0
$$513$$ 0 0
$$514$$ 0 0
$$515$$ − 687.308i − 1.33458i
$$516$$ 0 0
$$517$$ 623.538 1.20607
$$518$$ 0 0
$$519$$ 0 0
$$520$$ 0 0
$$521$$ 861.256i 1.65308i 0.562876 + 0.826541i $$0.309695\pi$$
−0.562876 + 0.826541i $$0.690305\pi$$
$$522$$ 0 0
$$523$$ −20.0000 −0.0382409 −0.0191205 0.999817i $$-0.506087\pi$$
−0.0191205 + 0.999817i $$0.506087\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ − 1102.27i − 2.09159i
$$528$$ 0 0
$$529$$ 313.000 0.591682
$$530$$ 0 0
$$531$$ 0 0
$$532$$ 0 0
$$533$$ − 749.544i − 1.40627i
$$534$$ 0 0
$$535$$ −1247.08 −2.33098
$$536$$ 0 0
$$537$$ 0 0
$$538$$ 0 0
$$539$$ − 500.632i − 0.928816i
$$540$$ 0 0
$$541$$ −363.731 −0.672330 −0.336165 0.941803i $$-0.609130\pi$$
−0.336165 + 0.941803i $$0.609130\pi$$
$$542$$ 0 0
$$543$$ 0 0
$$544$$ 0 0
$$545$$ 381.838i 0.700620i
$$546$$ 0 0
$$547$$ 760.000 1.38940 0.694698 0.719301i $$-0.255538\pi$$
0.694698 + 0.719301i $$0.255538\pi$$
$$548$$ 0 0
$$549$$ 0 0
$$550$$ 0 0
$$551$$ − 734.847i − 1.33366i
$$552$$ 0 0
$$553$$ −540.000 −0.976492
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 0 0
$$557$$ 992.043i 1.78105i 0.454937 + 0.890524i $$0.349662\pi$$
−0.454937 + 0.890524i $$0.650338\pi$$
$$558$$ 0 0
$$559$$ −415.692 −0.743635
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 0 0
$$563$$ − 212.132i − 0.376789i −0.982093 0.188394i $$-0.939672\pi$$
0.982093 0.188394i $$-0.0603283\pi$$
$$564$$ 0 0
$$565$$ −467.654 −0.827706
$$566$$ 0 0
$$567$$ 0 0
$$568$$ 0 0
$$569$$ − 479.418i − 0.842563i −0.906930 0.421282i $$-0.861580\pi$$
0.906930 0.421282i $$-0.138420\pi$$
$$570$$ 0 0
$$571$$ −280.000 −0.490368 −0.245184 0.969477i $$-0.578848\pi$$
−0.245184 + 0.969477i $$0.578848\pi$$
$$572$$ 0 0
$$573$$ 0 0
$$574$$ 0 0
$$575$$ − 426.211i − 0.741237i
$$576$$ 0 0
$$577$$ −650.000 −1.12652 −0.563258 0.826281i $$-0.690452\pi$$
−0.563258 + 0.826281i $$0.690452\pi$$
$$578$$ 0 0
$$579$$ 0 0
$$580$$ 0 0
$$581$$ 1322.72i 2.27663i
$$582$$ 0 0
$$583$$ 311.769 0.534767
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ 678.823i 1.15643i 0.815886 + 0.578213i $$0.196250\pi$$
−0.815886 + 0.578213i $$0.803750\pi$$
$$588$$ 0 0
$$589$$ −1039.23 −1.76440
$$590$$ 0 0
$$591$$ 0 0
$$592$$ 0 0
$$593$$ 445.477i 0.751226i 0.926777 + 0.375613i $$0.122568\pi$$
−0.926777 + 0.375613i $$0.877432\pi$$
$$594$$ 0 0
$$595$$ 1620.00 2.72269
$$596$$ 0 0
$$597$$ 0 0
$$598$$ 0 0
$$599$$ 367.423i 0.613395i 0.951807 + 0.306697i $$0.0992240\pi$$
−0.951807 + 0.306697i $$0.900776\pi$$
$$600$$ 0 0
$$601$$ 490.000 0.815308 0.407654 0.913137i $$-0.366347\pi$$
0.407654 + 0.913137i $$0.366347\pi$$
$$602$$ 0 0
$$603$$ 0 0
$$604$$ 0 0
$$605$$ 360.075i 0.595165i
$$606$$ 0 0
$$607$$ −322.161 −0.530744 −0.265372 0.964146i $$-0.585495\pi$$
−0.265372 + 0.964146i $$0.585495\pi$$
$$608$$ 0 0
$$609$$ 0 0
$$610$$ 0 0
$$611$$ − 763.675i − 1.24988i
$$612$$ 0 0
$$613$$ 374.123 0.610315 0.305157 0.952302i $$-0.401291\pi$$
0.305157 + 0.952302i $$0.401291\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ − 784.889i − 1.27210i −0.771646 0.636052i $$-0.780566\pi$$
0.771646 0.636052i $$-0.219434\pi$$
$$618$$ 0 0
$$619$$ 1112.00 1.79645 0.898223 0.439540i $$-0.144859\pi$$
0.898223 + 0.439540i $$0.144859\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ 0 0
$$623$$ 132.272i 0.212315i
$$624$$ 0 0
$$625$$ −509.000 −0.814400
$$626$$ 0 0
$$627$$ 0 0
$$628$$ 0 0
$$629$$ 881.816i 1.40193i
$$630$$ 0 0
$$631$$ 987.269 1.56461 0.782305 0.622896i $$-0.214044\pi$$
0.782305 + 0.622896i $$0.214044\pi$$
$$632$$ 0 0
$$633$$ 0 0
$$634$$ 0 0
$$635$$ 76.3675i 0.120264i
$$636$$ 0 0
$$637$$ −613.146 −0.962553
$$638$$ 0 0
$$639$$ 0 0
$$640$$ 0 0
$$641$$ 97.5807i 0.152232i 0.997099 + 0.0761160i $$0.0242519\pi$$
−0.997099 + 0.0761160i $$0.975748\pi$$
$$642$$ 0 0
$$643$$ −80.0000 −0.124417 −0.0622084 0.998063i $$-0.519814\pi$$
−0.0622084 + 0.998063i $$0.519814\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ 514.393i 0.795043i 0.917593 + 0.397522i $$0.130130\pi$$
−0.917593 + 0.397522i $$0.869870\pi$$
$$648$$ 0 0
$$649$$ −288.000 −0.443760
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 0 0
$$653$$ − 1080.22i − 1.65425i −0.562018 0.827125i $$-0.689975\pi$$
0.562018 0.827125i $$-0.310025\pi$$
$$654$$ 0 0
$$655$$ −374.123 −0.571180
$$656$$ 0 0
$$657$$ 0 0
$$658$$ 0 0
$$659$$ − 712.764i − 1.08158i −0.841156 0.540792i $$-0.818125\pi$$
0.841156 0.540792i $$-0.181875\pi$$
$$660$$ 0 0
$$661$$ −1247.08 −1.88665 −0.943326 0.331868i $$-0.892321\pi$$
−0.943326 + 0.331868i $$0.892321\pi$$
$$662$$ 0 0
$$663$$ 0 0
$$664$$ 0 0
$$665$$ − 1527.35i − 2.29677i
$$666$$ 0 0
$$667$$ −540.000 −0.809595
$$668$$ 0 0
$$669$$ 0 0
$$670$$ 0 0
$$671$$ 0 0
$$672$$ 0 0
$$673$$ −50.0000 −0.0742942 −0.0371471 0.999310i $$-0.511827\pi$$
−0.0371471 + 0.999310i $$0.511827\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ 0 0
$$677$$ − 169.015i − 0.249653i −0.992179 0.124826i $$-0.960163\pi$$
0.992179 0.124826i $$-0.0398374\pi$$
$$678$$ 0 0
$$679$$ 415.692 0.612212
$$680$$ 0 0
$$681$$ 0 0
$$682$$ 0 0
$$683$$ − 42.4264i − 0.0621177i −0.999518 0.0310589i $$-0.990112\pi$$
0.999518 0.0310589i $$-0.00988793\pi$$
$$684$$ 0 0
$$685$$ 779.423 1.13784
$$686$$ 0 0
$$687$$ 0 0
$$688$$ 0 0
$$689$$ − 381.838i − 0.554191i
$$690$$ 0 0
$$691$$ 400.000 0.578871 0.289436 0.957197i $$-0.406532\pi$$
0.289436 + 0.957197i $$0.406532\pi$$
$$692$$ 0 0
$$693$$ 0 0
$$694$$ 0 0
$$695$$ − 1263.94i − 1.81861i
$$696$$ 0 0
$$697$$ 1530.00 2.19512
$$698$$ 0 0
$$699$$ 0 0
$$700$$ 0 0
$$701$$ − 992.043i − 1.41518i −0.706622 0.707592i $$-0.749782\pi$$
0.706622 0.707592i $$-0.250218\pi$$
$$702$$ 0 0
$$703$$ 831.384 1.18262
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ − 1909.19i − 2.70041i
$$708$$ 0 0
$$709$$ 1195.12 1.68563 0.842817 0.538200i $$-0.180895\pi$$
0.842817 + 0.538200i $$0.180895\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ 0 0
$$713$$ 763.675i 1.07107i
$$714$$ 0 0
$$715$$ −648.000 −0.906294
$$716$$ 0 0
$$717$$ 0 0
$$718$$ 0 0
$$719$$ 661.362i 0.919836i 0.887961 + 0.459918i $$0.152121\pi$$
−0.887961 + 0.459918i $$0.847879\pi$$
$$720$$ 0 0
$$721$$ −972.000 −1.34813
$$722$$ 0 0
$$723$$ 0 0
$$724$$ 0 0
$$725$$ − 1065.53i − 1.46969i
$$726$$ 0 0
$$727$$ −93.5307 −0.128653 −0.0643265 0.997929i $$-0.520490\pi$$
−0.0643265 + 0.997929i $$0.520490\pi$$
$$728$$ 0 0
$$729$$ 0 0
$$730$$ 0 0
$$731$$ − 848.528i − 1.16078i
$$732$$ 0 0
$$733$$ 613.146 0.836488 0.418244 0.908335i $$-0.362646\pi$$
0.418244 + 0.908335i $$0.362646\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ 848.528i 1.15133i
$$738$$ 0 0
$$739$$ 920.000 1.24493 0.622463 0.782649i $$-0.286132\pi$$
0.622463 + 0.782649i $$0.286132\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 0 0
$$743$$ 1175.76i 1.58244i 0.611530 + 0.791221i $$0.290554\pi$$
−0.611530 + 0.791221i $$0.709446\pi$$
$$744$$ 0 0
$$745$$ −810.000 −1.08725
$$746$$ 0 0
$$747$$ 0 0
$$748$$ 0 0
$$749$$ 1763.63i 2.35465i
$$750$$ 0 0
$$751$$ 51.9615 0.0691898 0.0345949 0.999401i $$-0.488986\pi$$
0.0345949 + 0.999401i $$0.488986\pi$$
$$752$$ 0 0
$$753$$ 0 0
$$754$$ 0 0
$$755$$ − 1145.51i − 1.51724i
$$756$$ 0 0
$$757$$ 405.300 0.535403 0.267701 0.963502i $$-0.413736\pi$$
0.267701 + 0.963502i $$0.413736\pi$$
$$758$$ 0 0
$$759$$ 0 0
$$760$$ 0 0
$$761$$ − 1030.96i − 1.35475i −0.735640 0.677373i $$-0.763118\pi$$
0.735640 0.677373i $$-0.236882\pi$$
$$762$$ 0 0
$$763$$ 540.000 0.707733
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ 352.727i 0.459878i
$$768$$ 0 0
$$769$$ 890.000 1.15735 0.578674 0.815559i $$-0.303571\pi$$
0.578674 + 0.815559i $$0.303571\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ 0 0
$$773$$ − 933.256i − 1.20732i −0.797243 0.603658i $$-0.793709\pi$$
0.797243 0.603658i $$-0.206291\pi$$
$$774$$ 0 0
$$775$$ −1506.88 −1.94437
$$776$$ 0 0
$$777$$ 0 0
$$778$$ 0 0
$$779$$ − 1442.50i − 1.85173i
$$780$$ 0 0
$$781$$ 623.538 0.798384
$$782$$ 0 0
$$783$$ 0 0
$$784$$ 0 0
$$785$$ 1221.88i 1.55654i
$$786$$ 0 0
$$787$$ 100.000 0.127065 0.0635324 0.997980i $$-0.479763\pi$$
0.0635324 + 0.997980i $$0.479763\pi$$
$$788$$ 0 0
$$789$$ 0 0
$$790$$ 0 0
$$791$$ 661.362i 0.836109i
$$792$$ 0 0
$$793$$ 0 0
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 0 0
$$797$$ − 404.166i − 0.507109i −0.967321 0.253554i $$-0.918400\pi$$
0.967321 0.253554i $$-0.0815997\pi$$
$$798$$ 0 0
$$799$$ 1558.85 1.95100
$$800$$ 0 0
$$801$$ 0 0
$$802$$ 0 0
$$803$$ 169.706i 0.211340i
$$804$$ 0 0
$$805$$ −1122.37 −1.39425
$$806$$ 0 0
$$807$$ 0 0
$$808$$ 0 0
$$809$$ 1090.36i 1.34779i 0.738829 + 0.673893i $$0.235379\pi$$
−0.738829 + 0.673893i $$0.764621\pi$$
$$810$$ 0 0
$$811$$ 772.000 0.951911 0.475956 0.879469i $$-0.342102\pi$$
0.475956 + 0.879469i $$0.342102\pi$$
$$812$$ 0 0
$$813$$ 0 0
$$814$$ 0 0
$$815$$ 1175.76i 1.44264i
$$816$$ 0 0
$$817$$ −800.000 −0.979192
$$818$$ 0 0
$$819$$ 0 0
$$820$$ 0 0
$$821$$ − 477.650i − 0.581791i −0.956755 0.290896i $$-0.906047\pi$$
0.956755 0.290896i $$-0.0939532\pi$$
$$822$$ 0 0
$$823$$ 1132.76 1.37638 0.688190 0.725530i $$-0.258405\pi$$
0.688190 + 0.725530i $$0.258405\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ 763.675i 0.923428i 0.887029 + 0.461714i $$0.152765\pi$$
−0.887029 + 0.461714i $$0.847235\pi$$
$$828$$ 0 0
$$829$$ −363.731 −0.438758 −0.219379 0.975640i $$-0.570403\pi$$
−0.219379 + 0.975640i $$0.570403\pi$$
$$830$$ 0 0
$$831$$ 0 0
$$832$$ 0 0
$$833$$ − 1251.58i − 1.50250i
$$834$$ 0 0
$$835$$ 432.000 0.517365
$$836$$ 0 0
$$837$$ 0 0
$$838$$ 0 0
$$839$$ 808.332i 0.963447i 0.876323 + 0.481723i $$0.159989\pi$$
−0.876323 + 0.481723i $$0.840011\pi$$
$$840$$ 0 0
$$841$$ −509.000 −0.605232
$$842$$ 0 0
$$843$$ 0 0
$$844$$ 0 0
$$845$$ − 448.257i − 0.530481i
$$846$$ 0 0
$$847$$ 509.223 0.601208
$$848$$ 0 0
$$849$$ 0 0
$$850$$ 0 0
$$851$$ − 610.940i − 0.717909i
$$852$$ 0 0
$$853$$ 997.661 1.16959 0.584796 0.811181i $$-0.301175\pi$$
0.584796 + 0.811181i $$0.301175\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0 0
$$857$$ − 572.756i − 0.668327i −0.942515 0.334164i $$-0.891546\pi$$
0.942515 0.334164i $$-0.108454\pi$$
$$858$$ 0 0
$$859$$ 568.000 0.661234 0.330617 0.943765i $$-0.392743\pi$$
0.330617 + 0.943765i $$0.392743\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 0 0
$$863$$ − 823.029i − 0.953683i −0.878989 0.476842i $$-0.841782\pi$$
0.878989 0.476842i $$-0.158218\pi$$
$$864$$ 0 0
$$865$$ −270.000 −0.312139
$$866$$ 0 0
$$867$$ 0 0
$$868$$ 0 0
$$869$$ 440.908i 0.507374i
$$870$$ 0 0
$$871$$ 1039.23 1.19315
$$872$$ 0 0
$$873$$ 0 0
$$874$$ 0 0
$$875$$ − 305.470i − 0.349109i
$$876$$ 0 0
$$877$$ 41.5692 0.0473993 0.0236997 0.999719i $$-0.492455\pi$$
0.0236997 + 0.999719i $$0.492455\pi$$
$$878$$ 0 0
$$879$$ 0 0
$$880$$ 0 0
$$881$$ − 12.7279i − 0.0144471i −0.999974 0.00722357i $$-0.997701\pi$$
0.999974 0.00722357i $$-0.00229935\pi$$
$$882$$ 0 0
$$883$$ 520.000 0.588901 0.294451 0.955667i $$-0.404863\pi$$
0.294451 + 0.955667i $$0.404863\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 0 0
$$887$$ 88.1816i 0.0994156i 0.998764 + 0.0497078i $$0.0158290\pi$$
−0.998764 + 0.0497078i $$0.984171\pi$$
$$888$$ 0 0
$$889$$ 108.000 0.121485
$$890$$ 0 0
$$891$$ 0 0
$$892$$ 0 0
$$893$$ − 1469.69i − 1.64579i
$$894$$ 0 0
$$895$$ −374.123 −0.418014
$$896$$ 0 0
$$897$$ 0 0
$$898$$ 0 0
$$899$$ 1909.19i 2.12368i
$$900$$ 0 0
$$901$$ 779.423 0.865064
$$902$$ 0 0
$$903$$ 0 0
$$904$$ 0 0
$$905$$ 1909.19i 2.10960i
$$906$$ 0 0
$$907$$ 40.0000 0.0441014 0.0220507 0.999757i $$-0.492980\pi$$
0.0220507 + 0.999757i $$0.492980\pi$$
$$908$$ 0 0
$$909$$ 0 0
$$910$$ 0 0
$$911$$ 1175.76i 1.29062i 0.763921 + 0.645310i $$0.223272\pi$$
−0.763921 + 0.645310i $$0.776728\pi$$
$$912$$ 0 0
$$913$$ 1080.00 1.18291
$$914$$ 0 0
$$915$$ 0 0
$$916$$ 0 0
$$917$$ 529.090i 0.576979i
$$918$$ 0 0
$$919$$ 1714.73 1.86587 0.932933 0.360051i $$-0.117241\pi$$
0.932933 + 0.360051i $$0.117241\pi$$
$$920$$ 0 0
$$921$$ 0 0
$$922$$ 0 0
$$923$$ − 763.675i − 0.827384i
$$924$$ 0 0
$$925$$ 1205.51 1.30325
$$926$$ 0 0
$$927$$ 0 0
$$928$$ 0 0
$$929$$ − 1412.80i − 1.52077i −0.649470 0.760387i $$-0.725009\pi$$
0.649470 0.760387i $$-0.274991\pi$$
$$930$$ 0 0
$$931$$ −1180.00 −1.26745
$$932$$ 0 0
$$933$$ 0 0
$$934$$ 0 0
$$935$$ − 1322.72i − 1.41468i
$$936$$ 0 0
$$937$$ 470.000 0.501601 0.250800 0.968039i $$-0.419306\pi$$
0.250800 + 0.968039i $$0.419306\pi$$
$$938$$ 0 0
$$939$$ 0 0
$$940$$ 0 0
$$941$$ − 624.620i − 0.663783i −0.943318 0.331892i $$-0.892313\pi$$
0.943318 0.331892i $$-0.107687\pi$$
$$942$$ 0 0
$$943$$ −1060.02 −1.12409
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ − 339.411i − 0.358407i −0.983812 0.179203i $$-0.942648\pi$$
0.983812 0.179203i $$-0.0573520\pi$$
$$948$$ 0 0
$$949$$ 207.846 0.219016
$$950$$ 0 0
$$951$$ 0 0
$$952$$ 0 0
$$953$$ − 530.330i − 0.556485i −0.960511 0.278242i $$-0.910248\pi$$
0.960511 0.278242i $$-0.0897519\pi$$
$$954$$ 0 0
$$955$$ 2160.00 2.26178
$$956$$ 0 0
$$957$$ 0 0
$$958$$ 0 0
$$959$$ − 1102.27i − 1.14940i
$$960$$ 0 0
$$961$$ 1739.00 1.80957
$$962$$ 0 0
$$963$$ 0 0
$$964$$ 0 0
$$965$$ − 73.4847i − 0.0761499i
$$966$$ 0 0
$$967$$ 1028.84 1.06395 0.531974 0.846761i $$-0.321450\pi$$
0.531974 + 0.846761i $$0.321450\pi$$
$$968$$ 0 0
$$969$$ 0 0
$$970$$ 0 0
$$971$$ − 1603.72i − 1.65162i −0.563952 0.825808i $$-0.690720\pi$$
0.563952 0.825808i $$-0.309280\pi$$
$$972$$ 0 0
$$973$$ −1787.48 −1.83708
$$974$$ 0 0
$$975$$ 0 0
$$976$$ 0 0
$$977$$ 1378.86i 1.41132i 0.708551 + 0.705659i $$0.249349\pi$$
−0.708551 + 0.705659i $$0.750651\pi$$
$$978$$ 0 0
$$979$$ 108.000 0.110317
$$980$$ 0 0
$$981$$ 0 0
$$982$$ 0 0
$$983$$ 440.908i 0.448533i 0.974528 + 0.224267i $$0.0719986\pi$$
−0.974528 + 0.224267i $$0.928001\pi$$
$$984$$ 0 0
$$985$$ −162.000 −0.164467
$$986$$ 0 0
$$987$$ 0 0
$$988$$ 0 0
$$989$$ 587.878i 0.594416i
$$990$$ 0 0
$$991$$ 467.654 0.471901 0.235950 0.971765i $$-0.424180\pi$$
0.235950 + 0.971765i $$0.424180\pi$$
$$992$$ 0 0
$$993$$ 0 0
$$994$$ 0 0
$$995$$ − 381.838i − 0.383756i
$$996$$ 0 0
$$997$$ −249.415 −0.250166 −0.125083 0.992146i $$-0.539920\pi$$
−0.125083 + 0.992146i $$0.539920\pi$$
$$998$$ 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.e.g.1025.4 4
3.2 odd 2 inner 2304.3.e.g.1025.2 4
4.3 odd 2 2304.3.e.j.1025.3 4
8.3 odd 2 inner 2304.3.e.g.1025.1 4
8.5 even 2 2304.3.e.j.1025.2 4
12.11 even 2 2304.3.e.j.1025.1 4
16.3 odd 4 576.3.h.b.161.7 yes 8
16.5 even 4 576.3.h.b.161.2 yes 8
16.11 odd 4 576.3.h.b.161.4 yes 8
16.13 even 4 576.3.h.b.161.5 yes 8
24.5 odd 2 2304.3.e.j.1025.4 4
24.11 even 2 inner 2304.3.e.g.1025.3 4
48.5 odd 4 576.3.h.b.161.6 yes 8
48.11 even 4 576.3.h.b.161.8 yes 8
48.29 odd 4 576.3.h.b.161.1 8
48.35 even 4 576.3.h.b.161.3 yes 8

By twisted newform
Twist Min Dim Char Parity Ord Type
576.3.h.b.161.1 8 48.29 odd 4
576.3.h.b.161.2 yes 8 16.5 even 4
576.3.h.b.161.3 yes 8 48.35 even 4
576.3.h.b.161.4 yes 8 16.11 odd 4
576.3.h.b.161.5 yes 8 16.13 even 4
576.3.h.b.161.6 yes 8 48.5 odd 4
576.3.h.b.161.7 yes 8 16.3 odd 4
576.3.h.b.161.8 yes 8 48.11 even 4
2304.3.e.g.1025.1 4 8.3 odd 2 inner
2304.3.e.g.1025.2 4 3.2 odd 2 inner
2304.3.e.g.1025.3 4 24.11 even 2 inner
2304.3.e.g.1025.4 4 1.1 even 1 trivial
2304.3.e.j.1025.1 4 12.11 even 2
2304.3.e.j.1025.2 4 8.5 even 2
2304.3.e.j.1025.3 4 4.3 odd 2
2304.3.e.j.1025.4 4 24.5 odd 2