# Properties

 Label 8048.2.a.p.1.6 Level 8048 Weight 2 Character 8048.1 Self dual yes Analytic conductor 64.264 Analytic rank 1 Dimension 10 CM no Inner twists 1

# Learn more about

## Newspace parameters

 Level: $$N$$ $$=$$ $$8048 = 2^{4} \cdot 503$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 8048.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$64.2636035467$$ Analytic rank: $$1$$ Dimension: $$10$$ Coefficient field: $$\mathbb{Q}[x]/(x^{10} - \cdots)$$ Defining polynomial: $$x^{10} - 2 x^{9} - 9 x^{8} + 14 x^{7} + 27 x^{6} - 27 x^{5} - 34 x^{4} + 14 x^{3} + 17 x^{2} + x - 1$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 503) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.6 Root $$-0.489003$$ of defining polynomial Character $$\chi$$ $$=$$ 8048.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+1.48900 q^{3} -1.79865 q^{5} -0.552233 q^{7} -0.782869 q^{9} +O(q^{10})$$ $$q+1.48900 q^{3} -1.79865 q^{5} -0.552233 q^{7} -0.782869 q^{9} -4.33718 q^{11} +2.54873 q^{13} -2.67819 q^{15} +2.52304 q^{17} +5.11893 q^{19} -0.822276 q^{21} +3.78409 q^{23} -1.76486 q^{25} -5.63270 q^{27} +0.907287 q^{29} -0.380820 q^{31} -6.45807 q^{33} +0.993273 q^{35} -5.43266 q^{37} +3.79507 q^{39} +5.72459 q^{41} +9.21553 q^{43} +1.40811 q^{45} +8.81077 q^{47} -6.69504 q^{49} +3.75682 q^{51} -6.46357 q^{53} +7.80106 q^{55} +7.62210 q^{57} -3.40568 q^{59} -1.06109 q^{61} +0.432326 q^{63} -4.58427 q^{65} +0.253929 q^{67} +5.63451 q^{69} -11.7311 q^{71} -16.1271 q^{73} -2.62789 q^{75} +2.39513 q^{77} -7.76447 q^{79} -6.03851 q^{81} +16.3846 q^{83} -4.53807 q^{85} +1.35095 q^{87} -3.09992 q^{89} -1.40749 q^{91} -0.567042 q^{93} -9.20715 q^{95} -4.28605 q^{97} +3.39544 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$10q + 8q^{3} - q^{5} + 5q^{7} - 2q^{9} + O(q^{10})$$ $$10q + 8q^{3} - q^{5} + 5q^{7} - 2q^{9} + 3q^{11} - 18q^{13} + 2q^{15} - 11q^{17} + q^{21} + 2q^{23} - 27q^{25} + 2q^{27} - 9q^{29} + 22q^{31} - 10q^{33} + 6q^{35} - 35q^{37} - 8q^{39} - 4q^{41} + 20q^{43} + 2q^{45} - 7q^{47} - 27q^{49} - 9q^{51} - 24q^{53} + 11q^{55} - 23q^{57} - 17q^{59} - 4q^{61} - 10q^{63} - 16q^{65} + 6q^{67} - 2q^{69} + q^{71} - 31q^{73} - 30q^{75} + 3q^{77} + 10q^{79} - 6q^{81} - 22q^{83} - 6q^{85} - 25q^{87} + q^{89} - 10q^{91} - 6q^{93} - 39q^{95} - 57q^{97} - 35q^{99} + O(q^{100})$$

## Coefficient data

For each $$n$$ we display the coefficients of the $$q$$-expansion $$a_n$$, the Satake parameters $$\alpha_p$$, and the Satake angles $$\theta_p = \textrm{Arg}(\alpha_p)$$.

Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 0 0
$$3$$ 1.48900 0.859676 0.429838 0.902906i $$-0.358571\pi$$
0.429838 + 0.902906i $$0.358571\pi$$
$$4$$ 0 0
$$5$$ −1.79865 −0.804380 −0.402190 0.915556i $$-0.631751\pi$$
−0.402190 + 0.915556i $$0.631751\pi$$
$$6$$ 0 0
$$7$$ −0.552233 −0.208724 −0.104362 0.994539i $$-0.533280\pi$$
−0.104362 + 0.994539i $$0.533280\pi$$
$$8$$ 0 0
$$9$$ −0.782869 −0.260956
$$10$$ 0 0
$$11$$ −4.33718 −1.30771 −0.653854 0.756620i $$-0.726849\pi$$
−0.653854 + 0.756620i $$0.726849\pi$$
$$12$$ 0 0
$$13$$ 2.54873 0.706891 0.353445 0.935455i $$-0.385010\pi$$
0.353445 + 0.935455i $$0.385010\pi$$
$$14$$ 0 0
$$15$$ −2.67819 −0.691507
$$16$$ 0 0
$$17$$ 2.52304 0.611928 0.305964 0.952043i $$-0.401021\pi$$
0.305964 + 0.952043i $$0.401021\pi$$
$$18$$ 0 0
$$19$$ 5.11893 1.17436 0.587181 0.809455i $$-0.300238\pi$$
0.587181 + 0.809455i $$0.300238\pi$$
$$20$$ 0 0
$$21$$ −0.822276 −0.179435
$$22$$ 0 0
$$23$$ 3.78409 0.789036 0.394518 0.918888i $$-0.370912\pi$$
0.394518 + 0.918888i $$0.370912\pi$$
$$24$$ 0 0
$$25$$ −1.76486 −0.352972
$$26$$ 0 0
$$27$$ −5.63270 −1.08401
$$28$$ 0 0
$$29$$ 0.907287 0.168479 0.0842395 0.996446i $$-0.473154\pi$$
0.0842395 + 0.996446i $$0.473154\pi$$
$$30$$ 0 0
$$31$$ −0.380820 −0.0683972 −0.0341986 0.999415i $$-0.510888\pi$$
−0.0341986 + 0.999415i $$0.510888\pi$$
$$32$$ 0 0
$$33$$ −6.45807 −1.12421
$$34$$ 0 0
$$35$$ 0.993273 0.167894
$$36$$ 0 0
$$37$$ −5.43266 −0.893124 −0.446562 0.894753i $$-0.647352\pi$$
−0.446562 + 0.894753i $$0.647352\pi$$
$$38$$ 0 0
$$39$$ 3.79507 0.607697
$$40$$ 0 0
$$41$$ 5.72459 0.894031 0.447015 0.894526i $$-0.352487\pi$$
0.447015 + 0.894526i $$0.352487\pi$$
$$42$$ 0 0
$$43$$ 9.21553 1.40536 0.702678 0.711508i $$-0.251988\pi$$
0.702678 + 0.711508i $$0.251988\pi$$
$$44$$ 0 0
$$45$$ 1.40811 0.209908
$$46$$ 0 0
$$47$$ 8.81077 1.28518 0.642592 0.766209i $$-0.277859\pi$$
0.642592 + 0.766209i $$0.277859\pi$$
$$48$$ 0 0
$$49$$ −6.69504 −0.956434
$$50$$ 0 0
$$51$$ 3.75682 0.526060
$$52$$ 0 0
$$53$$ −6.46357 −0.887840 −0.443920 0.896066i $$-0.646413\pi$$
−0.443920 + 0.896066i $$0.646413\pi$$
$$54$$ 0 0
$$55$$ 7.80106 1.05190
$$56$$ 0 0
$$57$$ 7.62210 1.00957
$$58$$ 0 0
$$59$$ −3.40568 −0.443381 −0.221691 0.975117i $$-0.571157\pi$$
−0.221691 + 0.975117i $$0.571157\pi$$
$$60$$ 0 0
$$61$$ −1.06109 −0.135859 −0.0679294 0.997690i $$-0.521639\pi$$
−0.0679294 + 0.997690i $$0.521639\pi$$
$$62$$ 0 0
$$63$$ 0.432326 0.0544679
$$64$$ 0 0
$$65$$ −4.58427 −0.568609
$$66$$ 0 0
$$67$$ 0.253929 0.0310223 0.0155112 0.999880i $$-0.495062\pi$$
0.0155112 + 0.999880i $$0.495062\pi$$
$$68$$ 0 0
$$69$$ 5.63451 0.678316
$$70$$ 0 0
$$71$$ −11.7311 −1.39222 −0.696112 0.717933i $$-0.745088\pi$$
−0.696112 + 0.717933i $$0.745088\pi$$
$$72$$ 0 0
$$73$$ −16.1271 −1.88753 −0.943765 0.330617i $$-0.892743\pi$$
−0.943765 + 0.330617i $$0.892743\pi$$
$$74$$ 0 0
$$75$$ −2.62789 −0.303442
$$76$$ 0 0
$$77$$ 2.39513 0.272951
$$78$$ 0 0
$$79$$ −7.76447 −0.873571 −0.436785 0.899566i $$-0.643883\pi$$
−0.436785 + 0.899566i $$0.643883\pi$$
$$80$$ 0 0
$$81$$ −6.03851 −0.670945
$$82$$ 0 0
$$83$$ 16.3846 1.79844 0.899222 0.437492i $$-0.144133\pi$$
0.899222 + 0.437492i $$0.144133\pi$$
$$84$$ 0 0
$$85$$ −4.53807 −0.492223
$$86$$ 0 0
$$87$$ 1.35095 0.144837
$$88$$ 0 0
$$89$$ −3.09992 −0.328590 −0.164295 0.986411i $$-0.552535\pi$$
−0.164295 + 0.986411i $$0.552535\pi$$
$$90$$ 0 0
$$91$$ −1.40749 −0.147545
$$92$$ 0 0
$$93$$ −0.567042 −0.0587995
$$94$$ 0 0
$$95$$ −9.20715 −0.944634
$$96$$ 0 0
$$97$$ −4.28605 −0.435183 −0.217591 0.976040i $$-0.569820\pi$$
−0.217591 + 0.976040i $$0.569820\pi$$
$$98$$ 0 0
$$99$$ 3.39544 0.341255
$$100$$ 0 0
$$101$$ −3.73446 −0.371592 −0.185796 0.982588i $$-0.559486\pi$$
−0.185796 + 0.982588i $$0.559486\pi$$
$$102$$ 0 0
$$103$$ −2.83220 −0.279065 −0.139533 0.990217i $$-0.544560\pi$$
−0.139533 + 0.990217i $$0.544560\pi$$
$$104$$ 0 0
$$105$$ 1.47899 0.144334
$$106$$ 0 0
$$107$$ −10.0994 −0.976343 −0.488172 0.872748i $$-0.662336\pi$$
−0.488172 + 0.872748i $$0.662336\pi$$
$$108$$ 0 0
$$109$$ −6.86764 −0.657800 −0.328900 0.944365i $$-0.606678\pi$$
−0.328900 + 0.944365i $$0.606678\pi$$
$$110$$ 0 0
$$111$$ −8.08925 −0.767798
$$112$$ 0 0
$$113$$ 2.80558 0.263927 0.131963 0.991255i $$-0.457872\pi$$
0.131963 + 0.991255i $$0.457872\pi$$
$$114$$ 0 0
$$115$$ −6.80624 −0.634685
$$116$$ 0 0
$$117$$ −1.99532 −0.184468
$$118$$ 0 0
$$119$$ −1.39331 −0.127724
$$120$$ 0 0
$$121$$ 7.81112 0.710102
$$122$$ 0 0
$$123$$ 8.52393 0.768577
$$124$$ 0 0
$$125$$ 12.1676 1.08830
$$126$$ 0 0
$$127$$ 14.9147 1.32347 0.661734 0.749739i $$-0.269821\pi$$
0.661734 + 0.749739i $$0.269821\pi$$
$$128$$ 0 0
$$129$$ 13.7220 1.20815
$$130$$ 0 0
$$131$$ −13.5207 −1.18131 −0.590654 0.806925i $$-0.701130\pi$$
−0.590654 + 0.806925i $$0.701130\pi$$
$$132$$ 0 0
$$133$$ −2.82684 −0.245118
$$134$$ 0 0
$$135$$ 10.1313 0.871960
$$136$$ 0 0
$$137$$ −12.4479 −1.06350 −0.531750 0.846901i $$-0.678465\pi$$
−0.531750 + 0.846901i $$0.678465\pi$$
$$138$$ 0 0
$$139$$ −2.15443 −0.182736 −0.0913681 0.995817i $$-0.529124\pi$$
−0.0913681 + 0.995817i $$0.529124\pi$$
$$140$$ 0 0
$$141$$ 13.1193 1.10484
$$142$$ 0 0
$$143$$ −11.0543 −0.924407
$$144$$ 0 0
$$145$$ −1.63189 −0.135521
$$146$$ 0 0
$$147$$ −9.96894 −0.822224
$$148$$ 0 0
$$149$$ −8.59253 −0.703927 −0.351964 0.936014i $$-0.614486\pi$$
−0.351964 + 0.936014i $$0.614486\pi$$
$$150$$ 0 0
$$151$$ −0.527862 −0.0429568 −0.0214784 0.999769i $$-0.506837\pi$$
−0.0214784 + 0.999769i $$0.506837\pi$$
$$152$$ 0 0
$$153$$ −1.97521 −0.159687
$$154$$ 0 0
$$155$$ 0.684961 0.0550174
$$156$$ 0 0
$$157$$ 11.1711 0.891554 0.445777 0.895144i $$-0.352927\pi$$
0.445777 + 0.895144i $$0.352927\pi$$
$$158$$ 0 0
$$159$$ −9.62428 −0.763255
$$160$$ 0 0
$$161$$ −2.08970 −0.164691
$$162$$ 0 0
$$163$$ −20.0109 −1.56738 −0.783689 0.621153i $$-0.786664\pi$$
−0.783689 + 0.621153i $$0.786664\pi$$
$$164$$ 0 0
$$165$$ 11.6158 0.904289
$$166$$ 0 0
$$167$$ 11.9520 0.924877 0.462439 0.886651i $$-0.346975\pi$$
0.462439 + 0.886651i $$0.346975\pi$$
$$168$$ 0 0
$$169$$ −6.50397 −0.500305
$$170$$ 0 0
$$171$$ −4.00745 −0.306457
$$172$$ 0 0
$$173$$ 3.14561 0.239156 0.119578 0.992825i $$-0.461846\pi$$
0.119578 + 0.992825i $$0.461846\pi$$
$$174$$ 0 0
$$175$$ 0.974615 0.0736739
$$176$$ 0 0
$$177$$ −5.07106 −0.381164
$$178$$ 0 0
$$179$$ 6.02096 0.450028 0.225014 0.974356i $$-0.427757\pi$$
0.225014 + 0.974356i $$0.427757\pi$$
$$180$$ 0 0
$$181$$ −23.0475 −1.71311 −0.856553 0.516059i $$-0.827398\pi$$
−0.856553 + 0.516059i $$0.827398\pi$$
$$182$$ 0 0
$$183$$ −1.57997 −0.116795
$$184$$ 0 0
$$185$$ 9.77145 0.718412
$$186$$ 0 0
$$187$$ −10.9429 −0.800224
$$188$$ 0 0
$$189$$ 3.11056 0.226260
$$190$$ 0 0
$$191$$ 17.9774 1.30080 0.650398 0.759594i $$-0.274602\pi$$
0.650398 + 0.759594i $$0.274602\pi$$
$$192$$ 0 0
$$193$$ −10.4429 −0.751697 −0.375848 0.926681i $$-0.622649\pi$$
−0.375848 + 0.926681i $$0.622649\pi$$
$$194$$ 0 0
$$195$$ −6.82600 −0.488820
$$196$$ 0 0
$$197$$ −4.36026 −0.310656 −0.155328 0.987863i $$-0.549643\pi$$
−0.155328 + 0.987863i $$0.549643\pi$$
$$198$$ 0 0
$$199$$ −14.0205 −0.993888 −0.496944 0.867783i $$-0.665545\pi$$
−0.496944 + 0.867783i $$0.665545\pi$$
$$200$$ 0 0
$$201$$ 0.378101 0.0266692
$$202$$ 0 0
$$203$$ −0.501034 −0.0351657
$$204$$ 0 0
$$205$$ −10.2965 −0.719141
$$206$$ 0 0
$$207$$ −2.96244 −0.205904
$$208$$ 0 0
$$209$$ −22.2017 −1.53572
$$210$$ 0 0
$$211$$ 11.0580 0.761265 0.380633 0.924726i $$-0.375706\pi$$
0.380633 + 0.924726i $$0.375706\pi$$
$$212$$ 0 0
$$213$$ −17.4676 −1.19686
$$214$$ 0 0
$$215$$ −16.5755 −1.13044
$$216$$ 0 0
$$217$$ 0.210301 0.0142762
$$218$$ 0 0
$$219$$ −24.0133 −1.62267
$$220$$ 0 0
$$221$$ 6.43056 0.432566
$$222$$ 0 0
$$223$$ 15.5502 1.04132 0.520658 0.853765i $$-0.325687\pi$$
0.520658 + 0.853765i $$0.325687\pi$$
$$224$$ 0 0
$$225$$ 1.38166 0.0921104
$$226$$ 0 0
$$227$$ −1.27577 −0.0846759 −0.0423379 0.999103i $$-0.513481\pi$$
−0.0423379 + 0.999103i $$0.513481\pi$$
$$228$$ 0 0
$$229$$ −15.5232 −1.02580 −0.512901 0.858448i $$-0.671429\pi$$
−0.512901 + 0.858448i $$0.671429\pi$$
$$230$$ 0 0
$$231$$ 3.56636 0.234649
$$232$$ 0 0
$$233$$ −27.8010 −1.82130 −0.910651 0.413177i $$-0.864419\pi$$
−0.910651 + 0.413177i $$0.864419\pi$$
$$234$$ 0 0
$$235$$ −15.8475 −1.03378
$$236$$ 0 0
$$237$$ −11.5613 −0.750988
$$238$$ 0 0
$$239$$ −1.20369 −0.0778601 −0.0389301 0.999242i $$-0.512395\pi$$
−0.0389301 + 0.999242i $$0.512395\pi$$
$$240$$ 0 0
$$241$$ −25.6025 −1.64920 −0.824602 0.565713i $$-0.808601\pi$$
−0.824602 + 0.565713i $$0.808601\pi$$
$$242$$ 0 0
$$243$$ 7.90676 0.507219
$$244$$ 0 0
$$245$$ 12.0420 0.769337
$$246$$ 0 0
$$247$$ 13.0468 0.830146
$$248$$ 0 0
$$249$$ 24.3967 1.54608
$$250$$ 0 0
$$251$$ −13.2638 −0.837205 −0.418603 0.908170i $$-0.637480\pi$$
−0.418603 + 0.908170i $$0.637480\pi$$
$$252$$ 0 0
$$253$$ −16.4123 −1.03183
$$254$$ 0 0
$$255$$ −6.75720 −0.423152
$$256$$ 0 0
$$257$$ −16.1006 −1.00433 −0.502163 0.864773i $$-0.667462\pi$$
−0.502163 + 0.864773i $$0.667462\pi$$
$$258$$ 0 0
$$259$$ 3.00009 0.186417
$$260$$ 0 0
$$261$$ −0.710288 −0.0439657
$$262$$ 0 0
$$263$$ −6.77336 −0.417663 −0.208832 0.977952i $$-0.566966\pi$$
−0.208832 + 0.977952i $$0.566966\pi$$
$$264$$ 0 0
$$265$$ 11.6257 0.714161
$$266$$ 0 0
$$267$$ −4.61579 −0.282481
$$268$$ 0 0
$$269$$ 23.4853 1.43192 0.715962 0.698139i $$-0.245988\pi$$
0.715962 + 0.698139i $$0.245988\pi$$
$$270$$ 0 0
$$271$$ −17.5886 −1.06843 −0.534216 0.845348i $$-0.679393\pi$$
−0.534216 + 0.845348i $$0.679393\pi$$
$$272$$ 0 0
$$273$$ −2.09576 −0.126841
$$274$$ 0 0
$$275$$ 7.65452 0.461585
$$276$$ 0 0
$$277$$ −11.3448 −0.681645 −0.340823 0.940128i $$-0.610706\pi$$
−0.340823 + 0.940128i $$0.610706\pi$$
$$278$$ 0 0
$$279$$ 0.298132 0.0178487
$$280$$ 0 0
$$281$$ −9.36752 −0.558820 −0.279410 0.960172i $$-0.590139\pi$$
−0.279410 + 0.960172i $$0.590139\pi$$
$$282$$ 0 0
$$283$$ −26.8789 −1.59778 −0.798892 0.601474i $$-0.794580\pi$$
−0.798892 + 0.601474i $$0.794580\pi$$
$$284$$ 0 0
$$285$$ −13.7095 −0.812080
$$286$$ 0 0
$$287$$ −3.16131 −0.186606
$$288$$ 0 0
$$289$$ −10.6342 −0.625544
$$290$$ 0 0
$$291$$ −6.38195 −0.374117
$$292$$ 0 0
$$293$$ 17.2365 1.00697 0.503485 0.864004i $$-0.332051\pi$$
0.503485 + 0.864004i $$0.332051\pi$$
$$294$$ 0 0
$$295$$ 6.12561 0.356647
$$296$$ 0 0
$$297$$ 24.4301 1.41758
$$298$$ 0 0
$$299$$ 9.64462 0.557763
$$300$$ 0 0
$$301$$ −5.08912 −0.293332
$$302$$ 0 0
$$303$$ −5.56062 −0.319449
$$304$$ 0 0
$$305$$ 1.90853 0.109282
$$306$$ 0 0
$$307$$ −29.0001 −1.65512 −0.827561 0.561376i $$-0.810272\pi$$
−0.827561 + 0.561376i $$0.810272\pi$$
$$308$$ 0 0
$$309$$ −4.21716 −0.239906
$$310$$ 0 0
$$311$$ −27.2312 −1.54414 −0.772071 0.635536i $$-0.780779\pi$$
−0.772071 + 0.635536i $$0.780779\pi$$
$$312$$ 0 0
$$313$$ 5.53059 0.312607 0.156304 0.987709i $$-0.450042\pi$$
0.156304 + 0.987709i $$0.450042\pi$$
$$314$$ 0 0
$$315$$ −0.777603 −0.0438129
$$316$$ 0 0
$$317$$ 31.2030 1.75253 0.876267 0.481826i $$-0.160026\pi$$
0.876267 + 0.481826i $$0.160026\pi$$
$$318$$ 0 0
$$319$$ −3.93507 −0.220322
$$320$$ 0 0
$$321$$ −15.0380 −0.839339
$$322$$ 0 0
$$323$$ 12.9153 0.718626
$$324$$ 0 0
$$325$$ −4.49816 −0.249513
$$326$$ 0 0
$$327$$ −10.2259 −0.565495
$$328$$ 0 0
$$329$$ −4.86560 −0.268249
$$330$$ 0 0
$$331$$ −16.8025 −0.923551 −0.461775 0.886997i $$-0.652787\pi$$
−0.461775 + 0.886997i $$0.652787\pi$$
$$332$$ 0 0
$$333$$ 4.25307 0.233067
$$334$$ 0 0
$$335$$ −0.456729 −0.0249538
$$336$$ 0 0
$$337$$ 7.57943 0.412878 0.206439 0.978459i $$-0.433813\pi$$
0.206439 + 0.978459i $$0.433813\pi$$
$$338$$ 0 0
$$339$$ 4.17752 0.226892
$$340$$ 0 0
$$341$$ 1.65168 0.0894437
$$342$$ 0 0
$$343$$ 7.56285 0.408355
$$344$$ 0 0
$$345$$ −10.1345 −0.545624
$$346$$ 0 0
$$347$$ 28.0855 1.50771 0.753853 0.657043i $$-0.228193\pi$$
0.753853 + 0.657043i $$0.228193\pi$$
$$348$$ 0 0
$$349$$ 11.6230 0.622164 0.311082 0.950383i $$-0.399309\pi$$
0.311082 + 0.950383i $$0.399309\pi$$
$$350$$ 0 0
$$351$$ −14.3563 −0.766280
$$352$$ 0 0
$$353$$ −6.16556 −0.328160 −0.164080 0.986447i $$-0.552465\pi$$
−0.164080 + 0.986447i $$0.552465\pi$$
$$354$$ 0 0
$$355$$ 21.1001 1.11988
$$356$$ 0 0
$$357$$ −2.07464 −0.109802
$$358$$ 0 0
$$359$$ 10.3896 0.548344 0.274172 0.961681i $$-0.411596\pi$$
0.274172 + 0.961681i $$0.411596\pi$$
$$360$$ 0 0
$$361$$ 7.20342 0.379127
$$362$$ 0 0
$$363$$ 11.6308 0.610458
$$364$$ 0 0
$$365$$ 29.0069 1.51829
$$366$$ 0 0
$$367$$ −0.571294 −0.0298213 −0.0149107 0.999889i $$-0.504746\pi$$
−0.0149107 + 0.999889i $$0.504746\pi$$
$$368$$ 0 0
$$369$$ −4.48161 −0.233303
$$370$$ 0 0
$$371$$ 3.56940 0.185314
$$372$$ 0 0
$$373$$ 2.95332 0.152917 0.0764586 0.997073i $$-0.475639\pi$$
0.0764586 + 0.997073i $$0.475639\pi$$
$$374$$ 0 0
$$375$$ 18.1176 0.935590
$$376$$ 0 0
$$377$$ 2.31243 0.119096
$$378$$ 0 0
$$379$$ 26.2705 1.34942 0.674711 0.738082i $$-0.264268\pi$$
0.674711 + 0.738082i $$0.264268\pi$$
$$380$$ 0 0
$$381$$ 22.2081 1.13775
$$382$$ 0 0
$$383$$ −3.44503 −0.176033 −0.0880164 0.996119i $$-0.528053\pi$$
−0.0880164 + 0.996119i $$0.528053\pi$$
$$384$$ 0 0
$$385$$ −4.30800 −0.219556
$$386$$ 0 0
$$387$$ −7.21456 −0.366736
$$388$$ 0 0
$$389$$ −7.95150 −0.403157 −0.201578 0.979472i $$-0.564607\pi$$
−0.201578 + 0.979472i $$0.564607\pi$$
$$390$$ 0 0
$$391$$ 9.54742 0.482834
$$392$$ 0 0
$$393$$ −20.1323 −1.01554
$$394$$ 0 0
$$395$$ 13.9656 0.702683
$$396$$ 0 0
$$397$$ 21.7255 1.09037 0.545186 0.838315i $$-0.316459\pi$$
0.545186 + 0.838315i $$0.316459\pi$$
$$398$$ 0 0
$$399$$ −4.20917 −0.210722
$$400$$ 0 0
$$401$$ 14.1072 0.704481 0.352241 0.935909i $$-0.385420\pi$$
0.352241 + 0.935909i $$0.385420\pi$$
$$402$$ 0 0
$$403$$ −0.970607 −0.0483494
$$404$$ 0 0
$$405$$ 10.8612 0.539695
$$406$$ 0 0
$$407$$ 23.5624 1.16795
$$408$$ 0 0
$$409$$ −10.2905 −0.508834 −0.254417 0.967095i $$-0.581884\pi$$
−0.254417 + 0.967095i $$0.581884\pi$$
$$410$$ 0 0
$$411$$ −18.5350 −0.914266
$$412$$ 0 0
$$413$$ 1.88072 0.0925444
$$414$$ 0 0
$$415$$ −29.4702 −1.44663
$$416$$ 0 0
$$417$$ −3.20795 −0.157094
$$418$$ 0 0
$$419$$ −19.0965 −0.932925 −0.466463 0.884541i $$-0.654472\pi$$
−0.466463 + 0.884541i $$0.654472\pi$$
$$420$$ 0 0
$$421$$ 26.3025 1.28191 0.640953 0.767580i $$-0.278539\pi$$
0.640953 + 0.767580i $$0.278539\pi$$
$$422$$ 0 0
$$423$$ −6.89768 −0.335377
$$424$$ 0 0
$$425$$ −4.45283 −0.215994
$$426$$ 0 0
$$427$$ 0.585969 0.0283570
$$428$$ 0 0
$$429$$ −16.4599 −0.794691
$$430$$ 0 0
$$431$$ 16.5246 0.795963 0.397982 0.917393i $$-0.369711\pi$$
0.397982 + 0.917393i $$0.369711\pi$$
$$432$$ 0 0
$$433$$ 4.52467 0.217442 0.108721 0.994072i $$-0.465325\pi$$
0.108721 + 0.994072i $$0.465325\pi$$
$$434$$ 0 0
$$435$$ −2.42989 −0.116504
$$436$$ 0 0
$$437$$ 19.3705 0.926615
$$438$$ 0 0
$$439$$ 9.06783 0.432784 0.216392 0.976307i $$-0.430571\pi$$
0.216392 + 0.976307i $$0.430571\pi$$
$$440$$ 0 0
$$441$$ 5.24134 0.249588
$$442$$ 0 0
$$443$$ 12.6542 0.601219 0.300609 0.953747i $$-0.402810\pi$$
0.300609 + 0.953747i $$0.402810\pi$$
$$444$$ 0 0
$$445$$ 5.57566 0.264312
$$446$$ 0 0
$$447$$ −12.7943 −0.605150
$$448$$ 0 0
$$449$$ 34.8336 1.64390 0.821950 0.569560i $$-0.192886\pi$$
0.821950 + 0.569560i $$0.192886\pi$$
$$450$$ 0 0
$$451$$ −24.8286 −1.16913
$$452$$ 0 0
$$453$$ −0.785988 −0.0369290
$$454$$ 0 0
$$455$$ 2.53158 0.118683
$$456$$ 0 0
$$457$$ −14.3718 −0.672283 −0.336141 0.941812i $$-0.609122\pi$$
−0.336141 + 0.941812i $$0.609122\pi$$
$$458$$ 0 0
$$459$$ −14.2116 −0.663339
$$460$$ 0 0
$$461$$ −20.9323 −0.974914 −0.487457 0.873147i $$-0.662075\pi$$
−0.487457 + 0.873147i $$0.662075\pi$$
$$462$$ 0 0
$$463$$ −19.4204 −0.902541 −0.451271 0.892387i $$-0.649029\pi$$
−0.451271 + 0.892387i $$0.649029\pi$$
$$464$$ 0 0
$$465$$ 1.01991 0.0472971
$$466$$ 0 0
$$467$$ −22.0847 −1.02196 −0.510980 0.859593i $$-0.670717\pi$$
−0.510980 + 0.859593i $$0.670717\pi$$
$$468$$ 0 0
$$469$$ −0.140228 −0.00647512
$$470$$ 0 0
$$471$$ 16.6339 0.766448
$$472$$ 0 0
$$473$$ −39.9694 −1.83780
$$474$$ 0 0
$$475$$ −9.03420 −0.414518
$$476$$ 0 0
$$477$$ 5.06013 0.231688
$$478$$ 0 0
$$479$$ −25.4118 −1.16109 −0.580547 0.814227i $$-0.697161\pi$$
−0.580547 + 0.814227i $$0.697161\pi$$
$$480$$ 0 0
$$481$$ −13.8464 −0.631342
$$482$$ 0 0
$$483$$ −3.11156 −0.141581
$$484$$ 0 0
$$485$$ 7.70911 0.350053
$$486$$ 0 0
$$487$$ 36.7671 1.66608 0.833039 0.553214i $$-0.186599\pi$$
0.833039 + 0.553214i $$0.186599\pi$$
$$488$$ 0 0
$$489$$ −29.7964 −1.34744
$$490$$ 0 0
$$491$$ 12.6572 0.571212 0.285606 0.958347i $$-0.407805\pi$$
0.285606 + 0.958347i $$0.407805\pi$$
$$492$$ 0 0
$$493$$ 2.28913 0.103097
$$494$$ 0 0
$$495$$ −6.10721 −0.274499
$$496$$ 0 0
$$497$$ 6.47829 0.290591
$$498$$ 0 0
$$499$$ −2.39500 −0.107215 −0.0536074 0.998562i $$-0.517072\pi$$
−0.0536074 + 0.998562i $$0.517072\pi$$
$$500$$ 0 0
$$501$$ 17.7966 0.795095
$$502$$ 0 0
$$503$$ 1.00000 0.0445878
$$504$$ 0 0
$$505$$ 6.71697 0.298901
$$506$$ 0 0
$$507$$ −9.68443 −0.430101
$$508$$ 0 0
$$509$$ 6.46329 0.286480 0.143240 0.989688i $$-0.454248\pi$$
0.143240 + 0.989688i $$0.454248\pi$$
$$510$$ 0 0
$$511$$ 8.90589 0.393973
$$512$$ 0 0
$$513$$ −28.8334 −1.27303
$$514$$ 0 0
$$515$$ 5.09414 0.224475
$$516$$ 0 0
$$517$$ −38.2139 −1.68065
$$518$$ 0 0
$$519$$ 4.68382 0.205597
$$520$$ 0 0
$$521$$ 34.3036 1.50287 0.751434 0.659808i $$-0.229363\pi$$
0.751434 + 0.659808i $$0.229363\pi$$
$$522$$ 0 0
$$523$$ −30.3662 −1.32782 −0.663910 0.747812i $$-0.731104\pi$$
−0.663910 + 0.747812i $$0.731104\pi$$
$$524$$ 0 0
$$525$$ 1.45120 0.0633357
$$526$$ 0 0
$$527$$ −0.960825 −0.0418542
$$528$$ 0 0
$$529$$ −8.68070 −0.377422
$$530$$ 0 0
$$531$$ 2.66620 0.115703
$$532$$ 0 0
$$533$$ 14.5904 0.631982
$$534$$ 0 0
$$535$$ 18.1652 0.785351
$$536$$ 0 0
$$537$$ 8.96523 0.386878
$$538$$ 0 0
$$539$$ 29.0376 1.25074
$$540$$ 0 0
$$541$$ 39.1371 1.68263 0.841317 0.540542i $$-0.181781\pi$$
0.841317 + 0.540542i $$0.181781\pi$$
$$542$$ 0 0
$$543$$ −34.3178 −1.47272
$$544$$ 0 0
$$545$$ 12.3525 0.529121
$$546$$ 0 0
$$547$$ −24.0487 −1.02825 −0.514125 0.857715i $$-0.671883\pi$$
−0.514125 + 0.857715i $$0.671883\pi$$
$$548$$ 0 0
$$549$$ 0.830696 0.0354532
$$550$$ 0 0
$$551$$ 4.64434 0.197855
$$552$$ 0 0
$$553$$ 4.28779 0.182335
$$554$$ 0 0
$$555$$ 14.5497 0.617602
$$556$$ 0 0
$$557$$ −17.2987 −0.732971 −0.366486 0.930424i $$-0.619439\pi$$
−0.366486 + 0.930424i $$0.619439\pi$$
$$558$$ 0 0
$$559$$ 23.4879 0.993433
$$560$$ 0 0
$$561$$ −16.2940 −0.687934
$$562$$ 0 0
$$563$$ −5.63944 −0.237674 −0.118837 0.992914i $$-0.537917\pi$$
−0.118837 + 0.992914i $$0.537917\pi$$
$$564$$ 0 0
$$565$$ −5.04626 −0.212298
$$566$$ 0 0
$$567$$ 3.33466 0.140043
$$568$$ 0 0
$$569$$ −12.0350 −0.504535 −0.252268 0.967658i $$-0.581176\pi$$
−0.252268 + 0.967658i $$0.581176\pi$$
$$570$$ 0 0
$$571$$ −29.9404 −1.25297 −0.626484 0.779434i $$-0.715507\pi$$
−0.626484 + 0.779434i $$0.715507\pi$$
$$572$$ 0 0
$$573$$ 26.7683 1.11826
$$574$$ 0 0
$$575$$ −6.67839 −0.278508
$$576$$ 0 0
$$577$$ 17.6080 0.733032 0.366516 0.930412i $$-0.380551\pi$$
0.366516 + 0.930412i $$0.380551\pi$$
$$578$$ 0 0
$$579$$ −15.5495 −0.646216
$$580$$ 0 0
$$581$$ −9.04811 −0.375379
$$582$$ 0 0
$$583$$ 28.0337 1.16104
$$584$$ 0 0
$$585$$ 3.58889 0.148382
$$586$$ 0 0
$$587$$ 21.2268 0.876125 0.438062 0.898945i $$-0.355665\pi$$
0.438062 + 0.898945i $$0.355665\pi$$
$$588$$ 0 0
$$589$$ −1.94939 −0.0803231
$$590$$ 0 0
$$591$$ −6.49244 −0.267063
$$592$$ 0 0
$$593$$ 1.78783 0.0734175 0.0367087 0.999326i $$-0.488313\pi$$
0.0367087 + 0.999326i $$0.488313\pi$$
$$594$$ 0 0
$$595$$ 2.50607 0.102739
$$596$$ 0 0
$$597$$ −20.8766 −0.854422
$$598$$ 0 0
$$599$$ 4.15199 0.169646 0.0848228 0.996396i $$-0.472968\pi$$
0.0848228 + 0.996396i $$0.472968\pi$$
$$600$$ 0 0
$$601$$ −4.87221 −0.198742 −0.0993708 0.995050i $$-0.531683\pi$$
−0.0993708 + 0.995050i $$0.531683\pi$$
$$602$$ 0 0
$$603$$ −0.198793 −0.00809548
$$604$$ 0 0
$$605$$ −14.0495 −0.571192
$$606$$ 0 0
$$607$$ 22.0579 0.895303 0.447652 0.894208i $$-0.352261\pi$$
0.447652 + 0.894208i $$0.352261\pi$$
$$608$$ 0 0
$$609$$ −0.746041 −0.0302311
$$610$$ 0 0
$$611$$ 22.4563 0.908484
$$612$$ 0 0
$$613$$ −19.0369 −0.768895 −0.384447 0.923147i $$-0.625608\pi$$
−0.384447 + 0.923147i $$0.625608\pi$$
$$614$$ 0 0
$$615$$ −15.3316 −0.618228
$$616$$ 0 0
$$617$$ −22.0697 −0.888494 −0.444247 0.895904i $$-0.646529\pi$$
−0.444247 + 0.895904i $$0.646529\pi$$
$$618$$ 0 0
$$619$$ 5.69372 0.228850 0.114425 0.993432i $$-0.463497\pi$$
0.114425 + 0.993432i $$0.463497\pi$$
$$620$$ 0 0
$$621$$ −21.3146 −0.855327
$$622$$ 0 0
$$623$$ 1.71187 0.0685848
$$624$$ 0 0
$$625$$ −13.0609 −0.522438
$$626$$ 0 0
$$627$$ −33.0584 −1.32023
$$628$$ 0 0
$$629$$ −13.7069 −0.546528
$$630$$ 0 0
$$631$$ −37.6327 −1.49813 −0.749067 0.662494i $$-0.769498\pi$$
−0.749067 + 0.662494i $$0.769498\pi$$
$$632$$ 0 0
$$633$$ 16.4654 0.654442
$$634$$ 0 0
$$635$$ −26.8263 −1.06457
$$636$$ 0 0
$$637$$ −17.0639 −0.676095
$$638$$ 0 0
$$639$$ 9.18391 0.363310
$$640$$ 0 0
$$641$$ 6.63550 0.262086 0.131043 0.991377i $$-0.458167\pi$$
0.131043 + 0.991377i $$0.458167\pi$$
$$642$$ 0 0
$$643$$ 29.4970 1.16325 0.581625 0.813457i $$-0.302417\pi$$
0.581625 + 0.813457i $$0.302417\pi$$
$$644$$ 0 0
$$645$$ −24.6810 −0.971813
$$646$$ 0 0
$$647$$ 45.6726 1.79558 0.897788 0.440427i $$-0.145173\pi$$
0.897788 + 0.440427i $$0.145173\pi$$
$$648$$ 0 0
$$649$$ 14.7710 0.579813
$$650$$ 0 0
$$651$$ 0.313139 0.0122729
$$652$$ 0 0
$$653$$ −31.6691 −1.23931 −0.619655 0.784875i $$-0.712727\pi$$
−0.619655 + 0.784875i $$0.712727\pi$$
$$654$$ 0 0
$$655$$ 24.3190 0.950220
$$656$$ 0 0
$$657$$ 12.6254 0.492563
$$658$$ 0 0
$$659$$ −9.73186 −0.379099 −0.189550 0.981871i $$-0.560703\pi$$
−0.189550 + 0.981871i $$0.560703\pi$$
$$660$$ 0 0
$$661$$ −14.5544 −0.566100 −0.283050 0.959105i $$-0.591346\pi$$
−0.283050 + 0.959105i $$0.591346\pi$$
$$662$$ 0 0
$$663$$ 9.57513 0.371867
$$664$$ 0 0
$$665$$ 5.08449 0.197168
$$666$$ 0 0
$$667$$ 3.43325 0.132936
$$668$$ 0 0
$$669$$ 23.1542 0.895195
$$670$$ 0 0
$$671$$ 4.60214 0.177664
$$672$$ 0 0
$$673$$ −46.1914 −1.78055 −0.890273 0.455427i $$-0.849486\pi$$
−0.890273 + 0.455427i $$0.849486\pi$$
$$674$$ 0 0
$$675$$ 9.94095 0.382627
$$676$$ 0 0
$$677$$ −10.6087 −0.407725 −0.203863 0.978999i $$-0.565350\pi$$
−0.203863 + 0.978999i $$0.565350\pi$$
$$678$$ 0 0
$$679$$ 2.36690 0.0908333
$$680$$ 0 0
$$681$$ −1.89963 −0.0727939
$$682$$ 0 0
$$683$$ 35.9985 1.37745 0.688723 0.725025i $$-0.258172\pi$$
0.688723 + 0.725025i $$0.258172\pi$$
$$684$$ 0 0
$$685$$ 22.3895 0.855458
$$686$$ 0 0
$$687$$ −23.1141 −0.881858
$$688$$ 0 0
$$689$$ −16.4739 −0.627606
$$690$$ 0 0
$$691$$ −24.7115 −0.940070 −0.470035 0.882648i $$-0.655759\pi$$
−0.470035 + 0.882648i $$0.655759\pi$$
$$692$$ 0 0
$$693$$ −1.87508 −0.0712282
$$694$$ 0 0
$$695$$ 3.87506 0.146989
$$696$$ 0 0
$$697$$ 14.4434 0.547083
$$698$$ 0 0
$$699$$ −41.3957 −1.56573
$$700$$ 0 0
$$701$$ 28.8712 1.09045 0.545225 0.838290i $$-0.316444\pi$$
0.545225 + 0.838290i $$0.316444\pi$$
$$702$$ 0 0
$$703$$ −27.8094 −1.04885
$$704$$ 0 0
$$705$$ −23.5970 −0.888713
$$706$$ 0 0
$$707$$ 2.06229 0.0775603
$$708$$ 0 0
$$709$$ −31.6656 −1.18923 −0.594614 0.804012i $$-0.702695\pi$$
−0.594614 + 0.804012i $$0.702695\pi$$
$$710$$ 0 0
$$711$$ 6.07856 0.227964
$$712$$ 0 0
$$713$$ −1.44105 −0.0539679
$$714$$ 0 0
$$715$$ 19.8828 0.743575
$$716$$ 0 0
$$717$$ −1.79230 −0.0669345
$$718$$ 0 0
$$719$$ −30.1524 −1.12449 −0.562247 0.826969i $$-0.690063\pi$$
−0.562247 + 0.826969i $$0.690063\pi$$
$$720$$ 0 0
$$721$$ 1.56403 0.0582477
$$722$$ 0 0
$$723$$ −38.1222 −1.41778
$$724$$ 0 0
$$725$$ −1.60124 −0.0594685
$$726$$ 0 0
$$727$$ −15.1905 −0.563384 −0.281692 0.959505i $$-0.590896\pi$$
−0.281692 + 0.959505i $$0.590896\pi$$
$$728$$ 0 0
$$729$$ 29.8887 1.10699
$$730$$ 0 0
$$731$$ 23.2512 0.859977
$$732$$ 0 0
$$733$$ 39.8270 1.47104 0.735522 0.677501i $$-0.236937\pi$$
0.735522 + 0.677501i $$0.236937\pi$$
$$734$$ 0 0
$$735$$ 17.9306 0.661381
$$736$$ 0 0
$$737$$ −1.10134 −0.0405682
$$738$$ 0 0
$$739$$ 26.4367 0.972489 0.486245 0.873823i $$-0.338366\pi$$
0.486245 + 0.873823i $$0.338366\pi$$
$$740$$ 0 0
$$741$$ 19.4267 0.713657
$$742$$ 0 0
$$743$$ 26.5745 0.974923 0.487462 0.873144i $$-0.337923\pi$$
0.487462 + 0.873144i $$0.337923\pi$$
$$744$$ 0 0
$$745$$ 15.4549 0.566225
$$746$$ 0 0
$$747$$ −12.8270 −0.469316
$$748$$ 0 0
$$749$$ 5.57720 0.203787
$$750$$ 0 0
$$751$$ 30.1977 1.10193 0.550964 0.834529i $$-0.314260\pi$$
0.550964 + 0.834529i $$0.314260\pi$$
$$752$$ 0 0
$$753$$ −19.7499 −0.719725
$$754$$ 0 0
$$755$$ 0.949439 0.0345536
$$756$$ 0 0
$$757$$ 31.3606 1.13982 0.569911 0.821706i $$-0.306978\pi$$
0.569911 + 0.821706i $$0.306978\pi$$
$$758$$ 0 0
$$759$$ −24.4379 −0.887040
$$760$$ 0 0
$$761$$ 40.3426 1.46242 0.731210 0.682153i $$-0.238956\pi$$
0.731210 + 0.682153i $$0.238956\pi$$
$$762$$ 0 0
$$763$$ 3.79253 0.137299
$$764$$ 0 0
$$765$$ 3.55272 0.128449
$$766$$ 0 0
$$767$$ −8.68015 −0.313422
$$768$$ 0 0
$$769$$ −20.4304 −0.736739 −0.368369 0.929680i $$-0.620084\pi$$
−0.368369 + 0.929680i $$0.620084\pi$$
$$770$$ 0 0
$$771$$ −23.9738 −0.863395
$$772$$ 0 0
$$773$$ 45.6859 1.64321 0.821605 0.570058i $$-0.193079\pi$$
0.821605 + 0.570058i $$0.193079\pi$$
$$774$$ 0 0
$$775$$ 0.672094 0.0241423
$$776$$ 0 0
$$777$$ 4.46715 0.160258
$$778$$ 0 0
$$779$$ 29.3038 1.04992
$$780$$ 0 0
$$781$$ 50.8798 1.82062
$$782$$ 0 0
$$783$$ −5.11048 −0.182634
$$784$$ 0 0
$$785$$ −20.0930 −0.717149
$$786$$ 0 0
$$787$$ −15.1038 −0.538392 −0.269196 0.963085i $$-0.586758\pi$$
−0.269196 + 0.963085i $$0.586758\pi$$
$$788$$ 0 0
$$789$$ −10.0856 −0.359055
$$790$$ 0 0
$$791$$ −1.54933 −0.0550880
$$792$$ 0 0
$$793$$ −2.70444 −0.0960373
$$794$$ 0 0
$$795$$ 17.3107 0.613947
$$796$$ 0 0
$$797$$ 13.0697 0.462954 0.231477 0.972840i $$-0.425644\pi$$
0.231477 + 0.972840i $$0.425644\pi$$
$$798$$ 0 0
$$799$$ 22.2300 0.786440
$$800$$ 0 0
$$801$$ 2.42683 0.0857478
$$802$$ 0 0
$$803$$ 69.9460 2.46834
$$804$$ 0 0
$$805$$ 3.75863 0.132474
$$806$$ 0 0
$$807$$ 34.9697 1.23099
$$808$$ 0 0
$$809$$ −19.0282 −0.668996 −0.334498 0.942396i $$-0.608567\pi$$
−0.334498 + 0.942396i $$0.608567\pi$$
$$810$$ 0 0
$$811$$ −21.0749 −0.740042 −0.370021 0.929023i $$-0.620649\pi$$
−0.370021 + 0.929023i $$0.620649\pi$$
$$812$$ 0 0
$$813$$ −26.1895 −0.918506
$$814$$ 0 0
$$815$$ 35.9927 1.26077
$$816$$ 0 0
$$817$$ 47.1736 1.65040
$$818$$ 0 0
$$819$$ 1.10188 0.0385029
$$820$$ 0 0
$$821$$ 10.5704 0.368909 0.184454 0.982841i $$-0.440948\pi$$
0.184454 + 0.982841i $$0.440948\pi$$
$$822$$ 0 0
$$823$$ −40.5293 −1.41276 −0.706381 0.707832i $$-0.749673\pi$$
−0.706381 + 0.707832i $$0.749673\pi$$
$$824$$ 0 0
$$825$$ 11.3976 0.396814
$$826$$ 0 0
$$827$$ 47.1057 1.63803 0.819013 0.573775i $$-0.194522\pi$$
0.819013 + 0.573775i $$0.194522\pi$$
$$828$$ 0 0
$$829$$ −9.69487 −0.336717 −0.168358 0.985726i $$-0.553847\pi$$
−0.168358 + 0.985726i $$0.553847\pi$$
$$830$$ 0 0
$$831$$ −16.8925 −0.585994
$$832$$ 0 0
$$833$$ −16.8919 −0.585269
$$834$$ 0 0
$$835$$ −21.4975 −0.743953
$$836$$ 0 0
$$837$$ 2.14504 0.0741436
$$838$$ 0 0
$$839$$ −29.2517 −1.00988 −0.504940 0.863154i $$-0.668485\pi$$
−0.504940 + 0.863154i $$0.668485\pi$$
$$840$$ 0 0
$$841$$ −28.1768 −0.971615
$$842$$ 0 0
$$843$$ −13.9483 −0.480404
$$844$$ 0 0
$$845$$ 11.6984 0.402436
$$846$$ 0 0
$$847$$ −4.31356 −0.148216
$$848$$ 0 0
$$849$$ −40.0228 −1.37358
$$850$$ 0 0
$$851$$ −20.5577 −0.704708
$$852$$ 0 0
$$853$$ −31.0610 −1.06351 −0.531754 0.846899i $$-0.678467\pi$$
−0.531754 + 0.846899i $$0.678467\pi$$
$$854$$ 0 0
$$855$$ 7.20800 0.246508
$$856$$ 0 0
$$857$$ −27.3567 −0.934486 −0.467243 0.884129i $$-0.654753\pi$$
−0.467243 + 0.884129i $$0.654753\pi$$
$$858$$ 0 0
$$859$$ 37.0124 1.26285 0.631424 0.775438i $$-0.282471\pi$$
0.631424 + 0.775438i $$0.282471\pi$$
$$860$$ 0 0
$$861$$ −4.70719 −0.160421
$$862$$ 0 0
$$863$$ 37.6253 1.28078 0.640389 0.768051i $$-0.278773\pi$$
0.640389 + 0.768051i $$0.278773\pi$$
$$864$$ 0 0
$$865$$ −5.65785 −0.192373
$$866$$ 0 0
$$867$$ −15.8344 −0.537765
$$868$$ 0 0
$$869$$ 33.6759 1.14238
$$870$$ 0 0
$$871$$ 0.647196 0.0219294
$$872$$ 0 0
$$873$$ 3.35542 0.113564
$$874$$ 0 0
$$875$$ −6.71935 −0.227156
$$876$$ 0 0
$$877$$ 35.6727 1.20458 0.602291 0.798277i $$-0.294255\pi$$
0.602291 + 0.798277i $$0.294255\pi$$
$$878$$ 0 0
$$879$$ 25.6653 0.865668
$$880$$ 0 0
$$881$$ −30.3930 −1.02397 −0.511984 0.858995i $$-0.671089\pi$$
−0.511984 + 0.858995i $$0.671089\pi$$
$$882$$ 0 0
$$883$$ −23.6017 −0.794260 −0.397130 0.917762i $$-0.629994\pi$$
−0.397130 + 0.917762i $$0.629994\pi$$
$$884$$ 0 0
$$885$$ 9.12106 0.306601
$$886$$ 0 0
$$887$$ 18.1804 0.610437 0.305219 0.952282i $$-0.401270\pi$$
0.305219 + 0.952282i $$0.401270\pi$$
$$888$$ 0 0
$$889$$ −8.23639 −0.276240
$$890$$ 0 0
$$891$$ 26.1901 0.877401
$$892$$ 0 0
$$893$$ 45.1017 1.50927
$$894$$ 0 0
$$895$$ −10.8296 −0.361993
$$896$$ 0 0
$$897$$ 14.3609 0.479495
$$898$$ 0 0
$$899$$ −0.345513 −0.0115235
$$900$$ 0 0
$$901$$ −16.3079 −0.543294
$$902$$ 0 0
$$903$$ −7.57771 −0.252170
$$904$$ 0 0
$$905$$ 41.4543 1.37799
$$906$$ 0 0
$$907$$ 26.8615 0.891921 0.445961 0.895053i $$-0.352862\pi$$
0.445961 + 0.895053i $$0.352862\pi$$
$$908$$ 0 0
$$909$$ 2.92359 0.0969694
$$910$$ 0 0
$$911$$ −42.7019 −1.41478 −0.707388 0.706825i $$-0.750127\pi$$
−0.707388 + 0.706825i $$0.750127\pi$$
$$912$$ 0 0
$$913$$ −71.0630 −2.35184
$$914$$ 0 0
$$915$$ 2.84181 0.0939473
$$916$$ 0 0
$$917$$ 7.46656 0.246568
$$918$$ 0 0
$$919$$ 7.59767 0.250624 0.125312 0.992117i $$-0.460007\pi$$
0.125312 + 0.992117i $$0.460007\pi$$
$$920$$ 0 0
$$921$$ −43.1812 −1.42287
$$922$$ 0 0
$$923$$ −29.8994 −0.984151
$$924$$ 0 0
$$925$$ 9.58790 0.315248
$$926$$ 0 0
$$927$$ 2.21724 0.0728239
$$928$$ 0 0
$$929$$ −34.5765 −1.13442 −0.567210 0.823574i $$-0.691977\pi$$
−0.567210 + 0.823574i $$0.691977\pi$$
$$930$$ 0 0
$$931$$ −34.2714 −1.12320
$$932$$ 0 0
$$933$$ −40.5474 −1.32746
$$934$$ 0 0
$$935$$ 19.6824 0.643684
$$936$$ 0 0
$$937$$ 40.4353 1.32096 0.660482 0.750842i $$-0.270352\pi$$
0.660482 + 0.750842i $$0.270352\pi$$
$$938$$ 0 0
$$939$$ 8.23506 0.268741
$$940$$ 0 0
$$941$$ −22.3193 −0.727587 −0.363794 0.931480i $$-0.618519\pi$$
−0.363794 + 0.931480i $$0.618519\pi$$
$$942$$ 0 0
$$943$$ 21.6623 0.705423
$$944$$ 0 0
$$945$$ −5.59481 −0.181999
$$946$$ 0 0
$$947$$ 40.0277 1.30073 0.650363 0.759623i $$-0.274617\pi$$
0.650363 + 0.759623i $$0.274617\pi$$
$$948$$ 0 0
$$949$$ −41.1036 −1.33428
$$950$$ 0 0
$$951$$ 46.4613 1.50661
$$952$$ 0 0
$$953$$ −6.33099 −0.205081 −0.102540 0.994729i $$-0.532697\pi$$
−0.102540 + 0.994729i $$0.532697\pi$$
$$954$$ 0 0
$$955$$ −32.3349 −1.04633
$$956$$ 0 0
$$957$$ −5.85933 −0.189405
$$958$$ 0 0
$$959$$ 6.87416 0.221978
$$960$$ 0 0
$$961$$ −30.8550 −0.995322
$$962$$ 0 0
$$963$$ 7.90649 0.254783
$$964$$ 0 0
$$965$$ 18.7831 0.604650
$$966$$ 0 0
$$967$$ 12.8388 0.412867 0.206433 0.978461i $$-0.433814\pi$$
0.206433 + 0.978461i $$0.433814\pi$$
$$968$$ 0 0
$$969$$ 19.2309 0.617785
$$970$$ 0 0
$$971$$ −22.1505 −0.710842 −0.355421 0.934706i $$-0.615662\pi$$
−0.355421 + 0.934706i $$0.615662\pi$$
$$972$$ 0 0
$$973$$ 1.18975 0.0381415
$$974$$ 0 0
$$975$$ −6.69778 −0.214500
$$976$$ 0 0
$$977$$ 41.0297 1.31266 0.656328 0.754475i $$-0.272109\pi$$
0.656328 + 0.754475i $$0.272109\pi$$
$$978$$ 0 0
$$979$$ 13.4449 0.429701
$$980$$ 0 0
$$981$$ 5.37646 0.171657
$$982$$ 0 0
$$983$$ −43.9691 −1.40240 −0.701199 0.712966i $$-0.747351\pi$$
−0.701199 + 0.712966i $$0.747351\pi$$
$$984$$ 0 0
$$985$$ 7.84258 0.249885
$$986$$ 0 0
$$987$$ −7.24489 −0.230607
$$988$$ 0 0
$$989$$ 34.8724 1.10888
$$990$$ 0 0
$$991$$ −20.6965 −0.657446 −0.328723 0.944426i $$-0.606618\pi$$
−0.328723 + 0.944426i $$0.606618\pi$$
$$992$$ 0 0
$$993$$ −25.0190 −0.793955
$$994$$ 0 0
$$995$$ 25.2180 0.799464
$$996$$ 0 0
$$997$$ 52.9018 1.67542 0.837709 0.546117i $$-0.183895\pi$$
0.837709 + 0.546117i $$0.183895\pi$$
$$998$$ 0 0
$$999$$ 30.6006 0.968160
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 8048.2.a.p.1.6 10
4.3 odd 2 503.2.a.e.1.3 10
12.11 even 2 4527.2.a.k.1.8 10

By twisted newform
Twist Min Dim Char Parity Ord Type
503.2.a.e.1.3 10 4.3 odd 2
4527.2.a.k.1.8 10 12.11 even 2
8048.2.a.p.1.6 10 1.1 even 1 trivial