# Properties

 Label 9200.2.a.bd.1.1 Level $9200$ Weight $2$ Character 9200.1 Self dual yes Analytic conductor $73.462$ Analytic rank $0$ Dimension $1$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$9200 = 2^{4} \cdot 5^{2} \cdot 23$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 9200.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$73.4623698596$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 4600) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Character $$\chi$$ $$=$$ 9200.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+2.00000 q^{3} -1.00000 q^{7} +1.00000 q^{9} +O(q^{10})$$ $$q+2.00000 q^{3} -1.00000 q^{7} +1.00000 q^{9} +5.00000 q^{11} +1.00000 q^{13} -4.00000 q^{17} -7.00000 q^{19} -2.00000 q^{21} +1.00000 q^{23} -4.00000 q^{27} +5.00000 q^{29} -2.00000 q^{31} +10.0000 q^{33} -2.00000 q^{37} +2.00000 q^{39} +11.0000 q^{41} -1.00000 q^{43} +8.00000 q^{47} -6.00000 q^{49} -8.00000 q^{51} -14.0000 q^{57} +14.0000 q^{59} +10.0000 q^{61} -1.00000 q^{63} +8.00000 q^{67} +2.00000 q^{69} +10.0000 q^{71} +7.00000 q^{73} -5.00000 q^{77} -7.00000 q^{79} -11.0000 q^{81} +15.0000 q^{83} +10.0000 q^{87} +10.0000 q^{89} -1.00000 q^{91} -4.00000 q^{93} -4.00000 q^{97} +5.00000 q^{99} +O(q^{100})$$

## 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
$$3$$ 2.00000 1.15470 0.577350 0.816497i $$-0.304087\pi$$
0.577350 + 0.816497i $$0.304087\pi$$
$$4$$ 0 0
$$5$$ 0 0
$$6$$ 0 0
$$7$$ −1.00000 −0.377964 −0.188982 0.981981i $$-0.560519\pi$$
−0.188982 + 0.981981i $$0.560519\pi$$
$$8$$ 0 0
$$9$$ 1.00000 0.333333
$$10$$ 0 0
$$11$$ 5.00000 1.50756 0.753778 0.657129i $$-0.228229\pi$$
0.753778 + 0.657129i $$0.228229\pi$$
$$12$$ 0 0
$$13$$ 1.00000 0.277350 0.138675 0.990338i $$-0.455716\pi$$
0.138675 + 0.990338i $$0.455716\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 0 0
$$17$$ −4.00000 −0.970143 −0.485071 0.874475i $$-0.661206\pi$$
−0.485071 + 0.874475i $$0.661206\pi$$
$$18$$ 0 0
$$19$$ −7.00000 −1.60591 −0.802955 0.596040i $$-0.796740\pi$$
−0.802955 + 0.596040i $$0.796740\pi$$
$$20$$ 0 0
$$21$$ −2.00000 −0.436436
$$22$$ 0 0
$$23$$ 1.00000 0.208514
$$24$$ 0 0
$$25$$ 0 0
$$26$$ 0 0
$$27$$ −4.00000 −0.769800
$$28$$ 0 0
$$29$$ 5.00000 0.928477 0.464238 0.885710i $$-0.346328\pi$$
0.464238 + 0.885710i $$0.346328\pi$$
$$30$$ 0 0
$$31$$ −2.00000 −0.359211 −0.179605 0.983739i $$-0.557482\pi$$
−0.179605 + 0.983739i $$0.557482\pi$$
$$32$$ 0 0
$$33$$ 10.0000 1.74078
$$34$$ 0 0
$$35$$ 0 0
$$36$$ 0 0
$$37$$ −2.00000 −0.328798 −0.164399 0.986394i $$-0.552568\pi$$
−0.164399 + 0.986394i $$0.552568\pi$$
$$38$$ 0 0
$$39$$ 2.00000 0.320256
$$40$$ 0 0
$$41$$ 11.0000 1.71791 0.858956 0.512050i $$-0.171114\pi$$
0.858956 + 0.512050i $$0.171114\pi$$
$$42$$ 0 0
$$43$$ −1.00000 −0.152499 −0.0762493 0.997089i $$-0.524294\pi$$
−0.0762493 + 0.997089i $$0.524294\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 0 0
$$47$$ 8.00000 1.16692 0.583460 0.812142i $$-0.301699\pi$$
0.583460 + 0.812142i $$0.301699\pi$$
$$48$$ 0 0
$$49$$ −6.00000 −0.857143
$$50$$ 0 0
$$51$$ −8.00000 −1.12022
$$52$$ 0 0
$$53$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 0 0
$$57$$ −14.0000 −1.85435
$$58$$ 0 0
$$59$$ 14.0000 1.82264 0.911322 0.411693i $$-0.135063\pi$$
0.911322 + 0.411693i $$0.135063\pi$$
$$60$$ 0 0
$$61$$ 10.0000 1.28037 0.640184 0.768221i $$-0.278858\pi$$
0.640184 + 0.768221i $$0.278858\pi$$
$$62$$ 0 0
$$63$$ −1.00000 −0.125988
$$64$$ 0 0
$$65$$ 0 0
$$66$$ 0 0
$$67$$ 8.00000 0.977356 0.488678 0.872464i $$-0.337479\pi$$
0.488678 + 0.872464i $$0.337479\pi$$
$$68$$ 0 0
$$69$$ 2.00000 0.240772
$$70$$ 0 0
$$71$$ 10.0000 1.18678 0.593391 0.804914i $$-0.297789\pi$$
0.593391 + 0.804914i $$0.297789\pi$$
$$72$$ 0 0
$$73$$ 7.00000 0.819288 0.409644 0.912245i $$-0.365653\pi$$
0.409644 + 0.912245i $$0.365653\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 0 0
$$77$$ −5.00000 −0.569803
$$78$$ 0 0
$$79$$ −7.00000 −0.787562 −0.393781 0.919204i $$-0.628833\pi$$
−0.393781 + 0.919204i $$0.628833\pi$$
$$80$$ 0 0
$$81$$ −11.0000 −1.22222
$$82$$ 0 0
$$83$$ 15.0000 1.64646 0.823232 0.567705i $$-0.192169\pi$$
0.823232 + 0.567705i $$0.192169\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ 0 0
$$87$$ 10.0000 1.07211
$$88$$ 0 0
$$89$$ 10.0000 1.06000 0.529999 0.847998i $$-0.322192\pi$$
0.529999 + 0.847998i $$0.322192\pi$$
$$90$$ 0 0
$$91$$ −1.00000 −0.104828
$$92$$ 0 0
$$93$$ −4.00000 −0.414781
$$94$$ 0 0
$$95$$ 0 0
$$96$$ 0 0
$$97$$ −4.00000 −0.406138 −0.203069 0.979164i $$-0.565092\pi$$
−0.203069 + 0.979164i $$0.565092\pi$$
$$98$$ 0 0
$$99$$ 5.00000 0.502519
$$100$$ 0 0
$$101$$ 18.0000 1.79107 0.895533 0.444994i $$-0.146794\pi$$
0.895533 + 0.444994i $$0.146794\pi$$
$$102$$ 0 0
$$103$$ −11.0000 −1.08386 −0.541931 0.840423i $$-0.682307\pi$$
−0.541931 + 0.840423i $$0.682307\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$108$$ 0 0
$$109$$ 2.00000 0.191565 0.0957826 0.995402i $$-0.469465\pi$$
0.0957826 + 0.995402i $$0.469465\pi$$
$$110$$ 0 0
$$111$$ −4.00000 −0.379663
$$112$$ 0 0
$$113$$ 4.00000 0.376288 0.188144 0.982141i $$-0.439753\pi$$
0.188144 + 0.982141i $$0.439753\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ 0 0
$$117$$ 1.00000 0.0924500
$$118$$ 0 0
$$119$$ 4.00000 0.366679
$$120$$ 0 0
$$121$$ 14.0000 1.27273
$$122$$ 0 0
$$123$$ 22.0000 1.98367
$$124$$ 0 0
$$125$$ 0 0
$$126$$ 0 0
$$127$$ 22.0000 1.95218 0.976092 0.217357i $$-0.0697436\pi$$
0.976092 + 0.217357i $$0.0697436\pi$$
$$128$$ 0 0
$$129$$ −2.00000 −0.176090
$$130$$ 0 0
$$131$$ −2.00000 −0.174741 −0.0873704 0.996176i $$-0.527846\pi$$
−0.0873704 + 0.996176i $$0.527846\pi$$
$$132$$ 0 0
$$133$$ 7.00000 0.606977
$$134$$ 0 0
$$135$$ 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
$$139$$ −4.00000 −0.339276 −0.169638 0.985506i $$-0.554260\pi$$
−0.169638 + 0.985506i $$0.554260\pi$$
$$140$$ 0 0
$$141$$ 16.0000 1.34744
$$142$$ 0 0
$$143$$ 5.00000 0.418121
$$144$$ 0 0
$$145$$ 0 0
$$146$$ 0 0
$$147$$ −12.0000 −0.989743
$$148$$ 0 0
$$149$$ −2.00000 −0.163846 −0.0819232 0.996639i $$-0.526106\pi$$
−0.0819232 + 0.996639i $$0.526106\pi$$
$$150$$ 0 0
$$151$$ −10.0000 −0.813788 −0.406894 0.913475i $$-0.633388\pi$$
−0.406894 + 0.913475i $$0.633388\pi$$
$$152$$ 0 0
$$153$$ −4.00000 −0.323381
$$154$$ 0 0
$$155$$ 0 0
$$156$$ 0 0
$$157$$ −4.00000 −0.319235 −0.159617 0.987179i $$-0.551026\pi$$
−0.159617 + 0.987179i $$0.551026\pi$$
$$158$$ 0 0
$$159$$ 0 0
$$160$$ 0 0
$$161$$ −1.00000 −0.0788110
$$162$$ 0 0
$$163$$ −4.00000 −0.313304 −0.156652 0.987654i $$-0.550070\pi$$
−0.156652 + 0.987654i $$0.550070\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ 22.0000 1.70241 0.851206 0.524832i $$-0.175872\pi$$
0.851206 + 0.524832i $$0.175872\pi$$
$$168$$ 0 0
$$169$$ −12.0000 −0.923077
$$170$$ 0 0
$$171$$ −7.00000 −0.535303
$$172$$ 0 0
$$173$$ −9.00000 −0.684257 −0.342129 0.939653i $$-0.611148\pi$$
−0.342129 + 0.939653i $$0.611148\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 0 0
$$177$$ 28.0000 2.10461
$$178$$ 0 0
$$179$$ −16.0000 −1.19590 −0.597948 0.801535i $$-0.704017\pi$$
−0.597948 + 0.801535i $$0.704017\pi$$
$$180$$ 0 0
$$181$$ −12.0000 −0.891953 −0.445976 0.895045i $$-0.647144\pi$$
−0.445976 + 0.895045i $$0.647144\pi$$
$$182$$ 0 0
$$183$$ 20.0000 1.47844
$$184$$ 0 0
$$185$$ 0 0
$$186$$ 0 0
$$187$$ −20.0000 −1.46254
$$188$$ 0 0
$$189$$ 4.00000 0.290957
$$190$$ 0 0
$$191$$ 27.0000 1.95365 0.976826 0.214036i $$-0.0686611\pi$$
0.976826 + 0.214036i $$0.0686611\pi$$
$$192$$ 0 0
$$193$$ 6.00000 0.431889 0.215945 0.976406i $$-0.430717\pi$$
0.215945 + 0.976406i $$0.430717\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ −21.0000 −1.49619 −0.748094 0.663593i $$-0.769031\pi$$
−0.748094 + 0.663593i $$0.769031\pi$$
$$198$$ 0 0
$$199$$ 17.0000 1.20510 0.602549 0.798082i $$-0.294152\pi$$
0.602549 + 0.798082i $$0.294152\pi$$
$$200$$ 0 0
$$201$$ 16.0000 1.12855
$$202$$ 0 0
$$203$$ −5.00000 −0.350931
$$204$$ 0 0
$$205$$ 0 0
$$206$$ 0 0
$$207$$ 1.00000 0.0695048
$$208$$ 0 0
$$209$$ −35.0000 −2.42100
$$210$$ 0 0
$$211$$ −2.00000 −0.137686 −0.0688428 0.997628i $$-0.521931\pi$$
−0.0688428 + 0.997628i $$0.521931\pi$$
$$212$$ 0 0
$$213$$ 20.0000 1.37038
$$214$$ 0 0
$$215$$ 0 0
$$216$$ 0 0
$$217$$ 2.00000 0.135769
$$218$$ 0 0
$$219$$ 14.0000 0.946032
$$220$$ 0 0
$$221$$ −4.00000 −0.269069
$$222$$ 0 0
$$223$$ 20.0000 1.33930 0.669650 0.742677i $$-0.266444\pi$$
0.669650 + 0.742677i $$0.266444\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 0 0
$$227$$ −12.0000 −0.796468 −0.398234 0.917284i $$-0.630377\pi$$
−0.398234 + 0.917284i $$0.630377\pi$$
$$228$$ 0 0
$$229$$ −22.0000 −1.45380 −0.726900 0.686743i $$-0.759040\pi$$
−0.726900 + 0.686743i $$0.759040\pi$$
$$230$$ 0 0
$$231$$ −10.0000 −0.657952
$$232$$ 0 0
$$233$$ −15.0000 −0.982683 −0.491341 0.870967i $$-0.663493\pi$$
−0.491341 + 0.870967i $$0.663493\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ 0 0
$$237$$ −14.0000 −0.909398
$$238$$ 0 0
$$239$$ 16.0000 1.03495 0.517477 0.855697i $$-0.326871\pi$$
0.517477 + 0.855697i $$0.326871\pi$$
$$240$$ 0 0
$$241$$ −16.0000 −1.03065 −0.515325 0.856995i $$-0.672329\pi$$
−0.515325 + 0.856995i $$0.672329\pi$$
$$242$$ 0 0
$$243$$ −10.0000 −0.641500
$$244$$ 0 0
$$245$$ 0 0
$$246$$ 0 0
$$247$$ −7.00000 −0.445399
$$248$$ 0 0
$$249$$ 30.0000 1.90117
$$250$$ 0 0
$$251$$ 4.00000 0.252478 0.126239 0.992000i $$-0.459709\pi$$
0.126239 + 0.992000i $$0.459709\pi$$
$$252$$ 0 0
$$253$$ 5.00000 0.314347
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 0 0
$$257$$ −6.00000 −0.374270 −0.187135 0.982334i $$-0.559920\pi$$
−0.187135 + 0.982334i $$0.559920\pi$$
$$258$$ 0 0
$$259$$ 2.00000 0.124274
$$260$$ 0 0
$$261$$ 5.00000 0.309492
$$262$$ 0 0
$$263$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ 0 0
$$267$$ 20.0000 1.22398
$$268$$ 0 0
$$269$$ 17.0000 1.03651 0.518254 0.855227i $$-0.326582\pi$$
0.518254 + 0.855227i $$0.326582\pi$$
$$270$$ 0 0
$$271$$ −8.00000 −0.485965 −0.242983 0.970031i $$-0.578126\pi$$
−0.242983 + 0.970031i $$0.578126\pi$$
$$272$$ 0 0
$$273$$ −2.00000 −0.121046
$$274$$ 0 0
$$275$$ 0 0
$$276$$ 0 0
$$277$$ −1.00000 −0.0600842 −0.0300421 0.999549i $$-0.509564\pi$$
−0.0300421 + 0.999549i $$0.509564\pi$$
$$278$$ 0 0
$$279$$ −2.00000 −0.119737
$$280$$ 0 0
$$281$$ −28.0000 −1.67034 −0.835170 0.549992i $$-0.814631\pi$$
−0.835170 + 0.549992i $$0.814631\pi$$
$$282$$ 0 0
$$283$$ −28.0000 −1.66443 −0.832214 0.554455i $$-0.812927\pi$$
−0.832214 + 0.554455i $$0.812927\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ −11.0000 −0.649309
$$288$$ 0 0
$$289$$ −1.00000 −0.0588235
$$290$$ 0 0
$$291$$ −8.00000 −0.468968
$$292$$ 0 0
$$293$$ −12.0000 −0.701047 −0.350524 0.936554i $$-0.613996\pi$$
−0.350524 + 0.936554i $$0.613996\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ 0 0
$$297$$ −20.0000 −1.16052
$$298$$ 0 0
$$299$$ 1.00000 0.0578315
$$300$$ 0 0
$$301$$ 1.00000 0.0576390
$$302$$ 0 0
$$303$$ 36.0000 2.06815
$$304$$ 0 0
$$305$$ 0 0
$$306$$ 0 0
$$307$$ −16.0000 −0.913168 −0.456584 0.889680i $$-0.650927\pi$$
−0.456584 + 0.889680i $$0.650927\pi$$
$$308$$ 0 0
$$309$$ −22.0000 −1.25154
$$310$$ 0 0
$$311$$ −2.00000 −0.113410 −0.0567048 0.998391i $$-0.518059\pi$$
−0.0567048 + 0.998391i $$0.518059\pi$$
$$312$$ 0 0
$$313$$ 22.0000 1.24351 0.621757 0.783210i $$-0.286419\pi$$
0.621757 + 0.783210i $$0.286419\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ −5.00000 −0.280828 −0.140414 0.990093i $$-0.544843\pi$$
−0.140414 + 0.990093i $$0.544843\pi$$
$$318$$ 0 0
$$319$$ 25.0000 1.39973
$$320$$ 0 0
$$321$$ 0 0
$$322$$ 0 0
$$323$$ 28.0000 1.55796
$$324$$ 0 0
$$325$$ 0 0
$$326$$ 0 0
$$327$$ 4.00000 0.221201
$$328$$ 0 0
$$329$$ −8.00000 −0.441054
$$330$$ 0 0
$$331$$ 18.0000 0.989369 0.494685 0.869072i $$-0.335284\pi$$
0.494685 + 0.869072i $$0.335284\pi$$
$$332$$ 0 0
$$333$$ −2.00000 −0.109599
$$334$$ 0 0
$$335$$ 0 0
$$336$$ 0 0
$$337$$ −20.0000 −1.08947 −0.544735 0.838608i $$-0.683370\pi$$
−0.544735 + 0.838608i $$0.683370\pi$$
$$338$$ 0 0
$$339$$ 8.00000 0.434500
$$340$$ 0 0
$$341$$ −10.0000 −0.541530
$$342$$ 0 0
$$343$$ 13.0000 0.701934
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ 16.0000 0.858925 0.429463 0.903085i $$-0.358703\pi$$
0.429463 + 0.903085i $$0.358703\pi$$
$$348$$ 0 0
$$349$$ 21.0000 1.12410 0.562052 0.827102i $$-0.310012\pi$$
0.562052 + 0.827102i $$0.310012\pi$$
$$350$$ 0 0
$$351$$ −4.00000 −0.213504
$$352$$ 0 0
$$353$$ 11.0000 0.585471 0.292735 0.956193i $$-0.405434\pi$$
0.292735 + 0.956193i $$0.405434\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ 0 0
$$357$$ 8.00000 0.423405
$$358$$ 0 0
$$359$$ 33.0000 1.74167 0.870837 0.491572i $$-0.163578\pi$$
0.870837 + 0.491572i $$0.163578\pi$$
$$360$$ 0 0
$$361$$ 30.0000 1.57895
$$362$$ 0 0
$$363$$ 28.0000 1.46962
$$364$$ 0 0
$$365$$ 0 0
$$366$$ 0 0
$$367$$ 3.00000 0.156599 0.0782994 0.996930i $$-0.475051\pi$$
0.0782994 + 0.996930i $$0.475051\pi$$
$$368$$ 0 0
$$369$$ 11.0000 0.572637
$$370$$ 0 0
$$371$$ 0 0
$$372$$ 0 0
$$373$$ 4.00000 0.207112 0.103556 0.994624i $$-0.466978\pi$$
0.103556 + 0.994624i $$0.466978\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ 5.00000 0.257513
$$378$$ 0 0
$$379$$ 20.0000 1.02733 0.513665 0.857991i $$-0.328287\pi$$
0.513665 + 0.857991i $$0.328287\pi$$
$$380$$ 0 0
$$381$$ 44.0000 2.25419
$$382$$ 0 0
$$383$$ 19.0000 0.970855 0.485427 0.874277i $$-0.338664\pi$$
0.485427 + 0.874277i $$0.338664\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ −1.00000 −0.0508329
$$388$$ 0 0
$$389$$ 16.0000 0.811232 0.405616 0.914044i $$-0.367057\pi$$
0.405616 + 0.914044i $$0.367057\pi$$
$$390$$ 0 0
$$391$$ −4.00000 −0.202289
$$392$$ 0 0
$$393$$ −4.00000 −0.201773
$$394$$ 0 0
$$395$$ 0 0
$$396$$ 0 0
$$397$$ −2.00000 −0.100377 −0.0501886 0.998740i $$-0.515982\pi$$
−0.0501886 + 0.998740i $$0.515982\pi$$
$$398$$ 0 0
$$399$$ 14.0000 0.700877
$$400$$ 0 0
$$401$$ −6.00000 −0.299626 −0.149813 0.988714i $$-0.547867\pi$$
−0.149813 + 0.988714i $$0.547867\pi$$
$$402$$ 0 0
$$403$$ −2.00000 −0.0996271
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 0 0
$$407$$ −10.0000 −0.495682
$$408$$ 0 0
$$409$$ −11.0000 −0.543915 −0.271957 0.962309i $$-0.587671\pi$$
−0.271957 + 0.962309i $$0.587671\pi$$
$$410$$ 0 0
$$411$$ −12.0000 −0.591916
$$412$$ 0 0
$$413$$ −14.0000 −0.688895
$$414$$ 0 0
$$415$$ 0 0
$$416$$ 0 0
$$417$$ −8.00000 −0.391762
$$418$$ 0 0
$$419$$ 3.00000 0.146560 0.0732798 0.997311i $$-0.476653\pi$$
0.0732798 + 0.997311i $$0.476653\pi$$
$$420$$ 0 0
$$421$$ 6.00000 0.292422 0.146211 0.989253i $$-0.453292\pi$$
0.146211 + 0.989253i $$0.453292\pi$$
$$422$$ 0 0
$$423$$ 8.00000 0.388973
$$424$$ 0 0
$$425$$ 0 0
$$426$$ 0 0
$$427$$ −10.0000 −0.483934
$$428$$ 0 0
$$429$$ 10.0000 0.482805
$$430$$ 0 0
$$431$$ 32.0000 1.54139 0.770693 0.637207i $$-0.219910\pi$$
0.770693 + 0.637207i $$0.219910\pi$$
$$432$$ 0 0
$$433$$ −10.0000 −0.480569 −0.240285 0.970702i $$-0.577241\pi$$
−0.240285 + 0.970702i $$0.577241\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 0 0
$$437$$ −7.00000 −0.334855
$$438$$ 0 0
$$439$$ 32.0000 1.52728 0.763638 0.645644i $$-0.223411\pi$$
0.763638 + 0.645644i $$0.223411\pi$$
$$440$$ 0 0
$$441$$ −6.00000 −0.285714
$$442$$ 0 0
$$443$$ −30.0000 −1.42534 −0.712672 0.701498i $$-0.752515\pi$$
−0.712672 + 0.701498i $$0.752515\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ 0 0
$$447$$ −4.00000 −0.189194
$$448$$ 0 0
$$449$$ 30.0000 1.41579 0.707894 0.706319i $$-0.249646\pi$$
0.707894 + 0.706319i $$0.249646\pi$$
$$450$$ 0 0
$$451$$ 55.0000 2.58985
$$452$$ 0 0
$$453$$ −20.0000 −0.939682
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ −8.00000 −0.374224 −0.187112 0.982339i $$-0.559913\pi$$
−0.187112 + 0.982339i $$0.559913\pi$$
$$458$$ 0 0
$$459$$ 16.0000 0.746816
$$460$$ 0 0
$$461$$ 21.0000 0.978068 0.489034 0.872265i $$-0.337349\pi$$
0.489034 + 0.872265i $$0.337349\pi$$
$$462$$ 0 0
$$463$$ −8.00000 −0.371792 −0.185896 0.982569i $$-0.559519\pi$$
−0.185896 + 0.982569i $$0.559519\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ −27.0000 −1.24941 −0.624705 0.780860i $$-0.714781\pi$$
−0.624705 + 0.780860i $$0.714781\pi$$
$$468$$ 0 0
$$469$$ −8.00000 −0.369406
$$470$$ 0 0
$$471$$ −8.00000 −0.368621
$$472$$ 0 0
$$473$$ −5.00000 −0.229900
$$474$$ 0 0
$$475$$ 0 0
$$476$$ 0 0
$$477$$ 0 0
$$478$$ 0 0
$$479$$ 35.0000 1.59919 0.799595 0.600539i $$-0.205047\pi$$
0.799595 + 0.600539i $$0.205047\pi$$
$$480$$ 0 0
$$481$$ −2.00000 −0.0911922
$$482$$ 0 0
$$483$$ −2.00000 −0.0910032
$$484$$ 0 0
$$485$$ 0 0
$$486$$ 0 0
$$487$$ 20.0000 0.906287 0.453143 0.891438i $$-0.350303\pi$$
0.453143 + 0.891438i $$0.350303\pi$$
$$488$$ 0 0
$$489$$ −8.00000 −0.361773
$$490$$ 0 0
$$491$$ −6.00000 −0.270776 −0.135388 0.990793i $$-0.543228\pi$$
−0.135388 + 0.990793i $$0.543228\pi$$
$$492$$ 0 0
$$493$$ −20.0000 −0.900755
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ −10.0000 −0.448561
$$498$$ 0 0
$$499$$ −4.00000 −0.179065 −0.0895323 0.995984i $$-0.528537\pi$$
−0.0895323 + 0.995984i $$0.528537\pi$$
$$500$$ 0 0
$$501$$ 44.0000 1.96578
$$502$$ 0 0
$$503$$ −15.0000 −0.668817 −0.334408 0.942428i $$-0.608537\pi$$
−0.334408 + 0.942428i $$0.608537\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ 0 0
$$507$$ −24.0000 −1.06588
$$508$$ 0 0
$$509$$ 14.0000 0.620539 0.310270 0.950649i $$-0.399581\pi$$
0.310270 + 0.950649i $$0.399581\pi$$
$$510$$ 0 0
$$511$$ −7.00000 −0.309662
$$512$$ 0 0
$$513$$ 28.0000 1.23623
$$514$$ 0 0
$$515$$ 0 0
$$516$$ 0 0
$$517$$ 40.0000 1.75920
$$518$$ 0 0
$$519$$ −18.0000 −0.790112
$$520$$ 0 0
$$521$$ −10.0000 −0.438108 −0.219054 0.975713i $$-0.570297\pi$$
−0.219054 + 0.975713i $$0.570297\pi$$
$$522$$ 0 0
$$523$$ 1.00000 0.0437269 0.0218635 0.999761i $$-0.493040\pi$$
0.0218635 + 0.999761i $$0.493040\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ 8.00000 0.348485
$$528$$ 0 0
$$529$$ 1.00000 0.0434783
$$530$$ 0 0
$$531$$ 14.0000 0.607548
$$532$$ 0 0
$$533$$ 11.0000 0.476463
$$534$$ 0 0
$$535$$ 0 0
$$536$$ 0 0
$$537$$ −32.0000 −1.38090
$$538$$ 0 0
$$539$$ −30.0000 −1.29219
$$540$$ 0 0
$$541$$ −19.0000 −0.816874 −0.408437 0.912787i $$-0.633926\pi$$
−0.408437 + 0.912787i $$0.633926\pi$$
$$542$$ 0 0
$$543$$ −24.0000 −1.02994
$$544$$ 0 0
$$545$$ 0 0
$$546$$ 0 0
$$547$$ 32.0000 1.36822 0.684111 0.729378i $$-0.260191\pi$$
0.684111 + 0.729378i $$0.260191\pi$$
$$548$$ 0 0
$$549$$ 10.0000 0.426790
$$550$$ 0 0
$$551$$ −35.0000 −1.49105
$$552$$ 0 0
$$553$$ 7.00000 0.297670
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 0 0
$$557$$ 42.0000 1.77960 0.889799 0.456354i $$-0.150845\pi$$
0.889799 + 0.456354i $$0.150845\pi$$
$$558$$ 0 0
$$559$$ −1.00000 −0.0422955
$$560$$ 0 0
$$561$$ −40.0000 −1.68880
$$562$$ 0 0
$$563$$ −33.0000 −1.39078 −0.695392 0.718631i $$-0.744769\pi$$
−0.695392 + 0.718631i $$0.744769\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ 0 0
$$567$$ 11.0000 0.461957
$$568$$ 0 0
$$569$$ 12.0000 0.503066 0.251533 0.967849i $$-0.419065\pi$$
0.251533 + 0.967849i $$0.419065\pi$$
$$570$$ 0 0
$$571$$ −4.00000 −0.167395 −0.0836974 0.996491i $$-0.526673\pi$$
−0.0836974 + 0.996491i $$0.526673\pi$$
$$572$$ 0 0
$$573$$ 54.0000 2.25588
$$574$$ 0 0
$$575$$ 0 0
$$576$$ 0 0
$$577$$ −41.0000 −1.70685 −0.853426 0.521214i $$-0.825479\pi$$
−0.853426 + 0.521214i $$0.825479\pi$$
$$578$$ 0 0
$$579$$ 12.0000 0.498703
$$580$$ 0 0
$$581$$ −15.0000 −0.622305
$$582$$ 0 0
$$583$$ 0 0
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ −2.00000 −0.0825488 −0.0412744 0.999148i $$-0.513142\pi$$
−0.0412744 + 0.999148i $$0.513142\pi$$
$$588$$ 0 0
$$589$$ 14.0000 0.576860
$$590$$ 0 0
$$591$$ −42.0000 −1.72765
$$592$$ 0 0
$$593$$ −23.0000 −0.944497 −0.472248 0.881466i $$-0.656557\pi$$
−0.472248 + 0.881466i $$0.656557\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 0 0
$$597$$ 34.0000 1.39153
$$598$$ 0 0
$$599$$ −14.0000 −0.572024 −0.286012 0.958226i $$-0.592330\pi$$
−0.286012 + 0.958226i $$0.592330\pi$$
$$600$$ 0 0
$$601$$ −30.0000 −1.22373 −0.611863 0.790964i $$-0.709580\pi$$
−0.611863 + 0.790964i $$0.709580\pi$$
$$602$$ 0 0
$$603$$ 8.00000 0.325785
$$604$$ 0 0
$$605$$ 0 0
$$606$$ 0 0
$$607$$ −36.0000 −1.46119 −0.730597 0.682808i $$-0.760758\pi$$
−0.730597 + 0.682808i $$0.760758\pi$$
$$608$$ 0 0
$$609$$ −10.0000 −0.405220
$$610$$ 0 0
$$611$$ 8.00000 0.323645
$$612$$ 0 0
$$613$$ −14.0000 −0.565455 −0.282727 0.959200i $$-0.591239\pi$$
−0.282727 + 0.959200i $$0.591239\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ 8.00000 0.322068 0.161034 0.986949i $$-0.448517\pi$$
0.161034 + 0.986949i $$0.448517\pi$$
$$618$$ 0 0
$$619$$ 8.00000 0.321547 0.160774 0.986991i $$-0.448601\pi$$
0.160774 + 0.986991i $$0.448601\pi$$
$$620$$ 0 0
$$621$$ −4.00000 −0.160514
$$622$$ 0 0
$$623$$ −10.0000 −0.400642
$$624$$ 0 0
$$625$$ 0 0
$$626$$ 0 0
$$627$$ −70.0000 −2.79553
$$628$$ 0 0
$$629$$ 8.00000 0.318981
$$630$$ 0 0
$$631$$ −13.0000 −0.517522 −0.258761 0.965941i $$-0.583314\pi$$
−0.258761 + 0.965941i $$0.583314\pi$$
$$632$$ 0 0
$$633$$ −4.00000 −0.158986
$$634$$ 0 0
$$635$$ 0 0
$$636$$ 0 0
$$637$$ −6.00000 −0.237729
$$638$$ 0 0
$$639$$ 10.0000 0.395594
$$640$$ 0 0
$$641$$ 24.0000 0.947943 0.473972 0.880540i $$-0.342820\pi$$
0.473972 + 0.880540i $$0.342820\pi$$
$$642$$ 0 0
$$643$$ −5.00000 −0.197181 −0.0985904 0.995128i $$-0.531433\pi$$
−0.0985904 + 0.995128i $$0.531433\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ −28.0000 −1.10079 −0.550397 0.834903i $$-0.685524\pi$$
−0.550397 + 0.834903i $$0.685524\pi$$
$$648$$ 0 0
$$649$$ 70.0000 2.74774
$$650$$ 0 0
$$651$$ 4.00000 0.156772
$$652$$ 0 0
$$653$$ 25.0000 0.978326 0.489163 0.872192i $$-0.337302\pi$$
0.489163 + 0.872192i $$0.337302\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ 0 0
$$657$$ 7.00000 0.273096
$$658$$ 0 0
$$659$$ 3.00000 0.116863 0.0584317 0.998291i $$-0.481390\pi$$
0.0584317 + 0.998291i $$0.481390\pi$$
$$660$$ 0 0
$$661$$ 4.00000 0.155582 0.0777910 0.996970i $$-0.475213\pi$$
0.0777910 + 0.996970i $$0.475213\pi$$
$$662$$ 0 0
$$663$$ −8.00000 −0.310694
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 0 0
$$667$$ 5.00000 0.193601
$$668$$ 0 0
$$669$$ 40.0000 1.54649
$$670$$ 0 0
$$671$$ 50.0000 1.93023
$$672$$ 0 0
$$673$$ −23.0000 −0.886585 −0.443292 0.896377i $$-0.646190\pi$$
−0.443292 + 0.896377i $$0.646190\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ 0 0
$$677$$ −42.0000 −1.61419 −0.807096 0.590421i $$-0.798962\pi$$
−0.807096 + 0.590421i $$0.798962\pi$$
$$678$$ 0 0
$$679$$ 4.00000 0.153506
$$680$$ 0 0
$$681$$ −24.0000 −0.919682
$$682$$ 0 0
$$683$$ −6.00000 −0.229584 −0.114792 0.993390i $$-0.536620\pi$$
−0.114792 + 0.993390i $$0.536620\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ 0 0
$$687$$ −44.0000 −1.67870
$$688$$ 0 0
$$689$$ 0 0
$$690$$ 0 0
$$691$$ 8.00000 0.304334 0.152167 0.988355i $$-0.451375\pi$$
0.152167 + 0.988355i $$0.451375\pi$$
$$692$$ 0 0
$$693$$ −5.00000 −0.189934
$$694$$ 0 0
$$695$$ 0 0
$$696$$ 0 0
$$697$$ −44.0000 −1.66662
$$698$$ 0 0
$$699$$ −30.0000 −1.13470
$$700$$ 0 0
$$701$$ −48.0000 −1.81293 −0.906467 0.422276i $$-0.861231\pi$$
−0.906467 + 0.422276i $$0.861231\pi$$
$$702$$ 0 0
$$703$$ 14.0000 0.528020
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ −18.0000 −0.676960
$$708$$ 0 0
$$709$$ 26.0000 0.976450 0.488225 0.872718i $$-0.337644\pi$$
0.488225 + 0.872718i $$0.337644\pi$$
$$710$$ 0 0
$$711$$ −7.00000 −0.262521
$$712$$ 0 0
$$713$$ −2.00000 −0.0749006
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 0 0
$$717$$ 32.0000 1.19506
$$718$$ 0 0
$$719$$ −38.0000 −1.41716 −0.708580 0.705630i $$-0.750664\pi$$
−0.708580 + 0.705630i $$0.750664\pi$$
$$720$$ 0 0
$$721$$ 11.0000 0.409661
$$722$$ 0 0
$$723$$ −32.0000 −1.19009
$$724$$ 0 0
$$725$$ 0 0
$$726$$ 0 0
$$727$$ 40.0000 1.48352 0.741759 0.670667i $$-0.233992\pi$$
0.741759 + 0.670667i $$0.233992\pi$$
$$728$$ 0 0
$$729$$ 13.0000 0.481481
$$730$$ 0 0
$$731$$ 4.00000 0.147945
$$732$$ 0 0
$$733$$ −38.0000 −1.40356 −0.701781 0.712393i $$-0.747612\pi$$
−0.701781 + 0.712393i $$0.747612\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ 40.0000 1.47342
$$738$$ 0 0
$$739$$ 22.0000 0.809283 0.404642 0.914475i $$-0.367396\pi$$
0.404642 + 0.914475i $$0.367396\pi$$
$$740$$ 0 0
$$741$$ −14.0000 −0.514303
$$742$$ 0 0
$$743$$ 39.0000 1.43077 0.715386 0.698730i $$-0.246251\pi$$
0.715386 + 0.698730i $$0.246251\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ 0 0
$$747$$ 15.0000 0.548821
$$748$$ 0 0
$$749$$ 0 0
$$750$$ 0 0
$$751$$ −41.0000 −1.49611 −0.748056 0.663636i $$-0.769012\pi$$
−0.748056 + 0.663636i $$0.769012\pi$$
$$752$$ 0 0
$$753$$ 8.00000 0.291536
$$754$$ 0 0
$$755$$ 0 0
$$756$$ 0 0
$$757$$ 10.0000 0.363456 0.181728 0.983349i $$-0.441831\pi$$
0.181728 + 0.983349i $$0.441831\pi$$
$$758$$ 0 0
$$759$$ 10.0000 0.362977
$$760$$ 0 0
$$761$$ −23.0000 −0.833749 −0.416875 0.908964i $$-0.636875\pi$$
−0.416875 + 0.908964i $$0.636875\pi$$
$$762$$ 0 0
$$763$$ −2.00000 −0.0724049
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ 14.0000 0.505511
$$768$$ 0 0
$$769$$ −10.0000 −0.360609 −0.180305 0.983611i $$-0.557708\pi$$
−0.180305 + 0.983611i $$0.557708\pi$$
$$770$$ 0 0
$$771$$ −12.0000 −0.432169
$$772$$ 0 0
$$773$$ 6.00000 0.215805 0.107903 0.994161i $$-0.465587\pi$$
0.107903 + 0.994161i $$0.465587\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ 0 0
$$777$$ 4.00000 0.143499
$$778$$ 0 0
$$779$$ −77.0000 −2.75881
$$780$$ 0 0
$$781$$ 50.0000 1.78914
$$782$$ 0 0
$$783$$ −20.0000 −0.714742
$$784$$ 0 0
$$785$$ 0 0
$$786$$ 0 0
$$787$$ −47.0000 −1.67537 −0.837685 0.546154i $$-0.816091\pi$$
−0.837685 + 0.546154i $$0.816091\pi$$
$$788$$ 0 0
$$789$$ 0 0
$$790$$ 0 0
$$791$$ −4.00000 −0.142224
$$792$$ 0 0
$$793$$ 10.0000 0.355110
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 0 0
$$797$$ −54.0000 −1.91278 −0.956389 0.292096i $$-0.905647\pi$$
−0.956389 + 0.292096i $$0.905647\pi$$
$$798$$ 0 0
$$799$$ −32.0000 −1.13208
$$800$$ 0 0
$$801$$ 10.0000 0.353333
$$802$$ 0 0
$$803$$ 35.0000 1.23512
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 0 0
$$807$$ 34.0000 1.19686
$$808$$ 0 0
$$809$$ 17.0000 0.597688 0.298844 0.954302i $$-0.403399\pi$$
0.298844 + 0.954302i $$0.403399\pi$$
$$810$$ 0 0
$$811$$ 56.0000 1.96643 0.983213 0.182462i $$-0.0584065\pi$$
0.983213 + 0.182462i $$0.0584065\pi$$
$$812$$ 0 0
$$813$$ −16.0000 −0.561144
$$814$$ 0 0
$$815$$ 0 0
$$816$$ 0 0
$$817$$ 7.00000 0.244899
$$818$$ 0 0
$$819$$ −1.00000 −0.0349428
$$820$$ 0 0
$$821$$ 31.0000 1.08191 0.540954 0.841052i $$-0.318063\pi$$
0.540954 + 0.841052i $$0.318063\pi$$
$$822$$ 0 0
$$823$$ −36.0000 −1.25488 −0.627441 0.778664i $$-0.715897\pi$$
−0.627441 + 0.778664i $$0.715897\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ 41.0000 1.42571 0.712855 0.701312i $$-0.247402\pi$$
0.712855 + 0.701312i $$0.247402\pi$$
$$828$$ 0 0
$$829$$ −11.0000 −0.382046 −0.191023 0.981586i $$-0.561180\pi$$
−0.191023 + 0.981586i $$0.561180\pi$$
$$830$$ 0 0
$$831$$ −2.00000 −0.0693792
$$832$$ 0 0
$$833$$ 24.0000 0.831551
$$834$$ 0 0
$$835$$ 0 0
$$836$$ 0 0
$$837$$ 8.00000 0.276520
$$838$$ 0 0
$$839$$ −21.0000 −0.725001 −0.362500 0.931984i $$-0.618077\pi$$
−0.362500 + 0.931984i $$0.618077\pi$$
$$840$$ 0 0
$$841$$ −4.00000 −0.137931
$$842$$ 0 0
$$843$$ −56.0000 −1.92874
$$844$$ 0 0
$$845$$ 0 0
$$846$$ 0 0
$$847$$ −14.0000 −0.481046
$$848$$ 0 0
$$849$$ −56.0000 −1.92192
$$850$$ 0 0
$$851$$ −2.00000 −0.0685591
$$852$$ 0 0
$$853$$ −7.00000 −0.239675 −0.119838 0.992793i $$-0.538237\pi$$
−0.119838 + 0.992793i $$0.538237\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0 0
$$857$$ −58.0000 −1.98124 −0.990621 0.136637i $$-0.956370\pi$$
−0.990621 + 0.136637i $$0.956370\pi$$
$$858$$ 0 0
$$859$$ −30.0000 −1.02359 −0.511793 0.859109i $$-0.671019\pi$$
−0.511793 + 0.859109i $$0.671019\pi$$
$$860$$ 0 0
$$861$$ −22.0000 −0.749758
$$862$$ 0 0
$$863$$ −16.0000 −0.544646 −0.272323 0.962206i $$-0.587792\pi$$
−0.272323 + 0.962206i $$0.587792\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ 0 0
$$867$$ −2.00000 −0.0679236
$$868$$ 0 0
$$869$$ −35.0000 −1.18729
$$870$$ 0 0
$$871$$ 8.00000 0.271070
$$872$$ 0 0
$$873$$ −4.00000 −0.135379
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 0 0
$$877$$ 2.00000 0.0675352 0.0337676 0.999430i $$-0.489249\pi$$
0.0337676 + 0.999430i $$0.489249\pi$$
$$878$$ 0 0
$$879$$ −24.0000 −0.809500
$$880$$ 0 0
$$881$$ 14.0000 0.471672 0.235836 0.971793i $$-0.424217\pi$$
0.235836 + 0.971793i $$0.424217\pi$$
$$882$$ 0 0
$$883$$ −32.0000 −1.07689 −0.538443 0.842662i $$-0.680987\pi$$
−0.538443 + 0.842662i $$0.680987\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 0 0
$$887$$ −24.0000 −0.805841 −0.402921 0.915235i $$-0.632005\pi$$
−0.402921 + 0.915235i $$0.632005\pi$$
$$888$$ 0 0
$$889$$ −22.0000 −0.737856
$$890$$ 0 0
$$891$$ −55.0000 −1.84257
$$892$$ 0 0
$$893$$ −56.0000 −1.87397
$$894$$ 0 0
$$895$$ 0 0
$$896$$ 0 0
$$897$$ 2.00000 0.0667781
$$898$$ 0 0
$$899$$ −10.0000 −0.333519
$$900$$ 0 0
$$901$$ 0 0
$$902$$ 0 0
$$903$$ 2.00000 0.0665558
$$904$$ 0 0
$$905$$ 0 0
$$906$$ 0 0
$$907$$ −7.00000 −0.232431 −0.116216 0.993224i $$-0.537076\pi$$
−0.116216 + 0.993224i $$0.537076\pi$$
$$908$$ 0 0
$$909$$ 18.0000 0.597022
$$910$$ 0 0
$$911$$ −47.0000 −1.55718 −0.778590 0.627533i $$-0.784065\pi$$
−0.778590 + 0.627533i $$0.784065\pi$$
$$912$$ 0 0
$$913$$ 75.0000 2.48214
$$914$$ 0 0
$$915$$ 0 0
$$916$$ 0 0
$$917$$ 2.00000 0.0660458
$$918$$ 0 0
$$919$$ −32.0000 −1.05558 −0.527791 0.849374i $$-0.676980\pi$$
−0.527791 + 0.849374i $$0.676980\pi$$
$$920$$ 0 0
$$921$$ −32.0000 −1.05444
$$922$$ 0 0
$$923$$ 10.0000 0.329154
$$924$$ 0 0
$$925$$ 0 0
$$926$$ 0 0
$$927$$ −11.0000 −0.361287
$$928$$ 0 0
$$929$$ 27.0000 0.885841 0.442921 0.896561i $$-0.353942\pi$$
0.442921 + 0.896561i $$0.353942\pi$$
$$930$$ 0 0
$$931$$ 42.0000 1.37649
$$932$$ 0 0
$$933$$ −4.00000 −0.130954
$$934$$ 0 0
$$935$$ 0 0
$$936$$ 0 0
$$937$$ 48.0000 1.56809 0.784046 0.620703i $$-0.213153\pi$$
0.784046 + 0.620703i $$0.213153\pi$$
$$938$$ 0 0
$$939$$ 44.0000 1.43589
$$940$$ 0 0
$$941$$ −20.0000 −0.651981 −0.325991 0.945373i $$-0.605698\pi$$
−0.325991 + 0.945373i $$0.605698\pi$$
$$942$$ 0 0
$$943$$ 11.0000 0.358209
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ 38.0000 1.23483 0.617417 0.786636i $$-0.288179\pi$$
0.617417 + 0.786636i $$0.288179\pi$$
$$948$$ 0 0
$$949$$ 7.00000 0.227230
$$950$$ 0 0
$$951$$ −10.0000 −0.324272
$$952$$ 0 0
$$953$$ 38.0000 1.23094 0.615470 0.788160i $$-0.288966\pi$$
0.615470 + 0.788160i $$0.288966\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ 0 0
$$957$$ 50.0000 1.61627
$$958$$ 0 0
$$959$$ 6.00000 0.193750
$$960$$ 0 0
$$961$$ −27.0000 −0.870968
$$962$$ 0 0
$$963$$ 0 0
$$964$$ 0 0
$$965$$ 0 0
$$966$$ 0 0
$$967$$ −4.00000 −0.128631 −0.0643157 0.997930i $$-0.520486\pi$$
−0.0643157 + 0.997930i $$0.520486\pi$$
$$968$$ 0 0
$$969$$ 56.0000 1.79898
$$970$$ 0 0
$$971$$ 55.0000 1.76503 0.882517 0.470281i $$-0.155847\pi$$
0.882517 + 0.470281i $$0.155847\pi$$
$$972$$ 0 0
$$973$$ 4.00000 0.128234
$$974$$ 0 0
$$975$$ 0 0
$$976$$ 0 0
$$977$$ −8.00000 −0.255943 −0.127971 0.991778i $$-0.540847\pi$$
−0.127971 + 0.991778i $$0.540847\pi$$
$$978$$ 0 0
$$979$$ 50.0000 1.59801
$$980$$ 0 0
$$981$$ 2.00000 0.0638551
$$982$$ 0 0
$$983$$ 21.0000 0.669796 0.334898 0.942254i $$-0.391298\pi$$
0.334898 + 0.942254i $$0.391298\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ 0 0
$$987$$ −16.0000 −0.509286
$$988$$ 0 0
$$989$$ −1.00000 −0.0317982
$$990$$ 0 0
$$991$$ −28.0000 −0.889449 −0.444725 0.895667i $$-0.646698\pi$$
−0.444725 + 0.895667i $$0.646698\pi$$
$$992$$ 0 0
$$993$$ 36.0000 1.14243
$$994$$ 0 0
$$995$$ 0 0
$$996$$ 0 0
$$997$$ 23.0000 0.728417 0.364209 0.931317i $$-0.381339\pi$$
0.364209 + 0.931317i $$0.381339\pi$$
$$998$$ 0 0
$$999$$ 8.00000 0.253109
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 9200.2.a.bd.1.1 1
4.3 odd 2 4600.2.a.c.1.1 1
5.4 even 2 9200.2.a.i.1.1 1
20.3 even 4 4600.2.e.c.4049.1 2
20.7 even 4 4600.2.e.c.4049.2 2
20.19 odd 2 4600.2.a.n.1.1 yes 1

By twisted newform
Twist Min Dim Char Parity Ord Type
4600.2.a.c.1.1 1 4.3 odd 2
4600.2.a.n.1.1 yes 1 20.19 odd 2
4600.2.e.c.4049.1 2 20.3 even 4
4600.2.e.c.4049.2 2 20.7 even 4
9200.2.a.i.1.1 1 5.4 even 2
9200.2.a.bd.1.1 1 1.1 even 1 trivial