# Properties

 Label 2352.2.a.z.1.1 Level $2352$ Weight $2$ Character 2352.1 Self dual yes Analytic conductor $18.781$ Analytic rank $1$ Dimension $2$ CM no Inner twists $1$

# Related objects

## 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 1.1 Root $$-1.41421$$ of defining polynomial Character $$\chi$$ $$=$$ 2352.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-1.00000 q^{3} -3.41421 q^{5} +1.00000 q^{9} +O(q^{10})$$ $$q-1.00000 q^{3} -3.41421 q^{5} +1.00000 q^{9} -0.828427 q^{11} +4.24264 q^{13} +3.41421 q^{15} -7.41421 q^{17} +6.82843 q^{19} +4.82843 q^{23} +6.65685 q^{25} -1.00000 q^{27} +2.82843 q^{29} -2.82843 q^{31} +0.828427 q^{33} -1.65685 q^{37} -4.24264 q^{39} -10.2426 q^{41} +11.3137 q^{43} -3.41421 q^{45} -4.48528 q^{47} +7.41421 q^{51} -2.00000 q^{53} +2.82843 q^{55} -6.82843 q^{57} +8.48528 q^{59} -11.0711 q^{61} -14.4853 q^{65} -11.3137 q^{67} -4.82843 q^{69} +10.4853 q^{71} -7.75736 q^{73} -6.65685 q^{75} -13.6569 q^{79} +1.00000 q^{81} -4.00000 q^{83} +25.3137 q^{85} -2.82843 q^{87} -5.75736 q^{89} +2.82843 q^{93} -23.3137 q^{95} +0.242641 q^{97} -0.828427 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 2q^{3} - 4q^{5} + 2q^{9} + O(q^{10})$$ $$2q - 2q^{3} - 4q^{5} + 2q^{9} + 4q^{11} + 4q^{15} - 12q^{17} + 8q^{19} + 4q^{23} + 2q^{25} - 2q^{27} - 4q^{33} + 8q^{37} - 12q^{41} - 4q^{45} + 8q^{47} + 12q^{51} - 4q^{53} - 8q^{57} - 8q^{61} - 12q^{65} - 4q^{69} + 4q^{71} - 24q^{73} - 2q^{75} - 16q^{79} + 2q^{81} - 8q^{83} + 28q^{85} - 20q^{89} - 24q^{95} - 8q^{97} + 4q^{99} + O(q^{100})$$

## 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
$$5$$ −3.41421 −1.52688 −0.763441 0.645877i $$-0.776492\pi$$
−0.763441 + 0.645877i $$0.776492\pi$$
$$6$$ 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 $$-0.539858\pi$$
−0.124890 + 0.992171i $$0.539858\pi$$
$$12$$ 0 0
$$13$$ 4.24264 1.17670 0.588348 0.808608i $$-0.299778\pi$$
0.588348 + 0.808608i $$0.299778\pi$$
$$14$$ 0 0
$$15$$ 3.41421 0.881546
$$16$$ 0 0
$$17$$ −7.41421 −1.79821 −0.899105 0.437732i $$-0.855782\pi$$
−0.899105 + 0.437732i $$0.855782\pi$$
$$18$$ 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
$$25$$ 6.65685 1.33137
$$26$$ 0 0
$$27$$ −1.00000 −0.192450
$$28$$ 0 0
$$29$$ 2.82843 0.525226 0.262613 0.964901i $$-0.415416\pi$$
0.262613 + 0.964901i $$0.415416\pi$$
$$30$$ 0 0
$$31$$ −2.82843 −0.508001 −0.254000 0.967204i $$-0.581746\pi$$
−0.254000 + 0.967204i $$0.581746\pi$$
$$32$$ 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
$$39$$ −4.24264 −0.679366
$$40$$ 0 0
$$41$$ −10.2426 −1.59963 −0.799816 0.600245i $$-0.795070\pi$$
−0.799816 + 0.600245i $$0.795070\pi$$
$$42$$ 0 0
$$43$$ 11.3137 1.72532 0.862662 0.505781i $$-0.168795\pi$$
0.862662 + 0.505781i $$0.168795\pi$$
$$44$$ 0 0
$$45$$ −3.41421 −0.508961
$$46$$ 0 0
$$47$$ −4.48528 −0.654246 −0.327123 0.944982i $$-0.606079\pi$$
−0.327123 + 0.944982i $$0.606079\pi$$
$$48$$ 0 0
$$49$$ 0 0
$$50$$ 0 0
$$51$$ 7.41421 1.03820
$$52$$ 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
$$57$$ −6.82843 −0.904447
$$58$$ 0 0
$$59$$ 8.48528 1.10469 0.552345 0.833616i $$-0.313733\pi$$
0.552345 + 0.833616i $$0.313733\pi$$
$$60$$ 0 0
$$61$$ −11.0711 −1.41750 −0.708752 0.705457i $$-0.750742\pi$$
−0.708752 + 0.705457i $$0.750742\pi$$
$$62$$ 0 0
$$63$$ 0 0
$$64$$ 0 0
$$65$$ −14.4853 −1.79668
$$66$$ 0 0
$$67$$ −11.3137 −1.38219 −0.691095 0.722764i $$-0.742871\pi$$
−0.691095 + 0.722764i $$0.742871\pi$$
$$68$$ 0 0
$$69$$ −4.82843 −0.581274
$$70$$ 0 0
$$71$$ 10.4853 1.24437 0.622187 0.782869i $$-0.286244\pi$$
0.622187 + 0.782869i $$0.286244\pi$$
$$72$$ 0 0
$$73$$ −7.75736 −0.907930 −0.453965 0.891019i $$-0.649991\pi$$
−0.453965 + 0.891019i $$0.649991\pi$$
$$74$$ 0 0
$$75$$ −6.65685 −0.768667
$$76$$ 0 0
$$77$$ 0 0
$$78$$ 0 0
$$79$$ −13.6569 −1.53652 −0.768258 0.640140i $$-0.778876\pi$$
−0.768258 + 0.640140i $$0.778876\pi$$
$$80$$ 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
$$85$$ 25.3137 2.74566
$$86$$ 0 0
$$87$$ −2.82843 −0.303239
$$88$$ 0 0
$$89$$ −5.75736 −0.610279 −0.305139 0.952308i $$-0.598703\pi$$
−0.305139 + 0.952308i $$0.598703\pi$$
$$90$$ 0 0
$$91$$ 0 0
$$92$$ 0 0
$$93$$ 2.82843 0.293294
$$94$$ 0 0
$$95$$ −23.3137 −2.39194
$$96$$ 0 0
$$97$$ 0.242641 0.0246364 0.0123182 0.999924i $$-0.496079\pi$$
0.0123182 + 0.999924i $$0.496079\pi$$
$$98$$ 0 0
$$99$$ −0.828427 −0.0832601
$$100$$ 0 0
$$101$$ 10.7279 1.06747 0.533734 0.845652i $$-0.320788\pi$$
0.533734 + 0.845652i $$0.320788\pi$$
$$102$$ 0 0
$$103$$ 14.1421 1.39347 0.696733 0.717331i $$-0.254636\pi$$
0.696733 + 0.717331i $$0.254636\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ −12.8284 −1.24017 −0.620085 0.784534i $$-0.712902\pi$$
−0.620085 + 0.784534i $$0.712902\pi$$
$$108$$ 0 0
$$109$$ −3.31371 −0.317396 −0.158698 0.987327i $$-0.550730\pi$$
−0.158698 + 0.987327i $$0.550730\pi$$
$$110$$ 0 0
$$111$$ 1.65685 0.157262
$$112$$ 0 0
$$113$$ −10.0000 −0.940721 −0.470360 0.882474i $$-0.655876\pi$$
−0.470360 + 0.882474i $$0.655876\pi$$
$$114$$ 0 0
$$115$$ −16.4853 −1.53726
$$116$$ 0 0
$$117$$ 4.24264 0.392232
$$118$$ 0 0
$$119$$ 0 0
$$120$$ 0 0
$$121$$ −10.3137 −0.937610
$$122$$ 0 0
$$123$$ 10.2426 0.923548
$$124$$ 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
$$135$$ 3.41421 0.293849
$$136$$ 0 0
$$137$$ −9.17157 −0.783580 −0.391790 0.920055i $$-0.628144\pi$$
−0.391790 + 0.920055i $$0.628144\pi$$
$$138$$ 0 0
$$139$$ −20.9706 −1.77870 −0.889350 0.457227i $$-0.848843\pi$$
−0.889350 + 0.457227i $$0.848843\pi$$
$$140$$ 0 0
$$141$$ 4.48528 0.377729
$$142$$ 0 0
$$143$$ −3.51472 −0.293916
$$144$$ 0 0
$$145$$ −9.65685 −0.801958
$$146$$ 0 0
$$147$$ 0 0
$$148$$ 0 0
$$149$$ 1.31371 0.107623 0.0538116 0.998551i $$-0.482863\pi$$
0.0538116 + 0.998551i $$0.482863\pi$$
$$150$$ 0 0
$$151$$ 9.65685 0.785864 0.392932 0.919568i $$-0.371461\pi$$
0.392932 + 0.919568i $$0.371461\pi$$
$$152$$ 0 0
$$153$$ −7.41421 −0.599404
$$154$$ 0 0
$$155$$ 9.65685 0.775657
$$156$$ 0 0
$$157$$ −0.242641 −0.0193648 −0.00968242 0.999953i $$-0.503082\pi$$
−0.00968242 + 0.999953i $$0.503082\pi$$
$$158$$ 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 $$-0.571108\pi$$
−0.221540 + 0.975151i $$0.571108\pi$$
$$164$$ 0 0
$$165$$ −2.82843 −0.220193
$$166$$ 0 0
$$167$$ −14.8284 −1.14746 −0.573729 0.819045i $$-0.694504\pi$$
−0.573729 + 0.819045i $$0.694504\pi$$
$$168$$ 0 0
$$169$$ 5.00000 0.384615
$$170$$ 0 0
$$171$$ 6.82843 0.522183
$$172$$ 0 0
$$173$$ 16.5858 1.26099 0.630497 0.776192i $$-0.282851\pi$$
0.630497 + 0.776192i $$0.282851\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 0 0
$$177$$ −8.48528 −0.637793
$$178$$ 0 0
$$179$$ 6.48528 0.484733 0.242366 0.970185i $$-0.422076\pi$$
0.242366 + 0.970185i $$0.422076\pi$$
$$180$$ 0 0
$$181$$ 7.07107 0.525588 0.262794 0.964852i $$-0.415356\pi$$
0.262794 + 0.964852i $$0.415356\pi$$
$$182$$ 0 0
$$183$$ 11.0711 0.818397
$$184$$ 0 0
$$185$$ 5.65685 0.415900
$$186$$ 0 0
$$187$$ 6.14214 0.449157
$$188$$ 0 0
$$189$$ 0 0
$$190$$ 0 0
$$191$$ −12.1421 −0.878574 −0.439287 0.898347i $$-0.644769\pi$$
−0.439287 + 0.898347i $$0.644769\pi$$
$$192$$ 0 0
$$193$$ 2.00000 0.143963 0.0719816 0.997406i $$-0.477068\pi$$
0.0719816 + 0.997406i $$0.477068\pi$$
$$194$$ 0 0
$$195$$ 14.4853 1.03731
$$196$$ 0 0
$$197$$ −2.68629 −0.191390 −0.0956952 0.995411i $$-0.530507\pi$$
−0.0956952 + 0.995411i $$0.530507\pi$$
$$198$$ 0 0
$$199$$ −5.65685 −0.401004 −0.200502 0.979693i $$-0.564257\pi$$
−0.200502 + 0.979693i $$0.564257\pi$$
$$200$$ 0 0
$$201$$ 11.3137 0.798007
$$202$$ 0 0
$$203$$ 0 0
$$204$$ 0 0
$$205$$ 34.9706 2.44245
$$206$$ 0 0
$$207$$ 4.82843 0.335599
$$208$$ 0 0
$$209$$ −5.65685 −0.391293
$$210$$ 0 0
$$211$$ −1.65685 −0.114063 −0.0570313 0.998372i $$-0.518163\pi$$
−0.0570313 + 0.998372i $$0.518163\pi$$
$$212$$ 0 0
$$213$$ −10.4853 −0.718440
$$214$$ 0 0
$$215$$ −38.6274 −2.63437
$$216$$ 0 0
$$217$$ 0 0
$$218$$ 0 0
$$219$$ 7.75736 0.524194
$$220$$ 0 0
$$221$$ −31.4558 −2.11595
$$222$$ 0 0
$$223$$ 13.6569 0.914531 0.457265 0.889330i $$-0.348829\pi$$
0.457265 + 0.889330i $$0.348829\pi$$
$$224$$ 0 0
$$225$$ 6.65685 0.443790
$$226$$ 0 0
$$227$$ −5.17157 −0.343249 −0.171625 0.985162i $$-0.554902\pi$$
−0.171625 + 0.985162i $$0.554902\pi$$
$$228$$ 0 0
$$229$$ 25.4142 1.67942 0.839709 0.543036i $$-0.182725\pi$$
0.839709 + 0.543036i $$0.182725\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ −14.8284 −0.971443 −0.485721 0.874114i $$-0.661443\pi$$
−0.485721 + 0.874114i $$0.661443\pi$$
$$234$$ 0 0
$$235$$ 15.3137 0.998956
$$236$$ 0 0
$$237$$ 13.6569 0.887108
$$238$$ 0 0
$$239$$ 12.8284 0.829802 0.414901 0.909867i $$-0.363816\pi$$
0.414901 + 0.909867i $$0.363816\pi$$
$$240$$ 0 0
$$241$$ −13.8995 −0.895345 −0.447673 0.894198i $$-0.647747\pi$$
−0.447673 + 0.894198i $$0.647747\pi$$
$$242$$ 0 0
$$243$$ −1.00000 −0.0641500
$$244$$ 0 0
$$245$$ 0 0
$$246$$ 0 0
$$247$$ 28.9706 1.84335
$$248$$ 0 0
$$249$$ 4.00000 0.253490
$$250$$ 0 0
$$251$$ −6.14214 −0.387688 −0.193844 0.981032i $$-0.562096\pi$$
−0.193844 + 0.981032i $$0.562096\pi$$
$$252$$ 0 0
$$253$$ −4.00000 −0.251478
$$254$$ 0 0
$$255$$ −25.3137 −1.58521
$$256$$ 0 0
$$257$$ 3.41421 0.212973 0.106486 0.994314i $$-0.466040\pi$$
0.106486 + 0.994314i $$0.466040\pi$$
$$258$$ 0 0
$$259$$ 0 0
$$260$$ 0 0
$$261$$ 2.82843 0.175075
$$262$$ 0 0
$$263$$ 20.1421 1.24202 0.621009 0.783804i $$-0.286723\pi$$
0.621009 + 0.783804i $$0.286723\pi$$
$$264$$ 0 0
$$265$$ 6.82843 0.419467
$$266$$ 0 0
$$267$$ 5.75736 0.352345
$$268$$ 0 0
$$269$$ 0.100505 0.00612790 0.00306395 0.999995i $$-0.499025\pi$$
0.00306395 + 0.999995i $$0.499025\pi$$
$$270$$ 0 0
$$271$$ 6.14214 0.373108 0.186554 0.982445i $$-0.440268\pi$$
0.186554 + 0.982445i $$0.440268\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ 0 0
$$275$$ −5.51472 −0.332550
$$276$$ 0 0
$$277$$ −28.6274 −1.72005 −0.860027 0.510248i $$-0.829554\pi$$
−0.860027 + 0.510248i $$0.829554\pi$$
$$278$$ 0 0
$$279$$ −2.82843 −0.169334
$$280$$ 0 0
$$281$$ 1.17157 0.0698902 0.0349451 0.999389i $$-0.488874\pi$$
0.0349451 + 0.999389i $$0.488874\pi$$
$$282$$ 0 0
$$283$$ −26.1421 −1.55399 −0.776994 0.629508i $$-0.783257\pi$$
−0.776994 + 0.629508i $$0.783257\pi$$
$$284$$ 0 0
$$285$$ 23.3137 1.38098
$$286$$ 0 0
$$287$$ 0 0
$$288$$ 0 0
$$289$$ 37.9706 2.23356
$$290$$ 0 0
$$291$$ −0.242641 −0.0142238
$$292$$ 0 0
$$293$$ 1.75736 0.102666 0.0513330 0.998682i $$-0.483653\pi$$
0.0513330 + 0.998682i $$0.483653\pi$$
$$294$$ 0 0
$$295$$ −28.9706 −1.68673
$$296$$ 0 0
$$297$$ 0.828427 0.0480702
$$298$$ 0 0
$$299$$ 20.4853 1.18469
$$300$$ 0 0
$$301$$ 0 0
$$302$$ 0 0
$$303$$ −10.7279 −0.616303
$$304$$ 0 0
$$305$$ 37.7990 2.16436
$$306$$ 0 0
$$307$$ −11.5147 −0.657180 −0.328590 0.944473i $$-0.606573\pi$$
−0.328590 + 0.944473i $$0.606573\pi$$
$$308$$ 0 0
$$309$$ −14.1421 −0.804518
$$310$$ 0 0
$$311$$ 23.7990 1.34952 0.674758 0.738039i $$-0.264248\pi$$
0.674758 + 0.738039i $$0.264248\pi$$
$$312$$ 0 0
$$313$$ −28.7279 −1.62380 −0.811899 0.583798i $$-0.801566\pi$$
−0.811899 + 0.583798i $$0.801566\pi$$
$$314$$ 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
$$319$$ −2.34315 −0.131191
$$320$$ 0 0
$$321$$ 12.8284 0.716013
$$322$$ 0 0
$$323$$ −50.6274 −2.81698
$$324$$ 0 0
$$325$$ 28.2426 1.56662
$$326$$ 0 0
$$327$$ 3.31371 0.183248
$$328$$ 0 0
$$329$$ 0 0
$$330$$ 0 0
$$331$$ −7.31371 −0.401998 −0.200999 0.979591i $$-0.564419\pi$$
−0.200999 + 0.979591i $$0.564419\pi$$
$$332$$ 0 0
$$333$$ −1.65685 −0.0907951
$$334$$ 0 0
$$335$$ 38.6274 2.11044
$$336$$ 0 0
$$337$$ 10.3431 0.563427 0.281714 0.959499i $$-0.409097\pi$$
0.281714 + 0.959499i $$0.409097\pi$$
$$338$$ 0 0
$$339$$ 10.0000 0.543125
$$340$$ 0 0
$$341$$ 2.34315 0.126888
$$342$$ 0 0
$$343$$ 0 0
$$344$$ 0 0
$$345$$ 16.4853 0.887538
$$346$$ 0 0
$$347$$ 2.48528 0.133417 0.0667084 0.997773i $$-0.478750\pi$$
0.0667084 + 0.997773i $$0.478750\pi$$
$$348$$ 0 0
$$349$$ −10.5858 −0.566644 −0.283322 0.959025i $$-0.591437\pi$$
−0.283322 + 0.959025i $$0.591437\pi$$
$$350$$ 0 0
$$351$$ −4.24264 −0.226455
$$352$$ 0 0
$$353$$ 27.6985 1.47424 0.737121 0.675761i $$-0.236185\pi$$
0.737121 + 0.675761i $$0.236185\pi$$
$$354$$ 0 0
$$355$$ −35.7990 −1.90001
$$356$$ 0 0
$$357$$ 0 0
$$358$$ 0 0
$$359$$ −28.8284 −1.52151 −0.760753 0.649041i $$-0.775170\pi$$
−0.760753 + 0.649041i $$0.775170\pi$$
$$360$$ 0 0
$$361$$ 27.6274 1.45407
$$362$$ 0 0
$$363$$ 10.3137 0.541329
$$364$$ 0 0
$$365$$ 26.4853 1.38630
$$366$$ 0 0
$$367$$ 4.68629 0.244622 0.122311 0.992492i $$-0.460969\pi$$
0.122311 + 0.992492i $$0.460969\pi$$
$$368$$ 0 0
$$369$$ −10.2426 −0.533211
$$370$$ 0 0
$$371$$ 0 0
$$372$$ 0 0
$$373$$ 5.31371 0.275133 0.137567 0.990493i $$-0.456072\pi$$
0.137567 + 0.990493i $$0.456072\pi$$
$$374$$ 0 0
$$375$$ 5.65685 0.292119
$$376$$ 0 0
$$377$$ 12.0000 0.618031
$$378$$ 0 0
$$379$$ 23.3137 1.19754 0.598772 0.800919i $$-0.295655\pi$$
0.598772 + 0.800919i $$0.295655\pi$$
$$380$$ 0 0
$$381$$ 20.0000 1.02463
$$382$$ 0 0
$$383$$ −8.97056 −0.458374 −0.229187 0.973382i $$-0.573607\pi$$
−0.229187 + 0.973382i $$0.573607\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ 11.3137 0.575108
$$388$$ 0 0
$$389$$ −35.7990 −1.81508 −0.907540 0.419965i $$-0.862042\pi$$
−0.907540 + 0.419965i $$0.862042\pi$$
$$390$$ 0 0
$$391$$ −35.7990 −1.81043
$$392$$ 0 0
$$393$$ 4.00000 0.201773
$$394$$ 0 0
$$395$$ 46.6274 2.34608
$$396$$ 0 0
$$397$$ 16.2426 0.815195 0.407597 0.913162i $$-0.366367\pi$$
0.407597 + 0.913162i $$0.366367\pi$$
$$398$$ 0 0
$$399$$ 0 0
$$400$$ 0 0
$$401$$ 1.17157 0.0585056 0.0292528 0.999572i $$-0.490687\pi$$
0.0292528 + 0.999572i $$0.490687\pi$$
$$402$$ 0 0
$$403$$ −12.0000 −0.597763
$$404$$ 0 0
$$405$$ −3.41421 −0.169654
$$406$$ 0 0
$$407$$ 1.37258 0.0680364
$$408$$ 0 0
$$409$$ 8.24264 0.407572 0.203786 0.979015i $$-0.434675\pi$$
0.203786 + 0.979015i $$0.434675\pi$$
$$410$$ 0 0
$$411$$ 9.17157 0.452400
$$412$$ 0 0
$$413$$ 0 0
$$414$$ 0 0
$$415$$ 13.6569 0.670389
$$416$$ 0 0
$$417$$ 20.9706 1.02693
$$418$$ 0 0
$$419$$ 7.51472 0.367118 0.183559 0.983009i $$-0.441238\pi$$
0.183559 + 0.983009i $$0.441238\pi$$
$$420$$ 0 0
$$421$$ −25.3137 −1.23371 −0.616857 0.787075i $$-0.711594\pi$$
−0.616857 + 0.787075i $$0.711594\pi$$
$$422$$ 0 0
$$423$$ −4.48528 −0.218082
$$424$$ 0 0
$$425$$ −49.3553 −2.39409
$$426$$ 0 0
$$427$$ 0 0
$$428$$ 0 0
$$429$$ 3.51472 0.169692
$$430$$ 0 0
$$431$$ 8.14214 0.392193 0.196096 0.980585i $$-0.437173\pi$$
0.196096 + 0.980585i $$0.437173\pi$$
$$432$$ 0 0
$$433$$ −0.928932 −0.0446416 −0.0223208 0.999751i $$-0.507106\pi$$
−0.0223208 + 0.999751i $$0.507106\pi$$
$$434$$ 0 0
$$435$$ 9.65685 0.463011
$$436$$ 0 0
$$437$$ 32.9706 1.57720
$$438$$ 0 0
$$439$$ 12.6863 0.605484 0.302742 0.953073i $$-0.402098\pi$$
0.302742 + 0.953073i $$0.402098\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 0 0
$$443$$ 11.1716 0.530777 0.265389 0.964141i $$-0.414500\pi$$
0.265389 + 0.964141i $$0.414500\pi$$
$$444$$ 0 0
$$445$$ 19.6569 0.931824
$$446$$ 0 0
$$447$$ −1.31371 −0.0621363
$$448$$ 0 0
$$449$$ −16.6274 −0.784696 −0.392348 0.919817i $$-0.628337\pi$$
−0.392348 + 0.919817i $$0.628337\pi$$
$$450$$ 0 0
$$451$$ 8.48528 0.399556
$$452$$ 0 0
$$453$$ −9.65685 −0.453719
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ −24.6274 −1.15202 −0.576011 0.817442i $$-0.695391\pi$$
−0.576011 + 0.817442i $$0.695391\pi$$
$$458$$ 0 0
$$459$$ 7.41421 0.346066
$$460$$ 0 0
$$461$$ −16.3848 −0.763115 −0.381558 0.924345i $$-0.624612\pi$$
−0.381558 + 0.924345i $$0.624612\pi$$
$$462$$ 0 0
$$463$$ −1.65685 −0.0770005 −0.0385003 0.999259i $$-0.512258\pi$$
−0.0385003 + 0.999259i $$0.512258\pi$$
$$464$$ 0 0
$$465$$ −9.65685 −0.447826
$$466$$ 0 0
$$467$$ −18.8284 −0.871276 −0.435638 0.900122i $$-0.643477\pi$$
−0.435638 + 0.900122i $$0.643477\pi$$
$$468$$ 0 0
$$469$$ 0 0
$$470$$ 0 0
$$471$$ 0.242641 0.0111803
$$472$$ 0 0
$$473$$ −9.37258 −0.430952
$$474$$ 0 0
$$475$$ 45.4558 2.08566
$$476$$ 0 0
$$477$$ −2.00000 −0.0915737
$$478$$ 0 0
$$479$$ 22.8284 1.04306 0.521529 0.853234i $$-0.325362\pi$$
0.521529 + 0.853234i $$0.325362\pi$$
$$480$$ 0 0
$$481$$ −7.02944 −0.320515
$$482$$ 0 0
$$483$$ 0 0
$$484$$ 0 0
$$485$$ −0.828427 −0.0376169
$$486$$ 0 0
$$487$$ −4.97056 −0.225238 −0.112619 0.993638i $$-0.535924\pi$$
−0.112619 + 0.993638i $$0.535924\pi$$
$$488$$ 0 0
$$489$$ 5.65685 0.255812
$$490$$ 0 0
$$491$$ −19.1716 −0.865201 −0.432600 0.901586i $$-0.642404\pi$$
−0.432600 + 0.901586i $$0.642404\pi$$
$$492$$ 0 0
$$493$$ −20.9706 −0.944467
$$494$$ 0 0
$$495$$ 2.82843 0.127128
$$496$$ 0 0
$$497$$ 0 0
$$498$$ 0 0
$$499$$ 24.9706 1.11784 0.558918 0.829223i $$-0.311217\pi$$
0.558918 + 0.829223i $$0.311217\pi$$
$$500$$ 0 0
$$501$$ 14.8284 0.662485
$$502$$ 0 0
$$503$$ −19.3137 −0.861156 −0.430578 0.902553i $$-0.641690\pi$$
−0.430578 + 0.902553i $$0.641690\pi$$
$$504$$ 0 0
$$505$$ −36.6274 −1.62990
$$506$$ 0 0
$$507$$ −5.00000 −0.222058
$$508$$ 0 0
$$509$$ 36.3848 1.61273 0.806363 0.591420i $$-0.201433\pi$$
0.806363 + 0.591420i $$0.201433\pi$$
$$510$$ 0 0
$$511$$ 0 0
$$512$$ 0 0
$$513$$ −6.82843 −0.301482
$$514$$ 0 0
$$515$$ −48.2843 −2.12766
$$516$$ 0 0
$$517$$ 3.71573 0.163418
$$518$$ 0 0
$$519$$ −16.5858 −0.728035
$$520$$ 0 0
$$521$$ −14.2426 −0.623981 −0.311991 0.950085i $$-0.600996\pi$$
−0.311991 + 0.950085i $$0.600996\pi$$
$$522$$ 0 0
$$523$$ −4.97056 −0.217348 −0.108674 0.994077i $$-0.534660\pi$$
−0.108674 + 0.994077i $$0.534660\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ 20.9706 0.913492
$$528$$ 0 0
$$529$$ 0.313708 0.0136395
$$530$$ 0 0
$$531$$ 8.48528 0.368230
$$532$$ 0 0
$$533$$ −43.4558 −1.88228
$$534$$ 0 0
$$535$$ 43.7990 1.89360
$$536$$ 0 0
$$537$$ −6.48528 −0.279861
$$538$$ 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
$$547$$ 3.02944 0.129529 0.0647647 0.997901i $$-0.479370\pi$$
0.0647647 + 0.997901i $$0.479370\pi$$
$$548$$ 0 0
$$549$$ −11.0711 −0.472502
$$550$$ 0 0
$$551$$ 19.3137 0.822792
$$552$$ 0 0
$$553$$ 0 0
$$554$$ 0 0
$$555$$ −5.65685 −0.240120
$$556$$ 0 0
$$557$$ 32.6274 1.38247 0.691234 0.722631i $$-0.257067\pi$$
0.691234 + 0.722631i $$0.257067\pi$$
$$558$$ 0 0
$$559$$ 48.0000 2.03018
$$560$$ 0 0
$$561$$ −6.14214 −0.259321
$$562$$ 0 0
$$563$$ 1.85786 0.0782996 0.0391498 0.999233i $$-0.487535\pi$$
0.0391498 + 0.999233i $$0.487535\pi$$
$$564$$ 0 0
$$565$$ 34.1421 1.43637
$$566$$ 0 0
$$567$$ 0 0
$$568$$ 0 0
$$569$$ −5.85786 −0.245574 −0.122787 0.992433i $$-0.539183\pi$$
−0.122787 + 0.992433i $$0.539183\pi$$
$$570$$ 0 0
$$571$$ 41.6569 1.74329 0.871643 0.490142i $$-0.163055\pi$$
0.871643 + 0.490142i $$0.163055\pi$$
$$572$$ 0 0
$$573$$ 12.1421 0.507245
$$574$$ 0 0
$$575$$ 32.1421 1.34042
$$576$$ 0 0
$$577$$ −13.2132 −0.550073 −0.275036 0.961434i $$-0.588690\pi$$
−0.275036 + 0.961434i $$0.588690\pi$$
$$578$$ 0 0
$$579$$ −2.00000 −0.0831172
$$580$$ 0 0
$$581$$ 0 0
$$582$$ 0 0
$$583$$ 1.65685 0.0686199
$$584$$ 0 0
$$585$$ −14.4853 −0.598893
$$586$$ 0 0
$$587$$ −35.7990 −1.47758 −0.738791 0.673934i $$-0.764603\pi$$
−0.738791 + 0.673934i $$0.764603\pi$$
$$588$$ 0 0
$$589$$ −19.3137 −0.795807
$$590$$ 0 0
$$591$$ 2.68629 0.110499
$$592$$ 0 0
$$593$$ −47.4142 −1.94707 −0.973534 0.228541i $$-0.926604\pi$$
−0.973534 + 0.228541i $$0.926604\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 0 0
$$597$$ 5.65685 0.231520
$$598$$ 0 0
$$599$$ −23.4558 −0.958380 −0.479190 0.877711i $$-0.659069\pi$$
−0.479190 + 0.877711i $$0.659069\pi$$
$$600$$ 0 0
$$601$$ 12.2426 0.499388 0.249694 0.968325i $$-0.419670\pi$$
0.249694 + 0.968325i $$0.419670\pi$$
$$602$$ 0 0
$$603$$ −11.3137 −0.460730
$$604$$ 0 0
$$605$$ 35.2132 1.43162
$$606$$ 0 0
$$607$$ 24.9706 1.01352 0.506762 0.862086i $$-0.330842\pi$$
0.506762 + 0.862086i $$0.330842\pi$$
$$608$$ 0 0
$$609$$ 0 0
$$610$$ 0 0
$$611$$ −19.0294 −0.769849
$$612$$ 0 0
$$613$$ −10.3431 −0.417756 −0.208878 0.977942i $$-0.566981\pi$$
−0.208878 + 0.977942i $$0.566981\pi$$
$$614$$ 0 0
$$615$$ −34.9706 −1.41015
$$616$$ 0 0
$$617$$ −4.48528 −0.180571 −0.0902853 0.995916i $$-0.528778\pi$$
−0.0902853 + 0.995916i $$0.528778\pi$$
$$618$$ 0 0
$$619$$ 33.6569 1.35278 0.676392 0.736542i $$-0.263543\pi$$
0.676392 + 0.736542i $$0.263543\pi$$
$$620$$ 0 0
$$621$$ −4.82843 −0.193758
$$622$$ 0 0
$$623$$ 0 0
$$624$$ 0 0
$$625$$ −13.9706 −0.558823
$$626$$ 0 0
$$627$$ 5.65685 0.225913
$$628$$ 0 0
$$629$$ 12.2843 0.489806
$$630$$ 0 0
$$631$$ 3.02944 0.120600 0.0603000 0.998180i $$-0.480794\pi$$
0.0603000 + 0.998180i $$0.480794\pi$$
$$632$$ 0 0
$$633$$ 1.65685 0.0658540
$$634$$ 0 0
$$635$$ 68.2843 2.70978
$$636$$ 0 0
$$637$$ 0 0
$$638$$ 0 0
$$639$$ 10.4853 0.414791
$$640$$ 0 0
$$641$$ 35.1127 1.38687 0.693434 0.720520i $$-0.256097\pi$$
0.693434 + 0.720520i $$0.256097\pi$$
$$642$$ 0 0
$$643$$ 31.7990 1.25403 0.627015 0.779007i $$-0.284277\pi$$
0.627015 + 0.779007i $$0.284277\pi$$
$$644$$ 0 0
$$645$$ 38.6274 1.52095
$$646$$ 0 0
$$647$$ −0.201010 −0.00790252 −0.00395126 0.999992i $$-0.501258\pi$$
−0.00395126 + 0.999992i $$0.501258\pi$$
$$648$$ 0 0
$$649$$ −7.02944 −0.275930
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 0 0
$$653$$ 46.1421 1.80568 0.902841 0.429975i $$-0.141478\pi$$
0.902841 + 0.429975i $$0.141478\pi$$
$$654$$ 0 0
$$655$$ 13.6569 0.533617
$$656$$ 0 0
$$657$$ −7.75736 −0.302643
$$658$$ 0 0
$$659$$ −21.5147 −0.838094 −0.419047 0.907964i $$-0.637636\pi$$
−0.419047 + 0.907964i $$0.637636\pi$$
$$660$$ 0 0
$$661$$ −4.92893 −0.191713 −0.0958566 0.995395i $$-0.530559\pi$$
−0.0958566 + 0.995395i $$0.530559\pi$$
$$662$$ 0 0
$$663$$ 31.4558 1.22164
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 0 0
$$667$$ 13.6569 0.528796
$$668$$ 0 0
$$669$$ −13.6569 −0.528004
$$670$$ 0 0
$$671$$ 9.17157 0.354065
$$672$$ 0 0
$$673$$ −23.3137 −0.898677 −0.449339 0.893361i $$-0.648340\pi$$
−0.449339 + 0.893361i $$0.648340\pi$$
$$674$$ 0 0
$$675$$ −6.65685 −0.256222
$$676$$ 0 0
$$677$$ −27.6985 −1.06454 −0.532270 0.846575i $$-0.678661\pi$$
−0.532270 + 0.846575i $$0.678661\pi$$
$$678$$ 0 0
$$679$$ 0 0
$$680$$ 0 0
$$681$$ 5.17157 0.198175
$$682$$ 0 0
$$683$$ 1.79899 0.0688364 0.0344182 0.999408i $$-0.489042\pi$$
0.0344182 + 0.999408i $$0.489042\pi$$
$$684$$ 0 0
$$685$$ 31.3137 1.19644
$$686$$ 0 0
$$687$$ −25.4142 −0.969613
$$688$$ 0 0
$$689$$ −8.48528 −0.323263
$$690$$ 0 0
$$691$$ 8.68629 0.330442 0.165221 0.986257i $$-0.447166\pi$$
0.165221 + 0.986257i $$0.447166\pi$$
$$692$$ 0 0
$$693$$ 0 0
$$694$$ 0 0
$$695$$ 71.5980 2.71587
$$696$$ 0 0
$$697$$ 75.9411 2.87648
$$698$$ 0 0
$$699$$ 14.8284 0.560863
$$700$$ 0 0
$$701$$ −6.14214 −0.231985 −0.115993 0.993250i $$-0.537005\pi$$
−0.115993 + 0.993250i $$0.537005\pi$$
$$702$$ 0 0
$$703$$ −11.3137 −0.426705
$$704$$ 0 0
$$705$$ −15.3137 −0.576748
$$706$$ 0 0
$$707$$ 0 0
$$708$$ 0 0
$$709$$ 6.62742 0.248898 0.124449 0.992226i $$-0.460284\pi$$
0.124449 + 0.992226i $$0.460284\pi$$
$$710$$ 0 0
$$711$$ −13.6569 −0.512172
$$712$$ 0 0
$$713$$ −13.6569 −0.511453
$$714$$ 0 0
$$715$$ 12.0000 0.448775
$$716$$ 0 0
$$717$$ −12.8284 −0.479086
$$718$$ 0 0
$$719$$ −6.62742 −0.247161 −0.123580 0.992335i $$-0.539438\pi$$
−0.123580 + 0.992335i $$0.539438\pi$$
$$720$$ 0 0
$$721$$ 0 0
$$722$$ 0 0
$$723$$ 13.8995 0.516928
$$724$$ 0 0
$$725$$ 18.8284 0.699270
$$726$$ 0 0
$$727$$ −9.85786 −0.365608 −0.182804 0.983149i $$-0.558517\pi$$
−0.182804 + 0.983149i $$0.558517\pi$$
$$728$$ 0 0
$$729$$ 1.00000 0.0370370
$$730$$ 0 0
$$731$$ −83.8823 −3.10250
$$732$$ 0 0
$$733$$ 8.04163 0.297024 0.148512 0.988911i $$-0.452552\pi$$
0.148512 + 0.988911i $$0.452552\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ 9.37258 0.345244
$$738$$ 0 0
$$739$$ −32.9706 −1.21284 −0.606421 0.795144i $$-0.707395\pi$$
−0.606421 + 0.795144i $$0.707395\pi$$
$$740$$ 0 0
$$741$$ −28.9706 −1.06426
$$742$$ 0 0
$$743$$ 4.82843 0.177138 0.0885689 0.996070i $$-0.471771\pi$$
0.0885689 + 0.996070i $$0.471771\pi$$
$$744$$ 0 0
$$745$$ −4.48528 −0.164328
$$746$$ 0 0
$$747$$ −4.00000 −0.146352
$$748$$ 0 0
$$749$$ 0 0
$$750$$ 0 0
$$751$$ −20.2843 −0.740184 −0.370092 0.928995i $$-0.620674\pi$$
−0.370092 + 0.928995i $$0.620674\pi$$
$$752$$ 0 0
$$753$$ 6.14214 0.223832
$$754$$ 0 0
$$755$$ −32.9706 −1.19992
$$756$$ 0 0
$$757$$ 41.9411 1.52438 0.762188 0.647356i $$-0.224125\pi$$
0.762188 + 0.647356i $$0.224125\pi$$
$$758$$ 0 0
$$759$$ 4.00000 0.145191
$$760$$ 0 0
$$761$$ 31.8995 1.15636 0.578178 0.815911i $$-0.303764\pi$$
0.578178 + 0.815911i $$0.303764\pi$$
$$762$$ 0 0
$$763$$ 0 0
$$764$$ 0 0
$$765$$ 25.3137 0.915219
$$766$$ 0 0
$$767$$ 36.0000 1.29988
$$768$$ 0 0
$$769$$ −34.8701 −1.25745 −0.628723 0.777629i $$-0.716422\pi$$
−0.628723 + 0.777629i $$0.716422\pi$$
$$770$$ 0 0
$$771$$ −3.41421 −0.122960
$$772$$ 0 0
$$773$$ −33.3553 −1.19971 −0.599854 0.800109i $$-0.704775\pi$$
−0.599854 + 0.800109i $$0.704775\pi$$
$$774$$ 0 0
$$775$$ −18.8284 −0.676337
$$776$$ 0 0
$$777$$ 0 0
$$778$$ 0 0
$$779$$ −69.9411 −2.50590
$$780$$ 0 0
$$781$$ −8.68629 −0.310820
$$782$$ 0 0
$$783$$ −2.82843 −0.101080
$$784$$ 0 0
$$785$$ 0.828427 0.0295678
$$786$$ 0 0
$$787$$ 15.3137 0.545875 0.272937 0.962032i $$-0.412005\pi$$
0.272937 + 0.962032i $$0.412005\pi$$
$$788$$ 0 0
$$789$$ −20.1421 −0.717079
$$790$$ 0 0
$$791$$ 0 0
$$792$$ 0 0
$$793$$ −46.9706 −1.66797
$$794$$ 0 0
$$795$$ −6.82843 −0.242179
$$796$$ 0 0
$$797$$ 39.4142 1.39612 0.698062 0.716038i $$-0.254046\pi$$
0.698062 + 0.716038i $$0.254046\pi$$
$$798$$ 0 0
$$799$$ 33.2548 1.17647
$$800$$ 0 0
$$801$$ −5.75736 −0.203426
$$802$$ 0 0
$$803$$ 6.42641 0.226783
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 0 0
$$807$$ −0.100505 −0.00353795
$$808$$ 0 0
$$809$$ −4.62742 −0.162691 −0.0813457 0.996686i $$-0.525922\pi$$
−0.0813457 + 0.996686i $$0.525922\pi$$
$$810$$ 0 0
$$811$$ 25.6569 0.900934 0.450467 0.892793i $$-0.351257\pi$$
0.450467 + 0.892793i $$0.351257\pi$$
$$812$$ 0 0
$$813$$ −6.14214 −0.215414
$$814$$ 0 0
$$815$$ 19.3137 0.676530
$$816$$ 0 0
$$817$$ 77.2548 2.70280
$$818$$ 0 0
$$819$$ 0 0
$$820$$ 0 0
$$821$$ −10.6863 −0.372954 −0.186477 0.982459i $$-0.559707\pi$$
−0.186477 + 0.982459i $$0.559707\pi$$
$$822$$ 0 0
$$823$$ 24.9706 0.870419 0.435210 0.900329i $$-0.356674\pi$$
0.435210 + 0.900329i $$0.356674\pi$$
$$824$$ 0 0
$$825$$ 5.51472 0.191998
$$826$$ 0 0
$$827$$ −40.1421 −1.39588 −0.697939 0.716157i $$-0.745900\pi$$
−0.697939 + 0.716157i $$0.745900\pi$$
$$828$$ 0 0
$$829$$ −34.3848 −1.19423 −0.597116 0.802155i $$-0.703687\pi$$
−0.597116 + 0.802155i $$0.703687\pi$$
$$830$$ 0 0
$$831$$ 28.6274 0.993074
$$832$$ 0 0
$$833$$ 0 0
$$834$$ 0 0
$$835$$ 50.6274 1.75203
$$836$$ 0 0
$$837$$ 2.82843 0.0977647
$$838$$ 0 0
$$839$$ 23.7990 0.821632 0.410816 0.911718i $$-0.365244\pi$$
0.410816 + 0.911718i $$0.365244\pi$$
$$840$$ 0 0
$$841$$ −21.0000 −0.724138
$$842$$ 0 0
$$843$$ −1.17157 −0.0403511
$$844$$ 0 0
$$845$$ −17.0711 −0.587263
$$846$$ 0 0
$$847$$ 0 0
$$848$$ 0 0
$$849$$ 26.1421 0.897196
$$850$$ 0 0
$$851$$ −8.00000 −0.274236
$$852$$ 0 0
$$853$$ −4.92893 −0.168763 −0.0843817 0.996434i $$-0.526892\pi$$
−0.0843817 + 0.996434i $$0.526892\pi$$
$$854$$ 0 0
$$855$$ −23.3137 −0.797312
$$856$$ 0 0
$$857$$ 11.2132 0.383036 0.191518 0.981489i $$-0.438659\pi$$
0.191518 + 0.981489i $$0.438659\pi$$
$$858$$ 0 0
$$859$$ 2.54416 0.0868055 0.0434027 0.999058i $$-0.486180\pi$$
0.0434027 + 0.999058i $$0.486180\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 0 0
$$863$$ −54.0833 −1.84102 −0.920508 0.390724i $$-0.872225\pi$$
−0.920508 + 0.390724i $$0.872225\pi$$
$$864$$ 0 0
$$865$$ −56.6274 −1.92539
$$866$$ 0 0
$$867$$ −37.9706 −1.28955
$$868$$ 0 0
$$869$$ 11.3137 0.383791
$$870$$ 0 0
$$871$$ −48.0000 −1.62642
$$872$$ 0 0
$$873$$ 0.242641 0.00821214
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 0 0
$$877$$ −28.2843 −0.955092 −0.477546 0.878607i $$-0.658474\pi$$
−0.477546 + 0.878607i $$0.658474\pi$$
$$878$$ 0 0
$$879$$ −1.75736 −0.0592743
$$880$$ 0 0
$$881$$ −30.0416 −1.01213 −0.506064 0.862496i $$-0.668900\pi$$
−0.506064 + 0.862496i $$0.668900\pi$$
$$882$$ 0 0
$$883$$ 10.3431 0.348075 0.174037 0.984739i $$-0.444319\pi$$
0.174037 + 0.984739i $$0.444319\pi$$
$$884$$ 0 0
$$885$$ 28.9706 0.973835
$$886$$ 0 0
$$887$$ −39.7990 −1.33632 −0.668160 0.744018i $$-0.732918\pi$$
−0.668160 + 0.744018i $$0.732918\pi$$
$$888$$ 0 0
$$889$$ 0 0
$$890$$ 0 0
$$891$$ −0.828427 −0.0277534
$$892$$ 0 0
$$893$$ −30.6274 −1.02491
$$894$$ 0 0
$$895$$ −22.1421 −0.740130
$$896$$ 0 0
$$897$$ −20.4853 −0.683984
$$898$$ 0 0
$$899$$ −8.00000 −0.266815
$$900$$ 0 0
$$901$$ 14.8284 0.494007
$$902$$ 0 0
$$903$$ 0 0
$$904$$ 0 0
$$905$$ −24.1421 −0.802512
$$906$$ 0 0
$$907$$ −53.6569 −1.78165 −0.890823 0.454350i $$-0.849872\pi$$
−0.890823 + 0.454350i $$0.849872\pi$$
$$908$$ 0 0
$$909$$ 10.7279 0.355823
$$910$$ 0 0
$$911$$ −6.48528 −0.214867 −0.107433 0.994212i $$-0.534263\pi$$
−0.107433 + 0.994212i $$0.534263\pi$$
$$912$$ 0 0
$$913$$ 3.31371 0.109668
$$914$$ 0 0
$$915$$ −37.7990 −1.24960
$$916$$ 0 0
$$917$$ 0 0
$$918$$ 0 0
$$919$$ −31.3137 −1.03294 −0.516472 0.856304i $$-0.672755\pi$$
−0.516472 + 0.856304i $$0.672755\pi$$
$$920$$ 0 0
$$921$$ 11.5147 0.379423
$$922$$ 0 0
$$923$$ 44.4853 1.46425
$$924$$ 0 0
$$925$$ −11.0294 −0.362646
$$926$$ 0 0
$$927$$ 14.1421 0.464489
$$928$$ 0 0
$$929$$ 15.8995 0.521646 0.260823 0.965387i $$-0.416006\pi$$
0.260823 + 0.965387i $$0.416006\pi$$
$$930$$ 0 0
$$931$$ 0 0
$$932$$ 0 0
$$933$$ −23.7990 −0.779144
$$934$$ 0 0
$$935$$ −20.9706 −0.685811
$$936$$ 0 0
$$937$$ −51.3553 −1.67771 −0.838853 0.544358i $$-0.816773\pi$$
−0.838853 + 0.544358i $$0.816773\pi$$
$$938$$ 0 0
$$939$$ 28.7279 0.937500
$$940$$ 0 0
$$941$$ 14.7279 0.480117 0.240058 0.970758i $$-0.422833\pi$$
0.240058 + 0.970758i $$0.422833\pi$$
$$942$$ 0 0
$$943$$ −49.4558 −1.61050
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ 58.7696 1.90975 0.954877 0.297002i $$-0.0959867\pi$$
0.954877 + 0.297002i $$0.0959867\pi$$
$$948$$ 0 0
$$949$$ −32.9117 −1.06836
$$950$$ 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
$$955$$ 41.4558 1.34148
$$956$$ 0 0
$$957$$ 2.34315 0.0757431
$$958$$ 0 0
$$959$$ 0 0
$$960$$ 0 0
$$961$$ −23.0000 −0.741935
$$962$$ 0 0
$$963$$ −12.8284 −0.413390
$$964$$ 0 0
$$965$$ −6.82843 −0.219815
$$966$$ 0 0
$$967$$ 31.3137 1.00698 0.503490 0.864001i $$-0.332049\pi$$
0.503490 + 0.864001i $$0.332049\pi$$
$$968$$ 0 0
$$969$$ 50.6274 1.62639
$$970$$ 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
$$975$$ −28.2426 −0.904488
$$976$$ 0 0
$$977$$ −7.79899 −0.249512 −0.124756 0.992187i $$-0.539815\pi$$
−0.124756 + 0.992187i $$0.539815\pi$$
$$978$$ 0 0
$$979$$ 4.76955 0.152436
$$980$$ 0 0
$$981$$ −3.31371 −0.105799
$$982$$ 0 0
$$983$$ 1.37258 0.0437786 0.0218893 0.999760i $$-0.493032\pi$$
0.0218893 + 0.999760i $$0.493032\pi$$
$$984$$ 0 0
$$985$$ 9.17157 0.292231
$$986$$ 0 0
$$987$$ 0 0
$$988$$ 0 0
$$989$$ 54.6274 1.73705
$$990$$ 0 0
$$991$$ 18.6274 0.591719 0.295860 0.955231i $$-0.404394\pi$$
0.295860 + 0.955231i $$0.404394\pi$$
$$992$$ 0 0
$$993$$ 7.31371 0.232094
$$994$$ 0 0
$$995$$ 19.3137 0.612286
$$996$$ 0 0
$$997$$ 54.3848 1.72238 0.861192 0.508281i $$-0.169719\pi$$
0.861192 + 0.508281i $$0.169719\pi$$
$$998$$ 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 2352.2.a.z.1.1 2
3.2 odd 2 7056.2.a.cw.1.2 2
4.3 odd 2 1176.2.a.m.1.1 yes 2
7.2 even 3 2352.2.q.bg.1537.2 4
7.3 odd 6 2352.2.q.ba.961.1 4
7.4 even 3 2352.2.q.bg.961.2 4
7.5 odd 6 2352.2.q.ba.1537.1 4
7.6 odd 2 2352.2.a.bg.1.2 2
8.3 odd 2 9408.2.a.dr.1.2 2
8.5 even 2 9408.2.a.ed.1.2 2
12.11 even 2 3528.2.a.bm.1.2 2
21.20 even 2 7056.2.a.ce.1.1 2
28.3 even 6 1176.2.q.n.961.1 4
28.11 odd 6 1176.2.q.m.961.2 4
28.19 even 6 1176.2.q.n.361.1 4
28.23 odd 6 1176.2.q.m.361.2 4
28.27 even 2 1176.2.a.l.1.2 2
56.13 odd 2 9408.2.a.dh.1.1 2
56.27 even 2 9408.2.a.dv.1.1 2
84.11 even 6 3528.2.s.bc.3313.1 4
84.23 even 6 3528.2.s.bc.361.1 4
84.47 odd 6 3528.2.s.bl.361.2 4
84.59 odd 6 3528.2.s.bl.3313.2 4
84.83 odd 2 3528.2.a.bc.1.1 2

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