# Properties

 Label 5733.2.a.x.1.3 Level $5733$ Weight $2$ Character 5733.1 Self dual yes Analytic conductor $45.778$ Analytic rank $1$ Dimension $3$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$5733 = 3^{2} \cdot 7^{2} \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 5733.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$45.7782354788$$ Analytic rank: $$1$$ Dimension: $$3$$ Coefficient field: 3.3.316.1 Defining polynomial: $$x^{3} - x^{2} - 4 x + 2$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 91) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.3 Root $$-1.81361$$ of defining polynomial Character $$\chi$$ $$=$$ 5733.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+1.81361 q^{2} +1.28917 q^{4} +2.81361 q^{5} -1.28917 q^{8} +O(q^{10})$$ $$q+1.81361 q^{2} +1.28917 q^{4} +2.81361 q^{5} -1.28917 q^{8} +5.10278 q^{10} -3.10278 q^{11} -1.00000 q^{13} -4.91638 q^{16} -0.524438 q^{17} -0.813607 q^{19} +3.62721 q^{20} -5.62721 q^{22} -7.33804 q^{23} +2.91638 q^{25} -1.81361 q^{26} -8.28917 q^{29} -1.39194 q^{31} -6.33804 q^{32} -0.951124 q^{34} -6.15165 q^{37} -1.47556 q^{38} -3.62721 q^{40} -4.20555 q^{41} +6.75971 q^{43} -4.00000 q^{44} -13.3083 q^{46} -5.97028 q^{47} +5.28917 q^{50} -1.28917 q^{52} +2.49472 q^{53} -8.72999 q^{55} -15.0333 q^{58} -4.47054 q^{59} +2.00000 q^{61} -2.52444 q^{62} -1.66196 q^{64} -2.81361 q^{65} +10.0383 q^{67} -0.676089 q^{68} +8.72999 q^{71} +2.34307 q^{73} -11.1567 q^{74} -1.04888 q^{76} -13.5436 q^{79} -13.8328 q^{80} -7.62721 q^{82} +16.4791 q^{83} -1.47556 q^{85} +12.2594 q^{86} +4.00000 q^{88} -10.6464 q^{89} -9.45998 q^{92} -10.8277 q^{94} -2.28917 q^{95} +1.18639 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3q - q^{2} + 3q^{4} + 2q^{5} - 3q^{8} + O(q^{10})$$ $$3q - q^{2} + 3q^{4} + 2q^{5} - 3q^{8} + 8q^{10} - 2q^{11} - 3q^{13} - q^{16} + 4q^{17} + 4q^{19} - 2q^{20} - 4q^{22} - 10q^{23} - 5q^{25} + q^{26} - 24q^{29} + 4q^{31} - 7q^{32} - 14q^{34} - 10q^{38} + 2q^{40} + 2q^{41} + 10q^{43} - 12q^{44} - 18q^{46} - 8q^{47} + 15q^{50} - 3q^{52} - 8q^{53} - 6q^{55} + 12q^{58} - 4q^{59} + 6q^{61} - 2q^{62} - 17q^{64} - 2q^{65} - 12q^{67} + 22q^{68} + 6q^{71} + 10q^{73} - 30q^{74} + 8q^{76} - 14q^{79} - 14q^{80} - 10q^{82} - 12q^{83} - 10q^{85} + 26q^{86} + 12q^{88} + 2q^{89} + 12q^{92} + 10q^{94} - 6q^{95} + 10q^{97} + O(q^{100})$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 1.81361 1.28241 0.641207 0.767368i $$-0.278434\pi$$
0.641207 + 0.767368i $$0.278434\pi$$
$$3$$ 0 0
$$4$$ 1.28917 0.644584
$$5$$ 2.81361 1.25828 0.629142 0.777291i $$-0.283407\pi$$
0.629142 + 0.777291i $$0.283407\pi$$
$$6$$ 0 0
$$7$$ 0 0
$$8$$ −1.28917 −0.455790
$$9$$ 0 0
$$10$$ 5.10278 1.61364
$$11$$ −3.10278 −0.935522 −0.467761 0.883855i $$-0.654939\pi$$
−0.467761 + 0.883855i $$0.654939\pi$$
$$12$$ 0 0
$$13$$ −1.00000 −0.277350
$$14$$ 0 0
$$15$$ 0 0
$$16$$ −4.91638 −1.22910
$$17$$ −0.524438 −0.127195 −0.0635974 0.997976i $$-0.520257\pi$$
−0.0635974 + 0.997976i $$0.520257\pi$$
$$18$$ 0 0
$$19$$ −0.813607 −0.186654 −0.0933271 0.995636i $$-0.529750\pi$$
−0.0933271 + 0.995636i $$0.529750\pi$$
$$20$$ 3.62721 0.811069
$$21$$ 0 0
$$22$$ −5.62721 −1.19973
$$23$$ −7.33804 −1.53009 −0.765044 0.643978i $$-0.777283\pi$$
−0.765044 + 0.643978i $$0.777283\pi$$
$$24$$ 0 0
$$25$$ 2.91638 0.583276
$$26$$ −1.81361 −0.355677
$$27$$ 0 0
$$28$$ 0 0
$$29$$ −8.28917 −1.53926 −0.769630 0.638490i $$-0.779559\pi$$
−0.769630 + 0.638490i $$0.779559\pi$$
$$30$$ 0 0
$$31$$ −1.39194 −0.250000 −0.125000 0.992157i $$-0.539893\pi$$
−0.125000 + 0.992157i $$0.539893\pi$$
$$32$$ −6.33804 −1.12042
$$33$$ 0 0
$$34$$ −0.951124 −0.163116
$$35$$ 0 0
$$36$$ 0 0
$$37$$ −6.15165 −1.01133 −0.505663 0.862731i $$-0.668752\pi$$
−0.505663 + 0.862731i $$0.668752\pi$$
$$38$$ −1.47556 −0.239368
$$39$$ 0 0
$$40$$ −3.62721 −0.573513
$$41$$ −4.20555 −0.656797 −0.328398 0.944539i $$-0.606509\pi$$
−0.328398 + 0.944539i $$0.606509\pi$$
$$42$$ 0 0
$$43$$ 6.75971 1.03085 0.515423 0.856936i $$-0.327635\pi$$
0.515423 + 0.856936i $$0.327635\pi$$
$$44$$ −4.00000 −0.603023
$$45$$ 0 0
$$46$$ −13.3083 −1.96221
$$47$$ −5.97028 −0.870855 −0.435427 0.900224i $$-0.643403\pi$$
−0.435427 + 0.900224i $$0.643403\pi$$
$$48$$ 0 0
$$49$$ 0 0
$$50$$ 5.28917 0.748001
$$51$$ 0 0
$$52$$ −1.28917 −0.178776
$$53$$ 2.49472 0.342676 0.171338 0.985212i $$-0.445191\pi$$
0.171338 + 0.985212i $$0.445191\pi$$
$$54$$ 0 0
$$55$$ −8.72999 −1.17715
$$56$$ 0 0
$$57$$ 0 0
$$58$$ −15.0333 −1.97397
$$59$$ −4.47054 −0.582015 −0.291007 0.956721i $$-0.593990\pi$$
−0.291007 + 0.956721i $$0.593990\pi$$
$$60$$ 0 0
$$61$$ 2.00000 0.256074 0.128037 0.991769i $$-0.459132\pi$$
0.128037 + 0.991769i $$0.459132\pi$$
$$62$$ −2.52444 −0.320604
$$63$$ 0 0
$$64$$ −1.66196 −0.207744
$$65$$ −2.81361 −0.348985
$$66$$ 0 0
$$67$$ 10.0383 1.22638 0.613188 0.789937i $$-0.289887\pi$$
0.613188 + 0.789937i $$0.289887\pi$$
$$68$$ −0.676089 −0.0819878
$$69$$ 0 0
$$70$$ 0 0
$$71$$ 8.72999 1.03606 0.518029 0.855363i $$-0.326666\pi$$
0.518029 + 0.855363i $$0.326666\pi$$
$$72$$ 0 0
$$73$$ 2.34307 0.274235 0.137118 0.990555i $$-0.456216\pi$$
0.137118 + 0.990555i $$0.456216\pi$$
$$74$$ −11.1567 −1.29694
$$75$$ 0 0
$$76$$ −1.04888 −0.120314
$$77$$ 0 0
$$78$$ 0 0
$$79$$ −13.5436 −1.52377 −0.761887 0.647710i $$-0.775727\pi$$
−0.761887 + 0.647710i $$0.775727\pi$$
$$80$$ −13.8328 −1.54655
$$81$$ 0 0
$$82$$ −7.62721 −0.842285
$$83$$ 16.4791 1.80882 0.904410 0.426665i $$-0.140312\pi$$
0.904410 + 0.426665i $$0.140312\pi$$
$$84$$ 0 0
$$85$$ −1.47556 −0.160047
$$86$$ 12.2594 1.32197
$$87$$ 0 0
$$88$$ 4.00000 0.426401
$$89$$ −10.6464 −1.12851 −0.564256 0.825600i $$-0.690837\pi$$
−0.564256 + 0.825600i $$0.690837\pi$$
$$90$$ 0 0
$$91$$ 0 0
$$92$$ −9.45998 −0.986271
$$93$$ 0 0
$$94$$ −10.8277 −1.11680
$$95$$ −2.28917 −0.234864
$$96$$ 0 0
$$97$$ 1.18639 0.120460 0.0602300 0.998185i $$-0.480817\pi$$
0.0602300 + 0.998185i $$0.480817\pi$$
$$98$$ 0 0
$$99$$ 0 0
$$100$$ 3.75971 0.375971
$$101$$ 13.1028 1.30377 0.651887 0.758316i $$-0.273977\pi$$
0.651887 + 0.758316i $$0.273977\pi$$
$$102$$ 0 0
$$103$$ −4.41110 −0.434639 −0.217319 0.976101i $$-0.569731\pi$$
−0.217319 + 0.976101i $$0.569731\pi$$
$$104$$ 1.28917 0.126413
$$105$$ 0 0
$$106$$ 4.52444 0.439452
$$107$$ −0.578337 −0.0559100 −0.0279550 0.999609i $$-0.508900\pi$$
−0.0279550 + 0.999609i $$0.508900\pi$$
$$108$$ 0 0
$$109$$ 5.57331 0.533827 0.266913 0.963721i $$-0.413996\pi$$
0.266913 + 0.963721i $$0.413996\pi$$
$$110$$ −15.8328 −1.50959
$$111$$ 0 0
$$112$$ 0 0
$$113$$ −5.44584 −0.512302 −0.256151 0.966637i $$-0.582454\pi$$
−0.256151 + 0.966637i $$0.582454\pi$$
$$114$$ 0 0
$$115$$ −20.6464 −1.92528
$$116$$ −10.6861 −0.992183
$$117$$ 0 0
$$118$$ −8.10780 −0.746383
$$119$$ 0 0
$$120$$ 0 0
$$121$$ −1.37279 −0.124799
$$122$$ 3.62721 0.328392
$$123$$ 0 0
$$124$$ −1.79445 −0.161146
$$125$$ −5.86248 −0.524356
$$126$$ 0 0
$$127$$ −12.8816 −1.14306 −0.571530 0.820581i $$-0.693650\pi$$
−0.571530 + 0.820581i $$0.693650\pi$$
$$128$$ 9.66196 0.854004
$$129$$ 0 0
$$130$$ −5.10278 −0.447543
$$131$$ 9.04888 0.790604 0.395302 0.918551i $$-0.370640\pi$$
0.395302 + 0.918551i $$0.370640\pi$$
$$132$$ 0 0
$$133$$ 0 0
$$134$$ 18.2056 1.57272
$$135$$ 0 0
$$136$$ 0.676089 0.0579741
$$137$$ 6.25945 0.534781 0.267390 0.963588i $$-0.413839\pi$$
0.267390 + 0.963588i $$0.413839\pi$$
$$138$$ 0 0
$$139$$ 11.5733 0.981636 0.490818 0.871262i $$-0.336698\pi$$
0.490818 + 0.871262i $$0.336698\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 15.8328 1.32866
$$143$$ 3.10278 0.259467
$$144$$ 0 0
$$145$$ −23.3225 −1.93682
$$146$$ 4.24940 0.351683
$$147$$ 0 0
$$148$$ −7.93051 −0.651884
$$149$$ −8.52444 −0.698349 −0.349175 0.937058i $$-0.613538\pi$$
−0.349175 + 0.937058i $$0.613538\pi$$
$$150$$ 0 0
$$151$$ −11.9844 −0.975278 −0.487639 0.873045i $$-0.662142\pi$$
−0.487639 + 0.873045i $$0.662142\pi$$
$$152$$ 1.04888 0.0850751
$$153$$ 0 0
$$154$$ 0 0
$$155$$ −3.91638 −0.314571
$$156$$ 0 0
$$157$$ 12.8277 1.02377 0.511883 0.859055i $$-0.328948\pi$$
0.511883 + 0.859055i $$0.328948\pi$$
$$158$$ −24.5628 −1.95411
$$159$$ 0 0
$$160$$ −17.8328 −1.40980
$$161$$ 0 0
$$162$$ 0 0
$$163$$ −13.4600 −1.05427 −0.527133 0.849783i $$-0.676733\pi$$
−0.527133 + 0.849783i $$0.676733\pi$$
$$164$$ −5.42166 −0.423361
$$165$$ 0 0
$$166$$ 29.8867 2.31965
$$167$$ −2.02972 −0.157064 −0.0785322 0.996912i $$-0.525023\pi$$
−0.0785322 + 0.996912i $$0.525023\pi$$
$$168$$ 0 0
$$169$$ 1.00000 0.0769231
$$170$$ −2.67609 −0.205247
$$171$$ 0 0
$$172$$ 8.71440 0.664467
$$173$$ 20.2978 1.54321 0.771605 0.636102i $$-0.219454\pi$$
0.771605 + 0.636102i $$0.219454\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 15.2544 1.14985
$$177$$ 0 0
$$178$$ −19.3083 −1.44722
$$179$$ 11.0036 0.822445 0.411223 0.911535i $$-0.365102\pi$$
0.411223 + 0.911535i $$0.365102\pi$$
$$180$$ 0 0
$$181$$ −0.691675 −0.0514118 −0.0257059 0.999670i $$-0.508183\pi$$
−0.0257059 + 0.999670i $$0.508183\pi$$
$$182$$ 0 0
$$183$$ 0 0
$$184$$ 9.45998 0.697399
$$185$$ −17.3083 −1.27253
$$186$$ 0 0
$$187$$ 1.62721 0.118994
$$188$$ −7.69670 −0.561339
$$189$$ 0 0
$$190$$ −4.15165 −0.301192
$$191$$ −7.83276 −0.566759 −0.283379 0.959008i $$-0.591456\pi$$
−0.283379 + 0.959008i $$0.591456\pi$$
$$192$$ 0 0
$$193$$ −12.2056 −0.878575 −0.439287 0.898347i $$-0.644769\pi$$
−0.439287 + 0.898347i $$0.644769\pi$$
$$194$$ 2.15165 0.154480
$$195$$ 0 0
$$196$$ 0 0
$$197$$ 18.8222 1.34103 0.670513 0.741898i $$-0.266074\pi$$
0.670513 + 0.741898i $$0.266074\pi$$
$$198$$ 0 0
$$199$$ −21.6116 −1.53201 −0.766004 0.642836i $$-0.777758\pi$$
−0.766004 + 0.642836i $$0.777758\pi$$
$$200$$ −3.75971 −0.265851
$$201$$ 0 0
$$202$$ 23.7633 1.67198
$$203$$ 0 0
$$204$$ 0 0
$$205$$ −11.8328 −0.826436
$$206$$ −8.00000 −0.557386
$$207$$ 0 0
$$208$$ 4.91638 0.340890
$$209$$ 2.52444 0.174619
$$210$$ 0 0
$$211$$ −17.3764 −1.19624 −0.598119 0.801407i $$-0.704085\pi$$
−0.598119 + 0.801407i $$0.704085\pi$$
$$212$$ 3.21611 0.220884
$$213$$ 0 0
$$214$$ −1.04888 −0.0716997
$$215$$ 19.0192 1.29710
$$216$$ 0 0
$$217$$ 0 0
$$218$$ 10.1078 0.684586
$$219$$ 0 0
$$220$$ −11.2544 −0.758773
$$221$$ 0.524438 0.0352775
$$222$$ 0 0
$$223$$ 10.5486 0.706388 0.353194 0.935550i $$-0.385096\pi$$
0.353194 + 0.935550i $$0.385096\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ −9.87662 −0.656983
$$227$$ −6.95112 −0.461362 −0.230681 0.973029i $$-0.574095\pi$$
−0.230681 + 0.973029i $$0.574095\pi$$
$$228$$ 0 0
$$229$$ 21.0872 1.39348 0.696740 0.717323i $$-0.254633\pi$$
0.696740 + 0.717323i $$0.254633\pi$$
$$230$$ −37.4444 −2.46901
$$231$$ 0 0
$$232$$ 10.6861 0.701579
$$233$$ −6.08362 −0.398551 −0.199276 0.979943i $$-0.563859\pi$$
−0.199276 + 0.979943i $$0.563859\pi$$
$$234$$ 0 0
$$235$$ −16.7980 −1.09578
$$236$$ −5.76328 −0.375157
$$237$$ 0 0
$$238$$ 0 0
$$239$$ 14.2056 0.918881 0.459440 0.888209i $$-0.348050\pi$$
0.459440 + 0.888209i $$0.348050\pi$$
$$240$$ 0 0
$$241$$ −8.44082 −0.543721 −0.271860 0.962337i $$-0.587639\pi$$
−0.271860 + 0.962337i $$0.587639\pi$$
$$242$$ −2.48970 −0.160044
$$243$$ 0 0
$$244$$ 2.57834 0.165061
$$245$$ 0 0
$$246$$ 0 0
$$247$$ 0.813607 0.0517685
$$248$$ 1.79445 0.113948
$$249$$ 0 0
$$250$$ −10.6322 −0.672442
$$251$$ −23.9844 −1.51388 −0.756941 0.653483i $$-0.773307\pi$$
−0.756941 + 0.653483i $$0.773307\pi$$
$$252$$ 0 0
$$253$$ 22.7683 1.43143
$$254$$ −23.3622 −1.46588
$$255$$ 0 0
$$256$$ 20.8469 1.30293
$$257$$ 15.6116 0.973827 0.486913 0.873450i $$-0.338123\pi$$
0.486913 + 0.873450i $$0.338123\pi$$
$$258$$ 0 0
$$259$$ 0 0
$$260$$ −3.62721 −0.224950
$$261$$ 0 0
$$262$$ 16.4111 1.01388
$$263$$ 15.1708 0.935472 0.467736 0.883868i $$-0.345070\pi$$
0.467736 + 0.883868i $$0.345070\pi$$
$$264$$ 0 0
$$265$$ 7.01916 0.431183
$$266$$ 0 0
$$267$$ 0 0
$$268$$ 12.9411 0.790502
$$269$$ 23.2927 1.42018 0.710092 0.704109i $$-0.248653\pi$$
0.710092 + 0.704109i $$0.248653\pi$$
$$270$$ 0 0
$$271$$ −12.7456 −0.774238 −0.387119 0.922030i $$-0.626530\pi$$
−0.387119 + 0.922030i $$0.626530\pi$$
$$272$$ 2.57834 0.156335
$$273$$ 0 0
$$274$$ 11.3522 0.685810
$$275$$ −9.04888 −0.545668
$$276$$ 0 0
$$277$$ 8.12193 0.488000 0.244000 0.969775i $$-0.421540\pi$$
0.244000 + 0.969775i $$0.421540\pi$$
$$278$$ 20.9894 1.25886
$$279$$ 0 0
$$280$$ 0 0
$$281$$ −19.0333 −1.13543 −0.567715 0.823225i $$-0.692173\pi$$
−0.567715 + 0.823225i $$0.692173\pi$$
$$282$$ 0 0
$$283$$ −11.1466 −0.662598 −0.331299 0.943526i $$-0.607487\pi$$
−0.331299 + 0.943526i $$0.607487\pi$$
$$284$$ 11.2544 0.667827
$$285$$ 0 0
$$286$$ 5.62721 0.332744
$$287$$ 0 0
$$288$$ 0 0
$$289$$ −16.7250 −0.983821
$$290$$ −42.2978 −2.48381
$$291$$ 0 0
$$292$$ 3.02061 0.176768
$$293$$ −14.1758 −0.828161 −0.414080 0.910240i $$-0.635897\pi$$
−0.414080 + 0.910240i $$0.635897\pi$$
$$294$$ 0 0
$$295$$ −12.5783 −0.732339
$$296$$ 7.93051 0.460952
$$297$$ 0 0
$$298$$ −15.4600 −0.895572
$$299$$ 7.33804 0.424370
$$300$$ 0 0
$$301$$ 0 0
$$302$$ −21.7350 −1.25071
$$303$$ 0 0
$$304$$ 4.00000 0.229416
$$305$$ 5.62721 0.322213
$$306$$ 0 0
$$307$$ −13.5592 −0.773863 −0.386932 0.922108i $$-0.626465\pi$$
−0.386932 + 0.922108i $$0.626465\pi$$
$$308$$ 0 0
$$309$$ 0 0
$$310$$ −7.10278 −0.403411
$$311$$ 0.426686 0.0241952 0.0120976 0.999927i $$-0.496149\pi$$
0.0120976 + 0.999927i $$0.496149\pi$$
$$312$$ 0 0
$$313$$ 18.1517 1.02599 0.512996 0.858391i $$-0.328536\pi$$
0.512996 + 0.858391i $$0.328536\pi$$
$$314$$ 23.2645 1.31289
$$315$$ 0 0
$$316$$ −17.4600 −0.982200
$$317$$ −9.42166 −0.529173 −0.264587 0.964362i $$-0.585236\pi$$
−0.264587 + 0.964362i $$0.585236\pi$$
$$318$$ 0 0
$$319$$ 25.7194 1.44001
$$320$$ −4.67609 −0.261401
$$321$$ 0 0
$$322$$ 0 0
$$323$$ 0.426686 0.0237415
$$324$$ 0 0
$$325$$ −2.91638 −0.161772
$$326$$ −24.4111 −1.35201
$$327$$ 0 0
$$328$$ 5.42166 0.299361
$$329$$ 0 0
$$330$$ 0 0
$$331$$ −17.4005 −0.956420 −0.478210 0.878246i $$-0.658714\pi$$
−0.478210 + 0.878246i $$0.658714\pi$$
$$332$$ 21.2444 1.16594
$$333$$ 0 0
$$334$$ −3.68111 −0.201421
$$335$$ 28.2439 1.54313
$$336$$ 0 0
$$337$$ 22.0524 1.20127 0.600637 0.799522i $$-0.294914\pi$$
0.600637 + 0.799522i $$0.294914\pi$$
$$338$$ 1.81361 0.0986472
$$339$$ 0 0
$$340$$ −1.90225 −0.103164
$$341$$ 4.31889 0.233881
$$342$$ 0 0
$$343$$ 0 0
$$344$$ −8.71440 −0.469849
$$345$$ 0 0
$$346$$ 36.8122 1.97903
$$347$$ −25.3522 −1.36098 −0.680488 0.732759i $$-0.738232\pi$$
−0.680488 + 0.732759i $$0.738232\pi$$
$$348$$ 0 0
$$349$$ 5.70529 0.305397 0.152699 0.988273i $$-0.451204\pi$$
0.152699 + 0.988273i $$0.451204\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 19.6655 1.04818
$$353$$ −28.6761 −1.52627 −0.763137 0.646237i $$-0.776342\pi$$
−0.763137 + 0.646237i $$0.776342\pi$$
$$354$$ 0 0
$$355$$ 24.5628 1.30366
$$356$$ −13.7250 −0.727422
$$357$$ 0 0
$$358$$ 19.9561 1.05472
$$359$$ −11.0433 −0.582845 −0.291423 0.956594i $$-0.594129\pi$$
−0.291423 + 0.956594i $$0.594129\pi$$
$$360$$ 0 0
$$361$$ −18.3380 −0.965160
$$362$$ −1.25443 −0.0659312
$$363$$ 0 0
$$364$$ 0 0
$$365$$ 6.59247 0.345066
$$366$$ 0 0
$$367$$ −27.3466 −1.42748 −0.713741 0.700409i $$-0.753001\pi$$
−0.713741 + 0.700409i $$0.753001\pi$$
$$368$$ 36.0766 1.88062
$$369$$ 0 0
$$370$$ −31.3905 −1.63191
$$371$$ 0 0
$$372$$ 0 0
$$373$$ 16.1461 0.836014 0.418007 0.908444i $$-0.362729\pi$$
0.418007 + 0.908444i $$0.362729\pi$$
$$374$$ 2.95112 0.152599
$$375$$ 0 0
$$376$$ 7.69670 0.396927
$$377$$ 8.28917 0.426914
$$378$$ 0 0
$$379$$ 26.1305 1.34223 0.671117 0.741351i $$-0.265815\pi$$
0.671117 + 0.741351i $$0.265815\pi$$
$$380$$ −2.95112 −0.151389
$$381$$ 0 0
$$382$$ −14.2056 −0.726819
$$383$$ −21.0489 −1.07555 −0.537774 0.843089i $$-0.680735\pi$$
−0.537774 + 0.843089i $$0.680735\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ −22.1361 −1.12670
$$387$$ 0 0
$$388$$ 1.52946 0.0776466
$$389$$ 21.6061 1.09547 0.547736 0.836651i $$-0.315490\pi$$
0.547736 + 0.836651i $$0.315490\pi$$
$$390$$ 0 0
$$391$$ 3.84835 0.194619
$$392$$ 0 0
$$393$$ 0 0
$$394$$ 34.1361 1.71975
$$395$$ −38.1063 −1.91734
$$396$$ 0 0
$$397$$ 27.6952 1.38998 0.694992 0.719017i $$-0.255408\pi$$
0.694992 + 0.719017i $$0.255408\pi$$
$$398$$ −39.1950 −1.96467
$$399$$ 0 0
$$400$$ −14.3380 −0.716902
$$401$$ 2.57834 0.128756 0.0643780 0.997926i $$-0.479494\pi$$
0.0643780 + 0.997926i $$0.479494\pi$$
$$402$$ 0 0
$$403$$ 1.39194 0.0693376
$$404$$ 16.8917 0.840393
$$405$$ 0 0
$$406$$ 0 0
$$407$$ 19.0872 0.946117
$$408$$ 0 0
$$409$$ −15.1169 −0.747483 −0.373742 0.927533i $$-0.621925\pi$$
−0.373742 + 0.927533i $$0.621925\pi$$
$$410$$ −21.4600 −1.05983
$$411$$ 0 0
$$412$$ −5.68665 −0.280161
$$413$$ 0 0
$$414$$ 0 0
$$415$$ 46.3658 2.27601
$$416$$ 6.33804 0.310748
$$417$$ 0 0
$$418$$ 4.57834 0.223934
$$419$$ −9.99446 −0.488261 −0.244131 0.969742i $$-0.578503\pi$$
−0.244131 + 0.969742i $$0.578503\pi$$
$$420$$ 0 0
$$421$$ −25.9250 −1.26351 −0.631753 0.775170i $$-0.717664\pi$$
−0.631753 + 0.775170i $$0.717664\pi$$
$$422$$ −31.5139 −1.53407
$$423$$ 0 0
$$424$$ −3.21611 −0.156188
$$425$$ −1.52946 −0.0741898
$$426$$ 0 0
$$427$$ 0 0
$$428$$ −0.745574 −0.0360387
$$429$$ 0 0
$$430$$ 34.4933 1.66341
$$431$$ −30.6761 −1.47762 −0.738808 0.673916i $$-0.764611\pi$$
−0.738808 + 0.673916i $$0.764611\pi$$
$$432$$ 0 0
$$433$$ 3.51941 0.169132 0.0845661 0.996418i $$-0.473050\pi$$
0.0845661 + 0.996418i $$0.473050\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 7.18494 0.344096
$$437$$ 5.97028 0.285597
$$438$$ 0 0
$$439$$ −32.3517 −1.54406 −0.772030 0.635586i $$-0.780759\pi$$
−0.772030 + 0.635586i $$0.780759\pi$$
$$440$$ 11.2544 0.536534
$$441$$ 0 0
$$442$$ 0.951124 0.0452404
$$443$$ −15.4458 −0.733854 −0.366927 0.930250i $$-0.619590\pi$$
−0.366927 + 0.930250i $$0.619590\pi$$
$$444$$ 0 0
$$445$$ −29.9547 −1.41999
$$446$$ 19.1310 0.905881
$$447$$ 0 0
$$448$$ 0 0
$$449$$ 14.4705 0.682907 0.341453 0.939899i $$-0.389081\pi$$
0.341453 + 0.939899i $$0.389081\pi$$
$$450$$ 0 0
$$451$$ 13.0489 0.614448
$$452$$ −7.02061 −0.330222
$$453$$ 0 0
$$454$$ −12.6066 −0.591657
$$455$$ 0 0
$$456$$ 0 0
$$457$$ −34.6705 −1.62182 −0.810910 0.585171i $$-0.801027\pi$$
−0.810910 + 0.585171i $$0.801027\pi$$
$$458$$ 38.2439 1.78702
$$459$$ 0 0
$$460$$ −26.6167 −1.24101
$$461$$ 12.5400 0.584047 0.292024 0.956411i $$-0.405671\pi$$
0.292024 + 0.956411i $$0.405671\pi$$
$$462$$ 0 0
$$463$$ 12.1517 0.564735 0.282368 0.959306i $$-0.408880\pi$$
0.282368 + 0.959306i $$0.408880\pi$$
$$464$$ 40.7527 1.89190
$$465$$ 0 0
$$466$$ −11.0333 −0.511107
$$467$$ −37.0333 −1.71370 −0.856848 0.515569i $$-0.827581\pi$$
−0.856848 + 0.515569i $$0.827581\pi$$
$$468$$ 0 0
$$469$$ 0 0
$$470$$ −30.4650 −1.40525
$$471$$ 0 0
$$472$$ 5.76328 0.265276
$$473$$ −20.9739 −0.964379
$$474$$ 0 0
$$475$$ −2.37279 −0.108871
$$476$$ 0 0
$$477$$ 0 0
$$478$$ 25.7633 1.17838
$$479$$ 12.0086 0.548687 0.274343 0.961632i $$-0.411540\pi$$
0.274343 + 0.961632i $$0.411540\pi$$
$$480$$ 0 0
$$481$$ 6.15165 0.280491
$$482$$ −15.3083 −0.697275
$$483$$ 0 0
$$484$$ −1.76975 −0.0804434
$$485$$ 3.33804 0.151573
$$486$$ 0 0
$$487$$ 11.1184 0.503821 0.251911 0.967751i $$-0.418941\pi$$
0.251911 + 0.967751i $$0.418941\pi$$
$$488$$ −2.57834 −0.116716
$$489$$ 0 0
$$490$$ 0 0
$$491$$ −0.0594386 −0.00268243 −0.00134121 0.999999i $$-0.500427\pi$$
−0.00134121 + 0.999999i $$0.500427\pi$$
$$492$$ 0 0
$$493$$ 4.34715 0.195786
$$494$$ 1.47556 0.0663887
$$495$$ 0 0
$$496$$ 6.84333 0.307274
$$497$$ 0 0
$$498$$ 0 0
$$499$$ 10.2978 0.460991 0.230496 0.973073i $$-0.425965\pi$$
0.230496 + 0.973073i $$0.425965\pi$$
$$500$$ −7.55773 −0.337992
$$501$$ 0 0
$$502$$ −43.4983 −1.94142
$$503$$ −9.32391 −0.415733 −0.207866 0.978157i $$-0.566652\pi$$
−0.207866 + 0.978157i $$0.566652\pi$$
$$504$$ 0 0
$$505$$ 36.8661 1.64052
$$506$$ 41.2927 1.83569
$$507$$ 0 0
$$508$$ −16.6066 −0.736799
$$509$$ 39.6952 1.75946 0.879730 0.475473i $$-0.157723\pi$$
0.879730 + 0.475473i $$0.157723\pi$$
$$510$$ 0 0
$$511$$ 0 0
$$512$$ 18.4842 0.816892
$$513$$ 0 0
$$514$$ 28.3133 1.24885
$$515$$ −12.4111 −0.546898
$$516$$ 0 0
$$517$$ 18.5244 0.814704
$$518$$ 0 0
$$519$$ 0 0
$$520$$ 3.62721 0.159064
$$521$$ 22.3627 0.979729 0.489865 0.871798i $$-0.337046\pi$$
0.489865 + 0.871798i $$0.337046\pi$$
$$522$$ 0 0
$$523$$ 20.6550 0.903178 0.451589 0.892226i $$-0.350857\pi$$
0.451589 + 0.892226i $$0.350857\pi$$
$$524$$ 11.6655 0.509611
$$525$$ 0 0
$$526$$ 27.5139 1.19966
$$527$$ 0.729988 0.0317988
$$528$$ 0 0
$$529$$ 30.8469 1.34117
$$530$$ 12.7300 0.552955
$$531$$ 0 0
$$532$$ 0 0
$$533$$ 4.20555 0.182163
$$534$$ 0 0
$$535$$ −1.62721 −0.0703506
$$536$$ −12.9411 −0.558969
$$537$$ 0 0
$$538$$ 42.2439 1.82126
$$539$$ 0 0
$$540$$ 0 0
$$541$$ 5.62167 0.241695 0.120847 0.992671i $$-0.461439\pi$$
0.120847 + 0.992671i $$0.461439\pi$$
$$542$$ −23.1155 −0.992894
$$543$$ 0 0
$$544$$ 3.32391 0.142512
$$545$$ 15.6811 0.671705
$$546$$ 0 0
$$547$$ −10.3970 −0.444542 −0.222271 0.974985i $$-0.571347\pi$$
−0.222271 + 0.974985i $$0.571347\pi$$
$$548$$ 8.06949 0.344711
$$549$$ 0 0
$$550$$ −16.4111 −0.699772
$$551$$ 6.74412 0.287309
$$552$$ 0 0
$$553$$ 0 0
$$554$$ 14.7300 0.625817
$$555$$ 0 0
$$556$$ 14.9200 0.632747
$$557$$ −14.6550 −0.620951 −0.310475 0.950581i $$-0.600488\pi$$
−0.310475 + 0.950581i $$0.600488\pi$$
$$558$$ 0 0
$$559$$ −6.75971 −0.285905
$$560$$ 0 0
$$561$$ 0 0
$$562$$ −34.5189 −1.45609
$$563$$ −24.7456 −1.04290 −0.521451 0.853281i $$-0.674609\pi$$
−0.521451 + 0.853281i $$0.674609\pi$$
$$564$$ 0 0
$$565$$ −15.3225 −0.644621
$$566$$ −20.2156 −0.849725
$$567$$ 0 0
$$568$$ −11.2544 −0.472225
$$569$$ 20.5330 0.860789 0.430395 0.902641i $$-0.358374\pi$$
0.430395 + 0.902641i $$0.358374\pi$$
$$570$$ 0 0
$$571$$ −41.8953 −1.75326 −0.876631 0.481163i $$-0.840214\pi$$
−0.876631 + 0.481163i $$0.840214\pi$$
$$572$$ 4.00000 0.167248
$$573$$ 0 0
$$574$$ 0 0
$$575$$ −21.4005 −0.892464
$$576$$ 0 0
$$577$$ 20.1744 0.839870 0.419935 0.907554i $$-0.362053\pi$$
0.419935 + 0.907554i $$0.362053\pi$$
$$578$$ −30.3325 −1.26167
$$579$$ 0 0
$$580$$ −30.0666 −1.24845
$$581$$ 0 0
$$582$$ 0 0
$$583$$ −7.74055 −0.320581
$$584$$ −3.02061 −0.124994
$$585$$ 0 0
$$586$$ −25.7094 −1.06204
$$587$$ −18.7441 −0.773653 −0.386826 0.922153i $$-0.626429\pi$$
−0.386826 + 0.922153i $$0.626429\pi$$
$$588$$ 0 0
$$589$$ 1.13249 0.0466636
$$590$$ −22.8122 −0.939162
$$591$$ 0 0
$$592$$ 30.2439 1.24302
$$593$$ −2.98084 −0.122409 −0.0612043 0.998125i $$-0.519494\pi$$
−0.0612043 + 0.998125i $$0.519494\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ −10.9894 −0.450145
$$597$$ 0 0
$$598$$ 13.3083 0.544218
$$599$$ −7.47411 −0.305384 −0.152692 0.988274i $$-0.548794\pi$$
−0.152692 + 0.988274i $$0.548794\pi$$
$$600$$ 0 0
$$601$$ −21.4700 −0.875780 −0.437890 0.899028i $$-0.644274\pi$$
−0.437890 + 0.899028i $$0.644274\pi$$
$$602$$ 0 0
$$603$$ 0 0
$$604$$ −15.4499 −0.628649
$$605$$ −3.86248 −0.157032
$$606$$ 0 0
$$607$$ 22.9044 0.929660 0.464830 0.885400i $$-0.346116\pi$$
0.464830 + 0.885400i $$0.346116\pi$$
$$608$$ 5.15667 0.209131
$$609$$ 0 0
$$610$$ 10.2056 0.413211
$$611$$ 5.97028 0.241532
$$612$$ 0 0
$$613$$ −20.1461 −0.813694 −0.406847 0.913496i $$-0.633372\pi$$
−0.406847 + 0.913496i $$0.633372\pi$$
$$614$$ −24.5910 −0.992413
$$615$$ 0 0
$$616$$ 0 0
$$617$$ 13.7844 0.554939 0.277470 0.960734i $$-0.410504\pi$$
0.277470 + 0.960734i $$0.410504\pi$$
$$618$$ 0 0
$$619$$ 19.6655 0.790424 0.395212 0.918590i $$-0.370671\pi$$
0.395212 + 0.918590i $$0.370671\pi$$
$$620$$ −5.04888 −0.202768
$$621$$ 0 0
$$622$$ 0.773841 0.0310282
$$623$$ 0 0
$$624$$ 0 0
$$625$$ −31.0766 −1.24307
$$626$$ 32.9200 1.31575
$$627$$ 0 0
$$628$$ 16.5371 0.659903
$$629$$ 3.22616 0.128635
$$630$$ 0 0
$$631$$ −16.1672 −0.643608 −0.321804 0.946806i $$-0.604289\pi$$
−0.321804 + 0.946806i $$0.604289\pi$$
$$632$$ 17.4600 0.694521
$$633$$ 0 0
$$634$$ −17.0872 −0.678619
$$635$$ −36.2439 −1.43829
$$636$$ 0 0
$$637$$ 0 0
$$638$$ 46.6449 1.84669
$$639$$ 0 0
$$640$$ 27.1849 1.07458
$$641$$ 29.0036 1.14557 0.572786 0.819705i $$-0.305863\pi$$
0.572786 + 0.819705i $$0.305863\pi$$
$$642$$ 0 0
$$643$$ 39.2233 1.54681 0.773407 0.633910i $$-0.218551\pi$$
0.773407 + 0.633910i $$0.218551\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0.773841 0.0304464
$$647$$ −11.9844 −0.471156 −0.235578 0.971855i $$-0.575698\pi$$
−0.235578 + 0.971855i $$0.575698\pi$$
$$648$$ 0 0
$$649$$ 13.8711 0.544487
$$650$$ −5.28917 −0.207458
$$651$$ 0 0
$$652$$ −17.3522 −0.679564
$$653$$ −45.3311 −1.77394 −0.886971 0.461826i $$-0.847194\pi$$
−0.886971 + 0.461826i $$0.847194\pi$$
$$654$$ 0 0
$$655$$ 25.4600 0.994804
$$656$$ 20.6761 0.807266
$$657$$ 0 0
$$658$$ 0 0
$$659$$ −6.12193 −0.238477 −0.119238 0.992866i $$-0.538045\pi$$
−0.119238 + 0.992866i $$0.538045\pi$$
$$660$$ 0 0
$$661$$ 27.5280 1.07072 0.535358 0.844625i $$-0.320177\pi$$
0.535358 + 0.844625i $$0.320177\pi$$
$$662$$ −31.5577 −1.22653
$$663$$ 0 0
$$664$$ −21.2444 −0.824442
$$665$$ 0 0
$$666$$ 0 0
$$667$$ 60.8263 2.35520
$$668$$ −2.61665 −0.101241
$$669$$ 0 0
$$670$$ 51.2233 1.97893
$$671$$ −6.20555 −0.239563
$$672$$ 0 0
$$673$$ 27.9547 1.07757 0.538787 0.842442i $$-0.318883\pi$$
0.538787 + 0.842442i $$0.318883\pi$$
$$674$$ 39.9945 1.54053
$$675$$ 0 0
$$676$$ 1.28917 0.0495834
$$677$$ −12.6605 −0.486583 −0.243291 0.969953i $$-0.578227\pi$$
−0.243291 + 0.969953i $$0.578227\pi$$
$$678$$ 0 0
$$679$$ 0 0
$$680$$ 1.90225 0.0729479
$$681$$ 0 0
$$682$$ 7.83276 0.299932
$$683$$ 28.3033 1.08300 0.541498 0.840702i $$-0.317857\pi$$
0.541498 + 0.840702i $$0.317857\pi$$
$$684$$ 0 0
$$685$$ 17.6116 0.672906
$$686$$ 0 0
$$687$$ 0 0
$$688$$ −33.2333 −1.26701
$$689$$ −2.49472 −0.0950412
$$690$$ 0 0
$$691$$ −12.2353 −0.465452 −0.232726 0.972542i $$-0.574764\pi$$
−0.232726 + 0.972542i $$0.574764\pi$$
$$692$$ 26.1672 0.994729
$$693$$ 0 0
$$694$$ −45.9789 −1.74533
$$695$$ 32.5628 1.23518
$$696$$ 0 0
$$697$$ 2.20555 0.0835412
$$698$$ 10.3472 0.391646
$$699$$ 0 0
$$700$$ 0 0
$$701$$ −51.0419 −1.92783 −0.963913 0.266219i $$-0.914226\pi$$
−0.963913 + 0.266219i $$0.914226\pi$$
$$702$$ 0 0
$$703$$ 5.00502 0.188768
$$704$$ 5.15667 0.194349
$$705$$ 0 0
$$706$$ −52.0071 −1.95731
$$707$$ 0 0
$$708$$ 0 0
$$709$$ 42.5910 1.59954 0.799770 0.600307i $$-0.204955\pi$$
0.799770 + 0.600307i $$0.204955\pi$$
$$710$$ 44.5472 1.67183
$$711$$ 0 0
$$712$$ 13.7250 0.514365
$$713$$ 10.2141 0.382523
$$714$$ 0 0
$$715$$ 8.72999 0.326483
$$716$$ 14.1855 0.530135
$$717$$ 0 0
$$718$$ −20.0283 −0.747448
$$719$$ −42.4933 −1.58473 −0.792366 0.610046i $$-0.791151\pi$$
−0.792366 + 0.610046i $$0.791151\pi$$
$$720$$ 0 0
$$721$$ 0 0
$$722$$ −33.2580 −1.23773
$$723$$ 0 0
$$724$$ −0.891685 −0.0331392
$$725$$ −24.1744 −0.897814
$$726$$ 0 0
$$727$$ 3.75614 0.139307 0.0696537 0.997571i $$-0.477811\pi$$
0.0696537 + 0.997571i $$0.477811\pi$$
$$728$$ 0 0
$$729$$ 0 0
$$730$$ 11.9561 0.442517
$$731$$ −3.54505 −0.131118
$$732$$ 0 0
$$733$$ −45.7819 −1.69099 −0.845497 0.533980i $$-0.820696\pi$$
−0.845497 + 0.533980i $$0.820696\pi$$
$$734$$ −49.5960 −1.83062
$$735$$ 0 0
$$736$$ 46.5089 1.71434
$$737$$ −31.1466 −1.14730
$$738$$ 0 0
$$739$$ −14.0539 −0.516981 −0.258491 0.966014i $$-0.583225\pi$$
−0.258491 + 0.966014i $$0.583225\pi$$
$$740$$ −22.3133 −0.820255
$$741$$ 0 0
$$742$$ 0 0
$$743$$ −4.74557 −0.174098 −0.0870491 0.996204i $$-0.527744\pi$$
−0.0870491 + 0.996204i $$0.527744\pi$$
$$744$$ 0 0
$$745$$ −23.9844 −0.878721
$$746$$ 29.2827 1.07212
$$747$$ 0 0
$$748$$ 2.09775 0.0767014
$$749$$ 0 0
$$750$$ 0 0
$$751$$ 36.1008 1.31734 0.658669 0.752433i $$-0.271120\pi$$
0.658669 + 0.752433i $$0.271120\pi$$
$$752$$ 29.3522 1.07036
$$753$$ 0 0
$$754$$ 15.0333 0.547480
$$755$$ −33.7194 −1.22718
$$756$$ 0 0
$$757$$ −1.03474 −0.0376084 −0.0188042 0.999823i $$-0.505986\pi$$
−0.0188042 + 0.999823i $$0.505986\pi$$
$$758$$ 47.3905 1.72130
$$759$$ 0 0
$$760$$ 2.95112 0.107049
$$761$$ 29.8414 1.08175 0.540874 0.841104i $$-0.318094\pi$$
0.540874 + 0.841104i $$0.318094\pi$$
$$762$$ 0 0
$$763$$ 0 0
$$764$$ −10.0978 −0.365324
$$765$$ 0 0
$$766$$ −38.1744 −1.37930
$$767$$ 4.47054 0.161422
$$768$$ 0 0
$$769$$ −23.6358 −0.852329 −0.426164 0.904646i $$-0.640136\pi$$
−0.426164 + 0.904646i $$0.640136\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ −15.7350 −0.566315
$$773$$ 13.0278 0.468576 0.234288 0.972167i $$-0.424724\pi$$
0.234288 + 0.972167i $$0.424724\pi$$
$$774$$ 0 0
$$775$$ −4.05944 −0.145819
$$776$$ −1.52946 −0.0549045
$$777$$ 0 0
$$778$$ 39.1849 1.40485
$$779$$ 3.42166 0.122594
$$780$$ 0 0
$$781$$ −27.0872 −0.969256
$$782$$ 6.97939 0.249583
$$783$$ 0 0
$$784$$ 0 0
$$785$$ 36.0922 1.28819
$$786$$ 0 0
$$787$$ −46.2141 −1.64736 −0.823678 0.567058i $$-0.808082\pi$$
−0.823678 + 0.567058i $$0.808082\pi$$
$$788$$ 24.2650 0.864404
$$789$$ 0 0
$$790$$ −69.1099 −2.45882
$$791$$ 0 0
$$792$$ 0 0
$$793$$ −2.00000 −0.0710221
$$794$$ 50.2283 1.78253
$$795$$ 0 0
$$796$$ −27.8610 −0.987508
$$797$$ 53.1155 1.88145 0.940723 0.339176i $$-0.110148\pi$$
0.940723 + 0.339176i $$0.110148\pi$$
$$798$$ 0 0
$$799$$ 3.13104 0.110768
$$800$$ −18.4842 −0.653514
$$801$$ 0 0
$$802$$ 4.67609 0.165118
$$803$$ −7.27001 −0.256553
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 2.52444 0.0889195
$$807$$ 0 0
$$808$$ −16.8917 −0.594247
$$809$$ 54.4635 1.91484 0.957418 0.288705i $$-0.0932246\pi$$
0.957418 + 0.288705i $$0.0932246\pi$$
$$810$$ 0 0
$$811$$ −38.0978 −1.33779 −0.668897 0.743356i $$-0.733233\pi$$
−0.668897 + 0.743356i $$0.733233\pi$$
$$812$$ 0 0
$$813$$ 0 0
$$814$$ 34.6167 1.21331
$$815$$ −37.8711 −1.32657
$$816$$ 0 0
$$817$$ −5.49974 −0.192412
$$818$$ −27.4161 −0.958582
$$819$$ 0 0
$$820$$ −15.2544 −0.532708
$$821$$ 2.30330 0.0803858 0.0401929 0.999192i $$-0.487203\pi$$
0.0401929 + 0.999192i $$0.487203\pi$$
$$822$$ 0 0
$$823$$ 23.6172 0.823243 0.411621 0.911355i $$-0.364963\pi$$
0.411621 + 0.911355i $$0.364963\pi$$
$$824$$ 5.68665 0.198104
$$825$$ 0 0
$$826$$ 0 0
$$827$$ 48.1643 1.67484 0.837419 0.546562i $$-0.184064\pi$$
0.837419 + 0.546562i $$0.184064\pi$$
$$828$$ 0 0
$$829$$ −13.0716 −0.453996 −0.226998 0.973895i $$-0.572891\pi$$
−0.226998 + 0.973895i $$0.572891\pi$$
$$830$$ 84.0893 2.91878
$$831$$ 0 0
$$832$$ 1.66196 0.0576179
$$833$$ 0 0
$$834$$ 0 0
$$835$$ −5.71083 −0.197631
$$836$$ 3.25443 0.112557
$$837$$ 0 0
$$838$$ −18.1260 −0.626153
$$839$$ −17.6756 −0.610229 −0.305114 0.952316i $$-0.598695\pi$$
−0.305114 + 0.952316i $$0.598695\pi$$
$$840$$ 0 0
$$841$$ 39.7103 1.36932
$$842$$ −47.0177 −1.62034
$$843$$ 0 0
$$844$$ −22.4011 −0.771076
$$845$$ 2.81361 0.0967910
$$846$$ 0 0
$$847$$ 0 0
$$848$$ −12.2650 −0.421181
$$849$$ 0 0
$$850$$ −2.77384 −0.0951420
$$851$$ 45.1411 1.54742
$$852$$ 0 0
$$853$$ 5.48970 0.187964 0.0939818 0.995574i $$-0.470040\pi$$
0.0939818 + 0.995574i $$0.470040\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0.745574 0.0254832
$$857$$ −11.0489 −0.377422 −0.188711 0.982033i $$-0.560431\pi$$
−0.188711 + 0.982033i $$0.560431\pi$$
$$858$$ 0 0
$$859$$ −45.2616 −1.54430 −0.772152 0.635437i $$-0.780820\pi$$
−0.772152 + 0.635437i $$0.780820\pi$$
$$860$$ 24.5189 0.836087
$$861$$ 0 0
$$862$$ −55.6344 −1.89491
$$863$$ 5.90225 0.200915 0.100457 0.994941i $$-0.467969\pi$$
0.100457 + 0.994941i $$0.467969\pi$$
$$864$$ 0 0
$$865$$ 57.1099 1.94180
$$866$$ 6.38283 0.216898
$$867$$ 0 0
$$868$$ 0 0
$$869$$ 42.0227 1.42552
$$870$$ 0 0
$$871$$ −10.0383 −0.340135
$$872$$ −7.18494 −0.243313
$$873$$ 0 0
$$874$$ 10.8277 0.366254
$$875$$ 0 0
$$876$$ 0 0
$$877$$ −4.90727 −0.165707 −0.0828534 0.996562i $$-0.526403\pi$$
−0.0828534 + 0.996562i $$0.526403\pi$$
$$878$$ −58.6732 −1.98012
$$879$$ 0 0
$$880$$ 42.9200 1.44683
$$881$$ 44.2822 1.49190 0.745952 0.665999i $$-0.231995\pi$$
0.745952 + 0.665999i $$0.231995\pi$$
$$882$$ 0 0
$$883$$ −58.8605 −1.98081 −0.990407 0.138181i $$-0.955874\pi$$
−0.990407 + 0.138181i $$0.955874\pi$$
$$884$$ 0.676089 0.0227393
$$885$$ 0 0
$$886$$ −28.0127 −0.941104
$$887$$ −10.1289 −0.340096 −0.170048 0.985436i $$-0.554392\pi$$
−0.170048 + 0.985436i $$0.554392\pi$$
$$888$$ 0 0
$$889$$ 0 0
$$890$$ −54.3260 −1.82101
$$891$$ 0 0
$$892$$ 13.5989 0.455326
$$893$$ 4.85746 0.162549
$$894$$ 0 0
$$895$$ 30.9597 1.03487
$$896$$ 0 0
$$897$$ 0 0
$$898$$ 26.2439 0.875769
$$899$$ 11.5381 0.384816
$$900$$ 0 0
$$901$$ −1.30833 −0.0435866
$$902$$ 23.6655 0.787976
$$903$$ 0 0
$$904$$ 7.02061 0.233502
$$905$$ −1.94610 −0.0646906
$$906$$ 0 0
$$907$$ 37.9547 1.26026 0.630132 0.776488i $$-0.283001\pi$$
0.630132 + 0.776488i $$0.283001\pi$$
$$908$$ −8.96117 −0.297387
$$909$$ 0 0
$$910$$ 0 0
$$911$$ −5.57477 −0.184700 −0.0923501 0.995727i $$-0.529438\pi$$
−0.0923501 + 0.995727i $$0.529438\pi$$
$$912$$ 0 0
$$913$$ −51.1310 −1.69219
$$914$$ −62.8787 −2.07984
$$915$$ 0 0
$$916$$ 27.1849 0.898216
$$917$$ 0 0
$$918$$ 0 0
$$919$$ 15.7844 0.520679 0.260340 0.965517i $$-0.416165\pi$$
0.260340 + 0.965517i $$0.416165\pi$$
$$920$$ 26.6167 0.877525
$$921$$ 0 0
$$922$$ 22.7427 0.748990
$$923$$ −8.72999 −0.287351
$$924$$ 0 0
$$925$$ −17.9406 −0.589882
$$926$$ 22.0383 0.724224
$$927$$ 0 0
$$928$$ 52.5371 1.72462
$$929$$ −45.2630 −1.48503 −0.742516 0.669829i $$-0.766368\pi$$
−0.742516 + 0.669829i $$0.766368\pi$$
$$930$$ 0 0
$$931$$ 0 0
$$932$$ −7.84281 −0.256900
$$933$$ 0 0
$$934$$ −67.1638 −2.19767
$$935$$ 4.57834 0.149728
$$936$$ 0 0
$$937$$ −53.6188 −1.75165 −0.875824 0.482630i $$-0.839682\pi$$
−0.875824 + 0.482630i $$0.839682\pi$$
$$938$$ 0 0
$$939$$ 0 0
$$940$$ −21.6555 −0.706324
$$941$$ 20.7753 0.677255 0.338628 0.940920i $$-0.390037\pi$$
0.338628 + 0.940920i $$0.390037\pi$$
$$942$$ 0 0
$$943$$ 30.8605 1.00496
$$944$$ 21.9789 0.715351
$$945$$ 0 0
$$946$$ −38.0383 −1.23673
$$947$$ 10.8605 0.352919 0.176460 0.984308i $$-0.443535\pi$$
0.176460 + 0.984308i $$0.443535\pi$$
$$948$$ 0 0
$$949$$ −2.34307 −0.0760592
$$950$$ −4.30330 −0.139618
$$951$$ 0 0
$$952$$ 0 0
$$953$$ 25.7180 0.833087 0.416543 0.909116i $$-0.363241\pi$$
0.416543 + 0.909116i $$0.363241\pi$$
$$954$$ 0 0
$$955$$ −22.0383 −0.713143
$$956$$ 18.3133 0.592296
$$957$$ 0 0
$$958$$ 21.7789 0.703643
$$959$$ 0 0
$$960$$ 0 0
$$961$$ −29.0625 −0.937500
$$962$$ 11.1567 0.359706
$$963$$ 0 0
$$964$$ −10.8816 −0.350474
$$965$$ −34.3416 −1.10550
$$966$$ 0 0
$$967$$ 33.5038 1.07741 0.538705 0.842494i $$-0.318914\pi$$
0.538705 + 0.842494i $$0.318914\pi$$
$$968$$ 1.76975 0.0568820
$$969$$ 0 0
$$970$$ 6.05390 0.194379
$$971$$ 2.03831 0.0654126 0.0327063 0.999465i $$-0.489587\pi$$
0.0327063 + 0.999465i $$0.489587\pi$$
$$972$$ 0 0
$$973$$ 0 0
$$974$$ 20.1643 0.646107
$$975$$ 0 0
$$976$$ −9.83276 −0.314739
$$977$$ −15.1411 −0.484406 −0.242203 0.970226i $$-0.577870\pi$$
−0.242203 + 0.970226i $$0.577870\pi$$
$$978$$ 0 0
$$979$$ 33.0333 1.05575
$$980$$ 0 0
$$981$$ 0 0
$$982$$ −0.107798 −0.00343998
$$983$$ 49.3124 1.57282 0.786411 0.617704i $$-0.211937\pi$$
0.786411 + 0.617704i $$0.211937\pi$$
$$984$$ 0 0
$$985$$ 52.9583 1.68739
$$986$$ 7.88403 0.251079
$$987$$ 0 0
$$988$$ 1.04888 0.0333692
$$989$$ −49.6030 −1.57728
$$990$$ 0 0
$$991$$ 5.43171 0.172544 0.0862720 0.996272i $$-0.472505\pi$$
0.0862720 + 0.996272i $$0.472505\pi$$
$$992$$ 8.82220 0.280105
$$993$$ 0 0
$$994$$ 0 0
$$995$$ −60.8066 −1.92770
$$996$$ 0 0
$$997$$ 53.6061 1.69772 0.848861 0.528616i $$-0.177289\pi$$
0.848861 + 0.528616i $$0.177289\pi$$
$$998$$ 18.6761 0.591181
$$999$$ 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 5733.2.a.x.1.3 3
3.2 odd 2 637.2.a.j.1.1 3
7.6 odd 2 819.2.a.i.1.3 3
21.2 odd 6 637.2.e.i.508.3 6
21.5 even 6 637.2.e.j.508.3 6
21.11 odd 6 637.2.e.i.79.3 6
21.17 even 6 637.2.e.j.79.3 6
21.20 even 2 91.2.a.d.1.1 3
39.38 odd 2 8281.2.a.bg.1.3 3
84.83 odd 2 1456.2.a.t.1.3 3
105.104 even 2 2275.2.a.m.1.3 3
168.83 odd 2 5824.2.a.bs.1.1 3
168.125 even 2 5824.2.a.by.1.3 3
273.83 odd 4 1183.2.c.f.337.5 6
273.125 odd 4 1183.2.c.f.337.2 6
273.272 even 2 1183.2.a.i.1.3 3

By twisted newform
Twist Min Dim Char Parity Ord Type
91.2.a.d.1.1 3 21.20 even 2
637.2.a.j.1.1 3 3.2 odd 2
637.2.e.i.79.3 6 21.11 odd 6
637.2.e.i.508.3 6 21.2 odd 6
637.2.e.j.79.3 6 21.17 even 6
637.2.e.j.508.3 6 21.5 even 6
819.2.a.i.1.3 3 7.6 odd 2
1183.2.a.i.1.3 3 273.272 even 2
1183.2.c.f.337.2 6 273.125 odd 4
1183.2.c.f.337.5 6 273.83 odd 4
1456.2.a.t.1.3 3 84.83 odd 2
2275.2.a.m.1.3 3 105.104 even 2
5733.2.a.x.1.3 3 1.1 even 1 trivial
5824.2.a.bs.1.1 3 168.83 odd 2
5824.2.a.by.1.3 3 168.125 even 2
8281.2.a.bg.1.3 3 39.38 odd 2