# Properties

 Label 1521.2.a.p.1.2 Level $1521$ Weight $2$ Character 1521.1 Self dual yes Analytic conductor $12.145$ Analytic rank $0$ Dimension $3$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1521 = 3^{2} \cdot 13^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1521.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$12.1452461474$$ Analytic rank: $$0$$ Dimension: $$3$$ Coefficient field: $$\Q(\zeta_{14})^+$$ Defining polynomial: $$x^{3} - x^{2} - 2 x + 1$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 507) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.2 Root $$0.445042$$ of defining polynomial Character $$\chi$$ $$=$$ 1521.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-0.445042 q^{2} -1.80194 q^{4} +0.246980 q^{5} +1.75302 q^{7} +1.69202 q^{8} +O(q^{10})$$ $$q-0.445042 q^{2} -1.80194 q^{4} +0.246980 q^{5} +1.75302 q^{7} +1.69202 q^{8} -0.109916 q^{10} -5.65279 q^{11} -0.780167 q^{14} +2.85086 q^{16} +3.80194 q^{17} +5.58211 q^{19} -0.445042 q^{20} +2.51573 q^{22} -8.34481 q^{23} -4.93900 q^{25} -3.15883 q^{28} +5.93900 q^{29} +5.26875 q^{31} -4.65279 q^{32} -1.69202 q^{34} +0.432960 q^{35} +3.19806 q^{37} -2.48427 q^{38} +0.417895 q^{40} -0.445042 q^{41} +1.71379 q^{43} +10.1860 q^{44} +3.71379 q^{46} +6.73556 q^{47} -3.92692 q^{49} +2.19806 q^{50} +1.06100 q^{53} -1.39612 q^{55} +2.96615 q^{56} -2.64310 q^{58} +13.7017 q^{59} -8.51573 q^{61} -2.34481 q^{62} -3.63102 q^{64} +5.96077 q^{67} -6.85086 q^{68} -0.192685 q^{70} +5.71917 q^{71} +7.35690 q^{73} -1.42327 q^{74} -10.0586 q^{76} -9.90946 q^{77} +4.45473 q^{79} +0.704103 q^{80} +0.198062 q^{82} +10.1860 q^{83} +0.939001 q^{85} -0.762709 q^{86} -9.56465 q^{88} -0.137063 q^{89} +15.0368 q^{92} -2.99761 q^{94} +1.37867 q^{95} -13.6896 q^{97} +1.74764 q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3q - q^{2} - q^{4} - 4q^{5} + 10q^{7} + O(q^{10})$$ $$3q - q^{2} - q^{4} - 4q^{5} + 10q^{7} - q^{10} + q^{11} - q^{14} - 5q^{16} + 7q^{17} + 11q^{19} - q^{20} - 5q^{22} - 2q^{23} - 5q^{25} - q^{28} + 8q^{29} + 8q^{31} + 4q^{32} - 18q^{35} + 14q^{37} - 20q^{38} + 7q^{40} - q^{41} - 3q^{43} + 16q^{44} + 3q^{46} + 9q^{47} + 17q^{49} + 11q^{50} + 13q^{53} - 13q^{55} - 7q^{56} - 12q^{58} + 14q^{59} - 13q^{61} + 16q^{62} + 4q^{64} + 5q^{67} - 7q^{68} - 8q^{70} + 6q^{71} + 18q^{73} - 7q^{74} + q^{76} + 15q^{77} - 9q^{79} + 16q^{80} + 5q^{82} + 16q^{83} - 7q^{85} + 15q^{86} - 7q^{88} + 5q^{89} + 17q^{92} + 32q^{94} - 3q^{95} + 5q^{97} + 13q^{98} + O(q^{100})$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ −0.445042 −0.314692 −0.157346 0.987544i $$-0.550294\pi$$
−0.157346 + 0.987544i $$0.550294\pi$$
$$3$$ 0 0
$$4$$ −1.80194 −0.900969
$$5$$ 0.246980 0.110453 0.0552263 0.998474i $$-0.482412\pi$$
0.0552263 + 0.998474i $$0.482412\pi$$
$$6$$ 0 0
$$7$$ 1.75302 0.662579 0.331290 0.943529i $$-0.392516\pi$$
0.331290 + 0.943529i $$0.392516\pi$$
$$8$$ 1.69202 0.598220
$$9$$ 0 0
$$10$$ −0.109916 −0.0347586
$$11$$ −5.65279 −1.70438 −0.852191 0.523232i $$-0.824726\pi$$
−0.852191 + 0.523232i $$0.824726\pi$$
$$12$$ 0 0
$$13$$ 0 0
$$14$$ −0.780167 −0.208509
$$15$$ 0 0
$$16$$ 2.85086 0.712714
$$17$$ 3.80194 0.922105 0.461053 0.887373i $$-0.347472\pi$$
0.461053 + 0.887373i $$0.347472\pi$$
$$18$$ 0 0
$$19$$ 5.58211 1.28062 0.640311 0.768115i $$-0.278805\pi$$
0.640311 + 0.768115i $$0.278805\pi$$
$$20$$ −0.445042 −0.0995144
$$21$$ 0 0
$$22$$ 2.51573 0.536355
$$23$$ −8.34481 −1.74001 −0.870007 0.493039i $$-0.835886\pi$$
−0.870007 + 0.493039i $$0.835886\pi$$
$$24$$ 0 0
$$25$$ −4.93900 −0.987800
$$26$$ 0 0
$$27$$ 0 0
$$28$$ −3.15883 −0.596963
$$29$$ 5.93900 1.10284 0.551422 0.834226i $$-0.314085\pi$$
0.551422 + 0.834226i $$0.314085\pi$$
$$30$$ 0 0
$$31$$ 5.26875 0.946295 0.473148 0.880983i $$-0.343118\pi$$
0.473148 + 0.880983i $$0.343118\pi$$
$$32$$ −4.65279 −0.822505
$$33$$ 0 0
$$34$$ −1.69202 −0.290179
$$35$$ 0.432960 0.0731836
$$36$$ 0 0
$$37$$ 3.19806 0.525758 0.262879 0.964829i $$-0.415328\pi$$
0.262879 + 0.964829i $$0.415328\pi$$
$$38$$ −2.48427 −0.403002
$$39$$ 0 0
$$40$$ 0.417895 0.0660750
$$41$$ −0.445042 −0.0695039 −0.0347519 0.999396i $$-0.511064\pi$$
−0.0347519 + 0.999396i $$0.511064\pi$$
$$42$$ 0 0
$$43$$ 1.71379 0.261351 0.130675 0.991425i $$-0.458285\pi$$
0.130675 + 0.991425i $$0.458285\pi$$
$$44$$ 10.1860 1.53559
$$45$$ 0 0
$$46$$ 3.71379 0.547569
$$47$$ 6.73556 0.982483 0.491241 0.871024i $$-0.336543\pi$$
0.491241 + 0.871024i $$0.336543\pi$$
$$48$$ 0 0
$$49$$ −3.92692 −0.560988
$$50$$ 2.19806 0.310853
$$51$$ 0 0
$$52$$ 0 0
$$53$$ 1.06100 0.145739 0.0728697 0.997341i $$-0.476784\pi$$
0.0728697 + 0.997341i $$0.476784\pi$$
$$54$$ 0 0
$$55$$ −1.39612 −0.188253
$$56$$ 2.96615 0.396368
$$57$$ 0 0
$$58$$ −2.64310 −0.347057
$$59$$ 13.7017 1.78381 0.891905 0.452222i $$-0.149369\pi$$
0.891905 + 0.452222i $$0.149369\pi$$
$$60$$ 0 0
$$61$$ −8.51573 −1.09033 −0.545164 0.838330i $$-0.683533\pi$$
−0.545164 + 0.838330i $$0.683533\pi$$
$$62$$ −2.34481 −0.297792
$$63$$ 0 0
$$64$$ −3.63102 −0.453878
$$65$$ 0 0
$$66$$ 0 0
$$67$$ 5.96077 0.728224 0.364112 0.931355i $$-0.381373\pi$$
0.364112 + 0.931355i $$0.381373\pi$$
$$68$$ −6.85086 −0.830788
$$69$$ 0 0
$$70$$ −0.192685 −0.0230303
$$71$$ 5.71917 0.678740 0.339370 0.940653i $$-0.389786\pi$$
0.339370 + 0.940653i $$0.389786\pi$$
$$72$$ 0 0
$$73$$ 7.35690 0.861060 0.430530 0.902576i $$-0.358327\pi$$
0.430530 + 0.902576i $$0.358327\pi$$
$$74$$ −1.42327 −0.165452
$$75$$ 0 0
$$76$$ −10.0586 −1.15380
$$77$$ −9.90946 −1.12929
$$78$$ 0 0
$$79$$ 4.45473 0.501196 0.250598 0.968091i $$-0.419373\pi$$
0.250598 + 0.968091i $$0.419373\pi$$
$$80$$ 0.704103 0.0787211
$$81$$ 0 0
$$82$$ 0.198062 0.0218723
$$83$$ 10.1860 1.11806 0.559028 0.829149i $$-0.311174\pi$$
0.559028 + 0.829149i $$0.311174\pi$$
$$84$$ 0 0
$$85$$ 0.939001 0.101849
$$86$$ −0.762709 −0.0822450
$$87$$ 0 0
$$88$$ −9.56465 −1.01959
$$89$$ −0.137063 −0.0145287 −0.00726434 0.999974i $$-0.502312\pi$$
−0.00726434 + 0.999974i $$0.502312\pi$$
$$90$$ 0 0
$$91$$ 0 0
$$92$$ 15.0368 1.56770
$$93$$ 0 0
$$94$$ −2.99761 −0.309180
$$95$$ 1.37867 0.141448
$$96$$ 0 0
$$97$$ −13.6896 −1.38997 −0.694986 0.719024i $$-0.744589\pi$$
−0.694986 + 0.719024i $$0.744589\pi$$
$$98$$ 1.74764 0.176539
$$99$$ 0 0
$$100$$ 8.89977 0.889977
$$101$$ 5.41119 0.538434 0.269217 0.963080i $$-0.413235\pi$$
0.269217 + 0.963080i $$0.413235\pi$$
$$102$$ 0 0
$$103$$ 13.7560 1.35542 0.677710 0.735330i $$-0.262973\pi$$
0.677710 + 0.735330i $$0.262973\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ −0.472189 −0.0458630
$$107$$ 12.8170 1.23907 0.619533 0.784970i $$-0.287322\pi$$
0.619533 + 0.784970i $$0.287322\pi$$
$$108$$ 0 0
$$109$$ 12.1468 1.16345 0.581724 0.813386i $$-0.302378\pi$$
0.581724 + 0.813386i $$0.302378\pi$$
$$110$$ 0.621334 0.0592419
$$111$$ 0 0
$$112$$ 4.99761 0.472229
$$113$$ 1.63773 0.154064 0.0770322 0.997029i $$-0.475456\pi$$
0.0770322 + 0.997029i $$0.475456\pi$$
$$114$$ 0 0
$$115$$ −2.06100 −0.192189
$$116$$ −10.7017 −0.993629
$$117$$ 0 0
$$118$$ −6.09783 −0.561351
$$119$$ 6.66487 0.610968
$$120$$ 0 0
$$121$$ 20.9541 1.90492
$$122$$ 3.78986 0.343117
$$123$$ 0 0
$$124$$ −9.49396 −0.852583
$$125$$ −2.45473 −0.219558
$$126$$ 0 0
$$127$$ 10.7995 0.958305 0.479152 0.877732i $$-0.340944\pi$$
0.479152 + 0.877732i $$0.340944\pi$$
$$128$$ 10.9215 0.965337
$$129$$ 0 0
$$130$$ 0 0
$$131$$ −0.907542 −0.0792923 −0.0396462 0.999214i $$-0.512623\pi$$
−0.0396462 + 0.999214i $$0.512623\pi$$
$$132$$ 0 0
$$133$$ 9.78554 0.848514
$$134$$ −2.65279 −0.229166
$$135$$ 0 0
$$136$$ 6.43296 0.551622
$$137$$ 9.54825 0.815762 0.407881 0.913035i $$-0.366268\pi$$
0.407881 + 0.913035i $$0.366268\pi$$
$$138$$ 0 0
$$139$$ −4.09246 −0.347118 −0.173559 0.984823i $$-0.555527\pi$$
−0.173559 + 0.984823i $$0.555527\pi$$
$$140$$ −0.780167 −0.0659362
$$141$$ 0 0
$$142$$ −2.54527 −0.213594
$$143$$ 0 0
$$144$$ 0 0
$$145$$ 1.46681 0.121812
$$146$$ −3.27413 −0.270969
$$147$$ 0 0
$$148$$ −5.76271 −0.473692
$$149$$ 15.3884 1.26066 0.630332 0.776326i $$-0.282919\pi$$
0.630332 + 0.776326i $$0.282919\pi$$
$$150$$ 0 0
$$151$$ −3.67456 −0.299032 −0.149516 0.988759i $$-0.547772\pi$$
−0.149516 + 0.988759i $$0.547772\pi$$
$$152$$ 9.44504 0.766094
$$153$$ 0 0
$$154$$ 4.41013 0.355378
$$155$$ 1.30127 0.104521
$$156$$ 0 0
$$157$$ −4.87800 −0.389307 −0.194653 0.980872i $$-0.562358\pi$$
−0.194653 + 0.980872i $$0.562358\pi$$
$$158$$ −1.98254 −0.157723
$$159$$ 0 0
$$160$$ −1.14914 −0.0908479
$$161$$ −14.6286 −1.15290
$$162$$ 0 0
$$163$$ 8.63102 0.676034 0.338017 0.941140i $$-0.390244\pi$$
0.338017 + 0.941140i $$0.390244\pi$$
$$164$$ 0.801938 0.0626208
$$165$$ 0 0
$$166$$ −4.53319 −0.351844
$$167$$ −9.46980 −0.732795 −0.366397 0.930458i $$-0.619409\pi$$
−0.366397 + 0.930458i $$0.619409\pi$$
$$168$$ 0 0
$$169$$ 0 0
$$170$$ −0.417895 −0.0320511
$$171$$ 0 0
$$172$$ −3.08815 −0.235469
$$173$$ −4.77479 −0.363021 −0.181510 0.983389i $$-0.558099\pi$$
−0.181510 + 0.983389i $$0.558099\pi$$
$$174$$ 0 0
$$175$$ −8.65817 −0.654496
$$176$$ −16.1153 −1.21474
$$177$$ 0 0
$$178$$ 0.0609989 0.00457206
$$179$$ −3.43535 −0.256770 −0.128385 0.991724i $$-0.540979\pi$$
−0.128385 + 0.991724i $$0.540979\pi$$
$$180$$ 0 0
$$181$$ −13.4862 −1.00242 −0.501210 0.865326i $$-0.667112\pi$$
−0.501210 + 0.865326i $$0.667112\pi$$
$$182$$ 0 0
$$183$$ 0 0
$$184$$ −14.1196 −1.04091
$$185$$ 0.789856 0.0580714
$$186$$ 0 0
$$187$$ −21.4916 −1.57162
$$188$$ −12.1371 −0.885186
$$189$$ 0 0
$$190$$ −0.613564 −0.0445126
$$191$$ 1.30127 0.0941569 0.0470784 0.998891i $$-0.485009\pi$$
0.0470784 + 0.998891i $$0.485009\pi$$
$$192$$ 0 0
$$193$$ −9.19567 −0.661919 −0.330959 0.943645i $$-0.607372\pi$$
−0.330959 + 0.943645i $$0.607372\pi$$
$$194$$ 6.09246 0.437413
$$195$$ 0 0
$$196$$ 7.07606 0.505433
$$197$$ −4.11231 −0.292990 −0.146495 0.989211i $$-0.546799\pi$$
−0.146495 + 0.989211i $$0.546799\pi$$
$$198$$ 0 0
$$199$$ −24.7724 −1.75607 −0.878034 0.478598i $$-0.841145\pi$$
−0.878034 + 0.478598i $$0.841145\pi$$
$$200$$ −8.35690 −0.590922
$$201$$ 0 0
$$202$$ −2.40821 −0.169441
$$203$$ 10.4112 0.730722
$$204$$ 0 0
$$205$$ −0.109916 −0.00767688
$$206$$ −6.12200 −0.426540
$$207$$ 0 0
$$208$$ 0 0
$$209$$ −31.5545 −2.18267
$$210$$ 0 0
$$211$$ −5.93900 −0.408858 −0.204429 0.978881i $$-0.565534\pi$$
−0.204429 + 0.978881i $$0.565534\pi$$
$$212$$ −1.91185 −0.131307
$$213$$ 0 0
$$214$$ −5.70410 −0.389924
$$215$$ 0.423272 0.0288669
$$216$$ 0 0
$$217$$ 9.23623 0.626996
$$218$$ −5.40581 −0.366128
$$219$$ 0 0
$$220$$ 2.51573 0.169610
$$221$$ 0 0
$$222$$ 0 0
$$223$$ 14.2010 0.950972 0.475486 0.879723i $$-0.342272\pi$$
0.475486 + 0.879723i $$0.342272\pi$$
$$224$$ −8.15644 −0.544975
$$225$$ 0 0
$$226$$ −0.728857 −0.0484829
$$227$$ −16.0073 −1.06244 −0.531221 0.847233i $$-0.678267\pi$$
−0.531221 + 0.847233i $$0.678267\pi$$
$$228$$ 0 0
$$229$$ 1.84117 0.121668 0.0608339 0.998148i $$-0.480624\pi$$
0.0608339 + 0.998148i $$0.480624\pi$$
$$230$$ 0.917231 0.0604804
$$231$$ 0 0
$$232$$ 10.0489 0.659744
$$233$$ 23.4252 1.53464 0.767318 0.641267i $$-0.221591\pi$$
0.767318 + 0.641267i $$0.221591\pi$$
$$234$$ 0 0
$$235$$ 1.66355 0.108518
$$236$$ −24.6896 −1.60716
$$237$$ 0 0
$$238$$ −2.96615 −0.192267
$$239$$ −14.6015 −0.944491 −0.472246 0.881467i $$-0.656556\pi$$
−0.472246 + 0.881467i $$0.656556\pi$$
$$240$$ 0 0
$$241$$ 8.63102 0.555973 0.277987 0.960585i $$-0.410333\pi$$
0.277987 + 0.960585i $$0.410333\pi$$
$$242$$ −9.32544 −0.599462
$$243$$ 0 0
$$244$$ 15.3448 0.982351
$$245$$ −0.969869 −0.0619627
$$246$$ 0 0
$$247$$ 0 0
$$248$$ 8.91484 0.566093
$$249$$ 0 0
$$250$$ 1.09246 0.0690931
$$251$$ 3.80194 0.239976 0.119988 0.992775i $$-0.461714\pi$$
0.119988 + 0.992775i $$0.461714\pi$$
$$252$$ 0 0
$$253$$ 47.1715 2.96565
$$254$$ −4.80625 −0.301571
$$255$$ 0 0
$$256$$ 2.40150 0.150094
$$257$$ 20.5961 1.28475 0.642375 0.766391i $$-0.277949\pi$$
0.642375 + 0.766391i $$0.277949\pi$$
$$258$$ 0 0
$$259$$ 5.60627 0.348357
$$260$$ 0 0
$$261$$ 0 0
$$262$$ 0.403894 0.0249527
$$263$$ −0.332733 −0.0205172 −0.0102586 0.999947i $$-0.503265\pi$$
−0.0102586 + 0.999947i $$0.503265\pi$$
$$264$$ 0 0
$$265$$ 0.262045 0.0160973
$$266$$ −4.35498 −0.267021
$$267$$ 0 0
$$268$$ −10.7409 −0.656107
$$269$$ −27.3032 −1.66471 −0.832353 0.554247i $$-0.813006\pi$$
−0.832353 + 0.554247i $$0.813006\pi$$
$$270$$ 0 0
$$271$$ 27.9855 1.70000 0.850000 0.526783i $$-0.176602\pi$$
0.850000 + 0.526783i $$0.176602\pi$$
$$272$$ 10.8388 0.657197
$$273$$ 0 0
$$274$$ −4.24937 −0.256714
$$275$$ 27.9191 1.68359
$$276$$ 0 0
$$277$$ −2.10321 −0.126370 −0.0631849 0.998002i $$-0.520126\pi$$
−0.0631849 + 0.998002i $$0.520126\pi$$
$$278$$ 1.82132 0.109235
$$279$$ 0 0
$$280$$ 0.732578 0.0437799
$$281$$ −27.2349 −1.62470 −0.812349 0.583172i $$-0.801811\pi$$
−0.812349 + 0.583172i $$0.801811\pi$$
$$282$$ 0 0
$$283$$ 5.28382 0.314090 0.157045 0.987591i $$-0.449803\pi$$
0.157045 + 0.987591i $$0.449803\pi$$
$$284$$ −10.3056 −0.611524
$$285$$ 0 0
$$286$$ 0 0
$$287$$ −0.780167 −0.0460518
$$288$$ 0 0
$$289$$ −2.54527 −0.149722
$$290$$ −0.652793 −0.0383333
$$291$$ 0 0
$$292$$ −13.2567 −0.775788
$$293$$ −32.6625 −1.90816 −0.954081 0.299548i $$-0.903164\pi$$
−0.954081 + 0.299548i $$0.903164\pi$$
$$294$$ 0 0
$$295$$ 3.38404 0.197027
$$296$$ 5.41119 0.314519
$$297$$ 0 0
$$298$$ −6.84846 −0.396721
$$299$$ 0 0
$$300$$ 0 0
$$301$$ 3.00431 0.173166
$$302$$ 1.63533 0.0941029
$$303$$ 0 0
$$304$$ 15.9138 0.912717
$$305$$ −2.10321 −0.120430
$$306$$ 0 0
$$307$$ −20.7614 −1.18491 −0.592457 0.805602i $$-0.701842\pi$$
−0.592457 + 0.805602i $$0.701842\pi$$
$$308$$ 17.8562 1.01745
$$309$$ 0 0
$$310$$ −0.579121 −0.0328919
$$311$$ 11.3013 0.640836 0.320418 0.947276i $$-0.396177\pi$$
0.320418 + 0.947276i $$0.396177\pi$$
$$312$$ 0 0
$$313$$ −4.27173 −0.241453 −0.120726 0.992686i $$-0.538522\pi$$
−0.120726 + 0.992686i $$0.538522\pi$$
$$314$$ 2.17092 0.122512
$$315$$ 0 0
$$316$$ −8.02715 −0.451562
$$317$$ −15.4776 −0.869307 −0.434653 0.900598i $$-0.643129\pi$$
−0.434653 + 0.900598i $$0.643129\pi$$
$$318$$ 0 0
$$319$$ −33.5719 −1.87967
$$320$$ −0.896789 −0.0501320
$$321$$ 0 0
$$322$$ 6.51035 0.362808
$$323$$ 21.2228 1.18087
$$324$$ 0 0
$$325$$ 0 0
$$326$$ −3.84117 −0.212743
$$327$$ 0 0
$$328$$ −0.753020 −0.0415786
$$329$$ 11.8076 0.650973
$$330$$ 0 0
$$331$$ 6.06829 0.333544 0.166772 0.985996i $$-0.446666\pi$$
0.166772 + 0.985996i $$0.446666\pi$$
$$332$$ −18.3545 −1.00733
$$333$$ 0 0
$$334$$ 4.21446 0.230605
$$335$$ 1.47219 0.0804343
$$336$$ 0 0
$$337$$ 12.1239 0.660432 0.330216 0.943905i $$-0.392878\pi$$
0.330216 + 0.943905i $$0.392878\pi$$
$$338$$ 0 0
$$339$$ 0 0
$$340$$ −1.69202 −0.0917627
$$341$$ −29.7832 −1.61285
$$342$$ 0 0
$$343$$ −19.1551 −1.03428
$$344$$ 2.89977 0.156345
$$345$$ 0 0
$$346$$ 2.12498 0.114240
$$347$$ −23.1497 −1.24274 −0.621371 0.783516i $$-0.713424\pi$$
−0.621371 + 0.783516i $$0.713424\pi$$
$$348$$ 0 0
$$349$$ −22.1957 −1.18811 −0.594053 0.804426i $$-0.702473\pi$$
−0.594053 + 0.804426i $$0.702473\pi$$
$$350$$ 3.85325 0.205965
$$351$$ 0 0
$$352$$ 26.3013 1.40186
$$353$$ 5.07069 0.269885 0.134943 0.990853i $$-0.456915\pi$$
0.134943 + 0.990853i $$0.456915\pi$$
$$354$$ 0 0
$$355$$ 1.41252 0.0749687
$$356$$ 0.246980 0.0130899
$$357$$ 0 0
$$358$$ 1.52888 0.0808036
$$359$$ 16.6746 0.880050 0.440025 0.897986i $$-0.354970\pi$$
0.440025 + 0.897986i $$0.354970\pi$$
$$360$$ 0 0
$$361$$ 12.1599 0.639995
$$362$$ 6.00192 0.315454
$$363$$ 0 0
$$364$$ 0 0
$$365$$ 1.81700 0.0951063
$$366$$ 0 0
$$367$$ −1.17928 −0.0615577 −0.0307789 0.999526i $$-0.509799\pi$$
−0.0307789 + 0.999526i $$0.509799\pi$$
$$368$$ −23.7899 −1.24013
$$369$$ 0 0
$$370$$ −0.351519 −0.0182746
$$371$$ 1.85995 0.0965639
$$372$$ 0 0
$$373$$ −30.0925 −1.55813 −0.779064 0.626944i $$-0.784305\pi$$
−0.779064 + 0.626944i $$0.784305\pi$$
$$374$$ 9.56465 0.494576
$$375$$ 0 0
$$376$$ 11.3967 0.587741
$$377$$ 0 0
$$378$$ 0 0
$$379$$ −19.1631 −0.984345 −0.492172 0.870498i $$-0.663797\pi$$
−0.492172 + 0.870498i $$0.663797\pi$$
$$380$$ −2.48427 −0.127440
$$381$$ 0 0
$$382$$ −0.579121 −0.0296304
$$383$$ 15.3884 0.786308 0.393154 0.919473i $$-0.371384\pi$$
0.393154 + 0.919473i $$0.371384\pi$$
$$384$$ 0 0
$$385$$ −2.44743 −0.124733
$$386$$ 4.09246 0.208301
$$387$$ 0 0
$$388$$ 24.6679 1.25232
$$389$$ 24.0315 1.21844 0.609222 0.793000i $$-0.291482\pi$$
0.609222 + 0.793000i $$0.291482\pi$$
$$390$$ 0 0
$$391$$ −31.7265 −1.60448
$$392$$ −6.64443 −0.335594
$$393$$ 0 0
$$394$$ 1.83015 0.0922016
$$395$$ 1.10023 0.0553585
$$396$$ 0 0
$$397$$ 29.6015 1.48566 0.742828 0.669482i $$-0.233484\pi$$
0.742828 + 0.669482i $$0.233484\pi$$
$$398$$ 11.0248 0.552621
$$399$$ 0 0
$$400$$ −14.0804 −0.704019
$$401$$ 21.1032 1.05384 0.526922 0.849914i $$-0.323346\pi$$
0.526922 + 0.849914i $$0.323346\pi$$
$$402$$ 0 0
$$403$$ 0 0
$$404$$ −9.75063 −0.485112
$$405$$ 0 0
$$406$$ −4.63342 −0.229953
$$407$$ −18.0780 −0.896092
$$408$$ 0 0
$$409$$ −36.0224 −1.78119 −0.890596 0.454796i $$-0.849712\pi$$
−0.890596 + 0.454796i $$0.849712\pi$$
$$410$$ 0.0489173 0.00241586
$$411$$ 0 0
$$412$$ −24.7875 −1.22119
$$413$$ 24.0194 1.18192
$$414$$ 0 0
$$415$$ 2.51573 0.123492
$$416$$ 0 0
$$417$$ 0 0
$$418$$ 14.0431 0.686869
$$419$$ −5.96854 −0.291582 −0.145791 0.989315i $$-0.546573\pi$$
−0.145791 + 0.989315i $$0.546573\pi$$
$$420$$ 0 0
$$421$$ −2.09544 −0.102126 −0.0510628 0.998695i $$-0.516261\pi$$
−0.0510628 + 0.998695i $$0.516261\pi$$
$$422$$ 2.64310 0.128664
$$423$$ 0 0
$$424$$ 1.79523 0.0871842
$$425$$ −18.7778 −0.910856
$$426$$ 0 0
$$427$$ −14.9282 −0.722429
$$428$$ −23.0954 −1.11636
$$429$$ 0 0
$$430$$ −0.188374 −0.00908418
$$431$$ 2.88577 0.139003 0.0695014 0.997582i $$-0.477859\pi$$
0.0695014 + 0.997582i $$0.477859\pi$$
$$432$$ 0 0
$$433$$ −12.6485 −0.607847 −0.303924 0.952696i $$-0.598297\pi$$
−0.303924 + 0.952696i $$0.598297\pi$$
$$434$$ −4.11051 −0.197311
$$435$$ 0 0
$$436$$ −21.8877 −1.04823
$$437$$ −46.5816 −2.22830
$$438$$ 0 0
$$439$$ −10.9269 −0.521513 −0.260757 0.965405i $$-0.583972\pi$$
−0.260757 + 0.965405i $$0.583972\pi$$
$$440$$ −2.36227 −0.112617
$$441$$ 0 0
$$442$$ 0 0
$$443$$ 19.2403 0.914133 0.457067 0.889433i $$-0.348900\pi$$
0.457067 + 0.889433i $$0.348900\pi$$
$$444$$ 0 0
$$445$$ −0.0338518 −0.00160473
$$446$$ −6.32006 −0.299264
$$447$$ 0 0
$$448$$ −6.36526 −0.300730
$$449$$ 28.8200 1.36010 0.680050 0.733166i $$-0.261958\pi$$
0.680050 + 0.733166i $$0.261958\pi$$
$$450$$ 0 0
$$451$$ 2.51573 0.118461
$$452$$ −2.95108 −0.138807
$$453$$ 0 0
$$454$$ 7.12392 0.334342
$$455$$ 0 0
$$456$$ 0 0
$$457$$ −18.0707 −0.845311 −0.422656 0.906290i $$-0.638902\pi$$
−0.422656 + 0.906290i $$0.638902\pi$$
$$458$$ −0.819396 −0.0382879
$$459$$ 0 0
$$460$$ 3.71379 0.173156
$$461$$ 7.56763 0.352460 0.176230 0.984349i $$-0.443610\pi$$
0.176230 + 0.984349i $$0.443610\pi$$
$$462$$ 0 0
$$463$$ 35.3551 1.64309 0.821545 0.570143i $$-0.193112\pi$$
0.821545 + 0.570143i $$0.193112\pi$$
$$464$$ 16.9312 0.786013
$$465$$ 0 0
$$466$$ −10.4252 −0.482938
$$467$$ −13.0000 −0.601568 −0.300784 0.953692i $$-0.597248\pi$$
−0.300784 + 0.953692i $$0.597248\pi$$
$$468$$ 0 0
$$469$$ 10.4494 0.482506
$$470$$ −0.740348 −0.0341497
$$471$$ 0 0
$$472$$ 23.1836 1.06711
$$473$$ −9.68771 −0.445441
$$474$$ 0 0
$$475$$ −27.5700 −1.26500
$$476$$ −12.0097 −0.550463
$$477$$ 0 0
$$478$$ 6.49827 0.297224
$$479$$ 25.5265 1.16633 0.583167 0.812352i $$-0.301813\pi$$
0.583167 + 0.812352i $$0.301813\pi$$
$$480$$ 0 0
$$481$$ 0 0
$$482$$ −3.84117 −0.174960
$$483$$ 0 0
$$484$$ −37.7579 −1.71627
$$485$$ −3.38106 −0.153526
$$486$$ 0 0
$$487$$ 16.0073 0.725360 0.362680 0.931914i $$-0.381862\pi$$
0.362680 + 0.931914i $$0.381862\pi$$
$$488$$ −14.4088 −0.652256
$$489$$ 0 0
$$490$$ 0.431632 0.0194992
$$491$$ −20.7385 −0.935917 −0.467959 0.883750i $$-0.655010\pi$$
−0.467959 + 0.883750i $$0.655010\pi$$
$$492$$ 0 0
$$493$$ 22.5797 1.01694
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 15.0204 0.674438
$$497$$ 10.0258 0.449719
$$498$$ 0 0
$$499$$ −8.06770 −0.361160 −0.180580 0.983560i $$-0.557797\pi$$
−0.180580 + 0.983560i $$0.557797\pi$$
$$500$$ 4.42327 0.197815
$$501$$ 0 0
$$502$$ −1.69202 −0.0755186
$$503$$ 30.2422 1.34843 0.674216 0.738534i $$-0.264482\pi$$
0.674216 + 0.738534i $$0.264482\pi$$
$$504$$ 0 0
$$505$$ 1.33645 0.0594714
$$506$$ −20.9933 −0.933266
$$507$$ 0 0
$$508$$ −19.4601 −0.863403
$$509$$ −16.0495 −0.711382 −0.355691 0.934604i $$-0.615754\pi$$
−0.355691 + 0.934604i $$0.615754\pi$$
$$510$$ 0 0
$$511$$ 12.8968 0.570520
$$512$$ −22.9119 −1.01257
$$513$$ 0 0
$$514$$ −9.16613 −0.404301
$$515$$ 3.39745 0.149710
$$516$$ 0 0
$$517$$ −38.0747 −1.67452
$$518$$ −2.49502 −0.109625
$$519$$ 0 0
$$520$$ 0 0
$$521$$ 2.69309 0.117986 0.0589931 0.998258i $$-0.481211\pi$$
0.0589931 + 0.998258i $$0.481211\pi$$
$$522$$ 0 0
$$523$$ 35.3957 1.54774 0.773872 0.633342i $$-0.218317\pi$$
0.773872 + 0.633342i $$0.218317\pi$$
$$524$$ 1.63533 0.0714399
$$525$$ 0 0
$$526$$ 0.148080 0.00645659
$$527$$ 20.0315 0.872584
$$528$$ 0 0
$$529$$ 46.6359 2.02765
$$530$$ −0.116621 −0.00506569
$$531$$ 0 0
$$532$$ −17.6329 −0.764485
$$533$$ 0 0
$$534$$ 0 0
$$535$$ 3.16554 0.136858
$$536$$ 10.0858 0.435638
$$537$$ 0 0
$$538$$ 12.1511 0.523870
$$539$$ 22.1981 0.956138
$$540$$ 0 0
$$541$$ −34.7338 −1.49332 −0.746660 0.665205i $$-0.768344\pi$$
−0.746660 + 0.665205i $$0.768344\pi$$
$$542$$ −12.4547 −0.534976
$$543$$ 0 0
$$544$$ −17.6896 −0.758437
$$545$$ 3.00000 0.128506
$$546$$ 0 0
$$547$$ −26.1183 −1.11674 −0.558368 0.829593i $$-0.688572\pi$$
−0.558368 + 0.829593i $$0.688572\pi$$
$$548$$ −17.2054 −0.734976
$$549$$ 0 0
$$550$$ −12.4252 −0.529812
$$551$$ 33.1521 1.41233
$$552$$ 0 0
$$553$$ 7.80923 0.332082
$$554$$ 0.936017 0.0397676
$$555$$ 0 0
$$556$$ 7.37435 0.312742
$$557$$ −24.7748 −1.04974 −0.524871 0.851182i $$-0.675886\pi$$
−0.524871 + 0.851182i $$0.675886\pi$$
$$558$$ 0 0
$$559$$ 0 0
$$560$$ 1.23431 0.0521590
$$561$$ 0 0
$$562$$ 12.1207 0.511280
$$563$$ −5.26098 −0.221724 −0.110862 0.993836i $$-0.535361\pi$$
−0.110862 + 0.993836i $$0.535361\pi$$
$$564$$ 0 0
$$565$$ 0.404485 0.0170168
$$566$$ −2.35152 −0.0988417
$$567$$ 0 0
$$568$$ 9.67696 0.406036
$$569$$ 33.7458 1.41470 0.707350 0.706864i $$-0.249891\pi$$
0.707350 + 0.706864i $$0.249891\pi$$
$$570$$ 0 0
$$571$$ 23.0887 0.966234 0.483117 0.875556i $$-0.339505\pi$$
0.483117 + 0.875556i $$0.339505\pi$$
$$572$$ 0 0
$$573$$ 0 0
$$574$$ 0.347207 0.0144921
$$575$$ 41.2150 1.71879
$$576$$ 0 0
$$577$$ 3.57002 0.148622 0.0743110 0.997235i $$-0.476324\pi$$
0.0743110 + 0.997235i $$0.476324\pi$$
$$578$$ 1.13275 0.0471162
$$579$$ 0 0
$$580$$ −2.64310 −0.109749
$$581$$ 17.8562 0.740801
$$582$$ 0 0
$$583$$ −5.99761 −0.248396
$$584$$ 12.4480 0.515103
$$585$$ 0 0
$$586$$ 14.5362 0.600484
$$587$$ 11.4625 0.473108 0.236554 0.971618i $$-0.423982\pi$$
0.236554 + 0.971618i $$0.423982\pi$$
$$588$$ 0 0
$$589$$ 29.4107 1.21185
$$590$$ −1.50604 −0.0620027
$$591$$ 0 0
$$592$$ 9.11721 0.374715
$$593$$ 21.8538 0.897430 0.448715 0.893675i $$-0.351882\pi$$
0.448715 + 0.893675i $$0.351882\pi$$
$$594$$ 0 0
$$595$$ 1.64609 0.0674830
$$596$$ −27.7289 −1.13582
$$597$$ 0 0
$$598$$ 0 0
$$599$$ −27.0573 −1.10553 −0.552765 0.833337i $$-0.686427\pi$$
−0.552765 + 0.833337i $$0.686427\pi$$
$$600$$ 0 0
$$601$$ 10.8780 0.443723 0.221861 0.975078i $$-0.428787\pi$$
0.221861 + 0.975078i $$0.428787\pi$$
$$602$$ −1.33704 −0.0544939
$$603$$ 0 0
$$604$$ 6.62133 0.269418
$$605$$ 5.17523 0.210403
$$606$$ 0 0
$$607$$ −29.6359 −1.20289 −0.601443 0.798916i $$-0.705407\pi$$
−0.601443 + 0.798916i $$0.705407\pi$$
$$608$$ −25.9724 −1.05332
$$609$$ 0 0
$$610$$ 0.936017 0.0378982
$$611$$ 0 0
$$612$$ 0 0
$$613$$ 10.2343 0.413360 0.206680 0.978409i $$-0.433734\pi$$
0.206680 + 0.978409i $$0.433734\pi$$
$$614$$ 9.23968 0.372883
$$615$$ 0 0
$$616$$ −16.7670 −0.675563
$$617$$ −26.2828 −1.05810 −0.529052 0.848590i $$-0.677452\pi$$
−0.529052 + 0.848590i $$0.677452\pi$$
$$618$$ 0 0
$$619$$ 29.0834 1.16896 0.584479 0.811408i $$-0.301299\pi$$
0.584479 + 0.811408i $$0.301299\pi$$
$$620$$ −2.34481 −0.0941700
$$621$$ 0 0
$$622$$ −5.02954 −0.201666
$$623$$ −0.240275 −0.00962641
$$624$$ 0 0
$$625$$ 24.0887 0.963549
$$626$$ 1.90110 0.0759833
$$627$$ 0 0
$$628$$ 8.78986 0.350753
$$629$$ 12.1588 0.484804
$$630$$ 0 0
$$631$$ 25.4480 1.01307 0.506535 0.862219i $$-0.330926\pi$$
0.506535 + 0.862219i $$0.330926\pi$$
$$632$$ 7.53750 0.299826
$$633$$ 0 0
$$634$$ 6.88816 0.273564
$$635$$ 2.66727 0.105847
$$636$$ 0 0
$$637$$ 0 0
$$638$$ 14.9409 0.591517
$$639$$ 0 0
$$640$$ 2.69740 0.106624
$$641$$ 26.7409 1.05620 0.528102 0.849181i $$-0.322904\pi$$
0.528102 + 0.849181i $$0.322904\pi$$
$$642$$ 0 0
$$643$$ −32.9614 −1.29987 −0.649935 0.759990i $$-0.725204\pi$$
−0.649935 + 0.759990i $$0.725204\pi$$
$$644$$ 26.3599 1.03872
$$645$$ 0 0
$$646$$ −9.44504 −0.371610
$$647$$ −34.4946 −1.35612 −0.678060 0.735006i $$-0.737179\pi$$
−0.678060 + 0.735006i $$0.737179\pi$$
$$648$$ 0 0
$$649$$ −77.4529 −3.04029
$$650$$ 0 0
$$651$$ 0 0
$$652$$ −15.5526 −0.609085
$$653$$ −36.1517 −1.41472 −0.707362 0.706852i $$-0.750115\pi$$
−0.707362 + 0.706852i $$0.750115\pi$$
$$654$$ 0 0
$$655$$ −0.224144 −0.00875805
$$656$$ −1.26875 −0.0495364
$$657$$ 0 0
$$658$$ −5.25487 −0.204856
$$659$$ 6.81700 0.265553 0.132776 0.991146i $$-0.457611\pi$$
0.132776 + 0.991146i $$0.457611\pi$$
$$660$$ 0 0
$$661$$ −10.8944 −0.423743 −0.211871 0.977298i $$-0.567956\pi$$
−0.211871 + 0.977298i $$0.567956\pi$$
$$662$$ −2.70065 −0.104964
$$663$$ 0 0
$$664$$ 17.2349 0.668844
$$665$$ 2.41683 0.0937206
$$666$$ 0 0
$$667$$ −49.5599 −1.91897
$$668$$ 17.0640 0.660225
$$669$$ 0 0
$$670$$ −0.655186 −0.0253120
$$671$$ 48.1377 1.85833
$$672$$ 0 0
$$673$$ 20.7385 0.799412 0.399706 0.916643i $$-0.369112\pi$$
0.399706 + 0.916643i $$0.369112\pi$$
$$674$$ −5.39565 −0.207833
$$675$$ 0 0
$$676$$ 0 0
$$677$$ 25.5786 0.983067 0.491534 0.870859i $$-0.336436\pi$$
0.491534 + 0.870859i $$0.336436\pi$$
$$678$$ 0 0
$$679$$ −23.9982 −0.920966
$$680$$ 1.58881 0.0609281
$$681$$ 0 0
$$682$$ 13.2547 0.507551
$$683$$ −21.6310 −0.827688 −0.413844 0.910348i $$-0.635814\pi$$
−0.413844 + 0.910348i $$0.635814\pi$$
$$684$$ 0 0
$$685$$ 2.35822 0.0901031
$$686$$ 8.52483 0.325479
$$687$$ 0 0
$$688$$ 4.88577 0.186268
$$689$$ 0 0
$$690$$ 0 0
$$691$$ 2.62996 0.100048 0.0500242 0.998748i $$-0.484070\pi$$
0.0500242 + 0.998748i $$0.484070\pi$$
$$692$$ 8.60388 0.327070
$$693$$ 0 0
$$694$$ 10.3026 0.391081
$$695$$ −1.01075 −0.0383401
$$696$$ 0 0
$$697$$ −1.69202 −0.0640899
$$698$$ 9.87800 0.373888
$$699$$ 0 0
$$700$$ 15.6015 0.589681
$$701$$ −40.0925 −1.51427 −0.757136 0.653258i $$-0.773402\pi$$
−0.757136 + 0.653258i $$0.773402\pi$$
$$702$$ 0 0
$$703$$ 17.8519 0.673298
$$704$$ 20.5254 0.773581
$$705$$ 0 0
$$706$$ −2.25667 −0.0849308
$$707$$ 9.48593 0.356755
$$708$$ 0 0
$$709$$ 23.2097 0.871657 0.435829 0.900030i $$-0.356455\pi$$
0.435829 + 0.900030i $$0.356455\pi$$
$$710$$ −0.628630 −0.0235921
$$711$$ 0 0
$$712$$ −0.231914 −0.00869135
$$713$$ −43.9667 −1.64657
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 6.19029 0.231342
$$717$$ 0 0
$$718$$ −7.42088 −0.276945
$$719$$ 26.0146 0.970181 0.485090 0.874464i $$-0.338787\pi$$
0.485090 + 0.874464i $$0.338787\pi$$
$$720$$ 0 0
$$721$$ 24.1146 0.898073
$$722$$ −5.41166 −0.201401
$$723$$ 0 0
$$724$$ 24.3013 0.903150
$$725$$ −29.3327 −1.08939
$$726$$ 0 0
$$727$$ −16.5472 −0.613701 −0.306851 0.951758i $$-0.599275\pi$$
−0.306851 + 0.951758i $$0.599275\pi$$
$$728$$ 0 0
$$729$$ 0 0
$$730$$ −0.808643 −0.0299292
$$731$$ 6.51573 0.240993
$$732$$ 0 0
$$733$$ 18.8750 0.697165 0.348582 0.937278i $$-0.386663\pi$$
0.348582 + 0.937278i $$0.386663\pi$$
$$734$$ 0.524827 0.0193717
$$735$$ 0 0
$$736$$ 38.8267 1.43117
$$737$$ −33.6950 −1.24117
$$738$$ 0 0
$$739$$ 47.3239 1.74084 0.870419 0.492312i $$-0.163848\pi$$
0.870419 + 0.492312i $$0.163848\pi$$
$$740$$ −1.42327 −0.0523205
$$741$$ 0 0
$$742$$ −0.827757 −0.0303879
$$743$$ 8.88769 0.326058 0.163029 0.986621i $$-0.447874\pi$$
0.163029 + 0.986621i $$0.447874\pi$$
$$744$$ 0 0
$$745$$ 3.80061 0.139244
$$746$$ 13.3924 0.490331
$$747$$ 0 0
$$748$$ 38.7265 1.41598
$$749$$ 22.4685 0.820980
$$750$$ 0 0
$$751$$ 0.710808 0.0259377 0.0129689 0.999916i $$-0.495872\pi$$
0.0129689 + 0.999916i $$0.495872\pi$$
$$752$$ 19.2021 0.700229
$$753$$ 0 0
$$754$$ 0 0
$$755$$ −0.907542 −0.0330288
$$756$$ 0 0
$$757$$ 9.78554 0.355662 0.177831 0.984061i $$-0.443092\pi$$
0.177831 + 0.984061i $$0.443092\pi$$
$$758$$ 8.52840 0.309766
$$759$$ 0 0
$$760$$ 2.33273 0.0846171
$$761$$ 18.8810 0.684435 0.342218 0.939621i $$-0.388822\pi$$
0.342218 + 0.939621i $$0.388822\pi$$
$$762$$ 0 0
$$763$$ 21.2935 0.770877
$$764$$ −2.34481 −0.0848324
$$765$$ 0 0
$$766$$ −6.84846 −0.247445
$$767$$ 0 0
$$768$$ 0 0
$$769$$ −12.4349 −0.448413 −0.224207 0.974542i $$-0.571979\pi$$
−0.224207 + 0.974542i $$0.571979\pi$$
$$770$$ 1.08921 0.0392524
$$771$$ 0 0
$$772$$ 16.5700 0.596368
$$773$$ −45.6746 −1.64280 −0.821400 0.570353i $$-0.806807\pi$$
−0.821400 + 0.570353i $$0.806807\pi$$
$$774$$ 0 0
$$775$$ −26.0224 −0.934751
$$776$$ −23.1631 −0.831508
$$777$$ 0 0
$$778$$ −10.6950 −0.383435
$$779$$ −2.48427 −0.0890082
$$780$$ 0 0
$$781$$ −32.3293 −1.15683
$$782$$ 14.1196 0.504916
$$783$$ 0 0
$$784$$ −11.1951 −0.399824
$$785$$ −1.20477 −0.0430000
$$786$$ 0 0
$$787$$ −4.51871 −0.161075 −0.0805374 0.996752i $$-0.525664\pi$$
−0.0805374 + 0.996752i $$0.525664\pi$$
$$788$$ 7.41013 0.263975
$$789$$ 0 0
$$790$$ −0.489647 −0.0174209
$$791$$ 2.87097 0.102080
$$792$$ 0 0
$$793$$ 0 0
$$794$$ −13.1739 −0.467524
$$795$$ 0 0
$$796$$ 44.6383 1.58216
$$797$$ 28.7391 1.01799 0.508996 0.860769i $$-0.330017\pi$$
0.508996 + 0.860769i $$0.330017\pi$$
$$798$$ 0 0
$$799$$ 25.6082 0.905953
$$800$$ 22.9801 0.812471
$$801$$ 0 0
$$802$$ −9.39181 −0.331636
$$803$$ −41.5870 −1.46757
$$804$$ 0 0
$$805$$ −3.61297 −0.127341
$$806$$ 0 0
$$807$$ 0 0
$$808$$ 9.15585 0.322102
$$809$$ −5.42891 −0.190870 −0.0954352 0.995436i $$-0.530424\pi$$
−0.0954352 + 0.995436i $$0.530424\pi$$
$$810$$ 0 0
$$811$$ 0.629104 0.0220908 0.0110454 0.999939i $$-0.496484\pi$$
0.0110454 + 0.999939i $$0.496484\pi$$
$$812$$ −18.7603 −0.658358
$$813$$ 0 0
$$814$$ 8.04546 0.281993
$$815$$ 2.13169 0.0746697
$$816$$ 0 0
$$817$$ 9.56657 0.334692
$$818$$ 16.0315 0.560527
$$819$$ 0 0
$$820$$ 0.198062 0.00691663
$$821$$ −36.2640 −1.26562 −0.632811 0.774307i $$-0.718099\pi$$
−0.632811 + 0.774307i $$0.718099\pi$$
$$822$$ 0 0
$$823$$ 41.7396 1.45495 0.727476 0.686133i $$-0.240693\pi$$
0.727476 + 0.686133i $$0.240693\pi$$
$$824$$ 23.2755 0.810839
$$825$$ 0 0
$$826$$ −10.6896 −0.371940
$$827$$ −38.1997 −1.32833 −0.664167 0.747584i $$-0.731214\pi$$
−0.664167 + 0.747584i $$0.731214\pi$$
$$828$$ 0 0
$$829$$ 15.9788 0.554967 0.277484 0.960730i $$-0.410500\pi$$
0.277484 + 0.960730i $$0.410500\pi$$
$$830$$ −1.11960 −0.0388621
$$831$$ 0 0
$$832$$ 0 0
$$833$$ −14.9299 −0.517290
$$834$$ 0 0
$$835$$ −2.33885 −0.0809391
$$836$$ 56.8592 1.96652
$$837$$ 0 0
$$838$$ 2.65625 0.0917587
$$839$$ 4.63879 0.160149 0.0800744 0.996789i $$-0.474484\pi$$
0.0800744 + 0.996789i $$0.474484\pi$$
$$840$$ 0 0
$$841$$ 6.27173 0.216267
$$842$$ 0.932559 0.0321381
$$843$$ 0 0
$$844$$ 10.7017 0.368368
$$845$$ 0 0
$$846$$ 0 0
$$847$$ 36.7329 1.26216
$$848$$ 3.02475 0.103870
$$849$$ 0 0
$$850$$ 8.35690 0.286639
$$851$$ −26.6872 −0.914827
$$852$$ 0 0
$$853$$ 18.3884 0.629605 0.314803 0.949157i $$-0.398062\pi$$
0.314803 + 0.949157i $$0.398062\pi$$
$$854$$ 6.64370 0.227343
$$855$$ 0 0
$$856$$ 21.6866 0.741234
$$857$$ −28.1849 −0.962778 −0.481389 0.876507i $$-0.659868\pi$$
−0.481389 + 0.876507i $$0.659868\pi$$
$$858$$ 0 0
$$859$$ 33.3957 1.13944 0.569722 0.821837i $$-0.307051\pi$$
0.569722 + 0.821837i $$0.307051\pi$$
$$860$$ −0.762709 −0.0260082
$$861$$ 0 0
$$862$$ −1.28429 −0.0437431
$$863$$ 43.0640 1.46592 0.732958 0.680274i $$-0.238139\pi$$
0.732958 + 0.680274i $$0.238139\pi$$
$$864$$ 0 0
$$865$$ −1.17928 −0.0400966
$$866$$ 5.62910 0.191285
$$867$$ 0 0
$$868$$ −16.6431 −0.564904
$$869$$ −25.1817 −0.854230
$$870$$ 0 0
$$871$$ 0 0
$$872$$ 20.5526 0.695998
$$873$$ 0 0
$$874$$ 20.7308 0.701229
$$875$$ −4.30319 −0.145474
$$876$$ 0 0
$$877$$ −30.1702 −1.01877 −0.509387 0.860537i $$-0.670128\pi$$
−0.509387 + 0.860537i $$0.670128\pi$$
$$878$$ 4.86294 0.164116
$$879$$ 0 0
$$880$$ −3.98015 −0.134171
$$881$$ 3.56273 0.120031 0.0600157 0.998197i $$-0.480885\pi$$
0.0600157 + 0.998197i $$0.480885\pi$$
$$882$$ 0 0
$$883$$ −10.2088 −0.343554 −0.171777 0.985136i $$-0.554951\pi$$
−0.171777 + 0.985136i $$0.554951\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ −8.56273 −0.287670
$$887$$ 9.33645 0.313487 0.156744 0.987639i $$-0.449900\pi$$
0.156744 + 0.987639i $$0.449900\pi$$
$$888$$ 0 0
$$889$$ 18.9318 0.634953
$$890$$ 0.0150655 0.000504996 0
$$891$$ 0 0
$$892$$ −25.5894 −0.856797
$$893$$ 37.5986 1.25819
$$894$$ 0 0
$$895$$ −0.848462 −0.0283610
$$896$$ 19.1457 0.639613
$$897$$ 0 0
$$898$$ −12.8261 −0.428013
$$899$$ 31.2911 1.04362
$$900$$ 0 0
$$901$$ 4.03385 0.134387
$$902$$ −1.11960 −0.0372788
$$903$$ 0 0
$$904$$ 2.77107 0.0921644
$$905$$ −3.33081 −0.110720
$$906$$ 0 0
$$907$$ −41.0804 −1.36405 −0.682026 0.731328i $$-0.738901\pi$$
−0.682026 + 0.731328i $$0.738901\pi$$
$$908$$ 28.8442 0.957227
$$909$$ 0 0
$$910$$ 0 0
$$911$$ 18.9705 0.628519 0.314260 0.949337i $$-0.398244\pi$$
0.314260 + 0.949337i $$0.398244\pi$$
$$912$$ 0 0
$$913$$ −57.5792 −1.90559
$$914$$ 8.04221 0.266013
$$915$$ 0 0
$$916$$ −3.31767 −0.109619
$$917$$ −1.59094 −0.0525375
$$918$$ 0 0
$$919$$ 29.0019 0.956685 0.478343 0.878173i $$-0.341238\pi$$
0.478343 + 0.878173i $$0.341238\pi$$
$$920$$ −3.48725 −0.114971
$$921$$ 0 0
$$922$$ −3.36791 −0.110916
$$923$$ 0 0
$$924$$ 0 0
$$925$$ −15.7952 −0.519344
$$926$$ −15.7345 −0.517068
$$927$$ 0 0
$$928$$ −27.6329 −0.907096
$$929$$ 50.7211 1.66410 0.832052 0.554697i $$-0.187166\pi$$
0.832052 + 0.554697i $$0.187166\pi$$
$$930$$ 0 0
$$931$$ −21.9205 −0.718415
$$932$$ −42.2107 −1.38266
$$933$$ 0 0
$$934$$ 5.78554 0.189309
$$935$$ −5.30798 −0.173589
$$936$$ 0 0
$$937$$ 51.3051 1.67606 0.838032 0.545620i $$-0.183706\pi$$
0.838032 + 0.545620i $$0.183706\pi$$
$$938$$ −4.65040 −0.151841
$$939$$ 0 0
$$940$$ −2.99761 −0.0977712
$$941$$ 34.7036 1.13131 0.565653 0.824643i $$-0.308624\pi$$
0.565653 + 0.824643i $$0.308624\pi$$
$$942$$ 0 0
$$943$$ 3.71379 0.120938
$$944$$ 39.0616 1.27135
$$945$$ 0 0
$$946$$ 4.31144 0.140177
$$947$$ 13.0127 0.422855 0.211428 0.977394i $$-0.432189\pi$$
0.211428 + 0.977394i $$0.432189\pi$$
$$948$$ 0 0
$$949$$ 0 0
$$950$$ 12.2698 0.398085
$$951$$ 0 0
$$952$$ 11.2771 0.365493
$$953$$ −26.0151 −0.842711 −0.421355 0.906896i $$-0.638445\pi$$
−0.421355 + 0.906896i $$0.638445\pi$$
$$954$$ 0 0
$$955$$ 0.321388 0.0103999
$$956$$ 26.3110 0.850957
$$957$$ 0 0
$$958$$ −11.3604 −0.367036
$$959$$ 16.7383 0.540507
$$960$$ 0 0
$$961$$ −3.24027 −0.104525
$$962$$ 0 0
$$963$$ 0 0
$$964$$ −15.5526 −0.500914
$$965$$ −2.27114 −0.0731107
$$966$$ 0 0
$$967$$ 36.6644 1.17905 0.589524 0.807751i $$-0.299315\pi$$
0.589524 + 0.807751i $$0.299315\pi$$
$$968$$ 35.4547 1.13956
$$969$$ 0 0
$$970$$ 1.50471 0.0483134
$$971$$ 37.8465 1.21455 0.607277 0.794490i $$-0.292262\pi$$
0.607277 + 0.794490i $$0.292262\pi$$
$$972$$ 0 0
$$973$$ −7.17416 −0.229993
$$974$$ −7.12392 −0.228265
$$975$$ 0 0
$$976$$ −24.2771 −0.777091
$$977$$ −28.8998 −0.924586 −0.462293 0.886727i $$-0.652973\pi$$
−0.462293 + 0.886727i $$0.652973\pi$$
$$978$$ 0 0
$$979$$ 0.774791 0.0247624
$$980$$ 1.74764 0.0558264
$$981$$ 0 0
$$982$$ 9.22952 0.294526
$$983$$ 19.3991 0.618735 0.309368 0.950942i $$-0.399883\pi$$
0.309368 + 0.950942i $$0.399883\pi$$
$$984$$ 0 0
$$985$$ −1.01566 −0.0323615
$$986$$ −10.0489 −0.320023
$$987$$ 0 0
$$988$$ 0 0
$$989$$ −14.3013 −0.454754
$$990$$ 0 0
$$991$$ −5.18300 −0.164643 −0.0823217 0.996606i $$-0.526233\pi$$
−0.0823217 + 0.996606i $$0.526233\pi$$
$$992$$ −24.5144 −0.778333
$$993$$ 0 0
$$994$$ −4.46191 −0.141523
$$995$$ −6.11828 −0.193962
$$996$$ 0 0
$$997$$ −49.3642 −1.56338 −0.781690 0.623667i $$-0.785642\pi$$
−0.781690 + 0.623667i $$0.785642\pi$$
$$998$$ 3.59047 0.113654
$$999$$ 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 1521.2.a.p.1.2 3
3.2 odd 2 507.2.a.k.1.2 yes 3
12.11 even 2 8112.2.a.cf.1.1 3
13.5 odd 4 1521.2.b.m.1351.4 6
13.8 odd 4 1521.2.b.m.1351.3 6
13.12 even 2 1521.2.a.q.1.2 3
39.2 even 12 507.2.j.h.316.4 12
39.5 even 4 507.2.b.g.337.3 6
39.8 even 4 507.2.b.g.337.4 6
39.11 even 12 507.2.j.h.316.3 12
39.17 odd 6 507.2.e.k.484.2 6
39.20 even 12 507.2.j.h.361.4 12
39.23 odd 6 507.2.e.k.22.2 6
39.29 odd 6 507.2.e.j.22.2 6
39.32 even 12 507.2.j.h.361.3 12
39.35 odd 6 507.2.e.j.484.2 6
39.38 odd 2 507.2.a.j.1.2 3
156.155 even 2 8112.2.a.by.1.3 3

By twisted newform
Twist Min Dim Char Parity Ord Type
507.2.a.j.1.2 3 39.38 odd 2
507.2.a.k.1.2 yes 3 3.2 odd 2
507.2.b.g.337.3 6 39.5 even 4
507.2.b.g.337.4 6 39.8 even 4
507.2.e.j.22.2 6 39.29 odd 6
507.2.e.j.484.2 6 39.35 odd 6
507.2.e.k.22.2 6 39.23 odd 6
507.2.e.k.484.2 6 39.17 odd 6
507.2.j.h.316.3 12 39.11 even 12
507.2.j.h.316.4 12 39.2 even 12
507.2.j.h.361.3 12 39.32 even 12
507.2.j.h.361.4 12 39.20 even 12
1521.2.a.p.1.2 3 1.1 even 1 trivial
1521.2.a.q.1.2 3 13.12 even 2
1521.2.b.m.1351.3 6 13.8 odd 4
1521.2.b.m.1351.4 6 13.5 odd 4
8112.2.a.by.1.3 3 156.155 even 2
8112.2.a.cf.1.1 3 12.11 even 2