# Properties

 Label 9408.2.a.ee.1.1 Level $9408$ Weight $2$ Character 9408.1 Self dual yes Analytic conductor $75.123$ Analytic rank $0$ Dimension $2$ CM no Inner twists $1$

# Learn more about

## Newspace parameters

 Level: $$N$$ $$=$$ $$9408 = 2^{6} \cdot 3 \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 9408.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$75.1232582216$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\zeta_{8})^+$$ Defining polynomial: $$x^{2} - 2$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 1176) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Root $$-1.41421$$ of defining polynomial Character $$\chi$$ $$=$$ 9408.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+1.00000 q^{3} +0.585786 q^{5} +1.00000 q^{9} +O(q^{10})$$ $$q+1.00000 q^{3} +0.585786 q^{5} +1.00000 q^{9} +0.828427 q^{11} -1.41421 q^{13} +0.585786 q^{15} -2.24264 q^{17} +6.82843 q^{19} +4.82843 q^{23} -4.65685 q^{25} +1.00000 q^{27} -8.48528 q^{29} -5.17157 q^{31} +0.828427 q^{33} -1.65685 q^{37} -1.41421 q^{39} +0.585786 q^{41} +8.00000 q^{43} +0.585786 q^{45} -6.82843 q^{47} -2.24264 q^{51} +13.3137 q^{53} +0.485281 q^{55} +6.82843 q^{57} +5.17157 q^{59} +13.8995 q^{61} -0.828427 q^{65} +8.00000 q^{67} +4.82843 q^{69} -0.828427 q^{71} +11.0711 q^{73} -4.65685 q^{75} -2.34315 q^{79} +1.00000 q^{81} +15.3137 q^{83} -1.31371 q^{85} -8.48528 q^{87} +10.7279 q^{89} -5.17157 q^{93} +4.00000 q^{95} +7.75736 q^{97} +0.828427 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 2q^{3} + 4q^{5} + 2q^{9} + O(q^{10})$$ $$2q + 2q^{3} + 4q^{5} + 2q^{9} - 4q^{11} + 4q^{15} + 4q^{17} + 8q^{19} + 4q^{23} + 2q^{25} + 2q^{27} - 16q^{31} - 4q^{33} + 8q^{37} + 4q^{41} + 16q^{43} + 4q^{45} - 8q^{47} + 4q^{51} + 4q^{53} - 16q^{55} + 8q^{57} + 16q^{59} + 8q^{61} + 4q^{65} + 16q^{67} + 4q^{69} + 4q^{71} + 8q^{73} + 2q^{75} - 16q^{79} + 2q^{81} + 8q^{83} + 20q^{85} - 4q^{89} - 16q^{93} + 8q^{95} + 24q^{97} - 4q^{99} + O(q^{100})$$

## 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
$$3$$ 1.00000 0.577350
$$4$$ 0 0
$$5$$ 0.585786 0.261972 0.130986 0.991384i $$-0.458186\pi$$
0.130986 + 0.991384i $$0.458186\pi$$
$$6$$ 0 0
$$7$$ 0 0
$$8$$ 0 0
$$9$$ 1.00000 0.333333
$$10$$ 0 0
$$11$$ 0.828427 0.249780 0.124890 0.992171i $$-0.460142\pi$$
0.124890 + 0.992171i $$0.460142\pi$$
$$12$$ 0 0
$$13$$ −1.41421 −0.392232 −0.196116 0.980581i $$-0.562833\pi$$
−0.196116 + 0.980581i $$0.562833\pi$$
$$14$$ 0 0
$$15$$ 0.585786 0.151249
$$16$$ 0 0
$$17$$ −2.24264 −0.543920 −0.271960 0.962309i $$-0.587672\pi$$
−0.271960 + 0.962309i $$0.587672\pi$$
$$18$$ 0 0
$$19$$ 6.82843 1.56655 0.783274 0.621676i $$-0.213548\pi$$
0.783274 + 0.621676i $$0.213548\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 0 0
$$23$$ 4.82843 1.00680 0.503398 0.864054i $$-0.332083\pi$$
0.503398 + 0.864054i $$0.332083\pi$$
$$24$$ 0 0
$$25$$ −4.65685 −0.931371
$$26$$ 0 0
$$27$$ 1.00000 0.192450
$$28$$ 0 0
$$29$$ −8.48528 −1.57568 −0.787839 0.615882i $$-0.788800\pi$$
−0.787839 + 0.615882i $$0.788800\pi$$
$$30$$ 0 0
$$31$$ −5.17157 −0.928842 −0.464421 0.885615i $$-0.653738\pi$$
−0.464421 + 0.885615i $$0.653738\pi$$
$$32$$ 0 0
$$33$$ 0.828427 0.144211
$$34$$ 0 0
$$35$$ 0 0
$$36$$ 0 0
$$37$$ −1.65685 −0.272385 −0.136193 0.990682i $$-0.543487\pi$$
−0.136193 + 0.990682i $$0.543487\pi$$
$$38$$ 0 0
$$39$$ −1.41421 −0.226455
$$40$$ 0 0
$$41$$ 0.585786 0.0914845 0.0457422 0.998953i $$-0.485435\pi$$
0.0457422 + 0.998953i $$0.485435\pi$$
$$42$$ 0 0
$$43$$ 8.00000 1.21999 0.609994 0.792406i $$-0.291172\pi$$
0.609994 + 0.792406i $$0.291172\pi$$
$$44$$ 0 0
$$45$$ 0.585786 0.0873239
$$46$$ 0 0
$$47$$ −6.82843 −0.996028 −0.498014 0.867169i $$-0.665937\pi$$
−0.498014 + 0.867169i $$0.665937\pi$$
$$48$$ 0 0
$$49$$ 0 0
$$50$$ 0 0
$$51$$ −2.24264 −0.314033
$$52$$ 0 0
$$53$$ 13.3137 1.82878 0.914389 0.404836i $$-0.132671\pi$$
0.914389 + 0.404836i $$0.132671\pi$$
$$54$$ 0 0
$$55$$ 0.485281 0.0654353
$$56$$ 0 0
$$57$$ 6.82843 0.904447
$$58$$ 0 0
$$59$$ 5.17157 0.673281 0.336641 0.941633i $$-0.390709\pi$$
0.336641 + 0.941633i $$0.390709\pi$$
$$60$$ 0 0
$$61$$ 13.8995 1.77965 0.889824 0.456304i $$-0.150827\pi$$
0.889824 + 0.456304i $$0.150827\pi$$
$$62$$ 0 0
$$63$$ 0 0
$$64$$ 0 0
$$65$$ −0.828427 −0.102754
$$66$$ 0 0
$$67$$ 8.00000 0.977356 0.488678 0.872464i $$-0.337479\pi$$
0.488678 + 0.872464i $$0.337479\pi$$
$$68$$ 0 0
$$69$$ 4.82843 0.581274
$$70$$ 0 0
$$71$$ −0.828427 −0.0983162 −0.0491581 0.998791i $$-0.515654\pi$$
−0.0491581 + 0.998791i $$0.515654\pi$$
$$72$$ 0 0
$$73$$ 11.0711 1.29577 0.647885 0.761738i $$-0.275654\pi$$
0.647885 + 0.761738i $$0.275654\pi$$
$$74$$ 0 0
$$75$$ −4.65685 −0.537727
$$76$$ 0 0
$$77$$ 0 0
$$78$$ 0 0
$$79$$ −2.34315 −0.263624 −0.131812 0.991275i $$-0.542080\pi$$
−0.131812 + 0.991275i $$0.542080\pi$$
$$80$$ 0 0
$$81$$ 1.00000 0.111111
$$82$$ 0 0
$$83$$ 15.3137 1.68090 0.840449 0.541891i $$-0.182291\pi$$
0.840449 + 0.541891i $$0.182291\pi$$
$$84$$ 0 0
$$85$$ −1.31371 −0.142492
$$86$$ 0 0
$$87$$ −8.48528 −0.909718
$$88$$ 0 0
$$89$$ 10.7279 1.13716 0.568579 0.822629i $$-0.307493\pi$$
0.568579 + 0.822629i $$0.307493\pi$$
$$90$$ 0 0
$$91$$ 0 0
$$92$$ 0 0
$$93$$ −5.17157 −0.536267
$$94$$ 0 0
$$95$$ 4.00000 0.410391
$$96$$ 0 0
$$97$$ 7.75736 0.787641 0.393820 0.919187i $$-0.371153\pi$$
0.393820 + 0.919187i $$0.371153\pi$$
$$98$$ 0 0
$$99$$ 0.828427 0.0832601
$$100$$ 0 0
$$101$$ 3.41421 0.339727 0.169863 0.985468i $$-0.445667\pi$$
0.169863 + 0.985468i $$0.445667\pi$$
$$102$$ 0 0
$$103$$ −10.8284 −1.06696 −0.533478 0.845814i $$-0.679115\pi$$
−0.533478 + 0.845814i $$0.679115\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ −14.4853 −1.40035 −0.700173 0.713974i $$-0.746894\pi$$
−0.700173 + 0.713974i $$0.746894\pi$$
$$108$$ 0 0
$$109$$ 11.3137 1.08366 0.541828 0.840489i $$-0.317732\pi$$
0.541828 + 0.840489i $$0.317732\pi$$
$$110$$ 0 0
$$111$$ −1.65685 −0.157262
$$112$$ 0 0
$$113$$ 6.00000 0.564433 0.282216 0.959351i $$-0.408930\pi$$
0.282216 + 0.959351i $$0.408930\pi$$
$$114$$ 0 0
$$115$$ 2.82843 0.263752
$$116$$ 0 0
$$117$$ −1.41421 −0.130744
$$118$$ 0 0
$$119$$ 0 0
$$120$$ 0 0
$$121$$ −10.3137 −0.937610
$$122$$ 0 0
$$123$$ 0.585786 0.0528186
$$124$$ 0 0
$$125$$ −5.65685 −0.505964
$$126$$ 0 0
$$127$$ −15.3137 −1.35887 −0.679436 0.733735i $$-0.737775\pi$$
−0.679436 + 0.733735i $$0.737775\pi$$
$$128$$ 0 0
$$129$$ 8.00000 0.704361
$$130$$ 0 0
$$131$$ −7.31371 −0.639002 −0.319501 0.947586i $$-0.603515\pi$$
−0.319501 + 0.947586i $$0.603515\pi$$
$$132$$ 0 0
$$133$$ 0 0
$$134$$ 0 0
$$135$$ 0.585786 0.0504165
$$136$$ 0 0
$$137$$ 4.48528 0.383203 0.191602 0.981473i $$-0.438632\pi$$
0.191602 + 0.981473i $$0.438632\pi$$
$$138$$ 0 0
$$139$$ −1.65685 −0.140533 −0.0702663 0.997528i $$-0.522385\pi$$
−0.0702663 + 0.997528i $$0.522385\pi$$
$$140$$ 0 0
$$141$$ −6.82843 −0.575057
$$142$$ 0 0
$$143$$ −1.17157 −0.0979718
$$144$$ 0 0
$$145$$ −4.97056 −0.412783
$$146$$ 0 0
$$147$$ 0 0
$$148$$ 0 0
$$149$$ 10.0000 0.819232 0.409616 0.912258i $$-0.365663\pi$$
0.409616 + 0.912258i $$0.365663\pi$$
$$150$$ 0 0
$$151$$ 9.65685 0.785864 0.392932 0.919568i $$-0.371461\pi$$
0.392932 + 0.919568i $$0.371461\pi$$
$$152$$ 0 0
$$153$$ −2.24264 −0.181307
$$154$$ 0 0
$$155$$ −3.02944 −0.243330
$$156$$ 0 0
$$157$$ −13.8995 −1.10930 −0.554650 0.832084i $$-0.687148\pi$$
−0.554650 + 0.832084i $$0.687148\pi$$
$$158$$ 0 0
$$159$$ 13.3137 1.05585
$$160$$ 0 0
$$161$$ 0 0
$$162$$ 0 0
$$163$$ 13.6569 1.06969 0.534844 0.844951i $$-0.320370\pi$$
0.534844 + 0.844951i $$0.320370\pi$$
$$164$$ 0 0
$$165$$ 0.485281 0.0377791
$$166$$ 0 0
$$167$$ −1.17157 −0.0906590 −0.0453295 0.998972i $$-0.514434\pi$$
−0.0453295 + 0.998972i $$0.514434\pi$$
$$168$$ 0 0
$$169$$ −11.0000 −0.846154
$$170$$ 0 0
$$171$$ 6.82843 0.522183
$$172$$ 0 0
$$173$$ −3.41421 −0.259578 −0.129789 0.991542i $$-0.541430\pi$$
−0.129789 + 0.991542i $$0.541430\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 0 0
$$177$$ 5.17157 0.388719
$$178$$ 0 0
$$179$$ −17.7990 −1.33036 −0.665179 0.746684i $$-0.731645\pi$$
−0.665179 + 0.746684i $$0.731645\pi$$
$$180$$ 0 0
$$181$$ −9.89949 −0.735824 −0.367912 0.929861i $$-0.619927\pi$$
−0.367912 + 0.929861i $$0.619927\pi$$
$$182$$ 0 0
$$183$$ 13.8995 1.02748
$$184$$ 0 0
$$185$$ −0.970563 −0.0713572
$$186$$ 0 0
$$187$$ −1.85786 −0.135860
$$188$$ 0 0
$$189$$ 0 0
$$190$$ 0 0
$$191$$ 15.1716 1.09778 0.548888 0.835896i $$-0.315051\pi$$
0.548888 + 0.835896i $$0.315051\pi$$
$$192$$ 0 0
$$193$$ 24.6274 1.77272 0.886360 0.462996i $$-0.153226\pi$$
0.886360 + 0.462996i $$0.153226\pi$$
$$194$$ 0 0
$$195$$ −0.828427 −0.0593249
$$196$$ 0 0
$$197$$ −2.00000 −0.142494 −0.0712470 0.997459i $$-0.522698\pi$$
−0.0712470 + 0.997459i $$0.522698\pi$$
$$198$$ 0 0
$$199$$ −5.65685 −0.401004 −0.200502 0.979693i $$-0.564257\pi$$
−0.200502 + 0.979693i $$0.564257\pi$$
$$200$$ 0 0
$$201$$ 8.00000 0.564276
$$202$$ 0 0
$$203$$ 0 0
$$204$$ 0 0
$$205$$ 0.343146 0.0239663
$$206$$ 0 0
$$207$$ 4.82843 0.335599
$$208$$ 0 0
$$209$$ 5.65685 0.391293
$$210$$ 0 0
$$211$$ 14.3431 0.987423 0.493711 0.869626i $$-0.335640\pi$$
0.493711 + 0.869626i $$0.335640\pi$$
$$212$$ 0 0
$$213$$ −0.828427 −0.0567629
$$214$$ 0 0
$$215$$ 4.68629 0.319602
$$216$$ 0 0
$$217$$ 0 0
$$218$$ 0 0
$$219$$ 11.0711 0.748113
$$220$$ 0 0
$$221$$ 3.17157 0.213343
$$222$$ 0 0
$$223$$ 13.6569 0.914531 0.457265 0.889330i $$-0.348829\pi$$
0.457265 + 0.889330i $$0.348829\pi$$
$$224$$ 0 0
$$225$$ −4.65685 −0.310457
$$226$$ 0 0
$$227$$ −19.7990 −1.31411 −0.657053 0.753845i $$-0.728197\pi$$
−0.657053 + 0.753845i $$0.728197\pi$$
$$228$$ 0 0
$$229$$ 15.0711 0.995924 0.497962 0.867199i $$-0.334082\pi$$
0.497962 + 0.867199i $$0.334082\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ −28.4853 −1.86613 −0.933066 0.359704i $$-0.882878\pi$$
−0.933066 + 0.359704i $$0.882878\pi$$
$$234$$ 0 0
$$235$$ −4.00000 −0.260931
$$236$$ 0 0
$$237$$ −2.34315 −0.152204
$$238$$ 0 0
$$239$$ −3.17157 −0.205152 −0.102576 0.994725i $$-0.532708\pi$$
−0.102576 + 0.994725i $$0.532708\pi$$
$$240$$ 0 0
$$241$$ 21.8995 1.41067 0.705335 0.708874i $$-0.250796\pi$$
0.705335 + 0.708874i $$0.250796\pi$$
$$242$$ 0 0
$$243$$ 1.00000 0.0641500
$$244$$ 0 0
$$245$$ 0 0
$$246$$ 0 0
$$247$$ −9.65685 −0.614451
$$248$$ 0 0
$$249$$ 15.3137 0.970467
$$250$$ 0 0
$$251$$ 8.48528 0.535586 0.267793 0.963476i $$-0.413706\pi$$
0.267793 + 0.963476i $$0.413706\pi$$
$$252$$ 0 0
$$253$$ 4.00000 0.251478
$$254$$ 0 0
$$255$$ −1.31371 −0.0822676
$$256$$ 0 0
$$257$$ 30.2426 1.88648 0.943242 0.332106i $$-0.107759\pi$$
0.943242 + 0.332106i $$0.107759\pi$$
$$258$$ 0 0
$$259$$ 0 0
$$260$$ 0 0
$$261$$ −8.48528 −0.525226
$$262$$ 0 0
$$263$$ 24.8284 1.53099 0.765493 0.643444i $$-0.222495\pi$$
0.765493 + 0.643444i $$0.222495\pi$$
$$264$$ 0 0
$$265$$ 7.79899 0.479088
$$266$$ 0 0
$$267$$ 10.7279 0.656538
$$268$$ 0 0
$$269$$ 30.0416 1.83167 0.915835 0.401554i $$-0.131530\pi$$
0.915835 + 0.401554i $$0.131530\pi$$
$$270$$ 0 0
$$271$$ 13.1716 0.800116 0.400058 0.916490i $$-0.368990\pi$$
0.400058 + 0.916490i $$0.368990\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ 0 0
$$275$$ −3.85786 −0.232638
$$276$$ 0 0
$$277$$ 6.00000 0.360505 0.180253 0.983620i $$-0.442309\pi$$
0.180253 + 0.983620i $$0.442309\pi$$
$$278$$ 0 0
$$279$$ −5.17157 −0.309614
$$280$$ 0 0
$$281$$ −12.4853 −0.744809 −0.372405 0.928070i $$-0.621467\pi$$
−0.372405 + 0.928070i $$0.621467\pi$$
$$282$$ 0 0
$$283$$ −14.8284 −0.881458 −0.440729 0.897640i $$-0.645280\pi$$
−0.440729 + 0.897640i $$0.645280\pi$$
$$284$$ 0 0
$$285$$ 4.00000 0.236940
$$286$$ 0 0
$$287$$ 0 0
$$288$$ 0 0
$$289$$ −11.9706 −0.704151
$$290$$ 0 0
$$291$$ 7.75736 0.454744
$$292$$ 0 0
$$293$$ 1.07107 0.0625724 0.0312862 0.999510i $$-0.490040\pi$$
0.0312862 + 0.999510i $$0.490040\pi$$
$$294$$ 0 0
$$295$$ 3.02944 0.176381
$$296$$ 0 0
$$297$$ 0.828427 0.0480702
$$298$$ 0 0
$$299$$ −6.82843 −0.394898
$$300$$ 0 0
$$301$$ 0 0
$$302$$ 0 0
$$303$$ 3.41421 0.196141
$$304$$ 0 0
$$305$$ 8.14214 0.466217
$$306$$ 0 0
$$307$$ −11.5147 −0.657180 −0.328590 0.944473i $$-0.606573\pi$$
−0.328590 + 0.944473i $$0.606573\pi$$
$$308$$ 0 0
$$309$$ −10.8284 −0.616008
$$310$$ 0 0
$$311$$ 26.1421 1.48238 0.741192 0.671293i $$-0.234261\pi$$
0.741192 + 0.671293i $$0.234261\pi$$
$$312$$ 0 0
$$313$$ 17.4142 0.984310 0.492155 0.870508i $$-0.336209\pi$$
0.492155 + 0.870508i $$0.336209\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ −21.3137 −1.19710 −0.598549 0.801087i $$-0.704256\pi$$
−0.598549 + 0.801087i $$0.704256\pi$$
$$318$$ 0 0
$$319$$ −7.02944 −0.393573
$$320$$ 0 0
$$321$$ −14.4853 −0.808490
$$322$$ 0 0
$$323$$ −15.3137 −0.852078
$$324$$ 0 0
$$325$$ 6.58579 0.365314
$$326$$ 0 0
$$327$$ 11.3137 0.625650
$$328$$ 0 0
$$329$$ 0 0
$$330$$ 0 0
$$331$$ 8.68629 0.477442 0.238721 0.971088i $$-0.423272\pi$$
0.238721 + 0.971088i $$0.423272\pi$$
$$332$$ 0 0
$$333$$ −1.65685 −0.0907951
$$334$$ 0 0
$$335$$ 4.68629 0.256039
$$336$$ 0 0
$$337$$ −16.9706 −0.924445 −0.462223 0.886764i $$-0.652948\pi$$
−0.462223 + 0.886764i $$0.652948\pi$$
$$338$$ 0 0
$$339$$ 6.00000 0.325875
$$340$$ 0 0
$$341$$ −4.28427 −0.232006
$$342$$ 0 0
$$343$$ 0 0
$$344$$ 0 0
$$345$$ 2.82843 0.152277
$$346$$ 0 0
$$347$$ 4.14214 0.222361 0.111181 0.993800i $$-0.464537\pi$$
0.111181 + 0.993800i $$0.464537\pi$$
$$348$$ 0 0
$$349$$ 30.3848 1.62646 0.813230 0.581943i $$-0.197707\pi$$
0.813230 + 0.581943i $$0.197707\pi$$
$$350$$ 0 0
$$351$$ −1.41421 −0.0754851
$$352$$ 0 0
$$353$$ 15.8995 0.846245 0.423122 0.906073i $$-0.360934\pi$$
0.423122 + 0.906073i $$0.360934\pi$$
$$354$$ 0 0
$$355$$ −0.485281 −0.0257561
$$356$$ 0 0
$$357$$ 0 0
$$358$$ 0 0
$$359$$ −35.4558 −1.87129 −0.935644 0.352945i $$-0.885180\pi$$
−0.935644 + 0.352945i $$0.885180\pi$$
$$360$$ 0 0
$$361$$ 27.6274 1.45407
$$362$$ 0 0
$$363$$ −10.3137 −0.541329
$$364$$ 0 0
$$365$$ 6.48528 0.339455
$$366$$ 0 0
$$367$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$368$$ 0 0
$$369$$ 0.585786 0.0304948
$$370$$ 0 0
$$371$$ 0 0
$$372$$ 0 0
$$373$$ −14.6863 −0.760427 −0.380214 0.924899i $$-0.624150\pi$$
−0.380214 + 0.924899i $$0.624150\pi$$
$$374$$ 0 0
$$375$$ −5.65685 −0.292119
$$376$$ 0 0
$$377$$ 12.0000 0.618031
$$378$$ 0 0
$$379$$ 0.686292 0.0352524 0.0176262 0.999845i $$-0.494389\pi$$
0.0176262 + 0.999845i $$0.494389\pi$$
$$380$$ 0 0
$$381$$ −15.3137 −0.784545
$$382$$ 0 0
$$383$$ −24.9706 −1.27594 −0.637968 0.770063i $$-0.720225\pi$$
−0.637968 + 0.770063i $$0.720225\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ 8.00000 0.406663
$$388$$ 0 0
$$389$$ 14.1421 0.717035 0.358517 0.933523i $$-0.383282\pi$$
0.358517 + 0.933523i $$0.383282\pi$$
$$390$$ 0 0
$$391$$ −10.8284 −0.547617
$$392$$ 0 0
$$393$$ −7.31371 −0.368928
$$394$$ 0 0
$$395$$ −1.37258 −0.0690621
$$396$$ 0 0
$$397$$ 7.27208 0.364975 0.182488 0.983208i $$-0.441585\pi$$
0.182488 + 0.983208i $$0.441585\pi$$
$$398$$ 0 0
$$399$$ 0 0
$$400$$ 0 0
$$401$$ −12.4853 −0.623485 −0.311743 0.950167i $$-0.600913\pi$$
−0.311743 + 0.950167i $$0.600913\pi$$
$$402$$ 0 0
$$403$$ 7.31371 0.364322
$$404$$ 0 0
$$405$$ 0.585786 0.0291080
$$406$$ 0 0
$$407$$ −1.37258 −0.0680364
$$408$$ 0 0
$$409$$ −16.2426 −0.803147 −0.401573 0.915827i $$-0.631537\pi$$
−0.401573 + 0.915827i $$0.631537\pi$$
$$410$$ 0 0
$$411$$ 4.48528 0.221243
$$412$$ 0 0
$$413$$ 0 0
$$414$$ 0 0
$$415$$ 8.97056 0.440348
$$416$$ 0 0
$$417$$ −1.65685 −0.0811365
$$418$$ 0 0
$$419$$ 10.8284 0.529003 0.264502 0.964385i $$-0.414793\pi$$
0.264502 + 0.964385i $$0.414793\pi$$
$$420$$ 0 0
$$421$$ −6.68629 −0.325870 −0.162935 0.986637i $$-0.552096\pi$$
−0.162935 + 0.986637i $$0.552096\pi$$
$$422$$ 0 0
$$423$$ −6.82843 −0.332009
$$424$$ 0 0
$$425$$ 10.4437 0.506591
$$426$$ 0 0
$$427$$ 0 0
$$428$$ 0 0
$$429$$ −1.17157 −0.0565641
$$430$$ 0 0
$$431$$ −30.4853 −1.46842 −0.734212 0.678920i $$-0.762448\pi$$
−0.734212 + 0.678920i $$0.762448\pi$$
$$432$$ 0 0
$$433$$ −26.3848 −1.26797 −0.633986 0.773345i $$-0.718582\pi$$
−0.633986 + 0.773345i $$0.718582\pi$$
$$434$$ 0 0
$$435$$ −4.97056 −0.238320
$$436$$ 0 0
$$437$$ 32.9706 1.57720
$$438$$ 0 0
$$439$$ 3.31371 0.158155 0.0790773 0.996868i $$-0.474803\pi$$
0.0790773 + 0.996868i $$0.474803\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 0 0
$$443$$ −38.4853 −1.82849 −0.914245 0.405161i $$-0.867216\pi$$
−0.914245 + 0.405161i $$0.867216\pi$$
$$444$$ 0 0
$$445$$ 6.28427 0.297903
$$446$$ 0 0
$$447$$ 10.0000 0.472984
$$448$$ 0 0
$$449$$ −10.0000 −0.471929 −0.235965 0.971762i $$-0.575825\pi$$
−0.235965 + 0.971762i $$0.575825\pi$$
$$450$$ 0 0
$$451$$ 0.485281 0.0228510
$$452$$ 0 0
$$453$$ 9.65685 0.453719
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ 20.6274 0.964910 0.482455 0.875921i $$-0.339745\pi$$
0.482455 + 0.875921i $$0.339745\pi$$
$$458$$ 0 0
$$459$$ −2.24264 −0.104678
$$460$$ 0 0
$$461$$ 18.2426 0.849644 0.424822 0.905277i $$-0.360337\pi$$
0.424822 + 0.905277i $$0.360337\pi$$
$$462$$ 0 0
$$463$$ 20.9706 0.974585 0.487292 0.873239i $$-0.337985\pi$$
0.487292 + 0.873239i $$0.337985\pi$$
$$464$$ 0 0
$$465$$ −3.02944 −0.140487
$$466$$ 0 0
$$467$$ −10.8284 −0.501080 −0.250540 0.968106i $$-0.580608\pi$$
−0.250540 + 0.968106i $$0.580608\pi$$
$$468$$ 0 0
$$469$$ 0 0
$$470$$ 0 0
$$471$$ −13.8995 −0.640455
$$472$$ 0 0
$$473$$ 6.62742 0.304729
$$474$$ 0 0
$$475$$ −31.7990 −1.45904
$$476$$ 0 0
$$477$$ 13.3137 0.609593
$$478$$ 0 0
$$479$$ 43.1127 1.96987 0.984935 0.172926i $$-0.0553223\pi$$
0.984935 + 0.172926i $$0.0553223\pi$$
$$480$$ 0 0
$$481$$ 2.34315 0.106838
$$482$$ 0 0
$$483$$ 0 0
$$484$$ 0 0
$$485$$ 4.54416 0.206339
$$486$$ 0 0
$$487$$ −20.9706 −0.950267 −0.475133 0.879914i $$-0.657600\pi$$
−0.475133 + 0.879914i $$0.657600\pi$$
$$488$$ 0 0
$$489$$ 13.6569 0.617584
$$490$$ 0 0
$$491$$ −24.1421 −1.08952 −0.544760 0.838592i $$-0.683379\pi$$
−0.544760 + 0.838592i $$0.683379\pi$$
$$492$$ 0 0
$$493$$ 19.0294 0.857043
$$494$$ 0 0
$$495$$ 0.485281 0.0218118
$$496$$ 0 0
$$497$$ 0 0
$$498$$ 0 0
$$499$$ 28.2843 1.26618 0.633089 0.774079i $$-0.281787\pi$$
0.633089 + 0.774079i $$0.281787\pi$$
$$500$$ 0 0
$$501$$ −1.17157 −0.0523420
$$502$$ 0 0
$$503$$ 14.6274 0.652204 0.326102 0.945335i $$-0.394265\pi$$
0.326102 + 0.945335i $$0.394265\pi$$
$$504$$ 0 0
$$505$$ 2.00000 0.0889988
$$506$$ 0 0
$$507$$ −11.0000 −0.488527
$$508$$ 0 0
$$509$$ 37.0711 1.64315 0.821573 0.570103i $$-0.193097\pi$$
0.821573 + 0.570103i $$0.193097\pi$$
$$510$$ 0 0
$$511$$ 0 0
$$512$$ 0 0
$$513$$ 6.82843 0.301482
$$514$$ 0 0
$$515$$ −6.34315 −0.279512
$$516$$ 0 0
$$517$$ −5.65685 −0.248788
$$518$$ 0 0
$$519$$ −3.41421 −0.149867
$$520$$ 0 0
$$521$$ −30.7279 −1.34621 −0.673107 0.739545i $$-0.735041\pi$$
−0.673107 + 0.739545i $$0.735041\pi$$
$$522$$ 0 0
$$523$$ 9.65685 0.422265 0.211132 0.977457i $$-0.432285\pi$$
0.211132 + 0.977457i $$0.432285\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ 11.5980 0.505216
$$528$$ 0 0
$$529$$ 0.313708 0.0136395
$$530$$ 0 0
$$531$$ 5.17157 0.224427
$$532$$ 0 0
$$533$$ −0.828427 −0.0358832
$$534$$ 0 0
$$535$$ −8.48528 −0.366851
$$536$$ 0 0
$$537$$ −17.7990 −0.768083
$$538$$ 0 0
$$539$$ 0 0
$$540$$ 0 0
$$541$$ −36.6274 −1.57474 −0.787368 0.616483i $$-0.788557\pi$$
−0.787368 + 0.616483i $$0.788557\pi$$
$$542$$ 0 0
$$543$$ −9.89949 −0.424828
$$544$$ 0 0
$$545$$ 6.62742 0.283887
$$546$$ 0 0
$$547$$ −28.9706 −1.23869 −0.619346 0.785118i $$-0.712602\pi$$
−0.619346 + 0.785118i $$0.712602\pi$$
$$548$$ 0 0
$$549$$ 13.8995 0.593216
$$550$$ 0 0
$$551$$ −57.9411 −2.46837
$$552$$ 0 0
$$553$$ 0 0
$$554$$ 0 0
$$555$$ −0.970563 −0.0411981
$$556$$ 0 0
$$557$$ 23.9411 1.01442 0.507209 0.861823i $$-0.330677\pi$$
0.507209 + 0.861823i $$0.330677\pi$$
$$558$$ 0 0
$$559$$ −11.3137 −0.478519
$$560$$ 0 0
$$561$$ −1.85786 −0.0784391
$$562$$ 0 0
$$563$$ 21.1716 0.892275 0.446138 0.894964i $$-0.352799\pi$$
0.446138 + 0.894964i $$0.352799\pi$$
$$564$$ 0 0
$$565$$ 3.51472 0.147865
$$566$$ 0 0
$$567$$ 0 0
$$568$$ 0 0
$$569$$ 1.17157 0.0491149 0.0245574 0.999698i $$-0.492182\pi$$
0.0245574 + 0.999698i $$0.492182\pi$$
$$570$$ 0 0
$$571$$ 16.2843 0.681476 0.340738 0.940158i $$-0.389323\pi$$
0.340738 + 0.940158i $$0.389323\pi$$
$$572$$ 0 0
$$573$$ 15.1716 0.633802
$$574$$ 0 0
$$575$$ −22.4853 −0.937701
$$576$$ 0 0
$$577$$ 6.58579 0.274170 0.137085 0.990559i $$-0.456227\pi$$
0.137085 + 0.990559i $$0.456227\pi$$
$$578$$ 0 0
$$579$$ 24.6274 1.02348
$$580$$ 0 0
$$581$$ 0 0
$$582$$ 0 0
$$583$$ 11.0294 0.456793
$$584$$ 0 0
$$585$$ −0.828427 −0.0342512
$$586$$ 0 0
$$587$$ −21.1716 −0.873844 −0.436922 0.899499i $$-0.643931\pi$$
−0.436922 + 0.899499i $$0.643931\pi$$
$$588$$ 0 0
$$589$$ −35.3137 −1.45508
$$590$$ 0 0
$$591$$ −2.00000 −0.0822690
$$592$$ 0 0
$$593$$ −14.9289 −0.613058 −0.306529 0.951861i $$-0.599168\pi$$
−0.306529 + 0.951861i $$0.599168\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 0 0
$$597$$ −5.65685 −0.231520
$$598$$ 0 0
$$599$$ −12.1421 −0.496114 −0.248057 0.968745i $$-0.579792\pi$$
−0.248057 + 0.968745i $$0.579792\pi$$
$$600$$ 0 0
$$601$$ 3.75736 0.153266 0.0766329 0.997059i $$-0.475583\pi$$
0.0766329 + 0.997059i $$0.475583\pi$$
$$602$$ 0 0
$$603$$ 8.00000 0.325785
$$604$$ 0 0
$$605$$ −6.04163 −0.245627
$$606$$ 0 0
$$607$$ 47.5980 1.93194 0.965971 0.258650i $$-0.0832775\pi$$
0.965971 + 0.258650i $$0.0832775\pi$$
$$608$$ 0 0
$$609$$ 0 0
$$610$$ 0 0
$$611$$ 9.65685 0.390675
$$612$$ 0 0
$$613$$ 13.6569 0.551595 0.275798 0.961216i $$-0.411058\pi$$
0.275798 + 0.961216i $$0.411058\pi$$
$$614$$ 0 0
$$615$$ 0.343146 0.0138370
$$616$$ 0 0
$$617$$ −29.4558 −1.18585 −0.592924 0.805259i $$-0.702026\pi$$
−0.592924 + 0.805259i $$0.702026\pi$$
$$618$$ 0 0
$$619$$ 16.2843 0.654520 0.327260 0.944934i $$-0.393875\pi$$
0.327260 + 0.944934i $$0.393875\pi$$
$$620$$ 0 0
$$621$$ 4.82843 0.193758
$$622$$ 0 0
$$623$$ 0 0
$$624$$ 0 0
$$625$$ 19.9706 0.798823
$$626$$ 0 0
$$627$$ 5.65685 0.225913
$$628$$ 0 0
$$629$$ 3.71573 0.148156
$$630$$ 0 0
$$631$$ 48.2843 1.92217 0.961083 0.276259i $$-0.0890947\pi$$
0.961083 + 0.276259i $$0.0890947\pi$$
$$632$$ 0 0
$$633$$ 14.3431 0.570089
$$634$$ 0 0
$$635$$ −8.97056 −0.355986
$$636$$ 0 0
$$637$$ 0 0
$$638$$ 0 0
$$639$$ −0.828427 −0.0327721
$$640$$ 0 0
$$641$$ 30.8284 1.21765 0.608825 0.793305i $$-0.291641\pi$$
0.608825 + 0.793305i $$0.291641\pi$$
$$642$$ 0 0
$$643$$ 36.4853 1.43884 0.719420 0.694576i $$-0.244408\pi$$
0.719420 + 0.694576i $$0.244408\pi$$
$$644$$ 0 0
$$645$$ 4.68629 0.184523
$$646$$ 0 0
$$647$$ −9.17157 −0.360572 −0.180286 0.983614i $$-0.557702\pi$$
−0.180286 + 0.983614i $$0.557702\pi$$
$$648$$ 0 0
$$649$$ 4.28427 0.168172
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 0 0
$$653$$ 2.82843 0.110685 0.0553425 0.998467i $$-0.482375\pi$$
0.0553425 + 0.998467i $$0.482375\pi$$
$$654$$ 0 0
$$655$$ −4.28427 −0.167400
$$656$$ 0 0
$$657$$ 11.0711 0.431923
$$658$$ 0 0
$$659$$ −42.4853 −1.65499 −0.827496 0.561472i $$-0.810235\pi$$
−0.827496 + 0.561472i $$0.810235\pi$$
$$660$$ 0 0
$$661$$ 47.3553 1.84191 0.920955 0.389670i $$-0.127411\pi$$
0.920955 + 0.389670i $$0.127411\pi$$
$$662$$ 0 0
$$663$$ 3.17157 0.123174
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 0 0
$$667$$ −40.9706 −1.58639
$$668$$ 0 0
$$669$$ 13.6569 0.528004
$$670$$ 0 0
$$671$$ 11.5147 0.444521
$$672$$ 0 0
$$673$$ −7.31371 −0.281923 −0.140961 0.990015i $$-0.545019\pi$$
−0.140961 + 0.990015i $$0.545019\pi$$
$$674$$ 0 0
$$675$$ −4.65685 −0.179242
$$676$$ 0 0
$$677$$ 29.5563 1.13594 0.567971 0.823048i $$-0.307728\pi$$
0.567971 + 0.823048i $$0.307728\pi$$
$$678$$ 0 0
$$679$$ 0 0
$$680$$ 0 0
$$681$$ −19.7990 −0.758699
$$682$$ 0 0
$$683$$ 16.1421 0.617662 0.308831 0.951117i $$-0.400062\pi$$
0.308831 + 0.951117i $$0.400062\pi$$
$$684$$ 0 0
$$685$$ 2.62742 0.100388
$$686$$ 0 0
$$687$$ 15.0711 0.574997
$$688$$ 0 0
$$689$$ −18.8284 −0.717306
$$690$$ 0 0
$$691$$ 28.0000 1.06517 0.532585 0.846376i $$-0.321221\pi$$
0.532585 + 0.846376i $$0.321221\pi$$
$$692$$ 0 0
$$693$$ 0 0
$$694$$ 0 0
$$695$$ −0.970563 −0.0368155
$$696$$ 0 0
$$697$$ −1.31371 −0.0497603
$$698$$ 0 0
$$699$$ −28.4853 −1.07741
$$700$$ 0 0
$$701$$ 5.17157 0.195328 0.0976638 0.995219i $$-0.468863\pi$$
0.0976638 + 0.995219i $$0.468863\pi$$
$$702$$ 0 0
$$703$$ −11.3137 −0.426705
$$704$$ 0 0
$$705$$ −4.00000 −0.150649
$$706$$ 0 0
$$707$$ 0 0
$$708$$ 0 0
$$709$$ 8.00000 0.300446 0.150223 0.988652i $$-0.452001\pi$$
0.150223 + 0.988652i $$0.452001\pi$$
$$710$$ 0 0
$$711$$ −2.34315 −0.0878748
$$712$$ 0 0
$$713$$ −24.9706 −0.935155
$$714$$ 0 0
$$715$$ −0.686292 −0.0256658
$$716$$ 0 0
$$717$$ −3.17157 −0.118445
$$718$$ 0 0
$$719$$ 4.68629 0.174769 0.0873846 0.996175i $$-0.472149\pi$$
0.0873846 + 0.996175i $$0.472149\pi$$
$$720$$ 0 0
$$721$$ 0 0
$$722$$ 0 0
$$723$$ 21.8995 0.814451
$$724$$ 0 0
$$725$$ 39.5147 1.46754
$$726$$ 0 0
$$727$$ −25.4558 −0.944105 −0.472052 0.881570i $$-0.656487\pi$$
−0.472052 + 0.881570i $$0.656487\pi$$
$$728$$ 0 0
$$729$$ 1.00000 0.0370370
$$730$$ 0 0
$$731$$ −17.9411 −0.663576
$$732$$ 0 0
$$733$$ 11.7574 0.434268 0.217134 0.976142i $$-0.430329\pi$$
0.217134 + 0.976142i $$0.430329\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ 6.62742 0.244124
$$738$$ 0 0
$$739$$ −20.2843 −0.746169 −0.373084 0.927797i $$-0.621700\pi$$
−0.373084 + 0.927797i $$0.621700\pi$$
$$740$$ 0 0
$$741$$ −9.65685 −0.354753
$$742$$ 0 0
$$743$$ −11.1716 −0.409845 −0.204923 0.978778i $$-0.565694\pi$$
−0.204923 + 0.978778i $$0.565694\pi$$
$$744$$ 0 0
$$745$$ 5.85786 0.214616
$$746$$ 0 0
$$747$$ 15.3137 0.560299
$$748$$ 0 0
$$749$$ 0 0
$$750$$ 0 0
$$751$$ 29.6569 1.08219 0.541097 0.840960i $$-0.318009\pi$$
0.541097 + 0.840960i $$0.318009\pi$$
$$752$$ 0 0
$$753$$ 8.48528 0.309221
$$754$$ 0 0
$$755$$ 5.65685 0.205874
$$756$$ 0 0
$$757$$ 11.3137 0.411204 0.205602 0.978636i $$-0.434085\pi$$
0.205602 + 0.978636i $$0.434085\pi$$
$$758$$ 0 0
$$759$$ 4.00000 0.145191
$$760$$ 0 0
$$761$$ 9.75736 0.353704 0.176852 0.984237i $$-0.443409\pi$$
0.176852 + 0.984237i $$0.443409\pi$$
$$762$$ 0 0
$$763$$ 0 0
$$764$$ 0 0
$$765$$ −1.31371 −0.0474972
$$766$$ 0 0
$$767$$ −7.31371 −0.264083
$$768$$ 0 0
$$769$$ −15.0711 −0.543477 −0.271738 0.962371i $$-0.587599\pi$$
−0.271738 + 0.962371i $$0.587599\pi$$
$$770$$ 0 0
$$771$$ 30.2426 1.08916
$$772$$ 0 0
$$773$$ −3.41421 −0.122801 −0.0614004 0.998113i $$-0.519557\pi$$
−0.0614004 + 0.998113i $$0.519557\pi$$
$$774$$ 0 0
$$775$$ 24.0833 0.865096
$$776$$ 0 0
$$777$$ 0 0
$$778$$ 0 0
$$779$$ 4.00000 0.143315
$$780$$ 0 0
$$781$$ −0.686292 −0.0245574
$$782$$ 0 0
$$783$$ −8.48528 −0.303239
$$784$$ 0 0
$$785$$ −8.14214 −0.290605
$$786$$ 0 0
$$787$$ −10.6274 −0.378827 −0.189413 0.981897i $$-0.560659\pi$$
−0.189413 + 0.981897i $$0.560659\pi$$
$$788$$ 0 0
$$789$$ 24.8284 0.883915
$$790$$ 0 0
$$791$$ 0 0
$$792$$ 0 0
$$793$$ −19.6569 −0.698035
$$794$$ 0 0
$$795$$ 7.79899 0.276602
$$796$$ 0 0
$$797$$ −43.2132 −1.53069 −0.765345 0.643620i $$-0.777432\pi$$
−0.765345 + 0.643620i $$0.777432\pi$$
$$798$$ 0 0
$$799$$ 15.3137 0.541760
$$800$$ 0 0
$$801$$ 10.7279 0.379052
$$802$$ 0 0
$$803$$ 9.17157 0.323658
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 0 0
$$807$$ 30.0416 1.05752
$$808$$ 0 0
$$809$$ −30.0000 −1.05474 −0.527372 0.849635i $$-0.676823\pi$$
−0.527372 + 0.849635i $$0.676823\pi$$
$$810$$ 0 0
$$811$$ −20.9706 −0.736376 −0.368188 0.929751i $$-0.620022\pi$$
−0.368188 + 0.929751i $$0.620022\pi$$
$$812$$ 0 0
$$813$$ 13.1716 0.461947
$$814$$ 0 0
$$815$$ 8.00000 0.280228
$$816$$ 0 0
$$817$$ 54.6274 1.91117
$$818$$ 0 0
$$819$$ 0 0
$$820$$ 0 0
$$821$$ −10.0000 −0.349002 −0.174501 0.984657i $$-0.555831\pi$$
−0.174501 + 0.984657i $$0.555831\pi$$
$$822$$ 0 0
$$823$$ −2.34315 −0.0816769 −0.0408385 0.999166i $$-0.513003\pi$$
−0.0408385 + 0.999166i $$0.513003\pi$$
$$824$$ 0 0
$$825$$ −3.85786 −0.134314
$$826$$ 0 0
$$827$$ 19.4558 0.676546 0.338273 0.941048i $$-0.390157\pi$$
0.338273 + 0.941048i $$0.390157\pi$$
$$828$$ 0 0
$$829$$ 3.27208 0.113644 0.0568220 0.998384i $$-0.481903\pi$$
0.0568220 + 0.998384i $$0.481903\pi$$
$$830$$ 0 0
$$831$$ 6.00000 0.208138
$$832$$ 0 0
$$833$$ 0 0
$$834$$ 0 0
$$835$$ −0.686292 −0.0237501
$$836$$ 0 0
$$837$$ −5.17157 −0.178756
$$838$$ 0 0
$$839$$ −49.1716 −1.69759 −0.848796 0.528721i $$-0.822672\pi$$
−0.848796 + 0.528721i $$0.822672\pi$$
$$840$$ 0 0
$$841$$ 43.0000 1.48276
$$842$$ 0 0
$$843$$ −12.4853 −0.430016
$$844$$ 0 0
$$845$$ −6.44365 −0.221668
$$846$$ 0 0
$$847$$ 0 0
$$848$$ 0 0
$$849$$ −14.8284 −0.508910
$$850$$ 0 0
$$851$$ −8.00000 −0.274236
$$852$$ 0 0
$$853$$ 36.0416 1.23404 0.617021 0.786947i $$-0.288339\pi$$
0.617021 + 0.786947i $$0.288339\pi$$
$$854$$ 0 0
$$855$$ 4.00000 0.136797
$$856$$ 0 0
$$857$$ 43.6985 1.49271 0.746356 0.665547i $$-0.231802\pi$$
0.746356 + 0.665547i $$0.231802\pi$$
$$858$$ 0 0
$$859$$ −38.8284 −1.32481 −0.662404 0.749146i $$-0.730464\pi$$
−0.662404 + 0.749146i $$0.730464\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 0 0
$$863$$ −29.5147 −1.00469 −0.502346 0.864666i $$-0.667530\pi$$
−0.502346 + 0.864666i $$0.667530\pi$$
$$864$$ 0 0
$$865$$ −2.00000 −0.0680020
$$866$$ 0 0
$$867$$ −11.9706 −0.406542
$$868$$ 0 0
$$869$$ −1.94113 −0.0658482
$$870$$ 0 0
$$871$$ −11.3137 −0.383350
$$872$$ 0 0
$$873$$ 7.75736 0.262547
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 0 0
$$877$$ −20.2843 −0.684951 −0.342476 0.939527i $$-0.611265\pi$$
−0.342476 + 0.939527i $$0.611265\pi$$
$$878$$ 0 0
$$879$$ 1.07107 0.0361262
$$880$$ 0 0
$$881$$ 25.0711 0.844666 0.422333 0.906441i $$-0.361211\pi$$
0.422333 + 0.906441i $$0.361211\pi$$
$$882$$ 0 0
$$883$$ −18.3431 −0.617296 −0.308648 0.951176i $$-0.599877\pi$$
−0.308648 + 0.951176i $$0.599877\pi$$
$$884$$ 0 0
$$885$$ 3.02944 0.101833
$$886$$ 0 0
$$887$$ −21.4558 −0.720417 −0.360208 0.932872i $$-0.617294\pi$$
−0.360208 + 0.932872i $$0.617294\pi$$
$$888$$ 0 0
$$889$$ 0 0
$$890$$ 0 0
$$891$$ 0.828427 0.0277534
$$892$$ 0 0
$$893$$ −46.6274 −1.56033
$$894$$ 0 0
$$895$$ −10.4264 −0.348516
$$896$$ 0 0
$$897$$ −6.82843 −0.227995
$$898$$ 0 0
$$899$$ 43.8823 1.46356
$$900$$ 0 0
$$901$$ −29.8579 −0.994710
$$902$$ 0 0
$$903$$ 0 0
$$904$$ 0 0
$$905$$ −5.79899 −0.192765
$$906$$ 0 0
$$907$$ 7.02944 0.233409 0.116704 0.993167i $$-0.462767\pi$$
0.116704 + 0.993167i $$0.462767\pi$$
$$908$$ 0 0
$$909$$ 3.41421 0.113242
$$910$$ 0 0
$$911$$ −54.4853 −1.80518 −0.902589 0.430503i $$-0.858336\pi$$
−0.902589 + 0.430503i $$0.858336\pi$$
$$912$$ 0 0
$$913$$ 12.6863 0.419855
$$914$$ 0 0
$$915$$ 8.14214 0.269171
$$916$$ 0 0
$$917$$ 0 0
$$918$$ 0 0
$$919$$ −49.2548 −1.62477 −0.812384 0.583123i $$-0.801830\pi$$
−0.812384 + 0.583123i $$0.801830\pi$$
$$920$$ 0 0
$$921$$ −11.5147 −0.379423
$$922$$ 0 0
$$923$$ 1.17157 0.0385628
$$924$$ 0 0
$$925$$ 7.71573 0.253692
$$926$$ 0 0
$$927$$ −10.8284 −0.355652
$$928$$ 0 0
$$929$$ −60.8701 −1.99708 −0.998541 0.0540006i $$-0.982803\pi$$
−0.998541 + 0.0540006i $$0.982803\pi$$
$$930$$ 0 0
$$931$$ 0 0
$$932$$ 0 0
$$933$$ 26.1421 0.855855
$$934$$ 0 0
$$935$$ −1.08831 −0.0355916
$$936$$ 0 0
$$937$$ 6.10051 0.199295 0.0996474 0.995023i $$-0.468229\pi$$
0.0996474 + 0.995023i $$0.468229\pi$$
$$938$$ 0 0
$$939$$ 17.4142 0.568291
$$940$$ 0 0
$$941$$ −43.8995 −1.43108 −0.715541 0.698570i $$-0.753820\pi$$
−0.715541 + 0.698570i $$0.753820\pi$$
$$942$$ 0 0
$$943$$ 2.82843 0.0921063
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ −20.1421 −0.654531 −0.327266 0.944932i $$-0.606127\pi$$
−0.327266 + 0.944932i $$0.606127\pi$$
$$948$$ 0 0
$$949$$ −15.6569 −0.508243
$$950$$ 0 0
$$951$$ −21.3137 −0.691144
$$952$$ 0 0
$$953$$ −7.37258 −0.238821 −0.119411 0.992845i $$-0.538101\pi$$
−0.119411 + 0.992845i $$0.538101\pi$$
$$954$$ 0 0
$$955$$ 8.88730 0.287586
$$956$$ 0 0
$$957$$ −7.02944 −0.227229
$$958$$ 0 0
$$959$$ 0 0
$$960$$ 0 0
$$961$$ −4.25483 −0.137253
$$962$$ 0 0
$$963$$ −14.4853 −0.466782
$$964$$ 0 0
$$965$$ 14.4264 0.464402
$$966$$ 0 0
$$967$$ −34.6274 −1.11354 −0.556771 0.830666i $$-0.687960\pi$$
−0.556771 + 0.830666i $$0.687960\pi$$
$$968$$ 0 0
$$969$$ −15.3137 −0.491947
$$970$$ 0 0
$$971$$ −57.2548 −1.83740 −0.918698 0.394962i $$-0.870758\pi$$
−0.918698 + 0.394962i $$0.870758\pi$$
$$972$$ 0 0
$$973$$ 0 0
$$974$$ 0 0
$$975$$ 6.58579 0.210914
$$976$$ 0 0
$$977$$ 21.8579 0.699295 0.349648 0.936881i $$-0.386301\pi$$
0.349648 + 0.936881i $$0.386301\pi$$
$$978$$ 0 0
$$979$$ 8.88730 0.284039
$$980$$ 0 0
$$981$$ 11.3137 0.361219
$$982$$ 0 0
$$983$$ 14.6274 0.466542 0.233271 0.972412i $$-0.425057\pi$$
0.233271 + 0.972412i $$0.425057\pi$$
$$984$$ 0 0
$$985$$ −1.17157 −0.0373294
$$986$$ 0 0
$$987$$ 0 0
$$988$$ 0 0
$$989$$ 38.6274 1.22828
$$990$$ 0 0
$$991$$ 23.3137 0.740584 0.370292 0.928915i $$-0.379258\pi$$
0.370292 + 0.928915i $$0.379258\pi$$
$$992$$ 0 0
$$993$$ 8.68629 0.275651
$$994$$ 0 0
$$995$$ −3.31371 −0.105052
$$996$$ 0 0
$$997$$ −34.5858 −1.09534 −0.547671 0.836693i $$-0.684486\pi$$
−0.547671 + 0.836693i $$0.684486\pi$$
$$998$$ 0 0
$$999$$ −1.65685 −0.0524205
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 9408.2.a.ee.1.1 2
4.3 odd 2 9408.2.a.ds.1.1 2
7.6 odd 2 9408.2.a.dg.1.2 2
8.3 odd 2 2352.2.a.bd.1.2 2
8.5 even 2 1176.2.a.j.1.2 2
24.5 odd 2 3528.2.a.bl.1.1 2
24.11 even 2 7056.2.a.cx.1.1 2
28.27 even 2 9408.2.a.du.1.2 2
56.3 even 6 2352.2.q.be.961.2 4
56.5 odd 6 1176.2.q.k.361.2 4
56.11 odd 6 2352.2.q.bc.961.1 4
56.13 odd 2 1176.2.a.o.1.1 yes 2
56.19 even 6 2352.2.q.be.1537.2 4
56.27 even 2 2352.2.a.bb.1.1 2
56.37 even 6 1176.2.q.o.361.1 4
56.45 odd 6 1176.2.q.k.961.2 4
56.51 odd 6 2352.2.q.bc.1537.1 4
56.53 even 6 1176.2.q.o.961.1 4
168.5 even 6 3528.2.s.bm.361.1 4
168.53 odd 6 3528.2.s.bd.3313.2 4
168.83 odd 2 7056.2.a.cg.1.2 2
168.101 even 6 3528.2.s.bm.3313.1 4
168.125 even 2 3528.2.a.bb.1.2 2
168.149 odd 6 3528.2.s.bd.361.2 4

By twisted newform
Twist Min Dim Char Parity Ord Type
1176.2.a.j.1.2 2 8.5 even 2
1176.2.a.o.1.1 yes 2 56.13 odd 2
1176.2.q.k.361.2 4 56.5 odd 6
1176.2.q.k.961.2 4 56.45 odd 6
1176.2.q.o.361.1 4 56.37 even 6
1176.2.q.o.961.1 4 56.53 even 6
2352.2.a.bb.1.1 2 56.27 even 2
2352.2.a.bd.1.2 2 8.3 odd 2
2352.2.q.bc.961.1 4 56.11 odd 6
2352.2.q.bc.1537.1 4 56.51 odd 6
2352.2.q.be.961.2 4 56.3 even 6
2352.2.q.be.1537.2 4 56.19 even 6
3528.2.a.bb.1.2 2 168.125 even 2
3528.2.a.bl.1.1 2 24.5 odd 2
3528.2.s.bd.361.2 4 168.149 odd 6
3528.2.s.bd.3313.2 4 168.53 odd 6
3528.2.s.bm.361.1 4 168.5 even 6
3528.2.s.bm.3313.1 4 168.101 even 6
7056.2.a.cg.1.2 2 168.83 odd 2
7056.2.a.cx.1.1 2 24.11 even 2
9408.2.a.dg.1.2 2 7.6 odd 2
9408.2.a.ds.1.1 2 4.3 odd 2
9408.2.a.du.1.2 2 28.27 even 2
9408.2.a.ee.1.1 2 1.1 even 1 trivial