# Properties

 Label 1850.2.a.bb.1.2 Level $1850$ Weight $2$ Character 1850.1 Self dual yes Analytic conductor $14.772$ Analytic rank $1$ Dimension $3$ CM no Inner twists $1$

# Related objects

## 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: $$1$$ Dimension: $$3$$ Coefficient field: 3.3.148.1 Defining polynomial: $$x^{3} - x^{2} - 3x + 1$$ x^3 - x^2 - 3*x + 1 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.2 Root $$2.17009$$ of defining polynomial Character $$\chi$$ $$=$$ 1850.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+1.00000 q^{2} -1.53919 q^{3} +1.00000 q^{4} -1.53919 q^{6} +2.87936 q^{7} +1.00000 q^{8} -0.630898 q^{9} +O(q^{10})$$ $$q+1.00000 q^{2} -1.53919 q^{3} +1.00000 q^{4} -1.53919 q^{6} +2.87936 q^{7} +1.00000 q^{8} -0.630898 q^{9} -1.09171 q^{11} -1.53919 q^{12} -4.53919 q^{13} +2.87936 q^{14} +1.00000 q^{16} -2.80098 q^{17} -0.630898 q^{18} -5.04945 q^{19} -4.43188 q^{21} -1.09171 q^{22} -7.41855 q^{23} -1.53919 q^{24} -4.53919 q^{26} +5.58864 q^{27} +2.87936 q^{28} +6.68035 q^{29} +3.51026 q^{31} +1.00000 q^{32} +1.68035 q^{33} -2.80098 q^{34} -0.630898 q^{36} -1.00000 q^{37} -5.04945 q^{38} +6.98667 q^{39} -8.07838 q^{41} -4.43188 q^{42} -10.2329 q^{43} -1.09171 q^{44} -7.41855 q^{46} -8.68035 q^{47} -1.53919 q^{48} +1.29072 q^{49} +4.31124 q^{51} -4.53919 q^{52} +10.0989 q^{53} +5.58864 q^{54} +2.87936 q^{56} +7.77205 q^{57} +6.68035 q^{58} +10.2329 q^{59} +6.29791 q^{61} +3.51026 q^{62} -1.81658 q^{63} +1.00000 q^{64} +1.68035 q^{66} -13.2979 q^{67} -2.80098 q^{68} +11.4186 q^{69} +6.29791 q^{71} -0.630898 q^{72} -12.7093 q^{73} -1.00000 q^{74} -5.04945 q^{76} -3.14342 q^{77} +6.98667 q^{78} +2.58145 q^{79} -6.70928 q^{81} -8.07838 q^{82} -8.48360 q^{83} -4.43188 q^{84} -10.2329 q^{86} -10.2823 q^{87} -1.09171 q^{88} -6.51026 q^{89} -13.0700 q^{91} -7.41855 q^{92} -5.40295 q^{93} -8.68035 q^{94} -1.53919 q^{96} -3.07838 q^{97} +1.29072 q^{98} +0.688756 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q + 3 q^{2} - 3 q^{3} + 3 q^{4} - 3 q^{6} - 4 q^{7} + 3 q^{8} + 2 q^{9}+O(q^{10})$$ 3 * q + 3 * q^2 - 3 * q^3 + 3 * q^4 - 3 * q^6 - 4 * q^7 + 3 * q^8 + 2 * q^9 $$3 q + 3 q^{2} - 3 q^{3} + 3 q^{4} - 3 q^{6} - 4 q^{7} + 3 q^{8} + 2 q^{9} - q^{11} - 3 q^{12} - 12 q^{13} - 4 q^{14} + 3 q^{16} + q^{17} + 2 q^{18} + 3 q^{19} - q^{22} - 8 q^{23} - 3 q^{24} - 12 q^{26} - 3 q^{27} - 4 q^{28} - 2 q^{29} - 6 q^{31} + 3 q^{32} - 17 q^{33} + q^{34} + 2 q^{36} - 3 q^{37} + 3 q^{38} + 20 q^{39} - 21 q^{41} - 8 q^{43} - q^{44} - 8 q^{46} - 4 q^{47} - 3 q^{48} + 11 q^{49} - 13 q^{51} - 12 q^{52} - 6 q^{53} - 3 q^{54} - 4 q^{56} - q^{57} - 2 q^{58} + 8 q^{59} - 8 q^{61} - 6 q^{62} - 10 q^{63} + 3 q^{64} - 17 q^{66} - 13 q^{67} + q^{68} + 20 q^{69} - 8 q^{71} + 2 q^{72} - 31 q^{73} - 3 q^{74} + 3 q^{76} - 2 q^{77} + 20 q^{78} + 22 q^{79} - 13 q^{81} - 21 q^{82} - 7 q^{83} - 8 q^{86} + 10 q^{87} - q^{88} - 3 q^{89} + 12 q^{91} - 8 q^{92} + 12 q^{93} - 4 q^{94} - 3 q^{96} - 6 q^{97} + 11 q^{98} + 28 q^{99}+O(q^{100})$$ 3 * q + 3 * q^2 - 3 * q^3 + 3 * q^4 - 3 * q^6 - 4 * q^7 + 3 * q^8 + 2 * q^9 - q^11 - 3 * q^12 - 12 * q^13 - 4 * q^14 + 3 * q^16 + q^17 + 2 * q^18 + 3 * q^19 - q^22 - 8 * q^23 - 3 * q^24 - 12 * q^26 - 3 * q^27 - 4 * q^28 - 2 * q^29 - 6 * q^31 + 3 * q^32 - 17 * q^33 + q^34 + 2 * q^36 - 3 * q^37 + 3 * q^38 + 20 * q^39 - 21 * q^41 - 8 * q^43 - q^44 - 8 * q^46 - 4 * q^47 - 3 * q^48 + 11 * q^49 - 13 * q^51 - 12 * q^52 - 6 * q^53 - 3 * q^54 - 4 * q^56 - q^57 - 2 * q^58 + 8 * q^59 - 8 * q^61 - 6 * q^62 - 10 * q^63 + 3 * q^64 - 17 * q^66 - 13 * q^67 + q^68 + 20 * q^69 - 8 * q^71 + 2 * q^72 - 31 * q^73 - 3 * q^74 + 3 * q^76 - 2 * q^77 + 20 * q^78 + 22 * q^79 - 13 * q^81 - 21 * q^82 - 7 * q^83 - 8 * q^86 + 10 * q^87 - q^88 - 3 * q^89 + 12 * q^91 - 8 * q^92 + 12 * q^93 - 4 * q^94 - 3 * q^96 - 6 * q^97 + 11 * q^98 + 28 * q^99

## 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
$$3$$ −1.53919 −0.888651 −0.444326 0.895865i $$-0.646557\pi$$
−0.444326 + 0.895865i $$0.646557\pi$$
$$4$$ 1.00000 0.500000
$$5$$ 0 0
$$6$$ −1.53919 −0.628371
$$7$$ 2.87936 1.08830 0.544148 0.838989i $$-0.316853\pi$$
0.544148 + 0.838989i $$0.316853\pi$$
$$8$$ 1.00000 0.353553
$$9$$ −0.630898 −0.210299
$$10$$ 0 0
$$11$$ −1.09171 −0.329163 −0.164581 0.986364i $$-0.552627\pi$$
−0.164581 + 0.986364i $$0.552627\pi$$
$$12$$ −1.53919 −0.444326
$$13$$ −4.53919 −1.25894 −0.629472 0.777023i $$-0.716729\pi$$
−0.629472 + 0.777023i $$0.716729\pi$$
$$14$$ 2.87936 0.769542
$$15$$ 0 0
$$16$$ 1.00000 0.250000
$$17$$ −2.80098 −0.679338 −0.339669 0.940545i $$-0.610315\pi$$
−0.339669 + 0.940545i $$0.610315\pi$$
$$18$$ −0.630898 −0.148704
$$19$$ −5.04945 −1.15842 −0.579211 0.815177i $$-0.696639\pi$$
−0.579211 + 0.815177i $$0.696639\pi$$
$$20$$ 0 0
$$21$$ −4.43188 −0.967116
$$22$$ −1.09171 −0.232753
$$23$$ −7.41855 −1.54687 −0.773437 0.633873i $$-0.781464\pi$$
−0.773437 + 0.633873i $$0.781464\pi$$
$$24$$ −1.53919 −0.314186
$$25$$ 0 0
$$26$$ −4.53919 −0.890208
$$27$$ 5.58864 1.07553
$$28$$ 2.87936 0.544148
$$29$$ 6.68035 1.24051 0.620255 0.784401i $$-0.287029\pi$$
0.620255 + 0.784401i $$0.287029\pi$$
$$30$$ 0 0
$$31$$ 3.51026 0.630461 0.315231 0.949015i $$-0.397918\pi$$
0.315231 + 0.949015i $$0.397918\pi$$
$$32$$ 1.00000 0.176777
$$33$$ 1.68035 0.292511
$$34$$ −2.80098 −0.480365
$$35$$ 0 0
$$36$$ −0.630898 −0.105150
$$37$$ −1.00000 −0.164399
$$38$$ −5.04945 −0.819129
$$39$$ 6.98667 1.11876
$$40$$ 0 0
$$41$$ −8.07838 −1.26163 −0.630815 0.775933i $$-0.717279\pi$$
−0.630815 + 0.775933i $$0.717279\pi$$
$$42$$ −4.43188 −0.683854
$$43$$ −10.2329 −1.56050 −0.780249 0.625469i $$-0.784908\pi$$
−0.780249 + 0.625469i $$0.784908\pi$$
$$44$$ −1.09171 −0.164581
$$45$$ 0 0
$$46$$ −7.41855 −1.09381
$$47$$ −8.68035 −1.26616 −0.633079 0.774087i $$-0.718209\pi$$
−0.633079 + 0.774087i $$0.718209\pi$$
$$48$$ −1.53919 −0.222163
$$49$$ 1.29072 0.184389
$$50$$ 0 0
$$51$$ 4.31124 0.603695
$$52$$ −4.53919 −0.629472
$$53$$ 10.0989 1.38719 0.693595 0.720365i $$-0.256026\pi$$
0.693595 + 0.720365i $$0.256026\pi$$
$$54$$ 5.58864 0.760517
$$55$$ 0 0
$$56$$ 2.87936 0.384771
$$57$$ 7.77205 1.02943
$$58$$ 6.68035 0.877172
$$59$$ 10.2329 1.33221 0.666103 0.745860i $$-0.267961\pi$$
0.666103 + 0.745860i $$0.267961\pi$$
$$60$$ 0 0
$$61$$ 6.29791 0.806365 0.403183 0.915120i $$-0.367904\pi$$
0.403183 + 0.915120i $$0.367904\pi$$
$$62$$ 3.51026 0.445803
$$63$$ −1.81658 −0.228868
$$64$$ 1.00000 0.125000
$$65$$ 0 0
$$66$$ 1.68035 0.206836
$$67$$ −13.2979 −1.62460 −0.812299 0.583241i $$-0.801784\pi$$
−0.812299 + 0.583241i $$0.801784\pi$$
$$68$$ −2.80098 −0.339669
$$69$$ 11.4186 1.37463
$$70$$ 0 0
$$71$$ 6.29791 0.747425 0.373712 0.927545i $$-0.378085\pi$$
0.373712 + 0.927545i $$0.378085\pi$$
$$72$$ −0.630898 −0.0743520
$$73$$ −12.7093 −1.48751 −0.743754 0.668453i $$-0.766957\pi$$
−0.743754 + 0.668453i $$0.766957\pi$$
$$74$$ −1.00000 −0.116248
$$75$$ 0 0
$$76$$ −5.04945 −0.579211
$$77$$ −3.14342 −0.358226
$$78$$ 6.98667 0.791084
$$79$$ 2.58145 0.290436 0.145218 0.989400i $$-0.453612\pi$$
0.145218 + 0.989400i $$0.453612\pi$$
$$80$$ 0 0
$$81$$ −6.70928 −0.745475
$$82$$ −8.07838 −0.892108
$$83$$ −8.48360 −0.931196 −0.465598 0.884996i $$-0.654161\pi$$
−0.465598 + 0.884996i $$0.654161\pi$$
$$84$$ −4.43188 −0.483558
$$85$$ 0 0
$$86$$ −10.2329 −1.10344
$$87$$ −10.2823 −1.10238
$$88$$ −1.09171 −0.116377
$$89$$ −6.51026 −0.690086 −0.345043 0.938587i $$-0.612136\pi$$
−0.345043 + 0.938587i $$0.612136\pi$$
$$90$$ 0 0
$$91$$ −13.0700 −1.37010
$$92$$ −7.41855 −0.773437
$$93$$ −5.40295 −0.560260
$$94$$ −8.68035 −0.895309
$$95$$ 0 0
$$96$$ −1.53919 −0.157093
$$97$$ −3.07838 −0.312562 −0.156281 0.987713i $$-0.549951\pi$$
−0.156281 + 0.987713i $$0.549951\pi$$
$$98$$ 1.29072 0.130383
$$99$$ 0.688756 0.0692226
$$100$$ 0 0
$$101$$ −15.9155 −1.58365 −0.791825 0.610748i $$-0.790869\pi$$
−0.791825 + 0.610748i $$0.790869\pi$$
$$102$$ 4.31124 0.426877
$$103$$ 10.5886 1.04333 0.521665 0.853151i $$-0.325311\pi$$
0.521665 + 0.853151i $$0.325311\pi$$
$$104$$ −4.53919 −0.445104
$$105$$ 0 0
$$106$$ 10.0989 0.980892
$$107$$ 4.03612 0.390186 0.195093 0.980785i $$-0.437499\pi$$
0.195093 + 0.980785i $$0.437499\pi$$
$$108$$ 5.58864 0.537767
$$109$$ −4.49693 −0.430728 −0.215364 0.976534i $$-0.569094\pi$$
−0.215364 + 0.976534i $$0.569094\pi$$
$$110$$ 0 0
$$111$$ 1.53919 0.146093
$$112$$ 2.87936 0.272074
$$113$$ −3.58864 −0.337591 −0.168795 0.985651i $$-0.553988\pi$$
−0.168795 + 0.985651i $$0.553988\pi$$
$$114$$ 7.77205 0.727920
$$115$$ 0 0
$$116$$ 6.68035 0.620255
$$117$$ 2.86376 0.264755
$$118$$ 10.2329 0.942012
$$119$$ −8.06505 −0.739322
$$120$$ 0 0
$$121$$ −9.80817 −0.891652
$$122$$ 6.29791 0.570186
$$123$$ 12.4341 1.12115
$$124$$ 3.51026 0.315231
$$125$$ 0 0
$$126$$ −1.81658 −0.161834
$$127$$ −8.14116 −0.722411 −0.361205 0.932486i $$-0.617635\pi$$
−0.361205 + 0.932486i $$0.617635\pi$$
$$128$$ 1.00000 0.0883883
$$129$$ 15.7503 1.38674
$$130$$ 0 0
$$131$$ −1.46800 −0.128260 −0.0641298 0.997942i $$-0.520427\pi$$
−0.0641298 + 0.997942i $$0.520427\pi$$
$$132$$ 1.68035 0.146255
$$133$$ −14.5392 −1.26071
$$134$$ −13.2979 −1.14876
$$135$$ 0 0
$$136$$ −2.80098 −0.240182
$$137$$ 6.17727 0.527760 0.263880 0.964555i $$-0.414998\pi$$
0.263880 + 0.964555i $$0.414998\pi$$
$$138$$ 11.4186 0.972012
$$139$$ 13.0338 1.10552 0.552758 0.833342i $$-0.313575\pi$$
0.552758 + 0.833342i $$0.313575\pi$$
$$140$$ 0 0
$$141$$ 13.3607 1.12517
$$142$$ 6.29791 0.528509
$$143$$ 4.95547 0.414397
$$144$$ −0.630898 −0.0525748
$$145$$ 0 0
$$146$$ −12.7093 −1.05183
$$147$$ −1.98667 −0.163858
$$148$$ −1.00000 −0.0821995
$$149$$ −4.40522 −0.360890 −0.180445 0.983585i $$-0.557754\pi$$
−0.180445 + 0.983585i $$0.557754\pi$$
$$150$$ 0 0
$$151$$ 2.92162 0.237758 0.118879 0.992909i $$-0.462070\pi$$
0.118879 + 0.992909i $$0.462070\pi$$
$$152$$ −5.04945 −0.409564
$$153$$ 1.76713 0.142864
$$154$$ −3.14342 −0.253304
$$155$$ 0 0
$$156$$ 6.98667 0.559381
$$157$$ −22.1906 −1.77100 −0.885502 0.464636i $$-0.846185\pi$$
−0.885502 + 0.464636i $$0.846185\pi$$
$$158$$ 2.58145 0.205369
$$159$$ −15.5441 −1.23273
$$160$$ 0 0
$$161$$ −21.3607 −1.68346
$$162$$ −6.70928 −0.527130
$$163$$ 14.6248 1.14550 0.572750 0.819730i $$-0.305877\pi$$
0.572750 + 0.819730i $$0.305877\pi$$
$$164$$ −8.07838 −0.630815
$$165$$ 0 0
$$166$$ −8.48360 −0.658455
$$167$$ 6.34736 0.491174 0.245587 0.969375i $$-0.421019\pi$$
0.245587 + 0.969375i $$0.421019\pi$$
$$168$$ −4.43188 −0.341927
$$169$$ 7.60424 0.584941
$$170$$ 0 0
$$171$$ 3.18568 0.243615
$$172$$ −10.2329 −0.780249
$$173$$ 1.23513 0.0939054 0.0469527 0.998897i $$-0.485049\pi$$
0.0469527 + 0.998897i $$0.485049\pi$$
$$174$$ −10.2823 −0.779500
$$175$$ 0 0
$$176$$ −1.09171 −0.0822906
$$177$$ −15.7503 −1.18387
$$178$$ −6.51026 −0.487965
$$179$$ 1.76487 0.131912 0.0659562 0.997823i $$-0.478990\pi$$
0.0659562 + 0.997823i $$0.478990\pi$$
$$180$$ 0 0
$$181$$ 6.49693 0.482913 0.241456 0.970412i $$-0.422375\pi$$
0.241456 + 0.970412i $$0.422375\pi$$
$$182$$ −13.0700 −0.968810
$$183$$ −9.69368 −0.716577
$$184$$ −7.41855 −0.546903
$$185$$ 0 0
$$186$$ −5.40295 −0.396164
$$187$$ 3.05786 0.223613
$$188$$ −8.68035 −0.633079
$$189$$ 16.0917 1.17050
$$190$$ 0 0
$$191$$ 21.2039 1.53426 0.767131 0.641490i $$-0.221683\pi$$
0.767131 + 0.641490i $$0.221683\pi$$
$$192$$ −1.53919 −0.111081
$$193$$ −8.14342 −0.586177 −0.293088 0.956085i $$-0.594683\pi$$
−0.293088 + 0.956085i $$0.594683\pi$$
$$194$$ −3.07838 −0.221015
$$195$$ 0 0
$$196$$ 1.29072 0.0921946
$$197$$ 24.8443 1.77008 0.885041 0.465513i $$-0.154130\pi$$
0.885041 + 0.465513i $$0.154130\pi$$
$$198$$ 0.688756 0.0489478
$$199$$ −8.47027 −0.600441 −0.300221 0.953870i $$-0.597060\pi$$
−0.300221 + 0.953870i $$0.597060\pi$$
$$200$$ 0 0
$$201$$ 20.4680 1.44370
$$202$$ −15.9155 −1.11981
$$203$$ 19.2351 1.35004
$$204$$ 4.31124 0.301847
$$205$$ 0 0
$$206$$ 10.5886 0.737745
$$207$$ 4.68035 0.325307
$$208$$ −4.53919 −0.314736
$$209$$ 5.51253 0.381309
$$210$$ 0 0
$$211$$ 21.7370 1.49644 0.748218 0.663453i $$-0.230910\pi$$
0.748218 + 0.663453i $$0.230910\pi$$
$$212$$ 10.0989 0.693595
$$213$$ −9.69368 −0.664200
$$214$$ 4.03612 0.275903
$$215$$ 0 0
$$216$$ 5.58864 0.380259
$$217$$ 10.1073 0.686129
$$218$$ −4.49693 −0.304570
$$219$$ 19.5620 1.32188
$$220$$ 0 0
$$221$$ 12.7142 0.855249
$$222$$ 1.53919 0.103304
$$223$$ −16.6537 −1.11521 −0.557607 0.830105i $$-0.688280\pi$$
−0.557607 + 0.830105i $$0.688280\pi$$
$$224$$ 2.87936 0.192385
$$225$$ 0 0
$$226$$ −3.58864 −0.238713
$$227$$ 11.2123 0.744190 0.372095 0.928195i $$-0.378640\pi$$
0.372095 + 0.928195i $$0.378640\pi$$
$$228$$ 7.77205 0.514717
$$229$$ −7.62863 −0.504114 −0.252057 0.967712i $$-0.581107\pi$$
−0.252057 + 0.967712i $$0.581107\pi$$
$$230$$ 0 0
$$231$$ 4.83832 0.318338
$$232$$ 6.68035 0.438586
$$233$$ 0.0494483 0.00323947 0.00161973 0.999999i $$-0.499484\pi$$
0.00161973 + 0.999999i $$0.499484\pi$$
$$234$$ 2.86376 0.187210
$$235$$ 0 0
$$236$$ 10.2329 0.666103
$$237$$ −3.97334 −0.258096
$$238$$ −8.06505 −0.522779
$$239$$ −14.7070 −0.951317 −0.475659 0.879630i $$-0.657790\pi$$
−0.475659 + 0.879630i $$0.657790\pi$$
$$240$$ 0 0
$$241$$ −14.8999 −0.959786 −0.479893 0.877327i $$-0.659324\pi$$
−0.479893 + 0.877327i $$0.659324\pi$$
$$242$$ −9.80817 −0.630493
$$243$$ −6.43907 −0.413067
$$244$$ 6.29791 0.403183
$$245$$ 0 0
$$246$$ 12.4341 0.792772
$$247$$ 22.9204 1.45839
$$248$$ 3.51026 0.222902
$$249$$ 13.0579 0.827508
$$250$$ 0 0
$$251$$ 6.23513 0.393558 0.196779 0.980448i $$-0.436952\pi$$
0.196779 + 0.980448i $$0.436952\pi$$
$$252$$ −1.81658 −0.114434
$$253$$ 8.09890 0.509173
$$254$$ −8.14116 −0.510822
$$255$$ 0 0
$$256$$ 1.00000 0.0625000
$$257$$ −16.1568 −1.00783 −0.503915 0.863753i $$-0.668108\pi$$
−0.503915 + 0.863753i $$0.668108\pi$$
$$258$$ 15.7503 0.980572
$$259$$ −2.87936 −0.178915
$$260$$ 0 0
$$261$$ −4.21461 −0.260878
$$262$$ −1.46800 −0.0906933
$$263$$ 18.6537 1.15024 0.575118 0.818071i $$-0.304956\pi$$
0.575118 + 0.818071i $$0.304956\pi$$
$$264$$ 1.68035 0.103418
$$265$$ 0 0
$$266$$ −14.5392 −0.891455
$$267$$ 10.0205 0.613246
$$268$$ −13.2979 −0.812299
$$269$$ 21.8310 1.33106 0.665529 0.746372i $$-0.268206\pi$$
0.665529 + 0.746372i $$0.268206\pi$$
$$270$$ 0 0
$$271$$ 24.9783 1.51732 0.758661 0.651486i $$-0.225854\pi$$
0.758661 + 0.651486i $$0.225854\pi$$
$$272$$ −2.80098 −0.169835
$$273$$ 20.1171 1.21755
$$274$$ 6.17727 0.373183
$$275$$ 0 0
$$276$$ 11.4186 0.687316
$$277$$ 0.822726 0.0494328 0.0247164 0.999695i $$-0.492132\pi$$
0.0247164 + 0.999695i $$0.492132\pi$$
$$278$$ 13.0338 0.781718
$$279$$ −2.21461 −0.132585
$$280$$ 0 0
$$281$$ −1.65142 −0.0985153 −0.0492576 0.998786i $$-0.515686\pi$$
−0.0492576 + 0.998786i $$0.515686\pi$$
$$282$$ 13.3607 0.795618
$$283$$ −15.1340 −0.899621 −0.449811 0.893124i $$-0.648508\pi$$
−0.449811 + 0.893124i $$0.648508\pi$$
$$284$$ 6.29791 0.373712
$$285$$ 0 0
$$286$$ 4.95547 0.293023
$$287$$ −23.2606 −1.37303
$$288$$ −0.630898 −0.0371760
$$289$$ −9.15449 −0.538499
$$290$$ 0 0
$$291$$ 4.73820 0.277758
$$292$$ −12.7093 −0.743754
$$293$$ 25.4524 1.48695 0.743473 0.668766i $$-0.233177\pi$$
0.743473 + 0.668766i $$0.233177\pi$$
$$294$$ −1.98667 −0.115865
$$295$$ 0 0
$$296$$ −1.00000 −0.0581238
$$297$$ −6.10116 −0.354025
$$298$$ −4.40522 −0.255188
$$299$$ 33.6742 1.94743
$$300$$ 0 0
$$301$$ −29.4641 −1.69828
$$302$$ 2.92162 0.168120
$$303$$ 24.4969 1.40731
$$304$$ −5.04945 −0.289606
$$305$$ 0 0
$$306$$ 1.76713 0.101020
$$307$$ 27.8176 1.58764 0.793818 0.608155i $$-0.208090\pi$$
0.793818 + 0.608155i $$0.208090\pi$$
$$308$$ −3.14342 −0.179113
$$309$$ −16.2979 −0.927156
$$310$$ 0 0
$$311$$ 0.0650468 0.00368846 0.00184423 0.999998i $$-0.499413\pi$$
0.00184423 + 0.999998i $$0.499413\pi$$
$$312$$ 6.98667 0.395542
$$313$$ −22.5464 −1.27440 −0.637198 0.770700i $$-0.719907\pi$$
−0.637198 + 0.770700i $$0.719907\pi$$
$$314$$ −22.1906 −1.25229
$$315$$ 0 0
$$316$$ 2.58145 0.145218
$$317$$ 24.4775 1.37479 0.687395 0.726283i $$-0.258754\pi$$
0.687395 + 0.726283i $$0.258754\pi$$
$$318$$ −15.5441 −0.871670
$$319$$ −7.29299 −0.408329
$$320$$ 0 0
$$321$$ −6.21235 −0.346739
$$322$$ −21.3607 −1.19038
$$323$$ 14.1434 0.786961
$$324$$ −6.70928 −0.372738
$$325$$ 0 0
$$326$$ 14.6248 0.809990
$$327$$ 6.92162 0.382767
$$328$$ −8.07838 −0.446054
$$329$$ −24.9939 −1.37796
$$330$$ 0 0
$$331$$ −1.91321 −0.105160 −0.0525798 0.998617i $$-0.516744\pi$$
−0.0525798 + 0.998617i $$0.516744\pi$$
$$332$$ −8.48360 −0.465598
$$333$$ 0.630898 0.0345730
$$334$$ 6.34736 0.347312
$$335$$ 0 0
$$336$$ −4.43188 −0.241779
$$337$$ 12.6286 0.687925 0.343963 0.938983i $$-0.388231\pi$$
0.343963 + 0.938983i $$0.388231\pi$$
$$338$$ 7.60424 0.413616
$$339$$ 5.52359 0.300000
$$340$$ 0 0
$$341$$ −3.83218 −0.207524
$$342$$ 3.18568 0.172262
$$343$$ −16.4391 −0.887626
$$344$$ −10.2329 −0.551719
$$345$$ 0 0
$$346$$ 1.23513 0.0664012
$$347$$ 8.33403 0.447394 0.223697 0.974659i $$-0.428187\pi$$
0.223697 + 0.974659i $$0.428187\pi$$
$$348$$ −10.2823 −0.551190
$$349$$ 2.11837 0.113394 0.0566969 0.998391i $$-0.481943\pi$$
0.0566969 + 0.998391i $$0.481943\pi$$
$$350$$ 0 0
$$351$$ −25.3679 −1.35404
$$352$$ −1.09171 −0.0581883
$$353$$ −28.4534 −1.51442 −0.757212 0.653169i $$-0.773439\pi$$
−0.757212 + 0.653169i $$0.773439\pi$$
$$354$$ −15.7503 −0.837120
$$355$$ 0 0
$$356$$ −6.51026 −0.345043
$$357$$ 12.4136 0.656999
$$358$$ 1.76487 0.0932761
$$359$$ 2.65368 0.140056 0.0700280 0.997545i $$-0.477691\pi$$
0.0700280 + 0.997545i $$0.477691\pi$$
$$360$$ 0 0
$$361$$ 6.49693 0.341944
$$362$$ 6.49693 0.341471
$$363$$ 15.0966 0.792368
$$364$$ −13.0700 −0.685052
$$365$$ 0 0
$$366$$ −9.69368 −0.506697
$$367$$ 31.7575 1.65773 0.828864 0.559450i $$-0.188988\pi$$
0.828864 + 0.559450i $$0.188988\pi$$
$$368$$ −7.41855 −0.386719
$$369$$ 5.09663 0.265320
$$370$$ 0 0
$$371$$ 29.0784 1.50967
$$372$$ −5.40295 −0.280130
$$373$$ −25.1122 −1.30026 −0.650131 0.759822i $$-0.725286\pi$$
−0.650131 + 0.759822i $$0.725286\pi$$
$$374$$ 3.05786 0.158118
$$375$$ 0 0
$$376$$ −8.68035 −0.447655
$$377$$ −30.3234 −1.56173
$$378$$ 16.0917 0.827668
$$379$$ 22.5330 1.15744 0.578722 0.815525i $$-0.303551\pi$$
0.578722 + 0.815525i $$0.303551\pi$$
$$380$$ 0 0
$$381$$ 12.5308 0.641971
$$382$$ 21.2039 1.08489
$$383$$ 12.5886 0.643249 0.321625 0.946867i $$-0.395771\pi$$
0.321625 + 0.946867i $$0.395771\pi$$
$$384$$ −1.53919 −0.0785464
$$385$$ 0 0
$$386$$ −8.14342 −0.414489
$$387$$ 6.45589 0.328171
$$388$$ −3.07838 −0.156281
$$389$$ −0.879362 −0.0445854 −0.0222927 0.999751i $$-0.507097\pi$$
−0.0222927 + 0.999751i $$0.507097\pi$$
$$390$$ 0 0
$$391$$ 20.7792 1.05085
$$392$$ 1.29072 0.0651914
$$393$$ 2.25953 0.113978
$$394$$ 24.8443 1.25164
$$395$$ 0 0
$$396$$ 0.688756 0.0346113
$$397$$ 29.6814 1.48967 0.744833 0.667251i $$-0.232529\pi$$
0.744833 + 0.667251i $$0.232529\pi$$
$$398$$ −8.47027 −0.424576
$$399$$ 22.3786 1.12033
$$400$$ 0 0
$$401$$ −13.9867 −0.698461 −0.349230 0.937037i $$-0.613557\pi$$
−0.349230 + 0.937037i $$0.613557\pi$$
$$402$$ 20.4680 1.02085
$$403$$ −15.9337 −0.793716
$$404$$ −15.9155 −0.791825
$$405$$ 0 0
$$406$$ 19.2351 0.954624
$$407$$ 1.09171 0.0541140
$$408$$ 4.31124 0.213438
$$409$$ −21.9060 −1.08318 −0.541592 0.840642i $$-0.682178\pi$$
−0.541592 + 0.840642i $$0.682178\pi$$
$$410$$ 0 0
$$411$$ −9.50799 −0.468995
$$412$$ 10.5886 0.521665
$$413$$ 29.4641 1.44983
$$414$$ 4.68035 0.230026
$$415$$ 0 0
$$416$$ −4.53919 −0.222552
$$417$$ −20.0616 −0.982419
$$418$$ 5.51253 0.269627
$$419$$ −34.8648 −1.70326 −0.851629 0.524146i $$-0.824385\pi$$
−0.851629 + 0.524146i $$0.824385\pi$$
$$420$$ 0 0
$$421$$ −4.76487 −0.232225 −0.116113 0.993236i $$-0.537043\pi$$
−0.116113 + 0.993236i $$0.537043\pi$$
$$422$$ 21.7370 1.05814
$$423$$ 5.47641 0.266272
$$424$$ 10.0989 0.490446
$$425$$ 0 0
$$426$$ −9.69368 −0.469660
$$427$$ 18.1340 0.877564
$$428$$ 4.03612 0.195093
$$429$$ −7.62741 −0.368255
$$430$$ 0 0
$$431$$ 2.66597 0.128415 0.0642076 0.997937i $$-0.479548\pi$$
0.0642076 + 0.997937i $$0.479548\pi$$
$$432$$ 5.58864 0.268883
$$433$$ −32.3074 −1.55259 −0.776297 0.630368i $$-0.782904\pi$$
−0.776297 + 0.630368i $$0.782904\pi$$
$$434$$ 10.1073 0.485166
$$435$$ 0 0
$$436$$ −4.49693 −0.215364
$$437$$ 37.4596 1.79194
$$438$$ 19.5620 0.934707
$$439$$ −22.1568 −1.05748 −0.528742 0.848783i $$-0.677336\pi$$
−0.528742 + 0.848783i $$0.677336\pi$$
$$440$$ 0 0
$$441$$ −0.814315 −0.0387769
$$442$$ 12.7142 0.604753
$$443$$ −39.1061 −1.85799 −0.928993 0.370097i $$-0.879324\pi$$
−0.928993 + 0.370097i $$0.879324\pi$$
$$444$$ 1.53919 0.0730467
$$445$$ 0 0
$$446$$ −16.6537 −0.788575
$$447$$ 6.78047 0.320705
$$448$$ 2.87936 0.136037
$$449$$ 30.8915 1.45786 0.728929 0.684589i $$-0.240018\pi$$
0.728929 + 0.684589i $$0.240018\pi$$
$$450$$ 0 0
$$451$$ 8.81924 0.415282
$$452$$ −3.58864 −0.168795
$$453$$ −4.49693 −0.211284
$$454$$ 11.2123 0.526222
$$455$$ 0 0
$$456$$ 7.77205 0.363960
$$457$$ 0.438025 0.0204899 0.0102450 0.999948i $$-0.496739\pi$$
0.0102450 + 0.999948i $$0.496739\pi$$
$$458$$ −7.62863 −0.356462
$$459$$ −15.6537 −0.730651
$$460$$ 0 0
$$461$$ 0.523590 0.0243860 0.0121930 0.999926i $$-0.496119\pi$$
0.0121930 + 0.999926i $$0.496119\pi$$
$$462$$ 4.83832 0.225099
$$463$$ 29.1845 1.35632 0.678158 0.734916i $$-0.262778\pi$$
0.678158 + 0.734916i $$0.262778\pi$$
$$464$$ 6.68035 0.310127
$$465$$ 0 0
$$466$$ 0.0494483 0.00229065
$$467$$ −9.49079 −0.439181 −0.219591 0.975592i $$-0.570472\pi$$
−0.219591 + 0.975592i $$0.570472\pi$$
$$468$$ 2.86376 0.132378
$$469$$ −38.2895 −1.76804
$$470$$ 0 0
$$471$$ 34.1555 1.57380
$$472$$ 10.2329 0.471006
$$473$$ 11.1713 0.513657
$$474$$ −3.97334 −0.182501
$$475$$ 0 0
$$476$$ −8.06505 −0.369661
$$477$$ −6.37137 −0.291725
$$478$$ −14.7070 −0.672683
$$479$$ −6.12291 −0.279763 −0.139881 0.990168i $$-0.544672\pi$$
−0.139881 + 0.990168i $$0.544672\pi$$
$$480$$ 0 0
$$481$$ 4.53919 0.206969
$$482$$ −14.8999 −0.678671
$$483$$ 32.8781 1.49601
$$484$$ −9.80817 −0.445826
$$485$$ 0 0
$$486$$ −6.43907 −0.292082
$$487$$ −10.3018 −0.466819 −0.233409 0.972379i $$-0.574988\pi$$
−0.233409 + 0.972379i $$0.574988\pi$$
$$488$$ 6.29791 0.285093
$$489$$ −22.5103 −1.01795
$$490$$ 0 0
$$491$$ −10.5380 −0.475572 −0.237786 0.971318i $$-0.576422\pi$$
−0.237786 + 0.971318i $$0.576422\pi$$
$$492$$ 12.4341 0.560575
$$493$$ −18.7115 −0.842726
$$494$$ 22.9204 1.03124
$$495$$ 0 0
$$496$$ 3.51026 0.157615
$$497$$ 18.1340 0.813420
$$498$$ 13.0579 0.585137
$$499$$ −31.2123 −1.39726 −0.698628 0.715485i $$-0.746206\pi$$
−0.698628 + 0.715485i $$0.746206\pi$$
$$500$$ 0 0
$$501$$ −9.76979 −0.436482
$$502$$ 6.23513 0.278288
$$503$$ 36.1978 1.61398 0.806990 0.590565i $$-0.201095\pi$$
0.806990 + 0.590565i $$0.201095\pi$$
$$504$$ −1.81658 −0.0809170
$$505$$ 0 0
$$506$$ 8.09890 0.360040
$$507$$ −11.7044 −0.519809
$$508$$ −8.14116 −0.361205
$$509$$ −33.6092 −1.48970 −0.744850 0.667232i $$-0.767479\pi$$
−0.744850 + 0.667232i $$0.767479\pi$$
$$510$$ 0 0
$$511$$ −36.5946 −1.61885
$$512$$ 1.00000 0.0441942
$$513$$ −28.2195 −1.24592
$$514$$ −16.1568 −0.712644
$$515$$ 0 0
$$516$$ 15.7503 0.693369
$$517$$ 9.47641 0.416772
$$518$$ −2.87936 −0.126512
$$519$$ −1.90110 −0.0834492
$$520$$ 0 0
$$521$$ 5.42082 0.237490 0.118745 0.992925i $$-0.462113\pi$$
0.118745 + 0.992925i $$0.462113\pi$$
$$522$$ −4.21461 −0.184469
$$523$$ 31.3545 1.37104 0.685519 0.728054i $$-0.259575\pi$$
0.685519 + 0.728054i $$0.259575\pi$$
$$524$$ −1.46800 −0.0641298
$$525$$ 0 0
$$526$$ 18.6537 0.813339
$$527$$ −9.83218 −0.428297
$$528$$ 1.68035 0.0731277
$$529$$ 32.0349 1.39282
$$530$$ 0 0
$$531$$ −6.45589 −0.280162
$$532$$ −14.5392 −0.630354
$$533$$ 36.6693 1.58832
$$534$$ 10.0205 0.433630
$$535$$ 0 0
$$536$$ −13.2979 −0.574382
$$537$$ −2.71646 −0.117224
$$538$$ 21.8310 0.941199
$$539$$ −1.40910 −0.0606940
$$540$$ 0 0
$$541$$ −9.98440 −0.429263 −0.214631 0.976695i $$-0.568855\pi$$
−0.214631 + 0.976695i $$0.568855\pi$$
$$542$$ 24.9783 1.07291
$$543$$ −10.0000 −0.429141
$$544$$ −2.80098 −0.120091
$$545$$ 0 0
$$546$$ 20.1171 0.860934
$$547$$ −32.9649 −1.40948 −0.704739 0.709466i $$-0.748936\pi$$
−0.704739 + 0.709466i $$0.748936\pi$$
$$548$$ 6.17727 0.263880
$$549$$ −3.97334 −0.169578
$$550$$ 0 0
$$551$$ −33.7321 −1.43703
$$552$$ 11.4186 0.486006
$$553$$ 7.43293 0.316080
$$554$$ 0.822726 0.0349543
$$555$$ 0 0
$$556$$ 13.0338 0.552758
$$557$$ 21.7009 0.919495 0.459748 0.888050i $$-0.347940\pi$$
0.459748 + 0.888050i $$0.347940\pi$$
$$558$$ −2.21461 −0.0937521
$$559$$ 46.4489 1.96458
$$560$$ 0 0
$$561$$ −4.70662 −0.198714
$$562$$ −1.65142 −0.0696608
$$563$$ 1.05559 0.0444879 0.0222439 0.999753i $$-0.492919\pi$$
0.0222439 + 0.999753i $$0.492919\pi$$
$$564$$ 13.3607 0.562587
$$565$$ 0 0
$$566$$ −15.1340 −0.636128
$$567$$ −19.3184 −0.811298
$$568$$ 6.29791 0.264255
$$569$$ −41.4196 −1.73640 −0.868200 0.496215i $$-0.834723\pi$$
−0.868200 + 0.496215i $$0.834723\pi$$
$$570$$ 0 0
$$571$$ 21.4452 0.897454 0.448727 0.893669i $$-0.351878\pi$$
0.448727 + 0.893669i $$0.351878\pi$$
$$572$$ 4.95547 0.207199
$$573$$ −32.6369 −1.36342
$$574$$ −23.2606 −0.970878
$$575$$ 0 0
$$576$$ −0.630898 −0.0262874
$$577$$ −17.2667 −0.718823 −0.359411 0.933179i $$-0.617023\pi$$
−0.359411 + 0.933179i $$0.617023\pi$$
$$578$$ −9.15449 −0.380777
$$579$$ 12.5343 0.520906
$$580$$ 0 0
$$581$$ −24.4273 −1.01342
$$582$$ 4.73820 0.196405
$$583$$ −11.0251 −0.456611
$$584$$ −12.7093 −0.525914
$$585$$ 0 0
$$586$$ 25.4524 1.05143
$$587$$ 9.63090 0.397510 0.198755 0.980049i $$-0.436310\pi$$
0.198755 + 0.980049i $$0.436310\pi$$
$$588$$ −1.98667 −0.0819288
$$589$$ −17.7249 −0.730341
$$590$$ 0 0
$$591$$ −38.2401 −1.57299
$$592$$ −1.00000 −0.0410997
$$593$$ −2.78992 −0.114568 −0.0572842 0.998358i $$-0.518244\pi$$
−0.0572842 + 0.998358i $$0.518244\pi$$
$$594$$ −6.10116 −0.250334
$$595$$ 0 0
$$596$$ −4.40522 −0.180445
$$597$$ 13.0373 0.533583
$$598$$ 33.6742 1.37704
$$599$$ 25.2606 1.03212 0.516060 0.856553i $$-0.327398\pi$$
0.516060 + 0.856553i $$0.327398\pi$$
$$600$$ 0 0
$$601$$ 35.4040 1.44416 0.722080 0.691810i $$-0.243186\pi$$
0.722080 + 0.691810i $$0.243186\pi$$
$$602$$ −29.4641 −1.20087
$$603$$ 8.38962 0.341652
$$604$$ 2.92162 0.118879
$$605$$ 0 0
$$606$$ 24.4969 0.995120
$$607$$ −38.2485 −1.55246 −0.776229 0.630452i $$-0.782870\pi$$
−0.776229 + 0.630452i $$0.782870\pi$$
$$608$$ −5.04945 −0.204782
$$609$$ −29.6065 −1.19972
$$610$$ 0 0
$$611$$ 39.4017 1.59402
$$612$$ 1.76713 0.0714322
$$613$$ 46.3279 1.87117 0.935583 0.353107i $$-0.114875\pi$$
0.935583 + 0.353107i $$0.114875\pi$$
$$614$$ 27.8176 1.12263
$$615$$ 0 0
$$616$$ −3.14342 −0.126652
$$617$$ −23.7503 −0.956152 −0.478076 0.878319i $$-0.658666\pi$$
−0.478076 + 0.878319i $$0.658666\pi$$
$$618$$ −16.2979 −0.655598
$$619$$ −6.53797 −0.262783 −0.131392 0.991331i $$-0.541945\pi$$
−0.131392 + 0.991331i $$0.541945\pi$$
$$620$$ 0 0
$$621$$ −41.4596 −1.66372
$$622$$ 0.0650468 0.00260814
$$623$$ −18.7454 −0.751018
$$624$$ 6.98667 0.279691
$$625$$ 0 0
$$626$$ −22.5464 −0.901134
$$627$$ −8.48482 −0.338851
$$628$$ −22.1906 −0.885502
$$629$$ 2.80098 0.111683
$$630$$ 0 0
$$631$$ 27.1506 1.08085 0.540424 0.841393i $$-0.318264\pi$$
0.540424 + 0.841393i $$0.318264\pi$$
$$632$$ 2.58145 0.102685
$$633$$ −33.4573 −1.32981
$$634$$ 24.4775 0.972124
$$635$$ 0 0
$$636$$ −15.5441 −0.616364
$$637$$ −5.85884 −0.232136
$$638$$ −7.29299 −0.288732
$$639$$ −3.97334 −0.157183
$$640$$ 0 0
$$641$$ 31.9877 1.26344 0.631719 0.775197i $$-0.282350\pi$$
0.631719 + 0.775197i $$0.282350\pi$$
$$642$$ −6.21235 −0.245182
$$643$$ −26.2784 −1.03632 −0.518160 0.855284i $$-0.673383\pi$$
−0.518160 + 0.855284i $$0.673383\pi$$
$$644$$ −21.3607 −0.841729
$$645$$ 0 0
$$646$$ 14.1434 0.556466
$$647$$ −31.3679 −1.23320 −0.616599 0.787277i $$-0.711490\pi$$
−0.616599 + 0.787277i $$0.711490\pi$$
$$648$$ −6.70928 −0.263565
$$649$$ −11.1713 −0.438512
$$650$$ 0 0
$$651$$ −15.5571 −0.609729
$$652$$ 14.6248 0.572750
$$653$$ 25.3184 0.990787 0.495393 0.868669i $$-0.335024\pi$$
0.495393 + 0.868669i $$0.335024\pi$$
$$654$$ 6.92162 0.270657
$$655$$ 0 0
$$656$$ −8.07838 −0.315408
$$657$$ 8.01825 0.312822
$$658$$ −24.9939 −0.974362
$$659$$ −38.2072 −1.48834 −0.744172 0.667989i $$-0.767156\pi$$
−0.744172 + 0.667989i $$0.767156\pi$$
$$660$$ 0 0
$$661$$ 13.7899 0.536366 0.268183 0.963368i $$-0.413577\pi$$
0.268183 + 0.963368i $$0.413577\pi$$
$$662$$ −1.91321 −0.0743591
$$663$$ −19.5695 −0.760018
$$664$$ −8.48360 −0.329227
$$665$$ 0 0
$$666$$ 0.630898 0.0244468
$$667$$ −49.5585 −1.91891
$$668$$ 6.34736 0.245587
$$669$$ 25.6332 0.991035
$$670$$ 0 0
$$671$$ −6.87549 −0.265425
$$672$$ −4.43188 −0.170964
$$673$$ −4.41628 −0.170235 −0.0851176 0.996371i $$-0.527127\pi$$
−0.0851176 + 0.996371i $$0.527127\pi$$
$$674$$ 12.6286 0.486437
$$675$$ 0 0
$$676$$ 7.60424 0.292471
$$677$$ 38.1399 1.46584 0.732918 0.680317i $$-0.238158\pi$$
0.732918 + 0.680317i $$0.238158\pi$$
$$678$$ 5.52359 0.212132
$$679$$ −8.86376 −0.340160
$$680$$ 0 0
$$681$$ −17.2579 −0.661325
$$682$$ −3.83218 −0.146742
$$683$$ 22.6514 0.866732 0.433366 0.901218i $$-0.357326\pi$$
0.433366 + 0.901218i $$0.357326\pi$$
$$684$$ 3.18568 0.121808
$$685$$ 0 0
$$686$$ −16.4391 −0.627647
$$687$$ 11.7419 0.447982
$$688$$ −10.2329 −0.390124
$$689$$ −45.8408 −1.74640
$$690$$ 0 0
$$691$$ −15.8443 −0.602745 −0.301373 0.953506i $$-0.597445\pi$$
−0.301373 + 0.953506i $$0.597445\pi$$
$$692$$ 1.23513 0.0469527
$$693$$ 1.98318 0.0753347
$$694$$ 8.33403 0.316355
$$695$$ 0 0
$$696$$ −10.2823 −0.389750
$$697$$ 22.6274 0.857074
$$698$$ 2.11837 0.0801815
$$699$$ −0.0761103 −0.00287876
$$700$$ 0 0
$$701$$ −2.92284 −0.110394 −0.0551972 0.998475i $$-0.517579\pi$$
−0.0551972 + 0.998475i $$0.517579\pi$$
$$702$$ −25.3679 −0.957449
$$703$$ 5.04945 0.190444
$$704$$ −1.09171 −0.0411453
$$705$$ 0 0
$$706$$ −28.4534 −1.07086
$$707$$ −45.8264 −1.72348
$$708$$ −15.7503 −0.591933
$$709$$ −8.45136 −0.317397 −0.158699 0.987327i $$-0.550730\pi$$
−0.158699 + 0.987327i $$0.550730\pi$$
$$710$$ 0 0
$$711$$ −1.62863 −0.0610784
$$712$$ −6.51026 −0.243982
$$713$$ −26.0410 −0.975245
$$714$$ 12.4136 0.464568
$$715$$ 0 0
$$716$$ 1.76487 0.0659562
$$717$$ 22.6369 0.845389
$$718$$ 2.65368 0.0990346
$$719$$ −15.3763 −0.573439 −0.286719 0.958015i $$-0.592565\pi$$
−0.286719 + 0.958015i $$0.592565\pi$$
$$720$$ 0 0
$$721$$ 30.4885 1.13545
$$722$$ 6.49693 0.241791
$$723$$ 22.9337 0.852915
$$724$$ 6.49693 0.241456
$$725$$ 0 0
$$726$$ 15.0966 0.560288
$$727$$ 15.7275 0.583302 0.291651 0.956525i $$-0.405795\pi$$
0.291651 + 0.956525i $$0.405795\pi$$
$$728$$ −13.0700 −0.484405
$$729$$ 30.0388 1.11255
$$730$$ 0 0
$$731$$ 28.6621 1.06011
$$732$$ −9.69368 −0.358289
$$733$$ −15.7275 −0.580909 −0.290455 0.956889i $$-0.593807\pi$$
−0.290455 + 0.956889i $$0.593807\pi$$
$$734$$ 31.7575 1.17219
$$735$$ 0 0
$$736$$ −7.41855 −0.273451
$$737$$ 14.5174 0.534757
$$738$$ 5.09663 0.187610
$$739$$ 45.0472 1.65709 0.828544 0.559924i $$-0.189170\pi$$
0.828544 + 0.559924i $$0.189170\pi$$
$$740$$ 0 0
$$741$$ −35.2788 −1.29600
$$742$$ 29.0784 1.06750
$$743$$ 9.31965 0.341905 0.170952 0.985279i $$-0.445316\pi$$
0.170952 + 0.985279i $$0.445316\pi$$
$$744$$ −5.40295 −0.198082
$$745$$ 0 0
$$746$$ −25.1122 −0.919424
$$747$$ 5.35228 0.195830
$$748$$ 3.05786 0.111806
$$749$$ 11.6214 0.424638
$$750$$ 0 0
$$751$$ 11.6020 0.423362 0.211681 0.977339i $$-0.432106\pi$$
0.211681 + 0.977339i $$0.432106\pi$$
$$752$$ −8.68035 −0.316540
$$753$$ −9.59705 −0.349736
$$754$$ −30.3234 −1.10431
$$755$$ 0 0
$$756$$ 16.0917 0.585250
$$757$$ −38.3545 −1.39402 −0.697010 0.717062i $$-0.745487\pi$$
−0.697010 + 0.717062i $$0.745487\pi$$
$$758$$ 22.5330 0.818437
$$759$$ −12.4657 −0.452477
$$760$$ 0 0
$$761$$ −45.3318 −1.64328 −0.821638 0.570010i $$-0.806939\pi$$
−0.821638 + 0.570010i $$0.806939\pi$$
$$762$$ 12.5308 0.453942
$$763$$ −12.9483 −0.468759
$$764$$ 21.2039 0.767131
$$765$$ 0 0
$$766$$ 12.5886 0.454846
$$767$$ −46.4489 −1.67717
$$768$$ −1.53919 −0.0555407
$$769$$ −31.7454 −1.14477 −0.572384 0.819986i $$-0.693981\pi$$
−0.572384 + 0.819986i $$0.693981\pi$$
$$770$$ 0 0
$$771$$ 24.8683 0.895610
$$772$$ −8.14342 −0.293088
$$773$$ 38.0482 1.36850 0.684250 0.729248i $$-0.260130\pi$$
0.684250 + 0.729248i $$0.260130\pi$$
$$774$$ 6.45589 0.232052
$$775$$ 0 0
$$776$$ −3.07838 −0.110507
$$777$$ 4.43188 0.158993
$$778$$ −0.879362 −0.0315266
$$779$$ 40.7914 1.46150
$$780$$ 0 0
$$781$$ −6.87549 −0.246024
$$782$$ 20.7792 0.743064
$$783$$ 37.3340 1.33421
$$784$$ 1.29072 0.0460973
$$785$$ 0 0
$$786$$ 2.25953 0.0805947
$$787$$ 15.1506 0.540061 0.270031 0.962852i $$-0.412966\pi$$
0.270031 + 0.962852i $$0.412966\pi$$
$$788$$ 24.8443 0.885041
$$789$$ −28.7115 −1.02216
$$790$$ 0 0
$$791$$ −10.3330 −0.367399
$$792$$ 0.688756 0.0244739
$$793$$ −28.5874 −1.01517
$$794$$ 29.6814 1.05335
$$795$$ 0 0
$$796$$ −8.47027 −0.300221
$$797$$ −43.1350 −1.52792 −0.763960 0.645263i $$-0.776748\pi$$
−0.763960 + 0.645263i $$0.776748\pi$$
$$798$$ 22.3786 0.792192
$$799$$ 24.3135 0.860150
$$800$$ 0 0
$$801$$ 4.10731 0.145125
$$802$$ −13.9867 −0.493886
$$803$$ 13.8748 0.489632
$$804$$ 20.4680 0.721851
$$805$$ 0 0
$$806$$ −15.9337 −0.561242
$$807$$ −33.6020 −1.18285
$$808$$ −15.9155 −0.559905
$$809$$ 33.3874 1.17384 0.586918 0.809646i $$-0.300341\pi$$
0.586918 + 0.809646i $$0.300341\pi$$
$$810$$ 0 0
$$811$$ 10.9216 0.383510 0.191755 0.981443i $$-0.438582\pi$$
0.191755 + 0.981443i $$0.438582\pi$$
$$812$$ 19.2351 0.675021
$$813$$ −38.4463 −1.34837
$$814$$ 1.09171 0.0382644
$$815$$ 0 0
$$816$$ 4.31124 0.150924
$$817$$ 51.6703 1.80772
$$818$$ −21.9060 −0.765926
$$819$$ 8.24581 0.288132
$$820$$ 0 0
$$821$$ −32.7910 −1.14441 −0.572206 0.820110i $$-0.693912\pi$$
−0.572206 + 0.820110i $$0.693912\pi$$
$$822$$ −9.50799 −0.331629
$$823$$ −9.83218 −0.342728 −0.171364 0.985208i $$-0.554817\pi$$
−0.171364 + 0.985208i $$0.554817\pi$$
$$824$$ 10.5886 0.368873
$$825$$ 0 0
$$826$$ 29.4641 1.02519
$$827$$ −17.0228 −0.591940 −0.295970 0.955197i $$-0.595643\pi$$
−0.295970 + 0.955197i $$0.595643\pi$$
$$828$$ 4.68035 0.162653
$$829$$ 12.8260 0.445467 0.222733 0.974879i $$-0.428502\pi$$
0.222733 + 0.974879i $$0.428502\pi$$
$$830$$ 0 0
$$831$$ −1.26633 −0.0439285
$$832$$ −4.53919 −0.157368
$$833$$ −3.61530 −0.125263
$$834$$ −20.0616 −0.694675
$$835$$ 0 0
$$836$$ 5.51253 0.190655
$$837$$ 19.6176 0.678082
$$838$$ −34.8648 −1.20438
$$839$$ −11.5018 −0.397088 −0.198544 0.980092i $$-0.563621\pi$$
−0.198544 + 0.980092i $$0.563621\pi$$
$$840$$ 0 0
$$841$$ 15.6270 0.538863
$$842$$ −4.76487 −0.164208
$$843$$ 2.54184 0.0875457
$$844$$ 21.7370 0.748218
$$845$$ 0 0
$$846$$ 5.47641 0.188283
$$847$$ −28.2413 −0.970382
$$848$$ 10.0989 0.346798
$$849$$ 23.2940 0.799449
$$850$$ 0 0
$$851$$ 7.41855 0.254305
$$852$$ −9.69368 −0.332100
$$853$$ −40.5946 −1.38993 −0.694966 0.719042i $$-0.744581\pi$$
−0.694966 + 0.719042i $$0.744581\pi$$
$$854$$ 18.1340 0.620532
$$855$$ 0 0
$$856$$ 4.03612 0.137952
$$857$$ −38.0361 −1.29929 −0.649645 0.760238i $$-0.725082\pi$$
−0.649645 + 0.760238i $$0.725082\pi$$
$$858$$ −7.62741 −0.260395
$$859$$ −40.9506 −1.39721 −0.698607 0.715505i $$-0.746197\pi$$
−0.698607 + 0.715505i $$0.746197\pi$$
$$860$$ 0 0
$$861$$ 35.8024 1.22014
$$862$$ 2.66597 0.0908033
$$863$$ −33.0817 −1.12611 −0.563057 0.826418i $$-0.690375\pi$$
−0.563057 + 0.826418i $$0.690375\pi$$
$$864$$ 5.58864 0.190129
$$865$$ 0 0
$$866$$ −32.3074 −1.09785
$$867$$ 14.0905 0.478538
$$868$$ 10.1073 0.343064
$$869$$ −2.81819 −0.0956006
$$870$$ 0 0
$$871$$ 60.3617 2.04528
$$872$$ −4.49693 −0.152285
$$873$$ 1.94214 0.0657315
$$874$$ 37.4596 1.26709
$$875$$ 0 0
$$876$$ 19.5620 0.660938
$$877$$ −25.5057 −0.861267 −0.430634 0.902527i $$-0.641710\pi$$
−0.430634 + 0.902527i $$0.641710\pi$$
$$878$$ −22.1568 −0.747754
$$879$$ −39.1761 −1.32138
$$880$$ 0 0
$$881$$ 15.6514 0.527310 0.263655 0.964617i $$-0.415072\pi$$
0.263655 + 0.964617i $$0.415072\pi$$
$$882$$ −0.814315 −0.0274194
$$883$$ −56.1071 −1.88816 −0.944078 0.329723i $$-0.893045\pi$$
−0.944078 + 0.329723i $$0.893045\pi$$
$$884$$ 12.7142 0.427625
$$885$$ 0 0
$$886$$ −39.1061 −1.31379
$$887$$ 47.5052 1.59507 0.797534 0.603275i $$-0.206138\pi$$
0.797534 + 0.603275i $$0.206138\pi$$
$$888$$ 1.53919 0.0516518
$$889$$ −23.4413 −0.786197
$$890$$ 0 0
$$891$$ 7.32457 0.245382
$$892$$ −16.6537 −0.557607
$$893$$ 43.8310 1.46675
$$894$$ 6.78047 0.226773
$$895$$ 0 0
$$896$$ 2.87936 0.0961927
$$897$$ −51.8310 −1.73059
$$898$$ 30.8915 1.03086
$$899$$ 23.4497 0.782093
$$900$$ 0 0
$$901$$ −28.2868 −0.942372
$$902$$ 8.81924 0.293648
$$903$$ 45.3509 1.50918
$$904$$ −3.58864 −0.119356
$$905$$ 0 0
$$906$$ −4.49693 −0.149400
$$907$$ −15.8394 −0.525938 −0.262969 0.964804i $$-0.584702\pi$$
−0.262969 + 0.964804i $$0.584702\pi$$
$$908$$ 11.2123 0.372095
$$909$$ 10.0410 0.333040
$$910$$ 0 0
$$911$$ 11.9539 0.396049 0.198025 0.980197i $$-0.436547\pi$$
0.198025 + 0.980197i $$0.436547\pi$$
$$912$$ 7.77205 0.257358
$$913$$ 9.26162 0.306515
$$914$$ 0.438025 0.0144886
$$915$$ 0 0
$$916$$ −7.62863 −0.252057
$$917$$ −4.22690 −0.139585
$$918$$ −15.6537 −0.516649
$$919$$ −34.8781 −1.15052 −0.575262 0.817969i $$-0.695100\pi$$
−0.575262 + 0.817969i $$0.695100\pi$$
$$920$$ 0 0
$$921$$ −42.8166 −1.41085
$$922$$ 0.523590 0.0172435
$$923$$ −28.5874 −0.940966
$$924$$ 4.83832 0.159169
$$925$$ 0 0
$$926$$ 29.1845 0.959061
$$927$$ −6.68035 −0.219411
$$928$$ 6.68035 0.219293
$$929$$ −52.5439 −1.72391 −0.861955 0.506984i $$-0.830760\pi$$
−0.861955 + 0.506984i $$0.830760\pi$$
$$930$$ 0 0
$$931$$ −6.51745 −0.213601
$$932$$ 0.0494483 0.00161973
$$933$$ −0.100119 −0.00327776
$$934$$ −9.49079 −0.310548
$$935$$ 0 0
$$936$$ 2.86376 0.0936050
$$937$$ 13.3318 0.435530 0.217765 0.976001i $$-0.430123\pi$$
0.217765 + 0.976001i $$0.430123\pi$$
$$938$$ −38.2895 −1.25020
$$939$$ 34.7031 1.13249
$$940$$ 0 0
$$941$$ 14.5281 0.473603 0.236802 0.971558i $$-0.423901\pi$$
0.236802 + 0.971558i $$0.423901\pi$$
$$942$$ 34.1555 1.11285
$$943$$ 59.9299 1.95158
$$944$$ 10.2329 0.333051
$$945$$ 0 0
$$946$$ 11.1713 0.363211
$$947$$ 24.6803 0.802003 0.401002 0.916077i $$-0.368662\pi$$
0.401002 + 0.916077i $$0.368662\pi$$
$$948$$ −3.97334 −0.129048
$$949$$ 57.6898 1.87269
$$950$$ 0 0
$$951$$ −37.6754 −1.22171
$$952$$ −8.06505 −0.261390
$$953$$ 40.5936 1.31495 0.657477 0.753474i $$-0.271624\pi$$
0.657477 + 0.753474i $$0.271624\pi$$
$$954$$ −6.37137 −0.206281
$$955$$ 0 0
$$956$$ −14.7070 −0.475659
$$957$$ 11.2253 0.362862
$$958$$ −6.12291 −0.197822
$$959$$ 17.7866 0.574360
$$960$$ 0 0
$$961$$ −18.6781 −0.602519
$$962$$ 4.53919 0.146349
$$963$$ −2.54638 −0.0820558
$$964$$ −14.8999 −0.479893
$$965$$ 0 0
$$966$$ 32.8781 1.05784
$$967$$ 1.06400 0.0342160 0.0171080 0.999854i $$-0.494554\pi$$
0.0171080 + 0.999854i $$0.494554\pi$$
$$968$$ −9.80817 −0.315247
$$969$$ −21.7694 −0.699334
$$970$$ 0 0
$$971$$ 11.9333 0.382959 0.191480 0.981497i $$-0.438671\pi$$
0.191480 + 0.981497i $$0.438671\pi$$
$$972$$ −6.43907 −0.206533
$$973$$ 37.5292 1.20313
$$974$$ −10.3018 −0.330091
$$975$$ 0 0
$$976$$ 6.29791 0.201591
$$977$$ 19.1867 0.613838 0.306919 0.951736i $$-0.400702\pi$$
0.306919 + 0.951736i $$0.400702\pi$$
$$978$$ −22.5103 −0.719799
$$979$$ 7.10731 0.227151
$$980$$ 0 0
$$981$$ 2.83710 0.0905817
$$982$$ −10.5380 −0.336280
$$983$$ 22.9171 0.730942 0.365471 0.930823i $$-0.380908\pi$$
0.365471 + 0.930823i $$0.380908\pi$$
$$984$$ 12.4341 0.396386
$$985$$ 0 0
$$986$$ −18.7115 −0.595897
$$987$$ 38.4703 1.22452
$$988$$ 22.9204 0.729195
$$989$$ 75.9130 2.41389
$$990$$ 0 0
$$991$$ −52.2772 −1.66064 −0.830320 0.557287i $$-0.811842\pi$$
−0.830320 + 0.557287i $$0.811842\pi$$
$$992$$ 3.51026 0.111451
$$993$$ 2.94479 0.0934502
$$994$$ 18.1340 0.575175
$$995$$ 0 0
$$996$$ 13.0579 0.413754
$$997$$ −48.8892 −1.54834 −0.774168 0.632980i $$-0.781832\pi$$
−0.774168 + 0.632980i $$0.781832\pi$$
$$998$$ −31.2123 −0.988010
$$999$$ −5.58864 −0.176817
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.bb.1.2 yes 3
5.2 odd 4 1850.2.b.n.149.5 6
5.3 odd 4 1850.2.b.n.149.2 6
5.4 even 2 1850.2.a.ba.1.2 3

By twisted newform
Twist Min Dim Char Parity Ord Type
1850.2.a.ba.1.2 3 5.4 even 2
1850.2.a.bb.1.2 yes 3 1.1 even 1 trivial
1850.2.b.n.149.2 6 5.3 odd 4
1850.2.b.n.149.5 6 5.2 odd 4