# Properties

 Label 1521.2.a.s.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

 $$f(q)$$ $$=$$ $$q+2.35690 q^{2} +3.55496 q^{4} +3.69202 q^{5} +0.801938 q^{7} +3.66487 q^{8} +O(q^{10})$$ $$q+2.35690 q^{2} +3.55496 q^{4} +3.69202 q^{5} +0.801938 q^{7} +3.66487 q^{8} +8.70171 q^{10} -2.85086 q^{11} +1.89008 q^{14} +1.52781 q^{16} -2.93900 q^{17} +2.44504 q^{19} +13.1250 q^{20} -6.71917 q^{22} +7.78986 q^{23} +8.63102 q^{25} +2.85086 q^{28} -3.85086 q^{29} -2.34481 q^{31} -3.72886 q^{32} -6.92692 q^{34} +2.96077 q^{35} -7.44504 q^{37} +5.76271 q^{38} +13.5308 q^{40} -0.850855 q^{41} -1.61596 q^{43} -10.1347 q^{44} +18.3599 q^{46} +2.44504 q^{47} -6.35690 q^{49} +20.3424 q^{50} +9.96077 q^{53} -10.5254 q^{55} +2.93900 q^{56} -9.07606 q^{58} -5.38404 q^{59} -13.2567 q^{61} -5.52648 q^{62} -11.8442 q^{64} -14.3937 q^{67} -10.4480 q^{68} +6.97823 q^{70} +8.12498 q^{71} +11.8877 q^{73} -17.5472 q^{74} +8.69202 q^{76} -2.28621 q^{77} +5.40581 q^{79} +5.64071 q^{80} -2.00538 q^{82} +7.04892 q^{83} -10.8509 q^{85} -3.80864 q^{86} -10.4480 q^{88} +1.13169 q^{89} +27.6926 q^{92} +5.76271 q^{94} +9.02715 q^{95} +5.94438 q^{97} -14.9825 q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3q + 3q^{2} + 11q^{4} + 6q^{5} - 2q^{7} + 12q^{8} + O(q^{10})$$ $$3q + 3q^{2} + 11q^{4} + 6q^{5} - 2q^{7} + 12q^{8} - q^{10} + 5q^{11} + 5q^{14} + 11q^{16} + q^{17} + 7q^{19} + 15q^{20} - 9q^{22} + 11q^{25} - 5q^{28} + 2q^{29} + 16q^{31} + 22q^{32} + 8q^{34} - 4q^{35} - 22q^{37} + 3q^{40} + 11q^{41} - 15q^{43} + 16q^{44} + 7q^{46} + 7q^{47} - 15q^{49} - 3q^{50} + 17q^{53} + 3q^{55} - q^{56} - 12q^{58} - 6q^{59} - 13q^{61} + 2q^{62} - 11q^{67} + 13q^{68} + 24q^{70} - 6q^{73} - 15q^{74} + 21q^{76} - 15q^{77} + 3q^{79} - 20q^{80} - 3q^{82} + 12q^{83} - 19q^{85} - 29q^{86} + 13q^{88} + q^{89} - 7q^{92} + 21q^{95} + 5q^{97} - 29q^{98} + O(q^{100})$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 2.35690 1.66658 0.833289 0.552838i $$-0.186455\pi$$
0.833289 + 0.552838i $$0.186455\pi$$
$$3$$ 0 0
$$4$$ 3.55496 1.77748
$$5$$ 3.69202 1.65112 0.825561 0.564313i $$-0.190859\pi$$
0.825561 + 0.564313i $$0.190859\pi$$
$$6$$ 0 0
$$7$$ 0.801938 0.303104 0.151552 0.988449i $$-0.451573\pi$$
0.151552 + 0.988449i $$0.451573\pi$$
$$8$$ 3.66487 1.29573
$$9$$ 0 0
$$10$$ 8.70171 2.75172
$$11$$ −2.85086 −0.859565 −0.429783 0.902932i $$-0.641410\pi$$
−0.429783 + 0.902932i $$0.641410\pi$$
$$12$$ 0 0
$$13$$ 0 0
$$14$$ 1.89008 0.505146
$$15$$ 0 0
$$16$$ 1.52781 0.381953
$$17$$ −2.93900 −0.712812 −0.356406 0.934331i $$-0.615998\pi$$
−0.356406 + 0.934331i $$0.615998\pi$$
$$18$$ 0 0
$$19$$ 2.44504 0.560931 0.280466 0.959864i $$-0.409511\pi$$
0.280466 + 0.959864i $$0.409511\pi$$
$$20$$ 13.1250 2.93484
$$21$$ 0 0
$$22$$ −6.71917 −1.43253
$$23$$ 7.78986 1.62430 0.812149 0.583451i $$-0.198298\pi$$
0.812149 + 0.583451i $$0.198298\pi$$
$$24$$ 0 0
$$25$$ 8.63102 1.72620
$$26$$ 0 0
$$27$$ 0 0
$$28$$ 2.85086 0.538761
$$29$$ −3.85086 −0.715086 −0.357543 0.933897i $$-0.616385\pi$$
−0.357543 + 0.933897i $$0.616385\pi$$
$$30$$ 0 0
$$31$$ −2.34481 −0.421141 −0.210571 0.977579i $$-0.567532\pi$$
−0.210571 + 0.977579i $$0.567532\pi$$
$$32$$ −3.72886 −0.659175
$$33$$ 0 0
$$34$$ −6.92692 −1.18796
$$35$$ 2.96077 0.500462
$$36$$ 0 0
$$37$$ −7.44504 −1.22396 −0.611979 0.790874i $$-0.709626\pi$$
−0.611979 + 0.790874i $$0.709626\pi$$
$$38$$ 5.76271 0.934835
$$39$$ 0 0
$$40$$ 13.5308 2.13941
$$41$$ −0.850855 −0.132881 −0.0664406 0.997790i $$-0.521164\pi$$
−0.0664406 + 0.997790i $$0.521164\pi$$
$$42$$ 0 0
$$43$$ −1.61596 −0.246431 −0.123216 0.992380i $$-0.539321\pi$$
−0.123216 + 0.992380i $$0.539321\pi$$
$$44$$ −10.1347 −1.52786
$$45$$ 0 0
$$46$$ 18.3599 2.70702
$$47$$ 2.44504 0.356646 0.178323 0.983972i $$-0.442933\pi$$
0.178323 + 0.983972i $$0.442933\pi$$
$$48$$ 0 0
$$49$$ −6.35690 −0.908128
$$50$$ 20.3424 2.87685
$$51$$ 0 0
$$52$$ 0 0
$$53$$ 9.96077 1.36822 0.684109 0.729380i $$-0.260191\pi$$
0.684109 + 0.729380i $$0.260191\pi$$
$$54$$ 0 0
$$55$$ −10.5254 −1.41925
$$56$$ 2.93900 0.392741
$$57$$ 0 0
$$58$$ −9.07606 −1.19175
$$59$$ −5.38404 −0.700943 −0.350471 0.936573i $$-0.613979\pi$$
−0.350471 + 0.936573i $$0.613979\pi$$
$$60$$ 0 0
$$61$$ −13.2567 −1.69734 −0.848671 0.528921i $$-0.822597\pi$$
−0.848671 + 0.528921i $$0.822597\pi$$
$$62$$ −5.52648 −0.701864
$$63$$ 0 0
$$64$$ −11.8442 −1.48052
$$65$$ 0 0
$$66$$ 0 0
$$67$$ −14.3937 −1.75847 −0.879237 0.476384i $$-0.841947\pi$$
−0.879237 + 0.476384i $$0.841947\pi$$
$$68$$ −10.4480 −1.26701
$$69$$ 0 0
$$70$$ 6.97823 0.834058
$$71$$ 8.12498 0.964258 0.482129 0.876100i $$-0.339864\pi$$
0.482129 + 0.876100i $$0.339864\pi$$
$$72$$ 0 0
$$73$$ 11.8877 1.39135 0.695674 0.718357i $$-0.255106\pi$$
0.695674 + 0.718357i $$0.255106\pi$$
$$74$$ −17.5472 −2.03982
$$75$$ 0 0
$$76$$ 8.69202 0.997043
$$77$$ −2.28621 −0.260538
$$78$$ 0 0
$$79$$ 5.40581 0.608202 0.304101 0.952640i $$-0.401644\pi$$
0.304101 + 0.952640i $$0.401644\pi$$
$$80$$ 5.64071 0.630651
$$81$$ 0 0
$$82$$ −2.00538 −0.221457
$$83$$ 7.04892 0.773719 0.386860 0.922139i $$-0.373560\pi$$
0.386860 + 0.922139i $$0.373560\pi$$
$$84$$ 0 0
$$85$$ −10.8509 −1.17694
$$86$$ −3.80864 −0.410696
$$87$$ 0 0
$$88$$ −10.4480 −1.11376
$$89$$ 1.13169 0.119959 0.0599793 0.998200i $$-0.480897\pi$$
0.0599793 + 0.998200i $$0.480897\pi$$
$$90$$ 0 0
$$91$$ 0 0
$$92$$ 27.6926 2.88715
$$93$$ 0 0
$$94$$ 5.76271 0.594378
$$95$$ 9.02715 0.926166
$$96$$ 0 0
$$97$$ 5.94438 0.603560 0.301780 0.953378i $$-0.402419\pi$$
0.301780 + 0.953378i $$0.402419\pi$$
$$98$$ −14.9825 −1.51347
$$99$$ 0 0
$$100$$ 30.6829 3.06829
$$101$$ −4.62565 −0.460269 −0.230134 0.973159i $$-0.573917\pi$$
−0.230134 + 0.973159i $$0.573917\pi$$
$$102$$ 0 0
$$103$$ 1.20775 0.119003 0.0595016 0.998228i $$-0.481049\pi$$
0.0595016 + 0.998228i $$0.481049\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 23.4765 2.28024
$$107$$ 9.52111 0.920440 0.460220 0.887805i $$-0.347771\pi$$
0.460220 + 0.887805i $$0.347771\pi$$
$$108$$ 0 0
$$109$$ 1.78448 0.170922 0.0854611 0.996342i $$-0.472764\pi$$
0.0854611 + 0.996342i $$0.472764\pi$$
$$110$$ −24.8073 −2.36528
$$111$$ 0 0
$$112$$ 1.22521 0.115771
$$113$$ −4.95108 −0.465759 −0.232879 0.972506i $$-0.574815\pi$$
−0.232879 + 0.972506i $$0.574815\pi$$
$$114$$ 0 0
$$115$$ 28.7603 2.68191
$$116$$ −13.6896 −1.27105
$$117$$ 0 0
$$118$$ −12.6896 −1.16817
$$119$$ −2.35690 −0.216056
$$120$$ 0 0
$$121$$ −2.87263 −0.261148
$$122$$ −31.2446 −2.82875
$$123$$ 0 0
$$124$$ −8.33572 −0.748569
$$125$$ 13.4058 1.19905
$$126$$ 0 0
$$127$$ 5.67025 0.503153 0.251577 0.967837i $$-0.419051\pi$$
0.251577 + 0.967837i $$0.419051\pi$$
$$128$$ −20.4577 −1.80822
$$129$$ 0 0
$$130$$ 0 0
$$131$$ −18.2228 −1.59213 −0.796067 0.605208i $$-0.793090\pi$$
−0.796067 + 0.605208i $$0.793090\pi$$
$$132$$ 0 0
$$133$$ 1.96077 0.170020
$$134$$ −33.9245 −2.93063
$$135$$ 0 0
$$136$$ −10.7711 −0.923612
$$137$$ 9.45042 0.807404 0.403702 0.914891i $$-0.367723\pi$$
0.403702 + 0.914891i $$0.367723\pi$$
$$138$$ 0 0
$$139$$ 4.01507 0.340553 0.170277 0.985396i $$-0.445534\pi$$
0.170277 + 0.985396i $$0.445534\pi$$
$$140$$ 10.5254 0.889560
$$141$$ 0 0
$$142$$ 19.1497 1.60701
$$143$$ 0 0
$$144$$ 0 0
$$145$$ −14.2174 −1.18069
$$146$$ 28.0180 2.31879
$$147$$ 0 0
$$148$$ −26.4668 −2.17556
$$149$$ 19.4058 1.58979 0.794893 0.606750i $$-0.207527\pi$$
0.794893 + 0.606750i $$0.207527\pi$$
$$150$$ 0 0
$$151$$ 12.3623 1.00603 0.503014 0.864278i $$-0.332225\pi$$
0.503014 + 0.864278i $$0.332225\pi$$
$$152$$ 8.96077 0.726815
$$153$$ 0 0
$$154$$ −5.38835 −0.434206
$$155$$ −8.65710 −0.695355
$$156$$ 0 0
$$157$$ −18.6775 −1.49063 −0.745315 0.666712i $$-0.767701\pi$$
−0.745315 + 0.666712i $$0.767701\pi$$
$$158$$ 12.7409 1.01361
$$159$$ 0 0
$$160$$ −13.7670 −1.08838
$$161$$ 6.24698 0.492331
$$162$$ 0 0
$$163$$ −12.3394 −0.966499 −0.483250 0.875483i $$-0.660544\pi$$
−0.483250 + 0.875483i $$0.660544\pi$$
$$164$$ −3.02475 −0.236194
$$165$$ 0 0
$$166$$ 16.6136 1.28946
$$167$$ 11.4940 0.889429 0.444715 0.895672i $$-0.353305\pi$$
0.444715 + 0.895672i $$0.353305\pi$$
$$168$$ 0 0
$$169$$ 0 0
$$170$$ −25.5743 −1.96146
$$171$$ 0 0
$$172$$ −5.74466 −0.438026
$$173$$ −12.1142 −0.921028 −0.460514 0.887653i $$-0.652335\pi$$
−0.460514 + 0.887653i $$0.652335\pi$$
$$174$$ 0 0
$$175$$ 6.92154 0.523219
$$176$$ −4.35557 −0.328313
$$177$$ 0 0
$$178$$ 2.66727 0.199920
$$179$$ 0.538565 0.0402542 0.0201271 0.999797i $$-0.493593\pi$$
0.0201271 + 0.999797i $$0.493593\pi$$
$$180$$ 0 0
$$181$$ −23.2838 −1.73067 −0.865336 0.501192i $$-0.832895\pi$$
−0.865336 + 0.501192i $$0.832895\pi$$
$$182$$ 0 0
$$183$$ 0 0
$$184$$ 28.5488 2.10465
$$185$$ −27.4873 −2.02090
$$186$$ 0 0
$$187$$ 8.37867 0.612709
$$188$$ 8.69202 0.633931
$$189$$ 0 0
$$190$$ 21.2760 1.54353
$$191$$ −16.7657 −1.21312 −0.606561 0.795037i $$-0.707452\pi$$
−0.606561 + 0.795037i $$0.707452\pi$$
$$192$$ 0 0
$$193$$ −25.7439 −1.85309 −0.926544 0.376186i $$-0.877235\pi$$
−0.926544 + 0.376186i $$0.877235\pi$$
$$194$$ 14.0103 1.00588
$$195$$ 0 0
$$196$$ −22.5985 −1.61418
$$197$$ 21.4209 1.52617 0.763087 0.646296i $$-0.223683\pi$$
0.763087 + 0.646296i $$0.223683\pi$$
$$198$$ 0 0
$$199$$ 3.52781 0.250080 0.125040 0.992152i $$-0.460094\pi$$
0.125040 + 0.992152i $$0.460094\pi$$
$$200$$ 31.6316 2.23669
$$201$$ 0 0
$$202$$ −10.9022 −0.767074
$$203$$ −3.08815 −0.216745
$$204$$ 0 0
$$205$$ −3.14138 −0.219403
$$206$$ 2.84654 0.198328
$$207$$ 0 0
$$208$$ 0 0
$$209$$ −6.97046 −0.482157
$$210$$ 0 0
$$211$$ 1.21552 0.0836799 0.0418399 0.999124i $$-0.486678\pi$$
0.0418399 + 0.999124i $$0.486678\pi$$
$$212$$ 35.4101 2.43198
$$213$$ 0 0
$$214$$ 22.4403 1.53398
$$215$$ −5.96615 −0.406888
$$216$$ 0 0
$$217$$ −1.88040 −0.127650
$$218$$ 4.20583 0.284855
$$219$$ 0 0
$$220$$ −37.4174 −2.52268
$$221$$ 0 0
$$222$$ 0 0
$$223$$ 17.3884 1.16441 0.582205 0.813042i $$-0.302190\pi$$
0.582205 + 0.813042i $$0.302190\pi$$
$$224$$ −2.99031 −0.199799
$$225$$ 0 0
$$226$$ −11.6692 −0.776223
$$227$$ 17.4155 1.15591 0.577954 0.816070i $$-0.303851\pi$$
0.577954 + 0.816070i $$0.303851\pi$$
$$228$$ 0 0
$$229$$ −18.7603 −1.23972 −0.619858 0.784714i $$-0.712810\pi$$
−0.619858 + 0.784714i $$0.712810\pi$$
$$230$$ 67.7851 4.46962
$$231$$ 0 0
$$232$$ −14.1129 −0.926557
$$233$$ −3.95108 −0.258844 −0.129422 0.991590i $$-0.541312\pi$$
−0.129422 + 0.991590i $$0.541312\pi$$
$$234$$ 0 0
$$235$$ 9.02715 0.588866
$$236$$ −19.1400 −1.24591
$$237$$ 0 0
$$238$$ −5.55496 −0.360074
$$239$$ 0.818331 0.0529334 0.0264667 0.999650i $$-0.491574\pi$$
0.0264667 + 0.999650i $$0.491574\pi$$
$$240$$ 0 0
$$241$$ −6.03252 −0.388589 −0.194295 0.980943i $$-0.562242\pi$$
−0.194295 + 0.980943i $$0.562242\pi$$
$$242$$ −6.77048 −0.435223
$$243$$ 0 0
$$244$$ −47.1269 −3.01699
$$245$$ −23.4698 −1.49943
$$246$$ 0 0
$$247$$ 0 0
$$248$$ −8.59345 −0.545685
$$249$$ 0 0
$$250$$ 31.5961 1.99831
$$251$$ 26.8799 1.69665 0.848323 0.529479i $$-0.177613\pi$$
0.848323 + 0.529479i $$0.177613\pi$$
$$252$$ 0 0
$$253$$ −22.2078 −1.39619
$$254$$ 13.3642 0.838544
$$255$$ 0 0
$$256$$ −24.5284 −1.53303
$$257$$ 9.05323 0.564725 0.282362 0.959308i $$-0.408882\pi$$
0.282362 + 0.959308i $$0.408882\pi$$
$$258$$ 0 0
$$259$$ −5.97046 −0.370986
$$260$$ 0 0
$$261$$ 0 0
$$262$$ −42.9493 −2.65342
$$263$$ −23.1511 −1.42756 −0.713778 0.700372i $$-0.753017\pi$$
−0.713778 + 0.700372i $$0.753017\pi$$
$$264$$ 0 0
$$265$$ 36.7754 2.25909
$$266$$ 4.62133 0.283352
$$267$$ 0 0
$$268$$ −51.1691 −3.12565
$$269$$ 2.42088 0.147604 0.0738018 0.997273i $$-0.476487\pi$$
0.0738018 + 0.997273i $$0.476487\pi$$
$$270$$ 0 0
$$271$$ 21.4450 1.30269 0.651347 0.758780i $$-0.274204\pi$$
0.651347 + 0.758780i $$0.274204\pi$$
$$272$$ −4.49024 −0.272261
$$273$$ 0 0
$$274$$ 22.2737 1.34560
$$275$$ −24.6058 −1.48379
$$276$$ 0 0
$$277$$ −14.8073 −0.889685 −0.444843 0.895609i $$-0.646740\pi$$
−0.444843 + 0.895609i $$0.646740\pi$$
$$278$$ 9.46309 0.567558
$$279$$ 0 0
$$280$$ 10.8509 0.648463
$$281$$ 14.5036 0.865215 0.432608 0.901582i $$-0.357594\pi$$
0.432608 + 0.901582i $$0.357594\pi$$
$$282$$ 0 0
$$283$$ 25.6722 1.52605 0.763026 0.646368i $$-0.223713\pi$$
0.763026 + 0.646368i $$0.223713\pi$$
$$284$$ 28.8840 1.71395
$$285$$ 0 0
$$286$$ 0 0
$$287$$ −0.682333 −0.0402768
$$288$$ 0 0
$$289$$ −8.36227 −0.491898
$$290$$ −33.5090 −1.96772
$$291$$ 0 0
$$292$$ 42.2602 2.47309
$$293$$ −26.5230 −1.54949 −0.774746 0.632273i $$-0.782122\pi$$
−0.774746 + 0.632273i $$0.782122\pi$$
$$294$$ 0 0
$$295$$ −19.8780 −1.15734
$$296$$ −27.2851 −1.58592
$$297$$ 0 0
$$298$$ 45.7375 2.64950
$$299$$ 0 0
$$300$$ 0 0
$$301$$ −1.29590 −0.0746943
$$302$$ 29.1366 1.67662
$$303$$ 0 0
$$304$$ 3.73556 0.214249
$$305$$ −48.9439 −2.80252
$$306$$ 0 0
$$307$$ 8.24698 0.470680 0.235340 0.971913i $$-0.424380\pi$$
0.235340 + 0.971913i $$0.424380\pi$$
$$308$$ −8.12737 −0.463100
$$309$$ 0 0
$$310$$ −20.4039 −1.15886
$$311$$ 14.4179 0.817564 0.408782 0.912632i $$-0.365954\pi$$
0.408782 + 0.912632i $$0.365954\pi$$
$$312$$ 0 0
$$313$$ 14.2338 0.804544 0.402272 0.915520i $$-0.368221\pi$$
0.402272 + 0.915520i $$0.368221\pi$$
$$314$$ −44.0210 −2.48425
$$315$$ 0 0
$$316$$ 19.2174 1.08107
$$317$$ −6.84415 −0.384406 −0.192203 0.981355i $$-0.561563\pi$$
−0.192203 + 0.981355i $$0.561563\pi$$
$$318$$ 0 0
$$319$$ 10.9782 0.614663
$$320$$ −43.7289 −2.44452
$$321$$ 0 0
$$322$$ 14.7235 0.820507
$$323$$ −7.18598 −0.399839
$$324$$ 0 0
$$325$$ 0 0
$$326$$ −29.0828 −1.61075
$$327$$ 0 0
$$328$$ −3.11828 −0.172178
$$329$$ 1.96077 0.108101
$$330$$ 0 0
$$331$$ −9.44265 −0.519015 −0.259507 0.965741i $$-0.583560\pi$$
−0.259507 + 0.965741i $$0.583560\pi$$
$$332$$ 25.0586 1.37527
$$333$$ 0 0
$$334$$ 27.0901 1.48230
$$335$$ −53.1420 −2.90346
$$336$$ 0 0
$$337$$ 2.64310 0.143979 0.0719895 0.997405i $$-0.477065\pi$$
0.0719895 + 0.997405i $$0.477065\pi$$
$$338$$ 0 0
$$339$$ 0 0
$$340$$ −38.5743 −2.09199
$$341$$ 6.68473 0.361998
$$342$$ 0 0
$$343$$ −10.7114 −0.578361
$$344$$ −5.92228 −0.319308
$$345$$ 0 0
$$346$$ −28.5520 −1.53496
$$347$$ 10.1588 0.545355 0.272677 0.962106i $$-0.412091\pi$$
0.272677 + 0.962106i $$0.412091\pi$$
$$348$$ 0 0
$$349$$ −10.4397 −0.558822 −0.279411 0.960172i $$-0.590139\pi$$
−0.279411 + 0.960172i $$0.590139\pi$$
$$350$$ 16.3134 0.871986
$$351$$ 0 0
$$352$$ 10.6304 0.566604
$$353$$ 18.2911 0.973538 0.486769 0.873531i $$-0.338175\pi$$
0.486769 + 0.873531i $$0.338175\pi$$
$$354$$ 0 0
$$355$$ 29.9976 1.59211
$$356$$ 4.02310 0.213224
$$357$$ 0 0
$$358$$ 1.26934 0.0670867
$$359$$ −15.2731 −0.806081 −0.403041 0.915182i $$-0.632047\pi$$
−0.403041 + 0.915182i $$0.632047\pi$$
$$360$$ 0 0
$$361$$ −13.0218 −0.685356
$$362$$ −54.8775 −2.88430
$$363$$ 0 0
$$364$$ 0 0
$$365$$ 43.8896 2.29729
$$366$$ 0 0
$$367$$ −22.2717 −1.16258 −0.581288 0.813698i $$-0.697451\pi$$
−0.581288 + 0.813698i $$0.697451\pi$$
$$368$$ 11.9014 0.620405
$$369$$ 0 0
$$370$$ −64.7846 −3.36799
$$371$$ 7.98792 0.414712
$$372$$ 0 0
$$373$$ 4.12631 0.213652 0.106826 0.994278i $$-0.465931\pi$$
0.106826 + 0.994278i $$0.465931\pi$$
$$374$$ 19.7476 1.02113
$$375$$ 0 0
$$376$$ 8.96077 0.462116
$$377$$ 0 0
$$378$$ 0 0
$$379$$ 10.7071 0.549986 0.274993 0.961446i $$-0.411324\pi$$
0.274993 + 0.961446i $$0.411324\pi$$
$$380$$ 32.0911 1.64624
$$381$$ 0 0
$$382$$ −39.5150 −2.02176
$$383$$ 6.52648 0.333488 0.166744 0.986000i $$-0.446675\pi$$
0.166744 + 0.986000i $$0.446675\pi$$
$$384$$ 0 0
$$385$$ −8.44073 −0.430179
$$386$$ −60.6757 −3.08831
$$387$$ 0 0
$$388$$ 21.1320 1.07282
$$389$$ 11.7922 0.597891 0.298945 0.954270i $$-0.403365\pi$$
0.298945 + 0.954270i $$0.403365\pi$$
$$390$$ 0 0
$$391$$ −22.8944 −1.15782
$$392$$ −23.2972 −1.17669
$$393$$ 0 0
$$394$$ 50.4868 2.54349
$$395$$ 19.9584 1.00422
$$396$$ 0 0
$$397$$ 12.5429 0.629509 0.314754 0.949173i $$-0.398078\pi$$
0.314754 + 0.949173i $$0.398078\pi$$
$$398$$ 8.31468 0.416777
$$399$$ 0 0
$$400$$ 13.1866 0.659329
$$401$$ 17.8702 0.892397 0.446198 0.894934i $$-0.352778\pi$$
0.446198 + 0.894934i $$0.352778\pi$$
$$402$$ 0 0
$$403$$ 0 0
$$404$$ −16.4440 −0.818118
$$405$$ 0 0
$$406$$ −7.27844 −0.361223
$$407$$ 21.2247 1.05207
$$408$$ 0 0
$$409$$ 27.9119 1.38015 0.690076 0.723737i $$-0.257577\pi$$
0.690076 + 0.723737i $$0.257577\pi$$
$$410$$ −7.40389 −0.365652
$$411$$ 0 0
$$412$$ 4.29350 0.211526
$$413$$ −4.31767 −0.212459
$$414$$ 0 0
$$415$$ 26.0248 1.27750
$$416$$ 0 0
$$417$$ 0 0
$$418$$ −16.4286 −0.803551
$$419$$ 16.4034 0.801360 0.400680 0.916218i $$-0.368774\pi$$
0.400680 + 0.916218i $$0.368774\pi$$
$$420$$ 0 0
$$421$$ 3.03684 0.148006 0.0740032 0.997258i $$-0.476423\pi$$
0.0740032 + 0.997258i $$0.476423\pi$$
$$422$$ 2.86486 0.139459
$$423$$ 0 0
$$424$$ 36.5050 1.77284
$$425$$ −25.3666 −1.23046
$$426$$ 0 0
$$427$$ −10.6310 −0.514471
$$428$$ 33.8471 1.63606
$$429$$ 0 0
$$430$$ −14.0616 −0.678110
$$431$$ −3.33811 −0.160791 −0.0803955 0.996763i $$-0.525618\pi$$
−0.0803955 + 0.996763i $$0.525618\pi$$
$$432$$ 0 0
$$433$$ 11.9028 0.572010 0.286005 0.958228i $$-0.407673\pi$$
0.286005 + 0.958228i $$0.407673\pi$$
$$434$$ −4.43190 −0.212738
$$435$$ 0 0
$$436$$ 6.34375 0.303810
$$437$$ 19.0465 0.911119
$$438$$ 0 0
$$439$$ 3.71810 0.177455 0.0887277 0.996056i $$-0.471720\pi$$
0.0887277 + 0.996056i $$0.471720\pi$$
$$440$$ −38.5743 −1.83896
$$441$$ 0 0
$$442$$ 0 0
$$443$$ −1.45712 −0.0692300 −0.0346150 0.999401i $$-0.511021\pi$$
−0.0346150 + 0.999401i $$0.511021\pi$$
$$444$$ 0 0
$$445$$ 4.17821 0.198066
$$446$$ 40.9825 1.94058
$$447$$ 0 0
$$448$$ −9.49827 −0.448751
$$449$$ −12.1274 −0.572326 −0.286163 0.958181i $$-0.592380\pi$$
−0.286163 + 0.958181i $$0.592380\pi$$
$$450$$ 0 0
$$451$$ 2.42566 0.114220
$$452$$ −17.6009 −0.827876
$$453$$ 0 0
$$454$$ 41.0465 1.92641
$$455$$ 0 0
$$456$$ 0 0
$$457$$ 3.44803 0.161292 0.0806459 0.996743i $$-0.474302\pi$$
0.0806459 + 0.996743i $$0.474302\pi$$
$$458$$ −44.2161 −2.06608
$$459$$ 0 0
$$460$$ 102.242 4.76704
$$461$$ −6.75600 −0.314658 −0.157329 0.987546i $$-0.550288\pi$$
−0.157329 + 0.987546i $$0.550288\pi$$
$$462$$ 0 0
$$463$$ 7.45175 0.346312 0.173156 0.984894i $$-0.444604\pi$$
0.173156 + 0.984894i $$0.444604\pi$$
$$464$$ −5.88338 −0.273129
$$465$$ 0 0
$$466$$ −9.31229 −0.431384
$$467$$ −32.6098 −1.50900 −0.754502 0.656298i $$-0.772121\pi$$
−0.754502 + 0.656298i $$0.772121\pi$$
$$468$$ 0 0
$$469$$ −11.5429 −0.533001
$$470$$ 21.2760 0.981391
$$471$$ 0 0
$$472$$ −19.7318 −0.908232
$$473$$ 4.60686 0.211824
$$474$$ 0 0
$$475$$ 21.1032 0.968282
$$476$$ −8.37867 −0.384036
$$477$$ 0 0
$$478$$ 1.92872 0.0882177
$$479$$ 2.82908 0.129264 0.0646321 0.997909i $$-0.479413\pi$$
0.0646321 + 0.997909i $$0.479413\pi$$
$$480$$ 0 0
$$481$$ 0 0
$$482$$ −14.2180 −0.647614
$$483$$ 0 0
$$484$$ −10.2121 −0.464185
$$485$$ 21.9468 0.996552
$$486$$ 0 0
$$487$$ 41.2935 1.87119 0.935594 0.353079i $$-0.114865\pi$$
0.935594 + 0.353079i $$0.114865\pi$$
$$488$$ −48.5840 −2.19930
$$489$$ 0 0
$$490$$ −55.3159 −2.49892
$$491$$ 34.6698 1.56463 0.782313 0.622886i $$-0.214040\pi$$
0.782313 + 0.622886i $$0.214040\pi$$
$$492$$ 0 0
$$493$$ 11.3177 0.509722
$$494$$ 0 0
$$495$$ 0 0
$$496$$ −3.58243 −0.160856
$$497$$ 6.51573 0.292270
$$498$$ 0 0
$$499$$ −17.9409 −0.803146 −0.401573 0.915827i $$-0.631536\pi$$
−0.401573 + 0.915827i $$0.631536\pi$$
$$500$$ 47.6571 2.13129
$$501$$ 0 0
$$502$$ 63.3532 2.82759
$$503$$ 26.1812 1.16736 0.583681 0.811983i $$-0.301612\pi$$
0.583681 + 0.811983i $$0.301612\pi$$
$$504$$ 0 0
$$505$$ −17.0780 −0.759960
$$506$$ −52.3414 −2.32686
$$507$$ 0 0
$$508$$ 20.1575 0.894345
$$509$$ −5.50604 −0.244051 −0.122025 0.992527i $$-0.538939\pi$$
−0.122025 + 0.992527i $$0.538939\pi$$
$$510$$ 0 0
$$511$$ 9.53319 0.421723
$$512$$ −16.8955 −0.746681
$$513$$ 0 0
$$514$$ 21.3375 0.941158
$$515$$ 4.45904 0.196489
$$516$$ 0 0
$$517$$ −6.97046 −0.306560
$$518$$ −14.0718 −0.618277
$$519$$ 0 0
$$520$$ 0 0
$$521$$ 26.7211 1.17067 0.585336 0.810791i $$-0.300963\pi$$
0.585336 + 0.810791i $$0.300963\pi$$
$$522$$ 0 0
$$523$$ 36.5230 1.59704 0.798520 0.601968i $$-0.205617\pi$$
0.798520 + 0.601968i $$0.205617\pi$$
$$524$$ −64.7813 −2.82999
$$525$$ 0 0
$$526$$ −54.5646 −2.37913
$$527$$ 6.89141 0.300195
$$528$$ 0 0
$$529$$ 37.6819 1.63834
$$530$$ 86.6757 3.76495
$$531$$ 0 0
$$532$$ 6.97046 0.302208
$$533$$ 0 0
$$534$$ 0 0
$$535$$ 35.1521 1.51976
$$536$$ −52.7512 −2.27851
$$537$$ 0 0
$$538$$ 5.70576 0.245993
$$539$$ 18.1226 0.780595
$$540$$ 0 0
$$541$$ −18.4655 −0.793893 −0.396947 0.917842i $$-0.629930\pi$$
−0.396947 + 0.917842i $$0.629930\pi$$
$$542$$ 50.5437 2.17104
$$543$$ 0 0
$$544$$ 10.9591 0.469868
$$545$$ 6.58834 0.282213
$$546$$ 0 0
$$547$$ 39.8471 1.70374 0.851870 0.523753i $$-0.175468\pi$$
0.851870 + 0.523753i $$0.175468\pi$$
$$548$$ 33.5958 1.43514
$$549$$ 0 0
$$550$$ −57.9933 −2.47284
$$551$$ −9.41550 −0.401114
$$552$$ 0 0
$$553$$ 4.33513 0.184348
$$554$$ −34.8993 −1.48273
$$555$$ 0 0
$$556$$ 14.2734 0.605327
$$557$$ 9.20477 0.390018 0.195009 0.980801i $$-0.437526\pi$$
0.195009 + 0.980801i $$0.437526\pi$$
$$558$$ 0 0
$$559$$ 0 0
$$560$$ 4.52350 0.191153
$$561$$ 0 0
$$562$$ 34.1836 1.44195
$$563$$ 0.975246 0.0411017 0.0205509 0.999789i $$-0.493458\pi$$
0.0205509 + 0.999789i $$0.493458\pi$$
$$564$$ 0 0
$$565$$ −18.2795 −0.769024
$$566$$ 60.5066 2.54328
$$567$$ 0 0
$$568$$ 29.7770 1.24942
$$569$$ 16.8944 0.708250 0.354125 0.935198i $$-0.384779\pi$$
0.354125 + 0.935198i $$0.384779\pi$$
$$570$$ 0 0
$$571$$ −44.3226 −1.85484 −0.927421 0.374019i $$-0.877979\pi$$
−0.927421 + 0.374019i $$0.877979\pi$$
$$572$$ 0 0
$$573$$ 0 0
$$574$$ −1.60819 −0.0671244
$$575$$ 67.2344 2.80387
$$576$$ 0 0
$$577$$ −3.56704 −0.148498 −0.0742489 0.997240i $$-0.523656\pi$$
−0.0742489 + 0.997240i $$0.523656\pi$$
$$578$$ −19.7090 −0.819787
$$579$$ 0 0
$$580$$ −50.5424 −2.09866
$$581$$ 5.65279 0.234517
$$582$$ 0 0
$$583$$ −28.3967 −1.17607
$$584$$ 43.5669 1.80281
$$585$$ 0 0
$$586$$ −62.5120 −2.58235
$$587$$ −16.1172 −0.665229 −0.332614 0.943063i $$-0.607931\pi$$
−0.332614 + 0.943063i $$0.607931\pi$$
$$588$$ 0 0
$$589$$ −5.73317 −0.236231
$$590$$ −46.8504 −1.92880
$$591$$ 0 0
$$592$$ −11.3746 −0.467494
$$593$$ −42.8611 −1.76010 −0.880048 0.474885i $$-0.842490\pi$$
−0.880048 + 0.474885i $$0.842490\pi$$
$$594$$ 0 0
$$595$$ −8.70171 −0.356735
$$596$$ 68.9869 2.82581
$$597$$ 0 0
$$598$$ 0 0
$$599$$ −40.9420 −1.67284 −0.836422 0.548086i $$-0.815357\pi$$
−0.836422 + 0.548086i $$0.815357\pi$$
$$600$$ 0 0
$$601$$ 1.18705 0.0484206 0.0242103 0.999707i $$-0.492293\pi$$
0.0242103 + 0.999707i $$0.492293\pi$$
$$602$$ −3.05429 −0.124484
$$603$$ 0 0
$$604$$ 43.9474 1.78819
$$605$$ −10.6058 −0.431187
$$606$$ 0 0
$$607$$ 19.9922 0.811460 0.405730 0.913993i $$-0.367017\pi$$
0.405730 + 0.913993i $$0.367017\pi$$
$$608$$ −9.11721 −0.369752
$$609$$ 0 0
$$610$$ −115.356 −4.67062
$$611$$ 0 0
$$612$$ 0 0
$$613$$ 33.3618 1.34747 0.673735 0.738973i $$-0.264689\pi$$
0.673735 + 0.738973i $$0.264689\pi$$
$$614$$ 19.4373 0.784424
$$615$$ 0 0
$$616$$ −8.37867 −0.337586
$$617$$ 11.6233 0.467935 0.233967 0.972244i $$-0.424829\pi$$
0.233967 + 0.972244i $$0.424829\pi$$
$$618$$ 0 0
$$619$$ 16.5381 0.664722 0.332361 0.943152i $$-0.392155\pi$$
0.332361 + 0.943152i $$0.392155\pi$$
$$620$$ −30.7756 −1.23598
$$621$$ 0 0
$$622$$ 33.9815 1.36253
$$623$$ 0.907542 0.0363599
$$624$$ 0 0
$$625$$ 6.33944 0.253577
$$626$$ 33.5477 1.34083
$$627$$ 0 0
$$628$$ −66.3979 −2.64956
$$629$$ 21.8810 0.872452
$$630$$ 0 0
$$631$$ 36.4416 1.45072 0.725358 0.688372i $$-0.241674\pi$$
0.725358 + 0.688372i $$0.241674\pi$$
$$632$$ 19.8116 0.788064
$$633$$ 0 0
$$634$$ −16.1309 −0.640642
$$635$$ 20.9347 0.830768
$$636$$ 0 0
$$637$$ 0 0
$$638$$ 25.8745 1.02438
$$639$$ 0 0
$$640$$ −75.5303 −2.98560
$$641$$ −27.2067 −1.07460 −0.537300 0.843391i $$-0.680556\pi$$
−0.537300 + 0.843391i $$0.680556\pi$$
$$642$$ 0 0
$$643$$ 5.06962 0.199926 0.0999632 0.994991i $$-0.468127\pi$$
0.0999632 + 0.994991i $$0.468127\pi$$
$$644$$ 22.2078 0.875108
$$645$$ 0 0
$$646$$ −16.9366 −0.666362
$$647$$ −19.3207 −0.759573 −0.379787 0.925074i $$-0.624003\pi$$
−0.379787 + 0.925074i $$0.624003\pi$$
$$648$$ 0 0
$$649$$ 15.3491 0.602506
$$650$$ 0 0
$$651$$ 0 0
$$652$$ −43.8662 −1.71793
$$653$$ −35.2355 −1.37887 −0.689436 0.724347i $$-0.742141\pi$$
−0.689436 + 0.724347i $$0.742141\pi$$
$$654$$ 0 0
$$655$$ −67.2790 −2.62881
$$656$$ −1.29995 −0.0507544
$$657$$ 0 0
$$658$$ 4.62133 0.180158
$$659$$ 4.36168 0.169907 0.0849535 0.996385i $$-0.472926\pi$$
0.0849535 + 0.996385i $$0.472926\pi$$
$$660$$ 0 0
$$661$$ 15.4709 0.601747 0.300873 0.953664i $$-0.402722\pi$$
0.300873 + 0.953664i $$0.402722\pi$$
$$662$$ −22.2553 −0.864978
$$663$$ 0 0
$$664$$ 25.8334 1.00253
$$665$$ 7.23921 0.280725
$$666$$ 0 0
$$667$$ −29.9976 −1.16151
$$668$$ 40.8605 1.58094
$$669$$ 0 0
$$670$$ −125.250 −4.83883
$$671$$ 37.7928 1.45898
$$672$$ 0 0
$$673$$ −11.7409 −0.452580 −0.226290 0.974060i $$-0.572660\pi$$
−0.226290 + 0.974060i $$0.572660\pi$$
$$674$$ 6.22952 0.239952
$$675$$ 0 0
$$676$$ 0 0
$$677$$ −3.44504 −0.132404 −0.0662019 0.997806i $$-0.521088\pi$$
−0.0662019 + 0.997806i $$0.521088\pi$$
$$678$$ 0 0
$$679$$ 4.76702 0.182941
$$680$$ −39.7670 −1.52500
$$681$$ 0 0
$$682$$ 15.7552 0.603298
$$683$$ 20.4058 0.780807 0.390403 0.920644i $$-0.372336\pi$$
0.390403 + 0.920644i $$0.372336\pi$$
$$684$$ 0 0
$$685$$ 34.8911 1.33312
$$686$$ −25.2457 −0.963883
$$687$$ 0 0
$$688$$ −2.46888 −0.0941251
$$689$$ 0 0
$$690$$ 0 0
$$691$$ −27.8039 −1.05771 −0.528854 0.848713i $$-0.677378\pi$$
−0.528854 + 0.848713i $$0.677378\pi$$
$$692$$ −43.0656 −1.63711
$$693$$ 0 0
$$694$$ 23.9433 0.908876
$$695$$ 14.8237 0.562295
$$696$$ 0 0
$$697$$ 2.50066 0.0947194
$$698$$ −24.6052 −0.931321
$$699$$ 0 0
$$700$$ 24.6058 0.930012
$$701$$ −11.9715 −0.452158 −0.226079 0.974109i $$-0.572591\pi$$
−0.226079 + 0.974109i $$0.572591\pi$$
$$702$$ 0 0
$$703$$ −18.2034 −0.686556
$$704$$ 33.7660 1.27260
$$705$$ 0 0
$$706$$ 43.1102 1.62248
$$707$$ −3.70948 −0.139509
$$708$$ 0 0
$$709$$ −32.2664 −1.21179 −0.605894 0.795545i $$-0.707185\pi$$
−0.605894 + 0.795545i $$0.707185\pi$$
$$710$$ 70.7012 2.65337
$$711$$ 0 0
$$712$$ 4.14749 0.155434
$$713$$ −18.2658 −0.684058
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 1.91457 0.0715510
$$717$$ 0 0
$$718$$ −35.9970 −1.34340
$$719$$ −12.1086 −0.451574 −0.225787 0.974177i $$-0.572495\pi$$
−0.225787 + 0.974177i $$0.572495\pi$$
$$720$$ 0 0
$$721$$ 0.968541 0.0360704
$$722$$ −30.6910 −1.14220
$$723$$ 0 0
$$724$$ −82.7730 −3.07623
$$725$$ −33.2368 −1.23438
$$726$$ 0 0
$$727$$ −16.6200 −0.616402 −0.308201 0.951321i $$-0.599727\pi$$
−0.308201 + 0.951321i $$0.599727\pi$$
$$728$$ 0 0
$$729$$ 0 0
$$730$$ 103.443 3.82861
$$731$$ 4.74930 0.175659
$$732$$ 0 0
$$733$$ −17.7912 −0.657132 −0.328566 0.944481i $$-0.606565\pi$$
−0.328566 + 0.944481i $$0.606565\pi$$
$$734$$ −52.4922 −1.93752
$$735$$ 0 0
$$736$$ −29.0473 −1.07070
$$737$$ 41.0344 1.51152
$$738$$ 0 0
$$739$$ 27.3618 1.00652 0.503260 0.864135i $$-0.332134\pi$$
0.503260 + 0.864135i $$0.332134\pi$$
$$740$$ −97.7160 −3.59211
$$741$$ 0 0
$$742$$ 18.8267 0.691150
$$743$$ 8.38596 0.307651 0.153826 0.988098i $$-0.450841\pi$$
0.153826 + 0.988098i $$0.450841\pi$$
$$744$$ 0 0
$$745$$ 71.6467 2.62493
$$746$$ 9.72528 0.356068
$$747$$ 0 0
$$748$$ 29.7858 1.08908
$$749$$ 7.63533 0.278989
$$750$$ 0 0
$$751$$ −38.7778 −1.41502 −0.707511 0.706703i $$-0.750182\pi$$
−0.707511 + 0.706703i $$0.750182\pi$$
$$752$$ 3.73556 0.136222
$$753$$ 0 0
$$754$$ 0 0
$$755$$ 45.6418 1.66107
$$756$$ 0 0
$$757$$ 12.9729 0.471506 0.235753 0.971813i $$-0.424244\pi$$
0.235753 + 0.971813i $$0.424244\pi$$
$$758$$ 25.2355 0.916594
$$759$$ 0 0
$$760$$ 33.0834 1.20006
$$761$$ −5.15585 −0.186899 −0.0934497 0.995624i $$-0.529789\pi$$
−0.0934497 + 0.995624i $$0.529789\pi$$
$$762$$ 0 0
$$763$$ 1.43104 0.0518072
$$764$$ −59.6013 −2.15630
$$765$$ 0 0
$$766$$ 15.3822 0.555783
$$767$$ 0 0
$$768$$ 0 0
$$769$$ 35.5013 1.28021 0.640104 0.768288i $$-0.278891\pi$$
0.640104 + 0.768288i $$0.278891\pi$$
$$770$$ −19.8939 −0.716927
$$771$$ 0 0
$$772$$ −91.5186 −3.29383
$$773$$ −6.15585 −0.221411 −0.110705 0.993853i $$-0.535311\pi$$
−0.110705 + 0.993853i $$0.535311\pi$$
$$774$$ 0 0
$$775$$ −20.2381 −0.726976
$$776$$ 21.7854 0.782050
$$777$$ 0 0
$$778$$ 27.7931 0.996431
$$779$$ −2.08038 −0.0745372
$$780$$ 0 0
$$781$$ −23.1631 −0.828843
$$782$$ −53.9597 −1.92960
$$783$$ 0 0
$$784$$ −9.71214 −0.346862
$$785$$ −68.9579 −2.46121
$$786$$ 0 0
$$787$$ −14.2107 −0.506558 −0.253279 0.967393i $$-0.581509\pi$$
−0.253279 + 0.967393i $$0.581509\pi$$
$$788$$ 76.1503 2.71274
$$789$$ 0 0
$$790$$ 47.0398 1.67360
$$791$$ −3.97046 −0.141173
$$792$$ 0 0
$$793$$ 0 0
$$794$$ 29.5623 1.04913
$$795$$ 0 0
$$796$$ 12.5412 0.444512
$$797$$ −11.9022 −0.421596 −0.210798 0.977530i $$-0.567606\pi$$
−0.210798 + 0.977530i $$0.567606\pi$$
$$798$$ 0 0
$$799$$ −7.18598 −0.254222
$$800$$ −32.1839 −1.13787
$$801$$ 0 0
$$802$$ 42.1183 1.48725
$$803$$ −33.8901 −1.19596
$$804$$ 0 0
$$805$$ 23.0640 0.812899
$$806$$ 0 0
$$807$$ 0 0
$$808$$ −16.9524 −0.596384
$$809$$ −1.61596 −0.0568140 −0.0284070 0.999596i $$-0.509043\pi$$
−0.0284070 + 0.999596i $$0.509043\pi$$
$$810$$ 0 0
$$811$$ 51.9657 1.82476 0.912381 0.409342i $$-0.134242\pi$$
0.912381 + 0.409342i $$0.134242\pi$$
$$812$$ −10.9782 −0.385260
$$813$$ 0 0
$$814$$ 50.0245 1.75336
$$815$$ −45.5575 −1.59581
$$816$$ 0 0
$$817$$ −3.95108 −0.138231
$$818$$ 65.7853 2.30013
$$819$$ 0 0
$$820$$ −11.1675 −0.389985
$$821$$ 54.8327 1.91367 0.956837 0.290627i $$-0.0938638\pi$$
0.956837 + 0.290627i $$0.0938638\pi$$
$$822$$ 0 0
$$823$$ −46.8514 −1.63314 −0.816569 0.577247i $$-0.804127\pi$$
−0.816569 + 0.577247i $$0.804127\pi$$
$$824$$ 4.42626 0.154196
$$825$$ 0 0
$$826$$ −10.1763 −0.354078
$$827$$ 21.2021 0.737270 0.368635 0.929574i $$-0.379825\pi$$
0.368635 + 0.929574i $$0.379825\pi$$
$$828$$ 0 0
$$829$$ 18.2972 0.635489 0.317744 0.948176i $$-0.397075\pi$$
0.317744 + 0.948176i $$0.397075\pi$$
$$830$$ 61.3376 2.12906
$$831$$ 0 0
$$832$$ 0 0
$$833$$ 18.6829 0.647325
$$834$$ 0 0
$$835$$ 42.4359 1.46856
$$836$$ −24.7797 −0.857024
$$837$$ 0 0
$$838$$ 38.6612 1.33553
$$839$$ 21.1414 0.729881 0.364941 0.931031i $$-0.381089\pi$$
0.364941 + 0.931031i $$0.381089\pi$$
$$840$$ 0 0
$$841$$ −14.1709 −0.488652
$$842$$ 7.15751 0.246664
$$843$$ 0 0
$$844$$ 4.32113 0.148739
$$845$$ 0 0
$$846$$ 0 0
$$847$$ −2.30367 −0.0791549
$$848$$ 15.2182 0.522594
$$849$$ 0 0
$$850$$ −59.7864 −2.05066
$$851$$ −57.9958 −1.98807
$$852$$ 0 0
$$853$$ 7.13036 0.244139 0.122069 0.992522i $$-0.461047\pi$$
0.122069 + 0.992522i $$0.461047\pi$$
$$854$$ −25.0562 −0.857406
$$855$$ 0 0
$$856$$ 34.8937 1.19264
$$857$$ 44.7741 1.52945 0.764726 0.644355i $$-0.222874\pi$$
0.764726 + 0.644355i $$0.222874\pi$$
$$858$$ 0 0
$$859$$ 57.3782 1.95772 0.978859 0.204535i $$-0.0655681\pi$$
0.978859 + 0.204535i $$0.0655681\pi$$
$$860$$ −21.2094 −0.723235
$$861$$ 0 0
$$862$$ −7.86758 −0.267971
$$863$$ −6.67563 −0.227241 −0.113621 0.993524i $$-0.536245\pi$$
−0.113621 + 0.993524i $$0.536245\pi$$
$$864$$ 0 0
$$865$$ −44.7260 −1.52073
$$866$$ 28.0536 0.953299
$$867$$ 0 0
$$868$$ −6.68473 −0.226894
$$869$$ −15.4112 −0.522789
$$870$$ 0 0
$$871$$ 0 0
$$872$$ 6.53989 0.221469
$$873$$ 0 0
$$874$$ 44.8907 1.51845
$$875$$ 10.7506 0.363438
$$876$$ 0 0
$$877$$ −25.2983 −0.854263 −0.427131 0.904190i $$-0.640476\pi$$
−0.427131 + 0.904190i $$0.640476\pi$$
$$878$$ 8.76318 0.295743
$$879$$ 0 0
$$880$$ −16.0809 −0.542085
$$881$$ 35.3787 1.19194 0.595969 0.803008i $$-0.296768\pi$$
0.595969 + 0.803008i $$0.296768\pi$$
$$882$$ 0 0
$$883$$ −11.5851 −0.389869 −0.194935 0.980816i $$-0.562449\pi$$
−0.194935 + 0.980816i $$0.562449\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ −3.43429 −0.115377
$$887$$ 27.2892 0.916281 0.458141 0.888880i $$-0.348516\pi$$
0.458141 + 0.888880i $$0.348516\pi$$
$$888$$ 0 0
$$889$$ 4.54719 0.152508
$$890$$ 9.84761 0.330093
$$891$$ 0 0
$$892$$ 61.8149 2.06972
$$893$$ 5.97823 0.200054
$$894$$ 0 0
$$895$$ 1.98839 0.0664646
$$896$$ −16.4058 −0.548080
$$897$$ 0 0
$$898$$ −28.5830 −0.953826
$$899$$ 9.02954 0.301152
$$900$$ 0 0
$$901$$ −29.2747 −0.975282
$$902$$ 5.71704 0.190357
$$903$$ 0 0
$$904$$ −18.1451 −0.603497
$$905$$ −85.9643 −2.85755
$$906$$ 0 0
$$907$$ −30.4219 −1.01014 −0.505072 0.863077i $$-0.668534\pi$$
−0.505072 + 0.863077i $$0.668534\pi$$
$$908$$ 61.9114 2.05460
$$909$$ 0 0
$$910$$ 0 0
$$911$$ −53.5719 −1.77492 −0.887459 0.460887i $$-0.847531\pi$$
−0.887459 + 0.460887i $$0.847531\pi$$
$$912$$ 0 0
$$913$$ −20.0954 −0.665062
$$914$$ 8.12664 0.268805
$$915$$ 0 0
$$916$$ −66.6921 −2.20357
$$917$$ −14.6136 −0.482582
$$918$$ 0 0
$$919$$ −36.1672 −1.19305 −0.596523 0.802596i $$-0.703451\pi$$
−0.596523 + 0.802596i $$0.703451\pi$$
$$920$$ 105.403 3.47503
$$921$$ 0 0
$$922$$ −15.9232 −0.524403
$$923$$ 0 0
$$924$$ 0 0
$$925$$ −64.2583 −2.11280
$$926$$ 17.5630 0.577156
$$927$$ 0 0
$$928$$ 14.3593 0.471367
$$929$$ −13.3478 −0.437927 −0.218964 0.975733i $$-0.570268\pi$$
−0.218964 + 0.975733i $$0.570268\pi$$
$$930$$ 0 0
$$931$$ −15.5429 −0.509397
$$932$$ −14.0459 −0.460090
$$933$$ 0 0
$$934$$ −76.8580 −2.51487
$$935$$ 30.9342 1.01166
$$936$$ 0 0
$$937$$ 38.6872 1.26386 0.631928 0.775027i $$-0.282264\pi$$
0.631928 + 0.775027i $$0.282264\pi$$
$$938$$ −27.2054 −0.888286
$$939$$ 0 0
$$940$$ 32.0911 1.04670
$$941$$ 35.7275 1.16468 0.582342 0.812944i $$-0.302136\pi$$
0.582342 + 0.812944i $$0.302136\pi$$
$$942$$ 0 0
$$943$$ −6.62804 −0.215839
$$944$$ −8.22580 −0.267727
$$945$$ 0 0
$$946$$ 10.8579 0.353020
$$947$$ −17.8436 −0.579838 −0.289919 0.957051i $$-0.593628\pi$$
−0.289919 + 0.957051i $$0.593628\pi$$
$$948$$ 0 0
$$949$$ 0 0
$$950$$ 49.7381 1.61372
$$951$$ 0 0
$$952$$ −8.63773 −0.279950
$$953$$ −20.1691 −0.653342 −0.326671 0.945138i $$-0.605927\pi$$
−0.326671 + 0.945138i $$0.605927\pi$$
$$954$$ 0 0
$$955$$ −61.8993 −2.00301
$$956$$ 2.90913 0.0940881
$$957$$ 0 0
$$958$$ 6.66786 0.215429
$$959$$ 7.57865 0.244727
$$960$$ 0 0
$$961$$ −25.5018 −0.822640
$$962$$ 0 0
$$963$$ 0 0
$$964$$ −21.4454 −0.690709
$$965$$ −95.0471 −3.05967
$$966$$ 0 0
$$967$$ 32.7894 1.05444 0.527218 0.849730i $$-0.323235\pi$$
0.527218 + 0.849730i $$0.323235\pi$$
$$968$$ −10.5278 −0.338377
$$969$$ 0 0
$$970$$ 51.7263 1.66083
$$971$$ 26.9845 0.865973 0.432986 0.901401i $$-0.357460\pi$$
0.432986 + 0.901401i $$0.357460\pi$$
$$972$$ 0 0
$$973$$ 3.21983 0.103223
$$974$$ 97.3245 3.11848
$$975$$ 0 0
$$976$$ −20.2537 −0.648305
$$977$$ −16.9571 −0.542504 −0.271252 0.962508i $$-0.587438\pi$$
−0.271252 + 0.962508i $$0.587438\pi$$
$$978$$ 0 0
$$979$$ −3.22627 −0.103112
$$980$$ −83.4341 −2.66521
$$981$$ 0 0
$$982$$ 81.7131 2.60757
$$983$$ −32.6631 −1.04179 −0.520895 0.853621i $$-0.674402\pi$$
−0.520895 + 0.853621i $$0.674402\pi$$
$$984$$ 0 0
$$985$$ 79.0863 2.51990
$$986$$ 26.6746 0.849491
$$987$$ 0 0
$$988$$ 0 0
$$989$$ −12.5881 −0.400277
$$990$$ 0 0
$$991$$ 7.64310 0.242791 0.121396 0.992604i $$-0.461263\pi$$
0.121396 + 0.992604i $$0.461263\pi$$
$$992$$ 8.74348 0.277606
$$993$$ 0 0
$$994$$ 15.3569 0.487091
$$995$$ 13.0248 0.412912
$$996$$ 0 0
$$997$$ 36.1256 1.14411 0.572054 0.820216i $$-0.306147\pi$$
0.572054 + 0.820216i $$0.306147\pi$$
$$998$$ −42.2849 −1.33850
$$999$$ 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.s.1.2 3
3.2 odd 2 507.2.a.i.1.2 3
12.11 even 2 8112.2.a.cg.1.1 3
13.5 odd 4 1521.2.b.k.1351.2 6
13.8 odd 4 1521.2.b.k.1351.5 6
13.12 even 2 1521.2.a.n.1.2 3
39.2 even 12 507.2.j.i.316.2 12
39.5 even 4 507.2.b.f.337.5 6
39.8 even 4 507.2.b.f.337.2 6
39.11 even 12 507.2.j.i.316.5 12
39.17 odd 6 507.2.e.i.484.2 6
39.20 even 12 507.2.j.i.361.2 12
39.23 odd 6 507.2.e.i.22.2 6
39.29 odd 6 507.2.e.l.22.2 6
39.32 even 12 507.2.j.i.361.5 12
39.35 odd 6 507.2.e.l.484.2 6
39.38 odd 2 507.2.a.l.1.2 yes 3
156.155 even 2 8112.2.a.cp.1.3 3

By twisted newform
Twist Min Dim Char Parity Ord Type
507.2.a.i.1.2 3 3.2 odd 2
507.2.a.l.1.2 yes 3 39.38 odd 2
507.2.b.f.337.2 6 39.8 even 4
507.2.b.f.337.5 6 39.5 even 4
507.2.e.i.22.2 6 39.23 odd 6
507.2.e.i.484.2 6 39.17 odd 6
507.2.e.l.22.2 6 39.29 odd 6
507.2.e.l.484.2 6 39.35 odd 6
507.2.j.i.316.2 12 39.2 even 12
507.2.j.i.316.5 12 39.11 even 12
507.2.j.i.361.2 12 39.20 even 12
507.2.j.i.361.5 12 39.32 even 12
1521.2.a.n.1.2 3 13.12 even 2
1521.2.a.s.1.2 3 1.1 even 1 trivial
1521.2.b.k.1351.2 6 13.5 odd 4
1521.2.b.k.1351.5 6 13.8 odd 4
8112.2.a.cg.1.1 3 12.11 even 2
8112.2.a.cp.1.3 3 156.155 even 2