Properties

 Label 5054.2.a.bj.1.3 Level $5054$ Weight $2$ Character 5054.1 Self dual yes Analytic conductor $40.356$ Analytic rank $0$ Dimension $9$ 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: $$9$$ Coefficient field: $$\mathbb{Q}[x]/(x^{9} - \cdots)$$ Defining polynomial: $$x^{9} - 3x^{8} - 15x^{7} + 40x^{6} + 81x^{5} - 162x^{4} - 205x^{3} + 204x^{2} + 210x + 1$$ x^9 - 3*x^8 - 15*x^7 + 40*x^6 + 81*x^5 - 162*x^4 - 205*x^3 + 204*x^2 + 210*x + 1 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 1.3 Root $$2.18756$$ of defining polynomial Character $$\chi$$ $$=$$ 5054.1

$q$-expansion

 $$f(q)$$ $$=$$ $$q-1.00000 q^{2} -2.18756 q^{3} +1.00000 q^{4} -0.616139 q^{5} +2.18756 q^{6} -1.00000 q^{7} -1.00000 q^{8} +1.78540 q^{9} +O(q^{10})$$ $$q-1.00000 q^{2} -2.18756 q^{3} +1.00000 q^{4} -0.616139 q^{5} +2.18756 q^{6} -1.00000 q^{7} -1.00000 q^{8} +1.78540 q^{9} +0.616139 q^{10} +3.42998 q^{11} -2.18756 q^{12} -6.38112 q^{13} +1.00000 q^{14} +1.34784 q^{15} +1.00000 q^{16} +2.47999 q^{17} -1.78540 q^{18} -0.616139 q^{20} +2.18756 q^{21} -3.42998 q^{22} +2.70527 q^{23} +2.18756 q^{24} -4.62037 q^{25} +6.38112 q^{26} +2.65701 q^{27} -1.00000 q^{28} -6.00017 q^{29} -1.34784 q^{30} +7.72644 q^{31} -1.00000 q^{32} -7.50328 q^{33} -2.47999 q^{34} +0.616139 q^{35} +1.78540 q^{36} -4.26808 q^{37} +13.9591 q^{39} +0.616139 q^{40} +1.28851 q^{41} -2.18756 q^{42} -10.3584 q^{43} +3.42998 q^{44} -1.10005 q^{45} -2.70527 q^{46} +1.53750 q^{47} -2.18756 q^{48} +1.00000 q^{49} +4.62037 q^{50} -5.42512 q^{51} -6.38112 q^{52} +5.11208 q^{53} -2.65701 q^{54} -2.11335 q^{55} +1.00000 q^{56} +6.00017 q^{58} -12.7768 q^{59} +1.34784 q^{60} -5.54592 q^{61} -7.72644 q^{62} -1.78540 q^{63} +1.00000 q^{64} +3.93166 q^{65} +7.50328 q^{66} -4.16481 q^{67} +2.47999 q^{68} -5.91793 q^{69} -0.616139 q^{70} -0.0244707 q^{71} -1.78540 q^{72} +13.0675 q^{73} +4.26808 q^{74} +10.1073 q^{75} -3.42998 q^{77} -13.9591 q^{78} -2.96283 q^{79} -0.616139 q^{80} -11.1685 q^{81} -1.28851 q^{82} +12.9425 q^{83} +2.18756 q^{84} -1.52802 q^{85} +10.3584 q^{86} +13.1257 q^{87} -3.42998 q^{88} +12.4510 q^{89} +1.10005 q^{90} +6.38112 q^{91} +2.70527 q^{92} -16.9020 q^{93} -1.53750 q^{94} +2.18756 q^{96} +1.51714 q^{97} -1.00000 q^{98} +6.12389 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$9 q - 9 q^{2} - 3 q^{3} + 9 q^{4} + 3 q^{5} + 3 q^{6} - 9 q^{7} - 9 q^{8} + 12 q^{9}+O(q^{10})$$ 9 * q - 9 * q^2 - 3 * q^3 + 9 * q^4 + 3 * q^5 + 3 * q^6 - 9 * q^7 - 9 * q^8 + 12 * q^9 $$9 q - 9 q^{2} - 3 q^{3} + 9 q^{4} + 3 q^{5} + 3 q^{6} - 9 q^{7} - 9 q^{8} + 12 q^{9} - 3 q^{10} - 3 q^{11} - 3 q^{12} - 12 q^{13} + 9 q^{14} + 6 q^{15} + 9 q^{16} + 12 q^{17} - 12 q^{18} + 3 q^{20} + 3 q^{21} + 3 q^{22} + 6 q^{23} + 3 q^{24} + 12 q^{26} - 24 q^{27} - 9 q^{28} + 6 q^{29} - 6 q^{30} - 9 q^{31} - 9 q^{32} + 3 q^{33} - 12 q^{34} - 3 q^{35} + 12 q^{36} - 9 q^{37} + 33 q^{39} - 3 q^{40} - 3 q^{41} - 3 q^{42} + 21 q^{43} - 3 q^{44} + 18 q^{45} - 6 q^{46} + 21 q^{47} - 3 q^{48} + 9 q^{49} - 39 q^{51} - 12 q^{52} + 24 q^{54} + 24 q^{55} + 9 q^{56} - 6 q^{58} + 9 q^{59} + 6 q^{60} + 6 q^{61} + 9 q^{62} - 12 q^{63} + 9 q^{64} + 3 q^{65} - 3 q^{66} - 27 q^{67} + 12 q^{68} - 6 q^{69} + 3 q^{70} - 9 q^{71} - 12 q^{72} + 51 q^{73} + 9 q^{74} + 3 q^{75} + 3 q^{77} - 33 q^{78} - 24 q^{79} + 3 q^{80} - 3 q^{81} + 3 q^{82} + 9 q^{83} + 3 q^{84} + 6 q^{85} - 21 q^{86} - 3 q^{87} + 3 q^{88} - 9 q^{89} - 18 q^{90} + 12 q^{91} + 6 q^{92} + 3 q^{93} - 21 q^{94} + 3 q^{96} + 12 q^{97} - 9 q^{98} - 15 q^{99}+O(q^{100})$$ 9 * q - 9 * q^2 - 3 * q^3 + 9 * q^4 + 3 * q^5 + 3 * q^6 - 9 * q^7 - 9 * q^8 + 12 * q^9 - 3 * q^10 - 3 * q^11 - 3 * q^12 - 12 * q^13 + 9 * q^14 + 6 * q^15 + 9 * q^16 + 12 * q^17 - 12 * q^18 + 3 * q^20 + 3 * q^21 + 3 * q^22 + 6 * q^23 + 3 * q^24 + 12 * q^26 - 24 * q^27 - 9 * q^28 + 6 * q^29 - 6 * q^30 - 9 * q^31 - 9 * q^32 + 3 * q^33 - 12 * q^34 - 3 * q^35 + 12 * q^36 - 9 * q^37 + 33 * q^39 - 3 * q^40 - 3 * q^41 - 3 * q^42 + 21 * q^43 - 3 * q^44 + 18 * q^45 - 6 * q^46 + 21 * q^47 - 3 * q^48 + 9 * q^49 - 39 * q^51 - 12 * q^52 + 24 * q^54 + 24 * q^55 + 9 * q^56 - 6 * q^58 + 9 * q^59 + 6 * q^60 + 6 * q^61 + 9 * q^62 - 12 * q^63 + 9 * q^64 + 3 * q^65 - 3 * q^66 - 27 * q^67 + 12 * q^68 - 6 * q^69 + 3 * q^70 - 9 * q^71 - 12 * q^72 + 51 * q^73 + 9 * q^74 + 3 * q^75 + 3 * q^77 - 33 * q^78 - 24 * q^79 + 3 * q^80 - 3 * q^81 + 3 * q^82 + 9 * q^83 + 3 * q^84 + 6 * q^85 - 21 * q^86 - 3 * q^87 + 3 * q^88 - 9 * q^89 - 18 * q^90 + 12 * q^91 + 6 * q^92 + 3 * q^93 - 21 * q^94 + 3 * q^96 + 12 * q^97 - 9 * q^98 - 15 * 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
$$3$$ −2.18756 −1.26299 −0.631493 0.775382i $$-0.717558\pi$$
−0.631493 + 0.775382i $$0.717558\pi$$
$$4$$ 1.00000 0.500000
$$5$$ −0.616139 −0.275546 −0.137773 0.990464i $$-0.543994\pi$$
−0.137773 + 0.990464i $$0.543994\pi$$
$$6$$ 2.18756 0.893066
$$7$$ −1.00000 −0.377964
$$8$$ −1.00000 −0.353553
$$9$$ 1.78540 0.595133
$$10$$ 0.616139 0.194840
$$11$$ 3.42998 1.03418 0.517089 0.855931i $$-0.327015\pi$$
0.517089 + 0.855931i $$0.327015\pi$$
$$12$$ −2.18756 −0.631493
$$13$$ −6.38112 −1.76980 −0.884902 0.465776i $$-0.845775\pi$$
−0.884902 + 0.465776i $$0.845775\pi$$
$$14$$ 1.00000 0.267261
$$15$$ 1.34784 0.348010
$$16$$ 1.00000 0.250000
$$17$$ 2.47999 0.601486 0.300743 0.953705i $$-0.402765\pi$$
0.300743 + 0.953705i $$0.402765\pi$$
$$18$$ −1.78540 −0.420823
$$19$$ 0 0
$$20$$ −0.616139 −0.137773
$$21$$ 2.18756 0.477364
$$22$$ −3.42998 −0.731275
$$23$$ 2.70527 0.564088 0.282044 0.959401i $$-0.408988\pi$$
0.282044 + 0.959401i $$0.408988\pi$$
$$24$$ 2.18756 0.446533
$$25$$ −4.62037 −0.924074
$$26$$ 6.38112 1.25144
$$27$$ 2.65701 0.511341
$$28$$ −1.00000 −0.188982
$$29$$ −6.00017 −1.11420 −0.557101 0.830444i $$-0.688087\pi$$
−0.557101 + 0.830444i $$0.688087\pi$$
$$30$$ −1.34784 −0.246081
$$31$$ 7.72644 1.38771 0.693855 0.720115i $$-0.255911\pi$$
0.693855 + 0.720115i $$0.255911\pi$$
$$32$$ −1.00000 −0.176777
$$33$$ −7.50328 −1.30615
$$34$$ −2.47999 −0.425315
$$35$$ 0.616139 0.104147
$$36$$ 1.78540 0.297567
$$37$$ −4.26808 −0.701669 −0.350834 0.936438i $$-0.614102\pi$$
−0.350834 + 0.936438i $$0.614102\pi$$
$$38$$ 0 0
$$39$$ 13.9591 2.23524
$$40$$ 0.616139 0.0974202
$$41$$ 1.28851 0.201232 0.100616 0.994925i $$-0.467919\pi$$
0.100616 + 0.994925i $$0.467919\pi$$
$$42$$ −2.18756 −0.337547
$$43$$ −10.3584 −1.57964 −0.789819 0.613340i $$-0.789826\pi$$
−0.789819 + 0.613340i $$0.789826\pi$$
$$44$$ 3.42998 0.517089
$$45$$ −1.10005 −0.163986
$$46$$ −2.70527 −0.398871
$$47$$ 1.53750 0.224267 0.112134 0.993693i $$-0.464232\pi$$
0.112134 + 0.993693i $$0.464232\pi$$
$$48$$ −2.18756 −0.315746
$$49$$ 1.00000 0.142857
$$50$$ 4.62037 0.653419
$$51$$ −5.42512 −0.759669
$$52$$ −6.38112 −0.884902
$$53$$ 5.11208 0.702198 0.351099 0.936338i $$-0.385808\pi$$
0.351099 + 0.936338i $$0.385808\pi$$
$$54$$ −2.65701 −0.361573
$$55$$ −2.11335 −0.284964
$$56$$ 1.00000 0.133631
$$57$$ 0 0
$$58$$ 6.00017 0.787860
$$59$$ −12.7768 −1.66339 −0.831696 0.555232i $$-0.812630\pi$$
−0.831696 + 0.555232i $$0.812630\pi$$
$$60$$ 1.34784 0.174005
$$61$$ −5.54592 −0.710082 −0.355041 0.934851i $$-0.615533\pi$$
−0.355041 + 0.934851i $$0.615533\pi$$
$$62$$ −7.72644 −0.981259
$$63$$ −1.78540 −0.224939
$$64$$ 1.00000 0.125000
$$65$$ 3.93166 0.487662
$$66$$ 7.50328 0.923590
$$67$$ −4.16481 −0.508813 −0.254406 0.967097i $$-0.581880\pi$$
−0.254406 + 0.967097i $$0.581880\pi$$
$$68$$ 2.47999 0.300743
$$69$$ −5.91793 −0.712435
$$70$$ −0.616139 −0.0736427
$$71$$ −0.0244707 −0.00290414 −0.00145207 0.999999i $$-0.500462\pi$$
−0.00145207 + 0.999999i $$0.500462\pi$$
$$72$$ −1.78540 −0.210411
$$73$$ 13.0675 1.52943 0.764715 0.644368i $$-0.222880\pi$$
0.764715 + 0.644368i $$0.222880\pi$$
$$74$$ 4.26808 0.496155
$$75$$ 10.1073 1.16709
$$76$$ 0 0
$$77$$ −3.42998 −0.390883
$$78$$ −13.9591 −1.58055
$$79$$ −2.96283 −0.333344 −0.166672 0.986012i $$-0.553302\pi$$
−0.166672 + 0.986012i $$0.553302\pi$$
$$80$$ −0.616139 −0.0688865
$$81$$ −11.1685 −1.24095
$$82$$ −1.28851 −0.142292
$$83$$ 12.9425 1.42062 0.710312 0.703887i $$-0.248554\pi$$
0.710312 + 0.703887i $$0.248554\pi$$
$$84$$ 2.18756 0.238682
$$85$$ −1.52802 −0.165737
$$86$$ 10.3584 1.11697
$$87$$ 13.1257 1.40722
$$88$$ −3.42998 −0.365637
$$89$$ 12.4510 1.31980 0.659899 0.751354i $$-0.270599\pi$$
0.659899 + 0.751354i $$0.270599\pi$$
$$90$$ 1.10005 0.115956
$$91$$ 6.38112 0.668923
$$92$$ 2.70527 0.282044
$$93$$ −16.9020 −1.75266
$$94$$ −1.53750 −0.158581
$$95$$ 0 0
$$96$$ 2.18756 0.223266
$$97$$ 1.51714 0.154042 0.0770211 0.997029i $$-0.475459\pi$$
0.0770211 + 0.997029i $$0.475459\pi$$
$$98$$ −1.00000 −0.101015
$$99$$ 6.12389 0.615474
$$100$$ −4.62037 −0.462037
$$101$$ −9.92823 −0.987896 −0.493948 0.869491i $$-0.664447\pi$$
−0.493948 + 0.869491i $$0.664447\pi$$
$$102$$ 5.42512 0.537167
$$103$$ −8.34988 −0.822738 −0.411369 0.911469i $$-0.634949\pi$$
−0.411369 + 0.911469i $$0.634949\pi$$
$$104$$ 6.38112 0.625720
$$105$$ −1.34784 −0.131536
$$106$$ −5.11208 −0.496529
$$107$$ −13.5562 −1.31053 −0.655264 0.755400i $$-0.727443\pi$$
−0.655264 + 0.755400i $$0.727443\pi$$
$$108$$ 2.65701 0.255671
$$109$$ −0.558302 −0.0534756 −0.0267378 0.999642i $$-0.508512\pi$$
−0.0267378 + 0.999642i $$0.508512\pi$$
$$110$$ 2.11335 0.201500
$$111$$ 9.33667 0.886197
$$112$$ −1.00000 −0.0944911
$$113$$ −10.0986 −0.949999 −0.474999 0.879986i $$-0.657552\pi$$
−0.474999 + 0.879986i $$0.657552\pi$$
$$114$$ 0 0
$$115$$ −1.66682 −0.155432
$$116$$ −6.00017 −0.557101
$$117$$ −11.3929 −1.05327
$$118$$ 12.7768 1.17620
$$119$$ −2.47999 −0.227340
$$120$$ −1.34784 −0.123040
$$121$$ 0.764783 0.0695257
$$122$$ 5.54592 0.502104
$$123$$ −2.81869 −0.254153
$$124$$ 7.72644 0.693855
$$125$$ 5.92749 0.530171
$$126$$ 1.78540 0.159056
$$127$$ −4.44613 −0.394530 −0.197265 0.980350i $$-0.563206\pi$$
−0.197265 + 0.980350i $$0.563206\pi$$
$$128$$ −1.00000 −0.0883883
$$129$$ 22.6595 1.99506
$$130$$ −3.93166 −0.344829
$$131$$ 7.17183 0.626606 0.313303 0.949653i $$-0.398564\pi$$
0.313303 + 0.949653i $$0.398564\pi$$
$$132$$ −7.50328 −0.653077
$$133$$ 0 0
$$134$$ 4.16481 0.359785
$$135$$ −1.63709 −0.140898
$$136$$ −2.47999 −0.212658
$$137$$ −16.8073 −1.43594 −0.717971 0.696073i $$-0.754929\pi$$
−0.717971 + 0.696073i $$0.754929\pi$$
$$138$$ 5.91793 0.503768
$$139$$ −13.5566 −1.14986 −0.574929 0.818203i $$-0.694970\pi$$
−0.574929 + 0.818203i $$0.694970\pi$$
$$140$$ 0.616139 0.0520733
$$141$$ −3.36336 −0.283246
$$142$$ 0.0244707 0.00205353
$$143$$ −21.8871 −1.83029
$$144$$ 1.78540 0.148783
$$145$$ 3.69694 0.307014
$$146$$ −13.0675 −1.08147
$$147$$ −2.18756 −0.180427
$$148$$ −4.26808 −0.350834
$$149$$ 13.5990 1.11408 0.557038 0.830487i $$-0.311938\pi$$
0.557038 + 0.830487i $$0.311938\pi$$
$$150$$ −10.1073 −0.825259
$$151$$ 20.2325 1.64650 0.823250 0.567679i $$-0.192159\pi$$
0.823250 + 0.567679i $$0.192159\pi$$
$$152$$ 0 0
$$153$$ 4.42778 0.357964
$$154$$ 3.42998 0.276396
$$155$$ −4.76056 −0.382378
$$156$$ 13.9591 1.11762
$$157$$ −11.1097 −0.886648 −0.443324 0.896361i $$-0.646201\pi$$
−0.443324 + 0.896361i $$0.646201\pi$$
$$158$$ 2.96283 0.235710
$$159$$ −11.1830 −0.886866
$$160$$ 0.616139 0.0487101
$$161$$ −2.70527 −0.213205
$$162$$ 11.1685 0.877484
$$163$$ −10.2803 −0.805215 −0.402608 0.915373i $$-0.631896\pi$$
−0.402608 + 0.915373i $$0.631896\pi$$
$$164$$ 1.28851 0.100616
$$165$$ 4.62306 0.359905
$$166$$ −12.9425 −1.00453
$$167$$ 12.6748 0.980806 0.490403 0.871496i $$-0.336850\pi$$
0.490403 + 0.871496i $$0.336850\pi$$
$$168$$ −2.18756 −0.168774
$$169$$ 27.7187 2.13221
$$170$$ 1.52802 0.117194
$$171$$ 0 0
$$172$$ −10.3584 −0.789819
$$173$$ −6.24437 −0.474751 −0.237376 0.971418i $$-0.576287\pi$$
−0.237376 + 0.971418i $$0.576287\pi$$
$$174$$ −13.1257 −0.995057
$$175$$ 4.62037 0.349267
$$176$$ 3.42998 0.258545
$$177$$ 27.9499 2.10084
$$178$$ −12.4510 −0.933238
$$179$$ −17.7976 −1.33026 −0.665128 0.746729i $$-0.731623\pi$$
−0.665128 + 0.746729i $$0.731623\pi$$
$$180$$ −1.10005 −0.0819932
$$181$$ 5.92757 0.440593 0.220296 0.975433i $$-0.429298\pi$$
0.220296 + 0.975433i $$0.429298\pi$$
$$182$$ −6.38112 −0.473000
$$183$$ 12.1320 0.896823
$$184$$ −2.70527 −0.199435
$$185$$ 2.62973 0.193342
$$186$$ 16.9020 1.23932
$$187$$ 8.50633 0.622044
$$188$$ 1.53750 0.112134
$$189$$ −2.65701 −0.193269
$$190$$ 0 0
$$191$$ 16.3567 1.18353 0.591764 0.806111i $$-0.298432\pi$$
0.591764 + 0.806111i $$0.298432\pi$$
$$192$$ −2.18756 −0.157873
$$193$$ −19.9885 −1.43881 −0.719403 0.694593i $$-0.755584\pi$$
−0.719403 + 0.694593i $$0.755584\pi$$
$$194$$ −1.51714 −0.108924
$$195$$ −8.60072 −0.615911
$$196$$ 1.00000 0.0714286
$$197$$ −3.44430 −0.245396 −0.122698 0.992444i $$-0.539155\pi$$
−0.122698 + 0.992444i $$0.539155\pi$$
$$198$$ −6.12389 −0.435206
$$199$$ 3.13027 0.221899 0.110949 0.993826i $$-0.464611\pi$$
0.110949 + 0.993826i $$0.464611\pi$$
$$200$$ 4.62037 0.326710
$$201$$ 9.11076 0.642623
$$202$$ 9.92823 0.698548
$$203$$ 6.00017 0.421129
$$204$$ −5.42512 −0.379834
$$205$$ −0.793902 −0.0554485
$$206$$ 8.34988 0.581764
$$207$$ 4.82999 0.335708
$$208$$ −6.38112 −0.442451
$$209$$ 0 0
$$210$$ 1.34784 0.0930097
$$211$$ 16.6987 1.14959 0.574793 0.818299i $$-0.305083\pi$$
0.574793 + 0.818299i $$0.305083\pi$$
$$212$$ 5.11208 0.351099
$$213$$ 0.0535310 0.00366788
$$214$$ 13.5562 0.926684
$$215$$ 6.38221 0.435263
$$216$$ −2.65701 −0.180786
$$217$$ −7.72644 −0.524505
$$218$$ 0.558302 0.0378130
$$219$$ −28.5858 −1.93165
$$220$$ −2.11335 −0.142482
$$221$$ −15.8251 −1.06451
$$222$$ −9.33667 −0.626636
$$223$$ −14.5844 −0.976641 −0.488320 0.872664i $$-0.662390\pi$$
−0.488320 + 0.872664i $$0.662390\pi$$
$$224$$ 1.00000 0.0668153
$$225$$ −8.24921 −0.549947
$$226$$ 10.0986 0.671750
$$227$$ 4.73333 0.314162 0.157081 0.987586i $$-0.449792\pi$$
0.157081 + 0.987586i $$0.449792\pi$$
$$228$$ 0 0
$$229$$ 14.8611 0.982051 0.491026 0.871145i $$-0.336622\pi$$
0.491026 + 0.871145i $$0.336622\pi$$
$$230$$ 1.66682 0.109907
$$231$$ 7.50328 0.493679
$$232$$ 6.00017 0.393930
$$233$$ −4.36751 −0.286125 −0.143062 0.989714i $$-0.545695\pi$$
−0.143062 + 0.989714i $$0.545695\pi$$
$$234$$ 11.3929 0.744774
$$235$$ −0.947313 −0.0617959
$$236$$ −12.7768 −0.831696
$$237$$ 6.48135 0.421009
$$238$$ 2.47999 0.160754
$$239$$ −24.2217 −1.56677 −0.783385 0.621537i $$-0.786509\pi$$
−0.783385 + 0.621537i $$0.786509\pi$$
$$240$$ 1.34784 0.0870026
$$241$$ 17.1009 1.10157 0.550784 0.834648i $$-0.314329\pi$$
0.550784 + 0.834648i $$0.314329\pi$$
$$242$$ −0.764783 −0.0491621
$$243$$ 16.4608 1.05596
$$244$$ −5.54592 −0.355041
$$245$$ −0.616139 −0.0393637
$$246$$ 2.81869 0.179713
$$247$$ 0 0
$$248$$ −7.72644 −0.490629
$$249$$ −28.3125 −1.79423
$$250$$ −5.92749 −0.374887
$$251$$ 27.3096 1.72377 0.861883 0.507107i $$-0.169285\pi$$
0.861883 + 0.507107i $$0.169285\pi$$
$$252$$ −1.78540 −0.112470
$$253$$ 9.27904 0.583368
$$254$$ 4.44613 0.278975
$$255$$ 3.34263 0.209324
$$256$$ 1.00000 0.0625000
$$257$$ 9.96817 0.621797 0.310899 0.950443i $$-0.399370\pi$$
0.310899 + 0.950443i $$0.399370\pi$$
$$258$$ −22.6595 −1.41072
$$259$$ 4.26808 0.265206
$$260$$ 3.93166 0.243831
$$261$$ −10.7127 −0.663099
$$262$$ −7.17183 −0.443077
$$263$$ −24.9314 −1.53734 −0.768669 0.639647i $$-0.779080\pi$$
−0.768669 + 0.639647i $$0.779080\pi$$
$$264$$ 7.50328 0.461795
$$265$$ −3.14975 −0.193488
$$266$$ 0 0
$$267$$ −27.2371 −1.66689
$$268$$ −4.16481 −0.254406
$$269$$ 25.6665 1.56491 0.782456 0.622706i $$-0.213967\pi$$
0.782456 + 0.622706i $$0.213967\pi$$
$$270$$ 1.63709 0.0996299
$$271$$ 18.7964 1.14180 0.570900 0.821020i $$-0.306595\pi$$
0.570900 + 0.821020i $$0.306595\pi$$
$$272$$ 2.47999 0.150372
$$273$$ −13.9591 −0.844841
$$274$$ 16.8073 1.01536
$$275$$ −15.8478 −0.955658
$$276$$ −5.91793 −0.356218
$$277$$ 19.8745 1.19414 0.597071 0.802188i $$-0.296331\pi$$
0.597071 + 0.802188i $$0.296331\pi$$
$$278$$ 13.5566 0.813073
$$279$$ 13.7948 0.825872
$$280$$ −0.616139 −0.0368214
$$281$$ 21.2943 1.27031 0.635156 0.772384i $$-0.280936\pi$$
0.635156 + 0.772384i $$0.280936\pi$$
$$282$$ 3.36336 0.200285
$$283$$ −3.77466 −0.224380 −0.112190 0.993687i $$-0.535787\pi$$
−0.112190 + 0.993687i $$0.535787\pi$$
$$284$$ −0.0244707 −0.00145207
$$285$$ 0 0
$$286$$ 21.8871 1.29421
$$287$$ −1.28851 −0.0760584
$$288$$ −1.78540 −0.105206
$$289$$ −10.8496 −0.638214
$$290$$ −3.69694 −0.217092
$$291$$ −3.31883 −0.194553
$$292$$ 13.0675 0.764715
$$293$$ 16.9584 0.990721 0.495360 0.868688i $$-0.335036\pi$$
0.495360 + 0.868688i $$0.335036\pi$$
$$294$$ 2.18756 0.127581
$$295$$ 7.87226 0.458341
$$296$$ 4.26808 0.248077
$$297$$ 9.11349 0.528818
$$298$$ −13.5990 −0.787770
$$299$$ −17.2627 −0.998326
$$300$$ 10.1073 0.583546
$$301$$ 10.3584 0.597047
$$302$$ −20.2325 −1.16425
$$303$$ 21.7186 1.24770
$$304$$ 0 0
$$305$$ 3.41706 0.195660
$$306$$ −4.42778 −0.253119
$$307$$ 2.68585 0.153289 0.0766447 0.997058i $$-0.475579\pi$$
0.0766447 + 0.997058i $$0.475579\pi$$
$$308$$ −3.42998 −0.195441
$$309$$ 18.2658 1.03911
$$310$$ 4.76056 0.270382
$$311$$ 1.77079 0.100412 0.0502062 0.998739i $$-0.484012\pi$$
0.0502062 + 0.998739i $$0.484012\pi$$
$$312$$ −13.9591 −0.790276
$$313$$ 7.89729 0.446381 0.223191 0.974775i $$-0.428353\pi$$
0.223191 + 0.974775i $$0.428353\pi$$
$$314$$ 11.1097 0.626955
$$315$$ 1.10005 0.0619811
$$316$$ −2.96283 −0.166672
$$317$$ 11.6202 0.652656 0.326328 0.945257i $$-0.394189\pi$$
0.326328 + 0.945257i $$0.394189\pi$$
$$318$$ 11.1830 0.627109
$$319$$ −20.5805 −1.15229
$$320$$ −0.616139 −0.0344432
$$321$$ 29.6550 1.65518
$$322$$ 2.70527 0.150759
$$323$$ 0 0
$$324$$ −11.1685 −0.620475
$$325$$ 29.4832 1.63543
$$326$$ 10.2803 0.569373
$$327$$ 1.22132 0.0675389
$$328$$ −1.28851 −0.0711461
$$329$$ −1.53750 −0.0847650
$$330$$ −4.62306 −0.254491
$$331$$ 1.15352 0.0634032 0.0317016 0.999497i $$-0.489907\pi$$
0.0317016 + 0.999497i $$0.489907\pi$$
$$332$$ 12.9425 0.710312
$$333$$ −7.62023 −0.417586
$$334$$ −12.6748 −0.693534
$$335$$ 2.56610 0.140201
$$336$$ 2.18756 0.119341
$$337$$ 7.31544 0.398497 0.199249 0.979949i $$-0.436150\pi$$
0.199249 + 0.979949i $$0.436150\pi$$
$$338$$ −27.7187 −1.50770
$$339$$ 22.0913 1.19983
$$340$$ −1.52802 −0.0828685
$$341$$ 26.5016 1.43514
$$342$$ 0 0
$$343$$ −1.00000 −0.0539949
$$344$$ 10.3584 0.558486
$$345$$ 3.64627 0.196309
$$346$$ 6.24437 0.335700
$$347$$ −31.3704 −1.68405 −0.842025 0.539439i $$-0.818636\pi$$
−0.842025 + 0.539439i $$0.818636\pi$$
$$348$$ 13.1257 0.703611
$$349$$ −2.33966 −0.125239 −0.0626197 0.998037i $$-0.519946\pi$$
−0.0626197 + 0.998037i $$0.519946\pi$$
$$350$$ −4.62037 −0.246969
$$351$$ −16.9547 −0.904974
$$352$$ −3.42998 −0.182819
$$353$$ −6.64960 −0.353923 −0.176961 0.984218i $$-0.556627\pi$$
−0.176961 + 0.984218i $$0.556627\pi$$
$$354$$ −27.9499 −1.48552
$$355$$ 0.0150774 0.000800223 0
$$356$$ 12.4510 0.659899
$$357$$ 5.42512 0.287128
$$358$$ 17.7976 0.940633
$$359$$ 23.2605 1.22764 0.613821 0.789445i $$-0.289631\pi$$
0.613821 + 0.789445i $$0.289631\pi$$
$$360$$ 1.10005 0.0579780
$$361$$ 0 0
$$362$$ −5.92757 −0.311546
$$363$$ −1.67301 −0.0878100
$$364$$ 6.38112 0.334462
$$365$$ −8.05138 −0.421428
$$366$$ −12.1320 −0.634150
$$367$$ 10.6463 0.555733 0.277866 0.960620i $$-0.410373\pi$$
0.277866 + 0.960620i $$0.410373\pi$$
$$368$$ 2.70527 0.141022
$$369$$ 2.30051 0.119760
$$370$$ −2.62973 −0.136713
$$371$$ −5.11208 −0.265406
$$372$$ −16.9020 −0.876329
$$373$$ 28.8794 1.49532 0.747659 0.664083i $$-0.231178\pi$$
0.747659 + 0.664083i $$0.231178\pi$$
$$374$$ −8.50633 −0.439852
$$375$$ −12.9667 −0.669598
$$376$$ −1.53750 −0.0792904
$$377$$ 38.2878 1.97192
$$378$$ 2.65701 0.136662
$$379$$ 33.1311 1.70183 0.850916 0.525302i $$-0.176048\pi$$
0.850916 + 0.525302i $$0.176048\pi$$
$$380$$ 0 0
$$381$$ 9.72616 0.498286
$$382$$ −16.3567 −0.836880
$$383$$ 3.65421 0.186722 0.0933608 0.995632i $$-0.470239\pi$$
0.0933608 + 0.995632i $$0.470239\pi$$
$$384$$ 2.18756 0.111633
$$385$$ 2.11335 0.107706
$$386$$ 19.9885 1.01739
$$387$$ −18.4938 −0.940095
$$388$$ 1.51714 0.0770211
$$389$$ −9.25078 −0.469033 −0.234517 0.972112i $$-0.575351\pi$$
−0.234517 + 0.972112i $$0.575351\pi$$
$$390$$ 8.60072 0.435515
$$391$$ 6.70905 0.339291
$$392$$ −1.00000 −0.0505076
$$393$$ −15.6888 −0.791395
$$394$$ 3.44430 0.173521
$$395$$ 1.82552 0.0918517
$$396$$ 6.12389 0.307737
$$397$$ 19.2309 0.965172 0.482586 0.875849i $$-0.339698\pi$$
0.482586 + 0.875849i $$0.339698\pi$$
$$398$$ −3.13027 −0.156906
$$399$$ 0 0
$$400$$ −4.62037 −0.231019
$$401$$ 29.5430 1.47531 0.737654 0.675179i $$-0.235933\pi$$
0.737654 + 0.675179i $$0.235933\pi$$
$$402$$ −9.11076 −0.454403
$$403$$ −49.3034 −2.45597
$$404$$ −9.92823 −0.493948
$$405$$ 6.88138 0.341939
$$406$$ −6.00017 −0.297783
$$407$$ −14.6395 −0.725651
$$408$$ 5.42512 0.268583
$$409$$ 39.8965 1.97276 0.986378 0.164494i $$-0.0525991\pi$$
0.986378 + 0.164494i $$0.0525991\pi$$
$$410$$ 0.793902 0.0392080
$$411$$ 36.7668 1.81357
$$412$$ −8.34988 −0.411369
$$413$$ 12.7768 0.628703
$$414$$ −4.82999 −0.237381
$$415$$ −7.97439 −0.391447
$$416$$ 6.38112 0.312860
$$417$$ 29.6559 1.45225
$$418$$ 0 0
$$419$$ −34.5562 −1.68818 −0.844090 0.536201i $$-0.819859\pi$$
−0.844090 + 0.536201i $$0.819859\pi$$
$$420$$ −1.34784 −0.0657678
$$421$$ 28.8651 1.40680 0.703399 0.710795i $$-0.251665\pi$$
0.703399 + 0.710795i $$0.251665\pi$$
$$422$$ −16.6987 −0.812880
$$423$$ 2.74505 0.133469
$$424$$ −5.11208 −0.248265
$$425$$ −11.4585 −0.555818
$$426$$ −0.0535310 −0.00259358
$$427$$ 5.54592 0.268386
$$428$$ −13.5562 −0.655264
$$429$$ 47.8793 2.31164
$$430$$ −6.38221 −0.307777
$$431$$ 19.4879 0.938699 0.469350 0.883012i $$-0.344488\pi$$
0.469350 + 0.883012i $$0.344488\pi$$
$$432$$ 2.65701 0.127835
$$433$$ 1.44225 0.0693102 0.0346551 0.999399i $$-0.488967\pi$$
0.0346551 + 0.999399i $$0.488967\pi$$
$$434$$ 7.72644 0.370881
$$435$$ −8.08726 −0.387754
$$436$$ −0.558302 −0.0267378
$$437$$ 0 0
$$438$$ 28.5858 1.36588
$$439$$ −16.5525 −0.790006 −0.395003 0.918680i $$-0.629256\pi$$
−0.395003 + 0.918680i $$0.629256\pi$$
$$440$$ 2.11335 0.100750
$$441$$ 1.78540 0.0850190
$$442$$ 15.8251 0.752725
$$443$$ 3.08768 0.146700 0.0733500 0.997306i $$-0.476631\pi$$
0.0733500 + 0.997306i $$0.476631\pi$$
$$444$$ 9.33667 0.443099
$$445$$ −7.67152 −0.363665
$$446$$ 14.5844 0.690589
$$447$$ −29.7486 −1.40706
$$448$$ −1.00000 −0.0472456
$$449$$ 17.9081 0.845135 0.422568 0.906331i $$-0.361129\pi$$
0.422568 + 0.906331i $$0.361129\pi$$
$$450$$ 8.24921 0.388871
$$451$$ 4.41957 0.208109
$$452$$ −10.0986 −0.474999
$$453$$ −44.2598 −2.07951
$$454$$ −4.73333 −0.222146
$$455$$ −3.93166 −0.184319
$$456$$ 0 0
$$457$$ 31.2954 1.46394 0.731970 0.681337i $$-0.238601\pi$$
0.731970 + 0.681337i $$0.238601\pi$$
$$458$$ −14.8611 −0.694415
$$459$$ 6.58936 0.307565
$$460$$ −1.66682 −0.0777161
$$461$$ −11.6085 −0.540660 −0.270330 0.962768i $$-0.587133\pi$$
−0.270330 + 0.962768i $$0.587133\pi$$
$$462$$ −7.50328 −0.349084
$$463$$ 4.15132 0.192928 0.0964640 0.995336i $$-0.469247\pi$$
0.0964640 + 0.995336i $$0.469247\pi$$
$$464$$ −6.00017 −0.278551
$$465$$ 10.4140 0.482937
$$466$$ 4.36751 0.202321
$$467$$ −24.9196 −1.15314 −0.576570 0.817047i $$-0.695609\pi$$
−0.576570 + 0.817047i $$0.695609\pi$$
$$468$$ −11.3929 −0.526635
$$469$$ 4.16481 0.192313
$$470$$ 0.947313 0.0436963
$$471$$ 24.3030 1.11982
$$472$$ 12.7768 0.588098
$$473$$ −35.5291 −1.63363
$$474$$ −6.48135 −0.297698
$$475$$ 0 0
$$476$$ −2.47999 −0.113670
$$477$$ 9.12710 0.417901
$$478$$ 24.2217 1.10787
$$479$$ −18.9778 −0.867117 −0.433559 0.901125i $$-0.642742\pi$$
−0.433559 + 0.901125i $$0.642742\pi$$
$$480$$ −1.34784 −0.0615201
$$481$$ 27.2352 1.24182
$$482$$ −17.1009 −0.778926
$$483$$ 5.91793 0.269275
$$484$$ 0.764783 0.0347629
$$485$$ −0.934769 −0.0424457
$$486$$ −16.4608 −0.746677
$$487$$ −36.1508 −1.63815 −0.819074 0.573687i $$-0.805512\pi$$
−0.819074 + 0.573687i $$0.805512\pi$$
$$488$$ 5.54592 0.251052
$$489$$ 22.4887 1.01698
$$490$$ 0.616139 0.0278343
$$491$$ 3.08754 0.139339 0.0696693 0.997570i $$-0.477806\pi$$
0.0696693 + 0.997570i $$0.477806\pi$$
$$492$$ −2.81869 −0.127076
$$493$$ −14.8804 −0.670178
$$494$$ 0 0
$$495$$ −3.77317 −0.169591
$$496$$ 7.72644 0.346927
$$497$$ 0.0244707 0.00109766
$$498$$ 28.3125 1.26871
$$499$$ 30.9593 1.38593 0.692964 0.720972i $$-0.256304\pi$$
0.692964 + 0.720972i $$0.256304\pi$$
$$500$$ 5.92749 0.265085
$$501$$ −27.7268 −1.23874
$$502$$ −27.3096 −1.21889
$$503$$ 3.79268 0.169107 0.0845536 0.996419i $$-0.473054\pi$$
0.0845536 + 0.996419i $$0.473054\pi$$
$$504$$ 1.78540 0.0795280
$$505$$ 6.11717 0.272211
$$506$$ −9.27904 −0.412504
$$507$$ −60.6362 −2.69295
$$508$$ −4.44613 −0.197265
$$509$$ −36.0268 −1.59686 −0.798430 0.602087i $$-0.794336\pi$$
−0.798430 + 0.602087i $$0.794336\pi$$
$$510$$ −3.34263 −0.148014
$$511$$ −13.0675 −0.578071
$$512$$ −1.00000 −0.0441942
$$513$$ 0 0
$$514$$ −9.96817 −0.439677
$$515$$ 5.14469 0.226702
$$516$$ 22.6595 0.997530
$$517$$ 5.27359 0.231932
$$518$$ −4.26808 −0.187529
$$519$$ 13.6599 0.599604
$$520$$ −3.93166 −0.172415
$$521$$ −7.91028 −0.346555 −0.173278 0.984873i $$-0.555436\pi$$
−0.173278 + 0.984873i $$0.555436\pi$$
$$522$$ 10.7127 0.468882
$$523$$ 33.0876 1.44682 0.723409 0.690420i $$-0.242574\pi$$
0.723409 + 0.690420i $$0.242574\pi$$
$$524$$ 7.17183 0.313303
$$525$$ −10.1073 −0.441120
$$526$$ 24.9314 1.08706
$$527$$ 19.1615 0.834688
$$528$$ −7.50328 −0.326538
$$529$$ −15.6815 −0.681804
$$530$$ 3.14975 0.136817
$$531$$ −22.8116 −0.989939
$$532$$ 0 0
$$533$$ −8.22215 −0.356141
$$534$$ 27.2371 1.17867
$$535$$ 8.35251 0.361111
$$536$$ 4.16481 0.179892
$$537$$ 38.9333 1.68009
$$538$$ −25.6665 −1.10656
$$539$$ 3.42998 0.147740
$$540$$ −1.63709 −0.0704490
$$541$$ 9.32192 0.400781 0.200390 0.979716i $$-0.435779\pi$$
0.200390 + 0.979716i $$0.435779\pi$$
$$542$$ −18.7964 −0.807374
$$543$$ −12.9669 −0.556462
$$544$$ −2.47999 −0.106329
$$545$$ 0.343992 0.0147350
$$546$$ 13.9591 0.597393
$$547$$ 27.3456 1.16921 0.584607 0.811317i $$-0.301249\pi$$
0.584607 + 0.811317i $$0.301249\pi$$
$$548$$ −16.8073 −0.717971
$$549$$ −9.90167 −0.422593
$$550$$ 15.8478 0.675752
$$551$$ 0 0
$$552$$ 5.91793 0.251884
$$553$$ 2.96283 0.125992
$$554$$ −19.8745 −0.844386
$$555$$ −5.75269 −0.244188
$$556$$ −13.5566 −0.574929
$$557$$ −16.4963 −0.698970 −0.349485 0.936942i $$-0.613643\pi$$
−0.349485 + 0.936942i $$0.613643\pi$$
$$558$$ −13.7948 −0.583980
$$559$$ 66.0981 2.79565
$$560$$ 0.616139 0.0260366
$$561$$ −18.6081 −0.785633
$$562$$ −21.2943 −0.898246
$$563$$ 35.4681 1.49480 0.747401 0.664373i $$-0.231301\pi$$
0.747401 + 0.664373i $$0.231301\pi$$
$$564$$ −3.36336 −0.141623
$$565$$ 6.22216 0.261768
$$566$$ 3.77466 0.158661
$$567$$ 11.1685 0.469035
$$568$$ 0.0244707 0.00102677
$$569$$ 43.3131 1.81578 0.907889 0.419210i $$-0.137693\pi$$
0.907889 + 0.419210i $$0.137693\pi$$
$$570$$ 0 0
$$571$$ 1.18471 0.0495787 0.0247893 0.999693i $$-0.492108\pi$$
0.0247893 + 0.999693i $$0.492108\pi$$
$$572$$ −21.8871 −0.915147
$$573$$ −35.7811 −1.49478
$$574$$ 1.28851 0.0537814
$$575$$ −12.4994 −0.521260
$$576$$ 1.78540 0.0743916
$$577$$ 12.9555 0.539343 0.269671 0.962952i $$-0.413085\pi$$
0.269671 + 0.962952i $$0.413085\pi$$
$$578$$ 10.8496 0.451286
$$579$$ 43.7260 1.81719
$$580$$ 3.69694 0.153507
$$581$$ −12.9425 −0.536946
$$582$$ 3.31883 0.137570
$$583$$ 17.5343 0.726198
$$584$$ −13.0675 −0.540735
$$585$$ 7.01958 0.290224
$$586$$ −16.9584 −0.700545
$$587$$ 36.4405 1.50406 0.752030 0.659129i $$-0.229075\pi$$
0.752030 + 0.659129i $$0.229075\pi$$
$$588$$ −2.18756 −0.0902133
$$589$$ 0 0
$$590$$ −7.87226 −0.324096
$$591$$ 7.53460 0.309932
$$592$$ −4.26808 −0.175417
$$593$$ 31.6814 1.30100 0.650500 0.759506i $$-0.274559\pi$$
0.650500 + 0.759506i $$0.274559\pi$$
$$594$$ −9.11349 −0.373931
$$595$$ 1.52802 0.0626427
$$596$$ 13.5990 0.557038
$$597$$ −6.84763 −0.280255
$$598$$ 17.2627 0.705923
$$599$$ −5.21417 −0.213045 −0.106523 0.994310i $$-0.533972\pi$$
−0.106523 + 0.994310i $$0.533972\pi$$
$$600$$ −10.1073 −0.412630
$$601$$ 8.43231 0.343961 0.171981 0.985100i $$-0.444983\pi$$
0.171981 + 0.985100i $$0.444983\pi$$
$$602$$ −10.3584 −0.422176
$$603$$ −7.43585 −0.302811
$$604$$ 20.2325 0.823250
$$605$$ −0.471213 −0.0191575
$$606$$ −21.7186 −0.882256
$$607$$ −18.8481 −0.765022 −0.382511 0.923951i $$-0.624941\pi$$
−0.382511 + 0.923951i $$0.624941\pi$$
$$608$$ 0 0
$$609$$ −13.1257 −0.531880
$$610$$ −3.41706 −0.138353
$$611$$ −9.81096 −0.396909
$$612$$ 4.42778 0.178982
$$613$$ 14.8192 0.598542 0.299271 0.954168i $$-0.403256\pi$$
0.299271 + 0.954168i $$0.403256\pi$$
$$614$$ −2.68585 −0.108392
$$615$$ 1.73671 0.0700307
$$616$$ 3.42998 0.138198
$$617$$ 14.2858 0.575123 0.287562 0.957762i $$-0.407155\pi$$
0.287562 + 0.957762i $$0.407155\pi$$
$$618$$ −18.2658 −0.734760
$$619$$ −27.8881 −1.12092 −0.560459 0.828182i $$-0.689375\pi$$
−0.560459 + 0.828182i $$0.689375\pi$$
$$620$$ −4.76056 −0.191189
$$621$$ 7.18793 0.288442
$$622$$ −1.77079 −0.0710023
$$623$$ −12.4510 −0.498837
$$624$$ 13.9591 0.558810
$$625$$ 19.4497 0.777988
$$626$$ −7.89729 −0.315639
$$627$$ 0 0
$$628$$ −11.1097 −0.443324
$$629$$ −10.5848 −0.422044
$$630$$ −1.10005 −0.0438272
$$631$$ 26.3781 1.05010 0.525048 0.851073i $$-0.324047\pi$$
0.525048 + 0.851073i $$0.324047\pi$$
$$632$$ 2.96283 0.117855
$$633$$ −36.5293 −1.45191
$$634$$ −11.6202 −0.461498
$$635$$ 2.73944 0.108711
$$636$$ −11.1830 −0.443433
$$637$$ −6.38112 −0.252829
$$638$$ 20.5805 0.814789
$$639$$ −0.0436899 −0.00172835
$$640$$ 0.616139 0.0243550
$$641$$ −39.1472 −1.54622 −0.773110 0.634272i $$-0.781300\pi$$
−0.773110 + 0.634272i $$0.781300\pi$$
$$642$$ −29.6550 −1.17039
$$643$$ −6.20232 −0.244596 −0.122298 0.992493i $$-0.539026\pi$$
−0.122298 + 0.992493i $$0.539026\pi$$
$$644$$ −2.70527 −0.106603
$$645$$ −13.9614 −0.549731
$$646$$ 0 0
$$647$$ −2.11530 −0.0831610 −0.0415805 0.999135i $$-0.513239\pi$$
−0.0415805 + 0.999135i $$0.513239\pi$$
$$648$$ 11.1685 0.438742
$$649$$ −43.8240 −1.72024
$$650$$ −29.4832 −1.15642
$$651$$ 16.9020 0.662442
$$652$$ −10.2803 −0.402608
$$653$$ −20.2029 −0.790602 −0.395301 0.918552i $$-0.629360\pi$$
−0.395301 + 0.918552i $$0.629360\pi$$
$$654$$ −1.22132 −0.0477572
$$655$$ −4.41885 −0.172659
$$656$$ 1.28851 0.0503079
$$657$$ 23.3306 0.910215
$$658$$ 1.53750 0.0599379
$$659$$ 23.3082 0.907958 0.453979 0.891012i $$-0.350004\pi$$
0.453979 + 0.891012i $$0.350004\pi$$
$$660$$ 4.62306 0.179953
$$661$$ 28.4307 1.10583 0.552914 0.833238i $$-0.313516\pi$$
0.552914 + 0.833238i $$0.313516\pi$$
$$662$$ −1.15352 −0.0448329
$$663$$ 34.6184 1.34447
$$664$$ −12.9425 −0.502267
$$665$$ 0 0
$$666$$ 7.62023 0.295278
$$667$$ −16.2321 −0.628509
$$668$$ 12.6748 0.490403
$$669$$ 31.9041 1.23348
$$670$$ −2.56610 −0.0991372
$$671$$ −19.0224 −0.734351
$$672$$ −2.18756 −0.0843868
$$673$$ 9.48697 0.365696 0.182848 0.983141i $$-0.441468\pi$$
0.182848 + 0.983141i $$0.441468\pi$$
$$674$$ −7.31544 −0.281780
$$675$$ −12.2764 −0.472517
$$676$$ 27.7187 1.06610
$$677$$ −43.2866 −1.66364 −0.831820 0.555045i $$-0.812701\pi$$
−0.831820 + 0.555045i $$0.812701\pi$$
$$678$$ −22.0913 −0.848411
$$679$$ −1.51714 −0.0582225
$$680$$ 1.52802 0.0585969
$$681$$ −10.3544 −0.396782
$$682$$ −26.5016 −1.01480
$$683$$ 16.3891 0.627110 0.313555 0.949570i $$-0.398480\pi$$
0.313555 + 0.949570i $$0.398480\pi$$
$$684$$ 0 0
$$685$$ 10.3556 0.395668
$$686$$ 1.00000 0.0381802
$$687$$ −32.5096 −1.24032
$$688$$ −10.3584 −0.394910
$$689$$ −32.6208 −1.24275
$$690$$ −3.64627 −0.138811
$$691$$ 41.4257 1.57591 0.787953 0.615735i $$-0.211141\pi$$
0.787953 + 0.615735i $$0.211141\pi$$
$$692$$ −6.24437 −0.237376
$$693$$ −6.12389 −0.232627
$$694$$ 31.3704 1.19080
$$695$$ 8.35277 0.316839
$$696$$ −13.1257 −0.497528
$$697$$ 3.19550 0.121038
$$698$$ 2.33966 0.0885576
$$699$$ 9.55416 0.361372
$$700$$ 4.62037 0.174634
$$701$$ 46.1088 1.74150 0.870752 0.491722i $$-0.163632\pi$$
0.870752 + 0.491722i $$0.163632\pi$$
$$702$$ 16.9547 0.639913
$$703$$ 0 0
$$704$$ 3.42998 0.129272
$$705$$ 2.07230 0.0780473
$$706$$ 6.64960 0.250261
$$707$$ 9.92823 0.373390
$$708$$ 27.9499 1.05042
$$709$$ −27.0329 −1.01524 −0.507622 0.861580i $$-0.669475\pi$$
−0.507622 + 0.861580i $$0.669475\pi$$
$$710$$ −0.0150774 −0.000565843 0
$$711$$ −5.28983 −0.198384
$$712$$ −12.4510 −0.466619
$$713$$ 20.9021 0.782791
$$714$$ −5.42512 −0.203030
$$715$$ 13.4855 0.504330
$$716$$ −17.7976 −0.665128
$$717$$ 52.9862 1.97881
$$718$$ −23.2605 −0.868075
$$719$$ −47.8432 −1.78425 −0.892125 0.451789i $$-0.850786\pi$$
−0.892125 + 0.451789i $$0.850786\pi$$
$$720$$ −1.10005 −0.0409966
$$721$$ 8.34988 0.310966
$$722$$ 0 0
$$723$$ −37.4092 −1.39126
$$724$$ 5.92757 0.220296
$$725$$ 27.7230 1.02961
$$726$$ 1.67301 0.0620911
$$727$$ 4.90682 0.181984 0.0909919 0.995852i $$-0.470996\pi$$
0.0909919 + 0.995852i $$0.470996\pi$$
$$728$$ −6.38112 −0.236500
$$729$$ −2.50327 −0.0927136
$$730$$ 8.05138 0.297995
$$731$$ −25.6887 −0.950131
$$732$$ 12.1320 0.448412
$$733$$ 52.5699 1.94172 0.970858 0.239657i $$-0.0770349\pi$$
0.970858 + 0.239657i $$0.0770349\pi$$
$$734$$ −10.6463 −0.392962
$$735$$ 1.34784 0.0497158
$$736$$ −2.70527 −0.0997177
$$737$$ −14.2852 −0.526203
$$738$$ −2.30051 −0.0846828
$$739$$ 25.2192 0.927702 0.463851 0.885913i $$-0.346467\pi$$
0.463851 + 0.885913i $$0.346467\pi$$
$$740$$ 2.62973 0.0966709
$$741$$ 0 0
$$742$$ 5.11208 0.187670
$$743$$ −5.58183 −0.204777 −0.102389 0.994744i $$-0.532649\pi$$
−0.102389 + 0.994744i $$0.532649\pi$$
$$744$$ 16.9020 0.619658
$$745$$ −8.37889 −0.306979
$$746$$ −28.8794 −1.05735
$$747$$ 23.1075 0.845461
$$748$$ 8.50633 0.311022
$$749$$ 13.5562 0.495333
$$750$$ 12.9667 0.473477
$$751$$ −25.7524 −0.939718 −0.469859 0.882742i $$-0.655695\pi$$
−0.469859 + 0.882742i $$0.655695\pi$$
$$752$$ 1.53750 0.0560668
$$753$$ −59.7413 −2.17709
$$754$$ −38.2878 −1.39436
$$755$$ −12.4661 −0.453686
$$756$$ −2.65701 −0.0966344
$$757$$ −45.3483 −1.64821 −0.824106 0.566435i $$-0.808322\pi$$
−0.824106 + 0.566435i $$0.808322\pi$$
$$758$$ −33.1311 −1.20338
$$759$$ −20.2984 −0.736786
$$760$$ 0 0
$$761$$ −24.9534 −0.904560 −0.452280 0.891876i $$-0.649389\pi$$
−0.452280 + 0.891876i $$0.649389\pi$$
$$762$$ −9.72616 −0.352342
$$763$$ 0.558302 0.0202119
$$764$$ 16.3567 0.591764
$$765$$ −2.72813 −0.0986356
$$766$$ −3.65421 −0.132032
$$767$$ 81.5300 2.94388
$$768$$ −2.18756 −0.0789366
$$769$$ 12.9261 0.466126 0.233063 0.972462i $$-0.425125\pi$$
0.233063 + 0.972462i $$0.425125\pi$$
$$770$$ −2.11335 −0.0761597
$$771$$ −21.8059 −0.785321
$$772$$ −19.9885 −0.719403
$$773$$ −14.0205 −0.504281 −0.252141 0.967691i $$-0.581135\pi$$
−0.252141 + 0.967691i $$0.581135\pi$$
$$774$$ 18.4938 0.664748
$$775$$ −35.6990 −1.28235
$$776$$ −1.51714 −0.0544621
$$777$$ −9.33667 −0.334951
$$778$$ 9.25078 0.331657
$$779$$ 0 0
$$780$$ −8.60072 −0.307955
$$781$$ −0.0839340 −0.00300340
$$782$$ −6.70905 −0.239915
$$783$$ −15.9425 −0.569738
$$784$$ 1.00000 0.0357143
$$785$$ 6.84511 0.244312
$$786$$ 15.6888 0.559600
$$787$$ 4.13771 0.147494 0.0737468 0.997277i $$-0.476504\pi$$
0.0737468 + 0.997277i $$0.476504\pi$$
$$788$$ −3.44430 −0.122698
$$789$$ 54.5389 1.94164
$$790$$ −1.82552 −0.0649489
$$791$$ 10.0986 0.359066
$$792$$ −6.12389 −0.217603
$$793$$ 35.3892 1.25671
$$794$$ −19.2309 −0.682480
$$795$$ 6.89026 0.244372
$$796$$ 3.13027 0.110949
$$797$$ −53.5733 −1.89767 −0.948833 0.315780i $$-0.897734\pi$$
−0.948833 + 0.315780i $$0.897734\pi$$
$$798$$ 0 0
$$799$$ 3.81298 0.134894
$$800$$ 4.62037 0.163355
$$801$$ 22.2299 0.785456
$$802$$ −29.5430 −1.04320
$$803$$ 44.8212 1.58170
$$804$$ 9.11076 0.321312
$$805$$ 1.66682 0.0587478
$$806$$ 49.3034 1.73664
$$807$$ −56.1468 −1.97646
$$808$$ 9.92823 0.349274
$$809$$ 30.7139 1.07984 0.539922 0.841715i $$-0.318454\pi$$
0.539922 + 0.841715i $$0.318454\pi$$
$$810$$ −6.88138 −0.241787
$$811$$ 43.6163 1.53158 0.765788 0.643093i $$-0.222349\pi$$
0.765788 + 0.643093i $$0.222349\pi$$
$$812$$ 6.00017 0.210565
$$813$$ −41.1181 −1.44208
$$814$$ 14.6395 0.513113
$$815$$ 6.33410 0.221874
$$816$$ −5.42512 −0.189917
$$817$$ 0 0
$$818$$ −39.8965 −1.39495
$$819$$ 11.3929 0.398098
$$820$$ −0.793902 −0.0277243
$$821$$ −43.0095 −1.50104 −0.750521 0.660847i $$-0.770197\pi$$
−0.750521 + 0.660847i $$0.770197\pi$$
$$822$$ −36.7668 −1.28239
$$823$$ −3.41982 −0.119207 −0.0596037 0.998222i $$-0.518984\pi$$
−0.0596037 + 0.998222i $$0.518984\pi$$
$$824$$ 8.34988 0.290882
$$825$$ 34.6679 1.20698
$$826$$ −12.7768 −0.444560
$$827$$ 27.6387 0.961093 0.480547 0.876969i $$-0.340438\pi$$
0.480547 + 0.876969i $$0.340438\pi$$
$$828$$ 4.82999 0.167854
$$829$$ −25.7042 −0.892745 −0.446373 0.894847i $$-0.647284\pi$$
−0.446373 + 0.894847i $$0.647284\pi$$
$$830$$ 7.97439 0.276795
$$831$$ −43.4765 −1.50818
$$832$$ −6.38112 −0.221226
$$833$$ 2.47999 0.0859266
$$834$$ −29.6559 −1.02690
$$835$$ −7.80944 −0.270257
$$836$$ 0 0
$$837$$ 20.5292 0.709593
$$838$$ 34.5562 1.19372
$$839$$ −36.9844 −1.27684 −0.638422 0.769686i $$-0.720413\pi$$
−0.638422 + 0.769686i $$0.720413\pi$$
$$840$$ 1.34784 0.0465049
$$841$$ 7.00200 0.241448
$$842$$ −28.8651 −0.994757
$$843$$ −46.5825 −1.60439
$$844$$ 16.6987 0.574793
$$845$$ −17.0786 −0.587521
$$846$$ −2.74505 −0.0943766
$$847$$ −0.764783 −0.0262783
$$848$$ 5.11208 0.175550
$$849$$ 8.25727 0.283389
$$850$$ 11.4585 0.393023
$$851$$ −11.5463 −0.395803
$$852$$ 0.0535310 0.00183394
$$853$$ −39.1925 −1.34193 −0.670963 0.741491i $$-0.734119\pi$$
−0.670963 + 0.741491i $$0.734119\pi$$
$$854$$ −5.54592 −0.189777
$$855$$ 0 0
$$856$$ 13.5562 0.463342
$$857$$ 5.62471 0.192136 0.0960682 0.995375i $$-0.469373\pi$$
0.0960682 + 0.995375i $$0.469373\pi$$
$$858$$ −47.8793 −1.63457
$$859$$ −23.0096 −0.785076 −0.392538 0.919736i $$-0.628403\pi$$
−0.392538 + 0.919736i $$0.628403\pi$$
$$860$$ 6.38221 0.217631
$$861$$ 2.81869 0.0960607
$$862$$ −19.4879 −0.663760
$$863$$ −10.3667 −0.352888 −0.176444 0.984311i $$-0.556459\pi$$
−0.176444 + 0.984311i $$0.556459\pi$$
$$864$$ −2.65701 −0.0903932
$$865$$ 3.84740 0.130816
$$866$$ −1.44225 −0.0490097
$$867$$ 23.7342 0.806055
$$868$$ −7.72644 −0.262252
$$869$$ −10.1625 −0.344738
$$870$$ 8.08726 0.274184
$$871$$ 26.5762 0.900499
$$872$$ 0.558302 0.0189065
$$873$$ 2.70870 0.0916756
$$874$$ 0 0
$$875$$ −5.92749 −0.200386
$$876$$ −28.5858 −0.965825
$$877$$ 37.9086 1.28008 0.640040 0.768341i $$-0.278918\pi$$
0.640040 + 0.768341i $$0.278918\pi$$
$$878$$ 16.5525 0.558619
$$879$$ −37.0975 −1.25127
$$880$$ −2.11335 −0.0712409
$$881$$ 26.1648 0.881514 0.440757 0.897626i $$-0.354710\pi$$
0.440757 + 0.897626i $$0.354710\pi$$
$$882$$ −1.78540 −0.0601175
$$883$$ −9.63158 −0.324128 −0.162064 0.986780i $$-0.551815\pi$$
−0.162064 + 0.986780i $$0.551815\pi$$
$$884$$ −15.8251 −0.532257
$$885$$ −17.2210 −0.578878
$$886$$ −3.08768 −0.103733
$$887$$ −1.96127 −0.0658531 −0.0329266 0.999458i $$-0.510483\pi$$
−0.0329266 + 0.999458i $$0.510483\pi$$
$$888$$ −9.33667 −0.313318
$$889$$ 4.44613 0.149119
$$890$$ 7.67152 0.257150
$$891$$ −38.3079 −1.28336
$$892$$ −14.5844 −0.488320
$$893$$ 0 0
$$894$$ 29.7486 0.994942
$$895$$ 10.9658 0.366546
$$896$$ 1.00000 0.0334077
$$897$$ 37.7631 1.26087
$$898$$ −17.9081 −0.597601
$$899$$ −46.3599 −1.54619
$$900$$ −8.24921 −0.274974
$$901$$ 12.6779 0.422363
$$902$$ −4.41957 −0.147156
$$903$$ −22.6595 −0.754062
$$904$$ 10.0986 0.335875
$$905$$ −3.65221 −0.121403
$$906$$ 44.2598 1.47043
$$907$$ 43.4844 1.44387 0.721937 0.691958i $$-0.243252\pi$$
0.721937 + 0.691958i $$0.243252\pi$$
$$908$$ 4.73333 0.157081
$$909$$ −17.7259 −0.587930
$$910$$ 3.93166 0.130333
$$911$$ 4.93846 0.163618 0.0818092 0.996648i $$-0.473930\pi$$
0.0818092 + 0.996648i $$0.473930\pi$$
$$912$$ 0 0
$$913$$ 44.3926 1.46918
$$914$$ −31.2954 −1.03516
$$915$$ −7.47500 −0.247116
$$916$$ 14.8611 0.491026
$$917$$ −7.17183 −0.236835
$$918$$ −6.58936 −0.217481
$$919$$ 1.25165 0.0412883 0.0206441 0.999787i $$-0.493428\pi$$
0.0206441 + 0.999787i $$0.493428\pi$$
$$920$$ 1.66682 0.0549536
$$921$$ −5.87544 −0.193602
$$922$$ 11.6085 0.382304
$$923$$ 0.156150 0.00513975
$$924$$ 7.50328 0.246840
$$925$$ 19.7201 0.648394
$$926$$ −4.15132 −0.136421
$$927$$ −14.9079 −0.489639
$$928$$ 6.00017 0.196965
$$929$$ −38.7110 −1.27007 −0.635034 0.772484i $$-0.719014\pi$$
−0.635034 + 0.772484i $$0.719014\pi$$
$$930$$ −10.4140 −0.341488
$$931$$ 0 0
$$932$$ −4.36751 −0.143062
$$933$$ −3.87371 −0.126819
$$934$$ 24.9196 0.815394
$$935$$ −5.24108 −0.171402
$$936$$ 11.3929 0.372387
$$937$$ 18.5473 0.605913 0.302957 0.953004i $$-0.402026\pi$$
0.302957 + 0.953004i $$0.402026\pi$$
$$938$$ −4.16481 −0.135986
$$939$$ −17.2758 −0.563773
$$940$$ −0.947313 −0.0308979
$$941$$ −24.6283 −0.802859 −0.401429 0.915890i $$-0.631487\pi$$
−0.401429 + 0.915890i $$0.631487\pi$$
$$942$$ −24.3030 −0.791835
$$943$$ 3.48577 0.113512
$$944$$ −12.7768 −0.415848
$$945$$ 1.63709 0.0532544
$$946$$ 35.5291 1.15515
$$947$$ 20.7160 0.673178 0.336589 0.941652i $$-0.390727\pi$$
0.336589 + 0.941652i $$0.390727\pi$$
$$948$$ 6.48135 0.210505
$$949$$ −83.3851 −2.70679
$$950$$ 0 0
$$951$$ −25.4199 −0.824296
$$952$$ 2.47999 0.0803770
$$953$$ 42.6761 1.38242 0.691208 0.722656i $$-0.257079\pi$$
0.691208 + 0.722656i $$0.257079\pi$$
$$954$$ −9.12710 −0.295501
$$955$$ −10.0780 −0.326116
$$956$$ −24.2217 −0.783385
$$957$$ 45.0209 1.45532
$$958$$ 18.9778 0.613144
$$959$$ 16.8073 0.542735
$$960$$ 1.34784 0.0435013
$$961$$ 28.6979 0.925738
$$962$$ −27.2352 −0.878097
$$963$$ −24.2033 −0.779939
$$964$$ 17.1009 0.550784
$$965$$ 12.3157 0.396457
$$966$$ −5.91793 −0.190406
$$967$$ 46.5740 1.49772 0.748860 0.662728i $$-0.230602\pi$$
0.748860 + 0.662728i $$0.230602\pi$$
$$968$$ −0.764783 −0.0245811
$$969$$ 0 0
$$970$$ 0.934769 0.0300136
$$971$$ −16.9447 −0.543782 −0.271891 0.962328i $$-0.587649\pi$$
−0.271891 + 0.962328i $$0.587649\pi$$
$$972$$ 16.4608 0.527980
$$973$$ 13.5566 0.434606
$$974$$ 36.1508 1.15835
$$975$$ −64.4960 −2.06553
$$976$$ −5.54592 −0.177520
$$977$$ −10.1186 −0.323723 −0.161862 0.986813i $$-0.551750\pi$$
−0.161862 + 0.986813i $$0.551750\pi$$
$$978$$ −22.4887 −0.719110
$$979$$ 42.7066 1.36491
$$980$$ −0.616139 −0.0196818
$$981$$ −0.996792 −0.0318251
$$982$$ −3.08754 −0.0985273
$$983$$ −36.8594 −1.17563 −0.587816 0.808995i $$-0.700012\pi$$
−0.587816 + 0.808995i $$0.700012\pi$$
$$984$$ 2.81869 0.0898565
$$985$$ 2.12217 0.0676179
$$986$$ 14.8804 0.473887
$$987$$ 3.36336 0.107057
$$988$$ 0 0
$$989$$ −28.0222 −0.891055
$$990$$ 3.77317 0.119919
$$991$$ −50.6580 −1.60921 −0.804603 0.593813i $$-0.797622\pi$$
−0.804603 + 0.593813i $$0.797622\pi$$
$$992$$ −7.72644 −0.245315
$$993$$ −2.52339 −0.0800774
$$994$$ −0.0244707 −0.000776163 0
$$995$$ −1.92868 −0.0611432
$$996$$ −28.3125 −0.897114
$$997$$ −19.9246 −0.631019 −0.315510 0.948922i $$-0.602175\pi$$
−0.315510 + 0.948922i $$0.602175\pi$$
$$998$$ −30.9593 −0.979999
$$999$$ −11.3403 −0.358792
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.bj.1.3 9
19.14 odd 18 266.2.u.c.253.1 yes 18
19.15 odd 18 266.2.u.c.225.1 18
19.18 odd 2 5054.2.a.bk.1.7 9

By twisted newform
Twist Min Dim Char Parity Ord Type
266.2.u.c.225.1 18 19.15 odd 18
266.2.u.c.253.1 yes 18 19.14 odd 18
5054.2.a.bj.1.3 9 1.1 even 1 trivial
5054.2.a.bk.1.7 9 19.18 odd 2