# Properties

 Label 5445.2.a.s.1.1 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: $$\Q(\zeta_{12})^+$$ Defining polynomial: $$x^{2} - 3$$ 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 1.1 Root $$-1.73205$$ of defining polynomial Character $$\chi$$ $$=$$ 5445.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-1.73205 q^{2} +1.00000 q^{4} +1.00000 q^{5} -2.00000 q^{7} +1.73205 q^{8} +O(q^{10})$$ $$q-1.73205 q^{2} +1.00000 q^{4} +1.00000 q^{5} -2.00000 q^{7} +1.73205 q^{8} -1.73205 q^{10} -5.46410 q^{13} +3.46410 q^{14} -5.00000 q^{16} -5.46410 q^{19} +1.00000 q^{20} -6.92820 q^{23} +1.00000 q^{25} +9.46410 q^{26} -2.00000 q^{28} -3.46410 q^{29} -10.9282 q^{31} +5.19615 q^{32} -2.00000 q^{35} -4.92820 q^{37} +9.46410 q^{38} +1.73205 q^{40} +3.46410 q^{41} +4.92820 q^{43} +12.0000 q^{46} +6.92820 q^{47} -3.00000 q^{49} -1.73205 q^{50} -5.46410 q^{52} -0.928203 q^{53} -3.46410 q^{56} +6.00000 q^{58} +6.92820 q^{59} -2.00000 q^{61} +18.9282 q^{62} +1.00000 q^{64} -5.46410 q^{65} +8.00000 q^{67} +3.46410 q^{70} -13.8564 q^{71} +8.39230 q^{73} +8.53590 q^{74} -5.46410 q^{76} +6.53590 q^{79} -5.00000 q^{80} -6.00000 q^{82} +8.53590 q^{83} -8.53590 q^{86} -0.928203 q^{89} +10.9282 q^{91} -6.92820 q^{92} -12.0000 q^{94} -5.46410 q^{95} -10.0000 q^{97} +5.19615 q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 2q^{4} + 2q^{5} - 4q^{7} + O(q^{10})$$ $$2q + 2q^{4} + 2q^{5} - 4q^{7} - 4q^{13} - 10q^{16} - 4q^{19} + 2q^{20} + 2q^{25} + 12q^{26} - 4q^{28} - 8q^{31} - 4q^{35} + 4q^{37} + 12q^{38} - 4q^{43} + 24q^{46} - 6q^{49} - 4q^{52} + 12q^{53} + 12q^{58} - 4q^{61} + 24q^{62} + 2q^{64} - 4q^{65} + 16q^{67} - 4q^{73} + 24q^{74} - 4q^{76} + 20q^{79} - 10q^{80} - 12q^{82} + 24q^{83} - 24q^{86} + 12q^{89} + 8q^{91} - 24q^{94} - 4q^{95} - 20q^{97} + O(q^{100})$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ −1.73205 −1.22474 −0.612372 0.790569i $$-0.709785\pi$$
−0.612372 + 0.790569i $$0.709785\pi$$
$$3$$ 0 0
$$4$$ 1.00000 0.500000
$$5$$ 1.00000 0.447214
$$6$$ 0 0
$$7$$ −2.00000 −0.755929 −0.377964 0.925820i $$-0.623376\pi$$
−0.377964 + 0.925820i $$0.623376\pi$$
$$8$$ 1.73205 0.612372
$$9$$ 0 0
$$10$$ −1.73205 −0.547723
$$11$$ 0 0
$$12$$ 0 0
$$13$$ −5.46410 −1.51547 −0.757735 0.652563i $$-0.773694\pi$$
−0.757735 + 0.652563i $$0.773694\pi$$
$$14$$ 3.46410 0.925820
$$15$$ 0 0
$$16$$ −5.00000 −1.25000
$$17$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$18$$ 0 0
$$19$$ −5.46410 −1.25355 −0.626775 0.779200i $$-0.715626\pi$$
−0.626775 + 0.779200i $$0.715626\pi$$
$$20$$ 1.00000 0.223607
$$21$$ 0 0
$$22$$ 0 0
$$23$$ −6.92820 −1.44463 −0.722315 0.691564i $$-0.756922\pi$$
−0.722315 + 0.691564i $$0.756922\pi$$
$$24$$ 0 0
$$25$$ 1.00000 0.200000
$$26$$ 9.46410 1.85606
$$27$$ 0 0
$$28$$ −2.00000 −0.377964
$$29$$ −3.46410 −0.643268 −0.321634 0.946864i $$-0.604232\pi$$
−0.321634 + 0.946864i $$0.604232\pi$$
$$30$$ 0 0
$$31$$ −10.9282 −1.96276 −0.981382 0.192068i $$-0.938481\pi$$
−0.981382 + 0.192068i $$0.938481\pi$$
$$32$$ 5.19615 0.918559
$$33$$ 0 0
$$34$$ 0 0
$$35$$ −2.00000 −0.338062
$$36$$ 0 0
$$37$$ −4.92820 −0.810192 −0.405096 0.914274i $$-0.632762\pi$$
−0.405096 + 0.914274i $$0.632762\pi$$
$$38$$ 9.46410 1.53528
$$39$$ 0 0
$$40$$ 1.73205 0.273861
$$41$$ 3.46410 0.541002 0.270501 0.962720i $$-0.412811\pi$$
0.270501 + 0.962720i $$0.412811\pi$$
$$42$$ 0 0
$$43$$ 4.92820 0.751544 0.375772 0.926712i $$-0.377378\pi$$
0.375772 + 0.926712i $$0.377378\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 12.0000 1.76930
$$47$$ 6.92820 1.01058 0.505291 0.862949i $$-0.331385\pi$$
0.505291 + 0.862949i $$0.331385\pi$$
$$48$$ 0 0
$$49$$ −3.00000 −0.428571
$$50$$ −1.73205 −0.244949
$$51$$ 0 0
$$52$$ −5.46410 −0.757735
$$53$$ −0.928203 −0.127499 −0.0637493 0.997966i $$-0.520306\pi$$
−0.0637493 + 0.997966i $$0.520306\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ −3.46410 −0.462910
$$57$$ 0 0
$$58$$ 6.00000 0.787839
$$59$$ 6.92820 0.901975 0.450988 0.892530i $$-0.351072\pi$$
0.450988 + 0.892530i $$0.351072\pi$$
$$60$$ 0 0
$$61$$ −2.00000 −0.256074 −0.128037 0.991769i $$-0.540868\pi$$
−0.128037 + 0.991769i $$0.540868\pi$$
$$62$$ 18.9282 2.40388
$$63$$ 0 0
$$64$$ 1.00000 0.125000
$$65$$ −5.46410 −0.677738
$$66$$ 0 0
$$67$$ 8.00000 0.977356 0.488678 0.872464i $$-0.337479\pi$$
0.488678 + 0.872464i $$0.337479\pi$$
$$68$$ 0 0
$$69$$ 0 0
$$70$$ 3.46410 0.414039
$$71$$ −13.8564 −1.64445 −0.822226 0.569160i $$-0.807268\pi$$
−0.822226 + 0.569160i $$0.807268\pi$$
$$72$$ 0 0
$$73$$ 8.39230 0.982245 0.491122 0.871091i $$-0.336587\pi$$
0.491122 + 0.871091i $$0.336587\pi$$
$$74$$ 8.53590 0.992278
$$75$$ 0 0
$$76$$ −5.46410 −0.626775
$$77$$ 0 0
$$78$$ 0 0
$$79$$ 6.53590 0.735346 0.367673 0.929955i $$-0.380155\pi$$
0.367673 + 0.929955i $$0.380155\pi$$
$$80$$ −5.00000 −0.559017
$$81$$ 0 0
$$82$$ −6.00000 −0.662589
$$83$$ 8.53590 0.936937 0.468468 0.883480i $$-0.344806\pi$$
0.468468 + 0.883480i $$0.344806\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ −8.53590 −0.920450
$$87$$ 0 0
$$88$$ 0 0
$$89$$ −0.928203 −0.0983893 −0.0491947 0.998789i $$-0.515665\pi$$
−0.0491947 + 0.998789i $$0.515665\pi$$
$$90$$ 0 0
$$91$$ 10.9282 1.14559
$$92$$ −6.92820 −0.722315
$$93$$ 0 0
$$94$$ −12.0000 −1.23771
$$95$$ −5.46410 −0.560605
$$96$$ 0 0
$$97$$ −10.0000 −1.01535 −0.507673 0.861550i $$-0.669494\pi$$
−0.507673 + 0.861550i $$0.669494\pi$$
$$98$$ 5.19615 0.524891
$$99$$ 0 0
$$100$$ 1.00000 0.100000
$$101$$ −10.3923 −1.03407 −0.517036 0.855963i $$-0.672965\pi$$
−0.517036 + 0.855963i $$0.672965\pi$$
$$102$$ 0 0
$$103$$ 8.00000 0.788263 0.394132 0.919054i $$-0.371045\pi$$
0.394132 + 0.919054i $$0.371045\pi$$
$$104$$ −9.46410 −0.928032
$$105$$ 0 0
$$106$$ 1.60770 0.156153
$$107$$ 8.53590 0.825196 0.412598 0.910913i $$-0.364621\pi$$
0.412598 + 0.910913i $$0.364621\pi$$
$$108$$ 0 0
$$109$$ 10.0000 0.957826 0.478913 0.877862i $$-0.341031\pi$$
0.478913 + 0.877862i $$0.341031\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ 10.0000 0.944911
$$113$$ 12.9282 1.21618 0.608092 0.793867i $$-0.291935\pi$$
0.608092 + 0.793867i $$0.291935\pi$$
$$114$$ 0 0
$$115$$ −6.92820 −0.646058
$$116$$ −3.46410 −0.321634
$$117$$ 0 0
$$118$$ −12.0000 −1.10469
$$119$$ 0 0
$$120$$ 0 0
$$121$$ 0 0
$$122$$ 3.46410 0.313625
$$123$$ 0 0
$$124$$ −10.9282 −0.981382
$$125$$ 1.00000 0.0894427
$$126$$ 0 0
$$127$$ −8.92820 −0.792250 −0.396125 0.918197i $$-0.629645\pi$$
−0.396125 + 0.918197i $$0.629645\pi$$
$$128$$ −12.1244 −1.07165
$$129$$ 0 0
$$130$$ 9.46410 0.830057
$$131$$ 18.9282 1.65376 0.826882 0.562375i $$-0.190112\pi$$
0.826882 + 0.562375i $$0.190112\pi$$
$$132$$ 0 0
$$133$$ 10.9282 0.947595
$$134$$ −13.8564 −1.19701
$$135$$ 0 0
$$136$$ 0 0
$$137$$ 18.0000 1.53784 0.768922 0.639343i $$-0.220793\pi$$
0.768922 + 0.639343i $$0.220793\pi$$
$$138$$ 0 0
$$139$$ −12.3923 −1.05110 −0.525551 0.850762i $$-0.676141\pi$$
−0.525551 + 0.850762i $$0.676141\pi$$
$$140$$ −2.00000 −0.169031
$$141$$ 0 0
$$142$$ 24.0000 2.01404
$$143$$ 0 0
$$144$$ 0 0
$$145$$ −3.46410 −0.287678
$$146$$ −14.5359 −1.20300
$$147$$ 0 0
$$148$$ −4.92820 −0.405096
$$149$$ −15.4641 −1.26687 −0.633434 0.773796i $$-0.718355\pi$$
−0.633434 + 0.773796i $$0.718355\pi$$
$$150$$ 0 0
$$151$$ 20.3923 1.65950 0.829751 0.558134i $$-0.188482\pi$$
0.829751 + 0.558134i $$0.188482\pi$$
$$152$$ −9.46410 −0.767640
$$153$$ 0 0
$$154$$ 0 0
$$155$$ −10.9282 −0.877774
$$156$$ 0 0
$$157$$ −3.07180 −0.245156 −0.122578 0.992459i $$-0.539116\pi$$
−0.122578 + 0.992459i $$0.539116\pi$$
$$158$$ −11.3205 −0.900611
$$159$$ 0 0
$$160$$ 5.19615 0.410792
$$161$$ 13.8564 1.09204
$$162$$ 0 0
$$163$$ 9.85641 0.772013 0.386007 0.922496i $$-0.373854\pi$$
0.386007 + 0.922496i $$0.373854\pi$$
$$164$$ 3.46410 0.270501
$$165$$ 0 0
$$166$$ −14.7846 −1.14751
$$167$$ −10.3923 −0.804181 −0.402090 0.915600i $$-0.631716\pi$$
−0.402090 + 0.915600i $$0.631716\pi$$
$$168$$ 0 0
$$169$$ 16.8564 1.29665
$$170$$ 0 0
$$171$$ 0 0
$$172$$ 4.92820 0.375772
$$173$$ −12.0000 −0.912343 −0.456172 0.889892i $$-0.650780\pi$$
−0.456172 + 0.889892i $$0.650780\pi$$
$$174$$ 0 0
$$175$$ −2.00000 −0.151186
$$176$$ 0 0
$$177$$ 0 0
$$178$$ 1.60770 0.120502
$$179$$ −6.92820 −0.517838 −0.258919 0.965899i $$-0.583366\pi$$
−0.258919 + 0.965899i $$0.583366\pi$$
$$180$$ 0 0
$$181$$ 15.8564 1.17860 0.589299 0.807915i $$-0.299404\pi$$
0.589299 + 0.807915i $$0.299404\pi$$
$$182$$ −18.9282 −1.40305
$$183$$ 0 0
$$184$$ −12.0000 −0.884652
$$185$$ −4.92820 −0.362329
$$186$$ 0 0
$$187$$ 0 0
$$188$$ 6.92820 0.505291
$$189$$ 0 0
$$190$$ 9.46410 0.686598
$$191$$ 18.9282 1.36960 0.684798 0.728733i $$-0.259890\pi$$
0.684798 + 0.728733i $$0.259890\pi$$
$$192$$ 0 0
$$193$$ −24.3923 −1.75580 −0.877898 0.478847i $$-0.841055\pi$$
−0.877898 + 0.478847i $$0.841055\pi$$
$$194$$ 17.3205 1.24354
$$195$$ 0 0
$$196$$ −3.00000 −0.214286
$$197$$ −12.0000 −0.854965 −0.427482 0.904024i $$-0.640599\pi$$
−0.427482 + 0.904024i $$0.640599\pi$$
$$198$$ 0 0
$$199$$ −24.7846 −1.75693 −0.878467 0.477803i $$-0.841433\pi$$
−0.878467 + 0.477803i $$0.841433\pi$$
$$200$$ 1.73205 0.122474
$$201$$ 0 0
$$202$$ 18.0000 1.26648
$$203$$ 6.92820 0.486265
$$204$$ 0 0
$$205$$ 3.46410 0.241943
$$206$$ −13.8564 −0.965422
$$207$$ 0 0
$$208$$ 27.3205 1.89434
$$209$$ 0 0
$$210$$ 0 0
$$211$$ 8.39230 0.577750 0.288875 0.957367i $$-0.406719\pi$$
0.288875 + 0.957367i $$0.406719\pi$$
$$212$$ −0.928203 −0.0637493
$$213$$ 0 0
$$214$$ −14.7846 −1.01066
$$215$$ 4.92820 0.336101
$$216$$ 0 0
$$217$$ 21.8564 1.48371
$$218$$ −17.3205 −1.17309
$$219$$ 0 0
$$220$$ 0 0
$$221$$ 0 0
$$222$$ 0 0
$$223$$ 9.85641 0.660034 0.330017 0.943975i $$-0.392946\pi$$
0.330017 + 0.943975i $$0.392946\pi$$
$$224$$ −10.3923 −0.694365
$$225$$ 0 0
$$226$$ −22.3923 −1.48951
$$227$$ 15.4641 1.02639 0.513194 0.858272i $$-0.328462\pi$$
0.513194 + 0.858272i $$0.328462\pi$$
$$228$$ 0 0
$$229$$ −23.8564 −1.57648 −0.788238 0.615371i $$-0.789006\pi$$
−0.788238 + 0.615371i $$0.789006\pi$$
$$230$$ 12.0000 0.791257
$$231$$ 0 0
$$232$$ −6.00000 −0.393919
$$233$$ 12.0000 0.786146 0.393073 0.919507i $$-0.371412\pi$$
0.393073 + 0.919507i $$0.371412\pi$$
$$234$$ 0 0
$$235$$ 6.92820 0.451946
$$236$$ 6.92820 0.450988
$$237$$ 0 0
$$238$$ 0 0
$$239$$ −12.0000 −0.776215 −0.388108 0.921614i $$-0.626871\pi$$
−0.388108 + 0.921614i $$0.626871\pi$$
$$240$$ 0 0
$$241$$ −0.143594 −0.00924967 −0.00462484 0.999989i $$-0.501472\pi$$
−0.00462484 + 0.999989i $$0.501472\pi$$
$$242$$ 0 0
$$243$$ 0 0
$$244$$ −2.00000 −0.128037
$$245$$ −3.00000 −0.191663
$$246$$ 0 0
$$247$$ 29.8564 1.89972
$$248$$ −18.9282 −1.20194
$$249$$ 0 0
$$250$$ −1.73205 −0.109545
$$251$$ 1.85641 0.117175 0.0585877 0.998282i $$-0.481340\pi$$
0.0585877 + 0.998282i $$0.481340\pi$$
$$252$$ 0 0
$$253$$ 0 0
$$254$$ 15.4641 0.970304
$$255$$ 0 0
$$256$$ 19.0000 1.18750
$$257$$ 19.8564 1.23861 0.619304 0.785151i $$-0.287415\pi$$
0.619304 + 0.785151i $$0.287415\pi$$
$$258$$ 0 0
$$259$$ 9.85641 0.612447
$$260$$ −5.46410 −0.338869
$$261$$ 0 0
$$262$$ −32.7846 −2.02544
$$263$$ 20.5359 1.26630 0.633149 0.774030i $$-0.281762\pi$$
0.633149 + 0.774030i $$0.281762\pi$$
$$264$$ 0 0
$$265$$ −0.928203 −0.0570191
$$266$$ −18.9282 −1.16056
$$267$$ 0 0
$$268$$ 8.00000 0.488678
$$269$$ −19.8564 −1.21067 −0.605333 0.795972i $$-0.706960\pi$$
−0.605333 + 0.795972i $$0.706960\pi$$
$$270$$ 0 0
$$271$$ 11.6077 0.705117 0.352559 0.935790i $$-0.385312\pi$$
0.352559 + 0.935790i $$0.385312\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ −31.1769 −1.88347
$$275$$ 0 0
$$276$$ 0 0
$$277$$ −29.4641 −1.77033 −0.885163 0.465281i $$-0.845953\pi$$
−0.885163 + 0.465281i $$0.845953\pi$$
$$278$$ 21.4641 1.28733
$$279$$ 0 0
$$280$$ −3.46410 −0.207020
$$281$$ 3.46410 0.206651 0.103325 0.994648i $$-0.467052\pi$$
0.103325 + 0.994648i $$0.467052\pi$$
$$282$$ 0 0
$$283$$ 4.92820 0.292951 0.146476 0.989214i $$-0.453207\pi$$
0.146476 + 0.989214i $$0.453207\pi$$
$$284$$ −13.8564 −0.822226
$$285$$ 0 0
$$286$$ 0 0
$$287$$ −6.92820 −0.408959
$$288$$ 0 0
$$289$$ −17.0000 −1.00000
$$290$$ 6.00000 0.352332
$$291$$ 0 0
$$292$$ 8.39230 0.491122
$$293$$ −13.8564 −0.809500 −0.404750 0.914427i $$-0.632641\pi$$
−0.404750 + 0.914427i $$0.632641\pi$$
$$294$$ 0 0
$$295$$ 6.92820 0.403376
$$296$$ −8.53590 −0.496139
$$297$$ 0 0
$$298$$ 26.7846 1.55159
$$299$$ 37.8564 2.18929
$$300$$ 0 0
$$301$$ −9.85641 −0.568114
$$302$$ −35.3205 −2.03247
$$303$$ 0 0
$$304$$ 27.3205 1.56694
$$305$$ −2.00000 −0.114520
$$306$$ 0 0
$$307$$ −14.0000 −0.799022 −0.399511 0.916728i $$-0.630820\pi$$
−0.399511 + 0.916728i $$0.630820\pi$$
$$308$$ 0 0
$$309$$ 0 0
$$310$$ 18.9282 1.07505
$$311$$ −5.07180 −0.287595 −0.143798 0.989607i $$-0.545931\pi$$
−0.143798 + 0.989607i $$0.545931\pi$$
$$312$$ 0 0
$$313$$ 20.9282 1.18293 0.591466 0.806330i $$-0.298549\pi$$
0.591466 + 0.806330i $$0.298549\pi$$
$$314$$ 5.32051 0.300254
$$315$$ 0 0
$$316$$ 6.53590 0.367673
$$317$$ 24.9282 1.40011 0.700054 0.714090i $$-0.253159\pi$$
0.700054 + 0.714090i $$0.253159\pi$$
$$318$$ 0 0
$$319$$ 0 0
$$320$$ 1.00000 0.0559017
$$321$$ 0 0
$$322$$ −24.0000 −1.33747
$$323$$ 0 0
$$324$$ 0 0
$$325$$ −5.46410 −0.303094
$$326$$ −17.0718 −0.945519
$$327$$ 0 0
$$328$$ 6.00000 0.331295
$$329$$ −13.8564 −0.763928
$$330$$ 0 0
$$331$$ 9.85641 0.541757 0.270879 0.962614i $$-0.412686\pi$$
0.270879 + 0.962614i $$0.412686\pi$$
$$332$$ 8.53590 0.468468
$$333$$ 0 0
$$334$$ 18.0000 0.984916
$$335$$ 8.00000 0.437087
$$336$$ 0 0
$$337$$ −33.1769 −1.80726 −0.903631 0.428312i $$-0.859108\pi$$
−0.903631 + 0.428312i $$0.859108\pi$$
$$338$$ −29.1962 −1.58806
$$339$$ 0 0
$$340$$ 0 0
$$341$$ 0 0
$$342$$ 0 0
$$343$$ 20.0000 1.07990
$$344$$ 8.53590 0.460225
$$345$$ 0 0
$$346$$ 20.7846 1.11739
$$347$$ 22.3923 1.20208 0.601041 0.799218i $$-0.294753\pi$$
0.601041 + 0.799218i $$0.294753\pi$$
$$348$$ 0 0
$$349$$ 8.14359 0.435917 0.217958 0.975958i $$-0.430060\pi$$
0.217958 + 0.975958i $$0.430060\pi$$
$$350$$ 3.46410 0.185164
$$351$$ 0 0
$$352$$ 0 0
$$353$$ −12.9282 −0.688099 −0.344049 0.938952i $$-0.611799\pi$$
−0.344049 + 0.938952i $$0.611799\pi$$
$$354$$ 0 0
$$355$$ −13.8564 −0.735422
$$356$$ −0.928203 −0.0491947
$$357$$ 0 0
$$358$$ 12.0000 0.634220
$$359$$ −20.7846 −1.09697 −0.548485 0.836160i $$-0.684795\pi$$
−0.548485 + 0.836160i $$0.684795\pi$$
$$360$$ 0 0
$$361$$ 10.8564 0.571390
$$362$$ −27.4641 −1.44348
$$363$$ 0 0
$$364$$ 10.9282 0.572793
$$365$$ 8.39230 0.439273
$$366$$ 0 0
$$367$$ 20.0000 1.04399 0.521996 0.852948i $$-0.325188\pi$$
0.521996 + 0.852948i $$0.325188\pi$$
$$368$$ 34.6410 1.80579
$$369$$ 0 0
$$370$$ 8.53590 0.443760
$$371$$ 1.85641 0.0963798
$$372$$ 0 0
$$373$$ 20.3923 1.05587 0.527937 0.849284i $$-0.322966\pi$$
0.527937 + 0.849284i $$0.322966\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 12.0000 0.618853
$$377$$ 18.9282 0.974852
$$378$$ 0 0
$$379$$ −17.8564 −0.917222 −0.458611 0.888637i $$-0.651653\pi$$
−0.458611 + 0.888637i $$0.651653\pi$$
$$380$$ −5.46410 −0.280302
$$381$$ 0 0
$$382$$ −32.7846 −1.67741
$$383$$ 13.8564 0.708029 0.354015 0.935240i $$-0.384816\pi$$
0.354015 + 0.935240i $$0.384816\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 42.2487 2.15040
$$387$$ 0 0
$$388$$ −10.0000 −0.507673
$$389$$ −11.0718 −0.561362 −0.280681 0.959801i $$-0.590560\pi$$
−0.280681 + 0.959801i $$0.590560\pi$$
$$390$$ 0 0
$$391$$ 0 0
$$392$$ −5.19615 −0.262445
$$393$$ 0 0
$$394$$ 20.7846 1.04711
$$395$$ 6.53590 0.328857
$$396$$ 0 0
$$397$$ 2.00000 0.100377 0.0501886 0.998740i $$-0.484018\pi$$
0.0501886 + 0.998740i $$0.484018\pi$$
$$398$$ 42.9282 2.15180
$$399$$ 0 0
$$400$$ −5.00000 −0.250000
$$401$$ 7.85641 0.392330 0.196165 0.980571i $$-0.437151\pi$$
0.196165 + 0.980571i $$0.437151\pi$$
$$402$$ 0 0
$$403$$ 59.7128 2.97451
$$404$$ −10.3923 −0.517036
$$405$$ 0 0
$$406$$ −12.0000 −0.595550
$$407$$ 0 0
$$408$$ 0 0
$$409$$ 6.78461 0.335477 0.167739 0.985831i $$-0.446354\pi$$
0.167739 + 0.985831i $$0.446354\pi$$
$$410$$ −6.00000 −0.296319
$$411$$ 0 0
$$412$$ 8.00000 0.394132
$$413$$ −13.8564 −0.681829
$$414$$ 0 0
$$415$$ 8.53590 0.419011
$$416$$ −28.3923 −1.39205
$$417$$ 0 0
$$418$$ 0 0
$$419$$ −30.9282 −1.51094 −0.755471 0.655182i $$-0.772592\pi$$
−0.755471 + 0.655182i $$0.772592\pi$$
$$420$$ 0 0
$$421$$ 2.00000 0.0974740 0.0487370 0.998812i $$-0.484480\pi$$
0.0487370 + 0.998812i $$0.484480\pi$$
$$422$$ −14.5359 −0.707596
$$423$$ 0 0
$$424$$ −1.60770 −0.0780766
$$425$$ 0 0
$$426$$ 0 0
$$427$$ 4.00000 0.193574
$$428$$ 8.53590 0.412598
$$429$$ 0 0
$$430$$ −8.53590 −0.411638
$$431$$ 8.78461 0.423140 0.211570 0.977363i $$-0.432142\pi$$
0.211570 + 0.977363i $$0.432142\pi$$
$$432$$ 0 0
$$433$$ 0.143594 0.00690067 0.00345033 0.999994i $$-0.498902\pi$$
0.00345033 + 0.999994i $$0.498902\pi$$
$$434$$ −37.8564 −1.81717
$$435$$ 0 0
$$436$$ 10.0000 0.478913
$$437$$ 37.8564 1.81092
$$438$$ 0 0
$$439$$ −33.1769 −1.58345 −0.791724 0.610879i $$-0.790816\pi$$
−0.791724 + 0.610879i $$0.790816\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 0 0
$$443$$ −12.0000 −0.570137 −0.285069 0.958507i $$-0.592016\pi$$
−0.285069 + 0.958507i $$0.592016\pi$$
$$444$$ 0 0
$$445$$ −0.928203 −0.0440011
$$446$$ −17.0718 −0.808373
$$447$$ 0 0
$$448$$ −2.00000 −0.0944911
$$449$$ 26.7846 1.26404 0.632022 0.774950i $$-0.282225\pi$$
0.632022 + 0.774950i $$0.282225\pi$$
$$450$$ 0 0
$$451$$ 0 0
$$452$$ 12.9282 0.608092
$$453$$ 0 0
$$454$$ −26.7846 −1.25706
$$455$$ 10.9282 0.512322
$$456$$ 0 0
$$457$$ −12.3923 −0.579688 −0.289844 0.957074i $$-0.593603\pi$$
−0.289844 + 0.957074i $$0.593603\pi$$
$$458$$ 41.3205 1.93078
$$459$$ 0 0
$$460$$ −6.92820 −0.323029
$$461$$ 36.2487 1.68827 0.844135 0.536130i $$-0.180114\pi$$
0.844135 + 0.536130i $$0.180114\pi$$
$$462$$ 0 0
$$463$$ −28.0000 −1.30127 −0.650635 0.759390i $$-0.725497\pi$$
−0.650635 + 0.759390i $$0.725497\pi$$
$$464$$ 17.3205 0.804084
$$465$$ 0 0
$$466$$ −20.7846 −0.962828
$$467$$ −5.07180 −0.234695 −0.117347 0.993091i $$-0.537439\pi$$
−0.117347 + 0.993091i $$0.537439\pi$$
$$468$$ 0 0
$$469$$ −16.0000 −0.738811
$$470$$ −12.0000 −0.553519
$$471$$ 0 0
$$472$$ 12.0000 0.552345
$$473$$ 0 0
$$474$$ 0 0
$$475$$ −5.46410 −0.250710
$$476$$ 0 0
$$477$$ 0 0
$$478$$ 20.7846 0.950666
$$479$$ 12.0000 0.548294 0.274147 0.961688i $$-0.411605\pi$$
0.274147 + 0.961688i $$0.411605\pi$$
$$480$$ 0 0
$$481$$ 26.9282 1.22782
$$482$$ 0.248711 0.0113285
$$483$$ 0 0
$$484$$ 0 0
$$485$$ −10.0000 −0.454077
$$486$$ 0 0
$$487$$ −31.7128 −1.43704 −0.718522 0.695504i $$-0.755181\pi$$
−0.718522 + 0.695504i $$0.755181\pi$$
$$488$$ −3.46410 −0.156813
$$489$$ 0 0
$$490$$ 5.19615 0.234738
$$491$$ −30.9282 −1.39577 −0.697885 0.716210i $$-0.745875\pi$$
−0.697885 + 0.716210i $$0.745875\pi$$
$$492$$ 0 0
$$493$$ 0 0
$$494$$ −51.7128 −2.32667
$$495$$ 0 0
$$496$$ 54.6410 2.45345
$$497$$ 27.7128 1.24309
$$498$$ 0 0
$$499$$ 28.7846 1.28858 0.644288 0.764783i $$-0.277154\pi$$
0.644288 + 0.764783i $$0.277154\pi$$
$$500$$ 1.00000 0.0447214
$$501$$ 0 0
$$502$$ −3.21539 −0.143510
$$503$$ 31.1769 1.39011 0.695055 0.718957i $$-0.255380\pi$$
0.695055 + 0.718957i $$0.255380\pi$$
$$504$$ 0 0
$$505$$ −10.3923 −0.462451
$$506$$ 0 0
$$507$$ 0 0
$$508$$ −8.92820 −0.396125
$$509$$ −19.8564 −0.880120 −0.440060 0.897968i $$-0.645043\pi$$
−0.440060 + 0.897968i $$0.645043\pi$$
$$510$$ 0 0
$$511$$ −16.7846 −0.742507
$$512$$ −8.66025 −0.382733
$$513$$ 0 0
$$514$$ −34.3923 −1.51698
$$515$$ 8.00000 0.352522
$$516$$ 0 0
$$517$$ 0 0
$$518$$ −17.0718 −0.750092
$$519$$ 0 0
$$520$$ −9.46410 −0.415028
$$521$$ −6.00000 −0.262865 −0.131432 0.991325i $$-0.541958\pi$$
−0.131432 + 0.991325i $$0.541958\pi$$
$$522$$ 0 0
$$523$$ 22.0000 0.961993 0.480996 0.876723i $$-0.340275\pi$$
0.480996 + 0.876723i $$0.340275\pi$$
$$524$$ 18.9282 0.826882
$$525$$ 0 0
$$526$$ −35.5692 −1.55089
$$527$$ 0 0
$$528$$ 0 0
$$529$$ 25.0000 1.08696
$$530$$ 1.60770 0.0698338
$$531$$ 0 0
$$532$$ 10.9282 0.473798
$$533$$ −18.9282 −0.819871
$$534$$ 0 0
$$535$$ 8.53590 0.369039
$$536$$ 13.8564 0.598506
$$537$$ 0 0
$$538$$ 34.3923 1.48276
$$539$$ 0 0
$$540$$ 0 0
$$541$$ −27.8564 −1.19764 −0.598820 0.800883i $$-0.704364\pi$$
−0.598820 + 0.800883i $$0.704364\pi$$
$$542$$ −20.1051 −0.863589
$$543$$ 0 0
$$544$$ 0 0
$$545$$ 10.0000 0.428353
$$546$$ 0 0
$$547$$ −2.00000 −0.0855138 −0.0427569 0.999086i $$-0.513614\pi$$
−0.0427569 + 0.999086i $$0.513614\pi$$
$$548$$ 18.0000 0.768922
$$549$$ 0 0
$$550$$ 0 0
$$551$$ 18.9282 0.806369
$$552$$ 0 0
$$553$$ −13.0718 −0.555869
$$554$$ 51.0333 2.16820
$$555$$ 0 0
$$556$$ −12.3923 −0.525551
$$557$$ 3.21539 0.136240 0.0681202 0.997677i $$-0.478300\pi$$
0.0681202 + 0.997677i $$0.478300\pi$$
$$558$$ 0 0
$$559$$ −26.9282 −1.13894
$$560$$ 10.0000 0.422577
$$561$$ 0 0
$$562$$ −6.00000 −0.253095
$$563$$ 10.3923 0.437983 0.218992 0.975727i $$-0.429723\pi$$
0.218992 + 0.975727i $$0.429723\pi$$
$$564$$ 0 0
$$565$$ 12.9282 0.543894
$$566$$ −8.53590 −0.358791
$$567$$ 0 0
$$568$$ −24.0000 −1.00702
$$569$$ 5.32051 0.223047 0.111524 0.993762i $$-0.464427\pi$$
0.111524 + 0.993762i $$0.464427\pi$$
$$570$$ 0 0
$$571$$ −3.60770 −0.150977 −0.0754887 0.997147i $$-0.524052\pi$$
−0.0754887 + 0.997147i $$0.524052\pi$$
$$572$$ 0 0
$$573$$ 0 0
$$574$$ 12.0000 0.500870
$$575$$ −6.92820 −0.288926
$$576$$ 0 0
$$577$$ −18.7846 −0.782014 −0.391007 0.920388i $$-0.627873\pi$$
−0.391007 + 0.920388i $$0.627873\pi$$
$$578$$ 29.4449 1.22474
$$579$$ 0 0
$$580$$ −3.46410 −0.143839
$$581$$ −17.0718 −0.708257
$$582$$ 0 0
$$583$$ 0 0
$$584$$ 14.5359 0.601500
$$585$$ 0 0
$$586$$ 24.0000 0.991431
$$587$$ 18.9282 0.781251 0.390625 0.920550i $$-0.372259\pi$$
0.390625 + 0.920550i $$0.372259\pi$$
$$588$$ 0 0
$$589$$ 59.7128 2.46042
$$590$$ −12.0000 −0.494032
$$591$$ 0 0
$$592$$ 24.6410 1.01274
$$593$$ 8.78461 0.360741 0.180370 0.983599i $$-0.442270\pi$$
0.180370 + 0.983599i $$0.442270\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ −15.4641 −0.633434
$$597$$ 0 0
$$598$$ −65.5692 −2.68132
$$599$$ 37.8564 1.54677 0.773385 0.633936i $$-0.218562\pi$$
0.773385 + 0.633936i $$0.218562\pi$$
$$600$$ 0 0
$$601$$ 32.6410 1.33145 0.665727 0.746195i $$-0.268121\pi$$
0.665727 + 0.746195i $$0.268121\pi$$
$$602$$ 17.0718 0.695794
$$603$$ 0 0
$$604$$ 20.3923 0.829751
$$605$$ 0 0
$$606$$ 0 0
$$607$$ 18.7846 0.762444 0.381222 0.924484i $$-0.375503\pi$$
0.381222 + 0.924484i $$0.375503\pi$$
$$608$$ −28.3923 −1.15146
$$609$$ 0 0
$$610$$ 3.46410 0.140257
$$611$$ −37.8564 −1.53151
$$612$$ 0 0
$$613$$ 20.3923 0.823637 0.411819 0.911266i $$-0.364894\pi$$
0.411819 + 0.911266i $$0.364894\pi$$
$$614$$ 24.2487 0.978598
$$615$$ 0 0
$$616$$ 0 0
$$617$$ 36.9282 1.48667 0.743337 0.668917i $$-0.233242\pi$$
0.743337 + 0.668917i $$0.233242\pi$$
$$618$$ 0 0
$$619$$ 20.0000 0.803868 0.401934 0.915669i $$-0.368338\pi$$
0.401934 + 0.915669i $$0.368338\pi$$
$$620$$ −10.9282 −0.438887
$$621$$ 0 0
$$622$$ 8.78461 0.352231
$$623$$ 1.85641 0.0743754
$$624$$ 0 0
$$625$$ 1.00000 0.0400000
$$626$$ −36.2487 −1.44879
$$627$$ 0 0
$$628$$ −3.07180 −0.122578
$$629$$ 0 0
$$630$$ 0 0
$$631$$ −21.0718 −0.838855 −0.419427 0.907789i $$-0.637769\pi$$
−0.419427 + 0.907789i $$0.637769\pi$$
$$632$$ 11.3205 0.450306
$$633$$ 0 0
$$634$$ −43.1769 −1.71477
$$635$$ −8.92820 −0.354305
$$636$$ 0 0
$$637$$ 16.3923 0.649487
$$638$$ 0 0
$$639$$ 0 0
$$640$$ −12.1244 −0.479257
$$641$$ 12.9282 0.510633 0.255317 0.966857i $$-0.417820\pi$$
0.255317 + 0.966857i $$0.417820\pi$$
$$642$$ 0 0
$$643$$ 37.5692 1.48159 0.740793 0.671734i $$-0.234450\pi$$
0.740793 + 0.671734i $$0.234450\pi$$
$$644$$ 13.8564 0.546019
$$645$$ 0 0
$$646$$ 0 0
$$647$$ −27.7128 −1.08950 −0.544752 0.838597i $$-0.683376\pi$$
−0.544752 + 0.838597i $$0.683376\pi$$
$$648$$ 0 0
$$649$$ 0 0
$$650$$ 9.46410 0.371213
$$651$$ 0 0
$$652$$ 9.85641 0.386007
$$653$$ −19.8564 −0.777041 −0.388521 0.921440i $$-0.627014\pi$$
−0.388521 + 0.921440i $$0.627014\pi$$
$$654$$ 0 0
$$655$$ 18.9282 0.739586
$$656$$ −17.3205 −0.676252
$$657$$ 0 0
$$658$$ 24.0000 0.935617
$$659$$ −15.7128 −0.612084 −0.306042 0.952018i $$-0.599005\pi$$
−0.306042 + 0.952018i $$0.599005\pi$$
$$660$$ 0 0
$$661$$ 14.0000 0.544537 0.272268 0.962221i $$-0.412226\pi$$
0.272268 + 0.962221i $$0.412226\pi$$
$$662$$ −17.0718 −0.663514
$$663$$ 0 0
$$664$$ 14.7846 0.573754
$$665$$ 10.9282 0.423778
$$666$$ 0 0
$$667$$ 24.0000 0.929284
$$668$$ −10.3923 −0.402090
$$669$$ 0 0
$$670$$ −13.8564 −0.535320
$$671$$ 0 0
$$672$$ 0 0
$$673$$ 3.32051 0.127996 0.0639981 0.997950i $$-0.479615\pi$$
0.0639981 + 0.997950i $$0.479615\pi$$
$$674$$ 57.4641 2.21343
$$675$$ 0 0
$$676$$ 16.8564 0.648323
$$677$$ 8.78461 0.337620 0.168810 0.985649i $$-0.446008\pi$$
0.168810 + 0.985649i $$0.446008\pi$$
$$678$$ 0 0
$$679$$ 20.0000 0.767530
$$680$$ 0 0
$$681$$ 0 0
$$682$$ 0 0
$$683$$ 32.7846 1.25447 0.627234 0.778831i $$-0.284187\pi$$
0.627234 + 0.778831i $$0.284187\pi$$
$$684$$ 0 0
$$685$$ 18.0000 0.687745
$$686$$ −34.6410 −1.32260
$$687$$ 0 0
$$688$$ −24.6410 −0.939430
$$689$$ 5.07180 0.193220
$$690$$ 0 0
$$691$$ 47.7128 1.81508 0.907540 0.419965i $$-0.137958\pi$$
0.907540 + 0.419965i $$0.137958\pi$$
$$692$$ −12.0000 −0.456172
$$693$$ 0 0
$$694$$ −38.7846 −1.47224
$$695$$ −12.3923 −0.470067
$$696$$ 0 0
$$697$$ 0 0
$$698$$ −14.1051 −0.533887
$$699$$ 0 0
$$700$$ −2.00000 −0.0755929
$$701$$ 39.4641 1.49054 0.745269 0.666764i $$-0.232321\pi$$
0.745269 + 0.666764i $$0.232321\pi$$
$$702$$ 0 0
$$703$$ 26.9282 1.01562
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 22.3923 0.842746
$$707$$ 20.7846 0.781686
$$708$$ 0 0
$$709$$ −11.8564 −0.445277 −0.222638 0.974901i $$-0.571467\pi$$
−0.222638 + 0.974901i $$0.571467\pi$$
$$710$$ 24.0000 0.900704
$$711$$ 0 0
$$712$$ −1.60770 −0.0602509
$$713$$ 75.7128 2.83547
$$714$$ 0 0
$$715$$ 0 0
$$716$$ −6.92820 −0.258919
$$717$$ 0 0
$$718$$ 36.0000 1.34351
$$719$$ 5.07180 0.189146 0.0945731 0.995518i $$-0.469851\pi$$
0.0945731 + 0.995518i $$0.469851\pi$$
$$720$$ 0 0
$$721$$ −16.0000 −0.595871
$$722$$ −18.8038 −0.699807
$$723$$ 0 0
$$724$$ 15.8564 0.589299
$$725$$ −3.46410 −0.128654
$$726$$ 0 0
$$727$$ 8.00000 0.296704 0.148352 0.988935i $$-0.452603\pi$$
0.148352 + 0.988935i $$0.452603\pi$$
$$728$$ 18.9282 0.701526
$$729$$ 0 0
$$730$$ −14.5359 −0.537998
$$731$$ 0 0
$$732$$ 0 0
$$733$$ −53.9615 −1.99311 −0.996557 0.0829082i $$-0.973579\pi$$
−0.996557 + 0.0829082i $$0.973579\pi$$
$$734$$ −34.6410 −1.27862
$$735$$ 0 0
$$736$$ −36.0000 −1.32698
$$737$$ 0 0
$$738$$ 0 0
$$739$$ −17.4641 −0.642427 −0.321214 0.947007i $$-0.604091\pi$$
−0.321214 + 0.947007i $$0.604091\pi$$
$$740$$ −4.92820 −0.181164
$$741$$ 0 0
$$742$$ −3.21539 −0.118041
$$743$$ 25.6077 0.939455 0.469728 0.882811i $$-0.344352\pi$$
0.469728 + 0.882811i $$0.344352\pi$$
$$744$$ 0 0
$$745$$ −15.4641 −0.566561
$$746$$ −35.3205 −1.29318
$$747$$ 0 0
$$748$$ 0 0
$$749$$ −17.0718 −0.623790
$$750$$ 0 0
$$751$$ 26.9282 0.982624 0.491312 0.870984i $$-0.336517\pi$$
0.491312 + 0.870984i $$0.336517\pi$$
$$752$$ −34.6410 −1.26323
$$753$$ 0 0
$$754$$ −32.7846 −1.19395
$$755$$ 20.3923 0.742152
$$756$$ 0 0
$$757$$ 34.7846 1.26427 0.632134 0.774859i $$-0.282179\pi$$
0.632134 + 0.774859i $$0.282179\pi$$
$$758$$ 30.9282 1.12336
$$759$$ 0 0
$$760$$ −9.46410 −0.343299
$$761$$ −32.5359 −1.17943 −0.589713 0.807613i $$-0.700759\pi$$
−0.589713 + 0.807613i $$0.700759\pi$$
$$762$$ 0 0
$$763$$ −20.0000 −0.724049
$$764$$ 18.9282 0.684798
$$765$$ 0 0
$$766$$ −24.0000 −0.867155
$$767$$ −37.8564 −1.36692
$$768$$ 0 0
$$769$$ −50.4974 −1.82098 −0.910492 0.413527i $$-0.864297\pi$$
−0.910492 + 0.413527i $$0.864297\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ −24.3923 −0.877898
$$773$$ 4.14359 0.149035 0.0745174 0.997220i $$-0.476258\pi$$
0.0745174 + 0.997220i $$0.476258\pi$$
$$774$$ 0 0
$$775$$ −10.9282 −0.392553
$$776$$ −17.3205 −0.621770
$$777$$ 0 0
$$778$$ 19.1769 0.687526
$$779$$ −18.9282 −0.678173
$$780$$ 0 0
$$781$$ 0 0
$$782$$ 0 0
$$783$$ 0 0
$$784$$ 15.0000 0.535714
$$785$$ −3.07180 −0.109637
$$786$$ 0 0
$$787$$ −22.7846 −0.812184 −0.406092 0.913832i $$-0.633109\pi$$
−0.406092 + 0.913832i $$0.633109\pi$$
$$788$$ −12.0000 −0.427482
$$789$$ 0 0
$$790$$ −11.3205 −0.402766
$$791$$ −25.8564 −0.919348
$$792$$ 0 0
$$793$$ 10.9282 0.388072
$$794$$ −3.46410 −0.122936
$$795$$ 0 0
$$796$$ −24.7846 −0.878467
$$797$$ 52.6410 1.86464 0.932320 0.361634i $$-0.117781\pi$$
0.932320 + 0.361634i $$0.117781\pi$$
$$798$$ 0 0
$$799$$ 0 0
$$800$$ 5.19615 0.183712
$$801$$ 0 0
$$802$$ −13.6077 −0.480504
$$803$$ 0 0
$$804$$ 0 0
$$805$$ 13.8564 0.488374
$$806$$ −103.426 −3.64301
$$807$$ 0 0
$$808$$ −18.0000 −0.633238
$$809$$ −15.4641 −0.543689 −0.271844 0.962341i $$-0.587634\pi$$
−0.271844 + 0.962341i $$0.587634\pi$$
$$810$$ 0 0
$$811$$ −12.3923 −0.435153 −0.217576 0.976043i $$-0.569815\pi$$
−0.217576 + 0.976043i $$0.569815\pi$$
$$812$$ 6.92820 0.243132
$$813$$ 0 0
$$814$$ 0 0
$$815$$ 9.85641 0.345255
$$816$$ 0 0
$$817$$ −26.9282 −0.942099
$$818$$ −11.7513 −0.410874
$$819$$ 0 0
$$820$$ 3.46410 0.120972
$$821$$ 20.5359 0.716708 0.358354 0.933586i $$-0.383338\pi$$
0.358354 + 0.933586i $$0.383338\pi$$
$$822$$ 0 0
$$823$$ −33.5692 −1.17015 −0.585075 0.810979i $$-0.698935\pi$$
−0.585075 + 0.810979i $$0.698935\pi$$
$$824$$ 13.8564 0.482711
$$825$$ 0 0
$$826$$ 24.0000 0.835067
$$827$$ 22.3923 0.778657 0.389328 0.921099i $$-0.372707\pi$$
0.389328 + 0.921099i $$0.372707\pi$$
$$828$$ 0 0
$$829$$ 29.7128 1.03197 0.515984 0.856598i $$-0.327426\pi$$
0.515984 + 0.856598i $$0.327426\pi$$
$$830$$ −14.7846 −0.513181
$$831$$ 0 0
$$832$$ −5.46410 −0.189434
$$833$$ 0 0
$$834$$ 0 0
$$835$$ −10.3923 −0.359641
$$836$$ 0 0
$$837$$ 0 0
$$838$$ 53.5692 1.85052
$$839$$ −56.7846 −1.96042 −0.980211 0.197954i $$-0.936570\pi$$
−0.980211 + 0.197954i $$0.936570\pi$$
$$840$$ 0 0
$$841$$ −17.0000 −0.586207
$$842$$ −3.46410 −0.119381
$$843$$ 0 0
$$844$$ 8.39230 0.288875
$$845$$ 16.8564 0.579878
$$846$$ 0 0
$$847$$ 0 0
$$848$$ 4.64102 0.159373
$$849$$ 0 0
$$850$$ 0 0
$$851$$ 34.1436 1.17043
$$852$$ 0 0
$$853$$ −3.60770 −0.123525 −0.0617626 0.998091i $$-0.519672\pi$$
−0.0617626 + 0.998091i $$0.519672\pi$$
$$854$$ −6.92820 −0.237078
$$855$$ 0 0
$$856$$ 14.7846 0.505328
$$857$$ −37.8564 −1.29315 −0.646575 0.762850i $$-0.723799\pi$$
−0.646575 + 0.762850i $$0.723799\pi$$
$$858$$ 0 0
$$859$$ −7.71281 −0.263158 −0.131579 0.991306i $$-0.542005\pi$$
−0.131579 + 0.991306i $$0.542005\pi$$
$$860$$ 4.92820 0.168050
$$861$$ 0 0
$$862$$ −15.2154 −0.518238
$$863$$ 37.8564 1.28865 0.644324 0.764753i $$-0.277139\pi$$
0.644324 + 0.764753i $$0.277139\pi$$
$$864$$ 0 0
$$865$$ −12.0000 −0.408012
$$866$$ −0.248711 −0.00845155
$$867$$ 0 0
$$868$$ 21.8564 0.741855
$$869$$ 0 0
$$870$$ 0 0
$$871$$ −43.7128 −1.48115
$$872$$ 17.3205 0.586546
$$873$$ 0 0
$$874$$ −65.5692 −2.21791
$$875$$ −2.00000 −0.0676123
$$876$$ 0 0
$$877$$ 34.2487 1.15650 0.578248 0.815861i $$-0.303736\pi$$
0.578248 + 0.815861i $$0.303736\pi$$
$$878$$ 57.4641 1.93932
$$879$$ 0 0
$$880$$ 0 0
$$881$$ 0.928203 0.0312720 0.0156360 0.999878i $$-0.495023\pi$$
0.0156360 + 0.999878i $$0.495023\pi$$
$$882$$ 0 0
$$883$$ −4.00000 −0.134611 −0.0673054 0.997732i $$-0.521440\pi$$
−0.0673054 + 0.997732i $$0.521440\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 20.7846 0.698273
$$887$$ 12.2487 0.411271 0.205636 0.978629i $$-0.434074\pi$$
0.205636 + 0.978629i $$0.434074\pi$$
$$888$$ 0 0
$$889$$ 17.8564 0.598885
$$890$$ 1.60770 0.0538901
$$891$$ 0 0
$$892$$ 9.85641 0.330017
$$893$$ −37.8564 −1.26682
$$894$$ 0 0
$$895$$ −6.92820 −0.231584
$$896$$ 24.2487 0.810093
$$897$$ 0 0
$$898$$ −46.3923 −1.54813
$$899$$ 37.8564 1.26258
$$900$$ 0 0
$$901$$ 0 0
$$902$$ 0 0
$$903$$ 0 0
$$904$$ 22.3923 0.744757
$$905$$ 15.8564 0.527085
$$906$$ 0 0
$$907$$ 18.1436 0.602448 0.301224 0.953553i $$-0.402605\pi$$
0.301224 + 0.953553i $$0.402605\pi$$
$$908$$ 15.4641 0.513194
$$909$$ 0 0
$$910$$ −18.9282 −0.627464
$$911$$ −18.9282 −0.627119 −0.313560 0.949568i $$-0.601522\pi$$
−0.313560 + 0.949568i $$0.601522\pi$$
$$912$$ 0 0
$$913$$ 0 0
$$914$$ 21.4641 0.709969
$$915$$ 0 0
$$916$$ −23.8564 −0.788238
$$917$$ −37.8564 −1.25013
$$918$$ 0 0
$$919$$ 32.3923 1.06852 0.534262 0.845319i $$-0.320590\pi$$
0.534262 + 0.845319i $$0.320590\pi$$
$$920$$ −12.0000 −0.395628
$$921$$ 0 0
$$922$$ −62.7846 −2.06770
$$923$$ 75.7128 2.49212
$$924$$ 0 0
$$925$$ −4.92820 −0.162038
$$926$$ 48.4974 1.59372
$$927$$ 0 0
$$928$$ −18.0000 −0.590879
$$929$$ −2.78461 −0.0913601 −0.0456800 0.998956i $$-0.514545\pi$$
−0.0456800 + 0.998956i $$0.514545\pi$$
$$930$$ 0 0
$$931$$ 16.3923 0.537236
$$932$$ 12.0000 0.393073
$$933$$ 0 0
$$934$$ 8.78461 0.287441
$$935$$ 0 0
$$936$$ 0 0
$$937$$ 20.3923 0.666188 0.333094 0.942894i $$-0.391907\pi$$
0.333094 + 0.942894i $$0.391907\pi$$
$$938$$ 27.7128 0.904855
$$939$$ 0 0
$$940$$ 6.92820 0.225973
$$941$$ 27.4641 0.895304 0.447652 0.894208i $$-0.352260\pi$$
0.447652 + 0.894208i $$0.352260\pi$$
$$942$$ 0 0
$$943$$ −24.0000 −0.781548
$$944$$ −34.6410 −1.12747
$$945$$ 0 0
$$946$$ 0 0
$$947$$ 18.9282 0.615084 0.307542 0.951535i $$-0.400494\pi$$
0.307542 + 0.951535i $$0.400494\pi$$
$$948$$ 0 0
$$949$$ −45.8564 −1.48856
$$950$$ 9.46410 0.307056
$$951$$ 0 0
$$952$$ 0 0
$$953$$ −3.21539 −0.104157 −0.0520784 0.998643i $$-0.516585\pi$$
−0.0520784 + 0.998643i $$0.516585\pi$$
$$954$$ 0 0
$$955$$ 18.9282 0.612502
$$956$$ −12.0000 −0.388108
$$957$$ 0 0
$$958$$ −20.7846 −0.671520
$$959$$ −36.0000 −1.16250
$$960$$ 0 0
$$961$$ 88.4256 2.85244
$$962$$ −46.6410 −1.50377
$$963$$ 0 0
$$964$$ −0.143594 −0.00462484
$$965$$ −24.3923 −0.785216
$$966$$ 0 0
$$967$$ −22.7846 −0.732704 −0.366352 0.930476i $$-0.619393\pi$$
−0.366352 + 0.930476i $$0.619393\pi$$
$$968$$ 0 0
$$969$$ 0 0
$$970$$ 17.3205 0.556128
$$971$$ −25.8564 −0.829772 −0.414886 0.909873i $$-0.636178\pi$$
−0.414886 + 0.909873i $$0.636178\pi$$
$$972$$ 0 0
$$973$$ 24.7846 0.794558
$$974$$ 54.9282 1.76001
$$975$$ 0 0
$$976$$ 10.0000 0.320092
$$977$$ −47.5692 −1.52187 −0.760937 0.648826i $$-0.775260\pi$$
−0.760937 + 0.648826i $$0.775260\pi$$
$$978$$ 0 0
$$979$$ 0 0
$$980$$ −3.00000 −0.0958315
$$981$$ 0 0
$$982$$ 53.5692 1.70946
$$983$$ −24.0000 −0.765481 −0.382741 0.923856i $$-0.625020\pi$$
−0.382741 + 0.923856i $$0.625020\pi$$
$$984$$ 0 0
$$985$$ −12.0000 −0.382352
$$986$$ 0 0
$$987$$ 0 0
$$988$$ 29.8564 0.949859
$$989$$ −34.1436 −1.08570
$$990$$ 0 0
$$991$$ −7.21539 −0.229204 −0.114602 0.993411i $$-0.536559\pi$$
−0.114602 + 0.993411i $$0.536559\pi$$
$$992$$ −56.7846 −1.80291
$$993$$ 0 0
$$994$$ −48.0000 −1.52247
$$995$$ −24.7846 −0.785725
$$996$$ 0 0
$$997$$ −27.6077 −0.874344 −0.437172 0.899378i $$-0.644020\pi$$
−0.437172 + 0.899378i $$0.644020\pi$$
$$998$$ −49.8564 −1.57818
$$999$$ 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.s.1.1 2
3.2 odd 2 1815.2.a.i.1.2 2
11.10 odd 2 495.2.a.c.1.2 2
15.14 odd 2 9075.2.a.bh.1.1 2
33.32 even 2 165.2.a.b.1.1 2
44.43 even 2 7920.2.a.bz.1.2 2
55.32 even 4 2475.2.c.n.199.4 4
55.43 even 4 2475.2.c.n.199.1 4
55.54 odd 2 2475.2.a.r.1.1 2
132.131 odd 2 2640.2.a.x.1.2 2
165.32 odd 4 825.2.c.c.199.1 4
165.98 odd 4 825.2.c.c.199.4 4
165.164 even 2 825.2.a.e.1.2 2
231.230 odd 2 8085.2.a.bd.1.1 2

By twisted newform
Twist Min Dim Char Parity Ord Type
165.2.a.b.1.1 2 33.32 even 2
495.2.a.c.1.2 2 11.10 odd 2
825.2.a.e.1.2 2 165.164 even 2
825.2.c.c.199.1 4 165.32 odd 4
825.2.c.c.199.4 4 165.98 odd 4
1815.2.a.i.1.2 2 3.2 odd 2
2475.2.a.r.1.1 2 55.54 odd 2
2475.2.c.n.199.1 4 55.43 even 4
2475.2.c.n.199.4 4 55.32 even 4
2640.2.a.x.1.2 2 132.131 odd 2
5445.2.a.s.1.1 2 1.1 even 1 trivial
7920.2.a.bz.1.2 2 44.43 even 2
8085.2.a.bd.1.1 2 231.230 odd 2
9075.2.a.bh.1.1 2 15.14 odd 2