# Properties

 Label 4416.2.a.bs.1.2 Level $4416$ Weight $2$ Character 4416.1 Self dual yes Analytic conductor $35.262$ Analytic rank $0$ Dimension $3$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$4416 = 2^{6} \cdot 3 \cdot 23$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 4416.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$35.2619375326$$ Analytic rank: $$0$$ Dimension: $$3$$ Coefficient field: 3.3.148.1 Defining polynomial: $$x^{3} - x^{2} - 3 x + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: no (minimal twist has level 552) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.2 Root $$2.17009$$ of defining polynomial Character $$\chi$$ $$=$$ 4416.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+1.00000 q^{3} -1.07838 q^{5} -4.34017 q^{7} +1.00000 q^{9} +O(q^{10})$$ $$q+1.00000 q^{3} -1.07838 q^{5} -4.34017 q^{7} +1.00000 q^{9} -3.41855 q^{11} -2.00000 q^{13} -1.07838 q^{15} -2.34017 q^{17} +0.921622 q^{19} -4.34017 q^{21} +1.00000 q^{23} -3.83710 q^{25} +1.00000 q^{27} -6.68035 q^{29} +8.68035 q^{31} -3.41855 q^{33} +4.68035 q^{35} -7.26180 q^{37} -2.00000 q^{39} +8.83710 q^{41} +7.75872 q^{43} -1.07838 q^{45} +12.6803 q^{47} +11.8371 q^{49} -2.34017 q^{51} +11.9155 q^{53} +3.68649 q^{55} +0.921622 q^{57} -1.84324 q^{59} -4.73820 q^{61} -4.34017 q^{63} +2.15676 q^{65} -7.75872 q^{67} +1.00000 q^{69} -2.52359 q^{71} +2.00000 q^{73} -3.83710 q^{75} +14.8371 q^{77} +12.3402 q^{79} +1.00000 q^{81} -0.894960 q^{83} +2.52359 q^{85} -6.68035 q^{87} -8.49693 q^{89} +8.68035 q^{91} +8.68035 q^{93} -0.993857 q^{95} +8.15676 q^{97} -3.41855 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3q + 3q^{3} - 2q^{7} + 3q^{9} + O(q^{10})$$ $$3q + 3q^{3} - 2q^{7} + 3q^{9} + 4q^{11} - 6q^{13} + 4q^{17} + 6q^{19} - 2q^{21} + 3q^{23} + 17q^{25} + 3q^{27} + 2q^{29} + 4q^{31} + 4q^{33} - 8q^{35} - 14q^{37} - 6q^{39} - 2q^{41} - 2q^{43} + 16q^{47} + 7q^{49} + 4q^{51} + 4q^{53} + 24q^{55} + 6q^{57} - 12q^{59} - 22q^{61} - 2q^{63} + 2q^{67} + 3q^{69} + 8q^{71} + 6q^{73} + 17q^{75} + 16q^{77} + 26q^{79} + 3q^{81} - 4q^{83} - 8q^{85} + 2q^{87} - 8q^{89} + 4q^{91} + 4q^{93} + 32q^{95} + 18q^{97} + 4q^{99} + O(q^{100})$$

## 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
$$5$$ −1.07838 −0.482265 −0.241133 0.970492i $$-0.577519\pi$$
−0.241133 + 0.970492i $$0.577519\pi$$
$$6$$ 0 0
$$7$$ −4.34017 −1.64043 −0.820216 0.572055i $$-0.806147\pi$$
−0.820216 + 0.572055i $$0.806147\pi$$
$$8$$ 0 0
$$9$$ 1.00000 0.333333
$$10$$ 0 0
$$11$$ −3.41855 −1.03073 −0.515366 0.856970i $$-0.672344\pi$$
−0.515366 + 0.856970i $$0.672344\pi$$
$$12$$ 0 0
$$13$$ −2.00000 −0.554700 −0.277350 0.960769i $$-0.589456\pi$$
−0.277350 + 0.960769i $$0.589456\pi$$
$$14$$ 0 0
$$15$$ −1.07838 −0.278436
$$16$$ 0 0
$$17$$ −2.34017 −0.567575 −0.283788 0.958887i $$-0.591591\pi$$
−0.283788 + 0.958887i $$0.591591\pi$$
$$18$$ 0 0
$$19$$ 0.921622 0.211435 0.105717 0.994396i $$-0.466286\pi$$
0.105717 + 0.994396i $$0.466286\pi$$
$$20$$ 0 0
$$21$$ −4.34017 −0.947103
$$22$$ 0 0
$$23$$ 1.00000 0.208514
$$24$$ 0 0
$$25$$ −3.83710 −0.767420
$$26$$ 0 0
$$27$$ 1.00000 0.192450
$$28$$ 0 0
$$29$$ −6.68035 −1.24051 −0.620255 0.784401i $$-0.712971\pi$$
−0.620255 + 0.784401i $$0.712971\pi$$
$$30$$ 0 0
$$31$$ 8.68035 1.55904 0.779518 0.626380i $$-0.215464\pi$$
0.779518 + 0.626380i $$0.215464\pi$$
$$32$$ 0 0
$$33$$ −3.41855 −0.595093
$$34$$ 0 0
$$35$$ 4.68035 0.791123
$$36$$ 0 0
$$37$$ −7.26180 −1.19383 −0.596916 0.802304i $$-0.703607\pi$$
−0.596916 + 0.802304i $$0.703607\pi$$
$$38$$ 0 0
$$39$$ −2.00000 −0.320256
$$40$$ 0 0
$$41$$ 8.83710 1.38012 0.690062 0.723751i $$-0.257583\pi$$
0.690062 + 0.723751i $$0.257583\pi$$
$$42$$ 0 0
$$43$$ 7.75872 1.18319 0.591597 0.806234i $$-0.298498\pi$$
0.591597 + 0.806234i $$0.298498\pi$$
$$44$$ 0 0
$$45$$ −1.07838 −0.160755
$$46$$ 0 0
$$47$$ 12.6803 1.84962 0.924809 0.380431i $$-0.124224\pi$$
0.924809 + 0.380431i $$0.124224\pi$$
$$48$$ 0 0
$$49$$ 11.8371 1.69101
$$50$$ 0 0
$$51$$ −2.34017 −0.327690
$$52$$ 0 0
$$53$$ 11.9155 1.63672 0.818358 0.574708i $$-0.194884\pi$$
0.818358 + 0.574708i $$0.194884\pi$$
$$54$$ 0 0
$$55$$ 3.68649 0.497086
$$56$$ 0 0
$$57$$ 0.921622 0.122072
$$58$$ 0 0
$$59$$ −1.84324 −0.239970 −0.119985 0.992776i $$-0.538285\pi$$
−0.119985 + 0.992776i $$0.538285\pi$$
$$60$$ 0 0
$$61$$ −4.73820 −0.606665 −0.303332 0.952885i $$-0.598099\pi$$
−0.303332 + 0.952885i $$0.598099\pi$$
$$62$$ 0 0
$$63$$ −4.34017 −0.546810
$$64$$ 0 0
$$65$$ 2.15676 0.267513
$$66$$ 0 0
$$67$$ −7.75872 −0.947879 −0.473939 0.880557i $$-0.657168\pi$$
−0.473939 + 0.880557i $$0.657168\pi$$
$$68$$ 0 0
$$69$$ 1.00000 0.120386
$$70$$ 0 0
$$71$$ −2.52359 −0.299495 −0.149748 0.988724i $$-0.547846\pi$$
−0.149748 + 0.988724i $$0.547846\pi$$
$$72$$ 0 0
$$73$$ 2.00000 0.234082 0.117041 0.993127i $$-0.462659\pi$$
0.117041 + 0.993127i $$0.462659\pi$$
$$74$$ 0 0
$$75$$ −3.83710 −0.443070
$$76$$ 0 0
$$77$$ 14.8371 1.69084
$$78$$ 0 0
$$79$$ 12.3402 1.38838 0.694189 0.719793i $$-0.255763\pi$$
0.694189 + 0.719793i $$0.255763\pi$$
$$80$$ 0 0
$$81$$ 1.00000 0.111111
$$82$$ 0 0
$$83$$ −0.894960 −0.0982347 −0.0491173 0.998793i $$-0.515641\pi$$
−0.0491173 + 0.998793i $$0.515641\pi$$
$$84$$ 0 0
$$85$$ 2.52359 0.273722
$$86$$ 0 0
$$87$$ −6.68035 −0.716208
$$88$$ 0 0
$$89$$ −8.49693 −0.900673 −0.450336 0.892859i $$-0.648696\pi$$
−0.450336 + 0.892859i $$0.648696\pi$$
$$90$$ 0 0
$$91$$ 8.68035 0.909948
$$92$$ 0 0
$$93$$ 8.68035 0.900110
$$94$$ 0 0
$$95$$ −0.993857 −0.101968
$$96$$ 0 0
$$97$$ 8.15676 0.828193 0.414097 0.910233i $$-0.364098\pi$$
0.414097 + 0.910233i $$0.364098\pi$$
$$98$$ 0 0
$$99$$ −3.41855 −0.343577
$$100$$ 0 0
$$101$$ 6.31351 0.628218 0.314109 0.949387i $$-0.398294\pi$$
0.314109 + 0.949387i $$0.398294\pi$$
$$102$$ 0 0
$$103$$ 1.81658 0.178993 0.0894966 0.995987i $$-0.471474\pi$$
0.0894966 + 0.995987i $$0.471474\pi$$
$$104$$ 0 0
$$105$$ 4.68035 0.456755
$$106$$ 0 0
$$107$$ −5.94214 −0.574448 −0.287224 0.957863i $$-0.592732\pi$$
−0.287224 + 0.957863i $$0.592732\pi$$
$$108$$ 0 0
$$109$$ −14.7792 −1.41559 −0.707797 0.706416i $$-0.750311\pi$$
−0.707797 + 0.706416i $$0.750311\pi$$
$$110$$ 0 0
$$111$$ −7.26180 −0.689259
$$112$$ 0 0
$$113$$ −4.18342 −0.393543 −0.196771 0.980449i $$-0.563046\pi$$
−0.196771 + 0.980449i $$0.563046\pi$$
$$114$$ 0 0
$$115$$ −1.07838 −0.100559
$$116$$ 0 0
$$117$$ −2.00000 −0.184900
$$118$$ 0 0
$$119$$ 10.1568 0.931068
$$120$$ 0 0
$$121$$ 0.686489 0.0624081
$$122$$ 0 0
$$123$$ 8.83710 0.796815
$$124$$ 0 0
$$125$$ 9.52973 0.852365
$$126$$ 0 0
$$127$$ 20.9939 1.86290 0.931452 0.363865i $$-0.118543\pi$$
0.931452 + 0.363865i $$0.118543\pi$$
$$128$$ 0 0
$$129$$ 7.75872 0.683118
$$130$$ 0 0
$$131$$ 10.8371 0.946842 0.473421 0.880836i $$-0.343019\pi$$
0.473421 + 0.880836i $$0.343019\pi$$
$$132$$ 0 0
$$133$$ −4.00000 −0.346844
$$134$$ 0 0
$$135$$ −1.07838 −0.0928120
$$136$$ 0 0
$$137$$ −1.65983 −0.141809 −0.0709043 0.997483i $$-0.522588\pi$$
−0.0709043 + 0.997483i $$0.522588\pi$$
$$138$$ 0 0
$$139$$ −4.00000 −0.339276 −0.169638 0.985506i $$-0.554260\pi$$
−0.169638 + 0.985506i $$0.554260\pi$$
$$140$$ 0 0
$$141$$ 12.6803 1.06788
$$142$$ 0 0
$$143$$ 6.83710 0.571747
$$144$$ 0 0
$$145$$ 7.20394 0.598254
$$146$$ 0 0
$$147$$ 11.8371 0.976308
$$148$$ 0 0
$$149$$ 3.23513 0.265032 0.132516 0.991181i $$-0.457694\pi$$
0.132516 + 0.991181i $$0.457694\pi$$
$$150$$ 0 0
$$151$$ 17.3607 1.41279 0.706397 0.707816i $$-0.250320\pi$$
0.706397 + 0.707816i $$0.250320\pi$$
$$152$$ 0 0
$$153$$ −2.34017 −0.189192
$$154$$ 0 0
$$155$$ −9.36069 −0.751869
$$156$$ 0 0
$$157$$ 5.78539 0.461724 0.230862 0.972986i $$-0.425845\pi$$
0.230862 + 0.972986i $$0.425845\pi$$
$$158$$ 0 0
$$159$$ 11.9155 0.944959
$$160$$ 0 0
$$161$$ −4.34017 −0.342054
$$162$$ 0 0
$$163$$ 2.15676 0.168930 0.0844651 0.996426i $$-0.473082\pi$$
0.0844651 + 0.996426i $$0.473082\pi$$
$$164$$ 0 0
$$165$$ 3.68649 0.286993
$$166$$ 0 0
$$167$$ −5.47641 −0.423777 −0.211889 0.977294i $$-0.567961\pi$$
−0.211889 + 0.977294i $$0.567961\pi$$
$$168$$ 0 0
$$169$$ −9.00000 −0.692308
$$170$$ 0 0
$$171$$ 0.921622 0.0704782
$$172$$ 0 0
$$173$$ 9.31965 0.708560 0.354280 0.935139i $$-0.384726\pi$$
0.354280 + 0.935139i $$0.384726\pi$$
$$174$$ 0 0
$$175$$ 16.6537 1.25890
$$176$$ 0 0
$$177$$ −1.84324 −0.138547
$$178$$ 0 0
$$179$$ −4.36683 −0.326393 −0.163196 0.986594i $$-0.552180\pi$$
−0.163196 + 0.986594i $$0.552180\pi$$
$$180$$ 0 0
$$181$$ 3.94214 0.293017 0.146509 0.989209i $$-0.453196\pi$$
0.146509 + 0.989209i $$0.453196\pi$$
$$182$$ 0 0
$$183$$ −4.73820 −0.350258
$$184$$ 0 0
$$185$$ 7.83096 0.575744
$$186$$ 0 0
$$187$$ 8.00000 0.585018
$$188$$ 0 0
$$189$$ −4.34017 −0.315701
$$190$$ 0 0
$$191$$ −15.5174 −1.12280 −0.561402 0.827544i $$-0.689738\pi$$
−0.561402 + 0.827544i $$0.689738\pi$$
$$192$$ 0 0
$$193$$ −16.8371 −1.21196 −0.605981 0.795479i $$-0.707219\pi$$
−0.605981 + 0.795479i $$0.707219\pi$$
$$194$$ 0 0
$$195$$ 2.15676 0.154448
$$196$$ 0 0
$$197$$ −9.20394 −0.655753 −0.327877 0.944721i $$-0.606333\pi$$
−0.327877 + 0.944721i $$0.606333\pi$$
$$198$$ 0 0
$$199$$ −26.6947 −1.89234 −0.946169 0.323672i $$-0.895083\pi$$
−0.946169 + 0.323672i $$0.895083\pi$$
$$200$$ 0 0
$$201$$ −7.75872 −0.547258
$$202$$ 0 0
$$203$$ 28.9939 2.03497
$$204$$ 0 0
$$205$$ −9.52973 −0.665585
$$206$$ 0 0
$$207$$ 1.00000 0.0695048
$$208$$ 0 0
$$209$$ −3.15061 −0.217932
$$210$$ 0 0
$$211$$ 19.5174 1.34364 0.671818 0.740716i $$-0.265514\pi$$
0.671818 + 0.740716i $$0.265514\pi$$
$$212$$ 0 0
$$213$$ −2.52359 −0.172914
$$214$$ 0 0
$$215$$ −8.36683 −0.570613
$$216$$ 0 0
$$217$$ −37.6742 −2.55749
$$218$$ 0 0
$$219$$ 2.00000 0.135147
$$220$$ 0 0
$$221$$ 4.68035 0.314834
$$222$$ 0 0
$$223$$ −29.6742 −1.98713 −0.993566 0.113256i $$-0.963872\pi$$
−0.993566 + 0.113256i $$0.963872\pi$$
$$224$$ 0 0
$$225$$ −3.83710 −0.255807
$$226$$ 0 0
$$227$$ 13.2618 0.880216 0.440108 0.897945i $$-0.354940\pi$$
0.440108 + 0.897945i $$0.354940\pi$$
$$228$$ 0 0
$$229$$ 11.9421 0.789159 0.394579 0.918862i $$-0.370890\pi$$
0.394579 + 0.918862i $$0.370890\pi$$
$$230$$ 0 0
$$231$$ 14.8371 0.976210
$$232$$ 0 0
$$233$$ 5.68649 0.372534 0.186267 0.982499i $$-0.440361\pi$$
0.186267 + 0.982499i $$0.440361\pi$$
$$234$$ 0 0
$$235$$ −13.6742 −0.892007
$$236$$ 0 0
$$237$$ 12.3402 0.801580
$$238$$ 0 0
$$239$$ 21.6742 1.40199 0.700994 0.713167i $$-0.252740\pi$$
0.700994 + 0.713167i $$0.252740\pi$$
$$240$$ 0 0
$$241$$ 16.1568 1.04075 0.520374 0.853938i $$-0.325793\pi$$
0.520374 + 0.853938i $$0.325793\pi$$
$$242$$ 0 0
$$243$$ 1.00000 0.0641500
$$244$$ 0 0
$$245$$ −12.7649 −0.815517
$$246$$ 0 0
$$247$$ −1.84324 −0.117283
$$248$$ 0 0
$$249$$ −0.894960 −0.0567158
$$250$$ 0 0
$$251$$ 28.0989 1.77359 0.886793 0.462166i $$-0.152928\pi$$
0.886793 + 0.462166i $$0.152928\pi$$
$$252$$ 0 0
$$253$$ −3.41855 −0.214922
$$254$$ 0 0
$$255$$ 2.52359 0.158033
$$256$$ 0 0
$$257$$ −0.523590 −0.0326607 −0.0163303 0.999867i $$-0.505198\pi$$
−0.0163303 + 0.999867i $$0.505198\pi$$
$$258$$ 0 0
$$259$$ 31.5174 1.95840
$$260$$ 0 0
$$261$$ −6.68035 −0.413503
$$262$$ 0 0
$$263$$ −4.99386 −0.307934 −0.153967 0.988076i $$-0.549205\pi$$
−0.153967 + 0.988076i $$0.549205\pi$$
$$264$$ 0 0
$$265$$ −12.8494 −0.789332
$$266$$ 0 0
$$267$$ −8.49693 −0.520004
$$268$$ 0 0
$$269$$ 3.84324 0.234327 0.117163 0.993113i $$-0.462620\pi$$
0.117163 + 0.993113i $$0.462620\pi$$
$$270$$ 0 0
$$271$$ 24.6803 1.49922 0.749612 0.661877i $$-0.230240\pi$$
0.749612 + 0.661877i $$0.230240\pi$$
$$272$$ 0 0
$$273$$ 8.68035 0.525358
$$274$$ 0 0
$$275$$ 13.1173 0.791005
$$276$$ 0 0
$$277$$ −25.8843 −1.55524 −0.777618 0.628737i $$-0.783572\pi$$
−0.777618 + 0.628737i $$0.783572\pi$$
$$278$$ 0 0
$$279$$ 8.68035 0.519679
$$280$$ 0 0
$$281$$ 29.1773 1.74057 0.870285 0.492548i $$-0.163934\pi$$
0.870285 + 0.492548i $$0.163934\pi$$
$$282$$ 0 0
$$283$$ −2.28231 −0.135669 −0.0678347 0.997697i $$-0.521609\pi$$
−0.0678347 + 0.997697i $$0.521609\pi$$
$$284$$ 0 0
$$285$$ −0.993857 −0.0588710
$$286$$ 0 0
$$287$$ −38.3545 −2.26400
$$288$$ 0 0
$$289$$ −11.5236 −0.677858
$$290$$ 0 0
$$291$$ 8.15676 0.478157
$$292$$ 0 0
$$293$$ 7.60197 0.444112 0.222056 0.975034i $$-0.428723\pi$$
0.222056 + 0.975034i $$0.428723\pi$$
$$294$$ 0 0
$$295$$ 1.98771 0.115729
$$296$$ 0 0
$$297$$ −3.41855 −0.198364
$$298$$ 0 0
$$299$$ −2.00000 −0.115663
$$300$$ 0 0
$$301$$ −33.6742 −1.94095
$$302$$ 0 0
$$303$$ 6.31351 0.362702
$$304$$ 0 0
$$305$$ 5.10957 0.292573
$$306$$ 0 0
$$307$$ −10.8904 −0.621549 −0.310775 0.950484i $$-0.600588\pi$$
−0.310775 + 0.950484i $$0.600588\pi$$
$$308$$ 0 0
$$309$$ 1.81658 0.103342
$$310$$ 0 0
$$311$$ 11.5174 0.653095 0.326547 0.945181i $$-0.394115\pi$$
0.326547 + 0.945181i $$0.394115\pi$$
$$312$$ 0 0
$$313$$ −4.21008 −0.237968 −0.118984 0.992896i $$-0.537964\pi$$
−0.118984 + 0.992896i $$0.537964\pi$$
$$314$$ 0 0
$$315$$ 4.68035 0.263708
$$316$$ 0 0
$$317$$ −13.3197 −0.748106 −0.374053 0.927407i $$-0.622032\pi$$
−0.374053 + 0.927407i $$0.622032\pi$$
$$318$$ 0 0
$$319$$ 22.8371 1.27863
$$320$$ 0 0
$$321$$ −5.94214 −0.331658
$$322$$ 0 0
$$323$$ −2.15676 −0.120005
$$324$$ 0 0
$$325$$ 7.67420 0.425688
$$326$$ 0 0
$$327$$ −14.7792 −0.817294
$$328$$ 0 0
$$329$$ −55.0349 −3.03417
$$330$$ 0 0
$$331$$ 2.15676 0.118546 0.0592730 0.998242i $$-0.481122\pi$$
0.0592730 + 0.998242i $$0.481122\pi$$
$$332$$ 0 0
$$333$$ −7.26180 −0.397944
$$334$$ 0 0
$$335$$ 8.36683 0.457129
$$336$$ 0 0
$$337$$ 24.3545 1.32668 0.663338 0.748320i $$-0.269139\pi$$
0.663338 + 0.748320i $$0.269139\pi$$
$$338$$ 0 0
$$339$$ −4.18342 −0.227212
$$340$$ 0 0
$$341$$ −29.6742 −1.60695
$$342$$ 0 0
$$343$$ −20.9939 −1.13356
$$344$$ 0 0
$$345$$ −1.07838 −0.0580579
$$346$$ 0 0
$$347$$ 26.0410 1.39796 0.698978 0.715143i $$-0.253638\pi$$
0.698978 + 0.715143i $$0.253638\pi$$
$$348$$ 0 0
$$349$$ 0.523590 0.0280272 0.0140136 0.999902i $$-0.495539\pi$$
0.0140136 + 0.999902i $$0.495539\pi$$
$$350$$ 0 0
$$351$$ −2.00000 −0.106752
$$352$$ 0 0
$$353$$ 28.0410 1.49247 0.746237 0.665680i $$-0.231859\pi$$
0.746237 + 0.665680i $$0.231859\pi$$
$$354$$ 0 0
$$355$$ 2.72138 0.144436
$$356$$ 0 0
$$357$$ 10.1568 0.537553
$$358$$ 0 0
$$359$$ 9.84324 0.519507 0.259753 0.965675i $$-0.416359\pi$$
0.259753 + 0.965675i $$0.416359\pi$$
$$360$$ 0 0
$$361$$ −18.1506 −0.955295
$$362$$ 0 0
$$363$$ 0.686489 0.0360313
$$364$$ 0 0
$$365$$ −2.15676 −0.112890
$$366$$ 0 0
$$367$$ −17.3340 −0.904829 −0.452414 0.891808i $$-0.649437\pi$$
−0.452414 + 0.891808i $$0.649437\pi$$
$$368$$ 0 0
$$369$$ 8.83710 0.460041
$$370$$ 0 0
$$371$$ −51.7152 −2.68492
$$372$$ 0 0
$$373$$ −23.2618 −1.20445 −0.602225 0.798326i $$-0.705719\pi$$
−0.602225 + 0.798326i $$0.705719\pi$$
$$374$$ 0 0
$$375$$ 9.52973 0.492113
$$376$$ 0 0
$$377$$ 13.3607 0.688111
$$378$$ 0 0
$$379$$ −25.7998 −1.32524 −0.662622 0.748954i $$-0.730557\pi$$
−0.662622 + 0.748954i $$0.730557\pi$$
$$380$$ 0 0
$$381$$ 20.9939 1.07555
$$382$$ 0 0
$$383$$ 6.35455 0.324702 0.162351 0.986733i $$-0.448092\pi$$
0.162351 + 0.986733i $$0.448092\pi$$
$$384$$ 0 0
$$385$$ −16.0000 −0.815436
$$386$$ 0 0
$$387$$ 7.75872 0.394398
$$388$$ 0 0
$$389$$ −3.54864 −0.179923 −0.0899617 0.995945i $$-0.528674\pi$$
−0.0899617 + 0.995945i $$0.528674\pi$$
$$390$$ 0 0
$$391$$ −2.34017 −0.118348
$$392$$ 0 0
$$393$$ 10.8371 0.546659
$$394$$ 0 0
$$395$$ −13.3074 −0.669566
$$396$$ 0 0
$$397$$ 10.6270 0.533355 0.266677 0.963786i $$-0.414074\pi$$
0.266677 + 0.963786i $$0.414074\pi$$
$$398$$ 0 0
$$399$$ −4.00000 −0.200250
$$400$$ 0 0
$$401$$ 36.0677 1.80113 0.900567 0.434716i $$-0.143151\pi$$
0.900567 + 0.434716i $$0.143151\pi$$
$$402$$ 0 0
$$403$$ −17.3607 −0.864798
$$404$$ 0 0
$$405$$ −1.07838 −0.0535850
$$406$$ 0 0
$$407$$ 24.8248 1.23052
$$408$$ 0 0
$$409$$ −29.1506 −1.44141 −0.720703 0.693244i $$-0.756181\pi$$
−0.720703 + 0.693244i $$0.756181\pi$$
$$410$$ 0 0
$$411$$ −1.65983 −0.0818732
$$412$$ 0 0
$$413$$ 8.00000 0.393654
$$414$$ 0 0
$$415$$ 0.965105 0.0473752
$$416$$ 0 0
$$417$$ −4.00000 −0.195881
$$418$$ 0 0
$$419$$ 12.7792 0.624307 0.312153 0.950032i $$-0.398950\pi$$
0.312153 + 0.950032i $$0.398950\pi$$
$$420$$ 0 0
$$421$$ −28.2557 −1.37710 −0.688548 0.725191i $$-0.741752\pi$$
−0.688548 + 0.725191i $$0.741752\pi$$
$$422$$ 0 0
$$423$$ 12.6803 0.616540
$$424$$ 0 0
$$425$$ 8.97948 0.435569
$$426$$ 0 0
$$427$$ 20.5646 0.995192
$$428$$ 0 0
$$429$$ 6.83710 0.330098
$$430$$ 0 0
$$431$$ −37.8720 −1.82423 −0.912115 0.409935i $$-0.865552\pi$$
−0.912115 + 0.409935i $$0.865552\pi$$
$$432$$ 0 0
$$433$$ −12.8371 −0.616912 −0.308456 0.951239i $$-0.599812\pi$$
−0.308456 + 0.951239i $$0.599812\pi$$
$$434$$ 0 0
$$435$$ 7.20394 0.345402
$$436$$ 0 0
$$437$$ 0.921622 0.0440872
$$438$$ 0 0
$$439$$ 25.3607 1.21040 0.605200 0.796074i $$-0.293093\pi$$
0.605200 + 0.796074i $$0.293093\pi$$
$$440$$ 0 0
$$441$$ 11.8371 0.563671
$$442$$ 0 0
$$443$$ 37.9877 1.80485 0.902425 0.430846i $$-0.141785\pi$$
0.902425 + 0.430846i $$0.141785\pi$$
$$444$$ 0 0
$$445$$ 9.16290 0.434363
$$446$$ 0 0
$$447$$ 3.23513 0.153017
$$448$$ 0 0
$$449$$ −38.1978 −1.80267 −0.901333 0.433128i $$-0.857410\pi$$
−0.901333 + 0.433128i $$0.857410\pi$$
$$450$$ 0 0
$$451$$ −30.2101 −1.42254
$$452$$ 0 0
$$453$$ 17.3607 0.815676
$$454$$ 0 0
$$455$$ −9.36069 −0.438836
$$456$$ 0 0
$$457$$ 11.8432 0.554004 0.277002 0.960869i $$-0.410659\pi$$
0.277002 + 0.960869i $$0.410659\pi$$
$$458$$ 0 0
$$459$$ −2.34017 −0.109230
$$460$$ 0 0
$$461$$ 39.1917 1.82534 0.912669 0.408700i $$-0.134018\pi$$
0.912669 + 0.408700i $$0.134018\pi$$
$$462$$ 0 0
$$463$$ −25.3607 −1.17861 −0.589306 0.807910i $$-0.700599\pi$$
−0.589306 + 0.807910i $$0.700599\pi$$
$$464$$ 0 0
$$465$$ −9.36069 −0.434092
$$466$$ 0 0
$$467$$ −13.2618 −0.613683 −0.306841 0.951761i $$-0.599272\pi$$
−0.306841 + 0.951761i $$0.599272\pi$$
$$468$$ 0 0
$$469$$ 33.6742 1.55493
$$470$$ 0 0
$$471$$ 5.78539 0.266577
$$472$$ 0 0
$$473$$ −26.5236 −1.21956
$$474$$ 0 0
$$475$$ −3.53636 −0.162259
$$476$$ 0 0
$$477$$ 11.9155 0.545572
$$478$$ 0 0
$$479$$ 11.6865 0.533969 0.266985 0.963701i $$-0.413973\pi$$
0.266985 + 0.963701i $$0.413973\pi$$
$$480$$ 0 0
$$481$$ 14.5236 0.662219
$$482$$ 0 0
$$483$$ −4.34017 −0.197485
$$484$$ 0 0
$$485$$ −8.79606 −0.399409
$$486$$ 0 0
$$487$$ 5.04718 0.228710 0.114355 0.993440i $$-0.463520\pi$$
0.114355 + 0.993440i $$0.463520\pi$$
$$488$$ 0 0
$$489$$ 2.15676 0.0975319
$$490$$ 0 0
$$491$$ 0.680346 0.0307036 0.0153518 0.999882i $$-0.495113\pi$$
0.0153518 + 0.999882i $$0.495113\pi$$
$$492$$ 0 0
$$493$$ 15.6332 0.704082
$$494$$ 0 0
$$495$$ 3.68649 0.165695
$$496$$ 0 0
$$497$$ 10.9528 0.491301
$$498$$ 0 0
$$499$$ −25.1917 −1.12773 −0.563867 0.825866i $$-0.690687\pi$$
−0.563867 + 0.825866i $$0.690687\pi$$
$$500$$ 0 0
$$501$$ −5.47641 −0.244668
$$502$$ 0 0
$$503$$ 19.8843 0.886596 0.443298 0.896374i $$-0.353808\pi$$
0.443298 + 0.896374i $$0.353808\pi$$
$$504$$ 0 0
$$505$$ −6.80835 −0.302968
$$506$$ 0 0
$$507$$ −9.00000 −0.399704
$$508$$ 0 0
$$509$$ −33.4017 −1.48051 −0.740253 0.672329i $$-0.765294\pi$$
−0.740253 + 0.672329i $$0.765294\pi$$
$$510$$ 0 0
$$511$$ −8.68035 −0.383996
$$512$$ 0 0
$$513$$ 0.921622 0.0406906
$$514$$ 0 0
$$515$$ −1.95896 −0.0863222
$$516$$ 0 0
$$517$$ −43.3484 −1.90646
$$518$$ 0 0
$$519$$ 9.31965 0.409087
$$520$$ 0 0
$$521$$ 27.3874 1.19986 0.599931 0.800052i $$-0.295195\pi$$
0.599931 + 0.800052i $$0.295195\pi$$
$$522$$ 0 0
$$523$$ −28.0722 −1.22751 −0.613757 0.789495i $$-0.710342\pi$$
−0.613757 + 0.789495i $$0.710342\pi$$
$$524$$ 0 0
$$525$$ 16.6537 0.726826
$$526$$ 0 0
$$527$$ −20.3135 −0.884870
$$528$$ 0 0
$$529$$ 1.00000 0.0434783
$$530$$ 0 0
$$531$$ −1.84324 −0.0799900
$$532$$ 0 0
$$533$$ −17.6742 −0.765555
$$534$$ 0 0
$$535$$ 6.40787 0.277036
$$536$$ 0 0
$$537$$ −4.36683 −0.188443
$$538$$ 0 0
$$539$$ −40.4657 −1.74298
$$540$$ 0 0
$$541$$ 41.8843 1.80075 0.900373 0.435119i $$-0.143294\pi$$
0.900373 + 0.435119i $$0.143294\pi$$
$$542$$ 0 0
$$543$$ 3.94214 0.169173
$$544$$ 0 0
$$545$$ 15.9376 0.682692
$$546$$ 0 0
$$547$$ −31.2039 −1.33418 −0.667092 0.744975i $$-0.732461\pi$$
−0.667092 + 0.744975i $$0.732461\pi$$
$$548$$ 0 0
$$549$$ −4.73820 −0.202222
$$550$$ 0 0
$$551$$ −6.15676 −0.262287
$$552$$ 0 0
$$553$$ −53.5585 −2.27754
$$554$$ 0 0
$$555$$ 7.83096 0.332406
$$556$$ 0 0
$$557$$ −6.60811 −0.279995 −0.139997 0.990152i $$-0.544709\pi$$
−0.139997 + 0.990152i $$0.544709\pi$$
$$558$$ 0 0
$$559$$ −15.5174 −0.656318
$$560$$ 0 0
$$561$$ 8.00000 0.337760
$$562$$ 0 0
$$563$$ 29.2618 1.23324 0.616619 0.787262i $$-0.288502\pi$$
0.616619 + 0.787262i $$0.288502\pi$$
$$564$$ 0 0
$$565$$ 4.51130 0.189792
$$566$$ 0 0
$$567$$ −4.34017 −0.182270
$$568$$ 0 0
$$569$$ 11.3340 0.475147 0.237574 0.971370i $$-0.423648\pi$$
0.237574 + 0.971370i $$0.423648\pi$$
$$570$$ 0 0
$$571$$ 34.9627 1.46314 0.731571 0.681765i $$-0.238788\pi$$
0.731571 + 0.681765i $$0.238788\pi$$
$$572$$ 0 0
$$573$$ −15.5174 −0.648251
$$574$$ 0 0
$$575$$ −3.83710 −0.160018
$$576$$ 0 0
$$577$$ 1.15061 0.0479006 0.0239503 0.999713i $$-0.492376\pi$$
0.0239503 + 0.999713i $$0.492376\pi$$
$$578$$ 0 0
$$579$$ −16.8371 −0.699726
$$580$$ 0 0
$$581$$ 3.88428 0.161147
$$582$$ 0 0
$$583$$ −40.7337 −1.68702
$$584$$ 0 0
$$585$$ 2.15676 0.0891709
$$586$$ 0 0
$$587$$ 16.5113 0.681494 0.340747 0.940155i $$-0.389320\pi$$
0.340747 + 0.940155i $$0.389320\pi$$
$$588$$ 0 0
$$589$$ 8.00000 0.329634
$$590$$ 0 0
$$591$$ −9.20394 −0.378599
$$592$$ 0 0
$$593$$ 11.1629 0.458405 0.229203 0.973379i $$-0.426388\pi$$
0.229203 + 0.973379i $$0.426388\pi$$
$$594$$ 0 0
$$595$$ −10.9528 −0.449022
$$596$$ 0 0
$$597$$ −26.6947 −1.09254
$$598$$ 0 0
$$599$$ 32.0000 1.30748 0.653742 0.756717i $$-0.273198\pi$$
0.653742 + 0.756717i $$0.273198\pi$$
$$600$$ 0 0
$$601$$ 41.1506 1.67857 0.839284 0.543693i $$-0.182974\pi$$
0.839284 + 0.543693i $$0.182974\pi$$
$$602$$ 0 0
$$603$$ −7.75872 −0.315960
$$604$$ 0 0
$$605$$ −0.740294 −0.0300973
$$606$$ 0 0
$$607$$ 24.0000 0.974130 0.487065 0.873366i $$-0.338067\pi$$
0.487065 + 0.873366i $$0.338067\pi$$
$$608$$ 0 0
$$609$$ 28.9939 1.17489
$$610$$ 0 0
$$611$$ −25.3607 −1.02598
$$612$$ 0 0
$$613$$ −4.73820 −0.191374 −0.0956871 0.995411i $$-0.530505\pi$$
−0.0956871 + 0.995411i $$0.530505\pi$$
$$614$$ 0 0
$$615$$ −9.52973 −0.384276
$$616$$ 0 0
$$617$$ −34.4846 −1.38830 −0.694150 0.719831i $$-0.744219\pi$$
−0.694150 + 0.719831i $$0.744219\pi$$
$$618$$ 0 0
$$619$$ 41.0661 1.65059 0.825293 0.564705i $$-0.191010\pi$$
0.825293 + 0.564705i $$0.191010\pi$$
$$620$$ 0 0
$$621$$ 1.00000 0.0401286
$$622$$ 0 0
$$623$$ 36.8781 1.47749
$$624$$ 0 0
$$625$$ 8.90885 0.356354
$$626$$ 0 0
$$627$$ −3.15061 −0.125823
$$628$$ 0 0
$$629$$ 16.9939 0.677589
$$630$$ 0 0
$$631$$ 22.8638 0.910192 0.455096 0.890442i $$-0.349605\pi$$
0.455096 + 0.890442i $$0.349605\pi$$
$$632$$ 0 0
$$633$$ 19.5174 0.775749
$$634$$ 0 0
$$635$$ −22.6393 −0.898414
$$636$$ 0 0
$$637$$ −23.6742 −0.938006
$$638$$ 0 0
$$639$$ −2.52359 −0.0998317
$$640$$ 0 0
$$641$$ 34.7070 1.37084 0.685422 0.728146i $$-0.259618\pi$$
0.685422 + 0.728146i $$0.259618\pi$$
$$642$$ 0 0
$$643$$ −2.96266 −0.116836 −0.0584180 0.998292i $$-0.518606\pi$$
−0.0584180 + 0.998292i $$0.518606\pi$$
$$644$$ 0 0
$$645$$ −8.36683 −0.329444
$$646$$ 0 0
$$647$$ 14.4703 0.568885 0.284442 0.958693i $$-0.408192\pi$$
0.284442 + 0.958693i $$0.408192\pi$$
$$648$$ 0 0
$$649$$ 6.30122 0.247345
$$650$$ 0 0
$$651$$ −37.6742 −1.47657
$$652$$ 0 0
$$653$$ 10.0000 0.391330 0.195665 0.980671i $$-0.437313\pi$$
0.195665 + 0.980671i $$0.437313\pi$$
$$654$$ 0 0
$$655$$ −11.6865 −0.456629
$$656$$ 0 0
$$657$$ 2.00000 0.0780274
$$658$$ 0 0
$$659$$ 35.1338 1.36862 0.684309 0.729192i $$-0.260104\pi$$
0.684309 + 0.729192i $$0.260104\pi$$
$$660$$ 0 0
$$661$$ 42.2967 1.64515 0.822575 0.568656i $$-0.192537\pi$$
0.822575 + 0.568656i $$0.192537\pi$$
$$662$$ 0 0
$$663$$ 4.68035 0.181770
$$664$$ 0 0
$$665$$ 4.31351 0.167271
$$666$$ 0 0
$$667$$ −6.68035 −0.258664
$$668$$ 0 0
$$669$$ −29.6742 −1.14727
$$670$$ 0 0
$$671$$ 16.1978 0.625309
$$672$$ 0 0
$$673$$ 9.15061 0.352730 0.176365 0.984325i $$-0.443566\pi$$
0.176365 + 0.984325i $$0.443566\pi$$
$$674$$ 0 0
$$675$$ −3.83710 −0.147690
$$676$$ 0 0
$$677$$ 43.0037 1.65277 0.826383 0.563108i $$-0.190395\pi$$
0.826383 + 0.563108i $$0.190395\pi$$
$$678$$ 0 0
$$679$$ −35.4017 −1.35859
$$680$$ 0 0
$$681$$ 13.2618 0.508193
$$682$$ 0 0
$$683$$ −44.1978 −1.69118 −0.845591 0.533832i $$-0.820752\pi$$
−0.845591 + 0.533832i $$0.820752\pi$$
$$684$$ 0 0
$$685$$ 1.78992 0.0683893
$$686$$ 0 0
$$687$$ 11.9421 0.455621
$$688$$ 0 0
$$689$$ −23.8310 −0.907887
$$690$$ 0 0
$$691$$ −22.4703 −0.854809 −0.427405 0.904060i $$-0.640572\pi$$
−0.427405 + 0.904060i $$0.640572\pi$$
$$692$$ 0 0
$$693$$ 14.8371 0.563615
$$694$$ 0 0
$$695$$ 4.31351 0.163621
$$696$$ 0 0
$$697$$ −20.6803 −0.783324
$$698$$ 0 0
$$699$$ 5.68649 0.215083
$$700$$ 0 0
$$701$$ 24.4801 0.924601 0.462300 0.886723i $$-0.347024\pi$$
0.462300 + 0.886723i $$0.347024\pi$$
$$702$$ 0 0
$$703$$ −6.69263 −0.252417
$$704$$ 0 0
$$705$$ −13.6742 −0.515000
$$706$$ 0 0
$$707$$ −27.4017 −1.03055
$$708$$ 0 0
$$709$$ −32.5692 −1.22316 −0.611580 0.791182i $$-0.709466\pi$$
−0.611580 + 0.791182i $$0.709466\pi$$
$$710$$ 0 0
$$711$$ 12.3402 0.462793
$$712$$ 0 0
$$713$$ 8.68035 0.325082
$$714$$ 0 0
$$715$$ −7.37298 −0.275734
$$716$$ 0 0
$$717$$ 21.6742 0.809438
$$718$$ 0 0
$$719$$ −7.20394 −0.268661 −0.134331 0.990937i $$-0.542888\pi$$
−0.134331 + 0.990937i $$0.542888\pi$$
$$720$$ 0 0
$$721$$ −7.88428 −0.293626
$$722$$ 0 0
$$723$$ 16.1568 0.600876
$$724$$ 0 0
$$725$$ 25.6332 0.951992
$$726$$ 0 0
$$727$$ −16.6537 −0.617651 −0.308825 0.951119i $$-0.599936\pi$$
−0.308825 + 0.951119i $$0.599936\pi$$
$$728$$ 0 0
$$729$$ 1.00000 0.0370370
$$730$$ 0 0
$$731$$ −18.1568 −0.671552
$$732$$ 0 0
$$733$$ −46.2434 −1.70804 −0.854019 0.520242i $$-0.825842\pi$$
−0.854019 + 0.520242i $$0.825842\pi$$
$$734$$ 0 0
$$735$$ −12.7649 −0.470839
$$736$$ 0 0
$$737$$ 26.5236 0.977009
$$738$$ 0 0
$$739$$ 12.7337 0.468416 0.234208 0.972187i $$-0.424750\pi$$
0.234208 + 0.972187i $$0.424750\pi$$
$$740$$ 0 0
$$741$$ −1.84324 −0.0677133
$$742$$ 0 0
$$743$$ −20.3668 −0.747187 −0.373593 0.927593i $$-0.621874\pi$$
−0.373593 + 0.927593i $$0.621874\pi$$
$$744$$ 0 0
$$745$$ −3.48870 −0.127816
$$746$$ 0 0
$$747$$ −0.894960 −0.0327449
$$748$$ 0 0
$$749$$ 25.7899 0.942343
$$750$$ 0 0
$$751$$ 3.85762 0.140767 0.0703833 0.997520i $$-0.477578\pi$$
0.0703833 + 0.997520i $$0.477578\pi$$
$$752$$ 0 0
$$753$$ 28.0989 1.02398
$$754$$ 0 0
$$755$$ −18.7214 −0.681341
$$756$$ 0 0
$$757$$ 1.61634 0.0587470 0.0293735 0.999569i $$-0.490649\pi$$
0.0293735 + 0.999569i $$0.490649\pi$$
$$758$$ 0 0
$$759$$ −3.41855 −0.124086
$$760$$ 0 0
$$761$$ 20.0410 0.726487 0.363244 0.931694i $$-0.381669\pi$$
0.363244 + 0.931694i $$0.381669\pi$$
$$762$$ 0 0
$$763$$ 64.1445 2.32219
$$764$$ 0 0
$$765$$ 2.52359 0.0912406
$$766$$ 0 0
$$767$$ 3.68649 0.133111
$$768$$ 0 0
$$769$$ −23.1629 −0.835275 −0.417638 0.908614i $$-0.637142\pi$$
−0.417638 + 0.908614i $$0.637142\pi$$
$$770$$ 0 0
$$771$$ −0.523590 −0.0188566
$$772$$ 0 0
$$773$$ −41.2762 −1.48460 −0.742300 0.670067i $$-0.766265\pi$$
−0.742300 + 0.670067i $$0.766265\pi$$
$$774$$ 0 0
$$775$$ −33.3074 −1.19644
$$776$$ 0 0
$$777$$ 31.5174 1.13068
$$778$$ 0 0
$$779$$ 8.14447 0.291806
$$780$$ 0 0
$$781$$ 8.62702 0.308699
$$782$$ 0 0
$$783$$ −6.68035 −0.238736
$$784$$ 0 0
$$785$$ −6.23883 −0.222673
$$786$$ 0 0
$$787$$ 13.4329 0.478832 0.239416 0.970917i $$-0.423044\pi$$
0.239416 + 0.970917i $$0.423044\pi$$
$$788$$ 0 0
$$789$$ −4.99386 −0.177786
$$790$$ 0 0
$$791$$ 18.1568 0.645580
$$792$$ 0 0
$$793$$ 9.47641 0.336517
$$794$$ 0 0
$$795$$ −12.8494 −0.455721
$$796$$ 0 0
$$797$$ 5.07838 0.179885 0.0899427 0.995947i $$-0.471332\pi$$
0.0899427 + 0.995947i $$0.471332\pi$$
$$798$$ 0 0
$$799$$ −29.6742 −1.04980
$$800$$ 0 0
$$801$$ −8.49693 −0.300224
$$802$$ 0 0
$$803$$ −6.83710 −0.241276
$$804$$ 0 0
$$805$$ 4.68035 0.164961
$$806$$ 0 0
$$807$$ 3.84324 0.135289
$$808$$ 0 0
$$809$$ 8.35455 0.293730 0.146865 0.989157i $$-0.453082\pi$$
0.146865 + 0.989157i $$0.453082\pi$$
$$810$$ 0 0
$$811$$ 44.8781 1.57588 0.787942 0.615749i $$-0.211146\pi$$
0.787942 + 0.615749i $$0.211146\pi$$
$$812$$ 0 0
$$813$$ 24.6803 0.865578
$$814$$ 0 0
$$815$$ −2.32580 −0.0814691
$$816$$ 0 0
$$817$$ 7.15061 0.250168
$$818$$ 0 0
$$819$$ 8.68035 0.303316
$$820$$ 0 0
$$821$$ −49.3484 −1.72227 −0.861136 0.508375i $$-0.830246\pi$$
−0.861136 + 0.508375i $$0.830246\pi$$
$$822$$ 0 0
$$823$$ −10.0410 −0.350009 −0.175004 0.984568i $$-0.555994\pi$$
−0.175004 + 0.984568i $$0.555994\pi$$
$$824$$ 0 0
$$825$$ 13.1173 0.456687
$$826$$ 0 0
$$827$$ 54.0866 1.88078 0.940388 0.340104i $$-0.110462\pi$$
0.940388 + 0.340104i $$0.110462\pi$$
$$828$$ 0 0
$$829$$ 20.8371 0.723702 0.361851 0.932236i $$-0.382145\pi$$
0.361851 + 0.932236i $$0.382145\pi$$
$$830$$ 0 0
$$831$$ −25.8843 −0.897916
$$832$$ 0 0
$$833$$ −27.7009 −0.959778
$$834$$ 0 0
$$835$$ 5.90564 0.204373
$$836$$ 0 0
$$837$$ 8.68035 0.300037
$$838$$ 0 0
$$839$$ −8.68035 −0.299679 −0.149839 0.988710i $$-0.547876\pi$$
−0.149839 + 0.988710i $$0.547876\pi$$
$$840$$ 0 0
$$841$$ 15.6270 0.538863
$$842$$ 0 0
$$843$$ 29.1773 1.00492
$$844$$ 0 0
$$845$$ 9.70540 0.333876
$$846$$ 0 0
$$847$$ −2.97948 −0.102376
$$848$$ 0 0
$$849$$ −2.28231 −0.0783288
$$850$$ 0 0
$$851$$ −7.26180 −0.248931
$$852$$ 0 0
$$853$$ −38.0000 −1.30110 −0.650548 0.759465i $$-0.725461\pi$$
−0.650548 + 0.759465i $$0.725461\pi$$
$$854$$ 0 0
$$855$$ −0.993857 −0.0339892
$$856$$ 0 0
$$857$$ 18.0000 0.614868 0.307434 0.951569i $$-0.400530\pi$$
0.307434 + 0.951569i $$0.400530\pi$$
$$858$$ 0 0
$$859$$ 20.0000 0.682391 0.341196 0.939992i $$-0.389168\pi$$
0.341196 + 0.939992i $$0.389168\pi$$
$$860$$ 0 0
$$861$$ −38.3545 −1.30712
$$862$$ 0 0
$$863$$ −29.8432 −1.01588 −0.507938 0.861394i $$-0.669592\pi$$
−0.507938 + 0.861394i $$0.669592\pi$$
$$864$$ 0 0
$$865$$ −10.0501 −0.341714
$$866$$ 0 0
$$867$$ −11.5236 −0.391362
$$868$$ 0 0
$$869$$ −42.1855 −1.43105
$$870$$ 0 0
$$871$$ 15.5174 0.525789
$$872$$ 0 0
$$873$$ 8.15676 0.276064
$$874$$ 0 0
$$875$$ −41.3607 −1.39825
$$876$$ 0 0
$$877$$ −11.4764 −0.387531 −0.193765 0.981048i $$-0.562070\pi$$
−0.193765 + 0.981048i $$0.562070\pi$$
$$878$$ 0 0
$$879$$ 7.60197 0.256408
$$880$$ 0 0
$$881$$ −26.8227 −0.903681 −0.451840 0.892099i $$-0.649232\pi$$
−0.451840 + 0.892099i $$0.649232\pi$$
$$882$$ 0 0
$$883$$ −2.15676 −0.0725806 −0.0362903 0.999341i $$-0.511554\pi$$
−0.0362903 + 0.999341i $$0.511554\pi$$
$$884$$ 0 0
$$885$$ 1.98771 0.0668163
$$886$$ 0 0
$$887$$ −29.4140 −0.987626 −0.493813 0.869568i $$-0.664397\pi$$
−0.493813 + 0.869568i $$0.664397\pi$$
$$888$$ 0 0
$$889$$ −91.1170 −3.05597
$$890$$ 0 0
$$891$$ −3.41855 −0.114526
$$892$$ 0 0
$$893$$ 11.6865 0.391073
$$894$$ 0 0
$$895$$ 4.70910 0.157408
$$896$$ 0 0
$$897$$ −2.00000 −0.0667781
$$898$$ 0 0
$$899$$ −57.9877 −1.93400
$$900$$ 0 0
$$901$$ −27.8843 −0.928960
$$902$$ 0 0
$$903$$ −33.6742 −1.12061
$$904$$ 0 0
$$905$$ −4.25112 −0.141312
$$906$$ 0 0
$$907$$ −44.2700 −1.46996 −0.734981 0.678088i $$-0.762809\pi$$
−0.734981 + 0.678088i $$0.762809\pi$$
$$908$$ 0 0
$$909$$ 6.31351 0.209406
$$910$$ 0 0
$$911$$ −20.5113 −0.679570 −0.339785 0.940503i $$-0.610354\pi$$
−0.339785 + 0.940503i $$0.610354\pi$$
$$912$$ 0 0
$$913$$ 3.05947 0.101254
$$914$$ 0 0
$$915$$ 5.10957 0.168917
$$916$$ 0 0
$$917$$ −47.0349 −1.55323
$$918$$ 0 0
$$919$$ 2.01438 0.0664481 0.0332241 0.999448i $$-0.489423\pi$$
0.0332241 + 0.999448i $$0.489423\pi$$
$$920$$ 0 0
$$921$$ −10.8904 −0.358852
$$922$$ 0 0
$$923$$ 5.04718 0.166130
$$924$$ 0 0
$$925$$ 27.8642 0.916171
$$926$$ 0 0
$$927$$ 1.81658 0.0596644
$$928$$ 0 0
$$929$$ −4.21008 −0.138128 −0.0690641 0.997612i $$-0.522001\pi$$
−0.0690641 + 0.997612i $$0.522001\pi$$
$$930$$ 0 0
$$931$$ 10.9093 0.357539
$$932$$ 0 0
$$933$$ 11.5174 0.377064
$$934$$ 0 0
$$935$$ −8.62702 −0.282134
$$936$$ 0 0
$$937$$ −20.6393 −0.674257 −0.337128 0.941459i $$-0.609456\pi$$
−0.337128 + 0.941459i $$0.609456\pi$$
$$938$$ 0 0
$$939$$ −4.21008 −0.137391
$$940$$ 0 0
$$941$$ 52.5958 1.71457 0.857287 0.514838i $$-0.172148\pi$$
0.857287 + 0.514838i $$0.172148\pi$$
$$942$$ 0 0
$$943$$ 8.83710 0.287776
$$944$$ 0 0
$$945$$ 4.68035 0.152252
$$946$$ 0 0
$$947$$ 44.0288 1.43074 0.715371 0.698745i $$-0.246258\pi$$
0.715371 + 0.698745i $$0.246258\pi$$
$$948$$ 0 0
$$949$$ −4.00000 −0.129845
$$950$$ 0 0
$$951$$ −13.3197 −0.431919
$$952$$ 0 0
$$953$$ 6.39350 0.207106 0.103553 0.994624i $$-0.466979\pi$$
0.103553 + 0.994624i $$0.466979\pi$$
$$954$$ 0 0
$$955$$ 16.7337 0.541489
$$956$$ 0 0
$$957$$ 22.8371 0.738219
$$958$$ 0 0
$$959$$ 7.20394 0.232627
$$960$$ 0 0
$$961$$ 44.3484 1.43059
$$962$$ 0 0
$$963$$ −5.94214 −0.191483
$$964$$ 0 0
$$965$$ 18.1568 0.584487
$$966$$ 0 0
$$967$$ 39.0882 1.25699 0.628496 0.777813i $$-0.283671\pi$$
0.628496 + 0.777813i $$0.283671\pi$$
$$968$$ 0 0
$$969$$ −2.15676 −0.0692850
$$970$$ 0 0
$$971$$ −54.1933 −1.73914 −0.869572 0.493806i $$-0.835605\pi$$
−0.869572 + 0.493806i $$0.835605\pi$$
$$972$$ 0 0
$$973$$ 17.3607 0.556558
$$974$$ 0 0
$$975$$ 7.67420 0.245771
$$976$$ 0 0
$$977$$ 47.7009 1.52609 0.763043 0.646348i $$-0.223704\pi$$
0.763043 + 0.646348i $$0.223704\pi$$
$$978$$ 0 0
$$979$$ 29.0472 0.928352
$$980$$ 0 0
$$981$$ −14.7792 −0.471865
$$982$$ 0 0
$$983$$ −49.5585 −1.58067 −0.790335 0.612675i $$-0.790094\pi$$
−0.790335 + 0.612675i $$0.790094\pi$$
$$984$$ 0 0
$$985$$ 9.92532 0.316247
$$986$$ 0 0
$$987$$ −55.0349 −1.75178
$$988$$ 0 0
$$989$$ 7.75872 0.246713
$$990$$ 0 0
$$991$$ 20.2602 0.643586 0.321793 0.946810i $$-0.395714\pi$$
0.321793 + 0.946810i $$0.395714\pi$$
$$992$$ 0 0
$$993$$ 2.15676 0.0684426
$$994$$ 0 0
$$995$$ 28.7870 0.912609
$$996$$ 0 0
$$997$$ −1.57077 −0.0497468 −0.0248734 0.999691i $$-0.507918\pi$$
−0.0248734 + 0.999691i $$0.507918\pi$$
$$998$$ 0 0
$$999$$ −7.26180 −0.229753
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 4416.2.a.bs.1.2 3
4.3 odd 2 4416.2.a.bp.1.2 3
8.3 odd 2 552.2.a.g.1.2 3
8.5 even 2 1104.2.a.o.1.2 3
24.5 odd 2 3312.2.a.bf.1.2 3
24.11 even 2 1656.2.a.n.1.2 3

By twisted newform
Twist Min Dim Char Parity Ord Type
552.2.a.g.1.2 3 8.3 odd 2
1104.2.a.o.1.2 3 8.5 even 2
1656.2.a.n.1.2 3 24.11 even 2
3312.2.a.bf.1.2 3 24.5 odd 2
4416.2.a.bp.1.2 3 4.3 odd 2
4416.2.a.bs.1.2 3 1.1 even 1 trivial