# Properties

 Label 5445.2.a.m.1.2 Level $5445$ Weight $2$ Character 5445.1 Self dual yes Analytic conductor $43.479$ Analytic rank $0$ Dimension $2$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$5445 = 3^{2} \cdot 5 \cdot 11^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 5445.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$43.4785439006$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\zeta_{8})^+$$ Defining polynomial: $$x^{2} - 2$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 165) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.2 Root $$1.41421$$ of defining polynomial Character $$\chi$$ $$=$$ 5445.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+0.414214 q^{2} -1.82843 q^{4} +1.00000 q^{5} +4.82843 q^{7} -1.58579 q^{8} +O(q^{10})$$ $$q+0.414214 q^{2} -1.82843 q^{4} +1.00000 q^{5} +4.82843 q^{7} -1.58579 q^{8} +0.414214 q^{10} -5.65685 q^{13} +2.00000 q^{14} +3.00000 q^{16} -6.82843 q^{17} +1.17157 q^{19} -1.82843 q^{20} +4.00000 q^{23} +1.00000 q^{25} -2.34315 q^{26} -8.82843 q^{28} +0.828427 q^{29} +4.41421 q^{32} -2.82843 q^{34} +4.82843 q^{35} +0.343146 q^{37} +0.485281 q^{38} -1.58579 q^{40} -0.828427 q^{41} +3.17157 q^{43} +1.65685 q^{46} +4.00000 q^{47} +16.3137 q^{49} +0.414214 q^{50} +10.3431 q^{52} +13.3137 q^{53} -7.65685 q^{56} +0.343146 q^{58} +4.00000 q^{59} +0.343146 q^{61} -4.17157 q^{64} -5.65685 q^{65} +5.65685 q^{67} +12.4853 q^{68} +2.00000 q^{70} -13.6569 q^{71} +11.3137 q^{73} +0.142136 q^{74} -2.14214 q^{76} +8.48528 q^{79} +3.00000 q^{80} -0.343146 q^{82} -10.0000 q^{83} -6.82843 q^{85} +1.31371 q^{86} +7.65685 q^{89} -27.3137 q^{91} -7.31371 q^{92} +1.65685 q^{94} +1.17157 q^{95} +0.343146 q^{97} +6.75736 q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 2q^{2} + 2q^{4} + 2q^{5} + 4q^{7} - 6q^{8} + O(q^{10})$$ $$2q - 2q^{2} + 2q^{4} + 2q^{5} + 4q^{7} - 6q^{8} - 2q^{10} + 4q^{14} + 6q^{16} - 8q^{17} + 8q^{19} + 2q^{20} + 8q^{23} + 2q^{25} - 16q^{26} - 12q^{28} - 4q^{29} + 6q^{32} + 4q^{35} + 12q^{37} - 16q^{38} - 6q^{40} + 4q^{41} + 12q^{43} - 8q^{46} + 8q^{47} + 10q^{49} - 2q^{50} + 32q^{52} + 4q^{53} - 4q^{56} + 12q^{58} + 8q^{59} + 12q^{61} - 14q^{64} + 8q^{68} + 4q^{70} - 16q^{71} - 28q^{74} + 24q^{76} + 6q^{80} - 12q^{82} - 20q^{83} - 8q^{85} - 20q^{86} + 4q^{89} - 32q^{91} + 8q^{92} - 8q^{94} + 8q^{95} + 12q^{97} + 22q^{98} + O(q^{100})$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 0.414214 0.292893 0.146447 0.989219i $$-0.453216\pi$$
0.146447 + 0.989219i $$0.453216\pi$$
$$3$$ 0 0
$$4$$ −1.82843 −0.914214
$$5$$ 1.00000 0.447214
$$6$$ 0 0
$$7$$ 4.82843 1.82497 0.912487 0.409106i $$-0.134159\pi$$
0.912487 + 0.409106i $$0.134159\pi$$
$$8$$ −1.58579 −0.560660
$$9$$ 0 0
$$10$$ 0.414214 0.130986
$$11$$ 0 0
$$12$$ 0 0
$$13$$ −5.65685 −1.56893 −0.784465 0.620174i $$-0.787062\pi$$
−0.784465 + 0.620174i $$0.787062\pi$$
$$14$$ 2.00000 0.534522
$$15$$ 0 0
$$16$$ 3.00000 0.750000
$$17$$ −6.82843 −1.65614 −0.828068 0.560627i $$-0.810560\pi$$
−0.828068 + 0.560627i $$0.810560\pi$$
$$18$$ 0 0
$$19$$ 1.17157 0.268777 0.134389 0.990929i $$-0.457093\pi$$
0.134389 + 0.990929i $$0.457093\pi$$
$$20$$ −1.82843 −0.408849
$$21$$ 0 0
$$22$$ 0 0
$$23$$ 4.00000 0.834058 0.417029 0.908893i $$-0.363071\pi$$
0.417029 + 0.908893i $$0.363071\pi$$
$$24$$ 0 0
$$25$$ 1.00000 0.200000
$$26$$ −2.34315 −0.459529
$$27$$ 0 0
$$28$$ −8.82843 −1.66842
$$29$$ 0.828427 0.153835 0.0769175 0.997037i $$-0.475492\pi$$
0.0769175 + 0.997037i $$0.475492\pi$$
$$30$$ 0 0
$$31$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$32$$ 4.41421 0.780330
$$33$$ 0 0
$$34$$ −2.82843 −0.485071
$$35$$ 4.82843 0.816153
$$36$$ 0 0
$$37$$ 0.343146 0.0564128 0.0282064 0.999602i $$-0.491020\pi$$
0.0282064 + 0.999602i $$0.491020\pi$$
$$38$$ 0.485281 0.0787230
$$39$$ 0 0
$$40$$ −1.58579 −0.250735
$$41$$ −0.828427 −0.129379 −0.0646893 0.997905i $$-0.520606\pi$$
−0.0646893 + 0.997905i $$0.520606\pi$$
$$42$$ 0 0
$$43$$ 3.17157 0.483660 0.241830 0.970319i $$-0.422252\pi$$
0.241830 + 0.970319i $$0.422252\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 1.65685 0.244290
$$47$$ 4.00000 0.583460 0.291730 0.956501i $$-0.405769\pi$$
0.291730 + 0.956501i $$0.405769\pi$$
$$48$$ 0 0
$$49$$ 16.3137 2.33053
$$50$$ 0.414214 0.0585786
$$51$$ 0 0
$$52$$ 10.3431 1.43434
$$53$$ 13.3137 1.82878 0.914389 0.404836i $$-0.132671\pi$$
0.914389 + 0.404836i $$0.132671\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ −7.65685 −1.02319
$$57$$ 0 0
$$58$$ 0.343146 0.0450572
$$59$$ 4.00000 0.520756 0.260378 0.965507i $$-0.416153\pi$$
0.260378 + 0.965507i $$0.416153\pi$$
$$60$$ 0 0
$$61$$ 0.343146 0.0439353 0.0219677 0.999759i $$-0.493007\pi$$
0.0219677 + 0.999759i $$0.493007\pi$$
$$62$$ 0 0
$$63$$ 0 0
$$64$$ −4.17157 −0.521447
$$65$$ −5.65685 −0.701646
$$66$$ 0 0
$$67$$ 5.65685 0.691095 0.345547 0.938401i $$-0.387693\pi$$
0.345547 + 0.938401i $$0.387693\pi$$
$$68$$ 12.4853 1.51406
$$69$$ 0 0
$$70$$ 2.00000 0.239046
$$71$$ −13.6569 −1.62077 −0.810385 0.585897i $$-0.800742\pi$$
−0.810385 + 0.585897i $$0.800742\pi$$
$$72$$ 0 0
$$73$$ 11.3137 1.32417 0.662085 0.749429i $$-0.269672\pi$$
0.662085 + 0.749429i $$0.269672\pi$$
$$74$$ 0.142136 0.0165229
$$75$$ 0 0
$$76$$ −2.14214 −0.245720
$$77$$ 0 0
$$78$$ 0 0
$$79$$ 8.48528 0.954669 0.477334 0.878722i $$-0.341603\pi$$
0.477334 + 0.878722i $$0.341603\pi$$
$$80$$ 3.00000 0.335410
$$81$$ 0 0
$$82$$ −0.343146 −0.0378941
$$83$$ −10.0000 −1.09764 −0.548821 0.835940i $$-0.684923\pi$$
−0.548821 + 0.835940i $$0.684923\pi$$
$$84$$ 0 0
$$85$$ −6.82843 −0.740647
$$86$$ 1.31371 0.141661
$$87$$ 0 0
$$88$$ 0 0
$$89$$ 7.65685 0.811625 0.405812 0.913956i $$-0.366989\pi$$
0.405812 + 0.913956i $$0.366989\pi$$
$$90$$ 0 0
$$91$$ −27.3137 −2.86325
$$92$$ −7.31371 −0.762507
$$93$$ 0 0
$$94$$ 1.65685 0.170891
$$95$$ 1.17157 0.120201
$$96$$ 0 0
$$97$$ 0.343146 0.0348412 0.0174206 0.999848i $$-0.494455\pi$$
0.0174206 + 0.999848i $$0.494455\pi$$
$$98$$ 6.75736 0.682596
$$99$$ 0 0
$$100$$ −1.82843 −0.182843
$$101$$ 4.82843 0.480446 0.240223 0.970718i $$-0.422779\pi$$
0.240223 + 0.970718i $$0.422779\pi$$
$$102$$ 0 0
$$103$$ 19.3137 1.90304 0.951518 0.307593i $$-0.0995234\pi$$
0.951518 + 0.307593i $$0.0995234\pi$$
$$104$$ 8.97056 0.879636
$$105$$ 0 0
$$106$$ 5.51472 0.535637
$$107$$ −5.31371 −0.513696 −0.256848 0.966452i $$-0.582684\pi$$
−0.256848 + 0.966452i $$0.582684\pi$$
$$108$$ 0 0
$$109$$ 5.31371 0.508961 0.254480 0.967078i $$-0.418096\pi$$
0.254480 + 0.967078i $$0.418096\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ 14.4853 1.36873
$$113$$ −14.9706 −1.40831 −0.704156 0.710045i $$-0.748674\pi$$
−0.704156 + 0.710045i $$0.748674\pi$$
$$114$$ 0 0
$$115$$ 4.00000 0.373002
$$116$$ −1.51472 −0.140638
$$117$$ 0 0
$$118$$ 1.65685 0.152526
$$119$$ −32.9706 −3.02241
$$120$$ 0 0
$$121$$ 0 0
$$122$$ 0.142136 0.0128684
$$123$$ 0 0
$$124$$ 0 0
$$125$$ 1.00000 0.0894427
$$126$$ 0 0
$$127$$ −2.48528 −0.220533 −0.110267 0.993902i $$-0.535170\pi$$
−0.110267 + 0.993902i $$0.535170\pi$$
$$128$$ −10.5563 −0.933058
$$129$$ 0 0
$$130$$ −2.34315 −0.205507
$$131$$ −19.3137 −1.68745 −0.843723 0.536778i $$-0.819641\pi$$
−0.843723 + 0.536778i $$0.819641\pi$$
$$132$$ 0 0
$$133$$ 5.65685 0.490511
$$134$$ 2.34315 0.202417
$$135$$ 0 0
$$136$$ 10.8284 0.928530
$$137$$ −9.31371 −0.795724 −0.397862 0.917445i $$-0.630248\pi$$
−0.397862 + 0.917445i $$0.630248\pi$$
$$138$$ 0 0
$$139$$ 16.4853 1.39826 0.699132 0.714993i $$-0.253570\pi$$
0.699132 + 0.714993i $$0.253570\pi$$
$$140$$ −8.82843 −0.746138
$$141$$ 0 0
$$142$$ −5.65685 −0.474713
$$143$$ 0 0
$$144$$ 0 0
$$145$$ 0.828427 0.0687971
$$146$$ 4.68629 0.387840
$$147$$ 0 0
$$148$$ −0.627417 −0.0515734
$$149$$ 18.4853 1.51437 0.757187 0.653199i $$-0.226573\pi$$
0.757187 + 0.653199i $$0.226573\pi$$
$$150$$ 0 0
$$151$$ 0.485281 0.0394916 0.0197458 0.999805i $$-0.493714\pi$$
0.0197458 + 0.999805i $$0.493714\pi$$
$$152$$ −1.85786 −0.150693
$$153$$ 0 0
$$154$$ 0 0
$$155$$ 0 0
$$156$$ 0 0
$$157$$ 18.0000 1.43656 0.718278 0.695756i $$-0.244931\pi$$
0.718278 + 0.695756i $$0.244931\pi$$
$$158$$ 3.51472 0.279616
$$159$$ 0 0
$$160$$ 4.41421 0.348974
$$161$$ 19.3137 1.52213
$$162$$ 0 0
$$163$$ 15.3137 1.19946 0.599731 0.800202i $$-0.295274\pi$$
0.599731 + 0.800202i $$0.295274\pi$$
$$164$$ 1.51472 0.118280
$$165$$ 0 0
$$166$$ −4.14214 −0.321492
$$167$$ 9.31371 0.720716 0.360358 0.932814i $$-0.382654\pi$$
0.360358 + 0.932814i $$0.382654\pi$$
$$168$$ 0 0
$$169$$ 19.0000 1.46154
$$170$$ −2.82843 −0.216930
$$171$$ 0 0
$$172$$ −5.79899 −0.442169
$$173$$ −2.82843 −0.215041 −0.107521 0.994203i $$-0.534291\pi$$
−0.107521 + 0.994203i $$0.534291\pi$$
$$174$$ 0 0
$$175$$ 4.82843 0.364995
$$176$$ 0 0
$$177$$ 0 0
$$178$$ 3.17157 0.237719
$$179$$ 6.34315 0.474109 0.237054 0.971496i $$-0.423818\pi$$
0.237054 + 0.971496i $$0.423818\pi$$
$$180$$ 0 0
$$181$$ −14.0000 −1.04061 −0.520306 0.853980i $$-0.674182\pi$$
−0.520306 + 0.853980i $$0.674182\pi$$
$$182$$ −11.3137 −0.838628
$$183$$ 0 0
$$184$$ −6.34315 −0.467623
$$185$$ 0.343146 0.0252286
$$186$$ 0 0
$$187$$ 0 0
$$188$$ −7.31371 −0.533407
$$189$$ 0 0
$$190$$ 0.485281 0.0352060
$$191$$ 5.65685 0.409316 0.204658 0.978834i $$-0.434392\pi$$
0.204658 + 0.978834i $$0.434392\pi$$
$$192$$ 0 0
$$193$$ −2.34315 −0.168663 −0.0843317 0.996438i $$-0.526876\pi$$
−0.0843317 + 0.996438i $$0.526876\pi$$
$$194$$ 0.142136 0.0102047
$$195$$ 0 0
$$196$$ −29.8284 −2.13060
$$197$$ 8.48528 0.604551 0.302276 0.953221i $$-0.402254\pi$$
0.302276 + 0.953221i $$0.402254\pi$$
$$198$$ 0 0
$$199$$ −10.3431 −0.733206 −0.366603 0.930377i $$-0.619479\pi$$
−0.366603 + 0.930377i $$0.619479\pi$$
$$200$$ −1.58579 −0.112132
$$201$$ 0 0
$$202$$ 2.00000 0.140720
$$203$$ 4.00000 0.280745
$$204$$ 0 0
$$205$$ −0.828427 −0.0578599
$$206$$ 8.00000 0.557386
$$207$$ 0 0
$$208$$ −16.9706 −1.17670
$$209$$ 0 0
$$210$$ 0 0
$$211$$ −6.82843 −0.470088 −0.235044 0.971985i $$-0.575523\pi$$
−0.235044 + 0.971985i $$0.575523\pi$$
$$212$$ −24.3431 −1.67189
$$213$$ 0 0
$$214$$ −2.20101 −0.150458
$$215$$ 3.17157 0.216299
$$216$$ 0 0
$$217$$ 0 0
$$218$$ 2.20101 0.149071
$$219$$ 0 0
$$220$$ 0 0
$$221$$ 38.6274 2.59836
$$222$$ 0 0
$$223$$ −17.6569 −1.18239 −0.591195 0.806529i $$-0.701344\pi$$
−0.591195 + 0.806529i $$0.701344\pi$$
$$224$$ 21.3137 1.42408
$$225$$ 0 0
$$226$$ −6.20101 −0.412485
$$227$$ −14.0000 −0.929213 −0.464606 0.885517i $$-0.653804\pi$$
−0.464606 + 0.885517i $$0.653804\pi$$
$$228$$ 0 0
$$229$$ −2.00000 −0.132164 −0.0660819 0.997814i $$-0.521050\pi$$
−0.0660819 + 0.997814i $$0.521050\pi$$
$$230$$ 1.65685 0.109250
$$231$$ 0 0
$$232$$ −1.31371 −0.0862492
$$233$$ −13.1716 −0.862898 −0.431449 0.902137i $$-0.641998\pi$$
−0.431449 + 0.902137i $$0.641998\pi$$
$$234$$ 0 0
$$235$$ 4.00000 0.260931
$$236$$ −7.31371 −0.476082
$$237$$ 0 0
$$238$$ −13.6569 −0.885242
$$239$$ 6.34315 0.410304 0.205152 0.978730i $$-0.434231\pi$$
0.205152 + 0.978730i $$0.434231\pi$$
$$240$$ 0 0
$$241$$ 23.6569 1.52387 0.761936 0.647652i $$-0.224249\pi$$
0.761936 + 0.647652i $$0.224249\pi$$
$$242$$ 0 0
$$243$$ 0 0
$$244$$ −0.627417 −0.0401663
$$245$$ 16.3137 1.04224
$$246$$ 0 0
$$247$$ −6.62742 −0.421692
$$248$$ 0 0
$$249$$ 0 0
$$250$$ 0.414214 0.0261972
$$251$$ 12.9706 0.818695 0.409347 0.912379i $$-0.365756\pi$$
0.409347 + 0.912379i $$0.365756\pi$$
$$252$$ 0 0
$$253$$ 0 0
$$254$$ −1.02944 −0.0645926
$$255$$ 0 0
$$256$$ 3.97056 0.248160
$$257$$ 27.6569 1.72519 0.862594 0.505898i $$-0.168839\pi$$
0.862594 + 0.505898i $$0.168839\pi$$
$$258$$ 0 0
$$259$$ 1.65685 0.102952
$$260$$ 10.3431 0.641455
$$261$$ 0 0
$$262$$ −8.00000 −0.494242
$$263$$ 18.0000 1.10993 0.554964 0.831875i $$-0.312732\pi$$
0.554964 + 0.831875i $$0.312732\pi$$
$$264$$ 0 0
$$265$$ 13.3137 0.817855
$$266$$ 2.34315 0.143667
$$267$$ 0 0
$$268$$ −10.3431 −0.631808
$$269$$ 24.6274 1.50156 0.750780 0.660552i $$-0.229678\pi$$
0.750780 + 0.660552i $$0.229678\pi$$
$$270$$ 0 0
$$271$$ −27.7990 −1.68867 −0.844334 0.535817i $$-0.820004\pi$$
−0.844334 + 0.535817i $$0.820004\pi$$
$$272$$ −20.4853 −1.24210
$$273$$ 0 0
$$274$$ −3.85786 −0.233062
$$275$$ 0 0
$$276$$ 0 0
$$277$$ 13.6569 0.820561 0.410280 0.911959i $$-0.365431\pi$$
0.410280 + 0.911959i $$0.365431\pi$$
$$278$$ 6.82843 0.409542
$$279$$ 0 0
$$280$$ −7.65685 −0.457585
$$281$$ −16.8284 −1.00390 −0.501950 0.864897i $$-0.667384\pi$$
−0.501950 + 0.864897i $$0.667384\pi$$
$$282$$ 0 0
$$283$$ 3.17157 0.188530 0.0942652 0.995547i $$-0.469950\pi$$
0.0942652 + 0.995547i $$0.469950\pi$$
$$284$$ 24.9706 1.48173
$$285$$ 0 0
$$286$$ 0 0
$$287$$ −4.00000 −0.236113
$$288$$ 0 0
$$289$$ 29.6274 1.74279
$$290$$ 0.343146 0.0201502
$$291$$ 0 0
$$292$$ −20.6863 −1.21057
$$293$$ −1.17157 −0.0684440 −0.0342220 0.999414i $$-0.510895\pi$$
−0.0342220 + 0.999414i $$0.510895\pi$$
$$294$$ 0 0
$$295$$ 4.00000 0.232889
$$296$$ −0.544156 −0.0316284
$$297$$ 0 0
$$298$$ 7.65685 0.443550
$$299$$ −22.6274 −1.30858
$$300$$ 0 0
$$301$$ 15.3137 0.882667
$$302$$ 0.201010 0.0115668
$$303$$ 0 0
$$304$$ 3.51472 0.201583
$$305$$ 0.343146 0.0196485
$$306$$ 0 0
$$307$$ 8.82843 0.503865 0.251932 0.967745i $$-0.418934\pi$$
0.251932 + 0.967745i $$0.418934\pi$$
$$308$$ 0 0
$$309$$ 0 0
$$310$$ 0 0
$$311$$ −19.3137 −1.09518 −0.547590 0.836747i $$-0.684455\pi$$
−0.547590 + 0.836747i $$0.684455\pi$$
$$312$$ 0 0
$$313$$ 4.34315 0.245489 0.122745 0.992438i $$-0.460830\pi$$
0.122745 + 0.992438i $$0.460830\pi$$
$$314$$ 7.45584 0.420758
$$315$$ 0 0
$$316$$ −15.5147 −0.872771
$$317$$ −30.2843 −1.70093 −0.850467 0.526028i $$-0.823681\pi$$
−0.850467 + 0.526028i $$0.823681\pi$$
$$318$$ 0 0
$$319$$ 0 0
$$320$$ −4.17157 −0.233198
$$321$$ 0 0
$$322$$ 8.00000 0.445823
$$323$$ −8.00000 −0.445132
$$324$$ 0 0
$$325$$ −5.65685 −0.313786
$$326$$ 6.34315 0.351314
$$327$$ 0 0
$$328$$ 1.31371 0.0725374
$$329$$ 19.3137 1.06480
$$330$$ 0 0
$$331$$ 17.6569 0.970508 0.485254 0.874373i $$-0.338727\pi$$
0.485254 + 0.874373i $$0.338727\pi$$
$$332$$ 18.2843 1.00348
$$333$$ 0 0
$$334$$ 3.85786 0.211093
$$335$$ 5.65685 0.309067
$$336$$ 0 0
$$337$$ 19.3137 1.05208 0.526042 0.850458i $$-0.323675\pi$$
0.526042 + 0.850458i $$0.323675\pi$$
$$338$$ 7.87006 0.428075
$$339$$ 0 0
$$340$$ 12.4853 0.677109
$$341$$ 0 0
$$342$$ 0 0
$$343$$ 44.9706 2.42818
$$344$$ −5.02944 −0.271169
$$345$$ 0 0
$$346$$ −1.17157 −0.0629841
$$347$$ −6.68629 −0.358939 −0.179469 0.983764i $$-0.557438\pi$$
−0.179469 + 0.983764i $$0.557438\pi$$
$$348$$ 0 0
$$349$$ 22.9706 1.22959 0.614793 0.788688i $$-0.289240\pi$$
0.614793 + 0.788688i $$0.289240\pi$$
$$350$$ 2.00000 0.106904
$$351$$ 0 0
$$352$$ 0 0
$$353$$ −26.0000 −1.38384 −0.691920 0.721974i $$-0.743235\pi$$
−0.691920 + 0.721974i $$0.743235\pi$$
$$354$$ 0 0
$$355$$ −13.6569 −0.724831
$$356$$ −14.0000 −0.741999
$$357$$ 0 0
$$358$$ 2.62742 0.138863
$$359$$ 12.0000 0.633336 0.316668 0.948536i $$-0.397436\pi$$
0.316668 + 0.948536i $$0.397436\pi$$
$$360$$ 0 0
$$361$$ −17.6274 −0.927759
$$362$$ −5.79899 −0.304788
$$363$$ 0 0
$$364$$ 49.9411 2.61763
$$365$$ 11.3137 0.592187
$$366$$ 0 0
$$367$$ −1.65685 −0.0864871 −0.0432435 0.999065i $$-0.513769\pi$$
−0.0432435 + 0.999065i $$0.513769\pi$$
$$368$$ 12.0000 0.625543
$$369$$ 0 0
$$370$$ 0.142136 0.00738928
$$371$$ 64.2843 3.33747
$$372$$ 0 0
$$373$$ 34.6274 1.79294 0.896470 0.443105i $$-0.146123\pi$$
0.896470 + 0.443105i $$0.146123\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ −6.34315 −0.327123
$$377$$ −4.68629 −0.241356
$$378$$ 0 0
$$379$$ −0.686292 −0.0352524 −0.0176262 0.999845i $$-0.505611\pi$$
−0.0176262 + 0.999845i $$0.505611\pi$$
$$380$$ −2.14214 −0.109889
$$381$$ 0 0
$$382$$ 2.34315 0.119886
$$383$$ 8.00000 0.408781 0.204390 0.978889i $$-0.434479\pi$$
0.204390 + 0.978889i $$0.434479\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ −0.970563 −0.0494003
$$387$$ 0 0
$$388$$ −0.627417 −0.0318523
$$389$$ 12.3431 0.625822 0.312911 0.949782i $$-0.398696\pi$$
0.312911 + 0.949782i $$0.398696\pi$$
$$390$$ 0 0
$$391$$ −27.3137 −1.38131
$$392$$ −25.8701 −1.30664
$$393$$ 0 0
$$394$$ 3.51472 0.177069
$$395$$ 8.48528 0.426941
$$396$$ 0 0
$$397$$ 18.9706 0.952105 0.476053 0.879417i $$-0.342067\pi$$
0.476053 + 0.879417i $$0.342067\pi$$
$$398$$ −4.28427 −0.214751
$$399$$ 0 0
$$400$$ 3.00000 0.150000
$$401$$ 29.3137 1.46386 0.731928 0.681382i $$-0.238621\pi$$
0.731928 + 0.681382i $$0.238621\pi$$
$$402$$ 0 0
$$403$$ 0 0
$$404$$ −8.82843 −0.439231
$$405$$ 0 0
$$406$$ 1.65685 0.0822283
$$407$$ 0 0
$$408$$ 0 0
$$409$$ 8.34315 0.412542 0.206271 0.978495i $$-0.433867\pi$$
0.206271 + 0.978495i $$0.433867\pi$$
$$410$$ −0.343146 −0.0169468
$$411$$ 0 0
$$412$$ −35.3137 −1.73978
$$413$$ 19.3137 0.950365
$$414$$ 0 0
$$415$$ −10.0000 −0.490881
$$416$$ −24.9706 −1.22428
$$417$$ 0 0
$$418$$ 0 0
$$419$$ 3.02944 0.147998 0.0739988 0.997258i $$-0.476424\pi$$
0.0739988 + 0.997258i $$0.476424\pi$$
$$420$$ 0 0
$$421$$ −6.00000 −0.292422 −0.146211 0.989253i $$-0.546708\pi$$
−0.146211 + 0.989253i $$0.546708\pi$$
$$422$$ −2.82843 −0.137686
$$423$$ 0 0
$$424$$ −21.1127 −1.02532
$$425$$ −6.82843 −0.331227
$$426$$ 0 0
$$427$$ 1.65685 0.0801808
$$428$$ 9.71573 0.469627
$$429$$ 0 0
$$430$$ 1.31371 0.0633526
$$431$$ 10.3431 0.498212 0.249106 0.968476i $$-0.419863\pi$$
0.249106 + 0.968476i $$0.419863\pi$$
$$432$$ 0 0
$$433$$ −4.34315 −0.208718 −0.104359 0.994540i $$-0.533279\pi$$
−0.104359 + 0.994540i $$0.533279\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ −9.71573 −0.465299
$$437$$ 4.68629 0.224176
$$438$$ 0 0
$$439$$ 3.51472 0.167748 0.0838742 0.996476i $$-0.473271\pi$$
0.0838742 + 0.996476i $$0.473271\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 16.0000 0.761042
$$443$$ −12.0000 −0.570137 −0.285069 0.958507i $$-0.592016\pi$$
−0.285069 + 0.958507i $$0.592016\pi$$
$$444$$ 0 0
$$445$$ 7.65685 0.362970
$$446$$ −7.31371 −0.346314
$$447$$ 0 0
$$448$$ −20.1421 −0.951626
$$449$$ 2.97056 0.140190 0.0700948 0.997540i $$-0.477670\pi$$
0.0700948 + 0.997540i $$0.477670\pi$$
$$450$$ 0 0
$$451$$ 0 0
$$452$$ 27.3726 1.28750
$$453$$ 0 0
$$454$$ −5.79899 −0.272160
$$455$$ −27.3137 −1.28049
$$456$$ 0 0
$$457$$ 0.686292 0.0321034 0.0160517 0.999871i $$-0.494890\pi$$
0.0160517 + 0.999871i $$0.494890\pi$$
$$458$$ −0.828427 −0.0387099
$$459$$ 0 0
$$460$$ −7.31371 −0.341003
$$461$$ −28.1421 −1.31071 −0.655355 0.755321i $$-0.727481\pi$$
−0.655355 + 0.755321i $$0.727481\pi$$
$$462$$ 0 0
$$463$$ −28.9706 −1.34638 −0.673188 0.739471i $$-0.735076\pi$$
−0.673188 + 0.739471i $$0.735076\pi$$
$$464$$ 2.48528 0.115376
$$465$$ 0 0
$$466$$ −5.45584 −0.252737
$$467$$ −22.6274 −1.04707 −0.523536 0.852004i $$-0.675387\pi$$
−0.523536 + 0.852004i $$0.675387\pi$$
$$468$$ 0 0
$$469$$ 27.3137 1.26123
$$470$$ 1.65685 0.0764250
$$471$$ 0 0
$$472$$ −6.34315 −0.291967
$$473$$ 0 0
$$474$$ 0 0
$$475$$ 1.17157 0.0537555
$$476$$ 60.2843 2.76313
$$477$$ 0 0
$$478$$ 2.62742 0.120175
$$479$$ 3.02944 0.138419 0.0692093 0.997602i $$-0.477952\pi$$
0.0692093 + 0.997602i $$0.477952\pi$$
$$480$$ 0 0
$$481$$ −1.94113 −0.0885077
$$482$$ 9.79899 0.446332
$$483$$ 0 0
$$484$$ 0 0
$$485$$ 0.343146 0.0155814
$$486$$ 0 0
$$487$$ 20.9706 0.950267 0.475133 0.879914i $$-0.342400\pi$$
0.475133 + 0.879914i $$0.342400\pi$$
$$488$$ −0.544156 −0.0246328
$$489$$ 0 0
$$490$$ 6.75736 0.305266
$$491$$ 25.6569 1.15788 0.578939 0.815371i $$-0.303467\pi$$
0.578939 + 0.815371i $$0.303467\pi$$
$$492$$ 0 0
$$493$$ −5.65685 −0.254772
$$494$$ −2.74517 −0.123511
$$495$$ 0 0
$$496$$ 0 0
$$497$$ −65.9411 −2.95786
$$498$$ 0 0
$$499$$ −33.6569 −1.50669 −0.753344 0.657627i $$-0.771560\pi$$
−0.753344 + 0.657627i $$0.771560\pi$$
$$500$$ −1.82843 −0.0817697
$$501$$ 0 0
$$502$$ 5.37258 0.239790
$$503$$ −5.31371 −0.236927 −0.118463 0.992958i $$-0.537797\pi$$
−0.118463 + 0.992958i $$0.537797\pi$$
$$504$$ 0 0
$$505$$ 4.82843 0.214862
$$506$$ 0 0
$$507$$ 0 0
$$508$$ 4.54416 0.201614
$$509$$ −41.3137 −1.83120 −0.915599 0.402093i $$-0.868283\pi$$
−0.915599 + 0.402093i $$0.868283\pi$$
$$510$$ 0 0
$$511$$ 54.6274 2.41657
$$512$$ 22.7574 1.00574
$$513$$ 0 0
$$514$$ 11.4558 0.505296
$$515$$ 19.3137 0.851064
$$516$$ 0 0
$$517$$ 0 0
$$518$$ 0.686292 0.0301539
$$519$$ 0 0
$$520$$ 8.97056 0.393385
$$521$$ −12.6274 −0.553217 −0.276609 0.960983i $$-0.589211\pi$$
−0.276609 + 0.960983i $$0.589211\pi$$
$$522$$ 0 0
$$523$$ 26.4853 1.15812 0.579060 0.815285i $$-0.303420\pi$$
0.579060 + 0.815285i $$0.303420\pi$$
$$524$$ 35.3137 1.54269
$$525$$ 0 0
$$526$$ 7.45584 0.325090
$$527$$ 0 0
$$528$$ 0 0
$$529$$ −7.00000 −0.304348
$$530$$ 5.51472 0.239544
$$531$$ 0 0
$$532$$ −10.3431 −0.448432
$$533$$ 4.68629 0.202986
$$534$$ 0 0
$$535$$ −5.31371 −0.229732
$$536$$ −8.97056 −0.387469
$$537$$ 0 0
$$538$$ 10.2010 0.439797
$$539$$ 0 0
$$540$$ 0 0
$$541$$ 5.31371 0.228454 0.114227 0.993455i $$-0.463561\pi$$
0.114227 + 0.993455i $$0.463561\pi$$
$$542$$ −11.5147 −0.494600
$$543$$ 0 0
$$544$$ −30.1421 −1.29233
$$545$$ 5.31371 0.227614
$$546$$ 0 0
$$547$$ −20.1421 −0.861216 −0.430608 0.902539i $$-0.641701\pi$$
−0.430608 + 0.902539i $$0.641701\pi$$
$$548$$ 17.0294 0.727462
$$549$$ 0 0
$$550$$ 0 0
$$551$$ 0.970563 0.0413474
$$552$$ 0 0
$$553$$ 40.9706 1.74225
$$554$$ 5.65685 0.240337
$$555$$ 0 0
$$556$$ −30.1421 −1.27831
$$557$$ 10.8284 0.458815 0.229408 0.973330i $$-0.426321\pi$$
0.229408 + 0.973330i $$0.426321\pi$$
$$558$$ 0 0
$$559$$ −17.9411 −0.758829
$$560$$ 14.4853 0.612115
$$561$$ 0 0
$$562$$ −6.97056 −0.294035
$$563$$ 20.3431 0.857361 0.428681 0.903456i $$-0.358979\pi$$
0.428681 + 0.903456i $$0.358979\pi$$
$$564$$ 0 0
$$565$$ −14.9706 −0.629816
$$566$$ 1.31371 0.0552193
$$567$$ 0 0
$$568$$ 21.6569 0.908701
$$569$$ 15.4558 0.647943 0.323971 0.946067i $$-0.394982\pi$$
0.323971 + 0.946067i $$0.394982\pi$$
$$570$$ 0 0
$$571$$ −0.485281 −0.0203084 −0.0101542 0.999948i $$-0.503232\pi$$
−0.0101542 + 0.999948i $$0.503232\pi$$
$$572$$ 0 0
$$573$$ 0 0
$$574$$ −1.65685 −0.0691558
$$575$$ 4.00000 0.166812
$$576$$ 0 0
$$577$$ 14.0000 0.582828 0.291414 0.956597i $$-0.405874\pi$$
0.291414 + 0.956597i $$0.405874\pi$$
$$578$$ 12.2721 0.510451
$$579$$ 0 0
$$580$$ −1.51472 −0.0628953
$$581$$ −48.2843 −2.00317
$$582$$ 0 0
$$583$$ 0 0
$$584$$ −17.9411 −0.742409
$$585$$ 0 0
$$586$$ −0.485281 −0.0200468
$$587$$ 30.6274 1.26413 0.632064 0.774916i $$-0.282208\pi$$
0.632064 + 0.774916i $$0.282208\pi$$
$$588$$ 0 0
$$589$$ 0 0
$$590$$ 1.65685 0.0682116
$$591$$ 0 0
$$592$$ 1.02944 0.0423096
$$593$$ −17.1716 −0.705152 −0.352576 0.935783i $$-0.614694\pi$$
−0.352576 + 0.935783i $$0.614694\pi$$
$$594$$ 0 0
$$595$$ −32.9706 −1.35166
$$596$$ −33.7990 −1.38446
$$597$$ 0 0
$$598$$ −9.37258 −0.383273
$$599$$ −4.68629 −0.191477 −0.0957383 0.995407i $$-0.530521\pi$$
−0.0957383 + 0.995407i $$0.530521\pi$$
$$600$$ 0 0
$$601$$ −17.3137 −0.706241 −0.353120 0.935578i $$-0.614879\pi$$
−0.353120 + 0.935578i $$0.614879\pi$$
$$602$$ 6.34315 0.258527
$$603$$ 0 0
$$604$$ −0.887302 −0.0361038
$$605$$ 0 0
$$606$$ 0 0
$$607$$ −18.4853 −0.750294 −0.375147 0.926965i $$-0.622408\pi$$
−0.375147 + 0.926965i $$0.622408\pi$$
$$608$$ 5.17157 0.209735
$$609$$ 0 0
$$610$$ 0.142136 0.00575490
$$611$$ −22.6274 −0.915407
$$612$$ 0 0
$$613$$ −21.9411 −0.886194 −0.443097 0.896474i $$-0.646120\pi$$
−0.443097 + 0.896474i $$0.646120\pi$$
$$614$$ 3.65685 0.147579
$$615$$ 0 0
$$616$$ 0 0
$$617$$ −11.6569 −0.469287 −0.234644 0.972081i $$-0.575392\pi$$
−0.234644 + 0.972081i $$0.575392\pi$$
$$618$$ 0 0
$$619$$ −25.6569 −1.03124 −0.515618 0.856819i $$-0.672438\pi$$
−0.515618 + 0.856819i $$0.672438\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ −8.00000 −0.320771
$$623$$ 36.9706 1.48119
$$624$$ 0 0
$$625$$ 1.00000 0.0400000
$$626$$ 1.79899 0.0719021
$$627$$ 0 0
$$628$$ −32.9117 −1.31332
$$629$$ −2.34315 −0.0934273
$$630$$ 0 0
$$631$$ 34.3431 1.36718 0.683590 0.729867i $$-0.260418\pi$$
0.683590 + 0.729867i $$0.260418\pi$$
$$632$$ −13.4558 −0.535245
$$633$$ 0 0
$$634$$ −12.5442 −0.498192
$$635$$ −2.48528 −0.0986254
$$636$$ 0 0
$$637$$ −92.2843 −3.65644
$$638$$ 0 0
$$639$$ 0 0
$$640$$ −10.5563 −0.417276
$$641$$ 26.9706 1.06527 0.532637 0.846344i $$-0.321201\pi$$
0.532637 + 0.846344i $$0.321201\pi$$
$$642$$ 0 0
$$643$$ −29.9411 −1.18076 −0.590381 0.807124i $$-0.701023\pi$$
−0.590381 + 0.807124i $$0.701023\pi$$
$$644$$ −35.3137 −1.39156
$$645$$ 0 0
$$646$$ −3.31371 −0.130376
$$647$$ −27.3137 −1.07381 −0.536906 0.843642i $$-0.680407\pi$$
−0.536906 + 0.843642i $$0.680407\pi$$
$$648$$ 0 0
$$649$$ 0 0
$$650$$ −2.34315 −0.0919057
$$651$$ 0 0
$$652$$ −28.0000 −1.09656
$$653$$ 26.9706 1.05544 0.527720 0.849418i $$-0.323047\pi$$
0.527720 + 0.849418i $$0.323047\pi$$
$$654$$ 0 0
$$655$$ −19.3137 −0.754649
$$656$$ −2.48528 −0.0970339
$$657$$ 0 0
$$658$$ 8.00000 0.311872
$$659$$ 7.31371 0.284902 0.142451 0.989802i $$-0.454502\pi$$
0.142451 + 0.989802i $$0.454502\pi$$
$$660$$ 0 0
$$661$$ −13.3137 −0.517843 −0.258922 0.965898i $$-0.583367\pi$$
−0.258922 + 0.965898i $$0.583367\pi$$
$$662$$ 7.31371 0.284255
$$663$$ 0 0
$$664$$ 15.8579 0.615404
$$665$$ 5.65685 0.219363
$$666$$ 0 0
$$667$$ 3.31371 0.128307
$$668$$ −17.0294 −0.658889
$$669$$ 0 0
$$670$$ 2.34315 0.0905236
$$671$$ 0 0
$$672$$ 0 0
$$673$$ −29.6569 −1.14319 −0.571594 0.820537i $$-0.693675\pi$$
−0.571594 + 0.820537i $$0.693675\pi$$
$$674$$ 8.00000 0.308148
$$675$$ 0 0
$$676$$ −34.7401 −1.33616
$$677$$ −21.4558 −0.824615 −0.412308 0.911045i $$-0.635277\pi$$
−0.412308 + 0.911045i $$0.635277\pi$$
$$678$$ 0 0
$$679$$ 1.65685 0.0635842
$$680$$ 10.8284 0.415251
$$681$$ 0 0
$$682$$ 0 0
$$683$$ −24.0000 −0.918334 −0.459167 0.888350i $$-0.651852\pi$$
−0.459167 + 0.888350i $$0.651852\pi$$
$$684$$ 0 0
$$685$$ −9.31371 −0.355859
$$686$$ 18.6274 0.711198
$$687$$ 0 0
$$688$$ 9.51472 0.362745
$$689$$ −75.3137 −2.86922
$$690$$ 0 0
$$691$$ 20.0000 0.760836 0.380418 0.924815i $$-0.375780\pi$$
0.380418 + 0.924815i $$0.375780\pi$$
$$692$$ 5.17157 0.196594
$$693$$ 0 0
$$694$$ −2.76955 −0.105131
$$695$$ 16.4853 0.625322
$$696$$ 0 0
$$697$$ 5.65685 0.214269
$$698$$ 9.51472 0.360137
$$699$$ 0 0
$$700$$ −8.82843 −0.333683
$$701$$ 7.85786 0.296787 0.148394 0.988928i $$-0.452590\pi$$
0.148394 + 0.988928i $$0.452590\pi$$
$$702$$ 0 0
$$703$$ 0.402020 0.0151625
$$704$$ 0 0
$$705$$ 0 0
$$706$$ −10.7696 −0.405317
$$707$$ 23.3137 0.876802
$$708$$ 0 0
$$709$$ 29.3137 1.10090 0.550450 0.834868i $$-0.314456\pi$$
0.550450 + 0.834868i $$0.314456\pi$$
$$710$$ −5.65685 −0.212298
$$711$$ 0 0
$$712$$ −12.1421 −0.455046
$$713$$ 0 0
$$714$$ 0 0
$$715$$ 0 0
$$716$$ −11.5980 −0.433437
$$717$$ 0 0
$$718$$ 4.97056 0.185500
$$719$$ −31.5980 −1.17841 −0.589203 0.807985i $$-0.700558\pi$$
−0.589203 + 0.807985i $$0.700558\pi$$
$$720$$ 0 0
$$721$$ 93.2548 3.47299
$$722$$ −7.30152 −0.271734
$$723$$ 0 0
$$724$$ 25.5980 0.951341
$$725$$ 0.828427 0.0307670
$$726$$ 0 0
$$727$$ −33.9411 −1.25881 −0.629403 0.777079i $$-0.716701\pi$$
−0.629403 + 0.777079i $$0.716701\pi$$
$$728$$ 43.3137 1.60531
$$729$$ 0 0
$$730$$ 4.68629 0.173447
$$731$$ −21.6569 −0.801008
$$732$$ 0 0
$$733$$ 17.6569 0.652171 0.326085 0.945340i $$-0.394270\pi$$
0.326085 + 0.945340i $$0.394270\pi$$
$$734$$ −0.686292 −0.0253315
$$735$$ 0 0
$$736$$ 17.6569 0.650840
$$737$$ 0 0
$$738$$ 0 0
$$739$$ −47.1127 −1.73307 −0.866534 0.499118i $$-0.833658\pi$$
−0.866534 + 0.499118i $$0.833658\pi$$
$$740$$ −0.627417 −0.0230643
$$741$$ 0 0
$$742$$ 26.6274 0.977523
$$743$$ 47.6569 1.74836 0.874180 0.485602i $$-0.161399\pi$$
0.874180 + 0.485602i $$0.161399\pi$$
$$744$$ 0 0
$$745$$ 18.4853 0.677248
$$746$$ 14.3431 0.525140
$$747$$ 0 0
$$748$$ 0 0
$$749$$ −25.6569 −0.937481
$$750$$ 0 0
$$751$$ −36.2843 −1.32403 −0.662016 0.749490i $$-0.730299\pi$$
−0.662016 + 0.749490i $$0.730299\pi$$
$$752$$ 12.0000 0.437595
$$753$$ 0 0
$$754$$ −1.94113 −0.0706916
$$755$$ 0.485281 0.0176612
$$756$$ 0 0
$$757$$ 8.62742 0.313569 0.156784 0.987633i $$-0.449887\pi$$
0.156784 + 0.987633i $$0.449887\pi$$
$$758$$ −0.284271 −0.0103252
$$759$$ 0 0
$$760$$ −1.85786 −0.0673918
$$761$$ −23.1716 −0.839969 −0.419984 0.907531i $$-0.637964\pi$$
−0.419984 + 0.907531i $$0.637964\pi$$
$$762$$ 0 0
$$763$$ 25.6569 0.928840
$$764$$ −10.3431 −0.374202
$$765$$ 0 0
$$766$$ 3.31371 0.119729
$$767$$ −22.6274 −0.817029
$$768$$ 0 0
$$769$$ −33.3137 −1.20132 −0.600662 0.799503i $$-0.705096\pi$$
−0.600662 + 0.799503i $$0.705096\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ 4.28427 0.154194
$$773$$ 7.65685 0.275398 0.137699 0.990474i $$-0.456029\pi$$
0.137699 + 0.990474i $$0.456029\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ −0.544156 −0.0195341
$$777$$ 0 0
$$778$$ 5.11270 0.183299
$$779$$ −0.970563 −0.0347740
$$780$$ 0 0
$$781$$ 0 0
$$782$$ −11.3137 −0.404577
$$783$$ 0 0
$$784$$ 48.9411 1.74790
$$785$$ 18.0000 0.642448
$$786$$ 0 0
$$787$$ −8.14214 −0.290236 −0.145118 0.989414i $$-0.546356\pi$$
−0.145118 + 0.989414i $$0.546356\pi$$
$$788$$ −15.5147 −0.552689
$$789$$ 0 0
$$790$$ 3.51472 0.125048
$$791$$ −72.2843 −2.57013
$$792$$ 0 0
$$793$$ −1.94113 −0.0689314
$$794$$ 7.85786 0.278865
$$795$$ 0 0
$$796$$ 18.9117 0.670307
$$797$$ −1.02944 −0.0364645 −0.0182323 0.999834i $$-0.505804\pi$$
−0.0182323 + 0.999834i $$0.505804\pi$$
$$798$$ 0 0
$$799$$ −27.3137 −0.966290
$$800$$ 4.41421 0.156066
$$801$$ 0 0
$$802$$ 12.1421 0.428754
$$803$$ 0 0
$$804$$ 0 0
$$805$$ 19.3137 0.680719
$$806$$ 0 0
$$807$$ 0 0
$$808$$ −7.65685 −0.269367
$$809$$ −56.4264 −1.98385 −0.991923 0.126838i $$-0.959517\pi$$
−0.991923 + 0.126838i $$0.959517\pi$$
$$810$$ 0 0
$$811$$ 16.4853 0.578877 0.289438 0.957197i $$-0.406532\pi$$
0.289438 + 0.957197i $$0.406532\pi$$
$$812$$ −7.31371 −0.256661
$$813$$ 0 0
$$814$$ 0 0
$$815$$ 15.3137 0.536416
$$816$$ 0 0
$$817$$ 3.71573 0.129997
$$818$$ 3.45584 0.120831
$$819$$ 0 0
$$820$$ 1.51472 0.0528963
$$821$$ −7.17157 −0.250290 −0.125145 0.992138i $$-0.539940\pi$$
−0.125145 + 0.992138i $$0.539940\pi$$
$$822$$ 0 0
$$823$$ 16.0000 0.557725 0.278862 0.960331i $$-0.410043\pi$$
0.278862 + 0.960331i $$0.410043\pi$$
$$824$$ −30.6274 −1.06696
$$825$$ 0 0
$$826$$ 8.00000 0.278356
$$827$$ 18.6863 0.649786 0.324893 0.945751i $$-0.394672\pi$$
0.324893 + 0.945751i $$0.394672\pi$$
$$828$$ 0 0
$$829$$ −38.0000 −1.31979 −0.659897 0.751356i $$-0.729400\pi$$
−0.659897 + 0.751356i $$0.729400\pi$$
$$830$$ −4.14214 −0.143776
$$831$$ 0 0
$$832$$ 23.5980 0.818113
$$833$$ −111.397 −3.85968
$$834$$ 0 0
$$835$$ 9.31371 0.322314
$$836$$ 0 0
$$837$$ 0 0
$$838$$ 1.25483 0.0433475
$$839$$ −22.6274 −0.781185 −0.390593 0.920564i $$-0.627730\pi$$
−0.390593 + 0.920564i $$0.627730\pi$$
$$840$$ 0 0
$$841$$ −28.3137 −0.976335
$$842$$ −2.48528 −0.0856485
$$843$$ 0 0
$$844$$ 12.4853 0.429761
$$845$$ 19.0000 0.653620
$$846$$ 0 0
$$847$$ 0 0
$$848$$ 39.9411 1.37158
$$849$$ 0 0
$$850$$ −2.82843 −0.0970143
$$851$$ 1.37258 0.0470515
$$852$$ 0 0
$$853$$ 31.3137 1.07216 0.536080 0.844167i $$-0.319904\pi$$
0.536080 + 0.844167i $$0.319904\pi$$
$$854$$ 0.686292 0.0234844
$$855$$ 0 0
$$856$$ 8.42641 0.288009
$$857$$ −11.5147 −0.393335 −0.196668 0.980470i $$-0.563012\pi$$
−0.196668 + 0.980470i $$0.563012\pi$$
$$858$$ 0 0
$$859$$ −19.0294 −0.649276 −0.324638 0.945838i $$-0.605242\pi$$
−0.324638 + 0.945838i $$0.605242\pi$$
$$860$$ −5.79899 −0.197744
$$861$$ 0 0
$$862$$ 4.28427 0.145923
$$863$$ 43.3137 1.47442 0.737208 0.675666i $$-0.236144\pi$$
0.737208 + 0.675666i $$0.236144\pi$$
$$864$$ 0 0
$$865$$ −2.82843 −0.0961694
$$866$$ −1.79899 −0.0611322
$$867$$ 0 0
$$868$$ 0 0
$$869$$ 0 0
$$870$$ 0 0
$$871$$ −32.0000 −1.08428
$$872$$ −8.42641 −0.285354
$$873$$ 0 0
$$874$$ 1.94113 0.0656595
$$875$$ 4.82843 0.163231
$$876$$ 0 0
$$877$$ 42.6274 1.43943 0.719713 0.694272i $$-0.244273\pi$$
0.719713 + 0.694272i $$0.244273\pi$$
$$878$$ 1.45584 0.0491324
$$879$$ 0 0
$$880$$ 0 0
$$881$$ 13.0294 0.438973 0.219486 0.975616i $$-0.429562\pi$$
0.219486 + 0.975616i $$0.429562\pi$$
$$882$$ 0 0
$$883$$ −50.6274 −1.70375 −0.851874 0.523747i $$-0.824534\pi$$
−0.851874 + 0.523747i $$0.824534\pi$$
$$884$$ −70.6274 −2.37546
$$885$$ 0 0
$$886$$ −4.97056 −0.166989
$$887$$ −4.34315 −0.145829 −0.0729143 0.997338i $$-0.523230\pi$$
−0.0729143 + 0.997338i $$0.523230\pi$$
$$888$$ 0 0
$$889$$ −12.0000 −0.402467
$$890$$ 3.17157 0.106311
$$891$$ 0 0
$$892$$ 32.2843 1.08096
$$893$$ 4.68629 0.156821
$$894$$ 0 0
$$895$$ 6.34315 0.212028
$$896$$ −50.9706 −1.70281
$$897$$ 0 0
$$898$$ 1.23045 0.0410606
$$899$$ 0 0
$$900$$ 0 0
$$901$$ −90.9117 −3.02871
$$902$$ 0 0
$$903$$ 0 0
$$904$$ 23.7401 0.789584
$$905$$ −14.0000 −0.465376
$$906$$ 0 0
$$907$$ 7.02944 0.233409 0.116704 0.993167i $$-0.462767\pi$$
0.116704 + 0.993167i $$0.462767\pi$$
$$908$$ 25.5980 0.849499
$$909$$ 0 0
$$910$$ −11.3137 −0.375046
$$911$$ 15.0294 0.497947 0.248974 0.968510i $$-0.419907\pi$$
0.248974 + 0.968510i $$0.419907\pi$$
$$912$$ 0 0
$$913$$ 0 0
$$914$$ 0.284271 0.00940286
$$915$$ 0 0
$$916$$ 3.65685 0.120826
$$917$$ −93.2548 −3.07955
$$918$$ 0 0
$$919$$ −28.4853 −0.939643 −0.469821 0.882762i $$-0.655682\pi$$
−0.469821 + 0.882762i $$0.655682\pi$$
$$920$$ −6.34315 −0.209127
$$921$$ 0 0
$$922$$ −11.6569 −0.383898
$$923$$ 77.2548 2.54287
$$924$$ 0 0
$$925$$ 0.343146 0.0112826
$$926$$ −12.0000 −0.394344
$$927$$ 0 0
$$928$$ 3.65685 0.120042
$$929$$ −33.5980 −1.10231 −0.551157 0.834402i $$-0.685813\pi$$
−0.551157 + 0.834402i $$0.685813\pi$$
$$930$$ 0 0
$$931$$ 19.1127 0.626393
$$932$$ 24.0833 0.788873
$$933$$ 0 0
$$934$$ −9.37258 −0.306680
$$935$$ 0 0
$$936$$ 0 0
$$937$$ −44.9706 −1.46912 −0.734562 0.678541i $$-0.762612\pi$$
−0.734562 + 0.678541i $$0.762612\pi$$
$$938$$ 11.3137 0.369406
$$939$$ 0 0
$$940$$ −7.31371 −0.238547
$$941$$ 38.7696 1.26385 0.631926 0.775029i $$-0.282265\pi$$
0.631926 + 0.775029i $$0.282265\pi$$
$$942$$ 0 0
$$943$$ −3.31371 −0.107909
$$944$$ 12.0000 0.390567
$$945$$ 0 0
$$946$$ 0 0
$$947$$ 38.6274 1.25522 0.627611 0.778527i $$-0.284033\pi$$
0.627611 + 0.778527i $$0.284033\pi$$
$$948$$ 0 0
$$949$$ −64.0000 −2.07753
$$950$$ 0.485281 0.0157446
$$951$$ 0 0
$$952$$ 52.2843 1.69454
$$953$$ 27.7990 0.900498 0.450249 0.892903i $$-0.351335\pi$$
0.450249 + 0.892903i $$0.351335\pi$$
$$954$$ 0 0
$$955$$ 5.65685 0.183052
$$956$$ −11.5980 −0.375105
$$957$$ 0 0
$$958$$ 1.25483 0.0405418
$$959$$ −44.9706 −1.45218
$$960$$ 0 0
$$961$$ −31.0000 −1.00000
$$962$$ −0.804041 −0.0259233
$$963$$ 0 0
$$964$$ −43.2548 −1.39314
$$965$$ −2.34315 −0.0754285
$$966$$ 0 0
$$967$$ 39.4558 1.26881 0.634407 0.772999i $$-0.281244\pi$$
0.634407 + 0.772999i $$0.281244\pi$$
$$968$$ 0 0
$$969$$ 0 0
$$970$$ 0.142136 0.00456370
$$971$$ −10.6274 −0.341050 −0.170525 0.985353i $$-0.554546\pi$$
−0.170525 + 0.985353i $$0.554546\pi$$
$$972$$ 0 0
$$973$$ 79.5980 2.55179
$$974$$ 8.68629 0.278327
$$975$$ 0 0
$$976$$ 1.02944 0.0329515
$$977$$ −25.3137 −0.809857 −0.404929 0.914348i $$-0.632704\pi$$
−0.404929 + 0.914348i $$0.632704\pi$$
$$978$$ 0 0
$$979$$ 0 0
$$980$$ −29.8284 −0.952834
$$981$$ 0 0
$$982$$ 10.6274 0.339135
$$983$$ −14.6274 −0.466542 −0.233271 0.972412i $$-0.574943\pi$$
−0.233271 + 0.972412i $$0.574943\pi$$
$$984$$ 0 0
$$985$$ 8.48528 0.270364
$$986$$ −2.34315 −0.0746210
$$987$$ 0 0
$$988$$ 12.1177 0.385517
$$989$$ 12.6863 0.403401
$$990$$ 0 0
$$991$$ 14.6274 0.464655 0.232328 0.972638i $$-0.425366\pi$$
0.232328 + 0.972638i $$0.425366\pi$$
$$992$$ 0 0
$$993$$ 0 0
$$994$$ −27.3137 −0.866338
$$995$$ −10.3431 −0.327900
$$996$$ 0 0
$$997$$ 16.6863 0.528460 0.264230 0.964460i $$-0.414882\pi$$
0.264230 + 0.964460i $$0.414882\pi$$
$$998$$ −13.9411 −0.441299
$$999$$ 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 5445.2.a.m.1.2 2
3.2 odd 2 1815.2.a.k.1.1 2
11.10 odd 2 495.2.a.d.1.1 2
15.14 odd 2 9075.2.a.v.1.2 2
33.32 even 2 165.2.a.a.1.2 2
44.43 even 2 7920.2.a.cg.1.2 2
55.32 even 4 2475.2.c.m.199.2 4
55.43 even 4 2475.2.c.m.199.3 4
55.54 odd 2 2475.2.a.m.1.2 2
132.131 odd 2 2640.2.a.bb.1.2 2
165.32 odd 4 825.2.c.e.199.3 4
165.98 odd 4 825.2.c.e.199.2 4
165.164 even 2 825.2.a.g.1.1 2
231.230 odd 2 8085.2.a.ba.1.2 2

By twisted newform
Twist Min Dim Char Parity Ord Type
165.2.a.a.1.2 2 33.32 even 2
495.2.a.d.1.1 2 11.10 odd 2
825.2.a.g.1.1 2 165.164 even 2
825.2.c.e.199.2 4 165.98 odd 4
825.2.c.e.199.3 4 165.32 odd 4
1815.2.a.k.1.1 2 3.2 odd 2
2475.2.a.m.1.2 2 55.54 odd 2
2475.2.c.m.199.2 4 55.32 even 4
2475.2.c.m.199.3 4 55.43 even 4
2640.2.a.bb.1.2 2 132.131 odd 2
5445.2.a.m.1.2 2 1.1 even 1 trivial
7920.2.a.cg.1.2 2 44.43 even 2
8085.2.a.ba.1.2 2 231.230 odd 2
9075.2.a.v.1.2 2 15.14 odd 2