# Properties

 Label 289.2.b.a.288.2 Level $289$ Weight $2$ Character 289.288 Analytic conductor $2.308$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$289 = 17^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 289.b (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$2.30767661842$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(i)$$ Defining polynomial: $$x^{2} + 1$$ x^2 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 17) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 288.2 Root $$1.00000i$$ of defining polynomial Character $$\chi$$ $$=$$ 289.288 Dual form 289.2.b.a.288.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+1.00000 q^{2} -1.00000 q^{4} +2.00000i q^{5} +4.00000i q^{7} -3.00000 q^{8} +3.00000 q^{9} +O(q^{10})$$ $$q+1.00000 q^{2} -1.00000 q^{4} +2.00000i q^{5} +4.00000i q^{7} -3.00000 q^{8} +3.00000 q^{9} +2.00000i q^{10} -2.00000 q^{13} +4.00000i q^{14} -1.00000 q^{16} +3.00000 q^{18} +4.00000 q^{19} -2.00000i q^{20} +4.00000i q^{23} +1.00000 q^{25} -2.00000 q^{26} -4.00000i q^{28} -6.00000i q^{29} -4.00000i q^{31} +5.00000 q^{32} -8.00000 q^{35} -3.00000 q^{36} +2.00000i q^{37} +4.00000 q^{38} -6.00000i q^{40} -6.00000i q^{41} -4.00000 q^{43} +6.00000i q^{45} +4.00000i q^{46} -9.00000 q^{49} +1.00000 q^{50} +2.00000 q^{52} -6.00000 q^{53} -12.0000i q^{56} -6.00000i q^{58} +12.0000 q^{59} -10.0000i q^{61} -4.00000i q^{62} +12.0000i q^{63} +7.00000 q^{64} -4.00000i q^{65} +4.00000 q^{67} -8.00000 q^{70} +4.00000i q^{71} -9.00000 q^{72} +6.00000i q^{73} +2.00000i q^{74} -4.00000 q^{76} +12.0000i q^{79} -2.00000i q^{80} +9.00000 q^{81} -6.00000i q^{82} +4.00000 q^{83} -4.00000 q^{86} +10.0000 q^{89} +6.00000i q^{90} -8.00000i q^{91} -4.00000i q^{92} +8.00000i q^{95} -2.00000i q^{97} -9.00000 q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{2} - 2 q^{4} - 6 q^{8} + 6 q^{9}+O(q^{10})$$ 2 * q + 2 * q^2 - 2 * q^4 - 6 * q^8 + 6 * q^9 $$2 q + 2 q^{2} - 2 q^{4} - 6 q^{8} + 6 q^{9} - 4 q^{13} - 2 q^{16} + 6 q^{18} + 8 q^{19} + 2 q^{25} - 4 q^{26} + 10 q^{32} - 16 q^{35} - 6 q^{36} + 8 q^{38} - 8 q^{43} - 18 q^{49} + 2 q^{50} + 4 q^{52} - 12 q^{53} + 24 q^{59} + 14 q^{64} + 8 q^{67} - 16 q^{70} - 18 q^{72} - 8 q^{76} + 18 q^{81} + 8 q^{83} - 8 q^{86} + 20 q^{89} - 18 q^{98}+O(q^{100})$$ 2 * q + 2 * q^2 - 2 * q^4 - 6 * q^8 + 6 * q^9 - 4 * q^13 - 2 * q^16 + 6 * q^18 + 8 * q^19 + 2 * q^25 - 4 * q^26 + 10 * q^32 - 16 * q^35 - 6 * q^36 + 8 * q^38 - 8 * q^43 - 18 * q^49 + 2 * q^50 + 4 * q^52 - 12 * q^53 + 24 * q^59 + 14 * q^64 + 8 * q^67 - 16 * q^70 - 18 * q^72 - 8 * q^76 + 18 * q^81 + 8 * q^83 - 8 * q^86 + 20 * q^89 - 18 * q^98

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/289\mathbb{Z}\right)^\times$$.

 $$n$$ $$3$$ $$\chi(n)$$ $$-1$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 1.00000 0.707107 0.353553 0.935414i $$-0.384973\pi$$
0.353553 + 0.935414i $$0.384973\pi$$
$$3$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$4$$ −1.00000 −0.500000
$$5$$ 2.00000i 0.894427i 0.894427 + 0.447214i $$0.147584\pi$$
−0.894427 + 0.447214i $$0.852416\pi$$
$$6$$ 0 0
$$7$$ 4.00000i 1.51186i 0.654654 + 0.755929i $$0.272814\pi$$
−0.654654 + 0.755929i $$0.727186\pi$$
$$8$$ −3.00000 −1.06066
$$9$$ 3.00000 1.00000
$$10$$ 2.00000i 0.632456i
$$11$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$12$$ 0 0
$$13$$ −2.00000 −0.554700 −0.277350 0.960769i $$-0.589456\pi$$
−0.277350 + 0.960769i $$0.589456\pi$$
$$14$$ 4.00000i 1.06904i
$$15$$ 0 0
$$16$$ −1.00000 −0.250000
$$17$$ 0 0
$$18$$ 3.00000 0.707107
$$19$$ 4.00000 0.917663 0.458831 0.888523i $$-0.348268\pi$$
0.458831 + 0.888523i $$0.348268\pi$$
$$20$$ − 2.00000i − 0.447214i
$$21$$ 0 0
$$22$$ 0 0
$$23$$ 4.00000i 0.834058i 0.908893 + 0.417029i $$0.136929\pi$$
−0.908893 + 0.417029i $$0.863071\pi$$
$$24$$ 0 0
$$25$$ 1.00000 0.200000
$$26$$ −2.00000 −0.392232
$$27$$ 0 0
$$28$$ − 4.00000i − 0.755929i
$$29$$ − 6.00000i − 1.11417i −0.830455 0.557086i $$-0.811919\pi$$
0.830455 0.557086i $$-0.188081\pi$$
$$30$$ 0 0
$$31$$ − 4.00000i − 0.718421i −0.933257 0.359211i $$-0.883046\pi$$
0.933257 0.359211i $$-0.116954\pi$$
$$32$$ 5.00000 0.883883
$$33$$ 0 0
$$34$$ 0 0
$$35$$ −8.00000 −1.35225
$$36$$ −3.00000 −0.500000
$$37$$ 2.00000i 0.328798i 0.986394 + 0.164399i $$0.0525685\pi$$
−0.986394 + 0.164399i $$0.947432\pi$$
$$38$$ 4.00000 0.648886
$$39$$ 0 0
$$40$$ − 6.00000i − 0.948683i
$$41$$ − 6.00000i − 0.937043i −0.883452 0.468521i $$-0.844787\pi$$
0.883452 0.468521i $$-0.155213\pi$$
$$42$$ 0 0
$$43$$ −4.00000 −0.609994 −0.304997 0.952353i $$-0.598656\pi$$
−0.304997 + 0.952353i $$0.598656\pi$$
$$44$$ 0 0
$$45$$ 6.00000i 0.894427i
$$46$$ 4.00000i 0.589768i
$$47$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$48$$ 0 0
$$49$$ −9.00000 −1.28571
$$50$$ 1.00000 0.141421
$$51$$ 0 0
$$52$$ 2.00000 0.277350
$$53$$ −6.00000 −0.824163 −0.412082 0.911147i $$-0.635198\pi$$
−0.412082 + 0.911147i $$0.635198\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ − 12.0000i − 1.60357i
$$57$$ 0 0
$$58$$ − 6.00000i − 0.787839i
$$59$$ 12.0000 1.56227 0.781133 0.624364i $$-0.214642\pi$$
0.781133 + 0.624364i $$0.214642\pi$$
$$60$$ 0 0
$$61$$ − 10.0000i − 1.28037i −0.768221 0.640184i $$-0.778858\pi$$
0.768221 0.640184i $$-0.221142\pi$$
$$62$$ − 4.00000i − 0.508001i
$$63$$ 12.0000i 1.51186i
$$64$$ 7.00000 0.875000
$$65$$ − 4.00000i − 0.496139i
$$66$$ 0 0
$$67$$ 4.00000 0.488678 0.244339 0.969690i $$-0.421429\pi$$
0.244339 + 0.969690i $$0.421429\pi$$
$$68$$ 0 0
$$69$$ 0 0
$$70$$ −8.00000 −0.956183
$$71$$ 4.00000i 0.474713i 0.971423 + 0.237356i $$0.0762809\pi$$
−0.971423 + 0.237356i $$0.923719\pi$$
$$72$$ −9.00000 −1.06066
$$73$$ 6.00000i 0.702247i 0.936329 + 0.351123i $$0.114200\pi$$
−0.936329 + 0.351123i $$0.885800\pi$$
$$74$$ 2.00000i 0.232495i
$$75$$ 0 0
$$76$$ −4.00000 −0.458831
$$77$$ 0 0
$$78$$ 0 0
$$79$$ 12.0000i 1.35011i 0.737769 + 0.675053i $$0.235879\pi$$
−0.737769 + 0.675053i $$0.764121\pi$$
$$80$$ − 2.00000i − 0.223607i
$$81$$ 9.00000 1.00000
$$82$$ − 6.00000i − 0.662589i
$$83$$ 4.00000 0.439057 0.219529 0.975606i $$-0.429548\pi$$
0.219529 + 0.975606i $$0.429548\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ −4.00000 −0.431331
$$87$$ 0 0
$$88$$ 0 0
$$89$$ 10.0000 1.06000 0.529999 0.847998i $$-0.322192\pi$$
0.529999 + 0.847998i $$0.322192\pi$$
$$90$$ 6.00000i 0.632456i
$$91$$ − 8.00000i − 0.838628i
$$92$$ − 4.00000i − 0.417029i
$$93$$ 0 0
$$94$$ 0 0
$$95$$ 8.00000i 0.820783i
$$96$$ 0 0
$$97$$ − 2.00000i − 0.203069i −0.994832 0.101535i $$-0.967625\pi$$
0.994832 0.101535i $$-0.0323753\pi$$
$$98$$ −9.00000 −0.909137
$$99$$ 0 0
$$100$$ −1.00000 −0.100000
$$101$$ −10.0000 −0.995037 −0.497519 0.867453i $$-0.665755\pi$$
−0.497519 + 0.867453i $$0.665755\pi$$
$$102$$ 0 0
$$103$$ 8.00000 0.788263 0.394132 0.919054i $$-0.371045\pi$$
0.394132 + 0.919054i $$0.371045\pi$$
$$104$$ 6.00000 0.588348
$$105$$ 0 0
$$106$$ −6.00000 −0.582772
$$107$$ − 8.00000i − 0.773389i −0.922208 0.386695i $$-0.873617\pi$$
0.922208 0.386695i $$-0.126383\pi$$
$$108$$ 0 0
$$109$$ 6.00000i 0.574696i 0.957826 + 0.287348i $$0.0927736\pi$$
−0.957826 + 0.287348i $$0.907226\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ − 4.00000i − 0.377964i
$$113$$ − 14.0000i − 1.31701i −0.752577 0.658505i $$-0.771189\pi$$
0.752577 0.658505i $$-0.228811\pi$$
$$114$$ 0 0
$$115$$ −8.00000 −0.746004
$$116$$ 6.00000i 0.557086i
$$117$$ −6.00000 −0.554700
$$118$$ 12.0000 1.10469
$$119$$ 0 0
$$120$$ 0 0
$$121$$ 11.0000 1.00000
$$122$$ − 10.0000i − 0.905357i
$$123$$ 0 0
$$124$$ 4.00000i 0.359211i
$$125$$ 12.0000i 1.07331i
$$126$$ 12.0000i 1.06904i
$$127$$ −8.00000 −0.709885 −0.354943 0.934888i $$-0.615500\pi$$
−0.354943 + 0.934888i $$0.615500\pi$$
$$128$$ −3.00000 −0.265165
$$129$$ 0 0
$$130$$ − 4.00000i − 0.350823i
$$131$$ − 16.0000i − 1.39793i −0.715158 0.698963i $$-0.753645\pi$$
0.715158 0.698963i $$-0.246355\pi$$
$$132$$ 0 0
$$133$$ 16.0000i 1.38738i
$$134$$ 4.00000 0.345547
$$135$$ 0 0
$$136$$ 0 0
$$137$$ −6.00000 −0.512615 −0.256307 0.966595i $$-0.582506\pi$$
−0.256307 + 0.966595i $$0.582506\pi$$
$$138$$ 0 0
$$139$$ 8.00000i 0.678551i 0.940687 + 0.339276i $$0.110182\pi$$
−0.940687 + 0.339276i $$0.889818\pi$$
$$140$$ 8.00000 0.676123
$$141$$ 0 0
$$142$$ 4.00000i 0.335673i
$$143$$ 0 0
$$144$$ −3.00000 −0.250000
$$145$$ 12.0000 0.996546
$$146$$ 6.00000i 0.496564i
$$147$$ 0 0
$$148$$ − 2.00000i − 0.164399i
$$149$$ −10.0000 −0.819232 −0.409616 0.912258i $$-0.634337\pi$$
−0.409616 + 0.912258i $$0.634337\pi$$
$$150$$ 0 0
$$151$$ 16.0000 1.30206 0.651031 0.759051i $$-0.274337\pi$$
0.651031 + 0.759051i $$0.274337\pi$$
$$152$$ −12.0000 −0.973329
$$153$$ 0 0
$$154$$ 0 0
$$155$$ 8.00000 0.642575
$$156$$ 0 0
$$157$$ −2.00000 −0.159617 −0.0798087 0.996810i $$-0.525431\pi$$
−0.0798087 + 0.996810i $$0.525431\pi$$
$$158$$ 12.0000i 0.954669i
$$159$$ 0 0
$$160$$ 10.0000i 0.790569i
$$161$$ −16.0000 −1.26098
$$162$$ 9.00000 0.707107
$$163$$ 24.0000i 1.87983i 0.341415 + 0.939913i $$0.389094\pi$$
−0.341415 + 0.939913i $$0.610906\pi$$
$$164$$ 6.00000i 0.468521i
$$165$$ 0 0
$$166$$ 4.00000 0.310460
$$167$$ 4.00000i 0.309529i 0.987951 + 0.154765i $$0.0494619\pi$$
−0.987951 + 0.154765i $$0.950538\pi$$
$$168$$ 0 0
$$169$$ −9.00000 −0.692308
$$170$$ 0 0
$$171$$ 12.0000 0.917663
$$172$$ 4.00000 0.304997
$$173$$ − 22.0000i − 1.67263i −0.548250 0.836315i $$-0.684706\pi$$
0.548250 0.836315i $$-0.315294\pi$$
$$174$$ 0 0
$$175$$ 4.00000i 0.302372i
$$176$$ 0 0
$$177$$ 0 0
$$178$$ 10.0000 0.749532
$$179$$ −12.0000 −0.896922 −0.448461 0.893802i $$-0.648028\pi$$
−0.448461 + 0.893802i $$0.648028\pi$$
$$180$$ − 6.00000i − 0.447214i
$$181$$ − 2.00000i − 0.148659i −0.997234 0.0743294i $$-0.976318\pi$$
0.997234 0.0743294i $$-0.0236816\pi$$
$$182$$ − 8.00000i − 0.592999i
$$183$$ 0 0
$$184$$ − 12.0000i − 0.884652i
$$185$$ −4.00000 −0.294086
$$186$$ 0 0
$$187$$ 0 0
$$188$$ 0 0
$$189$$ 0 0
$$190$$ 8.00000i 0.580381i
$$191$$ −16.0000 −1.15772 −0.578860 0.815427i $$-0.696502\pi$$
−0.578860 + 0.815427i $$0.696502\pi$$
$$192$$ 0 0
$$193$$ 2.00000i 0.143963i 0.997406 + 0.0719816i $$0.0229323\pi$$
−0.997406 + 0.0719816i $$0.977068\pi$$
$$194$$ − 2.00000i − 0.143592i
$$195$$ 0 0
$$196$$ 9.00000 0.642857
$$197$$ − 18.0000i − 1.28245i −0.767354 0.641223i $$-0.778427\pi$$
0.767354 0.641223i $$-0.221573\pi$$
$$198$$ 0 0
$$199$$ 20.0000i 1.41776i 0.705328 + 0.708881i $$0.250800\pi$$
−0.705328 + 0.708881i $$0.749200\pi$$
$$200$$ −3.00000 −0.212132
$$201$$ 0 0
$$202$$ −10.0000 −0.703598
$$203$$ 24.0000 1.68447
$$204$$ 0 0
$$205$$ 12.0000 0.838116
$$206$$ 8.00000 0.557386
$$207$$ 12.0000i 0.834058i
$$208$$ 2.00000 0.138675
$$209$$ 0 0
$$210$$ 0 0
$$211$$ 8.00000i 0.550743i 0.961338 + 0.275371i $$0.0888008\pi$$
−0.961338 + 0.275371i $$0.911199\pi$$
$$212$$ 6.00000 0.412082
$$213$$ 0 0
$$214$$ − 8.00000i − 0.546869i
$$215$$ − 8.00000i − 0.545595i
$$216$$ 0 0
$$217$$ 16.0000 1.08615
$$218$$ 6.00000i 0.406371i
$$219$$ 0 0
$$220$$ 0 0
$$221$$ 0 0
$$222$$ 0 0
$$223$$ −24.0000 −1.60716 −0.803579 0.595198i $$-0.797074\pi$$
−0.803579 + 0.595198i $$0.797074\pi$$
$$224$$ 20.0000i 1.33631i
$$225$$ 3.00000 0.200000
$$226$$ − 14.0000i − 0.931266i
$$227$$ − 24.0000i − 1.59294i −0.604681 0.796468i $$-0.706699\pi$$
0.604681 0.796468i $$-0.293301\pi$$
$$228$$ 0 0
$$229$$ −6.00000 −0.396491 −0.198246 0.980152i $$-0.563524\pi$$
−0.198246 + 0.980152i $$0.563524\pi$$
$$230$$ −8.00000 −0.527504
$$231$$ 0 0
$$232$$ 18.0000i 1.18176i
$$233$$ 6.00000i 0.393073i 0.980497 + 0.196537i $$0.0629694\pi$$
−0.980497 + 0.196537i $$0.937031\pi$$
$$234$$ −6.00000 −0.392232
$$235$$ 0 0
$$236$$ −12.0000 −0.781133
$$237$$ 0 0
$$238$$ 0 0
$$239$$ −16.0000 −1.03495 −0.517477 0.855697i $$-0.673129\pi$$
−0.517477 + 0.855697i $$0.673129\pi$$
$$240$$ 0 0
$$241$$ − 18.0000i − 1.15948i −0.814801 0.579741i $$-0.803154\pi$$
0.814801 0.579741i $$-0.196846\pi$$
$$242$$ 11.0000 0.707107
$$243$$ 0 0
$$244$$ 10.0000i 0.640184i
$$245$$ − 18.0000i − 1.14998i
$$246$$ 0 0
$$247$$ −8.00000 −0.509028
$$248$$ 12.0000i 0.762001i
$$249$$ 0 0
$$250$$ 12.0000i 0.758947i
$$251$$ 12.0000 0.757433 0.378717 0.925513i $$-0.376365\pi$$
0.378717 + 0.925513i $$0.376365\pi$$
$$252$$ − 12.0000i − 0.755929i
$$253$$ 0 0
$$254$$ −8.00000 −0.501965
$$255$$ 0 0
$$256$$ −17.0000 −1.06250
$$257$$ −18.0000 −1.12281 −0.561405 0.827541i $$-0.689739\pi$$
−0.561405 + 0.827541i $$0.689739\pi$$
$$258$$ 0 0
$$259$$ −8.00000 −0.497096
$$260$$ 4.00000i 0.248069i
$$261$$ − 18.0000i − 1.11417i
$$262$$ − 16.0000i − 0.988483i
$$263$$ 16.0000 0.986602 0.493301 0.869859i $$-0.335790\pi$$
0.493301 + 0.869859i $$0.335790\pi$$
$$264$$ 0 0
$$265$$ − 12.0000i − 0.737154i
$$266$$ 16.0000i 0.981023i
$$267$$ 0 0
$$268$$ −4.00000 −0.244339
$$269$$ − 22.0000i − 1.34136i −0.741745 0.670682i $$-0.766002\pi$$
0.741745 0.670682i $$-0.233998\pi$$
$$270$$ 0 0
$$271$$ −16.0000 −0.971931 −0.485965 0.873978i $$-0.661532\pi$$
−0.485965 + 0.873978i $$0.661532\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ −6.00000 −0.362473
$$275$$ 0 0
$$276$$ 0 0
$$277$$ − 14.0000i − 0.841178i −0.907251 0.420589i $$-0.861823\pi$$
0.907251 0.420589i $$-0.138177\pi$$
$$278$$ 8.00000i 0.479808i
$$279$$ − 12.0000i − 0.718421i
$$280$$ 24.0000 1.43427
$$281$$ 6.00000 0.357930 0.178965 0.983855i $$-0.442725\pi$$
0.178965 + 0.983855i $$0.442725\pi$$
$$282$$ 0 0
$$283$$ − 16.0000i − 0.951101i −0.879688 0.475551i $$-0.842249\pi$$
0.879688 0.475551i $$-0.157751\pi$$
$$284$$ − 4.00000i − 0.237356i
$$285$$ 0 0
$$286$$ 0 0
$$287$$ 24.0000 1.41668
$$288$$ 15.0000 0.883883
$$289$$ 0 0
$$290$$ 12.0000 0.704664
$$291$$ 0 0
$$292$$ − 6.00000i − 0.351123i
$$293$$ 6.00000 0.350524 0.175262 0.984522i $$-0.443923\pi$$
0.175262 + 0.984522i $$0.443923\pi$$
$$294$$ 0 0
$$295$$ 24.0000i 1.39733i
$$296$$ − 6.00000i − 0.348743i
$$297$$ 0 0
$$298$$ −10.0000 −0.579284
$$299$$ − 8.00000i − 0.462652i
$$300$$ 0 0
$$301$$ − 16.0000i − 0.922225i
$$302$$ 16.0000 0.920697
$$303$$ 0 0
$$304$$ −4.00000 −0.229416
$$305$$ 20.0000 1.14520
$$306$$ 0 0
$$307$$ −12.0000 −0.684876 −0.342438 0.939540i $$-0.611253\pi$$
−0.342438 + 0.939540i $$0.611253\pi$$
$$308$$ 0 0
$$309$$ 0 0
$$310$$ 8.00000 0.454369
$$311$$ − 28.0000i − 1.58773i −0.608091 0.793867i $$-0.708065\pi$$
0.608091 0.793867i $$-0.291935\pi$$
$$312$$ 0 0
$$313$$ − 22.0000i − 1.24351i −0.783210 0.621757i $$-0.786419\pi$$
0.783210 0.621757i $$-0.213581\pi$$
$$314$$ −2.00000 −0.112867
$$315$$ −24.0000 −1.35225
$$316$$ − 12.0000i − 0.675053i
$$317$$ − 10.0000i − 0.561656i −0.959758 0.280828i $$-0.909391\pi$$
0.959758 0.280828i $$-0.0906090\pi$$
$$318$$ 0 0
$$319$$ 0 0
$$320$$ 14.0000i 0.782624i
$$321$$ 0 0
$$322$$ −16.0000 −0.891645
$$323$$ 0 0
$$324$$ −9.00000 −0.500000
$$325$$ −2.00000 −0.110940
$$326$$ 24.0000i 1.32924i
$$327$$ 0 0
$$328$$ 18.0000i 0.993884i
$$329$$ 0 0
$$330$$ 0 0
$$331$$ −4.00000 −0.219860 −0.109930 0.993939i $$-0.535063\pi$$
−0.109930 + 0.993939i $$0.535063\pi$$
$$332$$ −4.00000 −0.219529
$$333$$ 6.00000i 0.328798i
$$334$$ 4.00000i 0.218870i
$$335$$ 8.00000i 0.437087i
$$336$$ 0 0
$$337$$ 14.0000i 0.762629i 0.924445 + 0.381314i $$0.124528\pi$$
−0.924445 + 0.381314i $$0.875472\pi$$
$$338$$ −9.00000 −0.489535
$$339$$ 0 0
$$340$$ 0 0
$$341$$ 0 0
$$342$$ 12.0000 0.648886
$$343$$ − 8.00000i − 0.431959i
$$344$$ 12.0000 0.646997
$$345$$ 0 0
$$346$$ − 22.0000i − 1.18273i
$$347$$ 32.0000i 1.71785i 0.512101 + 0.858925i $$0.328867\pi$$
−0.512101 + 0.858925i $$0.671133\pi$$
$$348$$ 0 0
$$349$$ 18.0000 0.963518 0.481759 0.876304i $$-0.339998\pi$$
0.481759 + 0.876304i $$0.339998\pi$$
$$350$$ 4.00000i 0.213809i
$$351$$ 0 0
$$352$$ 0 0
$$353$$ −30.0000 −1.59674 −0.798369 0.602168i $$-0.794304\pi$$
−0.798369 + 0.602168i $$0.794304\pi$$
$$354$$ 0 0
$$355$$ −8.00000 −0.424596
$$356$$ −10.0000 −0.529999
$$357$$ 0 0
$$358$$ −12.0000 −0.634220
$$359$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$360$$ − 18.0000i − 0.948683i
$$361$$ −3.00000 −0.157895
$$362$$ − 2.00000i − 0.105118i
$$363$$ 0 0
$$364$$ 8.00000i 0.419314i
$$365$$ −12.0000 −0.628109
$$366$$ 0 0
$$367$$ 28.0000i 1.46159i 0.682598 + 0.730794i $$0.260850\pi$$
−0.682598 + 0.730794i $$0.739150\pi$$
$$368$$ − 4.00000i − 0.208514i
$$369$$ − 18.0000i − 0.937043i
$$370$$ −4.00000 −0.207950
$$371$$ − 24.0000i − 1.24602i
$$372$$ 0 0
$$373$$ 6.00000 0.310668 0.155334 0.987862i $$-0.450355\pi$$
0.155334 + 0.987862i $$0.450355\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ 12.0000i 0.618031i
$$378$$ 0 0
$$379$$ 8.00000i 0.410932i 0.978664 + 0.205466i $$0.0658711\pi$$
−0.978664 + 0.205466i $$0.934129\pi$$
$$380$$ − 8.00000i − 0.410391i
$$381$$ 0 0
$$382$$ −16.0000 −0.818631
$$383$$ 24.0000 1.22634 0.613171 0.789950i $$-0.289894\pi$$
0.613171 + 0.789950i $$0.289894\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 2.00000i 0.101797i
$$387$$ −12.0000 −0.609994
$$388$$ 2.00000i 0.101535i
$$389$$ −6.00000 −0.304212 −0.152106 0.988364i $$-0.548606\pi$$
−0.152106 + 0.988364i $$0.548606\pi$$
$$390$$ 0 0
$$391$$ 0 0
$$392$$ 27.0000 1.36371
$$393$$ 0 0
$$394$$ − 18.0000i − 0.906827i
$$395$$ −24.0000 −1.20757
$$396$$ 0 0
$$397$$ 6.00000i 0.301131i 0.988600 + 0.150566i $$0.0481095\pi$$
−0.988600 + 0.150566i $$0.951890\pi$$
$$398$$ 20.0000i 1.00251i
$$399$$ 0 0
$$400$$ −1.00000 −0.0500000
$$401$$ − 14.0000i − 0.699127i −0.936913 0.349563i $$-0.886330\pi$$
0.936913 0.349563i $$-0.113670\pi$$
$$402$$ 0 0
$$403$$ 8.00000i 0.398508i
$$404$$ 10.0000 0.497519
$$405$$ 18.0000i 0.894427i
$$406$$ 24.0000 1.19110
$$407$$ 0 0
$$408$$ 0 0
$$409$$ 26.0000 1.28562 0.642809 0.766027i $$-0.277769\pi$$
0.642809 + 0.766027i $$0.277769\pi$$
$$410$$ 12.0000 0.592638
$$411$$ 0 0
$$412$$ −8.00000 −0.394132
$$413$$ 48.0000i 2.36193i
$$414$$ 12.0000i 0.589768i
$$415$$ 8.00000i 0.392705i
$$416$$ −10.0000 −0.490290
$$417$$ 0 0
$$418$$ 0 0
$$419$$ 8.00000i 0.390826i 0.980721 + 0.195413i $$0.0626047\pi$$
−0.980721 + 0.195413i $$0.937395\pi$$
$$420$$ 0 0
$$421$$ 22.0000 1.07221 0.536107 0.844150i $$-0.319894\pi$$
0.536107 + 0.844150i $$0.319894\pi$$
$$422$$ 8.00000i 0.389434i
$$423$$ 0 0
$$424$$ 18.0000 0.874157
$$425$$ 0 0
$$426$$ 0 0
$$427$$ 40.0000 1.93574
$$428$$ 8.00000i 0.386695i
$$429$$ 0 0
$$430$$ − 8.00000i − 0.385794i
$$431$$ 12.0000i 0.578020i 0.957326 + 0.289010i $$0.0933260\pi$$
−0.957326 + 0.289010i $$0.906674\pi$$
$$432$$ 0 0
$$433$$ −2.00000 −0.0961139 −0.0480569 0.998845i $$-0.515303\pi$$
−0.0480569 + 0.998845i $$0.515303\pi$$
$$434$$ 16.0000 0.768025
$$435$$ 0 0
$$436$$ − 6.00000i − 0.287348i
$$437$$ 16.0000i 0.765384i
$$438$$ 0 0
$$439$$ 20.0000i 0.954548i 0.878755 + 0.477274i $$0.158375\pi$$
−0.878755 + 0.477274i $$0.841625\pi$$
$$440$$ 0 0
$$441$$ −27.0000 −1.28571
$$442$$ 0 0
$$443$$ 28.0000 1.33032 0.665160 0.746701i $$-0.268363\pi$$
0.665160 + 0.746701i $$0.268363\pi$$
$$444$$ 0 0
$$445$$ 20.0000i 0.948091i
$$446$$ −24.0000 −1.13643
$$447$$ 0 0
$$448$$ 28.0000i 1.32288i
$$449$$ 34.0000i 1.60456i 0.596948 + 0.802280i $$0.296380\pi$$
−0.596948 + 0.802280i $$0.703620\pi$$
$$450$$ 3.00000 0.141421
$$451$$ 0 0
$$452$$ 14.0000i 0.658505i
$$453$$ 0 0
$$454$$ − 24.0000i − 1.12638i
$$455$$ 16.0000 0.750092
$$456$$ 0 0
$$457$$ 6.00000 0.280668 0.140334 0.990104i $$-0.455182\pi$$
0.140334 + 0.990104i $$0.455182\pi$$
$$458$$ −6.00000 −0.280362
$$459$$ 0 0
$$460$$ 8.00000 0.373002
$$461$$ 2.00000 0.0931493 0.0465746 0.998915i $$-0.485169\pi$$
0.0465746 + 0.998915i $$0.485169\pi$$
$$462$$ 0 0
$$463$$ 32.0000 1.48717 0.743583 0.668644i $$-0.233125\pi$$
0.743583 + 0.668644i $$0.233125\pi$$
$$464$$ 6.00000i 0.278543i
$$465$$ 0 0
$$466$$ 6.00000i 0.277945i
$$467$$ −12.0000 −0.555294 −0.277647 0.960683i $$-0.589555\pi$$
−0.277647 + 0.960683i $$0.589555\pi$$
$$468$$ 6.00000 0.277350
$$469$$ 16.0000i 0.738811i
$$470$$ 0 0
$$471$$ 0 0
$$472$$ −36.0000 −1.65703
$$473$$ 0 0
$$474$$ 0 0
$$475$$ 4.00000 0.183533
$$476$$ 0 0
$$477$$ −18.0000 −0.824163
$$478$$ −16.0000 −0.731823
$$479$$ − 36.0000i − 1.64488i −0.568850 0.822441i $$-0.692612\pi$$
0.568850 0.822441i $$-0.307388\pi$$
$$480$$ 0 0
$$481$$ − 4.00000i − 0.182384i
$$482$$ − 18.0000i − 0.819878i
$$483$$ 0 0
$$484$$ −11.0000 −0.500000
$$485$$ 4.00000 0.181631
$$486$$ 0 0
$$487$$ 20.0000i 0.906287i 0.891438 + 0.453143i $$0.149697\pi$$
−0.891438 + 0.453143i $$0.850303\pi$$
$$488$$ 30.0000i 1.35804i
$$489$$ 0 0
$$490$$ − 18.0000i − 0.813157i
$$491$$ −20.0000 −0.902587 −0.451294 0.892375i $$-0.649037\pi$$
−0.451294 + 0.892375i $$0.649037\pi$$
$$492$$ 0 0
$$493$$ 0 0
$$494$$ −8.00000 −0.359937
$$495$$ 0 0
$$496$$ 4.00000i 0.179605i
$$497$$ −16.0000 −0.717698
$$498$$ 0 0
$$499$$ − 40.0000i − 1.79065i −0.445418 0.895323i $$-0.646945\pi$$
0.445418 0.895323i $$-0.353055\pi$$
$$500$$ − 12.0000i − 0.536656i
$$501$$ 0 0
$$502$$ 12.0000 0.535586
$$503$$ − 12.0000i − 0.535054i −0.963550 0.267527i $$-0.913794\pi$$
0.963550 0.267527i $$-0.0862064\pi$$
$$504$$ − 36.0000i − 1.60357i
$$505$$ − 20.0000i − 0.889988i
$$506$$ 0 0
$$507$$ 0 0
$$508$$ 8.00000 0.354943
$$509$$ −2.00000 −0.0886484 −0.0443242 0.999017i $$-0.514113\pi$$
−0.0443242 + 0.999017i $$0.514113\pi$$
$$510$$ 0 0
$$511$$ −24.0000 −1.06170
$$512$$ −11.0000 −0.486136
$$513$$ 0 0
$$514$$ −18.0000 −0.793946
$$515$$ 16.0000i 0.705044i
$$516$$ 0 0
$$517$$ 0 0
$$518$$ −8.00000 −0.351500
$$519$$ 0 0
$$520$$ 12.0000i 0.526235i
$$521$$ 26.0000i 1.13908i 0.821963 + 0.569540i $$0.192879\pi$$
−0.821963 + 0.569540i $$0.807121\pi$$
$$522$$ − 18.0000i − 0.787839i
$$523$$ −36.0000 −1.57417 −0.787085 0.616844i $$-0.788411\pi$$
−0.787085 + 0.616844i $$0.788411\pi$$
$$524$$ 16.0000i 0.698963i
$$525$$ 0 0
$$526$$ 16.0000 0.697633
$$527$$ 0 0
$$528$$ 0 0
$$529$$ 7.00000 0.304348
$$530$$ − 12.0000i − 0.521247i
$$531$$ 36.0000 1.56227
$$532$$ − 16.0000i − 0.693688i
$$533$$ 12.0000i 0.519778i
$$534$$ 0 0
$$535$$ 16.0000 0.691740
$$536$$ −12.0000 −0.518321
$$537$$ 0 0
$$538$$ − 22.0000i − 0.948487i
$$539$$ 0 0
$$540$$ 0 0
$$541$$ − 6.00000i − 0.257960i −0.991647 0.128980i $$-0.958830\pi$$
0.991647 0.128980i $$-0.0411703\pi$$
$$542$$ −16.0000 −0.687259
$$543$$ 0 0
$$544$$ 0 0
$$545$$ −12.0000 −0.514024
$$546$$ 0 0
$$547$$ 32.0000i 1.36822i 0.729378 + 0.684111i $$0.239809\pi$$
−0.729378 + 0.684111i $$0.760191\pi$$
$$548$$ 6.00000 0.256307
$$549$$ − 30.0000i − 1.28037i
$$550$$ 0 0
$$551$$ − 24.0000i − 1.02243i
$$552$$ 0 0
$$553$$ −48.0000 −2.04117
$$554$$ − 14.0000i − 0.594803i
$$555$$ 0 0
$$556$$ − 8.00000i − 0.339276i
$$557$$ 30.0000 1.27114 0.635570 0.772043i $$-0.280765\pi$$
0.635570 + 0.772043i $$0.280765\pi$$
$$558$$ − 12.0000i − 0.508001i
$$559$$ 8.00000 0.338364
$$560$$ 8.00000 0.338062
$$561$$ 0 0
$$562$$ 6.00000 0.253095
$$563$$ 4.00000 0.168580 0.0842900 0.996441i $$-0.473138\pi$$
0.0842900 + 0.996441i $$0.473138\pi$$
$$564$$ 0 0
$$565$$ 28.0000 1.17797
$$566$$ − 16.0000i − 0.672530i
$$567$$ 36.0000i 1.51186i
$$568$$ − 12.0000i − 0.503509i
$$569$$ 38.0000 1.59304 0.796521 0.604610i $$-0.206671\pi$$
0.796521 + 0.604610i $$0.206671\pi$$
$$570$$ 0 0
$$571$$ − 32.0000i − 1.33916i −0.742741 0.669579i $$-0.766474\pi$$
0.742741 0.669579i $$-0.233526\pi$$
$$572$$ 0 0
$$573$$ 0 0
$$574$$ 24.0000 1.00174
$$575$$ 4.00000i 0.166812i
$$576$$ 21.0000 0.875000
$$577$$ −14.0000 −0.582828 −0.291414 0.956597i $$-0.594126\pi$$
−0.291414 + 0.956597i $$0.594126\pi$$
$$578$$ 0 0
$$579$$ 0 0
$$580$$ −12.0000 −0.498273
$$581$$ 16.0000i 0.663792i
$$582$$ 0 0
$$583$$ 0 0
$$584$$ − 18.0000i − 0.744845i
$$585$$ − 12.0000i − 0.496139i
$$586$$ 6.00000 0.247858
$$587$$ −4.00000 −0.165098 −0.0825488 0.996587i $$-0.526306\pi$$
−0.0825488 + 0.996587i $$0.526306\pi$$
$$588$$ 0 0
$$589$$ − 16.0000i − 0.659269i
$$590$$ 24.0000i 0.988064i
$$591$$ 0 0
$$592$$ − 2.00000i − 0.0821995i
$$593$$ −18.0000 −0.739171 −0.369586 0.929197i $$-0.620500\pi$$
−0.369586 + 0.929197i $$0.620500\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 10.0000 0.409616
$$597$$ 0 0
$$598$$ − 8.00000i − 0.327144i
$$599$$ −24.0000 −0.980613 −0.490307 0.871550i $$-0.663115\pi$$
−0.490307 + 0.871550i $$0.663115\pi$$
$$600$$ 0 0
$$601$$ 10.0000i 0.407909i 0.978980 + 0.203954i $$0.0653794\pi$$
−0.978980 + 0.203954i $$0.934621\pi$$
$$602$$ − 16.0000i − 0.652111i
$$603$$ 12.0000 0.488678
$$604$$ −16.0000 −0.651031
$$605$$ 22.0000i 0.894427i
$$606$$ 0 0
$$607$$ − 20.0000i − 0.811775i −0.913923 0.405887i $$-0.866962\pi$$
0.913923 0.405887i $$-0.133038\pi$$
$$608$$ 20.0000 0.811107
$$609$$ 0 0
$$610$$ 20.0000 0.809776
$$611$$ 0 0
$$612$$ 0 0
$$613$$ −26.0000 −1.05013 −0.525065 0.851062i $$-0.675959\pi$$
−0.525065 + 0.851062i $$0.675959\pi$$
$$614$$ −12.0000 −0.484281
$$615$$ 0 0
$$616$$ 0 0
$$617$$ 6.00000i 0.241551i 0.992680 + 0.120775i $$0.0385381\pi$$
−0.992680 + 0.120775i $$0.961462\pi$$
$$618$$ 0 0
$$619$$ − 48.0000i − 1.92928i −0.263566 0.964641i $$-0.584899\pi$$
0.263566 0.964641i $$-0.415101\pi$$
$$620$$ −8.00000 −0.321288
$$621$$ 0 0
$$622$$ − 28.0000i − 1.12270i
$$623$$ 40.0000i 1.60257i
$$624$$ 0 0
$$625$$ −19.0000 −0.760000
$$626$$ − 22.0000i − 0.879297i
$$627$$ 0 0
$$628$$ 2.00000 0.0798087
$$629$$ 0 0
$$630$$ −24.0000 −0.956183
$$631$$ −16.0000 −0.636950 −0.318475 0.947931i $$-0.603171\pi$$
−0.318475 + 0.947931i $$0.603171\pi$$
$$632$$ − 36.0000i − 1.43200i
$$633$$ 0 0
$$634$$ − 10.0000i − 0.397151i
$$635$$ − 16.0000i − 0.634941i
$$636$$ 0 0
$$637$$ 18.0000 0.713186
$$638$$ 0 0
$$639$$ 12.0000i 0.474713i
$$640$$ − 6.00000i − 0.237171i
$$641$$ 30.0000i 1.18493i 0.805597 + 0.592464i $$0.201845\pi$$
−0.805597 + 0.592464i $$0.798155\pi$$
$$642$$ 0 0
$$643$$ − 32.0000i − 1.26196i −0.775800 0.630978i $$-0.782654\pi$$
0.775800 0.630978i $$-0.217346\pi$$
$$644$$ 16.0000 0.630488
$$645$$ 0 0
$$646$$ 0 0
$$647$$ 8.00000 0.314512 0.157256 0.987558i $$-0.449735\pi$$
0.157256 + 0.987558i $$0.449735\pi$$
$$648$$ −27.0000 −1.06066
$$649$$ 0 0
$$650$$ −2.00000 −0.0784465
$$651$$ 0 0
$$652$$ − 24.0000i − 0.939913i
$$653$$ 6.00000i 0.234798i 0.993085 + 0.117399i $$0.0374557\pi$$
−0.993085 + 0.117399i $$0.962544\pi$$
$$654$$ 0 0
$$655$$ 32.0000 1.25034
$$656$$ 6.00000i 0.234261i
$$657$$ 18.0000i 0.702247i
$$658$$ 0 0
$$659$$ 4.00000 0.155818 0.0779089 0.996960i $$-0.475176\pi$$
0.0779089 + 0.996960i $$0.475176\pi$$
$$660$$ 0 0
$$661$$ −38.0000 −1.47803 −0.739014 0.673690i $$-0.764708\pi$$
−0.739014 + 0.673690i $$0.764708\pi$$
$$662$$ −4.00000 −0.155464
$$663$$ 0 0
$$664$$ −12.0000 −0.465690
$$665$$ −32.0000 −1.24091
$$666$$ 6.00000i 0.232495i
$$667$$ 24.0000 0.929284
$$668$$ − 4.00000i − 0.154765i
$$669$$ 0 0
$$670$$ 8.00000i 0.309067i
$$671$$ 0 0
$$672$$ 0 0
$$673$$ 2.00000i 0.0770943i 0.999257 + 0.0385472i $$0.0122730\pi$$
−0.999257 + 0.0385472i $$0.987727\pi$$
$$674$$ 14.0000i 0.539260i
$$675$$ 0 0
$$676$$ 9.00000 0.346154
$$677$$ − 30.0000i − 1.15299i −0.817099 0.576497i $$-0.804419\pi$$
0.817099 0.576497i $$-0.195581\pi$$
$$678$$ 0 0
$$679$$ 8.00000 0.307012
$$680$$ 0 0
$$681$$ 0 0
$$682$$ 0 0
$$683$$ 40.0000i 1.53056i 0.643699 + 0.765279i $$0.277399\pi$$
−0.643699 + 0.765279i $$0.722601\pi$$
$$684$$ −12.0000 −0.458831
$$685$$ − 12.0000i − 0.458496i
$$686$$ − 8.00000i − 0.305441i
$$687$$ 0 0
$$688$$ 4.00000 0.152499
$$689$$ 12.0000 0.457164
$$690$$ 0 0
$$691$$ − 8.00000i − 0.304334i −0.988355 0.152167i $$-0.951375\pi$$
0.988355 0.152167i $$-0.0486252\pi$$
$$692$$ 22.0000i 0.836315i
$$693$$ 0 0
$$694$$ 32.0000i 1.21470i
$$695$$ −16.0000 −0.606915
$$696$$ 0 0
$$697$$ 0 0
$$698$$ 18.0000 0.681310
$$699$$ 0 0
$$700$$ − 4.00000i − 0.151186i
$$701$$ −18.0000 −0.679851 −0.339925 0.940452i $$-0.610402\pi$$
−0.339925 + 0.940452i $$0.610402\pi$$
$$702$$ 0 0
$$703$$ 8.00000i 0.301726i
$$704$$ 0 0
$$705$$ 0 0
$$706$$ −30.0000 −1.12906
$$707$$ − 40.0000i − 1.50435i
$$708$$ 0 0
$$709$$ 34.0000i 1.27690i 0.769665 + 0.638448i $$0.220423\pi$$
−0.769665 + 0.638448i $$0.779577\pi$$
$$710$$ −8.00000 −0.300235
$$711$$ 36.0000i 1.35011i
$$712$$ −30.0000 −1.12430
$$713$$ 16.0000 0.599205
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 12.0000 0.448461
$$717$$ 0 0
$$718$$ 0 0
$$719$$ − 4.00000i − 0.149175i −0.997214 0.0745874i $$-0.976236\pi$$
0.997214 0.0745874i $$-0.0237640\pi$$
$$720$$ − 6.00000i − 0.223607i
$$721$$ 32.0000i 1.19174i
$$722$$ −3.00000 −0.111648
$$723$$ 0 0
$$724$$ 2.00000i 0.0743294i
$$725$$ − 6.00000i − 0.222834i
$$726$$ 0 0
$$727$$ 40.0000 1.48352 0.741759 0.670667i $$-0.233992\pi$$
0.741759 + 0.670667i $$0.233992\pi$$
$$728$$ 24.0000i 0.889499i
$$729$$ 27.0000 1.00000
$$730$$ −12.0000 −0.444140
$$731$$ 0 0
$$732$$ 0 0
$$733$$ 50.0000 1.84679 0.923396 0.383849i $$-0.125402\pi$$
0.923396 + 0.383849i $$0.125402\pi$$
$$734$$ 28.0000i 1.03350i
$$735$$ 0 0
$$736$$ 20.0000i 0.737210i
$$737$$ 0 0
$$738$$ − 18.0000i − 0.662589i
$$739$$ −28.0000 −1.03000 −0.514998 0.857191i $$-0.672207\pi$$
−0.514998 + 0.857191i $$0.672207\pi$$
$$740$$ 4.00000 0.147043
$$741$$ 0 0
$$742$$ − 24.0000i − 0.881068i
$$743$$ − 12.0000i − 0.440237i −0.975473 0.220119i $$-0.929356\pi$$
0.975473 0.220119i $$-0.0706445\pi$$
$$744$$ 0 0
$$745$$ − 20.0000i − 0.732743i
$$746$$ 6.00000 0.219676
$$747$$ 12.0000 0.439057
$$748$$ 0 0
$$749$$ 32.0000 1.16925
$$750$$ 0 0
$$751$$ − 20.0000i − 0.729810i −0.931045 0.364905i $$-0.881101\pi$$
0.931045 0.364905i $$-0.118899\pi$$
$$752$$ 0 0
$$753$$ 0 0
$$754$$ 12.0000i 0.437014i
$$755$$ 32.0000i 1.16460i
$$756$$ 0 0
$$757$$ −22.0000 −0.799604 −0.399802 0.916602i $$-0.630921\pi$$
−0.399802 + 0.916602i $$0.630921\pi$$
$$758$$ 8.00000i 0.290573i
$$759$$ 0 0
$$760$$ − 24.0000i − 0.870572i
$$761$$ −22.0000 −0.797499 −0.398750 0.917060i $$-0.630556\pi$$
−0.398750 + 0.917060i $$0.630556\pi$$
$$762$$ 0 0
$$763$$ −24.0000 −0.868858
$$764$$ 16.0000 0.578860
$$765$$ 0 0
$$766$$ 24.0000 0.867155
$$767$$ −24.0000 −0.866590
$$768$$ 0 0
$$769$$ −14.0000 −0.504853 −0.252426 0.967616i $$-0.581229\pi$$
−0.252426 + 0.967616i $$0.581229\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ − 2.00000i − 0.0719816i
$$773$$ 26.0000 0.935155 0.467578 0.883952i $$-0.345127\pi$$
0.467578 + 0.883952i $$0.345127\pi$$
$$774$$ −12.0000 −0.431331
$$775$$ − 4.00000i − 0.143684i
$$776$$ 6.00000i 0.215387i
$$777$$ 0 0
$$778$$ −6.00000 −0.215110
$$779$$ − 24.0000i − 0.859889i
$$780$$ 0 0
$$781$$ 0 0
$$782$$ 0 0
$$783$$ 0 0
$$784$$ 9.00000 0.321429
$$785$$ − 4.00000i − 0.142766i
$$786$$ 0 0
$$787$$ 32.0000i 1.14068i 0.821410 + 0.570338i $$0.193188\pi$$
−0.821410 + 0.570338i $$0.806812\pi$$
$$788$$ 18.0000i 0.641223i
$$789$$ 0 0
$$790$$ −24.0000 −0.853882
$$791$$ 56.0000 1.99113
$$792$$ 0 0
$$793$$ 20.0000i 0.710221i
$$794$$ 6.00000i 0.212932i
$$795$$ 0 0
$$796$$ − 20.0000i − 0.708881i
$$797$$ 50.0000 1.77109 0.885545 0.464553i $$-0.153785\pi$$
0.885545 + 0.464553i $$0.153785\pi$$
$$798$$ 0 0
$$799$$ 0 0
$$800$$ 5.00000 0.176777
$$801$$ 30.0000 1.06000
$$802$$ − 14.0000i − 0.494357i
$$803$$ 0 0
$$804$$ 0 0
$$805$$ − 32.0000i − 1.12785i
$$806$$ 8.00000i 0.281788i
$$807$$ 0 0
$$808$$ 30.0000 1.05540
$$809$$ 26.0000i 0.914111i 0.889438 + 0.457056i $$0.151096\pi$$
−0.889438 + 0.457056i $$0.848904\pi$$
$$810$$ 18.0000i 0.632456i
$$811$$ − 40.0000i − 1.40459i −0.711886 0.702295i $$-0.752159\pi$$
0.711886 0.702295i $$-0.247841\pi$$
$$812$$ −24.0000 −0.842235
$$813$$ 0 0
$$814$$ 0 0
$$815$$ −48.0000 −1.68137
$$816$$ 0 0
$$817$$ −16.0000 −0.559769
$$818$$ 26.0000 0.909069
$$819$$ − 24.0000i − 0.838628i
$$820$$ −12.0000 −0.419058
$$821$$ 18.0000i 0.628204i 0.949389 + 0.314102i $$0.101703\pi$$
−0.949389 + 0.314102i $$0.898297\pi$$
$$822$$ 0 0
$$823$$ 20.0000i 0.697156i 0.937280 + 0.348578i $$0.113335\pi$$
−0.937280 + 0.348578i $$0.886665\pi$$
$$824$$ −24.0000 −0.836080
$$825$$ 0 0
$$826$$ 48.0000i 1.67013i
$$827$$ − 48.0000i − 1.66912i −0.550914 0.834562i $$-0.685721\pi$$
0.550914 0.834562i $$-0.314279\pi$$
$$828$$ − 12.0000i − 0.417029i
$$829$$ −34.0000 −1.18087 −0.590434 0.807086i $$-0.701044\pi$$
−0.590434 + 0.807086i $$0.701044\pi$$
$$830$$ 8.00000i 0.277684i
$$831$$ 0 0
$$832$$ −14.0000 −0.485363
$$833$$ 0 0
$$834$$ 0 0
$$835$$ −8.00000 −0.276851
$$836$$ 0 0
$$837$$ 0 0
$$838$$ 8.00000i 0.276355i
$$839$$ 20.0000i 0.690477i 0.938515 + 0.345238i $$0.112202\pi$$
−0.938515 + 0.345238i $$0.887798\pi$$
$$840$$ 0 0
$$841$$ −7.00000 −0.241379
$$842$$ 22.0000 0.758170
$$843$$ 0 0
$$844$$ − 8.00000i − 0.275371i
$$845$$ − 18.0000i − 0.619219i
$$846$$ 0 0
$$847$$ 44.0000i 1.51186i
$$848$$ 6.00000 0.206041
$$849$$ 0 0
$$850$$ 0 0
$$851$$ −8.00000 −0.274236
$$852$$ 0 0
$$853$$ − 14.0000i − 0.479351i −0.970853 0.239675i $$-0.922959\pi$$
0.970853 0.239675i $$-0.0770410\pi$$
$$854$$ 40.0000 1.36877
$$855$$ 24.0000i 0.820783i
$$856$$ 24.0000i 0.820303i
$$857$$ 10.0000i 0.341593i 0.985306 + 0.170797i $$0.0546341\pi$$
−0.985306 + 0.170797i $$0.945366\pi$$
$$858$$ 0 0
$$859$$ −52.0000 −1.77422 −0.887109 0.461561i $$-0.847290\pi$$
−0.887109 + 0.461561i $$0.847290\pi$$
$$860$$ 8.00000i 0.272798i
$$861$$ 0 0
$$862$$ 12.0000i 0.408722i
$$863$$ 16.0000 0.544646 0.272323 0.962206i $$-0.412208\pi$$
0.272323 + 0.962206i $$0.412208\pi$$
$$864$$ 0 0
$$865$$ 44.0000 1.49604
$$866$$ −2.00000 −0.0679628
$$867$$ 0 0
$$868$$ −16.0000 −0.543075
$$869$$ 0 0
$$870$$ 0 0
$$871$$ −8.00000 −0.271070
$$872$$ − 18.0000i − 0.609557i
$$873$$ − 6.00000i − 0.203069i
$$874$$ 16.0000i 0.541208i
$$875$$ −48.0000 −1.62270
$$876$$ 0 0
$$877$$ 6.00000i 0.202606i 0.994856 + 0.101303i $$0.0323011\pi$$
−0.994856 + 0.101303i $$0.967699\pi$$
$$878$$ 20.0000i 0.674967i
$$879$$ 0 0
$$880$$ 0 0
$$881$$ 46.0000i 1.54978i 0.632096 + 0.774890i $$0.282195\pi$$
−0.632096 + 0.774890i $$0.717805\pi$$
$$882$$ −27.0000 −0.909137
$$883$$ −12.0000 −0.403832 −0.201916 0.979403i $$-0.564717\pi$$
−0.201916 + 0.979403i $$0.564717\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 28.0000 0.940678
$$887$$ − 12.0000i − 0.402921i −0.979497 0.201460i $$-0.935431\pi$$
0.979497 0.201460i $$-0.0645687\pi$$
$$888$$ 0 0
$$889$$ − 32.0000i − 1.07325i
$$890$$ 20.0000i 0.670402i
$$891$$ 0 0
$$892$$ 24.0000 0.803579
$$893$$ 0 0
$$894$$ 0 0
$$895$$ − 24.0000i − 0.802232i
$$896$$ − 12.0000i − 0.400892i
$$897$$ 0 0
$$898$$ 34.0000i 1.13459i
$$899$$ −24.0000 −0.800445
$$900$$ −3.00000 −0.100000
$$901$$ 0 0
$$902$$ 0 0
$$903$$ 0 0
$$904$$ 42.0000i 1.39690i
$$905$$ 4.00000 0.132964
$$906$$ 0 0
$$907$$ 32.0000i 1.06254i 0.847202 + 0.531271i $$0.178286\pi$$
−0.847202 + 0.531271i $$0.821714\pi$$
$$908$$ 24.0000i 0.796468i
$$909$$ −30.0000 −0.995037
$$910$$ 16.0000 0.530395
$$911$$ − 4.00000i − 0.132526i −0.997802 0.0662630i $$-0.978892\pi$$
0.997802 0.0662630i $$-0.0211076\pi$$
$$912$$ 0 0
$$913$$ 0 0
$$914$$ 6.00000 0.198462
$$915$$ 0 0
$$916$$ 6.00000 0.198246
$$917$$ 64.0000 2.11347
$$918$$ 0 0
$$919$$ 24.0000 0.791687 0.395843 0.918318i $$-0.370452\pi$$
0.395843 + 0.918318i $$0.370452\pi$$
$$920$$ 24.0000 0.791257
$$921$$ 0 0
$$922$$ 2.00000 0.0658665
$$923$$ − 8.00000i − 0.263323i
$$924$$ 0 0
$$925$$ 2.00000i 0.0657596i
$$926$$ 32.0000 1.05159
$$927$$ 24.0000 0.788263
$$928$$ − 30.0000i − 0.984798i
$$929$$ − 30.0000i − 0.984268i −0.870519 0.492134i $$-0.836217\pi$$
0.870519 0.492134i $$-0.163783\pi$$
$$930$$ 0 0
$$931$$ −36.0000 −1.17985
$$932$$ − 6.00000i − 0.196537i
$$933$$ 0 0
$$934$$ −12.0000 −0.392652
$$935$$ 0 0
$$936$$ 18.0000 0.588348
$$937$$ −10.0000 −0.326686 −0.163343 0.986569i $$-0.552228\pi$$
−0.163343 + 0.986569i $$0.552228\pi$$
$$938$$ 16.0000i 0.522419i
$$939$$ 0 0
$$940$$ 0 0
$$941$$ 6.00000i 0.195594i 0.995206 + 0.0977972i $$0.0311797\pi$$
−0.995206 + 0.0977972i $$0.968820\pi$$
$$942$$ 0 0
$$943$$ 24.0000 0.781548
$$944$$ −12.0000 −0.390567
$$945$$ 0 0
$$946$$ 0 0
$$947$$ − 32.0000i − 1.03986i −0.854209 0.519930i $$-0.825958\pi$$
0.854209 0.519930i $$-0.174042\pi$$
$$948$$ 0 0
$$949$$ − 12.0000i − 0.389536i
$$950$$ 4.00000 0.129777
$$951$$ 0 0
$$952$$ 0 0
$$953$$ −22.0000 −0.712650 −0.356325 0.934362i $$-0.615970\pi$$
−0.356325 + 0.934362i $$0.615970\pi$$
$$954$$ −18.0000 −0.582772
$$955$$ − 32.0000i − 1.03550i
$$956$$ 16.0000 0.517477
$$957$$ 0 0
$$958$$ − 36.0000i − 1.16311i
$$959$$ − 24.0000i − 0.775000i
$$960$$ 0 0
$$961$$ 15.0000 0.483871
$$962$$ − 4.00000i − 0.128965i
$$963$$ − 24.0000i − 0.773389i
$$964$$ 18.0000i 0.579741i
$$965$$ −4.00000 −0.128765
$$966$$ 0 0
$$967$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$968$$ −33.0000 −1.06066
$$969$$ 0 0
$$970$$ 4.00000 0.128432
$$971$$ 12.0000 0.385098 0.192549 0.981287i $$-0.438325\pi$$
0.192549 + 0.981287i $$0.438325\pi$$
$$972$$ 0 0
$$973$$ −32.0000 −1.02587
$$974$$ 20.0000i 0.640841i
$$975$$ 0 0
$$976$$ 10.0000i 0.320092i
$$977$$ −18.0000 −0.575871 −0.287936 0.957650i $$-0.592969\pi$$
−0.287936 + 0.957650i $$0.592969\pi$$
$$978$$ 0 0
$$979$$ 0 0
$$980$$ 18.0000i 0.574989i
$$981$$ 18.0000i 0.574696i
$$982$$ −20.0000 −0.638226
$$983$$ − 12.0000i − 0.382741i −0.981518 0.191370i $$-0.938707\pi$$
0.981518 0.191370i $$-0.0612931\pi$$
$$984$$ 0 0
$$985$$ 36.0000 1.14706
$$986$$ 0 0
$$987$$ 0 0
$$988$$ 8.00000 0.254514
$$989$$ − 16.0000i − 0.508770i
$$990$$ 0 0
$$991$$ 12.0000i 0.381193i 0.981669 + 0.190596i $$0.0610421\pi$$
−0.981669 + 0.190596i $$0.938958\pi$$
$$992$$ − 20.0000i − 0.635001i
$$993$$ 0 0
$$994$$ −16.0000 −0.507489
$$995$$ −40.0000 −1.26809
$$996$$ 0 0
$$997$$ 46.0000i 1.45683i 0.685134 + 0.728417i $$0.259744\pi$$
−0.685134 + 0.728417i $$0.740256\pi$$
$$998$$ − 40.0000i − 1.26618i
$$999$$ 0 0
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 289.2.b.a.288.2 2
17.2 even 8 289.2.c.a.251.1 4
17.3 odd 16 289.2.d.d.110.1 8
17.4 even 4 289.2.a.a.1.1 1
17.5 odd 16 289.2.d.d.179.2 8
17.6 odd 16 289.2.d.d.134.2 8
17.7 odd 16 289.2.d.d.155.1 8
17.8 even 8 289.2.c.a.38.1 4
17.9 even 8 289.2.c.a.38.2 4
17.10 odd 16 289.2.d.d.155.2 8
17.11 odd 16 289.2.d.d.134.1 8
17.12 odd 16 289.2.d.d.179.1 8
17.13 even 4 17.2.a.a.1.1 1
17.14 odd 16 289.2.d.d.110.2 8
17.15 even 8 289.2.c.a.251.2 4
17.16 even 2 inner 289.2.b.a.288.1 2
51.38 odd 4 2601.2.a.g.1.1 1
51.47 odd 4 153.2.a.c.1.1 1
68.47 odd 4 272.2.a.b.1.1 1
68.55 odd 4 4624.2.a.d.1.1 1
85.4 even 4 7225.2.a.g.1.1 1
85.13 odd 4 425.2.b.b.324.2 2
85.47 odd 4 425.2.b.b.324.1 2
85.64 even 4 425.2.a.d.1.1 1
119.13 odd 4 833.2.a.a.1.1 1
119.30 even 12 833.2.e.b.18.1 2
119.47 odd 12 833.2.e.a.18.1 2
119.81 even 12 833.2.e.b.324.1 2
119.115 odd 12 833.2.e.a.324.1 2
136.13 even 4 1088.2.a.i.1.1 1
136.115 odd 4 1088.2.a.h.1.1 1
187.98 odd 4 2057.2.a.e.1.1 1
204.47 even 4 2448.2.a.o.1.1 1
221.64 even 4 2873.2.a.c.1.1 1
255.149 odd 4 3825.2.a.d.1.1 1
323.132 odd 4 6137.2.a.b.1.1 1
340.319 odd 4 6800.2.a.n.1.1 1
357.251 even 4 7497.2.a.l.1.1 1
391.183 odd 4 8993.2.a.a.1.1 1
408.149 odd 4 9792.2.a.n.1.1 1
408.251 even 4 9792.2.a.i.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
17.2.a.a.1.1 1 17.13 even 4
153.2.a.c.1.1 1 51.47 odd 4
272.2.a.b.1.1 1 68.47 odd 4
289.2.a.a.1.1 1 17.4 even 4
289.2.b.a.288.1 2 17.16 even 2 inner
289.2.b.a.288.2 2 1.1 even 1 trivial
289.2.c.a.38.1 4 17.8 even 8
289.2.c.a.38.2 4 17.9 even 8
289.2.c.a.251.1 4 17.2 even 8
289.2.c.a.251.2 4 17.15 even 8
289.2.d.d.110.1 8 17.3 odd 16
289.2.d.d.110.2 8 17.14 odd 16
289.2.d.d.134.1 8 17.11 odd 16
289.2.d.d.134.2 8 17.6 odd 16
289.2.d.d.155.1 8 17.7 odd 16
289.2.d.d.155.2 8 17.10 odd 16
289.2.d.d.179.1 8 17.12 odd 16
289.2.d.d.179.2 8 17.5 odd 16
425.2.a.d.1.1 1 85.64 even 4
425.2.b.b.324.1 2 85.47 odd 4
425.2.b.b.324.2 2 85.13 odd 4
833.2.a.a.1.1 1 119.13 odd 4
833.2.e.a.18.1 2 119.47 odd 12
833.2.e.a.324.1 2 119.115 odd 12
833.2.e.b.18.1 2 119.30 even 12
833.2.e.b.324.1 2 119.81 even 12
1088.2.a.h.1.1 1 136.115 odd 4
1088.2.a.i.1.1 1 136.13 even 4
2057.2.a.e.1.1 1 187.98 odd 4
2448.2.a.o.1.1 1 204.47 even 4
2601.2.a.g.1.1 1 51.38 odd 4
2873.2.a.c.1.1 1 221.64 even 4
3825.2.a.d.1.1 1 255.149 odd 4
4624.2.a.d.1.1 1 68.55 odd 4
6137.2.a.b.1.1 1 323.132 odd 4
6800.2.a.n.1.1 1 340.319 odd 4
7225.2.a.g.1.1 1 85.4 even 4
7497.2.a.l.1.1 1 357.251 even 4
8993.2.a.a.1.1 1 391.183 odd 4
9792.2.a.i.1.1 1 408.251 even 4
9792.2.a.n.1.1 1 408.149 odd 4