# Properties

 Label 896.2.b.f.449.4 Level $896$ Weight $2$ Character 896.449 Analytic conductor $7.155$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$896 = 2^{7} \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 896.b (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$7.15459602111$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: 4.0.2048.2 Defining polynomial: $$x^{4} + 4 x^{2} + 2$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2^{3}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 449.4 Root $$0.765367i$$ of defining polynomial Character $$\chi$$ $$=$$ 896.449 Dual form 896.2.b.f.449.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+2.61313i q^{3} -2.61313i q^{5} -1.00000 q^{7} -3.82843 q^{9} +O(q^{10})$$ $$q+2.61313i q^{3} -2.61313i q^{5} -1.00000 q^{7} -3.82843 q^{9} +2.16478i q^{11} +0.448342i q^{13} +6.82843 q^{15} -7.65685 q^{17} +4.77791i q^{19} -2.61313i q^{21} -6.82843 q^{23} -1.82843 q^{25} -2.16478i q^{27} +9.55582i q^{29} -5.65685 q^{31} -5.65685 q^{33} +2.61313i q^{35} -5.22625i q^{37} -1.17157 q^{39} -3.65685 q^{41} -2.16478i q^{43} +10.0042i q^{45} +8.00000 q^{47} +1.00000 q^{49} -20.0083i q^{51} +10.4525i q^{53} +5.65685 q^{55} -12.4853 q^{57} -0.448342i q^{59} -12.1689i q^{61} +3.82843 q^{63} +1.17157 q^{65} +3.06147i q^{67} -17.8435i q^{69} -2.34315 q^{71} +11.6569 q^{73} -4.77791i q^{75} -2.16478i q^{77} +2.34315 q^{79} -5.82843 q^{81} -13.0656i q^{83} +20.0083i q^{85} -24.9706 q^{87} -2.00000 q^{89} -0.448342i q^{91} -14.7821i q^{93} +12.4853 q^{95} -10.0000 q^{97} -8.28772i q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 4q^{7} - 4q^{9} + O(q^{10})$$ $$4q - 4q^{7} - 4q^{9} + 16q^{15} - 8q^{17} - 16q^{23} + 4q^{25} - 16q^{39} + 8q^{41} + 32q^{47} + 4q^{49} - 16q^{57} + 4q^{63} + 16q^{65} - 32q^{71} + 24q^{73} + 32q^{79} - 12q^{81} - 32q^{87} - 8q^{89} + 16q^{95} - 40q^{97} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/896\mathbb{Z}\right)^\times$$.

 $$n$$ $$127$$ $$129$$ $$645$$ $$\chi(n)$$ $$1$$ $$1$$ $$-1$$

## 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
$$3$$ 2.61313i 1.50869i 0.656479 + 0.754344i $$0.272045\pi$$
−0.656479 + 0.754344i $$0.727955\pi$$
$$4$$ 0 0
$$5$$ − 2.61313i − 1.16863i −0.811529 0.584313i $$-0.801364\pi$$
0.811529 0.584313i $$-0.198636\pi$$
$$6$$ 0 0
$$7$$ −1.00000 −0.377964
$$8$$ 0 0
$$9$$ −3.82843 −1.27614
$$10$$ 0 0
$$11$$ 2.16478i 0.652707i 0.945248 + 0.326354i $$0.105820\pi$$
−0.945248 + 0.326354i $$0.894180\pi$$
$$12$$ 0 0
$$13$$ 0.448342i 0.124348i 0.998065 + 0.0621738i $$0.0198033\pi$$
−0.998065 + 0.0621738i $$0.980197\pi$$
$$14$$ 0 0
$$15$$ 6.82843 1.76309
$$16$$ 0 0
$$17$$ −7.65685 −1.85706 −0.928530 0.371257i $$-0.878927\pi$$
−0.928530 + 0.371257i $$0.878927\pi$$
$$18$$ 0 0
$$19$$ 4.77791i 1.09613i 0.836436 + 0.548064i $$0.184635\pi$$
−0.836436 + 0.548064i $$0.815365\pi$$
$$20$$ 0 0
$$21$$ − 2.61313i − 0.570231i
$$22$$ 0 0
$$23$$ −6.82843 −1.42383 −0.711913 0.702268i $$-0.752171\pi$$
−0.711913 + 0.702268i $$0.752171\pi$$
$$24$$ 0 0
$$25$$ −1.82843 −0.365685
$$26$$ 0 0
$$27$$ − 2.16478i − 0.416613i
$$28$$ 0 0
$$29$$ 9.55582i 1.77447i 0.461316 + 0.887236i $$0.347377\pi$$
−0.461316 + 0.887236i $$0.652623\pi$$
$$30$$ 0 0
$$31$$ −5.65685 −1.01600 −0.508001 0.861357i $$-0.669615\pi$$
−0.508001 + 0.861357i $$0.669615\pi$$
$$32$$ 0 0
$$33$$ −5.65685 −0.984732
$$34$$ 0 0
$$35$$ 2.61313i 0.441699i
$$36$$ 0 0
$$37$$ − 5.22625i − 0.859191i −0.903022 0.429595i $$-0.858656\pi$$
0.903022 0.429595i $$-0.141344\pi$$
$$38$$ 0 0
$$39$$ −1.17157 −0.187602
$$40$$ 0 0
$$41$$ −3.65685 −0.571105 −0.285552 0.958363i $$-0.592177\pi$$
−0.285552 + 0.958363i $$0.592177\pi$$
$$42$$ 0 0
$$43$$ − 2.16478i − 0.330127i −0.986283 0.165063i $$-0.947217\pi$$
0.986283 0.165063i $$-0.0527828\pi$$
$$44$$ 0 0
$$45$$ 10.0042i 1.49133i
$$46$$ 0 0
$$47$$ 8.00000 1.16692 0.583460 0.812142i $$-0.301699\pi$$
0.583460 + 0.812142i $$0.301699\pi$$
$$48$$ 0 0
$$49$$ 1.00000 0.142857
$$50$$ 0 0
$$51$$ − 20.0083i − 2.80173i
$$52$$ 0 0
$$53$$ 10.4525i 1.43576i 0.696166 + 0.717881i $$0.254888\pi$$
−0.696166 + 0.717881i $$0.745112\pi$$
$$54$$ 0 0
$$55$$ 5.65685 0.762770
$$56$$ 0 0
$$57$$ −12.4853 −1.65372
$$58$$ 0 0
$$59$$ − 0.448342i − 0.0583691i −0.999574 0.0291845i $$-0.990709\pi$$
0.999574 0.0291845i $$-0.00929105\pi$$
$$60$$ 0 0
$$61$$ − 12.1689i − 1.55807i −0.626978 0.779037i $$-0.715708\pi$$
0.626978 0.779037i $$-0.284292\pi$$
$$62$$ 0 0
$$63$$ 3.82843 0.482336
$$64$$ 0 0
$$65$$ 1.17157 0.145316
$$66$$ 0 0
$$67$$ 3.06147i 0.374018i 0.982358 + 0.187009i $$0.0598793\pi$$
−0.982358 + 0.187009i $$0.940121\pi$$
$$68$$ 0 0
$$69$$ − 17.8435i − 2.14811i
$$70$$ 0 0
$$71$$ −2.34315 −0.278080 −0.139040 0.990287i $$-0.544402\pi$$
−0.139040 + 0.990287i $$0.544402\pi$$
$$72$$ 0 0
$$73$$ 11.6569 1.36433 0.682166 0.731198i $$-0.261038\pi$$
0.682166 + 0.731198i $$0.261038\pi$$
$$74$$ 0 0
$$75$$ − 4.77791i − 0.551706i
$$76$$ 0 0
$$77$$ − 2.16478i − 0.246700i
$$78$$ 0 0
$$79$$ 2.34315 0.263624 0.131812 0.991275i $$-0.457920\pi$$
0.131812 + 0.991275i $$0.457920\pi$$
$$80$$ 0 0
$$81$$ −5.82843 −0.647603
$$82$$ 0 0
$$83$$ − 13.0656i − 1.43414i −0.697002 0.717070i $$-0.745483\pi$$
0.697002 0.717070i $$-0.254517\pi$$
$$84$$ 0 0
$$85$$ 20.0083i 2.17021i
$$86$$ 0 0
$$87$$ −24.9706 −2.67713
$$88$$ 0 0
$$89$$ −2.00000 −0.212000 −0.106000 0.994366i $$-0.533804\pi$$
−0.106000 + 0.994366i $$0.533804\pi$$
$$90$$ 0 0
$$91$$ − 0.448342i − 0.0469990i
$$92$$ 0 0
$$93$$ − 14.7821i − 1.53283i
$$94$$ 0 0
$$95$$ 12.4853 1.28096
$$96$$ 0 0
$$97$$ −10.0000 −1.01535 −0.507673 0.861550i $$-0.669494\pi$$
−0.507673 + 0.861550i $$0.669494\pi$$
$$98$$ 0 0
$$99$$ − 8.28772i − 0.832947i
$$100$$ 0 0
$$101$$ 4.77791i 0.475420i 0.971336 + 0.237710i $$0.0763968\pi$$
−0.971336 + 0.237710i $$0.923603\pi$$
$$102$$ 0 0
$$103$$ −13.6569 −1.34565 −0.672825 0.739802i $$-0.734919\pi$$
−0.672825 + 0.739802i $$0.734919\pi$$
$$104$$ 0 0
$$105$$ −6.82843 −0.666386
$$106$$ 0 0
$$107$$ 13.5140i 1.30644i 0.757166 + 0.653222i $$0.226583\pi$$
−0.757166 + 0.653222i $$0.773417\pi$$
$$108$$ 0 0
$$109$$ − 11.3492i − 1.08705i −0.839391 0.543527i $$-0.817088\pi$$
0.839391 0.543527i $$-0.182912\pi$$
$$110$$ 0 0
$$111$$ 13.6569 1.29625
$$112$$ 0 0
$$113$$ 8.82843 0.830509 0.415254 0.909705i $$-0.363693\pi$$
0.415254 + 0.909705i $$0.363693\pi$$
$$114$$ 0 0
$$115$$ 17.8435i 1.66392i
$$116$$ 0 0
$$117$$ − 1.71644i − 0.158685i
$$118$$ 0 0
$$119$$ 7.65685 0.701903
$$120$$ 0 0
$$121$$ 6.31371 0.573973
$$122$$ 0 0
$$123$$ − 9.55582i − 0.861619i
$$124$$ 0 0
$$125$$ − 8.28772i − 0.741276i
$$126$$ 0 0
$$127$$ 4.48528 0.398004 0.199002 0.979999i $$-0.436230\pi$$
0.199002 + 0.979999i $$0.436230\pi$$
$$128$$ 0 0
$$129$$ 5.65685 0.498058
$$130$$ 0 0
$$131$$ 18.2919i 1.59817i 0.601219 + 0.799085i $$0.294682\pi$$
−0.601219 + 0.799085i $$0.705318\pi$$
$$132$$ 0 0
$$133$$ − 4.77791i − 0.414297i
$$134$$ 0 0
$$135$$ −5.65685 −0.486864
$$136$$ 0 0
$$137$$ 2.00000 0.170872 0.0854358 0.996344i $$-0.472772\pi$$
0.0854358 + 0.996344i $$0.472772\pi$$
$$138$$ 0 0
$$139$$ 19.5600i 1.65906i 0.558465 + 0.829528i $$0.311390\pi$$
−0.558465 + 0.829528i $$0.688610\pi$$
$$140$$ 0 0
$$141$$ 20.9050i 1.76052i
$$142$$ 0 0
$$143$$ −0.970563 −0.0811625
$$144$$ 0 0
$$145$$ 24.9706 2.07369
$$146$$ 0 0
$$147$$ 2.61313i 0.215527i
$$148$$ 0 0
$$149$$ 16.5754i 1.35791i 0.734179 + 0.678956i $$0.237567\pi$$
−0.734179 + 0.678956i $$0.762433\pi$$
$$150$$ 0 0
$$151$$ 18.1421 1.47639 0.738193 0.674590i $$-0.235679\pi$$
0.738193 + 0.674590i $$0.235679\pi$$
$$152$$ 0 0
$$153$$ 29.3137 2.36987
$$154$$ 0 0
$$155$$ 14.7821i 1.18732i
$$156$$ 0 0
$$157$$ 4.77791i 0.381319i 0.981656 + 0.190659i $$0.0610626\pi$$
−0.981656 + 0.190659i $$0.938937\pi$$
$$158$$ 0 0
$$159$$ −27.3137 −2.16612
$$160$$ 0 0
$$161$$ 6.82843 0.538155
$$162$$ 0 0
$$163$$ − 3.95815i − 0.310026i −0.987912 0.155013i $$-0.950458\pi$$
0.987912 0.155013i $$-0.0495420\pi$$
$$164$$ 0 0
$$165$$ 14.7821i 1.15078i
$$166$$ 0 0
$$167$$ −16.0000 −1.23812 −0.619059 0.785345i $$-0.712486\pi$$
−0.619059 + 0.785345i $$0.712486\pi$$
$$168$$ 0 0
$$169$$ 12.7990 0.984538
$$170$$ 0 0
$$171$$ − 18.2919i − 1.39882i
$$172$$ 0 0
$$173$$ − 16.1271i − 1.22612i −0.790036 0.613060i $$-0.789938\pi$$
0.790036 0.613060i $$-0.210062\pi$$
$$174$$ 0 0
$$175$$ 1.82843 0.138216
$$176$$ 0 0
$$177$$ 1.17157 0.0880608
$$178$$ 0 0
$$179$$ 22.1731i 1.65730i 0.559770 + 0.828648i $$0.310889\pi$$
−0.559770 + 0.828648i $$0.689111\pi$$
$$180$$ 0 0
$$181$$ 2.61313i 0.194232i 0.995273 + 0.0971161i $$0.0309618\pi$$
−0.995273 + 0.0971161i $$0.969038\pi$$
$$182$$ 0 0
$$183$$ 31.7990 2.35065
$$184$$ 0 0
$$185$$ −13.6569 −1.00407
$$186$$ 0 0
$$187$$ − 16.5754i − 1.21212i
$$188$$ 0 0
$$189$$ 2.16478i 0.157465i
$$190$$ 0 0
$$191$$ −19.3137 −1.39749 −0.698745 0.715370i $$-0.746258\pi$$
−0.698745 + 0.715370i $$0.746258\pi$$
$$192$$ 0 0
$$193$$ 13.7990 0.993273 0.496637 0.867959i $$-0.334568\pi$$
0.496637 + 0.867959i $$0.334568\pi$$
$$194$$ 0 0
$$195$$ 3.06147i 0.219236i
$$196$$ 0 0
$$197$$ − 6.12293i − 0.436241i −0.975922 0.218121i $$-0.930007\pi$$
0.975922 0.218121i $$-0.0699926\pi$$
$$198$$ 0 0
$$199$$ −8.97056 −0.635906 −0.317953 0.948106i $$-0.602995\pi$$
−0.317953 + 0.948106i $$0.602995\pi$$
$$200$$ 0 0
$$201$$ −8.00000 −0.564276
$$202$$ 0 0
$$203$$ − 9.55582i − 0.670687i
$$204$$ 0 0
$$205$$ 9.55582i 0.667407i
$$206$$ 0 0
$$207$$ 26.1421 1.81700
$$208$$ 0 0
$$209$$ −10.3431 −0.715450
$$210$$ 0 0
$$211$$ 9.18440i 0.632280i 0.948712 + 0.316140i $$0.102387\pi$$
−0.948712 + 0.316140i $$0.897613\pi$$
$$212$$ 0 0
$$213$$ − 6.12293i − 0.419537i
$$214$$ 0 0
$$215$$ −5.65685 −0.385794
$$216$$ 0 0
$$217$$ 5.65685 0.384012
$$218$$ 0 0
$$219$$ 30.4608i 2.05835i
$$220$$ 0 0
$$221$$ − 3.43289i − 0.230921i
$$222$$ 0 0
$$223$$ 8.97056 0.600713 0.300357 0.953827i $$-0.402894\pi$$
0.300357 + 0.953827i $$0.402894\pi$$
$$224$$ 0 0
$$225$$ 7.00000 0.466667
$$226$$ 0 0
$$227$$ − 14.3337i − 0.951363i −0.879618 0.475682i $$-0.842201\pi$$
0.879618 0.475682i $$-0.157799\pi$$
$$228$$ 0 0
$$229$$ 5.67459i 0.374988i 0.982266 + 0.187494i $$0.0600365\pi$$
−0.982266 + 0.187494i $$0.939964\pi$$
$$230$$ 0 0
$$231$$ 5.65685 0.372194
$$232$$ 0 0
$$233$$ −10.0000 −0.655122 −0.327561 0.944830i $$-0.606227\pi$$
−0.327561 + 0.944830i $$0.606227\pi$$
$$234$$ 0 0
$$235$$ − 20.9050i − 1.36369i
$$236$$ 0 0
$$237$$ 6.12293i 0.397727i
$$238$$ 0 0
$$239$$ 18.1421 1.17352 0.586759 0.809762i $$-0.300404\pi$$
0.586759 + 0.809762i $$0.300404\pi$$
$$240$$ 0 0
$$241$$ 1.31371 0.0846234 0.0423117 0.999104i $$-0.486528\pi$$
0.0423117 + 0.999104i $$0.486528\pi$$
$$242$$ 0 0
$$243$$ − 21.7248i − 1.39364i
$$244$$ 0 0
$$245$$ − 2.61313i − 0.166946i
$$246$$ 0 0
$$247$$ −2.14214 −0.136301
$$248$$ 0 0
$$249$$ 34.1421 2.16367
$$250$$ 0 0
$$251$$ − 0.819760i − 0.0517428i −0.999665 0.0258714i $$-0.991764\pi$$
0.999665 0.0258714i $$-0.00823604\pi$$
$$252$$ 0 0
$$253$$ − 14.7821i − 0.929341i
$$254$$ 0 0
$$255$$ −52.2843 −3.27417
$$256$$ 0 0
$$257$$ −9.31371 −0.580973 −0.290487 0.956879i $$-0.593817\pi$$
−0.290487 + 0.956879i $$0.593817\pi$$
$$258$$ 0 0
$$259$$ 5.22625i 0.324743i
$$260$$ 0 0
$$261$$ − 36.5838i − 2.26448i
$$262$$ 0 0
$$263$$ −2.34315 −0.144485 −0.0722423 0.997387i $$-0.523015\pi$$
−0.0722423 + 0.997387i $$0.523015\pi$$
$$264$$ 0 0
$$265$$ 27.3137 1.67787
$$266$$ 0 0
$$267$$ − 5.22625i − 0.319841i
$$268$$ 0 0
$$269$$ − 8.21080i − 0.500621i −0.968166 0.250311i $$-0.919467\pi$$
0.968166 0.250311i $$-0.0805327\pi$$
$$270$$ 0 0
$$271$$ 2.34315 0.142336 0.0711680 0.997464i $$-0.477327\pi$$
0.0711680 + 0.997464i $$0.477327\pi$$
$$272$$ 0 0
$$273$$ 1.17157 0.0709068
$$274$$ 0 0
$$275$$ − 3.95815i − 0.238685i
$$276$$ 0 0
$$277$$ − 14.7821i − 0.888169i −0.895985 0.444084i $$-0.853529\pi$$
0.895985 0.444084i $$-0.146471\pi$$
$$278$$ 0 0
$$279$$ 21.6569 1.29656
$$280$$ 0 0
$$281$$ −2.00000 −0.119310 −0.0596550 0.998219i $$-0.519000\pi$$
−0.0596550 + 0.998219i $$0.519000\pi$$
$$282$$ 0 0
$$283$$ − 20.8281i − 1.23810i −0.785351 0.619051i $$-0.787518\pi$$
0.785351 0.619051i $$-0.212482\pi$$
$$284$$ 0 0
$$285$$ 32.6256i 1.93257i
$$286$$ 0 0
$$287$$ 3.65685 0.215857
$$288$$ 0 0
$$289$$ 41.6274 2.44867
$$290$$ 0 0
$$291$$ − 26.1313i − 1.53184i
$$292$$ 0 0
$$293$$ 10.9008i 0.636834i 0.947951 + 0.318417i $$0.103151\pi$$
−0.947951 + 0.318417i $$0.896849\pi$$
$$294$$ 0 0
$$295$$ −1.17157 −0.0682116
$$296$$ 0 0
$$297$$ 4.68629 0.271926
$$298$$ 0 0
$$299$$ − 3.06147i − 0.177049i
$$300$$ 0 0
$$301$$ 2.16478i 0.124776i
$$302$$ 0 0
$$303$$ −12.4853 −0.717261
$$304$$ 0 0
$$305$$ −31.7990 −1.82080
$$306$$ 0 0
$$307$$ − 8.73606i − 0.498593i −0.968427 0.249297i $$-0.919801\pi$$
0.968427 0.249297i $$-0.0801994\pi$$
$$308$$ 0 0
$$309$$ − 35.6871i − 2.03017i
$$310$$ 0 0
$$311$$ −14.6274 −0.829445 −0.414722 0.909948i $$-0.636121\pi$$
−0.414722 + 0.909948i $$0.636121\pi$$
$$312$$ 0 0
$$313$$ −8.34315 −0.471582 −0.235791 0.971804i $$-0.575768\pi$$
−0.235791 + 0.971804i $$0.575768\pi$$
$$314$$ 0 0
$$315$$ − 10.0042i − 0.563671i
$$316$$ 0 0
$$317$$ 8.65914i 0.486346i 0.969983 + 0.243173i $$0.0781882\pi$$
−0.969983 + 0.243173i $$0.921812\pi$$
$$318$$ 0 0
$$319$$ −20.6863 −1.15821
$$320$$ 0 0
$$321$$ −35.3137 −1.97102
$$322$$ 0 0
$$323$$ − 36.5838i − 2.03558i
$$324$$ 0 0
$$325$$ − 0.819760i − 0.0454721i
$$326$$ 0 0
$$327$$ 29.6569 1.64003
$$328$$ 0 0
$$329$$ −8.00000 −0.441054
$$330$$ 0 0
$$331$$ 18.7402i 1.03006i 0.857173 + 0.515028i $$0.172218\pi$$
−0.857173 + 0.515028i $$0.827782\pi$$
$$332$$ 0 0
$$333$$ 20.0083i 1.09645i
$$334$$ 0 0
$$335$$ 8.00000 0.437087
$$336$$ 0 0
$$337$$ 11.1716 0.608554 0.304277 0.952584i $$-0.401585\pi$$
0.304277 + 0.952584i $$0.401585\pi$$
$$338$$ 0 0
$$339$$ 23.0698i 1.25298i
$$340$$ 0 0
$$341$$ − 12.2459i − 0.663151i
$$342$$ 0 0
$$343$$ −1.00000 −0.0539949
$$344$$ 0 0
$$345$$ −46.6274 −2.51034
$$346$$ 0 0
$$347$$ 12.6173i 0.677332i 0.940907 + 0.338666i $$0.109976\pi$$
−0.940907 + 0.338666i $$0.890024\pi$$
$$348$$ 0 0
$$349$$ 22.6215i 1.21090i 0.795884 + 0.605449i $$0.207007\pi$$
−0.795884 + 0.605449i $$0.792993\pi$$
$$350$$ 0 0
$$351$$ 0.970563 0.0518048
$$352$$ 0 0
$$353$$ 15.6569 0.833330 0.416665 0.909060i $$-0.363199\pi$$
0.416665 + 0.909060i $$0.363199\pi$$
$$354$$ 0 0
$$355$$ 6.12293i 0.324972i
$$356$$ 0 0
$$357$$ 20.0083i 1.05895i
$$358$$ 0 0
$$359$$ −6.82843 −0.360391 −0.180195 0.983631i $$-0.557673\pi$$
−0.180195 + 0.983631i $$0.557673\pi$$
$$360$$ 0 0
$$361$$ −3.82843 −0.201496
$$362$$ 0 0
$$363$$ 16.4985i 0.865947i
$$364$$ 0 0
$$365$$ − 30.4608i − 1.59439i
$$366$$ 0 0
$$367$$ −24.9706 −1.30345 −0.651726 0.758454i $$-0.725955\pi$$
−0.651726 + 0.758454i $$0.725955\pi$$
$$368$$ 0 0
$$369$$ 14.0000 0.728811
$$370$$ 0 0
$$371$$ − 10.4525i − 0.542667i
$$372$$ 0 0
$$373$$ 33.8937i 1.75495i 0.479622 + 0.877475i $$0.340774\pi$$
−0.479622 + 0.877475i $$0.659226\pi$$
$$374$$ 0 0
$$375$$ 21.6569 1.11836
$$376$$ 0 0
$$377$$ −4.28427 −0.220651
$$378$$ 0 0
$$379$$ 23.0698i 1.18502i 0.805565 + 0.592508i $$0.201862\pi$$
−0.805565 + 0.592508i $$0.798138\pi$$
$$380$$ 0 0
$$381$$ 11.7206i 0.600465i
$$382$$ 0 0
$$383$$ 19.3137 0.986884 0.493442 0.869779i $$-0.335738\pi$$
0.493442 + 0.869779i $$0.335738\pi$$
$$384$$ 0 0
$$385$$ −5.65685 −0.288300
$$386$$ 0 0
$$387$$ 8.28772i 0.421288i
$$388$$ 0 0
$$389$$ − 20.0083i − 1.01446i −0.861810 0.507231i $$-0.830669\pi$$
0.861810 0.507231i $$-0.169331\pi$$
$$390$$ 0 0
$$391$$ 52.2843 2.64413
$$392$$ 0 0
$$393$$ −47.7990 −2.41114
$$394$$ 0 0
$$395$$ − 6.12293i − 0.308078i
$$396$$ 0 0
$$397$$ 18.2919i 0.918043i 0.888425 + 0.459022i $$0.151800\pi$$
−0.888425 + 0.459022i $$0.848200\pi$$
$$398$$ 0 0
$$399$$ 12.4853 0.625046
$$400$$ 0 0
$$401$$ 9.51472 0.475142 0.237571 0.971370i $$-0.423649\pi$$
0.237571 + 0.971370i $$0.423649\pi$$
$$402$$ 0 0
$$403$$ − 2.53620i − 0.126337i
$$404$$ 0 0
$$405$$ 15.2304i 0.756805i
$$406$$ 0 0
$$407$$ 11.3137 0.560800
$$408$$ 0 0
$$409$$ 5.31371 0.262746 0.131373 0.991333i $$-0.458061\pi$$
0.131373 + 0.991333i $$0.458061\pi$$
$$410$$ 0 0
$$411$$ 5.22625i 0.257792i
$$412$$ 0 0
$$413$$ 0.448342i 0.0220614i
$$414$$ 0 0
$$415$$ −34.1421 −1.67597
$$416$$ 0 0
$$417$$ −51.1127 −2.50300
$$418$$ 0 0
$$419$$ 13.0656i 0.638298i 0.947705 + 0.319149i $$0.103397\pi$$
−0.947705 + 0.319149i $$0.896603\pi$$
$$420$$ 0 0
$$421$$ − 2.53620i − 0.123607i −0.998088 0.0618035i $$-0.980315\pi$$
0.998088 0.0618035i $$-0.0196852\pi$$
$$422$$ 0 0
$$423$$ −30.6274 −1.48916
$$424$$ 0 0
$$425$$ 14.0000 0.679100
$$426$$ 0 0
$$427$$ 12.1689i 0.588897i
$$428$$ 0 0
$$429$$ − 2.53620i − 0.122449i
$$430$$ 0 0
$$431$$ −6.82843 −0.328914 −0.164457 0.986384i $$-0.552587\pi$$
−0.164457 + 0.986384i $$0.552587\pi$$
$$432$$ 0 0
$$433$$ −12.3431 −0.593174 −0.296587 0.955006i $$-0.595848\pi$$
−0.296587 + 0.955006i $$0.595848\pi$$
$$434$$ 0 0
$$435$$ 65.2512i 3.12856i
$$436$$ 0 0
$$437$$ − 32.6256i − 1.56069i
$$438$$ 0 0
$$439$$ −0.970563 −0.0463224 −0.0231612 0.999732i $$-0.507373\pi$$
−0.0231612 + 0.999732i $$0.507373\pi$$
$$440$$ 0 0
$$441$$ −3.82843 −0.182306
$$442$$ 0 0
$$443$$ − 9.18440i − 0.436364i −0.975908 0.218182i $$-0.929987\pi$$
0.975908 0.218182i $$-0.0700127\pi$$
$$444$$ 0 0
$$445$$ 5.22625i 0.247748i
$$446$$ 0 0
$$447$$ −43.3137 −2.04867
$$448$$ 0 0
$$449$$ −24.6274 −1.16224 −0.581120 0.813818i $$-0.697385\pi$$
−0.581120 + 0.813818i $$0.697385\pi$$
$$450$$ 0 0
$$451$$ − 7.91630i − 0.372764i
$$452$$ 0 0
$$453$$ 47.4077i 2.22741i
$$454$$ 0 0
$$455$$ −1.17157 −0.0549242
$$456$$ 0 0
$$457$$ −10.4853 −0.490481 −0.245240 0.969462i $$-0.578867\pi$$
−0.245240 + 0.969462i $$0.578867\pi$$
$$458$$ 0 0
$$459$$ 16.5754i 0.773675i
$$460$$ 0 0
$$461$$ 10.3756i 0.483239i 0.970371 + 0.241619i $$0.0776786\pi$$
−0.970371 + 0.241619i $$0.922321\pi$$
$$462$$ 0 0
$$463$$ 8.00000 0.371792 0.185896 0.982569i $$-0.440481\pi$$
0.185896 + 0.982569i $$0.440481\pi$$
$$464$$ 0 0
$$465$$ −38.6274 −1.79130
$$466$$ 0 0
$$467$$ − 1.34502i − 0.0622403i −0.999516 0.0311202i $$-0.990093\pi$$
0.999516 0.0311202i $$-0.00990745\pi$$
$$468$$ 0 0
$$469$$ − 3.06147i − 0.141365i
$$470$$ 0 0
$$471$$ −12.4853 −0.575291
$$472$$ 0 0
$$473$$ 4.68629 0.215476
$$474$$ 0 0
$$475$$ − 8.73606i − 0.400838i
$$476$$ 0 0
$$477$$ − 40.0166i − 1.83224i
$$478$$ 0 0
$$479$$ −5.65685 −0.258468 −0.129234 0.991614i $$-0.541252\pi$$
−0.129234 + 0.991614i $$0.541252\pi$$
$$480$$ 0 0
$$481$$ 2.34315 0.106838
$$482$$ 0 0
$$483$$ 17.8435i 0.811909i
$$484$$ 0 0
$$485$$ 26.1313i 1.18656i
$$486$$ 0 0
$$487$$ 31.7990 1.44095 0.720475 0.693481i $$-0.243924\pi$$
0.720475 + 0.693481i $$0.243924\pi$$
$$488$$ 0 0
$$489$$ 10.3431 0.467733
$$490$$ 0 0
$$491$$ − 32.6256i − 1.47237i −0.676779 0.736187i $$-0.736625\pi$$
0.676779 0.736187i $$-0.263375\pi$$
$$492$$ 0 0
$$493$$ − 73.1675i − 3.29530i
$$494$$ 0 0
$$495$$ −21.6569 −0.973403
$$496$$ 0 0
$$497$$ 2.34315 0.105104
$$498$$ 0 0
$$499$$ − 11.7206i − 0.524686i −0.964975 0.262343i $$-0.915505\pi$$
0.964975 0.262343i $$-0.0844952\pi$$
$$500$$ 0 0
$$501$$ − 41.8100i − 1.86793i
$$502$$ 0 0
$$503$$ −5.65685 −0.252227 −0.126113 0.992016i $$-0.540250\pi$$
−0.126113 + 0.992016i $$0.540250\pi$$
$$504$$ 0 0
$$505$$ 12.4853 0.555588
$$506$$ 0 0
$$507$$ 33.4454i 1.48536i
$$508$$ 0 0
$$509$$ 3.50981i 0.155570i 0.996970 + 0.0777848i $$0.0247847\pi$$
−0.996970 + 0.0777848i $$0.975215\pi$$
$$510$$ 0 0
$$511$$ −11.6569 −0.515669
$$512$$ 0 0
$$513$$ 10.3431 0.456661
$$514$$ 0 0
$$515$$ 35.6871i 1.57256i
$$516$$ 0 0
$$517$$ 17.3183i 0.761657i
$$518$$ 0 0
$$519$$ 42.1421 1.84983
$$520$$ 0 0
$$521$$ 14.6863 0.643418 0.321709 0.946839i $$-0.395743\pi$$
0.321709 + 0.946839i $$0.395743\pi$$
$$522$$ 0 0
$$523$$ 7.46796i 0.326551i 0.986581 + 0.163276i $$0.0522060\pi$$
−0.986581 + 0.163276i $$0.947794\pi$$
$$524$$ 0 0
$$525$$ 4.77791i 0.208525i
$$526$$ 0 0
$$527$$ 43.3137 1.88677
$$528$$ 0 0
$$529$$ 23.6274 1.02728
$$530$$ 0 0
$$531$$ 1.71644i 0.0744873i
$$532$$ 0 0
$$533$$ − 1.63952i − 0.0710155i
$$534$$ 0 0
$$535$$ 35.3137 1.52674
$$536$$ 0 0
$$537$$ −57.9411 −2.50034
$$538$$ 0 0
$$539$$ 2.16478i 0.0932439i
$$540$$ 0 0
$$541$$ 31.3575i 1.34816i 0.738656 + 0.674082i $$0.235461\pi$$
−0.738656 + 0.674082i $$0.764539\pi$$
$$542$$ 0 0
$$543$$ −6.82843 −0.293036
$$544$$ 0 0
$$545$$ −29.6569 −1.27036
$$546$$ 0 0
$$547$$ − 43.9748i − 1.88023i −0.340862 0.940113i $$-0.610719\pi$$
0.340862 0.940113i $$-0.389281\pi$$
$$548$$ 0 0
$$549$$ 46.5879i 1.98832i
$$550$$ 0 0
$$551$$ −45.6569 −1.94505
$$552$$ 0 0
$$553$$ −2.34315 −0.0996407
$$554$$ 0 0
$$555$$ − 35.6871i − 1.51483i
$$556$$ 0 0
$$557$$ − 14.7821i − 0.626337i −0.949698 0.313168i $$-0.898610\pi$$
0.949698 0.313168i $$-0.101390\pi$$
$$558$$ 0 0
$$559$$ 0.970563 0.0410504
$$560$$ 0 0
$$561$$ 43.3137 1.82871
$$562$$ 0 0
$$563$$ − 17.7666i − 0.748774i −0.927273 0.374387i $$-0.877853\pi$$
0.927273 0.374387i $$-0.122147\pi$$
$$564$$ 0 0
$$565$$ − 23.0698i − 0.970553i
$$566$$ 0 0
$$567$$ 5.82843 0.244771
$$568$$ 0 0
$$569$$ −17.5147 −0.734255 −0.367128 0.930171i $$-0.619659\pi$$
−0.367128 + 0.930171i $$0.619659\pi$$
$$570$$ 0 0
$$571$$ − 18.7402i − 0.784254i −0.919911 0.392127i $$-0.871739\pi$$
0.919911 0.392127i $$-0.128261\pi$$
$$572$$ 0 0
$$573$$ − 50.4692i − 2.10838i
$$574$$ 0 0
$$575$$ 12.4853 0.520672
$$576$$ 0 0
$$577$$ −22.9706 −0.956277 −0.478139 0.878284i $$-0.658688\pi$$
−0.478139 + 0.878284i $$0.658688\pi$$
$$578$$ 0 0
$$579$$ 36.0585i 1.49854i
$$580$$ 0 0
$$581$$ 13.0656i 0.542054i
$$582$$ 0 0
$$583$$ −22.6274 −0.937132
$$584$$ 0 0
$$585$$ −4.48528 −0.185444
$$586$$ 0 0
$$587$$ − 33.4454i − 1.38044i −0.723600 0.690219i $$-0.757514\pi$$
0.723600 0.690219i $$-0.242486\pi$$
$$588$$ 0 0
$$589$$ − 27.0279i − 1.11367i
$$590$$ 0 0
$$591$$ 16.0000 0.658152
$$592$$ 0 0
$$593$$ −25.3137 −1.03951 −0.519755 0.854316i $$-0.673977\pi$$
−0.519755 + 0.854316i $$0.673977\pi$$
$$594$$ 0 0
$$595$$ − 20.0083i − 0.820261i
$$596$$ 0 0
$$597$$ − 23.4412i − 0.959385i
$$598$$ 0 0
$$599$$ −8.00000 −0.326871 −0.163436 0.986554i $$-0.552258\pi$$
−0.163436 + 0.986554i $$0.552258\pi$$
$$600$$ 0 0
$$601$$ 41.3137 1.68522 0.842611 0.538523i $$-0.181018\pi$$
0.842611 + 0.538523i $$0.181018\pi$$
$$602$$ 0 0
$$603$$ − 11.7206i − 0.477300i
$$604$$ 0 0
$$605$$ − 16.4985i − 0.670760i
$$606$$ 0 0
$$607$$ −27.3137 −1.10863 −0.554315 0.832307i $$-0.687020\pi$$
−0.554315 + 0.832307i $$0.687020\pi$$
$$608$$ 0 0
$$609$$ 24.9706 1.01186
$$610$$ 0 0
$$611$$ 3.58673i 0.145104i
$$612$$ 0 0
$$613$$ 26.1313i 1.05543i 0.849421 + 0.527716i $$0.176951\pi$$
−0.849421 + 0.527716i $$0.823049\pi$$
$$614$$ 0 0
$$615$$ −24.9706 −1.00691
$$616$$ 0 0
$$617$$ −41.7990 −1.68276 −0.841382 0.540441i $$-0.818257\pi$$
−0.841382 + 0.540441i $$0.818257\pi$$
$$618$$ 0 0
$$619$$ 33.9706i 1.36540i 0.730701 + 0.682698i $$0.239193\pi$$
−0.730701 + 0.682698i $$0.760807\pi$$
$$620$$ 0 0
$$621$$ 14.7821i 0.593184i
$$622$$ 0 0
$$623$$ 2.00000 0.0801283
$$624$$ 0 0
$$625$$ −30.7990 −1.23196
$$626$$ 0 0
$$627$$ − 27.0279i − 1.07939i
$$628$$ 0 0
$$629$$ 40.0166i 1.59557i
$$630$$ 0 0
$$631$$ −30.6274 −1.21926 −0.609629 0.792687i $$-0.708682\pi$$
−0.609629 + 0.792687i $$0.708682\pi$$
$$632$$ 0 0
$$633$$ −24.0000 −0.953914
$$634$$ 0 0
$$635$$ − 11.7206i − 0.465118i
$$636$$ 0 0
$$637$$ 0.448342i 0.0177639i
$$638$$ 0 0
$$639$$ 8.97056 0.354870
$$640$$ 0 0
$$641$$ 27.4558 1.08444 0.542220 0.840236i $$-0.317584\pi$$
0.542220 + 0.840236i $$0.317584\pi$$
$$642$$ 0 0
$$643$$ − 2.98454i − 0.117699i −0.998267 0.0588495i $$-0.981257\pi$$
0.998267 0.0588495i $$-0.0187432\pi$$
$$644$$ 0 0
$$645$$ − 14.7821i − 0.582044i
$$646$$ 0 0
$$647$$ −29.6569 −1.16593 −0.582966 0.812497i $$-0.698108\pi$$
−0.582966 + 0.812497i $$0.698108\pi$$
$$648$$ 0 0
$$649$$ 0.970563 0.0380979
$$650$$ 0 0
$$651$$ 14.7821i 0.579355i
$$652$$ 0 0
$$653$$ 22.5445i 0.882236i 0.897449 + 0.441118i $$0.145418\pi$$
−0.897449 + 0.441118i $$0.854582\pi$$
$$654$$ 0 0
$$655$$ 47.7990 1.86766
$$656$$ 0 0
$$657$$ −44.6274 −1.74108
$$658$$ 0 0
$$659$$ 10.0811i 0.392703i 0.980534 + 0.196352i $$0.0629094\pi$$
−0.980534 + 0.196352i $$0.937091\pi$$
$$660$$ 0 0
$$661$$ 0.448342i 0.0174385i 0.999962 + 0.00871923i $$0.00277545\pi$$
−0.999962 + 0.00871923i $$0.997225\pi$$
$$662$$ 0 0
$$663$$ 8.97056 0.348388
$$664$$ 0 0
$$665$$ −12.4853 −0.484158
$$666$$ 0 0
$$667$$ − 65.2512i − 2.52654i
$$668$$ 0 0
$$669$$ 23.4412i 0.906290i
$$670$$ 0 0
$$671$$ 26.3431 1.01697
$$672$$ 0 0
$$673$$ 28.6274 1.10351 0.551753 0.834008i $$-0.313959\pi$$
0.551753 + 0.834008i $$0.313959\pi$$
$$674$$ 0 0
$$675$$ 3.95815i 0.152349i
$$676$$ 0 0
$$677$$ 8.73606i 0.335754i 0.985808 + 0.167877i $$0.0536912\pi$$
−0.985808 + 0.167877i $$0.946309\pi$$
$$678$$ 0 0
$$679$$ 10.0000 0.383765
$$680$$ 0 0
$$681$$ 37.4558 1.43531
$$682$$ 0 0
$$683$$ 5.59767i 0.214189i 0.994249 + 0.107094i $$0.0341547\pi$$
−0.994249 + 0.107094i $$0.965845\pi$$
$$684$$ 0 0
$$685$$ − 5.22625i − 0.199685i
$$686$$ 0 0
$$687$$ −14.8284 −0.565740
$$688$$ 0 0
$$689$$ −4.68629 −0.178533
$$690$$ 0 0
$$691$$ − 35.7640i − 1.36053i −0.732968 0.680263i $$-0.761865\pi$$
0.732968 0.680263i $$-0.238135\pi$$
$$692$$ 0 0
$$693$$ 8.28772i 0.314824i
$$694$$ 0 0
$$695$$ 51.1127 1.93882
$$696$$ 0 0
$$697$$ 28.0000 1.06058
$$698$$ 0 0
$$699$$ − 26.1313i − 0.988375i
$$700$$ 0 0
$$701$$ 40.9133i 1.54528i 0.634847 + 0.772638i $$0.281063\pi$$
−0.634847 + 0.772638i $$0.718937\pi$$
$$702$$ 0 0
$$703$$ 24.9706 0.941783
$$704$$ 0 0
$$705$$ 54.6274 2.05739
$$706$$ 0 0
$$707$$ − 4.77791i − 0.179692i
$$708$$ 0 0
$$709$$ − 17.4721i − 0.656179i −0.944647 0.328090i $$-0.893595\pi$$
0.944647 0.328090i $$-0.106405\pi$$
$$710$$ 0 0
$$711$$ −8.97056 −0.336422
$$712$$ 0 0
$$713$$ 38.6274 1.44661
$$714$$ 0 0
$$715$$ 2.53620i 0.0948486i
$$716$$ 0 0
$$717$$ 47.4077i 1.77047i
$$718$$ 0 0
$$719$$ −3.31371 −0.123580 −0.0617902 0.998089i $$-0.519681\pi$$
−0.0617902 + 0.998089i $$0.519681\pi$$
$$720$$ 0 0
$$721$$ 13.6569 0.508608
$$722$$ 0 0
$$723$$ 3.43289i 0.127670i
$$724$$ 0 0
$$725$$ − 17.4721i − 0.648898i
$$726$$ 0 0
$$727$$ −20.6863 −0.767212 −0.383606 0.923497i $$-0.625318\pi$$
−0.383606 + 0.923497i $$0.625318\pi$$
$$728$$ 0 0
$$729$$ 39.2843 1.45497
$$730$$ 0 0
$$731$$ 16.5754i 0.613065i
$$732$$ 0 0
$$733$$ 21.7248i 0.802423i 0.915986 + 0.401211i $$0.131411\pi$$
−0.915986 + 0.401211i $$0.868589\pi$$
$$734$$ 0 0
$$735$$ 6.82843 0.251870
$$736$$ 0 0
$$737$$ −6.62742 −0.244124
$$738$$ 0 0
$$739$$ − 24.8632i − 0.914606i −0.889311 0.457303i $$-0.848815\pi$$
0.889311 0.457303i $$-0.151185\pi$$
$$740$$ 0 0
$$741$$ − 5.59767i − 0.205636i
$$742$$ 0 0
$$743$$ −27.5147 −1.00942 −0.504709 0.863290i $$-0.668400\pi$$
−0.504709 + 0.863290i $$0.668400\pi$$
$$744$$ 0 0
$$745$$ 43.3137 1.58689
$$746$$ 0 0
$$747$$ 50.0208i 1.83017i
$$748$$ 0 0
$$749$$ − 13.5140i − 0.493790i
$$750$$ 0 0
$$751$$ −45.4558 −1.65871 −0.829354 0.558724i $$-0.811291\pi$$
−0.829354 + 0.558724i $$0.811291\pi$$
$$752$$ 0 0
$$753$$ 2.14214 0.0780638
$$754$$ 0 0
$$755$$ − 47.4077i − 1.72534i
$$756$$ 0 0
$$757$$ 22.5445i 0.819395i 0.912221 + 0.409697i $$0.134366\pi$$
−0.912221 + 0.409697i $$0.865634\pi$$
$$758$$ 0 0
$$759$$ 38.6274 1.40209
$$760$$ 0 0
$$761$$ 18.9706 0.687682 0.343841 0.939028i $$-0.388272\pi$$
0.343841 + 0.939028i $$0.388272\pi$$
$$762$$ 0 0
$$763$$ 11.3492i 0.410868i
$$764$$ 0 0
$$765$$ − 76.6004i − 2.76949i
$$766$$ 0 0
$$767$$ 0.201010 0.00725806
$$768$$ 0 0
$$769$$ 35.2548 1.27132 0.635661 0.771968i $$-0.280728\pi$$
0.635661 + 0.771968i $$0.280728\pi$$
$$770$$ 0 0
$$771$$ − 24.3379i − 0.876508i
$$772$$ 0 0
$$773$$ − 8.73606i − 0.314214i −0.987582 0.157107i $$-0.949783\pi$$
0.987582 0.157107i $$-0.0502168\pi$$
$$774$$ 0 0
$$775$$ 10.3431 0.371537
$$776$$ 0 0
$$777$$ −13.6569 −0.489937
$$778$$ 0 0
$$779$$ − 17.4721i − 0.626004i
$$780$$ 0 0
$$781$$ − 5.07241i − 0.181505i
$$782$$ 0 0
$$783$$ 20.6863 0.739268
$$784$$ 0 0
$$785$$ 12.4853 0.445619
$$786$$ 0 0
$$787$$ − 7.46796i − 0.266204i −0.991102 0.133102i $$-0.957506\pi$$
0.991102 0.133102i $$-0.0424938\pi$$
$$788$$ 0 0
$$789$$ − 6.12293i − 0.217982i
$$790$$ 0 0
$$791$$ −8.82843 −0.313903
$$792$$ 0 0
$$793$$ 5.45584 0.193743
$$794$$ 0 0
$$795$$ 71.3742i 2.53138i
$$796$$ 0 0
$$797$$ 27.4763i 0.973260i 0.873608 + 0.486630i $$0.161774\pi$$
−0.873608 + 0.486630i $$0.838226\pi$$
$$798$$ 0 0
$$799$$ −61.2548 −2.16704
$$800$$ 0 0
$$801$$ 7.65685 0.270542
$$802$$ 0 0
$$803$$ 25.2346i 0.890509i
$$804$$ 0 0
$$805$$ − 17.8435i − 0.628902i
$$806$$ 0 0
$$807$$ 21.4558 0.755281
$$808$$ 0 0
$$809$$ −31.1716 −1.09593 −0.547967 0.836500i $$-0.684598\pi$$
−0.547967 + 0.836500i $$0.684598\pi$$
$$810$$ 0 0
$$811$$ 20.9819i 0.736775i 0.929672 + 0.368388i $$0.120090\pi$$
−0.929672 + 0.368388i $$0.879910\pi$$
$$812$$ 0 0
$$813$$ 6.12293i 0.214741i
$$814$$ 0 0
$$815$$ −10.3431 −0.362305
$$816$$ 0 0
$$817$$ 10.3431 0.361861
$$818$$ 0 0
$$819$$ 1.71644i 0.0599774i
$$820$$ 0 0
$$821$$ 8.65914i 0.302206i 0.988518 + 0.151103i $$0.0482825\pi$$
−0.988518 + 0.151103i $$0.951717\pi$$
$$822$$ 0 0
$$823$$ −40.9706 −1.42814 −0.714072 0.700072i $$-0.753151\pi$$
−0.714072 + 0.700072i $$0.753151\pi$$
$$824$$ 0 0
$$825$$ 10.3431 0.360102
$$826$$ 0 0
$$827$$ 10.9778i 0.381734i 0.981616 + 0.190867i $$0.0611300\pi$$
−0.981616 + 0.190867i $$0.938870\pi$$
$$828$$ 0 0
$$829$$ 6.19986i 0.215330i 0.994187 + 0.107665i $$0.0343374\pi$$
−0.994187 + 0.107665i $$0.965663\pi$$
$$830$$ 0 0
$$831$$ 38.6274 1.33997
$$832$$ 0 0
$$833$$ −7.65685 −0.265294
$$834$$ 0 0
$$835$$ 41.8100i 1.44690i
$$836$$ 0 0
$$837$$ 12.2459i 0.423279i
$$838$$ 0 0
$$839$$ −36.2843 −1.25267 −0.626336 0.779553i $$-0.715446\pi$$
−0.626336 + 0.779553i $$0.715446\pi$$
$$840$$ 0 0
$$841$$ −62.3137 −2.14875
$$842$$ 0 0
$$843$$ − 5.22625i − 0.180002i
$$844$$ 0 0
$$845$$ − 33.4454i − 1.15056i
$$846$$ 0 0
$$847$$ −6.31371 −0.216942
$$848$$ 0 0
$$849$$ 54.4264 1.86791
$$850$$ 0 0
$$851$$ 35.6871i 1.22334i
$$852$$ 0 0
$$853$$ − 47.3308i − 1.62057i −0.586033 0.810287i $$-0.699311\pi$$
0.586033 0.810287i $$-0.300689\pi$$
$$854$$ 0 0
$$855$$ −47.7990 −1.63469
$$856$$ 0 0
$$857$$ 0.627417 0.0214322 0.0107161 0.999943i $$-0.496589\pi$$
0.0107161 + 0.999943i $$0.496589\pi$$
$$858$$ 0 0
$$859$$ 3.50981i 0.119753i 0.998206 + 0.0598766i $$0.0190707\pi$$
−0.998206 + 0.0598766i $$0.980929\pi$$
$$860$$ 0 0
$$861$$ 9.55582i 0.325661i
$$862$$ 0 0
$$863$$ 51.3137 1.74674 0.873369 0.487058i $$-0.161930\pi$$
0.873369 + 0.487058i $$0.161930\pi$$
$$864$$ 0 0
$$865$$ −42.1421 −1.43288
$$866$$ 0 0
$$867$$ 108.778i 3.69428i
$$868$$ 0 0
$$869$$ 5.07241i 0.172070i
$$870$$ 0 0
$$871$$ −1.37258 −0.0465082
$$872$$ 0 0
$$873$$ 38.2843 1.29573
$$874$$ 0 0
$$875$$ 8.28772i 0.280176i
$$876$$ 0 0
$$877$$ 32.2542i 1.08915i 0.838713 + 0.544573i $$0.183308\pi$$
−0.838713 + 0.544573i $$0.816692\pi$$
$$878$$ 0 0
$$879$$ −28.4853 −0.960785
$$880$$ 0 0
$$881$$ 10.9706 0.369608 0.184804 0.982775i $$-0.440835\pi$$
0.184804 + 0.982775i $$0.440835\pi$$
$$882$$ 0 0
$$883$$ 51.7373i 1.74110i 0.492082 + 0.870549i $$0.336236\pi$$
−0.492082 + 0.870549i $$0.663764\pi$$
$$884$$ 0 0
$$885$$ − 3.06147i − 0.102910i
$$886$$ 0 0
$$887$$ 22.6274 0.759754 0.379877 0.925037i $$-0.375966\pi$$
0.379877 + 0.925037i $$0.375966\pi$$
$$888$$ 0 0
$$889$$ −4.48528 −0.150432
$$890$$ 0 0
$$891$$ − 12.6173i − 0.422695i
$$892$$ 0 0
$$893$$ 38.2233i 1.27909i
$$894$$ 0 0
$$895$$ 57.9411 1.93676
$$896$$ 0 0
$$897$$ 8.00000 0.267112
$$898$$ 0 0
$$899$$ − 54.0559i − 1.80286i
$$900$$ 0 0
$$901$$ − 80.0333i − 2.66630i
$$902$$ 0 0
$$903$$ −5.65685 −0.188248
$$904$$ 0 0
$$905$$ 6.82843 0.226985
$$906$$ 0 0
$$907$$ − 1.26810i − 0.0421066i −0.999778 0.0210533i $$-0.993298\pi$$
0.999778 0.0210533i $$-0.00670197\pi$$
$$908$$ 0 0
$$909$$ − 18.2919i − 0.606703i
$$910$$ 0 0
$$911$$ 38.8284 1.28644 0.643222 0.765680i $$-0.277597\pi$$
0.643222 + 0.765680i $$0.277597\pi$$
$$912$$ 0 0
$$913$$ 28.2843 0.936073
$$914$$ 0 0
$$915$$ − 83.0948i − 2.74703i
$$916$$ 0 0
$$917$$ − 18.2919i − 0.604051i
$$918$$ 0 0
$$919$$ −2.34315 −0.0772932 −0.0386466 0.999253i $$-0.512305\pi$$
−0.0386466 + 0.999253i $$0.512305\pi$$
$$920$$ 0 0
$$921$$ 22.8284 0.752222
$$922$$ 0 0
$$923$$ − 1.05053i − 0.0345786i
$$924$$ 0 0
$$925$$ 9.55582i 0.314193i
$$926$$ 0 0
$$927$$ 52.2843 1.71724
$$928$$ 0 0
$$929$$ 10.2843 0.337416 0.168708 0.985666i $$-0.446041\pi$$
0.168708 + 0.985666i $$0.446041\pi$$
$$930$$ 0 0
$$931$$ 4.77791i 0.156590i
$$932$$ 0 0
$$933$$ − 38.2233i − 1.25137i
$$934$$ 0 0
$$935$$ −43.3137 −1.41651
$$936$$ 0 0
$$937$$ −38.2843 −1.25069 −0.625346 0.780347i $$-0.715042\pi$$
−0.625346 + 0.780347i $$0.715042\pi$$
$$938$$ 0 0
$$939$$ − 21.8017i − 0.711471i
$$940$$ 0 0
$$941$$ 37.7749i 1.23143i 0.787970 + 0.615714i $$0.211132\pi$$
−0.787970 + 0.615714i $$0.788868\pi$$
$$942$$ 0 0
$$943$$ 24.9706 0.813153
$$944$$ 0 0
$$945$$ 5.65685 0.184017
$$946$$ 0 0
$$947$$ 48.3044i 1.56968i 0.619698 + 0.784841i $$0.287255\pi$$
−0.619698 + 0.784841i $$0.712745\pi$$
$$948$$ 0 0
$$949$$ 5.22625i 0.169651i
$$950$$ 0 0
$$951$$ −22.6274 −0.733744
$$952$$ 0 0
$$953$$ 37.3137 1.20871 0.604355 0.796715i $$-0.293431\pi$$
0.604355 + 0.796715i $$0.293431\pi$$
$$954$$ 0 0
$$955$$ 50.4692i 1.63314i
$$956$$ 0 0
$$957$$ − 54.0559i − 1.74738i
$$958$$ 0 0
$$959$$ −2.00000 −0.0645834
$$960$$ 0 0
$$961$$ 1.00000 0.0322581
$$962$$ 0 0
$$963$$ − 51.7373i − 1.66721i
$$964$$ 0 0
$$965$$ − 36.0585i − 1.16076i
$$966$$ 0 0
$$967$$ 29.8579 0.960164 0.480082 0.877224i $$-0.340607\pi$$
0.480082 + 0.877224i $$0.340607\pi$$
$$968$$ 0 0
$$969$$ 95.5980 3.07105
$$970$$ 0 0
$$971$$ 52.5570i 1.68663i 0.537416 + 0.843317i $$0.319401\pi$$
−0.537416 + 0.843317i $$0.680599\pi$$
$$972$$ 0 0
$$973$$ − 19.5600i − 0.627064i
$$974$$ 0 0
$$975$$ 2.14214 0.0686032
$$976$$ 0 0
$$977$$ 30.0000 0.959785 0.479893 0.877327i $$-0.340676\pi$$
0.479893 + 0.877327i $$0.340676\pi$$
$$978$$ 0 0
$$979$$ − 4.32957i − 0.138374i
$$980$$ 0 0
$$981$$ 43.4495i 1.38724i
$$982$$ 0 0
$$983$$ −23.0294 −0.734525 −0.367262 0.930117i $$-0.619705\pi$$
−0.367262 + 0.930117i $$0.619705\pi$$
$$984$$ 0 0
$$985$$ −16.0000 −0.509802
$$986$$ 0 0
$$987$$ − 20.9050i − 0.665414i
$$988$$ 0 0
$$989$$ 14.7821i 0.470043i
$$990$$ 0 0
$$991$$ 29.6569 0.942081 0.471041 0.882112i $$-0.343879\pi$$
0.471041 + 0.882112i $$0.343879\pi$$
$$992$$ 0 0
$$993$$ −48.9706 −1.55403
$$994$$ 0 0
$$995$$ 23.4412i 0.743136i
$$996$$ 0 0
$$997$$ − 0.0769232i − 0.00243618i −0.999999 0.00121809i $$-0.999612\pi$$
0.999999 0.00121809i $$-0.000387731\pi$$
$$998$$ 0 0
$$999$$ −11.3137 −0.357950
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 896.2.b.f.449.4 yes 4
4.3 odd 2 896.2.b.h.449.1 yes 4
8.3 odd 2 896.2.b.h.449.4 yes 4
8.5 even 2 inner 896.2.b.f.449.1 4
16.3 odd 4 1792.2.a.u.1.4 4
16.5 even 4 1792.2.a.w.1.4 4
16.11 odd 4 1792.2.a.u.1.1 4
16.13 even 4 1792.2.a.w.1.1 4

By twisted newform
Twist Min Dim Char Parity Ord Type
896.2.b.f.449.1 4 8.5 even 2 inner
896.2.b.f.449.4 yes 4 1.1 even 1 trivial
896.2.b.h.449.1 yes 4 4.3 odd 2
896.2.b.h.449.4 yes 4 8.3 odd 2
1792.2.a.u.1.1 4 16.11 odd 4
1792.2.a.u.1.4 4 16.3 odd 4
1792.2.a.w.1.1 4 16.13 even 4
1792.2.a.w.1.4 4 16.5 even 4