Properties

 Label 9025.2.a.bx.1.4 Level $9025$ Weight $2$ Character 9025.1 Self dual yes Analytic conductor $72.065$ Analytic rank $1$ Dimension $6$ CM no Inner twists $2$

Related objects

Newspace parameters

 Level: $$N$$ $$=$$ $$9025 = 5^{2} \cdot 19^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 9025.a (trivial)

Newform invariants

 Self dual: yes Analytic conductor: $$72.0649878242$$ Analytic rank: $$1$$ Dimension: $$6$$ Coefficient field: 6.6.66064384.1 Defining polynomial: $$x^{6} - 9 x^{4} + 13 x^{2} - 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 95) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

 Embedding label 1.4 Root $$-0.285442$$ of defining polynomial Character $$\chi$$ $$=$$ 9025.1

$q$-expansion

 $$f(q)$$ $$=$$ $$q+0.906968 q^{2} -3.21789 q^{3} -1.17741 q^{4} -2.91852 q^{6} +2.59637 q^{7} -2.88181 q^{8} +7.35482 q^{9} +O(q^{10})$$ $$q+0.906968 q^{2} -3.21789 q^{3} -1.17741 q^{4} -2.91852 q^{6} +2.59637 q^{7} -2.88181 q^{8} +7.35482 q^{9} +0.741113 q^{11} +3.78878 q^{12} +3.78878 q^{13} +2.35482 q^{14} -0.258887 q^{16} -3.16725 q^{17} +6.67058 q^{18} -8.35482 q^{21} +0.672165 q^{22} +0.570885 q^{23} +9.27334 q^{24} +3.43630 q^{26} -14.0133 q^{27} -3.05699 q^{28} -6.00000 q^{29} -5.83705 q^{31} +5.52881 q^{32} -2.38482 q^{33} -2.87259 q^{34} -8.65964 q^{36} +1.40396 q^{37} -12.1919 q^{39} +3.83705 q^{41} -7.57755 q^{42} -2.59637 q^{43} -0.872594 q^{44} +0.517774 q^{46} +5.08247 q^{47} +0.833070 q^{48} -0.258887 q^{49} +10.1919 q^{51} -4.46094 q^{52} +0.160905 q^{53} -12.7096 q^{54} -7.48223 q^{56} -5.44181 q^{58} -8.35482 q^{59} -8.57816 q^{61} -5.29401 q^{62} +19.0958 q^{63} +5.53223 q^{64} -2.16295 q^{66} +14.8464 q^{67} +3.72915 q^{68} -1.83705 q^{69} -3.64518 q^{71} -21.1952 q^{72} -10.8461 q^{73} +1.27334 q^{74} +1.92420 q^{77} -11.0576 q^{78} -1.83705 q^{79} +23.0289 q^{81} +3.48008 q^{82} -4.19876 q^{83} +9.83705 q^{84} -2.35482 q^{86} +19.3073 q^{87} -2.13574 q^{88} +16.9015 q^{89} +9.83705 q^{91} -0.672165 q^{92} +18.7830 q^{93} +4.60963 q^{94} -17.7911 q^{96} +3.78878 q^{97} -0.234802 q^{98} +5.45075 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6 q + 8 q^{4} + 14 q^{9} + O(q^{10})$$ $$6 q + 8 q^{4} + 14 q^{9} + 2 q^{11} - 16 q^{14} - 4 q^{16} - 20 q^{21} + 8 q^{24} + 8 q^{26} - 36 q^{29} + 8 q^{34} - 32 q^{36} - 8 q^{39} - 12 q^{41} + 20 q^{44} + 8 q^{46} - 4 q^{49} - 4 q^{51} - 16 q^{54} - 40 q^{56} - 20 q^{59} - 14 q^{61} - 12 q^{64} - 48 q^{66} + 24 q^{69} - 52 q^{71} - 40 q^{74} + 24 q^{79} + 38 q^{81} + 24 q^{84} + 16 q^{86} - 24 q^{89} + 24 q^{91} + 48 q^{94} - 64 q^{96} - 30 q^{99} + O(q^{100})$$

Coefficient data

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

Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 0.906968 0.641323 0.320661 0.947194i $$-0.396095\pi$$
0.320661 + 0.947194i $$0.396095\pi$$
$$3$$ −3.21789 −1.85785 −0.928925 0.370268i $$-0.879266\pi$$
−0.928925 + 0.370268i $$0.879266\pi$$
$$4$$ −1.17741 −0.588705
$$5$$ 0 0
$$6$$ −2.91852 −1.19148
$$7$$ 2.59637 0.981334 0.490667 0.871347i $$-0.336753\pi$$
0.490667 + 0.871347i $$0.336753\pi$$
$$8$$ −2.88181 −1.01887
$$9$$ 7.35482 2.45161
$$10$$ 0 0
$$11$$ 0.741113 0.223454 0.111727 0.993739i $$-0.464362\pi$$
0.111727 + 0.993739i $$0.464362\pi$$
$$12$$ 3.78878 1.09373
$$13$$ 3.78878 1.05082 0.525409 0.850850i $$-0.323912\pi$$
0.525409 + 0.850850i $$0.323912\pi$$
$$14$$ 2.35482 0.629352
$$15$$ 0 0
$$16$$ −0.258887 −0.0647218
$$17$$ −3.16725 −0.768171 −0.384086 0.923298i $$-0.625483\pi$$
−0.384086 + 0.923298i $$0.625483\pi$$
$$18$$ 6.67058 1.57227
$$19$$ 0 0
$$20$$ 0 0
$$21$$ −8.35482 −1.82317
$$22$$ 0.672165 0.143306
$$23$$ 0.570885 0.119038 0.0595189 0.998227i $$-0.481043\pi$$
0.0595189 + 0.998227i $$0.481043\pi$$
$$24$$ 9.27334 1.89291
$$25$$ 0 0
$$26$$ 3.43630 0.673913
$$27$$ −14.0133 −2.69687
$$28$$ −3.05699 −0.577716
$$29$$ −6.00000 −1.11417 −0.557086 0.830455i $$-0.688081\pi$$
−0.557086 + 0.830455i $$0.688081\pi$$
$$30$$ 0 0
$$31$$ −5.83705 −1.04836 −0.524182 0.851606i $$-0.675629\pi$$
−0.524182 + 0.851606i $$0.675629\pi$$
$$32$$ 5.52881 0.977365
$$33$$ −2.38482 −0.415144
$$34$$ −2.87259 −0.492646
$$35$$ 0 0
$$36$$ −8.65964 −1.44327
$$37$$ 1.40396 0.230809 0.115404 0.993319i $$-0.463184\pi$$
0.115404 + 0.993319i $$0.463184\pi$$
$$38$$ 0 0
$$39$$ −12.1919 −1.95226
$$40$$ 0 0
$$41$$ 3.83705 0.599246 0.299623 0.954058i $$-0.403139\pi$$
0.299623 + 0.954058i $$0.403139\pi$$
$$42$$ −7.57755 −1.16924
$$43$$ −2.59637 −0.395942 −0.197971 0.980208i $$-0.563435\pi$$
−0.197971 + 0.980208i $$0.563435\pi$$
$$44$$ −0.872594 −0.131548
$$45$$ 0 0
$$46$$ 0.517774 0.0763416
$$47$$ 5.08247 0.741354 0.370677 0.928762i $$-0.379126\pi$$
0.370677 + 0.928762i $$0.379126\pi$$
$$48$$ 0.833070 0.120243
$$49$$ −0.258887 −0.0369839
$$50$$ 0 0
$$51$$ 10.1919 1.42715
$$52$$ −4.46094 −0.618621
$$53$$ 0.160905 0.0221020 0.0110510 0.999939i $$-0.496482\pi$$
0.0110510 + 0.999939i $$0.496482\pi$$
$$54$$ −12.7096 −1.72956
$$55$$ 0 0
$$56$$ −7.48223 −0.999854
$$57$$ 0 0
$$58$$ −5.44181 −0.714544
$$59$$ −8.35482 −1.08770 −0.543852 0.839181i $$-0.683035\pi$$
−0.543852 + 0.839181i $$0.683035\pi$$
$$60$$ 0 0
$$61$$ −8.57816 −1.09832 −0.549160 0.835717i $$-0.685052\pi$$
−0.549160 + 0.835717i $$0.685052\pi$$
$$62$$ −5.29401 −0.672340
$$63$$ 19.0958 2.40584
$$64$$ 5.53223 0.691529
$$65$$ 0 0
$$66$$ −2.16295 −0.266241
$$67$$ 14.8464 1.81378 0.906888 0.421371i $$-0.138451\pi$$
0.906888 + 0.421371i $$0.138451\pi$$
$$68$$ 3.72915 0.452226
$$69$$ −1.83705 −0.221154
$$70$$ 0 0
$$71$$ −3.64518 −0.432603 −0.216302 0.976327i $$-0.569399\pi$$
−0.216302 + 0.976327i $$0.569399\pi$$
$$72$$ −21.1952 −2.49788
$$73$$ −10.8461 −1.26944 −0.634719 0.772743i $$-0.718884\pi$$
−0.634719 + 0.772743i $$0.718884\pi$$
$$74$$ 1.27334 0.148023
$$75$$ 0 0
$$76$$ 0 0
$$77$$ 1.92420 0.219283
$$78$$ −11.0576 −1.25203
$$79$$ −1.83705 −0.206684 −0.103342 0.994646i $$-0.532954\pi$$
−0.103342 + 0.994646i $$0.532954\pi$$
$$80$$ 0 0
$$81$$ 23.0289 2.55877
$$82$$ 3.48008 0.384310
$$83$$ −4.19876 −0.460873 −0.230437 0.973087i $$-0.574015\pi$$
−0.230437 + 0.973087i $$0.574015\pi$$
$$84$$ 9.83705 1.07331
$$85$$ 0 0
$$86$$ −2.35482 −0.253927
$$87$$ 19.3073 2.06996
$$88$$ −2.13574 −0.227671
$$89$$ 16.9015 1.79156 0.895778 0.444502i $$-0.146619\pi$$
0.895778 + 0.444502i $$0.146619\pi$$
$$90$$ 0 0
$$91$$ 9.83705 1.03120
$$92$$ −0.672165 −0.0700781
$$93$$ 18.7830 1.94770
$$94$$ 4.60963 0.475447
$$95$$ 0 0
$$96$$ −17.7911 −1.81580
$$97$$ 3.78878 0.384692 0.192346 0.981327i $$-0.438390\pi$$
0.192346 + 0.981327i $$0.438390\pi$$
$$98$$ −0.234802 −0.0237186
$$99$$ 5.45075 0.547821
$$100$$ 0 0
$$101$$ 8.35482 0.831336 0.415668 0.909517i $$-0.363548\pi$$
0.415668 + 0.909517i $$0.363548\pi$$
$$102$$ 9.24369 0.915262
$$103$$ −2.07612 −0.204566 −0.102283 0.994755i $$-0.532615\pi$$
−0.102283 + 0.994755i $$0.532615\pi$$
$$104$$ −10.9185 −1.07065
$$105$$ 0 0
$$106$$ 0.145935 0.0141745
$$107$$ 5.70399 0.551426 0.275713 0.961240i $$-0.411086\pi$$
0.275713 + 0.961240i $$0.411086\pi$$
$$108$$ 16.4994 1.58766
$$109$$ −1.64518 −0.157580 −0.0787899 0.996891i $$-0.525106\pi$$
−0.0787899 + 0.996891i $$0.525106\pi$$
$$110$$ 0 0
$$111$$ −4.51777 −0.428808
$$112$$ −0.672165 −0.0635137
$$113$$ 3.89006 0.365946 0.182973 0.983118i $$-0.441428\pi$$
0.182973 + 0.983118i $$0.441428\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ 7.06446 0.655918
$$117$$ 27.8658 2.57619
$$118$$ −7.57755 −0.697570
$$119$$ −8.22334 −0.753832
$$120$$ 0 0
$$121$$ −10.4508 −0.950068
$$122$$ −7.78011 −0.704378
$$123$$ −12.3472 −1.11331
$$124$$ 6.87259 0.617177
$$125$$ 0 0
$$126$$ 17.3193 1.54292
$$127$$ 14.4233 1.27986 0.639931 0.768432i $$-0.278963\pi$$
0.639931 + 0.768432i $$0.278963\pi$$
$$128$$ −6.04007 −0.533872
$$129$$ 8.35482 0.735601
$$130$$ 0 0
$$131$$ 9.96853 0.870954 0.435477 0.900200i $$-0.356580\pi$$
0.435477 + 0.900200i $$0.356580\pi$$
$$132$$ 2.80791 0.244397
$$133$$ 0 0
$$134$$ 13.4652 1.16322
$$135$$ 0 0
$$136$$ 9.12741 0.782669
$$137$$ −9.70431 −0.829095 −0.414548 0.910028i $$-0.636060\pi$$
−0.414548 + 0.910028i $$0.636060\pi$$
$$138$$ −1.66614 −0.141831
$$139$$ −13.4508 −1.14088 −0.570439 0.821340i $$-0.693227\pi$$
−0.570439 + 0.821340i $$0.693227\pi$$
$$140$$ 0 0
$$141$$ −16.3548 −1.37732
$$142$$ −3.30606 −0.277438
$$143$$ 2.80791 0.234809
$$144$$ −1.90407 −0.158672
$$145$$ 0 0
$$146$$ −9.83705 −0.814120
$$147$$ 0.833070 0.0687105
$$148$$ −1.65303 −0.135878
$$149$$ 15.0959 1.23671 0.618353 0.785900i $$-0.287800\pi$$
0.618353 + 0.785900i $$0.287800\pi$$
$$150$$ 0 0
$$151$$ −14.1919 −1.15492 −0.577459 0.816420i $$-0.695956\pi$$
−0.577459 + 0.816420i $$0.695956\pi$$
$$152$$ 0 0
$$153$$ −23.2946 −1.88325
$$154$$ 1.74519 0.140631
$$155$$ 0 0
$$156$$ 14.3548 1.14931
$$157$$ −7.57755 −0.604754 −0.302377 0.953188i $$-0.597780\pi$$
−0.302377 + 0.953188i $$0.597780\pi$$
$$158$$ −1.66614 −0.132551
$$159$$ −0.517774 −0.0410622
$$160$$ 0 0
$$161$$ 1.48223 0.116816
$$162$$ 20.8865 1.64100
$$163$$ 19.6757 1.54112 0.770559 0.637369i $$-0.219977\pi$$
0.770559 + 0.637369i $$0.219977\pi$$
$$164$$ −4.51777 −0.352779
$$165$$ 0 0
$$166$$ −3.80814 −0.295569
$$167$$ −10.7954 −0.835376 −0.417688 0.908590i $$-0.637160\pi$$
−0.417688 + 0.908590i $$0.637160\pi$$
$$168$$ 24.0770 1.85758
$$169$$ 1.35482 0.104217
$$170$$ 0 0
$$171$$ 0 0
$$172$$ 3.05699 0.233093
$$173$$ 20.3895 1.55018 0.775092 0.631848i $$-0.217703\pi$$
0.775092 + 0.631848i $$0.217703\pi$$
$$174$$ 17.5111 1.32752
$$175$$ 0 0
$$176$$ −0.191865 −0.0144623
$$177$$ 26.8849 2.02079
$$178$$ 15.3291 1.14897
$$179$$ −25.0645 −1.87341 −0.936703 0.350126i $$-0.886139\pi$$
−0.936703 + 0.350126i $$0.886139\pi$$
$$180$$ 0 0
$$181$$ 19.4193 1.44342 0.721712 0.692194i $$-0.243356\pi$$
0.721712 + 0.692194i $$0.243356\pi$$
$$182$$ 8.92188 0.661334
$$183$$ 27.6036 2.04051
$$184$$ −1.64518 −0.121284
$$185$$ 0 0
$$186$$ 17.0355 1.24911
$$187$$ −2.34729 −0.171651
$$188$$ −5.98414 −0.436439
$$189$$ −36.3837 −2.64653
$$190$$ 0 0
$$191$$ 11.4508 0.828547 0.414274 0.910152i $$-0.364036\pi$$
0.414274 + 0.910152i $$0.364036\pi$$
$$192$$ −17.8021 −1.28476
$$193$$ 3.78878 0.272722 0.136361 0.990659i $$-0.456459\pi$$
0.136361 + 0.990659i $$0.456459\pi$$
$$194$$ 3.43630 0.246712
$$195$$ 0 0
$$196$$ 0.304816 0.0217726
$$197$$ −2.28354 −0.162695 −0.0813477 0.996686i $$-0.525922\pi$$
−0.0813477 + 0.996686i $$0.525922\pi$$
$$198$$ 4.94366 0.351330
$$199$$ −19.4508 −1.37883 −0.689414 0.724368i $$-0.742132\pi$$
−0.689414 + 0.724368i $$0.742132\pi$$
$$200$$ 0 0
$$201$$ −47.7741 −3.36972
$$202$$ 7.57755 0.533155
$$203$$ −15.5782 −1.09337
$$204$$ −12.0000 −0.840168
$$205$$ 0 0
$$206$$ −1.88297 −0.131193
$$207$$ 4.19876 0.291834
$$208$$ −0.980865 −0.0680108
$$209$$ 0 0
$$210$$ 0 0
$$211$$ −11.2274 −0.772927 −0.386463 0.922305i $$-0.626303\pi$$
−0.386463 + 0.922305i $$0.626303\pi$$
$$212$$ −0.189451 −0.0130115
$$213$$ 11.7298 0.803712
$$214$$ 5.17334 0.353642
$$215$$ 0 0
$$216$$ 40.3837 2.74776
$$217$$ −15.1551 −1.02880
$$218$$ −1.49213 −0.101059
$$219$$ 34.9015 2.35843
$$220$$ 0 0
$$221$$ −12.0000 −0.807207
$$222$$ −4.09748 −0.275005
$$223$$ −4.03785 −0.270394 −0.135197 0.990819i $$-0.543167\pi$$
−0.135197 + 0.990819i $$0.543167\pi$$
$$224$$ 14.3548 0.959122
$$225$$ 0 0
$$226$$ 3.52815 0.234689
$$227$$ −11.2185 −0.744600 −0.372300 0.928112i $$-0.621431\pi$$
−0.372300 + 0.928112i $$0.621431\pi$$
$$228$$ 0 0
$$229$$ 16.1315 1.06600 0.532999 0.846116i $$-0.321065\pi$$
0.532999 + 0.846116i $$0.321065\pi$$
$$230$$ 0 0
$$231$$ −6.19186 −0.407395
$$232$$ 17.2908 1.13520
$$233$$ 2.12676 0.139329 0.0696644 0.997570i $$-0.477807\pi$$
0.0696644 + 0.997570i $$0.477807\pi$$
$$234$$ 25.2733 1.65217
$$235$$ 0 0
$$236$$ 9.83705 0.640337
$$237$$ 5.91141 0.383987
$$238$$ −7.45830 −0.483450
$$239$$ −14.4152 −0.932442 −0.466221 0.884668i $$-0.654385\pi$$
−0.466221 + 0.884668i $$0.654385\pi$$
$$240$$ 0 0
$$241$$ 0.162955 0.0104968 0.00524842 0.999986i $$-0.498329\pi$$
0.00524842 + 0.999986i $$0.498329\pi$$
$$242$$ −9.47849 −0.609301
$$243$$ −32.0645 −2.05694
$$244$$ 10.1000 0.646587
$$245$$ 0 0
$$246$$ −11.1985 −0.713990
$$247$$ 0 0
$$248$$ 16.8212 1.06815
$$249$$ 13.5111 0.856233
$$250$$ 0 0
$$251$$ 12.9330 0.816322 0.408161 0.912910i $$-0.366170\pi$$
0.408161 + 0.912910i $$0.366170\pi$$
$$252$$ −22.4836 −1.41633
$$253$$ 0.423090 0.0265995
$$254$$ 13.0815 0.820805
$$255$$ 0 0
$$256$$ −16.5426 −1.03391
$$257$$ −11.0445 −0.688938 −0.344469 0.938798i $$-0.611941\pi$$
−0.344469 + 0.938798i $$0.611941\pi$$
$$258$$ 7.57755 0.471758
$$259$$ 3.64518 0.226501
$$260$$ 0 0
$$261$$ −44.1289 −2.73151
$$262$$ 9.04113 0.558563
$$263$$ −17.8527 −1.10085 −0.550424 0.834885i $$-0.685534\pi$$
−0.550424 + 0.834885i $$0.685534\pi$$
$$264$$ 6.87259 0.422979
$$265$$ 0 0
$$266$$ 0 0
$$267$$ −54.3872 −3.32844
$$268$$ −17.4803 −1.06778
$$269$$ −24.9934 −1.52387 −0.761936 0.647652i $$-0.775751\pi$$
−0.761936 + 0.647652i $$0.775751\pi$$
$$270$$ 0 0
$$271$$ −23.8660 −1.44975 −0.724877 0.688879i $$-0.758103\pi$$
−0.724877 + 0.688879i $$0.758103\pi$$
$$272$$ 0.819960 0.0497174
$$273$$ −31.6545 −1.91582
$$274$$ −8.80150 −0.531718
$$275$$ 0 0
$$276$$ 2.16295 0.130195
$$277$$ 21.2315 1.27568 0.637840 0.770169i $$-0.279828\pi$$
0.637840 + 0.770169i $$0.279828\pi$$
$$278$$ −12.1994 −0.731671
$$279$$ −42.9304 −2.57018
$$280$$ 0 0
$$281$$ −3.83705 −0.228899 −0.114449 0.993429i $$-0.536510\pi$$
−0.114449 + 0.993429i $$0.536510\pi$$
$$282$$ −14.8333 −0.883310
$$283$$ −0.211545 −0.0125751 −0.00628753 0.999980i $$-0.502001\pi$$
−0.00628753 + 0.999980i $$0.502001\pi$$
$$284$$ 4.29187 0.254676
$$285$$ 0 0
$$286$$ 2.54668 0.150589
$$287$$ 9.96237 0.588060
$$288$$ 40.6634 2.39612
$$289$$ −6.96853 −0.409913
$$290$$ 0 0
$$291$$ −12.1919 −0.714700
$$292$$ 12.7703 0.747324
$$293$$ −14.9942 −0.875970 −0.437985 0.898982i $$-0.644308\pi$$
−0.437985 + 0.898982i $$0.644308\pi$$
$$294$$ 0.755568 0.0440656
$$295$$ 0 0
$$296$$ −4.04593 −0.235165
$$297$$ −10.3855 −0.602626
$$298$$ 13.6915 0.793129
$$299$$ 2.16295 0.125087
$$300$$ 0 0
$$301$$ −6.74111 −0.388551
$$302$$ −12.8716 −0.740675
$$303$$ −26.8849 −1.54450
$$304$$ 0 0
$$305$$ 0 0
$$306$$ −21.1274 −1.20777
$$307$$ 1.65303 0.0943434 0.0471717 0.998887i $$-0.484979\pi$$
0.0471717 + 0.998887i $$0.484979\pi$$
$$308$$ −2.26557 −0.129093
$$309$$ 6.68073 0.380053
$$310$$ 0 0
$$311$$ −0.741113 −0.0420247 −0.0210123 0.999779i $$-0.506689\pi$$
−0.0210123 + 0.999779i $$0.506689\pi$$
$$312$$ 35.1346 1.98911
$$313$$ 26.8849 1.51962 0.759812 0.650143i $$-0.225291\pi$$
0.759812 + 0.650143i $$0.225291\pi$$
$$314$$ −6.87259 −0.387843
$$315$$ 0 0
$$316$$ 2.16295 0.121676
$$317$$ −8.16155 −0.458398 −0.229199 0.973380i $$-0.573611\pi$$
−0.229199 + 0.973380i $$0.573611\pi$$
$$318$$ −0.469604 −0.0263341
$$319$$ −4.44668 −0.248966
$$320$$ 0 0
$$321$$ −18.3548 −1.02447
$$322$$ 1.34433 0.0749166
$$323$$ 0 0
$$324$$ −27.1145 −1.50636
$$325$$ 0 0
$$326$$ 17.8452 0.988354
$$327$$ 5.29401 0.292759
$$328$$ −11.0576 −0.610555
$$329$$ 13.1959 0.727516
$$330$$ 0 0
$$331$$ −8.00000 −0.439720 −0.219860 0.975531i $$-0.570560\pi$$
−0.219860 + 0.975531i $$0.570560\pi$$
$$332$$ 4.94366 0.271318
$$333$$ 10.3258 0.565852
$$334$$ −9.79112 −0.535746
$$335$$ 0 0
$$336$$ 2.16295 0.117999
$$337$$ −9.90275 −0.539437 −0.269718 0.962939i $$-0.586931\pi$$
−0.269718 + 0.962939i $$0.586931\pi$$
$$338$$ 1.22878 0.0668367
$$339$$ −12.5178 −0.679872
$$340$$ 0 0
$$341$$ −4.32591 −0.234261
$$342$$ 0 0
$$343$$ −18.8467 −1.01763
$$344$$ 7.48223 0.403415
$$345$$ 0 0
$$346$$ 18.4926 0.994169
$$347$$ 21.2781 1.14227 0.571133 0.820858i $$-0.306504\pi$$
0.571133 + 0.820858i $$0.306504\pi$$
$$348$$ −22.7327 −1.21860
$$349$$ −16.4152 −0.878686 −0.439343 0.898319i $$-0.644789\pi$$
−0.439343 + 0.898319i $$0.644789\pi$$
$$350$$ 0 0
$$351$$ −53.0934 −2.83391
$$352$$ 4.09748 0.218396
$$353$$ −23.8744 −1.27071 −0.635354 0.772221i $$-0.719146\pi$$
−0.635354 + 0.772221i $$0.719146\pi$$
$$354$$ 24.3837 1.29598
$$355$$ 0 0
$$356$$ −19.9000 −1.05470
$$357$$ 26.4618 1.40051
$$358$$ −22.7327 −1.20146
$$359$$ 2.22334 0.117343 0.0586717 0.998277i $$-0.481313\pi$$
0.0586717 + 0.998277i $$0.481313\pi$$
$$360$$ 0 0
$$361$$ 0 0
$$362$$ 17.6127 0.925701
$$363$$ 33.6294 1.76508
$$364$$ −11.5822 −0.607074
$$365$$ 0 0
$$366$$ 25.0355 1.30863
$$367$$ 4.52057 0.235972 0.117986 0.993015i $$-0.462356\pi$$
0.117986 + 0.993015i $$0.462356\pi$$
$$368$$ −0.147795 −0.00770433
$$369$$ 28.2208 1.46911
$$370$$ 0 0
$$371$$ 0.417768 0.0216894
$$372$$ −22.1153 −1.14662
$$373$$ 15.5186 0.803521 0.401760 0.915745i $$-0.368398\pi$$
0.401760 + 0.915745i $$0.368398\pi$$
$$374$$ −2.12892 −0.110084
$$375$$ 0 0
$$376$$ −14.6467 −0.755345
$$377$$ −22.7327 −1.17079
$$378$$ −32.9989 −1.69728
$$379$$ 18.9015 0.970905 0.485453 0.874263i $$-0.338655\pi$$
0.485453 + 0.874263i $$0.338655\pi$$
$$380$$ 0 0
$$381$$ −46.4126 −2.37779
$$382$$ 10.3855 0.531366
$$383$$ −13.7046 −0.700274 −0.350137 0.936699i $$-0.613865\pi$$
−0.350137 + 0.936699i $$0.613865\pi$$
$$384$$ 19.4363 0.991854
$$385$$ 0 0
$$386$$ 3.43630 0.174903
$$387$$ −19.0958 −0.970694
$$388$$ −4.46094 −0.226470
$$389$$ 12.7411 0.646000 0.323000 0.946399i $$-0.395309\pi$$
0.323000 + 0.946399i $$0.395309\pi$$
$$390$$ 0 0
$$391$$ −1.80814 −0.0914413
$$392$$ 0.746063 0.0376819
$$393$$ −32.0776 −1.61810
$$394$$ −2.07110 −0.104340
$$395$$ 0 0
$$396$$ −6.41777 −0.322505
$$397$$ 38.6522 1.93990 0.969950 0.243306i $$-0.0782319\pi$$
0.969950 + 0.243306i $$0.0782319\pi$$
$$398$$ −17.6412 −0.884274
$$399$$ 0 0
$$400$$ 0 0
$$401$$ −31.8660 −1.59131 −0.795655 0.605750i $$-0.792873\pi$$
−0.795655 + 0.605750i $$0.792873\pi$$
$$402$$ −43.3296 −2.16108
$$403$$ −22.1153 −1.10164
$$404$$ −9.83705 −0.489411
$$405$$ 0 0
$$406$$ −14.1289 −0.701206
$$407$$ 1.04049 0.0515751
$$408$$ −29.3710 −1.45408
$$409$$ −11.0645 −0.547102 −0.273551 0.961857i $$-0.588198\pi$$
−0.273551 + 0.961857i $$0.588198\pi$$
$$410$$ 0 0
$$411$$ 31.2274 1.54033
$$412$$ 2.44444 0.120429
$$413$$ −21.6922 −1.06740
$$414$$ 3.80814 0.187160
$$415$$ 0 0
$$416$$ 20.9474 1.02703
$$417$$ 43.2830 2.11958
$$418$$ 0 0
$$419$$ −25.7452 −1.25773 −0.628867 0.777513i $$-0.716481\pi$$
−0.628867 + 0.777513i $$0.716481\pi$$
$$420$$ 0 0
$$421$$ −27.4482 −1.33774 −0.668871 0.743378i $$-0.733222\pi$$
−0.668871 + 0.743378i $$0.733222\pi$$
$$422$$ −10.1829 −0.495696
$$423$$ 37.3806 1.81751
$$424$$ −0.463697 −0.0225191
$$425$$ 0 0
$$426$$ 10.6385 0.515439
$$427$$ −22.2720 −1.07782
$$428$$ −6.71593 −0.324627
$$429$$ −9.03555 −0.436240
$$430$$ 0 0
$$431$$ −1.74519 −0.0840627 −0.0420314 0.999116i $$-0.513383\pi$$
−0.0420314 + 0.999116i $$0.513383\pi$$
$$432$$ 3.62787 0.174546
$$433$$ −18.5208 −0.890052 −0.445026 0.895518i $$-0.646806\pi$$
−0.445026 + 0.895518i $$0.646806\pi$$
$$434$$ −13.7452 −0.659790
$$435$$ 0 0
$$436$$ 1.93705 0.0927679
$$437$$ 0 0
$$438$$ 31.6545 1.51251
$$439$$ −29.4482 −1.40549 −0.702743 0.711444i $$-0.748041\pi$$
−0.702743 + 0.711444i $$0.748041\pi$$
$$440$$ 0 0
$$441$$ −1.90407 −0.0906699
$$442$$ −10.8836 −0.517681
$$443$$ −11.7388 −0.557726 −0.278863 0.960331i $$-0.589958\pi$$
−0.278863 + 0.960331i $$0.589958\pi$$
$$444$$ 5.31927 0.252441
$$445$$ 0 0
$$446$$ −3.66220 −0.173410
$$447$$ −48.5771 −2.29762
$$448$$ 14.3637 0.678620
$$449$$ 7.06446 0.333392 0.166696 0.986008i $$-0.446690\pi$$
0.166696 + 0.986008i $$0.446690\pi$$
$$450$$ 0 0
$$451$$ 2.84368 0.133904
$$452$$ −4.58019 −0.215434
$$453$$ 45.6679 2.14566
$$454$$ −10.1748 −0.477529
$$455$$ 0 0
$$456$$ 0 0
$$457$$ 34.5000 1.61384 0.806920 0.590660i $$-0.201133\pi$$
0.806920 + 0.590660i $$0.201133\pi$$
$$458$$ 14.6307 0.683649
$$459$$ 44.3837 2.07166
$$460$$ 0 0
$$461$$ 8.03147 0.374063 0.187032 0.982354i $$-0.440113\pi$$
0.187032 + 0.982354i $$0.440113\pi$$
$$462$$ −5.61582 −0.261272
$$463$$ 25.3290 1.17714 0.588570 0.808447i $$-0.299691\pi$$
0.588570 + 0.808447i $$0.299691\pi$$
$$464$$ 1.55332 0.0721112
$$465$$ 0 0
$$466$$ 1.92890 0.0893547
$$467$$ −26.8759 −1.24367 −0.621834 0.783149i $$-0.713612\pi$$
−0.621834 + 0.783149i $$0.713612\pi$$
$$468$$ −32.8094 −1.51662
$$469$$ 38.5467 1.77992
$$470$$ 0 0
$$471$$ 24.3837 1.12354
$$472$$ 24.0770 1.10823
$$473$$ −1.92420 −0.0884748
$$474$$ 5.36146 0.246260
$$475$$ 0 0
$$476$$ 9.68224 0.443785
$$477$$ 1.18343 0.0541854
$$478$$ −13.0741 −0.597996
$$479$$ −28.9015 −1.32054 −0.660272 0.751027i $$-0.729559\pi$$
−0.660272 + 0.751027i $$0.729559\pi$$
$$480$$ 0 0
$$481$$ 5.31927 0.242538
$$482$$ 0.147795 0.00673187
$$483$$ −4.76964 −0.217026
$$484$$ 12.3048 0.559310
$$485$$ 0 0
$$486$$ −29.0815 −1.31916
$$487$$ −17.7294 −0.803395 −0.401697 0.915773i $$-0.631580\pi$$
−0.401697 + 0.915773i $$0.631580\pi$$
$$488$$ 24.7206 1.11905
$$489$$ −63.3141 −2.86316
$$490$$ 0 0
$$491$$ −35.1645 −1.58695 −0.793475 0.608603i $$-0.791730\pi$$
−0.793475 + 0.608603i $$0.791730\pi$$
$$492$$ 14.5377 0.655410
$$493$$ 19.0035 0.855875
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 1.51114 0.0678520
$$497$$ −9.46422 −0.424528
$$498$$ 12.2542 0.549122
$$499$$ 21.4508 0.960268 0.480134 0.877195i $$-0.340588\pi$$
0.480134 + 0.877195i $$0.340588\pi$$
$$500$$ 0 0
$$501$$ 34.7385 1.55200
$$502$$ 11.7298 0.523526
$$503$$ 5.34053 0.238122 0.119061 0.992887i $$-0.462012\pi$$
0.119061 + 0.992887i $$0.462012\pi$$
$$504$$ −55.0304 −2.45125
$$505$$ 0 0
$$506$$ 0.383729 0.0170588
$$507$$ −4.35966 −0.193619
$$508$$ −16.9821 −0.753461
$$509$$ −36.1919 −1.60418 −0.802088 0.597206i $$-0.796278\pi$$
−0.802088 + 0.597206i $$0.796278\pi$$
$$510$$ 0 0
$$511$$ −28.1604 −1.24574
$$512$$ −2.92346 −0.129200
$$513$$ 0 0
$$514$$ −10.0170 −0.441832
$$515$$ 0 0
$$516$$ −9.83705 −0.433052
$$517$$ 3.76668 0.165658
$$518$$ 3.30606 0.145260
$$519$$ −65.6111 −2.88001
$$520$$ 0 0
$$521$$ 2.77259 0.121469 0.0607346 0.998154i $$-0.480656\pi$$
0.0607346 + 0.998154i $$0.480656\pi$$
$$522$$ −40.0235 −1.75178
$$523$$ −20.5373 −0.898033 −0.449016 0.893524i $$-0.648226\pi$$
−0.449016 + 0.893524i $$0.648226\pi$$
$$524$$ −11.7370 −0.512735
$$525$$ 0 0
$$526$$ −16.1919 −0.705999
$$527$$ 18.4874 0.805323
$$528$$ 0.617399 0.0268688
$$529$$ −22.6741 −0.985830
$$530$$ 0 0
$$531$$ −61.4482 −2.66662
$$532$$ 0 0
$$533$$ 14.5377 0.629698
$$534$$ −49.3274 −2.13461
$$535$$ 0 0
$$536$$ −42.7845 −1.84801
$$537$$ 80.6547 3.48051
$$538$$ −22.6682 −0.977294
$$539$$ −0.191865 −0.00826419
$$540$$ 0 0
$$541$$ 35.4797 1.52539 0.762695 0.646758i $$-0.223876\pi$$
0.762695 + 0.646758i $$0.223876\pi$$
$$542$$ −21.6456 −0.929760
$$543$$ −62.4891 −2.68166
$$544$$ −17.5111 −0.750784
$$545$$ 0 0
$$546$$ −28.7096 −1.22866
$$547$$ 43.0756 1.84178 0.920890 0.389822i $$-0.127463\pi$$
0.920890 + 0.389822i $$0.127463\pi$$
$$548$$ 11.4260 0.488092
$$549$$ −63.0908 −2.69265
$$550$$ 0 0
$$551$$ 0 0
$$552$$ 5.29401 0.225328
$$553$$ −4.76964 −0.202826
$$554$$ 19.2563 0.818123
$$555$$ 0 0
$$556$$ 15.8370 0.671640
$$557$$ −40.4376 −1.71340 −0.856698 0.515818i $$-0.827488\pi$$
−0.856698 + 0.515818i $$0.827488\pi$$
$$558$$ −38.9365 −1.64831
$$559$$ −9.83705 −0.416063
$$560$$ 0 0
$$561$$ 7.55332 0.318902
$$562$$ −3.48008 −0.146798
$$563$$ 19.7173 0.830986 0.415493 0.909596i $$-0.363609\pi$$
0.415493 + 0.909596i $$0.363609\pi$$
$$564$$ 19.2563 0.810837
$$565$$ 0 0
$$566$$ −0.191865 −0.00806467
$$567$$ 59.7915 2.51101
$$568$$ 10.5047 0.440768
$$569$$ −18.6807 −0.783137 −0.391568 0.920149i $$-0.628067\pi$$
−0.391568 + 0.920149i $$0.628067\pi$$
$$570$$ 0 0
$$571$$ −29.9371 −1.25283 −0.626413 0.779491i $$-0.715478\pi$$
−0.626413 + 0.779491i $$0.715478\pi$$
$$572$$ −3.30606 −0.138233
$$573$$ −36.8473 −1.53932
$$574$$ 9.03555 0.377137
$$575$$ 0 0
$$576$$ 40.6885 1.69536
$$577$$ 0.156779 0.00652679 0.00326339 0.999995i $$-0.498961\pi$$
0.00326339 + 0.999995i $$0.498961\pi$$
$$578$$ −6.32023 −0.262887
$$579$$ −12.1919 −0.506677
$$580$$ 0 0
$$581$$ −10.9015 −0.452271
$$582$$ −11.0576 −0.458353
$$583$$ 0.119249 0.00493877
$$584$$ 31.2563 1.29340
$$585$$ 0 0
$$586$$ −13.5993 −0.561780
$$587$$ −31.1474 −1.28559 −0.642795 0.766038i $$-0.722225\pi$$
−0.642795 + 0.766038i $$0.722225\pi$$
$$588$$ −0.980865 −0.0404502
$$589$$ 0 0
$$590$$ 0 0
$$591$$ 7.34818 0.302264
$$592$$ −0.363466 −0.0149384
$$593$$ −28.8728 −1.18567 −0.592833 0.805326i $$-0.701991\pi$$
−0.592833 + 0.805326i $$0.701991\pi$$
$$594$$ −9.41928 −0.386478
$$595$$ 0 0
$$596$$ −17.7741 −0.728055
$$597$$ 62.5904 2.56165
$$598$$ 1.96173 0.0802211
$$599$$ −25.3274 −1.03485 −0.517425 0.855728i $$-0.673109\pi$$
−0.517425 + 0.855728i $$0.673109\pi$$
$$600$$ 0 0
$$601$$ 19.8370 0.809170 0.404585 0.914500i $$-0.367416\pi$$
0.404585 + 0.914500i $$0.367416\pi$$
$$602$$ −6.11397 −0.249187
$$603$$ 109.193 4.44667
$$604$$ 16.7096 0.679906
$$605$$ 0 0
$$606$$ −24.3837 −0.990521
$$607$$ −2.49921 −0.101440 −0.0507199 0.998713i $$-0.516152\pi$$
−0.0507199 + 0.998713i $$0.516152\pi$$
$$608$$ 0 0
$$609$$ 50.1289 2.03133
$$610$$ 0 0
$$611$$ 19.2563 0.779027
$$612$$ 27.4272 1.10868
$$613$$ 0.883711 0.0356927 0.0178464 0.999841i $$-0.494319\pi$$
0.0178464 + 0.999841i $$0.494319\pi$$
$$614$$ 1.49925 0.0605046
$$615$$ 0 0
$$616$$ −5.54517 −0.223421
$$617$$ 29.4085 1.18394 0.591971 0.805959i $$-0.298350\pi$$
0.591971 + 0.805959i $$0.298350\pi$$
$$618$$ 6.05921 0.243737
$$619$$ 30.3208 1.21870 0.609348 0.792903i $$-0.291431\pi$$
0.609348 + 0.792903i $$0.291431\pi$$
$$620$$ 0 0
$$621$$ −8.00000 −0.321029
$$622$$ −0.672165 −0.0269514
$$623$$ 43.8825 1.75811
$$624$$ 3.15632 0.126354
$$625$$ 0 0
$$626$$ 24.3837 0.974570
$$627$$ 0 0
$$628$$ 8.92188 0.356022
$$629$$ −4.44668 −0.177301
$$630$$ 0 0
$$631$$ 17.7767 0.707678 0.353839 0.935306i $$-0.384876\pi$$
0.353839 + 0.935306i $$0.384876\pi$$
$$632$$ 5.29401 0.210584
$$633$$ 36.1286 1.43598
$$634$$ −7.40226 −0.293981
$$635$$ 0 0
$$636$$ 0.609632 0.0241735
$$637$$ −0.980865 −0.0388633
$$638$$ −4.03299 −0.159668
$$639$$ −26.8096 −1.06057
$$640$$ 0 0
$$641$$ 32.6675 1.29029 0.645143 0.764062i $$-0.276798\pi$$
0.645143 + 0.764062i $$0.276798\pi$$
$$642$$ −16.6472 −0.657014
$$643$$ −31.8661 −1.25668 −0.628338 0.777941i $$-0.716264\pi$$
−0.628338 + 0.777941i $$0.716264\pi$$
$$644$$ −1.74519 −0.0687700
$$645$$ 0 0
$$646$$ 0 0
$$647$$ −21.2601 −0.835820 −0.417910 0.908488i $$-0.637237\pi$$
−0.417910 + 0.908488i $$0.637237\pi$$
$$648$$ −66.3649 −2.60706
$$649$$ −6.19186 −0.243052
$$650$$ 0 0
$$651$$ 48.7675 1.91135
$$652$$ −23.1663 −0.907263
$$653$$ −12.8340 −0.502234 −0.251117 0.967957i $$-0.580798\pi$$
−0.251117 + 0.967957i $$0.580798\pi$$
$$654$$ 4.80150 0.187753
$$655$$ 0 0
$$656$$ −0.993361 −0.0387842
$$657$$ −79.7710 −3.11216
$$658$$ 11.9683 0.466573
$$659$$ −20.3548 −0.792911 −0.396456 0.918054i $$-0.629760\pi$$
−0.396456 + 0.918054i $$0.629760\pi$$
$$660$$ 0 0
$$661$$ 30.7385 1.19559 0.597795 0.801649i $$-0.296043\pi$$
0.597795 + 0.801649i $$0.296043\pi$$
$$662$$ −7.25574 −0.282002
$$663$$ 38.6147 1.49967
$$664$$ 12.1000 0.469571
$$665$$ 0 0
$$666$$ 9.36520 0.362894
$$667$$ −3.42531 −0.132629
$$668$$ 12.7107 0.491790
$$669$$ 12.9934 0.502352
$$670$$ 0 0
$$671$$ −6.35738 −0.245424
$$672$$ −46.1922 −1.78190
$$673$$ 21.2094 0.817564 0.408782 0.912632i $$-0.365954\pi$$
0.408782 + 0.912632i $$0.365954\pi$$
$$674$$ −8.98147 −0.345953
$$675$$ 0 0
$$676$$ −1.59518 −0.0613530
$$677$$ 11.2650 0.432951 0.216475 0.976288i $$-0.430544\pi$$
0.216475 + 0.976288i $$0.430544\pi$$
$$678$$ −11.3532 −0.436018
$$679$$ 9.83705 0.377511
$$680$$ 0 0
$$681$$ 36.1000 1.38336
$$682$$ −3.92346 −0.150237
$$683$$ −12.3603 −0.472954 −0.236477 0.971637i $$-0.575993\pi$$
−0.236477 + 0.971637i $$0.575993\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ −17.0934 −0.652628
$$687$$ −51.9093 −1.98046
$$688$$ 0.672165 0.0256261
$$689$$ 0.609632 0.0232251
$$690$$ 0 0
$$691$$ 22.7493 0.865423 0.432711 0.901533i $$-0.357557\pi$$
0.432711 + 0.901533i $$0.357557\pi$$
$$692$$ −24.0068 −0.912601
$$693$$ 14.1521 0.537595
$$694$$ 19.2985 0.732561
$$695$$ 0 0
$$696$$ −55.6401 −2.10903
$$697$$ −12.1529 −0.460323
$$698$$ −14.8881 −0.563521
$$699$$ −6.84368 −0.258852
$$700$$ 0 0
$$701$$ −16.0289 −0.605404 −0.302702 0.953085i $$-0.597889\pi$$
−0.302702 + 0.953085i $$0.597889\pi$$
$$702$$ −48.1540 −1.81745
$$703$$ 0 0
$$704$$ 4.10001 0.154525
$$705$$ 0 0
$$706$$ −21.6533 −0.814934
$$707$$ 21.6922 0.815818
$$708$$ −31.6545 −1.18965
$$709$$ 31.4193 1.17998 0.589988 0.807412i $$-0.299133\pi$$
0.589988 + 0.807412i $$0.299133\pi$$
$$710$$ 0 0
$$711$$ −13.5111 −0.506707
$$712$$ −48.7069 −1.82537
$$713$$ −3.33228 −0.124795
$$714$$ 24.0000 0.898177
$$715$$ 0 0
$$716$$ 29.5111 1.10288
$$717$$ 46.3865 1.73234
$$718$$ 2.01650 0.0752550
$$719$$ 11.2589 0.419886 0.209943 0.977714i $$-0.432672\pi$$
0.209943 + 0.977714i $$0.432672\pi$$
$$720$$ 0 0
$$721$$ −5.39037 −0.200748
$$722$$ 0 0
$$723$$ −0.524371 −0.0195016
$$724$$ −22.8644 −0.849750
$$725$$ 0 0
$$726$$ 30.5008 1.13199
$$727$$ −48.9829 −1.81668 −0.908338 0.418237i $$-0.862648\pi$$
−0.908338 + 0.418237i $$0.862648\pi$$
$$728$$ −28.3485 −1.05066
$$729$$ 34.0934 1.26272
$$730$$ 0 0
$$731$$ 8.22334 0.304151
$$732$$ −32.5007 −1.20126
$$733$$ 35.9260 1.32696 0.663479 0.748195i $$-0.269079\pi$$
0.663479 + 0.748195i $$0.269079\pi$$
$$734$$ 4.10001 0.151334
$$735$$ 0 0
$$736$$ 3.15632 0.116343
$$737$$ 11.0029 0.405296
$$738$$ 25.5953 0.942177
$$739$$ 14.3523 0.527956 0.263978 0.964529i $$-0.414965\pi$$
0.263978 + 0.964529i $$0.414965\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 0.378902 0.0139099
$$743$$ 12.5629 0.460887 0.230443 0.973086i $$-0.425982\pi$$
0.230443 + 0.973086i $$0.425982\pi$$
$$744$$ −54.1289 −1.98446
$$745$$ 0 0
$$746$$ 14.0748 0.515316
$$747$$ −30.8811 −1.12988
$$748$$ 2.76372 0.101052
$$749$$ 14.8096 0.541133
$$750$$ 0 0
$$751$$ −26.4548 −0.965350 −0.482675 0.875799i $$-0.660335\pi$$
−0.482675 + 0.875799i $$0.660335\pi$$
$$752$$ −1.31578 −0.0479817
$$753$$ −41.6169 −1.51660
$$754$$ −20.6178 −0.750855
$$755$$ 0 0
$$756$$ 42.8386 1.55802
$$757$$ 15.7350 0.571897 0.285949 0.958245i $$-0.407691\pi$$
0.285949 + 0.958245i $$0.407691\pi$$
$$758$$ 17.1431 0.622664
$$759$$ −1.36146 −0.0494178
$$760$$ 0 0
$$761$$ 16.9619 0.614868 0.307434 0.951569i $$-0.400530\pi$$
0.307434 + 0.951569i $$0.400530\pi$$
$$762$$ −42.0948 −1.52493
$$763$$ −4.27149 −0.154638
$$764$$ −13.4822 −0.487770
$$765$$ 0 0
$$766$$ −12.4297 −0.449102
$$767$$ −31.6545 −1.14298
$$768$$ 53.2323 1.92086
$$769$$ 41.9974 1.51447 0.757233 0.653145i $$-0.226551\pi$$
0.757233 + 0.653145i $$0.226551\pi$$
$$770$$ 0 0
$$771$$ 35.5400 1.27994
$$772$$ −4.46094 −0.160553
$$773$$ −40.9579 −1.47315 −0.736576 0.676355i $$-0.763559\pi$$
−0.736576 + 0.676355i $$0.763559\pi$$
$$774$$ −17.3193 −0.622528
$$775$$ 0 0
$$776$$ −10.9185 −0.391952
$$777$$ −11.7298 −0.420804
$$778$$ 11.5558 0.414295
$$779$$ 0 0
$$780$$ 0 0
$$781$$ −2.70149 −0.0966669
$$782$$ −1.63992 −0.0586434
$$783$$ 84.0800 3.00477
$$784$$ 0.0670225 0.00239366
$$785$$ 0 0
$$786$$ −29.0934 −1.03773
$$787$$ −28.5379 −1.01727 −0.508634 0.860983i $$-0.669849\pi$$
−0.508634 + 0.860983i $$0.669849\pi$$
$$788$$ 2.68866 0.0957796
$$789$$ 57.4482 2.04521
$$790$$ 0 0
$$791$$ 10.1000 0.359115
$$792$$ −15.7080 −0.558160
$$793$$ −32.5007 −1.15413
$$794$$ 35.0563 1.24410
$$795$$ 0 0
$$796$$ 22.9015 0.811722
$$797$$ −33.2790 −1.17880 −0.589402 0.807840i $$-0.700636\pi$$
−0.589402 + 0.807840i $$0.700636\pi$$
$$798$$ 0 0
$$799$$ −16.0974 −0.569487
$$800$$ 0 0
$$801$$ 124.308 4.39219
$$802$$ −28.9014 −1.02054
$$803$$ −8.03817 −0.283661
$$804$$ 56.2497 1.98377
$$805$$ 0 0
$$806$$ −20.0578 −0.706507
$$807$$ 80.4259 2.83113
$$808$$ −24.0770 −0.847025
$$809$$ −34.4234 −1.21026 −0.605130 0.796126i $$-0.706879\pi$$
−0.605130 + 0.796126i $$0.706879\pi$$
$$810$$ 0 0
$$811$$ 11.2563 0.395263 0.197631 0.980276i $$-0.436675\pi$$
0.197631 + 0.980276i $$0.436675\pi$$
$$812$$ 18.3419 0.643675
$$813$$ 76.7980 2.69342
$$814$$ 0.943690 0.0330763
$$815$$ 0 0
$$816$$ −2.63854 −0.0923674
$$817$$ 0 0
$$818$$ −10.0351 −0.350869
$$819$$ 72.3497 2.52810
$$820$$ 0 0
$$821$$ −31.3167 −1.09296 −0.546480 0.837472i $$-0.684033\pi$$
−0.546480 + 0.837472i $$0.684033\pi$$
$$822$$ 28.3223 0.987852
$$823$$ −13.4800 −0.469882 −0.234941 0.972010i $$-0.575490\pi$$
−0.234941 + 0.972010i $$0.575490\pi$$
$$824$$ 5.98298 0.208427
$$825$$ 0 0
$$826$$ −19.6741 −0.684549
$$827$$ −15.9882 −0.555963 −0.277982 0.960586i $$-0.589665\pi$$
−0.277982 + 0.960586i $$0.589665\pi$$
$$828$$ −4.94366 −0.171804
$$829$$ 48.4837 1.68391 0.841955 0.539548i $$-0.181405\pi$$
0.841955 + 0.539548i $$0.181405\pi$$
$$830$$ 0 0
$$831$$ −68.3208 −2.37002
$$832$$ 20.9604 0.726670
$$833$$ 0.819960 0.0284099
$$834$$ 39.2563 1.35934
$$835$$ 0 0
$$836$$ 0 0
$$837$$ 81.7965 2.82730
$$838$$ −23.3501 −0.806614
$$839$$ 18.9934 0.655724 0.327862 0.944726i $$-0.393672\pi$$
0.327862 + 0.944726i $$0.393672\pi$$
$$840$$ 0 0
$$841$$ 7.00000 0.241379
$$842$$ −24.8946 −0.857925
$$843$$ 12.3472 0.425260
$$844$$ 13.2193 0.455026
$$845$$ 0 0
$$846$$ 33.9030 1.16561
$$847$$ −27.1340 −0.932334
$$848$$ −0.0416562 −0.00143048
$$849$$ 0.680729 0.0233626
$$850$$ 0 0
$$851$$ 0.801497 0.0274750
$$852$$ −13.8108 −0.473149
$$853$$ −44.1210 −1.51067 −0.755337 0.655337i $$-0.772527\pi$$
−0.755337 + 0.655337i $$0.772527\pi$$
$$854$$ −20.2000 −0.691230
$$855$$ 0 0
$$856$$ −16.4378 −0.561833
$$857$$ 32.4149 1.10727 0.553635 0.832759i $$-0.313240\pi$$
0.553635 + 0.832759i $$0.313240\pi$$
$$858$$ −8.19495 −0.279771
$$859$$ 6.29444 0.214763 0.107382 0.994218i $$-0.465753\pi$$
0.107382 + 0.994218i $$0.465753\pi$$
$$860$$ 0 0
$$861$$ −32.0578 −1.09253
$$862$$ −1.58283 −0.0539113
$$863$$ −17.4338 −0.593453 −0.296726 0.954963i $$-0.595895\pi$$
−0.296726 + 0.954963i $$0.595895\pi$$
$$864$$ −77.4771 −2.63582
$$865$$ 0 0
$$866$$ −16.7978 −0.570811
$$867$$ 22.4240 0.761557
$$868$$ 17.8438 0.605657
$$869$$ −1.36146 −0.0461843
$$870$$ 0 0
$$871$$ 56.2497 1.90595
$$872$$ 4.74109 0.160554
$$873$$ 27.8658 0.943113
$$874$$ 0 0
$$875$$ 0 0
$$876$$ −41.0934 −1.38842
$$877$$ 23.2904 0.786462 0.393231 0.919440i $$-0.371357\pi$$
0.393231 + 0.919440i $$0.371357\pi$$
$$878$$ −26.7086 −0.901370
$$879$$ 48.2497 1.62742
$$880$$ 0 0
$$881$$ 28.9619 0.975751 0.487875 0.872913i $$-0.337772\pi$$
0.487875 + 0.872913i $$0.337772\pi$$
$$882$$ −1.72693 −0.0581487
$$883$$ 37.3627 1.25735 0.628677 0.777667i $$-0.283597\pi$$
0.628677 + 0.777667i $$0.283597\pi$$
$$884$$ 14.1289 0.475207
$$885$$ 0 0
$$886$$ −10.6467 −0.357683
$$887$$ −23.0234 −0.773050 −0.386525 0.922279i $$-0.626325\pi$$
−0.386525 + 0.922279i $$0.626325\pi$$
$$888$$ 13.0194 0.436901
$$889$$ 37.4482 1.25597
$$890$$ 0 0
$$891$$ 17.0670 0.571767
$$892$$ 4.75420 0.159183
$$893$$ 0 0
$$894$$ −44.0578 −1.47351
$$895$$ 0 0
$$896$$ −15.6822 −0.523907
$$897$$ −6.96015 −0.232393
$$898$$ 6.40723 0.213812
$$899$$ 35.0223 1.16806
$$900$$ 0 0
$$901$$ −0.509626 −0.0169781
$$902$$ 2.57913 0.0858756
$$903$$ 21.6922 0.721870
$$904$$ −11.2104 −0.372852
$$905$$ 0 0
$$906$$ 41.4193 1.37606
$$907$$ −35.2693 −1.17110 −0.585549 0.810637i $$-0.699121\pi$$
−0.585549 + 0.810637i $$0.699121\pi$$
$$908$$ 13.2088 0.438350
$$909$$ 61.4482 2.03811
$$910$$ 0 0
$$911$$ 4.97260 0.164750 0.0823748 0.996601i $$-0.473750\pi$$
0.0823748 + 0.996601i $$0.473750\pi$$
$$912$$ 0 0
$$913$$ −3.11175 −0.102984
$$914$$ 31.2904 1.03499
$$915$$ 0 0
$$916$$ −18.9934 −0.627558
$$917$$ 25.8819 0.854697
$$918$$ 40.2546 1.32860
$$919$$ −48.7096 −1.60678 −0.803391 0.595451i $$-0.796973\pi$$
−0.803391 + 0.595451i $$0.796973\pi$$
$$920$$ 0 0
$$921$$ −5.31927 −0.175276
$$922$$ 7.28429 0.239895
$$923$$ −13.8108 −0.454587
$$924$$ 7.29036 0.239835
$$925$$ 0 0
$$926$$ 22.9726 0.754926
$$927$$ −15.2695 −0.501516
$$928$$ −33.1729 −1.08895
$$929$$ 32.5126 1.06671 0.533353 0.845893i $$-0.320932\pi$$
0.533353 + 0.845893i $$0.320932\pi$$
$$930$$ 0 0
$$931$$ 0 0
$$932$$ −2.50407 −0.0820235
$$933$$ 2.38482 0.0780755
$$934$$ −24.3756 −0.797593
$$935$$ 0 0
$$936$$ −80.3038 −2.62481
$$937$$ 0.385560 0.0125957 0.00629785 0.999980i $$-0.497995\pi$$
0.00629785 + 0.999980i $$0.497995\pi$$
$$938$$ 34.9606 1.14150
$$939$$ −86.5126 −2.82323
$$940$$ 0 0
$$941$$ −45.3482 −1.47831 −0.739154 0.673536i $$-0.764775\pi$$
−0.739154 + 0.673536i $$0.764775\pi$$
$$942$$ 22.1153 0.720554
$$943$$ 2.19051 0.0713329
$$944$$ 2.16295 0.0703982
$$945$$ 0 0
$$946$$ −1.74519 −0.0567409
$$947$$ −9.61202 −0.312349 −0.156174 0.987730i $$-0.549916\pi$$
−0.156174 + 0.987730i $$0.549916\pi$$
$$948$$ −6.96015 −0.226055
$$949$$ −41.0934 −1.33395
$$950$$ 0 0
$$951$$ 26.2630 0.851635
$$952$$ 23.6981 0.768059
$$953$$ −59.8421 −1.93848 −0.969238 0.246126i $$-0.920842\pi$$
−0.969238 + 0.246126i $$0.920842\pi$$
$$954$$ 1.07333 0.0347503
$$955$$ 0 0
$$956$$ 16.9726 0.548933
$$957$$ 14.3089 0.462542
$$958$$ −26.2127 −0.846895
$$959$$ −25.1959 −0.813619
$$960$$ 0 0
$$961$$ 3.07110 0.0990676
$$962$$ 4.82441 0.155545
$$963$$ 41.9518 1.35188
$$964$$ −0.191865 −0.00617954
$$965$$ 0 0
$$966$$ −4.32591 −0.139184
$$967$$ 0.368324 0.0118445 0.00592225 0.999982i $$-0.498115\pi$$
0.00592225 + 0.999982i $$0.498115\pi$$
$$968$$ 30.1171 0.967999
$$969$$ 0 0
$$970$$ 0 0
$$971$$ 45.8370 1.47098 0.735490 0.677535i $$-0.236952\pi$$
0.735490 + 0.677535i $$0.236952\pi$$
$$972$$ 37.7531 1.21093
$$973$$ −34.9231 −1.11958
$$974$$ −16.0800 −0.515235
$$975$$ 0 0
$$976$$ 2.22077 0.0710853
$$977$$ 29.0337 0.928872 0.464436 0.885607i $$-0.346257\pi$$
0.464436 + 0.885607i $$0.346257\pi$$
$$978$$ −57.4239 −1.83621
$$979$$ 12.5259 0.400330
$$980$$ 0 0
$$981$$ −12.1000 −0.386323
$$982$$ −31.8930 −1.01775
$$983$$ −11.0160 −0.351355 −0.175677 0.984448i $$-0.556212\pi$$
−0.175677 + 0.984448i $$0.556212\pi$$
$$984$$ 35.5822 1.13432
$$985$$ 0 0
$$986$$ 17.2356 0.548892
$$987$$ −42.4631 −1.35161
$$988$$ 0 0
$$989$$ −1.48223 −0.0471320
$$990$$ 0 0
$$991$$ 25.6822 0.815823 0.407912 0.913021i $$-0.366257\pi$$
0.407912 + 0.913021i $$0.366257\pi$$
$$992$$ −32.2719 −1.02463
$$993$$ 25.7431 0.816933
$$994$$ −8.58374 −0.272260
$$995$$ 0 0
$$996$$ −15.9081 −0.504069
$$997$$ −20.3854 −0.645611 −0.322805 0.946465i $$-0.604626\pi$$
−0.322805 + 0.946465i $$0.604626\pi$$
$$998$$ 19.4551 0.615842
$$999$$ −19.6741 −0.622461
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 9025.2.a.bx.1.4 6
5.2 odd 4 1805.2.b.e.1084.4 6
5.3 odd 4 1805.2.b.e.1084.3 6
5.4 even 2 inner 9025.2.a.bx.1.3 6
19.18 odd 2 475.2.a.j.1.3 6
57.56 even 2 4275.2.a.br.1.4 6
76.75 even 2 7600.2.a.ck.1.1 6
95.18 even 4 95.2.b.b.39.4 yes 6
95.37 even 4 95.2.b.b.39.3 6
95.94 odd 2 475.2.a.j.1.4 6
285.113 odd 4 855.2.c.d.514.3 6
285.227 odd 4 855.2.c.d.514.4 6
285.284 even 2 4275.2.a.br.1.3 6
380.227 odd 4 1520.2.d.h.609.6 6
380.303 odd 4 1520.2.d.h.609.1 6
380.379 even 2 7600.2.a.ck.1.6 6

By twisted newform
Twist Min Dim Char Parity Ord Type
95.2.b.b.39.3 6 95.37 even 4
95.2.b.b.39.4 yes 6 95.18 even 4
475.2.a.j.1.3 6 19.18 odd 2
475.2.a.j.1.4 6 95.94 odd 2
855.2.c.d.514.3 6 285.113 odd 4
855.2.c.d.514.4 6 285.227 odd 4
1520.2.d.h.609.1 6 380.303 odd 4
1520.2.d.h.609.6 6 380.227 odd 4
1805.2.b.e.1084.3 6 5.3 odd 4
1805.2.b.e.1084.4 6 5.2 odd 4
4275.2.a.br.1.3 6 285.284 even 2
4275.2.a.br.1.4 6 57.56 even 2
7600.2.a.ck.1.1 6 76.75 even 2
7600.2.a.ck.1.6 6 380.379 even 2
9025.2.a.bx.1.3 6 5.4 even 2 inner
9025.2.a.bx.1.4 6 1.1 even 1 trivial