# Properties

 Label 2352.2.a.z.1.2 Level $2352$ Weight $2$ Character 2352.1 Self dual yes Analytic conductor $18.781$ Analytic rank $1$ Dimension $2$ CM no Inner twists $1$

# Learn more about

## Newspace parameters

 Level: $$N$$ $$=$$ $$2352 = 2^{4} \cdot 3 \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2352.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$18.7808145554$$ Analytic rank: $$1$$ 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 1.2 Root $$1.41421$$ of defining polynomial Character $$\chi$$ $$=$$ 2352.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-1.00000 q^{3} -0.585786 q^{5} +1.00000 q^{9} +O(q^{10})$$ $$q-1.00000 q^{3} -0.585786 q^{5} +1.00000 q^{9} +4.82843 q^{11} -4.24264 q^{13} +0.585786 q^{15} -4.58579 q^{17} +1.17157 q^{19} -0.828427 q^{23} -4.65685 q^{25} -1.00000 q^{27} -2.82843 q^{29} +2.82843 q^{31} -4.82843 q^{33} +9.65685 q^{37} +4.24264 q^{39} -1.75736 q^{41} -11.3137 q^{43} -0.585786 q^{45} +12.4853 q^{47} +4.58579 q^{51} -2.00000 q^{53} -2.82843 q^{55} -1.17157 q^{57} -8.48528 q^{59} +3.07107 q^{61} +2.48528 q^{65} +11.3137 q^{67} +0.828427 q^{69} -6.48528 q^{71} -16.2426 q^{73} +4.65685 q^{75} -2.34315 q^{79} +1.00000 q^{81} -4.00000 q^{83} +2.68629 q^{85} +2.82843 q^{87} -14.2426 q^{89} -2.82843 q^{93} -0.686292 q^{95} -8.24264 q^{97} +4.82843 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 2q^{3} - 4q^{5} + 2q^{9} + O(q^{10})$$ $$2q - 2q^{3} - 4q^{5} + 2q^{9} + 4q^{11} + 4q^{15} - 12q^{17} + 8q^{19} + 4q^{23} + 2q^{25} - 2q^{27} - 4q^{33} + 8q^{37} - 12q^{41} - 4q^{45} + 8q^{47} + 12q^{51} - 4q^{53} - 8q^{57} - 8q^{61} - 12q^{65} - 4q^{69} + 4q^{71} - 24q^{73} - 2q^{75} - 16q^{79} + 2q^{81} - 8q^{83} + 28q^{85} - 20q^{89} - 24q^{95} - 8q^{97} + 4q^{99} + O(q^{100})$$

## 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 $$-0.541814\pi$$
−0.130986 + 0.991384i $$0.541814\pi$$
$$6$$ 0 0
$$7$$ 0 0
$$8$$ 0 0
$$9$$ 1.00000 0.333333
$$10$$ 0 0
$$11$$ 4.82843 1.45583 0.727913 0.685670i $$-0.240491\pi$$
0.727913 + 0.685670i $$0.240491\pi$$
$$12$$ 0 0
$$13$$ −4.24264 −1.17670 −0.588348 0.808608i $$-0.700222\pi$$
−0.588348 + 0.808608i $$0.700222\pi$$
$$14$$ 0 0
$$15$$ 0.585786 0.151249
$$16$$ 0 0
$$17$$ −4.58579 −1.11222 −0.556108 0.831110i $$-0.687706\pi$$
−0.556108 + 0.831110i $$0.687706\pi$$
$$18$$ 0 0
$$19$$ 1.17157 0.268777 0.134389 0.990929i $$-0.457093\pi$$
0.134389 + 0.990929i $$0.457093\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 0 0
$$23$$ −0.828427 −0.172739 −0.0863695 0.996263i $$-0.527527\pi$$
−0.0863695 + 0.996263i $$0.527527\pi$$
$$24$$ 0 0
$$25$$ −4.65685 −0.931371
$$26$$ 0 0
$$27$$ −1.00000 −0.192450
$$28$$ 0 0
$$29$$ −2.82843 −0.525226 −0.262613 0.964901i $$-0.584584\pi$$
−0.262613 + 0.964901i $$0.584584\pi$$
$$30$$ 0 0
$$31$$ 2.82843 0.508001 0.254000 0.967204i $$-0.418254\pi$$
0.254000 + 0.967204i $$0.418254\pi$$
$$32$$ 0 0
$$33$$ −4.82843 −0.840521
$$34$$ 0 0
$$35$$ 0 0
$$36$$ 0 0
$$37$$ 9.65685 1.58758 0.793789 0.608194i $$-0.208106\pi$$
0.793789 + 0.608194i $$0.208106\pi$$
$$38$$ 0 0
$$39$$ 4.24264 0.679366
$$40$$ 0 0
$$41$$ −1.75736 −0.274453 −0.137227 0.990540i $$-0.543819\pi$$
−0.137227 + 0.990540i $$0.543819\pi$$
$$42$$ 0 0
$$43$$ −11.3137 −1.72532 −0.862662 0.505781i $$-0.831205\pi$$
−0.862662 + 0.505781i $$0.831205\pi$$
$$44$$ 0 0
$$45$$ −0.585786 −0.0873239
$$46$$ 0 0
$$47$$ 12.4853 1.82117 0.910583 0.413327i $$-0.135633\pi$$
0.910583 + 0.413327i $$0.135633\pi$$
$$48$$ 0 0
$$49$$ 0 0
$$50$$ 0 0
$$51$$ 4.58579 0.642139
$$52$$ 0 0
$$53$$ −2.00000 −0.274721 −0.137361 0.990521i $$-0.543862\pi$$
−0.137361 + 0.990521i $$0.543862\pi$$
$$54$$ 0 0
$$55$$ −2.82843 −0.381385
$$56$$ 0 0
$$57$$ −1.17157 −0.155179
$$58$$ 0 0
$$59$$ −8.48528 −1.10469 −0.552345 0.833616i $$-0.686267\pi$$
−0.552345 + 0.833616i $$0.686267\pi$$
$$60$$ 0 0
$$61$$ 3.07107 0.393210 0.196605 0.980483i $$-0.437008\pi$$
0.196605 + 0.980483i $$0.437008\pi$$
$$62$$ 0 0
$$63$$ 0 0
$$64$$ 0 0
$$65$$ 2.48528 0.308261
$$66$$ 0 0
$$67$$ 11.3137 1.38219 0.691095 0.722764i $$-0.257129\pi$$
0.691095 + 0.722764i $$0.257129\pi$$
$$68$$ 0 0
$$69$$ 0.828427 0.0997309
$$70$$ 0 0
$$71$$ −6.48528 −0.769661 −0.384831 0.922987i $$-0.625740\pi$$
−0.384831 + 0.922987i $$0.625740\pi$$
$$72$$ 0 0
$$73$$ −16.2426 −1.90106 −0.950529 0.310637i $$-0.899458\pi$$
−0.950529 + 0.310637i $$0.899458\pi$$
$$74$$ 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
$$83$$ −4.00000 −0.439057 −0.219529 0.975606i $$-0.570452\pi$$
−0.219529 + 0.975606i $$0.570452\pi$$
$$84$$ 0 0
$$85$$ 2.68629 0.291369
$$86$$ 0 0
$$87$$ 2.82843 0.303239
$$88$$ 0 0
$$89$$ −14.2426 −1.50972 −0.754858 0.655888i $$-0.772294\pi$$
−0.754858 + 0.655888i $$0.772294\pi$$
$$90$$ 0 0
$$91$$ 0 0
$$92$$ 0 0
$$93$$ −2.82843 −0.293294
$$94$$ 0 0
$$95$$ −0.686292 −0.0704120
$$96$$ 0 0
$$97$$ −8.24264 −0.836913 −0.418457 0.908237i $$-0.637429\pi$$
−0.418457 + 0.908237i $$0.637429\pi$$
$$98$$ 0 0
$$99$$ 4.82843 0.485275
$$100$$ 0 0
$$101$$ −14.7279 −1.46548 −0.732742 0.680507i $$-0.761760\pi$$
−0.732742 + 0.680507i $$0.761760\pi$$
$$102$$ 0 0
$$103$$ −14.1421 −1.39347 −0.696733 0.717331i $$-0.745364\pi$$
−0.696733 + 0.717331i $$0.745364\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ −7.17157 −0.693302 −0.346651 0.937994i $$-0.612681\pi$$
−0.346651 + 0.937994i $$0.612681\pi$$
$$108$$ 0 0
$$109$$ 19.3137 1.84992 0.924959 0.380067i $$-0.124099\pi$$
0.924959 + 0.380067i $$0.124099\pi$$
$$110$$ 0 0
$$111$$ −9.65685 −0.916588
$$112$$ 0 0
$$113$$ −10.0000 −0.940721 −0.470360 0.882474i $$-0.655876\pi$$
−0.470360 + 0.882474i $$0.655876\pi$$
$$114$$ 0 0
$$115$$ 0.485281 0.0452527
$$116$$ 0 0
$$117$$ −4.24264 −0.392232
$$118$$ 0 0
$$119$$ 0 0
$$120$$ 0 0
$$121$$ 12.3137 1.11943
$$122$$ 0 0
$$123$$ 1.75736 0.158456
$$124$$ 0 0
$$125$$ 5.65685 0.505964
$$126$$ 0 0
$$127$$ −20.0000 −1.77471 −0.887357 0.461084i $$-0.847461\pi$$
−0.887357 + 0.461084i $$0.847461\pi$$
$$128$$ 0 0
$$129$$ 11.3137 0.996116
$$130$$ 0 0
$$131$$ −4.00000 −0.349482 −0.174741 0.984614i $$-0.555909\pi$$
−0.174741 + 0.984614i $$0.555909\pi$$
$$132$$ 0 0
$$133$$ 0 0
$$134$$ 0 0
$$135$$ 0.585786 0.0504165
$$136$$ 0 0
$$137$$ −14.8284 −1.26688 −0.633439 0.773793i $$-0.718357\pi$$
−0.633439 + 0.773793i $$0.718357\pi$$
$$138$$ 0 0
$$139$$ 12.9706 1.10015 0.550074 0.835116i $$-0.314599\pi$$
0.550074 + 0.835116i $$0.314599\pi$$
$$140$$ 0 0
$$141$$ −12.4853 −1.05145
$$142$$ 0 0
$$143$$ −20.4853 −1.71307
$$144$$ 0 0
$$145$$ 1.65685 0.137594
$$146$$ 0 0
$$147$$ 0 0
$$148$$ 0 0
$$149$$ −21.3137 −1.74609 −0.873044 0.487642i $$-0.837857\pi$$
−0.873044 + 0.487642i $$0.837857\pi$$
$$150$$ 0 0
$$151$$ −1.65685 −0.134833 −0.0674164 0.997725i $$-0.521476\pi$$
−0.0674164 + 0.997725i $$0.521476\pi$$
$$152$$ 0 0
$$153$$ −4.58579 −0.370739
$$154$$ 0 0
$$155$$ −1.65685 −0.133082
$$156$$ 0 0
$$157$$ 8.24264 0.657834 0.328917 0.944359i $$-0.393316\pi$$
0.328917 + 0.944359i $$0.393316\pi$$
$$158$$ 0 0
$$159$$ 2.00000 0.158610
$$160$$ 0 0
$$161$$ 0 0
$$162$$ 0 0
$$163$$ 5.65685 0.443079 0.221540 0.975151i $$-0.428892\pi$$
0.221540 + 0.975151i $$0.428892\pi$$
$$164$$ 0 0
$$165$$ 2.82843 0.220193
$$166$$ 0 0
$$167$$ −9.17157 −0.709718 −0.354859 0.934920i $$-0.615471\pi$$
−0.354859 + 0.934920i $$0.615471\pi$$
$$168$$ 0 0
$$169$$ 5.00000 0.384615
$$170$$ 0 0
$$171$$ 1.17157 0.0895924
$$172$$ 0 0
$$173$$ 19.4142 1.47604 0.738018 0.674781i $$-0.235762\pi$$
0.738018 + 0.674781i $$0.235762\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 0 0
$$177$$ 8.48528 0.637793
$$178$$ 0 0
$$179$$ −10.4853 −0.783707 −0.391853 0.920028i $$-0.628166\pi$$
−0.391853 + 0.920028i $$0.628166\pi$$
$$180$$ 0 0
$$181$$ −7.07107 −0.525588 −0.262794 0.964852i $$-0.584644\pi$$
−0.262794 + 0.964852i $$0.584644\pi$$
$$182$$ 0 0
$$183$$ −3.07107 −0.227020
$$184$$ 0 0
$$185$$ −5.65685 −0.415900
$$186$$ 0 0
$$187$$ −22.1421 −1.61919
$$188$$ 0 0
$$189$$ 0 0
$$190$$ 0 0
$$191$$ 16.1421 1.16800 0.584002 0.811752i $$-0.301486\pi$$
0.584002 + 0.811752i $$0.301486\pi$$
$$192$$ 0 0
$$193$$ 2.00000 0.143963 0.0719816 0.997406i $$-0.477068\pi$$
0.0719816 + 0.997406i $$0.477068\pi$$
$$194$$ 0 0
$$195$$ −2.48528 −0.177975
$$196$$ 0 0
$$197$$ −25.3137 −1.80353 −0.901764 0.432230i $$-0.857727\pi$$
−0.901764 + 0.432230i $$0.857727\pi$$
$$198$$ 0 0
$$199$$ 5.65685 0.401004 0.200502 0.979693i $$-0.435743\pi$$
0.200502 + 0.979693i $$0.435743\pi$$
$$200$$ 0 0
$$201$$ −11.3137 −0.798007
$$202$$ 0 0
$$203$$ 0 0
$$204$$ 0 0
$$205$$ 1.02944 0.0718990
$$206$$ 0 0
$$207$$ −0.828427 −0.0575797
$$208$$ 0 0
$$209$$ 5.65685 0.391293
$$210$$ 0 0
$$211$$ 9.65685 0.664805 0.332403 0.943138i $$-0.392141\pi$$
0.332403 + 0.943138i $$0.392141\pi$$
$$212$$ 0 0
$$213$$ 6.48528 0.444364
$$214$$ 0 0
$$215$$ 6.62742 0.451986
$$216$$ 0 0
$$217$$ 0 0
$$218$$ 0 0
$$219$$ 16.2426 1.09758
$$220$$ 0 0
$$221$$ 19.4558 1.30874
$$222$$ 0 0
$$223$$ 2.34315 0.156909 0.0784543 0.996918i $$-0.475002\pi$$
0.0784543 + 0.996918i $$0.475002\pi$$
$$224$$ 0 0
$$225$$ −4.65685 −0.310457
$$226$$ 0 0
$$227$$ −10.8284 −0.718708 −0.359354 0.933201i $$-0.617003\pi$$
−0.359354 + 0.933201i $$0.617003\pi$$
$$228$$ 0 0
$$229$$ 22.5858 1.49251 0.746255 0.665660i $$-0.231850\pi$$
0.746255 + 0.665660i $$0.231850\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ −9.17157 −0.600850 −0.300425 0.953805i $$-0.597128\pi$$
−0.300425 + 0.953805i $$0.597128\pi$$
$$234$$ 0 0
$$235$$ −7.31371 −0.477094
$$236$$ 0 0
$$237$$ 2.34315 0.152204
$$238$$ 0 0
$$239$$ 7.17157 0.463890 0.231945 0.972729i $$-0.425491\pi$$
0.231945 + 0.972729i $$0.425491\pi$$
$$240$$ 0 0
$$241$$ 5.89949 0.380020 0.190010 0.981782i $$-0.439148\pi$$
0.190010 + 0.981782i $$0.439148\pi$$
$$242$$ 0 0
$$243$$ −1.00000 −0.0641500
$$244$$ 0 0
$$245$$ 0 0
$$246$$ 0 0
$$247$$ −4.97056 −0.316269
$$248$$ 0 0
$$249$$ 4.00000 0.253490
$$250$$ 0 0
$$251$$ 22.1421 1.39760 0.698800 0.715317i $$-0.253718\pi$$
0.698800 + 0.715317i $$0.253718\pi$$
$$252$$ 0 0
$$253$$ −4.00000 −0.251478
$$254$$ 0 0
$$255$$ −2.68629 −0.168222
$$256$$ 0 0
$$257$$ 0.585786 0.0365404 0.0182702 0.999833i $$-0.494184\pi$$
0.0182702 + 0.999833i $$0.494184\pi$$
$$258$$ 0 0
$$259$$ 0 0
$$260$$ 0 0
$$261$$ −2.82843 −0.175075
$$262$$ 0 0
$$263$$ −8.14214 −0.502066 −0.251033 0.967979i $$-0.580770\pi$$
−0.251033 + 0.967979i $$0.580770\pi$$
$$264$$ 0 0
$$265$$ 1.17157 0.0719691
$$266$$ 0 0
$$267$$ 14.2426 0.871635
$$268$$ 0 0
$$269$$ 19.8995 1.21329 0.606647 0.794971i $$-0.292514\pi$$
0.606647 + 0.794971i $$0.292514\pi$$
$$270$$ 0 0
$$271$$ −22.1421 −1.34504 −0.672519 0.740079i $$-0.734788\pi$$
−0.672519 + 0.740079i $$0.734788\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ 0 0
$$275$$ −22.4853 −1.35591
$$276$$ 0 0
$$277$$ 16.6274 0.999045 0.499522 0.866301i $$-0.333509\pi$$
0.499522 + 0.866301i $$0.333509\pi$$
$$278$$ 0 0
$$279$$ 2.82843 0.169334
$$280$$ 0 0
$$281$$ 6.82843 0.407350 0.203675 0.979039i $$-0.434711\pi$$
0.203675 + 0.979039i $$0.434711\pi$$
$$282$$ 0 0
$$283$$ 2.14214 0.127337 0.0636684 0.997971i $$-0.479720\pi$$
0.0636684 + 0.997971i $$0.479720\pi$$
$$284$$ 0 0
$$285$$ 0.686292 0.0406524
$$286$$ 0 0
$$287$$ 0 0
$$288$$ 0 0
$$289$$ 4.02944 0.237026
$$290$$ 0 0
$$291$$ 8.24264 0.483192
$$292$$ 0 0
$$293$$ 10.2426 0.598381 0.299191 0.954193i $$-0.403283\pi$$
0.299191 + 0.954193i $$0.403283\pi$$
$$294$$ 0 0
$$295$$ 4.97056 0.289397
$$296$$ 0 0
$$297$$ −4.82843 −0.280174
$$298$$ 0 0
$$299$$ 3.51472 0.203261
$$300$$ 0 0
$$301$$ 0 0
$$302$$ 0 0
$$303$$ 14.7279 0.846097
$$304$$ 0 0
$$305$$ −1.79899 −0.103010
$$306$$ 0 0
$$307$$ −28.4853 −1.62574 −0.812870 0.582445i $$-0.802096\pi$$
−0.812870 + 0.582445i $$0.802096\pi$$
$$308$$ 0 0
$$309$$ 14.1421 0.804518
$$310$$ 0 0
$$311$$ −15.7990 −0.895879 −0.447939 0.894064i $$-0.647842\pi$$
−0.447939 + 0.894064i $$0.647842\pi$$
$$312$$ 0 0
$$313$$ −3.27208 −0.184949 −0.0924744 0.995715i $$-0.529478\pi$$
−0.0924744 + 0.995715i $$0.529478\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ −22.0000 −1.23564 −0.617822 0.786318i $$-0.711985\pi$$
−0.617822 + 0.786318i $$0.711985\pi$$
$$318$$ 0 0
$$319$$ −13.6569 −0.764637
$$320$$ 0 0
$$321$$ 7.17157 0.400278
$$322$$ 0 0
$$323$$ −5.37258 −0.298939
$$324$$ 0 0
$$325$$ 19.7574 1.09594
$$326$$ 0 0
$$327$$ −19.3137 −1.06805
$$328$$ 0 0
$$329$$ 0 0
$$330$$ 0 0
$$331$$ 15.3137 0.841718 0.420859 0.907126i $$-0.361729\pi$$
0.420859 + 0.907126i $$0.361729\pi$$
$$332$$ 0 0
$$333$$ 9.65685 0.529192
$$334$$ 0 0
$$335$$ −6.62742 −0.362094
$$336$$ 0 0
$$337$$ 21.6569 1.17972 0.589862 0.807504i $$-0.299182\pi$$
0.589862 + 0.807504i $$0.299182\pi$$
$$338$$ 0 0
$$339$$ 10.0000 0.543125
$$340$$ 0 0
$$341$$ 13.6569 0.739560
$$342$$ 0 0
$$343$$ 0 0
$$344$$ 0 0
$$345$$ −0.485281 −0.0261267
$$346$$ 0 0
$$347$$ −14.4853 −0.777611 −0.388805 0.921320i $$-0.627112\pi$$
−0.388805 + 0.921320i $$0.627112\pi$$
$$348$$ 0 0
$$349$$ −13.4142 −0.718046 −0.359023 0.933329i $$-0.616890\pi$$
−0.359023 + 0.933329i $$0.616890\pi$$
$$350$$ 0 0
$$351$$ 4.24264 0.226455
$$352$$ 0 0
$$353$$ −31.6985 −1.68714 −0.843570 0.537019i $$-0.819550\pi$$
−0.843570 + 0.537019i $$0.819550\pi$$
$$354$$ 0 0
$$355$$ 3.79899 0.201629
$$356$$ 0 0
$$357$$ 0 0
$$358$$ 0 0
$$359$$ −23.1716 −1.22295 −0.611474 0.791264i $$-0.709423\pi$$
−0.611474 + 0.791264i $$0.709423\pi$$
$$360$$ 0 0
$$361$$ −17.6274 −0.927759
$$362$$ 0 0
$$363$$ −12.3137 −0.646302
$$364$$ 0 0
$$365$$ 9.51472 0.498023
$$366$$ 0 0
$$367$$ 27.3137 1.42576 0.712882 0.701284i $$-0.247390\pi$$
0.712882 + 0.701284i $$0.247390\pi$$
$$368$$ 0 0
$$369$$ −1.75736 −0.0914845
$$370$$ 0 0
$$371$$ 0 0
$$372$$ 0 0
$$373$$ −17.3137 −0.896470 −0.448235 0.893916i $$-0.647947\pi$$
−0.448235 + 0.893916i $$0.647947\pi$$
$$374$$ 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
$$381$$ 20.0000 1.02463
$$382$$ 0 0
$$383$$ 24.9706 1.27594 0.637968 0.770063i $$-0.279775\pi$$
0.637968 + 0.770063i $$0.279775\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ −11.3137 −0.575108
$$388$$ 0 0
$$389$$ 3.79899 0.192616 0.0963082 0.995352i $$-0.469297\pi$$
0.0963082 + 0.995352i $$0.469297\pi$$
$$390$$ 0 0
$$391$$ 3.79899 0.192123
$$392$$ 0 0
$$393$$ 4.00000 0.201773
$$394$$ 0 0
$$395$$ 1.37258 0.0690621
$$396$$ 0 0
$$397$$ 7.75736 0.389331 0.194665 0.980870i $$-0.437638\pi$$
0.194665 + 0.980870i $$0.437638\pi$$
$$398$$ 0 0
$$399$$ 0 0
$$400$$ 0 0
$$401$$ 6.82843 0.340995 0.170498 0.985358i $$-0.445462\pi$$
0.170498 + 0.985358i $$0.445462\pi$$
$$402$$ 0 0
$$403$$ −12.0000 −0.597763
$$404$$ 0 0
$$405$$ −0.585786 −0.0291080
$$406$$ 0 0
$$407$$ 46.6274 2.31124
$$408$$ 0 0
$$409$$ −0.242641 −0.0119978 −0.00599890 0.999982i $$-0.501910\pi$$
−0.00599890 + 0.999982i $$0.501910\pi$$
$$410$$ 0 0
$$411$$ 14.8284 0.731432
$$412$$ 0 0
$$413$$ 0 0
$$414$$ 0 0
$$415$$ 2.34315 0.115021
$$416$$ 0 0
$$417$$ −12.9706 −0.635171
$$418$$ 0 0
$$419$$ 24.4853 1.19618 0.598092 0.801427i $$-0.295926\pi$$
0.598092 + 0.801427i $$0.295926\pi$$
$$420$$ 0 0
$$421$$ −2.68629 −0.130922 −0.0654609 0.997855i $$-0.520852\pi$$
−0.0654609 + 0.997855i $$0.520852\pi$$
$$422$$ 0 0
$$423$$ 12.4853 0.607055
$$424$$ 0 0
$$425$$ 21.3553 1.03589
$$426$$ 0 0
$$427$$ 0 0
$$428$$ 0 0
$$429$$ 20.4853 0.989039
$$430$$ 0 0
$$431$$ −20.1421 −0.970213 −0.485106 0.874455i $$-0.661219\pi$$
−0.485106 + 0.874455i $$0.661219\pi$$
$$432$$ 0 0
$$433$$ −15.0711 −0.724269 −0.362135 0.932126i $$-0.617952\pi$$
−0.362135 + 0.932126i $$0.617952\pi$$
$$434$$ 0 0
$$435$$ −1.65685 −0.0794401
$$436$$ 0 0
$$437$$ −0.970563 −0.0464283
$$438$$ 0 0
$$439$$ 35.3137 1.68543 0.842716 0.538359i $$-0.180956\pi$$
0.842716 + 0.538359i $$0.180956\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 0 0
$$443$$ 16.8284 0.799543 0.399771 0.916615i $$-0.369090\pi$$
0.399771 + 0.916615i $$0.369090\pi$$
$$444$$ 0 0
$$445$$ 8.34315 0.395503
$$446$$ 0 0
$$447$$ 21.3137 1.00810
$$448$$ 0 0
$$449$$ 28.6274 1.35101 0.675506 0.737355i $$-0.263925\pi$$
0.675506 + 0.737355i $$0.263925\pi$$
$$450$$ 0 0
$$451$$ −8.48528 −0.399556
$$452$$ 0 0
$$453$$ 1.65685 0.0778458
$$454$$ 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
$$459$$ 4.58579 0.214046
$$460$$ 0 0
$$461$$ 20.3848 0.949414 0.474707 0.880144i $$-0.342554\pi$$
0.474707 + 0.880144i $$0.342554\pi$$
$$462$$ 0 0
$$463$$ 9.65685 0.448792 0.224396 0.974498i $$-0.427959\pi$$
0.224396 + 0.974498i $$0.427959\pi$$
$$464$$ 0 0
$$465$$ 1.65685 0.0768348
$$466$$ 0 0
$$467$$ −13.1716 −0.609508 −0.304754 0.952431i $$-0.598574\pi$$
−0.304754 + 0.952431i $$0.598574\pi$$
$$468$$ 0 0
$$469$$ 0 0
$$470$$ 0 0
$$471$$ −8.24264 −0.379801
$$472$$ 0 0
$$473$$ −54.6274 −2.51177
$$474$$ 0 0
$$475$$ −5.45584 −0.250331
$$476$$ 0 0
$$477$$ −2.00000 −0.0915737
$$478$$ 0 0
$$479$$ 17.1716 0.784589 0.392295 0.919840i $$-0.371681\pi$$
0.392295 + 0.919840i $$0.371681\pi$$
$$480$$ 0 0
$$481$$ −40.9706 −1.86810
$$482$$ 0 0
$$483$$ 0 0
$$484$$ 0 0
$$485$$ 4.82843 0.219248
$$486$$ 0 0
$$487$$ 28.9706 1.31278 0.656391 0.754421i $$-0.272082\pi$$
0.656391 + 0.754421i $$0.272082\pi$$
$$488$$ 0 0
$$489$$ −5.65685 −0.255812
$$490$$ 0 0
$$491$$ −24.8284 −1.12049 −0.560246 0.828327i $$-0.689293\pi$$
−0.560246 + 0.828327i $$0.689293\pi$$
$$492$$ 0 0
$$493$$ 12.9706 0.584165
$$494$$ 0 0
$$495$$ −2.82843 −0.127128
$$496$$ 0 0
$$497$$ 0 0
$$498$$ 0 0
$$499$$ −8.97056 −0.401578 −0.200789 0.979635i $$-0.564350\pi$$
−0.200789 + 0.979635i $$0.564350\pi$$
$$500$$ 0 0
$$501$$ 9.17157 0.409756
$$502$$ 0 0
$$503$$ 3.31371 0.147751 0.0738755 0.997267i $$-0.476463\pi$$
0.0738755 + 0.997267i $$0.476463\pi$$
$$504$$ 0 0
$$505$$ 8.62742 0.383915
$$506$$ 0 0
$$507$$ −5.00000 −0.222058
$$508$$ 0 0
$$509$$ −0.384776 −0.0170549 −0.00852746 0.999964i $$-0.502714\pi$$
−0.00852746 + 0.999964i $$0.502714\pi$$
$$510$$ 0 0
$$511$$ 0 0
$$512$$ 0 0
$$513$$ −1.17157 −0.0517262
$$514$$ 0 0
$$515$$ 8.28427 0.365049
$$516$$ 0 0
$$517$$ 60.2843 2.65130
$$518$$ 0 0
$$519$$ −19.4142 −0.852189
$$520$$ 0 0
$$521$$ −5.75736 −0.252234 −0.126117 0.992015i $$-0.540252\pi$$
−0.126117 + 0.992015i $$0.540252\pi$$
$$522$$ 0 0
$$523$$ 28.9706 1.26679 0.633397 0.773827i $$-0.281660\pi$$
0.633397 + 0.773827i $$0.281660\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ −12.9706 −0.565007
$$528$$ 0 0
$$529$$ −22.3137 −0.970161
$$530$$ 0 0
$$531$$ −8.48528 −0.368230
$$532$$ 0 0
$$533$$ 7.45584 0.322948
$$534$$ 0 0
$$535$$ 4.20101 0.181626
$$536$$ 0 0
$$537$$ 10.4853 0.452473
$$538$$ 0 0
$$539$$ 0 0
$$540$$ 0 0
$$541$$ −18.0000 −0.773880 −0.386940 0.922105i $$-0.626468\pi$$
−0.386940 + 0.922105i $$0.626468\pi$$
$$542$$ 0 0
$$543$$ 7.07107 0.303449
$$544$$ 0 0
$$545$$ −11.3137 −0.484626
$$546$$ 0 0
$$547$$ 36.9706 1.58075 0.790374 0.612625i $$-0.209886\pi$$
0.790374 + 0.612625i $$0.209886\pi$$
$$548$$ 0 0
$$549$$ 3.07107 0.131070
$$550$$ 0 0
$$551$$ −3.31371 −0.141169
$$552$$ 0 0
$$553$$ 0 0
$$554$$ 0 0
$$555$$ 5.65685 0.240120
$$556$$ 0 0
$$557$$ −12.6274 −0.535041 −0.267520 0.963552i $$-0.586204\pi$$
−0.267520 + 0.963552i $$0.586204\pi$$
$$558$$ 0 0
$$559$$ 48.0000 2.03018
$$560$$ 0 0
$$561$$ 22.1421 0.934842
$$562$$ 0 0
$$563$$ 30.1421 1.27034 0.635170 0.772373i $$-0.280930\pi$$
0.635170 + 0.772373i $$0.280930\pi$$
$$564$$ 0 0
$$565$$ 5.85786 0.246442
$$566$$ 0 0
$$567$$ 0 0
$$568$$ 0 0
$$569$$ −34.1421 −1.43131 −0.715656 0.698453i $$-0.753872\pi$$
−0.715656 + 0.698453i $$0.753872\pi$$
$$570$$ 0 0
$$571$$ 30.3431 1.26982 0.634911 0.772586i $$-0.281037\pi$$
0.634911 + 0.772586i $$0.281037\pi$$
$$572$$ 0 0
$$573$$ −16.1421 −0.674347
$$574$$ 0 0
$$575$$ 3.85786 0.160884
$$576$$ 0 0
$$577$$ 29.2132 1.21616 0.608081 0.793875i $$-0.291940\pi$$
0.608081 + 0.793875i $$0.291940\pi$$
$$578$$ 0 0
$$579$$ −2.00000 −0.0831172
$$580$$ 0 0
$$581$$ 0 0
$$582$$ 0 0
$$583$$ −9.65685 −0.399946
$$584$$ 0 0
$$585$$ 2.48528 0.102754
$$586$$ 0 0
$$587$$ 3.79899 0.156801 0.0784005 0.996922i $$-0.475019\pi$$
0.0784005 + 0.996922i $$0.475019\pi$$
$$588$$ 0 0
$$589$$ 3.31371 0.136539
$$590$$ 0 0
$$591$$ 25.3137 1.04127
$$592$$ 0 0
$$593$$ −44.5858 −1.83092 −0.915459 0.402410i $$-0.868173\pi$$
−0.915459 + 0.402410i $$0.868173\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 0 0
$$597$$ −5.65685 −0.231520
$$598$$ 0 0
$$599$$ 27.4558 1.12182 0.560908 0.827878i $$-0.310452\pi$$
0.560908 + 0.827878i $$0.310452\pi$$
$$600$$ 0 0
$$601$$ 3.75736 0.153266 0.0766329 0.997059i $$-0.475583\pi$$
0.0766329 + 0.997059i $$0.475583\pi$$
$$602$$ 0 0
$$603$$ 11.3137 0.460730
$$604$$ 0 0
$$605$$ −7.21320 −0.293258
$$606$$ 0 0
$$607$$ −8.97056 −0.364104 −0.182052 0.983289i $$-0.558274\pi$$
−0.182052 + 0.983289i $$0.558274\pi$$
$$608$$ 0 0
$$609$$ 0 0
$$610$$ 0 0
$$611$$ −52.9706 −2.14296
$$612$$ 0 0
$$613$$ −21.6569 −0.874712 −0.437356 0.899288i $$-0.644085\pi$$
−0.437356 + 0.899288i $$0.644085\pi$$
$$614$$ 0 0
$$615$$ −1.02944 −0.0415109
$$616$$ 0 0
$$617$$ 12.4853 0.502639 0.251319 0.967904i $$-0.419136\pi$$
0.251319 + 0.967904i $$0.419136\pi$$
$$618$$ 0 0
$$619$$ 22.3431 0.898047 0.449023 0.893520i $$-0.351772\pi$$
0.449023 + 0.893520i $$0.351772\pi$$
$$620$$ 0 0
$$621$$ 0.828427 0.0332436
$$622$$ 0 0
$$623$$ 0 0
$$624$$ 0 0
$$625$$ 19.9706 0.798823
$$626$$ 0 0
$$627$$ −5.65685 −0.225913
$$628$$ 0 0
$$629$$ −44.2843 −1.76573
$$630$$ 0 0
$$631$$ 36.9706 1.47177 0.735887 0.677104i $$-0.236765\pi$$
0.735887 + 0.677104i $$0.236765\pi$$
$$632$$ 0 0
$$633$$ −9.65685 −0.383825
$$634$$ 0 0
$$635$$ 11.7157 0.464925
$$636$$ 0 0
$$637$$ 0 0
$$638$$ 0 0
$$639$$ −6.48528 −0.256554
$$640$$ 0 0
$$641$$ −27.1127 −1.07089 −0.535444 0.844571i $$-0.679856\pi$$
−0.535444 + 0.844571i $$0.679856\pi$$
$$642$$ 0 0
$$643$$ −7.79899 −0.307562 −0.153781 0.988105i $$-0.549145\pi$$
−0.153781 + 0.988105i $$0.549145\pi$$
$$644$$ 0 0
$$645$$ −6.62742 −0.260954
$$646$$ 0 0
$$647$$ −39.7990 −1.56466 −0.782330 0.622864i $$-0.785969\pi$$
−0.782330 + 0.622864i $$0.785969\pi$$
$$648$$ 0 0
$$649$$ −40.9706 −1.60824
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 0 0
$$653$$ 17.8579 0.698832 0.349416 0.936968i $$-0.386380\pi$$
0.349416 + 0.936968i $$0.386380\pi$$
$$654$$ 0 0
$$655$$ 2.34315 0.0915543
$$656$$ 0 0
$$657$$ −16.2426 −0.633686
$$658$$ 0 0
$$659$$ −38.4853 −1.49917 −0.749587 0.661906i $$-0.769748\pi$$
−0.749587 + 0.661906i $$0.769748\pi$$
$$660$$ 0 0
$$661$$ −19.0711 −0.741779 −0.370889 0.928677i $$-0.620947\pi$$
−0.370889 + 0.928677i $$0.620947\pi$$
$$662$$ 0 0
$$663$$ −19.4558 −0.755602
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 0 0
$$667$$ 2.34315 0.0907270
$$668$$ 0 0
$$669$$ −2.34315 −0.0905912
$$670$$ 0 0
$$671$$ 14.8284 0.572445
$$672$$ 0 0
$$673$$ −0.686292 −0.0264546 −0.0132273 0.999913i $$-0.504211\pi$$
−0.0132273 + 0.999913i $$0.504211\pi$$
$$674$$ 0 0
$$675$$ 4.65685 0.179242
$$676$$ 0 0
$$677$$ 31.6985 1.21827 0.609136 0.793066i $$-0.291516\pi$$
0.609136 + 0.793066i $$0.291516\pi$$
$$678$$ 0 0
$$679$$ 0 0
$$680$$ 0 0
$$681$$ 10.8284 0.414946
$$682$$ 0 0
$$683$$ −37.7990 −1.44634 −0.723169 0.690671i $$-0.757315\pi$$
−0.723169 + 0.690671i $$0.757315\pi$$
$$684$$ 0 0
$$685$$ 8.68629 0.331886
$$686$$ 0 0
$$687$$ −22.5858 −0.861701
$$688$$ 0 0
$$689$$ 8.48528 0.323263
$$690$$ 0 0
$$691$$ 31.3137 1.19123 0.595615 0.803270i $$-0.296908\pi$$
0.595615 + 0.803270i $$0.296908\pi$$
$$692$$ 0 0
$$693$$ 0 0
$$694$$ 0 0
$$695$$ −7.59798 −0.288208
$$696$$ 0 0
$$697$$ 8.05887 0.305252
$$698$$ 0 0
$$699$$ 9.17157 0.346901
$$700$$ 0 0
$$701$$ 22.1421 0.836297 0.418148 0.908379i $$-0.362679\pi$$
0.418148 + 0.908379i $$0.362679\pi$$
$$702$$ 0 0
$$703$$ 11.3137 0.426705
$$704$$ 0 0
$$705$$ 7.31371 0.275450
$$706$$ 0 0
$$707$$ 0 0
$$708$$ 0 0
$$709$$ −38.6274 −1.45068 −0.725342 0.688389i $$-0.758318\pi$$
−0.725342 + 0.688389i $$0.758318\pi$$
$$710$$ 0 0
$$711$$ −2.34315 −0.0878748
$$712$$ 0 0
$$713$$ −2.34315 −0.0877515
$$714$$ 0 0
$$715$$ 12.0000 0.448775
$$716$$ 0 0
$$717$$ −7.17157 −0.267827
$$718$$ 0 0
$$719$$ 38.6274 1.44056 0.720280 0.693684i $$-0.244013\pi$$
0.720280 + 0.693684i $$0.244013\pi$$
$$720$$ 0 0
$$721$$ 0 0
$$722$$ 0 0
$$723$$ −5.89949 −0.219405
$$724$$ 0 0
$$725$$ 13.1716 0.489180
$$726$$ 0 0
$$727$$ −38.1421 −1.41461 −0.707307 0.706907i $$-0.750090\pi$$
−0.707307 + 0.706907i $$0.750090\pi$$
$$728$$ 0 0
$$729$$ 1.00000 0.0370370
$$730$$ 0 0
$$731$$ 51.8823 1.91893
$$732$$ 0 0
$$733$$ −40.0416 −1.47897 −0.739486 0.673172i $$-0.764931\pi$$
−0.739486 + 0.673172i $$0.764931\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ 54.6274 2.01223
$$738$$ 0 0
$$739$$ 0.970563 0.0357027 0.0178514 0.999841i $$-0.494317\pi$$
0.0178514 + 0.999841i $$0.494317\pi$$
$$740$$ 0 0
$$741$$ 4.97056 0.182598
$$742$$ 0 0
$$743$$ −0.828427 −0.0303920 −0.0151960 0.999885i $$-0.504837\pi$$
−0.0151960 + 0.999885i $$0.504837\pi$$
$$744$$ 0 0
$$745$$ 12.4853 0.457425
$$746$$ 0 0
$$747$$ −4.00000 −0.146352
$$748$$ 0 0
$$749$$ 0 0
$$750$$ 0 0
$$751$$ 36.2843 1.32403 0.662016 0.749490i $$-0.269701\pi$$
0.662016 + 0.749490i $$0.269701\pi$$
$$752$$ 0 0
$$753$$ −22.1421 −0.806904
$$754$$ 0 0
$$755$$ 0.970563 0.0353224
$$756$$ 0 0
$$757$$ −25.9411 −0.942846 −0.471423 0.881907i $$-0.656260\pi$$
−0.471423 + 0.881907i $$0.656260\pi$$
$$758$$ 0 0
$$759$$ 4.00000 0.145191
$$760$$ 0 0
$$761$$ 12.1005 0.438643 0.219321 0.975653i $$-0.429616\pi$$
0.219321 + 0.975653i $$0.429616\pi$$
$$762$$ 0 0
$$763$$ 0 0
$$764$$ 0 0
$$765$$ 2.68629 0.0971231
$$766$$ 0 0
$$767$$ 36.0000 1.29988
$$768$$ 0 0
$$769$$ 18.8701 0.680472 0.340236 0.940340i $$-0.389493\pi$$
0.340236 + 0.940340i $$0.389493\pi$$
$$770$$ 0 0
$$771$$ −0.585786 −0.0210966
$$772$$ 0 0
$$773$$ 37.3553 1.34358 0.671789 0.740742i $$-0.265526\pi$$
0.671789 + 0.740742i $$0.265526\pi$$
$$774$$ 0 0
$$775$$ −13.1716 −0.473137
$$776$$ 0 0
$$777$$ 0 0
$$778$$ 0 0
$$779$$ −2.05887 −0.0737668
$$780$$ 0 0
$$781$$ −31.3137 −1.12049
$$782$$ 0 0
$$783$$ 2.82843 0.101080
$$784$$ 0 0
$$785$$ −4.82843 −0.172334
$$786$$ 0 0
$$787$$ −7.31371 −0.260706 −0.130353 0.991468i $$-0.541611\pi$$
−0.130353 + 0.991468i $$0.541611\pi$$
$$788$$ 0 0
$$789$$ 8.14214 0.289868
$$790$$ 0 0
$$791$$ 0 0
$$792$$ 0 0
$$793$$ −13.0294 −0.462689
$$794$$ 0 0
$$795$$ −1.17157 −0.0415514
$$796$$ 0 0
$$797$$ 36.5858 1.29594 0.647968 0.761668i $$-0.275619\pi$$
0.647968 + 0.761668i $$0.275619\pi$$
$$798$$ 0 0
$$799$$ −57.2548 −2.02553
$$800$$ 0 0
$$801$$ −14.2426 −0.503239
$$802$$ 0 0
$$803$$ −78.4264 −2.76761
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 0 0
$$807$$ −19.8995 −0.700495
$$808$$ 0 0
$$809$$ 40.6274 1.42838 0.714192 0.699950i $$-0.246794\pi$$
0.714192 + 0.699950i $$0.246794\pi$$
$$810$$ 0 0
$$811$$ 14.3431 0.503656 0.251828 0.967772i $$-0.418968\pi$$
0.251828 + 0.967772i $$0.418968\pi$$
$$812$$ 0 0
$$813$$ 22.1421 0.776559
$$814$$ 0 0
$$815$$ −3.31371 −0.116074
$$816$$ 0 0
$$817$$ −13.2548 −0.463728
$$818$$ 0 0
$$819$$ 0 0
$$820$$ 0 0
$$821$$ −33.3137 −1.16266 −0.581328 0.813669i $$-0.697467\pi$$
−0.581328 + 0.813669i $$0.697467\pi$$
$$822$$ 0 0
$$823$$ −8.97056 −0.312694 −0.156347 0.987702i $$-0.549972\pi$$
−0.156347 + 0.987702i $$0.549972\pi$$
$$824$$ 0 0
$$825$$ 22.4853 0.782837
$$826$$ 0 0
$$827$$ −11.8579 −0.412338 −0.206169 0.978516i $$-0.566100\pi$$
−0.206169 + 0.978516i $$0.566100\pi$$
$$828$$ 0 0
$$829$$ 2.38478 0.0828267 0.0414134 0.999142i $$-0.486814\pi$$
0.0414134 + 0.999142i $$0.486814\pi$$
$$830$$ 0 0
$$831$$ −16.6274 −0.576799
$$832$$ 0 0
$$833$$ 0 0
$$834$$ 0 0
$$835$$ 5.37258 0.185926
$$836$$ 0 0
$$837$$ −2.82843 −0.0977647
$$838$$ 0 0
$$839$$ −15.7990 −0.545442 −0.272721 0.962093i $$-0.587924\pi$$
−0.272721 + 0.962093i $$0.587924\pi$$
$$840$$ 0 0
$$841$$ −21.0000 −0.724138
$$842$$ 0 0
$$843$$ −6.82843 −0.235184
$$844$$ 0 0
$$845$$ −2.92893 −0.100758
$$846$$ 0 0
$$847$$ 0 0
$$848$$ 0 0
$$849$$ −2.14214 −0.0735179
$$850$$ 0 0
$$851$$ −8.00000 −0.274236
$$852$$ 0 0
$$853$$ −19.0711 −0.652981 −0.326490 0.945200i $$-0.605866\pi$$
−0.326490 + 0.945200i $$0.605866\pi$$
$$854$$ 0 0
$$855$$ −0.686292 −0.0234707
$$856$$ 0 0
$$857$$ −31.2132 −1.06622 −0.533111 0.846045i $$-0.678977\pi$$
−0.533111 + 0.846045i $$0.678977\pi$$
$$858$$ 0 0
$$859$$ 53.4558 1.82389 0.911945 0.410313i $$-0.134580\pi$$
0.911945 + 0.410313i $$0.134580\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 0 0
$$863$$ 42.0833 1.43253 0.716265 0.697828i $$-0.245850\pi$$
0.716265 + 0.697828i $$0.245850\pi$$
$$864$$ 0 0
$$865$$ −11.3726 −0.386679
$$866$$ 0 0
$$867$$ −4.02944 −0.136847
$$868$$ 0 0
$$869$$ −11.3137 −0.383791
$$870$$ 0 0
$$871$$ −48.0000 −1.62642
$$872$$ 0 0
$$873$$ −8.24264 −0.278971
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 0 0
$$877$$ 28.2843 0.955092 0.477546 0.878607i $$-0.341526\pi$$
0.477546 + 0.878607i $$0.341526\pi$$
$$878$$ 0 0
$$879$$ −10.2426 −0.345476
$$880$$ 0 0
$$881$$ 18.0416 0.607838 0.303919 0.952698i $$-0.401705\pi$$
0.303919 + 0.952698i $$0.401705\pi$$
$$882$$ 0 0
$$883$$ 21.6569 0.728811 0.364406 0.931240i $$-0.381272\pi$$
0.364406 + 0.931240i $$0.381272\pi$$
$$884$$ 0 0
$$885$$ −4.97056 −0.167084
$$886$$ 0 0
$$887$$ −0.201010 −0.00674926 −0.00337463 0.999994i $$-0.501074\pi$$
−0.00337463 + 0.999994i $$0.501074\pi$$
$$888$$ 0 0
$$889$$ 0 0
$$890$$ 0 0
$$891$$ 4.82843 0.161758
$$892$$ 0 0
$$893$$ 14.6274 0.489488
$$894$$ 0 0
$$895$$ 6.14214 0.205309
$$896$$ 0 0
$$897$$ −3.51472 −0.117353
$$898$$ 0 0
$$899$$ −8.00000 −0.266815
$$900$$ 0 0
$$901$$ 9.17157 0.305549
$$902$$ 0 0
$$903$$ 0 0
$$904$$ 0 0
$$905$$ 4.14214 0.137689
$$906$$ 0 0
$$907$$ −42.3431 −1.40598 −0.702991 0.711199i $$-0.748152\pi$$
−0.702991 + 0.711199i $$0.748152\pi$$
$$908$$ 0 0
$$909$$ −14.7279 −0.488494
$$910$$ 0 0
$$911$$ 10.4853 0.347393 0.173696 0.984799i $$-0.444429\pi$$
0.173696 + 0.984799i $$0.444429\pi$$
$$912$$ 0 0
$$913$$ −19.3137 −0.639190
$$914$$ 0 0
$$915$$ 1.79899 0.0594728
$$916$$ 0 0
$$917$$ 0 0
$$918$$ 0 0
$$919$$ −8.68629 −0.286534 −0.143267 0.989684i $$-0.545761\pi$$
−0.143267 + 0.989684i $$0.545761\pi$$
$$920$$ 0 0
$$921$$ 28.4853 0.938622
$$922$$ 0 0
$$923$$ 27.5147 0.905658
$$924$$ 0 0
$$925$$ −44.9706 −1.47862
$$926$$ 0 0
$$927$$ −14.1421 −0.464489
$$928$$ 0 0
$$929$$ −3.89949 −0.127938 −0.0639691 0.997952i $$-0.520376\pi$$
−0.0639691 + 0.997952i $$0.520376\pi$$
$$930$$ 0 0
$$931$$ 0 0
$$932$$ 0 0
$$933$$ 15.7990 0.517236
$$934$$ 0 0
$$935$$ 12.9706 0.424183
$$936$$ 0 0
$$937$$ 19.3553 0.632311 0.316156 0.948707i $$-0.397608\pi$$
0.316156 + 0.948707i $$0.397608\pi$$
$$938$$ 0 0
$$939$$ 3.27208 0.106780
$$940$$ 0 0
$$941$$ −10.7279 −0.349720 −0.174860 0.984593i $$-0.555947\pi$$
−0.174860 + 0.984593i $$0.555947\pi$$
$$942$$ 0 0
$$943$$ 1.45584 0.0474088
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ −14.7696 −0.479946 −0.239973 0.970780i $$-0.577139\pi$$
−0.239973 + 0.970780i $$0.577139\pi$$
$$948$$ 0 0
$$949$$ 68.9117 2.23697
$$950$$ 0 0
$$951$$ 22.0000 0.713399
$$952$$ 0 0
$$953$$ 2.00000 0.0647864 0.0323932 0.999475i $$-0.489687\pi$$
0.0323932 + 0.999475i $$0.489687\pi$$
$$954$$ 0 0
$$955$$ −9.45584 −0.305984
$$956$$ 0 0
$$957$$ 13.6569 0.441463
$$958$$ 0 0
$$959$$ 0 0
$$960$$ 0 0
$$961$$ −23.0000 −0.741935
$$962$$ 0 0
$$963$$ −7.17157 −0.231101
$$964$$ 0 0
$$965$$ −1.17157 −0.0377143
$$966$$ 0 0
$$967$$ 8.68629 0.279332 0.139666 0.990199i $$-0.455397\pi$$
0.139666 + 0.990199i $$0.455397\pi$$
$$968$$ 0 0
$$969$$ 5.37258 0.172592
$$970$$ 0 0
$$971$$ −20.0000 −0.641831 −0.320915 0.947108i $$-0.603990\pi$$
−0.320915 + 0.947108i $$0.603990\pi$$
$$972$$ 0 0
$$973$$ 0 0
$$974$$ 0 0
$$975$$ −19.7574 −0.632742
$$976$$ 0 0
$$977$$ 31.7990 1.01734 0.508670 0.860962i $$-0.330137\pi$$
0.508670 + 0.860962i $$0.330137\pi$$
$$978$$ 0 0
$$979$$ −68.7696 −2.19788
$$980$$ 0 0
$$981$$ 19.3137 0.616639
$$982$$ 0 0
$$983$$ 46.6274 1.48718 0.743592 0.668634i $$-0.233121\pi$$
0.743592 + 0.668634i $$0.233121\pi$$
$$984$$ 0 0
$$985$$ 14.8284 0.472473
$$986$$ 0 0
$$987$$ 0 0
$$988$$ 0 0
$$989$$ 9.37258 0.298031
$$990$$ 0 0
$$991$$ −26.6274 −0.845848 −0.422924 0.906165i $$-0.638996\pi$$
−0.422924 + 0.906165i $$0.638996\pi$$
$$992$$ 0 0
$$993$$ −15.3137 −0.485966
$$994$$ 0 0
$$995$$ −3.31371 −0.105052
$$996$$ 0 0
$$997$$ 17.6152 0.557880 0.278940 0.960309i $$-0.410017\pi$$
0.278940 + 0.960309i $$0.410017\pi$$
$$998$$ 0 0
$$999$$ −9.65685 −0.305529
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 2352.2.a.z.1.2 2
3.2 odd 2 7056.2.a.cw.1.1 2
4.3 odd 2 1176.2.a.m.1.2 yes 2
7.2 even 3 2352.2.q.bg.1537.1 4
7.3 odd 6 2352.2.q.ba.961.2 4
7.4 even 3 2352.2.q.bg.961.1 4
7.5 odd 6 2352.2.q.ba.1537.2 4
7.6 odd 2 2352.2.a.bg.1.1 2
8.3 odd 2 9408.2.a.dr.1.1 2
8.5 even 2 9408.2.a.ed.1.1 2
12.11 even 2 3528.2.a.bm.1.1 2
21.20 even 2 7056.2.a.ce.1.2 2
28.3 even 6 1176.2.q.n.961.2 4
28.11 odd 6 1176.2.q.m.961.1 4
28.19 even 6 1176.2.q.n.361.2 4
28.23 odd 6 1176.2.q.m.361.1 4
28.27 even 2 1176.2.a.l.1.1 2
56.13 odd 2 9408.2.a.dh.1.2 2
56.27 even 2 9408.2.a.dv.1.2 2
84.11 even 6 3528.2.s.bc.3313.2 4
84.23 even 6 3528.2.s.bc.361.2 4
84.47 odd 6 3528.2.s.bl.361.1 4
84.59 odd 6 3528.2.s.bl.3313.1 4
84.83 odd 2 3528.2.a.bc.1.2 2

By twisted newform
Twist Min Dim Char Parity Ord Type
1176.2.a.l.1.1 2 28.27 even 2
1176.2.a.m.1.2 yes 2 4.3 odd 2
1176.2.q.m.361.1 4 28.23 odd 6
1176.2.q.m.961.1 4 28.11 odd 6
1176.2.q.n.361.2 4 28.19 even 6
1176.2.q.n.961.2 4 28.3 even 6
2352.2.a.z.1.2 2 1.1 even 1 trivial
2352.2.a.bg.1.1 2 7.6 odd 2
2352.2.q.ba.961.2 4 7.3 odd 6
2352.2.q.ba.1537.2 4 7.5 odd 6
2352.2.q.bg.961.1 4 7.4 even 3
2352.2.q.bg.1537.1 4 7.2 even 3
3528.2.a.bc.1.2 2 84.83 odd 2
3528.2.a.bm.1.1 2 12.11 even 2
3528.2.s.bc.361.2 4 84.23 even 6
3528.2.s.bc.3313.2 4 84.11 even 6
3528.2.s.bl.361.1 4 84.47 odd 6
3528.2.s.bl.3313.1 4 84.59 odd 6
7056.2.a.ce.1.2 2 21.20 even 2
7056.2.a.cw.1.1 2 3.2 odd 2
9408.2.a.dh.1.2 2 56.13 odd 2
9408.2.a.dr.1.1 2 8.3 odd 2
9408.2.a.dv.1.2 2 56.27 even 2
9408.2.a.ed.1.1 2 8.5 even 2