# Properties

 Label 1850.2.a.bd.1.3 Level $1850$ Weight $2$ Character 1850.1 Self dual yes Analytic conductor $14.772$ Analytic rank $0$ Dimension $5$ CM no Inner twists $1$

# Learn more

## Newspace parameters

 Level: $$N$$ $$=$$ $$1850 = 2 \cdot 5^{2} \cdot 37$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1850.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$14.7723243739$$ Analytic rank: $$0$$ Dimension: $$5$$ Coefficient field: 5.5.1791440.1 Defining polynomial: $$x^{5} - 9 x^{3} + 13 x - 4$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 370) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.3 Root $$0.332924$$ of defining polynomial Character $$\chi$$ $$=$$ 1850.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-1.00000 q^{2} +0.332924 q^{3} +1.00000 q^{4} -0.332924 q^{6} -3.51336 q^{7} -1.00000 q^{8} -2.88916 q^{9} +O(q^{10})$$ $$q-1.00000 q^{2} +0.332924 q^{3} +1.00000 q^{4} -0.332924 q^{6} -3.51336 q^{7} -1.00000 q^{8} -2.88916 q^{9} -0.290044 q^{11} +0.332924 q^{12} +7.12558 q^{13} +3.51336 q^{14} +1.00000 q^{16} -6.17921 q^{17} +2.88916 q^{18} +5.83553 q^{19} -1.16968 q^{21} +0.290044 q^{22} -6.45973 q^{23} -0.332924 q^{24} -7.12558 q^{26} -1.96065 q^{27} -3.51336 q^{28} -1.18043 q^{29} +9.77838 q^{31} -1.00000 q^{32} -0.0965628 q^{33} +6.17921 q^{34} -2.88916 q^{36} -1.00000 q^{37} -5.83553 q^{38} +2.37228 q^{39} +1.64077 q^{41} +1.16968 q^{42} -5.34889 q^{43} -0.290044 q^{44} +6.45973 q^{46} +5.69256 q^{47} +0.332924 q^{48} +5.34367 q^{49} -2.05721 q^{51} +7.12558 q^{52} +9.32034 q^{53} +1.96065 q^{54} +3.51336 q^{56} +1.94279 q^{57} +1.18043 q^{58} +5.94279 q^{59} -1.27699 q^{61} -9.77838 q^{62} +10.1507 q^{63} +1.00000 q^{64} +0.0965628 q^{66} +6.04334 q^{67} -6.17921 q^{68} -2.15060 q^{69} +13.4995 q^{71} +2.88916 q^{72} +1.71348 q^{73} +1.00000 q^{74} +5.83553 q^{76} +1.01903 q^{77} -2.37228 q^{78} +8.25422 q^{79} +8.01474 q^{81} -1.64077 q^{82} +2.41807 q^{83} -1.16968 q^{84} +5.34889 q^{86} -0.392995 q^{87} +0.290044 q^{88} -10.2797 q^{89} -25.0347 q^{91} -6.45973 q^{92} +3.25546 q^{93} -5.69256 q^{94} -0.332924 q^{96} +15.1272 q^{97} -5.34367 q^{98} +0.837984 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$5 q - 5 q^{2} + 5 q^{4} + q^{7} - 5 q^{8} + 3 q^{9} + O(q^{10})$$ $$5 q - 5 q^{2} + 5 q^{4} + q^{7} - 5 q^{8} + 3 q^{9} + 3 q^{11} + 6 q^{13} - q^{14} + 5 q^{16} - 9 q^{17} - 3 q^{18} + 4 q^{19} + 16 q^{21} - 3 q^{22} - 6 q^{23} - 6 q^{26} + q^{28} + 11 q^{29} + 23 q^{31} - 5 q^{32} - 20 q^{33} + 9 q^{34} + 3 q^{36} - 5 q^{37} - 4 q^{38} + 20 q^{39} - 7 q^{41} - 16 q^{42} + 17 q^{43} + 3 q^{44} + 6 q^{46} - 12 q^{47} + 30 q^{49} - 20 q^{51} + 6 q^{52} + 7 q^{53} - q^{56} - 11 q^{58} + 20 q^{59} - 9 q^{61} - 23 q^{62} + 33 q^{63} + 5 q^{64} + 20 q^{66} - 12 q^{67} - 9 q^{68} + 16 q^{69} + 6 q^{71} - 3 q^{72} + 6 q^{73} + 5 q^{74} + 4 q^{76} + q^{77} - 20 q^{78} + 20 q^{79} - 7 q^{81} + 7 q^{82} - 12 q^{83} + 16 q^{84} - 17 q^{86} + 34 q^{87} - 3 q^{88} + 12 q^{89} + 16 q^{91} - 6 q^{92} + 4 q^{93} + 12 q^{94} - 3 q^{97} - 30 q^{98} - 11 q^{99} + O(q^{100})$$

## Coefficient data

For each $$n$$ we display the coefficients of the $$q$$-expansion $$a_n$$, the Satake parameters $$\alpha_p$$, and the Satake angles $$\theta_p = \textrm{Arg}(\alpha_p)$$.

Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ −1.00000 −0.707107
$$3$$ 0.332924 0.192214 0.0961070 0.995371i $$-0.469361\pi$$
0.0961070 + 0.995371i $$0.469361\pi$$
$$4$$ 1.00000 0.500000
$$5$$ 0 0
$$6$$ −0.332924 −0.135916
$$7$$ −3.51336 −1.32792 −0.663962 0.747766i $$-0.731126\pi$$
−0.663962 + 0.747766i $$0.731126\pi$$
$$8$$ −1.00000 −0.353553
$$9$$ −2.88916 −0.963054
$$10$$ 0 0
$$11$$ −0.290044 −0.0874516 −0.0437258 0.999044i $$-0.513923\pi$$
−0.0437258 + 0.999044i $$0.513923\pi$$
$$12$$ 0.332924 0.0961070
$$13$$ 7.12558 1.97628 0.988140 0.153559i $$-0.0490734\pi$$
0.988140 + 0.153559i $$0.0490734\pi$$
$$14$$ 3.51336 0.938984
$$15$$ 0 0
$$16$$ 1.00000 0.250000
$$17$$ −6.17921 −1.49868 −0.749339 0.662187i $$-0.769628\pi$$
−0.749339 + 0.662187i $$0.769628\pi$$
$$18$$ 2.88916 0.680982
$$19$$ 5.83553 1.33876 0.669381 0.742919i $$-0.266559\pi$$
0.669381 + 0.742919i $$0.266559\pi$$
$$20$$ 0 0
$$21$$ −1.16968 −0.255246
$$22$$ 0.290044 0.0618376
$$23$$ −6.45973 −1.34695 −0.673473 0.739212i $$-0.735198\pi$$
−0.673473 + 0.739212i $$0.735198\pi$$
$$24$$ −0.332924 −0.0679579
$$25$$ 0 0
$$26$$ −7.12558 −1.39744
$$27$$ −1.96065 −0.377326
$$28$$ −3.51336 −0.663962
$$29$$ −1.18043 −0.219201 −0.109600 0.993976i $$-0.534957\pi$$
−0.109600 + 0.993976i $$0.534957\pi$$
$$30$$ 0 0
$$31$$ 9.77838 1.75625 0.878124 0.478433i $$-0.158795\pi$$
0.878124 + 0.478433i $$0.158795\pi$$
$$32$$ −1.00000 −0.176777
$$33$$ −0.0965628 −0.0168094
$$34$$ 6.17921 1.05972
$$35$$ 0 0
$$36$$ −2.88916 −0.481527
$$37$$ −1.00000 −0.164399
$$38$$ −5.83553 −0.946648
$$39$$ 2.37228 0.379869
$$40$$ 0 0
$$41$$ 1.64077 0.256245 0.128123 0.991758i $$-0.459105\pi$$
0.128123 + 0.991758i $$0.459105\pi$$
$$42$$ 1.16968 0.180486
$$43$$ −5.34889 −0.815698 −0.407849 0.913049i $$-0.633721\pi$$
−0.407849 + 0.913049i $$0.633721\pi$$
$$44$$ −0.290044 −0.0437258
$$45$$ 0 0
$$46$$ 6.45973 0.952435
$$47$$ 5.69256 0.830346 0.415173 0.909743i $$-0.363721\pi$$
0.415173 + 0.909743i $$0.363721\pi$$
$$48$$ 0.332924 0.0480535
$$49$$ 5.34367 0.763382
$$50$$ 0 0
$$51$$ −2.05721 −0.288067
$$52$$ 7.12558 0.988140
$$53$$ 9.32034 1.28025 0.640123 0.768272i $$-0.278883\pi$$
0.640123 + 0.768272i $$0.278883\pi$$
$$54$$ 1.96065 0.266810
$$55$$ 0 0
$$56$$ 3.51336 0.469492
$$57$$ 1.94279 0.257329
$$58$$ 1.18043 0.154998
$$59$$ 5.94279 0.773686 0.386843 0.922146i $$-0.373566\pi$$
0.386843 + 0.922146i $$0.373566\pi$$
$$60$$ 0 0
$$61$$ −1.27699 −0.163502 −0.0817512 0.996653i $$-0.526051\pi$$
−0.0817512 + 0.996653i $$0.526051\pi$$
$$62$$ −9.77838 −1.24185
$$63$$ 10.1507 1.27886
$$64$$ 1.00000 0.125000
$$65$$ 0 0
$$66$$ 0.0965628 0.0118861
$$67$$ 6.04334 0.738312 0.369156 0.929368i $$-0.379647\pi$$
0.369156 + 0.929368i $$0.379647\pi$$
$$68$$ −6.17921 −0.749339
$$69$$ −2.15060 −0.258902
$$70$$ 0 0
$$71$$ 13.4995 1.60210 0.801050 0.598597i $$-0.204275\pi$$
0.801050 + 0.598597i $$0.204275\pi$$
$$72$$ 2.88916 0.340491
$$73$$ 1.71348 0.200548 0.100274 0.994960i $$-0.468028\pi$$
0.100274 + 0.994960i $$0.468028\pi$$
$$74$$ 1.00000 0.116248
$$75$$ 0 0
$$76$$ 5.83553 0.669381
$$77$$ 1.01903 0.116129
$$78$$ −2.37228 −0.268608
$$79$$ 8.25422 0.928672 0.464336 0.885659i $$-0.346293\pi$$
0.464336 + 0.885659i $$0.346293\pi$$
$$80$$ 0 0
$$81$$ 8.01474 0.890526
$$82$$ −1.64077 −0.181193
$$83$$ 2.41807 0.265418 0.132709 0.991155i $$-0.457632\pi$$
0.132709 + 0.991155i $$0.457632\pi$$
$$84$$ −1.16968 −0.127623
$$85$$ 0 0
$$86$$ 5.34889 0.576785
$$87$$ −0.392995 −0.0421335
$$88$$ 0.290044 0.0309188
$$89$$ −10.2797 −1.08965 −0.544823 0.838551i $$-0.683403\pi$$
−0.544823 + 0.838551i $$0.683403\pi$$
$$90$$ 0 0
$$91$$ −25.0347 −2.62435
$$92$$ −6.45973 −0.673473
$$93$$ 3.25546 0.337576
$$94$$ −5.69256 −0.587143
$$95$$ 0 0
$$96$$ −0.332924 −0.0339790
$$97$$ 15.1272 1.53594 0.767968 0.640489i $$-0.221268\pi$$
0.767968 + 0.640489i $$0.221268\pi$$
$$98$$ −5.34367 −0.539793
$$99$$ 0.837984 0.0842206
$$100$$ 0 0
$$101$$ 2.63730 0.262421 0.131210 0.991355i $$-0.458114\pi$$
0.131210 + 0.991355i $$0.458114\pi$$
$$102$$ 2.05721 0.203694
$$103$$ −1.72469 −0.169939 −0.0849695 0.996384i $$-0.527079\pi$$
−0.0849695 + 0.996384i $$0.527079\pi$$
$$104$$ −7.12558 −0.698720
$$105$$ 0 0
$$106$$ −9.32034 −0.905271
$$107$$ 10.0279 0.969438 0.484719 0.874670i $$-0.338922\pi$$
0.484719 + 0.874670i $$0.338922\pi$$
$$108$$ −1.96065 −0.188663
$$109$$ 4.87361 0.466807 0.233403 0.972380i $$-0.425014\pi$$
0.233403 + 0.972380i $$0.425014\pi$$
$$110$$ 0 0
$$111$$ −0.332924 −0.0315998
$$112$$ −3.51336 −0.331981
$$113$$ −15.9290 −1.49847 −0.749236 0.662303i $$-0.769579\pi$$
−0.749236 + 0.662303i $$0.769579\pi$$
$$114$$ −1.94279 −0.181959
$$115$$ 0 0
$$116$$ −1.18043 −0.109600
$$117$$ −20.5869 −1.90326
$$118$$ −5.94279 −0.547078
$$119$$ 21.7098 1.99013
$$120$$ 0 0
$$121$$ −10.9159 −0.992352
$$122$$ 1.27699 0.115614
$$123$$ 0.546253 0.0492540
$$124$$ 9.77838 0.878124
$$125$$ 0 0
$$126$$ −10.1507 −0.904292
$$127$$ 1.31265 0.116479 0.0582395 0.998303i $$-0.481451\pi$$
0.0582395 + 0.998303i $$0.481451\pi$$
$$128$$ −1.00000 −0.0883883
$$129$$ −1.78078 −0.156789
$$130$$ 0 0
$$131$$ 2.41562 0.211054 0.105527 0.994416i $$-0.466347\pi$$
0.105527 + 0.994416i $$0.466347\pi$$
$$132$$ −0.0965628 −0.00840471
$$133$$ −20.5023 −1.77778
$$134$$ −6.04334 −0.522065
$$135$$ 0 0
$$136$$ 6.17921 0.529862
$$137$$ 6.87366 0.587256 0.293628 0.955920i $$-0.405137\pi$$
0.293628 + 0.955920i $$0.405137\pi$$
$$138$$ 2.15060 0.183071
$$139$$ −9.80616 −0.831748 −0.415874 0.909422i $$-0.636524\pi$$
−0.415874 + 0.909422i $$0.636524\pi$$
$$140$$ 0 0
$$141$$ 1.89519 0.159604
$$142$$ −13.4995 −1.13286
$$143$$ −2.06673 −0.172829
$$144$$ −2.88916 −0.240763
$$145$$ 0 0
$$146$$ −1.71348 −0.141809
$$147$$ 1.77904 0.146733
$$148$$ −1.00000 −0.0821995
$$149$$ −9.07687 −0.743606 −0.371803 0.928312i $$-0.621260\pi$$
−0.371803 + 0.928312i $$0.621260\pi$$
$$150$$ 0 0
$$151$$ 20.1964 1.64356 0.821780 0.569805i $$-0.192981\pi$$
0.821780 + 0.569805i $$0.192981\pi$$
$$152$$ −5.83553 −0.473324
$$153$$ 17.8527 1.44331
$$154$$ −1.01903 −0.0821157
$$155$$ 0 0
$$156$$ 2.37228 0.189934
$$157$$ −17.2677 −1.37811 −0.689057 0.724707i $$-0.741975\pi$$
−0.689057 + 0.724707i $$0.741975\pi$$
$$158$$ −8.25422 −0.656670
$$159$$ 3.10297 0.246081
$$160$$ 0 0
$$161$$ 22.6953 1.78864
$$162$$ −8.01474 −0.629697
$$163$$ 16.6901 1.30727 0.653634 0.756810i $$-0.273243\pi$$
0.653634 + 0.756810i $$0.273243\pi$$
$$164$$ 1.64077 0.128123
$$165$$ 0 0
$$166$$ −2.41807 −0.187679
$$167$$ 1.47354 0.114026 0.0570130 0.998373i $$-0.481842\pi$$
0.0570130 + 0.998373i $$0.481842\pi$$
$$168$$ 1.16968 0.0902430
$$169$$ 37.7738 2.90568
$$170$$ 0 0
$$171$$ −16.8598 −1.28930
$$172$$ −5.34889 −0.407849
$$173$$ 13.8432 1.05248 0.526240 0.850336i $$-0.323601\pi$$
0.526240 + 0.850336i $$0.323601\pi$$
$$174$$ 0.392995 0.0297929
$$175$$ 0 0
$$176$$ −0.290044 −0.0218629
$$177$$ 1.97850 0.148713
$$178$$ 10.2797 0.770496
$$179$$ 21.7817 1.62804 0.814020 0.580836i $$-0.197274\pi$$
0.814020 + 0.580836i $$0.197274\pi$$
$$180$$ 0 0
$$181$$ 2.03100 0.150963 0.0754817 0.997147i $$-0.475951\pi$$
0.0754817 + 0.997147i $$0.475951\pi$$
$$182$$ 25.0347 1.85569
$$183$$ −0.425143 −0.0314275
$$184$$ 6.45973 0.476217
$$185$$ 0 0
$$186$$ −3.25546 −0.238702
$$187$$ 1.79224 0.131062
$$188$$ 5.69256 0.415173
$$189$$ 6.88845 0.501061
$$190$$ 0 0
$$191$$ −8.44668 −0.611180 −0.305590 0.952163i $$-0.598854\pi$$
−0.305590 + 0.952163i $$0.598854\pi$$
$$192$$ 0.332924 0.0240268
$$193$$ 3.64159 0.262127 0.131064 0.991374i $$-0.458161\pi$$
0.131064 + 0.991374i $$0.458161\pi$$
$$194$$ −15.1272 −1.08607
$$195$$ 0 0
$$196$$ 5.34367 0.381691
$$197$$ 10.1939 0.726288 0.363144 0.931733i $$-0.381703\pi$$
0.363144 + 0.931733i $$0.381703\pi$$
$$198$$ −0.837984 −0.0595530
$$199$$ −5.25299 −0.372375 −0.186187 0.982514i $$-0.559613\pi$$
−0.186187 + 0.982514i $$0.559613\pi$$
$$200$$ 0 0
$$201$$ 2.01198 0.141914
$$202$$ −2.63730 −0.185560
$$203$$ 4.14728 0.291082
$$204$$ −2.05721 −0.144033
$$205$$ 0 0
$$206$$ 1.72469 0.120165
$$207$$ 18.6632 1.29718
$$208$$ 7.12558 0.494070
$$209$$ −1.69256 −0.117077
$$210$$ 0 0
$$211$$ −4.81998 −0.331821 −0.165910 0.986141i $$-0.553056\pi$$
−0.165910 + 0.986141i $$0.553056\pi$$
$$212$$ 9.32034 0.640123
$$213$$ 4.49433 0.307946
$$214$$ −10.0279 −0.685496
$$215$$ 0 0
$$216$$ 1.96065 0.133405
$$217$$ −34.3549 −2.33216
$$218$$ −4.87361 −0.330082
$$219$$ 0.570460 0.0385481
$$220$$ 0 0
$$221$$ −44.0304 −2.96180
$$222$$ 0.332924 0.0223444
$$223$$ 3.70392 0.248033 0.124017 0.992280i $$-0.460422\pi$$
0.124017 + 0.992280i $$0.460422\pi$$
$$224$$ 3.51336 0.234746
$$225$$ 0 0
$$226$$ 15.9290 1.05958
$$227$$ 6.31512 0.419149 0.209575 0.977793i $$-0.432792\pi$$
0.209575 + 0.977793i $$0.432792\pi$$
$$228$$ 1.94279 0.128664
$$229$$ −20.5548 −1.35830 −0.679150 0.734000i $$-0.737651\pi$$
−0.679150 + 0.734000i $$0.737651\pi$$
$$230$$ 0 0
$$231$$ 0.339260 0.0223216
$$232$$ 1.18043 0.0774992
$$233$$ 21.4393 1.40453 0.702267 0.711914i $$-0.252171\pi$$
0.702267 + 0.711914i $$0.252171\pi$$
$$234$$ 20.5869 1.34581
$$235$$ 0 0
$$236$$ 5.94279 0.386843
$$237$$ 2.74803 0.178504
$$238$$ −21.7098 −1.40723
$$239$$ −20.7479 −1.34207 −0.671034 0.741426i $$-0.734150\pi$$
−0.671034 + 0.741426i $$0.734150\pi$$
$$240$$ 0 0
$$241$$ 10.7731 0.693957 0.346978 0.937873i $$-0.387208\pi$$
0.346978 + 0.937873i $$0.387208\pi$$
$$242$$ 10.9159 0.701699
$$243$$ 8.55024 0.548498
$$244$$ −1.27699 −0.0817512
$$245$$ 0 0
$$246$$ −0.546253 −0.0348278
$$247$$ 41.5815 2.64577
$$248$$ −9.77838 −0.620927
$$249$$ 0.805036 0.0510171
$$250$$ 0 0
$$251$$ −4.46812 −0.282025 −0.141013 0.990008i $$-0.545036\pi$$
−0.141013 + 0.990008i $$0.545036\pi$$
$$252$$ 10.1507 0.639431
$$253$$ 1.87361 0.117793
$$254$$ −1.31265 −0.0823631
$$255$$ 0 0
$$256$$ 1.00000 0.0625000
$$257$$ −1.24594 −0.0777194 −0.0388597 0.999245i $$-0.512373\pi$$
−0.0388597 + 0.999245i $$0.512373\pi$$
$$258$$ 1.78078 0.110866
$$259$$ 3.51336 0.218309
$$260$$ 0 0
$$261$$ 3.41046 0.211102
$$262$$ −2.41562 −0.149237
$$263$$ 16.3230 1.00652 0.503259 0.864135i $$-0.332134\pi$$
0.503259 + 0.864135i $$0.332134\pi$$
$$264$$ 0.0965628 0.00594303
$$265$$ 0 0
$$266$$ 20.5023 1.25708
$$267$$ −3.42236 −0.209445
$$268$$ 6.04334 0.369156
$$269$$ −18.5763 −1.13262 −0.566309 0.824193i $$-0.691629\pi$$
−0.566309 + 0.824193i $$0.691629\pi$$
$$270$$ 0 0
$$271$$ 16.4181 0.997327 0.498663 0.866796i $$-0.333824\pi$$
0.498663 + 0.866796i $$0.333824\pi$$
$$272$$ −6.17921 −0.374669
$$273$$ −8.33466 −0.504437
$$274$$ −6.87366 −0.415253
$$275$$ 0 0
$$276$$ −2.15060 −0.129451
$$277$$ 15.6104 0.937937 0.468968 0.883215i $$-0.344626\pi$$
0.468968 + 0.883215i $$0.344626\pi$$
$$278$$ 9.80616 0.588134
$$279$$ −28.2513 −1.69136
$$280$$ 0 0
$$281$$ −21.8665 −1.30445 −0.652224 0.758026i $$-0.726164\pi$$
−0.652224 + 0.758026i $$0.726164\pi$$
$$282$$ −1.89519 −0.112857
$$283$$ −2.24778 −0.133616 −0.0668082 0.997766i $$-0.521282\pi$$
−0.0668082 + 0.997766i $$0.521282\pi$$
$$284$$ 13.4995 0.801050
$$285$$ 0 0
$$286$$ 2.06673 0.122208
$$287$$ −5.76461 −0.340274
$$288$$ 2.88916 0.170245
$$289$$ 21.1826 1.24603
$$290$$ 0 0
$$291$$ 5.03622 0.295228
$$292$$ 1.71348 0.100274
$$293$$ −26.8794 −1.57031 −0.785157 0.619297i $$-0.787418\pi$$
−0.785157 + 0.619297i $$0.787418\pi$$
$$294$$ −1.77904 −0.103756
$$295$$ 0 0
$$296$$ 1.00000 0.0581238
$$297$$ 0.568674 0.0329978
$$298$$ 9.07687 0.525809
$$299$$ −46.0293 −2.66194
$$300$$ 0 0
$$301$$ 18.7926 1.08318
$$302$$ −20.1964 −1.16217
$$303$$ 0.878021 0.0504410
$$304$$ 5.83553 0.334691
$$305$$ 0 0
$$306$$ −17.8527 −1.02057
$$307$$ −23.9979 −1.36963 −0.684815 0.728717i $$-0.740117\pi$$
−0.684815 + 0.728717i $$0.740117\pi$$
$$308$$ 1.01903 0.0580645
$$309$$ −0.574192 −0.0326647
$$310$$ 0 0
$$311$$ −5.07563 −0.287812 −0.143906 0.989591i $$-0.545966\pi$$
−0.143906 + 0.989591i $$0.545966\pi$$
$$312$$ −2.37228 −0.134304
$$313$$ 8.96950 0.506986 0.253493 0.967337i $$-0.418420\pi$$
0.253493 + 0.967337i $$0.418420\pi$$
$$314$$ 17.2677 0.974474
$$315$$ 0 0
$$316$$ 8.25422 0.464336
$$317$$ 22.0181 1.23666 0.618330 0.785918i $$-0.287809\pi$$
0.618330 + 0.785918i $$0.287809\pi$$
$$318$$ −3.10297 −0.174006
$$319$$ 0.342377 0.0191695
$$320$$ 0 0
$$321$$ 3.33855 0.186339
$$322$$ −22.6953 −1.26476
$$323$$ −36.0589 −2.00637
$$324$$ 8.01474 0.445263
$$325$$ 0 0
$$326$$ −16.6901 −0.924379
$$327$$ 1.62254 0.0897268
$$328$$ −1.64077 −0.0905964
$$329$$ −20.0000 −1.10264
$$330$$ 0 0
$$331$$ −10.7134 −0.588864 −0.294432 0.955672i $$-0.595130\pi$$
−0.294432 + 0.955672i $$0.595130\pi$$
$$332$$ 2.41807 0.132709
$$333$$ 2.88916 0.158325
$$334$$ −1.47354 −0.0806286
$$335$$ 0 0
$$336$$ −1.16968 −0.0638114
$$337$$ −1.34097 −0.0730471 −0.0365235 0.999333i $$-0.511628\pi$$
−0.0365235 + 0.999333i $$0.511628\pi$$
$$338$$ −37.7738 −2.05463
$$339$$ −5.30315 −0.288027
$$340$$ 0 0
$$341$$ −2.83616 −0.153587
$$342$$ 16.8598 0.911673
$$343$$ 5.81926 0.314211
$$344$$ 5.34889 0.288393
$$345$$ 0 0
$$346$$ −13.8432 −0.744216
$$347$$ −0.883744 −0.0474419 −0.0237209 0.999719i $$-0.507551\pi$$
−0.0237209 + 0.999719i $$0.507551\pi$$
$$348$$ −0.392995 −0.0210667
$$349$$ −9.00521 −0.482038 −0.241019 0.970520i $$-0.577482\pi$$
−0.241019 + 0.970520i $$0.577482\pi$$
$$350$$ 0 0
$$351$$ −13.9707 −0.745702
$$352$$ 0.290044 0.0154594
$$353$$ −6.79030 −0.361411 −0.180706 0.983537i $$-0.557838\pi$$
−0.180706 + 0.983537i $$0.557838\pi$$
$$354$$ −1.97850 −0.105156
$$355$$ 0 0
$$356$$ −10.2797 −0.544823
$$357$$ 7.22771 0.382531
$$358$$ −21.7817 −1.15120
$$359$$ −10.2263 −0.539722 −0.269861 0.962899i $$-0.586978\pi$$
−0.269861 + 0.962899i $$0.586978\pi$$
$$360$$ 0 0
$$361$$ 15.0534 0.792286
$$362$$ −2.03100 −0.106747
$$363$$ −3.63416 −0.190744
$$364$$ −25.0347 −1.31217
$$365$$ 0 0
$$366$$ 0.425143 0.0222226
$$367$$ −17.1583 −0.895657 −0.447829 0.894119i $$-0.647802\pi$$
−0.447829 + 0.894119i $$0.647802\pi$$
$$368$$ −6.45973 −0.336737
$$369$$ −4.74045 −0.246778
$$370$$ 0 0
$$371$$ −32.7457 −1.70007
$$372$$ 3.25546 0.168788
$$373$$ −17.1601 −0.888515 −0.444257 0.895899i $$-0.646532\pi$$
−0.444257 + 0.895899i $$0.646532\pi$$
$$374$$ −1.79224 −0.0926747
$$375$$ 0 0
$$376$$ −5.69256 −0.293571
$$377$$ −8.41126 −0.433202
$$378$$ −6.88845 −0.354304
$$379$$ −27.5435 −1.41482 −0.707408 0.706805i $$-0.750136\pi$$
−0.707408 + 0.706805i $$0.750136\pi$$
$$380$$ 0 0
$$381$$ 0.437014 0.0223889
$$382$$ 8.44668 0.432170
$$383$$ 3.49954 0.178818 0.0894091 0.995995i $$-0.471502\pi$$
0.0894091 + 0.995995i $$0.471502\pi$$
$$384$$ −0.332924 −0.0169895
$$385$$ 0 0
$$386$$ −3.64159 −0.185352
$$387$$ 15.4538 0.785561
$$388$$ 15.1272 0.767968
$$389$$ −13.0015 −0.659203 −0.329602 0.944120i $$-0.606914\pi$$
−0.329602 + 0.944120i $$0.606914\pi$$
$$390$$ 0 0
$$391$$ 39.9160 2.01864
$$392$$ −5.34367 −0.269896
$$393$$ 0.804219 0.0405675
$$394$$ −10.1939 −0.513563
$$395$$ 0 0
$$396$$ 0.837984 0.0421103
$$397$$ −26.6658 −1.33832 −0.669160 0.743118i $$-0.733346\pi$$
−0.669160 + 0.743118i $$0.733346\pi$$
$$398$$ 5.25299 0.263309
$$399$$ −6.82572 −0.341713
$$400$$ 0 0
$$401$$ 20.3546 1.01646 0.508231 0.861221i $$-0.330300\pi$$
0.508231 + 0.861221i $$0.330300\pi$$
$$402$$ −2.01198 −0.100348
$$403$$ 69.6766 3.47084
$$404$$ 2.63730 0.131210
$$405$$ 0 0
$$406$$ −4.14728 −0.205826
$$407$$ 0.290044 0.0143770
$$408$$ 2.05721 0.101847
$$409$$ 29.9603 1.48144 0.740720 0.671814i $$-0.234485\pi$$
0.740720 + 0.671814i $$0.234485\pi$$
$$410$$ 0 0
$$411$$ 2.28841 0.112879
$$412$$ −1.72469 −0.0849695
$$413$$ −20.8791 −1.02740
$$414$$ −18.6632 −0.917246
$$415$$ 0 0
$$416$$ −7.12558 −0.349360
$$417$$ −3.26471 −0.159874
$$418$$ 1.69256 0.0827859
$$419$$ 23.3149 1.13901 0.569504 0.821988i $$-0.307135\pi$$
0.569504 + 0.821988i $$0.307135\pi$$
$$420$$ 0 0
$$421$$ 24.0289 1.17110 0.585548 0.810638i $$-0.300880\pi$$
0.585548 + 0.810638i $$0.300880\pi$$
$$422$$ 4.81998 0.234633
$$423$$ −16.4467 −0.799667
$$424$$ −9.32034 −0.452636
$$425$$ 0 0
$$426$$ −4.49433 −0.217751
$$427$$ 4.48654 0.217119
$$428$$ 10.0279 0.484719
$$429$$ −0.688066 −0.0332201
$$430$$ 0 0
$$431$$ −19.2917 −0.929247 −0.464624 0.885508i $$-0.653810\pi$$
−0.464624 + 0.885508i $$0.653810\pi$$
$$432$$ −1.96065 −0.0943316
$$433$$ −12.5816 −0.604634 −0.302317 0.953207i $$-0.597760\pi$$
−0.302317 + 0.953207i $$0.597760\pi$$
$$434$$ 34.3549 1.64909
$$435$$ 0 0
$$436$$ 4.87361 0.233403
$$437$$ −37.6959 −1.80324
$$438$$ −0.570460 −0.0272576
$$439$$ 14.0428 0.670225 0.335113 0.942178i $$-0.391226\pi$$
0.335113 + 0.942178i $$0.391226\pi$$
$$440$$ 0 0
$$441$$ −15.4387 −0.735178
$$442$$ 44.0304 2.09431
$$443$$ −24.5696 −1.16734 −0.583668 0.811992i $$-0.698383\pi$$
−0.583668 + 0.811992i $$0.698383\pi$$
$$444$$ −0.332924 −0.0157999
$$445$$ 0 0
$$446$$ −3.70392 −0.175386
$$447$$ −3.02191 −0.142932
$$448$$ −3.51336 −0.165990
$$449$$ −12.7574 −0.602061 −0.301030 0.953615i $$-0.597331\pi$$
−0.301030 + 0.953615i $$0.597331\pi$$
$$450$$ 0 0
$$451$$ −0.475896 −0.0224091
$$452$$ −15.9290 −0.749236
$$453$$ 6.72387 0.315915
$$454$$ −6.31512 −0.296383
$$455$$ 0 0
$$456$$ −1.94279 −0.0909795
$$457$$ 3.34889 0.156654 0.0783272 0.996928i $$-0.475042\pi$$
0.0783272 + 0.996928i $$0.475042\pi$$
$$458$$ 20.5548 0.960463
$$459$$ 12.1152 0.565491
$$460$$ 0 0
$$461$$ −0.859501 −0.0400310 −0.0200155 0.999800i $$-0.506372\pi$$
−0.0200155 + 0.999800i $$0.506372\pi$$
$$462$$ −0.339260 −0.0157838
$$463$$ 13.8711 0.644643 0.322321 0.946630i $$-0.395537\pi$$
0.322321 + 0.946630i $$0.395537\pi$$
$$464$$ −1.18043 −0.0548002
$$465$$ 0 0
$$466$$ −21.4393 −0.993155
$$467$$ 40.9281 1.89392 0.946962 0.321344i $$-0.104135\pi$$
0.946962 + 0.321344i $$0.104135\pi$$
$$468$$ −20.5869 −0.951632
$$469$$ −21.2324 −0.980422
$$470$$ 0 0
$$471$$ −5.74885 −0.264893
$$472$$ −5.94279 −0.273539
$$473$$ 1.55141 0.0713341
$$474$$ −2.74803 −0.126221
$$475$$ 0 0
$$476$$ 21.7098 0.995065
$$477$$ −26.9280 −1.23295
$$478$$ 20.7479 0.948986
$$479$$ 8.58990 0.392483 0.196241 0.980556i $$-0.437126\pi$$
0.196241 + 0.980556i $$0.437126\pi$$
$$480$$ 0 0
$$481$$ −7.12558 −0.324898
$$482$$ −10.7731 −0.490702
$$483$$ 7.55583 0.343802
$$484$$ −10.9159 −0.496176
$$485$$ 0 0
$$486$$ −8.55024 −0.387847
$$487$$ 25.8389 1.17087 0.585436 0.810718i $$-0.300923\pi$$
0.585436 + 0.810718i $$0.300923\pi$$
$$488$$ 1.27699 0.0578068
$$489$$ 5.55654 0.251275
$$490$$ 0 0
$$491$$ 11.4864 0.518376 0.259188 0.965827i $$-0.416545\pi$$
0.259188 + 0.965827i $$0.416545\pi$$
$$492$$ 0.546253 0.0246270
$$493$$ 7.29413 0.328511
$$494$$ −41.5815 −1.87084
$$495$$ 0 0
$$496$$ 9.77838 0.439062
$$497$$ −47.4287 −2.12747
$$498$$ −0.805036 −0.0360745
$$499$$ −9.74599 −0.436290 −0.218145 0.975916i $$-0.570001\pi$$
−0.218145 + 0.975916i $$0.570001\pi$$
$$500$$ 0 0
$$501$$ 0.490578 0.0219174
$$502$$ 4.46812 0.199422
$$503$$ 17.4803 0.779408 0.389704 0.920940i $$-0.372577\pi$$
0.389704 + 0.920940i $$0.372577\pi$$
$$504$$ −10.1507 −0.452146
$$505$$ 0 0
$$506$$ −1.87361 −0.0832920
$$507$$ 12.5758 0.558512
$$508$$ 1.31265 0.0582395
$$509$$ −16.3273 −0.723695 −0.361847 0.932237i $$-0.617854\pi$$
−0.361847 + 0.932237i $$0.617854\pi$$
$$510$$ 0 0
$$511$$ −6.02007 −0.266312
$$512$$ −1.00000 −0.0441942
$$513$$ −11.4414 −0.505151
$$514$$ 1.24594 0.0549559
$$515$$ 0 0
$$516$$ −1.78078 −0.0783943
$$517$$ −1.65109 −0.0726151
$$518$$ −3.51336 −0.154368
$$519$$ 4.60875 0.202301
$$520$$ 0 0
$$521$$ −4.78335 −0.209562 −0.104781 0.994495i $$-0.533414\pi$$
−0.104781 + 0.994495i $$0.533414\pi$$
$$522$$ −3.41046 −0.149272
$$523$$ −24.1224 −1.05480 −0.527399 0.849618i $$-0.676833\pi$$
−0.527399 + 0.849618i $$0.676833\pi$$
$$524$$ 2.41562 0.105527
$$525$$ 0 0
$$526$$ −16.3230 −0.711716
$$527$$ −60.4226 −2.63205
$$528$$ −0.0965628 −0.00420236
$$529$$ 18.7281 0.814264
$$530$$ 0 0
$$531$$ −17.1697 −0.745101
$$532$$ −20.5023 −0.888888
$$533$$ 11.6914 0.506412
$$534$$ 3.42236 0.148100
$$535$$ 0 0
$$536$$ −6.04334 −0.261033
$$537$$ 7.25166 0.312932
$$538$$ 18.5763 0.800881
$$539$$ −1.54990 −0.0667590
$$540$$ 0 0
$$541$$ 4.11745 0.177023 0.0885115 0.996075i $$-0.471789\pi$$
0.0885115 + 0.996075i $$0.471789\pi$$
$$542$$ −16.4181 −0.705217
$$543$$ 0.676171 0.0290173
$$544$$ 6.17921 0.264931
$$545$$ 0 0
$$546$$ 8.33466 0.356691
$$547$$ 9.22742 0.394536 0.197268 0.980350i $$-0.436793\pi$$
0.197268 + 0.980350i $$0.436793\pi$$
$$548$$ 6.87366 0.293628
$$549$$ 3.68944 0.157462
$$550$$ 0 0
$$551$$ −6.88845 −0.293458
$$552$$ 2.15060 0.0915357
$$553$$ −29.0000 −1.23321
$$554$$ −15.6104 −0.663222
$$555$$ 0 0
$$556$$ −9.80616 −0.415874
$$557$$ −2.25125 −0.0953885 −0.0476943 0.998862i $$-0.515187\pi$$
−0.0476943 + 0.998862i $$0.515187\pi$$
$$558$$ 28.2513 1.19597
$$559$$ −38.1139 −1.61205
$$560$$ 0 0
$$561$$ 0.596681 0.0251919
$$562$$ 21.8665 0.922384
$$563$$ −5.79316 −0.244153 −0.122076 0.992521i $$-0.538955\pi$$
−0.122076 + 0.992521i $$0.538955\pi$$
$$564$$ 1.89519 0.0798020
$$565$$ 0 0
$$566$$ 2.24778 0.0944811
$$567$$ −28.1586 −1.18255
$$568$$ −13.4995 −0.566428
$$569$$ −34.4807 −1.44551 −0.722753 0.691106i $$-0.757124\pi$$
−0.722753 + 0.691106i $$0.757124\pi$$
$$570$$ 0 0
$$571$$ −19.2152 −0.804133 −0.402066 0.915611i $$-0.631708\pi$$
−0.402066 + 0.915611i $$0.631708\pi$$
$$572$$ −2.06673 −0.0864144
$$573$$ −2.81211 −0.117477
$$574$$ 5.76461 0.240610
$$575$$ 0 0
$$576$$ −2.88916 −0.120382
$$577$$ −6.33354 −0.263669 −0.131834 0.991272i $$-0.542087\pi$$
−0.131834 + 0.991272i $$0.542087\pi$$
$$578$$ −21.1826 −0.881079
$$579$$ 1.21237 0.0503845
$$580$$ 0 0
$$581$$ −8.49555 −0.352455
$$582$$ −5.03622 −0.208758
$$583$$ −2.70331 −0.111960
$$584$$ −1.71348 −0.0709044
$$585$$ 0 0
$$586$$ 26.8794 1.11038
$$587$$ 21.9099 0.904320 0.452160 0.891937i $$-0.350654\pi$$
0.452160 + 0.891937i $$0.350654\pi$$
$$588$$ 1.77904 0.0733664
$$589$$ 57.0620 2.35120
$$590$$ 0 0
$$591$$ 3.39381 0.139603
$$592$$ −1.00000 −0.0410997
$$593$$ 9.40482 0.386210 0.193105 0.981178i $$-0.438144\pi$$
0.193105 + 0.981178i $$0.438144\pi$$
$$594$$ −0.568674 −0.0233330
$$595$$ 0 0
$$596$$ −9.07687 −0.371803
$$597$$ −1.74885 −0.0715756
$$598$$ 46.0293 1.88228
$$599$$ 16.4095 0.670474 0.335237 0.942134i $$-0.391184\pi$$
0.335237 + 0.942134i $$0.391184\pi$$
$$600$$ 0 0
$$601$$ −7.01392 −0.286104 −0.143052 0.989715i $$-0.545692\pi$$
−0.143052 + 0.989715i $$0.545692\pi$$
$$602$$ −18.7926 −0.765927
$$603$$ −17.4602 −0.711034
$$604$$ 20.1964 0.821780
$$605$$ 0 0
$$606$$ −0.878021 −0.0356672
$$607$$ −16.5621 −0.672234 −0.336117 0.941820i $$-0.609114\pi$$
−0.336117 + 0.941820i $$0.609114\pi$$
$$608$$ −5.83553 −0.236662
$$609$$ 1.38073 0.0559500
$$610$$ 0 0
$$611$$ 40.5628 1.64099
$$612$$ 17.8527 0.721653
$$613$$ −23.9370 −0.966804 −0.483402 0.875398i $$-0.660599\pi$$
−0.483402 + 0.875398i $$0.660599\pi$$
$$614$$ 23.9979 0.968475
$$615$$ 0 0
$$616$$ −1.01903 −0.0410578
$$617$$ 13.9489 0.561563 0.280781 0.959772i $$-0.409406\pi$$
0.280781 + 0.959772i $$0.409406\pi$$
$$618$$ 0.574192 0.0230974
$$619$$ 21.6942 0.871964 0.435982 0.899955i $$-0.356401\pi$$
0.435982 + 0.899955i $$0.356401\pi$$
$$620$$ 0 0
$$621$$ 12.6652 0.508238
$$622$$ 5.07563 0.203514
$$623$$ 36.1163 1.44697
$$624$$ 2.37228 0.0949671
$$625$$ 0 0
$$626$$ −8.96950 −0.358493
$$627$$ −0.563495 −0.0225038
$$628$$ −17.2677 −0.689057
$$629$$ 6.17921 0.246381
$$630$$ 0 0
$$631$$ −32.4902 −1.29342 −0.646708 0.762738i $$-0.723855\pi$$
−0.646708 + 0.762738i $$0.723855\pi$$
$$632$$ −8.25422 −0.328335
$$633$$ −1.60469 −0.0637806
$$634$$ −22.0181 −0.874451
$$635$$ 0 0
$$636$$ 3.10297 0.123041
$$637$$ 38.0768 1.50866
$$638$$ −0.342377 −0.0135549
$$639$$ −39.0024 −1.54291
$$640$$ 0 0
$$641$$ −28.1621 −1.11234 −0.556168 0.831070i $$-0.687729\pi$$
−0.556168 + 0.831070i $$0.687729\pi$$
$$642$$ −3.33855 −0.131762
$$643$$ −8.32452 −0.328287 −0.164144 0.986436i $$-0.552486\pi$$
−0.164144 + 0.986436i $$0.552486\pi$$
$$644$$ 22.6953 0.894321
$$645$$ 0 0
$$646$$ 36.0589 1.41872
$$647$$ −1.75827 −0.0691249 −0.0345624 0.999403i $$-0.511004\pi$$
−0.0345624 + 0.999403i $$0.511004\pi$$
$$648$$ −8.01474 −0.314849
$$649$$ −1.72367 −0.0676600
$$650$$ 0 0
$$651$$ −11.4376 −0.448275
$$652$$ 16.6901 0.653634
$$653$$ −24.7416 −0.968213 −0.484106 0.875009i $$-0.660855\pi$$
−0.484106 + 0.875009i $$0.660855\pi$$
$$654$$ −1.62254 −0.0634464
$$655$$ 0 0
$$656$$ 1.64077 0.0640613
$$657$$ −4.95053 −0.193138
$$658$$ 20.0000 0.779681
$$659$$ 18.2640 0.711466 0.355733 0.934588i $$-0.384231\pi$$
0.355733 + 0.934588i $$0.384231\pi$$
$$660$$ 0 0
$$661$$ 2.61857 0.101851 0.0509253 0.998702i $$-0.483783\pi$$
0.0509253 + 0.998702i $$0.483783\pi$$
$$662$$ 10.7134 0.416390
$$663$$ −14.6588 −0.569300
$$664$$ −2.41807 −0.0938394
$$665$$ 0 0
$$666$$ −2.88916 −0.111953
$$667$$ 7.62527 0.295252
$$668$$ 1.47354 0.0570130
$$669$$ 1.23313 0.0476754
$$670$$ 0 0
$$671$$ 0.370385 0.0142986
$$672$$ 1.16968 0.0451215
$$673$$ 48.4181 1.86638 0.933190 0.359384i $$-0.117013\pi$$
0.933190 + 0.359384i $$0.117013\pi$$
$$674$$ 1.34097 0.0516521
$$675$$ 0 0
$$676$$ 37.7738 1.45284
$$677$$ 7.49862 0.288195 0.144098 0.989563i $$-0.453972\pi$$
0.144098 + 0.989563i $$0.453972\pi$$
$$678$$ 5.30315 0.203666
$$679$$ −53.1473 −2.03961
$$680$$ 0 0
$$681$$ 2.10246 0.0805664
$$682$$ 2.83616 0.108602
$$683$$ 33.2987 1.27414 0.637070 0.770806i $$-0.280146\pi$$
0.637070 + 0.770806i $$0.280146\pi$$
$$684$$ −16.8598 −0.644650
$$685$$ 0 0
$$686$$ −5.81926 −0.222181
$$687$$ −6.84320 −0.261084
$$688$$ −5.34889 −0.203924
$$689$$ 66.4128 2.53012
$$690$$ 0 0
$$691$$ 38.3695 1.45964 0.729822 0.683638i $$-0.239603\pi$$
0.729822 + 0.683638i $$0.239603\pi$$
$$692$$ 13.8432 0.526240
$$693$$ −2.94414 −0.111839
$$694$$ 0.883744 0.0335465
$$695$$ 0 0
$$696$$ 0.392995 0.0148964
$$697$$ −10.1387 −0.384029
$$698$$ 9.00521 0.340852
$$699$$ 7.13766 0.269971
$$700$$ 0 0
$$701$$ −40.6339 −1.53472 −0.767360 0.641217i $$-0.778430\pi$$
−0.767360 + 0.641217i $$0.778430\pi$$
$$702$$ 13.9707 0.527291
$$703$$ −5.83553 −0.220091
$$704$$ −0.290044 −0.0109315
$$705$$ 0 0
$$706$$ 6.79030 0.255556
$$707$$ −9.26577 −0.348475
$$708$$ 1.97850 0.0743566
$$709$$ 11.1957 0.420464 0.210232 0.977651i $$-0.432578\pi$$
0.210232 + 0.977651i $$0.432578\pi$$
$$710$$ 0 0
$$711$$ −23.8478 −0.894361
$$712$$ 10.2797 0.385248
$$713$$ −63.1656 −2.36557
$$714$$ −7.22771 −0.270490
$$715$$ 0 0
$$716$$ 21.7817 0.814020
$$717$$ −6.90748 −0.257964
$$718$$ 10.2263 0.381641
$$719$$ 30.2226 1.12711 0.563556 0.826078i $$-0.309433\pi$$
0.563556 + 0.826078i $$0.309433\pi$$
$$720$$ 0 0
$$721$$ 6.05946 0.225666
$$722$$ −15.0534 −0.560231
$$723$$ 3.58663 0.133388
$$724$$ 2.03100 0.0754817
$$725$$ 0 0
$$726$$ 3.63416 0.134876
$$727$$ 19.1911 0.711758 0.355879 0.934532i $$-0.384182\pi$$
0.355879 + 0.934532i $$0.384182\pi$$
$$728$$ 25.0347 0.927847
$$729$$ −21.1976 −0.785097
$$730$$ 0 0
$$731$$ 33.0519 1.22247
$$732$$ −0.425143 −0.0157137
$$733$$ −5.39567 −0.199294 −0.0996468 0.995023i $$-0.531771\pi$$
−0.0996468 + 0.995023i $$0.531771\pi$$
$$734$$ 17.1583 0.633325
$$735$$ 0 0
$$736$$ 6.45973 0.238109
$$737$$ −1.75284 −0.0645665
$$738$$ 4.74045 0.174498
$$739$$ 24.7127 0.909072 0.454536 0.890728i $$-0.349805\pi$$
0.454536 + 0.890728i $$0.349805\pi$$
$$740$$ 0 0
$$741$$ 13.8435 0.508554
$$742$$ 32.7457 1.20213
$$743$$ −45.4537 −1.66753 −0.833767 0.552116i $$-0.813820\pi$$
−0.833767 + 0.552116i $$0.813820\pi$$
$$744$$ −3.25546 −0.119351
$$745$$ 0 0
$$746$$ 17.1601 0.628275
$$747$$ −6.98620 −0.255612
$$748$$ 1.79224 0.0655309
$$749$$ −35.2317 −1.28734
$$750$$ 0 0
$$751$$ 49.6360 1.81124 0.905621 0.424088i $$-0.139405\pi$$
0.905621 + 0.424088i $$0.139405\pi$$
$$752$$ 5.69256 0.207586
$$753$$ −1.48755 −0.0542093
$$754$$ 8.41126 0.306320
$$755$$ 0 0
$$756$$ 6.88845 0.250530
$$757$$ 27.9481 1.01579 0.507896 0.861419i $$-0.330424\pi$$
0.507896 + 0.861419i $$0.330424\pi$$
$$758$$ 27.5435 1.00043
$$759$$ 0.623769 0.0226414
$$760$$ 0 0
$$761$$ −53.4744 −1.93844 −0.969222 0.246188i $$-0.920822\pi$$
−0.969222 + 0.246188i $$0.920822\pi$$
$$762$$ −0.437014 −0.0158313
$$763$$ −17.1227 −0.619884
$$764$$ −8.44668 −0.305590
$$765$$ 0 0
$$766$$ −3.49954 −0.126444
$$767$$ 42.3458 1.52902
$$768$$ 0.332924 0.0120134
$$769$$ −8.46659 −0.305313 −0.152657 0.988279i $$-0.548783\pi$$
−0.152657 + 0.988279i $$0.548783\pi$$
$$770$$ 0 0
$$771$$ −0.414803 −0.0149388
$$772$$ 3.64159 0.131064
$$773$$ 0.530866 0.0190939 0.00954697 0.999954i $$-0.496961\pi$$
0.00954697 + 0.999954i $$0.496961\pi$$
$$774$$ −15.4538 −0.555475
$$775$$ 0 0
$$776$$ −15.1272 −0.543035
$$777$$ 1.16968 0.0419621
$$778$$ 13.0015 0.466127
$$779$$ 9.57477 0.343052
$$780$$ 0 0
$$781$$ −3.91546 −0.140106
$$782$$ −39.9160 −1.42739
$$783$$ 2.31441 0.0827102
$$784$$ 5.34367 0.190846
$$785$$ 0 0
$$786$$ −0.804219 −0.0286855
$$787$$ 2.16917 0.0773227 0.0386613 0.999252i $$-0.487691\pi$$
0.0386613 + 0.999252i $$0.487691\pi$$
$$788$$ 10.1939 0.363144
$$789$$ 5.43432 0.193467
$$790$$ 0 0
$$791$$ 55.9642 1.98986
$$792$$ −0.837984 −0.0297765
$$793$$ −9.09932 −0.323126
$$794$$ 26.6658 0.946336
$$795$$ 0 0
$$796$$ −5.25299 −0.186187
$$797$$ −26.2510 −0.929857 −0.464928 0.885348i $$-0.653920\pi$$
−0.464928 + 0.885348i $$0.653920\pi$$
$$798$$ 6.82572 0.241628
$$799$$ −35.1755 −1.24442
$$800$$ 0 0
$$801$$ 29.6997 1.04939
$$802$$ −20.3546 −0.718747
$$803$$ −0.496986 −0.0175382
$$804$$ 2.01198 0.0709569
$$805$$ 0 0
$$806$$ −69.6766 −2.45425
$$807$$ −6.18451 −0.217705
$$808$$ −2.63730 −0.0927798
$$809$$ 6.67443 0.234661 0.117330 0.993093i $$-0.462566\pi$$
0.117330 + 0.993093i $$0.462566\pi$$
$$810$$ 0 0
$$811$$ −8.24768 −0.289615 −0.144808 0.989460i $$-0.546256\pi$$
−0.144808 + 0.989460i $$0.546256\pi$$
$$812$$ 4.14728 0.145541
$$813$$ 5.46598 0.191700
$$814$$ −0.290044 −0.0101660
$$815$$ 0 0
$$816$$ −2.05721 −0.0720167
$$817$$ −31.2136 −1.09203
$$818$$ −29.9603 −1.04754
$$819$$ 72.3292 2.52739
$$820$$ 0 0
$$821$$ −9.61631 −0.335611 −0.167806 0.985820i $$-0.553668\pi$$
−0.167806 + 0.985820i $$0.553668\pi$$
$$822$$ −2.28841 −0.0798174
$$823$$ −32.7309 −1.14093 −0.570464 0.821322i $$-0.693237\pi$$
−0.570464 + 0.821322i $$0.693237\pi$$
$$824$$ 1.72469 0.0600825
$$825$$ 0 0
$$826$$ 20.8791 0.726478
$$827$$ 39.9956 1.39078 0.695391 0.718631i $$-0.255231\pi$$
0.695391 + 0.718631i $$0.255231\pi$$
$$828$$ 18.6632 0.648591
$$829$$ 36.8620 1.28027 0.640134 0.768263i $$-0.278879\pi$$
0.640134 + 0.768263i $$0.278879\pi$$
$$830$$ 0 0
$$831$$ 5.19708 0.180285
$$832$$ 7.12558 0.247035
$$833$$ −33.0197 −1.14406
$$834$$ 3.26471 0.113048
$$835$$ 0 0
$$836$$ −1.69256 −0.0585385
$$837$$ −19.1719 −0.662679
$$838$$ −23.3149 −0.805400
$$839$$ 36.7629 1.26919 0.634597 0.772843i $$-0.281166\pi$$
0.634597 + 0.772843i $$0.281166\pi$$
$$840$$ 0 0
$$841$$ −27.6066 −0.951951
$$842$$ −24.0289 −0.828089
$$843$$ −7.27991 −0.250733
$$844$$ −4.81998 −0.165910
$$845$$ 0 0
$$846$$ 16.4467 0.565450
$$847$$ 38.3514 1.31777
$$848$$ 9.32034 0.320062
$$849$$ −0.748340 −0.0256830
$$850$$ 0 0
$$851$$ 6.45973 0.221437
$$852$$ 4.49433 0.153973
$$853$$ −15.9547 −0.546278 −0.273139 0.961975i $$-0.588062\pi$$
−0.273139 + 0.961975i $$0.588062\pi$$
$$854$$ −4.48654 −0.153526
$$855$$ 0 0
$$856$$ −10.0279 −0.342748
$$857$$ 10.2157 0.348963 0.174481 0.984660i $$-0.444175\pi$$
0.174481 + 0.984660i $$0.444175\pi$$
$$858$$ 0.688066 0.0234902
$$859$$ 7.41899 0.253133 0.126566 0.991958i $$-0.459604\pi$$
0.126566 + 0.991958i $$0.459604\pi$$
$$860$$ 0 0
$$861$$ −1.91918 −0.0654055
$$862$$ 19.2917 0.657077
$$863$$ −9.35839 −0.318563 −0.159282 0.987233i $$-0.550918\pi$$
−0.159282 + 0.987233i $$0.550918\pi$$
$$864$$ 1.96065 0.0667025
$$865$$ 0 0
$$866$$ 12.5816 0.427541
$$867$$ 7.05220 0.239505
$$868$$ −34.3549 −1.16608
$$869$$ −2.39409 −0.0812139
$$870$$ 0 0
$$871$$ 43.0623 1.45911
$$872$$ −4.87361 −0.165041
$$873$$ −43.7050 −1.47919
$$874$$ 37.6959 1.27508
$$875$$ 0 0
$$876$$ 0.570460 0.0192741
$$877$$ 13.1486 0.443997 0.221998 0.975047i $$-0.428742\pi$$
0.221998 + 0.975047i $$0.428742\pi$$
$$878$$ −14.0428 −0.473921
$$879$$ −8.94882 −0.301836
$$880$$ 0 0
$$881$$ 26.7781 0.902178 0.451089 0.892479i $$-0.351036\pi$$
0.451089 + 0.892479i $$0.351036\pi$$
$$882$$ 15.4387 0.519849
$$883$$ 16.7245 0.562824 0.281412 0.959587i $$-0.409197\pi$$
0.281412 + 0.959587i $$0.409197\pi$$
$$884$$ −44.0304 −1.48090
$$885$$ 0 0
$$886$$ 24.5696 0.825432
$$887$$ −18.1469 −0.609312 −0.304656 0.952462i $$-0.598542\pi$$
−0.304656 + 0.952462i $$0.598542\pi$$
$$888$$ 0.332924 0.0111722
$$889$$ −4.61181 −0.154675
$$890$$ 0 0
$$891$$ −2.32463 −0.0778780
$$892$$ 3.70392 0.124017
$$893$$ 33.2191 1.11164
$$894$$ 3.02191 0.101068
$$895$$ 0 0
$$896$$ 3.51336 0.117373
$$897$$ −15.3243 −0.511663
$$898$$ 12.7574 0.425721
$$899$$ −11.5427 −0.384971
$$900$$ 0 0
$$901$$ −57.5923 −1.91868
$$902$$ 0.475896 0.0158456
$$903$$ 6.25650 0.208203
$$904$$ 15.9290 0.529790
$$905$$ 0 0
$$906$$ −6.72387 −0.223386
$$907$$ −50.1481 −1.66514 −0.832570 0.553920i $$-0.813131\pi$$
−0.832570 + 0.553920i $$0.813131\pi$$
$$908$$ 6.31512 0.209575
$$909$$ −7.61958 −0.252725
$$910$$ 0 0
$$911$$ 2.64691 0.0876960 0.0438480 0.999038i $$-0.486038\pi$$
0.0438480 + 0.999038i $$0.486038\pi$$
$$912$$ 1.94279 0.0643322
$$913$$ −0.701348 −0.0232112
$$914$$ −3.34889 −0.110771
$$915$$ 0 0
$$916$$ −20.5548 −0.679150
$$917$$ −8.48693 −0.280263
$$918$$ −12.1152 −0.399862
$$919$$ −39.5369 −1.30420 −0.652101 0.758132i $$-0.726112\pi$$
−0.652101 + 0.758132i $$0.726112\pi$$
$$920$$ 0 0
$$921$$ −7.98947 −0.263262
$$922$$ 0.859501 0.0283062
$$923$$ 96.1920 3.16620
$$924$$ 0.339260 0.0111608
$$925$$ 0 0
$$926$$ −13.8711 −0.455831
$$927$$ 4.98292 0.163660
$$928$$ 1.18043 0.0387496
$$929$$ −39.2999 −1.28939 −0.644693 0.764441i $$-0.723015\pi$$
−0.644693 + 0.764441i $$0.723015\pi$$
$$930$$ 0 0
$$931$$ 31.1832 1.02199
$$932$$ 21.4393 0.702267
$$933$$ −1.68980 −0.0553216
$$934$$ −40.9281 −1.33921
$$935$$ 0 0
$$936$$ 20.5869 0.672905
$$937$$ 27.7584 0.906826 0.453413 0.891301i $$-0.350206\pi$$
0.453413 + 0.891301i $$0.350206\pi$$
$$938$$ 21.2324 0.693263
$$939$$ 2.98617 0.0974499
$$940$$ 0 0
$$941$$ 48.2212 1.57197 0.785983 0.618249i $$-0.212158\pi$$
0.785983 + 0.618249i $$0.212158\pi$$
$$942$$ 5.74885 0.187308
$$943$$ −10.5989 −0.345149
$$944$$ 5.94279 0.193421
$$945$$ 0 0
$$946$$ −1.55141 −0.0504408
$$947$$ 26.5111 0.861495 0.430748 0.902472i $$-0.358250\pi$$
0.430748 + 0.902472i $$0.358250\pi$$
$$948$$ 2.74803 0.0892519
$$949$$ 12.2095 0.396339
$$950$$ 0 0
$$951$$ 7.33037 0.237703
$$952$$ −21.7098 −0.703617
$$953$$ 31.8487 1.03168 0.515840 0.856685i $$-0.327480\pi$$
0.515840 + 0.856685i $$0.327480\pi$$
$$954$$ 26.9280 0.871825
$$955$$ 0 0
$$956$$ −20.7479 −0.671034
$$957$$ 0.113986 0.00368464
$$958$$ −8.58990 −0.277527
$$959$$ −24.1496 −0.779832
$$960$$ 0 0
$$961$$ 64.6166 2.08441
$$962$$ 7.12558 0.229738
$$963$$ −28.9723 −0.933620
$$964$$ 10.7731 0.346978
$$965$$ 0 0
$$966$$ −7.55583 −0.243105
$$967$$ −19.4699 −0.626109 −0.313054 0.949735i $$-0.601352\pi$$
−0.313054 + 0.949735i $$0.601352\pi$$
$$968$$ 10.9159 0.350849
$$969$$ −12.0049 −0.385653
$$970$$ 0 0
$$971$$ −21.8178 −0.700168 −0.350084 0.936718i $$-0.613847\pi$$
−0.350084 + 0.936718i $$0.613847\pi$$
$$972$$ 8.55024 0.274249
$$973$$ 34.4525 1.10450
$$974$$ −25.8389 −0.827932
$$975$$ 0 0
$$976$$ −1.27699 −0.0408756
$$977$$ 41.8705 1.33956 0.669779 0.742561i $$-0.266389\pi$$
0.669779 + 0.742561i $$0.266389\pi$$
$$978$$ −5.55654 −0.177679
$$979$$ 2.98157 0.0952913
$$980$$ 0 0
$$981$$ −14.0806 −0.449560
$$982$$ −11.4864 −0.366547
$$983$$ 37.0175 1.18067 0.590337 0.807157i $$-0.298995\pi$$
0.590337 + 0.807157i $$0.298995\pi$$
$$984$$ −0.546253 −0.0174139
$$985$$ 0 0
$$986$$ −7.29413 −0.232292
$$987$$ −6.65849 −0.211942
$$988$$ 41.5815 1.32288
$$989$$ 34.5524 1.09870
$$990$$ 0 0
$$991$$ −28.4749 −0.904534 −0.452267 0.891883i $$-0.649385\pi$$
−0.452267 + 0.891883i $$0.649385\pi$$
$$992$$ −9.77838 −0.310464
$$993$$ −3.56677 −0.113188
$$994$$ 47.4287 1.50435
$$995$$ 0 0
$$996$$ 0.805036 0.0255085
$$997$$ 14.2442 0.451118 0.225559 0.974229i $$-0.427579\pi$$
0.225559 + 0.974229i $$0.427579\pi$$
$$998$$ 9.74599 0.308504
$$999$$ 1.96065 0.0620321
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 1850.2.a.bd.1.3 5
5.2 odd 4 370.2.b.d.149.3 10
5.3 odd 4 370.2.b.d.149.8 yes 10
5.4 even 2 1850.2.a.be.1.3 5
15.2 even 4 3330.2.d.p.1999.9 10
15.8 even 4 3330.2.d.p.1999.4 10

By twisted newform
Twist Min Dim Char Parity Ord Type
370.2.b.d.149.3 10 5.2 odd 4
370.2.b.d.149.8 yes 10 5.3 odd 4
1850.2.a.bd.1.3 5 1.1 even 1 trivial
1850.2.a.be.1.3 5 5.4 even 2
3330.2.d.p.1999.4 10 15.8 even 4
3330.2.d.p.1999.9 10 15.2 even 4