# Properties

 Label 5054.2.a.x.1.2 Level $5054$ Weight $2$ Character 5054.1 Self dual yes Analytic conductor $40.356$ Analytic rank $0$ Dimension $4$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$5054 = 2 \cdot 7 \cdot 19^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 5054.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$40.3563931816$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: 4.4.151572.1 Defining polynomial: $$x^{4} - x^{3} - 10x^{2} + 8x + 4$$ x^4 - x^3 - 10*x^2 + 8*x + 4 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 266) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.2 Root $$-0.352271$$ of defining polynomial Character $$\chi$$ $$=$$ 5054.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+1.00000 q^{2} -0.352271 q^{3} +1.00000 q^{4} +4.32518 q^{5} -0.352271 q^{6} +1.00000 q^{7} +1.00000 q^{8} -2.87591 q^{9} +O(q^{10})$$ $$q+1.00000 q^{2} -0.352271 q^{3} +1.00000 q^{4} +4.32518 q^{5} -0.352271 q^{6} +1.00000 q^{7} +1.00000 q^{8} -2.87591 q^{9} +4.32518 q^{10} +2.35227 q^{11} -0.352271 q^{12} +0.801544 q^{13} +1.00000 q^{14} -1.52363 q^{15} +1.00000 q^{16} +6.22818 q^{17} -2.87591 q^{18} +4.32518 q^{20} -0.352271 q^{21} +2.35227 q^{22} -1.77445 q^{23} -0.352271 q^{24} +13.7072 q^{25} +0.801544 q^{26} +2.06991 q^{27} +1.00000 q^{28} -5.83126 q^{29} -1.52363 q^{30} +5.12672 q^{31} +1.00000 q^{32} -0.828636 q^{33} +6.22818 q^{34} +4.32518 q^{35} -2.87591 q^{36} -10.6504 q^{37} -0.282361 q^{39} +4.32518 q^{40} +3.64773 q^{41} -0.352271 q^{42} +0.873277 q^{43} +2.35227 q^{44} -12.4388 q^{45} -1.77445 q^{46} -5.83126 q^{47} -0.352271 q^{48} +1.00000 q^{49} +13.7072 q^{50} -2.19400 q^{51} +0.801544 q^{52} -1.12672 q^{53} +2.06991 q^{54} +10.1740 q^{55} +1.00000 q^{56} -5.83126 q^{58} +14.2011 q^{59} -1.52363 q^{60} -2.72209 q^{61} +5.12672 q^{62} -2.87591 q^{63} +1.00000 q^{64} +3.46682 q^{65} -0.828636 q^{66} -4.58045 q^{67} +6.22818 q^{68} +0.625088 q^{69} +4.32518 q^{70} -14.0026 q^{71} -2.87591 q^{72} -9.23080 q^{73} -10.6504 q^{74} -4.82864 q^{75} +2.35227 q^{77} -0.282361 q^{78} +7.04727 q^{79} +4.32518 q^{80} +7.89855 q^{81} +3.64773 q^{82} +15.5560 q^{83} -0.352271 q^{84} +26.9380 q^{85} +0.873277 q^{86} +2.05418 q^{87} +2.35227 q^{88} -7.29546 q^{89} -12.4388 q^{90} +0.801544 q^{91} -1.77445 q^{92} -1.80600 q^{93} -5.83126 q^{94} -0.352271 q^{96} -3.05681 q^{97} +1.00000 q^{98} -6.76491 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 4 q^{2} + q^{3} + 4 q^{4} + q^{5} + q^{6} + 4 q^{7} + 4 q^{8} + 9 q^{9}+O(q^{10})$$ 4 * q + 4 * q^2 + q^3 + 4 * q^4 + q^5 + q^6 + 4 * q^7 + 4 * q^8 + 9 * q^9 $$4 q + 4 q^{2} + q^{3} + 4 q^{4} + q^{5} + q^{6} + 4 q^{7} + 4 q^{8} + 9 q^{9} + q^{10} + 7 q^{11} + q^{12} + 5 q^{13} + 4 q^{14} + 12 q^{15} + 4 q^{16} + 2 q^{17} + 9 q^{18} + q^{20} + q^{21} + 7 q^{22} + 5 q^{23} + q^{24} + 15 q^{25} + 5 q^{26} + q^{27} + 4 q^{28} - 4 q^{29} + 12 q^{30} + 6 q^{31} + 4 q^{32} - 19 q^{33} + 2 q^{34} + q^{35} + 9 q^{36} - 10 q^{37} - 6 q^{39} + q^{40} + 17 q^{41} + q^{42} + 18 q^{43} + 7 q^{44} - 19 q^{45} + 5 q^{46} - 4 q^{47} + q^{48} + 4 q^{49} + 15 q^{50} - 22 q^{51} + 5 q^{52} + 10 q^{53} + q^{54} - 10 q^{55} + 4 q^{56} - 4 q^{58} + 20 q^{59} + 12 q^{60} + 9 q^{61} + 6 q^{62} + 9 q^{63} + 4 q^{64} + 3 q^{65} - 19 q^{66} + 7 q^{67} + 2 q^{68} - 24 q^{69} + q^{70} - 21 q^{71} + 9 q^{72} + 21 q^{73} - 10 q^{74} - 35 q^{75} + 7 q^{77} - 6 q^{78} - 8 q^{79} + q^{80} + 40 q^{81} + 17 q^{82} - 12 q^{83} + q^{84} + 10 q^{85} + 18 q^{86} + 36 q^{87} + 7 q^{88} - 34 q^{89} - 19 q^{90} + 5 q^{91} + 5 q^{92} + 6 q^{93} - 4 q^{94} + q^{96} - 5 q^{97} + 4 q^{98} + 14 q^{99}+O(q^{100})$$ 4 * q + 4 * q^2 + q^3 + 4 * q^4 + q^5 + q^6 + 4 * q^7 + 4 * q^8 + 9 * q^9 + q^10 + 7 * q^11 + q^12 + 5 * q^13 + 4 * q^14 + 12 * q^15 + 4 * q^16 + 2 * q^17 + 9 * q^18 + q^20 + q^21 + 7 * q^22 + 5 * q^23 + q^24 + 15 * q^25 + 5 * q^26 + q^27 + 4 * q^28 - 4 * q^29 + 12 * q^30 + 6 * q^31 + 4 * q^32 - 19 * q^33 + 2 * q^34 + q^35 + 9 * q^36 - 10 * q^37 - 6 * q^39 + q^40 + 17 * q^41 + q^42 + 18 * q^43 + 7 * q^44 - 19 * q^45 + 5 * q^46 - 4 * q^47 + q^48 + 4 * q^49 + 15 * q^50 - 22 * q^51 + 5 * q^52 + 10 * q^53 + q^54 - 10 * q^55 + 4 * q^56 - 4 * q^58 + 20 * q^59 + 12 * q^60 + 9 * q^61 + 6 * q^62 + 9 * q^63 + 4 * q^64 + 3 * q^65 - 19 * q^66 + 7 * q^67 + 2 * q^68 - 24 * q^69 + q^70 - 21 * q^71 + 9 * q^72 + 21 * q^73 - 10 * q^74 - 35 * q^75 + 7 * q^77 - 6 * q^78 - 8 * q^79 + q^80 + 40 * q^81 + 17 * q^82 - 12 * q^83 + q^84 + 10 * q^85 + 18 * q^86 + 36 * q^87 + 7 * q^88 - 34 * q^89 - 19 * q^90 + 5 * q^91 + 5 * q^92 + 6 * q^93 - 4 * q^94 + q^96 - 5 * q^97 + 4 * q^98 + 14 * 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
$$3$$ −0.352271 −0.203384 −0.101692 0.994816i $$-0.532426\pi$$
−0.101692 + 0.994816i $$0.532426\pi$$
$$4$$ 1.00000 0.500000
$$5$$ 4.32518 1.93428 0.967139 0.254247i $$-0.0818276\pi$$
0.967139 + 0.254247i $$0.0818276\pi$$
$$6$$ −0.352271 −0.143814
$$7$$ 1.00000 0.377964
$$8$$ 1.00000 0.353553
$$9$$ −2.87591 −0.958635
$$10$$ 4.32518 1.36774
$$11$$ 2.35227 0.709236 0.354618 0.935011i $$-0.384611\pi$$
0.354618 + 0.935011i $$0.384611\pi$$
$$12$$ −0.352271 −0.101692
$$13$$ 0.801544 0.222308 0.111154 0.993803i $$-0.464545\pi$$
0.111154 + 0.993803i $$0.464545\pi$$
$$14$$ 1.00000 0.267261
$$15$$ −1.52363 −0.393401
$$16$$ 1.00000 0.250000
$$17$$ 6.22818 1.51055 0.755277 0.655405i $$-0.227502\pi$$
0.755277 + 0.655405i $$0.227502\pi$$
$$18$$ −2.87591 −0.677857
$$19$$ 0 0
$$20$$ 4.32518 0.967139
$$21$$ −0.352271 −0.0768718
$$22$$ 2.35227 0.501506
$$23$$ −1.77445 −0.369999 −0.184999 0.982739i $$-0.559228\pi$$
−0.184999 + 0.982739i $$0.559228\pi$$
$$24$$ −0.352271 −0.0719070
$$25$$ 13.7072 2.74143
$$26$$ 0.801544 0.157196
$$27$$ 2.06991 0.398354
$$28$$ 1.00000 0.188982
$$29$$ −5.83126 −1.08284 −0.541419 0.840753i $$-0.682113\pi$$
−0.541419 + 0.840753i $$0.682113\pi$$
$$30$$ −1.52363 −0.278176
$$31$$ 5.12672 0.920787 0.460393 0.887715i $$-0.347708\pi$$
0.460393 + 0.887715i $$0.347708\pi$$
$$32$$ 1.00000 0.176777
$$33$$ −0.828636 −0.144247
$$34$$ 6.22818 1.06812
$$35$$ 4.32518 0.731089
$$36$$ −2.87591 −0.479318
$$37$$ −10.6504 −1.75091 −0.875454 0.483302i $$-0.839437\pi$$
−0.875454 + 0.483302i $$0.839437\pi$$
$$38$$ 0 0
$$39$$ −0.282361 −0.0452139
$$40$$ 4.32518 0.683871
$$41$$ 3.64773 0.569680 0.284840 0.958575i $$-0.408060\pi$$
0.284840 + 0.958575i $$0.408060\pi$$
$$42$$ −0.352271 −0.0543566
$$43$$ 0.873277 0.133174 0.0665868 0.997781i $$-0.478789\pi$$
0.0665868 + 0.997781i $$0.478789\pi$$
$$44$$ 2.35227 0.354618
$$45$$ −12.4388 −1.85427
$$46$$ −1.77445 −0.261629
$$47$$ −5.83126 −0.850577 −0.425289 0.905058i $$-0.639827\pi$$
−0.425289 + 0.905058i $$0.639827\pi$$
$$48$$ −0.352271 −0.0508459
$$49$$ 1.00000 0.142857
$$50$$ 13.7072 1.93849
$$51$$ −2.19400 −0.307222
$$52$$ 0.801544 0.111154
$$53$$ −1.12672 −0.154767 −0.0773836 0.997001i $$-0.524657\pi$$
−0.0773836 + 0.997001i $$0.524657\pi$$
$$54$$ 2.06991 0.281679
$$55$$ 10.1740 1.37186
$$56$$ 1.00000 0.133631
$$57$$ 0 0
$$58$$ −5.83126 −0.765683
$$59$$ 14.2011 1.84882 0.924412 0.381396i $$-0.124557\pi$$
0.924412 + 0.381396i $$0.124557\pi$$
$$60$$ −1.52363 −0.196700
$$61$$ −2.72209 −0.348528 −0.174264 0.984699i $$-0.555755\pi$$
−0.174264 + 0.984699i $$0.555755\pi$$
$$62$$ 5.12672 0.651094
$$63$$ −2.87591 −0.362330
$$64$$ 1.00000 0.125000
$$65$$ 3.46682 0.430006
$$66$$ −0.828636 −0.101998
$$67$$ −4.58045 −0.559591 −0.279795 0.960060i $$-0.590267\pi$$
−0.279795 + 0.960060i $$0.590267\pi$$
$$68$$ 6.22818 0.755277
$$69$$ 0.625088 0.0752517
$$70$$ 4.32518 0.516958
$$71$$ −14.0026 −1.66181 −0.830903 0.556417i $$-0.812176\pi$$
−0.830903 + 0.556417i $$0.812176\pi$$
$$72$$ −2.87591 −0.338929
$$73$$ −9.23080 −1.08038 −0.540192 0.841542i $$-0.681648\pi$$
−0.540192 + 0.841542i $$0.681648\pi$$
$$74$$ −10.6504 −1.23808
$$75$$ −4.82864 −0.557563
$$76$$ 0 0
$$77$$ 2.35227 0.268066
$$78$$ −0.282361 −0.0319710
$$79$$ 7.04727 0.792880 0.396440 0.918061i $$-0.370246\pi$$
0.396440 + 0.918061i $$0.370246\pi$$
$$80$$ 4.32518 0.483570
$$81$$ 7.89855 0.877616
$$82$$ 3.64773 0.402824
$$83$$ 15.5560 1.70749 0.853745 0.520691i $$-0.174325\pi$$
0.853745 + 0.520691i $$0.174325\pi$$
$$84$$ −0.352271 −0.0384359
$$85$$ 26.9380 2.92183
$$86$$ 0.873277 0.0941679
$$87$$ 2.05418 0.220232
$$88$$ 2.35227 0.250753
$$89$$ −7.29546 −0.773317 −0.386659 0.922223i $$-0.626371\pi$$
−0.386659 + 0.922223i $$0.626371\pi$$
$$90$$ −12.4388 −1.31117
$$91$$ 0.801544 0.0840247
$$92$$ −1.77445 −0.184999
$$93$$ −1.80600 −0.187273
$$94$$ −5.83126 −0.601449
$$95$$ 0 0
$$96$$ −0.352271 −0.0359535
$$97$$ −3.05681 −0.310372 −0.155186 0.987885i $$-0.549598\pi$$
−0.155186 + 0.987885i $$0.549598\pi$$
$$98$$ 1.00000 0.101015
$$99$$ −6.76491 −0.679899
$$100$$ 13.7072 1.37072
$$101$$ 1.52363 0.151607 0.0758036 0.997123i $$-0.475848\pi$$
0.0758036 + 0.997123i $$0.475848\pi$$
$$102$$ −2.19400 −0.217239
$$103$$ 11.5236 1.13546 0.567729 0.823216i $$-0.307822\pi$$
0.567729 + 0.823216i $$0.307822\pi$$
$$104$$ 0.801544 0.0785979
$$105$$ −1.52363 −0.148691
$$106$$ −1.12672 −0.109437
$$107$$ −16.0053 −1.54729 −0.773643 0.633621i $$-0.781568\pi$$
−0.773643 + 0.633621i $$0.781568\pi$$
$$108$$ 2.06991 0.199177
$$109$$ 7.24035 0.693500 0.346750 0.937958i $$-0.387285\pi$$
0.346750 + 0.937958i $$0.387285\pi$$
$$110$$ 10.1740 0.970052
$$111$$ 3.75181 0.356106
$$112$$ 1.00000 0.0944911
$$113$$ −1.47637 −0.138885 −0.0694424 0.997586i $$-0.522122\pi$$
−0.0694424 + 0.997586i $$0.522122\pi$$
$$114$$ 0 0
$$115$$ −7.67482 −0.715681
$$116$$ −5.83126 −0.541419
$$117$$ −2.30517 −0.213113
$$118$$ 14.2011 1.30732
$$119$$ 6.22818 0.570936
$$120$$ −1.52363 −0.139088
$$121$$ −5.46682 −0.496984
$$122$$ −2.72209 −0.246446
$$123$$ −1.28499 −0.115864
$$124$$ 5.12672 0.460393
$$125$$ 37.6601 3.36842
$$126$$ −2.87591 −0.256206
$$127$$ 6.23080 0.552894 0.276447 0.961029i $$-0.410843\pi$$
0.276447 + 0.961029i $$0.410843\pi$$
$$128$$ 1.00000 0.0883883
$$129$$ −0.307630 −0.0270853
$$130$$ 3.46682 0.304060
$$131$$ −3.66528 −0.320237 −0.160118 0.987098i $$-0.551188\pi$$
−0.160118 + 0.987098i $$0.551188\pi$$
$$132$$ −0.828636 −0.0721235
$$133$$ 0 0
$$134$$ −4.58045 −0.395690
$$135$$ 8.95273 0.770528
$$136$$ 6.22818 0.534062
$$137$$ 3.39691 0.290218 0.145109 0.989416i $$-0.453647\pi$$
0.145109 + 0.989416i $$0.453647\pi$$
$$138$$ 0.625088 0.0532110
$$139$$ −5.19663 −0.440773 −0.220386 0.975413i $$-0.570732\pi$$
−0.220386 + 0.975413i $$0.570732\pi$$
$$140$$ 4.32518 0.365544
$$141$$ 2.05418 0.172994
$$142$$ −14.0026 −1.17507
$$143$$ 1.88545 0.157669
$$144$$ −2.87591 −0.239659
$$145$$ −25.2213 −2.09451
$$146$$ −9.23080 −0.763947
$$147$$ −0.352271 −0.0290548
$$148$$ −10.6504 −0.875454
$$149$$ 14.6504 1.20020 0.600102 0.799923i $$-0.295127\pi$$
0.600102 + 0.799923i $$0.295127\pi$$
$$150$$ −4.82864 −0.394257
$$151$$ −5.46682 −0.444884 −0.222442 0.974946i $$-0.571403\pi$$
−0.222442 + 0.974946i $$0.571403\pi$$
$$152$$ 0 0
$$153$$ −17.9116 −1.44807
$$154$$ 2.35227 0.189551
$$155$$ 22.1740 1.78106
$$156$$ −0.282361 −0.0226069
$$157$$ −3.45190 −0.275492 −0.137746 0.990468i $$-0.543986\pi$$
−0.137746 + 0.990468i $$0.543986\pi$$
$$158$$ 7.04727 0.560651
$$159$$ 0.396912 0.0314771
$$160$$ 4.32518 0.341935
$$161$$ −1.77445 −0.139846
$$162$$ 7.89855 0.620568
$$163$$ 8.65990 0.678296 0.339148 0.940733i $$-0.389861\pi$$
0.339148 + 0.940733i $$0.389861\pi$$
$$164$$ 3.64773 0.284840
$$165$$ −3.58400 −0.279014
$$166$$ 15.5560 1.20738
$$167$$ 6.05944 0.468894 0.234447 0.972129i $$-0.424672\pi$$
0.234447 + 0.972129i $$0.424672\pi$$
$$168$$ −0.352271 −0.0271783
$$169$$ −12.3575 −0.950579
$$170$$ 26.9380 2.06605
$$171$$ 0 0
$$172$$ 0.873277 0.0665868
$$173$$ −15.6259 −1.18801 −0.594007 0.804460i $$-0.702455\pi$$
−0.594007 + 0.804460i $$0.702455\pi$$
$$174$$ 2.05418 0.155727
$$175$$ 13.7072 1.03616
$$176$$ 2.35227 0.177309
$$177$$ −5.00263 −0.376021
$$178$$ −7.29546 −0.546818
$$179$$ −11.3995 −0.852042 −0.426021 0.904713i $$-0.640085\pi$$
−0.426021 + 0.904713i $$0.640085\pi$$
$$180$$ −12.4388 −0.927134
$$181$$ −3.27791 −0.243645 −0.121823 0.992552i $$-0.538874\pi$$
−0.121823 + 0.992552i $$0.538874\pi$$
$$182$$ 0.801544 0.0594144
$$183$$ 0.958913 0.0708849
$$184$$ −1.77445 −0.130814
$$185$$ −46.0647 −3.38674
$$186$$ −1.80600 −0.132422
$$187$$ 14.6504 1.07134
$$188$$ −5.83126 −0.425289
$$189$$ 2.06991 0.150564
$$190$$ 0 0
$$191$$ −3.23865 −0.234340 −0.117170 0.993112i $$-0.537382\pi$$
−0.117170 + 0.993112i $$0.537382\pi$$
$$192$$ −0.352271 −0.0254230
$$193$$ −12.0568 −0.867868 −0.433934 0.900945i $$-0.642875\pi$$
−0.433934 + 0.900945i $$0.642875\pi$$
$$194$$ −3.05681 −0.219466
$$195$$ −1.22126 −0.0874563
$$196$$ 1.00000 0.0714286
$$197$$ −20.9380 −1.49177 −0.745884 0.666075i $$-0.767973\pi$$
−0.745884 + 0.666075i $$0.767973\pi$$
$$198$$ −6.76491 −0.480761
$$199$$ 9.12672 0.646976 0.323488 0.946232i $$-0.395144\pi$$
0.323488 + 0.946232i $$0.395144\pi$$
$$200$$ 13.7072 0.969243
$$201$$ 1.61356 0.113812
$$202$$ 1.52363 0.107203
$$203$$ −5.83126 −0.409275
$$204$$ −2.19400 −0.153611
$$205$$ 15.7771 1.10192
$$206$$ 11.5236 0.802890
$$207$$ 5.10316 0.354694
$$208$$ 0.801544 0.0555771
$$209$$ 0 0
$$210$$ −1.52363 −0.105141
$$211$$ 4.00000 0.275371 0.137686 0.990476i $$-0.456034\pi$$
0.137686 + 0.990476i $$0.456034\pi$$
$$212$$ −1.12672 −0.0773836
$$213$$ 4.93272 0.337984
$$214$$ −16.0053 −1.09410
$$215$$ 3.77708 0.257595
$$216$$ 2.06991 0.140840
$$217$$ 5.12672 0.348025
$$218$$ 7.24035 0.490378
$$219$$ 3.25174 0.219732
$$220$$ 10.1740 0.685930
$$221$$ 4.99216 0.335809
$$222$$ 3.75181 0.251805
$$223$$ 12.2534 0.820551 0.410276 0.911962i $$-0.365432\pi$$
0.410276 + 0.911962i $$0.365432\pi$$
$$224$$ 1.00000 0.0668153
$$225$$ −39.4205 −2.62803
$$226$$ −1.47637 −0.0982064
$$227$$ 23.3531 1.55000 0.774999 0.631962i $$-0.217750\pi$$
0.774999 + 0.631962i $$0.217750\pi$$
$$228$$ 0 0
$$229$$ −25.8830 −1.71040 −0.855198 0.518302i $$-0.826564\pi$$
−0.855198 + 0.518302i $$0.826564\pi$$
$$230$$ −7.67482 −0.506063
$$231$$ −0.828636 −0.0545203
$$232$$ −5.83126 −0.382841
$$233$$ −2.60309 −0.170534 −0.0852670 0.996358i $$-0.527174\pi$$
−0.0852670 + 0.996358i $$0.527174\pi$$
$$234$$ −2.30517 −0.150693
$$235$$ −25.2213 −1.64525
$$236$$ 14.2011 0.924412
$$237$$ −2.48255 −0.161259
$$238$$ 6.22818 0.403713
$$239$$ −23.2781 −1.50573 −0.752867 0.658173i $$-0.771330\pi$$
−0.752867 + 0.658173i $$0.771330\pi$$
$$240$$ −1.52363 −0.0983502
$$241$$ 8.94319 0.576081 0.288041 0.957618i $$-0.406996\pi$$
0.288041 + 0.957618i $$0.406996\pi$$
$$242$$ −5.46682 −0.351421
$$243$$ −8.99216 −0.576847
$$244$$ −2.72209 −0.174264
$$245$$ 4.32518 0.276326
$$246$$ −1.28499 −0.0819279
$$247$$ 0 0
$$248$$ 5.12672 0.325547
$$249$$ −5.47992 −0.347276
$$250$$ 37.6601 2.38183
$$251$$ 27.3872 1.72867 0.864334 0.502918i $$-0.167740\pi$$
0.864334 + 0.502918i $$0.167740\pi$$
$$252$$ −2.87591 −0.181165
$$253$$ −4.17399 −0.262417
$$254$$ 6.23080 0.390955
$$255$$ −9.48946 −0.594253
$$256$$ 1.00000 0.0625000
$$257$$ −8.49209 −0.529722 −0.264861 0.964287i $$-0.585326\pi$$
−0.264861 + 0.964287i $$0.585326\pi$$
$$258$$ −0.307630 −0.0191522
$$259$$ −10.6504 −0.661781
$$260$$ 3.46682 0.215003
$$261$$ 16.7702 1.03805
$$262$$ −3.66528 −0.226442
$$263$$ −5.74918 −0.354510 −0.177255 0.984165i $$-0.556722\pi$$
−0.177255 + 0.984165i $$0.556722\pi$$
$$264$$ −0.828636 −0.0509990
$$265$$ −4.87328 −0.299363
$$266$$ 0 0
$$267$$ 2.56998 0.157280
$$268$$ −4.58045 −0.279795
$$269$$ 8.42218 0.513509 0.256755 0.966477i $$-0.417347\pi$$
0.256755 + 0.966477i $$0.417347\pi$$
$$270$$ 8.95273 0.544846
$$271$$ 20.7298 1.25925 0.629623 0.776901i $$-0.283209\pi$$
0.629623 + 0.776901i $$0.283209\pi$$
$$272$$ 6.22818 0.377639
$$273$$ −0.282361 −0.0170892
$$274$$ 3.39691 0.205215
$$275$$ 32.2430 1.94432
$$276$$ 0.625088 0.0376259
$$277$$ 17.4442 1.04812 0.524060 0.851682i $$-0.324417\pi$$
0.524060 + 0.851682i $$0.324417\pi$$
$$278$$ −5.19663 −0.311673
$$279$$ −14.7440 −0.882698
$$280$$ 4.32518 0.258479
$$281$$ −2.12409 −0.126713 −0.0633564 0.997991i $$-0.520180\pi$$
−0.0633564 + 0.997991i $$0.520180\pi$$
$$282$$ 2.05418 0.122325
$$283$$ −27.6696 −1.64479 −0.822394 0.568919i $$-0.807362\pi$$
−0.822394 + 0.568919i $$0.807362\pi$$
$$284$$ −14.0026 −0.830903
$$285$$ 0 0
$$286$$ 1.88545 0.111489
$$287$$ 3.64773 0.215319
$$288$$ −2.87591 −0.169464
$$289$$ 21.7902 1.28178
$$290$$ −25.2213 −1.48104
$$291$$ 1.07683 0.0631247
$$292$$ −9.23080 −0.540192
$$293$$ 13.2326 0.773058 0.386529 0.922277i $$-0.373674\pi$$
0.386529 + 0.922277i $$0.373674\pi$$
$$294$$ −0.352271 −0.0205449
$$295$$ 61.4222 3.57614
$$296$$ −10.6504 −0.619039
$$297$$ 4.86899 0.282527
$$298$$ 14.6504 0.848672
$$299$$ −1.42230 −0.0822538
$$300$$ −4.82864 −0.278781
$$301$$ 0.873277 0.0503349
$$302$$ −5.46682 −0.314580
$$303$$ −0.536732 −0.0308344
$$304$$ 0 0
$$305$$ −11.7735 −0.674150
$$306$$ −17.9116 −1.02394
$$307$$ −24.9398 −1.42339 −0.711695 0.702489i $$-0.752072\pi$$
−0.711695 + 0.702489i $$0.752072\pi$$
$$308$$ 2.35227 0.134033
$$309$$ −4.05944 −0.230934
$$310$$ 22.1740 1.25940
$$311$$ 15.9458 0.904204 0.452102 0.891966i $$-0.350674\pi$$
0.452102 + 0.891966i $$0.350674\pi$$
$$312$$ −0.282361 −0.0159855
$$313$$ −12.7203 −0.718992 −0.359496 0.933147i $$-0.617051\pi$$
−0.359496 + 0.933147i $$0.617051\pi$$
$$314$$ −3.45190 −0.194802
$$315$$ −12.4388 −0.700847
$$316$$ 7.04727 0.396440
$$317$$ 32.9091 1.84836 0.924178 0.381961i $$-0.124751\pi$$
0.924178 + 0.381961i $$0.124751\pi$$
$$318$$ 0.396912 0.0222577
$$319$$ −13.7167 −0.767989
$$320$$ 4.32518 0.241785
$$321$$ 5.63819 0.314693
$$322$$ −1.77445 −0.0988863
$$323$$ 0 0
$$324$$ 7.89855 0.438808
$$325$$ 10.9869 0.609444
$$326$$ 8.65990 0.479628
$$327$$ −2.55056 −0.141046
$$328$$ 3.64773 0.201412
$$329$$ −5.83126 −0.321488
$$330$$ −3.58400 −0.197293
$$331$$ 19.0079 1.04477 0.522384 0.852710i $$-0.325043\pi$$
0.522384 + 0.852710i $$0.325043\pi$$
$$332$$ 15.5560 0.853745
$$333$$ 30.6294 1.67848
$$334$$ 6.05944 0.331558
$$335$$ −19.8113 −1.08240
$$336$$ −0.352271 −0.0192180
$$337$$ −28.1914 −1.53568 −0.767842 0.640639i $$-0.778670\pi$$
−0.767842 + 0.640639i $$0.778670\pi$$
$$338$$ −12.3575 −0.672161
$$339$$ 0.520081 0.0282469
$$340$$ 26.9380 1.46092
$$341$$ 12.0594 0.653055
$$342$$ 0 0
$$343$$ 1.00000 0.0539949
$$344$$ 0.873277 0.0470840
$$345$$ 2.70362 0.145558
$$346$$ −15.6259 −0.840053
$$347$$ 22.7292 1.22017 0.610083 0.792338i $$-0.291136\pi$$
0.610083 + 0.792338i $$0.291136\pi$$
$$348$$ 2.05418 0.110116
$$349$$ −4.72981 −0.253181 −0.126590 0.991955i $$-0.540403\pi$$
−0.126590 + 0.991955i $$0.540403\pi$$
$$350$$ 13.7072 0.732679
$$351$$ 1.65912 0.0885575
$$352$$ 2.35227 0.125376
$$353$$ −5.76135 −0.306646 −0.153323 0.988176i $$-0.548997\pi$$
−0.153323 + 0.988176i $$0.548997\pi$$
$$354$$ −5.00263 −0.265887
$$355$$ −60.5639 −3.21440
$$356$$ −7.29546 −0.386659
$$357$$ −2.19400 −0.116119
$$358$$ −11.3995 −0.602484
$$359$$ −19.9511 −1.05298 −0.526489 0.850182i $$-0.676492\pi$$
−0.526489 + 0.850182i $$0.676492\pi$$
$$360$$ −12.4388 −0.655583
$$361$$ 0 0
$$362$$ −3.27791 −0.172283
$$363$$ 1.92580 0.101078
$$364$$ 0.801544 0.0420123
$$365$$ −39.9249 −2.08976
$$366$$ 0.958913 0.0501232
$$367$$ −1.24035 −0.0647456 −0.0323728 0.999476i $$-0.510306\pi$$
−0.0323728 + 0.999476i $$0.510306\pi$$
$$368$$ −1.77445 −0.0924997
$$369$$ −10.4905 −0.546115
$$370$$ −46.0647 −2.39479
$$371$$ −1.12672 −0.0584965
$$372$$ −1.80600 −0.0936365
$$373$$ −24.8733 −1.28789 −0.643945 0.765072i $$-0.722703\pi$$
−0.643945 + 0.765072i $$0.722703\pi$$
$$374$$ 14.6504 0.757552
$$375$$ −13.2665 −0.685081
$$376$$ −5.83126 −0.300725
$$377$$ −4.67402 −0.240724
$$378$$ 2.06991 0.106465
$$379$$ −32.4274 −1.66569 −0.832843 0.553510i $$-0.813288\pi$$
−0.832843 + 0.553510i $$0.813288\pi$$
$$380$$ 0 0
$$381$$ −2.19493 −0.112450
$$382$$ −3.23865 −0.165704
$$383$$ 14.5105 0.741454 0.370727 0.928742i $$-0.379109\pi$$
0.370727 + 0.928742i $$0.379109\pi$$
$$384$$ −0.352271 −0.0179767
$$385$$ 10.1740 0.518515
$$386$$ −12.0568 −0.613676
$$387$$ −2.51146 −0.127665
$$388$$ −3.05681 −0.155186
$$389$$ 25.5236 1.29410 0.647050 0.762448i $$-0.276003\pi$$
0.647050 + 0.762448i $$0.276003\pi$$
$$390$$ −1.22126 −0.0618409
$$391$$ −11.0516 −0.558903
$$392$$ 1.00000 0.0505076
$$393$$ 1.29117 0.0651309
$$394$$ −20.9380 −1.05484
$$395$$ 30.4807 1.53365
$$396$$ −6.76491 −0.339949
$$397$$ 11.6373 0.584057 0.292029 0.956410i $$-0.405670\pi$$
0.292029 + 0.956410i $$0.405670\pi$$
$$398$$ 9.12672 0.457481
$$399$$ 0 0
$$400$$ 13.7072 0.685358
$$401$$ 25.1740 1.25713 0.628565 0.777757i $$-0.283643\pi$$
0.628565 + 0.777757i $$0.283643\pi$$
$$402$$ 1.61356 0.0804770
$$403$$ 4.10929 0.204699
$$404$$ 1.52363 0.0758036
$$405$$ 34.1626 1.69755
$$406$$ −5.83126 −0.289401
$$407$$ −25.0525 −1.24181
$$408$$ −2.19400 −0.108619
$$409$$ −31.6277 −1.56389 −0.781945 0.623347i $$-0.785772\pi$$
−0.781945 + 0.623347i $$0.785772\pi$$
$$410$$ 15.7771 0.779174
$$411$$ −1.19663 −0.0590256
$$412$$ 11.5236 0.567729
$$413$$ 14.2011 0.698790
$$414$$ 5.10316 0.250806
$$415$$ 67.2824 3.30276
$$416$$ 0.801544 0.0392989
$$417$$ 1.83062 0.0896460
$$418$$ 0 0
$$419$$ −19.9205 −0.973182 −0.486591 0.873630i $$-0.661760\pi$$
−0.486591 + 0.873630i $$0.661760\pi$$
$$420$$ −1.52363 −0.0743457
$$421$$ −14.4274 −0.703150 −0.351575 0.936160i $$-0.614354\pi$$
−0.351575 + 0.936160i $$0.614354\pi$$
$$422$$ 4.00000 0.194717
$$423$$ 16.7702 0.815393
$$424$$ −1.12672 −0.0547185
$$425$$ 85.3707 4.14109
$$426$$ 4.93272 0.238991
$$427$$ −2.72209 −0.131731
$$428$$ −16.0053 −0.773643
$$429$$ −0.664189 −0.0320673
$$430$$ 3.77708 0.182147
$$431$$ 14.6504 0.705683 0.352841 0.935683i $$-0.385216\pi$$
0.352841 + 0.935683i $$0.385216\pi$$
$$432$$ 2.06991 0.0995886
$$433$$ 18.0043 0.865233 0.432616 0.901578i $$-0.357590\pi$$
0.432616 + 0.901578i $$0.357590\pi$$
$$434$$ 5.12672 0.246091
$$435$$ 8.88472 0.425989
$$436$$ 7.24035 0.346750
$$437$$ 0 0
$$438$$ 3.25174 0.155374
$$439$$ 21.6336 1.03252 0.516258 0.856433i $$-0.327325\pi$$
0.516258 + 0.856433i $$0.327325\pi$$
$$440$$ 10.1740 0.485026
$$441$$ −2.87591 −0.136948
$$442$$ 4.99216 0.237453
$$443$$ −3.45465 −0.164135 −0.0820677 0.996627i $$-0.526152\pi$$
−0.0820677 + 0.996627i $$0.526152\pi$$
$$444$$ 3.75181 0.178053
$$445$$ −31.5542 −1.49581
$$446$$ 12.2534 0.580217
$$447$$ −5.16089 −0.244102
$$448$$ 1.00000 0.0472456
$$449$$ 19.5069 0.920587 0.460294 0.887767i $$-0.347744\pi$$
0.460294 + 0.887767i $$0.347744\pi$$
$$450$$ −39.4205 −1.85830
$$451$$ 8.58045 0.404037
$$452$$ −1.47637 −0.0694424
$$453$$ 1.92580 0.0904821
$$454$$ 23.3531 1.09601
$$455$$ 3.46682 0.162527
$$456$$ 0 0
$$457$$ −13.2876 −0.621568 −0.310784 0.950480i $$-0.600592\pi$$
−0.310784 + 0.950480i $$0.600592\pi$$
$$458$$ −25.8830 −1.20943
$$459$$ 12.8918 0.601736
$$460$$ −7.67482 −0.357840
$$461$$ 16.1915 0.754115 0.377058 0.926190i $$-0.376936\pi$$
0.377058 + 0.926190i $$0.376936\pi$$
$$462$$ −0.828636 −0.0385517
$$463$$ 21.5010 0.999236 0.499618 0.866246i $$-0.333474\pi$$
0.499618 + 0.866246i $$0.333474\pi$$
$$464$$ −5.83126 −0.270710
$$465$$ −7.81125 −0.362238
$$466$$ −2.60309 −0.120586
$$467$$ −7.95536 −0.368130 −0.184065 0.982914i $$-0.558926\pi$$
−0.184065 + 0.982914i $$0.558926\pi$$
$$468$$ −2.30517 −0.106556
$$469$$ −4.58045 −0.211505
$$470$$ −25.2213 −1.16337
$$471$$ 1.21600 0.0560305
$$472$$ 14.2011 0.653658
$$473$$ 2.05418 0.0944515
$$474$$ −2.48255 −0.114027
$$475$$ 0 0
$$476$$ 6.22818 0.285468
$$477$$ 3.24035 0.148365
$$478$$ −23.2781 −1.06471
$$479$$ −26.6356 −1.21701 −0.608506 0.793549i $$-0.708231\pi$$
−0.608506 + 0.793549i $$0.708231\pi$$
$$480$$ −1.52363 −0.0695441
$$481$$ −8.53673 −0.389241
$$482$$ 8.94319 0.407351
$$483$$ 0.625088 0.0284425
$$484$$ −5.46682 −0.248492
$$485$$ −13.2213 −0.600347
$$486$$ −8.99216 −0.407893
$$487$$ −5.34964 −0.242415 −0.121208 0.992627i $$-0.538677\pi$$
−0.121208 + 0.992627i $$0.538677\pi$$
$$488$$ −2.72209 −0.123223
$$489$$ −3.05063 −0.137954
$$490$$ 4.32518 0.195392
$$491$$ 0.456352 0.0205949 0.0102974 0.999947i $$-0.496722\pi$$
0.0102974 + 0.999947i $$0.496722\pi$$
$$492$$ −1.28499 −0.0579318
$$493$$ −36.3181 −1.63569
$$494$$ 0 0
$$495$$ −29.2594 −1.31511
$$496$$ 5.12672 0.230197
$$497$$ −14.0026 −0.628104
$$498$$ −5.47992 −0.245561
$$499$$ −5.99046 −0.268170 −0.134085 0.990970i $$-0.542809\pi$$
−0.134085 + 0.990970i $$0.542809\pi$$
$$500$$ 37.6601 1.68421
$$501$$ −2.13456 −0.0953653
$$502$$ 27.3872 1.22235
$$503$$ 28.2929 1.26152 0.630758 0.775979i $$-0.282744\pi$$
0.630758 + 0.775979i $$0.282744\pi$$
$$504$$ −2.87591 −0.128103
$$505$$ 6.58999 0.293251
$$506$$ −4.17399 −0.185557
$$507$$ 4.35320 0.193332
$$508$$ 6.23080 0.276447
$$509$$ −19.0990 −0.846548 −0.423274 0.906002i $$-0.639119\pi$$
−0.423274 + 0.906002i $$0.639119\pi$$
$$510$$ −9.48946 −0.420200
$$511$$ −9.23080 −0.408347
$$512$$ 1.00000 0.0441942
$$513$$ 0 0
$$514$$ −8.49209 −0.374570
$$515$$ 49.8418 2.19629
$$516$$ −0.307630 −0.0135427
$$517$$ −13.7167 −0.603260
$$518$$ −10.6504 −0.467950
$$519$$ 5.50455 0.241623
$$520$$ 3.46682 0.152030
$$521$$ −24.1294 −1.05713 −0.528563 0.848894i $$-0.677269\pi$$
−0.528563 + 0.848894i $$0.677269\pi$$
$$522$$ 16.7702 0.734010
$$523$$ 11.9205 0.521249 0.260625 0.965440i $$-0.416072\pi$$
0.260625 + 0.965440i $$0.416072\pi$$
$$524$$ −3.66528 −0.160118
$$525$$ −4.82864 −0.210739
$$526$$ −5.74918 −0.250676
$$527$$ 31.9301 1.39090
$$528$$ −0.828636 −0.0360618
$$529$$ −19.8513 −0.863101
$$530$$ −4.87328 −0.211682
$$531$$ −40.8410 −1.77235
$$532$$ 0 0
$$533$$ 2.92382 0.126645
$$534$$ 2.56998 0.111214
$$535$$ −69.2256 −2.99288
$$536$$ −4.58045 −0.197845
$$537$$ 4.01573 0.173291
$$538$$ 8.42218 0.363106
$$539$$ 2.35227 0.101319
$$540$$ 8.95273 0.385264
$$541$$ 11.8411 0.509088 0.254544 0.967061i $$-0.418075\pi$$
0.254544 + 0.967061i $$0.418075\pi$$
$$542$$ 20.7298 0.890422
$$543$$ 1.15471 0.0495534
$$544$$ 6.22818 0.267031
$$545$$ 31.3158 1.34142
$$546$$ −0.282361 −0.0120839
$$547$$ −35.4747 −1.51679 −0.758394 0.651796i $$-0.774016\pi$$
−0.758394 + 0.651796i $$0.774016\pi$$
$$548$$ 3.39691 0.145109
$$549$$ 7.82847 0.334111
$$550$$ 32.2430 1.37485
$$551$$ 0 0
$$552$$ 0.625088 0.0266055
$$553$$ 7.04727 0.299680
$$554$$ 17.4442 0.741132
$$555$$ 16.2273 0.688808
$$556$$ −5.19663 −0.220386
$$557$$ 11.6976 0.495644 0.247822 0.968806i $$-0.420285\pi$$
0.247822 + 0.968806i $$0.420285\pi$$
$$558$$ −14.7440 −0.624162
$$559$$ 0.699970 0.0296056
$$560$$ 4.32518 0.182772
$$561$$ −5.16089 −0.217893
$$562$$ −2.12409 −0.0895995
$$563$$ 12.2562 0.516537 0.258268 0.966073i $$-0.416848\pi$$
0.258268 + 0.966073i $$0.416848\pi$$
$$564$$ 2.05418 0.0864968
$$565$$ −6.38554 −0.268642
$$566$$ −27.6696 −1.16304
$$567$$ 7.89855 0.331708
$$568$$ −14.0026 −0.587537
$$569$$ −5.80862 −0.243510 −0.121755 0.992560i $$-0.538852\pi$$
−0.121755 + 0.992560i $$0.538852\pi$$
$$570$$ 0 0
$$571$$ 37.2308 1.55806 0.779030 0.626986i $$-0.215712\pi$$
0.779030 + 0.626986i $$0.215712\pi$$
$$572$$ 1.88545 0.0788346
$$573$$ 1.14088 0.0476610
$$574$$ 3.64773 0.152253
$$575$$ −24.3227 −1.01433
$$576$$ −2.87591 −0.119829
$$577$$ −17.3444 −0.722058 −0.361029 0.932555i $$-0.617574\pi$$
−0.361029 + 0.932555i $$0.617574\pi$$
$$578$$ 21.7902 0.906352
$$579$$ 4.24726 0.176510
$$580$$ −25.2213 −1.04726
$$581$$ 15.5560 0.645371
$$582$$ 1.07683 0.0446359
$$583$$ −2.65036 −0.109767
$$584$$ −9.23080 −0.381973
$$585$$ −9.97025 −0.412219
$$586$$ 13.2326 0.546635
$$587$$ −33.7561 −1.39327 −0.696633 0.717428i $$-0.745319\pi$$
−0.696633 + 0.717428i $$0.745319\pi$$
$$588$$ −0.352271 −0.0145274
$$589$$ 0 0
$$590$$ 61.4222 2.52871
$$591$$ 7.37584 0.303401
$$592$$ −10.6504 −0.437727
$$593$$ −15.5079 −0.636833 −0.318417 0.947951i $$-0.603151\pi$$
−0.318417 + 0.947951i $$0.603151\pi$$
$$594$$ 4.86899 0.199777
$$595$$ 26.9380 1.10435
$$596$$ 14.6504 0.600102
$$597$$ −3.21508 −0.131584
$$598$$ −1.42230 −0.0581622
$$599$$ −38.1215 −1.55760 −0.778801 0.627271i $$-0.784172\pi$$
−0.778801 + 0.627271i $$0.784172\pi$$
$$600$$ −4.82864 −0.197128
$$601$$ −13.4799 −0.549857 −0.274929 0.961465i $$-0.588654\pi$$
−0.274929 + 0.961465i $$0.588654\pi$$
$$602$$ 0.873277 0.0355921
$$603$$ 13.1729 0.536443
$$604$$ −5.46682 −0.222442
$$605$$ −23.6450 −0.961305
$$606$$ −0.536732 −0.0218032
$$607$$ 6.25345 0.253820 0.126910 0.991914i $$-0.459494\pi$$
0.126910 + 0.991914i $$0.459494\pi$$
$$608$$ 0 0
$$609$$ 2.05418 0.0832398
$$610$$ −11.7735 −0.476696
$$611$$ −4.67402 −0.189090
$$612$$ −17.9116 −0.724035
$$613$$ 5.51999 0.222950 0.111475 0.993767i $$-0.464442\pi$$
0.111475 + 0.993767i $$0.464442\pi$$
$$614$$ −24.9398 −1.00649
$$615$$ −5.55781 −0.224112
$$616$$ 2.35227 0.0947757
$$617$$ 22.7823 0.917182 0.458591 0.888647i $$-0.348354\pi$$
0.458591 + 0.888647i $$0.348354\pi$$
$$618$$ −4.05944 −0.163295
$$619$$ −10.6879 −0.429584 −0.214792 0.976660i $$-0.568907\pi$$
−0.214792 + 0.976660i $$0.568907\pi$$
$$620$$ 22.1740 0.890529
$$621$$ −3.67296 −0.147391
$$622$$ 15.9458 0.639369
$$623$$ −7.29546 −0.292286
$$624$$ −0.282361 −0.0113035
$$625$$ 94.3507 3.77403
$$626$$ −12.7203 −0.508404
$$627$$ 0 0
$$628$$ −3.45190 −0.137746
$$629$$ −66.3323 −2.64484
$$630$$ −12.4388 −0.495574
$$631$$ −20.5069 −0.816366 −0.408183 0.912900i $$-0.633838\pi$$
−0.408183 + 0.912900i $$0.633838\pi$$
$$632$$ 7.04727 0.280325
$$633$$ −1.40908 −0.0560060
$$634$$ 32.9091 1.30699
$$635$$ 26.9493 1.06945
$$636$$ 0.396912 0.0157386
$$637$$ 0.801544 0.0317583
$$638$$ −13.7167 −0.543050
$$639$$ 40.2702 1.59307
$$640$$ 4.32518 0.170968
$$641$$ −22.3016 −0.880862 −0.440431 0.897787i $$-0.645174\pi$$
−0.440431 + 0.897787i $$0.645174\pi$$
$$642$$ 5.63819 0.222521
$$643$$ −2.79200 −0.110106 −0.0550529 0.998483i $$-0.517533\pi$$
−0.0550529 + 0.998483i $$0.517533\pi$$
$$644$$ −1.77445 −0.0699232
$$645$$ −1.33056 −0.0523906
$$646$$ 0 0
$$647$$ −5.69144 −0.223754 −0.111877 0.993722i $$-0.535686\pi$$
−0.111877 + 0.993722i $$0.535686\pi$$
$$648$$ 7.89855 0.310284
$$649$$ 33.4048 1.31125
$$650$$ 10.9869 0.430942
$$651$$ −1.80600 −0.0707825
$$652$$ 8.65990 0.339148
$$653$$ 0.485267 0.0189900 0.00949499 0.999955i $$-0.496978\pi$$
0.00949499 + 0.999955i $$0.496978\pi$$
$$654$$ −2.55056 −0.0997349
$$655$$ −15.8530 −0.619427
$$656$$ 3.64773 0.142420
$$657$$ 26.5469 1.03569
$$658$$ −5.83126 −0.227326
$$659$$ 17.8023 0.693481 0.346741 0.937961i $$-0.387288\pi$$
0.346741 + 0.937961i $$0.387288\pi$$
$$660$$ −3.58400 −0.139507
$$661$$ −19.8793 −0.773217 −0.386608 0.922244i $$-0.626353\pi$$
−0.386608 + 0.922244i $$0.626353\pi$$
$$662$$ 19.0079 0.738762
$$663$$ −1.75859 −0.0682981
$$664$$ 15.5560 0.603689
$$665$$ 0 0
$$666$$ 30.6294 1.18687
$$667$$ 10.3473 0.400649
$$668$$ 6.05944 0.234447
$$669$$ −4.31653 −0.166887
$$670$$ −19.8113 −0.765375
$$671$$ −6.40309 −0.247189
$$672$$ −0.352271 −0.0135891
$$673$$ 23.8339 0.918729 0.459365 0.888248i $$-0.348077\pi$$
0.459365 + 0.888248i $$0.348077\pi$$
$$674$$ −28.1914 −1.08589
$$675$$ 28.3726 1.09206
$$676$$ −12.3575 −0.475289
$$677$$ 24.7893 0.952728 0.476364 0.879248i $$-0.341954\pi$$
0.476364 + 0.879248i $$0.341954\pi$$
$$678$$ 0.520081 0.0199736
$$679$$ −3.05681 −0.117310
$$680$$ 26.9380 1.03302
$$681$$ −8.22661 −0.315244
$$682$$ 12.0594 0.461780
$$683$$ 39.3592 1.50604 0.753020 0.657998i $$-0.228597\pi$$
0.753020 + 0.657998i $$0.228597\pi$$
$$684$$ 0 0
$$685$$ 14.6922 0.561362
$$686$$ 1.00000 0.0381802
$$687$$ 9.11782 0.347867
$$688$$ 0.873277 0.0332934
$$689$$ −0.903118 −0.0344061
$$690$$ 2.70362 0.102925
$$691$$ 27.8226 1.05842 0.529211 0.848490i $$-0.322488\pi$$
0.529211 + 0.848490i $$0.322488\pi$$
$$692$$ −15.6259 −0.594007
$$693$$ −6.76491 −0.256978
$$694$$ 22.7292 0.862787
$$695$$ −22.4764 −0.852577
$$696$$ 2.05418 0.0778637
$$697$$ 22.7187 0.860532
$$698$$ −4.72981 −0.179026
$$699$$ 0.916992 0.0346838
$$700$$ 13.7072 0.518082
$$701$$ −37.5332 −1.41761 −0.708805 0.705404i $$-0.750765\pi$$
−0.708805 + 0.705404i $$0.750765\pi$$
$$702$$ 1.65912 0.0626196
$$703$$ 0 0
$$704$$ 2.35227 0.0886545
$$705$$ 8.88472 0.334618
$$706$$ −5.76135 −0.216831
$$707$$ 1.52363 0.0573022
$$708$$ −5.00263 −0.188010
$$709$$ 11.3802 0.427391 0.213696 0.976900i $$-0.431450\pi$$
0.213696 + 0.976900i $$0.431450\pi$$
$$710$$ −60.5639 −2.27292
$$711$$ −20.2673 −0.760082
$$712$$ −7.29546 −0.273409
$$713$$ −9.09712 −0.340690
$$714$$ −2.19400 −0.0821086
$$715$$ 8.15490 0.304976
$$716$$ −11.3995 −0.426021
$$717$$ 8.20019 0.306242
$$718$$ −19.9511 −0.744567
$$719$$ −40.3218 −1.50375 −0.751874 0.659306i $$-0.770850\pi$$
−0.751874 + 0.659306i $$0.770850\pi$$
$$720$$ −12.4388 −0.463567
$$721$$ 11.5236 0.429163
$$722$$ 0 0
$$723$$ −3.15042 −0.117166
$$724$$ −3.27791 −0.121823
$$725$$ −79.9301 −2.96853
$$726$$ 1.92580 0.0714732
$$727$$ 1.55781 0.0577758 0.0288879 0.999583i $$-0.490803\pi$$
0.0288879 + 0.999583i $$0.490803\pi$$
$$728$$ 0.801544 0.0297072
$$729$$ −20.5280 −0.760295
$$730$$ −39.9249 −1.47769
$$731$$ 5.43892 0.201166
$$732$$ 0.958913 0.0354424
$$733$$ −22.7825 −0.841489 −0.420745 0.907179i $$-0.638231\pi$$
−0.420745 + 0.907179i $$0.638231\pi$$
$$734$$ −1.24035 −0.0457821
$$735$$ −1.52363 −0.0562001
$$736$$ −1.77445 −0.0654072
$$737$$ −10.7745 −0.396882
$$738$$ −10.4905 −0.386162
$$739$$ −2.64081 −0.0971439 −0.0485719 0.998820i $$-0.515467\pi$$
−0.0485719 + 0.998820i $$0.515467\pi$$
$$740$$ −46.0647 −1.69337
$$741$$ 0 0
$$742$$ −1.12672 −0.0413633
$$743$$ 18.7955 0.689541 0.344770 0.938687i $$-0.387957\pi$$
0.344770 + 0.938687i $$0.387957\pi$$
$$744$$ −1.80600 −0.0662110
$$745$$ 63.3654 2.32153
$$746$$ −24.8733 −0.910675
$$747$$ −44.7375 −1.63686
$$748$$ 14.6504 0.535670
$$749$$ −16.0053 −0.584819
$$750$$ −13.2665 −0.484426
$$751$$ 42.6889 1.55774 0.778869 0.627186i $$-0.215794\pi$$
0.778869 + 0.627186i $$0.215794\pi$$
$$752$$ −5.83126 −0.212644
$$753$$ −9.64773 −0.351583
$$754$$ −4.67402 −0.170218
$$755$$ −23.6450 −0.860529
$$756$$ 2.06991 0.0752819
$$757$$ 43.8374 1.59330 0.796650 0.604441i $$-0.206604\pi$$
0.796650 + 0.604441i $$0.206604\pi$$
$$758$$ −32.4274 −1.17782
$$759$$ 1.47038 0.0533713
$$760$$ 0 0
$$761$$ 5.22462 0.189392 0.0946962 0.995506i $$-0.469812\pi$$
0.0946962 + 0.995506i $$0.469812\pi$$
$$762$$ −2.19493 −0.0795140
$$763$$ 7.24035 0.262118
$$764$$ −3.23865 −0.117170
$$765$$ −77.4711 −2.80097
$$766$$ 14.5105 0.524287
$$767$$ 11.3828 0.411009
$$768$$ −0.352271 −0.0127115
$$769$$ −32.9678 −1.18885 −0.594425 0.804151i $$-0.702620\pi$$
−0.594425 + 0.804151i $$0.702620\pi$$
$$770$$ 10.1740 0.366645
$$771$$ 2.99152 0.107737
$$772$$ −12.0568 −0.433934
$$773$$ −43.6906 −1.57144 −0.785721 0.618582i $$-0.787708\pi$$
−0.785721 + 0.618582i $$0.787708\pi$$
$$774$$ −2.51146 −0.0902727
$$775$$ 70.2729 2.52428
$$776$$ −3.05681 −0.109733
$$777$$ 3.75181 0.134595
$$778$$ 25.5236 0.915067
$$779$$ 0 0
$$780$$ −1.22126 −0.0437281
$$781$$ −32.9380 −1.17861
$$782$$ −11.0516 −0.395204
$$783$$ −12.0702 −0.431354
$$784$$ 1.00000 0.0357143
$$785$$ −14.9301 −0.532878
$$786$$ 1.29117 0.0460545
$$787$$ −26.7431 −0.953288 −0.476644 0.879097i $$-0.658147\pi$$
−0.476644 + 0.879097i $$0.658147\pi$$
$$788$$ −20.9380 −0.745884
$$789$$ 2.02527 0.0721015
$$790$$ 30.4807 1.08445
$$791$$ −1.47637 −0.0524935
$$792$$ −6.76491 −0.240381
$$793$$ −2.18188 −0.0774807
$$794$$ 11.6373 0.412991
$$795$$ 1.71671 0.0608856
$$796$$ 9.12672 0.323488
$$797$$ −7.16861 −0.253925 −0.126963 0.991907i $$-0.540523\pi$$
−0.126963 + 0.991907i $$0.540523\pi$$
$$798$$ 0 0
$$799$$ −36.3181 −1.28484
$$800$$ 13.7072 0.484622
$$801$$ 20.9810 0.741329
$$802$$ 25.1740 0.888925
$$803$$ −21.7134 −0.766248
$$804$$ 1.61356 0.0569058
$$805$$ −7.67482 −0.270502
$$806$$ 4.10929 0.144744
$$807$$ −2.96689 −0.104439
$$808$$ 1.52363 0.0536013
$$809$$ 1.58109 0.0555881 0.0277941 0.999614i $$-0.491152\pi$$
0.0277941 + 0.999614i $$0.491152\pi$$
$$810$$ 34.1626 1.20035
$$811$$ −1.41001 −0.0495121 −0.0247561 0.999694i $$-0.507881\pi$$
−0.0247561 + 0.999694i $$0.507881\pi$$
$$812$$ −5.83126 −0.204637
$$813$$ −7.30251 −0.256110
$$814$$ −25.0525 −0.878091
$$815$$ 37.4556 1.31201
$$816$$ −2.19400 −0.0768055
$$817$$ 0 0
$$818$$ −31.6277 −1.10584
$$819$$ −2.30517 −0.0805490
$$820$$ 15.7771 0.550960
$$821$$ −33.2114 −1.15909 −0.579543 0.814941i $$-0.696769\pi$$
−0.579543 + 0.814941i $$0.696769\pi$$
$$822$$ −1.19663 −0.0417374
$$823$$ −47.1198 −1.64249 −0.821247 0.570572i $$-0.806721\pi$$
−0.821247 + 0.570572i $$0.806721\pi$$
$$824$$ 11.5236 0.401445
$$825$$ −11.3583 −0.395444
$$826$$ 14.2011 0.494119
$$827$$ −16.1284 −0.560840 −0.280420 0.959877i $$-0.590474\pi$$
−0.280420 + 0.959877i $$0.590474\pi$$
$$828$$ 5.10316 0.177347
$$829$$ −46.9266 −1.62983 −0.814914 0.579582i $$-0.803216\pi$$
−0.814914 + 0.579582i $$0.803216\pi$$
$$830$$ 67.2824 2.33541
$$831$$ −6.14508 −0.213170
$$832$$ 0.801544 0.0277885
$$833$$ 6.22818 0.215794
$$834$$ 1.83062 0.0633893
$$835$$ 26.2082 0.906971
$$836$$ 0 0
$$837$$ 10.6119 0.366799
$$838$$ −19.9205 −0.688144
$$839$$ 8.19400 0.282888 0.141444 0.989946i $$-0.454825\pi$$
0.141444 + 0.989946i $$0.454825\pi$$
$$840$$ −1.52363 −0.0525704
$$841$$ 5.00365 0.172540
$$842$$ −14.4274 −0.497202
$$843$$ 0.748257 0.0257713
$$844$$ 4.00000 0.137686
$$845$$ −53.4485 −1.83868
$$846$$ 16.7702 0.576570
$$847$$ −5.46682 −0.187842
$$848$$ −1.12672 −0.0386918
$$849$$ 9.74720 0.334523
$$850$$ 85.3707 2.92819
$$851$$ 18.8985 0.647834
$$852$$ 4.93272 0.168992
$$853$$ −16.7791 −0.574504 −0.287252 0.957855i $$-0.592742\pi$$
−0.287252 + 0.957855i $$0.592742\pi$$
$$854$$ −2.72209 −0.0931480
$$855$$ 0 0
$$856$$ −16.0053 −0.547048
$$857$$ 9.47899 0.323796 0.161898 0.986807i $$-0.448238\pi$$
0.161898 + 0.986807i $$0.448238\pi$$
$$858$$ −0.664189 −0.0226750
$$859$$ −5.59354 −0.190849 −0.0954246 0.995437i $$-0.530421\pi$$
−0.0954246 + 0.995437i $$0.530421\pi$$
$$860$$ 3.77708 0.128797
$$861$$ −1.28499 −0.0437923
$$862$$ 14.6504 0.498993
$$863$$ 46.1241 1.57008 0.785042 0.619443i $$-0.212641\pi$$
0.785042 + 0.619443i $$0.212641\pi$$
$$864$$ 2.06991 0.0704198
$$865$$ −67.5848 −2.29795
$$866$$ 18.0043 0.611812
$$867$$ −7.67604 −0.260692
$$868$$ 5.12672 0.174012
$$869$$ 16.5771 0.562339
$$870$$ 8.88472 0.301220
$$871$$ −3.67143 −0.124402
$$872$$ 7.24035 0.245189
$$873$$ 8.79110 0.297534
$$874$$ 0 0
$$875$$ 37.6601 1.27314
$$876$$ 3.25174 0.109866
$$877$$ −16.7982 −0.567233 −0.283617 0.958938i $$-0.591534\pi$$
−0.283617 + 0.958938i $$0.591534\pi$$
$$878$$ 21.6336 0.730099
$$879$$ −4.66147 −0.157227
$$880$$ 10.1740 0.342965
$$881$$ −29.9153 −1.00787 −0.503937 0.863741i $$-0.668115\pi$$
−0.503937 + 0.863741i $$0.668115\pi$$
$$882$$ −2.87591 −0.0968368
$$883$$ −26.9284 −0.906214 −0.453107 0.891456i $$-0.649684\pi$$
−0.453107 + 0.891456i $$0.649684\pi$$
$$884$$ 4.99216 0.167904
$$885$$ −21.6373 −0.727329
$$886$$ −3.45465 −0.116061
$$887$$ −24.6907 −0.829033 −0.414516 0.910042i $$-0.636049\pi$$
−0.414516 + 0.910042i $$0.636049\pi$$
$$888$$ 3.75181 0.125903
$$889$$ 6.23080 0.208974
$$890$$ −31.5542 −1.05770
$$891$$ 18.5795 0.622437
$$892$$ 12.2534 0.410276
$$893$$ 0 0
$$894$$ −5.16089 −0.172606
$$895$$ −49.3050 −1.64809
$$896$$ 1.00000 0.0334077
$$897$$ 0.501035 0.0167291
$$898$$ 19.5069 0.650953
$$899$$ −29.8953 −0.997063
$$900$$ −39.4205 −1.31402
$$901$$ −7.01743 −0.233784
$$902$$ 8.58045 0.285698
$$903$$ −0.307630 −0.0102373
$$904$$ −1.47637 −0.0491032
$$905$$ −14.1775 −0.471278
$$906$$ 1.92580 0.0639805
$$907$$ 36.1383 1.19995 0.599975 0.800018i $$-0.295177\pi$$
0.599975 + 0.800018i $$0.295177\pi$$
$$908$$ 23.3531 0.774999
$$909$$ −4.38183 −0.145336
$$910$$ 3.46682 0.114924
$$911$$ 41.0210 1.35909 0.679543 0.733636i $$-0.262178\pi$$
0.679543 + 0.733636i $$0.262178\pi$$
$$912$$ 0 0
$$913$$ 36.5919 1.21101
$$914$$ −13.2876 −0.439515
$$915$$ 4.14747 0.137111
$$916$$ −25.8830 −0.855198
$$917$$ −3.66528 −0.121038
$$918$$ 12.8918 0.425492
$$919$$ −19.4521 −0.641664 −0.320832 0.947136i $$-0.603963\pi$$
−0.320832 + 0.947136i $$0.603963\pi$$
$$920$$ −7.67482 −0.253031
$$921$$ 8.78556 0.289494
$$922$$ 16.1915 0.533240
$$923$$ −11.2237 −0.369433
$$924$$ −0.828636 −0.0272601
$$925$$ −145.986 −4.80000
$$926$$ 21.5010 0.706566
$$927$$ −33.1409 −1.08849
$$928$$ −5.83126 −0.191421
$$929$$ 8.86009 0.290690 0.145345 0.989381i $$-0.453571\pi$$
0.145345 + 0.989381i $$0.453571\pi$$
$$930$$ −7.81125 −0.256141
$$931$$ 0 0
$$932$$ −2.60309 −0.0852670
$$933$$ −5.61725 −0.183900
$$934$$ −7.95536 −0.260307
$$935$$ 63.3654 2.07227
$$936$$ −2.30517 −0.0753467
$$937$$ −35.7413 −1.16762 −0.583809 0.811891i $$-0.698438\pi$$
−0.583809 + 0.811891i $$0.698438\pi$$
$$938$$ −4.58045 −0.149557
$$939$$ 4.48098 0.146231
$$940$$ −25.2213 −0.822627
$$941$$ −40.6128 −1.32394 −0.661970 0.749531i $$-0.730279\pi$$
−0.661970 + 0.749531i $$0.730279\pi$$
$$942$$ 1.21600 0.0396196
$$943$$ −6.47272 −0.210781
$$944$$ 14.2011 0.462206
$$945$$ 8.95273 0.291232
$$946$$ 2.05418 0.0667873
$$947$$ −41.3845 −1.34482 −0.672408 0.740181i $$-0.734740\pi$$
−0.672408 + 0.740181i $$0.734740\pi$$
$$948$$ −2.48255 −0.0806294
$$949$$ −7.39890 −0.240178
$$950$$ 0 0
$$951$$ −11.5929 −0.375926
$$952$$ 6.22818 0.201856
$$953$$ 5.45100 0.176575 0.0882877 0.996095i $$-0.471861\pi$$
0.0882877 + 0.996095i $$0.471861\pi$$
$$954$$ 3.24035 0.104910
$$955$$ −14.0077 −0.453279
$$956$$ −23.2781 −0.752867
$$957$$ 4.83200 0.156196
$$958$$ −26.6356 −0.860557
$$959$$ 3.39691 0.109692
$$960$$ −1.52363 −0.0491751
$$961$$ −4.71671 −0.152152
$$962$$ −8.53673 −0.275235
$$963$$ 46.0296 1.48328
$$964$$ 8.94319 0.288041
$$965$$ −52.1479 −1.67870
$$966$$ 0.625088 0.0201119
$$967$$ 1.62573 0.0522799 0.0261400 0.999658i $$-0.491678\pi$$
0.0261400 + 0.999658i $$0.491678\pi$$
$$968$$ −5.46682 −0.175710
$$969$$ 0 0
$$970$$ −13.2213 −0.424509
$$971$$ −35.1634 −1.12845 −0.564224 0.825622i $$-0.690824\pi$$
−0.564224 + 0.825622i $$0.690824\pi$$
$$972$$ −8.99216 −0.288424
$$973$$ −5.19663 −0.166596
$$974$$ −5.34964 −0.171414
$$975$$ −3.87037 −0.123951
$$976$$ −2.72209 −0.0871320
$$977$$ 37.2289 1.19106 0.595528 0.803334i $$-0.296943\pi$$
0.595528 + 0.803334i $$0.296943\pi$$
$$978$$ −3.05063 −0.0975484
$$979$$ −17.1609 −0.548465
$$980$$ 4.32518 0.138163
$$981$$ −20.8226 −0.664813
$$982$$ 0.456352 0.0145628
$$983$$ 1.29546 0.0413187 0.0206594 0.999787i $$-0.493423\pi$$
0.0206594 + 0.999787i $$0.493423\pi$$
$$984$$ −1.28499 −0.0409639
$$985$$ −90.5605 −2.88550
$$986$$ −36.3181 −1.15661
$$987$$ 2.05418 0.0653854
$$988$$ 0 0
$$989$$ −1.54959 −0.0492740
$$990$$ −29.2594 −0.929926
$$991$$ −13.7843 −0.437872 −0.218936 0.975739i $$-0.570259\pi$$
−0.218936 + 0.975739i $$0.570259\pi$$
$$992$$ 5.12672 0.162774
$$993$$ −6.69592 −0.212489
$$994$$ −14.0026 −0.444136
$$995$$ 39.4747 1.25143
$$996$$ −5.47992 −0.173638
$$997$$ −0.857580 −0.0271598 −0.0135799 0.999908i $$-0.504323\pi$$
−0.0135799 + 0.999908i $$0.504323\pi$$
$$998$$ −5.99046 −0.189625
$$999$$ −22.0453 −0.697482
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 5054.2.a.x.1.2 4
19.8 odd 6 266.2.f.d.197.2 8
19.12 odd 6 266.2.f.d.239.2 yes 8
19.18 odd 2 5054.2.a.w.1.3 4
57.8 even 6 2394.2.o.v.1261.4 8
57.50 even 6 2394.2.o.v.505.4 8

By twisted newform
Twist Min Dim Char Parity Ord Type
266.2.f.d.197.2 8 19.8 odd 6
266.2.f.d.239.2 yes 8 19.12 odd 6
2394.2.o.v.505.4 8 57.50 even 6
2394.2.o.v.1261.4 8 57.8 even 6
5054.2.a.w.1.3 4 19.18 odd 2
5054.2.a.x.1.2 4 1.1 even 1 trivial