# Properties

 Label 1216.3.e.c.1025.2 Level $1216$ Weight $3$ Character 1216.1025 Analytic conductor $33.134$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Learn more

## Newspace parameters

 Level: $$N$$ $$=$$ $$1216 = 2^{6} \cdot 19$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 1216.e (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$33.1336001462$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-2})$$ Defining polynomial: $$x^{2} + 2$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: no (minimal twist has level 152) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 1025.2 Root $$1.41421i$$ of defining polynomial Character $$\chi$$ $$=$$ 1216.1025 Dual form 1216.3.e.c.1025.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+5.65685i q^{3} -7.00000 q^{5} -11.0000 q^{7} -23.0000 q^{9} +O(q^{10})$$ $$q+5.65685i q^{3} -7.00000 q^{5} -11.0000 q^{7} -23.0000 q^{9} +3.00000 q^{11} +11.3137i q^{13} -39.5980i q^{15} -17.0000 q^{17} -19.0000 q^{19} -62.2254i q^{21} -2.00000 q^{23} +24.0000 q^{25} -79.1960i q^{27} +39.5980i q^{29} -5.65685i q^{31} +16.9706i q^{33} +77.0000 q^{35} +39.5980i q^{37} -64.0000 q^{39} +39.5980i q^{41} -21.0000 q^{43} +161.000 q^{45} +5.00000 q^{47} +72.0000 q^{49} -96.1665i q^{51} -5.65685i q^{53} -21.0000 q^{55} -107.480i q^{57} +33.9411i q^{59} -23.0000 q^{61} +253.000 q^{63} -79.1960i q^{65} +39.5980i q^{67} -11.3137i q^{69} -90.5097i q^{71} +39.0000 q^{73} +135.765i q^{75} -33.0000 q^{77} +96.1665i q^{79} +241.000 q^{81} -6.00000 q^{83} +119.000 q^{85} -224.000 q^{87} +118.794i q^{89} -124.451i q^{91} +32.0000 q^{93} +133.000 q^{95} -169.706i q^{97} -69.0000 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 14q^{5} - 22q^{7} - 46q^{9} + O(q^{10})$$ $$2q - 14q^{5} - 22q^{7} - 46q^{9} + 6q^{11} - 34q^{17} - 38q^{19} - 4q^{23} + 48q^{25} + 154q^{35} - 128q^{39} - 42q^{43} + 322q^{45} + 10q^{47} + 144q^{49} - 42q^{55} - 46q^{61} + 506q^{63} + 78q^{73} - 66q^{77} + 482q^{81} - 12q^{83} + 238q^{85} - 448q^{87} + 64q^{93} + 266q^{95} - 138q^{99} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1216\mathbb{Z}\right)^\times$$.

 $$n$$ $$191$$ $$705$$ $$837$$ $$\chi(n)$$ $$1$$ $$-1$$ $$1$$

## 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
$$3$$ 5.65685i 1.88562i 0.333333 + 0.942809i $$0.391827\pi$$
−0.333333 + 0.942809i $$0.608173\pi$$
$$4$$ 0 0
$$5$$ −7.00000 −1.40000 −0.700000 0.714143i $$-0.746817\pi$$
−0.700000 + 0.714143i $$0.746817\pi$$
$$6$$ 0 0
$$7$$ −11.0000 −1.57143 −0.785714 0.618590i $$-0.787704\pi$$
−0.785714 + 0.618590i $$0.787704\pi$$
$$8$$ 0 0
$$9$$ −23.0000 −2.55556
$$10$$ 0 0
$$11$$ 3.00000 0.272727 0.136364 0.990659i $$-0.456458\pi$$
0.136364 + 0.990659i $$0.456458\pi$$
$$12$$ 0 0
$$13$$ 11.3137i 0.870285i 0.900362 + 0.435143i $$0.143302\pi$$
−0.900362 + 0.435143i $$0.856698\pi$$
$$14$$ 0 0
$$15$$ − 39.5980i − 2.63987i
$$16$$ 0 0
$$17$$ −17.0000 −1.00000 −0.500000 0.866025i $$-0.666667\pi$$
−0.500000 + 0.866025i $$0.666667\pi$$
$$18$$ 0 0
$$19$$ −19.0000 −1.00000
$$20$$ 0 0
$$21$$ − 62.2254i − 2.96311i
$$22$$ 0 0
$$23$$ −2.00000 −0.0869565 −0.0434783 0.999054i $$-0.513844\pi$$
−0.0434783 + 0.999054i $$0.513844\pi$$
$$24$$ 0 0
$$25$$ 24.0000 0.960000
$$26$$ 0 0
$$27$$ − 79.1960i − 2.93318i
$$28$$ 0 0
$$29$$ 39.5980i 1.36545i 0.730677 + 0.682724i $$0.239205\pi$$
−0.730677 + 0.682724i $$0.760795\pi$$
$$30$$ 0 0
$$31$$ − 5.65685i − 0.182479i −0.995829 0.0912396i $$-0.970917\pi$$
0.995829 0.0912396i $$-0.0290829\pi$$
$$32$$ 0 0
$$33$$ 16.9706i 0.514259i
$$34$$ 0 0
$$35$$ 77.0000 2.20000
$$36$$ 0 0
$$37$$ 39.5980i 1.07022i 0.844784 + 0.535108i $$0.179729\pi$$
−0.844784 + 0.535108i $$0.820271\pi$$
$$38$$ 0 0
$$39$$ −64.0000 −1.64103
$$40$$ 0 0
$$41$$ 39.5980i 0.965804i 0.875674 + 0.482902i $$0.160417\pi$$
−0.875674 + 0.482902i $$0.839583\pi$$
$$42$$ 0 0
$$43$$ −21.0000 −0.488372 −0.244186 0.969728i $$-0.578521\pi$$
−0.244186 + 0.969728i $$0.578521\pi$$
$$44$$ 0 0
$$45$$ 161.000 3.57778
$$46$$ 0 0
$$47$$ 5.00000 0.106383 0.0531915 0.998584i $$-0.483061\pi$$
0.0531915 + 0.998584i $$0.483061\pi$$
$$48$$ 0 0
$$49$$ 72.0000 1.46939
$$50$$ 0 0
$$51$$ − 96.1665i − 1.88562i
$$52$$ 0 0
$$53$$ − 5.65685i − 0.106733i −0.998575 0.0533665i $$-0.983005\pi$$
0.998575 0.0533665i $$-0.0169952\pi$$
$$54$$ 0 0
$$55$$ −21.0000 −0.381818
$$56$$ 0 0
$$57$$ − 107.480i − 1.88562i
$$58$$ 0 0
$$59$$ 33.9411i 0.575273i 0.957740 + 0.287637i $$0.0928695\pi$$
−0.957740 + 0.287637i $$0.907130\pi$$
$$60$$ 0 0
$$61$$ −23.0000 −0.377049 −0.188525 0.982068i $$-0.560371\pi$$
−0.188525 + 0.982068i $$0.560371\pi$$
$$62$$ 0 0
$$63$$ 253.000 4.01587
$$64$$ 0 0
$$65$$ − 79.1960i − 1.21840i
$$66$$ 0 0
$$67$$ 39.5980i 0.591015i 0.955340 + 0.295507i $$0.0954887\pi$$
−0.955340 + 0.295507i $$0.904511\pi$$
$$68$$ 0 0
$$69$$ − 11.3137i − 0.163967i
$$70$$ 0 0
$$71$$ − 90.5097i − 1.27478i −0.770540 0.637392i $$-0.780013\pi$$
0.770540 0.637392i $$-0.219987\pi$$
$$72$$ 0 0
$$73$$ 39.0000 0.534247 0.267123 0.963662i $$-0.413927\pi$$
0.267123 + 0.963662i $$0.413927\pi$$
$$74$$ 0 0
$$75$$ 135.765i 1.81019i
$$76$$ 0 0
$$77$$ −33.0000 −0.428571
$$78$$ 0 0
$$79$$ 96.1665i 1.21730i 0.793440 + 0.608649i $$0.208288\pi$$
−0.793440 + 0.608649i $$0.791712\pi$$
$$80$$ 0 0
$$81$$ 241.000 2.97531
$$82$$ 0 0
$$83$$ −6.00000 −0.0722892 −0.0361446 0.999347i $$-0.511508\pi$$
−0.0361446 + 0.999347i $$0.511508\pi$$
$$84$$ 0 0
$$85$$ 119.000 1.40000
$$86$$ 0 0
$$87$$ −224.000 −2.57471
$$88$$ 0 0
$$89$$ 118.794i 1.33476i 0.744716 + 0.667382i $$0.232585\pi$$
−0.744716 + 0.667382i $$0.767415\pi$$
$$90$$ 0 0
$$91$$ − 124.451i − 1.36759i
$$92$$ 0 0
$$93$$ 32.0000 0.344086
$$94$$ 0 0
$$95$$ 133.000 1.40000
$$96$$ 0 0
$$97$$ − 169.706i − 1.74954i −0.484536 0.874771i $$-0.661012\pi$$
0.484536 0.874771i $$-0.338988\pi$$
$$98$$ 0 0
$$99$$ −69.0000 −0.696970
$$100$$ 0 0
$$101$$ −122.000 −1.20792 −0.603960 0.797014i $$-0.706411\pi$$
−0.603960 + 0.797014i $$0.706411\pi$$
$$102$$ 0 0
$$103$$ − 101.823i − 0.988576i −0.869298 0.494288i $$-0.835429\pi$$
0.869298 0.494288i $$-0.164571\pi$$
$$104$$ 0 0
$$105$$ 435.578i 4.14836i
$$106$$ 0 0
$$107$$ 158.392i 1.48030i 0.672443 + 0.740149i $$0.265245\pi$$
−0.672443 + 0.740149i $$0.734755\pi$$
$$108$$ 0 0
$$109$$ 118.794i 1.08985i 0.838484 + 0.544926i $$0.183442\pi$$
−0.838484 + 0.544926i $$0.816558\pi$$
$$110$$ 0 0
$$111$$ −224.000 −2.01802
$$112$$ 0 0
$$113$$ 50.9117i 0.450546i 0.974296 + 0.225273i $$0.0723274\pi$$
−0.974296 + 0.225273i $$0.927673\pi$$
$$114$$ 0 0
$$115$$ 14.0000 0.121739
$$116$$ 0 0
$$117$$ − 260.215i − 2.22406i
$$118$$ 0 0
$$119$$ 187.000 1.57143
$$120$$ 0 0
$$121$$ −112.000 −0.925620
$$122$$ 0 0
$$123$$ −224.000 −1.82114
$$124$$ 0 0
$$125$$ 7.00000 0.0560000
$$126$$ 0 0
$$127$$ 39.5980i 0.311795i 0.987773 + 0.155898i $$0.0498270\pi$$
−0.987773 + 0.155898i $$0.950173\pi$$
$$128$$ 0 0
$$129$$ − 118.794i − 0.920883i
$$130$$ 0 0
$$131$$ −149.000 −1.13740 −0.568702 0.822543i $$-0.692554\pi$$
−0.568702 + 0.822543i $$0.692554\pi$$
$$132$$ 0 0
$$133$$ 209.000 1.57143
$$134$$ 0 0
$$135$$ 554.372i 4.10646i
$$136$$ 0 0
$$137$$ 95.0000 0.693431 0.346715 0.937970i $$-0.387297\pi$$
0.346715 + 0.937970i $$0.387297\pi$$
$$138$$ 0 0
$$139$$ 155.000 1.11511 0.557554 0.830141i $$-0.311740\pi$$
0.557554 + 0.830141i $$0.311740\pi$$
$$140$$ 0 0
$$141$$ 28.2843i 0.200598i
$$142$$ 0 0
$$143$$ 33.9411i 0.237351i
$$144$$ 0 0
$$145$$ − 277.186i − 1.91163i
$$146$$ 0 0
$$147$$ 407.294i 2.77070i
$$148$$ 0 0
$$149$$ −63.0000 −0.422819 −0.211409 0.977398i $$-0.567805\pi$$
−0.211409 + 0.977398i $$0.567805\pi$$
$$150$$ 0 0
$$151$$ 124.451i 0.824177i 0.911144 + 0.412089i $$0.135201\pi$$
−0.911144 + 0.412089i $$0.864799\pi$$
$$152$$ 0 0
$$153$$ 391.000 2.55556
$$154$$ 0 0
$$155$$ 39.5980i 0.255471i
$$156$$ 0 0
$$157$$ 150.000 0.955414 0.477707 0.878519i $$-0.341468\pi$$
0.477707 + 0.878519i $$0.341468\pi$$
$$158$$ 0 0
$$159$$ 32.0000 0.201258
$$160$$ 0 0
$$161$$ 22.0000 0.136646
$$162$$ 0 0
$$163$$ −166.000 −1.01840 −0.509202 0.860647i $$-0.670060\pi$$
−0.509202 + 0.860647i $$0.670060\pi$$
$$164$$ 0 0
$$165$$ − 118.794i − 0.719963i
$$166$$ 0 0
$$167$$ − 209.304i − 1.25332i −0.779295 0.626658i $$-0.784423\pi$$
0.779295 0.626658i $$-0.215577\pi$$
$$168$$ 0 0
$$169$$ 41.0000 0.242604
$$170$$ 0 0
$$171$$ 437.000 2.55556
$$172$$ 0 0
$$173$$ − 56.5685i − 0.326986i −0.986545 0.163493i $$-0.947724\pi$$
0.986545 0.163493i $$-0.0522761\pi$$
$$174$$ 0 0
$$175$$ −264.000 −1.50857
$$176$$ 0 0
$$177$$ −192.000 −1.08475
$$178$$ 0 0
$$179$$ − 73.5391i − 0.410833i −0.978675 0.205416i $$-0.934145\pi$$
0.978675 0.205416i $$-0.0658549\pi$$
$$180$$ 0 0
$$181$$ 79.1960i 0.437547i 0.975776 + 0.218773i $$0.0702055\pi$$
−0.975776 + 0.218773i $$0.929794\pi$$
$$182$$ 0 0
$$183$$ − 130.108i − 0.710971i
$$184$$ 0 0
$$185$$ − 277.186i − 1.49830i
$$186$$ 0 0
$$187$$ −51.0000 −0.272727
$$188$$ 0 0
$$189$$ 871.156i 4.60929i
$$190$$ 0 0
$$191$$ 301.000 1.57592 0.787958 0.615729i $$-0.211138\pi$$
0.787958 + 0.615729i $$0.211138\pi$$
$$192$$ 0 0
$$193$$ 152.735i 0.791373i 0.918386 + 0.395687i $$0.129493\pi$$
−0.918386 + 0.395687i $$0.870507\pi$$
$$194$$ 0 0
$$195$$ 448.000 2.29744
$$196$$ 0 0
$$197$$ −90.0000 −0.456853 −0.228426 0.973561i $$-0.573358\pi$$
−0.228426 + 0.973561i $$0.573358\pi$$
$$198$$ 0 0
$$199$$ −147.000 −0.738693 −0.369347 0.929292i $$-0.620419\pi$$
−0.369347 + 0.929292i $$0.620419\pi$$
$$200$$ 0 0
$$201$$ −224.000 −1.11443
$$202$$ 0 0
$$203$$ − 435.578i − 2.14570i
$$204$$ 0 0
$$205$$ − 277.186i − 1.35213i
$$206$$ 0 0
$$207$$ 46.0000 0.222222
$$208$$ 0 0
$$209$$ −57.0000 −0.272727
$$210$$ 0 0
$$211$$ − 328.098i − 1.55496i −0.628905 0.777482i $$-0.716496\pi$$
0.628905 0.777482i $$-0.283504\pi$$
$$212$$ 0 0
$$213$$ 512.000 2.40376
$$214$$ 0 0
$$215$$ 147.000 0.683721
$$216$$ 0 0
$$217$$ 62.2254i 0.286753i
$$218$$ 0 0
$$219$$ 220.617i 1.00739i
$$220$$ 0 0
$$221$$ − 192.333i − 0.870285i
$$222$$ 0 0
$$223$$ − 356.382i − 1.59812i −0.601248 0.799062i $$-0.705330\pi$$
0.601248 0.799062i $$-0.294670\pi$$
$$224$$ 0 0
$$225$$ −552.000 −2.45333
$$226$$ 0 0
$$227$$ − 316.784i − 1.39552i −0.716330 0.697762i $$-0.754179\pi$$
0.716330 0.697762i $$-0.245821\pi$$
$$228$$ 0 0
$$229$$ 257.000 1.12227 0.561135 0.827724i $$-0.310365\pi$$
0.561135 + 0.827724i $$0.310365\pi$$
$$230$$ 0 0
$$231$$ − 186.676i − 0.808122i
$$232$$ 0 0
$$233$$ −177.000 −0.759657 −0.379828 0.925057i $$-0.624017\pi$$
−0.379828 + 0.925057i $$0.624017\pi$$
$$234$$ 0 0
$$235$$ −35.0000 −0.148936
$$236$$ 0 0
$$237$$ −544.000 −2.29536
$$238$$ 0 0
$$239$$ −363.000 −1.51883 −0.759414 0.650607i $$-0.774514\pi$$
−0.759414 + 0.650607i $$0.774514\pi$$
$$240$$ 0 0
$$241$$ 356.382i 1.47876i 0.673287 + 0.739381i $$0.264882\pi$$
−0.673287 + 0.739381i $$0.735118\pi$$
$$242$$ 0 0
$$243$$ 650.538i 2.67711i
$$244$$ 0 0
$$245$$ −504.000 −2.05714
$$246$$ 0 0
$$247$$ − 214.960i − 0.870285i
$$248$$ 0 0
$$249$$ − 33.9411i − 0.136310i
$$250$$ 0 0
$$251$$ −133.000 −0.529880 −0.264940 0.964265i $$-0.585352\pi$$
−0.264940 + 0.964265i $$0.585352\pi$$
$$252$$ 0 0
$$253$$ −6.00000 −0.0237154
$$254$$ 0 0
$$255$$ 673.166i 2.63987i
$$256$$ 0 0
$$257$$ − 135.765i − 0.528267i −0.964486 0.264133i $$-0.914914\pi$$
0.964486 0.264133i $$-0.0850859\pi$$
$$258$$ 0 0
$$259$$ − 435.578i − 1.68177i
$$260$$ 0 0
$$261$$ − 910.754i − 3.48948i
$$262$$ 0 0
$$263$$ 101.000 0.384030 0.192015 0.981392i $$-0.438498\pi$$
0.192015 + 0.981392i $$0.438498\pi$$
$$264$$ 0 0
$$265$$ 39.5980i 0.149426i
$$266$$ 0 0
$$267$$ −672.000 −2.51685
$$268$$ 0 0
$$269$$ − 356.382i − 1.32484i −0.749133 0.662420i $$-0.769530\pi$$
0.749133 0.662420i $$-0.230470\pi$$
$$270$$ 0 0
$$271$$ 142.000 0.523985 0.261993 0.965070i $$-0.415620\pi$$
0.261993 + 0.965070i $$0.415620\pi$$
$$272$$ 0 0
$$273$$ 704.000 2.57875
$$274$$ 0 0
$$275$$ 72.0000 0.261818
$$276$$ 0 0
$$277$$ −199.000 −0.718412 −0.359206 0.933258i $$-0.616952\pi$$
−0.359206 + 0.933258i $$0.616952\pi$$
$$278$$ 0 0
$$279$$ 130.108i 0.466336i
$$280$$ 0 0
$$281$$ 463.862i 1.65075i 0.564582 + 0.825377i $$0.309038\pi$$
−0.564582 + 0.825377i $$0.690962\pi$$
$$282$$ 0 0
$$283$$ 427.000 1.50883 0.754417 0.656395i $$-0.227920\pi$$
0.754417 + 0.656395i $$0.227920\pi$$
$$284$$ 0 0
$$285$$ 752.362i 2.63987i
$$286$$ 0 0
$$287$$ − 435.578i − 1.51769i
$$288$$ 0 0
$$289$$ 0 0
$$290$$ 0 0
$$291$$ 960.000 3.29897
$$292$$ 0 0
$$293$$ 237.588i 0.810880i 0.914122 + 0.405440i $$0.132882\pi$$
−0.914122 + 0.405440i $$0.867118\pi$$
$$294$$ 0 0
$$295$$ − 237.588i − 0.805383i
$$296$$ 0 0
$$297$$ − 237.588i − 0.799959i
$$298$$ 0 0
$$299$$ − 22.6274i − 0.0756770i
$$300$$ 0 0
$$301$$ 231.000 0.767442
$$302$$ 0 0
$$303$$ − 690.136i − 2.27768i
$$304$$ 0 0
$$305$$ 161.000 0.527869
$$306$$ 0 0
$$307$$ − 526.087i − 1.71364i −0.515616 0.856820i $$-0.672437\pi$$
0.515616 0.856820i $$-0.327563\pi$$
$$308$$ 0 0
$$309$$ 576.000 1.86408
$$310$$ 0 0
$$311$$ −235.000 −0.755627 −0.377814 0.925882i $$-0.623324\pi$$
−0.377814 + 0.925882i $$0.623324\pi$$
$$312$$ 0 0
$$313$$ 530.000 1.69329 0.846645 0.532158i $$-0.178619\pi$$
0.846645 + 0.532158i $$0.178619\pi$$
$$314$$ 0 0
$$315$$ −1771.00 −5.62222
$$316$$ 0 0
$$317$$ 214.960i 0.678109i 0.940767 + 0.339054i $$0.110107\pi$$
−0.940767 + 0.339054i $$0.889893\pi$$
$$318$$ 0 0
$$319$$ 118.794i 0.372395i
$$320$$ 0 0
$$321$$ −896.000 −2.79128
$$322$$ 0 0
$$323$$ 323.000 1.00000
$$324$$ 0 0
$$325$$ 271.529i 0.835474i
$$326$$ 0 0
$$327$$ −672.000 −2.05505
$$328$$ 0 0
$$329$$ −55.0000 −0.167173
$$330$$ 0 0
$$331$$ 96.1665i 0.290533i 0.989393 + 0.145267i $$0.0464040\pi$$
−0.989393 + 0.145267i $$0.953596\pi$$
$$332$$ 0 0
$$333$$ − 910.754i − 2.73500i
$$334$$ 0 0
$$335$$ − 277.186i − 0.827420i
$$336$$ 0 0
$$337$$ − 169.706i − 0.503578i −0.967782 0.251789i $$-0.918981\pi$$
0.967782 0.251789i $$-0.0810188\pi$$
$$338$$ 0 0
$$339$$ −288.000 −0.849558
$$340$$ 0 0
$$341$$ − 16.9706i − 0.0497670i
$$342$$ 0 0
$$343$$ −253.000 −0.737609
$$344$$ 0 0
$$345$$ 79.1960i 0.229554i
$$346$$ 0 0
$$347$$ −253.000 −0.729107 −0.364553 0.931183i $$-0.618778\pi$$
−0.364553 + 0.931183i $$0.618778\pi$$
$$348$$ 0 0
$$349$$ −351.000 −1.00573 −0.502865 0.864365i $$-0.667721\pi$$
−0.502865 + 0.864365i $$0.667721\pi$$
$$350$$ 0 0
$$351$$ 896.000 2.55271
$$352$$ 0 0
$$353$$ 98.0000 0.277620 0.138810 0.990319i $$-0.455672\pi$$
0.138810 + 0.990319i $$0.455672\pi$$
$$354$$ 0 0
$$355$$ 633.568i 1.78470i
$$356$$ 0 0
$$357$$ 1057.83i 2.96311i
$$358$$ 0 0
$$359$$ 61.0000 0.169916 0.0849582 0.996385i $$-0.472924\pi$$
0.0849582 + 0.996385i $$0.472924\pi$$
$$360$$ 0 0
$$361$$ 361.000 1.00000
$$362$$ 0 0
$$363$$ − 633.568i − 1.74537i
$$364$$ 0 0
$$365$$ −273.000 −0.747945
$$366$$ 0 0
$$367$$ 686.000 1.86921 0.934605 0.355688i $$-0.115753\pi$$
0.934605 + 0.355688i $$0.115753\pi$$
$$368$$ 0 0
$$369$$ − 910.754i − 2.46817i
$$370$$ 0 0
$$371$$ 62.2254i 0.167723i
$$372$$ 0 0
$$373$$ − 135.765i − 0.363980i −0.983300 0.181990i $$-0.941746\pi$$
0.983300 0.181990i $$-0.0582538\pi$$
$$374$$ 0 0
$$375$$ 39.5980i 0.105595i
$$376$$ 0 0
$$377$$ −448.000 −1.18833
$$378$$ 0 0
$$379$$ 79.1960i 0.208960i 0.994527 + 0.104480i $$0.0333179\pi$$
−0.994527 + 0.104480i $$0.966682\pi$$
$$380$$ 0 0
$$381$$ −224.000 −0.587927
$$382$$ 0 0
$$383$$ 16.9706i 0.0443096i 0.999755 + 0.0221548i $$0.00705266\pi$$
−0.999755 + 0.0221548i $$0.992947\pi$$
$$384$$ 0 0
$$385$$ 231.000 0.600000
$$386$$ 0 0
$$387$$ 483.000 1.24806
$$388$$ 0 0
$$389$$ −599.000 −1.53985 −0.769923 0.638137i $$-0.779705\pi$$
−0.769923 + 0.638137i $$0.779705\pi$$
$$390$$ 0 0
$$391$$ 34.0000 0.0869565
$$392$$ 0 0
$$393$$ − 842.871i − 2.14471i
$$394$$ 0 0
$$395$$ − 673.166i − 1.70422i
$$396$$ 0 0
$$397$$ 569.000 1.43325 0.716625 0.697459i $$-0.245686\pi$$
0.716625 + 0.697459i $$0.245686\pi$$
$$398$$ 0 0
$$399$$ 1182.28i 2.96311i
$$400$$ 0 0
$$401$$ − 16.9706i − 0.0423206i −0.999776 0.0211603i $$-0.993264\pi$$
0.999776 0.0211603i $$-0.00673604\pi$$
$$402$$ 0 0
$$403$$ 64.0000 0.158809
$$404$$ 0 0
$$405$$ −1687.00 −4.16543
$$406$$ 0 0
$$407$$ 118.794i 0.291877i
$$408$$ 0 0
$$409$$ 593.970i 1.45225i 0.687563 + 0.726124i $$0.258680\pi$$
−0.687563 + 0.726124i $$0.741320\pi$$
$$410$$ 0 0
$$411$$ 537.401i 1.30755i
$$412$$ 0 0
$$413$$ − 373.352i − 0.904001i
$$414$$ 0 0
$$415$$ 42.0000 0.101205
$$416$$ 0 0
$$417$$ 876.812i 2.10267i
$$418$$ 0 0
$$419$$ −230.000 −0.548926 −0.274463 0.961598i $$-0.588500\pi$$
−0.274463 + 0.961598i $$0.588500\pi$$
$$420$$ 0 0
$$421$$ 593.970i 1.41085i 0.708782 + 0.705427i $$0.249245\pi$$
−0.708782 + 0.705427i $$0.750755\pi$$
$$422$$ 0 0
$$423$$ −115.000 −0.271868
$$424$$ 0 0
$$425$$ −408.000 −0.960000
$$426$$ 0 0
$$427$$ 253.000 0.592506
$$428$$ 0 0
$$429$$ −192.000 −0.447552
$$430$$ 0 0
$$431$$ 475.176i 1.10250i 0.834341 + 0.551248i $$0.185848\pi$$
−0.834341 + 0.551248i $$0.814152\pi$$
$$432$$ 0 0
$$433$$ 593.970i 1.37175i 0.727717 + 0.685877i $$0.240581\pi$$
−0.727717 + 0.685877i $$0.759419\pi$$
$$434$$ 0 0
$$435$$ 1568.00 3.60460
$$436$$ 0 0
$$437$$ 38.0000 0.0869565
$$438$$ 0 0
$$439$$ − 118.794i − 0.270601i −0.990805 0.135301i $$-0.956800\pi$$
0.990805 0.135301i $$-0.0432000\pi$$
$$440$$ 0 0
$$441$$ −1656.00 −3.75510
$$442$$ 0 0
$$443$$ −653.000 −1.47404 −0.737020 0.675871i $$-0.763768\pi$$
−0.737020 + 0.675871i $$0.763768\pi$$
$$444$$ 0 0
$$445$$ − 831.558i − 1.86867i
$$446$$ 0 0
$$447$$ − 356.382i − 0.797275i
$$448$$ 0 0
$$449$$ − 147.078i − 0.327568i −0.986496 0.163784i $$-0.947630\pi$$
0.986496 0.163784i $$-0.0523701\pi$$
$$450$$ 0 0
$$451$$ 118.794i 0.263401i
$$452$$ 0 0
$$453$$ −704.000 −1.55408
$$454$$ 0 0
$$455$$ 871.156i 1.91463i
$$456$$ 0 0
$$457$$ −817.000 −1.78775 −0.893873 0.448320i $$-0.852022\pi$$
−0.893873 + 0.448320i $$0.852022\pi$$
$$458$$ 0 0
$$459$$ 1346.33i 2.93318i
$$460$$ 0 0
$$461$$ −463.000 −1.00434 −0.502169 0.864769i $$-0.667465\pi$$
−0.502169 + 0.864769i $$0.667465\pi$$
$$462$$ 0 0
$$463$$ 29.0000 0.0626350 0.0313175 0.999509i $$-0.490030\pi$$
0.0313175 + 0.999509i $$0.490030\pi$$
$$464$$ 0 0
$$465$$ −224.000 −0.481720
$$466$$ 0 0
$$467$$ 19.0000 0.0406852 0.0203426 0.999793i $$-0.493524\pi$$
0.0203426 + 0.999793i $$0.493524\pi$$
$$468$$ 0 0
$$469$$ − 435.578i − 0.928737i
$$470$$ 0 0
$$471$$ 848.528i 1.80155i
$$472$$ 0 0
$$473$$ −63.0000 −0.133192
$$474$$ 0 0
$$475$$ −456.000 −0.960000
$$476$$ 0 0
$$477$$ 130.108i 0.272762i
$$478$$ 0 0
$$479$$ 206.000 0.430063 0.215031 0.976607i $$-0.431015\pi$$
0.215031 + 0.976607i $$0.431015\pi$$
$$480$$ 0 0
$$481$$ −448.000 −0.931393
$$482$$ 0 0
$$483$$ 124.451i 0.257662i
$$484$$ 0 0
$$485$$ 1187.94i 2.44936i
$$486$$ 0 0
$$487$$ − 45.2548i − 0.0929257i −0.998920 0.0464629i $$-0.985205\pi$$
0.998920 0.0464629i $$-0.0147949\pi$$
$$488$$ 0 0
$$489$$ − 939.038i − 1.92032i
$$490$$ 0 0
$$491$$ −246.000 −0.501018 −0.250509 0.968114i $$-0.580598\pi$$
−0.250509 + 0.968114i $$0.580598\pi$$
$$492$$ 0 0
$$493$$ − 673.166i − 1.36545i
$$494$$ 0 0
$$495$$ 483.000 0.975758
$$496$$ 0 0
$$497$$ 995.606i 2.00323i
$$498$$ 0 0
$$499$$ −21.0000 −0.0420842 −0.0210421 0.999779i $$-0.506698\pi$$
−0.0210421 + 0.999779i $$0.506698\pi$$
$$500$$ 0 0
$$501$$ 1184.00 2.36327
$$502$$ 0 0
$$503$$ −386.000 −0.767396 −0.383698 0.923459i $$-0.625350\pi$$
−0.383698 + 0.923459i $$0.625350\pi$$
$$504$$ 0 0
$$505$$ 854.000 1.69109
$$506$$ 0 0
$$507$$ 231.931i 0.457458i
$$508$$ 0 0
$$509$$ 475.176i 0.933548i 0.884377 + 0.466774i $$0.154584\pi$$
−0.884377 + 0.466774i $$0.845416\pi$$
$$510$$ 0 0
$$511$$ −429.000 −0.839530
$$512$$ 0 0
$$513$$ 1504.72i 2.93318i
$$514$$ 0 0
$$515$$ 712.764i 1.38401i
$$516$$ 0 0
$$517$$ 15.0000 0.0290135
$$518$$ 0 0
$$519$$ 320.000 0.616570
$$520$$ 0 0
$$521$$ − 786.303i − 1.50922i −0.656175 0.754609i $$-0.727826\pi$$
0.656175 0.754609i $$-0.272174\pi$$
$$522$$ 0 0
$$523$$ 395.980i 0.757132i 0.925574 + 0.378566i $$0.123583\pi$$
−0.925574 + 0.378566i $$0.876417\pi$$
$$524$$ 0 0
$$525$$ − 1493.41i − 2.84459i
$$526$$ 0 0
$$527$$ 96.1665i 0.182479i
$$528$$ 0 0
$$529$$ −525.000 −0.992439
$$530$$ 0 0
$$531$$ − 780.646i − 1.47014i
$$532$$ 0 0
$$533$$ −448.000 −0.840525
$$534$$ 0 0
$$535$$ − 1108.74i − 2.07242i
$$536$$ 0 0
$$537$$ 416.000 0.774674
$$538$$ 0 0
$$539$$ 216.000 0.400742
$$540$$ 0 0
$$541$$ −7.00000 −0.0129390 −0.00646950 0.999979i $$-0.502059\pi$$
−0.00646950 + 0.999979i $$0.502059\pi$$
$$542$$ 0 0
$$543$$ −448.000 −0.825046
$$544$$ 0 0
$$545$$ − 831.558i − 1.52579i
$$546$$ 0 0
$$547$$ 712.764i 1.30304i 0.758631 + 0.651521i $$0.225869\pi$$
−0.758631 + 0.651521i $$0.774131\pi$$
$$548$$ 0 0
$$549$$ 529.000 0.963570
$$550$$ 0 0
$$551$$ − 752.362i − 1.36545i
$$552$$ 0 0
$$553$$ − 1057.83i − 1.91290i
$$554$$ 0 0
$$555$$ 1568.00 2.82523
$$556$$ 0 0
$$557$$ 1001.00 1.79713 0.898564 0.438843i $$-0.144612\pi$$
0.898564 + 0.438843i $$0.144612\pi$$
$$558$$ 0 0
$$559$$ − 237.588i − 0.425023i
$$560$$ 0 0
$$561$$ − 288.500i − 0.514259i
$$562$$ 0 0
$$563$$ − 322.441i − 0.572719i −0.958122 0.286359i $$-0.907555\pi$$
0.958122 0.286359i $$-0.0924451\pi$$
$$564$$ 0 0
$$565$$ − 356.382i − 0.630764i
$$566$$ 0 0
$$567$$ −2651.00 −4.67549
$$568$$ 0 0
$$569$$ − 712.764i − 1.25266i −0.779558 0.626330i $$-0.784556\pi$$
0.779558 0.626330i $$-0.215444\pi$$
$$570$$ 0 0
$$571$$ 746.000 1.30648 0.653240 0.757151i $$-0.273409\pi$$
0.653240 + 0.757151i $$0.273409\pi$$
$$572$$ 0 0
$$573$$ 1702.71i 2.97158i
$$574$$ 0 0
$$575$$ −48.0000 −0.0834783
$$576$$ 0 0
$$577$$ −25.0000 −0.0433276 −0.0216638 0.999765i $$-0.506896\pi$$
−0.0216638 + 0.999765i $$0.506896\pi$$
$$578$$ 0 0
$$579$$ −864.000 −1.49223
$$580$$ 0 0
$$581$$ 66.0000 0.113597
$$582$$ 0 0
$$583$$ − 16.9706i − 0.0291090i
$$584$$ 0 0
$$585$$ 1821.51i 3.11369i
$$586$$ 0 0
$$587$$ −869.000 −1.48041 −0.740204 0.672382i $$-0.765271\pi$$
−0.740204 + 0.672382i $$0.765271\pi$$
$$588$$ 0 0
$$589$$ 107.480i 0.182479i
$$590$$ 0 0
$$591$$ − 509.117i − 0.861450i
$$592$$ 0 0
$$593$$ 898.000 1.51433 0.757167 0.653221i $$-0.226583\pi$$
0.757167 + 0.653221i $$0.226583\pi$$
$$594$$ 0 0
$$595$$ −1309.00 −2.20000
$$596$$ 0 0
$$597$$ − 831.558i − 1.39289i
$$598$$ 0 0
$$599$$ 277.186i 0.462748i 0.972865 + 0.231374i $$0.0743220\pi$$
−0.972865 + 0.231374i $$0.925678\pi$$
$$600$$ 0 0
$$601$$ − 475.176i − 0.790642i −0.918543 0.395321i $$-0.870633\pi$$
0.918543 0.395321i $$-0.129367\pi$$
$$602$$ 0 0
$$603$$ − 910.754i − 1.51037i
$$604$$ 0 0
$$605$$ 784.000 1.29587
$$606$$ 0 0
$$607$$ 452.548i 0.745549i 0.927922 + 0.372775i $$0.121594\pi$$
−0.927922 + 0.372775i $$0.878406\pi$$
$$608$$ 0 0
$$609$$ 2464.00 4.04598
$$610$$ 0 0
$$611$$ 56.5685i 0.0925835i
$$612$$ 0 0
$$613$$ 585.000 0.954323 0.477162 0.878816i $$-0.341666\pi$$
0.477162 + 0.878816i $$0.341666\pi$$
$$614$$ 0 0
$$615$$ 1568.00 2.54959
$$616$$ 0 0
$$617$$ −873.000 −1.41491 −0.707455 0.706758i $$-0.750157\pi$$
−0.707455 + 0.706758i $$0.750157\pi$$
$$618$$ 0 0
$$619$$ 970.000 1.56704 0.783522 0.621364i $$-0.213421\pi$$
0.783522 + 0.621364i $$0.213421\pi$$
$$620$$ 0 0
$$621$$ 158.392i 0.255059i
$$622$$ 0 0
$$623$$ − 1306.73i − 2.09749i
$$624$$ 0 0
$$625$$ −649.000 −1.03840
$$626$$ 0 0
$$627$$ − 322.441i − 0.514259i
$$628$$ 0 0
$$629$$ − 673.166i − 1.07022i
$$630$$ 0 0
$$631$$ −259.000 −0.410460 −0.205230 0.978714i $$-0.565794\pi$$
−0.205230 + 0.978714i $$0.565794\pi$$
$$632$$ 0 0
$$633$$ 1856.00 2.93207
$$634$$ 0 0
$$635$$ − 277.186i − 0.436513i
$$636$$ 0 0
$$637$$ 814.587i 1.27879i
$$638$$ 0 0
$$639$$ 2081.72i 3.25778i
$$640$$ 0 0
$$641$$ 1063.49i 1.65911i 0.558426 + 0.829554i $$0.311405\pi$$
−0.558426 + 0.829554i $$0.688595\pi$$
$$642$$ 0 0
$$643$$ −645.000 −1.00311 −0.501555 0.865126i $$-0.667239\pi$$
−0.501555 + 0.865126i $$0.667239\pi$$
$$644$$ 0 0
$$645$$ 831.558i 1.28924i
$$646$$ 0 0
$$647$$ 29.0000 0.0448223 0.0224111 0.999749i $$-0.492866\pi$$
0.0224111 + 0.999749i $$0.492866\pi$$
$$648$$ 0 0
$$649$$ 101.823i 0.156893i
$$650$$ 0 0
$$651$$ −352.000 −0.540707
$$652$$ 0 0
$$653$$ −135.000 −0.206738 −0.103369 0.994643i $$-0.532962\pi$$
−0.103369 + 0.994643i $$0.532962\pi$$
$$654$$ 0 0
$$655$$ 1043.00 1.59237
$$656$$ 0 0
$$657$$ −897.000 −1.36530
$$658$$ 0 0
$$659$$ 780.646i 1.18459i 0.805721 + 0.592296i $$0.201778\pi$$
−0.805721 + 0.592296i $$0.798222\pi$$
$$660$$ 0 0
$$661$$ 995.606i 1.50621i 0.657899 + 0.753106i $$0.271445\pi$$
−0.657899 + 0.753106i $$0.728555\pi$$
$$662$$ 0 0
$$663$$ 1088.00 1.64103
$$664$$ 0 0
$$665$$ −1463.00 −2.20000
$$666$$ 0 0
$$667$$ − 79.1960i − 0.118735i
$$668$$ 0 0
$$669$$ 2016.00 3.01345
$$670$$ 0 0
$$671$$ −69.0000 −0.102832
$$672$$ 0 0
$$673$$ − 395.980i − 0.588380i −0.955747 0.294190i $$-0.904950\pi$$
0.955747 0.294190i $$-0.0950499\pi$$
$$674$$ 0 0
$$675$$ − 1900.70i − 2.81586i
$$676$$ 0 0
$$677$$ − 356.382i − 0.526413i −0.964739 0.263207i $$-0.915220\pi$$
0.964739 0.263207i $$-0.0847801\pi$$
$$678$$ 0 0
$$679$$ 1866.76i 2.74928i
$$680$$ 0 0
$$681$$ 1792.00 2.63142
$$682$$ 0 0
$$683$$ 520.431i 0.761977i 0.924580 + 0.380989i $$0.124416\pi$$
−0.924580 + 0.380989i $$0.875584\pi$$
$$684$$ 0 0
$$685$$ −665.000 −0.970803
$$686$$ 0 0
$$687$$ 1453.81i 2.11617i
$$688$$ 0 0
$$689$$ 64.0000 0.0928882
$$690$$ 0 0
$$691$$ 835.000 1.20839 0.604197 0.796835i $$-0.293494\pi$$
0.604197 + 0.796835i $$0.293494\pi$$
$$692$$ 0 0
$$693$$ 759.000 1.09524
$$694$$ 0 0
$$695$$ −1085.00 −1.56115
$$696$$ 0 0
$$697$$ − 673.166i − 0.965804i
$$698$$ 0 0
$$699$$ − 1001.26i − 1.43242i
$$700$$ 0 0
$$701$$ −1002.00 −1.42939 −0.714693 0.699438i $$-0.753434\pi$$
−0.714693 + 0.699438i $$0.753434\pi$$
$$702$$ 0 0
$$703$$ − 752.362i − 1.07022i
$$704$$ 0 0
$$705$$ − 197.990i − 0.280837i
$$706$$ 0 0
$$707$$ 1342.00 1.89816
$$708$$ 0 0
$$709$$ −250.000 −0.352609 −0.176305 0.984336i $$-0.556414\pi$$
−0.176305 + 0.984336i $$0.556414\pi$$
$$710$$ 0 0
$$711$$ − 2211.83i − 3.11087i
$$712$$ 0 0
$$713$$ 11.3137i 0.0158678i
$$714$$ 0 0
$$715$$ − 237.588i − 0.332291i
$$716$$ 0 0
$$717$$ − 2053.44i − 2.86393i
$$718$$ 0 0
$$719$$ −1171.00 −1.62865 −0.814325 0.580409i $$-0.802893\pi$$
−0.814325 + 0.580409i $$0.802893\pi$$
$$720$$ 0 0
$$721$$ 1120.06i 1.55348i
$$722$$ 0 0
$$723$$ −2016.00 −2.78838
$$724$$ 0 0
$$725$$ 950.352i 1.31083i
$$726$$ 0 0
$$727$$ −91.0000 −0.125172 −0.0625860 0.998040i $$-0.519935\pi$$
−0.0625860 + 0.998040i $$0.519935\pi$$
$$728$$ 0 0
$$729$$ −1511.00 −2.07270
$$730$$ 0 0
$$731$$ 357.000 0.488372
$$732$$ 0 0
$$733$$ −682.000 −0.930423 −0.465211 0.885200i $$-0.654022\pi$$
−0.465211 + 0.885200i $$0.654022\pi$$
$$734$$ 0 0
$$735$$ − 2851.05i − 3.87899i
$$736$$ 0 0
$$737$$ 118.794i 0.161186i
$$738$$ 0 0
$$739$$ −221.000 −0.299053 −0.149526 0.988758i $$-0.547775\pi$$
−0.149526 + 0.988758i $$0.547775\pi$$
$$740$$ 0 0
$$741$$ 1216.00 1.64103
$$742$$ 0 0
$$743$$ 288.500i 0.388290i 0.980973 + 0.194145i $$0.0621933\pi$$
−0.980973 + 0.194145i $$0.937807\pi$$
$$744$$ 0 0
$$745$$ 441.000 0.591946
$$746$$ 0 0
$$747$$ 138.000 0.184739
$$748$$ 0 0
$$749$$ − 1742.31i − 2.32618i
$$750$$ 0 0
$$751$$ − 927.724i − 1.23532i −0.786446 0.617659i $$-0.788081\pi$$
0.786446 0.617659i $$-0.211919\pi$$
$$752$$ 0 0
$$753$$ − 752.362i − 0.999152i
$$754$$ 0 0
$$755$$ − 871.156i − 1.15385i
$$756$$ 0 0
$$757$$ −895.000 −1.18230 −0.591149 0.806562i $$-0.701326\pi$$
−0.591149 + 0.806562i $$0.701326\pi$$
$$758$$ 0 0
$$759$$ − 33.9411i − 0.0447182i
$$760$$ 0 0
$$761$$ 95.0000 0.124836 0.0624179 0.998050i $$-0.480119\pi$$
0.0624179 + 0.998050i $$0.480119\pi$$
$$762$$ 0 0
$$763$$ − 1306.73i − 1.71263i
$$764$$ 0 0
$$765$$ −2737.00 −3.57778
$$766$$ 0 0
$$767$$ −384.000 −0.500652
$$768$$ 0 0
$$769$$ 679.000 0.882965 0.441482 0.897270i $$-0.354453\pi$$
0.441482 + 0.897270i $$0.354453\pi$$
$$770$$ 0 0
$$771$$ 768.000 0.996109
$$772$$ 0 0
$$773$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$774$$ 0 0
$$775$$ − 135.765i − 0.175180i
$$776$$ 0 0
$$777$$ 2464.00 3.17117
$$778$$ 0 0
$$779$$ − 752.362i − 0.965804i
$$780$$ 0 0
$$781$$ − 271.529i − 0.347668i
$$782$$ 0 0
$$783$$ 3136.00 4.00511
$$784$$ 0 0
$$785$$ −1050.00 −1.33758
$$786$$ 0 0
$$787$$ − 1391.59i − 1.76822i −0.467282 0.884108i $$-0.654767\pi$$
0.467282 0.884108i $$-0.345233\pi$$
$$788$$ 0 0
$$789$$ 571.342i 0.724135i
$$790$$ 0 0
$$791$$ − 560.029i − 0.708001i
$$792$$ 0 0
$$793$$ − 260.215i − 0.328140i
$$794$$ 0 0
$$795$$ −224.000 −0.281761
$$796$$ 0 0
$$797$$ 356.382i 0.447154i 0.974686 + 0.223577i $$0.0717734\pi$$
−0.974686 + 0.223577i $$0.928227\pi$$
$$798$$ 0 0
$$799$$ −85.0000 −0.106383
$$800$$ 0 0
$$801$$ − 2732.26i − 3.41106i
$$802$$ 0 0
$$803$$ 117.000 0.145704
$$804$$ 0 0
$$805$$ −154.000 −0.191304
$$806$$ 0 0
$$807$$ 2016.00 2.49814
$$808$$ 0 0
$$809$$ 455.000 0.562423 0.281211 0.959646i $$-0.409264\pi$$
0.281211 + 0.959646i $$0.409264\pi$$
$$810$$ 0 0
$$811$$ 475.176i 0.585913i 0.956126 + 0.292957i $$0.0946392\pi$$
−0.956126 + 0.292957i $$0.905361\pi$$
$$812$$ 0 0
$$813$$ 803.273i 0.988036i
$$814$$ 0 0
$$815$$ 1162.00 1.42577
$$816$$ 0 0
$$817$$ 399.000 0.488372
$$818$$ 0 0
$$819$$ 2862.37i 3.49496i
$$820$$ 0 0
$$821$$ −831.000 −1.01218 −0.506090 0.862481i $$-0.668910\pi$$
−0.506090 + 0.862481i $$0.668910\pi$$
$$822$$ 0 0
$$823$$ 109.000 0.132442 0.0662211 0.997805i $$-0.478906\pi$$
0.0662211 + 0.997805i $$0.478906\pi$$
$$824$$ 0 0
$$825$$ 407.294i 0.493689i
$$826$$ 0 0
$$827$$ 1238.85i 1.49801i 0.662567 + 0.749003i $$0.269467\pi$$
−0.662567 + 0.749003i $$0.730533\pi$$
$$828$$ 0 0
$$829$$ 667.509i 0.805198i 0.915377 + 0.402599i $$0.131893\pi$$
−0.915377 + 0.402599i $$0.868107\pi$$
$$830$$ 0 0
$$831$$ − 1125.71i − 1.35465i
$$832$$ 0 0
$$833$$ −1224.00 −1.46939
$$834$$ 0 0
$$835$$ 1465.13i 1.75464i
$$836$$ 0 0
$$837$$ −448.000 −0.535245
$$838$$ 0 0
$$839$$ − 316.784i − 0.377573i −0.982018 0.188787i $$-0.939545\pi$$
0.982018 0.188787i $$-0.0604554\pi$$
$$840$$ 0 0
$$841$$ −727.000 −0.864447
$$842$$ 0 0
$$843$$ −2624.00 −3.11269
$$844$$ 0 0
$$845$$ −287.000 −0.339645
$$846$$ 0 0
$$847$$ 1232.00 1.45455
$$848$$ 0 0
$$849$$ 2415.48i 2.84508i
$$850$$ 0 0
$$851$$ − 79.1960i − 0.0930622i
$$852$$ 0 0
$$853$$ 1126.00 1.32005 0.660023 0.751245i $$-0.270546\pi$$
0.660023 + 0.751245i $$0.270546\pi$$
$$854$$ 0 0
$$855$$ −3059.00 −3.57778
$$856$$ 0 0
$$857$$ − 203.647i − 0.237627i −0.992917 0.118814i $$-0.962091\pi$$
0.992917 0.118814i $$-0.0379091\pi$$
$$858$$ 0 0
$$859$$ 651.000 0.757858 0.378929 0.925426i $$-0.376292\pi$$
0.378929 + 0.925426i $$0.376292\pi$$
$$860$$ 0 0
$$861$$ 2464.00 2.86179
$$862$$ 0 0
$$863$$ − 39.5980i − 0.0458841i −0.999737 0.0229421i $$-0.992697\pi$$
0.999737 0.0229421i $$-0.00730332\pi$$
$$864$$ 0 0
$$865$$ 395.980i 0.457780i
$$866$$ 0 0
$$867$$ 0 0
$$868$$ 0 0
$$869$$ 288.500i 0.331990i
$$870$$ 0 0
$$871$$ −448.000 −0.514351
$$872$$ 0 0
$$873$$ 3903.23i 4.47105i
$$874$$ 0 0
$$875$$ −77.0000 −0.0880000
$$876$$ 0 0
$$877$$ − 1029.55i − 1.17394i −0.809608 0.586971i $$-0.800320\pi$$
0.809608 0.586971i $$-0.199680\pi$$
$$878$$ 0 0
$$879$$ −1344.00 −1.52901
$$880$$ 0 0
$$881$$ 391.000 0.443814 0.221907 0.975068i $$-0.428772\pi$$
0.221907 + 0.975068i $$0.428772\pi$$
$$882$$ 0 0
$$883$$ 995.000 1.12684 0.563420 0.826171i $$-0.309485\pi$$
0.563420 + 0.826171i $$0.309485\pi$$
$$884$$ 0 0
$$885$$ 1344.00 1.51864
$$886$$ 0 0
$$887$$ 475.176i 0.535711i 0.963459 + 0.267856i $$0.0863150\pi$$
−0.963459 + 0.267856i $$0.913685\pi$$
$$888$$ 0 0
$$889$$ − 435.578i − 0.489964i
$$890$$ 0 0
$$891$$ 723.000 0.811448
$$892$$ 0 0
$$893$$ −95.0000 −0.106383
$$894$$ 0 0
$$895$$ 514.774i 0.575166i
$$896$$ 0 0
$$897$$ 128.000 0.142698
$$898$$ 0 0
$$899$$ 224.000 0.249166
$$900$$ 0 0
$$901$$ 96.1665i 0.106733i
$$902$$ 0 0
$$903$$ 1306.73i 1.44710i
$$904$$ 0 0
$$905$$ − 554.372i − 0.612565i
$$906$$ 0 0
$$907$$ − 627.911i − 0.692294i −0.938180 0.346147i $$-0.887490\pi$$
0.938180 0.346147i $$-0.112510\pi$$
$$908$$ 0 0
$$909$$ 2806.00 3.08691
$$910$$ 0 0
$$911$$ 803.273i 0.881749i 0.897569 + 0.440874i $$0.145332\pi$$
−0.897569 + 0.440874i $$0.854668\pi$$
$$912$$ 0 0
$$913$$ −18.0000 −0.0197152
$$914$$ 0 0
$$915$$ 910.754i 0.995359i
$$916$$ 0 0
$$917$$ 1639.00 1.78735
$$918$$ 0 0
$$919$$ −1090.00 −1.18607 −0.593036 0.805176i $$-0.702071\pi$$
−0.593036 + 0.805176i $$0.702071\pi$$
$$920$$ 0 0
$$921$$ 2976.00 3.23127
$$922$$ 0 0
$$923$$ 1024.00 1.10943
$$924$$ 0 0
$$925$$ 950.352i 1.02741i
$$926$$ 0 0
$$927$$ 2341.94i 2.52636i
$$928$$ 0 0
$$929$$ 226.000 0.243272 0.121636 0.992575i $$-0.461186\pi$$
0.121636 + 0.992575i $$0.461186\pi$$
$$930$$ 0 0
$$931$$ −1368.00 −1.46939
$$932$$ 0 0
$$933$$ − 1329.36i − 1.42482i
$$934$$ 0 0
$$935$$ 357.000 0.381818
$$936$$ 0 0
$$937$$ 623.000 0.664888 0.332444 0.943123i $$-0.392127\pi$$
0.332444 + 0.943123i $$0.392127\pi$$
$$938$$ 0 0
$$939$$ 2998.13i 3.19290i
$$940$$ 0 0
$$941$$ 1549.98i 1.64716i 0.567200 + 0.823580i $$0.308027\pi$$
−0.567200 + 0.823580i $$0.691973\pi$$
$$942$$ 0 0
$$943$$ − 79.1960i − 0.0839830i
$$944$$ 0 0
$$945$$ − 6098.09i − 6.45300i
$$946$$ 0 0
$$947$$ 602.000 0.635692 0.317846 0.948142i $$-0.397041\pi$$
0.317846 + 0.948142i $$0.397041\pi$$
$$948$$ 0 0
$$949$$ 441.235i 0.464947i
$$950$$ 0 0
$$951$$ −1216.00 −1.27865
$$952$$ 0 0
$$953$$ 277.186i 0.290856i 0.989369 + 0.145428i $$0.0464559\pi$$
−0.989369 + 0.145428i $$0.953544\pi$$
$$954$$ 0 0
$$955$$ −2107.00 −2.20628
$$956$$ 0 0
$$957$$ −672.000 −0.702194
$$958$$ 0 0
$$959$$ −1045.00 −1.08968
$$960$$ 0 0
$$961$$ 929.000 0.966701
$$962$$ 0 0
$$963$$ − 3643.01i − 3.78298i
$$964$$ 0 0
$$965$$ − 1069.15i − 1.10792i
$$966$$ 0 0
$$967$$ −770.000 −0.796277 −0.398139 0.917325i $$-0.630344\pi$$
−0.398139 + 0.917325i $$0.630344\pi$$
$$968$$ 0 0
$$969$$ 1827.16i 1.88562i
$$970$$ 0 0
$$971$$ − 712.764i − 0.734051i −0.930211 0.367026i $$-0.880376\pi$$
0.930211 0.367026i $$-0.119624\pi$$
$$972$$ 0 0
$$973$$ −1705.00 −1.75231
$$974$$ 0 0
$$975$$ −1536.00 −1.57538
$$976$$ 0 0
$$977$$ 1340.67i 1.37224i 0.727490 + 0.686118i $$0.240687\pi$$
−0.727490 + 0.686118i $$0.759313\pi$$
$$978$$ 0 0
$$979$$ 356.382i 0.364026i
$$980$$ 0 0
$$981$$ − 2732.26i − 2.78518i
$$982$$ 0 0
$$983$$ − 554.372i − 0.563959i −0.959420 0.281980i $$-0.909009\pi$$
0.959420 0.281980i $$-0.0909910\pi$$
$$984$$ 0 0
$$985$$ 630.000 0.639594
$$986$$ 0 0
$$987$$ − 311.127i − 0.315225i
$$988$$ 0 0
$$989$$ 42.0000 0.0424671
$$990$$ 0 0
$$991$$ 639.225i 0.645030i 0.946564 + 0.322515i $$0.104528\pi$$
−0.946564 + 0.322515i $$0.895472\pi$$
$$992$$ 0 0
$$993$$ −544.000 −0.547835
$$994$$ 0 0
$$995$$ 1029.00 1.03417
$$996$$ 0 0
$$997$$ 473.000 0.474423 0.237212 0.971458i $$-0.423767\pi$$
0.237212 + 0.971458i $$0.423767\pi$$
$$998$$ 0 0
$$999$$ 3136.00 3.13914
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 1216.3.e.c.1025.2 2
4.3 odd 2 1216.3.e.d.1025.1 2
8.3 odd 2 152.3.e.a.113.2 yes 2
8.5 even 2 304.3.e.f.113.1 2
19.18 odd 2 inner 1216.3.e.c.1025.1 2
24.5 odd 2 2736.3.o.b.721.1 2
24.11 even 2 1368.3.o.a.721.1 2
76.75 even 2 1216.3.e.d.1025.2 2
152.37 odd 2 304.3.e.f.113.2 2
152.75 even 2 152.3.e.a.113.1 2
456.227 odd 2 1368.3.o.a.721.2 2
456.341 even 2 2736.3.o.b.721.2 2

By twisted newform
Twist Min Dim Char Parity Ord Type
152.3.e.a.113.1 2 152.75 even 2
152.3.e.a.113.2 yes 2 8.3 odd 2
304.3.e.f.113.1 2 8.5 even 2
304.3.e.f.113.2 2 152.37 odd 2
1216.3.e.c.1025.1 2 19.18 odd 2 inner
1216.3.e.c.1025.2 2 1.1 even 1 trivial
1216.3.e.d.1025.1 2 4.3 odd 2
1216.3.e.d.1025.2 2 76.75 even 2
1368.3.o.a.721.1 2 24.11 even 2
1368.3.o.a.721.2 2 456.227 odd 2
2736.3.o.b.721.1 2 24.5 odd 2
2736.3.o.b.721.2 2 456.341 even 2